id	sid	tid	token	lemma	pos
ejpam-5425	1	1	european	european	PROPN
ejpam-5425	1	2	journal	journal	PROPN
ejpam-5425	1	3	of	of	ADP
ejpam-5425	1	4	pure	pure	ADJ
ejpam-5425	1	5	and	and	CCONJ
ejpam-5425	1	6	applied	apply	VERB
ejpam-5425	1	7	mathematics	mathematic	NOUN
ejpam-5425	1	8	vol	vol	NOUN
ejpam-5425	1	9	.	.	PROPN
ejpam-5425	2	1	17	17	NUM
ejpam-5425	2	2	,	,	PUNCT
ejpam-5425	2	3	no	no	INTJ
ejpam-5425	2	4	.	.	NOUN
ejpam-5425	2	5	4	4	NUM
ejpam-5425	2	6	,	,	PUNCT
ejpam-5425	2	7	2024	2024	NUM
ejpam-5425	2	8	,	,	PUNCT
ejpam-5425	2	9	3254	3254	NUM
ejpam-5425	2	10	-	-	SYM
ejpam-5425	2	11	3267	3267	NUM
ejpam-5425	2	12	issn	issn	PROPN
ejpam-5425	2	13	1307	1307	NUM
ejpam-5425	2	14	-	-	SYM
ejpam-5425	2	15	5543	5543	NUM
ejpam-5425	2	16	–	–	PUNCT
ejpam-5425	2	17	ejpam.com	ejpam.com	X
ejpam-5425	2	18	published	publish	VERB
ejpam-5425	2	19	by	by	ADP
ejpam-5425	2	20	new	new	PROPN
ejpam-5425	2	21	york	york	PROPN
ejpam-5425	2	22	business	business	PROPN
ejpam-5425	2	23	global	global	PROPN
ejpam-5425	2	24	on	on	ADP
ejpam-5425	2	25	the	the	DET
ejpam-5425	2	26	periodicity	periodicity	NOUN
ejpam-5425	2	27	solutions	solution	NOUN
ejpam-5425	2	28	of	of	ADP
ejpam-5425	2	29	five	five	NUM
ejpam-5425	2	30	systems	system	NOUN
ejpam-5425	2	31	of	of	ADP
ejpam-5425	2	32	rational	rational	ADJ
ejpam-5425	2	33	systems	system	NOUN
ejpam-5425	2	34	of	of	ADP
ejpam-5425	2	35	difference	difference	NOUN
ejpam-5425	2	36	equations	equation	NOUN
ejpam-5425	2	37	of	of	ADP
ejpam-5425	2	38	order	order	NOUN
ejpam-5425	2	39	five	five	NUM
ejpam-5425	2	40	jawharah	jawharah	PROPN
ejpam-5425	2	41	ghuwayzi	ghuwayzi	PROPN
ejpam-5425	2	42	al	al	PROPN
ejpam-5425	2	43	-	-	PUNCT
ejpam-5425	2	44	juaid	juaid	PROPN
ejpam-5425	2	45	department	department	PROPN
ejpam-5425	2	46	of	of	ADP
ejpam-5425	2	47	mathematics	mathematics	PROPN
ejpam-5425	2	48	and	and	CCONJ
ejpam-5425	2	49	statistics	statistic	NOUN
ejpam-5425	2	50	,	,	PUNCT
ejpam-5425	2	51	collage	collage	NOUN
ejpam-5425	2	52	of	of	ADP
ejpam-5425	2	53	science	science	NOUN
ejpam-5425	2	54	,	,	PUNCT
ejpam-5425	2	55	taif	taif	PROPN
ejpam-5425	2	56	university	university	PROPN
ejpam-5425	2	57	,	,	PUNCT
ejpam-5425	2	58	p.o	p.o	PROPN
ejpam-5425	2	59	.	.	PROPN
ejpam-5425	2	60	box	box	PROPN
ejpam-5425	2	61	11099	11099	NUM
ejpam-5425	2	62	,	,	PUNCT
ejpam-5425	2	63	taif	taif	PROPN
ejpam-5425	2	64	21944	21944	NUM
ejpam-5425	2	65	,	,	PUNCT
ejpam-5425	3	1	saudi	saudi	PROPN
ejpam-5425	3	2	arabia	arabia	PROPN
ejpam-5425	3	3	abstract	abstract	NOUN
ejpam-5425	3	4	.	.	PUNCT
ejpam-5425	4	1	the	the	DET
ejpam-5425	4	2	primary	primary	ADJ
ejpam-5425	4	3	aim	aim	NOUN
ejpam-5425	4	4	of	of	ADP
ejpam-5425	4	5	this	this	DET
ejpam-5425	4	6	paper	paper	NOUN
ejpam-5425	4	7	is	be	AUX
ejpam-5425	4	8	to	to	PART
ejpam-5425	4	9	explore	explore	VERB
ejpam-5425	4	10	the	the	DET
ejpam-5425	4	11	behavior	behavior	NOUN
ejpam-5425	4	12	of	of	ADP
ejpam-5425	4	13	several	several	ADJ
ejpam-5425	4	14	nonlinear	nonlinear	ADJ
ejpam-5425	4	15	systems	system	NOUN
ejpam-5425	4	16	of	of	ADP
ejpam-5425	4	17	difference	difference	NOUN
ejpam-5425	4	18	equations	equation	NOUN
ejpam-5425	4	19	following	follow	VERB
ejpam-5425	4	20	tn+1	tn+1	PROPN
ejpam-5425	4	21	=	=	SYM
ejpam-5425	4	22	enen−4	enen−4	NOUN
ejpam-5425	4	23	tn−3(±1±	tn−3(±1±	NOUN
ejpam-5425	4	24	enen−4	enen−4	NOUN
ejpam-5425	4	25	)	)	PUNCT
ejpam-5425	4	26	,	,	PUNCT
ejpam-5425	4	27	en+1	en+1	ADJ
ejpam-5425	4	28	=	=	SYM
ejpam-5425	4	29	tntn−4	tntn−4	X
ejpam-5425	4	30	en−3(±1±	en−3(±1±	NOUN
ejpam-5425	4	31	tntn−4	tntn−4	NOUN
ejpam-5425	4	32	)	)	PUNCT
ejpam-5425	4	33	,	,	PUNCT
ejpam-5425	4	34	and	and	CCONJ
ejpam-5425	4	35	obtain	obtain	VERB
ejpam-5425	4	36	solution	solution	NOUN
ejpam-5425	4	37	expressions	expression	NOUN
ejpam-5425	4	38	for	for	ADP
ejpam-5425	4	39	them	they	PRON
ejpam-5425	4	40	.	.	PUNCT
ejpam-5425	5	1	moreover	moreover	ADV
ejpam-5425	5	2	,	,	PUNCT
ejpam-5425	5	3	we	we	PRON
ejpam-5425	5	4	utilize	utilize	VERB
ejpam-5425	5	5	matlab	matlab	PROPN
ejpam-5425	5	6	programming	programming	NOUN
ejpam-5425	5	7	to	to	PART
ejpam-5425	5	8	simulate	simulate	VERB
ejpam-5425	5	9	the	the	DET
ejpam-5425	5	10	dynamics	dynamic	NOUN
ejpam-5425	5	11	and	and	CCONJ
ejpam-5425	5	12	validate	validate	VERB
ejpam-5425	5	13	our	our	PRON
ejpam-5425	5	14	results	result	NOUN
ejpam-5425	5	15	.	.	PUNCT
ejpam-5425	6	1	2020	2020	NUM
ejpam-5425	6	2	mathematics	mathematic	NOUN
ejpam-5425	6	3	subject	subject	NOUN
ejpam-5425	6	4	classifications	classification	NOUN
ejpam-5425	6	5	:	:	PUNCT
ejpam-5425	6	6	39a10	39a10	NUM
ejpam-5425	6	7	key	key	ADJ
ejpam-5425	6	8	words	word	NOUN
ejpam-5425	6	9	and	and	CCONJ
ejpam-5425	6	10	phrases	phrase	NOUN
ejpam-5425	6	11	:	:	PUNCT
ejpam-5425	6	12	solutions	solution	NOUN
ejpam-5425	6	13	of	of	ADP
ejpam-5425	6	14	difference	difference	NOUN
ejpam-5425	6	15	equations	equation	NOUN
ejpam-5425	6	16	,	,	PUNCT
ejpam-5425	6	17	periodic	periodic	ADJ
ejpam-5425	6	18	solution	solution	NOUN
ejpam-5425	6	19	,	,	PUNCT
ejpam-5425	6	20	recursive	recursive	ADJ
ejpam-5425	6	21	sequences	sequence	NOUN
ejpam-5425	6	22	1	1	NUM
ejpam-5425	6	23	.	.	PUNCT
ejpam-5425	6	24	introduction	introduction	NOUN
ejpam-5425	6	25	the	the	DET
ejpam-5425	6	26	final	final	ADJ
ejpam-5425	6	27	25	25	NUM
ejpam-5425	6	28	years	year	NOUN
ejpam-5425	6	29	of	of	ADP
ejpam-5425	6	30	the	the	DET
ejpam-5425	6	31	20th	20th	ADJ
ejpam-5425	6	32	century	century	NOUN
ejpam-5425	6	33	saw	see	VERB
ejpam-5425	6	34	significant	significant	ADJ
ejpam-5425	6	35	advancements	advancement	NOUN
ejpam-5425	6	36	in	in	ADP
ejpam-5425	6	37	the	the	DET
ejpam-5425	6	38	theory	theory	NOUN
ejpam-5425	6	39	of	of	ADP
ejpam-5425	6	40	discrete	discrete	ADJ
ejpam-5425	6	41	dynamical	dynamical	ADJ
ejpam-5425	6	42	systems	system	NOUN
ejpam-5425	6	43	(	(	PUNCT
ejpam-5425	6	44	ddss	ddss	NOUN
ejpam-5425	6	45	)	)	PUNCT
ejpam-5425	6	46	and	and	CCONJ
ejpam-5425	6	47	difference	difference	NOUN
ejpam-5425	6	48	equations	equation	NOUN
ejpam-5425	6	49	(	(	PUNCT
ejpam-5425	6	50	des	des	PROPN
ejpam-5425	6	51	)	)	PUNCT
ejpam-5425	6	52	.	.	PUNCT
ejpam-5425	7	1	recently	recently	ADV
ejpam-5425	7	2	,	,	PUNCT
ejpam-5425	7	3	a	a	DET
ejpam-5425	7	4	wide	wide	ADJ
ejpam-5425	7	5	variety	variety	NOUN
ejpam-5425	7	6	of	of	ADP
ejpam-5425	7	7	fields	field	NOUN
ejpam-5425	7	8	,	,	PUNCT
ejpam-5425	7	9	including	include	VERB
ejpam-5425	7	10	biology	biology	NOUN
ejpam-5425	7	11	,	,	PUNCT
ejpam-5425	7	12	economics	economic	NOUN
ejpam-5425	7	13	,	,	PUNCT
ejpam-5425	7	14	physics	physics	NOUN
ejpam-5425	7	15	,	,	PUNCT
ejpam-5425	7	16	resource	resource	NOUN
ejpam-5425	7	17	management	management	NOUN
ejpam-5425	7	18	,	,	PUNCT
ejpam-5425	7	19	and	and	CCONJ
ejpam-5425	7	20	others	other	NOUN
ejpam-5425	7	21	,	,	PUNCT
ejpam-5425	7	22	have	have	AUX
ejpam-5425	7	23	seen	see	VERB
ejpam-5425	7	24	the	the	DET
ejpam-5425	7	25	use	use	NOUN
ejpam-5425	7	26	of	of	ADP
ejpam-5425	7	27	ddss	ddss	NOUN
ejpam-5425	7	28	and	and	CCONJ
ejpam-5425	7	29	des.a	des.a	PUNCT
ejpam-5425	7	30	key	key	ADJ
ejpam-5425	7	31	role	role	NOUN
ejpam-5425	7	32	in	in	ADP
ejpam-5425	7	33	practical	practical	ADJ
ejpam-5425	7	34	analysis	analysis	NOUN
ejpam-5425	7	35	is	be	AUX
ejpam-5425	7	36	played	play	VERB
ejpam-5425	7	37	by	by	ADP
ejpam-5425	7	38	the	the	DET
ejpam-5425	7	39	theory	theory	NOUN
ejpam-5425	7	40	of	of	ADP
ejpam-5425	7	41	difes	dife	NOUN
ejpam-5425	7	42	.	.	PUNCT
ejpam-5425	8	1	it	it	PRON
ejpam-5425	8	2	is	be	AUX
ejpam-5425	8	3	improbable	improbable	ADJ
ejpam-5425	8	4	that	that	SCONJ
ejpam-5425	8	5	the	the	DET
ejpam-5425	8	6	theory	theory	NOUN
ejpam-5425	8	7	of	of	ADP
ejpam-5425	8	8	des	des	PROPN
ejpam-5425	8	9	wo	will	AUX
ejpam-5425	8	10	n’t	not	PART
ejpam-5425	8	11	keep	keep	VERB
ejpam-5425	8	12	playing	play	VERB
ejpam-5425	8	13	a	a	DET
ejpam-5425	8	14	significant	significant	ADJ
ejpam-5425	8	15	part	part	NOUN
ejpam-5425	8	16	in	in	ADP
ejpam-5425	8	17	mathematics	mathematic	NOUN
ejpam-5425	8	18	as	as	ADP
ejpam-5425	8	19	a	a	DET
ejpam-5425	8	20	whole	whole	NOUN
ejpam-5425	8	21	.	.	PUNCT
ejpam-5425	9	1	in	in	ADP
ejpam-5425	9	2	applications	application	NOUN
ejpam-5425	9	3	,	,	PUNCT
ejpam-5425	9	4	nonlinear	nonlinear	ADJ
ejpam-5425	9	5	difference	difference	NOUN
ejpam-5425	9	6	equations	equation	NOUN
ejpam-5425	9	7	(	(	PUNCT
ejpam-5425	9	8	ndes	nde	NOUN
ejpam-5425	9	9	)	)	PUNCT
ejpam-5425	9	10	of	of	ADP
ejpam-5425	9	11	order	order	NOUN
ejpam-5425	9	12	greater	great	ADJ
ejpam-5425	9	13	than	than	ADP
ejpam-5425	9	14	one	one	NUM
ejpam-5425	9	15	are	be	AUX
ejpam-5425	9	16	crucial	crucial	ADJ
ejpam-5425	9	17	.	.	PUNCT
ejpam-5425	10	1	these	these	DET
ejpam-5425	10	2	equations	equation	NOUN
ejpam-5425	10	3	also	also	ADV
ejpam-5425	10	4	naturally	naturally	ADV
ejpam-5425	10	5	arise	arise	VERB
ejpam-5425	10	6	as	as	ADP
ejpam-5425	10	7	discrete	discrete	ADJ
ejpam-5425	10	8	analogs	analog	NOUN
ejpam-5425	10	9	and	and	CCONJ
ejpam-5425	10	10	numerical	numerical	ADJ
ejpam-5425	10	11	solutions	solution	NOUN
ejpam-5425	10	12	of	of	ADP
ejpam-5425	10	13	differential	differential	NOUN
ejpam-5425	10	14	and	and	CCONJ
ejpam-5425	10	15	delay	delay	VERB
ejpam-5425	10	16	differential	differential	ADJ
ejpam-5425	10	17	equations	equation	NOUN
ejpam-5425	10	18	,	,	PUNCT
ejpam-5425	10	19	which	which	PRON
ejpam-5425	10	20	model	model	VERB
ejpam-5425	10	21	a	a	DET
ejpam-5425	10	22	wide	wide	ADJ
ejpam-5425	10	23	range	range	NOUN
ejpam-5425	10	24	of	of	ADP
ejpam-5425	10	25	diverse	diverse	ADJ
ejpam-5425	10	26	phenomena	phenomenon	NOUN
ejpam-5425	10	27	in	in	ADP
ejpam-5425	10	28	biology	biology	NOUN
ejpam-5425	10	29	,	,	PUNCT
ejpam-5425	10	30	ecology	ecology	NOUN
ejpam-5425	10	31	,	,	PUNCT
ejpam-5425	10	32	psychology	psychology	NOUN
ejpam-5425	10	33	,	,	PUNCT
ejpam-5425	10	34	engineering	engineering	NOUN
ejpam-5425	10	35	,	,	PUNCT
ejpam-5425	10	36	physics	physics	NOUN
ejpam-5425	10	37	,	,	PUNCT
ejpam-5425	10	38	probability	probability	NOUN
ejpam-5425	10	39	theory	theory	NOUN
ejpam-5425	10	40	,	,	PUNCT
ejpam-5425	10	41	economics	economic	NOUN
ejpam-5425	10	42	,	,	PUNCT
ejpam-5425	10	43	genetics	genetic	NOUN
ejpam-5425	10	44	.	.	PUNCT
ejpam-5425	11	1	finding	find	VERB
ejpam-5425	11	2	out	out	ADP
ejpam-5425	11	3	how	how	SCONJ
ejpam-5425	11	4	a	a	DET
ejpam-5425	11	5	system	system	NOUN
ejpam-5425	11	6	of	of	ADP
ejpam-5425	11	7	higher	high	ADJ
ejpam-5425	11	8	-	-	PUNCT
ejpam-5425	11	9	order	order	NOUN
ejpam-5425	11	10	rational	rational	ADJ
ejpam-5425	11	11	difference	difference	NOUN
ejpam-5425	11	12	equations	equation	NOUN
ejpam-5425	11	13	(	(	PUNCT
ejpam-5425	11	14	rdes	rde	NOUN
ejpam-5425	11	15	)	)	PUNCT
ejpam-5425	11	16	behaves	behave	NOUN
ejpam-5425	11	17	and	and	CCONJ
ejpam-5425	11	18	talking	talk	VERB
ejpam-5425	11	19	about	about	ADP
ejpam-5425	11	20	how	how	SCONJ
ejpam-5425	11	21	stable	stable	ADJ
ejpam-5425	11	22	its	its	PRON
ejpam-5425	11	23	equilibrium	equilibrium	NOUN
ejpam-5425	11	24	points	point	NOUN
ejpam-5425	11	25	are	be	AUX
ejpam-5425	11	26	locally	locally	ADV
ejpam-5425	11	27	asymptotically	asymptotically	ADV
ejpam-5425	11	28	is	be	AUX
ejpam-5425	11	29	quite	quite	ADV
ejpam-5425	11	30	interesting	interesting	ADJ
ejpam-5425	11	31	.	.	PUNCT
ejpam-5425	12	1	many	many	ADJ
ejpam-5425	12	2	articles	article	NOUN
ejpam-5425	12	3	cover	cover	VERB
ejpam-5425	12	4	the	the	DET
ejpam-5425	12	5	des	des	X
ejpam-5425	12	6	system	system	NOUN
ejpam-5425	12	7	[	[	X
ejpam-5425	12	8	1–22	1–22	NOUN
ejpam-5425	12	9	]	]	PUNCT
ejpam-5425	12	10	.	.	PUNCT
ejpam-5425	13	1	for	for	ADP
ejpam-5425	13	2	example	example	NOUN
ejpam-5425	13	3	,	,	PUNCT
ejpam-5425	13	4	in	in	ADP
ejpam-5425	13	5	[	[	PUNCT
ejpam-5425	13	6	7	7	X
ejpam-5425	13	7	]	]	X
ejpam-5425	13	8	el	el	NOUN
ejpam-5425	13	9	-	-	PROPN
ejpam-5425	13	10	metwally	metwally	ADV
ejpam-5425	13	11	discussed	discuss	VERB
ejpam-5425	13	12	how	how	SCONJ
ejpam-5425	13	13	various	various	ADJ
ejpam-5425	13	14	systems	system	NOUN
ejpam-5425	13	15	of	of	ADP
ejpam-5425	13	16	third	third	ADJ
ejpam-5425	13	17	order	order	NOUN
ejpam-5425	13	18	rdes	rde	NOUN
ejpam-5425	13	19	with	with	ADP
ejpam-5425	13	20	initial	initial	ADJ
ejpam-5425	13	21	conditions	condition	NOUN
ejpam-5425	13	22	involving	involve	VERB
ejpam-5425	13	23	non	non	ADJ
ejpam-5425	13	24	-	-	ADJ
ejpam-5425	13	25	zero	zero	ADJ
ejpam-5425	13	26	real	real	ADJ
ejpam-5425	13	27	numbers	number	NOUN
ejpam-5425	13	28	should	should	AUX
ejpam-5425	13	29	be	be	AUX
ejpam-5425	13	30	solved	solve	VERB
ejpam-5425	13	31	.	.	PUNCT
ejpam-5425	14	1	additionally	additionally	ADV
ejpam-5425	14	2	,	,	PUNCT
ejpam-5425	14	3	doi	doi	INTJ
ejpam-5425	14	4	:	:	PUNCT
ejpam-5425	14	5	https://doi.org/10.29020/nybg.ejpam.v17i4.5425	https://doi.org/10.29020/nybg.ejpam.v17i4.5425	NUM
ejpam-5425	14	6	email	email	NOUN
ejpam-5425	14	7	address	address	NOUN
ejpam-5425	14	8	:	:	PUNCT
ejpam-5425	14	9	jo.gh@tu.edu.sa	jo.gh@tu.edu.sa	PROPN
ejpam-5425	14	10	(	(	PUNCT
ejpam-5425	14	11	j.	j.	PROPN
ejpam-5425	14	12	g.	g.	PROPN
ejpam-5425	14	13	al	al	PROPN
ejpam-5425	14	14	-	-	PUNCT
ejpam-5425	14	15	juaid	juaid	PROPN
ejpam-5425	14	16	)	)	PUNCT
ejpam-5425	14	17	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5425	15	1	3254	3254	NUM
ejpam-5425	15	2	copyright	copyright	NOUN
ejpam-5425	15	3	:	:	PUNCT
ejpam-5425	15	4	©	©	PROPN
ejpam-5425	15	5	2024	2024	NUM
ejpam-5425	15	6	the	the	DET
ejpam-5425	15	7	author(s	author(s	NOUN
ejpam-5425	15	8	)	)	PUNCT
ejpam-5425	15	9	.	.	PUNCT
ejpam-5425	16	1	(	(	PUNCT
ejpam-5425	16	2	cc	cc	NOUN
ejpam-5425	16	3	by	by	ADP
ejpam-5425	16	4	-	-	PUNCT
ejpam-5425	16	5	nc	nc	PROPN
ejpam-5425	16	6	4.0	4.0	NUM
ejpam-5425	16	7	)	)	PUNCT
ejpam-5425	16	8	j.	j.	PROPN
ejpam-5425	16	9	g.	g.	PROPN
ejpam-5425	16	10	al	al	PROPN
ejpam-5425	16	11	-	-	PUNCT
ejpam-5425	16	12	juaid	juaid	PROPN
ejpam-5425	16	13	/	/	SYM
ejpam-5425	16	14	eur	eur	PROPN
ejpam-5425	16	15	.	.	PUNCT
ejpam-5425	17	1	j.	j.	PROPN
ejpam-5425	17	2	pure	pure	PROPN
ejpam-5425	17	3	appl	appl	PROPN
ejpam-5425	17	4	.	.	PROPN
ejpam-5425	17	5	math	math	PROPN
ejpam-5425	17	6	,	,	PUNCT
ejpam-5425	17	7	17	17	NUM
ejpam-5425	17	8	(	(	PUNCT
ejpam-5425	17	9	4	4	NUM
ejpam-5425	17	10	)	)	PUNCT
ejpam-5425	17	11	(	(	PUNCT
ejpam-5425	17	12	2024	2024	NUM
ejpam-5425	17	13	)	)	PUNCT
ejpam-5425	17	14	,	,	PUNCT
ejpam-5425	17	15	3254	3254	NUM
ejpam-5425	17	16	-	-	SYM
ejpam-5425	17	17	3267	3267	NUM
ejpam-5425	17	18	3255	3255	NUM
ejpam-5425	17	19	he	he	PRON
ejpam-5425	17	20	showed	show	VERB
ejpam-5425	17	21	additional	additional	ADJ
ejpam-5425	17	22	numerical	numerical	ADJ
ejpam-5425	17	23	examples	example	NOUN
ejpam-5425	17	24	and	and	CCONJ
ejpam-5425	17	25	checked	check	VERB
ejpam-5425	17	26	into	into	ADP
ejpam-5425	17	27	some	some	PRON
ejpam-5425	17	28	of	of	ADP
ejpam-5425	17	29	the	the	DET
ejpam-5425	17	30	properties	property	NOUN
ejpam-5425	17	31	of	of	ADP
ejpam-5425	17	32	the	the	DET
ejpam-5425	17	33	obtained	obtain	VERB
ejpam-5425	17	34	solutions	solution	NOUN
ejpam-5425	17	35	tn+1	tn+1	NOUN
ejpam-5425	17	36	=	=	SYM
ejpam-5425	17	37	tn−1rn	tn−1rn	NUM
ejpam-5425	17	38	±tn−1	±tn−1	PROPN
ejpam-5425	17	39	±rn−2	±rn−2	PROPN
ejpam-5425	17	40	,	,	PUNCT
ejpam-5425	17	41	rn+1	rn+1	VERB
ejpam-5425	17	42	=	=	SYM
ejpam-5425	17	43	tnrn−1	tnrn−1	PROPN
ejpam-5425	17	44	±rn−1	±rn−1	ADV
ejpam-5425	17	45	±	±	NOUN
ejpam-5425	17	46	tn−2	tn−2	ADV
ejpam-5425	17	47	.	.	PUNCT
ejpam-5425	18	1	el	el	NOUN
ejpam-5425	18	2	-	-	NOUN
ejpam-5425	18	3	dessoky	dessoky	NOUN
ejpam-5425	19	1	[	[	X
ejpam-5425	19	2	6	6	NUM
ejpam-5425	19	3	]	]	PUNCT
ejpam-5425	19	4	studied	study	VERB
ejpam-5425	19	5	the	the	DET
ejpam-5425	19	6	presence	presence	NOUN
ejpam-5425	19	7	of	of	ADP
ejpam-5425	19	8	solutions	solution	NOUN
ejpam-5425	19	9	in	in	ADP
ejpam-5425	19	10	the	the	DET
ejpam-5425	19	11	case	case	NOUN
ejpam-5425	19	12	of	of	ADP
ejpam-5425	19	13	four	four	NUM
ejpam-5425	19	14	dimensions	dimension	NOUN
ejpam-5425	19	15	for	for	ADP
ejpam-5425	19	16	a	a	DET
ejpam-5425	19	17	class	class	NOUN
ejpam-5425	19	18	of	of	ADP
ejpam-5425	19	19	rational	rational	ADJ
ejpam-5425	19	20	systems	system	NOUN
ejpam-5425	19	21	of	of	ADP
ejpam-5425	19	22	differential	differential	ADJ
ejpam-5425	19	23	equations	equation	NOUN
ejpam-5425	19	24	(	(	PUNCT
ejpam-5425	19	25	sdes	sde	NOUN
ejpam-5425	19	26	)	)	PUNCT
ejpam-5425	19	27	of	of	ADP
ejpam-5425	19	28	order	order	NOUN
ejpam-5425	19	29	four	four	NUM
ejpam-5425	19	30	.	.	PUNCT
ejpam-5425	19	31	xn+1	xn+1	PROPN
ejpam-5425	20	1	=	=	SYM
ejpam-5425	20	2	xn−3	xn−3	PROPN
ejpam-5425	20	3	±1±	±1±	X
ejpam-5425	20	4	tnzn−1yn−2xn−3	tnzn−1yn−2xn−3	NOUN
ejpam-5425	20	5	,	,	PUNCT
ejpam-5425	20	6	yn+1	yn+1	PROPN
ejpam-5425	20	7	=	=	PROPN
ejpam-5425	20	8	yn−3	yn−3	PROPN
ejpam-5425	20	9	±1±xntn−1zn−2yn−3	±1±xntn−1zn−2yn−3	NOUN
ejpam-5425	20	10	.	.	PUNCT
ejpam-5425	21	1	zn+1	zn+1	X
ejpam-5425	21	2	=	=	SYM
ejpam-5425	21	3	zn−3	zn−3	PROPN
ejpam-5425	21	4	±1±	±1±	PART
ejpam-5425	21	5	ynxn−1tn−2zn−3	ynxn−1tn−2zn−3	PROPN
ejpam-5425	21	6	,	,	PUNCT
ejpam-5425	21	7	tn+1	tn+1	NOUN
ejpam-5425	21	8	=	=	SYM
ejpam-5425	21	9	tn−3	tn−3	PROPN
ejpam-5425	21	10	±1±	±1±	X
ejpam-5425	21	11	znyn−1xn−2tn−3	znyn−1xn−2tn−3	NOUN
ejpam-5425	21	12	,	,	PUNCT
ejpam-5425	21	13	elsayed	elsaye	VERB
ejpam-5425	21	14	and	and	CCONJ
ejpam-5425	21	15	alof	alof	ADJ
ejpam-5425	22	1	[	[	X
ejpam-5425	22	2	11	11	NUM
ejpam-5425	22	3	]	]	PUNCT
ejpam-5425	22	4	obtained	obtain	VERB
ejpam-5425	22	5	formulas	formula	NOUN
ejpam-5425	22	6	expressions	expression	NOUN
ejpam-5425	22	7	for	for	ADP
ejpam-5425	22	8	solutions	solution	NOUN
ejpam-5425	22	9	of	of	ADP
ejpam-5425	22	10	the	the	DET
ejpam-5425	22	11	following	follow	VERB
ejpam-5425	22	12	fractional	fractional	ADJ
ejpam-5425	22	13	sdes	sde	NOUN
ejpam-5425	22	14	xn+1	xn+1	PROPN
ejpam-5425	22	15	=	=	SYM
ejpam-5425	22	16	xn−3yn−4	xn−3yn−4	PROPN
ejpam-5425	22	17	yn(±1−xn−3yn−4rn−1tn−2	yn(±1−xn−3yn−4rn−1tn−2	PROPN
ejpam-5425	22	18	)	)	PUNCT
ejpam-5425	22	19	,	,	PUNCT
ejpam-5425	22	20	yn+1	yn+1	PROPN
ejpam-5425	22	21	=	=	SYM
ejpam-5425	22	22	yn−3rn−4	yn−3rn−4	PROPN
ejpam-5425	22	23	rn(±1−	rn(±1−	NUM
ejpam-5425	22	24	yn−3rn−4tn−1xn−2	yn−3rn−4tn−1xn−2	NUM
ejpam-5425	22	25	)	)	PUNCT
ejpam-5425	22	26	,	,	PUNCT
ejpam-5425	22	27	rn+1	rn+1	CCONJ
ejpam-5425	22	28	=	=	SYM
ejpam-5425	22	29	rn−3tn−4	rn−3tn−4	X
ejpam-5425	22	30	tn(±1±rn−3tn−4xn−1yn−2	tn(±1±rn−3tn−4xn−1yn−2	PROPN
ejpam-5425	22	31	)	)	PUNCT
ejpam-5425	22	32	,	,	PUNCT
ejpam-5425	22	33	tn+1	tn+1	NOUN
ejpam-5425	22	34	=	=	SYM
ejpam-5425	22	35	tn−3xn−4	tn−3xn−4	X
ejpam-5425	22	36	xn(±1±	xn(±1±	PROPN
ejpam-5425	22	37	tn−3xn−4yn−1rn−2	tn−3xn−4yn−1rn−2	PROPN
ejpam-5425	22	38	)	)	PUNCT
ejpam-5425	22	39	.	.	PUNCT
ejpam-5425	23	1	mansour	mansour	PROPN
ejpam-5425	23	2	et	et	PROPN
ejpam-5425	23	3	al	al	PROPN
ejpam-5425	23	4	.	.	PUNCT
ejpam-5425	24	1	[	[	X
ejpam-5425	24	2	20	20	NUM
ejpam-5425	24	3	]	]	PUNCT
ejpam-5425	24	4	checked	check	VERB
ejpam-5425	24	5	the	the	DET
ejpam-5425	24	6	behavior	behavior	NOUN
ejpam-5425	24	7	of	of	ADP
ejpam-5425	24	8	solutions	solution	NOUN
ejpam-5425	24	9	of	of	ADP
ejpam-5425	24	10	the	the	DET
ejpam-5425	24	11	sdes	sde	NOUN
ejpam-5425	24	12	wn+1	wn+1	VERB
ejpam-5425	24	13	=	=	SYM
ejpam-5425	24	14	wn−5	wn−5	PROPN
ejpam-5425	24	15	−1	−1	NOUN
ejpam-5425	24	16	+	+	ADJ
ejpam-5425	24	17	wn−5rn−2	wn−5rn−2	ADJ
ejpam-5425	24	18	,	,	PUNCT
ejpam-5425	24	19	rn+1	rn+1	X
ejpam-5425	24	20	=	=	VERB
ejpam-5425	24	21	rn−5	rn−5	NOUN
ejpam-5425	24	22	±1±rn−5wn−2	±1±rn−5wn−2	NOUN
ejpam-5425	24	23	.	.	PUNCT
ejpam-5425	25	1	des	de	NOUN
ejpam-5425	25	2	are	be	AUX
ejpam-5425	25	3	also	also	ADV
ejpam-5425	25	4	suitable	suitable	ADJ
ejpam-5425	25	5	models	model	NOUN
ejpam-5425	25	6	to	to	PART
ejpam-5425	25	7	describe	describe	VERB
ejpam-5425	25	8	circumstances	circumstance	NOUN
ejpam-5425	25	9	where	where	SCONJ
ejpam-5425	25	10	overlapping	overlap	VERB
ejpam-5425	25	11	generations	generation	NOUN
ejpam-5425	25	12	and	and	CCONJ
ejpam-5425	25	13	seasonal	seasonal	ADJ
ejpam-5425	25	14	population	population	NOUN
ejpam-5425	25	15	growth	growth	NOUN
ejpam-5425	25	16	occur	occur	VERB
ejpam-5425	25	17	.	.	PUNCT
ejpam-5425	26	1	the	the	DET
ejpam-5425	26	2	generalized	generalized	ADJ
ejpam-5425	26	3	beverton	beverton	PROPN
ejpam-5425	26	4	-	-	PUNCT
ejpam-5425	26	5	holt	holt	PROPN
ejpam-5425	26	6	stock	stock	PROPN
ejpam-5425	26	7	recruitment	recruitment	NOUN
ejpam-5425	26	8	mode	mode	NOUN
ejpam-5425	26	9	has	have	AUX
ejpam-5425	26	10	been	be	AUX
ejpam-5425	26	11	studied	study	VERB
ejpam-5425	26	12	by	by	ADP
ejpam-5425	26	13	researchers	researcher	NOUN
ejpam-5425	26	14	in	in	ADP
ejpam-5425	26	15	[	[	X
ejpam-5425	26	16	3	3	NUM
ejpam-5425	26	17	]	]	SYM
ejpam-5425	26	18	tn+1	tn+1	NOUN
ejpam-5425	26	19	=	=	NOUN
ejpam-5425	27	1	αtn	αtn	NOUN
ejpam-5425	28	1	+	+	CCONJ
ejpam-5425	28	2	βtn−1	βtn−1	PROPN
ejpam-5425	28	3	1	1	NUM
ejpam-5425	29	1	+	+	CCONJ
ejpam-5425	29	2	γtn−1	γtn−1	PROPN
ejpam-5425	29	3	+	+	CCONJ
ejpam-5425	29	4	ηtn	ηtn	INTJ
ejpam-5425	29	5	.	.	PUNCT
ejpam-5425	30	1	the	the	DET
ejpam-5425	30	2	following	follow	VERB
ejpam-5425	30	3	system	system	NOUN
ejpam-5425	30	4	of	of	ADP
ejpam-5425	30	5	discrete	discrete	ADJ
ejpam-5425	30	6	-	-	PUNCT
ejpam-5425	30	7	time	time	NOUN
ejpam-5425	30	8	two	two	NUM
ejpam-5425	30	9	-	-	PUNCT
ejpam-5425	30	10	predators	predator	NOUN
ejpam-5425	30	11	and	and	CCONJ
ejpam-5425	30	12	the	the	DET
ejpam-5425	30	13	one	one	NUM
ejpam-5425	30	14	-	-	PUNCT
ejpam-5425	30	15	prey	prey	NOUN
ejpam-5425	30	16	lot	lot	NOUN
ejpam-5425	30	17	was	be	AUX
ejpam-5425	30	18	explored	explore	VERB
ejpam-5425	30	19	dynamically	dynamically	ADV
ejpam-5425	30	20	by	by	ADP
ejpam-5425	30	21	khaliq	khaliq	PROPN
ejpam-5425	30	22	et	et	PROPN
ejpam-5425	30	23	al.[14	al.[14	PROPN
ejpam-5425	30	24	]	]	PUNCT
ejpam-5425	30	25	xn+1	xn+1	PROPN
ejpam-5425	31	1	=	=	PUNCT
ejpam-5425	31	2	axn	axn	PROPN
ejpam-5425	31	3	−	−	PROPN
ejpam-5425	31	4	cxnyn	cxnyn	NOUN
ejpam-5425	31	5	−	−	NOUN
ejpam-5425	31	6	exnzn	exnzn	NOUN
ejpam-5425	31	7	1	1	NUM
ejpam-5425	31	8	+	+	CCONJ
ejpam-5425	31	9	dxn	dxn	ADJ
ejpam-5425	31	10	,	,	PUNCT
ejpam-5425	31	11	yn+1	yn+1	PROPN
ejpam-5425	31	12	=	=	SYM
ejpam-5425	31	13	byn	byn	PROPN
ejpam-5425	31	14	+	+	CCONJ
ejpam-5425	31	15	txnyn	txnyn	NOUN
ejpam-5425	31	16	−	−	PROPN
ejpam-5425	31	17	pynzn	pynzn	NOUN
ejpam-5425	31	18	1	1	NUM
ejpam-5425	31	19	+	+	CCONJ
ejpam-5425	31	20	wyn	wyn	PROPN
ejpam-5425	31	21	,	,	PUNCT
ejpam-5425	31	22	zn+1	zn+1	PROPN
ejpam-5425	31	23	=	=	SYM
ejpam-5425	31	24	fzn	fzn	NOUN
ejpam-5425	31	25	+	+	CCONJ
ejpam-5425	31	26	rxnzn	rxnzn	NOUN
ejpam-5425	31	27	−	−	PROPN
ejpam-5425	31	28	hynzn	hynzn	NOUN
ejpam-5425	31	29	1	1	NUM
ejpam-5425	31	30	+	+	CCONJ
ejpam-5425	31	31	kzn	kzn	NOUN
ejpam-5425	31	32	.	.	PUNCT
ejpam-5425	32	1	din	din	VERB
ejpam-5425	32	2	and	and	CCONJ
ejpam-5425	32	3	elsayed	elsaye	VERB
ejpam-5425	32	4	[	[	X
ejpam-5425	32	5	5	5	NUM
ejpam-5425	32	6	]	]	PUNCT
ejpam-5425	32	7	evaluated	evaluate	VERB
ejpam-5425	32	8	the	the	DET
ejpam-5425	32	9	two	two	NUM
ejpam-5425	32	10	-	-	PUNCT
ejpam-5425	32	11	directional	directional	ADJ
ejpam-5425	32	12	interacting	interacting	ADJ
ejpam-5425	32	13	and	and	CCONJ
ejpam-5425	32	14	invasive	invasive	ADJ
ejpam-5425	32	15	species	specie	NOUN
ejpam-5425	32	16	model	model	NOUN
ejpam-5425	32	17	’s	’s	PART
ejpam-5425	32	18	boundedness	boundedness	PROPN
ejpam-5425	32	19	nature	nature	NOUN
ejpam-5425	32	20	,	,	PUNCT
ejpam-5425	32	21	persistence	persistence	NOUN
ejpam-5425	32	22	,	,	PUNCT
ejpam-5425	32	23	local	local	ADJ
ejpam-5425	32	24	and	and	CCONJ
ejpam-5425	32	25	global	global	ADJ
ejpam-5425	32	26	behavior	behavior	NOUN
ejpam-5425	32	27	rn+1	rn+1	X
ejpam-5425	32	28	=	=	SYM
ejpam-5425	32	29	η	η	PROPN
ejpam-5425	32	30	+	+	PROPN
ejpam-5425	32	31	αrn	αrn	PROPN
ejpam-5425	33	1	+	+	CCONJ
ejpam-5425	33	2	βrn−1e	βrn−1e	NOUN
ejpam-5425	33	3	−tn	−tn	NOUN
ejpam-5425	33	4	,	,	PUNCT
ejpam-5425	33	5	tn+1	tn+1	NOUN
ejpam-5425	33	6	=	=	SYM
ejpam-5425	33	7	γ	γ	X
ejpam-5425	33	8	+	+	NOUN
ejpam-5425	33	9	ctn	ctn	NOUN
ejpam-5425	33	10	+	+	CCONJ
ejpam-5425	34	1	wtn−1e	wtn−1e	PROPN
ejpam-5425	34	2	−rn	−rn	INTJ
ejpam-5425	34	3	.	.	PUNCT
ejpam-5425	35	1	in	in	ADP
ejpam-5425	35	2	a	a	DET
ejpam-5425	35	3	discrete	discrete	ADJ
ejpam-5425	35	4	-	-	PUNCT
ejpam-5425	35	5	time	time	NOUN
ejpam-5425	35	6	covid-19	covid-19	PROPN
ejpam-5425	35	7	epidemic	epidemic	NOUN
ejpam-5425	35	8	model	model	NOUN
ejpam-5425	35	9	,	,	PUNCT
ejpam-5425	35	10	the	the	DET
ejpam-5425	35	11	authors	author	NOUN
ejpam-5425	35	12	[	[	X
ejpam-5425	35	13	18	18	NUM
ejpam-5425	35	14	]	]	PUNCT
ejpam-5425	35	15	used	use	VERB
ejpam-5425	35	16	chaos	chaos	NOUN
ejpam-5425	35	17	management	management	NOUN
ejpam-5425	35	18	,	,	PUNCT
ejpam-5425	35	19	bifurcation	bifurcation	NOUN
ejpam-5425	35	20	analysis	analysis	NOUN
ejpam-5425	35	21	,	,	PUNCT
ejpam-5425	35	22	and	and	CCONJ
ejpam-5425	35	23	topological	topological	ADJ
ejpam-5425	35	24	classifications	classification	NOUN
ejpam-5425	35	25	to	to	PART
ejpam-5425	35	26	study	study	VERB
ejpam-5425	35	27	local	local	ADJ
ejpam-5425	35	28	dynamics	dynamic	NOUN
ejpam-5425	35	29	.	.	PUNCT
ejpam-5425	36	1	see	see	VERB
ejpam-5425	36	2	also	also	ADV
ejpam-5425	36	3	[	[	X
ejpam-5425	36	4	19	19	NUM
ejpam-5425	36	5	,	,	PUNCT
ejpam-5425	36	6	21	21	NUM
ejpam-5425	36	7	,	,	PUNCT
ejpam-5425	36	8	22	22	NUM
ejpam-5425	36	9	]	]	PUNCT
ejpam-5425	36	10	.	.	PUNCT
ejpam-5425	37	1	j.	j.	PROPN
ejpam-5425	37	2	g.	g.	PROPN
ejpam-5425	37	3	al	al	PROPN
ejpam-5425	37	4	-	-	PUNCT
ejpam-5425	37	5	juaid	juaid	PROPN
ejpam-5425	37	6	/	/	SYM
ejpam-5425	37	7	eur	eur	PROPN
ejpam-5425	37	8	.	.	PUNCT
ejpam-5425	38	1	j.	j.	PROPN
ejpam-5425	38	2	pure	pure	PROPN
ejpam-5425	38	3	appl	appl	PROPN
ejpam-5425	38	4	.	.	PROPN
ejpam-5425	38	5	math	math	PROPN
ejpam-5425	38	6	,	,	PUNCT
ejpam-5425	38	7	17	17	NUM
ejpam-5425	38	8	(	(	PUNCT
ejpam-5425	38	9	4	4	NUM
ejpam-5425	38	10	)	)	PUNCT
ejpam-5425	38	11	(	(	PUNCT
ejpam-5425	38	12	2024	2024	NUM
ejpam-5425	38	13	)	)	PUNCT
ejpam-5425	38	14	,	,	PUNCT
ejpam-5425	38	15	3254	3254	NUM
ejpam-5425	38	16	-	-	SYM
ejpam-5425	38	17	3267	3267	NUM
ejpam-5425	38	18	3256	3256	NUM
ejpam-5425	38	19	motivated	motivate	VERB
ejpam-5425	38	20	by	by	ADP
ejpam-5425	38	21	the	the	DET
ejpam-5425	38	22	aforementioned	aforementione	VERB
ejpam-5425	38	23	researches	research	NOUN
ejpam-5425	38	24	,	,	PUNCT
ejpam-5425	38	25	the	the	DET
ejpam-5425	38	26	objective	objective	NOUN
ejpam-5425	38	27	of	of	ADP
ejpam-5425	38	28	this	this	DET
ejpam-5425	38	29	work	work	NOUN
ejpam-5425	38	30	is	be	AUX
ejpam-5425	38	31	to	to	PART
ejpam-5425	38	32	ascertain	ascertain	VERB
ejpam-5425	38	33	whether	whether	SCONJ
ejpam-5425	38	34	solutions	solution	NOUN
ejpam-5425	38	35	to	to	ADP
ejpam-5425	38	36	the	the	DET
ejpam-5425	38	37	sdes	sde	NOUN
ejpam-5425	38	38	exist	exist	VERB
ejpam-5425	38	39	in	in	ADP
ejpam-5425	38	40	the	the	DET
ejpam-5425	38	41	two	two	NUM
ejpam-5425	38	42	-	-	PUNCT
ejpam-5425	38	43	dimensional	dimensional	ADJ
ejpam-5425	38	44	instances	instance	NOUN
ejpam-5425	38	45	tn+1	tn+1	NOUN
ejpam-5425	38	46	=	=	SYM
ejpam-5425	38	47	enen−4	enen−4	NOUN
ejpam-5425	38	48	tn−3(±1±	tn−3(±1±	NOUN
ejpam-5425	38	49	enen−4	enen−4	NOUN
ejpam-5425	38	50	)	)	PUNCT
ejpam-5425	38	51	,	,	PUNCT
ejpam-5425	38	52	en+1	en+1	ADJ
ejpam-5425	38	53	=	=	SYM
ejpam-5425	38	54	tntn−4	tntn−4	X
ejpam-5425	38	55	en−3(±1±	en−3(±1±	NOUN
ejpam-5425	38	56	tntn−4	tntn−4	NOUN
ejpam-5425	38	57	)	)	PUNCT
ejpam-5425	38	58	,	,	PUNCT
ejpam-5425	38	59	the	the	DET
ejpam-5425	38	60	initial	initial	ADJ
ejpam-5425	38	61	conditions	condition	NOUN
ejpam-5425	38	62	are	be	AUX
ejpam-5425	38	63	arbitrary	arbitrary	ADJ
ejpam-5425	38	64	nonzero	nonzero	ADJ
ejpam-5425	38	65	real	real	ADJ
ejpam-5425	38	66	numbers	number	NOUN
ejpam-5425	38	67	.	.	PUNCT
ejpam-5425	39	1	2	2	X
ejpam-5425	39	2	.	.	X
ejpam-5425	39	3	the	the	DET
ejpam-5425	39	4	system	system	NOUN
ejpam-5425	39	5	tn+1	tn+1	NOUN
ejpam-5425	39	6	=	=	SYM
ejpam-5425	39	7	enen−4	enen−4	NOUN
ejpam-5425	39	8	tn−3(1+enen−4	tn−3(1+enen−4	NUM
ejpam-5425	39	9	)	)	PUNCT
ejpam-5425	39	10	,	,	PUNCT
ejpam-5425	39	11	en+1	en+1	VERB
ejpam-5425	39	12	=	=	PROPN
ejpam-5425	39	13	tntn−4	tntn−4	NUM
ejpam-5425	39	14	en−3(−1+tntn−4	en−3(−1+tntn−4	NOUN
ejpam-5425	39	15	)	)	PUNCT
ejpam-5425	39	16	in	in	ADP
ejpam-5425	39	17	this	this	DET
ejpam-5425	39	18	section	section	NOUN
ejpam-5425	39	19	,	,	PUNCT
ejpam-5425	39	20	we	we	PRON
ejpam-5425	39	21	give	give	VERB
ejpam-5425	39	22	a	a	DET
ejpam-5425	39	23	specific	specific	ADJ
ejpam-5425	39	24	form	form	NOUN
ejpam-5425	39	25	the	the	DET
ejpam-5425	39	26	solutions	solution	NOUN
ejpam-5425	39	27	of	of	ADP
ejpam-5425	39	28	the	the	DET
ejpam-5425	39	29	sde	sde	PROPN
ejpam-5425	39	30	in	in	ADP
ejpam-5425	39	31	the	the	DET
ejpam-5425	39	32	form	form	NOUN
ejpam-5425	39	33	:	:	PUNCT
ejpam-5425	39	34	tn+1	tn+1	NOUN
ejpam-5425	39	35	=	=	SYM
ejpam-5425	39	36	enen−4	enen−4	X
ejpam-5425	39	37	tn−3(1	tn−3(1	NOUN
ejpam-5425	39	38	+	+	NUM
ejpam-5425	39	39	enen−4	enen−4	NOUN
ejpam-5425	39	40	)	)	PUNCT
ejpam-5425	39	41	,	,	PUNCT
ejpam-5425	39	42	en+1	en+1	ADJ
ejpam-5425	39	43	=	=	SYM
ejpam-5425	39	44	tntn−4	tntn−4	X
ejpam-5425	39	45	en−3(−1	en−3(−1	X
ejpam-5425	39	46	+	+	CCONJ
ejpam-5425	39	47	tntn−4	tntn−4	NOUN
ejpam-5425	39	48	)	)	PUNCT
ejpam-5425	39	49	.	.	PUNCT
ejpam-5425	40	1	(	(	PUNCT
ejpam-5425	40	2	1	1	X
ejpam-5425	40	3	)	)	PUNCT
ejpam-5425	40	4	theorem	theorem	NOUN
ejpam-5425	40	5	1	1	NUM
ejpam-5425	40	6	.	.	X
ejpam-5425	41	1	assume	assume	VERB
ejpam-5425	41	2	{	{	PUNCT
ejpam-5425	41	3	tn	tn	PROPN
ejpam-5425	41	4	,	,	PUNCT
ejpam-5425	41	5	en	en	ADP
ejpam-5425	41	6	}	}	PUNCT
ejpam-5425	41	7	are	be	AUX
ejpam-5425	41	8	solutions	solution	NOUN
ejpam-5425	41	9	of	of	ADP
ejpam-5425	41	10	eq.(1	eq.(1	ADJ
ejpam-5425	41	11	)	)	PUNCT
ejpam-5425	41	12	.	.	PUNCT
ejpam-5425	42	1	then	then	ADV
ejpam-5425	42	2	,	,	PUNCT
ejpam-5425	42	3	for	for	ADP
ejpam-5425	42	4	all	all	DET
ejpam-5425	42	5	n	n	NOUN
ejpam-5425	42	6	=	=	SYM
ejpam-5425	42	7	0	0	NUM
ejpam-5425	42	8	,	,	PUNCT
ejpam-5425	42	9	1	1	NUM
ejpam-5425	42	10	,	,	PUNCT
ejpam-5425	42	11	2	2	NUM
ejpam-5425	42	12	,	,	PUNCT
ejpam-5425	42	13	...	...	PUNCT
ejpam-5425	42	14	,	,	PUNCT
ejpam-5425	42	15	the	the	DET
ejpam-5425	42	16	following	follow	VERB
ejpam-5425	42	17	formulas	formula	NOUN
ejpam-5425	42	18	yield	yield	VERB
ejpam-5425	42	19	periodic	periodic	ADJ
ejpam-5425	42	20	solutions	solution	NOUN
ejpam-5425	42	21	of	of	ADP
ejpam-5425	42	22	eq	eq	PROPN
ejpam-5425	42	23	.	.	PUNCT
ejpam-5425	43	1	(	(	PUNCT
ejpam-5425	43	2	1	1	NUM
ejpam-5425	43	3	)	)	PUNCT
ejpam-5425	43	4	with	with	ADP
ejpam-5425	43	5	period	period	NOUN
ejpam-5425	43	6	eight	eight	NUM
ejpam-5425	43	7	.	.	PUNCT
ejpam-5425	44	1	t8n−4	t8n−4	NOUN
ejpam-5425	44	2	=	=	SYM
ejpam-5425	44	3	s	s	PROPN
ejpam-5425	44	4	,	,	PUNCT
ejpam-5425	44	5	t8n−3	t8n−3	PROPN
ejpam-5425	44	6	=	=	SYM
ejpam-5425	45	1	d	d	PROPN
ejpam-5425	45	2	,	,	PUNCT
ejpam-5425	45	3	t8n−2	t8n−2	PROPN
ejpam-5425	45	4	=	=	SYM
ejpam-5425	45	5	c	c	NOUN
ejpam-5425	45	6	,	,	PUNCT
ejpam-5425	45	7	t8n−1	t8n−1	PROPN
ejpam-5425	45	8	=	=	SYM
ejpam-5425	45	9	b	b	PROPN
ejpam-5425	45	10	,	,	PUNCT
ejpam-5425	45	11	t8n	t8n	NOUN
ejpam-5425	45	12	=	=	SYM
ejpam-5425	45	13	a	a	PRON
ejpam-5425	45	14	,	,	PUNCT
ejpam-5425	45	15	t8n+1	t8n+1	ADV
ejpam-5425	45	16	=	=	PUNCT
ejpam-5425	46	1	fw	fw	INTJ
ejpam-5425	46	2	d(1	d(1	PROPN
ejpam-5425	46	3	+	+	CCONJ
ejpam-5425	46	4	fw	fw	X
ejpam-5425	46	5	)	)	PUNCT
ejpam-5425	46	6	,	,	PUNCT
ejpam-5425	46	7	t8n+2	t8n+2	PROPN
ejpam-5425	47	1	=	=	PUNCT
ejpam-5425	47	2	as	as	ADP
ejpam-5425	47	3	c(−1	c(−1	PROPN
ejpam-5425	47	4	+	+	X
ejpam-5425	47	5	2as	2as	NUM
ejpam-5425	47	6	)	)	PUNCT
ejpam-5425	47	7	,	,	PUNCT
ejpam-5425	47	8	t8n+3	t8n+3	PUNCT
ejpam-5425	47	9	=	=	SYM
ejpam-5425	47	10	−fw	−fw	NOUN
ejpam-5425	47	11	b(1−	b(1−	PROPN
ejpam-5425	47	12	fw	fw	PROPN
ejpam-5425	47	13	)	)	PUNCT
ejpam-5425	47	14	,	,	PUNCT
ejpam-5425	47	15	e8n−4	e8n−4	NOUN
ejpam-5425	47	16	=	=	SYM
ejpam-5425	47	17	w	w	PROPN
ejpam-5425	47	18	,	,	PUNCT
ejpam-5425	47	19	e8n−3	e8n−3	PROPN
ejpam-5425	47	20	=	=	SYM
ejpam-5425	47	21	m	m	PROPN
ejpam-5425	47	22	,	,	PUNCT
ejpam-5425	47	23	e8n−2	e8n−2	PROPN
ejpam-5425	47	24	=	=	SYM
ejpam-5425	47	25	g	g	PROPN
ejpam-5425	47	26	,	,	PUNCT
ejpam-5425	47	27	e8n−1	e8n−1	PROPN
ejpam-5425	47	28	=	=	SYM
ejpam-5425	47	29	k	k	PROPN
ejpam-5425	47	30	,	,	PUNCT
ejpam-5425	47	31	e8n	e8n	X
ejpam-5425	47	32	=	=	SYM
ejpam-5425	47	33	f	f	X
ejpam-5425	47	34	,	,	PUNCT
ejpam-5425	47	35	e8n+1	e8n+1	NOUN
ejpam-5425	47	36	=	=	PUNCT
ejpam-5425	47	37	as	as	ADP
ejpam-5425	47	38	m(−1	m(−1	PROPN
ejpam-5425	47	39	+	+	CCONJ
ejpam-5425	47	40	as	as	ADP
ejpam-5425	47	41	)	)	PUNCT
ejpam-5425	47	42	,	,	PUNCT
ejpam-5425	47	43	e8n+2	e8n+2	NOUN
ejpam-5425	47	44	=	=	SYM
ejpam-5425	48	1	−fw	−fw	PROPN
ejpam-5425	48	2	g	g	NOUN
ejpam-5425	48	3	,	,	PUNCT
ejpam-5425	48	4	e8n+3	e8n+3	PROPN
ejpam-5425	48	5	=	=	PUNCT
ejpam-5425	48	6	as	as	ADP
ejpam-5425	48	7	k(1−	k(1−	ADJ
ejpam-5425	48	8	as	as	ADP
ejpam-5425	48	9	)	)	PUNCT
ejpam-5425	48	10	,	,	PUNCT
ejpam-5425	48	11	where	where	SCONJ
ejpam-5425	48	12	t0	t0	PROPN
ejpam-5425	48	13	=	=	PUNCT
ejpam-5425	48	14	a	a	PRON
ejpam-5425	48	15	,	,	PUNCT
ejpam-5425	48	16	t−1	t−1	PROPN
ejpam-5425	48	17	=	=	SYM
ejpam-5425	48	18	b	b	PROPN
ejpam-5425	48	19	,	,	PUNCT
ejpam-5425	48	20	t−2	t−2	PROPN
ejpam-5425	48	21	=	=	SYM
ejpam-5425	48	22	c	c	X
ejpam-5425	48	23	,	,	PUNCT
ejpam-5425	48	24	t−3	t−3	NOUN
ejpam-5425	48	25	=	=	SYM
ejpam-5425	48	26	d	d	PROPN
ejpam-5425	48	27	,	,	PUNCT
ejpam-5425	48	28	t−4	t−4	PROPN
ejpam-5425	48	29	=	=	SYM
ejpam-5425	48	30	s	s	PROPN
ejpam-5425	48	31	,	,	PUNCT
ejpam-5425	48	32	e0	e0	PROPN
ejpam-5425	48	33	=	=	SYM
ejpam-5425	48	34	f	f	X
ejpam-5425	48	35	,	,	PUNCT
ejpam-5425	48	36	e−1	e−1	PROPN
ejpam-5425	48	37	=	=	SYM
ejpam-5425	48	38	k	k	PROPN
ejpam-5425	48	39	,	,	PUNCT
ejpam-5425	48	40	e−2	e−2	PROPN
ejpam-5425	48	41	=	=	SYM
ejpam-5425	48	42	g	g	PROPN
ejpam-5425	48	43	,	,	PUNCT
ejpam-5425	48	44	e−3	e−3	PROPN
ejpam-5425	48	45	=	=	SYM
ejpam-5425	48	46	m	m	PROPN
ejpam-5425	48	47	,	,	PUNCT
ejpam-5425	48	48	and	and	CCONJ
ejpam-5425	48	49	e−4	e−4	PROPN
ejpam-5425	48	50	=	=	SYM
ejpam-5425	48	51	w.	w.	PROPN
ejpam-5425	48	52	also	also	ADV
ejpam-5425	48	53	,	,	PUNCT
ejpam-5425	48	54	where	where	SCONJ
ejpam-5425	48	55	e0e−4	e0e−4	PROPN
ejpam-5425	48	56	̸=	̸=	PROPN
ejpam-5425	48	57	±1	±1	VERB
ejpam-5425	48	58	and	and	CCONJ
ejpam-5425	48	59	t0t−4	t0t−4	PROPN
ejpam-5425	48	60	̸=	̸=	PROPN
ejpam-5425	48	61	±1	±1	VERB
ejpam-5425	48	62	.	.	PUNCT
ejpam-5425	49	1	proof	proof	NOUN
ejpam-5425	49	2	.	.	PUNCT
ejpam-5425	50	1	for	for	ADP
ejpam-5425	50	2	n	n	NOUN
ejpam-5425	50	3	=	=	SYM
ejpam-5425	50	4	0	0	NUM
ejpam-5425	50	5	,	,	PUNCT
ejpam-5425	50	6	the	the	DET
ejpam-5425	50	7	outcome	outcome	NOUN
ejpam-5425	50	8	is	be	AUX
ejpam-5425	50	9	valid	valid	ADJ
ejpam-5425	50	10	.	.	PUNCT
ejpam-5425	51	1	let	let	VERB
ejpam-5425	51	2	us	we	PRON
ejpam-5425	51	3	now	now	ADV
ejpam-5425	51	4	assume	assume	VERB
ejpam-5425	51	5	that	that	SCONJ
ejpam-5425	51	6	n	n	PROPN
ejpam-5425	51	7	>	>	X
ejpam-5425	51	8	0	0	PUNCT
ejpam-5425	52	1	and	and	CCONJ
ejpam-5425	52	2	that	that	SCONJ
ejpam-5425	52	3	n	n	ADV
ejpam-5425	52	4	−	−	PROPN
ejpam-5425	52	5	1	1	NUM
ejpam-5425	52	6	agrees	agree	VERB
ejpam-5425	52	7	with	with	ADP
ejpam-5425	52	8	our	our	PRON
ejpam-5425	52	9	assumption	assumption	NOUN
ejpam-5425	52	10	.	.	PUNCT
ejpam-5425	53	1	that	that	PRON
ejpam-5425	53	2	’s	’	VERB
ejpam-5425	53	3	t8n−12	t8n−12	NOUN
ejpam-5425	53	4	=	=	SYM
ejpam-5425	53	5	s	s	PROPN
ejpam-5425	53	6	,	,	PUNCT
ejpam-5425	53	7	t8n−11	t8n−11	PUNCT
ejpam-5425	54	1	=	=	PUNCT
ejpam-5425	54	2	d	d	PROPN
ejpam-5425	54	3	,	,	PUNCT
ejpam-5425	54	4	t8n−10	t8n−10	PROPN
ejpam-5425	54	5	=	=	SYM
ejpam-5425	54	6	c	c	X
ejpam-5425	54	7	,	,	PUNCT
ejpam-5425	54	8	t8n−9	t8n−9	NOUN
ejpam-5425	54	9	=	=	SYM
ejpam-5425	54	10	b	b	NOUN
ejpam-5425	54	11	,	,	PUNCT
ejpam-5425	54	12	t8n−8	t8n−8	PROPN
ejpam-5425	54	13	=	=	SYM
ejpam-5425	54	14	a	a	NOUN
ejpam-5425	54	15	,	,	PUNCT
ejpam-5425	54	16	t8n−7	t8n−7	NOUN
ejpam-5425	54	17	=	=	PUNCT
ejpam-5425	54	18	fw	fw	PROPN
ejpam-5425	55	1	d(1	d(1	PROPN
ejpam-5425	55	2	+	+	CCONJ
ejpam-5425	55	3	fw	fw	X
ejpam-5425	55	4	)	)	PUNCT
ejpam-5425	55	5	,	,	PUNCT
ejpam-5425	55	6	j.	j.	PROPN
ejpam-5425	55	7	g.	g.	PROPN
ejpam-5425	55	8	al	al	PROPN
ejpam-5425	55	9	-	-	PUNCT
ejpam-5425	55	10	juaid	juaid	PROPN
ejpam-5425	55	11	/	/	SYM
ejpam-5425	55	12	eur	eur	PROPN
ejpam-5425	55	13	.	.	PUNCT
ejpam-5425	56	1	j.	j.	PROPN
ejpam-5425	56	2	pure	pure	PROPN
ejpam-5425	56	3	appl	appl	PROPN
ejpam-5425	56	4	.	.	PROPN
ejpam-5425	56	5	math	math	PROPN
ejpam-5425	56	6	,	,	PUNCT
ejpam-5425	56	7	17	17	NUM
ejpam-5425	56	8	(	(	PUNCT
ejpam-5425	56	9	4	4	NUM
ejpam-5425	56	10	)	)	PUNCT
ejpam-5425	56	11	(	(	PUNCT
ejpam-5425	56	12	2024	2024	NUM
ejpam-5425	56	13	)	)	PUNCT
ejpam-5425	56	14	,	,	PUNCT
ejpam-5425	56	15	3254	3254	NUM
ejpam-5425	56	16	-	-	SYM
ejpam-5425	56	17	3267	3267	NUM
ejpam-5425	56	18	3257	3257	NUM
ejpam-5425	56	19	t8n−6	t8n−6	NOUN
ejpam-5425	56	20	=	=	PUNCT
ejpam-5425	56	21	as	as	ADP
ejpam-5425	56	22	c(−1	c(−1	PROPN
ejpam-5425	56	23	+	+	X
ejpam-5425	56	24	2as	2as	NUM
ejpam-5425	56	25	)	)	PUNCT
ejpam-5425	56	26	,	,	PUNCT
ejpam-5425	56	27	t8n−5	t8n−5	PROPN
ejpam-5425	56	28	=	=	SYM
ejpam-5425	56	29	−fw	−fw	PROPN
ejpam-5425	56	30	b(1−	b(1−	PROPN
ejpam-5425	56	31	fw	fw	PROPN
ejpam-5425	56	32	)	)	PUNCT
ejpam-5425	56	33	,	,	PUNCT
ejpam-5425	56	34	e8n−12	e8n−12	X
ejpam-5425	57	1	=	=	SYM
ejpam-5425	57	2	w	w	PROPN
ejpam-5425	57	3	,	,	PUNCT
ejpam-5425	57	4	e8n−11	e8n−11	VERB
ejpam-5425	57	5	=	=	PROPN
ejpam-5425	57	6	m	m	PROPN
ejpam-5425	57	7	,	,	PUNCT
ejpam-5425	57	8	e8n−10	e8n−10	PROPN
ejpam-5425	57	9	=	=	PUNCT
ejpam-5425	57	10	g	g	PROPN
ejpam-5425	57	11	,	,	PUNCT
ejpam-5425	57	12	e8n−9	e8n−9	PROPN
ejpam-5425	57	13	=	=	SYM
ejpam-5425	57	14	k	k	PROPN
ejpam-5425	57	15	,	,	PUNCT
ejpam-5425	57	16	e8n−8	e8n−8	PROPN
ejpam-5425	57	17	=	=	SYM
ejpam-5425	57	18	f	f	PROPN
ejpam-5425	57	19	,	,	PUNCT
ejpam-5425	57	20	e8n−7	e8n−7	NOUN
ejpam-5425	57	21	=	=	PUNCT
ejpam-5425	57	22	as	as	ADP
ejpam-5425	57	23	m(−1	m(−1	PROPN
ejpam-5425	57	24	+	+	CCONJ
ejpam-5425	57	25	as	as	ADP
ejpam-5425	57	26	)	)	PUNCT
ejpam-5425	57	27	,	,	PUNCT
ejpam-5425	57	28	e8n−6	e8n−6	PROPN
ejpam-5425	57	29	=	=	SYM
ejpam-5425	57	30	−fw	−fw	PROPN
ejpam-5425	57	31	g	g	NOUN
ejpam-5425	57	32	,	,	PUNCT
ejpam-5425	57	33	e8n−5	e8n−5	PROPN
ejpam-5425	57	34	=	=	PUNCT
ejpam-5425	57	35	as	as	ADP
ejpam-5425	57	36	k(1−	k(1−	ADJ
ejpam-5425	57	37	as	as	ADP
ejpam-5425	57	38	)	)	PUNCT
ejpam-5425	57	39	.	.	PUNCT
ejpam-5425	58	1	next	next	ADV
ejpam-5425	58	2	,	,	PUNCT
ejpam-5425	58	3	from	from	ADP
ejpam-5425	58	4	system	system	NOUN
ejpam-5425	58	5	(	(	PUNCT
ejpam-5425	58	6	1	1	X
ejpam-5425	58	7	)	)	PUNCT
ejpam-5425	58	8	we	we	PRON
ejpam-5425	58	9	have	have	VERB
ejpam-5425	58	10	t8n	t8n	NOUN
ejpam-5425	58	11	=	=	SYM
ejpam-5425	58	12	e8n−1e8n−5	e8n−1e8n−5	NOUN
ejpam-5425	58	13	t8n−4(1	t8n−4(1	NOUN
ejpam-5425	58	14	+	+	CCONJ
ejpam-5425	58	15	e8n−1e8n−5	e8n−1e8n−5	NOUN
ejpam-5425	58	16	)	)	PUNCT
ejpam-5425	59	1	=	=	PROPN
ejpam-5425	59	2	kas	kas	PROPN
ejpam-5425	59	3	k(1−as	k(1−as	PROPN
ejpam-5425	59	4	)	)	PUNCT
ejpam-5425	60	1	s(1	s(1	PROPN
ejpam-5425	60	2	+	+	NUM
ejpam-5425	60	3	kas	kas	PROPN
ejpam-5425	60	4	k(1−as	k(1−as	PROPN
ejpam-5425	60	5	)	)	PUNCT
ejpam-5425	60	6	)	)	PUNCT
ejpam-5425	61	1	=	=	PUNCT
ejpam-5425	61	2	a.	a.	NOUN
ejpam-5425	61	3	e8n	e8n	PUNCT
ejpam-5425	61	4	=	=	NOUN
ejpam-5425	61	5	t8n−1t8n−5	t8n−1t8n−5	PROPN
ejpam-5425	61	6	e8n−4(−1	e8n−4(−1	X
ejpam-5425	62	1	+	+	CCONJ
ejpam-5425	62	2	t8n−1t8n−5	t8n−1t8n−5	PROPN
ejpam-5425	62	3	)	)	PUNCT
ejpam-5425	63	1	=	=	PUNCT
ejpam-5425	63	2	−bfw	−bfw	NUM
ejpam-5425	63	3	b(1−fw	b(1−fw	PROPN
ejpam-5425	63	4	)	)	PUNCT
ejpam-5425	63	5	w(−1−	w(−1−	PROPN
ejpam-5425	63	6	fwb	fwb	PROPN
ejpam-5425	63	7	b(1−fw	b(1−fw	PROPN
ejpam-5425	63	8	)	)	PUNCT
ejpam-5425	63	9	)	)	PUNCT
ejpam-5425	64	1	=	=	PUNCT
ejpam-5425	64	2	f.	f.	PROPN
ejpam-5425	64	3	t8n−4	t8n−4	PROPN
ejpam-5425	64	4	=	=	PUNCT
ejpam-5425	64	5	e8n−5e8n−9	e8n−5e8n−9	NOUN
ejpam-5425	64	6	t8n−8(1	t8n−8(1	NOUN
ejpam-5425	64	7	+	+	CCONJ
ejpam-5425	64	8	e8n−5e8n−9	e8n−5e8n−9	NOUN
ejpam-5425	64	9	)	)	PUNCT
ejpam-5425	65	1	=	=	PUNCT
ejpam-5425	65	2	ask	ask	VERB
ejpam-5425	65	3	k(1−as	k(1−as	PROPN
ejpam-5425	65	4	)	)	PUNCT
ejpam-5425	66	1	a(1	a(1	PROPN
ejpam-5425	66	2	+	+	NUM
ejpam-5425	66	3	kas	kas	PROPN
ejpam-5425	66	4	k(1−as	k(1−as	PROPN
ejpam-5425	66	5	)	)	PUNCT
ejpam-5425	66	6	)	)	PUNCT
ejpam-5425	67	1	=	=	PUNCT
ejpam-5425	67	2	s.	s.	PROPN
ejpam-5425	67	3	e8n−4	e8n−4	PROPN
ejpam-5425	67	4	=	=	PROPN
ejpam-5425	67	5	t8n−5t8n−9	t8n−5t8n−9	PROPN
ejpam-5425	67	6	e8n−8(−1	e8n−8(−1	NOUN
ejpam-5425	67	7	+	+	X
ejpam-5425	67	8	t8n−5t8n−9	t8n−5t8n−9	NOUN
ejpam-5425	67	9	)	)	PUNCT
ejpam-5425	67	10	=	=	PUNCT
ejpam-5425	67	11	−bfw	−bfw	NUM
ejpam-5425	67	12	b(1−fw	b(1−fw	PROPN
ejpam-5425	67	13	)	)	PUNCT
ejpam-5425	67	14	f(−1−	f(−1−	NOUN
ejpam-5425	67	15	fwb	fwb	PROPN
ejpam-5425	67	16	b(1−fw	b(1−fw	PROPN
ejpam-5425	67	17	)	)	PUNCT
ejpam-5425	67	18	)	)	PUNCT
ejpam-5425	68	1	=	=	SYM
ejpam-5425	68	2	w.	w.	PROPN
ejpam-5425	68	3	similarly	similarly	ADV
ejpam-5425	68	4	,	,	PUNCT
ejpam-5425	68	5	we	we	PRON
ejpam-5425	68	6	can	can	AUX
ejpam-5425	68	7	prove	prove	VERB
ejpam-5425	68	8	the	the	DET
ejpam-5425	68	9	remaining	remain	VERB
ejpam-5425	68	10	relations	relation	NOUN
ejpam-5425	68	11	.	.	PUNCT
ejpam-5425	69	1	the	the	DET
ejpam-5425	69	2	proof	proof	NOUN
ejpam-5425	69	3	is	be	AUX
ejpam-5425	69	4	complete	complete	ADJ
ejpam-5425	69	5	.	.	PUNCT
ejpam-5425	70	1	example	example	NOUN
ejpam-5425	71	1	1	1	NUM
ejpam-5425	71	2	.	.	X
ejpam-5425	72	1	figure	figure	NOUN
ejpam-5425	72	2	(	(	PUNCT
ejpam-5425	72	3	1	1	X
ejpam-5425	72	4	)	)	PUNCT
ejpam-5425	72	5	demonstrates	demonstrate	VERB
ejpam-5425	72	6	the	the	DET
ejpam-5425	72	7	behavior	behavior	NOUN
ejpam-5425	72	8	of	of	ADP
ejpam-5425	72	9	the	the	DET
ejpam-5425	72	10	solutions	solution	NOUN
ejpam-5425	72	11	of	of	ADP
ejpam-5425	72	12	the	the	DET
ejpam-5425	72	13	sde	sde	PROPN
ejpam-5425	72	14	eq.(1	eq.(1	NUM
ejpam-5425	72	15	)	)	PUNCT
ejpam-5425	72	16	with	with	ADP
ejpam-5425	72	17	t0	t0	PROPN
ejpam-5425	72	18	=	=	SYM
ejpam-5425	73	1	1	1	NUM
ejpam-5425	73	2	,	,	PUNCT
ejpam-5425	73	3	t−1	t−1	NOUN
ejpam-5425	73	4	=	=	SYM
ejpam-5425	73	5	0.5	0.5	NUM
ejpam-5425	73	6	,	,	PUNCT
ejpam-5425	73	7	t−2	t−2	NOUN
ejpam-5425	73	8	=	=	PUNCT
ejpam-5425	73	9	0.2	0.2	NUM
ejpam-5425	73	10	,	,	PUNCT
ejpam-5425	73	11	t−3	t−3	NOUN
ejpam-5425	73	12	=	=	SYM
ejpam-5425	73	13	2	2	NUM
ejpam-5425	73	14	,	,	PUNCT
ejpam-5425	73	15	t−4	t−4	PROPN
ejpam-5425	73	16	=	=	SYM
ejpam-5425	73	17	3	3	NUM
ejpam-5425	73	18	,	,	PUNCT
ejpam-5425	73	19	e0	e0	PROPN
ejpam-5425	73	20	=	=	NOUN
ejpam-5425	73	21	0.1	0.1	NUM
ejpam-5425	73	22	,	,	PUNCT
ejpam-5425	73	23	e−1	e−1	PROPN
ejpam-5425	73	24	=	=	SYM
ejpam-5425	73	25	0.5	0.5	NUM
ejpam-5425	73	26	,	,	PUNCT
ejpam-5425	73	27	e−2	e−2	PROPN
ejpam-5425	73	28	=	=	SYM
ejpam-5425	73	29	1	1	NUM
ejpam-5425	73	30	,	,	PUNCT
ejpam-5425	73	31	e−3	e−3	PROPN
ejpam-5425	73	32	=	=	NUM
ejpam-5425	73	33	0.2	0.2	NUM
ejpam-5425	73	34	,	,	PUNCT
ejpam-5425	73	35	and	and	CCONJ
ejpam-5425	73	36	e−4	e−4	PROPN
ejpam-5425	73	37	=	=	SYM
ejpam-5425	73	38	0.3	0.3	NUM
ejpam-5425	73	39	.	.	PUNCT
ejpam-5425	74	1	3	3	X
ejpam-5425	74	2	.	.	X
ejpam-5425	75	1	the	the	DET
ejpam-5425	75	2	system	system	NOUN
ejpam-5425	75	3	tn+1	tn+1	NOUN
ejpam-5425	75	4	=	=	SYM
ejpam-5425	75	5	enen−4	enen−4	NOUN
ejpam-5425	75	6	tn−3(1+enen−4	tn−3(1+enen−4	NUM
ejpam-5425	75	7	)	)	PUNCT
ejpam-5425	75	8	,	,	PUNCT
ejpam-5425	75	9	en+1	en+1	ADJ
ejpam-5425	75	10	=	=	PROPN
ejpam-5425	75	11	tntn−4	tntn−4	NUM
ejpam-5425	75	12	en−3(1−tntn−4	en−3(1−tntn−4	NOUN
ejpam-5425	75	13	)	)	PUNCT
ejpam-5425	75	14	in	in	ADP
ejpam-5425	75	15	this	this	DET
ejpam-5425	75	16	section	section	NOUN
ejpam-5425	75	17	,	,	PUNCT
ejpam-5425	75	18	for	for	ADP
ejpam-5425	75	19	the	the	DET
ejpam-5425	75	20	aforementioned	aforementioned	ADJ
ejpam-5425	75	21	system	system	NOUN
ejpam-5425	75	22	,	,	PUNCT
ejpam-5425	75	23	we	we	PRON
ejpam-5425	75	24	provide	provide	VERB
ejpam-5425	75	25	the	the	DET
ejpam-5425	75	26	period	period	NOUN
ejpam-5425	75	27	eight	eight	NUM
ejpam-5425	75	28	periodic	periodic	ADJ
ejpam-5425	75	29	solutions	solution	NOUN
ejpam-5425	75	30	and	and	CCONJ
ejpam-5425	75	31	solution	solution	NOUN
ejpam-5425	75	32	expression	expression	NOUN
ejpam-5425	75	33	.	.	PUNCT
ejpam-5425	76	1	tn+1	tn+1	PROPN
ejpam-5425	76	2	=	=	PUNCT
ejpam-5425	77	1	enen−4	enen−4	X
ejpam-5425	77	2	tn−3(1	tn−3(1	NOUN
ejpam-5425	77	3	+	+	NUM
ejpam-5425	77	4	enen−4	enen−4	NOUN
ejpam-5425	77	5	)	)	PUNCT
ejpam-5425	77	6	,	,	PUNCT
ejpam-5425	77	7	en+1	en+1	ADJ
ejpam-5425	77	8	=	=	SYM
ejpam-5425	77	9	tntn−4	tntn−4	NUM
ejpam-5425	77	10	en−3(1−	en−3(1−	PROPN
ejpam-5425	77	11	tntn−4	tntn−4	NUM
ejpam-5425	77	12	)	)	PUNCT
ejpam-5425	77	13	.	.	PUNCT
ejpam-5425	78	1	(	(	PUNCT
ejpam-5425	78	2	2	2	X
ejpam-5425	78	3	)	)	PUNCT
ejpam-5425	78	4	theorem	theorem	NOUN
ejpam-5425	78	5	2	2	NUM
ejpam-5425	78	6	.	.	PUNCT
ejpam-5425	78	7	suppose	suppose	VERB
ejpam-5425	78	8	that	that	SCONJ
ejpam-5425	78	9	{	{	PUNCT
ejpam-5425	78	10	tn	tn	NOUN
ejpam-5425	78	11	,	,	PUNCT
ejpam-5425	78	12	en	en	ADP
ejpam-5425	78	13	}	}	PUNCT
ejpam-5425	78	14	be	be	AUX
ejpam-5425	78	15	solutions	solution	NOUN
ejpam-5425	78	16	of	of	ADP
ejpam-5425	78	17	the	the	DET
ejpam-5425	78	18	system	system	NOUN
ejpam-5425	78	19	of	of	ADP
ejpam-5425	78	20	sdes	sde	NOUN
ejpam-5425	78	21	(	(	PUNCT
ejpam-5425	78	22	2	2	NUM
ejpam-5425	78	23	)	)	PUNCT
ejpam-5425	78	24	.	.	PUNCT
ejpam-5425	79	1	then	then	ADV
ejpam-5425	79	2	for	for	ADP
ejpam-5425	79	3	n	n	NOUN
ejpam-5425	79	4	=	=	SYM
ejpam-5425	79	5	0	0	NUM
ejpam-5425	79	6	,	,	PUNCT
ejpam-5425	79	7	1	1	NUM
ejpam-5425	79	8	,	,	PUNCT
ejpam-5425	79	9	2	2	NUM
ejpam-5425	79	10	,	,	PUNCT
ejpam-5425	79	11	..	..	PUNCT
ejpam-5425	79	12	,	,	PUNCT
ejpam-5425	79	13	the	the	DET
ejpam-5425	79	14	solutions	solution	NOUN
ejpam-5425	79	15	of	of	ADP
ejpam-5425	79	16	eq.(2	eq.(2	NOUN
ejpam-5425	79	17	)	)	PUNCT
ejpam-5425	79	18	can	can	AUX
ejpam-5425	79	19	be	be	AUX
ejpam-5425	79	20	formed	form	VERB
ejpam-5425	79	21	as	as	SCONJ
ejpam-5425	79	22	follows	follow	VERB
ejpam-5425	79	23	t8n−4	t8n−4	PROPN
ejpam-5425	79	24	=	=	SYM
ejpam-5425	79	25	s	s	PROPN
ejpam-5425	79	26	,	,	PUNCT
ejpam-5425	79	27	t8n−3	t8n−3	PROPN
ejpam-5425	79	28	=	=	SYM
ejpam-5425	80	1	d	d	PROPN
ejpam-5425	80	2	,	,	PUNCT
ejpam-5425	80	3	t8n−2	t8n−2	PROPN
ejpam-5425	80	4	=	=	SYM
ejpam-5425	80	5	c	c	NOUN
ejpam-5425	80	6	,	,	PUNCT
ejpam-5425	80	7	t8n−1	t8n−1	PROPN
ejpam-5425	80	8	=	=	SYM
ejpam-5425	80	9	b	b	PROPN
ejpam-5425	80	10	,	,	PUNCT
ejpam-5425	80	11	j.	j.	PROPN
ejpam-5425	80	12	g.	g.	PROPN
ejpam-5425	80	13	al	al	PROPN
ejpam-5425	80	14	-	-	PUNCT
ejpam-5425	80	15	juaid	juaid	PROPN
ejpam-5425	80	16	/	/	SYM
ejpam-5425	80	17	eur	eur	PROPN
ejpam-5425	80	18	.	.	PUNCT
ejpam-5425	81	1	j.	j.	PROPN
ejpam-5425	81	2	pure	pure	PROPN
ejpam-5425	81	3	appl	appl	PROPN
ejpam-5425	81	4	.	.	PROPN
ejpam-5425	81	5	math	math	PROPN
ejpam-5425	81	6	,	,	PUNCT
ejpam-5425	81	7	17	17	NUM
ejpam-5425	81	8	(	(	PUNCT
ejpam-5425	81	9	4	4	NUM
ejpam-5425	81	10	)	)	PUNCT
ejpam-5425	81	11	(	(	PUNCT
ejpam-5425	81	12	2024	2024	NUM
ejpam-5425	81	13	)	)	PUNCT
ejpam-5425	81	14	,	,	PUNCT
ejpam-5425	81	15	3254	3254	NUM
ejpam-5425	81	16	-	-	SYM
ejpam-5425	81	17	3267	3267	NUM
ejpam-5425	81	18	3258	3258	NUM
ejpam-5425	81	19	figure	figure	NOUN
ejpam-5425	81	20	1	1	NUM
ejpam-5425	81	21	:	:	PUNCT
ejpam-5425	81	22	periodic	periodic	ADJ
ejpam-5425	81	23	solutions	solution	NOUN
ejpam-5425	81	24	of	of	ADP
ejpam-5425	81	25	eq.(1	eq.(1	ADJ
ejpam-5425	81	26	)	)	PUNCT
ejpam-5425	81	27	t8n	t8n	NOUN
ejpam-5425	81	28	=	=	PUNCT
ejpam-5425	81	29	a	a	NOUN
ejpam-5425	81	30	,	,	PUNCT
ejpam-5425	81	31	t8n+1	t8n+1	ADV
ejpam-5425	81	32	=	=	PUNCT
ejpam-5425	82	1	fw	fw	INTJ
ejpam-5425	82	2	d(1	d(1	PROPN
ejpam-5425	82	3	+	+	CCONJ
ejpam-5425	82	4	fw	fw	X
ejpam-5425	82	5	)	)	PUNCT
ejpam-5425	82	6	,	,	PUNCT
ejpam-5425	82	7	t8n+2	t8n+2	PROPN
ejpam-5425	83	1	=	=	PUNCT
ejpam-5425	83	2	as	as	ADP
ejpam-5425	83	3	c	c	PROPN
ejpam-5425	83	4	,	,	PUNCT
ejpam-5425	83	5	t8n+3	t8n+3	PUNCT
ejpam-5425	84	1	=	=	PUNCT
ejpam-5425	84	2	fw	fw	PROPN
ejpam-5425	84	3	b(1	b(1	PROPN
ejpam-5425	84	4	+	+	CCONJ
ejpam-5425	84	5	fw	fw	X
ejpam-5425	84	6	)	)	PUNCT
ejpam-5425	84	7	,	,	PUNCT
ejpam-5425	84	8	e8n−4	e8n−4	PROPN
ejpam-5425	84	9	=	=	SYM
ejpam-5425	84	10	w	w	PROPN
ejpam-5425	84	11	,	,	PUNCT
ejpam-5425	84	12	e8n−3	e8n−3	PROPN
ejpam-5425	84	13	=	=	SYM
ejpam-5425	84	14	m	m	PROPN
ejpam-5425	84	15	,	,	PUNCT
ejpam-5425	84	16	e8n−2	e8n−2	PROPN
ejpam-5425	84	17	=	=	SYM
ejpam-5425	84	18	g	g	PROPN
ejpam-5425	84	19	,	,	PUNCT
ejpam-5425	84	20	e8n−1	e8n−1	PROPN
ejpam-5425	84	21	=	=	SYM
ejpam-5425	84	22	k	k	PROPN
ejpam-5425	84	23	,	,	PUNCT
ejpam-5425	84	24	e8n	e8n	X
ejpam-5425	84	25	=	=	SYM
ejpam-5425	84	26	f	f	X
ejpam-5425	84	27	,	,	PUNCT
ejpam-5425	84	28	e8n+1	e8n+1	NOUN
ejpam-5425	84	29	=	=	PUNCT
ejpam-5425	84	30	as	as	ADP
ejpam-5425	84	31	m(1−	m(1−	ADJ
ejpam-5425	84	32	as	as	ADP
ejpam-5425	84	33	)	)	PUNCT
ejpam-5425	84	34	,	,	PUNCT
ejpam-5425	84	35	e8n+2	e8n+2	PROPN
ejpam-5425	84	36	=	=	PUNCT
ejpam-5425	85	1	fw	fw	PROPN
ejpam-5425	85	2	g	g	PROPN
ejpam-5425	85	3	,	,	PUNCT
ejpam-5425	85	4	e8n+3	e8n+3	PROPN
ejpam-5425	85	5	=	=	PUNCT
ejpam-5425	85	6	as	as	ADP
ejpam-5425	85	7	k(1−	k(1−	ADJ
ejpam-5425	85	8	as	as	ADP
ejpam-5425	85	9	)	)	PUNCT
ejpam-5425	85	10	,	,	PUNCT
ejpam-5425	85	11	where	where	SCONJ
ejpam-5425	85	12	e0e−4	e0e−4	PROPN
ejpam-5425	85	13	̸=	̸=	PROPN
ejpam-5425	85	14	−1	−1	NOUN
ejpam-5425	85	15	and	and	CCONJ
ejpam-5425	85	16	t0t−4	t0t−4	PROPN
ejpam-5425	85	17	̸=	̸=	PROPN
ejpam-5425	85	18	1	1	NUM
ejpam-5425	85	19	.	.	PUNCT
ejpam-5425	86	1	proof	proof	NOUN
ejpam-5425	86	2	.	.	PUNCT
ejpam-5425	87	1	for	for	ADP
ejpam-5425	87	2	n	n	NOUN
ejpam-5425	87	3	=	=	SYM
ejpam-5425	87	4	0	0	NUM
ejpam-5425	87	5	,	,	PUNCT
ejpam-5425	87	6	the	the	DET
ejpam-5425	87	7	outcome	outcome	NOUN
ejpam-5425	87	8	is	be	AUX
ejpam-5425	87	9	valid	valid	ADJ
ejpam-5425	87	10	.	.	PUNCT
ejpam-5425	88	1	assume	assume	VERB
ejpam-5425	88	2	for	for	ADP
ejpam-5425	88	3	the	the	DET
ejpam-5425	88	4	moment	moment	NOUN
ejpam-5425	88	5	that	that	SCONJ
ejpam-5425	88	6	n	n	CCONJ
ejpam-5425	88	7	>	>	X
ejpam-5425	88	8	0	0	PUNCT
ejpam-5425	89	1	and	and	CCONJ
ejpam-5425	89	2	that	that	PRON
ejpam-5425	89	3	n−	n−	NOUN
ejpam-5425	89	4	1	1	NUM
ejpam-5425	89	5	falls	fall	VERB
ejpam-5425	89	6	under	under	ADP
ejpam-5425	89	7	our	our	PRON
ejpam-5425	89	8	hypothesis	hypothesis	NOUN
ejpam-5425	89	9	.	.	PUNCT
ejpam-5425	90	1	that	that	PRON
ejpam-5425	90	2	’s	’	VERB
ejpam-5425	90	3	t8n−12	t8n−12	NOUN
ejpam-5425	90	4	=	=	SYM
ejpam-5425	90	5	s	s	PROPN
ejpam-5425	90	6	,	,	PUNCT
ejpam-5425	90	7	t8n−11	t8n−11	PUNCT
ejpam-5425	91	1	=	=	PUNCT
ejpam-5425	91	2	d	d	PROPN
ejpam-5425	91	3	,	,	PUNCT
ejpam-5425	91	4	t8n−10	t8n−10	PROPN
ejpam-5425	91	5	=	=	SYM
ejpam-5425	91	6	c	c	X
ejpam-5425	91	7	,	,	PUNCT
ejpam-5425	91	8	t8n−9	t8n−9	NOUN
ejpam-5425	91	9	=	=	SYM
ejpam-5425	91	10	b	b	NOUN
ejpam-5425	91	11	,	,	PUNCT
ejpam-5425	91	12	t8n−8	t8n−8	PROPN
ejpam-5425	91	13	=	=	SYM
ejpam-5425	91	14	a	a	NOUN
ejpam-5425	91	15	,	,	PUNCT
ejpam-5425	91	16	t8n−7	t8n−7	NOUN
ejpam-5425	91	17	=	=	PUNCT
ejpam-5425	91	18	fw	fw	PROPN
ejpam-5425	92	1	d(1	d(1	PROPN
ejpam-5425	92	2	+	+	CCONJ
ejpam-5425	92	3	fw	fw	X
ejpam-5425	92	4	)	)	PUNCT
ejpam-5425	92	5	,	,	PUNCT
ejpam-5425	92	6	t8n−6	t8n−6	PROPN
ejpam-5425	92	7	=	=	PUNCT
ejpam-5425	92	8	as	as	ADP
ejpam-5425	92	9	c	c	PROPN
ejpam-5425	92	10	,	,	PUNCT
ejpam-5425	92	11	t8n−5	t8n−5	PROPN
ejpam-5425	92	12	=	=	PUNCT
ejpam-5425	93	1	fw	fw	PROPN
ejpam-5425	93	2	b(1	b(1	PROPN
ejpam-5425	93	3	+	+	CCONJ
ejpam-5425	93	4	fw	fw	X
ejpam-5425	93	5	)	)	PUNCT
ejpam-5425	93	6	,	,	PUNCT
ejpam-5425	93	7	e8n−12	e8n−12	X
ejpam-5425	93	8	=	=	SYM
ejpam-5425	93	9	w	w	PROPN
ejpam-5425	93	10	,	,	PUNCT
ejpam-5425	93	11	e8n−11	e8n−11	VERB
ejpam-5425	93	12	=	=	PROPN
ejpam-5425	93	13	m	m	PROPN
ejpam-5425	93	14	,	,	PUNCT
ejpam-5425	93	15	j.	j.	PROPN
ejpam-5425	93	16	g.	g.	PROPN
ejpam-5425	93	17	al	al	PROPN
ejpam-5425	93	18	-	-	PUNCT
ejpam-5425	93	19	juaid	juaid	PROPN
ejpam-5425	93	20	/	/	SYM
ejpam-5425	93	21	eur	eur	PROPN
ejpam-5425	93	22	.	.	PUNCT
ejpam-5425	94	1	j.	j.	PROPN
ejpam-5425	94	2	pure	pure	PROPN
ejpam-5425	94	3	appl	appl	PROPN
ejpam-5425	94	4	.	.	PROPN
ejpam-5425	94	5	math	math	PROPN
ejpam-5425	94	6	,	,	PUNCT
ejpam-5425	94	7	17	17	NUM
ejpam-5425	94	8	(	(	PUNCT
ejpam-5425	94	9	4	4	NUM
ejpam-5425	94	10	)	)	PUNCT
ejpam-5425	94	11	(	(	PUNCT
ejpam-5425	94	12	2024	2024	NUM
ejpam-5425	94	13	)	)	PUNCT
ejpam-5425	94	14	,	,	PUNCT
ejpam-5425	94	15	3254	3254	NUM
ejpam-5425	94	16	-	-	SYM
ejpam-5425	94	17	3267	3267	NUM
ejpam-5425	94	18	3259	3259	NUM
ejpam-5425	94	19	e8n−10	e8n−10	X
ejpam-5425	95	1	=	=	PUNCT
ejpam-5425	95	2	g	g	PROPN
ejpam-5425	95	3	,	,	PUNCT
ejpam-5425	95	4	e8n−9	e8n−9	PROPN
ejpam-5425	95	5	=	=	SYM
ejpam-5425	95	6	k	k	PROPN
ejpam-5425	95	7	,	,	PUNCT
ejpam-5425	95	8	e8n−8	e8n−8	PROPN
ejpam-5425	95	9	=	=	SYM
ejpam-5425	95	10	f	f	PROPN
ejpam-5425	95	11	,	,	PUNCT
ejpam-5425	95	12	e8n−7	e8n−7	NOUN
ejpam-5425	95	13	=	=	PUNCT
ejpam-5425	95	14	as	as	ADP
ejpam-5425	95	15	m(−1	m(−1	PROPN
ejpam-5425	95	16	+	+	CCONJ
ejpam-5425	95	17	as	as	ADP
ejpam-5425	95	18	)	)	PUNCT
ejpam-5425	95	19	,	,	PUNCT
ejpam-5425	95	20	e8n−6	e8n−6	PROPN
ejpam-5425	95	21	=	=	PUNCT
ejpam-5425	95	22	fw	fw	PROPN
ejpam-5425	95	23	g	g	PROPN
ejpam-5425	95	24	,	,	PUNCT
ejpam-5425	95	25	e8n−5	e8n−5	PROPN
ejpam-5425	95	26	=	=	PUNCT
ejpam-5425	95	27	as	as	ADP
ejpam-5425	95	28	k(1−	k(1−	ADJ
ejpam-5425	95	29	as	as	ADP
ejpam-5425	95	30	)	)	PUNCT
ejpam-5425	95	31	.	.	PUNCT
ejpam-5425	96	1	next	next	ADV
ejpam-5425	96	2	,	,	PUNCT
ejpam-5425	96	3	from	from	ADP
ejpam-5425	96	4	system	system	NOUN
ejpam-5425	96	5	(	(	PUNCT
ejpam-5425	96	6	2	2	X
ejpam-5425	96	7	)	)	PUNCT
ejpam-5425	96	8	we	we	PRON
ejpam-5425	96	9	have	have	VERB
ejpam-5425	96	10	t8n+1	t8n+1	ADV
ejpam-5425	96	11	=	=	PUNCT
ejpam-5425	96	12	e8ne8n−4	e8ne8n−4	NOUN
ejpam-5425	96	13	t8n−3(1	t8n−3(1	NOUN
ejpam-5425	96	14	+	+	NUM
ejpam-5425	96	15	e8ne8n−4	e8ne8n−4	NOUN
ejpam-5425	96	16	)	)	PUNCT
ejpam-5425	97	1	=	=	PUNCT
ejpam-5425	98	1	fw	fw	INTJ
ejpam-5425	98	2	d(1	d(1	PROPN
ejpam-5425	98	3	+	+	CCONJ
ejpam-5425	98	4	fw	fw	X
ejpam-5425	98	5	)	)	PUNCT
ejpam-5425	98	6	.	.	PUNCT
ejpam-5425	99	1	e8n+1	e8n+1	PUNCT
ejpam-5425	100	1	=	=	NOUN
ejpam-5425	100	2	t8nt8n−4	t8nt8n−4	X
ejpam-5425	100	3	e8n−3(1−	e8n−3(1−	PROPN
ejpam-5425	100	4	t8nt8n−4	t8nt8n−4	NUM
ejpam-5425	100	5	)	)	PUNCT
ejpam-5425	100	6	=	=	PUNCT
ejpam-5425	100	7	as	as	ADP
ejpam-5425	100	8	m(1−	m(1−	ADJ
ejpam-5425	100	9	as	as	ADP
ejpam-5425	100	10	)	)	PUNCT
ejpam-5425	100	11	.	.	PUNCT
ejpam-5425	101	1	t8n	t8n	NOUN
ejpam-5425	101	2	=	=	PUNCT
ejpam-5425	101	3	e8n−1e8n−5	e8n−1e8n−5	NOUN
ejpam-5425	101	4	t8n−4(1	t8n−4(1	NOUN
ejpam-5425	101	5	+	+	CCONJ
ejpam-5425	101	6	e8n−1e8n−5	e8n−1e8n−5	NOUN
ejpam-5425	101	7	)	)	PUNCT
ejpam-5425	102	1	=	=	PUNCT
ejpam-5425	102	2	ask	ask	VERB
ejpam-5425	102	3	k(1−as	k(1−as	PROPN
ejpam-5425	102	4	)	)	PUNCT
ejpam-5425	102	5	s(1	s(1	PROPN
ejpam-5425	102	6	+	+	PROPN
ejpam-5425	102	7	kas	kas	PROPN
ejpam-5425	102	8	k(1−as	k(1−as	PROPN
ejpam-5425	102	9	)	)	PUNCT
ejpam-5425	102	10	)	)	PUNCT
ejpam-5425	103	1	=	=	PUNCT
ejpam-5425	103	2	a.	a.	NOUN
ejpam-5425	103	3	e8n	e8n	PUNCT
ejpam-5425	103	4	=	=	NOUN
ejpam-5425	103	5	t8n−1t8n−5	t8n−1t8n−5	PROPN
ejpam-5425	103	6	e8n−4(−1	e8n−4(−1	X
ejpam-5425	104	1	+	+	CCONJ
ejpam-5425	104	2	t8n−1t8n−5	t8n−1t8n−5	PROPN
ejpam-5425	104	3	)	)	PUNCT
ejpam-5425	105	1	=	=	PUNCT
ejpam-5425	105	2	bfw	bfw	PROPN
ejpam-5425	105	3	b(1+fw	b(1+fw	PROPN
ejpam-5425	105	4	)	)	PUNCT
ejpam-5425	105	5	w(1−	w(1−	PROPN
ejpam-5425	105	6	fwb	fwb	PROPN
ejpam-5425	105	7	b(1+fw	b(1+fw	PROPN
ejpam-5425	105	8	)	)	PUNCT
ejpam-5425	105	9	)	)	PUNCT
ejpam-5425	106	1	=	=	PUNCT
ejpam-5425	106	2	f.	f.	NOUN
ejpam-5425	106	3	we	we	PRON
ejpam-5425	106	4	can	can	AUX
ejpam-5425	106	5	confirm	confirm	VERB
ejpam-5425	106	6	the	the	DET
ejpam-5425	106	7	other	other	ADJ
ejpam-5425	106	8	forms	form	NOUN
ejpam-5425	106	9	using	use	VERB
ejpam-5425	106	10	the	the	DET
ejpam-5425	106	11	same	same	ADJ
ejpam-5425	106	12	method	method	NOUN
ejpam-5425	106	13	.	.	PUNCT
ejpam-5425	107	1	the	the	DET
ejpam-5425	107	2	evidence	evidence	NOUN
ejpam-5425	107	3	is	be	AUX
ejpam-5425	107	4	finished	finish	VERB
ejpam-5425	107	5	.	.	PUNCT
ejpam-5425	108	1	example	example	NOUN
ejpam-5425	108	2	2	2	NUM
ejpam-5425	108	3	.	.	X
ejpam-5425	108	4	figure	figure	NOUN
ejpam-5425	108	5	(	(	PUNCT
ejpam-5425	108	6	2	2	NUM
ejpam-5425	108	7	)	)	PUNCT
ejpam-5425	108	8	illustrates	illustrate	VERB
ejpam-5425	108	9	the	the	DET
ejpam-5425	108	10	behavior	behavior	NOUN
ejpam-5425	108	11	of	of	ADP
ejpam-5425	108	12	the	the	DET
ejpam-5425	108	13	solutions	solution	NOUN
ejpam-5425	108	14	of	of	ADP
ejpam-5425	108	15	the	the	DET
ejpam-5425	108	16	sdes	sde	NOUN
ejpam-5425	108	17	(	(	PUNCT
ejpam-5425	108	18	2	2	NUM
ejpam-5425	108	19	)	)	PUNCT
ejpam-5425	108	20	with	with	ADP
ejpam-5425	108	21	t0	t0	PROPN
ejpam-5425	108	22	=	=	SYM
ejpam-5425	109	1	2	2	NUM
ejpam-5425	109	2	,	,	PUNCT
ejpam-5425	109	3	t−1	t−1	NOUN
ejpam-5425	109	4	=	=	SYM
ejpam-5425	109	5	1.5	1.5	NUM
ejpam-5425	109	6	,	,	PUNCT
ejpam-5425	109	7	t−2	t−2	NOUN
ejpam-5425	109	8	=	=	PUNCT
ejpam-5425	109	9	0.2	0.2	NUM
ejpam-5425	109	10	,	,	PUNCT
ejpam-5425	109	11	t−3	t−3	NOUN
ejpam-5425	109	12	=	=	NUM
ejpam-5425	109	13	0.4	0.4	NUM
ejpam-5425	109	14	,	,	PUNCT
ejpam-5425	109	15	t−4	t−4	PROPN
ejpam-5425	109	16	=	=	SYM
ejpam-5425	109	17	1	1	NUM
ejpam-5425	109	18	,	,	PUNCT
ejpam-5425	109	19	e0	e0	PROPN
ejpam-5425	109	20	=	=	NOUN
ejpam-5425	109	21	0.1	0.1	NUM
ejpam-5425	109	22	,	,	PUNCT
ejpam-5425	109	23	e−1	e−1	PROPN
ejpam-5425	109	24	=	=	SYM
ejpam-5425	109	25	0.6	0.6	NUM
ejpam-5425	109	26	,	,	PUNCT
ejpam-5425	109	27	e−2	e−2	PROPN
ejpam-5425	109	28	=	=	SYM
ejpam-5425	109	29	3	3	NUM
ejpam-5425	109	30	,	,	PUNCT
ejpam-5425	109	31	e−3	e−3	PROPN
ejpam-5425	109	32	=	=	SYM
ejpam-5425	109	33	1	1	NUM
ejpam-5425	109	34	,	,	PUNCT
ejpam-5425	109	35	and	and	CCONJ
ejpam-5425	109	36	e−4	e−4	PROPN
ejpam-5425	109	37	=	=	SYM
ejpam-5425	109	38	0.3	0.3	NUM
ejpam-5425	109	39	.	.	PUNCT
ejpam-5425	110	1	figure	figure	NOUN
ejpam-5425	110	2	2	2	NUM
ejpam-5425	110	3	:	:	PUNCT
ejpam-5425	110	4	chart	chart	VERB
ejpam-5425	110	5	the	the	DET
ejpam-5425	110	6	behavior	behavior	NOUN
ejpam-5425	110	7	of	of	ADP
ejpam-5425	110	8	solutions	solution	NOUN
ejpam-5425	110	9	of	of	ADP
ejpam-5425	110	10	the	the	DET
ejpam-5425	110	11	sdes(2	sdes(2	NOUN
ejpam-5425	110	12	)	)	PUNCT
ejpam-5425	110	13	j.	j.	PROPN
ejpam-5425	110	14	g.	g.	PROPN
ejpam-5425	110	15	al	al	PROPN
ejpam-5425	110	16	-	-	PUNCT
ejpam-5425	110	17	juaid	juaid	PROPN
ejpam-5425	110	18	/	/	SYM
ejpam-5425	110	19	eur	eur	PROPN
ejpam-5425	110	20	.	.	PUNCT
ejpam-5425	111	1	j.	j.	PROPN
ejpam-5425	111	2	pure	pure	PROPN
ejpam-5425	111	3	appl	appl	PROPN
ejpam-5425	111	4	.	.	PROPN
ejpam-5425	111	5	math	math	PROPN
ejpam-5425	111	6	,	,	PUNCT
ejpam-5425	111	7	17	17	NUM
ejpam-5425	111	8	(	(	PUNCT
ejpam-5425	111	9	4	4	NUM
ejpam-5425	111	10	)	)	PUNCT
ejpam-5425	111	11	(	(	PUNCT
ejpam-5425	111	12	2024	2024	NUM
ejpam-5425	111	13	)	)	PUNCT
ejpam-5425	111	14	,	,	PUNCT
ejpam-5425	111	15	3254	3254	NUM
ejpam-5425	111	16	-	-	SYM
ejpam-5425	111	17	3267	3267	NUM
ejpam-5425	111	18	3260	3260	NUM
ejpam-5425	111	19	4	4	NUM
ejpam-5425	111	20	.	.	PUNCT
ejpam-5425	112	1	the	the	DET
ejpam-5425	112	2	system	system	NOUN
ejpam-5425	112	3	tn+1	tn+1	NOUN
ejpam-5425	112	4	=	=	SYM
ejpam-5425	112	5	enen−4	enen−4	NOUN
ejpam-5425	112	6	tn−3(1+enen−4	tn−3(1+enen−4	NUM
ejpam-5425	112	7	)	)	PUNCT
ejpam-5425	112	8	,	,	PUNCT
ejpam-5425	112	9	en+1	en+1	ADJ
ejpam-5425	112	10	=	=	VERB
ejpam-5425	112	11	tntn−4	tntn−4	NUM
ejpam-5425	112	12	en−3(−1−tntn−4	en−3(−1−tntn−4	NOUN
ejpam-5425	112	13	)	)	PUNCT
ejpam-5425	112	14	in	in	ADP
ejpam-5425	112	15	this	this	DET
ejpam-5425	112	16	section	section	NOUN
ejpam-5425	112	17	,	,	PUNCT
ejpam-5425	112	18	we	we	PRON
ejpam-5425	112	19	find	find	VERB
ejpam-5425	112	20	form	form	VERB
ejpam-5425	112	21	the	the	DET
ejpam-5425	112	22	solutions	solution	NOUN
ejpam-5425	112	23	of	of	ADP
ejpam-5425	112	24	the	the	DET
ejpam-5425	112	25	sdes	sde	NOUN
ejpam-5425	112	26	in	in	ADP
ejpam-5425	112	27	the	the	DET
ejpam-5425	112	28	form	form	NOUN
ejpam-5425	112	29	:	:	PUNCT
ejpam-5425	112	30	tn+1	tn+1	NOUN
ejpam-5425	112	31	=	=	SYM
ejpam-5425	112	32	enen−4	enen−4	X
ejpam-5425	112	33	tn−3(1	tn−3(1	NOUN
ejpam-5425	112	34	+	+	NUM
ejpam-5425	112	35	enen−4	enen−4	NOUN
ejpam-5425	112	36	)	)	PUNCT
ejpam-5425	112	37	,	,	PUNCT
ejpam-5425	112	38	en+1	en+1	ADJ
ejpam-5425	112	39	=	=	SYM
ejpam-5425	112	40	tntn−4	tntn−4	NUM
ejpam-5425	112	41	en−3(−1−	en−3(−1−	PROPN
ejpam-5425	112	42	tntn−4	tntn−4	NOUN
ejpam-5425	112	43	)	)	PUNCT
ejpam-5425	112	44	.	.	PUNCT
ejpam-5425	113	1	(	(	PUNCT
ejpam-5425	113	2	3	3	X
ejpam-5425	113	3	)	)	PUNCT
ejpam-5425	113	4	theorem	theorem	NOUN
ejpam-5425	113	5	3	3	NUM
ejpam-5425	113	6	.	.	PUNCT
ejpam-5425	113	7	suppose	suppose	VERB
ejpam-5425	113	8	that	that	SCONJ
ejpam-5425	113	9	{	{	PUNCT
ejpam-5425	113	10	tn	tn	NOUN
ejpam-5425	113	11	,	,	PUNCT
ejpam-5425	113	12	en	en	ADP
ejpam-5425	113	13	}	}	PUNCT
ejpam-5425	113	14	are	be	AUX
ejpam-5425	113	15	solutions	solution	NOUN
ejpam-5425	113	16	of	of	ADP
ejpam-5425	113	17	eq.(3	eq.(3	NOUN
ejpam-5425	113	18	)	)	PUNCT
ejpam-5425	113	19	.	.	PUNCT
ejpam-5425	114	1	after	after	ADP
ejpam-5425	114	2	that	that	PRON
ejpam-5425	114	3	,	,	PUNCT
ejpam-5425	114	4	all	all	PRON
ejpam-5425	114	5	of	of	ADP
ejpam-5425	114	6	solutions	solution	NOUN
ejpam-5425	114	7	of	of	ADP
ejpam-5425	114	8	eq.(3	eq.(3	NOUN
ejpam-5425	114	9	)	)	PUNCT
ejpam-5425	114	10	are	be	AUX
ejpam-5425	114	11	periodic	periodic	ADJ
ejpam-5425	114	12	with	with	ADP
ejpam-5425	114	13	period	period	NOUN
ejpam-5425	114	14	eight	eight	NUM
ejpam-5425	114	15	and	and	CCONJ
ejpam-5425	114	16	given	give	VERB
ejpam-5425	114	17	by	by	ADP
ejpam-5425	114	18	the	the	DET
ejpam-5425	114	19	following	follow	VERB
ejpam-5425	114	20	formulas	formula	NOUN
ejpam-5425	114	21	for	for	ADP
ejpam-5425	114	22	n=0,1,2	n=0,1,2	NUM
ejpam-5425	114	23	,	,	PUNCT
ejpam-5425	114	24	...	...	PUNCT
ejpam-5425	114	25	,	,	PUNCT
ejpam-5425	114	26	t8n−4	t8n−4	PROPN
ejpam-5425	114	27	=	=	SYM
ejpam-5425	114	28	s	s	PROPN
ejpam-5425	114	29	,	,	PUNCT
ejpam-5425	114	30	t8n−3	t8n−3	PROPN
ejpam-5425	114	31	=	=	SYM
ejpam-5425	115	1	d	d	PROPN
ejpam-5425	115	2	,	,	PUNCT
ejpam-5425	115	3	t8n−2	t8n−2	PROPN
ejpam-5425	115	4	=	=	SYM
ejpam-5425	115	5	c	c	NOUN
ejpam-5425	115	6	,	,	PUNCT
ejpam-5425	115	7	t8n−1	t8n−1	PROPN
ejpam-5425	115	8	=	=	SYM
ejpam-5425	115	9	b	b	PROPN
ejpam-5425	115	10	,	,	PUNCT
ejpam-5425	115	11	t8n	t8n	NOUN
ejpam-5425	115	12	=	=	SYM
ejpam-5425	115	13	a	a	PRON
ejpam-5425	115	14	,	,	PUNCT
ejpam-5425	115	15	t8n+1	t8n+1	ADV
ejpam-5425	115	16	=	=	PUNCT
ejpam-5425	116	1	fw	fw	INTJ
ejpam-5425	116	2	d(1	d(1	PROPN
ejpam-5425	116	3	+	+	CCONJ
ejpam-5425	116	4	fw	fw	X
ejpam-5425	116	5	)	)	PUNCT
ejpam-5425	116	6	,	,	PUNCT
ejpam-5425	116	7	t8n+2	t8n+2	PROPN
ejpam-5425	117	1	=	=	PUNCT
ejpam-5425	117	2	−as	−as	PROPN
ejpam-5425	117	3	c	c	NOUN
ejpam-5425	117	4	,	,	PUNCT
ejpam-5425	117	5	t8n+3	t8n+3	PUNCT
ejpam-5425	117	6	=	=	PUNCT
ejpam-5425	118	1	fw	fw	PROPN
ejpam-5425	118	2	b(−1−	b(−1−	PROPN
ejpam-5425	118	3	fw	fw	PROPN
ejpam-5425	118	4	)	)	PUNCT
ejpam-5425	118	5	,	,	PUNCT
ejpam-5425	118	6	e8n−4	e8n−4	PROPN
ejpam-5425	118	7	=	=	SYM
ejpam-5425	118	8	w	w	PROPN
ejpam-5425	118	9	,	,	PUNCT
ejpam-5425	118	10	e8n−3	e8n−3	PROPN
ejpam-5425	118	11	=	=	SYM
ejpam-5425	118	12	m	m	PROPN
ejpam-5425	118	13	,	,	PUNCT
ejpam-5425	118	14	e8n−2	e8n−2	PROPN
ejpam-5425	118	15	=	=	SYM
ejpam-5425	118	16	g	g	PROPN
ejpam-5425	118	17	,	,	PUNCT
ejpam-5425	118	18	e8n−1	e8n−1	PROPN
ejpam-5425	118	19	=	=	SYM
ejpam-5425	118	20	k	k	PROPN
ejpam-5425	118	21	,	,	PUNCT
ejpam-5425	118	22	e8n	e8n	X
ejpam-5425	118	23	=	=	SYM
ejpam-5425	118	24	f	f	X
ejpam-5425	118	25	,	,	PUNCT
ejpam-5425	118	26	e8n+1	e8n+1	NOUN
ejpam-5425	118	27	=	=	PUNCT
ejpam-5425	118	28	as	as	ADV
ejpam-5425	118	29	m(−1−	m(−1−	ADJ
ejpam-5425	118	30	as	as	ADP
ejpam-5425	118	31	)	)	PUNCT
ejpam-5425	118	32	,	,	PUNCT
ejpam-5425	118	33	e8n+2	e8n+2	NOUN
ejpam-5425	118	34	=	=	PUNCT
ejpam-5425	118	35	fw	fw	PROPN
ejpam-5425	118	36	g(−1−	g(−1−	PROPN
ejpam-5425	118	37	2fw	2fw	NOUN
ejpam-5425	118	38	)	)	PUNCT
ejpam-5425	118	39	,	,	PUNCT
ejpam-5425	118	40	e8n+3	e8n+3	X
ejpam-5425	118	41	=	=	SYM
ejpam-5425	118	42	−as	−as	PROPN
ejpam-5425	118	43	k(−1	k(−1	X
ejpam-5425	119	1	+	+	CCONJ
ejpam-5425	119	2	as	as	ADP
ejpam-5425	119	3	)	)	PUNCT
ejpam-5425	119	4	,	,	PUNCT
ejpam-5425	119	5	where	where	SCONJ
ejpam-5425	119	6	e0e−4	e0e−4	PROPN
ejpam-5425	119	7	̸=	̸=	PROPN
ejpam-5425	119	8	±1	±1	VERB
ejpam-5425	119	9	and	and	CCONJ
ejpam-5425	119	10	t0t−4	t0t−4	PROPN
ejpam-5425	119	11	̸=	̸=	PROPN
ejpam-5425	119	12	±1	±1	VERB
ejpam-5425	119	13	.	.	PUNCT
ejpam-5425	120	1	proof	proof	NOUN
ejpam-5425	120	2	.	.	PUNCT
ejpam-5425	121	1	the	the	DET
ejpam-5425	121	2	outcome	outcome	NOUN
ejpam-5425	121	3	is	be	AUX
ejpam-5425	121	4	valid	valid	ADJ
ejpam-5425	121	5	for	for	ADP
ejpam-5425	121	6	n	n	NOUN
ejpam-5425	121	7	=	=	SYM
ejpam-5425	121	8	0	0	X
ejpam-5425	121	9	.	.	PUNCT
ejpam-5425	122	1	let	let	VERB
ejpam-5425	122	2	us	we	PRON
ejpam-5425	122	3	now	now	ADV
ejpam-5425	122	4	assume	assume	VERB
ejpam-5425	122	5	that	that	SCONJ
ejpam-5425	122	6	n	n	PROPN
ejpam-5425	122	7	>	>	X
ejpam-5425	122	8	0	0	PUNCT
ejpam-5425	123	1	and	and	CCONJ
ejpam-5425	123	2	that	that	PRON
ejpam-5425	123	3	n−	n−	NOUN
ejpam-5425	123	4	1	1	NUM
ejpam-5425	123	5	is	be	AUX
ejpam-5425	123	6	consistent	consistent	ADJ
ejpam-5425	123	7	with	with	ADP
ejpam-5425	123	8	our	our	PRON
ejpam-5425	123	9	assumption	assumption	NOUN
ejpam-5425	123	10	.	.	PUNCT
ejpam-5425	124	1	that	that	PRON
ejpam-5425	124	2	is	be	AUX
ejpam-5425	124	3	t8n−12	t8n−12	X
ejpam-5425	124	4	=	=	SYM
ejpam-5425	124	5	s	s	PROPN
ejpam-5425	124	6	,	,	PUNCT
ejpam-5425	124	7	t8n−11	t8n−11	PUNCT
ejpam-5425	125	1	=	=	PUNCT
ejpam-5425	125	2	d	d	PROPN
ejpam-5425	125	3	,	,	PUNCT
ejpam-5425	125	4	t8n−10	t8n−10	PROPN
ejpam-5425	125	5	=	=	SYM
ejpam-5425	125	6	c	c	X
ejpam-5425	125	7	,	,	PUNCT
ejpam-5425	125	8	t8n−9	t8n−9	NOUN
ejpam-5425	125	9	=	=	SYM
ejpam-5425	125	10	b	b	NOUN
ejpam-5425	125	11	,	,	PUNCT
ejpam-5425	125	12	t8n−8	t8n−8	PROPN
ejpam-5425	125	13	=	=	SYM
ejpam-5425	125	14	a	a	NOUN
ejpam-5425	125	15	,	,	PUNCT
ejpam-5425	125	16	t8n−7	t8n−7	NOUN
ejpam-5425	125	17	=	=	PUNCT
ejpam-5425	125	18	fw	fw	PROPN
ejpam-5425	126	1	d(1	d(1	PROPN
ejpam-5425	126	2	+	+	CCONJ
ejpam-5425	126	3	fw	fw	X
ejpam-5425	126	4	)	)	PUNCT
ejpam-5425	126	5	,	,	PUNCT
ejpam-5425	126	6	t8n−6	t8n−6	PROPN
ejpam-5425	126	7	=	=	SYM
ejpam-5425	126	8	−as	−as	PROPN
ejpam-5425	126	9	c	c	NOUN
ejpam-5425	126	10	,	,	PUNCT
ejpam-5425	126	11	t8n−5	t8n−5	PROPN
ejpam-5425	126	12	=	=	PUNCT
ejpam-5425	127	1	fw	fw	PROPN
ejpam-5425	127	2	b(−1−	b(−1−	PROPN
ejpam-5425	127	3	fw	fw	PROPN
ejpam-5425	127	4	)	)	PUNCT
ejpam-5425	127	5	,	,	PUNCT
ejpam-5425	127	6	e8n−12	e8n−12	X
ejpam-5425	127	7	=	=	SYM
ejpam-5425	127	8	w	w	PROPN
ejpam-5425	127	9	,	,	PUNCT
ejpam-5425	127	10	e8n−11	e8n−11	VERB
ejpam-5425	127	11	=	=	PROPN
ejpam-5425	127	12	m	m	PROPN
ejpam-5425	127	13	,	,	PUNCT
ejpam-5425	127	14	e8n−10	e8n−10	PROPN
ejpam-5425	127	15	=	=	PUNCT
ejpam-5425	127	16	g	g	PROPN
ejpam-5425	127	17	,	,	PUNCT
ejpam-5425	127	18	e8n−9	e8n−9	PROPN
ejpam-5425	127	19	=	=	SYM
ejpam-5425	127	20	k	k	PROPN
ejpam-5425	127	21	,	,	PUNCT
ejpam-5425	127	22	e8n−8	e8n−8	PROPN
ejpam-5425	127	23	=	=	SYM
ejpam-5425	127	24	f	f	PROPN
ejpam-5425	127	25	,	,	PUNCT
ejpam-5425	127	26	e8n−7	e8n−7	NOUN
ejpam-5425	127	27	=	=	PUNCT
ejpam-5425	127	28	as	as	ADV
ejpam-5425	127	29	m(−1−	m(−1−	ADJ
ejpam-5425	127	30	as	as	ADP
ejpam-5425	127	31	)	)	PUNCT
ejpam-5425	127	32	,	,	PUNCT
ejpam-5425	127	33	e8n−6	e8n−6	PROPN
ejpam-5425	128	1	=	=	SYM
ejpam-5425	128	2	fw	fw	PROPN
ejpam-5425	128	3	g(−1−	g(−1−	PROPN
ejpam-5425	128	4	2fw	2fw	NOUN
ejpam-5425	128	5	)	)	PUNCT
ejpam-5425	128	6	,	,	PUNCT
ejpam-5425	128	7	e8n−5	e8n−5	PROPN
ejpam-5425	128	8	=	=	PUNCT
ejpam-5425	128	9	−as	−as	PROPN
ejpam-5425	128	10	k(−1	k(−1	X
ejpam-5425	128	11	+	+	CCONJ
ejpam-5425	128	12	as	as	ADP
ejpam-5425	128	13	)	)	PUNCT
ejpam-5425	128	14	.	.	PUNCT
ejpam-5425	129	1	j.	j.	PROPN
ejpam-5425	129	2	g.	g.	PROPN
ejpam-5425	129	3	al	al	PROPN
ejpam-5425	129	4	-	-	PUNCT
ejpam-5425	129	5	juaid	juaid	PROPN
ejpam-5425	129	6	/	/	SYM
ejpam-5425	129	7	eur	eur	PROPN
ejpam-5425	129	8	.	.	PUNCT
ejpam-5425	130	1	j.	j.	PROPN
ejpam-5425	130	2	pure	pure	PROPN
ejpam-5425	130	3	appl	appl	PROPN
ejpam-5425	130	4	.	.	PROPN
ejpam-5425	130	5	math	math	PROPN
ejpam-5425	130	6	,	,	PUNCT
ejpam-5425	130	7	17	17	NUM
ejpam-5425	130	8	(	(	PUNCT
ejpam-5425	130	9	4	4	NUM
ejpam-5425	130	10	)	)	PUNCT
ejpam-5425	130	11	(	(	PUNCT
ejpam-5425	130	12	2024	2024	NUM
ejpam-5425	130	13	)	)	PUNCT
ejpam-5425	130	14	,	,	PUNCT
ejpam-5425	130	15	3254	3254	NUM
ejpam-5425	130	16	-	-	SYM
ejpam-5425	130	17	3267	3267	NUM
ejpam-5425	130	18	3261	3261	NUM
ejpam-5425	130	19	next	next	ADV
ejpam-5425	130	20	,	,	PUNCT
ejpam-5425	130	21	from	from	ADP
ejpam-5425	130	22	system	system	NOUN
ejpam-5425	130	23	(	(	PUNCT
ejpam-5425	130	24	3	3	X
ejpam-5425	130	25	)	)	PUNCT
ejpam-5425	130	26	we	we	PRON
ejpam-5425	130	27	have	have	VERB
ejpam-5425	130	28	t8n+2	t8n+2	PROPN
ejpam-5425	131	1	=	=	PRON
ejpam-5425	131	2	e8n+1e8n−3	e8n+1e8n−3	PROPN
ejpam-5425	131	3	t8n−2(1	t8n−2(1	NOUN
ejpam-5425	131	4	+	+	CCONJ
ejpam-5425	131	5	e8n+1e8n−3	e8n+1e8n−3	PROPN
ejpam-5425	131	6	)	)	PUNCT
ejpam-5425	131	7	=	=	PROPN
ejpam-5425	131	8	mas	mas	PROPN
ejpam-5425	131	9	m(−1−as	m(−1−as	NOUN
ejpam-5425	131	10	)	)	PUNCT
ejpam-5425	131	11	c(1	c(1	PROPN
ejpam-5425	131	12	+	+	CCONJ
ejpam-5425	131	13	mas	mas	PROPN
ejpam-5425	131	14	m(−1−as	m(−1−as	NOUN
ejpam-5425	131	15	)	)	PUNCT
ejpam-5425	131	16	)	)	PUNCT
ejpam-5425	132	1	=	=	PUNCT
ejpam-5425	132	2	−as	−as	PROPN
ejpam-5425	132	3	c	c	NOUN
ejpam-5425	132	4	.	.	PUNCT
ejpam-5425	133	1	e8n+2	e8n+2	PROPN
ejpam-5425	134	1	=	=	X
ejpam-5425	134	2	t8n+1t8n−3	t8n+1t8n−3	PROPN
ejpam-5425	134	3	e8n−2(−1−	e8n−2(−1−	PROPN
ejpam-5425	134	4	t8n+1t8n−3	t8n+1t8n−3	PROPN
ejpam-5425	134	5	)	)	PUNCT
ejpam-5425	134	6	=	=	PUNCT
ejpam-5425	134	7	fwd	fwd	PROPN
ejpam-5425	134	8	d(1+fw	d(1+fw	PROPN
ejpam-5425	134	9	)	)	PUNCT
ejpam-5425	135	1	g(−1−	g(−1−	PROPN
ejpam-5425	135	2	fwd	fwd	PROPN
ejpam-5425	135	3	d(1+fw	d(1+fw	PROPN
ejpam-5425	135	4	)	)	PUNCT
ejpam-5425	135	5	)	)	PUNCT
ejpam-5425	136	1	=	=	PUNCT
ejpam-5425	136	2	fw	fw	PROPN
ejpam-5425	136	3	g(−1−	g(−1−	PROPN
ejpam-5425	136	4	2fw	2fw	NOUN
ejpam-5425	136	5	)	)	PUNCT
ejpam-5425	136	6	.	.	PUNCT
ejpam-5425	137	1	t8n+3	t8n+3	PUNCT
ejpam-5425	138	1	=	=	PRON
ejpam-5425	138	2	e8n+2e8n−2	e8n+2e8n−2	PROPN
ejpam-5425	138	3	t8n−1(1	t8n−1(1	VERB
ejpam-5425	138	4	+	+	CCONJ
ejpam-5425	138	5	e8n+2e8n−2	e8n+2e8n−2	ADJ
ejpam-5425	138	6	)	)	PUNCT
ejpam-5425	138	7	=	=	SYM
ejpam-5425	138	8	fwg	fwg	PROPN
ejpam-5425	138	9	g(−1−2fw	g(−1−2fw	X
ejpam-5425	138	10	)	)	PUNCT
ejpam-5425	138	11	b(1	b(1	PROPN
ejpam-5425	138	12	+	+	CCONJ
ejpam-5425	138	13	fwg	fwg	PROPN
ejpam-5425	138	14	g(−1−2fw	g(−1−2fw	NOUN
ejpam-5425	138	15	)	)	PUNCT
ejpam-5425	138	16	)	)	PUNCT
ejpam-5425	139	1	=	=	PUNCT
ejpam-5425	139	2	fw	fw	PROPN
ejpam-5425	139	3	b(−1−	b(−1−	PROPN
ejpam-5425	139	4	fw	fw	PROPN
ejpam-5425	139	5	)	)	PUNCT
ejpam-5425	139	6	.	.	PUNCT
ejpam-5425	140	1	e8n+3	e8n+3	X
ejpam-5425	141	1	=	=	SYM
ejpam-5425	141	2	t8n+2t8n−2	t8n+2t8n−2	PROPN
ejpam-5425	141	3	e8n−1(−1−	e8n−1(−1−	NOUN
ejpam-5425	141	4	t8n+2t8n−2	t8n+2t8n−2	PROPN
ejpam-5425	141	5	)	)	PUNCT
ejpam-5425	141	6	=	=	PUNCT
ejpam-5425	142	1	−asc	−asc	PROPN
ejpam-5425	142	2	c	c	PROPN
ejpam-5425	143	1	k(−1	k(−1	PROPN
ejpam-5425	143	2	+	+	CCONJ
ejpam-5425	143	3	asc	asc	PROPN
ejpam-5425	143	4	c	c	PROPN
ejpam-5425	143	5	=	=	PUNCT
ejpam-5425	143	6	−as	−as	PROPN
ejpam-5425	143	7	k(−1	k(−1	X
ejpam-5425	143	8	+	+	CCONJ
ejpam-5425	143	9	as	as	ADP
ejpam-5425	143	10	)	)	PUNCT
ejpam-5425	143	11	.	.	PUNCT
ejpam-5425	144	1	we	we	PRON
ejpam-5425	144	2	can	can	AUX
ejpam-5425	144	3	validate	validate	VERB
ejpam-5425	144	4	the	the	DET
ejpam-5425	144	5	other	other	ADJ
ejpam-5425	144	6	forms	form	NOUN
ejpam-5425	144	7	by	by	ADP
ejpam-5425	144	8	using	use	VERB
ejpam-5425	144	9	the	the	DET
ejpam-5425	144	10	same	same	ADJ
ejpam-5425	144	11	procedure	procedure	NOUN
ejpam-5425	144	12	.	.	PUNCT
ejpam-5425	145	1	the	the	DET
ejpam-5425	145	2	proof	proof	NOUN
ejpam-5425	145	3	is	be	AUX
ejpam-5425	145	4	complete	complete	ADJ
ejpam-5425	145	5	.	.	PUNCT
ejpam-5425	145	6	example	example	NOUN
ejpam-5425	146	1	3	3	NUM
ejpam-5425	146	2	.	.	PUNCT
ejpam-5425	146	3	the	the	DET
ejpam-5425	146	4	solution	solution	NOUN
ejpam-5425	146	5	is	be	AUX
ejpam-5425	146	6	periodic	periodic	ADJ
ejpam-5425	146	7	of	of	ADP
ejpam-5425	146	8	period	period	NOUN
ejpam-5425	146	9	eight	eight	NUM
ejpam-5425	146	10	when	when	SCONJ
ejpam-5425	146	11	theorem	theorem	ADJ
ejpam-5425	146	12	3	3	NUM
ejpam-5425	146	13	is	be	AUX
ejpam-5425	146	14	met	meet	VERB
ejpam-5425	146	15	and	and	CCONJ
ejpam-5425	146	16	the	the	DET
ejpam-5425	146	17	initial	initial	ADJ
ejpam-5425	146	18	values	value	NOUN
ejpam-5425	146	19	are	be	AUX
ejpam-5425	146	20	t0	t0	NOUN
ejpam-5425	146	21	=	=	SYM
ejpam-5425	146	22	3	3	NUM
ejpam-5425	146	23	,	,	PUNCT
ejpam-5425	146	24	t−1	t−1	NOUN
ejpam-5425	146	25	=	=	SYM
ejpam-5425	146	26	1	1	NUM
ejpam-5425	146	27	,	,	PUNCT
ejpam-5425	146	28	t−2	t−2	PROPN
ejpam-5425	146	29	=	=	SYM
ejpam-5425	146	30	0.1	0.1	NUM
ejpam-5425	146	31	,	,	PUNCT
ejpam-5425	146	32	t−3	t−3	NOUN
ejpam-5425	146	33	=	=	SYM
ejpam-5425	146	34	2	2	NUM
ejpam-5425	146	35	,	,	PUNCT
ejpam-5425	146	36	t−4	t−4	PROPN
ejpam-5425	146	37	=	=	SYM
ejpam-5425	146	38	0.2	0.2	NUM
ejpam-5425	146	39	,	,	PUNCT
ejpam-5425	146	40	e0	e0	PROPN
ejpam-5425	146	41	=	=	PROPN
ejpam-5425	146	42	1	1	NUM
ejpam-5425	146	43	,	,	PUNCT
ejpam-5425	146	44	e−1	e−1	PROPN
ejpam-5425	146	45	=	=	SYM
ejpam-5425	146	46	0.5	0.5	NUM
ejpam-5425	146	47	,	,	PUNCT
ejpam-5425	146	48	e−2	e−2	PROPN
ejpam-5425	146	49	=	=	SYM
ejpam-5425	146	50	0.3	0.3	NUM
ejpam-5425	146	51	,	,	PUNCT
ejpam-5425	146	52	e−3	e−3	PROPN
ejpam-5425	146	53	=	=	SYM
ejpam-5425	146	54	2	2	NUM
ejpam-5425	146	55	,	,	PUNCT
ejpam-5425	146	56	and	and	CCONJ
ejpam-5425	146	57	e−4	e−4	PROPN
ejpam-5425	146	58	=	=	PROPN
ejpam-5425	146	59	0.9	0.9	NUM
ejpam-5425	146	60	,	,	PUNCT
ejpam-5425	146	61	as	as	SCONJ
ejpam-5425	146	62	shown	show	VERB
ejpam-5425	146	63	in	in	ADP
ejpam-5425	146	64	figure	figure	NOUN
ejpam-5425	146	65	(	(	PUNCT
ejpam-5425	146	66	3	3	NUM
ejpam-5425	146	67	)	)	PUNCT
ejpam-5425	146	68	.	.	PUNCT
ejpam-5425	147	1	figure	figure	VERB
ejpam-5425	147	2	3	3	NUM
ejpam-5425	147	3	:	:	PUNCT
ejpam-5425	147	4	draw	draw	VERB
ejpam-5425	147	5	a	a	DET
ejpam-5425	147	6	graph	graph	NOUN
ejpam-5425	147	7	displaying	display	VERB
ejpam-5425	147	8	periodicity	periodicity	NOUN
ejpam-5425	147	9	of	of	ADP
ejpam-5425	147	10	the	the	DET
ejpam-5425	147	11	solutions	solution	NOUN
ejpam-5425	147	12	of	of	ADP
ejpam-5425	147	13	sdes	sde	NOUN
ejpam-5425	147	14	(	(	PUNCT
ejpam-5425	147	15	3	3	NUM
ejpam-5425	147	16	)	)	PUNCT
ejpam-5425	147	17	5	5	NUM
ejpam-5425	147	18	.	.	PUNCT
ejpam-5425	148	1	the	the	DET
ejpam-5425	148	2	system	system	NOUN
ejpam-5425	148	3	tn+1	tn+1	NOUN
ejpam-5425	148	4	=	=	SYM
ejpam-5425	148	5	enen−4	enen−4	NOUN
ejpam-5425	148	6	tn−3(−1+enen−4	tn−3(−1+enen−4	NOUN
ejpam-5425	148	7	)	)	PUNCT
ejpam-5425	148	8	,	,	PUNCT
ejpam-5425	148	9	en+1	en+1	ADJ
ejpam-5425	148	10	=	=	PROPN
ejpam-5425	148	11	tntn−4	tntn−4	NUM
ejpam-5425	148	12	en−3(−1+tntn−4	en−3(−1+tntn−4	NOUN
ejpam-5425	148	13	)	)	PUNCT
ejpam-5425	148	14	in	in	ADP
ejpam-5425	148	15	this	this	DET
ejpam-5425	148	16	section	section	NOUN
ejpam-5425	148	17	,	,	PUNCT
ejpam-5425	148	18	we	we	PRON
ejpam-5425	148	19	investigate	investigate	VERB
ejpam-5425	148	20	the	the	DET
ejpam-5425	148	21	solutions	solution	NOUN
ejpam-5425	148	22	of	of	ADP
ejpam-5425	148	23	the	the	DET
ejpam-5425	148	24	sdes	sde	NOUN
ejpam-5425	148	25	in	in	ADP
ejpam-5425	148	26	the	the	DET
ejpam-5425	148	27	form	form	NOUN
ejpam-5425	148	28	:	:	PUNCT
ejpam-5425	148	29	tn+1	tn+1	NOUN
ejpam-5425	148	30	=	=	SYM
ejpam-5425	148	31	enen−4	enen−4	X
ejpam-5425	148	32	tn−3(−1	tn−3(−1	PROPN
ejpam-5425	148	33	+	+	CCONJ
ejpam-5425	148	34	enen−4	enen−4	NOUN
ejpam-5425	148	35	)	)	PUNCT
ejpam-5425	148	36	,	,	PUNCT
ejpam-5425	148	37	en+1	en+1	ADJ
ejpam-5425	148	38	=	=	SYM
ejpam-5425	148	39	tntn−4	tntn−4	X
ejpam-5425	148	40	en−3(−1	en−3(−1	X
ejpam-5425	148	41	+	+	CCONJ
ejpam-5425	148	42	tntn−4	tntn−4	NOUN
ejpam-5425	148	43	)	)	PUNCT
ejpam-5425	148	44	.	.	PUNCT
ejpam-5425	149	1	(	(	PUNCT
ejpam-5425	149	2	4	4	X
ejpam-5425	149	3	)	)	PUNCT
ejpam-5425	149	4	j.	j.	PROPN
ejpam-5425	149	5	g.	g.	PROPN
ejpam-5425	149	6	al	al	PROPN
ejpam-5425	149	7	-	-	PUNCT
ejpam-5425	149	8	juaid	juaid	PROPN
ejpam-5425	149	9	/	/	SYM
ejpam-5425	149	10	eur	eur	PROPN
ejpam-5425	149	11	.	.	PUNCT
ejpam-5425	150	1	j.	j.	PROPN
ejpam-5425	150	2	pure	pure	PROPN
ejpam-5425	150	3	appl	appl	PROPN
ejpam-5425	150	4	.	.	PROPN
ejpam-5425	150	5	math	math	PROPN
ejpam-5425	150	6	,	,	PUNCT
ejpam-5425	150	7	17	17	NUM
ejpam-5425	150	8	(	(	PUNCT
ejpam-5425	150	9	4	4	NUM
ejpam-5425	150	10	)	)	PUNCT
ejpam-5425	150	11	(	(	PUNCT
ejpam-5425	150	12	2024	2024	NUM
ejpam-5425	150	13	)	)	PUNCT
ejpam-5425	150	14	,	,	PUNCT
ejpam-5425	150	15	3254	3254	NUM
ejpam-5425	150	16	-	-	SYM
ejpam-5425	150	17	3267	3267	NUM
ejpam-5425	150	18	3262	3262	NUM
ejpam-5425	150	19	theorem	theorem	VERB
ejpam-5425	150	20	4	4	NUM
ejpam-5425	150	21	.	.	X
ejpam-5425	151	1	consider	consider	VERB
ejpam-5425	151	2	{	{	PUNCT
ejpam-5425	151	3	tn	tn	NOUN
ejpam-5425	151	4	,	,	PUNCT
ejpam-5425	151	5	en	en	ADP
ejpam-5425	151	6	}	}	PUNCT
ejpam-5425	151	7	are	be	AUX
ejpam-5425	151	8	solutions	solution	NOUN
ejpam-5425	151	9	of	of	ADP
ejpam-5425	151	10	eq.(4	eq.(4	ADJ
ejpam-5425	151	11	)	)	PUNCT
ejpam-5425	151	12	.	.	PUNCT
ejpam-5425	152	1	afterwards	afterwards	ADV
ejpam-5425	152	2	,	,	PUNCT
ejpam-5425	152	3	for	for	ADP
ejpam-5425	152	4	all	all	DET
ejpam-5425	152	5	n	n	NOUN
ejpam-5425	152	6	=	=	SYM
ejpam-5425	152	7	0	0	NUM
ejpam-5425	152	8	,	,	PUNCT
ejpam-5425	152	9	1	1	NUM
ejpam-5425	152	10	,	,	PUNCT
ejpam-5425	152	11	2	2	NUM
ejpam-5425	152	12	,	,	PUNCT
ejpam-5425	152	13	...	...	PUNCT
ejpam-5425	152	14	,	,	PUNCT
ejpam-5425	152	15	the	the	DET
ejpam-5425	152	16	following	follow	VERB
ejpam-5425	152	17	formulas	formula	NOUN
ejpam-5425	152	18	obtain	obtain	VERB
ejpam-5425	152	19	periodic	periodic	ADJ
ejpam-5425	152	20	solutions	solution	NOUN
ejpam-5425	152	21	of	of	ADP
ejpam-5425	152	22	eq	eq	PROPN
ejpam-5425	152	23	.	.	PUNCT
ejpam-5425	153	1	(	(	PUNCT
ejpam-5425	153	2	4	4	NUM
ejpam-5425	153	3	)	)	PUNCT
ejpam-5425	153	4	with	with	ADP
ejpam-5425	153	5	period	period	NOUN
ejpam-5425	153	6	eight	eight	NUM
ejpam-5425	153	7	.	.	PUNCT
ejpam-5425	154	1	t8n−4	t8n−4	NOUN
ejpam-5425	154	2	=	=	SYM
ejpam-5425	154	3	s	s	PROPN
ejpam-5425	154	4	,	,	PUNCT
ejpam-5425	154	5	t8n−3	t8n−3	PROPN
ejpam-5425	154	6	=	=	SYM
ejpam-5425	155	1	d	d	PROPN
ejpam-5425	155	2	,	,	PUNCT
ejpam-5425	155	3	t8n−2	t8n−2	PROPN
ejpam-5425	155	4	=	=	SYM
ejpam-5425	155	5	c	c	NOUN
ejpam-5425	155	6	,	,	PUNCT
ejpam-5425	155	7	t8n−1	t8n−1	PROPN
ejpam-5425	155	8	=	=	SYM
ejpam-5425	155	9	b	b	PROPN
ejpam-5425	155	10	,	,	PUNCT
ejpam-5425	155	11	t8n	t8n	NOUN
ejpam-5425	155	12	=	=	SYM
ejpam-5425	155	13	a	a	PRON
ejpam-5425	155	14	,	,	PUNCT
ejpam-5425	155	15	t8n+1	t8n+1	ADV
ejpam-5425	155	16	=	=	PUNCT
ejpam-5425	156	1	fw	fw	PROPN
ejpam-5425	156	2	d(−1	d(−1	PROPN
ejpam-5425	156	3	+	+	CCONJ
ejpam-5425	156	4	fw	fw	X
ejpam-5425	156	5	)	)	PUNCT
ejpam-5425	156	6	,	,	PUNCT
ejpam-5425	156	7	t8n+2	t8n+2	PROPN
ejpam-5425	157	1	=	=	PUNCT
ejpam-5425	157	2	as	as	ADP
ejpam-5425	157	3	c	c	PROPN
ejpam-5425	157	4	,	,	PUNCT
ejpam-5425	157	5	t8n+3	t8n+3	PUNCT
ejpam-5425	158	1	=	=	PUNCT
ejpam-5425	158	2	fw	fw	PROPN
ejpam-5425	158	3	b(−1	b(−1	PROPN
ejpam-5425	158	4	+	+	CCONJ
ejpam-5425	158	5	fw	fw	X
ejpam-5425	158	6	)	)	PUNCT
ejpam-5425	158	7	,	,	PUNCT
ejpam-5425	158	8	e8n−4	e8n−4	PROPN
ejpam-5425	158	9	=	=	SYM
ejpam-5425	158	10	w	w	PROPN
ejpam-5425	158	11	,	,	PUNCT
ejpam-5425	158	12	e8n−3	e8n−3	PROPN
ejpam-5425	158	13	=	=	SYM
ejpam-5425	158	14	m	m	PROPN
ejpam-5425	158	15	,	,	PUNCT
ejpam-5425	158	16	e8n−2	e8n−2	PROPN
ejpam-5425	158	17	=	=	SYM
ejpam-5425	158	18	g	g	PROPN
ejpam-5425	158	19	,	,	PUNCT
ejpam-5425	158	20	e8n−1	e8n−1	PROPN
ejpam-5425	158	21	=	=	SYM
ejpam-5425	158	22	k	k	PROPN
ejpam-5425	158	23	,	,	PUNCT
ejpam-5425	158	24	e8n	e8n	X
ejpam-5425	158	25	=	=	SYM
ejpam-5425	158	26	f	f	X
ejpam-5425	158	27	,	,	PUNCT
ejpam-5425	158	28	e8n+1	e8n+1	NOUN
ejpam-5425	158	29	=	=	PUNCT
ejpam-5425	158	30	as	as	ADP
ejpam-5425	158	31	m(−1	m(−1	PROPN
ejpam-5425	158	32	+	+	CCONJ
ejpam-5425	158	33	as	as	ADP
ejpam-5425	158	34	)	)	PUNCT
ejpam-5425	158	35	,	,	PUNCT
ejpam-5425	158	36	e8n+2	e8n+2	PROPN
ejpam-5425	158	37	=	=	PUNCT
ejpam-5425	159	1	fw	fw	PROPN
ejpam-5425	159	2	g	g	PROPN
ejpam-5425	159	3	,	,	PUNCT
ejpam-5425	159	4	e8n+3	e8n+3	PROPN
ejpam-5425	159	5	=	=	PUNCT
ejpam-5425	159	6	as	as	ADP
ejpam-5425	159	7	k(−1	k(−1	PROPN
ejpam-5425	159	8	+	+	CCONJ
ejpam-5425	159	9	as	as	ADP
ejpam-5425	159	10	)	)	PUNCT
ejpam-5425	159	11	,	,	PUNCT
ejpam-5425	159	12	where	where	SCONJ
ejpam-5425	159	13	e0e−4	e0e−4	PROPN
ejpam-5425	159	14	̸=	̸=	PROPN
ejpam-5425	159	15	1	1	NUM
ejpam-5425	159	16	and	and	CCONJ
ejpam-5425	159	17	t0t−4	t0t−4	PROPN
ejpam-5425	159	18	̸=	̸=	PROPN
ejpam-5425	159	19	1	1	NUM
ejpam-5425	159	20	.	.	PUNCT
ejpam-5425	160	1	proof	proof	NOUN
ejpam-5425	160	2	.	.	PUNCT
ejpam-5425	161	1	the	the	DET
ejpam-5425	161	2	result	result	NOUN
ejpam-5425	161	3	is	be	AUX
ejpam-5425	161	4	valid	valid	ADJ
ejpam-5425	161	5	for	for	ADP
ejpam-5425	161	6	n	n	NOUN
ejpam-5425	161	7	=	=	SYM
ejpam-5425	161	8	0	0	X
ejpam-5425	161	9	.	.	PUNCT
ejpam-5425	162	1	let	let	VERB
ejpam-5425	162	2	us	we	PRON
ejpam-5425	162	3	now	now	ADV
ejpam-5425	162	4	assume	assume	VERB
ejpam-5425	162	5	that	that	SCONJ
ejpam-5425	162	6	n	n	PROPN
ejpam-5425	162	7	>	>	X
ejpam-5425	162	8	0	0	PUNCT
ejpam-5425	163	1	and	and	CCONJ
ejpam-5425	163	2	that	that	PRON
ejpam-5425	163	3	n−	n−	NOUN
ejpam-5425	163	4	1	1	NUM
ejpam-5425	163	5	agrees	agree	VERB
ejpam-5425	163	6	with	with	ADP
ejpam-5425	163	7	our	our	PRON
ejpam-5425	163	8	assumption	assumption	NOUN
ejpam-5425	163	9	.	.	PUNCT
ejpam-5425	164	1	that	that	PRON
ejpam-5425	164	2	is	be	AUX
ejpam-5425	164	3	t8n−12	t8n−12	X
ejpam-5425	164	4	=	=	SYM
ejpam-5425	164	5	s	s	PROPN
ejpam-5425	164	6	,	,	PUNCT
ejpam-5425	164	7	t8n−11	t8n−11	PUNCT
ejpam-5425	165	1	=	=	PUNCT
ejpam-5425	165	2	d	d	PROPN
ejpam-5425	165	3	,	,	PUNCT
ejpam-5425	165	4	t8n−10	t8n−10	PROPN
ejpam-5425	165	5	=	=	SYM
ejpam-5425	165	6	c	c	X
ejpam-5425	165	7	,	,	PUNCT
ejpam-5425	165	8	t8n−9	t8n−9	NOUN
ejpam-5425	165	9	=	=	SYM
ejpam-5425	165	10	b	b	NOUN
ejpam-5425	165	11	,	,	PUNCT
ejpam-5425	165	12	t8n−8	t8n−8	PROPN
ejpam-5425	165	13	=	=	SYM
ejpam-5425	165	14	a	a	NOUN
ejpam-5425	165	15	,	,	PUNCT
ejpam-5425	165	16	t8n−7	t8n−7	NOUN
ejpam-5425	165	17	=	=	PUNCT
ejpam-5425	165	18	fw	fw	PROPN
ejpam-5425	165	19	d(−1	d(−1	PROPN
ejpam-5425	165	20	+	+	CCONJ
ejpam-5425	165	21	fw	fw	X
ejpam-5425	165	22	)	)	PUNCT
ejpam-5425	165	23	,	,	PUNCT
ejpam-5425	165	24	t8n−6	t8n−6	PROPN
ejpam-5425	165	25	=	=	PUNCT
ejpam-5425	165	26	as	as	ADP
ejpam-5425	165	27	c	c	PROPN
ejpam-5425	165	28	,	,	PUNCT
ejpam-5425	165	29	t8n−5	t8n−5	PROPN
ejpam-5425	165	30	=	=	PUNCT
ejpam-5425	166	1	fw	fw	PROPN
ejpam-5425	166	2	b(−1	b(−1	PROPN
ejpam-5425	166	3	+	+	CCONJ
ejpam-5425	166	4	fw	fw	X
ejpam-5425	166	5	)	)	PUNCT
ejpam-5425	166	6	,	,	PUNCT
ejpam-5425	166	7	e8n−12	e8n−12	X
ejpam-5425	166	8	=	=	SYM
ejpam-5425	166	9	w	w	PROPN
ejpam-5425	166	10	,	,	PUNCT
ejpam-5425	166	11	e8n−11	e8n−11	VERB
ejpam-5425	166	12	=	=	PROPN
ejpam-5425	166	13	m	m	PROPN
ejpam-5425	166	14	,	,	PUNCT
ejpam-5425	166	15	e8n−10	e8n−10	PROPN
ejpam-5425	166	16	=	=	PUNCT
ejpam-5425	166	17	g	g	PROPN
ejpam-5425	166	18	,	,	PUNCT
ejpam-5425	166	19	e8n−9	e8n−9	PROPN
ejpam-5425	166	20	=	=	SYM
ejpam-5425	166	21	k	k	PROPN
ejpam-5425	166	22	,	,	PUNCT
ejpam-5425	166	23	e8n−8	e8n−8	PROPN
ejpam-5425	166	24	=	=	SYM
ejpam-5425	166	25	f	f	PROPN
ejpam-5425	166	26	,	,	PUNCT
ejpam-5425	166	27	e8n−7	e8n−7	NOUN
ejpam-5425	166	28	=	=	PUNCT
ejpam-5425	166	29	as	as	ADP
ejpam-5425	166	30	m(−1	m(−1	PROPN
ejpam-5425	166	31	+	+	CCONJ
ejpam-5425	166	32	as	as	ADP
ejpam-5425	166	33	)	)	PUNCT
ejpam-5425	166	34	,	,	PUNCT
ejpam-5425	166	35	e8n−6	e8n−6	PROPN
ejpam-5425	167	1	=	=	PUNCT
ejpam-5425	167	2	fw	fw	PROPN
ejpam-5425	167	3	g	g	PROPN
ejpam-5425	167	4	,	,	PUNCT
ejpam-5425	167	5	e8n−5	e8n−5	PROPN
ejpam-5425	167	6	=	=	PUNCT
ejpam-5425	167	7	as	as	ADP
ejpam-5425	167	8	k(−1	k(−1	PROPN
ejpam-5425	167	9	+	+	CCONJ
ejpam-5425	167	10	as	as	ADP
ejpam-5425	167	11	)	)	PUNCT
ejpam-5425	167	12	.	.	PUNCT
ejpam-5425	168	1	next	next	ADV
ejpam-5425	168	2	,	,	PUNCT
ejpam-5425	168	3	from	from	ADP
ejpam-5425	168	4	system	system	NOUN
ejpam-5425	168	5	(	(	PUNCT
ejpam-5425	168	6	4	4	X
ejpam-5425	168	7	)	)	PUNCT
ejpam-5425	168	8	we	we	PRON
ejpam-5425	168	9	have	have	VERB
ejpam-5425	168	10	t8n−4	t8n−4	NOUN
ejpam-5425	168	11	=	=	SYM
ejpam-5425	168	12	e8n−5e8n−9	e8n−5e8n−9	NOUN
ejpam-5425	168	13	t8n−8(−1	t8n−8(−1	VERB
ejpam-5425	168	14	+	+	CCONJ
ejpam-5425	168	15	e8n−5e8n−9	e8n−5e8n−9	NOUN
ejpam-5425	168	16	)	)	PUNCT
ejpam-5425	169	1	=	=	PRON
ejpam-5425	169	2	ask	ask	VERB
ejpam-5425	169	3	k(−1+as	k(−1+as	PROPN
ejpam-5425	169	4	)	)	PUNCT
ejpam-5425	169	5	a(−1	a(−1	PROPN
ejpam-5425	170	1	+	+	CCONJ
ejpam-5425	170	2	ask	ask	VERB
ejpam-5425	170	3	k(−1+as	k(−1+as	PROPN
ejpam-5425	170	4	)	)	PUNCT
ejpam-5425	170	5	)	)	PUNCT
ejpam-5425	171	1	=	=	PUNCT
ejpam-5425	171	2	s.	s.	PROPN
ejpam-5425	171	3	j.	j.	PROPN
ejpam-5425	171	4	g.	g.	PROPN
ejpam-5425	171	5	al	al	PROPN
ejpam-5425	171	6	-	-	PUNCT
ejpam-5425	171	7	juaid	juaid	PROPN
ejpam-5425	171	8	/	/	SYM
ejpam-5425	171	9	eur	eur	PROPN
ejpam-5425	171	10	.	.	PUNCT
ejpam-5425	172	1	j.	j.	PROPN
ejpam-5425	172	2	pure	pure	PROPN
ejpam-5425	172	3	appl	appl	PROPN
ejpam-5425	172	4	.	.	PROPN
ejpam-5425	172	5	math	math	PROPN
ejpam-5425	172	6	,	,	PUNCT
ejpam-5425	172	7	17	17	NUM
ejpam-5425	172	8	(	(	PUNCT
ejpam-5425	172	9	4	4	NUM
ejpam-5425	172	10	)	)	PUNCT
ejpam-5425	172	11	(	(	PUNCT
ejpam-5425	172	12	2024	2024	NUM
ejpam-5425	172	13	)	)	PUNCT
ejpam-5425	172	14	,	,	PUNCT
ejpam-5425	172	15	3254	3254	NUM
ejpam-5425	172	16	-	-	SYM
ejpam-5425	172	17	3267	3267	NUM
ejpam-5425	172	18	3263	3263	NUM
ejpam-5425	172	19	e8n−4	e8n−4	NOUN
ejpam-5425	172	20	=	=	PUNCT
ejpam-5425	172	21	t8n−5t8n−9	t8n−5t8n−9	PROPN
ejpam-5425	172	22	e8n−8(−1	e8n−8(−1	NOUN
ejpam-5425	172	23	+	+	X
ejpam-5425	172	24	t8n−5t8n−9	t8n−5t8n−9	NOUN
ejpam-5425	172	25	)	)	PUNCT
ejpam-5425	172	26	=	=	SYM
ejpam-5425	173	1	fwb	fwb	PROPN
ejpam-5425	173	2	b(−1+fw	b(−1+fw	NOUN
ejpam-5425	173	3	)	)	PUNCT
ejpam-5425	174	1	f(−1	f(−1	PROPN
ejpam-5425	175	1	+	+	CCONJ
ejpam-5425	175	2	fwb	fwb	PROPN
ejpam-5425	175	3	b(−1+fw	b(−1+fw	NUM
ejpam-5425	175	4	)	)	PUNCT
ejpam-5425	175	5	)	)	PUNCT
ejpam-5425	176	1	=	=	SYM
ejpam-5425	176	2	w.	w.	PROPN
ejpam-5425	176	3	t8n	t8n	PROPN
ejpam-5425	176	4	=	=	PUNCT
ejpam-5425	176	5	e8n−1e8n−5	e8n−1e8n−5	PROPN
ejpam-5425	176	6	t8n−4(−1	t8n−4(−1	NOUN
ejpam-5425	176	7	+	+	CCONJ
ejpam-5425	176	8	e8n−1e8n−5	e8n−1e8n−5	PROPN
ejpam-5425	176	9	)	)	PUNCT
ejpam-5425	177	1	=	=	PRON
ejpam-5425	177	2	ask	ask	VERB
ejpam-5425	177	3	k(−1+as	k(−1+as	PROPN
ejpam-5425	177	4	)	)	PUNCT
ejpam-5425	177	5	s(−1	s(−1	PROPN
ejpam-5425	178	1	+	+	CCONJ
ejpam-5425	178	2	kas	kas	PROPN
ejpam-5425	178	3	k(−1+as	k(−1+as	PROPN
ejpam-5425	178	4	)	)	PUNCT
ejpam-5425	178	5	)	)	PUNCT
ejpam-5425	179	1	=	=	PUNCT
ejpam-5425	179	2	a.	a.	NOUN
ejpam-5425	179	3	e8n	e8n	PUNCT
ejpam-5425	179	4	=	=	NOUN
ejpam-5425	179	5	t8n−1t8n−5	t8n−1t8n−5	PROPN
ejpam-5425	179	6	e8n−4(−1	e8n−4(−1	X
ejpam-5425	180	1	+	+	CCONJ
ejpam-5425	180	2	t8n−1t8n−5	t8n−1t8n−5	PROPN
ejpam-5425	180	3	)	)	PUNCT
ejpam-5425	181	1	=	=	SYM
ejpam-5425	181	2	bfw	bfw	PROPN
ejpam-5425	181	3	b(−1+fw	b(−1+fw	NOUN
ejpam-5425	181	4	)	)	PUNCT
ejpam-5425	181	5	w(−1	w(−1	PUNCT
ejpam-5425	182	1	+	+	CCONJ
ejpam-5425	182	2	fwb	fwb	PROPN
ejpam-5425	182	3	b(−1+fw	b(−1+fw	NUM
ejpam-5425	182	4	)	)	PUNCT
ejpam-5425	182	5	)	)	PUNCT
ejpam-5425	183	1	=	=	SYM
ejpam-5425	183	2	f.	f.	PROPN
ejpam-5425	183	3	also	also	ADV
ejpam-5425	183	4	,	,	PUNCT
ejpam-5425	183	5	we	we	PRON
ejpam-5425	183	6	can	can	AUX
ejpam-5425	183	7	prove	prove	VERB
ejpam-5425	183	8	the	the	DET
ejpam-5425	183	9	other	other	ADJ
ejpam-5425	183	10	relations	relation	NOUN
ejpam-5425	183	11	.	.	PUNCT
ejpam-5425	184	1	this	this	PRON
ejpam-5425	184	2	completes	complete	VERB
ejpam-5425	184	3	the	the	DET
ejpam-5425	184	4	proof	proof	NOUN
ejpam-5425	184	5	.	.	PUNCT
ejpam-5425	185	1	example	example	NOUN
ejpam-5425	186	1	4	4	X
ejpam-5425	186	2	.	.	PUNCT
ejpam-5425	187	1	we	we	PRON
ejpam-5425	187	2	assume	assume	VERB
ejpam-5425	187	3	that	that	SCONJ
ejpam-5425	187	4	t0	t0	PROPN
ejpam-5425	187	5	=	=	PUNCT
ejpam-5425	187	6	0.1	0.1	NUM
ejpam-5425	187	7	,	,	PUNCT
ejpam-5425	187	8	t−1	t−1	NOUN
ejpam-5425	187	9	=	=	SYM
ejpam-5425	187	10	2	2	NUM
ejpam-5425	187	11	,	,	PUNCT
ejpam-5425	187	12	t−2	t−2	PROPN
ejpam-5425	187	13	=	=	SYM
ejpam-5425	187	14	0.5	0.5	NUM
ejpam-5425	187	15	,	,	PUNCT
ejpam-5425	187	16	t−3	t−3	NOUN
ejpam-5425	187	17	=	=	SYM
ejpam-5425	187	18	1	1	NUM
ejpam-5425	187	19	,	,	PUNCT
ejpam-5425	187	20	t−4	t−4	PROPN
ejpam-5425	187	21	=	=	PUNCT
ejpam-5425	187	22	0.3	0.3	NUM
ejpam-5425	187	23	,	,	PUNCT
ejpam-5425	187	24	e0	e0	PROPN
ejpam-5425	187	25	=	=	NOUN
ejpam-5425	187	26	0.1	0.1	NUM
ejpam-5425	187	27	,	,	PUNCT
ejpam-5425	187	28	e−1	e−1	PROPN
ejpam-5425	187	29	=	=	SYM
ejpam-5425	187	30	1	1	NUM
ejpam-5425	187	31	,	,	PUNCT
ejpam-5425	187	32	e−2	e−2	PROPN
ejpam-5425	187	33	=	=	SYM
ejpam-5425	187	34	0.6	0.6	NUM
ejpam-5425	187	35	,	,	PUNCT
ejpam-5425	187	36	e−3	e−3	PROPN
ejpam-5425	187	37	=	=	PROPN
ejpam-5425	187	38	1.1	1.1	NUM
ejpam-5425	187	39	,	,	PUNCT
ejpam-5425	187	40	and	and	CCONJ
ejpam-5425	187	41	e−4	e−4	PROPN
ejpam-5425	187	42	=	=	PROPN
ejpam-5425	187	43	0.7	0.7	NUM
ejpam-5425	187	44	for	for	ADP
ejpam-5425	187	45	the	the	DET
ejpam-5425	187	46	eq.(4	eq.(4	ADJ
ejpam-5425	187	47	)	)	PUNCT
ejpam-5425	187	48	.	.	PUNCT
ejpam-5425	188	1	(	(	PUNCT
ejpam-5425	188	2	see	see	VERB
ejpam-5425	188	3	fig	fig	NOUN
ejpam-5425	188	4	.	.	PUNCT
ejpam-5425	189	1	4	4	NUM
ejpam-5425	189	2	)	)	PUNCT
ejpam-5425	189	3	.	.	PUNCT
ejpam-5425	190	1	figure	figure	VERB
ejpam-5425	190	2	4	4	NUM
ejpam-5425	190	3	:	:	PUNCT
ejpam-5425	190	4	sketch	sketch	VERB
ejpam-5425	190	5	the	the	DET
ejpam-5425	190	6	periodicity	periodicity	NOUN
ejpam-5425	190	7	of	of	ADP
ejpam-5425	190	8	the	the	DET
ejpam-5425	190	9	solution	solution	NOUN
ejpam-5425	190	10	of	of	ADP
ejpam-5425	190	11	eq.(4	eq.(4	ADJ
ejpam-5425	190	12	)	)	PUNCT
ejpam-5425	190	13	6	6	NUM
ejpam-5425	190	14	.	.	PUNCT
ejpam-5425	191	1	the	the	DET
ejpam-5425	191	2	system	system	NOUN
ejpam-5425	191	3	tn+1	tn+1	NOUN
ejpam-5425	191	4	=	=	SYM
ejpam-5425	191	5	enen−4	enen−4	NOUN
ejpam-5425	191	6	tn−3(−1−enen−4	tn−3(−1−enen−4	NOUN
ejpam-5425	191	7	)	)	PUNCT
ejpam-5425	191	8	,	,	PUNCT
ejpam-5425	191	9	en+1	en+1	ADJ
ejpam-5425	191	10	=	=	PROPN
ejpam-5425	191	11	tntn−4	tntn−4	NUM
ejpam-5425	191	12	en−3(1−tntn−4	en−3(1−tntn−4	NOUN
ejpam-5425	191	13	)	)	PUNCT
ejpam-5425	191	14	in	in	ADP
ejpam-5425	191	15	this	this	DET
ejpam-5425	191	16	section	section	NOUN
ejpam-5425	191	17	,	,	PUNCT
ejpam-5425	191	18	we	we	PRON
ejpam-5425	191	19	obtain	obtain	VERB
ejpam-5425	191	20	the	the	DET
ejpam-5425	191	21	form	form	NOUN
ejpam-5425	191	22	of	of	ADP
ejpam-5425	191	23	the	the	DET
ejpam-5425	191	24	solutions	solution	NOUN
ejpam-5425	191	25	of	of	ADP
ejpam-5425	191	26	the	the	DET
ejpam-5425	191	27	sdes	sde	NOUN
ejpam-5425	191	28	tn+1	tn+1	VERB
ejpam-5425	191	29	=	=	SYM
ejpam-5425	191	30	enen−4	enen−4	PUNCT
ejpam-5425	191	31	tn−3(−1−	tn−3(−1−	NOUN
ejpam-5425	191	32	enen−4	enen−4	NOUN
ejpam-5425	191	33	)	)	PUNCT
ejpam-5425	191	34	,	,	PUNCT
ejpam-5425	191	35	en+1	en+1	ADJ
ejpam-5425	191	36	=	=	SYM
ejpam-5425	191	37	tntn−4	tntn−4	NUM
ejpam-5425	191	38	en−3(1−	en−3(1−	PROPN
ejpam-5425	191	39	tntn−4	tntn−4	NUM
ejpam-5425	191	40	)	)	PUNCT
ejpam-5425	191	41	.	.	PUNCT
ejpam-5425	192	1	(	(	PUNCT
ejpam-5425	192	2	5	5	X
ejpam-5425	192	3	)	)	PUNCT
ejpam-5425	192	4	theorem	theorem	NOUN
ejpam-5425	192	5	5	5	NUM
ejpam-5425	192	6	.	.	PUNCT
ejpam-5425	193	1	let	let	VERB
ejpam-5425	193	2	that	that	PRON
ejpam-5425	193	3	{	{	PUNCT
ejpam-5425	193	4	tn	tn	NOUN
ejpam-5425	193	5	,	,	PUNCT
ejpam-5425	193	6	en	en	ADP
ejpam-5425	193	7	}	}	PUNCT
ejpam-5425	193	8	are	be	AUX
ejpam-5425	193	9	solutions	solution	NOUN
ejpam-5425	193	10	of	of	ADP
ejpam-5425	193	11	eq	eq	PROPN
ejpam-5425	193	12	.	.	PUNCT
ejpam-5425	194	1	(	(	PUNCT
ejpam-5425	194	2	5	5	NUM
ejpam-5425	194	3	)	)	PUNCT
ejpam-5425	194	4	.	.	PUNCT
ejpam-5425	195	1	next	next	ADV
ejpam-5425	195	2	,	,	PUNCT
ejpam-5425	195	3	each	each	PRON
ejpam-5425	195	4	of	of	ADP
ejpam-5425	195	5	solutions	solution	NOUN
ejpam-5425	195	6	eq.(5	eq.(5	NOUN
ejpam-5425	195	7	)	)	PUNCT
ejpam-5425	195	8	are	be	AUX
ejpam-5425	195	9	periodic	periodic	ADJ
ejpam-5425	195	10	,	,	PUNCT
ejpam-5425	195	11	having	have	VERB
ejpam-5425	195	12	a	a	DET
ejpam-5425	195	13	period	period	NOUN
ejpam-5425	195	14	of	of	ADP
ejpam-5425	195	15	eight	eight	NUM
ejpam-5425	195	16	,	,	PUNCT
ejpam-5425	195	17	and	and	CCONJ
ejpam-5425	195	18	is	be	AUX
ejpam-5425	195	19	given	give	VERB
ejpam-5425	195	20	by	by	ADP
ejpam-5425	195	21	formulas	formula	NOUN
ejpam-5425	195	22	for	for	ADP
ejpam-5425	195	23	n	n	NOUN
ejpam-5425	195	24	=	=	SYM
ejpam-5425	195	25	0	0	NUM
ejpam-5425	195	26	,	,	PUNCT
ejpam-5425	195	27	1	1	NUM
ejpam-5425	195	28	,	,	PUNCT
ejpam-5425	195	29	2	2	NUM
ejpam-5425	195	30	,	,	PUNCT
ejpam-5425	195	31	...	...	PUNCT
ejpam-5425	195	32	,	,	PUNCT
ejpam-5425	195	33	t8n−4	t8n−4	PROPN
ejpam-5425	195	34	=	=	SYM
ejpam-5425	195	35	s	s	PROPN
ejpam-5425	195	36	,	,	PUNCT
ejpam-5425	195	37	t8n−3	t8n−3	PROPN
ejpam-5425	195	38	=	=	SYM
ejpam-5425	196	1	d	d	PROPN
ejpam-5425	196	2	,	,	PUNCT
ejpam-5425	196	3	t8n−2	t8n−2	PROPN
ejpam-5425	196	4	=	=	SYM
ejpam-5425	196	5	c	c	NOUN
ejpam-5425	196	6	,	,	PUNCT
ejpam-5425	196	7	t8n−1	t8n−1	PROPN
ejpam-5425	196	8	=	=	SYM
ejpam-5425	196	9	b	b	PROPN
ejpam-5425	196	10	,	,	PUNCT
ejpam-5425	196	11	j.	j.	PROPN
ejpam-5425	196	12	g.	g.	PROPN
ejpam-5425	196	13	al	al	PROPN
ejpam-5425	196	14	-	-	PUNCT
ejpam-5425	196	15	juaid	juaid	PROPN
ejpam-5425	196	16	/	/	SYM
ejpam-5425	196	17	eur	eur	PROPN
ejpam-5425	196	18	.	.	PUNCT
ejpam-5425	197	1	j.	j.	PROPN
ejpam-5425	197	2	pure	pure	PROPN
ejpam-5425	197	3	appl	appl	PROPN
ejpam-5425	197	4	.	.	PROPN
ejpam-5425	197	5	math	math	PROPN
ejpam-5425	197	6	,	,	PUNCT
ejpam-5425	197	7	17	17	NUM
ejpam-5425	197	8	(	(	PUNCT
ejpam-5425	197	9	4	4	NUM
ejpam-5425	197	10	)	)	PUNCT
ejpam-5425	197	11	(	(	PUNCT
ejpam-5425	197	12	2024	2024	NUM
ejpam-5425	197	13	)	)	PUNCT
ejpam-5425	197	14	,	,	PUNCT
ejpam-5425	197	15	3254	3254	NUM
ejpam-5425	197	16	-	-	SYM
ejpam-5425	197	17	3267	3267	NUM
ejpam-5425	197	18	3264	3264	NUM
ejpam-5425	197	19	t8n	t8n	PROPN
ejpam-5425	197	20	=	=	SYM
ejpam-5425	197	21	a	a	NOUN
ejpam-5425	197	22	,	,	PUNCT
ejpam-5425	197	23	t8n+1	t8n+1	ADV
ejpam-5425	197	24	=	=	PUNCT
ejpam-5425	197	25	fw	fw	PROPN
ejpam-5425	197	26	d(−1−	d(−1−	PROPN
ejpam-5425	197	27	fw	fw	PROPN
ejpam-5425	197	28	)	)	PUNCT
ejpam-5425	197	29	,	,	PUNCT
ejpam-5425	197	30	t8n+2	t8n+2	PROPN
ejpam-5425	198	1	=	=	PUNCT
ejpam-5425	198	2	−as	−as	PROPN
ejpam-5425	198	3	c	c	NOUN
ejpam-5425	198	4	,	,	PUNCT
ejpam-5425	198	5	t8n+3	t8n+3	SCONJ
ejpam-5425	198	6	=	=	PUNCT
ejpam-5425	199	1	fw	fw	PROPN
ejpam-5425	199	2	b(1	b(1	PROPN
ejpam-5425	199	3	+	+	CCONJ
ejpam-5425	199	4	fw	fw	X
ejpam-5425	199	5	)	)	PUNCT
ejpam-5425	199	6	,	,	PUNCT
ejpam-5425	199	7	e8n−4	e8n−4	PROPN
ejpam-5425	199	8	=	=	SYM
ejpam-5425	199	9	w	w	PROPN
ejpam-5425	199	10	,	,	PUNCT
ejpam-5425	199	11	e8n−3	e8n−3	PROPN
ejpam-5425	199	12	=	=	SYM
ejpam-5425	199	13	m	m	PROPN
ejpam-5425	199	14	,	,	PUNCT
ejpam-5425	199	15	e8n−2	e8n−2	PROPN
ejpam-5425	199	16	=	=	SYM
ejpam-5425	199	17	g	g	PROPN
ejpam-5425	199	18	,	,	PUNCT
ejpam-5425	199	19	e8n−1	e8n−1	PROPN
ejpam-5425	199	20	=	=	SYM
ejpam-5425	199	21	k	k	PROPN
ejpam-5425	199	22	,	,	PUNCT
ejpam-5425	199	23	e8n	e8n	X
ejpam-5425	199	24	=	=	SYM
ejpam-5425	199	25	f	f	X
ejpam-5425	199	26	,	,	PUNCT
ejpam-5425	199	27	e8n+1	e8n+1	NOUN
ejpam-5425	199	28	=	=	PUNCT
ejpam-5425	199	29	as	as	ADP
ejpam-5425	199	30	m(1−	m(1−	ADJ
ejpam-5425	199	31	as	as	ADP
ejpam-5425	199	32	)	)	PUNCT
ejpam-5425	199	33	,	,	PUNCT
ejpam-5425	199	34	e8n+2	e8n+2	NOUN
ejpam-5425	199	35	=	=	PUNCT
ejpam-5425	200	1	fw	fw	PROPN
ejpam-5425	200	2	g(−1−	g(−1−	PROPN
ejpam-5425	200	3	2fw	2fw	NOUN
ejpam-5425	200	4	)	)	PUNCT
ejpam-5425	200	5	,	,	PUNCT
ejpam-5425	201	1	e8n+3	e8n+3	NOUN
ejpam-5425	201	2	=	=	SYM
ejpam-5425	201	3	−as	−as	PROPN
ejpam-5425	201	4	k(1	k(1	PROPN
ejpam-5425	201	5	+	+	CCONJ
ejpam-5425	201	6	as	as	ADP
ejpam-5425	201	7	)	)	PUNCT
ejpam-5425	201	8	,	,	PUNCT
ejpam-5425	201	9	where	where	SCONJ
ejpam-5425	201	10	e0e−4	e0e−4	PROPN
ejpam-5425	201	11	̸=	̸=	PROPN
ejpam-5425	201	12	±1	±1	VERB
ejpam-5425	201	13	and	and	CCONJ
ejpam-5425	201	14	t0t−4	t0t−4	PROPN
ejpam-5425	201	15	̸=	̸=	PROPN
ejpam-5425	201	16	±1	±1	VERB
ejpam-5425	201	17	.	.	PUNCT
ejpam-5425	202	1	proof	proof	NOUN
ejpam-5425	202	2	.	.	PUNCT
ejpam-5425	203	1	for	for	ADP
ejpam-5425	203	2	n	n	NOUN
ejpam-5425	203	3	=	=	SYM
ejpam-5425	203	4	0	0	NUM
ejpam-5425	203	5	,	,	PUNCT
ejpam-5425	203	6	the	the	DET
ejpam-5425	203	7	result	result	NOUN
ejpam-5425	203	8	is	be	AUX
ejpam-5425	203	9	true	true	ADJ
ejpam-5425	203	10	.	.	PUNCT
ejpam-5425	204	1	now	now	ADV
ejpam-5425	204	2	,	,	PUNCT
ejpam-5425	204	3	let	let	VERB
ejpam-5425	204	4	’s	’s	NOUN
ejpam-5425	204	5	suppose	suppose	VERB
ejpam-5425	204	6	that	that	SCONJ
ejpam-5425	204	7	n	n	PROPN
ejpam-5425	204	8	>	>	X
ejpam-5425	204	9	0	0	PUNCT
ejpam-5425	205	1	and	and	CCONJ
ejpam-5425	205	2	that	that	SCONJ
ejpam-5425	205	3	n	n	CCONJ
ejpam-5425	205	4	−	−	PROPN
ejpam-5425	205	5	1	1	NUM
ejpam-5425	205	6	supports	support	VERB
ejpam-5425	205	7	our	our	PRON
ejpam-5425	205	8	hypothesis	hypothesis	NOUN
ejpam-5425	205	9	.	.	PUNCT
ejpam-5425	206	1	that	that	PRON
ejpam-5425	206	2	’s	’	VERB
ejpam-5425	206	3	t8n−12	t8n−12	NOUN
ejpam-5425	206	4	=	=	SYM
ejpam-5425	206	5	s	s	PROPN
ejpam-5425	206	6	,	,	PUNCT
ejpam-5425	206	7	t8n−11	t8n−11	PUNCT
ejpam-5425	207	1	=	=	PUNCT
ejpam-5425	207	2	d	d	PROPN
ejpam-5425	207	3	,	,	PUNCT
ejpam-5425	207	4	t8n−10	t8n−10	PROPN
ejpam-5425	207	5	=	=	SYM
ejpam-5425	207	6	c	c	X
ejpam-5425	207	7	,	,	PUNCT
ejpam-5425	207	8	t8n−9	t8n−9	NOUN
ejpam-5425	207	9	=	=	SYM
ejpam-5425	207	10	b	b	NOUN
ejpam-5425	207	11	,	,	PUNCT
ejpam-5425	207	12	t8n−8	t8n−8	PROPN
ejpam-5425	207	13	=	=	SYM
ejpam-5425	207	14	a	a	NOUN
ejpam-5425	207	15	,	,	PUNCT
ejpam-5425	207	16	t8n−7	t8n−7	NOUN
ejpam-5425	207	17	=	=	PUNCT
ejpam-5425	207	18	fw	fw	PROPN
ejpam-5425	207	19	d(−1−	d(−1−	PROPN
ejpam-5425	207	20	fw	fw	PROPN
ejpam-5425	207	21	)	)	PUNCT
ejpam-5425	207	22	,	,	PUNCT
ejpam-5425	207	23	t8n−6	t8n−6	PROPN
ejpam-5425	207	24	=	=	SYM
ejpam-5425	207	25	−as	−as	PROPN
ejpam-5425	207	26	c	c	NOUN
ejpam-5425	207	27	,	,	PUNCT
ejpam-5425	207	28	t8n−5	t8n−5	PROPN
ejpam-5425	207	29	=	=	PUNCT
ejpam-5425	208	1	fw	fw	PROPN
ejpam-5425	208	2	b(1	b(1	PROPN
ejpam-5425	208	3	+	+	CCONJ
ejpam-5425	208	4	fw	fw	X
ejpam-5425	208	5	)	)	PUNCT
ejpam-5425	208	6	,	,	PUNCT
ejpam-5425	208	7	e8n−12	e8n−12	X
ejpam-5425	208	8	=	=	SYM
ejpam-5425	208	9	w	w	PROPN
ejpam-5425	208	10	,	,	PUNCT
ejpam-5425	208	11	e8n−11	e8n−11	VERB
ejpam-5425	208	12	=	=	PROPN
ejpam-5425	208	13	m	m	PROPN
ejpam-5425	208	14	,	,	PUNCT
ejpam-5425	208	15	e8n−10	e8n−10	PROPN
ejpam-5425	208	16	=	=	PUNCT
ejpam-5425	208	17	g	g	PROPN
ejpam-5425	208	18	,	,	PUNCT
ejpam-5425	208	19	e8n−9	e8n−9	PROPN
ejpam-5425	208	20	=	=	SYM
ejpam-5425	208	21	k	k	PROPN
ejpam-5425	208	22	,	,	PUNCT
ejpam-5425	208	23	e8n−8	e8n−8	PROPN
ejpam-5425	208	24	=	=	SYM
ejpam-5425	208	25	f	f	PROPN
ejpam-5425	208	26	,	,	PUNCT
ejpam-5425	208	27	e8n−7	e8n−7	NOUN
ejpam-5425	208	28	=	=	PUNCT
ejpam-5425	208	29	as	as	ADP
ejpam-5425	208	30	m(1−	m(1−	ADJ
ejpam-5425	208	31	as	as	ADP
ejpam-5425	208	32	)	)	PUNCT
ejpam-5425	208	33	,	,	PUNCT
ejpam-5425	208	34	e8n−6	e8n−6	PROPN
ejpam-5425	208	35	=	=	SYM
ejpam-5425	208	36	fw	fw	PROPN
ejpam-5425	208	37	g(−1−	g(−1−	PROPN
ejpam-5425	208	38	2fw	2fw	NOUN
ejpam-5425	208	39	)	)	PUNCT
ejpam-5425	208	40	,	,	PUNCT
ejpam-5425	208	41	e8n−5	e8n−5	PROPN
ejpam-5425	208	42	=	=	PUNCT
ejpam-5425	208	43	−as	−as	PROPN
ejpam-5425	208	44	k(1	k(1	NOUN
ejpam-5425	208	45	+	+	CCONJ
ejpam-5425	208	46	as	as	ADP
ejpam-5425	208	47	)	)	PUNCT
ejpam-5425	208	48	.	.	PUNCT
ejpam-5425	209	1	next	next	ADV
ejpam-5425	209	2	,	,	PUNCT
ejpam-5425	209	3	from	from	ADP
ejpam-5425	209	4	system	system	NOUN
ejpam-5425	209	5	(	(	PUNCT
ejpam-5425	209	6	5	5	X
ejpam-5425	209	7	)	)	PUNCT
ejpam-5425	209	8	we	we	PRON
ejpam-5425	209	9	have	have	VERB
ejpam-5425	209	10	t8n+1	t8n+1	ADV
ejpam-5425	209	11	=	=	NOUN
ejpam-5425	209	12	e8ne8n−4	e8ne8n−4	NOUN
ejpam-5425	209	13	t8n−3(−1−	t8n−3(−1−	NOUN
ejpam-5425	209	14	e8ne8n−4	e8ne8n−4	NOUN
ejpam-5425	209	15	)	)	PUNCT
ejpam-5425	210	1	=	=	PUNCT
ejpam-5425	210	2	fw	fw	PROPN
ejpam-5425	210	3	d(−1−	d(−1−	PROPN
ejpam-5425	210	4	fw	fw	PROPN
ejpam-5425	210	5	)	)	PUNCT
ejpam-5425	210	6	.	.	PUNCT
ejpam-5425	211	1	e8n+1	e8n+1	PUNCT
ejpam-5425	212	1	=	=	NOUN
ejpam-5425	212	2	t8nt8n−4	t8nt8n−4	X
ejpam-5425	212	3	e8n−3(1−	e8n−3(1−	PROPN
ejpam-5425	212	4	t8nt8n−4	t8nt8n−4	NUM
ejpam-5425	212	5	)	)	PUNCT
ejpam-5425	212	6	=	=	PUNCT
ejpam-5425	212	7	as	as	ADP
ejpam-5425	212	8	m(1−	m(1−	ADJ
ejpam-5425	212	9	as	as	ADP
ejpam-5425	212	10	)	)	PUNCT
ejpam-5425	212	11	.	.	PUNCT
ejpam-5425	213	1	t8n+3	t8n+3	PUNCT
ejpam-5425	214	1	=	=	PRON
ejpam-5425	214	2	e8n+2e8n−2	e8n+2e8n−2	ADP
ejpam-5425	214	3	t8n−1(−1−	t8n−1(−1−	ADP
ejpam-5425	214	4	e8n+2e8n−2	e8n+2e8n−2	PROPN
ejpam-5425	214	5	)	)	PUNCT
ejpam-5425	214	6	=	=	SYM
ejpam-5425	214	7	fwg	fwg	PROPN
ejpam-5425	214	8	g(−1−2fw	g(−1−2fw	NOUN
ejpam-5425	214	9	)	)	PUNCT
ejpam-5425	214	10	b(−1−	b(−1−	PROPN
ejpam-5425	214	11	fwg	fwg	PROPN
ejpam-5425	214	12	g(−1−2fw	g(−1−2fw	PROPN
ejpam-5425	214	13	)	)	PUNCT
ejpam-5425	214	14	)	)	PUNCT
ejpam-5425	215	1	=	=	PUNCT
ejpam-5425	215	2	fw	fw	PROPN
ejpam-5425	215	3	b(1	b(1	PROPN
ejpam-5425	215	4	+	+	CCONJ
ejpam-5425	215	5	fw	fw	X
ejpam-5425	215	6	)	)	PUNCT
ejpam-5425	215	7	.	.	PUNCT
ejpam-5425	216	1	j.	j.	PROPN
ejpam-5425	216	2	g.	g.	PROPN
ejpam-5425	216	3	al	al	PROPN
ejpam-5425	216	4	-	-	PUNCT
ejpam-5425	216	5	juaid	juaid	PROPN
ejpam-5425	216	6	/	/	SYM
ejpam-5425	216	7	eur	eur	PROPN
ejpam-5425	216	8	.	.	PUNCT
ejpam-5425	217	1	j.	j.	PROPN
ejpam-5425	217	2	pure	pure	PROPN
ejpam-5425	217	3	appl	appl	PROPN
ejpam-5425	217	4	.	.	PROPN
ejpam-5425	217	5	math	math	PROPN
ejpam-5425	217	6	,	,	PUNCT
ejpam-5425	217	7	17	17	NUM
ejpam-5425	217	8	(	(	PUNCT
ejpam-5425	217	9	4	4	NUM
ejpam-5425	217	10	)	)	PUNCT
ejpam-5425	217	11	(	(	PUNCT
ejpam-5425	217	12	2024	2024	NUM
ejpam-5425	217	13	)	)	PUNCT
ejpam-5425	217	14	,	,	PUNCT
ejpam-5425	217	15	3254	3254	NUM
ejpam-5425	217	16	-	-	SYM
ejpam-5425	217	17	3267	3267	NUM
ejpam-5425	217	18	3265	3265	NUM
ejpam-5425	217	19	e8n+3	e8n+3	NOUN
ejpam-5425	217	20	=	=	SYM
ejpam-5425	218	1	t8n+2t8n−2	t8n+2t8n−2	PROPN
ejpam-5425	218	2	e8n−1(1−	e8n−1(1−	PROPN
ejpam-5425	218	3	t8n+2t8n−2	t8n+2t8n−2	PROPN
ejpam-5425	218	4	)	)	PUNCT
ejpam-5425	218	5	=	=	PUNCT
ejpam-5425	218	6	−asc	−asc	PROPN
ejpam-5425	218	7	c	c	PROPN
ejpam-5425	218	8	k(1−	k(1−	PROPN
ejpam-5425	218	9	−asc	−asc	PROPN
ejpam-5425	218	10	c	c	NOUN
ejpam-5425	218	11	)	)	PUNCT
ejpam-5425	219	1	=	=	PUNCT
ejpam-5425	219	2	−as	−as	PROPN
ejpam-5425	219	3	k(1	k(1	NOUN
ejpam-5425	219	4	+	+	CCONJ
ejpam-5425	219	5	as	as	ADP
ejpam-5425	219	6	)	)	PUNCT
ejpam-5425	219	7	.	.	PUNCT
ejpam-5425	220	1	we	we	PRON
ejpam-5425	220	2	can	can	AUX
ejpam-5425	220	3	prove	prove	VERB
ejpam-5425	220	4	the	the	DET
ejpam-5425	220	5	other	other	ADJ
ejpam-5425	220	6	relations	relation	NOUN
ejpam-5425	220	7	as	as	ADV
ejpam-5425	220	8	well	well	ADV
ejpam-5425	220	9	.	.	PUNCT
ejpam-5425	221	1	this	this	PRON
ejpam-5425	221	2	brings	bring	VERB
ejpam-5425	221	3	the	the	DET
ejpam-5425	221	4	proof	proof	NOUN
ejpam-5425	221	5	to	to	ADP
ejpam-5425	221	6	a	a	DET
ejpam-5425	221	7	close	close	NOUN
ejpam-5425	221	8	.	.	PUNCT
ejpam-5425	221	9	example	example	NOUN
ejpam-5425	222	1	5	5	NUM
ejpam-5425	222	2	.	.	X
ejpam-5425	222	3	see	see	VERB
ejpam-5425	222	4	figure	figure	NOUN
ejpam-5425	222	5	(	(	PUNCT
ejpam-5425	222	6	5	5	NUM
ejpam-5425	222	7	)	)	PUNCT
ejpam-5425	222	8	where	where	SCONJ
ejpam-5425	222	9	we	we	PRON
ejpam-5425	222	10	takes	take	VERB
ejpam-5425	222	11	system	system	NOUN
ejpam-5425	222	12	(	(	PUNCT
ejpam-5425	222	13	5	5	NUM
ejpam-5425	222	14	)	)	PUNCT
ejpam-5425	222	15	with	with	ADP
ejpam-5425	222	16	t0	t0	PROPN
ejpam-5425	222	17	=	=	SYM
ejpam-5425	222	18	0.5	0.5	NUM
ejpam-5425	222	19	,	,	PUNCT
ejpam-5425	222	20	t−1	t−1	NOUN
ejpam-5425	222	21	=	=	SYM
ejpam-5425	222	22	4	4	NUM
ejpam-5425	222	23	,	,	PUNCT
ejpam-5425	222	24	t−2	t−2	NOUN
ejpam-5425	222	25	=	=	SYM
ejpam-5425	222	26	7	7	NUM
ejpam-5425	222	27	,	,	PUNCT
ejpam-5425	222	28	t−3	t−3	NOUN
ejpam-5425	222	29	=	=	NUM
ejpam-5425	222	30	0.5	0.5	NUM
ejpam-5425	222	31	,	,	PUNCT
ejpam-5425	222	32	t−4	t−4	PROPN
ejpam-5425	222	33	=	=	SYM
ejpam-5425	222	34	1	1	NUM
ejpam-5425	222	35	,	,	PUNCT
ejpam-5425	222	36	e0	e0	PROPN
ejpam-5425	222	37	=	=	NOUN
ejpam-5425	222	38	1.2	1.2	NUM
ejpam-5425	222	39	,	,	PUNCT
ejpam-5425	222	40	e−1	e−1	PROPN
ejpam-5425	222	41	=	=	SYM
ejpam-5425	222	42	0.5	0.5	NUM
ejpam-5425	222	43	,	,	PUNCT
ejpam-5425	222	44	e−2	e−2	PROPN
ejpam-5425	222	45	=	=	SYM
ejpam-5425	222	46	3	3	NUM
ejpam-5425	222	47	,	,	PUNCT
ejpam-5425	222	48	e−3	e−3	PROPN
ejpam-5425	222	49	=	=	SYM
ejpam-5425	222	50	4	4	NUM
ejpam-5425	222	51	,	,	PUNCT
ejpam-5425	222	52	and	and	CCONJ
ejpam-5425	222	53	e−4	e−4	PROPN
ejpam-5425	222	54	=	=	PROPN
ejpam-5425	222	55	0.2	0.2	NUM
ejpam-5425	222	56	.	.	PUNCT
ejpam-5425	223	1	figure	figure	NOUN
ejpam-5425	223	2	5	5	NUM
ejpam-5425	223	3	:	:	PUNCT
ejpam-5425	223	4	chart	chart	VERB
ejpam-5425	223	5	the	the	DET
ejpam-5425	223	6	behavior	behavior	NOUN
ejpam-5425	223	7	of	of	ADP
ejpam-5425	223	8	the	the	DET
ejpam-5425	223	9	solutions	solution	NOUN
ejpam-5425	223	10	of	of	ADP
ejpam-5425	223	11	system	system	NOUN
ejpam-5425	223	12	(	(	PUNCT
ejpam-5425	223	13	5	5	NUM
ejpam-5425	223	14	)	)	PUNCT
ejpam-5425	223	15	remark	remark	NOUN
ejpam-5425	223	16	1	1	NUM
ejpam-5425	223	17	the	the	DET
ejpam-5425	223	18	solutions	solution	NOUN
ejpam-5425	223	19	of	of	ADP
ejpam-5425	223	20	the	the	DET
ejpam-5425	223	21	following	follow	VERB
ejpam-5425	223	22	systems	system	NOUN
ejpam-5425	223	23	can	can	AUX
ejpam-5425	223	24	be	be	AUX
ejpam-5425	223	25	also	also	ADV
ejpam-5425	223	26	obtained	obtain	VERB
ejpam-5425	223	27	.	.	PUNCT
ejpam-5425	224	1	tn+1	tn+1	PROPN
ejpam-5425	224	2	=	=	SYM
ejpam-5425	224	3	enen−4	enen−4	X
ejpam-5425	224	4	tn−3(1	tn−3(1	NOUN
ejpam-5425	224	5	+	+	NUM
ejpam-5425	224	6	enen−4	enen−4	NOUN
ejpam-5425	224	7	)	)	PUNCT
ejpam-5425	224	8	,	,	PUNCT
ejpam-5425	224	9	en+1	en+1	ADJ
ejpam-5425	224	10	=	=	PROPN
ejpam-5425	224	11	tntn−4	tntn−4	PART
ejpam-5425	224	12	en−3(1	en−3(1	NOUN
ejpam-5425	224	13	+	+	CCONJ
ejpam-5425	224	14	tntn−4	tntn−4	NOUN
ejpam-5425	224	15	)	)	PUNCT
ejpam-5425	224	16	.	.	PUNCT
ejpam-5425	225	1	tn+1	tn+1	PROPN
ejpam-5425	225	2	=	=	SYM
ejpam-5425	226	1	enen−4	enen−4	X
ejpam-5425	226	2	tn−3(−1	tn−3(−1	PROPN
ejpam-5425	226	3	+	+	CCONJ
ejpam-5425	226	4	enen−4	enen−4	NOUN
ejpam-5425	226	5	)	)	PUNCT
ejpam-5425	226	6	,	,	PUNCT
ejpam-5425	226	7	en+1	en+1	ADJ
ejpam-5425	226	8	=	=	SYM
ejpam-5425	226	9	tntn−4	tntn−4	NUM
ejpam-5425	226	10	en−3(−1−	en−3(−1−	PROPN
ejpam-5425	226	11	tntn−4	tntn−4	NOUN
ejpam-5425	226	12	)	)	PUNCT
ejpam-5425	226	13	.	.	PUNCT
ejpam-5425	227	1	tn+1	tn+1	PROPN
ejpam-5425	227	2	=	=	SYM
ejpam-5425	228	1	enen−4	enen−4	X
ejpam-5425	228	2	tn−3(−1	tn−3(−1	PROPN
ejpam-5425	228	3	+	+	CCONJ
ejpam-5425	228	4	enen−4	enen−4	NOUN
ejpam-5425	228	5	)	)	PUNCT
ejpam-5425	228	6	,	,	PUNCT
ejpam-5425	228	7	en+1	en+1	ADJ
ejpam-5425	228	8	=	=	SYM
ejpam-5425	228	9	tntn−4	tntn−4	NUM
ejpam-5425	228	10	en−3(1−	en−3(1−	PROPN
ejpam-5425	228	11	tntn−4	tntn−4	NUM
ejpam-5425	228	12	)	)	PUNCT
ejpam-5425	228	13	.	.	PUNCT
ejpam-5425	229	1	7	7	X
ejpam-5425	229	2	.	.	X
ejpam-5425	229	3	conclusion	conclusion	NOUN
ejpam-5425	229	4	this	this	DET
ejpam-5425	229	5	study	study	NOUN
ejpam-5425	229	6	investigates	investigate	VERB
ejpam-5425	229	7	the	the	DET
ejpam-5425	229	8	solutions	solution	NOUN
ejpam-5425	229	9	of	of	ADP
ejpam-5425	229	10	five	five	NUM
ejpam-5425	229	11	sdes	sde	NOUN
ejpam-5425	229	12	.	.	PUNCT
ejpam-5425	230	1	across	across	ADP
ejpam-5425	230	2	five	five	NUM
ejpam-5425	230	3	sections	section	NOUN
ejpam-5425	230	4	,	,	PUNCT
ejpam-5425	230	5	we	we	PRON
ejpam-5425	230	6	derive	derive	VERB
ejpam-5425	230	7	periodic	periodic	ADJ
ejpam-5425	230	8	solutions	solution	NOUN
ejpam-5425	230	9	with	with	ADP
ejpam-5425	230	10	a	a	DET
ejpam-5425	230	11	period	period	NOUN
ejpam-5425	230	12	of	of	ADP
ejpam-5425	230	13	eight	eight	NUM
ejpam-5425	230	14	for	for	ADP
ejpam-5425	230	15	each	each	DET
ejpam-5425	230	16	system	system	NOUN
ejpam-5425	230	17	.	.	PUNCT
ejpam-5425	231	1	to	to	PART
ejpam-5425	231	2	corroborate	corroborate	VERB
ejpam-5425	231	3	our	our	PRON
ejpam-5425	231	4	theoretical	theoretical	ADJ
ejpam-5425	231	5	findings	finding	NOUN
ejpam-5425	231	6	,	,	PUNCT
ejpam-5425	231	7	numerical	numerical	ADJ
ejpam-5425	231	8	examples	example	NOUN
ejpam-5425	231	9	are	be	AUX
ejpam-5425	231	10	presented	present	VERB
ejpam-5425	231	11	for	for	ADP
ejpam-5425	231	12	each	each	DET
ejpam-5425	231	13	system	system	NOUN
ejpam-5425	231	14	,	,	PUNCT
ejpam-5425	231	15	with	with	ADP
ejpam-5425	231	16	figures	figure	NOUN
ejpam-5425	231	17	1	1	NUM
ejpam-5425	231	18	-	-	SYM
ejpam-5425	231	19	5	5	NUM
ejpam-5425	231	20	providing	provide	VERB
ejpam-5425	231	21	visual	visual	ADJ
ejpam-5425	231	22	confirmation	confirmation	NOUN
ejpam-5425	231	23	of	of	ADP
ejpam-5425	231	24	the	the	DET
ejpam-5425	231	25	results	result	NOUN
ejpam-5425	231	26	.	.	PUNCT
ejpam-5425	232	1	references	reference	NOUN
ejpam-5425	232	2	3266	3266	NUM
ejpam-5425	232	3	references	reference	NOUN
ejpam-5425	232	4	[	[	X
ejpam-5425	232	5	1	1	NUM
ejpam-5425	232	6	]	]	PUNCT
ejpam-5425	232	7	j.	j.	PROPN
ejpam-5425	232	8	g.	g.	PROPN
ejpam-5425	232	9	al	al	PROPN
ejpam-5425	232	10	-	-	PUNCT
ejpam-5425	232	11	juaid	juaid	PROPN
ejpam-5425	232	12	,	,	PUNCT
ejpam-5425	232	13	e.	e.	PROPN
ejpam-5425	232	14	m.	m.	PROPN
ejpam-5425	232	15	elsayed	elsaye	VERB
ejpam-5425	232	16	,	,	PUNCT
ejpam-5425	232	17	and	and	CCONJ
ejpam-5425	232	18	h.	h.	PROPN
ejpam-5425	232	19	malaikah	malaikah	PROPN
ejpam-5425	232	20	.	.	PUNCT
ejpam-5425	233	1	behavior	behavior	NOUN
ejpam-5425	233	2	and	and	CCONJ
ejpam-5425	233	3	formula	formula	NOUN
ejpam-5425	233	4	of	of	ADP
ejpam-5425	233	5	the	the	DET
ejpam-5425	233	6	solutions	solution	NOUN
ejpam-5425	233	7	of	of	ADP
ejpam-5425	233	8	rational	rational	ADJ
ejpam-5425	233	9	difference	difference	NOUN
ejpam-5425	233	10	equations	equation	NOUN
ejpam-5425	233	11	of	of	ADP
ejpam-5425	233	12	order	order	NOUN
ejpam-5425	233	13	six	six	NUM
ejpam-5425	233	14	.	.	PUNCT
ejpam-5425	233	15	annals	annal	NOUN
ejpam-5425	233	16	of	of	ADP
ejpam-5425	233	17	communications	communication	NOUN
ejpam-5425	233	18	in	in	ADP
ejpam-5425	233	19	mathematics	mathematic	NOUN
ejpam-5425	233	20	,	,	PUNCT
ejpam-5425	233	21	6(2):72–85	6(2):72–85	NUM
ejpam-5425	233	22	,	,	PUNCT
ejpam-5425	233	23	2023	2023	NUM
ejpam-5425	233	24	.	.	PUNCT
ejpam-5425	234	1	[	[	X
ejpam-5425	234	2	2	2	NUM
ejpam-5425	234	3	]	]	PUNCT
ejpam-5425	234	4	a.	a.	PROPN
ejpam-5425	234	5	al	al	PROPN
ejpam-5425	234	6	-	-	PUNCT
ejpam-5425	234	7	khedhairi	khedhairi	PROPN
ejpam-5425	234	8	,	,	PUNCT
ejpam-5425	234	9	a.	a.	NOUN
ejpam-5425	234	10	a.	a.	NOUN
ejpam-5425	234	11	elsadany	elsadany	PROPN
ejpam-5425	234	12	,	,	PUNCT
ejpam-5425	234	13	and	and	CCONJ
ejpam-5425	234	14	a.	a.	NOUN
ejpam-5425	234	15	elsonbaty	elsonbaty	NOUN
ejpam-5425	234	16	.	.	PUNCT
ejpam-5425	235	1	on	on	ADP
ejpam-5425	235	2	the	the	DET
ejpam-5425	235	3	dynamics	dynamic	NOUN
ejpam-5425	235	4	of	of	ADP
ejpam-5425	235	5	a	a	DET
ejpam-5425	235	6	discrete	discrete	ADJ
ejpam-5425	235	7	fractional	fractional	ADJ
ejpam-5425	235	8	-	-	PUNCT
ejpam-5425	235	9	order	order	NOUN
ejpam-5425	235	10	cournot	cournot	NOUN
ejpam-5425	235	11	–	–	PUNCT
ejpam-5425	235	12	bertrand	bertrand	PROPN
ejpam-5425	235	13	competition	competition	NOUN
ejpam-5425	235	14	duopoly	duopoly	ADJ
ejpam-5425	235	15	game	game	NOUN
ejpam-5425	235	16	.	.	PUNCT
ejpam-5425	236	1	math	math	NOUN
ejpam-5425	236	2	.	.	PUNCT
ejpam-5425	237	1	probl	probl	PROPN
ejpam-5425	237	2	.	.	PUNCT
ejpam-5425	238	1	eng	eng	PROPN
ejpam-5425	238	2	.	.	PROPN
ejpam-5425	238	3	,	,	PUNCT
ejpam-5425	238	4	2022:8249215	2022:8249215	NUM
ejpam-5425	238	5	,	,	PUNCT
ejpam-5425	238	6	2022	2022	NUM
ejpam-5425	238	7	.	.	PUNCT
ejpam-5425	239	1	[	[	X
ejpam-5425	239	2	3	3	X
ejpam-5425	239	3	]	]	PUNCT
ejpam-5425	239	4	r.	r.	PROPN
ejpam-5425	239	5	j.	j.	PROPN
ejpam-5425	239	6	beverton	beverton	PROPN
ejpam-5425	239	7	and	and	CCONJ
ejpam-5425	239	8	s.	s.	PROPN
ejpam-5425	239	9	j.	j.	PROPN
ejpam-5425	239	10	holt	holt	PROPN
ejpam-5425	239	11	.	.	PUNCT
ejpam-5425	240	1	on	on	ADP
ejpam-5425	240	2	the	the	DET
ejpam-5425	240	3	dynamics	dynamic	NOUN
ejpam-5425	240	4	of	of	ADP
ejpam-5425	240	5	exploited	exploit	VERB
ejpam-5425	240	6	fish	fish	NOUN
ejpam-5425	240	7	populations	population	NOUN
ejpam-5425	240	8	.	.	PUNCT
ejpam-5425	241	1	fish	fish	NOUN
ejpam-5425	241	2	invest	invest	PROPN
ejpam-5425	241	3	:	:	PUNCT
ejpam-5425	241	4	london	london	PROPN
ejpam-5425	241	5	,	,	PUNCT
ejpam-5425	241	6	uk	uk	PROPN
ejpam-5425	241	7	,	,	PUNCT
ejpam-5425	241	8	1957	1957	NUM
ejpam-5425	241	9	.	.	PUNCT
ejpam-5425	242	1	[	[	X
ejpam-5425	242	2	4	4	NUM
ejpam-5425	242	3	]	]	X
ejpam-5425	242	4	q.	q.	PROPN
ejpam-5425	242	5	din	din	PROPN
ejpam-5425	242	6	,	,	PUNCT
ejpam-5425	242	7	e.	e.	PROPN
ejpam-5425	242	8	m.	m.	PROPN
ejpam-5425	242	9	elabbasy	elabbasy	PROPN
ejpam-5425	242	10	,	,	PUNCT
ejpam-5425	242	11	a.	a.	NOUN
ejpam-5425	242	12	a.	a.	NOUN
ejpam-5425	242	13	elsadany	elsadany	PROPN
ejpam-5425	242	14	,	,	PUNCT
ejpam-5425	242	15	and	and	CCONJ
ejpam-5425	242	16	s.	s.	PROPN
ejpam-5425	242	17	ibrahim	ibrahim	PROPN
ejpam-5425	242	18	.	.	PUNCT
ejpam-5425	243	1	bifurcation	bifurcation	NOUN
ejpam-5425	243	2	analysis	analysis	NOUN
ejpam-5425	243	3	and	and	CCONJ
ejpam-5425	243	4	chaos	chaos	NOUN
ejpam-5425	243	5	control	control	NOUN
ejpam-5425	243	6	of	of	ADP
ejpam-5425	243	7	a	a	DET
ejpam-5425	243	8	second	second	ADJ
ejpam-5425	243	9	-	-	PUNCT
ejpam-5425	243	10	order	order	NOUN
ejpam-5425	243	11	exponential	exponential	ADJ
ejpam-5425	243	12	difference	difference	NOUN
ejpam-5425	243	13	equations	equation	NOUN
ejpam-5425	243	14	.	.	PUNCT
ejpam-5425	244	1	dynam	dynam	PROPN
ejpam-5425	244	2	.	.	PUNCT
ejpam-5425	245	1	syst	syst	PROPN
ejpam-5425	245	2	.	.	PUNCT
ejpam-5425	245	3	appl	appl	PROPN
ejpam-5425	245	4	.	.	PROPN
ejpam-5425	245	5	,	,	PUNCT
ejpam-5425	245	6	33(15):5003–5022	33(15):5003–5022	PROPN
ejpam-5425	245	7	,	,	PUNCT
ejpam-5425	245	8	2019	2019	NUM
ejpam-5425	245	9	.	.	PUNCT
ejpam-5425	246	1	[	[	X
ejpam-5425	246	2	5	5	NUM
ejpam-5425	246	3	]	]	PUNCT
ejpam-5425	246	4	q.	q.	PROPN
ejpam-5425	246	5	din	din	PROPN
ejpam-5425	246	6	and	and	CCONJ
ejpam-5425	246	7	e.	e.	PROPN
ejpam-5425	246	8	m.	m.	PROPN
ejpam-5425	246	9	elsayed	elsaye	VERB
ejpam-5425	246	10	.	.	PUNCT
ejpam-5425	247	1	stability	stability	NOUN
ejpam-5425	247	2	analysis	analysis	NOUN
ejpam-5425	247	3	of	of	ADP
ejpam-5425	247	4	a	a	DET
ejpam-5425	247	5	discrete	discrete	ADJ
ejpam-5425	247	6	ecological	ecological	ADJ
ejpam-5425	247	7	model	model	NOUN
ejpam-5425	247	8	.	.	PUNCT
ejpam-5425	248	1	comput	comput	NOUN
ejpam-5425	248	2	.	.	PUNCT
ejpam-5425	249	1	ecol	ecol	PROPN
ejpam-5425	249	2	.	.	PUNCT
ejpam-5425	250	1	softw	softw	PROPN
ejpam-5425	250	2	.	.	PUNCT
ejpam-5425	250	3	,	,	PUNCT
ejpam-5425	250	4	4:89–103	4:89–103	NUM
ejpam-5425	250	5	,	,	PUNCT
ejpam-5425	250	6	2014	2014	NUM
ejpam-5425	250	7	.	.	PUNCT
ejpam-5425	251	1	[	[	X
ejpam-5425	251	2	6	6	NUM
ejpam-5425	251	3	]	]	PUNCT
ejpam-5425	251	4	m.	m.	NOUN
ejpam-5425	251	5	m.	m.	PROPN
ejpam-5425	251	6	el	el	PROPN
ejpam-5425	251	7	-	-	NOUN
ejpam-5425	251	8	dessoky	dessoky	NOUN
ejpam-5425	251	9	.	.	PUNCT
ejpam-5425	252	1	on	on	ADP
ejpam-5425	252	2	a	a	DET
ejpam-5425	252	3	solvable	solvable	NOUN
ejpam-5425	252	4	for	for	ADP
ejpam-5425	252	5	some	some	DET
ejpam-5425	252	6	systems	system	NOUN
ejpam-5425	252	7	of	of	ADP
ejpam-5425	252	8	rational	rational	ADJ
ejpam-5425	252	9	difference	difference	NOUN
ejpam-5425	252	10	equations	equation	NOUN
ejpam-5425	252	11	.	.	PUNCT
ejpam-5425	253	1	j.	j.	PROPN
ejpam-5425	253	2	nonlinear	nonlinear	PROPN
ejpam-5425	253	3	sci	sci	PROPN
ejpam-5425	253	4	.	.	PUNCT
ejpam-5425	253	5	appl	appl	PROPN
ejpam-5425	253	6	.	.	PROPN
ejpam-5425	253	7	,	,	PUNCT
ejpam-5425	253	8	9:3744–3759	9:3744–3759	NUM
ejpam-5425	253	9	,	,	PUNCT
ejpam-5425	253	10	2016	2016	NUM
ejpam-5425	253	11	.	.	PUNCT
ejpam-5425	254	1	[	[	X
ejpam-5425	254	2	7	7	X
ejpam-5425	254	3	]	]	X
ejpam-5425	254	4	h.	h.	PROPN
ejpam-5425	254	5	el	el	PROPN
ejpam-5425	254	6	-	-	PROPN
ejpam-5425	254	7	metwally	metwally	ADV
ejpam-5425	254	8	.	.	PUNCT
ejpam-5425	255	1	solutions	solution	NOUN
ejpam-5425	255	2	form	form	NOUN
ejpam-5425	255	3	for	for	ADP
ejpam-5425	255	4	some	some	DET
ejpam-5425	255	5	rational	rational	ADJ
ejpam-5425	255	6	systems	system	NOUN
ejpam-5425	255	7	of	of	ADP
ejpam-5425	255	8	difference	difference	NOUN
ejpam-5425	255	9	equations	equation	NOUN
ejpam-5425	255	10	.	.	PUNCT
ejpam-5425	256	1	discrete	discrete	ADJ
ejpam-5425	256	2	dyn	dyn	PROPN
ejpam-5425	256	3	,	,	PUNCT
ejpam-5425	256	4	nat	nat	PROPN
ejpam-5425	256	5	.	.	PUNCT
ejpam-5425	257	1	soc	soc	PROPN
ejpam-5425	257	2	.	.	PUNCT
ejpam-5425	257	3	,	,	PUNCT
ejpam-5425	257	4	page	page	NOUN
ejpam-5425	257	5	10	10	NUM
ejpam-5425	257	6	,	,	PUNCT
ejpam-5425	257	7	2013	2013	NUM
ejpam-5425	257	8	.	.	PUNCT
ejpam-5425	258	1	[	[	X
ejpam-5425	258	2	8	8	NUM
ejpam-5425	258	3	]	]	X
ejpam-5425	258	4	e.	e.	PROPN
ejpam-5425	258	5	m.	m.	PROPN
ejpam-5425	258	6	elsayed	elsaye	VERB
ejpam-5425	258	7	and	and	CCONJ
ejpam-5425	258	8	j.	j.	PROPN
ejpam-5425	258	9	g.	g.	PROPN
ejpam-5425	258	10	al	al	PROPN
ejpam-5425	258	11	-	-	PUNCT
ejpam-5425	258	12	juaid	juaid	PROPN
ejpam-5425	258	13	.	.	PUNCT
ejpam-5425	259	1	the	the	DET
ejpam-5425	259	2	form	form	NOUN
ejpam-5425	259	3	of	of	ADP
ejpam-5425	259	4	solutions	solution	NOUN
ejpam-5425	259	5	and	and	CCONJ
ejpam-5425	259	6	periodic	periodic	ADJ
ejpam-5425	259	7	nature	nature	NOUN
ejpam-5425	259	8	for	for	ADP
ejpam-5425	259	9	some	some	DET
ejpam-5425	259	10	system	system	NOUN
ejpam-5425	259	11	of	of	ADP
ejpam-5425	259	12	difference	difference	NOUN
ejpam-5425	259	13	equations	equation	NOUN
ejpam-5425	259	14	.	.	PUNCT
ejpam-5425	260	1	fundamental	fundamental	ADJ
ejpam-5425	260	2	journal	journal	NOUN
ejpam-5425	260	3	of	of	ADP
ejpam-5425	260	4	mathematics	mathematic	NOUN
ejpam-5425	260	5	and	and	CCONJ
ejpam-5425	260	6	applications	application	NOUN
ejpam-5425	260	7	,	,	PUNCT
ejpam-5425	260	8	6(1):24–34	6(1):24–34	NUM
ejpam-5425	260	9	,	,	PUNCT
ejpam-5425	260	10	2023	2023	NUM
ejpam-5425	260	11	.	.	PUNCT
ejpam-5425	261	1	[	[	X
ejpam-5425	261	2	9	9	NUM
ejpam-5425	261	3	]	]	X
ejpam-5425	261	4	e.	e.	PROPN
ejpam-5425	261	5	m.	m.	PROPN
ejpam-5425	261	6	elsayed	elsaye	VERB
ejpam-5425	261	7	,	,	PUNCT
ejpam-5425	261	8	j.	j.	PROPN
ejpam-5425	261	9	g.	g.	PROPN
ejpam-5425	261	10	al	al	PROPN
ejpam-5425	261	11	-	-	PUNCT
ejpam-5425	261	12	juaid	juaid	PROPN
ejpam-5425	261	13	,	,	PUNCT
ejpam-5425	261	14	and	and	CCONJ
ejpam-5425	261	15	h.	h.	PROPN
ejpam-5425	261	16	malaikah	malaikah	PROPN
ejpam-5425	261	17	.	.	PUNCT
ejpam-5425	262	1	on	on	ADP
ejpam-5425	262	2	the	the	DET
ejpam-5425	262	3	solutions	solution	NOUN
ejpam-5425	262	4	of	of	ADP
ejpam-5425	262	5	systems	system	NOUN
ejpam-5425	262	6	of	of	ADP
ejpam-5425	262	7	rational	rational	ADJ
ejpam-5425	262	8	difference	difference	NOUN
ejpam-5425	262	9	equations	equation	NOUN
ejpam-5425	262	10	.	.	PUNCT
ejpam-5425	263	1	journal	journal	NOUN
ejpam-5425	263	2	of	of	ADP
ejpam-5425	263	3	progressive	progressive	ADJ
ejpam-5425	263	4	research	research	NOUN
ejpam-5425	263	5	in	in	ADP
ejpam-5425	263	6	mathematics	mathematic	NOUN
ejpam-5425	263	7	,	,	PUNCT
ejpam-5425	263	8	19(2):49	19(2):49	NUM
ejpam-5425	263	9	–	–	PUNCT
ejpam-5425	263	10	59	59	NUM
ejpam-5425	263	11	,	,	PUNCT
ejpam-5425	263	12	2022	2022	NUM
ejpam-5425	263	13	.	.	PUNCT
ejpam-5425	264	1	[	[	X
ejpam-5425	264	2	10	10	NUM
ejpam-5425	264	3	]	]	X
ejpam-5425	264	4	e.	e.	PROPN
ejpam-5425	264	5	m.	m.	PROPN
ejpam-5425	264	6	elsayed	elsaye	VERB
ejpam-5425	264	7	,	,	PUNCT
ejpam-5425	264	8	j.	j.	PROPN
ejpam-5425	264	9	g.	g.	PROPN
ejpam-5425	264	10	al	al	PROPN
ejpam-5425	264	11	-	-	PUNCT
ejpam-5425	264	12	juaid	juaid	PROPN
ejpam-5425	264	13	,	,	PUNCT
ejpam-5425	264	14	and	and	CCONJ
ejpam-5425	264	15	h.	h.	PROPN
ejpam-5425	264	16	malaikah	malaikah	PROPN
ejpam-5425	264	17	.	.	PUNCT
ejpam-5425	265	1	on	on	ADP
ejpam-5425	265	2	the	the	DET
ejpam-5425	265	3	dynamical	dynamical	ADJ
ejpam-5425	265	4	behaviors	behavior	NOUN
ejpam-5425	265	5	of	of	ADP
ejpam-5425	265	6	a	a	DET
ejpam-5425	265	7	quadratic	quadratic	ADJ
ejpam-5425	265	8	difference	difference	NOUN
ejpam-5425	265	9	equation	equation	NOUN
ejpam-5425	265	10	of	of	ADP
ejpam-5425	265	11	order	order	NOUN
ejpam-5425	265	12	three	three	NUM
ejpam-5425	265	13	.	.	PUNCT
ejpam-5425	266	1	european	european	PROPN
ejpam-5425	266	2	journal	journal	PROPN
ejpam-5425	266	3	of	of	ADP
ejpam-5425	266	4	mathematics	mathematic	NOUN
ejpam-5425	266	5	and	and	CCONJ
ejpam-5425	266	6	applications	application	NOUN
ejpam-5425	266	7	,	,	PUNCT
ejpam-5425	266	8	3	3	NUM
ejpam-5425	266	9	,	,	PUNCT
ejpam-5425	266	10	2023	2023	NUM
ejpam-5425	266	11	.	.	PUNCT
ejpam-5425	267	1	article	article	NOUN
ejpam-5425	267	2	i	i	PROPN
ejpam-5425	267	3	d	d	PROPN
ejpam-5425	267	4	1	1	X
ejpam-5425	267	5	.	.	PUNCT
ejpam-5425	268	1	[	[	X
ejpam-5425	268	2	11	11	NUM
ejpam-5425	268	3	]	]	X
ejpam-5425	268	4	e.	e.	PROPN
ejpam-5425	268	5	m.	m.	PROPN
ejpam-5425	268	6	elsayed	elsaye	VERB
ejpam-5425	268	7	and	and	CCONJ
ejpam-5425	268	8	b.	b.	PROPN
ejpam-5425	268	9	s.	s.	PROPN
ejpam-5425	268	10	alof	alof	PROPN
ejpam-5425	268	11	.	.	PUNCT
ejpam-5425	269	1	the	the	DET
ejpam-5425	269	2	periodic	periodic	ADJ
ejpam-5425	269	3	nature	nature	NOUN
ejpam-5425	269	4	and	and	CCONJ
ejpam-5425	269	5	expression	expression	NOUN
ejpam-5425	269	6	on	on	ADP
ejpam-5425	269	7	solutions	solution	NOUN
ejpam-5425	269	8	of	of	ADP
ejpam-5425	269	9	some	some	DET
ejpam-5425	269	10	rational	rational	ADJ
ejpam-5425	269	11	systems	system	NOUN
ejpam-5425	269	12	of	of	ADP
ejpam-5425	269	13	difference	difference	NOUN
ejpam-5425	269	14	equations	equation	NOUN
ejpam-5425	269	15	.	.	PUNCT
ejpam-5425	270	1	alexandria	alexandria	PROPN
ejpam-5425	270	2	engineering	engineering	PROPN
ejpam-5425	270	3	journal	journal	PROPN
ejpam-5425	270	4	,	,	PUNCT
ejpam-5425	270	5	74:269–283	74:269–283	PROPN
ejpam-5425	270	6	,	,	PUNCT
ejpam-5425	270	7	2023	2023	NUM
ejpam-5425	270	8	.	.	PUNCT
ejpam-5425	271	1	[	[	X
ejpam-5425	271	2	12	12	NUM
ejpam-5425	271	3	]	]	X
ejpam-5425	271	4	e.	e.	PROPN
ejpam-5425	271	5	m.	m.	PROPN
ejpam-5425	271	6	elsayed	elsaye	VERB
ejpam-5425	271	7	,	,	PUNCT
ejpam-5425	271	8	q.	q.	PROPN
ejpam-5425	271	9	din	din	PROPN
ejpam-5425	271	10	,	,	PUNCT
ejpam-5425	271	11	and	and	CCONJ
ejpam-5425	271	12	n.	n.	PROPN
ejpam-5425	271	13	a.	a.	NOUN
ejpam-5425	271	14	bukhary	bukhary	PROPN
ejpam-5425	271	15	.	.	PUNCT
ejpam-5425	272	1	theoretical	theoretical	ADJ
ejpam-5425	272	2	and	and	CCONJ
ejpam-5425	272	3	numerical	numerical	ADJ
ejpam-5425	272	4	analysis	analysis	NOUN
ejpam-5425	272	5	of	of	ADP
ejpam-5425	272	6	solutions	solution	NOUN
ejpam-5425	272	7	of	of	ADP
ejpam-5425	272	8	some	some	DET
ejpam-5425	272	9	systems	system	NOUN
ejpam-5425	272	10	of	of	ADP
ejpam-5425	272	11	nonlinear	nonlinear	ADJ
ejpam-5425	272	12	difference	difference	NOUN
ejpam-5425	272	13	equations	equation	NOUN
ejpam-5425	272	14	.	.	PUNCT
ejpam-5425	273	1	aims	aim	VERB
ejpam-5425	273	2	math	math	NOUN
ejpam-5425	273	3	.	.	PUNCT
ejpam-5425	273	4	,	,	PUNCT
ejpam-5425	273	5	7(8):15532	7(8):15532	NOUN
ejpam-5425	273	6	–	–	PUNCT
ejpam-5425	273	7	15549	15549	NUM
ejpam-5425	273	8	,	,	PUNCT
ejpam-5425	273	9	2022	2022	NUM
ejpam-5425	273	10	.	.	PUNCT
ejpam-5425	274	1	[	[	X
ejpam-5425	274	2	13	13	NUM
ejpam-5425	274	3	]	]	PUNCT
ejpam-5425	274	4	t.	t.	PROPN
ejpam-5425	274	5	f.	f.	PROPN
ejpam-5425	274	6	ibrahim	ibrahim	PROPN
ejpam-5425	274	7	.	.	PUNCT
ejpam-5425	275	1	asymptotic	asymptotic	ADJ
ejpam-5425	275	2	behavior	behavior	NOUN
ejpam-5425	275	3	of	of	ADP
ejpam-5425	275	4	a	a	DET
ejpam-5425	275	5	difference	difference	NOUN
ejpam-5425	275	6	equation	equation	NOUN
ejpam-5425	275	7	model	model	NOUN
ejpam-5425	275	8	in	in	ADP
ejpam-5425	275	9	exponential	exponential	ADJ
ejpam-5425	275	10	form	form	NOUN
ejpam-5425	275	11	.	.	PUNCT
ejpam-5425	276	1	math	math	NOUN
ejpam-5425	276	2	.	.	PUNCT
ejpam-5425	277	1	methods	method	NOUN
ejpam-5425	277	2	appl	appl	PROPN
ejpam-5425	277	3	.	.	PUNCT
ejpam-5425	278	1	sci	sci	PROPN
ejpam-5425	278	2	.	.	PROPN
ejpam-5425	278	3	,	,	PUNCT
ejpam-5425	278	4	45:10736–10748	45:10736–10748	PROPN
ejpam-5425	278	5	,	,	PUNCT
ejpam-5425	278	6	2022	2022	NUM
ejpam-5425	278	7	.	.	PUNCT
ejpam-5425	279	1	references	reference	NOUN
ejpam-5425	279	2	3267	3267	NUM
ejpam-5425	279	3	[	[	X
ejpam-5425	279	4	14	14	NUM
ejpam-5425	279	5	]	]	PUNCT
ejpam-5425	279	6	a.	a.	NOUN
ejpam-5425	279	7	khaliq	khaliq	PROPN
ejpam-5425	279	8	,	,	PUNCT
ejpam-5425	279	9	t.	t.	PROPN
ejpam-5425	279	10	f.	f.	PROPN
ejpam-5425	279	11	ibrahim	ibrahim	PROPN
ejpam-5425	279	12	,	,	PUNCT
ejpam-5425	279	13	a.	a.	NOUN
ejpam-5425	279	14	m.	m.	NOUN
ejpam-5425	279	15	alotaibi	alotaibi	PROPN
ejpam-5425	279	16	,	,	PUNCT
ejpam-5425	279	17	m.	m.	NOUN
ejpam-5425	279	18	shoaib	shoaib	PROPN
ejpam-5425	279	19	,	,	PUNCT
ejpam-5425	279	20	and	and	CCONJ
ejpam-5425	279	21	m.	m.	PROPN
ejpam-5425	279	22	el	el	PROPN
ejpam-5425	279	23	-	-	PROPN
ejpam-5425	279	24	moneam	moneam	NOUN
ejpam-5425	279	25	.	.	PUNCT
ejpam-5425	280	1	dynamical	dynamical	ADJ
ejpam-5425	280	2	analysis	analysis	NOUN
ejpam-5425	280	3	of	of	ADP
ejpam-5425	280	4	discrete	discrete	ADJ
ejpam-5425	280	5	-	-	PUNCT
ejpam-5425	280	6	time	time	NOUN
ejpam-5425	280	7	two	two	NUM
ejpam-5425	280	8	-	-	PUNCT
ejpam-5425	280	9	predators	predator	NOUN
ejpam-5425	280	10	one	one	NUM
ejpam-5425	280	11	-	-	PUNCT
ejpam-5425	280	12	prey	prey	NOUN
ejpam-5425	280	13	lotka	lotka	PROPN
ejpam-5425	280	14	–	–	PUNCT
ejpam-5425	280	15	volterra	volterra	PROPN
ejpam-5425	280	16	model	model	NOUN
ejpam-5425	280	17	.	.	PUNCT
ejpam-5425	281	1	mathematics	mathematic	NOUN
ejpam-5425	281	2	,	,	PUNCT
ejpam-5425	281	3	10:4015	10:4015	NUM
ejpam-5425	281	4	,	,	PUNCT
ejpam-5425	281	5	2022	2022	NUM
ejpam-5425	281	6	.	.	PUNCT
ejpam-5425	282	1	[	[	X
ejpam-5425	282	2	15	15	NUM
ejpam-5425	282	3	]	]	PUNCT
ejpam-5425	282	4	a.	a.	NOUN
ejpam-5425	282	5	khaliq	khaliq	PROPN
ejpam-5425	282	6	,	,	PUNCT
ejpam-5425	282	7	i.	i.	PROPN
ejpam-5425	282	8	mustafa	mustafa	PROPN
ejpam-5425	282	9	,	,	PUNCT
ejpam-5425	282	10	t.	t.	PROPN
ejpam-5425	282	11	f.	f.	PROPN
ejpam-5425	282	12	ibrahim	ibrahim	PROPN
ejpam-5425	282	13	,	,	PUNCT
ejpam-5425	282	14	w.	w.	PROPN
ejpam-5425	282	15	m.	m.	PROPN
ejpam-5425	282	16	osman	osman	PROPN
ejpam-5425	282	17	,	,	PUNCT
ejpam-5425	282	18	b.	b.	PROPN
ejpam-5425	282	19	r.	r.	PROPN
ejpam-5425	282	20	al	al	PROPN
ejpam-5425	282	21	-	-	PUNCT
ejpam-5425	282	22	sinan	sinan	PROPN
ejpam-5425	282	23	,	,	PUNCT
ejpam-5425	282	24	a.	a.	PROPN
ejpam-5425	282	25	a.	a.	PROPN
ejpam-5425	282	26	dawood	dawood	PROPN
ejpam-5425	282	27	,	,	PUNCT
ejpam-5425	282	28	and	and	CCONJ
ejpam-5425	282	29	m.	m.	PROPN
ejpam-5425	282	30	y.	y.	PROPN
ejpam-5425	282	31	juma	juma	PROPN
ejpam-5425	282	32	.	.	PUNCT
ejpam-5425	283	1	stability	stability	NOUN
ejpam-5425	283	2	and	and	CCONJ
ejpam-5425	283	3	bifurcation	bifurcation	NOUN
ejpam-5425	283	4	analysis	analysis	NOUN
ejpam-5425	283	5	of	of	ADP
ejpam-5425	283	6	fifth	fifth	ADJ
ejpam-5425	283	7	-	-	PUNCT
ejpam-5425	283	8	order	order	NOUN
ejpam-5425	283	9	nonlinear	nonlinear	ADJ
ejpam-5425	283	10	fractional	fractional	ADJ
ejpam-5425	283	11	difference	difference	NOUN
ejpam-5425	283	12	equation	equation	NOUN
ejpam-5425	283	13	.	.	PUNCT
ejpam-5425	284	1	fractal	fractal	ADJ
ejpam-5425	284	2	fract	fract	PROPN
ejpam-5425	284	3	.	.	PUNCT
ejpam-5425	284	4	,	,	PUNCT
ejpam-5425	285	1	7:113	7:113	NUM
ejpam-5425	285	2	,	,	PUNCT
ejpam-5425	285	3	2023	2023	NUM
ejpam-5425	285	4	.	.	PUNCT
ejpam-5425	286	1	[	[	X
ejpam-5425	286	2	16	16	NUM
ejpam-5425	286	3	]	]	PUNCT
ejpam-5425	286	4	a.	a.	PROPN
ejpam-5425	286	5	q.	q.	PROPN
ejpam-5425	286	6	khan	khan	PROPN
ejpam-5425	286	7	and	and	CCONJ
ejpam-5425	286	8	h.	h.	PROPN
ejpam-5425	286	9	s.	s.	PROPN
ejpam-5425	286	10	alayachi	alayachi	PROPN
ejpam-5425	286	11	.	.	PUNCT
ejpam-5425	287	1	bifurcation	bifurcation	NOUN
ejpam-5425	287	2	and	and	CCONJ
ejpam-5425	287	3	chaos	chaos	NOUN
ejpam-5425	287	4	in	in	ADP
ejpam-5425	287	5	a	a	DET
ejpam-5425	287	6	phytoplanktonzooplankton	phytoplanktonzooplankton	NOUN
ejpam-5425	287	7	model	model	NOUN
ejpam-5425	287	8	with	with	ADP
ejpam-5425	287	9	holling	holle	VERB
ejpam-5425	287	10	type	type	NOUN
ejpam-5425	287	11	-	-	PUNCT
ejpam-5425	287	12	ii	ii	NOUN
ejpam-5425	287	13	response	response	NOUN
ejpam-5425	287	14	and	and	CCONJ
ejpam-5425	287	15	toxicity	toxicity	NOUN
ejpam-5425	287	16	.	.	PUNCT
ejpam-5425	288	1	international	international	ADJ
ejpam-5425	288	2	journal	journal	NOUN
ejpam-5425	288	3	of	of	ADP
ejpam-5425	288	4	bifurcation	bifurcation	NOUN
ejpam-5425	288	5	and	and	CCONJ
ejpam-5425	288	6	chaos	chaos	NOUN
ejpam-5425	288	7	,	,	PUNCT
ejpam-5425	288	8	32(12	32(12	NUM
ejpam-5425	288	9	)	)	PUNCT
ejpam-5425	288	10	,	,	PUNCT
ejpam-5425	288	11	2022	2022	NUM
ejpam-5425	288	12	.	.	PUNCT
ejpam-5425	289	1	article	article	NOUN
ejpam-5425	289	2	2250176	2250176	NUM
ejpam-5425	289	3	.	.	PUNCT
ejpam-5425	290	1	[	[	X
ejpam-5425	290	2	17	17	NUM
ejpam-5425	290	3	]	]	PUNCT
ejpam-5425	290	4	a.	a.	PROPN
ejpam-5425	290	5	q.	q.	PROPN
ejpam-5425	290	6	khan	khan	PROPN
ejpam-5425	290	7	,	,	PUNCT
ejpam-5425	290	8	f.	f.	PROPN
ejpam-5425	290	9	nazir	nazir	PROPN
ejpam-5425	290	10	,	,	PUNCT
ejpam-5425	290	11	and	and	CCONJ
ejpam-5425	290	12	m.	m.	PROPN
ejpam-5425	290	13	b.	b.	PROPN
ejpam-5425	290	14	almatrafi	almatrafi	PROPN
ejpam-5425	290	15	.	.	PUNCT
ejpam-5425	291	1	bifurcation	bifurcation	NOUN
ejpam-5425	291	2	analysis	analysis	NOUN
ejpam-5425	291	3	of	of	ADP
ejpam-5425	291	4	a	a	DET
ejpam-5425	291	5	discrete	discrete	ADJ
ejpam-5425	291	6	phytoplankton	phytoplankton	NOUN
ejpam-5425	291	7	-	-	PUNCT
ejpam-5425	291	8	zooplankton	zooplankton	NOUN
ejpam-5425	291	9	model	model	NOUN
ejpam-5425	291	10	with	with	ADP
ejpam-5425	291	11	linear	linear	ADJ
ejpam-5425	291	12	predational	predational	ADJ
ejpam-5425	291	13	response	response	NOUN
ejpam-5425	291	14	function	function	NOUN
ejpam-5425	291	15	and	and	CCONJ
ejpam-5425	291	16	toxic	toxic	ADJ
ejpam-5425	291	17	substance	substance	NOUN
ejpam-5425	291	18	distribution	distribution	NOUN
ejpam-5425	291	19	.	.	PUNCT
ejpam-5425	292	1	int	int	NOUN
ejpam-5425	292	2	.	.	PUNCT
ejpam-5425	293	1	j.	j.	PROPN
ejpam-5425	293	2	biomath	biomath	PROPN
ejpam-5425	293	3	.	.	PUNCT
ejpam-5425	293	4	,	,	PUNCT
ejpam-5425	293	5	16(4	16(4	PROPN
ejpam-5425	293	6	)	)	PUNCT
ejpam-5425	293	7	,	,	PUNCT
ejpam-5425	293	8	2023	2023	NUM
ejpam-5425	293	9	.	.	PUNCT
ejpam-5425	294	1	article	article	NOUN
ejpam-5425	294	2	2250095	2250095	NUM
ejpam-5425	294	3	.	.	PUNCT
ejpam-5425	295	1	[	[	X
ejpam-5425	295	2	18	18	NUM
ejpam-5425	295	3	]	]	PUNCT
ejpam-5425	295	4	a.	a.	PROPN
ejpam-5425	295	5	q.	q.	PROPN
ejpam-5425	295	6	khan	khan	PROPN
ejpam-5425	295	7	,	,	PUNCT
ejpam-5425	295	8	m.	m.	NOUN
ejpam-5425	295	9	tasneem	tasneem	PROPN
ejpam-5425	295	10	,	,	PUNCT
ejpam-5425	295	11	b.	b.	PROPN
ejpam-5425	295	12	younis	younis	PROPN
ejpam-5425	295	13	,	,	PUNCT
ejpam-5425	295	14	and	and	CCONJ
ejpam-5425	295	15	t.	t.	PROPN
ejpam-5425	295	16	f.	f.	PROPN
ejpam-5425	295	17	ibrahim	ibrahim	PROPN
ejpam-5425	295	18	.	.	PUNCT
ejpam-5425	296	1	dynamical	dynamical	ADJ
ejpam-5425	296	2	analysis	analysis	NOUN
ejpam-5425	296	3	of	of	ADP
ejpam-5425	296	4	a	a	DET
ejpam-5425	296	5	discrete	discrete	ADJ
ejpam-5425	296	6	-	-	PUNCT
ejpam-5425	296	7	time	time	NOUN
ejpam-5425	296	8	covid-19	covid-19	PROPN
ejpam-5425	296	9	epidemic	epidemic	NOUN
ejpam-5425	296	10	model	model	NOUN
ejpam-5425	296	11	.	.	PUNCT
ejpam-5425	297	1	math	math	NOUN
ejpam-5425	297	2	.	.	PUNCT
ejpam-5425	298	1	meth	meth	NOUN
ejpam-5425	298	2	.	.	PUNCT
ejpam-5425	299	1	appl	appl	PROPN
ejpam-5425	299	2	.	.	PUNCT
ejpam-5425	300	1	sci	sci	PROPN
ejpam-5425	300	2	.	.	PROPN
ejpam-5425	300	3	,	,	PUNCT
ejpam-5425	300	4	46:4789–4814	46:4789–4814	PROPN
ejpam-5425	300	5	,	,	PUNCT
ejpam-5425	300	6	2022	2022	NUM
ejpam-5425	300	7	.	.	PUNCT
ejpam-5425	301	1	[	[	X
ejpam-5425	301	2	19	19	NUM
ejpam-5425	301	3	]	]	X
ejpam-5425	301	4	c.	c.	PROPN
ejpam-5425	301	5	ma	ma	PROPN
ejpam-5425	301	6	,	,	PUNCT
ejpam-5425	301	7	b.	b.	PROPN
ejpam-5425	301	8	shiri	shiri	PROPN
ejpam-5425	301	9	,	,	PUNCT
ejpam-5425	301	10	g.	g.	PROPN
ejpam-5425	301	11	wu	wu	PROPN
ejpam-5425	301	12	,	,	PUNCT
ejpam-5425	301	13	and	and	CCONJ
ejpam-5425	301	14	d.	d.	PROPN
ejpam-5425	301	15	baleanu	baleanu	PROPN
ejpam-5425	301	16	.	.	PUNCT
ejpam-5425	302	1	new	new	ADJ
ejpam-5425	302	2	fractional	fractional	ADJ
ejpam-5425	302	3	signal	signal	NOUN
ejpam-5425	302	4	smoothing	smoothing	NOUN
ejpam-5425	302	5	equations	equation	NOUN
ejpam-5425	302	6	with	with	ADP
ejpam-5425	302	7	short	short	ADJ
ejpam-5425	302	8	memory	memory	NOUN
ejpam-5425	302	9	and	and	CCONJ
ejpam-5425	302	10	variable	variable	ADJ
ejpam-5425	302	11	order	order	NOUN
ejpam-5425	302	12	.	.	PUNCT
ejpam-5425	303	1	optik	optik	PROPN
ejpam-5425	303	2	,	,	PUNCT
ejpam-5425	303	3	218:164507	218:164507	NUM
ejpam-5425	303	4	,	,	PUNCT
ejpam-5425	303	5	2020	2020	NUM
ejpam-5425	303	6	.	.	PUNCT
ejpam-5425	304	1	[	[	X
ejpam-5425	304	2	20	20	NUM
ejpam-5425	304	3	]	]	PUNCT
ejpam-5425	304	4	m.	m.	NOUN
ejpam-5425	304	5	mansour	mansour	PROPN
ejpam-5425	304	6	,	,	PUNCT
ejpam-5425	304	7	m.	m.	PROPN
ejpam-5425	304	8	m.	m.	PROPN
ejpam-5425	304	9	el	el	PROPN
ejpam-5425	304	10	-	-	NOUN
ejpam-5425	304	11	dessoky	dessoky	NOUN
ejpam-5425	304	12	,	,	PUNCT
ejpam-5425	304	13	and	and	CCONJ
ejpam-5425	304	14	e.	e.	PROPN
ejpam-5425	304	15	m.	m.	PROPN
ejpam-5425	304	16	elsayed	elsaye	VERB
ejpam-5425	304	17	.	.	PUNCT
ejpam-5425	305	1	the	the	DET
ejpam-5425	305	2	form	form	NOUN
ejpam-5425	305	3	of	of	ADP
ejpam-5425	305	4	the	the	DET
ejpam-5425	305	5	solutions	solution	NOUN
ejpam-5425	305	6	and	and	CCONJ
ejpam-5425	305	7	periodicity	periodicity	NOUN
ejpam-5425	305	8	of	of	ADP
ejpam-5425	305	9	some	some	DET
ejpam-5425	305	10	systems	system	NOUN
ejpam-5425	305	11	of	of	ADP
ejpam-5425	305	12	difference	difference	NOUN
ejpam-5425	305	13	equations	equation	NOUN
ejpam-5425	305	14	.	.	PUNCT
ejpam-5425	306	1	dis	dis	PROPN
ejpam-5425	306	2	.	.	PUNCT
ejpam-5425	307	1	dyn	dyn	PROPN
ejpam-5425	307	2	.	.	PUNCT
ejpam-5425	308	1	nat	nat	PROPN
ejpam-5425	308	2	.	.	PUNCT
ejpam-5425	309	1	soc	soc	PROPN
ejpam-5425	309	2	.	.	PUNCT
ejpam-5425	309	3	,	,	PUNCT
ejpam-5425	309	4	2012:1–17	2012:1–17	NUM
ejpam-5425	309	5	,	,	PUNCT
ejpam-5425	309	6	2012	2012	NUM
ejpam-5425	309	7	.	.	PUNCT
ejpam-5425	310	1	[	[	X
ejpam-5425	310	2	21	21	NUM
ejpam-5425	310	3	]	]	X
ejpam-5425	310	4	n.	n.	PROPN
ejpam-5425	310	5	l.	l.	PROPN
ejpam-5425	310	6	wang	wang	PROPN
ejpam-5425	310	7	,	,	PUNCT
ejpam-5425	310	8	p.	p.	PROPN
ejpam-5425	310	9	agarwal	agarwal	PROPN
ejpam-5425	310	10	,	,	PUNCT
ejpam-5425	310	11	and	and	CCONJ
ejpam-5425	310	12	s.	s.	PROPN
ejpam-5425	310	13	kanemitsu	kanemitsu	PROPN
ejpam-5425	310	14	.	.	PUNCT
ejpam-5425	311	1	limiting	limit	VERB
ejpam-5425	311	2	values	value	NOUN
ejpam-5425	311	3	and	and	CCONJ
ejpam-5425	311	4	functional	functional	ADJ
ejpam-5425	311	5	and	and	CCONJ
ejpam-5425	311	6	difference	difference	NOUN
ejpam-5425	311	7	equations	equation	NOUN
ejpam-5425	311	8	.	.	PUNCT
ejpam-5425	312	1	mathematics	mathematic	NOUN
ejpam-5425	312	2	,	,	PUNCT
ejpam-5425	312	3	8:407	8:407	NUM
ejpam-5425	312	4	,	,	PUNCT
ejpam-5425	312	5	2020	2020	NUM
ejpam-5425	312	6	.	.	PUNCT
ejpam-5425	313	1	[	[	X
ejpam-5425	313	2	22	22	NUM
ejpam-5425	313	3	]	]	X
ejpam-5425	313	4	g.	g.	PROPN
ejpam-5425	313	5	wu	wu	PROPN
ejpam-5425	313	6	,	,	PUNCT
ejpam-5425	313	7	z.	z.	PROPN
ejpam-5425	313	8	deng	deng	PROPN
ejpam-5425	313	9	,	,	PUNCT
ejpam-5425	313	10	d.	d.	PROPN
ejpam-5425	313	11	baleanu	baleanu	PROPN
ejpam-5425	313	12	,	,	PUNCT
ejpam-5425	313	13	and	and	CCONJ
ejpam-5425	313	14	d.	d.	PROPN
ejpam-5425	313	15	zeng	zeng	PROPN
ejpam-5425	313	16	.	.	PUNCT
ejpam-5425	314	1	new	new	ADJ
ejpam-5425	314	2	variable	variable	ADJ
ejpam-5425	314	3	-	-	PUNCT
ejpam-5425	314	4	order	order	NOUN
ejpam-5425	314	5	fractional	fractional	ADJ
ejpam-5425	314	6	chaotic	chaotic	ADJ
ejpam-5425	314	7	systems	system	NOUN
ejpam-5425	314	8	for	for	ADP
ejpam-5425	314	9	fast	fast	ADJ
ejpam-5425	314	10	image	image	NOUN
ejpam-5425	314	11	encryption	encryption	NOUN
ejpam-5425	314	12	.	.	PUNCT
ejpam-5425	315	1	chaos	chaos	NOUN
ejpam-5425	315	2	,	,	PUNCT
ejpam-5425	315	3	29(8	29(8	NOUN
ejpam-5425	315	4	)	)	PUNCT
ejpam-5425	315	5	,	,	PUNCT
ejpam-5425	315	6	2019	2019	NUM
ejpam-5425	315	7	.	.	PUNCT
ejpam-5425	316	1	article	article	NOUN
ejpam-5425	316	2	083103	083103	NUM
ejpam-5425	316	3	.	.	PUNCT