id	sid	tid	token	lemma	pos
iajs-3432	1	1	386	386	NUM
iajs-3432	1	2	©	©	NOUN
iajs-3432	1	3	published	publish	VERB
iajs-3432	1	4	by	by	ADP
iajs-3432	1	5	college	college	NOUN
iajs-3432	1	6	of	of	ADP
iajs-3432	1	7	education	education	NOUN
iajs-3432	1	8	for	for	ADP
iajs-3432	1	9	pure	pure	ADJ
iajs-3432	1	10	science	science	NOUN
iajs-3432	1	11	(	(	PUNCT
iajs-3432	1	12	ibn	ibn	PROPN
iajs-3432	1	13	al	al	PROPN
iajs-3432	1	14	-	-	PUNCT
iajs-3432	1	15	haitham	haitham	PROPN
iajs-3432	1	16	)	)	PUNCT
iajs-3432	1	17	,	,	PUNCT
iajs-3432	1	18	university	university	NOUN
iajs-3432	1	19	of	of	ADP
iajs-3432	1	20	baghdad	baghdad	PROPN
iajs-3432	1	21	.	.	PUNCT
iajs-3432	2	1	this	this	PRON
iajs-3432	2	2	is	be	AUX
iajs-3432	2	3	an	an	DET
iajs-3432	2	4	open	open	ADJ
iajs-3432	2	5	-	-	PUNCT
iajs-3432	2	6	access	access	NOUN
iajs-3432	2	7	article	article	NOUN
iajs-3432	2	8	distributed	distribute	VERB
iajs-3432	2	9	under	under	ADP
iajs-3432	2	10	the	the	DET
iajs-3432	2	11	terms	term	NOUN
iajs-3432	2	12	of	of	ADP
iajs-3432	2	13	the	the	DET
iajs-3432	2	14	creative	creative	ADJ
iajs-3432	2	15	commons	common	NOUN
iajs-3432	2	16	attribution	attribution	NOUN
iajs-3432	2	17	4.0	4.0	NUM
iajs-3432	2	18	international	international	ADJ
iajs-3432	2	19	license	license	NOUN
iajs-3432	2	20	a	a	DET
iajs-3432	2	21	mathematical	mathematical	ADJ
iajs-3432	2	22	approach	approach	NOUN
iajs-3432	2	23	to	to	ADP
iajs-3432	2	24	oscillation	oscillation	NOUN
iajs-3432	2	25	of	of	ADP
iajs-3432	2	26	a	a	DET
iajs-3432	2	27	discrete	discrete	ADJ
iajs-3432	2	28	hematopoiesis	hematopoiesis	NOUN
iajs-3432	2	29	model	model	NOUN
iajs-3432	2	30	with	with	ADP
iajs-3432	2	31	positive	positive	ADJ
iajs-3432	2	32	and	and	CCONJ
iajs-3432	2	33	negative	negative	ADJ
iajs-3432	2	34	coefficients	coefficient	NOUN
iajs-3432	2	35	iman	iman	PROPN
iajs-3432	2	36	sabeeh	sabeeh	PROPN
iajs-3432	2	37	hadeed1	hadeed1	PROPN
iajs-3432	2	38	*	*	PUNCT
iajs-3432	2	39	,	,	PUNCT
iajs-3432	2	40	hussain	hussain	PROPN
iajs-3432	2	41	ali	ali	PROPN
iajs-3432	2	42	mohamad2	mohamad2	PROPN
iajs-3432	2	43	1department	1department	NUM
iajs-3432	2	44	of	of	ADP
iajs-3432	2	45	mathematics	mathematic	NOUN
iajs-3432	2	46	,	,	PUNCT
iajs-3432	2	47	college	college	NOUN
iajs-3432	2	48	of	of	ADP
iajs-3432	2	49	sciences	science	NOUN
iajs-3432	2	50	,	,	PUNCT
iajs-3432	2	51	university	university	NOUN
iajs-3432	2	52	of	of	ADP
iajs-3432	2	53	baghdad	baghdad	PROPN
iajs-3432	2	54	,	,	PUNCT
iajs-3432	2	55	baghdad	baghdad	PROPN
iajs-3432	2	56	,	,	PUNCT
iajs-3432	2	57	iraq	iraq	PROPN
iajs-3432	2	58	.	.	PUNCT
iajs-3432	3	1	2department	2department	NUM
iajs-3432	3	2	of	of	ADP
iajs-3432	3	3	mathematics	mathematic	NOUN
iajs-3432	3	4	,	,	PUNCT
iajs-3432	3	5	college	college	NOUN
iajs-3432	3	6	of	of	ADP
iajs-3432	3	7	sciences	science	NOUN
iajs-3432	3	8	for	for	ADP
iajs-3432	3	9	women	woman	NOUN
iajs-3432	3	10	,	,	PUNCT
iajs-3432	3	11	university	university	NOUN
iajs-3432	3	12	of	of	ADP
iajs-3432	3	13	baghdad	baghdad	PROPN
iajs-3432	3	14	,	,	PUNCT
iajs-3432	3	15	baghdad	baghdad	PROPN
iajs-3432	3	16	,	,	PUNCT
iajs-3432	3	17	iraq	iraq	PROPN
iajs-3432	3	18	.	.	PUNCT
iajs-3432	4	1	*	*	PUNCT
iajs-3432	4	2	corresponding	correspond	VERB
iajs-3432	4	3	author	author	NOUN
iajs-3432	4	4	.	.	PUNCT
iajs-3432	5	1	received:19	received:19	X
iajs-3432	5	2	april	april	PROPN
iajs-3432	5	3	2023	2023	NUM
iajs-3432	5	4	accepted:12	accepted:12	X
iajs-3432	5	5	june	june	PROPN
iajs-3432	5	6	2023	2023	NUM
iajs-3432	5	7	published:20	published:20	NOUN
iajs-3432	5	8	july	july	PROPN
iajs-3432	5	9	2024	2024	NUM
iajs-3432	5	10	doi.org/10.30526/37.3.3432	doi.org/10.30526/37.3.3432	NOUN
iajs-3432	5	11	abstract	abstract	NOUN
iajs-3432	5	12	in	in	ADP
iajs-3432	5	13	this	this	DET
iajs-3432	5	14	paper	paper	NOUN
iajs-3432	5	15	,	,	PUNCT
iajs-3432	5	16	we	we	PRON
iajs-3432	5	17	investigate	investigate	VERB
iajs-3432	5	18	a	a	DET
iajs-3432	5	19	mathematical	mathematical	ADJ
iajs-3432	5	20	model	model	NOUN
iajs-3432	5	21	of	of	ADP
iajs-3432	5	22	hematopoiesis	hematopoiesis	NOUN
iajs-3432	5	23	,	,	PUNCT
iajs-3432	5	24	a	a	DET
iajs-3432	5	25	process	process	NOUN
iajs-3432	5	26	responsible	responsible	ADJ
iajs-3432	5	27	for	for	ADP
iajs-3432	5	28	the	the	DET
iajs-3432	5	29	regular	regular	ADJ
iajs-3432	5	30	replacement	replacement	NOUN
iajs-3432	5	31	of	of	ADP
iajs-3432	5	32	circulating	circulate	VERB
iajs-3432	5	33	blood	blood	NOUN
iajs-3432	5	34	cells	cell	NOUN
iajs-3432	5	35	.	.	PUNCT
iajs-3432	6	1	since	since	SCONJ
iajs-3432	6	2	differential	differential	ADJ
iajs-3432	6	3	delay	delay	NOUN
iajs-3432	6	4	equations	equation	NOUN
iajs-3432	6	5	are	be	AUX
iajs-3432	6	6	difficult	difficult	ADJ
iajs-3432	6	7	to	to	PART
iajs-3432	6	8	control	control	VERB
iajs-3432	6	9	analytically	analytically	ADV
iajs-3432	6	10	,	,	PUNCT
iajs-3432	6	11	numerous	numerous	ADJ
iajs-3432	6	12	studies	study	NOUN
iajs-3432	6	13	have	have	AUX
iajs-3432	6	14	considered	consider	VERB
iajs-3432	6	15	the	the	DET
iajs-3432	6	16	models	model	NOUN
iajs-3432	6	17	as	as	ADP
iajs-3432	6	18	difference	difference	NOUN
iajs-3432	6	19	equations	equation	NOUN
iajs-3432	6	20	.	.	PUNCT
iajs-3432	7	1	the	the	DET
iajs-3432	7	2	main	main	ADJ
iajs-3432	7	3	objective	objective	NOUN
iajs-3432	7	4	of	of	ADP
iajs-3432	7	5	this	this	DET
iajs-3432	7	6	work	work	NOUN
iajs-3432	7	7	is	be	AUX
iajs-3432	7	8	to	to	PART
iajs-3432	7	9	provide	provide	VERB
iajs-3432	7	10	the	the	DET
iajs-3432	7	11	necessary	necessary	ADJ
iajs-3432	7	12	and	and	CCONJ
iajs-3432	7	13	sufficient	sufficient	ADJ
iajs-3432	7	14	conditions	condition	NOUN
iajs-3432	7	15	for	for	ADP
iajs-3432	7	16	oscillation	oscillation	NOUN
iajs-3432	7	17	.	.	PUNCT
iajs-3432	8	1	we	we	PRON
iajs-3432	8	2	address	address	VERB
iajs-3432	8	3	our	our	PRON
iajs-3432	8	4	problem	problem	NOUN
iajs-3432	8	5	mainly	mainly	ADV
iajs-3432	8	6	on	on	ADP
iajs-3432	8	7	the	the	DET
iajs-3432	8	8	basis	basis	NOUN
iajs-3432	8	9	of	of	ADP
iajs-3432	8	10	oscillatory	oscillatory	ADJ
iajs-3432	8	11	behavior	behavior	NOUN
iajs-3432	8	12	.	.	PUNCT
iajs-3432	9	1	moreover	moreover	ADV
iajs-3432	9	2	,	,	PUNCT
iajs-3432	9	3	the	the	DET
iajs-3432	9	4	latest	late	ADJ
iajs-3432	9	5	findings	finding	NOUN
iajs-3432	9	6	on	on	ADP
iajs-3432	9	7	the	the	DET
iajs-3432	9	8	qualitative	qualitative	ADJ
iajs-3432	9	9	behavior	behavior	NOUN
iajs-3432	9	10	of	of	ADP
iajs-3432	9	11	the	the	DET
iajs-3432	9	12	biological	biological	ADJ
iajs-3432	9	13	mathematical	mathematical	ADJ
iajs-3432	9	14	model	model	NOUN
iajs-3432	9	15	of	of	ADP
iajs-3432	9	16	discrete	discrete	ADJ
iajs-3432	9	17	hematopoiesis	hematopoiesis	NOUN
iajs-3432	9	18	are	be	AUX
iajs-3432	9	19	taken	take	VERB
iajs-3432	9	20	into	into	ADP
iajs-3432	9	21	account	account	NOUN
iajs-3432	9	22	.	.	PUNCT
iajs-3432	10	1	more	more	ADV
iajs-3432	10	2	specifically	specifically	ADV
iajs-3432	10	3	,	,	PUNCT
iajs-3432	10	4	we	we	PRON
iajs-3432	10	5	explain	explain	VERB
iajs-3432	10	6	the	the	DET
iajs-3432	10	7	mathematical	mathematical	ADJ
iajs-3432	10	8	differential	differential	ADJ
iajs-3432	10	9	equation	equation	NOUN
iajs-3432	10	10	of	of	ADP
iajs-3432	10	11	discrete	discrete	ADJ
iajs-3432	10	12	hematopoiesis	hematopoiesis	NOUN
iajs-3432	10	13	.	.	PUNCT
iajs-3432	11	1	moreover	moreover	ADV
iajs-3432	11	2	,	,	PUNCT
iajs-3432	11	3	certain	certain	ADJ
iajs-3432	11	4	significant	significant	ADJ
iajs-3432	11	5	,	,	PUNCT
iajs-3432	11	6	necessary	necessary	ADJ
iajs-3432	11	7	and	and	CCONJ
iajs-3432	11	8	sufficient	sufficient	ADJ
iajs-3432	11	9	criteria	criterion	NOUN
iajs-3432	11	10	for	for	ADP
iajs-3432	11	11	the	the	DET
iajs-3432	11	12	solution	solution	NOUN
iajs-3432	11	13	of	of	ADP
iajs-3432	11	14	this	this	DET
iajs-3432	11	15	discrete	discrete	ADJ
iajs-3432	11	16	problem	problem	NOUN
iajs-3432	11	17	are	be	AUX
iajs-3432	11	18	found	find	VERB
iajs-3432	11	19	,	,	PUNCT
iajs-3432	11	20	which	which	PRON
iajs-3432	11	21	guarantee	guarantee	VERB
iajs-3432	11	22	either	either	CCONJ
iajs-3432	11	23	the	the	DET
iajs-3432	11	24	convergence	convergence	NOUN
iajs-3432	11	25	of	of	ADP
iajs-3432	11	26	the	the	DET
iajs-3432	11	27	non	non	ADJ
iajs-3432	11	28	-	-	ADJ
iajs-3432	11	29	oscillating	oscillating	ADJ
iajs-3432	11	30	solutions	solution	NOUN
iajs-3432	11	31	towards	towards	ADP
iajs-3432	11	32	zero	zero	NUM
iajs-3432	11	33	or	or	CCONJ
iajs-3432	11	34	the	the	DET
iajs-3432	11	35	oscillation	oscillation	NOUN
iajs-3432	11	36	of	of	ADP
iajs-3432	11	37	all	all	DET
iajs-3432	11	38	solutions	solution	NOUN
iajs-3432	11	39	of	of	ADP
iajs-3432	11	40	the	the	DET
iajs-3432	11	41	discrete	discrete	ADJ
iajs-3432	11	42	hematopoiesis	hematopoiesis	NOUN
iajs-3432	11	43	model	model	NOUN
iajs-3432	11	44	to	to	ADP
iajs-3432	11	45	the	the	DET
iajs-3432	11	46	nonlinear	nonlinear	ADJ
iajs-3432	11	47	lag	lag	NOUN
iajs-3432	11	48	difference	difference	NOUN
iajs-3432	11	49	with	with	ADP
iajs-3432	11	50	positive	positive	ADJ
iajs-3432	11	51	and	and	CCONJ
iajs-3432	11	52	negative	negative	ADJ
iajs-3432	11	53	coefficients	coefficient	NOUN
iajs-3432	11	54	.	.	PUNCT
iajs-3432	12	1	some	some	DET
iajs-3432	12	2	numerical	numerical	ADJ
iajs-3432	12	3	examples	example	NOUN
iajs-3432	12	4	are	be	AUX
iajs-3432	12	5	also	also	ADV
iajs-3432	12	6	given	give	VERB
iajs-3432	12	7	to	to	PART
iajs-3432	12	8	illustrate	illustrate	VERB
iajs-3432	12	9	the	the	DET
iajs-3432	12	10	most	most	ADV
iajs-3432	12	11	important	important	ADJ
iajs-3432	12	12	results	result	NOUN
iajs-3432	12	13	.	.	PUNCT
iajs-3432	13	1	keywords	keyword	NOUN
iajs-3432	13	2	:	:	PUNCT
iajs-3432	13	3	oscillation	oscillation	NOUN
iajs-3432	13	4	,	,	PUNCT
iajs-3432	13	5	delay	delay	VERB
iajs-3432	13	6	differential	differential	ADJ
iajs-3432	13	7	equations	equation	NOUN
iajs-3432	13	8	,	,	PUNCT
iajs-3432	13	9	difference	difference	NOUN
iajs-3432	13	10	equations	equation	NOUN
iajs-3432	13	11	,	,	PUNCT
iajs-3432	13	12	delay	delay	VERB
iajs-3432	13	13	differential	differential	ADJ
iajs-3432	13	14	equations	equation	NOUN
iajs-3432	13	15	,	,	PUNCT
iajs-3432	13	16	hematopoiesis	hematopoiesis	NOUN
iajs-3432	13	17	model	model	NOUN
iajs-3432	13	18	.	.	PUNCT
iajs-3432	14	1	1	1	X
iajs-3432	14	2	.	.	X
iajs-3432	14	3	introduction	introduction	NOUN
iajs-3432	14	4	the	the	DET
iajs-3432	14	5	oscillatory	oscillatory	ADJ
iajs-3432	14	6	behavior	behavior	NOUN
iajs-3432	14	7	of	of	ADP
iajs-3432	14	8	difference	difference	NOUN
iajs-3432	14	9	equations	equation	NOUN
iajs-3432	14	10	'	'	PART
iajs-3432	14	11	and	and	CCONJ
iajs-3432	14	12	dynamic	dynamic	ADJ
iajs-3432	14	13	equations	equation	NOUN
iajs-3432	14	14	'	'	PART
iajs-3432	14	15	solutions	solution	NOUN
iajs-3432	14	16	has	have	AUX
iajs-3432	14	17	recently	recently	ADV
iajs-3432	14	18	been	be	AUX
iajs-3432	14	19	the	the	DET
iajs-3432	14	20	subject	subject	NOUN
iajs-3432	14	21	of	of	ADP
iajs-3432	14	22	a	a	DET
iajs-3432	14	23	lot	lot	NOUN
iajs-3432	14	24	of	of	ADP
iajs-3432	14	25	research	research	NOUN
iajs-3432	14	26	.	.	PUNCT
iajs-3432	15	1	numerous	numerous	ADJ
iajs-3432	15	2	recent	recent	ADJ
iajs-3432	15	3	works	work	NOUN
iajs-3432	15	4	have	have	AUX
iajs-3432	15	5	been	be	AUX
iajs-3432	15	6	focused	focus	VERB
iajs-3432	15	7	on	on	ADP
iajs-3432	15	8	oscillations	oscillation	NOUN
iajs-3432	15	9	of	of	ADP
iajs-3432	15	10	delay	delay	NOUN
iajs-3432	15	11	differential	differential	ADJ
iajs-3432	15	12	equation	equation	NOUN
iajs-3432	15	13	(	(	PUNCT
iajs-3432	15	14	dde	dde	PROPN
iajs-3432	15	15	)	)	PUNCT
iajs-3432	15	16	solutions	solution	NOUN
iajs-3432	15	17	among	among	ADP
iajs-3432	15	18	these	these	DET
iajs-3432	15	19	investigations	investigation	NOUN
iajs-3432	15	20	[	[	X
iajs-3432	15	21	1	1	NUM
iajs-3432	15	22	-	-	SYM
iajs-3432	15	23	12	12	NUM
iajs-3432	15	24	]	]	PUNCT
iajs-3432	15	25	.	.	PUNCT
iajs-3432	16	1	this	this	DET
iajs-3432	16	2	discovery	discovery	NOUN
iajs-3432	16	3	has	have	AUX
iajs-3432	16	4	garnered	garner	VERB
iajs-3432	16	5	a	a	DET
iajs-3432	16	6	lot	lot	NOUN
iajs-3432	16	7	of	of	ADP
iajs-3432	16	8	interest	interest	NOUN
iajs-3432	16	9	since	since	SCONJ
iajs-3432	16	10	it	it	PRON
iajs-3432	16	11	has	have	VERB
iajs-3432	16	12	numerous	numerous	ADJ
iajs-3432	16	13	practical	practical	ADJ
iajs-3432	16	14	applications	application	NOUN
iajs-3432	16	15	in	in	ADP
iajs-3432	16	16	mathematical	mathematical	ADJ
iajs-3432	16	17	models	model	NOUN
iajs-3432	16	18	of	of	ADP
iajs-3432	16	19	biology	biology	NOUN
iajs-3432	16	20	,	,	PUNCT
iajs-3432	16	21	ecology	ecology	NOUN
iajs-3432	16	22	,	,	PUNCT
iajs-3432	16	23	and	and	CCONJ
iajs-3432	16	24	the	the	DET
iajs-3432	16	25	transmission	transmission	NOUN
iajs-3432	16	26	of	of	ADP
iajs-3432	16	27	various	various	ADJ
iajs-3432	16	28	infectious	infectious	ADJ
iajs-3432	16	29	diseases	disease	NOUN
iajs-3432	16	30	in	in	ADP
iajs-3432	16	31	people	people	NOUN
iajs-3432	16	32	,	,	PUNCT
iajs-3432	16	33	and	and	CCONJ
iajs-3432	16	34	other	other	ADJ
iajs-3432	16	35	areas	area	NOUN
iajs-3432	16	36	.	.	PUNCT
iajs-3432	17	1	the	the	DET
iajs-3432	17	2	reader	reader	NOUN
iajs-3432	17	3	can	can	AUX
iajs-3432	17	4	turn	turn	VERB
iajs-3432	17	5	to	to	ADP
iajs-3432	17	6	[	[	PUNCT
iajs-3432	17	7	13	13	NUM
iajs-3432	17	8	-	-	SYM
iajs-3432	17	9	22	22	NUM
iajs-3432	17	10	]	]	PUNCT
iajs-3432	17	11	and	and	CCONJ
iajs-3432	17	12	the	the	DET
iajs-3432	17	13	references	reference	NOUN
iajs-3432	17	14	therein	therein	ADV
iajs-3432	17	15	for	for	ADP
iajs-3432	17	16	more	more	ADJ
iajs-3432	17	17	details	detail	NOUN
iajs-3432	17	18	on	on	ADP
iajs-3432	17	19	this	this	DET
iajs-3432	17	20	study	study	NOUN
iajs-3432	17	21	.	.	PUNCT
iajs-3432	18	1	the	the	DET
iajs-3432	18	2	behavior	behavior	NOUN
iajs-3432	18	3	of	of	ADP
iajs-3432	18	4	the	the	DET
iajs-3432	18	5	solutions	solution	NOUN
iajs-3432	18	6	for	for	ADP
iajs-3432	18	7	dde	dde	PROPN
iajs-3432	18	8	has	have	AUX
iajs-3432	18	9	received	receive	VERB
iajs-3432	18	10	a	a	DET
iajs-3432	18	11	lot	lot	NOUN
iajs-3432	18	12	of	of	ADP
iajs-3432	18	13	attention	attention	NOUN
iajs-3432	18	14	in	in	ADP
iajs-3432	18	15	relation	relation	NOUN
iajs-3432	18	16	to	to	ADP
iajs-3432	18	17	the	the	DET
iajs-3432	18	18	study	study	NOUN
iajs-3432	18	19	of	of	ADP
iajs-3432	18	20	the	the	DET
iajs-3432	18	21	oscillations	oscillation	NOUN
iajs-3432	18	22	of	of	ADP
iajs-3432	18	23	the	the	DET
iajs-3432	18	24	analytical	analytical	ADJ
iajs-3432	18	25	solutions	solution	NOUN
iajs-3432	18	26	.	.	PUNCT
iajs-3432	19	1	to	to	ADP
iajs-3432	19	2	the	the	DET
iajs-3432	19	3	best	good	ADJ
iajs-3432	19	4	of	of	ADP
iajs-3432	19	5	our	our	PRON
iajs-3432	19	6	knowledge	knowledge	NOUN
iajs-3432	19	7	,	,	PUNCT
iajs-3432	19	8	there	there	PRON
iajs-3432	19	9	have	have	AUX
iajs-3432	19	10	been	be	AUX
iajs-3432	19	11	very	very	ADV
iajs-3432	19	12	few	few	ADJ
iajs-3432	19	13	studies	study	NOUN
iajs-3432	19	14	that	that	PRON
iajs-3432	19	15	address	address	VERB
iajs-3432	19	16	the	the	DET
iajs-3432	19	17	oscillations	oscillation	NOUN
iajs-3432	19	18	of	of	ADP
iajs-3432	19	19	nonlinear	nonlinear	ADJ
iajs-3432	19	20	dde	dde	PROPN
iajs-3432	19	21	solutions	solution	NOUN
iajs-3432	19	22	.	.	PUNCT
iajs-3432	20	1	in	in	ADP
iajs-3432	20	2	our	our	PRON
iajs-3432	20	3	work	work	NOUN
iajs-3432	20	4	,	,	PUNCT
iajs-3432	20	5	we	we	PRON
iajs-3432	20	6	focus	focus	VERB
iajs-3432	20	7	on	on	ADP
iajs-3432	20	8	this	this	DET
iajs-3432	20	9	subject	subject	NOUN
iajs-3432	20	10	.	.	PUNCT
iajs-3432	21	1	an	an	DET
iajs-3432	21	2	equation	equation	NOUN
iajs-3432	21	3	or	or	CCONJ
iajs-3432	21	4	a	a	DET
iajs-3432	21	5	system	system	NOUN
iajs-3432	21	6	of	of	ADP
iajs-3432	21	7	equations	equation	NOUN
iajs-3432	21	8	used	use	VERB
iajs-3432	21	9	to	to	PART
iajs-3432	21	10	describe	describe	VERB
iajs-3432	21	11	a	a	DET
iajs-3432	21	12	natural	natural	ADJ
iajs-3432	21	13	occurrence	occurrence	NOUN
iajs-3432	21	14	is	be	AUX
iajs-3432	21	15	known	know	VERB
iajs-3432	21	16	as	as	ADP
iajs-3432	21	17	a	a	DET
iajs-3432	21	18	mathematical	mathematical	ADJ
iajs-3432	21	19	model	model	NOUN
iajs-3432	21	20	.	.	PUNCT
iajs-3432	22	1	numerous	numerous	ADJ
iajs-3432	22	2	scholars	scholar	NOUN
iajs-3432	22	3	investigate	investigate	VERB
iajs-3432	22	4	how	how	SCONJ
iajs-3432	22	5	nonlinear	nonlinear	ADJ
iajs-3432	22	6	delay	delay	NOUN
iajs-3432	22	7	mathematical	mathematical	ADJ
iajs-3432	22	8	models	model	NOUN
iajs-3432	22	9	behave	behave	VERB
iajs-3432	22	10	qualitatively	qualitatively	ADV
iajs-3432	22	11	in	in	ADP
iajs-3432	22	12	single	single	ADJ
iajs-3432	22	13	species	specie	NOUN
iajs-3432	22	14	as	as	ADV
iajs-3432	22	15	well	well	ADV
iajs-3432	22	16	as	as	ADP
iajs-3432	22	17	in	in	ADP
iajs-3432	22	18	species	specie	NOUN
iajs-3432	22	19	that	that	PRON
iajs-3432	22	20	interact	interact	VERB
iajs-3432	22	21	.	.	PUNCT
iajs-3432	23	1	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
iajs-3432	23	2	https://orcid.org/0000-0002-1059-7790	https://orcid.org/0000-0002-1059-7790	NOUN
iajs-3432	23	3	mailto:%20iyman.sobaih1103a@sc.uobaghdad.edu.iq	mailto:%20iyman.sobaih1103a@sc.uobaghdad.edu.iq	VERB
iajs-3432	23	4	https://orcid.org/0000-0002-4684-786x	https://orcid.org/0000-0002-4684-786x	NOUN
iajs-3432	23	5	mailto:hussainam_math@scw.uobaghdad.edu.iq	mailto:hussainam_math@scw.uobaghdad.edu.iq	NOUN
iajs-3432	23	6	.	.	PUNCT
iajs-3432	24	1	ihjpas	ihjpas	PROPN
iajs-3432	24	2	.	.	PUNCT
iajs-3432	25	1	2024	2024	NUM
iajs-3432	25	2	,	,	PUNCT
iajs-3432	25	3	37	37	NUM
iajs-3432	25	4	(	(	PUNCT
iajs-3432	25	5	3	3	NUM
iajs-3432	25	6	)	)	PUNCT
iajs-3432	25	7	387	387	NUM
iajs-3432	25	8	there	there	PRON
iajs-3432	25	9	has	have	AUX
iajs-3432	25	10	been	be	AUX
iajs-3432	25	11	a	a	DET
iajs-3432	25	12	great	great	ADJ
iajs-3432	25	13	deal	deal	NOUN
iajs-3432	25	14	of	of	ADP
iajs-3432	25	15	research	research	NOUN
iajs-3432	25	16	done	do	VERB
iajs-3432	25	17	on	on	ADP
iajs-3432	25	18	the	the	DET
iajs-3432	25	19	qualitative	qualitative	ADJ
iajs-3432	25	20	analysis	analysis	NOUN
iajs-3432	25	21	of	of	ADP
iajs-3432	25	22	delay	delay	NOUN
iajs-3432	25	23	models	model	NOUN
iajs-3432	25	24	with	with	ADP
iajs-3432	25	25	constant	constant	ADJ
iajs-3432	25	26	coefficients	coefficient	NOUN
iajs-3432	25	27	(	(	PUNCT
iajs-3432	25	28	autonomous	autonomous	ADJ
iajs-3432	25	29	models	model	NOUN
iajs-3432	25	30	)	)	PUNCT
iajs-3432	25	31	.	.	PUNCT
iajs-3432	26	1	we	we	PRON
iajs-3432	26	2	are	be	AUX
iajs-3432	26	3	aware	aware	ADJ
iajs-3432	26	4	that	that	SCONJ
iajs-3432	26	5	a	a	DET
iajs-3432	26	6	variety	variety	NOUN
iajs-3432	26	7	of	of	ADP
iajs-3432	26	8	biological	biological	ADJ
iajs-3432	26	9	and	and	CCONJ
iajs-3432	26	10	ecological	ecological	ADJ
iajs-3432	26	11	dynamical	dynamical	ADJ
iajs-3432	26	12	systems	system	NOUN
iajs-3432	26	13	heavily	heavily	ADV
iajs-3432	26	14	depend	depend	VERB
iajs-3432	26	15	on	on	ADP
iajs-3432	26	16	the	the	DET
iajs-3432	26	17	environment	environment	NOUN
iajs-3432	26	18	.	.	PUNCT
iajs-3432	27	1	for	for	ADP
iajs-3432	27	2	instance	instance	NOUN
iajs-3432	27	3	,	,	PUNCT
iajs-3432	27	4	consider	consider	VERB
iajs-3432	27	5	the	the	DET
iajs-3432	27	6	physical	physical	ADJ
iajs-3432	27	7	environment	environment	NOUN
iajs-3432	27	8	's	's	PART
iajs-3432	27	9	elements	element	NOUN
iajs-3432	27	10	,	,	PUNCT
iajs-3432	27	11	like	like	ADP
iajs-3432	27	12	temperature	temperature	NOUN
iajs-3432	27	13	and	and	CCONJ
iajs-3432	27	14	humidity	humidity	NOUN
iajs-3432	27	15	,	,	PUNCT
iajs-3432	27	16	as	as	ADV
iajs-3432	27	17	well	well	ADV
iajs-3432	27	18	as	as	ADP
iajs-3432	27	19	the	the	DET
iajs-3432	27	20	accessibility	accessibility	NOUN
iajs-3432	27	21	of	of	ADP
iajs-3432	27	22	resources	resource	NOUN
iajs-3432	27	23	like	like	ADP
iajs-3432	27	24	food	food	NOUN
iajs-3432	27	25	,	,	PUNCT
iajs-3432	27	26	water	water	NOUN
iajs-3432	27	27	,	,	PUNCT
iajs-3432	27	28	and	and	CCONJ
iajs-3432	27	29	other	other	ADJ
iajs-3432	27	30	essentials	essential	NOUN
iajs-3432	27	31	,	,	PUNCT
iajs-3432	27	32	typically	typically	ADV
iajs-3432	27	33	change	change	VERB
iajs-3432	27	34	with	with	ADP
iajs-3432	27	35	time	time	NOUN
iajs-3432	27	36	on	on	ADP
iajs-3432	27	37	a	a	DET
iajs-3432	27	38	seasonal	seasonal	ADJ
iajs-3432	27	39	or	or	CCONJ
iajs-3432	27	40	daily	daily	ADJ
iajs-3432	27	41	basis	basis	NOUN
iajs-3432	27	42	.	.	PUNCT
iajs-3432	28	1	therefore	therefore	ADV
iajs-3432	28	2	,	,	PUNCT
iajs-3432	28	3	nonautonomous	nonautonomous	ADJ
iajs-3432	28	4	systems	system	NOUN
iajs-3432	28	5	would	would	AUX
iajs-3432	28	6	have	have	VERB
iajs-3432	28	7	more	more	ADV
iajs-3432	28	8	accurate	accurate	ADJ
iajs-3432	28	9	representations	representation	NOUN
iajs-3432	28	10	[	[	X
iajs-3432	28	11	2,20	2,20	NOUN
iajs-3432	28	12	]	]	X
iajs-3432	28	13	.	.	PUNCT
iajs-3432	29	1	studying	study	VERB
iajs-3432	29	2	the	the	DET
iajs-3432	29	3	oscillation	oscillation	NOUN
iajs-3432	29	4	and	and	CCONJ
iajs-3432	29	5	nonoscillation	nonoscillation	NOUN
iajs-3432	29	6	of	of	ADP
iajs-3432	29	7	a	a	DET
iajs-3432	29	8	certain	certain	ADJ
iajs-3432	29	9	kind	kind	NOUN
iajs-3432	29	10	of	of	ADP
iajs-3432	29	11	nonautonomous	nonautonomous	ADJ
iajs-3432	29	12	hematopoiesis	hematopoiesis	NOUN
iajs-3432	29	13	delay	delay	NOUN
iajs-3432	29	14	model	model	NOUN
iajs-3432	29	15	in	in	ADP
iajs-3432	29	16	biology	biology	NOUN
iajs-3432	29	17	is	be	AUX
iajs-3432	29	18	one	one	NUM
iajs-3432	29	19	of	of	ADP
iajs-3432	29	20	the	the	DET
iajs-3432	29	21	goals	goal	NOUN
iajs-3432	29	22	of	of	ADP
iajs-3432	29	23	our	our	PRON
iajs-3432	29	24	work	work	NOUN
iajs-3432	29	25	.	.	PUNCT
iajs-3432	30	1	many	many	ADJ
iajs-3432	30	2	authors	author	NOUN
iajs-3432	30	3	looked	look	VERB
iajs-3432	30	4	for	for	ADP
iajs-3432	30	5	sufficient	sufficient	ADJ
iajs-3432	30	6	conditions	condition	NOUN
iajs-3432	30	7	to	to	PART
iajs-3432	30	8	ensure	ensure	VERB
iajs-3432	30	9	oscillatory	oscillatory	ADJ
iajs-3432	30	10	properties	property	NOUN
iajs-3432	30	11	for	for	ADP
iajs-3432	30	12	different	different	ADJ
iajs-3432	30	13	differential	differential	ADJ
iajs-3432	30	14	equations	equation	NOUN
iajs-3432	30	15	.	.	PUNCT
iajs-3432	31	1	hematopoiesis	hematopoiesis	NOUN
iajs-3432	31	2	refers	refer	VERB
iajs-3432	31	3	to	to	ADP
iajs-3432	31	4	the	the	DET
iajs-3432	31	5	process	process	NOUN
iajs-3432	31	6	of	of	ADP
iajs-3432	31	7	producing	produce	VERB
iajs-3432	31	8	blood	blood	NOUN
iajs-3432	31	9	cells	cell	NOUN
iajs-3432	31	10	.	.	PUNCT
iajs-3432	32	1	in	in	ADP
iajs-3432	32	2	1978	1978	NUM
iajs-3432	32	3	,	,	PUNCT
iajs-3432	32	4	mackey	mackey	PROPN
iajs-3432	32	5	and	and	CCONJ
iajs-3432	32	6	glass	glass	NOUN
iajs-3432	32	7	proposed	propose	VERB
iajs-3432	32	8	the	the	DET
iajs-3432	32	9	first	first	ADJ
iajs-3432	32	10	mathematical	mathematical	ADJ
iajs-3432	32	11	models	model	NOUN
iajs-3432	32	12	of	of	ADP
iajs-3432	32	13	hematopoiesis	hematopoiesis	NOUN
iajs-3432	32	14	dynamics	dynamic	NOUN
iajs-3432	32	15	[	[	X
iajs-3432	32	16	23	23	NUM
iajs-3432	32	17	]	]	PUNCT
iajs-3432	32	18	.	.	PUNCT
iajs-3432	33	1	in	in	ADP
iajs-3432	33	2	saker	saker	PROPN
iajs-3432	33	3	,	,	PUNCT
iajs-3432	33	4	s.	s.	PROPN
iajs-3432	33	5	h.	h.	PROPN
iajs-3432	34	1	[	[	X
iajs-3432	34	2	24	24	NUM
iajs-3432	34	3	]	]	PUNCT
iajs-3432	34	4	,	,	PUNCT
iajs-3432	34	5	consider	consider	VERB
iajs-3432	34	6	the	the	DET
iajs-3432	34	7	following	follow	VERB
iajs-3432	34	8	equation	equation	NOUN
iajs-3432	34	9	of	of	ADP
iajs-3432	34	10	the	the	DET
iajs-3432	34	11	hematopoiesis	hematopoiesis	NOUN
iajs-3432	34	12	model	model	NOUN
iajs-3432	34	13	with	with	ADP
iajs-3432	34	14	positive	positive	ADJ
iajs-3432	34	15	and	and	CCONJ
iajs-3432	34	16	negative	negative	ADJ
iajs-3432	34	17	coefficients	coefficient	NOUN
iajs-3432	34	18	:	:	PUNCT
iajs-3432	34	19	ℋ′(𝑡	ℋ′(𝑡	X
iajs-3432	34	20	)	)	PUNCT
iajs-3432	34	21	=	=	SYM
iajs-3432	34	22	𝛽(𝑡	𝛽(𝑡	PROPN
iajs-3432	34	23	)	)	PUNCT
iajs-3432	34	24	1	1	NUM
iajs-3432	35	1	+	+	CCONJ
iajs-3432	35	2	ℋ𝑚(𝑡	ℋ𝑚(𝑡	NUM
iajs-3432	35	3	−	−	PRON
iajs-3432	35	4	𝜏	𝜏	NOUN
iajs-3432	35	5	)	)	PUNCT
iajs-3432	35	6	−	−	NOUN
iajs-3432	35	7	𝛿(𝑡)ℋ(𝑡	𝛿(𝑡)ℋ(𝑡	NOUN
iajs-3432	35	8	)	)	PUNCT
iajs-3432	35	9	,	,	PUNCT
iajs-3432	35	10	𝑡	𝑡	PROPN
iajs-3432	35	11	≥	≥	NOUN
iajs-3432	35	12	0	0	NUM
iajs-3432	35	13	.	.	PUNCT
iajs-3432	36	1	(	(	PUNCT
iajs-3432	36	2	1	1	X
iajs-3432	36	3	)	)	PUNCT
iajs-3432	36	4	where	where	SCONJ
iajs-3432	36	5	𝛽(𝑡	𝛽(𝑡	PROPN
iajs-3432	36	6	)	)	PUNCT
iajs-3432	36	7	,	,	PUNCT
iajs-3432	36	8	𝛿(𝑡	𝛿(𝑡	PROPN
iajs-3432	36	9	)	)	PUNCT
iajs-3432	36	10	∈	∈	PROPN
iajs-3432	36	11	∁([0	∁([0	NOUN
iajs-3432	36	12	,	,	PUNCT
iajs-3432	36	13	∞	∞	PROPN
iajs-3432	36	14	)	)	PUNCT
iajs-3432	36	15	,	,	PUNCT
iajs-3432	36	16	𝑅+	𝑅+	PROPN
iajs-3432	36	17	)	)	PUNCT
iajs-3432	36	18	,	,	PUNCT
iajs-3432	36	19	𝜏	𝜏	X
iajs-3432	36	20	∈	∈	PROPN
iajs-3432	37	1	[	[	X
iajs-3432	37	2	0	0	NUM
iajs-3432	37	3	,	,	PUNCT
iajs-3432	37	4	∞	∞	PROPN
iajs-3432	37	5	)	)	PUNCT
iajs-3432	37	6	,	,	PUNCT
iajs-3432	37	7	𝑚	𝑚	PROPN
iajs-3432	37	8	∈	∈	PROPN
iajs-3432	37	9	𝑁.	𝑁.	PROPN
iajs-3432	37	10	equation	equation	NOUN
iajs-3432	37	11	(	(	PUNCT
iajs-3432	37	12	1	1	X
iajs-3432	37	13	)	)	PUNCT
iajs-3432	37	14	has	have	VERB
iajs-3432	37	15	a	a	DET
iajs-3432	37	16	unique	unique	ADJ
iajs-3432	37	17	positive	positive	ADJ
iajs-3432	37	18	equilibrium	equilibrium	NOUN
iajs-3432	37	19	point	point	NOUN
iajs-3432	37	20	𝐾	𝐾	PROPN
iajs-3432	37	21	,	,	PUNCT
iajs-3432	37	22	and	and	CCONJ
iajs-3432	37	23	satisfies	satisfy	VERB
iajs-3432	37	24	the	the	DET
iajs-3432	37	25	equation	equation	NOUN
iajs-3432	37	26	𝛽	𝛽	NOUN
iajs-3432	37	27	1	1	NUM
iajs-3432	37	28	+	+	CCONJ
iajs-3432	37	29	𝐾𝑚	𝐾𝑚	PROPN
iajs-3432	37	30	=	=	PUNCT
iajs-3432	37	31	𝛿𝐾.	𝛿𝐾.	NOUN
iajs-3432	37	32	(	(	PUNCT
iajs-3432	37	33	2	2	X
iajs-3432	37	34	)	)	PUNCT
iajs-3432	37	35	since	since	SCONJ
iajs-3432	37	36	differential	differential	ADJ
iajs-3432	37	37	delay	delay	NOUN
iajs-3432	37	38	equations	equation	NOUN
iajs-3432	37	39	are	be	AUX
iajs-3432	37	40	challenging	challenge	VERB
iajs-3432	37	41	to	to	PART
iajs-3432	37	42	control	control	VERB
iajs-3432	37	43	analytically	analytically	ADV
iajs-3432	37	44	,	,	PUNCT
iajs-3432	37	45	numerous	numerous	ADJ
iajs-3432	37	46	studies	study	NOUN
iajs-3432	37	47	have	have	AUX
iajs-3432	37	48	looked	look	VERB
iajs-3432	37	49	at	at	ADP
iajs-3432	37	50	the	the	DET
iajs-3432	37	51	models	model	NOUN
iajs-3432	37	52	as	as	ADP
iajs-3432	37	53	difference	difference	NOUN
iajs-3432	37	54	equations	equation	NOUN
iajs-3432	37	55	.	.	PUNCT
iajs-3432	38	1	so	so	ADV
iajs-3432	38	2	,	,	PUNCT
iajs-3432	38	3	there	there	PRON
iajs-3432	38	4	are	be	VERB
iajs-3432	38	5	a	a	DET
iajs-3432	38	6	number	number	NOUN
iajs-3432	38	7	of	of	ADP
iajs-3432	38	8	analytical	analytical	ADJ
iajs-3432	38	9	results	result	NOUN
iajs-3432	38	10	concerning	concern	VERB
iajs-3432	38	11	the	the	DET
iajs-3432	38	12	oscillation	oscillation	NOUN
iajs-3432	38	13	,	,	PUNCT
iajs-3432	38	14	global	global	ADJ
iajs-3432	38	15	attractivity	attractivity	NOUN
iajs-3432	38	16	,	,	PUNCT
iajs-3432	38	17	and	and	CCONJ
iajs-3432	38	18	periodicity	periodicity	NOUN
iajs-3432	38	19	of	of	ADP
iajs-3432	38	20	equation	equation	NOUN
iajs-3432	38	21	(	(	PUNCT
iajs-3432	38	22	1	1	X
iajs-3432	38	23	)	)	PUNCT
iajs-3432	39	1	[	[	X
iajs-3432	39	2	20,23	20,23	NUM
iajs-3432	39	3	]	]	PUNCT
iajs-3432	39	4	.	.	PUNCT
iajs-3432	40	1	the	the	DET
iajs-3432	40	2	corresponding	corresponding	ADJ
iajs-3432	40	3	nonlinear	nonlinear	ADJ
iajs-3432	40	4	first	first	ADJ
iajs-3432	40	5	order	order	NOUN
iajs-3432	40	6	delay	delay	NOUN
iajs-3432	40	7	difference	difference	NOUN
iajs-3432	40	8	equation	equation	NOUN
iajs-3432	40	9	with	with	ADP
iajs-3432	40	10	positive	positive	ADJ
iajs-3432	40	11	and	and	CCONJ
iajs-3432	40	12	negative	negative	ADJ
iajs-3432	40	13	coefficients	coefficient	NOUN
iajs-3432	40	14	of	of	ADP
iajs-3432	40	15	equation	equation	NOUN
iajs-3432	40	16	(	(	PUNCT
iajs-3432	40	17	1	1	NUM
iajs-3432	40	18	)	)	PUNCT
iajs-3432	40	19	in	in	ADP
iajs-3432	40	20	the	the	DET
iajs-3432	40	21	discrete	discrete	ADJ
iajs-3432	40	22	hematopoiesis	hematopoiesis	NOUN
iajs-3432	40	23	model	model	NOUN
iajs-3432	40	24	is	be	AUX
iajs-3432	40	25	:	:	PUNCT
iajs-3432	40	26	∆𝐻𝑛	∆𝐻𝑛	ADJ
iajs-3432	40	27	+	+	NUM
iajs-3432	40	28	𝛿𝑛𝐻𝑛	𝛿𝑛𝐻𝑛	NOUN
iajs-3432	40	29	−	−	NOUN
iajs-3432	40	30	𝛽𝑛𝐺(𝐻𝑛−𝑙	𝛽𝑛𝐺(𝐻𝑛−𝑙	NUM
iajs-3432	40	31	)	)	PUNCT
iajs-3432	40	32	=	=	SYM
iajs-3432	41	1	𝑓𝑛	𝑓𝑛	NOUN
iajs-3432	41	2	.	.	PUNCT
iajs-3432	42	1	(	(	PUNCT
iajs-3432	42	2	3	3	X
iajs-3432	42	3	)	)	PUNCT
iajs-3432	42	4	where𝛿𝑛	where𝛿𝑛	NOUN
iajs-3432	42	5	,	,	PUNCT
iajs-3432	42	6	𝛽𝑛	𝛽𝑛	PROPN
iajs-3432	42	7	and	and	CCONJ
iajs-3432	42	8	𝑓𝑛	𝑓𝑛	PRON
iajs-3432	42	9	are	be	AUX
iajs-3432	42	10	infinite	infinite	ADJ
iajs-3432	42	11	sequences	sequence	NOUN
iajs-3432	42	12	of	of	ADP
iajs-3432	42	13	real	real	ADJ
iajs-3432	42	14	numbers	number	NOUN
iajs-3432	42	15	and	and	CCONJ
iajs-3432	42	16	∆	∆	PROPN
iajs-3432	42	17	is	be	AUX
iajs-3432	42	18	the	the	DET
iajs-3432	42	19	forward	forward	ADJ
iajs-3432	42	20	difference	difference	NOUN
iajs-3432	42	21	operator	operator	NOUN
iajs-3432	42	22	.	.	PUNCT
iajs-3432	43	1	and	and	CCONJ
iajs-3432	43	2	𝐺(𝐻𝑛	𝐺(𝐻𝑛	NUM
iajs-3432	43	3	)	)	PUNCT
iajs-3432	44	1	=	=	NOUN
iajs-3432	44	2	1	1	NUM
iajs-3432	44	3	1+𝐻𝑛	1+𝐻𝑛	NUM
iajs-3432	44	4	𝑚	𝑚	PROPN
iajs-3432	44	5	∈	∈	PROPN
iajs-3432	44	6	(	(	PUNCT
iajs-3432	44	7	𝑅	𝑅	PROPN
iajs-3432	44	8	,	,	PUNCT
iajs-3432	44	9	𝑅	𝑅	PROPN
iajs-3432	44	10	)	)	PUNCT
iajs-3432	44	11	is	be	AUX
iajs-3432	44	12	a	a	DET
iajs-3432	44	13	flux	flux	NOUN
iajs-3432	44	14	function	function	NOUN
iajs-3432	44	15	that	that	PRON
iajs-3432	44	16	depends	depend	VERB
iajs-3432	44	17	on	on	ADP
iajs-3432	44	18	the	the	DET
iajs-3432	44	19	size	size	NOUN
iajs-3432	44	20	of	of	ADP
iajs-3432	44	21	cells	cell	NOUN
iajs-3432	44	22	𝐻𝑛	𝐻𝑛	ADJ
iajs-3432	44	23	and	and	CCONJ
iajs-3432	44	24	𝐻𝑛−𝑙at	𝐻𝑛−𝑙at	PROPN
iajs-3432	44	25	times	time	NOUN
iajs-3432	44	26	n	n	PROPN
iajs-3432	44	27	and	and	CCONJ
iajs-3432	44	28	n	n	CCONJ
iajs-3432	44	29	–	–	PUNCT
iajs-3432	44	30	l	l	NOUN
iajs-3432	44	31	,	,	PUNCT
iajs-3432	44	32	respectively	respectively	ADV
iajs-3432	44	33	,	,	PUNCT
iajs-3432	44	34	and	and	CCONJ
iajs-3432	44	35	l	l	NOUN
iajs-3432	44	36	is	be	AUX
iajs-3432	44	37	the	the	DET
iajs-3432	44	38	time	time	NOUN
iajs-3432	44	39	of	of	ADP
iajs-3432	44	40	maturation	maturation	NOUN
iajs-3432	44	41	,	,	PUNCT
iajs-3432	44	42	such	such	ADJ
iajs-3432	44	43	as	as	ADP
iajs-3432	44	44	that	that	DET
iajs-3432	44	45	𝐻𝑛𝐺(𝐻𝑛	𝐻𝑛𝐺(𝐻𝑛	PROPN
iajs-3432	44	46	)	)	PUNCT
iajs-3432	44	47	>	>	X
iajs-3432	45	1	0	0	X
iajs-3432	45	2	.	.	PUNCT
iajs-3432	46	1	there	there	PRON
iajs-3432	46	2	are	be	VERB
iajs-3432	46	3	some	some	DET
iajs-3432	46	4	studies	study	NOUN
iajs-3432	46	5	on	on	ADP
iajs-3432	46	6	the	the	DET
iajs-3432	46	7	discrete	discrete	ADJ
iajs-3432	46	8	hematopoiesis	hematopoiesis	NOUN
iajs-3432	46	9	model	model	NOUN
iajs-3432	46	10	,	,	PUNCT
iajs-3432	46	11	for	for	ADP
iajs-3432	46	12	instance	instance	NOUN
iajs-3432	46	13	.	.	PUNCT
iajs-3432	47	1	wang	wang	PROPN
iajs-3432	47	2	et	et	PROPN
iajs-3432	47	3	.	.	PUNCT
iajs-3432	48	1	al	al	PROPN
iajs-3432	48	2	.	.	PROPN
iajs-3432	49	1	(	(	PUNCT
iajs-3432	49	2	2013	2013	NUM
iajs-3432	49	3	)	)	PUNCT
iajs-3432	50	1	[	[	X
iajs-3432	50	2	23	23	NUM
iajs-3432	50	3	]	]	X
iajs-3432	50	4	:	:	PUNCT
iajs-3432	50	5	with	with	ADP
iajs-3432	50	6	the	the	DET
iajs-3432	50	7	assistance	assistance	NOUN
iajs-3432	50	8	of	of	ADP
iajs-3432	50	9	two	two	NUM
iajs-3432	50	10	𝜃-methods	𝜃-method	NOUN
iajs-3432	50	11	,	,	PUNCT
iajs-3432	50	12	it	it	PRON
iajs-3432	50	13	was	be	AUX
iajs-3432	50	14	possible	possible	ADJ
iajs-3432	50	15	to	to	PART
iajs-3432	50	16	discuss	discuss	VERB
iajs-3432	50	17	the	the	DET
iajs-3432	50	18	conditions	condition	NOUN
iajs-3432	50	19	under	under	ADP
iajs-3432	50	20	which	which	PRON
iajs-3432	50	21	the	the	DET
iajs-3432	50	22	numerical	numerical	ADJ
iajs-3432	50	23	solutions	solution	NOUN
iajs-3432	50	24	fluctuate	fluctuate	NOUN
iajs-3432	50	25	for	for	ADP
iajs-3432	50	26	the	the	DET
iajs-3432	50	27	nonlinear	nonlinear	ADJ
iajs-3432	50	28	delay	delay	NOUN
iajs-3432	50	29	differential	differential	ADJ
iajs-3432	50	30	equations	equation	NOUN
iajs-3432	50	31	in	in	ADP
iajs-3432	50	32	the	the	DET
iajs-3432	50	33	hematopoiesis	hematopoiesis	NOUN
iajs-3432	50	34	model	model	NOUN
iajs-3432	50	35	.	.	PUNCT
iajs-3432	51	1	additionally	additionally	ADV
iajs-3432	51	2	,	,	PUNCT
iajs-3432	51	3	it	it	PRON
iajs-3432	51	4	has	have	AUX
iajs-3432	51	5	been	be	AUX
iajs-3432	51	6	established	establish	VERB
iajs-3432	51	7	that	that	SCONJ
iajs-3432	51	8	every	every	DET
iajs-3432	51	9	non	non	ADJ
iajs-3432	51	10	-	-	ADJ
iajs-3432	51	11	oscillatory	oscillatory	ADJ
iajs-3432	51	12	numerical	numerical	ADJ
iajs-3432	51	13	solution	solution	NOUN
iajs-3432	51	14	tends	tend	VERB
iajs-3432	51	15	to	to	ADP
iajs-3432	51	16	an	an	DET
iajs-3432	51	17	equilibrium	equilibrium	NOUN
iajs-3432	51	18	point	point	NOUN
iajs-3432	51	19	of	of	ADP
iajs-3432	51	20	(	(	PUNCT
iajs-3432	51	21	3	3	NUM
iajs-3432	51	22	)	)	PUNCT
iajs-3432	51	23	.	.	PUNCT
iajs-3432	52	1	[	[	X
iajs-3432	52	2	25	25	NUM
iajs-3432	52	3	]	]	PUNCT
iajs-3432	52	4	:	:	PUNCT
iajs-3432	52	5	established	establish	VERB
iajs-3432	52	6	sufficient	sufficient	ADJ
iajs-3432	52	7	conditions	condition	NOUN
iajs-3432	52	8	for	for	ADP
iajs-3432	52	9	the	the	DET
iajs-3432	52	10	existence	existence	NOUN
iajs-3432	52	11	of	of	ADP
iajs-3432	52	12	at	at	ADV
iajs-3432	52	13	least	least	ADV
iajs-3432	52	14	three	three	NUM
iajs-3432	52	15	positive	positive	ADJ
iajs-3432	52	16	t	t	NOUN
iajs-3432	52	17	-	-	PUNCT
iajs-3432	52	18	periodic	periodic	NOUN
iajs-3432	52	19	solutions	solution	NOUN
iajs-3432	52	20	for	for	ADP
iajs-3432	52	21	a	a	DET
iajs-3432	52	22	discrete	discrete	ADJ
iajs-3432	52	23	delay	delay	NOUN
iajs-3432	52	24	hematopoiesis	hematopoiesis	NOUN
iajs-3432	52	25	model	model	NOUN
iajs-3432	52	26	.	.	PUNCT
iajs-3432	53	1	wei	wei	PROPN
iajs-3432	53	2	li	li	PROPN
iajs-3432	53	3	and	and	CCONJ
iajs-3432	53	4	xianyi	xianyi	PROPN
iajs-3432	53	5	li	li	PROPN
iajs-3432	53	6	(	(	PUNCT
iajs-3432	53	7	2018	2018	NUM
iajs-3432	53	8	)	)	PUNCT
iajs-3432	54	1	[	[	X
iajs-3432	54	2	15	15	NUM
iajs-3432	54	3	]	]	X
iajs-3432	54	4	:	:	PUNCT
iajs-3432	54	5	they	they	PRON
iajs-3432	54	6	derived	derive	VERB
iajs-3432	54	7	a	a	DET
iajs-3432	54	8	semi	semi	ADJ
iajs-3432	54	9	-	-	ADJ
iajs-3432	54	10	discrete	discrete	ADJ
iajs-3432	54	11	system	system	NOUN
iajs-3432	54	12	for	for	ADP
iajs-3432	54	13	a	a	DET
iajs-3432	54	14	nonlinear	nonlinear	ADJ
iajs-3432	54	15	model	model	NOUN
iajs-3432	54	16	of	of	ADP
iajs-3432	54	17	blood	blood	NOUN
iajs-3432	54	18	cell	cell	NOUN
iajs-3432	54	19	production	production	NOUN
iajs-3432	54	20	.	.	PUNCT
iajs-3432	55	1	in	in	ADP
iajs-3432	55	2	[	[	X
iajs-3432	55	3	15	15	NUM
iajs-3432	55	4	]	]	PUNCT
iajs-3432	55	5	,	,	PUNCT
iajs-3432	55	6	the	the	DET
iajs-3432	55	7	authors	author	NOUN
iajs-3432	55	8	discovered	discover	VERB
iajs-3432	55	9	a	a	DET
iajs-3432	55	10	few	few	ADJ
iajs-3432	55	11	prerequisites	prerequisite	NOUN
iajs-3432	55	12	for	for	ADP
iajs-3432	55	13	the	the	DET
iajs-3432	55	14	oscillation	oscillation	NOUN
iajs-3432	55	15	of	of	ADP
iajs-3432	55	16	all	all	DET
iajs-3432	55	17	∆𝑌(𝑛	∆𝑌(𝑛	NUM
iajs-3432	55	18	)	)	PUNCT
iajs-3432	56	1	+	+	NUM
iajs-3432	56	2	𝑝(𝑛)𝑌(𝜏(𝑛	𝑝(𝑛)𝑌(𝜏(𝑛	NOUN
iajs-3432	56	3	)	)	PUNCT
iajs-3432	56	4	)	)	PUNCT
iajs-3432	57	1	=	=	SYM
iajs-3432	57	2	0	0	NUM
iajs-3432	57	3	solutions	solution	NOUN
iajs-3432	57	4	of	of	ADP
iajs-3432	57	5	the	the	DET
iajs-3432	57	6	linear	linear	ADJ
iajs-3432	57	7	difference	difference	NOUN
iajs-3432	57	8	equation	equation	NOUN
iajs-3432	57	9	with	with	ADP
iajs-3432	57	10	varying	vary	VERB
iajs-3432	57	11	delays	delay	NOUN
iajs-3432	57	12	.	.	PUNCT
iajs-3432	58	1	every	every	DET
iajs-3432	58	2	solution	solution	NOUN
iajs-3432	58	3	of	of	ADP
iajs-3432	58	4	first	first	ADJ
iajs-3432	58	5	order	order	NOUN
iajs-3432	58	6	linear	linear	ADJ
iajs-3432	58	7	difference	difference	NOUN
iajs-3432	58	8	equations	equation	NOUN
iajs-3432	58	9	with	with	ADP
iajs-3432	58	10	positive	positive	ADJ
iajs-3432	58	11	and	and	CCONJ
iajs-3432	58	12	negative	negative	ADJ
iajs-3432	58	13	coefficients	coefficient	NOUN
iajs-3432	58	14	of	of	ADP
iajs-3432	58	15	the	the	DET
iajs-3432	58	16	form	form	NOUN
iajs-3432	58	17	is	be	AUX
iajs-3432	58	18	given	give	VERB
iajs-3432	58	19	sufficient	sufficient	ADJ
iajs-3432	58	20	conditions	condition	NOUN
iajs-3432	58	21	to	to	PART
iajs-3432	58	22	oscillate	oscillate	VERB
iajs-3432	58	23	[	[	X
iajs-3432	58	24	26	26	NUM
iajs-3432	58	25	]	]	PUNCT
iajs-3432	58	26	.	.	PUNCT
iajs-3432	59	1	ψ𝑛+1	ψ𝑛+1	X
iajs-3432	60	1	−	−	NOUN
iajs-3432	60	2	ψ𝑛	ψ𝑛	PROPN
iajs-3432	61	1	+	+	NUM
iajs-3432	61	2	ℛψ𝑛−𝑘	ℛψ𝑛−𝑘	NOUN
iajs-3432	61	3	−	−	PROPN
iajs-3432	61	4	𝑄ψ𝑛−𝑙	𝑄ψ𝑛−𝑙	NOUN
iajs-3432	61	5	=	=	SYM
iajs-3432	61	6	0	0	NUM
iajs-3432	61	7	,	,	PUNCT
iajs-3432	61	8	𝑛	𝑛	DET
iajs-3432	61	9	∈	∈	NOUN
iajs-3432	61	10	𝑁.	𝑁.	PROPN
iajs-3432	61	11	(	(	PUNCT
iajs-3432	61	12	4	4	NUM
iajs-3432	61	13	)	)	PUNCT
iajs-3432	61	14	every	every	DET
iajs-3432	61	15	solution	solution	NOUN
iajs-3432	61	16	to	to	ADP
iajs-3432	61	17	equation	equation	NOUN
iajs-3432	61	18	(	(	PUNCT
iajs-3432	61	19	4	4	X
iajs-3432	61	20	)	)	PUNCT
iajs-3432	61	21	oscillates	oscillate	NOUN
iajs-3432	61	22	if	if	SCONJ
iajs-3432	61	23	𝑄(𝑘	𝑄(𝑘	PROPN
iajs-3432	61	24	−	−	PROPN
iajs-3432	61	25	𝑙	𝑙	X
iajs-3432	61	26	)	)	PUNCT
iajs-3432	61	27	<	<	X
iajs-3432	61	28	1	1	NUM
iajs-3432	61	29	and	and	CCONJ
iajs-3432	61	30	(	(	PUNCT
iajs-3432	61	31	ℛ−	ℛ−	NOUN
iajs-3432	61	32	𝑄)(𝑘	𝑄)(𝑘	NOUN
iajs-3432	61	33	+	+	CCONJ
iajs-3432	61	34	1)𝑘+1	1)𝑘+1	NUM
iajs-3432	61	35	𝑘𝑘	𝑘𝑘	X
iajs-3432	61	36	>	>	X
iajs-3432	61	37	1	1	X
iajs-3432	61	38	.	.	PUNCT
iajs-3432	62	1	the	the	DET
iajs-3432	62	2	authors	author	NOUN
iajs-3432	62	3	discovered	discover	VERB
iajs-3432	62	4	sufficient	sufficient	ADJ
iajs-3432	62	5	conditions	condition	NOUN
iajs-3432	62	6	for	for	ADP
iajs-3432	62	7	the	the	DET
iajs-3432	62	8	oscillation	oscillation	NOUN
iajs-3432	62	9	of	of	ADP
iajs-3432	62	10	each	each	DET
iajs-3432	62	11	solution	solution	NOUN
iajs-3432	62	12	of	of	ADP
iajs-3432	62	13	first	first	ADJ
iajs-3432	62	14	-	-	PUNCT
iajs-3432	62	15	order	order	NOUN
iajs-3432	62	16	linear	linear	ADJ
iajs-3432	62	17	difference	difference	NOUN
iajs-3432	62	18	equations	equation	NOUN
iajs-3432	62	19	with	with	ADP
iajs-3432	62	20	multiple	multiple	ADJ
iajs-3432	62	21	positive	positive	ADJ
iajs-3432	62	22	and	and	CCONJ
iajs-3432	62	23	negative	negative	ADJ
iajs-3432	62	24	coefficients	coefficient	NOUN
iajs-3432	62	25	in	in	ADP
iajs-3432	62	26	[	[	X
iajs-3432	62	27	27	27	NUM
iajs-3432	62	28	]	]	PUNCT
iajs-3432	62	29	,	,	PUNCT
iajs-3432	62	30	extending	extend	VERB
iajs-3432	62	31	the	the	DET
iajs-3432	62	32	findings	finding	NOUN
iajs-3432	62	33	from	from	ADP
iajs-3432	62	34	[	[	X
iajs-3432	62	35	26	26	NUM
iajs-3432	62	36	]	]	PUNCT
iajs-3432	62	37	.	.	PUNCT
iajs-3432	63	1	ihjpas	ihjpas	PROPN
iajs-3432	63	2	.	.	PUNCT
iajs-3432	64	1	2024	2024	NUM
iajs-3432	64	2	,	,	PUNCT
iajs-3432	64	3	37	37	NUM
iajs-3432	64	4	(	(	PUNCT
iajs-3432	64	5	3	3	NUM
iajs-3432	64	6	)	)	PUNCT
iajs-3432	64	7	388	388	NUM
iajs-3432	64	8	ψ𝑛+1	ψ𝑛+1	NUM
iajs-3432	64	9	−	−	NOUN
iajs-3432	65	1	ψ𝑛	ψ𝑛	PRON
iajs-3432	65	2	+	+	CCONJ
iajs-3432	65	3	∑	∑	ADJ
iajs-3432	65	4	ℛ𝑖ψ𝑛−𝑘𝑖	ℛ𝑖ψ𝑛−𝑘𝑖	ADJ
iajs-3432	65	5	−	−	NOUN
iajs-3432	65	6	𝑄𝑖ψ𝑛−𝑙𝑖	𝑄𝑖ψ𝑛−𝑙𝑖	NOUN
iajs-3432	65	7	𝑚	𝑚	X
iajs-3432	65	8	𝑖=1	𝑖=1	PUNCT
iajs-3432	65	9	=	=	SYM
iajs-3432	65	10	0	0	NUM
iajs-3432	65	11	,	,	PUNCT
iajs-3432	65	12	𝑛	𝑛	DET
iajs-3432	65	13	∈	∈	NOUN
iajs-3432	65	14	𝑁0	𝑁0	VERB
iajs-3432	65	15	.	.	PUNCT
iajs-3432	66	1	(	(	PUNCT
iajs-3432	66	2	5	5	X
iajs-3432	66	3	)	)	PUNCT
iajs-3432	66	4	it	it	PRON
iajs-3432	66	5	is	be	AUX
iajs-3432	66	6	proved	prove	VERB
iajs-3432	66	7	that	that	SCONJ
iajs-3432	66	8	if	if	SCONJ
iajs-3432	66	9	∑	∑	PART
iajs-3432	66	10	𝑄𝑖(𝑘𝑖	𝑄𝑖(𝑘𝑖	VERB
iajs-3432	66	11	−	−	PROPN
iajs-3432	66	12	𝑙𝑖	𝑙𝑖	NOUN
iajs-3432	66	13	)	)	PUNCT
iajs-3432	66	14	𝑚	𝑚	ADP
iajs-3432	66	15	𝑖=0	𝑖=0	PROPN
iajs-3432	66	16	<	<	X
iajs-3432	66	17	1	1	NUM
iajs-3432	66	18	and	and	CCONJ
iajs-3432	66	19	∑	∑	PROPN
iajs-3432	66	20	(	(	PUNCT
iajs-3432	66	21	ℛ𝑖−	ℛ𝑖−	PROPN
iajs-3432	66	22	𝑄𝑖)(𝑘𝑖	𝑄𝑖)(𝑘𝑖	PROPN
iajs-3432	66	23	+	+	CCONJ
iajs-3432	66	24	1)𝑘𝑖+1	1)𝑘𝑖+1	NUM
iajs-3432	66	25	𝑘𝑖	𝑘𝑖	ADP
iajs-3432	66	26	𝑘𝑖	𝑘𝑖	NOUN
iajs-3432	66	27	𝑚	𝑚	X
iajs-3432	66	28	𝑖=0	𝑖=0	PROPN
iajs-3432	66	29	>	>	X
iajs-3432	66	30	1	1	NUM
iajs-3432	66	31	,	,	PUNCT
iajs-3432	66	32	then	then	ADV
iajs-3432	66	33	every	every	DET
iajs-3432	66	34	solution	solution	NOUN
iajs-3432	66	35	of	of	ADP
iajs-3432	66	36	equation	equation	NOUN
iajs-3432	66	37	(	(	PUNCT
iajs-3432	66	38	5	5	NUM
iajs-3432	66	39	)	)	PUNCT
iajs-3432	66	40	oscillates	oscillate	NOUN
iajs-3432	66	41	.	.	PUNCT
iajs-3432	67	1	according	accord	VERB
iajs-3432	67	2	to	to	ADP
iajs-3432	67	3	mohamad	mohamad	PROPN
iajs-3432	67	4	(	(	PUNCT
iajs-3432	67	5	2015	2015	NUM
iajs-3432	67	6	)	)	PUNCT
iajs-3432	68	1	[	[	X
iajs-3432	68	2	4	4	NUM
iajs-3432	68	3	]	]	PUNCT
iajs-3432	68	4	,	,	PUNCT
iajs-3432	68	5	all	all	PRON
iajs-3432	68	6	of	of	ADP
iajs-3432	68	7	the	the	DET
iajs-3432	68	8	solutions	solution	NOUN
iajs-3432	68	9	to	to	ADP
iajs-3432	68	10	the	the	DET
iajs-3432	68	11	first	first	ADJ
iajs-3432	68	12	-	-	PUNCT
iajs-3432	68	13	order	order	NOUN
iajs-3432	68	14	neutral	neutral	ADJ
iajs-3432	68	15	difference	difference	NOUN
iajs-3432	68	16	equation	equation	NOUN
iajs-3432	68	17	with	with	ADP
iajs-3432	68	18	positive	positive	ADJ
iajs-3432	68	19	and	and	CCONJ
iajs-3432	68	20	negative	negative	ADJ
iajs-3432	68	21	coefficients	coefficient	NOUN
iajs-3432	68	22	will	will	AUX
iajs-3432	68	23	oscillate	oscillate	VERB
iajs-3432	68	24	if	if	SCONJ
iajs-3432	68	25	the	the	DET
iajs-3432	68	26	necessary	necessary	ADJ
iajs-3432	68	27	and	and	CCONJ
iajs-3432	68	28	sufficient	sufficient	ADJ
iajs-3432	68	29	requirements	requirement	NOUN
iajs-3432	68	30	are	be	AUX
iajs-3432	68	31	satisfied	satisfied	ADJ
iajs-3432	68	32	.	.	PUNCT
iajs-3432	69	1	the	the	DET
iajs-3432	69	2	authors	author	NOUN
iajs-3432	69	3	of	of	ADP
iajs-3432	69	4	[	[	X
iajs-3432	69	5	28	28	NUM
iajs-3432	69	6	]	]	PUNCT
iajs-3432	69	7	provided	provide	VERB
iajs-3432	69	8	sufficient	sufficient	ADJ
iajs-3432	69	9	conditions	condition	NOUN
iajs-3432	69	10	to	to	PART
iajs-3432	69	11	ensure	ensure	VERB
iajs-3432	69	12	that	that	SCONJ
iajs-3432	69	13	all	all	DET
iajs-3432	69	14	solutions	solution	NOUN
iajs-3432	69	15	to	to	ADP
iajs-3432	69	16	equation	equation	NOUN
iajs-3432	69	17	(	(	PUNCT
iajs-3432	69	18	4	4	X
iajs-3432	69	19	)	)	PUNCT
iajs-3432	69	20	are	be	AUX
iajs-3432	69	21	oscillatory	oscillatory	ADJ
iajs-3432	69	22	or	or	CCONJ
iajs-3432	69	23	tend	tend	VERB
iajs-3432	69	24	to	to	PART
iajs-3432	69	25	zero	zero	NUM
iajs-3432	69	26	.	.	PUNCT
iajs-3432	70	1	the	the	DET
iajs-3432	70	2	oscillations	oscillation	NOUN
iajs-3432	70	3	and	and	CCONJ
iajs-3432	70	4	global	global	ADJ
iajs-3432	70	5	attractivity	attractivity	NOUN
iajs-3432	70	6	of	of	ADP
iajs-3432	70	7	equation	equation	NOUN
iajs-3432	70	8	(	(	PUNCT
iajs-3432	70	9	3	3	NUM
iajs-3432	70	10	)	)	PUNCT
iajs-3432	70	11	with	with	ADP
iajs-3432	70	12	periodic	periodic	ADJ
iajs-3432	70	13	time	time	NOUN
iajs-3432	70	14	coefficients	coefficient	NOUN
iajs-3432	70	15	were	be	AUX
iajs-3432	70	16	investigated	investigate	VERB
iajs-3432	70	17	by	by	ADP
iajs-3432	70	18	agarwal	agarwal	PROPN
iajs-3432	70	19	and	and	CCONJ
iajs-3432	70	20	saker	saker	PROPN
iajs-3432	70	21	[	[	X
iajs-3432	70	22	24	24	NUM
iajs-3432	70	23	]	]	PUNCT
iajs-3432	70	24	.	.	PUNCT
iajs-3432	71	1	a	a	DET
iajs-3432	71	2	mathematical	mathematical	ADJ
iajs-3432	71	3	model	model	NOUN
iajs-3432	71	4	of	of	ADP
iajs-3432	71	5	hematopoietic	hematopoietic	ADJ
iajs-3432	71	6	stem	stem	NOUN
iajs-3432	71	7	cell	cell	NOUN
iajs-3432	71	8	dynamics	dynamic	NOUN
iajs-3432	71	9	is	be	AUX
iajs-3432	71	10	also	also	ADV
iajs-3432	71	11	proposed	propose	VERB
iajs-3432	71	12	and	and	CCONJ
iajs-3432	71	13	investigated	investigate	VERB
iajs-3432	71	14	in	in	ADP
iajs-3432	71	15	[	[	X
iajs-3432	71	16	29	29	NUM
iajs-3432	71	17	]	]	PUNCT
iajs-3432	71	18	,	,	PUNCT
iajs-3432	71	19	which	which	PRON
iajs-3432	71	20	considers	consider	VERB
iajs-3432	71	21	the	the	DET
iajs-3432	71	22	dynamics	dynamic	NOUN
iajs-3432	71	23	of	of	ADP
iajs-3432	71	24	two	two	NUM
iajs-3432	71	25	cell	cell	NOUN
iajs-3432	71	26	populations	population	NOUN
iajs-3432	71	27	:	:	PUNCT
iajs-3432	71	28	an	an	DET
iajs-3432	71	29	immature	immature	ADJ
iajs-3432	71	30	population	population	NOUN
iajs-3432	71	31	and	and	CCONJ
iajs-3432	71	32	a	a	DET
iajs-3432	71	33	mature	mature	ADJ
iajs-3432	71	34	population	population	NOUN
iajs-3432	71	35	.	.	PUNCT
iajs-3432	72	1	our	our	PRON
iajs-3432	72	2	study	study	NOUN
iajs-3432	72	3	includes	include	VERB
iajs-3432	72	4	recent	recent	ADJ
iajs-3432	72	5	findings	finding	NOUN
iajs-3432	72	6	on	on	ADP
iajs-3432	72	7	the	the	DET
iajs-3432	72	8	qualitative	qualitative	ADJ
iajs-3432	72	9	behavior	behavior	NOUN
iajs-3432	72	10	of	of	ADP
iajs-3432	72	11	a	a	DET
iajs-3432	72	12	biological	biological	ADJ
iajs-3432	72	13	mathematical	mathematical	ADJ
iajs-3432	72	14	model	model	NOUN
iajs-3432	72	15	of	of	ADP
iajs-3432	72	16	discrete	discrete	ADJ
iajs-3432	72	17	hematopoiesis	hematopoiesis	NOUN
iajs-3432	72	18	.	.	PUNCT
iajs-3432	73	1	let	let	VERB
iajs-3432	73	2	's	us	PRON
iajs-3432	73	3	introduce	introduce	VERB
iajs-3432	73	4	an	an	DET
iajs-3432	73	5	invariant	invariant	ADJ
iajs-3432	73	6	oscillation	oscillation	NOUN
iajs-3432	73	7	transformation	transformation	NOUN
iajs-3432	73	8	based	base	VERB
iajs-3432	73	9	on	on	ADP
iajs-3432	73	10	the	the	DET
iajs-3432	73	11	analogous	analogous	ADJ
iajs-3432	73	12	procedure	procedure	NOUN
iajs-3432	73	13	in	in	ADP
iajs-3432	73	14	[	[	X
iajs-3432	73	15	20	20	NUM
iajs-3432	73	16	]	]	PUNCT
iajs-3432	73	17	.	.	PUNCT
iajs-3432	74	1	𝐻𝑛	𝐻𝑛	ADJ
iajs-3432	74	2	=	=	SYM
iajs-3432	74	3	𝑈𝑛	𝑈𝑛	PROPN
iajs-3432	74	4	−	−	PROPN
iajs-3432	74	5	𝐾	𝐾	PROPN
iajs-3432	74	6	,	,	PUNCT
iajs-3432	74	7	(	(	PUNCT
iajs-3432	74	8	6	6	NUM
iajs-3432	74	9	)	)	PUNCT
iajs-3432	74	10	where	where	SCONJ
iajs-3432	74	11	𝐾	𝐾	PROPN
iajs-3432	74	12	is	be	AUX
iajs-3432	74	13	the	the	DET
iajs-3432	74	14	unique	unique	ADJ
iajs-3432	74	15	positive	positive	ADJ
iajs-3432	74	16	equilibrium	equilibrium	NOUN
iajs-3432	74	17	point	point	NOUN
iajs-3432	74	18	of	of	ADP
iajs-3432	74	19	equation	equation	NOUN
iajs-3432	74	20	(	(	PUNCT
iajs-3432	74	21	1	1	NUM
iajs-3432	74	22	)	)	PUNCT
iajs-3432	74	23	.	.	PUNCT
iajs-3432	75	1	then	then	ADV
iajs-3432	75	2	equation	equation	NOUN
iajs-3432	75	3	(	(	PUNCT
iajs-3432	75	4	3	3	X
iajs-3432	75	5	)	)	PUNCT
iajs-3432	75	6	can	can	AUX
iajs-3432	75	7	be	be	AUX
iajs-3432	75	8	reduced	reduce	VERB
iajs-3432	75	9	to	to	ADP
iajs-3432	75	10	∆𝑈𝑛	∆𝑈𝑛	PROPN
iajs-3432	75	11	+	+	CCONJ
iajs-3432	75	12	𝛿𝑛𝑈𝑛	𝛿𝑛𝑈𝑛	ADP
iajs-3432	75	13	−	−	NOUN
iajs-3432	75	14	𝛽𝑛𝑉(𝑈𝑛−𝑙	𝛽𝑛𝑉(𝑈𝑛−𝑙	NOUN
iajs-3432	75	15	)	)	PUNCT
iajs-3432	75	16	=	=	SYM
iajs-3432	75	17	𝜉𝑛	𝜉𝑛	X
iajs-3432	75	18	,	,	PUNCT
iajs-3432	75	19	(	(	PUNCT
iajs-3432	75	20	7	7	NUM
iajs-3432	75	21	)	)	PUNCT
iajs-3432	75	22	where	where	SCONJ
iajs-3432	75	23	𝑉(𝑈𝑛	𝑉(𝑈𝑛	NOUN
iajs-3432	75	24	)	)	PUNCT
iajs-3432	75	25	=	=	SYM
iajs-3432	75	26	1	1	NUM
iajs-3432	75	27	1+(𝑈𝑛+𝐾)𝑚	1+(𝑈𝑛+𝐾)𝑚	NUM
iajs-3432	75	28	and	and	CCONJ
iajs-3432	75	29	𝜉𝑛	𝜉𝑛	X
iajs-3432	75	30	=	=	PUNCT
iajs-3432	75	31	𝑓𝑛	𝑓𝑛	ADJ
iajs-3432	75	32	−	−	PROPN
iajs-3432	75	33	𝛿𝑛𝐾.	𝛿𝑛𝐾.	PROPN
iajs-3432	75	34	then	then	ADV
iajs-3432	75	35	𝐻𝑛oscillates	𝐻𝑛oscillate	VERB
iajs-3432	75	36	about	about	ADP
iajs-3432	75	37	k	k	PROPN
iajs-3432	75	38	if	if	SCONJ
iajs-3432	75	39	and	and	CCONJ
iajs-3432	75	40	only	only	ADV
iajs-3432	75	41	if	if	SCONJ
iajs-3432	75	42	𝑈𝑛	𝑈𝑛	PROPN
iajs-3432	75	43	oscillates	oscillate	NOUN
iajs-3432	75	44	about	about	ADP
iajs-3432	75	45	zero	zero	NUM
iajs-3432	75	46	.	.	PUNCT
iajs-3432	76	1	the	the	DET
iajs-3432	76	2	following	follow	VERB
iajs-3432	76	3	arguments	argument	NOUN
iajs-3432	76	4	are	be	AUX
iajs-3432	76	5	established	establish	VERB
iajs-3432	76	6	in	in	ADP
iajs-3432	76	7	this	this	DET
iajs-3432	76	8	article	article	NOUN
iajs-3432	76	9	:	:	PUNCT
iajs-3432	76	10	𝐴1	𝐴1	PROPN
iajs-3432	76	11	:	:	PUNCT
iajs-3432	76	12	∑	∑	PUNCT
iajs-3432	76	13	𝛽𝑛	𝛽𝑛	PROPN
iajs-3432	76	14	∞	∞	NUM
iajs-3432	76	15	𝑛=0	𝑛=0	X
iajs-3432	76	16	<	<	X
iajs-3432	76	17	∞	∞	PROPN
iajs-3432	76	18	.	.	PUNCT
iajs-3432	77	1	𝐴2	𝐴2	NOUN
iajs-3432	77	2	:	:	PUNCT
iajs-3432	77	3	𝑉(휁𝑛	𝑉(휁𝑛	PROPN
iajs-3432	77	4	)	)	PUNCT
iajs-3432	77	5	휁𝑛	휁𝑛	PROPN
iajs-3432	77	6	≤	≤	NUM
iajs-3432	77	7	𝜆2	𝜆2	NOUN
iajs-3432	77	8	.	.	PUNCT
iajs-3432	78	1	𝐴3	𝐴3	PROPN
iajs-3432	78	2	:	:	PUNCT
iajs-3432	78	3	there	there	PRON
iajs-3432	78	4	exists	exist	VERB
iajs-3432	78	5	a	a	DET
iajs-3432	78	6	sequence	sequence	NOUN
iajs-3432	78	7	𝐹𝑛	𝐹𝑛	PROPN
iajs-3432	78	8	such	such	ADJ
iajs-3432	78	9	that	that	SCONJ
iajs-3432	78	10	∆𝐹𝑛	∆𝐹𝑛	ADV
iajs-3432	78	11	=	=	PUNCT
iajs-3432	78	12	𝜉𝑛	𝜉𝑛	X
iajs-3432	78	13	and	and	CCONJ
iajs-3432	78	14	lim	lim	PROPN
iajs-3432	78	15	𝑛→∞	𝑛→∞	NUM
iajs-3432	78	16	𝐹𝑛	𝐹𝑛	PROPN
iajs-3432	78	17	=	=	PUNCT
iajs-3432	78	18	0	0	PROPN
iajs-3432	78	19	.	.	PUNCT
iajs-3432	79	1	𝐴4	𝐴4	PROPN
iajs-3432	79	2	:	:	PUNCT
iajs-3432	79	3	∑	∑	PUNCT
iajs-3432	79	4	𝛿𝑛	𝛿𝑛	ADP
iajs-3432	79	5	∞	∞	NUM
iajs-3432	79	6	𝑛=𝑛∗	𝑛=𝑛∗	X
iajs-3432	79	7	<	<	X
iajs-3432	79	8	∞	∞	PROPN
iajs-3432	79	9	,	,	PUNCT
iajs-3432	79	10	𝑛∗	𝑛∗	PROPN
iajs-3432	79	11	≥	≥	NUM
iajs-3432	79	12	𝑛0	𝑛0	VERB
iajs-3432	79	13	.	.	PUNCT
iajs-3432	80	1	for	for	ADP
iajs-3432	80	2	the	the	DET
iajs-3432	80	3	conditions	condition	NOUN
iajs-3432	80	4	produced	produce	VERB
iajs-3432	80	5	𝐴1	𝐴1	PROPN
iajs-3432	80	6	,	,	PUNCT
iajs-3432	80	7	𝐴2	𝐴2	PROPN
iajs-3432	80	8	,	,	PUNCT
iajs-3432	80	9	𝐴3	𝐴3	PROPN
iajs-3432	80	10	is	be	AUX
iajs-3432	80	11	the	the	DET
iajs-3432	80	12	assertion	assertion	NOUN
iajs-3432	80	13	that	that	SCONJ
iajs-3432	80	14	there	there	PRON
iajs-3432	80	15	is	be	VERB
iajs-3432	80	16	no	no	DET
iajs-3432	80	17	non	non	ADJ
iajs-3432	80	18	-	-	ADJ
iajs-3432	80	19	oscillating	oscillating	ADJ
iajs-3432	80	20	solution	solution	NOUN
iajs-3432	80	21	that	that	PRON
iajs-3432	80	22	achieves	achieve	VERB
iajs-3432	80	23	the	the	DET
iajs-3432	80	24	equation	equation	NOUN
iajs-3432	80	25	of	of	ADP
iajs-3432	80	26	hematopoiesis	hematopoiesis	NOUN
iajs-3432	80	27	or	or	CCONJ
iajs-3432	80	28	the	the	DET
iajs-3432	80	29	resulting	result	VERB
iajs-3432	80	30	inequality	inequality	NOUN
iajs-3432	80	31	from	from	ADP
iajs-3432	80	32	it	it	PRON
iajs-3432	80	33	.	.	PUNCT
iajs-3432	81	1	in	in	ADP
iajs-3432	81	2	return	return	NOUN
iajs-3432	81	3	,	,	PUNCT
iajs-3432	81	4	the	the	DET
iajs-3432	81	5	biological	biological	ADJ
iajs-3432	81	6	interpretation	interpretation	NOUN
iajs-3432	81	7	of	of	ADP
iajs-3432	81	8	these	these	DET
iajs-3432	81	9	conditions	condition	NOUN
iajs-3432	81	10	is	be	AUX
iajs-3432	81	11	like	like	ADP
iajs-3432	81	12	the	the	DET
iajs-3432	81	13	doses	dose	NOUN
iajs-3432	81	14	that	that	PRON
iajs-3432	81	15	are	be	AUX
iajs-3432	81	16	given	give	VERB
iajs-3432	81	17	to	to	ADP
iajs-3432	81	18	blood	blood	NOUN
iajs-3432	81	19	cells	cell	NOUN
iajs-3432	81	20	in	in	ADP
iajs-3432	81	21	order	order	NOUN
iajs-3432	81	22	to	to	PART
iajs-3432	81	23	control	control	VERB
iajs-3432	81	24	the	the	DET
iajs-3432	81	25	number	number	NOUN
iajs-3432	81	26	of	of	ADP
iajs-3432	81	27	cells	cell	NOUN
iajs-3432	81	28	produced	produce	VERB
iajs-3432	81	29	.	.	PUNCT
iajs-3432	82	1	a	a	DET
iajs-3432	82	2	sequence	sequence	NOUN
iajs-3432	82	3	,	,	PUNCT
iajs-3432	82	4	𝑈𝑛	𝑈𝑛	ADP
iajs-3432	82	5	satisfying	satisfy	VERB
iajs-3432	82	6	equation	equation	NOUN
iajs-3432	82	7	(	(	PUNCT
iajs-3432	82	8	7	7	NUM
iajs-3432	82	9	)	)	PUNCT
iajs-3432	82	10	for	for	ADP
iajs-3432	82	11	𝑛	𝑛	PROPN
iajs-3432	82	12	>	>	SYM
iajs-3432	82	13	0	0	NUM
iajs-3432	82	14	is	be	AUX
iajs-3432	82	15	referred	refer	VERB
iajs-3432	82	16	to	to	ADP
iajs-3432	82	17	as	as	ADP
iajs-3432	82	18	a	a	DET
iajs-3432	82	19	solution	solution	NOUN
iajs-3432	82	20	of	of	ADP
iajs-3432	82	21	equation(7	equation(7	PROPN
iajs-3432	82	22	)	)	PUNCT
iajs-3432	82	23	.	.	PUNCT
iajs-3432	83	1	if	if	SCONJ
iajs-3432	83	2	there	there	PRON
iajs-3432	83	3	is	be	VERB
iajs-3432	83	4	𝑛	𝑛	PRON
iajs-3432	83	5	>	>	X
iajs-3432	83	6	𝑛𝑗	𝑛𝑗	ADP
iajs-3432	83	7	such	such	ADJ
iajs-3432	83	8	that	that	SCONJ
iajs-3432	83	9	,	,	PUNCT
iajs-3432	83	10	𝑈𝑛.	𝑈𝑛.	PROPN
iajs-3432	83	11	𝑈𝑛+1	𝑈𝑛+1	PROPN
iajs-3432	83	12	>	>	X
iajs-3432	83	13	0	0	PROPN
iajs-3432	83	14	,	,	PUNCT
iajs-3432	83	15	then	then	ADV
iajs-3432	83	16	a	a	DET
iajs-3432	83	17	nontrivial	nontrivial	ADJ
iajs-3432	83	18	solution	solution	NOUN
iajs-3432	83	19	,	,	PUNCT
iajs-3432	83	20	𝑈𝑛.	𝑈𝑛.	PROPN
iajs-3432	83	21	is	be	AUX
iajs-3432	83	22	said	say	VERB
iajs-3432	83	23	to	to	PART
iajs-3432	83	24	be	be	AUX
iajs-3432	83	25	oscillatory	oscillatory	ADJ
iajs-3432	83	26	(	(	PUNCT
iajs-3432	83	27	oscillating	oscillate	VERB
iajs-3432	83	28	about	about	ADP
iajs-3432	83	29	zero	zero	NUM
iajs-3432	83	30	)	)	PUNCT
iajs-3432	83	31	.	.	PUNCT
iajs-3432	84	1	otherwise	otherwise	ADV
iajs-3432	84	2	,	,	PUNCT
iajs-3432	84	3	it	it	PRON
iajs-3432	84	4	is	be	AUX
iajs-3432	84	5	argued	argue	VERB
iajs-3432	84	6	that	that	SCONJ
iajs-3432	84	7	the	the	DET
iajs-3432	84	8	solution	solution	NOUN
iajs-3432	84	9	is	be	AUX
iajs-3432	84	10	nonoscillatory	nonoscillatory	ADJ
iajs-3432	84	11	[	[	X
iajs-3432	84	12	30	30	NUM
iajs-3432	84	13	]	]	PUNCT
iajs-3432	84	14	.	.	PUNCT
iajs-3432	85	1	in	in	ADP
iajs-3432	85	2	this	this	DET
iajs-3432	85	3	work	work	NOUN
iajs-3432	85	4	,	,	PUNCT
iajs-3432	85	5	several	several	ADJ
iajs-3432	85	6	new	new	ADJ
iajs-3432	85	7	necessary	necessary	ADJ
iajs-3432	85	8	conditions	condition	NOUN
iajs-3432	85	9	are	be	AUX
iajs-3432	85	10	found	find	VERB
iajs-3432	85	11	to	to	PART
iajs-3432	85	12	cause	cause	VERB
iajs-3432	85	13	oscillations	oscillation	NOUN
iajs-3432	85	14	or	or	CCONJ
iajs-3432	85	15	convergence	convergence	NOUN
iajs-3432	85	16	to	to	ADP
iajs-3432	85	17	equilibrium	equilibrium	NOUN
iajs-3432	85	18	𝐾	𝐾	PROPN
iajs-3432	85	19	in	in	ADP
iajs-3432	85	20	the	the	DET
iajs-3432	85	21	solutions	solution	NOUN
iajs-3432	85	22	and	and	CCONJ
iajs-3432	85	23	bounded	bounded	ADJ
iajs-3432	85	24	solutions	solution	NOUN
iajs-3432	85	25	of	of	ADP
iajs-3432	85	26	equation	equation	NOUN
iajs-3432	85	27	(	(	PUNCT
iajs-3432	85	28	3	3	NUM
iajs-3432	85	29	)	)	PUNCT
iajs-3432	85	30	.	.	PUNCT
iajs-3432	86	1	the	the	DET
iajs-3432	86	2	theoretical	theoretical	ADJ
iajs-3432	86	3	results	result	NOUN
iajs-3432	86	4	are	be	AUX
iajs-3432	86	5	supported	support	VERB
iajs-3432	86	6	by	by	ADP
iajs-3432	86	7	a	a	DET
iajs-3432	86	8	few	few	ADJ
iajs-3432	86	9	examples	example	NOUN
iajs-3432	86	10	.	.	PUNCT
iajs-3432	87	1	ihjpas	ihjpas	PROPN
iajs-3432	87	2	.	.	PUNCT
iajs-3432	88	1	2024	2024	NUM
iajs-3432	88	2	,	,	PUNCT
iajs-3432	88	3	37	37	NUM
iajs-3432	88	4	(	(	PUNCT
iajs-3432	88	5	3	3	NUM
iajs-3432	88	6	)	)	PUNCT
iajs-3432	88	7	389	389	NUM
iajs-3432	88	8	2	2	NUM
iajs-3432	88	9	.	.	PUNCT
iajs-3432	88	10	background	background	NOUN
iajs-3432	88	11	definition	definition	NOUN
iajs-3432	88	12	1	1	NUM
iajs-3432	88	13	[	[	X
iajs-3432	88	14	30	30	NUM
iajs-3432	88	15	]	]	X
iajs-3432	88	16	a	a	DET
iajs-3432	88	17	point	point	NOUN
iajs-3432	88	18	𝐾	𝐾	NOUN
iajs-3432	88	19	in	in	ADP
iajs-3432	88	20	the	the	DET
iajs-3432	88	21	domain	domain	NOUN
iajs-3432	88	22	of	of	ADP
iajs-3432	88	23	𝐻𝑛	𝐻𝑛	PROPN
iajs-3432	88	24	is	be	AUX
iajs-3432	88	25	said	say	VERB
iajs-3432	88	26	to	to	PART
iajs-3432	88	27	be	be	AUX
iajs-3432	88	28	an	an	DET
iajs-3432	88	29	equilibrium	equilibrium	NOUN
iajs-3432	88	30	point	point	NOUN
iajs-3432	88	31	of	of	ADP
iajs-3432	88	32	(	(	PUNCT
iajs-3432	88	33	3	3	X
iajs-3432	88	34	)	)	PUNCT
iajs-3432	88	35	if	if	SCONJ
iajs-3432	88	36	it	it	PRON
iajs-3432	88	37	is	be	AUX
iajs-3432	88	38	a	a	DET
iajs-3432	88	39	fixed	fixed	ADJ
iajs-3432	88	40	point	point	NOUN
iajs-3432	88	41	of	of	ADP
iajs-3432	88	42	𝐻𝑛	𝐻𝑛	PROPN
iajs-3432	88	43	,	,	PUNCT
iajs-3432	88	44	that	that	ADV
iajs-3432	88	45	is	is	ADV
iajs-3432	88	46	,	,	PUNCT
iajs-3432	88	47	𝐻𝑛(𝐾	𝐻𝑛(𝐾	NOUN
iajs-3432	88	48	)	)	PUNCT
iajs-3432	89	1	=	=	SYM
iajs-3432	89	2	𝐾	𝐾	PROPN
iajs-3432	89	3	.	.	PUNCT
iajs-3432	90	1	definition	definition	NOUN
iajs-3432	90	2	2	2	NUM
iajs-3432	91	1	[	[	X
iajs-3432	91	2	20	20	NUM
iajs-3432	91	3	]	]	X
iajs-3432	91	4	if	if	SCONJ
iajs-3432	91	5	a	a	DET
iajs-3432	91	6	solution	solution	NOUN
iajs-3432	91	7	to	to	ADP
iajs-3432	91	8	equation	equation	NOUN
iajs-3432	91	9	(	(	PUNCT
iajs-3432	91	10	3	3	NUM
iajs-3432	91	11	)	)	PUNCT
iajs-3432	91	12	,	,	PUNCT
iajs-3432	91	13	𝐻𝑛	𝐻𝑛	ADJ
iajs-3432	91	14	oscillates	oscillate	NOUN
iajs-3432	91	15	around	around	ADP
iajs-3432	91	16	equilibrium	equilibrium	NOUN
iajs-3432	91	17	𝐾	𝐾	PROPN
iajs-3432	91	18	,	,	PUNCT
iajs-3432	91	19	then	then	ADV
iajs-3432	91	20	𝐻𝑛	𝐻𝑛	ADJ
iajs-3432	91	21	−	−	PROPN
iajs-3432	91	22	𝑘	𝑘	PROPN
iajs-3432	91	23	𝐾	𝐾	PROPN
iajs-3432	91	24	oscillate	oscillate	VERB
iajs-3432	91	25	about	about	ADP
iajs-3432	91	26	zeros	zero	NOUN
iajs-3432	91	27	.	.	PUNCT
iajs-3432	92	1	if	if	SCONJ
iajs-3432	92	2	not	not	PART
iajs-3432	92	3	,	,	PUNCT
iajs-3432	92	4	𝐻𝑛	𝐻𝑛	PROPN
iajs-3432	92	5	is	be	AUX
iajs-3432	92	6	regarded	regard	VERB
iajs-3432	92	7	as	as	ADP
iajs-3432	92	8	non	non	ADJ
iajs-3432	92	9	-	-	ADJ
iajs-3432	92	10	oscillatory	oscillatory	ADJ
iajs-3432	92	11	.	.	PUNCT
iajs-3432	93	1	when	when	SCONJ
iajs-3432	93	2	𝐾	𝐾	PROPN
iajs-3432	93	3	=	=	SYM
iajs-3432	93	4	0	0	NUM
iajs-3432	93	5	,	,	PUNCT
iajs-3432	93	6	we	we	PRON
iajs-3432	93	7	say	say	VERB
iajs-3432	93	8	that	that	SCONJ
iajs-3432	94	1	𝐻𝑛	𝐻𝑛	ADJ
iajs-3432	94	2	simply	simply	ADV
iajs-3432	94	3	oscillates	oscillate	NOUN
iajs-3432	94	4	or	or	CCONJ
iajs-3432	94	5	oscillates	oscillate	NOUN
iajs-3432	94	6	about	about	ADP
iajs-3432	94	7	zero	zero	NUM
iajs-3432	94	8	.	.	PUNCT
iajs-3432	95	1	based	base	VERB
iajs-3432	95	2	on	on	ADP
iajs-3432	95	3	theorem	theorem	NOUN
iajs-3432	95	4	1	1	NUM
iajs-3432	95	5	in	in	ADP
iajs-3432	95	6	[	[	PUNCT
iajs-3432	95	7	4	4	NUM
iajs-3432	95	8	]	]	PUNCT
iajs-3432	95	9	,	,	PUNCT
iajs-3432	95	10	the	the	DET
iajs-3432	95	11	result	result	NOUN
iajs-3432	95	12	follows	follow	VERB
iajs-3432	95	13	.	.	PUNCT
iajs-3432	96	1	3	3	X
iajs-3432	96	2	.	.	NOUN
iajs-3432	96	3	results	result	NOUN
iajs-3432	96	4	and	and	CCONJ
iajs-3432	96	5	discussion	discussion	NOUN
iajs-3432	96	6	in	in	ADP
iajs-3432	96	7	this	this	DET
iajs-3432	96	8	section	section	NOUN
iajs-3432	96	9	,	,	PUNCT
iajs-3432	96	10	some	some	DET
iajs-3432	96	11	sufficient	sufficient	ADJ
iajs-3432	96	12	conditions	condition	NOUN
iajs-3432	96	13	are	be	AUX
iajs-3432	96	14	established	establish	VERB
iajs-3432	96	15	for	for	ADP
iajs-3432	96	16	the	the	DET
iajs-3432	96	17	oscillation	oscillation	NOUN
iajs-3432	96	18	of	of	ADP
iajs-3432	96	19	all	all	DET
iajs-3432	96	20	solutions	solution	NOUN
iajs-3432	96	21	to	to	ADP
iajs-3432	96	22	equation	equation	NOUN
iajs-3432	96	23	(	(	PUNCT
iajs-3432	96	24	3	3	NUM
iajs-3432	96	25	)	)	PUNCT
iajs-3432	96	26	.	.	PUNCT
iajs-3432	97	1	it	it	PRON
iajs-3432	97	2	is	be	AUX
iajs-3432	97	3	simple	simple	ADJ
iajs-3432	97	4	to	to	PART
iajs-3432	97	5	demonstrate	demonstrate	VERB
iajs-3432	97	6	that	that	SCONJ
iajs-3432	97	7	𝐻𝑛	𝐻𝑛	ADJ
iajs-3432	97	8	oscillates	oscillate	NOUN
iajs-3432	97	9	about	about	ADP
iajs-3432	97	10	k	k	PROPN
iajs-3432	97	11	if	if	SCONJ
iajs-3432	97	12	and	and	CCONJ
iajs-3432	97	13	only	only	ADV
iajs-3432	97	14	if	if	SCONJ
iajs-3432	97	15	𝑈𝑛	𝑈𝑛	PROPN
iajs-3432	97	16	.we	.we	PUNCT
iajs-3432	97	17	just	just	ADV
iajs-3432	97	18	need	need	VERB
iajs-3432	97	19	to	to	PART
iajs-3432	97	20	take	take	VERB
iajs-3432	97	21	into	into	ADP
iajs-3432	97	22	account	account	NOUN
iajs-3432	97	23	the	the	DET
iajs-3432	97	24	oscillations	oscillation	NOUN
iajs-3432	97	25	of	of	ADP
iajs-3432	97	26	equation	equation	NOUN
iajs-3432	97	27	(	(	PUNCT
iajs-3432	97	28	3	3	NUM
iajs-3432	97	29	)	)	PUNCT
iajs-3432	97	30	in	in	ADP
iajs-3432	97	31	order	order	NOUN
iajs-3432	97	32	to	to	PART
iajs-3432	97	33	investigate	investigate	VERB
iajs-3432	97	34	the	the	DET
iajs-3432	97	35	oscillations	oscillation	NOUN
iajs-3432	97	36	of	of	ADP
iajs-3432	97	37	equation	equation	NOUN
iajs-3432	97	38	(	(	PUNCT
iajs-3432	97	39	7	7	NUM
iajs-3432	97	40	)	)	PUNCT
iajs-3432	97	41	.	.	PUNCT
iajs-3432	98	1	let	let	VERB
iajs-3432	98	2	the	the	DET
iajs-3432	98	3	sequence	sequence	NOUN
iajs-3432	98	4	𝑍𝑛	𝑍𝑛	PROPN
iajs-3432	98	5	be	be	AUX
iajs-3432	98	6	defined	define	VERB
iajs-3432	98	7	as	as	ADP
iajs-3432	98	8	𝑍𝑛	𝑍𝑛	PROPN
iajs-3432	98	9	=	=	SYM
iajs-3432	98	10	𝑈𝑛	𝑈𝑛	PROPN
iajs-3432	98	11	+	+	CCONJ
iajs-3432	98	12	∑	∑	PUNCT
iajs-3432	98	13	𝛽𝑖	𝛽𝑖	ADJ
iajs-3432	98	14	𝑛+𝑙−1	𝑛+𝑙−1	ADP
iajs-3432	98	15	𝑖=𝑛	𝑖=𝑛	PROPN
iajs-3432	98	16	𝑉(𝑈𝑖−𝑙	𝑉(𝑈𝑖−𝑙	PROPN
iajs-3432	98	17	)	)	PUNCT
iajs-3432	98	18	−	−	PROPN
iajs-3432	99	1	𝐹𝑛	𝐹𝑛	PROPN
iajs-3432	99	2	.	.	PUNCT
iajs-3432	100	1	(	(	PUNCT
iajs-3432	100	2	8)	8)	NUM
iajs-3432	100	3	the	the	DET
iajs-3432	100	4	following	following	ADJ
iajs-3432	100	5	lemma	lemma	PROPN
iajs-3432	100	6	helps	help	VERB
iajs-3432	100	7	to	to	PART
iajs-3432	100	8	demonstrate	demonstrate	VERB
iajs-3432	100	9	the	the	DET
iajs-3432	100	10	main	main	ADJ
iajs-3432	100	11	findings	finding	NOUN
iajs-3432	100	12	.	.	PUNCT
iajs-3432	101	1	lemma	lemma	PROPN
iajs-3432	101	2	1	1	X
iajs-3432	101	3	.	.	PUNCT
iajs-3432	101	4	suppose	suppose	VERB
iajs-3432	101	5	,	,	PUNCT
iajs-3432	101	6	𝛿𝑛	𝛿𝑛	ADP
iajs-3432	101	7	−	−	PROPN
iajs-3432	101	8	𝛽𝑛+𝑙𝜆2	𝛽𝑛+𝑙𝜆2	PROPN
iajs-3432	101	9	≥	≥	NOUN
iajs-3432	101	10	0	0	PUNCT
iajs-3432	101	11	and	and	CCONJ
iajs-3432	101	12	assume	assume	VERB
iajs-3432	101	13	that	that	SCONJ
iajs-3432	101	14	(	(	PUNCT
iajs-3432	101	15	𝐴1	𝐴1	PROPN
iajs-3432	101	16	)	)	PUNCT
iajs-3432	102	1	−	−	PROPN
iajs-3432	102	2	(	(	PUNCT
iajs-3432	102	3	𝐴3	𝐴3	PROPN
iajs-3432	102	4	)	)	PUNCT
iajs-3432	102	5	hold	hold	VERB
iajs-3432	102	6	.	.	PUNCT
iajs-3432	103	1	let	let	VERB
iajs-3432	103	2	𝐻𝑛	𝐻𝑛	PROPN
iajs-3432	103	3	be	be	AUX
iajs-3432	103	4	a	a	DET
iajs-3432	103	5	nonoscillatory	nonoscillatory	ADJ
iajs-3432	103	6	solution	solution	NOUN
iajs-3432	103	7	to	to	ADP
iajs-3432	103	8	𝐾	𝐾	PROPN
iajs-3432	103	9	of	of	ADP
iajs-3432	103	10	equation	equation	NOUN
iajs-3432	103	11	(	(	PUNCT
iajs-3432	103	12	3	3	NUM
iajs-3432	103	13	)	)	PUNCT
iajs-3432	103	14	.	.	PUNCT
iajs-3432	104	1	then	then	ADV
iajs-3432	104	2	,	,	PUNCT
iajs-3432	104	3	𝑍𝑛	𝑍𝑛	PROPN
iajs-3432	104	4	≥	≥	NUM
iajs-3432	104	5	0	0	NUM
iajs-3432	104	6	and	and	CCONJ
iajs-3432	104	7	a	a	DET
iajs-3432	104	8	nonincreasing	nonincrease	VERB
iajs-3432	104	9	sequence	sequence	NOUN
iajs-3432	104	10	.	.	PUNCT
iajs-3432	105	1	proof	proof	NOUN
iajs-3432	105	2	.	.	PUNCT
iajs-3432	106	1	let	let	VERB
iajs-3432	106	2	𝐻𝑛	𝐻𝑛	PROPN
iajs-3432	106	3	be	be	AUX
iajs-3432	106	4	a	a	DET
iajs-3432	106	5	nonoscillatory	nonoscillatory	ADJ
iajs-3432	106	6	solution	solution	NOUN
iajs-3432	106	7	to	to	ADP
iajs-3432	106	8	𝐾	𝐾	PROPN
iajs-3432	106	9	of	of	ADP
iajs-3432	106	10	equation	equation	NOUN
iajs-3432	106	11	(	(	PUNCT
iajs-3432	106	12	3	3	NUM
iajs-3432	106	13	)	)	PUNCT
iajs-3432	106	14	,	,	PUNCT
iajs-3432	106	15	that	that	PRON
iajs-3432	106	16	is	be	AUX
iajs-3432	106	17	𝐻𝑛	𝐻𝑛	PROPN
iajs-3432	106	18	>	>	X
iajs-3432	106	19	𝐾	𝐾	PROPN
iajs-3432	106	20	,	,	PUNCT
iajs-3432	106	21	𝑛	𝑛	DET
iajs-3432	106	22	≥	≥	NOUN
iajs-3432	106	23	𝑛0	𝑛0	VERB
iajs-3432	106	24	,	,	PUNCT
iajs-3432	106	25	(	(	PUNCT
iajs-3432	106	26	the	the	DET
iajs-3432	106	27	proof	proof	NOUN
iajs-3432	106	28	of	of	ADP
iajs-3432	106	29	the	the	DET
iajs-3432	106	30	case	case	NOUN
iajs-3432	106	31	0	0	PUNCT
iajs-3432	107	1	<	<	X
iajs-3432	108	1	𝐻𝑛	𝐻𝑛	PROPN
iajs-3432	108	2	<	<	X
iajs-3432	108	3	𝐾	𝐾	PROPN
iajs-3432	108	4	is	be	AUX
iajs-3432	108	5	similar	similar	ADJ
iajs-3432	108	6	and	and	CCONJ
iajs-3432	108	7	will	will	AUX
iajs-3432	108	8	be	be	AUX
iajs-3432	108	9	omitted	omit	VERB
iajs-3432	108	10	)	)	PUNCT
iajs-3432	108	11	,	,	PUNCT
iajs-3432	108	12	hence	hence	ADV
iajs-3432	108	13	𝑈𝑛	𝑈𝑛	PROPN
iajs-3432	108	14	>	>	X
iajs-3432	108	15	0	0	PUNCT
iajs-3432	109	1	eventually	eventually	ADV
iajs-3432	109	2	.	.	PUNCT
iajs-3432	110	1	from	from	ADP
iajs-3432	110	2	equation	equation	NOUN
iajs-3432	110	3	(	(	PUNCT
iajs-3432	110	4	8)	8)	NUM
iajs-3432	110	5	,	,	PUNCT
iajs-3432	110	6	we	we	PRON
iajs-3432	110	7	obtain	obtain	VERB
iajs-3432	110	8	∆𝑍𝑛	∆𝑍𝑛	NOUN
iajs-3432	110	9	=	=	SYM
iajs-3432	110	10	−𝛿𝑛𝑈𝑛	−𝛿𝑛𝑈𝑛	PROPN
iajs-3432	110	11	+	+	CCONJ
iajs-3432	110	12	𝛽𝑛+𝑙𝑉(𝑈𝑛	𝛽𝑛+𝑙𝑉(𝑈𝑛	NOUN
iajs-3432	110	13	)	)	PUNCT
iajs-3432	110	14	≤	≤	PUNCT
iajs-3432	111	1	−(𝛿𝑛	−(𝛿𝑛	SCONJ
iajs-3432	111	2	−	−	PRON
iajs-3432	111	3	𝛽𝑛+𝑙𝜆2)𝑈𝑛	𝛽𝑛+𝑙𝜆2)𝑈𝑛	VERB
iajs-3432	111	4	≤	≤	NOUN
iajs-3432	111	5	0	0	NUM
iajs-3432	111	6	.	.	PUNCT
iajs-3432	112	1	(	(	PUNCT
iajs-3432	112	2	9	9	X
iajs-3432	112	3	)	)	PUNCT
iajs-3432	112	4	hence	hence	ADV
iajs-3432	112	5	,	,	PUNCT
iajs-3432	112	6	𝑍𝑛	𝑍𝑛	PROPN
iajs-3432	112	7	is	be	AUX
iajs-3432	112	8	a	a	DET
iajs-3432	112	9	nonincreasing	nonincrease	VERB
iajs-3432	112	10	sequence	sequence	NOUN
iajs-3432	112	11	,	,	PUNCT
iajs-3432	112	12	and	and	CCONJ
iajs-3432	112	13	lim	lim	PROPN
iajs-3432	112	14	𝑛→∞	𝑛→∞	NUM
iajs-3432	112	15	𝑍𝑛	𝑍𝑛	PROPN
iajs-3432	112	16	=	=	SYM
iajs-3432	112	17	𝐿	𝐿	PROPN
iajs-3432	112	18	,	,	PUNCT
iajs-3432	112	19	where	where	SCONJ
iajs-3432	112	20	−∞	−∞	ADP
iajs-3432	112	21	≤	≤	PROPN
iajs-3432	112	22	𝐿	𝐿	PROPN
iajs-3432	112	23	<	<	X
iajs-3432	112	24	∞.	∞.	PROPN
iajs-3432	112	25	we	we	PRON
iajs-3432	112	26	claim	claim	VERB
iajs-3432	112	27	that	that	SCONJ
iajs-3432	112	28	𝐿	𝐿	PROPN
iajs-3432	112	29	≥	≥	NOUN
iajs-3432	112	30	0	0	NUM
iajs-3432	112	31	.	.	PUNCT
iajs-3432	113	1	otherwise	otherwise	ADV
iajs-3432	113	2	𝐿	𝐿	PROPN
iajs-3432	113	3	<	<	X
iajs-3432	113	4	0	0	NUM
iajs-3432	113	5	,	,	PUNCT
iajs-3432	113	6	so	so	SCONJ
iajs-3432	113	7	there	there	PRON
iajs-3432	113	8	exists	exist	VERB
iajs-3432	113	9	𝑛1	𝑛1	ADJ
iajs-3432	113	10	≥	≥	NOUN
iajs-3432	113	11	𝑛0	𝑛0	VERB
iajs-3432	113	12	and	and	CCONJ
iajs-3432	113	13	𝛼	𝛼	ADJ
iajs-3432	113	14	<	<	X
iajs-3432	113	15	0	0	NUM
iajs-3432	113	16	,	,	PUNCT
iajs-3432	113	17	such	such	ADJ
iajs-3432	113	18	that	that	SCONJ
iajs-3432	113	19	𝑍𝑛	𝑍𝑛	PROPN
iajs-3432	113	20	≤	≤	NOUN
iajs-3432	114	1	𝛼	𝛼	X
iajs-3432	114	2	<	<	X
iajs-3432	114	3	0	0	NUM
iajs-3432	114	4	for	for	ADP
iajs-3432	114	5	𝑛	𝑛	PROPN
iajs-3432	114	6	≥	≥	NOUN
iajs-3432	114	7	𝑛1	𝑛1	NOUN
iajs-3432	114	8	.	.	PUNCT
iajs-3432	115	1	from	from	ADP
iajs-3432	115	2	equation	equation	NOUN
iajs-3432	115	3	(	(	PUNCT
iajs-3432	115	4	8)	8)	NUM
iajs-3432	115	5	,	,	PUNCT
iajs-3432	115	6	we	we	PRON
iajs-3432	115	7	get	get	VERB
iajs-3432	115	8	𝑈𝑛	𝑈𝑛	PROPN
iajs-3432	115	9	=	=	SYM
iajs-3432	115	10	𝑍𝑛	𝑍𝑛	PROPN
iajs-3432	115	11	−	−	PROPN
iajs-3432	115	12	∑	∑	PUNCT
iajs-3432	115	13	𝛽𝑖	𝛽𝑖	VERB
iajs-3432	115	14	𝑛+𝑙−1	𝑛+𝑙−1	PROPN
iajs-3432	115	15	𝑖=𝑛	𝑖=𝑛	PROPN
iajs-3432	115	16	𝑉(𝑈𝑖−𝑙	𝑉(𝑈𝑖−𝑙	PROPN
iajs-3432	115	17	)	)	PUNCT
iajs-3432	115	18	+	+	NUM
iajs-3432	116	1	𝐹𝑛	𝐹𝑛	PROPN
iajs-3432	116	2	,	,	PUNCT
iajs-3432	116	3	𝑈𝑛	𝑈𝑛	VERB
iajs-3432	116	4	≤	≤	NOUN
iajs-3432	116	5	𝛼	𝛼	PRON
iajs-3432	116	6	−	−	NOUN
iajs-3432	116	7	∑	∑	PUNCT
iajs-3432	116	8	𝛽𝑖	𝛽𝑖	VERB
iajs-3432	116	9	𝑛+𝑙−1	𝑛+𝑙−1	PROPN
iajs-3432	116	10	𝑖=𝑛	𝑖=𝑛	PROPN
iajs-3432	116	11	𝑉(𝑈𝑖−𝑙	𝑉(𝑈𝑖−𝑙	PROPN
iajs-3432	116	12	)	)	PUNCT
iajs-3432	116	13	+	+	NUM
iajs-3432	117	1	𝐹𝑛	𝐹𝑛	PROPN
iajs-3432	117	2	,	,	PUNCT
iajs-3432	117	3	𝑈𝑛	𝑈𝑛	VERB
iajs-3432	117	4	≤	≤	NUM
iajs-3432	117	5	𝛼	𝛼	NOUN
iajs-3432	117	6	+	+	NOUN
iajs-3432	117	7	𝐹𝑛	𝐹𝑛	PROPN
iajs-3432	117	8	<	<	X
iajs-3432	117	9	𝛼	𝛼	X
iajs-3432	118	1	+	+	CCONJ
iajs-3432	119	1	휀	휀	X
iajs-3432	119	2	,	,	PUNCT
iajs-3432	119	3	휀	휀	NOUN
iajs-3432	119	4	>	>	X
iajs-3432	119	5	0	0	NUM
iajs-3432	119	6	,	,	PUNCT
iajs-3432	119	7	𝑛	𝑛	DET
iajs-3432	119	8	≥	≥	NOUN
iajs-3432	119	9	𝑛2	𝑛2	PROPN
iajs-3432	119	10	≥	≥	PROPN
iajs-3432	119	11	𝑛1	𝑛1	PROPN
iajs-3432	119	12	.	.	PUNCT
iajs-3432	120	1	(	(	PUNCT
iajs-3432	120	2	10	10	NUM
iajs-3432	120	3	)	)	PUNCT
iajs-3432	120	4	since	since	SCONJ
iajs-3432	120	5	휀	휀	NOUN
iajs-3432	120	6	is	be	AUX
iajs-3432	120	7	arbitrary	arbitrary	ADJ
iajs-3432	120	8	,	,	PUNCT
iajs-3432	120	9	then	then	ADV
iajs-3432	120	10	equation	equation	NOUN
iajs-3432	120	11	(	(	PUNCT
iajs-3432	120	12	10	10	NUM
iajs-3432	120	13	)	)	PUNCT
iajs-3432	120	14	leads	lead	VERB
iajs-3432	120	15	to	to	ADP
iajs-3432	120	16	𝑈𝑛	𝑈𝑛	PROPN
iajs-3432	120	17	≤	≤	NUM
iajs-3432	120	18	𝛼	𝛼	NUM
iajs-3432	120	19	which	which	PRON
iajs-3432	120	20	is	be	AUX
iajs-3432	120	21	a	a	DET
iajs-3432	120	22	contradiction	contradiction	NOUN
iajs-3432	120	23	since	since	SCONJ
iajs-3432	120	24	𝑈𝑛	𝑈𝑛	PROPN
iajs-3432	120	25	is	be	AUX
iajs-3432	120	26	positive	positive	ADJ
iajs-3432	120	27	.	.	PUNCT
iajs-3432	121	1	so	so	ADV
iajs-3432	121	2	,	,	PUNCT
iajs-3432	121	3	since	since	SCONJ
iajs-3432	121	4	our	our	PRON
iajs-3432	121	5	claim	claim	NOUN
iajs-3432	121	6	has	have	AUX
iajs-3432	121	7	been	be	AUX
iajs-3432	121	8	proven	prove	VERB
iajs-3432	121	9	,	,	PUNCT
iajs-3432	121	10	𝐿	𝐿	PROPN
iajs-3432	121	11	≥	≥	NOUN
iajs-3432	121	12	0	0	NUM
iajs-3432	121	13	,	,	PUNCT
iajs-3432	121	14	or	or	CCONJ
iajs-3432	121	15	𝑍𝑛	𝑍𝑛	PROPN
iajs-3432	121	16	≥	≥	NOUN
iajs-3432	121	17	0	0	NUM
iajs-3432	121	18	,	,	PUNCT
iajs-3432	121	19	follows	follow	VERB
iajs-3432	121	20	.	.	PUNCT
iajs-3432	122	1	theorem	theorem	NOUN
iajs-3432	122	2	1	1	X
iajs-3432	122	3	.	.	PUNCT
iajs-3432	122	4	assume	assume	VERB
iajs-3432	122	5	𝛿𝑛	𝛿𝑛	INTJ
iajs-3432	122	6	−	−	PROPN
iajs-3432	122	7	𝛽𝑛+𝑙𝜆2	𝛽𝑛+𝑙𝜆2	PROPN
iajs-3432	122	8	≥	≥	NOUN
iajs-3432	122	9	0	0	PUNCT
iajs-3432	123	1	and	and	CCONJ
iajs-3432	123	2	let	let	VERB
iajs-3432	123	3	(	(	PUNCT
iajs-3432	123	4	𝐴1	𝐴1	PROPN
iajs-3432	123	5	)	)	PUNCT
iajs-3432	124	1	−	−	PROPN
iajs-3432	124	2	(	(	PUNCT
iajs-3432	124	3	𝐴3	𝐴3	PROPN
iajs-3432	124	4	)	)	PUNCT
iajs-3432	124	5	hold	hold	VERB
iajs-3432	124	6	,	,	PUNCT
iajs-3432	124	7	and	and	CCONJ
iajs-3432	124	8	limsup	limsup	NOUN
iajs-3432	124	9	𝑛→∞	𝑛→∞	NUM
iajs-3432	124	10	∑	∑	PROPN
iajs-3432	124	11	𝛿𝑖	𝛿𝑖	PROPN
iajs-3432	124	12	𝑛−1	𝑛−1	PROPN
iajs-3432	124	13	𝑖=𝑛∗	𝑖=𝑛∗	PUNCT
iajs-3432	124	14	=	=	SYM
iajs-3432	124	15	∞	∞	PROPN
iajs-3432	124	16	,	,	PUNCT
iajs-3432	124	17	𝑛∗	𝑛∗	PROPN
iajs-3432	124	18	≥	≥	PRON
iajs-3432	124	19	𝑛0	𝑛0	VERB
iajs-3432	124	20	.	.	PUNCT
iajs-3432	125	1	(	(	PUNCT
iajs-3432	125	2	11	11	NUM
iajs-3432	125	3	)	)	PUNCT
iajs-3432	125	4	then	then	ADV
iajs-3432	125	5	every	every	DET
iajs-3432	125	6	solution	solution	NOUN
iajs-3432	125	7	of	of	ADP
iajs-3432	125	8	equation	equation	NOUN
iajs-3432	125	9	(	(	PUNCT
iajs-3432	125	10	3	3	X
iajs-3432	125	11	)	)	PUNCT
iajs-3432	125	12	either	either	CCONJ
iajs-3432	125	13	oscillates	oscillate	VERB
iajs-3432	125	14	about	about	ADP
iajs-3432	125	15	a	a	DET
iajs-3432	125	16	unique	unique	ADJ
iajs-3432	125	17	equilibrium	equilibrium	NOUN
iajs-3432	125	18	𝐾	𝐾	NOUN
iajs-3432	125	19	or	or	CCONJ
iajs-3432	125	20	nonoscillatory	nonoscillatory	NOUN
iajs-3432	125	21	tends	tend	VERB
iajs-3432	125	22	to	to	ADP
iajs-3432	125	23	𝐾	𝐾	PROPN
iajs-3432	125	24	as	as	ADP
iajs-3432	125	25	𝑡	𝑡	PROPN
iajs-3432	125	26	→	→	SYM
iajs-3432	125	27	∞.	∞.	PROPN
iajs-3432	125	28	proof	proof	NOUN
iajs-3432	125	29	.	.	PUNCT
iajs-3432	126	1	assume	assume	VERB
iajs-3432	126	2	that	that	SCONJ
iajs-3432	126	3	equation	equation	NOUN
iajs-3432	126	4	(	(	PUNCT
iajs-3432	126	5	3	3	X
iajs-3432	126	6	)	)	PUNCT
iajs-3432	126	7	posse	posse	NOUN
iajs-3432	126	8	nonoscillatory	nonoscillatory	ADJ
iajs-3432	126	9	solution	solution	NOUN
iajs-3432	126	10	𝐻𝑛	𝐻𝑛	ADJ
iajs-3432	126	11	about	about	ADP
iajs-3432	126	12	𝐾	𝐾	PROPN
iajs-3432	126	13	,	,	PUNCT
iajs-3432	126	14	let	let	VERB
iajs-3432	126	15	𝐻𝑛	𝐻𝑛	PROPN
iajs-3432	126	16	>	>	X
iajs-3432	126	17	𝐾	𝐾	PROPN
iajs-3432	126	18	,	,	PUNCT
iajs-3432	126	19	𝑛	𝑛	DET
iajs-3432	126	20	≥	≥	NOUN
iajs-3432	126	21	𝑛0	𝑛0	VERB
iajs-3432	126	22	,	,	PUNCT
iajs-3432	126	23	hence	hence	ADV
iajs-3432	126	24	𝑈𝑛	𝑈𝑛	PROPN
iajs-3432	126	25	>	>	X
iajs-3432	126	26	0	0	PUNCT
iajs-3432	127	1	eventually	eventually	ADV
iajs-3432	127	2	.	.	PUNCT
iajs-3432	128	1	then	then	ADV
iajs-3432	128	2	by	by	ADP
iajs-3432	128	3	lemma	lemma	PROPN
iajs-3432	128	4	1	1	NUM
iajs-3432	128	5	:	:	PUNCT
iajs-3432	128	6	𝑍𝑛	𝑍𝑛	PROPN
iajs-3432	128	7	≥	≥	NUM
iajs-3432	128	8	0	0	NUM
iajs-3432	128	9	and	and	CCONJ
iajs-3432	128	10	∆𝑍𝑛	∆𝑍𝑛	NOUN
iajs-3432	128	11	≤	≤	NOUN
iajs-3432	128	12	0	0	NUM
iajs-3432	128	13	,	,	PUNCT
iajs-3432	128	14	hence	hence	ADV
iajs-3432	128	15	lim	lim	NOUN
iajs-3432	128	16	𝑛→∞	𝑛→∞	NUM
iajs-3432	128	17	𝑍𝑛	𝑍𝑛	PROPN
iajs-3432	128	18	=	=	SYM
iajs-3432	128	19	𝐿	𝐿	PROPN
iajs-3432	128	20	,	,	PUNCT
iajs-3432	128	21	where	where	SCONJ
iajs-3432	128	22	−∞	−∞	ADP
iajs-3432	128	23	≤	≤	PROPN
iajs-3432	128	24	𝐿	𝐿	PROPN
iajs-3432	128	25	<	<	X
iajs-3432	128	26	∞.	∞.	PROPN
iajs-3432	128	27	we	we	PRON
iajs-3432	128	28	look	look	VERB
iajs-3432	128	29	at	at	ADP
iajs-3432	128	30	two	two	NUM
iajs-3432	128	31	situations	situation	NOUN
iajs-3432	128	32	:	:	PUNCT
iajs-3432	128	33	case	case	NOUN
iajs-3432	128	34	1	1	NUM
iajs-3432	128	35	.	.	PUNCT
iajs-3432	129	1	if	if	SCONJ
iajs-3432	129	2	𝑈𝑛	𝑈𝑛	PROPN
iajs-3432	129	3	is	be	AUX
iajs-3432	129	4	unbounded	unbounded	ADJ
iajs-3432	129	5	.	.	PUNCT
iajs-3432	130	1	so	so	ADV
iajs-3432	130	2	there	there	PRON
iajs-3432	130	3	exists	exist	VERB
iajs-3432	130	4	a	a	DET
iajs-3432	130	5	subsequence	subsequence	NOUN
iajs-3432	130	6	𝑛𝑗	𝑛𝑗	ADP
iajs-3432	130	7	of	of	ADP
iajs-3432	130	8	𝑛	𝑛	PRON
iajs-3432	130	9	such	such	ADJ
iajs-3432	130	10	that	that	SCONJ
iajs-3432	130	11	lim	lim	PROPN
iajs-3432	130	12	𝑗→∞	𝑗→∞	NUM
iajs-3432	130	13	𝑛𝑗	𝑛𝑗	ADP
iajs-3432	130	14	=	=	SYM
iajs-3432	130	15	∞	∞	PROPN
iajs-3432	130	16	,	,	PUNCT
iajs-3432	130	17	lim	lim	PROPN
iajs-3432	130	18	𝑗→∞	𝑗→∞	NUM
iajs-3432	131	1	𝑈𝑛𝑗	𝑈𝑛𝑗	PROPN
iajs-3432	131	2	=	=	SYM
iajs-3432	131	3	∞	∞	PROPN
iajs-3432	131	4	and	and	CCONJ
iajs-3432	131	5	𝑈𝑛𝑗	𝑈𝑛𝑗	PROPN
iajs-3432	131	6	=	=	SYM
iajs-3432	131	7	max	max	PROPN
iajs-3432	131	8	{	{	PUNCT
iajs-3432	131	9	𝑈𝑠	𝑈𝑠	NOUN
iajs-3432	131	10	:	:	PUNCT
iajs-3432	131	11	𝑛0	𝑛0	VERB
iajs-3432	131	12	<	<	X
iajs-3432	131	13	𝑠	𝑠	X
iajs-3432	131	14	<	<	X
iajs-3432	131	15	𝑛𝑗	𝑛𝑗	ADP
iajs-3432	131	16	}	}	PUNCT
iajs-3432	131	17	.	.	PUNCT
iajs-3432	132	1	by	by	ADP
iajs-3432	132	2	using	use	VERB
iajs-3432	132	3	(	(	PUNCT
iajs-3432	132	4	𝐴4	𝐴4	PROPN
iajs-3432	132	5	)	)	PUNCT
iajs-3432	132	6	,	,	PUNCT
iajs-3432	132	7	we	we	PRON
iajs-3432	132	8	obtain	obtain	VERB
iajs-3432	132	9	from	from	ADP
iajs-3432	132	10	equation	equation	NOUN
iajs-3432	132	11	(	(	PUNCT
iajs-3432	132	12	8)	8)	NUM
iajs-3432	132	13	;	;	PUNCT
iajs-3432	132	14	ihjpas	ihjpa	NOUN
iajs-3432	132	15	.	.	PUNCT
iajs-3432	133	1	2024	2024	NUM
iajs-3432	133	2	,	,	PUNCT
iajs-3432	133	3	37	37	NUM
iajs-3432	133	4	(	(	PUNCT
iajs-3432	133	5	3	3	NUM
iajs-3432	133	6	)	)	PUNCT
iajs-3432	133	7	390	390	NUM
iajs-3432	133	8	𝑍𝑛𝑗	𝑍𝑛𝑗	PROPN
iajs-3432	133	9	=	=	SYM
iajs-3432	133	10	𝑈𝑛𝑗	𝑈𝑛𝑗	PROPN
iajs-3432	133	11	+	+	CCONJ
iajs-3432	133	12	∑	∑	PUNCT
iajs-3432	133	13	𝛽𝑖	𝛽𝑖	VERB
iajs-3432	133	14	𝑛𝑗+𝑙−1	𝑛𝑗+𝑙−1	PROPN
iajs-3432	133	15	𝑖=𝑛𝑗	𝑖=𝑛𝑗	PUNCT
iajs-3432	133	16	𝑉(𝑈𝑖−𝑙	𝑉(𝑈𝑖−𝑙	NUM
iajs-3432	133	17	)	)	PUNCT
iajs-3432	133	18	−	−	PROPN
iajs-3432	134	1	𝐹𝑛𝑗	𝐹𝑛𝑗	NOUN
iajs-3432	134	2	.	.	PUNCT
iajs-3432	135	1	≥	≥	X
iajs-3432	136	1	𝑈𝑛𝑗	𝑈𝑛𝑗	PROPN
iajs-3432	136	2	−	−	PROPN
iajs-3432	136	3	𝐹𝑛𝑗	𝐹𝑛𝑗	NOUN
iajs-3432	136	4	hence	hence	ADV
iajs-3432	136	5	lim	lim	NOUN
iajs-3432	136	6	𝑗→∞	𝑗→∞	NUM
iajs-3432	136	7	𝑍𝑛𝑗	𝑍𝑛𝑗	PROPN
iajs-3432	136	8	=	=	SYM
iajs-3432	136	9	∞	∞	PROPN
iajs-3432	136	10	leads	lead	VERB
iajs-3432	136	11	to	to	ADP
iajs-3432	136	12	a	a	DET
iajs-3432	136	13	contradiction	contradiction	NOUN
iajs-3432	136	14	.	.	PUNCT
iajs-3432	137	1	case	case	NOUN
iajs-3432	137	2	2	2	NUM
iajs-3432	137	3	.	.	PUNCT
iajs-3432	138	1	if	if	SCONJ
iajs-3432	138	2	𝑈𝑛	𝑈𝑛	PROPN
iajs-3432	138	3	is	be	AUX
iajs-3432	138	4	bounded	bound	VERB
iajs-3432	138	5	.	.	PUNCT
iajs-3432	139	1	from	from	ADP
iajs-3432	139	2	equation	equation	NOUN
iajs-3432	139	3	(	(	PUNCT
iajs-3432	139	4	9	9	NUM
iajs-3432	139	5	)	)	PUNCT
iajs-3432	139	6	and	and	CCONJ
iajs-3432	139	7	condition	condition	NOUN
iajs-3432	139	8	(	(	PUNCT
iajs-3432	139	9	𝐴2	𝐴2	PROPN
iajs-3432	139	10	)	)	PUNCT
iajs-3432	139	11	,	,	PUNCT
iajs-3432	139	12	we	we	PRON
iajs-3432	139	13	have	have	VERB
iajs-3432	139	14	∆𝑍𝑛	∆𝑍𝑛	NOUN
iajs-3432	139	15	≤	≤	ADJ
iajs-3432	140	1	−	−	PROPN
iajs-3432	140	2	(	(	PUNCT
iajs-3432	140	3	𝛿𝑛	𝛿𝑛	INTJ
iajs-3432	140	4	−	−	PROPN
iajs-3432	140	5	𝛽𝑛+𝑙𝜆2)𝑈𝑛	𝛽𝑛+𝑙𝜆2)𝑈𝑛	PROPN
iajs-3432	140	6	.	.	PUNCT
iajs-3432	141	1	let	let	VERB
iajs-3432	141	2	liminf	liminf	VERB
iajs-3432	141	3	𝑛→∞	𝑛→∞	VERB
iajs-3432	141	4	𝑈𝑛	𝑈𝑛	VERB
iajs-3432	141	5	=	=	SYM
iajs-3432	141	6	𝑙	𝑙	PRON
iajs-3432	141	7	≥	≥	NOUN
iajs-3432	141	8	0	0	NUM
iajs-3432	141	9	.	.	PUNCT
iajs-3432	142	1	by	by	ADP
iajs-3432	142	2	taking	take	VERB
iajs-3432	142	3	summation	summation	NOUN
iajs-3432	142	4	to	to	ADP
iajs-3432	142	5	both	both	DET
iajs-3432	142	6	sides	side	NOUN
iajs-3432	142	7	of	of	ADP
iajs-3432	142	8	the	the	DET
iajs-3432	142	9	above	above	ADJ
iajs-3432	142	10	inequality	inequality	NOUN
iajs-3432	142	11	,	,	PUNCT
iajs-3432	142	12	it	it	PRON
iajs-3432	142	13	yields	yield	VERB
iajs-3432	142	14	:	:	PUNCT
iajs-3432	142	15	∑	∑	PUNCT
iajs-3432	142	16	∆𝑍𝑖	∆𝑍𝑖	PROPN
iajs-3432	142	17	≤	≤	NOUN
iajs-3432	142	18	−	−	NOUN
iajs-3432	142	19	∑	∑	PROPN
iajs-3432	142	20	(	(	PUNCT
iajs-3432	142	21	𝛿𝑖	𝛿𝑖	NOUN
iajs-3432	142	22	−	−	NOUN
iajs-3432	142	23	𝛽𝑖+𝑙𝜆2)𝑈𝑖	𝛽𝑖+𝑙𝜆2)𝑈𝑖	VERB
iajs-3432	142	24	𝑛−1	𝑛−1	NUM
iajs-3432	142	25	𝑖=𝑛0	𝑖=𝑛0	NOUN
iajs-3432	142	26	𝑛−1	𝑛−1	PROPN
iajs-3432	142	27	𝑖=𝑛0	𝑖=𝑛0	NOUN
iajs-3432	142	28	.	.	PUNCT
iajs-3432	143	1	then	then	ADV
iajs-3432	143	2	𝑍𝑛	𝑍𝑛	PROPN
iajs-3432	143	3	−	−	PROPN
iajs-3432	143	4	𝑍𝑛0	𝑍𝑛0	VERB
iajs-3432	143	5	≤	≤	ADV
iajs-3432	143	6	−	−	PROPN
iajs-3432	143	7	∑	∑	PROPN
iajs-3432	143	8	(	(	PUNCT
iajs-3432	143	9	𝛿𝑖	𝛿𝑖	NOUN
iajs-3432	143	10	−	−	NOUN
iajs-3432	144	1	𝛽𝑖+𝑙𝜆2)𝑈𝑖	𝛽𝑖+𝑙𝜆2)𝑈𝑖	VERB
iajs-3432	144	2	𝑛−1	𝑛−1	PROPN
iajs-3432	144	3	𝑖=𝑛0	𝑖=𝑛0	NUM
iajs-3432	144	4	.	.	PUNCT
iajs-3432	145	1	(	(	PUNCT
iajs-3432	145	2	12	12	NUM
iajs-3432	145	3	)	)	PUNCT
iajs-3432	145	4	we	we	PRON
iajs-3432	145	5	claim	claim	VERB
iajs-3432	145	6	that	that	SCONJ
iajs-3432	145	7	𝑙	𝑙	X
iajs-3432	145	8	=	=	SYM
iajs-3432	145	9	0	0	NUM
iajs-3432	145	10	,	,	PUNCT
iajs-3432	145	11	otherwise	otherwise	ADV
iajs-3432	145	12	if	if	SCONJ
iajs-3432	145	13	𝑙	𝑙	PROPN
iajs-3432	145	14	>	>	X
iajs-3432	145	15	0	0	NUM
iajs-3432	145	16	,	,	PUNCT
iajs-3432	145	17	then	then	ADV
iajs-3432	145	18	there	there	PRON
iajs-3432	145	19	exists	exist	VERB
iajs-3432	145	20	𝑛1	𝑛1	ADJ
iajs-3432	145	21	≥	≥	NOUN
iajs-3432	145	22	𝑛0	𝑛0	VERB
iajs-3432	145	23	large	large	ADJ
iajs-3432	145	24	enough	enough	ADV
iajs-3432	145	25	such	such	ADJ
iajs-3432	145	26	that	that	SCONJ
iajs-3432	145	27	𝑈𝑛	𝑈𝑛	PROPN
iajs-3432	145	28	≥	≥	NUM
iajs-3432	145	29	𝑙	𝑙	NUM
iajs-3432	145	30	,	,	PUNCT
iajs-3432	145	31	𝑛	𝑛	DET
iajs-3432	145	32	≥	≥	NOUN
iajs-3432	145	33	𝑛1	𝑛1	NOUN
iajs-3432	145	34	.	.	PUNCT
iajs-3432	146	1	from	from	ADP
iajs-3432	146	2	equation	equation	NOUN
iajs-3432	146	3	(	(	PUNCT
iajs-3432	146	4	12	12	NUM
iajs-3432	146	5	)	)	PUNCT
iajs-3432	146	6	,	,	PUNCT
iajs-3432	146	7	we	we	PRON
iajs-3432	146	8	have	have	VERB
iajs-3432	146	9	𝑍𝑛	𝑍𝑛	PROPN
iajs-3432	146	10	−	−	PROPN
iajs-3432	146	11	𝑍𝑛0	𝑍𝑛0	VERB
iajs-3432	146	12	≤	≤	NUM
iajs-3432	146	13	−𝑙	−𝑙	NOUN
iajs-3432	146	14	∑	∑	PROPN
iajs-3432	146	15	(	(	PUNCT
iajs-3432	146	16	𝛿𝑖	𝛿𝑖	PROPN
iajs-3432	146	17	−	−	PROPN
iajs-3432	146	18	𝛽𝑖+𝑙𝜆2	𝛽𝑖+𝑙𝜆2	PROPN
iajs-3432	146	19	)	)	PUNCT
iajs-3432	146	20	𝑛−1	𝑛−1	NUM
iajs-3432	146	21	𝑖=𝑛1	𝑖=𝑛1	NOUN
iajs-3432	146	22	.	.	PUNCT
iajs-3432	147	1	let	let	VERB
iajs-3432	147	2	𝑛	𝑛	PRON
iajs-3432	147	3	→	→	SYM
iajs-3432	147	4	∞	∞	PROPN
iajs-3432	147	5	,	,	PUNCT
iajs-3432	147	6	in	in	ADP
iajs-3432	147	7	virtue	virtue	NOUN
iajs-3432	147	8	of	of	ADP
iajs-3432	147	9	equation	equation	NOUN
iajs-3432	147	10	(	(	PUNCT
iajs-3432	147	11	11	11	NUM
iajs-3432	147	12	)	)	PUNCT
iajs-3432	147	13	and	and	CCONJ
iajs-3432	147	14	𝐴1	𝐴1	PROPN
iajs-3432	147	15	,	,	PUNCT
iajs-3432	147	16	the	the	DET
iajs-3432	147	17	last	last	ADJ
iajs-3432	147	18	inequality	inequality	NOUN
iajs-3432	147	19	leads	lead	VERB
iajs-3432	147	20	to	to	ADP
iajs-3432	147	21	lim	lim	PROPN
iajs-3432	147	22	𝑛→∞	𝑛→∞	NUM
iajs-3432	148	1	𝑍𝑛	𝑍𝑛	PROPN
iajs-3432	148	2	=	=	SYM
iajs-3432	148	3	−∞	−∞	PRON
iajs-3432	148	4	which	which	PRON
iajs-3432	148	5	is	be	AUX
iajs-3432	148	6	a	a	DET
iajs-3432	148	7	contradiction	contradiction	NOUN
iajs-3432	148	8	.	.	PUNCT
iajs-3432	149	1	hence	hence	ADV
iajs-3432	149	2	,	,	PUNCT
iajs-3432	149	3	liminf	liminf	INTJ
iajs-3432	149	4	𝑛→∞	𝑛→∞	PUNCT
iajs-3432	149	5	𝑈𝑛	𝑈𝑛	NOUN
iajs-3432	149	6	=	=	SYM
iajs-3432	149	7	0	0	NUM
iajs-3432	149	8	,	,	PUNCT
iajs-3432	149	9	then	then	ADV
iajs-3432	149	10	there	there	PRON
iajs-3432	149	11	exists	exist	VERB
iajs-3432	149	12	a	a	DET
iajs-3432	149	13	subsequence	subsequence	NOUN
iajs-3432	149	14	𝑛𝑗	𝑛𝑗	ADP
iajs-3432	149	15	such	such	ADJ
iajs-3432	149	16	that	that	SCONJ
iajs-3432	149	17	lim	lim	PROPN
iajs-3432	149	18	𝑗→∞	𝑗→∞	NUM
iajs-3432	150	1	𝑈𝑛𝑗	𝑈𝑛𝑗	PROPN
iajs-3432	150	2	=	=	NOUN
iajs-3432	150	3	0	0	NUM
iajs-3432	150	4	.	.	PUNCT
iajs-3432	151	1	from	from	ADP
iajs-3432	151	2	equation	equation	NOUN
iajs-3432	151	3	(	(	PUNCT
iajs-3432	151	4	8)	8)	NUM
iajs-3432	151	5	,	,	PUNCT
iajs-3432	151	6	it	it	PRON
iajs-3432	151	7	can	can	AUX
iajs-3432	151	8	be	be	AUX
iajs-3432	151	9	easily	easily	ADV
iajs-3432	151	10	got	get	VERB
iajs-3432	151	11	𝑍𝑛	𝑍𝑛	PROPN
iajs-3432	151	12	≥	≥	NOUN
iajs-3432	151	13	𝑈𝑛	𝑈𝑛	PROPN
iajs-3432	151	14	,	,	PUNCT
iajs-3432	151	15	𝑛	𝑛	DET
iajs-3432	151	16	≥	≥	NOUN
iajs-3432	151	17	𝑛2	𝑛2	NOUN
iajs-3432	151	18	≥	≥	NOUN
iajs-3432	151	19	𝑛0	𝑛0	VERB
iajs-3432	151	20	.	.	PUNCT
iajs-3432	152	1	so	so	ADV
iajs-3432	152	2	from	from	ADP
iajs-3432	152	3	equation	equation	NOUN
iajs-3432	152	4	(	(	PUNCT
iajs-3432	152	5	8)	8)	NUM
iajs-3432	152	6	,	,	PUNCT
iajs-3432	152	7	it	it	PRON
iajs-3432	152	8	follows	follow	VERB
iajs-3432	152	9	𝑍𝑛	𝑍𝑛	PROPN
iajs-3432	152	10	≤	≤	PRON
iajs-3432	152	11	𝑈𝑛	𝑈𝑛	PROPN
iajs-3432	152	12	+	+	CCONJ
iajs-3432	152	13	∑	∑	PUNCT
iajs-3432	152	14	𝛽𝑖	𝛽𝑖	ADJ
iajs-3432	152	15	𝑛+𝑙−1	𝑛+𝑙−1	PROPN
iajs-3432	152	16	𝑖=𝑛	𝑖=𝑛	PROPN
iajs-3432	152	17	𝜆2𝑈𝑖−𝑙	𝜆2𝑈𝑖−𝑙	NOUN
iajs-3432	152	18	−	−	PROPN
iajs-3432	153	1	𝐹𝑛	𝐹𝑛	PROPN
iajs-3432	153	2	,	,	PUNCT
iajs-3432	153	3	≤	≤	NUM
iajs-3432	153	4	𝑈𝑛	𝑈𝑛	PROPN
iajs-3432	153	5	+	+	CCONJ
iajs-3432	153	6	𝜆2	𝜆2	NOUN
iajs-3432	153	7	∑	∑	PUNCT
iajs-3432	153	8	𝛽𝑖	𝛽𝑖	VERB
iajs-3432	153	9	𝑛+𝑙−1	𝑛+𝑙−1	PROPN
iajs-3432	153	10	𝑖=𝑛	𝑖=𝑛	PROPN
iajs-3432	153	11	𝑍𝑖−𝑙	𝑍𝑖−𝑙	PROPN
iajs-3432	154	1	−	−	PROPN
iajs-3432	154	2	𝐹𝑛	𝐹𝑛	PROPN
iajs-3432	154	3	,	,	PUNCT
iajs-3432	154	4	therefore	therefore	ADV
iajs-3432	154	5	𝑍𝑛𝑗	𝑍𝑛𝑗	PROPN
iajs-3432	154	6	≤	≤	PUNCT
iajs-3432	154	7	𝑈𝑛𝑗	𝑈𝑛𝑗	PROPN
iajs-3432	154	8	+	+	NUM
iajs-3432	154	9	𝜆2𝑍𝑛−𝑙	𝜆2𝑍𝑛−𝑙	NOUN
iajs-3432	154	10	∑	∑	PUNCT
iajs-3432	154	11	𝛽𝑖	𝛽𝑖	VERB
iajs-3432	154	12	𝑛+𝑙−1	𝑛+𝑙−1	PROPN
iajs-3432	154	13	𝑖=𝑛	𝑖=𝑛	PROPN
iajs-3432	154	14	−	−	PUNCT
iajs-3432	155	1	𝐹𝑛𝑗	𝐹𝑛𝑗	NOUN
iajs-3432	155	2	≤	≤	X
iajs-3432	156	1	𝑈𝑛𝑗	𝑈𝑛𝑗	PROPN
iajs-3432	156	2	+	+	CCONJ
iajs-3432	156	3	𝜇2	𝜇2	NOUN
iajs-3432	156	4	∑	∑	PUNCT
iajs-3432	156	5	𝛽𝑖	𝛽𝑖	VERB
iajs-3432	156	6	𝑛+𝑙−1	𝑛+𝑙−1	PROPN
iajs-3432	156	7	𝑖=𝑛	𝑖=𝑛	PROPN
iajs-3432	156	8	−	−	PROPN
iajs-3432	157	1	𝐹𝑛𝑗	𝐹𝑛𝑗	NOUN
iajs-3432	157	2	.	.	PUNCT
iajs-3432	158	1	where	where	SCONJ
iajs-3432	158	2	𝜆2𝑍𝑛−𝑙	𝜆2𝑍𝑛−𝑙	NOUN
iajs-3432	158	3	≤	≤	PROPN
iajs-3432	158	4	𝜇2	𝜇2	NOUN
iajs-3432	158	5	.	.	PUNCT
iajs-3432	159	1	then	then	ADV
iajs-3432	159	2	by	by	ADP
iajs-3432	159	3	𝐴1	𝐴1	PROPN
iajs-3432	159	4	and	and	CCONJ
iajs-3432	159	5	𝐴3	𝐴3	PROPN
iajs-3432	159	6	,	,	PUNCT
iajs-3432	159	7	we	we	PRON
iajs-3432	159	8	obtain	obtain	VERB
iajs-3432	159	9	lim	lim	PROPN
iajs-3432	159	10	𝑗→∞	𝑗→∞	NUM
iajs-3432	160	1	𝑍𝑛𝑗	𝑍𝑛𝑗	PROPN
iajs-3432	160	2	≤	≤	PROPN
iajs-3432	160	3	lim	lim	PROPN
iajs-3432	160	4	𝑗→∞	𝑗→∞	NUM
iajs-3432	161	1	𝑈𝑛𝑗	𝑈𝑛𝑗	PROPN
iajs-3432	161	2	=	=	NOUN
iajs-3432	161	3	0	0	PUNCT
iajs-3432	162	1	thus	thus	ADV
iajs-3432	162	2	,	,	PUNCT
iajs-3432	162	3	lim	lim	PROPN
iajs-3432	162	4	𝑛→∞	𝑛→∞	NUM
iajs-3432	162	5	𝑍𝑛	𝑍𝑛	PROPN
iajs-3432	162	6	=	=	SYM
iajs-3432	162	7	0	0	NUM
iajs-3432	162	8	implies	imply	VERB
iajs-3432	162	9	that	that	SCONJ
iajs-3432	162	10	lim	lim	PROPN
iajs-3432	162	11	𝑛→∞	𝑛→∞	NUM
iajs-3432	162	12	𝑈𝑛	𝑈𝑛	PROPN
iajs-3432	162	13	=	=	SYM
iajs-3432	162	14	0	0	NUM
iajs-3432	162	15	that	that	PRON
iajs-3432	162	16	is	be	AUX
iajs-3432	162	17	lim	lim	NOUN
iajs-3432	162	18	𝑛→∞	𝑛→∞	NUM
iajs-3432	163	1	𝐻𝑛	𝐻𝑛	PROPN
iajs-3432	163	2	=	=	SYM
iajs-3432	163	3	𝐾.	𝐾.	NOUN
iajs-3432	163	4	the	the	DET
iajs-3432	163	5	proof	proof	NOUN
iajs-3432	163	6	is	be	AUX
iajs-3432	163	7	finished	finish	VERB
iajs-3432	163	8	.	.	PUNCT
iajs-3432	164	1	in	in	ADP
iajs-3432	164	2	the	the	DET
iajs-3432	164	3	next	next	ADJ
iajs-3432	164	4	result	result	NOUN
iajs-3432	164	5	,	,	PUNCT
iajs-3432	164	6	the	the	DET
iajs-3432	164	7	sequence	sequence	NOUN
iajs-3432	164	8	𝑊𝑛	𝑊𝑛	PROPN
iajs-3432	164	9	will	will	AUX
iajs-3432	164	10	be	be	AUX
iajs-3432	164	11	used	use	VERB
iajs-3432	164	12	:	:	PUNCT
iajs-3432	164	13	𝑊𝑛	𝑊𝑛	PROPN
iajs-3432	164	14	=	=	SYM
iajs-3432	164	15	𝑈𝑛	𝑈𝑛	PROPN
iajs-3432	164	16	+	+	PROPN
iajs-3432	164	17	∑	∑	PROPN
iajs-3432	164	18	𝛿𝑖	𝛿𝑖	PROPN
iajs-3432	164	19	𝑛−1	𝑛−1	NUM
iajs-3432	164	20	𝑖=𝑛−𝑙	𝑖=𝑛−𝑙	PUNCT
iajs-3432	165	1	𝑈𝑖	𝑈𝑖	PROPN
iajs-3432	165	2	−	−	PROPN
iajs-3432	165	3	𝐹𝑛	𝐹𝑛	PROPN
iajs-3432	165	4	.	.	PUNCT
iajs-3432	166	1	(	(	PUNCT
iajs-3432	166	2	13	13	X
iajs-3432	166	3	)	)	PUNCT
iajs-3432	166	4	lemma	lemma	PROPN
iajs-3432	166	5	2	2	NUM
iajs-3432	166	6	:	:	PUNCT
iajs-3432	166	7	assume	assume	VERB
iajs-3432	166	8	,	,	PUNCT
iajs-3432	166	9	𝛽𝑛𝜆2	𝛽𝑛𝜆2	ADJ
iajs-3432	166	10	−	−	PROPN
iajs-3432	166	11	𝛿𝑛−𝑙	𝛿𝑛−𝑙	NOUN
iajs-3432	166	12	≤	≤	NOUN
iajs-3432	166	13	0	0	PUNCT
iajs-3432	167	1	and	and	CCONJ
iajs-3432	167	2	suppose	suppose	VERB
iajs-3432	167	3	that	that	SCONJ
iajs-3432	167	4	(	(	PUNCT
iajs-3432	167	5	𝐴1	𝐴1	PROPN
iajs-3432	167	6	)	)	PUNCT
iajs-3432	168	1	−	−	PROPN
iajs-3432	168	2	(	(	PUNCT
iajs-3432	168	3	𝐴3	𝐴3	PROPN
iajs-3432	168	4	)	)	PUNCT
iajs-3432	168	5	hold	hold	VERB
iajs-3432	168	6	.	.	PUNCT
iajs-3432	169	1	let	let	VERB
iajs-3432	169	2	𝐻𝑛	𝐻𝑛	PROPN
iajs-3432	169	3	be	be	AUX
iajs-3432	169	4	a	a	DET
iajs-3432	169	5	nonoscillatory	nonoscillatory	ADJ
iajs-3432	169	6	solution	solution	NOUN
iajs-3432	169	7	to	to	ADP
iajs-3432	169	8	𝐾	𝐾	PROPN
iajs-3432	169	9	of	of	ADP
iajs-3432	169	10	equation	equation	NOUN
iajs-3432	169	11	(	(	PUNCT
iajs-3432	169	12	3	3	NUM
iajs-3432	169	13	)	)	PUNCT
iajs-3432	169	14	.	.	PUNCT
iajs-3432	170	1	then	then	ADV
iajs-3432	170	2	,	,	PUNCT
iajs-3432	170	3	𝑊𝑛	𝑊𝑛	PROPN
iajs-3432	170	4	≥	≥	NUM
iajs-3432	170	5	0	0	NUM
iajs-3432	170	6	and	and	CCONJ
iajs-3432	170	7	a	a	DET
iajs-3432	170	8	nonincreasing	nonincrease	VERB
iajs-3432	170	9	sequence	sequence	NOUN
iajs-3432	170	10	.	.	PUNCT
iajs-3432	171	1	proof	proof	NOUN
iajs-3432	171	2	.	.	PUNCT
iajs-3432	172	1	let	let	VERB
iajs-3432	172	2	𝐻𝑛	𝐻𝑛	PROPN
iajs-3432	172	3	be	be	AUX
iajs-3432	172	4	a	a	DET
iajs-3432	172	5	nonoscillatory	nonoscillatory	ADJ
iajs-3432	172	6	solution	solution	NOUN
iajs-3432	172	7	to	to	ADP
iajs-3432	172	8	𝐾	𝐾	PROPN
iajs-3432	172	9	of	of	ADP
iajs-3432	172	10	equation	equation	NOUN
iajs-3432	172	11	(	(	PUNCT
iajs-3432	172	12	3	3	NUM
iajs-3432	172	13	)	)	PUNCT
iajs-3432	172	14	.	.	PUNCT
iajs-3432	173	1	let	let	VERB
iajs-3432	173	2	𝐻𝑛	𝐻𝑛	VERB
iajs-3432	173	3	>	>	X
iajs-3432	173	4	𝐾	𝐾	PROPN
iajs-3432	173	5	,	,	PUNCT
iajs-3432	173	6	𝑛	𝑛	DET
iajs-3432	173	7	≥	≥	NOUN
iajs-3432	173	8	𝑛0	𝑛0	VERB
iajs-3432	173	9	.	.	PUNCT
iajs-3432	174	1	from	from	ADP
iajs-3432	174	2	equation	equation	NOUN
iajs-3432	174	3	(	(	PUNCT
iajs-3432	174	4	13	13	NUM
iajs-3432	174	5	)	)	PUNCT
iajs-3432	174	6	and	and	CCONJ
iajs-3432	174	7	equation	equation	NOUN
iajs-3432	174	8	(	(	PUNCT
iajs-3432	174	9	3	3	NUM
iajs-3432	174	10	)	)	PUNCT
iajs-3432	174	11	,	,	PUNCT
iajs-3432	174	12	we	we	PRON
iajs-3432	174	13	have	have	VERB
iajs-3432	174	14	∆𝑊𝑛	∆𝑊𝑛	PROPN
iajs-3432	174	15	=	=	SYM
iajs-3432	174	16	𝛽𝑛𝑉(𝑈𝑛−𝑙	𝛽𝑛𝑉(𝑈𝑛−𝑙	NOUN
iajs-3432	174	17	)	)	PUNCT
iajs-3432	174	18	−	−	PROPN
iajs-3432	174	19	𝛿𝑛−𝑙𝑈𝑛−𝑙	𝛿𝑛−𝑙𝑈𝑛−𝑙	PROPN
iajs-3432	174	20	≤	≤	NOUN
iajs-3432	174	21	(	(	PUNCT
iajs-3432	174	22	𝛽𝑛𝜆2	𝛽𝑛𝜆2	ADJ
iajs-3432	174	23	−	−	PROPN
iajs-3432	174	24	𝛿𝑛−𝑙)𝑈𝑛−𝑙	𝛿𝑛−𝑙)𝑈𝑛−𝑙	PROPN
iajs-3432	174	25	≤	≤	NOUN
iajs-3432	174	26	0	0	NUM
iajs-3432	174	27	.	.	PUNCT
iajs-3432	175	1	(	(	PUNCT
iajs-3432	175	2	14	14	NUM
iajs-3432	175	3	)	)	PUNCT
iajs-3432	175	4	hence	hence	ADV
iajs-3432	175	5	,	,	PUNCT
iajs-3432	175	6	𝑊𝑛	𝑊𝑛	PROPN
iajs-3432	175	7	is	be	AUX
iajs-3432	175	8	a	a	DET
iajs-3432	175	9	nonincreasing	nonincrease	VERB
iajs-3432	175	10	sequence	sequence	NOUN
iajs-3432	175	11	and	and	CCONJ
iajs-3432	175	12	lim	lim	PROPN
iajs-3432	175	13	𝑛→∞	𝑛→∞	AUX
iajs-3432	176	1	𝑊𝑛	𝑊𝑛	PROPN
iajs-3432	176	2	=	=	SYM
iajs-3432	176	3	𝐿	𝐿	PROPN
iajs-3432	176	4	,	,	PUNCT
iajs-3432	176	5	where	where	SCONJ
iajs-3432	176	6	−∞	−∞	ADP
iajs-3432	176	7	≤	≤	PROPN
iajs-3432	176	8	𝐿	𝐿	PROPN
iajs-3432	176	9	<	<	X
iajs-3432	176	10	∞.	∞.	PROPN
iajs-3432	176	11	we	we	PRON
iajs-3432	176	12	claim	claim	VERB
iajs-3432	176	13	that	that	SCONJ
iajs-3432	176	14	𝐿	𝐿	PROPN
iajs-3432	176	15	≥	≥	NOUN
iajs-3432	176	16	0	0	NUM
iajs-3432	176	17	.	.	PUNCT
iajs-3432	177	1	otherwise	otherwise	ADV
iajs-3432	177	2	,	,	PUNCT
iajs-3432	177	3	𝐿	𝐿	PROPN
iajs-3432	177	4	<	<	X
iajs-3432	177	5	0	0	NUM
iajs-3432	177	6	,	,	PUNCT
iajs-3432	177	7	and	and	CCONJ
iajs-3432	177	8	then	then	ADV
iajs-3432	177	9	there	there	PRON
iajs-3432	177	10	exists	exist	VERB
iajs-3432	177	11	𝑛1	𝑛1	ADJ
iajs-3432	177	12	≥	≥	NOUN
iajs-3432	177	13	𝑛0	𝑛0	VERB
iajs-3432	177	14	and	and	CCONJ
iajs-3432	177	15	𝛼	𝛼	ADJ
iajs-3432	177	16	<	<	X
iajs-3432	177	17	0	0	NUM
iajs-3432	177	18	,	,	PUNCT
iajs-3432	177	19	such	such	ADJ
iajs-3432	177	20	that	that	SCONJ
iajs-3432	178	1	𝑊𝑛	𝑊𝑛	PROPN
iajs-3432	178	2	≤	≤	NOUN
iajs-3432	178	3	𝛼	𝛼	X
iajs-3432	178	4	<	<	X
iajs-3432	178	5	0	0	NUM
iajs-3432	178	6	for	for	ADP
iajs-3432	178	7	𝑛	𝑛	PROPN
iajs-3432	178	8	≥	≥	NOUN
iajs-3432	178	9	𝑛1	𝑛1	NOUN
iajs-3432	178	10	.	.	PUNCT
iajs-3432	179	1	from	from	ADP
iajs-3432	179	2	equation	equation	NOUN
iajs-3432	179	3	(	(	PUNCT
iajs-3432	179	4	13	13	NUM
iajs-3432	179	5	)	)	PUNCT
iajs-3432	179	6	,	,	PUNCT
iajs-3432	179	7	we	we	PRON
iajs-3432	179	8	get	get	VERB
iajs-3432	180	1	𝑈𝑛	𝑈𝑛	PROPN
iajs-3432	180	2	=	=	SYM
iajs-3432	180	3	𝑊𝑛	𝑊𝑛	PROPN
iajs-3432	180	4	−	−	PROPN
iajs-3432	180	5	∑	∑	VERB
iajs-3432	180	6	𝛿𝑖	𝛿𝑖	PROPN
iajs-3432	180	7	𝑛−1	𝑛−1	NUM
iajs-3432	180	8	𝑖=𝑛−𝑙	𝑖=𝑛−𝑙	PUNCT
iajs-3432	181	1	𝑈𝑖	𝑈𝑖	PROPN
iajs-3432	181	2	+	+	CCONJ
iajs-3432	181	3	𝐹𝑛	𝐹𝑛	PROPN
iajs-3432	181	4	≤	≤	NUM
iajs-3432	181	5	𝛼	𝛼	NOUN
iajs-3432	182	1	+	+	NOUN
iajs-3432	182	2	𝐹𝑛	𝐹𝑛	PROPN
iajs-3432	182	3	,	,	PUNCT
iajs-3432	182	4	𝑈𝑛	𝑈𝑛	VERB
iajs-3432	182	5	<	<	X
iajs-3432	182	6	𝛼	𝛼	NOUN
iajs-3432	182	7	+	+	CCONJ
iajs-3432	182	8	휀	휀	X
iajs-3432	182	9	,	,	PUNCT
iajs-3432	182	10	𝑛	𝑛	DET
iajs-3432	182	11	≥	≥	NOUN
iajs-3432	182	12	𝑛2	𝑛2	PROPN
iajs-3432	182	13	≥	≥	NOUN
iajs-3432	182	14	𝑛1	𝑛1	PROPN
iajs-3432	182	15	,	,	PUNCT
iajs-3432	182	16	휀	휀	NOUN
iajs-3432	182	17	>	>	X
iajs-3432	182	18	0	0	X
iajs-3432	182	19	.	.	PUNCT
iajs-3432	183	1	since	since	SCONJ
iajs-3432	183	2	휀	휀	NOUN
iajs-3432	183	3	is	be	AUX
iajs-3432	183	4	arbitrary	arbitrary	ADJ
iajs-3432	183	5	,	,	PUNCT
iajs-3432	183	6	the	the	DET
iajs-3432	183	7	last	last	ADJ
iajs-3432	183	8	inequality	inequality	NOUN
iajs-3432	183	9	leads	lead	VERB
iajs-3432	183	10	to	to	ADP
iajs-3432	183	11	𝑈𝑛	𝑈𝑛	PROPN
iajs-3432	183	12	≤	≤	NUM
iajs-3432	183	13	𝛼	𝛼	NOUN
iajs-3432	183	14	.	.	PUNCT
iajs-3432	184	1	this	this	PRON
iajs-3432	184	2	is	be	AUX
iajs-3432	184	3	a	a	DET
iajs-3432	184	4	contradiction	contradiction	NOUN
iajs-3432	184	5	in	in	ADP
iajs-3432	184	6	terms	term	NOUN
iajs-3432	184	7	of	of	ADP
iajs-3432	184	8	the	the	DET
iajs-3432	184	9	fact	fact	NOUN
iajs-3432	184	10	ihjpas	ihjpa	VERB
iajs-3432	184	11	.	.	PUNCT
iajs-3432	185	1	2024	2024	NUM
iajs-3432	185	2	,	,	PUNCT
iajs-3432	185	3	37	37	NUM
iajs-3432	185	4	(	(	PUNCT
iajs-3432	185	5	3	3	NUM
iajs-3432	185	6	)	)	PUNCT
iajs-3432	185	7	391	391	NUM
iajs-3432	185	8	that	that	PRON
iajs-3432	185	9	𝑈𝑛	𝑈𝑛	PROPN
iajs-3432	185	10	is	be	AUX
iajs-3432	185	11	positive	positive	ADJ
iajs-3432	185	12	.	.	PUNCT
iajs-3432	186	1	so	so	ADV
iajs-3432	186	2	,	,	PUNCT
iajs-3432	186	3	since	since	SCONJ
iajs-3432	186	4	our	our	PRON
iajs-3432	186	5	claim	claim	NOUN
iajs-3432	186	6	has	have	AUX
iajs-3432	186	7	been	be	AUX
iajs-3432	186	8	proven,𝐿	proven,𝐿	NOUN
iajs-3432	186	9	≥	≥	NOUN
iajs-3432	186	10	0	0	NUM
iajs-3432	186	11	or	or	CCONJ
iajs-3432	186	12	𝑊𝑛	𝑊𝑛	PROPN
iajs-3432	186	13	≥	≥	NOUN
iajs-3432	186	14	0	0	NUM
iajs-3432	186	15	.	.	PUNCT
iajs-3432	187	1	theorem	theorem	ADJ
iajs-3432	187	2	2	2	NUM
iajs-3432	187	3	assume	assume	VERB
iajs-3432	187	4	𝛽𝑛𝜆2	𝛽𝑛𝜆2	ADJ
iajs-3432	187	5	−	−	PROPN
iajs-3432	187	6	𝛿𝑛−𝑙	𝛿𝑛−𝑙	NOUN
iajs-3432	187	7	≤	≤	NOUN
iajs-3432	187	8	𝛾	𝛾	ADP
iajs-3432	187	9	<	<	X
iajs-3432	187	10	0	0	PUNCT
iajs-3432	187	11	and	and	CCONJ
iajs-3432	187	12	let	let	VERB
iajs-3432	187	13	(	(	PUNCT
iajs-3432	187	14	𝐴1	𝐴1	PROPN
iajs-3432	187	15	)	)	PUNCT
iajs-3432	188	1	−	−	PROPN
iajs-3432	188	2	(	(	PUNCT
iajs-3432	188	3	𝐴4	𝐴4	PROPN
iajs-3432	188	4	)	)	PUNCT
iajs-3432	188	5	hold	hold	VERB
iajs-3432	188	6	.	.	PUNCT
iajs-3432	189	1	then	then	ADV
iajs-3432	189	2	,	,	PUNCT
iajs-3432	189	3	every	every	DET
iajs-3432	189	4	solution	solution	NOUN
iajs-3432	189	5	of	of	ADP
iajs-3432	189	6	equation	equation	NOUN
iajs-3432	189	7	(	(	PUNCT
iajs-3432	189	8	3	3	X
iajs-3432	189	9	)	)	PUNCT
iajs-3432	189	10	either	either	CCONJ
iajs-3432	189	11	oscillates	oscillate	NOUN
iajs-3432	189	12	about	about	ADP
iajs-3432	189	13	a	a	DET
iajs-3432	189	14	positive	positive	ADJ
iajs-3432	189	15	equilibrium	equilibrium	NOUN
iajs-3432	189	16	𝐾	𝐾	NOUN
iajs-3432	189	17	or	or	CCONJ
iajs-3432	189	18	nonoscillatory	nonoscillatory	NOUN
iajs-3432	189	19	tends	tend	VERB
iajs-3432	189	20	to	to	ADP
iajs-3432	189	21	𝐾	𝐾	PROPN
iajs-3432	189	22	as	as	ADP
iajs-3432	189	23	𝑡	𝑡	PROPN
iajs-3432	189	24	→	→	SYM
iajs-3432	189	25	∞.	∞.	PROPN
iajs-3432	189	26	proof	proof	NOUN
iajs-3432	189	27	.	.	PUNCT
iajs-3432	190	1	let	let	VERB
iajs-3432	190	2	𝐻𝑛	𝐻𝑛	PROPN
iajs-3432	190	3	be	be	AUX
iajs-3432	190	4	a	a	DET
iajs-3432	190	5	nonoscillatory	nonoscillatory	ADJ
iajs-3432	190	6	solution	solution	NOUN
iajs-3432	190	7	to	to	ADP
iajs-3432	190	8	𝐾	𝐾	PROPN
iajs-3432	190	9	of	of	ADP
iajs-3432	190	10	equation	equation	NOUN
iajs-3432	190	11	(	(	PUNCT
iajs-3432	190	12	3	3	NUM
iajs-3432	190	13	)	)	PUNCT
iajs-3432	190	14	;	;	PUNCT
iajs-3432	190	15	assume	assume	VERB
iajs-3432	190	16	𝐻𝑛	𝐻𝑛	VERB
iajs-3432	190	17	>	>	X
iajs-3432	190	18	𝐾	𝐾	PROPN
iajs-3432	190	19	eventually	eventually	ADV
iajs-3432	190	20	,	,	PUNCT
iajs-3432	190	21	hence	hence	ADV
iajs-3432	190	22	𝑈𝑛	𝑈𝑛	PROPN
iajs-3432	190	23	>	>	X
iajs-3432	190	24	0	0	NUM
iajs-3432	190	25	,	,	PUNCT
iajs-3432	190	26	𝑛	𝑛	DET
iajs-3432	190	27	≥	≥	NOUN
iajs-3432	190	28	𝑛0	𝑛0	VERB
iajs-3432	190	29	.	.	PUNCT
iajs-3432	191	1	from	from	ADP
iajs-3432	191	2	lemma	lemma	PROPN
iajs-3432	191	3	2	2	NUM
iajs-3432	191	4	,	,	PUNCT
iajs-3432	191	5	it	it	PRON
iajs-3432	191	6	yields	yield	VERB
iajs-3432	191	7	𝑊𝑛	𝑊𝑛	PROPN
iajs-3432	191	8	≥	≥	NOUN
iajs-3432	191	9	0	0	NUM
iajs-3432	192	1	and	and	CCONJ
iajs-3432	192	2	decreasing	decrease	VERB
iajs-3432	192	3	sequence	sequence	NOUN
iajs-3432	192	4	.	.	PUNCT
iajs-3432	193	1	this	this	PRON
iajs-3432	193	2	means	mean	VERB
iajs-3432	193	3	𝑊𝑛	𝑊𝑛	PROPN
iajs-3432	193	4	is	be	AUX
iajs-3432	193	5	a	a	DET
iajs-3432	193	6	bounded	bounded	ADJ
iajs-3432	193	7	sequence	sequence	NOUN
iajs-3432	193	8	.	.	PUNCT
iajs-3432	194	1	hence	hence	ADV
iajs-3432	194	2	,	,	PUNCT
iajs-3432	194	3	from	from	ADP
iajs-3432	194	4	equation	equation	NOUN
iajs-3432	194	5	(	(	PUNCT
iajs-3432	194	6	13	13	NUM
iajs-3432	194	7	)	)	PUNCT
iajs-3432	194	8	,	,	PUNCT
iajs-3432	194	9	it	it	PRON
iajs-3432	194	10	follows	follow	VERB
iajs-3432	194	11	that	that	SCONJ
iajs-3432	194	12	there	there	PRON
iajs-3432	194	13	exists	exist	VERB
iajs-3432	194	14	𝑛1	𝑛1	ADJ
iajs-3432	194	15	≥	≥	NOUN
iajs-3432	194	16	𝑛0	𝑛0	VERB
iajs-3432	194	17	such	such	ADJ
iajs-3432	194	18	that	that	SCONJ
iajs-3432	195	1	𝑊𝑛	𝑊𝑛	PROPN
iajs-3432	195	2	≥	≥	PRON
iajs-3432	195	3	𝑈𝑛	𝑈𝑛	PROPN
iajs-3432	195	4	,	,	PUNCT
iajs-3432	195	5	𝑛	𝑛	DET
iajs-3432	195	6	≥	≥	NOUN
iajs-3432	195	7	𝑛1	𝑛1	NOUN
iajs-3432	195	8	.	.	PUNCT
iajs-3432	196	1	(	(	PUNCT
iajs-3432	196	2	15	15	NUM
iajs-3432	196	3	)	)	PUNCT
iajs-3432	196	4	which	which	PRON
iajs-3432	196	5	means	mean	VERB
iajs-3432	196	6	𝑈𝑛	𝑈𝑛	PROPN
iajs-3432	196	7	is	be	AUX
iajs-3432	196	8	bounded	bound	VERB
iajs-3432	196	9	.	.	PUNCT
iajs-3432	197	1	let	let	VERB
iajs-3432	197	2	liminf	liminf	INTJ
iajs-3432	197	3	𝑛→∞	𝑛→∞	VERB
iajs-3432	197	4	𝑈𝑛	𝑈𝑛	VERB
iajs-3432	197	5	=	=	SYM
iajs-3432	197	6	𝑙	𝑙	PRON
iajs-3432	197	7	≥	≥	NOUN
iajs-3432	197	8	0	0	NUM
iajs-3432	197	9	.	.	PUNCT
iajs-3432	198	1	we	we	PRON
iajs-3432	198	2	claim	claim	VERB
iajs-3432	198	3	that	that	SCONJ
iajs-3432	198	4	𝑙	𝑙	X
iajs-3432	198	5	=	=	SYM
iajs-3432	198	6	0	0	NUM
iajs-3432	198	7	otherwise	otherwise	ADV
iajs-3432	199	1	𝑙	𝑙	X
iajs-3432	199	2	>	>	X
iajs-3432	199	3	0	0	NUM
iajs-3432	199	4	.	.	PUNCT
iajs-3432	200	1	then	then	ADV
iajs-3432	200	2	,	,	PUNCT
iajs-3432	200	3	there	there	PRON
iajs-3432	200	4	exists	exist	VERB
iajs-3432	200	5	𝑛1	𝑛1	ADJ
iajs-3432	200	6	≥	≥	NOUN
iajs-3432	200	7	𝑛0	𝑛0	VERB
iajs-3432	200	8	large	large	ADJ
iajs-3432	200	9	enough	enough	ADV
iajs-3432	200	10	,	,	PUNCT
iajs-3432	200	11	such	such	ADJ
iajs-3432	200	12	that	that	SCONJ
iajs-3432	200	13	𝑈𝑛	𝑈𝑛	PROPN
iajs-3432	200	14	≥	≥	NUM
iajs-3432	200	15	𝑙	𝑙	NUM
iajs-3432	200	16	,	,	PUNCT
iajs-3432	200	17	𝑛	𝑛	DET
iajs-3432	200	18	≥	≥	NOUN
iajs-3432	200	19	𝑛1	𝑛1	NOUN
iajs-3432	200	20	.	.	PUNCT
iajs-3432	201	1	taking	take	VERB
iajs-3432	201	2	summation	summation	NOUN
iajs-3432	201	3	to	to	ADP
iajs-3432	201	4	both	both	DET
iajs-3432	201	5	sides	side	NOUN
iajs-3432	201	6	of	of	ADP
iajs-3432	201	7	equation	equation	NOUN
iajs-3432	201	8	(	(	PUNCT
iajs-3432	201	9	14	14	NUM
iajs-3432	201	10	)	)	PUNCT
iajs-3432	201	11	,	,	PUNCT
iajs-3432	201	12	we	we	PRON
iajs-3432	201	13	get	get	VERB
iajs-3432	201	14	∑	∑	PROPN
iajs-3432	201	15	∆𝑊𝑛	∆𝑊𝑛	PROPN
iajs-3432	201	16	≤	≤	NOUN
iajs-3432	201	17	∑	∑	PUNCT
iajs-3432	201	18	(	(	PUNCT
iajs-3432	201	19	𝛽𝑖𝜆2	𝛽𝑖𝜆2	NOUN
iajs-3432	201	20	−	−	PROPN
iajs-3432	201	21	𝛿𝑖−𝑙)𝑈𝑖−𝑙	𝛿𝑖−𝑙)𝑈𝑖−𝑙	PROPN
iajs-3432	201	22	𝑛−1	𝑛−1	PROPN
iajs-3432	201	23	𝑖=𝑛0	𝑖=𝑛0	NOUN
iajs-3432	201	24	𝑛−1	𝑛−1	PROPN
iajs-3432	201	25	𝑖=𝑛0	𝑖=𝑛0	NOUN
iajs-3432	201	26	,	,	PUNCT
iajs-3432	202	1	𝑊𝑛	𝑊𝑛	PROPN
iajs-3432	202	2	−	−	PROPN
iajs-3432	202	3	𝑊𝑛0	𝑊𝑛0	NOUN
iajs-3432	202	4	≤	≤	NUM
iajs-3432	202	5	∑	∑	PUNCT
iajs-3432	202	6	(	(	PUNCT
iajs-3432	202	7	𝛽𝑖𝜆2	𝛽𝑖𝜆2	NOUN
iajs-3432	202	8	−	−	PROPN
iajs-3432	202	9	𝛿𝑖−𝑙)𝑈𝑖−𝑙	𝛿𝑖−𝑙)𝑈𝑖−𝑙	PROPN
iajs-3432	202	10	𝑛−1	𝑛−1	PROPN
iajs-3432	202	11	𝑖=𝑛0	𝑖=𝑛0	PROPN
iajs-3432	202	12	.	.	PUNCT
iajs-3432	203	1	(	(	PUNCT
iajs-3432	203	2	16	16	NUM
iajs-3432	203	3	)	)	PUNCT
iajs-3432	203	4	from	from	ADP
iajs-3432	203	5	equation	equation	NOUN
iajs-3432	203	6	(	(	PUNCT
iajs-3432	203	7	16	16	NUM
iajs-3432	203	8	)	)	PUNCT
iajs-3432	203	9	,	,	PUNCT
iajs-3432	203	10	we	we	PRON
iajs-3432	203	11	have	have	VERB
iajs-3432	203	12	𝑊𝑛	𝑊𝑛	PRON
iajs-3432	203	13	−	−	PROPN
iajs-3432	203	14	𝑊𝑛0	𝑊𝑛0	NOUN
iajs-3432	203	15	≤	≤	NOUN
iajs-3432	204	1	𝑙	𝑙	X
iajs-3432	204	2	∑	∑	PROPN
iajs-3432	204	3	(	(	PUNCT
iajs-3432	204	4	𝛽𝑖𝜆2	𝛽𝑖𝜆2	PROPN
iajs-3432	204	5	−	−	PROPN
iajs-3432	204	6	𝛿𝑖−𝑙	𝛿𝑖−𝑙	PROPN
iajs-3432	204	7	)	)	PUNCT
iajs-3432	205	1	𝑛−1	𝑛−1	PROPN
iajs-3432	205	2	𝑖=𝑛0	𝑖=𝑛0	NOUN
iajs-3432	205	3	≤	≤	NOUN
iajs-3432	205	4	𝑙𝛾(𝑛	𝑙𝛾(𝑛	VERB
iajs-3432	205	5	−	−	PROPN
iajs-3432	205	6	1	1	NUM
iajs-3432	205	7	)	)	PUNCT
iajs-3432	205	8	.	.	PUNCT
iajs-3432	206	1	as	as	ADP
iajs-3432	206	2	𝑛	𝑛	PROPN
iajs-3432	206	3	→	→	SYM
iajs-3432	206	4	∞	∞	PROPN
iajs-3432	206	5	,	,	PUNCT
iajs-3432	206	6	the	the	DET
iajs-3432	206	7	last	last	ADJ
iajs-3432	206	8	inequality	inequality	NOUN
iajs-3432	206	9	leads	lead	VERB
iajs-3432	206	10	to	to	ADP
iajs-3432	206	11	lim	lim	PROPN
iajs-3432	206	12	𝑛→∞	𝑛→∞	PUNCT
iajs-3432	207	1	𝑊𝑛	𝑊𝑛	PROPN
iajs-3432	207	2	=	=	SYM
iajs-3432	207	3	−∞	−∞	PRON
iajs-3432	207	4	which	which	PRON
iajs-3432	207	5	is	be	AUX
iajs-3432	207	6	a	a	DET
iajs-3432	207	7	contradiction	contradiction	NOUN
iajs-3432	207	8	.	.	PUNCT
iajs-3432	208	1	hence	hence	ADV
iajs-3432	208	2	,	,	PUNCT
iajs-3432	208	3	liminf	liminf	INTJ
iajs-3432	208	4	𝑛→∞	𝑛→∞	PUNCT
iajs-3432	208	5	𝑈𝑛	𝑈𝑛	NOUN
iajs-3432	208	6	=	=	SYM
iajs-3432	208	7	0	0	X
iajs-3432	208	8	.	.	PUNCT
iajs-3432	209	1	therefore	therefore	ADV
iajs-3432	209	2	,	,	PUNCT
iajs-3432	209	3	there	there	PRON
iajs-3432	209	4	exists	exist	VERB
iajs-3432	209	5	a	a	DET
iajs-3432	209	6	subsequence	subsequence	NOUN
iajs-3432	209	7	𝑛𝑗	𝑛𝑗	ADP
iajs-3432	209	8	such	such	ADJ
iajs-3432	209	9	that	that	SCONJ
iajs-3432	209	10	lim	lim	PROPN
iajs-3432	209	11	𝑗→∞	𝑗→∞	NUM
iajs-3432	210	1	𝑈𝑛𝑗	𝑈𝑛𝑗	PROPN
iajs-3432	210	2	=	=	SYM
iajs-3432	210	3	0	0	X
iajs-3432	210	4	.	.	PUNCT
iajs-3432	211	1	let	let	VERB
iajs-3432	211	2	lim	lim	PROPN
iajs-3432	211	3	𝑛→∞	𝑛→∞	VERB
iajs-3432	212	1	𝑊𝑛	𝑊𝑛	PROPN
iajs-3432	212	2	=	=	PUNCT
iajs-3432	212	3	𝐿.	𝐿.	VERB
iajs-3432	212	4	from	from	ADP
iajs-3432	212	5	equation	equation	NOUN
iajs-3432	212	6	(	(	PUNCT
iajs-3432	212	7	13	13	NUM
iajs-3432	212	8	)	)	PUNCT
iajs-3432	212	9	,	,	PUNCT
iajs-3432	212	10	it	it	PRON
iajs-3432	212	11	follows	follow	VERB
iajs-3432	212	12	𝑈𝑛	𝑈𝑛	PROPN
iajs-3432	212	13	=	=	SYM
iajs-3432	212	14	𝑊𝑛	𝑊𝑛	PROPN
iajs-3432	212	15	−	−	PROPN
iajs-3432	212	16	∑	∑	VERB
iajs-3432	212	17	𝛿𝑖	𝛿𝑖	PROPN
iajs-3432	212	18	𝑛−1	𝑛−1	NUM
iajs-3432	212	19	𝑖=𝑛−𝑙	𝑖=𝑛−𝑙	PUNCT
iajs-3432	213	1	𝑈𝑖	𝑈𝑖	PROPN
iajs-3432	213	2	+	+	CCONJ
iajs-3432	213	3	𝐹𝑛	𝐹𝑛	PROPN
iajs-3432	213	4	,	,	PUNCT
iajs-3432	213	5	≥	≥	X
iajs-3432	214	1	𝑊𝑛	𝑊𝑛	ADP
iajs-3432	214	2	−	−	PROPN
iajs-3432	214	3	∑	∑	VERB
iajs-3432	214	4	𝛿𝑖	𝛿𝑖	PROPN
iajs-3432	214	5	𝑛−1	𝑛−1	NUM
iajs-3432	214	6	𝑖=𝑛−𝑙	𝑖=𝑛−𝑙	PUNCT
iajs-3432	215	1	𝑊𝑖	𝑊𝑖	PROPN
iajs-3432	215	2	−	−	PROPN
iajs-3432	215	3	𝐹𝑛	𝐹𝑛	PROPN
iajs-3432	215	4	,	,	PUNCT
iajs-3432	215	5	𝑈𝑛𝑗	𝑈𝑛𝑗	PROPN
iajs-3432	215	6	≥	≥	NOUN
iajs-3432	215	7	𝑊𝑛𝑗	𝑊𝑛𝑗	PROPN
iajs-3432	215	8	−	−	PROPN
iajs-3432	215	9	𝑊𝑛𝑗−𝑙	𝑊𝑛𝑗−𝑙	PROPN
iajs-3432	215	10	∑	∑	PUNCT
iajs-3432	215	11	𝛿𝑖	𝛿𝑖	ADP
iajs-3432	215	12	𝑛𝑗−1	𝑛𝑗−1	PROPN
iajs-3432	215	13	𝑖=𝑛𝑗−𝑙	𝑖=𝑛𝑗−𝑙	X
iajs-3432	215	14	−	−	PROPN
iajs-3432	216	1	𝐹𝑛𝑗	𝐹𝑛𝑗	INTJ
iajs-3432	216	2	,	,	PUNCT
iajs-3432	216	3	lim	lim	PROPN
iajs-3432	216	4	𝑗→∞	𝑗→∞	NUM
iajs-3432	216	5	𝑊𝑛𝑗	𝑊𝑛𝑗	PROPN
iajs-3432	216	6	≤	≤	PROPN
iajs-3432	216	7	lim	lim	PROPN
iajs-3432	216	8	𝑗→∞	𝑗→∞	NUM
iajs-3432	217	1	𝑈𝑛𝑗	𝑈𝑛𝑗	PROPN
iajs-3432	217	2	=	=	NOUN
iajs-3432	217	3	0	0	NUM
iajs-3432	217	4	.	.	PUNCT
iajs-3432	218	1	thus	thus	ADV
iajs-3432	218	2	,	,	PUNCT
iajs-3432	218	3	lim	lim	PROPN
iajs-3432	218	4	𝑛→∞	𝑛→∞	AUX
iajs-3432	218	5	𝑊𝑛	𝑊𝑛	PROPN
iajs-3432	218	6	=	=	SYM
iajs-3432	218	7	0	0	NUM
iajs-3432	218	8	implies	imply	VERB
iajs-3432	218	9	that	that	SCONJ
iajs-3432	218	10	lim	lim	PROPN
iajs-3432	218	11	𝑛→∞	𝑛→∞	NUM
iajs-3432	218	12	𝑈𝑛	𝑈𝑛	PROPN
iajs-3432	218	13	=	=	PRON
iajs-3432	218	14	0,that	0,that	PRON
iajs-3432	218	15	is	be	AUX
iajs-3432	218	16	lim	lim	NOUN
iajs-3432	218	17	𝑛→∞	𝑛→∞	NUM
iajs-3432	219	1	𝐻𝑛	𝐻𝑛	PROPN
iajs-3432	219	2	=	=	SYM
iajs-3432	219	3	𝐾	𝐾	PROPN
iajs-3432	219	4	and	and	CCONJ
iajs-3432	219	5	the	the	DET
iajs-3432	219	6	proof	proof	NOUN
iajs-3432	219	7	is	be	AUX
iajs-3432	219	8	finished	finish	VERB
iajs-3432	219	9	.	.	PUNCT
iajs-3432	220	1	we	we	PRON
iajs-3432	220	2	provide	provide	VERB
iajs-3432	220	3	examples	example	NOUN
iajs-3432	220	4	to	to	PART
iajs-3432	220	5	discuss	discuss	VERB
iajs-3432	220	6	and	and	CCONJ
iajs-3432	220	7	illustrate	illustrate	VERB
iajs-3432	220	8	the	the	DET
iajs-3432	220	9	previous	previous	ADJ
iajs-3432	220	10	results	result	NOUN
iajs-3432	220	11	.	.	PUNCT
iajs-3432	221	1	in	in	ADP
iajs-3432	221	2	the	the	DET
iajs-3432	221	3	following	following	NOUN
iajs-3432	221	4	,	,	PUNCT
iajs-3432	221	5	we	we	PRON
iajs-3432	221	6	discuss	discuss	VERB
iajs-3432	221	7	the	the	DET
iajs-3432	221	8	accuracy	accuracy	NOUN
iajs-3432	221	9	of	of	ADP
iajs-3432	221	10	the	the	DET
iajs-3432	221	11	numerical	numerical	ADJ
iajs-3432	221	12	solution	solution	NOUN
iajs-3432	221	13	and	and	CCONJ
iajs-3432	221	14	the	the	DET
iajs-3432	221	15	oscillatory	oscillatory	ADJ
iajs-3432	221	16	behavior	behavior	NOUN
iajs-3432	221	17	of	of	ADP
iajs-3432	221	18	equation	equation	NOUN
iajs-3432	221	19	(	(	PUNCT
iajs-3432	221	20	17	17	NUM
iajs-3432	221	21	)	)	PUNCT
iajs-3432	221	22	and	and	CCONJ
iajs-3432	221	23	(	(	PUNCT
iajs-3432	221	24	18	18	NUM
iajs-3432	221	25	)	)	PUNCT
iajs-3432	221	26	.	.	PUNCT
iajs-3432	222	1	therefore	therefore	ADV
iajs-3432	222	2	,	,	PUNCT
iajs-3432	222	3	in	in	ADP
iajs-3432	222	4	comparison	comparison	NOUN
iajs-3432	222	5	to	to	ADP
iajs-3432	222	6	the	the	DET
iajs-3432	222	7	exponential	exponential	ADJ
iajs-3432	222	8	θ	θ	NOUN
iajs-3432	222	9	-	-	NOUN
iajs-3432	222	10	method	method	NOUN
iajs-3432	222	11	in	in	ADP
iajs-3432	222	12	[	[	X
iajs-3432	222	13	25	25	NUM
iajs-3432	222	14	]	]	PUNCT
iajs-3432	222	15	,	,	PUNCT
iajs-3432	222	16	our	our	PRON
iajs-3432	222	17	results	result	NOUN
iajs-3432	222	18	have	have	VERB
iajs-3432	222	19	higher	high	ADJ
iajs-3432	222	20	accuracy	accuracy	NOUN
iajs-3432	222	21	.	.	PUNCT
iajs-3432	223	1	example	example	NOUN
iajs-3432	224	1	1	1	NUM
iajs-3432	224	2	consider	consider	VERB
iajs-3432	224	3	the	the	DET
iajs-3432	224	4	nonlinear	nonlinear	ADJ
iajs-3432	224	5	first	first	ADJ
iajs-3432	224	6	-	-	PUNCT
iajs-3432	224	7	order	order	NOUN
iajs-3432	224	8	difference	difference	NOUN
iajs-3432	224	9	equation	equation	NOUN
iajs-3432	224	10	∆𝑈𝑛	∆𝑈𝑛	PROPN
iajs-3432	224	11	+	+	NOUN
iajs-3432	224	12	𝑛	𝑛	PROPN
iajs-3432	224	13	64	64	NUM
iajs-3432	224	14	𝑈𝑛	𝑈𝑛	PROPN
iajs-3432	224	15	−	−	PROPN
iajs-3432	224	16	6	6	NUM
iajs-3432	224	17	(	(	PUNCT
iajs-3432	224	18	1	1	NUM
iajs-3432	224	19	𝑒	𝑒	NOUN
iajs-3432	224	20	)	)	PUNCT
iajs-3432	224	21	𝑛+4	𝑛+4	NUM
iajs-3432	224	22	1	1	NUM
iajs-3432	224	23	1	1	NUM
iajs-3432	224	24	+	+	NUM
iajs-3432	224	25	𝑈𝑛−2	𝑈𝑛−2	NOUN
iajs-3432	224	26	=	=	SYM
iajs-3432	224	27	2(−1)𝑛3𝑛−2𝑒−2𝑛	2(−1)𝑛3𝑛−2𝑒−2𝑛	NUM
iajs-3432	224	28	,	,	PUNCT
iajs-3432	224	29	𝑛	𝑛	DET
iajs-3432	224	30	≥	≥	NUM
iajs-3432	224	31	1	1	NUM
iajs-3432	224	32	.	.	PUNCT
iajs-3432	225	1	(	(	PUNCT
iajs-3432	225	2	17	17	NUM
iajs-3432	225	3	)	)	PUNCT
iajs-3432	225	4	where	where	SCONJ
iajs-3432	225	5	𝑙	𝑙	NOUN
iajs-3432	225	6	=	=	SYM
iajs-3432	225	7	2	2	NUM
iajs-3432	225	8	,	,	PUNCT
iajs-3432	225	9	𝛿𝑛	𝛿𝑛	ADP
iajs-3432	225	10	=	=	PUNCT
iajs-3432	225	11	𝑛	𝑛	PROPN
iajs-3432	225	12	64	64	NUM
iajs-3432	225	13	,	,	PUNCT
iajs-3432	225	14	𝛽𝑛	𝛽𝑛	PROPN
iajs-3432	225	15	=	=	SYM
iajs-3432	225	16	6	6	NUM
iajs-3432	225	17	(	(	PUNCT
iajs-3432	225	18	1	1	NUM
iajs-3432	225	19	𝑒	𝑒	NOUN
iajs-3432	225	20	)	)	PUNCT
iajs-3432	225	21	𝑛+4	𝑛+4	PROPN
iajs-3432	225	22	,	,	PUNCT
iajs-3432	225	23	𝑉(𝑈𝑛	𝑉(𝑈𝑛	NOUN
iajs-3432	225	24	)	)	PUNCT
iajs-3432	225	25	=	=	SYM
iajs-3432	225	26	1	1	NUM
iajs-3432	225	27	1+𝑈𝑛	1+𝑈𝑛	NUM
iajs-3432	225	28	,	,	PUNCT
iajs-3432	225	29	𝜆2	𝜆2	NOUN
iajs-3432	225	30	=	=	SYM
iajs-3432	225	31	2	2	NUM
iajs-3432	225	32	,	,	PUNCT
iajs-3432	225	33	𝑓𝑛	𝑓𝑛	NOUN
iajs-3432	225	34	=	=	NOUN
iajs-3432	225	35	2(−1)𝑛3𝑛−2𝑒−2𝑛.	2(−1)𝑛3𝑛−2𝑒−2𝑛.	NUM
iajs-3432	225	36	to	to	PART
iajs-3432	225	37	show	show	VERB
iajs-3432	225	38	that	that	SCONJ
iajs-3432	225	39	all	all	DET
iajs-3432	225	40	conditions	condition	NOUN
iajs-3432	225	41	of	of	ADP
iajs-3432	225	42	theorem	theorem	NOUN
iajs-3432	225	43	1	1	NUM
iajs-3432	225	44	are	be	AUX
iajs-3432	225	45	satisfying	satisfy	VERB
iajs-3432	225	46	:	:	PUNCT
iajs-3432	225	47	(	(	PUNCT
iajs-3432	225	48	𝛿𝑛	𝛿𝑛	INTJ
iajs-3432	225	49	−	−	PROPN
iajs-3432	225	50	𝛽𝑛+𝑙𝜆2	𝛽𝑛+𝑙𝜆2	PROPN
iajs-3432	225	51	)	)	PUNCT
iajs-3432	225	52	=	=	SYM
iajs-3432	225	53	𝑛	𝑛	PRON
iajs-3432	225	54	64	64	NUM
iajs-3432	225	55	−	−	NOUN
iajs-3432	225	56	12	12	NUM
iajs-3432	225	57	(	(	PUNCT
iajs-3432	225	58	1	1	NUM
iajs-3432	225	59	𝑒	𝑒	X
iajs-3432	225	60	)	)	PUNCT
iajs-3432	226	1	𝑛+6	𝑛+6	PROPN
iajs-3432	226	2	>	>	PUNCT
iajs-3432	226	3	0	0	NUM
iajs-3432	226	4	,	,	PUNCT
iajs-3432	226	5	𝑛	𝑛	DET
iajs-3432	226	6	≥	≥	NOUN
iajs-3432	226	7	1	1	NUM
iajs-3432	226	8	.	.	PUNCT
iajs-3432	226	9	limsup	limsup	NOUN
iajs-3432	226	10	𝑛→∞	𝑛→∞	NUM
iajs-3432	226	11	∑	∑	PUNCT
iajs-3432	226	12	𝛿𝑖	𝛿𝑖	ADP
iajs-3432	226	13	𝑛−1	𝑛−1	PROPN
iajs-3432	226	14	𝑖=𝑛0	𝑖=𝑛0	NOUN
iajs-3432	227	1	=	=	SYM
iajs-3432	227	2	lim	lim	PROPN
iajs-3432	227	3	𝑛→∞	𝑛→∞	NUM
iajs-3432	227	4	∑	∑	PROPN
iajs-3432	227	5	𝑖	𝑖	SYM
iajs-3432	227	6	64	64	NUM
iajs-3432	227	7	𝑛−1	𝑛−1	NUM
iajs-3432	227	8	𝑖=𝑛0	𝑖=𝑛0	NUM
iajs-3432	227	9	=	=	SYM
iajs-3432	227	10	∞	∞	PROPN
iajs-3432	227	11	.	.	PUNCT
iajs-3432	228	1	ihjpas	ihjpas	PROPN
iajs-3432	228	2	.	.	PUNCT
iajs-3432	229	1	2024	2024	NUM
iajs-3432	229	2	,	,	PUNCT
iajs-3432	229	3	37	37	NUM
iajs-3432	229	4	(	(	PUNCT
iajs-3432	229	5	3	3	NUM
iajs-3432	229	6	)	)	PUNCT
iajs-3432	229	7	392	392	NUM
iajs-3432	229	8	that	that	PRON
iajs-3432	229	9	is	be	AUX
iajs-3432	229	10	,	,	PUNCT
iajs-3432	229	11	the	the	DET
iajs-3432	229	12	analytic	analytic	ADJ
iajs-3432	229	13	solutions	solution	NOUN
iajs-3432	229	14	of	of	ADP
iajs-3432	229	15	equation	equation	NOUN
iajs-3432	229	16	(	(	PUNCT
iajs-3432	229	17	17	17	NUM
iajs-3432	229	18	)	)	PUNCT
iajs-3432	229	19	are	be	AUX
iajs-3432	229	20	oscillatory	oscillatory	ADJ
iajs-3432	229	21	about	about	ADP
iajs-3432	229	22	0	0	NUM
iajs-3432	230	1	as	as	ADP
iajs-3432	230	2	𝑡	𝑡	PROPN
iajs-3432	230	3	→	→	SYM
iajs-3432	230	4	∞	∞	NUM
iajs-3432	230	5	which	which	PRON
iajs-3432	230	6	is	be	AUX
iajs-3432	230	7	illustrated	illustrate	VERB
iajs-3432	230	8	in	in	ADP
iajs-3432	230	9	figure	figure	NOUN
iajs-3432	230	10	1	1	NUM
iajs-3432	230	11	since	since	SCONJ
iajs-3432	230	12	all	all	DET
iajs-3432	230	13	conditions	condition	NOUN
iajs-3432	230	14	of	of	ADP
iajs-3432	230	15	theorem	theorem	ADJ
iajs-3432	230	16	1	1	NUM
iajs-3432	230	17	holds	hold	NOUN
iajs-3432	230	18	.	.	PUNCT
iajs-3432	231	1	so	so	ADV
iajs-3432	231	2	,	,	PUNCT
iajs-3432	231	3	all	all	DET
iajs-3432	231	4	the	the	DET
iajs-3432	231	5	solutions	solution	NOUN
iajs-3432	231	6	of	of	ADP
iajs-3432	231	7	the	the	DET
iajs-3432	231	8	corresponding	corresponding	ADJ
iajs-3432	231	9	hematopoiesis	hematopoiesis	NOUN
iajs-3432	231	10	model	model	NOUN
iajs-3432	231	11	of	of	ADP
iajs-3432	231	12	equation	equation	NOUN
iajs-3432	231	13	(	(	PUNCT
iajs-3432	231	14	17	17	NUM
iajs-3432	231	15	)	)	PUNCT
iajs-3432	231	16	are	be	AUX
iajs-3432	231	17	oscillatory	oscillatory	ADJ
iajs-3432	231	18	around	around	ADP
iajs-3432	231	19	equilibrium	equilibrium	NOUN
iajs-3432	231	20	𝐾	𝐾	PROPN
iajs-3432	231	21	as	as	ADP
iajs-3432	231	22	𝑡	𝑡	PROPN
iajs-3432	231	23	→	→	SYM
iajs-3432	231	24	∞.	∞.	PROPN
iajs-3432	231	25	the	the	DET
iajs-3432	231	26	matlab	matlab	PROPN
iajs-3432	231	27	solver	solver	VERB
iajs-3432	231	28	ode23	ode23	PROPN
iajs-3432	231	29	that	that	PRON
iajs-3432	231	30	allows	allow	VERB
iajs-3432	231	31	to	to	PART
iajs-3432	231	32	numerically	numerically	ADV
iajs-3432	231	33	solve	solve	VERB
iajs-3432	231	34	delay	delay	NOUN
iajs-3432	231	35	differential	differential	ADJ
iajs-3432	231	36	equation	equation	NOUN
iajs-3432	231	37	(	(	PUNCT
iajs-3432	231	38	17	17	NUM
iajs-3432	231	39	)	)	PUNCT
iajs-3432	231	40	is	be	AUX
iajs-3432	231	41	used	use	VERB
iajs-3432	231	42	to	to	PART
iajs-3432	231	43	perform	perform	VERB
iajs-3432	231	44	the	the	DET
iajs-3432	231	45	numerical	numerical	ADJ
iajs-3432	231	46	result	result	NOUN
iajs-3432	231	47	.	.	PUNCT
iajs-3432	232	1	figure	figure	NOUN
iajs-3432	232	2	1	1	NUM
iajs-3432	232	3	.	.	PUNCT
iajs-3432	233	1	the	the	DET
iajs-3432	233	2	solution	solution	NOUN
iajs-3432	233	3	of	of	ADP
iajs-3432	233	4	equation	equation	NOUN
iajs-3432	233	5	(	(	PUNCT
iajs-3432	233	6	17	17	NUM
iajs-3432	233	7	)	)	PUNCT
iajs-3432	233	8	is	be	AUX
iajs-3432	233	9	oscillatory	oscillatory	ADJ
iajs-3432	233	10	about	about	ADP
iajs-3432	233	11	0	0	NUM
iajs-3432	233	12	as	as	SCONJ
iajs-3432	233	13	𝑡	𝑡	PROPN
iajs-3432	233	14	→	→	SYM
iajs-3432	233	15	∞.	∞.	PROPN
iajs-3432	233	16	example	example	NOUN
iajs-3432	233	17	2	2	NUM
iajs-3432	233	18	consider	consider	VERB
iajs-3432	233	19	the	the	DET
iajs-3432	233	20	nonlinear	nonlinear	ADJ
iajs-3432	233	21	hematopoiesis	hematopoiesis	NOUN
iajs-3432	233	22	difference	difference	NOUN
iajs-3432	233	23	equation	equation	NOUN
iajs-3432	233	24	∆𝐻𝑛	∆𝐻𝑛	VERB
iajs-3432	233	25	+	+	CCONJ
iajs-3432	233	26	3	3	NUM
iajs-3432	233	27	(	(	PUNCT
iajs-3432	233	28	1	1	NUM
iajs-3432	233	29	𝑒	𝑒	NOUN
iajs-3432	233	30	)	)	PUNCT
iajs-3432	233	31	𝑛+4	𝑛+4	PROPN
iajs-3432	234	1	𝐻𝑛	𝐻𝑛	PROPN
iajs-3432	234	2	−	−	PROPN
iajs-3432	234	3	𝑛(1	𝑛(1	PROPN
iajs-3432	234	4	+	+	CCONJ
iajs-3432	234	5	3𝑒	3𝑒	ADJ
iajs-3432	234	6	+	+	NUM
iajs-3432	234	7	𝑒3	𝑒3	NOUN
iajs-3432	234	8	)	)	PUNCT
iajs-3432	234	9	1	1	NUM
iajs-3432	234	10	1	1	NUM
iajs-3432	234	11	+	+	NUM
iajs-3432	234	12	𝐻𝑛−2	𝐻𝑛−2	NOUN
iajs-3432	234	13	=	=	SYM
iajs-3432	234	14	2	2	NUM
iajs-3432	234	15	9	9	NUM
iajs-3432	234	16	(	(	PUNCT
iajs-3432	234	17	1	1	NUM
iajs-3432	234	18	2	2	NUM
iajs-3432	234	19	)	)	PUNCT
iajs-3432	234	20	𝑛	𝑛	PROPN
iajs-3432	234	21	.	.	PUNCT
iajs-3432	235	1	(	(	PUNCT
iajs-3432	235	2	−3	−3	NOUN
iajs-3432	235	3	2	2	X
iajs-3432	235	4	)	)	PUNCT
iajs-3432	235	5	𝑛	𝑛	PROPN
iajs-3432	235	6	.	.	PUNCT
iajs-3432	236	1	(	(	PUNCT
iajs-3432	236	2	18	18	NUM
iajs-3432	236	3	)	)	PUNCT
iajs-3432	236	4	where	where	SCONJ
iajs-3432	236	5	l=2	l=2	PUNCT
iajs-3432	236	6	,	,	PUNCT
iajs-3432	236	7	𝜆2	𝜆2	NOUN
iajs-3432	236	8	=	=	SYM
iajs-3432	236	9	1	1	NUM
iajs-3432	236	10	,	,	PUNCT
iajs-3432	236	11	𝛿𝑛	𝛿𝑛	ADP
iajs-3432	236	12	=	=	SYM
iajs-3432	236	13	3	3	NUM
iajs-3432	236	14	(	(	PUNCT
iajs-3432	236	15	1	1	NUM
iajs-3432	236	16	𝑒	𝑒	NOUN
iajs-3432	236	17	)	)	PUNCT
iajs-3432	236	18	𝑛+4	𝑛+4	PROPN
iajs-3432	236	19	,	,	PUNCT
iajs-3432	236	20	𝛽𝑛	𝛽𝑛	PROPN
iajs-3432	236	21	=	=	SYM
iajs-3432	236	22	𝑛(1	𝑛(1	PROPN
iajs-3432	236	23	+	+	CCONJ
iajs-3432	236	24	3𝑒	3𝑒	ADJ
iajs-3432	236	25	+	+	NUM
iajs-3432	236	26	𝑒3	𝑒3	NOUN
iajs-3432	236	27	)	)	PUNCT
iajs-3432	236	28	,	,	PUNCT
iajs-3432	236	29	𝑓𝑛	𝑓𝑛	NOUN
iajs-3432	236	30	=	=	VERB
iajs-3432	236	31	−𝑒	−𝑒	PROPN
iajs-3432	236	32	(	(	PUNCT
iajs-3432	236	33	1	1	NUM
iajs-3432	236	34	𝑒	𝑒	NOUN
iajs-3432	236	35	)	)	PUNCT
iajs-3432	236	36	𝑛	𝑛	PROPN
iajs-3432	236	37	.	.	PUNCT
iajs-3432	237	1	(	(	PUNCT
iajs-3432	237	2	−1	−1	NOUN
iajs-3432	237	3	𝑒	𝑒	X
iajs-3432	237	4	)	)	PUNCT
iajs-3432	237	5	𝑛	𝑛	NOUN
iajs-3432	237	6	,	,	PUNCT
iajs-3432	237	7	𝐺(𝐻𝑛	𝐺(𝐻𝑛	NUM
iajs-3432	237	8	)	)	PUNCT
iajs-3432	237	9	=	=	SYM
iajs-3432	237	10	1	1	NUM
iajs-3432	237	11	1+𝐻𝑛−2	1+𝐻𝑛−2	NUM
iajs-3432	237	12	.	.	PUNCT
iajs-3432	238	1	it	it	PRON
iajs-3432	238	2	's	be	AUX
iajs-3432	238	3	easy	easy	ADJ
iajs-3432	238	4	to	to	PART
iajs-3432	238	5	show	show	VERB
iajs-3432	238	6	that	that	SCONJ
iajs-3432	238	7	all	all	DET
iajs-3432	238	8	condition	condition	NOUN
iajs-3432	238	9	of	of	ADP
iajs-3432	238	10	theorem	theorem	ADJ
iajs-3432	238	11	2	2	NUM
iajs-3432	238	12	are	be	AUX
iajs-3432	238	13	satisfies	satisfie	NOUN
iajs-3432	238	14	:	:	PUNCT
iajs-3432	238	15	(	(	PUNCT
iajs-3432	238	16	𝛽𝑛𝜆2	𝛽𝑛𝜆2	ADJ
iajs-3432	238	17	−	−	PROPN
iajs-3432	238	18	𝛿𝑛−𝑙	𝛿𝑛−𝑙	NOUN
iajs-3432	238	19	)	)	PUNCT
iajs-3432	238	20	=	=	SYM
iajs-3432	239	1	3	3	NUM
iajs-3432	239	2	(	(	PUNCT
iajs-3432	239	3	1	1	NUM
iajs-3432	239	4	𝑒	𝑒	X
iajs-3432	239	5	)	)	PUNCT
iajs-3432	239	6	𝑛+2	𝑛+2	NOUN
iajs-3432	239	7	−	−	PROPN
iajs-3432	239	8	𝑛(1	𝑛(1	PROPN
iajs-3432	239	9	+	+	CCONJ
iajs-3432	239	10	3𝑒	3𝑒	ADJ
iajs-3432	239	11	+	+	NUM
iajs-3432	239	12	𝑒3	𝑒3	NOUN
iajs-3432	239	13	)	)	PUNCT
iajs-3432	239	14	<	<	X
iajs-3432	239	15	0	0	X
iajs-3432	239	16	.	.	X
iajs-3432	239	17	limsup	limsup	PROPN
iajs-3432	239	18	𝑛→∞	𝑛→∞	NUM
iajs-3432	239	19	(	(	PUNCT
iajs-3432	239	20	𝛽𝑛	𝛽𝑛	NOUN
iajs-3432	239	21	)	)	PUNCT
iajs-3432	239	22	=	=	SYM
iajs-3432	239	23	limsup	limsup	NOUN
iajs-3432	239	24	𝑛→∞	𝑛→∞	PUNCT
iajs-3432	239	25	(	(	PUNCT
iajs-3432	239	26	𝑛(1	𝑛(1	PROPN
iajs-3432	239	27	+	+	CCONJ
iajs-3432	239	28	3𝑒	3𝑒	ADJ
iajs-3432	239	29	+	+	NUM
iajs-3432	239	30	𝑒3	𝑒3	NOUN
iajs-3432	239	31	)	)	PUNCT
iajs-3432	239	32	)	)	PUNCT
iajs-3432	240	1	=	=	SYM
iajs-3432	240	2	∞.	∞.	PROPN
iajs-3432	240	3	since	since	SCONJ
iajs-3432	240	4	all	all	DET
iajs-3432	240	5	conditions	condition	NOUN
iajs-3432	240	6	of	of	ADP
iajs-3432	240	7	theorem	theorem	ADJ
iajs-3432	240	8	2	2	NUM
iajs-3432	240	9	hold	hold	NOUN
iajs-3432	240	10	,	,	PUNCT
iajs-3432	240	11	the	the	DET
iajs-3432	240	12	solution	solution	NOUN
iajs-3432	240	13	is	be	AUX
iajs-3432	240	14	non	non	ADJ
iajs-3432	240	15	-	-	ADJ
iajs-3432	240	16	oscillatory	oscillatory	ADJ
iajs-3432	240	17	and	and	CCONJ
iajs-3432	240	18	tends	tend	VERB
iajs-3432	240	19	to	to	ADP
iajs-3432	240	20	equilibrium	equilibrium	NOUN
iajs-3432	240	21	𝐾	𝐾	NOUN
iajs-3432	240	22	=	=	PROPN
iajs-3432	240	23	10	10	NUM
iajs-3432	240	24	,	,	PUNCT
iajs-3432	240	25	as	as	ADP
iajs-3432	240	26	𝑡	𝑡	PROPN
iajs-3432	240	27	→	→	SYM
iajs-3432	240	28	∞	∞	PROPN
iajs-3432	240	29	,	,	PUNCT
iajs-3432	240	30	as	as	SCONJ
iajs-3432	240	31	illustrated	illustrate	VERB
iajs-3432	240	32	in	in	ADP
iajs-3432	240	33	figure	figure	NOUN
iajs-3432	240	34	2	2	NUM
iajs-3432	240	35	.	.	PUNCT
iajs-3432	241	1	the	the	DET
iajs-3432	241	2	matlab	matlab	PROPN
iajs-3432	241	3	solver	solver	VERB
iajs-3432	241	4	ode23	ode23	PROPN
iajs-3432	241	5	that	that	PRON
iajs-3432	241	6	allows	allow	VERB
iajs-3432	241	7	to	to	PART
iajs-3432	241	8	numerically	numerically	ADV
iajs-3432	241	9	solve	solve	VERB
iajs-3432	241	10	delay	delay	NOUN
iajs-3432	241	11	differential	differential	ADJ
iajs-3432	241	12	equation	equation	NOUN
iajs-3432	241	13	(	(	PUNCT
iajs-3432	241	14	18	18	NUM
iajs-3432	241	15	)	)	PUNCT
iajs-3432	241	16	is	be	AUX
iajs-3432	241	17	used	use	VERB
iajs-3432	241	18	to	to	PART
iajs-3432	241	19	perform	perform	VERB
iajs-3432	241	20	the	the	DET
iajs-3432	241	21	numerical	numerical	ADJ
iajs-3432	241	22	result	result	NOUN
iajs-3432	241	23	.	.	PUNCT
iajs-3432	242	1	ihjpas	ihjpas	PROPN
iajs-3432	242	2	.	.	PUNCT
iajs-3432	243	1	2024	2024	NUM
iajs-3432	243	2	,	,	PUNCT
iajs-3432	243	3	37	37	NUM
iajs-3432	243	4	(	(	PUNCT
iajs-3432	243	5	3	3	NUM
iajs-3432	243	6	)	)	PUNCT
iajs-3432	243	7	393	393	NUM
iajs-3432	243	8	figure	figure	NOUN
iajs-3432	243	9	2	2	NUM
iajs-3432	243	10	.	.	PUNCT
iajs-3432	243	11	non	non	ADJ
iajs-3432	243	12	-	-	ADJ
iajs-3432	243	13	oscillatory	oscillatory	ADJ
iajs-3432	243	14	solution	solution	NOUN
iajs-3432	243	15	for	for	ADP
iajs-3432	243	16	example2	example2	PROPN
iajs-3432	243	17	of	of	ADP
iajs-3432	243	18	(	(	PUNCT
iajs-3432	243	19	18	18	NUM
iajs-3432	243	20	)	)	PUNCT
iajs-3432	243	21	tends	tend	VERB
iajs-3432	243	22	to	to	ADP
iajs-3432	243	23	equilibrium	equilibrium	NOUN
iajs-3432	243	24	k=10	k=10	PROPN
iajs-3432	243	25	as	as	SCONJ
iajs-3432	243	26	𝑡	𝑡	PROPN
iajs-3432	243	27	→	→	SYM
iajs-3432	243	28	∞.	∞.	PROPN
iajs-3432	243	29	the	the	DET
iajs-3432	243	30	matlap	matlap	NOUN
iajs-3432	243	31	program	program	NOUN
iajs-3432	243	32	is	be	AUX
iajs-3432	243	33	used	use	VERB
iajs-3432	243	34	to	to	PART
iajs-3432	243	35	illustrate	illustrate	VERB
iajs-3432	243	36	the	the	DET
iajs-3432	243	37	numerical	numerical	PROPN
iajs-3432	243	38	oscillation	oscillation	NOUN
iajs-3432	243	39	solutions	solution	NOUN
iajs-3432	243	40	of	of	ADP
iajs-3432	243	41	equations	equation	NOUN
iajs-3432	243	42	(	(	PUNCT
iajs-3432	243	43	17	17	NUM
iajs-3432	243	44	)	)	PUNCT
iajs-3432	243	45	and	and	CCONJ
iajs-3432	243	46	(	(	PUNCT
iajs-3432	243	47	18	18	NUM
iajs-3432	243	48	)	)	PUNCT
iajs-3432	243	49	.	.	PUNCT
iajs-3432	244	1	5	5	X
iajs-3432	244	2	.	.	X
iajs-3432	244	3	conclusions	conclusion	NOUN
iajs-3432	244	4	homeostasis	homeostasis	NOUN
iajs-3432	244	5	is	be	AUX
iajs-3432	244	6	a	a	DET
iajs-3432	244	7	reasonably	reasonably	ADV
iajs-3432	244	8	stable	stable	ADJ
iajs-3432	244	9	internal	internal	ADJ
iajs-3432	244	10	state	state	NOUN
iajs-3432	244	11	of	of	ADP
iajs-3432	244	12	physical	physical	ADJ
iajs-3432	244	13	and	and	CCONJ
iajs-3432	244	14	chemical	chemical	NOUN
iajs-3432	244	15	conditions	condition	NOUN
iajs-3432	244	16	that	that	PRON
iajs-3432	244	17	is	be	AUX
iajs-3432	244	18	regulated	regulate	VERB
iajs-3432	244	19	by	by	ADP
iajs-3432	244	20	living	live	VERB
iajs-3432	244	21	systems	system	NOUN
iajs-3432	244	22	through	through	ADP
iajs-3432	244	23	a	a	DET
iajs-3432	244	24	self	self	NOUN
iajs-3432	244	25	-	-	PUNCT
iajs-3432	244	26	regulating	regulate	VERB
iajs-3432	244	27	mechanism	mechanism	NOUN
iajs-3432	244	28	,	,	PUNCT
iajs-3432	244	29	despite	despite	SCONJ
iajs-3432	244	30	the	the	DET
iajs-3432	244	31	changes	change	NOUN
iajs-3432	244	32	necessary	necessary	ADJ
iajs-3432	244	33	for	for	ADP
iajs-3432	244	34	existence	existence	NOUN
iajs-3432	244	35	.	.	PUNCT
iajs-3432	245	1	the	the	DET
iajs-3432	245	2	blood	blood	NOUN
iajs-3432	245	3	maintains	maintain	VERB
iajs-3432	245	4	homeostasis	homeostasis	NOUN
iajs-3432	245	5	.	.	PUNCT
iajs-3432	246	1	part	part	NOUN
iajs-3432	246	2	of	of	ADP
iajs-3432	246	3	this	this	DET
iajs-3432	246	4	process	process	NOUN
iajs-3432	246	5	that	that	PRON
iajs-3432	246	6	allows	allow	VERB
iajs-3432	246	7	us	we	PRON
iajs-3432	246	8	to	to	PART
iajs-3432	246	9	adapt	adapt	VERB
iajs-3432	246	10	to	to	PART
iajs-3432	246	11	change	change	VERB
iajs-3432	246	12	and	and	CCONJ
iajs-3432	246	13	maintain	maintain	VERB
iajs-3432	246	14	life	life	NOUN
iajs-3432	246	15	are	be	AUX
iajs-3432	246	16	negative	negative	ADJ
iajs-3432	246	17	feedback	feedback	NOUN
iajs-3432	246	18	loops	loop	NOUN
iajs-3432	246	19	.	.	PUNCT
iajs-3432	247	1	mathematically	mathematically	ADV
iajs-3432	247	2	,	,	PUNCT
iajs-3432	247	3	homeostasis	homeostasis	NOUN
iajs-3432	247	4	is	be	AUX
iajs-3432	247	5	the	the	DET
iajs-3432	247	6	stability	stability	NOUN
iajs-3432	247	7	of	of	ADP
iajs-3432	247	8	a	a	DET
iajs-3432	247	9	state	state	NOUN
iajs-3432	247	10	of	of	ADP
iajs-3432	247	11	equilibrium	equilibrium	NOUN
iajs-3432	247	12	or	or	CCONJ
iajs-3432	247	13	oscillation	oscillation	NOUN
iajs-3432	247	14	.	.	PUNCT
iajs-3432	248	1	the	the	DET
iajs-3432	248	2	discrete	discrete	ADJ
iajs-3432	248	3	hematopoiesis	hematopoiesis	NOUN
iajs-3432	248	4	model	model	NOUN
iajs-3432	248	5	(	(	PUNCT
iajs-3432	248	6	1	1	NUM
iajs-3432	248	7	)	)	PUNCT
iajs-3432	248	8	,	,	PUNCT
iajs-3432	248	9	which	which	PRON
iajs-3432	248	10	has	have	VERB
iajs-3432	248	11	positive	positive	ADJ
iajs-3432	248	12	and	and	CCONJ
iajs-3432	248	13	negative	negative	ADJ
iajs-3432	248	14	coefficients	coefficient	NOUN
iajs-3432	248	15	oscillating	oscillate	VERB
iajs-3432	248	16	around	around	ADP
iajs-3432	248	17	the	the	DET
iajs-3432	248	18	equilibrium	equilibrium	NOUN
iajs-3432	248	19	k	k	PROPN
iajs-3432	248	20	,	,	PUNCT
iajs-3432	248	21	has	have	VERB
iajs-3432	248	22	the	the	DET
iajs-3432	248	23	primary	primary	ADJ
iajs-3432	248	24	purpose	purpose	NOUN
iajs-3432	248	25	of	of	ADP
iajs-3432	248	26	identifying	identify	VERB
iajs-3432	248	27	appropriate	appropriate	ADJ
iajs-3432	248	28	conditions	condition	NOUN
iajs-3432	248	29	for	for	ADP
iajs-3432	248	30	this	this	DET
iajs-3432	248	31	oscillation	oscillation	NOUN
iajs-3432	248	32	.	.	PUNCT
iajs-3432	249	1	the	the	DET
iajs-3432	249	2	hematopoiesis	hematopoiesis	NOUN
iajs-3432	249	3	model	model	NOUN
iajs-3432	249	4	is	be	AUX
iajs-3432	249	5	therefore	therefore	ADV
iajs-3432	249	6	analyzed	analyze	VERB
iajs-3432	249	7	as	as	ADP
iajs-3432	249	8	a	a	DET
iajs-3432	249	9	difference	difference	NOUN
iajs-3432	249	10	equation	equation	NOUN
iajs-3432	249	11	(	(	PUNCT
iajs-3432	249	12	3	3	NUM
iajs-3432	249	13	)	)	PUNCT
iajs-3432	249	14	.	.	PUNCT
iajs-3432	250	1	its	its	PRON
iajs-3432	250	2	oscillation	oscillation	NOUN
iajs-3432	250	3	is	be	AUX
iajs-3432	250	4	guaranteed	guarantee	VERB
iajs-3432	250	5	by	by	ADP
iajs-3432	250	6	sufficient	sufficient	ADJ
iajs-3432	250	7	conditions	condition	NOUN
iajs-3432	250	8	,	,	PUNCT
iajs-3432	250	9	which	which	PRON
iajs-3432	250	10	is	be	AUX
iajs-3432	250	11	a	a	DET
iajs-3432	250	12	new	new	ADJ
iajs-3432	250	13	discovery	discovery	NOUN
iajs-3432	250	14	for	for	ADP
iajs-3432	250	15	the	the	DET
iajs-3432	250	16	oscillatory	oscillatory	ADJ
iajs-3432	250	17	behavior	behavior	NOUN
iajs-3432	250	18	of	of	ADP
iajs-3432	250	19	the	the	DET
iajs-3432	250	20	presented	present	VERB
iajs-3432	250	21	model	model	NOUN
iajs-3432	250	22	.	.	PUNCT
iajs-3432	251	1	it	it	PRON
iajs-3432	251	2	is	be	AUX
iajs-3432	251	3	also	also	ADV
iajs-3432	251	4	necessary	necessary	ADJ
iajs-3432	251	5	to	to	PART
iajs-3432	251	6	find	find	VERB
iajs-3432	251	7	suitable	suitable	ADJ
iajs-3432	251	8	conditions	condition	NOUN
iajs-3432	251	9	to	to	PART
iajs-3432	251	10	guarantee	guarantee	VERB
iajs-3432	251	11	the	the	DET
iajs-3432	251	12	convergence	convergence	NOUN
iajs-3432	251	13	of	of	ADP
iajs-3432	251	14	non	non	ADJ
iajs-3432	251	15	-	-	ADJ
iajs-3432	251	16	oscillating	oscillating	ADJ
iajs-3432	251	17	solutions	solution	NOUN
iajs-3432	251	18	to	to	ADP
iajs-3432	251	19	the	the	DET
iajs-3432	251	20	equilibrium	equilibrium	NOUN
iajs-3432	251	21	k.	k.	PROPN
iajs-3432	252	1	we	we	PRON
iajs-3432	252	2	also	also	ADV
iajs-3432	252	3	show	show	VERB
iajs-3432	252	4	that	that	SCONJ
iajs-3432	252	5	non	non	ADJ
iajs-3432	252	6	-	-	ADJ
iajs-3432	252	7	oscillatory	oscillatory	ADJ
iajs-3432	252	8	numerical	numerical	ADJ
iajs-3432	252	9	solutions	solution	NOUN
iajs-3432	252	10	,	,	PUNCT
iajs-3432	252	11	as	as	SCONJ
iajs-3432	252	12	shown	show	VERB
iajs-3432	252	13	in	in	ADP
iajs-3432	252	14	example	example	NOUN
iajs-3432	252	15	2	2	NUM
iajs-3432	252	16	,	,	PUNCT
iajs-3432	252	17	can	can	AUX
iajs-3432	252	18	retain	retain	VERB
iajs-3432	252	19	the	the	DET
iajs-3432	252	20	associated	associate	VERB
iajs-3432	252	21	properties	property	NOUN
iajs-3432	252	22	of	of	ADP
iajs-3432	252	23	the	the	DET
iajs-3432	252	24	analytical	analytical	ADJ
iajs-3432	252	25	solutions	solution	NOUN
iajs-3432	252	26	.	.	PUNCT
iajs-3432	253	1	it	it	PRON
iajs-3432	253	2	is	be	AUX
iajs-3432	253	3	clear	clear	ADJ
iajs-3432	253	4	that	that	SCONJ
iajs-3432	253	5	the	the	DET
iajs-3432	253	6	technique	technique	NOUN
iajs-3432	253	7	used	use	VERB
iajs-3432	253	8	here	here	ADV
iajs-3432	253	9	can	can	AUX
iajs-3432	253	10	be	be	AUX
iajs-3432	253	11	applied	apply	VERB
iajs-3432	253	12	to	to	ADP
iajs-3432	253	13	models	model	NOUN
iajs-3432	253	14	that	that	PRON
iajs-3432	253	15	are	be	AUX
iajs-3432	253	16	periodic	periodic	ADJ
iajs-3432	253	17	or	or	CCONJ
iajs-3432	253	18	nearly	nearly	ADV
iajs-3432	253	19	periodic	periodic	ADJ
iajs-3432	253	20	as	as	ADV
iajs-3432	253	21	long	long	ADV
iajs-3432	253	22	as	as	SCONJ
iajs-3432	253	23	a	a	DET
iajs-3432	253	24	positive	positive	ADJ
iajs-3432	253	25	nearly	nearly	ADV
iajs-3432	253	26	periodic	periodic	ADJ
iajs-3432	253	27	solution	solution	NOUN
iajs-3432	253	28	exists	exist	VERB
iajs-3432	253	29	.	.	PUNCT
iajs-3432	254	1	acknowledgment	acknowledgment	NOUN
iajs-3432	254	2	the	the	DET
iajs-3432	254	3	authors	author	NOUN
iajs-3432	254	4	would	would	AUX
iajs-3432	254	5	like	like	VERB
iajs-3432	254	6	to	to	PART
iajs-3432	254	7	thank	thank	VERB
iajs-3432	254	8	prof	prof	PROPN
iajs-3432	254	9	.	.	PUNCT
iajs-3432	255	1	dinesh	dinesh	PROPN
iajs-3432	255	2	verma	verma	PROPN
iajs-3432	255	3	for	for	ADP
iajs-3432	255	4	his	his	PRON
iajs-3432	255	5	guidance	guidance	NOUN
iajs-3432	255	6	.	.	PUNCT
iajs-3432	256	1	conflict	conflict	NOUN
iajs-3432	256	2	of	of	ADP
iajs-3432	256	3	interest	interest	NOUN
iajs-3432	256	4	“	"	PUNCT
iajs-3432	256	5	conflict	conflict	NOUN
iajs-3432	256	6	of	of	ADP
iajs-3432	256	7	interest	interest	NOUN
iajs-3432	256	8	:	:	PUNCT
iajs-3432	256	9	the	the	DET
iajs-3432	256	10	authors	author	NOUN
iajs-3432	256	11	declare	declare	VERB
iajs-3432	256	12	that	that	SCONJ
iajs-3432	256	13	they	they	PRON
iajs-3432	256	14	have	have	VERB
iajs-3432	256	15	no	no	DET
iajs-3432	256	16	conflicts	conflict	NOUN
iajs-3432	256	17	of	of	ADP
iajs-3432	256	18	interest	interest	NOUN
iajs-3432	256	19	.	.	PUNCT
iajs-3432	256	20	”	"	PUNCT
iajs-3432	257	1	funding	fund	VERB
iajs-3432	257	2	there	there	PRON
iajs-3432	257	3	is	be	VERB
iajs-3432	257	4	no	no	DET
iajs-3432	257	5	financial	financial	ADJ
iajs-3432	257	6	support	support	NOUN
iajs-3432	257	7	in	in	ADP
iajs-3432	257	8	preparation	preparation	NOUN
iajs-3432	257	9	for	for	ADP
iajs-3432	257	10	the	the	DET
iajs-3432	257	11	publication	publication	NOUN
iajs-3432	257	12	.	.	PUNCT
iajs-3432	258	1	references	reference	NOUN
iajs-3432	258	2	1	1	NUM
iajs-3432	258	3	.	.	PUNCT
iajs-3432	258	4	knauer	knauer	PROPN
iajs-3432	258	5	,	,	PUNCT
iajs-3432	258	6	f.	f.	PROPN
iajs-3432	258	7	;	;	PUNCT
iajs-3432	258	8	stiehl	stiehl	PROPN
iajs-3432	258	9	,	,	PUNCT
iajs-3432	258	10	t.	t.	PROPN
iajs-3432	258	11	;	;	PUNCT
iajs-3432	258	12	marciniak	marciniak	PROPN
iajs-3432	258	13	-	-	NOUN
iajs-3432	258	14	czochra	czochra	PROPN
iajs-3432	258	15	,	,	PUNCT
iajs-3432	258	16	a.	a.	NOUN
iajs-3432	258	17	oscillations	oscillation	NOUN
iajs-3432	258	18	in	in	ADP
iajs-3432	258	19	a	a	DET
iajs-3432	258	20	white	white	ADJ
iajs-3432	258	21	blood	blood	NOUN
iajs-3432	258	22	cell	cell	NOUN
iajs-3432	258	23	production	production	NOUN
iajs-3432	258	24	model	model	NOUN
iajs-3432	258	25	with	with	ADP
iajs-3432	258	26	multiple	multiple	ADJ
iajs-3432	258	27	differentiation	differentiation	NOUN
iajs-3432	258	28	stages	stage	NOUN
iajs-3432	258	29	.	.	PUNCT
iajs-3432	259	1	journal	journal	NOUN
iajs-3432	259	2	of	of	ADP
iajs-3432	259	3	mathematical	mathematical	ADJ
iajs-3432	259	4	biology	biology	NOUN
iajs-3432	259	5	.	.	PUNCT
iajs-3432	260	1	2020	2020	NUM
iajs-3432	260	2	,	,	PUNCT
iajs-3432	260	3	80	80	NUM
iajs-3432	260	4	,	,	PUNCT
iajs-3432	260	5	575	575	NUM
iajs-3432	260	6	-	-	SYM
iajs-3432	260	7	600	600	NUM
iajs-3432	260	8	.	.	PUNCT
iajs-3432	261	1	https://doi.org/10.1007/s00285-019-01432-6	https://doi.org/10.1007/s00285-019-01432-6	PROPN
iajs-3432	261	2	.	.	PUNCT
iajs-3432	261	3	ihjpas	ihjpas	PROPN
iajs-3432	261	4	.	.	PUNCT
iajs-3432	262	1	2024	2024	NUM
iajs-3432	262	2	,	,	PUNCT
iajs-3432	262	3	37	37	NUM
iajs-3432	262	4	(	(	PUNCT
iajs-3432	262	5	3	3	NUM
iajs-3432	262	6	)	)	PUNCT
iajs-3432	262	7	394	394	NUM
iajs-3432	262	8	2	2	NUM
iajs-3432	262	9	.	.	PUNCT
iajs-3432	263	1	shao	shao	PROPN
iajs-3432	263	2	,	,	PUNCT
iajs-3432	263	3	j.	j.	PROPN
iajs-3432	263	4	pseudo	pseudo	NOUN
iajs-3432	263	5	almost	almost	ADV
iajs-3432	263	6	periodic	periodic	ADJ
iajs-3432	263	7	solutions	solution	NOUN
iajs-3432	263	8	for	for	ADP
iajs-3432	263	9	a	a	DET
iajs-3432	263	10	lasota	lasota	ADJ
iajs-3432	263	11	–	–	PUNCT
iajs-3432	263	12	wazewska	wazewska	NOUN
iajs-3432	263	13	model	model	NOUN
iajs-3432	263	14	with	with	ADP
iajs-3432	263	15	an	an	DET
iajs-3432	263	16	oscillating	oscillate	VERB
iajs-3432	263	17	death	death	NOUN
iajs-3432	263	18	rate	rate	NOUN
iajs-3432	263	19	.	.	PUNCT
iajs-3432	264	1	applied	apply	VERB
iajs-3432	264	2	mathematics	mathematics	NOUN
iajs-3432	264	3	letters	letter	NOUN
iajs-3432	264	4	.	.	PUNCT
iajs-3432	265	1	2015	2015	NUM
iajs-3432	265	2	,	,	PUNCT
iajs-3432	265	3	43	43	NUM
iajs-3432	265	4	,	,	PUNCT
iajs-3432	265	5	90	90	NUM
iajs-3432	265	6	-	-	SYM
iajs-3432	265	7	95	95	NUM
iajs-3432	265	8	.	.	PUNCT
iajs-3432	265	9	https://doi.org/10.1016/j.aml.2014.12.006	https://doi.org/10.1016/j.aml.2014.12.006	PROPN
iajs-3432	265	10	.	.	PUNCT
iajs-3432	266	1	3	3	X
iajs-3432	266	2	.	.	X
iajs-3432	266	3	santra	santra	PROPN
iajs-3432	266	4	,	,	PUNCT
iajs-3432	266	5	s.	s.	PROPN
iajs-3432	266	6	2016	2016	NUM
iajs-3432	266	7	.	.	PUNCT
iajs-3432	267	1	existence	existence	NOUN
iajs-3432	267	2	of	of	ADP
iajs-3432	267	3	positive	positive	ADJ
iajs-3432	267	4	solution	solution	NOUN
iajs-3432	267	5	and	and	CCONJ
iajs-3432	267	6	new	new	ADJ
iajs-3432	267	7	oscillation	oscillation	NOUN
iajs-3432	267	8	criteria	criterion	NOUN
iajs-3432	267	9	for	for	ADP
iajs-3432	267	10	nonlinear	nonlinear	ADJ
iajs-3432	267	11	first	first	ADJ
iajs-3432	267	12	-	-	PUNCT
iajs-3432	267	13	order	order	NOUN
iajs-3432	267	14	neutral	neutral	ADJ
iajs-3432	267	15	delay	delay	NOUN
iajs-3432	267	16	differential	differential	PROPN
iajs-3432	267	17	equations	equation	NOUN
iajs-3432	267	18	.	.	PUNCT
iajs-3432	268	1	differ	differ	VERB
iajs-3432	268	2	.	.	PUNCT
iajs-3432	269	1	equ	equ	PROPN
iajs-3432	269	2	.	.	PUNCT
iajs-3432	269	3	appl	appl	PROPN
iajs-3432	269	4	.	.	PROPN
iajs-3432	269	5	2016	2016	NUM
iajs-3432	269	6	,	,	PUNCT
iajs-3432	269	7	8(1	8(1	NOUN
iajs-3432	269	8	)	)	PUNCT
iajs-3432	269	9	,	,	PUNCT
iajs-3432	269	10	33-51.http://doi.org/10.7153/dea-0803	33-51.http://doi.org/10.7153/dea-0803	NOUN
iajs-3432	269	11	.	.	PUNCT
iajs-3432	270	1	4	4	NUM
iajs-3432	270	2	.	.	X
iajs-3432	270	3	mohamad	mohamad	PROPN
iajs-3432	270	4	,	,	PUNCT
iajs-3432	270	5	h.	h.	PROPN
iajs-3432	270	6	a.	a.	PROPN
iajs-3432	270	7	;	;	PUNCT
iajs-3432	270	8	mohi	mohi	PROPN
iajs-3432	270	9	,	,	PUNCT
iajs-3432	270	10	h.	h.	PROPN
iajs-3432	270	11	m.	m.	NOUN
iajs-3432	270	12	oscillation	oscillation	NOUN
iajs-3432	270	13	of	of	ADP
iajs-3432	270	14	first	first	ADJ
iajs-3432	270	15	-	-	PUNCT
iajs-3432	270	16	order	order	NOUN
iajs-3432	270	17	neutral	neutral	ADJ
iajs-3432	270	18	difference	difference	NOUN
iajs-3432	270	19	equations	equation	NOUN
iajs-3432	270	20	with	with	ADP
iajs-3432	270	21	positive	positive	ADJ
iajs-3432	270	22	and	and	CCONJ
iajs-3432	270	23	negative	negative	ADJ
iajs-3432	270	24	coefficients	coefficient	NOUN
iajs-3432	270	25	.	.	PUNCT
iajs-3432	271	1	international	international	ADJ
iajs-3432	271	2	journal	journal	NOUN
iajs-3432	271	3	of	of	ADP
iajs-3432	271	4	difference	difference	NOUN
iajs-3432	271	5	equations	equation	NOUN
iajs-3432	271	6	.	.	PUNCT
iajs-3432	272	1	2015,10(2),213–220	2015,10(2),213–220	ADJ
iajs-3432	272	2	.	.	PUNCT
iajs-3432	273	1	5	5	NUM
iajs-3432	273	2	.	.	X
iajs-3432	273	3	mushtt	mushtt	PROPN
iajs-3432	273	4	,	,	PUNCT
iajs-3432	273	5	i.	i.	PROPN
iajs-3432	273	6	z.	z.	PROPN
iajs-3432	273	7	oscillation	oscillation	NOUN
iajs-3432	273	8	of	of	ADP
iajs-3432	273	9	solution	solution	NOUN
iajs-3432	273	10	of	of	ADP
iajs-3432	273	11	third	third	ADJ
iajs-3432	273	12	order	order	NOUN
iajs-3432	273	13	non	non	ADJ
iajs-3432	273	14	-	-	ADJ
iajs-3432	273	15	linear	linear	ADJ
iajs-3432	273	16	neutral	neutral	ADJ
iajs-3432	273	17	difference	difference	NOUN
iajs-3432	273	18	equations	equation	NOUN
iajs-3432	273	19	.	.	PUNCT
iajs-3432	274	1	international	international	ADJ
iajs-3432	274	2	journal	journal	NOUN
iajs-3432	274	3	of	of	ADP
iajs-3432	274	4	advanced	advanced	ADJ
iajs-3432	274	5	scientific	scientific	ADJ
iajs-3432	274	6	and	and	CCONJ
iajs-3432	274	7	technical	technical	ADJ
iajs-3432	274	8	research	research	NOUN
iajs-3432	274	9	.	.	PUNCT
iajs-3432	275	1	2020	2020	NUM
iajs-3432	275	2	,	,	PUNCT
iajs-3432	275	3	10(4),1	10(4),1	NUM
iajs-3432	275	4	-	-	SYM
iajs-3432	275	5	10	10	NUM
iajs-3432	275	6	.	.	PUNCT
iajs-3432	276	1	https://doi.org/10.1080/09720502.2021.1892275	https://doi.org/10.1080/09720502.2021.1892275	X
iajs-3432	276	2	.	.	PROPN
iajs-3432	277	1	6	6	NUM
iajs-3432	277	2	.	.	X
iajs-3432	277	3	koplatadze	koplatadze	PROPN
iajs-3432	277	4	,	,	PUNCT
iajs-3432	277	5	r.	r.	PROPN
iajs-3432	277	6	on	on	ADP
iajs-3432	277	7	asymptotic	asymptotic	ADJ
iajs-3432	277	8	behavior	behavior	NOUN
iajs-3432	277	9	of	of	ADP
iajs-3432	277	10	solutions	solution	NOUN
iajs-3432	277	11	of	of	ADP
iajs-3432	277	12	first	first	ADJ
iajs-3432	277	13	order	order	NOUN
iajs-3432	277	14	difference	difference	NOUN
iajs-3432	277	15	equations	equation	NOUN
iajs-3432	277	16	with	with	ADP
iajs-3432	277	17	several	several	ADJ
iajs-3432	277	18	delay	delay	NOUN
iajs-3432	277	19	.	.	PUNCT
iajs-3432	278	1	bulletin	bulletin	NOUN
iajs-3432	278	2	.	.	PUNCT
iajs-3432	279	1	tbilisi	tbilisi	PROPN
iajs-3432	279	2	international	international	ADJ
iajs-3432	279	3	centre	centre	NOUN
iajs-3432	279	4	of	of	ADP
iajs-3432	279	5	mathematics	mathematic	NOUN
iajs-3432	279	6	and	and	CCONJ
iajs-3432	279	7	informatics	informatic	NOUN
iajs-3432	279	8	.	.	PUNCT
iajs-3432	280	1	2017,21	2017,21	NUM
iajs-3432	280	2	(	(	PUNCT
iajs-3432	280	3	2),117–123	2),117–123	NUM
iajs-3432	280	4	.	.	PUNCT
iajs-3432	281	1	7	7	X
iajs-3432	281	2	.	.	X
iajs-3432	281	3	agarwal	agarwal	PROPN
iajs-3432	281	4	,	,	PUNCT
iajs-3432	281	5	r.	r.	PROPN
iajs-3432	281	6	p.	p.	PROPN
iajs-3432	281	7	;	;	PUNCT
iajs-3432	282	1	karakoc	karakoc	PROPN
iajs-3432	282	2	,	,	PUNCT
iajs-3432	282	3	f.	f.	PROPN
iajs-3432	282	4	oscillation	oscillation	NOUN
iajs-3432	282	5	of	of	ADP
iajs-3432	282	6	impulsive	impulsive	ADJ
iajs-3432	282	7	partial	partial	ADJ
iajs-3432	282	8	difference	difference	NOUN
iajs-3432	282	9	equations	equation	NOUN
iajs-3432	282	10	with	with	ADP
iajs-3432	282	11	continuous	continuous	ADJ
iajs-3432	282	12	variables	variable	NOUN
iajs-3432	282	13	.	.	PUNCT
iajs-3432	283	1	mathematical	mathematical	ADJ
iajs-3432	283	2	and	and	CCONJ
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iajs-3432	283	4	modelling	modelling	NOUN
iajs-3432	283	5	.	.	PUNCT
iajs-3432	284	1	2009	2009	NUM
iajs-3432	284	2	,	,	PUNCT
iajs-3432	284	3	50	50	NUM
iajs-3432	284	4	,	,	PUNCT
iajs-3432	284	5	9	9	NUM
iajs-3432	284	6	-	-	SYM
iajs-3432	284	7	10	10	NUM
iajs-3432	284	8	,	,	PUNCT
iajs-3432	284	9	1262	1262	NUM
iajs-3432	284	10	-	-	SYM
iajs-3432	284	11	1278	1278	NUM
iajs-3432	284	12	.	.	PUNCT
iajs-3432	285	1	https://doi.org/10.1016/j.mcm.2009.07.013	https://doi.org/10.1016/j.mcm.2009.07.013	PROPN
iajs-3432	285	2	.	.	PROPN
iajs-3432	285	3	8	8	NUM
iajs-3432	285	4	.	.	X
iajs-3432	285	5	sharba	sharba	NOUN
iajs-3432	285	6	,	,	PUNCT
iajs-3432	285	7	b.	b.	PROPN
iajs-3432	285	8	a.	a.	PROPN
iajs-3432	285	9	;	;	PUNCT
iajs-3432	285	10	jaddoa	jaddoa	PROPN
iajs-3432	285	11	,	,	PUNCT
iajs-3432	285	12	a.	a.	PROPN
iajs-3432	285	13	f.	f.	PROPN
iajs-3432	285	14	on	on	ADP
iajs-3432	285	15	the	the	DET
iajs-3432	285	16	existence	existence	NOUN
iajs-3432	285	17	and	and	CCONJ
iajs-3432	285	18	oscillatory	oscillatory	ADJ
iajs-3432	285	19	solutions	solution	NOUN
iajs-3432	285	20	of	of	ADP
iajs-3432	285	21	multiple	multiple	ADJ
iajs-3432	285	22	delay	delay	NOUN
iajs-3432	285	23	differential	differential	ADJ
iajs-3432	285	24	equation	equation	NOUN
iajs-3432	285	25	.	.	PUNCT
iajs-3432	286	1	iraqi	iraqi	ADJ
iajs-3432	286	2	journal	journal	PROPN
iajs-3432	286	3	of	of	ADP
iajs-3432	286	4	science	science	NOUN
iajs-3432	286	5	.	.	PUNCT
iajs-3432	287	1	2023	2023	NUM
iajs-3432	287	2	,	,	PUNCT
iajs-3432	287	3	878	878	NUM
iajs-3432	287	4	-	-	SYM
iajs-3432	287	5	892	892	NUM
iajs-3432	287	6	.	.	PUNCT
iajs-3432	288	1	https://doi.org/10.24996/ijs.2023.64.2.33	https://doi.org/10.24996/ijs.2023.64.2.33	PROPN
iajs-3432	288	2	.	.	PUNCT
iajs-3432	288	3	9	9	X
iajs-3432	288	4	.	.	X
iajs-3432	288	5	bhuyan	bhuyan	PROPN
iajs-3432	288	6	,	,	PUNCT
iajs-3432	288	7	a.k	a.k	PROPN
iajs-3432	288	8	.	.	PROPN
iajs-3432	288	9	;	;	PUNCT
iajs-3432	288	10	padhy	padhy	PROPN
iajs-3432	288	11	,	,	PUNCT
iajs-3432	288	12	l.n	l.n	PROPN
iajs-3432	288	13	.	.	PROPN
iajs-3432	288	14	;	;	PUNCT
iajs-3432	288	15	rath	rath	PROPN
iajs-3432	288	16	,	,	PUNCT
iajs-3432	288	17	r.	r.	PROPN
iajs-3432	288	18	impact	impact	NOUN
iajs-3432	288	19	of	of	ADP
iajs-3432	288	20	different	different	ADJ
iajs-3432	288	21	types	type	NOUN
iajs-3432	288	22	of	of	ADP
iajs-3432	288	23	non	non	ADJ
iajs-3432	288	24	-	-	NOUN
iajs-3432	288	25	linearity	linearity	NOUN
iajs-3432	288	26	on	on	ADP
iajs-3432	288	27	the	the	DET
iajs-3432	288	28	oscillatory	oscillatory	ADJ
iajs-3432	288	29	behavior	behavior	NOUN
iajs-3432	288	30	of	of	ADP
iajs-3432	288	31	higher	high	ADJ
iajs-3432	288	32	order	order	NOUN
iajs-3432	288	33	neutral	neutral	ADJ
iajs-3432	288	34	difference	difference	NOUN
iajs-3432	288	35	equations	equation	NOUN
iajs-3432	288	36	.	.	PUNCT
iajs-3432	289	1	mathematica	mathematica	PROPN
iajs-3432	289	2	slovaca	slovaca	PROPN
iajs-3432	289	3	.	.	PROPN
iajs-3432	290	1	2021	2021	NUM
iajs-3432	290	2	,	,	PUNCT
iajs-3432	290	3	71(4	71(4	NUM
iajs-3432	290	4	)	)	PUNCT
iajs-3432	290	5	,	,	PUNCT
iajs-3432	290	6	941	941	NUM
iajs-3432	290	7	-	-	SYM
iajs-3432	290	8	960	960	NUM
iajs-3432	290	9	.	.	PUNCT
iajs-3432	291	1	https://doi.org/10.1515/ms-2021-0032	https://doi.org/10.1515/ms-2021-0032	X
iajs-3432	291	2	.	.	PROPN
iajs-3432	292	1	10	10	NUM
iajs-3432	292	2	.	.	PUNCT
iajs-3432	293	1	chatzarakis	chatzarakis	PROPN
iajs-3432	293	2	,	,	PUNCT
iajs-3432	293	3	e.	e.	PROPN
iajs-3432	293	4	g.	g.	PROPN
iajs-3432	293	5	;	;	PUNCT
iajs-3432	293	6	philos	philos	PROPN
iajs-3432	293	7	,	,	PUNCT
iajs-3432	293	8	g.	g.	PROPN
iajs-3432	293	9	c.	c.	PROPN
iajs-3432	293	10	,	,	PUNCT
iajs-3432	293	11	stavroulakis	stavroulakis	PROPN
iajs-3432	293	12	,	,	PUNCT
iajs-3432	293	13	p.	p.	NOUN
iajs-3432	293	14	i.	i.	NOUN
iajs-3432	293	15	on	on	ADP
iajs-3432	293	16	the	the	DET
iajs-3432	293	17	oscillation	oscillation	NOUN
iajs-3432	293	18	of	of	ADP
iajs-3432	293	19	solutions	solution	NOUN
iajs-3432	293	20	to	to	PART
iajs-3432	293	21	linear	linear	VERB
iajs-3432	293	22	difference	difference	NOUN
iajs-3432	293	23	equation	equation	NOUN
iajs-3432	293	24	with	with	ADP
iajs-3432	293	25	variable	variable	ADJ
iajs-3432	293	26	delay	delay	NOUN
iajs-3432	293	27	.	.	PUNCT
iajs-3432	294	1	electronic	electronic	ADJ
iajs-3432	294	2	journal	journal	NOUN
iajs-3432	294	3	of	of	ADP
iajs-3432	294	4	differential	differential	ADJ
iajs-3432	294	5	equations	equation	NOUN
iajs-3432	294	6	.	.	PUNCT
iajs-3432	295	1	2008	2008	NUM
iajs-3432	295	2	,	,	PUNCT
iajs-3432	295	3	50	50	NUM
iajs-3432	295	4	,	,	PUNCT
iajs-3432	295	5	1–15	1–15	NUM
iajs-3432	295	6	.	.	PUNCT
iajs-3432	296	1	11	11	NUM
iajs-3432	296	2	.	.	PUNCT
iajs-3432	297	1	mushtt	mushtt	PROPN
iajs-3432	297	2	,	,	PUNCT
iajs-3432	297	3	i.	i.	PROPN
iajs-3432	297	4	z.	z.	PROPN
iajs-3432	297	5	;	;	PUNCT
iajs-3432	297	6	hameed	hameed	PROPN
iajs-3432	297	7	,	,	PUNCT
iajs-3432	297	8	d.	d.	PROPN
iajs-3432	297	9	m.	m.	PROPN
iajs-3432	297	10	;	;	PUNCT
iajs-3432	297	11	alwan	alwan	PROPN
iajs-3432	297	12	,	,	PUNCT
iajs-3432	297	13	b.	b.	PROPN
iajs-3432	297	14	m.	m.	NOUN
iajs-3432	297	15	on	on	ADP
iajs-3432	297	16	oscillating	oscillate	VERB
iajs-3432	297	17	outcomes	outcome	NOUN
iajs-3432	297	18	for	for	ADP
iajs-3432	297	19	non	non	ADJ
iajs-3432	297	20	-	-	ADJ
iajs-3432	297	21	homogenous	homogenous	ADJ
iajs-3432	297	22	neutral	neutral	ADJ
iajs-3432	297	23	difference	difference	NOUN
iajs-3432	297	24	equations	equation	NOUN
iajs-3432	297	25	of	of	ADP
iajs-3432	297	26	a	a	DET
iajs-3432	297	27	third	third	ADJ
iajs-3432	297	28	order	order	NOUN
iajs-3432	297	29	.	.	PUNCT
iajs-3432	298	1	journal	journal	NOUN
iajs-3432	298	2	of	of	ADP
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iajs-3432	298	4	mathematics	mathematic	NOUN
iajs-3432	298	5	.	.	PUNCT
iajs-3432	299	1	2021	2021	NUM
iajs-3432	299	2	,	,	PUNCT
iajs-3432	299	3	24(5	24(5	NUM
iajs-3432	299	4	)	)	PUNCT
iajs-3432	299	5	,	,	PUNCT
iajs-3432	299	6	1387	1387	NUM
iajs-3432	299	7	-	-	SYM
iajs-3432	299	8	1395	1395	NUM
iajs-3432	299	9	.	.	PUNCT
iajs-3432	300	1	https://doi.org/10.1080/09720502.2021.1892275	https://doi.org/10.1080/09720502.2021.1892275	X
iajs-3432	300	2	.	.	PROPN
iajs-3432	301	1	12	12	NUM
iajs-3432	301	2	.	.	PUNCT
iajs-3432	302	1	hadeed	hadeed	ADJ
iajs-3432	302	2	,	,	PUNCT
iajs-3432	302	3	i.s	i.s	PROPN
iajs-3432	302	4	.	.	PROPN
iajs-3432	302	5	;	;	PUNCT
iajs-3432	302	6	mohamad	mohamad	PROPN
iajs-3432	302	7	,	,	PUNCT
iajs-3432	302	8	h.a	h.a	PROPN
iajs-3432	302	9	.	.	PROPN
iajs-3432	302	10	oscillation	oscillation	NOUN
iajs-3432	302	11	of	of	ADP
iajs-3432	302	12	the	the	DET
iajs-3432	302	13	impulsive	impulsive	ADJ
iajs-3432	302	14	hematopoiesis	hematopoiesis	NOUN
iajs-3432	302	15	model	model	NOUN
iajs-3432	302	16	with	with	ADP
iajs-3432	302	17	positive	positive	ADJ
iajs-3432	302	18	and	and	CCONJ
iajs-3432	302	19	negative	negative	ADJ
iajs-3432	302	20	coefficients	coefficient	NOUN
iajs-3432	302	21	.	.	PUNCT
iajs-3432	303	1	baghdad	baghdad	PROPN
iajs-3432	303	2	science	science	PROPN
iajs-3432	303	3	journal	journal	PROPN
iajs-3432	303	4	.	.	PUNCT
iajs-3432	304	1	2023	2023	NUM
iajs-3432	304	2	.	.	PUNCT
iajs-3432	305	1	https://doi.org/10.21123/bsj.2023.8796	https://doi.org/10.21123/bsj.2023.8796	AUX
iajs-3432	305	2	13	13	NUM
iajs-3432	305	3	.	.	PUNCT
iajs-3432	306	1	ma	ma	PROPN
iajs-3432	306	2	,	,	PUNCT
iajs-3432	306	3	r.	r.	PROPN
iajs-3432	306	4	;	;	PUNCT
iajs-3432	306	5	wang	wang	PROPN
iajs-3432	306	6	,	,	PUNCT
iajs-3432	306	7	j.	j.	PROPN
iajs-3432	306	8	;	;	PUNCT
iajs-3432	306	9	li	li	PROPN
iajs-3432	306	10	,	,	PUNCT
iajs-3432	306	11	m.	m.	NOUN
iajs-3432	306	12	,	,	PUNCT
iajs-3432	306	13	2022	2022	NUM
iajs-3432	306	14	.	.	PUNCT
iajs-3432	307	1	almost	almost	ADV
iajs-3432	307	2	periodic	periodic	ADJ
iajs-3432	307	3	solutions	solution	NOUN
iajs-3432	307	4	for	for	ADP
iajs-3432	307	5	two	two	NUM
iajs-3432	307	6	non	non	ADJ
iajs-3432	307	7	-	-	ADJ
iajs-3432	307	8	instantaneous	instantaneous	ADJ
iajs-3432	307	9	impulsive	impulsive	ADJ
iajs-3432	307	10	biological	biological	ADJ
iajs-3432	307	11	models	model	NOUN
iajs-3432	307	12	.	.	PUNCT
iajs-3432	308	1	qualitative	qualitative	ADJ
iajs-3432	308	2	theory	theory	NOUN
iajs-3432	308	3	of	of	ADP
iajs-3432	308	4	dynamical	dynamical	ADJ
iajs-3432	308	5	systems	system	NOUN
iajs-3432	308	6	.	.	PUNCT
iajs-3432	309	1	2022	2022	NUM
iajs-3432	309	2	,	,	PUNCT
iajs-3432	309	3	21(3	21(3	NUM
iajs-3432	309	4	)	)	PUNCT
iajs-3432	309	5	,	,	PUNCT
iajs-3432	309	6	84	84	NUM
iajs-3432	309	7	.	.	PUNCT
iajs-3432	310	1	https://doi.org/10.1007/s12346-022-00603-z	https://doi.org/10.1007/s12346-022-00603-z	NOUN
iajs-3432	310	2	.	.	PUNCT
iajs-3432	311	1	14	14	NUM
iajs-3432	311	2	.	.	X
iajs-3432	311	3	saker	saker	PROPN
iajs-3432	311	4	,	,	PUNCT
iajs-3432	311	5	s.	s.	PROPN
iajs-3432	311	6	h.	h.	PROPN
iajs-3432	311	7	qualitative	qualitative	VERB
iajs-3432	311	8	analysis	analysis	NOUN
iajs-3432	311	9	of	of	ADP
iajs-3432	311	10	discrete	discrete	ADJ
iajs-3432	311	11	nonlinear	nonlinear	ADJ
iajs-3432	311	12	delay	delay	NOUN
iajs-3432	311	13	survival	survival	PROPN
iajs-3432	311	14	red	red	ADJ
iajs-3432	311	15	blood	blood	NOUN
iajs-3432	311	16	cells	cell	NOUN
iajs-3432	311	17	model	model	NOUN
iajs-3432	311	18	.	.	PUNCT
iajs-3432	312	1	nonlinear	nonlinear	ADJ
iajs-3432	312	2	analysis	analysis	NOUN
iajs-3432	312	3	:	:	PUNCT
iajs-3432	312	4	real	real	ADJ
iajs-3432	312	5	world	world	NOUN
iajs-3432	312	6	applications	application	NOUN
iajs-3432	312	7	.	.	PUNCT
iajs-3432	313	1	2008	2008	NUM
iajs-3432	313	2	,	,	PUNCT
iajs-3432	313	3	9	9	NUM
iajs-3432	313	4	,	,	PUNCT
iajs-3432	313	5	471	471	NUM
iajs-3432	313	6	-	-	SYM
iajs-3432	313	7	489	489	NUM
iajs-3432	313	8	.	.	PUNCT
iajs-3432	314	1	https://doi.org/10.1016/j.nonrwa.2006.11.013	https://doi.org/10.1016/j.nonrwa.2006.11.013	PROPN
iajs-3432	314	2	.	.	PUNCT
iajs-3432	315	1	15	15	NUM
iajs-3432	315	2	.	.	PUNCT
iajs-3432	316	1	li	li	PROPN
iajs-3432	316	2	,	,	PUNCT
iajs-3432	316	3	w.	w.	PROPN
iajs-3432	316	4	;	;	PUNCT
iajs-3432	316	5	xianyi	xianyi	PROPN
iajs-3432	316	6	,	,	PUNCT
iajs-3432	316	7	l.	l.	PROPN
iajs-3432	316	8	neimark	neimark	PROPN
iajs-3432	316	9	-	-	PUNCT
iajs-3432	316	10	sacker	sacker	NOUN
iajs-3432	316	11	bifurcation	bifurcation	NOUN
iajs-3432	316	12	of	of	ADP
iajs-3432	316	13	a	a	DET
iajs-3432	316	14	semi	semi	ADJ
iajs-3432	316	15	-	-	ADJ
iajs-3432	316	16	discrete	discrete	ADJ
iajs-3432	316	17	hematopoiesis	hematopoiesis	NOUN
iajs-3432	316	18	model	model	NOUN
iajs-3432	316	19	.	.	PUNCT
iajs-3432	317	1	journal	journal	PROPN
iajs-3432	317	2	of	of	ADP
iajs-3432	317	3	applied	apply	VERB
iajs-3432	317	4	analysis	analysis	NOUN
iajs-3432	317	5	and	and	CCONJ
iajs-3432	317	6	computation	computation	NOUN
iajs-3432	317	7	.	.	PUNCT
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iajs-3432	318	2	,	,	PUNCT
iajs-3432	318	3	8(6),1679	8(6),1679	NOUN
iajs-3432	318	4	-	-	SYM
iajs-3432	318	5	1693	1693	NUM
iajs-3432	318	6	.	.	PUNCT
iajs-3432	319	1	http://doi.org/10.11948/2018.1679	http://doi.org/10.11948/2018.1679	NOUN
iajs-3432	319	2	16	16	NUM
iajs-3432	319	3	.	.	PUNCT
iajs-3432	320	1	al	al	PROPN
iajs-3432	320	2	-	-	PUNCT
iajs-3432	320	3	nassir	nassir	PROPN
iajs-3432	320	4	,	,	PUNCT
iajs-3432	320	5	s.	s.	PROPN
iajs-3432	320	6	the	the	DET
iajs-3432	320	7	dynamics	dynamic	NOUN
iajs-3432	320	8	of	of	ADP
iajs-3432	320	9	biological	biological	ADJ
iajs-3432	320	10	models	model	NOUN
iajs-3432	320	11	with	with	ADP
iajs-3432	320	12	optimal	optimal	ADJ
iajs-3432	320	13	harvesting	harvesting	NOUN
iajs-3432	320	14	.	.	PUNCT
iajs-3432	321	1	iraqi	iraqi	ADJ
iajs-3432	321	2	journal	journal	PROPN
iajs-3432	321	3	of	of	ADP
iajs-3432	321	4	science	science	NOUN
iajs-3432	321	5	.	.	PUNCT
iajs-3432	322	1	2021	2021	NUM
iajs-3432	322	2	,	,	PUNCT
iajs-3432	322	3	62(9	62(9	NUM
iajs-3432	322	4	)	)	PUNCT
iajs-3432	322	5	,	,	PUNCT
iajs-3432	322	6	3039	3039	NUM
iajs-3432	322	7	-	-	SYM
iajs-3432	322	8	3051	3051	NUM
iajs-3432	322	9	.	.	PUNCT
iajs-3432	323	1	https://doi.org/10.24996/ijs.2021.62.9.19	https://doi.org/10.24996/ijs.2021.62.9.19	PROPN
iajs-3432	323	2	17	17	NUM
iajs-3432	323	3	.	.	PUNCT
iajs-3432	324	1	tipsri	tipsri	PROPN
iajs-3432	324	2	,	,	PUNCT
iajs-3432	324	3	s.	s.	PROPN
iajs-3432	324	4	;	;	PUNCT
iajs-3432	324	5	chinviriyasit	chinviriyasit	ADJ
iajs-3432	324	6	,	,	PUNCT
iajs-3432	324	7	w.	w.	NOUN
iajs-3432	324	8	the	the	DET
iajs-3432	324	9	effect	effect	NOUN
iajs-3432	324	10	of	of	ADP
iajs-3432	324	11	time	time	NOUN
iajs-3432	324	12	delay	delay	NOUN
iajs-3432	324	13	on	on	ADP
iajs-3432	324	14	the	the	DET
iajs-3432	324	15	dynamics	dynamic	NOUN
iajs-3432	324	16	of	of	ADP
iajs-3432	324	17	an	an	DET
iajs-3432	324	18	seir	seir	ADJ
iajs-3432	324	19	model	model	NOUN
iajs-3432	324	20	with	with	ADP
iajs-3432	324	21	nonlinear	nonlinear	ADJ
iajs-3432	324	22	incidence	incidence	NOUN
iajs-3432	324	23	.	.	PUNCT
iajs-3432	325	1	chaos	chaos	NOUN
iajs-3432	325	2	solitons	soliton	NOUN
iajs-3432	325	3	&	&	CCONJ
iajs-3432	325	4	fractals	fractal	NOUN
iajs-3432	325	5	.	.	PUNCT
iajs-3432	326	1	2015	2015	NUM
iajs-3432	326	2	,	,	PUNCT
iajs-3432	326	3	75	75	NUM
iajs-3432	326	4	,	,	PUNCT
iajs-3432	326	5	153–172	153–172	NUM
iajs-3432	326	6	.	.	PUNCT
iajs-3432	327	1	https://doi.org/10.1016/j.chaos.2015.02.017	https://doi.org/10.1016/j.chaos.2015.02.017	NOUN
iajs-3432	327	2	.	.	PUNCT
iajs-3432	327	3	18	18	NUM
iajs-3432	327	4	.	.	PUNCT
iajs-3432	328	1	naji	naji	PROPN
iajs-3432	328	2	,	,	PUNCT
iajs-3432	328	3	r.	r.	PROPN
iajs-3432	328	4	k.	k.	PROPN
iajs-3432	328	5	;	;	PUNCT
iajs-3432	328	6	mohsen	mohsen	PROPN
iajs-3432	328	7	,	,	PUNCT
iajs-3432	328	8	a.	a.	NOUN
iajs-3432	328	9	a.	a.	NOUN
iajs-3432	328	10	stability	stability	NOUN
iajs-3432	328	11	and	and	CCONJ
iajs-3432	328	12	bifurcation	bifurcation	NOUN
iajs-3432	328	13	of	of	ADP
iajs-3432	328	14	a	a	DET
iajs-3432	328	15	delay	delay	NOUN
iajs-3432	328	16	cancer	cancer	NOUN
iajs-3432	328	17	model	model	NOUN
iajs-3432	328	18	in	in	ADP
iajs-3432	328	19	the	the	DET
iajs-3432	328	20	polluted	polluted	ADJ
iajs-3432	328	21	environment	environment	NOUN
iajs-3432	328	22	.	.	PUNCT
iajs-3432	329	1	advances	advance	NOUN
iajs-3432	329	2	in	in	ADP
iajs-3432	329	3	systems	system	NOUN
iajs-3432	329	4	science	science	NOUN
iajs-3432	329	5	and	and	CCONJ
iajs-3432	329	6	applications	application	NOUN
iajs-3432	329	7	.	.	PUNCT
iajs-3432	330	1	2022	2022	NUM
iajs-3432	330	2	,	,	PUNCT
iajs-3432	330	3	22(3	22(3	NUM
iajs-3432	330	4	)	)	PUNCT
iajs-3432	330	5	,	,	PUNCT
iajs-3432	330	6	1	1	NUM
iajs-3432	330	7	-	-	SYM
iajs-3432	330	8	17	17	NUM
iajs-3432	330	9	.	.	PUNCT
iajs-3432	331	1	https://doi.org/10.25728/assa.2022.22.3.983	https://doi.org/10.25728/assa.2022.22.3.983	PROPN
iajs-3432	331	2	.	.	PUNCT
iajs-3432	332	1	19	19	NUM
iajs-3432	332	2	.	.	PUNCT
iajs-3432	332	3	song	song	PROPN
iajs-3432	332	4	,	,	PUNCT
iajs-3432	332	5	y.	y.	PROPN
iajs-3432	332	6	;	;	PUNCT
iajs-3432	332	7	wei	wei	PROPN
iajs-3432	332	8	,	,	PUNCT
iajs-3432	332	9	j.	j.	PROPN
iajs-3432	332	10	;	;	PUNCT
iajs-3432	332	11	han	han	PROPN
iajs-3432	332	12	,	,	PUNCT
iajs-3432	332	13	m.	m.	NOUN
iajs-3432	332	14	2	2	NUM
iajs-3432	332	15	.	.	PUNCT
iajs-3432	332	16	local	local	ADJ
iajs-3432	332	17	and	and	CCONJ
iajs-3432	332	18	global	global	ADJ
iajs-3432	332	19	hopf	hopf	ADJ
iajs-3432	332	20	bifurcation	bifurcation	NOUN
iajs-3432	332	21	in	in	ADP
iajs-3432	332	22	a	a	DET
iajs-3432	332	23	delayed	delay	VERB
iajs-3432	332	24	hematopoiesis	hematopoiesis	NOUN
iajs-3432	332	25	model	model	NOUN
iajs-3432	332	26	.	.	PUNCT
iajs-3432	333	1	international	international	ADJ
iajs-3432	333	2	journal	journal	PROPN
iajs-3432	333	3	of	of	ADP
iajs-3432	333	4	bifurcation	bifurcation	NOUN
iajs-3432	333	5	and	and	CCONJ
iajs-3432	333	6	chaos	chaos	NOUN
iajs-3432	333	7	.	.	PUNCT
iajs-3432	334	1	2004	2004	NUM
iajs-3432	334	2	,	,	PUNCT
iajs-3432	334	3	14(11	14(11	NUM
iajs-3432	334	4	)	)	PUNCT
iajs-3432	334	5	,	,	PUNCT
iajs-3432	334	6	3909	3909	NUM
iajs-3432	334	7	-	-	SYM
iajs-3432	334	8	3919	3919	NUM
iajs-3432	334	9	.	.	PUNCT
iajs-3432	335	1	https://doi.org/10.1142/s0218127404011697	https://doi.org/10.1142/s0218127404011697	NUM
iajs-3432	335	2	.	.	PROPN
iajs-3432	335	3	20	20	NUM
iajs-3432	335	4	.	.	PUNCT
iajs-3432	336	1	hadeed	hadeed	PROPN
iajs-3432	336	2	,	,	PUNCT
iajs-3432	336	3	i.	i.	PROPN
iajs-3432	336	4	s.	s.	PROPN
iajs-3432	336	5	;	;	PUNCT
iajs-3432	336	6	mohamad	mohamad	PROPN
iajs-3432	336	7	,	,	PUNCT
iajs-3432	336	8	h.	h.	PROPN
iajs-3432	336	9	a.	a.	NOUN
iajs-3432	336	10	oscillation	oscillation	NOUN
iajs-3432	336	11	of	of	ADP
iajs-3432	336	12	the	the	DET
iajs-3432	336	13	solutions	solution	NOUN
iajs-3432	336	14	for	for	ADP
iajs-3432	336	15	hematopoiesis	hematopoiesis	NOUN
iajs-3432	336	16	models	model	NOUN
iajs-3432	336	17	.	.	PUNCT
iajs-3432	337	1	iraqi	iraqi	ADJ
iajs-3432	337	2	journal	journal	PROPN
iajs-3432	337	3	of	of	ADP
iajs-3432	337	4	science	science	NOUN
iajs-3432	337	5	2023	2023	NUM
iajs-3432	337	6	,	,	PUNCT
iajs-3432	337	7	64(10	64(10	NUM
iajs-3432	337	8	)	)	PUNCT
iajs-3432	337	9	,	,	PUNCT
iajs-3432	337	10	5165	5165	NUM
iajs-3432	337	11	-	-	SYM
iajs-3432	337	12	5172	5172	NUM
iajs-3432	337	13	.	.	PUNCT
iajs-3432	338	1	https://doi.org/10.24996/ijs.2023.64.10.24	https://doi.org/10.24996/ijs.2023.64.10.24	PRON
iajs-3432	338	2	21	21	NUM
iajs-3432	338	3	.	.	PUNCT
iajs-3432	339	1	mohsen	mohsen	PROPN
iajs-3432	339	2	,	,	PUNCT
iajs-3432	339	3	a.	a.	PROPN
iajs-3432	339	4	;	;	PUNCT
iajs-3432	339	5	hattaf	hattaf	NOUN
iajs-3432	339	6	,	,	PUNCT
iajs-3432	339	7	k.	k.	PROPN
iajs-3432	339	8	;	;	PUNCT
iajs-3432	339	9	hassan	hassan	PROPN
iajs-3432	339	10	,	,	PUNCT
iajs-3432	339	11	a.	a.	PROPN
iajs-3432	339	12	h.	h.	PROPN
iajs-3432	339	13	dynamical	dynamical	ADJ
iajs-3432	339	14	analysis	analysis	NOUN
iajs-3432	339	15	of	of	ADP
iajs-3432	339	16	transmission	transmission	NOUN
iajs-3432	339	17	of	of	ADP
iajs-3432	339	18	hepatitis	hepatitis	PROPN
iajs-3432	339	19	b	b	PROPN
iajs-3432	339	20	and	and	CCONJ
iajs-3432	339	21	c	c	PROPN
iajs-3432	339	22	viruses	virus	NOUN
iajs-3432	339	23	with	with	ADP
iajs-3432	339	24	external	external	ADJ
iajs-3432	339	25	source	source	NOUN
iajs-3432	339	26	of	of	ADP
iajs-3432	339	27	disease	disease	NOUN
iajs-3432	339	28	by	by	ADP
iajs-3432	339	29	mathematical	mathematical	ADJ
iajs-3432	339	30	model	model	NOUN
iajs-3432	339	31	2021	2021	NUM
iajs-3432	339	32	,	,	PUNCT
iajs-3432	339	33	doi	doi	NOUN
iajs-3432	339	34	:	:	PUNCT
iajs-3432	339	35	https://doi.org/10.21203/rs.3.rs-730763/v1	https://doi.org/10.21203/rs.3.rs-730763/v1	PROPN
iajs-3432	339	36	.	.	PUNCT
iajs-3432	340	1	22	22	NUM
iajs-3432	340	2	.	.	PUNCT
iajs-3432	341	1	mahdi	mahdi	PROPN
iajs-3432	341	2	,	,	PUNCT
iajs-3432	341	3	w.	w.	PROPN
iajs-3432	341	4	a.	a.	PROPN
iajs-3432	341	5	;	;	PUNCT
iajs-3432	341	6	al	al	PROPN
iajs-3432	341	7	-	-	PUNCT
iajs-3432	341	8	nassir	nassir	PROPN
iajs-3432	341	9	,	,	PUNCT
iajs-3432	341	10	s.	s.	PROPN
iajs-3432	341	11	dynamical	dynamical	ADJ
iajs-3432	341	12	behaviors	behavior	NOUN
iajs-3432	341	13	of	of	ADP
iajs-3432	341	14	stage	stage	NOUN
iajs-3432	341	15	-	-	PUNCT
iajs-3432	341	16	structured	structure	VERB
iajs-3432	341	17	fractional	fractional	ADJ
iajs-3432	341	18	-	-	PUNCT
iajs-3432	341	19	order	order	NOUN
iajs-3432	341	20	prey	prey	NOUN
iajs-3432	341	21	-	-	PUNCT
iajs-3432	341	22	predator	predator	NOUN
iajs-3432	341	23	model	model	NOUN
iajs-3432	341	24	with	with	ADP
iajs-3432	341	25	crowley	crowley	PROPN
iajs-3432	341	26	–	–	PUNCT
iajs-3432	341	27	martin	martin	PROPN
iajs-3432	341	28	functional	functional	ADJ
iajs-3432	341	29	response	response	NOUN
iajs-3432	341	30	.	.	PUNCT
iajs-3432	342	1	international	international	ADJ
iajs-3432	342	2	journal	journal	PROPN
iajs-3432	342	3	of	of	ADP
iajs-3432	342	4	differential	differential	ADJ
iajs-3432	342	5	equations	equation	NOUN
iajs-3432	342	6	.	.	PUNCT
iajs-3432	343	1	2022	2022	NUM
iajs-3432	343	2	,	,	PUNCT
iajs-3432	343	3	2022(1	2022(1	NUM
iajs-3432	343	4	)	)	PUNCT
iajs-3432	343	5	.	.	PUNCT
iajs-3432	344	1	https://doi.org/10.1155/2022/6818454	https://doi.org/10.1155/2022/6818454	PROPN
iajs-3432	344	2	.	.	PUNCT
iajs-3432	345	1	23	23	NUM
iajs-3432	345	2	.	.	X
iajs-3432	346	1	wang	wang	PROPN
iajs-3432	346	2	,	,	PUNCT
iajs-3432	346	3	q.	q.	PROPN
iajs-3432	346	4	;	;	PUNCT
iajs-3432	346	5	wen	wen	PROPN
iajs-3432	346	6	,	,	PUNCT
iajs-3432	346	7	j.	j.	PROPN
iajs-3432	346	8	;	;	PUNCT
iajs-3432	346	9	qiu	qiu	PROPN
iajs-3432	346	10	,	,	PUNCT
iajs-3432	346	11	s.	s.	PROPN
iajs-3432	346	12	;	;	PUNCT
iajs-3432	346	13	guo	guo	PROPN
iajs-3432	346	14	,	,	PUNCT
iajs-3432	346	15	c.	c.	PROPN
iajs-3432	346	16	numerical	numerical	PROPN
iajs-3432	346	17	oscillations	oscillation	NOUN
iajs-3432	346	18	for	for	ADP
iajs-3432	346	19	first	first	ADJ
iajs-3432	346	20	-	-	PUNCT
iajs-3432	346	21	order	order	NOUN
iajs-3432	346	22	nonlinear	nonlinear	ADJ
iajs-3432	346	23	delay	delay	NOUN
iajs-3432	346	24	differential	differential	ADJ
iajs-3432	346	25	equations	equation	NOUN
iajs-3432	346	26	in	in	ADP
iajs-3432	346	27	a	a	DET
iajs-3432	346	28	hematopoiesis	hematopoiesis	NOUN
iajs-3432	346	29	model	model	NOUN
iajs-3432	346	30	.	.	PUNCT
iajs-3432	347	1	advances	advance	NOUN
iajs-3432	347	2	in	in	ADP
iajs-3432	347	3	difference	difference	NOUN
iajs-3432	347	4	equations	equation	NOUN
iajs-3432	347	5	.	.	PUNCT
iajs-3432	348	1	2013	2013	NUM
iajs-3432	348	2	,	,	PUNCT
iajs-3432	348	3	163	163	NUM
iajs-3432	348	4	,	,	PUNCT
iajs-3432	348	5	1	1	NUM
iajs-3432	348	6	-	-	SYM
iajs-3432	348	7	17	17	NUM
iajs-3432	348	8	.	.	PUNCT
iajs-3432	349	1	https://doi.org/10.1186/1687-1847-2013-163	https://doi.org/10.1186/1687-1847-2013-163	PROPN
iajs-3432	349	2	.	.	PUNCT
iajs-3432	350	1	https://doi.org/10.1016/j.aml.2014.12.006	https://doi.org/10.1016/j.aml.2014.12.006	PROPN
iajs-3432	351	1	https://doi.org/10.24996/ijs.2023.64.2.33	https://doi.org/10.24996/ijs.2023.64.2.33	INTJ
iajs-3432	351	2	https://doi.org/10.1515/ms-2021-0032	https://doi.org/10.1515/ms-2021-0032	X
iajs-3432	351	3	https://doi.org/10.1080/09720502.2021.1892275	https://doi.org/10.1080/09720502.2021.1892275	INTJ
iajs-3432	351	4	https://doi.org/10.21123/bsj.2023.8796	https://doi.org/10.21123/bsj.2023.8796	AUX
iajs-3432	351	5	https://doi.org/10.1007/s12346-022-00603-z	https://doi.org/10.1007/s12346-022-00603-z	VERB
iajs-3432	351	6	https://doi.org/10.1016/j.nonrwa.2006.11.013	https://doi.org/10.1016/j.nonrwa.2006.11.013	PROPN
iajs-3432	351	7	http://doi.org/10.11948/2018.1679	http://doi.org/10.11948/2018.1679	NOUN
iajs-3432	351	8	https://doi.org/10.24996/ijs.2021.62.9.19	https://doi.org/10.24996/ijs.2021.62.9.19	NOUN
iajs-3432	352	1	https://doi.org/10.1142/s0218127404011697	https://doi.org/10.1142/s0218127404011697	NUM
iajs-3432	352	2	https://doi.org/10.24996/ijs.2023.64.10.24	https://doi.org/10.24996/ijs.2023.64.10.24	NOUN
iajs-3432	352	3	https://doi.org/10.21203/rs.3.rs-730763/v1	https://doi.org/10.21203/rs.3.rs-730763/v1	PUNCT
iajs-3432	352	4	https://doi.org/10.1155/2022/6818454	https://doi.org/10.1155/2022/6818454	PROPN
iajs-3432	352	5	ihjpas	ihjpas	PROPN
iajs-3432	352	6	.	.	PUNCT
iajs-3432	353	1	2024	2024	NUM
iajs-3432	353	2	,	,	PUNCT
iajs-3432	353	3	37	37	NUM
iajs-3432	353	4	(	(	PUNCT
iajs-3432	353	5	3	3	NUM
iajs-3432	353	6	)	)	PUNCT
iajs-3432	353	7	395	395	NUM
iajs-3432	353	8	24	24	NUM
iajs-3432	353	9	.	.	PUNCT
iajs-3432	353	10	saker	saker	PROPN
iajs-3432	353	11	,	,	PUNCT
iajs-3432	353	12	s.	s.	PROPN
iajs-3432	353	13	h.	h.	PROPN
iajs-3432	353	14	;	;	PUNCT
iajs-3432	353	15	agarwal	agarwal	PROPN
iajs-3432	353	16	,	,	PUNCT
iajs-3432	353	17	s.	s.	PROPN
iajs-3432	353	18	oscillation	oscillation	PROPN
iajs-3432	353	19	and	and	CCONJ
iajs-3432	353	20	global	global	ADJ
iajs-3432	353	21	attractivity	attractivity	NOUN
iajs-3432	353	22	in	in	ADP
iajs-3432	353	23	a	a	DET
iajs-3432	353	24	nonlinear	nonlinear	ADJ
iajs-3432	353	25	delay	delay	NOUN
iajs-3432	353	26	periodic	periodic	ADJ
iajs-3432	353	27	model	model	NOUN
iajs-3432	353	28	of	of	ADP
iajs-3432	353	29	population	population	NOUN
iajs-3432	353	30	dynamics	dynamic	NOUN
iajs-3432	353	31	.	.	PUNCT
iajs-3432	354	1	applicable	applicable	ADJ
iajs-3432	354	2	analysis	analysis	NOUN
iajs-3432	354	3	.	.	PUNCT
iajs-3432	355	1	2002	2002	NUM
iajs-3432	355	2	,	,	PUNCT
iajs-3432	355	3	81(14	81(14	NUM
iajs-3432	355	4	)	)	PUNCT
iajs-3432	355	5	,	,	PUNCT
iajs-3432	355	6	787	787	NUM
iajs-3432	355	7	-	-	SYM
iajs-3432	355	8	799	799	NUM
iajs-3432	355	9	.	.	PUNCT
iajs-3432	356	1	https://doi.org/10.1080/0003681021000004429	https://doi.org/10.1080/0003681021000004429	NOUN
iajs-3432	356	2	.	.	PUNCT
iajs-3432	357	1	25	25	NUM
iajs-3432	357	2	.	.	PUNCT
iajs-3432	357	3	;	;	PUNCT
iajs-3432	357	4	pati	pati	PROPN
iajs-3432	357	5	,	,	PUNCT
iajs-3432	357	6	s.	s.	PROPN
iajs-3432	357	7	;	;	PUNCT
iajs-3432	357	8	padhi	padhi	PROPN
iajs-3432	357	9	,	,	PUNCT
iajs-3432	357	10	s.	s.	PROPN
iajs-3432	357	11	;	;	PUNCT
iajs-3432	357	12	srinivasu	srinivasu	PROPN
iajs-3432	357	13	,	,	PUNCT
iajs-3432	357	14	p.	p.	NOUN
iajs-3432	357	15	conditions	condition	NOUN
iajs-3432	357	16	for	for	ADP
iajs-3432	357	17	existence	existence	NOUN
iajs-3432	357	18	of	of	ADP
iajs-3432	357	19	positive	positive	ADJ
iajs-3432	357	20	solutions	solution	NOUN
iajs-3432	357	21	of	of	ADP
iajs-3432	357	22	first	first	ADJ
iajs-3432	357	23	order	order	NOUN
iajs-3432	357	24	boundary	boundary	ADJ
iajs-3432	357	25	value	value	NOUN
iajs-3432	357	26	problems	problem	NOUN
iajs-3432	357	27	with	with	ADP
iajs-3432	357	28	delay	delay	NOUN
iajs-3432	357	29	and	and	CCONJ
iajs-3432	357	30	nonlinear	nonlinear	ADJ
iajs-3432	357	31	nonlocal	nonlocal	ADJ
iajs-3432	357	32	boundary	boundary	ADJ
iajs-3432	357	33	conditions	condition	NOUN
iajs-3432	357	34	and	and	CCONJ
iajs-3432	357	35	application	application	NOUN
iajs-3432	357	36	to	to	ADP
iajs-3432	357	37	hematopoiesis	hematopoiesis	NOUN
iajs-3432	357	38	.	.	PUNCT
iajs-3432	358	1	communications	communication	NOUN
iajs-3432	358	2	in	in	ADP
iajs-3432	358	3	applied	apply	VERB
iajs-3432	358	4	analysis	analysis	NOUN
iajs-3432	358	5	2017	2017	NUM
iajs-3432	358	6	,	,	PUNCT
iajs-3432	358	7	21(1	21(1	NUM
iajs-3432	358	8	)	)	PUNCT
iajs-3432	358	9	.	.	PUNCT
iajs-3432	359	1	26	26	NUM
iajs-3432	359	2	.	.	PUNCT
iajs-3432	359	3	ladas	ladas	PROPN
iajs-3432	359	4	,	,	PUNCT
iajs-3432	359	5	g.	g.	PROPN
iajs-3432	359	6	oscillations	oscillation	NOUN
iajs-3432	359	7	of	of	ADP
iajs-3432	359	8	difference	difference	NOUN
iajs-3432	359	9	equations	equation	NOUN
iajs-3432	359	10	with	with	ADP
iajs-3432	359	11	positive	positive	ADJ
iajs-3432	359	12	and	and	CCONJ
iajs-3432	359	13	negative	negative	ADJ
iajs-3432	359	14	coefficients	coefficient	NOUN
iajs-3432	359	15	.	.	PUNCT
iajs-3432	360	1	rocky	rocky	ADJ
iajs-3432	360	2	mountain	mountain	PROPN
iajs-3432	360	3	j.	j.	PROPN
iajs-3432	360	4	math	math	PROPN
iajs-3432	360	5	1990	1990	NUM
iajs-3432	360	6	,	,	PUNCT
iajs-3432	360	7	20(4	20(4	NOUN
iajs-3432	360	8	)	)	PUNCT
iajs-3432	360	9	,	,	PUNCT
iajs-3432	360	10	1051–1061	1051–1061	NUM
iajs-3432	360	11	.	.	PUNCT
iajs-3432	361	1	27	27	NUM
iajs-3432	361	2	.	.	X
iajs-3432	361	3	ögunmez	ögunmez	PROPN
iajs-3432	361	4	,	,	PUNCT
iajs-3432	361	5	h.	h.	PROPN
iajs-3432	361	6	;	;	PUNCT
iajs-3432	361	7	öcalan	öcalan	NOUN
iajs-3432	361	8	,	,	PUNCT
iajs-3432	361	9	ö.	ö.	VERB
iajs-3432	361	10	oscillation	oscillation	NOUN
iajs-3432	361	11	of	of	ADP
iajs-3432	361	12	difference	difference	NOUN
iajs-3432	361	13	equations	equation	NOUN
iajs-3432	361	14	with	with	ADP
iajs-3432	361	15	several	several	ADJ
iajs-3432	361	16	positive	positive	ADJ
iajs-3432	361	17	and	and	CCONJ
iajs-3432	361	18	negative	negative	ADJ
iajs-3432	361	19	coefficients	coefficient	NOUN
iajs-3432	361	20	.	.	PUNCT
iajs-3432	362	1	fasciculi	fasciculi	PROPN
iajs-3432	362	2	mathematici	mathematici	PROPN
iajs-3432	362	3	2013	2013	NUM
iajs-3432	362	4	,	,	PUNCT
iajs-3432	362	5	15	15	NUM
iajs-3432	362	6	,	,	PUNCT
iajs-3432	362	7	115–122	115–122	NUM
iajs-3432	362	8	.	.	PUNCT
iajs-3432	363	1	https://doi.org/10.24996/ijs.2021.62.9.19	https://doi.org/10.24996/ijs.2021.62.9.19	PROPN
iajs-3432	364	1	28	28	NUM
iajs-3432	364	2	.	.	PUNCT
iajs-3432	365	1	rath	rath	PROPN
iajs-3432	365	2	,	,	PUNCT
iajs-3432	365	3	r.	r.	PROPN
iajs-3432	365	4	n.	n.	PROPN
iajs-3432	365	5	;	;	PUNCT
iajs-3432	365	6	padhy	padhy	PROPN
iajs-3432	365	7	,	,	PUNCT
iajs-3432	365	8	l.	l.	PROPN
iajs-3432	365	9	n.	n.	PROPN
iajs-3432	365	10	oscillations	oscillations	PROPN
iajs-3432	365	11	and	and	CCONJ
iajs-3432	365	12	non	non	NOUN
iajs-3432	365	13	-	-	NOUN
iajs-3432	365	14	oscillations	oscillation	NOUN
iajs-3432	365	15	of	of	ADP
iajs-3432	365	16	neutral	neutral	ADJ
iajs-3432	365	17	difference	difference	NOUN
iajs-3432	365	18	equations	equation	NOUN
iajs-3432	365	19	of	of	ADP
iajs-3432	365	20	first	first	ADJ
iajs-3432	365	21	order	order	NOUN
iajs-3432	365	22	with	with	ADP
iajs-3432	365	23	positive	positive	ADJ
iajs-3432	365	24	and	and	CCONJ
iajs-3432	365	25	negative	negative	ADJ
iajs-3432	365	26	coefficients	coefficient	NOUN
iajs-3432	365	27	.	.	PUNCT
iajs-3432	366	1	fasciculi	fasciculi	PROPN
iajs-3432	366	2	mathematici	mathematici	PROPN
iajs-3432	366	3	2007	2007	NUM
iajs-3432	366	4	,	,	PUNCT
iajs-3432	366	5	37	37	NUM
iajs-3432	366	6	,	,	PUNCT
iajs-3432	366	7	57–65	57–65	NUM
iajs-3432	366	8	.	.	PUNCT
iajs-3432	367	1	https://doi.org/10.24996/ijs.2021.62.9.19	https://doi.org/10.24996/ijs.2021.62.9.19	PROPN
iajs-3432	367	2	29	29	NUM
iajs-3432	367	3	.	.	PUNCT
iajs-3432	368	1	adimy	adimy	PROPN
iajs-3432	368	2	,	,	PUNCT
iajs-3432	368	3	m	m	PROPN
iajs-3432	368	4	;	;	PUNCT
iajs-3432	368	5	crauste	crauste	PROPN
iajs-3432	368	6	,	,	PUNCT
iajs-3432	368	7	f.	f.	PROPN
iajs-3432	368	8	;	;	PUNCT
iajs-3432	368	9	el	el	PROPN
iajs-3432	368	10	abdllaoui	abdllaoui	PROPN
iajs-3432	368	11	,	,	PUNCT
iajs-3432	368	12	a.	a.	NOUN
iajs-3432	368	13	discrete	discrete	NOUN
iajs-3432	368	14	-	-	PUNCT
iajs-3432	368	15	maturity	maturity	NOUN
iajs-3432	368	16	structured	structured	ADJ
iajs-3432	368	17	model	model	NOUN
iajs-3432	368	18	of	of	ADP
iajs-3432	368	19	cell	cell	NOUN
iajs-3432	368	20	differentiation	differentiation	NOUN
iajs-3432	368	21	with	with	ADP
iajs-3432	368	22	applications	application	NOUN
iajs-3432	368	23	to	to	PART
iajs-3432	368	24	acute	acute	VERB
iajs-3432	368	25	myelogenous	myelogenous	ADJ
iajs-3432	368	26	leukemia	leukemia	NOUN
iajs-3432	368	27	.	.	PUNCT
iajs-3432	369	1	journal	journal	NOUN
iajs-3432	369	2	of	of	ADP
iajs-3432	369	3	biological	biological	ADJ
iajs-3432	369	4	systems	system	NOUN
iajs-3432	369	5	2008	2008	NUM
iajs-3432	369	6	,	,	PUNCT
iajs-3432	369	7	16(3	16(3	NOUN
iajs-3432	369	8	)	)	PUNCT
iajs-3432	369	9	,	,	PUNCT
iajs-3432	369	10	395	395	NUM
iajs-3432	369	11	-	-	SYM
iajs-3432	369	12	424	424	NUM
iajs-3432	369	13	.	.	PUNCT
iajs-3432	370	1	https://doi.org/10.1142/s0218339008002599	https://doi.org/10.1142/s0218339008002599	NUM
iajs-3432	370	2	.	.	NOUN
iajs-3432	370	3	30	30	NUM
iajs-3432	370	4	.	.	X
iajs-3432	370	5	elaydi	elaydi	PROPN
iajs-3432	370	6	,	,	PUNCT
iajs-3432	370	7	s.	s.	PROPN
iajs-3432	370	8	n.	n.	PROPN
iajs-3432	370	9	an	an	DET
iajs-3432	370	10	introduction	introduction	NOUN
iajs-3432	370	11	to	to	ADP
iajs-3432	370	12	difference	difference	NOUN
iajs-3432	370	13	equations	equation	NOUN
iajs-3432	370	14	.	.	PUNCT
iajs-3432	371	1	springer	springer	NOUN
iajs-3432	371	2	-	-	PUNCT
iajs-3432	371	3	verlag	verlag	PROPN
iajs-3432	371	4	,	,	PUNCT
iajs-3432	371	5	new	new	PROPN
iajs-3432	371	6	york	york	PROPN
iajs-3432	371	7	,	,	PUNCT
iajs-3432	371	8	inc	inc	PROPN
iajs-3432	371	9	.	.	PROPN
iajs-3432	371	10	2005	2005	NUM
iajs-3432	371	11	.	.	PUNCT
iajs-3432	372	1	https://doi.org/10.24996/ijs.2021.62.9.19	https://doi.org/10.24996/ijs.2021.62.9.19	INTJ
iajs-3432	372	2	https://doi.org/10.1080/0003681021000004429	https://doi.org/10.1080/0003681021000004429	ADJ
iajs-3432	373	1	https://doi.org/10.24996/ijs.2021.62.9.19	https://doi.org/10.24996/ijs.2021.62.9.19	INTJ
iajs-3432	373	2	https://doi.org/10.24996/ijs.2021.62.9.19	https://doi.org/10.24996/ijs.2021.62.9.19	INTJ
iajs-3432	374	1	https://doi.org/10.1142/s0218339008002599	https://doi.org/10.1142/s0218339008002599	NUM
iajs-3432	374	2	https://doi.org/10.24996/ijs.2021.62.9.19	https://doi.org/10.24996/ijs.2021.62.9.19	INTJ