id	sid	tid	token	lemma	pos
cana-328	1	1	communications	communication	NOUN
cana-328	1	2	on	on	ADP
cana-328	1	3	applied	apply	VERB
cana-328	1	4	nonlinear	nonlinear	ADJ
cana-328	1	5	analysis	analysis	NOUN
cana-328	1	6	issn	issn	NOUN
cana-328	1	7	:	:	PUNCT
cana-328	1	8	1074	1074	NUM
cana-328	1	9	-	-	PUNCT
cana-328	1	10	133x	133x	NUM
cana-328	1	11	vol	vol	NOUN
cana-328	1	12	31	31	NUM
cana-328	1	13	no	no	NOUN
cana-328	1	14	.	.	NOUN
cana-328	1	15	1	1	NUM
cana-328	1	16	(	(	PUNCT
cana-328	1	17	2024	2024	NUM
cana-328	1	18	)	)	PUNCT
cana-328	1	19	89	89	NUM
cana-328	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-328	1	21	oscillatory	oscillatory	ADJ
cana-328	1	22	and	and	CCONJ
cana-328	1	23	property	property	NOUN
cana-328	1	24	conditions	condition	NOUN
cana-328	1	25	for	for	ADP
cana-328	1	26	third	third	ADJ
cana-328	1	27	order	order	NOUN
cana-328	1	28	difference	difference	NOUN
cana-328	1	29	equation	equation	NOUN
cana-328	1	30	with	with	ADP
cana-328	1	31	negative	negative	ADJ
cana-328	1	32	middle	middle	ADJ
cana-328	1	33	-	-	PUNCT
cana-328	1	34	term	term	NOUN
cana-328	1	35	s.	s.	PROPN
cana-328	1	36	kaleeswari	kaleeswari	PROPN
cana-328	1	37	1	1	NUM
cana-328	1	38	,	,	PUNCT
cana-328	1	39	s.	s.	PROPN
cana-328	1	40	rangasri	rangasri	VERB
cana-328	1	41	2	2	NUM
cana-328	1	42	1,2department	1,2department	NUM
cana-328	1	43	of	of	ADP
cana-328	1	44	mathematics	mathematic	NOUN
cana-328	1	45	,	,	PUNCT
cana-328	1	46	nallamuthu	nallamuthu	ADJ
cana-328	1	47	gounder	gounder	PROPN
cana-328	1	48	mahalingam	mahalingam	PROPN
cana-328	1	49	college	college	PROPN
cana-328	1	50	,	,	PUNCT
cana-328	1	51	pollachi	pollachi	NOUN
cana-328	1	52	,	,	PUNCT
cana-328	1	53	tamilnadu	tamilnadu	PROPN
cana-328	1	54	,	,	PUNCT
cana-328	1	55	india	india	PROPN
cana-328	1	56	.	.	PUNCT
cana-328	2	1	e	e	X
cana-328	2	2	-	-	NOUN
cana-328	2	3	mail	mail	NOUN
cana-328	2	4	:	:	PUNCT
cana-328	2	5	kaleesdesika@gmail.com	kaleesdesika@gmail.com	X
cana-328	2	6	,	,	PUNCT
cana-328	2	7	e	e	NOUN
cana-328	2	8	-	-	NOUN
cana-328	2	9	mail	mail	NOUN
cana-328	2	10	:	:	PUNCT
cana-328	2	11	rangasrisuresh97@gmail.com	rangasrisuresh97@gmail.com	PROPN
cana-328	2	12	.	.	PROPN
cana-328	2	13	article	article	NOUN
cana-328	2	14	history	history	NOUN
cana-328	2	15	:	:	PUNCT
cana-328	2	16	received	receive	VERB
cana-328	2	17	:	:	PUNCT
cana-328	2	18	18	18	NUM
cana-328	2	19	-	-	SYM
cana-328	2	20	09	09	NUM
cana-328	2	21	-	-	PUNCT
cana-328	2	22	2023	2023	NUM
cana-328	2	23	revised	revise	VERB
cana-328	2	24	:	:	PUNCT
cana-328	2	25	25	25	NUM
cana-328	2	26	-	-	SYM
cana-328	2	27	10	10	NUM
cana-328	2	28	-	-	PUNCT
cana-328	2	29	2023	2023	NUM
cana-328	2	30	accepted	accept	VERB
cana-328	2	31	:	:	PUNCT
cana-328	2	32	19	19	NUM
cana-328	2	33	-	-	SYM
cana-328	2	34	11	11	NUM
cana-328	2	35	-	-	SYM
cana-328	2	36	2023	2023	NUM
cana-328	2	37	abstract	abstract	NOUN
cana-328	2	38	:	:	PUNCT
cana-328	2	39	this	this	DET
cana-328	2	40	paper	paper	NOUN
cana-328	2	41	presents	present	VERB
cana-328	2	42	a	a	DET
cana-328	2	43	detailed	detailed	ADJ
cana-328	2	44	analysis	analysis	NOUN
cana-328	2	45	of	of	ADP
cana-328	2	46	the	the	DET
cana-328	2	47	oscillation	oscillation	NOUN
cana-328	2	48	and	and	CCONJ
cana-328	2	49	property	property	NOUN
cana-328	2	50	a	a	DET
cana-328	2	51	conditions	condition	NOUN
cana-328	2	52	for	for	ADP
cana-328	2	53	third	third	ADJ
cana-328	2	54	-	-	PUNCT
cana-328	2	55	order	order	NOUN
cana-328	2	56	advanced	advanced	ADJ
cana-328	2	57	difference	difference	NOUN
cana-328	2	58	equations	equation	NOUN
cana-328	2	59	with	with	ADP
cana-328	2	60	negative	negative	ADJ
cana-328	2	61	middle	middle	ADJ
cana-328	2	62	terms	term	NOUN
cana-328	2	63	.	.	PUNCT
cana-328	3	1	the	the	DET
cana-328	3	2	authors	author	NOUN
cana-328	3	3	employ	employ	VERB
cana-328	3	4	the	the	DET
cana-328	3	5	generalized	generalized	ADJ
cana-328	3	6	riccati	riccati	NOUN
cana-328	3	7	transformation	transformation	NOUN
cana-328	3	8	technique	technique	NOUN
cana-328	3	9	and	and	CCONJ
cana-328	3	10	summation	summation	NOUN
cana-328	3	11	by	by	ADP
cana-328	3	12	parts	part	NOUN
cana-328	3	13	to	to	PART
cana-328	3	14	establish	establish	VERB
cana-328	3	15	sufficient	sufficient	ADJ
cana-328	3	16	criteria	criterion	NOUN
cana-328	3	17	for	for	SCONJ
cana-328	3	18	the	the	DET
cana-328	3	19	solutions	solution	NOUN
cana-328	3	20	to	to	PART
cana-328	3	21	be	be	AUX
cana-328	3	22	oscillatory	oscillatory	ADJ
cana-328	3	23	or	or	CCONJ
cana-328	3	24	have	have	VERB
cana-328	3	25	property	property	NOUN
cana-328	3	26	a.	a.	NOUN
cana-328	3	27	the	the	DET
cana-328	3	28	main	main	ADJ
cana-328	3	29	results	result	NOUN
cana-328	3	30	and	and	CCONJ
cana-328	3	31	their	their	PRON
cana-328	3	32	proofs	proof	NOUN
cana-328	3	33	are	be	AUX
cana-328	3	34	presented	present	VERB
cana-328	3	35	clearly	clearly	ADV
cana-328	3	36	and	and	CCONJ
cana-328	3	37	concisely	concisely	ADV
cana-328	3	38	.	.	PUNCT
cana-328	4	1	the	the	DET
cana-328	4	2	difference	difference	NOUN
cana-328	4	3	equations	equation	NOUN
cana-328	4	4	have	have	AUX
cana-328	4	5	been	be	AUX
cana-328	4	6	a	a	DET
cana-328	4	7	topic	topic	NOUN
cana-328	4	8	of	of	ADP
cana-328	4	9	interest	interest	NOUN
cana-328	4	10	in	in	ADP
cana-328	4	11	various	various	ADJ
cana-328	4	12	scientific	scientific	ADJ
cana-328	4	13	fields	field	NOUN
cana-328	4	14	such	such	ADJ
cana-328	4	15	as	as	ADP
cana-328	4	16	economics	economic	NOUN
cana-328	4	17	,	,	PUNCT
cana-328	4	18	physics	physics	NOUN
cana-328	4	19	,	,	PUNCT
cana-328	4	20	and	and	CCONJ
cana-328	4	21	mathematical	mathematical	ADJ
cana-328	4	22	biology	biology	NOUN
cana-328	4	23	.	.	PUNCT
cana-328	5	1	the	the	DET
cana-328	5	2	results	result	NOUN
cana-328	5	3	presented	present	VERB
cana-328	5	4	in	in	ADP
cana-328	5	5	this	this	DET
cana-328	5	6	paper	paper	NOUN
cana-328	5	7	can	can	AUX
cana-328	5	8	be	be	AUX
cana-328	5	9	used	use	VERB
cana-328	5	10	to	to	PART
cana-328	5	11	inform	inform	VERB
cana-328	5	12	the	the	DET
cana-328	5	13	development	development	NOUN
cana-328	5	14	of	of	ADP
cana-328	5	15	more	more	ADV
cana-328	5	16	accurate	accurate	ADJ
cana-328	5	17	models	model	NOUN
cana-328	5	18	for	for	ADP
cana-328	5	19	real	real	ADJ
cana-328	5	20	-	-	PUNCT
cana-328	5	21	world	world	NOUN
cana-328	5	22	problems	problem	NOUN
cana-328	5	23	in	in	ADP
cana-328	5	24	various	various	ADJ
cana-328	5	25	scientific	scientific	ADJ
cana-328	5	26	fields	field	NOUN
cana-328	5	27	.	.	PUNCT
cana-328	6	1	additionally	additionally	ADV
cana-328	6	2	,	,	PUNCT
cana-328	6	3	this	this	DET
cana-328	6	4	paper	paper	NOUN
cana-328	6	5	explores	explore	VERB
cana-328	6	6	the	the	DET
cana-328	6	7	economic	economic	ADJ
cana-328	6	8	theory	theory	NOUN
cana-328	6	9	of	of	ADP
cana-328	6	10	business	business	NOUN
cana-328	6	11	cycles	cycle	NOUN
cana-328	6	12	and	and	CCONJ
cana-328	6	13	its	its	PRON
cana-328	6	14	mathematical	mathematical	ADJ
cana-328	6	15	foundations	foundation	NOUN
cana-328	6	16	,	,	PUNCT
cana-328	6	17	specifically	specifically	ADV
cana-328	6	18	samuelson	samuelson	PROPN
cana-328	6	19	's	's	PART
cana-328	6	20	model	model	NOUN
cana-328	6	21	,	,	PUNCT
cana-328	6	22	which	which	PRON
cana-328	6	23	outlines	outline	VERB
cana-328	6	24	five	five	NUM
cana-328	6	25	potential	potential	ADJ
cana-328	6	26	trajectories	trajectory	NOUN
cana-328	6	27	or	or	CCONJ
cana-328	6	28	trends	trend	NOUN
cana-328	6	29	that	that	PRON
cana-328	6	30	economic	economic	ADJ
cana-328	6	31	activity	activity	NOUN
cana-328	6	32	can	can	AUX
cana-328	6	33	follow	follow	VERB
cana-328	6	34	based	base	VERB
cana-328	6	35	on	on	ADP
cana-328	6	36	different	different	ADJ
cana-328	6	37	combinations	combination	NOUN
cana-328	6	38	of	of	ADP
cana-328	6	39	the	the	DET
cana-328	6	40	marginal	marginal	ADJ
cana-328	6	41	propensity	propensity	NOUN
cana-328	6	42	to	to	PART
cana-328	6	43	consume	consume	VERB
cana-328	6	44	multiplier	multipli	ADJ
cana-328	6	45	parameter(α	parameter(α	NOUN
cana-328	6	46	)	)	PUNCT
cana-328	6	47	and	and	CCONJ
cana-328	6	48	the	the	DET
cana-328	6	49	accelerator	accelerator	NOUN
cana-328	6	50	coefficient	coefficient	NOUN
cana-328	6	51	(	(	PUNCT
cana-328	6	52	β	β	NOUN
cana-328	6	53	)	)	PUNCT
cana-328	6	54	.	.	PUNCT
cana-328	7	1	keywords	keyword	NOUN
cana-328	7	2	:	:	PUNCT
cana-328	7	3	advanced	advanced	ADJ
cana-328	7	4	equation	equation	NOUN
cana-328	7	5	,	,	PUNCT
cana-328	7	6	property	property	NOUN
cana-328	7	7	a	a	NOUN
cana-328	7	8	,	,	PUNCT
cana-328	7	9	oscillation	oscillation	NOUN
cana-328	7	10	criteria	criterion	NOUN
cana-328	7	11	,	,	PUNCT
cana-328	7	12	riccati	riccati	PROPN
cana-328	7	13	transformation	transformation	NOUN
cana-328	7	14	,	,	PUNCT
cana-328	7	15	semi	semi	ADJ
cana-328	7	16	-	-	ADJ
cana-328	7	17	canonical	canonical	ADJ
cana-328	7	18	.	.	PUNCT
cana-328	8	1	1	1	X
cana-328	8	2	.	.	X
cana-328	8	3	introduction	introduction	NOUN
cana-328	8	4	in	in	ADP
cana-328	8	5	this	this	DET
cana-328	8	6	article	article	NOUN
cana-328	8	7	,	,	PUNCT
cana-328	8	8	we	we	PRON
cana-328	8	9	will	will	AUX
cana-328	8	10	discuss	discuss	VERB
cana-328	8	11	the	the	DET
cana-328	8	12	economic	economic	ADJ
cana-328	8	13	theory	theory	NOUN
cana-328	8	14	of	of	ADP
cana-328	8	15	business	business	NOUN
cana-328	8	16	cycles	cycle	NOUN
cana-328	8	17	and	and	CCONJ
cana-328	8	18	its	its	PRON
cana-328	8	19	mathematical	mathematical	ADJ
cana-328	8	20	foundations	foundation	NOUN
cana-328	8	21	.	.	PUNCT
cana-328	9	1	specifically	specifically	ADV
cana-328	9	2	,	,	PUNCT
cana-328	9	3	we	we	PRON
cana-328	9	4	will	will	AUX
cana-328	9	5	explore	explore	VERB
cana-328	9	6	samuelson	samuelson	PROPN
cana-328	9	7	's	's	PART
cana-328	9	8	model	model	NOUN
cana-328	9	9	,	,	PUNCT
cana-328	9	10	which	which	PRON
cana-328	9	11	outlines	outline	VERB
cana-328	9	12	five	five	NUM
cana-328	9	13	potential	potential	ADJ
cana-328	9	14	trajectories	trajectory	NOUN
cana-328	9	15	or	or	CCONJ
cana-328	9	16	trends	trend	NOUN
cana-328	9	17	that	that	PRON
cana-328	9	18	economic	economic	ADJ
cana-328	9	19	activity	activity	NOUN
cana-328	9	20	can	can	AUX
cana-328	9	21	follow	follow	VERB
cana-328	9	22	.	.	PUNCT
cana-328	10	1	these	these	DET
cana-328	10	2	paths	path	NOUN
cana-328	10	3	are	be	AUX
cana-328	10	4	determined	determine	VERB
cana-328	10	5	by	by	ADP
cana-328	10	6	different	different	ADJ
cana-328	10	7	combinations	combination	NOUN
cana-328	10	8	of	of	ADP
cana-328	10	9	the	the	DET
cana-328	10	10	marginal	marginal	ADJ
cana-328	10	11	propensity	propensity	NOUN
cana-328	10	12	to	to	PART
cana-328	10	13	consume	consume	VERB
cana-328	10	14	(	(	PUNCT
cana-328	10	15	α	α	NOUN
cana-328	10	16	)	)	PUNCT
cana-328	10	17	and	and	CCONJ
cana-328	10	18	the	the	DET
cana-328	10	19	accelerator	accelerator	NOUN
cana-328	10	20	coefficient	coefficient	NOUN
cana-328	10	21	(	(	PUNCT
cana-328	10	22	β	β	NOUN
cana-328	10	23	)	)	PUNCT
cana-328	10	24	.	.	PUNCT
cana-328	11	1	each	each	DET
cana-328	11	2	combination	combination	NOUN
cana-328	11	3	results	result	VERB
cana-328	11	4	in	in	ADP
cana-328	11	5	a	a	DET
cana-328	11	6	unique	unique	ADJ
cana-328	11	7	pattern	pattern	NOUN
cana-328	11	8	of	of	ADP
cana-328	11	9	economic	economic	ADJ
cana-328	11	10	movement	movement	NOUN
cana-328	11	11	.	.	PUNCT
cana-328	12	1	we	we	PRON
cana-328	12	2	will	will	AUX
cana-328	12	3	examine	examine	VERB
cana-328	12	4	the	the	DET
cana-328	12	5	model	model	NOUN
cana-328	12	6	in	in	ADP
cana-328	12	7	detail	detail	NOUN
cana-328	12	8	and	and	CCONJ
cana-328	12	9	highlight	highlight	VERB
cana-328	12	10	its	its	PRON
cana-328	12	11	key	key	ADJ
cana-328	12	12	features	feature	NOUN
cana-328	12	13	,	,	PUNCT
cana-328	12	14	including	include	VERB
cana-328	12	15	the	the	DET
cana-328	12	16	dynamic	dynamic	NOUN
cana-328	12	17	of	of	ADP
cana-328	12	18	economic	economic	ADJ
cana-328	12	19	cycles	cycle	NOUN
cana-328	12	20	and	and	CCONJ
cana-328	12	21	their	their	PRON
cana-328	12	22	relationship	relationship	NOUN
cana-328	12	23	to	to	ADP
cana-328	12	24	changes	change	NOUN
cana-328	12	25	in	in	ADP
cana-328	12	26	consumption	consumption	NOUN
cana-328	12	27	and	and	CCONJ
cana-328	12	28	investment	investment	NOUN
cana-328	12	29	.	.	PUNCT
cana-328	13	1	the	the	DET
cana-328	13	2	article	article	NOUN
cana-328	13	3	will	will	AUX
cana-328	13	4	also	also	ADV
cana-328	13	5	delve	delve	VERB
cana-328	13	6	into	into	ADP
cana-328	13	7	the	the	DET
cana-328	13	8	mathematical	mathematical	ADJ
cana-328	13	9	foundations	foundation	NOUN
cana-328	13	10	of	of	ADP
cana-328	13	11	the	the	DET
cana-328	13	12	model	model	NOUN
cana-328	13	13	,	,	PUNCT
cana-328	13	14	including	include	VERB
cana-328	13	15	second	second	ADJ
cana-328	13	16	-	-	PUNCT
cana-328	13	17	order	order	NOUN
cana-328	13	18	difference	difference	NOUN
cana-328	13	19	equations	equation	NOUN
cana-328	13	20	in	in	ADP
cana-328	13	21	the	the	DET
cana-328	13	22	context	context	NOUN
cana-328	13	23	of	of	ADP
cana-328	13	24	business	business	NOUN
cana-328	13	25	cycles	cycle	NOUN
cana-328	13	26	.	.	PUNCT
cana-328	14	1	we	we	PRON
cana-328	14	2	will	will	AUX
cana-328	14	3	explore	explore	VERB
cana-328	14	4	the	the	DET
cana-328	14	5	oscillatory	oscillatory	ADJ
cana-328	14	6	behavior	behavior	NOUN
cana-328	14	7	and	and	CCONJ
cana-328	14	8	property	property	NOUN
cana-328	14	9	a	a	DET
cana-328	14	10	conditions	condition	NOUN
cana-328	14	11	for	for	ADP
cana-328	14	12	third	third	ADJ
cana-328	14	13	-	-	PUNCT
cana-328	14	14	order	order	NOUN
cana-328	14	15	difference	difference	NOUN
cana-328	14	16	equations	equation	NOUN
cana-328	14	17	,	,	PUNCT
cana-328	14	18	which	which	PRON
cana-328	14	19	have	have	VERB
cana-328	14	20	applications	application	NOUN
cana-328	14	21	in	in	ADP
cana-328	14	22	economics	economic	NOUN
cana-328	14	23	,	,	PUNCT
cana-328	14	24	physics	physics	NOUN
cana-328	14	25	,	,	PUNCT
cana-328	14	26	and	and	CCONJ
cana-328	14	27	mathematical	mathematical	ADJ
cana-328	14	28	biology	biology	NOUN
cana-328	14	29	.	.	PUNCT
cana-328	15	1	the	the	DET
cana-328	15	2	paper	paper	NOUN
cana-328	15	3	will	will	AUX
cana-328	15	4	highlight	highlight	VERB
cana-328	15	5	the	the	DET
cana-328	15	6	connection	connection	NOUN
cana-328	15	7	between	between	ADP
cana-328	15	8	these	these	DET
cana-328	15	9	equations	equation	NOUN
cana-328	15	10	and	and	CCONJ
cana-328	15	11	samuelson	samuelson	PROPN
cana-328	15	12	's	's	PART
cana-328	15	13	model	model	NOUN
cana-328	15	14	,	,	PUNCT
cana-328	15	15	demonstrating	demonstrate	VERB
cana-328	15	16	how	how	SCONJ
cana-328	15	17	they	they	PRON
cana-328	15	18	can	can	AUX
cana-328	15	19	be	be	AUX
cana-328	15	20	used	use	VERB
cana-328	15	21	to	to	PART
cana-328	15	22	gain	gain	VERB
cana-328	15	23	insights	insight	NOUN
cana-328	15	24	into	into	ADP
cana-328	15	25	the	the	DET
cana-328	15	26	behavior	behavior	NOUN
cana-328	15	27	of	of	ADP
cana-328	15	28	economic	economic	ADJ
cana-328	15	29	cycles	cycle	NOUN
cana-328	15	30	.	.	PUNCT
cana-328	16	1	finally	finally	ADV
cana-328	16	2	,	,	PUNCT
cana-328	16	3	we	we	PRON
cana-328	16	4	will	will	AUX
cana-328	16	5	present	present	VERB
cana-328	16	6	practical	practical	ADJ
cana-328	16	7	examples	example	NOUN
cana-328	16	8	to	to	PART
cana-328	16	9	demonstrate	demonstrate	VERB
cana-328	16	10	the	the	DET
cana-328	16	11	main	main	ADJ
cana-328	16	12	results	result	NOUN
cana-328	16	13	,	,	PUNCT
cana-328	16	14	providing	provide	VERB
cana-328	16	15	to	to	PART
cana-328	16	16	apply	apply	VERB
cana-328	16	17	in	in	ADP
cana-328	16	18	realworld	realworld	PROPN
cana-328	16	19	scenarios	scenario	NOUN
cana-328	16	20	.	.	PUNCT
cana-328	17	1	this	this	DET
cana-328	17	2	chapter	chapter	NOUN
cana-328	17	3	aims	aim	VERB
cana-328	17	4	to	to	PART
cana-328	17	5	provide	provide	VERB
cana-328	17	6	readers	reader	NOUN
cana-328	17	7	with	with	ADP
cana-328	17	8	a	a	DET
cana-328	17	9	comprehensive	comprehensive	ADJ
cana-328	17	10	overview	overview	NOUN
cana-328	17	11	of	of	ADP
cana-328	17	12	samuelson	samuelson	PROPN
cana-328	17	13	's	's	PART
cana-328	17	14	communications	communication	NOUN
cana-328	17	15	on	on	ADP
cana-328	17	16	applied	apply	VERB
cana-328	17	17	nonlinear	nonlinear	ADJ
cana-328	17	18	analysis	analysis	NOUN
cana-328	17	19	issn	issn	NOUN
cana-328	17	20	:	:	PUNCT
cana-328	17	21	1074	1074	NUM
cana-328	17	22	-	-	PUNCT
cana-328	17	23	133x	133x	NUM
cana-328	17	24	vol	vol	NOUN
cana-328	17	25	31	31	NUM
cana-328	17	26	no	no	NOUN
cana-328	17	27	.	.	NOUN
cana-328	17	28	1	1	NUM
cana-328	17	29	(	(	PUNCT
cana-328	17	30	2024	2024	NUM
cana-328	17	31	)	)	PUNCT
cana-328	17	32	90	90	NUM
cana-328	17	33	https://internationalpubls.com	https://internationalpubls.com	X
cana-328	17	34	model	model	NOUN
cana-328	17	35	and	and	CCONJ
cana-328	17	36	its	its	PRON
cana-328	17	37	mathematical	mathematical	ADJ
cana-328	17	38	foundations	foundation	NOUN
cana-328	17	39	,	,	PUNCT
cana-328	17	40	highlighting	highlight	VERB
cana-328	17	41	its	its	PRON
cana-328	17	42	relevance	relevance	NOUN
cana-328	17	43	and	and	CCONJ
cana-328	17	44	potential	potential	ADJ
cana-328	17	45	implications	implication	NOUN
cana-328	17	46	for	for	ADP
cana-328	17	47	various	various	ADJ
cana-328	17	48	fields	field	NOUN
cana-328	17	49	.	.	PUNCT
cana-328	18	1	third	third	ADJ
cana-328	18	2	-	-	PUNCT
cana-328	18	3	order	order	NOUN
cana-328	18	4	advanced	advanced	ADJ
cana-328	18	5	difference	difference	NOUN
cana-328	18	6	equation	equation	NOUN
cana-328	18	7	𝛥	𝛥	PROPN
cana-328	18	8	(	(	PUNCT
cana-328	18	9	𝑎(𝜑)𝛥(𝑏(𝜑)(𝛥𝑦(𝜑	𝑎(𝜑)𝛥(𝑏(𝜑)(𝛥𝑦(𝜑	PROPN
cana-328	18	10	)	)	PUNCT
cana-328	18	11	)	)	PUNCT
cana-328	18	12	𝜅	𝜅	X
cana-328	18	13	)	)	PUNCT
cana-328	18	14	)	)	PUNCT
cana-328	19	1	−	−	PROPN
cana-328	19	2	𝑝(𝜑)(𝛥𝑦(𝜑+	𝑝(𝜑)(𝛥𝑦(𝜑+	PROPN
cana-328	19	3	1	1	NUM
cana-328	19	4	)	)	PUNCT
cana-328	19	5	)	)	PUNCT
cana-328	20	1	𝜅	𝜅	X
cana-328	20	2	+	+	X
cana-328	20	3	𝑞(𝜑)𝑓(𝑦(𝜎(𝜑	𝑞(𝜑)𝑓(𝑦(𝜎(𝜑	NOUN
cana-328	20	4	)	)	PUNCT
cana-328	20	5	)	)	PUNCT
cana-328	20	6	)	)	PUNCT
cana-328	21	1	=	=	PUNCT
cana-328	21	2	0,𝜑	0,𝜑	PROPN
cana-328	21	3	≥	≥	PROPN
cana-328	21	4	𝜑0	𝜑0	NOUN
cana-328	21	5	(	(	PUNCT
cana-328	21	6	1	1	NUM
cana-328	21	7	)	)	PUNCT
cana-328	21	8	where	where	SCONJ
cana-328	21	9	{	{	PUNCT
cana-328	21	10	𝑎(𝜑	𝑎(𝜑	NOUN
cana-328	21	11	)	)	PUNCT
cana-328	21	12	}	}	PUNCT
cana-328	21	13	,	,	PUNCT
cana-328	21	14	{	{	PUNCT
cana-328	21	15	𝑏(𝜑	𝑏(𝜑	NOUN
cana-328	21	16	)	)	PUNCT
cana-328	21	17	}	}	PUNCT
cana-328	21	18	,	,	PUNCT
cana-328	21	19	{	{	PUNCT
cana-328	21	20	𝑝(𝜑	𝑝(𝜑	PROPN
cana-328	21	21	)	)	PUNCT
cana-328	21	22	}	}	PUNCT
cana-328	21	23	and	and	CCONJ
cana-328	21	24	{	{	PUNCT
cana-328	21	25	𝑞(𝜑	𝑞(𝜑	NUM
cana-328	21	26	)	)	PUNCT
cana-328	21	27	}	}	PUNCT
cana-328	21	28	are	be	AUX
cana-328	21	29	the	the	DET
cana-328	21	30	positive	positive	ADJ
cana-328	21	31	real	real	ADJ
cana-328	21	32	sequences	sequence	NOUN
cana-328	21	33	for	for	ADP
cana-328	21	34	𝜑	𝜑	PRON
cana-328	21	35	∈	∈	PROPN
cana-328	21	36	𝑁.	𝑁.	PROPN
cana-328	21	37	terms	term	NOUN
cana-328	21	38	considered	consider	VERB
cana-328	21	39	:	:	PUNCT
cana-328	21	40	h1	h1	PROPN
cana-328	21	41	)	)	PUNCT
cana-328	21	42	𝜅	𝜅	PRON
cana-328	21	43	is	be	AUX
cana-328	21	44	a	a	DET
cana-328	21	45	quotient	quotient	NOUN
cana-328	21	46	of	of	ADP
cana-328	21	47	odd	odd	ADJ
cana-328	21	48	positive	positive	ADJ
cana-328	21	49	integers	integer	NOUN
cana-328	21	50	;	;	PUNCT
cana-328	21	51	h2	h2	NOUN
cana-328	21	52	)	)	PUNCT
cana-328	21	53	𝜎(𝜑	𝜎(𝜑	PROPN
cana-328	21	54	)	)	PUNCT
cana-328	21	55	is	be	AUX
cana-328	21	56	an	an	DET
cana-328	21	57	increasing	increase	VERB
cana-328	21	58	sequence	sequence	NOUN
cana-328	21	59	such	such	ADJ
cana-328	21	60	that	that	SCONJ
cana-328	21	61	𝜎(𝜑	𝜎(𝜑	NOUN
cana-328	21	62	)	)	PUNCT
cana-328	21	63	≥	≥	X
cana-328	21	64	𝜑+	𝜑+	X
cana-328	21	65	1	1	NUM
cana-328	21	66	for	for	ADP
cana-328	21	67	all	all	DET
cana-328	21	68	𝜑	𝜑	PRON
cana-328	21	69	≥	≥	NOUN
cana-328	21	70	𝜑0	𝜑0	NOUN
cana-328	21	71	;	;	PUNCT
cana-328	21	72	h3	h3	NOUN
cana-328	21	73	)	)	PUNCT
cana-328	21	74	∑𝑠=𝜑0	∑𝑠=𝜑0	NUM
cana-328	21	75	∞	∞	NUM
cana-328	21	76	  	  	SPACE
cana-328	21	77	1	1	NUM
cana-328	21	78	𝑎(𝑠	𝑎(𝑠	PROPN
cana-328	21	79	)	)	PUNCT
cana-328	21	80	<	<	X
cana-328	22	1	∞,∑𝑠=𝜑0	∞,∑𝑠=𝜑0	PROPN
cana-328	22	2	∞	∞	NUM
cana-328	22	3	  	  	SPACE
cana-328	22	4	1	1	NUM
cana-328	22	5	𝑏(𝑠	𝑏(𝑠	PROPN
cana-328	22	6	)	)	PUNCT
cana-328	22	7	=	=	SYM
cana-328	22	8	∞	∞	PROPN
cana-328	22	9	;	;	PUNCT
cana-328	22	10	h4	h4	NOUN
cana-328	22	11	)	)	PUNCT
cana-328	22	12	function	function	VERB
cana-328	22	13	𝑓	𝑓	NOUN
cana-328	22	14	:	:	PUNCT
cana-328	22	15	𝑅	𝑅	PROPN
cana-328	22	16	→	→	SYM
cana-328	22	17	𝑅	𝑅	PROPN
cana-328	22	18	is	be	AUX
cana-328	22	19	continuous	continuous	ADJ
cana-328	22	20	such	such	ADJ
cana-328	22	21	that	that	SCONJ
cana-328	22	22	𝑢𝑓(𝑢	𝑢𝑓(𝑢	ADV
cana-328	22	23	)	)	PUNCT
cana-328	22	24	>	>	X
cana-328	22	25	0	0	PUNCT
cana-328	22	26	for	for	ADP
cana-328	22	27	𝑢	𝑢	PRON
cana-328	22	28	≠	≠	PROPN
cana-328	22	29	0	0	NUM
cana-328	22	30	and	and	CCONJ
cana-328	22	31	𝑓(𝑢𝑣	𝑓(𝑢𝑣	ADJ
cana-328	22	32	)	)	PUNCT
cana-328	22	33	≥	≥	NOUN
cana-328	22	34	𝑓(𝑢)𝑓(𝑣	𝑓(𝑢)𝑓(𝑣	NOUN
cana-328	22	35	)	)	PUNCT
cana-328	22	36	:	:	PUNCT
cana-328	22	37	a	a	DET
cana-328	22	38	solution	solution	NOUN
cana-328	22	39	of	of	ADP
cana-328	22	40	(	(	PUNCT
cana-328	22	41	1	1	NUM
cana-328	22	42	)	)	PUNCT
cana-328	22	43	is	be	AUX
cana-328	22	44	a	a	DET
cana-328	22	45	real	real	ADJ
cana-328	22	46	sequence	sequence	NOUN
cana-328	22	47	{	{	PUNCT
cana-328	22	48	𝑦(𝜑	𝑦(𝜑	NOUN
cana-328	22	49	)	)	PUNCT
cana-328	22	50	}	}	PUNCT
cana-328	22	51	defined	define	VERB
cana-328	22	52	for	for	ADP
cana-328	22	53	all	all	DET
cana-328	22	54	𝜑	𝜑	PRON
cana-328	22	55	≥	≥	NOUN
cana-328	22	56	𝜑0	𝜑0	NOUN
cana-328	22	57	.	.	PUNCT
cana-328	23	1	a	a	DET
cana-328	23	2	nontrivial	nontrivial	ADJ
cana-328	23	3	solution	solution	NOUN
cana-328	23	4	{	{	PUNCT
cana-328	23	5	𝑦(𝜑	𝑦(𝜑	NOUN
cana-328	23	6	)	)	PUNCT
cana-328	23	7	}	}	PUNCT
cana-328	23	8	of	of	ADP
cana-328	23	9	(	(	PUNCT
cana-328	23	10	1	1	X
cana-328	23	11	)	)	PUNCT
cana-328	23	12	is	be	AUX
cana-328	23	13	said	say	VERB
cana-328	23	14	to	to	PART
cana-328	23	15	be	be	AUX
cana-328	23	16	oscillatory	oscillatory	ADJ
cana-328	23	17	if	if	SCONJ
cana-328	23	18	it	it	PRON
cana-328	23	19	is	be	AUX
cana-328	23	20	neither	neither	CCONJ
cana-328	23	21	positive	positive	ADJ
cana-328	23	22	nor	nor	CCONJ
cana-328	23	23	negative	negative	ADJ
cana-328	23	24	,	,	PUNCT
cana-328	23	25	otherwise	otherwise	ADV
cana-328	23	26	it	it	PRON
cana-328	23	27	is	be	AUX
cana-328	23	28	non	non	ADJ
cana-328	23	29	-	-	ADJ
cana-328	23	30	oscillatory	oscillatory	ADJ
cana-328	23	31	.	.	PUNCT
cana-328	24	1	by	by	ADP
cana-328	24	2	property	property	NOUN
cana-328	24	3	a	a	PRON
cana-328	24	4	of	of	ADP
cana-328	24	5	(	(	PUNCT
cana-328	24	6	1	1	X
cana-328	24	7	)	)	PUNCT
cana-328	24	8	it	it	PRON
cana-328	24	9	is	be	AUX
cana-328	24	10	meant	mean	VERB
cana-328	24	11	that	that	SCONJ
cana-328	24	12	every	every	DET
cana-328	24	13	solution	solution	NOUN
cana-328	24	14	𝑦(𝜑	𝑦(𝜑	NOUN
cana-328	24	15	)	)	PUNCT
cana-328	24	16	of	of	ADP
cana-328	24	17	(	(	PUNCT
cana-328	24	18	1	1	X
cana-328	24	19	)	)	PUNCT
cana-328	24	20	is	be	AUX
cana-328	24	21	strictly	strictly	ADV
cana-328	24	22	increasing	increase	VERB
cana-328	24	23	𝑦(𝜑	𝑦(𝜑	NOUN
cana-328	24	24	)	)	PUNCT
cana-328	24	25	)	)	PUNCT
cana-328	25	1	>	>	X
cana-328	25	2	0	0	NUM
cana-328	25	3	,	,	PUNCT
cana-328	25	4	𝑑(𝜑)𝛥(𝑦(𝜑))𝜅	𝑑(𝜑)𝛥(𝑦(𝜑))𝜅	PROPN
cana-328	25	5	<	<	X
cana-328	25	6	0	0	NUM
cana-328	25	7	,	,	PUNCT
cana-328	25	8	𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥(𝑦(𝜑))𝜅	𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥(𝑦(𝜑))𝜅	NOUN
cana-328	25	9	)	)	PUNCT
cana-328	25	10	)	)	PUNCT
cana-328	26	1	>	>	X
cana-328	26	2	0	0	NUM
cana-328	26	3	,	,	PUNCT
cana-328	26	4	𝛥	𝛥	PROPN
cana-328	26	5	(	(	PUNCT
cana-328	26	6	𝑐(𝜑)𝛥	𝑐(𝜑)𝛥	ADJ
cana-328	26	7	(	(	PUNCT
cana-328	26	8	𝑑(𝜑)(𝛥(𝑦(𝜑	𝑑(𝜑)(𝛥(𝑦(𝜑	PROPN
cana-328	26	9	)	)	PUNCT
cana-328	26	10	)	)	PUNCT
cana-328	26	11	𝜅	𝜅	X
cana-328	26	12	)	)	PUNCT
cana-328	26	13	)	)	PUNCT
cana-328	26	14	)	)	PUNCT
cana-328	27	1	<	<	X
cana-328	27	2	0	0	NUM
cana-328	27	3	recent	recent	ADJ
cana-328	27	4	research	research	NOUN
cana-328	27	5	has	have	AUX
cana-328	27	6	explored	explore	VERB
cana-328	27	7	the	the	DET
cana-328	27	8	oscillation	oscillation	NOUN
cana-328	27	9	criteria	criterion	NOUN
cana-328	27	10	of	of	ADP
cana-328	27	11	difference	difference	NOUN
cana-328	27	12	equations	equation	NOUN
cana-328	27	13	,	,	PUNCT
cana-328	27	14	primarily	primarily	ADV
cana-328	27	15	focused	focus	VERB
cana-328	27	16	on	on	ADP
cana-328	27	17	[	[	X
cana-328	27	18	1][4	1][4	NUM
cana-328	27	19	]	]	PUNCT
cana-328	27	20	,	,	PUNCT
cana-328	27	21	[	[	X
cana-328	27	22	15	15	NUM
cana-328	27	23	]	]	PUNCT
cana-328	27	24	.	.	PUNCT
cana-328	28	1	however	however	ADV
cana-328	28	2	,	,	PUNCT
cana-328	28	3	third	third	ADJ
cana-328	28	4	-	-	PUNCT
cana-328	28	5	order	order	NOUN
cana-328	28	6	equations	equation	NOUN
cana-328	28	7	are	be	AUX
cana-328	28	8	also	also	ADV
cana-328	28	9	essential	essential	ADJ
cana-328	28	10	,	,	PUNCT
cana-328	28	11	as	as	SCONJ
cana-328	28	12	they	they	PRON
cana-328	28	13	have	have	VERB
cana-328	28	14	applications	application	NOUN
cana-328	28	15	in	in	ADP
cana-328	28	16	various	various	ADJ
cana-328	28	17	fields	field	NOUN
cana-328	28	18	,	,	PUNCT
cana-328	28	19	including	include	VERB
cana-328	28	20	economics	economic	NOUN
cana-328	28	21	,	,	PUNCT
cana-328	28	22	physics	physics	NOUN
cana-328	28	23	,	,	PUNCT
cana-328	28	24	and	and	CCONJ
cana-328	28	25	mathematical	mathematical	ADJ
cana-328	28	26	biology	biology	NOUN
cana-328	28	27	(	(	PUNCT
cana-328	28	28	[	[	X
cana-328	28	29	7],[10	7],[10	X
cana-328	28	30	]	]	X
cana-328	28	31	,	,	PUNCT
cana-328	28	32	[	[	X
cana-328	28	33	12	12	NUM
cana-328	28	34	]	]	PUNCT
cana-328	28	35	)	)	PUNCT
cana-328	28	36	.	.	PUNCT
cana-328	29	1	for	for	ADP
cana-328	29	2	instance	instance	NOUN
cana-328	29	3	,	,	PUNCT
cana-328	29	4	samuelson	samuelson	PROPN
cana-328	29	5	's	's	PART
cana-328	29	6	business	business	NOUN
cana-328	29	7	cycles	cycle	NOUN
cana-328	29	8	theory	theory	NOUN
cana-328	29	9	,	,	PUNCT
cana-328	29	10	a	a	DET
cana-328	29	11	central	central	ADJ
cana-328	29	12	economic	economic	ADJ
cana-328	29	13	theory	theory	NOUN
cana-328	29	14	related	relate	VERB
cana-328	29	15	to	to	ADP
cana-328	29	16	the	the	DET
cana-328	29	17	business	business	NOUN
cana-328	29	18	cycle	cycle	NOUN
cana-328	29	19	,	,	PUNCT
cana-328	29	20	can	can	AUX
cana-328	29	21	be	be	AUX
cana-328	29	22	modelled	model	VERB
cana-328	29	23	by	by	ADP
cana-328	29	24	linear	linear	PROPN
cana-328	29	25	second	second	ADJ
cana-328	29	26	and	and	CCONJ
cana-328	29	27	third	third	ADJ
cana-328	29	28	-	-	PUNCT
cana-328	29	29	order	order	NOUN
cana-328	29	30	difference	difference	NOUN
cana-328	29	31	equations	equation	NOUN
cana-328	29	32	.	.	PUNCT
cana-328	30	1	this	this	DET
cana-328	30	2	capability	capability	NOUN
cana-328	30	3	allows	allow	VERB
cana-328	30	4	advanced	advanced	ADJ
cana-328	30	5	arguments	argument	NOUN
cana-328	30	6	to	to	PART
cana-328	30	7	depict	depict	VERB
cana-328	30	8	events	event	NOUN
cana-328	30	9	dependent	dependent	ADJ
cana-328	30	10	on	on	ADP
cana-328	30	11	another	another	DET
cana-328	30	12	action	action	NOUN
cana-328	30	13	.	.	PUNCT
cana-328	31	1	recent	recent	ADJ
cana-328	31	2	studies	study	NOUN
cana-328	31	3	have	have	AUX
cana-328	31	4	investigated	investigate	VERB
cana-328	31	5	the	the	DET
cana-328	31	6	oscillatory	oscillatory	ADJ
cana-328	31	7	criteria	criterion	NOUN
cana-328	31	8	and	and	CCONJ
cana-328	31	9	property	property	NOUN
cana-328	31	10	a	a	DET
cana-328	31	11	conditions	condition	NOUN
cana-328	31	12	for	for	ADP
cana-328	31	13	difference	difference	NOUN
cana-328	31	14	equations	equation	NOUN
cana-328	31	15	,	,	PUNCT
cana-328	31	16	and	and	CCONJ
cana-328	31	17	their	their	PRON
cana-328	31	18	findings	finding	NOUN
cana-328	31	19	are	be	AUX
cana-328	31	20	available	available	ADJ
cana-328	31	21	in	in	ADP
cana-328	31	22	the	the	DET
cana-328	31	23	literature	literature	NOUN
cana-328	31	24	(	(	PUNCT
cana-328	31	25	[	[	X
cana-328	31	26	6],[8],[9],[11	6],[8],[9],[11	NUM
cana-328	31	27	]	]	X
cana-328	31	28	)	)	PUNCT
cana-328	31	29	.	.	PUNCT
cana-328	32	1	this	this	DET
cana-328	32	2	paper	paper	NOUN
cana-328	32	3	's	's	PART
cana-328	32	4	results	result	NOUN
cana-328	32	5	and	and	CCONJ
cana-328	32	6	recent	recent	ADJ
cana-328	32	7	research	research	NOUN
cana-328	32	8	can	can	AUX
cana-328	32	9	inform	inform	VERB
cana-328	32	10	the	the	DET
cana-328	32	11	development	development	NOUN
cana-328	32	12	of	of	ADP
cana-328	32	13	more	more	ADV
cana-328	32	14	accurate	accurate	ADJ
cana-328	32	15	models	model	NOUN
cana-328	32	16	for	for	ADP
cana-328	32	17	real	real	ADJ
cana-328	32	18	-	-	PUNCT
cana-328	32	19	world	world	NOUN
cana-328	32	20	problems	problem	NOUN
cana-328	32	21	.	.	PUNCT
cana-328	33	1	in	in	ADP
cana-328	33	2	[	[	X
cana-328	33	3	13	13	NUM
cana-328	33	4	]	]	PUNCT
cana-328	33	5	,	,	PUNCT
cana-328	33	6	the	the	DET
cana-328	33	7	authors	author	NOUN
cana-328	33	8	used	use	VERB
cana-328	33	9	advanced	advanced	ADJ
cana-328	33	10	arguments	argument	NOUN
cana-328	33	11	to	to	PART
cana-328	33	12	demonstrate	demonstrate	VERB
cana-328	33	13	the	the	DET
cana-328	33	14	oscillatory	oscillatory	ADJ
cana-328	33	15	and	and	CCONJ
cana-328	33	16	property	property	NOUN
cana-328	33	17	b	b	NOUN
cana-328	33	18	conditions	condition	NOUN
cana-328	33	19	for	for	ADP
cana-328	33	20	difference	difference	NOUN
cana-328	33	21	equation	equation	NOUN
cana-328	33	22	,	,	PUNCT
cana-328	33	23	𝛥	𝛥	PROPN
cana-328	33	24	(	(	PUNCT
cana-328	33	25	𝑎𝑛(𝛥(𝑏𝑛𝛥𝑥𝑛	𝑎𝑛(𝛥(𝑏𝑛𝛥𝑥𝑛	PROPN
cana-328	33	26	)	)	PUNCT
cana-328	33	27	)	)	PUNCT
cana-328	33	28	𝛼	𝛼	X
cana-328	33	29	)	)	PUNCT
cana-328	33	30	−	−	PROPN
cana-328	33	31	𝑝𝑛𝑓(𝑥𝜎(𝑛	𝑝𝑛𝑓(𝑥𝜎(𝑛	NOUN
cana-328	33	32	)	)	PUNCT
cana-328	33	33	)	)	PUNCT
cana-328	34	1	=	=	SYM
cana-328	34	2	0,𝑛	0,𝑛	ADJ
cana-328	34	3	≥	≥	NOUN
cana-328	34	4	𝑛0	𝑛0	VERB
cana-328	34	5	where	where	SCONJ
cana-328	34	6	𝑎𝑛	𝑎𝑛	PROPN
cana-328	34	7	,	,	PUNCT
cana-328	34	8	𝑏𝑛	𝑏𝑛	NOUN
cana-328	34	9	,	,	PUNCT
cana-328	34	10	and	and	CCONJ
cana-328	34	11	𝑝𝑛	𝑝𝑛	NOUN
cana-328	34	12	are	be	AUX
cana-328	34	13	the	the	DET
cana-328	34	14	positive	positive	ADJ
cana-328	34	15	real	real	ADJ
cana-328	34	16	sequences	sequence	NOUN
cana-328	34	17	.	.	PUNCT
cana-328	35	1	in	in	ADP
cana-328	35	2	[	[	X
cana-328	35	3	14	14	NUM
cana-328	35	4	]	]	PUNCT
cana-328	35	5	,	,	PUNCT
cana-328	35	6	the	the	DET
cana-328	35	7	oscillatory	oscillatory	ADJ
cana-328	35	8	behavior	behavior	NOUN
cana-328	35	9	of	of	ADP
cana-328	35	10	difference	difference	NOUN
cana-328	35	11	equation	equation	NOUN
cana-328	35	12	was	be	AUX
cana-328	35	13	investigated	investigate	VERB
cana-328	35	14	𝛥	𝛥	PROPN
cana-328	35	15	(	(	PUNCT
cana-328	35	16	𝑐(𝑙)𝛥(𝑑(𝑙)(𝛥𝑥(𝑙	𝑐(𝑙)𝛥(𝑑(𝑙)(𝛥𝑥(𝑙	NOUN
cana-328	35	17	)	)	PUNCT
cana-328	35	18	)	)	PUNCT
cana-328	35	19	𝛼	𝛼	X
cana-328	35	20	)	)	PUNCT
cana-328	35	21	)	)	PUNCT
cana-328	36	1	−	−	PROPN
cana-328	36	2	𝑎(𝑙)(𝛥𝑥(𝑙	𝑎(𝑙)(𝛥𝑥(𝑙	ADJ
cana-328	36	3	+	+	CCONJ
cana-328	36	4	1	1	NUM
cana-328	36	5	)	)	PUNCT
cana-328	36	6	)	)	PUNCT
cana-328	37	1	𝛼	𝛼	ADP
cana-328	37	2	−	−	PROPN
cana-328	37	3	𝑏(𝑙)𝑔(𝑥(𝜎(𝑙	𝑏(𝑙)𝑔(𝑥(𝜎(𝑙	NOUN
cana-328	37	4	)	)	PUNCT
cana-328	37	5	)	)	PUNCT
cana-328	37	6	)	)	PUNCT
cana-328	38	1	=	=	PUNCT
cana-328	38	2	0	0	NUM
cana-328	38	3	,	,	PUNCT
cana-328	38	4	𝑙	𝑙	PRON
cana-328	38	5	≥	≥	NOUN
cana-328	38	6	𝑙0	𝑙0	NOUN
cana-328	38	7	proceeding	proceed	VERB
cana-328	38	8	the	the	DET
cana-328	38	9	manner	manner	NOUN
cana-328	38	10	,	,	PUNCT
cana-328	38	11	the	the	DET
cana-328	38	12	aim	aim	NOUN
cana-328	38	13	of	of	ADP
cana-328	38	14	this	this	DET
cana-328	38	15	paper	paper	NOUN
cana-328	38	16	is	be	AUX
cana-328	38	17	to	to	PART
cana-328	38	18	commence	commence	VERB
cana-328	38	19	sufficient	sufficient	ADJ
cana-328	38	20	condition	condition	NOUN
cana-328	38	21	to	to	PART
cana-328	38	22	ensure	ensure	VERB
cana-328	38	23	that	that	SCONJ
cana-328	38	24	solution	solution	NOUN
cana-328	38	25	of	of	ADP
cana-328	38	26	(	(	PUNCT
cana-328	38	27	1	1	NUM
cana-328	38	28	)	)	PUNCT
cana-328	38	29	are	be	AUX
cana-328	38	30	oscillatory	oscillatory	ADJ
cana-328	38	31	and	and	CCONJ
cana-328	38	32	also	also	ADV
cana-328	38	33	it	it	PRON
cana-328	38	34	is	be	AUX
cana-328	38	35	closely	closely	ADV
cana-328	38	36	relevant	relevant	ADJ
cana-328	38	37	to	to	ADP
cana-328	38	38	the	the	DET
cana-328	38	39	auxiliary	auxiliary	ADJ
cana-328	38	40	second	second	ADJ
cana-328	38	41	order	order	NOUN
cana-328	38	42	equation	equation	NOUN
cana-328	38	43	communications	communication	NOUN
cana-328	38	44	on	on	ADP
cana-328	38	45	applied	apply	VERB
cana-328	38	46	nonlinear	nonlinear	ADJ
cana-328	38	47	analysis	analysis	NOUN
cana-328	38	48	issn	issn	NOUN
cana-328	38	49	:	:	PUNCT
cana-328	38	50	1074	1074	NUM
cana-328	38	51	-	-	PUNCT
cana-328	38	52	133x	133x	NUM
cana-328	38	53	vol	vol	NOUN
cana-328	38	54	31	31	NUM
cana-328	38	55	no	no	NOUN
cana-328	38	56	.	.	NOUN
cana-328	38	57	1	1	NUM
cana-328	38	58	(	(	PUNCT
cana-328	38	59	2024	2024	NUM
cana-328	38	60	)	)	PUNCT
cana-328	38	61	91	91	NUM
cana-328	38	62	https://internationalpubls.com	https://internationalpubls.com	X
cana-328	38	63	𝛥(𝑎(𝜑)𝛥(𝑧(𝜑	𝛥(𝑎(𝜑)𝛥(𝑧(𝜑	NUM
cana-328	38	64	)	)	PUNCT
cana-328	38	65	)	)	PUNCT
cana-328	38	66	)	)	PUNCT
cana-328	39	1	−	−	PROPN
cana-328	40	1	𝑝(𝜑	𝑝(𝜑	NUM
cana-328	40	2	)	)	PUNCT
cana-328	40	3	𝑏(𝜑+1	𝑏(𝜑+1	PROPN
cana-328	40	4	)	)	PUNCT
cana-328	40	5	𝑥(𝜑	𝑥(𝜑	PROPN
cana-328	41	1	+	+	CCONJ
cana-328	41	2	1	1	X
cana-328	41	3	)	)	PUNCT
cana-328	41	4	(	(	PUNCT
cana-328	41	5	2	2	X
cana-328	41	6	)	)	PUNCT
cana-328	41	7	we	we	PRON
cana-328	41	8	also	also	ADV
cana-328	41	9	derive	derive	VERB
cana-328	41	10	a	a	DET
cana-328	41	11	solution	solution	NOUN
cana-328	41	12	for	for	ADP
cana-328	41	13	equation	equation	NOUN
cana-328	41	14	(	(	PUNCT
cana-328	41	15	1	1	NUM
cana-328	41	16	)	)	PUNCT
cana-328	41	17	that	that	PRON
cana-328	41	18	possesses	possess	VERB
cana-328	41	19	property	property	NOUN
cana-328	41	20	a	a	PRON
cana-328	41	21	,	,	PUNCT
cana-328	41	22	and	and	CCONJ
cana-328	41	23	its	its	PRON
cana-328	41	24	corresponding	corresponding	ADJ
cana-328	41	25	secondorder	secondorder	ADJ
cana-328	41	26	equation	equation	NOUN
cana-328	41	27	aligns	align	VERB
cana-328	41	28	with	with	ADP
cana-328	41	29	samuelson	samuelson	PROPN
cana-328	41	30	’s	’s	PART
cana-328	41	31	business	business	NOUN
cana-328	41	32	cycle	cycle	NOUN
cana-328	41	33	model	model	NOUN
cana-328	41	34	.	.	PUNCT
cana-328	42	1	2	2	X
cana-328	42	2	.	.	X
cana-328	42	3	main	main	ADJ
cana-328	42	4	results	result	NOUN
cana-328	42	5	let	let	VERB
cana-328	42	6	us	we	PRON
cana-328	42	7	define	define	VERB
cana-328	42	8	:	:	PUNCT
cana-328	42	9	𝑐(𝜑	𝑐(𝜑	NOUN
cana-328	42	10	)	)	PUNCT
cana-328	43	1	=	=	PUNCT
cana-328	44	1	𝑎(𝜑)𝑧(𝜑)𝑧(𝜑	𝑎(𝜑)𝑧(𝜑)𝑧(𝜑	X
cana-328	44	2	+	+	NOUN
cana-328	44	3	1	1	NUM
cana-328	44	4	)	)	PUNCT
cana-328	44	5	,	,	PUNCT
cana-328	44	6	𝑑(𝜑	𝑑(𝜑	ADJ
cana-328	44	7	)	)	PUNCT
cana-328	44	8	=	=	SYM
cana-328	44	9	𝑏(𝜑	𝑏(𝜑	NOUN
cana-328	44	10	)	)	PUNCT
cana-328	44	11	𝑧(𝜑	𝑧(𝜑	NOUN
cana-328	44	12	)	)	PUNCT
cana-328	44	13	,	,	PUNCT
cana-328	44	14	𝑄(𝜑	𝑄(𝜑	NUM
cana-328	44	15	)	)	PUNCT
cana-328	44	16	=	=	PUNCT
cana-328	44	17	𝑞(𝜑)𝑧(𝜑	𝑞(𝜑)𝑧(𝜑	PROPN
cana-328	44	18	+	+	CCONJ
cana-328	44	19	1	1	NUM
cana-328	44	20	)	)	PUNCT
cana-328	44	21	,	,	PUNCT
cana-328	44	22	𝐶(𝜑	𝐶(𝜑	NOUN
cana-328	44	23	)	)	PUNCT
cana-328	44	24	=	=	PUNCT
cana-328	44	25	∑	∑	PROPN
cana-328	44	26	1	1	NUM
cana-328	44	27	𝑐(𝑠	𝑐(𝑠	NOUN
cana-328	44	28	)	)	PUNCT
cana-328	44	29	𝜑−1	𝜑−1	PROPN
cana-328	44	30	𝑠=𝜑1	𝑠=𝜑1	PROPN
cana-328	44	31	   	   	SPACE
cana-328	44	32	,	,	PUNCT
cana-328	44	33	𝐷(𝜑	𝐷(𝜑	PRON
cana-328	44	34	)	)	PUNCT
cana-328	44	35	=	=	PUNCT
cana-328	45	1	∑	∑	PUNCT
cana-328	45	2	1	1	NUM
cana-328	45	3	𝑑	𝑑	PRON
cana-328	45	4	1	1	NUM
cana-328	45	5	𝜅(𝑠	𝜅(𝑠	NUM
cana-328	45	6	)	)	PUNCT
cana-328	45	7	𝜑−1	𝜑−1	PROPN
cana-328	45	8	𝑠=𝜑1	𝑠=𝜑1	PROPN
cana-328	45	9	   	   	SPACE
cana-328	45	10	,	,	PUNCT
cana-328	45	11	𝐸(𝜑	𝐸(𝜑	NUM
cana-328	45	12	)	)	PUNCT
cana-328	45	13	=	=	SYM
cana-328	46	1	∑	∑	PUNCT
cana-328	46	2	1	1	NUM
cana-328	46	3	𝑑	𝑑	PRON
cana-328	46	4	1	1	NUM
cana-328	46	5	𝜅(𝑠	𝜅(𝑠	NUM
cana-328	46	6	)	)	PUNCT
cana-328	46	7	𝜑−1	𝜑−1	PROPN
cana-328	46	8	𝑠=𝜑1	𝑠=𝜑1	PROPN
cana-328	46	9	  	  	SPACE
cana-328	46	10	(	(	PUNCT
cana-328	46	11	∑	∑	PROPN
cana-328	46	12	1	1	NUM
cana-328	46	13	𝑐(𝑠	𝑐(𝑠	NOUN
cana-328	46	14	)	)	PUNCT
cana-328	47	1	𝜑−1	𝜑−1	PROPN
cana-328	47	2	𝑠=𝜑1	𝑠=𝜑1	PROPN
cana-328	47	3	  	  	SPACE
cana-328	47	4	)	)	PUNCT
cana-328	47	5	1	1	NUM
cana-328	47	6	𝜅	𝜅	PROPN
cana-328	47	7	.	.	PUNCT
cana-328	48	1	theorem	theorem	VERB
cana-328	48	2	2.1	2.1	NUM
cana-328	48	3	.	.	PUNCT
cana-328	49	1	assume	assume	VERB
cana-328	49	2	that	that	SCONJ
cana-328	49	3	∑	∑	PROPN
cana-328	49	4	1	1	NUM
cana-328	49	5	𝑑	𝑑	PRON
cana-328	49	6	1	1	NUM
cana-328	49	7	𝜅(𝑠	𝜅(𝑠	NUM
cana-328	49	8	)	)	PUNCT
cana-328	49	9	∞	∞	PROPN
cana-328	50	1	𝑠=𝜑0	𝑠=𝜑0	NOUN
cana-328	50	2	  	  	SPACE
cana-328	50	3	=	=	SYM
cana-328	50	4	∞	∞	PROPN
cana-328	50	5	,	,	PUNCT
cana-328	50	6	(	(	PUNCT
cana-328	50	7	3	3	NUM
cana-328	50	8	)	)	PUNCT
cana-328	50	9	and	and	CCONJ
cana-328	50	10	𝐿𝑦	𝐿𝑦	PROPN
cana-328	50	11	(	(	PUNCT
cana-328	50	12	𝜑	𝜑	NOUN
cana-328	50	13	)	)	PUNCT
cana-328	50	14	=	=	SYM
cana-328	50	15	𝛥(𝑎(𝜑)𝛥(𝑏(𝜑)(𝛥𝑦(𝜑))𝜅	𝛥(𝑎(𝜑)𝛥(𝑏(𝜑)(𝛥𝑦(𝜑))𝜅	PROPN
cana-328	50	16	)	)	PUNCT
cana-328	50	17	)	)	PUNCT
cana-328	51	1	−	−	PROPN
cana-328	52	1	𝑝(𝜑)(𝛥𝑦(𝜑	𝑝(𝜑)(𝛥𝑦(𝜑	NOUN
cana-328	52	2	+	+	CCONJ
cana-328	52	3	1))𝜅	1))𝜅	NUM
cana-328	52	4	then	then	ADV
cana-328	52	5	𝐿𝑦	𝐿𝑦	PROPN
cana-328	52	6	(	(	PUNCT
cana-328	52	7	𝜑	𝜑	NOUN
cana-328	52	8	)	)	PUNCT
cana-328	52	9	can	can	AUX
cana-328	52	10	be	be	AUX
cana-328	52	11	written	write	VERB
cana-328	52	12	as	as	ADP
cana-328	52	13	𝐿𝑦	𝐿𝑦	PROPN
cana-328	52	14	(	(	PUNCT
cana-328	52	15	𝜑	𝜑	NOUN
cana-328	52	16	)	)	PUNCT
cana-328	52	17	=	=	SYM
cana-328	53	1	1	1	NUM
cana-328	53	2	𝑧(𝜑	𝑧(𝜑	NOUN
cana-328	53	3	+	+	CCONJ
cana-328	53	4	1	1	X
cana-328	53	5	)	)	PUNCT
cana-328	53	6	𝛥	𝛥	PROPN
cana-328	53	7	(	(	PUNCT
cana-328	53	8	𝑎(𝜑)𝑧(𝜑)𝑧(𝜑	𝑎(𝜑)𝑧(𝜑)𝑧(𝜑	X
cana-328	53	9	+	+	NUM
cana-328	53	10	1)𝛥	1)𝛥	NOUN
cana-328	53	11	(	(	PUNCT
cana-328	53	12	𝑏(𝜑	𝑏(𝜑	NOUN
cana-328	53	13	)	)	PUNCT
cana-328	53	14	𝑧(𝜑	𝑧(𝜑	NOUN
cana-328	53	15	)	)	PUNCT
cana-328	53	16	(	(	PUNCT
cana-328	53	17	𝛥𝑦(𝜑))𝜅	𝛥𝑦(𝜑))𝜅	NOUN
cana-328	53	18	)	)	PUNCT
cana-328	53	19	)	)	PUNCT
cana-328	53	20	.	.	PUNCT
cana-328	54	1	proof	proof	NOUN
cana-328	54	2	.	.	PUNCT
cana-328	55	1	1	1	NUM
cana-328	55	2	𝑧(𝜑+1	𝑧(𝜑+1	SYM
cana-328	55	3	)	)	PUNCT
cana-328	55	4	𝛥(𝑎(𝜑)𝑧(𝜑)𝑧(𝜑	𝛥(𝑎(𝜑)𝑧(𝜑)𝑧(𝜑	NOUN
cana-328	56	1	+	+	CCONJ
cana-328	56	2	1)𝛥	1)𝛥	NOUN
cana-328	56	3	(	(	PUNCT
cana-328	56	4	𝑏(𝜑	𝑏(𝜑	NOUN
cana-328	56	5	)	)	PUNCT
cana-328	56	6	𝑧(𝜑	𝑧(𝜑	NOUN
cana-328	56	7	)	)	PUNCT
cana-328	56	8	(	(	PUNCT
cana-328	56	9	𝛥𝑦(𝜑))𝜅	𝛥𝑦(𝜑))𝜅	NOUN
cana-328	56	10	)	)	PUNCT
cana-328	56	11	)	)	PUNCT
cana-328	57	1	=	=	SYM
cana-328	57	2	1	1	NUM
cana-328	57	3	𝑧(𝜑+1	𝑧(𝜑+1	X
cana-328	57	4	)	)	PUNCT
cana-328	58	1	[	[	X
cana-328	58	2	𝛥(𝑎(𝜑)(𝛥(𝑏(𝜑)(𝛥𝑦(𝜑))𝜅𝑧(𝜑	𝛥(𝑎(𝜑)(𝛥(𝑏(𝜑)(𝛥𝑦(𝜑))𝜅𝑧(𝜑	X
cana-328	58	3	+	+	NUM
cana-328	58	4	1	1	NUM
cana-328	58	5	)	)	PUNCT
cana-328	58	6	)	)	PUNCT
cana-328	59	1	−	−	ADP
cana-328	60	1	𝑏(𝜑	𝑏(𝜑	NOUN
cana-328	60	2	+	+	NOUN
cana-328	60	3	1)(𝛥𝑦(𝜑	1)(𝛥𝑦(𝜑	NUM
cana-328	60	4	+	+	CCONJ
cana-328	60	5	1))𝜅𝛥𝑧(𝜑	1))𝜅𝛥𝑧(𝜑	NUM
cana-328	60	6	)	)	PUNCT
cana-328	60	7	)	)	PUNCT
cana-328	60	8	)	)	PUNCT
cana-328	60	9	]	]	PUNCT
cana-328	61	1	=	=	PUNCT
cana-328	61	2	1	1	NUM
cana-328	61	3	𝑧(𝜑+1	𝑧(𝜑+1	X
cana-328	61	4	)	)	PUNCT
cana-328	62	1	[	[	X
cana-328	62	2	𝑧(𝜑	𝑧(𝜑	X
cana-328	62	3	+	+	CCONJ
cana-328	62	4	1)𝛥(𝑎(𝜑)𝛥(𝑏(𝜑)(𝛥𝑦(𝜑	1)𝛥(𝑎(𝜑)𝛥(𝑏(𝜑)(𝛥𝑦(𝜑	NUM
cana-328	62	5	+	+	SYM
cana-328	62	6	1))𝜅	1))𝜅	NUM
cana-328	62	7	)	)	PUNCT
cana-328	62	8	)	)	PUNCT
cana-328	62	9	−	−	ADP
cana-328	62	10	𝑏(𝜑+1	𝑏(𝜑+1	NOUN
cana-328	62	11	)	)	PUNCT
cana-328	62	12	𝑧(𝜑+1	𝑧(𝜑+1	X
cana-328	62	13	)	)	PUNCT
cana-328	62	14	(	(	PUNCT
cana-328	62	15	𝛥𝑦(𝜑	𝛥𝑦(𝜑	ADV
cana-328	62	16	+	+	NOUN
cana-328	62	17	1))𝜅𝛥(𝑎(𝜑𝛥𝑧(𝜑	1))𝜅𝛥(𝑎(𝜑𝛥𝑧(𝜑	NUM
cana-328	62	18	)	)	PUNCT
cana-328	62	19	)	)	PUNCT
cana-328	62	20	)	)	PUNCT
cana-328	62	21	]	]	PUNCT
cana-328	63	1	=	=	PUNCT
cana-328	63	2	𝛥(𝑎(𝜑)𝛥(𝑏(𝜑)(𝛥𝑦(𝜑	𝛥(𝑎(𝜑)𝛥(𝑏(𝜑)(𝛥𝑦(𝜑	NOUN
cana-328	63	3	+	+	NUM
cana-328	63	4	1))𝜅	1))𝜅	NUM
cana-328	63	5	)	)	PUNCT
cana-328	63	6	)	)	PUNCT
cana-328	64	1	−	−	ADP
cana-328	65	1	𝑝(𝜑)(𝛥𝑦(𝜑	𝑝(𝜑)(𝛥𝑦(𝜑	NOUN
cana-328	66	1	+	+	CCONJ
cana-328	66	2	1))𝜅.	1))𝜅.	NUM
cana-328	66	3	hence	hence	ADV
cana-328	66	4	the	the	DET
cana-328	66	5	proof	proof	NOUN
cana-328	66	6	.	.	PUNCT
cana-328	67	1	corollary	corollary	ADJ
cana-328	67	2	2.2	2.2	NUM
cana-328	67	3	.	.	PUNCT
cana-328	68	1	assume	assume	VERB
cana-328	68	2	that	that	SCONJ
cana-328	68	3	the	the	DET
cana-328	68	4	solution	solution	NOUN
cana-328	68	5	{	{	PUNCT
cana-328	68	6	𝑧(𝜑	𝑧(𝜑	NOUN
cana-328	68	7	)	)	PUNCT
cana-328	68	8	}	}	PUNCT
cana-328	68	9	of	of	ADP
cana-328	68	10	(	(	PUNCT
cana-328	68	11	2	2	X
cana-328	68	12	)	)	PUNCT
cana-328	68	13	is	be	AUX
cana-328	68	14	positive	positive	ADJ
cana-328	68	15	.	.	PUNCT
cana-328	69	1	then	then	ADV
cana-328	69	2	the	the	DET
cana-328	69	3	semi	semi	ADJ
cana-328	69	4	-	-	ADJ
cana-328	69	5	canonical	canonical	ADJ
cana-328	69	6	difference	difference	NOUN
cana-328	69	7	equation	equation	NOUN
cana-328	69	8	(	(	PUNCT
cana-328	69	9	1	1	X
cana-328	69	10	)	)	PUNCT
cana-328	69	11	is	be	AUX
cana-328	69	12	in	in	ADP
cana-328	69	13	equivalent	equivalent	ADJ
cana-328	69	14	canonical	canonical	ADJ
cana-328	69	15	form	form	NOUN
cana-328	69	16	𝛥	𝛥	PROPN
cana-328	69	17	(	(	PUNCT
cana-328	69	18	𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑	𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑	ADJ
cana-328	69	19	)	)	PUNCT
cana-328	69	20	)	)	PUNCT
cana-328	69	21	𝜅	𝜅	X
cana-328	69	22	)	)	PUNCT
cana-328	69	23	)	)	PUNCT
cana-328	70	1	+	+	PROPN
cana-328	70	2	𝑄(𝜑)𝑓(𝑦(𝜎(𝜑	𝑄(𝜑)𝑓(𝑦(𝜎(𝜑	NOUN
cana-328	70	3	)	)	PUNCT
cana-328	70	4	)	)	PUNCT
cana-328	70	5	)	)	PUNCT
cana-328	71	1	=	=	PUNCT
cana-328	71	2	0	0	X
cana-328	71	3	.	.	PUNCT
cana-328	71	4	(	(	PUNCT
cana-328	71	5	4	4	X
cana-328	71	6	)	)	PUNCT
cana-328	71	7	lemma	lemma	PROPN
cana-328	71	8	2.3	2.3	NUM
cana-328	71	9	.	.	PUNCT
cana-328	72	1	(	(	PUNCT
cana-328	72	2	see	see	VERB
cana-328	72	3	[	[	X
cana-328	72	4	2	2	NUM
cana-328	72	5	]	]	PUNCT
cana-328	72	6	)	)	PUNCT
cana-328	72	7	if	if	SCONJ
cana-328	72	8	𝑎(𝜑	𝑎(𝜑	NOUN
cana-328	72	9	)	)	PUNCT
cana-328	72	10	≥	≥	NOUN
cana-328	72	11	1,∑	1,∑	NUM
cana-328	72	12	𝑝(𝜑	𝑝(𝜑	NUM
cana-328	72	13	)	)	PUNCT
cana-328	72	14	𝑏(𝜑+1	𝑏(𝜑+1	PROPN
cana-328	72	15	)	)	PUNCT
cana-328	72	16	∞	∞	NUM
cana-328	72	17	𝜑=𝜑0	𝜑=𝜑0	NUM
cana-328	72	18	  	  	SPACE
cana-328	72	19	<	<	X
cana-328	72	20	∞	∞	PROPN
cana-328	72	21	,	,	PUNCT
cana-328	72	22	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
cana-328	72	23	 	 	SPACE
cana-328	72	24	𝑠𝑢𝑝𝜑→∞	𝑠𝑢𝑝𝜑→∞	PROPN
cana-328	72	25	 	 	SPACE
cana-328	72	26	𝑛	𝑛	PROPN
cana-328	72	27	∑	∑	PROPN
cana-328	72	28	𝑝(𝑠	𝑝(𝑠	PROPN
cana-328	72	29	)	)	PUNCT
cana-328	72	30	𝑏(𝑠+1	𝑏(𝑠+1	NUM
cana-328	72	31	)	)	PUNCT
cana-328	72	32	∞	∞	NUM
cana-328	72	33	𝑠=𝜑	𝑠=𝜑	PUNCT
cana-328	72	34	  	  	SPACE
cana-328	72	35	<	<	X
cana-328	72	36	1	1	NUM
cana-328	72	37	4	4	NUM
cana-328	72	38	,	,	PUNCT
cana-328	72	39	(	(	PUNCT
cana-328	72	40	5	5	NUM
cana-328	72	41	)	)	PUNCT
cana-328	72	42	then	then	ADV
cana-328	72	43	the	the	DET
cana-328	72	44	solution	solution	NOUN
cana-328	72	45	of	of	ADP
cana-328	72	46	equation	equation	NOUN
cana-328	72	47	(	(	PUNCT
cana-328	72	48	2	2	X
cana-328	72	49	)	)	PUNCT
cana-328	72	50	is	be	AUX
cana-328	72	51	positive	positive	ADJ
cana-328	72	52	.	.	PUNCT
cana-328	73	1	communications	communication	NOUN
cana-328	73	2	on	on	ADP
cana-328	73	3	applied	apply	VERB
cana-328	73	4	nonlinear	nonlinear	ADJ
cana-328	73	5	analysis	analysis	NOUN
cana-328	73	6	issn	issn	NOUN
cana-328	73	7	:	:	PUNCT
cana-328	73	8	1074	1074	NUM
cana-328	73	9	-	-	PUNCT
cana-328	73	10	133x	133x	NUM
cana-328	73	11	vol	vol	NOUN
cana-328	73	12	31	31	NUM
cana-328	73	13	no	no	NOUN
cana-328	73	14	.	.	NOUN
cana-328	73	15	1	1	NUM
cana-328	73	16	(	(	PUNCT
cana-328	73	17	2024	2024	NUM
cana-328	73	18	)	)	PUNCT
cana-328	73	19	92	92	NUM
cana-328	73	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-328	73	21	lemma	lemma	PROPN
cana-328	73	22	2.4	2.4	NUM
cana-328	73	23	.	.	PUNCT
cana-328	74	1	(	(	PUNCT
cana-328	74	2	see	see	VERB
cana-328	74	3	[	[	X
cana-328	74	4	2	2	NUM
cana-328	74	5	]	]	PUNCT
cana-328	74	6	)	)	PUNCT
cana-328	75	1	assume	assume	VERB
cana-328	75	2	that	that	SCONJ
cana-328	75	3	(	(	PUNCT
cana-328	75	4	5	5	X
cana-328	75	5	)	)	PUNCT
cana-328	75	6	holds	hold	VERB
cana-328	75	7	.	.	PUNCT
cana-328	76	1	then	then	ADV
cana-328	76	2	∃	∃	PROPN
cana-328	76	3	a	a	DET
cana-328	76	4	non	non	ADJ
cana-328	76	5	-	-	ADJ
cana-328	76	6	oscillatory	oscillatory	ADJ
cana-328	76	7	solution	solution	NOUN
cana-328	76	8	to	to	ADP
cana-328	76	9	the	the	DET
cana-328	76	10	equation	equation	NOUN
cana-328	76	11	(	(	PUNCT
cana-328	76	12	2	2	X
cana-328	76	13	)	)	PUNCT
cana-328	76	14	fulfilling	fulfil	VERB
cana-328	76	15	∑	∑	PROPN
cana-328	76	16	1	1	NUM
cana-328	76	17	𝑎(𝜑)𝑧(𝜑)𝑧(𝜑+1	𝑎(𝜑)𝑧(𝜑)𝑧(𝜑+1	NUM
cana-328	76	18	)	)	PUNCT
cana-328	76	19	∞	∞	PROPN
cana-328	76	20	𝜑=𝑛0	𝜑=𝑛0	PROPN
cana-328	76	21	  	  	SPACE
cana-328	76	22	=	=	SYM
cana-328	76	23	∞.	∞.	PROPN
cana-328	76	24	(	(	PUNCT
cana-328	76	25	6	6	NUM
cana-328	76	26	)	)	PUNCT
cana-328	76	27	remark	remark	NOUN
cana-328	76	28	:	:	PUNCT
cana-328	76	29	from	from	ADP
cana-328	76	30	(	(	PUNCT
cana-328	76	31	3	3	NUM
cana-328	76	32	)	)	PUNCT
cana-328	76	33	∑	∑	ADV
cana-328	76	34	𝑧(𝑠	𝑧(𝑠	PROPN
cana-328	76	35	)	)	PUNCT
cana-328	76	36	𝑏(𝑠	𝑏(𝑠	PROPN
cana-328	76	37	)	)	PUNCT
cana-328	76	38	∞	∞	PROPN
cana-328	77	1	𝑠=𝜑0	𝑠=𝜑0	NUM
cana-328	77	2	  	  	SPACE
cana-328	77	3	=	=	SYM
cana-328	77	4	∞	∞	PROPN
cana-328	77	5	,	,	PUNCT
cana-328	77	6	(	(	PUNCT
cana-328	77	7	7	7	NUM
cana-328	77	8	)	)	PUNCT
cana-328	77	9	where	where	SCONJ
cana-328	77	10	{	{	PUNCT
cana-328	77	11	𝑦(𝜑	𝑦(𝜑	NOUN
cana-328	77	12	)	)	PUNCT
cana-328	77	13	}	}	PUNCT
cana-328	77	14	is	be	AUX
cana-328	77	15	a	a	DET
cana-328	77	16	positive	positive	ADJ
cana-328	77	17	solution	solution	NOUN
cana-328	77	18	.	.	PUNCT
cana-328	78	1	in	in	ADP
cana-328	78	2	the	the	DET
cana-328	78	3	continuation	continuation	NOUN
cana-328	78	4	,	,	PUNCT
cana-328	78	5	we	we	PRON
cana-328	78	6	examine	examine	VERB
cana-328	78	7	the	the	DET
cana-328	78	8	positive	positive	ADJ
cana-328	78	9	solution	solution	NOUN
cana-328	78	10	of	of	ADP
cana-328	78	11	(	(	PUNCT
cana-328	78	12	4	4	NUM
cana-328	78	13	)	)	PUNCT
cana-328	78	14	without	without	ADP
cana-328	78	15	losing	lose	VERB
cana-328	78	16	generality	generality	NOUN
cana-328	78	17	.	.	PUNCT
cana-328	79	1	so	so	ADV
cana-328	79	2	,	,	PUNCT
cana-328	79	3	we	we	PRON
cana-328	79	4	introduce	introduce	VERB
cana-328	79	5	the	the	DET
cana-328	79	6	following	follow	VERB
cana-328	79	7	classes	class	NOUN
cana-328	79	8	:	:	PUNCT
cana-328	79	9	𝑦(𝜑	𝑦(𝜑	NOUN
cana-328	79	10	)	)	PUNCT
cana-328	79	11	∈	∈	PROPN
cana-328	79	12	𝑁0	𝑁0	VERB
cana-328	79	13	:	:	PUNCT
cana-328	79	14	𝑦(𝜑	𝑦(𝜑	ADJ
cana-328	79	15	)	)	PUNCT
cana-328	79	16	)	)	PUNCT
cana-328	79	17	>	>	X
cana-328	79	18	0	0	NUM
cana-328	79	19	,	,	PUNCT
cana-328	79	20	𝑑(𝜑)𝛥(𝑦(𝜑	𝑑(𝜑)𝛥(𝑦(𝜑	PROPN
cana-328	79	21	)	)	PUNCT
cana-328	79	22	)	)	PUNCT
cana-328	80	1	𝜅	𝜅	ADP
cana-328	80	2	<	<	X
cana-328	80	3	0	0	NUM
cana-328	80	4	,	,	PUNCT
cana-328	80	5	𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥(𝑦(𝜑))𝜅	𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥(𝑦(𝜑))𝜅	NOUN
cana-328	80	6	)	)	PUNCT
cana-328	80	7	)	)	PUNCT
cana-328	80	8	>	>	X
cana-328	81	1	0	0	NUM
cana-328	81	2	,	,	PUNCT
cana-328	81	3	𝛥	𝛥	PROPN
cana-328	81	4	(	(	PUNCT
cana-328	81	5	𝑐(𝜑)𝛥	𝑐(𝜑)𝛥	ADJ
cana-328	81	6	(	(	PUNCT
cana-328	81	7	𝑑(𝜑)(𝛥(𝑦(𝜑	𝑑(𝜑)(𝛥(𝑦(𝜑	PROPN
cana-328	81	8	)	)	PUNCT
cana-328	81	9	)	)	PUNCT
cana-328	81	10	𝜅	𝜅	X
cana-328	81	11	)	)	PUNCT
cana-328	81	12	)	)	PUNCT
cana-328	81	13	)	)	PUNCT
cana-328	82	1	<	<	X
cana-328	82	2	0	0	PUNCT
cana-328	82	3	𝑦(𝜑	𝑦(𝜑	NOUN
cana-328	82	4	)	)	PUNCT
cana-328	82	5	∈	∈	PROPN
cana-328	82	6	𝑁1	𝑁1	VERB
cana-328	82	7	:	:	PUNCT
cana-328	82	8	𝑦(𝜑	𝑦(𝜑	NUM
cana-328	82	9	)	)	PUNCT
cana-328	82	10	)	)	PUNCT
cana-328	83	1	>	>	X
cana-328	83	2	0	0	NUM
cana-328	83	3	,	,	PUNCT
cana-328	83	4	𝑑(𝜑)𝛥(𝑦(𝜑	𝑑(𝜑)𝛥(𝑦(𝜑	PROPN
cana-328	83	5	)	)	PUNCT
cana-328	83	6	)	)	PUNCT
cana-328	84	1	𝜅	𝜅	ADP
cana-328	84	2	>	>	X
cana-328	84	3	0	0	NUM
cana-328	84	4	,	,	PUNCT
cana-328	84	5	𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥(𝑦(𝜑))𝜅	𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥(𝑦(𝜑))𝜅	NOUN
cana-328	84	6	)	)	PUNCT
cana-328	84	7	)	)	PUNCT
cana-328	84	8	>	>	X
cana-328	85	1	0	0	NUM
cana-328	85	2	,	,	PUNCT
cana-328	85	3	𝛥	𝛥	PROPN
cana-328	85	4	(	(	PUNCT
cana-328	85	5	𝑐(𝜑)𝛥	𝑐(𝜑)𝛥	ADJ
cana-328	85	6	(	(	PUNCT
cana-328	85	7	𝑑(𝜑)(𝛥(𝑦(𝜑	𝑑(𝜑)(𝛥(𝑦(𝜑	PROPN
cana-328	85	8	)	)	PUNCT
cana-328	85	9	)	)	PUNCT
cana-328	85	10	𝜅	𝜅	X
cana-328	85	11	)	)	PUNCT
cana-328	85	12	)	)	PUNCT
cana-328	85	13	)	)	PUNCT
cana-328	86	1	<	<	X
cana-328	86	2	0	0	X
cana-328	86	3	.	.	PUNCT
cana-328	87	1	for	for	ADP
cana-328	87	2	all	all	DET
cana-328	87	3	𝜑	𝜑	PRON
cana-328	87	4	≥	≥	NOUN
cana-328	87	5	𝜑2	𝜑2	VERB
cana-328	87	6	≥	≥	NOUN
cana-328	87	7	𝜑1	𝜑1	VERB
cana-328	87	8	.	.	PUNCT
cana-328	88	1	lemma	lemma	PROPN
cana-328	88	2	2.5	2.5	NUM
cana-328	88	3	.	.	PUNCT
cana-328	88	4	suppose	suppose	VERB
cana-328	88	5	that	that	SCONJ
cana-328	88	6	{	{	PUNCT
cana-328	88	7	𝑦(𝜑	𝑦(𝜑	NOUN
cana-328	88	8	)	)	PUNCT
cana-328	88	9	}	}	PUNCT
cana-328	88	10	is	be	AUX
cana-328	88	11	a	a	DET
cana-328	88	12	positive	positive	ADJ
cana-328	88	13	solution	solution	NOUN
cana-328	88	14	of	of	ADP
cana-328	88	15	(	(	PUNCT
cana-328	88	16	4	4	NUM
cana-328	88	17	)	)	PUNCT
cana-328	88	18	which	which	PRON
cana-328	88	19	satisfies	satisfy	VERB
cana-328	88	20	case	case	NOUN
cana-328	88	21	𝑁1	𝑁1	PROPN
cana-328	88	22	and	and	CCONJ
cana-328	88	23	(	(	PUNCT
cana-328	88	24	∑	∑	PROPN
cana-328	88	25	1	1	NUM
cana-328	88	26	𝑐(𝑠	𝑐(𝑠	NOUN
cana-328	88	27	)	)	PUNCT
cana-328	88	28	𝜑−1	𝜑−1	PROPN
cana-328	88	29	𝑠=𝜑3	𝑠=𝜑3	PROPN
cana-328	88	30	  	  	SPACE
cana-328	88	31	∑	∑	PROPN
cana-328	88	32	 	 	SPACE
cana-328	88	33	𝑄(𝑡)𝑓(𝐷(𝜎(𝑡)))∞	𝑄(𝑡)𝑓(𝐷(𝜎(𝑡)))∞	PROPN
cana-328	88	34	𝑡=𝑠	𝑡=𝑠	PROPN
cana-328	88	35	 	 	SPACE
cana-328	88	36	)	)	PUNCT
cana-328	88	37	1	1	NUM
cana-328	88	38	𝜅	𝜅	X
cana-328	88	39	=	=	X
cana-328	88	40	∞.	∞.	PROPN
cana-328	88	41	(	(	PUNCT
cana-328	88	42	8)	8)	NUM
cana-328	88	43	then	then	ADV
cana-328	88	44	(	(	PUNCT
cana-328	88	45	i	i	NOUN
cana-328	88	46	)	)	PUNCT
cana-328	88	47	{	{	PUNCT
cana-328	88	48	𝑦(𝜑	𝑦(𝜑	NOUN
cana-328	88	49	)	)	PUNCT
cana-328	88	50	𝐷(𝜑	𝐷(𝜑	NOUN
cana-328	88	51	)	)	PUNCT
cana-328	88	52	}	}	PUNCT
cana-328	88	53	is	be	AUX
cana-328	88	54	increasing	increase	VERB
cana-328	88	55	∀	∀	X
cana-328	88	56	𝜑	𝜑	NOUN
cana-328	88	57	≥	≥	NOUN
cana-328	88	58	𝑁	𝑁	PROPN
cana-328	88	59	,	,	PUNCT
cana-328	88	60	(	(	PUNCT
cana-328	88	61	ii	ii	NOUN
cana-328	88	62	)	)	PUNCT
cana-328	88	63	{	{	PUNCT
cana-328	88	64	𝑦(𝜑	𝑦(𝜑	NOUN
cana-328	88	65	)	)	PUNCT
cana-328	88	66	𝐸(𝜑	𝐸(𝜑	NOUN
cana-328	88	67	)	)	PUNCT
cana-328	88	68	}	}	PUNCT
cana-328	88	69	is	be	AUX
cana-328	88	70	decreasing	decrease	VERB
cana-328	88	71	∀	∀	NOUN
cana-328	88	72	𝜑	𝜑	NOUN
cana-328	88	73	≥	≥	NOUN
cana-328	88	74	𝑁	𝑁	PROPN
cana-328	88	75	,	,	PUNCT
cana-328	88	76	(	(	PUNCT
cana-328	88	77	iii	iii	NOUN
cana-328	88	78	)	)	PUNCT
cana-328	88	79	{	{	PUNCT
cana-328	88	80	(	(	PUNCT
cana-328	88	81	𝑑(𝜑)(𝛥𝑦(𝜑))𝜅	𝑑(𝜑)(𝛥𝑦(𝜑))𝜅	NOUN
cana-328	88	82	𝐶(𝜑	𝐶(𝜑	NOUN
cana-328	88	83	)	)	PUNCT
cana-328	88	84	)	)	PUNCT
cana-328	88	85	}	}	PUNCT
cana-328	88	86	is	be	AUX
cana-328	88	87	decreasing	decrease	VERB
cana-328	88	88	∀	∀	NOUN
cana-328	88	89	𝜑	𝜑	ADP
cana-328	88	90	≥	≥	NOUN
cana-328	88	91	𝑁.	𝑁.	PROPN
cana-328	88	92	proof	proof	NOUN
cana-328	88	93	.	.	PUNCT
cana-328	89	1	let	let	AUX
cana-328	89	2	{	{	PUNCT
cana-328	89	3	𝑦(𝜑	𝑦(𝜑	VERB
cana-328	89	4	)	)	PUNCT
cana-328	89	5	}	}	PUNCT
cana-328	89	6	be	be	AUX
cana-328	89	7	a	a	DET
cana-328	89	8	positive	positive	ADJ
cana-328	89	9	solution	solution	NOUN
cana-328	89	10	to	to	ADP
cana-328	89	11	(	(	PUNCT
cana-328	89	12	4	4	X
cana-328	89	13	)	)	PUNCT
cana-328	89	14	that	that	PRON
cana-328	89	15	fulfills	fulfill	VERB
cana-328	89	16	{	{	PUNCT
cana-328	89	17	𝑦(𝜑	𝑦(𝜑	NOUN
cana-328	89	18	)	)	PUNCT
cana-328	89	19	}	}	PUNCT
cana-328	89	20	∈	∈	PROPN
cana-328	89	21	𝑁1	𝑁1	NOUN
cana-328	89	22	for	for	ADP
cana-328	89	23	all	all	PRON
cana-328	89	24	𝜑	𝜑	PRON
cana-328	89	25	∈	∈	NOUN
cana-328	89	26	𝑁.	𝑁.	PROPN
cana-328	89	27	since	since	SCONJ
cana-328	89	28	𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑))𝜅	𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑))𝜅	NOUN
cana-328	89	29	)	)	PUNCT
cana-328	89	30	is	be	AUX
cana-328	89	31	decreasing	decrease	VERB
cana-328	89	32	,	,	PUNCT
cana-328	89	33	we	we	PRON
cana-328	89	34	get	get	VERB
cana-328	89	35	𝑑(𝜑)(𝛥𝑦(𝜑))𝜅	𝑑(𝜑)(𝛥𝑦(𝜑))𝜅	NOUN
cana-328	89	36	=	=	SYM
cana-328	89	37	𝑑(𝜑1)(𝛥𝑦(𝜑1	𝑑(𝜑1)(𝛥𝑦(𝜑1	NOUN
cana-328	89	38	)	)	PUNCT
cana-328	89	39	)	)	PUNCT
cana-328	90	1	𝜅	𝜅	X
cana-328	90	2	+	+	CCONJ
cana-328	90	3	∑	∑	PUNCT
cana-328	90	4	𝑐(𝑖)𝛥(𝑑(𝑖)(𝛥𝑦(𝑖))𝜅	𝑐(𝑖)𝛥(𝑑(𝑖)(𝛥𝑦(𝑖))𝜅	PROPN
cana-328	90	5	)	)	PUNCT
cana-328	90	6	𝑐(𝑖	𝑐(𝑖	PROPN
cana-328	90	7	)	)	PUNCT
cana-328	90	8	𝜑−1	𝜑−1	PROPN
cana-328	90	9	𝑖=𝜑1	𝑖=𝜑1	PROPN
cana-328	90	10	   	   	SPACE
cana-328	90	11	≥	≥	NOUN
cana-328	90	12	𝐶(𝜑)𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑))𝜅	𝐶(𝜑)𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑))𝜅	NOUN
cana-328	90	13	)	)	PUNCT
cana-328	90	14	.	.	PUNCT
cana-328	91	1	this	this	PRON
cana-328	91	2	gives	give	VERB
cana-328	91	3	𝛥	𝛥	PROPN
cana-328	91	4	(	(	PUNCT
cana-328	91	5	𝑑(𝜑)(𝛥𝑦(𝜑))𝜅	𝑑(𝜑)(𝛥𝑦(𝜑))𝜅	NOUN
cana-328	91	6	𝐶(𝜑	𝐶(𝜑	NOUN
cana-328	91	7	)	)	PUNCT
cana-328	91	8	)	)	PUNCT
cana-328	92	1	=	=	SYM
cana-328	92	2	𝐶(𝜑)𝑐(𝜑)𝛥(𝑑(𝜑(𝛥𝑦(𝜑	𝐶(𝜑)𝑐(𝜑)𝛥(𝑑(𝜑(𝛥𝑦(𝜑	NOUN
cana-328	92	3	)	)	PUNCT
cana-328	92	4	)	)	PUNCT
cana-328	93	1	𝜅	𝜅	X
cana-328	93	2	)	)	PUNCT
cana-328	93	3	)	)	PUNCT
cana-328	93	4	−𝑑(𝜑)(𝛥𝑦(𝜑	−𝑑(𝜑)(𝛥𝑦(𝜑	PROPN
cana-328	93	5	)	)	PUNCT
cana-328	93	6	)	)	PUNCT
cana-328	94	1	𝜅	𝜅	PRON
cana-328	94	2	𝑐(𝜑)𝐶(𝜑)𝐶(𝜑+1	𝑐(𝜑)𝐶(𝜑)𝐶(𝜑+1	NOUN
cana-328	94	3	)	)	PUNCT
cana-328	94	4	≤	≤	NOUN
cana-328	94	5	0	0	NUM
cana-328	94	6	.	.	PUNCT
cana-328	95	1	so	so	ADV
cana-328	95	2	(	(	PUNCT
cana-328	95	3	𝑑(𝜑)(𝛥𝑦(𝜑))𝜅	𝑑(𝜑)(𝛥𝑦(𝜑))𝜅	NOUN
cana-328	95	4	𝐶(𝜑	𝐶(𝜑	NOUN
cana-328	95	5	)	)	PUNCT
cana-328	95	6	)	)	PUNCT
cana-328	95	7	is	be	AUX
cana-328	95	8	decreasing	decrease	VERB
cana-328	95	9	and	and	CCONJ
cana-328	95	10	further	far	ADV
cana-328	95	11	,	,	PUNCT
cana-328	95	12	this	this	DET
cana-328	95	13	fact	fact	NOUN
cana-328	95	14	yields	yield	VERB
cana-328	95	15	communications	communication	NOUN
cana-328	95	16	on	on	ADP
cana-328	95	17	applied	apply	VERB
cana-328	95	18	nonlinear	nonlinear	ADJ
cana-328	95	19	analysis	analysis	NOUN
cana-328	95	20	issn	issn	NOUN
cana-328	95	21	:	:	PUNCT
cana-328	95	22	1074	1074	NUM
cana-328	95	23	-	-	PUNCT
cana-328	95	24	133x	133x	NUM
cana-328	95	25	vol	vol	NOUN
cana-328	95	26	31	31	NUM
cana-328	95	27	no	no	NOUN
cana-328	95	28	.	.	NOUN
cana-328	95	29	1	1	NUM
cana-328	95	30	(	(	PUNCT
cana-328	95	31	2024	2024	NUM
cana-328	95	32	)	)	PUNCT
cana-328	95	33	93	93	NUM
cana-328	95	34	https://internationalpubls.com	https://internationalpubls.com	X
cana-328	95	35	𝑦(𝜑	𝑦(𝜑	NOUN
cana-328	95	36	)	)	PUNCT
cana-328	95	37	=	=	SYM
cana-328	95	38	𝑦(𝜑1	𝑦(𝜑1	ADV
cana-328	95	39	)	)	PUNCT
cana-328	96	1	+	+	CCONJ
cana-328	96	2	(	(	PUNCT
cana-328	96	3	∑	∑	PART
cana-328	96	4	𝐶	𝐶	PROPN
cana-328	96	5	1	1	NUM
cana-328	96	6	𝜅(𝑠)𝑑	𝜅(𝑠)𝑑	ADP
cana-328	96	7	1	1	NUM
cana-328	96	8	𝜅(𝑠)(𝛥𝑦(𝑠	𝜅(𝑠)(𝛥𝑦(𝑠	NUM
cana-328	96	9	)	)	PUNCT
cana-328	96	10	)	)	PUNCT
cana-328	96	11	𝐶	𝐶	PROPN
cana-328	96	12	1	1	NUM
cana-328	96	13	𝜅(𝑠)𝑑(𝑠	𝜅(𝑠)𝑑(𝑠	NOUN
cana-328	96	14	)	)	PUNCT
cana-328	96	15	𝜑−1	𝜑−1	PROPN
cana-328	96	16	𝑠=𝜑1	𝑠=𝜑1	PROPN
cana-328	96	17	  	  	SPACE
cana-328	96	18	)	)	PUNCT
cana-328	96	19	≥	≥	NOUN
cana-328	96	20	𝐸(𝜑	𝐸(𝜑	NOUN
cana-328	96	21	)	)	PUNCT
cana-328	96	22	(	(	PUNCT
cana-328	96	23	𝑑	𝑑	PROPN
cana-328	96	24	1	1	NUM
cana-328	96	25	𝜅(𝜑)(𝛥𝑦(𝜑	𝜅(𝜑)(𝛥𝑦(𝜑	NOUN
cana-328	96	26	)	)	PUNCT
cana-328	96	27	)	)	PUNCT
cana-328	96	28	𝐶	𝐶	PROPN
cana-328	96	29	1	1	NUM
cana-328	96	30	𝜅(𝜑	𝜅(𝜑	PROPN
cana-328	96	31	)	)	PUNCT
cana-328	96	32	)	)	PUNCT
cana-328	96	33	.	.	PUNCT
cana-328	97	1	(	(	PUNCT
cana-328	97	2	9	9	X
cana-328	97	3	)	)	PUNCT
cana-328	97	4	hence	hence	ADV
cana-328	97	5	𝛥	𝛥	PROPN
cana-328	97	6	(	(	PUNCT
cana-328	97	7	𝑦(𝜑	𝑦(𝜑	PROPN
cana-328	97	8	)	)	PUNCT
cana-328	97	9	𝐸(𝜑	𝐸(𝜑	NOUN
cana-328	97	10	)	)	PUNCT
cana-328	97	11	)	)	PUNCT
cana-328	98	1	=	=	PUNCT
cana-328	99	1	𝐸(𝜑)𝑑	𝐸(𝜑)𝑑	ADJ
cana-328	99	2	1	1	NUM
cana-328	99	3	𝜅(𝜑)𝛥𝑦(𝜑)−𝑦(𝜑)𝐶	𝜅(𝜑)𝛥𝑦(𝜑)−𝑦(𝜑)𝐶	NOUN
cana-328	99	4	1	1	NUM
cana-328	99	5	𝜅(𝜑	𝜅(𝜑	PROPN
cana-328	99	6	)	)	PUNCT
cana-328	99	7	𝑑	𝑑	PROPN
cana-328	99	8	1	1	NUM
cana-328	99	9	𝜅(𝜑)𝐸(𝜑)𝐸(𝜑+1	𝜅(𝜑)𝐸(𝜑)𝐸(𝜑+1	NOUN
cana-328	99	10	)	)	PUNCT
cana-328	99	11	≤	≤	ADV
cana-328	99	12	0	0	NUM
cana-328	99	13	,	,	PUNCT
cana-328	99	14	which	which	PRON
cana-328	99	15	gives	give	VERB
cana-328	99	16	𝑦(𝜑	𝑦(𝜑	NOUN
cana-328	99	17	)	)	PUNCT
cana-328	99	18	𝐸(𝜑	𝐸(𝜑	NUM
cana-328	99	19	)	)	PUNCT
cana-328	99	20	is	be	AUX
cana-328	99	21	decreasing	decrease	VERB
cana-328	99	22	.	.	PUNCT
cana-328	100	1	next	next	ADV
cana-328	100	2	,	,	PUNCT
cana-328	100	3	since	since	SCONJ
cana-328	100	4	{	{	PUNCT
cana-328	100	5	𝑑	𝑑	PROPN
cana-328	100	6	1	1	NUM
cana-328	100	7	𝜅(𝜑)𝛥𝑦(𝜑	𝜅(𝜑)𝛥𝑦(𝜑	NOUN
cana-328	100	8	)	)	PUNCT
cana-328	100	9	}	}	PUNCT
cana-328	100	10	is	be	AUX
cana-328	100	11	increasing	increase	VERB
cana-328	100	12	for	for	ADP
cana-328	100	13	all	all	PRON
cana-328	100	14	𝜑	𝜑	DET
cana-328	100	15	≥	≥	NOUN
cana-328	100	16	𝑁	𝑁	PROPN
cana-328	100	17	,	,	PUNCT
cana-328	100	18	it	it	PRON
cana-328	100	19	is	be	AUX
cana-328	100	20	apparent	apparent	ADJ
cana-328	100	21	∀	∀	NOUN
cana-328	100	22	𝜑	𝜑	NOUN
cana-328	100	23	≥	≥	NOUN
cana-328	100	24	𝜑2	𝜑2	NOUN
cana-328	100	25	≥	≥	NOUN
cana-328	100	26	𝜑1	𝜑1	VERB
cana-328	100	27	𝑦(𝜑	𝑦(𝜑	NOUN
cana-328	100	28	)	)	PUNCT
cana-328	100	29	=	=	SYM
cana-328	100	30	𝑦(𝜑2	𝑦(𝜑2	NOUN
cana-328	100	31	)	)	PUNCT
cana-328	101	1	+	+	CCONJ
cana-328	101	2	∑	∑	PROPN
cana-328	101	3	𝑑	𝑑	PROPN
cana-328	101	4	1	1	NUM
cana-328	101	5	𝜅(𝑠	𝜅(𝑠	NUM
cana-328	101	6	)	)	PUNCT
cana-328	101	7	𝑑	𝑑	PART
cana-328	101	8	1	1	NUM
cana-328	101	9	𝜅(𝑠	𝜅(𝑠	NUM
cana-328	101	10	)	)	PUNCT
cana-328	101	11	𝛥𝑦(𝑠	𝛥𝑦(𝑠	NUM
cana-328	101	12	)	)	PUNCT
cana-328	101	13	𝜑−1	𝜑−1	PROPN
cana-328	101	14	𝑠=𝜑2	𝑠=𝜑2	NOUN
cana-328	101	15	   	   	SPACE
cana-328	101	16	≤	≤	NUM
cana-328	101	17	𝑦(𝜑2	𝑦(𝜑2	NOUN
cana-328	101	18	)	)	PUNCT
cana-328	102	1	+	+	CCONJ
cana-328	102	2	𝑑	𝑑	PROPN
cana-328	102	3	1	1	NUM
cana-328	102	4	𝜅(𝜑)𝛥𝑦(𝜑)∑	𝜅(𝜑)𝛥𝑦(𝜑)∑	NOUN
cana-328	102	5	1	1	NUM
cana-328	102	6	𝑑	𝑑	NOUN
cana-328	102	7	1	1	NUM
cana-328	102	8	𝜅(𝑠	𝜅(𝑠	NUM
cana-328	102	9	)	)	PUNCT
cana-328	102	10	𝜑−1	𝜑−1	PROPN
cana-328	102	11	𝑠=𝜑2	𝑠=𝜑2	NOUN
cana-328	102	12	   	   	SPACE
cana-328	102	13	=	=	SYM
cana-328	102	14	𝑦(𝜑2	𝑦(𝜑2	NOUN
cana-328	102	15	)	)	PUNCT
cana-328	102	16	−	−	PROPN
cana-328	102	17	𝑑	𝑑	PROPN
cana-328	102	18	1	1	NUM
cana-328	102	19	𝜅(𝜑)𝛥𝑦(𝜑)∑	𝜅(𝜑)𝛥𝑦(𝜑)∑	NUM
cana-328	102	20	1	1	NUM
cana-328	102	21	𝑑	𝑑	NOUN
cana-328	102	22	1	1	NUM
cana-328	102	23	𝜅(𝑠	𝜅(𝑠	NUM
cana-328	102	24	)	)	PUNCT
cana-328	102	25	𝜑1−1	𝜑1−1	PRON
cana-328	102	26	𝑠=𝜑2	𝑠=𝜑2	VERB
cana-328	102	27	  	  	SPACE
cana-328	103	1	+	+	CCONJ
cana-328	103	2	𝑑	𝑑	PROPN
cana-328	103	3	1	1	NUM
cana-328	103	4	𝜅(𝜑)𝛥𝑦(𝜑)∑	𝜅(𝜑)𝛥𝑦(𝜑)∑	NOUN
cana-328	103	5	1	1	NUM
cana-328	103	6	𝑑	𝑑	NOUN
cana-328	103	7	1	1	NUM
cana-328	103	8	𝜅(𝑠	𝜅(𝑠	NUM
cana-328	103	9	)	)	PUNCT
cana-328	103	10	𝜑−1	𝜑−1	PROPN
cana-328	103	11	𝑠=𝜑2	𝑠=𝜑2	NOUN
cana-328	103	12	   	   	SPACE
cana-328	103	13	.	.	PUNCT
cana-328	104	1	(	(	PUNCT
cana-328	104	2	10	10	NUM
cana-328	104	3	)	)	PUNCT
cana-328	104	4	we	we	PRON
cana-328	104	5	claim	claim	VERB
cana-328	104	6	that	that	SCONJ
cana-328	104	7	𝑑	𝑑	PROPN
cana-328	104	8	1	1	NUM
cana-328	104	9	𝜅(𝜑)𝛥𝑦(𝜑	𝜅(𝜑)𝛥𝑦(𝜑	NOUN
cana-328	104	10	)	)	PUNCT
cana-328	104	11	→	→	SYM
cana-328	104	12	∞	∞	PROPN
cana-328	104	13	as	as	ADP
cana-328	104	14	𝜑	𝜑	PROPN
cana-328	104	15	→	→	SYM
cana-328	104	16	∞.	∞.	PROPN
cana-328	104	17	suppose	suppose	VERB
cana-328	104	18	𝑑	𝑑	PRON
cana-328	104	19	1	1	NUM
cana-328	104	20	𝜅(𝜑)𝛥𝑦(𝜑	𝜅(𝜑)𝛥𝑦(𝜑	NOUN
cana-328	104	21	)	)	PUNCT
cana-328	104	22	→	→	SYM
cana-328	104	23	2𝑤	2𝑤	NUM
cana-328	104	24	as	as	ADP
cana-328	104	25	𝜑	𝜑	X
cana-328	104	26	→	→	SYM
cana-328	104	27	∞	∞	PROPN
cana-328	104	28	,	,	PUNCT
cana-328	104	29	we	we	PRON
cana-328	104	30	have	have	VERB
cana-328	104	31	𝑑	𝑑	PROPN
cana-328	104	32	1	1	NUM
cana-328	104	33	𝜅(𝜑)𝛥𝑦(𝜑	𝜅(𝜑)𝛥𝑦(𝜑	NOUN
cana-328	104	34	)	)	PUNCT
cana-328	104	35	≥	≥	NOUN
cana-328	104	36	𝑤.	𝑤.	NOUN
cana-328	104	37	this	this	PRON
cana-328	104	38	gives	give	VERB
cana-328	104	39	𝑦(𝜑	𝑦(𝜑	NOUN
cana-328	104	40	)	)	PUNCT
cana-328	104	41	≥	≥	NOUN
cana-328	104	42	𝑤𝐷(𝜑	𝑤𝐷(𝜑	NUM
cana-328	104	43	)	)	PUNCT
cana-328	104	44	,	,	PUNCT
cana-328	104	45	summing	sum	VERB
cana-328	104	46	(	(	PUNCT
cana-328	104	47	4	4	NUM
cana-328	104	48	)	)	PUNCT
cana-328	104	49	from	from	ADP
cana-328	104	50	𝜑	𝜑	PRON
cana-328	104	51	to	to	ADP
cana-328	104	52	∞	∞	PROPN
cana-328	104	53	,	,	PUNCT
cana-328	104	54	we	we	PRON
cana-328	104	55	have	have	VERB
cana-328	104	56	𝛥(𝑑(𝜑)(𝛥𝑦(𝜑	𝛥(𝑑(𝜑)(𝛥𝑦(𝜑	NOUN
cana-328	104	57	)	)	PUNCT
cana-328	104	58	)	)	PUNCT
cana-328	105	1	𝜅	𝜅	X
cana-328	105	2	)	)	PUNCT
cana-328	105	3	≥	≥	NOUN
cana-328	105	4	1	1	NUM
cana-328	105	5	𝑐(𝜑	𝑐(𝜑	NOUN
cana-328	105	6	)	)	PUNCT
cana-328	105	7	𝑓(𝑤)∑	𝑓(𝑤)∑	X
cana-328	105	8	𝑄(𝑠)𝑓	𝑄(𝑠)𝑓	ADJ
cana-328	105	9	(	(	PUNCT
cana-328	105	10	𝐷(𝜎(𝑠)))∞	𝐷(𝜎(𝑠)))∞	NOUN
cana-328	105	11	𝑠=𝜑	𝑠=𝜑	NOUN
cana-328	105	12	 	 	SPACE
cana-328	105	13	.	.	PUNCT
cana-328	106	1	again	again	ADV
cana-328	106	2	summing	sum	VERB
cana-328	106	3	from	from	ADP
cana-328	106	4	𝜑3	𝜑3	PROPN
cana-328	106	5	to	to	ADP
cana-328	106	6	𝜑−	𝜑−	PROPN
cana-328	106	7	1	1	NUM
cana-328	106	8	,	,	PUNCT
cana-328	106	9	we	we	PRON
cana-328	106	10	obtain	obtain	VERB
cana-328	106	11	𝑑	𝑑	PROPN
cana-328	106	12	1	1	NUM
cana-328	106	13	𝜅(𝜑)𝛥𝑦(𝜑	𝜅(𝜑)𝛥𝑦(𝜑	NOUN
cana-328	106	14	)	)	PUNCT
cana-328	106	15	≥	≥	NOUN
cana-328	106	16	𝑓	𝑓	DET
cana-328	106	17	1	1	NUM
cana-328	106	18	𝜅(𝑤	𝜅(𝑤	PROPN
cana-328	106	19	)	)	PUNCT
cana-328	106	20	(	(	PUNCT
cana-328	106	21	∑	∑	ADV
cana-328	106	22	1	1	NUM
cana-328	106	23	𝑐(𝑠	𝑐(𝑠	NOUN
cana-328	106	24	)	)	PUNCT
cana-328	106	25	𝜑−1	𝜑−1	PROPN
cana-328	106	26	𝑠=𝜑3	𝑠=𝜑3	PROPN
cana-328	106	27	  	  	SPACE
cana-328	106	28	∑	∑	PROPN
cana-328	106	29	𝑄(𝑡)𝑓(𝐷(𝜎(𝑡)))∞	𝑄(𝑡)𝑓(𝐷(𝜎(𝑡)))∞	PROPN
cana-328	106	30	𝑡=𝑠	𝑡=𝑠	PROPN
cana-328	106	31	  	  	SPACE
cana-328	106	32	)	)	PUNCT
cana-328	106	33	1	1	NUM
cana-328	106	34	𝜅	𝜅	PROPN
cana-328	106	35	,	,	PUNCT
cana-328	106	36	(	(	PUNCT
cana-328	106	37	11	11	NUM
cana-328	106	38	)	)	PUNCT
cana-328	106	39	for	for	ADP
cana-328	106	40	all	all	DET
cana-328	106	41	𝜑	𝜑	PRON
cana-328	106	42	≥	≥	NOUN
cana-328	106	43	𝜑3	𝜑3	PROPN
cana-328	106	44	.	.	PUNCT
cana-328	107	1	2𝑤	2𝑤	NUM
cana-328	107	2	≥	≥	X
cana-328	107	3	𝑓	𝑓	PRON
cana-328	107	4	1	1	NUM
cana-328	107	5	𝜅(𝑤	𝜅(𝑤	PROPN
cana-328	107	6	)	)	PUNCT
cana-328	107	7	(	(	PUNCT
cana-328	107	8	∑	∑	PROPN
cana-328	107	9	1	1	NUM
cana-328	107	10	𝑎(𝑠	𝑎(𝑠	PROPN
cana-328	107	11	)	)	PUNCT
cana-328	107	12	𝜑−1	𝜑−1	PROPN
cana-328	107	13	𝑠=𝜑3	𝑠=𝜑3	PROPN
cana-328	107	14	  	  	SPACE
cana-328	107	15	∑	∑	PROPN
cana-328	107	16	𝑄(𝑡)𝑓(𝐵(𝜎(𝑡)))∞	𝑄(𝑡)𝑓(𝐵(𝜎(𝑡)))∞	PROPN
cana-328	107	17	𝑡=𝑠	𝑡=𝑠	PROPN
cana-328	107	18	  	  	SPACE
cana-328	107	19	)	)	PUNCT
cana-328	107	20	1	1	NUM
cana-328	107	21	𝜅	𝜅	PROPN
cana-328	107	22	,	,	PUNCT
cana-328	107	23	which	which	PRON
cana-328	107	24	contradicts	contradict	VERB
cana-328	107	25	(	(	PUNCT
cana-328	107	26	8)	8)	NUM
cana-328	107	27	so	so	CCONJ
cana-328	107	28	𝑑	𝑑	NOUN
cana-328	107	29	1	1	NUM
cana-328	107	30	𝜅(𝜑)𝛥𝑦(𝜑	𝜅(𝜑)𝛥𝑦(𝜑	NOUN
cana-328	107	31	)	)	PUNCT
cana-328	107	32	→	→	SYM
cana-328	107	33	∞	∞	PROPN
cana-328	107	34	as	as	ADP
cana-328	107	35	𝜑	𝜑	X
cana-328	107	36	→∞.	→∞.	PROPN
cana-328	107	37	hence	hence	ADV
cana-328	107	38	,	,	PUNCT
cana-328	107	39	(	(	PUNCT
cana-328	107	40	10	10	NUM
cana-328	107	41	)	)	PUNCT
cana-328	107	42	becomes	become	VERB
cana-328	107	43	𝑦(𝜑	𝑦(𝜑	NOUN
cana-328	107	44	)	)	PUNCT
cana-328	107	45	≤	≤	NOUN
cana-328	108	1	𝑑	𝑑	VERB
cana-328	108	2	1	1	NUM
cana-328	108	3	𝜅(𝜑)𝛥𝑦(𝜑)𝐷(𝜑	𝜅(𝜑)𝛥𝑦(𝜑)𝐷(𝜑	NUM
cana-328	108	4	)	)	PUNCT
cana-328	108	5	.	.	PUNCT
cana-328	109	1	since	since	SCONJ
cana-328	109	2	𝛥	𝛥	PROPN
cana-328	109	3	(	(	PUNCT
cana-328	109	4	𝑦(𝜑	𝑦(𝜑	PROPN
cana-328	109	5	)	)	PUNCT
cana-328	109	6	𝐷(𝜑	𝐷(𝜑	NOUN
cana-328	109	7	)	)	PUNCT
cana-328	109	8	)	)	PUNCT
cana-328	110	1	=	=	PUNCT
cana-328	110	2	𝑑	𝑑	PROPN
cana-328	110	3	1	1	NUM
cana-328	110	4	𝜅(𝜑)𝐷(𝜑)𝛥𝑦(𝜑)−𝑦(𝜑	𝜅(𝜑)𝐷(𝜑)𝛥𝑦(𝜑)−𝑦(𝜑	NUM
cana-328	110	5	)	)	PUNCT
cana-328	110	6	𝑑	𝑑	PROPN
cana-328	110	7	1	1	NUM
cana-328	110	8	𝜅(𝜑)𝐷(𝜑)𝐷(𝜑+1	𝜅(𝜑)𝐷(𝜑)𝐷(𝜑+1	PROPN
cana-328	110	9	)	)	PUNCT
cana-328	110	10	)	)	PUNCT
cana-328	110	11	≥	≥	NOUN
cana-328	110	12	0	0	NUM
cana-328	110	13	,	,	PUNCT
cana-328	110	14	𝑦(𝜑	𝑦(𝜑	NOUN
cana-328	110	15	)	)	PUNCT
cana-328	110	16	𝐷(𝜑	𝐷(𝜑	NOUN
cana-328	110	17	)	)	PUNCT
cana-328	110	18	is	be	AUX
cana-328	110	19	increasing	increase	VERB
cana-328	110	20	.	.	PUNCT
cana-328	111	1	this	this	PRON
cana-328	111	2	concludes	conclude	VERB
cana-328	111	3	the	the	DET
cana-328	111	4	proof	proof	NOUN
cana-328	111	5	.	.	PUNCT
cana-328	112	1	theorem	theorem	VERB
cana-328	112	2	2.6	2.6	NUM
cana-328	112	3	.	.	PUNCT
cana-328	113	1	let	let	VERB
cana-328	113	2	(	(	PUNCT
cana-328	113	3	8)	8)	NUM
cana-328	113	4	hold	hold	NOUN
cana-328	113	5	.	.	PUNCT
cana-328	114	1	assume	assume	VERB
cana-328	114	2	that	that	SCONJ
cana-328	114	3	(	(	PUNCT
cana-328	114	4	2	2	X
cana-328	114	5	)	)	PUNCT
cana-328	114	6	has	have	VERB
cana-328	114	7	a	a	DET
cana-328	114	8	positive	positive	ADJ
cana-328	114	9	solution	solution	NOUN
cana-328	114	10	{	{	PUNCT
cana-328	114	11	𝑧(𝜑	𝑧(𝜑	NOUN
cana-328	114	12	)	)	PUNCT
cana-328	114	13	}	}	PUNCT
cana-328	114	14	and	and	CCONJ
cana-328	114	15	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
cana-328	114	16	 	 	SPACE
cana-328	114	17	𝑠𝑢𝑝𝑢→∞	𝑠𝑢𝑝𝑢→∞	PROPN
cana-328	114	18	  	  	SPACE
cana-328	114	19	𝑢𝜅	𝑢𝜅	PROPN
cana-328	114	20	𝑓(𝑢	𝑓(𝑢	PROPN
cana-328	114	21	)	)	PUNCT
cana-328	115	1	=	=	SYM
cana-328	115	2	𝐾	𝐾	PROPN
cana-328	115	3	<	<	X
cana-328	115	4	∞.	∞.	PROPN
cana-328	115	5	(	(	PUNCT
cana-328	115	6	12	12	NUM
cana-328	115	7	)	)	PUNCT
cana-328	115	8	communications	communication	NOUN
cana-328	115	9	on	on	ADP
cana-328	115	10	applied	apply	VERB
cana-328	115	11	nonlinear	nonlinear	ADJ
cana-328	115	12	analysis	analysis	NOUN
cana-328	115	13	issn	issn	NOUN
cana-328	115	14	:	:	PUNCT
cana-328	115	15	1074	1074	NUM
cana-328	115	16	-	-	PUNCT
cana-328	115	17	133x	133x	NUM
cana-328	115	18	vol	vol	NOUN
cana-328	115	19	31	31	NUM
cana-328	115	20	no	no	NOUN
cana-328	115	21	.	.	NOUN
cana-328	115	22	1	1	NUM
cana-328	115	23	(	(	PUNCT
cana-328	115	24	2024	2024	NUM
cana-328	115	25	)	)	PUNCT
cana-328	115	26	94	94	NUM
cana-328	115	27	https://internationalpubls.com	https://internationalpubls.com	X
cana-328	115	28	if	if	SCONJ
cana-328	115	29	𝑙𝑖𝑚	𝑙𝑖𝑚	PROPN
cana-328	115	30	 	 	SPACE
cana-328	115	31	𝑠𝑢𝑝𝜑→∞	𝑠𝑢𝑝𝜑→∞	PROPN
cana-328	115	32	  	  	SPACE
cana-328	115	33	(	(	PUNCT
cana-328	115	34	𝐸𝜅(𝜎(𝜑	𝐸𝜅(𝜎(𝜑	PROPN
cana-328	115	35	)	)	PUNCT
cana-328	115	36	)	)	PUNCT
cana-328	115	37	𝐶𝜅(𝜎(𝜑	𝐶𝜅(𝜎(𝜑	NOUN
cana-328	115	38	)	)	PUNCT
cana-328	115	39	)	)	PUNCT
cana-328	116	1	𝑓	𝑓	X
cana-328	116	2	(	(	PUNCT
cana-328	116	3	1	1	NUM
cana-328	116	4	𝐸(𝜎(𝜑	𝐸(𝜎(𝜑	NOUN
cana-328	116	5	)	)	PUNCT
cana-328	116	6	)	)	PUNCT
cana-328	116	7	)	)	PUNCT
cana-328	116	8	∑	∑	PUNCT
cana-328	116	9	𝐶(𝑠	𝐶(𝑠	PROPN
cana-328	116	10	+	+	NUM
cana-328	116	11	1)𝑄(𝑠)𝑓	1)𝑄(𝑠)𝑓	NUM
cana-328	116	12	(	(	PUNCT
cana-328	116	13	𝐸(𝜎(𝑠	𝐸(𝜎(𝑠	PROPN
cana-328	116	14	)	)	PUNCT
cana-328	116	15	)	)	PUNCT
cana-328	116	16	)	)	PUNCT
cana-328	117	1	𝜎(𝜑)−1	𝜎(𝜑)−1	PROPN
cana-328	117	2	𝑠=𝜑1	𝑠=𝜑1	PROPN
cana-328	118	1	+	+	NOUN
cana-328	118	2	𝐸𝜅(𝜎(𝜑))𝑓	𝐸𝜅(𝜎(𝜑))𝑓	PROPN
cana-328	118	3	(	(	PUNCT
cana-328	118	4	1	1	NUM
cana-328	118	5	𝐸(𝜎(𝜑	𝐸(𝜎(𝜑	NOUN
cana-328	118	6	)	)	PUNCT
cana-328	118	7	)	)	PUNCT
cana-328	118	8	)	)	PUNCT
cana-328	119	1	∑	∑	PROPN
cana-328	119	2	𝑄(𝑡)𝑓	𝑄(𝑡)𝑓	NOUN
cana-328	119	3	(	(	PUNCT
cana-328	119	4	𝐸(𝜎(𝑡	𝐸(𝜎(𝑡	NOUN
cana-328	119	5	)	)	PUNCT
cana-328	119	6	)	)	PUNCT
cana-328	119	7	)	)	PUNCT
cana-328	119	8	𝜑−1	𝜑−1	PROPN
cana-328	119	9	𝑡=𝜎(𝜑	𝑡=𝜎(𝜑	NOUN
cana-328	119	10	)	)	PUNCT
cana-328	120	1	+	+	ADJ
cana-328	120	2	 	 	SPACE
cana-328	120	3	𝐸𝜅(𝜎(𝜑))𝑓	𝐸𝜅(𝜎(𝜑))𝑓	NOUN
cana-328	120	4	(	(	PUNCT
cana-328	120	5	1	1	NUM
cana-328	120	6	𝐷(𝜎(𝜑	𝐷(𝜎(𝜑	NOUN
cana-328	120	7	)	)	PUNCT
cana-328	120	8	)	)	PUNCT
cana-328	120	9	)	)	PUNCT
cana-328	121	1	∑	∑	ADP
cana-328	121	2	 	 	SPACE
cana-328	121	3	𝑄(𝑡)𝑓(𝐷(𝜎(𝑡)))∞	𝑄(𝑡)𝑓(𝐷(𝜎(𝑡)))∞	PROPN
cana-328	121	4	𝑡=𝜑	𝑡=𝜑	X
cana-328	121	5	)	)	PUNCT
cana-328	121	6	>	>	PUNCT
cana-328	122	1	𝐾	𝐾	PROPN
cana-328	122	2	,	,	PUNCT
cana-328	122	3	(	(	PUNCT
cana-328	122	4	13	13	NUM
cana-328	122	5	)	)	PUNCT
cana-328	122	6	then	then	ADV
cana-328	122	7	𝑦(𝜑	𝑦(𝜑	PROPN
cana-328	122	8	)	)	PUNCT
cana-328	122	9	∉	∉	PROPN
cana-328	122	10	𝑁1	𝑁1	PROPN
cana-328	122	11	which	which	PRON
cana-328	122	12	implies	imply	VERB
cana-328	122	13	(	(	PUNCT
cana-328	122	14	1	1	X
cana-328	122	15	)	)	PUNCT
cana-328	122	16	has	have	VERB
cana-328	122	17	property	property	NOUN
cana-328	122	18	𝐴.	𝐴.	NOUN
cana-328	122	19	proof	proof	NOUN
cana-328	122	20	:	:	PUNCT
cana-328	122	21	assume	assume	VERB
cana-328	122	22	that	that	SCONJ
cana-328	122	23	{	{	PUNCT
cana-328	122	24	𝑦(𝜑	𝑦(𝜑	NOUN
cana-328	122	25	)	)	PUNCT
cana-328	122	26	}	}	PUNCT
cana-328	122	27	is	be	AUX
cana-328	122	28	a	a	DET
cana-328	122	29	positive	positive	ADJ
cana-328	122	30	solution	solution	NOUN
cana-328	122	31	of	of	ADP
cana-328	122	32	(	(	PUNCT
cana-328	122	33	1	1	NUM
cana-328	122	34	)	)	PUNCT
cana-328	122	35	and	and	CCONJ
cana-328	122	36	so	so	ADV
cana-328	122	37	a	a	DET
cana-328	122	38	solution	solution	NOUN
cana-328	122	39	of	of	ADP
cana-328	122	40	(	(	PUNCT
cana-328	122	41	4	4	NUM
cana-328	122	42	)	)	PUNCT
cana-328	122	43	which	which	PRON
cana-328	122	44	belonging	belong	VERB
cana-328	122	45	to	to	ADP
cana-328	122	46	either	either	CCONJ
cana-328	122	47	𝑁0	𝑁0	ADJ
cana-328	122	48	or	or	CCONJ
cana-328	122	49	𝑁1	𝑁1	NOUN
cana-328	122	50	for	for	ADP
cana-328	122	51	all	all	DET
cana-328	122	52	𝜑	𝜑	PRON
cana-328	122	53	≥	≥	NOUN
cana-328	122	54	𝑁.	𝑁.	PROPN
cana-328	123	1	now	now	ADV
cana-328	123	2	we	we	PRON
cana-328	123	3	take	take	VERB
cana-328	123	4	𝑦(𝜑	𝑦(𝜑	ADV
cana-328	123	5	)	)	PUNCT
cana-328	123	6	∈	∈	PROPN
cana-328	123	7	𝑁1	𝑁1	PROPN
cana-328	123	8	,	,	PUNCT
cana-328	123	9	summation	summation	NOUN
cana-328	123	10	of	of	ADP
cana-328	123	11	(	(	PUNCT
cana-328	123	12	4	4	NUM
cana-328	123	13	)	)	PUNCT
cana-328	123	14	from	from	ADP
cana-328	123	15	𝜑	𝜑	PRON
cana-328	123	16	to	to	ADP
cana-328	123	17	∞	∞	PROPN
cana-328	123	18	gives	give	VERB
cana-328	123	19	𝛥(𝑑(𝜑)(𝛥𝑦(𝜑	𝛥(𝑑(𝜑)(𝛥𝑦(𝜑	NOUN
cana-328	123	20	)	)	PUNCT
cana-328	123	21	)	)	PUNCT
cana-328	124	1	𝜅	𝜅	X
cana-328	124	2	)	)	PUNCT
cana-328	124	3	≥	≥	NOUN
cana-328	124	4	1	1	NUM
cana-328	124	5	𝑐(𝜑	𝑐(𝜑	NOUN
cana-328	124	6	)	)	PUNCT
cana-328	124	7	∑	∑	PUNCT
cana-328	124	8	𝑄(𝑠)𝑓	𝑄(𝑠)𝑓	ADJ
cana-328	124	9	(	(	PUNCT
cana-328	124	10	𝑦(𝜎(𝑠)))∞	𝑦(𝜎(𝑠)))∞	NOUN
cana-328	124	11	𝑠=𝜑	𝑠=𝜑	PRON
cana-328	124	12	 	 	SPACE
cana-328	124	13	.	.	PUNCT
cana-328	125	1	summing	sum	VERB
cana-328	125	2	the	the	DET
cana-328	125	3	aforementioned	aforementioned	ADJ
cana-328	125	4	inequality	inequality	NOUN
cana-328	125	5	from	from	ADP
cana-328	125	6	𝜑1	𝜑1	NOUN
cana-328	125	7	to	to	ADP
cana-328	125	8	𝜑−	𝜑−	PROPN
cana-328	125	9	1	1	NUM
cana-328	125	10	,	,	PUNCT
cana-328	125	11	we	we	PRON
cana-328	125	12	get	get	VERB
cana-328	125	13	𝑑(𝜑)(𝛥𝑦(𝜑))𝜅	𝑑(𝜑)(𝛥𝑦(𝜑))𝜅	NOUN
cana-328	125	14	≥	≥	NOUN
cana-328	125	15	∑	∑	NOUN
cana-328	125	16	1	1	NUM
cana-328	125	17	𝑐(𝑠	𝑐(𝑠	NOUN
cana-328	125	18	)	)	PUNCT
cana-328	125	19	𝜑−1	𝜑−1	PROPN
cana-328	125	20	𝑠=𝜑1	𝑠=𝜑1	PROPN
cana-328	125	21	  	  	SPACE
cana-328	125	22	∑	∑	PROPN
cana-328	125	23	𝑄(𝑡)𝑓	𝑄(𝑡)𝑓	NOUN
cana-328	125	24	(	(	PUNCT
cana-328	125	25	𝑦(𝜎(𝑡	𝑦(𝜎(𝑡	NOUN
cana-328	125	26	)	)	PUNCT
cana-328	125	27	)	)	PUNCT
cana-328	125	28	)	)	PUNCT
cana-328	126	1	∞	∞	PROPN
cana-328	126	2	𝑡=𝑠	𝑡=𝑠	PUNCT
cana-328	126	3	=	=	PUNCT
cana-328	126	4	∑	∑	PUNCT
cana-328	126	5	𝐶(𝑠	𝐶(𝑠	PROPN
cana-328	126	6	+	+	NUM
cana-328	126	7	1)𝑄(𝑠)𝑓	1)𝑄(𝑠)𝑓	NUM
cana-328	126	8	(	(	PUNCT
cana-328	126	9	𝑦(𝜎(𝑠	𝑦(𝜎(𝑠	PROPN
cana-328	126	10	)	)	PUNCT
cana-328	126	11	)	)	PUNCT
cana-328	126	12	)	)	PUNCT
cana-328	127	1	𝜑−1	𝜑−1	PROPN
cana-328	127	2	𝑠=𝜑1	𝑠=𝜑1	PROPN
cana-328	127	3	  	  	SPACE
cana-328	127	4	+	+	CCONJ
cana-328	127	5	𝐶(𝜑)∑	𝐶(𝜑)∑	ADJ
cana-328	127	6	𝑄(𝑡)𝑓(𝑦(𝜎(𝑡)))∞	𝑄(𝑡)𝑓(𝑦(𝜎(𝑡)))∞	NOUN
cana-328	127	7	𝑡=𝜑	𝑡=𝜑	PUNCT
cana-328	127	8	   	   	SPACE
cana-328	127	9	.	.	PUNCT
cana-328	128	1	from	from	ADP
cana-328	128	2	(	(	PUNCT
cana-328	128	3	9	9	NUM
cana-328	128	4	)	)	PUNCT
cana-328	128	5	,	,	PUNCT
cana-328	128	6	we	we	PRON
cana-328	128	7	obtain	obtain	VERB
cana-328	128	8	𝐶(𝜑)𝑦𝜅(𝜑	𝐶(𝜑)𝑦𝜅(𝜑	ADV
cana-328	128	9	)	)	PUNCT
cana-328	128	10	𝐸𝜅(𝜑	𝐸𝜅(𝜑	NOUN
cana-328	128	11	)	)	PUNCT
cana-328	128	12	≥	≥	NOUN
cana-328	128	13	∑	∑	PUNCT
cana-328	128	14	𝐶(𝑠	𝐶(𝑠	PROPN
cana-328	128	15	+	+	PROPN
cana-328	128	16	1)𝑄(𝑠)𝑓(𝑦(𝜎(𝑠	1)𝑄(𝑠)𝑓(𝑦(𝜎(𝑠	NUM
cana-328	128	17	)	)	PUNCT
cana-328	128	18	)	)	PUNCT
cana-328	128	19	)	)	PUNCT
cana-328	129	1	𝜑−1	𝜑−1	PROPN
cana-328	129	2	𝑠=𝜑1	𝑠=𝜑1	PROPN
cana-328	129	3	 	 	SPACE
cana-328	129	4	+	+	CCONJ
cana-328	129	5	𝐶(𝜑)∑	𝐶(𝜑)∑	ADJ
cana-328	129	6	𝑄(𝑡)𝑓(𝑦(𝜎(𝑡)))∞	𝑄(𝑡)𝑓(𝑦(𝜎(𝑡)))∞	NOUN
cana-328	129	7	𝑡=𝜑	𝑡=𝜑	PUNCT
cana-328	129	8	 	 	SPACE
cana-328	129	9	.	.	PUNCT
cana-328	130	1	or	or	CCONJ
cana-328	130	2	𝐶(𝜎(𝜑))𝑦𝜅(𝜎(𝜑	𝐶(𝜎(𝜑))𝑦𝜅(𝜎(𝜑	NOUN
cana-328	130	3	)	)	PUNCT
cana-328	130	4	)	)	PUNCT
cana-328	131	1	𝐸𝜅(𝜎(𝜑	𝐸𝜅(𝜎(𝜑	NUM
cana-328	131	2	)	)	PUNCT
cana-328	131	3	)	)	PUNCT
cana-328	132	1	≥	≥	X
cana-328	132	2	∑	∑	PUNCT
cana-328	132	3	𝐶(𝑠	𝐶(𝑠	PROPN
cana-328	132	4	+	+	NUM
cana-328	132	5	1)𝑄(𝑠)𝑓	1)𝑄(𝑠)𝑓	NUM
cana-328	132	6	(	(	PUNCT
cana-328	132	7	𝑦(𝜎(𝑠	𝑦(𝜎(𝑠	PROPN
cana-328	132	8	)	)	PUNCT
cana-328	132	9	)	)	PUNCT
cana-328	132	10	)	)	PUNCT
cana-328	133	1	𝜎(𝜑)−1	𝜎(𝜑)−1	PROPN
cana-328	133	2	𝑠=𝜑1	𝑠=𝜑1	PROPN
cana-328	133	3	  	  	SPACE
cana-328	133	4	+	+	ADV
cana-328	133	5	𝐶(𝜎(𝜑	𝐶(𝜎(𝜑	NOUN
cana-328	133	6	)	)	PUNCT
cana-328	133	7	)	)	PUNCT
cana-328	134	1	∑	∑	PROPN
cana-328	134	2	𝑄(𝑡)𝑓	𝑄(𝑡)𝑓	NOUN
cana-328	134	3	(	(	PUNCT
cana-328	134	4	𝑦(𝜎(𝑡	𝑦(𝜎(𝑡	NOUN
cana-328	134	5	)	)	PUNCT
cana-328	134	6	)	)	PUNCT
cana-328	134	7	)	)	PUNCT
cana-328	135	1	𝜑−1	𝜑−1	PROPN
cana-328	135	2	𝑡=𝜎(𝜑	𝑡=𝜎(𝜑	NOUN
cana-328	135	3	)	)	PUNCT
cana-328	135	4	   	   	SPACE
cana-328	136	1	+	+	PROPN
cana-328	136	2	𝐶(𝜎(𝜑))∑	𝐶(𝜎(𝜑))∑	PROPN
cana-328	136	3	𝑄(𝑡)𝑓	𝑄(𝑡)𝑓	NOUN
cana-328	136	4	(	(	PUNCT
cana-328	136	5	𝑦(𝜎(𝑡)))∞	𝑦(𝜎(𝑡)))∞	PROPN
cana-328	136	6	𝑡=𝜑	𝑡=𝜑	X
cana-328	136	7	 	 	SPACE
cana-328	136	8	.	.	PUNCT
cana-328	137	1	using	use	VERB
cana-328	137	2	the	the	DET
cana-328	137	3	monotonic	monotonic	ADJ
cana-328	137	4	properties	property	NOUN
cana-328	137	5	(	(	PUNCT
cana-328	137	6	i)-(ii	i)-(ii	PROPN
cana-328	137	7	)	)	PUNCT
cana-328	137	8	of	of	ADP
cana-328	137	9	lemma	lemma	PROPN
cana-328	137	10	2.5	2.5	NUM
cana-328	137	11	and	and	CCONJ
cana-328	137	12	applying	apply	VERB
cana-328	137	13	𝑓	𝑓	PRON
cana-328	137	14	,	,	PUNCT
cana-328	137	15	we	we	PRON
cana-328	137	16	get	get	VERB
cana-328	137	17	𝐶(𝜎(𝜑))𝑦𝜅(𝜎(𝜑	𝐶(𝜎(𝜑))𝑦𝜅(𝜎(𝜑	NOUN
cana-328	137	18	)	)	PUNCT
cana-328	137	19	)	)	PUNCT
cana-328	137	20	𝐸𝜅(𝜎(𝜑	𝐸𝜅(𝜎(𝜑	NUM
cana-328	137	21	)	)	PUNCT
cana-328	137	22	)	)	PUNCT
cana-328	138	1	≥	≥	AUX
cana-328	138	2	𝑓	𝑓	PROPN
cana-328	138	3	(	(	PUNCT
cana-328	138	4	𝑦(𝜎(𝜑	𝑦(𝜎(𝜑	NOUN
cana-328	138	5	)	)	PUNCT
cana-328	138	6	)	)	PUNCT
cana-328	138	7	𝐸(𝜎(𝜑	𝐸(𝜎(𝜑	ADV
cana-328	138	8	)	)	PUNCT
cana-328	138	9	)	)	PUNCT
cana-328	138	10	)	)	PUNCT
cana-328	139	1	∑	∑	PUNCT
cana-328	139	2	𝐶(𝑠	𝐶(𝑠	PROPN
cana-328	139	3	+	+	NUM
cana-328	139	4	1)𝑄(𝑠)𝑓	1)𝑄(𝑠)𝑓	NUM
cana-328	139	5	(	(	PUNCT
cana-328	139	6	𝐸(𝜎(𝑠)))𝜎(𝜑)−1	𝐸(𝜎(𝑠)))𝜎(𝜑)−1	NOUN
cana-328	139	7	𝑠=𝜑1	𝑠=𝜑1	PROPN
cana-328	139	8	   	   	SPACE
cana-328	139	9	+	+	NOUN
cana-328	139	10	𝐶(𝜎(𝜑))𝑓	𝐶(𝜎(𝜑))𝑓	PROPN
cana-328	139	11	(	(	PUNCT
cana-328	139	12	𝑦(𝜎(𝜑	𝑦(𝜎(𝜑	NOUN
cana-328	139	13	)	)	PUNCT
cana-328	139	14	)	)	PUNCT
cana-328	139	15	𝐸(𝜎(𝜑	𝐸(𝜎(𝜑	ADV
cana-328	139	16	)	)	PUNCT
cana-328	139	17	)	)	PUNCT
cana-328	139	18	)	)	PUNCT
cana-328	139	19	∑	∑	PROPN
cana-328	140	1	𝑄(𝑡)𝑓	𝑄(𝑡)𝑓	NOUN
cana-328	140	2	(	(	PUNCT
cana-328	140	3	𝐸(𝜎(𝑡	𝐸(𝜎(𝑡	NOUN
cana-328	140	4	)	)	PUNCT
cana-328	140	5	)	)	PUNCT
cana-328	140	6	)	)	PUNCT
cana-328	141	1	𝜑−1	𝜑−1	PROPN
cana-328	141	2	𝑡=𝜎(𝜑	𝑡=𝜎(𝜑	NOUN
cana-328	141	3	)	)	PUNCT
cana-328	141	4	   	   	SPACE
cana-328	142	1	+	+	NOUN
cana-328	142	2	𝐶(𝜎(𝜑))𝑓	𝐶(𝜎(𝜑))𝑓	PROPN
cana-328	142	3	(	(	PUNCT
cana-328	142	4	𝑦(𝜎(𝜑	𝑦(𝜎(𝜑	NOUN
cana-328	142	5	)	)	PUNCT
cana-328	142	6	)	)	PUNCT
cana-328	142	7	𝐷(𝜎(𝜑	𝐷(𝜎(𝜑	NOUN
cana-328	142	8	)	)	PUNCT
cana-328	142	9	)	)	PUNCT
cana-328	142	10	)	)	PUNCT
cana-328	143	1	∑	∑	PROPN
cana-328	143	2	𝑄(𝑡)𝑓	𝑄(𝑡)𝑓	NOUN
cana-328	143	3	(	(	PUNCT
cana-328	143	4	𝐷(𝜎(𝑡)))∞	𝐷(𝜎(𝑡)))∞	ADJ
cana-328	143	5	𝑡=𝜑	𝑡=𝜑	NUM
cana-328	143	6	,	,	PUNCT
cana-328	143	7	or	or	CCONJ
cana-328	143	8	𝑦𝜅(𝜎(𝜑	𝑦𝜅(𝜎(𝜑	NOUN
cana-328	143	9	)	)	PUNCT
cana-328	143	10	)	)	PUNCT
cana-328	143	11	𝑓(𝑦(𝜎(𝜑	𝑓(𝑦(𝜎(𝜑	PROPN
cana-328	143	12	)	)	PUNCT
cana-328	143	13	)	)	PUNCT
cana-328	143	14	)	)	PUNCT
cana-328	143	15	≥	≥	NOUN
cana-328	143	16	𝐸𝜅(𝜎(𝜑	𝐸𝜅(𝜎(𝜑	NOUN
cana-328	143	17	)	)	PUNCT
cana-328	143	18	)	)	PUNCT
cana-328	143	19	𝐶(𝜎(𝜑	𝐶(𝜎(𝜑	NOUN
cana-328	143	20	)	)	PUNCT
cana-328	143	21	)	)	PUNCT
cana-328	144	1	𝑓	𝑓	X
cana-328	144	2	(	(	PUNCT
cana-328	144	3	1	1	NUM
cana-328	144	4	𝐸(𝜎(𝜑	𝐸(𝜎(𝜑	NOUN
cana-328	144	5	)	)	PUNCT
cana-328	144	6	)	)	PUNCT
cana-328	144	7	)	)	PUNCT
cana-328	144	8	∑	∑	PUNCT
cana-328	144	9	𝐶(𝑠	𝐶(𝑠	PROPN
cana-328	144	10	+	+	NUM
cana-328	144	11	1)𝑄(𝑠)𝑓	1)𝑄(𝑠)𝑓	NUM
cana-328	144	12	(	(	PUNCT
cana-328	144	13	𝐸(𝜎(𝑠)))𝜎(𝜑)−1	𝐸(𝜎(𝑠)))𝜎(𝜑)−1	NOUN
cana-328	144	14	𝑠=𝜑1	𝑠=𝜑1	PROPN
cana-328	144	15	  	  	SPACE
cana-328	144	16	+	+	PROPN
cana-328	144	17	𝐸𝜅(𝜎(𝜑))𝑓	𝐸𝜅(𝜎(𝜑))𝑓	PROPN
cana-328	144	18	(	(	PUNCT
cana-328	144	19	1	1	NUM
cana-328	144	20	𝐸(𝜎(𝜑	𝐸(𝜎(𝜑	NOUN
cana-328	144	21	)	)	PUNCT
cana-328	144	22	)	)	PUNCT
cana-328	144	23	)	)	PUNCT
cana-328	145	1	∑	∑	PROPN
cana-328	145	2	𝑄(𝑡)𝑓	𝑄(𝑡)𝑓	NOUN
cana-328	145	3	(	(	PUNCT
cana-328	145	4	𝐸(𝜎(𝑡	𝐸(𝜎(𝑡	NOUN
cana-328	145	5	)	)	PUNCT
cana-328	145	6	)	)	PUNCT
cana-328	145	7	)	)	PUNCT
cana-328	145	8	𝜑−1	𝜑−1	PROPN
cana-328	145	9	𝑡=𝜎(𝜑	𝑡=𝜎(𝜑	NOUN
cana-328	145	10	)	)	PUNCT
cana-328	145	11	   	   	SPACE
cana-328	146	1	+	+	NOUN
cana-328	146	2	𝐸𝜅(𝜎(𝜑))𝑓	𝐸𝜅(𝜎(𝜑))𝑓	PROPN
cana-328	146	3	(	(	PUNCT
cana-328	146	4	1	1	NUM
cana-328	146	5	𝐷(𝜎(𝜑	𝐷(𝜎(𝜑	NOUN
cana-328	146	6	)	)	PUNCT
cana-328	146	7	)	)	PUNCT
cana-328	146	8	)	)	PUNCT
cana-328	146	9	∑	∑	ADP
cana-328	146	10	 	 	SPACE
cana-328	146	11	𝑄(𝑡)𝑓	𝑄(𝑡)𝑓	NOUN
cana-328	146	12	(	(	PUNCT
cana-328	146	13	𝐷(𝜎(𝑡)))∞	𝐷(𝜎(𝑡)))∞	X
cana-328	146	14	𝑡=𝜑	𝑡=𝜑	X
cana-328	146	15	.	.	PUNCT
cana-328	147	1	communications	communication	NOUN
cana-328	147	2	on	on	ADP
cana-328	147	3	applied	apply	VERB
cana-328	147	4	nonlinear	nonlinear	ADJ
cana-328	147	5	analysis	analysis	NOUN
cana-328	147	6	issn	issn	NOUN
cana-328	147	7	:	:	PUNCT
cana-328	147	8	1074	1074	NUM
cana-328	147	9	-	-	PUNCT
cana-328	147	10	133x	133x	NUM
cana-328	147	11	vol	vol	NOUN
cana-328	147	12	31	31	NUM
cana-328	147	13	no	no	NOUN
cana-328	147	14	.	.	NOUN
cana-328	147	15	1	1	NUM
cana-328	147	16	(	(	PUNCT
cana-328	147	17	2024	2024	NUM
cana-328	147	18	)	)	PUNCT
cana-328	147	19	95	95	NUM
cana-328	147	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-328	147	21	taking	take	VERB
cana-328	147	22	limsup	limsup	NOUN
cana-328	147	23	as	as	ADP
cana-328	147	24	𝜑	𝜑	PROPN
cana-328	147	25	→	→	SYM
cana-328	147	26	∞	∞	NUM
cana-328	147	27	on	on	ADP
cana-328	147	28	both	both	DET
cana-328	147	29	side	side	NOUN
cana-328	147	30	of	of	ADP
cana-328	147	31	above	above	ADP
cana-328	147	32	inequality	inequality	NOUN
cana-328	147	33	,	,	PUNCT
cana-328	147	34	we	we	PRON
cana-328	147	35	get	get	VERB
cana-328	147	36	contradiction	contradiction	NOUN
cana-328	147	37	with	with	ADP
cana-328	147	38	(	(	PUNCT
cana-328	147	39	13	13	NUM
cana-328	147	40	)	)	PUNCT
cana-328	147	41	.	.	PUNCT
cana-328	148	1	hence	hence	ADV
cana-328	148	2	𝑦(𝜑	𝑦(𝜑	ADV
cana-328	148	3	)	)	PUNCT
cana-328	148	4	∉	∉	PROPN
cana-328	148	5	𝑁1	𝑁1	PROPN
cana-328	148	6	,	,	PUNCT
cana-328	148	7	so	so	ADV
cana-328	148	8	𝑦(𝜑	𝑦(𝜑	ADJ
cana-328	148	9	)	)	PUNCT
cana-328	148	10	satisfies	satisfy	VERB
cana-328	148	11	the	the	DET
cana-328	148	12	case	case	NOUN
cana-328	148	13	𝑁0	𝑁0	VERB
cana-328	148	14	.	.	PUNCT
cana-328	149	1	therefore	therefore	ADV
cana-328	149	2	(	(	PUNCT
cana-328	149	3	1	1	X
cana-328	149	4	)	)	PUNCT
cana-328	149	5	has	have	VERB
cana-328	149	6	property	property	NOUN
cana-328	149	7	a.	a.	NOUN
cana-328	149	8	corollary	corollary	NOUN
cana-328	149	9	2.7	2.7	NUM
cana-328	149	10	.	.	PUNCT
cana-328	150	1	let	let	VERB
cana-328	150	2	(	(	PUNCT
cana-328	150	3	8)	8)	NUM
cana-328	150	4	hold	hold	NOUN
cana-328	150	5	,	,	PUNCT
cana-328	150	6	𝑓(𝑢	𝑓(𝑢	PROPN
cana-328	150	7	)	)	PUNCT
cana-328	150	8	=	=	SYM
cana-328	151	1	𝑢𝜅	𝑢𝜅	PROPN
cana-328	151	2	and	and	CCONJ
cana-328	151	3	(	(	PUNCT
cana-328	151	4	2	2	X
cana-328	151	5	)	)	PUNCT
cana-328	151	6	have	have	VERB
cana-328	151	7	a	a	DET
cana-328	151	8	positive	positive	ADJ
cana-328	151	9	solution	solution	NOUN
cana-328	151	10	{	{	PUNCT
cana-328	151	11	𝑧(𝜑	𝑧(𝜑	NOUN
cana-328	151	12	)	)	PUNCT
cana-328	151	13	}	}	PUNCT
cana-328	151	14	.	.	PUNCT
cana-328	152	1	if	if	SCONJ
cana-328	152	2	𝑙𝑖𝑚	𝑙𝑖𝑚	PROPN
cana-328	152	3	 	 	SPACE
cana-328	152	4	𝑠𝑢𝑝𝜑→∞	𝑠𝑢𝑝𝜑→∞	PROPN
cana-328	152	5	  	  	SPACE
cana-328	152	6	(	(	PUNCT
cana-328	152	7	1	1	NUM
cana-328	152	8	𝐶(𝜎(𝜑	𝐶(𝜎(𝜑	NOUN
cana-328	152	9	)	)	PUNCT
cana-328	152	10	)	)	PUNCT
cana-328	152	11	∑	∑	PUNCT
cana-328	152	12	𝐶(𝑠	𝐶(𝑠	PROPN
cana-328	152	13	+	+	NOUN
cana-328	152	14	1)𝑄(𝑠)𝐸𝜅(𝜎(𝑠	1)𝑄(𝑠)𝐸𝜅(𝜎(𝑠	NUM
cana-328	152	15	)	)	PUNCT
cana-328	152	16	)	)	PUNCT
cana-328	152	17	𝜎(𝜑)−1	𝜎(𝜑)−1	PROPN
cana-328	152	18	𝑠=𝜑1	𝑠=𝜑1	PROPN
cana-328	152	19	   	   	SPACE
cana-328	152	20	+	+	PROPN
cana-328	152	21	∑	∑	PROPN
cana-328	152	22	𝑄(𝑡)𝑓	𝑄(𝑡)𝑓	NOUN
cana-328	152	23	(	(	PUNCT
cana-328	152	24	𝐸(𝜎(𝑡	𝐸(𝜎(𝑡	NOUN
cana-328	152	25	)	)	PUNCT
cana-328	152	26	)	)	PUNCT
cana-328	152	27	)	)	PUNCT
cana-328	153	1	𝜑−1	𝜑−1	PROPN
cana-328	153	2	𝑡=𝜎(𝜑	𝑡=𝜎(𝜑	NOUN
cana-328	153	3	)	)	PUNCT
cana-328	153	4	   	   	SPACE
cana-328	153	5	+	+	PUNCT
cana-328	153	6	𝐸𝜅(𝜎(𝜑	𝐸𝜅(𝜎(𝜑	NUM
cana-328	153	7	)	)	PUNCT
cana-328	153	8	)	)	PUNCT
cana-328	153	9	𝐷𝜅(𝜎(𝜑	𝐷𝜅(𝜎(𝜑	NUM
cana-328	153	10	)	)	PUNCT
cana-328	153	11	)	)	PUNCT
cana-328	153	12	∑	∑	PUNCT
cana-328	153	13	𝑄(𝑡)𝑓(𝐷(𝜎(𝑡)))∞	𝑄(𝑡)𝑓(𝐷(𝜎(𝑡)))∞	PROPN
cana-328	153	14	𝑡=𝜑	𝑡=𝜑	X
cana-328	153	15	   	   	SPACE
cana-328	153	16	)	)	PUNCT
cana-328	153	17	>	>	X
cana-328	154	1	1	1	X
cana-328	154	2	.	.	PUNCT
cana-328	154	3	(	(	PUNCT
cana-328	154	4	14	14	NUM
cana-328	154	5	)	)	PUNCT
cana-328	154	6	then	then	ADV
cana-328	154	7	𝑦(𝜑	𝑦(𝜑	PROPN
cana-328	154	8	)	)	PUNCT
cana-328	154	9	∉	∉	PROPN
cana-328	154	10	𝑁1	𝑁1	PROPN
cana-328	154	11	which	which	PRON
cana-328	154	12	implies	imply	VERB
cana-328	154	13	(	(	PUNCT
cana-328	154	14	1	1	X
cana-328	154	15	)	)	PUNCT
cana-328	154	16	has	have	VERB
cana-328	154	17	property	property	NOUN
cana-328	154	18	𝐴	𝐴	PROPN
cana-328	154	19	throughout	throughout	ADP
cana-328	154	20	the	the	DET
cana-328	154	21	proof	proof	NOUN
cana-328	154	22	of	of	ADP
cana-328	154	23	the	the	DET
cana-328	154	24	following	following	NOUN
cana-328	154	25	theorem	theorem	NOUN
cana-328	154	26	,	,	PUNCT
cana-328	154	27	we	we	PRON
cana-328	154	28	use	use	VERB
cana-328	154	29	𝐻(𝜑	𝐻(𝜑	PROPN
cana-328	154	30	,	,	PUNCT
cana-328	154	31	𝑗	𝑗	NOUN
cana-328	154	32	):	):	PUNCT
cana-328	154	33	𝜑	𝜑	PRON
cana-328	154	34	,	,	PUNCT
cana-328	154	35	𝑗	𝑗	PROPN
cana-328	154	36	∈	∈	PROPN
cana-328	154	37	𝑁	𝑁	PROPN
cana-328	154	38	,	,	PUNCT
cana-328	154	39	𝜑	𝜑	PRON
cana-328	154	40	≥	≥	NOUN
cana-328	154	41	𝑗	𝑗	X
cana-328	154	42	≥	≥	NOUN
cana-328	154	43	0	0	NUM
cana-328	154	44	to	to	PART
cana-328	154	45	denote	denote	VERB
cana-328	154	46	the	the	DET
cana-328	154	47	double	double	ADJ
cana-328	154	48	sequence	sequence	NOUN
cana-328	154	49	satisfying	satisfy	VERB
cana-328	154	50	𝐻(𝜑	𝐻(𝜑	PROPN
cana-328	154	51	,	,	PUNCT
cana-328	154	52	𝜑	𝜑	NOUN
cana-328	154	53	)	)	PUNCT
cana-328	154	54	=	=	SYM
cana-328	154	55	0	0	NUM
cana-328	155	1	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-328	155	2	𝜑	𝜑	PROPN
cana-328	155	3	≥	≥	NOUN
cana-328	155	4	𝜑0	𝜑0	NOUN
cana-328	155	5	;	;	PUNCT
cana-328	155	6	𝐻(𝜑	𝐻(𝜑	PROPN
cana-328	155	7	,	,	PUNCT
cana-328	155	8	𝑗	𝑗	NOUN
cana-328	155	9	)	)	PUNCT
cana-328	155	10	>	>	X
cana-328	155	11	0	0	PUNCT
cana-328	156	1	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-328	156	2	𝜑	𝜑	X
cana-328	156	3	>	>	X
cana-328	156	4	𝑗	𝑗	PROPN
cana-328	156	5	≥	≥	NOUN
cana-328	156	6	𝜑0	𝜑0	NOUN
cana-328	156	7	;	;	PUNCT
cana-328	156	8	𝛥2𝐻(𝜑	𝛥2𝐻(𝜑	PROPN
cana-328	156	9	,	,	PUNCT
cana-328	156	10	𝑗	𝑗	NOUN
cana-328	156	11	)	)	PUNCT
cana-328	156	12	=	=	SYM
cana-328	156	13	𝐻(𝜑	𝐻(𝜑	PROPN
cana-328	156	14	,	,	PUNCT
cana-328	156	15	𝑗	𝑗	NOUN
cana-328	156	16	+	+	ADJ
cana-328	156	17	1	1	NUM
cana-328	156	18	)	)	PUNCT
cana-328	156	19	−	−	PROPN
cana-328	157	1	𝐻(𝜑	𝐻(𝜑	SYM
cana-328	157	2	,	,	PUNCT
cana-328	157	3	𝑗	𝑗	NOUN
cana-328	157	4	)	)	PUNCT
cana-328	157	5	<	<	X
cana-328	157	6	0	0	NUM
cana-328	158	1	𝑓𝑜𝑟	𝑓𝑜𝑟	NOUN
cana-328	158	2	𝜑	𝜑	X
cana-328	158	3	>	>	X
cana-328	158	4	𝑗	𝑗	PROPN
cana-328	158	5	≥	≥	NOUN
cana-328	158	6	𝜑0	𝜑0	NOUN
cana-328	158	7	.	.	PUNCT
cana-328	159	1	theorem	theorem	VERB
cana-328	159	2	2.8.(h1)-(h3	2.8.(h1)-(h3	NUM
cana-328	159	3	)	)	PUNCT
cana-328	159	4	hold	hold	VERB
cana-328	159	5	and	and	CCONJ
cana-328	159	6	(	(	PUNCT
cana-328	159	7	2	2	X
cana-328	159	8	)	)	PUNCT
cana-328	159	9	has	have	VERB
cana-328	159	10	a	a	DET
cana-328	159	11	positive	positive	ADJ
cana-328	159	12	solution	solution	NOUN
cana-328	159	13	{	{	PUNCT
cana-328	159	14	𝑧(𝜑	𝑧(𝜑	NOUN
cana-328	159	15	)	)	PUNCT
cana-328	159	16	}	}	PUNCT
cana-328	159	17	.	.	PUNCT
cana-328	160	1	if	if	SCONJ
cana-328	160	2	𝑙𝑖𝑚	𝑙𝑖𝑚	PROPN
cana-328	160	3	 	 	SPACE
cana-328	160	4	𝑠𝑢𝑝𝜑→∞	𝑠𝑢𝑝𝜑→∞	PROPN
cana-328	160	5	  	  	SPACE
cana-328	160	6	1	1	NUM
cana-328	160	7	𝐻(𝜑,𝑀	𝐻(𝜑,𝑀	NOUN
cana-328	160	8	)	)	PUNCT
cana-328	160	9	∑	∑	ADP
cana-328	160	10	 	 	SPACE
cana-328	160	11	[	[	X
cana-328	160	12	𝐻(𝜑	𝐻(𝜑	PROPN
cana-328	160	13	,	,	PUNCT
cana-328	160	14	𝑗	𝑗	NOUN
cana-328	160	15	)	)	PUNCT
cana-328	160	16	𝜌(𝑗)𝑄(𝑗)𝑓(𝑦(𝜎(𝑗	𝜌(𝑗)𝑄(𝑗)𝑓(𝑦(𝜎(𝑗	PROPN
cana-328	160	17	)	)	PUNCT
cana-328	160	18	)	)	PUNCT
cana-328	160	19	)	)	PUNCT
cana-328	160	20	𝑦(𝜎(𝑗))𝜅	𝑦(𝜎(𝑗))𝜅	NOUN
cana-328	160	21	−	−	PROPN
cana-328	160	22	ℎ2(𝜑,𝑗)𝜌(𝑗+1)𝑐(𝑗)𝑑(𝜎(𝑗	ℎ2(𝜑,𝑗)𝜌(𝑗+1)𝑐(𝑗)𝑑(𝜎(𝑗	PROPN
cana-328	160	23	)	)	PUNCT
cana-328	160	24	)	)	PUNCT
cana-328	160	25	4(𝜎(𝑗)−𝑗)𝐻(𝜑,𝑗	4(𝜎(𝑗)−𝑗)𝐻(𝜑,𝑗	NUM
cana-328	160	26	)	)	PUNCT
cana-328	160	27	]	]	PUNCT
cana-328	161	1	𝜑−1	𝜑−1	PROPN
cana-328	161	2	𝑗=𝑀	𝑗=𝑀	NOUN
cana-328	162	1	=	=	SYM
cana-328	162	2	∞	∞	PROPN
cana-328	162	3	,	,	PUNCT
cana-328	162	4	(	(	PUNCT
cana-328	162	5	15	15	NUM
cana-328	162	6	)	)	PUNCT
cana-328	162	7	and	and	CCONJ
cana-328	162	8	∑	∑	ADP
cana-328	162	9	(	(	PUNCT
cana-328	162	10	1	1	NUM
cana-328	162	11	𝑑(𝑡	𝑑(𝑡	PROPN
cana-328	162	12	)	)	PUNCT
cana-328	162	13	∑	∑	ADV
cana-328	162	14	1	1	NUM
cana-328	162	15	𝑐(𝑗	𝑐(𝑗	NOUN
cana-328	162	16	)	)	PUNCT
cana-328	162	17	∞	∞	NUM
cana-328	163	1	𝑗=𝑡	𝑗=𝑡	PROPN
cana-328	163	2	  	  	SPACE
cana-328	163	3	∑	∑	PROPN
cana-328	163	4	𝑄(𝑖)𝑓(𝑦(𝜎(𝑖)))∞	𝑄(𝑖)𝑓(𝑦(𝜎(𝑖)))∞	PROPN
cana-328	163	5	𝑖=𝑗	𝑖=𝑗	PROPN
cana-328	163	6	  	  	SPACE
cana-328	163	7	)	)	PUNCT
cana-328	163	8	1	1	NUM
cana-328	163	9	𝜅∞	𝜅∞	X
cana-328	163	10	𝑡=𝜑	𝑡=𝜑	X
cana-328	163	11	  	  	SPACE
cana-328	163	12	=	=	SYM
cana-328	163	13	∞	∞	PROPN
cana-328	163	14	,	,	PUNCT
cana-328	163	15	(	(	PUNCT
cana-328	163	16	16	16	NUM
cana-328	163	17	)	)	PUNCT
cana-328	163	18	then	then	ADV
cana-328	163	19	(	(	PUNCT
cana-328	163	20	1	1	X
cana-328	163	21	)	)	PUNCT
cana-328	163	22	is	be	AUX
cana-328	163	23	oscillatory	oscillatory	ADJ
cana-328	163	24	.	.	PUNCT
cana-328	164	1	proof	proof	NOUN
cana-328	164	2	.	.	PUNCT
cana-328	165	1	let	let	AUX
cana-328	165	2	𝑦(𝜑	𝑦(𝜑	NOUN
cana-328	165	3	)	)	PUNCT
cana-328	165	4	be	be	AUX
cana-328	165	5	a	a	DET
cana-328	165	6	nonoscillatory	nonoscillatory	ADJ
cana-328	165	7	solution	solution	NOUN
cana-328	165	8	of	of	ADP
cana-328	165	9	(	(	PUNCT
cana-328	165	10	4	4	NUM
cana-328	165	11	)	)	PUNCT
cana-328	165	12	.	.	PUNCT
cana-328	166	1	without	without	ADP
cana-328	166	2	loss	loss	NOUN
cana-328	166	3	of	of	ADP
cana-328	166	4	generality	generality	NOUN
cana-328	166	5	,	,	PUNCT
cana-328	166	6	𝑦(𝜑	𝑦(𝜑	NOUN
cana-328	166	7	)	)	PUNCT
cana-328	166	8	is	be	AUX
cana-328	166	9	eventually	eventually	ADV
cana-328	166	10	positive	positive	ADJ
cana-328	166	11	such	such	ADJ
cana-328	166	12	that	that	DET
cana-328	166	13	𝑦(𝜑	𝑦(𝜑	NOUN
cana-328	166	14	)	)	PUNCT
cana-328	166	15	∈	∈	NOUN
cana-328	166	16	𝑁0	𝑁0	ADJ
cana-328	166	17	or	or	CCONJ
cana-328	166	18	𝑦(𝜑	𝑦(𝜑	NOUN
cana-328	166	19	)	)	PUNCT
cana-328	166	20	∈	∈	PROPN
cana-328	166	21	𝑁1	𝑁1	PROPN
cana-328	166	22	.	.	PUNCT
cana-328	167	1	if	if	SCONJ
cana-328	167	2	𝑦(𝜑	𝑦(𝜑	NOUN
cana-328	167	3	)	)	PUNCT
cana-328	167	4	∈	∈	PROPN
cana-328	167	5	𝑁1	𝑁1	PROPN
cana-328	167	6	,	,	PUNCT
cana-328	167	7	then	then	ADV
cana-328	167	8	we	we	PRON
cana-328	167	9	have	have	VERB
cana-328	167	10	to	to	PART
cana-328	167	11	define	define	VERB
cana-328	167	12	for	for	ADP
cana-328	167	13	𝜌(𝜑	𝜌(𝜑	NOUN
cana-328	167	14	)	)	PUNCT
cana-328	167	15	>	>	X
cana-328	167	16	0	0	NUM
cana-328	167	17	,	,	PUNCT
cana-328	167	18	𝑤(𝜑	𝑤(𝜑	PROPN
cana-328	167	19	)	)	PUNCT
cana-328	167	20	=	=	SYM
cana-328	167	21	𝜌(𝜑	𝜌(𝜑	NOUN
cana-328	167	22	)	)	PUNCT
cana-328	167	23	𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑))𝜅	𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑))𝜅	NOUN
cana-328	167	24	)	)	PUNCT
cana-328	167	25	(	(	PUNCT
cana-328	167	26	𝑦(𝜎(𝜑)))𝜅	𝑦(𝜎(𝜑)))𝜅	PROPN
cana-328	167	27	,	,	PUNCT
cana-328	167	28	then	then	ADV
cana-328	167	29	𝑤(𝜑	𝑤(𝜑	NOUN
cana-328	167	30	)	)	PUNCT
cana-328	167	31	>	>	X
cana-328	167	32	0	0	PUNCT
cana-328	168	1	for	for	ADP
cana-328	168	2	𝜑	𝜑	PROPN
cana-328	168	3	≥	≥	NOUN
cana-328	168	4	𝜑1	𝜑1	VERB
cana-328	168	5	,	,	PUNCT
cana-328	168	6	and	and	CCONJ
cana-328	168	7	𝛥𝑤(𝜑	𝛥𝑤(𝜑	NUM
cana-328	168	8	)	)	PUNCT
cana-328	168	9	=	=	SYM
cana-328	168	10	𝛥(𝜌(𝜑	𝛥(𝜌(𝜑	NOUN
cana-328	168	11	)	)	PUNCT
cana-328	168	12	𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑	𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑	ADJ
cana-328	168	13	)	)	PUNCT
cana-328	168	14	)	)	PUNCT
cana-328	168	15	𝜅	𝜅	X
cana-328	168	16	)	)	PUNCT
cana-328	168	17	(	(	PUNCT
cana-328	168	18	𝑦(𝜎(𝜑	𝑦(𝜎(𝜑	NOUN
cana-328	168	19	)	)	PUNCT
cana-328	168	20	)	)	PUNCT
cana-328	168	21	)	)	PUNCT
cana-328	169	1	𝜅	𝜅	X
cana-328	169	2	)	)	PUNCT
cana-328	169	3	=	=	SYM
cana-328	169	4	𝛥𝜌(𝜑	𝛥𝜌(𝜑	NOUN
cana-328	169	5	)	)	PUNCT
cana-328	169	6	𝑐(𝜑+	𝑐(𝜑+	NOUN
cana-328	170	1	1)𝛥(𝑑(𝜑	1)𝛥(𝑑(𝜑	NUM
cana-328	170	2	+	+	CCONJ
cana-328	170	3	1)(𝛥𝑦(𝜑+	1)(𝛥𝑦(𝜑+	NUM
cana-328	170	4	1	1	NUM
cana-328	170	5	)	)	PUNCT
cana-328	170	6	)	)	PUNCT
cana-328	171	1	𝜅	𝜅	X
cana-328	171	2	)	)	PUNCT
cana-328	171	3	(	(	PUNCT
cana-328	171	4	𝑦(𝜎(𝜑+	𝑦(𝜎(𝜑+	NOUN
cana-328	171	5	1)))𝜅	1)))𝜅	NUM
cana-328	171	6	+	+	CCONJ
cana-328	171	7	𝜌(𝜑)𝛥	𝜌(𝜑)𝛥	X
cana-328	171	8	(	(	PUNCT
cana-328	171	9	𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑	𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑	ADJ
cana-328	171	10	)	)	PUNCT
cana-328	171	11	)	)	PUNCT
cana-328	172	1	𝜅	𝜅	X
cana-328	172	2	)	)	PUNCT
cana-328	172	3	(	(	PUNCT
cana-328	172	4	𝑥(𝜎(𝜑)))𝜅	𝑥(𝜎(𝜑)))𝜅	NOUN
cana-328	172	5	)	)	PUNCT
cana-328	173	1	=	=	PUNCT
cana-328	173	2	𝑤(𝜑+	𝑤(𝜑+	PUNCT
cana-328	173	3	1	1	NUM
cana-328	173	4	)	)	PUNCT
cana-328	173	5	𝜌(𝜑+	𝜌(𝜑+	X
cana-328	173	6	1	1	X
cana-328	173	7	)	)	PUNCT
cana-328	173	8	𝛥𝜌(𝜑	𝛥𝜌(𝜑	NOUN
cana-328	173	9	)	)	PUNCT
cana-328	174	1	+	+	CCONJ
cana-328	174	2	𝜌(𝜑	𝜌(𝜑	NOUN
cana-328	174	3	)	)	PUNCT
cana-328	174	4	𝛥	𝛥	PROPN
cana-328	174	5	(	(	PUNCT
cana-328	174	6	𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑	𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑	ADJ
cana-328	174	7	)	)	PUNCT
cana-328	174	8	)	)	PUNCT
cana-328	174	9	𝜅	𝜅	X
cana-328	174	10	)	)	PUNCT
cana-328	174	11	)	)	PUNCT
cana-328	174	12	(	(	PUNCT
cana-328	174	13	𝑦(𝜎(𝜑)))𝜅	𝑦(𝜎(𝜑)))𝜅	NUM
cana-328	174	14	−	−	NOUN
cana-328	174	15	𝜌(𝜑)𝑤(𝜑+	𝜌(𝜑)𝑤(𝜑+	SYM
cana-328	174	16	1	1	X
cana-328	174	17	)	)	PUNCT
cana-328	174	18	(	(	PUNCT
cana-328	174	19	𝑦(𝜎(𝜑)))𝜅𝜌(𝜑+	𝑦(𝜎(𝜑)))𝜅𝜌(𝜑+	SYM
cana-328	174	20	1	1	NUM
cana-328	174	21	)	)	PUNCT
cana-328	174	22	𝛥(𝑦(𝜎(𝜑	𝛥(𝑦(𝜎(𝜑	NUM
cana-328	174	23	)	)	PUNCT
cana-328	174	24	)	)	PUNCT
cana-328	174	25	)	)	PUNCT
cana-328	175	1	𝜅	𝜅	X
cana-328	175	2	.	.	PUNCT
cana-328	176	1	from	from	ADP
cana-328	176	2	(	(	PUNCT
cana-328	176	3	4	4	NUM
cana-328	176	4	)	)	PUNCT
cana-328	176	5	,	,	PUNCT
cana-328	176	6	we	we	PRON
cana-328	176	7	have	have	VERB
cana-328	176	8	communications	communication	NOUN
cana-328	176	9	on	on	ADP
cana-328	176	10	applied	apply	VERB
cana-328	176	11	nonlinear	nonlinear	ADJ
cana-328	176	12	analysis	analysis	NOUN
cana-328	176	13	issn	issn	NOUN
cana-328	176	14	:	:	PUNCT
cana-328	176	15	1074	1074	NUM
cana-328	176	16	-	-	PUNCT
cana-328	176	17	133x	133x	NUM
cana-328	176	18	vol	vol	NOUN
cana-328	176	19	31	31	NUM
cana-328	176	20	no	no	NOUN
cana-328	176	21	.	.	NOUN
cana-328	176	22	1	1	NUM
cana-328	176	23	(	(	PUNCT
cana-328	176	24	2024	2024	NUM
cana-328	176	25	)	)	PUNCT
cana-328	176	26	96	96	NUM
cana-328	176	27	https://internationalpubls.com	https://internationalpubls.com	X
cana-328	176	28	𝛥𝑤(𝜑	𝛥𝑤(𝜑	PROPN
cana-328	176	29	)	)	PUNCT
cana-328	176	30	=	=	SYM
cana-328	177	1	𝑤(𝜑+1	𝑤(𝜑+1	PROPN
cana-328	177	2	)	)	PUNCT
cana-328	177	3	𝜌(𝜑+1	𝜌(𝜑+1	PUNCT
cana-328	177	4	)	)	PUNCT
cana-328	178	1	𝛥𝜌(𝜑	𝛥𝜌(𝜑	NOUN
cana-328	178	2	)	)	PUNCT
cana-328	178	3	−	−	NOUN
cana-328	178	4	𝜌(𝜑)𝑄(𝜑)𝑓(𝑦(𝜎(𝜑	𝜌(𝜑)𝑄(𝜑)𝑓(𝑦(𝜎(𝜑	NOUN
cana-328	178	5	)	)	PUNCT
cana-328	178	6	)	)	PUNCT
cana-328	178	7	)	)	PUNCT
cana-328	178	8	𝑦(𝜎(𝜑))𝜅	𝑦(𝜎(𝜑))𝜅	NOUN
cana-328	178	9	−	−	PROPN
cana-328	178	10	𝜌(𝜑)𝑤(𝜑+1	𝜌(𝜑)𝑤(𝜑+1	PROPN
cana-328	178	11	)	)	PUNCT
cana-328	178	12	(	(	PUNCT
cana-328	178	13	𝑦(𝜎(𝜑)))𝜅𝜌(𝜑+1	𝑦(𝜎(𝜑)))𝜅𝜌(𝜑+1	PROPN
cana-328	178	14	)	)	PUNCT
cana-328	178	15	𝛥(𝑦(𝜎(𝜑)))𝜅.	𝛥(𝑦(𝜎(𝜑)))𝜅.	PROPN
cana-328	178	16	(	(	PUNCT
cana-328	178	17	17	17	NUM
cana-328	178	18	)	)	PUNCT
cana-328	178	19	since	since	SCONJ
cana-328	178	20	𝛥(𝑑(𝜑)𝛥(𝑦(𝜑	𝛥(𝑑(𝜑)𝛥(𝑦(𝜑	NUM
cana-328	178	21	)	)	PUNCT
cana-328	178	22	)	)	PUNCT
cana-328	179	1	𝜅	𝜅	X
cana-328	179	2	)	)	PUNCT
cana-328	179	3	is	be	AUX
cana-328	179	4	decreasing	decrease	VERB
cana-328	179	5	,	,	PUNCT
cana-328	179	6	so	so	SCONJ
cana-328	179	7	𝑑(𝜑)𝛥(𝑦(𝜑))𝜅	𝑑(𝜑)𝛥(𝑦(𝜑))𝜅	PROPN
cana-328	179	8	=	=	SYM
cana-328	179	9	𝑑(𝑗)𝛥(𝑦(𝑗))𝜅	𝑑(𝑗)𝛥(𝑦(𝑗))𝜅	PROPN
cana-328	179	10	+	+	CCONJ
cana-328	179	11	∑𝑠=𝑗	∑𝑠=𝑗	PROPN
cana-328	179	12	𝜑−1	𝜑−1	PROPN
cana-328	179	13	 	 	SPACE
cana-328	179	14	𝛥(𝑑(𝑠)𝛥𝑦(𝑠	𝛥(𝑑(𝑠)𝛥𝑦(𝑠	PROPN
cana-328	179	15	)	)	PUNCT
cana-328	179	16	)	)	PUNCT
cana-328	179	17	,	,	PUNCT
cana-328	179	18	𝜑	𝜑	PROPN
cana-328	179	19	≥	≥	VERB
cana-328	179	20	𝑗	𝑗	X
cana-328	179	21	≥	≥	NOUN
cana-328	179	22	𝜑2	𝜑2	PROPN
cana-328	179	23	,	,	PUNCT
cana-328	179	24	which	which	PRON
cana-328	179	25	implies	imply	VERB
cana-328	179	26	𝑑(𝜑)(𝛥𝑦(𝜑))𝜅	𝑑(𝜑)(𝛥𝑦(𝜑))𝜅	NUM
cana-328	179	27	≥	≥	NUM
cana-328	179	28	(	(	PUNCT
cana-328	179	29	𝜑	𝜑	NOUN
cana-328	179	30	−	−	PROPN
cana-328	179	31	𝑗)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑))𝜅	𝑗)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑))𝜅	NOUN
cana-328	179	32	)	)	PUNCT
cana-328	179	33	.	.	PUNCT
cana-328	180	1	and	and	CCONJ
cana-328	180	2	so	so	ADV
cana-328	180	3	(	(	PUNCT
cana-328	180	4	𝛥𝑦(𝜎(𝜑)))𝜅	𝛥𝑦(𝜎(𝜑)))𝜅	NOUN
cana-328	180	5	≥	≥	X
cana-328	180	6	(	(	PUNCT
cana-328	180	7	𝜎(𝜑)−𝑗)𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑))𝜅	𝜎(𝜑)−𝑗)𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑))𝜅	PROPN
cana-328	180	8	)	)	PUNCT
cana-328	180	9	𝑐(𝜑)𝑑(𝜎(𝜑	𝑐(𝜑)𝑑(𝜎(𝜑	NUM
cana-328	180	10	)	)	PUNCT
cana-328	180	11	)	)	PUNCT
cana-328	180	12	.	.	PUNCT
cana-328	181	1	(	(	PUNCT
cana-328	181	2	18	18	NUM
cana-328	181	3	)	)	PUNCT
cana-328	181	4	from	from	ADP
cana-328	181	5	(	(	PUNCT
cana-328	181	6	17	17	NUM
cana-328	181	7	)	)	PUNCT
cana-328	181	8	and	and	CCONJ
cana-328	181	9	(	(	PUNCT
cana-328	181	10	18	18	NUM
cana-328	181	11	)	)	PUNCT
cana-328	181	12	,	,	PUNCT
cana-328	181	13	we	we	PRON
cana-328	181	14	carry	carry	VERB
cana-328	181	15	𝛥𝑤(𝜑	𝛥𝑤(𝜑	PROPN
cana-328	181	16	)	)	PUNCT
cana-328	181	17	≤	≤	NOUN
cana-328	181	18	𝑤(𝜑+1	𝑤(𝜑+1	NUM
cana-328	181	19	)	)	PUNCT
cana-328	181	20	𝜌(𝜑+1	𝜌(𝜑+1	PUNCT
cana-328	181	21	)	)	PUNCT
cana-328	182	1	𝛥𝜌(𝜑)−	𝛥𝜌(𝜑)−	NOUN
cana-328	182	2	𝜌(𝜑)𝑄(𝜑)𝑓(𝑦(𝜎(𝜑	𝜌(𝜑)𝑄(𝜑)𝑓(𝑦(𝜎(𝜑	NOUN
cana-328	182	3	)	)	PUNCT
cana-328	182	4	)	)	PUNCT
cana-328	182	5	)	)	PUNCT
cana-328	182	6	𝑦(𝜎(𝜑))𝜅	𝑦(𝜎(𝜑))𝜅	NOUN
cana-328	182	7	−	−	PROPN
cana-328	182	8	𝜌(𝜑)𝑤(𝜑+1	𝜌(𝜑)𝑤(𝜑+1	PROPN
cana-328	182	9	)	)	PUNCT
cana-328	182	10	(	(	PUNCT
cana-328	182	11	𝑦(𝜎(𝜑)))𝜅𝜌(𝜑+1	𝑦(𝜎(𝜑)))𝜅𝜌(𝜑+1	PROPN
cana-328	182	12	)	)	PUNCT
cana-328	182	13	(	(	PUNCT
cana-328	182	14	𝜎(𝜑)−𝑗)𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑	𝜎(𝜑)−𝑗)𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑	NUM
cana-328	182	15	)	)	PUNCT
cana-328	182	16	)	)	PUNCT
cana-328	183	1	𝜅	𝜅	X
cana-328	183	2	)	)	PUNCT
cana-328	183	3	𝑐(𝜑)𝑑(𝜎(𝜑	𝑐(𝜑)𝑑(𝜎(𝜑	NUM
cana-328	183	4	)	)	PUNCT
cana-328	183	5	)	)	PUNCT
cana-328	183	6	,	,	PUNCT
cana-328	183	7	𝛥𝑤(𝜑	𝛥𝑤(𝜑	PROPN
cana-328	183	8	)	)	PUNCT
cana-328	183	9	≤	≤	NOUN
cana-328	183	10	−	−	NOUN
cana-328	183	11	𝜌(𝜑)𝑄(𝜑)𝑓(𝑦(𝜎(𝜑	𝜌(𝜑)𝑄(𝜑)𝑓(𝑦(𝜎(𝜑	NOUN
cana-328	183	12	)	)	PUNCT
cana-328	183	13	)	)	PUNCT
cana-328	183	14	)	)	PUNCT
cana-328	183	15	𝑦(𝜎(𝜑))𝜅	𝑦(𝜎(𝜑))𝜅	NOUN
cana-328	183	16	+	+	CCONJ
cana-328	183	17	𝛥𝜌(𝜑	𝛥𝜌(𝜑	NOUN
cana-328	183	18	)	)	PUNCT
cana-328	183	19	𝜌(𝜑+1	𝜌(𝜑+1	PUNCT
cana-328	183	20	)	)	PUNCT
cana-328	183	21	𝑤(𝜑+	𝑤(𝜑+	X
cana-328	183	22	1	1	NUM
cana-328	183	23	)	)	PUNCT
cana-328	183	24	−	−	PROPN
cana-328	183	25	(	(	PUNCT
cana-328	183	26	𝜎(𝜑)−𝑗	𝜎(𝜑)−𝑗	NOUN
cana-328	183	27	)	)	PUNCT
cana-328	183	28	𝜌(𝜑+1)𝑐(𝜑)𝑑(𝜎(𝜑	𝜌(𝜑+1)𝑐(𝜑)𝑑(𝜎(𝜑	NUM
cana-328	183	29	)	)	PUNCT
cana-328	183	30	)	)	PUNCT
cana-328	184	1	𝑤2(𝜑+	𝑤2(𝜑+	PROPN
cana-328	184	2	1	1	NUM
cana-328	184	3	)	)	PUNCT
cana-328	184	4	.	.	PUNCT
cana-328	185	1	multiplying	multiply	VERB
cana-328	185	2	both	both	DET
cana-328	185	3	side	side	NOUN
cana-328	185	4	by	by	ADP
cana-328	185	5	𝐻(𝜑	𝐻(𝜑	PROPN
cana-328	185	6	,	,	PUNCT
cana-328	185	7	𝑗	𝑗	NOUN
cana-328	185	8	)	)	PUNCT
cana-328	185	9	and	and	CCONJ
cana-328	185	10	then	then	ADV
cana-328	185	11	summing	sum	VERB
cana-328	185	12	up	up	ADP
cana-328	185	13	from	from	ADP
cana-328	185	14	𝑀	𝑀	PROPN
cana-328	185	15	to	to	ADP
cana-328	185	16	𝜑−	𝜑−	PROPN
cana-328	185	17	1	1	NUM
cana-328	185	18	for	for	ADP
cana-328	185	19	all	all	DET
cana-328	185	20	𝜑	𝜑	NOUN
cana-328	185	21	≥𝑀	≥𝑀	NOUN
cana-328	185	22	,	,	PUNCT
cana-328	185	23	we	we	PRON
cana-328	185	24	obtain	obtain	VERB
cana-328	185	25	∑	∑	ADV
cana-328	185	26	𝐻(𝜑	𝐻(𝜑	PROPN
cana-328	185	27	,	,	PUNCT
cana-328	185	28	𝑗	𝑗	NOUN
cana-328	185	29	)	)	PUNCT
cana-328	185	30	𝜌(𝑗)𝑄(𝑗)𝑓(𝑦(𝜎(𝑗	𝜌(𝑗)𝑄(𝑗)𝑓(𝑦(𝜎(𝑗	PROPN
cana-328	185	31	)	)	PUNCT
cana-328	185	32	)	)	PUNCT
cana-328	185	33	)	)	PUNCT
cana-328	185	34	𝑦(𝜎(𝑗))𝜅	𝑦(𝜎(𝑗))𝜅	NOUN
cana-328	185	35	𝜑−1	𝜑−1	PROPN
cana-328	185	36	𝑗=𝑀	𝑗=𝑀	PRON
cana-328	185	37	   	   	SPACE
cana-328	185	38	≤	≤	NOUN
cana-328	185	39	∑	∑	PUNCT
cana-328	185	40	𝐻(𝜑	𝐻(𝜑	PROPN
cana-328	185	41	,	,	PUNCT
cana-328	185	42	𝑗	𝑗	NOUN
cana-328	185	43	)	)	PUNCT
cana-328	185	44	𝛥𝜌(𝑗	𝛥𝜌(𝑗	ADJ
cana-328	185	45	)	)	PUNCT
cana-328	185	46	𝜌(𝑗+1	𝜌(𝑗+1	NOUN
cana-328	185	47	)	)	PUNCT
cana-328	185	48	𝑤(𝑗	𝑤(𝑗	NOUN
cana-328	186	1	+	+	CCONJ
cana-328	186	2	1	1	X
cana-328	186	3	)	)	PUNCT
cana-328	186	4	𝜑−1	𝜑−1	PROPN
cana-328	186	5	𝑗=𝑀	𝑗=𝑀	ADJ
cana-328	186	6	   	   	SPACE
cana-328	186	7	−	−	PROPN
cana-328	186	8	∑	∑	PROPN
cana-328	186	9	𝐻(𝜑	𝐻(𝜑	PROPN
cana-328	186	10	,	,	PUNCT
cana-328	186	11	𝑗	𝑗	NOUN
cana-328	186	12	)	)	PUNCT
cana-328	186	13	(	(	PUNCT
cana-328	186	14	𝜎(𝑗)−𝑗	𝜎(𝑗)−𝑗	PROPN
cana-328	186	15	)	)	PUNCT
cana-328	186	16	𝜌(𝑗+1)𝑐(𝑗)𝑑(𝜎(𝑗	𝜌(𝑗+1)𝑐(𝑗)𝑑(𝜎(𝑗	NUM
cana-328	186	17	)	)	PUNCT
cana-328	186	18	)	)	PUNCT
cana-328	187	1	𝑤2(𝑗	𝑤2(𝑗	VERB
cana-328	187	2	+	+	NOUN
cana-328	187	3	1	1	X
cana-328	187	4	)	)	PUNCT
cana-328	187	5	𝜑−1	𝜑−1	PROPN
cana-328	187	6	𝑗=𝑀	𝑗=𝑀	NOUN
cana-328	187	7	  	  	SPACE
cana-328	187	8	−	−	PROPN
cana-328	187	9	∑	∑	PROPN
cana-328	187	10	𝐻(𝜑	𝐻(𝜑	PROPN
cana-328	187	11	,	,	PUNCT
cana-328	187	12	𝑗)𝑤(𝑗	𝑗)𝑤(𝑗	ADJ
cana-328	187	13	)	)	PUNCT
cana-328	187	14	𝜑−1	𝜑−1	PROPN
cana-328	187	15	𝑗=𝑀	𝑗=𝑀	PROPN
cana-328	187	16	 	 	SPACE
cana-328	187	17	.	.	PUNCT
cana-328	188	1	(	(	PUNCT
cana-328	188	2	19	19	NUM
cana-328	188	3	)	)	PUNCT
cana-328	188	4	using	use	VERB
cana-328	188	5	summation	summation	NOUN
cana-328	188	6	by	by	ADP
cana-328	188	7	parts	part	NOUN
cana-328	188	8	,	,	PUNCT
cana-328	188	9	we	we	PRON
cana-328	188	10	see	see	VERB
cana-328	188	11	that	that	SCONJ
cana-328	188	12	−∑	−∑	PROPN
cana-328	188	13	 	 	SPACE
cana-328	188	14	𝑤(𝑗)𝐻(𝜑	𝑤(𝑗)𝐻(𝜑	PROPN
cana-328	188	15	,	,	PUNCT
cana-328	188	16	𝑗	𝑗	NOUN
cana-328	188	17	)	)	PUNCT
cana-328	188	18	𝜑−1	𝜑−1	PROPN
cana-328	188	19	𝑗=𝑀	𝑗=𝑀	NUM
cana-328	188	20	=	=	SYM
cana-328	188	21	𝐻(𝜑,𝑀)𝑤(𝑀	𝐻(𝜑,𝑀)𝑤(𝑀	X
cana-328	188	22	)	)	PUNCT
cana-328	188	23	+	+	CCONJ
cana-328	188	24	∑	∑	PUNCT
cana-328	188	25	𝑤(𝑗	𝑤(𝑗	PROPN
cana-328	188	26	+	+	CCONJ
cana-328	188	27	1)𝛥2𝐻(𝜑	1)𝛥2𝐻(𝜑	NUM
cana-328	188	28	,	,	PUNCT
cana-328	188	29	𝑗	𝑗	X
cana-328	188	30	)	)	PUNCT
cana-328	188	31	𝜑−1	𝜑−1	PROPN
cana-328	188	32	𝑗=𝑀	𝑗=𝑀	PROPN
cana-328	188	33	 	 	SPACE
cana-328	188	34	.	.	PUNCT
cana-328	189	1	(	(	PUNCT
cana-328	189	2	20	20	NUM
cana-328	189	3	)	)	PUNCT
cana-328	189	4	substituting	substitute	VERB
cana-328	189	5	(	(	PUNCT
cana-328	189	6	20	20	NUM
cana-328	189	7	)	)	PUNCT
cana-328	189	8	in	in	ADP
cana-328	189	9	(	(	PUNCT
cana-328	189	10	19	19	NUM
cana-328	189	11	)	)	PUNCT
cana-328	189	12	,	,	PUNCT
cana-328	189	13	we	we	PRON
cana-328	189	14	carry	carry	VERB
cana-328	189	15	∑	∑	PROPN
cana-328	189	16	𝐻(𝜑	𝐻(𝜑	PROPN
cana-328	189	17	,	,	PUNCT
cana-328	189	18	𝑗	𝑗	NOUN
cana-328	189	19	)	)	PUNCT
cana-328	189	20	𝜌(𝑗)𝑄(𝑗)𝑓(𝑦(𝜎(𝑗	𝜌(𝑗)𝑄(𝑗)𝑓(𝑦(𝜎(𝑗	PROPN
cana-328	189	21	)	)	PUNCT
cana-328	189	22	)	)	PUNCT
cana-328	189	23	)	)	PUNCT
cana-328	190	1	𝑦(𝜎(𝑗))𝜅	𝑦(𝜎(𝑗))𝜅	NOUN
cana-328	191	1	𝜑−1	𝜑−1	PROPN
cana-328	191	2	𝑗=𝑀	𝑗=𝑀	PRON
cana-328	191	3	   	   	SPACE
cana-328	191	4	≤	≤	NOUN
cana-328	191	5	𝐻(𝜑,𝑀)𝑤(𝑀	𝐻(𝜑,𝑀)𝑤(𝑀	NUM
cana-328	191	6	)	)	PUNCT
cana-328	191	7	+	+	CCONJ
cana-328	191	8	∑	∑	PUNCT
cana-328	191	9	ℎ(𝜑	ℎ(𝜑	NOUN
cana-328	191	10	,	,	PUNCT
cana-328	191	11	𝑗)𝑤(𝑗	𝑗)𝑤(𝑗	VERB
cana-328	191	12	+	+	CCONJ
cana-328	191	13	1	1	X
cana-328	191	14	)	)	PUNCT
cana-328	191	15	𝜑−1	𝜑−1	PROPN
cana-328	191	16	𝑗=𝑀	𝑗=𝑀	ADJ
cana-328	191	17	   	   	SPACE
cana-328	191	18	−∑	−∑	PROPN
cana-328	191	19	𝐻(𝜑	𝐻(𝜑	PROPN
cana-328	191	20	,	,	PUNCT
cana-328	191	21	𝑗	𝑗	NOUN
cana-328	191	22	)	)	PUNCT
cana-328	191	23	(	(	PUNCT
cana-328	191	24	𝜎(𝑗)−𝑗	𝜎(𝑗)−𝑗	PROPN
cana-328	191	25	)	)	PUNCT
cana-328	191	26	𝜌(𝑗+1)𝑐(𝑗)𝑑(𝜎(𝑗	𝜌(𝑗+1)𝑐(𝑗)𝑑(𝜎(𝑗	NUM
cana-328	191	27	)	)	PUNCT
cana-328	191	28	)	)	PUNCT
cana-328	192	1	𝑤2(𝑗	𝑤2(𝑗	VERB
cana-328	192	2	+	+	NOUN
cana-328	192	3	1	1	X
cana-328	192	4	)	)	PUNCT
cana-328	192	5	𝜑−1	𝜑−1	PROPN
cana-328	192	6	𝑗=𝑀	𝑗=𝑀	PROPN
cana-328	192	7	   	   	SPACE
cana-328	192	8	,	,	PUNCT
cana-328	192	9	where	where	SCONJ
cana-328	192	10	ℎ(𝜑	ℎ(𝜑	X
cana-328	192	11	,	,	PUNCT
cana-328	192	12	𝑗	𝑗	NOUN
cana-328	192	13	)	)	PUNCT
cana-328	192	14	=	=	SYM
cana-328	192	15	𝛥2𝐻(𝜑	𝛥2𝐻(𝜑	PROPN
cana-328	192	16	,	,	PUNCT
cana-328	192	17	𝑗	𝑗	NOUN
cana-328	192	18	)	)	PUNCT
cana-328	192	19	+	+	NUM
cana-328	192	20	𝐻(𝜑	𝐻(𝜑	PROPN
cana-328	192	21	,	,	PUNCT
cana-328	192	22	𝑗	𝑗	NOUN
cana-328	192	23	)	)	PUNCT
cana-328	192	24	𝛥𝜌(𝑗	𝛥𝜌(𝑗	ADJ
cana-328	192	25	)	)	PUNCT
cana-328	192	26	𝜌(𝑗+1	𝜌(𝑗+1	NOUN
cana-328	192	27	)	)	PUNCT
cana-328	192	28	.	.	PUNCT
cana-328	193	1	∑	∑	PUNCT
cana-328	193	2	𝐻(𝜑	𝐻(𝜑	PROPN
cana-328	193	3	,	,	PUNCT
cana-328	193	4	𝑗	𝑗	NOUN
cana-328	193	5	)	)	PUNCT
cana-328	193	6	𝜌(𝑗)𝑄(𝑗)𝑓(𝑦(𝜎(𝑗	𝜌(𝑗)𝑄(𝑗)𝑓(𝑦(𝜎(𝑗	PROPN
cana-328	193	7	)	)	PUNCT
cana-328	193	8	)	)	PUNCT
cana-328	193	9	)	)	PUNCT
cana-328	193	10	𝑦(𝜎(𝑗))𝜅	𝑦(𝜎(𝑗))𝜅	NOUN
cana-328	194	1	𝜑−1	𝜑−1	PROPN
cana-328	194	2	𝑗=𝑀	𝑗=𝑀	PRON
cana-328	194	3	   	   	SPACE
cana-328	194	4	≤	≤	NOUN
cana-328	194	5	𝐻(𝜑,𝑀)𝑤(𝑀	𝐻(𝜑,𝑀)𝑤(𝑀	NUM
cana-328	194	6	)	)	PUNCT
cana-328	194	7	+	+	CCONJ
cana-328	194	8	∑	∑	PUNCT
cana-328	194	9	ℎ2(𝜑,𝑗)𝜌(𝑗+1)𝑐(𝑗)𝑑(𝜎(𝑗	ℎ2(𝜑,𝑗)𝜌(𝑗+1)𝑐(𝑗)𝑑(𝜎(𝑗	ADJ
cana-328	194	10	)	)	PUNCT
cana-328	194	11	)	)	PUNCT
cana-328	194	12	4(𝜎(𝑗)−𝑗)𝐻(𝜑,𝑗	4(𝜎(𝑗)−𝑗)𝐻(𝜑,𝑗	NUM
cana-328	194	13	)	)	PUNCT
cana-328	194	14	𝜑−1	𝜑−1	PROPN
cana-328	194	15	𝑗=𝑀	𝑗=𝑀	PROPN
cana-328	194	16	   	   	SPACE
cana-328	194	17	,	,	PUNCT
cana-328	194	18	or	or	CCONJ
cana-328	194	19	∑	∑	ADP
cana-328	194	20	[	[	X
cana-328	194	21	𝐻(𝜑	𝐻(𝜑	NOUN
cana-328	194	22	,	,	PUNCT
cana-328	194	23	𝑗	𝑗	NOUN
cana-328	194	24	)	)	PUNCT
cana-328	194	25	𝜌(𝑗)𝑄(𝑗)𝑓(𝑦(𝜎(𝑗	𝜌(𝑗)𝑄(𝑗)𝑓(𝑦(𝜎(𝑗	PROPN
cana-328	194	26	)	)	PUNCT
cana-328	194	27	)	)	PUNCT
cana-328	194	28	)	)	PUNCT
cana-328	194	29	𝑦(𝜎(𝑗))𝜅	𝑦(𝜎(𝑗))𝜅	NOUN
cana-328	194	30	−	−	PROPN
cana-328	194	31	ℎ2(𝜑,𝑗)𝜌(𝑗+1)𝑐(𝑗)𝑑(𝜎(𝑗	ℎ2(𝜑,𝑗)𝜌(𝑗+1)𝑐(𝑗)𝑑(𝜎(𝑗	PROPN
cana-328	194	32	)	)	PUNCT
cana-328	194	33	)	)	PUNCT
cana-328	194	34	4(𝜎(𝑗)−𝑗)𝐻(𝜑,𝑗	4(𝜎(𝑗)−𝑗)𝐻(𝜑,𝑗	NUM
cana-328	194	35	)	)	PUNCT
cana-328	194	36	]	]	PUNCT
cana-328	195	1	𝜑−1	𝜑−1	PROPN
cana-328	195	2	𝑗=𝑀	𝑗=𝑀	PROPN
cana-328	195	3	  	  	SPACE
cana-328	195	4	≤	≤	NOUN
cana-328	195	5	𝐻(𝜑,𝑀)𝑤(𝑀	𝐻(𝜑,𝑀)𝑤(𝑀	NUM
cana-328	195	6	)	)	PUNCT
cana-328	195	7	.	.	PUNCT
cana-328	196	1	(	(	PUNCT
cana-328	196	2	21	21	X
cana-328	196	3	)	)	PUNCT
cana-328	196	4	taking	take	VERB
cana-328	196	5	𝑙𝑖𝑚𝑠𝑢𝑝	𝑙𝑖𝑚𝑠𝑢𝑝	ADV
cana-328	196	6	as	as	ADP
cana-328	196	7	𝜑	𝜑	PROPN
cana-328	196	8	→∞	→∞	PROPN
cana-328	196	9	,	,	PUNCT
cana-328	196	10	we	we	PRON
cana-328	196	11	obtain	obtain	VERB
cana-328	196	12	𝑙𝑖𝑚	𝑙𝑖𝑚	ADJ
cana-328	196	13	 	 	SPACE
cana-328	196	14	𝑠𝑢𝑝𝜑→∞	𝑠𝑢𝑝𝜑→∞	PROPN
cana-328	196	15	  	  	SPACE
cana-328	196	16	1	1	NUM
cana-328	196	17	𝐻(𝜑,𝑀	𝐻(𝜑,𝑀	NOUN
cana-328	196	18	)	)	PUNCT
cana-328	196	19	∑	∑	PUNCT
cana-328	197	1	[	[	X
cana-328	197	2	𝐻(𝜑	𝐻(𝜑	SYM
cana-328	197	3	,	,	PUNCT
cana-328	197	4	𝑗	𝑗	NOUN
cana-328	197	5	)	)	PUNCT
cana-328	197	6	𝜌(𝑗)𝑄(𝑗)𝑓(𝑦(𝜎(𝑗	𝜌(𝑗)𝑄(𝑗)𝑓(𝑦(𝜎(𝑗	PROPN
cana-328	197	7	)	)	PUNCT
cana-328	197	8	)	)	PUNCT
cana-328	197	9	)	)	PUNCT
cana-328	197	10	𝑦(𝜎(𝑗))𝜅	𝑦(𝜎(𝑗))𝜅	NOUN
cana-328	197	11	−	−	PROPN
cana-328	197	12	ℎ2(𝜑,𝑗)𝜌(𝑗+1)𝑐(𝑗)𝑑(𝜎(𝑗	ℎ2(𝜑,𝑗)𝜌(𝑗+1)𝑐(𝑗)𝑑(𝜎(𝑗	PROPN
cana-328	197	13	)	)	PUNCT
cana-328	197	14	)	)	PUNCT
cana-328	198	1	4(𝜎(𝑗)−𝑗)𝐻(𝜑,𝑗	4(𝜎(𝑗)−𝑗)𝐻(𝜑,𝑗	NUM
cana-328	198	2	)	)	PUNCT
cana-328	198	3	]	]	PUNCT
cana-328	199	1	𝜑−1	𝜑−1	PROPN
cana-328	199	2	𝑗=𝑀	𝑗=𝑀	PROPN
cana-328	199	3	  	  	SPACE
cana-328	199	4	≤	≤	PROPN
cana-328	199	5	𝑤(𝑀	𝑤(𝑀	NUM
cana-328	199	6	)	)	PUNCT
cana-328	199	7	.	.	PUNCT
cana-328	200	1	which	which	PRON
cana-328	200	2	contradicts	contradict	VERB
cana-328	200	3	(	(	PUNCT
cana-328	200	4	15	15	NUM
cana-328	200	5	)	)	PUNCT
cana-328	200	6	.	.	PUNCT
cana-328	201	1	communications	communication	NOUN
cana-328	201	2	on	on	ADP
cana-328	201	3	applied	apply	VERB
cana-328	201	4	nonlinear	nonlinear	ADJ
cana-328	201	5	analysis	analysis	NOUN
cana-328	201	6	issn	issn	NOUN
cana-328	201	7	:	:	PUNCT
cana-328	201	8	1074	1074	NUM
cana-328	201	9	-	-	PUNCT
cana-328	201	10	133x	133x	NUM
cana-328	201	11	vol	vol	NOUN
cana-328	201	12	31	31	NUM
cana-328	201	13	no	no	NOUN
cana-328	201	14	.	.	NOUN
cana-328	201	15	1	1	NUM
cana-328	201	16	(	(	PUNCT
cana-328	201	17	2024	2024	NUM
cana-328	201	18	)	)	PUNCT
cana-328	201	19	97	97	NUM
cana-328	201	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-328	202	1	now	now	ADV
cana-328	202	2	we	we	PRON
cana-328	202	3	take	take	VERB
cana-328	202	4	𝑦(𝜑	𝑦(𝜑	ADV
cana-328	202	5	)	)	PUNCT
cana-328	202	6	∈	∈	PROPN
cana-328	202	7	𝑁0	𝑁0	VERB
cana-328	202	8	,	,	PUNCT
cana-328	202	9	summing	sum	VERB
cana-328	202	10	(	(	PUNCT
cana-328	202	11	4	4	NUM
cana-328	202	12	)	)	PUNCT
cana-328	202	13	from	from	ADP
cana-328	202	14	𝜑	𝜑	PRON
cana-328	202	15	to	to	ADP
cana-328	202	16	∞	∞	PROPN
cana-328	202	17	,	,	PUNCT
cana-328	202	18	we	we	PRON
cana-328	202	19	have	have	VERB
cana-328	202	20	𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑	𝑐(𝜑)𝛥(𝑑(𝜑)(𝛥𝑦(𝜑	ADJ
cana-328	202	21	)	)	PUNCT
cana-328	202	22	)	)	PUNCT
cana-328	203	1	𝜅	𝜅	X
cana-328	203	2	)	)	PUNCT
cana-328	203	3	≥	≥	NOUN
cana-328	203	4	∑	∑	PROPN
cana-328	203	5	𝑄(𝑡)𝑓	𝑄(𝑡)𝑓	NOUN
cana-328	203	6	(	(	PUNCT
cana-328	203	7	𝑦(𝜎(𝑡	𝑦(𝜎(𝑡	NOUN
cana-328	203	8	)	)	PUNCT
cana-328	203	9	)	)	PUNCT
cana-328	203	10	)	)	PUNCT
cana-328	204	1	∞	∞	NUM
cana-328	204	2	𝑡=𝜑	𝑡=𝜑	PUNCT
cana-328	204	3	,	,	PUNCT
cana-328	204	4	𝛥(𝑑(𝜑)(𝛥𝑦(𝜑	𝛥(𝑑(𝜑)(𝛥𝑦(𝜑	NOUN
cana-328	204	5	)	)	PUNCT
cana-328	204	6	)	)	PUNCT
cana-328	205	1	𝜅	𝜅	X
cana-328	205	2	)	)	PUNCT
cana-328	205	3	   	   	SPACE
cana-328	205	4	≥	≥	NOUN
cana-328	205	5	1	1	NUM
cana-328	205	6	𝑐(𝜑	𝑐(𝜑	NOUN
cana-328	205	7	)	)	PUNCT
cana-328	205	8	∑	∑	DET
cana-328	205	9	  	  	SPACE
cana-328	205	10	𝑄(𝑡)𝑓(𝑦(𝜎(𝑡	𝑄(𝑡)𝑓(𝑦(𝜎(𝑡	PROPN
cana-328	205	11	)	)	PUNCT
cana-328	205	12	)	)	PUNCT
cana-328	205	13	)	)	PUNCT
cana-328	206	1	∞	∞	NUM
cana-328	206	2	𝑡=𝜑	𝑡=𝜑	PUNCT
cana-328	206	3	.	.	PUNCT
cana-328	207	1	again	again	ADV
cana-328	207	2	summing	sum	VERB
cana-328	207	3	from	from	ADP
cana-328	207	4	𝜑	𝜑	PRON
cana-328	207	5	to	to	ADP
cana-328	207	6	∞	∞	PROPN
cana-328	207	7	(	(	PUNCT
cana-328	207	8	𝛥𝑦(𝜑))𝜅	𝛥𝑦(𝜑))𝜅	NOUN
cana-328	207	9	≤	≤	NOUN
cana-328	207	10	−	−	ADP
cana-328	207	11	1	1	NUM
cana-328	207	12	𝑑(𝜑	𝑑(𝜑	VERB
cana-328	207	13	)	)	PUNCT
cana-328	207	14	∑	∑	PROPN
cana-328	207	15	1	1	NUM
cana-328	207	16	𝑐(𝑡	𝑐(𝑡	PROPN
cana-328	207	17	)	)	PUNCT
cana-328	207	18	∞	∞	PUNCT
cana-328	207	19	𝑡=𝜑	𝑡=𝜑	PUNCT
cana-328	207	20	  	  	SPACE
cana-328	207	21	∑	∑	PROPN
cana-328	207	22	𝑄(𝑗)𝑓	𝑄(𝑗)𝑓	PROPN
cana-328	207	23	(	(	PUNCT
cana-328	207	24	𝑦(𝜎(𝑗)))∞	𝑦(𝜎(𝑗)))∞	PROPN
cana-328	207	25	𝑗=𝑡	𝑗=𝑡	PROPN
cana-328	207	26	  	  	SPACE
cana-328	207	27	,	,	PUNCT
cana-328	207	28	𝛥𝑦(𝜑	𝛥𝑦(𝜑	NOUN
cana-328	207	29	)	)	PUNCT
cana-328	207	30	≤	≤	NOUN
cana-328	208	1	−	−	PROPN
cana-328	208	2	(	(	PUNCT
cana-328	208	3	1	1	NUM
cana-328	208	4	𝑑(𝜑1	𝑑(𝜑1	ADJ
cana-328	208	5	)	)	PUNCT
cana-328	208	6	∑	∑	PROPN
cana-328	208	7	1	1	NUM
cana-328	208	8	𝑐(𝑡	𝑐(𝑡	PROPN
cana-328	208	9	)	)	PUNCT
cana-328	208	10	∞	∞	PUNCT
cana-328	208	11	𝑡=𝜑	𝑡=𝜑	PUNCT
cana-328	208	12	  	  	SPACE
cana-328	208	13	∑	∑	PROPN
cana-328	208	14	𝑄(𝑗)𝑓(𝑦(𝜎(𝑗)))∞	𝑄(𝑗)𝑓(𝑦(𝜎(𝑗)))∞	PROPN
cana-328	208	15	𝑗=𝑡	𝑗=𝑡	PROPN
cana-328	208	16	  	  	SPACE
cana-328	208	17	)	)	PUNCT
cana-328	208	18	1	1	NUM
cana-328	208	19	𝜅	𝜅	NOUN
cana-328	208	20	.	.	PUNCT
cana-328	209	1	summing	sum	VERB
cana-328	209	2	once	once	ADV
cana-328	209	3	again	again	ADV
cana-328	209	4	from	from	ADP
cana-328	209	5	𝜑1	𝜑1	NOUN
cana-328	209	6	to	to	ADP
cana-328	209	7	∞	∞	NUM
cana-328	209	8	,	,	PUNCT
cana-328	209	9	we	we	PRON
cana-328	209	10	obtain	obtain	VERB
cana-328	209	11	𝑦(𝜑1	𝑦(𝜑1	ADV
cana-328	209	12	)	)	PUNCT
cana-328	209	13	≥	≥	NOUN
cana-328	209	14	∑	∑	PROPN
cana-328	209	15	1	1	NUM
cana-328	209	16	𝑑(𝑡	𝑑(𝑡	PROPN
cana-328	209	17	)	)	PUNCT
cana-328	209	18	(	(	PUNCT
cana-328	209	19	∑	∑	ADV
cana-328	209	20	1	1	NUM
cana-328	209	21	𝑐(𝑗	𝑐(𝑗	NOUN
cana-328	209	22	)	)	PUNCT
cana-328	209	23	∞	∞	NUM
cana-328	210	1	𝑗=𝑡	𝑗=𝑡	PROPN
cana-328	210	2	  	  	SPACE
cana-328	210	3	∑	∑	PROPN
cana-328	210	4	𝑄(𝑖)𝑓(𝑦(𝜎(𝑖)))∞	𝑄(𝑖)𝑓(𝑦(𝜎(𝑖)))∞	PROPN
cana-328	210	5	𝑖=𝑗	𝑖=𝑗	PROPN
cana-328	210	6	  	  	SPACE
cana-328	210	7	)	)	PUNCT
cana-328	210	8	1	1	NUM
cana-328	210	9	𝜅∞	𝜅∞	NOUN
cana-328	210	10	𝑡=𝜑1	𝑡=𝜑1	PROPN
cana-328	210	11	,	,	PUNCT
cana-328	210	12	which	which	PRON
cana-328	210	13	contradicts	contradict	VERB
cana-328	210	14	(	(	PUNCT
cana-328	210	15	16	16	NUM
cana-328	210	16	)	)	PUNCT
cana-328	210	17	.	.	PUNCT
cana-328	211	1	therefore	therefore	ADV
cana-328	211	2	,	,	PUNCT
cana-328	211	3	every	every	DET
cana-328	211	4	solution	solution	NOUN
cana-328	211	5	of	of	ADP
cana-328	211	6	(	(	PUNCT
cana-328	211	7	1	1	NUM
cana-328	211	8	)	)	PUNCT
cana-328	211	9	is	be	AUX
cana-328	211	10	oscillatory	oscillatory	ADJ
cana-328	211	11	.	.	PUNCT
cana-328	212	1	hence	hence	ADV
cana-328	212	2	the	the	DET
cana-328	212	3	proof	proof	NOUN
cana-328	212	4	.	.	PUNCT
cana-328	213	1	3	3	NUM
cana-328	213	2	examples	example	NOUN
cana-328	213	3	example	example	NOUN
cana-328	213	4	3.1	3.1	NUM
cana-328	213	5	.	.	PUNCT
cana-328	214	1	look	look	VERB
cana-328	214	2	into	into	ADP
cana-328	214	3	the	the	DET
cana-328	214	4	equation	equation	NOUN
cana-328	214	5	𝛥(𝜑2𝛥(𝜑(𝛥𝑦(𝜑))))−	𝛥(𝜑2𝛥(𝜑(𝛥𝑦(𝜑))))−	X
cana-328	214	6	(	(	PUNCT
cana-328	214	7	2𝜑+1)(𝜑+1	2𝜑+1)(𝜑+1	NUM
cana-328	214	8	)	)	PUNCT
cana-328	214	9	𝜑+2	𝜑+2	NUM
cana-328	214	10	(	(	PUNCT
cana-328	214	11	𝛥(𝜑	𝛥(𝜑	X
cana-328	214	12	+	+	PROPN
cana-328	214	13	1	1	NUM
cana-328	214	14	)	)	PUNCT
cana-328	214	15	)	)	PUNCT
cana-328	215	1	+	+	CCONJ
cana-328	215	2	𝜑+1	𝜑+1	X
cana-328	215	3	𝜑+2	𝜑+2	NUM
cana-328	215	4	𝑓(𝑦(3𝜑	𝑓(𝑦(3𝜑	NUM
cana-328	215	5	)	)	PUNCT
cana-328	215	6	)	)	PUNCT
cana-328	215	7	)	)	PUNCT
cana-328	216	1	=	=	PUNCT
cana-328	216	2	0,𝜑	0,𝜑	NOUN
cana-328	216	3	≥	≥	NOUN
cana-328	216	4	1	1	NUM
cana-328	216	5	.	.	PUNCT
cana-328	217	1	(	(	PUNCT
cana-328	217	2	22	22	NUM
cana-328	217	3	)	)	PUNCT
cana-328	217	4	here	here	ADV
cana-328	217	5	𝑎(𝜑	𝑎(𝜑	NUM
cana-328	217	6	)	)	PUNCT
cana-328	217	7	=	=	SYM
cana-328	217	8	𝜑2	𝜑2	VERB
cana-328	217	9	,	,	PUNCT
cana-328	217	10	𝑏(𝜑	𝑏(𝜑	NOUN
cana-328	217	11	)	)	PUNCT
cana-328	217	12	=	=	SYM
cana-328	217	13	𝜑	𝜑	NOUN
cana-328	217	14	,	,	PUNCT
cana-328	217	15	𝑝(𝜑	𝑝(𝜑	PROPN
cana-328	217	16	)	)	PUNCT
cana-328	217	17	=	=	SYM
cana-328	217	18	(	(	PUNCT
cana-328	217	19	2𝜑+1)(𝜑+1	2𝜑+1)(𝜑+1	NUM
cana-328	217	20	)	)	PUNCT
cana-328	217	21	𝜑+2	𝜑+2	NUM
cana-328	217	22	,	,	PUNCT
cana-328	217	23	𝑞(𝜑	𝑞(𝜑	NUM
cana-328	217	24	)	)	PUNCT
cana-328	217	25	=	=	PUNCT
cana-328	218	1	𝜑+1	𝜑+1	X
cana-328	218	2	𝜑+2	𝜑+2	NUM
cana-328	218	3	,	,	PUNCT
cana-328	218	4	𝜅	𝜅	X
cana-328	218	5	=	=	SYM
cana-328	218	6	1	1	NUM
cana-328	218	7	,	,	PUNCT
cana-328	218	8	𝜎(𝜑	𝜎(𝜑	NOUN
cana-328	218	9	)	)	PUNCT
cana-328	218	10	=	=	SYM
cana-328	218	11	3𝜑	3𝜑	NUM
cana-328	218	12	,	,	PUNCT
cana-328	218	13	𝑓(𝑦	𝑓(𝑦	PROPN
cana-328	218	14	)	)	PUNCT
cana-328	219	1	=	=	SYM
cana-328	219	2	𝑦𝜅	𝑦𝜅	NOUN
cana-328	219	3	,	,	PUNCT
cana-328	219	4	𝑐(𝜑	𝑐(𝜑	PROPN
cana-328	219	5	)	)	PUNCT
cana-328	219	6	=	=	PUNCT
cana-328	220	1	𝜑2(𝜑	𝜑2(𝜑	PROPN
cana-328	221	1	+	+	NUM
cana-328	221	2	1)(𝜑	1)(𝜑	NUM
cana-328	221	3	+	+	CCONJ
cana-328	221	4	2	2	NUM
cana-328	221	5	)	)	PUNCT
cana-328	221	6	,	,	PUNCT
cana-328	221	7	𝑑(𝜑	𝑑(𝜑	ADJ
cana-328	221	8	)	)	PUNCT
cana-328	221	9	=	=	SYM
cana-328	221	10	𝜑	𝜑	PROPN
cana-328	221	11	𝜑+1	𝜑+1	X
cana-328	221	12	,	,	PUNCT
cana-328	221	13	𝑄(𝜑	𝑄(𝜑	NUM
cana-328	221	14	)	)	PUNCT
cana-328	221	15	=	=	PUNCT
cana-328	221	16	𝜑	𝜑	PROPN
cana-328	221	17	+	+	NOUN
cana-328	221	18	1	1	NUM
cana-328	221	19	,	,	PUNCT
cana-328	221	20	𝐶(𝜑	𝐶(𝜑	NOUN
cana-328	221	21	)	)	PUNCT
cana-328	221	22	≈	≈	PROPN
cana-328	221	23	1	1	NUM
cana-328	221	24	𝜑2(𝜑+1)(𝜑+2	𝜑2(𝜑+1)(𝜑+2	NUM
cana-328	221	25	)	)	PUNCT
cana-328	221	26	,	,	PUNCT
cana-328	221	27	𝐷(𝜑	𝐷(𝜑	PRON
cana-328	221	28	)	)	PUNCT
cana-328	222	1	≈	≈	PROPN
cana-328	222	2	𝜑+1	𝜑+1	PUNCT
cana-328	222	3	𝜑	𝜑	NOUN
cana-328	222	4	,	,	PUNCT
cana-328	222	5	𝐸(𝜑	𝐸(𝜑	NOUN
cana-328	222	6	)	)	PUNCT
cana-328	223	1	≈	≈	PROPN
cana-328	223	2	1	1	NUM
cana-328	223	3	𝜑3(𝜑+2	𝜑3(𝜑+2	PROPN
cana-328	223	4	)	)	PUNCT
cana-328	223	5	.	.	PUNCT
cana-328	224	1	now	now	ADV
cana-328	224	2	∑	∑	ADP
cana-328	224	3	1	1	NUM
cana-328	224	4	𝑎(𝑠	𝑎(𝑠	PROPN
cana-328	224	5	)	)	PUNCT
cana-328	224	6	∞	∞	PROPN
cana-328	224	7	𝑠=1	𝑠=1	PUNCT
cana-328	224	8	  	  	SPACE
cana-328	224	9	=	=	SYM
cana-328	224	10	∑	∑	PROPN
cana-328	224	11	1	1	NUM
cana-328	224	12	𝑠2	𝑠2	NOUN
cana-328	224	13	∞	∞	PROPN
cana-328	224	14	𝑠−1	𝑠−1	PROPN
cana-328	224	15	  	  	SPACE
cana-328	224	16	<	<	X
cana-328	224	17	∞	∞	PROPN
cana-328	224	18	,	,	PUNCT
cana-328	224	19	∑	∑	PROPN
cana-328	224	20	1	1	NUM
cana-328	224	21	𝑏	𝑏	NOUN
cana-328	224	22	1	1	NUM
cana-328	224	23	𝜅(𝑠	𝜅(𝑠	NUM
cana-328	224	24	)	)	PUNCT
cana-328	224	25	∞	∞	PROPN
cana-328	224	26	𝑠=1	𝑠=1	PUNCT
cana-328	224	27	  	  	SPACE
cana-328	224	28	=	=	SYM
cana-328	224	29	∑	∑	PUNCT
cana-328	224	30	  	  	SPACE
cana-328	224	31	1	1	NUM
cana-328	224	32	𝑠	𝑠	NUM
cana-328	224	33	∞	∞	NUM
cana-328	224	34	𝑠=1	𝑠=1	PUNCT
cana-328	224	35	=	=	SYM
cana-328	224	36	∞	∞	PROPN
cana-328	224	37	,	,	PUNCT
cana-328	224	38	and	and	CCONJ
cana-328	224	39	the	the	DET
cana-328	224	40	auxiliary	auxiliary	ADJ
cana-328	224	41	second	second	ADJ
cana-328	224	42	order	order	NOUN
cana-328	224	43	equation	equation	NOUN
cana-328	224	44	(	(	PUNCT
cana-328	224	45	4	4	X
cana-328	224	46	)	)	PUNCT
cana-328	224	47	becomes	become	VERB
cana-328	224	48	,	,	PUNCT
cana-328	224	49	𝛥(𝜑2𝛥𝑧(𝜑))−	𝛥(𝜑2𝛥𝑧(𝜑))−	NUM
cana-328	224	50	(	(	PUNCT
cana-328	224	51	2𝜑+1)(𝜑+1	2𝜑+1)(𝜑+1	NUM
cana-328	224	52	)	)	PUNCT
cana-328	224	53	(	(	PUNCT
cana-328	224	54	𝜑+2	𝜑+2	NUM
cana-328	224	55	)	)	PUNCT
cana-328	224	56	𝑧(𝜑+	𝑧(𝜑+	PROPN
cana-328	224	57	1	1	NUM
cana-328	224	58	)	)	PUNCT
cana-328	224	59	=	=	SYM
cana-328	225	1	0	0	X
cana-328	225	2	.	.	PUNCT
cana-328	226	1	(	(	PUNCT
cana-328	226	2	23	23	NUM
cana-328	226	3	)	)	PUNCT
cana-328	226	4	it	it	PRON
cana-328	226	5	has	have	VERB
cana-328	226	6	a	a	DET
cana-328	226	7	nonoscillatory	nonoscillatory	ADJ
cana-328	226	8	solution	solution	NOUN
cana-328	226	9	𝑧(𝜑	𝑧(𝜑	NOUN
cana-328	226	10	)	)	PUNCT
cana-328	227	1	=	=	PUNCT
cana-328	227	2	𝜑	𝜑	PROPN
cana-328	228	1	+	+	NOUN
cana-328	229	1	1	1	X
cana-328	229	2	.	.	PUNCT
cana-328	230	1	moreover	moreover	ADV
cana-328	230	2	,	,	PUNCT
cana-328	230	3	the	the	DET
cana-328	230	4	positive	positive	ADJ
cana-328	230	5	sequence	sequence	NOUN
cana-328	230	6	𝜌(𝜑	𝜌(𝜑	NOUN
cana-328	230	7	)	)	PUNCT
cana-328	231	1	=	=	SYM
cana-328	231	2	𝑙	𝑙	NOUN
cana-328	231	3	which	which	PRON
cana-328	231	4	implies	imply	VERB
cana-328	231	5	that	that	SCONJ
cana-328	231	6	𝛥𝜌(𝜑	𝛥𝜌(𝜑	NOUN
cana-328	231	7	)	)	PUNCT
cana-328	231	8	=	=	SYM
cana-328	231	9	1	1	NUM
cana-328	231	10	and	and	CCONJ
cana-328	231	11	𝐻(𝜑	𝐻(𝜑	PROPN
cana-328	231	12	,	,	PUNCT
cana-328	231	13	𝜑	𝜑	NOUN
cana-328	231	14	)	)	PUNCT
cana-328	231	15	=	=	SYM
cana-328	231	16	0	0	NUM
cana-328	231	17	,	,	PUNCT
cana-328	231	18	𝜑	𝜑	PROPN
cana-328	231	19	≥	≥	NOUN
cana-328	231	20	𝜑0	𝜑0	NOUN
cana-328	231	21	,	,	PUNCT
cana-328	231	22	𝐻(𝜑	𝐻(𝜑	PROPN
cana-328	231	23	,	,	PUNCT
cana-328	231	24	𝑗	𝑗	NOUN
cana-328	231	25	)	)	PUNCT
cana-328	231	26	=	=	SYM
cana-328	232	1	𝜑	𝜑	PROPN
cana-328	232	2	−	−	NOUN
cana-328	232	3	𝑗	𝑗	INTJ
cana-328	232	4	>	>	X
cana-328	232	5	0	0	PROPN
cana-328	232	6	,	,	PUNCT
cana-328	232	7	𝜑	𝜑	X
cana-328	232	8	>	>	X
cana-328	232	9	𝑗	𝑗	PROPN
cana-328	232	10	≥	≥	NOUN
cana-328	232	11	𝜑0	𝜑0	NOUN
cana-328	232	12	,	,	PUNCT
cana-328	232	13	𝛥2[𝐻(𝜑	𝛥2[𝐻(𝜑	NOUN
cana-328	232	14	,	,	PUNCT
cana-328	232	15	𝑗	𝑗	NOUN
cana-328	232	16	)	)	PUNCT
cana-328	232	17	]	]	PUNCT
cana-328	233	1	=	=	PUNCT
cana-328	233	2	𝐻(𝜑	𝐻(𝜑	PROPN
cana-328	233	3	,	,	PUNCT
cana-328	233	4	𝑗	𝑗	NOUN
cana-328	233	5	+	+	NOUN
cana-328	233	6	1)−𝐻(𝜑	1)−𝐻(𝜑	NUM
cana-328	233	7	,	,	PUNCT
cana-328	233	8	𝑗	𝑗	NOUN
cana-328	233	9	)	)	PUNCT
cana-328	233	10	=	=	SYM
cana-328	233	11	−1	−1	NOUN
cana-328	233	12	<	<	X
cana-328	233	13	0,𝜑	0,𝜑	X
cana-328	233	14	>	>	X
cana-328	233	15	𝑗	𝑗	PROPN
cana-328	233	16	≥	≥	NOUN
cana-328	233	17	𝜑0	𝜑0	NOUN
cana-328	233	18	,	,	PUNCT
cana-328	233	19	ℎ(𝜑	ℎ(𝜑	PROPN
cana-328	233	20	,	,	PUNCT
cana-328	233	21	𝑗	𝑗	NOUN
cana-328	233	22	)	)	PUNCT
cana-328	233	23	=	=	SYM
cana-328	234	1	1	1	NUM
cana-328	234	2	𝜑+1	𝜑+1	NOUN
cana-328	235	1	[	[	X
cana-328	235	2	(	(	PUNCT
cana-328	235	3	𝜑	𝜑	PROPN
cana-328	235	4	−	−	PROPN
cana-328	235	5	𝑗	𝑗	NOUN
cana-328	235	6	)	)	PUNCT
cana-328	235	7	−	−	PROPN
cana-328	235	8	(	(	PUNCT
cana-328	235	9	𝜑	𝜑	PROPN
cana-328	235	10	+	+	NOUN
cana-328	235	11	1	1	NUM
cana-328	235	12	)	)	PUNCT
cana-328	235	13	]	]	PUNCT
cana-328	235	14	.	.	PUNCT
cana-328	236	1	from	from	ADP
cana-328	236	2	the	the	DET
cana-328	236	3	above	above	ADJ
cana-328	236	4	values	value	NOUN
cana-328	236	5	,	,	PUNCT
cana-328	236	6	(	(	PUNCT
cana-328	236	7	15	15	NUM
cana-328	236	8	)	)	PUNCT
cana-328	236	9	becomes	become	VERB
cana-328	236	10	communications	communication	NOUN
cana-328	236	11	on	on	ADP
cana-328	236	12	applied	apply	VERB
cana-328	236	13	nonlinear	nonlinear	ADJ
cana-328	236	14	analysis	analysis	NOUN
cana-328	236	15	issn	issn	NOUN
cana-328	236	16	:	:	PUNCT
cana-328	236	17	1074	1074	NUM
cana-328	236	18	-	-	PUNCT
cana-328	236	19	133x	133x	NUM
cana-328	236	20	vol	vol	NOUN
cana-328	236	21	31	31	NUM
cana-328	236	22	no	no	NOUN
cana-328	236	23	.	.	NOUN
cana-328	236	24	1	1	NUM
cana-328	236	25	(	(	PUNCT
cana-328	236	26	2024	2024	NUM
cana-328	236	27	)	)	PUNCT
cana-328	236	28	98	98	NUM
cana-328	236	29	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-328	236	30	𝑙𝑖𝑚	𝑙𝑖𝑚	PROPN
cana-328	236	31	 	 	SPACE
cana-328	236	32	𝑠𝑢𝑝𝜑→∞	𝑠𝑢𝑝𝜑→∞	PROPN
cana-328	236	33	  	  	SPACE
cana-328	236	34	1	1	NUM
cana-328	236	35	𝐻(𝜑,𝑀	𝐻(𝜑,𝑀	NOUN
cana-328	236	36	)	)	PUNCT
cana-328	236	37	∑	∑	PUNCT
cana-328	237	1	[	[	X
cana-328	237	2	𝑗(𝜑	𝑗(𝜑	NUM
cana-328	237	3	−	−	PROPN
cana-328	237	4	𝑗)(𝑗	𝑗)(𝑗	NOUN
cana-328	238	1	+	+	CCONJ
cana-328	238	2	1	1	X
cana-328	238	3	)	)	PUNCT
cana-328	238	4	−	−	PROPN
cana-328	238	5	3𝑗3(2(𝑗+1)+(𝜑−𝑗	3𝑗3(2(𝑗+1)+(𝜑−𝑗	NUM
cana-328	238	6	)	)	PUNCT
cana-328	238	7	8𝑗(3𝑗+1	8𝑗(3𝑗+1	NUM
cana-328	238	8	)	)	PUNCT
cana-328	238	9	]	]	PUNCT
cana-328	239	1	𝜑−1	𝜑−1	PROPN
cana-328	239	2	𝑗=𝑀	𝑗=𝑀	PROPN
cana-328	239	3	  	  	SPACE
cana-328	239	4	=	=	SYM
cana-328	239	5	∞	∞	PROPN
cana-328	239	6	,	,	PUNCT
cana-328	239	7	and	and	CCONJ
cana-328	239	8	(	(	PUNCT
cana-328	239	9	16	16	NUM
cana-328	239	10	)	)	PUNCT
cana-328	239	11	becomes	become	VERB
cana-328	239	12	∑	∑	PUNCT
cana-328	239	13	(	(	PUNCT
cana-328	239	14	𝑡+1	𝑡+1	NUM
cana-328	239	15	𝑡	𝑡	X
cana-328	239	16	∑	∑	PROPN
cana-328	239	17	1	1	NUM
cana-328	239	18	𝑗2(𝑗+2)(𝑗+1	𝑗2(𝑗+2)(𝑗+1	PROPN
cana-328	239	19	)	)	PUNCT
cana-328	239	20	∞	∞	PROPN
cana-328	240	1	𝑗=𝑡	𝑗=𝑡	PROPN
cana-328	240	2	  	  	SPACE
cana-328	240	3	∑	∑	PUNCT
cana-328	240	4	  	  	SPACE
cana-328	240	5	(	(	PUNCT
cana-328	240	6	𝑖	𝑖	PROPN
cana-328	240	7	+	+	PROPN
cana-328	240	8	1)∞	1)∞	NUM
cana-328	240	9	𝑖=𝑗	𝑖=𝑗	NUM
cana-328	240	10	)	)	PUNCT
cana-328	240	11	∞	∞	PROPN
cana-328	240	12	𝑡=𝜑	𝑡=𝜑	PUNCT
cana-328	240	13	  	  	SPACE
cana-328	240	14	=	=	SYM
cana-328	240	15	∞.	∞.	PROPN
cana-328	240	16	therefore	therefore	ADV
cana-328	240	17	,	,	PUNCT
cana-328	240	18	conditions	condition	NOUN
cana-328	240	19	of	of	ADP
cana-328	240	20	theorem	theorem	ADJ
cana-328	240	21	2.8	2.8	NUM
cana-328	240	22	are	be	AUX
cana-328	240	23	satisfied	satisfied	ADJ
cana-328	240	24	,	,	PUNCT
cana-328	240	25	so	so	CCONJ
cana-328	240	26	every	every	DET
cana-328	240	27	solution	solution	NOUN
cana-328	240	28	of	of	ADP
cana-328	240	29	(	(	PUNCT
cana-328	240	30	22	22	NUM
cana-328	240	31	)	)	PUNCT
cana-328	240	32	is	be	AUX
cana-328	240	33	oscillatory	oscillatory	ADJ
cana-328	240	34	.	.	PUNCT
cana-328	240	35	example	example	NOUN
cana-328	241	1	3.2	3.2	NUM
cana-328	241	2	.	.	PUNCT
cana-328	242	1	look	look	VERB
cana-328	242	2	over	over	ADP
cana-328	242	3	the	the	DET
cana-328	242	4	equation	equation	NOUN
cana-328	242	5	𝛥(𝜑2𝛥(𝜑(𝛥𝑦(𝜑))))−	𝛥(𝜑2𝛥(𝜑(𝛥𝑦(𝜑))))−	X
cana-328	242	6	(	(	PUNCT
cana-328	242	7	2𝜑+1)(𝜑+1	2𝜑+1)(𝜑+1	NUM
cana-328	242	8	)	)	PUNCT
cana-328	242	9	𝜑+2	𝜑+2	NUM
cana-328	242	10	(	(	PUNCT
cana-328	242	11	𝛥(𝜑	𝛥(𝜑	X
cana-328	242	12	+	+	PROPN
cana-328	242	13	1	1	NUM
cana-328	242	14	)	)	PUNCT
cana-328	242	15	)	)	PUNCT
cana-328	243	1	+	+	CCONJ
cana-328	243	2	8𝜑2	8𝜑2	NUM
cana-328	243	3	𝜑+1	𝜑+1	PRON
cana-328	243	4	𝑓(𝑦(3𝜑	𝑓(𝑦(3𝜑	NUM
cana-328	243	5	)	)	PUNCT
cana-328	243	6	)	)	PUNCT
cana-328	243	7	)	)	PUNCT
cana-328	244	1	=	=	PUNCT
cana-328	244	2	0,𝜑	0,𝜑	NOUN
cana-328	244	3	≥	≥	NUM
cana-328	244	4	1	1	NUM
cana-328	244	5	(	(	PUNCT
cana-328	244	6	24	24	NUM
cana-328	244	7	)	)	PUNCT
cana-328	244	8	here	here	ADV
cana-328	244	9	𝑎(𝜑	𝑎(𝜑	NUM
cana-328	244	10	)	)	PUNCT
cana-328	244	11	=	=	SYM
cana-328	244	12	𝜑2	𝜑2	VERB
cana-328	244	13	,	,	PUNCT
cana-328	244	14	𝑏(𝜑	𝑏(𝜑	NOUN
cana-328	244	15	)	)	PUNCT
cana-328	244	16	=	=	SYM
cana-328	244	17	𝜑	𝜑	NOUN
cana-328	244	18	,	,	PUNCT
cana-328	244	19	𝑝(𝜑	𝑝(𝜑	PROPN
cana-328	244	20	)	)	PUNCT
cana-328	244	21	=	=	SYM
cana-328	244	22	(	(	PUNCT
cana-328	244	23	2𝜑+1)(𝜑+1	2𝜑+1)(𝜑+1	NUM
cana-328	244	24	)	)	PUNCT
cana-328	244	25	𝜑+2	𝜑+2	NUM
cana-328	244	26	,	,	PUNCT
cana-328	244	27	𝑞(𝜑	𝑞(𝜑	NUM
cana-328	244	28	)	)	PUNCT
cana-328	244	29	=	=	SYM
cana-328	244	30	8𝜑2	8𝜑2	NUM
cana-328	245	1	𝜑+1	𝜑+1	NUM
cana-328	245	2	,	,	PUNCT
cana-328	245	3	𝜅	𝜅	X
cana-328	245	4	=	=	SYM
cana-328	245	5	1	1	NUM
cana-328	245	6	,	,	PUNCT
cana-328	245	7	𝜎(𝜑	𝜎(𝜑	NOUN
cana-328	245	8	)	)	PUNCT
cana-328	245	9	=	=	PUNCT
cana-328	245	10	𝜑	𝜑	PROPN
cana-328	245	11	+	+	NOUN
cana-328	245	12	1	1	NUM
cana-328	245	13	,	,	PUNCT
cana-328	245	14	𝑓(𝑦	𝑓(𝑦	ADV
cana-328	245	15	)	)	PUNCT
cana-328	246	1	=	=	SYM
cana-328	246	2	𝑦𝜅	𝑦𝜅	NOUN
cana-328	246	3	,	,	PUNCT
cana-328	246	4	𝑐(𝜑	𝑐(𝜑	PROPN
cana-328	246	5	)	)	PUNCT
cana-328	246	6	=	=	PUNCT
cana-328	247	1	𝜑2(𝜑	𝜑2(𝜑	PROPN
cana-328	248	1	+	+	NUM
cana-328	248	2	1)(𝜑	1)(𝜑	NUM
cana-328	248	3	+	+	CCONJ
cana-328	248	4	2	2	NUM
cana-328	248	5	)	)	PUNCT
cana-328	248	6	,	,	PUNCT
cana-328	248	7	𝑑(𝜑	𝑑(𝜑	ADJ
cana-328	248	8	)	)	PUNCT
cana-328	248	9	=	=	SYM
cana-328	248	10	𝜑	𝜑	PROPN
cana-328	248	11	𝜑+1	𝜑+1	X
cana-328	248	12	,	,	PUNCT
cana-328	248	13	𝑄(𝜑	𝑄(𝜑	NUM
cana-328	248	14	)	)	PUNCT
cana-328	248	15	=	=	PUNCT
cana-328	248	16	𝜑	𝜑	PROPN
cana-328	248	17	+	+	NOUN
cana-328	248	18	1	1	NUM
cana-328	248	19	,	,	PUNCT
cana-328	248	20	𝐶(𝜑	𝐶(𝜑	NOUN
cana-328	248	21	)	)	PUNCT
cana-328	248	22	≈	≈	PROPN
cana-328	248	23	1	1	NUM
cana-328	248	24	𝜑2(𝜑+1)(𝜑+2	𝜑2(𝜑+1)(𝜑+2	NUM
cana-328	248	25	)	)	PUNCT
cana-328	248	26	,	,	PUNCT
cana-328	248	27	𝐷(𝜑	𝐷(𝜑	PRON
cana-328	248	28	)	)	PUNCT
cana-328	249	1	≈	≈	PROPN
cana-328	249	2	𝜑+1	𝜑+1	PUNCT
cana-328	249	3	𝜑	𝜑	NOUN
cana-328	249	4	,	,	PUNCT
cana-328	249	5	𝐸(𝜑	𝐸(𝜑	NOUN
cana-328	249	6	)	)	PUNCT
cana-328	250	1	≈	≈	PROPN
cana-328	250	2	1	1	NUM
cana-328	250	3	𝜑3(𝜑+2	𝜑3(𝜑+2	PROPN
cana-328	250	4	)	)	PUNCT
cana-328	250	5	.	.	PUNCT
cana-328	251	1	now	now	ADV
cana-328	251	2	∑	∑	ADP
cana-328	251	3	1	1	NUM
cana-328	251	4	𝑎(𝑠	𝑎(𝑠	PROPN
cana-328	251	5	)	)	PUNCT
cana-328	251	6	∞	∞	PROPN
cana-328	251	7	𝑠=1	𝑠=1	PUNCT
cana-328	251	8	  	  	SPACE
cana-328	251	9	=	=	SYM
cana-328	251	10	∑	∑	PROPN
cana-328	251	11	1	1	NUM
cana-328	251	12	𝑠2	𝑠2	NOUN
cana-328	251	13	∞	∞	PROPN
cana-328	251	14	𝑠−1	𝑠−1	PROPN
cana-328	251	15	  	  	SPACE
cana-328	251	16	<	<	X
cana-328	251	17	∞	∞	PROPN
cana-328	251	18	,	,	PUNCT
cana-328	251	19	∑	∑	PROPN
cana-328	251	20	1	1	NUM
cana-328	251	21	𝑏	𝑏	PROPN
cana-328	251	22	1	1	NUM
cana-328	251	23	𝛼(𝑠	𝛼(𝑠	PROPN
cana-328	251	24	)	)	PUNCT
cana-328	251	25	∞	∞	PROPN
cana-328	251	26	𝑠=1	𝑠=1	PUNCT
cana-328	251	27	  	  	SPACE
cana-328	251	28	=	=	SYM
cana-328	251	29	∑	∑	PROPN
cana-328	251	30	1	1	NUM
cana-328	251	31	𝑠	𝑠	NUM
cana-328	251	32	∞	∞	PROPN
cana-328	251	33	𝑠=1	𝑠=1	PUNCT
cana-328	251	34	  	  	SPACE
cana-328	251	35	=	=	SYM
cana-328	251	36	∞	∞	PROPN
cana-328	251	37	,	,	PUNCT
cana-328	251	38	and	and	CCONJ
cana-328	251	39	equation	equation	NOUN
cana-328	251	40	(	(	PUNCT
cana-328	251	41	4	4	X
cana-328	251	42	)	)	PUNCT
cana-328	251	43	becomes	become	VERB
cana-328	251	44	,	,	PUNCT
cana-328	251	45	𝛥(𝜑2𝛥𝑧(𝜑))−	𝛥(𝜑2𝛥𝑧(𝜑))−	NUM
cana-328	251	46	(	(	PUNCT
cana-328	251	47	2𝜑+1)(𝜑+1	2𝜑+1)(𝜑+1	NUM
cana-328	251	48	)	)	PUNCT
cana-328	251	49	(	(	PUNCT
cana-328	251	50	𝜑+2	𝜑+2	NUM
cana-328	251	51	)	)	PUNCT
cana-328	251	52	𝑧(𝜑+	𝑧(𝜑+	PROPN
cana-328	251	53	1	1	NUM
cana-328	251	54	)	)	PUNCT
cana-328	251	55	=	=	SYM
cana-328	252	1	0	0	X
cana-328	252	2	.	.	PUNCT
cana-328	253	1	(	(	PUNCT
cana-328	253	2	25	25	NUM
cana-328	253	3	)	)	PUNCT
cana-328	253	4	it	it	PRON
cana-328	253	5	has	have	VERB
cana-328	253	6	a	a	DET
cana-328	253	7	nonoscillatory	nonoscillatory	ADJ
cana-328	253	8	solution	solution	NOUN
cana-328	253	9	𝑧(𝜑	𝑧(𝜑	NOUN
cana-328	253	10	)	)	PUNCT
cana-328	254	1	=	=	PUNCT
cana-328	254	2	𝜑	𝜑	PROPN
cana-328	255	1	+	+	NOUN
cana-328	256	1	1	1	X
cana-328	256	2	.	.	PUNCT
cana-328	256	3	moreover	moreover	ADV
cana-328	256	4	,	,	PUNCT
cana-328	256	5	(	(	PUNCT
cana-328	256	6	8)	8)	NUM
cana-328	256	7	becomes	become	VERB
cana-328	256	8	∑	∑	PROPN
cana-328	256	9	1	1	NUM
cana-328	256	10	𝜑2(𝜑+1)(𝜑+2	𝜑2(𝜑+1)(𝜑+2	NOUN
cana-328	256	11	)	)	PUNCT
cana-328	256	12	𝜑−1	𝜑−1	PROPN
cana-328	256	13	𝑠=𝜑3	𝑠=𝜑3	PROPN
cana-328	256	14	 	 	SPACE
cana-328	256	15	∑	∑	PUNCT
cana-328	256	16	8𝑡(𝑡	8𝑡(𝑡	PROPN
cana-328	256	17	+	+	CCONJ
cana-328	256	18	2)∞	2)∞	NUM
cana-328	256	19	𝑡=𝑠	𝑡=𝑠	PROPN
cana-328	256	20	  	  	SPACE
cana-328	256	21	=	=	SYM
cana-328	256	22	∞	∞	PROPN
cana-328	256	23	,	,	PUNCT
cana-328	256	24	and	and	CCONJ
cana-328	256	25	from	from	ADP
cana-328	256	26	corollary	corollary	ADJ
cana-328	256	27	2.7	2.7	NUM
cana-328	256	28	,	,	PUNCT
cana-328	256	29	(	(	PUNCT
cana-328	256	30	14	14	NUM
cana-328	256	31	)	)	PUNCT
cana-328	256	32	becomes	become	VERB
cana-328	256	33	𝑙𝑖𝑚	𝑙𝑖𝑚	PROPN
cana-328	256	34	 	 	SPACE
cana-328	256	35	𝑠𝑢𝑝𝜑→∞	𝑠𝑢𝑝𝜑→∞	PROPN
cana-328	256	36	  	  	SPACE
cana-328	257	1	[	[	X
cana-328	257	2	(	(	PUNCT
cana-328	257	3	𝜑	𝜑	NOUN
cana-328	257	4	+	+	NOUN
cana-328	257	5	1	1	NUM
cana-328	257	6	)	)	PUNCT
cana-328	257	7	2(𝜑	2(𝜑	NUM
cana-328	258	1	+	+	CCONJ
cana-328	258	2	2)(𝜑	2)(𝜑	NUM
cana-328	258	3	+	+	SYM
cana-328	258	4	3)∑	3)∑	NUM
cana-328	258	5	(	(	PUNCT
cana-328	258	6	𝜑	𝜑	X
cana-328	258	7	+	+	CCONJ
cana-328	258	8	1)2(𝜑	1)2(𝜑	PROPN
cana-328	259	1	+	+	CCONJ
cana-328	259	2	2)(𝜑	2)(𝜑	NUM
cana-328	259	3	+	+	CCONJ
cana-328	259	4	3	3	X
cana-328	259	5	)	)	PUNCT
cana-328	259	6	𝜑−1	𝜑−1	PROPN
cana-328	259	7	𝑠=𝜑1	𝑠=𝜑1	PROPN
cana-328	259	8	  	  	SPACE
cana-328	260	1	+	+	CCONJ
cana-328	260	2	∑	∑	PROPN
cana-328	260	3	(	(	PUNCT
cana-328	260	4	1	1	NUM
cana-328	260	5	(	(	PUNCT
cana-328	260	6	𝑡+1)(𝑡+3	𝑡+1)(𝑡+3	NOUN
cana-328	260	7	)	)	PUNCT
cana-328	260	8	)	)	PUNCT
cana-328	261	1	+	+	CCONJ
cana-328	261	2	𝜑−1	𝜑−1	PROPN
cana-328	261	3	𝑡=𝜑+1	𝑡=𝜑+1	NOUN
cana-328	261	4	(	(	PUNCT
cana-328	261	5	𝜑+1)3(𝜑+3	𝜑+1)3(𝜑+3	PROPN
cana-328	261	6	)	)	PUNCT
cana-328	261	7	𝜑+2	𝜑+2	NUM
cana-328	261	8	  	  	SPACE
cana-328	261	9	∑	∑	PUNCT
cana-328	261	10	𝑡	𝑡	PROPN
cana-328	261	11	+	+	NOUN
cana-328	261	12	2∞	2∞	NUM
cana-328	261	13	𝑡=𝜑	𝑡=𝜑	X
cana-328	261	14	]	]	PUNCT
cana-328	261	15	   	   	SPACE
cana-328	261	16	>	>	X
cana-328	261	17	1	1	X
cana-328	261	18	.	.	PUNCT
cana-328	262	1	so	so	ADV
cana-328	262	2	every	every	DET
cana-328	262	3	solution	solution	NOUN
cana-328	262	4	of	of	ADP
cana-328	262	5	(	(	PUNCT
cana-328	262	6	24	24	NUM
cana-328	262	7	)	)	PUNCT
cana-328	262	8	are	be	AUX
cana-328	262	9	oscillatory	oscillatory	ADJ
cana-328	262	10	.	.	PUNCT
cana-328	263	1	4	4	NUM
cana-328	263	2	application	application	NOUN
cana-328	263	3	samuelson	samuelson	NOUN
cana-328	263	4	is	be	AUX
cana-328	263	5	the	the	DET
cana-328	263	6	originator	originator	NOUN
cana-328	263	7	of	of	ADP
cana-328	263	8	the	the	DET
cana-328	263	9	business	business	NOUN
cana-328	263	10	cycle	cycle	NOUN
cana-328	263	11	model	model	NOUN
cana-328	263	12	[	[	X
cana-328	263	13	12	12	NUM
cana-328	263	14	]	]	PUNCT
cana-328	263	15	.	.	PUNCT
cana-328	264	1	this	this	DET
cana-328	264	2	model	model	NOUN
cana-328	264	3	is	be	AUX
cana-328	264	4	built	build	VERB
cana-328	264	5	on	on	ADP
cana-328	264	6	a	a	DET
cana-328	264	7	set	set	NOUN
cana-328	264	8	of	of	ADP
cana-328	264	9	underlying	underlie	VERB
cana-328	264	10	assumptions	assumption	NOUN
cana-328	264	11	.	.	PUNCT
cana-328	265	1	yearly	yearly	ADJ
cana-328	265	2	income	income	NOUN
cana-328	265	3	𝑦(𝜑	𝑦(𝜑	NOUN
cana-328	265	4	)	)	PUNCT
cana-328	265	5	is	be	AUX
cana-328	265	6	equal	equal	ADJ
cana-328	265	7	to	to	ADP
cana-328	265	8	the	the	DET
cana-328	265	9	sum	sum	NOUN
cana-328	265	10	of	of	ADP
cana-328	265	11	capital	capital	NOUN
cana-328	265	12	investment	investment	NOUN
cana-328	265	13	𝐺(𝜑	𝐺(𝜑	NUM
cana-328	265	14	)	)	PUNCT
cana-328	265	15	,	,	PUNCT
cana-328	265	16	consumption	consumption	NOUN
cana-328	265	17	𝐶(𝜑	𝐶(𝜑	NOUN
cana-328	265	18	)	)	PUNCT
cana-328	265	19	and	and	CCONJ
cana-328	265	20	private	private	ADJ
cana-328	265	21	investment	investment	NOUN
cana-328	265	22	𝐼(𝜑	𝐼(𝜑	PROPN
cana-328	265	23	)	)	PUNCT
cana-328	265	24	𝑦(𝜑	𝑦(𝜑	NOUN
cana-328	265	25	)	)	PUNCT
cana-328	265	26	=	=	SYM
cana-328	265	27	𝐺(𝜑	𝐺(𝜑	NUM
cana-328	265	28	)	)	PUNCT
cana-328	265	29	+	+	CCONJ
cana-328	265	30	𝐶(𝜑	𝐶(𝜑	NUM
cana-328	265	31	)	)	PUNCT
cana-328	265	32	+	+	NUM
cana-328	265	33	𝐼(𝜑	𝐼(𝜑	NOUN
cana-328	265	34	)	)	PUNCT
cana-328	265	35	.	.	PUNCT
cana-328	266	1	(	(	PUNCT
cana-328	266	2	26	26	NUM
cana-328	266	3	)	)	PUNCT
cana-328	266	4	capital	capital	NOUN
cana-328	266	5	investment	investment	NOUN
cana-328	266	6	𝐺(𝜑	𝐺(𝜑	NUM
cana-328	266	7	)	)	PUNCT
cana-328	266	8	remains	remain	VERB
cana-328	266	9	constant	constant	ADJ
cana-328	266	10	.	.	PUNCT
cana-328	267	1	𝐺(𝜑	𝐺(𝜑	NUM
cana-328	267	2	)	)	PUNCT
cana-328	267	3	=	=	SYM
cana-328	267	4	𝐺.	𝐺.	NOUN
cana-328	267	5	(	(	PUNCT
cana-328	267	6	27	27	NUM
cana-328	267	7	)	)	PUNCT
cana-328	267	8	communications	communication	NOUN
cana-328	267	9	on	on	ADP
cana-328	267	10	applied	apply	VERB
cana-328	267	11	nonlinear	nonlinear	ADJ
cana-328	267	12	analysis	analysis	NOUN
cana-328	267	13	issn	issn	NOUN
cana-328	267	14	:	:	PUNCT
cana-328	267	15	1074	1074	NUM
cana-328	267	16	-	-	PUNCT
cana-328	267	17	133x	133x	NUM
cana-328	267	18	vol	vol	NOUN
cana-328	267	19	31	31	NUM
cana-328	267	20	no	no	NOUN
cana-328	267	21	.	.	NOUN
cana-328	267	22	1	1	NUM
cana-328	267	23	(	(	PUNCT
cana-328	267	24	2024	2024	NUM
cana-328	267	25	)	)	PUNCT
cana-328	267	26	99	99	NUM
cana-328	267	27	https://internationalpubls.com	https://internationalpubls.com	X
cana-328	267	28	consumption	consumption	NOUN
cana-328	267	29	𝐶(𝜑	𝐶(𝜑	NOUN
cana-328	267	30	)	)	PUNCT
cana-328	267	31	depends	depend	VERB
cana-328	267	32	on	on	ADP
cana-328	267	33	the	the	DET
cana-328	267	34	previous	previous	ADJ
cana-328	267	35	year	year	NOUN
cana-328	267	36	income	income	NOUN
cana-328	267	37	and	and	CCONJ
cana-328	267	38	on	on	ADP
cana-328	267	39	marginal	marginal	ADJ
cana-328	267	40	tendency	tendency	NOUN
cana-328	267	41	to	to	PART
cana-328	267	42	consume	consume	VERB
cana-328	267	43	,	,	PUNCT
cana-328	267	44	it	it	PRON
cana-328	267	45	is	be	AUX
cana-328	267	46	denoted	denote	VERB
cana-328	267	47	as	as	ADP
cana-328	267	48	𝛼.	𝛼.	NOUN
cana-328	267	49	𝐶(𝜑	𝐶(𝜑	NOUN
cana-328	267	50	)	)	PUNCT
cana-328	267	51	=	=	SYM
cana-328	267	52	𝛼𝑦(𝜑	𝛼𝑦(𝜑	NUM
cana-328	267	53	−	−	PROPN
cana-328	267	54	1	1	NUM
cana-328	267	55	)	)	PUNCT
cana-328	267	56	,	,	PUNCT
cana-328	267	57	(	(	PUNCT
cana-328	267	58	28	28	NUM
cana-328	267	59	)	)	PUNCT
cana-328	267	60	where	where	SCONJ
cana-328	267	61	the	the	DET
cana-328	267	62	multiplier	multipli	ADJ
cana-328	267	63	parameter	parameter	NOUN
cana-328	267	64	0	0	PUNCT
cana-328	267	65	<	<	X
cana-328	267	66	𝛼	𝛼	X
cana-328	267	67	<	<	X
cana-328	267	68	1	1	NUM
cana-328	267	69	.	.	PUNCT
cana-328	267	70	private	private	ADJ
cana-328	267	71	investment	investment	NOUN
cana-328	267	72	i(r	i(r	PROPN
cana-328	267	73	)	)	PUNCT
cana-328	267	74	depends	depend	VERB
cana-328	267	75	on	on	ADP
cana-328	267	76	consumption	consumption	NOUN
cana-328	267	77	changes	change	NOUN
cana-328	267	78	and	and	CCONJ
cana-328	267	79	on	on	ADP
cana-328	267	80	the	the	DET
cana-328	267	81	accelerator	accelerator	NOUN
cana-328	267	82	factor	factor	NOUN
cana-328	267	83	𝛽	𝛽	PROPN
cana-328	267	84	,	,	PUNCT
cana-328	267	85	where	where	SCONJ
cana-328	267	86	𝛽	𝛽	NOUN
cana-328	267	87	>	>	X
cana-328	267	88	0	0	PUNCT
cana-328	267	89	𝐼(𝜑	𝐼(𝜑	NOUN
cana-328	267	90	)	)	PUNCT
cana-328	267	91	=	=	SYM
cana-328	267	92	𝛽(𝐶(𝜑	𝛽(𝐶(𝜑	NOUN
cana-328	267	93	)	)	PUNCT
cana-328	267	94	−	−	NOUN
cana-328	267	95	𝐶(𝜑	𝐶(𝜑	NOUN
cana-328	267	96	−	−	PROPN
cana-328	267	97	1	1	NUM
cana-328	267	98	)	)	PUNCT
cana-328	267	99	)	)	PUNCT
cana-328	268	1	=	=	PUNCT
cana-328	269	1	𝛼𝛽(𝑦(𝜑	𝛼𝛽(𝑦(𝜑	NUM
cana-328	269	2	−	−	NUM
cana-328	269	3	1	1	NUM
cana-328	269	4	)	)	PUNCT
cana-328	269	5	−	−	PROPN
cana-328	269	6	𝑦(𝜑	𝑦(𝜑	ADJ
cana-328	269	7	−	−	NOUN
cana-328	269	8	2	2	NUM
cana-328	269	9	)	)	PUNCT
cana-328	269	10	)	)	PUNCT
cana-328	269	11	.	.	PUNCT
cana-328	270	1	(	(	PUNCT
cana-328	270	2	29	29	NUM
cana-328	270	3	)	)	PUNCT
cana-328	270	4	substitute	substitute	NOUN
cana-328	270	5	(	(	PUNCT
cana-328	270	6	27	27	NUM
cana-328	270	7	)	)	PUNCT
cana-328	270	8	,	,	PUNCT
cana-328	270	9	(	(	PUNCT
cana-328	270	10	28	28	NUM
cana-328	270	11	)	)	PUNCT
cana-328	270	12	and	and	CCONJ
cana-328	270	13	(	(	PUNCT
cana-328	270	14	29	29	NUM
cana-328	270	15	)	)	PUNCT
cana-328	270	16	,	,	PUNCT
cana-328	270	17	the	the	DET
cana-328	270	18	yearly	yearly	ADJ
cana-328	270	19	income	income	NOUN
cana-328	270	20	𝑦(𝜑	𝑦(𝜑	NOUN
cana-328	270	21	)	)	PUNCT
cana-328	270	22	can	can	AUX
cana-328	270	23	be	be	AUX
cana-328	270	24	determined	determine	VERB
cana-328	270	25	as	as	ADP
cana-328	270	26	a	a	DET
cana-328	270	27	second	second	ADJ
cana-328	270	28	-	-	PUNCT
cana-328	270	29	order	order	NOUN
cana-328	270	30	difference	difference	NOUN
cana-328	270	31	equation	equation	NOUN
cana-328	270	32	𝑦(𝜑	𝑦(𝜑	NOUN
cana-328	270	33	)	)	PUNCT
cana-328	271	1	=	=	SYM
cana-328	271	2	𝐺	𝐺	PROPN
cana-328	271	3	+	+	X
cana-328	271	4	𝛼𝑦(𝜑	𝛼𝑦(𝜑	ADV
cana-328	271	5	−	−	PROPN
cana-328	271	6	1	1	NUM
cana-328	271	7	)	)	PUNCT
cana-328	271	8	+	+	CCONJ
cana-328	272	1	𝛼𝛽𝑦(𝜑	𝛼𝛽𝑦(𝜑	NUM
cana-328	273	1	−	−	NOUN
cana-328	273	2	1	1	NUM
cana-328	273	3	)	)	PUNCT
cana-328	273	4	−	−	PROPN
cana-328	274	1	𝛼𝛽𝑦(𝜑	𝛼𝛽𝑦(𝜑	NUM
cana-328	275	1	−	−	NOUN
cana-328	275	2	2	2	NUM
cana-328	275	3	)	)	PUNCT
cana-328	275	4	.	.	PUNCT
cana-328	276	1	(	(	PUNCT
cana-328	276	2	30	30	NUM
cana-328	276	3	)	)	PUNCT
cana-328	276	4	the	the	DET
cana-328	276	5	model	model	NOUN
cana-328	276	6	developed	develop	VERB
cana-328	276	7	by	by	ADP
cana-328	276	8	paul	paul	PROPN
cana-328	276	9	samuelson	samuelson	PROPN
cana-328	276	10	is	be	AUX
cana-328	276	11	a	a	DET
cana-328	276	12	cornerstone	cornerstone	NOUN
cana-328	276	13	in	in	ADP
cana-328	276	14	the	the	DET
cana-328	276	15	field	field	NOUN
cana-328	276	16	of	of	ADP
cana-328	276	17	economics	economic	NOUN
cana-328	276	18	.	.	PUNCT
cana-328	277	1	it	it	PRON
cana-328	277	2	delineates	delineate	VERB
cana-328	277	3	five	five	NUM
cana-328	277	4	possible	possible	ADJ
cana-328	277	5	paths	path	NOUN
cana-328	277	6	or	or	CCONJ
cana-328	277	7	trends	trend	NOUN
cana-328	277	8	that	that	PRON
cana-328	277	9	economic	economic	ADJ
cana-328	277	10	activities	activity	NOUN
cana-328	277	11	might	might	AUX
cana-328	277	12	follow	follow	VERB
cana-328	277	13	.	.	PUNCT
cana-328	278	1	these	these	DET
cana-328	278	2	trajectories	trajectory	NOUN
cana-328	278	3	are	be	AUX
cana-328	278	4	shaped	shape	VERB
cana-328	278	5	by	by	ADP
cana-328	278	6	the	the	DET
cana-328	278	7	interplay	interplay	NOUN
cana-328	278	8	of	of	ADP
cana-328	278	9	two	two	NUM
cana-328	278	10	fundamental	fundamental	ADJ
cana-328	278	11	parameters	parameter	NOUN
cana-328	278	12	:	:	PUNCT
cana-328	278	13	the	the	DET
cana-328	278	14	marginal	marginal	ADJ
cana-328	278	15	propensity	propensity	NOUN
cana-328	278	16	to	to	PART
cana-328	278	17	consume	consume	VERB
cana-328	278	18	(	(	PUNCT
cana-328	278	19	α	α	NOUN
cana-328	278	20	)	)	PUNCT
cana-328	278	21	and	and	CCONJ
cana-328	278	22	the	the	DET
cana-328	278	23	accelerator	accelerator	NOUN
cana-328	278	24	coefficient	coefficient	NOUN
cana-328	278	25	(	(	PUNCT
cana-328	278	26	β	β	NOUN
cana-328	278	27	)	)	PUNCT
cana-328	278	28	.	.	PUNCT
cana-328	279	1	the	the	DET
cana-328	279	2	marginal	marginal	ADJ
cana-328	279	3	propensity	propensity	NOUN
cana-328	279	4	to	to	PART
cana-328	279	5	consume	consume	VERB
cana-328	279	6	(	(	PUNCT
cana-328	279	7	α	α	X
cana-328	279	8	)	)	PUNCT
cana-328	279	9	quantifies	quantifie	NOUN
cana-328	279	10	the	the	DET
cana-328	279	11	proportion	proportion	NOUN
cana-328	279	12	of	of	ADP
cana-328	279	13	additional	additional	ADJ
cana-328	279	14	income	income	NOUN
cana-328	279	15	consumers	consumer	NOUN
cana-328	279	16	spend	spend	VERB
cana-328	279	17	rather	rather	ADV
cana-328	279	18	than	than	ADP
cana-328	279	19	save	save	VERB
cana-328	279	20	.	.	PUNCT
cana-328	280	1	conversely	conversely	ADV
cana-328	280	2	,	,	PUNCT
cana-328	280	3	the	the	DET
cana-328	280	4	accelerator	accelerator	NOUN
cana-328	280	5	coefficient	coefficient	NOUN
cana-328	280	6	(	(	PUNCT
cana-328	280	7	β	β	NOUN
cana-328	280	8	)	)	PUNCT
cana-328	280	9	is	be	AUX
cana-328	280	10	a	a	DET
cana-328	280	11	metric	metric	NOUN
cana-328	280	12	that	that	PRON
cana-328	280	13	gauges	gauge	VERB
cana-328	280	14	the	the	DET
cana-328	280	15	sensitivity	sensitivity	NOUN
cana-328	280	16	of	of	ADP
cana-328	280	17	investment	investment	NOUN
cana-328	280	18	expenditures	expenditure	NOUN
cana-328	280	19	to	to	ADP
cana-328	280	20	shifts	shift	NOUN
cana-328	280	21	in	in	ADP
cana-328	280	22	gdp	gdp	NOUN
cana-328	280	23	.	.	PUNCT
cana-328	281	1	each	each	DET
cana-328	281	2	unique	unique	ADJ
cana-328	281	3	pairing	pairing	NOUN
cana-328	281	4	of	of	ADP
cana-328	281	5	α	α	PROPN
cana-328	281	6	and	and	CCONJ
cana-328	281	7	β	β	PROPN
cana-328	281	8	yields	yield	VERB
cana-328	281	9	a	a	DET
cana-328	281	10	specific	specific	ADJ
cana-328	281	11	pattern	pattern	NOUN
cana-328	281	12	of	of	ADP
cana-328	281	13	economic	economic	ADJ
cana-328	281	14	activity	activity	NOUN
cana-328	281	15	,	,	PUNCT
cana-328	281	16	which	which	PRON
cana-328	281	17	can	can	AUX
cana-328	281	18	range	range	VERB
cana-328	281	19	from	from	ADP
cana-328	281	20	stable	stable	ADJ
cana-328	281	21	growth	growth	NOUN
cana-328	281	22	to	to	ADP
cana-328	281	23	cyclical	cyclical	ADJ
cana-328	281	24	oscillations	oscillation	NOUN
cana-328	281	25	.	.	PUNCT
cana-328	282	1	this	this	DET
cana-328	282	2	spectrum	spectrum	NOUN
cana-328	282	3	of	of	ADP
cana-328	282	4	outcomes	outcome	NOUN
cana-328	282	5	underscores	underscore	VERB
cana-328	282	6	the	the	DET
cana-328	282	7	intricate	intricate	ADJ
cana-328	282	8	relationship	relationship	NOUN
cana-328	282	9	between	between	ADP
cana-328	282	10	consumption	consumption	NOUN
cana-328	282	11	,	,	PUNCT
cana-328	282	12	investment	investment	NOUN
cana-328	282	13	,	,	PUNCT
cana-328	282	14	and	and	CCONJ
cana-328	282	15	aggregate	aggregate	VERB
cana-328	282	16	economic	economic	ADJ
cana-328	282	17	activity	activity	NOUN
cana-328	282	18	.	.	PUNCT
cana-328	283	1	samuelson	samuelson	PROPN
cana-328	283	2	’s	’s	PART
cana-328	283	3	model	model	NOUN
cana-328	283	4	offers	offer	VERB
cana-328	283	5	a	a	DET
cana-328	283	6	holistic	holistic	ADJ
cana-328	283	7	blueprint	blueprint	NOUN
cana-328	283	8	for	for	ADP
cana-328	283	9	deciphering	decipher	VERB
cana-328	283	10	the	the	DET
cana-328	283	11	mechanics	mechanic	NOUN
cana-328	283	12	of	of	ADP
cana-328	283	13	economic	economic	ADJ
cana-328	283	14	cycles	cycle	NOUN
cana-328	283	15	.	.	PUNCT
cana-328	284	1	it	it	PRON
cana-328	284	2	equips	equip	VERB
cana-328	284	3	economists	economist	NOUN
cana-328	284	4	with	with	ADP
cana-328	284	5	the	the	DET
cana-328	284	6	tools	tool	NOUN
cana-328	284	7	to	to	PART
cana-328	284	8	scrutinize	scrutinize	VERB
cana-328	284	9	and	and	CCONJ
cana-328	284	10	forecast	forecast	VERB
cana-328	284	11	the	the	DET
cana-328	284	12	impact	impact	NOUN
cana-328	284	13	of	of	ADP
cana-328	284	14	alterations	alteration	NOUN
cana-328	284	15	in	in	ADP
cana-328	284	16	consumer	consumer	NOUN
cana-328	284	17	behavior	behavior	NOUN
cana-328	284	18	and	and	CCONJ
cana-328	284	19	investment	investment	NOUN
cana-328	284	20	on	on	ADP
cana-328	284	21	the	the	DET
cana-328	284	22	course	course	NOUN
cana-328	284	23	of	of	ADP
cana-328	284	24	the	the	DET
cana-328	284	25	economy	economy	NOUN
cana-328	284	26	.	.	PUNCT
cana-328	285	1	this	this	DET
cana-328	285	2	model	model	NOUN
cana-328	285	3	is	be	AUX
cana-328	285	4	particularly	particularly	ADV
cana-328	285	5	instrumental	instrumental	ADJ
cana-328	285	6	in	in	ADP
cana-328	285	7	policy	policy	NOUN
cana-328	285	8	formulation	formulation	NOUN
cana-328	285	9	,	,	PUNCT
cana-328	285	10	as	as	SCONJ
cana-328	285	11	it	it	PRON
cana-328	285	12	aids	aid	VERB
cana-328	285	13	in	in	ADP
cana-328	285	14	devising	devise	VERB
cana-328	285	15	strategies	strategy	NOUN
cana-328	285	16	to	to	PART
cana-328	285	17	navigate	navigate	VERB
cana-328	285	18	economic	economic	ADJ
cana-328	285	19	cycles	cycle	NOUN
cana-328	285	20	effectively	effectively	ADV
cana-328	285	21	.	.	PUNCT
cana-328	286	1	in	in	ADP
cana-328	286	2	essence	essence	NOUN
cana-328	286	3	,	,	PUNCT
cana-328	286	4	samuelson	samuelson	PROPN
cana-328	286	5	’s	’s	PART
cana-328	286	6	model	model	NOUN
cana-328	286	7	is	be	AUX
cana-328	286	8	a	a	DET
cana-328	286	9	potent	potent	ADJ
cana-328	286	10	instrument	instrument	NOUN
cana-328	286	11	furnishing	furnish	VERB
cana-328	286	12	invaluable	invaluable	ADJ
cana-328	286	13	insights	insight	NOUN
cana-328	286	14	into	into	ADP
cana-328	286	15	economic	economic	ADJ
cana-328	286	16	cycle	cycle	NOUN
cana-328	286	17	dynamics	dynamic	NOUN
cana-328	286	18	.	.	PUNCT
cana-328	287	1	it	it	PRON
cana-328	287	2	is	be	AUX
cana-328	287	3	indispensable	indispensable	ADJ
cana-328	287	4	in	in	ADP
cana-328	287	5	economic	economic	ADJ
cana-328	287	6	prognostication	prognostication	NOUN
cana-328	287	7	and	and	CCONJ
cana-328	287	8	policy	policy	NOUN
cana-328	287	9	design	design	NOUN
cana-328	287	10	,	,	PUNCT
cana-328	287	11	contributing	contribute	VERB
cana-328	287	12	significantly	significantly	ADV
cana-328	287	13	to	to	ADP
cana-328	287	14	our	our	PRON
cana-328	287	15	understanding	understanding	NOUN
cana-328	287	16	of	of	ADP
cana-328	287	17	economic	economic	ADJ
cana-328	287	18	phenomena	phenomenon	NOUN
cana-328	287	19	.	.	PUNCT
cana-328	288	1	this	this	DET
cana-328	288	2	model	model	NOUN
cana-328	288	3	’s	’s	PART
cana-328	288	4	versatility	versatility	NOUN
cana-328	288	5	and	and	CCONJ
cana-328	288	6	predictive	predictive	ADJ
cana-328	288	7	power	power	NOUN
cana-328	288	8	make	make	VERB
cana-328	288	9	it	it	PRON
cana-328	288	10	an	an	DET
cana-328	288	11	essential	essential	ADJ
cana-328	288	12	tool	tool	NOUN
cana-328	288	13	in	in	ADP
cana-328	288	14	the	the	DET
cana-328	288	15	economist	economist	NOUN
cana-328	288	16	’s	’s	PART
cana-328	288	17	toolkit	toolkit	NOUN
cana-328	288	18	.	.	PUNCT
cana-328	289	1	its	its	PRON
cana-328	289	2	ability	ability	NOUN
cana-328	289	3	to	to	PART
cana-328	289	4	capture	capture	VERB
cana-328	289	5	the	the	DET
cana-328	289	6	nuanced	nuanced	ADJ
cana-328	289	7	interdependencies	interdependency	NOUN
cana-328	289	8	within	within	ADP
cana-328	289	9	an	an	DET
cana-328	289	10	economy	economy	NOUN
cana-328	289	11	underscores	underscore	VERB
cana-328	289	12	its	its	PRON
cana-328	289	13	enduring	endure	VERB
cana-328	289	14	relevance	relevance	NOUN
cana-328	289	15	in	in	ADP
cana-328	289	16	economic	economic	ADJ
cana-328	289	17	analysis	analysis	NOUN
cana-328	289	18	and	and	CCONJ
cana-328	289	19	policy	policy	NOUN
cana-328	289	20	planning	planning	NOUN
cana-328	289	21	.	.	PUNCT
cana-328	290	1	table	table	NOUN
cana-328	290	2	1	1	NUM
cana-328	290	3	:	:	SYM
cana-328	290	4	five	five	NUM
cana-328	290	5	different	different	ADJ
cana-328	290	6	path	path	NOUN
cana-328	290	7	cycle	cycle	NOUN
cana-328	290	8	case	case	NOUN
cana-328	290	9	values	value	VERB
cana-328	290	10	behavior	behavior	NOUN
cana-328	290	11	of	of	ADP
cana-328	290	12	the	the	DET
cana-328	290	13	cycles	cycle	NOUN
cana-328	290	14	1	1	NUM
cana-328	290	15	𝛼	𝛼	NOUN
cana-328	290	16	=	=	SYM
cana-328	290	17	0.5	0.5	NUM
cana-328	290	18	,	,	PUNCT
cana-328	290	19	𝛽	𝛽	NOUN
cana-328	290	20	=	=	SYM
cana-328	290	21	0	0	NUM
cana-328	290	22	cycle	cycle	NOUN
cana-328	290	23	less	less	ADJ
cana-328	290	24	path	path	NOUN
cana-328	290	25	2	2	NUM
cana-328	290	26	𝛼	𝛼	NOUN
cana-328	290	27	=	=	SYM
cana-328	290	28	0.5	0.5	NUM
cana-328	290	29	,	,	PUNCT
cana-328	290	30	𝛽	𝛽	NOUN
cana-328	290	31	=	=	SYM
cana-328	290	32	1	1	NUM
cana-328	290	33	damped	damp	VERB
cana-328	290	34	fluctuations	fluctuation	NOUN
cana-328	290	35	3	3	NUM
cana-328	290	36	𝛼	𝛼	NOUN
cana-328	290	37	=	=	SYM
cana-328	290	38	0.5	0.5	NUM
cana-328	290	39	,	,	PUNCT
cana-328	290	40	𝛽	𝛽	NOUN
cana-328	290	41	=	=	SYM
cana-328	290	42	2	2	NUM
cana-328	290	43	fluctuation	fluctuation	NOUN
cana-328	290	44	of	of	ADP
cana-328	290	45	constant	constant	ADJ
cana-328	290	46	amplitude	amplitude	NOUN
cana-328	290	47	4	4	NUM
cana-328	290	48	𝛼	𝛼	NOUN
cana-328	290	49	=	=	SYM
cana-328	290	50	0.5	0.5	NUM
cana-328	290	51	,	,	PUNCT
cana-328	290	52	𝛽	𝛽	NOUN
cana-328	290	53	=	=	SYM
cana-328	290	54	3	3	NUM
cana-328	290	55	explosive	explosive	ADJ
cana-328	290	56	cycles	cycle	NOUN
cana-328	290	57	5	5	NUM
cana-328	290	58	𝛼	𝛼	NOUN
cana-328	290	59	=	=	SYM
cana-328	290	60	0.5	0.5	NUM
cana-328	290	61	,	,	PUNCT
cana-328	290	62	𝛽	𝛽	NOUN
cana-328	290	63	=	=	SYM
cana-328	290	64	4	4	NUM
cana-328	290	65	cycle	cycle	NOUN
cana-328	290	66	less	less	ADV
cana-328	290	67	explosive	explosive	ADJ
cana-328	290	68	path	path	NOUN
cana-328	290	69	communications	communication	NOUN
cana-328	290	70	on	on	ADP
cana-328	290	71	applied	apply	VERB
cana-328	290	72	nonlinear	nonlinear	ADJ
cana-328	290	73	analysis	analysis	NOUN
cana-328	290	74	issn	issn	NOUN
cana-328	290	75	:	:	PUNCT
cana-328	290	76	1074	1074	NUM
cana-328	290	77	-	-	PUNCT
cana-328	290	78	133x	133x	NUM
cana-328	290	79	vol	vol	NOUN
cana-328	290	80	31	31	NUM
cana-328	290	81	no	no	NOUN
cana-328	290	82	.	.	NOUN
cana-328	290	83	1	1	NUM
cana-328	290	84	(	(	PUNCT
cana-328	290	85	2024	2024	NUM
cana-328	290	86	)	)	PUNCT
cana-328	290	87	100	100	NUM
cana-328	290	88	https://internationalpubls.com	https://internationalpubls.com	X
cana-328	290	89	samuelson	samuelson	PROPN
cana-328	290	90	’s	’s	PART
cana-328	290	91	model	model	NOUN
cana-328	290	92	,	,	PUNCT
cana-328	290	93	a	a	DET
cana-328	290	94	cornerstone	cornerstone	NOUN
cana-328	290	95	in	in	ADP
cana-328	290	96	economic	economic	ADJ
cana-328	290	97	theory	theory	NOUN
cana-328	290	98	,	,	PUNCT
cana-328	290	99	illustrates	illustrate	VERB
cana-328	290	100	the	the	DET
cana-328	290	101	behavior	behavior	NOUN
cana-328	290	102	of	of	ADP
cana-328	290	103	economic	economic	ADJ
cana-328	290	104	cycles	cycle	NOUN
cana-328	290	105	based	base	VERB
cana-328	290	106	on	on	ADP
cana-328	290	107	varying	vary	VERB
cana-328	290	108	values	value	NOUN
cana-328	290	109	of	of	ADP
cana-328	290	110	the	the	DET
cana-328	290	111	marginal	marginal	ADJ
cana-328	290	112	propensity	propensity	NOUN
cana-328	290	113	to	to	PART
cana-328	290	114	consume	consume	VERB
cana-328	290	115	(	(	PUNCT
cana-328	290	116	α	α	NOUN
cana-328	290	117	)	)	PUNCT
cana-328	290	118	and	and	CCONJ
cana-328	290	119	the	the	DET
cana-328	290	120	accelerator	accelerator	NOUN
cana-328	290	121	coefficient	coefficient	NOUN
cana-328	290	122	(	(	PUNCT
cana-328	290	123	β	β	NOUN
cana-328	290	124	)	)	PUNCT
cana-328	290	125	.	.	PUNCT
cana-328	291	1	1	1	X
cana-328	291	2	.	.	X
cana-328	291	3	when	when	SCONJ
cana-328	291	4	α=0.5	α=0.5	NOUN
cana-328	291	5	and	and	CCONJ
cana-328	291	6	β=0	β=0	ADP
cana-328	291	7	,	,	PUNCT
cana-328	291	8	the	the	DET
cana-328	291	9	economy	economy	NOUN
cana-328	291	10	follows	follow	VERB
cana-328	291	11	a	a	DET
cana-328	291	12	cycle	cycle	NOUN
cana-328	291	13	-	-	PUNCT
cana-328	291	14	less	less	ADJ
cana-328	291	15	path	path	NOUN
cana-328	291	16	,	,	PUNCT
cana-328	291	17	indicating	indicate	VERB
cana-328	291	18	stability	stability	NOUN
cana-328	291	19	without	without	ADP
cana-328	291	20	fluctuations	fluctuation	NOUN
cana-328	291	21	.	.	PUNCT
cana-328	292	1	this	this	PRON
cana-328	292	2	suggests	suggest	VERB
cana-328	292	3	a	a	DET
cana-328	292	4	balanced	balanced	ADJ
cana-328	292	5	economy	economy	NOUN
cana-328	292	6	where	where	SCONJ
cana-328	292	7	changes	change	NOUN
cana-328	292	8	in	in	ADP
cana-328	292	9	income	income	NOUN
cana-328	292	10	do	do	AUX
cana-328	292	11	not	not	PART
cana-328	292	12	significantly	significantly	ADV
cana-328	292	13	affect	affect	VERB
cana-328	292	14	consumption	consumption	NOUN
cana-328	292	15	or	or	CCONJ
cana-328	292	16	investment	investment	NOUN
cana-328	292	17	patterns	pattern	NOUN
cana-328	292	18	.	.	PUNCT
cana-328	293	1	2	2	X
cana-328	293	2	.	.	X
cana-328	293	3	with	with	ADP
cana-328	293	4	α=0.5	α=0.5	NOUN
cana-328	293	5	and	and	CCONJ
cana-328	293	6	β=1	β=1	ADP
cana-328	293	7	,	,	PUNCT
cana-328	293	8	the	the	DET
cana-328	293	9	economy	economy	NOUN
cana-328	293	10	experiences	experience	VERB
cana-328	293	11	damped	damped	VERB
cana-328	293	12	fluctuations	fluctuation	NOUN
cana-328	293	13	,	,	PUNCT
cana-328	293	14	suggesting	suggest	VERB
cana-328	293	15	cycles	cycle	NOUN
cana-328	293	16	gradually	gradually	ADV
cana-328	293	17	diminishing	diminish	VERB
cana-328	293	18	over	over	ADP
cana-328	293	19	time	time	NOUN
cana-328	293	20	.	.	PUNCT
cana-328	294	1	this	this	PRON
cana-328	294	2	indicates	indicate	VERB
cana-328	294	3	an	an	DET
cana-328	294	4	economy	economy	NOUN
cana-328	294	5	that	that	PRON
cana-328	294	6	,	,	PUNCT
cana-328	294	7	while	while	SCONJ
cana-328	294	8	initially	initially	ADV
cana-328	294	9	reactive	reactive	ADJ
cana-328	294	10	to	to	ADP
cana-328	294	11	changes	change	NOUN
cana-328	294	12	in	in	ADP
cana-328	294	13	income	income	NOUN
cana-328	294	14	,	,	PUNCT
cana-328	294	15	eventually	eventually	ADV
cana-328	294	16	stabilizes	stabilize	VERB
cana-328	294	17	.	.	PUNCT
cana-328	295	1	3	3	X
cana-328	295	2	.	.	X
cana-328	295	3	for	for	ADP
cana-328	295	4	α=0.5	α=0.5	PRON
cana-328	295	5	and	and	CCONJ
cana-328	295	6	β=2	β=2	PROPN
cana-328	295	7	,	,	PUNCT
cana-328	295	8	the	the	DET
cana-328	295	9	economy	economy	NOUN
cana-328	295	10	undergoes	undergo	VERB
cana-328	295	11	fluctuations	fluctuation	NOUN
cana-328	295	12	of	of	ADP
cana-328	295	13	constant	constant	ADJ
cana-328	295	14	amplitude	amplitude	NOUN
cana-328	295	15	,	,	PUNCT
cana-328	295	16	indicating	indicate	VERB
cana-328	295	17	regular	regular	ADJ
cana-328	295	18	cyclical	cyclical	ADJ
cana-328	295	19	patterns	pattern	NOUN
cana-328	295	20	.	.	PUNCT
cana-328	296	1	this	this	PRON
cana-328	296	2	represents	represent	VERB
cana-328	296	3	an	an	DET
cana-328	296	4	economy	economy	NOUN
cana-328	296	5	with	with	ADP
cana-328	296	6	consistent	consistent	ADJ
cana-328	296	7	cycles	cycle	NOUN
cana-328	296	8	,	,	PUNCT
cana-328	296	9	reflecting	reflect	VERB
cana-328	296	10	a	a	DET
cana-328	296	11	direct	direct	ADJ
cana-328	296	12	,	,	PUNCT
cana-328	296	13	proportional	proportional	ADJ
cana-328	296	14	relationship	relationship	NOUN
cana-328	296	15	between	between	ADP
cana-328	296	16	income	income	NOUN
cana-328	296	17	changes	change	NOUN
cana-328	296	18	and	and	CCONJ
cana-328	296	19	consumption	consumption	NOUN
cana-328	296	20	or	or	CCONJ
cana-328	296	21	investment	investment	NOUN
cana-328	296	22	.	.	PUNCT
cana-328	297	1	4	4	X
cana-328	297	2	.	.	X
cana-328	297	3	when	when	SCONJ
cana-328	297	4	α=0.5	α=0.5	NOUN
cana-328	297	5	and	and	CCONJ
cana-328	297	6	β=3	β=3	ADP
cana-328	297	7	,	,	PUNCT
cana-328	297	8	the	the	DET
cana-328	297	9	economy	economy	NOUN
cana-328	297	10	exhibits	exhibit	VERB
cana-328	297	11	explosive	explosive	ADJ
cana-328	297	12	cycles	cycle	NOUN
cana-328	297	13	,	,	PUNCT
cana-328	297	14	suggesting	suggest	VERB
cana-328	297	15	rapid	rapid	ADJ
cana-328	297	16	and	and	CCONJ
cana-328	297	17	potentially	potentially	ADV
cana-328	297	18	unstable	unstable	ADJ
cana-328	297	19	growth	growth	NOUN
cana-328	297	20	.	.	PUNCT
cana-328	298	1	this	this	DET
cana-328	298	2	scenario	scenario	NOUN
cana-328	298	3	might	might	AUX
cana-328	298	4	occur	occur	VERB
cana-328	298	5	when	when	SCONJ
cana-328	298	6	an	an	DET
cana-328	298	7	increase	increase	NOUN
cana-328	298	8	in	in	ADP
cana-328	298	9	income	income	NOUN
cana-328	298	10	significantly	significantly	ADV
cana-328	298	11	boosts	boost	VERB
cana-328	298	12	consumption	consumption	NOUN
cana-328	298	13	and	and	CCONJ
cana-328	298	14	investment	investment	NOUN
cana-328	298	15	,	,	PUNCT
cana-328	298	16	leading	lead	VERB
cana-328	298	17	to	to	AUX
cana-328	298	18	accelerated	accelerate	VERB
cana-328	298	19	economic	economic	ADJ
cana-328	298	20	growth	growth	NOUN
cana-328	298	21	.	.	PUNCT
cana-328	299	1	5	5	X
cana-328	299	2	.	.	X
cana-328	299	3	lastly	lastly	ADV
cana-328	299	4	,	,	PUNCT
cana-328	299	5	with	with	ADP
cana-328	299	6	α=0.5	α=0.5	NOUN
cana-328	299	7	and	and	CCONJ
cana-328	299	8	β=4	β=4	ADP
cana-328	299	9	,	,	PUNCT
cana-328	299	10	the	the	DET
cana-328	299	11	economy	economy	NOUN
cana-328	299	12	follows	follow	VERB
cana-328	299	13	a	a	DET
cana-328	299	14	cycle	cycle	NOUN
cana-328	299	15	-	-	PUNCT
cana-328	299	16	less	less	ADV
cana-328	299	17	explosive	explosive	ADJ
cana-328	299	18	path	path	NOUN
cana-328	299	19	,	,	PUNCT
cana-328	299	20	indicating	indicate	VERB
cana-328	299	21	steady	steady	ADJ
cana-328	299	22	,	,	PUNCT
cana-328	299	23	rapid	rapid	ADJ
cana-328	299	24	growth	growth	NOUN
cana-328	299	25	without	without	ADP
cana-328	299	26	cyclical	cyclical	ADJ
cana-328	299	27	fluctuations	fluctuation	NOUN
cana-328	299	28	.	.	PUNCT
cana-328	300	1	this	this	PRON
cana-328	300	2	could	could	AUX
cana-328	300	3	represent	represent	VERB
cana-328	300	4	an	an	DET
cana-328	300	5	economy	economy	NOUN
cana-328	300	6	where	where	SCONJ
cana-328	300	7	income	income	NOUN
cana-328	300	8	changes	change	NOUN
cana-328	300	9	lead	lead	VERB
cana-328	300	10	to	to	ADP
cana-328	300	11	increased	increase	VERB
cana-328	300	12	consumption	consumption	NOUN
cana-328	300	13	and	and	CCONJ
cana-328	300	14	investment	investment	NOUN
cana-328	300	15	but	but	CCONJ
cana-328	300	16	without	without	ADP
cana-328	300	17	the	the	DET
cana-328	300	18	cyclical	cyclical	ADJ
cana-328	300	19	patterns	pattern	NOUN
cana-328	300	20	seen	see	VERB
cana-328	300	21	in	in	ADP
cana-328	300	22	other	other	ADJ
cana-328	300	23	scenarios	scenario	NOUN
cana-328	300	24	.	.	PUNCT
cana-328	301	1	these	these	DET
cana-328	301	2	patterns	pattern	NOUN
cana-328	301	3	provide	provide	VERB
cana-328	301	4	valuable	valuable	ADJ
cana-328	301	5	insights	insight	NOUN
cana-328	301	6	into	into	ADP
cana-328	301	7	how	how	SCONJ
cana-328	301	8	changes	change	NOUN
cana-328	301	9	in	in	ADP
cana-328	301	10	consumption	consumption	NOUN
cana-328	301	11	and	and	CCONJ
cana-328	301	12	investment	investment	NOUN
cana-328	301	13	behaviors	behavior	NOUN
cana-328	301	14	,	,	PUNCT
cana-328	301	15	driven	drive	VERB
cana-328	301	16	by	by	ADP
cana-328	301	17	shifts	shift	NOUN
cana-328	301	18	in	in	ADP
cana-328	301	19	income	income	NOUN
cana-328	301	20	,	,	PUNCT
cana-328	301	21	can	can	AUX
cana-328	301	22	impact	impact	VERB
cana-328	301	23	the	the	DET
cana-328	301	24	cyclical	cyclical	ADJ
cana-328	301	25	behavior	behavior	NOUN
cana-328	301	26	of	of	ADP
cana-328	301	27	an	an	DET
cana-328	301	28	economy	economy	NOUN
cana-328	301	29	.	.	PUNCT
cana-328	302	1	understanding	understand	VERB
cana-328	302	2	these	these	DET
cana-328	302	3	dynamics	dynamic	NOUN
cana-328	302	4	is	be	AUX
cana-328	302	5	crucial	crucial	ADJ
cana-328	302	6	for	for	SCONJ
cana-328	302	7	economists	economist	NOUN
cana-328	302	8	and	and	CCONJ
cana-328	302	9	policymakers	policymaker	NOUN
cana-328	302	10	to	to	PART
cana-328	302	11	effectively	effectively	ADV
cana-328	302	12	manage	manage	VERB
cana-328	302	13	economic	economic	ADJ
cana-328	302	14	stability	stability	NOUN
cana-328	302	15	and	and	CCONJ
cana-328	302	16	growth	growth	NOUN
cana-328	302	17	.	.	PUNCT
cana-328	303	1	thus	thus	ADV
cana-328	303	2	,	,	PUNCT
cana-328	303	3	samuelson	samuelson	PROPN
cana-328	303	4	’s	’s	PART
cana-328	303	5	model	model	NOUN
cana-328	303	6	is	be	AUX
cana-328	303	7	a	a	DET
cana-328	303	8	powerful	powerful	ADJ
cana-328	303	9	tool	tool	NOUN
cana-328	303	10	in	in	ADP
cana-328	303	11	economic	economic	ADJ
cana-328	303	12	analysis	analysis	NOUN
cana-328	303	13	and	and	CCONJ
cana-328	303	14	policy	policy	NOUN
cana-328	303	15	planning	planning	NOUN
cana-328	303	16	.	.	PUNCT
cana-328	304	1	based	base	VERB
cana-328	304	2	on	on	ADP
cana-328	304	3	these	these	DET
cana-328	304	4	assumptions	assumption	NOUN
cana-328	304	5	,	,	PUNCT
cana-328	304	6	the	the	DET
cana-328	304	7	nonoscillatory	nonoscillatory	ADJ
cana-328	304	8	second	second	ADJ
cana-328	304	9	-	-	PUNCT
cana-328	304	10	order	order	NOUN
cana-328	304	11	difference	difference	NOUN
cana-328	304	12	equation	equation	NOUN
cana-328	304	13	(	(	PUNCT
cana-328	304	14	25	25	NUM
cana-328	304	15	)	)	PUNCT
cana-328	304	16	is	be	AUX
cana-328	304	17	𝑦(𝜑	𝑦(𝜑	NOUN
cana-328	304	18	)	)	PUNCT
cana-328	305	1	=	=	SYM
cana-328	305	2	𝐺	𝐺	NOUN
cana-328	305	3	+	+	CCONJ
cana-328	305	4	(	(	PUNCT
cana-328	305	5	1	1	NUM
cana-328	305	6	+	+	CCONJ
cana-328	305	7	(	(	PUNCT
cana-328	305	8	2𝜑−3)(𝜑−1	2𝜑−3)(𝜑−1	NUM
cana-328	305	9	)	)	PUNCT
cana-328	305	10	𝜑	𝜑	NOUN
cana-328	305	11	)	)	PUNCT
cana-328	305	12	𝑦(𝜑	𝑦(𝜑	ADV
cana-328	305	13	−	−	NOUN
cana-328	305	14	1	1	NUM
cana-328	305	15	)	)	PUNCT
cana-328	305	16	+	+	CCONJ
cana-328	305	17	(	(	PUNCT
cana-328	305	18	𝜑−2)2	𝜑−2)2	PROPN
cana-328	305	19	(	(	PUNCT
cana-328	305	20	𝜑−1)2	𝜑−1)2	PROPN
cana-328	305	21	𝑦(𝜑	𝑦(𝜑	PROPN
cana-328	305	22	−	−	PROPN
cana-328	305	23	1	1	NUM
cana-328	305	24	)	)	PUNCT
cana-328	305	25	−	−	PROPN
cana-328	305	26	(	(	PUNCT
cana-328	305	27	𝜑−2)2	𝜑−2)2	PROPN
cana-328	305	28	(	(	PUNCT
cana-328	305	29	𝜑−1)2	𝜑−1)2	PROPN
cana-328	305	30	𝑦(𝜑	𝑦(𝜑	PROPN
cana-328	305	31	−	−	PROPN
cana-328	305	32	2	2	NUM
cana-328	305	33	)	)	PUNCT
cana-328	305	34	.	.	PUNCT
cana-328	306	1	(	(	PUNCT
cana-328	306	2	31	31	NUM
cana-328	306	3	)	)	PUNCT
cana-328	306	4	it	it	PRON
cana-328	306	5	is	be	AUX
cana-328	306	6	a	a	DET
cana-328	306	7	corresponding	corresponding	ADJ
cana-328	306	8	equation	equation	NOUN
cana-328	306	9	of	of	ADP
cana-328	306	10	(	(	PUNCT
cana-328	306	11	30	30	NUM
cana-328	306	12	)	)	PUNCT
cana-328	306	13	.	.	PUNCT
cana-328	307	1	here	here	ADV
cana-328	307	2	𝑦(𝜑	𝑦(𝜑	NOUN
cana-328	307	3	)	)	PUNCT
cana-328	307	4	denotes	denote	VERB
cana-328	307	5	the	the	DET
cana-328	307	6	current	current	ADJ
cana-328	307	7	year	year	NOUN
cana-328	307	8	income	income	NOUN
cana-328	307	9	of	of	ADP
cana-328	307	10	the	the	DET
cana-328	307	11	business	business	NOUN
cana-328	307	12	and	and	CCONJ
cana-328	307	13	𝛼	𝛼	NOUN
cana-328	307	14	=	=	PUNCT
cana-328	307	15	(	(	PUNCT
cana-328	307	16	1	1	NUM
cana-328	307	17	+	+	CCONJ
cana-328	307	18	(	(	PUNCT
cana-328	307	19	2𝜑−3)(𝜑−1	2𝜑−3)(𝜑−1	NUM
cana-328	307	20	)	)	PUNCT
cana-328	307	21	𝜑	𝜑	NOUN
cana-328	307	22	)	)	PUNCT
cana-328	308	1	=	=	SYM
cana-328	308	2	𝑇ℎ𝑒𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑟	𝑇ℎ𝑒𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑒𝑟	PROPN
cana-328	308	3	,	,	PUNCT
cana-328	308	4	and	and	CCONJ
cana-328	308	5	𝛽	𝛽	NOUN
cana-328	308	6	=	=	SYM
cana-328	308	7	(	(	PUNCT
cana-328	308	8	𝜑(𝜑−2)2	𝜑(𝜑−2)2	NOUN
cana-328	308	9	[	[	X
cana-328	308	10	𝜑+(2𝜑−3)(𝜑−1)](𝜑−1)2	𝜑+(2𝜑−3)(𝜑−1)](𝜑−1)2	NOUN
cana-328	308	11	)	)	PUNCT
cana-328	309	1	=	=	SYM
cana-328	309	2	𝑇ℎ𝑒𝐴𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑜𝑟	𝑇ℎ𝑒𝐴𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑜𝑟	NOUN
cana-328	309	3	,	,	PUNCT
cana-328	309	4	illustration	illustration	NOUN
cana-328	309	5	:	:	PUNCT
cana-328	309	6	let	let	VERB
cana-328	309	7	's	us	PRON
cana-328	309	8	consider	consider	VERB
cana-328	309	9	an	an	DET
cana-328	309	10	illustration	illustration	NOUN
cana-328	309	11	to	to	PART
cana-328	309	12	understand	understand	VERB
cana-328	309	13	this	this	DET
cana-328	309	14	equation	equation	NOUN
cana-328	309	15	better	well	ADV
cana-328	309	16	.	.	PUNCT
cana-328	310	1	suppose	suppose	VERB
cana-328	310	2	a	a	DET
cana-328	310	3	steel	steel	NOUN
cana-328	310	4	company	company	NOUN
cana-328	310	5	owner	owner	NOUN
cana-328	310	6	invests	invest	VERB
cana-328	310	7	an	an	DET
cana-328	310	8	autonomous	autonomous	ADJ
cana-328	310	9	investment	investment	NOUN
cana-328	310	10	of	of	ADP
cana-328	310	11	rs	rs	NOUN
cana-328	310	12	.	.	PROPN
cana-328	310	13	2	2	NUM
cana-328	310	14	crore	crore	NUM
cana-328	310	15	.	.	PUNCT
cana-328	311	1	the	the	DET
cana-328	311	2	previous	previous	ADJ
cana-328	311	3	two	two	NUM
cana-328	311	4	years	year	NOUN
cana-328	311	5	'	'	PART
cana-328	311	6	income	income	NOUN
cana-328	311	7	is	be	AUX
cana-328	311	8	unknown	unknown	ADJ
cana-328	311	9	,	,	PUNCT
cana-328	311	10	and	and	CCONJ
cana-328	311	11	we	we	PRON
cana-328	311	12	refer	refer	VERB
cana-328	311	13	to	to	ADP
cana-328	311	14	the	the	DET
cana-328	311	15	path	path	NOUN
cana-328	311	16	cycle	cycle	NOUN
cana-328	311	17	table	table	NOUN
cana-328	311	18	to	to	PART
cana-328	311	19	determine	determine	VERB
cana-328	311	20	the	the	DET
cana-328	311	21	values	value	NOUN
cana-328	311	22	of	of	ADP
cana-328	311	23	autonomous	autonomous	ADJ
cana-328	311	24	investment	investment	NOUN
cana-328	311	25	,	,	PUNCT
cana-328	311	26	current	current	ADJ
cana-328	311	27	consumption	consumption	NOUN
cana-328	311	28	,	,	PUNCT
cana-328	311	29	induced	induced	ADJ
cana-328	311	30	investment	investment	NOUN
cana-328	311	31	,	,	PUNCT
cana-328	311	32	and	and	CCONJ
cana-328	311	33	yearly	yearly	ADJ
cana-328	311	34	income	income	NOUN
cana-328	311	35	for	for	ADP
cana-328	311	36	the	the	DET
cana-328	311	37	next	next	ADJ
cana-328	311	38	five	five	NUM
cana-328	311	39	years	year	NOUN
cana-328	311	40	and	and	CCONJ
cana-328	311	41	from	from	ADP
cana-328	311	42	the	the	DET
cana-328	311	43	path	path	NOUN
cana-328	311	44	cycle	cycle	NOUN
cana-328	311	45	table	table	NOUN
cana-328	311	46	we	we	PRON
cana-328	311	47	take	take	VERB
cana-328	311	48	𝛼	𝛼	NOUN
cana-328	311	49	=	=	SYM
cana-328	311	50	0.5	0.5	NUM
cana-328	311	51	,	,	PUNCT
cana-328	311	52	𝛽	𝛽	NOUN
cana-328	311	53	=	=	SYM
cana-328	311	54	1	1	NUM
cana-328	311	55	communications	communication	NOUN
cana-328	311	56	on	on	ADP
cana-328	311	57	applied	apply	VERB
cana-328	311	58	nonlinear	nonlinear	ADJ
cana-328	311	59	analysis	analysis	NOUN
cana-328	311	60	issn	issn	NOUN
cana-328	311	61	:	:	PUNCT
cana-328	311	62	1074	1074	NUM
cana-328	311	63	-	-	PUNCT
cana-328	311	64	133x	133x	NUM
cana-328	311	65	vol	vol	NOUN
cana-328	311	66	31	31	NUM
cana-328	311	67	no	no	NOUN
cana-328	311	68	.	.	NOUN
cana-328	311	69	1	1	NUM
cana-328	311	70	(	(	PUNCT
cana-328	311	71	2024	2024	NUM
cana-328	311	72	)	)	PUNCT
cana-328	311	73	101	101	NUM
cana-328	311	74	https://internationalpubls.com	https://internationalpubls.com	X
cana-328	311	75	table	table	NOUN
cana-328	311	76	2	2	NUM
cana-328	311	77	:	:	PUNCT
cana-328	311	78	business	business	NOUN
cana-328	311	79	income	income	NOUN
cana-328	311	80	time	time	NOUN
cana-328	311	81	period	period	NOUN
cana-328	311	82	(	(	PUNCT
cana-328	311	83	𝜑	𝜑	NOUN
cana-328	311	84	)	)	PUNCT
cana-328	311	85	autonomous	autonomous	ADJ
cana-328	311	86	investment	investment	NOUN
cana-328	311	87	𝐺(𝜑	𝐺(𝜑	NUM
cana-328	311	88	)	)	PUNCT
cana-328	311	89	current	current	ADJ
cana-328	311	90	consumption	consumption	NOUN
cana-328	311	91	𝐶(𝜑	𝐶(𝜑	NOUN
cana-328	311	92	)	)	PUNCT
cana-328	311	93	induced	induce	VERB
cana-328	311	94	investment	investment	NOUN
cana-328	311	95	𝐼(𝜑	𝐼(𝜑	NOUN
cana-328	311	96	)	)	PUNCT
cana-328	311	97	yearly	yearly	ADJ
cana-328	311	98	income	income	NOUN
cana-328	311	99	𝑌(𝜑	𝑌(𝜑	NOUN
cana-328	311	100	)	)	PUNCT
cana-328	311	101	1	1	NUM
cana-328	312	1	2,00,00,000	2,00,00,000	NUM
cana-328	312	2	0	0	NUM
cana-328	312	3	0	0	NUM
cana-328	312	4	2,00,00,000	2,00,00,000	NUM
cana-328	312	5	2	2	NUM
cana-328	312	6	2,00,00,000	2,00,00,000	NUM
cana-328	312	7	1,00,00,000	1,00,00,000	NUM
cana-328	312	8	1,00,00,000	1,00,00,000	NUM
cana-328	312	9	4,00,00,000	4,00,00,000	NUM
cana-328	312	10	3	3	NUM
cana-328	312	11	2,00,00,000	2,00,00,000	NUM
cana-328	312	12	2,00,00,000	2,00,00,000	NUM
cana-328	312	13	1,00,00,000	1,00,00,000	NUM
cana-328	312	14	5,00,00,000	5,00,00,000	NUM
cana-328	312	15	4	4	NUM
cana-328	312	16	2,00,00,000	2,00,00,000	NUM
cana-328	312	17	2,50,00,000	2,50,00,000	NUM
cana-328	312	18	50,00,000	50,00,000	NUM
cana-328	312	19	5,00,00,000	5,00,00,000	NUM
cana-328	312	20	5	5	NUM
cana-328	312	21	2,00,00,000	2,00,00,000	NUM
cana-328	312	22	2,50,00,000	2,50,00,000	NUM
cana-328	312	23	0	0	NUM
cana-328	312	24	4,50,00,000	4,50,00,000	NUM
cana-328	312	25	the	the	DET
cana-328	312	26	data	datum	NOUN
cana-328	312	27	represents	represent	VERB
cana-328	312	28	a	a	DET
cana-328	312	29	business	business	NOUN
cana-328	312	30	’s	’s	PART
cana-328	312	31	income	income	NOUN
cana-328	312	32	over	over	ADP
cana-328	312	33	five	five	NUM
cana-328	312	34	years	year	NOUN
cana-328	312	35	.	.	PUNCT
cana-328	313	1	the	the	DET
cana-328	313	2	autonomous	autonomous	ADJ
cana-328	313	3	investment	investment	NOUN
cana-328	313	4	,	,	PUNCT
cana-328	313	5	g(φ	g(φ	PROPN
cana-328	313	6	)	)	PUNCT
cana-328	313	7	,	,	PUNCT
cana-328	313	8	remains	remain	VERB
cana-328	313	9	constant	constant	ADJ
cana-328	313	10	at	at	ADP
cana-328	313	11	2,00,00,000	2,00,00,000	NUM
cana-328	313	12	.	.	PUNCT
cana-328	314	1	in	in	ADP
cana-328	314	2	the	the	DET
cana-328	314	3	first	first	ADJ
cana-328	314	4	year	year	NOUN
cana-328	314	5	,	,	PUNCT
cana-328	314	6	no	no	DET
cana-328	314	7	consumption	consumption	NOUN
cana-328	314	8	or	or	CCONJ
cana-328	314	9	induced	induce	VERB
cana-328	314	10	investment	investment	NOUN
cana-328	314	11	leads	lead	VERB
cana-328	314	12	to	to	ADP
cana-328	314	13	a	a	DET
cana-328	314	14	yearly	yearly	ADJ
cana-328	314	15	income	income	NOUN
cana-328	314	16	,	,	PUNCT
cana-328	314	17	y(φ	y(φ	PROPN
cana-328	314	18	)	)	PUNCT
cana-328	314	19	,	,	PUNCT
cana-328	314	20	of	of	ADP
cana-328	314	21	2,00,00,000	2,00,00,000	NUM
cana-328	314	22	.	.	PUNCT
cana-328	315	1	in	in	ADP
cana-328	315	2	the	the	DET
cana-328	315	3	second	second	ADJ
cana-328	315	4	year	year	NOUN
cana-328	315	5	,	,	PUNCT
cana-328	315	6	consumption	consumption	NOUN
cana-328	315	7	,	,	PUNCT
cana-328	315	8	c(φ	c(φ	NOUN
cana-328	315	9	)	)	PUNCT
cana-328	315	10	,	,	PUNCT
cana-328	315	11	increases	increase	VERB
cana-328	315	12	to	to	ADP
cana-328	315	13	1,00,00,000	1,00,00,000	NUM
cana-328	315	14	and	and	CCONJ
cana-328	315	15	induced	induced	ADJ
cana-328	315	16	investment	investment	NOUN
cana-328	315	17	,	,	PUNCT
cana-328	315	18	i(φ	i(φ	PROPN
cana-328	315	19	)	)	PUNCT
cana-328	315	20	,	,	PUNCT
cana-328	315	21	to	to	ADP
cana-328	315	22	1,00,00,000	1,00,00,000	NUM
cana-328	315	23	,	,	PUNCT
cana-328	315	24	doubling	double	VERB
cana-328	315	25	the	the	DET
cana-328	315	26	yearly	yearly	ADJ
cana-328	315	27	income	income	NOUN
cana-328	315	28	to	to	ADP
cana-328	315	29	4,00,00,000	4,00,00,000	NUM
cana-328	315	30	.	.	PUNCT
cana-328	316	1	in	in	ADP
cana-328	316	2	the	the	DET
cana-328	316	3	third	third	ADJ
cana-328	316	4	year	year	NOUN
cana-328	316	5	,	,	PUNCT
cana-328	316	6	consumption	consumption	NOUN
cana-328	316	7	doubles	double	VERB
cana-328	316	8	,	,	PUNCT
cana-328	316	9	and	and	CCONJ
cana-328	316	10	the	the	DET
cana-328	316	11	yearly	yearly	ADJ
cana-328	316	12	income	income	NOUN
cana-328	316	13	increases	increase	NOUN
cana-328	316	14	to	to	ADP
cana-328	316	15	5,00,00,000	5,00,00,000	NUM
cana-328	316	16	.	.	PUNCT
cana-328	317	1	in	in	ADP
cana-328	317	2	the	the	DET
cana-328	317	3	fourth	fourth	ADJ
cana-328	317	4	year	year	NOUN
cana-328	317	5	,	,	PUNCT
cana-328	317	6	consumption	consumption	NOUN
cana-328	317	7	increases	increase	VERB
cana-328	317	8	further	far	ADV
cana-328	317	9	,	,	PUNCT
cana-328	317	10	but	but	CCONJ
cana-328	317	11	induced	induce	VERB
cana-328	317	12	investment	investment	NOUN
cana-328	317	13	decreases	decrease	VERB
cana-328	317	14	,	,	PUNCT
cana-328	317	15	keeping	keep	VERB
cana-328	317	16	the	the	DET
cana-328	317	17	yearly	yearly	ADJ
cana-328	317	18	income	income	NOUN
cana-328	317	19	steady	steady	ADJ
cana-328	317	20	.	.	PUNCT
cana-328	318	1	in	in	ADP
cana-328	318	2	the	the	DET
cana-328	318	3	fifth	fifth	ADJ
cana-328	318	4	year	year	NOUN
cana-328	318	5	,	,	PUNCT
cana-328	318	6	induced	induce	VERB
cana-328	318	7	investment	investment	NOUN
cana-328	318	8	drops	drop	NOUN
cana-328	318	9	to	to	ADP
cana-328	318	10	zero	zero	NUM
cana-328	318	11	,	,	PUNCT
cana-328	318	12	reducing	reduce	VERB
cana-328	318	13	the	the	DET
cana-328	318	14	yearly	yearly	ADJ
cana-328	318	15	income	income	NOUN
cana-328	318	16	to	to	ADP
cana-328	318	17	4,50,00,000	4,50,00,000	NUM
cana-328	318	18	.	.	PUNCT
cana-328	319	1	this	this	PRON
cana-328	319	2	suggests	suggest	VERB
cana-328	319	3	a	a	DET
cana-328	319	4	need	need	NOUN
cana-328	319	5	for	for	ADP
cana-328	319	6	strategies	strategy	NOUN
cana-328	319	7	to	to	PART
cana-328	319	8	maintain	maintain	VERB
cana-328	319	9	or	or	CCONJ
cana-328	319	10	increase	increase	VERB
cana-328	319	11	induced	induced	ADJ
cana-328	319	12	investment	investment	NOUN
cana-328	319	13	.	.	PUNCT
cana-328	320	1	the	the	DET
cana-328	320	2	nonoscillatory	nonoscillatory	ADJ
cana-328	320	3	second	second	ADJ
cana-328	320	4	-	-	PUNCT
cana-328	320	5	order	order	NOUN
cana-328	320	6	difference	difference	NOUN
cana-328	320	7	equation	equation	NOUN
cana-328	320	8	(	(	PUNCT
cana-328	320	9	25	25	NUM
cana-328	320	10	)	)	PUNCT
cana-328	320	11	can	can	AUX
cana-328	320	12	be	be	AUX
cana-328	320	13	written	write	VERB
cana-328	320	14	as	as	ADP
cana-328	320	15	a	a	DET
cana-328	320	16	corresponding	corresponding	ADJ
cana-328	320	17	equation	equation	NOUN
cana-328	320	18	.	.	PUNCT
cana-328	321	1	here	here	ADV
cana-328	321	2	,	,	PUNCT
cana-328	321	3	the	the	DET
cana-328	321	4	notation	notation	NOUN
cana-328	321	5	"	"	PUNCT
cana-328	321	6	φ	φ	PROPN
cana-328	321	7	"	"	PUNCT
cana-328	321	8	denotes	denote	VERB
cana-328	321	9	the	the	DET
cana-328	321	10	current	current	ADJ
cana-328	321	11	year	year	NOUN
cana-328	321	12	's	's	PART
cana-328	321	13	income	income	NOUN
cana-328	321	14	of	of	ADP
cana-328	321	15	a	a	DET
cana-328	321	16	business	business	NOUN
cana-328	321	17	,	,	PUNCT
cana-328	321	18	"	"	PUNCT
cana-328	321	19	φ	φ	PROPN
cana-328	321	20	-1	-1	PUNCT
cana-328	321	21	"	"	PUNCT
cana-328	321	22	represents	represent	VERB
cana-328	321	23	the	the	DET
cana-328	321	24	income	income	NOUN
cana-328	321	25	of	of	ADP
cana-328	321	26	the	the	DET
cana-328	321	27	previous	previous	ADJ
cana-328	321	28	year	year	NOUN
cana-328	321	29	,	,	PUNCT
cana-328	321	30	and	and	CCONJ
cana-328	321	31	"	"	PUNCT
cana-328	321	32	φ	φ	PROPN
cana-328	321	33	-2	-2	PUNCT
cana-328	321	34	"	"	PUNCT
cana-328	321	35	represents	represent	VERB
cana-328	321	36	the	the	DET
cana-328	321	37	income	income	NOUN
cana-328	321	38	of	of	ADP
cana-328	321	39	the	the	DET
cana-328	321	40	year	year	NOUN
cana-328	321	41	before	before	ADP
cana-328	321	42	that	that	PRON
cana-328	321	43	.	.	PUNCT
cana-328	322	1	as	as	ADP
cana-328	322	2	per	per	ADP
cana-328	322	3	the	the	DET
cana-328	322	4	table	table	NOUN
cana-328	322	5	,	,	PUNCT
cana-328	322	6	in	in	ADP
cana-328	322	7	the	the	DET
cana-328	322	8	first	first	ADJ
cana-328	322	9	year	year	NOUN
cana-328	322	10	,	,	PUNCT
cana-328	322	11	there	there	PRON
cana-328	322	12	is	be	VERB
cana-328	322	13	no	no	DET
cana-328	322	14	autonomous	autonomous	ADJ
cana-328	322	15	investment	investment	NOUN
cana-328	322	16	or	or	CCONJ
cana-328	322	17	consumption	consumption	NOUN
cana-328	322	18	,	,	PUNCT
cana-328	322	19	and	and	CCONJ
cana-328	322	20	hence	hence	ADV
cana-328	322	21	,	,	PUNCT
cana-328	322	22	the	the	DET
cana-328	322	23	yearly	yearly	ADJ
cana-328	322	24	income	income	NOUN
cana-328	322	25	is	be	AUX
cana-328	322	26	rs	rs	ADJ
cana-328	322	27	.	.	PROPN
cana-328	322	28	2	2	NUM
cana-328	322	29	crore	crore	NUM
cana-328	322	30	.	.	PUNCT
cana-328	323	1	in	in	ADP
cana-328	323	2	the	the	DET
cana-328	323	3	second	second	ADJ
cana-328	323	4	year	year	NOUN
cana-328	323	5	,	,	PUNCT
cana-328	323	6	the	the	DET
cana-328	323	7	autonomous	autonomous	ADJ
cana-328	323	8	investment	investment	NOUN
cana-328	323	9	is	be	AUX
cana-328	323	10	still	still	ADV
cana-328	323	11	rs	rs	ADJ
cana-328	323	12	.	.	PROPN
cana-328	323	13	2	2	NUM
cana-328	323	14	crore	crore	NUM
cana-328	323	15	,	,	PUNCT
cana-328	323	16	but	but	CCONJ
cana-328	323	17	there	there	PRON
cana-328	323	18	is	be	VERB
cana-328	323	19	a	a	DET
cana-328	323	20	consumption	consumption	NOUN
cana-328	323	21	of	of	ADP
cana-328	323	22	rs	rs	NOUN
cana-328	323	23	.	.	PROPN
cana-328	323	24	3	3	NUM
cana-328	323	25	crore	crore	NUM
cana-328	323	26	,	,	PUNCT
cana-328	323	27	resulting	result	VERB
cana-328	323	28	in	in	ADP
cana-328	323	29	an	an	DET
cana-328	323	30	induced	induced	ADJ
cana-328	323	31	investment	investment	NOUN
cana-328	323	32	of	of	ADP
cana-328	323	33	rs	rs	NOUN
cana-328	323	34	.	.	PROPN
cana-328	323	35	1.5	1.5	NUM
cana-328	323	36	crore	crore	NOUN
cana-328	323	37	.	.	PUNCT
cana-328	324	1	this	this	PRON
cana-328	324	2	brings	bring	VERB
cana-328	324	3	the	the	DET
cana-328	324	4	yearly	yearly	ADJ
cana-328	324	5	income	income	NOUN
cana-328	324	6	to	to	AUX
cana-328	324	7	rs	rs	PROPN
cana-328	324	8	.	.	PUNCT
cana-328	324	9	4.5	4.5	NUM
cana-328	324	10	crore	crore	NUM
cana-328	324	11	.	.	PUNCT
cana-328	325	1	similarly	similarly	ADV
cana-328	325	2	,	,	PUNCT
cana-328	325	3	in	in	ADP
cana-328	325	4	the	the	DET
cana-328	325	5	third	third	ADJ
cana-328	325	6	year	year	NOUN
cana-328	325	7	,	,	PUNCT
cana-328	325	8	with	with	ADP
cana-328	325	9	a	a	DET
cana-328	325	10	consumption	consumption	NOUN
cana-328	325	11	of	of	ADP
cana-328	325	12	rs	rs	NOUN
cana-328	325	13	.	.	PROPN
cana-328	325	14	4	4	NUM
cana-328	325	15	crore	crore	NUM
cana-328	325	16	,	,	PUNCT
cana-328	325	17	the	the	DET
cana-328	325	18	induced	induced	ADJ
cana-328	325	19	investment	investment	NOUN
cana-328	325	20	is	be	AUX
cana-328	325	21	rs	rs	ADJ
cana-328	325	22	.	.	PUNCT
cana-328	325	23	4	4	NUM
cana-328	325	24	crore	crore	NUM
cana-328	325	25	,	,	PUNCT
cana-328	325	26	leading	lead	VERB
cana-328	325	27	to	to	ADP
cana-328	325	28	a	a	DET
cana-328	325	29	yearly	yearly	ADJ
cana-328	325	30	income	income	NOUN
cana-328	325	31	of	of	ADP
cana-328	325	32	rs	rs	NOUN
cana-328	325	33	.	.	PUNCT
cana-328	325	34	10	10	NUM
cana-328	325	35	crore	crore	NUM
cana-328	325	36	.	.	PUNCT
cana-328	326	1	by	by	ADP
cana-328	326	2	using	use	VERB
cana-328	326	3	the	the	DET
cana-328	326	4	second	second	ADJ
cana-328	326	5	-	-	PUNCT
cana-328	326	6	order	order	NOUN
cana-328	326	7	difference	difference	NOUN
cana-328	326	8	equation	equation	NOUN
cana-328	326	9	,	,	PUNCT
cana-328	326	10	we	we	PRON
cana-328	326	11	can	can	AUX
cana-328	326	12	determine	determine	VERB
cana-328	326	13	the	the	DET
cana-328	326	14	yearly	yearly	ADJ
cana-328	326	15	income	income	NOUN
cana-328	326	16	for	for	ADP
cana-328	326	17	the	the	DET
cana-328	326	18	remaining	remain	VERB
cana-328	326	19	two	two	NUM
cana-328	326	20	years	year	NOUN
cana-328	326	21	based	base	VERB
cana-328	326	22	on	on	ADP
cana-328	326	23	the	the	DET
cana-328	326	24	values	value	NOUN
cana-328	326	25	of	of	ADP
cana-328	326	26	consumption	consumption	NOUN
cana-328	326	27	and	and	CCONJ
cana-328	326	28	induced	induced	ADJ
cana-328	326	29	investment	investment	NOUN
cana-328	326	30	.	.	PUNCT
cana-328	327	1	this	this	DET
cana-328	327	2	equation	equation	NOUN
cana-328	327	3	helps	help	VERB
cana-328	327	4	to	to	PART
cana-328	327	5	provide	provide	VERB
cana-328	327	6	a	a	DET
cana-328	327	7	better	well	ADJ
cana-328	327	8	understanding	understanding	NOUN
cana-328	327	9	of	of	ADP
cana-328	327	10	the	the	DET
cana-328	327	11	dynamics	dynamic	NOUN
cana-328	327	12	of	of	ADP
cana-328	327	13	economic	economic	ADJ
cana-328	327	14	cycles	cycle	NOUN
cana-328	327	15	and	and	CCONJ
cana-328	327	16	how	how	SCONJ
cana-328	327	17	changes	change	NOUN
cana-328	327	18	in	in	ADP
cana-328	327	19	consumption	consumption	NOUN
cana-328	327	20	and	and	CCONJ
cana-328	327	21	investment	investment	NOUN
cana-328	327	22	can	can	AUX
cana-328	327	23	impact	impact	VERB
cana-328	327	24	them	they	PRON
cana-328	327	25	.	.	PUNCT
cana-328	328	1	figure	figure	VERB
cana-328	328	2	1	1	NUM
cana-328	328	3	:	:	PUNCT
cana-328	328	4	communications	communication	NOUN
cana-328	328	5	on	on	ADP
cana-328	328	6	applied	apply	VERB
cana-328	328	7	nonlinear	nonlinear	ADJ
cana-328	328	8	analysis	analysis	NOUN
cana-328	328	9	issn	issn	NOUN
cana-328	328	10	:	:	PUNCT
cana-328	328	11	1074	1074	NUM
cana-328	328	12	-	-	PUNCT
cana-328	328	13	133x	133x	NUM
cana-328	328	14	vol	vol	NOUN
cana-328	328	15	31	31	NUM
cana-328	328	16	no	no	NOUN
cana-328	328	17	.	.	NOUN
cana-328	328	18	1	1	NUM
cana-328	328	19	(	(	PUNCT
cana-328	328	20	2024	2024	NUM
cana-328	328	21	)	)	PUNCT
cana-328	328	22	102	102	NUM
cana-328	328	23	https://internationalpubls.com	https://internationalpubls.com	X
cana-328	328	24	this	this	DET
cana-328	328	25	graph	graph	NOUN
cana-328	328	26	indicates	indicate	VERB
cana-328	328	27	the	the	DET
cana-328	328	28	five	five	NUM
cana-328	328	29	-	-	PUNCT
cana-328	328	30	year	year	NOUN
cana-328	328	31	income	income	NOUN
cana-328	328	32	of	of	ADP
cana-328	328	33	the	the	DET
cana-328	328	34	business	business	NOUN
cana-328	328	35	.	.	PUNCT
cana-328	329	1	it	it	PRON
cana-328	329	2	will	will	AUX
cana-328	329	3	be	be	AUX
cana-328	329	4	changed	change	VERB
cana-328	329	5	based	base	VERB
cana-328	329	6	on	on	ADP
cana-328	329	7	the	the	DET
cana-328	329	8	consumption	consumption	NOUN
cana-328	329	9	.	.	PUNCT
cana-328	330	1	5	5	NUM
cana-328	330	2	conclusion	conclusion	NOUN
cana-328	330	3	the	the	DET
cana-328	330	4	article	article	NOUN
cana-328	330	5	shows	show	VERB
cana-328	330	6	that	that	SCONJ
cana-328	330	7	property	property	NOUN
cana-328	330	8	a	a	DET
cana-328	330	9	solution	solution	NOUN
cana-328	330	10	is	be	AUX
cana-328	330	11	closely	closely	ADV
cana-328	330	12	related	relate	VERB
cana-328	330	13	to	to	ADP
cana-328	330	14	samuelson	samuelson	PROPN
cana-328	330	15	’s	’s	PART
cana-328	330	16	business	business	PROPN
cana-328	330	17	cycle	cycle	NOUN
cana-328	330	18	model	model	NOUN
cana-328	330	19	,	,	PUNCT
cana-328	330	20	which	which	PRON
cana-328	330	21	is	be	AUX
cana-328	330	22	a	a	DET
cana-328	330	23	prominent	prominent	ADJ
cana-328	330	24	economic	economic	ADJ
cana-328	330	25	theory	theory	NOUN
cana-328	330	26	of	of	ADP
cana-328	330	27	the	the	DET
cana-328	330	28	fluctuations	fluctuation	NOUN
cana-328	330	29	in	in	ADP
cana-328	330	30	output	output	NOUN
cana-328	330	31	and	and	CCONJ
cana-328	330	32	employment	employment	NOUN
cana-328	330	33	.	.	PUNCT
cana-328	331	1	the	the	DET
cana-328	331	2	authors	author	NOUN
cana-328	331	3	illustrate	illustrate	VERB
cana-328	331	4	the	the	DET
cana-328	331	5	importance	importance	NOUN
cana-328	331	6	of	of	ADP
cana-328	331	7	their	their	PRON
cana-328	331	8	main	main	ADJ
cana-328	331	9	results	result	NOUN
cana-328	331	10	by	by	ADP
cana-328	331	11	giving	give	VERB
cana-328	331	12	some	some	DET
cana-328	331	13	examples	example	NOUN
cana-328	331	14	and	and	CCONJ
cana-328	331	15	comparing	compare	VERB
cana-328	331	16	them	they	PRON
cana-328	331	17	with	with	ADP
cana-328	331	18	existing	exist	VERB
cana-328	331	19	literature	literature	NOUN
cana-328	331	20	.	.	PUNCT
cana-328	332	1	they	they	PRON
cana-328	332	2	also	also	ADV
cana-328	332	3	suggest	suggest	VERB
cana-328	332	4	some	some	DET
cana-328	332	5	possible	possible	ADJ
cana-328	332	6	directions	direction	NOUN
cana-328	332	7	for	for	ADP
cana-328	332	8	future	future	ADJ
cana-328	332	9	research	research	NOUN
cana-328	332	10	on	on	ADP
cana-328	332	11	this	this	DET
cana-328	332	12	topic	topic	NOUN
cana-328	332	13	.	.	PUNCT
cana-328	333	1	the	the	DET
cana-328	333	2	research	research	NOUN
cana-328	333	3	paper	paper	NOUN
cana-328	333	4	discussed	discuss	VERB
cana-328	333	5	the	the	DET
cana-328	333	6	oscillatory	oscillatory	ADJ
cana-328	333	7	behavior	behavior	NOUN
cana-328	333	8	and	and	CCONJ
cana-328	333	9	property	property	NOUN
cana-328	333	10	a	a	DET
cana-328	333	11	condition	condition	NOUN
cana-328	333	12	for	for	ADP
cana-328	333	13	a	a	DET
cana-328	333	14	positive	positive	ADJ
cana-328	333	15	middle	middle	ADJ
cana-328	333	16	term	term	NOUN
cana-328	333	17	.	.	PUNCT
cana-328	334	1	this	this	DET
cana-328	334	2	equation	equation	NOUN
cana-328	334	3	has	have	VERB
cana-328	334	4	applications	application	NOUN
cana-328	334	5	in	in	ADP
cana-328	334	6	economics	economic	NOUN
cana-328	334	7	,	,	PUNCT
cana-328	334	8	physics	physics	NOUN
cana-328	334	9	,	,	PUNCT
cana-328	334	10	and	and	CCONJ
cana-328	334	11	mathematical	mathematical	ADJ
cana-328	334	12	biology	biology	NOUN
cana-328	334	13	.	.	PUNCT
cana-328	335	1	the	the	DET
cana-328	335	2	article	article	NOUN
cana-328	335	3	establishes	establish	VERB
cana-328	335	4	new	new	ADJ
cana-328	335	5	oscillatory	oscillatory	ADJ
cana-328	335	6	behavior	behavior	NOUN
cana-328	335	7	and	and	CCONJ
cana-328	335	8	property	property	NOUN
cana-328	335	9	a	a	DET
cana-328	335	10	conditions	condition	NOUN
cana-328	335	11	for	for	ADP
cana-328	335	12	the	the	DET
cana-328	335	13	equation	equation	NOUN
cana-328	335	14	,	,	PUNCT
cana-328	335	15	which	which	PRON
cana-328	335	16	can	can	AUX
cana-328	335	17	be	be	AUX
cana-328	335	18	considered	consider	VERB
cana-328	335	19	an	an	DET
cana-328	335	20	improvement	improvement	NOUN
cana-328	335	21	to	to	ADP
cana-328	335	22	previous	previous	ADJ
cana-328	335	23	results	result	NOUN
cana-328	335	24	.	.	PUNCT
cana-328	336	1	the	the	DET
cana-328	336	2	authors	author	NOUN
cana-328	336	3	use	use	VERB
cana-328	336	4	a	a	DET
cana-328	336	5	generalized	generalized	ADJ
cana-328	336	6	riccati	riccati	NOUN
cana-328	336	7	transformation	transformation	NOUN
cana-328	336	8	technique	technique	NOUN
cana-328	336	9	and	and	CCONJ
cana-328	336	10	some	some	DET
cana-328	336	11	auxiliary	auxiliary	ADJ
cana-328	336	12	second	second	ADJ
cana-328	336	13	-	-	PUNCT
cana-328	336	14	order	order	NOUN
cana-328	336	15	difference	difference	NOUN
cana-328	336	16	equations	equation	NOUN
cana-328	336	17	.	.	PUNCT
cana-328	337	1	additionally	additionally	ADV
cana-328	337	2	,	,	PUNCT
cana-328	337	3	the	the	DET
cana-328	337	4	article	article	NOUN
cana-328	337	5	derives	derive	VERB
cana-328	337	6	a	a	DET
cana-328	337	7	solution	solution	NOUN
cana-328	337	8	for	for	ADP
cana-328	337	9	the	the	DET
cana-328	337	10	equation	equation	NOUN
cana-328	337	11	that	that	PRON
cana-328	337	12	possesses	possess	VERB
cana-328	337	13	property	property	NOUN
cana-328	337	14	a	a	PRON
cana-328	337	15	and	and	CCONJ
cana-328	337	16	oscillatory	oscillatory	ADJ
cana-328	337	17	.	.	PUNCT
cana-328	338	1	the	the	DET
cana-328	338	2	article	article	NOUN
cana-328	338	3	also	also	ADV
cana-328	338	4	highlights	highlight	VERB
cana-328	338	5	the	the	DET
cana-328	338	6	connection	connection	NOUN
cana-328	338	7	between	between	ADP
cana-328	338	8	their	their	PRON
cana-328	338	9	property	property	NOUN
cana-328	338	10	a	a	DET
cana-328	338	11	solution	solution	NOUN
cana-328	338	12	and	and	CCONJ
cana-328	338	13	samuelson	samuelson	PROPN
cana-328	338	14	’s	’s	PART
cana-328	338	15	business	business	NOUN
cana-328	338	16	cycle	cycle	NOUN
cana-328	338	17	model	model	NOUN
cana-328	338	18	.	.	PUNCT
cana-328	339	1	this	this	DET
cana-328	339	2	model	model	NOUN
cana-328	339	3	is	be	AUX
cana-328	339	4	a	a	DET
cana-328	339	5	well	well	ADV
cana-328	339	6	-	-	PUNCT
cana-328	339	7	known	know	VERB
cana-328	339	8	economic	economic	ADJ
cana-328	339	9	theory	theory	NOUN
cana-328	339	10	that	that	PRON
cana-328	339	11	explains	explain	VERB
cana-328	339	12	the	the	DET
cana-328	339	13	fluctuations	fluctuation	NOUN
cana-328	339	14	in	in	ADP
cana-328	339	15	output	output	NOUN
cana-328	339	16	and	and	CCONJ
cana-328	339	17	employment	employment	NOUN
cana-328	339	18	.	.	PUNCT
cana-328	340	1	by	by	ADP
cana-328	340	2	drawing	draw	VERB
cana-328	340	3	on	on	ADP
cana-328	340	4	this	this	DET
cana-328	340	5	connection	connection	NOUN
cana-328	340	6	,	,	PUNCT
cana-328	340	7	the	the	DET
cana-328	340	8	authors	author	NOUN
cana-328	340	9	illustrate	illustrate	VERB
cana-328	340	10	the	the	DET
cana-328	340	11	importance	importance	NOUN
cana-328	340	12	of	of	ADP
cana-328	340	13	their	their	PRON
cana-328	340	14	main	main	ADJ
cana-328	340	15	results	result	NOUN
cana-328	340	16	by	by	ADP
cana-328	340	17	giving	give	VERB
cana-328	340	18	some	some	DET
cana-328	340	19	examples	example	NOUN
cana-328	340	20	and	and	CCONJ
cana-328	340	21	comparing	compare	VERB
cana-328	340	22	them	they	PRON
cana-328	340	23	with	with	ADP
cana-328	340	24	existing	exist	VERB
cana-328	340	25	literature	literature	NOUN
cana-328	340	26	.	.	PUNCT
cana-328	341	1	the	the	DET
cana-328	341	2	article	article	NOUN
cana-328	341	3	also	also	ADV
cana-328	341	4	suggests	suggest	VERB
cana-328	341	5	possible	possible	ADJ
cana-328	341	6	directions	direction	NOUN
cana-328	341	7	for	for	ADP
cana-328	341	8	future	future	ADJ
cana-328	341	9	research	research	NOUN
cana-328	341	10	on	on	ADP
cana-328	341	11	this	this	DET
cana-328	341	12	topic	topic	NOUN
cana-328	341	13	,	,	PUNCT
cana-328	341	14	highlighting	highlight	VERB
cana-328	341	15	the	the	DET
cana-328	341	16	potential	potential	NOUN
cana-328	341	17	for	for	ADP
cana-328	341	18	further	further	ADJ
cana-328	341	19	exploration	exploration	NOUN
cana-328	341	20	and	and	CCONJ
cana-328	341	21	applications	application	NOUN
cana-328	341	22	of	of	ADP
cana-328	341	23	the	the	DET
cana-328	341	24	results	result	NOUN
cana-328	341	25	in	in	ADP
cana-328	341	26	various	various	ADJ
cana-328	341	27	fields	field	NOUN
cana-328	341	28	.	.	PUNCT
cana-328	342	1	the	the	DET
cana-328	342	2	application	application	NOUN
cana-328	342	3	of	of	ADP
cana-328	342	4	samuelson	samuelson	PROPN
cana-328	342	5	's	's	PART
cana-328	342	6	business	business	NOUN
cana-328	342	7	cycle	cycle	NOUN
cana-328	342	8	model	model	NOUN
cana-328	342	9	in	in	ADP
cana-328	342	10	a	a	DET
cana-328	342	11	second	second	ADJ
cana-328	342	12	-	-	PUNCT
cana-328	342	13	order	order	NOUN
cana-328	342	14	difference	difference	NOUN
cana-328	342	15	equation	equation	NOUN
cana-328	342	16	(	(	PUNCT
cana-328	342	17	2	2	X
cana-328	342	18	)	)	PUNCT
cana-328	342	19	is	be	AUX
cana-328	342	20	presented	present	VERB
cana-328	342	21	,	,	PUNCT
cana-328	342	22	and	and	CCONJ
cana-328	342	23	the	the	DET
cana-328	342	24	yearly	yearly	ADJ
cana-328	342	25	income	income	NOUN
cana-328	342	26	is	be	AUX
cana-328	342	27	calculated	calculate	VERB
cana-328	342	28	based	base	VERB
cana-328	342	29	on	on	ADP
cana-328	342	30	the	the	DET
cana-328	342	31	model	model	NOUN
cana-328	342	32	.	.	PUNCT
cana-328	343	1	this	this	PRON
cana-328	343	2	demonstrates	demonstrate	VERB
cana-328	343	3	the	the	DET
cana-328	343	4	practical	practical	ADJ
cana-328	343	5	relevance	relevance	NOUN
cana-328	343	6	of	of	ADP
cana-328	343	7	the	the	DET
cana-328	343	8	research	research	NOUN
cana-328	343	9	and	and	CCONJ
cana-328	343	10	its	its	PRON
cana-328	343	11	potential	potential	ADJ
cana-328	343	12	implications	implication	NOUN
cana-328	343	13	for	for	ADP
cana-328	343	14	real	real	ADJ
cana-328	343	15	-	-	PUNCT
cana-328	343	16	world	world	NOUN
cana-328	343	17	scenarios	scenario	NOUN
cana-328	343	18	.	.	PUNCT
cana-328	344	1	references	reference	NOUN
cana-328	344	2	[	[	X
cana-328	344	3	1	1	NUM
cana-328	344	4	]	]	PUNCT
cana-328	344	5	agarwal	agarwal	PROPN
cana-328	344	6	,	,	PUNCT
cana-328	344	7	r.p	r.p	PROPN
cana-328	344	8	.	.	PROPN
cana-328	344	9	,	,	PUNCT
cana-328	344	10	difference	difference	NOUN
cana-328	344	11	equation	equation	NOUN
cana-328	344	12	and	and	CCONJ
cana-328	344	13	inequalities	inequality	NOUN
cana-328	344	14	,	,	PUNCT
cana-328	344	15	marcel	marcel	PROPN
cana-328	344	16	dekker	dekker	PROPN
cana-328	344	17	,	,	PUNCT
cana-328	344	18	new	new	PROPN
cana-328	344	19	york	york	PROPN
cana-328	344	20	,	,	PUNCT
cana-328	344	21	2000	2000	NUM
cana-328	344	22	.	.	PUNCT
cana-328	345	1	[	[	X
cana-328	345	2	2	2	NUM
cana-328	345	3	]	]	PUNCT
cana-328	345	4	agarwal	agarwal	PROPN
cana-328	345	5	,	,	PUNCT
cana-328	345	6	r.p	r.p	PROPN
cana-328	345	7	.	.	PROPN
cana-328	345	8	,	,	PUNCT
cana-328	345	9	bohner	bohner	NOUN
cana-328	345	10	,	,	PUNCT
cana-328	345	11	m.	m.	NOUN
cana-328	345	12	,	,	PUNCT
cana-328	345	13	grace	grace	NOUN
cana-328	345	14	,	,	PUNCT
cana-328	345	15	s.r	s.r	PROPN
cana-328	345	16	.	.	PROPN
cana-328	345	17	,	,	PUNCT
cana-328	345	18	o'regan	o'regan	PROPN
cana-328	345	19	,	,	PUNCT
cana-328	345	20	d.	d.	PROPN
cana-328	345	21	,	,	PUNCT
cana-328	345	22	discrete	discrete	VERB
cana-328	345	23	oscillation	oscillation	NOUN
cana-328	345	24	theory	theory	NOUN
cana-328	345	25	,	,	PUNCT
cana-328	345	26	hindawi	hindawi	ADJ
cana-328	345	27	,	,	PUNCT
cana-328	345	28	new	new	PROPN
cana-328	345	29	york	york	PROPN
cana-328	345	30	,	,	PUNCT
cana-328	345	31	2005	2005	NUM
cana-328	345	32	.	.	PUNCT
cana-328	346	1	[	[	X
cana-328	346	2	3	3	NUM
cana-328	346	3	]	]	PUNCT
cana-328	346	4	elaydi	elaydi	PROPN
cana-328	346	5	,	,	PUNCT
cana-328	346	6	s.	s.	PROPN
cana-328	346	7	an	an	DET
cana-328	346	8	introduction	introduction	NOUN
cana-328	346	9	to	to	ADP
cana-328	346	10	difference	difference	NOUN
cana-328	346	11	equations	equation	NOUN
cana-328	346	12	,	,	PUNCT
cana-328	346	13	springer	springer	NOUN
cana-328	346	14	-	-	PUNCT
cana-328	346	15	verlag	verlag	PROPN
cana-328	346	16	,	,	PUNCT
cana-328	346	17	new	new	PROPN
cana-328	346	18	york	york	PROPN
cana-328	346	19	,	,	PUNCT
cana-328	346	20	1996	1996	NUM
cana-328	346	21	.	.	PUNCT
cana-328	347	1	[	[	X
cana-328	347	2	4	4	NUM
cana-328	347	3	]	]	X
cana-328	347	4	gyori	gyori	NOUN
cana-328	347	5	,	,	PUNCT
cana-328	347	6	i.	i.	PROPN
cana-328	347	7	,	,	PUNCT
cana-328	347	8	ladas	ladas	PROPN
cana-328	347	9	,	,	PUNCT
cana-328	347	10	g.	g.	PROPN
cana-328	347	11	,	,	PUNCT
cana-328	347	12	oscillation	oscillation	NOUN
cana-328	347	13	theory	theory	NOUN
cana-328	347	14	of	of	ADP
cana-328	347	15	delay	delay	NOUN
cana-328	347	16	differential	differential	ADJ
cana-328	347	17	equations	equation	NOUN
cana-328	347	18	with	with	ADP
cana-328	347	19	applications	application	NOUN
cana-328	347	20	,	,	PUNCT
cana-328	347	21	clarendon	clarendon	PROPN
cana-328	347	22	press	press	NOUN
cana-328	347	23	,	,	PUNCT
cana-328	347	24	oxford	oxford	NOUN
cana-328	347	25	,	,	PUNCT
cana-328	347	26	1991	1991	NUM
cana-328	347	27	.	.	PUNCT
cana-328	348	1	[	[	X
cana-328	348	2	5	5	NUM
cana-328	348	3	]	]	X
cana-328	348	4	grace	grace	NOUN
cana-328	348	5	,	,	PUNCT
cana-328	348	6	s.r	s.r	PROPN
cana-328	348	7	.	.	PROPN
cana-328	348	8	,	,	PUNCT
cana-328	348	9	kaleeswari	kaleeswari	PROPN
cana-328	348	10	,	,	PUNCT
cana-328	348	11	s.	s.	PROPN
cana-328	348	12	,	,	PUNCT
cana-328	348	13	oscillation	oscillation	NOUN
cana-328	348	14	of	of	ADP
cana-328	348	15	even	even	ADJ
cana-328	348	16	-	-	PUNCT
cana-328	348	17	order	order	NOUN
cana-328	348	18	nonlinear	nonlinear	ADJ
cana-328	348	19	difference	difference	NOUN
cana-328	348	20	equations	equation	NOUN
cana-328	348	21	with	with	ADP
cana-328	348	22	an	an	DET
cana-328	348	23	advanced	advanced	ADJ
cana-328	348	24	argument	argument	NOUN
cana-328	348	25	,	,	PUNCT
cana-328	348	26	dynamics	dynamic	NOUN
cana-328	348	27	of	of	ADP
cana-328	348	28	continuous	continuous	ADJ
cana-328	348	29	,	,	PUNCT
cana-328	348	30	discrete	discrete	ADJ
cana-328	348	31	and	and	CCONJ
cana-328	348	32	impulsive	impulsive	ADJ
cana-328	348	33	system	system	NOUN
cana-328	348	34	series	series	NOUN
cana-328	348	35	a	a	NOUN
cana-328	348	36	,	,	PUNCT
cana-328	348	37	30	30	NUM
cana-328	348	38	,	,	PUNCT
cana-328	348	39	3a	3a	NUM
cana-328	348	40	,	,	PUNCT
cana-328	348	41	243	243	NUM
cana-328	348	42	-	-	SYM
cana-328	348	43	251	251	NUM
cana-328	348	44	,	,	PUNCT
cana-328	348	45	2023	2023	NUM
cana-328	348	46	.	.	PUNCT
cana-328	349	1	[	[	X
cana-328	349	2	6	6	NUM
cana-328	349	3	]	]	PUNCT
cana-328	349	4	john	john	PROPN
cana-328	349	5	r.	r.	PROPN
cana-328	349	6	graef	graef	PROPN
cana-328	349	7	,	,	PUNCT
cana-328	349	8	thandapani	thandapani	PROPN
cana-328	349	9	,	,	PUNCT
cana-328	349	10	e.	e.	PROPN
cana-328	349	11	,	,	PUNCT
cana-328	349	12	oscillatory	oscillatory	ADJ
cana-328	349	13	and	and	CCONJ
cana-328	349	14	asymptotic	asymptotic	ADJ
cana-328	349	15	behavior	behavior	NOUN
cana-328	349	16	of	of	ADP
cana-328	349	17	solutions	solution	NOUN
cana-328	349	18	of	of	ADP
cana-328	349	19	third	third	ADJ
cana-328	349	20	order	order	NOUN
cana-328	349	21	delay	delay	NOUN
cana-328	349	22	difference	difference	NOUN
cana-328	349	23	equations	equation	NOUN
cana-328	349	24	,	,	PUNCT
cana-328	349	25	funkcialaj	funkcialaj	PROPN
cana-328	349	26	ekvacioj	ekvacioj	PROPN
cana-328	349	27	,	,	PUNCT
cana-328	349	28	42	42	NUM
cana-328	349	29	,	,	PUNCT
cana-328	349	30	355	355	NUM
cana-328	349	31	-	-	SYM
cana-328	349	32	369	369	NUM
cana-328	349	33	,	,	PUNCT
cana-328	349	34	1999	1999	NUM
cana-328	349	35	.	.	PUNCT
cana-328	350	1	[	[	X
cana-328	350	2	7	7	NUM
cana-328	350	3	]	]	X
cana-328	350	4	kaleeswari	kaleeswari	NOUN
cana-328	350	5	,	,	PUNCT
cana-328	350	6	s.	s.	PROPN
cana-328	350	7	,	,	PUNCT
cana-328	350	8	selvaraj	selvaraj	PROPN
cana-328	350	9	,	,	PUNCT
cana-328	350	10	b.	b.	PROPN
cana-328	350	11	,	,	PUNCT
cana-328	350	12	thiyagarajan	thiyagarajan	NOUN
cana-328	350	13	,	,	PUNCT
cana-328	350	14	m.	m.	NOUN
cana-328	350	15	,	,	PUNCT
cana-328	350	16	a	a	DET
cana-328	350	17	new	new	ADJ
cana-328	350	18	creation	creation	NOUN
cana-328	350	19	of	of	ADP
cana-328	350	20	mask	mask	NOUN
cana-328	350	21	from	from	ADP
cana-328	350	22	difference	difference	NOUN
cana-328	350	23	operator	operator	NOUN
cana-328	350	24	to	to	PART
cana-328	350	25	image	image	VERB
cana-328	350	26	analysis	analysis	NOUN
cana-328	350	27	,	,	PUNCT
cana-328	350	28	journal	journal	NOUN
cana-328	350	29	of	of	ADP
cana-328	350	30	theoretical	theoretical	ADJ
cana-328	350	31	and	and	CCONJ
cana-328	350	32	applied	apply	VERB
cana-328	350	33	information	information	NOUN
cana-328	350	34	technology	technology	NOUN
cana-328	350	35	,	,	PUNCT
cana-328	350	36	69(1),2014	69(1),2014	NUM
cana-328	350	37	[	[	X
cana-328	350	38	8	8	NUM
cana-328	350	39	]	]	X
cana-328	350	40	kaleeswari	kaleeswari	NOUN
cana-328	350	41	,	,	PUNCT
cana-328	350	42	s.	s.	PROPN
cana-328	350	43	,	,	PUNCT
cana-328	350	44	selvaraj	selvaraj	PROPN
cana-328	350	45	,	,	PUNCT
cana-328	350	46	b.	b.	PROPN
cana-328	350	47	,	,	PUNCT
cana-328	350	48	on	on	ADP
cana-328	350	49	the	the	DET
cana-328	350	50	oscillation	oscillation	NOUN
cana-328	350	51	of	of	ADP
cana-328	350	52	certain	certain	ADJ
cana-328	350	53	odd	odd	ADJ
cana-328	350	54	-	-	PUNCT
cana-328	350	55	order	order	NOUN
cana-328	350	56	nonlinear	nonlinear	ADJ
cana-328	350	57	neutral	neutral	ADJ
cana-328	350	58	difference	difference	NOUN
cana-328	350	59	equations	equation	NOUN
cana-328	350	60	,	,	PUNCT
cana-328	350	61	applied	apply	VERB
cana-328	350	62	sciences	science	NOUN
cana-328	350	63	,	,	PUNCT
cana-328	350	64	18	18	NUM
cana-328	350	65	,	,	PUNCT
cana-328	350	66	pp	pp	ADJ
cana-328	350	67	.	.	PUNCT
cana-328	351	1	50	50	NUM
cana-328	351	2	-	-	SYM
cana-328	351	3	59	59	NUM
cana-328	351	4	,	,	PUNCT
cana-328	351	5	2016	2016	NUM
cana-328	351	6	.	.	PUNCT
cana-328	352	1	communications	communication	NOUN
cana-328	352	2	on	on	ADP
cana-328	352	3	applied	apply	VERB
cana-328	352	4	nonlinear	nonlinear	ADJ
cana-328	352	5	analysis	analysis	NOUN
cana-328	352	6	issn	issn	NOUN
cana-328	352	7	:	:	PUNCT
cana-328	352	8	1074	1074	NUM
cana-328	352	9	-	-	PUNCT
cana-328	352	10	133x	133x	NUM
cana-328	352	11	vol	vol	NOUN
cana-328	352	12	31	31	NUM
cana-328	352	13	no	no	NOUN
cana-328	352	14	.	.	NOUN
cana-328	352	15	1	1	NUM
cana-328	352	16	(	(	PUNCT
cana-328	352	17	2024	2024	NUM
cana-328	352	18	)	)	PUNCT
cana-328	352	19	103	103	NUM
cana-328	352	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-328	353	1	[	[	X
cana-328	353	2	9	9	NUM
cana-328	353	3	]	]	PUNCT
cana-328	353	4	kaleeswari	kaleeswari	NOUN
cana-328	353	5	,	,	PUNCT
cana-328	353	6	s.	s.	PROPN
cana-328	353	7	,	,	PUNCT
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cana-328	353	9	criteria	criterion	NOUN
cana-328	353	10	for	for	ADP
cana-328	353	11	mixed	mixed	ADJ
cana-328	353	12	neutral	neutral	ADJ
cana-328	353	13	difference	difference	NOUN
cana-328	353	14	equations	equation	NOUN
cana-328	353	15	,	,	PUNCT
cana-328	353	16	asian	asian	ADJ
cana-328	353	17	journal	journal	NOUN
cana-328	353	18	of	of	ADP
cana-328	353	19	mathematics	mathematics	PROPN
cana-328	353	20	and	and	CCONJ
cana-328	353	21	computer	computer	NOUN
cana-328	353	22	research	research	NOUN
cana-328	353	23	,	,	PUNCT
cana-328	353	24	25(6	25(6	NUM
cana-328	353	25	)	)	PUNCT
cana-328	353	26	,	,	PUNCT
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cana-328	353	28	,	,	PUNCT
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cana-328	353	32	-	-	SYM
cana-328	353	33	339	339	NUM
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cana-328	353	35	.	.	PUNCT
cana-328	354	1	[	[	X
cana-328	354	2	10	10	NUM
cana-328	354	3	]	]	X
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cana-328	354	5	,	,	PUNCT
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cana-328	354	7	,	,	PUNCT
cana-328	354	8	selvaraj	selvaraj	PROPN
cana-328	354	9	,	,	PUNCT
cana-328	354	10	b.	b.	PROPN
cana-328	354	11	,	,	PUNCT
cana-328	354	12	thyagarajan	thyagarajan	PROPN
cana-328	354	13	,	,	PUNCT
cana-328	354	14	m.	m.	NOUN
cana-328	354	15	,	,	PUNCT
cana-328	354	16	removing	remove	VERB
cana-328	354	17	noise	noise	NOUN
cana-328	354	18	through	through	ADP
cana-328	354	19	a	a	DET
cana-328	354	20	nonlinear	nonlinear	ADJ
cana-328	354	21	difference	difference	NOUN
cana-328	354	22	operator	operator	NOUN
cana-328	354	23	,	,	PUNCT
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cana-328	354	25	journal	journal	NOUN
cana-328	354	26	of	of	ADP
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cana-328	354	29	research	research	NOUN
cana-328	354	30	,	,	PUNCT
cana-328	354	31	9	9	NUM
cana-328	354	32	(	(	PUNCT
cana-328	354	33	21	21	NUM
cana-328	354	34	)	)	PUNCT
cana-328	354	35	,	,	PUNCT
cana-328	354	36	5100	5100	NUM
cana-328	354	37	-	-	SYM
cana-328	354	38	5106	5106	NUM
cana-328	354	39	,	,	PUNCT
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cana-328	355	1	[	[	X
cana-328	355	2	11	11	NUM
cana-328	355	3	]	]	PUNCT
cana-328	355	4	kaleeswari	kaleeswari	NOUN
cana-328	355	5	,	,	PUNCT
cana-328	355	6	s.	s.	PROPN
cana-328	355	7	,	,	PUNCT
cana-328	355	8	gowri	gowri	PROPN
cana-328	355	9	,	,	PUNCT
cana-328	355	10	j.	j.	PROPN
cana-328	355	11	,	,	PUNCT
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cana-328	355	19	,	,	PUNCT
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cana-328	355	21	chemical	chemical	PROPN
cana-328	355	22	bulletin	bulletin	PROPN
cana-328	355	23	,	,	PUNCT
cana-328	355	24	12(special	12(special	ADJ
cana-328	355	25	issue	issue	NOUN
cana-328	355	26	9	9	NUM
cana-328	355	27	)	)	PUNCT
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cana-328	355	34	.	.	PUNCT
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cana-328	356	2	12	12	NUM
cana-328	356	3	]	]	X
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cana-328	356	5	,	,	PUNCT
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cana-328	356	7	,	,	PUNCT
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cana-328	356	16	of	of	ADP
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cana-328	357	3	,	,	PUNCT
cana-328	357	4	vol.21	vol.21	PROPN
cana-328	357	5	,	,	PUNCT
cana-328	357	6	no.2	no.2	PROPN
cana-328	357	7	,	,	PUNCT
cana-328	357	8	1939	1939	NUM
cana-328	357	9	.	.	PUNCT
cana-328	358	1	[	[	X
cana-328	358	2	13	13	NUM
cana-328	358	3	]	]	X
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cana-328	358	5	,	,	PUNCT
cana-328	358	6	r.	r.	PROPN
cana-328	358	7	,	,	PUNCT
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cana-328	358	9	,	,	PUNCT
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cana-328	358	11	,	,	PUNCT
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cana-328	358	13	and	and	CCONJ
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cana-328	358	20	difference	difference	NOUN
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cana-328	358	30	and	and	CCONJ
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cana-328	358	32	mathematics	mathematic	NOUN
cana-328	358	33	,	,	PUNCT
cana-328	358	34	113	113	NUM
cana-328	358	35	,	,	PUNCT
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cana-328	358	37	,	,	PUNCT
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cana-328	358	40	360	360	NUM
cana-328	358	41	,	,	PUNCT
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cana-328	358	43	.	.	PUNCT
cana-328	359	1	[	[	X
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cana-328	359	3	]	]	SYM
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cana-328	359	5	,	,	PUNCT
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cana-328	359	7	,	,	PUNCT
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cana-328	359	13	,	,	PUNCT
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cana-328	359	18	3𝑟𝑑	3𝑟𝑑	ADJ
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cana-328	359	20	advanced	advanced	ADJ
cana-328	359	21	difference	difference	NOUN
cana-328	359	22	equations	equation	NOUN
cana-328	359	23	with	with	ADP
cana-328	359	24	a	a	DET
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cana-328	359	26	middle	middle	ADJ
cana-328	359	27	term	term	NOUN
cana-328	359	28	,	,	PUNCT
cana-328	359	29	journal	journal	NOUN
cana-328	359	30	of	of	ADP
cana-328	359	31	mathematics	mathematic	NOUN
cana-328	359	32	and	and	CCONJ
cana-328	359	33	computer	computer	NOUN
cana-328	359	34	science	science	NOUN
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cana-328	359	37	,	,	PUNCT
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cana-328	359	39	-	-	SYM
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cana-328	359	41	,	,	PUNCT
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cana-328	359	43	.	.	PUNCT
cana-328	360	1	[	[	X
cana-328	360	2	15	15	NUM
cana-328	360	3	]	]	X
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cana-328	360	5	,	,	PUNCT
cana-328	360	6	g.k	g.k	PROPN
cana-328	360	7	.	.	PROPN
cana-328	360	8	,	,	PUNCT
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cana-328	360	13	,	,	PUNCT
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cana-328	360	15	equations	equation	NOUN
cana-328	360	16	introduction	introduction	NOUN
cana-328	360	17	with	with	ADP
cana-328	360	18	applications	application	NOUN
cana-328	360	19	,	,	PUNCT
cana-328	360	20	second	second	ADJ
cana-328	360	21	edition	edition	NOUN
cana-328	360	22	,	,	PUNCT
cana-328	360	23	academic	academic	ADJ
cana-328	360	24	press	press	NOUN
cana-328	360	25	,	,	PUNCT
cana-328	360	26	1991	1991	NUM
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