id	sid	tid	token	lemma	pos
cana-852	1	1	communications	communication	NOUN
cana-852	1	2	on	on	ADP
cana-852	1	3	applied	apply	VERB
cana-852	1	4	nonlinear	nonlinear	ADJ
cana-852	1	5	analysis	analysis	NOUN
cana-852	1	6	issn	issn	NOUN
cana-852	1	7	:	:	PUNCT
cana-852	1	8	1074	1074	NUM
cana-852	1	9	-	-	PUNCT
cana-852	1	10	133x	133x	NUM
cana-852	1	11	vol	vol	NOUN
cana-852	1	12	31	31	NUM
cana-852	1	13	no	no	NOUN
cana-852	1	14	.	.	PUNCT
cana-852	2	1	4s	4s	NUM
cana-852	2	2	(	(	PUNCT
cana-852	2	3	2024	2024	NUM
cana-852	2	4	)	)	PUNCT
cana-852	2	5	286	286	NUM
cana-852	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-852	2	7	on	on	ADP
cana-852	2	8	the	the	DET
cana-852	2	9	oscillation	oscillation	NOUN
cana-852	2	10	of	of	ADP
cana-852	2	11	a	a	DET
cana-852	2	12	class	class	NOUN
cana-852	2	13	of	of	ADP
cana-852	2	14	conformable	conformable	ADJ
cana-852	2	15	schrodinger	schrodinger	NOUN
cana-852	2	16	equations	equation	NOUN
cana-852	2	17	1	1	NUM
cana-852	2	18	n.	n.	NOUN
cana-852	2	19	sasikala	sasikala	PROPN
cana-852	2	20	,	,	PUNCT
cana-852	2	21	2	2	NUM
cana-852	2	22	v.	v.	ADP
cana-852	2	23	sadhasivam	sadhasivam	VERB
cana-852	2	24	1	1	NUM
cana-852	2	25	post	post	NOUN
cana-852	2	26	graduate	graduate	NOUN
cana-852	2	27	and	and	CCONJ
cana-852	2	28	research	research	NOUN
cana-852	2	29	department	department	PROPN
cana-852	2	30	of	of	ADP
cana-852	2	31	mathematics	mathematic	NOUN
cana-852	2	32	,	,	PUNCT
cana-852	2	33	thiruvalluvar	thiruvalluvar	NOUN
cana-852	2	34	government	government	NOUN
cana-852	2	35	arts	arts	PROPN
cana-852	2	36	college	college	PROPN
cana-852	2	37	,	,	PUNCT
cana-852	2	38	rasipuram	rasipuram	VERB
cana-852	2	39	637	637	NUM
cana-852	2	40	401	401	NUM
cana-852	2	41	,	,	PUNCT
cana-852	2	42	namakkal	namakkal	NOUN
cana-852	2	43	dt	dt	PROPN
cana-852	2	44	.	.	PUNCT
cana-852	3	1	tamil	tamil	PROPN
cana-852	3	2	nadu	nadu	PROPN
cana-852	3	3	,	,	PUNCT
cana-852	3	4	india	india	PROPN
cana-852	3	5	,	,	PUNCT
cana-852	3	6	e	e	NOUN
cana-852	3	7	-	-	NOUN
cana-852	3	8	mail	mail	NOUN
cana-852	3	9	:	:	PUNCT
cana-852	4	1	krishivyattish@gmail.com	krishivyattish@gmail.com	X
cana-852	4	2	.	.	PUNCT
cana-852	5	1	2associate	2associate	NUM
cana-852	5	2	professor	professor	NOUN
cana-852	5	3	&	&	CCONJ
cana-852	5	4	head	head	PROPN
cana-852	5	5	,	,	PUNCT
cana-852	5	6	post	post	VERB
cana-852	5	7	graduate	graduate	NOUN
cana-852	5	8	and	and	CCONJ
cana-852	5	9	research	research	PROPN
cana-852	5	10	department	department	PROPN
cana-852	5	11	of	of	ADP
cana-852	5	12	mathematics	mathematic	NOUN
cana-852	5	13	,	,	PUNCT
cana-852	5	14	thiruvalluvar	thiruvalluvar	NOUN
cana-852	5	15	government	government	NOUN
cana-852	5	16	arts	arts	PROPN
cana-852	5	17	college	college	PROPN
cana-852	5	18	,	,	PUNCT
cana-852	5	19	rasipuram	rasipuram	VERB
cana-852	5	20	637	637	NUM
cana-852	5	21	401	401	NUM
cana-852	5	22	,	,	PUNCT
cana-852	5	23	namakkal	namakkal	NOUN
cana-852	5	24	dt	dt	PROPN
cana-852	5	25	.	.	PUNCT
cana-852	6	1	tamil	tamil	PROPN
cana-852	6	2	nadu	nadu	PROPN
cana-852	6	3	,	,	PUNCT
cana-852	6	4	india	india	PROPN
cana-852	6	5	.	.	PUNCT
cana-852	7	1	e	e	X
cana-852	7	2	-	-	NOUN
cana-852	7	3	mail	mail	NOUN
cana-852	7	4	ovsadha@gmail.com	ovsadha@gmail.com	SYM
cana-852	7	5	.	.	PUNCT
cana-852	7	6	article	article	PROPN
cana-852	7	7	history	history	NOUN
cana-852	7	8	:	:	PUNCT
cana-852	7	9	received	receive	VERB
cana-852	7	10	:	:	PUNCT
cana-852	7	11	20	20	NUM
cana-852	7	12	-	-	PUNCT
cana-852	7	13	04	04	NUM
cana-852	7	14	-	-	PUNCT
cana-852	7	15	2024	2024	NUM
cana-852	7	16	revised	revise	VERB
cana-852	7	17	:	:	PUNCT
cana-852	7	18	12	12	NUM
cana-852	7	19	-	-	PUNCT
cana-852	7	20	06	06	NUM
cana-852	7	21	-	-	PUNCT
cana-852	7	22	2024	2024	NUM
cana-852	7	23	accepted	accept	VERB
cana-852	7	24	:	:	PUNCT
cana-852	7	25	22	22	NUM
cana-852	7	26	-	-	SYM
cana-852	7	27	06	06	NUM
cana-852	7	28	-	-	PUNCT
cana-852	7	29	2024	2024	NUM
cana-852	7	30	abstract	abstract	NOUN
cana-852	7	31	in	in	ADP
cana-852	7	32	this	this	DET
cana-852	7	33	article	article	NOUN
cana-852	7	34	,	,	PUNCT
cana-852	7	35	we	we	PRON
cana-852	7	36	have	have	AUX
cana-852	7	37	derived	derive	VERB
cana-852	7	38	a	a	DET
cana-852	7	39	new	new	ADJ
cana-852	7	40	oscillation	oscillation	NOUN
cana-852	7	41	criteria	criterion	NOUN
cana-852	7	42	for	for	ADP
cana-852	7	43	a	a	DET
cana-852	7	44	class	class	NOUN
cana-852	7	45	of	of	ADP
cana-852	7	46	conformable	conformable	ADJ
cana-852	7	47	schrodinger	schrodinger	NOUN
cana-852	7	48	equations	equation	NOUN
cana-852	7	49	.	.	PUNCT
cana-852	8	1	based	base	VERB
cana-852	8	2	on	on	ADP
cana-852	8	3	the	the	DET
cana-852	8	4	generalized	generalize	VERB
cana-852	8	5	riccati	riccati	NOUN
cana-852	8	6	technique	technique	NOUN
cana-852	8	7	,	,	PUNCT
cana-852	8	8	the	the	DET
cana-852	8	9	results	result	NOUN
cana-852	8	10	were	be	AUX
cana-852	8	11	obtained	obtain	VERB
cana-852	8	12	here	here	ADV
cana-852	8	13	.	.	PUNCT
cana-852	9	1	also	also	ADV
cana-852	9	2	we	we	PRON
cana-852	9	3	have	have	AUX
cana-852	9	4	extended	extend	VERB
cana-852	9	5	the	the	DET
cana-852	9	6	hartman	hartman	PROPN
cana-852	9	7	-	-	PUNCT
cana-852	9	8	winter	winter	NOUN
cana-852	9	9	oscillation	oscillation	NOUN
cana-852	9	10	criteria	criterion	NOUN
cana-852	9	11	to	to	PART
cana-852	9	12	conformable	conformable	VERB
cana-852	9	13	schrodinger	schrodinger	NOUN
cana-852	9	14	equation	equation	NOUN
cana-852	9	15	.	.	PUNCT
cana-852	10	1	keywords	keyword	NOUN
cana-852	10	2	:	:	PUNCT
cana-852	10	3	.	.	PUNCT
cana-852	11	1	oscillation	oscillation	NOUN
cana-852	11	2	,	,	PUNCT
cana-852	11	3	conformable	conformable	ADJ
cana-852	11	4	schrodinger	schrodinger	NOUN
cana-852	11	5	equation	equation	NOUN
cana-852	11	6	,	,	PUNCT
cana-852	11	7	elliptic	elliptic	ADJ
cana-852	11	8	partial	partial	ADJ
cana-852	11	9	differential	differential	NOUN
cana-852	11	10	equation	equation	NOUN
cana-852	11	11	.	.	PUNCT
cana-852	12	1	ams	am	NOUN
cana-852	12	2	subject	subject	ADJ
cana-852	12	3	classification	classification	NOUN
cana-852	12	4	:	:	PUNCT
cana-852	12	5	34a08	34a08	NUM
cana-852	12	6	,	,	PUNCT
cana-852	12	7	34a34	34a34	NUM
cana-852	12	8	,	,	PUNCT
cana-852	12	9	34k11	34k11	NUM
cana-852	12	10	,	,	PUNCT
cana-852	12	11	35b05	35b05	NUM
cana-852	12	12	,	,	PUNCT
cana-852	12	13	35r20	35r20	NUM
cana-852	12	14	.	.	PUNCT
cana-852	13	1	1	1	X
cana-852	13	2	.	.	X
cana-852	13	3	introduction	introduction	NOUN
cana-852	13	4	the	the	DET
cana-852	13	5	area	area	NOUN
cana-852	13	6	of	of	ADP
cana-852	13	7	research	research	NOUN
cana-852	13	8	that	that	PRON
cana-852	13	9	has	have	AUX
cana-852	13	10	grown	grow	VERB
cana-852	13	11	the	the	DET
cana-852	13	12	fastest	fast	ADJ
cana-852	13	13	in	in	ADP
cana-852	13	14	recent	recent	ADJ
cana-852	13	15	years	year	NOUN
cana-852	13	16	is	be	AUX
cana-852	13	17	differential	differential	ADJ
cana-852	13	18	equations	equation	NOUN
cana-852	13	19	in	in	ADP
cana-852	13	20	fractional	fractional	ADJ
cana-852	13	21	calculus	calculus	NOUN
cana-852	13	22	.	.	PUNCT
cana-852	14	1	although	although	SCONJ
cana-852	14	2	there	there	PRON
cana-852	14	3	are	be	VERB
cana-852	14	4	various	various	ADJ
cana-852	14	5	fractional	fractional	ADJ
cana-852	14	6	derivative	derivative	ADJ
cana-852	14	7	notions	notion	NOUN
cana-852	14	8	,	,	PUNCT
cana-852	14	9	including	include	VERB
cana-852	14	10	riemann	riemann	PROPN
cana-852	14	11	-	-	PUNCT
cana-852	14	12	liouville	liouville	PROPN
cana-852	14	13	and	and	CCONJ
cana-852	14	14	caputo	caputo	PROPN
cana-852	14	15	fractional	fractional	ADJ
cana-852	14	16	derivatives	derivative	NOUN
cana-852	14	17	,	,	PUNCT
cana-852	14	18	which	which	PRON
cana-852	14	19	are	be	AUX
cana-852	14	20	based	base	VERB
cana-852	14	21	on	on	ADP
cana-852	14	22	singular	singular	ADJ
cana-852	14	23	integrals	integral	NOUN
cana-852	14	24	and	and	CCONJ
cana-852	14	25	non	non	ADJ
cana-852	14	26	-	-	ADJ
cana-852	14	27	locality	locality	ADJ
cana-852	14	28	,	,	PUNCT
cana-852	14	29	they	they	PRON
cana-852	14	30	are	be	AUX
cana-852	14	31	commonly	commonly	ADV
cana-852	14	32	utilized	utilize	VERB
cana-852	14	33	.	.	PUNCT
cana-852	15	1	in	in	ADP
cana-852	15	2	2014	2014	NUM
cana-852	15	3	,	,	PUNCT
cana-852	15	4	khalil	khalil	PROPN
cana-852	15	5	et	et	PROPN
cana-852	15	6	al	al	PROPN
cana-852	15	7	.	.	PUNCT
cana-852	16	1	[	[	X
cana-852	16	2	13	13	NUM
cana-852	16	3	]	]	PUNCT
cana-852	16	4	developed	develop	VERB
cana-852	16	5	the	the	DET
cana-852	16	6	conformable	conformable	ADJ
cana-852	16	7	fractional	fractional	ADJ
cana-852	16	8	derivative	derivative	NOUN
cana-852	16	9	,	,	PUNCT
cana-852	16	10	which	which	PRON
cana-852	16	11	was	be	AUX
cana-852	16	12	based	base	VERB
cana-852	16	13	on	on	ADP
cana-852	16	14	a	a	DET
cana-852	16	15	limit	limit	NOUN
cana-852	16	16	concept	concept	NOUN
cana-852	16	17	similar	similar	ADJ
cana-852	16	18	to	to	ADP
cana-852	16	19	that	that	PRON
cana-852	16	20	of	of	ADP
cana-852	16	21	integer	integer	NOUN
cana-852	16	22	order	order	NOUN
cana-852	16	23	derivatives	derivative	NOUN
cana-852	16	24	.	.	PUNCT
cana-852	17	1	the	the	DET
cana-852	17	2	conformable	conformable	ADJ
cana-852	17	3	derivative	derivative	NOUN
cana-852	17	4	of	of	ADP
cana-852	17	5	khalil	khalil	PROPN
cana-852	17	6	was	be	AUX
cana-852	17	7	quickly	quickly	ADV
cana-852	17	8	made	make	VERB
cana-852	17	9	general	general	ADJ
cana-852	17	10	by	by	ADP
cana-852	17	11	katugampola	katugampola	PROPN
cana-852	17	12	fractional	fractional	PROPN
cana-852	17	13	derivative	derivative	ADJ
cana-852	17	14	or	or	CCONJ
cana-852	17	15	alpha	alpha	NOUN
cana-852	17	16	fractional	fractional	ADJ
cana-852	17	17	derivative	derivative	ADJ
cana-852	18	1	[	[	X
cana-852	18	2	11	11	NUM
cana-852	18	3	,	,	PUNCT
cana-852	18	4	12	12	NUM
cana-852	18	5	]	]	PUNCT
cana-852	18	6	.	.	PUNCT
cana-852	19	1	it	it	PRON
cana-852	19	2	has	have	VERB
cana-852	19	3	wide	wide	ADJ
cana-852	19	4	application	application	NOUN
cana-852	19	5	in	in	ADP
cana-852	19	6	biophysics	biophysic	NOUN
cana-852	19	7	,	,	PUNCT
cana-852	19	8	quantum	quantum	NOUN
cana-852	19	9	mechanics	mechanic	NOUN
cana-852	19	10	,	,	PUNCT
cana-852	19	11	wave	wave	NOUN
cana-852	19	12	theory	theory	NOUN
cana-852	19	13	and	and	CCONJ
cana-852	19	14	polymers	polymer	NOUN
cana-852	19	15	and	and	CCONJ
cana-852	19	16	it	it	PRON
cana-852	19	17	is	be	AUX
cana-852	19	18	a	a	DET
cana-852	19	19	crucial	crucial	ADJ
cana-852	19	20	tool	tool	NOUN
cana-852	19	21	for	for	ADP
cana-852	19	22	simulating	simulate	VERB
cana-852	19	23	a	a	DET
cana-852	19	24	variety	variety	NOUN
cana-852	19	25	of	of	ADP
cana-852	19	26	physical	physical	ADJ
cana-852	19	27	phenomena	phenomenon	NOUN
cana-852	19	28	,	,	PUNCT
cana-852	19	29	including	include	VERB
cana-852	19	30	electromagnetic	electromagnetic	ADJ
cana-852	19	31	waves	wave	NOUN
cana-852	19	32	and	and	CCONJ
cana-852	19	33	viscoelastic	viscoelastic	NOUN
cana-852	19	34	systems	system	NOUN
cana-852	20	1	[	[	X
cana-852	20	2	9	9	NUM
cana-852	20	3	,	,	PUNCT
cana-852	20	4	14	14	NUM
cana-852	20	5	]	]	PUNCT
cana-852	20	6	.	.	PUNCT
cana-852	21	1	numerous	numerous	ADJ
cana-852	21	2	studies	study	NOUN
cana-852	21	3	have	have	AUX
cana-852	21	4	been	be	AUX
cana-852	21	5	done	do	VERB
cana-852	21	6	in	in	ADP
cana-852	21	7	the	the	DET
cana-852	21	8	literature	literature	NOUN
cana-852	21	9	on	on	ADP
cana-852	21	10	the	the	DET
cana-852	21	11	oscillation	oscillation	NOUN
cana-852	21	12	of	of	ADP
cana-852	21	13	conformable	conformable	ADJ
cana-852	21	14	fractional	fractional	ADJ
cana-852	21	15	differential	differential	ADJ
cana-852	21	16	equations	equation	NOUN
cana-852	21	17	[	[	X
cana-852	21	18	2	2	NUM
cana-852	21	19	,	,	PUNCT
cana-852	21	20	4	4	NUM
cana-852	21	21	,	,	PUNCT
cana-852	21	22	8	8	NUM
cana-852	21	23	,	,	PUNCT
cana-852	21	24	17	17	NUM
cana-852	21	25	]	]	PUNCT
cana-852	21	26	.	.	PUNCT
cana-852	22	1	in	in	ADP
cana-852	22	2	mathematically	mathematically	ADV
cana-852	22	3	oriented	orient	VERB
cana-852	22	4	sciences	science	NOUN
cana-852	22	5	like	like	ADP
cana-852	22	6	physics	physics	NOUN
cana-852	22	7	and	and	CCONJ
cana-852	22	8	engineering	engineering	NOUN
cana-852	22	9	,	,	PUNCT
cana-852	22	10	conformable	conformable	ADJ
cana-852	22	11	partial	partial	ADJ
cana-852	22	12	differential	differential	NOUN
cana-852	22	13	equations	equation	NOUN
cana-852	22	14	are	be	AUX
cana-852	22	15	widely	widely	ADV
cana-852	22	16	encountered	encounter	VERB
cana-852	22	17	[	[	X
cana-852	22	18	19	19	NUM
cana-852	22	19	,	,	PUNCT
cana-852	22	20	20	20	NUM
cana-852	22	21	]	]	PUNCT
cana-852	22	22	.	.	PUNCT
cana-852	23	1	for	for	ADP
cana-852	23	2	instance	instance	NOUN
cana-852	23	3	,	,	PUNCT
cana-852	23	4	they	they	PRON
cana-852	23	5	form	form	VERB
cana-852	23	6	the	the	DET
cana-852	23	7	basis	basis	NOUN
cana-852	23	8	of	of	ADP
cana-852	23	9	current	current	ADJ
cana-852	23	10	scientific	scientific	ADJ
cana-852	23	11	understanding	understanding	NOUN
cana-852	23	12	of	of	ADP
cana-852	23	13	diffusion	diffusion	NOUN
cana-852	23	14	,	,	PUNCT
cana-852	23	15	electrostatics	electrostatic	NOUN
cana-852	23	16	,	,	PUNCT
cana-852	23	17	materials	material	NOUN
cana-852	23	18	,	,	PUNCT
cana-852	23	19	dynamical	dynamical	ADJ
cana-852	23	20	theory	theory	NOUN
cana-852	23	21	,	,	PUNCT
cana-852	23	22	hydrodynamics	hydrodynamic	NOUN
cana-852	23	23	,	,	PUNCT
cana-852	23	24	electrodynamics	electrodynamic	NOUN
cana-852	23	25	,	,	PUNCT
cana-852	23	26	viscoelasticity	viscoelasticity	NOUN
cana-852	23	27	and	and	CCONJ
cana-852	23	28	quantum	quantum	NOUN
cana-852	23	29	mechanics	mechanic	NOUN
cana-852	23	30	.	.	PUNCT
cana-852	24	1	moreover	moreover	ADV
cana-852	24	2	,	,	PUNCT
cana-852	24	3	fractional	fractional	ADJ
cana-852	24	4	partial	partial	ADJ
cana-852	24	5	differential	differential	NOUN
cana-852	24	6	equations	equation	NOUN
cana-852	24	7	have	have	AUX
cana-852	24	8	gained	gain	VERB
cana-852	24	9	popularity	popularity	NOUN
cana-852	24	10	in	in	ADP
cana-852	24	11	recent	recent	ADJ
cana-852	24	12	years	year	NOUN
cana-852	24	13	as	as	ADP
cana-852	24	14	a	a	DET
cana-852	24	15	tool	tool	NOUN
cana-852	24	16	for	for	ADP
cana-852	24	17	mathematical	mathematical	ADJ
cana-852	24	18	modelling	modelling	NOUN
cana-852	24	19	.	.	PUNCT
cana-852	25	1	the	the	DET
cana-852	25	2	oscillation	oscillation	NOUN
cana-852	25	3	of	of	ADP
cana-852	25	4	conformable	conformable	ADJ
cana-852	25	5	partial	partial	ADJ
cana-852	25	6	differential	differential	NOUN
cana-852	25	7	equations	equation	NOUN
cana-852	25	8	has	have	AUX
cana-852	25	9	been	be	AUX
cana-852	25	10	researched	research	VERB
cana-852	25	11	by	by	ADP
cana-852	25	12	numerous	numerous	ADJ
cana-852	25	13	authors	author	NOUN
cana-852	25	14	,	,	PUNCT
cana-852	25	15	see	see	VERB
cana-852	25	16	[	[	X
cana-852	25	17	5	5	NUM
cana-852	25	18	,	,	PUNCT
cana-852	25	19	6	6	NUM
cana-852	25	20	]	]	PUNCT
cana-852	25	21	.	.	PUNCT
cana-852	26	1	the	the	DET
cana-852	26	2	concept	concept	NOUN
cana-852	26	3	of	of	ADP
cana-852	26	4	elliptic	elliptic	ADJ
cana-852	26	5	equation	equation	NOUN
cana-852	26	6	has	have	AUX
cana-852	26	7	undergone	undergo	VERB
cana-852	26	8	an	an	DET
cana-852	26	9	important	important	ADJ
cana-852	26	10	growth	growth	NOUN
cana-852	26	11	over	over	ADP
cana-852	26	12	the	the	DET
cana-852	26	13	last	last	ADJ
cana-852	26	14	two	two	NUM
cana-852	26	15	centuries	century	NOUN
cana-852	26	16	.	.	PUNCT
cana-852	27	1	together	together	ADV
cana-852	27	2	with	with	ADP
cana-852	27	3	electro	electro	ADJ
cana-852	27	4	statistics	statistic	NOUN
cana-852	27	5	heat	heat	NOUN
cana-852	27	6	and	and	CCONJ
cana-852	27	7	mass	mass	NOUN
cana-852	27	8	diffusion	diffusion	NOUN
cana-852	27	9	,	,	PUNCT
cana-852	27	10	hydrodynamics	hydrodynamic	NOUN
cana-852	27	11	and	and	CCONJ
cana-852	27	12	many	many	ADJ
cana-852	27	13	other	other	ADJ
cana-852	27	14	applications	application	NOUN
cana-852	27	15	it	it	PRON
cana-852	27	16	has	have	AUX
cana-852	27	17	become	become	VERB
cana-852	27	18	one	one	NUM
cana-852	27	19	of	of	ADP
cana-852	27	20	the	the	DET
cana-852	27	21	most	most	ADV
cana-852	27	22	richly	richly	ADV
cana-852	27	23	enhanced	enhance	VERB
cana-852	27	24	field	field	NOUN
cana-852	27	25	of	of	ADP
cana-852	27	26	mathematics	mathematic	NOUN
cana-852	27	27	.	.	PUNCT
cana-852	28	1	numerous	numerous	ADJ
cana-852	28	2	authors	author	NOUN
cana-852	28	3	have	have	AUX
cana-852	28	4	been	be	AUX
cana-852	28	5	communications	communication	NOUN
cana-852	28	6	on	on	ADP
cana-852	28	7	applied	apply	VERB
cana-852	28	8	nonlinear	nonlinear	ADJ
cana-852	28	9	analysis	analysis	NOUN
cana-852	28	10	issn	issn	NOUN
cana-852	28	11	:	:	PUNCT
cana-852	28	12	1074	1074	NUM
cana-852	28	13	-	-	PUNCT
cana-852	28	14	133x	133x	NUM
cana-852	28	15	vol	vol	NOUN
cana-852	28	16	31	31	NUM
cana-852	28	17	no	no	NOUN
cana-852	28	18	.	.	PUNCT
cana-852	29	1	4s	4s	NUM
cana-852	29	2	(	(	PUNCT
cana-852	29	3	2024	2024	NUM
cana-852	29	4	)	)	PUNCT
cana-852	29	5	287	287	NUM
cana-852	29	6	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-852	29	7	motivated	motivate	VERB
cana-852	29	8	by	by	ADP
cana-852	29	9	the	the	DET
cana-852	29	10	oscillation	oscillation	NOUN
cana-852	29	11	theory	theory	NOUN
cana-852	29	12	of	of	ADP
cana-852	29	13	elliptic	elliptic	ADJ
cana-852	29	14	equations	equation	NOUN
cana-852	29	15	in	in	ADP
cana-852	29	16	recent	recent	ADJ
cana-852	29	17	years	year	NOUN
cana-852	30	1	[	[	X
cana-852	30	2	1	1	NUM
cana-852	30	3	,	,	PUNCT
cana-852	30	4	7	7	NUM
cana-852	30	5	]	]	PUNCT
cana-852	30	6	.	.	PUNCT
cana-852	31	1	over	over	ADP
cana-852	31	2	the	the	DET
cana-852	31	3	past	past	ADJ
cana-852	31	4	few	few	ADJ
cana-852	31	5	years	year	NOUN
cana-852	31	6	,	,	PUNCT
cana-852	31	7	there	there	PRON
cana-852	31	8	has	have	AUX
cana-852	31	9	been	be	AUX
cana-852	31	10	a	a	DET
cana-852	31	11	lot	lot	NOUN
cana-852	31	12	of	of	ADP
cana-852	31	13	attention	attention	NOUN
cana-852	31	14	paid	pay	VERB
cana-852	31	15	to	to	ADP
cana-852	31	16	the	the	DET
cana-852	31	17	issue	issue	NOUN
cana-852	31	18	of	of	ADP
cana-852	31	19	oscillation	oscillation	NOUN
cana-852	31	20	and	and	CCONJ
cana-852	31	21	non	non	NOUN
cana-852	31	22	-	-	NOUN
cana-852	31	23	oscillation	oscillation	NOUN
cana-852	31	24	of	of	ADP
cana-852	31	25	differential	differential	ADJ
cana-852	31	26	equation	equation	NOUN
cana-852	31	27	solutions	solution	NOUN
cana-852	31	28	[	[	X
cana-852	31	29	10	10	NUM
cana-852	31	30	,	,	PUNCT
cana-852	31	31	18	18	NUM
cana-852	31	32	]	]	PUNCT
cana-852	31	33	.	.	PUNCT
cana-852	32	1	a	a	DET
cana-852	32	2	linear	linear	ADJ
cana-852	32	3	partial	partial	ADJ
cana-852	32	4	differential	differential	NOUN
cana-852	32	5	equation	equation	NOUN
cana-852	32	6	called	call	VERB
cana-852	32	7	the	the	DET
cana-852	32	8	schrodinger	schrodinger	NOUN
cana-852	32	9	equation	equation	NOUN
cana-852	32	10	controls	control	VERB
cana-852	32	11	how	how	SCONJ
cana-852	32	12	a	a	DET
cana-852	32	13	quantum	quantum	ADJ
cana-852	32	14	mechanical	mechanical	ADJ
cana-852	32	15	system	system	NOUN
cana-852	32	16	behaves	behave	VERB
cana-852	32	17	in	in	ADP
cana-852	32	18	terms	term	NOUN
cana-852	32	19	of	of	ADP
cana-852	32	20	its	its	PRON
cana-852	32	21	wave	wave	NOUN
cana-852	32	22	function	function	NOUN
cana-852	32	23	.	.	PUNCT
cana-852	33	1	the	the	DET
cana-852	33	2	schrodinger	schrodinger	PROPN
cana-852	33	3	equation	equation	NOUN
cana-852	33	4	is	be	AUX
cana-852	33	5	the	the	DET
cana-852	33	6	cornerstone	cornerstone	NOUN
cana-852	33	7	of	of	ADP
cana-852	33	8	quantum	quantum	ADJ
cana-852	33	9	mechanics	mechanic	NOUN
cana-852	33	10	,	,	PUNCT
cana-852	33	11	the	the	DET
cana-852	33	12	study	study	NOUN
cana-852	33	13	of	of	ADP
cana-852	33	14	microscopic	microscopic	ADJ
cana-852	33	15	events	event	NOUN
cana-852	33	16	.	.	PUNCT
cana-852	34	1	the	the	DET
cana-852	34	2	schrodinger	schrodinger	PROPN
cana-852	34	3	equation	equation	NOUN
cana-852	34	4	,	,	PUNCT
cana-852	34	5	created	create	VERB
cana-852	34	6	in	in	ADP
cana-852	34	7	1926	1926	NUM
cana-852	34	8	by	by	ADP
cana-852	34	9	the	the	DET
cana-852	34	10	austrian	austrian	ADJ
cana-852	34	11	scientist	scientist	NOUN
cana-852	34	12	erwin	erwin	PROPN
cana-852	34	13	schrodinger	schrodinger	PROPN
cana-852	34	14	,	,	PUNCT
cana-852	34	15	is	be	AUX
cana-852	34	16	as	as	ADV
cana-852	34	17	essential	essential	ADJ
cana-852	34	18	to	to	ADP
cana-852	34	19	understanding	understand	VERB
cana-852	34	20	quantum	quantum	ADJ
cana-852	34	21	mechanics	mechanic	NOUN
cana-852	34	22	as	as	ADP
cana-852	34	23	newton	newton	PROPN
cana-852	34	24	's	's	PART
cana-852	34	25	laws	law	NOUN
cana-852	34	26	of	of	ADP
cana-852	34	27	motion	motion	NOUN
cana-852	34	28	are	be	AUX
cana-852	34	29	to	to	ADP
cana-852	34	30	understanding	understand	VERB
cana-852	34	31	large	large	ADJ
cana-852	34	32	-	-	PUNCT
cana-852	34	33	scale	scale	NOUN
cana-852	34	34	classical	classical	ADJ
cana-852	34	35	mechanics	mechanic	NOUN
cana-852	34	36	occurrences	occurrence	NOUN
cana-852	34	37	.	.	PUNCT
cana-852	35	1	later	later	ADV
cana-852	35	2	,	,	PUNCT
cana-852	35	3	noussair	noussair	PROPN
cana-852	35	4	[	[	X
cana-852	35	5	21	21	NUM
cana-852	35	6	]	]	PUNCT
cana-852	35	7	used	use	VERB
cana-852	35	8	the	the	DET
cana-852	35	9	n	n	ADV
cana-852	35	10	-	-	PUNCT
cana-852	35	11	dimensional	dimensional	ADJ
cana-852	35	12	emden	emden	ADJ
cana-852	35	13	-	-	PUNCT
cana-852	35	14	fowler	fowler	PROPN
cana-852	35	15	method	method	NOUN
cana-852	35	16	to	to	PART
cana-852	35	17	generate	generate	VERB
cana-852	35	18	solutions	solution	NOUN
cana-852	35	19	to	to	ADP
cana-852	35	20	the	the	DET
cana-852	35	21	nonlinear	nonlinear	ADJ
cana-852	35	22	schrodinger	schrodinger	PROPN
cana-852	35	23	equation	equation	NOUN
cana-852	35	24	in	in	ADP
cana-852	35	25	the	the	DET
cana-852	35	26	outer	outer	ADJ
cana-852	35	27	domain	domain	NOUN
cana-852	35	28	bn	bn	NOUN
cana-852	35	29	.	.	NOUN
cana-852	35	30	e.mullerpfeiffer	e.mullerpfeiffer	X
cana-852	36	1	[	[	X
cana-852	36	2	22	22	NUM
cana-852	36	3	]	]	PUNCT
cana-852	36	4	also	also	ADV
cana-852	36	5	obtained	obtain	VERB
cana-852	36	6	oscillation	oscillation	NOUN
cana-852	36	7	criteria	criterion	NOUN
cana-852	36	8	for	for	ADP
cana-852	36	9	the	the	DET
cana-852	36	10	schrodinger	schrodinger	ADJ
cana-852	36	11	equation	equation	NOUN
cana-852	36	12	in	in	ADP
cana-852	36	13	sobolev	sobolev	NOUN
cana-852	36	14	space	space	NOUN
cana-852	36	15	and	and	CCONJ
cana-852	36	16	generalized	generalize	VERB
cana-852	36	17	the	the	DET
cana-852	36	18	derivative	derivative	NOUN
cana-852	36	19	of	of	ADP
cana-852	36	20	order	order	NOUN
cana-852	36	21	2	2	NUM
cana-852	36	22	summable	summable	ADJ
cana-852	36	23	on	on	ADP
cana-852	36	24	every	every	DET
cana-852	36	25	compact	compact	ADJ
cana-852	36	26	subdomain	subdomain	NOUN
cana-852	36	27	of	of	ADP
cana-852	36	28	g.	g.	NOUN
cana-852	36	29	by	by	ADP
cana-852	36	30	using	use	VERB
cana-852	36	31	the	the	DET
cana-852	36	32	emden	emden	ADJ
cana-852	36	33	-	-	PUNCT
cana-852	36	34	fowler	fowler	PROPN
cana-852	36	35	equation	equation	NOUN
cana-852	36	36	,	,	PUNCT
cana-852	36	37	hirashi	hirashi	NOUN
cana-852	36	38	onose	onose	NOUN
cana-852	37	1	[	[	X
cana-852	37	2	23	23	NUM
cana-852	37	3	]	]	PUNCT
cana-852	37	4	has	have	AUX
cana-852	37	5	explored	explore	VERB
cana-852	37	6	some	some	DET
cana-852	37	7	conclusions	conclusion	NOUN
cana-852	37	8	on	on	ADP
cana-852	37	9	the	the	DET
cana-852	37	10	sublinear	sublinear	NOUN
cana-852	37	11	schrodinger	schrodinger	PROPN
cana-852	37	12	equation	equation	NOUN
cana-852	37	13	.	.	PUNCT
cana-852	38	1	swanson	swanson	PROPN
cana-852	39	1	[	[	X
cana-852	39	2	24	24	NUM
cana-852	39	3	]	]	PUNCT
cana-852	39	4	used	use	VERB
cana-852	39	5	a	a	DET
cana-852	39	6	modified	modify	VERB
cana-852	39	7	sublinear	sublinear	NOUN
cana-852	39	8	hypothesis	hypothesis	NOUN
cana-852	39	9	to	to	PART
cana-852	39	10	build	build	VERB
cana-852	39	11	the	the	DET
cana-852	39	12	article	article	NOUN
cana-852	39	13	in	in	ADP
cana-852	39	14	an	an	DET
cana-852	39	15	emden	emden	ADJ
cana-852	39	16	-	-	PUNCT
cana-852	39	17	fowler	fowler	ADJ
cana-852	39	18	type	type	NOUN
cana-852	39	19	sublinear	sublinear	NOUN
cana-852	39	20	equation	equation	NOUN
cana-852	39	21	.	.	PUNCT
cana-852	40	1	zhang	zhang	PROPN
cana-852	41	1	[	[	X
cana-852	41	2	25	25	NUM
cana-852	41	3	]	]	PUNCT
cana-852	41	4	has	have	AUX
cana-852	41	5	introduced	introduce	VERB
cana-852	41	6	unique	unique	ADJ
cana-852	41	7	standards	standard	NOUN
cana-852	41	8	for	for	ADP
cana-852	41	9	the	the	DET
cana-852	41	10	absence	absence	NOUN
cana-852	41	11	of	of	ADP
cana-852	41	12	positive	positive	ADJ
cana-852	41	13	results	result	NOUN
cana-852	41	14	by	by	ADP
cana-852	41	15	using	use	VERB
cana-852	41	16	the	the	DET
cana-852	41	17	perturbed	perturb	VERB
cana-852	41	18	schrodinger	schrodinger	PROPN
cana-852	41	19	equation	equation	NOUN
cana-852	41	20	.	.	PUNCT
cana-852	42	1	in	in	ADP
cana-852	42	2	this	this	DET
cana-852	42	3	paper	paper	NOUN
cana-852	42	4	,	,	PUNCT
cana-852	42	5	we	we	PRON
cana-852	42	6	are	be	AUX
cana-852	42	7	concerned	concerned	ADJ
cana-852	42	8	with	with	ADP
cana-852	42	9	conformable	conformable	ADJ
cana-852	42	10	elliptic	elliptic	ADJ
cana-852	42	11	equations	equation	NOUN
cana-852	42	12	is	be	AUX
cana-852	42	13	of	of	ADP
cana-852	42	14	the	the	DET
cana-852	42	15	type	type	NOUN
cana-852	42	16	δ𝑥	δ𝑥	ADP
cana-852	42	17	𝛼𝑢	𝛼𝑢	NOUN
cana-852	43	1	+	+	CCONJ
cana-852	43	2	𝑝(𝑥)𝑢	𝑝(𝑥)𝑢	NOUN
cana-852	43	3	=	=	SYM
cana-852	43	4	0	0	NUM
cana-852	43	5	,	,	PUNCT
cana-852	43	6	⬚	⬚	VERB
cana-852	43	7	δ𝑥	δ𝑥	ADP
cana-852	43	8	𝛼𝑢	𝛼𝑢	NOUN
cana-852	43	9	=	=	SYM
cana-852	43	10	∑	∑	PART
cana-852	43	11	 	 	SPACE
cana-852	43	12	𝑛	𝑛	DET
cana-852	43	13	𝑖=1	𝑖=1	PROPN
cana-852	43	14	∂2𝛼𝑢	∂2𝛼𝑢	NOUN
cana-852	43	15	∂𝑥𝑖	∂𝑥𝑖	PROPN
cana-852	43	16	2𝛼	2𝛼	PROPN
cana-852	43	17	(	(	PUNCT
cana-852	43	18	1.1	1.1	NUM
cana-852	43	19	)	)	PUNCT
cana-852	43	20	where	where	SCONJ
cana-852	43	21	𝛼	𝛼	X
cana-852	43	22	∈	∈	PROPN
cana-852	43	23	(	(	PUNCT
cana-852	43	24	0,1	0,1	NUM
cana-852	43	25	)	)	PUNCT
cana-852	43	26	,	,	PUNCT
cana-852	43	27	𝑥	𝑥	PROPN
cana-852	43	28	=	=	SYM
cana-852	43	29	(	(	PUNCT
cana-852	43	30	𝑥1	𝑥1	NOUN
cana-852	43	31	,	,	PUNCT
cana-852	43	32	𝑥2	𝑥2	NOUN
cana-852	43	33	,	,	PUNCT
cana-852	43	34	…	…	PUNCT
cana-852	43	35	𝑥𝑛	𝑥𝑛	NOUN
cana-852	43	36	)	)	PUNCT
cana-852	43	37	,	,	PUNCT
cana-852	43	38	δ𝑥	δ𝑥	SCONJ
cana-852	43	39	𝛼	𝛼	NOUN
cana-852	43	40	is	be	AUX
cana-852	43	41	the	the	DET
cana-852	43	42	conformable	conformable	ADJ
cana-852	43	43	nabla	nabla	NOUN
cana-852	43	44	operator	operator	NOUN
cana-852	43	45	and	and	CCONJ
cana-852	43	46	𝑝(𝑥	𝑝(𝑥	NOUN
cana-852	43	47	)	)	PUNCT
cana-852	43	48	:	:	PUNCT
cana-852	44	1	ℝ𝑛	ℝ𝑛	ADP
cana-852	44	2	→	→	SYM
cana-852	44	3	ℝ	ℝ	PROPN
cana-852	44	4	is	be	AUX
cana-852	44	5	potential	potential	ADJ
cana-852	44	6	function	function	NOUN
cana-852	44	7	and	and	CCONJ
cana-852	44	8	each	each	DET
cana-852	44	9	compact	compact	ADJ
cana-852	44	10	subset	subset	NOUN
cana-852	44	11	of	of	ADP
cana-852	44	12	ω	ω	PROPN
cana-852	44	13	.	.	PUNCT
cana-852	45	1	define	define	VERB
cana-852	45	2	the	the	DET
cana-852	45	3	set	set	NOUN
cana-852	45	4	ω(𝑎	ω(𝑎	NUM
cana-852	45	5	)	)	PUNCT
cana-852	45	6	=	=	PRON
cana-852	45	7	{	{	PUNCT
cana-852	45	8	𝑥	𝑥	PUNCT
cana-852	45	9	∈	∈	PROPN
cana-852	46	1	ℝ𝑛	ℝ𝑛	NOUN
cana-852	46	2	:	:	PUNCT
cana-852	46	3	𝑎	𝑎	PROPN
cana-852	46	4	≤	≤	ADJ
cana-852	46	5	𝑟	𝑟	NOUN
cana-852	46	6	}	}	PUNCT
cana-852	46	7	,	,	PUNCT
cana-852	46	8	ω(𝑎	ω(𝑎	PROPN
cana-852	46	9	,	,	PUNCT
cana-852	46	10	𝑏	𝑏	NOUN
cana-852	46	11	)	)	PUNCT
cana-852	46	12	=	=	PRON
cana-852	46	13	{	{	PUNCT
cana-852	46	14	𝑥	𝑥	PUNCT
cana-852	46	15	∈	∈	PROPN
cana-852	47	1	ℝ𝑛	ℝ𝑛	NOUN
cana-852	47	2	:	:	PUNCT
cana-852	47	3	𝑎	𝑎	PROPN
cana-852	47	4	≤	≤	PUNCT
cana-852	47	5	𝑟	𝑟	NOUN
cana-852	47	6	≤	≤	NOUN
cana-852	47	7	𝑏	𝑏	NOUN
cana-852	47	8	}	}	PUNCT
cana-852	47	9	,	,	PUNCT
cana-852	47	10	𝑆(𝑎	𝑆(𝑎	NUM
cana-852	47	11	)	)	PUNCT
cana-852	47	12	=	=	NOUN
cana-852	47	13	{	{	PUNCT
cana-852	47	14	𝑥	𝑥	PUNCT
cana-852	47	15	∈	∈	PROPN
cana-852	48	1	ℝ𝑛	ℝ𝑛	NOUN
cana-852	48	2	:	:	PUNCT
cana-852	48	3	𝑟	𝑟	NOUN
cana-852	48	4	=	=	SYM
cana-852	48	5	𝑎	𝑎	X
cana-852	48	6	}	}	PUNCT
cana-852	48	7	,	,	PUNCT
cana-852	48	8	𝑢(𝑥	𝑢(𝑥	PROPN
cana-852	48	9	):	):	PUNCT
cana-852	48	10	ω	ω	PROPN
cana-852	48	11	→	→	SYM
cana-852	48	12	ℝ	ℝ	PROPN
cana-852	48	13	is	be	AUX
cana-852	48	14	almost	almost	ADV
cana-852	48	15	always	always	ADV
cana-852	48	16	absolutely	absolutely	ADV
cana-852	48	17	continuous	continuous	ADJ
cana-852	48	18	in	in	ADP
cana-852	48	19	𝛼-fractional	𝛼-fractional	ADJ
cana-852	48	20	derivative	derivative	NOUN
cana-852	48	21	of	of	ADP
cana-852	48	22	compact	compact	ADJ
cana-852	48	23	subsets	subset	NOUN
cana-852	48	24	that	that	PRON
cana-852	48	25	fulfills	fulfill	VERB
cana-852	48	26	equation	equation	NOUN
cana-852	48	27	(	(	PUNCT
cana-852	48	28	1.1	1.1	NUM
cana-852	48	29	)	)	PUNCT
cana-852	48	30	on	on	ADP
cana-852	48	31	ω	ω	PROPN
cana-852	48	32	is	be	AUX
cana-852	48	33	almost	almost	ADV
cana-852	48	34	everywhere	everywhere	ADV
cana-852	48	35	.	.	PUNCT
cana-852	49	1	a	a	DET
cana-852	49	2	constrained	constrained	ADJ
cana-852	49	3	area	area	NOUN
cana-852	49	4	𝐺	𝐺	PROPN
cana-852	49	5	⊂	⊂	PROPN
cana-852	49	6	ω	ω	PROPN
cana-852	49	7	is	be	AUX
cana-852	49	8	a	a	DET
cana-852	49	9	nodal	nodal	ADJ
cana-852	49	10	domain	domain	NOUN
cana-852	49	11	for	for	ADP
cana-852	49	12	(	(	PUNCT
cana-852	49	13	1.1	1.1	NUM
cana-852	49	14	)	)	PUNCT
cana-852	49	15	if	if	SCONJ
cana-852	49	16	there	there	PRON
cana-852	49	17	exists	exist	VERB
cana-852	49	18	a	a	DET
cana-852	49	19	nontrivial	nontrivial	ADJ
cana-852	49	20	function	function	NOUN
cana-852	49	21	u	u	PROPN
cana-852	49	22	∈	∈	PROPN
cana-852	49	23	𝐶2(𝐺	𝐶2(𝐺	NOUN
cana-852	49	24	;	;	PUNCT
cana-852	49	25	ℝ	ℝ	PROPN
cana-852	49	26	)	)	PUNCT
cana-852	49	27	∩	∩	ADJ
cana-852	49	28	𝐶(𝐺‾	𝐶(𝐺‾	NOUN
cana-852	49	29	;	;	PUNCT
cana-852	49	30	ℝ	ℝ	PROPN
cana-852	49	31	)	)	PUNCT
cana-852	49	32	such	such	ADJ
cana-852	49	33	that	that	SCONJ
cana-852	49	34	1.1	1.1	NUM
cana-852	49	35	is	be	AUX
cana-852	49	36	equal	equal	ADJ
cana-852	49	37	to	to	ADP
cana-852	49	38	zero	zero	NUM
cana-852	49	39	and	and	CCONJ
cana-852	49	40	u	u	X
cana-852	49	41	=	=	NOUN
cana-852	49	42	0	0	NUM
cana-852	49	43	on	on	ADP
cana-852	49	44	∂𝐺.	∂𝐺.	NOUN
cana-852	49	45	if	if	SCONJ
cana-852	49	46	for	for	ADP
cana-852	49	47	each	each	DET
cana-852	49	48	r	r	NOUN
cana-852	49	49	>	>	SYM
cana-852	49	50	0	0	NUM
cana-852	49	51	equation	equation	NOUN
cana-852	49	52	(	(	PUNCT
cana-852	49	53	1.1	1.1	NUM
cana-852	49	54	)	)	PUNCT
cana-852	49	55	has	have	VERB
cana-852	49	56	a	a	DET
cana-852	49	57	nodal	nodal	ADJ
cana-852	49	58	domain	domain	NOUN
cana-852	49	59	contained	contain	VERB
cana-852	49	60	and	and	CCONJ
cana-852	49	61	enclosed	enclose	VERB
cana-852	49	62	in	in	ADV
cana-852	49	63	ω𝑟	ω𝑟	ADP
cana-852	49	64	=	=	PROPN
cana-852	49	65	ω	ω	PROPN
cana-852	49	66	∩	∩	X
cana-852	49	67	{	{	PUNCT
cana-852	49	68	𝑥	𝑥	X
cana-852	49	69	∈	∈	PROPN
cana-852	50	1	ℝ𝑛	ℝ𝑛	NOUN
cana-852	50	2	:	:	PUNCT
cana-852	50	3	|𝑥|	|𝑥|	INTJ
cana-852	50	4	>	>	X
cana-852	50	5	𝑟	𝑟	X
cana-852	50	6	}	}	PUNCT
cana-852	50	7	,	,	PUNCT
cana-852	50	8	then	then	ADV
cana-852	50	9	equation	equation	NOUN
cana-852	50	10	(	(	PUNCT
cana-852	50	11	1.1	1.1	NUM
cana-852	50	12	)	)	PUNCT
cana-852	50	13	is	be	AUX
cana-852	50	14	called	call	VERB
cana-852	50	15	nodally	nodally	ADV
cana-852	50	16	oscillatory	oscillatory	ADJ
cana-852	50	17	.	.	PUNCT
cana-852	51	1	if	if	SCONJ
cana-852	51	2	the	the	DET
cana-852	51	3	function	function	NOUN
cana-852	51	4	f(x	f(x	PROPN
cana-852	51	5	)	)	PUNCT
cana-852	51	6	has	have	VERB
cana-852	51	7	zero	zero	NUM
cana-852	51	8	outside	outside	ADP
cana-852	51	9	of	of	ADP
cana-852	51	10	arbitrary	arbitrary	ADJ
cana-852	51	11	ball	ball	NOUN
cana-852	51	12	in	in	ADP
cana-852	51	13	ℝ𝑛	ℝ𝑛	PROPN
cana-852	51	14	that	that	PRON
cana-852	51	15	is	be	AUX
cana-852	51	16	centered	center	VERB
cana-852	51	17	in	in	ADP
cana-852	51	18	the	the	DET
cana-852	51	19	origin	origin	NOUN
cana-852	51	20	,	,	PUNCT
cana-852	51	21	is	be	AUX
cana-852	51	22	said	say	VERB
cana-852	51	23	to	to	PART
cana-852	51	24	be	be	AUX
cana-852	51	25	oscillatory	oscillatory	ADJ
cana-852	51	26	;	;	PUNCT
cana-852	51	27	if	if	SCONJ
cana-852	51	28	not	not	PART
cana-852	51	29	,	,	PUNCT
cana-852	51	30	it	it	PRON
cana-852	51	31	is	be	AUX
cana-852	51	32	said	say	VERB
cana-852	51	33	non	non	ADJ
cana-852	51	34	-	-	ADJ
cana-852	51	35	oscillatory	oscillatory	ADJ
cana-852	51	36	.	.	PUNCT
cana-852	52	1	we	we	PRON
cana-852	52	2	get	get	VERB
cana-852	52	3	the	the	DET
cana-852	52	4	function	function	NOUN
cana-852	52	5	p(t	p(t	NOUN
cana-852	52	6	)	)	PUNCT
cana-852	52	7	form	form	VERB
cana-852	52	8	the	the	DET
cana-852	52	9	hartman	hartman	PROPN
cana-852	52	10	-	-	PUNCT
cana-852	52	11	winter	winter	NOUN
cana-852	52	12	theorem	theorem	NOUN
cana-852	52	13	𝑃(𝑟	𝑃(𝑟	NOUN
cana-852	52	14	)	)	PUNCT
cana-852	52	15	=	=	SYM
cana-852	52	16	1	1	NUM
cana-852	52	17	𝑟𝛼	𝑟𝛼	NUM
cana-852	52	18	∫	∫	PROPN
cana-852	52	19	  	  	SPACE
cana-852	52	20	𝑟	𝑟	PRON
cana-852	52	21	1	1	NUM
cana-852	52	22	∫	∫	NOUN
cana-852	52	23	  	  	SPACE
cana-852	52	24	ω(1,𝑟	ω(1,𝑟	NUM
cana-852	52	25	)	)	PUNCT
cana-852	52	26	𝑟1−𝑛+𝜆𝑝(𝑥)𝑑𝛼𝑥𝑑𝛼𝑟	𝑟1−𝑛+𝜆𝑝(𝑥)𝑑𝛼𝑥𝑑𝛼𝑟	NOUN
cana-852	52	27	(	(	PUNCT
cana-852	52	28	1.2	1.2	NUM
cana-852	52	29	)	)	PUNCT
cana-852	52	30	we	we	PRON
cana-852	52	31	distinguish	distinguish	VERB
cana-852	52	32	two	two	NUM
cana-852	52	33	cases	case	NOUN
cana-852	52	34	,	,	PUNCT
cana-852	52	35	(	(	PUNCT
cana-852	52	36	i	i	NOUN
cana-852	52	37	)	)	PUNCT
cana-852	52	38	there	there	PRON
cana-852	52	39	is	be	VERB
cana-852	52	40	a	a	DET
cana-852	52	41	finite	finite	ADJ
cana-852	52	42	limit	limit	NOUN
cana-852	52	43	lim	lim	PROPN
cana-852	52	44	𝑟→∞	𝑟→∞	NUM
cana-852	52	45	 	 	SPACE
cana-852	52	46	𝑃(𝑟	𝑃(𝑟	NOUN
cana-852	52	47	)	)	PUNCT
cana-852	52	48	=	=	NOUN
cana-852	52	49	𝑃0	𝑃0	NOUN
cana-852	52	50	(	(	PUNCT
cana-852	52	51	1.3	1.3	NUM
cana-852	52	52	)	)	PUNCT
cana-852	52	53	communications	communication	NOUN
cana-852	52	54	on	on	ADP
cana-852	52	55	applied	apply	VERB
cana-852	52	56	nonlinear	nonlinear	ADJ
cana-852	52	57	analysis	analysis	NOUN
cana-852	52	58	issn	issn	NOUN
cana-852	52	59	:	:	PUNCT
cana-852	52	60	1074	1074	NUM
cana-852	52	61	-	-	PUNCT
cana-852	52	62	133x	133x	NUM
cana-852	52	63	vol	vol	NOUN
cana-852	52	64	31	31	NUM
cana-852	52	65	no	no	NOUN
cana-852	52	66	.	.	PUNCT
cana-852	53	1	4s	4s	NUM
cana-852	53	2	(	(	PUNCT
cana-852	53	3	2024	2024	NUM
cana-852	53	4	)	)	PUNCT
cana-852	53	5	288	288	NUM
cana-852	53	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-852	53	7	(	(	PUNCT
cana-852	53	8	ii	ii	NOUN
cana-852	53	9	)	)	PUNCT
cana-852	53	10	the	the	DET
cana-852	53	11	(	(	PUNCT
cana-852	53	12	1.3	1.3	NUM
cana-852	53	13	)	)	PUNCT
cana-852	53	14	condition	condition	NOUN
cana-852	53	15	is	be	AUX
cana-852	53	16	fails	fail	VERB
cana-852	53	17	to	to	PART
cana-852	53	18	hold	hold	VERB
cana-852	53	19	and	and	CCONJ
cana-852	53	20	lim	lim	PROPN
cana-852	53	21	 	 	SPACE
cana-852	53	22	inf𝑟→∞	inf𝑟→∞	PROPN
cana-852	53	23	 	 	SPACE
cana-852	53	24	𝑃(𝑟	𝑃(𝑟	ADP
cana-852	53	25	)	)	PUNCT
cana-852	53	26	>	>	X
cana-852	53	27	−∞.	−∞.	PROPN
cana-852	53	28	to	to	ADP
cana-852	53	29	my	my	PRON
cana-852	53	30	knowledge	knowledge	NOUN
cana-852	53	31	,	,	PUNCT
cana-852	53	32	aware	aware	ADJ
cana-852	53	33	,	,	PUNCT
cana-852	53	34	there	there	PRON
cana-852	53	35	is	be	VERB
cana-852	53	36	no	no	DET
cana-852	53	37	literature	literature	NOUN
cana-852	53	38	exists	exist	VERB
cana-852	53	39	on	on	ADP
cana-852	53	40	the	the	DET
cana-852	53	41	oscillation	oscillation	NOUN
cana-852	53	42	of	of	ADP
cana-852	53	43	the	the	DET
cana-852	53	44	conformable	conformable	ADJ
cana-852	53	45	elliptic	elliptic	ADJ
cana-852	53	46	equation	equation	NOUN
cana-852	53	47	.	.	PUNCT
cana-852	54	1	inspired	inspire	VERB
cana-852	54	2	by	by	ADP
cana-852	54	3	robert	robert	PROPN
cana-852	54	4	marik	marik	PROPN
cana-852	55	1	[	[	X
cana-852	55	2	16	16	NUM
cana-852	55	3	]	]	PUNCT
cana-852	55	4	and	and	CCONJ
cana-852	55	5	lomtatidze	lomtatidze	VERB
cana-852	55	6	[	[	X
cana-852	55	7	3	3	NUM
cana-852	55	8	,	,	PUNCT
cana-852	55	9	15	15	NUM
cana-852	55	10	]	]	PUNCT
cana-852	55	11	we	we	PRON
cana-852	55	12	investigating	investigate	VERB
cana-852	55	13	the	the	DET
cana-852	55	14	following	follow	VERB
cana-852	55	15	conformable	conformable	ADJ
cana-852	55	16	elliptic	elliptic	ADJ
cana-852	55	17	equation	equation	NOUN
cana-852	55	18	of	of	ADP
cana-852	55	19	the	the	DET
cana-852	55	20	form	form	NOUN
cana-852	55	21	𝑀(𝑟	𝑀(𝑟	NOUN
cana-852	55	22	)	)	PUNCT
cana-852	55	23	=	=	SYM
cana-852	56	1	𝑟	𝑟	NOUN
cana-852	56	2	(	(	PUNCT
cana-852	56	3	𝛼𝑃𝑜	𝛼𝑃𝑜	NOUN
cana-852	56	4	−	−	PROPN
cana-852	56	5	∫	∫	PROPN
cana-852	56	6	  	  	SPACE
cana-852	56	7	ω(1,𝑟	ω(1,𝑟	NUM
cana-852	56	8	)	)	PUNCT
cana-852	56	9	  	  	SPACE
cana-852	56	10	𝑟1−𝑛+𝜆𝑝(𝑥)𝑑𝛼𝑥	𝑟1−𝑛+𝜆𝑝(𝑥)𝑑𝛼𝑥	PROPN
cana-852	56	11	)	)	PUNCT
cana-852	56	12	(	(	PUNCT
cana-852	56	13	1.4	1.4	NUM
cana-852	56	14	)	)	PUNCT
cana-852	56	15	,	,	PUNCT
cana-852	56	16	𝑁(𝑟	𝑁(𝑟	PROPN
cana-852	56	17	)	)	PUNCT
cana-852	56	18	=	=	SYM
cana-852	56	19	1	1	NUM
cana-852	56	20	𝑟	𝑟	NUM
cana-852	56	21	∫	∫	PROPN
cana-852	56	22	  	  	SPACE
cana-852	56	23	ω(1,𝑟	ω(1,𝑟	NUM
cana-852	56	24	)	)	PUNCT
cana-852	56	25	  	  	SPACE
cana-852	56	26	𝑟3−𝑛+𝜆𝑝(𝑥)𝑑𝛼𝑥.	𝑟3−𝑛+𝜆𝑝(𝑥)𝑑𝛼𝑥.	PROPN
cana-852	56	27	(	(	PUNCT
cana-852	56	28	1.5	1.5	NUM
cana-852	56	29	)	)	SYM
cana-852	56	30	2	2	NUM
cana-852	56	31	.	.	PUNCT
cana-852	56	32	preliminaries	preliminary	NOUN
cana-852	56	33	in	in	ADP
cana-852	56	34	order	order	NOUN
cana-852	56	35	to	to	PART
cana-852	56	36	make	make	VERB
cana-852	56	37	our	our	PRON
cana-852	56	38	method	method	NOUN
cana-852	56	39	clear	clear	ADJ
cana-852	56	40	,	,	PUNCT
cana-852	56	41	we	we	PRON
cana-852	56	42	provide	provide	VERB
cana-852	56	43	certain	certain	ADJ
cana-852	56	44	fundamental	fundamental	ADJ
cana-852	56	45	definitions	definition	NOUN
cana-852	56	46	,	,	PUNCT
cana-852	56	47	properties	property	NOUN
cana-852	56	48	and	and	CCONJ
cana-852	56	49	lemmas	lemmas	ADJ
cana-852	56	50	definition	definition	NOUN
cana-852	56	51	2.1	2.1	NUM
cana-852	56	52	.	.	PUNCT
cana-852	57	1	given	give	VERB
cana-852	57	2	𝑢	𝑢	NOUN
cana-852	57	3	:	:	PUNCT
cana-852	57	4	[	[	X
cana-852	57	5	0	0	NUM
cana-852	57	6	,	,	PUNCT
cana-852	57	7	∞	∞	PROPN
cana-852	57	8	)	)	PUNCT
cana-852	57	9	→	→	SYM
cana-852	57	10	ℝ.	ℝ.	PROPN
cana-852	57	11	conformable	conformable	ADJ
cana-852	57	12	fractional	fractional	ADJ
cana-852	57	13	derivative	derivative	NOUN
cana-852	57	14	of	of	ADP
cana-852	57	15	𝑢	𝑢	NOUN
cana-852	57	16	of	of	ADP
cana-852	57	17	order	order	NOUN
cana-852	57	18	𝛼	𝛼	NOUN
cana-852	57	19	is	be	AUX
cana-852	57	20	given	give	VERB
cana-852	57	21	by	by	ADP
cana-852	57	22	𝑇𝛼(𝑢)(𝑥	𝑇𝛼(𝑢)(𝑥	PROPN
cana-852	57	23	)	)	PUNCT
cana-852	58	1	=	=	VERB
cana-852	58	2	lim	lim	PROPN
cana-852	58	3	𝜖→0	𝜖→0	PUNCT
cana-852	58	4	  	  	SPACE
cana-852	58	5	𝑢(𝑥	𝑢(𝑥	PROPN
cana-852	58	6	+	+	CCONJ
cana-852	58	7	𝜖𝑥1−𝛼	𝜖𝑥1−𝛼	NOUN
cana-852	58	8	)	)	PUNCT
cana-852	58	9	−	−	PROPN
cana-852	59	1	𝑢(𝑥	𝑢(𝑥	PROPN
cana-852	59	2	)	)	PUNCT
cana-852	60	1	𝜖	𝜖	PART
cana-852	60	2	⬚	⬚	PROPN
cana-852	60	3	∀𝑥	∀𝑥	X
cana-852	60	4	>	>	X
cana-852	60	5	0	0	NUM
cana-852	60	6	,	,	PUNCT
cana-852	60	7	⬚	⬚	NOUN
cana-852	60	8	𝛼	𝛼	PRON
cana-852	60	9	∈	∈	NOUN
cana-852	60	10	(	(	PUNCT
cana-852	60	11	0,1	0,1	NUM
cana-852	60	12	)	)	PUNCT
cana-852	60	13	.	.	PUNCT
cana-852	61	1	if	if	SCONJ
cana-852	61	2	𝑢	𝑢	PRON
cana-852	61	3	can	can	AUX
cana-852	61	4	be	be	AUX
cana-852	61	5	𝛼-differentiable	𝛼-differentiable	NOUN
cana-852	61	6	in	in	ADP
cana-852	61	7	some	some	PRON
cana-852	61	8	(	(	PUNCT
cana-852	61	9	0	0	NUM
cana-852	61	10	,	,	PUNCT
cana-852	61	11	𝑎	𝑎	NOUN
cana-852	61	12	)	)	PUNCT
cana-852	61	13	,	,	PUNCT
cana-852	61	14	𝑎	𝑎	X
cana-852	61	15	>	>	SYM
cana-852	61	16	0	0	NUM
cana-852	61	17	and	and	CCONJ
cana-852	61	18	lim𝑥→0	lim𝑥→0	PROPN
cana-852	61	19	+	+	CCONJ
cana-852	61	20	 	 	SPACE
cana-852	61	21	𝑢𝛼(𝑥	𝑢𝛼(𝑥	NUM
cana-852	61	22	)	)	PUNCT
cana-852	61	23	exists	exist	VERB
cana-852	61	24	,	,	PUNCT
cana-852	61	25	then	then	ADV
cana-852	61	26	we	we	PRON
cana-852	61	27	define	define	VERB
cana-852	61	28	𝑢𝛼(0	𝑢𝛼(0	PROPN
cana-852	61	29	)	)	PUNCT
cana-852	61	30	=	=	PROPN
cana-852	61	31	lim	lim	PROPN
cana-852	61	32	𝑥→0	𝑥→0	PROPN
cana-852	61	33	+	+	NOUN
cana-852	61	34	 	 	SPACE
cana-852	61	35	𝑢𝛼(𝑥	𝑢𝛼(𝑥	NUM
cana-852	61	36	)	)	PUNCT
cana-852	61	37	.	.	PUNCT
cana-852	62	1	definition	definition	NOUN
cana-852	62	2	2.2	2.2	NUM
cana-852	62	3	.	.	PUNCT
cana-852	63	1	𝐼𝛼	𝐼𝛼	PROPN
cana-852	63	2	𝑎(𝑢)(𝑥	𝑎(𝑢)(𝑥	ADV
cana-852	63	3	)	)	PUNCT
cana-852	63	4	=	=	PRON
cana-852	63	5	𝐼1	𝐼1	NOUN
cana-852	63	6	𝛼(𝑥𝛼−1)(𝑢	𝛼(𝑥𝛼−1)(𝑢	NUM
cana-852	63	7	)	)	PUNCT
cana-852	63	8	=	=	SYM
cana-852	64	1	∫	∫	PROPN
cana-852	64	2	𝑎	𝑎	X
cana-852	64	3	𝑥	𝑥	NOUN
cana-852	64	4	  	  	SPACE
cana-852	64	5	𝑢(𝑥	𝑢(𝑥	PROPN
cana-852	64	6	)	)	PUNCT
cana-852	65	1	𝑥1−𝛼	𝑥1−𝛼	PROPN
cana-852	65	2	𝑑𝑥	𝑑𝑥	VERB
cana-852	65	3	,	,	PUNCT
cana-852	65	4	where	where	SCONJ
cana-852	65	5	the	the	DET
cana-852	65	6	integral	integral	ADJ
cana-852	65	7	is	be	AUX
cana-852	65	8	the	the	DET
cana-852	65	9	standard	standard	ADJ
cana-852	65	10	riemann	riemann	PROPN
cana-852	65	11	improper	improper	ADJ
cana-852	65	12	integral	integral	ADJ
cana-852	65	13	and	and	CCONJ
cana-852	65	14	𝛼	𝛼	NOUN
cana-852	65	15	∈	∈	PROPN
cana-852	65	16	(	(	PUNCT
cana-852	65	17	0,1	0,1	NOUN
cana-852	65	18	)	)	PUNCT
cana-852	65	19	.	.	PUNCT
cana-852	66	1	properties	property	NOUN
cana-852	66	2	2.1	2.1	NUM
cana-852	66	3	.	.	PUNCT
cana-852	67	1	let	let	VERB
cana-852	67	2	𝛼	𝛼	PRON
cana-852	67	3	∈	∈	PROPN
cana-852	67	4	(	(	PUNCT
cana-852	67	5	0,1	0,1	NOUN
cana-852	67	6	]	]	PUNCT
cana-852	67	7	and	and	CCONJ
cana-852	67	8	at	at	ADP
cana-852	67	9	some	some	DET
cana-852	67	10	point	point	NOUN
cana-852	68	1	𝑥	𝑥	INTJ
cana-852	68	2	>	>	X
cana-852	68	3	0	0	PUNCT
cana-852	68	4	.	.	PUNCT
cana-852	69	1	𝑢and	𝑢and	NOUN
cana-852	69	2	𝑣	𝑣	PROPN
cana-852	69	3	will	will	AUX
cana-852	69	4	eventually	eventually	ADV
cana-852	69	5	be	be	AUX
cana-852	69	6	𝛼	𝛼	NOUN
cana-852	69	7	differentiable	differentiable	ADJ
cana-852	69	8	.	.	PUNCT
cana-852	70	1	then	then	ADV
cana-852	70	2	,	,	PUNCT
cana-852	70	3	(	(	PUNCT
cana-852	70	4	1	1	X
cana-852	70	5	)	)	PUNCT
cana-852	70	6	𝑇𝛼(𝑎1𝑢	𝑇𝛼(𝑎1𝑢	NOUN
cana-852	70	7	+	+	CCONJ
cana-852	70	8	𝑎2𝑣	𝑎2𝑣	NOUN
cana-852	70	9	)	)	PUNCT
cana-852	70	10	=	=	PUNCT
cana-852	70	11	𝑎1𝑇𝛼(𝑢	𝑎1𝑇𝛼(𝑢	ADJ
cana-852	70	12	)	)	PUNCT
cana-852	70	13	+	+	PUNCT
cana-852	70	14	𝑎2𝑇𝛼(𝑣	𝑎2𝑇𝛼(𝑣	PROPN
cana-852	70	15	)	)	PUNCT
cana-852	70	16	,	,	PUNCT
cana-852	70	17	⬚	⬚	PROPN
cana-852	70	18	∀𝑎1	∀𝑎1	PROPN
cana-852	70	19	,	,	PUNCT
cana-852	70	20	𝑎2	𝑎2	PROPN
cana-852	70	21	∈	∈	PROPN
cana-852	70	22	ℝ	ℝ	PROPN
cana-852	70	23	(	(	PUNCT
cana-852	70	24	2	2	NUM
cana-852	70	25	)	)	PUNCT
cana-852	70	26	𝑇𝛼(𝑢𝑣	𝑇𝛼(𝑢𝑣	NUM
cana-852	70	27	)	)	PUNCT
cana-852	70	28	=	=	SYM
cana-852	70	29	𝑢𝑇𝛼(𝑣	𝑢𝑇𝛼(𝑣	NOUN
cana-852	70	30	)	)	PUNCT
cana-852	70	31	+	+	CCONJ
cana-852	70	32	𝑣𝑇𝛼(𝑢	𝑣𝑇𝛼(𝑢	NOUN
cana-852	70	33	)	)	PUNCT
cana-852	70	34	(	(	PUNCT
cana-852	70	35	3	3	X
cana-852	70	36	)	)	PUNCT
cana-852	70	37	𝑇𝛼(𝑥𝑝	𝑇𝛼(𝑥𝑝	NOUN
cana-852	70	38	)	)	PUNCT
cana-852	70	39	=	=	SYM
cana-852	70	40	𝑝𝑥𝑝−𝛼	𝑝𝑥𝑝−𝛼	PROPN
cana-852	70	41	,	,	PUNCT
cana-852	70	42	⬚	⬚	SYM
cana-852	70	43	∀𝑝	∀𝑝	NOUN
cana-852	70	44	∈	∈	PROPN
cana-852	70	45	ℝ	ℝ	PROPN
cana-852	70	46	(	(	PUNCT
cana-852	70	47	4	4	NUM
cana-852	70	48	)	)	PUNCT
cana-852	70	49	𝑇𝛼(𝑎	𝑇𝛼(𝑎	PROPN
cana-852	70	50	)	)	PUNCT
cana-852	70	51	=	=	SYM
cana-852	70	52	0	0	NUM
cana-852	70	53	,	,	PUNCT
cana-852	70	54	⬚	⬚	NOUN
cana-852	70	55	𝑢(𝑥	𝑢(𝑥	NUM
cana-852	70	56	)	)	PUNCT
cana-852	70	57	=	=	NOUN
cana-852	71	1	𝑎	𝑎	NOUN
cana-852	71	2	for	for	ADP
cana-852	71	3	every	every	DET
cana-852	71	4	constant	constant	ADJ
cana-852	71	5	functions	function	NOUN
cana-852	71	6	.	.	PUNCT
cana-852	72	1	(	(	PUNCT
cana-852	72	2	5	5	X
cana-852	72	3	)	)	PUNCT
cana-852	72	4	𝑇𝛼	𝑇𝛼	NOUN
cana-852	72	5	(	(	PUNCT
cana-852	72	6	𝑢	𝑢	NOUN
cana-852	72	7	𝑣	𝑣	NOUN
cana-852	72	8	)	)	PUNCT
cana-852	72	9	=	=	SYM
cana-852	72	10	𝑣𝑇𝛼(𝑢)−𝑢𝑇𝛼(𝑣	𝑣𝑇𝛼(𝑢)−𝑢𝑇𝛼(𝑣	PROPN
cana-852	72	11	)	)	PUNCT
cana-852	72	12	𝑣2	𝑣2	NOUN
cana-852	72	13	(	(	PUNCT
cana-852	72	14	6	6	NUM
cana-852	72	15	)	)	PUNCT
cana-852	72	16	if	if	SCONJ
cana-852	72	17	𝑢	𝑢	NOUN
cana-852	72	18	is	be	AUX
cana-852	72	19	differential	differential	ADJ
cana-852	72	20	,	,	PUNCT
cana-852	72	21	then	then	ADV
cana-852	72	22	𝑇𝛼(𝑢(𝑥	𝑇𝛼(𝑢(𝑥	PROPN
cana-852	72	23	)	)	PUNCT
cana-852	72	24	)	)	PUNCT
cana-852	73	1	=	=	PUNCT
cana-852	73	2	𝑥1−𝛼	𝑥1−𝛼	NOUN
cana-852	73	3	𝑑𝑢(𝑥	𝑑𝑢(𝑥	AUX
cana-852	73	4	)	)	PUNCT
cana-852	73	5	𝑑𝑥	𝑑𝑥	VERB
cana-852	73	6	.	.	PUNCT
cana-852	74	1	proof	proof	NOUN
cana-852	74	2	.	.	PUNCT
cana-852	75	1	refer	refer	VERB
cana-852	75	2	[	[	X
cana-852	75	3	13	13	NUM
cana-852	75	4	]	]	PUNCT
cana-852	75	5	definition	definition	NOUN
cana-852	75	6	2.3	2.3	NUM
cana-852	75	7	.	.	PUNCT
cana-852	76	1	let	let	VERB
cana-852	76	2	𝑢	𝑢	PRON
cana-852	76	3	be	be	AUX
cana-852	76	4	a	a	DET
cana-852	76	5	function	function	NOUN
cana-852	76	6	with	with	ADP
cana-852	76	7	𝑚	𝑚	PROPN
cana-852	76	8	variable	variable	PROPN
cana-852	76	9	𝑥1	𝑥1	PROPN
cana-852	76	10	,	,	PUNCT
cana-852	76	11	…	…	PUNCT
cana-852	76	12	.	.	PUNCT
cana-852	76	13	.	.	PUNCT
cana-852	77	1	.	.	PUNCT
cana-852	78	1	,	,	PUNCT
cana-852	78	2	𝑥𝑚	𝑥𝑚	NOUN
cana-852	78	3	,	,	PUNCT
cana-852	78	4	and	and	CCONJ
cana-852	78	5	the	the	DET
cana-852	78	6	conformable	conformable	ADJ
cana-852	78	7	partial	partial	ADJ
cana-852	78	8	derivative	derivative	NOUN
cana-852	78	9	of	of	ADP
cana-852	78	10	𝑢	𝑢	NOUN
cana-852	78	11	of	of	ADP
cana-852	78	12	order	order	NOUN
cana-852	78	13	0	0	PUNCT
cana-852	78	14	<	<	X
cana-852	78	15	𝛼	𝛼	PRON
cana-852	78	16	≤	≤	NUM
cana-852	78	17	1	1	NUM
cana-852	78	18	in	in	ADP
cana-852	78	19	𝑥𝑖	𝑥𝑖	PROPN
cana-852	78	20	is	be	AUX
cana-852	78	21	defined	define	VERB
cana-852	78	22	as	as	SCONJ
cana-852	78	23	follows	follow	VERB
cana-852	78	24	∂𝛼	∂𝛼	PROPN
cana-852	78	25	∂𝑥𝑖	∂𝑥𝑖	PROPN
cana-852	78	26	𝛼	𝛼	NOUN
cana-852	78	27	𝑢(𝑥1	𝑢(𝑥1	ADV
cana-852	78	28	,	,	PUNCT
cana-852	78	29	…	…	PUNCT
cana-852	78	30	…	…	PUNCT
cana-852	78	31	,	,	PUNCT
cana-852	78	32	𝑥𝑚	𝑥𝑚	X
cana-852	78	33	)	)	PUNCT
cana-852	79	1	=	=	SYM
cana-852	79	2	lim	lim	PROPN
cana-852	79	3	𝜖→0	𝜖→0	NOUN
cana-852	79	4	  	  	SPACE
cana-852	79	5	𝑢(𝑥1	𝑢(𝑥1	ADV
cana-852	79	6	,	,	PUNCT
cana-852	79	7	.	.	PUNCT
cana-852	79	8	.	.	PUNCT
cana-852	80	1	,	,	PUNCT
cana-852	80	2	𝑥𝑖−1	𝑥𝑖−1	PROPN
cana-852	80	3	,	,	PUNCT
cana-852	80	4	𝑥𝑖	𝑥𝑖	PROPN
cana-852	80	5	+	+	CCONJ
cana-852	80	6	𝜖𝑥𝑖	𝜖𝑥𝑖	X
cana-852	80	7	1−𝛼	1−𝛼	NUM
cana-852	80	8	…	…	PUNCT
cana-852	80	9	.	.	PUNCT
cana-852	80	10	,	,	PUNCT
cana-852	80	11	𝑥𝑚	𝑥𝑚	NOUN
cana-852	80	12	)	)	PUNCT
cana-852	80	13	−	−	NOUN
cana-852	80	14	𝑢(𝑥1	𝑢(𝑥1	ADV
cana-852	80	15	,	,	PUNCT
cana-852	80	16	…	…	PUNCT
cana-852	80	17	…	…	PUNCT
cana-852	80	18	,	,	PUNCT
cana-852	80	19	𝑥𝑚	𝑥𝑚	X
cana-852	80	20	)	)	PUNCT
cana-852	80	21	𝜖	𝜖	PROPN
cana-852	80	22	lemma	lemma	PROPN
cana-852	80	23	2.1	2.1	NUM
cana-852	80	24	.	.	PUNCT
cana-852	81	1	if	if	SCONJ
cana-852	81	2	r⃗	r⃗	ADJ
cana-852	81	3	=	=	PUNCT
cana-852	81	4	xi⃗⃗⃗	xi⃗⃗⃗	NOUN
cana-852	82	1	⃗	⃗	PROPN
cana-852	82	2	+	+	NUM
cana-852	82	3	yj⃗⃗⃗	yj⃗⃗⃗	PROPN
cana-852	82	4	⃗	⃗	PROPN
cana-852	82	5	+	+	PROPN
cana-852	82	6	zk⃗⃗	zk⃗⃗	PROPN
cana-852	82	7	⃗⃗	⃗⃗	PROPN
cana-852	82	8	and	and	CCONJ
cana-852	82	9	𝑟	𝑟	NOUN
cana-852	82	10	=	=	SYM
cana-852	82	11	|r⃗|	|r⃗|	X
cana-852	82	12	then	then	ADV
cana-852	82	13	grad𝛼	grad𝛼	PROPN
cana-852	82	14	𝑓(𝑟	𝑓(𝑟	PROPN
cana-852	82	15	)	)	PUNCT
cana-852	83	1	=	=	SYM
cana-852	83	2	𝑟1−𝛼grad	𝑟1−𝛼grad	NOUN
cana-852	83	3	𝑓(𝑟	𝑓(𝑟	NOUN
cana-852	83	4	)	)	PUNCT
cana-852	83	5	.	.	PUNCT
cana-852	84	1	communications	communication	NOUN
cana-852	84	2	on	on	ADP
cana-852	84	3	applied	apply	VERB
cana-852	84	4	nonlinear	nonlinear	ADJ
cana-852	84	5	analysis	analysis	NOUN
cana-852	84	6	issn	issn	NOUN
cana-852	84	7	:	:	PUNCT
cana-852	84	8	1074	1074	NUM
cana-852	84	9	-	-	PUNCT
cana-852	84	10	133x	133x	NUM
cana-852	84	11	vol	vol	NOUN
cana-852	84	12	31	31	NUM
cana-852	84	13	no	no	NOUN
cana-852	84	14	.	.	PUNCT
cana-852	85	1	4s	4s	NUM
cana-852	85	2	(	(	PUNCT
cana-852	85	3	2024	2024	NUM
cana-852	85	4	)	)	PUNCT
cana-852	85	5	289	289	NUM
cana-852	86	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-852	86	2	proof	proof	NOUN
cana-852	86	3	.	.	PUNCT
cana-852	87	1	if	if	SCONJ
cana-852	87	2	𝑓	𝑓	PRON
cana-852	87	3	is	be	AUX
cana-852	87	4	differentiable	differentiable	ADJ
cana-852	87	5	then	then	ADV
cana-852	87	6	by	by	ADP
cana-852	87	7	using	use	VERB
cana-852	87	8	the	the	DET
cana-852	87	9	properties	property	NOUN
cana-852	87	10	of	of	ADP
cana-852	87	11	2.1	2.1	NUM
cana-852	87	12	we	we	PRON
cana-852	87	13	get	get	VERB
cana-852	87	14	grad𝛼	grad𝛼	ADJ
cana-852	87	15	𝑓(𝑟	𝑓(𝑟	NOUN
cana-852	87	16	)	)	PUNCT
cana-852	87	17	=	=	SYM
cana-852	87	18	𝑟1−𝛼grad	𝑟1−𝛼grad	NOUN
cana-852	87	19	𝑓(𝑟	𝑓(𝑟	NOUN
cana-852	87	20	)	)	PUNCT
cana-852	87	21	.	.	PUNCT
cana-852	88	1	hence	hence	ADV
cana-852	88	2	the	the	DET
cana-852	88	3	proof	proof	NOUN
cana-852	88	4	is	be	AUX
cana-852	88	5	complete	complete	ADJ
cana-852	88	6	.	.	PUNCT
cana-852	89	1	first	first	ADV
cana-852	89	2	we	we	PRON
cana-852	89	3	introduce	introduce	VERB
cana-852	89	4	the	the	DET
cana-852	89	5	riccati	riccati	PROPN
cana-852	89	6	method	method	NOUN
cana-852	89	7	.	.	PUNCT
cana-852	90	1	there	there	PRON
cana-852	90	2	exists	exist	VERB
cana-852	90	3	a	a	PRON
cana-852	90	4	ω𝑟	ω𝑟	ADP
cana-852	90	5	=	=	PRON
cana-852	90	6	{	{	PUNCT
cana-852	90	7	𝑥	𝑥	X
cana-852	90	8	∈	∈	PROPN
cana-852	91	1	ℝ𝑛	ℝ𝑛	NOUN
cana-852	91	2	:	:	PUNCT
cana-852	91	3	∥	∥	PUNCT
cana-852	91	4	𝑥	𝑥	NOUN
cana-852	91	5	∥≥	∥≥	ADJ
cana-852	91	6	𝑟	𝑟	NOUN
cana-852	91	7	}	}	PUNCT
cana-852	91	8	and	and	CCONJ
cana-852	91	9	a	a	DET
cana-852	91	10	solution	solution	NOUN
cana-852	91	11	𝑢	𝑢	ADP
cana-852	91	12	of	of	ADP
cana-852	91	13	(	(	PUNCT
cana-852	91	14	1.1	1.1	NUM
cana-852	91	15	)	)	PUNCT
cana-852	91	16	that	that	PRON
cana-852	91	17	is	be	AUX
cana-852	91	18	non	non	X
cana-852	91	19	negative	negative	ADJ
cana-852	91	20	on	on	ADP
cana-852	91	21	ω𝑟.	ω𝑟.	NOUN
cana-852	91	22	let	let	VERB
cana-852	91	23	�	�	PRON
cana-852	91	24	⃗⃗⃗⃗	⃗⃗⃗⃗	NUM
cana-852	91	25	�	�	PROPN
cana-852	92	1	=	=	PRON
cana-852	93	1	grad𝛼u	grad𝛼u	PUNCT
cana-852	94	1	u	u	NOUN
cana-852	94	2	be	be	VERB
cana-852	94	3	the	the	DET
cana-852	94	4	vector	vector	NOUN
cana-852	94	5	function	function	NOUN
cana-852	94	6	representing	represent	VERB
cana-852	94	7	the	the	DET
cana-852	94	8	solution	solution	NOUN
cana-852	94	9	to	to	ADP
cana-852	94	10	the	the	DET
cana-852	94	11	riccati	riccati	PROPN
cana-852	94	12	equation	equation	NOUN
cana-852	94	13	defined	define	VERB
cana-852	94	14	on	on	ADP
cana-852	94	15	the	the	DET
cana-852	94	16	set	set	NOUN
cana-852	94	17	ω𝑟.	ω𝑟.	NOUN
cana-852	94	18	div𝛼	div𝛼	NOUN
cana-852	94	19	�	�	PROPN
cana-852	94	20	⃗⃗⃗⃗	⃗⃗⃗⃗	NUM
cana-852	94	21	�	�	PROPN
cana-852	94	22	+	+	CCONJ
cana-852	94	23	𝑝(𝑥)+∥	𝑝(𝑥)+∥	ADV
cana-852	94	24	𝑊	𝑊	NOUN
cana-852	94	25	∥2=	∥2=	NOUN
cana-852	94	26	0	0	NUM
cana-852	94	27	(	(	PUNCT
cana-852	94	28	2.1	2.1	NUM
cana-852	94	29	)	)	PUNCT
cana-852	94	30	the	the	DET
cana-852	94	31	operator	operator	NOUN
cana-852	94	32	𝑑𝑖𝑣𝛼	𝑑𝑖𝑣𝛼	NOUN
cana-852	94	33	is	be	AUX
cana-852	94	34	typical	typical	ADJ
cana-852	94	35	divergent	divergent	ADJ
cana-852	94	36	operator	operator	NOUN
cana-852	94	37	,	,	PUNCT
cana-852	94	38	i.e.	i.e.	X
cana-852	94	39	for	for	ADP
cana-852	94	40	�	�	PROPN
cana-852	94	41	⃗⃗⃗⃗	⃗⃗⃗⃗	PROPN
cana-852	94	42	�	�	PROPN
cana-852	94	43	=	=	SYM
cana-852	94	44	(	(	PUNCT
cana-852	94	45	𝑊1	𝑊1	PROPN
cana-852	94	46	,	,	PUNCT
cana-852	94	47	…	…	PUNCT
cana-852	94	48	.	.	PUNCT
cana-852	94	49	,	,	PUNCT
cana-852	95	1	𝑊𝑛	𝑊𝑛	NOUN
cana-852	95	2	)	)	PUNCT
cana-852	95	3	where	where	SCONJ
cana-852	95	4	the	the	DET
cana-852	95	5	common	common	ADJ
cana-852	95	6	euclidean	euclidean	ADJ
cana-852	95	7	norm	norm	NOUN
cana-852	95	8	in	in	ADP
cana-852	95	9	ℝ𝑛	ℝ𝑛	PROPN
cana-852	95	10	is	be	AUX
cana-852	95	11	represented	represent	VERB
cana-852	95	12	by	by	ADP
cana-852	95	13	∥.∥.	∥.∥.	PROPN
cana-852	95	14	lemma	lemma	PROPN
cana-852	95	15	2.2	2.2	NUM
cana-852	95	16	.	.	PUNCT
cana-852	96	1	let	let	VERB
cana-852	96	2	equation	equation	NOUN
cana-852	96	3	(	(	PUNCT
cana-852	96	4	1.1	1.1	NUM
cana-852	96	5	)	)	PUNCT
cana-852	96	6	be	be	AUX
cana-852	96	7	non	non	ADJ
cana-852	96	8	-	-	ADJ
cana-852	96	9	oscillatory	oscillatory	ADJ
cana-852	96	10	,	,	PUNCT
cana-852	96	11	i.e	i.e	PROPN
cana-852	96	12	,	,	PUNCT
cana-852	96	13	(	(	PUNCT
cana-852	96	14	1.1	1.1	NUM
cana-852	96	15	)	)	PUNCT
cana-852	96	16	has	have	VERB
cana-852	96	17	a	a	DET
cana-852	96	18	positive	positive	ADJ
cana-852	96	19	solution	solution	NOUN
cana-852	96	20	on	on	ADP
cana-852	96	21	ω𝑎	ω𝑎	NUM
cana-852	96	22	for	for	ADP
cana-852	96	23	some	some	DET
cana-852	96	24	𝑎	𝑎	PROPN
cana-852	96	25	≤	≤	NUM
cana-852	96	26	1	1	NUM
cana-852	96	27	.	.	PUNCT
cana-852	97	1	the	the	DET
cana-852	97	2	below	below	ADJ
cana-852	97	3	statements	statement	NOUN
cana-852	97	4	are	be	AUX
cana-852	97	5	equivalent	equivalent	ADJ
cana-852	97	6	:	:	PUNCT
cana-852	97	7	i	i	X
cana-852	97	8	)	)	PUNCT
cana-852	97	9	its	its	PRON
cana-852	97	10	∫	∫	PROPN
cana-852	97	11	  	  	SPACE
cana-852	97	12	ω(𝑎,∞	ω(𝑎,∞	PROPN
cana-852	97	13	)	)	PUNCT
cana-852	98	1	𝑟1−𝑛+𝜆	𝑟1−𝑛+𝜆	NOUN
cana-852	98	2	∥	∥	PUNCT
cana-852	98	3	𝑊	𝑊	NOUN
cana-852	98	4	∥2	∥2	NOUN
cana-852	98	5	𝑑𝛼𝑥	𝑑𝛼𝑥	VERB
cana-852	98	6	<	<	X
cana-852	98	7	∞	∞	PROPN
cana-852	98	8	(	(	PUNCT
cana-852	98	9	2.2	2.2	NUM
cana-852	98	10	)	)	PUNCT
cana-852	98	11	ii	ii	NOUN
cana-852	98	12	)	)	PUNCT
cana-852	98	13	there	there	PRON
cana-852	98	14	's	be	VERB
cana-852	98	15	a	a	DET
cana-852	98	16	finite	finite	ADJ
cana-852	98	17	limit	limit	NOUN
cana-852	98	18	lim	lim	PROPN
cana-852	98	19	𝑟→∞	𝑟→∞	NUM
cana-852	98	20	 	 	SPACE
cana-852	98	21	𝑃(𝑟	𝑃(𝑟	NOUN
cana-852	98	22	)	)	PUNCT
cana-852	98	23	=	=	NOUN
cana-852	98	24	𝑃0	𝑃0	NOUN
cana-852	98	25	(	(	PUNCT
cana-852	98	26	2.3	2.3	NUM
cana-852	98	27	)	)	PUNCT
cana-852	98	28	iii	iii	NOUN
cana-852	98	29	)	)	PUNCT
cana-852	98	30	its	its	PRON
cana-852	98	31	holds	hold	NOUN
cana-852	98	32	lim	lim	PROPN
cana-852	98	33	inf	inf	PROPN
cana-852	98	34	𝑟→∞	𝑟→∞	NUM
cana-852	98	35	 	 	SPACE
cana-852	98	36	𝑃(𝑟	𝑃(𝑟	NOUN
cana-852	98	37	)	)	PUNCT
cana-852	98	38	>	>	X
cana-852	99	1	−∞	−∞	X
cana-852	99	2	(	(	PUNCT
cana-852	99	3	2.4	2.4	NUM
cana-852	99	4	)	)	PUNCT
cana-852	99	5	proof	proof	NOUN
cana-852	99	6	.	.	PUNCT
cana-852	100	1	let	let	VERB
cana-852	100	2	equation	equation	NOUN
cana-852	100	3	(	(	PUNCT
cana-852	100	4	1.1	1.1	NUM
cana-852	100	5	)	)	PUNCT
cana-852	100	6	be	be	AUX
cana-852	100	7	non	non	ADJ
cana-852	100	8	-	-	ADJ
cana-852	100	9	oscillatory	oscillatory	ADJ
cana-852	100	10	.	.	PUNCT
cana-852	101	1	there	there	PRON
cana-852	101	2	is	be	VERB
cana-852	101	3	a	a	DET
cana-852	101	4	number	number	NOUN
cana-852	101	5	a	a	DET
cana-852	101	6	∈	∈	NOUN
cana-852	101	7	ℝ+and	ℝ+and	VERB
cana-852	101	8	a	a	DET
cana-852	101	9	solution	solution	NOUN
cana-852	101	10	u	u	NOUN
cana-852	101	11	of	of	ADP
cana-852	101	12	(	(	PUNCT
cana-852	101	13	1.1	1.1	NUM
cana-852	101	14	)	)	PUNCT
cana-852	101	15	that	that	PRON
cana-852	101	16	is	be	AUX
cana-852	101	17	non	non	X
cana-852	101	18	negative	negative	ADJ
cana-852	101	19	on	on	ADP
cana-852	101	20	ω𝑎.	ω𝑎.	PROPN
cana-852	101	21	let	let	VERB
cana-852	101	22	w⃗⃗⃗⃗	w⃗⃗⃗⃗	PROPN
cana-852	101	23	=	=	PUNCT
cana-852	102	1	grad𝛼	grad𝛼	VERB
cana-852	102	2	𝑢	𝑢	PRON
cana-852	102	3	𝑢	𝑢	NOUN
cana-852	102	4	be	be	VERB
cana-852	102	5	the	the	DET
cana-852	102	6	vector	vector	NOUN
cana-852	102	7	function	function	NOUN
cana-852	102	8	representing	represent	VERB
cana-852	102	9	the	the	DET
cana-852	102	10	solution	solution	NOUN
cana-852	102	11	of	of	ADP
cana-852	102	12	riccati	riccati	NOUN
cana-852	102	13	equation	equation	NOUN
cana-852	102	14	defined	define	VERB
cana-852	102	15	on	on	ADP
cana-852	102	16	ω𝑎	ω𝑎	NUM
cana-852	102	17	and	and	CCONJ
cana-852	102	18	using	use	VERB
cana-852	102	19	the	the	DET
cana-852	102	20	gauss	gauss	PROPN
cana-852	102	21	divergence	divergence	NOUN
cana-852	102	22	theorem	theorem	NOUN
cana-852	102	23	and	and	CCONJ
cana-852	102	24	the	the	DET
cana-852	102	25	identity	identity	NOUN
cana-852	102	26	∫	∫	PROPN
cana-852	102	27	  	  	SPACE
cana-852	102	28	𝑆(𝑟	𝑆(𝑟	NOUN
cana-852	102	29	)	)	PUNCT
cana-852	102	30	 	 	SPACE
cana-852	102	31	𝑟𝛼−𝑛+𝜆⟨𝑊	𝑟𝛼−𝑛+𝜆⟨𝑊	NOUN
cana-852	102	32	,	,	PUNCT
cana-852	102	33	𝑒𝑖⟩𝑑𝑆	𝑒𝑖⟩𝑑𝑆	PUNCT
cana-852	102	34	−	−	PROPN
cana-852	102	35	∫	∫	PROPN
cana-852	102	36	  	  	SPACE
cana-852	102	37	𝑆(𝑎	𝑆(𝑎	NOUN
cana-852	102	38	)	)	PUNCT
cana-852	102	39	  	  	SPACE
cana-852	102	40	𝑟𝛼−𝑛+𝜆⟨𝑊	𝑟𝛼−𝑛+𝜆⟨𝑊	NOUN
cana-852	102	41	,	,	PUNCT
cana-852	102	42	𝑒𝑖⟩𝑑𝑆	𝑒𝑖⟩𝑑𝑆	PUNCT
cana-852	103	1	+	+	NUM
cana-852	103	2	∫	∫	PROPN
cana-852	103	3	  	  	SPACE
cana-852	103	4	ω(𝑎,𝑟	ω(𝑎,𝑟	NUM
cana-852	103	5	)	)	PUNCT
cana-852	103	6	  	  	SPACE
cana-852	104	1	𝑟𝛼−𝑛+𝜆𝑝(𝑥)𝑑𝑥	𝑟𝛼−𝑛+𝜆𝑝(𝑥)𝑑𝑥	PROPN
cana-852	104	2	+	+	NUM
cana-852	104	3	∫	∫	PROPN
cana-852	104	4	  	  	SPACE
cana-852	104	5	ω(𝑎,𝑟	ω(𝑎,𝑟	NUM
cana-852	104	6	)	)	PUNCT
cana-852	104	7	  	  	SPACE
cana-852	105	1	𝑟𝛼−𝑛+𝜆	𝑟𝛼−𝑛+𝜆	PRON
cana-852	105	2	∥	∥	NOUN
cana-852	105	3	𝑊	𝑊	NOUN
cana-852	105	4	∥2	∥2	NOUN
cana-852	105	5	𝑑𝑥	𝑑𝑥	NOUN
cana-852	105	6	−	−	PROPN
cana-852	105	7	∫	∫	PROPN
cana-852	105	8	  	  	SPACE
cana-852	105	9	ω(𝑎,𝑟	ω(𝑎,𝑟	NUM
cana-852	105	10	)	)	PUNCT
cana-852	105	11	  	  	SPACE
cana-852	105	12	(	(	PUNCT
cana-852	105	13	𝛼	𝛼	NOUN
cana-852	105	14	−	−	PROPN
cana-852	105	15	𝑛	𝑛	PROPN
cana-852	105	16	+	+	PROPN
cana-852	105	17	𝜆)𝑟−𝑛+𝜆⟨𝑊	𝜆)𝑟−𝑛+𝜆⟨𝑊	PROPN
cana-852	105	18	,	,	PUNCT
cana-852	105	19	𝑒𝑖⟩𝑑𝑥.	𝑒𝑖⟩𝑑𝑥.	PROPN
cana-852	105	20	(	(	PUNCT
cana-852	105	21	2.5	2.5	NUM
cana-852	105	22	)	)	PUNCT
cana-852	105	23	i	i	NOUN
cana-852	105	24	)	)	PUNCT
cana-852	105	25	⇒	⇒	PROPN
cana-852	105	26	ii	ii	PROPN
cana-852	105	27	)	)	PUNCT
cana-852	105	28	if	if	SCONJ
cana-852	105	29	(	(	PUNCT
cana-852	105	30	2.2	2.2	NUM
cana-852	105	31	)	)	PUNCT
cana-852	105	32	holds	hold	VERB
cana-852	105	33	.	.	PUNCT
cana-852	106	1	then	then	ADV
cana-852	106	2	the	the	DET
cana-852	106	3	cauchy	cauchy	PROPN
cana-852	106	4	inequality	inequality	NOUN
cana-852	106	5	gives	give	VERB
cana-852	106	6	∫	∫	PROPN
cana-852	106	7	  	  	SPACE
cana-852	106	8	ω(𝑎,𝑟	ω(𝑎,𝑟	NUM
cana-852	106	9	)	)	PUNCT
cana-852	106	10	  	  	SPACE
cana-852	106	11	𝑟−𝑛+𝜆	𝑟−𝑛+𝜆	PRON
cana-852	107	1	∥	∥	PUNCT
cana-852	107	2	𝑊	𝑊	AUX
cana-852	107	3	∥	∥	PRON
cana-852	107	4	𝑑𝑥≤	𝑑𝑥≤	PUNCT
cana-852	107	5	(	(	PUNCT
cana-852	107	6	∫	∫	PROPN
cana-852	107	7	  	  	SPACE
cana-852	107	8	ω(𝑎,𝑟	ω(𝑎,𝑟	NUM
cana-852	107	9	)	)	PUNCT
cana-852	107	10	  	  	SPACE
cana-852	107	11	𝑟𝛼−𝑛+𝜆	𝑟𝛼−𝑛+𝜆	PRON
cana-852	107	12	∥	∥	PUNCT
cana-852	107	13	𝑊	𝑊	NOUN
cana-852	107	14	∥2	∥2	PRON
cana-852	107	15	𝑑𝑥	𝑑𝑥	NOUN
cana-852	107	16	)	)	PUNCT
cana-852	107	17	1	1	NUM
cana-852	107	18	2	2	NUM
cana-852	107	19	(	(	PUNCT
cana-852	107	20	∫	∫	PROPN
cana-852	107	21	  	  	SPACE
cana-852	107	22	ω(𝑎,𝑟	ω(𝑎,𝑟	NUM
cana-852	107	23	)	)	PUNCT
cana-852	107	24	  	  	SPACE
cana-852	107	25	𝑟−𝛼−𝑛+𝜆𝑑𝑥	𝑟−𝛼−𝑛+𝜆𝑑𝑥	PROPN
cana-852	107	26	)	)	PUNCT
cana-852	107	27	1	1	NUM
cana-852	107	28	2	2	NUM
cana-852	107	29	=	=	SYM
cana-852	107	30	(	(	PUNCT
cana-852	107	31	∫	∫	PROPN
cana-852	107	32	  	  	SPACE
cana-852	107	33	ω(𝑎,𝑟	ω(𝑎,𝑟	NUM
cana-852	107	34	)	)	PUNCT
cana-852	107	35	  	  	SPACE
cana-852	108	1	𝑟𝛼−𝑛+𝜆	𝑟𝛼−𝑛+𝜆	PRON
cana-852	108	2	∥	∥	PUNCT
cana-852	108	3	𝑊	𝑊	NOUN
cana-852	108	4	∥2	∥2	PRON
cana-852	108	5	𝑑𝑥	𝑑𝑥	NOUN
cana-852	108	6	)	)	PUNCT
cana-852	108	7	1	1	NUM
cana-852	108	8	2	2	NUM
cana-852	108	9	(	(	PUNCT
cana-852	108	10	𝜔𝑛	𝜔𝑛	ADP
cana-852	108	11	∫	∫	PROPN
cana-852	108	12	  	  	SPACE
cana-852	108	13	𝑟	𝑟	NOUN
cana-852	108	14	𝑎	𝑎	X
cana-852	108	15	  	  	SPACE
cana-852	108	16	𝑟𝛼+𝜆−3𝑑𝑟	𝑟𝛼+𝜆−3𝑑𝑟	NOUN
cana-852	108	17	)	)	PUNCT
cana-852	108	18	1	1	NUM
cana-852	108	19	2	2	NUM
cana-852	108	20	.	.	PUNCT
cana-852	109	1	communications	communication	NOUN
cana-852	109	2	on	on	ADP
cana-852	109	3	applied	apply	VERB
cana-852	109	4	nonlinear	nonlinear	ADJ
cana-852	109	5	analysis	analysis	NOUN
cana-852	109	6	issn	issn	NOUN
cana-852	109	7	:	:	PUNCT
cana-852	109	8	1074	1074	NUM
cana-852	109	9	-	-	PUNCT
cana-852	109	10	133x	133x	NUM
cana-852	109	11	vol	vol	NOUN
cana-852	109	12	31	31	NUM
cana-852	109	13	no	no	NOUN
cana-852	109	14	.	.	PUNCT
cana-852	110	1	4s	4s	NUM
cana-852	110	2	(	(	PUNCT
cana-852	110	3	2024	2024	NUM
cana-852	110	4	)	)	PUNCT
cana-852	110	5	290	290	NUM
cana-852	110	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-852	110	7	here	here	ADV
cana-852	110	8	,	,	PUNCT
cana-852	110	9	𝜔𝑛	𝜔𝑛	ADJ
cana-852	110	10	is	be	AUX
cana-852	110	11	represented	represent	VERB
cana-852	110	12	as	as	ADP
cana-852	110	13	the	the	DET
cana-852	110	14	measure	measure	NOUN
cana-852	110	15	of	of	ADP
cana-852	110	16	the	the	DET
cana-852	110	17	sphere	sphere	NOUN
cana-852	110	18	in	in	ADP
cana-852	110	19	ℝ𝑛	ℝ𝑛	PROPN
cana-852	110	20	and	and	CCONJ
cana-852	110	21	𝜔𝑛	𝜔𝑛	NOUN
cana-852	110	22	=	=	VERB
cana-852	110	23	2π	2π	NOUN
cana-852	110	24	𝑛	𝑛	ADP
cana-852	110	25	2	2	NUM
cana-852	110	26	γ	γ	NOUN
cana-852	110	27	n	n	PRON
cana-852	110	28	2	2	NUM
cana-852	110	29	,	,	PUNCT
cana-852	110	30	∫	∫	PROPN
cana-852	110	31	  	  	SPACE
cana-852	110	32	ω(𝑎,∞	ω(𝑎,∞	PROPN
cana-852	110	33	)	)	PUNCT
cana-852	110	34	𝑟−𝑛+𝜆⟨𝑊	𝑟−𝑛+𝜆⟨𝑊	NOUN
cana-852	110	35	,	,	PUNCT
cana-852	110	36	𝑒𝑖⟩𝑑𝑥	𝑒𝑖⟩𝑑𝑥	SCONJ
cana-852	110	37	<	<	X
cana-852	110	38	∞	∞	PROPN
cana-852	110	39	(	(	PUNCT
cana-852	110	40	2.6	2.6	NUM
cana-852	110	41	)	)	PUNCT
cana-852	110	42	not	not	PART
cana-852	110	43	diverges	diverge	NOUN
cana-852	110	44	.	.	PUNCT
cana-852	111	1	evaluate	evaluate	VERB
cana-852	111	2	of	of	ADP
cana-852	111	3	(	(	PUNCT
cana-852	111	4	2.5	2.5	NUM
cana-852	111	5	)	)	PUNCT
cana-852	111	6	and	and	CCONJ
cana-852	111	7	(	(	PUNCT
cana-852	111	8	2.6	2.6	NUM
cana-852	111	9	)	)	PUNCT
cana-852	111	10	gives	give	VERB
cana-852	111	11	�	�	PROPN
cana-852	111	12	̂	̂	PROPN
cana-852	111	13	�	�	PROPN
cana-852	111	14	−	−	PROPN
cana-852	112	1	∫	∫	PROPN
cana-852	112	2	  	  	SPACE
cana-852	112	3	ω(1,𝑟	ω(1,𝑟	NUM
cana-852	112	4	)	)	PUNCT
cana-852	112	5	  	  	SPACE
cana-852	113	1	𝑟𝛼−𝑛+𝜆𝑝(𝑥)𝑑𝑥	𝑟𝛼−𝑛+𝜆𝑝(𝑥)𝑑𝑥	NUM
cana-852	113	2	=	=	SYM
cana-852	113	3	∫	∫	PROPN
cana-852	113	4	  	  	SPACE
cana-852	113	5	𝑆(𝑟	𝑆(𝑟	NOUN
cana-852	113	6	)	)	PUNCT
cana-852	113	7	  	  	SPACE
cana-852	113	8	𝑟𝛼−𝑛+𝜆⟨𝑊	𝑟𝛼−𝑛+𝜆⟨𝑊	NOUN
cana-852	113	9	,	,	PUNCT
cana-852	113	10	𝑒𝑖⟩𝑑𝑆	𝑒𝑖⟩𝑑𝑆	PUNCT
cana-852	113	11	+	+	NUM
cana-852	113	12	∫	∫	PROPN
cana-852	113	13	  	  	SPACE
cana-852	113	14	ω(𝑟,∞	ω(𝑟,∞	NOUN
cana-852	113	15	)	)	PUNCT
cana-852	113	16	  	  	SPACE
cana-852	113	17	(	(	PUNCT
cana-852	113	18	𝛼	𝛼	NOUN
cana-852	113	19	−	−	PROPN
cana-852	113	20	𝑛	𝑛	PROPN
cana-852	113	21	+	+	PROPN
cana-852	113	22	𝜆)𝑟−𝑛+𝜆⟨𝑊	𝜆)𝑟−𝑛+𝜆⟨𝑊	NOUN
cana-852	113	23	,	,	PUNCT
cana-852	113	24	𝑒𝑖⟩𝑑𝑥	𝑒𝑖⟩𝑑𝑥	X
cana-852	113	25	−	−	PROPN
cana-852	113	26	∫	∫	PROPN
cana-852	113	27	  	  	SPACE
cana-852	113	28	ω(𝑟,∞	ω(𝑟,∞	NOUN
cana-852	113	29	)	)	PUNCT
cana-852	113	30	  	  	SPACE
cana-852	113	31	𝑟𝛼−𝑛+𝜆	𝑟𝛼−𝑛+𝜆	PRON
cana-852	113	32	∥	∥	PUNCT
cana-852	113	33	𝑊	𝑊	NOUN
cana-852	113	34	∥2	∥2	PRON
cana-852	113	35	𝑑𝑥	𝑑𝑥	NOUN
cana-852	113	36	,	,	PUNCT
cana-852	113	37	(	(	PUNCT
cana-852	113	38	2.7	2.7	NUM
cana-852	113	39	)	)	PUNCT
cana-852	113	40	where	where	SCONJ
cana-852	113	41	�	�	PROPN
cana-852	113	42	̂	̂	VERB
cana-852	113	43	�	�	PROPN
cana-852	113	44	=	=	SYM
cana-852	113	45	∫	∫	PROPN
cana-852	113	46	  	  	SPACE
cana-852	113	47	𝑆(𝑎	𝑆(𝑎	NOUN
cana-852	113	48	)	)	PUNCT
cana-852	113	49	  	  	SPACE
cana-852	113	50	𝑟𝛼−𝑛+𝜆⟨𝑊	𝑟𝛼−𝑛+𝜆⟨𝑊	NOUN
cana-852	113	51	,	,	PUNCT
cana-852	113	52	𝑒𝑖⟩𝑑𝑆	𝑒𝑖⟩𝑑𝑆	PUNCT
cana-852	113	53	+	+	NUM
cana-852	113	54	∫	∫	PROPN
cana-852	113	55	  	  	SPACE
cana-852	113	56	ω(𝑎,∞	ω(𝑎,∞	PROPN
cana-852	113	57	)	)	PUNCT
cana-852	113	58	  	  	SPACE
cana-852	113	59	(	(	PUNCT
cana-852	113	60	𝛼	𝛼	NOUN
cana-852	113	61	−	−	PROPN
cana-852	113	62	𝑛	𝑛	PROPN
cana-852	113	63	+	+	PROPN
cana-852	113	64	𝜆)𝑟−𝑛+𝜆⟨𝑊	𝜆)𝑟−𝑛+𝜆⟨𝑊	NOUN
cana-852	113	65	,	,	PUNCT
cana-852	113	66	𝑒𝑖⟩𝑑𝑥	𝑒𝑖⟩𝑑𝑥	X
cana-852	113	67	+	+	NUM
cana-852	113	68	∫	∫	PROPN
cana-852	113	69	  	  	SPACE
cana-852	113	70	ω(1,𝑎	ω(1,𝑎	NUM
cana-852	113	71	)	)	PUNCT
cana-852	113	72	  	  	SPACE
cana-852	114	1	𝑟𝛼−𝑛+𝜆𝑝(𝑥)𝑑𝑥	𝑟𝛼−𝑛+𝜆𝑝(𝑥)𝑑𝑥	NOUN
cana-852	115	1	−	−	PROPN
cana-852	115	2	∫	∫	PROPN
cana-852	115	3	  	  	SPACE
cana-852	115	4	ω(𝑎,∞	ω(𝑎,∞	PROPN
cana-852	115	5	)	)	PUNCT
cana-852	115	6	  	  	SPACE
cana-852	116	1	𝑟𝛼−𝑛+𝜆	𝑟𝛼−𝑛+𝜆	DET
cana-852	116	2	∥	∥	PUNCT
cana-852	116	3	𝑊	𝑊	NOUN
cana-852	116	4	∥2	∥2	PRON
cana-852	116	5	𝑑𝑥	𝑑𝑥	NOUN
cana-852	116	6	,	,	PUNCT
cana-852	116	7	is	be	AUX
cana-852	116	8	a	a	DET
cana-852	116	9	finite	finite	ADJ
cana-852	116	10	number	number	NOUN
cana-852	116	11	.	.	PUNCT
cana-852	117	1	we	we	PRON
cana-852	117	2	will	will	AUX
cana-852	117	3	show	show	VERB
cana-852	117	4	that	that	SCONJ
cana-852	117	5	�	�	PROPN
cana-852	117	6	̂	̂	VERB
cana-852	117	7	�	�	NOUN
cana-852	117	8	=	=	SYM
cana-852	117	9	𝛼𝑃0	𝛼𝑃0	PROPN
cana-852	117	10	.	.	PUNCT
cana-852	118	1	(	(	PUNCT
cana-852	118	2	2.8	2.8	NUM
cana-852	118	3	)	)	PUNCT
cana-852	118	4	then	then	ADV
cana-852	118	5	it	it	PRON
cana-852	118	6	follows	follow	VERB
cana-852	118	7	that	that	SCONJ
cana-852	118	8	�	�	PROPN
cana-852	118	9	̂	̂	SYM
cana-852	118	10	�	�	PROPN
cana-852	118	11	actually	actually	ADV
cana-852	118	12	does	do	AUX
cana-852	118	13	not	not	PART
cana-852	118	14	always	always	ADV
cana-852	118	15	depend	depend	VERB
cana-852	118	16	on	on	ADP
cana-852	118	17	the	the	DET
cana-852	118	18	choice	choice	NOUN
cana-852	118	19	of	of	ADP
cana-852	118	20	the	the	DET
cana-852	118	21	number	number	NOUN
cana-852	118	22	for	for	ADP
cana-852	118	23	𝑎.	𝑎.	NOUN
cana-852	118	24	using	use	VERB
cana-852	118	25	(	(	PUNCT
cana-852	118	26	2.7	2.7	NUM
cana-852	118	27	)	)	PUNCT
cana-852	118	28	and	and	CCONJ
cana-852	118	29	the	the	DET
cana-852	118	30	inequality	inequality	NOUN
cana-852	118	31	|𝑏	|𝑏	NOUN
cana-852	119	1	+	+	CCONJ
cana-852	119	2	𝑐	𝑐	X
cana-852	119	3	+	+	PUNCT
cana-852	119	4	𝑑|2	𝑑|2	PROPN
cana-852	119	5	≤	≤	PUNCT
cana-852	119	6	4|𝑏|2	4|𝑏|2	NOUN
cana-852	120	1	+	+	CCONJ
cana-852	120	2	4|𝑐|2	4|𝑐|2	NUM
cana-852	121	1	+	+	CCONJ
cana-852	121	2	4|𝑑|2	4|𝑑|2	NUM
cana-852	121	3	.	.	PUNCT
cana-852	122	1	taking	take	VERB
cana-852	122	2	integration	integration	NOUN
cana-852	122	3	from	from	ADP
cana-852	122	4	𝑎	𝑎	PROPN
cana-852	122	5	→	→	SYM
cana-852	122	6	𝑅	𝑅	NOUN
cana-852	122	7	and	and	CCONJ
cana-852	122	8	multiply	multiply	ADV
cana-852	122	9	by	by	ADP
cana-852	122	10	1	1	NUM
cana-852	122	11	𝑅𝛼	𝑅𝛼	PROPN
cana-852	122	12	on	on	ADP
cana-852	122	13	both	both	DET
cana-852	122	14	side	side	NOUN
cana-852	122	15	1	1	NUM
cana-852	122	16	𝑅𝛼	𝑅𝛼	PROPN
cana-852	122	17	∫	∫	PROPN
cana-852	122	18	  	  	SPACE
cana-852	122	19	𝑅	𝑅	PROPN
cana-852	122	20	𝑎	𝑎	NOUN
cana-852	122	21	  	  	SPACE
cana-852	122	22	|	|	ADV
cana-852	122	23	�	�	PROPN
cana-852	122	24	̂	̂	PROPN
cana-852	122	25	�	�	PROPN
cana-852	122	26	−	−	PROPN
cana-852	122	27	∫	∫	PROPN
cana-852	122	28	  	  	SPACE
cana-852	122	29	ω(1,𝑟	ω(1,𝑟	NUM
cana-852	122	30	)	)	PUNCT
cana-852	122	31	  	  	SPACE
cana-852	122	32	𝑟𝛼−𝑛+𝜆𝑝(𝑥)𝑑𝑥|	𝑟𝛼−𝑛+𝜆𝑝(𝑥)𝑑𝑥|	NOUN
cana-852	122	33	2	2	NUM
cana-852	122	34	𝑑𝛼𝑟	𝑑𝛼𝑟	ADP
cana-852	122	35	=	=	SYM
cana-852	122	36	4	4	NUM
cana-852	122	37	𝑅𝛼	𝑅𝛼	PROPN
cana-852	122	38	∫	∫	PROPN
cana-852	122	39	  	  	SPACE
cana-852	122	40	𝑅	𝑅	PROPN
cana-852	122	41	𝑎	𝑎	PROPN
cana-852	122	42	  	  	SPACE
cana-852	122	43	|∫	|∫	NOUN
cana-852	122	44	  	  	SPACE
cana-852	122	45	𝑠(𝑟	𝑠(𝑟	NOUN
cana-852	122	46	)	)	PUNCT
cana-852	122	47	 	 	SPACE
cana-852	122	48	𝑟𝛼−𝑛+𝜆⟨𝑊	𝑟𝛼−𝑛+𝜆⟨𝑊	SYM
cana-852	122	49	,	,	PUNCT
cana-852	122	50	𝑒𝑖⟩𝑑𝑆|	𝑒𝑖⟩𝑑𝑆|	ADP
cana-852	122	51	2	2	NUM
cana-852	122	52	𝑑𝛼𝑟	𝑑𝛼𝑟	PROPN
cana-852	122	53	(	(	PUNCT
cana-852	122	54	2.9	2.9	NUM
cana-852	122	55	)	)	PUNCT
cana-852	123	1	+	+	NUM
cana-852	124	1	4(𝛼	4(𝛼	NUM
cana-852	124	2	−	−	NOUN
cana-852	124	3	𝑛	𝑛	DET
cana-852	124	4	+	+	CCONJ
cana-852	124	5	𝜆)2	𝜆)2	PROPN
cana-852	124	6	𝑅𝛼	𝑅𝛼	PROPN
cana-852	124	7	∫	∫	PROPN
cana-852	124	8	  	  	SPACE
cana-852	124	9	𝑅	𝑅	PROPN
cana-852	124	10	𝑎	𝑎	PROPN
cana-852	124	11	  	  	SPACE
cana-852	124	12	|∫	|∫	NOUN
cana-852	124	13	  	  	SPACE
cana-852	124	14	ω(𝑟,∞	ω(𝑟,∞	NOUN
cana-852	124	15	)	)	PUNCT
cana-852	124	16	  	  	SPACE
cana-852	124	17	𝑟−𝑛+𝜆⟨𝑊	𝑟−𝑛+𝜆⟨𝑊	NOUN
cana-852	124	18	,	,	PUNCT
cana-852	124	19	𝑒𝑖⟩𝑑𝑥|	𝑒𝑖⟩𝑑𝑥|	PROPN
cana-852	124	20	2	2	NUM
cana-852	124	21	𝑑𝛼𝑟	𝑑𝛼𝑟	ADP
cana-852	124	22	(	(	PUNCT
cana-852	124	23	2.10	2.10	NUM
cana-852	124	24	)	)	PUNCT
cana-852	125	1	+	+	CCONJ
cana-852	125	2	4	4	NUM
cana-852	125	3	𝑅𝛼	𝑅𝛼	NUM
cana-852	125	4	∫	∫	PROPN
cana-852	125	5	  	  	SPACE
cana-852	125	6	𝑅	𝑅	PROPN
cana-852	125	7	𝑎	𝑎	PROPN
cana-852	125	8	  	  	SPACE
cana-852	125	9	|∫	|∫	NOUN
cana-852	125	10	  	  	SPACE
cana-852	125	11	ω(𝑟,∞	ω(𝑟,∞	NOUN
cana-852	125	12	)	)	PUNCT
cana-852	125	13	  	  	SPACE
cana-852	126	1	𝑟𝛼−𝑛+𝜆	𝑟𝛼−𝑛+𝜆	DET
cana-852	126	2	∥	∥	PROPN
cana-852	126	3	𝑊	𝑊	NOUN
cana-852	126	4	∥2	∥2	NOUN
cana-852	126	5	𝑑𝑥|	𝑑𝑥|	PROPN
cana-852	126	6	2	2	NUM
cana-852	126	7	𝑑𝛼𝑟	𝑑𝛼𝑟	ADP
cana-852	126	8	=	=	SYM
cana-852	126	9	0	0	PROPN
cana-852	126	10	.	.	PUNCT
cana-852	127	1	(	(	PUNCT
cana-852	127	2	2.11	2.11	NUM
cana-852	127	3	)	)	PUNCT
cana-852	127	4	the	the	DET
cana-852	127	5	terms	term	NOUN
cana-852	127	6	(	(	PUNCT
cana-852	127	7	2.10	2.10	NUM
cana-852	127	8	)	)	PUNCT
cana-852	127	9	and	and	CCONJ
cana-852	127	10	(	(	PUNCT
cana-852	127	11	2.11	2.11	NUM
cana-852	127	12	)	)	PUNCT
cana-852	127	13	tends	tend	VERB
cana-852	127	14	to	to	ADP
cana-852	127	15	zero	zero	NUM
cana-852	127	16	for	for	ADP
cana-852	127	17	𝑟	𝑟	NOUN
cana-852	127	18	→	→	SYM
cana-852	127	19	∞	∞	PROPN
cana-852	127	20	,	,	PUNCT
cana-852	127	21	according	accord	VERB
cana-852	127	22	to	to	ADP
cana-852	127	23	the	the	DET
cana-852	127	24	l'hospital	l'hospital	ADJ
cana-852	127	25	rule	rule	NOUN
cana-852	127	26	,	,	PUNCT
cana-852	127	27	(	(	PUNCT
cana-852	127	28	2.2	2.2	NUM
cana-852	127	29	)	)	PUNCT
cana-852	127	30	and	and	CCONJ
cana-852	127	31	(	(	PUNCT
cana-852	127	32	2.6	2.6	NUM
cana-852	127	33	)	)	PUNCT
cana-852	127	34	.	.	PUNCT
cana-852	128	1	if	if	SCONJ
cana-852	128	2	follows	follow	VERB
cana-852	128	3	from	from	ADP
cana-852	128	4	the	the	DET
cana-852	128	5	cauchy	cauchy	ADJ
cana-852	128	6	inequality	inequality	NOUN
cana-852	128	7	1	1	NUM
cana-852	128	8	𝑅𝛼	𝑅𝛼	PROPN
cana-852	128	9	∫	∫	PROPN
cana-852	128	10	  	  	SPACE
cana-852	128	11	𝑅	𝑅	PROPN
cana-852	128	12	𝑎	𝑎	PROPN
cana-852	128	13	  	  	SPACE
cana-852	128	14	|∫	|∫	NOUN
cana-852	128	15	  	  	SPACE
cana-852	128	16	𝑆(𝑟	𝑆(𝑟	X
cana-852	128	17	)	)	PUNCT
cana-852	128	18	  	  	SPACE
cana-852	128	19	𝑟𝛼−𝑛+𝜆⟨𝑊	𝑟𝛼−𝑛+𝜆⟨𝑊	NOUN
cana-852	128	20	,	,	PUNCT
cana-852	128	21	𝑒𝑖⟩𝑑𝑆|	𝑒𝑖⟩𝑑𝑆|	ADP
cana-852	128	22	2	2	NUM
cana-852	128	23	𝑑𝛼𝑟	𝑑𝛼𝑟	PRON
cana-852	128	24	≤	≤	NUM
cana-852	128	25	1	1	NUM
cana-852	128	26	𝑅𝛼	𝑅𝛼	PROPN
cana-852	128	27	∫	∫	PROPN
cana-852	128	28	  	  	SPACE
cana-852	128	29	𝑅	𝑅	PROPN
cana-852	128	30	𝑎	𝑎	PROPN
cana-852	128	31	 	 	SPACE
cana-852	128	32	(	(	PUNCT
cana-852	128	33	∫	∫	PROPN
cana-852	128	34	  	  	SPACE
cana-852	128	35	𝑆(𝑟	𝑆(𝑟	NOUN
cana-852	128	36	)	)	PUNCT
cana-852	128	37	  	  	SPACE
cana-852	128	38	𝑟𝛼−𝑛+𝜆	𝑟𝛼−𝑛+𝜆	DET
cana-852	128	39	∥	∥	NUM
cana-852	128	40	𝑊	𝑊	VERB
cana-852	128	41	∥2	∥2	NOUN
cana-852	128	42	𝑑𝑆	𝑑𝑆	ADJ
cana-852	128	43	)	)	PUNCT
cana-852	128	44	(	(	PUNCT
cana-852	128	45	𝜔𝑛	𝜔𝑛	NOUN
cana-852	128	46	∫	∫	PROPN
cana-852	128	47	  	  	SPACE
cana-852	128	48	𝑆(𝑟	𝑆(𝑟	X
cana-852	128	49	)	)	PUNCT
cana-852	128	50	  	  	SPACE
cana-852	128	51	𝑟𝛼+𝜆−1𝑑𝑆	𝑟𝛼+𝜆−1𝑑𝑆	PROPN
cana-852	128	52	)	)	PUNCT
cana-852	128	53	𝑑𝛼𝑟.	𝑑𝛼𝑟.	NOUN
cana-852	128	54	and	and	CCONJ
cana-852	128	55	the	the	DET
cana-852	128	56	term	term	NOUN
cana-852	128	57	(	(	PUNCT
cana-852	128	58	2.9	2.9	NUM
cana-852	128	59	)	)	PUNCT
cana-852	128	60	tends	tend	VERB
cana-852	128	61	to	to	ADP
cana-852	128	62	zero	zero	NUM
cana-852	128	63	by	by	ADP
cana-852	128	64	using	use	VERB
cana-852	128	65	(	(	PUNCT
cana-852	128	66	2.2	2.2	NUM
cana-852	128	67	)	)	PUNCT
cana-852	128	68	.	.	PUNCT
cana-852	129	1	hence	hence	ADV
cana-852	129	2	1	1	NUM
cana-852	129	3	𝑅𝛼	𝑅𝛼	PROPN
cana-852	129	4	∫	∫	PROPN
cana-852	129	5	  	  	SPACE
cana-852	129	6	𝑅	𝑅	PROPN
cana-852	129	7	𝑎	𝑎	PROPN
cana-852	129	8	|	|	NOUN
cana-852	129	9	�	�	NOUN
cana-852	129	10	̂	̂	SYM
cana-852	129	11	�	�	PROPN
cana-852	129	12	−	−	PROPN
cana-852	129	13	∫	∫	PROPN
cana-852	129	14	  	  	SPACE
cana-852	129	15	ω(1,𝑟	ω(1,𝑟	NUM
cana-852	129	16	)	)	PUNCT
cana-852	129	17	  	  	SPACE
cana-852	130	1	𝑟𝛼−𝑛+𝜆𝑝(𝑥)𝑑𝑥|	𝑟𝛼−𝑛+𝜆𝑝(𝑥)𝑑𝑥|	PROPN
cana-852	130	2	2	2	NUM
cana-852	130	3	𝑑𝛼𝑟	𝑑𝛼𝑟	ADP
cana-852	130	4	→	→	SYM
cana-852	130	5	0	0	NUM
cana-852	130	6	⬚	⬚	NOUN
cana-852	130	7	for	for	ADP
cana-852	130	8	⬚	⬚	PROPN
cana-852	130	9	𝑅	𝑅	PROPN
cana-852	130	10	→	→	SYM
cana-852	130	11	∞	∞	PROPN
cana-852	130	12	(	(	PUNCT
cana-852	130	13	2.12	2.12	NUM
cana-852	130	14	)	)	PUNCT
cana-852	130	15	communications	communication	NOUN
cana-852	130	16	on	on	ADP
cana-852	130	17	applied	apply	VERB
cana-852	130	18	nonlinear	nonlinear	ADJ
cana-852	130	19	analysis	analysis	NOUN
cana-852	130	20	issn	issn	NOUN
cana-852	130	21	:	:	PUNCT
cana-852	130	22	1074	1074	NUM
cana-852	130	23	-	-	PUNCT
cana-852	130	24	133x	133x	NUM
cana-852	130	25	vol	vol	NOUN
cana-852	130	26	31	31	NUM
cana-852	130	27	no	no	NOUN
cana-852	130	28	.	.	PUNCT
cana-852	131	1	4s	4s	NUM
cana-852	131	2	(	(	PUNCT
cana-852	131	3	2024	2024	NUM
cana-852	131	4	)	)	PUNCT
cana-852	131	5	291	291	NUM
cana-852	131	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-852	132	1	therefore	therefore	ADV
cana-852	132	2	|	|	ADV
cana-852	132	3	1	1	NUM
cana-852	133	1	𝑅𝛼	𝑅𝛼	PROPN
cana-852	133	2	∫	∫	NOUN
cana-852	133	3	𝑎	𝑎	PROPN
cana-852	133	4	𝑅	𝑅	PROPN
cana-852	133	5	  	  	SPACE
cana-852	133	6	(	(	PUNCT
cana-852	133	7	�	�	PROPN
cana-852	133	8	̂	̂	PROPN
cana-852	133	9	�	�	PROPN
cana-852	133	10	−	−	PROPN
cana-852	133	11	∫	∫	PROPN
cana-852	133	12	ω(1,𝑟	ω(1,𝑟	NUM
cana-852	133	13	)	)	PUNCT
cana-852	133	14	 	 	SPACE
cana-852	133	15	𝑟𝛼−𝑛+𝜆𝑝(𝑥)𝑑𝑥	𝑟𝛼−𝑛+𝜆𝑝(𝑥)𝑑𝑥	NOUN
cana-852	133	16	)	)	PUNCT
cana-852	133	17	𝑑𝛼𝑟|	𝑑𝛼𝑟|	VERB
cana-852	133	18	≤	≤	NUM
cana-852	133	19	(	(	PUNCT
cana-852	133	20	1	1	NUM
cana-852	133	21	𝑅𝛼	𝑅𝛼	PROPN
cana-852	133	22	∫	∫	NOUN
cana-852	133	23	𝑎	𝑎	PROPN
cana-852	133	24	𝑅	𝑅	PROPN
cana-852	133	25	  	  	SPACE
cana-852	133	26	|	|	ADV
cana-852	133	27	�	�	PROPN
cana-852	133	28	̂	̂	PROPN
cana-852	133	29	�	�	PROPN
cana-852	133	30	−	−	PROPN
cana-852	133	31	∫	∫	PROPN
cana-852	133	32	ω(1,𝑟	ω(1,𝑟	NUM
cana-852	133	33	)	)	PUNCT
cana-852	133	34	 	 	SPACE
cana-852	133	35	𝑟𝛼−𝑛+𝜆𝑝(𝑥)𝑑𝑥|	𝑟𝛼−𝑛+𝜆𝑝(𝑥)𝑑𝑥|	PROPN
cana-852	133	36	2	2	NUM
cana-852	133	37	𝑑𝛼𝑟	𝑑𝛼𝑟	NOUN
cana-852	133	38	)	)	PUNCT
cana-852	133	39	1	1	NUM
cana-852	133	40	2	2	NUM
cana-852	133	41	and	and	CCONJ
cana-852	133	42	from	from	ADP
cana-852	133	43	(	(	PUNCT
cana-852	133	44	2.12	2.12	NUM
cana-852	133	45	)	)	PUNCT
cana-852	133	46	it	it	PRON
cana-852	133	47	follows	follow	VERB
cana-852	133	48	that	that	SCONJ
cana-852	133	49	(	(	PUNCT
cana-852	133	50	2.3	2.3	NUM
cana-852	133	51	)	)	PUNCT
cana-852	133	52	,	,	PUNCT
cana-852	133	53	�	�	PROPN
cana-852	133	54	̂	̂	SYM
cana-852	133	55	�	�	NOUN
cana-852	133	56	=	=	SYM
cana-852	133	57	𝛼𝑃0	𝛼𝑃0	PROPN
cana-852	133	58	⬚	⬚	PROPN
cana-852	133	59	holds	hold	VERB
cana-852	133	60	.	.	PUNCT
cana-852	133	61	ii	ii	PROPN
cana-852	133	62	)	)	PUNCT
cana-852	133	63	⇒	⇒	PROPN
cana-852	133	64	iii	iii	PROPN
cana-852	133	65	)	)	PUNCT
cana-852	133	66	is	be	AUX
cana-852	133	67	trivial	trivial	ADJ
cana-852	133	68	.	.	PUNCT
cana-852	134	1	iii	iii	X
cana-852	134	2	)	)	PUNCT
cana-852	134	3	⇒	⇒	NOUN
cana-852	134	4	i	i	PRON
cana-852	134	5	)	)	PUNCT
cana-852	134	6	if	if	SCONJ
cana-852	134	7	the	the	DET
cana-852	134	8	(	(	PUNCT
cana-852	134	9	2.4	2.4	NUM
cana-852	134	10	)	)	PUNCT
cana-852	134	11	holds	hold	NOUN
cana-852	134	12	and	and	CCONJ
cana-852	134	13	equation	equation	NOUN
cana-852	134	14	(	(	PUNCT
cana-852	134	15	2.2	2.2	NUM
cana-852	134	16	)	)	PUNCT
cana-852	134	17	does	do	AUX
cana-852	134	18	not	not	PART
cana-852	134	19	hold	hold	VERB
cana-852	134	20	.	.	PUNCT
cana-852	135	1	denoting	denote	VERB
cana-852	135	2	𝜒(𝑟	𝜒(𝑟	NOUN
cana-852	135	3	):	):	PUNCT
cana-852	135	4	=	=	X
cana-852	135	5	∫	∫	PROPN
cana-852	135	6	  	  	SPACE
cana-852	135	7	𝑟	𝑟	NOUN
cana-852	135	8	𝑎	𝑎	DET
cana-852	135	9	∫	∫	NOUN
cana-852	135	10	  	  	SPACE
cana-852	135	11	ω(𝑎,𝑟	ω(𝑎,𝑟	NUM
cana-852	135	12	)	)	PUNCT
cana-852	136	1	𝑟𝛼−𝑛+𝜆	𝑟𝛼−𝑛+𝜆	CCONJ
cana-852	136	2	∥	∥	PUNCT
cana-852	136	3	𝑊	𝑊	NOUN
cana-852	136	4	∥2	∥2	NOUN
cana-852	136	5	𝑑𝑥𝑑𝛼𝑟	𝑑𝑥𝑑𝛼𝑟	ADP
cana-852	136	6	this	this	DET
cana-852	136	7	function	function	NOUN
cana-852	136	8	satisfies	satisfy	VERB
cana-852	136	9	lim	lim	PROPN
cana-852	136	10	𝑟→∞	𝑟→∞	NUM
cana-852	136	11	  	  	SPACE
cana-852	136	12	𝜒(𝑟	𝜒(𝑟	NOUN
cana-852	136	13	)	)	PUNCT
cana-852	136	14	𝑟	𝑟	NOUN
cana-852	136	15	→	→	SYM
cana-852	136	16	∞	∞	NUM
cana-852	136	17	⬚	⬚	PROPN
cana-852	136	18	for	for	ADP
cana-852	136	19	⬚	⬚	NOUN
cana-852	136	20	𝑟	𝑟	NOUN
cana-852	136	21	→	→	SYM
cana-852	136	22	∞	∞	PROPN
cana-852	136	23	(	(	PUNCT
cana-852	136	24	2.13	2.13	NUM
cana-852	136	25	)	)	PUNCT
cana-852	136	26	∫	∫	PROPN
cana-852	136	27	  	  	SPACE
cana-852	136	28	ω(𝑎,∞	ω(𝑎,∞	PROPN
cana-852	136	29	)	)	PUNCT
cana-852	137	1	𝑟𝛼−𝑛+𝜆	𝑟𝛼−𝑛+𝜆	PRON
cana-852	137	2	∥	∥	PUNCT
cana-852	137	3	𝑊	𝑊	NOUN
cana-852	137	4	∥2	∥2	NOUN
cana-852	137	5	𝑑𝑥	𝑑𝑥	NOUN
cana-852	137	6	=	=	SYM
cana-852	137	7	∞.	∞.	PROPN
cana-852	137	8	from	from	ADP
cana-852	137	9	(	(	PUNCT
cana-852	137	10	2.5	2.5	NUM
cana-852	137	11	)	)	PUNCT
cana-852	137	12	we	we	PRON
cana-852	137	13	integrate	integrate	VERB
cana-852	137	14	from	from	ADP
cana-852	137	15	𝑎	𝑎	ADJ
cana-852	137	16	→	→	SYM
cana-852	137	17	r	r	NOUN
cana-852	137	18	and	and	CCONJ
cana-852	137	19	multiply	multiply	ADV
cana-852	137	20	by	by	ADP
cana-852	137	21	1	1	NUM
cana-852	137	22	𝑅𝑎	𝑅𝑎	PROPN
cana-852	137	23	we	we	PRON
cana-852	137	24	get	get	VERB
cana-852	137	25	∣	∣	ADJ
cana-852	137	26	1	1	NUM
cana-852	137	27	𝑅𝛼	𝑅𝛼	PROPN
cana-852	137	28	𝜒(𝑅	𝜒(𝑅	NUM
cana-852	137	29	)	)	PUNCT
cana-852	137	30	−	−	PROPN
cana-852	137	31	1	1	NUM
cana-852	137	32	𝑅𝛼	𝑅𝛼	PROPN
cana-852	137	33	∫	∫	PROPN
cana-852	137	34	  	  	SPACE
cana-852	137	35	𝑅	𝑅	PROPN
cana-852	137	36	𝑎	𝑎	PROPN
cana-852	137	37	 	 	SPACE
cana-852	137	38	∫	∫	PROPN
cana-852	137	39	  	  	SPACE
cana-852	137	40	ω(𝑎,𝑟	ω(𝑎,𝑟	NUM
cana-852	137	41	)	)	PUNCT
cana-852	137	42	  	  	SPACE
cana-852	137	43	(	(	PUNCT
cana-852	137	44	𝛼	𝛼	NOUN
cana-852	137	45	−	−	PROPN
cana-852	137	46	𝑛	𝑛	PROPN
cana-852	137	47	+	+	PROPN
cana-852	137	48	𝜆)𝑟−𝑛+𝜆⟨𝑊	𝜆)𝑟−𝑛+𝜆⟨𝑊	NOUN
cana-852	137	49	,	,	PUNCT
cana-852	137	50	𝑒𝑖⟩𝑑𝑥𝑑𝛼𝑟	𝑒𝑖⟩𝑑𝑥𝑑𝛼𝑟	PROPN
cana-852	137	51	+	+	ADJ
cana-852	137	52	1	1	NUM
cana-852	137	53	𝑅𝛼	𝑅𝛼	PROPN
cana-852	137	54	∫	∫	PROPN
cana-852	137	55	  	  	SPACE
cana-852	137	56	𝑅	𝑅	PROPN
cana-852	137	57	𝑎	𝑎	PROPN
cana-852	137	58	 	 	SPACE
cana-852	137	59	∫	∫	PROPN
cana-852	137	60	  	  	SPACE
cana-852	137	61	𝑠(𝑟	𝑠(𝑟	NOUN
cana-852	137	62	)	)	PUNCT
cana-852	137	63	  	  	SPACE
cana-852	137	64	𝑟𝛼−𝑛+𝜆⟨𝑊	𝑟𝛼−𝑛+𝜆⟨𝑊	NOUN
cana-852	137	65	,	,	PUNCT
cana-852	137	66	𝑒𝑖⟩𝑑𝑆𝑑𝛼𝑟|	𝑒𝑖⟩𝑑𝑆𝑑𝛼𝑟|	PROPN
cana-852	137	67	=	=	PUNCT
cana-852	138	1	|	|	ADV
cana-852	138	2	−	−	NOUN
cana-852	138	3	𝑃(𝑅	𝑃(𝑅	NOUN
cana-852	138	4	)	)	PUNCT
cana-852	138	5	+	+	CCONJ
cana-852	138	6	1	1	NUM
cana-852	138	7	𝑅𝛼	𝑅𝛼	PROPN
cana-852	138	8	∫	∫	PROPN
cana-852	138	9	  	  	SPACE
cana-852	138	10	𝑅	𝑅	PROPN
cana-852	138	11	𝑎	𝑎	PROPN
cana-852	138	12	 	 	SPACE
cana-852	138	13	∫	∫	PROPN
cana-852	138	14	  	  	SPACE
cana-852	138	15	𝑠(𝑎	𝑠(𝑎	PROPN
cana-852	138	16	)	)	PUNCT
cana-852	138	17	  	  	SPACE
cana-852	138	18	𝑟𝛼−𝑛+𝜆⟨𝑊	𝑟𝛼−𝑛+𝜆⟨𝑊	NOUN
cana-852	138	19	,	,	PUNCT
cana-852	138	20	𝑒𝑖⟩𝑑𝑆𝑑𝛼𝑟	𝑒𝑖⟩𝑑𝑆𝑑𝛼𝑟	NOUN
cana-852	138	21	∣	∣	VERB
cana-852	138	22	if	if	SCONJ
cana-852	138	23	iii	iii	NOUN
cana-852	138	24	)	)	PUNCT
cana-852	138	25	holds	hold	NOUN
cana-852	138	26	,	,	PUNCT
cana-852	138	27	lim𝑟→∞	lim𝑟→∞	PROPN
cana-852	138	28	 	 	SPACE
cana-852	138	29	inf𝑃(𝑟	inf𝑃(𝑟	PROPN
cana-852	138	30	)	)	PUNCT
cana-852	138	31	>	>	X
cana-852	139	1	−∞	−∞	ADP
cana-852	139	2	hold	hold	NOUN
cana-852	139	3	.	.	PUNCT
cana-852	140	1	less	less	ADJ
cana-852	140	2	than	than	ADP
cana-852	140	3	1	1	NUM
cana-852	140	4	4𝑅𝛼	4𝑅𝛼	NUM
cana-852	140	5	𝜒(𝑅	𝜒(𝑅	NUM
cana-852	140	6	)	)	PUNCT
cana-852	140	7	and	and	CCONJ
cana-852	140	8	the	the	DET
cana-852	140	9	right	right	ADJ
cana-852	140	10	-	-	PUNCT
cana-852	140	11	hand	hand	NOUN
cana-852	140	12	side	side	NOUN
cana-852	140	13	is	be	AUX
cana-852	140	14	bounded	bound	VERB
cana-852	140	15	from	from	ADP
cana-852	140	16	above	above	ADP
cana-852	140	17	if	if	SCONJ
cana-852	140	18	(	(	PUNCT
cana-852	140	19	2.4	2.4	NUM
cana-852	140	20	)	)	PUNCT
cana-852	140	21	is	be	AUX
cana-852	140	22	valid	valid	ADJ
cana-852	140	23	.	.	PUNCT
cana-852	141	1	hence	hence	ADV
cana-852	141	2	3𝜒(𝑅	3𝜒(𝑅	NUM
cana-852	141	3	)	)	PUNCT
cana-852	141	4	4𝑅𝛼	4𝑅𝛼	NOUN
cana-852	141	5	≤	≤	NUM
cana-852	141	6	|	|	ADV
cana-852	141	7	1	1	NUM
cana-852	141	8	𝑅𝛼	𝑅𝛼	PROPN
cana-852	141	9	∫	∫	PROPN
cana-852	141	10	  	  	SPACE
cana-852	141	11	𝑅	𝑅	PROPN
cana-852	141	12	𝑎	𝑎	PROPN
cana-852	141	13	 	 	SPACE
cana-852	141	14	∫	∫	PROPN
cana-852	141	15	  	  	SPACE
cana-852	141	16	ω(𝑎,𝑟	ω(𝑎,𝑟	NUM
cana-852	141	17	)	)	PUNCT
cana-852	141	18	  	  	SPACE
cana-852	141	19	(	(	PUNCT
cana-852	141	20	𝛼	𝛼	NOUN
cana-852	141	21	−	−	PROPN
cana-852	141	22	𝑛	𝑛	PROPN
cana-852	141	23	+	+	PROPN
cana-852	141	24	𝜆)𝑟−𝑛+𝜆⟨𝑊	𝜆)𝑟−𝑛+𝜆⟨𝑊	PROPN
cana-852	141	25	,	,	PUNCT
cana-852	141	26	𝑒𝑖⟩𝑑𝑥𝑑𝛼𝑟|	𝑒𝑖⟩𝑑𝑥𝑑𝛼𝑟|	NOUN
cana-852	141	27	+	+	CCONJ
cana-852	141	28	|	|	ADV
cana-852	141	29	1	1	NUM
cana-852	141	30	𝑅𝛼	𝑅𝛼	PROPN
cana-852	141	31	∫	∫	PROPN
cana-852	141	32	  	  	SPACE
cana-852	141	33	𝑅	𝑅	PROPN
cana-852	141	34	𝑎	𝑎	PROPN
cana-852	141	35	 	 	SPACE
cana-852	141	36	∫	∫	PROPN
cana-852	141	37	  	  	SPACE
cana-852	141	38	𝑠(𝑟	𝑠(𝑟	NOUN
cana-852	141	39	)	)	PUNCT
cana-852	141	40	  	  	SPACE
cana-852	141	41	𝑟𝛼−𝑛+𝜆⟨𝑊	𝑟𝛼−𝑛+𝜆⟨𝑊	NUM
cana-852	141	42	,	,	PUNCT
cana-852	141	43	𝑒𝑖⟩𝑑𝑆𝑑𝛼𝑟|	𝑒𝑖⟩𝑑𝑆𝑑𝛼𝑟|	PROPN
cana-852	141	44	.	.	PROPN
cana-852	142	1	(	(	PUNCT
cana-852	142	2	2.14	2.14	NUM
cana-852	142	3	)	)	PUNCT
cana-852	142	4	by	by	ADP
cana-852	142	5	the	the	DET
cana-852	142	6	cauchy	cauchy	PROPN
cana-852	142	7	inequality	inequality	NOUN
cana-852	142	8	,	,	PUNCT
cana-852	142	9	1	1	NUM
cana-852	142	10	𝑅𝛼	𝑅𝛼	PROPN
cana-852	142	11	∫	∫	PROPN
cana-852	142	12	  	  	SPACE
cana-852	142	13	𝑅	𝑅	PROPN
cana-852	142	14	𝑎	𝑎	PROPN
cana-852	142	15	 	 	SPACE
cana-852	142	16	∫	∫	PROPN
cana-852	142	17	  	  	SPACE
cana-852	142	18	𝑠(𝑟	𝑠(𝑟	NOUN
cana-852	142	19	)	)	PUNCT
cana-852	142	20	 	 	SPACE
cana-852	142	21	𝑟𝛼−𝑛+𝜆⟨𝑊	𝑟𝛼−𝑛+𝜆⟨𝑊	NOUN
cana-852	142	22	,	,	PUNCT
cana-852	142	23	𝑒𝑖⟩𝑑𝑆𝑑𝛼𝑟	𝑒𝑖⟩𝑑𝑆𝑑𝛼𝑟	VERB
cana-852	142	24	≤	≤	NUM
cana-852	142	25	(	(	PUNCT
cana-852	142	26	𝑇𝛼𝜒(𝑅	𝑇𝛼𝜒(𝑅	NUM
cana-852	142	27	)	)	PUNCT
cana-852	142	28	)	)	PUNCT
cana-852	142	29	1	1	NUM
cana-852	142	30	2	2	NUM
cana-852	142	31	(	(	PUNCT
cana-852	142	32	𝜔𝑛𝑅𝜆+2𝛼	𝜔𝑛𝑅𝜆+2𝛼	X
cana-852	142	33	(	(	PUNCT
cana-852	142	34	𝜆	𝜆	ADP
cana-852	142	35	+	+	ADJ
cana-852	142	36	𝛼)(𝜆	𝛼)(𝜆	NUM
cana-852	142	37	+	+	NUM
cana-852	142	38	2𝛼	2𝛼	NUM
cana-852	142	39	)	)	PUNCT
cana-852	142	40	)	)	PUNCT
cana-852	142	41	1	1	NUM
cana-852	142	42	2	2	NUM
cana-852	142	43	1	1	NUM
cana-852	142	44	𝑅𝛼	𝑅𝛼	PROPN
cana-852	142	45	∫	∫	PROPN
cana-852	142	46	  	  	SPACE
cana-852	142	47	𝑅	𝑅	PROPN
cana-852	142	48	𝑎	𝑎	PROPN
cana-852	142	49	 	 	SPACE
cana-852	142	50	∫	∫	PROPN
cana-852	142	51	  	  	SPACE
cana-852	142	52	ω(𝑎,𝑟	ω(𝑎,𝑟	NUM
cana-852	142	53	)	)	PUNCT
cana-852	142	54	  	  	SPACE
cana-852	143	1	(	(	PUNCT
cana-852	143	2	𝛼	𝛼	NOUN
cana-852	143	3	−	−	PROPN
cana-852	143	4	𝑛	𝑛	PROPN
cana-852	143	5	+	+	PROPN
cana-852	143	6	𝜆)𝑟−𝑛+𝜆⟨𝑊	𝜆)𝑟−𝑛+𝜆⟨𝑊	PROPN
cana-852	143	7	,	,	PUNCT
cana-852	143	8	𝑒𝑖⟩𝑑𝑥𝑑𝛼𝑟	𝑒𝑖⟩𝑑𝑥𝑑𝛼𝑟	PROPN
cana-852	143	9	≤	≤	X
cana-852	143	10	(	(	PUNCT
cana-852	143	11	𝛼	𝛼	NOUN
cana-852	143	12	−	−	NOUN
cana-852	143	13	𝑛	𝑛	PROPN
cana-852	143	14	+	+	NUM
cana-852	143	15	𝜆)(𝜒(𝑅	𝜆)(𝜒(𝑅	NUM
cana-852	143	16	)	)	PUNCT
cana-852	143	17	)	)	PUNCT
cana-852	143	18	1	1	NUM
cana-852	143	19	2	2	NUM
cana-852	143	20	(	(	PUNCT
cana-852	143	21	𝜔𝑛𝑅𝜆	𝜔𝑛𝑅𝜆	PROPN
cana-852	143	22	𝜆(𝜆	𝜆(𝜆	PROPN
cana-852	143	23	−	−	NUM
cana-852	143	24	𝛼	𝛼	NOUN
cana-852	143	25	)	)	PUNCT
cana-852	143	26	)	)	PUNCT
cana-852	143	27	1	1	NUM
cana-852	143	28	2	2	NUM
cana-852	143	29	.	.	PUNCT
cana-852	144	1	communications	communication	NOUN
cana-852	144	2	on	on	ADP
cana-852	144	3	applied	apply	VERB
cana-852	144	4	nonlinear	nonlinear	ADJ
cana-852	144	5	analysis	analysis	NOUN
cana-852	144	6	issn	issn	NOUN
cana-852	144	7	:	:	PUNCT
cana-852	144	8	1074	1074	NUM
cana-852	144	9	-	-	PUNCT
cana-852	144	10	133x	133x	NUM
cana-852	144	11	vol	vol	NOUN
cana-852	144	12	31	31	NUM
cana-852	144	13	no	no	NOUN
cana-852	144	14	.	.	PUNCT
cana-852	145	1	4s	4s	NUM
cana-852	145	2	(	(	PUNCT
cana-852	145	3	2024	2024	NUM
cana-852	145	4	)	)	PUNCT
cana-852	145	5	292	292	NUM
cana-852	145	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-852	145	7	from	from	ADP
cana-852	145	8	(	(	PUNCT
cana-852	145	9	2.13	2.13	NUM
cana-852	145	10	)	)	PUNCT
cana-852	145	11	it	it	PRON
cana-852	145	12	follows	follow	VERB
cana-852	145	13	that	that	SCONJ
cana-852	145	14	𝜒(𝑅	𝜒(𝑅	NUM
cana-852	145	15	)	)	PUNCT
cana-852	145	16	42(𝛼−𝑛+𝜆)2𝑅𝜆	42(𝛼−𝑛+𝜆)2𝑅𝜆	NUM
cana-852	145	17	≤	≤	NUM
cana-852	145	18	𝜔𝑛	𝜔𝑛	ADP
cana-852	145	19	𝜆(𝜆−𝛼	𝜆(𝜆−𝛼	PROPN
cana-852	145	20	)	)	PUNCT
cana-852	145	21	(	(	PUNCT
cana-852	145	22	2.15	2.15	NUM
cana-852	145	23	)	)	PUNCT
cana-852	145	24	for	for	ADP
cana-852	145	25	𝑅	𝑅	PROPN
cana-852	145	26	large	large	ADJ
cana-852	145	27	enough	enough	ADV
cana-852	145	28	,	,	PUNCT
cana-852	145	29	therefore	therefore	ADV
cana-852	145	30	1	1	NUM
cana-852	145	31	𝑅𝛼	𝑅𝛼	PROPN
cana-852	145	32	∫	∫	PROPN
cana-852	145	33	  	  	SPACE
cana-852	145	34	𝑅	𝑅	PROPN
cana-852	145	35	𝑎	𝑎	PROPN
cana-852	145	36	∫	∫	PROPN
cana-852	145	37	  	  	SPACE
cana-852	145	38	ω(𝑎,𝑟	ω(𝑎,𝑟	NUM
cana-852	145	39	)	)	PUNCT
cana-852	145	40	(	(	PUNCT
cana-852	145	41	𝛼	𝛼	NOUN
cana-852	145	42	−	−	PROPN
cana-852	145	43	𝑛	𝑛	PROPN
cana-852	145	44	+	+	PROPN
cana-852	145	45	𝜆)𝑟−𝑛+𝜆⟨𝑊	𝜆)𝑟−𝑛+𝜆⟨𝑊	NOUN
cana-852	145	46	,	,	PUNCT
cana-852	145	47	𝑒𝑖⟩𝑑𝑥𝑑𝛼𝑟	𝑒𝑖⟩𝑑𝑥𝑑𝛼𝑟	PROPN
cana-852	145	48	=	=	SYM
cana-852	145	49	𝜒(𝑅	𝜒(𝑅	PROPN
cana-852	145	50	)	)	PUNCT
cana-852	145	51	4	4	NUM
cana-852	145	52	(	(	PUNCT
cana-852	145	53	2.16	2.16	NUM
cana-852	145	54	)	)	PUNCT
cana-852	145	55	combining	combine	VERB
cana-852	145	56	the	the	DET
cana-852	145	57	above	above	ADJ
cana-852	145	58	computations	computation	NOUN
cana-852	145	59	above	above	ADV
cana-852	145	60	for	for	ADP
cana-852	145	61	r	r	NOUN
cana-852	145	62	large	large	ADJ
cana-852	145	63	enough	enough	ADV
cana-852	145	64	,	,	PUNCT
cana-852	145	65	we	we	PRON
cana-852	145	66	obtain	obtain	VERB
cana-852	145	67	𝜒2(𝑅	𝜒2(𝑅	NOUN
cana-852	145	68	)	)	PUNCT
cana-852	145	69	4	4	NUM
cana-852	145	70	≤	≤	NOUN
cana-852	145	71	𝜔𝑛	𝜔𝑛	X
cana-852	145	72	(	(	PUNCT
cana-852	145	73	𝜆	𝜆	ADP
cana-852	145	74	+	+	ADJ
cana-852	145	75	𝛼)(𝜆	𝛼)(𝜆	NUM
cana-852	145	76	+	+	NUM
cana-852	145	77	2𝛼	2𝛼	NUM
cana-852	145	78	)	)	PUNCT
cana-852	146	1	𝑇𝛼𝜒(𝑅)𝑅𝜆+2𝛼	𝑇𝛼𝜒(𝑅)𝑅𝜆+2𝛼	NOUN
cana-852	146	2	and	and	CCONJ
cana-852	146	3	from	from	ADP
cana-852	146	4	here	here	ADV
cana-852	146	5	we	we	PRON
cana-852	146	6	get	get	VERB
cana-852	146	7	4𝜔𝑛	4𝜔𝑛	ADJ
cana-852	146	8	𝑇𝛼𝜒(𝑅	𝑇𝛼𝜒(𝑅	NOUN
cana-852	146	9	)	)	PUNCT
cana-852	146	10	𝜒2(𝑅	𝜒2(𝑅	NOUN
cana-852	146	11	)	)	PUNCT
cana-852	146	12	≥	≥	NOUN
cana-852	146	13	(	(	PUNCT
cana-852	146	14	𝜆	𝜆	ADP
cana-852	146	15	+	+	ADJ
cana-852	146	16	𝛼)(𝜆	𝛼)(𝜆	NUM
cana-852	146	17	+	+	NUM
cana-852	146	18	2𝛼	2𝛼	NUM
cana-852	146	19	)	)	PUNCT
cana-852	146	20	𝑅𝜆+2𝛼	𝑅𝜆+2𝛼	X
cana-852	146	21	for	for	ADP
cana-852	146	22	𝑅	𝑅	PROPN
cana-852	146	23	large	large	ADJ
cana-852	146	24	enough	enough	ADV
cana-852	146	25	.	.	PUNCT
cana-852	147	1	integration	integration	NOUN
cana-852	147	2	from	from	ADP
cana-852	147	3	𝑟1	𝑟1	PROPN
cana-852	147	4	→	→	SYM
cana-852	147	5	∞	∞	PROPN
cana-852	147	6	gives	give	VERB
cana-852	147	7	a	a	DET
cana-852	147	8	divergent	divergent	ADJ
cana-852	147	9	integral	integral	NOUN
cana-852	147	10	on	on	ADP
cana-852	147	11	the	the	DET
cana-852	147	12	right	right	ADJ
cana-852	147	13	-	-	PUNCT
cana-852	147	14	hand	hand	NOUN
cana-852	147	15	side	side	NOUN
cana-852	147	16	and	and	CCONJ
cana-852	147	17	convergent	convergent	NOUN
cana-852	147	18	integral	integral	ADJ
cana-852	147	19	on	on	ADP
cana-852	147	20	the	the	DET
cana-852	147	21	left	left	ADJ
cana-852	147	22	-	-	PUNCT
cana-852	147	23	hand	hand	NOUN
cana-852	147	24	side	side	NOUN
cana-852	147	25	.	.	PUNCT
cana-852	148	1	this	this	PRON
cana-852	148	2	contradicts	contradict	VERB
cana-852	148	3	our	our	PRON
cana-852	148	4	proof	proof	NOUN
cana-852	148	5	.	.	PUNCT
cana-852	149	1	introducing	introduce	VERB
cana-852	149	2	this	this	DET
cana-852	149	3	function	function	NOUN
cana-852	149	4	𝜌(𝑟	𝜌(𝑟	NOUN
cana-852	149	5	)	)	PUNCT
cana-852	149	6	is	be	AUX
cana-852	149	7	defined	define	VERB
cana-852	149	8	𝜌(𝑟	𝜌(𝑟	NOUN
cana-852	149	9	)	)	PUNCT
cana-852	150	1	=	=	SYM
cana-852	150	2	∫	∫	PROPN
cana-852	150	3	  	  	SPACE
cana-852	150	4	𝑆(𝑟	𝑆(𝑟	NUM
cana-852	150	5	)	)	PUNCT
cana-852	150	6	𝑟𝛼−𝑛+𝜆⟨𝑊	𝑟𝛼−𝑛+𝜆⟨𝑊	NOUN
cana-852	150	7	,	,	PUNCT
cana-852	150	8	𝑒𝑖⟩𝑑𝑆	𝑒𝑖⟩𝑑𝑆	PUNCT
cana-852	150	9	(	(	PUNCT
cana-852	150	10	2.17	2.17	NUM
cana-852	150	11	)	)	PUNCT
cana-852	150	12	lemma	lemma	PROPN
cana-852	150	13	:	:	PUNCT
cana-852	150	14	2.3	2.3	NUM
cana-852	150	15	.	.	PUNCT
cana-852	151	1	let	let	VERB
cana-852	151	2	(	(	PUNCT
cana-852	151	3	2.3	2.3	NUM
cana-852	151	4	)	)	PUNCT
cana-852	151	5	holds	hold	VERB
cana-852	151	6	.	.	PUNCT
cana-852	152	1	let	let	VERB
cana-852	152	2	the	the	DET
cana-852	152	3	equation	equation	NOUN
cana-852	152	4	(	(	PUNCT
cana-852	152	5	1.1	1.1	NUM
cana-852	152	6	)	)	PUNCT
cana-852	152	7	have	have	VERB
cana-852	152	8	a	a	DET
cana-852	152	9	non	non	ADJ
cana-852	152	10	-	-	ADJ
cana-852	152	11	oscillatory	oscillatory	ADJ
cana-852	152	12	solution	solution	NOUN
cana-852	152	13	.	.	PUNCT
cana-852	153	1	then	then	ADV
cana-852	153	2	,	,	PUNCT
cana-852	153	3	𝑀(𝑟	𝑀(𝑟	PROPN
cana-852	153	4	)	)	PUNCT
cana-852	153	5	−	−	PROPN
cana-852	153	6	(	(	PUNCT
cana-852	153	7	(	(	PUNCT
cana-852	153	8	𝛼	𝛼	NOUN
cana-852	153	9	−	−	NOUN
cana-852	153	10	𝑛	𝑛	PROPN
cana-852	153	11	+	+	CCONJ
cana-852	153	12	𝜆	𝜆	X
cana-852	153	13	)	)	PUNCT
cana-852	153	14	𝛼	𝛼	NOUN
cana-852	154	1	+	+	NOUN
cana-852	154	2	1	1	NUM
cana-852	154	3	)	)	PUNCT
cana-852	154	4	𝑔	𝑔	NOUN
cana-852	154	5	+	+	NOUN
cana-852	154	6	𝑟1−(𝛼+𝜆)𝑔2	𝑟1−(𝛼+𝜆)𝑔2	NUM
cana-852	154	7	𝜔𝑛(𝜆	𝜔𝑛(𝜆	NUM
cana-852	154	8	+	+	CCONJ
cana-852	154	9	𝛼	𝛼	X
cana-852	154	10	)	)	PUNCT
cana-852	154	11	≤	≤	NOUN
cana-852	154	12	0	0	NUM
cana-852	154	13	and	and	CCONJ
cana-852	154	14	𝑟−1(𝑟𝑁(𝑟	𝑟−1(𝑟𝑁(𝑟	NUM
cana-852	154	15	)	)	PUNCT
cana-852	154	16	−	−	NUM
cana-852	155	1	𝜏𝜖𝑁(𝜏𝜖	𝜏𝜖𝑁(𝜏𝜖	NOUN
cana-852	155	2	)	)	PUNCT
cana-852	155	3	−	−	PROPN
cana-852	155	4	𝜏𝜖	𝜏𝜖	NOUN
cana-852	155	5	2𝜌(𝜏𝜖	2𝜌(𝜏𝜖	NUM
cana-852	155	6	)	)	PUNCT
cana-852	155	7	)	)	PUNCT
cana-852	156	1	−	−	PROPN
cana-852	156	2	𝐺	𝐺	NOUN
cana-852	156	3	(	(	PUNCT
cana-852	156	4	(	(	PUNCT
cana-852	156	5	𝛼	𝛼	NOUN
cana-852	156	6	−	−	PROPN
cana-852	156	7	𝑛	𝑛	PROPN
cana-852	156	8	+	+	PROPN
cana-852	156	9	𝜆)𝑟(1−𝛼	𝜆)𝑟(1−𝛼	NUM
cana-852	156	10	)	)	PUNCT
cana-852	156	11	2	2	NUM
cana-852	156	12	−	−	NOUN
cana-852	156	13	𝛼	𝛼	PRON
cana-852	156	14	+	+	CCONJ
cana-852	156	15	(	(	PUNCT
cana-852	156	16	2𝑟𝛼−1	2𝑟𝛼−1	NOUN
cana-852	156	17	)	)	PUNCT
cana-852	156	18	𝛼	𝛼	NOUN
cana-852	156	19	−	−	PROPN
cana-852	156	20	1	1	NUM
cana-852	156	21	)	)	PUNCT
cana-852	156	22	+	+	NUM
cana-852	156	23	𝑟1−(𝛼+𝜆)𝐺2	𝑟1−(𝛼+𝜆)𝐺2	NOUN
cana-852	156	24	𝜔𝑛(2	𝜔𝑛(2	ADP
cana-852	156	25	−	−	NOUN
cana-852	156	26	𝛼	𝛼	INTJ
cana-852	156	27	−	−	NOUN
cana-852	156	28	𝜆	𝜆	NOUN
cana-852	156	29	)	)	PUNCT
cana-852	156	30	≤	≤	NOUN
cana-852	156	31	0	0	NUM
cana-852	156	32	.	.	PUNCT
cana-852	156	33	are	be	AUX
cana-852	156	34	solvable	solvable	ADJ
cana-852	156	35	.	.	PUNCT
cana-852	157	1	proof	proof	NOUN
cana-852	157	2	.	.	PUNCT
cana-852	158	1	let	let	VERB
cana-852	158	2	�	�	PRON
cana-852	158	3	⃗⃗⃗⃗	⃗⃗⃗⃗	NUM
cana-852	158	4	�	�	PROPN
cana-852	158	5	represent	represent	VERB
cana-852	158	6	the	the	DET
cana-852	158	7	solution	solution	NOUN
cana-852	158	8	of	of	ADP
cana-852	158	9	(	(	PUNCT
cana-852	158	10	2.1	2.1	NUM
cana-852	158	11	)	)	PUNCT
cana-852	158	12	,	,	PUNCT
cana-852	158	13	which	which	PRON
cana-852	158	14	is	be	AUX
cana-852	158	15	based	base	VERB
cana-852	158	16	on	on	ADP
cana-852	158	17	𝑊𝑎	𝑊𝑎	PROPN
cana-852	158	18	for	for	ADP
cana-852	158	19	each	each	DET
cana-852	158	20	𝑎	𝑎	PRON
cana-852	158	21	∈	∈	NOUN
cana-852	158	22	ℝ.	ℝ.	PROPN
cana-852	158	23	cauchy	cauchy	NOUN
cana-852	158	24	inequality	inequality	NOUN
cana-852	158	25	gives	give	VERB
cana-852	158	26	us	we	PRON
cana-852	158	27	𝜌2(𝑟	𝜌2(𝑟	NUM
cana-852	158	28	)	)	PUNCT
cana-852	158	29	=	=	PUNCT
cana-852	158	30	𝜔𝑛𝑟𝜆+𝛼−1	𝜔𝑛𝑟𝜆+𝛼−1	PROPN
cana-852	158	31	∫	∫	PROPN
cana-852	158	32	  	  	SPACE
cana-852	158	33	𝑆(𝑟	𝑆(𝑟	NOUN
cana-852	158	34	)	)	PUNCT
cana-852	158	35	𝑟𝛼−𝑛+𝜆	𝑟𝛼−𝑛+𝜆	CCONJ
cana-852	158	36	∥	∥	NUM
cana-852	158	37	𝑊	𝑊	NOUN
cana-852	158	38	∥2	∥2	NOUN
cana-852	158	39	𝑑𝑆	𝑑𝑆	ADJ
cana-852	158	40	(	(	PUNCT
cana-852	158	41	2.18	2.18	NUM
cana-852	158	42	)	)	PUNCT
cana-852	158	43	introducing	introduce	VERB
cana-852	158	44	the	the	DET
cana-852	158	45	notation	notation	NOUN
cana-852	158	46	𝑔	𝑔	PROPN
cana-852	158	47	=	=	PUNCT
cana-852	158	48	liminf𝑟𝜌(𝑟)	liminf𝑟𝜌(𝑟)	PROPN
cana-852	158	49	⬚	⬚	PROPN
cana-852	158	50	𝐺	𝐺	NOUN
cana-852	158	51	=	=	SYM
cana-852	158	52	limsup𝑟𝜌(𝑟	limsup𝑟𝜌(𝑟	PROPN
cana-852	158	53	)	)	PUNCT
cana-852	158	54	obviously	obviously	ADV
cana-852	158	55	,	,	PUNCT
cana-852	158	56	to	to	ADP
cana-852	158	57	any	any	DET
cana-852	158	58	0	0	PUNCT
cana-852	158	59	<	<	X
cana-852	158	60	𝜖	𝜖	X
cana-852	158	61	<	<	X
cana-852	158	62	𝑚𝑖𝑛{𝑔	𝑚𝑖𝑛{𝑔	NOUN
cana-852	158	63	,	,	PUNCT
cana-852	158	64	1	1	NUM
cana-852	158	65	−	−	PROPN
cana-852	158	66	𝐺	𝐺	NOUN
cana-852	158	67	}	}	PUNCT
cana-852	158	68	there	there	PRON
cana-852	158	69	exists	exist	VERB
cana-852	158	70	𝜏𝜖	𝜏𝜖	NOUN
cana-852	158	71	>	>	X
cana-852	158	72	𝑟0	𝑟0	NOUN
cana-852	158	73	and	and	CCONJ
cana-852	158	74	𝑟𝜖	𝑟𝜖	PRON
cana-852	158	75	>	>	X
cana-852	158	76	𝜏𝜖	𝜏𝜖	NOUN
cana-852	159	1	such	such	ADJ
cana-852	159	2	that	that	SCONJ
cana-852	159	3	𝑔−∈	𝑔−∈	NOUN
cana-852	159	4	<	<	X
cana-852	159	5	𝑟𝜌(𝑟	𝑟𝜌(𝑟	NOUN
cana-852	159	6	)	)	PUNCT
cana-852	159	7	<	<	X
cana-852	159	8	𝐺	𝐺	PROPN
cana-852	159	9	+	+	CCONJ
cana-852	159	10	𝜖	𝜖	X
cana-852	159	11	(	(	PUNCT
cana-852	159	12	2.19	2.19	NUM
cana-852	159	13	)	)	PUNCT
cana-852	159	14	communications	communication	NOUN
cana-852	159	15	on	on	ADP
cana-852	159	16	applied	apply	VERB
cana-852	159	17	nonlinear	nonlinear	ADJ
cana-852	159	18	analysis	analysis	NOUN
cana-852	159	19	issn	issn	NOUN
cana-852	159	20	:	:	PUNCT
cana-852	159	21	1074	1074	NUM
cana-852	159	22	-	-	PUNCT
cana-852	159	23	133x	133x	NUM
cana-852	159	24	vol	vol	NOUN
cana-852	159	25	31	31	NUM
cana-852	159	26	no	no	NOUN
cana-852	159	27	.	.	PUNCT
cana-852	160	1	4s	4s	NUM
cana-852	160	2	(	(	PUNCT
cana-852	160	3	2024	2024	NUM
cana-852	160	4	)	)	PUNCT
cana-852	160	5	293	293	NUM
cana-852	160	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-852	160	7	from	from	ADP
cana-852	160	8	(	(	PUNCT
cana-852	160	9	2.7	2.7	NUM
cana-852	160	10	)	)	PUNCT
cana-852	160	11	,	,	PUNCT
cana-852	160	12	(	(	PUNCT
cana-852	160	13	2.8	2.8	NUM
cana-852	160	14	)	)	PUNCT
cana-852	160	15	we	we	PRON
cana-852	160	16	easily	easily	ADV
cana-852	160	17	find	find	VERB
cana-852	160	18	that	that	SCONJ
cana-852	160	19	𝑟𝜌(𝑟	𝑟𝜌(𝑟	X
cana-852	160	20	)	)	PUNCT
cana-852	160	21	=	=	SYM
cana-852	161	1	𝑀(𝑟	𝑀(𝑟	NUM
cana-852	161	2	)	)	PUNCT
cana-852	161	3	−	−	PROPN
cana-852	162	1	(	(	PUNCT
cana-852	162	2	𝛼	𝛼	NOUN
cana-852	162	3	−	−	PROPN
cana-852	162	4	𝑛	𝑛	PROPN
cana-852	162	5	+	+	CCONJ
cana-852	162	6	𝜆)𝑟	𝜆)𝑟	X
cana-852	162	7	∫	∫	PROPN
cana-852	162	8	  	  	SPACE
cana-852	162	9	∞	∞	PROPN
cana-852	162	10	𝑟	𝑟	X
cana-852	162	11	𝜌(𝑠)𝑠−𝛼𝑑𝑠	𝜌(𝑠)𝑠−𝛼𝑑𝑠	NOUN
cana-852	163	1	+	+	NOUN
cana-852	163	2	𝑟	𝑟	NOUN
cana-852	163	3	𝜔𝑛	𝜔𝑛	ADP
cana-852	163	4	∫	∫	PROPN
cana-852	163	5	  	  	SPACE
cana-852	163	6	∞	∞	NOUN
cana-852	164	1	𝑟	𝑟	NOUN
cana-852	164	2	𝜌2(𝑠)𝑠1−(𝜆+𝛼)𝑑𝑠	𝜌2(𝑠)𝑠1−(𝜆+𝛼)𝑑𝑠	NOUN
cana-852	164	3	(	(	PUNCT
cana-852	164	4	2.20	2.20	NUM
cana-852	164	5	)	)	PUNCT
cana-852	164	6	taking	take	VERB
cana-852	164	7	the	the	DET
cana-852	164	8	new	new	ADJ
cana-852	164	9	account	account	NOUN
cana-852	164	10	and	and	CCONJ
cana-852	164	11	the	the	DET
cana-852	164	12	above	above	ADJ
cana-852	164	13	argument	argument	NOUN
cana-852	164	14	,	,	PUNCT
cana-852	164	15	we	we	PRON
cana-852	164	16	get	get	VERB
cana-852	164	17	𝑔	𝑔	PROPN
cana-852	164	18	−	−	PROPN
cana-852	164	19	𝜖	𝜖	X
cana-852	164	20	≥	≥	X
cana-852	164	21	𝑀(𝑟	𝑀(𝑟	NOUN
cana-852	164	22	)	)	PUNCT
cana-852	164	23	−	−	PROPN
cana-852	164	24	(	(	PUNCT
cana-852	164	25	𝛼	𝛼	NOUN
cana-852	164	26	−	−	NOUN
cana-852	164	27	𝑛	𝑛	PROPN
cana-852	164	28	+	+	CCONJ
cana-852	164	29	𝜆	𝜆	X
cana-852	164	30	)	)	PUNCT
cana-852	164	31	𝛼	𝛼	NOUN
cana-852	164	32	(	(	PUNCT
cana-852	164	33	𝑔	𝑔	PROPN
cana-852	164	34	−	−	PROPN
cana-852	164	35	𝜖	𝜖	PROPN
cana-852	164	36	)	)	PUNCT
cana-852	164	37	+	+	CCONJ
cana-852	164	38	𝑟1−(𝛼+𝜆)(𝑔	𝑟1−(𝛼+𝜆)(𝑔	NUM
cana-852	164	39	−	−	NUM
cana-852	164	40	𝜖)2	𝜖)2	NUM
cana-852	164	41	𝜔𝑛(𝜆	𝜔𝑛(𝜆	X
cana-852	164	42	+	+	CCONJ
cana-852	164	43	𝛼	𝛼	X
cana-852	164	44	)	)	PUNCT
cana-852	164	45	taking	take	VERB
cana-852	164	46	differentiation	differentiation	NOUN
cana-852	164	47	,	,	PUNCT
cana-852	164	48	multiply	multiply	ADV
cana-852	164	49	by	by	ADP
cana-852	164	50	𝑟2	𝑟2	NOUN
cana-852	164	51	and	and	CCONJ
cana-852	164	52	integrating	integrate	VERB
cana-852	164	53	from	from	ADP
cana-852	164	54	𝜏	𝜏	PRON
cana-852	164	55	→	→	SYM
cana-852	164	56	r	r	NOUN
cana-852	164	57	we	we	PRON
cana-852	164	58	get	get	VERB
cana-852	164	59	𝑅𝜌(𝑅	𝑅𝜌(𝑅	NOUN
cana-852	164	60	)	)	PUNCT
cana-852	165	1	=	=	NOUN
cana-852	165	2	𝑅−1(𝜏2𝜌(𝜏	𝑅−1(𝜏2𝜌(𝜏	NOUN
cana-852	165	3	)	)	PUNCT
cana-852	165	4	−	−	PROPN
cana-852	165	5	𝑅𝑁(𝑅	𝑅𝑁(𝑅	NOUN
cana-852	165	6	)	)	PUNCT
cana-852	166	1	+	+	CCONJ
cana-852	166	2	𝜏2𝑁(𝜏	𝜏2𝑁(𝜏	NOUN
cana-852	166	3	)	)	PUNCT
cana-852	166	4	)	)	PUNCT
cana-852	167	1	+	+	CCONJ
cana-852	167	2	𝑅−1	𝑅−1	NOUN
cana-852	167	3	∫	∫	PROPN
cana-852	167	4	  	  	SPACE
cana-852	167	5	𝑅	𝑅	PROPN
cana-852	167	6	𝜏	𝜏	PROPN
cana-852	167	7	 	 	SPACE
cana-852	167	8	2𝑠𝛼𝜌(𝑠)𝑑𝑠	2𝑠𝛼𝜌(𝑠)𝑑𝑠	NUM
cana-852	167	9	+	+	CCONJ
cana-852	167	10	𝑅−1	𝑅−1	NOUN
cana-852	167	11	∫	∫	PROPN
cana-852	167	12	  	  	SPACE
cana-852	167	13	𝑅	𝑅	PROPN
cana-852	167	14	𝜏	𝜏	PROPN
cana-852	167	15	  	  	SPACE
cana-852	167	16	(	(	PUNCT
cana-852	167	17	𝛼	𝛼	NOUN
cana-852	167	18	−	−	NOUN
cana-852	167	19	𝑛	𝑛	NOUN
cana-852	167	20	+	+	CCONJ
cana-852	168	1	𝜆)𝑠2−𝛼𝜌(𝑠)𝑑𝑠	𝜆)𝑠2−𝛼𝜌(𝑠)𝑑𝑠	NOUN
cana-852	168	2	−	−	NOUN
cana-852	168	3	𝑅−1	𝑅−1	X
cana-852	168	4	∫	∫	PROPN
cana-852	168	5	  	  	SPACE
cana-852	168	6	𝑅	𝑅	PROPN
cana-852	168	7	𝜏	𝜏	NOUN
cana-852	168	8	  	  	SPACE
cana-852	168	9	𝑠(3−𝜆−𝛼)𝜌2(𝑠	𝑠(3−𝜆−𝛼)𝜌2(𝑠	NOUN
cana-852	168	10	)	)	PUNCT
cana-852	168	11	𝜔𝑛	𝜔𝑛	ADP
cana-852	168	12	𝑑𝑠	𝑑𝑠	ADP
cana-852	168	13	here	here	ADV
cana-852	168	14	substituting	substitute	VERB
cana-852	168	15	𝑅	𝑅	PROPN
cana-852	168	16	=	=	PUNCT
cana-852	168	17	𝑟	𝑟	NOUN
cana-852	168	18	and	and	CCONJ
cana-852	168	19	𝜏	𝜏	X
cana-852	168	20	=	=	NOUN
cana-852	168	21	𝜏𝜖	𝜏𝜖	NOUN
cana-852	168	22	𝑟𝜌(𝑟	𝑟𝜌(𝑟	NOUN
cana-852	168	23	)	)	PUNCT
cana-852	169	1	=	=	SYM
cana-852	169	2	𝑟−1(𝜏𝜖	𝑟−1(𝜏𝜖	NUM
cana-852	169	3	2𝜌(𝜏𝜖	2𝜌(𝜏𝜖	NUM
cana-852	169	4	)	)	PUNCT
cana-852	170	1	−	−	PROPN
cana-852	171	1	𝑟𝑁(𝑟	𝑟𝑁(𝑟	NUM
cana-852	171	2	)	)	PUNCT
cana-852	171	3	+	+	NUM
cana-852	171	4	𝜏𝜖	𝜏𝜖	NOUN
cana-852	171	5	2𝑁(𝜏𝜖	2𝑁(𝜏𝜖	NUM
cana-852	171	6	)	)	PUNCT
cana-852	171	7	)	)	PUNCT
cana-852	172	1	+	+	CCONJ
cana-852	172	2	𝑟−1	𝑟−1	PROPN
cana-852	172	3	∫	∫	NOUN
cana-852	172	4	  	  	SPACE
cana-852	172	5	𝑟	𝑟	NOUN
cana-852	172	6	𝜏	𝜏	X
cana-852	172	7	 	 	SPACE
cana-852	172	8	2𝑠𝛼𝜌(𝑠)𝑑𝑠	2𝑠𝛼𝜌(𝑠)𝑑𝑠	NUM
cana-852	172	9	+	+	CCONJ
cana-852	172	10	𝑟−1	𝑟−1	PROPN
cana-852	172	11	∫	∫	NOUN
cana-852	172	12	  	  	SPACE
cana-852	172	13	𝑟	𝑟	NOUN
cana-852	172	14	𝜏	𝜏	X
cana-852	172	15	  	  	SPACE
cana-852	172	16	(	(	PUNCT
cana-852	172	17	𝛼	𝛼	NOUN
cana-852	172	18	−	−	NOUN
cana-852	172	19	𝑛	𝑛	NOUN
cana-852	172	20	+	+	CCONJ
cana-852	173	1	𝜆)𝑠2−𝛼𝜌(𝑠)𝑑𝑠	𝜆)𝑠2−𝛼𝜌(𝑠)𝑑𝑠	NOUN
cana-852	173	2	−	−	PROPN
cana-852	173	3	𝑟−1	𝑟−1	PROPN
cana-852	173	4	∫	∫	NOUN
cana-852	173	5	  	  	SPACE
cana-852	173	6	𝑟	𝑟	NOUN
cana-852	173	7	𝜏	𝜏	X
cana-852	173	8	  	  	SPACE
cana-852	173	9	𝑠(3−𝜆−𝛼)𝜌2(𝑠	𝑠(3−𝜆−𝛼)𝜌2(𝑠	NOUN
cana-852	173	10	)	)	PUNCT
cana-852	173	11	𝜔𝑛	𝜔𝑛	ADP
cana-852	173	12	𝑑𝑠	𝑑𝑠	X
cana-852	173	13	(	(	PUNCT
cana-852	173	14	2.21	2.21	NUM
cana-852	173	15	)	)	PUNCT
cana-852	173	16	𝐺	𝐺	NOUN
cana-852	173	17	+	+	CCONJ
cana-852	173	18	𝜖	𝜖	X
cana-852	173	19	≤𝑟−1(𝜏𝜖	≤𝑟−1(𝜏𝜖	ADP
cana-852	173	20	2𝜌(𝜏𝜖	2𝜌(𝜏𝜖	NUM
cana-852	173	21	)	)	PUNCT
cana-852	174	1	−	−	PROPN
cana-852	175	1	𝑟𝑁(𝑟	𝑟𝑁(𝑟	NUM
cana-852	175	2	)	)	PUNCT
cana-852	175	3	+	+	CCONJ
cana-852	175	4	𝜏𝜖𝑁(𝜏𝜖	𝜏𝜖𝑁(𝜏𝜖	NOUN
cana-852	175	5	)	)	PUNCT
cana-852	175	6	)	)	PUNCT
cana-852	176	1	+	+	CCONJ
cana-852	176	2	(	(	PUNCT
cana-852	176	3	𝐺	𝐺	PROPN
cana-852	176	4	+	+	CCONJ
cana-852	176	5	𝜖	𝜖	PROPN
cana-852	176	6	)	)	PUNCT
cana-852	176	7	(	(	PUNCT
cana-852	176	8	2𝑟(𝛼−1	2𝑟(𝛼−1	NUM
cana-852	176	9	)	)	PUNCT
cana-852	176	10	𝛼	𝛼	NOUN
cana-852	177	1	+	+	CCONJ
cana-852	177	2	(	(	PUNCT
cana-852	177	3	𝛼	𝛼	NOUN
cana-852	177	4	−	−	NOUN
cana-852	177	5	𝑛	𝑛	NOUN
cana-852	177	6	+	+	CCONJ
cana-852	178	1	𝜆)𝑟1−𝛼	𝜆)𝑟1−𝛼	NOUN
cana-852	178	2	2	2	NUM
cana-852	178	3	−	−	NOUN
cana-852	178	4	𝛼	𝛼	PRON
cana-852	178	5	−	−	PROPN
cana-852	178	6	(	(	PUNCT
cana-852	178	7	𝐺	𝐺	PROPN
cana-852	178	8	+	+	PROPN
cana-852	178	9	𝜖)𝑟𝑡1−(𝜆+𝛼	𝜖)𝑟𝑡1−(𝜆+𝛼	PROPN
cana-852	178	10	)	)	PUNCT
cana-852	178	11	𝜔𝑛(2	𝜔𝑛(2	PROPN
cana-852	178	12	−	−	PROPN
cana-852	178	13	𝛼	𝛼	INTJ
cana-852	178	14	−	−	NOUN
cana-852	178	15	𝜆	𝜆	NOUN
cana-852	178	16	)	)	PUNCT
cana-852	178	17	)	)	PUNCT
cana-852	178	18	hence	hence	ADV
cana-852	178	19	𝑀(𝑟	𝑀(𝑟	NUM
cana-852	178	20	)	)	PUNCT
cana-852	178	21	−	−	PROPN
cana-852	178	22	(	(	PUNCT
cana-852	178	23	(	(	PUNCT
cana-852	178	24	𝛼	𝛼	NOUN
cana-852	178	25	−	−	NOUN
cana-852	178	26	𝑛	𝑛	PROPN
cana-852	178	27	+	+	CCONJ
cana-852	178	28	𝜆	𝜆	X
cana-852	178	29	)	)	PUNCT
cana-852	178	30	𝛼	𝛼	NOUN
cana-852	178	31	+	+	NOUN
cana-852	178	32	1	1	NUM
cana-852	178	33	)	)	PUNCT
cana-852	178	34	𝑔	𝑔	NOUN
cana-852	178	35	+	+	NOUN
cana-852	178	36	𝑟1−(𝛼+𝜆)𝑔2	𝑟1−(𝛼+𝜆)𝑔2	NUM
cana-852	178	37	𝜔𝑛(𝜆	𝜔𝑛(𝜆	NUM
cana-852	178	38	+	+	CCONJ
cana-852	178	39	𝛼	𝛼	X
cana-852	178	40	)	)	PUNCT
cana-852	178	41	≤	≤	NOUN
cana-852	178	42	0	0	NUM
cana-852	178	43	and	and	CCONJ
cana-852	178	44	𝑟−1(𝑟𝑁(𝑟	𝑟−1(𝑟𝑁(𝑟	NUM
cana-852	178	45	)	)	PUNCT
cana-852	179	1	−	−	NUM
cana-852	179	2	𝜏𝜖𝑁(𝜏𝜖	𝜏𝜖𝑁(𝜏𝜖	NOUN
cana-852	179	3	)	)	PUNCT
cana-852	179	4	−	−	PROPN
cana-852	179	5	𝜏𝜖	𝜏𝜖	NOUN
cana-852	179	6	2𝜌(𝜏𝜖	2𝜌(𝜏𝜖	NUM
cana-852	179	7	)	)	PUNCT
cana-852	179	8	)	)	PUNCT
cana-852	180	1	−	−	PROPN
cana-852	180	2	𝐺	𝐺	NOUN
cana-852	180	3	(	(	PUNCT
cana-852	180	4	(	(	PUNCT
cana-852	180	5	𝛼	𝛼	NOUN
cana-852	180	6	−	−	PROPN
cana-852	180	7	𝑛	𝑛	PROPN
cana-852	180	8	+	+	PROPN
cana-852	180	9	𝜆)𝑟(1−𝛼	𝜆)𝑟(1−𝛼	NUM
cana-852	180	10	)	)	PUNCT
cana-852	180	11	2	2	NUM
cana-852	180	12	−	−	NOUN
cana-852	180	13	𝛼	𝛼	PRON
cana-852	180	14	+	+	CCONJ
cana-852	180	15	(	(	PUNCT
cana-852	180	16	2𝑟𝛼−1	2𝑟𝛼−1	NOUN
cana-852	180	17	)	)	PUNCT
cana-852	180	18	𝛼	𝛼	NOUN
cana-852	180	19	−	−	PROPN
cana-852	180	20	1	1	NUM
cana-852	180	21	)	)	PUNCT
cana-852	180	22	+	+	NUM
cana-852	180	23	𝑟1−(𝛼+𝜆)𝐺2	𝑟1−(𝛼+𝜆)𝐺2	NOUN
cana-852	180	24	𝜔𝑛(2	𝜔𝑛(2	ADP
cana-852	180	25	−	−	NOUN
cana-852	180	26	𝛼	𝛼	INTJ
cana-852	180	27	−	−	NOUN
cana-852	180	28	𝜆	𝜆	NOUN
cana-852	180	29	)	)	PUNCT
cana-852	180	30	≤	≤	NOUN
cana-852	180	31	0	0	NUM
cana-852	181	1	hence	hence	ADV
cana-852	181	2	it	it	PRON
cana-852	181	3	is	be	AUX
cana-852	181	4	proved	prove	VERB
cana-852	181	5	.	.	PUNCT
cana-852	182	1	the	the	DET
cana-852	182	2	hartman	hartman	PROPN
cana-852	182	3	winter	winter	PROPN
cana-852	182	4	theorem	theorem	PROPN
cana-852	182	5	and	and	CCONJ
cana-852	182	6	newly	newly	ADV
cana-852	182	7	discovered	discover	VERB
cana-852	182	8	oscillation	oscillation	NOUN
cana-852	182	9	requirements	requirement	NOUN
cana-852	182	10	for	for	ADP
cana-852	182	11	conformable	conformable	ADJ
cana-852	182	12	elliptic	elliptic	ADJ
cana-852	182	13	equations	equation	NOUN
cana-852	182	14	are	be	AUX
cana-852	182	15	covered	cover	VERB
cana-852	182	16	in	in	ADP
cana-852	182	17	the	the	DET
cana-852	182	18	following	follow	VERB
cana-852	182	19	session	session	NOUN
cana-852	182	20	.	.	PUNCT
cana-852	183	1	3	3	X
cana-852	183	2	.	.	X
cana-852	183	3	main	main	ADJ
cana-852	183	4	results	result	NOUN
cana-852	183	5	in	in	ADP
cana-852	183	6	this	this	DET
cana-852	183	7	section	section	NOUN
cana-852	183	8	,	,	PUNCT
cana-852	183	9	the	the	DET
cana-852	183	10	following	follow	VERB
cana-852	183	11	results	result	NOUN
cana-852	183	12	has	have	AUX
cana-852	183	13	been	be	AUX
cana-852	183	14	established	establish	VERB
cana-852	183	15	.	.	PUNCT
cana-852	184	1	the	the	DET
cana-852	184	2	below	below	ADJ
cana-852	184	3	theorem	theorem	NOUN
cana-852	184	4	is	be	AUX
cana-852	184	5	an	an	DET
cana-852	184	6	oscillation	oscillation	NOUN
cana-852	184	7	criterion	criterion	NOUN
cana-852	184	8	of	of	ADP
cana-852	184	9	the	the	DET
cana-852	184	10	hartman	hartman	PROPN
cana-852	184	11	winter	winter	PROPN
cana-852	184	12	type	type	NOUN
cana-852	184	13	.	.	PUNCT
cana-852	185	1	theorem	theorem	VERB
cana-852	185	2	3.1	3.1	NUM
cana-852	185	3	.	.	PUNCT
cana-852	186	1	if−∞	if−∞	PROPN
cana-852	186	2	<	<	X
cana-852	186	3	lim	lim	PROPN
cana-852	186	4	𝑟→∞	𝑟→∞	NUM
cana-852	186	5	 	 	SPACE
cana-852	186	6	inf𝑃(𝑟	inf𝑃(𝑟	PROPN
cana-852	186	7	)	)	PUNCT
cana-852	186	8	<	<	X
cana-852	186	9	lim	lim	PROPN
cana-852	186	10	𝑟→∞	𝑟→∞	NUM
cana-852	186	11	 	 	SPACE
cana-852	186	12	sup𝑃(𝑟	sup𝑃(𝑟	PROPN
cana-852	186	13	)	)	PUNCT
cana-852	186	14	≤	≤	NOUN
cana-852	186	15	∞	∞	PROPN
cana-852	186	16	(	(	PUNCT
cana-852	186	17	3.1	3.1	NUM
cana-852	186	18	)	)	PUNCT
cana-852	186	19	communications	communication	NOUN
cana-852	186	20	on	on	ADP
cana-852	186	21	applied	apply	VERB
cana-852	186	22	nonlinear	nonlinear	ADJ
cana-852	186	23	analysis	analysis	NOUN
cana-852	186	24	issn	issn	NOUN
cana-852	186	25	:	:	PUNCT
cana-852	186	26	1074	1074	NUM
cana-852	186	27	-	-	PUNCT
cana-852	186	28	133x	133x	NUM
cana-852	186	29	vol	vol	NOUN
cana-852	186	30	31	31	NUM
cana-852	186	31	no	no	NOUN
cana-852	186	32	.	.	PUNCT
cana-852	187	1	4s	4s	NUM
cana-852	187	2	(	(	PUNCT
cana-852	187	3	2024	2024	NUM
cana-852	187	4	)	)	PUNCT
cana-852	187	5	294	294	NUM
cana-852	187	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-852	187	7	or	or	CCONJ
cana-852	187	8	if	if	SCONJ
cana-852	187	9	lim	lim	PROPN
cana-852	187	10	𝑟→∞	𝑟→∞	NUM
cana-852	187	11	 	 	SPACE
cana-852	187	12	𝑃(𝑟	𝑃(𝑟	NOUN
cana-852	187	13	)	)	PUNCT
cana-852	187	14	=	=	SYM
cana-852	187	15	∞	∞	PROPN
cana-852	187	16	(	(	PUNCT
cana-852	187	17	3.2	3.2	NUM
cana-852	187	18	)	)	PUNCT
cana-852	187	19	then	then	ADV
cana-852	187	20	(	(	PUNCT
cana-852	187	21	1.1	1.1	NUM
cana-852	187	22	)	)	PUNCT
cana-852	187	23	is	be	AUX
cana-852	187	24	oscillatory	oscillatory	ADJ
cana-852	187	25	proof	proof	NOUN
cana-852	187	26	.	.	PUNCT
cana-852	188	1	assume	assume	VERB
cana-852	188	2	3.1	3.1	NUM
cana-852	188	3	is	be	AUX
cana-852	188	4	true	true	ADJ
cana-852	188	5	by	by	ADP
cana-852	188	6	contradiction	contradiction	NOUN
cana-852	188	7	and	and	CCONJ
cana-852	188	8	∃𝑟	∃𝑟	VERB
cana-852	188	9	there	there	PRON
cana-852	188	10	is	be	VERB
cana-852	188	11	a	a	DET
cana-852	188	12	non	non	ADJ
cana-852	188	13	negative	negative	ADJ
cana-852	188	14	solution	solution	NOUN
cana-852	188	15	of	of	ADP
cana-852	188	16	1.1	1.1	NUM
cana-852	188	17	on	on	ADP
cana-852	188	18	ω𝑟	ω𝑟	ADP
cana-852	188	19	exists	exist	NOUN
cana-852	188	20	.	.	PUNCT
cana-852	189	1	it	it	PRON
cana-852	189	2	follows	follow	VERB
cana-852	189	3	that	that	SCONJ
cana-852	189	4	the	the	DET
cana-852	189	5	riccati	riccati	PROPN
cana-852	189	6	equations	equation	NOUN
cana-852	189	7	corresponding	correspond	VERB
cana-852	189	8	solution	solution	NOUN
cana-852	189	9	is	be	AUX
cana-852	189	10	defined	define	VERB
cana-852	189	11	on	on	ADP
cana-852	189	12	𝑅.	𝑅.	SYM
cana-852	189	13	lemma	lemma	PROPN
cana-852	189	14	2.1	2.1	NUM
cana-852	189	15	’	'	PUNCT
cana-852	189	16	s	s	PART
cana-852	189	17	(	(	PUNCT
cana-852	189	18	iii	iii	X
cana-852	189	19	)	)	PUNCT
cana-852	189	20	⇒	⇒	NOUN
cana-852	189	21	(	(	PUNCT
cana-852	189	22	ii	ii	X
cana-852	189	23	)	)	PUNCT
cana-852	189	24	portion	portion	NOUN
cana-852	189	25	and	and	CCONJ
cana-852	189	26	the	the	DET
cana-852	189	27	first	first	ADJ
cana-852	189	28	inequality	inequality	NOUN
cana-852	189	29	in	in	ADP
cana-852	189	30	3.1	3.1	NUM
cana-852	189	31	indicates	indicate	VERB
cana-852	189	32	the	the	DET
cana-852	189	33	existence	existence	NOUN
cana-852	189	34	of	of	ADP
cana-852	189	35	a	a	DET
cana-852	189	36	finite	finite	ADJ
cana-852	189	37	limit	limit	NOUN
cana-852	189	38	lim𝑟→∞	lim𝑟→∞	PRON
cana-852	189	39	 	 	SPACE
cana-852	189	40	𝑃(𝑟	𝑃(𝑟	ADP
cana-852	189	41	)	)	PUNCT
cana-852	189	42	which	which	PRON
cana-852	189	43	is	be	AUX
cana-852	189	44	in	in	ADP
cana-852	189	45	opposition	opposition	NOUN
cana-852	189	46	to	to	ADP
cana-852	189	47	3.1	3.1	NUM
cana-852	189	48	.	.	PUNCT
cana-852	190	1	the	the	DET
cana-852	190	2	same	same	ADJ
cana-852	190	3	proof	proof	NOUN
cana-852	190	4	applies	apply	VERB
cana-852	190	5	to	to	ADP
cana-852	190	6	3.2	3.2	NUM
cana-852	190	7	theorem	theorem	VERB
cana-852	190	8	3.2	3.2	NUM
cana-852	190	9	.	.	PUNCT
cana-852	191	1	let	let	VERB
cana-852	191	2	equation	equation	NOUN
cana-852	191	3	(	(	PUNCT
cana-852	191	4	1.1	1.1	NUM
cana-852	191	5	)	)	PUNCT
cana-852	191	6	have	have	VERB
cana-852	191	7	oscillatory	oscillatory	ADJ
cana-852	191	8	solution	solution	NOUN
cana-852	191	9	𝑢.	𝑢.	NOUN
cana-852	191	10	then	then	ADV
cana-852	192	1	𝑀(𝑟	𝑀(𝑟	AUX
cana-852	192	2	)	)	PUNCT
cana-852	192	3	>	>	X
cana-852	193	1	𝐵	𝐵	PROPN
cana-852	193	2	+	+	CCONJ
cana-852	193	3	𝜖	𝜖	PROPN
cana-852	193	4	+	+	CCONJ
cana-852	193	5	(	(	PUNCT
cana-852	193	6	𝛼	𝛼	NOUN
cana-852	193	7	−	−	PROPN
cana-852	193	8	𝑛	𝑛	PROPN
cana-852	193	9	+	+	NUM
cana-852	193	10	𝜆)2𝜔𝑛	𝜆)2𝜔𝑛	ADJ
cana-852	193	11	4(𝛼	4(𝛼	NUM
cana-852	193	12	−	−	NOUN
cana-852	193	13	𝜆	𝜆	NOUN
cana-852	193	14	)	)	PUNCT
cana-852	193	15	𝑟𝜆−𝛼+1	𝑟𝜆−𝛼+1	NOUN
cana-852	193	16	and	and	CCONJ
cana-852	193	17	𝑁(𝑟	𝑁(𝑟	PROPN
cana-852	193	18	)	)	PUNCT
cana-852	193	19	>	>	X
cana-852	193	20	𝑟−1(𝑟𝜖	𝑟−1(𝑟𝜖	PROPN
cana-852	193	21	2𝜌(𝑟𝜖	2𝜌(𝑟𝜖	NUM
cana-852	193	22	)	)	PUNCT
cana-852	194	1	+	+	CCONJ
cana-852	194	2	𝑟𝜖	𝑟𝜖	PROPN
cana-852	194	3	2𝑁(𝑟𝜖	2𝑁(𝑟𝜖	NOUN
cana-852	194	4	)	)	PUNCT
cana-852	194	5	)	)	PUNCT
cana-852	195	1	−	−	PROPN
cana-852	196	1	𝜔𝑛𝑟−1	𝜔𝑛𝑟−1	NUM
cana-852	196	2	4	4	NUM
cana-852	196	3	(	(	PUNCT
cana-852	196	4	(	(	PUNCT
cana-852	196	5	𝛼	𝛼	NOUN
cana-852	196	6	−	−	PROPN
cana-852	196	7	𝑛	𝑛	PROPN
cana-852	196	8	+	+	CCONJ
cana-852	196	9	𝜆)2𝑟𝜆+2−𝛼	𝜆)2𝑟𝜆+2−𝛼	PROPN
cana-852	196	10	𝜆	𝜆	PRON
cana-852	196	11	−	−	PROPN
cana-852	196	12	𝛼	𝛼	NOUN
cana-852	196	13	+	+	NOUN
cana-852	196	14	2	2	NUM
cana-852	196	15	+	+	NUM
cana-852	196	16	4𝑟𝜆+3𝛼−2	4𝑟𝜆+3𝛼−2	NUM
cana-852	196	17	𝜆	𝜆	NOUN
cana-852	197	1	+	+	ADJ
cana-852	197	2	3𝛼	3𝛼	PRON
cana-852	197	3	−	−	NOUN
cana-852	197	4	2	2	NUM
cana-852	198	1	+	+	NUM
cana-852	198	2	4(𝛼	4(𝛼	NUM
cana-852	198	3	−	−	NOUN
cana-852	198	4	𝑛	𝑛	PROPN
cana-852	199	1	+	+	CCONJ
cana-852	200	1	𝜆)𝑟𝛼+𝜆	𝜆)𝑟𝛼+𝜆	NUM
cana-852	200	2	𝛼	𝛼	NOUN
cana-852	200	3	+	+	NOUN
cana-852	200	4	𝜆	𝜆	X
cana-852	200	5	)	)	PUNCT
cana-852	200	6	−	−	PROPN
cana-852	200	7	𝐴	𝐴	PROPN
cana-852	200	8	+	+	CCONJ
cana-852	200	9	𝜖	𝜖	PROPN
cana-852	200	10	are	be	AUX
cana-852	200	11	oscillatory	oscillatory	ADJ
cana-852	200	12	.	.	PUNCT
cana-852	201	1	moreover	moreover	ADV
cana-852	201	2	,	,	PUNCT
cana-852	201	3	lim	lim	PROPN
cana-852	201	4	 	 	SPACE
cana-852	201	5	inf	inf	PROPN
cana-852	201	6	𝑟𝜌(𝑟	𝑟𝜌(𝑟	NUM
cana-852	201	7	)	)	PUNCT
cana-852	201	8	≥	≥	PROPN
cana-852	201	9	𝐴	𝐴	PROPN
cana-852	201	10	,	,	PUNCT
cana-852	201	11	⬚	⬚	PROPN
cana-852	201	12	liminf𝑟𝜌(𝑟	liminf𝑟𝜌(𝑟	NOUN
cana-852	201	13	)	)	PUNCT
cana-852	201	14	≤	≤	NOUN
cana-852	201	15	𝐵	𝐵	NOUN
cana-852	201	16	where	where	SCONJ
cana-852	201	17	𝐴	𝐴	PROPN
cana-852	201	18	is	be	AUX
cana-852	201	19	the	the	DET
cana-852	201	20	least	least	ADJ
cana-852	201	21	non	non	ADJ
cana-852	201	22	-	-	ADJ
cana-852	201	23	negative	negative	ADJ
cana-852	201	24	root	root	NOUN
cana-852	201	25	of	of	ADP
cana-852	201	26	equation	equation	NOUN
cana-852	201	27	and	and	CCONJ
cana-852	201	28	𝐵	𝐵	NOUN
cana-852	201	29	is	be	AUX
cana-852	201	30	the	the	DET
cana-852	201	31	largest	large	ADJ
cana-852	201	32	root	root	NOUN
cana-852	201	33	of	of	ADP
cana-852	201	34	equation	equation	NOUN
cana-852	201	35	.	.	PUNCT
cana-852	202	1	proof	proof	NOUN
cana-852	202	2	.	.	PUNCT
cana-852	203	1	assume	assume	VERB
cana-852	203	2	the	the	DET
cana-852	203	3	contradiction	contradiction	NOUN
cana-852	203	4	.	.	PUNCT
cana-852	204	1	let	let	VERB
cana-852	204	2	equation	equation	NOUN
cana-852	204	3	(	(	PUNCT
cana-852	204	4	1.1	1.1	NUM
cana-852	204	5	)	)	PUNCT
cana-852	204	6	have	have	VERB
cana-852	204	7	the	the	DET
cana-852	204	8	non	non	ADJ
cana-852	204	9	-	-	ADJ
cana-852	204	10	oscillatory	oscillatory	ADJ
cana-852	204	11	solution	solution	NOUN
cana-852	204	12	.	.	PUNCT
cana-852	205	1	from	from	ADP
cana-852	205	2	lemma	lemma	PROPN
cana-852	205	3	2.1	2.1	NUM
cana-852	205	4	,	,	PUNCT
cana-852	205	5	(	(	PUNCT
cana-852	205	6	2.5	2.5	NUM
cana-852	205	7	)	)	PUNCT
cana-852	205	8	and	and	CCONJ
cana-852	205	9	(	(	PUNCT
cana-852	205	10	2.7	2.7	NUM
cana-852	205	11	)	)	PUNCT
cana-852	205	12	there	there	PRON
cana-852	205	13	exists	exist	VERB
cana-852	205	14	𝑟𝜖	𝑟𝜖	PROPN
cana-852	205	15	>	>	PUNCT
cana-852	205	16	𝑟0	𝑟0	NOUN
cana-852	205	17	such	such	ADJ
cana-852	205	18	that	that	SCONJ
cana-852	205	19	𝐴	𝐴	PROPN
cana-852	205	20	−	−	PROPN
cana-852	205	21	𝜖	𝜖	X
cana-852	205	22	<	<	X
cana-852	205	23	𝑟𝜌(𝑟	𝑟𝜌(𝑟	X
cana-852	205	24	)	)	PUNCT
cana-852	205	25	<	<	X
cana-852	205	26	𝐵	𝐵	PROPN
cana-852	205	27	+	+	CCONJ
cana-852	205	28	𝜖	𝜖	PROPN
cana-852	205	29	for	for	ADP
cana-852	205	30	𝑟	𝑟	X
cana-852	205	31	>	>	PUNCT
cana-852	205	32	𝑟𝜖	𝑟𝜖	X
cana-852	205	33	integrating	integrate	VERB
cana-852	205	34	from	from	ADP
cana-852	205	35	𝑟	𝑟	NOUN
cana-852	205	36	→	→	SYM
cana-852	205	37	∞	∞	PROPN
cana-852	205	38	and	and	CCONJ
cana-852	205	39	taking	take	VERB
cana-852	205	40	into	into	ADP
cana-852	205	41	account	account	NOUN
cana-852	205	42	of	of	ADP
cana-852	205	43	(	(	PUNCT
cana-852	205	44	2.5	2.5	NUM
cana-852	205	45	)	)	PUNCT
cana-852	205	46	and	and	CCONJ
cana-852	205	47	(	(	PUNCT
cana-852	205	48	2.7	2.7	NUM
cana-852	205	49	)	)	PUNCT
cana-852	205	50	we	we	PRON
cana-852	205	51	get	get	VERB
cana-852	205	52	that	that	PRON
cana-852	205	53	𝑟𝜌(𝑟	𝑟𝜌(𝑟	PUNCT
cana-852	205	54	)	)	PUNCT
cana-852	205	55	=	=	SYM
cana-852	206	1	𝑀(𝑟	𝑀(𝑟	NUM
cana-852	206	2	)	)	PUNCT
cana-852	206	3	−	−	PROPN
cana-852	207	1	(	(	PUNCT
cana-852	207	2	𝛼	𝛼	NOUN
cana-852	207	3	−	−	PROPN
cana-852	207	4	𝑛	𝑛	PROPN
cana-852	207	5	+	+	CCONJ
cana-852	207	6	𝜆)𝑟	𝜆)𝑟	X
cana-852	207	7	∫	∫	PROPN
cana-852	207	8	  	  	SPACE
cana-852	207	9	∞	∞	PROPN
cana-852	207	10	𝑟	𝑟	PRON
cana-852	207	11	 	 	SPACE
cana-852	207	12	𝜌(𝑠)𝑠−𝛼𝑑𝑠	𝜌(𝑠)𝑠−𝛼𝑑𝑠	NOUN
cana-852	207	13	+	+	NUM
cana-852	207	14	𝑟	𝑟	NOUN
cana-852	207	15	𝜔𝑛	𝜔𝑛	ADP
cana-852	207	16	∫	∫	PROPN
cana-852	207	17	  	  	SPACE
cana-852	207	18	∞	∞	PROPN
cana-852	208	1	𝑟	𝑟	NOUN
cana-852	208	2	 	 	SPACE
cana-852	208	3	𝜌2(𝑠)𝑠1−(𝜆+𝛼)𝑑𝑠	𝜌2(𝑠)𝑠1−(𝜆+𝛼)𝑑𝑠	PROPN
cana-852	208	4	𝑀(𝑟	𝑀(𝑟	NUM
cana-852	208	5	)	)	PUNCT
cana-852	208	6	=	=	SYM
cana-852	208	7	𝑟𝜌(𝑟	𝑟𝜌(𝑟	X
cana-852	208	8	)	)	PUNCT
cana-852	208	9	−	−	NOUN
cana-852	209	1	𝑟	𝑟	NOUN
cana-852	209	2	(	(	PUNCT
cana-852	209	3	𝑟	𝑟	NOUN
cana-852	209	4	𝜔𝑛	𝜔𝑛	ADP
cana-852	209	5	∫	∫	PROPN
cana-852	209	6	  	  	SPACE
cana-852	209	7	∞	∞	PROPN
cana-852	209	8	𝑟	𝑟	NOUN
cana-852	209	9	 	 	SPACE
cana-852	209	10	𝜌2(𝑠)𝑠1−(𝜆+𝛼)𝑑𝑠	𝜌2(𝑠)𝑠1−(𝜆+𝛼)𝑑𝑠	PROPN
cana-852	209	11	−	−	PROPN
cana-852	209	12	(	(	PUNCT
cana-852	209	13	𝛼	𝛼	NOUN
cana-852	209	14	−	−	NOUN
cana-852	209	15	𝑛	𝑛	PROPN
cana-852	209	16	+	+	CCONJ
cana-852	209	17	𝜆	𝜆	X
cana-852	209	18	)	)	PUNCT
cana-852	209	19	∫	∫	PROPN
cana-852	209	20	  	  	SPACE
cana-852	209	21	∞	∞	PROPN
cana-852	209	22	𝑟	𝑟	PRON
cana-852	209	23	 	 	SPACE
cana-852	209	24	𝜌(𝑠)𝑠−𝛼𝑑𝑠	𝜌(𝑠)𝑠−𝛼𝑑𝑠	NOUN
cana-852	209	25	)	)	PUNCT
cana-852	209	26	𝑀(𝑟	𝑀(𝑟	NOUN
cana-852	209	27	)	)	PUNCT
cana-852	209	28	<	<	X
cana-852	209	29	𝐵	𝐵	PROPN
cana-852	209	30	+	+	CCONJ
cana-852	209	31	𝜖	𝜖	PROPN
cana-852	209	32	+	+	CCONJ
cana-852	209	33	(	(	PUNCT
cana-852	209	34	𝛼	𝛼	NOUN
cana-852	209	35	−	−	PROPN
cana-852	209	36	𝑛	𝑛	PROPN
cana-852	209	37	+	+	NUM
cana-852	209	38	𝜆)2𝜔𝑛	𝜆)2𝜔𝑛	ADJ
cana-852	209	39	4(𝛼	4(𝛼	NUM
cana-852	209	40	−	−	NOUN
cana-852	209	41	𝜆	𝜆	NOUN
cana-852	209	42	)	)	PUNCT
cana-852	209	43	𝑟𝜆−𝛼+1	𝑟𝜆−𝛼+1	PROPN
cana-852	209	44	(	(	PUNCT
cana-852	209	45	3.3	3.3	NUM
cana-852	209	46	)	)	PUNCT
cana-852	209	47	similarly	similarly	ADV
cana-852	209	48	,	,	PUNCT
cana-852	209	49	from	from	ADP
cana-852	209	50	(	(	PUNCT
cana-852	209	51	2.5	2.5	NUM
cana-852	209	52	)	)	PUNCT
cana-852	209	53	𝑟𝑁(𝑟	𝑟𝑁(𝑟	NUM
cana-852	209	54	)	)	PUNCT
cana-852	210	1	=	=	NOUN
cana-852	210	2	𝜏𝜖	𝜏𝜖	NOUN
cana-852	210	3	2𝜌(𝜏𝜖	2𝜌(𝜏𝜖	NUM
cana-852	210	4	)	)	PUNCT
cana-852	211	1	+	+	CCONJ
cana-852	212	1	𝜏𝜖𝑁(𝜏𝜖	𝜏𝜖𝑁(𝜏𝜖	NOUN
cana-852	212	2	)	)	PUNCT
cana-852	213	1	−	−	NUM
cana-852	213	2	∫	∫	PROPN
cana-852	213	3	  	  	SPACE
cana-852	213	4	𝜏𝜖	𝜏𝜖	NOUN
cana-852	213	5	𝑟	𝑟	X
cana-852	213	6	 	 	SPACE
cana-852	213	7	(	(	PUNCT
cana-852	213	8	𝑠3−𝛼−𝜆𝜌2(𝑠	𝑠3−𝛼−𝜆𝜌2(𝑠	PROPN
cana-852	213	9	)	)	PUNCT
cana-852	213	10	𝜔𝑛	𝜔𝑛	NOUN
cana-852	213	11	−	−	PROPN
cana-852	213	12	𝜌(𝑠)((𝛼	𝜌(𝑠)((𝛼	NOUN
cana-852	213	13	−	−	NOUN
cana-852	213	14	𝑛	𝑛	PROPN
cana-852	213	15	+	+	CCONJ
cana-852	213	16	𝜆)𝑠2−𝛼	𝜆)𝑠2−𝛼	NOUN
cana-852	213	17	+	+	CCONJ
cana-852	213	18	2𝑠𝛼	2𝑠𝛼	NOUN
cana-852	213	19	)	)	PUNCT
cana-852	213	20	)	)	PUNCT
cana-852	213	21	𝑑𝑠	𝑑𝑠	ADP
cana-852	213	22	−	−	PROPN
cana-852	213	23	𝑟2𝜌(𝑟	𝑟2𝜌(𝑟	PROPN
cana-852	213	24	)	)	PUNCT
cana-852	213	25	𝑁(𝑟	𝑁(𝑟	PROPN
cana-852	213	26	)	)	PUNCT
cana-852	213	27	<	<	X
cana-852	213	28	𝑟−1(𝑟𝜖	𝑟−1(𝑟𝜖	PROPN
cana-852	213	29	2𝜌(𝑟𝜖	2𝜌(𝑟𝜖	NUM
cana-852	213	30	)	)	PUNCT
cana-852	214	1	+	+	CCONJ
cana-852	214	2	𝑟𝜖	𝑟𝜖	PROPN
cana-852	214	3	2𝑁(𝑟𝜖	2𝑁(𝑟𝜖	NOUN
cana-852	214	4	)	)	PUNCT
cana-852	214	5	)	)	PUNCT
cana-852	215	1	−	−	PROPN
cana-852	216	1	𝜔𝑛𝑟−1	𝜔𝑛𝑟−1	NUM
cana-852	216	2	4	4	NUM
cana-852	216	3	(	(	PUNCT
cana-852	216	4	(	(	PUNCT
cana-852	216	5	𝛼	𝛼	NOUN
cana-852	216	6	−	−	PROPN
cana-852	216	7	𝑛	𝑛	PROPN
cana-852	216	8	+	+	CCONJ
cana-852	216	9	𝜆)2𝑟𝜆+2−𝛼	𝜆)2𝑟𝜆+2−𝛼	PROPN
cana-852	216	10	𝜆	𝜆	PRON
cana-852	216	11	−	−	PROPN
cana-852	216	12	𝛼	𝛼	NOUN
cana-852	216	13	+	+	NOUN
cana-852	216	14	2	2	NUM
cana-852	216	15	+	+	NUM
cana-852	216	16	4𝑟𝜆+3𝛼−2	4𝑟𝜆+3𝛼−2	NUM
cana-852	216	17	𝜆	𝜆	NOUN
cana-852	217	1	+	+	ADJ
cana-852	217	2	3𝛼	3𝛼	PRON
cana-852	217	3	−	−	NOUN
cana-852	217	4	2	2	NUM
cana-852	218	1	+	+	NUM
cana-852	218	2	4(𝛼	4(𝛼	NUM
cana-852	218	3	−	−	NOUN
cana-852	218	4	𝑛	𝑛	PROPN
cana-852	219	1	+	+	CCONJ
cana-852	220	1	𝜆)𝑟𝛼+𝜆	𝜆)𝑟𝛼+𝜆	NUM
cana-852	220	2	𝛼	𝛼	NOUN
cana-852	220	3	+	+	NOUN
cana-852	220	4	𝜆	𝜆	X
cana-852	220	5	)	)	PUNCT
cana-852	221	1	−	−	PROPN
cana-852	221	2	𝐴	𝐴	PROPN
cana-852	221	3	+	+	CCONJ
cana-852	221	4	𝜖	𝜖	PROPN
cana-852	221	5	communications	communication	NOUN
cana-852	221	6	on	on	ADP
cana-852	221	7	applied	apply	VERB
cana-852	221	8	nonlinear	nonlinear	ADJ
cana-852	221	9	analysis	analysis	NOUN
cana-852	221	10	issn	issn	NOUN
cana-852	221	11	:	:	PUNCT
cana-852	221	12	1074	1074	NUM
cana-852	221	13	-	-	PUNCT
cana-852	221	14	133x	133x	NUM
cana-852	221	15	vol	vol	NOUN
cana-852	221	16	31	31	NUM
cana-852	221	17	no	no	NOUN
cana-852	221	18	.	.	PUNCT
cana-852	222	1	4s	4s	NUM
cana-852	222	2	(	(	PUNCT
cana-852	222	3	2024	2024	NUM
cana-852	222	4	)	)	PUNCT
cana-852	222	5	295	295	NUM
cana-852	222	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-852	222	7	which	which	PRON
cana-852	222	8	contradicts	contradict	VERB
cana-852	222	9	the	the	DET
cana-852	222	10	theorem	theorem	NOUN
cana-852	223	1	and	and	CCONJ
cana-852	223	2	it	it	PRON
cana-852	223	3	is	be	AUX
cana-852	223	4	proved	prove	VERB
cana-852	223	5	.	.	PUNCT
cana-852	224	1	theorem	theorem	VERB
cana-852	224	2	3.3	3.3	NUM
cana-852	224	3	.	.	PUNCT
cana-852	225	1	let	let	VERB
cana-852	225	2	equation	equation	NOUN
cana-852	225	3	(	(	PUNCT
cana-852	225	4	1.1	1.1	NUM
cana-852	225	5	)	)	PUNCT
cana-852	225	6	have	have	VERB
cana-852	225	7	oscillatory	oscillatory	ADJ
cana-852	225	8	solution	solution	NOUN
cana-852	225	9	𝑢.	𝑢.	NOUN
cana-852	225	10	then	then	ADV
cana-852	226	1	𝑀(𝑟	𝑀(𝑟	AUX
cana-852	226	2	)	)	PUNCT
cana-852	226	3	>	>	X
cana-852	227	1	𝐵	𝐵	PROPN
cana-852	227	2	+	+	CCONJ
cana-852	227	3	𝜖	𝜖	PROPN
cana-852	227	4	+	+	CCONJ
cana-852	227	5	(	(	PUNCT
cana-852	227	6	𝐴	𝐴	PROPN
cana-852	227	7	+	+	CCONJ
cana-852	227	8	𝜖	𝜖	PROPN
cana-852	227	9	)	)	PUNCT
cana-852	227	10	(	(	PUNCT
cana-852	227	11	(	(	PUNCT
cana-852	227	12	𝛼	𝛼	NOUN
cana-852	227	13	−	−	PROPN
cana-852	227	14	𝑛	𝑛	PROPN
cana-852	227	15	+	+	PROPN
cana-852	227	16	𝜆)𝑟(1−𝛼	𝜆)𝑟(1−𝛼	NUM
cana-852	227	17	)	)	PUNCT
cana-852	227	18	𝛼	𝛼	PROPN
cana-852	227	19	−	−	PROPN
cana-852	227	20	(	(	PUNCT
cana-852	227	21	𝐴	𝐴	PROPN
cana-852	227	22	+	+	CCONJ
cana-852	227	23	𝜖)𝑟1−(𝜆+𝛼	𝜖)𝑟1−(𝜆+𝛼	NOUN
cana-852	227	24	)	)	PUNCT
cana-852	227	25	𝜔𝑛(𝜆	𝜔𝑛(𝜆	PUNCT
cana-852	227	26	+	+	CCONJ
cana-852	227	27	𝛼	𝛼	X
cana-852	227	28	)	)	PUNCT
cana-852	227	29	)	)	PUNCT
cana-852	227	30	and	and	CCONJ
cana-852	227	31	𝑁(𝑟	𝑁(𝑟	PROPN
cana-852	227	32	)	)	PUNCT
cana-852	227	33	>	>	PUNCT
cana-852	228	1	𝜖	𝜖	PROPN
cana-852	228	2	−	−	PROPN
cana-852	228	3	𝐴	𝐴	PROPN
cana-852	228	4	+	+	CCONJ
cana-852	228	5	𝑟−1(𝑟𝜖	𝑟−1(𝑟𝜖	PROPN
cana-852	228	6	2𝜌(𝑟𝜖	2𝜌(𝑟𝜖	NUM
cana-852	228	7	)	)	PUNCT
cana-852	229	1	+	+	CCONJ
cana-852	229	2	𝑟𝜖𝑁(𝑟𝜖	𝑟𝜖𝑁(𝑟𝜖	NOUN
cana-852	229	3	)	)	PUNCT
cana-852	229	4	)	)	PUNCT
cana-852	230	1	+	+	CCONJ
cana-852	230	2	(	(	PUNCT
cana-852	230	3	𝐵	𝐵	NOUN
cana-852	230	4	+	+	NOUN
cana-852	230	5	𝜖)𝑟−1	𝜖)𝑟−1	PROPN
cana-852	230	6	(	(	PUNCT
cana-852	230	7	(	(	PUNCT
cana-852	230	8	𝛼	𝛼	NOUN
cana-852	230	9	−	−	NOUN
cana-852	230	10	𝑛	𝑛	PROPN
cana-852	230	11	+	+	CCONJ
cana-852	230	12	𝜆	𝜆	X
cana-852	230	13	)	)	PUNCT
cana-852	230	14	𝑟2−𝛼	𝑟2−𝛼	PROPN
cana-852	230	15	(	(	PUNCT
cana-852	230	16	2−𝛼	2−𝛼	NUM
cana-852	230	17	)	)	PUNCT
cana-852	231	1	+	+	CCONJ
cana-852	231	2	2𝑟𝛼	2𝑟𝛼	NOUN
cana-852	231	3	𝛼	𝛼	X
cana-852	231	4	−	−	PROPN
cana-852	231	5	(	(	PUNCT
cana-852	231	6	𝐵+𝜖)𝑟2−𝛼−𝜆	𝐵+𝜖)𝑟2−𝛼−𝜆	PROPN
cana-852	231	7	𝜔𝑛(2−𝛼−𝜆	𝜔𝑛(2−𝛼−𝜆	PROPN
cana-852	231	8	)	)	PUNCT
cana-852	231	9	)	)	PUNCT
cana-852	231	10	are	be	AUX
cana-852	231	11	oscillatory	oscillatory	ADJ
cana-852	231	12	.	.	PUNCT
cana-852	232	1	proof	proof	NOUN
cana-852	232	2	.	.	PUNCT
cana-852	233	1	assume	assume	VERB
cana-852	233	2	the	the	DET
cana-852	233	3	contradiction	contradiction	NOUN
cana-852	233	4	.	.	PUNCT
cana-852	234	1	let	let	VERB
cana-852	234	2	equation	equation	NOUN
cana-852	234	3	(	(	PUNCT
cana-852	234	4	1.1	1.1	NUM
cana-852	234	5	)	)	PUNCT
cana-852	234	6	have	have	VERB
cana-852	234	7	the	the	DET
cana-852	234	8	non	non	ADJ
cana-852	234	9	-	-	ADJ
cana-852	234	10	oscillatory	oscillatory	ADJ
cana-852	234	11	solution	solution	NOUN
cana-852	234	12	.	.	PUNCT
cana-852	235	1	by	by	ADP
cana-852	235	2	lemma	lemma	PROPN
cana-852	235	3	2.1	2.1	NUM
cana-852	235	4	,	,	PUNCT
cana-852	235	5	(	(	PUNCT
cana-852	235	6	2.5	2.5	NUM
cana-852	235	7	)	)	PUNCT
cana-852	235	8	and	and	CCONJ
cana-852	235	9	(	(	PUNCT
cana-852	235	10	2.7	2.7	NUM
cana-852	235	11	)	)	PUNCT
cana-852	235	12	𝑀(𝑟	𝑀(𝑟	NOUN
cana-852	235	13	)	)	PUNCT
cana-852	235	14	=	=	SYM
cana-852	235	15	𝑟𝜌(𝑟	𝑟𝜌(𝑟	NUM
cana-852	235	16	)	)	PUNCT
cana-852	235	17	)	)	PUNCT
cana-852	236	1	+	+	CCONJ
cana-852	236	2	(	(	PUNCT
cana-852	236	3	𝛼	𝛼	AUX
cana-852	236	4	−	−	PROPN
cana-852	236	5	𝑛	𝑛	PROPN
cana-852	236	6	+	+	CCONJ
cana-852	236	7	𝜆)𝑟	𝜆)𝑟	X
cana-852	236	8	∫	∫	PROPN
cana-852	236	9	  	  	SPACE
cana-852	236	10	∞	∞	PROPN
cana-852	236	11	𝑟	𝑟	PRON
cana-852	236	12	 	 	SPACE
cana-852	236	13	𝜌(𝑠)𝑠−𝛼𝑑𝑠	𝜌(𝑠)𝑠−𝛼𝑑𝑠	NOUN
cana-852	236	14	−	−	NOUN
cana-852	236	15	𝑟	𝑟	NOUN
cana-852	236	16	𝜔𝑛	𝜔𝑛	ADP
cana-852	236	17	∫	∫	PROPN
cana-852	236	18	  	  	SPACE
cana-852	236	19	∞	∞	PROPN
cana-852	236	20	𝑟	𝑟	NOUN
cana-852	236	21	 	 	SPACE
cana-852	236	22	𝜌(𝑠)2𝑠1−(𝜆+𝛼)𝑑𝑠	𝜌(𝑠)2𝑠1−(𝜆+𝛼)𝑑𝑠	X
cana-852	236	23	𝑀(𝑟	𝑀(𝑟	NOUN
cana-852	236	24	)	)	PUNCT
cana-852	236	25	<	<	X
cana-852	236	26	𝐵	𝐵	PROPN
cana-852	236	27	+	+	CCONJ
cana-852	236	28	𝜖	𝜖	PROPN
cana-852	236	29	+	+	CCONJ
cana-852	236	30	(	(	PUNCT
cana-852	236	31	𝐴	𝐴	PROPN
cana-852	236	32	+	+	CCONJ
cana-852	236	33	𝜖	𝜖	PROPN
cana-852	236	34	)	)	PUNCT
cana-852	236	35	(	(	PUNCT
cana-852	236	36	(	(	PUNCT
cana-852	236	37	𝛼	𝛼	NOUN
cana-852	236	38	−	−	PROPN
cana-852	236	39	𝑛	𝑛	PROPN
cana-852	236	40	+	+	PROPN
cana-852	236	41	𝜆)𝑟(1−𝛼	𝜆)𝑟(1−𝛼	NUM
cana-852	236	42	)	)	PUNCT
cana-852	237	1	𝛼	𝛼	PROPN
cana-852	237	2	−	−	PROPN
cana-852	237	3	(	(	PUNCT
cana-852	237	4	𝐴	𝐴	PROPN
cana-852	237	5	+	+	CCONJ
cana-852	237	6	𝜖)𝑟1−(𝜆+𝛼	𝜖)𝑟1−(𝜆+𝛼	NOUN
cana-852	237	7	)	)	PUNCT
cana-852	237	8	𝜔𝑛(𝜆	𝜔𝑛(𝜆	PUNCT
cana-852	237	9	+	+	CCONJ
cana-852	237	10	𝛼	𝛼	X
cana-852	237	11	)	)	PUNCT
cana-852	237	12	)	)	PUNCT
cana-852	238	1	(	(	PUNCT
cana-852	238	2	3.4	3.4	NUM
cana-852	238	3	)	)	PUNCT
cana-852	238	4	similarly	similarly	ADV
cana-852	238	5	,	,	PUNCT
cana-852	238	6	we	we	PRON
cana-852	238	7	obtain	obtain	VERB
cana-852	238	8	from	from	ADP
cana-852	238	9	(	(	PUNCT
cana-852	238	10	2.5	2.5	NUM
cana-852	238	11	)	)	PUNCT
cana-852	238	12	−𝑁(𝑟	−𝑁(𝑟	PROPN
cana-852	238	13	)	)	PUNCT
cana-852	238	14	=	=	SYM
cana-852	238	15	𝑡𝜌(𝑟	𝑡𝜌(𝑟	X
cana-852	238	16	)	)	PUNCT
cana-852	238	17	−	−	NUM
cana-852	238	18	𝑟−1(𝜏2𝜌(𝜏	𝑟−1(𝜏2𝜌(𝜏	NOUN
cana-852	238	19	)	)	PUNCT
cana-852	238	20	+	+	X
cana-852	238	21	𝜏𝑁(𝜏	𝜏𝑁(𝜏	NUM
cana-852	238	22	)	)	PUNCT
cana-852	238	23	)	)	PUNCT
cana-852	239	1	+	+	CCONJ
cana-852	239	2	𝑟−1	𝑟−1	PROPN
cana-852	239	3	∫	∫	NOUN
cana-852	239	4	  	  	SPACE
cana-852	239	5	𝑟	𝑟	NOUN
cana-852	240	1	𝜏	𝜏	X
cana-852	240	2	 	 	SPACE
cana-852	240	3	(	(	PUNCT
cana-852	240	4	𝑠(3−𝜆−𝛼)𝜌(𝑠)2	𝑠(3−𝜆−𝛼)𝜌(𝑠)2	PROPN
cana-852	240	5	𝜔𝑛	𝜔𝑛	NOUN
cana-852	240	6	−	−	PROPN
cana-852	240	7	(	(	PUNCT
cana-852	240	8	𝛼	𝛼	NOUN
cana-852	240	9	−	−	PROPN
cana-852	240	10	𝑛	𝑛	PROPN
cana-852	240	11	+	+	NUM
cana-852	240	12	𝜆)𝑠2−𝛼𝜌(𝑠	𝜆)𝑠2−𝛼𝜌(𝑠	NOUN
cana-852	240	13	)	)	PUNCT
cana-852	240	14	−	−	PROPN
cana-852	240	15	2𝑠𝛼𝜌(𝑠	2𝑠𝛼𝜌(𝑠	NUM
cana-852	240	16	)	)	PUNCT
cana-852	240	17	)	)	PUNCT
cana-852	240	18	𝑑𝑠	𝑑𝑠	ADP
cana-852	240	19	𝑁(𝑟	𝑁(𝑟	PROPN
cana-852	240	20	)	)	PUNCT
cana-852	240	21	<	<	X
cana-852	240	22	𝜖	𝜖	X
cana-852	240	23	−	−	PROPN
cana-852	240	24	𝐴	𝐴	PROPN
cana-852	240	25	+	+	CCONJ
cana-852	240	26	𝑟−1(𝑡𝜖	𝑟−1(𝑡𝜖	PROPN
cana-852	240	27	2𝜌(𝑟𝜖	2𝜌(𝑟𝜖	NUM
cana-852	240	28	)	)	PUNCT
cana-852	241	1	+	+	CCONJ
cana-852	241	2	𝑟𝜖𝑁(𝑟𝜖	𝑟𝜖𝑁(𝑟𝜖	NOUN
cana-852	241	3	)	)	PUNCT
cana-852	241	4	)	)	PUNCT
cana-852	242	1	⬚	⬚	PROPN
cana-852	242	2	+	+	CCONJ
cana-852	242	3	(	(	PUNCT
cana-852	242	4	𝐵	𝐵	NOUN
cana-852	242	5	+	+	X
cana-852	242	6	𝜖)𝑟−1	𝜖)𝑟−1	PROPN
cana-852	242	7	(	(	PUNCT
cana-852	242	8	(	(	PUNCT
cana-852	242	9	𝛼	𝛼	NOUN
cana-852	242	10	−	−	NOUN
cana-852	242	11	𝑛	𝑛	PROPN
cana-852	242	12	+	+	CCONJ
cana-852	242	13	𝜆	𝜆	X
cana-852	242	14	)	)	PUNCT
cana-852	242	15	𝑟2−𝛼	𝑟2−𝛼	PROPN
cana-852	242	16	(	(	PUNCT
cana-852	242	17	2	2	NUM
cana-852	242	18	−	−	NOUN
cana-852	242	19	𝛼	𝛼	NOUN
cana-852	242	20	)	)	PUNCT
cana-852	242	21	+	+	CCONJ
cana-852	242	22	2𝑟𝛼	2𝑟𝛼	ADJ
cana-852	242	23	𝛼	𝛼	X
cana-852	242	24	−	−	PROPN
cana-852	242	25	(	(	PUNCT
cana-852	242	26	𝐵	𝐵	NOUN
cana-852	242	27	+	+	CCONJ
cana-852	242	28	𝜖)𝑟2−𝛼−𝜆	𝜖)𝑟2−𝛼−𝜆	X
cana-852	242	29	𝜔𝑛(2	𝜔𝑛(2	ADP
cana-852	242	30	−	−	PROPN
cana-852	242	31	𝛼	𝛼	INTJ
cana-852	242	32	−	−	NOUN
cana-852	242	33	𝜆	𝜆	NOUN
cana-852	242	34	)	)	PUNCT
cana-852	242	35	)	)	PUNCT
cana-852	242	36	(	(	PUNCT
cana-852	242	37	3.5	3.5	NUM
cana-852	242	38	)	)	PUNCT
cana-852	242	39	which	which	PRON
cana-852	242	40	contradicts	contradict	VERB
cana-852	242	41	the	the	DET
cana-852	242	42	theorem	theorem	NOUN
cana-852	242	43	and	and	CCONJ
cana-852	242	44	it	it	PRON
cana-852	242	45	is	be	AUX
cana-852	242	46	proved	prove	VERB
cana-852	242	47	.	.	PUNCT
cana-852	243	1	4	4	X
cana-852	243	2	.	.	X
cana-852	243	3	conclusion	conclusion	NOUN
cana-852	243	4	using	use	VERB
cana-852	243	5	the	the	DET
cana-852	243	6	traditional	traditional	ADJ
cana-852	243	7	riccati	riccati	NOUN
cana-852	243	8	substitution	substitution	NOUN
cana-852	243	9	,	,	PUNCT
cana-852	243	10	this	this	DET
cana-852	243	11	work	work	NOUN
cana-852	243	12	presents	present	VERB
cana-852	243	13	some	some	DET
cana-852	243	14	oscillation	oscillation	NOUN
cana-852	243	15	results	result	NOUN
cana-852	243	16	for	for	ADP
cana-852	243	17	the	the	DET
cana-852	243	18	class	class	NOUN
cana-852	243	19	of	of	ADP
cana-852	243	20	conformable	conformable	ADJ
cana-852	243	21	schrodinger	schrodinger	NOUN
cana-852	243	22	equations	equation	NOUN
cana-852	243	23	.	.	PUNCT
cana-852	244	1	the	the	DET
cana-852	244	2	result	result	NOUN
cana-852	244	3	demonstrates	demonstrate	VERB
cana-852	244	4	that	that	SCONJ
cana-852	244	5	hartman	hartman	PROPN
cana-852	244	6	-	-	PUNCT
cana-852	244	7	winter	winter	NOUN
cana-852	244	8	criteria	criterion	NOUN
cana-852	244	9	may	may	AUX
cana-852	244	10	be	be	AUX
cana-852	244	11	effectively	effectively	ADV
cana-852	244	12	applied	apply	VERB
cana-852	244	13	in	in	ADP
cana-852	244	14	the	the	DET
cana-852	244	15	theorem	theorem	NOUN
cana-852	244	16	to	to	PART
cana-852	244	17	derive	derive	VERB
cana-852	244	18	oscillation	oscillation	NOUN
cana-852	244	19	criterion	criterion	NOUN
cana-852	244	20	.	.	PUNCT
cana-852	245	1	our	our	PRON
cana-852	245	2	newly	newly	ADV
cana-852	245	3	obtained	obtain	VERB
cana-852	245	4	results	result	NOUN
cana-852	245	5	in	in	ADP
cana-852	245	6	this	this	DET
cana-852	245	7	study	study	NOUN
cana-852	245	8	have	have	AUX
cana-852	245	9	improved	improve	VERB
cana-852	245	10	,	,	PUNCT
cana-852	245	11	extending	extend	VERB
cana-852	245	12	and	and	CCONJ
cana-852	245	13	adopting	adopt	VERB
cana-852	245	14	a	a	DET
cana-852	245	15	broad	broad	ADJ
cana-852	245	16	perspective	perspective	NOUN
cana-852	245	17	of	of	ADP
cana-852	245	18	certain	certain	ADJ
cana-852	245	19	known	know	VERB
cana-852	245	20	results	result	NOUN
cana-852	245	21	that	that	PRON
cana-852	245	22	are	be	AUX
cana-852	245	23	already	already	ADV
cana-852	245	24	there	there	ADV
cana-852	245	25	in	in	ADP
cana-852	245	26	the	the	DET
cana-852	245	27	literature	literature	NOUN
cana-852	245	28	.	.	PUNCT
cana-852	246	1	references	reference	NOUN
cana-852	246	2	[	[	X
cana-852	246	3	1	1	NUM
cana-852	246	4	]	]	PUNCT
cana-852	246	5	w.	w.	PROPN
cana-852	246	6	allegretto	allegretto	PROPN
cana-852	246	7	,	,	PUNCT
cana-852	246	8	on	on	ADP
cana-852	246	9	the	the	DET
cana-852	246	10	equivalence	equivalence	NOUN
cana-852	246	11	of	of	ADP
cana-852	246	12	two	two	NUM
cana-852	246	13	type	type	NOUN
cana-852	246	14	of	of	ADP
cana-852	246	15	oscillation	oscillation	NOUN
cana-852	246	16	for	for	ADP
cana-852	246	17	elliptic	elliptic	ADJ
cana-852	246	18	operators	operator	NOUN
cana-852	246	19	,	,	PUNCT
cana-852	246	20	pacific	pacific	PROPN
cana-852	246	21	.	.	PUNCT
cana-852	247	1	j.	j.	PROPN
cana-852	247	2	math	math	PROPN
cana-852	247	3	,	,	PUNCT
cana-852	247	4	55	55	NUM
cana-852	247	5	(	(	PUNCT
cana-852	247	6	1974	1974	NUM
cana-852	247	7	)	)	PUNCT
cana-852	247	8	,	,	PUNCT
cana-852	247	9	319	319	NUM
cana-852	247	10	-	-	SYM
cana-852	247	11	328	328	NUM
cana-852	247	12	.	.	PUNCT
cana-852	248	1	[	[	X
cana-852	248	2	2	2	NUM
cana-852	248	3	]	]	PUNCT
cana-852	248	4	a.	a.	NOUN
cana-852	248	5	atangana	atangana	PROPN
cana-852	248	6	,	,	PUNCT
cana-852	248	7	d.	d.	PROPN
cana-852	248	8	baleanu	baleanu	PROPN
cana-852	248	9	,	,	PUNCT
cana-852	248	10	a.	a.	NOUN
cana-852	248	11	alsaedi	alsaedi	PROPN
cana-852	248	12	,	,	PUNCT
cana-852	248	13	new	new	ADJ
cana-852	248	14	properties	property	NOUN
cana-852	248	15	of	of	ADP
cana-852	248	16	conformable	conformable	ADJ
cana-852	248	17	derivatives	derivative	NOUN
cana-852	248	18	,	,	PUNCT
cana-852	248	19	open	open	ADJ
cana-852	248	20	mathematics	mathematic	NOUN
cana-852	248	21	,	,	PUNCT
cana-852	248	22	7(2	7(2	NUM
cana-852	248	23	)	)	PUNCT
cana-852	248	24	(	(	PUNCT
cana-852	248	25	2015	2015	NUM
cana-852	248	26	)	)	PUNCT
cana-852	248	27	,	,	PUNCT
cana-852	248	28	889	889	NUM
cana-852	248	29	-	-	SYM
cana-852	248	30	898	898	NUM
cana-852	248	31	.	.	PUNCT
cana-852	249	1	[	[	X
cana-852	249	2	3	3	X
cana-852	249	3	]	]	X
cana-852	249	4	t.	t.	NOUN
cana-852	249	5	chantladze	chantladze	PROPN
cana-852	249	6	,	,	PUNCT
cana-852	249	7	n.	n.	PROPN
cana-852	249	8	kandelaki	kandelaki	PROPN
cana-852	249	9	,	,	PUNCT
cana-852	249	10	a.	a.	NOUN
cana-852	249	11	lomtatide	lomtatide	NOUN
cana-852	249	12	,	,	PUNCT
cana-852	249	13	oscillation	oscillation	NOUN
cana-852	249	14	and	and	CCONJ
cana-852	249	15	nonoscillation	nonoscillation	NOUN
cana-852	249	16	criteria	criterion	NOUN
cana-852	249	17	of	of	ADP
cana-852	249	18	a	a	DET
cana-852	249	19	second	second	ADJ
cana-852	249	20	order	order	NOUN
cana-852	249	21	linear	linear	NOUN
cana-852	249	22	equation	equation	NOUN
cana-852	249	23	,	,	PUNCT
cana-852	249	24	georgian	georgian	PROPN
cana-852	249	25	math	math	NOUN
cana-852	249	26	,	,	PUNCT
cana-852	249	27	6(5	6(5	NUM
cana-852	249	28	)	)	PUNCT
cana-852	249	29	(	(	PUNCT
cana-852	249	30	1999	1999	NUM
cana-852	249	31	)	)	PUNCT
cana-852	249	32	,	,	PUNCT
cana-852	249	33	401	401	NUM
cana-852	249	34	-	-	SYM
cana-852	249	35	414	414	NUM
cana-852	249	36	.	.	PUNCT
cana-852	250	1	[	[	X
cana-852	250	2	4	4	X
cana-852	250	3	]	]	X
cana-852	250	4	g.	g.	PROPN
cana-852	250	5	e.	e.	PROPN
cana-852	250	6	chatzarakis	chatzarakis	PROPN
cana-852	250	7	,	,	PUNCT
cana-852	250	8	m.	m.	NOUN
cana-852	250	9	deepa	deepa	PROPN
cana-852	250	10	,	,	PUNCT
cana-852	250	11	n.	n.	NOUN
cana-852	250	12	nagajothi	nagajothi	ADV
cana-852	250	13	and	and	CCONJ
cana-852	250	14	v.	v.	ADP
cana-852	250	15	sadhasivam	sadhasivam	NOUN
cana-852	250	16	,	,	PUNCT
cana-852	250	17	oscillatory	oscillatory	ADJ
cana-852	250	18	properties	property	NOUN
cana-852	250	19	of	of	ADP
cana-852	250	20	a	a	DET
cana-852	250	21	certain	certain	ADJ
cana-852	250	22	class	class	NOUN
cana-852	250	23	of	of	ADP
cana-852	250	24	mixed	mixed	ADJ
cana-852	250	25	fractional	fractional	ADJ
cana-852	250	26	differential	differential	ADJ
cana-852	250	27	equations	equation	NOUN
cana-852	250	28	,	,	PUNCT
cana-852	250	29	applied	apply	VERB
cana-852	250	30	mathematics	mathematic	NOUN
cana-852	250	31	and	and	CCONJ
cana-852	250	32	information	information	NOUN
cana-852	250	33	sciences	science	NOUN
cana-852	250	34	,	,	PUNCT
cana-852	250	35	14(1	14(1	NUM
cana-852	250	36	)	)	PUNCT
cana-852	250	37	(	(	PUNCT
cana-852	250	38	2020	2020	NUM
cana-852	250	39	)	)	PUNCT
cana-852	250	40	,	,	PUNCT
cana-852	250	41	109	109	NUM
cana-852	250	42	-	-	SYM
cana-852	250	43	117	117	NUM
cana-852	250	44	.	.	PUNCT
cana-852	251	1	communications	communication	NOUN
cana-852	251	2	on	on	ADP
cana-852	251	3	applied	apply	VERB
cana-852	251	4	nonlinear	nonlinear	ADJ
cana-852	251	5	analysis	analysis	NOUN
cana-852	251	6	issn	issn	NOUN
cana-852	251	7	:	:	PUNCT
cana-852	251	8	1074	1074	NUM
cana-852	251	9	-	-	PUNCT
cana-852	251	10	133x	133x	NUM
cana-852	251	11	vol	vol	NOUN
cana-852	251	12	31	31	NUM
cana-852	251	13	no	no	NOUN
cana-852	251	14	.	.	PUNCT
cana-852	252	1	4s	4s	NUM
cana-852	252	2	(	(	PUNCT
cana-852	252	3	2024	2024	NUM
cana-852	252	4	)	)	PUNCT
cana-852	252	5	296	296	NUM
cana-852	252	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-852	252	7	[	[	X
cana-852	252	8	5	5	NUM
cana-852	252	9	]	]	X
cana-852	252	10	chatzarakis	chatzaraki	NOUN
cana-852	252	11	,	,	PUNCT
cana-852	252	12	g.e	g.e	PROPN
cana-852	252	13	.	.	PROPN
cana-852	252	14	and	and	CCONJ
cana-852	252	15	logaarasi	logaarasi	PROPN
cana-852	252	16	,	,	PUNCT
cana-852	252	17	k.	k.	PROPN
cana-852	252	18	,	,	PUNCT
cana-852	252	19	2023	2023	NUM
cana-852	252	20	.	.	PUNCT
cana-852	253	1	forced	force	VERB
cana-852	253	2	oscillation	oscillation	NOUN
cana-852	253	3	of	of	ADP
cana-852	253	4	impulsive	impulsive	ADJ
cana-852	253	5	fractional	fractional	ADJ
cana-852	253	6	partial	partial	ADJ
cana-852	253	7	differential	differential	NOUN
cana-852	253	8	equations	equation	NOUN
cana-852	253	9	.	.	PUNCT
cana-852	254	1	partial	partial	ADJ
cana-852	254	2	differential	differential	ADJ
cana-852	254	3	equations	equation	NOUN
cana-852	254	4	in	in	ADP
cana-852	254	5	applied	applied	ADJ
cana-852	254	6	mathematics	mathematic	NOUN
cana-852	254	7	,	,	PUNCT
cana-852	254	8	7	7	NUM
cana-852	254	9	,	,	PUNCT
cana-852	254	10	p.100478	p.100478	NOUN
cana-852	254	11	.	.	PUNCT
cana-852	255	1	[	[	X
cana-852	255	2	6	6	NUM
cana-852	255	3	]	]	PUNCT
cana-852	255	4	g.	g.	PROPN
cana-852	255	5	e.	e.	PROPN
cana-852	255	6	chatzarakis	chatzarakis	PROPN
cana-852	255	7	,	,	PUNCT
cana-852	255	8	k.	k.	PROPN
cana-852	255	9	logaarasi	logaarasi	PROPN
cana-852	255	10	,	,	PUNCT
cana-852	255	11	t.	t.	PROPN
cana-852	255	12	raja	raja	PROPN
cana-852	255	13	,	,	PUNCT
cana-852	255	14	and	and	CCONJ
cana-852	255	15	v.	v.	ADP
cana-852	255	16	sadhasivam	sadhasivam	NOUN
cana-852	255	17	,	,	PUNCT
cana-852	255	18	on	on	ADP
cana-852	255	19	the	the	DET
cana-852	255	20	oscillation	oscillation	NOUN
cana-852	255	21	of	of	ADP
cana-852	255	22	conformable	conformable	ADJ
cana-852	255	23	impulsive	impulsive	ADJ
cana-852	255	24	vector	vector	NOUN
cana-852	255	25	partial	partial	ADJ
cana-852	255	26	differential	differential	NOUN
cana-852	255	27	equations	equation	NOUN
cana-852	255	28	,	,	PUNCT
cana-852	255	29	tatra	tatra	PROPN
cana-852	255	30	mt	mt	PROPN
cana-852	255	31	.	.	PROPN
cana-852	255	32	math	math	PROPN
cana-852	255	33	.	.	PUNCT
cana-852	256	1	publ	publ	PROPN
cana-852	256	2	.	.	PUNCT
cana-852	257	1	(	(	PUNCT
cana-852	257	2	2019	2019	NUM
cana-852	257	3	)	)	PUNCT
cana-852	257	4	.	.	PUNCT
cana-852	258	1	[	[	X
cana-852	258	2	7	7	X
cana-852	258	3	]	]	PUNCT
cana-852	258	4	f.fiedler	f.fiedler	NOUN
cana-852	258	5	,	,	PUNCT
cana-852	258	6	oscillation	oscillation	NOUN
cana-852	258	7	criteria	criterion	NOUN
cana-852	258	8	of	of	ADP
cana-852	258	9	nehari	nehari	NOUN
cana-852	258	10	-	-	PUNCT
cana-852	258	11	type	type	NOUN
cana-852	258	12	for	for	ADP
cana-852	258	13	sturm	sturm	NOUN
cana-852	258	14	-	-	PUNCT
cana-852	258	15	liouville	liouville	NOUN
cana-852	258	16	operators	operator	NOUN
cana-852	258	17	and	and	CCONJ
cana-852	258	18	elliptic	elliptic	ADJ
cana-852	258	19	operators	operator	NOUN
cana-852	258	20	of	of	ADP
cana-852	258	21	second	second	ADJ
cana-852	258	22	order	order	NOUN
cana-852	258	23	and	and	CCONJ
cana-852	258	24	lower	low	ADJ
cana-852	258	25	spectrum	spectrum	NOUN
cana-852	258	26	,	,	PUNCT
cana-852	258	27	proc	proc	NOUN
cana-852	258	28	.	.	PROPN
cana-852	259	1	of	of	ADP
cana-852	259	2	roy	roy	PROPN
cana-852	259	3	.	.	PROPN
cana-852	259	4	soc	soc	PROPN
cana-852	259	5	.	.	PUNCT
cana-852	260	1	of	of	ADP
cana-852	260	2	edinb	edinb	NOUN
cana-852	260	3	,	,	PUNCT
cana-852	260	4	109a	109a	NUM
cana-852	260	5	(	(	PUNCT
cana-852	260	6	1988	1988	NUM
cana-852	260	7	)	)	PUNCT
cana-852	260	8	127	127	NUM
cana-852	260	9	-	-	SYM
cana-852	260	10	144	144	NUM
cana-852	260	11	.	.	PUNCT
cana-852	261	1	[	[	X
cana-852	261	2	8	8	NUM
cana-852	261	3	]	]	PUNCT
cana-852	261	4	t.	t.	PROPN
cana-852	261	5	gayathri	gayathri	PROPN
cana-852	261	6	,	,	PUNCT
cana-852	261	7	m.	m.	NOUN
cana-852	261	8	deepa	deepa	PROPN
cana-852	261	9	,	,	PUNCT
cana-852	261	10	m.	m.	NOUN
cana-852	261	11	s.	s.	PROPN
cana-852	261	12	kumar	kumar	PROPN
cana-852	261	13	and	and	CCONJ
cana-852	261	14	v.	v.	ADP
cana-852	261	15	sadhasivam	sadhasivam	NOUN
cana-852	261	16	,	,	PUNCT
cana-852	261	17	hille	hille	PROPN
cana-852	261	18	and	and	CCONJ
cana-852	261	19	nehari	nehari	NOUN
cana-852	261	20	type	type	NOUN
cana-852	261	21	oscillation	oscillation	NOUN
cana-852	261	22	criteria	criterion	NOUN
cana-852	261	23	for	for	ADP
cana-852	261	24	conformable	conformable	ADJ
cana-852	261	25	fractional	fractional	ADJ
cana-852	261	26	differential	differential	NOUN
cana-852	261	27	equation	equation	NOUN
cana-852	261	28	,	,	PUNCT
cana-852	261	29	iraqi	iraqi	ADJ
cana-852	261	30	journal	journal	NOUN
cana-852	261	31	of	of	ADP
cana-852	261	32	science	science	NOUN
cana-852	261	33	,	,	PUNCT
cana-852	261	34	(	(	PUNCT
cana-852	261	35	2021	2021	NUM
cana-852	261	36	)	)	PUNCT
cana-852	261	37	,	,	PUNCT
cana-852	261	38	578	578	NUM
cana-852	261	39	-	-	SYM
cana-852	261	40	587	587	NUM
cana-852	261	41	.	.	PUNCT
cana-852	262	1	[	[	X
cana-852	262	2	9	9	NUM
cana-852	262	3	]	]	SYM
cana-852	262	4	h.hilfer	h.hilfer	NUM
cana-852	262	5	,	,	PUNCT
cana-852	262	6	applications	application	NOUN
cana-852	262	7	of	of	ADP
cana-852	262	8	fractional	fractional	ADJ
cana-852	262	9	calculus	calculus	NOUN
cana-852	262	10	in	in	ADP
cana-852	262	11	physics	physics	PROPN
cana-852	262	12	,	,	PUNCT
cana-852	262	13	world	world	NOUN
cana-852	262	14	scientific	scientific	ADJ
cana-852	262	15	publishing	publishing	NOUN
cana-852	262	16	company	company	NOUN
cana-852	262	17	,	,	PUNCT
cana-852	262	18	singapore	singapore	PROPN
cana-852	262	19	,	,	PUNCT
cana-852	262	20	2000	2000	NUM
cana-852	262	21	.	.	PUNCT
cana-852	263	1	[	[	X
cana-852	263	2	10	10	NUM
cana-852	263	3	]	]	X
cana-852	263	4	e.	e.	PROPN
cana-852	263	5	hille	hille	PROPN
cana-852	263	6	,	,	PUNCT
cana-852	263	7	non	non	ADJ
cana-852	263	8	-	-	ADJ
cana-852	263	9	oscillation	oscillation	ADJ
cana-852	263	10	theorems	theorem	NOUN
cana-852	263	11	,	,	PUNCT
cana-852	263	12	trans	trans	PROPN
cana-852	263	13	.	.	PROPN
cana-852	264	1	amer	amer	PROPN
cana-852	264	2	.	.	PUNCT
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cana-852	264	4	.	.	PUNCT
cana-852	265	1	soc	soc	PROPN
cana-852	265	2	.	.	PUNCT
cana-852	266	1	64	64	NUM
cana-852	266	2	(	(	PUNCT
cana-852	266	3	1948	1948	NUM
cana-852	266	4	)	)	PUNCT
cana-852	266	5	,	,	PUNCT
cana-852	266	6	234	234	NUM
cana-852	266	7	-	-	SYM
cana-852	266	8	252	252	NUM
cana-852	266	9	.	.	PUNCT
cana-852	267	1	[	[	X
cana-852	267	2	11	11	NUM
cana-852	267	3	]	]	PUNCT
cana-852	267	4	un	un	PROPN
cana-852	267	5	katugampola	katugampola	PROPN
cana-852	267	6	,	,	PUNCT
cana-852	267	7	a	a	DET
cana-852	267	8	new	new	ADJ
cana-852	267	9	fractional	fractional	ADJ
cana-852	267	10	derivative	derivative	NOUN
cana-852	267	11	with	with	ADP
cana-852	267	12	classical	classical	ADJ
cana-852	267	13	properties	property	NOUN
cana-852	267	14	,	,	PUNCT
cana-852	267	15	e	e	NOUN
cana-852	267	16	-	-	NOUN
cana-852	267	17	print	print	NOUN
cana-852	267	18	arxiv:14140.6535	arxiv:14140.6535	NOUN
cana-852	267	19	,	,	PUNCT
cana-852	267	20	(	(	PUNCT
cana-852	267	21	2014	2014	NUM
cana-852	267	22	)	)	PUNCT
cana-852	267	23	.	.	PUNCT
cana-852	268	1	[	[	X
cana-852	268	2	12	12	NUM
cana-852	268	3	]	]	PUNCT
cana-852	268	4	un	un	PROPN
cana-852	268	5	katugampola	katugampola	PROPN
cana-852	268	6	,	,	PUNCT
cana-852	268	7	a	a	DET
cana-852	268	8	new	new	ADJ
cana-852	268	9	approach	approach	NOUN
cana-852	268	10	to	to	ADP
cana-852	268	11	generalized	generalized	ADJ
cana-852	268	12	fractional	fractional	ADJ
cana-852	268	13	derivatives	derivative	NOUN
cana-852	268	14	,	,	PUNCT
cana-852	268	15	bull.math	bull.math	NOUN
cana-852	268	16	.	.	PUNCT
cana-852	269	1	anal	anal	PROPN
cana-852	269	2	.	.	PUNCT
cana-852	269	3	appl	appl	PROPN
cana-852	269	4	.	.	PROPN
cana-852	269	5	,	,	PUNCT
cana-852	269	6	6(4	6(4	NUM
cana-852	269	7	)	)	PUNCT
cana-852	269	8	(	(	PUNCT
cana-852	269	9	2014	2014	NUM
cana-852	269	10	)	)	PUNCT
cana-852	269	11	115	115	NUM
cana-852	269	12	.	.	PUNCT
cana-852	270	1	[	[	X
cana-852	270	2	13	13	NUM
cana-852	270	3	]	]	PUNCT
cana-852	270	4	r.	r.	PROPN
cana-852	270	5	r.	r.	PROPN
cana-852	270	6	khalil	khalil	PROPN
cana-852	270	7	,	,	PUNCT
cana-852	270	8	m.	m.	PROPN
cana-852	270	9	al	al	PROPN
cana-852	270	10	.	.	PROPN
cana-852	270	11	horani	horani	PROPN
cana-852	270	12	,	,	PUNCT
cana-852	270	13	a.	a.	PROPN
cana-852	270	14	yousef	yousef	PROPN
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cana-852	270	16	m.	m.	NOUN
cana-852	270	17	sababheh	sababheh	PROPN
cana-852	270	18	,	,	PUNCT
cana-852	270	19	a	a	DET
cana-852	270	20	new	new	ADJ
cana-852	270	21	definition	definition	NOUN
cana-852	270	22	of	of	ADP
cana-852	270	23	fractional	fractional	ADJ
cana-852	270	24	derivative	derivative	NOUN
cana-852	270	25	,	,	PUNCT
cana-852	270	26	j.	j.	PROPN
cana-852	270	27	com	com	PROPN
cana-852	270	28	.	.	PUNCT
cana-852	270	29	appl	appl	PROPN
cana-852	270	30	.	.	PROPN
cana-852	270	31	math	math	PROPN
cana-852	270	32	.	.	PUNCT
cana-852	270	33	,	,	PUNCT
cana-852	270	34	264	264	NUM
cana-852	270	35	(	(	PUNCT
cana-852	270	36	2014	2014	NUM
cana-852	270	37	)	)	PUNCT
cana-852	270	38	,	,	PUNCT
cana-852	270	39	65	65	NUM
cana-852	270	40	-	-	SYM
cana-852	270	41	70	70	NUM
cana-852	270	42	.	.	PUNCT
cana-852	271	1	[	[	X
cana-852	271	2	14	14	NUM
cana-852	271	3	]	]	SYM
cana-852	271	4	a.a.kilbas	a.a.kilbas	ADJ
cana-852	271	5	,	,	PUNCT
cana-852	271	6	h.m.srivastava	h.m.srivastava	NOUN
cana-852	271	7	and	and	CCONJ
cana-852	271	8	j.j.trujillo	j.j.trujillo	PROPN
cana-852	271	9	,	,	PUNCT
cana-852	271	10	theory	theory	NOUN
cana-852	271	11	and	and	CCONJ
cana-852	271	12	applications	application	NOUN
cana-852	271	13	of	of	ADP
cana-852	271	14	fractional	fractional	ADJ
cana-852	271	15	differential	differential	ADJ
cana-852	271	16	equations	equation	NOUN
cana-852	271	17	,	,	PUNCT
cana-852	271	18	elsevier	elsevier	PROPN
cana-852	271	19	science	science	PROPN
cana-852	271	20	b.v	b.v	PROPN
cana-852	271	21	.	.	PROPN
cana-852	271	22	,	,	PUNCT
cana-852	271	23	amsterdam	amsterdam	PROPN
cana-852	271	24	,	,	PUNCT
cana-852	271	25	the	the	DET
cana-852	271	26	netherlands	netherlands	PROPN
cana-852	271	27	,	,	PUNCT
cana-852	271	28	204	204	NUM
cana-852	271	29	(	(	PUNCT
cana-852	271	30	2006	2006	NUM
cana-852	271	31	)	)	PUNCT
cana-852	271	32	.	.	PUNCT
cana-852	272	1	[	[	X
cana-852	272	2	15	15	NUM
cana-852	272	3	]	]	X
cana-852	272	4	a.	a.	NOUN
cana-852	272	5	lomtatidze	lomtatidze	PROPN
cana-852	272	6	,	,	PUNCT
cana-852	272	7	oscillation	oscillation	NOUN
cana-852	272	8	and	and	CCONJ
cana-852	272	9	non	non	ADJ
cana-852	272	10	-	-	ADJ
cana-852	272	11	oscillation	oscillation	ADJ
cana-852	272	12	criteria	criterion	NOUN
cana-852	272	13	for	for	ADP
cana-852	272	14	secondorder	secondorder	ADJ
cana-852	272	15	linear	linear	PROPN
cana-852	272	16	differential	differential	NOUN
cana-852	272	17	equations	equation	NOUN
cana-852	272	18	,	,	PUNCT
cana-852	272	19	georgian	georgian	PROPN
cana-852	272	20	mathematical	mathematical	ADJ
cana-852	272	21	journal	journal	NOUN
cana-852	272	22	,	,	PUNCT
cana-852	272	23	4(2	4(2	NUM
cana-852	272	24	)	)	PUNCT
cana-852	272	25	(	(	PUNCT
cana-852	272	26	1997	1997	NUM
cana-852	272	27	)	)	PUNCT
cana-852	272	28	,	,	PUNCT
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cana-852	272	30	-	-	SYM
cana-852	272	31	138	138	NUM
cana-852	272	32	.	.	PUNCT
cana-852	273	1	[	[	X
cana-852	273	2	16	16	NUM
cana-852	273	3	]	]	X
cana-852	273	4	robert	robert	PROPN
cana-852	273	5	marik	marik	PROPN
cana-852	273	6	,	,	PUNCT
cana-852	273	7	oscillation	oscillation	NOUN
cana-852	273	8	criteria	criterion	NOUN
cana-852	273	9	for	for	ADP
cana-852	273	10	the	the	DET
cana-852	273	11	schrodinger	schrodinger	PROPN
cana-852	273	12	pde	pde	PROPN
cana-852	273	13	,	,	PUNCT
cana-852	273	14	adv	adv	PROPN
cana-852	273	15	.	.	PUNCT
cana-852	273	16	math	math	PROPN
cana-852	273	17	.	.	PUNCT
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cana-852	274	2	.	.	PUNCT
cana-852	274	3	appl	appl	PROPN
cana-852	274	4	.	.	PROPN
cana-852	274	5	,	,	PUNCT
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cana-852	274	7	(	(	PUNCT
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cana-852	274	9	)	)	PUNCT
cana-852	274	10	,	,	PUNCT
cana-852	274	11	491511	491511	NUM
cana-852	274	12	.	.	PUNCT
cana-852	275	1	[	[	X
cana-852	275	2	17	17	NUM
cana-852	275	3	]	]	PUNCT
cana-852	275	4	v.	v.	CCONJ
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cana-852	275	6	,	,	PUNCT
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cana-852	275	10	k.	k.	PROPN
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cana-852	275	12	,	,	PUNCT
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cana-852	275	15	oscillation	oscillation	NOUN
cana-852	275	16	of	of	ADP
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cana-852	275	18	fractional	fractional	ADJ
cana-852	275	19	differential	differential	ADJ
cana-852	275	20	nonlinear	nonlinear	PROPN
cana-852	275	21	differential	differential	ADJ
cana-852	275	22	equations	equation	NOUN
cana-852	275	23	,	,	PUNCT
cana-852	275	24	international	international	ADJ
cana-852	275	25	journal	journal	NOUN
cana-852	275	26	of	of	ADP
cana-852	275	27	mathematical	mathematical	ADJ
cana-852	275	28	archive	archive	NOUN
cana-852	275	29	,	,	PUNCT
cana-852	275	30	9	9	NUM
cana-852	275	31	(	(	PUNCT
cana-852	275	32	13	13	NUM
cana-852	275	33	)	)	PUNCT
cana-852	275	34	(	(	PUNCT
cana-852	275	35	2018	2018	NUM
cana-852	275	36	)	)	PUNCT
cana-852	275	37	,	,	PUNCT
cana-852	275	38	189	189	NUM
cana-852	275	39	-	-	SYM
cana-852	275	40	193	193	NUM
cana-852	275	41	.	.	PUNCT
cana-852	276	1	[	[	X
cana-852	276	2	18	18	NUM
cana-852	276	3	]	]	X
cana-852	276	4	c.	c.	PROPN
cana-852	276	5	swanson	swanson	PROPN
cana-852	276	6	,	,	PUNCT
cana-852	276	7	comparison	comparison	NOUN
cana-852	276	8	and	and	CCONJ
cana-852	276	9	oscillation	oscillation	NOUN
cana-852	276	10	theory	theory	NOUN
cana-852	276	11	of	of	ADP
cana-852	276	12	linear	linear	PROPN
cana-852	276	13	differential	differential	ADJ
cana-852	276	14	equations	equation	NOUN
cana-852	276	15	,	,	PUNCT
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cana-852	276	17	press	press	NOUN
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cana-852	276	21	,	,	PUNCT
cana-852	276	22	(	(	PUNCT
cana-852	276	23	1968	1968	NUM
cana-852	276	24	)	)	PUNCT
cana-852	276	25	.	.	PUNCT
cana-852	277	1	[	[	X
cana-852	277	2	19	19	NUM
cana-852	277	3	]	]	PUNCT
cana-852	277	4	j.	j.	PROPN
cana-852	277	5	wu	wu	PROPN
cana-852	277	6	,	,	PUNCT
cana-852	277	7	theory	theory	NOUN
cana-852	277	8	and	and	CCONJ
cana-852	277	9	applications	application	NOUN
cana-852	277	10	of	of	ADP
cana-852	277	11	partial	partial	ADJ
cana-852	277	12	functional	functional	ADJ
cana-852	277	13	differential	differential	NOUN
cana-852	277	14	equations	equation	NOUN
cana-852	277	15	,	,	PUNCT
cana-852	277	16	springer	springer	NOUN
cana-852	277	17	,	,	PUNCT
cana-852	277	18	newyork	newyork	PROPN
cana-852	277	19	,	,	PUNCT
cana-852	277	20	(	(	PUNCT
cana-852	277	21	1996	1996	NUM
cana-852	277	22	)	)	PUNCT
cana-852	277	23	.	.	PUNCT
cana-852	278	1	[	[	X
cana-852	278	2	20	20	NUM
cana-852	278	3	]	]	X
cana-852	278	4	n.	n.	PROPN
cana-852	278	5	yoshida	yoshida	PROPN
cana-852	278	6	,	,	PUNCT
cana-852	278	7	oscillation	oscillation	NOUN
cana-852	278	8	theory	theory	NOUN
cana-852	278	9	of	of	ADP
cana-852	278	10	partial	partial	ADJ
cana-852	278	11	differential	differential	NOUN
cana-852	278	12	equations	equation	NOUN
cana-852	278	13	,	,	PUNCT
cana-852	278	14	world	world	NOUN
cana-852	278	15	scientific	scientific	PROPN
cana-852	278	16	,	,	PUNCT
cana-852	278	17	singapore	singapore	PROPN
cana-852	278	18	,	,	PUNCT
cana-852	278	19	(	(	PUNCT
cana-852	278	20	2008	2008	NUM
cana-852	278	21	)	)	PUNCT
cana-852	278	22	.	.	PUNCT
cana-852	279	1	[	[	X
cana-852	279	2	21	21	NUM
cana-852	279	3	]	]	X
cana-852	279	4	noussair	noussair	PROPN
cana-852	279	5	,	,	PUNCT
cana-852	279	6	e.	e.	PROPN
cana-852	279	7	,	,	PUNCT
cana-852	279	8	swanson	swanson	PROPN
cana-852	279	9	,	,	PUNCT
cana-852	279	10	c.	c.	PROPN
cana-852	279	11	,	,	PUNCT
cana-852	279	12	oscillation	oscillation	NOUN
cana-852	279	13	theory	theory	NOUN
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cana-852	279	15	semilinear	semilinear	NOUN
cana-852	279	16	schrodingerequations	schrodingerequation	NOUN
cana-852	279	17	and	and	CCONJ
cana-852	279	18	inequalities	inequality	NOUN
cana-852	279	19	.	.	PUNCT
cana-852	280	1	proceedings	proceeding	NOUN
cana-852	280	2	of	of	ADP
cana-852	280	3	the	the	DET
cana-852	280	4	royal	royal	ADJ
cana-852	280	5	society	society	NOUN
cana-852	280	6	of	of	ADP
cana-852	280	7	edinburgh	edinburgh	PROPN
cana-852	280	8	section	section	PROPN
cana-852	280	9	a	a	DET
cana-852	280	10	:	:	PUNCT
cana-852	280	11	mathematics	mathematic	NOUN
cana-852	280	12	,	,	PUNCT
cana-852	280	13	75(1	75(1	NOUN
cana-852	280	14	)	)	PUNCT
cana-852	280	15	(	(	PUNCT
cana-852	280	16	1976	1976	NUM
cana-852	280	17	)	)	PUNCT
cana-852	280	18	,	,	PUNCT
cana-852	280	19	67	67	NUM
cana-852	280	20	-	-	SYM
cana-852	280	21	81	81	NUM
cana-852	280	22	.	.	PUNCT
cana-852	281	1	[	[	X
cana-852	281	2	22	22	NUM
cana-852	281	3	]	]	PUNCT
cana-852	281	4	muller	muller	PROPN
cana-852	281	5	-	-	PUNCT
cana-852	281	6	pfeiffer	pfeiffer	PROPN
cana-852	281	7	,	,	PUNCT
cana-852	281	8	e.	e.	PROPN
cana-852	281	9	oscillation	oscillation	PROPN
cana-852	281	10	criteria	criterion	NOUN
cana-852	281	11	of	of	ADP
cana-852	281	12	nehari	nehari	NOUN
cana-852	281	13	-	-	PUNCT
cana-852	281	14	type	type	NOUN
cana-852	281	15	for	for	ADP
cana-852	281	16	the	the	DET
cana-852	281	17	schrodinger	schrodinger	PROPN
cana-852	281	18	equation	equation	NOUN
cana-852	281	19	,	,	PUNCT
cana-852	281	20	mathematische	mathematische	NOUN
cana-852	281	21	nachrichten	nachrichten	NOUN
cana-852	281	22	,	,	PUNCT
cana-852	281	23	96(1	96(1	NUM
cana-852	281	24	)	)	PUNCT
cana-852	281	25	(	(	PUNCT
cana-852	281	26	1980	1980	NUM
cana-852	281	27	)	)	PUNCT
cana-852	281	28	,	,	PUNCT
cana-852	281	29	185	185	NUM
cana-852	281	30	-	-	SYM
cana-852	281	31	194	194	NUM
cana-852	281	32	.	.	PUNCT
cana-852	282	1	[	[	X
cana-852	282	2	23	23	NUM
cana-852	282	3	]	]	X
cana-852	282	4	onose	onose	PROPN
cana-852	282	5	,	,	PUNCT
cana-852	282	6	hiroshi	hiroshi	PROPN
cana-852	282	7	,	,	PUNCT
cana-852	282	8	oscillation	oscillation	NOUN
cana-852	282	9	criteria	criterion	NOUN
cana-852	282	10	for	for	ADP
cana-852	282	11	the	the	DET
cana-852	282	12	sub	sub	ADJ
cana-852	282	13	-	-	ADJ
cana-852	282	14	linear	linear	ADJ
cana-852	282	15	schrodinger	schrodinger	NOUN
cana-852	282	16	equation	equation	NOUN
cana-852	282	17	,	,	PUNCT
cana-852	282	18	proceedings	proceeding	NOUN
cana-852	282	19	of	of	ADP
cana-852	282	20	the	the	DET
cana-852	282	21	american	american	PROPN
cana-852	282	22	mathematical	mathematical	PROPN
cana-852	282	23	society	society	NOUN
cana-852	282	24	,	,	PUNCT
cana-852	282	25	85(1	85(1	NOUN
cana-852	282	26	)	)	PUNCT
cana-852	282	27	(	(	PUNCT
cana-852	282	28	1982	1982	NUM
cana-852	282	29	)	)	PUNCT
cana-852	282	30	,	,	PUNCT
cana-852	282	31	69	69	NUM
cana-852	282	32	-	-	SYM
cana-852	282	33	72	72	NUM
cana-852	282	34	.	.	PUNCT
cana-852	283	1	[	[	X
cana-852	283	2	24	24	NUM
cana-852	283	3	]	]	PUNCT
cana-852	283	4	c.	c.	PROPN
cana-852	283	5	swanson	swanson	PROPN
cana-852	283	6	,	,	PUNCT
cana-852	283	7	criteria	criterion	NOUN
cana-852	283	8	for	for	ADP
cana-852	283	9	oscillatory	oscillatory	ADJ
cana-852	283	10	sub	sub	ADJ
cana-852	283	11	-	-	ADJ
cana-852	283	12	linear	linear	ADJ
cana-852	283	13	schrodinger	schrodinger	PROPN
cana-852	283	14	equations	equation	NOUN
cana-852	283	15	,	,	PUNCT
cana-852	283	16	pacific	pacific	PROPN
cana-852	283	17	journal	journal	NOUN
cana-852	283	18	of	of	ADP
cana-852	283	19	mathematics	mathematic	NOUN
cana-852	283	20	,	,	PUNCT
cana-852	283	21	104(2	104(2	NUM
cana-852	283	22	)	)	PUNCT
cana-852	283	23	(	(	PUNCT
cana-852	283	24	1983	1983	NUM
cana-852	283	25	)	)	PUNCT
cana-852	283	26	,	,	PUNCT
cana-852	283	27	483	483	NUM
cana-852	283	28	-	-	SYM
cana-852	283	29	493	493	NUM
cana-852	283	30	.	.	PUNCT
cana-852	284	1	[	[	X
cana-852	284	2	25	25	NUM
cana-852	284	3	]	]	X
cana-852	284	4	ming	ming	PROPN
cana-852	284	5	-	-	PUNCT
cana-852	284	6	po	po	NOUN
cana-852	284	7	chen	chen	PROPN
cana-852	284	8	,	,	PUNCT
cana-852	284	9	b.g.zhang	b.g.zhang	PROPN
cana-852	284	10	,	,	PUNCT
cana-852	284	11	oscillation	oscillation	NOUN
cana-852	284	12	criteria	criterion	NOUN
cana-852	284	13	for	for	ADP
cana-852	284	14	a	a	DET
cana-852	284	15	class	class	NOUN
cana-852	284	16	of	of	ADP
cana-852	284	17	perturbed	perturb	VERB
cana-852	284	18	schrodinger	schrodinger	PROPN
cana-852	284	19	equations	equation	NOUN
cana-852	284	20	,	,	PUNCT
cana-852	284	21	hiroshima	hiroshima	PROPN
cana-852	284	22	mathematical	mathematical	PROPN
cana-852	284	23	journal	journal	PROPN
cana-852	284	24	,	,	PUNCT
cana-852	284	25	25(1	25(1	NUM
cana-852	284	26	)	)	PUNCT
cana-852	284	27	(	(	PUNCT
cana-852	284	28	1995	1995	NUM
cana-852	284	29	)	)	PUNCT
cana-852	284	30	,	,	PUNCT
cana-852	284	31	207	207	NUM
cana-852	284	32	-	-	SYM
cana-852	284	33	214	214	NUM
cana-852	284	34	.	.	PUNCT