id	sid	tid	token	lemma	pos
cana-5362	1	1	communications	communication	NOUN
cana-5362	1	2	on	on	ADP
cana-5362	1	3	applied	apply	VERB
cana-5362	1	4	nonlinear	nonlinear	ADJ
cana-5362	1	5	analysis	analysis	NOUN
cana-5362	1	6	issn	issn	NOUN
cana-5362	1	7	:	:	PUNCT
cana-5362	1	8	1074	1074	NUM
cana-5362	1	9	-	-	PUNCT
cana-5362	1	10	133x	133x	NUM
cana-5362	1	11	vol	vol	VERB
cana-5362	1	12	32	32	NUM
cana-5362	1	13	no	no	NOUN
cana-5362	1	14	.	.	PUNCT
cana-5362	2	1	10s	10	NOUN
cana-5362	2	2	(	(	PUNCT
cana-5362	2	3	2025	2025	NUM
cana-5362	2	4	)	)	PUNCT
cana-5362	2	5	1985	1985	NUM
cana-5362	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5362	2	7	a	a	DET
cana-5362	2	8	new	new	ADJ
cana-5362	2	9	stochastic	stochastic	ADJ
cana-5362	2	10	partial	partial	ADJ
cana-5362	2	11	differential	differential	NOUN
cana-5362	2	12	model	model	NOUN
cana-5362	2	13	for	for	ADP
cana-5362	2	14	image	image	NOUN
cana-5362	2	15	restoration	restoration	NOUN
cana-5362	2	16	1radhia	1radhia	NUM
cana-5362	2	17	halilou	halilou	NOUN
cana-5362	2	18	,	,	PUNCT
cana-5362	2	19	2fatma	2fatma	PROPN
cana-5362	2	20	zohra	zohra	PROPN
cana-5362	2	21	nouri	nouri	PROPN
cana-5362	2	22	and	and	CCONJ
cana-5362	2	23	3mohamed	3mohamed	NUM
cana-5362	2	24	lakhdar	lakhdar	NOUN
cana-5362	2	25	hadji	hadji	PROPN
cana-5362	2	26	,	,	PUNCT
cana-5362	2	27	1,2,3mathematical	1,2,3mathematical	NUM
cana-5362	2	28	modeling	modeling	NOUN
cana-5362	2	29	and	and	CCONJ
cana-5362	2	30	numerical	numerical	PROPN
cana-5362	2	31	simulation	simulation	PROPN
cana-5362	2	32	laboratory	laboratory	PROPN
cana-5362	2	33	,	,	PUNCT
cana-5362	2	34	department	department	NOUN
cana-5362	2	35	of	of	ADP
cana-5362	2	36	mathematics	mathematics	PROPN
cana-5362	2	37	,	,	PUNCT
cana-5362	2	38	badji	badji	PROPN
cana-5362	2	39	mokhtar	mokhtar	PROPN
cana-5362	2	40	university	university	PROPN
cana-5362	2	41	,	,	PUNCT
cana-5362	2	42	po	po	PROPN
cana-5362	2	43	box	box	PROPN
cana-5362	2	44	.	.	PUNCT
cana-5362	3	1	12	12	NUM
cana-5362	3	2	,	,	PUNCT
cana-5362	3	3	annaba	annaba	PROPN
cana-5362	3	4	23000	23000	NUM
cana-5362	3	5	-	-	PUNCT
cana-5362	3	6	algeria	algeria	PROPN
cana-5362	3	7	,	,	PUNCT
cana-5362	3	8	1e	1e	NOUN
cana-5362	3	9	-	-	PUNCT
cana-5362	3	10	mail	mail	NOUN
cana-5362	3	11	:	:	PUNCT
cana-5362	3	12	radhia.halilou@univ-annaba.dz	radhia.halilou@univ-annaba.dz	PROPN
cana-5362	3	13	,	,	PUNCT
cana-5362	3	14	2email	2email	NUM
cana-5362	3	15	:	:	PUNCT
cana-5362	3	16	tassili.nan09@gmail.com	tassili.nan09@gmail.com	X
cana-5362	3	17	,	,	PUNCT
cana-5362	3	18	3e	3e	NOUN
cana-5362	3	19	-	-	NOUN
cana-5362	3	20	mail	mail	NOUN
cana-5362	3	21	:	:	PUNCT
cana-5362	3	22	ml_hadji@yahoo.fr	ml_hadji@yahoo.fr	NOUN
cana-5362	3	23	,	,	PUNCT
cana-5362	3	24	article	article	NOUN
cana-5362	3	25	history	history	NOUN
cana-5362	3	26	:	:	PUNCT
cana-5362	3	27	received	receive	VERB
cana-5362	3	28	:	:	PUNCT
cana-5362	3	29	12	12	NUM
cana-5362	3	30	-	-	SYM
cana-5362	3	31	01	01	NUM
cana-5362	3	32	-	-	PUNCT
cana-5362	3	33	2025	2025	NUM
cana-5362	3	34	revised	revise	VERB
cana-5362	3	35	:	:	PUNCT
cana-5362	3	36	15	15	NUM
cana-5362	3	37	-	-	NUM
cana-5362	3	38	02	02	NUM
cana-5362	3	39	-	-	PUNCT
cana-5362	3	40	2025	2025	NUM
cana-5362	3	41	accepted	accept	VERB
cana-5362	3	42	:	:	PUNCT
cana-5362	3	43	01	01	NUM
cana-5362	3	44	-	-	SYM
cana-5362	3	45	03	03	NUM
cana-5362	3	46	-	-	PUNCT
cana-5362	3	47	2025	2025	NUM
cana-5362	3	48	abstract	abstract	NOUN
cana-5362	3	49	:	:	PUNCT
cana-5362	3	50	in	in	ADP
cana-5362	3	51	this	this	DET
cana-5362	3	52	paper	paper	NOUN
cana-5362	3	53	,	,	PUNCT
cana-5362	3	54	we	we	PRON
cana-5362	3	55	propose	propose	VERB
cana-5362	3	56	a	a	DET
cana-5362	3	57	stochastic	stochastic	ADJ
cana-5362	3	58	partial	partial	ADJ
cana-5362	3	59	differential	differential	NOUN
cana-5362	3	60	equation	equation	NOUN
cana-5362	3	61	model	model	NOUN
cana-5362	3	62	for	for	ADP
cana-5362	3	63	the	the	DET
cana-5362	3	64	restoration	restoration	NOUN
cana-5362	3	65	of	of	ADP
cana-5362	3	66	noisy	noisy	ADJ
cana-5362	3	67	images	image	NOUN
cana-5362	3	68	.	.	PUNCT
cana-5362	4	1	this	this	DET
cana-5362	4	2	proposed	propose	VERB
cana-5362	4	3	model	model	NOUN
cana-5362	4	4	is	be	AUX
cana-5362	4	5	based	base	VERB
cana-5362	4	6	on	on	ADP
cana-5362	4	7	the	the	DET
cana-5362	4	8	well	well	ADV
cana-5362	4	9	-	-	PUNCT
cana-5362	4	10	known	know	VERB
cana-5362	4	11	perona	perona	NOUN
cana-5362	4	12	-	-	PUNCT
cana-5362	4	13	malik	malik	PROPN
cana-5362	4	14	one	one	NOUN
cana-5362	4	15	,	,	PUNCT
cana-5362	4	16	to	to	PART
cana-5362	4	17	which	which	PRON
cana-5362	4	18	we	we	PRON
cana-5362	4	19	add	add	VERB
cana-5362	4	20	a	a	DET
cana-5362	4	21	stochastic	stochastic	ADJ
cana-5362	4	22	process	process	NOUN
cana-5362	4	23	.	.	PUNCT
cana-5362	5	1	the	the	DET
cana-5362	5	2	mathematical	mathematical	ADJ
cana-5362	5	3	analysis	analysis	NOUN
cana-5362	5	4	of	of	ADP
cana-5362	5	5	the	the	DET
cana-5362	5	6	proposed	propose	VERB
cana-5362	5	7	model	model	NOUN
cana-5362	5	8	is	be	AUX
cana-5362	5	9	carried	carry	VERB
cana-5362	5	10	out	out	ADP
cana-5362	5	11	within	within	ADP
cana-5362	5	12	a	a	DET
cana-5362	5	13	well	well	ADV
cana-5362	5	14	-	-	PUNCT
cana-5362	5	15	defined	define	VERB
cana-5362	5	16	framework	framework	NOUN
cana-5362	5	17	,	,	PUNCT
cana-5362	5	18	considering	consider	VERB
cana-5362	5	19	the	the	DET
cana-5362	5	20	specific	specific	ADJ
cana-5362	5	21	conditions	condition	NOUN
cana-5362	5	22	associated	associate	VERB
cana-5362	5	23	with	with	ADP
cana-5362	5	24	an	an	DET
cana-5362	5	25	anisotropic	anisotropic	NOUN
cana-5362	5	26	diffusion	diffusion	NOUN
cana-5362	5	27	.	.	PUNCT
cana-5362	6	1	for	for	ADP
cana-5362	6	2	the	the	DET
cana-5362	6	3	numerical	numerical	PROPN
cana-5362	6	4	approximation	approximation	NOUN
cana-5362	6	5	,	,	PUNCT
cana-5362	6	6	we	we	PRON
cana-5362	6	7	employ	employ	VERB
cana-5362	6	8	a	a	DET
cana-5362	6	9	stable	stable	ADJ
cana-5362	6	10	finite	finite	ADJ
cana-5362	6	11	difference	difference	NOUN
cana-5362	6	12	scheme	scheme	NOUN
cana-5362	6	13	,	,	PUNCT
cana-5362	6	14	ensuring	ensure	VERB
cana-5362	6	15	its	its	PRON
cana-5362	6	16	stability	stability	NOUN
cana-5362	6	17	through	through	ADP
cana-5362	6	18	fourier	fourier	ADJ
cana-5362	6	19	analysis	analysis	NOUN
cana-5362	6	20	.	.	PUNCT
cana-5362	7	1	the	the	DET
cana-5362	7	2	obtained	obtain	VERB
cana-5362	7	3	numerical	numerical	ADJ
cana-5362	7	4	results	result	NOUN
cana-5362	7	5	demonstrate	demonstrate	VERB
cana-5362	7	6	that	that	SCONJ
cana-5362	7	7	our	our	PRON
cana-5362	7	8	approach	approach	NOUN
cana-5362	7	9	enhances	enhance	VERB
cana-5362	7	10	images	image	NOUN
cana-5362	7	11	while	while	SCONJ
cana-5362	7	12	preserving	preserve	VERB
cana-5362	7	13	their	their	PRON
cana-5362	7	14	important	important	ADJ
cana-5362	7	15	structure	structure	NOUN
cana-5362	7	16	details	detail	NOUN
cana-5362	7	17	,	,	PUNCT
cana-5362	7	18	proving	prove	VERB
cana-5362	7	19	efficiency	efficiency	NOUN
cana-5362	7	20	and	and	CCONJ
cana-5362	7	21	competiveness	competiveness	NOUN
cana-5362	7	22	compare	compare	VERB
cana-5362	7	23	to	to	ADP
cana-5362	7	24	other	other	ADJ
cana-5362	7	25	approaches	approach	NOUN
cana-5362	7	26	for	for	ADP
cana-5362	7	27	image	image	NOUN
cana-5362	7	28	restoration	restoration	NOUN
cana-5362	7	29	.	.	PUNCT
cana-5362	8	1	keywords	keyword	NOUN
cana-5362	8	2	:	:	PUNCT
cana-5362	8	3	stochastic	stochastic	ADJ
cana-5362	8	4	partial	partial	ADJ
cana-5362	8	5	differential	differential	NOUN
cana-5362	8	6	equations	equation	NOUN
cana-5362	8	7	,	,	PUNCT
cana-5362	8	8	image	image	NOUN
cana-5362	8	9	denoising	denoising	NOUN
cana-5362	8	10	,	,	PUNCT
cana-5362	8	11	anisotropic	anisotropic	NOUN
cana-5362	8	12	diffusion	diffusion	NOUN
cana-5362	8	13	.	.	PUNCT
cana-5362	9	1	1	1	X
cana-5362	9	2	.	.	X
cana-5362	9	3	introduction	introduction	NOUN
cana-5362	9	4	mathematics	mathematics	PROPN
cana-5362	9	5	plays	play	VERB
cana-5362	9	6	an	an	DET
cana-5362	9	7	important	important	ADJ
cana-5362	9	8	role	role	NOUN
cana-5362	9	9	in	in	ADP
cana-5362	9	10	image	image	NOUN
cana-5362	9	11	processing	processing	NOUN
cana-5362	9	12	,	,	PUNCT
cana-5362	9	13	providing	provide	VERB
cana-5362	9	14	effective	effective	ADJ
cana-5362	9	15	tools	tool	NOUN
cana-5362	9	16	for	for	ADP
cana-5362	9	17	various	various	ADJ
cana-5362	9	18	tasks	task	NOUN
cana-5362	9	19	such	such	ADJ
cana-5362	9	20	as	as	ADP
cana-5362	9	21	noise	noise	NOUN
cana-5362	9	22	removal	removal	NOUN
cana-5362	9	23	,	,	PUNCT
cana-5362	9	24	edge	edge	NOUN
cana-5362	9	25	detection	detection	NOUN
cana-5362	9	26	,	,	PUNCT
cana-5362	9	27	and	and	CCONJ
cana-5362	9	28	feature	feature	NOUN
cana-5362	9	29	enhancement	enhancement	NOUN
cana-5362	9	30	.	.	PUNCT
cana-5362	10	1	various	various	ADJ
cana-5362	10	2	techniques	technique	NOUN
cana-5362	10	3	,	,	PUNCT
cana-5362	10	4	including	include	VERB
cana-5362	10	5	,	,	PUNCT
cana-5362	10	6	variational	variational	ADJ
cana-5362	10	7	methods	method	NOUN
cana-5362	10	8	,	,	PUNCT
cana-5362	10	9	fourier	fourier	NOUN
cana-5362	10	10	-	-	PUNCT
cana-5362	10	11	based	base	VERB
cana-5362	10	12	approaches	approach	NOUN
cana-5362	10	13	,	,	PUNCT
cana-5362	10	14	deterministic	deterministic	ADJ
cana-5362	10	15	and	and	CCONJ
cana-5362	10	16	stochastic	stochastic	ADJ
cana-5362	10	17	differential	differential	ADJ
cana-5362	10	18	equations	equation	NOUN
cana-5362	10	19	have	have	AUX
cana-5362	10	20	been	be	AUX
cana-5362	10	21	utilized	utilize	VERB
cana-5362	10	22	to	to	PART
cana-5362	10	23	tackle	tackle	VERB
cana-5362	10	24	these	these	DET
cana-5362	10	25	problems	problem	NOUN
cana-5362	10	26	effectively	effectively	ADV
cana-5362	10	27	.	.	PUNCT
cana-5362	11	1	recently	recently	ADV
cana-5362	11	2	,	,	PUNCT
cana-5362	11	3	stochastic	stochastic	ADJ
cana-5362	11	4	partial	partial	ADJ
cana-5362	11	5	differential	differential	NOUN
cana-5362	11	6	equations	equation	NOUN
cana-5362	11	7	(	(	PUNCT
cana-5362	11	8	spdes	spde	NOUN
cana-5362	11	9	)	)	PUNCT
cana-5362	11	10	have	have	AUX
cana-5362	11	11	gained	gain	VERB
cana-5362	11	12	attention	attention	NOUN
cana-5362	11	13	in	in	ADP
cana-5362	11	14	image	image	NOUN
cana-5362	11	15	processing	processing	NOUN
cana-5362	11	16	as	as	ADP
cana-5362	11	17	a	a	DET
cana-5362	11	18	promising	promising	ADJ
cana-5362	11	19	framework	framework	NOUN
cana-5362	11	20	for	for	ADP
cana-5362	11	21	removing	remove	VERB
cana-5362	11	22	noise	noise	NOUN
cana-5362	11	23	from	from	ADP
cana-5362	11	24	images	image	NOUN
cana-5362	11	25	and	and	CCONJ
cana-5362	11	26	enhancing	enhance	VERB
cana-5362	11	27	their	their	PRON
cana-5362	11	28	structures	structure	NOUN
cana-5362	11	29	.	.	PUNCT
cana-5362	12	1	the	the	DET
cana-5362	12	2	spdes	spde	NOUN
cana-5362	12	3	are	be	AUX
cana-5362	12	4	build	build	VERB
cana-5362	12	5	on	on	ADP
cana-5362	12	6	the	the	DET
cana-5362	12	7	foundational	foundational	ADJ
cana-5362	12	8	work	work	NOUN
cana-5362	12	9	of	of	ADP
cana-5362	12	10	stochastic	stochastic	ADJ
cana-5362	12	11	differential	differential	ADJ
cana-5362	12	12	equations	equation	NOUN
cana-5362	12	13	(	(	PUNCT
cana-5362	12	14	sdes	sde	NOUN
cana-5362	12	15	)	)	PUNCT
cana-5362	12	16	.	.	PUNCT
cana-5362	13	1	early	early	ADJ
cana-5362	13	2	research	research	NOUN
cana-5362	13	3	in	in	ADP
cana-5362	13	4	this	this	DET
cana-5362	13	5	field	field	NOUN
cana-5362	13	6	focused	focus	VERB
cana-5362	13	7	on	on	ADP
cana-5362	13	8	employing	employ	VERB
cana-5362	13	9	different	different	ADJ
cana-5362	13	10	sdes	sde	NOUN
cana-5362	13	11	using	use	VERB
cana-5362	13	12	various	various	ADJ
cana-5362	13	13	diffusion	diffusion	NOUN
cana-5362	13	14	processes	process	NOUN
cana-5362	13	15	for	for	ADP
cana-5362	13	16	image	image	NOUN
cana-5362	13	17	noise	noise	NOUN
cana-5362	13	18	reduction	reduction	NOUN
cana-5362	13	19	.	.	PUNCT
cana-5362	14	1	the	the	DET
cana-5362	14	2	authors	author	NOUN
cana-5362	14	3	descombes	descombe	VERB
cana-5362	14	4	and	and	CCONJ
cana-5362	14	5	zhizhina	zhizhina	NOUN
cana-5362	14	6	in	in	ADP
cana-5362	14	7	2003	2003	NUM
cana-5362	14	8	[	[	X
cana-5362	14	9	10	10	NUM
cana-5362	14	10	]	]	PUNCT
cana-5362	14	11	explored	explore	VERB
cana-5362	14	12	the	the	DET
cana-5362	14	13	application	application	NOUN
cana-5362	14	14	of	of	ADP
cana-5362	14	15	sdes	sde	NOUN
cana-5362	14	16	,	,	PUNCT
cana-5362	14	17	demonstrating	demonstrate	VERB
cana-5362	14	18	their	their	PRON
cana-5362	14	19	potential	potential	NOUN
cana-5362	14	20	in	in	ADP
cana-5362	14	21	image	image	NOUN
cana-5362	14	22	denoising	denoise	VERB
cana-5362	14	23	.	.	PUNCT
cana-5362	15	1	later	later	ADV
cana-5362	15	2	,	,	PUNCT
cana-5362	15	3	barbu	barbu	PROPN
cana-5362	15	4	in	in	ADP
cana-5362	15	5	2016	2016	NUM
cana-5362	15	6	[	[	X
cana-5362	15	7	1	1	X
cana-5362	15	8	]	]	PUNCT
cana-5362	15	9	introduced	introduce	VERB
cana-5362	15	10	a	a	DET
cana-5362	15	11	novel	novel	ADJ
cana-5362	15	12	sde	sde	NOUN
cana-5362	15	13	-	-	PUNCT
cana-5362	15	14	based	base	VERB
cana-5362	15	15	image	image	NOUN
cana-5362	15	16	restoration	restoration	NOUN
cana-5362	15	17	technique	technique	NOUN
cana-5362	15	18	,	,	PUNCT
cana-5362	15	19	which	which	PRON
cana-5362	15	20	significantly	significantly	ADV
cana-5362	15	21	improved	improve	VERB
cana-5362	15	22	noise	noise	NOUN
cana-5362	15	23	reduction	reduction	NOUN
cana-5362	15	24	while	while	SCONJ
cana-5362	15	25	preserving	preserve	VERB
cana-5362	15	26	important	important	ADJ
cana-5362	15	27	image	image	NOUN
cana-5362	15	28	structures	structure	NOUN
cana-5362	15	29	.	.	PUNCT
cana-5362	16	1	more	more	ADV
cana-5362	16	2	recent	recent	ADJ
cana-5362	16	3	advancements	advancement	NOUN
cana-5362	16	4	include	include	VERB
cana-5362	16	5	the	the	DET
cana-5362	16	6	use	use	NOUN
cana-5362	16	7	of	of	ADP
cana-5362	16	8	backward	backward	ADJ
cana-5362	16	9	stochastic	stochastic	ADJ
cana-5362	16	10	differential	differential	ADJ
cana-5362	16	11	equations	equation	NOUN
cana-5362	16	12	(	(	PUNCT
cana-5362	16	13	bsdes	bsdes	PROPN
cana-5362	16	14	)	)	PUNCT
cana-5362	16	15	for	for	ADP
cana-5362	16	16	image	image	NOUN
cana-5362	16	17	denoising	denoise	VERB
cana-5362	16	18	by	by	ADP
cana-5362	16	19	borkowski	borkowski	PROPN
cana-5362	16	20	et	et	PROPN
cana-5362	16	21	al	al	PROPN
cana-5362	17	1	[	[	X
cana-5362	17	2	5	5	NUM
cana-5362	17	3	]	]	PUNCT
cana-5362	17	4	,	,	PUNCT
cana-5362	17	5	[	[	X
cana-5362	17	6	2	2	NUM
cana-5362	17	7	]	]	PUNCT
cana-5362	17	8	and	and	CCONJ
cana-5362	17	9	[	[	X
cana-5362	17	10	3	3	NUM
cana-5362	17	11	]	]	PUNCT
cana-5362	17	12	applied	apply	VERB
cana-5362	17	13	bsdes	bsde	NOUN
cana-5362	17	14	to	to	PART
cana-5362	17	15	reconstruct	reconstruct	VERB
cana-5362	17	16	rgb	rgb	PROPN
cana-5362	17	17	images	image	NOUN
cana-5362	17	18	affected	affect	VERB
cana-5362	17	19	by	by	ADP
cana-5362	17	20	additive	additive	ADJ
cana-5362	17	21	gaussian	gaussian	ADJ
cana-5362	17	22	noise	noise	NOUN
cana-5362	17	23	,	,	PUNCT
cana-5362	17	24	achieving	achieve	VERB
cana-5362	17	25	a	a	DET
cana-5362	17	26	great	great	ADJ
cana-5362	17	27	success	success	NOUN
cana-5362	17	28	in	in	ADP
cana-5362	17	29	smoothing	smooth	VERB
cana-5362	17	30	noisy	noisy	ADJ
cana-5362	17	31	pixels	pixel	NOUN
cana-5362	17	32	while	while	SCONJ
cana-5362	17	33	enhancing	enhance	VERB
cana-5362	17	34	edges	edge	NOUN
cana-5362	17	35	.	.	PUNCT
cana-5362	18	1	mailto:radhia.halilou@univ-annaba.dz	mailto:radhia.halilou@univ-annaba.dz	PROPN
cana-5362	18	2	mailto:tassili.nan09@gmail.com	mailto:tassili.nan09@gmail.com	PROPN
cana-5362	18	3	communications	communication	NOUN
cana-5362	18	4	on	on	ADP
cana-5362	18	5	applied	apply	VERB
cana-5362	18	6	nonlinear	nonlinear	ADJ
cana-5362	18	7	analysis	analysis	NOUN
cana-5362	18	8	issn	issn	NOUN
cana-5362	18	9	:	:	PUNCT
cana-5362	18	10	1074	1074	NUM
cana-5362	18	11	-	-	PUNCT
cana-5362	18	12	133x	133x	NUM
cana-5362	18	13	vol	vol	VERB
cana-5362	18	14	32	32	NUM
cana-5362	18	15	no	no	NOUN
cana-5362	18	16	.	.	PUNCT
cana-5362	19	1	10s	10	NOUN
cana-5362	19	2	(	(	PUNCT
cana-5362	19	3	2025	2025	NUM
cana-5362	19	4	)	)	PUNCT
cana-5362	19	5	1986	1986	NUM
cana-5362	19	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5362	19	7	the	the	DET
cana-5362	19	8	evolution	evolution	NOUN
cana-5362	19	9	of	of	ADP
cana-5362	19	10	spdes	spde	NOUN
cana-5362	19	11	in	in	ADP
cana-5362	19	12	image	image	NOUN
cana-5362	19	13	denoising	denoising	NOUN
cana-5362	19	14	reflects	reflect	VERB
cana-5362	19	15	a	a	DET
cana-5362	19	16	broader	broad	ADJ
cana-5362	19	17	trend	trend	NOUN
cana-5362	19	18	in	in	ADP
cana-5362	19	19	applying	apply	VERB
cana-5362	19	20	stochastic	stochastic	ADJ
cana-5362	19	21	models	model	NOUN
cana-5362	19	22	to	to	ADP
cana-5362	19	23	image	image	NOUN
cana-5362	19	24	processing	processing	NOUN
cana-5362	19	25	tasks	task	NOUN
cana-5362	19	26	.	.	PUNCT
cana-5362	20	1	by	by	ADP
cana-5362	20	2	integrating	integrate	VERB
cana-5362	20	3	randomness	randomness	NOUN
cana-5362	20	4	into	into	ADP
cana-5362	20	5	the	the	DET
cana-5362	20	6	model	model	NOUN
cana-5362	20	7	,	,	PUNCT
cana-5362	20	8	the	the	DET
cana-5362	20	9	spde	spde	ADV
cana-5362	20	10	-	-	PUNCT
cana-5362	20	11	based	base	VERB
cana-5362	20	12	approaches	approach	NOUN
cana-5362	20	13	offer	offer	VERB
cana-5362	20	14	enhanced	enhance	VERB
cana-5362	20	15	flexibility	flexibility	NOUN
cana-5362	20	16	and	and	CCONJ
cana-5362	20	17	robustness	robustness	NOUN
cana-5362	20	18	,	,	PUNCT
cana-5362	20	19	leading	lead	VERB
cana-5362	20	20	to	to	ADP
cana-5362	20	21	more	more	ADV
cana-5362	20	22	effective	effective	ADJ
cana-5362	20	23	noise	noise	NOUN
cana-5362	20	24	reduction	reduction	NOUN
cana-5362	20	25	and	and	CCONJ
cana-5362	20	26	better	well	ADJ
cana-5362	20	27	preservation	preservation	NOUN
cana-5362	20	28	of	of	ADP
cana-5362	20	29	critical	critical	ADJ
cana-5362	20	30	image	image	NOUN
cana-5362	20	31	features	feature	NOUN
cana-5362	20	32	.	.	PUNCT
cana-5362	21	1	in	in	ADP
cana-5362	21	2	this	this	DET
cana-5362	21	3	work	work	NOUN
cana-5362	21	4	,	,	PUNCT
cana-5362	21	5	we	we	PRON
cana-5362	21	6	focus	focus	VERB
cana-5362	21	7	on	on	ADP
cana-5362	21	8	the	the	DET
cana-5362	21	9	restoration	restoration	NOUN
cana-5362	21	10	of	of	ADP
cana-5362	21	11	noisy	noisy	ADJ
cana-5362	21	12	images	image	NOUN
cana-5362	21	13	,	,	PUNCT
cana-5362	21	14	and	and	CCONJ
cana-5362	21	15	propose	propose	VERB
cana-5362	21	16	a	a	DET
cana-5362	21	17	novel	novel	ADJ
cana-5362	21	18	spde	spde	NOUN
cana-5362	21	19	model	model	NOUN
cana-5362	21	20	that	that	PRON
cana-5362	21	21	extends	extend	VERB
cana-5362	21	22	peronamalik	peronamalik	NOUN
cana-5362	21	23	(	(	PUNCT
cana-5362	21	24	pm	pm	NOUN
cana-5362	21	25	)	)	PUNCT
cana-5362	22	1	[	[	X
cana-5362	22	2	17	17	NUM
cana-5362	22	3	]	]	PUNCT
cana-5362	22	4	model	model	NOUN
cana-5362	22	5	by	by	ADP
cana-5362	22	6	including	include	VERB
cana-5362	22	7	a	a	DET
cana-5362	22	8	stochastic	stochastic	ADJ
cana-5362	22	9	process	process	NOUN
cana-5362	22	10	.	.	PUNCT
cana-5362	23	1	the	the	DET
cana-5362	23	2	pm	pm	NOUN
cana-5362	23	3	model	model	NOUN
cana-5362	23	4	[	[	X
cana-5362	23	5	17	17	NUM
cana-5362	23	6	]	]	PUNCT
cana-5362	23	7	,	,	PUNCT
cana-5362	23	8	though	though	SCONJ
cana-5362	23	9	effective	effective	ADJ
cana-5362	23	10	,	,	PUNCT
cana-5362	23	11	is	be	AUX
cana-5362	23	12	known	know	VERB
cana-5362	23	13	to	to	PART
cana-5362	23	14	be	be	AUX
cana-5362	23	15	ill	ill	ADV
cana-5362	23	16	-	-	PUNCT
cana-5362	23	17	posed	pose	VERB
cana-5362	23	18	,	,	PUNCT
cana-5362	23	19	which	which	PRON
cana-5362	23	20	limits	limit	VERB
cana-5362	23	21	its	its	PRON
cana-5362	23	22	applicability	applicability	NOUN
cana-5362	23	23	.	.	PUNCT
cana-5362	24	1	to	to	PART
cana-5362	24	2	address	address	VERB
cana-5362	24	3	the	the	DET
cana-5362	24	4	well	well	ADV
cana-5362	24	5	-	-	PUNCT
cana-5362	24	6	known	know	VERB
cana-5362	24	7	ill	ill	ADJ
cana-5362	24	8	-	-	PUNCT
cana-5362	24	9	posedness	posedness	NOUN
cana-5362	24	10	of	of	ADP
cana-5362	24	11	the	the	DET
cana-5362	24	12	pm	pm	NOUN
cana-5362	24	13	model	model	NOUN
cana-5362	24	14	,	,	PUNCT
cana-5362	24	15	we	we	PRON
cana-5362	24	16	adopt	adopt	VERB
cana-5362	24	17	the	the	DET
cana-5362	24	18	regularization	regularization	NOUN
cana-5362	24	19	approach	approach	NOUN
cana-5362	24	20	of	of	ADP
cana-5362	24	21	catté	catté	NOUN
cana-5362	24	22	et	et	PROPN
cana-5362	24	23	al	al	PROPN
cana-5362	24	24	.	.	PROPN
cana-5362	24	25	introduced	introduce	VERB
cana-5362	24	26	in	in	ADP
cana-5362	24	27	1992	1992	NUM
cana-5362	24	28	[	[	X
cana-5362	24	29	8	8	NUM
cana-5362	24	30	]	]	PUNCT
cana-5362	24	31	and	and	CCONJ
cana-5362	24	32	[	[	X
cana-5362	24	33	15	15	NUM
cana-5362	24	34	]	]	PUNCT
cana-5362	24	35	,	,	PUNCT
cana-5362	24	36	replacing	replace	VERB
cana-5362	24	37	the	the	DET
cana-5362	24	38	gradient	gradient	NOUN
cana-5362	24	39	|∇u|	|∇u|	NOUN
cana-5362	24	40	by	by	ADP
cana-5362	24	41	|∇gσ	|∇gσ	X
cana-5362	24	42	∗	∗	NOUN
cana-5362	24	43	u|	u|	NOUN
cana-5362	24	44	,	,	PUNCT
cana-5362	24	45	to	to	PART
cana-5362	24	46	ensure	ensure	VERB
cana-5362	24	47	the	the	DET
cana-5362	24	48	consistency	consistency	NOUN
cana-5362	24	49	and	and	CCONJ
cana-5362	24	50	well	well	ADV
cana-5362	24	51	-	-	PUNCT
cana-5362	24	52	posedness	posedness	NOUN
cana-5362	24	53	of	of	ADP
cana-5362	24	54	the	the	DET
cana-5362	24	55	model	model	NOUN
cana-5362	24	56	.	.	PUNCT
cana-5362	25	1	the	the	DET
cana-5362	25	2	main	main	ADJ
cana-5362	25	3	idea	idea	NOUN
cana-5362	25	4	here	here	ADV
cana-5362	25	5	is	be	AUX
cana-5362	25	6	that	that	SCONJ
cana-5362	25	7	this	this	DET
cana-5362	25	8	proposed	propose	VERB
cana-5362	25	9	spde	spde	NOUN
cana-5362	25	10	captures	capture	VERB
cana-5362	25	11	the	the	DET
cana-5362	25	12	stochastic	stochastic	ADJ
cana-5362	25	13	nature	nature	NOUN
cana-5362	25	14	of	of	ADP
cana-5362	25	15	noise	noise	NOUN
cana-5362	25	16	more	more	ADV
cana-5362	25	17	effectively	effectively	ADV
cana-5362	25	18	.	.	PUNCT
cana-5362	26	1	we	we	PRON
cana-5362	26	2	build	build	VERB
cana-5362	26	3	the	the	DET
cana-5362	26	4	mathematical	mathematical	ADJ
cana-5362	26	5	study	study	NOUN
cana-5362	26	6	of	of	ADP
cana-5362	26	7	this	this	DET
cana-5362	26	8	spde	spde	NOUN
cana-5362	26	9	by	by	ADP
cana-5362	26	10	combining	combine	VERB
cana-5362	26	11	the	the	DET
cana-5362	26	12	deterministic	deterministic	ADJ
cana-5362	26	13	analysis	analysis	NOUN
cana-5362	26	14	of	of	ADP
cana-5362	26	15	catté	catté	NOUN
cana-5362	26	16	et	et	NOUN
cana-5362	26	17	al	al	PROPN
cana-5362	26	18	.	.	PUNCT
cana-5362	27	1	[	[	X
cana-5362	27	2	8	8	NUM
cana-5362	27	3	]	]	PUNCT
cana-5362	27	4	and	and	CCONJ
cana-5362	27	5	the	the	DET
cana-5362	27	6	weak	weak	ADJ
cana-5362	27	7	solution	solution	NOUN
cana-5362	27	8	theory	theory	NOUN
cana-5362	27	9	developed	develop	VERB
cana-5362	27	10	by	by	ADP
cana-5362	27	11	benssoussen	benssoussen	NOUN
cana-5362	27	12	and	and	CCONJ
cana-5362	27	13	temam	temam	NOUN
cana-5362	27	14	(	(	PUNCT
cana-5362	27	15	1971	1971	NUM
cana-5362	27	16	)	)	PUNCT
cana-5362	28	1	[	[	X
cana-5362	28	2	4	4	NUM
cana-5362	28	3	]	]	PUNCT
cana-5362	28	4	.	.	PUNCT
cana-5362	29	1	to	to	PART
cana-5362	29	2	approximate	approximate	VERB
cana-5362	29	3	the	the	DET
cana-5362	29	4	solution	solution	NOUN
cana-5362	29	5	,	,	PUNCT
cana-5362	29	6	we	we	PRON
cana-5362	29	7	employ	employ	VERB
cana-5362	29	8	an	an	DET
cana-5362	29	9	appropriate	appropriate	ADJ
cana-5362	29	10	finite	finite	ADJ
cana-5362	29	11	difference	difference	NOUN
cana-5362	29	12	scheme	scheme	NOUN
cana-5362	29	13	and	and	CCONJ
cana-5362	29	14	verify	verify	VERB
cana-5362	29	15	its	its	PRON
cana-5362	29	16	stability	stability	NOUN
cana-5362	29	17	via	via	ADP
cana-5362	29	18	fourier	fourier	ADJ
cana-5362	29	19	analysis	analysis	NOUN
cana-5362	29	20	.	.	PUNCT
cana-5362	30	1	numerical	numerical	ADJ
cana-5362	30	2	experiments	experiment	NOUN
cana-5362	30	3	demonstrate	demonstrate	VERB
cana-5362	30	4	the	the	DET
cana-5362	30	5	efficiency	efficiency	NOUN
cana-5362	30	6	of	of	ADP
cana-5362	30	7	the	the	DET
cana-5362	30	8	proposed	propose	VERB
cana-5362	30	9	model	model	NOUN
cana-5362	30	10	in	in	ADP
cana-5362	30	11	enhancing	enhance	VERB
cana-5362	30	12	image	image	NOUN
cana-5362	30	13	quality	quality	NOUN
cana-5362	30	14	,	,	PUNCT
cana-5362	30	15	while	while	SCONJ
cana-5362	30	16	preserving	preserve	VERB
cana-5362	30	17	structure	structure	NOUN
cana-5362	30	18	details	detail	NOUN
cana-5362	30	19	,	,	PUNCT
cana-5362	30	20	showing	show	VERB
cana-5362	30	21	significant	significant	ADJ
cana-5362	30	22	improvements	improvement	NOUN
cana-5362	30	23	in	in	ADP
cana-5362	30	24	noise	noise	NOUN
cana-5362	30	25	removal	removal	NOUN
cana-5362	30	26	and	and	CCONJ
cana-5362	30	27	feature	feature	NOUN
cana-5362	30	28	refinement	refinement	NOUN
cana-5362	30	29	.	.	PUNCT
cana-5362	31	1	this	this	DET
cana-5362	31	2	paper	paper	NOUN
cana-5362	31	3	is	be	AUX
cana-5362	31	4	organized	organize	VERB
cana-5362	31	5	as	as	SCONJ
cana-5362	31	6	follows	follow	VERB
cana-5362	31	7	:	:	PUNCT
cana-5362	31	8	section	section	NOUN
cana-5362	31	9	2	2	NUM
cana-5362	31	10	introduces	introduce	NOUN
cana-5362	31	11	the	the	DET
cana-5362	31	12	proposed	propose	VERB
cana-5362	31	13	spde	spde	NOUN
cana-5362	31	14	model	model	NOUN
cana-5362	31	15	.	.	PUNCT
cana-5362	32	1	section	section	NOUN
cana-5362	32	2	3	3	NUM
cana-5362	32	3	provides	provide	VERB
cana-5362	32	4	the	the	DET
cana-5362	32	5	mathematical	mathematical	ADJ
cana-5362	32	6	analysis	analysis	NOUN
cana-5362	32	7	of	of	ADP
cana-5362	32	8	the	the	DET
cana-5362	32	9	model	model	NOUN
cana-5362	32	10	.	.	PUNCT
cana-5362	33	1	in	in	ADP
cana-5362	33	2	section	section	NOUN
cana-5362	33	3	4	4	NUM
cana-5362	33	4	,	,	PUNCT
cana-5362	33	5	the	the	DET
cana-5362	33	6	numerical	numerical	ADJ
cana-5362	33	7	discretisation	discretisation	NOUN
cana-5362	33	8	is	be	AUX
cana-5362	33	9	detailed	detail	VERB
cana-5362	33	10	with	with	ADP
cana-5362	33	11	a	a	DET
cana-5362	33	12	stability	stability	NOUN
cana-5362	33	13	analysis	analysis	NOUN
cana-5362	33	14	,	,	PUNCT
cana-5362	33	15	and	and	CCONJ
cana-5362	33	16	results	result	NOUN
cana-5362	33	17	.	.	PUNCT
cana-5362	34	1	finally	finally	ADV
cana-5362	34	2	,	,	PUNCT
cana-5362	34	3	concluding	conclude	VERB
cana-5362	34	4	remarks	remark	NOUN
cana-5362	34	5	are	be	AUX
cana-5362	34	6	presented	present	VERB
cana-5362	34	7	in	in	ADP
cana-5362	34	8	section	section	NOUN
cana-5362	34	9	5	5	NUM
cana-5362	34	10	.	.	SYM
cana-5362	34	11	2	2	NUM
cana-5362	34	12	.	.	NUM
cana-5362	34	13	proposed	propose	VERB
cana-5362	34	14	model	model	NOUN
cana-5362	34	15	in	in	ADP
cana-5362	34	16	this	this	DET
cana-5362	34	17	section	section	NOUN
cana-5362	34	18	,	,	PUNCT
cana-5362	34	19	we	we	PRON
cana-5362	34	20	use	use	VERB
cana-5362	34	21	a	a	DET
cana-5362	34	22	2d	2d	NUM
cana-5362	34	23	standard	standard	ADJ
cana-5362	34	24	brownian	brownian	ADJ
cana-5362	34	25	motion	motion	NOUN
cana-5362	34	26	to	to	PART
cana-5362	34	27	perturb	perturb	VERB
cana-5362	34	28	a	a	DET
cana-5362	34	29	regularised	regularise	VERB
cana-5362	34	30	pde	pde	NOUN
cana-5362	34	31	due	due	ADP
cana-5362	34	32	to	to	ADP
cana-5362	34	33	pm	pm	NOUN
cana-5362	34	34	[	[	X
cana-5362	34	35	17	17	NUM
cana-5362	34	36	]	]	PUNCT
cana-5362	34	37	proposed	propose	VERB
cana-5362	34	38	by	by	ADP
cana-5362	34	39	f.	f.	PROPN
cana-5362	34	40	catté	catté	PROPN
cana-5362	34	41	,	,	PUNCT
cana-5362	34	42	p.l	p.l	PROPN
cana-5362	34	43	.	.	PROPN
cana-5362	34	44	lions	lion	NOUN
cana-5362	34	45	,	,	PUNCT
cana-5362	34	46	j.m	j.m	PROPN
cana-5362	34	47	.	.	PROPN
cana-5362	34	48	morel	morel	PROPN
cana-5362	34	49	and	and	CCONJ
cana-5362	34	50	t.	t.	PROPN
cana-5362	34	51	coll	coll	PROPN
cana-5362	35	1	[	[	X
cana-5362	35	2	8	8	NUM
cana-5362	35	3	]	]	PUNCT
cana-5362	35	4	and	and	CCONJ
cana-5362	35	5	[	[	X
cana-5362	35	6	15	15	NUM
cana-5362	35	7	]	]	PUNCT
cana-5362	35	8	.	.	PUNCT
cana-5362	36	1	the	the	DET
cana-5362	36	2	problem	problem	NOUN
cana-5362	36	3	can	can	AUX
cana-5362	36	4	be	be	AUX
cana-5362	36	5	written	write	VERB
cana-5362	36	6	as	as	SCONJ
cana-5362	36	7	follows	follow	VERB
cana-5362	36	8	{	{	PUNCT
cana-5362	36	9	𝜕𝑢	𝜕𝑢	NOUN
cana-5362	36	10	𝜕𝑡	𝜕𝑡	NOUN
cana-5362	36	11	=	=	SYM
cana-5362	36	12	𝑑𝑖𝑣(𝑔(|𝛻𝐺𝜎	𝑑𝑖𝑣(𝑔(|𝛻𝐺𝜎	PROPN
cana-5362	36	13	∗	∗	NOUN
cana-5362	36	14	𝑢|)𝛻𝑢	𝑢|)𝛻𝑢	NOUN
cana-5362	36	15	)	)	PUNCT
cana-5362	37	1	+	+	NUM
cana-5362	37	2	𝑑𝑊𝑡	𝑑𝑊𝑡	NOUN
cana-5362	37	3	𝑖𝑛	𝑖𝑛	X
cana-5362	37	4	]	]	X
cana-5362	37	5	0	0	NUM
cana-5362	37	6	,	,	PUNCT
cana-5362	37	7	𝑇	𝑇	PROPN
cana-5362	37	8	[	[	PUNCT
cana-5362	37	9	×	×	PROPN
cana-5362	37	10	𝐷	𝐷	PROPN
cana-5362	37	11	,	,	PUNCT
cana-5362	37	12	𝑢(0	𝑢(0	PROPN
cana-5362	37	13	,	,	PUNCT
cana-5362	37	14	𝑥	𝑥	PROPN
cana-5362	37	15	)	)	PUNCT
cana-5362	37	16	=	=	SYM
cana-5362	37	17	𝑢0(𝑥	𝑢0(𝑥	PROPN
cana-5362	37	18	)	)	PUNCT
cana-5362	37	19	,	,	PUNCT
cana-5362	37	20	∀𝑥	∀𝑥	PROPN
cana-5362	37	21	∈	∈	PROPN
cana-5362	37	22	𝐷	𝐷	PROPN
cana-5362	37	23	,	,	PUNCT
cana-5362	37	24	(	(	PUNCT
cana-5362	37	25	1	1	X
cana-5362	37	26	)	)	PUNCT
cana-5362	37	27	with	with	ADP
cana-5362	37	28	𝑔(|𝛻𝐺𝜎	𝑔(|𝛻𝐺𝜎	ADJ
cana-5362	37	29	∗	∗	NOUN
cana-5362	37	30	𝑢|	𝑢|	NOUN
cana-5362	37	31	)	)	PUNCT
cana-5362	37	32	=	=	PUNCT
cana-5362	38	1	𝑒	𝑒	PROPN
cana-5362	38	2	−	−	NOUN
cana-5362	38	3	|𝛻𝐺𝜎∗𝑢|	|𝛻𝐺𝜎∗𝑢|	SYM
cana-5362	38	4	2	2	NUM
cana-5362	38	5	𝑘2	𝑘2	PROPN
cana-5362	38	6	𝑜𝑟	𝑜𝑟	ADP
cana-5362	38	7	𝑔(|𝛻𝐺𝜎	𝑔(|𝛻𝐺𝜎	NOUN
cana-5362	38	8	∗	∗	NOUN
cana-5362	38	9	𝑢|	𝑢|	PROPN
cana-5362	38	10	)	)	PUNCT
cana-5362	38	11	=	=	SYM
cana-5362	38	12	1	1	NUM
cana-5362	38	13	1	1	NUM
cana-5362	38	14	+	+	NUM
cana-5362	38	15	|𝛻𝐺𝜎∗𝑢|	|𝛻𝐺𝜎∗𝑢|	PROPN
cana-5362	38	16	2	2	NUM
cana-5362	38	17	𝑘2	𝑘2	PROPN
cana-5362	38	18	,	,	PUNCT
cana-5362	38	19	𝑘	𝑘	X
cana-5362	38	20	>	>	X
cana-5362	38	21	0	0	PUNCT
cana-5362	39	1	(	(	PUNCT
cana-5362	39	2	2	2	NUM
cana-5362	39	3	)	)	PUNCT
cana-5362	39	4	in	in	ADP
cana-5362	39	5	the	the	DET
cana-5362	39	6	problem	problem	NOUN
cana-5362	39	7	(	(	PUNCT
cana-5362	39	8	1	1	X
cana-5362	39	9	)	)	PUNCT
cana-5362	39	10	we	we	PRON
cana-5362	39	11	have	have	VERB
cana-5362	39	12	the	the	DET
cana-5362	39	13	following	follow	VERB
cana-5362	39	14	denotes	denote	NOUN
cana-5362	39	15	•	•	ADP
cana-5362	39	16	𝑢0	𝑢0	PROPN
cana-5362	39	17	∶	∶	PROPN
cana-5362	39	18	𝐷	𝐷	PROPN
cana-5362	39	19	⊂	⊂	NOUN
cana-5362	39	20	ℝ	ℝ	PROPN
cana-5362	39	21	2	2	NUM
cana-5362	39	22	→	→	SYM
cana-5362	39	23	ℝ	ℝ	PROPN
cana-5362	39	24	be	be	AUX
cana-5362	39	25	the	the	DET
cana-5362	39	26	initial	initial	ADJ
cana-5362	39	27	image	image	NOUN
cana-5362	39	28	.	.	PUNCT
cana-5362	40	1	•	•	NUM
cana-5362	40	2	𝐷	𝐷	PROPN
cana-5362	40	3	⊂	⊂	PROPN
cana-5362	40	4	ℝ2	ℝ2	VERB
cana-5362	40	5	is	be	AUX
cana-5362	40	6	a	a	DET
cana-5362	40	7	bounded	bounded	ADJ
cana-5362	40	8	domain	domain	NOUN
cana-5362	40	9	with	with	ADP
cana-5362	40	10	a	a	DET
cana-5362	40	11	lipschitz	lipschitz	NOUN
cana-5362	40	12	boundary	boundary	NOUN
cana-5362	40	13	and	and	CCONJ
cana-5362	40	14	0	0	NUM
cana-5362	40	15	<	<	X
cana-5362	40	16	𝑇	𝑇	PROPN
cana-5362	40	17	<	<	X
cana-5362	40	18	∞.	∞.	PROPN
cana-5362	40	19	•	•	NOUN
cana-5362	40	20	gσ	gσ	VERB
cana-5362	40	21	is	be	AUX
cana-5362	40	22	a	a	DET
cana-5362	40	23	gaussian	gaussian	ADJ
cana-5362	40	24	filter	filter	NOUN
cana-5362	40	25	(	(	PUNCT
cana-5362	40	26	gf	gf	NOUN
cana-5362	40	27	)	)	PUNCT
cana-5362	40	28	,	,	PUNCT
cana-5362	40	29	𝐺𝜎(𝑥	𝐺𝜎(𝑥	NOUN
cana-5362	40	30	)	)	PUNCT
cana-5362	40	31	=	=	SYM
cana-5362	40	32	1	1	NUM
cana-5362	40	33	√2𝜋𝜎	√2𝜋𝜎	PROPN
cana-5362	40	34	exp	exp	NOUN
cana-5362	40	35	(	(	PUNCT
cana-5362	40	36	−	−	PROPN
cana-5362	40	37	|𝑥|2	|𝑥|2	ADJ
cana-5362	40	38	4𝜎	4𝜎	PROPN
cana-5362	40	39	)	)	PUNCT
cana-5362	40	40	,	,	PUNCT
cana-5362	40	41	𝜎	𝜎	PROPN
cana-5362	40	42	>	>	X
cana-5362	40	43	0	0	PROPN
cana-5362	40	44	,	,	PUNCT
cana-5362	40	45	𝑥	𝑥	NOUN
cana-5362	40	46	=	=	SYM
cana-5362	40	47	(	(	PUNCT
cana-5362	40	48	𝑥1	𝑥1	NOUN
cana-5362	40	49	,	,	PUNCT
cana-5362	40	50	𝑥2	𝑥2	NOUN
cana-5362	40	51	)	)	PUNCT
cana-5362	40	52	∈	∈	PROPN
cana-5362	40	53	𝑅	𝑅	PROPN
cana-5362	40	54	2	2	NUM
cana-5362	40	55	communications	communication	NOUN
cana-5362	40	56	on	on	ADP
cana-5362	40	57	applied	apply	VERB
cana-5362	40	58	nonlinear	nonlinear	ADJ
cana-5362	40	59	analysis	analysis	NOUN
cana-5362	40	60	issn	issn	NOUN
cana-5362	40	61	:	:	PUNCT
cana-5362	40	62	1074	1074	NUM
cana-5362	40	63	-	-	PUNCT
cana-5362	40	64	133x	133x	NUM
cana-5362	40	65	vol	vol	VERB
cana-5362	40	66	32	32	NUM
cana-5362	40	67	no	no	NOUN
cana-5362	40	68	.	.	PUNCT
cana-5362	41	1	10s	10	NOUN
cana-5362	41	2	(	(	PUNCT
cana-5362	41	3	2025	2025	NUM
cana-5362	41	4	)	)	PUNCT
cana-5362	41	5	1987	1987	NUM
cana-5362	41	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5362	41	7	•	•	X
cana-5362	41	8	(	(	PUNCT
cana-5362	41	9	𝐷	𝐷	PROPN
cana-5362	41	10	,	,	PUNCT
cana-5362	41	11	ℱ	ℱ	PROPN
cana-5362	41	12	,	,	PUNCT
cana-5362	41	13	𝒫	𝒫	NOUN
cana-5362	41	14	)	)	PUNCT
cana-5362	41	15	a	a	DET
cana-5362	41	16	completely	completely	ADV
cana-5362	41	17	regular	regular	ADJ
cana-5362	41	18	topological	topological	ADJ
cana-5362	41	19	space	space	NOUN
cana-5362	41	20	equipped	equip	VERB
cana-5362	41	21	with	with	ADP
cana-5362	41	22	its	its	PRON
cana-5362	41	23	borel	borel	PROPN
cana-5362	41	24	σ	σ	PROPN
cana-5362	41	25	-	-	PUNCT
cana-5362	41	26	algebra	algebra	PROPN
cana-5362	41	27	ℱ	ℱ	PROPN
cana-5362	41	28	,	,	PUNCT
cana-5362	41	29	and	and	CCONJ
cana-5362	41	30	𝒫	𝒫	NOUN
cana-5362	41	31	is	be	AUX
cana-5362	41	32	a	a	DET
cana-5362	41	33	radon	radon	ADJ
cana-5362	41	34	probability	probability	NOUN
cana-5362	41	35	measure	measure	NOUN
cana-5362	41	36	(	(	PUNCT
cana-5362	41	37	i.e.	i.e.	X
cana-5362	41	38	,	,	PUNCT
cana-5362	41	39	an	an	DET
cana-5362	41	40	abstract	abstract	ADJ
cana-5362	41	41	measure	measure	NOUN
cana-5362	41	42	on	on	ADP
cana-5362	41	43	ℱ	ℱ	PROPN
cana-5362	41	44	that	that	PRON
cana-5362	41	45	is	be	AUX
cana-5362	41	46	inner	inner	ADJ
cana-5362	41	47	regular	regular	ADV
cana-5362	41	48	)	)	PUNCT
cana-5362	42	1	[	[	X
cana-5362	42	2	19	19	NUM
cana-5362	42	3	]	]	PUNCT
cana-5362	42	4	and	and	CCONJ
cana-5362	42	5	[	[	X
cana-5362	42	6	13	13	NUM
cana-5362	42	7	]	]	PUNCT
cana-5362	42	8	.	.	PUNCT
cana-5362	43	1	•	•	INTJ
cana-5362	43	2	𝑊𝑡	𝑊𝑡	PROPN
cana-5362	43	3	be	be	VERB
cana-5362	43	4	a	a	DET
cana-5362	43	5	wiener	wiener	NOUN
cana-5362	43	6	process	process	NOUN
cana-5362	43	7	defined	define	VERB
cana-5362	43	8	on	on	ADP
cana-5362	43	9	(	(	PUNCT
cana-5362	43	10	𝐷	𝐷	NOUN
cana-5362	43	11	,	,	PUNCT
cana-5362	43	12	ℱ,𝒫	ℱ,𝒫	ADJ
cana-5362	43	13	)	)	PUNCT
cana-5362	43	14	and	and	CCONJ
cana-5362	43	15	taking	take	VERB
cana-5362	43	16	values	value	NOUN
cana-5362	43	17	in	in	ADP
cana-5362	43	18	the	the	DET
cana-5362	43	19	separable	separable	ADJ
cana-5362	43	20	hilbert	hilbert	PROPN
cana-5362	43	21	space	space	PROPN
cana-5362	43	22	𝐻	𝐻	PROPN
cana-5362	43	23	,	,	PUNCT
cana-5362	43	24	with	with	SCONJ
cana-5362	43	25	incremental	incremental	ADJ
cana-5362	43	26	covariance	covariance	NOUN
cana-5362	43	27	operator	operator	NOUN
cana-5362	43	28	w.	w.	NOUN
cana-5362	43	29	let	let	VERB
cana-5362	43	30	(	(	PUNCT
cana-5362	43	31	ℱ𝑡)𝑡>0	ℱ𝑡)𝑡>0	NOUN
cana-5362	43	32	be	be	AUX
cana-5362	43	33	the	the	DET
cana-5362	43	34	σ	σ	PROPN
cana-5362	43	35	-	-	PUNCT
cana-5362	43	36	algebra	algebra	NOUN
cana-5362	43	37	generated	generate	VERB
cana-5362	43	38	by	by	ADP
cana-5362	43	39	𝑊𝑠	𝑊𝑠	PROPN
cana-5362	43	40	,	,	PUNCT
cana-5362	43	41	0	0	NUM
cana-5362	43	42	≤	≤	NUM
cana-5362	43	43	𝑠	𝑠	X
cana-5362	43	44	≤	≤	NUM
cana-5362	43	45	𝑡	𝑡	PROPN
cana-5362	43	46	then	then	ADV
cana-5362	43	47	𝑊𝑡	𝑊𝑡	PROPN
cana-5362	43	48	is	be	AUX
cana-5362	43	49	a	a	DET
cana-5362	43	50	martingale	martingale	NOUN
cana-5362	43	51	relative	relative	ADJ
cana-5362	43	52	to	to	ADP
cana-5362	43	53	(	(	PUNCT
cana-5362	43	54	(	(	PUNCT
cana-5362	43	55	ℱ𝑡)𝑡>0	ℱ𝑡)𝑡>0	NOUN
cana-5362	43	56	and	and	CCONJ
cana-5362	43	57	we	we	PRON
cana-5362	43	58	have	have	VERB
cana-5362	43	59	the	the	DET
cana-5362	43	60	following	follow	VERB
cana-5362	43	61	representation	representation	NOUN
cana-5362	43	62	of	of	ADP
cana-5362	43	63	𝑊𝑡	𝑊𝑡	PROPN
cana-5362	43	64	:	:	PUNCT
cana-5362	43	65	𝑊𝑡	𝑊𝑡	PROPN
cana-5362	43	66	=	=	NOUN
cana-5362	43	67	∑𝛽𝑡	∑𝛽𝑡	X
cana-5362	43	68	𝑖	𝑖	NOUN
cana-5362	43	69	∞	∞	NUM
cana-5362	43	70	𝑖=1	𝑖=1	PUNCT
cana-5362	43	71	𝑒𝑖	𝑒𝑖	ADP
cana-5362	43	72	where	where	SCONJ
cana-5362	43	73	𝑒𝑖	𝑒𝑖	PROPN
cana-5362	43	74	is	be	AUX
cana-5362	43	75	an	an	DET
cana-5362	43	76	orthonormal	orthonormal	ADJ
cana-5362	43	77	set	set	NOUN
cana-5362	43	78	of	of	ADP
cana-5362	43	79	eigenvectors	eigenvector	NOUN
cana-5362	43	80	of	of	ADP
cana-5362	43	81	𝑤,𝛽𝑡	𝑤,𝛽𝑡	PUNCT
cana-5362	43	82	𝑖	𝑖	NOUN
cana-5362	43	83	are	be	AUX
cana-5362	43	84	mutually	mutually	ADV
cana-5362	43	85	independent	independent	ADJ
cana-5362	43	86	real	real	ADJ
cana-5362	43	87	wiener	wiener	NOUN
cana-5362	43	88	processes	process	NOUN
cana-5362	43	89	with	with	ADP
cana-5362	43	90	incremental	incremental	ADJ
cana-5362	43	91	covariance	covariance	NOUN
cana-5362	43	92	λ	λ	PROPN
cana-5362	43	93	>	>	X
cana-5362	43	94	0	0	PROPN
cana-5362	43	95	,	,	PUNCT
cana-5362	43	96	𝑤𝑒𝑖	𝑤𝑒𝑖	NOUN
cana-5362	43	97	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-5362	43	98	𝑡𝑟𝑤	𝑡𝑟𝑤	NOUN
cana-5362	43	99	=	=	SYM
cana-5362	43	100	∑	∑	PUNCT
cana-5362	43	101	λ𝑖	λ𝑖	ADV
cana-5362	43	102	∞	∞	PROPN
cana-5362	43	103	𝑖=1	𝑖=1	PUNCT
cana-5362	44	1	(	(	PUNCT
cana-5362	44	2	tr	tr	VERB
cana-5362	44	3	denotes	denote	VERB
cana-5362	44	4	the	the	DET
cana-5362	44	5	trace	trace	NOUN
cana-5362	44	6	of	of	ADP
cana-5362	44	7	an	an	DET
cana-5362	44	8	operator	operator	NOUN
cana-5362	44	9	,	,	PUNCT
cana-5362	44	10	see	see	VERB
cana-5362	44	11	[	[	X
cana-5362	44	12	6	6	NUM
cana-5362	44	13	]	]	PUNCT
cana-5362	44	14	and	and	CCONJ
cana-5362	44	15	[	[	X
cana-5362	44	16	3	3	X
cana-5362	44	17	]	]	PUNCT
cana-5362	44	18	for	for	ADP
cana-5362	44	19	more	more	ADJ
cana-5362	44	20	details	detail	NOUN
cana-5362	44	21	on	on	ADP
cana-5362	44	22	stochastic	stochastic	ADJ
cana-5362	44	23	analysis	analysis	NOUN
cana-5362	44	24	)	)	PUNCT
cana-5362	44	25	.	.	PUNCT
cana-5362	45	1	in	in	ADP
cana-5362	45	2	the	the	DET
cana-5362	45	3	next	next	ADJ
cana-5362	45	4	sections	section	NOUN
cana-5362	45	5	,	,	PUNCT
cana-5362	45	6	we	we	PRON
cana-5362	45	7	consider	consider	VERB
cana-5362	45	8	processes	process	NOUN
cana-5362	45	9	as	as	ADP
cana-5362	45	10	:	:	PUNCT
cana-5362	45	11	𝑊	𝑊	NOUN
cana-5362	45	12	=	=	SYM
cana-5362	45	13	𝑊(𝑡	𝑊(𝑡	NOUN
cana-5362	45	14	,	,	PUNCT
cana-5362	45	15	𝑥	𝑥	NOUN
cana-5362	45	16	)	)	PUNCT
cana-5362	45	17	of	of	ADP
cana-5362	45	18	the	the	DET
cana-5362	45	19	form	form	NOUN
cana-5362	45	20	of	of	ADP
cana-5362	45	21	the	the	DET
cana-5362	45	22	following	follow	VERB
cana-5362	45	23	proposition	proposition	NOUN
cana-5362	45	24	.	.	PUNCT
cana-5362	46	1	proposition	proposition	NOUN
cana-5362	46	2	1	1	NUM
cana-5362	46	3	under	under	ADP
cana-5362	46	4	the	the	DET
cana-5362	46	5	previous	previous	ADJ
cana-5362	46	6	assumptions	assumption	NOUN
cana-5362	46	7	,	,	PUNCT
cana-5362	46	8	𝑊𝑡	𝑊𝑡	PROPN
cana-5362	46	9	∈	∈	PROPN
cana-5362	46	10	𝐶([0	𝐶([0	PROPN
cana-5362	46	11	,	,	PUNCT
cana-5362	46	12	𝑇	𝑇	PROPN
cana-5362	46	13	]	]	PUNCT
cana-5362	46	14	;	;	PUNCT
cana-5362	46	15	𝐿	𝐿	PROPN
cana-5362	46	16	2(𝐷,𝒫;𝐻	2(𝐷,𝒫;𝐻	NUM
cana-5362	46	17	)	)	PUNCT
cana-5362	46	18	)	)	PUNCT
cana-5362	47	1	(	(	PUNCT
cana-5362	47	2	3	3	X
cana-5362	47	3	)	)	PUNCT
cana-5362	47	4	proof	proof	NOUN
cana-5362	47	5	.	.	PUNCT
cana-5362	48	1	for	for	ADP
cana-5362	48	2	the	the	DET
cana-5362	48	3	details	detail	NOUN
cana-5362	48	4	of	of	ADP
cana-5362	48	5	the	the	DET
cana-5362	48	6	proof	proof	NOUN
cana-5362	48	7	of	of	ADP
cana-5362	48	8	this	this	DET
cana-5362	48	9	result	result	NOUN
cana-5362	48	10	,	,	PUNCT
cana-5362	48	11	the	the	DET
cana-5362	48	12	reader	reader	NOUN
cana-5362	48	13	is	be	AUX
cana-5362	48	14	referred	refer	VERB
cana-5362	48	15	to	to	ADP
cana-5362	48	16	[	[	X
cana-5362	48	17	4	4	NUM
cana-5362	48	18	]	]	SYM
cana-5362	48	19	3	3	NUM
cana-5362	48	20	.	.	NOUN
cana-5362	48	21	mathematical	mathematical	ADJ
cana-5362	48	22	study	study	NOUN
cana-5362	48	23	in	in	ADP
cana-5362	48	24	this	this	DET
cana-5362	48	25	section	section	NOUN
cana-5362	48	26	,	,	PUNCT
cana-5362	48	27	we	we	PRON
cana-5362	48	28	investigate	investigate	VERB
cana-5362	48	29	the	the	DET
cana-5362	48	30	well	well	ADJ
cana-5362	48	31	posedness	posedness	NOUN
cana-5362	48	32	of	of	ADP
cana-5362	48	33	(	(	PUNCT
cana-5362	48	34	1	1	NUM
cana-5362	48	35	)	)	PUNCT
cana-5362	48	36	,	,	PUNCT
cana-5362	48	37	according	accord	VERB
cana-5362	48	38	to	to	ADP
cana-5362	48	39	c.	c.	PROPN
cana-5362	48	40	catté	catté	PROPN
cana-5362	48	41	et	et	PROPN
cana-5362	48	42	al	al	PROPN
cana-5362	48	43	.	.	PUNCT
cana-5362	49	1	[	[	X
cana-5362	49	2	8	8	NUM
cana-5362	49	3	]	]	PUNCT
cana-5362	49	4	and	and	CCONJ
cana-5362	49	5	[	[	X
cana-5362	49	6	15	15	NUM
cana-5362	49	7	]	]	PUNCT
cana-5362	49	8	,	,	PUNCT
cana-5362	49	9	a.	a.	PROPN
cana-5362	49	10	bensoussan	bensoussan	PROPN
cana-5362	49	11	,	,	PUNCT
cana-5362	49	12	r.	r.	NOUN
cana-5362	49	13	temam	temam	NOUN
cana-5362	49	14	[	[	X
cana-5362	49	15	4	4	NUM
cana-5362	49	16	]	]	PUNCT
cana-5362	49	17	,	,	PUNCT
cana-5362	49	18	e.	e.	PROPN
cana-5362	49	19	pardoux	pardoux	PROPN
cana-5362	49	20	[	[	X
cana-5362	49	21	16	16	NUM
cana-5362	49	22	]	]	PUNCT
cana-5362	49	23	and	and	CCONJ
cana-5362	49	24	t.c	t.c	PROPN
cana-5362	49	25	.	.	PROPN
cana-5362	49	26	garrido	garrido	PROPN
cana-5362	50	1	[	[	X
cana-5362	50	2	11	11	NUM
cana-5362	50	3	]	]	PUNCT
cana-5362	50	4	.	.	PUNCT
cana-5362	51	1	the	the	DET
cana-5362	51	2	variational	variational	ADJ
cana-5362	51	3	method	method	NOUN
cana-5362	51	4	we	we	PRON
cana-5362	51	5	will	will	AUX
cana-5362	51	6	employ	employ	VERB
cana-5362	51	7	is	be	AUX
cana-5362	51	8	defined	define	VERB
cana-5362	51	9	within	within	ADP
cana-5362	51	10	two	two	NUM
cana-5362	51	11	fundamental	fundamental	ADJ
cana-5362	51	12	spaces	space	NOUN
cana-5362	51	13	:	:	PUNCT
cana-5362	51	14	a	a	DET
cana-5362	51	15	real	real	ADJ
cana-5362	51	16	,	,	PUNCT
cana-5362	51	17	reflexive	reflexive	ADJ
cana-5362	51	18	,	,	PUNCT
cana-5362	51	19	and	and	CCONJ
cana-5362	51	20	separable	separable	ADJ
cana-5362	51	21	banach	banach	NOUN
cana-5362	51	22	space	space	NOUN
cana-5362	51	23	𝑉	𝑉	PROPN
cana-5362	51	24	,	,	PUNCT
cana-5362	51	25	and	and	CCONJ
cana-5362	51	26	a	a	DET
cana-5362	51	27	real	real	ADJ
cana-5362	51	28	hilbert	hilbert	NOUN
cana-5362	51	29	space	space	NOUN
cana-5362	51	30	𝐻.	𝐻.	PROPN
cana-5362	51	31	we	we	PRON
cana-5362	51	32	identify	identify	VERB
cana-5362	51	33	𝐻	𝐻	PROPN
cana-5362	51	34	with	with	ADP
cana-5362	51	35	its	its	PRON
cana-5362	51	36	dual	dual	ADJ
cana-5362	51	37	and	and	CCONJ
cana-5362	51	38	denote	denote	VERB
cana-5362	51	39	the	the	DET
cana-5362	51	40	dual	dual	ADJ
cana-5362	51	41	space	space	NOUN
cana-5362	51	42	of	of	ADP
cana-5362	51	43	𝑉	𝑉	PROPN
cana-5362	51	44	𝑏𝑦	𝑏𝑦	NOUN
cana-5362	51	45	𝑉	𝑉	PROPN
cana-5362	51	46	′.	′.	NOUN
cana-5362	51	47	the	the	DET
cana-5362	51	48	embedding	embed	VERB
cana-5362	51	49	𝑉	𝑉	PROPN
cana-5362	51	50	↪	↪	PROPN
cana-5362	51	51	𝐻	𝐻	PROPN
cana-5362	51	52	is	be	AUX
cana-5362	51	53	continuous	continuous	ADJ
cana-5362	51	54	,	,	PUNCT
cana-5362	51	55	and	and	CCONJ
cana-5362	51	56	𝑉	𝑉	PROPN
cana-5362	51	57	is	be	AUX
cana-5362	51	58	dense	dense	ADJ
cana-5362	51	59	in	in	ADP
cana-5362	51	60	𝐻.	𝐻.	PROPN
cana-5362	51	61	these	these	DET
cana-5362	51	62	relationships	relationship	NOUN
cana-5362	51	63	are	be	AUX
cana-5362	51	64	summarised	summarise	VERB
cana-5362	51	65	as	as	ADP
cana-5362	51	66	:	:	PUNCT
cana-5362	51	67	𝑉	𝑉	PROPN
cana-5362	51	68	⊂	⊂	PROPN
cana-5362	51	69	𝐻	𝐻	PROPN
cana-5362	51	70	⊂	⊂	PUNCT
cana-5362	51	71	𝑉	𝑉	PROPN
cana-5362	52	1	′	′	NOUN
cana-5362	53	1	we	we	PRON
cana-5362	53	2	will	will	AUX
cana-5362	53	3	denote	denote	VERB
cana-5362	53	4	by	by	ADP
cana-5362	53	5	∥.	∥.	X
cana-5362	53	6	∥	∥	NOUN
cana-5362	53	7	,	,	PUNCT
cana-5362	53	8	|	|	ADV
cana-5362	53	9	.	.	PUNCT
cana-5362	54	1	|	|	ADV
cana-5362	55	1	and	and	CCONJ
cana-5362	55	2	∥.∥∗	∥.∥∗	PRON
cana-5362	55	3	the	the	DET
cana-5362	55	4	norms	norm	NOUN
cana-5362	55	5	in	in	ADP
cana-5362	55	6	𝑉	𝑉	PROPN
cana-5362	55	7	,	,	PUNCT
cana-5362	55	8	𝐻	𝐻	PROPN
cana-5362	55	9	and	and	CCONJ
cana-5362	55	10	𝑉	𝑉	PROPN
cana-5362	55	11	′	′	NOUN
cana-5362	55	12	respectively	respectively	ADV
cana-5362	55	13	;	;	PUNCT
cana-5362	55	14	by	by	ADP
cana-5362	55	15	⟨.	⟨.	PRON
cana-5362	55	16	,	,	PUNCT
cana-5362	55	17	.	.	PUNCT
cana-5362	56	1	⟩	⟩	NOUN
cana-5362	56	2	the	the	DET
cana-5362	56	3	duality	duality	NOUN
cana-5362	56	4	product	product	NOUN
cana-5362	56	5	between	between	ADP
cana-5362	56	6	𝑉	𝑉	PROPN
cana-5362	56	7	,	,	PUNCT
cana-5362	56	8	𝑉	𝑉	PROPN
cana-5362	56	9	′.	′.	NOUN
cana-5362	56	10	we	we	PRON
cana-5362	56	11	introduce	introduce	VERB
cana-5362	56	12	𝐻	𝐻	PROPN
cana-5362	56	13	,	,	PUNCT
cana-5362	56	14	𝑉	𝑉	PROPN
cana-5362	56	15	,	,	PUNCT
cana-5362	56	16	ℋ	ℋ	PROPN
cana-5362	56	17	and	and	CCONJ
cana-5362	56	18	𝒱	𝒱	PROPN
cana-5362	56	19	as	as	SCONJ
cana-5362	56	20	follow	follow	VERB
cana-5362	56	21	•	•	NUM
cana-5362	56	22	𝐻	𝐻	PROPN
cana-5362	56	23	=	=	SYM
cana-5362	56	24	𝐿2(𝐷	𝐿2(𝐷	PROPN
cana-5362	56	25	)	)	PUNCT
cana-5362	56	26	,	,	PUNCT
cana-5362	56	27	the	the	DET
cana-5362	56	28	space	space	NOUN
cana-5362	56	29	of	of	ADP
cana-5362	56	30	square	square	ADJ
cana-5362	56	31	-	-	PUNCT
cana-5362	56	32	integrable	integrable	ADJ
cana-5362	56	33	functions	function	NOUN
cana-5362	56	34	over	over	ADP
cana-5362	56	35	d	d	PROPN
cana-5362	56	36	,	,	PUNCT
cana-5362	56	37	with	with	ADP
cana-5362	56	38	the	the	DET
cana-5362	56	39	inner	inner	ADJ
cana-5362	56	40	product	product	NOUN
cana-5362	56	41	(	(	PUNCT
cana-5362	56	42	𝑢,𝓋	𝑢,𝓋	NOUN
cana-5362	56	43	)	)	PUNCT
cana-5362	56	44	=	=	SYM
cana-5362	56	45	∫𝐷𝑢(𝑥)𝓋(𝑥)𝑑𝑥	∫𝐷𝑢(𝑥)𝓋(𝑥)𝑑𝑥	PUNCT
cana-5362	56	46			NOUN
cana-5362	56	47	𝑉	𝑉	PROPN
cana-5362	56	48	=	=	PROPN
cana-5362	56	49	𝐻0	𝐻0	PROPN
cana-5362	56	50	1(𝐷	1(𝐷	PROPN
cana-5362	56	51	)	)	PUNCT
cana-5362	56	52	=	=	PRON
cana-5362	56	53	{	{	PUNCT
cana-5362	56	54	𝓋	𝓋	NOUN
cana-5362	56	55	∈	∈	PROPN
cana-5362	56	56	𝐻	𝐻	PROPN
cana-5362	56	57	,	,	PUNCT
cana-5362	56	58	𝜕𝓋	𝜕𝓋	ADV
cana-5362	56	59	𝜕𝑥	𝜕𝑥	X
cana-5362	56	60	∈	∈	PROPN
cana-5362	56	61	𝐻	𝐻	PROPN
cana-5362	56	62	,	,	PUNCT
cana-5362	56	63	𝑤𝑖𝑡ℎ	𝑤𝑖𝑡ℎ	NOUN
cana-5362	56	64	𝓋	𝓋	NOUN
cana-5362	56	65	=	=	SYM
cana-5362	56	66	0	0	NUM
cana-5362	56	67	𝑜𝑛	𝑜𝑛	PROPN
cana-5362	56	68	𝑡ℎ𝑒	𝑡ℎ𝑒	VERB
cana-5362	56	69	𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦	𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦	PROPN
cana-5362	56	70	γ	γ	PROPN
cana-5362	56	71	}	}	PUNCT
cana-5362	56	72	•	•	NUM
cana-5362	56	73	ℋ	ℋ	PROPN
cana-5362	56	74	=	=	SYM
cana-5362	56	75	𝐿2(𝐷,𝒫;𝐻	𝐿2(𝐷,𝒫;𝐻	NOUN
cana-5362	56	76	)	)	PUNCT
cana-5362	56	77	,	,	PUNCT
cana-5362	56	78	hilbert	hilbert	NOUN
cana-5362	56	79	space	space	NOUN
cana-5362	56	80	with	with	ADP
cana-5362	56	81	scalar	scalar	ADJ
cana-5362	56	82	product	product	NOUN
cana-5362	56	83	(	(	PUNCT
cana-5362	56	84	𝑢,𝓋)ℋ	𝑢,𝓋)ℋ	PROPN
cana-5362	56	85	=	=	SYM
cana-5362	56	86	𝐸(𝑢	𝐸(𝑢	X
cana-5362	56	87	(	(	PUNCT
cana-5362	56	88	.	.	PUNCT
cana-5362	56	89	)	)	PUNCT
cana-5362	56	90	,	,	PUNCT
cana-5362	56	91	𝓋	𝓋	PROPN
cana-5362	56	92	(	(	PUNCT
cana-5362	56	93	.	.	PUNCT
cana-5362	56	94	)	)	PUNCT
cana-5362	56	95	)	)	PUNCT
cana-5362	57	1	=	=	PRON
cana-5362	57	2	(	(	PUNCT
cana-5362	57	3	𝑢(𝑥	𝑢(𝑥	PROPN
cana-5362	57	4	)	)	PUNCT
cana-5362	57	5	,	,	PUNCT
cana-5362	57	6	𝓋(𝑥))𝑑𝒫(𝑥	𝓋(𝑥))𝑑𝒫(𝑥	VERB
cana-5362	57	7	)	)	PUNCT
cana-5362	57	8	(	(	PUNCT
cana-5362	57	9	4	4	NUM
cana-5362	57	10	)	)	PUNCT
cana-5362	57	11	•	•	NUM
cana-5362	57	12	𝒱	𝒱	NOUN
cana-5362	57	13	=	=	PUNCT
cana-5362	57	14	𝐿2(𝐷,𝒫	𝐿2(𝐷,𝒫	NOUN
cana-5362	57	15	;	;	PUNCT
cana-5362	57	16	𝑉	𝑉	PROPN
cana-5362	57	17	)	)	PUNCT
cana-5362	57	18	,	,	PUNCT
cana-5362	57	19	banach	banach	NOUN
cana-5362	57	20	space	space	NOUN
cana-5362	57	21	equipped	equip	VERB
cana-5362	57	22	with	with	ADP
cana-5362	57	23	the	the	DET
cana-5362	57	24	norm	norm	NOUN
cana-5362	57	25	‖𝑢‖𝒱	‖𝑢‖𝒱	PUNCT
cana-5362	58	1	=	=	PUNCT
cana-5362	58	2	{	{	PUNCT
cana-5362	58	3	∫𝐷‖𝑢‖𝑉	∫𝐷‖𝑢‖𝑉	NUM
cana-5362	58	4	2𝑑𝒫(𝑥	2𝑑𝒫(𝑥	NUM
cana-5362	58	5	)	)	PUNCT
cana-5362	58	6	}	}	PUNCT
cana-5362	58	7	1	1	NUM
cana-5362	58	8	2	2	NUM
cana-5362	58	9	(	(	PUNCT
cana-5362	58	10	5	5	NUM
cana-5362	58	11	)	)	PUNCT
cana-5362	58	12	communications	communication	NOUN
cana-5362	58	13	on	on	ADP
cana-5362	58	14	applied	apply	VERB
cana-5362	58	15	nonlinear	nonlinear	ADJ
cana-5362	58	16	analysis	analysis	NOUN
cana-5362	58	17	issn	issn	NOUN
cana-5362	58	18	:	:	PUNCT
cana-5362	58	19	1074	1074	NUM
cana-5362	58	20	-	-	PUNCT
cana-5362	58	21	133x	133x	NUM
cana-5362	58	22	vol	vol	VERB
cana-5362	58	23	32	32	NUM
cana-5362	58	24	no	no	NOUN
cana-5362	58	25	.	.	PUNCT
cana-5362	59	1	10s	10	NOUN
cana-5362	59	2	(	(	PUNCT
cana-5362	59	3	2025	2025	NUM
cana-5362	59	4	)	)	PUNCT
cana-5362	59	5	1988	1988	NUM
cana-5362	59	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5362	59	7	•	•	NUM
cana-5362	59	8	𝒱′	𝒱′	NOUN
cana-5362	59	9	is	be	AUX
cana-5362	59	10	the	the	DET
cana-5362	59	11	dual	dual	ADJ
cana-5362	59	12	space	space	NOUN
cana-5362	59	13	of	of	ADP
cana-5362	59	14	𝒱	𝒱	PROPN
cana-5362	59	15	,	,	PUNCT
cana-5362	59	16	by	by	ADP
cana-5362	59	17	the	the	DET
cana-5362	59	18	riesz	riesz	PROPN
cana-5362	59	19	representation	representation	NOUN
cana-5362	59	20	theorem	theorem	NOUN
cana-5362	59	21	,	,	PUNCT
cana-5362	59	22	the	the	DET
cana-5362	59	23	norm	norm	NOUN
cana-5362	59	24	of	of	ADP
cana-5362	59	25	𝒜(u	𝒜(u	PROPN
cana-5362	59	26	)	)	PUNCT
cana-5362	59	27	in	in	ADP
cana-5362	59	28	𝒱′	𝒱′	NUM
cana-5362	59	29	is	be	AUX
cana-5362	59	30	‖𝒜(𝑢)‖𝒱′	‖𝒜(𝑢)‖𝒱′	PUNCT
cana-5362	59	31	=	=	PRON
cana-5362	59	32	𝑠𝑢𝑝	𝑠𝑢𝑝	VERB
cana-5362	59	33	‖𝓋‖𝒱≤1	‖𝓋‖𝒱≤1	ADJ
cana-5362	59	34	|〈𝐴(𝑢	|〈𝐴(𝑢	PROPN
cana-5362	59	35	)	)	PUNCT
cana-5362	59	36	,	,	PUNCT
cana-5362	59	37	𝓋〉|	𝓋〉|	PROPN
cana-5362	59	38	(	(	PUNCT
cana-5362	59	39	6	6	NUM
cana-5362	59	40	)	)	PUNCT
cana-5362	59	41	we	we	PRON
cana-5362	59	42	introduce	introduce	VERB
cana-5362	59	43	the	the	DET
cana-5362	59	44	following	following	ADJ
cana-5362	59	45	notation	notation	NOUN
cana-5362	59	46	𝒜(𝑢	𝒜(𝑢	PUNCT
cana-5362	59	47	)	)	PUNCT
cana-5362	60	1	=	=	SYM
cana-5362	60	2	−𝑑𝑖𝑣	−𝑑𝑖𝑣	NOUN
cana-5362	60	3	(	(	PUNCT
cana-5362	60	4	𝑔(|𝛻𝐺𝜎	𝑔(|𝛻𝐺𝜎	NOUN
cana-5362	60	5	∗	∗	NOUN
cana-5362	60	6	𝑢|)𝛻𝑢	𝑢|)𝛻𝑢	NOUN
cana-5362	60	7	)	)	PUNCT
cana-5362	60	8	(	(	PUNCT
cana-5362	60	9	7	7	X
cana-5362	60	10	)	)	PUNCT
cana-5362	60	11	the	the	DET
cana-5362	60	12	next	next	ADJ
cana-5362	60	13	lemma	lemma	PROPN
cana-5362	60	14	plays	play	VERB
cana-5362	60	15	a	a	DET
cana-5362	60	16	crucial	crucial	ADJ
cana-5362	60	17	role	role	NOUN
cana-5362	60	18	in	in	ADP
cana-5362	60	19	establishing	establish	VERB
cana-5362	60	20	the	the	DET
cana-5362	60	21	well	well	NOUN
cana-5362	60	22	-	-	PUNCT
cana-5362	60	23	posedness	posedness	NOUN
cana-5362	60	24	of	of	ADP
cana-5362	60	25	the	the	DET
cana-5362	60	26	proposed	propose	VERB
cana-5362	60	27	model	model	NOUN
cana-5362	60	28	(	(	PUNCT
cana-5362	60	29	1	1	NUM
cana-5362	60	30	)	)	PUNCT
cana-5362	60	31	.	.	PUNCT
cana-5362	61	1	lemma	lemma	PROPN
cana-5362	61	2	1	1	NUM
cana-5362	61	3	let	let	VERB
cana-5362	61	4	𝒜	𝒜	NOUN
cana-5362	61	5	(	(	PUNCT
cana-5362	61	6	.	.	PUNCT
cana-5362	61	7	)	)	PUNCT
cana-5362	62	1	∶	∶	NOUN
cana-5362	62	2	𝒱	𝒱	NOUN
cana-5362	62	3	→	→	SYM
cana-5362	62	4	𝒱′	𝒱′	X
cana-5362	62	5	be	be	AUX
cana-5362	62	6	a	a	DET
cana-5362	62	7	nonlinear	nonlinear	ADJ
cana-5362	62	8	operator	operator	NOUN
cana-5362	62	9	defined	define	VERB
cana-5362	62	10	almost	almost	ADV
cana-5362	62	11	everywhere	everywhere	ADV
cana-5362	62	12	in	in	ADP
cana-5362	62	13	t.	t.	NOUN
cana-5362	62	14	we	we	PRON
cana-5362	62	15	assume	assume	VERB
cana-5362	62	16	the	the	DET
cana-5362	62	17	following	follow	VERB
cana-5362	62	18	hypotheses	hypothesis	NOUN
cana-5362	62	19	:	:	PUNCT
cana-5362	63	1	1	1	X
cana-5362	63	2	.	.	X
cana-5362	63	3	coercivity	coercivity	NOUN
cana-5362	63	4	:	:	PUNCT
cana-5362	63	5	∃	∃	PROPN
cana-5362	63	6	𝜌	𝜌	X
cana-5362	63	7	>	>	X
cana-5362	63	8	0	0	NUM
cana-5362	63	9	such	such	ADJ
cana-5362	63	10	that	that	SCONJ
cana-5362	63	11	〈	〈	NOUN
cana-5362	63	12	𝒜(𝑢),𝓋〉𝒱,𝒱′	𝒜(𝑢),𝓋〉𝒱,𝒱′	SYM
cana-5362	63	13	≥	≥	NOUN
cana-5362	63	14	𝜌‖𝑢‖𝒱	𝜌‖𝑢‖𝒱	PROPN
cana-5362	63	15	2	2	NUM
cana-5362	63	16	,	,	PUNCT
cana-5362	63	17	∀𝑢	∀𝑢	NOUN
cana-5362	63	18	,	,	PUNCT
cana-5362	63	19	𝓋	𝓋	PROPN
cana-5362	63	20	∈	∈	PROPN
cana-5362	63	21	𝒱	𝒱	PROPN
cana-5362	63	22	,	,	PUNCT
cana-5362	63	23	𝑎.	𝑎.	PROPN
cana-5362	63	24	𝑒.	𝑒.	PROPN
cana-5362	64	1	𝑡.	𝑡.	PROPN
cana-5362	64	2	(	(	PUNCT
cana-5362	64	3	8)	8)	NUM
cana-5362	64	4	2	2	NUM
cana-5362	64	5	.	.	PUNCT
cana-5362	65	1	monotonicity	monotonicity	NOUN
cana-5362	65	2	:	:	PUNCT
cana-5362	65	3	〈	〈	NOUN
cana-5362	65	4	𝒜(𝑢	𝒜(𝑢	NUM
cana-5362	65	5	)	)	PUNCT
cana-5362	65	6	−𝒜(𝓋	−𝒜(𝓋	PROPN
cana-5362	65	7	)	)	PUNCT
cana-5362	65	8	,	,	PUNCT
cana-5362	65	9	𝑢	𝑢	PRON
cana-5362	65	10	−	−	PROPN
cana-5362	65	11	𝓋〉𝒱,𝒱′	𝓋〉𝒱,𝒱′	PRON
cana-5362	65	12	≥	≥	NOUN
cana-5362	65	13	0	0	NUM
cana-5362	65	14	,	,	PUNCT
cana-5362	65	15	∀𝑢,𝓋	∀𝑢,𝓋	PROPN
cana-5362	65	16	∈	∈	PROPN
cana-5362	65	17	𝒱	𝒱	PROPN
cana-5362	65	18	,	,	PUNCT
cana-5362	65	19	𝑎.	𝑎.	PROPN
cana-5362	65	20	𝑒.	𝑒.	PROPN
cana-5362	66	1	𝑡.	𝑡.	NOUN
cana-5362	66	2	(	(	PUNCT
cana-5362	66	3	9	9	NUM
cana-5362	66	4	)	)	SYM
cana-5362	66	5	3	3	NUM
cana-5362	66	6	.	.	PUNCT
cana-5362	67	1	boundedness	boundedness	NOUN
cana-5362	67	2	:	:	PUNCT
cana-5362	67	3	∃𝛽	∃𝛽	PROPN
cana-5362	67	4	>	>	X
cana-5362	67	5	0	0	NUM
cana-5362	67	6	:	:	PUNCT
cana-5362	67	7	‖𝒜(𝑢)‖𝒱′	‖𝒜(𝑢)‖𝒱′	PUNCT
cana-5362	67	8	≤	≤	NOUN
cana-5362	67	9	𝛽‖𝑢‖𝒱	𝛽‖𝑢‖𝒱	VERB
cana-5362	67	10	,	,	PUNCT
cana-5362	67	11	∀𝑢	∀𝑢	PRON
cana-5362	67	12	∈	∈	PROPN
cana-5362	67	13	𝒱	𝒱	PROPN
cana-5362	67	14	,	,	PUNCT
cana-5362	67	15	𝑎.	𝑎.	PROPN
cana-5362	67	16	𝑒.	𝑒.	PROPN
cana-5362	68	1	𝑡.	𝑡.	PROPN
cana-5362	68	2	(	(	PUNCT
cana-5362	68	3	10	10	NUM
cana-5362	68	4	)	)	PUNCT
cana-5362	68	5	4	4	NUM
cana-5362	68	6	.	.	PUNCT
cana-5362	69	1	hemi	hemi	NOUN
cana-5362	69	2	-	-	PUNCT
cana-5362	69	3	continuity	continuity	NOUN
cana-5362	69	4	:	:	PUNCT
cana-5362	69	5	∀𝑢,𝓋	∀𝑢,𝓋	X
cana-5362	69	6	∈	∈	PROPN
cana-5362	69	7	𝒱	𝒱	PROPN
cana-5362	69	8	,	,	PUNCT
cana-5362	70	1	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-5362	70	2	𝜓	𝜓	PROPN
cana-5362	70	3	∈	∈	PROPN
cana-5362	70	4	𝒱′	𝒱′	PROPN
cana-5362	70	5	𝑎.	𝑎.	NOUN
cana-5362	70	6	𝑒.	𝑒.	PROPN
cana-5362	71	1	𝑡	𝑡	PROPN
cana-5362	71	2	∈	∈	PROPN
cana-5362	72	1	[	[	X
cana-5362	72	2	0	0	NUM
cana-5362	72	3	,	,	PUNCT
cana-5362	72	4	𝑇	𝑇	PROPN
cana-5362	72	5	]	]	X
cana-5362	72	6	𝛩	𝛩	PROPN
cana-5362	72	7	∈	∈	NOUN
cana-5362	72	8	ℝ	ℝ	PROPN
cana-5362	72	9	⟶	⟶	NOUN
cana-5362	72	10	⟨𝒜(𝑢	⟨𝒜(𝑢	NOUN
cana-5362	73	1	+	+	CCONJ
cana-5362	73	2	𝛩𝓋),𝜓⟩𝒱,𝒱′	𝛩𝓋),𝜓⟩𝒱,𝒱′	PROPN
cana-5362	73	3	𝑖𝑠	𝑖𝑠	PROPN
cana-5362	73	4	𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠	𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠	NOUN
cana-5362	73	5	𝑖𝑛	𝑖𝑛	PRON
cana-5362	73	6	ℝ	ℝ	PROPN
cana-5362	73	7	(	(	PUNCT
cana-5362	73	8	11	11	NUM
cana-5362	73	9	)	)	PUNCT
cana-5362	73	10	proof	proof	NOUN
cana-5362	73	11	.	.	PUNCT
cana-5362	74	1	according	accord	VERB
cana-5362	74	2	to	to	ADP
cana-5362	74	3	[	[	X
cana-5362	74	4	8	8	NUM
cana-5362	74	5	]	]	PUNCT
cana-5362	74	6	,	,	PUNCT
cana-5362	74	7	𝑔	𝑔	NOUN
cana-5362	74	8	,	,	PUNCT
cana-5362	74	9	𝐺	𝐺	PROPN
cana-5362	74	10	are	be	AUX
cana-5362	74	11	infinitely	infinitely	ADV
cana-5362	74	12	differentiable	differentiable	ADJ
cana-5362	74	13	in	in	ADP
cana-5362	74	14	𝐷.	𝐷.	PROPN
cana-5362	74	15	so	so	ADV
cana-5362	74	16	,	,	PUNCT
cana-5362	74	17	g(.)∈	g(.)∈	PROPN
cana-5362	74	18	𝐿∞(0,t	𝐿∞(0,t	PROPN
cana-5362	74	19	;	;	PUNCT
cana-5362	74	20	𝐶∞(d	𝐶∞(d	PROPN
cana-5362	74	21	)	)	PUNCT
cana-5362	74	22	)	)	PUNCT
cana-5362	74	23	.	.	PUNCT
cana-5362	75	1	thus	thus	ADV
cana-5362	75	2	,	,	PUNCT
cana-5362	75	3	since	since	SCONJ
cana-5362	75	4	𝑔	𝑔	PROPN
cana-5362	75	5	is	be	AUX
cana-5362	75	6	decreasing	decrease	VERB
cana-5362	75	7	,	,	PUNCT
cana-5362	75	8	there	there	PRON
cana-5362	75	9	exists	exist	VERB
cana-5362	75	10	a	a	DET
cana-5362	75	11	constant	constant	ADJ
cana-5362	75	12	𝜌	𝜌	ADP
cana-5362	75	13	such	such	ADJ
cana-5362	75	14	that	that	PRON
cana-5362	75	15	𝑔(|𝛻𝐺𝜎	𝑔(|𝛻𝐺𝜎	VERB
cana-5362	75	16	∗	∗	NOUN
cana-5362	75	17	𝜛|	𝜛|	PROPN
cana-5362	75	18	)	)	PUNCT
cana-5362	75	19	≥	≥	AUX
cana-5362	75	20	𝜌	𝜌	X
cana-5362	75	21	,	,	PUNCT
cana-5362	75	22	𝑎.	𝑎.	PROPN
cana-5362	75	23	𝑒.	𝑒.	PROPN
cana-5362	75	24	∈]0	∈]0	ADV
cana-5362	75	25	,	,	PUNCT
cana-5362	75	26	𝑇[×	𝑇[×	PROPN
cana-5362	75	27	𝐷	𝐷	PROPN
cana-5362	75	28	,	,	PUNCT
cana-5362	75	29	(	(	PUNCT
cana-5362	75	30	12	12	NUM
cana-5362	75	31	)	)	PUNCT
cana-5362	75	32	where	where	SCONJ
cana-5362	75	33	‖𝜛‖𝐿∞(0,𝑇:𝐿2(𝐷	‖𝜛‖𝐿∞(0,𝑇:𝐿2(𝐷	PROPN
cana-5362	75	34	)	)	PUNCT
cana-5362	75	35	)	)	PUNCT
cana-5362	76	1	≤	≤	NUM
cana-5362	76	2	‖𝑢0‖𝐿2(𝐷	‖𝑢0‖𝐿2(𝐷	PROPN
cana-5362	76	3	)	)	PUNCT
cana-5362	76	4	.	.	PUNCT
cana-5362	77	1	1	1	X
cana-5362	77	2	.	.	X
cana-5362	77	3	coercivity	coercivity	NOUN
cana-5362	77	4	〈	〈	PROPN
cana-5362	77	5	𝒜(𝑢	𝒜(𝑢	NUM
cana-5362	77	6	)	)	PUNCT
cana-5362	77	7	,	,	PUNCT
cana-5362	77	8	𝑢	𝑢	X
cana-5362	77	9	〉	〉	NOUN
cana-5362	77	10	=	=	SYM
cana-5362	77	11	−[𝑑𝑖𝑣(𝑔(|𝛻𝐺𝜎	−[𝑑𝑖𝑣(𝑔(|𝛻𝐺𝜎	NOUN
cana-5362	77	12	∗	∗	NOUN
cana-5362	77	13	𝑢|)𝛻𝑢)𝑢]𝑑𝒫(𝑥	𝑢|)𝛻𝑢)𝑢]𝑑𝒫(𝑥	NOUN
cana-5362	77	14	)	)	PUNCT
cana-5362	77	15	=	=	SYM
cana-5362	77	16	𝑔(|𝛻𝐺𝜎	𝑔(|𝛻𝐺𝜎	NOUN
cana-5362	77	17	∗	∗	NOUN
cana-5362	77	18	𝑢|)(𝛻𝑢	𝑢|)(𝛻𝑢	NUM
cana-5362	77	19	)	)	PUNCT
cana-5362	77	20	2𝑑𝒫(𝑥	2𝑑𝒫(𝑥	NUM
cana-5362	77	21	)	)	PUNCT
cana-5362	77	22	≥	≥	NOUN
cana-5362	77	23	𝜌‖𝑢‖𝒱	𝜌‖𝑢‖𝒱	X
cana-5362	77	24	2	2	NUM
cana-5362	77	25	(	(	PUNCT
cana-5362	77	26	13	13	NUM
cana-5362	77	27	)	)	PUNCT
cana-5362	77	28	hence	hence	ADV
cana-5362	77	29	,	,	PUNCT
cana-5362	77	30	(	(	PUNCT
cana-5362	77	31	8)	8)	NUM
cana-5362	77	32	hold	hold	NOUN
cana-5362	77	33	.	.	PUNCT
cana-5362	78	1	2	2	X
cana-5362	78	2	.	.	X
cana-5362	78	3	monotonicity	monotonicity	NOUN
cana-5362	78	4	⟨𝒜(𝑢	⟨𝒜(𝑢	NOUN
cana-5362	78	5	)	)	PUNCT
cana-5362	78	6	−𝒜(𝓋	−𝒜(𝓋	PROPN
cana-5362	78	7	)	)	PUNCT
cana-5362	78	8	,	,	PUNCT
cana-5362	78	9	𝑢	𝑢	PRON
cana-5362	78	10	−	−	PROPN
cana-5362	78	11	𝓋⟩	𝓋⟩	NOUN
cana-5362	78	12	=	=	PUNCT
cana-5362	78	13	−∫𝐷[(𝑑𝑖𝑣(𝑔(|𝛻𝐺𝜎	−∫𝐷[(𝑑𝑖𝑣(𝑔(|𝛻𝐺𝜎	NOUN
cana-5362	78	14	∗	∗	NOUN
cana-5362	78	15	𝑢|)𝛻𝑢	𝑢|)𝛻𝑢	NOUN
cana-5362	78	16	)	)	PUNCT
cana-5362	78	17	−	−	PROPN
cana-5362	78	18	𝑑𝑖𝑣(𝑔(|𝛻𝐺𝜎	𝑑𝑖𝑣(𝑔(|𝛻𝐺𝜎	PROPN
cana-5362	78	19	∗	∗	PROPN
cana-5362	78	20	𝓋	𝓋	PROPN
cana-5362	78	21	|)𝛻𝓋))(𝑢	|)𝛻𝓋))(𝑢	PROPN
cana-5362	78	22	−	−	PROPN
cana-5362	78	23	𝓋)]𝑑𝒫(𝑥	𝓋)]𝑑𝒫(𝑥	VERB
cana-5362	78	24	)	)	PUNCT
cana-5362	78	25	=	=	NOUN
cana-5362	78	26	−∫𝐷(𝑑𝑖𝑣(𝑔(|𝛻𝐺𝜎	−∫𝐷(𝑑𝑖𝑣(𝑔(|𝛻𝐺𝜎	NOUN
cana-5362	78	27	∗	∗	NOUN
cana-5362	78	28	𝑢|)𝛻𝑢)𝑢)𝑑𝒫(𝑥	𝑢|)𝛻𝑢)𝑢)𝑑𝒫(𝑥	NOUN
cana-5362	78	29	)	)	PUNCT
cana-5362	79	1	+	+	NUM
cana-5362	79	2	∫𝐷(𝑑𝑖𝑣(𝑔(|𝛻𝐺𝜎𝓋|)𝛻𝓋)𝑢)𝑑𝒫(𝑥	∫𝐷(𝑑𝑖𝑣(𝑔(|𝛻𝐺𝜎𝓋|)𝛻𝓋)𝑢)𝑑𝒫(𝑥	NOUN
cana-5362	79	3	)	)	PUNCT
cana-5362	79	4	+	+	ADP
cana-5362	79	5	∫𝐷(𝑑𝑖𝑣(𝑔(|𝛻𝐺𝜎	∫𝐷(𝑑𝑖𝑣(𝑔(|𝛻𝐺𝜎	X
cana-5362	79	6	∗	∗	NOUN
cana-5362	79	7	𝑢|)𝛻𝑢)𝓋)𝑑𝒫(𝑥	𝑢|)𝛻𝑢)𝓋)𝑑𝒫(𝑥	NOUN
cana-5362	79	8	)	)	PUNCT
cana-5362	80	1	−	−	NOUN
cana-5362	80	2	∫𝐷(𝑑𝑖𝑣(𝑔(|𝛻𝐺𝜎	∫𝐷(𝑑𝑖𝑣(𝑔(|𝛻𝐺𝜎	X
cana-5362	80	3	∗	∗	X
cana-5362	80	4	𝓋|)𝛻𝓋)𝓋)𝑑𝒫(𝑥	𝓋|)𝛻𝓋)𝓋)𝑑𝒫(𝑥	NOUN
cana-5362	80	5	)	)	PUNCT
cana-5362	80	6	=	=	PUNCT
cana-5362	81	1	∫𝐷𝑔(|𝛻𝐺𝜎	∫𝐷𝑔(|𝛻𝐺𝜎	X
cana-5362	81	2	∗	∗	NOUN
cana-5362	81	3	𝑢|)(𝛻𝑢	𝑢|)(𝛻𝑢	NUM
cana-5362	81	4	)	)	PUNCT
cana-5362	81	5	2𝑑𝒫(𝑥	2𝑑𝒫(𝑥	NUM
cana-5362	81	6	)	)	PUNCT
cana-5362	81	7	−	−	ADP
cana-5362	81	8	∫𝐷𝑔(|𝛻𝐺𝜎	∫𝐷𝑔(|𝛻𝐺𝜎	PROPN
cana-5362	81	9	∗	∗	NOUN
cana-5362	81	10	𝓋|)𝛻𝑢𝛻𝓋𝑑𝒫(𝑥	𝓋|)𝛻𝑢𝛻𝓋𝑑𝒫(𝑥	NOUN
cana-5362	81	11	)	)	PUNCT
cana-5362	81	12	−∫𝐷𝑔(|𝛻𝐺𝜎	−∫𝐷𝑔(|𝛻𝐺𝜎	NUM
cana-5362	81	13	∗	∗	NOUN
cana-5362	81	14	𝑢|)𝛻𝑢𝛻𝓋𝑑𝒫(𝑥	𝑢|)𝛻𝑢𝛻𝓋𝑑𝒫(𝑥	NOUN
cana-5362	81	15	)	)	PUNCT
cana-5362	82	1	+	+	CCONJ
cana-5362	82	2	∫𝐷𝑔(|𝛻𝐺𝜎	∫𝐷𝑔(|𝛻𝐺𝜎	PROPN
cana-5362	82	3	∗	∗	NOUN
cana-5362	82	4	𝓋|)(𝛻𝓋	𝓋|)(𝛻𝓋	PROPN
cana-5362	82	5	)	)	PUNCT
cana-5362	82	6	2𝑑𝒫(𝑥	2𝑑𝒫(𝑥	NUM
cana-5362	82	7	)	)	PUNCT
cana-5362	82	8	communications	communication	NOUN
cana-5362	82	9	on	on	ADP
cana-5362	82	10	applied	apply	VERB
cana-5362	82	11	nonlinear	nonlinear	ADJ
cana-5362	82	12	analysis	analysis	NOUN
cana-5362	82	13	issn	issn	NOUN
cana-5362	82	14	:	:	PUNCT
cana-5362	82	15	1074	1074	NUM
cana-5362	82	16	-	-	PUNCT
cana-5362	82	17	133x	133x	NUM
cana-5362	82	18	vol	vol	VERB
cana-5362	82	19	32	32	NUM
cana-5362	82	20	no	no	NOUN
cana-5362	82	21	.	.	PUNCT
cana-5362	83	1	10s	10	NOUN
cana-5362	83	2	(	(	PUNCT
cana-5362	83	3	2025	2025	NUM
cana-5362	83	4	)	)	PUNCT
cana-5362	83	5	1989	1989	NUM
cana-5362	83	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5362	83	7	=	=	PUNCT
cana-5362	83	8	∫𝐷𝑔(|𝛻𝐺𝜎	∫𝐷𝑔(|𝛻𝐺𝜎	X
cana-5362	83	9	∗	∗	NOUN
cana-5362	83	10	𝑢|)(𝛻𝑢	𝑢|)(𝛻𝑢	NUM
cana-5362	83	11	2	2	NUM
cana-5362	83	12	−	−	ADP
cana-5362	83	13	𝛻𝑢𝛻𝓋)𝑑𝒫	𝛻𝑢𝛻𝓋)𝑑𝒫	NOUN
cana-5362	83	14	+	+	CCONJ
cana-5362	83	15	∫𝐷𝑔(|𝛻𝐺𝜎	∫𝐷𝑔(|𝛻𝐺𝜎	PROPN
cana-5362	83	16	∗	∗	NOUN
cana-5362	83	17	𝓋|)(𝛻𝓋	𝓋|)(𝛻𝓋	VERB
cana-5362	83	18	2	2	NUM
cana-5362	83	19	−	−	NOUN
cana-5362	83	20	𝛻𝑢𝛻𝓋)𝑑𝒫(𝑥	𝛻𝑢𝛻𝓋)𝑑𝒫(𝑥	VERB
cana-5362	83	21	)	)	PUNCT
cana-5362	83	22	≥	≥	PROPN
cana-5362	83	23	min	min	PROPN
cana-5362	83	24	ρ	ρ	PROPN
cana-5362	83	25	,	,	PUNCT
cana-5362	83	26	ρ′≥0	ρ′≥0	X
cana-5362	83	27	(	(	PUNCT
cana-5362	83	28	ρ	ρ	NOUN
cana-5362	83	29	,	,	PUNCT
cana-5362	83	30	ρ′	ρ′	NUM
cana-5362	83	31	)	)	PUNCT
cana-5362	83	32	(	(	PUNCT
cana-5362	83	33	∫𝐷(𝛻𝑢	∫𝐷(𝛻𝑢	VERB
cana-5362	83	34	−	−	PROPN
cana-5362	83	35	𝛻𝓋	𝛻𝓋	NOUN
cana-5362	83	36	)	)	PUNCT
cana-5362	83	37	2𝑑𝒫(𝑥	2𝑑𝒫(𝑥	NUM
cana-5362	83	38	)	)	PUNCT
cana-5362	83	39	)	)	PUNCT
cana-5362	84	1	=	=	NUM
cana-5362	84	2	min	min	NOUN
cana-5362	84	3	ρ	ρ	PROPN
cana-5362	84	4	,	,	PUNCT
cana-5362	84	5	ρ′≥0	ρ′≥0	X
cana-5362	84	6	(	(	PUNCT
cana-5362	84	7	ρ	ρ	NOUN
cana-5362	84	8	,	,	PUNCT
cana-5362	84	9	ρ′	ρ′	NUM
cana-5362	84	10	)	)	PUNCT
cana-5362	84	11	‖u	‖u	NOUN
cana-5362	84	12	−	−	PROPN
cana-5362	84	13	𝓋‖𝒱	𝓋‖𝒱	PROPN
cana-5362	84	14	2	2	NUM
cana-5362	84	15	≥	≥	NOUN
cana-5362	84	16	0	0	NUM
cana-5362	84	17	,	,	PUNCT
cana-5362	84	18	∀u	∀u	NOUN
cana-5362	84	19	,	,	PUNCT
cana-5362	84	20	𝓋	𝓋	PROPN
cana-5362	84	21	∈	∈	PROPN
cana-5362	84	22	𝒱	𝒱	PROPN
cana-5362	84	23	(	(	PUNCT
cana-5362	84	24	14	14	NUM
cana-5362	84	25	)	)	PUNCT
cana-5362	84	26	then	then	ADV
cana-5362	84	27	,	,	PUNCT
cana-5362	84	28	the	the	DET
cana-5362	84	29	condition	condition	NOUN
cana-5362	84	30	(	(	PUNCT
cana-5362	84	31	9	9	X
cana-5362	84	32	)	)	PUNCT
cana-5362	84	33	hold	hold	NOUN
cana-5362	84	34	.	.	PUNCT
cana-5362	85	1	3	3	X
cana-5362	85	2	.	.	X
cana-5362	85	3	boundedness	boundedness	PROPN
cana-5362	85	4	now	now	ADV
cana-5362	85	5	,	,	PUNCT
cana-5362	85	6	we	we	PRON
cana-5362	85	7	verify	verify	VERB
cana-5362	85	8	the	the	DET
cana-5362	85	9	boundedness	boundedness	NOUN
cana-5362	85	10	condition	condition	NOUN
cana-5362	85	11	,	,	PUNCT
cana-5362	85	12	we	we	PRON
cana-5362	85	13	use	use	VERB
cana-5362	85	14	the	the	DET
cana-5362	85	15	weak	weak	ADJ
cana-5362	85	16	form	form	NOUN
cana-5362	85	17	of	of	ADP
cana-5362	85	18	𝒜(u	𝒜(u	PRON
cana-5362	85	19	)	)	PUNCT
cana-5362	85	20	and	and	CCONJ
cana-5362	85	21	the	the	DET
cana-5362	85	22	cauchyschwartz	cauchyschwartz	NOUN
cana-5362	85	23	inequality	inequality	NOUN
cana-5362	85	24	,	,	PUNCT
cana-5362	85	25	we	we	PRON
cana-5362	85	26	obtain	obtain	VERB
cana-5362	85	27	‖𝒜(𝑢)‖𝒱′	‖𝒜(𝑢)‖𝒱′	X
cana-5362	85	28	=	=	PUNCT
cana-5362	85	29	sup	sup	NOUN
cana-5362	85	30	‖v‖𝒱′≤1	‖v‖𝒱′≤1	VERB
cana-5362	85	31	|⟨𝒜(𝑢),𝓋⟩|	|⟨𝒜(𝑢),𝓋⟩|	ADJ
cana-5362	85	32	=	=	SYM
cana-5362	85	33	sup	sup	NUM
cana-5362	85	34	‖𝓋‖𝒱′≤1	‖𝓋‖𝒱′≤1	PROPN
cana-5362	85	35	|∫𝐷	|∫𝐷	PROPN
cana-5362	85	36	(	(	PUNCT
cana-5362	85	37	𝑔(|𝛻𝐺𝜎	𝑔(|𝛻𝐺𝜎	NOUN
cana-5362	85	38	∗	∗	NOUN
cana-5362	85	39	𝑢|)𝛻𝑢𝛻𝓋)𝑑𝒫|	𝑢|)𝛻𝑢𝛻𝓋)𝑑𝒫|	PROPN
cana-5362	85	40	≤	≤	NOUN
cana-5362	85	41	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-5362	85	42	‖𝓋‖𝒱′≤1	‖𝓋‖𝒱′≤1	VERB
cana-5362	85	43	∥	∥	PUNCT
cana-5362	85	44	𝑔(|𝛻𝐺𝜎	𝑔(|𝛻𝐺𝜎	NUM
cana-5362	85	45	∗	∗	NOUN
cana-5362	85	46	𝑢|	𝑢|	NOUN
cana-5362	85	47	)	)	PUNCT
cana-5362	85	48	∥𝐿2(𝐷)∥	∥𝐿2(𝐷)∥	ADP
cana-5362	85	49	𝛻𝑢	𝛻𝑢	X
cana-5362	85	50	∥𝐿2(𝐷)∥	∥𝐿2(𝐷)∥	X
cana-5362	85	51	𝛻𝓋	𝛻𝓋	X
cana-5362	85	52	∥𝐿2(𝐷	∥𝐿2(𝐷	PROPN
cana-5362	85	53	)	)	PUNCT
cana-5362	85	54	(	(	PUNCT
cana-5362	85	55	15	15	NUM
cana-5362	85	56	)	)	PUNCT
cana-5362	85	57	by	by	ADP
cana-5362	85	58	using	use	VERB
cana-5362	85	59	the	the	DET
cana-5362	85	60	poincaré	poincaré	PROPN
cana-5362	85	61	inequality	inequality	NOUN
cana-5362	85	62	,	,	PUNCT
cana-5362	85	63	we	we	PRON
cana-5362	85	64	obtain	obtain	VERB
cana-5362	85	65	‖𝒜(𝑢)‖𝒱′	‖𝒜(𝑢)‖𝒱′	PUNCT
cana-5362	85	66	≤	≤	PROPN
cana-5362	85	67	𝛽‖𝑢‖𝒱	𝛽‖𝑢‖𝒱	ADJ
cana-5362	85	68	,	,	PUNCT
cana-5362	85	69	𝛽	𝛽	NOUN
cana-5362	85	70	>	>	X
cana-5362	85	71	0	0	NUM
cana-5362	85	72	.	.	PUNCT
cana-5362	86	1	(	(	PUNCT
cana-5362	86	2	16	16	NUM
cana-5362	86	3	)	)	PUNCT
cana-5362	86	4	hence	hence	ADV
cana-5362	86	5	,	,	PUNCT
cana-5362	86	6	(	(	PUNCT
cana-5362	86	7	10	10	NUM
cana-5362	86	8	)	)	PUNCT
cana-5362	86	9	holds	hold	NOUN
cana-5362	86	10	.	.	PUNCT
cana-5362	87	1	4	4	X
cana-5362	87	2	.	.	X
cana-5362	87	3	hemi	hemi	NOUN
cana-5362	87	4	-	-	PUNCT
cana-5362	87	5	continuity	continuity	NOUN
cana-5362	87	6	:	:	PUNCT
cana-5362	87	7	in	in	ADP
cana-5362	87	8	[	[	X
cana-5362	87	9	8	8	NUM
cana-5362	87	10	]	]	PUNCT
cana-5362	87	11	,	,	PUNCT
cana-5362	87	12	we	we	PRON
cana-5362	87	13	find	find	VERB
cana-5362	87	14	the	the	DET
cana-5362	87	15	weak	weak	ADJ
cana-5362	87	16	continuity	continuity	NOUN
cana-5362	87	17	of	of	ADP
cana-5362	87	18	the	the	DET
cana-5362	87	19	deterministic	deterministic	ADJ
cana-5362	87	20	case	case	NOUN
cana-5362	87	21	of	of	ADP
cana-5362	87	22	(	(	PUNCT
cana-5362	87	23	1	1	NUM
cana-5362	87	24	)	)	PUNCT
cana-5362	87	25	,	,	PUNCT
cana-5362	87	26	for	for	ADP
cana-5362	87	27	𝛩	𝛩	PRON
cana-5362	87	28	→	→	SYM
cana-5362	87	29	0	0	NUM
cana-5362	87	30	⟨𝒜(𝑢(𝑥	⟨𝒜(𝑢(𝑥	PROPN
cana-5362	87	31	)	)	PUNCT
cana-5362	88	1	+	+	CCONJ
cana-5362	88	2	𝛩𝓋(𝑥	𝛩𝓋(𝑥	ADJ
cana-5362	88	3	)	)	PUNCT
cana-5362	88	4	)	)	PUNCT
cana-5362	88	5	,	,	PUNCT
cana-5362	88	6	𝜓(𝑥)⟩	𝜓(𝑥)⟩	PROPN
cana-5362	88	7	⟶	⟶	NOUN
cana-5362	88	8	⟨𝒜(𝑢(𝑥	⟨𝒜(𝑢(𝑥	PROPN
cana-5362	88	9	)	)	PUNCT
cana-5362	88	10	)	)	PUNCT
cana-5362	88	11	,	,	PUNCT
cana-5362	88	12	𝜓(𝑥)⟩	𝜓(𝑥)⟩	PROPN
cana-5362	88	13	𝑎.	𝑎.	PROPN
cana-5362	88	14	𝑠	𝑠	PROPN
cana-5362	88	15	(	(	PUNCT
cana-5362	88	16	17	17	NUM
cana-5362	88	17	)	)	PUNCT
cana-5362	88	18	by	by	ADP
cana-5362	88	19	using	use	VERB
cana-5362	88	20	the	the	DET
cana-5362	88	21	boundedness	boundedness	NOUN
cana-5362	88	22	(	(	PUNCT
cana-5362	88	23	14	14	NUM
cana-5362	88	24	)	)	PUNCT
cana-5362	88	25	,	,	PUNCT
cana-5362	88	26	we	we	PRON
cana-5362	88	27	obtain	obtain	VERB
cana-5362	88	28	|⟨𝒜(𝑢(𝑥	|⟨𝒜(𝑢(𝑥	NUM
cana-5362	88	29	)	)	PUNCT
cana-5362	89	1	+	+	CCONJ
cana-5362	89	2	𝛩𝓋(𝑥	𝛩𝓋(𝑥	ADJ
cana-5362	89	3	)	)	PUNCT
cana-5362	89	4	)	)	PUNCT
cana-5362	89	5	,	,	PUNCT
cana-5362	89	6	𝜓(𝑥)⟩|	𝜓(𝑥)⟩|	NOUN
cana-5362	89	7	<	<	X
cana-5362	89	8	‖𝒜(𝑢(𝑥	‖𝒜(𝑢(𝑥	PROPN
cana-5362	89	9	)	)	PUNCT
cana-5362	89	10	)	)	PUNCT
cana-5362	90	1	+	+	CCONJ
cana-5362	90	2	𝛩𝓋(𝑥))‖𝒱‖𝜓(𝑥)‖𝒱	𝛩𝓋(𝑥))‖𝒱‖𝜓(𝑥)‖𝒱	VERB
cana-5362	90	3	≤	≤	NUM
cana-5362	90	4	𝛽‖𝑢(𝑥	𝛽‖𝑢(𝑥	PROPN
cana-5362	90	5	)	)	PUNCT
cana-5362	90	6	+	+	SYM
cana-5362	90	7	𝛩𝓋(𝑥)‖𝒱′	𝛩𝓋(𝑥)‖𝒱′	NUM
cana-5362	90	8	𝑝−1‖𝜓(𝑥)‖𝒱	𝑝−1‖𝜓(𝑥)‖𝒱	NOUN
cana-5362	90	9	≤	≤	NUM
cana-5362	90	10	𝛽	𝛽	NOUN
cana-5362	90	11	(	(	PUNCT
cana-5362	90	12	‖𝑢(𝑥)‖𝒱′	‖𝑢(𝑥)‖𝒱′	NUM
cana-5362	90	13	+	+	NUM
cana-5362	90	14	‖𝓋(𝑥)‖𝒱′	‖𝓋(𝑥)‖𝒱′	NUM
cana-5362	90	15	)	)	PUNCT
cana-5362	90	16	𝑝−1	𝑝−1	PROPN
cana-5362	90	17	‖𝜓(𝑥)‖𝒱	‖𝜓(𝑥)‖𝒱	PROPN
cana-5362	90	18	(	(	PUNCT
cana-5362	90	19	18	18	NUM
cana-5362	90	20	)	)	PUNCT
cana-5362	90	21	(	(	PUNCT
cana-5362	90	22	by	by	ADP
cana-5362	90	23	limiting	limit	VERB
cana-5362	90	24	to	to	ADP
cana-5362	90	25	|𝛩|	|𝛩|	NOUN
cana-5362	90	26	≤	≤	NUM
cana-5362	90	27	1	1	NUM
cana-5362	90	28	,	,	PUNCT
cana-5362	90	29	it	it	PRON
cana-5362	90	30	is	be	AUX
cana-5362	90	31	sufficient	sufficient	ADJ
cana-5362	90	32	)	)	PUNCT
cana-5362	90	33	.	.	PUNCT
cana-5362	91	1	according	accord	VERB
cana-5362	91	2	to	to	ADP
cana-5362	91	3	lebesgue	lebesgue	PROPN
cana-5362	91	4	’s	’s	PART
cana-5362	91	5	theorem	theorem	PROPN
cana-5362	91	6	,	,	PUNCT
cana-5362	91	7	we	we	PRON
cana-5362	91	8	obtain	obtain	VERB
cana-5362	91	9	∫𝐷⟨𝒜(𝑢(𝑥	∫𝐷⟨𝒜(𝑢(𝑥	NOUN
cana-5362	91	10	)	)	PUNCT
cana-5362	92	1	+	+	CCONJ
cana-5362	92	2	𝛩𝓋(𝑥	𝛩𝓋(𝑥	ADJ
cana-5362	92	3	)	)	PUNCT
cana-5362	92	4	)	)	PUNCT
cana-5362	92	5	,	,	PUNCT
cana-5362	92	6	𝜓(𝑥)⟩𝑑𝒫(𝑥	𝜓(𝑥)⟩𝑑𝒫(𝑥	NOUN
cana-5362	92	7	)	)	PUNCT
cana-5362	92	8	→	→	SYM
cana-5362	92	9	∫𝐷⟨𝒜(𝑢(𝑥	∫𝐷⟨𝒜(𝑢(𝑥	NUM
cana-5362	92	10	)	)	PUNCT
cana-5362	92	11	)	)	PUNCT
cana-5362	92	12	,	,	PUNCT
cana-5362	92	13	𝜓(𝑥)⟩𝑑𝒫(𝑥	𝜓(𝑥)⟩𝑑𝒫(𝑥	NOUN
cana-5362	92	14	)	)	PUNCT
cana-5362	92	15	(	(	PUNCT
cana-5362	92	16	19	19	NUM
cana-5362	92	17	)	)	PUNCT
cana-5362	92	18	hence	hence	ADV
cana-5362	92	19	,	,	PUNCT
cana-5362	92	20	(	(	PUNCT
cana-5362	92	21	10	10	NUM
cana-5362	92	22	)	)	PUNCT
cana-5362	92	23	holds	hold	VERB
cana-5362	92	24	.	.	PUNCT
cana-5362	93	1	after	after	ADP
cana-5362	93	2	proving	prove	VERB
cana-5362	93	3	lemma	lemma	PROPN
cana-5362	93	4	1	1	NUM
cana-5362	93	5	,	,	PUNCT
cana-5362	93	6	we	we	PRON
cana-5362	93	7	need	need	VERB
cana-5362	93	8	to	to	PART
cana-5362	93	9	prove	prove	VERB
cana-5362	93	10	the	the	DET
cana-5362	93	11	following	follow	VERB
cana-5362	93	12	propositions	proposition	NOUN
cana-5362	93	13	2	2	NUM
cana-5362	93	14	and	and	CCONJ
cana-5362	93	15	3	3	NUM
cana-5362	93	16	to	to	PART
cana-5362	93	17	verify	verify	VERB
cana-5362	93	18	existence	existence	NOUN
cana-5362	93	19	and	and	CCONJ
cana-5362	93	20	uniqueness	uniqueness	NOUN
cana-5362	93	21	in	in	ADP
cana-5362	93	22	the	the	DET
cana-5362	93	23	weak	weak	ADJ
cana-5362	93	24	sense	sense	NOUN
cana-5362	93	25	.	.	PUNCT
cana-5362	94	1	proposition	proposition	NOUN
cana-5362	94	2	2	2	NUM
cana-5362	94	3	if	if	SCONJ
cana-5362	94	4	t	t	NOUN
cana-5362	94	5	→	→	SYM
cana-5362	94	6	u(t	u(t	NOUN
cana-5362	94	7	)	)	PUNCT
cana-5362	94	8	is	be	AUX
cana-5362	94	9	a	a	DET
cana-5362	94	10	measurable	measurable	ADJ
cana-5362	94	11	mapping	mapping	NOUN
cana-5362	94	12	with	with	ADP
cana-5362	94	13	values	value	NOUN
cana-5362	94	14	in	in	ADP
cana-5362	94	15	𝒱	𝒱	PROPN
cana-5362	94	16	,	,	PUNCT
cana-5362	94	17	then	then	ADV
cana-5362	94	18	𝑡	𝑡	X
cana-5362	94	19	→	→	SYM
cana-5362	94	20	𝒜(𝑢(𝑡	𝒜(𝑢(𝑡	NUM
cana-5362	94	21	)	)	PUNCT
cana-5362	94	22	)	)	PUNCT
cana-5362	94	23	is	be	AUX
cana-5362	94	24	measurable	measurable	ADJ
cana-5362	94	25	with	with	ADP
cana-5362	94	26	values	value	NOUN
cana-5362	94	27	in	in	ADP
cana-5362	94	28	𝒱′.	𝒱′.	ADJ
cana-5362	94	29	communications	communication	NOUN
cana-5362	94	30	on	on	ADP
cana-5362	94	31	applied	apply	VERB
cana-5362	94	32	nonlinear	nonlinear	ADJ
cana-5362	94	33	analysis	analysis	NOUN
cana-5362	94	34	issn	issn	NOUN
cana-5362	94	35	:	:	PUNCT
cana-5362	94	36	1074	1074	NUM
cana-5362	94	37	-	-	PUNCT
cana-5362	94	38	133x	133x	NUM
cana-5362	94	39	vol	vol	VERB
cana-5362	94	40	32	32	NUM
cana-5362	94	41	no	no	NOUN
cana-5362	94	42	.	.	PUNCT
cana-5362	95	1	10s	10	NOUN
cana-5362	95	2	(	(	PUNCT
cana-5362	95	3	2025	2025	NUM
cana-5362	95	4	)	)	PUNCT
cana-5362	95	5	1990	1990	NUM
cana-5362	95	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5362	95	7	proof	proof	NOUN
cana-5362	95	8	.	.	PUNCT
cana-5362	96	1	we	we	PRON
cana-5362	96	2	have	have	AUX
cana-5362	96	3	proved	prove	VERB
cana-5362	96	4	above	above	ADV
cana-5362	96	5	that	that	SCONJ
cana-5362	96	6	the	the	DET
cana-5362	96	7	operator	operator	NOUN
cana-5362	96	8	𝒜	𝒜	NOUN
cana-5362	96	9	∶	∶	NOUN
cana-5362	96	10	𝒱	𝒱	NOUN
cana-5362	96	11	→	→	SYM
cana-5362	96	12	𝒱′	𝒱′	NUM
cana-5362	96	13	is	be	AUX
cana-5362	96	14	hemi	hemi	NOUN
cana-5362	96	15	-	-	ADJ
cana-5362	96	16	continuous	continuous	ADJ
cana-5362	96	17	,	,	PUNCT
cana-5362	96	18	monotone	monotone	ADJ
cana-5362	96	19	,	,	PUNCT
cana-5362	96	20	and	and	CCONJ
cana-5362	96	21	coercive	coercive	ADJ
cana-5362	96	22	.	.	PUNCT
cana-5362	97	1	consequently	consequently	ADV
cana-5362	97	2	,	,	PUNCT
cana-5362	97	3	𝒜	𝒜	NOUN
cana-5362	97	4	is	be	AUX
cana-5362	97	5	continuous	continuous	ADJ
cana-5362	97	6	from	from	ADP
cana-5362	97	7	strong	strong	ADJ
cana-5362	97	8	𝒱	𝒱	PROPN
cana-5362	97	9	to	to	PART
cana-5362	97	10	weak	weak	VERB
cana-5362	97	11	𝒱′.	𝒱′.	NOUN
cana-5362	97	12	hence	hence	ADV
cana-5362	97	13	,	,	PUNCT
cana-5362	97	14	the	the	DET
cana-5362	97	15	proposition	proposition	NOUN
cana-5362	97	16	2	2	NUM
cana-5362	97	17	holds	hold	NOUN
cana-5362	97	18	.	.	PUNCT
cana-5362	98	1	(	(	PUNCT
cana-5362	98	2	see	see	VERB
cana-5362	98	3	[	[	X
cana-5362	98	4	4	4	NUM
cana-5362	98	5	]	]	PUNCT
cana-5362	98	6	,	,	PUNCT
cana-5362	98	7	p.105	p.105	NOUN
cana-5362	98	8	)	)	PUNCT
cana-5362	98	9	proposition	proposition	NOUN
cana-5362	98	10	3	3	NUM
cana-5362	98	11	if	if	SCONJ
cana-5362	98	12	𝑢	𝑢	PROPN
cana-5362	98	13	∈	∈	PROPN
cana-5362	98	14	𝐿2(𝒱	𝐿2(𝒱	PROPN
cana-5362	98	15	)	)	PUNCT
cana-5362	98	16	,	,	PUNCT
cana-5362	98	17	then	then	ADV
cana-5362	98	18	{	{	PUNCT
cana-5362	98	19	𝑡	𝑡	PROPN
cana-5362	98	20	→	→	SYM
cana-5362	98	21	𝒜(𝑢(𝑡	𝒜(𝑢(𝑡	NUM
cana-5362	98	22	)	)	PUNCT
cana-5362	98	23	)	)	PUNCT
cana-5362	98	24	}	}	PUNCT
cana-5362	98	25	∈	∈	PROPN
cana-5362	98	26	𝐿2(𝒱′	𝐿2(𝒱′	PROPN
cana-5362	98	27	)	)	PUNCT
cana-5362	98	28	𝑎.	𝑎.	PROPN
cana-5362	98	29	𝑒.	𝑒.	PROPN
cana-5362	99	1	𝑡.	𝑡.	NOUN
cana-5362	99	2	(	(	PUNCT
cana-5362	99	3	20	20	NUM
cana-5362	99	4	)	)	PUNCT
cana-5362	99	5	proof	proof	NOUN
cana-5362	99	6	.	.	PUNCT
cana-5362	100	1	if	if	SCONJ
cana-5362	100	2	𝑢	𝑢	PRON
cana-5362	100	3	∈	∈	PROPN
cana-5362	100	4	𝐿2(𝒱	𝐿2(𝒱	PROPN
cana-5362	100	5	)	)	PUNCT
cana-5362	100	6	,	,	PUNCT
cana-5362	100	7	then	then	ADV
cana-5362	100	8	by	by	ADP
cana-5362	100	9	the	the	DET
cana-5362	100	10	proposition	proposition	NOUN
cana-5362	100	11	2	2	NUM
cana-5362	100	12	,	,	PUNCT
cana-5362	100	13	𝒜(𝑢(𝑡	𝒜(𝑢(𝑡	NOUN
cana-5362	100	14	)	)	PUNCT
cana-5362	100	15	)	)	PUNCT
cana-5362	100	16	is	be	AUX
cana-5362	100	17	measurable	measurable	ADJ
cana-5362	100	18	with	with	ADP
cana-5362	100	19	values	value	NOUN
cana-5362	100	20	in	in	ADP
cana-5362	100	21	𝒱′	𝒱′	NOUN
cana-5362	100	22	,	,	PUNCT
cana-5362	100	23	and	and	CCONJ
cana-5362	100	24	by	by	ADP
cana-5362	100	25	(	(	PUNCT
cana-5362	100	26	14	14	NUM
cana-5362	100	27	)	)	PUNCT
cana-5362	100	28	,	,	PUNCT
cana-5362	100	29	‖𝒜(𝑢)‖𝒱′	‖𝒜(𝑢)‖𝒱′	PROPN
cana-5362	100	30	≤	≤	X
cana-5362	100	31	𝛽‖𝑢‖𝒱.	𝛽‖𝑢‖𝒱.	NOUN
cana-5362	100	32	raising	raise	VERB
cana-5362	100	33	this	this	DET
cana-5362	100	34	inequality	inequality	NOUN
cana-5362	100	35	to	to	ADP
cana-5362	100	36	the	the	DET
cana-5362	100	37	power	power	NOUN
cana-5362	100	38	2	2	NUM
cana-5362	100	39	and	and	CCONJ
cana-5362	100	40	integrating	integrating	NOUN
cana-5362	100	41	,	,	PUNCT
cana-5362	100	42	it	it	PRON
cana-5362	100	43	follows	follow	VERB
cana-5362	100	44	that	that	SCONJ
cana-5362	100	45	𝓐(𝑢(𝑡	𝓐(𝑢(𝑡	NOUN
cana-5362	100	46	)	)	PUNCT
cana-5362	100	47	)	)	PUNCT
cana-5362	101	1	∈	∈	PROPN
cana-5362	101	2	𝐿2	𝐿2	NOUN
cana-5362	101	3	(	(	PUNCT
cana-5362	101	4	𝒱′	𝒱′	NOUN
cana-5362	101	5	)	)	PUNCT
cana-5362	101	6	.	.	PUNCT
cana-5362	102	1	the	the	DET
cana-5362	102	2	other	other	ADJ
cana-5362	102	3	properties	property	NOUN
cana-5362	102	4	stated	state	VERB
cana-5362	102	5	above	above	ADP
cana-5362	102	6	,	,	PUNCT
cana-5362	102	7	such	such	ADJ
cana-5362	102	8	as	as	ADP
cana-5362	102	9	monotonicity	monotonicity	NOUN
cana-5362	102	10	,	,	PUNCT
cana-5362	102	11	hemi	hemi	NOUN
cana-5362	102	12	-	-	PUNCT
cana-5362	102	13	continuity	continuity	NOUN
cana-5362	102	14	,	,	PUNCT
cana-5362	102	15	and	and	CCONJ
cana-5362	102	16	coercivity	coercivity	NOUN
cana-5362	102	17	,	,	PUNCT
cana-5362	102	18	are	be	AUX
cana-5362	102	19	similarly	similarly	ADV
cana-5362	102	20	verified	verify	VERB
cana-5362	102	21	using	use	VERB
cana-5362	102	22	the	the	DET
cana-5362	102	23	assumptions	assumption	NOUN
cana-5362	102	24	and	and	CCONJ
cana-5362	102	25	lebesgue	lebesgue	PROPN
cana-5362	102	26	’s	’s	PART
cana-5362	102	27	theorem	theorem	PROPN
cana-5362	102	28	.	.	PUNCT
cana-5362	103	1	therefore	therefore	ADV
cana-5362	103	2	,	,	PUNCT
cana-5362	103	3	all	all	DET
cana-5362	103	4	hypotheses	hypothesis	NOUN
cana-5362	103	5	are	be	AUX
cana-5362	103	6	satisfied	satisfied	ADJ
cana-5362	103	7	(	(	PUNCT
cana-5362	103	8	see	see	VERB
cana-5362	103	9	[	[	X
cana-5362	103	10	4	4	NUM
cana-5362	103	11	]	]	NUM
cana-5362	103	12	)	)	PUNCT
cana-5362	103	13	.	.	PUNCT
cana-5362	104	1	theorem	theorem	NOUN
cana-5362	104	2	1	1	NUM
cana-5362	104	3	under	under	ADP
cana-5362	104	4	the	the	DET
cana-5362	104	5	hypotheses	hypothesis	NOUN
cana-5362	104	6	of	of	ADP
cana-5362	104	7	lemma	lemma	PROPN
cana-5362	104	8	1	1	NUM
cana-5362	104	9	of	of	ADP
cana-5362	104	10	𝒜	𝒜	NOUN
cana-5362	104	11	(	(	PUNCT
cana-5362	104	12	.	.	PUNCT
cana-5362	104	13	)	)	PUNCT
cana-5362	104	14	,	,	PUNCT
cana-5362	104	15	and	and	CCONJ
cana-5362	104	16	propositions	proposition	NOUN
cana-5362	104	17	1	1	NUM
cana-5362	104	18	,	,	PUNCT
cana-5362	104	19	2	2	NUM
cana-5362	104	20	and	and	CCONJ
cana-5362	104	21	3	3	NUM
cana-5362	104	22	,	,	PUNCT
cana-5362	104	23	then	then	ADV
cana-5362	104	24	there	there	PRON
cana-5362	104	25	exists	exist	VERB
cana-5362	104	26	a	a	DET
cana-5362	104	27	unique	unique	ADJ
cana-5362	104	28	solution	solution	NOUN
cana-5362	105	1	𝑢	𝑢	X
cana-5362	105	2	∈	∈	PROPN
cana-5362	105	3	𝐿2(𝒱	𝐿2(𝒱	PROPN
cana-5362	105	4	)	)	PUNCT
cana-5362	105	5	∩	∩	NOUN
cana-5362	105	6	𝐶(𝐻	𝐶(𝐻	NOUN
cana-5362	105	7	)	)	PUNCT
cana-5362	105	8	,	,	PUNCT
cana-5362	105	9	for	for	ADP
cana-5362	105	10	𝑡	𝑡	PROPN
cana-5362	105	11	∈	∈	PROPN
cana-5362	105	12	]	]	X
cana-5362	105	13	0	0	NUM
cana-5362	105	14	,	,	PUNCT
cana-5362	105	15	𝑇	𝑇	PROPN
cana-5362	105	16	[	[	NOUN
cana-5362	105	17	,	,	PUNCT
cana-5362	105	18	verifying	verifying	NOUN
cana-5362	105	19	{	{	PUNCT
cana-5362	105	20	𝜕𝑢	𝜕𝑢	NOUN
cana-5362	105	21	𝜕𝑡	𝜕𝑡	NOUN
cana-5362	105	22	+	+	NOUN
cana-5362	105	23	𝒜(𝑢	𝒜(𝑢	NUM
cana-5362	105	24	)	)	PUNCT
cana-5362	106	1	=	=	PUNCT
cana-5362	107	1	𝜕𝑊	𝜕𝑊	NUM
cana-5362	107	2	𝜕𝑡	𝜕𝑡	NOUN
cana-5362	107	3	𝑖𝑛	𝑖𝑛	X
cana-5362	107	4	]	]	X
cana-5362	107	5	0	0	NUM
cana-5362	107	6	,	,	PUNCT
cana-5362	107	7	𝑇	𝑇	PROPN
cana-5362	107	8	[	[	PUNCT
cana-5362	107	9	×	×	PROPN
cana-5362	107	10	𝐷	𝐷	PROPN
cana-5362	107	11	𝑢(0	𝑢(0	PROPN
cana-5362	107	12	,	,	PUNCT
cana-5362	107	13	𝑥	𝑥	NOUN
cana-5362	107	14	)	)	PUNCT
cana-5362	107	15	=	=	SYM
cana-5362	107	16	𝑢0(𝑥	𝑢0(𝑥	PROPN
cana-5362	107	17	)	)	PUNCT
cana-5362	107	18	,	,	PUNCT
cana-5362	107	19	∀𝑥	∀𝑥	PROPN
cana-5362	107	20	∈	∈	PROPN
cana-5362	107	21	𝐷	𝐷	PROPN
cana-5362	107	22	(	(	PUNCT
cana-5362	107	23	21	21	NUM
cana-5362	107	24	)	)	PUNCT
cana-5362	107	25	proof	proof	NOUN
cana-5362	107	26	.	.	PUNCT
cana-5362	108	1	let	let	VERB
cana-5362	108	2	us	we	PRON
cana-5362	108	3	consider	consider	VERB
cana-5362	108	4	the	the	DET
cana-5362	108	5	operator	operator	NOUN
cana-5362	108	6	𝒜𝑊	𝒜𝑊	PROPN
cana-5362	108	7	∶	∶	NOUN
cana-5362	108	8	𝒱	𝒱	PROPN
cana-5362	108	9	→	→	SYM
cana-5362	108	10	𝒱′	𝒱′	NUM
cana-5362	108	11	defined	define	VERB
cana-5362	108	12	by	by	ADP
cana-5362	108	13	𝒜𝑊(𝓋	𝒜𝑊(𝓋	NOUN
cana-5362	108	14	)	)	PUNCT
cana-5362	109	1	=	=	PRON
cana-5362	109	2	𝒜(𝓋	𝒜(𝓋	X
cana-5362	109	3	+	+	X
cana-5362	109	4	𝑊𝑡	𝑊𝑡	PROPN
cana-5362	109	5	)	)	PUNCT
cana-5362	109	6	,	,	PUNCT
cana-5362	109	7	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-5362	109	8	𝓋	𝓋	PROPN
cana-5362	109	9	∈	∈	PROPN
cana-5362	109	10	𝒱	𝒱	PROPN
cana-5362	109	11	(	(	PUNCT
cana-5362	109	12	22	22	NUM
cana-5362	109	13	)	)	PUNCT
cana-5362	109	14	so	so	ADV
cana-5362	109	15	,	,	PUNCT
cana-5362	109	16	(	(	PUNCT
cana-5362	109	17	21	21	NUM
cana-5362	109	18	)	)	PUNCT
cana-5362	109	19	became	become	VERB
cana-5362	109	20	as	as	ADP
cana-5362	109	21	follow	follow	VERB
cana-5362	109	22	{	{	PUNCT
cana-5362	109	23	𝜕𝓋	𝜕𝓋	ADP
cana-5362	109	24	𝜕𝑡	𝜕𝑡	PROPN
cana-5362	109	25	+	+	NOUN
cana-5362	109	26	𝒜𝑊(𝓋(𝑥	𝒜𝑊(𝓋(𝑥	ADJ
cana-5362	109	27	)	)	PUNCT
cana-5362	109	28	)	)	PUNCT
cana-5362	110	1	=	=	PUNCT
cana-5362	110	2	0	0	NUM
cana-5362	110	3	𝑢(0	𝑢(0	PROPN
cana-5362	110	4	,	,	PUNCT
cana-5362	110	5	𝑥	𝑥	NOUN
cana-5362	110	6	)	)	PUNCT
cana-5362	110	7	=	=	SYM
cana-5362	110	8	𝑢0(𝑥	𝑢0(𝑥	PROPN
cana-5362	110	9	)	)	PUNCT
cana-5362	110	10	,	,	PUNCT
cana-5362	110	11	∀𝑥	∀𝑥	PROPN
cana-5362	110	12	∈	∈	PROPN
cana-5362	110	13	𝐷	𝐷	PROPN
cana-5362	110	14	(	(	PUNCT
cana-5362	110	15	23	23	NUM
cana-5362	110	16	)	)	PUNCT
cana-5362	110	17	the	the	DET
cana-5362	110	18	necessary	necessary	ADJ
cana-5362	110	19	and	and	CCONJ
cana-5362	110	20	sufficient	sufficient	ADJ
cana-5362	110	21	condition	condition	NOUN
cana-5362	110	22	for	for	ADP
cana-5362	110	23	(	(	PUNCT
cana-5362	110	24	21	21	NUM
cana-5362	110	25	)	)	PUNCT
cana-5362	110	26	to	to	PART
cana-5362	110	27	have	have	VERB
cana-5362	110	28	a	a	DET
cana-5362	110	29	unique	unique	ADJ
cana-5362	110	30	solution	solution	NOUN
cana-5362	110	31	is	be	AUX
cana-5362	110	32	that	that	SCONJ
cana-5362	110	33	(	(	PUNCT
cana-5362	110	34	23	23	NUM
cana-5362	110	35	)	)	PUNCT
cana-5362	110	36	has	have	VERB
cana-5362	110	37	a	a	DET
cana-5362	110	38	unique	unique	ADJ
cana-5362	110	39	solution	solution	NOUN
cana-5362	110	40	in	in	ADP
cana-5362	110	41	𝐿2(𝒱	𝐿2(𝒱	PROPN
cana-5362	110	42	)	)	PUNCT
cana-5362	110	43	∩	∩	PROPN
cana-5362	110	44	𝐿∞(ℋ	𝐿∞(ℋ	PROPN
cana-5362	110	45	)	)	PUNCT
cana-5362	110	46	,	,	PUNCT
cana-5362	110	47	such	such	ADJ
cana-5362	110	48	that	that	DET
cana-5362	110	49	𝜕𝓋	𝜕𝓋	ADV
cana-5362	110	50	𝜕𝑡	𝜕𝑡	PROPN
cana-5362	110	51	∈	∈	PROPN
cana-5362	110	52	𝐿2(𝒱′	𝐿2(𝒱′	PROPN
cana-5362	110	53	)	)	PUNCT
cana-5362	111	1	+	+	SYM
cana-5362	112	1	𝐿1(ℋ	𝐿1(ℋ	PROPN
cana-5362	112	2	)	)	PUNCT
cana-5362	112	3	indeed	indeed	ADV
cana-5362	112	4	,	,	PUNCT
cana-5362	112	5	if	if	SCONJ
cana-5362	112	6	𝓋	𝓋	PROPN
cana-5362	112	7	∈	∈	PROPN
cana-5362	112	8	𝐿2(𝒱)∩	𝐿2(𝒱)∩	PROPN
cana-5362	112	9	𝐿∞(ℋ	𝐿∞(ℋ	PROPN
cana-5362	112	10	)	)	PUNCT
cana-5362	112	11	,	,	PUNCT
cana-5362	112	12	𝜕𝓋	𝜕𝓋	ADP
cana-5362	112	13	𝜕𝑡	𝜕𝑡	PROPN
cana-5362	112	14	∈	∈	PROPN
cana-5362	112	15	𝐿2(𝒱′)∩	𝐿2(𝒱′)∩	VERB
cana-5362	112	16	𝐿1(ℋ	𝐿1(ℋ	PROPN
cana-5362	112	17	)	)	PUNCT
cana-5362	112	18	,	,	PUNCT
cana-5362	112	19	then	then	ADV
cana-5362	112	20	𝓋	𝓋	PROPN
cana-5362	112	21	∈	∈	PROPN
cana-5362	112	22	𝐶(ℋ	𝐶(ℋ	NOUN
cana-5362	112	23	)	)	PUNCT
cana-5362	112	24	if	if	SCONJ
cana-5362	112	25	we	we	PRON
cana-5362	112	26	set	set	VERB
cana-5362	112	27	𝑢	𝑢	NOUN
cana-5362	112	28	=	=	PUNCT
cana-5362	112	29	𝓋	𝓋	PROPN
cana-5362	112	30	+	+	CCONJ
cana-5362	112	31	𝑊	𝑊	PROPN
cana-5362	112	32	,	,	PUNCT
cana-5362	112	33	then	then	ADV
cana-5362	112	34	𝑢	𝑢	PROPN
cana-5362	112	35	∈	∈	PROPN
cana-5362	112	36	𝐿2(𝒱	𝐿2(𝒱	PROPN
cana-5362	112	37	)	)	PUNCT
cana-5362	112	38	∩	∩	NOUN
cana-5362	112	39	𝐶(ℋ	𝐶(ℋ	NOUN
cana-5362	112	40	)	)	PUNCT
cana-5362	112	41	(	(	PUNCT
cana-5362	112	42	thanks	thank	NOUN
cana-5362	112	43	to	to	ADP
cana-5362	112	44	(	(	PUNCT
cana-5362	112	45	3	3	NUM
cana-5362	112	46	)	)	PUNCT
cana-5362	112	47	)	)	PUNCT
cana-5362	112	48	,	,	PUNCT
cana-5362	112	49	and	and	CCONJ
cana-5362	112	50	it	it	PRON
cana-5362	112	51	is	be	AUX
cana-5362	112	52	clear	clear	ADJ
cana-5362	112	53	that	that	SCONJ
cana-5362	112	54	u	u	NOUN
cana-5362	112	55	is	be	AUX
cana-5362	112	56	a	a	DET
cana-5362	112	57	solution	solution	NOUN
cana-5362	112	58	of	of	ADP
cana-5362	112	59	(	(	PUNCT
cana-5362	112	60	1	1	NUM
cana-5362	112	61	)	)	PUNCT
cana-5362	112	62	.	.	PUNCT
cana-5362	113	1	the	the	DET
cana-5362	113	2	reverse	reverse	ADJ
cana-5362	113	3	implication	implication	NOUN
cana-5362	113	4	shows	show	VERB
cana-5362	113	5	that	that	SCONJ
cana-5362	113	6	if	if	SCONJ
cana-5362	113	7	u	u	NOUN
cana-5362	113	8	is	be	AUX
cana-5362	113	9	a	a	DET
cana-5362	113	10	solution	solution	NOUN
cana-5362	113	11	of	of	ADP
cana-5362	113	12	(	(	PUNCT
cana-5362	113	13	21	21	NUM
cana-5362	113	14	)	)	PUNCT
cana-5362	113	15	,	,	PUNCT
cana-5362	113	16	then	then	ADV
cana-5362	113	17	𝓋	𝓋	VERB
cana-5362	113	18	=	=	SYM
cana-5362	113	19	𝑢	𝑢	PRON
cana-5362	113	20	−	−	PROPN
cana-5362	113	21	𝑊	𝑊	NOUN
cana-5362	113	22	is	be	AUX
cana-5362	113	23	a	a	DET
cana-5362	113	24	solution	solution	NOUN
cana-5362	113	25	of	of	ADP
cana-5362	113	26	(	(	PUNCT
cana-5362	113	27	23	23	NUM
cana-5362	113	28	)	)	PUNCT
cana-5362	113	29	,	,	PUNCT
cana-5362	113	30	and	and	CCONJ
cana-5362	113	31	𝜕𝓋	𝜕𝓋	ADV
cana-5362	113	32	𝜕𝑡	𝜕𝑡	NOUN
cana-5362	113	33	=	=	SYM
cana-5362	113	34	−𝒜(𝑢	−𝒜(𝑢	PROPN
cana-5362	113	35	)	)	PUNCT
cana-5362	113	36	∈	∈	PROPN
cana-5362	113	37	𝐿2(𝒱′	𝐿2(𝒱′	PROPN
cana-5362	113	38	)	)	PUNCT
cana-5362	113	39	(	(	PUNCT
cana-5362	113	40	24	24	NUM
cana-5362	113	41	)	)	PUNCT
cana-5362	113	42	lemma	lemma	PROPN
cana-5362	113	43	2	2	NUM
cana-5362	113	44	let	let	VERB
cana-5362	113	45	𝒜𝑊	𝒜𝑊	PROPN
cana-5362	113	46	∶	∶	NOUN
cana-5362	113	47	𝒱	𝒱	PROPN
cana-5362	113	48	→	→	SYM
cana-5362	113	49	𝒱′	𝒱′	NOUN
cana-5362	113	50	satisfied	satisfied	ADJ
cana-5362	113	51	for	for	ADP
cana-5362	113	52	a.e	a.e	PROPN
cana-5362	113	53	.	.	PUNCT
cana-5362	113	54	𝑡	𝑡	PROPN
cana-5362	113	55	∈	∈	PROPN
cana-5362	114	1	[	[	X
cana-5362	114	2	0	0	NUM
cana-5362	114	3	,	,	PUNCT
cana-5362	114	4	𝑇	𝑇	PROPN
cana-5362	114	5	]	]	PUNCT
cana-5362	114	6	1	1	NUM
cana-5362	114	7	.	.	PUNCT
cana-5362	115	1	𝒜𝑊	𝒜𝑊	PROPN
cana-5362	115	2	(	(	PUNCT
cana-5362	115	3	.	.	PUNCT
cana-5362	115	4	)	)	PUNCT
cana-5362	115	5	is	be	AUX
cana-5362	115	6	hemi	hemi	NOUN
cana-5362	115	7	-	-	PUNCT
cana-5362	115	8	continuous	continuous	ADJ
cana-5362	115	9	;	;	PUNCT
cana-5362	115	10	communications	communication	NOUN
cana-5362	115	11	on	on	ADP
cana-5362	115	12	applied	apply	VERB
cana-5362	115	13	nonlinear	nonlinear	ADJ
cana-5362	115	14	analysis	analysis	NOUN
cana-5362	115	15	issn	issn	NOUN
cana-5362	115	16	:	:	PUNCT
cana-5362	115	17	1074	1074	NUM
cana-5362	115	18	-	-	PUNCT
cana-5362	115	19	133x	133x	NUM
cana-5362	115	20	vol	vol	VERB
cana-5362	115	21	32	32	NUM
cana-5362	115	22	no	no	NOUN
cana-5362	115	23	.	.	PUNCT
cana-5362	115	24	10s	10	NOUN
cana-5362	115	25	(	(	PUNCT
cana-5362	115	26	2025	2025	NUM
cana-5362	115	27	)	)	PUNCT
cana-5362	115	28	1991	1991	NUM
cana-5362	115	29	https://internationalpubls.com	https://internationalpubls.com	X
cana-5362	115	30	2	2	X
cana-5362	115	31	.	.	PUNCT
cana-5362	116	1	𝒜𝑊	𝒜𝑊	PROPN
cana-5362	116	2	(	(	PUNCT
cana-5362	116	3	.	.	PUNCT
cana-5362	116	4	)	)	PUNCT
cana-5362	116	5	is	be	AUX
cana-5362	116	6	monotone	monotone	ADJ
cana-5362	116	7	;	;	PUNCT
cana-5362	117	1	3	3	X
cana-5362	117	2	.	.	X
cana-5362	118	1	if	if	SCONJ
cana-5362	118	2	𝑢	𝑢	PROPN
cana-5362	118	3	∈	∈	PROPN
cana-5362	118	4	𝐿2(𝒱	𝐿2(𝒱	PROPN
cana-5362	118	5	)	)	PUNCT
cana-5362	118	6	,	,	PUNCT
cana-5362	118	7	then	then	ADV
cana-5362	118	8	{	{	PUNCT
cana-5362	118	9	𝑡	𝑡	PROPN
cana-5362	118	10	→	→	SYM
cana-5362	118	11	𝒜𝑊	𝒜𝑊	PROPN
cana-5362	118	12	(	(	PUNCT
cana-5362	118	13	𝑢(𝑥	𝑢(𝑥	PROPN
cana-5362	118	14	)	)	PUNCT
cana-5362	118	15	)	)	PUNCT
cana-5362	119	1	∈	∈	PROPN
cana-5362	119	2	𝐿	𝐿	PROPN
cana-5362	119	3	2(𝒱′	2(𝒱′	PROPN
cana-5362	119	4	)	)	PUNCT
cana-5362	119	5	}	}	PUNCT
cana-5362	119	6	proof	proof	NOUN
cana-5362	119	7	.	.	PUNCT
cana-5362	120	1	let	let	VERB
cana-5362	120	2	𝑢,𝓋	𝑢,𝓋	NOUN
cana-5362	120	3	∈	∈	PROPN
cana-5362	120	4	𝒱	𝒱	PROPN
cana-5362	120	5	and	and	CCONJ
cana-5362	120	6	𝜓	𝜓	ADP
cana-5362	120	7	∈	∈	NOUN
cana-5362	120	8	𝒱′	𝒱′	NOUN
cana-5362	120	9	;	;	PUNCT
cana-5362	120	10	we	we	PRON
cana-5362	120	11	have	have	VERB
cana-5362	120	12	𝛩	𝛩	NUM
cana-5362	120	13	∈	∈	PROPN
cana-5362	120	14	𝑅	𝑅	PROPN
cana-5362	120	15	⟨𝒜𝑊	⟨𝒜𝑊	PROPN
cana-5362	120	16	(	(	PUNCT
cana-5362	120	17	𝑢	𝑢	X
cana-5362	120	18	+	+	X
cana-5362	120	19	𝛩𝓋),𝜓⟩	𝛩𝓋),𝜓⟩	ADJ
cana-5362	120	20	𝒱,𝒱′	𝒱,𝒱′	NOUN
cana-5362	120	21	=	=	PUNCT
cana-5362	121	1	⟨𝒜(𝑢	⟨𝒜(𝑢	PROPN
cana-5362	122	1	+	+	CCONJ
cana-5362	122	2	𝑊𝑡	𝑊𝑡	PROPN
cana-5362	122	3	+	+	CCONJ
cana-5362	122	4	𝛩𝓋	𝛩𝓋	NOUN
cana-5362	122	5	)	)	PUNCT
cana-5362	122	6	,	,	PUNCT
cana-5362	122	7	𝜓⟩𝒱,𝒱′	𝜓⟩𝒱,𝒱′	PROPN
cana-5362	122	8	(	(	PUNCT
cana-5362	122	9	24	24	NUM
cana-5362	122	10	)	)	PUNCT
cana-5362	122	11	and	and	CCONJ
cana-5362	122	12	according	accord	VERB
cana-5362	122	13	to	to	ADP
cana-5362	122	14	(	(	PUNCT
cana-5362	122	15	19	19	NUM
cana-5362	122	16	)	)	PUNCT
cana-5362	122	17	,	,	PUNCT
cana-5362	122	18	when	when	SCONJ
cana-5362	122	19	𝛩	𝛩	PROPN
cana-5362	122	20	⟶	⟶	NOUN
cana-5362	122	21	0	0	NUM
cana-5362	122	22	,	,	PUNCT
cana-5362	122	23	the	the	DET
cana-5362	122	24	second	second	ADJ
cana-5362	122	25	member	member	NOUN
cana-5362	122	26	of	of	ADP
cana-5362	122	27	(	(	PUNCT
cana-5362	122	28	25	25	NUM
cana-5362	122	29	)	)	PUNCT
cana-5362	122	30	converges	converge	VERB
cana-5362	122	31	to	to	ADP
cana-5362	122	32	⟨𝒜(𝑢	⟨𝒜(𝑢	PROPN
cana-5362	122	33	+	+	CCONJ
cana-5362	122	34	𝑊𝑡	𝑊𝑡	PROPN
cana-5362	122	35	)	)	PUNCT
cana-5362	122	36	,	,	PUNCT
cana-5362	122	37	𝜓⟩𝒱,𝒱′	𝜓⟩𝒱,𝒱′	AUX
cana-5362	122	38	=	=	PUNCT
cana-5362	122	39	⟨𝒜𝑊	⟨𝒜𝑊	PROPN
cana-5362	122	40	(	(	PUNCT
cana-5362	122	41	𝑢	𝑢	NOUN
cana-5362	122	42	)	)	PUNCT
cana-5362	122	43	,	,	PUNCT
cana-5362	122	44	𝜓⟩𝒱,𝒱′.	𝜓⟩𝒱,𝒱′.	X
cana-5362	122	45	(	(	PUNCT
cana-5362	122	46	25	25	NUM
cana-5362	122	47	)	)	PUNCT
cana-5362	122	48	hence	hence	ADV
cana-5362	122	49	,	,	PUNCT
cana-5362	122	50	𝒜𝑊	𝒜𝑊	PROPN
cana-5362	122	51	(	(	PUNCT
cana-5362	122	52	.	.	PUNCT
cana-5362	122	53	)	)	PUNCT
cana-5362	122	54	is	be	AUX
cana-5362	122	55	hemi	hemi	NOUN
cana-5362	122	56	-	-	PUNCT
cana-5362	122	57	continuous	continuous	ADJ
cana-5362	122	58	.	.	PUNCT
cana-5362	123	1	to	to	PART
cana-5362	123	2	prove	prove	VERB
cana-5362	123	3	the	the	DET
cana-5362	123	4	monotonicity	monotonicity	NOUN
cana-5362	123	5	of	of	ADP
cana-5362	123	6	𝒜𝑊	𝒜𝑊	PROPN
cana-5362	123	7	(	(	PUNCT
cana-5362	123	8	𝑢	𝑢	NOUN
cana-5362	123	9	)	)	PUNCT
cana-5362	123	10	,	,	PUNCT
cana-5362	123	11	we	we	PRON
cana-5362	123	12	write	write	VERB
cana-5362	123	13	⟨𝒜𝑊	⟨𝒜𝑊	PROPN
cana-5362	123	14	(	(	PUNCT
cana-5362	123	15	𝑢	𝑢	X
cana-5362	123	16	)	)	PUNCT
cana-5362	123	17	–	–	PUNCT
cana-5362	123	18	𝒜𝑊	𝒜𝑊	PROPN
cana-5362	123	19	(	(	PUNCT
cana-5362	123	20	𝓋	𝓋	NOUN
cana-5362	123	21	)	)	PUNCT
cana-5362	123	22	,	,	PUNCT
cana-5362	123	23	𝑢	𝑢	X
cana-5362	123	24	–	–	PUNCT
cana-5362	123	25	𝓋⟩	𝓋⟩	PUNCT
cana-5362	123	26	=	=	PUNCT
cana-5362	124	1	⟨𝒜(𝑢	⟨𝒜(𝑢	PROPN
cana-5362	125	1	+	+	CCONJ
cana-5362	125	2	𝑊𝑡	𝑊𝑡	PROPN
cana-5362	125	3	)	)	PUNCT
cana-5362	125	4	–	–	PUNCT
cana-5362	125	5	𝒜(𝓋+	𝒜(𝓋+	PUNCT
cana-5362	125	6	𝑊𝑡	𝑊𝑡	PROPN
cana-5362	125	7	)	)	PUNCT
cana-5362	125	8	,	,	PUNCT
cana-5362	125	9	(	(	PUNCT
cana-5362	125	10	𝑢	𝑢	X
cana-5362	125	11	+	+	X
cana-5362	125	12	𝑊𝑡	𝑊𝑡	PROPN
cana-5362	125	13	)	)	PUNCT
cana-5362	125	14	−	−	PROPN
cana-5362	126	1	(	(	PUNCT
cana-5362	126	2	𝓋+	𝓋+	PROPN
cana-5362	126	3	𝑊𝑡)⟩	𝑊𝑡)⟩	NOUN
cana-5362	126	4	≥	≥	NOUN
cana-5362	126	5	0	0	NUM
cana-5362	126	6	,	,	PUNCT
cana-5362	126	7	based	base	VERB
cana-5362	126	8	on	on	ADP
cana-5362	126	9	(	(	PUNCT
cana-5362	126	10	9	9	NUM
cana-5362	126	11	)	)	PUNCT
cana-5362	126	12	.	.	PUNCT
cana-5362	127	1	finally	finally	ADV
cana-5362	127	2	,	,	PUNCT
cana-5362	127	3	if	if	SCONJ
cana-5362	127	4	𝑢	𝑢	PROPN
cana-5362	127	5	∈	∈	PROPN
cana-5362	127	6	𝐿2(𝒱	𝐿2(𝒱	PROPN
cana-5362	127	7	)	)	PUNCT
cana-5362	127	8	and	and	CCONJ
cana-5362	127	9	according	accord	VERB
cana-5362	127	10	to	to	ADP
cana-5362	127	11	(	(	PUNCT
cana-5362	127	12	3	3	NUM
cana-5362	127	13	)	)	PUNCT
cana-5362	127	14	,	,	PUNCT
cana-5362	127	15	then	then	ADV
cana-5362	127	16	𝑢	𝑢	X
cana-5362	127	17	(	(	PUNCT
cana-5362	127	18	.	.	PUNCT
cana-5362	127	19	)	)	PUNCT
cana-5362	128	1	+	+	CCONJ
cana-5362	128	2	𝑊	𝑊	NOUN
cana-5362	128	3	(	(	PUNCT
cana-5362	128	4	.	.	PUNCT
cana-5362	128	5	)	)	PUNCT
cana-5362	128	6	∈	∈	PROPN
cana-5362	128	7	𝐿2(𝒱	𝐿2(𝒱	PROPN
cana-5362	128	8	)	)	PUNCT
cana-5362	128	9	,	,	PUNCT
cana-5362	128	10	and	and	CCONJ
cana-5362	128	11	therefore	therefore	ADV
cana-5362	128	12	𝒜𝑊	𝒜𝑊	PROPN
cana-5362	128	13	𝑢	𝑢	PRON
cana-5362	128	14	(	(	PUNCT
cana-5362	128	15	.	.	PUNCT
cana-5362	128	16	)	)	PUNCT
cana-5362	129	1	=	=	PUNCT
cana-5362	129	2	𝒜(𝑢	𝒜(𝑢	PROPN
cana-5362	129	3	(	(	PUNCT
cana-5362	129	4	.	.	PUNCT
cana-5362	129	5	)	)	PUNCT
cana-5362	130	1	+	+	CCONJ
cana-5362	130	2	𝑊	𝑊	PROPN
cana-5362	130	3	(	(	PUNCT
cana-5362	130	4	.	.	PUNCT
cana-5362	130	5	)	)	PUNCT
cana-5362	130	6	)	)	PUNCT
cana-5362	131	1	∈	∈	PROPN
cana-5362	131	2	𝐿	𝐿	PROPN
cana-5362	131	3	2(𝒱′	2(𝒱′	PROPN
cana-5362	131	4	)	)	PUNCT
cana-5362	131	5	(	(	PUNCT
cana-5362	131	6	26	26	NUM
cana-5362	131	7	)	)	PUNCT
cana-5362	131	8	3.1	3.1	NUM
cana-5362	131	9	approximation	approximation	NOUN
cana-5362	131	10	let	let	VERB
cana-5362	131	11	𝑁	𝑁	PROPN
cana-5362	131	12	an	an	DET
cana-5362	131	13	integer	integer	NOUN
cana-5362	131	14	intended	intend	VERB
cana-5362	131	15	to	to	PART
cana-5362	131	16	approach	approach	VERB
cana-5362	131	17	infinity	infinity	NOUN
cana-5362	131	18	and	and	CCONJ
cana-5362	131	19	𝑘	𝑘	PRON
cana-5362	131	20	=	=	SYM
cana-5362	131	21	𝑇	𝑇	PROPN
cana-5362	131	22	𝑁	𝑁	PROPN
cana-5362	131	23	.	.	PUNCT
cana-5362	132	1	we	we	PRON
cana-5362	132	2	consider	consider	VERB
cana-5362	132	3	a	a	DET
cana-5362	132	4	partition	partition	NOUN
cana-5362	132	5	of	of	ADP
cana-5362	132	6	the	the	DET
cana-5362	132	7	interval	interval	NOUN
cana-5362	132	8	[	[	X
cana-5362	132	9	0	0	NUM
cana-5362	132	10	,	,	PUNCT
cana-5362	132	11	𝑇	𝑇	PROPN
cana-5362	132	12	]	]	X
cana-5362	132	13	,	,	PUNCT
cana-5362	132	14	0	0	NUM
cana-5362	132	15	,	,	PUNCT
cana-5362	132	16	𝑘	𝑘	NOUN
cana-5362	132	17	,	,	PUNCT
cana-5362	132	18	2𝑘	2𝑘	NUM
cana-5362	132	19	,	,	PUNCT
cana-5362	132	20	.	.	PUNCT
cana-5362	132	21	.	.	PUNCT
cana-5362	133	1	.	.	PUNCT
cana-5362	134	1	,	,	PUNCT
cana-5362	134	2	𝑁𝑘.	𝑁𝑘.	INTJ
cana-5362	134	3	we	we	PRON
cana-5362	134	4	propose	propose	VERB
cana-5362	134	5	𝑊𝑛	𝑊𝑛	PROPN
cana-5362	134	6	=	=	PUNCT
cana-5362	134	7	𝑊(𝑛𝑘	𝑊(𝑛𝑘	PROPN
cana-5362	134	8	)	)	PUNCT
cana-5362	134	9	∈	∈	PROPN
cana-5362	134	10	𝒱	𝒱	PROPN
cana-5362	134	11	(	(	PUNCT
cana-5362	134	12	27	27	NUM
cana-5362	134	13	)	)	PUNCT
cana-5362	134	14	and	and	CCONJ
cana-5362	134	15	introduce	introduce	VERB
cana-5362	134	16	the	the	DET
cana-5362	134	17	family	family	NOUN
cana-5362	134	18	of	of	ADP
cana-5362	134	19	operators	operator	NOUN
cana-5362	134	20	𝒜𝑊	𝒜𝑊	VERB
cana-5362	134	21	𝑛	𝑛	DET
cana-5362	134	22	∶	∶	NOUN
cana-5362	134	23	𝒱	𝒱	PROPN
cana-5362	134	24	⟶	⟶	NOUN
cana-5362	134	25	𝒱′	𝒱′	NOUN
cana-5362	134	26	defined	define	VERB
cana-5362	134	27	by	by	ADP
cana-5362	134	28	𝒜𝑊	𝒜𝑊	PROPN
cana-5362	134	29	𝑛	𝑛	PRON
cana-5362	134	30	𝜓	𝜓	NOUN
cana-5362	134	31	=	=	SYM
cana-5362	134	32	1	1	NUM
cana-5362	134	33	𝑘	𝑘	DET
cana-5362	134	34	∫	∫	PROPN
cana-5362	135	1	𝒜(𝜓	𝒜(𝜓	X
cana-5362	136	1	+	+	ADJ
cana-5362	136	2	𝑊𝑛)𝑑𝑡	𝑊𝑛)𝑑𝑡	PROPN
cana-5362	136	3	𝑛𝑘	𝑛𝑘	X
cana-5362	136	4	(	(	PUNCT
cana-5362	136	5	𝑛−1)𝑘	𝑛−1)𝑘	X
cana-5362	136	6	(	(	PUNCT
cana-5362	136	7	28	28	NUM
cana-5362	136	8	)	)	PUNCT
cana-5362	136	9	consider	consider	VERB
cana-5362	136	10	the	the	DET
cana-5362	136	11	recurrence	recurrence	NOUN
cana-5362	136	12	relations	relation	NOUN
cana-5362	136	13	{	{	PUNCT
cana-5362	136	14	𝓋𝑛−𝓋𝑛−1	𝓋𝑛−𝓋𝑛−1	NOUN
cana-5362	136	15	𝑘	𝑘	X
cana-5362	137	1	+	+	NOUN
cana-5362	137	2	𝒜𝑊𝓋	𝒜𝑊𝓋	NOUN
cana-5362	137	3	𝑛	𝑛	VERB
cana-5362	137	4	=	=	SYM
cana-5362	137	5	0	0	PUNCT
cana-5362	138	1	𝓋(0	𝓋(0	PROPN
cana-5362	138	2	,	,	PUNCT
cana-5362	138	3	𝑥	𝑥	NOUN
cana-5362	138	4	)	)	PUNCT
cana-5362	138	5	=	=	VERB
cana-5362	138	6	𝑢0	𝑢0	PROPN
cana-5362	138	7	−𝑊0	−𝑊0	PROPN
cana-5362	138	8	(	(	PUNCT
cana-5362	138	9	29	29	NUM
cana-5362	138	10	)	)	PUNCT
cana-5362	138	11	first	first	ADV
cana-5362	138	12	,	,	PUNCT
cana-5362	138	13	note	note	VERB
cana-5362	138	14	that	that	SCONJ
cana-5362	138	15	(	(	PUNCT
cana-5362	138	16	30	30	NUM
cana-5362	138	17	)	)	PUNCT
cana-5362	138	18	uniquely	uniquely	ADV
cana-5362	138	19	defines	define	VERB
cana-5362	138	20	a	a	DET
cana-5362	138	21	sequence	sequence	NOUN
cana-5362	138	22	𝓋𝑛	𝓋𝑛	X
cana-5362	138	23	of	of	ADP
cana-5362	138	24	elements	element	NOUN
cana-5362	138	25	in	in	ADP
cana-5362	138	26	𝒱	𝒱	PROPN
cana-5362	138	27	(	(	PUNCT
cana-5362	138	28	except	except	SCONJ
cana-5362	138	29	for	for	ADP
cana-5362	138	30	𝑛	𝑛	PROPN
cana-5362	138	31	=	=	SYM
cana-5362	138	32	0	0	NUM
cana-5362	138	33	,	,	PUNCT
cana-5362	138	34	where	where	SCONJ
cana-5362	138	35	𝓋𝑛	𝓋𝑛	DET
cana-5362	138	36	∈	∈	PROPN
cana-5362	138	37	ℋ	ℋ	PROPN
cana-5362	138	38	)	)	PUNCT
cana-5362	138	39	.	.	PUNCT
cana-5362	139	1	indeed	indeed	ADV
cana-5362	139	2	,	,	PUNCT
cana-5362	139	3	introduce	introduce	VERB
cana-5362	139	4	𝒜𝑛	𝒜𝑛	PROPN
cana-5362	139	5	𝒱	𝒱	PROPN
cana-5362	139	6	⟶	⟶	NOUN
cana-5362	139	7	𝒱′	𝒱′	NOUN
cana-5362	139	8	,	,	PUNCT
cana-5362	139	9	defined	define	VERB
cana-5362	139	10	as	as	ADP
cana-5362	139	11	𝒜𝑛𝜓	𝒜𝑛𝜓	PROPN
cana-5362	139	12	=	=	NOUN
cana-5362	139	13	1	1	NUM
cana-5362	139	14	𝑘	𝑘	DET
cana-5362	139	15	∫	∫	PROPN
cana-5362	139	16	𝒜𝜓𝑑𝑡	𝒜𝜓𝑑𝑡	PROPN
cana-5362	139	17	,	,	PUNCT
cana-5362	139	18	∀	∀	X
cana-5362	139	19	𝜓	𝜓	X
cana-5362	139	20	𝑛𝑘	𝑛𝑘	PROPN
cana-5362	139	21	(	(	PUNCT
cana-5362	139	22	𝑛−1)𝑘	𝑛−1)𝑘	NOUN
cana-5362	139	23	∈	∈	PROPN
cana-5362	139	24	𝒱	𝒱	PROPN
cana-5362	139	25	(	(	PUNCT
cana-5362	139	26	30	30	NUM
cana-5362	139	27	)	)	PUNCT
cana-5362	139	28	then	then	ADV
cana-5362	139	29	(	(	PUNCT
cana-5362	139	30	30	30	NUM
cana-5362	139	31	)	)	PUNCT
cana-5362	139	32	can	can	AUX
cana-5362	139	33	be	be	AUX
cana-5362	139	34	written	write	VERB
cana-5362	139	35	as	as	ADP
cana-5362	139	36	𝓋𝑛−𝓋𝑛−1	𝓋𝑛−𝓋𝑛−1	NOUN
cana-5362	139	37	𝑘	𝑘	PROPN
cana-5362	140	1	+	+	ADJ
cana-5362	140	2	𝒜𝑛(𝓋𝑛	𝒜𝑛(𝓋𝑛	PROPN
cana-5362	140	3	+	+	ADJ
cana-5362	140	4	𝑊𝑛	𝑊𝑛	PROPN
cana-5362	140	5	)	)	PUNCT
cana-5362	140	6	=	=	SYM
cana-5362	140	7	0	0	NUM
cana-5362	140	8	(	(	PUNCT
cana-5362	140	9	32	32	NUM
cana-5362	140	10	)	)	PUNCT
cana-5362	140	11	communications	communication	NOUN
cana-5362	140	12	on	on	ADP
cana-5362	140	13	applied	apply	VERB
cana-5362	140	14	nonlinear	nonlinear	ADJ
cana-5362	140	15	analysis	analysis	NOUN
cana-5362	140	16	issn	issn	NOUN
cana-5362	140	17	:	:	PUNCT
cana-5362	140	18	1074	1074	NUM
cana-5362	140	19	-	-	PUNCT
cana-5362	140	20	133x	133x	NUM
cana-5362	140	21	vol	vol	VERB
cana-5362	140	22	32	32	NUM
cana-5362	140	23	no	no	NOUN
cana-5362	140	24	.	.	PUNCT
cana-5362	141	1	10s	10	NOUN
cana-5362	141	2	(	(	PUNCT
cana-5362	141	3	2025	2025	NUM
cana-5362	141	4	)	)	PUNCT
cana-5362	141	5	1992	1992	NUM
cana-5362	141	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5362	141	7	but	but	CCONJ
cana-5362	141	8	then	then	ADV
cana-5362	141	9	,	,	PUNCT
cana-5362	141	10	by	by	ADP
cana-5362	141	11	setting	set	VERB
cana-5362	141	12	𝑢𝑛	𝑢𝑛	NOUN
cana-5362	141	13	=	=	PUNCT
cana-5362	141	14	𝓋𝑛	𝓋𝑛	X
cana-5362	141	15	+	+	X
cana-5362	142	1	𝑊𝑛	𝑊𝑛	PROPN
cana-5362	142	2	,	,	PUNCT
cana-5362	142	3	we	we	PRON
cana-5362	142	4	see	see	VERB
cana-5362	142	5	that	that	SCONJ
cana-5362	142	6	𝑢𝑛	𝑢𝑛	NOUN
cana-5362	142	7	must	must	AUX
cana-5362	142	8	satisfy	satisfy	VERB
cana-5362	142	9	the	the	DET
cana-5362	142	10	recurrence	recurrence	NOUN
cana-5362	142	11	relations	relation	NOUN
cana-5362	142	12	{	{	PUNCT
cana-5362	142	13	𝑢𝑛−𝑢𝑛−1	𝑢𝑛−𝑢𝑛−1	NOUN
cana-5362	142	14	𝑘	𝑘	X
cana-5362	143	1	+	+	ADJ
cana-5362	143	2	𝒜𝑛𝑢𝑛	𝒜𝑛𝑢𝑛	NOUN
cana-5362	143	3	=	=	SYM
cana-5362	143	4	𝑊𝑛−𝑊𝑛−1	𝑊𝑛−𝑊𝑛−1	NOUN
cana-5362	143	5	𝑘	𝑘	DET
cana-5362	143	6	𝑢0	𝑢0	PROPN
cana-5362	143	7	=	=	SYM
cana-5362	143	8	𝑢0	𝑢0	PROPN
cana-5362	143	9	(	(	PUNCT
cana-5362	143	10	33	33	NUM
cana-5362	143	11	)	)	PUNCT
cana-5362	143	12	it	it	PRON
cana-5362	143	13	follows	follow	VERB
cana-5362	143	14	from	from	ADP
cana-5362	143	15	the	the	DET
cana-5362	143	16	properties	property	NOUN
cana-5362	143	17	of	of	ADP
cana-5362	143	18	𝒜	𝒜	NOUN
cana-5362	143	19	(	(	PUNCT
cana-5362	143	20	.	.	PUNCT
cana-5362	143	21	)	)	PUNCT
cana-5362	144	1	(	(	PUNCT
cana-5362	144	2	lemma	lemma	PROPN
cana-5362	144	3	1	1	NUM
cana-5362	144	4	)	)	PUNCT
cana-5362	144	5	that	that	SCONJ
cana-5362	144	6	𝒜𝑛	𝒜𝑛	PROPN
cana-5362	144	7	is	be	AUX
cana-5362	144	8	monotone	monotone	ADJ
cana-5362	144	9	,	,	PUNCT
cana-5362	144	10	hemi	hemi	NOUN
cana-5362	144	11	-	-	PUNCT
cana-5362	144	12	continuous	continuous	ADJ
cana-5362	144	13	,	,	PUNCT
cana-5362	144	14	and	and	CCONJ
cana-5362	144	15	coercive	coercive	ADJ
cana-5362	144	16	from	from	ADP
cana-5362	144	17	𝒱	𝒱	PROPN
cana-5362	144	18	to	to	ADP
cana-5362	144	19	𝒱′	𝒱′	NOUN
cana-5362	144	20	,	,	PUNCT
cana-5362	144	21	and	and	CCONJ
cana-5362	144	22	consequently	consequently	ADV
cana-5362	144	23	(	(	PUNCT
cana-5362	144	24	cf	cf	NOUN
cana-5362	144	25	.	.	PUNCT
cana-5362	145	1	lions	lion	NOUN
cana-5362	146	1	[	[	X
cana-5362	146	2	11	11	NUM
cana-5362	146	3	]	]	PUNCT
cana-5362	146	4	)	)	PUNCT
cana-5362	146	5	(	(	PUNCT
cana-5362	146	6	𝐼	𝐼	PROPN
cana-5362	146	7	+	+	CCONJ
cana-5362	146	8	𝑘𝑛	𝑘𝑛	NOUN
cana-5362	146	9	)	)	PUNCT
cana-5362	146	10	is	be	AUX
cana-5362	146	11	invertible	invertible	ADJ
cana-5362	146	12	.	.	PUNCT
cana-5362	147	1	thus	thus	ADV
cana-5362	147	2	,	,	PUNCT
cana-5362	147	3	in	in	ADP
cana-5362	147	4	(	(	PUNCT
cana-5362	147	5	33	33	NUM
cana-5362	147	6	)	)	PUNCT
cana-5362	147	7	,	,	PUNCT
cana-5362	147	8	when	when	SCONJ
cana-5362	147	9	𝑢𝑛−1	𝑢𝑛−1	NOUN
cana-5362	147	10	is	be	AUX
cana-5362	147	11	known	know	VERB
cana-5362	147	12	,	,	PUNCT
cana-5362	147	13	𝑢𝑛	𝑢𝑛	PROPN
cana-5362	147	14	is	be	AUX
cana-5362	147	15	uniquely	uniquely	ADV
cana-5362	147	16	defined	define	VERB
cana-5362	147	17	as	as	ADP
cana-5362	147	18	an	an	DET
cana-5362	147	19	element	element	NOUN
cana-5362	147	20	of	of	ADP
cana-5362	147	21	𝒱.	𝒱.	PROPN
cana-5362	147	22	we	we	PRON
cana-5362	147	23	now	now	ADV
cana-5362	147	24	introduce	introduce	VERB
cana-5362	147	25	the	the	DET
cana-5362	147	26	step	step	NOUN
cana-5362	147	27	functions	function	NOUN
cana-5362	147	28	.	.	PUNCT
cana-5362	148	1	{	{	PUNCT
cana-5362	148	2	𝑊𝑘(𝑡	𝑊𝑘(𝑡	NOUN
cana-5362	148	3	)	)	PUNCT
cana-5362	148	4	=	=	SYM
cana-5362	148	5	𝑊	𝑊	PROPN
cana-5362	148	6	𝑛	𝑛	PRON
cana-5362	148	7	𝑑𝑎𝑛𝑠	𝑑𝑎𝑛𝑠	NOUN
cana-5362	149	1	[	[	X
cana-5362	149	2	𝑛𝑘	𝑛𝑘	NOUN
cana-5362	149	3	,	,	PUNCT
cana-5362	149	4	(	(	PUNCT
cana-5362	149	5	𝑛	𝑛	PROPN
cana-5362	149	6	+	+	SYM
cana-5362	149	7	1)𝑘	1)𝑘	NOUN
cana-5362	149	8	[	[	PUNCT
cana-5362	149	9	𝑢𝑘(𝑡	𝑢𝑘(𝑡	NOUN
cana-5362	149	10	)	)	PUNCT
cana-5362	149	11	=	=	PRON
cana-5362	149	12	𝑢	𝑢	PART
cana-5362	149	13	𝑛	𝑛	DET
cana-5362	149	14	𝑑𝑎𝑛𝑠	𝑑𝑎𝑛𝑠	NOUN
cana-5362	150	1	[	[	X
cana-5362	150	2	𝑛𝑘	𝑛𝑘	NOUN
cana-5362	150	3	,	,	PUNCT
cana-5362	150	4	(	(	PUNCT
cana-5362	150	5	𝑛	𝑛	PROPN
cana-5362	150	6	+	+	ADJ
cana-5362	150	7	1)𝑘	1)𝑘	NOUN
cana-5362	150	8	[	[	PUNCT
cana-5362	150	9	𝓋𝑘(𝑡	𝓋𝑘(𝑡	X
cana-5362	150	10	)	)	PUNCT
cana-5362	150	11	=	=	SYM
cana-5362	150	12	𝑣	𝑣	PART
cana-5362	150	13	𝑛	𝑛	PRON
cana-5362	150	14	𝑑𝑎𝑛𝑠	𝑑𝑎𝑛𝑠	NOUN
cana-5362	151	1	[	[	X
cana-5362	151	2	𝑛𝑘	𝑛𝑘	NOUN
cana-5362	151	3	,	,	PUNCT
cana-5362	151	4	(	(	PUNCT
cana-5362	151	5	𝑛	𝑛	PROPN
cana-5362	151	6	+	+	ADJ
cana-5362	151	7	1)𝑘	1)𝑘	NOUN
cana-5362	151	8	[	[	PUNCT
cana-5362	151	9	(	(	PUNCT
cana-5362	151	10	34	34	NUM
cana-5362	151	11	)	)	PUNCT
cana-5362	151	12	lemma	lemma	PROPN
cana-5362	151	13	3	3	NUM
cana-5362	151	14	𝑢𝑘	𝑢𝑘	PROPN
cana-5362	151	15	(	(	PUNCT
cana-5362	151	16	.	.	PUNCT
cana-5362	151	17	)	)	PUNCT
cana-5362	152	1	and	and	CCONJ
cana-5362	152	2	v𝑘	v𝑘	PRON
cana-5362	152	3	(	(	PUNCT
cana-5362	152	4	.	.	PUNCT
cana-5362	152	5	)	)	PUNCT
cana-5362	152	6	remain	remain	VERB
cana-5362	152	7	,	,	PUNCT
cana-5362	152	8	as	as	ADP
cana-5362	152	9	𝑘	𝑘	DET
cana-5362	152	10	⟶	⟶	NOUN
cana-5362	152	11	0	0	NUM
cana-5362	152	12	,	,	PUNCT
cana-5362	152	13	within	within	ADP
cana-5362	152	14	bounded	bounded	ADJ
cana-5362	152	15	subsets	subset	NOUN
cana-5362	152	16	of	of	ADP
cana-5362	152	17	𝐿∞(ℋ	𝐿∞(ℋ	PROPN
cana-5362	152	18	)	)	PUNCT
cana-5362	152	19	and	and	CCONJ
cana-5362	152	20	𝐿2(𝒱	𝐿2(𝒱	PROPN
cana-5362	152	21	)	)	PUNCT
cana-5362	152	22	.	.	PUNCT
cana-5362	153	1	proof	proof	NOUN
cana-5362	153	2	.	.	PUNCT
cana-5362	154	1	let	let	VERB
cana-5362	154	2	us	we	PRON
cana-5362	154	3	consider	consider	VERB
cana-5362	154	4	relation	relation	NOUN
cana-5362	154	5	(	(	PUNCT
cana-5362	154	6	32	32	NUM
cana-5362	154	7	)	)	PUNCT
cana-5362	154	8	,	,	PUNCT
cana-5362	154	9	which	which	PRON
cana-5362	154	10	is	be	AUX
cana-5362	154	11	written	write	VERB
cana-5362	154	12	as	as	ADP
cana-5362	154	13	𝓋𝑛	𝓋𝑛	PROPN
cana-5362	154	14	−𝓋𝑛−1	−𝓋𝑛−1	NOUN
cana-5362	155	1	+	+	PROPN
cana-5362	155	2	𝑘𝒜𝑛𝑢𝑛	𝑘𝒜𝑛𝑢𝑛	PROPN
cana-5362	155	3	=	=	SYM
cana-5362	155	4	0	0	NUM
cana-5362	155	5	(	(	PUNCT
cana-5362	155	6	35	35	NUM
cana-5362	155	7	)	)	PUNCT
cana-5362	155	8	so	so	ADV
cana-5362	155	9	,	,	PUNCT
cana-5362	155	10	(	(	PUNCT
cana-5362	155	11	𝓋𝑛	𝓋𝑛	INTJ
cana-5362	155	12	−𝓋𝑛−1	−𝓋𝑛−1	NOUN
cana-5362	155	13	,	,	PUNCT
cana-5362	155	14	𝓋𝑛	𝓋𝑛	PROPN
cana-5362	155	15	)	)	PUNCT
cana-5362	155	16	+	+	CCONJ
cana-5362	155	17	𝑘⟨𝒜𝑛𝑢𝑛	𝑘⟨𝒜𝑛𝑢𝑛	ADJ
cana-5362	155	18	,	,	PUNCT
cana-5362	155	19	𝑢𝑛	𝑢𝑛	NOUN
cana-5362	155	20	−	−	PROPN
cana-5362	155	21	𝑊𝑛⟩	𝑊𝑛⟩	NOUN
cana-5362	155	22	=	=	SYM
cana-5362	155	23	0	0	NUM
cana-5362	155	24	(	(	PUNCT
cana-5362	155	25	36	36	NUM
cana-5362	155	26	)	)	PUNCT
cana-5362	155	27	let	let	VERB
cana-5362	155	28	(	(	PUNCT
cana-5362	155	29	𝓋𝑛	𝓋𝑛	INTJ
cana-5362	155	30	−𝓋𝑛−1	−𝓋𝑛−1	NOUN
cana-5362	155	31	,	,	PUNCT
cana-5362	155	32	𝓋𝑛	𝓋𝑛	PROPN
cana-5362	155	33	)	)	PUNCT
cana-5362	156	1	+	+	CCONJ
cana-5362	156	2	𝑘⟨𝒜𝑛𝑢𝑛	𝑘⟨𝒜𝑛𝑢𝑛	ADJ
cana-5362	156	3	,	,	PUNCT
cana-5362	156	4	𝑢𝑛	𝑢𝑛	PROPN
cana-5362	156	5	⟩	⟩	PROPN
cana-5362	156	6	=	=	SYM
cana-5362	156	7	𝑘⟨	𝑘⟨	PROPN
cana-5362	157	1	𝑊𝑛	𝑊𝑛	PROPN
cana-5362	157	2	,	,	PUNCT
cana-5362	157	3	𝒜𝑛𝑢𝑛⟩	𝒜𝑛𝑢𝑛⟩	X
cana-5362	157	4	(	(	PUNCT
cana-5362	157	5	37	37	NUM
cana-5362	157	6	)	)	PUNCT
cana-5362	157	7	but	but	CCONJ
cana-5362	157	8	(	(	PUNCT
cana-5362	157	9	𝓋𝑛	𝓋𝑛	INTJ
cana-5362	157	10	−𝓋𝑛−1	−𝓋𝑛−1	NOUN
cana-5362	157	11	,	,	PUNCT
cana-5362	157	12	𝓋𝑛	𝓋𝑛	PROPN
cana-5362	157	13	)	)	PUNCT
cana-5362	157	14	=	=	SYM
cana-5362	157	15	1	1	NUM
cana-5362	157	16	2	2	NUM
cana-5362	157	17	(	(	PUNCT
cana-5362	157	18	|𝓋𝑛|2	|𝓋𝑛|2	PUNCT
cana-5362	157	19	−	−	PROPN
cana-5362	157	20	|𝓋𝑛−1|2	|𝓋𝑛−1|2	PROPN
cana-5362	157	21	)	)	PUNCT
cana-5362	158	1	+	+	CCONJ
cana-5362	158	2	1	1	NUM
cana-5362	158	3	2	2	NUM
cana-5362	158	4	|𝓋𝑛	|𝓋𝑛	NUM
cana-5362	158	5	−𝓋𝑛−1|2	−𝓋𝑛−1|2	X
cana-5362	158	6	(	(	PUNCT
cana-5362	158	7	38	38	NUM
cana-5362	158	8	)	)	PUNCT
cana-5362	158	9	thus	thus	ADV
cana-5362	158	10	(	(	PUNCT
cana-5362	158	11	37	37	NUM
cana-5362	158	12	)	)	PUNCT
cana-5362	158	13	implies	imply	VERB
cana-5362	158	14	|𝓋𝑛|2	|𝓋𝑛|2	PROPN
cana-5362	158	15	−	−	PROPN
cana-5362	158	16	|𝓋𝑛−1|2	|𝓋𝑛−1|2	PROPN
cana-5362	158	17	+	+	CCONJ
cana-5362	158	18	|𝓋𝑛	|𝓋𝑛	X
cana-5362	158	19	−𝓋𝑛−1|2	−𝓋𝑛−1|2	X
cana-5362	158	20	+	+	CCONJ
cana-5362	158	21	2𝑘	2𝑘	NUM
cana-5362	158	22	⟨𝒜𝑛𝑢𝑛	⟨𝒜𝑛𝑢𝑛	NOUN
cana-5362	158	23	,	,	PUNCT
cana-5362	158	24	𝑢𝑛	𝑢𝑛	PROPN
cana-5362	158	25	⟩	⟩	NOUN
cana-5362	158	26	=	=	PROPN
cana-5362	159	1	2𝑘⟨	2𝑘⟨	NUM
cana-5362	160	1	𝑊𝑛	𝑊𝑛	PROPN
cana-5362	160	2	,	,	PUNCT
cana-5362	160	3	𝒜𝑛𝑢𝑛⟩	𝒜𝑛𝑢𝑛⟩	X
cana-5362	160	4	(	(	PUNCT
cana-5362	160	5	39	39	NUM
cana-5362	160	6	)	)	PUNCT
cana-5362	160	7	according	accord	VERB
cana-5362	160	8	to	to	ADP
cana-5362	160	9	properties	property	NOUN
cana-5362	160	10	(	(	PUNCT
cana-5362	160	11	8)	8)	NUM
cana-5362	160	12	and	and	CCONJ
cana-5362	160	13	(	(	PUNCT
cana-5362	160	14	10	10	NUM
cana-5362	160	15	)	)	PUNCT
cana-5362	160	16	of	of	ADP
cana-5362	160	17	𝒜	𝒜	NOUN
cana-5362	160	18	,	,	PUNCT
cana-5362	160	19	which	which	PRON
cana-5362	160	20	lead	lead	VERB
cana-5362	160	21	to	to	ADP
cana-5362	160	22	the	the	DET
cana-5362	160	23	same	same	ADJ
cana-5362	160	24	for	for	ADP
cana-5362	160	25	𝒜𝑛	𝒜𝑛	PROPN
cana-5362	160	26	,	,	PUNCT
cana-5362	160	27	we	we	PRON
cana-5362	160	28	deduce	deduce	VERB
cana-5362	160	29	from	from	ADP
cana-5362	160	30	(	(	PUNCT
cana-5362	160	31	39	39	NUM
cana-5362	160	32	)	)	PUNCT
cana-5362	160	33	the	the	DET
cana-5362	160	34	following	follow	VERB
cana-5362	160	35	estimate	estimate	NOUN
cana-5362	160	36	|𝓋𝑛|2	|𝓋𝑛|2	PUNCT
cana-5362	160	37	−	−	PROPN
cana-5362	160	38	|𝓋𝑛−1|2	|𝓋𝑛−1|2	PROPN
cana-5362	160	39	+	+	CCONJ
cana-5362	160	40	2𝑘𝜌‖𝑢𝑛‖2	2𝑘𝜌‖𝑢𝑛‖2	NUM
cana-5362	160	41	≤	≤	NUM
cana-5362	160	42	2𝑘𝛽‖𝑊𝑛‖𝒱‖𝑢	2𝑘𝛽‖𝑊𝑛‖𝒱‖𝑢	NUM
cana-5362	160	43	𝑛‖𝒱	𝑛‖𝒱	NOUN
cana-5362	160	44	(	(	PUNCT
cana-5362	160	45	40	40	NUM
cana-5362	160	46	)	)	PUNCT
cana-5362	160	47	we	we	PRON
cana-5362	160	48	then	then	ADV
cana-5362	160	49	use	use	VERB
cana-5362	160	50	the	the	DET
cana-5362	160	51	following	follow	VERB
cana-5362	160	52	classic	classic	ADJ
cana-5362	160	53	inequality	inequality	NOUN
cana-5362	160	54	:	:	PUNCT
cana-5362	160	55	if	if	SCONJ
cana-5362	160	56	𝑖	𝑖	X
cana-5362	160	57	,	,	PUNCT
cana-5362	160	58	𝑗	𝑗	INTJ
cana-5362	160	59	>	>	X
cana-5362	160	60	0	0	NUM
cana-5362	160	61	satisfy	satisfy	NOUN
cana-5362	160	62	1	1	NUM
cana-5362	160	63	𝑖	𝑖	SYM
cana-5362	161	1	+	+	NOUN
cana-5362	161	2	1	1	NUM
cana-5362	161	3	𝑗	𝑗	NOUN
cana-5362	161	4	=	=	SYM
cana-5362	161	5	1	1	NUM
cana-5362	161	6	,	,	PUNCT
cana-5362	161	7	then	then	ADV
cana-5362	161	8	𝑎𝑏	𝑎𝑏	PROPN
cana-5362	161	9	≤	≤	ADV
cana-5362	161	10	𝑎𝑖𝑐𝑖	𝑎𝑖𝑐𝑖	ADJ
cana-5362	161	11	𝑖	𝑖	X
cana-5362	161	12	+	+	NUM
cana-5362	161	13	𝑏𝑗	𝑏𝑗	X
cana-5362	161	14	𝑗𝑐𝑗	𝑗𝑐𝑗	NOUN
cana-5362	161	15	,	,	PUNCT
cana-5362	161	16	∀𝑎	∀𝑎	PROPN
cana-5362	161	17	,	,	PUNCT
cana-5362	161	18	𝑏	𝑏	NOUN
cana-5362	161	19	,	,	PUNCT
cana-5362	161	20	𝑐	𝑐	NOUN
cana-5362	161	21	>	>	X
cana-5362	161	22	0	0	NUM
cana-5362	161	23	.	.	PUNCT
cana-5362	162	1	therefore	therefore	ADV
cana-5362	162	2	,	,	PUNCT
cana-5362	162	3	we	we	PRON
cana-5362	162	4	have	have	VERB
cana-5362	162	5	‖𝑊𝑛‖𝒱‖𝑢	‖𝑊𝑛‖𝒱‖𝑢	ADP
cana-5362	162	6	𝑛‖𝒱	𝑛‖𝒱	NOUN
cana-5362	162	7	𝑝−1	𝑝−1	PROPN
cana-5362	162	8	≤	≤	NUM
cana-5362	162	9	‖𝑢𝑛‖𝒱	‖𝑢𝑛‖𝒱	PROPN
cana-5362	162	10	𝑝	𝑝	PROPN
cana-5362	162	11	𝑙𝑝	𝑙𝑝	INTJ
cana-5362	162	12	′	′	NUM
cana-5362	162	13	𝑝′	𝑝′	PUNCT
cana-5362	163	1	+	+	CCONJ
cana-5362	163	2	‖𝑊𝑛‖𝒱	‖𝑊𝑛‖𝒱	PROPN
cana-5362	163	3	𝑝	𝑝	PROPN
cana-5362	163	4	1	1	NUM
cana-5362	163	5	𝑝𝑙𝑝	𝑝𝑙𝑝	NOUN
cana-5362	163	6	,	,	PUNCT
cana-5362	163	7	∀𝑙	∀𝑙	NOUN
cana-5362	163	8	>	>	X
cana-5362	163	9	0	0	PROPN
cana-5362	163	10	.	.	PUNCT
cana-5362	164	1	(	(	PUNCT
cana-5362	164	2	41	41	NUM
cana-5362	164	3	)	)	PUNCT
cana-5362	164	4	communications	communication	NOUN
cana-5362	164	5	on	on	ADP
cana-5362	164	6	applied	apply	VERB
cana-5362	164	7	nonlinear	nonlinear	ADJ
cana-5362	164	8	analysis	analysis	NOUN
cana-5362	164	9	issn	issn	NOUN
cana-5362	164	10	:	:	PUNCT
cana-5362	164	11	1074	1074	NUM
cana-5362	164	12	-	-	PUNCT
cana-5362	164	13	133x	133x	NUM
cana-5362	164	14	vol	vol	VERB
cana-5362	164	15	32	32	NUM
cana-5362	164	16	no	no	NOUN
cana-5362	164	17	.	.	PUNCT
cana-5362	165	1	10s	10	NOUN
cana-5362	165	2	(	(	PUNCT
cana-5362	165	3	2025	2025	NUM
cana-5362	165	4	)	)	PUNCT
cana-5362	165	5	1993	1993	NUM
cana-5362	165	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5362	165	7	let	let	VERB
cana-5362	165	8	us	we	PRON
cana-5362	165	9	choose	choose	VERB
cana-5362	165	10	𝑙	𝑙	PRON
cana-5362	165	11	such	such	ADJ
cana-5362	165	12	that	that	DET
cana-5362	165	13	𝐶	𝐶	PROPN
cana-5362	165	14	=	=	PROPN
cana-5362	165	15	2𝜌	2𝜌	NOUN
cana-5362	165	16	+	+	CCONJ
cana-5362	165	17	2𝛽	2𝛽	NOUN
cana-5362	165	18	𝑙2	𝑙2	NOUN
cana-5362	165	19	2	2	NUM
cana-5362	165	20	.	.	PUNCT
cana-5362	166	1	then	then	ADV
cana-5362	166	2	,	,	PUNCT
cana-5362	166	3	taking	take	VERB
cana-5362	166	4	into	into	ADP
cana-5362	166	5	account	account	NOUN
cana-5362	166	6	(	(	PUNCT
cana-5362	166	7	41	41	NUM
cana-5362	166	8	)	)	PUNCT
cana-5362	166	9	,	,	PUNCT
cana-5362	166	10	we	we	PRON
cana-5362	166	11	deduce	deduce	VERB
cana-5362	166	12	from	from	ADP
cana-5362	166	13	(	(	PUNCT
cana-5362	166	14	36	36	NUM
cana-5362	166	15	)	)	PUNCT
cana-5362	166	16	the	the	DET
cana-5362	166	17	following	follow	VERB
cana-5362	166	18	inequality	inequality	NOUN
cana-5362	166	19	|𝑣𝑛|2	|𝑣𝑛|2	PUNCT
cana-5362	166	20	−	−	PROPN
cana-5362	166	21	|𝑣𝑛−1|2	|𝑣𝑛−1|2	PROPN
cana-5362	167	1	+	+	NUM
cana-5362	167	2	𝑘𝐶‖𝑢𝑛‖2	𝑘𝐶‖𝑢𝑛‖2	NUM
cana-5362	167	3	≤	≤	NUM
cana-5362	167	4	2𝑘𝛽	2𝑘𝛽	ADJ
cana-5362	167	5	2𝑙2	2𝑙2	NUM
cana-5362	168	1	‖𝑊𝑛‖𝒱	‖𝑊𝑛‖𝒱	NOUN
cana-5362	168	2	(	(	PUNCT
cana-5362	168	3	42	42	NUM
cana-5362	168	4	)	)	PUNCT
cana-5362	168	5	since	since	SCONJ
cana-5362	168	6	𝑊𝑡	𝑊𝑡	PROPN
cana-5362	168	7	∈	∈	PROPN
cana-5362	168	8	𝐶(𝒱	𝐶(𝒱	PROPN
cana-5362	168	9	)	)	PUNCT
cana-5362	169	1	‖𝑊𝑛‖𝒱	‖𝑊𝑛‖𝒱	NOUN
cana-5362	169	2	≤	≤	NOUN
cana-5362	169	3	𝐶1	𝐶1	NUM
cana-5362	169	4	,	,	PUNCT
cana-5362	169	5	∀𝑛	∀𝑛	NOUN
cana-5362	169	6	(	(	PUNCT
cana-5362	169	7	43	43	NUM
cana-5362	169	8	)	)	PUNCT
cana-5362	169	9	from	from	ADP
cana-5362	169	10	(	(	PUNCT
cana-5362	169	11	43	43	NUM
cana-5362	169	12	)	)	PUNCT
cana-5362	169	13	,	,	PUNCT
cana-5362	169	14	we	we	PRON
cana-5362	169	15	finally	finally	ADV
cana-5362	169	16	obtain	obtain	VERB
cana-5362	169	17	the	the	DET
cana-5362	169	18	upper	upper	ADJ
cana-5362	169	19	bound	bind	VERB
cana-5362	169	20	|𝓋𝑛|ℋ	|𝓋𝑛|ℋ	PROPN
cana-5362	169	21	2	2	NUM
cana-5362	169	22	−	−	NOUN
cana-5362	169	23	|𝓋𝑛−1|ℋ	|𝓋𝑛−1|ℋ	NUM
cana-5362	169	24	2	2	NUM
cana-5362	169	25	+	+	CCONJ
cana-5362	169	26	𝑘𝐶‖𝑢𝑛‖𝒱	𝑘𝐶‖𝑢𝑛‖𝒱	PROPN
cana-5362	169	27	2	2	NUM
cana-5362	169	28	≤	≤	NUM
cana-5362	169	29	𝑘𝐶2	𝑘𝐶2	PROPN
cana-5362	169	30	,	,	PUNCT
cana-5362	169	31	(	(	PUNCT
cana-5362	169	32	44	44	NUM
cana-5362	169	33	)	)	PUNCT
cana-5362	169	34	where	where	SCONJ
cana-5362	169	35	𝐶2	𝐶2	ADV
cana-5362	169	36	=	=	NOUN
cana-5362	169	37	2𝐶1	2𝐶1	NUM
cana-5362	170	1	𝛽	𝛽	NOUN
cana-5362	170	2	2𝑙2	2𝑙2	NUM
cana-5362	170	3	.	.	PUNCT
cana-5362	171	1	by	by	ADP
cana-5362	171	2	summing	sum	VERB
cana-5362	171	3	,	,	PUNCT
cana-5362	171	4	we	we	PRON
cana-5362	171	5	easily	easily	ADV
cana-5362	171	6	deduce	deduce	VERB
cana-5362	171	7	the	the	DET
cana-5362	171	8	following	follow	VERB
cana-5362	171	9	bounds	bound	NOUN
cana-5362	171	10	from	from	ADP
cana-5362	171	11	(	(	PUNCT
cana-5362	171	12	44	44	NUM
cana-5362	171	13	)	)	PUNCT
cana-5362	171	14	.	.	PUNCT
cana-5362	172	1	{	{	PUNCT
cana-5362	172	2	∀𝑛	∀𝑛	NOUN
cana-5362	172	3	,	,	PUNCT
cana-5362	172	4	|𝓋𝑛|ℋ	|𝓋𝑛|ℋ	PROPN
cana-5362	172	5	2	2	NUM
cana-5362	172	6	≤	≤	NOUN
cana-5362	172	7	|𝑢0	|𝑢0	VERB
cana-5362	172	8	−𝑊0|ℋ	−𝑊0|ℋ	NOUN
cana-5362	172	9	2	2	NUM
cana-5362	172	10	+	+	CCONJ
cana-5362	172	11	𝑇𝐶2	𝑇𝐶2	PROPN
cana-5362	172	12	𝐶	𝐶	PROPN
cana-5362	172	13	∑	∑	PROPN
cana-5362	172	14	𝑘𝑁	𝑘𝑁	PROPN
cana-5362	172	15	𝑛=1	𝑛=1	PROPN
cana-5362	172	16	‖𝑢𝑛‖𝒱	‖𝑢𝑛‖𝒱	PROPN
cana-5362	172	17	2	2	NUM
cana-5362	172	18	≤	≤	NOUN
cana-5362	172	19	|𝑢0	|𝑢0	VERB
cana-5362	172	20	−𝑊0|ℋ	−𝑊0|ℋ	NOUN
cana-5362	172	21	2	2	NUM
cana-5362	172	22	+	+	CCONJ
cana-5362	172	23	𝑇𝐶2	𝑇𝐶2	PROPN
cana-5362	172	24	(	(	PUNCT
cana-5362	172	25	45	45	NUM
cana-5362	172	26	)	)	PUNCT
cana-5362	172	27	but	but	CCONJ
cana-5362	172	28	(	(	PUNCT
cana-5362	172	29	45	45	NUM
cana-5362	172	30	)	)	PUNCT
cana-5362	172	31	means	mean	VERB
cana-5362	172	32	that	that	SCONJ
cana-5362	172	33	uk	uk	PROPN
cana-5362	172	34	remains	remain	VERB
cana-5362	172	35	bounded	bounded	ADJ
cana-5362	172	36	in	in	ADP
cana-5362	172	37	𝐿2(𝒱	𝐿2(𝒱	PROPN
cana-5362	172	38	)	)	PUNCT
cana-5362	172	39	and	and	CCONJ
cana-5362	172	40	𝓋k	𝓋k	NOUN
cana-5362	172	41	remains	remain	NOUN
cana-5362	172	42	bounded	bound	VERB
cana-5362	172	43	in	in	ADP
cana-5362	172	44	𝐿∞(ℋ	𝐿∞(ℋ	PROPN
cana-5362	172	45	)	)	PUNCT
cana-5362	172	46	.	.	PUNCT
cana-5362	173	1	since	since	SCONJ
cana-5362	173	2	wk	wk	X
cana-5362	173	3	remains	remain	VERB
cana-5362	173	4	bounded	bound	VERB
cana-5362	173	5	in	in	ADP
cana-5362	173	6	𝐿(𝒱	𝐿(𝒱	PROPN
cana-5362	173	7	)	)	PUNCT
cana-5362	173	8	,	,	PUNCT
cana-5362	173	9	the	the	DET
cana-5362	173	10	lemma	lemma	PROPN
cana-5362	173	11	follows	follow	VERB
cana-5362	173	12	.	.	PUNCT
cana-5362	174	1	let	let	VERB
cana-5362	174	2	t	t	NOUN
cana-5362	174	3	be	be	AUX
cana-5362	174	4	a	a	DET
cana-5362	174	5	fixed	fix	VERB
cana-5362	174	6	value	value	NOUN
cana-5362	174	7	in	in	ADP
cana-5362	174	8	[	[	X
cana-5362	174	9	0	0	NUM
cana-5362	174	10	,	,	PUNCT
cana-5362	174	11	t	t	PROPN
cana-5362	174	12	]	]	PUNCT
cana-5362	174	13	,	,	PUNCT
cana-5362	174	14	and	and	CCONJ
cana-5362	174	15	define	define	VERB
cana-5362	174	16	nt	not	PART
cana-5362	174	17	=	=	PROPN
cana-5362	174	18	t	t	PROPN
cana-5362	174	19	k	k	PROPN
cana-5362	174	20	.	.	PUNCT
cana-5362	175	1	by	by	ADP
cana-5362	175	2	summing	sum	VERB
cana-5362	175	3	the	the	DET
cana-5362	175	4	discret	discret	ADJ
cana-5362	175	5	relations	relation	NOUN
cana-5362	175	6	for	for	ADP
cana-5362	175	7	n	n	NUM
cana-5362	175	8	from	from	ADP
cana-5362	175	9	1	1	NUM
cana-5362	175	10	to	to	ADP
cana-5362	175	11	nt	not	PART
cana-5362	175	12	,	,	PUNCT
cana-5362	175	13	we	we	PRON
cana-5362	175	14	obtain	obtain	VERB
cana-5362	175	15	𝓋𝑛𝑡	𝓋𝑛𝑡	NOUN
cana-5362	176	1	+	+	CCONJ
cana-5362	176	2	𝑘∑	𝑘∑	PROPN
cana-5362	176	3	𝒜𝑛𝑢𝑛	𝒜𝑛𝑢𝑛	PROPN
cana-5362	176	4	=	=	PUNCT
cana-5362	176	5	𝑢0	𝑢0	PROPN
cana-5362	176	6	−𝑊0	−𝑊0	PROPN
cana-5362	176	7	𝑛𝑡	𝑛𝑡	NOUN
cana-5362	176	8	𝑛=1	𝑛=1	NOUN
cana-5362	176	9	,	,	PUNCT
cana-5362	176	10	(	(	PUNCT
cana-5362	176	11	46	46	NUM
cana-5362	176	12	)	)	PUNCT
cana-5362	176	13	which	which	PRON
cana-5362	176	14	is	be	AUX
cana-5362	176	15	also	also	ADV
cana-5362	176	16	written	write	VERB
cana-5362	176	17	as	as	ADP
cana-5362	176	18	𝓋𝑘(𝑡	𝓋𝑘(𝑡	NOUN
cana-5362	176	19	)	)	PUNCT
cana-5362	177	1	+	+	CCONJ
cana-5362	177	2	∫	∫	PROPN
cana-5362	177	3	𝒜𝑘𝑢𝑘	𝒜𝑘𝑢𝑘	ADJ
cana-5362	177	4	𝑛𝑡𝑘	𝑛𝑡𝑘	NOUN
cana-5362	177	5	0	0	PUNCT
cana-5362	178	1	(	(	PUNCT
cana-5362	178	2	𝑠)𝑑𝑠	𝑠)𝑑𝑠	PROPN
cana-5362	178	3	=	=	SYM
cana-5362	178	4	𝑢0	𝑢0	PROPN
cana-5362	178	5	−𝑊0	−𝑊0	PROPN
cana-5362	178	6	(	(	PUNCT
cana-5362	178	7	47	47	NUM
cana-5362	178	8	)	)	PUNCT
cana-5362	178	9	also	also	ADV
cana-5362	178	10	,	,	PUNCT
cana-5362	178	11	𝒜k	𝒜k	PROPN
cana-5362	178	12	can	can	AUX
cana-5362	178	13	be	be	AUX
cana-5362	178	14	replaced	replace	VERB
cana-5362	178	15	by	by	ADP
cana-5362	178	16	𝒜	𝒜	NOUN
cana-5362	178	17	in	in	ADP
cana-5362	178	18	this	this	DET
cana-5362	178	19	equality	equality	NOUN
cana-5362	178	20	(	(	PUNCT
cana-5362	178	21	see	see	VERB
cana-5362	178	22	[	[	X
cana-5362	178	23	9	9	NUM
cana-5362	178	24	]	]	NUM
cana-5362	178	25	)	)	PUNCT
cana-5362	178	26	.	.	PUNCT
cana-5362	179	1	now	now	ADV
cana-5362	179	2	,	,	PUNCT
cana-5362	179	3	let	let	VERB
cana-5362	179	4	χ	χ	PRON
cana-5362	179	5	∈	∈	PROPN
cana-5362	179	6	𝒱	𝒱	PROPN
cana-5362	179	7	,	,	PUNCT
cana-5362	179	8	(	(	PUNCT
cana-5362	179	9	47	47	NUM
cana-5362	179	10	)	)	PUNCT
cana-5362	179	11	gives	give	VERB
cana-5362	179	12	the	the	DET
cana-5362	179	13	following	follow	VERB
cana-5362	179	14	(	(	PUNCT
cana-5362	179	15	𝓋𝑘(𝑡	𝓋𝑘(𝑡	NOUN
cana-5362	179	16	)	)	PUNCT
cana-5362	179	17	,	,	PUNCT
cana-5362	179	18	𝜒	𝜒	X
cana-5362	179	19	)	)	PUNCT
cana-5362	179	20	+	+	NUM
cana-5362	179	21	∫	∫	PROPN
cana-5362	179	22	〈	〈	NOUN
cana-5362	179	23	𝒜𝑢𝑘(𝑠	𝒜𝑢𝑘(𝑠	PROPN
cana-5362	179	24	)	)	PUNCT
cana-5362	179	25	,	,	PUNCT
cana-5362	179	26	𝜁𝑛𝑡𝑘(𝑠)𝜒〉𝑑𝑠	𝜁𝑛𝑡𝑘(𝑠)𝜒〉𝑑𝑠	PROPN
cana-5362	179	27	𝑇	𝑇	PROPN
cana-5362	179	28	0	0	NUM
cana-5362	179	29	=	=	SYM
cana-5362	179	30	𝑢0	𝑢0	PROPN
cana-5362	179	31	,	,	PUNCT
cana-5362	179	32	(	(	PUNCT
cana-5362	179	33	48	48	NUM
cana-5362	179	34	)	)	PUNCT
cana-5362	179	35	where	where	SCONJ
cana-5362	179	36	ζntk(s	ζntk(s	NOUN
cana-5362	179	37	)	)	PUNCT
cana-5362	179	38	denotes	denote	VERB
cana-5362	179	39	the	the	DET
cana-5362	179	40	characteristic	characteristic	ADJ
cana-5362	179	41	function	function	NOUN
cana-5362	179	42	of	of	ADP
cana-5362	179	43	]	]	X
cana-5362	179	44	0	0	NUM
cana-5362	179	45	,	,	PUNCT
cana-5362	179	46	ntk	ntk	PROPN
cana-5362	179	47	[	[	X
cana-5362	179	48	.	.	PUNCT
cana-5362	180	1	lemma	lemma	PROPN
cana-5362	180	2	3	3	NUM
cana-5362	180	3	allows	allow	VERB
cana-5362	180	4	us	we	PRON
cana-5362	180	5	to	to	PART
cana-5362	180	6	extract	extract	VERB
cana-5362	180	7	sequences	sequence	NOUN
cana-5362	180	8	uk	uk	PROPN
cana-5362	180	9	and	and	CCONJ
cana-5362	180	10	𝓋k	𝓋k	NOUN
cana-5362	180	11	from	from	ADP
cana-5362	180	12	subsequences	subsequence	NOUN
cana-5362	180	13	denoted	denote	VERB
cana-5362	180	14	in	in	ADP
cana-5362	180	15	the	the	DET
cana-5362	180	16	same	same	ADJ
cana-5362	180	17	way	way	NOUN
cana-5362	180	18	,	,	PUNCT
cana-5362	180	19	which	which	PRON
cana-5362	180	20	converge	converge	VERB
cana-5362	180	21	to	to	ADP
cana-5362	180	22	elements	element	NOUN
cana-5362	180	23	u	u	NOUN
cana-5362	180	24	and	and	CCONJ
cana-5362	180	25	𝓋	𝓋	PROPN
cana-5362	180	26	in	in	ADP
cana-5362	180	27	weak	weak	ADJ
cana-5362	180	28	𝐿2(𝒱	𝐿2(𝒱	PROPN
cana-5362	180	29	)	)	PUNCT
cana-5362	180	30	and	and	CCONJ
cana-5362	180	31	weak	weak	ADJ
cana-5362	180	32	-	-	PUNCT
cana-5362	180	33	star	star	NOUN
cana-5362	180	34	𝐿∞(ℋ	𝐿∞(ℋ	PROPN
cana-5362	180	35	)	)	PUNCT
cana-5362	180	36	.	.	PUNCT
cana-5362	181	1	it	it	PRON
cana-5362	181	2	is	be	AUX
cana-5362	181	3	evident	evident	ADJ
cana-5362	181	4	that	that	SCONJ
cana-5362	181	5	u	u	NOUN
cana-5362	181	6	=	=	X
cana-5362	181	7	𝓋	𝓋	PROPN
cana-5362	181	8	+	+	NOUN
cana-5362	181	9	w.	w.	NOUN
cana-5362	181	10	furthermore	furthermore	ADV
cana-5362	181	11	,	,	PUNCT
cana-5362	181	12	according	accord	VERB
cana-5362	181	13	to	to	ADP
cana-5362	181	14	property	property	NOUN
cana-5362	181	15	(	(	PUNCT
cana-5362	181	16	10	10	NUM
cana-5362	181	17	)	)	PUNCT
cana-5362	181	18	of	of	ADP
cana-5362	181	19	𝒜	𝒜	NOUN
cana-5362	181	20	,	,	PUNCT
cana-5362	181	21	we	we	PRON
cana-5362	181	22	deduce	deduce	VERB
cana-5362	181	23	from	from	ADP
cana-5362	181	24	lemma	lemma	PROPN
cana-5362	181	25	3	3	NUM
cana-5362	181	26	that	that	PRON
cana-5362	181	27	𝒜uk(s	𝒜uk(s	PROPN
cana-5362	181	28	)	)	PUNCT
cana-5362	181	29	remains	remain	VERB
cana-5362	181	30	bounded	bound	VERB
cana-5362	181	31	in	in	ADP
cana-5362	181	32	𝐿2(𝒱′	𝐿2(𝒱′	PROPN
cana-5362	181	33	)	)	PUNCT
cana-5362	181	34	and	and	CCONJ
cana-5362	181	35	by	by	ADP
cana-5362	181	36	extracting	extract	VERB
cana-5362	181	37	a	a	DET
cana-5362	181	38	new	new	ADJ
cana-5362	181	39	subsequence	subsequence	NOUN
cana-5362	181	40	if	if	SCONJ
cana-5362	181	41	necessary	necessary	ADJ
cana-5362	181	42	,	,	PUNCT
cana-5362	181	43	we	we	PRON
cana-5362	181	44	can	can	AUX
cana-5362	181	45	always	always	ADV
cana-5362	181	46	assume	assume	VERB
cana-5362	181	47	that	that	SCONJ
cana-5362	181	48	:	:	PUNCT
cana-5362	181	49	𝒜uk	𝒜uk	PROPN
cana-5362	181	50	(	(	PUNCT
cana-5362	181	51	.	.	PUNCT
cana-5362	181	52	)	)	PUNCT
cana-5362	182	1	→	→	PUNCT
cana-5362	182	2	𝛹(s	𝛹(s	ADV
cana-5362	182	3	)	)	PUNCT
cana-5362	182	4	weakly	weakly	ADV
cana-5362	182	5	in	in	ADP
cana-5362	182	6	𝐿2(𝒱′	𝐿2(𝒱′	PROPN
cana-5362	182	7	)	)	PUNCT
cana-5362	182	8	.	.	PUNCT
cana-5362	183	1	we	we	PRON
cana-5362	183	2	can	can	AUX
cana-5362	183	3	then	then	ADV
cana-5362	183	4	take	take	VERB
cana-5362	183	5	the	the	DET
cana-5362	183	6	limit	limit	NOUN
cana-5362	183	7	in	in	ADP
cana-5362	183	8	(	(	PUNCT
cana-5362	183	9	48	48	NUM
cana-5362	183	10	)	)	PUNCT
cana-5362	183	11	and	and	CCONJ
cana-5362	183	12	obtain	obtain	VERB
cana-5362	183	13	(	(	PUNCT
cana-5362	183	14	𝓋𝑘(𝑡	𝓋𝑘(𝑡	NOUN
cana-5362	183	15	)	)	PUNCT
cana-5362	183	16	,	,	PUNCT
cana-5362	183	17	𝜒	𝜒	X
cana-5362	183	18	)	)	PUNCT
cana-5362	183	19	+	+	NUM
cana-5362	183	20	∫	∫	PROPN
cana-5362	183	21	〈	〈	PROPN
cana-5362	183	22	𝛹(𝑠	𝛹(𝑠	NOUN
cana-5362	183	23	)	)	PUNCT
cana-5362	183	24	,	,	PUNCT
cana-5362	183	25	𝜒〉𝑑𝑠	𝜒〉𝑑𝑠	PUNCT
cana-5362	183	26	=	=	SYM
cana-5362	183	27	𝑢0	𝑢0	PROPN
cana-5362	183	28	𝑡	𝑡	PROPN
cana-5362	183	29	0	0	NUM
cana-5362	183	30	.	.	PUNCT
cana-5362	184	1	(	(	PUNCT
cana-5362	184	2	49	49	NUM
cana-5362	184	3	)	)	PUNCT
cana-5362	184	4	then	then	ADV
cana-5362	184	5	communications	communication	NOUN
cana-5362	184	6	on	on	ADP
cana-5362	184	7	applied	apply	VERB
cana-5362	184	8	nonlinear	nonlinear	ADJ
cana-5362	184	9	analysis	analysis	NOUN
cana-5362	184	10	issn	issn	NOUN
cana-5362	184	11	:	:	PUNCT
cana-5362	184	12	1074	1074	NUM
cana-5362	184	13	-	-	PUNCT
cana-5362	184	14	133x	133x	NUM
cana-5362	184	15	vol	vol	VERB
cana-5362	184	16	32	32	NUM
cana-5362	184	17	no	no	NOUN
cana-5362	184	18	.	.	PUNCT
cana-5362	185	1	10s	10	NOUN
cana-5362	185	2	(	(	PUNCT
cana-5362	185	3	2025	2025	NUM
cana-5362	185	4	)	)	PUNCT
cana-5362	185	5	1994	1994	NUM
cana-5362	185	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5362	185	7	𝑑𝓋	𝑑𝓋	ADP
cana-5362	185	8	𝑑𝑡	𝑑𝑡	ADP
cana-5362	185	9	+	+	PROPN
cana-5362	185	10	𝛹(𝑠	𝛹(𝑠	NOUN
cana-5362	185	11	)	)	PUNCT
cana-5362	185	12	=	=	SYM
cana-5362	185	13	0	0	NUM
cana-5362	185	14	,	,	PUNCT
cana-5362	185	15	𝑎.	𝑎.	PROPN
cana-5362	185	16	𝑒.	𝑒.	PROPN
cana-5362	186	1	𝑡	𝑡	PROPN
cana-5362	186	2	∈	∈	PROPN
cana-5362	187	1	[	[	X
cana-5362	187	2	0	0	NUM
cana-5362	187	3	,	,	PUNCT
cana-5362	187	4	𝑇	𝑇	PROPN
cana-5362	187	5	]	]	PUNCT
cana-5362	187	6	(	(	PUNCT
cana-5362	187	7	equality	equality	NOUN
cana-5362	187	8	in	in	ADP
cana-5362	187	9	𝒱′	𝒱′	NOUN
cana-5362	187	10	)	)	PUNCT
cana-5362	187	11	.	.	PUNCT
cana-5362	188	1	(	(	PUNCT
cana-5362	188	2	50	50	NUM
cana-5362	188	3	)	)	PUNCT
cana-5362	188	4	then	then	ADV
cana-5362	188	5	,	,	PUNCT
cana-5362	188	6	we	we	PRON
cana-5362	188	7	have	have	VERB
cana-5362	188	8	lemma	lemma	PROPN
cana-5362	188	9	4	4	NUM
cana-5362	188	10	ψ(t	ψ(t	PROPN
cana-5362	188	11	)	)	PUNCT
cana-5362	188	12	=	=	SYM
cana-5362	188	13	𝒜u(t	𝒜u(t	NOUN
cana-5362	188	14	)	)	PUNCT
cana-5362	188	15	,	,	PUNCT
cana-5362	188	16	a.	a.	PROPN
cana-5362	188	17	e.	e.	PROPN
cana-5362	188	18	t	t	PROPN
cana-5362	188	19	∈	∈	PROPN
cana-5362	189	1	[	[	X
cana-5362	189	2	0	0	NUM
cana-5362	189	3	,	,	PUNCT
cana-5362	189	4	t	t	PROPN
cana-5362	189	5	]	]	X
cana-5362	189	6	(	(	PUNCT
cana-5362	189	7	51	51	NUM
cana-5362	189	8	)	)	PUNCT
cana-5362	189	9	proof	proof	NOUN
cana-5362	189	10	.	.	PUNCT
cana-5362	190	1	first	first	ADV
cana-5362	190	2	,	,	PUNCT
cana-5362	190	3	let	let	VERB
cana-5362	190	4	’s	’s	PRON
cana-5362	190	5	prove	prove	VERB
cana-5362	190	6	that	that	SCONJ
cana-5362	190	7	∫	∫	PROPN
cana-5362	190	8	〈	〈	NOUN
cana-5362	190	9	𝒜𝑢𝑘(𝑠	𝒜𝑢𝑘(𝑠	PROPN
cana-5362	190	10	)	)	PUNCT
cana-5362	190	11	,	,	PUNCT
cana-5362	190	12	𝑢𝑘(𝑠)〉𝑑𝑠	𝑢𝑘(𝑠)〉𝑑𝑠	PROPN
cana-5362	190	13	𝑘→0	𝑘→0	PUNCT
cana-5362	190	14	→	→	SYM
cana-5362	190	15	∫	∫	PROPN
cana-5362	190	16	〈	〈	PROPN
cana-5362	190	17	𝛹(𝑠	𝛹(𝑠	NOUN
cana-5362	190	18	)	)	PUNCT
cana-5362	190	19	,	,	PUNCT
cana-5362	190	20	𝑢(𝑠)〉𝑑𝑠.	𝑢(𝑠)〉𝑑𝑠.	PROPN
cana-5362	190	21	𝑇	𝑇	PROPN
cana-5362	190	22	0	0	NUM
cana-5362	190	23	𝑇	𝑇	PROPN
cana-5362	190	24	0	0	NUM
cana-5362	190	25	(	(	PUNCT
cana-5362	190	26	52	52	NUM
cana-5362	190	27	)	)	PUNCT
cana-5362	190	28	indeed	indeed	ADV
cana-5362	190	29	,	,	PUNCT
cana-5362	190	30	(	(	PUNCT
cana-5362	190	31	50	50	NUM
cana-5362	190	32	)	)	PUNCT
cana-5362	190	33	shows	show	VERB
cana-5362	190	34	that	that	PRON
cana-5362	190	35	d𝓋	d𝓋	VERB
cana-5362	190	36	dt	dt	ADP
cana-5362	190	37	∈	∈	PROPN
cana-5362	190	38	𝐿2	𝐿2	PROPN
cana-5362	190	39	(	(	PUNCT
cana-5362	190	40	𝒱′	𝒱′	NOUN
cana-5362	190	41	)	)	PUNCT
cana-5362	190	42	.	.	PUNCT
cana-5362	191	1	since	since	SCONJ
cana-5362	191	2	𝓋	𝓋	PROPN
cana-5362	191	3	∈	∈	PROPN
cana-5362	191	4	𝐿2	𝐿2	PROPN
cana-5362	191	5	(	(	PUNCT
cana-5362	191	6	𝒱	𝒱	PROPN
cana-5362	191	7	)	)	PUNCT
cana-5362	191	8	,	,	PUNCT
cana-5362	191	9	it	it	PRON
cana-5362	191	10	follows	follow	VERB
cana-5362	191	11	(	(	PUNCT
cana-5362	191	12	cf	cf	NOUN
cana-5362	191	13	.	.	PUNCT
cana-5362	192	1	[	[	X
cana-5362	192	2	16	16	NUM
cana-5362	192	3	]	]	PUNCT
cana-5362	192	4	)	)	PUNCT
cana-5362	192	5	that	that	SCONJ
cana-5362	192	6	we	we	PRON
cana-5362	192	7	have	have	AUX
cana-5362	192	8	𝑑	𝑑	NOUN
cana-5362	192	9	𝑑𝑡	𝑑𝑡	ADP
cana-5362	192	10	|𝓋(𝑡)|ℋ	|𝓋(𝑡)|ℋ	NOUN
cana-5362	192	11	2	2	NUM
cana-5362	192	12	=	=	SYM
cana-5362	192	13	2	2	NUM
cana-5362	192	14	〈	〈	NOUN
cana-5362	192	15	𝓋(𝑡	𝓋(𝑡	PROPN
cana-5362	192	16	)	)	PUNCT
cana-5362	192	17	,	,	PUNCT
cana-5362	192	18	𝑑𝓋	𝑑𝓋	ADP
cana-5362	192	19	𝑑𝑡	𝑑𝑡	ADP
cana-5362	192	20	(	(	PUNCT
cana-5362	192	21	𝑡)〉𝒱,𝒱′	𝑡)〉𝒱,𝒱′	PROPN
cana-5362	192	22	𝑎.	𝑎.	PROPN
cana-5362	192	23	𝑒.	𝑒.	PROPN
cana-5362	192	24	𝑡.	𝑡.	PROPN
cana-5362	192	25	(	(	PUNCT
cana-5362	192	26	53	53	NUM
cana-5362	192	27	)	)	PUNCT
cana-5362	192	28	let	let	VERB
cana-5362	192	29	|𝓋(𝑇)|2	|𝓋(𝑇)|2	X
cana-5362	192	30	=	=	SYM
cana-5362	192	31	|𝓋(0)|2	|𝓋(0)|2	X
cana-5362	192	32	+	+	CCONJ
cana-5362	192	33	2∫	2∫	NUM
cana-5362	192	34	〈	〈	PROPN
cana-5362	192	35	𝓋(𝑠	𝓋(𝑠	PROPN
cana-5362	192	36	)	)	PUNCT
cana-5362	193	1	−	−	PROPN
cana-5362	194	1	𝛹(𝑠)〉𝒱,𝒱′𝑑𝑠.	𝛹(𝑠)〉𝒱,𝒱′𝑑𝑠.	PROPN
cana-5362	194	2	𝑇	𝑇	PROPN
cana-5362	194	3	0	0	NUM
cana-5362	194	4	(	(	PUNCT
cana-5362	194	5	54	54	NUM
cana-5362	194	6	)	)	PUNCT
cana-5362	194	7	next	next	ADV
cana-5362	194	8	,	,	PUNCT
cana-5362	194	9	consider	consider	VERB
cana-5362	194	10	the	the	DET
cana-5362	194	11	equalities	equality	NOUN
cana-5362	194	12	(	(	PUNCT
cana-5362	194	13	39	39	NUM
cana-5362	194	14	)	)	PUNCT
cana-5362	194	15	,	,	PUNCT
cana-5362	194	16	which	which	PRON
cana-5362	194	17	we	we	PRON
cana-5362	194	18	sum	sum	VERB
cana-5362	194	19	over	over	ADP
cana-5362	194	20	n	n	ADV
cana-5362	194	21	from	from	ADP
cana-5362	194	22	1	1	NUM
cana-5362	194	23	to	to	PART
cana-5362	194	24	n.	n.	VERB
cana-5362	194	25	from	from	ADP
cana-5362	194	26	this	this	PRON
cana-5362	194	27	,	,	PUNCT
cana-5362	194	28	we	we	PRON
cana-5362	194	29	easily	easily	ADV
cana-5362	194	30	deduce	deduce	VERB
cana-5362	194	31	the	the	DET
cana-5362	194	32	inequality	inequality	NOUN
cana-5362	194	33	|𝓋𝑘(𝑇)|	|𝓋𝑘(𝑇)|	NUM
cana-5362	194	34	2	2	NUM
cana-5362	194	35	+	+	CCONJ
cana-5362	194	36	2∫	2∫	NUM
cana-5362	194	37	〈	〈	ADJ
cana-5362	194	38	𝒜𝑢𝑘	𝒜𝑢𝑘	PROPN
cana-5362	194	39	,	,	PUNCT
cana-5362	194	40	𝑢𝑘(𝑠)〉𝑑𝑠	𝑢𝑘(𝑠)〉𝑑𝑠	PROPN
cana-5362	194	41	≤	≤	NUM
cana-5362	194	42	|𝓋(0)|	|𝓋(0)|	NOUN
cana-5362	194	43	2𝑇	2𝑇	NOUN
cana-5362	194	44	0	0	NUM
cana-5362	195	1	+	+	CCONJ
cana-5362	195	2	2∫	2∫	NUM
cana-5362	195	3	〈	〈	NOUN
cana-5362	195	4	𝑊𝑘(𝑠),𝒜𝑢𝑘〉𝑑𝑠	𝑊𝑘(𝑠),𝒜𝑢𝑘〉𝑑𝑠	NOUN
cana-5362	195	5	𝑇	𝑇	PROPN
cana-5362	195	6	0	0	NUM
cana-5362	195	7	.	.	PUNCT
cana-5362	196	1	(	(	PUNCT
cana-5362	196	2	55	55	NUM
cana-5362	196	3	)	)	PUNCT
cana-5362	196	4	by	by	ADP
cana-5362	196	5	taking	take	VERB
cana-5362	196	6	the	the	DET
cana-5362	196	7	upper	upper	ADJ
cana-5362	196	8	limit	limit	NOUN
cana-5362	196	9	,	,	PUNCT
cana-5362	196	10	we	we	PRON
cana-5362	196	11	obtain	obtain	VERB
cana-5362	196	12	𝑙𝑖𝑚	𝑙𝑖𝑚	ADJ
cana-5362	196	13	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-5362	196	14	𝑘→0	𝑘→0	PUNCT
cana-5362	197	1	∫	∫	PROPN
cana-5362	198	1	〈	〈	PROPN
cana-5362	198	2	𝒜𝑢𝑘	𝒜𝑢𝑘	PROPN
cana-5362	198	3	,	,	PUNCT
cana-5362	198	4	𝑢𝑘(𝑠)〉𝑑𝑠	𝑢𝑘(𝑠)〉𝑑𝑠	PROPN
cana-5362	198	5	≤	≤	ADV
cana-5362	198	6	1	1	NUM
cana-5362	198	7	2	2	NUM
cana-5362	198	8	|𝓋(0)|2	|𝓋(0)|2	X
cana-5362	198	9	+	+	CCONJ
cana-5362	199	1	∫	∫	PROPN
cana-5362	199	2	〈	〈	PRON
cana-5362	199	3	𝑊𝑘(𝑠),𝛹(𝑠)〉𝑑𝑠	𝑊𝑘(𝑠),𝛹(𝑠)〉𝑑𝑠	PROPN
cana-5362	199	4	−	−	PROPN
cana-5362	199	5	𝑙𝑖𝑚	𝑙𝑖𝑚	NOUN
cana-5362	200	1	𝑖𝑛𝑓	𝑖𝑛𝑓	NOUN
cana-5362	200	2	𝑘→0	𝑘→0	NUM
cana-5362	201	1	1	1	NUM
cana-5362	201	2	2	2	NUM
cana-5362	201	3	|𝓋𝑘(𝑇)|	|𝓋𝑘(𝑇)|	NUM
cana-5362	201	4	2𝑇	2𝑇	NOUN
cana-5362	201	5	0	0	NUM
cana-5362	201	6	𝑇	𝑇	PROPN
cana-5362	201	7	0	0	NUM
cana-5362	201	8	,	,	PUNCT
cana-5362	201	9	(	(	PUNCT
cana-5362	201	10	56	56	NUM
cana-5362	201	11	)	)	PUNCT
cana-5362	201	12	but	but	CCONJ
cana-5362	201	13	,	,	PUNCT
cana-5362	201	14	1	1	NUM
cana-5362	201	15	2	2	NUM
cana-5362	201	16	|𝓋(𝑇)|2	|𝓋(𝑇)|2	NOUN
cana-5362	201	17	≤	≤	NUM
cana-5362	201	18	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
cana-5362	202	1	𝑖𝑛𝑓	𝑖𝑛𝑓	NOUN
cana-5362	202	2	𝑘→0	𝑘→0	NUM
cana-5362	203	1	1	1	NUM
cana-5362	203	2	2	2	NUM
cana-5362	203	3	|𝓋𝑘(𝑇)|	|𝓋𝑘(𝑇)|	NUM
cana-5362	203	4	2	2	NUM
cana-5362	203	5	.	.	PUNCT
cana-5362	203	6	(	(	PUNCT
cana-5362	203	7	57	57	NUM
cana-5362	203	8	)	)	PUNCT
cana-5362	203	9	and	and	CCONJ
cana-5362	203	10	considering	consider	VERB
cana-5362	203	11	(	(	PUNCT
cana-5362	203	12	54	54	NUM
cana-5362	203	13	)	)	PUNCT
cana-5362	203	14	,	,	PUNCT
cana-5362	203	15	we	we	PRON
cana-5362	203	16	then	then	ADV
cana-5362	203	17	deduce	deduce	VERB
cana-5362	203	18	from	from	ADP
cana-5362	203	19	(	(	PUNCT
cana-5362	203	20	56	56	NUM
cana-5362	203	21	)	)	PUNCT
cana-5362	203	22	𝑙𝑖𝑚	𝑙𝑖𝑚	NOUN
cana-5362	203	23	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-5362	203	24	k→0	k→0	X
cana-5362	203	25	∫	∫	PROPN
cana-5362	203	26	〈	〈	X
cana-5362	203	27	𝒜uk	𝒜uk	PROPN
cana-5362	203	28	,	,	PUNCT
cana-5362	203	29	uk(s)〉ds	uk(s)〉ds	ADJ
cana-5362	203	30	≤	≤	ADV
cana-5362	203	31	1	1	NUM
cana-5362	203	32	2	2	NUM
cana-5362	203	33	|𝓋(0)|2	|𝓋(0)|2	X
cana-5362	203	34	+	+	CCONJ
cana-5362	203	35	∫	∫	PROPN
cana-5362	203	36	〈	〈	ADJ
cana-5362	203	37	wk(s),ψ(𝑠)〉ds	wk(s),ψ(𝑠)〉ds	X
cana-5362	203	38	−	−	PROPN
cana-5362	203	39	t	t	NOUN
cana-5362	203	40	0	0	NUM
cana-5362	203	41	t	t	NOUN
cana-5362	203	42	0	0	NUM
cana-5362	203	43	1	1	NUM
cana-5362	203	44	2	2	NUM
cana-5362	203	45	|𝓋(0)|2	|𝓋(0)|2	SYM
cana-5362	203	46	−	−	PROPN
cana-5362	203	47	∫	∫	PROPN
cana-5362	203	48	〈	〈	NOUN
cana-5362	203	49	𝑢(𝑠	𝑢(𝑠	NOUN
cana-5362	203	50	)	)	PUNCT
cana-5362	203	51	−	−	PROPN
cana-5362	203	52	𝑊(𝑠),𝛹(𝑠)〉𝑑𝑠	𝑊(𝑠),𝛹(𝑠)〉𝑑𝑠	NOUN
cana-5362	203	53	𝑇	𝑇	PROPN
cana-5362	203	54	0	0	NUM
cana-5362	203	55	=	=	SYM
cana-5362	203	56	∫	∫	PROPN
cana-5362	203	57	〈	〈	PROPN
cana-5362	203	58	𝑢(𝑠),𝛹(𝑠)〉𝑑𝑠	𝑢(𝑠),𝛹(𝑠)〉𝑑𝑠	PROPN
cana-5362	203	59	𝑇	𝑇	PROPN
cana-5362	203	60	0	0	NUM
cana-5362	203	61	(	(	PUNCT
cana-5362	203	62	58	58	NUM
cana-5362	203	63	)	)	PUNCT
cana-5362	203	64	and	and	CCONJ
cana-5362	203	65	as	as	SCONJ
cana-5362	203	66	𝒜	𝒜	NOUN
cana-5362	203	67	is	be	AUX
cana-5362	203	68	monotone	monotone	ADJ
cana-5362	203	69	,	,	PUNCT
cana-5362	203	70	we	we	PRON
cana-5362	203	71	deduce	deduce	VERB
cana-5362	203	72	that	that	SCONJ
cana-5362	203	73	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
cana-5362	204	1	𝑖𝑛𝑓	𝑖𝑛𝑓	NOUN
cana-5362	204	2	𝑘→0	𝑘→0	X
cana-5362	205	1	∫	∫	PROPN
cana-5362	205	2	〈	〈	NOUN
cana-5362	205	3	𝒜𝑢𝑘(𝑠	𝒜𝑢𝑘(𝑠	PROPN
cana-5362	205	4	)	)	PUNCT
cana-5362	205	5	,	,	PUNCT
cana-5362	205	6	𝑢𝑘(𝑠)〉𝑑𝑠	𝑢𝑘(𝑠)〉𝑑𝑠	PROPN
cana-5362	205	7	≥	≥	NUM
cana-5362	205	8	∫	∫	PROPN
cana-5362	205	9	〈	〈	PROPN
cana-5362	205	10	𝛹(𝑠	𝛹(𝑠	NOUN
cana-5362	205	11	)	)	PUNCT
cana-5362	205	12	,	,	PUNCT
cana-5362	205	13	𝑢(𝑠)〉𝑑𝑠	𝑢(𝑠)〉𝑑𝑠	NOUN
cana-5362	205	14	𝑇	𝑇	PROPN
cana-5362	205	15	0	0	NUM
cana-5362	205	16	𝑇	𝑇	PROPN
cana-5362	205	17	0	0	NUM
cana-5362	205	18	,	,	PUNCT
cana-5362	205	19	(	(	PUNCT
cana-5362	205	20	59	59	NUM
cana-5362	205	21	)	)	PUNCT
cana-5362	205	22	which	which	PRON
cana-5362	205	23	,	,	PUNCT
cana-5362	205	24	compared	compare	VERB
cana-5362	205	25	with	with	ADP
cana-5362	205	26	(	(	PUNCT
cana-5362	205	27	58	58	NUM
cana-5362	205	28	)	)	PUNCT
cana-5362	205	29	,	,	PUNCT
cana-5362	205	30	clearly	clearly	ADV
cana-5362	205	31	shows	show	VERB
cana-5362	205	32	(	(	PUNCT
cana-5362	205	33	52	52	NUM
cana-5362	205	34	)	)	PUNCT
cana-5362	205	35	.	.	PUNCT
cana-5362	206	1	to	to	PART
cana-5362	206	2	prove	prove	VERB
cana-5362	206	3	lemma	lemma	PROPN
cana-5362	206	4	4	4	NUM
cana-5362	206	5	.	.	PUNCT
cana-5362	206	6	from	from	ADP
cana-5362	206	7	(	(	PUNCT
cana-5362	206	8	52	52	NUM
cana-5362	206	9	)	)	PUNCT
cana-5362	206	10	,	,	PUNCT
cana-5362	206	11	we	we	PRON
cana-5362	206	12	use	use	VERB
cana-5362	206	13	a	a	DET
cana-5362	206	14	classical	classical	ADJ
cana-5362	206	15	technique	technique	NOUN
cana-5362	206	16	based	base	VERB
cana-5362	206	17	on	on	ADP
cana-5362	206	18	the	the	DET
cana-5362	206	19	monotonicity	monotonicity	NOUN
cana-5362	206	20	and	and	CCONJ
cana-5362	206	21	hemi	hemi	NOUN
cana-5362	206	22	-	-	PUNCT
cana-5362	206	23	continuity	continuity	NOUN
cana-5362	206	24	of	of	ADP
cana-5362	206	25	𝒜	𝒜	NOUN
cana-5362	206	26	(	(	PUNCT
cana-5362	206	27	cf	cf	NOUN
cana-5362	206	28	.	.	PUNCT
cana-5362	207	1	lions	lion	NOUN
cana-5362	208	1	[	[	X
cana-5362	208	2	12	12	NUM
cana-5362	208	3	]	]	PUNCT
cana-5362	208	4	,	,	PUNCT
cana-5362	208	5	brézis	brézi	NOUN
cana-5362	208	6	[	[	X
cana-5362	208	7	7	7	NUM
cana-5362	208	8	]	]	PUNCT
cana-5362	208	9	,	,	PUNCT
cana-5362	208	10	da	da	X
cana-5362	208	11	.	.	PUNCT
cana-5362	208	12	prato[9	prato[9	PART
cana-5362	208	13	]	]	PUNCT
cana-5362	208	14	and	and	CCONJ
cana-5362	208	15	minty	minty	ADJ
cana-5362	208	16	[	[	X
cana-5362	208	17	14	14	NUM
cana-5362	208	18	]	]	PUNCT
cana-5362	208	19	)	)	PUNCT
cana-5362	208	20	.	.	PUNCT
cana-5362	209	1	according	accord	VERB
cana-5362	209	2	to	to	ADP
cana-5362	209	3	the	the	DET
cana-5362	209	4	monotonicity	monotonicity	NOUN
cana-5362	209	5	of	of	ADP
cana-5362	209	6	𝒜	𝒜	NOUN
cana-5362	209	7	,	,	PUNCT
cana-5362	209	8	we	we	PRON
cana-5362	209	9	have	have	AUX
cana-5362	209	10	,	,	PUNCT
cana-5362	209	11	for	for	SCONJ
cana-5362	209	12	all	all	DET
cana-5362	209	13	∅	∅	NOUN
cana-5362	209	14	∈	∈	PROPN
cana-5362	209	15	𝐿2(𝒱	𝐿2(𝒱	PROPN
cana-5362	209	16	)	)	PUNCT
cana-5362	209	17	communications	communication	NOUN
cana-5362	209	18	on	on	ADP
cana-5362	209	19	applied	apply	VERB
cana-5362	209	20	nonlinear	nonlinear	ADJ
cana-5362	209	21	analysis	analysis	NOUN
cana-5362	209	22	issn	issn	NOUN
cana-5362	209	23	:	:	PUNCT
cana-5362	209	24	1074	1074	NUM
cana-5362	209	25	-	-	PUNCT
cana-5362	209	26	133x	133x	NUM
cana-5362	209	27	vol	vol	VERB
cana-5362	209	28	32	32	NUM
cana-5362	209	29	no	no	NOUN
cana-5362	209	30	.	.	PUNCT
cana-5362	210	1	10s	10	NOUN
cana-5362	210	2	(	(	PUNCT
cana-5362	210	3	2025	2025	NUM
cana-5362	210	4	)	)	PUNCT
cana-5362	210	5	1995	1995	NUM
cana-5362	210	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5362	210	7	∫	∫	PROPN
cana-5362	211	1	〈	〈	PROPN
cana-5362	211	2	𝒜𝑢𝑘(𝑠	𝒜𝑢𝑘(𝑠	PROPN
cana-5362	211	3	)	)	PUNCT
cana-5362	211	4	−𝒜∅(𝑠	−𝒜∅(𝑠	PROPN
cana-5362	211	5	)	)	PUNCT
cana-5362	211	6	,	,	PUNCT
cana-5362	211	7	𝑢𝑘(𝑠	𝑢𝑘(𝑠	NUM
cana-5362	211	8	)	)	PUNCT
cana-5362	212	1	−	−	PROPN
cana-5362	212	2	∅(𝑠)〉𝑑𝑠	∅(𝑠)〉𝑑𝑠	NOUN
cana-5362	212	3	≥	≥	NOUN
cana-5362	212	4	0	0	NUM
cana-5362	212	5	𝑇	𝑇	PROPN
cana-5362	212	6	0	0	NUM
cana-5362	212	7	(	(	PUNCT
cana-5362	212	8	60	60	NUM
cana-5362	212	9	)	)	PUNCT
cana-5362	212	10	thus	thus	ADV
cana-5362	212	11	,	,	PUNCT
cana-5362	212	12	by	by	ADP
cana-5362	212	13	passing	pass	VERB
cana-5362	212	14	to	to	ADP
cana-5362	212	15	the	the	DET
cana-5362	212	16	limit	limit	NOUN
cana-5362	212	17	,	,	PUNCT
cana-5362	212	18	from	from	ADP
cana-5362	212	19	(	(	PUNCT
cana-5362	212	20	52	52	NUM
cana-5362	212	21	)	)	PUNCT
cana-5362	212	22	,	,	PUNCT
cana-5362	212	23	it	it	PRON
cana-5362	212	24	follows	follow	VERB
cana-5362	212	25	that	that	SCONJ
cana-5362	212	26	∫	∫	PROPN
cana-5362	212	27	〈	〈	PROPN
cana-5362	212	28	𝛹(𝑠	𝛹(𝑠	NOUN
cana-5362	212	29	)	)	PUNCT
cana-5362	212	30	−𝒜∅(𝑠	−𝒜∅(𝑠	PROPN
cana-5362	212	31	)	)	PUNCT
cana-5362	212	32	,	,	PUNCT
cana-5362	212	33	𝑢(𝑠	𝑢(𝑠	NOUN
cana-5362	212	34	)	)	PUNCT
cana-5362	213	1	−	−	PROPN
cana-5362	214	1	∅(𝑠)〉𝑑𝑠	∅(𝑠)〉𝑑𝑠	NOUN
cana-5362	214	2	≥	≥	NOUN
cana-5362	214	3	0	0	NUM
cana-5362	214	4	𝑇	𝑇	PROPN
cana-5362	214	5	0	0	NUM
cana-5362	214	6	(	(	PUNCT
cana-5362	214	7	61	61	NUM
cana-5362	214	8	)	)	PUNCT
cana-5362	214	9	we	we	PRON
cana-5362	214	10	then	then	ADV
cana-5362	214	11	choose	choose	VERB
cana-5362	214	12	,	,	PUNCT
cana-5362	214	13	∅(t	∅(t	PROPN
cana-5362	214	14	)	)	PUNCT
cana-5362	214	15	=	=	SYM
cana-5362	214	16	u(t	u(t	NOUN
cana-5362	214	17	)	)	PUNCT
cana-5362	214	18	−	−	NOUN
cana-5362	214	19	γφ(t	γφ(t	NUM
cana-5362	214	20	)	)	PUNCT
cana-5362	214	21	,	,	PUNCT
cana-5362	214	22	where	where	SCONJ
cana-5362	214	23	φ	φ	PROPN
cana-5362	214	24	∈	∈	PROPN
cana-5362	214	25	𝐿2(𝒱	𝐿2(𝒱	PROPN
cana-5362	214	26	)	)	PUNCT
cana-5362	214	27	and	and	CCONJ
cana-5362	214	28	λ	λ	X
cana-5362	214	29	>	>	X
cana-5362	214	30	0	0	NUM
cana-5362	214	31	are	be	AUX
cana-5362	214	32	arbitrary	arbitrary	ADJ
cana-5362	214	33	;	;	PUNCT
cana-5362	214	34	(	(	PUNCT
cana-5362	214	35	61	61	NUM
cana-5362	214	36	)	)	PUNCT
cana-5362	214	37	became	become	VERB
cana-5362	214	38	as	as	ADP
cana-5362	214	39	follow	follow	NOUN
cana-5362	214	40	∫	∫	PROPN
cana-5362	214	41	〈	〈	PROPN
cana-5362	214	42	𝛹(𝑠	𝛹(𝑠	NOUN
cana-5362	214	43	)	)	PUNCT
cana-5362	214	44	−𝒜(𝑢(𝑠	−𝒜(𝑢(𝑠	NOUN
cana-5362	214	45	)	)	PUNCT
cana-5362	214	46	−	−	NUM
cana-5362	214	47	𝛾𝜑(𝑠	𝛾𝜑(𝑠	NUM
cana-5362	214	48	)	)	PUNCT
cana-5362	214	49	)	)	PUNCT
cana-5362	214	50	,	,	PUNCT
cana-5362	214	51	𝛾𝜑(𝑠)〉𝑑𝑠	𝛾𝜑(𝑠)〉𝑑𝑠	NOUN
cana-5362	214	52	≥	≥	NOUN
cana-5362	214	53	0	0	NUM
cana-5362	214	54	𝑇	𝑇	PROPN
cana-5362	214	55	0	0	NUM
cana-5362	214	56	(	(	PUNCT
cana-5362	214	57	62	62	NUM
cana-5362	214	58	)	)	PUNCT
cana-5362	214	59	we	we	PRON
cana-5362	214	60	devise	devise	VERB
cana-5362	214	61	then	then	ADV
cana-5362	214	62	by	by	ADP
cana-5362	214	63	γ	γ	NOUN
cana-5362	214	64	,	,	PUNCT
cana-5362	214	65	we	we	PRON
cana-5362	214	66	obtain	obtain	VERB
cana-5362	214	67	∫	∫	PROPN
cana-5362	214	68	〈	〈	PROPN
cana-5362	214	69	𝛹(𝑠	𝛹(𝑠	NOUN
cana-5362	214	70	)	)	PUNCT
cana-5362	214	71	−𝒜(𝑢(𝑠	−𝒜(𝑢(𝑠	NOUN
cana-5362	214	72	)	)	PUNCT
cana-5362	214	73	−	−	NUM
cana-5362	214	74	𝛾𝜑(𝑠	𝛾𝜑(𝑠	NUM
cana-5362	214	75	)	)	PUNCT
cana-5362	214	76	)	)	PUNCT
cana-5362	214	77	,	,	PUNCT
cana-5362	214	78	𝜑(𝑠)〉𝑑𝑠	𝜑(𝑠)〉𝑑𝑠	NOUN
cana-5362	214	79	≥	≥	NOUN
cana-5362	214	80	0	0	NUM
cana-5362	214	81	𝑇	𝑇	PROPN
cana-5362	214	82	0	0	NUM
cana-5362	214	83	(	(	PUNCT
cana-5362	214	84	63	63	NUM
cana-5362	214	85	)	)	PUNCT
cana-5362	214	86	we	we	PRON
cana-5362	214	87	then	then	ADV
cana-5362	214	88	let	let	VERB
cana-5362	214	89	γ	γ	NOUN
cana-5362	214	90	tend	tend	VERB
cana-5362	214	91	to	to	PART
cana-5362	214	92	0	0	NUM
cana-5362	214	93	in	in	ADP
cana-5362	214	94	(	(	PUNCT
cana-5362	214	95	63	63	NUM
cana-5362	214	96	)	)	PUNCT
cana-5362	214	97	.	.	PUNCT
cana-5362	215	1	the	the	DET
cana-5362	215	2	property	property	NOUN
cana-5362	215	3	of	of	ADP
cana-5362	215	4	hemi	hemi	NOUN
cana-5362	215	5	-	-	PUNCT
cana-5362	215	6	continuity	continuity	NOUN
cana-5362	215	7	leads	lead	VERB
cana-5362	215	8	to	to	ADP
cana-5362	215	9	∫	∫	PROPN
cana-5362	215	10	〈	〈	PROPN
cana-5362	215	11	𝛹(𝑠	𝛹(𝑠	NOUN
cana-5362	215	12	)	)	PUNCT
cana-5362	215	13	−𝒜𝑢(𝑠	−𝒜𝑢(𝑠	NOUN
cana-5362	215	14	)	)	PUNCT
cana-5362	215	15	,	,	PUNCT
cana-5362	215	16	𝜑(𝑠)〉𝑑𝑠	𝜑(𝑠)〉𝑑𝑠	NOUN
cana-5362	215	17	≥	≥	NOUN
cana-5362	215	18	0	0	NUM
cana-5362	215	19	𝑇	𝑇	PROPN
cana-5362	215	20	0	0	NUM
cana-5362	215	21	.	.	PUNCT
cana-5362	216	1	(	(	PUNCT
cana-5362	216	2	64	64	NUM
cana-5362	216	3	)	)	PUNCT
cana-5362	217	1	so	so	ADV
cana-5362	217	2	,	,	PUNCT
cana-5362	217	3	the	the	DET
cana-5362	217	4	result	result	NOUN
cana-5362	217	5	holds	hold	VERB
cana-5362	217	6	.	.	PUNCT
cana-5362	217	7	3.2	3.2	NUM
cana-5362	217	8	prove	prove	NOUN
cana-5362	217	9	of	of	ADP
cana-5362	217	10	the	the	DET
cana-5362	217	11	theorem	theorem	NOUN
cana-5362	217	12	1	1	NUM
cana-5362	217	13	we	we	PRON
cana-5362	217	14	summarize	summarize	VERB
cana-5362	217	15	the	the	DET
cana-5362	217	16	results	result	NOUN
cana-5362	217	17	obtained	obtain	VERB
cana-5362	217	18	.	.	PUNCT
cana-5362	218	1	there	there	PRON
cana-5362	218	2	exists	exist	VERB
cana-5362	218	3	𝓋	𝓋	PROPN
cana-5362	218	4	∈	∈	PROPN
cana-5362	218	5	𝐿2(𝒱	𝐿2(𝒱	PROPN
cana-5362	218	6	)	)	PUNCT
cana-5362	218	7	,	,	PUNCT
cana-5362	218	8	d𝓋	d𝓋	VERB
cana-5362	218	9	dt	dt	X
cana-5362	218	10	∈	∈	PROPN
cana-5362	218	11	𝐿2(𝒱′	𝐿2(𝒱′	PROPN
cana-5362	218	12	)	)	PUNCT
cana-5362	218	13	and	and	CCONJ
cana-5362	218	14	u	u	PROPN
cana-5362	218	15	∈	∈	PROPN
cana-5362	218	16	𝐿2(𝒱	𝐿2(𝒱	PROPN
cana-5362	218	17	)	)	PUNCT
cana-5362	218	18	such	such	ADJ
cana-5362	218	19	that	that	DET
cana-5362	218	20	u	u	NOUN
cana-5362	218	21	=	=	PROPN
cana-5362	218	22	𝓋+w	𝓋+w	PROPN
cana-5362	218	23	.	.	PUNCT
cana-5362	219	1	moreover	moreover	ADV
cana-5362	219	2	(	(	PUNCT
cana-5362	219	3	50	50	NUM
cana-5362	219	4	)	)	PUNCT
cana-5362	219	5	and	and	CCONJ
cana-5362	219	6	the	the	DET
cana-5362	219	7	lemma	lemma	PROPN
cana-5362	219	8	4	4	NUM
cana-5362	219	9	give	give	VERB
cana-5362	219	10	𝑑𝓋	𝑑𝓋	ADP
cana-5362	219	11	𝑑𝑡	𝑑𝑡	ADP
cana-5362	219	12	+	+	NOUN
cana-5362	219	13	𝒜𝑢(𝑡	𝒜𝑢(𝑡	X
cana-5362	219	14	)	)	PUNCT
cana-5362	219	15	=	=	SYM
cana-5362	219	16	0	0	NUM
cana-5362	220	1	(	(	PUNCT
cana-5362	220	2	65	65	NUM
cana-5362	220	3	)	)	PUNCT
cana-5362	220	4	finally	finally	ADV
cana-5362	220	5	,	,	PUNCT
cana-5362	220	6	𝓋(0	𝓋(0	PROPN
cana-5362	220	7	)	)	PUNCT
cana-5362	220	8	=	=	SYM
cana-5362	220	9	u(0	u(0	PROPN
cana-5362	220	10	)	)	PUNCT
cana-5362	220	11	−	−	PROPN
cana-5362	220	12	w0	w0	PROPN
cana-5362	220	13	(	(	PUNCT
cana-5362	220	14	according	accord	VERB
cana-5362	220	15	to	to	ADP
cana-5362	220	16	(	(	PUNCT
cana-5362	220	17	49	49	NUM
cana-5362	220	18	)	)	PUNCT
cana-5362	220	19	)	)	PUNCT
cana-5362	220	20	.	.	PUNCT
cana-5362	221	1	then	then	ADV
cana-5362	221	2	𝓋	𝓋	X
cana-5362	221	3	is	be	AUX
cana-5362	221	4	a	a	DET
cana-5362	221	5	solution	solution	NOUN
cana-5362	221	6	of	of	ADP
cana-5362	221	7	(	(	PUNCT
cana-5362	221	8	23	23	NUM
cana-5362	221	9	)	)	PUNCT
cana-5362	221	10	,	,	PUNCT
cana-5362	221	11	and	and	CCONJ
cana-5362	221	12	consequently	consequently	ADV
cana-5362	221	13	u	u	NOUN
cana-5362	221	14	of	of	ADP
cana-5362	221	15	(	(	PUNCT
cana-5362	221	16	21	21	NUM
cana-5362	221	17	)	)	PUNCT
cana-5362	221	18	.	.	PUNCT
cana-5362	222	1	uniqueness	uniqueness	PROPN
cana-5362	222	2	follows	follow	VERB
cana-5362	222	3	from	from	ADP
cana-5362	222	4	the	the	DET
cana-5362	222	5	monotonicity	monotonicity	NOUN
cana-5362	222	6	property	property	NOUN
cana-5362	222	7	of	of	ADP
cana-5362	222	8	the	the	DET
cana-5362	222	9	operator	operator	NOUN
cana-5362	222	10	𝒜wt	𝒜wt	PROPN
cana-5362	222	11	.	.	PUNCT
cana-5362	223	1	indeed	indeed	ADV
cana-5362	223	2	,	,	PUNCT
cana-5362	223	3	suppose	suppose	VERB
cana-5362	223	4	that	that	SCONJ
cana-5362	223	5	(	(	PUNCT
cana-5362	223	6	23	23	NUM
cana-5362	223	7	)	)	PUNCT
cana-5362	223	8	has	have	VERB
cana-5362	223	9	two	two	NUM
cana-5362	223	10	solutions	solution	NOUN
cana-5362	223	11	𝓋1	𝓋1	NOUN
cana-5362	223	12	,	,	PUNCT
cana-5362	223	13	𝓋2	𝓋2	PROPN
cana-5362	223	14	,	,	PUNCT
cana-5362	223	15	and	and	CCONJ
cana-5362	223	16	let	let	VERB
cana-5362	223	17	χ	χ	X
cana-5362	223	18	=	=	PUNCT
cana-5362	223	19	𝓋1	𝓋1	PROPN
cana-5362	223	20	−	−	PROPN
cana-5362	223	21	𝓋2	𝓋2	PROPN
cana-5362	223	22	.	.	PUNCT
cana-5362	224	1	we	we	PRON
cana-5362	224	2	then	then	ADV
cana-5362	224	3	have	have	VERB
cana-5362	224	4	by	by	ADP
cana-5362	224	5	subtraction	subtraction	NOUN
cana-5362	224	6	,	,	PUNCT
cana-5362	224	7	{	{	PUNCT
cana-5362	224	8	𝑑𝜒	𝑑𝜒	X
cana-5362	224	9	𝑑𝑡	𝑑𝑡	ADP
cana-5362	224	10	+	+	PROPN
cana-5362	224	11	𝒜𝑊𝓋1	𝒜𝑊𝓋1	X
cana-5362	224	12	−𝒜𝑊𝓋	−𝒜𝑊𝓋	X
cana-5362	224	13	=	=	PUNCT
cana-5362	224	14	0	0	NUM
cana-5362	224	15	𝑎.	𝑎.	PROPN
cana-5362	224	16	𝑒.	𝑒.	PROPN
cana-5362	225	1	𝑡	𝑡	ADP
cana-5362	225	2	𝜒	𝜒	X
cana-5362	225	3	(	(	PUNCT
cana-5362	225	4	0	0	NUM
cana-5362	225	5	)	)	PUNCT
cana-5362	225	6	=	=	SYM
cana-5362	225	7	0	0	PUNCT
cana-5362	225	8	(	(	PUNCT
cana-5362	225	9	66	66	NUM
cana-5362	225	10	)	)	PUNCT
cana-5362	225	11	and	and	CCONJ
cana-5362	225	12	thus	thus	ADV
cana-5362	225	13	,	,	PUNCT
cana-5362	225	14	by	by	ADP
cana-5362	225	15	multiplying	multiply	VERB
cana-5362	225	16	by	by	ADP
cana-5362	225	17	χ	χ	X
cana-5362	225	18	and	and	CCONJ
cana-5362	225	19	integrating	integrating	NOUN
cana-5362	225	20	,	,	PUNCT
cana-5362	225	21	we	we	PRON
cana-5362	225	22	get	get	VERB
cana-5362	225	23	∀t	∀t	PROPN
cana-5362	225	24	∈	∈	NOUN
cana-5362	226	1	[	[	X
cana-5362	226	2	0	0	NUM
cana-5362	226	3	,	,	PUNCT
cana-5362	226	4	t	t	PROPN
cana-5362	226	5	]	]	X
cana-5362	226	6	1	1	NUM
cana-5362	226	7	2	2	NUM
cana-5362	226	8	|𝜒	|𝜒	X
cana-5362	226	9	(	(	PUNCT
cana-5362	226	10	𝑡)|2	𝑡)|2	NOUN
cana-5362	226	11	+	+	CCONJ
cana-5362	226	12	∫	∫	PROPN
cana-5362	226	13	〈	〈	PROPN
cana-5362	226	14	𝒜𝑊𝑠𝓋1(𝑠	𝒜𝑊𝑠𝓋1(𝑠	PROPN
cana-5362	226	15	)	)	PUNCT
cana-5362	226	16	−	−	PROPN
cana-5362	227	1	𝓋(𝑠	𝓋(𝑠	PROPN
cana-5362	227	2	)	)	PUNCT
cana-5362	227	3	,	,	PUNCT
cana-5362	227	4	𝓋1(𝑠	𝓋1(𝑠	X
cana-5362	227	5	)	)	PUNCT
cana-5362	227	6	−	−	PROPN
cana-5362	227	7	𝓋2(𝑠	𝓋2(𝑠	NUM
cana-5362	227	8	)	)	PUNCT
cana-5362	227	9	〉	〉	NOUN
cana-5362	227	10	𝑡	𝑡	X
cana-5362	227	11	0	0	NUM
cana-5362	227	12	𝑑𝑠	𝑑𝑠	NOUN
cana-5362	227	13	=	=	SYM
cana-5362	227	14	0	0	PROPN
cana-5362	227	15	,	,	PUNCT
cana-5362	227	16	(	(	PUNCT
cana-5362	227	17	67	67	NUM
cana-5362	227	18	)	)	PUNCT
cana-5362	227	19	and	and	CCONJ
cana-5362	227	20	since	since	SCONJ
cana-5362	227	21	𝒜𝑊𝑡	𝒜𝑊𝑡	NOUN
cana-5362	227	22	is	be	AUX
cana-5362	227	23	monotone	monotone	ADJ
cana-5362	227	24	,	,	PUNCT
cana-5362	227	25	we	we	PRON
cana-5362	227	26	see	see	VERB
cana-5362	227	27	that	that	SCONJ
cana-5362	227	28	χ(t	χ(t	NOUN
cana-5362	227	29	)	)	PUNCT
cana-5362	227	30	=	=	SYM
cana-5362	227	31	0	0	NUM
cana-5362	227	32	,	,	PUNCT
cana-5362	227	33	∀t	∀t	PROPN
cana-5362	227	34	.	.	PROPN
cana-5362	227	35	3.3	3.3	NUM
cana-5362	227	36	energy	energy	NOUN
cana-5362	227	37	estimates	estimate	NOUN
cana-5362	227	38	and	and	CCONJ
cana-5362	227	39	stability	stability	NOUN
cana-5362	227	40	analysis	analysis	NOUN
cana-5362	227	41	this	this	DET
cana-5362	227	42	section	section	NOUN
cana-5362	227	43	establishes	establish	VERB
cana-5362	227	44	energy	energy	NOUN
cana-5362	227	45	estimates	estimate	NOUN
cana-5362	227	46	to	to	PART
cana-5362	227	47	verify	verify	VERB
cana-5362	227	48	the	the	DET
cana-5362	227	49	stability	stability	NOUN
cana-5362	227	50	of	of	ADP
cana-5362	227	51	solutions	solution	NOUN
cana-5362	227	52	.	.	PUNCT
cana-5362	228	1	we	we	PRON
cana-5362	228	2	will	will	AUX
cana-5362	228	3	need	need	VERB
cana-5362	228	4	to	to	PART
cana-5362	228	5	consider	consider	VERB
cana-5362	228	6	process	process	NOUN
cana-5362	228	7	of	of	ADP
cana-5362	228	8	the	the	DET
cana-5362	228	9	form	form	NOUN
cana-5362	228	10	𝑊	𝑊	PROPN
cana-5362	228	11	,	,	PUNCT
cana-5362	228	12	which	which	PRON
cana-5362	228	13	satisfy	satisfy	VERB
cana-5362	228	14	condition	condition	NOUN
cana-5362	228	15	(	(	PUNCT
cana-5362	228	16	3	3	NUM
cana-5362	228	17	)	)	PUNCT
cana-5362	228	18	.	.	PUNCT
cana-5362	229	1	as	as	ADP
cana-5362	229	2	for	for	ADP
cana-5362	229	3	the	the	DET
cana-5362	229	4	correlations	correlation	NOUN
cana-5362	229	5	between	between	ADP
cana-5362	229	6	𝑊,𝑢	𝑊,𝑢	NOUN
cana-5362	229	7	,	,	PUNCT
cana-5362	229	8	we	we	PRON
cana-5362	229	9	will	will	AUX
cana-5362	229	10	assume	assume	VERB
cana-5362	229	11	the	the	DET
cana-5362	229	12	following	follow	VERB
cana-5362	229	13	hypothesis	hypothesis	NOUN
cana-5362	229	14	.	.	PUNCT
cana-5362	230	1	communications	communication	NOUN
cana-5362	230	2	on	on	ADP
cana-5362	230	3	applied	apply	VERB
cana-5362	230	4	nonlinear	nonlinear	ADJ
cana-5362	230	5	analysis	analysis	NOUN
cana-5362	230	6	issn	issn	NOUN
cana-5362	230	7	:	:	PUNCT
cana-5362	230	8	1074	1074	NUM
cana-5362	230	9	-	-	PUNCT
cana-5362	230	10	133x	133x	NUM
cana-5362	230	11	vol	vol	VERB
cana-5362	230	12	32	32	NUM
cana-5362	230	13	no	no	NOUN
cana-5362	230	14	.	.	PUNCT
cana-5362	231	1	10s	10	NOUN
cana-5362	231	2	(	(	PUNCT
cana-5362	231	3	2025	2025	NUM
cana-5362	231	4	)	)	PUNCT
cana-5362	231	5	1996	1996	NUM
cana-5362	231	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5362	231	7	hypothesis	hypothesis	NOUN
cana-5362	231	8	:	:	PUNCT
cana-5362	231	9	∀𝑡1	∀𝑡1	ADJ
cana-5362	231	10	,	,	PUNCT
cana-5362	231	11	𝑡2	𝑡2	ADJ
cana-5362	231	12	𝑤𝑖𝑡ℎ	𝑤𝑖𝑡ℎ	NOUN
cana-5362	231	13	0	0	NUM
cana-5362	231	14	≤	≤	NUM
cana-5362	231	15	𝑡1	𝑡1	NOUN
cana-5362	231	16	≤	≤	NOUN
cana-5362	231	17	𝑡2	𝑡2	ADJ
cana-5362	231	18	≤	≤	NOUN
cana-5362	231	19	𝑇,𝑊𝑡1	𝑇,𝑊𝑡1	PUNCT
cana-5362	231	20	−	−	PROPN
cana-5362	231	21	𝑊𝑡2	𝑊𝑡2	NOUN
cana-5362	231	22	is	be	AUX
cana-5362	231	23	a	a	DET
cana-5362	231	24	random	random	ADJ
cana-5362	231	25	variable	variable	NOUN
cana-5362	231	26	taking	take	VERB
cana-5362	231	27	values	value	NOUN
cana-5362	231	28	in	in	ADP
cana-5362	231	29	𝐻	𝐻	PROPN
cana-5362	231	30	independent	independent	ADJ
cana-5362	231	31	of	of	ADP
cana-5362	231	32	the	the	DET
cana-5362	231	33	random	random	ADJ
cana-5362	231	34	variable	variable	NOUN
cana-5362	231	35	{	{	PUNCT
cana-5362	231	36	𝑢0	𝑢0	PROPN
cana-5362	231	37	,	,	PUNCT
cana-5362	231	38	𝑊(𝑡𝑗1	𝑊(𝑡𝑗1	NOUN
cana-5362	231	39	)	)	PUNCT
cana-5362	231	40	,	,	PUNCT
cana-5362	231	41	…	…	PUNCT
cana-5362	231	42	,	,	PUNCT
cana-5362	231	43	𝑊(𝑡𝑗𝑞	𝑊(𝑡𝑗𝑞	NOUN
cana-5362	231	44	)	)	PUNCT
cana-5362	231	45	}	}	PUNCT
cana-5362	231	46	taking	take	VERB
cana-5362	231	47	values	value	NOUN
cana-5362	231	48	in	in	ADP
cana-5362	231	49	𝐻	𝐻	PROPN
cana-5362	232	1	×	×	NOUN
cana-5362	233	1	𝐻𝑞	𝐻𝑞	PROPN
cana-5362	233	2	∀𝑞	∀𝑞	NOUN
cana-5362	233	3	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-5362	233	4	𝑡𝑗1	𝑡𝑗1	NOUN
cana-5362	233	5	,	,	PUNCT
cana-5362	233	6	.	.	PUNCT
cana-5362	233	7	.	.	PUNCT
cana-5362	233	8	.	.	PUNCT
cana-5362	234	1	,	,	PUNCT
cana-5362	234	2	𝑡𝑗𝑞	𝑡𝑗𝑞	ADJ
cana-5362	234	3	≤	≤	NUM
cana-5362	234	4	𝑡1	𝑡1	NOUN
cana-5362	234	5	.	.	PUNCT
cana-5362	235	1	based	base	VERB
cana-5362	235	2	on	on	ADP
cana-5362	235	3	the	the	DET
cana-5362	235	4	previously	previously	ADV
cana-5362	235	5	stated	state	VERB
cana-5362	235	6	hypotheses	hypothesis	NOUN
cana-5362	235	7	,	,	PUNCT
cana-5362	235	8	we	we	PRON
cana-5362	235	9	derive	derive	VERB
cana-5362	235	10	the	the	DET
cana-5362	235	11	following	follow	VERB
cana-5362	235	12	theorem	theorem	NOUN
cana-5362	235	13	.	.	PUNCT
cana-5362	235	14	theorem	theorem	NOUN
cana-5362	235	15	2	2	NUM
cana-5362	235	16	under	under	ADP
cana-5362	235	17	the	the	DET
cana-5362	235	18	assumptions	assumption	NOUN
cana-5362	235	19	stated	state	VERB
cana-5362	235	20	in	in	ADP
cana-5362	235	21	lemma	lemma	PROPN
cana-5362	235	22	2	2	NUM
cana-5362	235	23	and	and	CCONJ
cana-5362	235	24	hypothesis	hypothesis	NOUN
cana-5362	235	25	above	above	ADV
cana-5362	235	26	,	,	PUNCT
cana-5362	235	27	the	the	DET
cana-5362	235	28	following	follow	VERB
cana-5362	235	29	energy	energy	NOUN
cana-5362	235	30	equality	equality	NOUN
cana-5362	235	31	hold	hold	VERB
cana-5362	235	32	𝐸‖𝑢(𝑡)‖𝐻	𝐸‖𝑢(𝑡)‖𝐻	NOUN
cana-5362	235	33	2	2	NUM
cana-5362	235	34	+	+	CCONJ
cana-5362	235	35	2𝐸	2𝐸	ADJ
cana-5362	235	36	∫	∫	PROPN
cana-5362	235	37	〈	〈	NOUN
cana-5362	235	38	𝒜(𝑠)𝑢(𝑠	𝒜(𝑠)𝑢(𝑠	NOUN
cana-5362	235	39	)	)	PUNCT
cana-5362	235	40	,	,	PUNCT
cana-5362	235	41	𝑢(𝑠	𝑢(𝑠	NOUN
cana-5362	235	42	)	)	PUNCT
cana-5362	235	43	〉	〉	NOUN
cana-5362	235	44	𝑡	𝑡	X
cana-5362	235	45	0	0	NUM
cana-5362	235	46	𝑑𝑠	𝑑𝑠	NOUN
cana-5362	235	47	=	=	NOUN
cana-5362	235	48	𝐸‖𝑢(0)‖𝐻	𝐸‖𝑢(0)‖𝐻	PROPN
cana-5362	235	49	2	2	NUM
cana-5362	235	50	+	+	NUM
cana-5362	235	51	𝐸‖𝑊(𝑡)‖𝐻	𝐸‖𝑊(𝑡)‖𝐻	X
cana-5362	235	52	2	2	NUM
cana-5362	235	53	∀𝑡	∀𝑡	NOUN
cana-5362	235	54	∈	∈	PROPN
cana-5362	236	1	[	[	X
cana-5362	236	2	0	0	NUM
cana-5362	236	3	,	,	PUNCT
cana-5362	236	4	𝑇	𝑇	PROPN
cana-5362	236	5	]	]	PUNCT
cana-5362	236	6	.	.	PUNCT
cana-5362	237	1	(	(	PUNCT
cana-5362	237	2	68	68	NUM
cana-5362	237	3	)	)	PUNCT
cana-5362	237	4	proof	proof	NOUN
cana-5362	237	5	.	.	PUNCT
cana-5362	238	1	see	see	VERB
cana-5362	238	2	[	[	X
cana-5362	238	3	4	4	X
cana-5362	238	4	]	]	PUNCT
cana-5362	238	5	for	for	ADP
cana-5362	238	6	details	detail	NOUN
cana-5362	238	7	.	.	PUNCT
cana-5362	239	1	now	now	ADV
cana-5362	239	2	,	,	PUNCT
cana-5362	239	3	we	we	PRON
cana-5362	239	4	prove	prove	VERB
cana-5362	239	5	the	the	DET
cana-5362	239	6	following	follow	VERB
cana-5362	239	7	stability	stability	NOUN
cana-5362	239	8	result	result	NOUN
cana-5362	239	9	.	.	PUNCT
cana-5362	240	1	proposition	proposition	NOUN
cana-5362	240	2	4	4	NUM
cana-5362	240	3	consider	consider	VERB
cana-5362	240	4	𝑊	𝑊	PROPN
cana-5362	240	5	∈	∈	PROPN
cana-5362	240	6	𝐶([0	𝐶([0	PROPN
cana-5362	240	7	,	,	PUNCT
cana-5362	240	8	𝑇	𝑇	PROPN
cana-5362	240	9	]	]	PUNCT
cana-5362	240	10	;	;	PUNCT
cana-5362	240	11	𝐿2(𝐷,𝒫,𝐻	𝐿2(𝐷,𝒫,𝐻	NUM
cana-5362	240	12	)	)	PUNCT
cana-5362	240	13	)	)	PUNCT
cana-5362	240	14	,	,	PUNCT
cana-5362	240	15	𝑢0	𝑢0	PROPN
cana-5362	240	16	∈	∈	PROPN
cana-5362	240	17	𝐻0	𝐻0	PROPN
cana-5362	240	18	1(𝐷	1(𝐷	PROPN
cana-5362	240	19	)	)	PUNCT
cana-5362	240	20	and	and	CCONJ
cana-5362	240	21	𝑢	𝑢	PROPN
cana-5362	240	22	associated	associated	ADJ
cana-5362	240	23	solution	solution	NOUN
cana-5362	240	24	,	,	PUNCT
cana-5362	240	25	then	then	ADV
cana-5362	240	26	for	for	ADP
cana-5362	240	27	any	any	DET
cana-5362	240	28	𝑡	𝑡	NOUN
cana-5362	240	29	,	,	PUNCT
cana-5362	240	30	𝐸‖𝑢(𝑡)‖𝐻	𝐸‖𝑢(𝑡)‖𝐻	NOUN
cana-5362	240	31	2	2	NUM
cana-5362	240	32	≤	≤	NOUN
cana-5362	240	33	𝐸‖𝑢(0)‖𝐻	𝐸‖𝑢(0)‖𝐻	NOUN
cana-5362	240	34	2	2	NUM
cana-5362	240	35	+	+	NUM
cana-5362	240	36	𝐸‖𝑊(𝑡)‖𝐻	𝐸‖𝑊(𝑡)‖𝐻	X
cana-5362	240	37	2	2	NUM
cana-5362	240	38	∀𝑡	∀𝑡	NOUN
cana-5362	240	39	∈	∈	PROPN
cana-5362	241	1	[	[	X
cana-5362	241	2	0	0	NUM
cana-5362	241	3	,	,	PUNCT
cana-5362	241	4	𝑇	𝑇	PROPN
cana-5362	241	5	]	]	PUNCT
cana-5362	241	6	.	.	PUNCT
cana-5362	242	1	(	(	PUNCT
cana-5362	242	2	69	69	NUM
cana-5362	242	3	)	)	PUNCT
cana-5362	242	4	proof	proof	NOUN
cana-5362	242	5	.	.	PUNCT
cana-5362	243	1	since	since	SCONJ
cana-5362	243	2	the	the	DET
cana-5362	243	3	operator	operator	NOUN
cana-5362	243	4	𝒜(𝑡	𝒜(𝑡	CCONJ
cana-5362	243	5	)	)	PUNCT
cana-5362	243	6	verify	verify	VERB
cana-5362	243	7	the	the	DET
cana-5362	243	8	coersivity	coersivity	NOUN
cana-5362	243	9	condition	condition	NOUN
cana-5362	243	10	,	,	PUNCT
cana-5362	243	11	i.e.	i.e.	X
cana-5362	243	12	⟨𝒜(𝑡)𝑢(𝑡	⟨𝒜(𝑡)𝑢(𝑡	NOUN
cana-5362	243	13	)	)	PUNCT
cana-5362	243	14	,	,	PUNCT
cana-5362	243	15	𝑢(𝑡)⟩	𝑢(𝑡)⟩	PROPN
cana-5362	243	16	≥	≥	X
cana-5362	243	17	𝜌‖𝑢(𝑡)‖2	𝜌‖𝑢(𝑡)‖2	ADJ
cana-5362	243	18	,	,	PUNCT
cana-5362	243	19	the	the	DET
cana-5362	243	20	energy	energy	NOUN
cana-5362	243	21	equality	equality	NOUN
cana-5362	243	22	(	(	PUNCT
cana-5362	243	23	68	68	NUM
cana-5362	243	24	)	)	PUNCT
cana-5362	243	25	became	become	VERB
cana-5362	243	26	as	as	ADP
cana-5362	243	27	follow	follow	VERB
cana-5362	243	28	𝐸‖𝑢(𝑡)‖𝐻	𝐸‖𝑢(𝑡)‖𝐻	NOUN
cana-5362	243	29	2	2	NUM
cana-5362	243	30	+	+	CCONJ
cana-5362	243	31	2𝜌𝐸	2𝜌𝐸	NUM
cana-5362	243	32	∫	∫	NUM
cana-5362	243	33	‖𝑢(𝑠)‖2	‖𝑢(𝑠)‖2	NOUN
cana-5362	244	1	𝑡	𝑡	PROPN
cana-5362	244	2	0	0	NUM
cana-5362	244	3	𝑑𝑠	𝑑𝑠	ADP
cana-5362	244	4	≤	≤	ADJ
cana-5362	244	5	𝐸‖𝑢(0)‖𝐻	𝐸‖𝑢(0)‖𝐻	NOUN
cana-5362	244	6	2	2	NUM
cana-5362	244	7	+	+	NUM
cana-5362	244	8	𝐸‖𝑊(𝑡)‖𝐻	𝐸‖𝑊(𝑡)‖𝐻	X
cana-5362	244	9	2	2	NUM
cana-5362	244	10	(	(	PUNCT
cana-5362	244	11	70	70	NUM
cana-5362	244	12	)	)	PUNCT
cana-5362	244	13	we	we	PRON
cana-5362	244	14	know	know	VERB
cana-5362	244	15	2𝜌𝐸	2𝜌𝐸	NUM
cana-5362	244	16	∫	∫	NOUN
cana-5362	244	17	‖𝑢(𝑠)‖2	‖𝑢(𝑠)‖2	NOUN
cana-5362	245	1	𝑡	𝑡	PROPN
cana-5362	245	2	0	0	PUNCT
cana-5362	245	3	𝑑𝑠	𝑑𝑠	NOUN
cana-5362	245	4	≥	≥	NOUN
cana-5362	245	5	0	0	NUM
cana-5362	245	6	,	,	PUNCT
cana-5362	245	7	we	we	PRON
cana-5362	245	8	substitute	substitute	VERB
cana-5362	245	9	that	that	PRON
cana-5362	245	10	in	in	ADP
cana-5362	245	11	(	(	PUNCT
cana-5362	245	12	65	65	NUM
cana-5362	245	13	)	)	PUNCT
cana-5362	245	14	,	,	PUNCT
cana-5362	245	15	we	we	PRON
cana-5362	245	16	obtain	obtain	VERB
cana-5362	245	17	(	(	PUNCT
cana-5362	245	18	69	69	NUM
cana-5362	245	19	)	)	PUNCT
cana-5362	245	20	.	.	PUNCT
cana-5362	246	1	hence	hence	ADV
cana-5362	246	2	,	,	PUNCT
cana-5362	246	3	the	the	DET
cana-5362	246	4	proposition	proposition	NOUN
cana-5362	246	5	4	4	NUM
cana-5362	246	6	hold	hold	NOUN
cana-5362	246	7	.	.	PUNCT
cana-5362	247	1	4	4	X
cana-5362	247	2	.	.	X
cana-5362	247	3	numerical	numerical	PROPN
cana-5362	247	4	scheme	scheme	PROPN
cana-5362	247	5	the	the	DET
cana-5362	247	6	spde	spde	ADV
cana-5362	247	7	-	-	PUNCT
cana-5362	247	8	based	base	VERB
cana-5362	247	9	denoising	denoising	NOUN
cana-5362	247	10	scheme	scheme	NOUN
cana-5362	247	11	is	be	AUX
cana-5362	247	12	approximated	approximate	VERB
cana-5362	247	13	by	by	ADP
cana-5362	247	14	applying	apply	VERB
cana-5362	247	15	a	a	DET
cana-5362	247	16	finite	finite	ADJ
cana-5362	247	17	-	-	PUNCT
cana-5362	247	18	difference	difference	NOUN
cana-5362	247	19	based	base	VERB
cana-5362	247	20	method	method	NOUN
cana-5362	247	21	.	.	PUNCT
cana-5362	248	1	thus	thus	ADV
cana-5362	248	2	,	,	PUNCT
cana-5362	248	3	we	we	PRON
cana-5362	248	4	put	put	VERB
cana-5362	248	5	a	a	DET
cana-5362	248	6	space	space	NOUN
cana-5362	248	7	grid	grid	NOUN
cana-5362	248	8	size	size	NOUN
cana-5362	248	9	of	of	ADP
cana-5362	248	10	∆𝑥	∆𝑥	PROPN
cana-5362	248	11	=	=	SYM
cana-5362	248	12	∆𝑦	∆𝑦	PROPN
cana-5362	248	13	=	=	SYM
cana-5362	248	14	1	1	NUM
cana-5362	248	15	and	and	CCONJ
cana-5362	248	16	a	a	DET
cana-5362	248	17	time	time	NOUN
cana-5362	248	18	step	step	NOUN
cana-5362	248	19	∆𝑡	∆𝑡	PROPN
cana-5362	249	1	=	=	SYM
cana-5362	249	2	𝑇	𝑇	PROPN
cana-5362	249	3	𝑁	𝑁	PROPN
cana-5362	249	4	,	,	PUNCT
cana-5362	249	5	where	where	SCONJ
cana-5362	249	6	𝑇	𝑇	PROPN
cana-5362	249	7	and	and	CCONJ
cana-5362	249	8	𝑁	𝑁	PROPN
cana-5362	249	9	are	be	AUX
cana-5362	249	10	the	the	DET
cana-5362	249	11	final	final	ADJ
cana-5362	249	12	time	time	NOUN
cana-5362	249	13	and	and	CCONJ
cana-5362	249	14	the	the	DET
cana-5362	249	15	number	number	NOUN
cana-5362	249	16	of	of	ADP
cana-5362	249	17	iterations	iteration	NOUN
cana-5362	249	18	respectively	respectively	ADV
cana-5362	249	19	.	.	PUNCT
cana-5362	250	1	𝑢𝑖,𝑗	𝑢𝑖,𝑗	AUX
cana-5362	250	2	𝑛+1	𝑛+1	PROPN
cana-5362	250	3	=	=	PUNCT
cana-5362	250	4	𝑢𝑖,𝑗	𝑢𝑖,𝑗	X
cana-5362	250	5	𝑛	𝑛	NOUN
cana-5362	250	6	+	+	PUNCT
cana-5362	250	7	∆𝑡	∆𝑡	PROPN
cana-5362	251	1	+	+	CCONJ
cana-5362	251	2	∆𝑡	∆𝑡	PROPN
cana-5362	251	3	4	4	NUM
cana-5362	252	1	[	[	X
cana-5362	252	2	(	(	PUNCT
cana-5362	252	3	𝑔𝑖+1,𝑗	𝑔𝑖+1,𝑗	NOUN
cana-5362	252	4	𝑛	𝑛	PROPN
cana-5362	252	5	−	−	PROPN
cana-5362	252	6	𝑔𝑖−1,𝑗	𝑔𝑖−1,𝑗	ADJ
cana-5362	252	7	𝑛	𝑛	PROPN
cana-5362	252	8	)	)	PUNCT
cana-5362	252	9	(	(	PUNCT
cana-5362	252	10	𝑢𝑖+1,𝑗	𝑢𝑖+1,𝑗	NOUN
cana-5362	252	11	𝑛	𝑛	PRON
cana-5362	252	12	−	−	PROPN
cana-5362	252	13	𝑢𝑖−1,𝑗	𝑢𝑖−1,𝑗	NOUN
cana-5362	252	14	𝑛	𝑛	PROPN
cana-5362	252	15	)	)	PUNCT
cana-5362	252	16	]	]	PUNCT
cana-5362	253	1	+	+	CCONJ
cana-5362	253	2	∆𝑡	∆𝑡	NOUN
cana-5362	253	3	4	4	NUM
cana-5362	254	1	[	[	X
cana-5362	254	2	(	(	PUNCT
cana-5362	254	3	𝑔𝑖,𝑗+1	𝑔𝑖,𝑗+1	X
cana-5362	254	4	𝑛	𝑛	DET
cana-5362	254	5	−	−	PROPN
cana-5362	254	6	𝑔𝑖,𝑗−1	𝑔𝑖,𝑗−1	NOUN
cana-5362	254	7	𝑛	𝑛	PROPN
cana-5362	254	8	)	)	PUNCT
cana-5362	254	9	(	(	PUNCT
cana-5362	254	10	𝑢𝑖,𝑗+1	𝑢𝑖,𝑗+1	X
cana-5362	254	11	𝑛	𝑛	DET
cana-5362	254	12	−	−	PROPN
cana-5362	254	13	𝑢𝑖,𝑗−1	𝑢𝑖,𝑗−1	PROPN
cana-5362	254	14	𝑛	𝑛	PRON
cana-5362	254	15	)	)	PUNCT
cana-5362	254	16	+	+	CCONJ
cana-5362	254	17	(	(	PUNCT
cana-5362	254	18	𝑊𝑖,𝑗	𝑊𝑖,𝑗	PROPN
cana-5362	254	19	𝑛+1	𝑛+1	PROPN
cana-5362	254	20	−𝑊𝑖,𝑗	−𝑊𝑖,𝑗	PROPN
cana-5362	254	21	𝑛	𝑛	PRON
cana-5362	254	22	)	)	PUNCT
cana-5362	254	23	]	]	PUNCT
cana-5362	254	24	(	(	PUNCT
cana-5362	254	25	71	71	NUM
cana-5362	254	26	)	)	PUNCT
cana-5362	254	27	iterative	iterative	NOUN
cana-5362	254	28	algorithm	algorithm	NOUN
cana-5362	254	29	given	give	VERB
cana-5362	254	30	by	by	ADP
cana-5362	254	31	(	(	PUNCT
cana-5362	254	32	71	71	NUM
cana-5362	254	33	)	)	PUNCT
cana-5362	254	34	,	,	PUNCT
cana-5362	254	35	it	it	PRON
cana-5362	254	36	begins	begin	VERB
cana-5362	254	37	by	by	ADP
cana-5362	254	38	inputting	inputte	VERB
cana-5362	254	39	the	the	DET
cana-5362	254	40	initial	initial	ADJ
cana-5362	254	41	conditions	condition	NOUN
cana-5362	254	42	which	which	PRON
cana-5362	254	43	is	be	AUX
cana-5362	254	44	the	the	DET
cana-5362	254	45	noisy	noisy	ADJ
cana-5362	254	46	image	image	NOUN
cana-5362	254	47	𝑢0	𝑢0	PROPN
cana-5362	254	48	.	.	PUNCT
cana-5362	255	1	then	then	ADV
cana-5362	255	2	,	,	PUNCT
cana-5362	255	3	we	we	PRON
cana-5362	255	4	define	define	VERB
cana-5362	255	5	the	the	DET
cana-5362	255	6	continuous	continuous	ADJ
cana-5362	255	7	wiener	wiener	NOUN
cana-5362	255	8	process	process	NOUN
cana-5362	255	9	𝑊𝑡	𝑊𝑡	PROPN
cana-5362	255	10	,	,	PUNCT
cana-5362	255	11	𝑎.	𝑎.	PROPN
cana-5362	255	12	𝑒.	𝑒.	PROPN
cana-5362	255	13	𝑡	𝑡	PROPN
cana-5362	255	14	and	and	CCONJ
cana-5362	255	15	the	the	DET
cana-5362	255	16	function	function	NOUN
cana-5362	255	17	g.	g.	PROPN
cana-5362	255	18	next	next	ADV
cana-5362	255	19	,	,	PUNCT
cana-5362	255	20	we	we	PRON
cana-5362	255	21	repeat	repeat	VERB
cana-5362	255	22	𝑁	𝑁	PROPN
cana-5362	255	23	+	+	CCONJ
cana-5362	255	24	1	1	NUM
cana-5362	255	25	times	time	NOUN
cana-5362	255	26	by	by	ADP
cana-5362	255	27	using	use	VERB
cana-5362	255	28	the	the	DET
cana-5362	255	29	numerical	numerical	ADJ
cana-5362	255	30	scheme	scheme	NOUN
cana-5362	255	31	(	(	PUNCT
cana-5362	255	32	71	71	NUM
cana-5362	255	33	)	)	PUNCT
cana-5362	255	34	.	.	PUNCT
cana-5362	256	1	finally	finally	ADV
cana-5362	256	2	,	,	PUNCT
cana-5362	256	3	we	we	PRON
cana-5362	256	4	get	get	VERB
cana-5362	256	5	the	the	DET
cana-5362	256	6	restored	restore	VERB
cana-5362	256	7	image	image	NOUN
cana-5362	256	8	𝑢𝑁+1	𝑢𝑁+1	PROPN
cana-5362	256	9	without	without	ADP
cana-5362	256	10	noise	noise	NOUN
cana-5362	256	11	.	.	PUNCT
cana-5362	257	1	communications	communication	NOUN
cana-5362	257	2	on	on	ADP
cana-5362	257	3	applied	apply	VERB
cana-5362	257	4	nonlinear	nonlinear	ADJ
cana-5362	257	5	analysis	analysis	NOUN
cana-5362	257	6	issn	issn	NOUN
cana-5362	257	7	:	:	PUNCT
cana-5362	257	8	1074	1074	NUM
cana-5362	257	9	-	-	PUNCT
cana-5362	257	10	133x	133x	NUM
cana-5362	257	11	vol	vol	VERB
cana-5362	257	12	32	32	NUM
cana-5362	257	13	no	no	NOUN
cana-5362	257	14	.	.	PUNCT
cana-5362	258	1	10s	10	NOUN
cana-5362	258	2	(	(	PUNCT
cana-5362	258	3	2025	2025	NUM
cana-5362	258	4	)	)	PUNCT
cana-5362	258	5	1997	1997	NUM
cana-5362	258	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5362	258	7	4.1	4.1	NUM
cana-5362	258	8	stability	stability	NOUN
cana-5362	258	9	analysis	analysis	NOUN
cana-5362	258	10	in	in	ADP
cana-5362	258	11	this	this	DET
cana-5362	258	12	section	section	NOUN
cana-5362	258	13	,	,	PUNCT
cana-5362	258	14	we	we	PRON
cana-5362	258	15	study	study	VERB
cana-5362	258	16	the	the	DET
cana-5362	258	17	stability	stability	NOUN
cana-5362	258	18	of	of	ADP
cana-5362	258	19	the	the	DET
cana-5362	258	20	numerical	numerical	ADJ
cana-5362	258	21	scheme	scheme	NOUN
cana-5362	258	22	using	use	VERB
cana-5362	258	23	the	the	DET
cana-5362	258	24	fourier	fourier	NOUN
cana-5362	258	25	transform	transform	NOUN
cana-5362	258	26	method	method	NOUN
cana-5362	258	27	.	.	PUNCT
cana-5362	259	1	specifically	specifically	ADV
cana-5362	259	2	,	,	PUNCT
cana-5362	259	3	we	we	PRON
cana-5362	259	4	implement	implement	VERB
cana-5362	259	5	this	this	DET
cana-5362	259	6	change	change	NOUN
cana-5362	259	7	in	in	ADP
cana-5362	259	8	(	(	PUNCT
cana-5362	259	9	71	71	NUM
cana-5362	259	10	)	)	PUNCT
cana-5362	259	11	,	,	PUNCT
cana-5362	259	12	given	give	VERB
cana-5362	259	13	by	by	ADP
cana-5362	259	14	:	:	PUNCT
cana-5362	259	15	𝑢𝑖,𝑗	𝑢𝑖,𝑗	NOUN
cana-5362	259	16	𝑛	𝑛	PROPN
cana-5362	259	17	=	=	SYM
cana-5362	259	18	�	�	PROPN
cana-5362	259	19	̂	̂	NUM
cana-5362	259	20	�	�	NOUN
cana-5362	259	21	𝑛𝑒𝐼𝜋(𝑘𝑖+𝑚𝑗	𝑛𝑒𝐼𝜋(𝑘𝑖+𝑚𝑗	NOUN
cana-5362	259	22	)	)	PUNCT
cana-5362	259	23	(	(	PUNCT
cana-5362	259	24	72	72	X
cana-5362	259	25	)	)	PUNCT
cana-5362	259	26	definition	definition	NOUN
cana-5362	259	27	1	1	NUM
cana-5362	259	28	scheme	scheme	NOUN
cana-5362	259	29	(	(	PUNCT
cana-5362	259	30	67	67	NUM
cana-5362	259	31	)	)	PUNCT
cana-5362	259	32	said	say	VERB
cana-5362	259	33	to	to	PART
cana-5362	259	34	be	be	AUX
cana-5362	259	35	stable	stable	ADJ
cana-5362	259	36	,	,	PUNCT
cana-5362	259	37	if	if	SCONJ
cana-5362	259	38	there	there	PRON
cana-5362	259	39	exists	exist	VERB
cana-5362	259	40	a	a	DET
cana-5362	259	41	constant	constant	ADJ
cana-5362	259	42	𝐶	𝐶	NOUN
cana-5362	259	43	such	such	ADJ
cana-5362	259	44	that	that	SCONJ
cana-5362	259	45	|	|	NOUN
cana-5362	259	46	�	�	PROPN
cana-5362	259	47	̂	̂	NOUN
cana-5362	259	48	�	�	PROPN
cana-5362	259	49	𝑛+1	𝑛+1	PROPN
cana-5362	259	50	�	�	PROPN
cana-5362	259	51	̂	̂	NOUN
cana-5362	259	52	�	�	NOUN
cana-5362	259	53	𝑛	𝑛	PRON
cana-5362	259	54	|	|	ADV
cana-5362	259	55	≤	≤	ADJ
cana-5362	259	56	𝐶∆𝑡	𝐶∆𝑡	PROPN
cana-5362	259	57	(	(	PUNCT
cana-5362	259	58	73	73	NUM
cana-5362	259	59	)	)	PUNCT
cana-5362	259	60	let	let	VERB
cana-5362	259	61	us	we	PRON
cana-5362	259	62	find	find	VERB
cana-5362	259	63	the	the	DET
cana-5362	259	64	stability	stability	NOUN
cana-5362	259	65	condition	condition	NOUN
cana-5362	259	66	for	for	ADP
cana-5362	259	67	(	(	PUNCT
cana-5362	259	68	71	71	NUM
cana-5362	259	69	)	)	PUNCT
cana-5362	259	70	,	,	PUNCT
cana-5362	259	71	i.e.	i.e.	X
cana-5362	259	72	find	find	VERB
cana-5362	259	73	the	the	DET
cana-5362	259	74	constant	constant	ADJ
cana-5362	259	75	c	c	NOUN
cana-5362	259	76	in	in	ADP
cana-5362	259	77	(	(	PUNCT
cana-5362	259	78	73	73	NUM
cana-5362	259	79	)	)	PUNCT
cana-5362	259	80	.	.	PUNCT
cana-5362	260	1	proposition	proposition	NOUN
cana-5362	260	2	5	5	NUM
cana-5362	260	3	if	if	SCONJ
cana-5362	260	4	(	(	PUNCT
cana-5362	260	5	2	2	NUM
cana-5362	260	6	)	)	PUNCT
cana-5362	260	7	satisfied	satisfied	ADJ
cana-5362	260	8	𝑔	𝑔	NOUN
cana-5362	260	9	,	,	PUNCT
cana-5362	260	10	then	then	ADV
cana-5362	260	11	∆𝑡	∆𝑡	PROPN
cana-5362	260	12	≤	≤	NUM
cana-5362	260	13	2	2	NUM
cana-5362	260	14	8−𝜉𝑛	8−𝜉𝑛	NUM
cana-5362	260	15	,	,	PUNCT
cana-5362	260	16	𝜉𝑛	𝜉𝑛	AUX
cana-5362	260	17	=	=	PUNCT
cana-5362	260	18	𝑊𝑖,𝑗	𝑊𝑖,𝑗	PROPN
cana-5362	260	19	𝑛+1	𝑛+1	PROPN
cana-5362	260	20	−𝑊𝑖,𝑗	−𝑊𝑖,𝑗	PROPN
cana-5362	260	21	𝑛	𝑛	X
cana-5362	260	22	(	(	PUNCT
cana-5362	260	23	74	74	NUM
cana-5362	260	24	)	)	PUNCT
cana-5362	260	25	proof	proof	NOUN
cana-5362	260	26	.	.	PUNCT
cana-5362	261	1	we	we	PRON
cana-5362	261	2	substitute	substitute	VERB
cana-5362	261	3	(	(	PUNCT
cana-5362	261	4	72	72	NUM
cana-5362	261	5	)	)	PUNCT
cana-5362	261	6	in	in	ADP
cana-5362	261	7	(	(	PUNCT
cana-5362	261	8	71	71	NUM
cana-5362	261	9	)	)	PUNCT
cana-5362	261	10	as	as	ADP
cana-5362	261	11	follow	follow	VERB
cana-5362	261	12	�	�	PROPN
cana-5362	261	13	̂	̂	NOUN
cana-5362	261	14	�	�	NOUN
cana-5362	261	15	𝑛+1𝑒𝐼𝜋(𝑘𝑖+𝑚𝑗	𝑛+1𝑒𝐼𝜋(𝑘𝑖+𝑚𝑗	NOUN
cana-5362	261	16	)	)	PUNCT
cana-5362	261	17	=	=	SYM
cana-5362	261	18	�	�	PROPN
cana-5362	261	19	̂	̂	NUM
cana-5362	261	20	�	�	NOUN
cana-5362	261	21	𝑛𝑒𝐼𝜋(𝑘𝑖+𝑚𝑗	𝑛𝑒𝐼𝜋(𝑘𝑖+𝑚𝑗	NOUN
cana-5362	261	22	)	)	PUNCT
cana-5362	261	23	+	+	PUNCT
cana-5362	262	1	∆𝑡𝑔𝑖,𝑗	∆𝑡𝑔𝑖,𝑗	PROPN
cana-5362	262	2	𝑛	𝑛	PRON
cana-5362	262	3	(	(	PUNCT
cana-5362	262	4	�	�	PROPN
cana-5362	262	5	̂	̂	NOUN
cana-5362	262	6	�	�	NOUN
cana-5362	262	7	𝑛𝑒𝐼𝜋(𝑘(𝑖+1)+𝑚𝑗	𝑛𝑒𝐼𝜋(𝑘(𝑖+1)+𝑚𝑗	NOUN
cana-5362	262	8	)	)	PUNCT
cana-5362	262	9	+	+	NUM
cana-5362	262	10	�	�	PROPN
cana-5362	262	11	̂	̂	NOUN
cana-5362	262	12	�	�	NOUN
cana-5362	262	13	𝑛𝑒𝐼𝜋(𝑘(𝑖−1)+𝑚𝑗	𝑛𝑒𝐼𝜋(𝑘(𝑖−1)+𝑚𝑗	NUM
cana-5362	262	14	)	)	PUNCT
cana-5362	262	15	)	)	PUNCT
cana-5362	263	1	+	+	VERB
cana-5362	263	2	∆𝑡𝑔𝑖,𝑗	∆𝑡𝑔𝑖,𝑗	PROPN
cana-5362	263	3	𝑛	𝑛	PRON
cana-5362	263	4	(	(	PUNCT
cana-5362	263	5	�	�	PROPN
cana-5362	263	6	̂	̂	NOUN
cana-5362	263	7	�	�	NOUN
cana-5362	263	8	𝑛𝑒𝐼𝜋(𝑘𝑖+𝑚(𝑗+1	𝑛𝑒𝐼𝜋(𝑘𝑖+𝑚(𝑗+1	NUM
cana-5362	263	9	)	)	PUNCT
cana-5362	263	10	)	)	PUNCT
cana-5362	264	1	+	+	CCONJ
cana-5362	264	2	�	�	PROPN
cana-5362	264	3	̂	̂	NUM
cana-5362	264	4	�	�	NOUN
cana-5362	264	5	𝑛𝑒𝐼𝜋(𝑘𝑖+𝑚(𝑗−1	𝑛𝑒𝐼𝜋(𝑘𝑖+𝑚(𝑗−1	NOUN
cana-5362	264	6	)	)	PUNCT
cana-5362	264	7	)	)	PUNCT
cana-5362	265	1	−	−	ADP
cana-5362	265	2	4	4	NUM
cana-5362	265	3	�	�	PROPN
cana-5362	265	4	̂	̂	NUM
cana-5362	265	5	�	�	NOUN
cana-5362	265	6	𝑛𝑒𝐼𝜋(𝑘𝑖+𝑚𝑗	𝑛𝑒𝐼𝜋(𝑘𝑖+𝑚𝑗	NOUN
cana-5362	265	7	)	)	PUNCT
cana-5362	265	8	)	)	PUNCT
cana-5362	266	1	+	+	CCONJ
cana-5362	266	2	∆𝑡	∆𝑡	NOUN
cana-5362	266	3	4	4	NUM
cana-5362	267	1	[	[	X
cana-5362	267	2	(	(	PUNCT
cana-5362	267	3	𝑔𝑖+1,𝑗	𝑔𝑖+1,𝑗	NOUN
cana-5362	267	4	𝑛	𝑛	PROPN
cana-5362	267	5	−	−	PROPN
cana-5362	267	6	𝑔𝑖−1,𝑗	𝑔𝑖−1,𝑗	ADJ
cana-5362	267	7	𝑛	𝑛	PROPN
cana-5362	267	8	)	)	PUNCT
cana-5362	267	9	(	(	PUNCT
cana-5362	267	10	�	�	NOUN
cana-5362	267	11	̂	̂	SYM
cana-5362	267	12	�	�	NOUN
cana-5362	267	13	𝑛𝑒𝐼𝜋(𝑘(𝑖+1)+𝑚𝑗	𝑛𝑒𝐼𝜋(𝑘(𝑖+1)+𝑚𝑗	NOUN
cana-5362	267	14	)	)	PUNCT
cana-5362	267	15	+	+	NUM
cana-5362	267	16	�	�	PROPN
cana-5362	267	17	̂	̂	NOUN
cana-5362	267	18	�	�	NOUN
cana-5362	267	19	𝑛𝑒𝐼𝜋(𝑘(𝑖−1)+𝑚𝑗	𝑛𝑒𝐼𝜋(𝑘(𝑖−1)+𝑚𝑗	NUM
cana-5362	267	20	)	)	PUNCT
cana-5362	267	21	)	)	PUNCT
cana-5362	267	22	]	]	PUNCT
cana-5362	268	1	+	+	CCONJ
cana-5362	268	2	∆𝑡	∆𝑡	PROPN
cana-5362	268	3	4	4	NUM
cana-5362	268	4	(	(	PUNCT
cana-5362	268	5	𝑔𝑖,𝑗+1	𝑔𝑖,𝑗+1	X
cana-5362	268	6	𝑛	𝑛	PRON
cana-5362	268	7	−	−	PROPN
cana-5362	268	8	𝑔𝑖,𝑗−1	𝑔𝑖,𝑗−1	NOUN
cana-5362	268	9	𝑛	𝑛	X
cana-5362	268	10	)	)	PUNCT
cana-5362	268	11	(	(	PUNCT
cana-5362	268	12	�	�	NOUN
cana-5362	268	13	̂	̂	NOUN
cana-5362	268	14	�	�	NOUN
cana-5362	268	15	𝑛𝑒𝐼𝜋(𝑘𝑖+𝑚(𝑗+1	𝑛𝑒𝐼𝜋(𝑘𝑖+𝑚(𝑗+1	NUM
cana-5362	268	16	)	)	PUNCT
cana-5362	268	17	)	)	PUNCT
cana-5362	269	1	+	+	CCONJ
cana-5362	269	2	�	�	PROPN
cana-5362	269	3	̂	̂	NUM
cana-5362	269	4	�	�	NOUN
cana-5362	269	5	𝑛𝑒𝐼𝜋(𝑘𝑖+𝑚(𝑗−1	𝑛𝑒𝐼𝜋(𝑘𝑖+𝑚(𝑗−1	NOUN
cana-5362	269	6	)	)	PUNCT
cana-5362	269	7	)	)	PUNCT
cana-5362	269	8	)	)	PUNCT
cana-5362	270	1	+	+	ADP
cana-5362	270	2	∆𝑡(𝑊𝑖,𝑗	∆𝑡(𝑊𝑖,𝑗	NOUN
cana-5362	270	3	𝑛+1	𝑛+1	VERB
cana-5362	270	4	−𝑊𝑖,𝑗	−𝑊𝑖,𝑗	PROPN
cana-5362	270	5	𝑛	𝑛	PRON
cana-5362	270	6	)	)	PUNCT
cana-5362	270	7	�	�	PROPN
cana-5362	270	8	̂	̂	NOUN
cana-5362	270	9	�	�	NOUN
cana-5362	270	10	𝑛+1𝑒𝐼𝜋(𝑘𝑖+𝑚𝑗	𝑛+1𝑒𝐼𝜋(𝑘𝑖+𝑚𝑗	NOUN
cana-5362	270	11	)	)	PUNCT
cana-5362	270	12	=	=	SYM
cana-5362	270	13	�	�	PROPN
cana-5362	270	14	̂	̂	NUM
cana-5362	270	15	�	�	NOUN
cana-5362	270	16	𝑛𝑒𝐼𝜋(𝑘𝑖+𝑚𝑗	𝑛𝑒𝐼𝜋(𝑘𝑖+𝑚𝑗	NOUN
cana-5362	270	17	)	)	PUNCT
cana-5362	270	18	(	(	PUNCT
cana-5362	270	19	1	1	NUM
cana-5362	270	20	+	+	NUM
cana-5362	270	21	∆𝑡	∆𝑡	PROPN
cana-5362	270	22	(	(	PUNCT
cana-5362	270	23	𝑔𝑖,𝑗	𝑔𝑖,𝑗	NOUN
cana-5362	270	24	𝑛	𝑛	PRON
cana-5362	270	25	(	(	PUNCT
cana-5362	270	26	𝑒𝐼𝜋𝑘	𝑒𝐼𝜋𝑘	PROPN
cana-5362	270	27	+	+	NUM
cana-5362	270	28	𝑒−𝐼𝜋𝑘	𝑒−𝐼𝜋𝑘	NUM
cana-5362	270	29	+	+	CCONJ
cana-5362	271	1	𝑒𝐼𝜋𝑚	𝑒𝐼𝜋𝑚	PROPN
cana-5362	271	2	+	+	CCONJ
cana-5362	271	3	𝑒−𝐼𝜋𝑚	𝑒−𝐼𝜋𝑚	NOUN
cana-5362	271	4	−	−	NOUN
cana-5362	271	5	4	4	NUM
cana-5362	271	6	)	)	PUNCT
cana-5362	271	7	)	)	PUNCT
cana-5362	271	8	)	)	PUNCT
cana-5362	272	1	+	+	VERB
cana-5362	272	2	�	�	PROPN
cana-5362	272	3	̂	̂	SYM
cana-5362	272	4	�	�	NOUN
cana-5362	272	5	𝑛𝑒𝐼𝜋(𝑘𝑖+𝑚𝑗	𝑛𝑒𝐼𝜋(𝑘𝑖+𝑚𝑗	NOUN
cana-5362	272	6	)	)	PUNCT
cana-5362	272	7	(	(	PUNCT
cana-5362	272	8	∆𝑡	∆𝑡	PROPN
cana-5362	272	9	4	4	NUM
cana-5362	272	10	(	(	PUNCT
cana-5362	272	11	𝑔𝑖+1,𝑗	𝑔𝑖+1,𝑗	NOUN
cana-5362	272	12	𝑛	𝑛	PRON
cana-5362	272	13	−	−	PROPN
cana-5362	272	14	𝑔𝑖−1,𝑗	𝑔𝑖−1,𝑗	ADJ
cana-5362	272	15	𝑛	𝑛	PROPN
cana-5362	272	16	)	)	PUNCT
cana-5362	272	17	(	(	PUNCT
cana-5362	272	18	𝑒𝐼𝜋𝑘	𝑒𝐼𝜋𝑘	PROPN
cana-5362	272	19	+	+	NUM
cana-5362	272	20	𝑒−𝐼𝜋𝑘	𝑒−𝐼𝜋𝑘	NUM
cana-5362	272	21	)	)	PUNCT
cana-5362	272	22	)	)	PUNCT
cana-5362	273	1	+	+	VERB
cana-5362	273	2	�	�	PROPN
cana-5362	273	3	̂	̂	SYM
cana-5362	273	4	�	�	NOUN
cana-5362	273	5	𝑛𝑒𝐼𝜋(𝑘𝑖+𝑚𝑗	𝑛𝑒𝐼𝜋(𝑘𝑖+𝑚𝑗	NOUN
cana-5362	273	6	)	)	PUNCT
cana-5362	273	7	(	(	PUNCT
cana-5362	273	8	∆𝑡	∆𝑡	PROPN
cana-5362	273	9	4	4	NUM
cana-5362	273	10	(	(	PUNCT
cana-5362	273	11	𝑔𝑖,𝑗+1	𝑔𝑖,𝑗+1	X
cana-5362	273	12	𝑛	𝑛	PRON
cana-5362	273	13	−	−	PROPN
cana-5362	273	14	𝑔𝑖,𝑗−1	𝑔𝑖,𝑗−1	NOUN
cana-5362	273	15	𝑛	𝑛	PROPN
cana-5362	273	16	)	)	PUNCT
cana-5362	273	17	(	(	PUNCT
cana-5362	273	18	𝑒𝐼𝜋𝑚	𝑒𝐼𝜋𝑚	PROPN
cana-5362	273	19	+	+	NUM
cana-5362	273	20	𝑒−𝐼𝜋𝑚	𝑒−𝐼𝜋𝑚	NUM
cana-5362	273	21	)	)	PUNCT
cana-5362	273	22	)	)	PUNCT
cana-5362	274	1	+	+	ADP
cana-5362	274	2	∆𝑡(𝑊𝑖,𝑗	∆𝑡(𝑊𝑖,𝑗	NOUN
cana-5362	274	3	𝑛+1	𝑛+1	VERB
cana-5362	274	4	−𝑊𝑖,𝑗	−𝑊𝑖,𝑗	PROPN
cana-5362	274	5	𝑛	𝑛	PRON
cana-5362	274	6	)	)	PUNCT
cana-5362	274	7	�	�	PROPN
cana-5362	274	8	̂	̂	NOUN
cana-5362	274	9	�	�	NOUN
cana-5362	274	10	𝑛+1𝑒𝐼𝜋(𝑘𝑖+𝑚𝑗	𝑛+1𝑒𝐼𝜋(𝑘𝑖+𝑚𝑗	NOUN
cana-5362	274	11	)	)	PUNCT
cana-5362	274	12	�	�	PROPN
cana-5362	274	13	̂	̂	VERB
cana-5362	274	14	�	�	NOUN
cana-5362	274	15	𝑛𝑒𝐼𝜋(𝑘𝑖+𝑚𝑗	𝑛𝑒𝐼𝜋(𝑘𝑖+𝑚𝑗	NOUN
cana-5362	274	16	)	)	PUNCT
cana-5362	274	17	=	=	SYM
cana-5362	274	18	1	1	NUM
cana-5362	274	19	+	+	NUM
cana-5362	274	20	∆𝑡(𝑔𝑖,𝑗	∆𝑡(𝑔𝑖,𝑗	NOUN
cana-5362	274	21	𝑛	𝑛	X
cana-5362	274	22	(	(	PUNCT
cana-5362	274	23	𝑒𝐼𝜋𝑘	𝑒𝐼𝜋𝑘	PROPN
cana-5362	274	24	+	+	NUM
cana-5362	274	25	𝑒−𝐼𝜋𝑘	𝑒−𝐼𝜋𝑘	NUM
cana-5362	274	26	+	+	CCONJ
cana-5362	274	27	𝑒𝐼𝜋𝑚	𝑒𝐼𝜋𝑚	PROPN
cana-5362	274	28	+	+	CCONJ
cana-5362	274	29	𝑒−𝐼𝜋𝑚	𝑒−𝐼𝜋𝑚	NOUN
cana-5362	274	30	−	−	NOUN
cana-5362	274	31	4	4	NUM
cana-5362	274	32	)	)	PUNCT
cana-5362	274	33	)	)	PUNCT
cana-5362	275	1	+	+	CCONJ
cana-5362	275	2	∆𝑡	∆𝑡	PROPN
cana-5362	275	3	4	4	NUM
cana-5362	275	4	(	(	PUNCT
cana-5362	275	5	𝑔𝑖+1,𝑗	𝑔𝑖+1,𝑗	NOUN
cana-5362	275	6	𝑛	𝑛	PRON
cana-5362	275	7	−	−	PROPN
cana-5362	275	8	𝑔𝑖−1,𝑗	𝑔𝑖−1,𝑗	ADJ
cana-5362	275	9	𝑛	𝑛	PROPN
cana-5362	275	10	)	)	PUNCT
cana-5362	275	11	(	(	PUNCT
cana-5362	275	12	𝑒𝐼𝜋𝑘	𝑒𝐼𝜋𝑘	PROPN
cana-5362	275	13	+	+	NUM
cana-5362	275	14	𝑒−𝐼𝜋𝑘	𝑒−𝐼𝜋𝑘	NUM
cana-5362	275	15	)	)	PUNCT
cana-5362	276	1	+	+	CCONJ
cana-5362	276	2	∆𝑡	∆𝑡	PROPN
cana-5362	276	3	4	4	NUM
cana-5362	276	4	(	(	PUNCT
cana-5362	276	5	𝑔𝑖,𝑗+1	𝑔𝑖,𝑗+1	X
cana-5362	276	6	𝑛	𝑛	PRON
cana-5362	276	7	−	−	PROPN
cana-5362	276	8	𝑔𝑖,𝑗−1	𝑔𝑖,𝑗−1	NOUN
cana-5362	276	9	𝑛	𝑛	PROPN
cana-5362	276	10	)	)	PUNCT
cana-5362	276	11	(	(	PUNCT
cana-5362	276	12	𝑒𝐼𝜋𝑚	𝑒𝐼𝜋𝑚	PROPN
cana-5362	276	13	+	+	NUM
cana-5362	276	14	𝑒−𝐼𝜋𝑚	𝑒−𝐼𝜋𝑚	NUM
cana-5362	276	15	)	)	PUNCT
cana-5362	277	1	+	+	NOUN
cana-5362	277	2	∆𝑡	∆𝑡	PROPN
cana-5362	277	3	(	(	PUNCT
cana-5362	277	4	𝑊𝑖,𝑗	𝑊𝑖,𝑗	PROPN
cana-5362	277	5	𝑛+1−𝑊𝑖,𝑗	𝑛+1−𝑊𝑖,𝑗	VERB
cana-5362	277	6	𝑛	𝑛	DET
cana-5362	277	7	�	�	PROPN
cana-5362	277	8	̂	̂	NUM
cana-5362	277	9	�	�	NOUN
cana-5362	277	10	𝑛𝑒𝐼𝜋(𝑘𝑖+𝑚𝑗	𝑛𝑒𝐼𝜋(𝑘𝑖+𝑚𝑗	NOUN
cana-5362	277	11	)	)	PUNCT
cana-5362	277	12	)	)	PUNCT
cana-5362	277	13	,	,	PUNCT
cana-5362	277	14	such	such	ADJ
cana-5362	277	15	as	as	ADP
cana-5362	277	16	𝑔𝑖,𝑗	𝑔𝑖,𝑗	NOUN
cana-5362	277	17	𝑛	𝑛	DET
cana-5362	277	18	≤	≤	NUM
cana-5362	277	19	1	1	NUM
cana-5362	277	20	,	,	PUNCT
cana-5362	277	21	𝑔𝑖+1,𝑗	𝑔𝑖+1,𝑗	NOUN
cana-5362	277	22	𝑛	𝑛	PROPN
cana-5362	277	23	≤	≤	ADJ
cana-5362	277	24	1	1	NUM
cana-5362	277	25	,	,	PUNCT
cana-5362	277	26	𝑔𝑖−1,𝑗	𝑔𝑖−1,𝑗	ADV
cana-5362	277	27	𝑛	𝑛	PRON
cana-5362	277	28	≤	≤	NUM
cana-5362	277	29	1	1	NUM
cana-5362	277	30	,	,	PUNCT
cana-5362	277	31	𝑔𝑖,𝑗+1	𝑔𝑖,𝑗+1	NOUN
cana-5362	277	32	𝑛	𝑛	PRON
cana-5362	277	33	≤	≤	NUM
cana-5362	277	34	1	1	NUM
cana-5362	277	35	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-5362	277	36	𝑔𝑖,𝑗−1	𝑔𝑖,𝑗−1	NOUN
cana-5362	277	37	𝑛	𝑛	PROPN
cana-5362	277	38	,	,	PUNCT
cana-5362	277	39	communications	communication	NOUN
cana-5362	277	40	on	on	ADP
cana-5362	277	41	applied	apply	VERB
cana-5362	277	42	nonlinear	nonlinear	ADJ
cana-5362	277	43	analysis	analysis	NOUN
cana-5362	277	44	issn	issn	NOUN
cana-5362	277	45	:	:	PUNCT
cana-5362	277	46	1074	1074	NUM
cana-5362	277	47	-	-	PUNCT
cana-5362	277	48	133x	133x	NUM
cana-5362	277	49	vol	vol	VERB
cana-5362	277	50	32	32	NUM
cana-5362	277	51	no	no	NOUN
cana-5362	277	52	.	.	PUNCT
cana-5362	278	1	10s	10	NOUN
cana-5362	278	2	(	(	PUNCT
cana-5362	278	3	2025	2025	NUM
cana-5362	278	4	)	)	PUNCT
cana-5362	278	5	1998	1998	NUM
cana-5362	278	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5362	278	7	and	and	CCONJ
cana-5362	278	8	{	{	PUNCT
cana-5362	278	9	𝑒𝐼𝜋𝑘	𝑒𝐼𝜋𝑘	PROPN
cana-5362	278	10	+	+	NUM
cana-5362	278	11	𝑒−𝐼𝜋𝑘	𝑒−𝐼𝜋𝑘	PUNCT
cana-5362	278	12	=	=	SYM
cana-5362	278	13	2	2	NUM
cana-5362	278	14	𝑐𝑜𝑠(𝜋𝑘	𝑐𝑜𝑠(𝜋𝑘	NUM
cana-5362	278	15	)	)	PUNCT
cana-5362	279	1	𝑒𝐼𝜋𝑚	𝑒𝐼𝜋𝑚	PROPN
cana-5362	279	2	+	+	CCONJ
cana-5362	279	3	𝑒−𝐼𝜋𝑚	𝑒−𝐼𝜋𝑚	NOUN
cana-5362	279	4	=	=	SYM
cana-5362	279	5	2𝑐𝑜𝑠(𝜋𝑚	2𝑐𝑜𝑠(𝜋𝑚	NUM
cana-5362	279	6	)	)	PUNCT
cana-5362	279	7	(	(	PUNCT
cana-5362	279	8	75	75	NUM
cana-5362	279	9	)	)	PUNCT
cana-5362	279	10	⟹	⟹	NOUN
cana-5362	279	11	{	{	PUNCT
cana-5362	279	12	2𝑐𝑜𝑠(𝜋𝑘	2𝑐𝑜𝑠(𝜋𝑘	NUM
cana-5362	279	13	)	)	PUNCT
cana-5362	279	14	−	−	PROPN
cana-5362	279	15	2	2	NUM
cana-5362	279	16	=	=	SYM
cana-5362	279	17	−4𝑠𝑖𝑛	−4𝑠𝑖𝑛	NOUN
cana-5362	279	18	2	2	NUM
cana-5362	279	19	(	(	PUNCT
cana-5362	279	20	𝜋𝑘	𝜋𝑘	NOUN
cana-5362	279	21	2	2	NUM
cana-5362	279	22	)	)	PUNCT
cana-5362	279	23	2	2	NUM
cana-5362	279	24	𝑐𝑜𝑠(𝜋𝑚	𝑐𝑜𝑠(𝜋𝑚	NOUN
cana-5362	279	25	)	)	PUNCT
cana-5362	279	26	−	−	NOUN
cana-5362	279	27	2	2	NUM
cana-5362	279	28	=	=	SYM
cana-5362	279	29	−4𝑠𝑖𝑛	−4𝑠𝑖𝑛	NOUN
cana-5362	279	30	2	2	NUM
cana-5362	279	31	(	(	PUNCT
cana-5362	279	32	𝜋𝑚	𝜋𝑚	PROPN
cana-5362	279	33	2	2	NUM
cana-5362	279	34	)	)	PUNCT
cana-5362	279	35	,	,	PUNCT
cana-5362	279	36	(	(	PUNCT
cana-5362	279	37	76	76	NUM
cana-5362	279	38	)	)	PUNCT
cana-5362	279	39	with	with	ADP
cana-5362	279	40	sin2	sin2	NOUN
cana-5362	279	41	(	(	PUNCT
cana-5362	279	42	𝜋𝑘	𝜋𝑘	NOUN
cana-5362	279	43	2	2	NUM
cana-5362	279	44	)	)	PUNCT
cana-5362	279	45	≤	≤	NUM
cana-5362	279	46	1	1	NUM
cana-5362	279	47	,	,	PUNCT
cana-5362	279	48	sin	sin	NOUN
cana-5362	279	49	2	2	NUM
cana-5362	279	50	(	(	PUNCT
cana-5362	279	51	𝜋𝑚	𝜋𝑚	PROPN
cana-5362	279	52	2	2	NUM
cana-5362	279	53	)	)	PUNCT
cana-5362	279	54	≤	≤	NOUN
cana-5362	279	55	1	1	NUM
cana-5362	279	56	|	|	NOUN
cana-5362	279	57	�	�	PROPN
cana-5362	279	58	̂	̂	NOUN
cana-5362	279	59	�	�	PROPN
cana-5362	279	60	𝑛+1	𝑛+1	PROPN
cana-5362	279	61	�	�	PROPN
cana-5362	279	62	̂	̂	NOUN
cana-5362	279	63	�	�	NOUN
cana-5362	279	64	𝑛	𝑛	PRON
cana-5362	279	65	|	|	ADV
cana-5362	279	66	≤	≤	PUNCT
cana-5362	279	67	|1	|1	PRON
cana-5362	280	1	−	−	NOUN
cana-5362	280	2	8∆𝑡	8∆𝑡	NUM
cana-5362	281	1	+	+	CCONJ
cana-5362	281	2	∆𝑡	∆𝑡	PROPN
cana-5362	281	3	(	(	PUNCT
cana-5362	281	4	𝑊𝑖,𝑗	𝑊𝑖,𝑗	PROPN
cana-5362	281	5	𝑛+1	𝑛+1	PROPN
cana-5362	281	6	−𝑊𝑖,𝑗	−𝑊𝑖,𝑗	PROPN
cana-5362	281	7	𝑛	𝑛	PRON
cana-5362	281	8	�	�	PROPN
cana-5362	281	9	̂	̂	NUM
cana-5362	281	10	�	�	NOUN
cana-5362	281	11	𝑛𝑒𝐼𝜋(𝑘𝑖+𝑚𝑗	𝑛𝑒𝐼𝜋(𝑘𝑖+𝑚𝑗	NOUN
cana-5362	281	12	)	)	PUNCT
cana-5362	281	13	)	)	PUNCT
cana-5362	281	14	|	|	ADV
cana-5362	281	15	≤	≤	NOUN
cana-5362	281	16	|1	|1	PRON
cana-5362	281	17	−	−	NOUN
cana-5362	281	18	8∆𝑡	8∆𝑡	NUM
cana-5362	281	19	+	+	CCONJ
cana-5362	281	20	∆𝑡(𝑊𝑖,𝑗	∆𝑡(𝑊𝑖,𝑗	NOUN
cana-5362	281	21	𝑛+1	𝑛+1	ADP
cana-5362	281	22	−𝑊𝑖,𝑗	−𝑊𝑖,𝑗	PROPN
cana-5362	281	23	𝑛	𝑛	PRON
cana-5362	281	24	)	)	PUNCT
cana-5362	281	25	|	|	ADV
cana-5362	281	26	≤	≤	PUNCT
cana-5362	281	27	|1	|1	PRON
cana-5362	281	28	−	−	PROPN
cana-5362	282	1	∆𝑡(8	∆𝑡(8	NOUN
cana-5362	282	2	−	−	PROPN
cana-5362	282	3	𝜉𝑛)|	𝜉𝑛)|	PROPN
cana-5362	282	4	|1	|1	NUM
cana-5362	282	5	−	−	PROPN
cana-5362	283	1	∆𝑡(8	∆𝑡(8	NOUN
cana-5362	283	2	−	−	PROPN
cana-5362	283	3	𝜉𝑛)|	𝜉𝑛)|	PROPN
cana-5362	283	4	≤	≤	NUM
cana-5362	283	5	1	1	NUM
cana-5362	283	6	⟹	⟹	NUM
cana-5362	283	7	−1	−1	NOUN
cana-5362	283	8	≤	≤	NOUN
cana-5362	283	9	1	1	NUM
cana-5362	283	10	−	−	NOUN
cana-5362	283	11	∆𝑡(8	∆𝑡(8	NOUN
cana-5362	283	12	−	−	NOUN
cana-5362	283	13	𝜉𝑛	𝜉𝑛	NOUN
cana-5362	283	14	)	)	PUNCT
cana-5362	283	15	≤	≤	NUM
cana-5362	283	16	1	1	NUM
cana-5362	283	17	⟹	⟹	NUM
cana-5362	283	18	∆𝑡	∆𝑡	PROPN
cana-5362	283	19	≤	≤	NUM
cana-5362	283	20	2	2	NUM
cana-5362	283	21	8−𝜉𝑛	8−𝜉𝑛	NUM
cana-5362	283	22	(	(	PUNCT
cana-5362	283	23	77	77	NUM
cana-5362	283	24	)	)	PUNCT
cana-5362	283	25	then	then	ADV
cana-5362	283	26	proposition	proposition	VERB
cana-5362	283	27	5	5	NUM
cana-5362	283	28	hold	hold	NOUN
cana-5362	283	29	.	.	PUNCT
cana-5362	284	1	if	if	SCONJ
cana-5362	284	2	we	we	PRON
cana-5362	284	3	consider	consider	VERB
cana-5362	284	4	∆𝑡	∆𝑡	NOUN
cana-5362	284	5	≤	≤	NUM
cana-5362	284	6	2	2	NUM
cana-5362	284	7	8−𝜉𝑛	8−𝜉𝑛	NUM
cana-5362	284	8	=	=	SYM
cana-5362	284	9	1	1	NUM
cana-5362	284	10	4−	4−	NUM
cana-5362	284	11	1	1	NUM
cana-5362	284	12	2	2	NUM
cana-5362	284	13	𝜉𝑛	𝜉𝑛	NOUN
cana-5362	284	14	=	=	SYM
cana-5362	284	15	1	1	NUM
cana-5362	284	16	𝐶	𝐶	PROPN
cana-5362	284	17	.	.	PUNCT
cana-5362	285	1	𝐶	𝐶	PROPN
cana-5362	285	2	=	=	NOUN
cana-5362	285	3	4	4	NUM
cana-5362	285	4	−	−	NOUN
cana-5362	285	5	1	1	NUM
cana-5362	285	6	2	2	NUM
cana-5362	285	7	𝜉𝑛	𝜉𝑛	NOUN
cana-5362	285	8	(	(	PUNCT
cana-5362	285	9	78	78	NUM
cana-5362	285	10	)	)	PUNCT
cana-5362	285	11	we	we	PRON
cana-5362	285	12	obtain	obtain	VERB
cana-5362	285	13	stability	stability	NOUN
cana-5362	285	14	conditions	condition	NOUN
cana-5362	285	15	(	(	PUNCT
cana-5362	285	16	78	78	NUM
cana-5362	285	17	)	)	PUNCT
cana-5362	285	18	for	for	ADP
cana-5362	285	19	a	a	DET
cana-5362	285	20	good	good	ADJ
cana-5362	285	21	choice	choice	NOUN
cana-5362	285	22	of	of	ADP
cana-5362	285	23	the	the	DET
cana-5362	285	24	time	time	NOUN
cana-5362	285	25	discretisation	discretisation	NOUN
cana-5362	285	26	parameter	parameter	NOUN
cana-5362	285	27	to	to	PART
cana-5362	285	28	solve	solve	VERB
cana-5362	285	29	(	(	PUNCT
cana-5362	285	30	71	71	NUM
cana-5362	285	31	)	)	PUNCT
cana-5362	285	32	.	.	PUNCT
cana-5362	286	1	4.2	4.2	NUM
cana-5362	286	2	numerical	numerical	ADJ
cana-5362	286	3	results	result	NOUN
cana-5362	286	4	and	and	CCONJ
cana-5362	286	5	comments	comment	NOUN
cana-5362	286	6	in	in	ADP
cana-5362	286	7	this	this	DET
cana-5362	286	8	section	section	NOUN
cana-5362	286	9	,	,	PUNCT
cana-5362	286	10	we	we	PRON
cana-5362	286	11	present	present	VERB
cana-5362	286	12	the	the	DET
cana-5362	286	13	obtained	obtain	VERB
cana-5362	286	14	results	result	NOUN
cana-5362	286	15	from	from	ADP
cana-5362	286	16	our	our	PRON
cana-5362	286	17	numerical	numerical	ADJ
cana-5362	286	18	experimentations	experimentation	NOUN
cana-5362	286	19	,	,	PUNCT
cana-5362	286	20	using	use	VERB
cana-5362	286	21	matlab	matlab	PROPN
cana-5362	286	22	r2022b	r2022b	PROPN
cana-5362	286	23	.	.	PUNCT
cana-5362	287	1	we	we	PRON
cana-5362	287	2	tested	test	VERB
cana-5362	287	3	different	different	ADJ
cana-5362	287	4	approaches	approach	NOUN
cana-5362	287	5	to	to	PART
cana-5362	287	6	evaluate	evaluate	VERB
cana-5362	287	7	the	the	DET
cana-5362	287	8	performance	performance	NOUN
cana-5362	287	9	of	of	ADP
cana-5362	287	10	our	our	PRON
cana-5362	287	11	proposed	propose	VERB
cana-5362	287	12	spde	spde	ADJ
cana-5362	287	13	model	model	NOUN
cana-5362	287	14	for	for	ADP
cana-5362	287	15	image	image	NOUN
cana-5362	287	16	restoration	restoration	NOUN
cana-5362	287	17	.	.	PUNCT
cana-5362	288	1	to	to	PART
cana-5362	288	2	measure	measure	VERB
cana-5362	288	3	the	the	DET
cana-5362	288	4	quality	quality	NOUN
cana-5362	288	5	of	of	ADP
cana-5362	288	6	the	the	DET
cana-5362	288	7	restored	restore	VERB
cana-5362	288	8	images	image	NOUN
cana-5362	288	9	,	,	PUNCT
cana-5362	288	10	we	we	PRON
cana-5362	288	11	calculated	calculate	VERB
cana-5362	288	12	the	the	DET
cana-5362	288	13	peak	peak	NOUN
cana-5362	288	14	signal	signal	NOUN
cana-5362	288	15	-	-	PUNCT
cana-5362	288	16	to	to	ADP
cana-5362	288	17	-	-	PUNCT
cana-5362	288	18	noise	noise	NOUN
cana-5362	288	19	ratio	ratio	NOUN
cana-5362	288	20	(	(	PUNCT
cana-5362	288	21	psnr	psnr	NOUN
cana-5362	288	22	)	)	PUNCT
cana-5362	288	23	and	and	CCONJ
cana-5362	288	24	the	the	DET
cana-5362	288	25	structural	structural	ADJ
cana-5362	288	26	similarity	similarity	NOUN
cana-5362	288	27	index	index	NOUN
cana-5362	288	28	(	(	PUNCT
cana-5362	288	29	ssim	ssim	NOUN
cana-5362	288	30	)	)	PUNCT
cana-5362	288	31	.	.	PUNCT
cana-5362	289	1	both	both	PRON
cana-5362	289	2	gaussian	gaussian	ADJ
cana-5362	289	3	and	and	CCONJ
cana-5362	289	4	salt	salt	NOUN
cana-5362	289	5	&	&	CCONJ
cana-5362	289	6	pepper	pepper	PROPN
cana-5362	289	7	noises	noise	NOUN
cana-5362	289	8	were	be	AUX
cana-5362	289	9	considered	consider	VERB
cana-5362	289	10	,	,	PUNCT
cana-5362	289	11	where	where	SCONJ
cana-5362	289	12	the	the	DET
cana-5362	289	13	tests	test	NOUN
cana-5362	289	14	were	be	AUX
cana-5362	289	15	carried	carry	VERB
cana-5362	289	16	out	out	ADP
cana-5362	289	17	by	by	ADP
cana-5362	289	18	varying	vary	VERB
cana-5362	289	19	the	the	DET
cana-5362	289	20	standard	standard	ADJ
cana-5362	289	21	deviation	deviation	NOUN
cana-5362	289	22	(	(	PUNCT
cana-5362	289	23	σ	σ	NOUN
cana-5362	289	24	)	)	PUNCT
cana-5362	289	25	of	of	ADP
cana-5362	289	26	the	the	DET
cana-5362	289	27	gaussian	gaussian	ADJ
cana-5362	289	28	filter	filter	NOUN
cana-5362	289	29	,	,	PUNCT
cana-5362	289	30	while	while	SCONJ
cana-5362	289	31	keeping	keep	VERB
cana-5362	289	32	the	the	DET
cana-5362	289	33	noise	noise	NOUN
cana-5362	289	34	variance	variance	NOUN
cana-5362	289	35	fixed	fix	VERB
cana-5362	289	36	at	at	ADP
cana-5362	289	37	γ	γ	X
cana-5362	289	38	=	=	SYM
cana-5362	289	39	0.1	0.1	NUM
cana-5362	289	40	&	&	CCONJ
cana-5362	289	41	0.01	0.01	NUM
cana-5362	289	42	.	.	PUNCT
cana-5362	290	1	note	note	VERB
cana-5362	290	2	that	that	SCONJ
cana-5362	290	3	we	we	PRON
cana-5362	290	4	can	can	AUX
cana-5362	290	5	evaluate	evaluate	VERB
cana-5362	290	6	how	how	SCONJ
cana-5362	290	7	well	well	ADV
cana-5362	290	8	the	the	DET
cana-5362	290	9	model	model	NOUN
cana-5362	290	10	is	be	AUX
cana-5362	290	11	adapted	adapt	VERB
cana-5362	290	12	to	to	ADP
cana-5362	290	13	different	different	ADJ
cana-5362	290	14	smoothing	smoothing	NOUN
cana-5362	290	15	conditions	condition	NOUN
cana-5362	290	16	in	in	ADP
cana-5362	290	17	image	image	NOUN
cana-5362	290	18	denoising	denoising	NOUN
cana-5362	290	19	,	,	PUNCT
cana-5362	290	20	as	as	SCONJ
cana-5362	290	21	shown	show	VERB
cana-5362	290	22	in	in	ADP
cana-5362	290	23	the	the	DET
cana-5362	290	24	results	result	NOUN
cana-5362	290	25	in	in	ADP
cana-5362	290	26	table	table	NOUN
cana-5362	290	27	3	3	NUM
cana-5362	290	28	.	.	PUNCT
cana-5362	290	29	table	table	NOUN
cana-5362	290	30	1	1	NUM
cana-5362	290	31	.	.	PUNCT
cana-5362	291	1	psnr	psnr	NOUN
cana-5362	291	2	values	value	NOUN
cana-5362	291	3	for	for	ADP
cana-5362	291	4	denoised	denoise	VERB
cana-5362	291	5	images	image	NOUN
cana-5362	291	6	with	with	ADP
cana-5362	291	7	gaussian	gaussian	ADJ
cana-5362	291	8	noise	noise	NOUN
cana-5362	291	9	model	model	PROPN
cana-5362	291	10	pde	pde	PROPN
cana-5362	291	11	kolmogorov	kolmogorov	PROPN
cana-5362	291	12	sde	sde	PROPN
cana-5362	291	13	barbu	barbu	PROPN
cana-5362	291	14	sde	sde	PROPN
cana-5362	291	15	borkowski	borkowski	PROPN
cana-5362	291	16	pm1	pm1	PROPN
cana-5362	291	17	pm2	pm2	PROPN
cana-5362	291	18	spde	spde	PROPN
cana-5362	291	19	1	1	NUM
cana-5362	291	20	spde	spde	NOUN
cana-5362	291	21	2	2	NUM
cana-5362	291	22	γ	γ	X
cana-5362	291	23	=	=	SYM
cana-5362	291	24	0.1	0.1	NUM
cana-5362	291	25	21.3094	21.3094	NUM
cana-5362	291	26	20.1880	20.1880	NUM
cana-5362	291	27	24.6288	24.6288	NUM
cana-5362	291	28	24.1495	24.1495	NUM
cana-5362	291	29	24.1601	24.1601	NUM
cana-5362	291	30	27.6852	27.6852	NUM
cana-5362	291	31	27.7182	27.7182	NUM
cana-5362	291	32	γ	γ	X
cana-5362	291	33	=	=	NOUN
cana-5362	291	34	0.01	0.01	NUM
cana-5362	291	35	24.3094	24.3094	NUM
cana-5362	291	36	30.1259	30.1259	NUM
cana-5362	291	37	30.0948	30.0948	NUM
cana-5362	291	38	30.2702	30.2702	NUM
cana-5362	291	39	30.1051	30.1051	NUM
cana-5362	291	40	33.3760	33.3760	NUM
cana-5362	291	41	33.4251	33.4251	NUM
cana-5362	291	42	communications	communication	NOUN
cana-5362	291	43	on	on	ADP
cana-5362	291	44	applied	apply	VERB
cana-5362	291	45	nonlinear	nonlinear	ADJ
cana-5362	291	46	analysis	analysis	NOUN
cana-5362	291	47	issn	issn	NOUN
cana-5362	291	48	:	:	PUNCT
cana-5362	291	49	1074	1074	NUM
cana-5362	291	50	-	-	PUNCT
cana-5362	291	51	133x	133x	NUM
cana-5362	291	52	vol	vol	VERB
cana-5362	291	53	32	32	NUM
cana-5362	291	54	no	no	NOUN
cana-5362	291	55	.	.	PUNCT
cana-5362	292	1	10s	10	NOUN
cana-5362	292	2	(	(	PUNCT
cana-5362	292	3	2025	2025	NUM
cana-5362	292	4	)	)	PUNCT
cana-5362	292	5	1999	1999	NUM
cana-5362	292	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5362	292	7	table	table	NOUN
cana-5362	292	8	2	2	NUM
cana-5362	292	9	.	.	PUNCT
cana-5362	292	10	ssim	ssim	NOUN
cana-5362	292	11	values	value	NOUN
cana-5362	292	12	for	for	ADP
cana-5362	292	13	denoised	denoise	VERB
cana-5362	292	14	images	image	NOUN
cana-5362	292	15	with	with	ADP
cana-5362	292	16	gaussian	gaussian	ADJ
cana-5362	292	17	noise	noise	NOUN
cana-5362	292	18	models	model	NOUN
cana-5362	292	19	pde	pde	PROPN
cana-5362	292	20	kolmogorov	kolmogorov	PROPN
cana-5362	292	21	sde	sde	PROPN
cana-5362	292	22	barbu	barbu	PROPN
cana-5362	292	23	sde	sde	PROPN
cana-5362	292	24	borkowski	borkowski	PROPN
cana-5362	292	25	pm1	pm1	PROPN
cana-5362	292	26	pm2	pm2	PROPN
cana-5362	292	27	spde	spde	PROPN
cana-5362	292	28	1	1	NUM
cana-5362	292	29	spde	spde	NOUN
cana-5362	292	30	2	2	NUM
cana-5362	292	31	γ	γ	X
cana-5362	292	32	=	=	SYM
cana-5362	292	33	0.1	0.1	NUM
cana-5362	292	34	0.6008	0.6008	NUM
cana-5362	292	35	0.5391	0.5391	NUM
cana-5362	292	36	0.6104	0.6104	NUM
cana-5362	292	37	0.6769	0.6769	NUM
cana-5362	292	38	0.6766	0.6766	NUM
cana-5362	292	39	0.7452	0.7452	NUM
cana-5362	292	40	0.7559	0.7559	NUM
cana-5362	292	41	γ	γ	X
cana-5362	292	42	=	=	NOUN
cana-5362	292	43	0.01	0.01	NUM
cana-5362	292	44	0.6017	0.6017	NUM
cana-5362	292	45	0.7111	0.7111	NUM
cana-5362	292	46	0.8160	0.8160	NUM
cana-5362	292	47	0.8035	0.8035	NUM
cana-5362	292	48	0.8040	0.8040	NUM
cana-5362	292	49	0.8913	0.8913	NUM
cana-5362	292	50	0.8926	0.8926	NUM
cana-5362	292	51	table	table	NOUN
cana-5362	292	52	3	3	NUM
cana-5362	292	53	.	.	PUNCT
cana-5362	292	54	impact	impact	NOUN
cana-5362	292	55	of	of	ADP
cana-5362	292	56	gaussian	gaussian	ADJ
cana-5362	292	57	filter	filter	NOUN
cana-5362	292	58	variance	variance	NOUN
cana-5362	292	59	on	on	ADP
cana-5362	292	60	image	image	NOUN
cana-5362	292	61	quality	quality	NOUN
cana-5362	292	62	metrics	metric	NOUN
cana-5362	292	63	under	under	ADP
cana-5362	292	64	fixed	fix	VERB
cana-5362	292	65	noise	noise	NOUN
cana-5362	292	66	level	level	NOUN
cana-5362	292	67	γ	γ	X
cana-5362	292	68	=	=	SYM
cana-5362	292	69	0.1	0.1	NUM
cana-5362	292	70	&	&	CCONJ
cana-5362	292	71	0.01	0.01	NUM
cana-5362	292	72	𝜎	𝜎	PROPN
cana-5362	292	73	0.45	0.45	NUM
cana-5362	292	74	0.9	0.9	NUM
cana-5362	292	75	1	1	NUM
cana-5362	292	76	1.6	1.6	NUM
cana-5362	292	77	psnr	psnr	NOUN
cana-5362	292	78	0.1	0.1	NUM
cana-5362	292	79	22.7798	22.7798	NUM
cana-5362	292	80	25.2683	25.2683	NUM
cana-5362	292	81	25.1102	25.1102	NUM
cana-5362	292	82	24.6648	24.6648	NUM
cana-5362	292	83	0.01	0.01	NUM
cana-5362	292	84	33.5837	33.5837	NUM
cana-5362	292	85	26.3696	26.3696	NUM
cana-5362	292	86	27.9571	27.9571	NUM
cana-5362	292	87	26.2707	26.2707	NUM
cana-5362	292	88	ssim	ssim	NOUN
cana-5362	293	1	0.1	0.1	NUM
cana-5362	293	2	0.5513	0.5513	NUM
cana-5362	293	3	0.7466	0.7466	NUM
cana-5362	293	4	0.7153	0.7153	NUM
cana-5362	293	5	0.7104	0.7104	NUM
cana-5362	293	6	0.01	0.01	NUM
cana-5362	293	7	0.8915	0.8915	NUM
cana-5362	293	8	0.8404	0.8404	NUM
cana-5362	293	9	0.8381	0.8381	NUM
cana-5362	293	10	0.8188	0.8188	NUM
cana-5362	293	11	table	table	NOUN
cana-5362	293	12	4	4	NUM
cana-5362	293	13	.	.	PUNCT
cana-5362	293	14	performance	performance	NOUN
cana-5362	293	15	of	of	ADP
cana-5362	293	16	our	our	PRON
cana-5362	293	17	spde	spde	ADJ
cana-5362	293	18	model	model	NOUN
cana-5362	293	19	with	with	ADP
cana-5362	293	20	salt	salt	NOUN
cana-5362	293	21	&	&	CCONJ
cana-5362	293	22	pepper	pepper	NOUN
cana-5362	293	23	noise	noise	NOUN
cana-5362	293	24	compared	compare	VERB
cana-5362	293	25	to	to	ADP
cana-5362	293	26	other	other	ADJ
cana-5362	293	27	approaches	approach	NOUN
cana-5362	293	28	under	under	ADP
cana-5362	293	29	a	a	DET
cana-5362	293	30	fixed	fix	VERB
cana-5362	293	31	noise	noise	NOUN
cana-5362	293	32	level	level	NOUN
cana-5362	293	33	γ	γ	X
cana-5362	293	34	=	=	SYM
cana-5362	293	35	0.1	0.1	NUM
cana-5362	293	36	and	and	CCONJ
cana-5362	293	37	σ	σ	NUM
cana-5362	293	38	=	=	NOUN
cana-5362	293	39	0.45	0.45	NUM
cana-5362	293	40	.	.	PUNCT
cana-5362	294	1	model	model	PROPN
cana-5362	294	2	pde	pde	PROPN
cana-5362	294	3	kolmogorov	kolmogorov	PROPN
cana-5362	294	4	sde	sde	PROPN
cana-5362	294	5	barbu	barbu	PROPN
cana-5362	294	6	sde	sde	PROPN
cana-5362	294	7	borkowski	borkowski	PROPN
cana-5362	294	8	pm	pm	VERB
cana-5362	294	9	1	1	NUM
cana-5362	294	10	pm	pm	NOUN
cana-5362	294	11	2	2	NUM
cana-5362	294	12	spde	spde	NOUN
cana-5362	294	13	1	1	NUM
cana-5362	294	14	spde	spde	NOUN
cana-5362	294	15	2	2	NUM
cana-5362	294	16	psnr	psnr	NOUN
cana-5362	294	17	22.9429	22.9429	NUM
cana-5362	294	18	34.3444	34.3444	NUM
cana-5362	294	19	30.9117	30.9117	NUM
cana-5362	294	20	31.6016	31.6016	NUM
cana-5362	294	21	32.2173	32.2173	NUM
cana-5362	294	22	35.4783	35.4783	NUM
cana-5362	294	23	35.9521	35.9521	NUM
cana-5362	294	24	ssim	ssim	NOUN
cana-5362	294	25	0.7748	0.7748	NUM
cana-5362	294	26	0.9739	0.9739	NUM
cana-5362	294	27	0.8999	0.8999	NUM
cana-5362	294	28	0.9613	0.9613	NUM
cana-5362	294	29	0.9642	0.9642	NUM
cana-5362	294	30	0.9861	0.9861	NUM
cana-5362	294	31	0.9881	0.9881	NUM
cana-5362	294	32	we	we	PRON
cana-5362	294	33	denote	denote	VERB
cana-5362	294	34	:	:	PUNCT
cana-5362	294	35	•	•	NUM
cana-5362	294	36	pde	pde	NOUN
cana-5362	294	37	kolmogorov	kolmogorov	PROPN
cana-5362	294	38	:	:	PUNCT
cana-5362	294	39	the	the	DET
cana-5362	294	40	partial	partial	ADJ
cana-5362	294	41	differential	differential	NOUN
cana-5362	294	42	equation	equation	NOUN
cana-5362	294	43	(	(	PUNCT
cana-5362	294	44	pde	pde	NOUN
cana-5362	294	45	)	)	PUNCT
cana-5362	294	46	related	relate	VERB
cana-5362	294	47	to	to	ADP
cana-5362	294	48	sde	sde	PROPN
cana-5362	294	49	of	of	ADP
cana-5362	294	50	barbu	barbu	PROPN
cana-5362	295	1	[	[	X
cana-5362	295	2	1	1	NUM
cana-5362	295	3	]	]	PUNCT
cana-5362	295	4	.	.	PUNCT
cana-5362	296	1	•	•	NUM
cana-5362	296	2	sde	sde	PROPN
cana-5362	296	3	barbu	barbu	PROPN
cana-5362	296	4	:	:	PUNCT
cana-5362	296	5	the	the	DET
cana-5362	296	6	model	model	NOUN
cana-5362	296	7	introduced	introduce	VERB
cana-5362	296	8	by	by	ADP
cana-5362	296	9	barbu	barbu	PROPN
cana-5362	296	10	in	in	ADP
cana-5362	296	11	2016	2016	NUM
cana-5362	296	12	[	[	X
cana-5362	296	13	17	17	NUM
cana-5362	296	14	]	]	PUNCT
cana-5362	296	15	.	.	PUNCT
cana-5362	297	1	•	•	NUM
cana-5362	297	2	pm	pm	NOUN
cana-5362	297	3	1	1	NUM
cana-5362	297	4	and	and	CCONJ
cana-5362	297	5	pm	pm	VERB
cana-5362	297	6	2	2	NUM
cana-5362	297	7	:	:	PUNCT
cana-5362	297	8	the	the	DET
cana-5362	297	9	model	model	NOUN
cana-5362	297	10	of	of	ADP
cana-5362	297	11	pm	pm	NOUN
cana-5362	297	12	[	[	X
cana-5362	297	13	17	17	NUM
cana-5362	297	14	]	]	PUNCT
cana-5362	297	15	with	with	ADP
cana-5362	297	16	their	their	PRON
cana-5362	297	17	decreasing	decrease	VERB
cana-5362	297	18	functions	function	NOUN
cana-5362	297	19	(	(	PUNCT
cana-5362	297	20	2	2	NUM
cana-5362	297	21	)	)	PUNCT
cana-5362	297	22	(	(	PUNCT
cana-5362	297	23	fractional	fractional	ADJ
cana-5362	297	24	and	and	CCONJ
cana-5362	297	25	exponential	exponential	NOUN
cana-5362	297	26	respectively	respectively	ADV
cana-5362	297	27	)	)	PUNCT
cana-5362	297	28	.	.	PUNCT
cana-5362	298	1	•	•	NUM
cana-5362	298	2	sde	sde	PROPN
cana-5362	298	3	borkowski	borkowski	PROPN
cana-5362	298	4	:	:	PUNCT
cana-5362	298	5	borkowski	borkowski	PROPN
cana-5362	298	6	’s	’s	PART
cana-5362	298	7	model	model	NOUN
cana-5362	298	8	,	,	PUNCT
cana-5362	298	9	introduced	introduce	VERB
cana-5362	298	10	in	in	ADP
cana-5362	298	11	2013	2013	NUM
cana-5362	298	12	[	[	X
cana-5362	298	13	6	6	NUM
cana-5362	298	14	]	]	PUNCT
cana-5362	298	15	.	.	PUNCT
cana-5362	299	1	•	•	NUM
cana-5362	299	2	spde	spde	NOUN
cana-5362	299	3	1	1	NUM
cana-5362	299	4	and	and	CCONJ
cana-5362	299	5	spde	spde	ADJ
cana-5362	299	6	2	2	NUM
cana-5362	299	7	:	:	PUNCT
cana-5362	299	8	the	the	DET
cana-5362	299	9	used	use	VERB
cana-5362	299	10	of	of	ADP
cana-5362	299	11	exponential	exponential	ADJ
cana-5362	299	12	and	and	CCONJ
cana-5362	299	13	fractional	fractional	ADJ
cana-5362	299	14	functions	function	NOUN
cana-5362	299	15	(	(	PUNCT
cana-5362	299	16	2	2	NUM
cana-5362	299	17	)	)	PUNCT
cana-5362	299	18	respectively	respectively	ADV
cana-5362	299	19	.	.	PUNCT
cana-5362	300	1	communications	communication	NOUN
cana-5362	300	2	on	on	ADP
cana-5362	300	3	applied	apply	VERB
cana-5362	300	4	nonlinear	nonlinear	ADJ
cana-5362	300	5	analysis	analysis	NOUN
cana-5362	300	6	issn	issn	NOUN
cana-5362	300	7	:	:	PUNCT
cana-5362	300	8	1074	1074	NUM
cana-5362	300	9	-	-	PUNCT
cana-5362	300	10	133x	133x	NUM
cana-5362	300	11	vol	vol	VERB
cana-5362	300	12	32	32	NUM
cana-5362	300	13	no	no	NOUN
cana-5362	300	14	.	.	PUNCT
cana-5362	301	1	10s	10	NOUN
cana-5362	301	2	(	(	PUNCT
cana-5362	301	3	2025	2025	NUM
cana-5362	301	4	)	)	PUNCT
cana-5362	301	5	2000	2000	NUM
cana-5362	301	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5362	301	7	figure	figure	NOUN
cana-5362	301	8	1	1	NUM
cana-5362	301	9	.	.	PUNCT
cana-5362	301	10	restored	restore	VERB
cana-5362	301	11	cameraman	cameraman	ADJ
cana-5362	301	12	image	image	NOUN
cana-5362	301	13	by	by	ADP
cana-5362	301	14	using	use	VERB
cana-5362	301	15	different	different	ADJ
cana-5362	301	16	approaches	approach	NOUN
cana-5362	301	17	after	after	ADP
cana-5362	301	18	additive	additive	ADJ
cana-5362	301	19	gaussian	gaussian	ADJ
cana-5362	301	20	noise	noise	NOUN
cana-5362	301	21	with	with	ADP
cana-5362	301	22	∆𝒕	∆𝒕	PROPN
cana-5362	301	23	=	=	SYM
cana-5362	301	24	𝑻	𝑻	PROPN
cana-5362	301	25	𝑵	𝑵	PROPN
cana-5362	301	26	,	,	PUNCT
cana-5362	301	27	n=100	n=100	PROPN
cana-5362	301	28	,	,	PUNCT
cana-5362	301	29	t=1	t=1	ADV
cana-5362	301	30	,	,	PUNCT
cana-5362	301	31	𝜸	𝜸	X
cana-5362	301	32	=	=	X
cana-5362	301	33	𝟎.	𝟎.	NOUN
cana-5362	301	34	𝟏	𝟏	NUM
cana-5362	301	35	,	,	PUNCT
cana-5362	301	36	𝝈	𝝈	X
cana-5362	301	37	=	=	PUNCT
cana-5362	301	38	𝟎.	𝟎.	NOUN
cana-5362	301	39	𝟗.	𝟗.	NOUN
cana-5362	301	40	communications	communication	NOUN
cana-5362	301	41	on	on	ADP
cana-5362	301	42	applied	apply	VERB
cana-5362	301	43	nonlinear	nonlinear	ADJ
cana-5362	301	44	analysis	analysis	NOUN
cana-5362	301	45	issn	issn	NOUN
cana-5362	301	46	:	:	PUNCT
cana-5362	301	47	1074	1074	NUM
cana-5362	301	48	-	-	PUNCT
cana-5362	301	49	133x	133x	NUM
cana-5362	301	50	vol	vol	VERB
cana-5362	301	51	32	32	NUM
cana-5362	301	52	no	no	NOUN
cana-5362	301	53	.	.	PUNCT
cana-5362	302	1	10s	10	NOUN
cana-5362	302	2	(	(	PUNCT
cana-5362	302	3	2025	2025	NUM
cana-5362	302	4	)	)	PUNCT
cana-5362	302	5	2001	2001	NUM
cana-5362	302	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5362	302	7	figure	figure	NOUN
cana-5362	302	8	2	2	NUM
cana-5362	302	9	.	.	PUNCT
cana-5362	302	10	restored	restore	VERB
cana-5362	302	11	image	image	NOUN
cana-5362	302	12	results	result	NOUN
cana-5362	302	13	after	after	ADP
cana-5362	302	14	’	'	PUNCT
cana-5362	302	15	salt	salt	NOUN
cana-5362	302	16	&	&	CCONJ
cana-5362	302	17	pepper	pepper	NOUN
cana-5362	302	18	’	'	PUNCT
cana-5362	302	19	noise	noise	NOUN
cana-5362	302	20	application	application	NOUN
cana-5362	302	21	with	with	ADP
cana-5362	302	22	n	n	NOUN
cana-5362	302	23	=	=	SYM
cana-5362	302	24	5	5	NUM
cana-5362	302	25	,	,	PUNCT
cana-5362	302	26	t	t	NOUN
cana-5362	302	27	=	=	SYM
cana-5362	302	28	1	1	NUM
cana-5362	302	29	,	,	PUNCT
cana-5362	302	30	γ	γ	X
cana-5362	302	31	=	=	SYM
cana-5362	302	32	0.1	0.1	NUM
cana-5362	302	33	and	and	CCONJ
cana-5362	302	34	σ	σ	NUM
cana-5362	302	35	=	=	NOUN
cana-5362	302	36	0.45	0.45	NUM
cana-5362	302	37	communications	communication	NOUN
cana-5362	302	38	on	on	ADP
cana-5362	302	39	applied	apply	VERB
cana-5362	302	40	nonlinear	nonlinear	ADJ
cana-5362	302	41	analysis	analysis	NOUN
cana-5362	302	42	issn	issn	NOUN
cana-5362	302	43	:	:	PUNCT
cana-5362	302	44	1074	1074	NUM
cana-5362	302	45	-	-	PUNCT
cana-5362	302	46	133x	133x	NUM
cana-5362	302	47	vol	vol	VERB
cana-5362	302	48	32	32	NUM
cana-5362	302	49	no	no	NOUN
cana-5362	302	50	.	.	PUNCT
cana-5362	303	1	10s	10	NOUN
cana-5362	303	2	(	(	PUNCT
cana-5362	303	3	2025	2025	NUM
cana-5362	303	4	)	)	PUNCT
cana-5362	303	5	2002	2002	NUM
cana-5362	304	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5362	304	2	comments	comment	NOUN
cana-5362	304	3	on	on	ADP
cana-5362	304	4	numerical	numerical	ADJ
cana-5362	304	5	results	result	NOUN
cana-5362	304	6	the	the	DET
cana-5362	304	7	numerical	numerical	ADJ
cana-5362	304	8	results	result	NOUN
cana-5362	304	9	are	be	AUX
cana-5362	304	10	resumed	resume	VERB
cana-5362	304	11	in	in	ADP
cana-5362	304	12	the	the	DET
cana-5362	304	13	tables	table	NOUN
cana-5362	304	14	1,2,3	1,2,3	NUM
cana-5362	304	15	and	and	CCONJ
cana-5362	304	16	4	4	NUM
cana-5362	304	17	as	as	ADV
cana-5362	304	18	well	well	ADV
cana-5362	304	19	as	as	ADP
cana-5362	304	20	the	the	DET
cana-5362	304	21	figures	figure	NOUN
cana-5362	304	22	1	1	NUM
cana-5362	304	23	and	and	CCONJ
cana-5362	304	24	2	2	NUM
cana-5362	304	25	.	.	PUNCT
cana-5362	304	26	by	by	ADP
cana-5362	304	27	closely	closely	ADV
cana-5362	304	28	observing	observe	VERB
cana-5362	304	29	tables	table	NOUN
cana-5362	304	30	1,2	1,2	NUM
cana-5362	304	31	,	,	PUNCT
cana-5362	304	32	3	3	NUM
cana-5362	304	33	,	,	PUNCT
cana-5362	304	34	and	and	CCONJ
cana-5362	304	35	4	4	NUM
cana-5362	304	36	and	and	CCONJ
cana-5362	304	37	figures	figure	NOUN
cana-5362	304	38	1	1	NUM
cana-5362	304	39	&	&	CCONJ
cana-5362	304	40	2	2	NUM
cana-5362	304	41	,	,	PUNCT
cana-5362	304	42	we	we	PRON
cana-5362	304	43	notice	notice	VERB
cana-5362	304	44	that	that	SCONJ
cana-5362	304	45			PRON
cana-5362	304	46	the	the	DET
cana-5362	304	47	restoration	restoration	NOUN
cana-5362	304	48	performance	performance	NOUN
cana-5362	304	49	under	under	ADP
cana-5362	304	50	gaussian	gaussian	ADJ
cana-5362	304	51	noise	noise	NOUN
cana-5362	304	52	varies	vary	VERB
cana-5362	304	53	according	accord	VERB
cana-5362	304	54	the	the	DET
cana-5362	304	55	parameters	parameter	NOUN
cana-5362	304	56	choices	choice	NOUN
cana-5362	304	57	.	.	PUNCT
cana-5362	305	1	as	as	SCONJ
cana-5362	305	2	shown	show	VERB
cana-5362	305	3	in	in	ADP
cana-5362	305	4	tables	table	NOUN
cana-5362	305	5	1	1	NUM
cana-5362	305	6	&	&	CCONJ
cana-5362	305	7	2	2	NUM
cana-5362	305	8	,	,	PUNCT
cana-5362	305	9	barbu	barbu	PROPN
cana-5362	305	10	and	and	CCONJ
cana-5362	305	11	its	its	PRON
cana-5362	305	12	related	related	ADJ
cana-5362	305	13	pde	pde	NOUN
cana-5362	305	14	models	model	NOUN
cana-5362	305	15	(	(	PUNCT
cana-5362	305	16	kolmogorov	kolmogorov	PROPN
cana-5362	305	17	’s	’s	PART
cana-5362	305	18	pde	pde	NOUN
cana-5362	305	19	)	)	PUNCT
cana-5362	306	1	[	[	X
cana-5362	306	2	1	1	X
cana-5362	306	3	]	]	PUNCT
cana-5362	306	4	exhibit	exhibit	VERB
cana-5362	306	5	less	less	ADV
cana-5362	306	6	performant	performant	ADJ
cana-5362	306	7	compare	compare	VERB
cana-5362	306	8	to	to	ADP
cana-5362	306	9	pm1	pm1	NUM
cana-5362	306	10	,	,	PUNCT
cana-5362	306	11	pm2	pm2	PROPN
cana-5362	307	1	[	[	X
cana-5362	307	2	17	17	NUM
cana-5362	307	3	]	]	PUNCT
cana-5362	307	4	and	and	CCONJ
cana-5362	307	5	sde	sde	PROPN
cana-5362	307	6	borkowski	borkowski	PROPN
cana-5362	307	7	models	model	NOUN
cana-5362	307	8	[	[	X
cana-5362	307	9	5	5	NUM
cana-5362	307	10	]	]	PUNCT
cana-5362	307	11	,	,	PUNCT
cana-5362	307	12	as	as	SCONJ
cana-5362	307	13	reflected	reflect	VERB
cana-5362	307	14	in	in	ADP
cana-5362	307	15	their	their	PRON
cana-5362	307	16	psnr	psnr	NOUN
cana-5362	307	17	and	and	CCONJ
cana-5362	307	18	ssim	ssim	NOUN
cana-5362	307	19	values	value	NOUN
cana-5362	307	20	.	.	PUNCT
cana-5362	308	1	specifically	specifically	ADV
cana-5362	308	2	,	,	PUNCT
cana-5362	308	3	when	when	SCONJ
cana-5362	308	4	γ	γ	X
cana-5362	308	5	=	=	SYM
cana-5362	308	6	0.1	0.1	NUM
cana-5362	308	7	,	,	PUNCT
cana-5362	308	8	barbu	barbu	PROPN
cana-5362	308	9	’s	’s	PROPN
cana-5362	308	10	sde	sde	PROPN
cana-5362	308	11	model	model	NOUN
cana-5362	308	12	[	[	X
cana-5362	308	13	1	1	NUM
cana-5362	308	14	]	]	PUNCT
cana-5362	308	15	achieved	achieve	VERB
cana-5362	308	16	a	a	DET
cana-5362	308	17	psnr	psnr	NOUN
cana-5362	308	18	of	of	ADP
cana-5362	308	19	20.1880	20.1880	NUM
cana-5362	308	20	db	db	PROPN
cana-5362	308	21	and	and	CCONJ
cana-5362	308	22	an	an	DET
cana-5362	308	23	ssim	ssim	NOUN
cana-5362	308	24	of	of	ADP
cana-5362	308	25	0.5391	0.5391	NUM
cana-5362	308	26	,	,	PUNCT
cana-5362	308	27	which	which	PRON
cana-5362	308	28	is	be	AUX
cana-5362	308	29	lower	low	ADJ
cana-5362	308	30	than	than	ADP
cana-5362	308	31	pm1	pm1	NOUN
cana-5362	308	32	(	(	PUNCT
cana-5362	308	33	where	where	SCONJ
cana-5362	308	34	psnr=24.1495	psnr=24.1495	PROPN
cana-5362	308	35	db	db	X
cana-5362	308	36	and	and	CCONJ
cana-5362	308	37	ssim=0.6769	ssim=0.6769	NOUN
cana-5362	308	38	)	)	PUNCT
cana-5362	308	39	and	and	CCONJ
cana-5362	308	40	pm2	pm2	NOUN
cana-5362	308	41	(	(	PUNCT
cana-5362	308	42	with	with	ADP
cana-5362	308	43	psnr=24.1601	psnr=24.1601	DET
cana-5362	308	44	db	db	PROPN
cana-5362	308	45	and	and	CCONJ
cana-5362	308	46	ssim=0.6766	ssim=0.6766	NOUN
cana-5362	308	47	)	)	PUNCT
cana-5362	308	48	.	.	PUNCT
cana-5362	309	1	this	this	DET
cana-5362	309	2	discrepancy	discrepancy	NOUN
cana-5362	309	3	underscores	underscore	VERB
cana-5362	309	4	the	the	DET
cana-5362	309	5	important	important	ADJ
cana-5362	309	6	role	role	NOUN
cana-5362	309	7	of	of	ADP
cana-5362	309	8	the	the	DET
cana-5362	309	9	diffusion	diffusion	NOUN
cana-5362	309	10	in	in	ADP
cana-5362	309	11	image	image	NOUN
cana-5362	309	12	restoration	restoration	NOUN
cana-5362	309	13	,	,	PUNCT
cana-5362	309	14	that	that	PRON
cana-5362	309	15	has	have	AUX
cana-5362	309	16	been	be	AUX
cana-5362	309	17	neglected	neglect	VERB
cana-5362	309	18	by	by	ADP
cana-5362	309	19	barbu	barbu	PROPN
cana-5362	309	20	et	et	PROPN
cana-5362	309	21	al	al	PROPN
cana-5362	309	22	who	who	PRON
cana-5362	309	23	relied	rely	VERB
cana-5362	309	24	solely	solely	ADV
cana-5362	309	25	on	on	ADP
cana-5362	309	26	the	the	DET
cana-5362	309	27	drift	drift	NOUN
cana-5362	309	28	term	term	NOUN
cana-5362	309	29	in	in	ADP
cana-5362	309	30	their	their	PRON
cana-5362	309	31	model	model	NOUN
cana-5362	309	32	.	.	PUNCT
cana-5362	310	1	in	in	ADP
cana-5362	310	2	contrast	contrast	NOUN
cana-5362	310	3	,	,	PUNCT
cana-5362	310	4	our	our	PRON
cana-5362	310	5	proposed	propose	VERB
cana-5362	310	6	model	model	NOUN
cana-5362	310	7	,	,	PUNCT
cana-5362	310	8	which	which	PRON
cana-5362	310	9	integrates	integrate	VERB
cana-5362	310	10	(	(	PUNCT
cana-5362	310	11	combines	combine	NOUN
cana-5362	310	12	)	)	PUNCT
cana-5362	310	13	both	both	DET
cana-5362	310	14	sdes	sde	NOUN
cana-5362	310	15	and	and	CCONJ
cana-5362	310	16	pdes	pde	NOUN
cana-5362	310	17	,	,	PUNCT
cana-5362	310	18	achieves	achieve	VERB
cana-5362	310	19	better	well	ADJ
cana-5362	310	20	qualitative	qualitative	ADJ
cana-5362	310	21	image	image	NOUN
cana-5362	310	22	restoration	restoration	NOUN
cana-5362	310	23	results	result	NOUN
cana-5362	310	24	,	,	PUNCT
cana-5362	310	25	surpassing	surpass	VERB
cana-5362	310	26	models	model	NOUN
cana-5362	310	27	that	that	PRON
cana-5362	310	28	employ	employ	VERB
cana-5362	310	29	either	either	CCONJ
cana-5362	310	30	sdes	sde	NOUN
cana-5362	310	31	or	or	CCONJ
cana-5362	310	32	pdes	pde	NOUN
cana-5362	310	33	.	.	PUNCT
cana-5362	311	1	as	as	SCONJ
cana-5362	311	2	indicated	indicate	VERB
cana-5362	311	3	in	in	ADP
cana-5362	311	4	tables	table	NOUN
cana-5362	311	5	1	1	NUM
cana-5362	311	6	and	and	CCONJ
cana-5362	311	7	2	2	NUM
cana-5362	311	8	for	for	ADP
cana-5362	311	9	our	our	PRON
cana-5362	311	10	approach	approach	NOUN
cana-5362	311	11	,	,	PUNCT
cana-5362	311	12	we	we	PRON
cana-5362	311	13	can	can	AUX
cana-5362	311	14	confirm	confirm	VERB
cana-5362	311	15	its	its	PRON
cana-5362	311	16	effectiveness	effectiveness	NOUN
cana-5362	311	17	and	and	CCONJ
cana-5362	311	18	further	far	ADV
cana-5362	311	19	highlight	highlight	VERB
cana-5362	311	20	the	the	DET
cana-5362	311	21	advantage	advantage	NOUN
cana-5362	311	22	of	of	ADP
cana-5362	311	23	spdes	spde	NOUN
cana-5362	311	24	in	in	ADP
cana-5362	311	25	enhancing	enhance	VERB
cana-5362	311	26	image	image	NOUN
cana-5362	311	27	restoration	restoration	NOUN
cana-5362	311	28	.	.	PUNCT
cana-5362	312	1	•	•	ADP
cana-5362	312	2	our	our	PRON
cana-5362	312	3	proposed	propose	VERB
cana-5362	312	4	model	model	NOUN
cana-5362	312	5	incorporates	incorporate	VERB
cana-5362	312	6	regularisation	regularisation	NOUN
cana-5362	312	7	through	through	ADP
cana-5362	312	8	the	the	DET
cana-5362	312	9	convolution	convolution	NOUN
cana-5362	312	10	of	of	ADP
cana-5362	312	11	the	the	DET
cana-5362	312	12	functions	function	NOUN
cana-5362	312	13	𝑔	𝑔	NOUN
cana-5362	312	14	and	and	CCONJ
cana-5362	312	15	𝐺𝜎	𝐺𝜎	PROPN
cana-5362	312	16	,	,	PUNCT
cana-5362	312	17	which	which	PRON
cana-5362	312	18	plays	play	VERB
cana-5362	312	19	a	a	DET
cana-5362	312	20	crucial	crucial	ADJ
cana-5362	312	21	role	role	NOUN
cana-5362	312	22	in	in	ADP
cana-5362	312	23	enhancing	enhance	VERB
cana-5362	312	24	image	image	NOUN
cana-5362	312	25	quality	quality	NOUN
cana-5362	312	26	while	while	SCONJ
cana-5362	312	27	preserving	preserve	VERB
cana-5362	312	28	image	image	NOUN
cana-5362	312	29	details	detail	NOUN
cana-5362	312	30	.	.	PUNCT
cana-5362	313	1	the	the	DET
cana-5362	313	2	precise	precise	ADJ
cana-5362	313	3	selection	selection	NOUN
cana-5362	313	4	of	of	ADP
cana-5362	313	5	the	the	DET
cana-5362	313	6	parameter	parameter	NOUN
cana-5362	313	7	σ	σ	PROPN
cana-5362	313	8	is	be	AUX
cana-5362	313	9	essential	essential	ADJ
cana-5362	313	10	for	for	ADP
cana-5362	313	11	obtaining	obtain	VERB
cana-5362	313	12	optimal	optimal	ADJ
cana-5362	313	13	restoration	restoration	NOUN
cana-5362	313	14	results	result	NOUN
cana-5362	313	15	,	,	PUNCT
cana-5362	313	16	as	as	SCONJ
cana-5362	313	17	demonstrated	demonstrate	VERB
cana-5362	313	18	by	by	ADP
cana-5362	313	19	the	the	DET
cana-5362	313	20	qualitative	qualitative	ADJ
cana-5362	313	21	evaluations	evaluation	NOUN
cana-5362	313	22	in	in	ADP
cana-5362	313	23	table	table	NOUN
cana-5362	313	24	3	3	NUM
cana-5362	313	25	.	.	PUNCT
cana-5362	313	26	specifically	specifically	ADV
cana-5362	313	27	,	,	PUNCT
cana-5362	313	28	selecting	select	VERB
cana-5362	313	29	σ	σ	X
cana-5362	313	30	=	=	SYM
cana-5362	313	31	0.9	0.9	NUM
cana-5362	313	32	with	with	ADP
cana-5362	313	33	γ	γ	X
cana-5362	313	34	=	=	SYM
cana-5362	313	35	0.1	0.1	NUM
cana-5362	313	36	leads	lead	VERB
cana-5362	313	37	to	to	ADP
cana-5362	313	38	an	an	DET
cana-5362	313	39	increase	increase	NOUN
cana-5362	313	40	in	in	ADP
cana-5362	313	41	psnr	psnr	NOUN
cana-5362	313	42	to	to	ADP
cana-5362	313	43	25.2683	25.2683	NUM
cana-5362	313	44	db	db	NOUN
cana-5362	313	45	and	and	CCONJ
cana-5362	313	46	ssim	ssim	VERB
cana-5362	313	47	to	to	ADP
cana-5362	313	48	0.7466	0.7466	NUM
cana-5362	313	49	,	,	PUNCT
cana-5362	313	50	and	and	CCONJ
cana-5362	313	51	σ	σ	NUM
cana-5362	313	52	=	=	NOUN
cana-5362	313	53	0.45	0.45	NUM
cana-5362	313	54	with	with	ADP
cana-5362	313	55	γ	γ	X
cana-5362	313	56	=	=	NOUN
cana-5362	313	57	0.01	0.01	NUM
cana-5362	313	58	to	to	ADP
cana-5362	313	59	higher	high	ADJ
cana-5362	313	60	psnr=33.5837	psnr=33.5837	NOUN
cana-5362	313	61	db	db	PROPN
cana-5362	313	62	and	and	CCONJ
cana-5362	313	63	ssim=0.8915	ssim=0.8915	PRON
cana-5362	313	64	,	,	PUNCT
cana-5362	313	65	reinforcing	reinforce	VERB
cana-5362	313	66	the	the	DET
cana-5362	313	67	importance	importance	NOUN
cana-5362	313	68	of	of	ADP
cana-5362	313	69	careful	careful	ADJ
cana-5362	313	70	parameter	parameter	NOUN
cana-5362	313	71	choices	choice	NOUN
cana-5362	313	72	in	in	ADP
cana-5362	313	73	order	order	NOUN
cana-5362	313	74	to	to	PART
cana-5362	313	75	achieve	achieve	VERB
cana-5362	313	76	a	a	DET
cana-5362	313	77	better	well	ADJ
cana-5362	313	78	denoising	denoising	NOUN
cana-5362	313	79	performance	performance	NOUN
cana-5362	313	80	.	.	PUNCT
cana-5362	314	1	•	•	NUM
cana-5362	314	2	for	for	ADP
cana-5362	314	3	salt	salt	NOUN
cana-5362	314	4	-	-	PUNCT
cana-5362	314	5	and	and	CCONJ
cana-5362	314	6	-	-	PUNCT
cana-5362	314	7	pepper	pepper	NOUN
cana-5362	314	8	noise	noise	NOUN
cana-5362	314	9	,	,	PUNCT
cana-5362	314	10	all	all	DET
cana-5362	314	11	models	model	NOUN
cana-5362	314	12	give	give	VERB
cana-5362	314	13	good	good	ADJ
cana-5362	314	14	performance	performance	NOUN
cana-5362	314	15	improvements	improvement	NOUN
cana-5362	314	16	,	,	PUNCT
cana-5362	314	17	achieving	achieve	VERB
cana-5362	314	18	higher	high	ADJ
cana-5362	314	19	psnr	psnr	NOUN
cana-5362	314	20	and	and	CCONJ
cana-5362	314	21	ssim	ssim	NOUN
cana-5362	314	22	values	value	NOUN
cana-5362	314	23	.	.	PUNCT
cana-5362	315	1	however	however	ADV
cana-5362	315	2	,	,	PUNCT
cana-5362	315	3	while	while	SCONJ
cana-5362	315	4	barbu	barbu	PROPN
cana-5362	315	5	sde	sde	PROPN
cana-5362	315	6	model	model	NOUN
cana-5362	315	7	perform	perform	VERB
cana-5362	315	8	significantly	significantly	ADV
cana-5362	315	9	better	well	ADV
cana-5362	315	10	with	with	ADP
cana-5362	315	11	salt	salt	NOUN
cana-5362	315	12	&	&	CCONJ
cana-5362	315	13	pepper	pepper	NOUN
cana-5362	315	14	noise	noise	NOUN
cana-5362	315	15	than	than	ADP
cana-5362	315	16	the	the	DET
cana-5362	315	17	white	white	PROPN
cana-5362	315	18	gaussian	gaussian	PROPN
cana-5362	315	19	noise	noise	NOUN
cana-5362	315	20	,	,	PUNCT
cana-5362	315	21	more	more	ADV
cana-5362	315	22	competitive	competitive	ADJ
cana-5362	315	23	results	result	NOUN
cana-5362	315	24	are	be	AUX
cana-5362	315	25	obtained	obtain	VERB
cana-5362	315	26	by	by	ADP
cana-5362	315	27	our	our	PRON
cana-5362	315	28	proposed	propose	VERB
cana-5362	315	29	approach	approach	NOUN
cana-5362	315	30	,	,	PUNCT
cana-5362	315	31	as	as	SCONJ
cana-5362	315	32	shown	show	VERB
cana-5362	315	33	in	in	ADP
cana-5362	315	34	table	table	NOUN
cana-5362	315	35	4	4	NUM
cana-5362	315	36	for	for	ADP
cana-5362	315	37	the	the	DET
cana-5362	315	38	psnr	psnr	NOUN
cana-5362	315	39	and	and	CCONJ
cana-5362	315	40	ssim	ssim	NOUN
cana-5362	315	41	values	value	NOUN
cana-5362	315	42	,	,	PUNCT
cana-5362	315	43	and	and	CCONJ
cana-5362	315	44	visually	visually	ADV
cana-5362	315	45	in	in	ADP
cana-5362	315	46	figure	figure	NOUN
cana-5362	315	47	2	2	NUM
cana-5362	315	48	.	.	NOUN
cana-5362	315	49	5	5	NUM
cana-5362	315	50	.	.	X
cana-5362	315	51	conclusion	conclusion	NOUN
cana-5362	315	52	in	in	ADP
cana-5362	315	53	this	this	DET
cana-5362	315	54	work	work	NOUN
cana-5362	315	55	,	,	PUNCT
cana-5362	315	56	we	we	PRON
cana-5362	315	57	introduced	introduce	VERB
cana-5362	315	58	an	an	DET
cana-5362	315	59	image	image	NOUN
cana-5362	315	60	restoration	restoration	NOUN
cana-5362	315	61	technique	technique	NOUN
cana-5362	315	62	based	base	VERB
cana-5362	315	63	on	on	ADP
cana-5362	315	64	spdes	spde	NOUN
cana-5362	315	65	,	,	PUNCT
cana-5362	315	66	integrating	integrate	VERB
cana-5362	315	67	the	the	DET
cana-5362	315	68	pm	pm	NOUN
cana-5362	315	69	equation	equation	NOUN
cana-5362	315	70	with	with	ADP
cana-5362	315	71	stochastic	stochastic	ADJ
cana-5362	315	72	perturbations	perturbation	NOUN
cana-5362	315	73	to	to	PART
cana-5362	315	74	enhance	enhance	VERB
cana-5362	315	75	noise	noise	NOUN
cana-5362	315	76	removal	removal	NOUN
cana-5362	315	77	while	while	SCONJ
cana-5362	315	78	preserving	preserve	VERB
cana-5362	315	79	image	image	NOUN
cana-5362	315	80	details	detail	NOUN
cana-5362	315	81	.	.	PUNCT
cana-5362	316	1	the	the	DET
cana-5362	316	2	stochastic	stochastic	ADJ
cana-5362	316	3	component	component	NOUN
cana-5362	316	4	introduces	introduce	NOUN
cana-5362	316	5	controlled	control	VERB
cana-5362	316	6	randomness	randomness	NOUN
cana-5362	316	7	,	,	PUNCT
cana-5362	316	8	preventing	prevent	VERB
cana-5362	316	9	excessive	excessive	ADJ
cana-5362	316	10	smoothing	smoothing	NOUN
cana-5362	316	11	while	while	SCONJ
cana-5362	316	12	maintaining	maintain	VERB
cana-5362	316	13	important	important	ADJ
cana-5362	316	14	image	image	NOUN
cana-5362	316	15	structures	structure	NOUN
cana-5362	316	16	.	.	PUNCT
cana-5362	317	1	this	this	DET
cana-5362	317	2	approach	approach	NOUN
cana-5362	317	3	allows	allow	VERB
cana-5362	317	4	adaptive	adaptive	ADJ
cana-5362	317	5	noise	noise	NOUN
cana-5362	317	6	reduction	reduction	NOUN
cana-5362	317	7	,	,	PUNCT
cana-5362	317	8	making	make	VERB
cana-5362	317	9	the	the	DET
cana-5362	317	10	restoration	restoration	NOUN
cana-5362	317	11	process	process	NOUN
cana-5362	317	12	more	more	ADV
cana-5362	317	13	resilient	resilient	ADJ
cana-5362	317	14	to	to	ADP
cana-5362	317	15	varying	vary	VERB
cana-5362	317	16	noise	noise	NOUN
cana-5362	317	17	levels	level	NOUN
cana-5362	317	18	.	.	PUNCT
cana-5362	318	1	through	through	ADP
cana-5362	318	2	a	a	DET
cana-5362	318	3	detailed	detailed	ADJ
cana-5362	318	4	mathematical	mathematical	ADJ
cana-5362	318	5	analysis	analysis	NOUN
cana-5362	318	6	,	,	PUNCT
cana-5362	318	7	we	we	PRON
cana-5362	318	8	established	establish	VERB
cana-5362	318	9	the	the	DET
cana-5362	318	10	theoretical	theoretical	ADJ
cana-5362	318	11	and	and	CCONJ
cana-5362	318	12	numerical	numerical	ADJ
cana-5362	318	13	stability	stability	NOUN
cana-5362	318	14	of	of	ADP
cana-5362	318	15	the	the	DET
cana-5362	318	16	proposed	propose	VERB
cana-5362	318	17	model	model	NOUN
cana-5362	318	18	,	,	PUNCT
cana-5362	318	19	ensuring	ensure	VERB
cana-5362	318	20	its	its	PRON
cana-5362	318	21	robustness	robustness	NOUN
cana-5362	318	22	under	under	ADP
cana-5362	318	23	different	different	ADJ
cana-5362	318	24	conditions	condition	NOUN
cana-5362	318	25	.	.	PUNCT
cana-5362	319	1	numerical	numerical	PROPN
cana-5362	319	2	experimentations	experimentation	NOUN
cana-5362	319	3	confirm	confirm	VERB
cana-5362	319	4	its	its	PRON
cana-5362	319	5	effectiveness	effectiveness	NOUN
cana-5362	319	6	,	,	PUNCT
cana-5362	319	7	according	accord	VERB
cana-5362	319	8	to	to	ADP
cana-5362	319	9	the	the	DET
cana-5362	319	10	obtained	obtain	VERB
cana-5362	319	11	results	result	NOUN
cana-5362	319	12	for	for	ADP
cana-5362	319	13	the	the	DET
cana-5362	319	14	psnr	psnr	NOUN
cana-5362	319	15	and	and	CCONJ
cana-5362	319	16	ssim	ssim	NOUN
cana-5362	319	17	values	value	NOUN
cana-5362	319	18	,	,	PUNCT
cana-5362	319	19	highlighting	highlight	VERB
cana-5362	319	20	the	the	DET
cana-5362	319	21	potential	potential	NOUN
cana-5362	319	22	of	of	ADP
cana-5362	319	23	spde	spde	NOUN
cana-5362	319	24	-	-	PUNCT
cana-5362	319	25	based	base	VERB
cana-5362	319	26	models	model	NOUN
cana-5362	319	27	for	for	ADP
cana-5362	319	28	image	image	NOUN
cana-5362	319	29	restoration	restoration	NOUN
cana-5362	319	30	.	.	PUNCT
cana-5362	320	1	communications	communication	NOUN
cana-5362	320	2	on	on	ADP
cana-5362	320	3	applied	apply	VERB
cana-5362	320	4	nonlinear	nonlinear	ADJ
cana-5362	320	5	analysis	analysis	NOUN
cana-5362	320	6	issn	issn	NOUN
cana-5362	320	7	:	:	PUNCT
cana-5362	320	8	1074	1074	NUM
cana-5362	320	9	-	-	PUNCT
cana-5362	320	10	133x	133x	NUM
cana-5362	320	11	vol	vol	VERB
cana-5362	320	12	32	32	NUM
cana-5362	320	13	no	no	NOUN
cana-5362	320	14	.	.	PUNCT
cana-5362	321	1	10s	10	NOUN
cana-5362	321	2	(	(	PUNCT
cana-5362	321	3	2025	2025	NUM
cana-5362	321	4	)	)	PUNCT
cana-5362	321	5	2003	2003	NUM
cana-5362	322	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-5362	322	2	references	reference	NOUN
cana-5362	322	3	[	[	X
cana-5362	322	4	1	1	NUM
cana-5362	322	5	]	]	PUNCT
cana-5362	322	6	t.	t.	PROPN
cana-5362	322	7	barbu	barbu	PROPN
cana-5362	322	8	,	,	PUNCT
cana-5362	322	9	et	et	PROPN
cana-5362	322	10	a.	a.	NOUN
cana-5362	322	11	favini	favini	PROPN
cana-5362	322	12	,	,	PUNCT
cana-5362	322	13	"	"	PUNCT
cana-5362	322	14	novel	novel	ADJ
cana-5362	322	15	stochastic	stochastic	ADJ
cana-5362	322	16	differential	differential	NOUN
cana-5362	322	17	model	model	NOUN
cana-5362	322	18	for	for	ADP
cana-5362	322	19	image	image	NOUN
cana-5362	322	20	restoration	restoration	NOUN
cana-5362	322	21	"	"	PUNCT
cana-5362	322	22	,	,	PUNCT
cana-5362	322	23	processings	processing	NOUN
cana-5362	322	24	of	of	ADP
cana-5362	322	25	the	the	DET
cana-5362	322	26	romanian	romanian	PROPN
cana-5362	322	27	academy	academy	PROPN
cana-5362	322	28	-	-	PUNCT
cana-5362	322	29	series	series	PROPN
cana-5362	322	30	a	a	DET
cana-5362	322	31	:	:	PUNCT
cana-5362	322	32	mathematics	mathematic	NOUN
cana-5362	322	33	,	,	PUNCT
cana-5362	322	34	physics	physics	NOUN
cana-5362	322	35	,	,	PUNCT
cana-5362	322	36	technical	technical	ADJ
cana-5362	322	37	sciences	science	NOUN
cana-5362	322	38	,	,	PUNCT
cana-5362	322	39	information	information	NOUN
cana-5362	322	40	science	science	NOUN
cana-5362	322	41	.	.	PUNCT
cana-5362	323	1	vol	vol	NOUN
cana-5362	323	2	.	.	PROPN
cana-5362	324	1	17	17	NUM
cana-5362	324	2	,	,	PUNCT
cana-5362	324	3	no	no	INTJ
cana-5362	324	4	.	.	NOUN
cana-5362	324	5	2	2	NUM
cana-5362	324	6	,	,	PUNCT
cana-5362	324	7	pp	pp	ADJ
cana-5362	324	8	.	.	PUNCT
cana-5362	325	1	109	109	NUM
cana-5362	325	2	-	-	SYM
cana-5362	325	3	116	116	NUM
cana-5362	325	4	,	,	PUNCT
cana-5362	325	5	2016	2016	NUM
cana-5362	325	6	.	.	PUNCT
cana-5362	326	1	[	[	X
cana-5362	326	2	2	2	NUM
cana-5362	326	3	]	]	PUNCT
cana-5362	326	4	m.	m.	NOUN
cana-5362	326	5	benseghir	benseghir	NOUN
cana-5362	326	6	and	and	CCONJ
cana-5362	326	7	f.z	f.z	PROPN
cana-5362	326	8	.	.	PROPN
cana-5362	326	9	nouri	nouri	PROPN
cana-5362	326	10	,	,	PUNCT
cana-5362	326	11	"	"	PUNCT
cana-5362	326	12	a	a	DET
cana-5362	326	13	study	study	NOUN
cana-5362	326	14	of	of	ADP
cana-5362	326	15	a	a	DET
cana-5362	326	16	stochastic	stochastic	ADJ
cana-5362	326	17	differential	differential	ADJ
cana-5362	326	18	equation	equation	NOUN
cana-5362	326	19	with	with	ADP
cana-5362	326	20	reflection	reflection	NOUN
cana-5362	326	21	for	for	ADP
cana-5362	326	22	image	image	NOUN
cana-5362	326	23	processing	processing	NOUN
cana-5362	326	24	"	"	PUNCT
cana-5362	326	25	,	,	PUNCT
cana-5362	326	26	journal	journal	NOUN
cana-5362	326	27	of	of	ADP
cana-5362	326	28	applied	apply	VERB
cana-5362	326	29	probability	probability	NOUN
cana-5362	326	30	and	and	CCONJ
cana-5362	326	31	statistics	statistic	NOUN
cana-5362	326	32	2022	2022	NUM
cana-5362	326	33	,	,	PUNCT
cana-5362	326	34	vol	vol	NOUN
cana-5362	326	35	.	.	PROPN
cana-5362	326	36	17	17	NUM
cana-5362	326	37	,	,	PUNCT
cana-5362	326	38	no	no	INTJ
cana-5362	326	39	.	.	NOUN
cana-5362	326	40	2	2	NUM
cana-5362	326	41	,	,	PUNCT
cana-5362	326	42	pp	pp	ADJ
cana-5362	326	43	.	.	PUNCT
cana-5362	327	1	035	035	NUM
cana-5362	327	2	-	-	PUNCT
cana-5362	327	3	046	046	NUM
cana-5362	327	4	.	.	PUNCT
cana-5362	328	1	[	[	X
cana-5362	328	2	3	3	X
cana-5362	328	3	]	]	X
cana-5362	328	4	m.	m.	NOUN
cana-5362	328	5	benseghir	benseghir	NOUN
cana-5362	328	6	,	,	PUNCT
cana-5362	328	7	f.-z	f.-z	PROPN
cana-5362	328	8	.	.	PUNCT
cana-5362	329	1	nouri	nouri	PROPN
cana-5362	329	2	,	,	PUNCT
cana-5362	329	3	and	and	CCONJ
cana-5362	329	4	p.-c	p.-c	NOUN
cana-5362	329	5	.	.	PUNCT
cana-5362	330	1	tauber	tauber	PROPN
cana-5362	330	2	,	,	PUNCT
cana-5362	330	3	"	"	PUNCT
cana-5362	330	4	a	a	DET
cana-5362	330	5	new	new	ADJ
cana-5362	330	6	partial	partial	ADJ
cana-5362	330	7	differential	differential	NOUN
cana-5362	330	8	equation	equation	NOUN
cana-5362	330	9	for	for	ADP
cana-5362	330	10	image	image	NOUN
cana-5362	330	11	inpainting	inpainte	VERB
cana-5362	330	12	"	"	PUNCT
cana-5362	330	13	,	,	PUNCT
cana-5362	330	14	bol	bol	NOUN
cana-5362	330	15	.	.	PUNCT
cana-5362	331	1	soc	soc	PROPN
cana-5362	331	2	.	.	PUNCT
cana-5362	332	1	paran	paran	PROPN
cana-5362	332	2	.	.	PUNCT
cana-5362	333	1	mat	mat	PROPN
cana-5362	333	2	.	.	PUNCT
cana-5362	334	1	(	(	PUNCT
cana-5362	334	2	3s	3s	NOUN
cana-5362	334	3	.	.	PUNCT
cana-5362	334	4	)	)	PUNCT
cana-5362	335	1	v.	v.	ADP
cana-5362	335	2	39	39	NUM
cana-5362	335	3	3	3	NUM
cana-5362	335	4	,	,	PUNCT
cana-5362	335	5	137–155	137–155	NUM
cana-5362	335	6	,	,	PUNCT
cana-5362	335	7	2021	2021	NUM
cana-5362	335	8	.	.	PUNCT
cana-5362	336	1	[	[	X
cana-5362	336	2	4	4	NUM
cana-5362	336	3	]	]	PUNCT
cana-5362	336	4	a.	a.	NOUN
cana-5362	336	5	bensoussan	bensoussan	PROPN
cana-5362	336	6	and	and	CCONJ
cana-5362	336	7	r.	r.	PROPN
cana-5362	336	8	temam	temam	NOUN
cana-5362	336	9	,	,	PUNCT
cana-5362	336	10	"	"	PUNCT
cana-5362	336	11	equations	equation	NOUN
cana-5362	336	12	aux	aux	PROPN
cana-5362	336	13	dérivées	dérivées	VERB
cana-5362	336	14	partielles	partielles	PROPN
cana-5362	336	15	stochastiques	stochastique	NOUN
cana-5362	336	16	non	non	PROPN
cana-5362	336	17	lineaires	lineaire	NOUN
cana-5362	336	18	(	(	PUNCT
cana-5362	336	19	1	1	NUM
cana-5362	336	20	)	)	PUNCT
cana-5362	336	21	"	"	PUNCT
cana-5362	336	22	,	,	PUNCT
cana-5362	336	23	1971	1971	NUM
cana-5362	336	24	.	.	PUNCT
cana-5362	337	1	[	[	X
cana-5362	337	2	5	5	X
cana-5362	337	3	]	]	PUNCT
cana-5362	337	4	d.	d.	PROPN
cana-5362	337	5	borkowski	borkowski	PROPN
cana-5362	337	6	,	,	PUNCT
cana-5362	337	7	and	and	CCONJ
cana-5362	337	8	k.	k.	PROPN
cana-5362	337	9	jańczak	jańczak	PROPN
cana-5362	337	10	-	-	PUNCT
cana-5362	337	11	borkowska	borkowska	NOUN
cana-5362	337	12	,	,	PUNCT
cana-5362	337	13	"	"	PUNCT
cana-5362	337	14	image	image	NOUN
cana-5362	337	15	denoising	denoising	NOUN
cana-5362	337	16	using	use	VERB
cana-5362	337	17	backward	backward	ADJ
cana-5362	337	18	stochastic	stochastic	ADJ
cana-5362	337	19	differential	differential	ADJ
cana-5362	337	20	equations	equation	NOUN
cana-5362	337	21	"	"	PUNCT
cana-5362	337	22	,	,	PUNCT
cana-5362	337	23	advances	advance	VERB
cana-5362	337	24	in	in	ADP
cana-5362	337	25	intelligent	intelligent	ADJ
cana-5362	337	26	systems	system	NOUN
cana-5362	337	27	and	and	CCONJ
cana-5362	337	28	computing	computing	NOUN
cana-5362	337	29	,	,	PUNCT
cana-5362	337	30	pp	pp	ADJ
cana-5362	337	31	.	.	PUNCT
cana-5362	338	1	185	185	NUM
cana-5362	338	2	-	-	SYM
cana-5362	338	3	194	194	NUM
cana-5362	338	4	,	,	PUNCT
cana-5362	338	5	2017	2017	NUM
cana-5362	338	6	.	.	PUNCT
cana-5362	339	1	[	[	X
cana-5362	339	2	6	6	NUM
cana-5362	339	3	]	]	PUNCT
cana-5362	339	4	d.	d.	PROPN
cana-5362	339	5	borkowski	borkowski	PROPN
cana-5362	339	6	and	and	CCONJ
cana-5362	339	7	k.	k.	PROPN
cana-5362	339	8	jańczak	jańczak	PROPN
cana-5362	339	9	-	-	PUNCT
cana-5362	339	10	borkowska	borkowska	PROPN
cana-5362	339	11	,	,	PUNCT
cana-5362	339	12	"	"	PUNCT
cana-5362	339	13	image	image	NOUN
cana-5362	339	14	restoration	restoration	NOUN
cana-5362	339	15	using	use	VERB
cana-5362	339	16	anisotopic	anisotopic	ADJ
cana-5362	339	17	stochastic	stochastic	ADJ
cana-5362	339	18	diffusion	diffusion	NOUN
cana-5362	339	19	collaborated	collaborate	VERB
cana-5362	339	20	with	with	ADP
cana-5362	339	21	non	non	PRON
cana-5362	339	22	local	local	ADJ
cana-5362	339	23	means	mean	NOUN
cana-5362	339	24	"	"	PUNCT
cana-5362	339	25	.	.	PUNCT
cana-5362	340	1	in	in	ADP
cana-5362	340	2	ifip	ifip	PROPN
cana-5362	340	3	internatinal	internatinal	ADJ
cana-5362	340	4	conference	conference	NOUN
cana-5362	340	5	on	on	ADP
cana-5362	340	6	computer	computer	NOUN
cana-5362	340	7	information	information	NOUN
cana-5362	340	8	systems	system	NOUN
cana-5362	340	9	and	and	CCONJ
cana-5362	340	10	industial	industial	ADJ
cana-5362	340	11	management	management	NOUN
cana-5362	340	12	.	.	PUNCT
cana-5362	341	1	springer	springer	NOUN
cana-5362	341	2	,	,	PUNCT
cana-5362	341	3	berlin	berlin	PROPN
cana-5362	341	4	,	,	PUNCT
cana-5362	341	5	heidelberg	heidelberg	PROPN
cana-5362	341	6	,	,	PUNCT
cana-5362	341	7	2013	2013	NUM
cana-5362	341	8	.	.	PUNCT
cana-5362	342	1	[	[	X
cana-5362	342	2	7	7	X
cana-5362	342	3	]	]	X
cana-5362	342	4	h.	h.	NOUN
cana-5362	342	5	brézis	brézi	NOUN
cana-5362	342	6	and	and	CCONJ
cana-5362	342	7	m.	m.	NOUN
cana-5362	342	8	sibony	sibony	NOUN
cana-5362	342	9	,	,	PUNCT
cana-5362	342	10	"	"	PUNCT
cana-5362	342	11	méthodes	méthode	NOUN
cana-5362	342	12	d’approximation	d’approximation	NOUN
cana-5362	342	13	et	et	NOUN
cana-5362	342	14	d’itération	d’itération	NOUN
cana-5362	342	15	pour	pour	VERB
cana-5362	342	16	les	les	X
cana-5362	342	17	opérateurs	opérateurs	PROPN
cana-5362	342	18	monotones	monotone	NOUN
cana-5362	342	19	"	"	PUNCT
cana-5362	342	20	,	,	PUNCT
cana-5362	342	21	mémoire	mémoire	NOUN
cana-5362	342	22	présenté	présenté	PROPN
cana-5362	342	23	par	par	PROPN
cana-5362	342	24	j.l	j.l	PROPN
cana-5362	342	25	.	.	PROPN
cana-5362	342	26	lions	lion	NOUN
cana-5362	342	27	,	,	PUNCT
cana-5362	342	28	pour	pour	VERB
cana-5362	342	29	les	les	X
cana-5362	342	30	opérateurs	opérateur	NOUN
cana-5362	342	31	monotones	monotone	NOUN
cana-5362	342	32	.	.	PUNCT
cana-5362	343	1	[	[	X
cana-5362	343	2	8	8	NUM
cana-5362	343	3	]	]	X
cana-5362	343	4	c.	c.	NOUN
cana-5362	343	5	catté	catté	PROPN
cana-5362	343	6	,	,	PUNCT
cana-5362	343	7	p.l	p.l	PROPN
cana-5362	343	8	.	.	PROPN
cana-5362	343	9	lions	lion	NOUN
cana-5362	343	10	,	,	PUNCT
cana-5362	343	11	j.m	j.m	PROPN
cana-5362	343	12	.	.	PROPN
cana-5362	343	13	morel	morel	PROPN
cana-5362	343	14	and	and	CCONJ
cana-5362	343	15	t.	t.	PROPN
cana-5362	343	16	coll	coll	PROPN
cana-5362	343	17	,	,	PUNCT
cana-5362	343	18	"	"	PUNCT
cana-5362	343	19	image	image	NOUN
cana-5362	343	20	selective	selective	ADJ
cana-5362	343	21	smoothing	smoothing	NOUN
cana-5362	343	22	and	and	CCONJ
cana-5362	343	23	edge	edge	NOUN
cana-5362	343	24	detection	detection	NOUN
cana-5362	343	25	by	by	ADP
cana-5362	343	26	nonlinear	nonlinear	ADJ
cana-5362	343	27	diffusion	diffusion	NOUN
cana-5362	343	28	"	"	PUNCT
cana-5362	343	29	,	,	PUNCT
cana-5362	343	30	siam	siam	PROPN
cana-5362	343	31	j.	j.	PROPN
cana-5362	343	32	numer	numer	PROPN
cana-5362	343	33	.	.	PROPN
cana-5362	344	1	anal	anal	PROPN
cana-5362	344	2	,	,	PUNCT
cana-5362	344	3	vol.29	vol.29	NOUN
cana-5362	344	4	,	,	PUNCT
cana-5362	344	5	no	no	INTJ
cana-5362	344	6	.	.	NOUN
cana-5362	344	7	1	1	NUM
cana-5362	344	8	,	,	PUNCT
cana-5362	344	9	pp.182	pp.182	NOUN
cana-5362	344	10	-	-	PUNCT
cana-5362	344	11	193	193	NUM
cana-5362	344	12	,	,	PUNCT
cana-5362	344	13	1992	1992	NUM
cana-5362	344	14	.	.	PUNCT
cana-5362	345	1	[	[	X
cana-5362	345	2	9	9	NUM
cana-5362	345	3	]	]	X
cana-5362	345	4	g.	g.	PROPN
cana-5362	345	5	da	da	PROPN
cana-5362	345	6	prato	prato	PROPN
cana-5362	345	7	and	and	CCONJ
cana-5362	345	8	j.	j.	PROPN
cana-5362	345	9	zabczyk	zabczyk	PROPN
cana-5362	345	10	,	,	PUNCT
cana-5362	345	11	"	"	PUNCT
cana-5362	345	12	stochastic	stochastic	ADJ
cana-5362	345	13	equations	equation	NOUN
cana-5362	345	14	in	in	ADP
cana-5362	345	15	infinite	infinite	ADJ
cana-5362	345	16	dimensions	dimension	NOUN
cana-5362	345	17	"	"	PUNCT
cana-5362	345	18	,	,	PUNCT
cana-5362	345	19	cambridge	cambridge	PROPN
cana-5362	345	20	university	university	PROPN
cana-5362	345	21	press	press	NOUN
cana-5362	345	22	,	,	PUNCT
cana-5362	345	23	series	series	NOUN
cana-5362	345	24	encyclopedia	encyclopedia	NOUN
cana-5362	345	25	of	of	ADP
cana-5362	345	26	mathematics	mathematic	NOUN
cana-5362	345	27	and	and	CCONJ
cana-5362	345	28	its	its	PRON
cana-5362	345	29	applications	application	NOUN
cana-5362	345	30	,	,	PUNCT
cana-5362	345	31	44	44	NUM
cana-5362	345	32	,	,	PUNCT
cana-5362	345	33	1992	1992	NUM
cana-5362	345	34	.	.	PUNCT
cana-5362	346	1	[	[	X
cana-5362	346	2	10	10	NUM
cana-5362	346	3	]	]	PUNCT
cana-5362	346	4	x.	x.	NOUN
cana-5362	346	5	descombes	descombe	NOUN
cana-5362	346	6	and	and	CCONJ
cana-5362	346	7	e.	e.	PROPN
cana-5362	346	8	zhizhina	zhizhina	PROPN
cana-5362	346	9	,	,	PUNCT
cana-5362	346	10	"	"	PUNCT
cana-5362	346	11	image	image	NOUN
cana-5362	346	12	denoising	denoising	NOUN
cana-5362	346	13	using	use	VERB
cana-5362	346	14	stochastic	stochastic	ADJ
cana-5362	346	15	differential	differential	ADJ
cana-5362	346	16	equations	equation	NOUN
cana-5362	346	17	"	"	PUNCT
cana-5362	346	18	,	,	PUNCT
cana-5362	346	19	inria	inria	NOUN
cana-5362	346	20	.	.	PUNCT
cana-5362	347	1	[	[	X
cana-5362	347	2	11	11	NUM
cana-5362	347	3	]	]	X
cana-5362	347	4	t.c	t.c	PROPN
cana-5362	347	5	.	.	PROPN
cana-5362	347	6	garrido	garrido	PROPN
cana-5362	347	7	,	,	PUNCT
cana-5362	347	8	"	"	PUNCT
cana-5362	347	9	existence	existence	NOUN
cana-5362	347	10	and	and	CCONJ
cana-5362	347	11	uniqueness	uniqueness	NOUN
cana-5362	347	12	of	of	ADP
cana-5362	347	13	solutions	solution	NOUN
cana-5362	347	14	for	for	ADP
cana-5362	347	15	non	non	ADJ
cana-5362	347	16	-	-	ADJ
cana-5362	347	17	linear	linear	ADJ
cana-5362	347	18	stochastic	stochastic	ADJ
cana-5362	347	19	partial	partial	ADJ
cana-5362	347	20	differential	differential	NOUN
cana-5362	347	21	equations	equation	NOUN
cana-5362	347	22	"	"	PUNCT
cana-5362	347	23	,	,	PUNCT
cana-5362	347	24	departamento	departamento	PROPN
cana-5362	347	25	de	de	PROPN
cana-5362	347	26	anàlisis	anàlisis	PROPN
cana-5362	347	27	matemàtico	matemàtico	PROPN
cana-5362	347	28	,	,	PUNCT
cana-5362	347	29	universidad	universidad	PROPN
cana-5362	347	30	de	de	PROPN
cana-5362	347	31	sevilla	sevilla	PROPN
cana-5362	347	32	,	,	PUNCT
cana-5362	347	33	apartado	apartado	X
cana-5362	347	34	de	de	X
cana-5362	347	35	correos	correos	PROPN
cana-5362	347	36	1.160	1.160	NUM
cana-5362	347	37	.	.	PUNCT
cana-5362	348	1	41080	41080	NUM
cana-5362	348	2	-	-	PUNCT
cana-5362	348	3	sevilla	sevilla	PROPN
cana-5362	348	4	,	,	PUNCT
cana-5362	348	5	spain	spain	PROPN
cana-5362	348	6	.	.	PUNCT
cana-5362	349	1	[	[	X
cana-5362	349	2	12	12	NUM
cana-5362	349	3	]	]	X
cana-5362	349	4	j.l	j.l	PROPN
cana-5362	349	5	.	.	PROPN
cana-5362	349	6	lions	lion	NOUN
cana-5362	349	7	,	,	PUNCT
cana-5362	349	8	"	"	PUNCT
cana-5362	349	9	quelques	quelques	X
cana-5362	349	10	méthodes	méthode	NOUN
cana-5362	349	11	de	de	ADP
cana-5362	349	12	résolutions	résolutions	PROPN
cana-5362	349	13	des	des	X
cana-5362	349	14	problèmes	problèmes	PROPN
cana-5362	349	15	aux	aux	PROPN
cana-5362	349	16	limites	limites	PROPN
cana-5362	349	17	non	non	X
cana-5362	349	18	linéaires	linéaires	PROPN
cana-5362	349	19	"	"	PUNCT
cana-5362	349	20	,	,	PUNCT
cana-5362	349	21	dunod	dunod	PROPN
cana-5362	349	22	gauthier	gauthier	PROPN
cana-5362	349	23	villars	villar	NOUN
cana-5362	349	24	,	,	PUNCT
cana-5362	349	25	paris	paris	PROPN
cana-5362	349	26	,	,	PUNCT
cana-5362	349	27	1969	1969	NUM
cana-5362	349	28	.	.	PUNCT
cana-5362	350	1	[	[	X
cana-5362	350	2	13	13	NUM
cana-5362	350	3	]	]	X
cana-5362	350	4	p.a	p.a	PROPN
cana-5362	350	5	.	.	PROPN
cana-5362	350	6	mayer	mayer	PROPN
cana-5362	350	7	,	,	PUNCT
cana-5362	350	8	"	"	PUNCT
cana-5362	350	9	probabilités	probabilités	PROPN
cana-5362	350	10	et	et	PROPN
cana-5362	350	11	potentiel	potentiel	PROPN
cana-5362	350	12	,	,	PUNCT
cana-5362	350	13	herman	herman	PROPN
cana-5362	350	14	"	"	PUNCT
cana-5362	350	15	,	,	PUNCT
cana-5362	350	16	paris	paris	PROPN
cana-5362	350	17	,	,	PUNCT
cana-5362	350	18	(	(	PUNCT
cana-5362	350	19	1966	1966	NUM
cana-5362	350	20	)	)	PUNCT
cana-5362	350	21	.	.	PUNCT
cana-5362	351	1	[	[	X
cana-5362	351	2	14	14	NUM
cana-5362	351	3	]	]	X
cana-5362	351	4	g.j	g.j	PROPN
cana-5362	351	5	.	.	PROPN
cana-5362	351	6	minty	minty	PROPN
cana-5362	351	7	,	,	PUNCT
cana-5362	351	8	"	"	PUNCT
cana-5362	351	9	monotone	monotone	ADJ
cana-5362	351	10	nonlinear	nonlinear	ADJ
cana-5362	351	11	operators	operator	NOUN
cana-5362	351	12	in	in	ADP
cana-5362	351	13	hilbert	hilbert	PROPN
cana-5362	351	14	spaces	space	NOUN
cana-5362	351	15	"	"	PUNCT
cana-5362	351	16	,	,	PUNCT
cana-5362	351	17	duke	duke	PROPN
cana-5362	351	18	math	math	PROPN
cana-5362	351	19	.	.	PUNCT
cana-5362	352	1	j.	j.	PROPN
cana-5362	352	2	29,341	29,341	NUM
cana-5362	352	3	-	-	SYM
cana-5362	352	4	346	346	NUM
cana-5362	352	5	,	,	PUNCT
cana-5362	352	6	1962	1962	NUM
cana-5362	352	7	.	.	PUNCT
cana-5362	353	1	communications	communication	NOUN
cana-5362	353	2	on	on	ADP
cana-5362	353	3	applied	apply	VERB
cana-5362	353	4	nonlinear	nonlinear	ADJ
cana-5362	353	5	analysis	analysis	NOUN
cana-5362	353	6	issn	issn	NOUN
cana-5362	353	7	:	:	PUNCT
cana-5362	353	8	1074	1074	NUM
cana-5362	353	9	-	-	PUNCT
cana-5362	353	10	133x	133x	NUM
cana-5362	353	11	vol	vol	VERB
cana-5362	353	12	32	32	NUM
cana-5362	353	13	no	no	NOUN
cana-5362	353	14	.	.	PUNCT
cana-5362	354	1	10s	10	NOUN
cana-5362	354	2	(	(	PUNCT
cana-5362	354	3	2025	2025	NUM
cana-5362	354	4	)	)	PUNCT
cana-5362	354	5	2004	2004	NUM
cana-5362	354	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5362	355	1	[	[	X
cana-5362	355	2	15	15	NUM
cana-5362	355	3	]	]	X
cana-5362	355	4	f.z	f.z	PROPN
cana-5362	355	5	.	.	PROPN
cana-5362	355	6	nouri	nouri	PROPN
cana-5362	355	7	,	,	PUNCT
cana-5362	355	8	"	"	PUNCT
cana-5362	355	9	uniqueness	uniqueness	NOUN
cana-5362	355	10	and	and	CCONJ
cana-5362	355	11	existence	existence	NOUN
cana-5362	355	12	results	result	VERB
cana-5362	355	13	for	for	ADP
cana-5362	355	14	a	a	DET
cana-5362	355	15	partial	partial	ADJ
cana-5362	355	16	differential	differential	NOUN
cana-5362	355	17	equation	equation	NOUN
cana-5362	355	18	in	in	ADP
cana-5362	355	19	image	image	NOUN
cana-5362	355	20	inpainting	inpainte	VERB
cana-5362	355	21	"	"	PUNCT
cana-5362	355	22	,	,	PUNCT
cana-5362	355	23	commun	commun	PROPN
cana-5362	355	24	.	.	PUNCT
cana-5362	356	1	optim	optim	PROPN
cana-5362	356	2	.	.	PUNCT
cana-5362	357	1	theory	theory	NOUN
cana-5362	357	2	2020	2020	NUM
cana-5362	357	3	(	(	PUNCT
cana-5362	357	4	2020	2020	NUM
cana-5362	357	5	)	)	PUNCT
cana-5362	357	6	,	,	PUNCT
cana-5362	357	7	article	article	NOUN
cana-5362	357	8	i	i	PROPN
cana-5362	357	9	d	d	PROPN
cana-5362	357	10	7	7	NUM
cana-5362	357	11	,	,	PUNCT
cana-5362	357	12	pp	pp	ADJ
cana-5362	357	13	.	.	PUNCT
cana-5362	358	1	1	1	NUM
cana-5362	358	2	-	-	SYM
cana-5362	358	3	17	17	NUM
cana-5362	358	4	.	.	PUNCT
cana-5362	359	1	[	[	X
cana-5362	359	2	16	16	NUM
cana-5362	359	3	]	]	X
cana-5362	359	4	e.	e.	PROPN
cana-5362	359	5	pardoux	pardoux	PROPN
cana-5362	359	6	,	,	PUNCT
cana-5362	359	7	"	"	PUNCT
cana-5362	359	8	stochastic	stochastic	ADJ
cana-5362	359	9	partial	partial	ADJ
cana-5362	359	10	differential	differential	NOUN
cana-5362	359	11	equations	equation	NOUN
cana-5362	359	12	and	and	CCONJ
cana-5362	359	13	filtering	filtering	NOUN
cana-5362	359	14	of	of	ADP
cana-5362	359	15	diffusion	diffusion	NOUN
cana-5362	359	16	processes	process	NOUN
cana-5362	359	17	"	"	PUNCT
cana-5362	359	18	,	,	PUNCT
cana-5362	359	19	stochastic	stochastic	ADJ
cana-5362	359	20	3	3	NUM
cana-5362	359	21	,	,	PUNCT
cana-5362	359	22	127	127	NUM
cana-5362	359	23	-	-	SYM
cana-5362	359	24	167	167	NUM
cana-5362	359	25	,	,	PUNCT
cana-5362	359	26	(	(	PUNCT
cana-5362	359	27	1979	1979	NUM
cana-5362	359	28	)	)	PUNCT
cana-5362	359	29	.	.	PUNCT
cana-5362	360	1	[	[	X
cana-5362	360	2	17	17	NUM
cana-5362	360	3	]	]	X
cana-5362	360	4	p.	p.	NOUN
cana-5362	360	5	perona	perona	PROPN
cana-5362	360	6	and	and	CCONJ
cana-5362	360	7	j.	j.	PROPN
cana-5362	360	8	malik	malik	PROPN
cana-5362	360	9	,	,	PUNCT
cana-5362	360	10	"	"	PUNCT
cana-5362	360	11	scale	scale	NOUN
cana-5362	360	12	-	-	PUNCT
cana-5362	360	13	space	space	NOUN
cana-5362	360	14	and	and	CCONJ
cana-5362	360	15	edge	edge	NOUN
cana-5362	360	16	detection	detection	NOUN
cana-5362	360	17	using	use	VERB
cana-5362	360	18	anisotropic	anisotropic	NOUN
cana-5362	360	19	diffusion	diffusion	NOUN
cana-5362	360	20	"	"	PUNCT
cana-5362	360	21	,	,	PUNCT
cana-5362	360	22	ieee	ieee	PROPN
cana-5362	360	23	trans	tran	NOUN
cana-5362	360	24	.	.	PUNCT
cana-5362	361	1	pattern	pattern	PROPN
cana-5362	361	2	anal	anal	PROPN
cana-5362	361	3	.	.	PUNCT
cana-5362	362	1	machine	machine	NOUN
cana-5362	362	2	intell	intell	PROPN
cana-5362	362	3	.	.	PUNCT
cana-5362	362	4	,	,	PUNCT
cana-5362	362	5	vol	vol	NOUN
cana-5362	362	6	.	.	PROPN
cana-5362	363	1	12	12	NUM
cana-5362	363	2	,	,	PUNCT
cana-5362	363	3	pp	pp	ADJ
cana-5362	363	4	.	.	PUNCT
cana-5362	364	1	629	629	NUM
cana-5362	364	2	-	-	SYM
cana-5362	364	3	639	639	NUM
cana-5362	364	4	,	,	PUNCT
cana-5362	364	5	1990	1990	NUM
cana-5362	364	6	.	.	PUNCT
cana-5362	365	1	[	[	X
cana-5362	365	2	18	18	NUM
cana-5362	365	3	]	]	X
cana-5362	365	4	c.	c.	PROPN
cana-5362	365	5	prévot	prévot	PROPN
cana-5362	365	6	and	and	CCONJ
cana-5362	365	7	m.	m.	NOUN
cana-5362	365	8	röckner	röckner	NOUN
cana-5362	365	9	,	,	PUNCT
cana-5362	365	10	"	"	PUNCT
cana-5362	365	11	a	a	DET
cana-5362	365	12	concise	concise	ADJ
cana-5362	365	13	cours	cour	NOUN
cana-5362	365	14	on	on	ADP
cana-5362	365	15	stochastic	stochastic	ADJ
cana-5362	365	16	partial	partial	ADJ
cana-5362	365	17	differential	differential	NOUN
cana-5362	365	18	equations	equation	NOUN
cana-5362	365	19	"	"	PUNCT
cana-5362	365	20	,	,	PUNCT
cana-5362	365	21	springer	springer	NOUN
cana-5362	365	22	,	,	PUNCT
cana-5362	365	23	lecture	lecture	NOUN
cana-5362	365	24	notes	note	NOUN
cana-5362	365	25	in	in	ADP
cana-5362	365	26	mathematics	mathematic	NOUN
cana-5362	365	27	,	,	PUNCT
cana-5362	365	28	vol	vol	NOUN
cana-5362	365	29	.	.	PROPN
cana-5362	365	30	1905	1905	NUM
cana-5362	365	31	,	,	PUNCT
cana-5362	365	32	2007	2007	NUM
cana-5362	365	33	.	.	PUNCT
cana-5362	366	1	[	[	X
cana-5362	366	2	19	19	NUM
cana-5362	366	3	]	]	PUNCT
cana-5362	366	4	l.	l.	PROPN
cana-5362	366	5	schwartz	schwartz	PROPN
cana-5362	366	6	,	,	PUNCT
cana-5362	366	7	"	"	PUNCT
cana-5362	366	8	radon	radon	NOUN
cana-5362	366	9	measures	measure	NOUN
cana-5362	366	10	on	on	ADP
cana-5362	366	11	arbitrary	arbitrary	ADJ
cana-5362	366	12	topological	topological	ADJ
cana-5362	366	13	spaces	space	NOUN
cana-5362	366	14	"	"	PUNCT
cana-5362	366	15	,	,	PUNCT
cana-5362	366	16	à	à	X
cana-5362	366	17	paraître	paraître	PROPN
cana-5362	366	18	,	,	PUNCT
cana-5362	366	19	tata	tata	PROPN
cana-5362	366	20	institute	institute	PROPN
cana-5362	366	21	of	of	ADP
cana-5362	366	22	fundamental	fundamental	ADJ
cana-5362	366	23	research	research	PROPN
cana-5362	366	24	bombay	bombay	PROPN
cana-5362	366	25	.	.	PUNCT