id	sid	tid	token	lemma	pos
m-1152	1	1	stability	stability	NOUN
m-1152	1	2	in	in	ADP
m-1152	1	3	semigroups	semigroup	NOUN
m-1152	1	4	of	of	ADP
m-1152	1	5	bounded	bounded	ADJ
m-1152	1	6	linear	linear	PROPN
m-1152	1	7	operators	operator	NOUN
m-1152	1	8	:	:	PUNCT
m-1152	1	9	bridging	bridge	VERB
m-1152	1	10	algebraic	algebraic	ADJ
m-1152	1	11	and	and	CCONJ
m-1152	1	12	analytic	analytic	ADJ
m-1152	1	13	notions	notion	NOUN
m-1152	1	14	1otobong	1otobong	NUM
m-1152	1	15	j.	j.	PROPN
m-1152	1	16	tom∗	tom∗	PROPN
m-1152	1	17	,	,	PUNCT
m-1152	1	18	and	and	CCONJ
m-1152	1	19	2otobong	2otobong	NUM
m-1152	1	20	g.	g.	NOUN
m-1152	1	21	udoaka	udoaka	ADV
m-1152	1	22	.	.	PUNCT
m-1152	2	1	1	1	NUM
m-1152	2	2	fed	feed	VERB
m-1152	2	3	.	.	PUNCT
m-1152	3	1	uni	uni	PROPN
m-1152	3	2	.	.	PUNCT
m-1152	3	3	tech	tech	PROPN
m-1152	3	4	.	.	PUNCT
m-1152	3	5	,	,	PUNCT
m-1152	3	6	ikot	ikot	PROPN
m-1152	3	7	abasi	abasi	PROPN
m-1152	3	8	&	&	CCONJ
m-1152	3	9	akwa	akwa	PROPN
m-1152	3	10	ibom	ibom	ADJ
m-1152	3	11	state	state	NOUN
m-1152	3	12	uni	uni	PROPN
m-1152	3	13	.	.	PROPN
m-1152	3	14	,	,	PUNCT
m-1152	3	15	ikot	ikot	PROPN
m-1152	3	16	akpaden	akpaden	VERB
m-1152	3	17	.	.	PUNCT
m-1152	4	1	2akwa	2akwa	NUM
m-1152	4	2	ibom	ibom	ADJ
m-1152	4	3	state	state	NOUN
m-1152	4	4	university	university	NOUN
m-1152	4	5	,	,	PUNCT
m-1152	4	6	ikot	ikot	PROPN
m-1152	4	7	akpaden	akpaden	VERB
m-1152	4	8	.	.	PUNCT
m-1152	5	1	abstract	abstract	ADJ
m-1152	5	2	stability	stability	NOUN
m-1152	5	3	is	be	AUX
m-1152	5	4	a	a	DET
m-1152	5	5	cornerstone	cornerstone	NOUN
m-1152	5	6	in	in	ADP
m-1152	5	7	the	the	DET
m-1152	5	8	theory	theory	NOUN
m-1152	5	9	of	of	ADP
m-1152	5	10	semigroups	semigroup	NOUN
m-1152	5	11	,	,	PUNCT
m-1152	5	12	shaping	shape	VERB
m-1152	5	13	the	the	DET
m-1152	5	14	study	study	NOUN
m-1152	5	15	of	of	ADP
m-1152	5	16	evolution	evolution	NOUN
m-1152	5	17	equations	equation	NOUN
m-1152	5	18	,	,	PUNCT
m-1152	5	19	operator	operator	NOUN
m-1152	5	20	theory	theory	NOUN
m-1152	5	21	,	,	PUNCT
m-1152	5	22	and	and	CCONJ
m-1152	5	23	algebraic	algebraic	ADJ
m-1152	5	24	structures	structure	NOUN
m-1152	5	25	.	.	PUNCT
m-1152	6	1	yet	yet	ADV
m-1152	6	2	,	,	PUNCT
m-1152	6	3	algebraic	algebraic	ADJ
m-1152	6	4	and	and	CCONJ
m-1152	6	5	analytic	analytic	ADJ
m-1152	6	6	perspectives	perspective	NOUN
m-1152	6	7	on	on	ADP
m-1152	6	8	stability	stability	NOUN
m-1152	6	9	have	have	AUX
m-1152	6	10	traditionally	traditionally	ADV
m-1152	6	11	developed	develop	VERB
m-1152	6	12	in	in	ADP
m-1152	6	13	isolation	isolation	NOUN
m-1152	6	14	.	.	PUNCT
m-1152	7	1	this	this	DET
m-1152	7	2	paper	paper	NOUN
m-1152	7	3	builds	build	VERB
m-1152	7	4	a	a	DET
m-1152	7	5	novel	novel	ADJ
m-1152	7	6	bridge	bridge	NOUN
m-1152	7	7	between	between	ADP
m-1152	7	8	the	the	DET
m-1152	7	9	two	two	NUM
m-1152	7	10	.	.	PUNCT
m-1152	8	1	tom	tom	PROPN
m-1152	8	2	,	,	PUNCT
m-1152	8	3	udoaka	udoaka	ADV
m-1152	8	4	and	and	CCONJ
m-1152	8	5	udo	udo	PROPN
m-1152	8	6	�	�	PROPN
m-1152	8	7	a	a	PRON
m-1152	8	8	(	(	PUNCT
m-1152	8	9	2025	2025	NUM
m-1152	8	10	)	)	PUNCT
m-1152	8	11	established	establish	VERB
m-1152	8	12	that	that	SCONJ
m-1152	8	13	every	every	DET
m-1152	8	14	strongly	strongly	ADV
m-1152	8	15	continuous	continuous	ADJ
m-1152	8	16	(	(	PUNCT
m-1152	8	17	c0	c0	NOUN
m-1152	8	18	)	)	PUNCT
m-1152	8	19	semigroup	semigroup	NOUN
m-1152	8	20	of	of	ADP
m-1152	8	21	bounded	bounded	PROPN
m-1152	8	22	linear	linear	PROPN
m-1152	8	23	operators	operator	NOUN
m-1152	8	24	is	be	AUX
m-1152	8	25	stable	stable	ADJ
m-1152	8	26	in	in	ADP
m-1152	8	27	the	the	DET
m-1152	8	28	sense	sense	NOUN
m-1152	8	29	of	of	ADP
m-1152	8	30	koch	koch	PROPN
m-1152	8	31	and	and	CCONJ
m-1152	8	32	wallace	wallace	PROPN
m-1152	8	33	(	(	PUNCT
m-1152	8	34	kw	kw	PROPN
m-1152	8	35	)	)	PUNCT
m-1152	8	36	,	,	PUNCT
m-1152	8	37	a	a	DET
m-1152	8	38	universal	universal	ADJ
m-1152	8	39	algebraic	algebraic	ADJ
m-1152	8	40	property	property	NOUN
m-1152	8	41	that	that	PRON
m-1152	8	42	forces	force	VERB
m-1152	8	43	green	green	PROPN
m-1152	8	44	's	's	PART
m-1152	8	45	relations	relation	NOUN
m-1152	8	46	to	to	PART
m-1152	8	47	collapse	collapse	VERB
m-1152	8	48	(	(	PUNCT
m-1152	8	49	d	d	X
m-1152	8	50	=	=	SYM
m-1152	8	51	j	j	PROPN
m-1152	8	52	=	=	PUNCT
m-1152	8	53	l	l	NOUN
m-1152	8	54	=	=	SYM
m-1152	8	55	r	r	NOUN
m-1152	8	56	)	)	PUNCT
m-1152	8	57	.	.	PUNCT
m-1152	9	1	this	this	DET
m-1152	9	2	recognition	recognition	NOUN
m-1152	9	3	is	be	AUX
m-1152	9	4	new	new	ADJ
m-1152	9	5	in	in	ADP
m-1152	9	6	operator	operator	NOUN
m-1152	9	7	semigroup	semigroup	PROPN
m-1152	9	8	theory	theory	NOUN
m-1152	9	9	,	,	PUNCT
m-1152	9	10	where	where	SCONJ
m-1152	9	11	stability	stability	NOUN
m-1152	9	12	has	have	AUX
m-1152	9	13	typically	typically	ADV
m-1152	9	14	been	be	AUX
m-1152	9	15	studied	study	VERB
m-1152	9	16	only	only	ADV
m-1152	9	17	in	in	ADP
m-1152	9	18	analytic	analytic	ADJ
m-1152	9	19	terms	term	NOUN
m-1152	9	20	.	.	PUNCT
m-1152	10	1	we	we	PRON
m-1152	10	2	further	far	ADV
m-1152	10	3	provide	provide	VERB
m-1152	10	4	precise	precise	ADJ
m-1152	10	5	spectral	spectral	ADJ
m-1152	10	6	conditions	condition	NOUN
m-1152	10	7	under	under	ADP
m-1152	10	8	which	which	PRON
m-1152	10	9	kw	kw	VERB
m-1152	10	10	-	-	PUNCT
m-1152	10	11	stability	stability	NOUN
m-1152	10	12	aligns	align	VERB
m-1152	10	13	with	with	ADP
m-1152	10	14	analytic	analytic	ADJ
m-1152	10	15	stability	stability	NOUN
m-1152	10	16	notions	notion	NOUN
m-1152	10	17	�	�	NOUN
m-1152	10	18	strong	strong	ADJ
m-1152	10	19	,	,	PUNCT
m-1152	10	20	asymptotic	asymptotic	ADJ
m-1152	10	21	,	,	PUNCT
m-1152	10	22	exponential	exponential	NOUN
m-1152	10	23	,	,	PUNCT
m-1152	10	24	and	and	CCONJ
m-1152	10	25	uniform	uniform	NOUN
m-1152	10	26	�	�	PROPN
m-1152	10	27	thereby	thereby	ADV
m-1152	10	28	unifying	unify	VERB
m-1152	10	29	algebraic	algebraic	ADJ
m-1152	10	30	semigroup	semigroup	ADJ
m-1152	10	31	stability	stability	NOUN
m-1152	10	32	with	with	ADP
m-1152	10	33	spectral	spectral	ADJ
m-1152	10	34	/	/	SYM
m-1152	10	35	operator	operator	NOUN
m-1152	10	36	-	-	PUNCT
m-1152	10	37	theoretic	theoretic	NOUN
m-1152	10	38	stability	stability	NOUN
m-1152	10	39	.	.	PUNCT
m-1152	11	1	illustrative	illustrative	ADJ
m-1152	11	2	examples	example	NOUN
m-1152	11	3	,	,	PUNCT
m-1152	11	4	including	include	VERB
m-1152	11	5	the	the	DET
m-1152	11	6	translation	translation	NOUN
m-1152	11	7	,	,	PUNCT
m-1152	11	8	right	right	ADJ
m-1152	11	9	shift	shift	NOUN
m-1152	11	10	,	,	PUNCT
m-1152	11	11	heat	heat	NOUN
m-1152	11	12	,	,	PUNCT
m-1152	11	13	and	and	CCONJ
m-1152	11	14	damped	damp	VERB
m-1152	11	15	wave	wave	NOUN
m-1152	11	16	semigroups	semigroup	NOUN
m-1152	11	17	,	,	PUNCT
m-1152	11	18	demonstrate	demonstrate	VERB
m-1152	11	19	the	the	DET
m-1152	11	20	stability	stability	NOUN
m-1152	11	21	�	�	PROPN
m-1152	11	22	gap	gap	PROPN
m-1152	11	23	�	�	PROPN
m-1152	11	24	and	and	CCONJ
m-1152	11	25	the	the	DET
m-1152	11	26	exact	exact	ADJ
m-1152	11	27	conditions	condition	NOUN
m-1152	11	28	under	under	ADP
m-1152	11	29	which	which	PRON
m-1152	11	30	the	the	DET
m-1152	11	31	two	two	NUM
m-1152	11	32	approaches	approach	NOUN
m-1152	11	33	coincide	coincide	NOUN
m-1152	11	34	.	.	PUNCT
m-1152	12	1	the	the	DET
m-1152	12	2	study	study	NOUN
m-1152	12	3	is	be	AUX
m-1152	12	4	signi	signi	ADJ
m-1152	12	5	�	�	NOUN
m-1152	12	6	cant	cant	VERB
m-1152	12	7	because	because	SCONJ
m-1152	12	8	it	it	PRON
m-1152	12	9	supplies	supply	VERB
m-1152	12	10	a	a	DET
m-1152	12	11	universal	universal	ADJ
m-1152	12	12	structural	structural	ADJ
m-1152	12	13	property	property	NOUN
m-1152	12	14	of	of	ADP
m-1152	12	15	operator	operator	NOUN
m-1152	12	16	semigroups	semigroup	NOUN
m-1152	12	17	,	,	PUNCT
m-1152	12	18	a	a	DET
m-1152	12	19	spectral	spectral	ADJ
m-1152	12	20	criterion	criterion	NOUN
m-1152	12	21	for	for	ADP
m-1152	12	22	analytic	analytic	ADJ
m-1152	12	23	decay	decay	NOUN
m-1152	12	24	,	,	PUNCT
m-1152	12	25	and	and	CCONJ
m-1152	12	26	practical	practical	ADJ
m-1152	12	27	insights	insight	NOUN
m-1152	12	28	for	for	ADP
m-1152	12	29	evolution	evolution	NOUN
m-1152	12	30	equations	equation	NOUN
m-1152	12	31	,	,	PUNCT
m-1152	12	32	control	control	NOUN
m-1152	12	33	design	design	NOUN
m-1152	12	34	,	,	PUNCT
m-1152	12	35	and	and	CCONJ
m-1152	12	36	numerical	numerical	ADJ
m-1152	12	37	discretization	discretization	NOUN
m-1152	12	38	.	.	PUNCT
m-1152	13	1	keywords	keyword	NOUN
m-1152	13	2	:	:	PUNCT
m-1152	13	3	semigroup	semigroup	PROPN
m-1152	13	4	theory	theory	NOUN
m-1152	13	5	;	;	PUNCT
m-1152	13	6	kw	kw	NOUN
m-1152	13	7	-	-	NOUN
m-1152	13	8	stability	stability	NOUN
m-1152	13	9	;	;	PUNCT
m-1152	13	10	analytic	analytic	ADJ
m-1152	13	11	stability	stability	NOUN
m-1152	13	12	;	;	PUNCT
m-1152	13	13	spectral	spectral	ADJ
m-1152	13	14	bound	bind	VERB
m-1152	13	15	;	;	PUNCT
m-1152	13	16	green	green	PROPN
m-1152	13	17	's	's	PART
m-1152	13	18	relations	relation	NOUN
m-1152	13	19	;	;	PUNCT
m-1152	13	20	evolution	evolution	NOUN
m-1152	13	21	equations	equation	NOUN
m-1152	13	22	.	.	PUNCT
m-1152	14	1	1	1	NUM
m-1152	14	2	introduction	introduction	NOUN
m-1152	14	3	the	the	DET
m-1152	14	4	concept	concept	NOUN
m-1152	14	5	of	of	ADP
m-1152	14	6	stability	stability	NOUN
m-1152	14	7	is	be	AUX
m-1152	14	8	central	central	ADJ
m-1152	14	9	to	to	ADP
m-1152	14	10	mathematics	mathematic	NOUN
m-1152	14	11	,	,	PUNCT
m-1152	14	12	capturing	capture	VERB
m-1152	14	13	how	how	SCONJ
m-1152	14	14	systems	system	NOUN
m-1152	14	15	behave	behave	VERB
m-1152	14	16	under	under	ADP
m-1152	14	17	iteration	iteration	NOUN
m-1152	14	18	,	,	PUNCT
m-1152	14	19	evolution	evolution	NOUN
m-1152	14	20	,	,	PUNCT
m-1152	14	21	or	or	CCONJ
m-1152	14	22	perturbation	perturbation	NOUN
m-1152	14	23	.	.	PUNCT
m-1152	15	1	within	within	ADP
m-1152	15	2	algebraic	algebraic	PROPN
m-1152	15	3	semigroup	semigroup	PROPN
m-1152	15	4	theory	theory	NOUN
m-1152	15	5	,	,	PUNCT
m-1152	15	6	stability	stability	NOUN
m-1152	15	7	was	be	AUX
m-1152	15	8	formally	formally	ADV
m-1152	15	9	introduced	introduce	VERB
m-1152	15	10	by	by	ADP
m-1152	15	11	koch	koch	PROPN
m-1152	15	12	and	and	CCONJ
m-1152	15	13	wallace	wallace	PROPN
m-1152	15	14	(	(	PUNCT
m-1152	15	15	1956	1956	NUM
m-1152	15	16	)	)	PUNCT
m-1152	16	1	[	[	X
m-1152	16	2	1	1	NUM
m-1152	16	3	]	]	PUNCT
m-1152	16	4	,	,	PUNCT
m-1152	16	5	who	who	PRON
m-1152	16	6	de	de	PROPN
m-1152	16	7	�	�	PROPN
m-1152	16	8	ned	ne	VERB
m-1152	16	9	a	a	DET
m-1152	16	10	semigroup	semigroup	NOUN
m-1152	16	11	s	s	VERB
m-1152	16	12	to	to	PART
m-1152	16	13	be	be	AUX
m-1152	16	14	stable	stable	ADJ
m-1152	16	15	if	if	SCONJ
m-1152	16	16	as	as	ADP
m-1152	16	17	⊆	⊆	NUM
m-1152	16	18	abs	ab	NOUN
m-1152	16	19	=	=	NOUN
m-1152	16	20	⇒	⇒	NOUN
m-1152	16	21	as	as	ADP
m-1152	16	22	=	=	NOUN
m-1152	16	23	abs	ab	NOUN
m-1152	16	24	,	,	PUNCT
m-1152	16	25	sa	sa	PROPN
m-1152	16	26	⊆	⊆	NUM
m-1152	16	27	sab	sab	ADJ
m-1152	16	28	=	=	NOUN
m-1152	16	29	⇒	⇒	NOUN
m-1152	16	30	sa	sa	X
m-1152	17	1	=	=	PUNCT
m-1152	17	2	sab	sab	PROPN
m-1152	17	3	.	.	PUNCT
m-1152	18	1	this	this	DET
m-1152	18	2	condition	condition	NOUN
m-1152	18	3	enforces	enforce	VERB
m-1152	18	4	the	the	DET
m-1152	18	5	collapse	collapse	NOUN
m-1152	18	6	of	of	ADP
m-1152	18	7	green	green	PROPN
m-1152	18	8	's	's	PART
m-1152	18	9	relations	relation	NOUN
m-1152	18	10	�	�	PROPN
m-1152	18	11	fundamental	fundamental	ADJ
m-1152	18	12	equivalence	equivalence	NOUN
m-1152	18	13	relations	relation	NOUN
m-1152	18	14	describing	describe	VERB
m-1152	18	15	semigroup	semigroup	PROPN
m-1152	18	16	structure	structure	NOUN
m-1152	18	17	�	�	PROPN
m-1152	18	18	so	so	SCONJ
m-1152	18	19	that	that	SCONJ
m-1152	18	20	d	d	NOUN
m-1152	18	21	=	=	SYM
m-1152	18	22	j	j	PROPN
m-1152	18	23	=	=	SYM
m-1152	18	24	l	l	PROPN
m-1152	18	25	=	=	PUNCT
m-1152	18	26	r.	r.	NOUN
m-1152	18	27	stability	stability	NOUN
m-1152	18	28	in	in	ADP
m-1152	18	29	this	this	DET
m-1152	18	30	sense	sense	NOUN
m-1152	18	31	ijo	ijo	PROPN
m-1152	18	32	international	international	PROPN
m-1152	18	33	journal	journal	PROPN
m-1152	18	34	of	of	ADP
m-1152	18	35	mathematics	mathematics	PROPN
m-1152	18	36	(	(	PUNCT
m-1152	18	37	issn	issn	PROPN
m-1152	18	38	:	:	PUNCT
m-1152	18	39	2992	2992	NUM
m-1152	18	40	-	-	SYM
m-1152	18	41	4421	4421	NUM
m-1152	18	42	)	)	PUNCT
m-1152	18	43	volume	volume	NOUN
m-1152	18	44	08	08	NUM
m-1152	19	1	|	|	ADV
m-1152	19	2	issue	issue	NOUN
m-1152	19	3	9	9	NUM
m-1152	19	4	|	|	CCONJ
m-1152	19	5	september	september	PROPN
m-1152	19	6	2025	2025	NUM
m-1152	19	7	|	|	ADV
m-1152	19	8	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1152	19	9	10	10	NUM
m-1152	19	10	has	have	VERB
m-1152	19	11	deep	deep	ADJ
m-1152	19	12	structural	structural	ADJ
m-1152	19	13	implications	implication	NOUN
m-1152	19	14	,	,	PUNCT
m-1152	19	15	simplifying	simplify	VERB
m-1152	19	16	semigroup	semigroup	ADJ
m-1152	19	17	decompositions	decomposition	NOUN
m-1152	19	18	and	and	CCONJ
m-1152	19	19	embedding	embed	VERB
m-1152	19	20	properties	property	NOUN
m-1152	19	21	[	[	X
m-1152	19	22	4	4	NUM
m-1152	19	23	,	,	PUNCT
m-1152	19	24	20	20	NUM
m-1152	19	25	,	,	PUNCT
m-1152	19	26	21	21	NUM
m-1152	19	27	,	,	PUNCT
m-1152	19	28	22	22	NUM
m-1152	19	29	,	,	PUNCT
m-1152	19	30	23	23	NUM
m-1152	19	31	,	,	PUNCT
m-1152	19	32	24	24	NUM
m-1152	19	33	,	,	PUNCT
m-1152	19	34	25	25	NUM
m-1152	19	35	,	,	PUNCT
m-1152	19	36	26	26	NUM
m-1152	19	37	]	]	PUNCT
m-1152	19	38	.	.	PUNCT
m-1152	20	1	in	in	ADP
m-1152	20	2	operator	operator	NOUN
m-1152	20	3	theory	theory	NOUN
m-1152	20	4	and	and	CCONJ
m-1152	20	5	functional	functional	ADJ
m-1152	20	6	analysis	analysis	NOUN
m-1152	20	7	,	,	PUNCT
m-1152	20	8	a	a	DET
m-1152	20	9	di	di	NOUN
m-1152	20	10	�	�	PROPN
m-1152	20	11	erent	erent	NOUN
m-1152	20	12	approach	approach	NOUN
m-1152	20	13	to	to	ADP
m-1152	20	14	stability	stability	NOUN
m-1152	20	15	has	have	AUX
m-1152	20	16	developed	develop	VERB
m-1152	20	17	through	through	ADP
m-1152	20	18	the	the	DET
m-1152	20	19	study	study	NOUN
m-1152	20	20	of	of	ADP
m-1152	20	21	c0	c0	PROPN
m-1152	20	22	-	-	PUNCT
m-1152	20	23	semigroups	semigroup	NOUN
m-1152	20	24	,	,	PUNCT
m-1152	20	25	which	which	PRON
m-1152	20	26	arise	arise	VERB
m-1152	20	27	naturally	naturally	ADV
m-1152	20	28	in	in	ADP
m-1152	20	29	solving	solve	VERB
m-1152	20	30	the	the	DET
m-1152	20	31	abstract	abstract	ADJ
m-1152	20	32	cauchy	cauchy	PROPN
m-1152	20	33	problem	problem	NOUN
m-1152	20	34	du	du	PROPN
m-1152	20	35	dt	dt	PROPN
m-1152	20	36	=	=	SYM
m-1152	20	37	uau	uau	PROPN
m-1152	20	38	,	,	PUNCT
m-1152	20	39	(	(	PUNCT
m-1152	20	40	0	0	NUM
m-1152	20	41	)	)	PUNCT
m-1152	20	42	=	=	SYM
m-1152	21	1	u0	u0	ADJ
m-1152	21	2	,	,	PUNCT
m-1152	21	3	where	where	SCONJ
m-1152	21	4	a	a	PRON
m-1152	21	5	is	be	AUX
m-1152	21	6	the	the	DET
m-1152	21	7	generator	generator	NOUN
m-1152	22	1	[	[	X
m-1152	22	2	2	2	NUM
m-1152	22	3	,	,	PUNCT
m-1152	22	4	6	6	NUM
m-1152	22	5	,	,	PUNCT
m-1152	22	6	11	11	NUM
m-1152	22	7	,	,	PUNCT
m-1152	22	8	12	12	NUM
m-1152	22	9	,	,	PUNCT
m-1152	22	10	13	13	NUM
m-1152	22	11	,	,	PUNCT
m-1152	22	12	15	15	NUM
m-1152	22	13	]	]	PUNCT
m-1152	22	14	.	.	PUNCT
m-1152	23	1	here	here	ADV
m-1152	23	2	,	,	PUNCT
m-1152	23	3	stability	stability	NOUN
m-1152	23	4	is	be	AUX
m-1152	23	5	measured	measure	VERB
m-1152	23	6	analytically	analytically	ADV
m-1152	23	7	in	in	ADP
m-1152	23	8	terms	term	NOUN
m-1152	23	9	of	of	ADP
m-1152	23	10	operator	operator	NOUN
m-1152	23	11	norms	norm	NOUN
m-1152	23	12	and	and	CCONJ
m-1152	23	13	spectral	spectral	ADJ
m-1152	23	14	conditions	condition	NOUN
m-1152	23	15	.	.	PUNCT
m-1152	24	1	classical	classical	ADJ
m-1152	24	2	notions	notion	NOUN
m-1152	24	3	include	include	VERB
m-1152	24	4	asymptotic	asymptotic	ADJ
m-1152	24	5	stability	stability	NOUN
m-1152	24	6	,	,	PUNCT
m-1152	24	7	strong	strong	ADJ
m-1152	24	8	stability	stability	NOUN
m-1152	24	9	,	,	PUNCT
m-1152	24	10	exponential	exponential	ADJ
m-1152	24	11	stability	stability	NOUN
m-1152	24	12	,	,	PUNCT
m-1152	24	13	and	and	CCONJ
m-1152	24	14	uniform	uniform	ADJ
m-1152	24	15	stability	stability	NOUN
m-1152	24	16	,	,	PUNCT
m-1152	24	17	each	each	PRON
m-1152	24	18	re	re	VERB
m-1152	24	19	�	�	PROPN
m-1152	24	20	ecting	ecte	VERB
m-1152	24	21	different	different	ADJ
m-1152	24	22	aspects	aspect	NOUN
m-1152	24	23	of	of	ADP
m-1152	24	24	long	long	ADJ
m-1152	24	25	-	-	PUNCT
m-1152	24	26	time	time	NOUN
m-1152	24	27	behaviour	behaviour	NOUN
m-1152	24	28	.	.	PUNCT
m-1152	25	1	these	these	DET
m-1152	25	2	notions	notion	NOUN
m-1152	25	3	are	be	AUX
m-1152	25	4	closely	closely	ADV
m-1152	25	5	tied	tie	VERB
m-1152	25	6	to	to	ADP
m-1152	25	7	the	the	DET
m-1152	25	8	spectrum	spectrum	NOUN
m-1152	25	9	of	of	ADP
m-1152	25	10	the	the	DET
m-1152	25	11	generator	generator	NOUN
m-1152	25	12	and	and	CCONJ
m-1152	25	13	results	result	NOUN
m-1152	25	14	such	such	ADJ
m-1152	25	15	as	as	ADP
m-1152	25	16	the	the	DET
m-1152	25	17	gearhart	gearhart	PROPN
m-1152	25	18	�	�	PROPN
m-1152	25	19	prüss	prüss	PROPN
m-1152	25	20	theorem	theorem	VERB
m-1152	25	21	[	[	X
m-1152	25	22	5	5	NUM
m-1152	25	23	,	,	PUNCT
m-1152	25	24	7	7	NUM
m-1152	25	25	,	,	PUNCT
m-1152	25	26	8	8	NUM
m-1152	25	27	,	,	PUNCT
m-1152	25	28	9	9	NUM
m-1152	25	29	,	,	PUNCT
m-1152	25	30	10	10	NUM
m-1152	25	31	,	,	PUNCT
m-1152	25	32	17	17	NUM
m-1152	25	33	,	,	PUNCT
m-1152	25	34	32	32	NUM
m-1152	25	35	]	]	PUNCT
m-1152	25	36	.	.	PUNCT
m-1152	26	1	although	although	SCONJ
m-1152	26	2	both	both	DET
m-1152	26	3	traditions	tradition	NOUN
m-1152	26	4	revolve	revolve	VERB
m-1152	26	5	around	around	ADP
m-1152	26	6	the	the	DET
m-1152	26	7	idea	idea	NOUN
m-1152	26	8	of	of	ADP
m-1152	26	9	stability	stability	NOUN
m-1152	26	10	,	,	PUNCT
m-1152	26	11	they	they	PRON
m-1152	26	12	have	have	AUX
m-1152	26	13	historically	historically	ADV
m-1152	26	14	evolved	evolve	VERB
m-1152	26	15	in	in	ADP
m-1152	26	16	relative	relative	ADJ
m-1152	26	17	isolation	isolation	NOUN
m-1152	26	18	:	:	PUNCT
m-1152	26	19	the	the	DET
m-1152	26	20	algebraic	algebraic	ADJ
m-1152	26	21	approach	approach	NOUN
m-1152	26	22	is	be	AUX
m-1152	26	23	structural	structural	ADJ
m-1152	26	24	and	and	CCONJ
m-1152	26	25	norm	norm	NOUN
m-1152	26	26	-	-	PUNCT
m-1152	26	27	free	free	ADJ
m-1152	26	28	,	,	PUNCT
m-1152	26	29	while	while	SCONJ
m-1152	26	30	the	the	DET
m-1152	26	31	analytic	analytic	ADJ
m-1152	26	32	approach	approach	NOUN
m-1152	26	33	is	be	AUX
m-1152	26	34	spectral	spectral	ADJ
m-1152	26	35	and	and	CCONJ
m-1152	26	36	dynamical	dynamical	ADJ
m-1152	26	37	.	.	PUNCT
m-1152	27	1	a	a	DET
m-1152	27	2	recent	recent	ADJ
m-1152	27	3	advance	advance	NOUN
m-1152	27	4	has	have	AUX
m-1152	27	5	begun	begin	VERB
m-1152	27	6	to	to	PART
m-1152	27	7	bridge	bridge	VERB
m-1152	27	8	this	this	DET
m-1152	27	9	divide	divide	NOUN
m-1152	27	10	.	.	PUNCT
m-1152	28	1	tom	tom	PROPN
m-1152	28	2	,	,	PUNCT
m-1152	28	3	udoaka	udoaka	ADV
m-1152	28	4	,	,	PUNCT
m-1152	28	5	and	and	CCONJ
m-1152	28	6	udo	udo	PROPN
m-1152	28	7	�	�	PROPN
m-1152	28	8	a	a	PRON
m-1152	28	9	(	(	PUNCT
m-1152	28	10	2025	2025	NUM
m-1152	28	11	)	)	PUNCT
m-1152	29	1	[	[	X
m-1152	29	2	3	3	NUM
m-1152	29	3	]	]	PUNCT
m-1152	29	4	introduced	introduce	VERB
m-1152	29	5	kw	kw	NOUN
m-1152	29	6	-	-	NOUN
m-1152	29	7	stability	stability	NOUN
m-1152	29	8	into	into	ADP
m-1152	29	9	the	the	DET
m-1152	29	10	setting	setting	NOUN
m-1152	29	11	of	of	ADP
m-1152	29	12	semigroups	semigroup	NOUN
m-1152	29	13	of	of	ADP
m-1152	29	14	bounded	bounded	ADJ
m-1152	29	15	linear	linear	PROPN
m-1152	29	16	operators	operator	NOUN
m-1152	29	17	,	,	PUNCT
m-1152	29	18	proving	prove	VERB
m-1152	29	19	that	that	SCONJ
m-1152	29	20	every	every	DET
m-1152	29	21	strongly	strongly	ADV
m-1152	29	22	continuous	continuous	ADJ
m-1152	29	23	(	(	PUNCT
m-1152	29	24	c0	c0	NOUN
m-1152	29	25	)	)	PUNCT
m-1152	29	26	semigroup	semigroup	NOUN
m-1152	29	27	on	on	ADP
m-1152	29	28	a	a	DET
m-1152	29	29	banach	banach	NOUN
m-1152	29	30	space	space	NOUN
m-1152	29	31	is	be	AUX
m-1152	29	32	stable	stable	ADJ
m-1152	29	33	in	in	ADP
m-1152	29	34	the	the	DET
m-1152	29	35	sense	sense	NOUN
m-1152	29	36	of	of	ADP
m-1152	29	37	koch	koch	PROPN
m-1152	29	38	�	�	PROPN
m-1152	29	39	wallace	wallace	PROPN
m-1152	29	40	.	.	PUNCT
m-1152	30	1	this	this	DET
m-1152	30	2	recognition	recognition	NOUN
m-1152	30	3	provides	provide	VERB
m-1152	30	4	a	a	DET
m-1152	30	5	new	new	ADJ
m-1152	30	6	link	link	NOUN
m-1152	30	7	between	between	ADP
m-1152	30	8	classical	classical	ADJ
m-1152	30	9	semigroup	semigroup	NOUN
m-1152	30	10	stability	stability	NOUN
m-1152	30	11	theory	theory	NOUN
m-1152	30	12	and	and	CCONJ
m-1152	30	13	operator	operator	NOUN
m-1152	30	14	semigroup	semigroup	NOUN
m-1152	30	15	analysis	analysis	NOUN
m-1152	30	16	,	,	PUNCT
m-1152	30	17	placing	place	VERB
m-1152	30	18	algebraic	algebraic	ADJ
m-1152	30	19	stability	stability	NOUN
m-1152	30	20	at	at	ADP
m-1152	30	21	the	the	DET
m-1152	30	22	foundation	foundation	NOUN
m-1152	30	23	of	of	ADP
m-1152	30	24	analytic	analytic	ADJ
m-1152	30	25	operator	operator	NOUN
m-1152	30	26	theory	theory	NOUN
m-1152	30	27	.	.	PUNCT
m-1152	31	1	their	their	PRON
m-1152	31	2	work	work	NOUN
m-1152	31	3	suggests	suggest	VERB
m-1152	31	4	further	further	ADJ
m-1152	31	5	directions	direction	NOUN
m-1152	31	6	for	for	ADP
m-1152	31	7	stability	stability	NOUN
m-1152	31	8	research	research	NOUN
m-1152	31	9	,	,	PUNCT
m-1152	31	10	including	include	VERB
m-1152	31	11	the	the	DET
m-1152	31	12	study	study	NOUN
m-1152	31	13	of	of	ADP
m-1152	31	14	unbounded	unbounded	ADJ
m-1152	31	15	operator	operator	NOUN
m-1152	31	16	semigroups	semigroup	NOUN
m-1152	31	17	,	,	PUNCT
m-1152	31	18	hypersemigroups	hypersemigroup	NOUN
m-1152	31	19	,	,	PUNCT
m-1152	31	20	and	and	CCONJ
m-1152	31	21	semigroups	semigroup	NOUN
m-1152	31	22	arising	arise	VERB
m-1152	31	23	in	in	ADP
m-1152	31	24	stochastic	stochastic	ADJ
m-1152	31	25	analysis	analysis	NOUN
m-1152	31	26	[	[	X
m-1152	31	27	7	7	NUM
m-1152	31	28	,	,	PUNCT
m-1152	31	29	9	9	NUM
m-1152	31	30	,	,	PUNCT
m-1152	31	31	16	16	NUM
m-1152	31	32	,	,	PUNCT
m-1152	31	33	32	32	NUM
m-1152	31	34	]	]	PUNCT
m-1152	31	35	.	.	PUNCT
m-1152	32	1	in	in	ADP
m-1152	32	2	what	what	PRON
m-1152	32	3	follows	follow	VERB
m-1152	32	4	,	,	PUNCT
m-1152	32	5	we	we	PRON
m-1152	32	6	continue	continue	VERB
m-1152	32	7	this	this	DET
m-1152	32	8	line	line	NOUN
m-1152	32	9	of	of	ADP
m-1152	32	10	investigation	investigation	NOUN
m-1152	32	11	by	by	ADP
m-1152	32	12	examining	examine	VERB
m-1152	32	13	how	how	SCONJ
m-1152	32	14	kw	kw	NOUN
m-1152	32	15	-	-	PUNCT
m-1152	32	16	stability	stability	NOUN
m-1152	32	17	interacts	interact	VERB
m-1152	32	18	with	with	ADP
m-1152	32	19	analytic	analytic	ADJ
m-1152	32	20	stability	stability	NOUN
m-1152	32	21	notions	notion	NOUN
m-1152	32	22	.	.	PUNCT
m-1152	33	1	in	in	ADP
m-1152	33	2	particular	particular	ADJ
m-1152	33	3	,	,	PUNCT
m-1152	33	4	we	we	PRON
m-1152	33	5	identify	identify	VERB
m-1152	33	6	the	the	DET
m-1152	33	7	spectral	spectral	ADJ
m-1152	33	8	conditions	condition	NOUN
m-1152	33	9	under	under	ADP
m-1152	33	10	which	which	PRON
m-1152	33	11	algebraic	algebraic	ADJ
m-1152	33	12	and	and	CCONJ
m-1152	33	13	analytic	analytic	ADJ
m-1152	33	14	stability	stability	NOUN
m-1152	33	15	coincide	coincide	NOUN
m-1152	33	16	and	and	CCONJ
m-1152	33	17	illustrate	illustrate	VERB
m-1152	33	18	this	this	DET
m-1152	33	19	interplay	interplay	NOUN
m-1152	33	20	with	with	ADP
m-1152	33	21	canonical	canonical	ADJ
m-1152	33	22	examples	example	NOUN
m-1152	33	23	such	such	ADJ
m-1152	33	24	as	as	ADP
m-1152	33	25	translation	translation	NOUN
m-1152	33	26	,	,	PUNCT
m-1152	33	27	shift	shift	NOUN
m-1152	33	28	,	,	PUNCT
m-1152	33	29	heat	heat	NOUN
m-1152	33	30	,	,	PUNCT
m-1152	33	31	and	and	CCONJ
m-1152	33	32	damped	damp	VERB
m-1152	33	33	wave	wave	NOUN
m-1152	33	34	semigroups	semigroup	NOUN
m-1152	33	35	.	.	PUNCT
m-1152	34	1	2	2	NUM
m-1152	34	2	preliminaries	preliminary	NOUN
m-1152	34	3	de	de	X
m-1152	34	4	�	�	NOUN
m-1152	34	5	nition	nition	NOUN
m-1152	34	6	2.1	2.1	NUM
m-1152	34	7	(	(	PUNCT
m-1152	34	8	normed	normed	ADJ
m-1152	34	9	linear	linear	ADJ
m-1152	34	10	space	space	NOUN
m-1152	34	11	)	)	PUNCT
m-1152	34	12	.	.	PUNCT
m-1152	35	1	a	a	DET
m-1152	35	2	normed	normed	ADJ
m-1152	35	3	linear	linear	ADJ
m-1152	35	4	space	space	NOUN
m-1152	35	5	is	be	AUX
m-1152	35	6	a	a	DET
m-1152	35	7	pair	pair	NOUN
m-1152	35	8	(	(	PUNCT
m-1152	35	9	x	x	X
m-1152	35	10	,	,	PUNCT
m-1152	35	11	∥	∥	X
m-1152	35	12	·	·	PUNCT
m-1152	35	13	∥	∥	X
m-1152	35	14	)	)	PUNCT
m-1152	35	15	where	where	SCONJ
m-1152	35	16	x	x	PRON
m-1152	35	17	is	be	AUX
m-1152	35	18	a	a	DET
m-1152	35	19	vector	vector	NOUN
m-1152	35	20	space	space	NOUN
m-1152	35	21	over	over	ADP
m-1152	35	22	the	the	DET
m-1152	35	23	�	�	PROPN
m-1152	35	24	eld	eld	NOUN
m-1152	35	25	r	r	NOUN
m-1152	35	26	or	or	CCONJ
m-1152	35	27	c	c	NOUN
m-1152	35	28	,	,	PUNCT
m-1152	35	29	and	and	CCONJ
m-1152	35	30	∥	∥	PRON
m-1152	35	31	·	·	PUNCT
m-1152	36	1	∥	∥	X
m-1152	36	2	:	:	PUNCT
m-1152	37	1	x	x	X
m-1152	37	2	→	→	PUNCT
m-1152	37	3	[	[	X
m-1152	37	4	0,∞	0,∞	NUM
m-1152	37	5	)	)	PUNCT
m-1152	37	6	is	be	AUX
m-1152	37	7	a	a	DET
m-1152	37	8	function	function	NOUN
m-1152	37	9	,	,	PUNCT
m-1152	37	10	called	call	VERB
m-1152	37	11	a	a	DET
m-1152	37	12	norm	norm	NOUN
m-1152	37	13	,	,	PUNCT
m-1152	37	14	satisfying	satisfy	VERB
m-1152	37	15	the	the	DET
m-1152	37	16	following	follow	VERB
m-1152	37	17	properties	property	NOUN
m-1152	37	18	for	for	ADP
m-1152	37	19	all	all	DET
m-1152	37	20	x	x	NOUN
m-1152	37	21	,	,	PUNCT
m-1152	37	22	y	y	PROPN
m-1152	37	23	∈	∈	PROPN
m-1152	37	24	x	x	X
m-1152	37	25	and	and	CCONJ
m-1152	37	26	all	all	DET
m-1152	37	27	scalars	scalar	VERB
m-1152	37	28	α	α	X
m-1152	37	29	:	:	PUNCT
m-1152	37	30	1	1	NUM
m-1152	37	31	.	.	X
m-1152	37	32	positivity	positivity	NOUN
m-1152	37	33	:	:	PUNCT
m-1152	37	34	∥x∥	∥x∥	NOUN
m-1152	37	35	≥	≥	NOUN
m-1152	37	36	0	0	NUM
m-1152	37	37	,	,	PUNCT
m-1152	37	38	and	and	CCONJ
m-1152	37	39	∥x∥	∥x∥	NOUN
m-1152	38	1	=	=	SYM
m-1152	38	2	0	0	PUNCT
m-1152	38	3	if	if	SCONJ
m-1152	38	4	and	and	CCONJ
m-1152	38	5	only	only	ADV
m-1152	38	6	if	if	SCONJ
m-1152	38	7	x	x	PROPN
m-1152	38	8	=	=	NOUN
m-1152	38	9	0	0	NUM
m-1152	38	10	.	.	NOUN
m-1152	39	1	2	2	NUM
m-1152	39	2	.	.	X
m-1152	39	3	homogeneity	homogeneity	NOUN
m-1152	39	4	(	(	PUNCT
m-1152	39	5	absolute	absolute	ADJ
m-1152	39	6	scalability	scalability	NOUN
m-1152	39	7	):	):	PUNCT
m-1152	39	8	∥αx∥	∥αx∥	ADV
m-1152	39	9	=	=	SYM
m-1152	39	10	|α|	|α|	NUM
m-1152	39	11	∥x∥.	∥x∥.	NOUN
m-1152	39	12	3	3	NUM
m-1152	39	13	.	.	PUNCT
m-1152	39	14	triangle	triangle	NOUN
m-1152	39	15	inequality	inequality	NOUN
m-1152	39	16	:	:	PUNCT
m-1152	39	17	∥x+	∥x+	X
m-1152	39	18	y∥	y∥	VERB
m-1152	39	19	≤	≤	NOUN
m-1152	39	20	∥x∥+	∥x∥+	ADP
m-1152	39	21	∥y∥.	∥y∥.	X
m-1152	39	22	de	de	PROPN
m-1152	39	23	�	�	PROPN
m-1152	39	24	nition	nition	NOUN
m-1152	39	25	2.2	2.2	NUM
m-1152	39	26	(	(	PUNCT
m-1152	39	27	banach	banach	NOUN
m-1152	39	28	space	space	NOUN
m-1152	39	29	)	)	PUNCT
m-1152	39	30	.	.	PUNCT
m-1152	40	1	a	a	DET
m-1152	40	2	banach	banach	NOUN
m-1152	40	3	space	space	NOUN
m-1152	40	4	is	be	AUX
m-1152	40	5	a	a	DET
m-1152	40	6	vector	vector	NOUN
m-1152	40	7	space	space	NOUN
m-1152	40	8	x	x	NOUN
m-1152	40	9	over	over	ADP
m-1152	40	10	the	the	DET
m-1152	40	11	�	�	PROPN
m-1152	40	12	eld	eld	NOUN
m-1152	40	13	r	r	NOUN
m-1152	40	14	or	or	CCONJ
m-1152	40	15	c	c	PROPN
m-1152	40	16	together	together	ADV
m-1152	40	17	with	with	ADP
m-1152	40	18	a	a	DET
m-1152	40	19	norm	norm	NOUN
m-1152	40	20	∥	∥	X
m-1152	40	21	·	·	PUNCT
m-1152	40	22	∥	∥	X
m-1152	41	1	:	:	PUNCT
m-1152	41	2	x	x	X
m-1152	41	3	→	→	PUNCT
m-1152	41	4	[	[	X
m-1152	41	5	0,∞	0,∞	NUM
m-1152	41	6	)	)	PUNCT
m-1152	41	7	such	such	ADJ
m-1152	41	8	that	that	SCONJ
m-1152	41	9	(	(	PUNCT
m-1152	41	10	x	x	X
m-1152	41	11	,	,	PUNCT
m-1152	41	12	∥	∥	X
m-1152	41	13	·	·	PUNCT
m-1152	41	14	∥	∥	X
m-1152	41	15	)	)	PUNCT
m-1152	41	16	is	be	AUX
m-1152	41	17	complete	complete	ADJ
m-1152	41	18	;	;	PUNCT
m-1152	41	19	that	that	ADV
m-1152	41	20	is	is	ADV
m-1152	41	21	,	,	PUNCT
m-1152	41	22	every	every	DET
m-1152	41	23	cauchy	cauchy	ADJ
m-1152	41	24	sequence	sequence	NOUN
m-1152	41	25	{	{	PUNCT
m-1152	41	26	xn	xn	NOUN
m-1152	41	27	}	}	PUNCT
m-1152	41	28	in	in	ADP
m-1152	41	29	x	x	SYM
m-1152	41	30	converges	converge	NOUN
m-1152	41	31	to	to	ADP
m-1152	41	32	some	some	DET
m-1152	41	33	x	x	SYM
m-1152	41	34	∈	∈	PROPN
m-1152	41	35	x	x	PUNCT
m-1152	41	36	with	with	ADP
m-1152	41	37	respect	respect	NOUN
m-1152	41	38	to	to	ADP
m-1152	41	39	the	the	DET
m-1152	41	40	norm	norm	NOUN
m-1152	41	41	∥	∥	X
m-1152	41	42	·	·	PUNCT
m-1152	41	43	∥.	∥.	ADV
m-1152	41	44	formally	formally	ADV
m-1152	41	45	,	,	PUNCT
m-1152	41	46	for	for	ADP
m-1152	41	47	every	every	DET
m-1152	41	48	sequence	sequence	NOUN
m-1152	41	49	{	{	PUNCT
m-1152	41	50	xn	xn	NOUN
m-1152	41	51	}	}	PUNCT
m-1152	41	52	in	in	ADP
m-1152	41	53	x	x	SYM
m-1152	41	54	,	,	PUNCT
m-1152	41	55	if	if	SCONJ
m-1152	41	56	lim	lim	PROPN
m-1152	41	57	m	m	PROPN
m-1152	41	58	,	,	PUNCT
m-1152	41	59	n→∞	n→∞	X
m-1152	41	60	∥xn	∥xn	PROPN
m-1152	41	61	−	−	PROPN
m-1152	41	62	xm∥	xm∥	PROPN
m-1152	41	63	=	=	SYM
m-1152	41	64	0	0	PROPN
m-1152	41	65	,	,	PUNCT
m-1152	41	66	then	then	ADV
m-1152	41	67	there	there	PRON
m-1152	41	68	exists	exist	VERB
m-1152	41	69	x	x	X
m-1152	41	70	∈	∈	PROPN
m-1152	41	71	x	x	X
m-1152	41	72	such	such	ADJ
m-1152	41	73	that	that	SCONJ
m-1152	41	74	lim	lim	PROPN
m-1152	41	75	n→∞	n→∞	PRON
m-1152	41	76	∥xn	∥xn	PROPN
m-1152	41	77	−	−	PROPN
m-1152	41	78	x∥	x∥	PROPN
m-1152	42	1	=	=	SYM
m-1152	42	2	0	0	X
m-1152	42	3	.	.	PUNCT
m-1152	43	1	ijo	ijo	PROPN
m-1152	43	2	international	international	PROPN
m-1152	43	3	journal	journal	PROPN
m-1152	43	4	of	of	ADP
m-1152	43	5	mathematics	mathematics	PROPN
m-1152	43	6	(	(	PUNCT
m-1152	43	7	issn	issn	PROPN
m-1152	43	8	:	:	PUNCT
m-1152	43	9	2992	2992	NUM
m-1152	43	10	-	-	SYM
m-1152	43	11	4421	4421	NUM
m-1152	43	12	)	)	PUNCT
m-1152	43	13	volume	volume	NOUN
m-1152	43	14	08	08	NUM
m-1152	44	1	|	|	ADV
m-1152	44	2	issue	issue	NOUN
m-1152	44	3	9	9	NUM
m-1152	44	4	|	|	CCONJ
m-1152	44	5	september	september	PROPN
m-1152	44	6	2025	2025	NUM
m-1152	45	1	|	|	ADV
m-1152	45	2	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1152	45	3	11	11	NUM
m-1152	45	4	de	de	X
m-1152	45	5	�	�	PROPN
m-1152	45	6	nition	nition	NOUN
m-1152	45	7	2.3	2.3	NUM
m-1152	45	8	(	(	PUNCT
m-1152	45	9	bounded	bound	VERB
m-1152	45	10	linear	linear	ADJ
m-1152	45	11	operator	operator	NOUN
m-1152	45	12	)	)	PUNCT
m-1152	45	13	.	.	PUNCT
m-1152	46	1	let	let	VERB
m-1152	46	2	x	x	PRON
m-1152	46	3	and	and	CCONJ
m-1152	46	4	y	y	PROPN
m-1152	46	5	be	be	AUX
m-1152	46	6	normed	norme	VERB
m-1152	46	7	linear	linear	ADJ
m-1152	46	8	spaces	space	NOUN
m-1152	46	9	.	.	PUNCT
m-1152	47	1	a	a	DET
m-1152	47	2	mapping	mapping	NOUN
m-1152	47	3	t	t	NOUN
m-1152	47	4	:	:	PUNCT
m-1152	47	5	x	x	X
m-1152	47	6	→	→	SYM
m-1152	47	7	y	y	PROPN
m-1152	47	8	is	be	AUX
m-1152	47	9	called	call	VERB
m-1152	47	10	a	a	DET
m-1152	47	11	linear	linear	ADJ
m-1152	47	12	operator	operator	NOUN
m-1152	47	13	if	if	SCONJ
m-1152	47	14	t	t	PROPN
m-1152	47	15	(	(	PUNCT
m-1152	47	16	αx+	αx+	VERB
m-1152	47	17	βy	βy	ADP
m-1152	47	18	)	)	PUNCT
m-1152	48	1	=	=	SYM
m-1152	48	2	αt	αt	PROPN
m-1152	48	3	(	(	PUNCT
m-1152	48	4	x	x	X
m-1152	48	5	)	)	PUNCT
m-1152	48	6	+	+	CCONJ
m-1152	48	7	βt	βt	PRON
m-1152	48	8	(	(	PUNCT
m-1152	48	9	y	y	NOUN
m-1152	48	10	)	)	PUNCT
m-1152	48	11	,	,	PUNCT
m-1152	48	12	∀	∀	X
m-1152	48	13	x	x	NOUN
m-1152	48	14	,	,	PUNCT
m-1152	48	15	y	y	PROPN
m-1152	48	16	∈	∈	PROPN
m-1152	48	17	x	x	PROPN
m-1152	48	18	,	,	PUNCT
m-1152	48	19	α	α	X
m-1152	48	20	,	,	PUNCT
m-1152	48	21	β	β	X
m-1152	48	22	∈	∈	NOUN
m-1152	48	23	r	r	NOUN
m-1152	48	24	or	or	CCONJ
m-1152	48	25	c.	c.	NOUN
m-1152	48	26	the	the	DET
m-1152	48	27	operator	operator	NOUN
m-1152	48	28	t	t	PROPN
m-1152	48	29	is	be	AUX
m-1152	48	30	said	say	VERB
m-1152	48	31	to	to	PART
m-1152	48	32	be	be	AUX
m-1152	48	33	bounded	bound	VERB
m-1152	48	34	if	if	SCONJ
m-1152	48	35	there	there	PRON
m-1152	48	36	exists	exist	VERB
m-1152	48	37	a	a	DET
m-1152	48	38	constant	constant	ADJ
m-1152	48	39	m	m	NOUN
m-1152	48	40	>	>	X
m-1152	48	41	0	0	NUM
m-1152	48	42	such	such	ADJ
m-1152	48	43	that	that	SCONJ
m-1152	48	44	∥t	∥t	PROPN
m-1152	48	45	(	(	PUNCT
m-1152	48	46	x)∥y	x)∥y	PROPN
m-1152	48	47	≤	≤	PROPN
m-1152	48	48	m∥x∥x	m∥x∥x	NOUN
m-1152	48	49	,	,	PUNCT
m-1152	48	50	∀	∀	X
m-1152	48	51	x	x	SYM
m-1152	48	52	∈	∈	NOUN
m-1152	48	53	x.	x.	NOUN
m-1152	48	54	equivalently	equivalently	PROPN
m-1152	48	55	,	,	PUNCT
m-1152	48	56	t	t	PROPN
m-1152	48	57	is	be	AUX
m-1152	48	58	bounded	bound	VERB
m-1152	48	59	if	if	SCONJ
m-1152	48	60	and	and	CCONJ
m-1152	48	61	only	only	ADV
m-1152	48	62	if	if	SCONJ
m-1152	48	63	it	it	PRON
m-1152	48	64	is	be	AUX
m-1152	48	65	continuous	continuous	ADJ
m-1152	48	66	at	at	ADP
m-1152	48	67	0	0	NUM
m-1152	48	68	(	(	PUNCT
m-1152	48	69	and	and	CCONJ
m-1152	48	70	hence	hence	ADV
m-1152	48	71	continuous	continuous	ADJ
m-1152	48	72	everywhere	everywhere	ADV
m-1152	48	73	)	)	PUNCT
m-1152	48	74	.	.	PUNCT
m-1152	49	1	de	de	PROPN
m-1152	49	2	�	�	PROPN
m-1152	49	3	nition	nition	NOUN
m-1152	49	4	2.4	2.4	NUM
m-1152	49	5	(	(	PUNCT
m-1152	49	6	operator	operator	NOUN
m-1152	49	7	semigroup	semigroup	NOUN
m-1152	49	8	)	)	PUNCT
m-1152	49	9	.	.	PUNCT
m-1152	50	1	let	let	VERB
m-1152	50	2	x	x	PRON
m-1152	50	3	be	be	AUX
m-1152	50	4	a	a	DET
m-1152	50	5	banach	banach	NOUN
m-1152	50	6	space	space	NOUN
m-1152	50	7	and	and	CCONJ
m-1152	50	8	b(x	b(x	NOUN
m-1152	50	9	)	)	PUNCT
m-1152	50	10	the	the	DET
m-1152	50	11	algebra	algebra	NOUN
m-1152	50	12	of	of	ADP
m-1152	50	13	all	all	DET
m-1152	50	14	bounded	bound	VERB
m-1152	50	15	linear	linear	PROPN
m-1152	50	16	operators	operator	NOUN
m-1152	50	17	on	on	ADP
m-1152	50	18	x.	x.	PROPN
m-1152	50	19	a	a	DET
m-1152	50	20	family	family	NOUN
m-1152	50	21	{	{	PUNCT
m-1152	50	22	t	t	PROPN
m-1152	50	23	(	(	PUNCT
m-1152	50	24	t)}t≥0	t)}t≥0	NOUN
m-1152	50	25	⊆	⊆	NUM
m-1152	50	26	b(x	b(x	NOUN
m-1152	50	27	)	)	PUNCT
m-1152	50	28	is	be	AUX
m-1152	50	29	called	call	VERB
m-1152	50	30	a	a	DET
m-1152	50	31	strongly	strongly	ADV
m-1152	50	32	continuous	continuous	ADJ
m-1152	50	33	semigroup	semigroup	NOUN
m-1152	50	34	(	(	PUNCT
m-1152	50	35	c0	c0	NOUN
m-1152	50	36	-	-	PUNCT
m-1152	50	37	semigroup	semigroup	NOUN
m-1152	50	38	)	)	PUNCT
m-1152	51	1	if	if	SCONJ
m-1152	51	2	:	:	PUNCT
m-1152	51	3	1	1	X
m-1152	51	4	.	.	X
m-1152	51	5	t	t	PROPN
m-1152	51	6	(	(	PUNCT
m-1152	51	7	0	0	NUM
m-1152	51	8	)	)	PUNCT
m-1152	51	9	=	=	SYM
m-1152	51	10	i	i	PRON
m-1152	51	11	(	(	PUNCT
m-1152	51	12	the	the	DET
m-1152	51	13	identity	identity	NOUN
m-1152	51	14	operator	operator	NOUN
m-1152	51	15	)	)	PUNCT
m-1152	51	16	,	,	PUNCT
m-1152	51	17	2	2	X
m-1152	51	18	.	.	X
m-1152	51	19	t	t	PROPN
m-1152	51	20	(	(	PUNCT
m-1152	51	21	t+	t+	NOUN
m-1152	51	22	s	s	NOUN
m-1152	51	23	)	)	PUNCT
m-1152	51	24	=	=	SYM
m-1152	51	25	t	t	PROPN
m-1152	51	26	(	(	PUNCT
m-1152	51	27	t)t	t)t	X
m-1152	51	28	(	(	PUNCT
m-1152	51	29	s	s	X
m-1152	51	30	)	)	PUNCT
m-1152	51	31	for	for	ADP
m-1152	51	32	all	all	DET
m-1152	51	33	t	t	PROPN
m-1152	51	34	,	,	PUNCT
m-1152	51	35	s	s	VERB
m-1152	51	36	≥	≥	NOUN
m-1152	51	37	0	0	NUM
m-1152	51	38	,	,	PUNCT
m-1152	51	39	3	3	NUM
m-1152	51	40	.	.	X
m-1152	52	1	for	for	ADP
m-1152	52	2	every	every	DET
m-1152	52	3	x	x	SYM
m-1152	52	4	∈	∈	PROPN
m-1152	52	5	x	x	NOUN
m-1152	52	6	,	,	PUNCT
m-1152	52	7	limt→0	limt→0	PROPN
m-1152	52	8	+	+	SYM
m-1152	52	9	t	t	PROPN
m-1152	52	10	(	(	PUNCT
m-1152	52	11	t)x	t)x	X
m-1152	52	12	=	=	SYM
m-1152	52	13	x.	x.	NOUN
m-1152	52	14	for	for	ADP
m-1152	52	15	more	more	ADJ
m-1152	52	16	about	about	ADP
m-1152	52	17	this	this	PRON
m-1152	52	18	,	,	PUNCT
m-1152	52	19	the	the	DET
m-1152	52	20	reader	reader	NOUN
m-1152	52	21	is	be	AUX
m-1152	52	22	referred	refer	VERB
m-1152	52	23	to	to	ADP
m-1152	52	24	[	[	X
m-1152	52	25	3	3	NUM
m-1152	52	26	]	]	PUNCT
m-1152	52	27	.	.	PUNCT
m-1152	53	1	de	de	PROPN
m-1152	53	2	�	�	PROPN
m-1152	53	3	nition	nition	NOUN
m-1152	53	4	2.5	2.5	NUM
m-1152	53	5	(	(	PUNCT
m-1152	53	6	koch	koch	PROPN
m-1152	53	7	and	and	CCONJ
m-1152	53	8	wallace	wallace	PROPN
m-1152	53	9	stability	stability	NOUN
m-1152	53	10	[	[	X
m-1152	53	11	1	1	NUM
m-1152	53	12	]	]	NUM
m-1152	53	13	)	)	PUNCT
m-1152	53	14	.	.	PUNCT
m-1152	54	1	a	a	DET
m-1152	54	2	semigroup	semigroup	NOUN
m-1152	54	3	s	s	VERB
m-1152	54	4	is	be	AUX
m-1152	54	5	called	call	VERB
m-1152	54	6	stable	stable	ADJ
m-1152	54	7	if	if	SCONJ
m-1152	54	8	for	for	ADP
m-1152	54	9	all	all	DET
m-1152	54	10	a	a	PRON
m-1152	54	11	,	,	PUNCT
m-1152	54	12	b	b	X
m-1152	54	13	∈	∈	PROPN
m-1152	54	14	s	s	PART
m-1152	54	15	:	:	PUNCT
m-1152	54	16	�	�	PROPN
m-1152	54	17	(	(	PUNCT
m-1152	54	18	right	right	ADJ
m-1152	54	19	stability	stability	NOUN
m-1152	54	20	):	):	PUNCT
m-1152	54	21	as	as	ADP
m-1152	54	22	⊆	⊆	NUM
m-1152	54	23	abs	ab	NOUN
m-1152	54	24	=	=	NOUN
m-1152	54	25	⇒	⇒	NOUN
m-1152	54	26	as	as	SCONJ
m-1152	54	27	=	=	NOUN
m-1152	54	28	abs	ab	NOUN
m-1152	54	29	,	,	PUNCT
m-1152	54	30	�	�	PROPN
m-1152	54	31	(	(	PUNCT
m-1152	54	32	left	leave	VERB
m-1152	54	33	stability	stability	NOUN
m-1152	54	34	):	):	PUNCT
m-1152	54	35	sa	sa	PROPN
m-1152	54	36	⊆	⊆	NUM
m-1152	54	37	sab	sab	ADJ
m-1152	54	38	=	=	NOUN
m-1152	54	39	⇒	⇒	NOUN
m-1152	54	40	sa	sa	X
m-1152	55	1	=	=	PUNCT
m-1152	55	2	sab	sab	PROPN
m-1152	55	3	.	.	PUNCT
m-1152	56	1	this	this	DET
m-1152	56	2	de	de	PROPN
m-1152	56	3	�	�	PROPN
m-1152	56	4	nition	nition	NOUN
m-1152	56	5	was	be	AUX
m-1152	56	6	also	also	ADV
m-1152	56	7	given	give	VERB
m-1152	56	8	by	by	ADP
m-1152	56	9	east	east	NOUN
m-1152	56	10	using	use	VERB
m-1152	56	11	green	green	PROPN
m-1152	56	12	's	's	PART
m-1152	56	13	relation	relation	NOUN
m-1152	56	14	in	in	ADP
m-1152	56	15	[	[	X
m-1152	56	16	3	3	NUM
m-1152	56	17	,	,	PUNCT
m-1152	56	18	33	33	NUM
m-1152	56	19	]	]	PUNCT
m-1152	56	20	.	.	PUNCT
m-1152	57	1	equivalently	equivalently	ADV
m-1152	57	2	,	,	PUNCT
m-1152	57	3	if	if	SCONJ
m-1152	57	4	a	a	DET
m-1152	57	5	≤j	≤j	PROPN
m-1152	57	6	ab	ab	PROPN
m-1152	57	7	,	,	PUNCT
m-1152	57	8	then	then	ADV
m-1152	57	9	arab	arab	PROPN
m-1152	57	10	,	,	PUNCT
m-1152	57	11	and	and	CCONJ
m-1152	57	12	if	if	SCONJ
m-1152	57	13	a	a	DET
m-1152	57	14	≤j	≤j	PROPN
m-1152	57	15	ba	ba	PROPN
m-1152	57	16	,	,	PUNCT
m-1152	57	17	then	then	ADV
m-1152	57	18	alba	alba	PROPN
m-1152	57	19	.	.	PUNCT
m-1152	58	1	3	3	NUM
m-1152	58	2	main	main	ADJ
m-1152	58	3	results	result	NOUN
m-1152	58	4	proposition	proposition	NOUN
m-1152	58	5	3.1	3.1	NUM
m-1152	58	6	(	(	PUNCT
m-1152	58	7	kw	kw	NOUN
m-1152	58	8	-	-	PUNCT
m-1152	58	9	stability	stability	NOUN
m-1152	58	10	of	of	ADP
m-1152	58	11	c0−semigroups	c0−semigroup	NOUN
m-1152	58	12	[	[	X
m-1152	58	13	3	3	NUM
m-1152	58	14	]	]	NUM
m-1152	58	15	)	)	PUNCT
m-1152	58	16	.	.	PUNCT
m-1152	59	1	every	every	DET
m-1152	59	2	c0	c0	PROPN
m-1152	59	3	-	-	PUNCT
m-1152	59	4	semigroup	semigroup	NOUN
m-1152	59	5	of	of	ADP
m-1152	59	6	bounded	bound	VERB
m-1152	59	7	linear	linear	PROPN
m-1152	59	8	operators	operator	NOUN
m-1152	59	9	is	be	AUX
m-1152	59	10	stable	stable	ADJ
m-1152	59	11	in	in	ADP
m-1152	59	12	the	the	DET
m-1152	59	13	sense	sense	NOUN
m-1152	59	14	of	of	ADP
m-1152	59	15	koch	koch	PROPN
m-1152	59	16	�	�	PROPN
m-1152	59	17	wallace	wallace	PROPN
m-1152	59	18	.	.	PUNCT
m-1152	60	1	proof	proof	NOUN
m-1152	60	2	.	.	PUNCT
m-1152	61	1	let	let	VERB
m-1152	61	2	s	s	PRON
m-1152	61	3	=	=	X
m-1152	61	4	{	{	PUNCT
m-1152	61	5	t	t	PROPN
m-1152	61	6	(	(	PUNCT
m-1152	61	7	t	t	PROPN
m-1152	61	8	)	)	PUNCT
m-1152	61	9	:	:	PUNCT
m-1152	62	1	t	t	X
m-1152	62	2	≥	≥	NOUN
m-1152	62	3	0	0	NUM
m-1152	62	4	}	}	PUNCT
m-1152	62	5	.	.	PUNCT
m-1152	63	1	pick	pick	VERB
m-1152	63	2	a	a	DET
m-1152	63	3	=	=	X
m-1152	63	4	t	t	PROPN
m-1152	63	5	(	(	PUNCT
m-1152	63	6	t	t	PROPN
m-1152	63	7	)	)	PUNCT
m-1152	63	8	,	,	PUNCT
m-1152	63	9	b	b	X
m-1152	63	10	=	=	SYM
m-1152	63	11	t	t	PROPN
m-1152	63	12	(	(	PUNCT
m-1152	63	13	s	s	NOUN
m-1152	63	14	)	)	PUNCT
m-1152	63	15	with	with	ADP
m-1152	63	16	t	t	PROPN
m-1152	63	17	,	,	PUNCT
m-1152	63	18	s	s	PART
m-1152	63	19	≥	≥	NOUN
m-1152	63	20	0	0	NUM
m-1152	63	21	.	.	PUNCT
m-1152	64	1	by	by	ADP
m-1152	64	2	the	the	DET
m-1152	64	3	semigroup	semigroup	PROPN
m-1152	64	4	law	law	NOUN
m-1152	64	5	,	,	PUNCT
m-1152	64	6	ab	ab	PROPN
m-1152	64	7	=	=	SYM
m-1152	64	8	t	t	PROPN
m-1152	64	9	(	(	PUNCT
m-1152	64	10	t)t	t)t	X
m-1152	64	11	(	(	PUNCT
m-1152	64	12	s	s	X
m-1152	64	13	)	)	PUNCT
m-1152	64	14	=	=	SYM
m-1152	64	15	t	t	PROPN
m-1152	64	16	(	(	PUNCT
m-1152	64	17	t+	t+	NOUN
m-1152	64	18	s	s	NOUN
m-1152	64	19	)	)	PUNCT
m-1152	64	20	=	=	SYM
m-1152	64	21	t	t	PROPN
m-1152	64	22	(	(	PUNCT
m-1152	64	23	s+	s+	PROPN
m-1152	64	24	t	t	PROPN
m-1152	64	25	)	)	PUNCT
m-1152	64	26	=	=	SYM
m-1152	64	27	t	t	PROPN
m-1152	64	28	(	(	PUNCT
m-1152	64	29	s)t	s)t	X
m-1152	64	30	(	(	PUNCT
m-1152	64	31	t	t	NOUN
m-1152	64	32	)	)	PUNCT
m-1152	65	1	=	=	SYM
m-1152	65	2	ba	ba	PROPN
m-1152	65	3	,	,	PUNCT
m-1152	65	4	so	so	ADV
m-1152	65	5	s	s	VERB
m-1152	65	6	is	be	AUX
m-1152	65	7	commutative	commutative	ADJ
m-1152	65	8	.	.	PUNCT
m-1152	66	1	suppose	suppose	VERB
m-1152	66	2	as	as	ADP
m-1152	66	3	⊆	⊆	NUM
m-1152	66	4	abs	ab	NOUN
m-1152	66	5	.	.	PUNCT
m-1152	67	1	for	for	ADP
m-1152	67	2	x	x	X
m-1152	67	3	=	=	SYM
m-1152	67	4	t	t	PROPN
m-1152	67	5	(	(	PUNCT
m-1152	67	6	u	u	NOUN
m-1152	67	7	)	)	PUNCT
m-1152	67	8	∈	∈	PROPN
m-1152	67	9	s	s	PROPN
m-1152	67	10	,	,	PUNCT
m-1152	67	11	(	(	PUNCT
m-1152	67	12	ab)x	ab)x	PROPN
m-1152	67	13	=	=	SYM
m-1152	67	14	t	t	PROPN
m-1152	67	15	(	(	PUNCT
m-1152	67	16	t+	t+	NOUN
m-1152	67	17	s)t	s)t	X
m-1152	67	18	(	(	PUNCT
m-1152	67	19	u	u	NOUN
m-1152	67	20	)	)	PUNCT
m-1152	67	21	=	=	SYM
m-1152	67	22	t	t	PROPN
m-1152	67	23	(	(	PUNCT
m-1152	67	24	t+	t+	NOUN
m-1152	67	25	s+	s+	PUNCT
m-1152	67	26	u	u	NOUN
m-1152	67	27	)	)	PUNCT
m-1152	67	28	.	.	PUNCT
m-1152	68	1	by	by	ADP
m-1152	68	2	commutativity	commutativity	NOUN
m-1152	68	3	,	,	PUNCT
m-1152	68	4	(	(	PUNCT
m-1152	68	5	ab)x	ab)x	NOUN
m-1152	68	6	=	=	SYM
m-1152	68	7	at	at	ADP
m-1152	68	8	(	(	PUNCT
m-1152	68	9	s+u	s+u	ADJ
m-1152	68	10	)	)	PUNCT
m-1152	68	11	∈	∈	PROPN
m-1152	68	12	as	as	ADP
m-1152	68	13	.	.	PUNCT
m-1152	69	1	thus	thus	ADV
m-1152	69	2	(	(	PUNCT
m-1152	69	3	ab)s	ab)s	PROPN
m-1152	69	4	⊆	⊆	NUM
m-1152	69	5	as	as	ADP
m-1152	69	6	.	.	PUNCT
m-1152	69	7	combined	combine	VERB
m-1152	69	8	with	with	ADP
m-1152	69	9	as	as	ADP
m-1152	69	10	⊆	⊆	NUM
m-1152	69	11	abs	ab	NOUN
m-1152	69	12	,	,	PUNCT
m-1152	69	13	we	we	PRON
m-1152	69	14	get	get	VERB
m-1152	69	15	as	as	ADP
m-1152	69	16	=	=	NOUN
m-1152	69	17	abs	ab	NOUN
m-1152	69	18	.	.	PUNCT
m-1152	70	1	similarly	similarly	ADV
m-1152	70	2	,	,	PUNCT
m-1152	70	3	if	if	SCONJ
m-1152	70	4	sa	sa	PROPN
m-1152	70	5	⊆	⊆	NUM
m-1152	70	6	sab	sab	ADJ
m-1152	70	7	,	,	PUNCT
m-1152	70	8	then	then	ADV
m-1152	70	9	for	for	ADP
m-1152	70	10	x	x	PROPN
m-1152	70	11	=	=	SYM
m-1152	70	12	t	t	PROPN
m-1152	70	13	(	(	PUNCT
m-1152	70	14	u	u	NOUN
m-1152	70	15	)	)	PUNCT
m-1152	70	16	∈	∈	PROPN
m-1152	70	17	s	s	PROPN
m-1152	70	18	,	,	PUNCT
m-1152	70	19	x(ab	x(ab	PROPN
m-1152	70	20	)	)	PUNCT
m-1152	70	21	=	=	SYM
m-1152	70	22	t	t	PROPN
m-1152	70	23	(	(	PUNCT
m-1152	70	24	u)t	u)t	X
m-1152	70	25	(	(	PUNCT
m-1152	70	26	t+	t+	NOUN
m-1152	70	27	s	s	NOUN
m-1152	70	28	)	)	PUNCT
m-1152	70	29	=	=	SYM
m-1152	70	30	t	t	PROPN
m-1152	70	31	(	(	PUNCT
m-1152	70	32	u+	u+	NUM
m-1152	70	33	t+	t+	PRON
m-1152	70	34	s	s	NOUN
m-1152	70	35	)	)	PUNCT
m-1152	70	36	=	=	SYM
m-1152	70	37	t	t	PROPN
m-1152	70	38	(	(	PUNCT
m-1152	70	39	u+	u+	NUM
m-1152	70	40	t)t	t)t	X
m-1152	70	41	(	(	PUNCT
m-1152	70	42	s	s	X
m-1152	70	43	)	)	PUNCT
m-1152	70	44	=	=	SYM
m-1152	70	45	(	(	PUNCT
m-1152	70	46	xa)b	xa)b	PROPN
m-1152	70	47	∈	∈	PROPN
m-1152	70	48	sa	sa	PROPN
m-1152	70	49	,	,	PUNCT
m-1152	70	50	so	so	ADV
m-1152	70	51	sab	sab	VERB
m-1152	70	52	⊆	⊆	NUM
m-1152	70	53	sa	sa	NOUN
m-1152	70	54	,	,	PUNCT
m-1152	70	55	hence	hence	ADV
m-1152	70	56	sa	sa	PROPN
m-1152	71	1	=	=	PUNCT
m-1152	71	2	sab	sab	PROPN
m-1152	71	3	.	.	PUNCT
m-1152	72	1	therefore	therefore	ADV
m-1152	72	2	s	s	VERB
m-1152	72	3	satis	satis	PROPN
m-1152	72	4	�	�	PROPN
m-1152	72	5	es	es	NOUN
m-1152	72	6	kw	kw	NOUN
m-1152	72	7	-	-	NOUN
m-1152	72	8	stability	stability	NOUN
m-1152	72	9	.	.	PUNCT
m-1152	73	1	ijo	ijo	PROPN
m-1152	73	2	international	international	PROPN
m-1152	73	3	journal	journal	PROPN
m-1152	73	4	of	of	ADP
m-1152	73	5	mathematics	mathematics	PROPN
m-1152	73	6	(	(	PUNCT
m-1152	73	7	issn	issn	PROPN
m-1152	73	8	:	:	PUNCT
m-1152	73	9	2992	2992	NUM
m-1152	73	10	-	-	SYM
m-1152	73	11	4421	4421	NUM
m-1152	73	12	)	)	PUNCT
m-1152	73	13	volume	volume	NOUN
m-1152	73	14	08	08	NUM
m-1152	74	1	|	|	ADV
m-1152	74	2	issue	issue	NOUN
m-1152	74	3	9	9	NUM
m-1152	74	4	|	|	CCONJ
m-1152	74	5	september	september	PROPN
m-1152	74	6	2025	2025	NUM
m-1152	74	7	|	|	ADV
m-1152	74	8	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1152	74	9	12	12	NUM
m-1152	74	10	theorem	theorem	VERB
m-1152	74	11	3.2	3.2	NUM
m-1152	74	12	(	(	PUNCT
m-1152	74	13	equivalence	equivalence	NOUN
m-1152	74	14	of	of	ADP
m-1152	74	15	green	green	PROPN
m-1152	74	16	's	's	PART
m-1152	74	17	relations	relation	NOUN
m-1152	74	18	[	[	X
m-1152	74	19	3	3	NUM
m-1152	74	20	]	]	NUM
m-1152	74	21	)	)	PUNCT
m-1152	74	22	.	.	PUNCT
m-1152	75	1	for	for	ADP
m-1152	75	2	a	a	DET
m-1152	75	3	c0	c0	NOUN
m-1152	75	4	-	-	PUNCT
m-1152	75	5	semigroup	semigroup	NOUN
m-1152	75	6	of	of	ADP
m-1152	75	7	bounded	bounded	ADJ
m-1152	75	8	linear	linear	PROPN
m-1152	75	9	operators	operator	NOUN
m-1152	75	10	,	,	PUNCT
m-1152	75	11	d	d	PROPN
m-1152	75	12	=	=	SYM
m-1152	75	13	j	j	PROPN
m-1152	75	14	=	=	SYM
m-1152	75	15	l	l	PROPN
m-1152	75	16	=	=	SYM
m-1152	75	17	r.	r.	NOUN
m-1152	75	18	proof	proof	NOUN
m-1152	75	19	.	.	PUNCT
m-1152	76	1	since	since	SCONJ
m-1152	76	2	s	s	PROPN
m-1152	76	3	is	be	AUX
m-1152	76	4	commutative	commutative	ADJ
m-1152	76	5	,	,	PUNCT
m-1152	76	6	left	left	ADJ
m-1152	76	7	and	and	CCONJ
m-1152	76	8	right	right	ADJ
m-1152	76	9	ideals	ideal	NOUN
m-1152	76	10	coincide	coincide	VERB
m-1152	76	11	:	:	PUNCT
m-1152	76	12	sa	sa	PROPN
m-1152	76	13	=	=	PUNCT
m-1152	76	14	as	as	SCONJ
m-1152	76	15	,	,	PUNCT
m-1152	76	16	so	so	ADV
m-1152	76	17	l	l	NOUN
m-1152	76	18	=	=	SYM
m-1152	76	19	r.	r.	NOUN
m-1152	76	20	for	for	ADP
m-1152	76	21	any	any	DET
m-1152	76	22	a	a	DET
m-1152	76	23	∈	∈	PROPN
m-1152	76	24	s	s	NOUN
m-1152	76	25	,	,	PUNCT
m-1152	76	26	sas	sas	X
m-1152	76	27	=	=	SYM
m-1152	76	28	{	{	PUNCT
m-1152	76	29	xay	xay	PROPN
m-1152	76	30	:	:	PUNCT
m-1152	76	31	x	x	X
m-1152	76	32	,	,	PUNCT
m-1152	76	33	y	y	PROPN
m-1152	76	34	∈	∈	PROPN
m-1152	76	35	s	s	PART
m-1152	76	36	}	}	PUNCT
m-1152	76	37	.	.	PUNCT
m-1152	77	1	but	but	CCONJ
m-1152	77	2	commutativity	commutativity	NOUN
m-1152	77	3	gives	give	VERB
m-1152	77	4	xay	xay	PROPN
m-1152	77	5	=	=	PUNCT
m-1152	77	6	(	(	PUNCT
m-1152	77	7	xy)a	xy)a	PROPN
m-1152	77	8	∈	∈	PROPN
m-1152	77	9	sa	sa	PROPN
m-1152	77	10	,	,	PUNCT
m-1152	77	11	so	so	ADV
m-1152	77	12	sas	sa	VERB
m-1152	77	13	⊆	⊆	NUM
m-1152	77	14	sa	sa	NOUN
m-1152	77	15	.	.	PUNCT
m-1152	78	1	conversely	conversely	ADV
m-1152	78	2	,	,	PUNCT
m-1152	78	3	for	for	ADP
m-1152	78	4	xa	xa	PROPN
m-1152	78	5	∈	∈	PROPN
m-1152	78	6	sa	sa	PROPN
m-1152	78	7	,	,	PUNCT
m-1152	78	8	write	write	VERB
m-1152	78	9	xa	xa	PROPN
m-1152	78	10	=	=	PRON
m-1152	78	11	(	(	PUNCT
m-1152	78	12	x)a	x)a	PUNCT
m-1152	78	13	·	·	PUNCT
m-1152	78	14	i	i	PRON
m-1152	78	15	∈	∈	PROPN
m-1152	78	16	sas	sas	VERB
m-1152	78	17	.	.	PUNCT
m-1152	79	1	thus	thus	ADV
m-1152	79	2	sa	sa	X
m-1152	79	3	=	=	SYM
m-1152	79	4	sas	sas	PROPN
m-1152	79	5	,	,	PUNCT
m-1152	79	6	so	so	SCONJ
m-1152	79	7	j	j	PROPN
m-1152	79	8	=	=	PROPN
m-1152	79	9	l.	l.	PROPN
m-1152	80	1	finally	finally	ADV
m-1152	80	2	,	,	PUNCT
m-1152	80	3	d	d	PROPN
m-1152	80	4	=	=	PUNCT
m-1152	80	5	l	l	NOUN
m-1152	80	6	◦	◦	NOUN
m-1152	80	7	r	r	NOUN
m-1152	80	8	and	and	CCONJ
m-1152	80	9	l	l	NOUN
m-1152	81	1	=	=	NOUN
m-1152	81	2	r	r	AUX
m-1152	81	3	imply	imply	NOUN
m-1152	81	4	d	d	NOUN
m-1152	81	5	=	=	PUNCT
m-1152	81	6	l	l	NOUN
m-1152	82	1	=	=	PUNCT
m-1152	83	1	r	r	NOUN
m-1152	83	2	=	=	SYM
m-1152	83	3	j	j	PROPN
m-1152	83	4	.	.	PUNCT
m-1152	84	1	4	4	NUM
m-1152	84	2	illustrative	illustrative	ADJ
m-1152	84	3	examples	example	NOUN
m-1152	84	4	example	example	NOUN
m-1152	84	5	4.1	4.1	NUM
m-1152	84	6	(	(	PUNCT
m-1152	84	7	translation	translation	NOUN
m-1152	84	8	semigroup	semigroup	PROPN
m-1152	84	9	)	)	PUNCT
m-1152	84	10	.	.	PUNCT
m-1152	85	1	let	let	VERB
m-1152	85	2	x	x	PUNCT
m-1152	85	3	=	=	SYM
m-1152	85	4	c0(r	c0(r	PROPN
m-1152	85	5	)	)	PUNCT
m-1152	85	6	,	,	PUNCT
m-1152	85	7	the	the	DET
m-1152	85	8	banach	banach	NOUN
m-1152	85	9	space	space	NOUN
m-1152	85	10	of	of	ADP
m-1152	85	11	continuous	continuous	ADJ
m-1152	85	12	functions	function	NOUN
m-1152	85	13	on	on	ADP
m-1152	85	14	r	r	NOUN
m-1152	85	15	vanishing	vanish	VERB
m-1152	85	16	at	at	ADP
m-1152	85	17	in	in	ADP
m-1152	85	18	�	�	PROPN
m-1152	85	19	nity	nity	NOUN
m-1152	85	20	,	,	PUNCT
m-1152	85	21	equipped	equip	VERB
m-1152	85	22	with	with	ADP
m-1152	85	23	the	the	DET
m-1152	85	24	supremum	supremum	ADJ
m-1152	85	25	norm	norm	NOUN
m-1152	85	26	∥f∥∞	∥f∥∞	PUNCT
m-1152	86	1	=	=	SYM
m-1152	86	2	sup	sup	NOUN
m-1152	86	3	x∈r	x∈r	PROPN
m-1152	86	4	|f(x)|	|f(x)|	PROPN
m-1152	86	5	.	.	PUNCT
m-1152	86	6	de	de	PROPN
m-1152	86	7	�	�	PROPN
m-1152	86	8	ne	ne	PROPN
m-1152	86	9	a	a	DET
m-1152	86	10	family	family	NOUN
m-1152	86	11	of	of	ADP
m-1152	86	12	operators	operator	NOUN
m-1152	86	13	{	{	PUNCT
m-1152	86	14	t	t	PROPN
m-1152	86	15	(	(	PUNCT
m-1152	86	16	t)}t≥0	t)}t≥0	NOUN
m-1152	86	17	by	by	ADP
m-1152	86	18	(	(	PUNCT
m-1152	86	19	t	t	PROPN
m-1152	86	20	(	(	PUNCT
m-1152	86	21	t)f)(x	t)f)(x	PROPN
m-1152	86	22	)	)	PUNCT
m-1152	86	23	=	=	PUNCT
m-1152	86	24	f(x+	f(x+	NOUN
m-1152	86	25	t	t	PROPN
m-1152	86	26	)	)	PUNCT
m-1152	86	27	,	,	PUNCT
m-1152	86	28	f	f	PROPN
m-1152	86	29	∈	∈	PROPN
m-1152	86	30	x	x	PROPN
m-1152	86	31	,	,	PUNCT
m-1152	86	32	t	t	PROPN
m-1152	86	33	≥	≥	NUM
m-1152	86	34	0	0	NUM
m-1152	86	35	,	,	PUNCT
m-1152	86	36	x	x	PROPN
m-1152	86	37	∈	∈	PROPN
m-1152	86	38	r.	r.	NOUN
m-1152	86	39	for	for	ADP
m-1152	86	40	strong	strong	ADJ
m-1152	86	41	continuity	continuity	NOUN
m-1152	86	42	.	.	PUNCT
m-1152	87	1	we	we	PRON
m-1152	87	2	verify	verify	VERB
m-1152	87	3	that	that	SCONJ
m-1152	87	4	{	{	PUNCT
m-1152	87	5	t	t	NOUN
m-1152	87	6	(	(	PUNCT
m-1152	87	7	t)}t≥0	t)}t≥0	NOUN
m-1152	87	8	is	be	AUX
m-1152	87	9	a	a	DET
m-1152	87	10	strongly	strongly	ADV
m-1152	87	11	continuous	continuous	ADJ
m-1152	87	12	semigroup	semigroup	NOUN
m-1152	87	13	.	.	PUNCT
m-1152	88	1	for	for	ADP
m-1152	88	2	�	�	PROPN
m-1152	88	3	xed	xed	PROPN
m-1152	88	4	f	f	PROPN
m-1152	88	5	∈	∈	PROPN
m-1152	88	6	x	x	SYM
m-1152	88	7	,	,	PUNCT
m-1152	88	8	∥t	∥t	PROPN
m-1152	88	9	(	(	PUNCT
m-1152	88	10	t)f	t)f	SYM
m-1152	88	11	−	−	PROPN
m-1152	89	1	f∥∞	f∥∞	NOUN
m-1152	89	2	=	=	SYM
m-1152	89	3	sup	sup	NOUN
m-1152	89	4	x∈r	x∈r	PROPN
m-1152	89	5	|f(x+	|f(x+	PROPN
m-1152	89	6	t)−	t)−	PROPN
m-1152	89	7	f(x)|	f(x)|	VERB
m-1152	89	8	.	.	PUNCT
m-1152	90	1	since	since	SCONJ
m-1152	90	2	f	f	PROPN
m-1152	90	3	is	be	AUX
m-1152	90	4	uniformly	uniformly	ADV
m-1152	90	5	continuous	continuous	ADJ
m-1152	90	6	on	on	ADP
m-1152	90	7	r	r	NOUN
m-1152	90	8	(	(	PUNCT
m-1152	90	9	as	as	ADP
m-1152	90	10	every	every	DET
m-1152	90	11	f	f	PROPN
m-1152	90	12	∈	∈	PROPN
m-1152	90	13	c0(r	c0(r	PROPN
m-1152	90	14	)	)	PUNCT
m-1152	90	15	is	be	AUX
m-1152	90	16	uniformly	uniformly	ADV
m-1152	90	17	continuous	continuous	ADJ
m-1152	90	18	)	)	PUNCT
m-1152	90	19	,	,	PUNCT
m-1152	90	20	the	the	DET
m-1152	90	21	right	right	ADJ
m-1152	90	22	-	-	PUNCT
m-1152	90	23	hand	hand	NOUN
m-1152	90	24	side	side	NOUN
m-1152	90	25	tends	tend	VERB
m-1152	90	26	to	to	ADP
m-1152	90	27	zero	zero	NUM
m-1152	90	28	as	as	ADP
m-1152	90	29	t	t	PROPN
m-1152	90	30	→	→	SYM
m-1152	90	31	0	0	NUM
m-1152	90	32	.	.	PUNCT
m-1152	91	1	hence	hence	ADV
m-1152	91	2	,	,	PUNCT
m-1152	91	3	lim	lim	PROPN
m-1152	91	4	t→0	t→0	AUX
m-1152	91	5	+	+	CCONJ
m-1152	91	6	∥t	∥t	ADJ
m-1152	91	7	(	(	PUNCT
m-1152	91	8	t)f	t)f	SYM
m-1152	91	9	−	−	PROPN
m-1152	91	10	f∥∞	f∥∞	NOUN
m-1152	91	11	=	=	SYM
m-1152	91	12	0	0	NUM
m-1152	91	13	,	,	PUNCT
m-1152	91	14	so	so	ADV
m-1152	91	15	{	{	PUNCT
m-1152	91	16	t	t	PROPN
m-1152	91	17	(	(	PUNCT
m-1152	91	18	t)}t≥0	t)}t≥0	NOUN
m-1152	91	19	is	be	AUX
m-1152	91	20	a	a	DET
m-1152	91	21	c0	c0	NOUN
m-1152	91	22	-	-	PUNCT
m-1152	91	23	semigroup	semigroup	NOUN
m-1152	91	24	.	.	PUNCT
m-1152	92	1	for	for	ADP
m-1152	92	2	in	in	ADP
m-1152	92	3	�	�	PROPN
m-1152	92	4	nitesimal	nitesimal	NOUN
m-1152	92	5	generator	generator	PROPN
m-1152	92	6	.	.	PUNCT
m-1152	93	1	let	let	VERB
m-1152	93	2	a	a	DET
m-1152	93	3	denote	denote	NOUN
m-1152	93	4	the	the	DET
m-1152	93	5	generator	generator	NOUN
m-1152	93	6	of	of	ADP
m-1152	93	7	{	{	PUNCT
m-1152	93	8	t	t	PROPN
m-1152	93	9	(	(	PUNCT
m-1152	93	10	t)}t≥0	t)}t≥0	NOUN
m-1152	93	11	.	.	PUNCT
m-1152	93	12	by	by	ADP
m-1152	93	13	de	de	PROPN
m-1152	93	14	�	�	PROPN
m-1152	93	15	nition	nition	NOUN
m-1152	93	16	,	,	PUNCT
m-1152	93	17	af	af	PROPN
m-1152	93	18	=	=	PROPN
m-1152	93	19	lim	lim	PROPN
m-1152	93	20	t→0	t→0	PROPN
m-1152	93	21	+	+	PROPN
m-1152	93	22	t	t	PROPN
m-1152	93	23	(	(	PUNCT
m-1152	93	24	t)f	t)f	SYM
m-1152	93	25	−	−	PROPN
m-1152	94	1	f	f	X
m-1152	94	2	t	t	PROPN
m-1152	94	3	f	f	PROPN
m-1152	94	4	,	,	PUNCT
m-1152	94	5	∈	∈	PROPN
m-1152	94	6	d(a	d(a	PROPN
m-1152	94	7	)	)	PUNCT
m-1152	94	8	,	,	PUNCT
m-1152	94	9	where	where	SCONJ
m-1152	94	10	the	the	DET
m-1152	94	11	domain	domain	NOUN
m-1152	94	12	consists	consist	VERB
m-1152	94	13	of	of	ADP
m-1152	94	14	those	those	DET
m-1152	94	15	f	f	PROPN
m-1152	94	16	∈	∈	PROPN
m-1152	94	17	x	x	PUNCT
m-1152	94	18	for	for	ADP
m-1152	94	19	which	which	PRON
m-1152	94	20	the	the	DET
m-1152	94	21	above	above	ADJ
m-1152	94	22	limit	limit	NOUN
m-1152	94	23	exists	exist	VERB
m-1152	94	24	in	in	ADP
m-1152	94	25	x.	x.	NOUN
m-1152	94	26	a	a	DET
m-1152	94	27	direct	direct	ADJ
m-1152	94	28	computation	computation	NOUN
m-1152	94	29	shows	show	VERB
m-1152	94	30	that	that	SCONJ
m-1152	94	31	t	t	PROPN
m-1152	94	32	(	(	PUNCT
m-1152	94	33	t)f(x)−	t)f(x)−	PROPN
m-1152	94	34	f(x	f(x	PROPN
m-1152	94	35	)	)	PUNCT
m-1152	94	36	t	t	NOUN
m-1152	94	37	=	=	PUNCT
m-1152	94	38	f(x+	f(x+	NOUN
m-1152	94	39	t)−	t)−	PROPN
m-1152	94	40	f(x	f(x	PROPN
m-1152	94	41	)	)	PUNCT
m-1152	94	42	t	t	PROPN
m-1152	94	43	.	.	PUNCT
m-1152	95	1	thus	thus	ADV
m-1152	95	2	,	,	PUNCT
m-1152	95	3	the	the	DET
m-1152	95	4	limit	limit	NOUN
m-1152	95	5	exists	exist	VERB
m-1152	95	6	precisely	precisely	ADV
m-1152	95	7	when	when	SCONJ
m-1152	95	8	f	f	PROPN
m-1152	95	9	is	be	AUX
m-1152	95	10	continuously	continuously	ADV
m-1152	95	11	di	di	ADJ
m-1152	95	12	�	�	PROPN
m-1152	95	13	erentiable	erentiable	ADJ
m-1152	95	14	with	with	ADP
m-1152	95	15	derivative	derivative	ADJ
m-1152	95	16	vanishing	vanishing	NOUN
m-1152	95	17	at	at	ADP
m-1152	95	18	in	in	ADP
m-1152	95	19	�	�	NOUN
m-1152	95	20	nity	nity	NOUN
m-1152	95	21	.	.	PUNCT
m-1152	96	1	therefore	therefore	ADV
m-1152	96	2	,	,	PUNCT
m-1152	96	3	af	af	PROPN
m-1152	96	4	=	=	SYM
m-1152	96	5	f	f	PROPN
m-1152	96	6	′	′	NOUN
m-1152	96	7	,	,	PUNCT
m-1152	96	8	d(a	d(a	PROPN
m-1152	96	9	)	)	PUNCT
m-1152	96	10	=	=	PRON
m-1152	97	1	{	{	PUNCT
m-1152	97	2	f	f	PROPN
m-1152	97	3	∈	∈	PROPN
m-1152	97	4	c0(r	c0(r	PROPN
m-1152	97	5	)	)	PUNCT
m-1152	97	6	:	:	PUNCT
m-1152	98	1	f	f	X
m-1152	98	2	′	′	NUM
m-1152	98	3	∈	∈	PROPN
m-1152	98	4	c0(r	c0(r	NOUN
m-1152	98	5	)	)	PUNCT
m-1152	98	6	}	}	PUNCT
m-1152	98	7	.	.	PUNCT
m-1152	99	1	ijo	ijo	PROPN
m-1152	99	2	international	international	PROPN
m-1152	99	3	journal	journal	PROPN
m-1152	99	4	of	of	ADP
m-1152	99	5	mathematics	mathematics	PROPN
m-1152	99	6	(	(	PUNCT
m-1152	99	7	issn	issn	PROPN
m-1152	99	8	:	:	PUNCT
m-1152	99	9	2992	2992	NUM
m-1152	99	10	-	-	SYM
m-1152	99	11	4421	4421	NUM
m-1152	99	12	)	)	PUNCT
m-1152	99	13	volume	volume	NOUN
m-1152	99	14	08	08	NUM
m-1152	100	1	|	|	ADV
m-1152	100	2	issue	issue	NOUN
m-1152	100	3	9	9	NUM
m-1152	100	4	|	|	CCONJ
m-1152	100	5	september	september	PROPN
m-1152	100	6	2025	2025	NUM
m-1152	100	7	|	|	ADV
m-1152	100	8	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1152	100	9	13	13	NUM
m-1152	100	10	for	for	ADP
m-1152	100	11	kw	kw	NOUN
m-1152	100	12	-	-	NOUN
m-1152	100	13	stability	stability	NOUN
m-1152	100	14	.	.	PUNCT
m-1152	101	1	by	by	ADP
m-1152	101	2	proposition	proposition	NOUN
m-1152	101	3	3.1	3.1	NUM
m-1152	101	4	,	,	PUNCT
m-1152	101	5	every	every	DET
m-1152	101	6	c0	c0	NOUN
m-1152	101	7	-	-	PUNCT
m-1152	101	8	semigroup	semigroup	NOUN
m-1152	101	9	on	on	ADP
m-1152	101	10	a	a	DET
m-1152	101	11	banach	banach	NOUN
m-1152	101	12	space	space	NOUN
m-1152	101	13	is	be	AUX
m-1152	101	14	stable	stable	ADJ
m-1152	101	15	in	in	ADP
m-1152	101	16	the	the	DET
m-1152	101	17	sense	sense	NOUN
m-1152	101	18	of	of	ADP
m-1152	101	19	koch	koch	PROPN
m-1152	101	20	�	�	PROPN
m-1152	101	21	wallace	wallace	PROPN
m-1152	101	22	.	.	PUNCT
m-1152	102	1	explicitly	explicitly	ADV
m-1152	102	2	,	,	PUNCT
m-1152	102	3	for	for	SCONJ
m-1152	102	4	f	f	PROPN
m-1152	102	5	∈	∈	PROPN
m-1152	102	6	x	x	X
m-1152	102	7	and	and	CCONJ
m-1152	102	8	t	t	PROPN
m-1152	102	9	,	,	PUNCT
m-1152	102	10	s	s	VERB
m-1152	102	11	≥	≥	NOUN
m-1152	102	12	0	0	NUM
m-1152	102	13	,	,	PUNCT
m-1152	102	14	t	t	PROPN
m-1152	102	15	(	(	PUNCT
m-1152	102	16	t+	t+	NOUN
m-1152	102	17	s)f	s)f	NOUN
m-1152	102	18	=	=	SYM
m-1152	102	19	t	t	PROPN
m-1152	102	20	(	(	PUNCT
m-1152	102	21	t)t	t)t	X
m-1152	102	22	(	(	PUNCT
m-1152	102	23	s)f	s)f	NUM
m-1152	102	24	,	,	PUNCT
m-1152	102	25	and	and	CCONJ
m-1152	102	26	the	the	DET
m-1152	102	27	kw	kw	VERB
m-1152	102	28	-	-	PUNCT
m-1152	102	29	condition	condition	NOUN
m-1152	102	30	t	t	NOUN
m-1152	102	31	(	(	PUNCT
m-1152	102	32	t)x	t)x	X
m-1152	102	33	⊆	⊆	NUM
m-1152	102	34	t	t	NOUN
m-1152	102	35	(	(	PUNCT
m-1152	102	36	t+	t+	NOUN
m-1152	102	37	s)x	s)x	X
m-1152	102	38	⇒	⇒	PROPN
m-1152	102	39	t	t	PROPN
m-1152	102	40	(	(	PUNCT
m-1152	102	41	t)x	t)x	X
m-1152	102	42	=	=	SYM
m-1152	102	43	t	t	PROPN
m-1152	102	44	(	(	PUNCT
m-1152	102	45	t+	t+	NOUN
m-1152	102	46	s)x	s)x	NOUN
m-1152	102	47	is	be	AUX
m-1152	102	48	satis	satis	NOUN
m-1152	102	49	�	�	NOUN
m-1152	102	50	ed	ed	NOUN
m-1152	102	51	.	.	PUNCT
m-1152	103	1	thus	thus	ADV
m-1152	103	2	the	the	DET
m-1152	103	3	translation	translation	NOUN
m-1152	103	4	semigroup	semigroup	NOUN
m-1152	103	5	is	be	AUX
m-1152	103	6	kw	kw	VERB
m-1152	103	7	-	-	ADJ
m-1152	103	8	stable	stable	ADJ
m-1152	103	9	.	.	PUNCT
m-1152	104	1	for	for	ADP
m-1152	104	2	analytic	analytic	ADJ
m-1152	104	3	behavior	behavior	NOUN
m-1152	104	4	.	.	PUNCT
m-1152	105	1	we	we	PRON
m-1152	105	2	compute	compute	VERB
m-1152	105	3	the	the	DET
m-1152	105	4	operator	operator	NOUN
m-1152	105	5	norm	norm	NOUN
m-1152	105	6	:	:	PUNCT
m-1152	105	7	∥t	∥t	ADJ
m-1152	105	8	(	(	PUNCT
m-1152	105	9	t)∥	t)∥	X
m-1152	105	10	sup=	sup=	ADJ
m-1152	105	11	∥f∥∞=1	∥f∥∞=1	NOUN
m-1152	105	12	∥t	∥t	PROPN
m-1152	105	13	(	(	PUNCT
m-1152	105	14	t)f∥∞.	t)f∥∞.	INTJ
m-1152	105	15	but	but	CCONJ
m-1152	105	16	for	for	ADP
m-1152	105	17	any	any	DET
m-1152	105	18	f	f	PROPN
m-1152	105	19	∈	∈	PROPN
m-1152	105	20	x	x	SYM
m-1152	105	21	,	,	PUNCT
m-1152	105	22	∥t	∥t	PROPN
m-1152	105	23	(	(	PUNCT
m-1152	105	24	t)f∥∞	t)f∥∞	PROPN
m-1152	105	25	=	=	PUNCT
m-1152	105	26	sup	sup	NOUN
m-1152	105	27	x∈r	x∈r	PROPN
m-1152	105	28	|f(x+	|f(x+	ADV
m-1152	105	29	t)|	t)|	NOUN
m-1152	105	30	=	=	PUNCT
m-1152	105	31	sup	sup	NOUN
m-1152	105	32	y∈r	y∈r	NOUN
m-1152	105	33	|f(y)|	|f(y)|	PROPN
m-1152	105	34	=	=	PUNCT
m-1152	105	35	∥f∥∞.	∥f∥∞.	NOUN
m-1152	105	36	hence	hence	ADV
m-1152	105	37	,	,	PUNCT
m-1152	105	38	∥t	∥t	PROPN
m-1152	105	39	(	(	PUNCT
m-1152	105	40	t)∥	t)∥	NUM
m-1152	105	41	=	=	SYM
m-1152	105	42	1	1	NUM
m-1152	105	43	for	for	ADP
m-1152	105	44	all	all	DET
m-1152	105	45	t	t	PROPN
m-1152	105	46	≥	≥	NOUN
m-1152	105	47	0	0	NUM
m-1152	105	48	.	.	PUNCT
m-1152	106	1	therefore	therefore	ADV
m-1152	106	2	,	,	PUNCT
m-1152	106	3	there	there	PRON
m-1152	106	4	is	be	VERB
m-1152	106	5	no	no	DET
m-1152	106	6	decay	decay	NOUN
m-1152	106	7	as	as	ADP
m-1152	106	8	t	t	PROPN
m-1152	106	9	→	→	SYM
m-1152	106	10	∞	∞	PROPN
m-1152	106	11	,	,	PUNCT
m-1152	106	12	and	and	CCONJ
m-1152	106	13	the	the	DET
m-1152	106	14	semigroup	semigroup	NOUN
m-1152	106	15	fails	fail	VERB
m-1152	106	16	to	to	PART
m-1152	106	17	be	be	AUX
m-1152	106	18	analytically	analytically	ADV
m-1152	106	19	stable	stable	ADJ
m-1152	106	20	(	(	PUNCT
m-1152	106	21	in	in	ADP
m-1152	106	22	the	the	DET
m-1152	106	23	sense	sense	NOUN
m-1152	106	24	of	of	ADP
m-1152	106	25	uniform	uniform	ADJ
m-1152	106	26	exponential	exponential	ADJ
m-1152	106	27	stability	stability	NOUN
m-1152	106	28	)	)	PUNCT
m-1152	106	29	.	.	PUNCT
m-1152	107	1	graphical	graphical	ADJ
m-1152	107	2	interpretation	interpretation	NOUN
m-1152	107	3	.	.	PUNCT
m-1152	108	1	the	the	DET
m-1152	108	2	operator	operator	NOUN
m-1152	108	3	t	t	PROPN
m-1152	108	4	(	(	PUNCT
m-1152	108	5	t	t	NOUN
m-1152	108	6	)	)	PUNCT
m-1152	108	7	acts	act	VERB
m-1152	108	8	as	as	ADP
m-1152	108	9	a	a	DET
m-1152	108	10	horizontal	horizontal	ADJ
m-1152	108	11	shift	shift	NOUN
m-1152	108	12	of	of	ADP
m-1152	108	13	the	the	DET
m-1152	108	14	function	function	NOUN
m-1152	108	15	graph	graph	NOUN
m-1152	108	16	.	.	PUNCT
m-1152	109	1	for	for	ADP
m-1152	109	2	example	example	NOUN
m-1152	109	3	,	,	PUNCT
m-1152	109	4	if	if	SCONJ
m-1152	109	5	f(x	f(x	PROPN
m-1152	109	6	)	)	PUNCT
m-1152	109	7	=	=	PRON
m-1152	109	8	e−x2	e−x2	PRON
m-1152	109	9	is	be	AUX
m-1152	109	10	a	a	DET
m-1152	109	11	bell	bell	NOUN
m-1152	109	12	-	-	PUNCT
m-1152	109	13	shaped	shape	VERB
m-1152	109	14	curve	curve	NOUN
m-1152	109	15	centered	center	VERB
m-1152	109	16	at	at	ADP
m-1152	109	17	the	the	DET
m-1152	109	18	origin	origin	NOUN
m-1152	109	19	,	,	PUNCT
m-1152	109	20	then	then	ADV
m-1152	109	21	t	t	PROPN
m-1152	109	22	(	(	PUNCT
m-1152	109	23	1)f(x	1)f(x	PROPN
m-1152	109	24	)	)	PUNCT
m-1152	109	25	=	=	SYM
m-1152	109	26	f(x+1	f(x+1	NOUN
m-1152	109	27	)	)	PUNCT
m-1152	109	28	is	be	AUX
m-1152	109	29	the	the	DET
m-1152	109	30	same	same	ADJ
m-1152	109	31	curve	curve	NOUN
m-1152	109	32	shifted	shift	VERB
m-1152	109	33	left	leave	VERB
m-1152	109	34	by	by	ADP
m-1152	109	35	one	one	NUM
m-1152	109	36	unit	unit	NOUN
m-1152	109	37	.	.	PUNCT
m-1152	110	1	importantly	importantly	ADV
m-1152	110	2	,	,	PUNCT
m-1152	110	3	the	the	DET
m-1152	110	4	height	height	NOUN
m-1152	110	5	of	of	ADP
m-1152	110	6	the	the	DET
m-1152	110	7	curve	curve	NOUN
m-1152	110	8	is	be	AUX
m-1152	110	9	unchanged	unchanged	ADJ
m-1152	110	10	,	,	PUNCT
m-1152	110	11	so	so	ADV
m-1152	110	12	∥t	∥t	ADJ
m-1152	110	13	(	(	PUNCT
m-1152	110	14	t)f∥∞	t)f∥∞	PROPN
m-1152	110	15	=	=	SYM
m-1152	110	16	∥f∥∞	∥f∥∞	ADP
m-1152	110	17	for	for	ADP
m-1152	110	18	all	all	DET
m-1152	110	19	t	t	PROPN
m-1152	110	20	≥	≥	NOUN
m-1152	110	21	0	0	NUM
m-1152	110	22	.	.	PUNCT
m-1152	111	1	this	this	PRON
m-1152	111	2	shows	show	VERB
m-1152	111	3	why	why	SCONJ
m-1152	111	4	the	the	DET
m-1152	111	5	semigroup	semigroup	NOUN
m-1152	111	6	is	be	AUX
m-1152	111	7	kw	kw	VERB
m-1152	111	8	-	-	ADJ
m-1152	111	9	stable	stable	ADJ
m-1152	111	10	(	(	PUNCT
m-1152	111	11	algebraically	algebraically	ADV
m-1152	111	12	the	the	DET
m-1152	111	13	orbits	orbit	NOUN
m-1152	111	14	are	be	AUX
m-1152	111	15	preserved	preserve	VERB
m-1152	111	16	)	)	PUNCT
m-1152	111	17	but	but	CCONJ
m-1152	111	18	not	not	PART
m-1152	111	19	analytically	analytically	ADV
m-1152	111	20	stable	stable	ADJ
m-1152	111	21	(	(	PUNCT
m-1152	111	22	no	no	DET
m-1152	111	23	decay	decay	NOUN
m-1152	111	24	in	in	ADP
m-1152	111	25	norm	norm	NOUN
m-1152	111	26	)	)	PUNCT
m-1152	111	27	.	.	PUNCT
m-1152	112	1	x	x	PUNCT
m-1152	112	2	f(x	f(x	PROPN
m-1152	112	3	)	)	PUNCT
m-1152	112	4	f(x	f(x	PROPN
m-1152	112	5	)	)	PUNCT
m-1152	113	1	=	=	PRON
m-1152	113	2	e−x2	e−x2	NUM
m-1152	113	3	t	t	PROPN
m-1152	113	4	(	(	PUNCT
m-1152	113	5	1)f(x	1)f(x	NUM
m-1152	113	6	)	)	PUNCT
m-1152	113	7	=	=	PUNCT
m-1152	114	1	f(x+	f(x+	NUM
m-1152	114	2	1	1	NUM
m-1152	114	3	)	)	PUNCT
m-1152	114	4	0−1	0−1	NUM
m-1152	114	5	figure	figure	NOUN
m-1152	114	6	1	1	NUM
m-1152	114	7	:	:	PUNCT
m-1152	114	8	translation	translation	NOUN
m-1152	114	9	semigroup	semigroup	PROPN
m-1152	114	10	illustrated	illustrate	VERB
m-1152	114	11	on	on	ADP
m-1152	114	12	f(x	f(x	PROPN
m-1152	114	13	)	)	PUNCT
m-1152	115	1	=	=	PRON
m-1152	115	2	e−x2	e−x2	NOUN
m-1152	115	3	.	.	PUNCT
m-1152	116	1	the	the	DET
m-1152	116	2	original	original	ADJ
m-1152	116	3	function	function	NOUN
m-1152	116	4	(	(	PUNCT
m-1152	116	5	blue	blue	ADJ
m-1152	116	6	)	)	PUNCT
m-1152	116	7	and	and	CCONJ
m-1152	116	8	its	its	PRON
m-1152	116	9	translated	translate	VERB
m-1152	116	10	version	version	NOUN
m-1152	116	11	(	(	PUNCT
m-1152	116	12	red	red	PROPN
m-1152	116	13	dashed	dash	VERB
m-1152	116	14	)	)	PUNCT
m-1152	116	15	have	have	VERB
m-1152	116	16	identical	identical	ADJ
m-1152	116	17	amplitude	amplitude	NOUN
m-1152	116	18	but	but	CCONJ
m-1152	116	19	shifted	shift	VERB
m-1152	116	20	position	position	NOUN
m-1152	116	21	,	,	PUNCT
m-1152	116	22	explaining	explain	VERB
m-1152	116	23	why	why	SCONJ
m-1152	116	24	kw	kw	NOUN
m-1152	116	25	-	-	PUNCT
m-1152	116	26	stability	stability	NOUN
m-1152	116	27	holds	hold	VERB
m-1152	116	28	while	while	SCONJ
m-1152	116	29	analytic	analytic	ADJ
m-1152	116	30	stability	stability	NOUN
m-1152	116	31	fails	fail	VERB
m-1152	116	32	.	.	PUNCT
m-1152	117	1	example	example	NOUN
m-1152	117	2	4.2	4.2	NUM
m-1152	117	3	(	(	PUNCT
m-1152	117	4	right	right	ADJ
m-1152	117	5	shift	shift	NOUN
m-1152	117	6	on	on	ADP
m-1152	117	7	ℓ2	ℓ2	NOUN
m-1152	117	8	)	)	PUNCT
m-1152	117	9	.	.	PUNCT
m-1152	118	1	let	let	VERB
m-1152	118	2	x	x	PUNCT
m-1152	118	3	=	=	SYM
m-1152	118	4	ℓ2(n	ℓ2(n	PROPN
m-1152	118	5	)	)	PUNCT
m-1152	118	6	with	with	ADP
m-1152	118	7	norm	norm	NOUN
m-1152	118	8	∥x∥2	∥x∥2	NOUN
m-1152	118	9	=	=	SYM
m-1152	118	10	∑	∑	PUNCT
m-1152	118	11	k≥1	k≥1	PROPN
m-1152	118	12	|xk|2	|xk|2	PROPN
m-1152	118	13	.	.	PUNCT
m-1152	118	14	de	de	PROPN
m-1152	118	15	�	�	PROPN
m-1152	118	16	ne	ne	PROPN
m-1152	118	17	t	t	PROPN
m-1152	118	18	(	(	PUNCT
m-1152	118	19	n	n	CCONJ
m-1152	118	20	)	)	PUNCT
m-1152	118	21	for	for	ADP
m-1152	118	22	n	n	PRON
m-1152	118	23	∈	∈	PROPN
m-1152	118	24	n	n	X
m-1152	118	25	by	by	ADP
m-1152	118	26	t	t	PROPN
m-1152	118	27	(	(	PUNCT
m-1152	118	28	n)(x1	n)(x1	PROPN
m-1152	118	29	,	,	PUNCT
m-1152	118	30	x2	x2	PROPN
m-1152	118	31	,	,	PUNCT
m-1152	118	32	x3	x3	ADJ
m-1152	118	33	,	,	PUNCT
m-1152	118	34	.	.	PUNCT
m-1152	118	35	.	.	PUNCT
m-1152	118	36	.	.	PUNCT
m-1152	118	37	)	)	PUNCT
m-1152	119	1	=	=	PUNCT
m-1152	119	2	(	(	PUNCT
m-1152	119	3	0	0	NUM
m-1152	119	4	,	,	PUNCT
m-1152	119	5	.	.	PUNCT
m-1152	119	6	.	.	PUNCT
m-1152	120	1	.	.	PUNCT
m-1152	121	1	,	,	PUNCT
m-1152	121	2	0︸	0︸	PUNCT
m-1152	122	1	︷︷	︷︷	VERB
m-1152	122	2	︸	︸	PUNCT
m-1152	122	3	n	n	PROPN
m-1152	122	4	,	,	PUNCT
m-1152	122	5	x1	x1	PROPN
m-1152	122	6	,	,	PUNCT
m-1152	122	7	x2	x2	PROPN
m-1152	122	8	,	,	PUNCT
m-1152	122	9	x3	x3	ADJ
m-1152	122	10	,	,	PUNCT
m-1152	122	11	.	.	PUNCT
m-1152	122	12	.	.	PUNCT
m-1152	122	13	.	.	PUNCT
m-1152	122	14	)	)	PUNCT
m-1152	122	15	.	.	PUNCT
m-1152	123	1	for	for	ADP
m-1152	123	2	semigroup	semigroup	ADJ
m-1152	123	3	property	property	NOUN
m-1152	123	4	.	.	PUNCT
m-1152	124	1	for	for	ADP
m-1152	124	2	m	m	PROPN
m-1152	124	3	,	,	PUNCT
m-1152	124	4	n	n	PROPN
m-1152	124	5	∈	∈	PROPN
m-1152	124	6	n	n	CCONJ
m-1152	124	7	,	,	PUNCT
m-1152	124	8	t	t	PROPN
m-1152	124	9	(	(	PUNCT
m-1152	124	10	m)t	m)t	PROPN
m-1152	124	11	(	(	PUNCT
m-1152	124	12	n	n	CCONJ
m-1152	124	13	)	)	PUNCT
m-1152	124	14	=	=	SYM
m-1152	124	15	t	t	PROPN
m-1152	124	16	(	(	PUNCT
m-1152	124	17	m+	m+	NOUN
m-1152	124	18	n	n	CCONJ
m-1152	124	19	)	)	PUNCT
m-1152	124	20	,	,	PUNCT
m-1152	124	21	ijo	ijo	PROPN
m-1152	124	22	international	international	PROPN
m-1152	124	23	journal	journal	PROPN
m-1152	124	24	of	of	ADP
m-1152	124	25	mathematics	mathematics	PROPN
m-1152	124	26	(	(	PUNCT
m-1152	124	27	issn	issn	PROPN
m-1152	124	28	:	:	PUNCT
m-1152	124	29	2992	2992	NUM
m-1152	124	30	-	-	SYM
m-1152	124	31	4421	4421	NUM
m-1152	124	32	)	)	PUNCT
m-1152	124	33	volume	volume	NOUN
m-1152	124	34	08	08	NUM
m-1152	125	1	|	|	ADV
m-1152	125	2	issue	issue	NOUN
m-1152	125	3	9	9	NUM
m-1152	125	4	|	|	CCONJ
m-1152	125	5	september	september	PROPN
m-1152	125	6	2025	2025	NUM
m-1152	125	7	|	|	ADV
m-1152	125	8	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1152	125	9	14	14	NUM
m-1152	126	1	so	so	ADV
m-1152	126	2	{	{	PUNCT
m-1152	126	3	t	t	PROPN
m-1152	126	4	(	(	PUNCT
m-1152	126	5	n)}n∈n	n)}n∈n	PROPN
m-1152	126	6	is	be	AUX
m-1152	126	7	a	a	DET
m-1152	126	8	(	(	PUNCT
m-1152	126	9	discrete	discrete	NOUN
m-1152	126	10	)	)	PUNCT
m-1152	126	11	semigroup	semigroup	NOUN
m-1152	126	12	.	.	PUNCT
m-1152	127	1	for	for	ADP
m-1152	127	2	kw	kw	NOUN
m-1152	127	3	-	-	NOUN
m-1152	127	4	stability	stability	NOUN
m-1152	127	5	.	.	PUNCT
m-1152	128	1	the	the	DET
m-1152	128	2	orbit	orbit	NOUN
m-1152	128	3	structure	structure	NOUN
m-1152	128	4	is	be	AUX
m-1152	128	5	preserved	preserve	VERB
m-1152	128	6	under	under	ADP
m-1152	128	7	shifts	shift	NOUN
m-1152	128	8	:	:	PUNCT
m-1152	128	9	t	t	PROPN
m-1152	128	10	(	(	PUNCT
m-1152	128	11	m)t	m)t	PROPN
m-1152	128	12	(	(	PUNCT
m-1152	128	13	n	n	CCONJ
m-1152	128	14	)	)	PUNCT
m-1152	128	15	=	=	SYM
m-1152	128	16	t	t	PROPN
m-1152	128	17	(	(	PUNCT
m-1152	128	18	m+	m+	NOUN
m-1152	128	19	n	n	CCONJ
m-1152	128	20	)	)	PUNCT
m-1152	128	21	and	and	CCONJ
m-1152	128	22	the	the	DET
m-1152	128	23	kw	kw	VERB
m-1152	128	24	-	-	PUNCT
m-1152	128	25	condition	condition	NOUN
m-1152	128	26	t	t	NOUN
m-1152	128	27	(	(	PUNCT
m-1152	128	28	n)x	n)x	NOUN
m-1152	128	29	⊆	⊆	NUM
m-1152	128	30	t	t	NOUN
m-1152	128	31	(	(	PUNCT
m-1152	128	32	n+m)x	n+m)x	X
m-1152	128	33	⇒	⇒	PROPN
m-1152	128	34	t	t	PROPN
m-1152	128	35	(	(	PUNCT
m-1152	128	36	n)x	n)x	ADP
m-1152	128	37	=	=	SYM
m-1152	128	38	t	t	PROPN
m-1152	128	39	(	(	PUNCT
m-1152	128	40	n+m)x	n+m)x	NOUN
m-1152	128	41	holds	hold	NOUN
m-1152	128	42	.	.	PUNCT
m-1152	129	1	for	for	ADP
m-1152	129	2	isometry	isometry	NOUN
m-1152	129	3	/	/	SYM
m-1152	129	4	norm	norm	NOUN
m-1152	129	5	.	.	PUNCT
m-1152	130	1	for	for	ADP
m-1152	130	2	any	any	DET
m-1152	130	3	x	x	PROPN
m-1152	130	4	∈	∈	PROPN
m-1152	130	5	ℓ2	ℓ2	NOUN
m-1152	130	6	,	,	PUNCT
m-1152	130	7	∥t	∥t	PROPN
m-1152	130	8	(	(	PUNCT
m-1152	130	9	n)x∥2	n)x∥2	NOUN
m-1152	130	10	=	=	SYM
m-1152	130	11	∑	∑	ADP
m-1152	130	12	k≥1	k≥1	NOUN
m-1152	130	13	|(t	|(t	NOUN
m-1152	130	14	(	(	PUNCT
m-1152	130	15	n)x)k|2	n)x)k|2	VERB
m-1152	130	16	=	=	SYM
m-1152	130	17	∑	∑	PUNCT
m-1152	130	18	k≥1	k≥1	NOUN
m-1152	130	19	|xk|2	|xk|2	PUNCT
m-1152	130	20	=	=	SYM
m-1152	130	21	∥x∥2	∥x∥2	NOUN
m-1152	130	22	,	,	PUNCT
m-1152	130	23	so	so	ADV
m-1152	130	24	∥t	∥t	ADJ
m-1152	130	25	(	(	PUNCT
m-1152	130	26	n)∥	n)∥	PUNCT
m-1152	130	27	=	=	SYM
m-1152	130	28	1	1	NUM
m-1152	130	29	for	for	ADP
m-1152	130	30	all	all	DET
m-1152	130	31	n	n	NUM
m-1152	130	32	:	:	PUNCT
m-1152	130	33	each	each	DET
m-1152	130	34	t	t	NOUN
m-1152	130	35	(	(	PUNCT
m-1152	130	36	n	n	CCONJ
m-1152	130	37	)	)	PUNCT
m-1152	130	38	is	be	AUX
m-1152	130	39	an	an	DET
m-1152	130	40	isometry	isometry	NOUN
m-1152	130	41	.	.	PUNCT
m-1152	131	1	for	for	ADP
m-1152	131	2	analytic	analytic	ADJ
m-1152	131	3	behaviour	behaviour	NOUN
m-1152	131	4	.	.	PUNCT
m-1152	132	1	since	since	SCONJ
m-1152	132	2	there	there	PRON
m-1152	132	3	is	be	VERB
m-1152	132	4	no	no	DET
m-1152	132	5	decay	decay	NOUN
m-1152	132	6	(	(	PUNCT
m-1152	132	7	∥t	∥t	PROPN
m-1152	132	8	(	(	PUNCT
m-1152	132	9	n)∥	n)∥	NUM
m-1152	132	10	=	=	SYM
m-1152	132	11	1	1	NUM
m-1152	132	12	always	always	ADV
m-1152	132	13	)	)	PUNCT
m-1152	132	14	,	,	PUNCT
m-1152	132	15	the	the	DET
m-1152	132	16	semigroup	semigroup	NOUN
m-1152	132	17	is	be	AUX
m-1152	132	18	not	not	PART
m-1152	132	19	analytically	analytically	ADV
m-1152	132	20	(	(	PUNCT
m-1152	132	21	exponentially	exponentially	ADV
m-1152	132	22	)	)	PUNCT
m-1152	132	23	stable	stable	ADJ
m-1152	132	24	.	.	PUNCT
m-1152	133	1	interpretation	interpretation	NOUN
m-1152	133	2	.	.	PUNCT
m-1152	134	1	the	the	DET
m-1152	134	2	right	right	ADJ
m-1152	134	3	-	-	PUNCT
m-1152	134	4	shift	shift	NOUN
m-1152	134	5	moves	move	NOUN
m-1152	134	6	entries	entry	NOUN
m-1152	134	7	to	to	ADP
m-1152	134	8	the	the	DET
m-1152	134	9	right	right	ADJ
m-1152	134	10	while	while	SCONJ
m-1152	134	11	preserving	preserve	VERB
m-1152	134	12	total	total	ADJ
m-1152	134	13	ℓ2	ℓ2	NOUN
m-1152	134	14	energy	energy	NOUN
m-1152	134	15	.	.	PUNCT
m-1152	135	1	algebraic	algebraic	ADJ
m-1152	135	2	stability	stability	NOUN
m-1152	135	3	(	(	PUNCT
m-1152	135	4	kw	kw	INTJ
m-1152	135	5	)	)	PUNCT
m-1152	135	6	holds	hold	NOUN
m-1152	135	7	but	but	CCONJ
m-1152	135	8	analytic	analytic	ADJ
m-1152	135	9	decay	decay	NOUN
m-1152	135	10	does	do	VERB
m-1152	135	11	not	not	PART
m-1152	135	12	.	.	PUNCT
m-1152	136	1	x1	x1	NUM
m-1152	137	1	x2	x2	NOUN
m-1152	137	2	x3	x3	PROPN
m-1152	137	3	·	·	PUNCT
m-1152	137	4	·	·	PUNCT
m-1152	137	5	·	·	PUNCT
m-1152	137	6	0	0	PUNCT
m-1152	138	1	x1	x1	NUM
m-1152	139	1	x2	x2	NOUN
m-1152	139	2	x3	x3	PROPN
m-1152	139	3	·	·	PUNCT
m-1152	139	4	·	·	PUNCT
m-1152	139	5	·	·	PUNCT
m-1152	140	1	original	original	ADJ
m-1152	140	2	after	after	ADP
m-1152	140	3	t	t	PROPN
m-1152	140	4	(	(	PUNCT
m-1152	140	5	1	1	NUM
m-1152	140	6	)	)	PUNCT
m-1152	140	7	figure	figure	NOUN
m-1152	140	8	2	2	NUM
m-1152	140	9	:	:	PUNCT
m-1152	140	10	schematic	schematic	ADJ
m-1152	140	11	of	of	ADP
m-1152	140	12	the	the	DET
m-1152	140	13	right	right	ADJ
m-1152	140	14	shift	shift	NOUN
m-1152	140	15	t	t	PROPN
m-1152	140	16	(	(	PUNCT
m-1152	140	17	1	1	NUM
m-1152	140	18	)	)	PUNCT
m-1152	140	19	on	on	ADP
m-1152	140	20	ℓ2	ℓ2	NOUN
m-1152	140	21	:	:	PUNCT
m-1152	140	22	each	each	DET
m-1152	140	23	component	component	NOUN
m-1152	140	24	moves	move	VERB
m-1152	140	25	one	one	NUM
m-1152	140	26	box	box	NOUN
m-1152	140	27	to	to	ADP
m-1152	140	28	the	the	DET
m-1152	140	29	right	right	NOUN
m-1152	140	30	and	and	CCONJ
m-1152	140	31	a	a	DET
m-1152	140	32	0	0	NUM
m-1152	140	33	is	be	AUX
m-1152	140	34	inserted	insert	VERB
m-1152	140	35	at	at	ADP
m-1152	140	36	the	the	DET
m-1152	140	37	left	left	NOUN
m-1152	140	38	.	.	PUNCT
m-1152	141	1	example	example	NOUN
m-1152	141	2	4.3	4.3	NUM
m-1152	141	3	(	(	PUNCT
m-1152	141	4	heat	heat	NOUN
m-1152	141	5	semigroup	semigroup	PROPN
m-1152	141	6	)	)	PUNCT
m-1152	141	7	.	.	PUNCT
m-1152	142	1	let	let	VERB
m-1152	142	2	x	x	PUNCT
m-1152	142	3	=	=	SYM
m-1152	142	4	l2(rn	l2(rn	PROPN
m-1152	142	5	)	)	PUNCT
m-1152	142	6	,	,	PUNCT
m-1152	142	7	the	the	DET
m-1152	142	8	hilbert	hilbert	NOUN
m-1152	142	9	space	space	NOUN
m-1152	142	10	of	of	ADP
m-1152	142	11	square	square	ADJ
m-1152	142	12	-	-	PUNCT
m-1152	142	13	integrable	integrable	ADJ
m-1152	142	14	functions	function	NOUN
m-1152	142	15	on	on	ADP
m-1152	142	16	rn	rn	PROPN
m-1152	142	17	.	.	PUNCT
m-1152	142	18	de	de	PROPN
m-1152	142	19	�	�	PROPN
m-1152	142	20	ne	ne	PROPN
m-1152	142	21	,	,	PUNCT
m-1152	142	22	for	for	ADP
m-1152	142	23	t	t	PROPN
m-1152	142	24	>	>	X
m-1152	142	25	0	0	NUM
m-1152	142	26	,	,	PUNCT
m-1152	142	27	(	(	PUNCT
m-1152	142	28	t	t	PROPN
m-1152	142	29	(	(	PUNCT
m-1152	142	30	t)f)(x	t)f)(x	PROPN
m-1152	142	31	)	)	PUNCT
m-1152	142	32	=	=	PRON
m-1152	143	1	(	(	PUNCT
m-1152	143	2	gt	gt	INTJ
m-1152	143	3	∗	∗	NOUN
m-1152	143	4	f)(x	f)(x	PROPN
m-1152	143	5	)	)	PUNCT
m-1152	143	6	,	,	PUNCT
m-1152	143	7	gt(x	gt(x	PUNCT
m-1152	143	8	)	)	PUNCT
m-1152	144	1	=	=	SYM
m-1152	145	1	(	(	PUNCT
m-1152	145	2	4πt)−n/2e−	4πt)−n/2e−	PROPN
m-1152	145	3	|x|2	|x|2	PROPN
m-1152	145	4	4	4	NUM
m-1152	145	5	t	t	NOUN
m-1152	145	6	.	.	PUNCT
m-1152	146	1	here	here	ADV
m-1152	146	2	gt	gt	PROPN
m-1152	146	3	is	be	AUX
m-1152	146	4	the	the	DET
m-1152	146	5	gaussian	gaussian	ADJ
m-1152	146	6	heat	heat	NOUN
m-1152	146	7	kernel	kernel	NOUN
m-1152	146	8	,	,	PUNCT
m-1152	146	9	representing	represent	VERB
m-1152	146	10	the	the	DET
m-1152	146	11	fundamental	fundamental	ADJ
m-1152	146	12	solution	solution	NOUN
m-1152	146	13	of	of	ADP
m-1152	146	14	the	the	DET
m-1152	146	15	heat	heat	NOUN
m-1152	146	16	equation	equation	NOUN
m-1152	146	17	.	.	PUNCT
m-1152	147	1	for	for	ADP
m-1152	147	2	strong	strong	ADJ
m-1152	147	3	continuity	continuity	NOUN
m-1152	147	4	.	.	PUNCT
m-1152	148	1	for	for	ADP
m-1152	148	2	each	each	DET
m-1152	148	3	f	f	PROPN
m-1152	148	4	∈	∈	PROPN
m-1152	148	5	l2(rn	l2(rn	PROPN
m-1152	148	6	)	)	PUNCT
m-1152	148	7	,	,	PUNCT
m-1152	148	8	the	the	DET
m-1152	148	9	convolution	convolution	NOUN
m-1152	148	10	t	t	PROPN
m-1152	148	11	(	(	PUNCT
m-1152	148	12	t)f	t)f	SYM
m-1152	148	13	=	=	PUNCT
m-1152	148	14	gt	gt	PROPN
m-1152	148	15	∗	∗	X
m-1152	148	16	f	f	PROPN
m-1152	148	17	de	de	PROPN
m-1152	148	18	�	�	PROPN
m-1152	148	19	nes	ne	NOUN
m-1152	148	20	a	a	DET
m-1152	148	21	continuous	continuous	ADJ
m-1152	148	22	function	function	NOUN
m-1152	148	23	of	of	ADP
m-1152	148	24	t.	t.	PROPN
m-1152	148	25	as	as	ADP
m-1152	148	26	t	t	PROPN
m-1152	148	27	→	→	SYM
m-1152	148	28	0	0	NUM
m-1152	148	29	+	+	PROPN
m-1152	148	30	,	,	PUNCT
m-1152	148	31	gt	gt	PROPN
m-1152	148	32	tends	tend	VERB
m-1152	148	33	to	to	ADP
m-1152	148	34	the	the	DET
m-1152	148	35	dirac	dirac	NOUN
m-1152	148	36	delta	delta	PROPN
m-1152	148	37	distribution	distribution	NOUN
m-1152	148	38	δ	δ	PROPN
m-1152	148	39	,	,	PUNCT
m-1152	148	40	so	so	SCONJ
m-1152	148	41	t	t	PROPN
m-1152	148	42	(	(	PUNCT
m-1152	148	43	t)f	t)f	PUNCT
m-1152	148	44	→	→	SYM
m-1152	148	45	f	f	PROPN
m-1152	148	46	in	in	ADP
m-1152	148	47	l2	l2	NOUN
m-1152	148	48	,	,	PUNCT
m-1152	148	49	ensuring	ensure	VERB
m-1152	148	50	lim	lim	NOUN
m-1152	148	51	t→0	t→0	PROPN
m-1152	148	52	+	+	CCONJ
m-1152	148	53	∥t	∥t	ADJ
m-1152	148	54	(	(	PUNCT
m-1152	148	55	t)f	t)f	NOUN
m-1152	148	56	−	−	PROPN
m-1152	148	57	f∥2	f∥2	NOUN
m-1152	149	1	=	=	SYM
m-1152	149	2	0	0	X
m-1152	149	3	.	.	PUNCT
m-1152	150	1	hence	hence	ADV
m-1152	150	2	{	{	PUNCT
m-1152	150	3	t	t	PROPN
m-1152	150	4	(	(	PUNCT
m-1152	150	5	t)}t≥0	t)}t≥0	NOUN
m-1152	150	6	is	be	AUX
m-1152	150	7	a	a	DET
m-1152	150	8	strongly	strongly	ADV
m-1152	150	9	continuous	continuous	ADJ
m-1152	150	10	semigroup	semigroup	NOUN
m-1152	150	11	(	(	PUNCT
m-1152	150	12	a	a	DET
m-1152	150	13	c0	c0	NOUN
m-1152	150	14	-	-	PUNCT
m-1152	150	15	semigroup	semigroup	NOUN
m-1152	150	16	)	)	PUNCT
m-1152	150	17	.	.	PUNCT
m-1152	151	1	for	for	ADP
m-1152	151	2	the	the	DET
m-1152	151	3	generator	generator	NOUN
m-1152	151	4	.	.	PUNCT
m-1152	152	1	we	we	PRON
m-1152	152	2	recall	recall	VERB
m-1152	152	3	that	that	PRON
m-1152	152	4	t	t	PROPN
m-1152	152	5	(	(	PUNCT
m-1152	152	6	t	t	NOUN
m-1152	152	7	)	)	PUNCT
m-1152	152	8	solves	solve	VERB
m-1152	152	9	the	the	DET
m-1152	152	10	cauchy	cauchy	PROPN
m-1152	152	11	problem	problem	NOUN
m-1152	152	12	for	for	ADP
m-1152	152	13	the	the	DET
m-1152	152	14	heat	heat	NOUN
m-1152	152	15	equation	equation	NOUN
m-1152	152	16	:	:	PUNCT
m-1152	152	17	∂u	∂u	PROPN
m-1152	152	18	∂t	∂t	PROPN
m-1152	152	19	=	=	PUNCT
m-1152	152	20	∆	∆	PROPN
m-1152	153	1	uu	uu	PROPN
m-1152	153	2	,	,	PUNCT
m-1152	153	3	(	(	PUNCT
m-1152	153	4	0	0	NUM
m-1152	153	5	,	,	PUNCT
m-1152	153	6	x	x	NOUN
m-1152	153	7	)	)	PUNCT
m-1152	153	8	=	=	SYM
m-1152	153	9	f(x	f(x	PROPN
m-1152	153	10	)	)	PUNCT
m-1152	153	11	.	.	PUNCT
m-1152	154	1	thus	thus	ADV
m-1152	154	2	,	,	PUNCT
m-1152	154	3	the	the	DET
m-1152	154	4	generator	generator	NOUN
m-1152	154	5	is	be	AUX
m-1152	154	6	the	the	DET
m-1152	154	7	laplacian	laplacian	ADJ
m-1152	154	8	operator	operator	NOUN
m-1152	154	9	af	af	PROPN
m-1152	154	10	=	=	SYM
m-1152	154	11	∆f	∆f	PROPN
m-1152	154	12	,	,	PUNCT
m-1152	154	13	d(a	d(a	PROPN
m-1152	154	14	)	)	PUNCT
m-1152	154	15	=	=	SYM
m-1152	154	16	h2(rn	h2(rn	PROPN
m-1152	154	17	)	)	PUNCT
m-1152	154	18	=	=	PRON
m-1152	154	19	{	{	PUNCT
m-1152	154	20	f	f	PROPN
m-1152	154	21	∈	∈	PROPN
m-1152	154	22	l2(rn	l2(rn	PROPN
m-1152	154	23	)	)	PUNCT
m-1152	154	24	:	:	PUNCT
m-1152	154	25	∆f	∆f	PROPN
m-1152	154	26	∈	∈	PROPN
m-1152	154	27	l2(rn	l2(rn	PROPN
m-1152	154	28	)	)	PUNCT
m-1152	154	29	}	}	PUNCT
m-1152	154	30	.	.	PUNCT
m-1152	155	1	this	this	PRON
m-1152	155	2	follows	follow	VERB
m-1152	155	3	by	by	ADP
m-1152	155	4	di	di	PROPN
m-1152	155	5	�	�	PROPN
m-1152	155	6	erentiating	erentiating	ADJ
m-1152	155	7	t	t	PROPN
m-1152	155	8	(	(	PUNCT
m-1152	155	9	t)f	t)f	SYM
m-1152	155	10	at	at	ADP
m-1152	155	11	t	t	NOUN
m-1152	155	12	=	=	SYM
m-1152	155	13	0	0	NUM
m-1152	155	14	under	under	ADP
m-1152	155	15	the	the	DET
m-1152	155	16	fourier	fourier	NOUN
m-1152	155	17	transform	transform	NOUN
m-1152	155	18	.	.	PUNCT
m-1152	156	1	ijo	ijo	PROPN
m-1152	156	2	international	international	PROPN
m-1152	156	3	journal	journal	PROPN
m-1152	156	4	of	of	ADP
m-1152	156	5	mathematics	mathematics	PROPN
m-1152	156	6	(	(	PUNCT
m-1152	156	7	issn	issn	PROPN
m-1152	156	8	:	:	PUNCT
m-1152	156	9	2992	2992	NUM
m-1152	156	10	-	-	SYM
m-1152	156	11	4421	4421	NUM
m-1152	156	12	)	)	PUNCT
m-1152	156	13	volume	volume	NOUN
m-1152	156	14	08	08	NUM
m-1152	157	1	|	|	ADV
m-1152	157	2	issue	issue	NOUN
m-1152	157	3	9	9	NUM
m-1152	157	4	|	|	CCONJ
m-1152	157	5	september	september	PROPN
m-1152	157	6	2025	2025	NUM
m-1152	158	1	|	|	ADV
m-1152	158	2	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1152	158	3	15	15	NUM
m-1152	158	4	for	for	ADP
m-1152	158	5	kw	kw	NOUN
m-1152	158	6	-	-	NOUN
m-1152	158	7	stability	stability	NOUN
m-1152	158	8	.	.	PUNCT
m-1152	159	1	by	by	ADP
m-1152	159	2	proposition	proposition	NOUN
m-1152	159	3	3.1	3.1	NUM
m-1152	159	4	,	,	PUNCT
m-1152	159	5	every	every	DET
m-1152	159	6	c0	c0	NOUN
m-1152	159	7	-	-	PUNCT
m-1152	159	8	semigroup	semigroup	PROPN
m-1152	159	9	is	be	AUX
m-1152	159	10	kw	kw	VERB
m-1152	159	11	-	-	ADJ
m-1152	159	12	stable	stable	ADJ
m-1152	159	13	in	in	ADP
m-1152	159	14	the	the	DET
m-1152	159	15	algebraic	algebraic	ADJ
m-1152	159	16	sense	sense	NOUN
m-1152	159	17	.	.	PUNCT
m-1152	160	1	in	in	ADP
m-1152	160	2	particular	particular	ADJ
m-1152	160	3	,	,	PUNCT
m-1152	160	4	for	for	ADP
m-1152	160	5	the	the	DET
m-1152	160	6	heat	heat	NOUN
m-1152	160	7	semigroup	semigroup	NOUN
m-1152	160	8	,	,	PUNCT
m-1152	160	9	{	{	PUNCT
m-1152	160	10	t	t	NOUN
m-1152	160	11	(	(	PUNCT
m-1152	160	12	t)f	t)f	NOUN
m-1152	160	13	:	:	PUNCT
m-1152	160	14	t	t	X
m-1152	160	15	≥	≥	NOUN
m-1152	160	16	0	0	NUM
m-1152	160	17	}	}	PUNCT
m-1152	160	18	=	=	SYM
m-1152	160	19	{	{	PUNCT
m-1152	160	20	t	t	PROPN
m-1152	160	21	(	(	PUNCT
m-1152	160	22	s+	s+	ADV
m-1152	160	23	t)f	t)f	PUNCT
m-1152	160	24	:	:	PUNCT
m-1152	160	25	t	t	X
m-1152	160	26	≥	≥	NOUN
m-1152	160	27	0	0	NUM
m-1152	160	28	}	}	PUNCT
m-1152	160	29	,	,	PUNCT
m-1152	160	30	for	for	SCONJ
m-1152	160	31	all	all	DET
m-1152	160	32	s	s	PART
m-1152	160	33	≥	≥	NOUN
m-1152	160	34	0	0	NUM
m-1152	160	35	,	,	PUNCT
m-1152	160	36	re	re	VERB
m-1152	160	37	�	�	VERB
m-1152	160	38	ecting	ecte	VERB
m-1152	160	39	the	the	DET
m-1152	160	40	invariance	invariance	NOUN
m-1152	160	41	of	of	ADP
m-1152	160	42	reachable	reachable	ADJ
m-1152	160	43	states	state	NOUN
m-1152	160	44	.	.	PUNCT
m-1152	161	1	for	for	ADP
m-1152	161	2	analytic	analytic	ADJ
m-1152	161	3	stability	stability	NOUN
m-1152	161	4	.	.	PUNCT
m-1152	162	1	the	the	DET
m-1152	162	2	fourier	fourier	NOUN
m-1152	162	3	transform	transform	NOUN
m-1152	162	4	of	of	ADP
m-1152	162	5	gt	gt	PROPN
m-1152	162	6	is	be	AUX
m-1152	162	7	given	give	VERB
m-1152	162	8	by	by	ADP
m-1152	162	9	ĝt(ξ	ĝt(ξ	ADV
m-1152	162	10	)	)	PUNCT
m-1152	163	1	=	=	SYM
m-1152	163	2	e−t|ξ|2	e−t|ξ|2	PROPN
m-1152	163	3	,	,	PUNCT
m-1152	163	4	so	so	CCONJ
m-1152	163	5	in	in	ADP
m-1152	163	6	the	the	DET
m-1152	163	7	fourier	fourier	NOUN
m-1152	163	8	domain	domain	NOUN
m-1152	163	9	,	,	PUNCT
m-1152	163	10	t	t	PROPN
m-1152	163	11	(	(	PUNCT
m-1152	163	12	t)f	t)f	ADP
m-1152	163	13	has	have	VERB
m-1152	163	14	the	the	DET
m-1152	163	15	multiplier	multipli	ADJ
m-1152	163	16	e−t|ξ|2	e−t|ξ|2	PROPN
m-1152	163	17	,	,	PUNCT
m-1152	163	18	which	which	PRON
m-1152	163	19	decays	decay	VERB
m-1152	163	20	exponentially	exponentially	ADV
m-1152	163	21	in	in	ADP
m-1152	163	22	t	t	PROPN
m-1152	163	23	for	for	ADP
m-1152	163	24	each	each	DET
m-1152	163	25	ξ	ξ	PROPN
m-1152	163	26	̸=	̸=	PROPN
m-1152	163	27	0	0	NUM
m-1152	163	28	.	.	PUNCT
m-1152	164	1	since	since	SCONJ
m-1152	164	2	the	the	DET
m-1152	164	3	spectrum	spectrum	NOUN
m-1152	164	4	of	of	ADP
m-1152	164	5	∆	∆	PROPN
m-1152	164	6	is	be	AUX
m-1152	164	7	σ(∆	σ(∆	NOUN
m-1152	164	8	)	)	PUNCT
m-1152	165	1	=	=	SYM
m-1152	165	2	(	(	PUNCT
m-1152	165	3	−∞	−∞	NOUN
m-1152	165	4	,	,	PUNCT
m-1152	165	5	0	0	NUM
m-1152	165	6	]	]	PUNCT
m-1152	165	7	,	,	PUNCT
m-1152	165	8	the	the	DET
m-1152	165	9	spectral	spectral	ADJ
m-1152	165	10	bound	bind	VERB
m-1152	165	11	is	be	AUX
m-1152	165	12	strictly	strictly	ADV
m-1152	165	13	negative	negative	ADJ
m-1152	165	14	.	.	PUNCT
m-1152	166	1	therefore	therefore	ADV
m-1152	166	2	,	,	PUNCT
m-1152	166	3	there	there	PRON
m-1152	166	4	exists	exist	VERB
m-1152	166	5	ω	ω	PROPN
m-1152	166	6	>	>	X
m-1152	166	7	0	0	NUM
m-1152	167	1	such	such	ADJ
m-1152	167	2	that	that	SCONJ
m-1152	167	3	∥t	∥t	PROPN
m-1152	167	4	(	(	PUNCT
m-1152	167	5	t)f∥2	t)f∥2	ADJ
m-1152	167	6	≤	≤	NOUN
m-1152	167	7	e−ωt∥f∥2	e−ωt∥f∥2	NOUN
m-1152	167	8	,	,	PUNCT
m-1152	167	9	∀t	∀t	PROPN
m-1152	167	10	≥	≥	NOUN
m-1152	167	11	0	0	NUM
m-1152	167	12	.	.	PUNCT
m-1152	168	1	hence	hence	ADV
m-1152	168	2	the	the	DET
m-1152	168	3	semigroup	semigroup	NOUN
m-1152	168	4	is	be	AUX
m-1152	168	5	not	not	PART
m-1152	168	6	only	only	ADV
m-1152	168	7	contractive	contractive	ADJ
m-1152	168	8	but	but	CCONJ
m-1152	168	9	also	also	ADV
m-1152	168	10	exponentially	exponentially	ADV
m-1152	168	11	stable	stable	ADJ
m-1152	168	12	.	.	PUNCT
m-1152	169	1	interpretation	interpretation	NOUN
m-1152	169	2	.	.	PUNCT
m-1152	170	1	in	in	ADP
m-1152	170	2	this	this	DET
m-1152	170	3	example	example	NOUN
m-1152	170	4	,	,	PUNCT
m-1152	170	5	algebraic	algebraic	ADJ
m-1152	170	6	stability	stability	NOUN
m-1152	170	7	(	(	PUNCT
m-1152	170	8	kw	kw	NOUN
m-1152	170	9	-	-	NOUN
m-1152	170	10	stability	stability	NOUN
m-1152	170	11	)	)	PUNCT
m-1152	170	12	and	and	CCONJ
m-1152	170	13	analytic	analytic	ADJ
m-1152	170	14	stability	stability	NOUN
m-1152	170	15	(	(	PUNCT
m-1152	170	16	exponential	exponential	ADJ
m-1152	170	17	decay	decay	NOUN
m-1152	170	18	of	of	ADP
m-1152	170	19	norms	norm	NOUN
m-1152	170	20	)	)	PUNCT
m-1152	170	21	coincide	coincide	NOUN
m-1152	170	22	.	.	PUNCT
m-1152	171	1	unlike	unlike	ADP
m-1152	171	2	the	the	DET
m-1152	171	3	translation	translation	NOUN
m-1152	171	4	and	and	CCONJ
m-1152	171	5	shift	shift	NOUN
m-1152	171	6	semigroups	semigroup	NOUN
m-1152	171	7	,	,	PUNCT
m-1152	171	8	which	which	PRON
m-1152	171	9	preserve	preserve	VERB
m-1152	171	10	norm	norm	NOUN
m-1152	171	11	without	without	ADP
m-1152	171	12	decay	decay	NOUN
m-1152	171	13	,	,	PUNCT
m-1152	171	14	the	the	DET
m-1152	171	15	heat	heat	NOUN
m-1152	171	16	semigroup	semigroup	NOUN
m-1152	171	17	smooths	smooth	NOUN
m-1152	171	18	and	and	CCONJ
m-1152	171	19	dissipates	dissipate	VERB
m-1152	171	20	initial	initial	ADJ
m-1152	171	21	data	datum	NOUN
m-1152	171	22	over	over	ADP
m-1152	171	23	time	time	NOUN
m-1152	171	24	.	.	PUNCT
m-1152	172	1	physically	physically	ADV
m-1152	172	2	,	,	PUNCT
m-1152	172	3	this	this	PRON
m-1152	172	4	corresponds	correspond	VERB
m-1152	172	5	to	to	ADP
m-1152	172	6	the	the	DET
m-1152	172	7	di	di	PROPN
m-1152	172	8	�	�	PROPN
m-1152	172	9	usion	usion	NOUN
m-1152	172	10	of	of	ADP
m-1152	172	11	heat	heat	NOUN
m-1152	172	12	:	:	PUNCT
m-1152	172	13	local	local	ADJ
m-1152	172	14	peaks	peak	NOUN
m-1152	172	15	�	�	PROPN
m-1152	172	16	atten	atten	VERB
m-1152	172	17	,	,	PUNCT
m-1152	172	18	energy	energy	NOUN
m-1152	172	19	spreads	spread	VERB
m-1152	172	20	out	out	ADP
m-1152	172	21	,	,	PUNCT
m-1152	172	22	and	and	CCONJ
m-1152	172	23	the	the	DET
m-1152	172	24	system	system	NOUN
m-1152	172	25	relaxes	relax	VERB
m-1152	172	26	exponentially	exponentially	ADV
m-1152	172	27	fast	fast	ADV
m-1152	172	28	.	.	PUNCT
m-1152	173	1	−6	−6	INTJ
m-1152	174	1	−5	−5	NOUN
m-1152	175	1	−4	−4	X
m-1152	175	2	−3	−3	X
m-1152	175	3	−2	−2	PROPN
m-1152	175	4	−1	−1	NOUN
m-1152	175	5	0	0	NUM
m-1152	175	6	1	1	NUM
m-1152	175	7	2	2	NUM
m-1152	175	8	3	3	NUM
m-1152	175	9	4	4	NUM
m-1152	175	10	5	5	NUM
m-1152	175	11	6	6	NUM
m-1152	175	12	0	0	NUM
m-1152	175	13	0.2	0.2	NUM
m-1152	175	14	0.4	0.4	NUM
m-1152	175	15	x	x	SYM
m-1152	175	16	g	g	PROPN
m-1152	175	17	t	t	PROPN
m-1152	175	18	(	(	PUNCT
m-1152	175	19	x	x	SYM
m-1152	175	20	)	)	PUNCT
m-1152	175	21	t	t	NOUN
m-1152	175	22	=	=	SYM
m-1152	175	23	0.5	0.5	NUM
m-1152	175	24	t	t	NOUN
m-1152	175	25	=	=	SYM
m-1152	175	26	1	1	NUM
m-1152	175	27	t	t	NOUN
m-1152	175	28	=	=	SYM
m-1152	175	29	2	2	NUM
m-1152	175	30	figure	figure	NOUN
m-1152	175	31	3	3	NUM
m-1152	175	32	:	:	PUNCT
m-1152	175	33	gaussian	gaussian	ADJ
m-1152	175	34	heat	heat	NOUN
m-1152	175	35	kernel	kernel	PROPN
m-1152	175	36	gt(x	gt(x	PROPN
m-1152	175	37	)	)	PUNCT
m-1152	175	38	at	at	ADP
m-1152	175	39	di	di	PROPN
m-1152	175	40	�	�	PROPN
m-1152	175	41	erent	erent	NOUN
m-1152	175	42	times	time	NOUN
m-1152	175	43	t	t	PROPN
m-1152	175	44	=	=	SYM
m-1152	175	45	0.5	0.5	NUM
m-1152	175	46	(	(	PUNCT
m-1152	175	47	blue	blue	NOUN
m-1152	175	48	)	)	PUNCT
m-1152	175	49	,	,	PUNCT
m-1152	175	50	t	t	PROPN
m-1152	175	51	=	=	SYM
m-1152	175	52	1	1	NUM
m-1152	175	53	(	(	PUNCT
m-1152	175	54	red	red	NOUN
m-1152	175	55	)	)	PUNCT
m-1152	175	56	,	,	PUNCT
m-1152	175	57	and	and	CCONJ
m-1152	175	58	t	t	X
m-1152	175	59	=	=	SYM
m-1152	175	60	2	2	NUM
m-1152	175	61	(	(	PUNCT
m-1152	175	62	green	green	NOUN
m-1152	175	63	)	)	PUNCT
m-1152	175	64	.	.	PUNCT
m-1152	176	1	graphical	graphical	ADJ
m-1152	176	2	interpretation	interpretation	NOUN
m-1152	176	3	(	(	PUNCT
m-1152	176	4	figure	figure	NOUN
m-1152	176	5	3	3	NUM
m-1152	176	6	)	)	PUNCT
m-1152	176	7	.	.	PUNCT
m-1152	177	1	the	the	DET
m-1152	177	2	curves	curve	NOUN
m-1152	177	3	show	show	VERB
m-1152	177	4	the	the	DET
m-1152	177	5	gaussian	gaussian	ADJ
m-1152	177	6	kernel	kernel	NOUN
m-1152	177	7	gt(x	gt(x	PROPN
m-1152	177	8	)	)	PUNCT
m-1152	177	9	for	for	ADP
m-1152	177	10	di	di	NOUN
m-1152	177	11	�	�	PROPN
m-1152	177	12	erent	erent	NOUN
m-1152	177	13	times	time	NOUN
m-1152	177	14	:	:	PUNCT
m-1152	177	15	�	�	PROPN
m-1152	177	16	at	at	ADP
m-1152	177	17	t	t	PROPN
m-1152	177	18	=	=	SYM
m-1152	177	19	0.5	0.5	NUM
m-1152	177	20	(	(	PUNCT
m-1152	177	21	blue	blue	ADJ
m-1152	177	22	):	):	PUNCT
m-1152	177	23	the	the	DET
m-1152	177	24	kernel	kernel	NOUN
m-1152	177	25	is	be	AUX
m-1152	177	26	tall	tall	ADJ
m-1152	177	27	and	and	CCONJ
m-1152	177	28	narrow	narrow	ADJ
m-1152	177	29	,	,	PUNCT
m-1152	177	30	concentrated	concentrate	VERB
m-1152	177	31	near	near	ADP
m-1152	177	32	x	x	X
m-1152	177	33	=	=	SYM
m-1152	177	34	0	0	X
m-1152	177	35	.	.	PUNCT
m-1152	178	1	heat	heat	NOUN
m-1152	178	2	is	be	AUX
m-1152	178	3	still	still	ADV
m-1152	178	4	localized	localize	VERB
m-1152	178	5	.	.	PUNCT
m-1152	179	1	�	�	PROPN
m-1152	179	2	at	at	ADP
m-1152	179	3	t	t	PROPN
m-1152	179	4	=	=	SYM
m-1152	179	5	1	1	NUM
m-1152	179	6	(	(	PUNCT
m-1152	179	7	red	red	ADJ
m-1152	179	8	):	):	PUNCT
m-1152	179	9	the	the	DET
m-1152	179	10	peak	peak	NOUN
m-1152	179	11	is	be	AUX
m-1152	179	12	lower	low	ADJ
m-1152	179	13	but	but	CCONJ
m-1152	179	14	wider	wide	ADJ
m-1152	179	15	,	,	PUNCT
m-1152	179	16	showing	show	VERB
m-1152	179	17	partial	partial	ADJ
m-1152	179	18	di	di	NOUN
m-1152	179	19	�	�	PROPN
m-1152	179	20	usion	usion	NOUN
m-1152	179	21	and	and	CCONJ
m-1152	179	22	�	�	NOUN
m-1152	179	23	attening	attening	NOUN
m-1152	179	24	of	of	ADP
m-1152	179	25	the	the	DET
m-1152	179	26	initial	initial	ADJ
m-1152	179	27	concentration	concentration	NOUN
m-1152	179	28	.	.	PUNCT
m-1152	180	1	�	�	PROPN
m-1152	180	2	at	at	ADP
m-1152	180	3	t	t	PROPN
m-1152	180	4	=	=	SYM
m-1152	180	5	2	2	NUM
m-1152	180	6	(	(	PUNCT
m-1152	180	7	green	green	ADJ
m-1152	180	8	):	):	PUNCT
m-1152	180	9	the	the	DET
m-1152	180	10	kernel	kernel	NOUN
m-1152	180	11	is	be	AUX
m-1152	180	12	very	very	ADJ
m-1152	180	13	�	�	PROPN
m-1152	180	14	at	at	ADP
m-1152	180	15	and	and	CCONJ
m-1152	180	16	spread	spread	VERB
m-1152	180	17	out	out	ADP
m-1152	180	18	,	,	PUNCT
m-1152	180	19	indicating	indicate	VERB
m-1152	180	20	that	that	SCONJ
m-1152	180	21	heat	heat	NOUN
m-1152	180	22	has	have	AUX
m-1152	180	23	dissipated	dissipate	VERB
m-1152	180	24	signi	signi	NOUN
m-1152	180	25	�	�	NOUN
m-1152	180	26	cantly	cantly	ADV
m-1152	180	27	.	.	PUNCT
m-1152	181	1	thus	thus	ADV
m-1152	181	2	,	,	PUNCT
m-1152	181	3	as	as	ADP
m-1152	181	4	t	t	PROPN
m-1152	181	5	→	→	SYM
m-1152	181	6	∞	∞	PROPN
m-1152	181	7	,	,	PUNCT
m-1152	181	8	the	the	DET
m-1152	181	9	peak	peak	NOUN
m-1152	181	10	decays	decay	VERB
m-1152	181	11	while	while	SCONJ
m-1152	181	12	the	the	DET
m-1152	181	13	width	width	NOUN
m-1152	181	14	grows	grow	VERB
m-1152	181	15	like	like	ADP
m-1152	181	16	√	√	PROPN
m-1152	181	17	t.	t.	NOUN
m-1152	181	18	this	this	PRON
m-1152	181	19	illustrates	illustrate	VERB
m-1152	181	20	exponential	exponential	ADJ
m-1152	181	21	stability	stability	NOUN
m-1152	181	22	in	in	ADP
m-1152	181	23	the	the	DET
m-1152	181	24	semigroup	semigroup	ADJ
m-1152	181	25	sense	sense	NOUN
m-1152	181	26	and	and	CCONJ
m-1152	181	27	di	di	NOUN
m-1152	181	28	�	�	PROPN
m-1152	181	29	usion	usion	NOUN
m-1152	181	30	in	in	ADP
m-1152	181	31	the	the	DET
m-1152	181	32	physical	physical	ADJ
m-1152	181	33	sense	sense	NOUN
m-1152	181	34	.	.	PUNCT
m-1152	182	1	for	for	ADP
m-1152	182	2	detailed	detailed	ADJ
m-1152	182	3	expositions	exposition	NOUN
m-1152	182	4	of	of	ADP
m-1152	182	5	heat	heat	NOUN
m-1152	182	6	semigroups	semigroup	NOUN
m-1152	182	7	and	and	CCONJ
m-1152	182	8	their	their	PRON
m-1152	182	9	stability	stability	NOUN
m-1152	182	10	properties	property	NOUN
m-1152	182	11	,	,	PUNCT
m-1152	182	12	see	see	VERB
m-1152	182	13	[	[	X
m-1152	182	14	16	16	NUM
m-1152	182	15	,	,	PUNCT
m-1152	182	16	18	18	NUM
m-1152	182	17	,	,	PUNCT
m-1152	182	18	19	19	NUM
m-1152	182	19	,	,	PUNCT
m-1152	182	20	27	27	NUM
m-1152	182	21	,	,	PUNCT
m-1152	182	22	28	28	NUM
m-1152	182	23	,	,	PUNCT
m-1152	182	24	29	29	NUM
m-1152	182	25	,	,	PUNCT
m-1152	182	26	30	30	NUM
m-1152	182	27	,	,	PUNCT
m-1152	182	28	31	31	NUM
m-1152	182	29	]	]	PUNCT
m-1152	182	30	.	.	PUNCT
m-1152	183	1	ijo	ijo	PROPN
m-1152	183	2	international	international	PROPN
m-1152	183	3	journal	journal	PROPN
m-1152	183	4	of	of	ADP
m-1152	183	5	mathematics	mathematics	PROPN
m-1152	183	6	(	(	PUNCT
m-1152	183	7	issn	issn	PROPN
m-1152	183	8	:	:	PUNCT
m-1152	183	9	2992	2992	NUM
m-1152	183	10	-	-	SYM
m-1152	183	11	4421	4421	NUM
m-1152	183	12	)	)	PUNCT
m-1152	183	13	volume	volume	NOUN
m-1152	183	14	08	08	NUM
m-1152	184	1	|	|	ADV
m-1152	184	2	issue	issue	NOUN
m-1152	184	3	9	9	NUM
m-1152	184	4	|	|	CCONJ
m-1152	184	5	september	september	PROPN
m-1152	184	6	2025	2025	NUM
m-1152	185	1	|	|	ADV
m-1152	185	2	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1152	185	3	16	16	NUM
m-1152	185	4	4.1	4.1	NUM
m-1152	185	5	example	example	NOUN
m-1152	185	6	3.4	3.4	NUM
m-1152	185	7	(	(	PUNCT
m-1152	185	8	damped	damped	ADJ
m-1152	185	9	wave	wave	NOUN
m-1152	185	10	equation	equation	NOUN
m-1152	185	11	)	)	PUNCT
m-1152	185	12	we	we	PRON
m-1152	185	13	consider	consider	VERB
m-1152	185	14	the	the	DET
m-1152	185	15	damped	damped	NOUN
m-1152	185	16	wave	wave	NOUN
m-1152	185	17	equation	equation	NOUN
m-1152	185	18	on	on	ADP
m-1152	185	19	the	the	DET
m-1152	185	20	bounded	bounded	ADJ
m-1152	185	21	interval	interval	NOUN
m-1152	185	22	(	(	PUNCT
m-1152	185	23	0	0	NUM
m-1152	185	24	,	,	PUNCT
m-1152	185	25	l	l	NOUN
m-1152	185	26	)	)	PUNCT
m-1152	185	27	with	with	ADP
m-1152	185	28	homogeneous	homogeneous	ADJ
m-1152	185	29	dirichlet	dirichlet	PROPN
m-1152	185	30	boundary	boundary	ADJ
m-1152	185	31	conditions	condition	NOUN
m-1152	185	32	:	:	PUNCT
m-1152	186	1	utt(x	utt(x	PROPN
m-1152	186	2	,	,	PUNCT
m-1152	186	3	t	t	PROPN
m-1152	186	4	)	)	PUNCT
m-1152	186	5	+	+	PROPN
m-1152	186	6	αut(x	αut(x	PROPN
m-1152	186	7	,	,	PUNCT
m-1152	186	8	t)−	t)−	PROPN
m-1152	186	9	uxx(x	uxx(x	PROPN
m-1152	186	10	,	,	PUNCT
m-1152	186	11	t	t	PROPN
m-1152	186	12	)	)	PUNCT
m-1152	186	13	=	=	SYM
m-1152	186	14	0	0	NUM
m-1152	186	15	,	,	PUNCT
m-1152	186	16	x	x	SYM
m-1152	186	17	∈	∈	PROPN
m-1152	186	18	(	(	PUNCT
m-1152	186	19	0	0	NUM
m-1152	186	20	,	,	PUNCT
m-1152	186	21	l	l	NOUN
m-1152	186	22	)	)	PUNCT
m-1152	186	23	,	,	PUNCT
m-1152	186	24	t	t	PROPN
m-1152	186	25	>	>	X
m-1152	186	26	0	0	PROPN
m-1152	186	27	,	,	PUNCT
m-1152	186	28	u(0	u(0	PROPN
m-1152	186	29	,	,	PUNCT
m-1152	186	30	t	t	PROPN
m-1152	186	31	)	)	PUNCT
m-1152	186	32	=	=	SYM
m-1152	187	1	u(l	u(l	PROPN
m-1152	187	2	,	,	PUNCT
m-1152	187	3	t	t	PROPN
m-1152	187	4	)	)	PUNCT
m-1152	187	5	=	=	SYM
m-1152	187	6	0	0	NUM
m-1152	187	7	t	t	PROPN
m-1152	187	8	,	,	PUNCT
m-1152	187	9	≥	≥	NOUN
m-1152	187	10	0	0	NUM
m-1152	187	11	,	,	PUNCT
m-1152	187	12	u(x	u(x	NOUN
m-1152	187	13	,	,	PUNCT
m-1152	187	14	0	0	NUM
m-1152	187	15	)	)	PUNCT
m-1152	187	16	=	=	SYM
m-1152	187	17	u0(x	u0(x	NUM
m-1152	187	18	)	)	PUNCT
m-1152	187	19	u	u	NOUN
m-1152	187	20	,	,	PUNCT
m-1152	187	21	t(x	t(x	PROPN
m-1152	187	22	,	,	PUNCT
m-1152	187	23	0	0	NUM
m-1152	187	24	)	)	PUNCT
m-1152	187	25	=	=	SYM
m-1152	187	26	v0(x	v0(x	NOUN
m-1152	187	27	)	)	PUNCT
m-1152	187	28	x	x	NOUN
m-1152	187	29	,	,	PUNCT
m-1152	187	30	∈	∈	PROPN
m-1152	187	31	(	(	PUNCT
m-1152	187	32	0	0	NUM
m-1152	187	33	,	,	PUNCT
m-1152	187	34	l	l	NOUN
m-1152	187	35	)	)	PUNCT
m-1152	187	36	,	,	PUNCT
m-1152	187	37	(	(	PUNCT
m-1152	187	38	1	1	X
m-1152	187	39	)	)	PUNCT
m-1152	187	40	where	where	SCONJ
m-1152	187	41	α	α	X
m-1152	187	42	∈	∈	PROPN
m-1152	187	43	r	r	NOUN
m-1152	187	44	is	be	AUX
m-1152	187	45	the	the	PRON
m-1152	187	46	(	(	PUNCT
m-1152	187	47	constant	constant	ADJ
m-1152	187	48	)	)	PUNCT
m-1152	187	49	damping	damp	VERB
m-1152	187	50	coe	coe	PROPN
m-1152	187	51	�	�	PROPN
m-1152	187	52	cient	cient	PROPN
m-1152	187	53	.	.	PUNCT
m-1152	188	1	state	state	NOUN
m-1152	188	2	space	space	NOUN
m-1152	188	3	and	and	CCONJ
m-1152	188	4	energy	energy	NOUN
m-1152	188	5	.	.	PUNCT
m-1152	189	1	set	set	VERB
m-1152	189	2	x	x	PUNCT
m-1152	190	1	=	=	PRON
m-1152	190	2	h1	h1	NOUN
m-1152	190	3	0	0	NUM
m-1152	190	4	(	(	PUNCT
m-1152	190	5	0	0	NUM
m-1152	190	6	,	,	PUNCT
m-1152	190	7	l)×	l)×	NOUN
m-1152	190	8	l2(0	l2(0	NOUN
m-1152	190	9	,	,	PUNCT
m-1152	190	10	l	l	NOUN
m-1152	190	11	)	)	PUNCT
m-1152	190	12	,	,	PUNCT
m-1152	190	13	with	with	ADP
m-1152	190	14	state	state	NOUN
m-1152	190	15	variable	variable	ADJ
m-1152	190	16	u(t	u(t	NOUN
m-1152	190	17	)	)	PUNCT
m-1152	190	18	=	=	PRON
m-1152	190	19	(	(	PUNCT
m-1152	190	20	u	u	NOUN
m-1152	190	21	(	(	PUNCT
m-1152	190	22	·	·	PUNCT
m-1152	190	23	,	,	PUNCT
m-1152	190	24	t	t	PROPN
m-1152	190	25	)	)	PUNCT
m-1152	190	26	,	,	PUNCT
m-1152	190	27	ut	ut	PROPN
m-1152	190	28	(	(	PUNCT
m-1152	190	29	·	·	PUNCT
m-1152	190	30	,	,	PUNCT
m-1152	190	31	t))⊤.	t))⊤.	NOUN
m-1152	190	32	we	we	PRON
m-1152	190	33	equip	equip	VERB
m-1152	190	34	x	x	PUNCT
m-1152	190	35	with	with	ADP
m-1152	190	36	the	the	DET
m-1152	190	37	energy	energy	NOUN
m-1152	190	38	inner	inner	ADJ
m-1152	190	39	product	product	NOUN
m-1152	190	40	〈	〈	PROPN
m-1152	190	41	(	(	PUNCT
m-1152	190	42	u1	u1	NOUN
m-1152	190	43	,	,	PUNCT
m-1152	190	44	v1	v1	NOUN
m-1152	190	45	)	)	PUNCT
m-1152	190	46	,	,	PUNCT
m-1152	190	47	(	(	PUNCT
m-1152	190	48	u2	u2	NOUN
m-1152	190	49	,	,	PUNCT
m-1152	190	50	v2	v2	NOUN
m-1152	190	51	)	)	PUNCT
m-1152	190	52	〉	〉	NOUN
m-1152	190	53	x	x	X
m-1152	190	54	:	:	PUNCT
m-1152	190	55	=	=	NUM
m-1152	190	56	∫	∫	PROPN
m-1152	190	57	l	l	NOUN
m-1152	190	58	0	0	NUM
m-1152	190	59	u′	u′	PROPN
m-1152	190	60	1(x)u	1(x)u	NOUN
m-1152	190	61	′	′	NUM
m-1152	190	62	2(x	2(x	NUM
m-1152	190	63	)	)	PUNCT
m-1152	191	1	dx+	dx+	ADJ
m-1152	191	2	∫	∫	PROPN
m-1152	191	3	l	l	NOUN
m-1152	191	4	0	0	NUM
m-1152	191	5	v1(x)v2(x	v1(x)v2(x	NOUN
m-1152	191	6	)	)	PUNCT
m-1152	191	7	dx	dx	PROPN
m-1152	191	8	,	,	PUNCT
m-1152	191	9	and	and	CCONJ
m-1152	191	10	corresponding	correspond	VERB
m-1152	191	11	norm	norm	NOUN
m-1152	191	12	∥(u	∥(u	NOUN
m-1152	191	13	,	,	PUNCT
m-1152	191	14	v)∥2x	v)∥2x	NOUN
m-1152	191	15	=	=	SYM
m-1152	191	16	∥u′∥2l2(0,l	∥u′∥2l2(0,l	PROPN
m-1152	191	17	)	)	PUNCT
m-1152	191	18	+	+	NUM
m-1152	191	19	∥v∥2l2(0,l	∥v∥2l2(0,l	NOUN
m-1152	191	20	)	)	PUNCT
m-1152	191	21	.	.	PUNCT
m-1152	192	1	the	the	DET
m-1152	192	2	physical	physical	ADJ
m-1152	192	3	energy	energy	NOUN
m-1152	192	4	associated	associate	VERB
m-1152	192	5	to	to	ADP
m-1152	192	6	a	a	DET
m-1152	192	7	solution	solution	NOUN
m-1152	192	8	of	of	ADP
m-1152	192	9	(	(	PUNCT
m-1152	192	10	1	1	NUM
m-1152	192	11	)	)	PUNCT
m-1152	192	12	is	be	AUX
m-1152	192	13	e(t	e(t	NOUN
m-1152	192	14	)	)	PUNCT
m-1152	193	1	=	=	SYM
m-1152	193	2	1	1	NUM
m-1152	193	3	2	2	NUM
m-1152	193	4	(	(	PUNCT
m-1152	193	5	∥ut	∥ut	PROPN
m-1152	193	6	(	(	PUNCT
m-1152	193	7	·	·	PUNCT
m-1152	193	8	,	,	PUNCT
m-1152	193	9	t)∥2l2	t)∥2l2	PROPN
m-1152	193	10	+	+	CCONJ
m-1152	193	11	∥ux	∥ux	PROPN
m-1152	193	12	(	(	PUNCT
m-1152	193	13	·	·	PUNCT
m-1152	193	14	,	,	PUNCT
m-1152	193	15	t)∥2l2	t)∥2l2	PROPN
m-1152	193	16	)	)	PUNCT
m-1152	193	17	.	.	PUNCT
m-1152	194	1	first	first	ADJ
m-1152	194	2	-	-	PUNCT
m-1152	194	3	order	order	NOUN
m-1152	194	4	formulation	formulation	NOUN
m-1152	194	5	and	and	CCONJ
m-1152	194	6	generator	generator	NOUN
m-1152	194	7	.	.	PUNCT
m-1152	195	1	write	write	VERB
m-1152	195	2	(	(	PUNCT
m-1152	195	3	1	1	NUM
m-1152	195	4	)	)	PUNCT
m-1152	195	5	as	as	ADP
m-1152	195	6	a	a	DET
m-1152	195	7	�	�	NOUN
m-1152	195	8	rst	rst	ADJ
m-1152	195	9	-	-	PUNCT
m-1152	195	10	order	order	NOUN
m-1152	195	11	system	system	NOUN
m-1152	195	12	u	u	NOUN
m-1152	195	13	′(t	′(t	NOUN
m-1152	195	14	)	)	PUNCT
m-1152	195	15	=	=	SYM
m-1152	195	16	au(t	au(t	PROPN
m-1152	195	17	)	)	PUNCT
m-1152	195	18	by	by	ADP
m-1152	195	19	setting	set	VERB
m-1152	195	20	u	u	NOUN
m-1152	195	21	=	=	PUNCT
m-1152	195	22	(	(	PUNCT
m-1152	195	23	u	u	NOUN
m-1152	195	24	,	,	PUNCT
m-1152	195	25	v)⊤	v)⊤	PROPN
m-1152	195	26	with	with	ADP
m-1152	195	27	v	v	NOUN
m-1152	195	28	=	=	SYM
m-1152	195	29	ut	ut	PROPN
m-1152	195	30	.	.	PROPN
m-1152	195	31	de	de	PROPN
m-1152	195	32	�	�	PROPN
m-1152	195	33	ne	ne	PROPN
m-1152	195	34	a	a	DET
m-1152	195	35	(	(	PUNCT
m-1152	195	36	u	u	NOUN
m-1152	195	37	v	v	NOUN
m-1152	195	38	)	)	PUNCT
m-1152	195	39	=	=	SYM
m-1152	195	40	(	(	PUNCT
m-1152	195	41	v	v	NUM
m-1152	195	42	uxx	uxx	X
m-1152	195	43	−	−	NOUN
m-1152	195	44	αv	αv	NOUN
m-1152	195	45	)	)	PUNCT
m-1152	195	46	,	,	PUNCT
m-1152	195	47	d(a	d(a	PROPN
m-1152	195	48	)	)	PUNCT
m-1152	195	49	=	=	PRON
m-1152	195	50	(	(	PUNCT
m-1152	195	51	h2(0	h2(0	NOUN
m-1152	195	52	,	,	PUNCT
m-1152	195	53	l	l	NOUN
m-1152	195	54	)	)	PUNCT
m-1152	196	1	∩h1	∩h1	NOUN
m-1152	196	2	0	0	NUM
m-1152	196	3	(	(	PUNCT
m-1152	196	4	0	0	NUM
m-1152	196	5	,	,	PUNCT
m-1152	196	6	l	l	NOUN
m-1152	196	7	)	)	PUNCT
m-1152	196	8	)	)	PUNCT
m-1152	197	1	×h1	×h1	PROPN
m-1152	197	2	0	0	NUM
m-1152	197	3	(	(	PUNCT
m-1152	197	4	0	0	NUM
m-1152	197	5	,	,	PUNCT
m-1152	197	6	l	l	NOUN
m-1152	197	7	)	)	PUNCT
m-1152	197	8	.	.	PUNCT
m-1152	198	1	equivalently	equivalently	ADV
m-1152	198	2	,	,	PUNCT
m-1152	198	3	in	in	ADP
m-1152	198	4	matrix	matrix	NOUN
m-1152	198	5	form	form	NOUN
m-1152	198	6	,	,	PUNCT
m-1152	198	7	a	a	DET
m-1152	198	8	=	=	X
m-1152	198	9	(	(	PUNCT
m-1152	198	10	0	0	NUM
m-1152	198	11	i	i	NOUN
m-1152	198	12	∂xx	∂xx	PROPN
m-1152	198	13	−αi	−αi	NOUN
m-1152	198	14	)	)	PUNCT
m-1152	198	15	,	,	PUNCT
m-1152	198	16	d(a	d(a	PROPN
m-1152	198	17	)	)	PUNCT
m-1152	198	18	=	=	PUNCT
m-1152	199	1	(	(	PUNCT
m-1152	199	2	h2	h2	NOUN
m-1152	199	3	∩h1	∩h1	NOUN
m-1152	199	4	0	0	NUM
m-1152	199	5	)	)	PUNCT
m-1152	199	6	×h1	×h1	PROPN
m-1152	199	7	0	0	NUM
m-1152	199	8	.	.	PUNCT
m-1152	200	1	generation	generation	NOUN
m-1152	200	2	of	of	ADP
m-1152	200	3	a	a	DET
m-1152	200	4	c0	c0	NOUN
m-1152	200	5	-	-	PUNCT
m-1152	200	6	semigroup	semigroup	PROPN
m-1152	200	7	(	(	PUNCT
m-1152	200	8	sketch	sketch	NOUN
m-1152	200	9	)	)	PUNCT
m-1152	200	10	.	.	PUNCT
m-1152	201	1	the	the	DET
m-1152	201	2	operator	operator	NOUN
m-1152	201	3	∂xx	∂xx	PROPN
m-1152	201	4	with	with	ADP
m-1152	201	5	dirichlet	dirichlet	PROPN
m-1152	201	6	boundary	boundary	PROPN
m-1152	201	7	conditions	condition	NOUN
m-1152	201	8	is	be	AUX
m-1152	201	9	self	self	NOUN
m-1152	201	10	-	-	PUNCT
m-1152	201	11	adjoint	adjoint	NOUN
m-1152	201	12	and	and	CCONJ
m-1152	201	13	has	have	VERB
m-1152	201	14	compact	compact	ADJ
m-1152	201	15	inverse	inverse	NOUN
m-1152	201	16	on	on	ADP
m-1152	201	17	l2(0	l2(0	PROPN
m-1152	201	18	,	,	PUNCT
m-1152	201	19	l	l	NOUN
m-1152	201	20	)	)	PUNCT
m-1152	201	21	.	.	PUNCT
m-1152	202	1	standard	standard	ADJ
m-1152	202	2	results	result	NOUN
m-1152	202	3	for	for	ADP
m-1152	202	4	second	second	ADJ
m-1152	202	5	-	-	PUNCT
m-1152	202	6	order	order	NOUN
m-1152	202	7	hyperbolic	hyperbolic	ADJ
m-1152	202	8	operators	operator	NOUN
m-1152	202	9	with	with	ADP
m-1152	202	10	bounded	bounded	ADJ
m-1152	202	11	damping	damp	VERB
m-1152	202	12	(	(	PUNCT
m-1152	202	13	see	see	VERB
m-1152	202	14	[	[	X
m-1152	202	15	7	7	NUM
m-1152	202	16	,	,	PUNCT
m-1152	202	17	9	9	NUM
m-1152	202	18	,	,	PUNCT
m-1152	202	19	10	10	NUM
m-1152	202	20	,	,	PUNCT
m-1152	202	21	32	32	NUM
m-1152	202	22	]	]	PUNCT
m-1152	202	23	)	)	PUNCT
m-1152	202	24	imply	imply	VERB
m-1152	202	25	that	that	SCONJ
m-1152	202	26	a	a	PRON
m-1152	202	27	with	with	ADP
m-1152	202	28	domain	domain	NOUN
m-1152	202	29	above	above	ADV
m-1152	202	30	is	be	AUX
m-1152	202	31	the	the	DET
m-1152	202	32	generator	generator	NOUN
m-1152	202	33	of	of	ADP
m-1152	202	34	a	a	DET
m-1152	202	35	c0	c0	NOUN
m-1152	202	36	-	-	PUNCT
m-1152	202	37	semigroup	semigroup	PROPN
m-1152	202	38	{	{	PUNCT
m-1152	202	39	t	t	NOUN
m-1152	202	40	(	(	PUNCT
m-1152	202	41	t)}t≥0	t)}t≥0	NOUN
m-1152	202	42	on	on	ADP
m-1152	202	43	x.	x.	NOUN
m-1152	202	44	in	in	ADP
m-1152	202	45	particular	particular	ADJ
m-1152	202	46	:	:	PUNCT
m-1152	202	47	for	for	ADP
m-1152	202	48	α	α	DET
m-1152	202	49	≥	≥	NOUN
m-1152	202	50	0	0	NUM
m-1152	203	1	the	the	DET
m-1152	203	2	semigroup	semigroup	NOUN
m-1152	203	3	is	be	AUX
m-1152	203	4	contractive	contractive	ADJ
m-1152	203	5	with	with	ADP
m-1152	203	6	respect	respect	NOUN
m-1152	203	7	to	to	ADP
m-1152	203	8	a	a	DET
m-1152	203	9	suitable	suitable	ADJ
m-1152	203	10	equivalent	equivalent	ADJ
m-1152	203	11	energy	energy	NOUN
m-1152	203	12	norm	norm	NOUN
m-1152	203	13	(	(	PUNCT
m-1152	203	14	damping	damp	VERB
m-1152	203	15	is	be	AUX
m-1152	203	16	nonnegative	nonnegative	ADJ
m-1152	203	17	)	)	PUNCT
m-1152	203	18	.	.	PUNCT
m-1152	204	1	for	for	ADP
m-1152	204	2	α	α	PRON
m-1152	204	3	<	<	X
m-1152	204	4	0	0	NUM
m-1152	205	1	the	the	DET
m-1152	205	2	operator	operator	NOUN
m-1152	205	3	has	have	VERB
m-1152	205	4	a	a	DET
m-1152	205	5	component	component	NOUN
m-1152	205	6	that	that	PRON
m-1152	205	7	can	can	AUX
m-1152	205	8	generate	generate	VERB
m-1152	205	9	growth	growth	NOUN
m-1152	205	10	(	(	PUNCT
m-1152	205	11	negative	negative	ADJ
m-1152	205	12	damping	damp	VERB
m-1152	205	13	gives	give	VERB
m-1152	205	14	energy	energy	NOUN
m-1152	205	15	injection	injection	NOUN
m-1152	205	16	)	)	PUNCT
m-1152	205	17	.	.	PUNCT
m-1152	206	1	we	we	PRON
m-1152	206	2	therefore	therefore	ADV
m-1152	206	3	treat	treat	VERB
m-1152	206	4	{	{	PUNCT
m-1152	206	5	t	t	PROPN
m-1152	206	6	(	(	PUNCT
m-1152	206	7	t	t	PROPN
m-1152	206	8	)	)	PUNCT
m-1152	206	9	}	}	PUNCT
m-1152	206	10	as	as	ADP
m-1152	206	11	the	the	DET
m-1152	206	12	evolution	evolution	NOUN
m-1152	206	13	operator	operator	NOUN
m-1152	206	14	for	for	ADP
m-1152	206	15	(	(	PUNCT
m-1152	206	16	1	1	NUM
m-1152	206	17	)	)	PUNCT
m-1152	206	18	.	.	PUNCT
m-1152	207	1	ijo	ijo	PROPN
m-1152	207	2	international	international	PROPN
m-1152	207	3	journal	journal	PROPN
m-1152	207	4	of	of	ADP
m-1152	207	5	mathematics	mathematics	PROPN
m-1152	207	6	(	(	PUNCT
m-1152	207	7	issn	issn	PROPN
m-1152	207	8	:	:	PUNCT
m-1152	207	9	2992	2992	NUM
m-1152	207	10	-	-	SYM
m-1152	207	11	4421	4421	NUM
m-1152	207	12	)	)	PUNCT
m-1152	207	13	volume	volume	NOUN
m-1152	207	14	08	08	NUM
m-1152	208	1	|	|	ADV
m-1152	208	2	issue	issue	NOUN
m-1152	208	3	9	9	NUM
m-1152	208	4	|	|	CCONJ
m-1152	208	5	september	september	PROPN
m-1152	208	6	2025	2025	NUM
m-1152	208	7	|	|	ADV
m-1152	208	8	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1152	208	9	17	17	NUM
m-1152	208	10	modal	modal	NOUN
m-1152	208	11	decomposition	decomposition	NOUN
m-1152	208	12	and	and	CCONJ
m-1152	208	13	spectrum	spectrum	NOUN
m-1152	208	14	.	.	PUNCT
m-1152	209	1	let	let	AUX
m-1152	209	2	{	{	PUNCT
m-1152	209	3	φn}n≥1	φn}n≥1	NUM
m-1152	209	4	denote	denote	VERB
m-1152	209	5	the	the	DET
m-1152	209	6	dirichlet	dirichlet	PROPN
m-1152	209	7	laplacian	laplacian	ADJ
m-1152	209	8	eigenfunctions	eigenfunction	NOUN
m-1152	209	9	φn(x	φn(x	PUNCT
m-1152	209	10	)	)	PUNCT
m-1152	209	11	=	=	SYM
m-1152	209	12	sin	sin	NOUN
m-1152	209	13	(	(	PUNCT
m-1152	209	14	nπx	nπx	NOUN
m-1152	209	15	l	l	NOUN
m-1152	209	16	)	)	PUNCT
m-1152	209	17	,	,	PUNCT
m-1152	209	18	−φ′′	−φ′′	ADP
m-1152	209	19	n	n	PROPN
m-1152	209	20	=	=	PROPN
m-1152	209	21	ω2	ω2	PROPN
m-1152	209	22	nφn	nφn	PROPN
m-1152	209	23	,	,	PUNCT
m-1152	209	24	ωn	ωn	PROPN
m-1152	209	25	=	=	NUM
m-1152	209	26	nπ	nπ	NOUN
m-1152	209	27	l	l	NOUN
m-1152	209	28	,	,	PUNCT
m-1152	209	29	n	n	PROPN
m-1152	209	30	∈	∈	PROPN
m-1152	209	31	n.	n.	NOUN
m-1152	209	32	expand	expand	VERB
m-1152	209	33	the	the	DET
m-1152	209	34	solution	solution	NOUN
m-1152	209	35	as	as	ADP
m-1152	209	36	u(x	u(x	NOUN
m-1152	209	37	,	,	PUNCT
m-1152	209	38	t	t	NOUN
m-1152	209	39	)	)	PUNCT
m-1152	209	40	=	=	SYM
m-1152	209	41	∑	∑	PUNCT
m-1152	209	42	n≥1	n≥1	VERB
m-1152	209	43	qn(t)φn(x	qn(t)φn(x	NOUN
m-1152	209	44	)	)	PUNCT
m-1152	209	45	.	.	PUNCT
m-1152	210	1	each	each	DET
m-1152	210	2	modal	modal	PROPN
m-1152	210	3	coe	coe	PROPN
m-1152	210	4	�	�	PROPN
m-1152	210	5	cient	cient	PROPN
m-1152	210	6	satis	satis	PROPN
m-1152	210	7	�	�	PROPN
m-1152	210	8	es	es	ADP
m-1152	210	9	the	the	DET
m-1152	210	10	scalar	scalar	ADJ
m-1152	210	11	ode	ode	PROPN
m-1152	210	12	q′′n(t	q′′n(t	PROPN
m-1152	210	13	)	)	PUNCT
m-1152	211	1	+	+	NUM
m-1152	211	2	αq′n(t	αq′n(t	X
m-1152	211	3	)	)	PUNCT
m-1152	212	1	+	+	CCONJ
m-1152	212	2	ω2	ω2	ADJ
m-1152	212	3	nqn(t	nqn(t	NUM
m-1152	212	4	)	)	PUNCT
m-1152	212	5	=	=	SYM
m-1152	213	1	0	0	X
m-1152	213	2	.	.	PUNCT
m-1152	214	1	the	the	DET
m-1152	214	2	characteristic	characteristic	ADJ
m-1152	214	3	equation	equation	NOUN
m-1152	214	4	is	be	AUX
m-1152	214	5	λ2	λ2	NOUN
m-1152	214	6	+	+	CCONJ
m-1152	214	7	αλ+	αλ+	NOUN
m-1152	215	1	ω2	ω2	ADJ
m-1152	215	2	n	n	NOUN
m-1152	215	3	=	=	NOUN
m-1152	215	4	0	0	NUM
m-1152	215	5	with	with	ADP
m-1152	215	6	roots	root	NOUN
m-1152	215	7	λ±	λ±	PROPN
m-1152	215	8	n	n	NOUN
m-1152	215	9	=	=	SYM
m-1152	215	10	−α±	−α±	NOUN
m-1152	215	11	√	√	NOUN
m-1152	215	12	α2	α2	PROPN
m-1152	215	13	−	−	PROPN
m-1152	215	14	4ω2	4ω2	NUM
m-1152	215	15	n	n	PRON
m-1152	215	16	2	2	NUM
m-1152	215	17	.	.	PUNCT
m-1152	216	1	hence	hence	ADV
m-1152	216	2	ℜ(λ±	ℜ(λ±	NOUN
m-1152	216	3	n	n	PRON
m-1152	216	4	)	)	PUNCT
m-1152	216	5	≤	≤	NUM
m-1152	216	6	−α	−α	NOUN
m-1152	216	7	2	2	NUM
m-1152	216	8	for	for	ADP
m-1152	216	9	every	every	DET
m-1152	216	10	n	n	PRON
m-1152	216	11	≥	≥	NOUN
m-1152	216	12	1	1	NUM
m-1152	216	13	,	,	PUNCT
m-1152	216	14	and	and	CCONJ
m-1152	216	15	the	the	DET
m-1152	216	16	spectral	spectral	ADJ
m-1152	216	17	bound	bind	VERB
m-1152	216	18	of	of	ADP
m-1152	216	19	a	a	DET
m-1152	216	20	satis	satis	NOUN
m-1152	216	21	�	�	PROPN
m-1152	216	22	es	es	NOUN
m-1152	216	23	s(a	s(a	PROPN
m-1152	216	24	)	)	PUNCT
m-1152	216	25	:	:	PUNCT
m-1152	217	1	=	=	SYM
m-1152	217	2	sup{ℜλ	sup{ℜλ	NOUN
m-1152	217	3	:	:	PUNCT
m-1152	217	4	λ	λ	X
m-1152	217	5	∈	∈	PROPN
m-1152	217	6	σ(a	σ(a	PROPN
m-1152	217	7	)	)	PUNCT
m-1152	217	8	}	}	PUNCT
m-1152	218	1	=	=	SYM
m-1152	218	2	−α	−α	NOUN
m-1152	218	3	2	2	NUM
m-1152	218	4	.	.	PUNCT
m-1152	219	1	(	(	PUNCT
m-1152	219	2	here	here	ADV
m-1152	219	3	we	we	PRON
m-1152	219	4	used	use	VERB
m-1152	219	5	that	that	SCONJ
m-1152	219	6	the	the	DET
m-1152	219	7	full	full	ADJ
m-1152	219	8	spectrum	spectrum	NOUN
m-1152	219	9	of	of	ADP
m-1152	219	10	a	a	DET
m-1152	219	11	consists	consist	NOUN
m-1152	219	12	of	of	ADP
m-1152	219	13	these	these	DET
m-1152	219	14	modal	modal	NOUN
m-1152	219	15	eigenvalues	eigenvalue	NOUN
m-1152	219	16	due	due	ADJ
m-1152	219	17	to	to	ADP
m-1152	219	18	compactness	compactness	NOUN
m-1152	219	19	of	of	ADP
m-1152	219	20	the	the	DET
m-1152	219	21	spatial	spatial	ADJ
m-1152	219	22	resolvent	resolvent	NOUN
m-1152	219	23	and	and	CCONJ
m-1152	219	24	separation	separation	NOUN
m-1152	219	25	of	of	ADP
m-1152	219	26	variables	variable	NOUN
m-1152	219	27	.	.	PUNCT
m-1152	219	28	)	)	PUNCT
m-1152	219	29	energy	energy	NOUN
m-1152	219	30	identity	identity	NOUN
m-1152	219	31	and	and	CCONJ
m-1152	219	32	exponential	exponential	ADJ
m-1152	219	33	decay	decay	NOUN
m-1152	219	34	for	for	ADP
m-1152	219	35	α	α	PROPN
m-1152	219	36	>	>	X
m-1152	219	37	0	0	PROPN
m-1152	219	38	.	.	PUNCT
m-1152	220	1	multiply	multiply	ADV
m-1152	220	2	(	(	PUNCT
m-1152	220	3	1	1	X
m-1152	220	4	)	)	PUNCT
m-1152	220	5	by	by	ADP
m-1152	220	6	ut	ut	PROPN
m-1152	220	7	and	and	CCONJ
m-1152	220	8	integrate	integrate	VERB
m-1152	220	9	over	over	ADP
m-1152	220	10	(	(	PUNCT
m-1152	220	11	0	0	NUM
m-1152	220	12	,	,	PUNCT
m-1152	220	13	l	l	NOUN
m-1152	220	14	)	)	PUNCT
m-1152	220	15	to	to	PART
m-1152	220	16	obtain	obtain	VERB
m-1152	220	17	the	the	DET
m-1152	220	18	standard	standard	ADJ
m-1152	220	19	energy	energy	NOUN
m-1152	220	20	balance	balance	NOUN
m-1152	220	21	:	:	PUNCT
m-1152	220	22	d	d	X
m-1152	220	23	dt	dt	X
m-1152	220	24	e(t	e(t	NOUN
m-1152	220	25	)	)	PUNCT
m-1152	221	1	=	=	SYM
m-1152	221	2	−α	−α	PROPN
m-1152	221	3	∫	∫	PROPN
m-1152	221	4	l	l	NOUN
m-1152	221	5	0	0	NUM
m-1152	221	6	|ut(x	|ut(x	NUM
m-1152	221	7	,	,	PUNCT
m-1152	221	8	t)|2	t)|2	PROPN
m-1152	221	9	dx	dx	PROPN
m-1152	221	10	≤	≤	PROPN
m-1152	221	11	0	0	NUM
m-1152	221	12	.	.	PUNCT
m-1152	222	1	thus	thus	ADV
m-1152	222	2	energy	energy	NOUN
m-1152	222	3	is	be	AUX
m-1152	222	4	nonincreasing	nonincrease	VERB
m-1152	222	5	.	.	PUNCT
m-1152	223	1	to	to	PART
m-1152	223	2	obtain	obtain	VERB
m-1152	223	3	exponential	exponential	ADJ
m-1152	223	4	decay	decay	NOUN
m-1152	223	5	we	we	PRON
m-1152	223	6	combine	combine	VERB
m-1152	223	7	this	this	DET
m-1152	223	8	dissipation	dissipation	NOUN
m-1152	223	9	with	with	ADP
m-1152	223	10	a	a	DET
m-1152	223	11	coercivity	coercivity	NOUN
m-1152	223	12	(	(	PUNCT
m-1152	223	13	poincaré	poincaré	PROPN
m-1152	223	14	)	)	PUNCT
m-1152	223	15	inequality	inequality	NOUN
m-1152	223	16	:	:	PUNCT
m-1152	223	17	for	for	ADP
m-1152	223	18	u	u	PROPN
m-1152	223	19	∈	∈	PROPN
m-1152	223	20	h1	h1	NOUN
m-1152	223	21	0	0	NUM
m-1152	223	22	(	(	PUNCT
m-1152	223	23	0	0	NUM
m-1152	223	24	,	,	PUNCT
m-1152	223	25	l	l	NOUN
m-1152	223	26	)	)	PUNCT
m-1152	223	27	,	,	PUNCT
m-1152	223	28	∥u∥l2	∥u∥l2	ADJ
m-1152	223	29	≤	≤	NUM
m-1152	223	30	1	1	NUM
m-1152	223	31	ω1	ω1	X
m-1152	223	32	∥u′∥l2	∥u′∥l2	ADJ
m-1152	223	33	,	,	PUNCT
m-1152	223	34	ω1	ω1	PROPN
m-1152	223	35	=	=	PROPN
m-1152	223	36	π	π	PROPN
m-1152	223	37	l	l	NOUN
m-1152	223	38	.	.	PUNCT
m-1152	224	1	using	use	VERB
m-1152	224	2	the	the	DET
m-1152	224	3	energy	energy	NOUN
m-1152	224	4	e(t	e(t	NOUN
m-1152	224	5	)	)	PUNCT
m-1152	224	6	and	and	CCONJ
m-1152	224	7	the	the	DET
m-1152	224	8	modal	modal	ADJ
m-1152	224	9	spectral	spectral	ADJ
m-1152	224	10	gap	gap	NOUN
m-1152	224	11	one	one	PRON
m-1152	224	12	can	can	AUX
m-1152	224	13	show	show	VERB
m-1152	224	14	(	(	PUNCT
m-1152	224	15	standard	standard	ADJ
m-1152	224	16	multiplier	multipli	ADJ
m-1152	224	17	or	or	CCONJ
m-1152	224	18	resolvent	resolvent	ADJ
m-1152	224	19	estimates	estimate	NOUN
m-1152	224	20	;	;	PUNCT
m-1152	224	21	see	see	VERB
m-1152	224	22	[	[	X
m-1152	224	23	27	27	NUM
m-1152	224	24	,	,	PUNCT
m-1152	224	25	28	28	NUM
m-1152	224	26	,	,	PUNCT
m-1152	224	27	29	29	NUM
m-1152	224	28	,	,	PUNCT
m-1152	224	29	30	30	NUM
m-1152	224	30	,	,	PUNCT
m-1152	224	31	31	31	NUM
m-1152	224	32	]	]	PUNCT
m-1152	224	33	)	)	PUNCT
m-1152	224	34	that	that	SCONJ
m-1152	224	35	there	there	PRON
m-1152	224	36	exist	exist	VERB
m-1152	224	37	constants	constant	NOUN
m-1152	224	38	m	m	VERB
m-1152	224	39	≥	≥	NOUN
m-1152	224	40	1	1	NUM
m-1152	224	41	and	and	CCONJ
m-1152	224	42	γ	γ	X
m-1152	224	43	>	>	X
m-1152	224	44	0	0	PUNCT
m-1152	224	45	(	(	PUNCT
m-1152	224	46	depending	depend	VERB
m-1152	224	47	on	on	ADP
m-1152	224	48	α	α	NOUN
m-1152	224	49	and	and	CCONJ
m-1152	224	50	l	l	NOUN
m-1152	224	51	)	)	PUNCT
m-1152	224	52	such	such	ADJ
m-1152	224	53	that	that	SCONJ
m-1152	224	54	∥t	∥t	PROPN
m-1152	224	55	(	(	PUNCT
m-1152	224	56	t)∥l(x	t)∥l(x	PROPN
m-1152	224	57	)	)	PUNCT
m-1152	224	58	≤	≤	NOUN
m-1152	224	59	me−γt	me−γt	NOUN
m-1152	224	60	,	,	PUNCT
m-1152	224	61	t	t	PROPN
m-1152	224	62	≥	≥	NUM
m-1152	224	63	0	0	NUM
m-1152	224	64	.	.	PUNCT
m-1152	225	1	a	a	DET
m-1152	225	2	simple	simple	ADJ
m-1152	225	3	modal	modal	NOUN
m-1152	225	4	estimate	estimate	NOUN
m-1152	225	5	gives	give	VERB
m-1152	225	6	a	a	DET
m-1152	225	7	concrete	concrete	NOUN
m-1152	225	8	lower	lower	ADV
m-1152	225	9	bound	bind	VERB
m-1152	225	10	γ	γ	PROPN
m-1152	225	11	≥	≥	PROPN
m-1152	225	12	α/2	α/2	NUM
m-1152	225	13	in	in	ADP
m-1152	225	14	the	the	DET
m-1152	225	15	case	case	NOUN
m-1152	225	16	of	of	ADP
m-1152	225	17	uniform	uniform	ADJ
m-1152	225	18	damping	damp	VERB
m-1152	225	19	(	(	PUNCT
m-1152	225	20	constant	constant	ADJ
m-1152	225	21	α	α	NOUN
m-1152	225	22	)	)	PUNCT
m-1152	225	23	;	;	PUNCT
m-1152	225	24	more	more	ADV
m-1152	225	25	careful	careful	ADJ
m-1152	225	26	resolvent	resolvent	ADJ
m-1152	225	27	estimates	estimate	NOUN
m-1152	225	28	may	may	AUX
m-1152	225	29	yield	yield	VERB
m-1152	225	30	the	the	DET
m-1152	225	31	optimal	optimal	ADJ
m-1152	225	32	rate	rate	NOUN
m-1152	225	33	γ	γ	X
m-1152	225	34	=	=	SYM
m-1152	225	35	α/2	α/2	NUM
m-1152	225	36	when	when	SCONJ
m-1152	225	37	the	the	DET
m-1152	225	38	poincaré	poincaré	PROPN
m-1152	225	39	constant	constant	ADJ
m-1152	225	40	is	be	AUX
m-1152	225	41	accounted	account	VERB
m-1152	225	42	for	for	ADP
m-1152	225	43	.	.	PUNCT
m-1152	226	1	non	non	ADJ
m-1152	226	2	-	-	NOUN
m-1152	226	3	decay	decay	NOUN
m-1152	226	4	when	when	SCONJ
m-1152	226	5	α	α	NOUN
m-1152	226	6	=	=	NOUN
m-1152	226	7	0	0	PROPN
m-1152	226	8	.	.	PUNCT
m-1152	227	1	if	if	SCONJ
m-1152	227	2	α	α	NOUN
m-1152	227	3	=	=	SYM
m-1152	227	4	0	0	NUM
m-1152	227	5	equation	equation	NOUN
m-1152	227	6	(	(	PUNCT
m-1152	227	7	1	1	X
m-1152	227	8	)	)	PUNCT
m-1152	227	9	reduces	reduce	VERB
m-1152	227	10	to	to	ADP
m-1152	227	11	the	the	DET
m-1152	227	12	undamped	undampe	VERB
m-1152	227	13	wave	wave	NOUN
m-1152	227	14	equation	equation	NOUN
m-1152	227	15	.	.	PUNCT
m-1152	228	1	modal	modal	ADJ
m-1152	228	2	eigenvalues	eigenvalue	NOUN
m-1152	228	3	are	be	AUX
m-1152	228	4	purely	purely	ADV
m-1152	228	5	imaginary	imaginary	ADJ
m-1152	228	6	λ±	λ±	PROPN
m-1152	228	7	n	n	NOUN
m-1152	228	8	=	=	PUNCT
m-1152	228	9	±iωn	±iωn	NOUN
m-1152	228	10	,	,	PUNCT
m-1152	228	11	so	so	SCONJ
m-1152	228	12	s(a	s(a	NOUN
m-1152	228	13	)	)	PUNCT
m-1152	229	1	=	=	SYM
m-1152	229	2	0	0	X
m-1152	229	3	.	.	PUNCT
m-1152	230	1	energy	energy	NOUN
m-1152	230	2	is	be	AUX
m-1152	230	3	conserved	conserve	VERB
m-1152	230	4	(	(	PUNCT
m-1152	230	5	de	de	NOUN
m-1152	230	6	/	/	SYM
m-1152	230	7	dt	dt	NOUN
m-1152	230	8	=	=	NOUN
m-1152	230	9	0	0	NUM
m-1152	230	10	)	)	PUNCT
m-1152	230	11	and	and	CCONJ
m-1152	230	12	no	no	DET
m-1152	230	13	decay	decay	NOUN
m-1152	230	14	of	of	ADP
m-1152	230	15	the	the	DET
m-1152	230	16	norm	norm	NOUN
m-1152	230	17	occurs	occur	VERB
m-1152	230	18	in	in	ADP
m-1152	230	19	general	general	ADJ
m-1152	230	20	(	(	PUNCT
m-1152	230	21	solutions	solution	NOUN
m-1152	230	22	persist	persist	VERB
m-1152	230	23	as	as	ADP
m-1152	230	24	undamped	undampe	VERB
m-1152	230	25	oscillations	oscillation	NOUN
m-1152	230	26	)	)	PUNCT
m-1152	230	27	.	.	PUNCT
m-1152	231	1	thus	thus	ADV
m-1152	231	2	analytic	analytic	ADJ
m-1152	231	3	stability	stability	NOUN
m-1152	231	4	fails	fail	VERB
m-1152	231	5	when	when	SCONJ
m-1152	231	6	α	α	NOUN
m-1152	231	7	=	=	NOUN
m-1152	231	8	0	0	X
m-1152	231	9	.	.	PUNCT
m-1152	231	10	ijo	ijo	PROPN
m-1152	231	11	international	international	PROPN
m-1152	231	12	journal	journal	PROPN
m-1152	231	13	of	of	ADP
m-1152	231	14	mathematics	mathematics	PROPN
m-1152	231	15	(	(	PUNCT
m-1152	231	16	issn	issn	PROPN
m-1152	231	17	:	:	PUNCT
m-1152	231	18	2992	2992	NUM
m-1152	231	19	-	-	SYM
m-1152	231	20	4421	4421	NUM
m-1152	231	21	)	)	PUNCT
m-1152	231	22	volume	volume	NOUN
m-1152	231	23	08	08	NUM
m-1152	232	1	|	|	ADV
m-1152	232	2	issue	issue	NOUN
m-1152	232	3	9	9	NUM
m-1152	232	4	|	|	CCONJ
m-1152	232	5	september	september	PROPN
m-1152	232	6	2025	2025	NUM
m-1152	232	7	|	|	ADV
m-1152	232	8	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1152	232	9	18	18	NUM
m-1152	232	10	instability	instability	NOUN
m-1152	232	11	for	for	ADP
m-1152	232	12	α	α	PRON
m-1152	232	13	<	<	X
m-1152	232	14	0	0	NUM
m-1152	232	15	.	.	PUNCT
m-1152	233	1	if	if	SCONJ
m-1152	233	2	α	α	PRON
m-1152	233	3	<	<	X
m-1152	233	4	0	0	NUM
m-1152	234	1	the	the	DET
m-1152	234	2	modal	modal	ADJ
m-1152	234	3	real	real	ADJ
m-1152	234	4	parts	part	NOUN
m-1152	234	5	satisfy	satisfy	VERB
m-1152	234	6	ℜ(λ±	ℜ(λ±	NOUN
m-1152	234	7	n	n	PART
m-1152	234	8	)	)	PUNCT
m-1152	234	9	≥	≥	X
m-1152	234	10	−α/2	−α/2	ADV
m-1152	234	11	>	>	X
m-1152	234	12	0	0	PUNCT
m-1152	235	1	(	(	PUNCT
m-1152	235	2	note	note	NOUN
m-1152	235	3	sign	sign	NOUN
m-1152	235	4	)	)	PUNCT
m-1152	235	5	,	,	PUNCT
m-1152	235	6	and	and	CCONJ
m-1152	235	7	high	high	ADJ
m-1152	235	8	modes	mode	NOUN
m-1152	235	9	may	may	AUX
m-1152	235	10	exhibit	exhibit	VERB
m-1152	235	11	growth	growth	NOUN
m-1152	235	12	;	;	PUNCT
m-1152	235	13	hence	hence	ADV
m-1152	235	14	the	the	DET
m-1152	235	15	semigroup	semigroup	NOUN
m-1152	235	16	is	be	AUX
m-1152	235	17	not	not	PART
m-1152	235	18	stable	stable	ADJ
m-1152	235	19	and	and	CCONJ
m-1152	235	20	solutions	solution	NOUN
m-1152	235	21	typically	typically	ADV
m-1152	235	22	grow	grow	VERB
m-1152	235	23	exponentially	exponentially	ADV
m-1152	235	24	[	[	X
m-1152	235	25	6	6	NUM
m-1152	235	26	,	,	PUNCT
m-1152	235	27	7	7	NUM
m-1152	235	28	,	,	PUNCT
m-1152	235	29	11	11	NUM
m-1152	235	30	,	,	PUNCT
m-1152	235	31	19	19	NUM
m-1152	235	32	]	]	PUNCT
m-1152	235	33	.	.	PUNCT
m-1152	236	1	remarks	remark	NOUN
m-1152	236	2	.	.	PUNCT
m-1152	237	1	�	�	PROPN
m-1152	237	2	the	the	DET
m-1152	237	3	exponential	exponential	ADJ
m-1152	237	4	decay	decay	NOUN
m-1152	237	5	argument	argument	NOUN
m-1152	237	6	above	above	ADV
m-1152	237	7	uses	use	VERB
m-1152	237	8	that	that	SCONJ
m-1152	237	9	damping	damp	VERB
m-1152	237	10	is	be	AUX
m-1152	237	11	uniform	uniform	ADJ
m-1152	237	12	(	(	PUNCT
m-1152	237	13	constant	constant	ADJ
m-1152	237	14	α	α	PROPN
m-1152	237	15	>	>	X
m-1152	237	16	0	0	NUM
m-1152	237	17	)	)	PUNCT
m-1152	237	18	and	and	CCONJ
m-1152	237	19	the	the	DET
m-1152	237	20	spatial	spatial	ADJ
m-1152	237	21	domain	domain	NOUN
m-1152	237	22	is	be	AUX
m-1152	237	23	bounded	bound	VERB
m-1152	237	24	so	so	SCONJ
m-1152	237	25	the	the	DET
m-1152	237	26	laplacian	laplacian	NOUN
m-1152	237	27	has	have	VERB
m-1152	237	28	compact	compact	ADJ
m-1152	237	29	resolvent	resolvent	NOUN
m-1152	237	30	.	.	PUNCT
m-1152	238	1	for	for	ADP
m-1152	238	2	localized	localized	ADJ
m-1152	238	3	damping	damp	VERB
m-1152	238	4	(	(	PUNCT
m-1152	238	5	e.g.	e.g.	ADV
m-1152	238	6	α(x	α(x	NOUN
m-1152	238	7	)	)	PUNCT
m-1152	238	8	≥	≥	NOUN
m-1152	238	9	0	0	NUM
m-1152	238	10	supported	support	VERB
m-1152	238	11	only	only	ADV
m-1152	238	12	on	on	ADP
m-1152	238	13	a	a	DET
m-1152	238	14	subregion	subregion	NOUN
m-1152	238	15	)	)	PUNCT
m-1152	238	16	exponential	exponential	ADJ
m-1152	238	17	decay	decay	NOUN
m-1152	238	18	may	may	AUX
m-1152	238	19	fail	fail	VERB
m-1152	238	20	or	or	CCONJ
m-1152	238	21	require	require	VERB
m-1152	238	22	geometric	geometric	ADJ
m-1152	238	23	control	control	NOUN
m-1152	238	24	/	/	SYM
m-1152	238	25	observability	observability	NOUN
m-1152	238	26	conditions	condition	NOUN
m-1152	238	27	(	(	PUNCT
m-1152	238	28	see	see	VERB
m-1152	238	29	bardos	bardos	PROPN
m-1152	238	30	�	�	PROPN
m-1152	238	31	lebeau	lebeau	PROPN
m-1152	238	32	�	�	PROPN
m-1152	238	33	rauch	rauch	NOUN
m-1152	238	34	-	-	PUNCT
m-1152	238	35	type	type	NOUN
m-1152	238	36	results	result	NOUN
m-1152	238	37	)	)	PUNCT
m-1152	238	38	.	.	PUNCT
m-1152	239	1	�	�	PROPN
m-1152	239	2	the	the	DET
m-1152	239	3	modal	modal	ADJ
m-1152	239	4	description	description	NOUN
m-1152	239	5	also	also	ADV
m-1152	239	6	explains	explain	VERB
m-1152	239	7	why	why	SCONJ
m-1152	239	8	s(a	s(a	NOUN
m-1152	239	9	)	)	PUNCT
m-1152	239	10	=	=	PUNCT
m-1152	240	1	−α/2	−α/2	NOUN
m-1152	240	2	in	in	ADP
m-1152	240	3	this	this	DET
m-1152	240	4	uniform	uniform	ADJ
m-1152	240	5	case	case	NOUN
m-1152	240	6	:	:	PUNCT
m-1152	240	7	the	the	DET
m-1152	240	8	real	real	ADJ
m-1152	240	9	parts	part	NOUN
m-1152	240	10	of	of	ADP
m-1152	240	11	all	all	DET
m-1152	240	12	modal	modal	ADJ
m-1152	240	13	eigenvalues	eigenvalue	NOUN
m-1152	240	14	are	be	AUX
m-1152	240	15	bounded	bound	VERB
m-1152	240	16	above	above	ADV
m-1152	240	17	by	by	ADP
m-1152	240	18	−α/2	−α/2	PROPN
m-1152	240	19	.	.	PUNCT
m-1152	241	1	thus	thus	ADV
m-1152	241	2	the	the	DET
m-1152	241	3	spectral	spectral	ADJ
m-1152	241	4	criterion	criterion	NOUN
m-1152	241	5	s(a	s(a	PROPN
m-1152	241	6	)	)	PUNCT
m-1152	241	7	<	<	X
m-1152	241	8	0	0	PUNCT
m-1152	241	9	is	be	AUX
m-1152	241	10	equivalent	equivalent	ADJ
m-1152	241	11	to	to	ADP
m-1152	241	12	α	α	PROPN
m-1152	241	13	>	>	X
m-1152	241	14	0	0	PUNCT
m-1152	242	1	here	here	ADV
m-1152	242	2	.	.	PUNCT
m-1152	243	1	conclusion	conclusion	NOUN
m-1152	243	2	for	for	ADP
m-1152	243	3	example	example	NOUN
m-1152	243	4	3.4	3.4	NUM
m-1152	243	5	.	.	PUNCT
m-1152	244	1	with	with	ADP
m-1152	244	2	the	the	DET
m-1152	244	3	state	state	NOUN
m-1152	244	4	space	space	NOUN
m-1152	244	5	x	x	PUNCT
m-1152	244	6	=	=	PRON
m-1152	244	7	h1	h1	NOUN
m-1152	244	8	0	0	NUM
m-1152	244	9	(	(	PUNCT
m-1152	244	10	0	0	NUM
m-1152	244	11	,	,	PUNCT
m-1152	244	12	l	l	NOUN
m-1152	244	13	)	)	PUNCT
m-1152	244	14	×	×	NOUN
m-1152	244	15	l2(0	l2(0	NOUN
m-1152	244	16	,	,	PUNCT
m-1152	244	17	l	l	NOUN
m-1152	244	18	)	)	PUNCT
m-1152	244	19	and	and	CCONJ
m-1152	244	20	generator	generator	NOUN
m-1152	244	21	a	a	DET
m-1152	244	22	de	de	X
m-1152	244	23	�	�	NOUN
m-1152	244	24	ned	ned	ADJ
m-1152	244	25	above	above	ADV
m-1152	244	26	,	,	PUNCT
m-1152	244	27	the	the	DET
m-1152	244	28	semigroup	semigroup	PROPN
m-1152	244	29	{	{	PUNCT
m-1152	244	30	t	t	PROPN
m-1152	244	31	(	(	PUNCT
m-1152	244	32	t	t	NOUN
m-1152	244	33	)	)	PUNCT
m-1152	244	34	}	}	PUNCT
m-1152	244	35	satis	satis	PROPN
m-1152	244	36	�	�	PROPN
m-1152	244	37	es	es	NOUN
m-1152	244	38	:	:	PUNCT
m-1152	244	39	�	�	PROPN
m-1152	244	40	kw	kw	NOUN
m-1152	244	41	-	-	NOUN
m-1152	244	42	stability	stability	NOUN
m-1152	244	43	for	for	ADP
m-1152	244	44	all	all	PRON
m-1152	244	45	α	α	PRON
m-1152	244	46	∈	∈	NOUN
m-1152	244	47	r	r	NOUN
m-1152	244	48	(	(	PUNCT
m-1152	244	49	algebraic	algebraic	ADJ
m-1152	244	50	property	property	NOUN
m-1152	244	51	of	of	ADP
m-1152	244	52	the	the	DET
m-1152	244	53	one	one	NUM
m-1152	244	54	-	-	PUNCT
m-1152	244	55	parameter	parameter	NOUN
m-1152	244	56	family	family	NOUN
m-1152	244	57	)	)	PUNCT
m-1152	244	58	.	.	PUNCT
m-1152	245	1	�	�	PROPN
m-1152	245	2	exponential	exponential	PROPN
m-1152	245	3	(	(	PUNCT
m-1152	245	4	analytic	analytic	ADJ
m-1152	245	5	)	)	PUNCT
m-1152	245	6	stability	stability	NOUN
m-1152	245	7	if	if	SCONJ
m-1152	245	8	and	and	CCONJ
m-1152	245	9	only	only	ADV
m-1152	245	10	if	if	SCONJ
m-1152	245	11	α	α	PROPN
m-1152	245	12	>	>	X
m-1152	245	13	0	0	PUNCT
m-1152	246	1	(	(	PUNCT
m-1152	246	2	spectral	spectral	ADJ
m-1152	246	3	bound	bind	VERB
m-1152	246	4	negative	negative	NOUN
m-1152	246	5	)	)	PUNCT
m-1152	246	6	.	.	PUNCT
m-1152	247	1	�	�	PROPN
m-1152	247	2	conservation	conservation	NOUN
m-1152	247	3	of	of	ADP
m-1152	247	4	energy	energy	NOUN
m-1152	247	5	(	(	PUNCT
m-1152	247	6	no	no	DET
m-1152	247	7	decay	decay	NOUN
m-1152	247	8	)	)	PUNCT
m-1152	248	1	when	when	SCONJ
m-1152	248	2	α	α	PROPN
m-1152	248	3	=	=	SYM
m-1152	248	4	0	0	PROPN
m-1152	248	5	.	.	PUNCT
m-1152	248	6	�	�	PROPN
m-1152	248	7	instability	instability	NOUN
m-1152	248	8	when	when	SCONJ
m-1152	248	9	α	α	X
m-1152	248	10	<	<	X
m-1152	248	11	0	0	PUNCT
m-1152	249	1	[	[	X
m-1152	249	2	6	6	NUM
m-1152	249	3	,	,	PUNCT
m-1152	249	4	11	11	NUM
m-1152	249	5	,	,	PUNCT
m-1152	249	6	12	12	NUM
m-1152	249	7	,	,	PUNCT
m-1152	249	8	13	13	NUM
m-1152	249	9	,	,	PUNCT
m-1152	249	10	24	24	NUM
m-1152	249	11	]	]	PUNCT
m-1152	249	12	.	.	PUNCT
m-1152	250	1	for	for	ADP
m-1152	250	2	more	more	ADJ
m-1152	250	3	about	about	ADP
m-1152	250	4	pde	pde	NOUN
m-1152	250	5	:	:	PUNCT
m-1152	250	6	see	see	VERB
m-1152	250	7	[	[	X
m-1152	250	8	8	8	NUM
m-1152	250	9	,	,	PUNCT
m-1152	250	10	14	14	NUM
m-1152	250	11	,	,	PUNCT
m-1152	250	12	18	18	NUM
m-1152	250	13	,	,	PUNCT
m-1152	250	14	19	19	NUM
m-1152	250	15	]	]	PUNCT
m-1152	250	16	for	for	ADP
m-1152	250	17	generation	generation	NOUN
m-1152	250	18	results	result	NOUN
m-1152	250	19	,	,	PUNCT
m-1152	250	20	spectral	spectral	ADJ
m-1152	250	21	mapping	mapping	NOUN
m-1152	250	22	,	,	PUNCT
m-1152	250	23	and	and	CCONJ
m-1152	250	24	standard	standard	ADJ
m-1152	250	25	energy	energy	NOUN
m-1152	250	26	/	/	SYM
m-1152	250	27	multiplier	multipli	ADJ
m-1152	250	28	proofs	proof	NOUN
m-1152	250	29	;	;	PUNCT
m-1152	250	30	for	for	ADP
m-1152	250	31	localized	localized	ADJ
m-1152	250	32	damping	damp	VERB
m-1152	250	33	and	and	CCONJ
m-1152	250	34	geometric	geometric	ADJ
m-1152	250	35	control	control	NOUN
m-1152	250	36	see	see	VERB
m-1152	250	37	the	the	DET
m-1152	250	38	survey	survey	NOUN
m-1152	250	39	by	by	ADP
m-1152	250	40	lebeau	lebeau	NOUN
m-1152	250	41	and	and	CCONJ
m-1152	250	42	rauch	rauch	PROPN
m-1152	250	43	and	and	CCONJ
m-1152	250	44	the	the	DET
m-1152	250	45	literature	literature	NOUN
m-1152	250	46	cited	cite	VERB
m-1152	250	47	therein	therein	ADV
m-1152	250	48	.	.	PUNCT
m-1152	251	1	5	5	NUM
m-1152	251	2	conditions	condition	NOUN
m-1152	251	3	for	for	ADP
m-1152	251	4	coincidence	coincidence	NOUN
m-1152	251	5	theorem	theorem	VERB
m-1152	251	6	5.1	5.1	NUM
m-1152	251	7	(	(	PUNCT
m-1152	251	8	coincidence	coincidence	NOUN
m-1152	251	9	of	of	ADP
m-1152	251	10	stability	stability	NOUN
m-1152	251	11	)	)	PUNCT
m-1152	251	12	.	.	PUNCT
m-1152	252	1	let	let	AUX
m-1152	252	2	{	{	PUNCT
m-1152	252	3	t	t	PROPN
m-1152	252	4	(	(	PUNCT
m-1152	252	5	t	t	NOUN
m-1152	252	6	)	)	PUNCT
m-1152	252	7	}	}	PUNCT
m-1152	252	8	be	be	AUX
m-1152	252	9	a	a	DET
m-1152	252	10	c0	c0	NOUN
m-1152	252	11	-	-	PUNCT
m-1152	252	12	semigroup	semigroup	PROPN
m-1152	252	13	with	with	ADP
m-1152	252	14	generator	generator	PROPN
m-1152	252	15	a.	a.	PROPN
m-1152	252	16	kw	kw	PROPN
m-1152	252	17	-	-	NOUN
m-1152	252	18	stability	stability	NOUN
m-1152	252	19	and	and	CCONJ
m-1152	252	20	analytic	analytic	ADJ
m-1152	252	21	stability	stability	NOUN
m-1152	252	22	yield	yield	VERB
m-1152	252	23	the	the	DET
m-1152	252	24	same	same	ADJ
m-1152	252	25	conclusion	conclusion	NOUN
m-1152	252	26	i	i	PRON
m-1152	252	27	�	�	PROPN
m-1152	252	28	s(a	s(a	PROPN
m-1152	252	29	)	)	PUNCT
m-1152	252	30	<	<	X
m-1152	252	31	0	0	NUM
m-1152	252	32	and	and	CCONJ
m-1152	252	33	σ(t	σ(t	PROPN
m-1152	252	34	(	(	PUNCT
m-1152	252	35	t	t	PROPN
m-1152	252	36	)	)	PUNCT
m-1152	252	37	)	)	PUNCT
m-1152	252	38	\	\	NOUN
m-1152	253	1	{	{	PUNCT
m-1152	253	2	0	0	NUM
m-1152	253	3	}	}	PUNCT
m-1152	253	4	=	=	SYM
m-1152	253	5	etσ(a	etσ(a	PROPN
m-1152	253	6	)	)	PUNCT
m-1152	253	7	.	.	PUNCT
m-1152	254	1	proof	proof	NOUN
m-1152	254	2	.	.	PUNCT
m-1152	255	1	if	if	SCONJ
m-1152	255	2	s(a	s(a	NOUN
m-1152	255	3	)	)	PUNCT
m-1152	255	4	<	<	X
m-1152	255	5	0	0	NUM
m-1152	255	6	,	,	PUNCT
m-1152	255	7	then	then	ADV
m-1152	255	8	r(t	r(t	NOUN
m-1152	255	9	(	(	PUNCT
m-1152	255	10	t	t	NOUN
m-1152	255	11	)	)	PUNCT
m-1152	255	12	)	)	PUNCT
m-1152	256	1	=	=	PUNCT
m-1152	256	2	ets(a	ets(a	PROPN
m-1152	256	3	)	)	PUNCT
m-1152	256	4	<	<	X
m-1152	256	5	1	1	NUM
m-1152	256	6	for	for	ADP
m-1152	256	7	t	t	PROPN
m-1152	256	8	>	>	X
m-1152	256	9	0	0	X
m-1152	256	10	.	.	PUNCT
m-1152	256	11	by	by	ADP
m-1152	256	12	spectral	spectral	ADJ
m-1152	256	13	mapping	mapping	NOUN
m-1152	256	14	and	and	CCONJ
m-1152	256	15	gearhart	gearhart	PROPN
m-1152	256	16	�	�	PROPN
m-1152	256	17	prüss	prüss	PROPN
m-1152	256	18	theorem	theorem	NOUN
m-1152	256	19	,	,	PUNCT
m-1152	256	20	exponential	exponential	ADJ
m-1152	256	21	stability	stability	NOUN
m-1152	256	22	follows	follow	VERB
m-1152	256	23	.	.	PUNCT
m-1152	257	1	conversely	conversely	ADV
m-1152	257	2	,	,	PUNCT
m-1152	257	3	if	if	SCONJ
m-1152	257	4	s(a	s(a	PROPN
m-1152	257	5	)	)	PUNCT
m-1152	257	6	≥	≥	NOUN
m-1152	257	7	0	0	NUM
m-1152	257	8	,	,	PUNCT
m-1152	257	9	analytic	analytic	ADJ
m-1152	257	10	decay	decay	NOUN
m-1152	257	11	fails	fail	VERB
m-1152	257	12	although	although	SCONJ
m-1152	257	13	kw	kw	NOUN
m-1152	257	14	-	-	PUNCT
m-1152	257	15	stability	stability	NOUN
m-1152	257	16	holds	hold	VERB
m-1152	257	17	(	(	PUNCT
m-1152	257	18	examples	example	NOUN
m-1152	257	19	4.1	4.1	NUM
m-1152	257	20	,	,	PUNCT
m-1152	257	21	4.2	4.2	NUM
m-1152	257	22	)	)	PUNCT
m-1152	257	23	.	.	PUNCT
m-1152	258	1	thus	thus	ADV
m-1152	258	2	both	both	DET
m-1152	258	3	s(a	s(a	NOUN
m-1152	258	4	)	)	PUNCT
m-1152	258	5	<	<	X
m-1152	258	6	0	0	NUM
m-1152	259	1	and	and	CCONJ
m-1152	259	2	spectral	spectral	ADJ
m-1152	259	3	mapping	mapping	NOUN
m-1152	259	4	are	be	AUX
m-1152	259	5	necessary	necessary	ADJ
m-1152	259	6	and	and	CCONJ
m-1152	259	7	su	su	PROPN
m-1152	259	8	�	�	PROPN
m-1152	259	9	cient	cient	PROPN
m-1152	259	10	.	.	PUNCT
m-1152	260	1	5.1	5.1	NUM
m-1152	260	2	spectral	spectral	ADJ
m-1152	260	3	-	-	PUNCT
m-1152	260	4	bound	bind	VERB
m-1152	260	5	table	table	NOUN
m-1152	260	6	spectral	spectral	ADJ
m-1152	260	7	bound	bind	VERB
m-1152	260	8	s(a	s(a	PROPN
m-1152	260	9	)	)	PUNCT
m-1152	260	10	kw	kw	VERB
m-1152	260	11	-	-	PUNCT
m-1152	260	12	stabilityanalytic	stabilityanalytic	ADJ
m-1152	260	13	conclusionspectral	conclusionspectral	ADJ
m-1152	260	14	mapping	mapping	NOUN
m-1152	260	15	s(a	s(a	PROPN
m-1152	260	16	)	)	PUNCT
m-1152	261	1	<	<	X
m-1152	261	2	0	0	PUNCT
m-1152	261	3	always	always	ADV
m-1152	261	4	trueexponentially	trueexponentially	ADV
m-1152	261	5	stableholds	stablehold	VERB
m-1152	261	6	s(a	s(a	PROPN
m-1152	261	7	)	)	PUNCT
m-1152	262	1	=	=	PUNCT
m-1152	262	2	0	0	PUNCT
m-1152	262	3	always	always	ADV
m-1152	262	4	trueneutral	trueneutral	ADJ
m-1152	262	5	(	(	PUNCT
m-1152	262	6	no	no	PRON
m-1152	262	7	decay)holds	decay)hold	NOUN
m-1152	262	8	s(a	s(a	PROPN
m-1152	262	9	)	)	PUNCT
m-1152	262	10	>	>	X
m-1152	262	11	0	0	PUNCT
m-1152	263	1	always	always	ADV
m-1152	263	2	trueunstable	trueunstable	ADJ
m-1152	263	3	(	(	PUNCT
m-1152	263	4	growth)holds	growth)holds	PROPN
m-1152	263	5	ijo	ijo	PROPN
m-1152	263	6	international	international	PROPN
m-1152	263	7	journal	journal	PROPN
m-1152	263	8	of	of	ADP
m-1152	263	9	mathematics	mathematics	PROPN
m-1152	263	10	(	(	PUNCT
m-1152	263	11	issn	issn	PROPN
m-1152	263	12	:	:	PUNCT
m-1152	263	13	2992	2992	NUM
m-1152	263	14	-	-	SYM
m-1152	263	15	4421	4421	NUM
m-1152	263	16	)	)	PUNCT
m-1152	263	17	volume	volume	NOUN
m-1152	263	18	08	08	NUM
m-1152	264	1	|	|	ADV
m-1152	264	2	issue	issue	NOUN
m-1152	264	3	9	9	NUM
m-1152	264	4	|	|	CCONJ
m-1152	264	5	september	september	PROPN
m-1152	264	6	2025	2025	NUM
m-1152	264	7	|	|	ADV
m-1152	264	8	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1152	264	9	19	19	NUM
m-1152	264	10	stability	stability	NOUN
m-1152	264	11	hierarchy	hierarchy	NOUN
m-1152	264	12	diagram5.2	diagram5.2	NOUN
m-1152	264	13	exponential	exponential	ADJ
m-1152	264	14	stability	stability	NOUN
m-1152	264	15	uniform	uniform	ADJ
m-1152	264	16	stability	stability	NOUN
m-1152	264	17	strong	strong	ADJ
m-1152	264	18	stability	stability	NOUN
m-1152	264	19	asymptotic	asymptotic	ADJ
m-1152	264	20	stability	stability	PROPN
m-1152	264	21	koch	koch	PROPN
m-1152	264	22	�	�	PROPN
m-1152	264	23	wallace	wallace	PROPN
m-1152	264	24	stability	stability	NOUN
m-1152	264	25	(	(	PUNCT
m-1152	264	26	always	always	ADV
m-1152	264	27	holds	hold	VERB
m-1152	264	28	)	)	PUNCT
m-1152	264	29	figure	figure	NOUN
m-1152	264	30	4	4	NUM
m-1152	264	31	:	:	PUNCT
m-1152	264	32	hierarchy	hierarchy	NOUN
m-1152	264	33	of	of	ADP
m-1152	264	34	analytic	analytic	ADJ
m-1152	264	35	stability	stability	NOUN
m-1152	264	36	notions	notion	NOUN
m-1152	264	37	with	with	ADP
m-1152	264	38	koch	koch	PROPN
m-1152	264	39	�	�	PROPN
m-1152	264	40	wallace	wallace	PROPN
m-1152	264	41	stability	stability	NOUN
m-1152	264	42	in	in	ADP
m-1152	264	43	parallel	parallel	NOUN
m-1152	264	44	.	.	PUNCT
m-1152	265	1	6	6	NUM
m-1152	265	2	conclusion	conclusion	NOUN
m-1152	265	3	this	this	DET
m-1152	265	4	work	work	NOUN
m-1152	265	5	has	have	AUX
m-1152	265	6	revealed	reveal	VERB
m-1152	265	7	a	a	DET
m-1152	265	8	fundamental	fundamental	ADJ
m-1152	265	9	connection	connection	NOUN
m-1152	265	10	between	between	ADP
m-1152	265	11	algebraic	algebraic	ADJ
m-1152	265	12	and	and	CCONJ
m-1152	265	13	analytic	analytic	ADJ
m-1152	265	14	stability	stability	NOUN
m-1152	265	15	in	in	ADP
m-1152	265	16	semigroups	semigroup	NOUN
m-1152	265	17	of	of	ADP
m-1152	265	18	bounded	bounded	ADJ
m-1152	265	19	linear	linear	PROPN
m-1152	265	20	operators	operator	NOUN
m-1152	265	21	.	.	PUNCT
m-1152	266	1	we	we	PRON
m-1152	266	2	showed	show	VERB
m-1152	266	3	that	that	SCONJ
m-1152	266	4	every	every	DET
m-1152	266	5	c0−semigroup	c0−semigroup	NOUN
m-1152	266	6	is	be	AUX
m-1152	266	7	stable	stable	ADJ
m-1152	266	8	in	in	ADP
m-1152	266	9	the	the	DET
m-1152	266	10	sense	sense	NOUN
m-1152	266	11	of	of	ADP
m-1152	266	12	koch	koch	PROPN
m-1152	266	13	�	�	PROPN
m-1152	266	14	wallace	wallace	PROPN
m-1152	266	15	,	,	PUNCT
m-1152	266	16	a	a	DET
m-1152	266	17	universal	universal	ADJ
m-1152	266	18	algebraic	algebraic	ADJ
m-1152	266	19	property	property	NOUN
m-1152	266	20	that	that	PRON
m-1152	266	21	enforces	enforce	VERB
m-1152	266	22	the	the	DET
m-1152	266	23	collapse	collapse	NOUN
m-1152	266	24	of	of	ADP
m-1152	266	25	green	green	PROPN
m-1152	266	26	's	's	PART
m-1152	266	27	relations	relation	NOUN
m-1152	266	28	.	.	PUNCT
m-1152	267	1	at	at	ADP
m-1152	267	2	the	the	DET
m-1152	267	3	same	same	ADJ
m-1152	267	4	time	time	NOUN
m-1152	267	5	,	,	PUNCT
m-1152	267	6	we	we	PRON
m-1152	267	7	identi	identi	VERB
m-1152	267	8	�	�	PROPN
m-1152	267	9	ed	ed	ADJ
m-1152	267	10	precise	precise	ADJ
m-1152	267	11	spectral	spectral	ADJ
m-1152	267	12	conditions	condition	NOUN
m-1152	267	13	under	under	ADP
m-1152	267	14	which	which	PRON
m-1152	267	15	this	this	DET
m-1152	267	16	algebraic	algebraic	ADJ
m-1152	267	17	stability	stability	NOUN
m-1152	267	18	coincides	coincide	VERB
m-1152	267	19	with	with	ADP
m-1152	267	20	analytic	analytic	ADJ
m-1152	267	21	stability	stability	NOUN
m-1152	267	22	in	in	ADP
m-1152	267	23	the	the	DET
m-1152	267	24	form	form	NOUN
m-1152	267	25	of	of	ADP
m-1152	267	26	decay	decay	NOUN
m-1152	267	27	properties	property	NOUN
m-1152	267	28	such	such	ADJ
m-1152	267	29	as	as	ADP
m-1152	267	30	strong	strong	ADJ
m-1152	267	31	,	,	PUNCT
m-1152	267	32	asymptotic	asymptotic	ADJ
m-1152	267	33	,	,	PUNCT
m-1152	267	34	and	and	CCONJ
m-1152	267	35	exponential	exponential	ADJ
m-1152	267	36	stability	stability	NOUN
m-1152	267	37	.	.	PUNCT
m-1152	268	1	the	the	DET
m-1152	268	2	examples	example	NOUN
m-1152	268	3	of	of	ADP
m-1152	268	4	the	the	DET
m-1152	268	5	translation	translation	NOUN
m-1152	268	6	semigroup	semigroup	NOUN
m-1152	268	7	,	,	PUNCT
m-1152	268	8	right	right	ADJ
m-1152	268	9	shift	shift	NOUN
m-1152	268	10	,	,	PUNCT
m-1152	268	11	heat	heat	NOUN
m-1152	268	12	semigroup	semigroup	NOUN
m-1152	268	13	,	,	PUNCT
m-1152	268	14	and	and	CCONJ
m-1152	268	15	damped	damp	VERB
m-1152	268	16	wave	wave	NOUN
m-1152	268	17	semigroup	semigroup	PROPN
m-1152	268	18	illustrate	illustrate	VERB
m-1152	268	19	the	the	DET
m-1152	268	20	subtle	subtle	ADJ
m-1152	268	21	boundary	boundary	NOUN
m-1152	268	22	between	between	ADP
m-1152	268	23	structural	structural	ADJ
m-1152	268	24	invariance	invariance	NOUN
m-1152	268	25	and	and	CCONJ
m-1152	268	26	spectral	spectral	ADJ
m-1152	268	27	decay	decay	NOUN
m-1152	268	28	,	,	PUNCT
m-1152	268	29	giving	give	VERB
m-1152	268	30	rise	rise	NOUN
m-1152	268	31	to	to	ADP
m-1152	268	32	what	what	PRON
m-1152	268	33	may	may	AUX
m-1152	268	34	be	be	AUX
m-1152	268	35	described	describe	VERB
m-1152	268	36	as	as	ADP
m-1152	268	37	a	a	DET
m-1152	268	38	stability	stability	NOUN
m-1152	268	39	gap	gap	NOUN
m-1152	268	40	.	.	PUNCT
m-1152	269	1	this	this	DET
m-1152	269	2	recognition	recognition	NOUN
m-1152	269	3	clari	clari	PROPN
m-1152	269	4	�	�	PROPN
m-1152	269	5	es	es	NOUN
m-1152	269	6	why	why	SCONJ
m-1152	269	7	operator	operator	NOUN
m-1152	269	8	semigroups	semigroup	NOUN
m-1152	269	9	can	can	AUX
m-1152	269	10	exhibit	exhibit	VERB
m-1152	269	11	robust	robust	ADJ
m-1152	269	12	algebraic	algebraic	ADJ
m-1152	269	13	structure	structure	NOUN
m-1152	269	14	while	while	SCONJ
m-1152	269	15	displaying	display	VERB
m-1152	269	16	very	very	ADV
m-1152	269	17	di	di	NOUN
m-1152	269	18	�	�	NOUN
m-1152	269	19	erent	erent	NOUN
m-1152	269	20	analytic	analytic	ADJ
m-1152	269	21	behavior	behavior	NOUN
m-1152	269	22	depending	depend	VERB
m-1152	269	23	on	on	ADP
m-1152	269	24	their	their	PRON
m-1152	269	25	spectral	spectral	ADJ
m-1152	269	26	placement	placement	NOUN
m-1152	269	27	.	.	PUNCT
m-1152	270	1	beyond	beyond	ADP
m-1152	270	2	its	its	PRON
m-1152	270	3	theoretical	theoretical	ADJ
m-1152	270	4	interest	interest	NOUN
m-1152	270	5	,	,	PUNCT
m-1152	270	6	the	the	DET
m-1152	270	7	study	study	NOUN
m-1152	270	8	o	o	NOUN
m-1152	270	9	�	�	NOUN
m-1152	270	10	ers	er	NOUN
m-1152	270	11	insight	insight	VERB
m-1152	270	12	into	into	ADP
m-1152	270	13	the	the	DET
m-1152	270	14	analysis	analysis	NOUN
m-1152	270	15	of	of	ADP
m-1152	270	16	evolution	evolution	NOUN
m-1152	270	17	equations	equation	NOUN
m-1152	270	18	,	,	PUNCT
m-1152	270	19	the	the	DET
m-1152	270	20	design	design	NOUN
m-1152	270	21	of	of	ADP
m-1152	270	22	stable	stable	ADJ
m-1152	270	23	control	control	NOUN
m-1152	270	24	systems	system	NOUN
m-1152	270	25	,	,	PUNCT
m-1152	270	26	and	and	CCONJ
m-1152	270	27	the	the	DET
m-1152	270	28	assessment	assessment	NOUN
m-1152	270	29	of	of	ADP
m-1152	270	30	numerical	numerical	ADJ
m-1152	270	31	schemes	scheme	NOUN
m-1152	270	32	where	where	SCONJ
m-1152	270	33	stability	stability	NOUN
m-1152	270	34	properties	property	NOUN
m-1152	270	35	are	be	AUX
m-1152	270	36	decisive	decisive	ADJ
m-1152	270	37	.	.	PUNCT
m-1152	271	1	by	by	ADP
m-1152	271	2	showing	show	VERB
m-1152	271	3	that	that	SCONJ
m-1152	271	4	kw	kw	NOUN
m-1152	271	5	-	-	PUNCT
m-1152	271	6	stability	stability	NOUN
m-1152	271	7	is	be	AUX
m-1152	271	8	always	always	ADV
m-1152	271	9	present	present	ADJ
m-1152	271	10	while	while	SCONJ
m-1152	271	11	analytic	analytic	ADJ
m-1152	271	12	stability	stability	NOUN
m-1152	271	13	is	be	AUX
m-1152	271	14	conditional	conditional	ADJ
m-1152	271	15	,	,	PUNCT
m-1152	271	16	we	we	PRON
m-1152	271	17	provide	provide	VERB
m-1152	271	18	a	a	DET
m-1152	271	19	uni	uni	PROPN
m-1152	271	20	�	�	PROPN
m-1152	271	21	ed	ed	NOUN
m-1152	271	22	framework	framework	NOUN
m-1152	271	23	that	that	PRON
m-1152	271	24	advances	advance	NOUN
m-1152	271	25	semigroup	semigroup	PROPN
m-1152	271	26	theory	theory	NOUN
m-1152	271	27	and	and	CCONJ
m-1152	271	28	strengthens	strengthen	VERB
m-1152	271	29	its	its	PRON
m-1152	271	30	applications	application	NOUN
m-1152	271	31	in	in	ADP
m-1152	271	32	mathematics	mathematic	NOUN
m-1152	271	33	,	,	PUNCT
m-1152	271	34	physics	physics	NOUN
m-1152	271	35	,	,	PUNCT
m-1152	271	36	and	and	CCONJ
m-1152	271	37	engineering	engineering	NOUN
m-1152	271	38	.	.	PUNCT
m-1152	272	1	references	reference	NOUN
m-1152	272	2	[	[	X
m-1152	272	3	1	1	NUM
m-1152	272	4	]	]	X
m-1152	272	5	s.b	s.b	PROPN
m-1152	272	6	.	.	PROPN
m-1152	272	7	koch	koch	PROPN
m-1152	272	8	and	and	CCONJ
m-1152	272	9	a.d	a.d	PROPN
m-1152	272	10	.	.	PROPN
m-1152	272	11	wallace	wallace	PROPN
m-1152	272	12	,	,	PUNCT
m-1152	272	13	�	�	NOUN
m-1152	272	14	stability	stability	NOUN
m-1152	272	15	in	in	ADP
m-1152	272	16	semigroups	semigroup	NOUN
m-1152	272	17	,	,	PUNCT
m-1152	272	18	�	�	PROPN
m-1152	272	19	duke	duke	PROPN
m-1152	272	20	math	math	PROPN
m-1152	272	21	.	.	PUNCT
m-1152	273	1	j.	j.	PROPN
m-1152	273	2	,	,	PUNCT
m-1152	273	3	vol	vol	NOUN
m-1152	273	4	.	.	PROPN
m-1152	273	5	23	23	NUM
m-1152	273	6	,	,	PUNCT
m-1152	273	7	pp	pp	ADJ
m-1152	273	8	.	.	PUNCT
m-1152	273	9	193	193	NUM
m-1152	273	10	�	�	PROPN
m-1152	273	11	202	202	NUM
m-1152	273	12	,	,	PUNCT
m-1152	273	13	1956	1956	NUM
m-1152	273	14	.	.	PUNCT
m-1152	274	1	[	[	X
m-1152	274	2	2	2	X
m-1152	274	3	]	]	PUNCT
m-1152	274	4	e.	e.	PROPN
m-1152	274	5	hille	hille	PROPN
m-1152	274	6	and	and	CCONJ
m-1152	274	7	r.	r.	PROPN
m-1152	274	8	s.	s.	PROPN
m-1152	274	9	phillips	phillips	PROPN
m-1152	274	10	,	,	PUNCT
m-1152	274	11	functional	functional	ADJ
m-1152	274	12	analysis	analysis	NOUN
m-1152	274	13	and	and	CCONJ
m-1152	274	14	semi	semi	NOUN
m-1152	274	15	-	-	NOUN
m-1152	274	16	groups	group	NOUN
m-1152	274	17	,	,	PUNCT
m-1152	274	18	american	american	PROPN
m-1152	274	19	mathematical	mathematical	ADJ
m-1152	274	20	society	society	NOUN
m-1152	274	21	,	,	PUNCT
m-1152	274	22	1957	1957	NUM
m-1152	274	23	.	.	PUNCT
m-1152	275	1	[	[	X
m-1152	275	2	3	3	X
m-1152	275	3	]	]	X
m-1152	275	4	o.	o.	PROPN
m-1152	275	5	j.	j.	PROPN
m-1152	275	6	tom	tom	PROPN
m-1152	275	7	,	,	PUNCT
m-1152	275	8	o.	o.	PROPN
m-1152	275	9	g.	g.	PROPN
m-1152	275	10	udouaka	udouaka	PROPN
m-1152	275	11	,	,	PUNCT
m-1152	275	12	and	and	CCONJ
m-1152	275	13	e.	e.	PROPN
m-1152	275	14	s.	s.	PROPN
m-1152	275	15	udo	udo	PROPN
m-1152	275	16	�	�	PROPN
m-1152	275	17	a	a	DET
m-1152	275	18	kw	kw	PROPN
m-1152	275	19	�	�	NOUN
m-1152	275	20	stability	stability	NOUN
m-1152	275	21	in	in	ADP
m-1152	275	22	semigroup	semigroup	NOUN
m-1152	275	23	of	of	ADP
m-1152	275	24	bounded	bounded	PROPN
m-1152	275	25	linear	linear	PROPN
m-1152	275	26	operators	operator	NOUN
m-1152	275	27	,	,	PUNCT
m-1152	275	28	international	international	ADJ
m-1152	275	29	journal	journal	NOUN
m-1152	275	30	of	of	ADP
m-1152	275	31	applied	apply	VERB
m-1152	275	32	science	science	NOUN
m-1152	275	33	and	and	CCONJ
m-1152	275	34	mathematical	mathematical	ADJ
m-1152	275	35	theory	theory	NOUN
m-1152	275	36	eissn	eissn	PROPN
m-1152	275	37	2489	2489	NUM
m-1152	275	38	-	-	PUNCT
m-1152	275	39	009x	009x	NOUN
m-1152	275	40	p	p	PROPN
m-1152	275	41	-	-	PUNCT
m-1152	275	42	issn	issn	VERB
m-1152	275	43	2695	2695	NUM
m-1152	275	44	-	-	SYM
m-1152	275	45	1908	1908	NUM
m-1152	275	46	,	,	PUNCT
m-1152	275	47	vol	vol	NOUN
m-1152	275	48	.	.	PROPN
m-1152	275	49	11	11	NUM
m-1152	275	50	no	no	NOUN
m-1152	275	51	.	.	NOUN
m-1152	275	52	7	7	NUM
m-1152	275	53	,	,	PUNCT
m-1152	275	54	pp	pp	ADJ
m-1152	275	55	.	.	PUNCT
m-1152	275	56	9	9	NUM
m-1152	275	57	�	�	SYM
m-1152	275	58	14	14	NUM
m-1152	275	59	2025	2025	NUM
m-1152	275	60	www.iiardjournals.org	www.iiardjournals.org	NOUN
m-1152	276	1	[	[	X
m-1152	276	2	4	4	NUM
m-1152	276	3	]	]	PUNCT
m-1152	276	4	j.	j.	PROPN
m-1152	276	5	east	east	PROPN
m-1152	276	6	and	and	CCONJ
m-1152	276	7	p.	p.	PROPN
m-1152	276	8	higgins	higgins	PROPN
m-1152	276	9	,	,	PUNCT
m-1152	276	10	stability	stability	NOUN
m-1152	276	11	in	in	ADP
m-1152	276	12	semigroups	semigroup	NOUN
m-1152	276	13	:	:	PUNCT
m-1152	276	14	green	green	PROPN
m-1152	276	15	's	's	PART
m-1152	276	16	relations	relation	NOUN
m-1152	276	17	and	and	CCONJ
m-1152	276	18	beyond	beyond	ADP
m-1152	276	19	,	,	PUNCT
m-1152	276	20	semigroup	semigroup	PROPN
m-1152	276	21	forum	forum	PROPN
m-1152	276	22	,	,	PUNCT
m-1152	276	23	101:1	101:1	NUM
m-1152	276	24	�	�	NOUN
m-1152	276	25	25	25	NUM
m-1152	276	26	,	,	PUNCT
m-1152	276	27	2020	2020	NUM
m-1152	276	28	.	.	PUNCT
m-1152	277	1	ijo	ijo	PROPN
m-1152	277	2	international	international	PROPN
m-1152	277	3	journal	journal	PROPN
m-1152	277	4	of	of	ADP
m-1152	277	5	mathematics	mathematics	PROPN
m-1152	277	6	(	(	PUNCT
m-1152	277	7	issn	issn	PROPN
m-1152	277	8	:	:	PUNCT
m-1152	277	9	2992	2992	NUM
m-1152	277	10	-	-	SYM
m-1152	277	11	4421	4421	NUM
m-1152	277	12	)	)	PUNCT
m-1152	277	13	volume	volume	NOUN
m-1152	277	14	08	08	NUM
m-1152	278	1	|	|	ADV
m-1152	278	2	issue	issue	NOUN
m-1152	278	3	9	9	NUM
m-1152	278	4	|	|	CCONJ
m-1152	278	5	september	september	PROPN
m-1152	278	6	2025	2025	NUM
m-1152	279	1	|	|	ADV
m-1152	279	2	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1152	279	3	20	20	NUM
m-1152	280	1	[	[	X
m-1152	280	2	5	5	NUM
m-1152	280	3	]	]	PUNCT
m-1152	280	4	a.	a.	NOUN
m-1152	280	5	bátkai	bátkai	NOUN
m-1152	280	6	and	and	CCONJ
m-1152	280	7	s.	s.	PROPN
m-1152	280	8	piazzera	piazzera	PROPN
m-1152	280	9	,	,	PUNCT
m-1152	280	10	semigroups	semigroups	X
m-1152	280	11	for	for	ADP
m-1152	280	12	delay	delay	NOUN
m-1152	280	13	equations	equation	NOUN
m-1152	280	14	,	,	PUNCT
m-1152	280	15	research	research	NOUN
m-1152	280	16	notes	note	NOUN
m-1152	280	17	in	in	ADP
m-1152	280	18	mathematics	mathematics	NOUN
m-1152	280	19	10	10	NUM
m-1152	280	20	,	,	PUNCT
m-1152	280	21	a	a	DET
m-1152	280	22	k	k	PROPN
m-1152	280	23	peters	peters	PROPN
m-1152	280	24	,	,	PUNCT
m-1152	280	25	2005	2005	NUM
m-1152	280	26	.	.	PUNCT
m-1152	281	1	[	[	X
m-1152	281	2	6	6	NUM
m-1152	281	3	]	]	PUNCT
m-1152	281	4	j.	j.	PROPN
m-1152	281	5	glück	glück	PROPN
m-1152	281	6	and	and	CCONJ
m-1152	281	7	a.	a.	NOUN
m-1152	281	8	mironchenko	mironchenko	PROPN
m-1152	281	9	,	,	PUNCT
m-1152	281	10	stability	stability	NOUN
m-1152	281	11	criteria	criterion	NOUN
m-1152	281	12	for	for	ADP
m-1152	281	13	positive	positive	ADJ
m-1152	281	14	semigroups	semigroup	NOUN
m-1152	281	15	on	on	ADP
m-1152	281	16	ordered	order	VERB
m-1152	281	17	banach	banach	NOUN
m-1152	281	18	spaces	space	NOUN
m-1152	281	19	,	,	PUNCT
m-1152	281	20	j.	j.	PROPN
m-1152	281	21	evol	evol	PROPN
m-1152	281	22	.	.	PUNCT
m-1152	282	1	equ	equ	PROPN
m-1152	282	2	,	,	PUNCT
m-1152	282	3	vol	vol	NOUN
m-1152	282	4	.	.	PROPN
m-1152	282	5	25	25	NUM
m-1152	282	6	,	,	PUNCT
m-1152	282	7	no	no	INTJ
m-1152	282	8	.	.	NOUN
m-1152	282	9	12	12	NUM
m-1152	282	10	,	,	PUNCT
m-1152	282	11	1424	1424	NUM
m-1152	282	12	-	-	PUNCT
m-1152	282	13	3199/25/010001	3199/25/010001	NUM
m-1152	282	14	-	-	PUNCT
m-1152	282	15	49	49	NUM
m-1152	282	16	,	,	PUNCT
m-1152	282	17	2024	2024	NUM
m-1152	282	18	.	.	PUNCT
m-1152	283	1	https://doi.org/10.1007/s00028-024-01044-8	https://doi.org/10.1007/s00028-024-01044-8	NUM
m-1152	283	2	.	.	PUNCT
m-1152	284	1	[	[	X
m-1152	284	2	7	7	X
m-1152	284	3	]	]	X
m-1152	284	4	j.	j.	PROPN
m-1152	284	5	mui	mui	PROPN
m-1152	284	6	,	,	PUNCT
m-1152	284	7	spectral	spectral	ADJ
m-1152	284	8	properties	property	NOUN
m-1152	284	9	of	of	ADP
m-1152	284	10	locally	locally	ADV
m-1152	284	11	eventually	eventually	ADV
m-1152	284	12	positive	positive	ADJ
m-1152	284	13	operator	operator	NOUN
m-1152	284	14	semigroups	semigroup	NOUN
m-1152	284	15	,	,	PUNCT
m-1152	284	16	semigroup	semigroup	PROPN
m-1152	284	17	forum	forum	PROPN
m-1152	284	18	,	,	PUNCT
m-1152	284	19	vol	vol	NOUN
m-1152	284	20	.	.	PROPN
m-1152	284	21	106	106	NUM
m-1152	284	22	,	,	PUNCT
m-1152	284	23	no	no	INTJ
m-1152	284	24	.	.	NOUN
m-1152	284	25	2	2	NUM
m-1152	284	26	,	,	PUNCT
m-1152	284	27	pp	pp	ADJ
m-1152	284	28	.	.	PUNCT
m-1152	284	29	460	460	NUM
m-1152	284	30	�	�	PROPN
m-1152	284	31	480	480	NUM
m-1152	284	32	,	,	PUNCT
m-1152	284	33	2023	2023	NUM
m-1152	284	34	.	.	PUNCT
m-1152	285	1	[	[	X
m-1152	285	2	8	8	NUM
m-1152	285	3	]	]	X
m-1152	285	4	r.	r.	PROPN
m-1152	285	5	c.	c.	PROPN
m-1152	285	6	penney	penney	PROPN
m-1152	285	7	,	,	PUNCT
m-1152	285	8	self	self	NOUN
m-1152	285	9	-	-	PUNCT
m-1152	285	10	dual	dual	ADJ
m-1152	285	11	cones	cone	NOUN
m-1152	285	12	in	in	ADP
m-1152	285	13	hilbert	hilbert	NOUN
m-1152	285	14	space	space	NOUN
m-1152	285	15	,	,	PUNCT
m-1152	285	16	j.	j.	PROPN
m-1152	285	17	funct	funct	PROPN
m-1152	285	18	.	.	PUNCT
m-1152	286	1	anal	anal	PROPN
m-1152	286	2	.	.	PUNCT
m-1152	286	3	,	,	PUNCT
m-1152	286	4	vol	vol	NOUN
m-1152	286	5	.	.	PROPN
m-1152	286	6	21	21	NUM
m-1152	286	7	,	,	PUNCT
m-1152	286	8	pp	pp	ADJ
m-1152	286	9	.	.	PUNCT
m-1152	287	1	305	305	NUM
m-1152	287	2	�	�	SYM
m-1152	287	3	315	315	NUM
m-1152	287	4	,	,	PUNCT
m-1152	287	5	1976	1976	NUM
m-1152	287	6	.	.	PUNCT
m-1152	288	1	[	[	X
m-1152	288	2	9	9	NUM
m-1152	288	3	]	]	X
m-1152	288	4	h.	h.	PROPN
m-1152	288	5	h.	h.	PROPN
m-1152	288	6	schaefer	schaefer	PROPN
m-1152	288	7	,	,	PUNCT
m-1152	288	8	halbgeordnete	halbgeordnete	ADJ
m-1152	288	9	lokalkonvexe	lokalkonvexe	PROPN
m-1152	288	10	vektorräume	vektorräume	PROPN
m-1152	288	11	,	,	PUNCT
m-1152	288	12	iii	iii	PROPN
m-1152	288	13	.	.	PROPN
m-1152	288	14	math	math	NOUN
m-1152	288	15	.	.	PUNCT
m-1152	289	1	ann	ann	PROPN
m-1152	289	2	.	.	PROPN
m-1152	289	3	,	,	PUNCT
m-1152	289	4	141	141	NUM
m-1152	289	5	,	,	PUNCT
m-1152	289	6	pp	pp	ADJ
m-1152	289	7	.	.	PUNCT
m-1152	289	8	113	113	NUM
m-1152	289	9	�	�	PROPN
m-1152	289	10	142	142	NUM
m-1152	289	11	,	,	PUNCT
m-1152	289	12	1960	1960	NUM
m-1152	289	13	.	.	PUNCT
m-1152	290	1	[	[	X
m-1152	290	2	10	10	NUM
m-1152	290	3	]	]	X
m-1152	290	4	h.	h.	PROPN
m-1152	290	5	h.	h.	PROPN
m-1152	290	6	schaefer	schaefer	PROPN
m-1152	290	7	,	,	PUNCT
m-1152	290	8	invariant	invariant	ADJ
m-1152	290	9	ideals	ideal	NOUN
m-1152	290	10	of	of	ADP
m-1152	290	11	positive	positive	ADJ
m-1152	290	12	operators	operator	NOUN
m-1152	290	13	in	in	ADP
m-1152	290	14	c(x	c(x	NOUN
m-1152	290	15	)	)	PUNCT
m-1152	290	16	,	,	PUNCT
m-1152	290	17	i	i	PRON
m-1152	290	18	,	,	PUNCT
m-1152	290	19	ill	ill	PROPN
m-1152	290	20	.	.	PUNCT
m-1152	291	1	j.	j.	PROPN
m-1152	291	2	math	math	PROPN
m-1152	291	3	.	.	PUNCT
m-1152	291	4	,	,	PUNCT
m-1152	291	5	11	11	NUM
m-1152	291	6	,	,	PUNCT
m-1152	291	7	pp	pp	ADJ
m-1152	291	8	.	.	PUNCT
m-1152	292	1	703	703	NUM
m-1152	292	2	�	�	NOUN
m-1152	292	3	715	715	NUM
m-1152	292	4	,	,	PUNCT
m-1152	292	5	1967	1967	NUM
m-1152	292	6	.	.	PUNCT
m-1152	293	1	[	[	X
m-1152	293	2	11	11	NUM
m-1152	293	3	]	]	X
m-1152	293	4	h.	h.	PROPN
m-1152	293	5	vogt	vogt	PROPN
m-1152	293	6	,	,	PUNCT
m-1152	293	7	stability	stability	NOUN
m-1152	293	8	of	of	ADP
m-1152	293	9	uniformly	uniformly	ADV
m-1152	293	10	eventually	eventually	ADV
m-1152	293	11	positive	positive	ADJ
m-1152	293	12	c0−semigroups	c0−semigroup	NOUN
m-1152	293	13	on	on	ADP
m-1152	293	14	lp−spaces	lp−space	NOUN
m-1152	293	15	,	,	PUNCT
m-1152	293	16	proc	proc	NOUN
m-1152	293	17	.	.	PUNCT
m-1152	293	18	am	be	AUX
m-1152	293	19	.	.	PUNCT
m-1152	294	1	math	math	NOUN
m-1152	294	2	.	.	PUNCT
m-1152	295	1	soc	soc	PROPN
m-1152	295	2	.	.	PUNCT
m-1152	296	1	,	,	PUNCT
m-1152	296	2	vol	vol	NOUN
m-1152	296	3	.	.	PROPN
m-1152	297	1	150	150	NUM
m-1152	297	2	no	no	NOUN
m-1152	297	3	.	.	NOUN
m-1152	297	4	8	8	NUM
m-1152	297	5	,	,	PUNCT
m-1152	297	6	pp	pp	ADJ
m-1152	297	7	.	.	PUNCT
m-1152	298	1	3513	3513	NUM
m-1152	298	2	�	�	NOUN
m-1152	298	3	3515	3515	NUM
m-1152	298	4	,	,	PUNCT
m-1152	298	5	2022	2022	NUM
m-1152	298	6	.	.	PUNCT
m-1152	299	1	[	[	X
m-1152	299	2	12	12	NUM
m-1152	299	3	]	]	X
m-1152	299	4	l.	l.	PROPN
m-1152	299	5	weis	weis	PROPN
m-1152	299	6	,	,	PUNCT
m-1152	299	7	the	the	DET
m-1152	299	8	stability	stability	NOUN
m-1152	299	9	of	of	ADP
m-1152	299	10	positive	positive	ADJ
m-1152	299	11	semigroups	semigroup	NOUN
m-1152	299	12	on	on	ADP
m-1152	299	13	lp	lp	ADJ
m-1152	299	14	spaces	space	NOUN
m-1152	299	15	,	,	PUNCT
m-1152	299	16	proc	proc	NOUN
m-1152	299	17	.	.	PUNCT
m-1152	299	18	am	be	AUX
m-1152	299	19	.	.	PUNCT
m-1152	300	1	math	math	NOUN
m-1152	300	2	.	.	PUNCT
m-1152	301	1	soc	soc	PROPN
m-1152	301	2	.	.	PUNCT
m-1152	302	1	,	,	PUNCT
m-1152	302	2	vol	vol	NOUN
m-1152	302	3	.	.	PROPN
m-1152	303	1	123	123	NUM
m-1152	303	2	,	,	PUNCT
m-1152	303	3	no	no	INTJ
m-1152	303	4	.	.	NOUN
m-1152	303	5	10	10	NUM
m-1152	303	6	,	,	PUNCT
m-1152	303	7	pp	pp	ADJ
m-1152	303	8	.	.	PUNCT
m-1152	304	1	3089	3089	NUM
m-1152	304	2	�	�	SYM
m-1152	304	3	3094	3094	NUM
m-1152	304	4	,	,	PUNCT
m-1152	304	5	1995	1995	NUM
m-1152	304	6	.	.	PUNCT
m-1152	305	1	[	[	X
m-1152	305	2	13	13	NUM
m-1152	305	3	]	]	X
m-1152	305	4	l.	l.	PROPN
m-1152	305	5	weis	weis	PROPN
m-1152	305	6	,	,	PUNCT
m-1152	305	7	a	a	DET
m-1152	305	8	short	short	ADJ
m-1152	305	9	proof	proof	NOUN
m-1152	305	10	for	for	ADP
m-1152	305	11	the	the	DET
m-1152	305	12	stability	stability	NOUN
m-1152	305	13	theorem	theorem	VERB
m-1152	305	14	for	for	ADP
m-1152	305	15	positive	positive	ADJ
m-1152	305	16	semigroups	semigroup	NOUN
m-1152	305	17	on	on	ADP
m-1152	305	18	lp(µ	lp(µ	NOUN
m-1152	305	19	)	)	PUNCT
m-1152	305	20	.	.	PUNCT
m-1152	306	1	proc	proc	PROPN
m-1152	306	2	.	.	PUNCT
m-1152	307	1	am	be	AUX
m-1152	307	2	.	.	PUNCT
m-1152	308	1	math	math	NOUN
m-1152	308	2	.	.	PUNCT
m-1152	309	1	soc	soc	PROPN
m-1152	309	2	.	.	PUNCT
m-1152	309	3	,	,	PUNCT
m-1152	309	4	vol	vol	NOUN
m-1152	309	5	.	.	PROPN
m-1152	310	1	126	126	NUM
m-1152	310	2	,	,	PUNCT
m-1152	310	3	no	no	INTJ
m-1152	310	4	.	.	NOUN
m-1152	310	5	11	11	NUM
m-1152	310	6	,	,	PUNCT
m-1152	310	7	pp	pp	ADJ
m-1152	310	8	.	.	PUNCT
m-1152	311	1	3253	3253	NUM
m-1152	311	2	�	�	PROPN
m-1152	311	3	3256	3256	NUM
m-1152	311	4	,	,	PUNCT
m-1152	311	5	1998	1998	NUM
m-1152	311	6	.	.	PUNCT
m-1152	312	1	[	[	X
m-1152	312	2	14	14	NUM
m-1152	312	3	]	]	PUNCT
m-1152	312	4	a.	a.	PROPN
m-1152	312	5	w.	w.	PROPN
m-1152	312	6	wickstead	wickstead	PROPN
m-1152	312	7	,	,	PUNCT
m-1152	312	8	compact	compact	ADJ
m-1152	312	9	subsets	subset	NOUN
m-1152	312	10	of	of	ADP
m-1152	312	11	partially	partially	ADV
m-1152	312	12	ordered	order	VERB
m-1152	312	13	banach	banach	NOUN
m-1152	312	14	spaces	space	NOUN
m-1152	312	15	,	,	PUNCT
m-1152	312	16	math	math	NOUN
m-1152	312	17	.	.	PUNCT
m-1152	313	1	ann	ann	PROPN
m-1152	313	2	.	.	PROPN
m-1152	313	3	,	,	PUNCT
m-1152	313	4	212	212	NUM
m-1152	313	5	,	,	PUNCT
m-1152	313	6	pp	pp	ADJ
m-1152	313	7	.	.	PUNCT
m-1152	314	1	271	271	NUM
m-1152	314	2	�	�	NOUN
m-1152	314	3	284	284	NUM
m-1152	314	4	,	,	PUNCT
m-1152	314	5	1975	1975	NUM
m-1152	314	6	.	.	PUNCT
m-1152	315	1	[	[	X
m-1152	315	2	15	15	NUM
m-1152	315	3	]	]	X
m-1152	315	4	p.	p.	NOUN
m-1152	315	5	p.	p.	PROPN
m-1152	315	6	zabre	zabre	PROPN
m-1152	315	7	�	�	PROPN
m-1152	315	8	�	�	PROPN
m-1152	315	9	ko	ko	PROPN
m-1152	315	10	and	and	CCONJ
m-1152	315	11	s.	s.	PROPN
m-1152	315	12	v.	v.	PROPN
m-1152	315	13	smickih	smickih	PROPN
m-1152	315	14	,	,	PUNCT
m-1152	315	15	a	a	DET
m-1152	315	16	theorem	theorem	NOUN
m-1152	315	17	of	of	ADP
m-1152	315	18	m.	m.	NOUN
m-1152	315	19	g.	g.	PROPN
m-1152	315	20	kre	kre	PROPN
m-1152	315	21	�	�	PROPN
m-1152	315	22	�	�	PROPN
m-1152	315	23	n	n	NOUN
m-1152	315	24	and	and	CCONJ
m-1152	315	25	m.	m.	PROPN
m-1152	315	26	a.	a.	PROPN
m-1152	315	27	rutman	rutman	PROPN
m-1152	315	28	,	,	PUNCT
m-1152	315	29	funktsional	funktsional	ADJ
m-1152	315	30	.	.	PUNCT
m-1152	316	1	anal	anal	NOUN
m-1152	316	2	.	.	PUNCT
m-1152	317	1	i	i	PRON
m-1152	317	2	prilozhen	prilozhen	VERB
m-1152	317	3	,	,	PUNCT
m-1152	317	4	vol	vol	NOUN
m-1152	317	5	.	.	PROPN
m-1152	317	6	13	13	NUM
m-1152	317	7	,	,	PUNCT
m-1152	317	8	no	no	INTJ
m-1152	317	9	.	.	NOUN
m-1152	317	10	3	3	NUM
m-1152	317	11	,	,	PUNCT
m-1152	317	12	pp	pp	ADJ
m-1152	317	13	.	.	PUNCT
m-1152	318	1	81	81	NUM
m-1152	318	2	�	�	PROPN
m-1152	318	3	82	82	NUM
m-1152	318	4	,	,	PUNCT
m-1152	318	5	1979	1979	NUM
m-1152	318	6	.	.	PUNCT
m-1152	319	1	[	[	X
m-1152	319	2	16	16	NUM
m-1152	319	3	]	]	PUNCT
m-1152	319	4	b.	b.	PROPN
m-1152	319	5	simon	simon	PROPN
m-1152	319	6	,	,	PUNCT
m-1152	319	7	the	the	DET
m-1152	319	8	bound	bound	ADJ
m-1152	319	9	state	state	NOUN
m-1152	319	10	of	of	ADP
m-1152	319	11	weakly	weakly	ADV
m-1152	319	12	coupled	couple	VERB
m-1152	319	13	schrödinger	schrödinger	ADJ
m-1152	319	14	operators	operator	NOUN
m-1152	319	15	in	in	ADP
m-1152	319	16	one	one	NUM
m-1152	319	17	and	and	CCONJ
m-1152	319	18	two	two	NUM
m-1152	319	19	dimensions	dimension	NOUN
m-1152	319	20	,	,	PUNCT
m-1152	319	21	ann	ann	PROPN
m-1152	319	22	.	.	PUNCT
m-1152	319	23	phys	phys	PROPN
m-1152	319	24	.	.	PUNCT
m-1152	319	25	,	,	PUNCT
m-1152	319	26	97	97	NUM
m-1152	319	27	,	,	PUNCT
m-1152	319	28	pp	pp	ADJ
m-1152	319	29	.	.	PUNCT
m-1152	319	30	279	279	NUM
m-1152	319	31	�	�	NOUN
m-1152	319	32	288	288	NUM
m-1152	319	33	,	,	PUNCT
m-1152	319	34	1976	1976	NUM
m-1152	319	35	.	.	PUNCT
m-1152	320	1	[	[	X
m-1152	320	2	17	17	NUM
m-1152	320	3	]	]	X
m-1152	320	4	josep	josep	PROPN
m-1152	320	5	martinez	martinez	PROPN
m-1152	320	6	and	and	CCONJ
m-1152	320	7	josé	josé	PROPN
m-1152	320	8	m.	m.	PROPN
m-1152	320	9	mazon	mazon	PROPN
m-1152	320	10	.	.	PUNCT
m-1152	321	1	c0−semigroups	c0−semigroup	NOUN
m-1152	321	2	norm	norm	VERB
m-1152	321	3	continuous	continuous	ADJ
m-1152	321	4	at	at	ADP
m-1152	321	5	in	in	ADP
m-1152	321	6	�	�	PROPN
m-1152	321	7	nity	nity	NOUN
m-1152	321	8	,	,	PUNCT
m-1152	321	9	semigroup	semigroup	PROPN
m-1152	321	10	forum	forum	PROPN
m-1152	321	11	,	,	PUNCT
m-1152	321	12	vol	vol	NOUN
m-1152	321	13	.	.	PROPN
m-1152	322	1	52	52	NUM
m-1152	322	2	,	,	PUNCT
m-1152	322	3	no	no	INTJ
m-1152	322	4	.	.	NOUN
m-1152	322	5	2	2	NUM
m-1152	322	6	,	,	PUNCT
m-1152	322	7	pp	pp	ADJ
m-1152	322	8	.	.	PUNCT
m-1152	323	1	213	213	NUM
m-1152	323	2	�	�	NOUN
m-1152	323	3	224	224	NUM
m-1152	323	4	,	,	PUNCT
m-1152	323	5	1996	1996	NUM
m-1152	323	6	.	.	PUNCT
m-1152	324	1	[	[	X
m-1152	324	2	18	18	NUM
m-1152	324	3	]	]	X
m-1152	324	4	desheng	desheng	PROPN
m-1152	324	5	li	li	PROPN
m-1152	324	6	and	and	CCONJ
m-1152	324	7	mo	mo	PROPN
m-1152	324	8	jia	jia	PROPN
m-1152	324	9	,	,	PUNCT
m-1152	324	10	a	a	DET
m-1152	324	11	dynamical	dynamical	ADJ
m-1152	324	12	approach	approach	NOUN
m-1152	324	13	to	to	ADP
m-1152	324	14	the	the	DET
m-1152	324	15	perron	perron	PROPN
m-1152	324	16	-	-	PUNCT
m-1152	324	17	frobenius	frobenius	NOUN
m-1152	324	18	theory	theory	NOUN
m-1152	324	19	and	and	CCONJ
m-1152	324	20	generalized	generalized	ADJ
m-1152	324	21	krein	krein	NOUN
m-1152	324	22	-	-	PUNCT
m-1152	324	23	rutman	rutman	NOUN
m-1152	324	24	type	type	NOUN
m-1152	324	25	theorems	theorem	NOUN
m-1152	324	26	,	,	PUNCT
m-1152	324	27	j.	j.	PROPN
m-1152	324	28	math	math	PROPN
m-1152	324	29	.	.	PUNCT
m-1152	325	1	anal	anal	PROPN
m-1152	325	2	.	.	PUNCT
m-1152	326	1	appl	appl	PROPN
m-1152	326	2	.	.	PROPN
m-1152	326	3	,	,	PUNCT
m-1152	326	4	vol	vol	NOUN
m-1152	326	5	.	.	PUNCT
m-1152	327	1	496(2):paper	496(2):paper	NUM
m-1152	327	2	no	no	NOUN
m-1152	327	3	.	.	PUNCT
m-1152	328	1	124828	124828	NUM
m-1152	328	2	,	,	PUNCT
m-1152	328	3	22	22	NUM
m-1152	328	4	,	,	PUNCT
m-1152	328	5	2021	2021	NUM
m-1152	328	6	.	.	PUNCT
m-1152	329	1	[	[	X
m-1152	329	2	19	19	NUM
m-1152	329	3	]	]	X
m-1152	329	4	matthias	matthias	PROPN
m-1152	329	5	keller	keller	PROPN
m-1152	329	6	,	,	PUNCT
m-1152	329	7	daniel	daniel	PROPN
m-1152	329	8	lenz	lenz	PROPN
m-1152	329	9	,	,	PUNCT
m-1152	329	10	hendrik	hendrik	PROPN
m-1152	329	11	vogt	vogt	PROPN
m-1152	329	12	,	,	PUNCT
m-1152	329	13	and	and	CCONJ
m-1152	329	14	radosª	radosª	VERB
m-1152	329	15	aw	aw	INTJ
m-1152	329	16	wojciechowski	wojciechowski	ADJ
m-1152	329	17	,	,	PUNCT
m-1152	329	18	note	note	VERB
m-1152	329	19	on	on	ADP
m-1152	329	20	basic	basic	ADJ
m-1152	329	21	features	feature	NOUN
m-1152	329	22	of	of	ADP
m-1152	329	23	large	large	ADJ
m-1152	329	24	time	time	NOUN
m-1152	329	25	behaviour	behaviour	NOUN
m-1152	329	26	of	of	ADP
m-1152	329	27	heat	heat	NOUN
m-1152	329	28	kernels	kernel	NOUN
m-1152	329	29	,	,	PUNCT
m-1152	329	30	j.	j.	PROPN
m-1152	329	31	reine	reine	PROPN
m-1152	329	32	angew	angew	PROPN
m-1152	329	33	.	.	PUNCT
m-1152	330	1	math	math	NOUN
m-1152	330	2	.	.	PUNCT
m-1152	330	3	,	,	PUNCT
m-1152	330	4	708	708	NUM
m-1152	330	5	,	,	PUNCT
m-1152	330	6	pp	pp	ADJ
m-1152	330	7	.	.	PUNCT
m-1152	331	1	73-	73-	NUM
m-1152	331	2	�	�	NOUN
m-1152	331	3	95	95	NUM
m-1152	331	4	,	,	PUNCT
m-1152	331	5	2015	2015	NUM
m-1152	331	6	.	.	PUNCT
m-1152	332	1	[	[	X
m-1152	332	2	20	20	NUM
m-1152	332	3	]	]	X
m-1152	332	4	o.	o.	PROPN
m-1152	332	5	g.	g.	PROPN
m-1152	332	6	udoaka	udoaka	PROPN
m-1152	332	7	.	.	PUNCT
m-1152	333	1	rank	rank	NOUN
m-1152	333	2	of	of	ADP
m-1152	333	3	some	some	DET
m-1152	333	4	semigroups	semigroup	NOUN
m-1152	333	5	,	,	PUNCT
m-1152	333	6	international	international	ADJ
m-1152	333	7	journal	journal	NOUN
m-1152	333	8	of	of	ADP
m-1152	333	9	applied	apply	VERB
m-1152	333	10	science	science	NOUN
m-1152	333	11	and	and	CCONJ
m-1152	333	12	mathematical	mathematical	ADJ
m-1152	333	13	theory	theory	NOUN
m-1152	333	14	,	,	PUNCT
m-1152	333	15	vol	vol	NOUN
m-1152	333	16	.	.	PROPN
m-1152	334	1	9	9	NUM
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m-1152	334	3	.	.	NOUN
m-1152	335	1	3	3	NUM
m-1152	335	2	,	,	PUNCT
m-1152	335	3	pp	pp	ADJ
m-1152	335	4	.	.	PUNCT
m-1152	336	1	90	90	NUM
m-1152	336	2	-	-	SYM
m-1152	336	3	100	100	NUM
m-1152	336	4	,	,	PUNCT
m-1152	336	5	2023	2023	NUM
m-1152	336	6	.	.	PUNCT
m-1152	337	1	ijo	ijo	PROPN
m-1152	337	2	international	international	PROPN
m-1152	337	3	journal	journal	PROPN
m-1152	337	4	of	of	ADP
m-1152	337	5	mathematics	mathematics	PROPN
m-1152	337	6	(	(	PUNCT
m-1152	337	7	issn	issn	PROPN
m-1152	337	8	:	:	PUNCT
m-1152	337	9	2992	2992	NUM
m-1152	337	10	-	-	SYM
m-1152	337	11	4421	4421	NUM
m-1152	337	12	)	)	PUNCT
m-1152	337	13	volume	volume	NOUN
m-1152	337	14	08	08	NUM
m-1152	338	1	|	|	ADV
m-1152	338	2	issue	issue	NOUN
m-1152	338	3	9	9	NUM
m-1152	338	4	|	|	CCONJ
m-1152	338	5	september	september	PROPN
m-1152	338	6	2025	2025	NUM
m-1152	339	1	|	|	ADV
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m-1152	339	3	21	21	NUM
m-1152	340	1	[	[	X
m-1152	340	2	21	21	NUM
m-1152	340	3	]	]	PUNCT
m-1152	340	4	m.	m.	PROPN
m-1152	340	5	n.	n.	PROPN
m-1152	340	6	john	john	PROPN
m-1152	340	7	,	,	PUNCT
m-1152	340	8	and	and	CCONJ
m-1152	340	9	o.	o.	PROPN
m-1152	340	10	g.	g.	PROPN
m-1152	340	11	udoaka	udoaka	ADV
m-1152	340	12	,	,	PUNCT
m-1152	340	13	algebraic	algebraic	ADJ
m-1152	340	14	and	and	CCONJ
m-1152	340	15	topological	topological	ADJ
m-1152	340	16	analysis	analysis	NOUN
m-1152	340	17	of	of	ADP
m-1152	340	18	enveloping	envelop	VERB
m-1152	340	19	semigroups	semigroup	NOUN
m-1152	340	20	in	in	ADP
m-1152	340	21	transformation	transformation	NOUN
m-1152	340	22	groups	group	NOUN
m-1152	340	23	:	:	PUNCT
m-1152	340	24	proximal	proximal	ADJ
m-1152	340	25	equivalence	equivalence	NOUN
m-1152	340	26	and	and	CCONJ
m-1152	340	27	homomorphic	homomorphic	ADJ
m-1152	340	28	image	image	NOUN
m-1152	340	29	,	,	PUNCT
m-1152	340	30	ijo	ijo	PROPN
m-1152	340	31	international	international	PROPN
m-1152	340	32	journal	journal	PROPN
m-1152	340	33	of	of	ADP
m-1152	340	34	mathematics	mathematics	PROPN
m-1152	340	35	,	,	PUNCT
m-1152	340	36	vol	vol	NOUN
m-1152	340	37	.	.	PROPN
m-1152	340	38	6	6	NUM
m-1152	340	39	,	,	PUNCT
m-1152	340	40	no	no	INTJ
m-1152	340	41	.	.	NOUN
m-1152	340	42	12	12	NUM
m-1152	340	43	,	,	PUNCT
m-1152	340	44	pp	pp	ADJ
m-1152	340	45	.	.	PUNCT
m-1152	340	46	9	9	NUM
m-1152	340	47	�	�	NOUN
m-1152	340	48	23	23	NUM
m-1152	340	49	,	,	PUNCT
m-1152	340	50	2023	2023	NUM
m-1152	340	51	.	.	PUNCT
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m-1152	341	4	;	;	PUNCT
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m-1152	341	6	.	.	PUNCT
m-1152	342	1	[	[	X
m-1152	342	2	22	22	NUM
m-1152	342	3	]	]	PUNCT
m-1152	342	4	r.	r.	PROPN
m-1152	342	5	u.	u.	PROPN
m-1152	342	6	ndubuisi	ndubuisi	PROPN
m-1152	342	7	,	,	PUNCT
m-1152	342	8	o.	o.	PROPN
m-1152	342	9	g.	g.	PROPN
m-1152	342	10	udoaka	udoaka	PROPN
m-1152	342	11	,	,	PUNCT
m-1152	342	12	k	k	PROPN
m-1152	342	13	p	p	X
m-1152	342	14	shum	shum	NOUN
m-1152	342	15	,	,	PUNCT
m-1152	342	16	and	and	CCONJ
m-1152	342	17	r	r	NOUN
m-1152	342	18	b	b	PROPN
m-1152	342	19	abubakar	abubakar	PROPN
m-1152	342	20	,	,	PUNCT
m-1152	342	21	on	on	ADP
m-1152	342	22	homomorphisms	homomorphism	NOUN
m-1152	342	23	(	(	PUNCT
m-1152	342	24	good	good	ADJ
m-1152	342	25	homomorphisms	homomorphism	NOUN
m-1152	342	26	)	)	PUNCT
m-1152	342	27	between	between	ADP
m-1152	342	28	completely	completely	ADV
m-1152	342	29	j	j	NOUN
m-1152	342	30	◦	◦	NOUN
m-1152	342	31	simple	simple	ADJ
m-1152	342	32	semigroups	semigroup	NOUN
m-1152	342	33	,	,	PUNCT
m-1152	342	34	canadian	canadian	ADJ
m-1152	342	35	journal	journal	NOUN
m-1152	342	36	of	of	ADP
m-1152	342	37	pure	pure	ADJ
m-1152	342	38	and	and	CCONJ
m-1152	342	39	applied	applied	ADJ
m-1152	342	40	sciences	science	NOUN
m-1152	342	41	,	,	PUNCT
m-1152	342	42	vol	vol	NOUN
m-1152	342	43	.	.	PROPN
m-1152	342	44	13	13	NUM
m-1152	342	45	,	,	PUNCT
m-1152	342	46	no	no	INTJ
m-1152	342	47	.	.	NOUN
m-1152	342	48	2	2	NUM
m-1152	342	49	,	,	PUNCT
m-1152	342	50	pp	pp	ADJ
m-1152	342	51	.	.	PUNCT
m-1152	343	1	4793	4793	NUM
m-1152	343	2	-	-	SYM
m-1152	343	3	4797	4797	NUM
m-1152	343	4	,	,	PUNCT
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m-1152	343	6	.	.	PUNCT
m-1152	344	1	[	[	X
m-1152	344	2	23	23	NUM
m-1152	344	3	]	]	PUNCT
m-1152	344	4	r.	r.	PROPN
m-1152	344	5	u.	u.	PROPN
m-1152	344	6	ndubisi	ndubisi	PROPN
m-1152	344	7	and	and	CCONJ
m-1152	344	8	o.	o.	PROPN
m-1152	344	9	g.	g.	PROPN
m-1152	344	10	udoaka	udoaka	PROPN
m-1152	344	11	,	,	PUNCT
m-1152	344	12	a	a	DET
m-1152	344	13	structure	structure	NOUN
m-1152	344	14	theorem	theorem	VERB
m-1152	344	15	for	for	ADP
m-1152	344	16	left	left	ADJ
m-1152	344	17	restriction	restriction	NOUN
m-1152	344	18	semigroups	semigroup	NOUN
m-1152	344	19	of	of	ADP
m-1152	344	20	type	type	NOUN
m-1152	344	21	f	f	PROPN
m-1152	344	22	,	,	PUNCT
m-1152	344	23	international	international	ADJ
m-1152	344	24	journal	journal	NOUN
m-1152	344	25	of	of	ADP
m-1152	344	26	semigroup	semigroup	PROPN
m-1152	344	27	theory	theory	NOUN
m-1152	344	28	appl	appl	PROPN
m-1152	344	29	.	.	PROPN
m-1152	344	30	,	,	PUNCT
m-1152	344	31	vol	vol	NOUN
m-1152	344	32	.	.	PROPN
m-1152	344	33	2	2	NUM
m-1152	344	34	,	,	PUNCT
m-1152	344	35	2018	2018	NUM
m-1152	344	36	.	.	PUNCT
m-1152	345	1	[	[	X
m-1152	345	2	24	24	NUM
m-1152	345	3	]	]	X
m-1152	345	4	o.	o.	PROPN
m-1152	345	5	j.	j.	PROPN
m-1152	345	6	tom	tom	PROPN
m-1152	345	7	,	,	PUNCT
m-1152	345	8	and	and	CCONJ
m-1152	345	9	o.	o.	PROPN
m-1152	345	10	g.	g.	PROPN
m-1152	345	11	udoaka	udoaka	PROPN
m-1152	345	12	,	,	PUNCT
m-1152	345	13	semigroup	semigroup	ADJ
m-1152	345	14	approach	approach	NOUN
m-1152	345	15	for	for	ADP
m-1152	345	16	the	the	DET
m-1152	345	17	solution	solution	NOUN
m-1152	345	18	of	of	ADP
m-1152	345	19	boundary	boundary	ADJ
m-1152	345	20	layer	layer	NOUN
m-1152	345	21	euation	euation	NOUN
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m-1152	345	23	sinc	sinc	ADJ
m-1152	345	24	function	function	NOUN
m-1152	345	25	term	term	NOUN
m-1152	345	26	,	,	PUNCT
m-1152	345	27	ijo	ijo	PROPN
m-1152	345	28	international	international	PROPN
m-1152	345	29	journal	journal	PROPN
m-1152	345	30	of	of	ADP
m-1152	345	31	mathematics	mathematics	PROPN
m-1152	345	32	,	,	PUNCT
m-1152	345	33	vol	vol	NOUN
m-1152	345	34	.	.	NOUN
m-1152	345	35	8	8	NUM
m-1152	345	36	,	,	PUNCT
m-1152	345	37	issue	issue	NOUN
m-1152	345	38	4	4	NUM
m-1152	345	39	,	,	PUNCT
m-1152	345	40	pp	pp	ADJ
m-1152	345	41	.	.	PUNCT
m-1152	346	1	22	22	NUM
m-1152	346	2	�	�	PROPN
m-1152	346	3	37	37	NUM
m-1152	346	4	.	.	PUNCT
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m-1152	346	6	.	.	PUNCT
m-1152	347	1	[	[	X
m-1152	347	2	25	25	NUM
m-1152	347	3	]	]	X
m-1152	347	4	a.h	a.h	PROPN
m-1152	347	5	.	.	PROPN
m-1152	347	6	cli	cli	PROPN
m-1152	347	7	�	�	PROPN
m-1152	347	8	ord	ord	PROPN
m-1152	347	9	and	and	CCONJ
m-1152	347	10	g.b	g.b	PROPN
m-1152	347	11	.	.	PROPN
m-1152	347	12	preston	preston	PROPN
m-1152	347	13	,	,	PUNCT
m-1152	347	14	the	the	DET
m-1152	347	15	algebraic	algebraic	ADJ
m-1152	347	16	theory	theory	NOUN
m-1152	347	17	of	of	ADP
m-1152	347	18	semigroups	semigroup	NOUN
m-1152	347	19	,	,	PUNCT
m-1152	347	20	vols	vol	NOUN
m-1152	347	21	.	.	PUNCT
m-1152	348	1	i	i	PRON
m-1152	348	2	&	&	CCONJ
m-1152	348	3	ii	ii	PROPN
m-1152	348	4	,	,	PUNCT
m-1152	348	5	ams	ams	PROPN
m-1152	348	6	mathematical	mathematical	ADJ
m-1152	348	7	surveys	survey	NOUN
m-1152	348	8	,	,	PUNCT
m-1152	348	9	1961/1967	1961/1967	NUM
m-1152	348	10	.	.	PUNCT
m-1152	349	1	[	[	X
m-1152	349	2	26	26	NUM
m-1152	349	3	]	]	X
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m-1152	349	5	.	.	PROPN
m-1152	349	6	green	green	PROPN
m-1152	349	7	,	,	PUNCT
m-1152	349	8	�	�	PROPN
m-1152	349	9	on	on	ADP
m-1152	349	10	the	the	DET
m-1152	349	11	structure	structure	NOUN
m-1152	349	12	of	of	ADP
m-1152	349	13	semigroups	semigroup	NOUN
m-1152	349	14	,	,	PUNCT
m-1152	349	15	�	�	PROPN
m-1152	349	16	annals	annal	NOUN
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m-1152	349	18	mathematics	mathematic	NOUN
m-1152	349	19	,	,	PUNCT
m-1152	349	20	vol	vol	NOUN
m-1152	349	21	.	.	PROPN
m-1152	350	1	54	54	NUM
m-1152	350	2	,	,	PUNCT
m-1152	350	3	pp	pp	ADJ
m-1152	350	4	.	.	PUNCT
m-1152	351	1	163	163	NUM
m-1152	351	2	�	�	PROPN
m-1152	351	3	172	172	NUM
m-1152	351	4	,	,	PUNCT
m-1152	351	5	1951	1951	NUM
m-1152	351	6	.	.	PUNCT
m-1152	352	1	[	[	X
m-1152	352	2	27	27	NUM
m-1152	352	3	]	]	PUNCT
m-1152	352	4	d.	d.	PROPN
m-1152	352	5	daners	daners	PROPN
m-1152	352	6	,	,	PUNCT
m-1152	352	7	j.	j.	PROPN
m-1152	352	8	glück	glück	PROPN
m-1152	352	9	,	,	PUNCT
m-1152	352	10	and	and	CCONJ
m-1152	352	11	j.	j.	PROPN
m-1152	352	12	b.	b.	PROPN
m-1152	352	13	kennedy	kennedy	PROPN
m-1152	352	14	,	,	PUNCT
m-1152	352	15	eventually	eventually	ADV
m-1152	352	16	positive	positive	ADJ
m-1152	352	17	semigroups	semigroup	NOUN
m-1152	352	18	of	of	ADP
m-1152	352	19	linear	linear	PROPN
m-1152	352	20	operators	operator	NOUN
m-1152	352	21	,	,	PUNCT
m-1152	352	22	j.	j.	PROPN
m-1152	352	23	math	math	PROPN
m-1152	352	24	.	.	PUNCT
m-1152	353	1	anal	anal	PROPN
m-1152	353	2	.	.	PUNCT
m-1152	354	1	appl	appl	PROPN
m-1152	354	2	.	.	PROPN
m-1152	354	3	,	,	PUNCT
m-1152	354	4	433(2	433(2	NUM
m-1152	354	5	)	)	PUNCT
m-1152	354	6	,	,	PUNCT
m-1152	354	7	pp	pp	ADJ
m-1152	354	8	.	.	PUNCT
m-1152	355	1	1561	1561	NUM
m-1152	355	2	�	�	PROPN
m-1152	355	3	1593	1593	NUM
m-1152	355	4	,	,	PUNCT
m-1152	355	5	2016	2016	NUM
m-1152	355	6	.	.	PUNCT
m-1152	356	1	[	[	X
m-1152	356	2	28	28	NUM
m-1152	356	3	]	]	X
m-1152	356	4	m.	m.	NOUN
m-1152	356	5	d.	d.	PROPN
m-1152	356	6	donsker	donsker	PROPN
m-1152	356	7	and	and	CCONJ
m-1152	356	8	s.	s.	PROPN
m-1152	356	9	r.	r.	PROPN
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m-1152	356	11	varadhan	varadhan	PROPN
m-1152	356	12	,	,	PUNCT
m-1152	356	13	on	on	ADP
m-1152	356	14	a	a	DET
m-1152	356	15	variational	variational	ADJ
m-1152	356	16	formula	formula	NOUN
m-1152	356	17	for	for	ADP
m-1152	356	18	the	the	DET
m-1152	356	19	principal	principal	ADJ
m-1152	356	20	eigenvalue	eigenvalue	NOUN
m-1152	356	21	for	for	ADP
m-1152	356	22	operators	operator	NOUN
m-1152	356	23	with	with	ADP
m-1152	356	24	maximum	maximum	ADJ
m-1152	356	25	principle	principle	NOUN
m-1152	356	26	,	,	PUNCT
m-1152	356	27	proc	proc	PROPN
m-1152	356	28	.	.	PUNCT
m-1152	357	1	natl	natl	PROPN
m-1152	357	2	.	.	PUNCT
m-1152	358	1	acad	acad	PROPN
m-1152	358	2	.	.	PUNCT
m-1152	359	1	sci	sci	PROPN
m-1152	359	2	.	.	PROPN
m-1152	359	3	usa	usa	PROPN
m-1152	359	4	,	,	PUNCT
m-1152	359	5	72	72	NUM
m-1152	359	6	,	,	PUNCT
m-1152	359	7	pp	pp	ADJ
m-1152	359	8	.	.	PUNCT
m-1152	360	1	780	780	NUM
m-1152	360	2	�	�	PROPN
m-1152	360	3	783	783	NUM
m-1152	360	4	,	,	PUNCT
m-1152	360	5	1975	1975	NUM
m-1152	360	6	.	.	PUNCT
m-1152	361	1	[	[	X
m-1152	361	2	29	29	NUM
m-1152	361	3	]	]	PUNCT
m-1152	361	4	m.	m.	NOUN
m-1152	361	5	d.	d.	PROPN
m-1152	361	6	donsker	donsker	PROPN
m-1152	361	7	and	and	CCONJ
m-1152	361	8	s.	s.	PROPN
m-1152	361	9	r.	r.	PROPN
m-1152	361	10	srinivasa	srinivasa	PROPN
m-1152	361	11	varadhan	varadhan	PROPN
m-1152	361	12	.	.	PUNCT
m-1152	362	1	on	on	ADP
m-1152	362	2	the	the	DET
m-1152	362	3	principal	principal	ADJ
m-1152	362	4	eigenvalue	eigenvalue	NOUN
m-1152	362	5	of	of	ADP
m-1152	362	6	secondorder	secondorder	PROPN
m-1152	362	7	elliptic	elliptic	PROPN
m-1152	362	8	di	di	PROPN
m-1152	362	9	�	�	PROPN
m-1152	362	10	erential	erential	ADJ
m-1152	362	11	operators	operator	NOUN
m-1152	362	12	,	,	PUNCT
m-1152	362	13	commun	commun	PROPN
m-1152	362	14	.	.	PUNCT
m-1152	363	1	pure	pure	ADJ
m-1152	363	2	appl	appl	PROPN
m-1152	363	3	.	.	PUNCT
m-1152	363	4	math	math	PROPN
m-1152	363	5	.	.	PUNCT
m-1152	363	6	,	,	PUNCT
m-1152	363	7	29	29	NUM
m-1152	363	8	,	,	PUNCT
m-1152	363	9	pp	pp	ADJ
m-1152	363	10	.	.	PUNCT
m-1152	364	1	595	595	NUM
m-1152	364	2	�	�	NOUN
m-1152	364	3	621	621	NUM
m-1152	364	4	,	,	PUNCT
m-1152	364	5	1976	1976	NUM
m-1152	364	6	.	.	PUNCT
m-1152	365	1	[	[	X
m-1152	365	2	30	30	NUM
m-1152	365	3	]	]	X
m-1152	365	4	s.	s.	PROPN
m-1152	365	5	friedland	friedland	PROPN
m-1152	365	6	,	,	PUNCT
m-1152	365	7	characterizations	characterization	NOUN
m-1152	365	8	of	of	ADP
m-1152	365	9	the	the	DET
m-1152	365	10	spectral	spectral	ADJ
m-1152	365	11	radius	radius	NOUN
m-1152	365	12	of	of	ADP
m-1152	365	13	positive	positive	ADJ
m-1152	365	14	operators	operator	NOUN
m-1152	365	15	,	,	PUNCT
m-1152	365	16	linear	linear	PROPN
m-1152	365	17	algebra	algebra	PROPN
m-1152	365	18	appl	appl	NOUN
m-1152	365	19	.	.	PROPN
m-1152	365	20	,	,	PUNCT
m-1152	365	21	134	134	NUM
m-1152	365	22	,	,	PUNCT
m-1152	365	23	pp	pp	ADJ
m-1152	365	24	.	.	PUNCT
m-1152	366	1	93	93	NUM
m-1152	366	2	�	�	PROPN
m-1152	366	3	105	105	NUM
m-1152	366	4	,	,	PUNCT
m-1152	366	5	1990	1990	NUM
m-1152	366	6	.	.	PUNCT
m-1152	367	1	[	[	X
m-1152	367	2	31	31	NUM
m-1152	367	3	]	]	PUNCT
m-1152	367	4	s.	s.	PROPN
m-1152	367	5	friedland	friedland	PROPN
m-1152	367	6	,	,	PUNCT
m-1152	367	7	the	the	DET
m-1152	367	8	collatz	collatz	NOUN
m-1152	367	9	-	-	PUNCT
m-1152	367	10	wielandt	wielandt	PROPN
m-1152	367	11	quotient	quotient	NOUN
m-1152	367	12	for	for	ADP
m-1152	367	13	pairs	pair	NOUN
m-1152	367	14	of	of	ADP
m-1152	367	15	nonnegative	nonnegative	ADJ
m-1152	367	16	operators	operator	NOUN
m-1152	367	17	,	,	PUNCT
m-1152	367	18	appl	appl	PROPN
m-1152	367	19	.	.	PROPN
m-1152	367	20	math	math	PROPN
m-1152	367	21	.	.	PUNCT
m-1152	367	22	,	,	PUNCT
m-1152	367	23	praha	praha	PROPN
m-1152	367	24	,	,	PUNCT
m-1152	367	25	65(5	65(5	NUM
m-1152	367	26	)	)	PUNCT
m-1152	367	27	,	,	PUNCT
m-1152	367	28	pp	pp	PROPN
m-1152	367	29	.	.	PUNCT
m-1152	368	1	557	557	NUM
m-1152	368	2	�	�	NOUN
m-1152	368	3	597	597	NUM
m-1152	368	4	,	,	PUNCT
m-1152	368	5	2020	2020	NUM
m-1152	368	6	.	.	PUNCT
m-1152	369	1	[	[	X
m-1152	369	2	32	32	NUM
m-1152	369	3	]	]	PUNCT
m-1152	369	4	j.	j.	PROPN
m-1152	369	5	mui	mui	PROPN
m-1152	369	6	,	,	PUNCT
m-1152	369	7	spectral	spectral	ADJ
m-1152	369	8	properties	property	NOUN
m-1152	369	9	of	of	ADP
m-1152	369	10	locally	locally	ADV
m-1152	369	11	eventually	eventually	ADV
m-1152	369	12	positive	positive	ADJ
m-1152	369	13	operator	operator	NOUN
m-1152	369	14	semigroups	semigroup	NOUN
m-1152	369	15	,	,	PUNCT
m-1152	369	16	semigroup	semigroup	PROPN
m-1152	369	17	forum	forum	PROPN
m-1152	369	18	,	,	PUNCT
m-1152	369	19	106(2	106(2	NUM
m-1152	369	20	)	)	PUNCT
m-1152	369	21	,	,	PUNCT
m-1152	369	22	pp	pp	PROPN
m-1152	369	23	.	.	PUNCT
m-1152	370	1	460	460	NUM
m-1152	370	2	�	�	PROPN
m-1152	370	3	480	480	NUM
m-1152	370	4	,	,	PUNCT
m-1152	370	5	2023	2023	NUM
m-1152	370	6	.	.	PUNCT
m-1152	371	1	[	[	X
m-1152	371	2	33	33	NUM
m-1152	371	3	]	]	PUNCT
m-1152	371	4	j.	j.	PROPN
m-1152	371	5	east	east	PROPN
m-1152	371	6	and	and	CCONJ
m-1152	371	7	p.	p.	PROPN
m-1152	371	8	m.	m.	PROPN
m-1152	371	9	higgins	higgins	PROPN
m-1152	371	10	,	,	PUNCT
m-1152	371	11	�	�	PROPN
m-1152	371	12	green	green	PROPN
m-1152	371	13	's	's	PART
m-1152	371	14	relations	relation	NOUN
m-1152	371	15	and	and	CCONJ
m-1152	371	16	stability	stability	NOUN
m-1152	371	17	for	for	ADP
m-1152	371	18	subsemigroups	subsemigroup	NOUN
m-1152	371	19	,	,	PUNCT
m-1152	371	20	�	�	PROPN
m-1152	371	21	semigroup	semigroup	PROPN
m-1152	371	22	forum	forum	PROPN
m-1152	371	23	,	,	PUNCT
m-1152	371	24	vol	vol	NOUN
m-1152	371	25	.	.	PROPN
m-1152	372	1	101	101	NUM
m-1152	372	2	,	,	PUNCT
m-1152	372	3	no	no	DET
m-1152	372	4	1	1	NUM
m-1152	372	5	,	,	PUNCT
m-1152	372	6	pp	pp	ADJ
m-1152	372	7	.	.	PUNCT
m-1152	373	1	77	77	NUM
m-1152	373	2	�	�	NOUN
m-1152	373	3	86	86	NUM
m-1152	373	4	,	,	PUNCT
m-1152	373	5	2020	2020	NUM
m-1152	373	6	.	.	PUNCT
m-1152	374	1	ijo	ijo	PROPN
m-1152	374	2	international	international	PROPN
m-1152	374	3	journal	journal	PROPN
m-1152	374	4	of	of	ADP
m-1152	374	5	mathematics	mathematics	PROPN
m-1152	374	6	(	(	PUNCT
m-1152	374	7	issn	issn	PROPN
m-1152	374	8	:	:	PUNCT
m-1152	374	9	2992	2992	NUM
m-1152	374	10	-	-	SYM
m-1152	374	11	4421	4421	NUM
m-1152	374	12	)	)	PUNCT
m-1152	374	13	volume	volume	NOUN
m-1152	374	14	08	08	NUM
m-1152	375	1	|	|	ADV
m-1152	375	2	issue	issue	NOUN
m-1152	375	3	9	9	NUM
m-1152	375	4	|	|	CCONJ
m-1152	375	5	september	september	PROPN
m-1152	375	6	2025	2025	NUM
m-1152	375	7	|	|	ADV
m-1152	375	8	http://ijojournals.com/index.php/m/index	http://ijojournals.com/index.php/m/index	PROPN
m-1152	375	9	22	22	NUM
m-1152	375	10	introduction	introduction	NOUN
m-1152	375	11	preliminaries	preliminary	NOUN
m-1152	375	12	main	main	ADJ
m-1152	375	13	results	result	NOUN
m-1152	375	14	illustrative	illustrative	ADJ
m-1152	375	15	examples	example	NOUN
m-1152	375	16	example	example	NOUN
m-1152	375	17	3.4	3.4	NUM
m-1152	375	18	(	(	PUNCT
m-1152	375	19	damped	damped	ADJ
m-1152	375	20	wave	wave	NOUN
m-1152	375	21	equation	equation	NOUN
m-1152	375	22	)	)	PUNCT
m-1152	375	23	conditions	condition	NOUN
m-1152	375	24	for	for	ADP
m-1152	375	25	coincidence	coincidence	NOUN
m-1152	375	26	spectral	spectral	ADJ
m-1152	375	27	-	-	PUNCT
m-1152	375	28	bound	bind	VERB
m-1152	375	29	table	table	NOUN
m-1152	375	30	stability	stability	NOUN
m-1152	375	31	hierarchy	hierarchy	NOUN
m-1152	375	32	diagram	diagram	NOUN
m-1152	375	33	conclusion	conclusion	NOUN