id	sid	tid	token	lemma	pos
cana-6282	1	1	communications	communication	NOUN
cana-6282	1	2	on	on	ADP
cana-6282	1	3	applied	apply	VERB
cana-6282	1	4	nonlinear	nonlinear	ADJ
cana-6282	1	5	analysis	analysis	NOUN
cana-6282	1	6	issn	issn	NOUN
cana-6282	1	7	:	:	PUNCT
cana-6282	1	8	1074	1074	NUM
cana-6282	1	9	-	-	PUNCT
cana-6282	1	10	133x	133x	NUM
cana-6282	1	11	vol	vol	VERB
cana-6282	1	12	32	32	NUM
cana-6282	1	13	no	no	NOUN
cana-6282	1	14	.	.	PUNCT
cana-6282	2	1	10s	10	NOUN
cana-6282	2	2	(	(	PUNCT
cana-6282	2	3	2025	2025	NUM
cana-6282	2	4	)	)	PUNCT
cana-6282	2	5	3684	3684	NUM
cana-6282	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-6282	2	7	a	a	DET
cana-6282	2	8	hyperstability	hyperstability	NOUN
cana-6282	2	9	approach	approach	NOUN
cana-6282	2	10	to	to	ADP
cana-6282	2	11	generalized	generalized	ADJ
cana-6282	2	12	functional	functional	ADJ
cana-6282	2	13	equations	equation	NOUN
cana-6282	2	14	on	on	ADP
cana-6282	2	15	involutive	involutive	ADJ
cana-6282	2	16	semigroups	semigroup	NOUN
cana-6282	2	17	ismaail	ismaail	PROPN
cana-6282	2	18	essalih	essalih	PROPN
cana-6282	2	19	*	*	PROPN
cana-6282	2	20	,	,	PUNCT
cana-6282	2	21	nordine	nordine	ADJ
cana-6282	2	22	bounader	bounader	NOUN
cana-6282	2	23	,	,	PUNCT
cana-6282	2	24	and	and	CCONJ
cana-6282	2	25	ahmed	ahmed	PROPN
cana-6282	2	26	akkaoui	akkaoui	PROPN
cana-6282	2	27	department	department	PROPN
cana-6282	2	28	of	of	ADP
cana-6282	2	29	mathematics	mathematic	NOUN
cana-6282	2	30	,	,	PUNCT
cana-6282	2	31	faculty	faculty	NOUN
cana-6282	2	32	of	of	ADP
cana-6282	2	33	sciences	science	NOUN
cana-6282	2	34	,	,	PUNCT
cana-6282	2	35	ibn	ibn	PROPN
cana-6282	2	36	tofail	tofail	NOUN
cana-6282	2	37	university	university	PROPN
cana-6282	2	38	,	,	PUNCT
cana-6282	2	39	kénitra	kénitra	PROPN
cana-6282	2	40	,	,	PUNCT
cana-6282	2	41	morocco	morocco	PROPN
cana-6282	2	42	.	.	PUNCT
cana-6282	3	1	e	e	X
cana-6282	3	2	-	-	NOUN
cana-6282	3	3	mail	mail	NOUN
cana-6282	3	4	address	address	NOUN
cana-6282	3	5	:	:	PUNCT
cana-6282	3	6	essalih.ismaail@gmail.com	essalih.ismaail@gmail.com	NUM
cana-6282	3	7	,	,	PUNCT
cana-6282	3	8	n.bounader@live.fr	n.bounader@live.fr	X
cana-6282	3	9	,	,	PUNCT
cana-6282	3	10	ahmed.maths78@gmail.com	ahmed.maths78@gmail.com	X
cana-6282	3	11	*	*	PUNCT
cana-6282	3	12	corresponding	correspond	VERB
cana-6282	3	13	author	author	NOUN
cana-6282	3	14	article	article	NOUN
cana-6282	3	15	history	history	NOUN
cana-6282	3	16	:	:	PUNCT
cana-6282	3	17	received:04/09/2025	received:04/09/2025	ADJ
cana-6282	3	18	revised:03/10/2025	revised:03/10/2025	PROPN
cana-6282	3	19	accepted:27/11/2025	accepted:27/11/2025	PROPN
cana-6282	3	20	abstract	abstract	NOUN
cana-6282	3	21	:	:	PUNCT
cana-6282	3	22	in	in	ADP
cana-6282	3	23	this	this	DET
cana-6282	3	24	paper	paper	NOUN
cana-6282	3	25	,	,	PUNCT
cana-6282	3	26	we	we	PRON
cana-6282	3	27	examine	examine	VERB
cana-6282	3	28	the	the	DET
cana-6282	3	29	hyperstability	hyperstability	NOUN
cana-6282	3	30	of	of	ADP
cana-6282	3	31	a	a	DET
cana-6282	3	32	general	general	ADJ
cana-6282	3	33	functional	functional	ADJ
cana-6282	3	34	equation	equation	NOUN
cana-6282	3	35	involving	involve	VERB
cana-6282	3	36	involutions	involution	NOUN
cana-6282	3	37	on	on	ADP
cana-6282	3	38	semigroups	semigroup	NOUN
cana-6282	3	39	,	,	PUNCT
cana-6282	3	40	where	where	SCONJ
cana-6282	3	41	𝑓	𝑓	DET
cana-6282	3	42	∶	∶	NOUN
cana-6282	3	43	𝑆2	𝑆2	ADV
cana-6282	3	44	→	→	PUNCT
cana-6282	3	45	𝑋	𝑋	PROPN
cana-6282	3	46	and	and	CCONJ
cana-6282	3	47	(	(	PUNCT
cana-6282	3	48	𝑆	𝑆	PROPN
cana-6282	3	49	,	,	PUNCT
cana-6282	3	50	·	·	PUNCT
cana-6282	3	51	)	)	PUNCT
cana-6282	3	52	is	be	AUX
cana-6282	3	53	an	an	DET
cana-6282	3	54	arbitrary	arbitrary	ADJ
cana-6282	3	55	semigroup	semigroup	NOUN
cana-6282	3	56	equipped	equip	VERB
cana-6282	3	57	with	with	ADP
cana-6282	3	58	involutive	involutive	ADJ
cana-6282	3	59	mappings	mapping	NOUN
cana-6282	3	60	𝜎	𝜎	PROPN
cana-6282	3	61	and	and	CCONJ
cana-6282	3	62	𝜏	𝜏	NOUN
cana-6282	3	63	.	.	PUNCT
cana-6282	4	1	the	the	DET
cana-6282	4	2	functional	functional	ADJ
cana-6282	4	3	equation	equation	NOUN
cana-6282	4	4	studied	study	VERB
cana-6282	4	5	in	in	ADP
cana-6282	4	6	this	this	DET
cana-6282	4	7	paper	paper	NOUN
cana-6282	4	8	emerges	emerge	VERB
cana-6282	4	9	naturally	naturally	ADV
cana-6282	4	10	from	from	ADP
cana-6282	4	11	the	the	DET
cana-6282	4	12	theory	theory	NOUN
cana-6282	4	13	of	of	ADP
cana-6282	4	14	additive	additive	ADJ
cana-6282	4	15	and	and	CCONJ
cana-6282	4	16	quadratic	quadratic	ADJ
cana-6282	4	17	equations	equation	NOUN
cana-6282	4	18	,	,	PUNCT
cana-6282	4	19	particularly	particularly	ADV
cana-6282	4	20	as	as	ADP
cana-6282	4	21	a	a	DET
cana-6282	4	22	hybrid	hybrid	ADJ
cana-6282	4	23	form	form	NOUN
cana-6282	4	24	combining	combine	VERB
cana-6282	4	25	structural	structural	ADJ
cana-6282	4	26	properties	property	NOUN
cana-6282	4	27	of	of	ADP
cana-6282	4	28	both	both	PRON
cana-6282	4	29	.	.	PUNCT
cana-6282	5	1	such	such	ADJ
cana-6282	5	2	equations	equation	NOUN
cana-6282	5	3	typically	typically	ADV
cana-6282	5	4	appear	appear	VERB
cana-6282	5	5	when	when	SCONJ
cana-6282	5	6	analyzing	analyze	VERB
cana-6282	5	7	mappings	mapping	NOUN
cana-6282	5	8	that	that	PRON
cana-6282	5	9	preserve	preserve	VERB
cana-6282	5	10	symmetrical	symmetrical	ADJ
cana-6282	5	11	or	or	CCONJ
cana-6282	5	12	involutive	involutive	ADJ
cana-6282	5	13	relationships	relationship	NOUN
cana-6282	5	14	within	within	ADP
cana-6282	5	15	algebraic	algebraic	ADJ
cana-6282	5	16	structures	structure	NOUN
cana-6282	5	17	,	,	PUNCT
cana-6282	5	18	especially	especially	ADV
cana-6282	5	19	in	in	ADP
cana-6282	5	20	semigroups	semigroup	NOUN
cana-6282	5	21	endowed	endow	VERB
cana-6282	5	22	with	with	ADP
cana-6282	5	23	additional	additional	ADJ
cana-6282	5	24	symmetries	symmetry	NOUN
cana-6282	5	25	.	.	PUNCT
cana-6282	6	1	by	by	ADP
cana-6282	6	2	employing	employ	VERB
cana-6282	6	3	a	a	DET
cana-6282	6	4	technique	technique	NOUN
cana-6282	6	5	inspired	inspire	VERB
cana-6282	6	6	by	by	ADP
cana-6282	6	7	maksa	maksa	ADJ
cana-6282	6	8	and	and	CCONJ
cana-6282	6	9	páles	pále	NOUN
cana-6282	6	10	,	,	PUNCT
cana-6282	6	11	we	we	PRON
cana-6282	6	12	derive	derive	VERB
cana-6282	6	13	sufficient	sufficient	ADJ
cana-6282	6	14	asymptotic	asymptotic	ADJ
cana-6282	6	15	conditions	condition	NOUN
cana-6282	6	16	ensuring	ensure	VERB
cana-6282	6	17	hyperstability	hyperstability	NOUN
cana-6282	6	18	.	.	PUNCT
cana-6282	7	1	additionally	additionally	ADV
cana-6282	7	2	,	,	PUNCT
cana-6282	7	3	we	we	PRON
cana-6282	7	4	extend	extend	VERB
cana-6282	7	5	the	the	DET
cana-6282	7	6	result	result	NOUN
cana-6282	7	7	to	to	ADP
cana-6282	7	8	an	an	DET
cana-6282	7	9	inhomogeneous	inhomogeneous	ADJ
cana-6282	7	10	variant	variant	NOUN
cana-6282	7	11	incorporating	incorporate	VERB
cana-6282	7	12	a	a	DET
cana-6282	7	13	perturbation	perturbation	NOUN
cana-6282	7	14	term	term	NOUN
cana-6282	7	15	𝐹	𝐹	PROPN
cana-6282	7	16	∶	∶	PROPN
cana-6282	7	17	𝑆4	𝑆4	PROPN
cana-6282	7	18	→	→	PUNCT
cana-6282	7	19	𝑋.	𝑋.	PROPN
cana-6282	7	20	these	these	DET
cana-6282	7	21	findings	finding	NOUN
cana-6282	7	22	contribute	contribute	VERB
cana-6282	7	23	to	to	ADP
cana-6282	7	24	the	the	DET
cana-6282	7	25	broader	broad	ADJ
cana-6282	7	26	understanding	understanding	NOUN
cana-6282	7	27	of	of	ADP
cana-6282	7	28	stability	stability	NOUN
cana-6282	7	29	phenomena	phenomenon	NOUN
cana-6282	7	30	in	in	ADP
cana-6282	7	31	functional	functional	ADJ
cana-6282	7	32	equations	equation	NOUN
cana-6282	7	33	on	on	ADP
cana-6282	7	34	algebraic	algebraic	ADJ
cana-6282	7	35	structures	structure	NOUN
cana-6282	7	36	.	.	PUNCT
cana-6282	8	1	keywords	keyword	NOUN
cana-6282	8	2	:	:	PUNCT
cana-6282	8	3	semigroups	semigroup	NOUN
cana-6282	8	4	,	,	PUNCT
cana-6282	8	5	involutions	involution	NOUN
cana-6282	8	6	,	,	PUNCT
cana-6282	8	7	stability	stability	NOUN
cana-6282	8	8	,	,	PUNCT
cana-6282	8	9	hyperstability	hyperstability	NOUN
cana-6282	8	10	,	,	PUNCT
cana-6282	8	11	functional	functional	ADJ
cana-6282	8	12	equations	equation	NOUN
cana-6282	8	13	.	.	PUNCT
cana-6282	9	1	mathematics	mathematic	NOUN
cana-6282	9	2	subject	subject	ADJ
cana-6282	9	3	classification	classification	NOUN
cana-6282	9	4	:	:	PUNCT
cana-6282	9	5	39b62	39b62	NUM
cana-6282	9	6	,	,	PUNCT
cana-6282	9	7	39b82	39b82	NUM
cana-6282	9	8	,	,	PUNCT
cana-6282	9	9	41a60	41a60	NUM
cana-6282	9	10	,	,	PUNCT
cana-6282	9	11	46b06	46b06	NUM
cana-6282	9	12	1	1	NUM
cana-6282	9	13	.	.	PUNCT
cana-6282	9	14	introduction	introduction	NOUN
cana-6282	9	15	one	one	NUM
cana-6282	9	16	important	important	ADJ
cana-6282	9	17	aspect	aspect	NOUN
cana-6282	9	18	of	of	ADP
cana-6282	9	19	functional	functional	ADJ
cana-6282	9	20	equations	equation	NOUN
cana-6282	9	21	is	be	AUX
cana-6282	9	22	their	their	PRON
cana-6282	9	23	stability	stability	NOUN
cana-6282	9	24	.	.	PUNCT
cana-6282	10	1	in	in	ADP
cana-6282	10	2	mathematics	mathematic	NOUN
cana-6282	10	3	,	,	PUNCT
cana-6282	10	4	the	the	DET
cana-6282	10	5	stability	stability	NOUN
cana-6282	10	6	of	of	ADP
cana-6282	10	7	an	an	DET
cana-6282	10	8	equation	equation	NOUN
cana-6282	10	9	refers	refer	VERB
cana-6282	10	10	to	to	ADP
cana-6282	10	11	the	the	DET
cana-6282	10	12	sensitivity	sensitivity	NOUN
cana-6282	10	13	of	of	ADP
cana-6282	10	14	its	its	PRON
cana-6282	10	15	solutions	solution	NOUN
cana-6282	10	16	to	to	ADP
cana-6282	10	17	small	small	ADJ
cana-6282	10	18	perturbations	perturbation	NOUN
cana-6282	10	19	in	in	ADP
cana-6282	10	20	the	the	DET
cana-6282	10	21	equation	equation	NOUN
cana-6282	10	22	itself	itself	PRON
cana-6282	10	23	.	.	PUNCT
cana-6282	11	1	in	in	ADP
cana-6282	11	2	the	the	DET
cana-6282	11	3	context	context	NOUN
cana-6282	11	4	of	of	ADP
cana-6282	11	5	functional	functional	ADJ
cana-6282	11	6	equations	equation	NOUN
cana-6282	11	7	,	,	PUNCT
cana-6282	11	8	stability	stability	NOUN
cana-6282	11	9	refers	refer	VERB
cana-6282	11	10	to	to	ADP
cana-6282	11	11	the	the	DET
cana-6282	11	12	extent	extent	NOUN
cana-6282	11	13	to	to	PART
cana-6282	11	14	which	which	PRON
cana-6282	11	15	the	the	DET
cana-6282	11	16	solutions	solution	NOUN
cana-6282	11	17	of	of	ADP
cana-6282	11	18	an	an	DET
cana-6282	11	19	equation	equation	NOUN
cana-6282	11	20	remain	remain	VERB
cana-6282	11	21	close	close	ADJ
cana-6282	11	22	to	to	ADP
cana-6282	11	23	each	each	DET
cana-6282	11	24	other	other	ADJ
cana-6282	11	25	under	under	ADP
cana-6282	11	26	small	small	ADJ
cana-6282	11	27	changes	change	NOUN
cana-6282	11	28	to	to	ADP
cana-6282	11	29	the	the	DET
cana-6282	11	30	equation	equation	NOUN
cana-6282	11	31	.	.	PUNCT
cana-6282	12	1	stability	stability	NOUN
cana-6282	12	2	is	be	AUX
cana-6282	12	3	an	an	DET
cana-6282	12	4	essential	essential	ADJ
cana-6282	12	5	concept	concept	NOUN
cana-6282	12	6	in	in	ADP
cana-6282	12	7	functional	functional	ADJ
cana-6282	12	8	analysis	analysis	NOUN
cana-6282	12	9	,	,	PUNCT
cana-6282	12	10	as	as	SCONJ
cana-6282	12	11	it	it	PRON
cana-6282	12	12	can	can	AUX
cana-6282	12	13	help	help	VERB
cana-6282	12	14	ensure	ensure	VERB
cana-6282	12	15	the	the	DET
cana-6282	12	16	existence	existence	NOUN
cana-6282	12	17	and	and	CCONJ
cana-6282	12	18	uniqueness	uniqueness	NOUN
cana-6282	12	19	of	of	ADP
cana-6282	12	20	solutions	solution	NOUN
cana-6282	12	21	to	to	ADP
cana-6282	12	22	functional	functional	ADJ
cana-6282	12	23	equations	equation	NOUN
cana-6282	12	24	.	.	PUNCT
cana-6282	13	1	for	for	ADP
cana-6282	13	2	example	example	NOUN
cana-6282	13	3	,	,	PUNCT
cana-6282	13	4	if	if	SCONJ
cana-6282	13	5	a	a	DET
cana-6282	13	6	functional	functional	ADJ
cana-6282	13	7	equation	equation	NOUN
cana-6282	13	8	is	be	AUX
cana-6282	13	9	stable	stable	ADJ
cana-6282	13	10	,	,	PUNCT
cana-6282	13	11	it	it	PRON
cana-6282	13	12	may	may	AUX
cana-6282	13	13	be	be	AUX
cana-6282	13	14	possible	possible	ADJ
cana-6282	13	15	to	to	PART
cana-6282	13	16	use	use	VERB
cana-6282	13	17	fixed	fix	VERB
cana-6282	13	18	-	-	PUNCT
cana-6282	13	19	point	point	NOUN
cana-6282	13	20	theorems	theorem	NOUN
cana-6282	13	21	to	to	PART
cana-6282	13	22	show	show	VERB
cana-6282	13	23	that	that	SCONJ
cana-6282	13	24	the	the	DET
cana-6282	13	25	equation	equation	NOUN
cana-6282	13	26	has	have	VERB
cana-6282	13	27	a	a	DET
cana-6282	13	28	unique	unique	ADJ
cana-6282	13	29	solution	solution	NOUN
cana-6282	13	30	.	.	PUNCT
cana-6282	14	1	the	the	DET
cana-6282	14	2	study	study	NOUN
cana-6282	14	3	of	of	ADP
cana-6282	14	4	the	the	DET
cana-6282	14	5	stability	stability	NOUN
cana-6282	14	6	of	of	ADP
cana-6282	14	7	functional	functional	ADJ
cana-6282	14	8	equations	equation	NOUN
cana-6282	14	9	is	be	AUX
cana-6282	14	10	an	an	DET
cana-6282	14	11	active	active	ADJ
cana-6282	14	12	area	area	NOUN
cana-6282	14	13	of	of	ADP
cana-6282	14	14	research	research	NOUN
cana-6282	14	15	in	in	ADP
cana-6282	14	16	mathematics	mathematic	NOUN
cana-6282	14	17	,	,	PUNCT
cana-6282	14	18	with	with	ADP
cana-6282	14	19	many	many	ADJ
cana-6282	14	20	open	open	ADJ
cana-6282	14	21	problems	problem	NOUN
cana-6282	14	22	and	and	CCONJ
cana-6282	14	23	challenges	challenge	NOUN
cana-6282	14	24	.	.	PUNCT
cana-6282	15	1	researchers	researcher	NOUN
cana-6282	15	2	are	be	AUX
cana-6282	15	3	interested	interested	ADJ
cana-6282	15	4	in	in	ADP
cana-6282	15	5	understanding	understand	VERB
cana-6282	15	6	the	the	DET
cana-6282	15	7	stability	stability	NOUN
cana-6282	15	8	properties	property	NOUN
cana-6282	15	9	of	of	ADP
cana-6282	15	10	different	different	ADJ
cana-6282	15	11	classes	class	NOUN
cana-6282	15	12	of	of	ADP
cana-6282	15	13	functional	functional	ADJ
cana-6282	15	14	equations	equation	NOUN
cana-6282	15	15	,	,	PUNCT
cana-6282	15	16	as	as	ADV
cana-6282	15	17	well	well	ADV
cana-6282	15	18	as	as	ADP
cana-6282	15	19	developing	develop	VERB
cana-6282	15	20	new	new	ADJ
cana-6282	15	21	techniques	technique	NOUN
cana-6282	15	22	and	and	CCONJ
cana-6282	15	23	methods	method	NOUN
cana-6282	15	24	for	for	ADP
cana-6282	15	25	analyzing	analyze	VERB
cana-6282	15	26	the	the	DET
cana-6282	15	27	stability	stability	NOUN
cana-6282	15	28	of	of	ADP
cana-6282	15	29	these	these	DET
cana-6282	15	30	equations	equation	NOUN
cana-6282	15	31	.	.	PUNCT
cana-6282	16	1	the	the	DET
cana-6282	16	2	idea	idea	NOUN
cana-6282	16	3	of	of	ADP
cana-6282	16	4	studying	study	VERB
cana-6282	16	5	the	the	DET
cana-6282	16	6	stability	stability	NOUN
cana-6282	16	7	of	of	ADP
cana-6282	16	8	functional	functional	ADJ
cana-6282	16	9	equations	equation	NOUN
cana-6282	16	10	emerged	emerge	VERB
cana-6282	16	11	in	in	ADP
cana-6282	16	12	1940	1940	NUM
cana-6282	16	13	when	when	SCONJ
cana-6282	16	14	the	the	DET
cana-6282	16	15	mathematician	mathematician	ADJ
cana-6282	16	16	s.	s.	PROPN
cana-6282	16	17	m.	m.	PROPN
cana-6282	16	18	ulam	ulam	PROPN
cana-6282	16	19	put	put	VERB
cana-6282	16	20	together	together	ADV
cana-6282	16	21	a	a	DET
cana-6282	16	22	list	list	NOUN
cana-6282	16	23	of	of	ADP
cana-6282	16	24	unsolved	unsolved	ADJ
cana-6282	16	25	problems	problem	NOUN
cana-6282	16	26	that	that	PRON
cana-6282	16	27	included	include	VERB
cana-6282	16	28	the	the	DET
cana-6282	16	29	following	follow	VERB
cana-6282	16	30	problem	problem	NOUN
cana-6282	16	31	:	:	PUNCT
cana-6282	17	1	mailto:essalih.ismaail@gmail.com	mailto:essalih.ismaail@gmail.com	X
cana-6282	17	2	mailto:n.bounader@live.fr	mailto:n.bounader@live.fr	PROPN
cana-6282	17	3	mailto:ahmed.maths78@gmail.com	mailto:ahmed.maths78@gmail.com	X
cana-6282	17	4	communications	communication	NOUN
cana-6282	17	5	on	on	ADP
cana-6282	17	6	applied	apply	VERB
cana-6282	17	7	nonlinear	nonlinear	ADJ
cana-6282	17	8	analysis	analysis	NOUN
cana-6282	17	9	issn	issn	NOUN
cana-6282	17	10	:	:	PUNCT
cana-6282	17	11	1074	1074	NUM
cana-6282	17	12	-	-	PUNCT
cana-6282	17	13	133x	133x	NUM
cana-6282	17	14	vol	vol	VERB
cana-6282	17	15	32	32	NUM
cana-6282	17	16	no	no	NOUN
cana-6282	17	17	.	.	PUNCT
cana-6282	18	1	10s	10	NOUN
cana-6282	18	2	(	(	PUNCT
cana-6282	18	3	2025	2025	NUM
cana-6282	18	4	)	)	PUNCT
cana-6282	18	5	3685	3685	NUM
cana-6282	18	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-6282	18	7	ulam	ulam	PROPN
cana-6282	18	8	’s	’s	PART
cana-6282	18	9	problem	problem	NOUN
cana-6282	18	10	:	:	PUNCT
cana-6282	19	1	[	[	X
cana-6282	19	2	17	17	NUM
cana-6282	19	3	]	]	X
cana-6282	19	4	let	let	ADJ
cana-6282	19	5	(	(	PUNCT
cana-6282	19	6	𝐺1,∗1	𝐺1,∗1	VERB
cana-6282	19	7	)	)	PUNCT
cana-6282	19	8	be	be	AUX
cana-6282	19	9	a	a	DET
cana-6282	19	10	group	group	NOUN
cana-6282	19	11	and	and	CCONJ
cana-6282	19	12	let	let	VERB
cana-6282	19	13	(	(	PUNCT
cana-6282	19	14	𝐺2,∗2	𝐺2,∗2	NOUN
cana-6282	19	15	)	)	PUNCT
cana-6282	19	16	be	be	AUX
cana-6282	19	17	a	a	DET
cana-6282	19	18	metric	metric	ADJ
cana-6282	19	19	group	group	NOUN
cana-6282	19	20	with	with	ADP
cana-6282	19	21	a	a	DET
cana-6282	19	22	metric	metric	ADJ
cana-6282	19	23	𝑑	𝑑	NOUN
cana-6282	19	24	(	(	PUNCT
cana-6282	19	25	.	.	PUNCT
cana-6282	19	26	,	,	PUNCT
cana-6282	19	27	.	.	PUNCT
cana-6282	19	28	)	)	PUNCT
cana-6282	19	29	.	.	PUNCT
cana-6282	20	1	𝐺𝑖𝑣𝑒𝑛휀	𝐺𝑖𝑣𝑒𝑛휀	PROPN
cana-6282	20	2	>	>	SYM
cana-6282	20	3	0	0	NUM
cana-6282	20	4	,	,	PUNCT
cana-6282	20	5	does	do	AUX
cana-6282	20	6	there	there	PRON
cana-6282	20	7	exists	exist	VERB
cana-6282	20	8	a	a	DET
cana-6282	20	9	𝛿	𝛿	PROPN
cana-6282	20	10	>	>	X
cana-6282	20	11	0	0	NUM
cana-6282	20	12	such	such	ADJ
cana-6282	20	13	that	that	SCONJ
cana-6282	20	14	if	if	SCONJ
cana-6282	20	15	a	a	DET
cana-6282	20	16	mapping	mapping	NOUN
cana-6282	20	17	ℎ	ℎ	NOUN
cana-6282	20	18	:	:	PUNCT
cana-6282	20	19	𝐺1	𝐺1	X
cana-6282	20	20	→	→	PUNCT
cana-6282	20	21	𝐺2	𝐺2	ADJ
cana-6282	20	22	satisfies	satisfy	VERB
cana-6282	20	23	the	the	DET
cana-6282	20	24	inequality	inequality	NOUN
cana-6282	20	25	𝑑(ℎ(𝑥	𝑑(ℎ(𝑥	PRON
cana-6282	20	26	∗1	∗1	PROPN
cana-6282	20	27	𝑦	𝑦	NOUN
cana-6282	20	28	)	)	PUNCT
cana-6282	20	29	,	,	PUNCT
cana-6282	20	30	ℎ(𝑥	ℎ(𝑥	NUM
cana-6282	20	31	)	)	PUNCT
cana-6282	20	32	∗2	∗2	NOUN
cana-6282	20	33	ℎ(𝑦	ℎ(𝑦	NOUN
cana-6282	20	34	)	)	PUNCT
cana-6282	20	35	)	)	PUNCT
cana-6282	21	1	<	<	X
cana-6282	21	2	휀	휀	X
cana-6282	21	3	for	for	ADP
cana-6282	21	4	all	all	PRON
cana-6282	21	5	𝑥	𝑥	PROPN
cana-6282	21	6	,	,	PUNCT
cana-6282	21	7	𝑦	𝑦	PROPN
cana-6282	21	8	∈	∈	NOUN
cana-6282	21	9	𝐺1	𝐺1	NOUN
cana-6282	21	10	,	,	PUNCT
cana-6282	21	11	then	then	ADV
cana-6282	21	12	there	there	PRON
cana-6282	21	13	exists	exist	VERB
cana-6282	21	14	a	a	DET
cana-6282	21	15	homomorphism	homomorphism	NOUN
cana-6282	21	16	𝐻	𝐻	NOUN
cana-6282	21	17	:	:	PUNCT
cana-6282	21	18	𝐺1	𝐺1	PROPN
cana-6282	21	19	→	→	PUNCT
cana-6282	21	20	𝐺2	𝐺2	NOUN
cana-6282	21	21	with	with	ADP
cana-6282	21	22	𝑑(ℎ(𝑥	𝑑(ℎ(𝑥	NUM
cana-6282	21	23	)	)	PUNCT
cana-6282	21	24	,	,	PUNCT
cana-6282	21	25	𝐻(𝑥	𝐻(𝑥	NOUN
cana-6282	21	26	)	)	PUNCT
cana-6282	21	27	)	)	PUNCT
cana-6282	22	1	<	<	X
cana-6282	22	2	𝛿	𝛿	X
cana-6282	22	3	for	for	ADP
cana-6282	22	4	all	all	PRON
cana-6282	22	5	𝑥	𝑥	DET
cana-6282	22	6	∈	∈	NOUN
cana-6282	22	7	𝐺1	𝐺1	NOUN
cana-6282	22	8	?	?	PUNCT
cana-6282	23	1	this	this	DET
cana-6282	23	2	question	question	NOUN
cana-6282	23	3	aroused	arouse	VERB
cana-6282	23	4	the	the	DET
cana-6282	23	5	attention	attention	NOUN
cana-6282	23	6	of	of	ADP
cana-6282	23	7	many	many	ADJ
cana-6282	23	8	mathematicians	mathematician	NOUN
cana-6282	23	9	.	.	PUNCT
cana-6282	24	1	almost	almost	ADV
cana-6282	24	2	a	a	PRON
cana-6282	24	3	year	year	NOUN
cana-6282	24	4	later	later	ADV
cana-6282	24	5	,	,	PUNCT
cana-6282	24	6	d.	d.	PROPN
cana-6282	24	7	h.	h.	PROPN
cana-6282	24	8	hayes	hayes	PROPN
cana-6282	24	9	published	publish	VERB
cana-6282	24	10	a	a	DET
cana-6282	24	11	paper	paper	NOUN
cana-6282	24	12	in	in	ADP
cana-6282	24	13	which	which	PRON
cana-6282	24	14	he	he	PRON
cana-6282	24	15	presented	present	VERB
cana-6282	24	16	an	an	DET
cana-6282	24	17	answer	answer	NOUN
cana-6282	24	18	to	to	ADP
cana-6282	24	19	ulam	ulam	PROPN
cana-6282	24	20	's	's	PART
cana-6282	24	21	problem	problem	NOUN
cana-6282	24	22	in	in	ADP
cana-6282	24	23	the	the	DET
cana-6282	24	24	case	case	NOUN
cana-6282	24	25	where	where	SCONJ
cana-6282	24	26	𝐺1	𝐺1	PROPN
cana-6282	24	27	and	and	CCONJ
cana-6282	24	28	𝐺2	𝐺2	NOUN
cana-6282	24	29	are	be	AUX
cana-6282	24	30	banach	banach	NOUN
cana-6282	24	31	spaces	space	NOUN
cana-6282	24	32	to	to	PART
cana-6282	24	33	investigate	investigate	VERB
cana-6282	24	34	the	the	DET
cana-6282	24	35	stability	stability	NOUN
cana-6282	24	36	of	of	ADP
cana-6282	24	37	cauchy	cauchy	ADJ
cana-6282	24	38	functional	functional	ADJ
cana-6282	24	39	equation	equation	NOUN
cana-6282	24	40	.	.	PUNCT
cana-6282	25	1	theorem	theorem	VERB
cana-6282	25	2	1.1	1.1	NUM
cana-6282	25	3	.	.	PUNCT
cana-6282	26	1	[	[	X
cana-6282	26	2	14	14	NUM
cana-6282	26	3	]	]	PUNCT
cana-6282	26	4	]	]	PUNCT
cana-6282	26	5	let	let	VERB
cana-6282	26	6	𝐸1	𝐸1	NOUN
cana-6282	26	7	and	and	CCONJ
cana-6282	26	8	𝐸2	𝐸2	ADJ
cana-6282	26	9	be	be	AUX
cana-6282	26	10	two	two	NUM
cana-6282	26	11	banach	banach	NOUN
cana-6282	26	12	spaces	space	NOUN
cana-6282	26	13	and	and	CCONJ
cana-6282	26	14	𝑓	𝑓	DET
cana-6282	26	15	:	:	PUNCT
cana-6282	26	16	𝐸1	𝐸1	PROPN
cana-6282	26	17	→	→	SYM
cana-6282	26	18	𝐸2	𝐸2	PROPN
cana-6282	26	19	be	be	AUX
cana-6282	26	20	a	a	DET
cana-6282	26	21	function	function	NOUN
cana-6282	26	22	such	such	ADJ
cana-6282	26	23	that	that	DET
cana-6282	26	24	‖𝑓(𝑥	‖𝑓(𝑥	NOUN
cana-6282	27	1	+	+	NUM
cana-6282	27	2	𝑦	𝑦	X
cana-6282	27	3	)	)	PUNCT
cana-6282	27	4	−	−	NOUN
cana-6282	27	5	𝑓(𝑥	𝑓(𝑥	NOUN
cana-6282	27	6	)	)	PUNCT
cana-6282	27	7	−	−	NUM
cana-6282	27	8	𝑓(𝑦)‖	𝑓(𝑦)‖	PROPN
cana-6282	27	9	≤	≤	ADJ
cana-6282	27	10	𝛿	𝛿	NOUN
cana-6282	27	11	for	for	ADP
cana-6282	27	12	some	some	DET
cana-6282	27	13	𝛿	𝛿	NOUN
cana-6282	27	14	>	>	X
cana-6282	27	15	0	0	PUNCT
cana-6282	27	16	and	and	CCONJ
cana-6282	27	17	for	for	ADP
cana-6282	27	18	all	all	DET
cana-6282	27	19	𝑥	𝑥	PROPN
cana-6282	27	20	,	,	PUNCT
cana-6282	27	21	𝑦	𝑦	PROPN
cana-6282	27	22	∈	∈	PROPN
cana-6282	27	23	𝐸1	𝐸1	NOUN
cana-6282	27	24	.	.	PUNCT
cana-6282	28	1	then	then	ADV
cana-6282	28	2	the	the	DET
cana-6282	28	3	limit	limit	NOUN
cana-6282	28	4	𝐴(𝑥	𝐴(𝑥	NOUN
cana-6282	28	5	):	):	PUNCT
cana-6282	28	6	=	=	PUNCT
cana-6282	28	7	lim	lim	PROPN
cana-6282	28	8	𝑛→∞	𝑛→∞	NUM
cana-6282	28	9	 	 	SPACE
cana-6282	28	10	2−𝑛𝑓(2𝑛𝑥	2−𝑛𝑓(2𝑛𝑥	NUM
cana-6282	28	11	)	)	PUNCT
cana-6282	28	12	exists	exist	VERB
cana-6282	28	13	for	for	ADP
cana-6282	28	14	each	each	DET
cana-6282	28	15	𝑥	𝑥	PROPN
cana-6282	28	16	∈	∈	PROPN
cana-6282	28	17	𝐸1	𝐸1	NOUN
cana-6282	28	18	,	,	PUNCT
cana-6282	28	19	and	and	CCONJ
cana-6282	28	20	𝐴	𝐴	PROPN
cana-6282	28	21	:	:	PUNCT
cana-6282	28	22	𝐸1	𝐸1	PROPN
cana-6282	28	23	→	→	SYM
cana-6282	28	24	𝐸2	𝐸2	PROPN
cana-6282	28	25	is	be	AUX
cana-6282	28	26	the	the	DET
cana-6282	28	27	unique	unique	ADJ
cana-6282	28	28	additive	additive	ADJ
cana-6282	28	29	function	function	NOUN
cana-6282	28	30	such	such	ADJ
cana-6282	28	31	that	that	PRON
cana-6282	28	32	‖𝑓(𝑥	‖𝑓(𝑥	NOUN
cana-6282	28	33	)	)	PUNCT
cana-6282	29	1	−	−	ADP
cana-6282	29	2	𝐴(𝑥)‖	𝐴(𝑥)‖	NOUN
cana-6282	29	3	≤	≤	VERB
cana-6282	29	4	𝛿	𝛿	NOUN
cana-6282	29	5	for	for	ADP
cana-6282	29	6	all	all	PRON
cana-6282	29	7	𝑥	𝑥	DET
cana-6282	29	8	∈	∈	PROPN
cana-6282	29	9	𝐸1	𝐸1	NOUN
cana-6282	29	10	.	.	PUNCT
cana-6282	30	1	moreover	moreover	ADV
cana-6282	30	2	,	,	PUNCT
cana-6282	30	3	if	if	SCONJ
cana-6282	30	4	𝑓(𝑡𝑥	𝑓(𝑡𝑥	NOUN
cana-6282	30	5	)	)	PUNCT
cana-6282	30	6	is	be	AUX
cana-6282	30	7	continuous	continuous	ADJ
cana-6282	30	8	in	in	ADP
cana-6282	30	9	𝑡	𝑡	PROPN
cana-6282	30	10	for	for	ADP
cana-6282	30	11	each	each	DET
cana-6282	30	12	fixed	fix	VERB
cana-6282	30	13	𝑥	𝑥	DET
cana-6282	30	14	∈	∈	PROPN
cana-6282	30	15	𝐸1	𝐸1	NOUN
cana-6282	30	16	,	,	PUNCT
cana-6282	30	17	then	then	ADV
cana-6282	30	18	the	the	DET
cana-6282	30	19	function	function	NOUN
cana-6282	30	20	𝐴	𝐴	PROPN
cana-6282	30	21	is	be	AUX
cana-6282	30	22	linear	linear	ADJ
cana-6282	30	23	.	.	PUNCT
cana-6282	31	1	overall	overall	ADV
cana-6282	31	2	,	,	PUNCT
cana-6282	31	3	hyers	hyer	NOUN
cana-6282	31	4	'	'	PART
cana-6282	31	5	contribution	contribution	NOUN
cana-6282	31	6	to	to	ADP
cana-6282	31	7	the	the	DET
cana-6282	31	8	theory	theory	NOUN
cana-6282	31	9	of	of	ADP
cana-6282	31	10	functional	functional	ADJ
cana-6282	31	11	equations	equation	NOUN
cana-6282	31	12	had	have	AUX
cana-6282	31	13	significantly	significantly	ADV
cana-6282	31	14	impacted	impact	VERB
cana-6282	31	15	mathematics	mathematic	NOUN
cana-6282	31	16	and	and	CCONJ
cana-6282	31	17	opened	open	VERB
cana-6282	31	18	up	up	ADP
cana-6282	31	19	new	new	ADJ
cana-6282	31	20	areas	area	NOUN
cana-6282	31	21	of	of	ADP
cana-6282	31	22	research	research	NOUN
cana-6282	31	23	in	in	ADP
cana-6282	31	24	this	this	DET
cana-6282	31	25	field	field	NOUN
cana-6282	31	26	,	,	PUNCT
cana-6282	31	27	and	and	CCONJ
cana-6282	31	28	it	it	PRON
cana-6282	31	29	was	be	AUX
cana-6282	31	30	only	only	ADV
cana-6282	31	31	the	the	DET
cana-6282	31	32	beginning	beginning	NOUN
cana-6282	31	33	of	of	ADP
cana-6282	31	34	a	a	DET
cana-6282	31	35	long	long	ADJ
cana-6282	31	36	list	list	NOUN
cana-6282	31	37	of	of	ADP
cana-6282	31	38	necessary	necessary	ADJ
cana-6282	31	39	studies	study	NOUN
cana-6282	31	40	and	and	CCONJ
cana-6282	31	41	results	result	NOUN
cana-6282	31	42	in	in	ADP
cana-6282	31	43	the	the	DET
cana-6282	31	44	stability	stability	NOUN
cana-6282	31	45	of	of	ADP
cana-6282	31	46	functional	functional	ADJ
cana-6282	31	47	equations	equation	NOUN
cana-6282	31	48	.	.	PUNCT
cana-6282	32	1	starting	start	VERB
cana-6282	32	2	with	with	ADP
cana-6282	32	3	d.	d.	PROPN
cana-6282	32	4	g.	g.	PROPN
cana-6282	32	5	bourgin	bourgin	PROPN
cana-6282	32	6	[	[	X
cana-6282	32	7	8	8	NUM
cana-6282	32	8	]	]	PUNCT
cana-6282	32	9	,	,	PUNCT
cana-6282	32	10	[	[	X
cana-6282	32	11	9	9	NUM
cana-6282	32	12	]	]	PUNCT
cana-6282	32	13	and	and	CCONJ
cana-6282	32	14	t.	t.	PROPN
cana-6282	32	15	aoki	aoki	PROPN
cana-6282	33	1	[	[	X
cana-6282	33	2	4	4	X
cana-6282	33	3	]	]	PUNCT
cana-6282	33	4	who	who	PRON
cana-6282	33	5	addressed	address	VERB
cana-6282	33	6	the	the	DET
cana-6282	33	7	stability	stability	NOUN
cana-6282	33	8	problem	problem	NOUN
cana-6282	33	9	with	with	ADP
cana-6282	33	10	unbounded	unbounded	ADJ
cana-6282	33	11	cauchy	cauchy	NOUN
cana-6282	33	12	differences	difference	NOUN
cana-6282	33	13	,	,	PUNCT
cana-6282	33	14	in	in	ADP
cana-6282	33	15	the	the	DET
cana-6282	33	16	scenario	scenario	NOUN
cana-6282	33	17	where	where	SCONJ
cana-6282	33	18	the	the	DET
cana-6282	33	19	relevant	relevant	ADJ
cana-6282	33	20	inequality	inequality	NOUN
cana-6282	33	21	is	be	AUX
cana-6282	33	22	not	not	PART
cana-6282	33	23	bounded	bound	VERB
cana-6282	33	24	,	,	PUNCT
cana-6282	33	25	th	th	X
cana-6282	33	26	.	.	PUNCT
cana-6282	33	27	m.	m.	NOUN
cana-6282	33	28	rassias	rassias	PROPN
cana-6282	34	1	[	[	X
cana-6282	34	2	16	16	NUM
cana-6282	34	3	]	]	X
cana-6282	34	4	generalized	generalize	VERB
cana-6282	34	5	hyers	hyer	NOUN
cana-6282	34	6	'	'	PART
cana-6282	34	7	theorem	theorem	NOUN
cana-6282	34	8	by	by	ADP
cana-6282	34	9	proving	prove	VERB
cana-6282	34	10	the	the	DET
cana-6282	34	11	existence	existence	NOUN
cana-6282	34	12	of	of	ADP
cana-6282	34	13	singular	singular	ADJ
cana-6282	34	14	linear	linear	PROPN
cana-6282	34	15	mappings	mapping	NOUN
cana-6282	34	16	close	close	ADJ
cana-6282	34	17	to	to	PART
cana-6282	34	18	approximate	approximate	ADJ
cana-6282	34	19	additive	additive	ADJ
cana-6282	34	20	mappings	mapping	NOUN
cana-6282	34	21	.	.	PUNCT
cana-6282	35	1	these	these	DET
cana-6282	35	2	results	result	NOUN
cana-6282	35	3	can	can	AUX
cana-6282	35	4	be	be	AUX
cana-6282	35	5	incorporated	incorporate	VERB
cana-6282	35	6	into	into	ADP
cana-6282	35	7	the	the	DET
cana-6282	35	8	following	follow	VERB
cana-6282	35	9	theorem	theorem	NOUN
cana-6282	35	10	.	.	PUNCT
cana-6282	35	11	theorem	theorem	PROPN
cana-6282	35	12	1.2	1.2	NUM
cana-6282	35	13	.	.	PUNCT
cana-6282	36	1	let	let	VERB
cana-6282	36	2	𝑋	𝑋	NOUN
cana-6282	36	3	and	and	CCONJ
cana-6282	36	4	𝑌	𝑌	PROPN
cana-6282	36	5	be	be	VERB
cana-6282	36	6	normed	normed	ADJ
cana-6282	36	7	space	space	NOUN
cana-6282	36	8	and	and	CCONJ
cana-6282	36	9	banach	banach	NOUN
cana-6282	36	10	space	space	NOUN
cana-6282	36	11	,	,	PUNCT
cana-6282	36	12	respectively	respectively	ADV
cana-6282	36	13	.	.	PUNCT
cana-6282	37	1	let	let	VERB
cana-6282	37	2	𝑐	𝑐	NOUN
cana-6282	37	3	and	and	CCONJ
cana-6282	37	4	𝑝	𝑝	PROPN
cana-6282	37	5	be	be	AUX
cana-6282	37	6	two	two	NUM
cana-6282	37	7	real	real	ADJ
cana-6282	37	8	numbers	number	NOUN
cana-6282	37	9	such	such	ADJ
cana-6282	37	10	that	that	SCONJ
cana-6282	37	11	𝑐	𝑐	PROPN
cana-6282	37	12	≥	≥	NUM
cana-6282	37	13	0	0	NUM
cana-6282	37	14	and	and	CCONJ
cana-6282	37	15	𝑝	𝑝	NOUN
cana-6282	37	16	≠	≠	PROPN
cana-6282	37	17	1	1	NUM
cana-6282	37	18	.	.	PUNCT
cana-6282	38	1	consider	consider	VERB
cana-6282	38	2	the	the	DET
cana-6282	38	3	operator	operator	NOUN
cana-6282	38	4	𝑓	𝑓	X
cana-6282	38	5	:	:	PUNCT
cana-6282	38	6	𝑋	𝑋	PROPN
cana-6282	38	7	→	→	SYM
cana-6282	38	8	𝑌	𝑌	PROPN
cana-6282	38	9	that	that	PRON
cana-6282	38	10	fulfills	fulfill	VERB
cana-6282	38	11	the	the	DET
cana-6282	38	12	inequality	inequality	NOUN
cana-6282	38	13	‖𝑓(𝑥	‖𝑓(𝑥	X
cana-6282	39	1	+	+	PUNCT
cana-6282	39	2	𝑦	𝑦	X
cana-6282	39	3	)	)	PUNCT
cana-6282	39	4	−	−	NOUN
cana-6282	39	5	𝑓(𝑥	𝑓(𝑥	NOUN
cana-6282	39	6	)	)	PUNCT
cana-6282	39	7	−	−	NUM
cana-6282	39	8	𝑓(𝑦)‖	𝑓(𝑦)‖	PROPN
cana-6282	39	9	≤	≤	PROPN
cana-6282	39	10	𝑐(‖𝑥‖𝑝	𝑐(‖𝑥‖𝑝	PROPN
cana-6282	39	11	+	+	CCONJ
cana-6282	39	12	‖𝑦‖𝑝	‖𝑦‖𝑝	NOUN
cana-6282	39	13	)	)	PUNCT
cana-6282	39	14	for	for	ADP
cana-6282	39	15	all	all	PRON
cana-6282	39	16	𝑥	𝑥	PROPN
cana-6282	39	17	,	,	PUNCT
cana-6282	39	18	𝑦	𝑦	NOUN
cana-6282	39	19	∈	∈	NOUN
cana-6282	39	20	𝑋	𝑋	NOUN
cana-6282	39	21	∖	∖	X
cana-6282	39	22	{	{	PUNCT
cana-6282	39	23	0	0	NUM
cana-6282	39	24	}	}	PUNCT
cana-6282	39	25	.	.	PUNCT
cana-6282	40	1	then	then	ADV
cana-6282	40	2	there	there	PRON
cana-6282	40	3	exists	exist	VERB
cana-6282	40	4	an	an	DET
cana-6282	40	5	additive	additive	ADJ
cana-6282	40	6	mapping	mapping	NOUN
cana-6282	40	7	𝑇	𝑇	PROPN
cana-6282	40	8	:	:	PUNCT
cana-6282	40	9	𝑋	𝑋	PROPN
cana-6282	40	10	→	→	SYM
cana-6282	40	11	𝑌	𝑌	PROPN
cana-6282	40	12	with	with	ADP
cana-6282	40	13	‖𝑓(𝑥	‖𝑓(𝑥	NOUN
cana-6282	40	14	)	)	PUNCT
cana-6282	40	15	−	−	ADP
cana-6282	40	16	𝑇(𝑥)‖	𝑇(𝑥)‖	ADJ
cana-6282	40	17	≤	≤	ADV
cana-6282	40	18	𝑐	𝑐	ADP
cana-6282	40	19	|1	|1	NUM
cana-6282	40	20	−	−	NUM
cana-6282	40	21	2𝑝−1|	2𝑝−1|	NOUN
cana-6282	40	22	‖𝑥‖𝑝	‖𝑥‖𝑝	NOUN
cana-6282	40	23	for	for	ADP
cana-6282	40	24	all	all	PRON
cana-6282	40	25	𝑥	𝑥	DET
cana-6282	40	26	∈	∈	NOUN
cana-6282	40	27	𝑋	𝑋	NOUN
cana-6282	40	28	∖	∖	PROPN
cana-6282	40	29	{	{	PUNCT
cana-6282	40	30	0	0	NUM
cana-6282	40	31	}	}	PUNCT
cana-6282	40	32	.	.	PUNCT
cana-6282	41	1	in	in	ADP
cana-6282	41	2	the	the	DET
cana-6282	41	3	spirit	spirit	NOUN
cana-6282	41	4	of	of	ADP
cana-6282	41	5	rassias	rassias	PROPN
cana-6282	41	6	'	'	PART
cana-6282	41	7	approach	approach	NOUN
cana-6282	41	8	,	,	PUNCT
cana-6282	41	9	g.	g.	PROPN
cana-6282	41	10	l.	l.	PROPN
cana-6282	41	11	forti	forti	PROPN
cana-6282	42	1	[	[	X
cana-6282	42	2	12	12	NUM
cana-6282	42	3	]	]	PUNCT
cana-6282	42	4	and	and	CCONJ
cana-6282	42	5	p.	p.	NOUN
cana-6282	42	6	găvruţă	găvruţă	NOUN
cana-6282	42	7	[	[	X
cana-6282	42	8	13	13	NUM
cana-6282	42	9	]	]	PUNCT
cana-6282	42	10	generalized	generalize	VERB
cana-6282	42	11	all	all	PRON
cana-6282	42	12	of	of	ADP
cana-6282	42	13	the	the	DET
cana-6282	42	14	aforementioned	aforementioned	ADJ
cana-6282	42	15	stability	stability	NOUN
cana-6282	42	16	results	result	NOUN
cana-6282	42	17	by	by	ADP
cana-6282	42	18	swapping	swap	VERB
cana-6282	42	19	out	out	ADP
cana-6282	42	20	the	the	DET
cana-6282	42	21	cauchy	cauchy	ADJ
cana-6282	42	22	differences	difference	NOUN
cana-6282	42	23	for	for	ADP
cana-6282	42	24	a	a	DET
cana-6282	42	25	control	control	NOUN
cana-6282	42	26	function	function	NOUN
cana-6282	42	27	𝜑.	𝜑.	VERB
cana-6282	42	28	at	at	ADP
cana-6282	42	29	the	the	DET
cana-6282	42	30	same	same	ADJ
cana-6282	42	31	time	time	NOUN
cana-6282	42	32	,	,	PUNCT
cana-6282	42	33	a	a	DET
cana-6282	42	34	special	special	ADJ
cana-6282	42	35	kind	kind	NOUN
cana-6282	42	36	of	of	ADP
cana-6282	42	37	stability	stability	NOUN
cana-6282	42	38	known	know	VERB
cana-6282	42	39	as	as	ADP
cana-6282	42	40	hyperstability	hyperstability	NOUN
cana-6282	42	41	emerged	emerge	VERB
cana-6282	42	42	.	.	PUNCT
cana-6282	43	1	the	the	DET
cana-6282	43	2	hyperstability	hyperstability	NOUN
cana-6282	43	3	of	of	ADP
cana-6282	43	4	communications	communication	NOUN
cana-6282	43	5	on	on	ADP
cana-6282	43	6	applied	apply	VERB
cana-6282	43	7	nonlinear	nonlinear	ADJ
cana-6282	43	8	analysis	analysis	NOUN
cana-6282	43	9	issn	issn	NOUN
cana-6282	43	10	:	:	PUNCT
cana-6282	43	11	1074	1074	NUM
cana-6282	43	12	-	-	PUNCT
cana-6282	43	13	133x	133x	NUM
cana-6282	43	14	vol	vol	VERB
cana-6282	43	15	32	32	NUM
cana-6282	43	16	no	no	NOUN
cana-6282	43	17	.	.	PUNCT
cana-6282	44	1	10s	10	NOUN
cana-6282	44	2	(	(	PUNCT
cana-6282	44	3	2025	2025	NUM
cana-6282	44	4	)	)	PUNCT
cana-6282	44	5	3686	3686	NUM
cana-6282	44	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-6282	44	7	functional	functional	ADJ
cana-6282	44	8	equations	equation	NOUN
cana-6282	44	9	refers	refer	VERB
cana-6282	44	10	to	to	ADP
cana-6282	44	11	the	the	DET
cana-6282	44	12	property	property	NOUN
cana-6282	44	13	that	that	DET
cana-6282	44	14	small	small	ADJ
cana-6282	44	15	perturbations	perturbation	NOUN
cana-6282	44	16	of	of	ADP
cana-6282	44	17	a	a	DET
cana-6282	44	18	given	give	VERB
cana-6282	44	19	functional	functional	ADJ
cana-6282	44	20	equation	equation	NOUN
cana-6282	44	21	lead	lead	NOUN
cana-6282	44	22	to	to	ADP
cana-6282	44	23	solutions	solution	NOUN
cana-6282	44	24	that	that	PRON
cana-6282	44	25	are	be	AUX
cana-6282	44	26	close	close	ADJ
cana-6282	44	27	to	to	ADP
cana-6282	44	28	the	the	DET
cana-6282	44	29	original	original	ADJ
cana-6282	44	30	one	one	NOUN
cana-6282	44	31	.	.	PUNCT
cana-6282	45	1	in	in	ADP
cana-6282	45	2	other	other	ADJ
cana-6282	45	3	words	word	NOUN
cana-6282	45	4	,	,	PUNCT
cana-6282	45	5	if	if	SCONJ
cana-6282	45	6	a	a	DET
cana-6282	45	7	function	function	NOUN
cana-6282	45	8	satisfies	satisfy	VERB
cana-6282	45	9	a	a	DET
cana-6282	45	10	functional	functional	ADJ
cana-6282	45	11	equation	equation	NOUN
cana-6282	45	12	,	,	PUNCT
cana-6282	45	13	then	then	ADV
cana-6282	45	14	nearby	nearby	ADJ
cana-6282	45	15	functions	function	NOUN
cana-6282	45	16	also	also	ADV
cana-6282	45	17	satisfy	satisfy	VERB
cana-6282	45	18	the	the	DET
cana-6282	45	19	same	same	ADJ
cana-6282	45	20	equation	equation	NOUN
cana-6282	45	21	.	.	PUNCT
cana-6282	46	1	the	the	DET
cana-6282	46	2	following	follow	VERB
cana-6282	46	3	definition	definition	NOUN
cana-6282	46	4	helps	help	VERB
cana-6282	46	5	us	we	PRON
cana-6282	46	6	distinguish	distinguish	VERB
cana-6282	46	7	between	between	ADP
cana-6282	46	8	three	three	NUM
cana-6282	46	9	key	key	ADJ
cana-6282	46	10	ideas	idea	NOUN
cana-6282	46	11	in	in	ADP
cana-6282	46	12	the	the	DET
cana-6282	46	13	theory	theory	NOUN
cana-6282	46	14	of	of	ADP
cana-6282	46	15	the	the	DET
cana-6282	46	16	stability	stability	NOUN
cana-6282	46	17	of	of	ADP
cana-6282	46	18	functional	functional	ADJ
cana-6282	46	19	equations	equation	NOUN
cana-6282	46	20	.	.	PUNCT
cana-6282	47	1	definition	definition	NOUN
cana-6282	47	2	1.3	1.3	NUM
cana-6282	47	3	.	.	PUNCT
cana-6282	48	1	let	let	VERB
cana-6282	48	2	𝑋	𝑋	NOUN
cana-6282	48	3	be	be	AUX
cana-6282	48	4	a	a	DET
cana-6282	48	5	nonempty	nonempty	ADJ
cana-6282	48	6	set	set	NOUN
cana-6282	48	7	,	,	PUNCT
cana-6282	48	8	(	(	PUNCT
cana-6282	48	9	𝑌	𝑌	PROPN
cana-6282	48	10	,	,	PUNCT
cana-6282	48	11	𝑑	𝑑	NOUN
cana-6282	48	12	)	)	PUNCT
cana-6282	48	13	be	be	AUX
cana-6282	48	14	a	a	DET
cana-6282	48	15	metric	metric	ADJ
cana-6282	48	16	space	space	NOUN
cana-6282	48	17	,	,	PUNCT
cana-6282	48	18	and	and	CCONJ
cana-6282	48	19	𝜑	𝜑	X
cana-6282	48	20	:	:	PUNCT
cana-6282	48	21	𝑋	𝑋	NOUN
cana-6282	48	22	×	×	NOUN
cana-6282	48	23	𝑋	𝑋	PROPN
cana-6282	48	24	⟶	⟶	NOUN
cana-6282	48	25	[	[	X
cana-6282	48	26	0	0	NUM
cana-6282	48	27	,	,	PUNCT
cana-6282	48	28	∞	∞	PROPN
cana-6282	48	29	)	)	PUNCT
cana-6282	48	30	and	and	CCONJ
cana-6282	48	31	𝜓	𝜓	NOUN
cana-6282	48	32	:	:	PUNCT
cana-6282	48	33	𝑋	𝑋	NOUN
cana-6282	48	34	⟶	⟶	NOUN
cana-6282	48	35	[	[	X
cana-6282	48	36	0	0	NUM
cana-6282	48	37	,	,	PUNCT
cana-6282	48	38	∞	∞	PROPN
cana-6282	48	39	)	)	PUNCT
cana-6282	48	40	be	be	VERB
cana-6282	48	41	two	two	NUM
cana-6282	48	42	functions	function	NOUN
cana-6282	48	43	.	.	PUNCT
cana-6282	49	1	let	let	VERB
cana-6282	49	2	𝑔𝑖	𝑔𝑖	VERB
cana-6282	49	3	:	:	PUNCT
cana-6282	49	4	𝑋	𝑋	PROPN
cana-6282	49	5	×	×	PROPN
cana-6282	49	6	𝑋	𝑋	PROPN
cana-6282	49	7	⟶	⟶	NOUN
cana-6282	49	8	𝑋	𝑋	NOUN
cana-6282	49	9	for	for	ADP
cana-6282	49	10	𝑖	𝑖	ADP
cana-6282	49	11	∈	∈	PROPN
cana-6282	49	12	{	{	PUNCT
cana-6282	49	13	1	1	NUM
cana-6282	49	14	,	,	PUNCT
cana-6282	49	15	…	…	PUNCT
cana-6282	49	16	,	,	PUNCT
cana-6282	49	17	4	4	X
cana-6282	49	18	}	}	PUNCT
cana-6282	49	19	and	and	CCONJ
cana-6282	49	20	𝐹1	𝐹1	NOUN
cana-6282	49	21	,	,	PUNCT
cana-6282	49	22	𝐹2	𝐹2	NOUN
cana-6282	49	23	:	:	PUNCT
cana-6282	49	24	𝑌	𝑌	PROPN
cana-6282	49	25	×	×	NOUN
cana-6282	49	26	𝑌	𝑌	PROPN
cana-6282	49	27	⟶	⟶	NOUN
cana-6282	49	28	𝑌.	𝑌.	PROPN
cana-6282	49	29	then	then	ADV
cana-6282	49	30	:	:	PUNCT
cana-6282	49	31	(	(	PUNCT
cana-6282	49	32	1	1	X
cana-6282	49	33	)	)	PUNCT
cana-6282	49	34	the	the	DET
cana-6282	49	35	following	follow	VERB
cana-6282	49	36	equation	equation	NOUN
cana-6282	49	37	𝐹1(𝑓(𝑔1(𝑥	𝐹1(𝑓(𝑔1(𝑥	NUM
cana-6282	49	38	,	,	PUNCT
cana-6282	49	39	𝑦	𝑦	NOUN
cana-6282	49	40	)	)	PUNCT
cana-6282	49	41	)	)	PUNCT
cana-6282	49	42	,	,	PUNCT
cana-6282	49	43	𝑓(𝑔2(𝑥	𝑓(𝑔2(𝑥	PROPN
cana-6282	49	44	,	,	PUNCT
cana-6282	49	45	𝑦	𝑦	NOUN
cana-6282	49	46	)	)	PUNCT
cana-6282	49	47	)	)	PUNCT
cana-6282	49	48	)	)	PUNCT
cana-6282	50	1	=	=	PUNCT
cana-6282	50	2	𝐹2(𝑓(𝑔3(𝑥	𝐹2(𝑓(𝑔3(𝑥	PROPN
cana-6282	50	3	,	,	PUNCT
cana-6282	50	4	𝑦	𝑦	NOUN
cana-6282	50	5	)	)	PUNCT
cana-6282	50	6	)	)	PUNCT
cana-6282	50	7	,	,	PUNCT
cana-6282	50	8	𝑓(𝑔4(𝑥	𝑓(𝑔4(𝑥	PROPN
cana-6282	50	9	,	,	PUNCT
cana-6282	50	10	𝑦	𝑦	NOUN
cana-6282	50	11	)	)	PUNCT
cana-6282	50	12	)	)	PUNCT
cana-6282	50	13	)	)	PUNCT
cana-6282	50	14	(	(	PUNCT
cana-6282	50	15	1.1	1.1	NUM
cana-6282	50	16	)	)	PUNCT
cana-6282	50	17	is	be	AUX
cana-6282	50	18	called	call	VERB
cana-6282	50	19	a	a	DET
cana-6282	50	20	functional	functional	ADJ
cana-6282	50	21	equation	equation	NOUN
cana-6282	50	22	that	that	PRON
cana-6282	50	23	specifies	specify	VERB
cana-6282	50	24	the	the	DET
cana-6282	50	25	unknown	unknown	ADJ
cana-6282	50	26	function	function	NOUN
cana-6282	50	27	𝑓	𝑓	NOUN
cana-6282	50	28	:	:	PUNCT
cana-6282	50	29	𝑋	𝑋	PROPN
cana-6282	50	30	⟶	⟶	NOUN
cana-6282	50	31	𝑌.	𝑌.	PROPN
cana-6282	50	32	(	(	PUNCT
cana-6282	50	33	2	2	NUM
cana-6282	50	34	)	)	PUNCT
cana-6282	50	35	if	if	SCONJ
cana-6282	50	36	for	for	ADP
cana-6282	50	37	every	every	DET
cana-6282	50	38	function	function	NOUN
cana-6282	50	39	𝑓	𝑓	NOUN
cana-6282	50	40	:	:	PUNCT
cana-6282	50	41	𝑋	𝑋	PROPN
cana-6282	50	42	⟶	⟶	NOUN
cana-6282	50	43	𝑌	𝑌	PROPN
cana-6282	50	44	satisfying	satisfy	VERB
cana-6282	50	45	the	the	DET
cana-6282	50	46	inequality	inequality	NOUN
cana-6282	50	47	𝑑	𝑑	PROPN
cana-6282	50	48	(	(	PUNCT
cana-6282	50	49	𝐹1(𝑓(𝑔1(𝑥	𝐹1(𝑓(𝑔1(𝑥	PROPN
cana-6282	50	50	,	,	PUNCT
cana-6282	50	51	𝑦	𝑦	NOUN
cana-6282	50	52	)	)	PUNCT
cana-6282	50	53	)	)	PUNCT
cana-6282	50	54	,	,	PUNCT
cana-6282	50	55	𝑓(𝑔2(𝑥	𝑓(𝑔2(𝑥	PROPN
cana-6282	50	56	,	,	PUNCT
cana-6282	50	57	𝑦	𝑦	NOUN
cana-6282	50	58	)	)	PUNCT
cana-6282	50	59	)	)	PUNCT
cana-6282	50	60	)	)	PUNCT
cana-6282	50	61	,	,	PUNCT
cana-6282	50	62	𝐺2(𝑓(𝑔3(𝑥	𝐺2(𝑓(𝑔3(𝑥	PROPN
cana-6282	50	63	,	,	PUNCT
cana-6282	50	64	𝑦	𝑦	NOUN
cana-6282	50	65	)	)	PUNCT
cana-6282	50	66	)	)	PUNCT
cana-6282	50	67	,	,	PUNCT
cana-6282	50	68	𝑓(𝑔4(𝑥	𝑓(𝑔4(𝑥	PROPN
cana-6282	50	69	,	,	PUNCT
cana-6282	50	70	𝑦	𝑦	NOUN
cana-6282	50	71	)	)	PUNCT
cana-6282	50	72	)	)	PUNCT
cana-6282	50	73	)	)	PUNCT
cana-6282	50	74	)	)	PUNCT
cana-6282	50	75	≤	≤	NUM
cana-6282	50	76	𝜑(𝑥	𝜑(𝑥	NOUN
cana-6282	50	77	,	,	PUNCT
cana-6282	50	78	𝑦	𝑦	NOUN
cana-6282	50	79	)	)	PUNCT
cana-6282	50	80	(	(	PUNCT
cana-6282	50	81	1.2	1.2	NUM
cana-6282	50	82	)	)	PUNCT
cana-6282	50	83	for	for	ADP
cana-6282	50	84	all	all	PRON
cana-6282	50	85	𝑥	𝑥	PROPN
cana-6282	50	86	,	,	PUNCT
cana-6282	50	87	𝑦	𝑦	NOUN
cana-6282	50	88	∈	∈	PROPN
cana-6282	50	89	𝑋	𝑋	NOUN
cana-6282	50	90	,	,	PUNCT
cana-6282	50	91	there	there	PRON
cana-6282	50	92	exists	exist	VERB
cana-6282	50	93	a	a	DET
cana-6282	50	94	function	function	NOUN
cana-6282	50	95	𝑇	𝑇	PROPN
cana-6282	50	96	:	:	PUNCT
cana-6282	50	97	𝑋	𝑋	PROPN
cana-6282	50	98	→	→	SYM
cana-6282	50	99	𝑌	𝑌	PROPN
cana-6282	50	100	that	that	PRON
cana-6282	50	101	satisfies	satisfy	VERB
cana-6282	50	102	the	the	DET
cana-6282	50	103	equation	equation	NOUN
cana-6282	50	104	(	(	PUNCT
cana-6282	50	105	1.1	1.1	NUM
cana-6282	50	106	)	)	PUNCT
cana-6282	50	107	such	such	ADJ
cana-6282	50	108	that	that	DET
cana-6282	50	109	𝑑(𝑓(𝑥	𝑑(𝑓(𝑥	NOUN
cana-6282	50	110	)	)	PUNCT
cana-6282	50	111	,	,	PUNCT
cana-6282	50	112	𝑇(𝑥	𝑇(𝑥	NOUN
cana-6282	50	113	)	)	PUNCT
cana-6282	50	114	)	)	PUNCT
cana-6282	50	115	≤	≤	PUNCT
cana-6282	50	116	𝜓(𝑥	𝜓(𝑥	ADV
cana-6282	50	117	)	)	PUNCT
cana-6282	50	118	(	(	PUNCT
cana-6282	50	119	1.3	1.3	NUM
cana-6282	50	120	)	)	PUNCT
cana-6282	50	121	for	for	ADP
cana-6282	50	122	all	all	PRON
cana-6282	50	123	𝑥	𝑥	DET
cana-6282	50	124	∈	∈	PROPN
cana-6282	50	125	𝑋	𝑋	PROPN
cana-6282	50	126	,	,	PUNCT
cana-6282	50	127	then	then	ADV
cana-6282	50	128	we	we	PRON
cana-6282	50	129	say	say	VERB
cana-6282	50	130	that	that	SCONJ
cana-6282	50	131	the	the	DET
cana-6282	50	132	functional	functional	ADJ
cana-6282	50	133	equation	equation	NOUN
cana-6282	50	134	(	(	PUNCT
cana-6282	50	135	1.1	1.1	NUM
cana-6282	50	136	)	)	PUNCT
cana-6282	50	137	is	be	AUX
cana-6282	50	138	generalized	generalize	VERB
cana-6282	50	139	hyers	hyer	NOUN
cana-6282	50	140	-	-	PUNCT
cana-6282	50	141	ulam	ulam	ADJ
cana-6282	50	142	-	-	PUNCT
cana-6282	50	143	rassias	rassias	NOUN
cana-6282	50	144	stable	stable	ADJ
cana-6282	50	145	on	on	ADP
cana-6282	50	146	(	(	PUNCT
cana-6282	50	147	𝑋	𝑋	PROPN
cana-6282	50	148	,	,	PUNCT
cana-6282	50	149	𝑌	𝑌	PROPN
cana-6282	50	150	)	)	PUNCT
cana-6282	50	151	with	with	ADP
cana-6282	50	152	control	control	NOUN
cana-6282	50	153	functions	function	NOUN
cana-6282	50	154	𝜑	𝜑	NOUN
cana-6282	50	155	and	and	CCONJ
cana-6282	50	156	𝜓.	𝜓.	PROPN
cana-6282	50	157	when	when	SCONJ
cana-6282	50	158	𝜑(𝑥	𝜑(𝑥	NOUN
cana-6282	50	159	,	,	PUNCT
cana-6282	50	160	𝑦	𝑦	NOUN
cana-6282	50	161	)	)	PUNCT
cana-6282	50	162	in	in	ADP
cana-6282	50	163	(	(	PUNCT
cana-6282	50	164	1.2	1.2	NUM
cana-6282	50	165	)	)	PUNCT
cana-6282	50	166	and	and	CCONJ
cana-6282	50	167	𝜓(𝑥	𝜓(𝑥	ADV
cana-6282	50	168	)	)	PUNCT
cana-6282	50	169	in	in	ADP
cana-6282	50	170	(	(	PUNCT
cana-6282	50	171	1.3	1.3	NUM
cana-6282	50	172	)	)	PUNCT
cana-6282	50	173	are	be	AUX
cana-6282	50	174	replaced	replace	VERB
cana-6282	50	175	by	by	ADP
cana-6282	50	176	the	the	DET
cana-6282	50	177	reals	real	NOUN
cana-6282	50	178	𝛿	𝛿	PROPN
cana-6282	50	179	>	>	X
cana-6282	50	180	0	0	PUNCT
cana-6282	50	181	and	and	CCONJ
cana-6282	50	182	휀	휀	ADJ
cana-6282	50	183	>	>	X
cana-6282	50	184	0	0	NUM
cana-6282	50	185	respectively	respectively	ADV
cana-6282	50	186	,	,	PUNCT
cana-6282	50	187	then	then	ADV
cana-6282	50	188	we	we	PRON
cana-6282	50	189	say	say	VERB
cana-6282	50	190	that	that	SCONJ
cana-6282	50	191	corresponding	correspond	VERB
cana-6282	50	192	phenomenon	phenomenon	NOUN
cana-6282	50	193	of	of	ADP
cana-6282	50	194	the	the	DET
cana-6282	50	195	functional	functional	ADJ
cana-6282	50	196	equation	equation	NOUN
cana-6282	50	197	(	(	PUNCT
cana-6282	50	198	1.1	1.1	NUM
cana-6282	50	199	)	)	PUNCT
cana-6282	50	200	is	be	AUX
cana-6282	50	201	hyers	hyer	NOUN
cana-6282	50	202	-	-	PUNCT
cana-6282	50	203	ulam	ulam	X
cana-6282	50	204	stable	stable	ADJ
cana-6282	50	205	on	on	ADP
cana-6282	50	206	(	(	PUNCT
cana-6282	50	207	𝑋	𝑋	PROPN
cana-6282	50	208	,	,	PUNCT
cana-6282	50	209	𝑌	𝑌	PROPN
cana-6282	50	210	)	)	PUNCT
cana-6282	50	211	with	with	ADP
cana-6282	50	212	controls	control	NOUN
cana-6282	50	213	휀	휀	NOUN
cana-6282	50	214	and	and	CCONJ
cana-6282	50	215	𝛿.	𝛿.	ADJ
cana-6282	50	216	(	(	PUNCT
cana-6282	50	217	3	3	X
cana-6282	50	218	)	)	PUNCT
cana-6282	50	219	if	if	SCONJ
cana-6282	50	220	for	for	ADP
cana-6282	50	221	every	every	DET
cana-6282	50	222	function	function	NOUN
cana-6282	50	223	𝑓	𝑓	NOUN
cana-6282	50	224	:	:	PUNCT
cana-6282	50	225	𝑋	𝑋	PROPN
cana-6282	50	226	⟶	⟶	NOUN
cana-6282	50	227	𝑌	𝑌	PROPN
cana-6282	50	228	satisfying	satisfy	VERB
cana-6282	50	229	the	the	DET
cana-6282	50	230	inequality	inequality	NOUN
cana-6282	50	231	(	(	PUNCT
cana-6282	50	232	1.2	1.2	NUM
cana-6282	50	233	)	)	PUNCT
cana-6282	50	234	,	,	PUNCT
cana-6282	50	235	and	and	CCONJ
cana-6282	50	236	either	either	CCONJ
cana-6282	50	237	𝑓	𝑓	PRON
cana-6282	50	238	is	be	AUX
cana-6282	50	239	bounded	bound	VERB
cana-6282	50	240	or	or	CCONJ
cana-6282	50	241	it	it	PRON
cana-6282	50	242	is	be	AUX
cana-6282	50	243	a	a	DET
cana-6282	50	244	solution	solution	NOUN
cana-6282	50	245	to	to	ADP
cana-6282	50	246	equation	equation	NOUN
cana-6282	50	247	(	(	PUNCT
cana-6282	50	248	1.1	1.1	NUM
cana-6282	50	249	)	)	PUNCT
cana-6282	50	250	,	,	PUNCT
cana-6282	50	251	then	then	ADV
cana-6282	50	252	we	we	PRON
cana-6282	50	253	say	say	VERB
cana-6282	50	254	that	that	SCONJ
cana-6282	50	255	the	the	DET
cana-6282	50	256	functional	functional	ADJ
cana-6282	50	257	equation	equation	NOUN
cana-6282	50	258	(	(	PUNCT
cana-6282	50	259	1.1	1.1	NUM
cana-6282	50	260	)	)	PUNCT
cana-6282	50	261	is	be	AUX
cana-6282	50	262	superstable	superstable	ADJ
cana-6282	50	263	on	on	ADP
cana-6282	50	264	(	(	PUNCT
cana-6282	50	265	𝑋	𝑋	PROPN
cana-6282	50	266	,	,	PUNCT
cana-6282	50	267	𝑌	𝑌	PROPN
cana-6282	50	268	)	)	PUNCT
cana-6282	50	269	with	with	ADP
cana-6282	50	270	control	control	NOUN
cana-6282	50	271	function	function	NOUN
cana-6282	50	272	𝜑.	𝜑.	NOUN
cana-6282	50	273	(	(	PUNCT
cana-6282	50	274	4	4	X
cana-6282	50	275	)	)	PUNCT
cana-6282	50	276	if	if	SCONJ
cana-6282	50	277	for	for	ADP
cana-6282	50	278	every	every	DET
cana-6282	50	279	function	function	NOUN
cana-6282	50	280	𝑓	𝑓	NOUN
cana-6282	50	281	:	:	PUNCT
cana-6282	50	282	𝑋	𝑋	PROPN
cana-6282	50	283	⟶	⟶	NOUN
cana-6282	50	284	𝑌	𝑌	PROPN
cana-6282	50	285	satisfying	satisfy	VERB
cana-6282	50	286	the	the	DET
cana-6282	50	287	inequality	inequality	NOUN
cana-6282	50	288	(	(	PUNCT
cana-6282	50	289	1.2	1.2	NUM
cana-6282	50	290	)	)	PUNCT
cana-6282	50	291	and	and	CCONJ
cana-6282	50	292	𝑓	𝑓	PRON
cana-6282	50	293	is	be	AUX
cana-6282	50	294	a	a	DET
cana-6282	50	295	solution	solution	NOUN
cana-6282	50	296	of	of	ADP
cana-6282	50	297	equation	equation	NOUN
cana-6282	50	298	(	(	PUNCT
cana-6282	50	299	1.1	1.1	NUM
cana-6282	50	300	)	)	PUNCT
cana-6282	50	301	,	,	PUNCT
cana-6282	50	302	then	then	ADV
cana-6282	50	303	we	we	PRON
cana-6282	50	304	say	say	VERB
cana-6282	50	305	that	that	SCONJ
cana-6282	50	306	the	the	DET
cana-6282	50	307	functional	functional	ADJ
cana-6282	50	308	equation	equation	NOUN
cana-6282	50	309	(	(	PUNCT
cana-6282	50	310	1.1	1.1	NUM
cana-6282	50	311	)	)	PUNCT
cana-6282	50	312	is	be	AUX
cana-6282	50	313	hyperstable	hyperstable	ADJ
cana-6282	50	314	on	on	ADP
cana-6282	50	315	(	(	PUNCT
cana-6282	50	316	𝑋	𝑋	PROPN
cana-6282	50	317	,	,	PUNCT
cana-6282	50	318	𝑌	𝑌	PROPN
cana-6282	50	319	)	)	PUNCT
cana-6282	50	320	with	with	ADP
cana-6282	50	321	control	control	NOUN
cana-6282	50	322	𝜑.	𝜑.	VERB
cana-6282	50	323	the	the	DET
cana-6282	50	324	use	use	NOUN
cana-6282	50	325	of	of	ADP
cana-6282	50	326	hyperstability	hyperstability	NOUN
cana-6282	50	327	in	in	ADP
cana-6282	50	328	functional	functional	ADJ
cana-6282	50	329	equations	equation	NOUN
cana-6282	50	330	is	be	AUX
cana-6282	50	331	mainly	mainly	ADV
cana-6282	50	332	theoretical	theoretical	ADJ
cana-6282	50	333	,	,	PUNCT
cana-6282	50	334	as	as	SCONJ
cana-6282	50	335	it	it	PRON
cana-6282	50	336	allows	allow	VERB
cana-6282	50	337	us	we	PRON
cana-6282	50	338	to	to	PART
cana-6282	50	339	extend	extend	VERB
cana-6282	50	340	the	the	DET
cana-6282	50	341	validity	validity	NOUN
cana-6282	50	342	of	of	ADP
cana-6282	50	343	known	know	VERB
cana-6282	50	344	solutions	solution	NOUN
cana-6282	50	345	to	to	ADP
cana-6282	50	346	nearby	nearby	ADJ
cana-6282	50	347	cases	case	NOUN
cana-6282	50	348	that	that	PRON
cana-6282	50	349	may	may	AUX
cana-6282	50	350	be	be	AUX
cana-6282	50	351	of	of	ADP
cana-6282	50	352	interest	interest	NOUN
cana-6282	50	353	.	.	PUNCT
cana-6282	51	1	for	for	ADP
cana-6282	51	2	example	example	NOUN
cana-6282	51	3	,	,	PUNCT
cana-6282	51	4	if	if	SCONJ
cana-6282	51	5	we	we	PRON
cana-6282	51	6	have	have	VERB
cana-6282	51	7	a	a	DET
cana-6282	51	8	solution	solution	NOUN
cana-6282	51	9	to	to	ADP
cana-6282	51	10	a	a	DET
cana-6282	51	11	functional	functional	ADJ
cana-6282	51	12	equation	equation	NOUN
cana-6282	51	13	that	that	PRON
cana-6282	51	14	describes	describe	VERB
cana-6282	51	15	a	a	DET
cana-6282	51	16	certain	certain	ADJ
cana-6282	51	17	phenomenon	phenomenon	NOUN
cana-6282	51	18	,	,	PUNCT
cana-6282	51	19	we	we	PRON
cana-6282	51	20	can	can	AUX
cana-6282	51	21	use	use	VERB
cana-6282	51	22	hyperstability	hyperstability	NOUN
cana-6282	51	23	to	to	PART
cana-6282	51	24	conclude	conclude	VERB
cana-6282	51	25	that	that	SCONJ
cana-6282	51	26	nearby	nearby	ADJ
cana-6282	51	27	phenomena	phenomenon	NOUN
cana-6282	51	28	are	be	AUX
cana-6282	51	29	likely	likely	ADJ
cana-6282	51	30	to	to	PART
cana-6282	51	31	exhibit	exhibit	VERB
cana-6282	51	32	similar	similar	ADJ
cana-6282	51	33	behavior	behavior	NOUN
cana-6282	51	34	.	.	PUNCT
cana-6282	52	1	hyperstability	hyperstability	NOUN
cana-6282	52	2	can	can	AUX
cana-6282	52	3	also	also	ADV
cana-6282	52	4	be	be	AUX
cana-6282	52	5	used	use	VERB
cana-6282	52	6	in	in	ADP
cana-6282	52	7	the	the	DET
cana-6282	52	8	study	study	NOUN
cana-6282	52	9	of	of	ADP
cana-6282	52	10	the	the	DET
cana-6282	52	11	stability	stability	NOUN
cana-6282	52	12	of	of	ADP
cana-6282	52	13	numerical	numerical	ADJ
cana-6282	52	14	methods	method	NOUN
cana-6282	52	15	for	for	ADP
cana-6282	52	16	solving	solve	VERB
cana-6282	52	17	functional	functional	ADJ
cana-6282	52	18	equations	equation	NOUN
cana-6282	52	19	.	.	PUNCT
cana-6282	53	1	if	if	SCONJ
cana-6282	53	2	we	we	PRON
cana-6282	53	3	know	know	VERB
cana-6282	53	4	that	that	SCONJ
cana-6282	53	5	the	the	DET
cana-6282	53	6	equation	equation	NOUN
cana-6282	53	7	is	be	AUX
cana-6282	53	8	hyperstable	hyperstable	ADJ
cana-6282	53	9	,	,	PUNCT
cana-6282	53	10	then	then	ADV
cana-6282	53	11	small	small	ADJ
cana-6282	53	12	errors	error	NOUN
cana-6282	53	13	in	in	ADP
cana-6282	53	14	the	the	DET
cana-6282	53	15	numerical	numerical	ADJ
cana-6282	53	16	solution	solution	NOUN
cana-6282	53	17	will	will	AUX
cana-6282	53	18	not	not	PART
cana-6282	53	19	significantly	significantly	ADV
cana-6282	53	20	affect	affect	VERB
cana-6282	53	21	the	the	DET
cana-6282	53	22	accuracy	accuracy	NOUN
cana-6282	53	23	of	of	ADP
cana-6282	53	24	the	the	DET
cana-6282	53	25	result	result	NOUN
cana-6282	53	26	.	.	PUNCT
cana-6282	54	1	overall	overall	ADV
cana-6282	54	2	,	,	PUNCT
cana-6282	54	3	hyperstability	hyperstability	NOUN
cana-6282	54	4	is	be	AUX
cana-6282	54	5	a	a	DET
cana-6282	54	6	useful	useful	ADJ
cana-6282	54	7	property	property	NOUN
cana-6282	54	8	of	of	ADP
cana-6282	54	9	functional	functional	ADJ
cana-6282	54	10	equations	equation	NOUN
cana-6282	54	11	that	that	PRON
cana-6282	54	12	allows	allow	VERB
cana-6282	54	13	us	we	PRON
cana-6282	54	14	to	to	PART
cana-6282	54	15	extend	extend	VERB
cana-6282	54	16	the	the	DET
cana-6282	54	17	range	range	NOUN
cana-6282	54	18	of	of	ADP
cana-6282	54	19	their	their	PRON
cana-6282	54	20	applicability	applicability	NOUN
cana-6282	54	21	and	and	CCONJ
cana-6282	54	22	better	well	ADV
cana-6282	54	23	understand	understand	VERB
cana-6282	54	24	their	their	PRON
cana-6282	54	25	behavior	behavior	NOUN
cana-6282	54	26	.	.	PUNCT
cana-6282	55	1	although	although	SCONJ
cana-6282	55	2	the	the	DET
cana-6282	55	3	term	term	NOUN
cana-6282	55	4	"	"	PUNCT
cana-6282	55	5	hyperstability	hyperstability	NOUN
cana-6282	55	6	"	"	PUNCT
cana-6282	55	7	was	be	AUX
cana-6282	55	8	first	first	ADV
cana-6282	55	9	used	use	VERB
cana-6282	55	10	in	in	ADP
cana-6282	55	11	2001	2001	NUM
cana-6282	55	12	[	[	X
cana-6282	55	13	15	15	NUM
cana-6282	55	14	]	]	PUNCT
cana-6282	55	15	,	,	PUNCT
cana-6282	55	16	the	the	DET
cana-6282	55	17	first	first	ADJ
cana-6282	55	18	hyperstability	hyperstability	NOUN
cana-6282	55	19	finding	finding	NOUN
cana-6282	55	20	appears	appear	VERB
cana-6282	55	21	to	to	PART
cana-6282	55	22	have	have	AUX
cana-6282	55	23	been	be	AUX
cana-6282	55	24	published	publish	VERB
cana-6282	55	25	in	in	ADP
cana-6282	55	26	1949	1949	NUM
cana-6282	56	1	[	[	X
cana-6282	56	2	8	8	NUM
cana-6282	56	3	]	]	PUNCT
cana-6282	56	4	.	.	PUNCT
cana-6282	57	1	let	let	VERB
cana-6282	57	2	𝑋	𝑋	NOUN
cana-6282	57	3	be	be	AUX
cana-6282	57	4	a	a	DET
cana-6282	57	5	real	real	ADV
cana-6282	57	6	normed	normed	ADJ
cana-6282	57	7	space	space	NOUN
cana-6282	57	8	and	and	CCONJ
cana-6282	57	9	(	(	PUNCT
cana-6282	57	10	𝑆,⋅	𝑆,⋅	PROPN
cana-6282	57	11	)	)	PUNCT
cana-6282	57	12	be	be	AUX
cana-6282	57	13	an	an	DET
cana-6282	57	14	arbitrary	arbitrary	ADJ
cana-6282	57	15	semigroup	semigroup	NOUN
cana-6282	57	16	.	.	PUNCT
cana-6282	58	1	in	in	ADP
cana-6282	58	2	2001	2001	NUM
cana-6282	58	3	,	,	PUNCT
cana-6282	58	4	one	one	NUM
cana-6282	58	5	of	of	ADP
cana-6282	58	6	the	the	DET
cana-6282	58	7	important	important	ADJ
cana-6282	58	8	results	result	NOUN
cana-6282	58	9	was	be	AUX
cana-6282	58	10	shown	show	VERB
cana-6282	58	11	in	in	ADP
cana-6282	58	12	stability	stability	NOUN
cana-6282	58	13	by	by	ADP
cana-6282	58	14	gy	gy	PROPN
cana-6282	58	15	.	.	PROPN
cana-6282	58	16	maksa	maksa	PROPN
cana-6282	58	17	and	and	CCONJ
cana-6282	58	18	zs	zs	PROPN
cana-6282	58	19	.	.	PUNCT
cana-6282	58	20	páles	pále	NOUN
cana-6282	59	1	[	[	X
cana-6282	59	2	15	15	NUM
cana-6282	59	3	]	]	PUNCT
cana-6282	59	4	.	.	PUNCT
cana-6282	60	1	they	they	PRON
cana-6282	60	2	investigated	investigate	VERB
cana-6282	60	3	the	the	DET
cana-6282	60	4	hyperstability	hyperstability	NOUN
cana-6282	60	5	of	of	ADP
cana-6282	60	6	the	the	DET
cana-6282	60	7	following	follow	VERB
cana-6282	60	8	particular	particular	ADJ
cana-6282	60	9	class	class	NOUN
cana-6282	60	10	of	of	ADP
cana-6282	60	11	linear	linear	ADJ
cana-6282	60	12	functional	functional	ADJ
cana-6282	60	13	equations	equation	NOUN
cana-6282	60	14	communications	communication	NOUN
cana-6282	60	15	on	on	ADP
cana-6282	60	16	applied	apply	VERB
cana-6282	60	17	nonlinear	nonlinear	ADJ
cana-6282	60	18	analysis	analysis	NOUN
cana-6282	60	19	issn	issn	NOUN
cana-6282	60	20	:	:	PUNCT
cana-6282	60	21	1074	1074	NUM
cana-6282	60	22	-	-	PUNCT
cana-6282	60	23	133x	133x	NUM
cana-6282	60	24	vol	vol	VERB
cana-6282	60	25	32	32	NUM
cana-6282	60	26	no	no	NOUN
cana-6282	60	27	.	.	PUNCT
cana-6282	60	28	10s	10	NOUN
cana-6282	60	29	(	(	PUNCT
cana-6282	60	30	2025	2025	NUM
cana-6282	60	31	)	)	PUNCT
cana-6282	60	32	3687	3687	NUM
cana-6282	60	33	https://internationalpubls.com	https://internationalpubls.com	X
cana-6282	60	34	𝑓(𝑥	𝑓(𝑥	NOUN
cana-6282	60	35	)	)	PUNCT
cana-6282	60	36	+	+	CCONJ
cana-6282	60	37	𝑓(𝑦	𝑓(𝑦	PROPN
cana-6282	60	38	)	)	PUNCT
cana-6282	60	39	=	=	SYM
cana-6282	60	40	1	1	NUM
cana-6282	60	41	𝑛	𝑛	PRON
cana-6282	60	42	∑	∑	ADP
cana-6282	60	43	  	  	SPACE
cana-6282	60	44	𝑛	𝑛	PRON
cana-6282	60	45	𝑖=1	𝑖=1	PROPN
cana-6282	60	46	 	 	SPACE
cana-6282	60	47	𝑓(𝑥𝜑𝑖(𝑦	𝑓(𝑥𝜑𝑖(𝑦	NOUN
cana-6282	60	48	)	)	PUNCT
cana-6282	60	49	)	)	PUNCT
cana-6282	60	50	(	(	PUNCT
cana-6282	60	51	1.4	1.4	NUM
cana-6282	60	52	)	)	PUNCT
cana-6282	60	53	where	where	SCONJ
cana-6282	60	54	𝑓	𝑓	X
cana-6282	60	55	:	:	PUNCT
cana-6282	60	56	𝑆	𝑆	PROPN
cana-6282	60	57	→	→	SYM
cana-6282	60	58	𝑋	𝑋	PROPN
cana-6282	60	59	and	and	CCONJ
cana-6282	60	60	where	where	SCONJ
cana-6282	60	61	𝜑1	𝜑1	VERB
cana-6282	60	62	,	,	PUNCT
cana-6282	60	63	⋯	⋯	PROPN
cana-6282	60	64	,	,	PUNCT
cana-6282	60	65	𝜑𝑛	𝜑𝑛	PROPN
cana-6282	60	66	:	:	PUNCT
cana-6282	60	67	𝑆	𝑆	PROPN
cana-6282	60	68	→	→	SYM
cana-6282	60	69	𝑆	𝑆	PROPN
cana-6282	60	70	are	be	AUX
cana-6282	60	71	pairwise	pairwise	NOUN
cana-6282	60	72	distinct	distinct	ADJ
cana-6282	60	73	automorphisms	automorphism	NOUN
cana-6282	60	74	of	of	ADP
cana-6282	60	75	𝑆.	𝑆.	PROPN
cana-6282	60	76	let	let	VERB
cana-6282	60	77	𝜎	𝜎	NOUN
cana-6282	60	78	:	:	PUNCT
cana-6282	60	79	𝑆	𝑆	PROPN
cana-6282	60	80	→	→	SYM
cana-6282	60	81	𝑆	𝑆	PROPN
cana-6282	60	82	be	be	AUX
cana-6282	60	83	an	an	DET
cana-6282	60	84	involution	involution	NOUN
cana-6282	60	85	,	,	PUNCT
cana-6282	60	86	that	that	ADV
cana-6282	60	87	is	is	ADV
cana-6282	60	88	,	,	PUNCT
cana-6282	60	89	𝜎(𝜎(𝑥	𝜎(𝜎(𝑥	PROPN
cana-6282	60	90	)	)	PUNCT
cana-6282	60	91	)	)	PUNCT
cana-6282	61	1	=	=	PUNCT
cana-6282	61	2	𝑥	𝑥	PROPN
cana-6282	61	3	and	and	CCONJ
cana-6282	61	4	𝜎(𝑥	𝜎(𝑥	PROPN
cana-6282	61	5	⋅	⋅	PROPN
cana-6282	61	6	𝑦	𝑦	NOUN
cana-6282	61	7	)	)	PUNCT
cana-6282	61	8	=	=	SYM
cana-6282	61	9	𝜎(𝑥	𝜎(𝑥	NOUN
cana-6282	61	10	)	)	PUNCT
cana-6282	61	11	⋅	⋅	PROPN
cana-6282	61	12	𝜎(𝑦	𝜎(𝑦	PROPN
cana-6282	61	13	)	)	PUNCT
cana-6282	61	14	for	for	ADP
cana-6282	61	15	all	all	PRON
cana-6282	61	16	𝑥	𝑥	PROPN
cana-6282	61	17	,	,	PUNCT
cana-6282	61	18	𝑦	𝑦	NOUN
cana-6282	61	19	∈	∈	NOUN
cana-6282	61	20	𝑆.	𝑆.	NOUN
cana-6282	61	21	in	in	ADP
cana-6282	61	22	2018	2018	NUM
cana-6282	61	23	,	,	PUNCT
cana-6282	61	24	jaehyeong	jaehyeong	ADP
cana-6282	61	25	bae	bae	PROPN
cana-6282	61	26	and	and	CCONJ
cana-6282	61	27	won	win	VERB
cana-6282	61	28	-	-	PUNCT
cana-6282	61	29	gil	gil	NOUN
cana-6282	61	30	park	park	NOUN
cana-6282	62	1	[	[	X
cana-6282	62	2	7	7	X
cana-6282	62	3	]	]	PUNCT
cana-6282	62	4	introduced	introduce	VERB
cana-6282	62	5	the	the	DET
cana-6282	62	6	following	follow	VERB
cana-6282	62	7	functional	functional	ADJ
cana-6282	62	8	equation	equation	NOUN
cana-6282	62	9	𝑓(𝑥1	𝑓(𝑥1	ADV
cana-6282	62	10	⋅	⋅	PROPN
cana-6282	62	11	𝑥2	𝑥2	NOUN
cana-6282	62	12	,	,	PUNCT
cana-6282	62	13	𝑥3	𝑥3	ADJ
cana-6282	62	14	⋅	⋅	PROPN
cana-6282	62	15	𝑥4	𝑥4	NOUN
cana-6282	62	16	)	)	PUNCT
cana-6282	62	17	+	+	CCONJ
cana-6282	62	18	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	62	19	⋅	⋅	PROPN
cana-6282	62	20	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	62	21	)	)	PUNCT
cana-6282	62	22	,	,	PUNCT
cana-6282	62	23	𝑥3	𝑥3	ADJ
cana-6282	62	24	⋅	⋅	PROPN
cana-6282	62	25	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	62	26	)	)	PUNCT
cana-6282	62	27	)	)	PUNCT
cana-6282	63	1	=	=	SYM
cana-6282	63	2	2𝑓(𝑥1	2𝑓(𝑥1	NUM
cana-6282	63	3	,	,	PUNCT
cana-6282	63	4	𝑥3	𝑥3	NOUN
cana-6282	63	5	)	)	PUNCT
cana-6282	63	6	+	+	CCONJ
cana-6282	63	7	2𝑓(𝑥2	2𝑓(𝑥2	NUM
cana-6282	63	8	,	,	PUNCT
cana-6282	63	9	𝑥4	𝑥4	ADJ
cana-6282	63	10	)	)	PUNCT
cana-6282	63	11	,	,	PUNCT
cana-6282	63	12	(	(	PUNCT
cana-6282	63	13	1.5	1.5	NUM
cana-6282	63	14	)	)	PUNCT
cana-6282	63	15	for	for	ADP
cana-6282	63	16	all	all	DET
cana-6282	63	17	𝑥1	𝑥1	NOUN
cana-6282	63	18	,	,	PUNCT
cana-6282	63	19	𝑥2	𝑥2	NOUN
cana-6282	63	20	,	,	PUNCT
cana-6282	63	21	𝑥3	𝑥3	NOUN
cana-6282	63	22	,	,	PUNCT
cana-6282	63	23	𝑥4	𝑥4	NOUN
cana-6282	63	24	∈	∈	PROPN
cana-6282	63	25	𝑋	𝑋	PROPN
cana-6282	63	26	,	,	PUNCT
cana-6282	63	27	where	where	SCONJ
cana-6282	63	28	𝑓	𝑓	X
cana-6282	63	29	:	:	PUNCT
cana-6282	63	30	𝑋	𝑋	NOUN
cana-6282	63	31	×	×	NOUN
cana-6282	63	32	𝑋	𝑋	PROPN
cana-6282	63	33	→	→	SYM
cana-6282	63	34	𝑌	𝑌	PROPN
cana-6282	63	35	with	with	ADP
cana-6282	63	36	𝑌	𝑌	PROPN
cana-6282	63	37	is	be	AUX
cana-6282	63	38	a	a	DET
cana-6282	63	39	banach	banach	NOUN
cana-6282	63	40	space	space	NOUN
cana-6282	63	41	and	and	CCONJ
cana-6282	63	42	where	where	SCONJ
cana-6282	63	43	𝜎	𝜎	PROPN
cana-6282	63	44	and	and	CCONJ
cana-6282	63	45	𝜏	𝜏	NOUN
cana-6282	63	46	are	be	AUX
cana-6282	63	47	involutions	involution	NOUN
cana-6282	63	48	on	on	ADP
cana-6282	63	49	𝑋.	𝑋.	PROPN
cana-6282	63	50	when	when	SCONJ
cana-6282	63	51	𝜎(𝑥	𝜎(𝑥	NOUN
cana-6282	63	52	)	)	PUNCT
cana-6282	63	53	=	=	SYM
cana-6282	63	54	𝜏(𝑥	𝜏(𝑥	NOUN
cana-6282	63	55	)	)	PUNCT
cana-6282	64	1	=	=	SYM
cana-6282	64	2	𝑥−1	𝑥−1	NOUN
cana-6282	64	3	,	,	PUNCT
cana-6282	64	4	the	the	DET
cana-6282	64	5	following	follow	VERB
cana-6282	64	6	quadratic	quadratic	ADJ
cana-6282	64	7	equation	equation	NOUN
cana-6282	64	8	is	be	AUX
cana-6282	64	9	obtained	obtain	VERB
cana-6282	64	10	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	64	11	⋅	⋅	ADJ
cana-6282	64	12	𝑥2	𝑥2	NOUN
cana-6282	64	13	,	,	PUNCT
cana-6282	64	14	𝑥3	𝑥3	ADJ
cana-6282	64	15	⋅	⋅	PROPN
cana-6282	64	16	𝑥4	𝑥4	NOUN
cana-6282	64	17	)	)	PUNCT
cana-6282	64	18	+	+	CCONJ
cana-6282	64	19	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	64	20	⋅	⋅	ADJ
cana-6282	64	21	𝑥2	𝑥2	NOUN
cana-6282	64	22	−1	−1	NOUN
cana-6282	64	23	,	,	PUNCT
cana-6282	64	24	𝑥3	𝑥3	ADJ
cana-6282	64	25	⋅	⋅	PROPN
cana-6282	64	26	𝑥4	𝑥4	NOUN
cana-6282	64	27	−1	−1	NOUN
cana-6282	64	28	)	)	PUNCT
cana-6282	64	29	=	=	SYM
cana-6282	64	30	2𝑓(𝑥1	2𝑓(𝑥1	NUM
cana-6282	64	31	,	,	PUNCT
cana-6282	64	32	𝑥3	𝑥3	NOUN
cana-6282	64	33	)	)	PUNCT
cana-6282	64	34	+	+	CCONJ
cana-6282	64	35	2𝑓(𝑥2	2𝑓(𝑥2	NUM
cana-6282	64	36	,	,	PUNCT
cana-6282	64	37	𝑥4	𝑥4	ADJ
cana-6282	64	38	)	)	PUNCT
cana-6282	64	39	(	(	PUNCT
cana-6282	64	40	1.6	1.6	NUM
cana-6282	64	41	)	)	PUNCT
cana-6282	64	42	the	the	DET
cana-6282	64	43	stability	stability	NOUN
cana-6282	64	44	of	of	ADP
cana-6282	64	45	eq	eq	PROPN
cana-6282	64	46	.	.	PUNCT
cana-6282	65	1	(	(	PUNCT
cana-6282	65	2	1.6	1.6	NUM
cana-6282	65	3	)	)	PUNCT
cana-6282	65	4	was	be	AUX
cana-6282	65	5	investigated	investigate	VERB
cana-6282	65	6	in	in	ADP
cana-6282	65	7	[	[	X
cana-6282	65	8	5	5	NUM
cana-6282	65	9	]	]	PUNCT
cana-6282	65	10	and	and	CCONJ
cana-6282	65	11	[	[	X
cana-6282	65	12	6	6	NUM
cana-6282	65	13	]	]	PUNCT
cana-6282	65	14	.	.	PUNCT
cana-6282	66	1	we	we	PRON
cana-6282	66	2	observe	observe	VERB
cana-6282	66	3	that	that	SCONJ
cana-6282	66	4	the	the	DET
cana-6282	66	5	functional	functional	ADJ
cana-6282	66	6	equation	equation	NOUN
cana-6282	66	7	(	(	PUNCT
cana-6282	66	8	1.6	1.6	NUM
cana-6282	66	9	)	)	PUNCT
cana-6282	66	10	,	,	PUNCT
cana-6282	66	11	can	can	AUX
cana-6282	66	12	be	be	AUX
cana-6282	66	13	reformulated	reformulate	VERB
cana-6282	66	14	in	in	ADP
cana-6282	66	15	a	a	DET
cana-6282	66	16	more	more	ADV
cana-6282	66	17	compact	compact	ADJ
cana-6282	66	18	and	and	CCONJ
cana-6282	66	19	structured	structure	VERB
cana-6282	66	20	form	form	NOUN
cana-6282	66	21	by	by	ADP
cana-6282	66	22	introducing	introduce	VERB
cana-6282	66	23	the	the	DET
cana-6282	66	24	mapping	mapping	NOUN
cana-6282	66	25	𝜑(𝑥2	𝜑(𝑥2	NOUN
cana-6282	66	26	,	,	PUNCT
cana-6282	66	27	𝑥4	𝑥4	NUM
cana-6282	66	28	):	):	PUNCT
cana-6282	66	29	=	=	SYM
cana-6282	66	30	(	(	PUNCT
cana-6282	66	31	𝜎(𝑥2	𝜎(𝑥2	NOUN
cana-6282	66	32	)	)	PUNCT
cana-6282	66	33	,	,	PUNCT
cana-6282	66	34	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	66	35	)	)	PUNCT
cana-6282	66	36	)	)	PUNCT
cana-6282	66	37	where	where	SCONJ
cana-6282	66	38	𝜎	𝜎	PRON
cana-6282	66	39	and	and	CCONJ
cana-6282	66	40	𝜏	𝜏	NOUN
cana-6282	66	41	are	be	AUX
cana-6282	66	42	involutions	involution	NOUN
cana-6282	66	43	on	on	ADP
cana-6282	66	44	a	a	DET
cana-6282	66	45	semigroup	semigroup	NOUN
cana-6282	66	46	𝑆.	𝑆.	NOUN
cana-6282	66	47	let	let	VERB
cana-6282	66	48	𝒮	𝒮	NOUN
cana-6282	66	49	:	:	PUNCT
cana-6282	66	50	=	=	SYM
cana-6282	66	51	𝑆	𝑆	PROPN
cana-6282	66	52	×	×	PROPN
cana-6282	66	53	𝑆	𝑆	PROPN
cana-6282	66	54	denote	denote	VERB
cana-6282	66	55	the	the	DET
cana-6282	66	56	direct	direct	ADJ
cana-6282	66	57	product	product	NOUN
cana-6282	66	58	semigroup	semigroup	NOUN
cana-6282	66	59	equipped	equip	VERB
cana-6282	66	60	with	with	ADP
cana-6282	66	61	the	the	DET
cana-6282	66	62	componentwise	componentwise	NOUN
cana-6282	66	63	multiplication	multiplication	NOUN
cana-6282	66	64	(	(	PUNCT
cana-6282	66	65	𝑥1	𝑥1	NOUN
cana-6282	66	66	,	,	PUNCT
cana-6282	66	67	𝑥3)(𝑥2	𝑥3)(𝑥2	NUM
cana-6282	66	68	,	,	PUNCT
cana-6282	66	69	𝑥4	𝑥4	NUM
cana-6282	66	70	):	):	PUNCT
cana-6282	66	71	=	=	SYM
cana-6282	66	72	(	(	PUNCT
cana-6282	66	73	𝑥1𝑥2	𝑥1𝑥2	NOUN
cana-6282	66	74	,	,	PUNCT
cana-6282	66	75	𝑥3𝑥4	𝑥3𝑥4	NOUN
cana-6282	66	76	)	)	PUNCT
cana-6282	66	77	for	for	ADP
cana-6282	66	78	𝑋	𝑋	PROPN
cana-6282	66	79	=	=	SYM
cana-6282	66	80	(	(	PUNCT
cana-6282	66	81	𝑥1	𝑥1	NOUN
cana-6282	66	82	,	,	PUNCT
cana-6282	66	83	𝑥3	𝑥3	NOUN
cana-6282	66	84	)	)	PUNCT
cana-6282	66	85	and	and	CCONJ
cana-6282	66	86	𝑌	𝑌	PROPN
cana-6282	66	87	=	=	SYM
cana-6282	66	88	(	(	PUNCT
cana-6282	66	89	𝑥2	𝑥2	NOUN
cana-6282	66	90	,	,	PUNCT
cana-6282	66	91	𝑥4	𝑥4	NOUN
cana-6282	66	92	)	)	PUNCT
cana-6282	66	93	in	in	ADP
cana-6282	66	94	𝒮	𝒮	PROPN
cana-6282	66	95	,	,	PUNCT
cana-6282	66	96	the	the	DET
cana-6282	66	97	original	original	ADJ
cana-6282	66	98	equation	equation	NOUN
cana-6282	66	99	can	can	AUX
cana-6282	66	100	then	then	ADV
cana-6282	66	101	be	be	AUX
cana-6282	66	102	expressed	express	VERB
cana-6282	66	103	as	as	ADP
cana-6282	66	104	𝑓(𝑋𝑌	𝑓(𝑋𝑌	PROPN
cana-6282	66	105	)	)	PUNCT
cana-6282	66	106	+	+	NUM
cana-6282	66	107	𝑓(𝑋𝜑(𝑌	𝑓(𝑋𝜑(𝑌	X
cana-6282	66	108	)	)	PUNCT
cana-6282	66	109	)	)	PUNCT
cana-6282	67	1	=	=	SYM
cana-6282	67	2	2𝑓(𝑋	2𝑓(𝑋	NUM
cana-6282	67	3	)	)	PUNCT
cana-6282	68	1	+	+	CCONJ
cana-6282	68	2	2𝑓(𝑌	2𝑓(𝑌	NUM
cana-6282	68	3	)	)	PUNCT
cana-6282	68	4	,	,	PUNCT
cana-6282	69	1	∀𝑋	∀𝑋	PROPN
cana-6282	69	2	,	,	PUNCT
cana-6282	69	3	𝑌	𝑌	PROPN
cana-6282	69	4	∈	∈	PROPN
cana-6282	69	5	𝒮	𝒮	PROPN
cana-6282	69	6	(	(	PUNCT
cana-6282	69	7	1.7	1.7	NUM
cana-6282	69	8	)	)	PUNCT
cana-6282	69	9	this	this	DET
cana-6282	69	10	equation	equation	NOUN
cana-6282	69	11	exhibits	exhibit	VERB
cana-6282	69	12	a	a	DET
cana-6282	69	13	symmetric	symmetric	ADJ
cana-6282	69	14	quadratic	quadratic	ADJ
cana-6282	69	15	structure	structure	NOUN
cana-6282	69	16	on	on	ADP
cana-6282	69	17	the	the	DET
cana-6282	69	18	semigroup	semigroup	PROPN
cana-6282	69	19	𝒮	𝒮	PROPN
cana-6282	69	20	with	with	ADP
cana-6282	69	21	an	an	DET
cana-6282	69	22	involutive	involutive	ADJ
cana-6282	69	23	automorphism	automorphism	NOUN
cana-6282	69	24	𝜑.	𝜑.	VERB
cana-6282	69	25	such	such	ADJ
cana-6282	69	26	reformulations	reformulation	NOUN
cana-6282	69	27	are	be	AUX
cana-6282	69	28	useful	useful	ADJ
cana-6282	69	29	for	for	ADP
cana-6282	69	30	highlighting	highlight	VERB
cana-6282	69	31	the	the	DET
cana-6282	69	32	algebraic	algebraic	ADJ
cana-6282	69	33	structure	structure	NOUN
cana-6282	69	34	underlying	underlie	VERB
cana-6282	69	35	the	the	DET
cana-6282	69	36	equation	equation	NOUN
cana-6282	69	37	and	and	CCONJ
cana-6282	69	38	for	for	ADP
cana-6282	69	39	enabling	enable	VERB
cana-6282	69	40	more	more	ADJ
cana-6282	69	41	general	general	ADJ
cana-6282	69	42	approaches	approach	NOUN
cana-6282	69	43	to	to	ADP
cana-6282	69	44	stability	stability	NOUN
cana-6282	69	45	analysis	analysis	NOUN
cana-6282	69	46	.	.	PUNCT
cana-6282	70	1	the	the	DET
cana-6282	70	2	equation	equation	NOUN
cana-6282	70	3	(	(	PUNCT
cana-6282	70	4	1.7	1.7	NUM
cana-6282	70	5	)	)	PUNCT
cana-6282	70	6	is	be	AUX
cana-6282	70	7	the	the	DET
cana-6282	70	8	quadratic	quadratic	ADJ
cana-6282	70	9	type	type	NOUN
cana-6282	70	10	,	,	PUNCT
cana-6282	70	11	its	its	PRON
cana-6282	70	12	central	central	ADJ
cana-6282	70	13	solutions	solution	NOUN
cana-6282	70	14	are	be	AUX
cana-6282	70	15	determined	determined	ADJ
cana-6282	70	16	on	on	ADP
cana-6282	70	17	arbitrary	arbitrary	ADJ
cana-6282	70	18	semigroups	semigroup	NOUN
cana-6282	70	19	and	and	CCONJ
cana-6282	70	20	its	its	PRON
cana-6282	70	21	all	all	DET
cana-6282	70	22	solutions	solution	NOUN
cana-6282	70	23	on	on	ADP
cana-6282	70	24	abelian	abelian	ADJ
cana-6282	70	25	semigroups	semigroup	NOUN
cana-6282	70	26	in	in	ADP
cana-6282	70	27	several	several	ADJ
cana-6282	70	28	works	work	NOUN
cana-6282	70	29	(	(	PUNCT
cana-6282	70	30	see	see	VERB
cana-6282	70	31	for	for	ADP
cana-6282	70	32	example	example	NOUN
cana-6282	70	33	[	[	X
cana-6282	70	34	1	1	NUM
cana-6282	70	35	]	]	PUNCT
cana-6282	70	36	,	,	PUNCT
cana-6282	70	37	[	[	X
cana-6282	70	38	2	2	NUM
cana-6282	70	39	]	]	PUNCT
cana-6282	70	40	,	,	PUNCT
cana-6282	70	41	and	and	CCONJ
cana-6282	70	42	[	[	X
cana-6282	70	43	11	11	NUM
cana-6282	70	44	]	]	NUM
cana-6282	70	45	)	)	PUNCT
cana-6282	70	46	.	.	PUNCT
cana-6282	71	1	the	the	DET
cana-6282	71	2	primary	primary	ADJ
cana-6282	71	3	objective	objective	NOUN
cana-6282	71	4	of	of	ADP
cana-6282	71	5	this	this	DET
cana-6282	71	6	work	work	NOUN
cana-6282	71	7	is	be	AUX
cana-6282	71	8	to	to	PART
cana-6282	71	9	investigate	investigate	VERB
cana-6282	71	10	the	the	DET
cana-6282	71	11	hyperstability	hyperstability	NOUN
cana-6282	71	12	of	of	ADP
cana-6282	71	13	a	a	DET
cana-6282	71	14	general	general	ADJ
cana-6282	71	15	functional	functional	ADJ
cana-6282	71	16	equation	equation	NOUN
cana-6282	71	17	of	of	ADP
cana-6282	71	18	the	the	DET
cana-6282	71	19	form	form	NOUN
cana-6282	71	20	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	71	21	⋅	⋅	ADJ
cana-6282	71	22	𝑥2	𝑥2	NOUN
cana-6282	71	23	,	,	PUNCT
cana-6282	71	24	𝑥3	𝑥3	ADJ
cana-6282	71	25	⋅	⋅	PROPN
cana-6282	71	26	𝑥4	𝑥4	NOUN
cana-6282	71	27	)	)	PUNCT
cana-6282	71	28	+	+	CCONJ
cana-6282	71	29	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	71	30	⋅	⋅	PROPN
cana-6282	71	31	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	71	32	)	)	PUNCT
cana-6282	71	33	,	,	PUNCT
cana-6282	71	34	𝑥3	𝑥3	ADJ
cana-6282	71	35	⋅	⋅	PROPN
cana-6282	71	36	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	71	37	)	)	PUNCT
cana-6282	71	38	)	)	PUNCT
cana-6282	72	1	=	=	SYM
cana-6282	72	2	2𝑓(𝑥1	2𝑓(𝑥1	NUM
cana-6282	72	3	,	,	PUNCT
cana-6282	72	4	𝑥3	𝑥3	NOUN
cana-6282	72	5	)	)	PUNCT
cana-6282	72	6	+	+	CCONJ
cana-6282	72	7	2𝑓(𝑥2	2𝑓(𝑥2	NUM
cana-6282	72	8	,	,	PUNCT
cana-6282	72	9	𝑥4	𝑥4	ADJ
cana-6282	72	10	)	)	PUNCT
cana-6282	72	11	,	,	PUNCT
cana-6282	72	12	where	where	SCONJ
cana-6282	72	13	𝑓	𝑓	X
cana-6282	72	14	:	:	PUNCT
cana-6282	72	15	𝑆2	𝑆2	PROPN
cana-6282	72	16	→	→	SYM
cana-6282	72	17	𝑋	𝑋	PROPN
cana-6282	72	18	,	,	PUNCT
cana-6282	72	19	and	and	CCONJ
cana-6282	72	20	𝜎	𝜎	X
cana-6282	72	21	,	,	PUNCT
cana-6282	72	22	𝜏	𝜏	NOUN
cana-6282	72	23	are	be	AUX
cana-6282	72	24	involutive	involutive	ADJ
cana-6282	72	25	mappings	mapping	NOUN
cana-6282	72	26	on	on	ADP
cana-6282	72	27	a	a	DET
cana-6282	72	28	semigroup	semigroup	NOUN
cana-6282	72	29	𝑆.	𝑆.	NOUN
cana-6282	72	30	our	our	PRON
cana-6282	72	31	approach	approach	NOUN
cana-6282	72	32	is	be	AUX
cana-6282	72	33	inspired	inspire	VERB
cana-6282	72	34	by	by	ADP
cana-6282	72	35	the	the	DET
cana-6282	72	36	asymptotic	asymptotic	ADJ
cana-6282	72	37	technique	technique	NOUN
cana-6282	72	38	introduced	introduce	VERB
cana-6282	72	39	by	by	ADP
cana-6282	72	40	maksa	maksa	ADJ
cana-6282	72	41	and	and	CCONJ
cana-6282	72	42	páles	pále	NOUN
cana-6282	72	43	[	[	X
cana-6282	72	44	15	15	NUM
cana-6282	72	45	]	]	X
cana-6282	72	46	,	,	PUNCT
cana-6282	72	47	which	which	PRON
cana-6282	72	48	has	have	AUX
cana-6282	72	49	proven	prove	VERB
cana-6282	72	50	effective	effective	ADJ
cana-6282	72	51	in	in	ADP
cana-6282	72	52	establishing	establish	VERB
cana-6282	72	53	hyperstability	hyperstability	NOUN
cana-6282	72	54	results	result	NOUN
cana-6282	72	55	for	for	ADP
cana-6282	72	56	linear	linear	ADJ
cana-6282	72	57	-	-	PUNCT
cana-6282	72	58	type	type	NOUN
cana-6282	72	59	functional	functional	ADJ
cana-6282	72	60	equations	equation	NOUN
cana-6282	72	61	.	.	PUNCT
cana-6282	73	1	the	the	DET
cana-6282	73	2	main	main	ADJ
cana-6282	73	3	contribution	contribution	NOUN
cana-6282	73	4	of	of	ADP
cana-6282	73	5	this	this	DET
cana-6282	73	6	paper	paper	NOUN
cana-6282	73	7	lies	lie	VERB
cana-6282	73	8	in	in	ADP
cana-6282	73	9	extending	extend	VERB
cana-6282	73	10	this	this	DET
cana-6282	73	11	method	method	NOUN
cana-6282	73	12	to	to	ADP
cana-6282	73	13	a	a	DET
cana-6282	73	14	more	more	ADV
cana-6282	73	15	general	general	ADJ
cana-6282	73	16	setting	setting	NOUN
cana-6282	73	17	involving	involve	VERB
cana-6282	73	18	products	product	NOUN
cana-6282	73	19	and	and	CCONJ
cana-6282	73	20	involutions	involution	NOUN
cana-6282	73	21	on	on	ADP
cana-6282	73	22	semigroups	semigroup	NOUN
cana-6282	73	23	.	.	PUNCT
cana-6282	74	1	specifically	specifically	ADV
cana-6282	74	2	,	,	PUNCT
cana-6282	74	3	we	we	PRON
cana-6282	74	4	provide	provide	VERB
cana-6282	74	5	sufficient	sufficient	ADJ
cana-6282	74	6	asymptotic	asymptotic	ADJ
cana-6282	74	7	conditions	condition	NOUN
cana-6282	74	8	under	under	ADP
cana-6282	74	9	which	which	PRON
cana-6282	74	10	the	the	DET
cana-6282	74	11	equation	equation	NOUN
cana-6282	74	12	admits	admit	VERB
cana-6282	74	13	hyperstability	hyperstability	NOUN
cana-6282	74	14	on	on	ADP
cana-6282	74	15	arbitrary	arbitrary	ADJ
cana-6282	74	16	semigroups	semigroup	NOUN
cana-6282	74	17	.	.	PUNCT
cana-6282	75	1	moreover	moreover	ADV
cana-6282	75	2	,	,	PUNCT
cana-6282	75	3	we	we	PRON
cana-6282	75	4	go	go	VERB
cana-6282	75	5	beyond	beyond	ADP
cana-6282	75	6	the	the	DET
cana-6282	75	7	homogeneous	homogeneous	ADJ
cana-6282	75	8	case	case	NOUN
cana-6282	75	9	by	by	ADP
cana-6282	75	10	analyzing	analyze	VERB
cana-6282	75	11	the	the	DET
cana-6282	75	12	hyperstability	hyperstability	NOUN
cana-6282	75	13	of	of	ADP
cana-6282	75	14	an	an	DET
cana-6282	75	15	associated	associated	ADJ
cana-6282	75	16	inhomogeneous	inhomogeneous	ADJ
cana-6282	75	17	functional	functional	ADJ
cana-6282	75	18	equation	equation	NOUN
cana-6282	75	19	:	:	PUNCT
cana-6282	75	20	communications	communication	NOUN
cana-6282	75	21	on	on	ADP
cana-6282	75	22	applied	apply	VERB
cana-6282	75	23	nonlinear	nonlinear	ADJ
cana-6282	75	24	analysis	analysis	NOUN
cana-6282	75	25	issn	issn	NOUN
cana-6282	75	26	:	:	PUNCT
cana-6282	75	27	1074	1074	NUM
cana-6282	75	28	-	-	PUNCT
cana-6282	75	29	133x	133x	NUM
cana-6282	75	30	vol	vol	VERB
cana-6282	75	31	32	32	NUM
cana-6282	75	32	no	no	NOUN
cana-6282	75	33	.	.	PUNCT
cana-6282	76	1	10s	10	NOUN
cana-6282	76	2	(	(	PUNCT
cana-6282	76	3	2025	2025	NUM
cana-6282	76	4	)	)	PUNCT
cana-6282	76	5	3688	3688	NUM
cana-6282	76	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-6282	76	7	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	76	8	⋅	⋅	ADJ
cana-6282	76	9	𝑥2	𝑥2	NOUN
cana-6282	76	10	,	,	PUNCT
cana-6282	76	11	𝑥3	𝑥3	ADJ
cana-6282	76	12	⋅	⋅	PROPN
cana-6282	76	13	𝑥4	𝑥4	NOUN
cana-6282	76	14	)	)	PUNCT
cana-6282	76	15	+	+	CCONJ
cana-6282	76	16	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	76	17	⋅	⋅	PROPN
cana-6282	76	18	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	76	19	)	)	PUNCT
cana-6282	76	20	,	,	PUNCT
cana-6282	76	21	𝑥3	𝑥3	ADJ
cana-6282	76	22	⋅	⋅	PROPN
cana-6282	76	23	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	76	24	)	)	PUNCT
cana-6282	76	25	)	)	PUNCT
cana-6282	77	1	=	=	SYM
cana-6282	77	2	2𝑓(𝑥1	2𝑓(𝑥1	NUM
cana-6282	77	3	,	,	PUNCT
cana-6282	77	4	𝑥3	𝑥3	NOUN
cana-6282	77	5	)	)	PUNCT
cana-6282	77	6	+	+	CCONJ
cana-6282	77	7	2𝑓(𝑥2	2𝑓(𝑥2	NUM
cana-6282	77	8	,	,	PUNCT
cana-6282	77	9	𝑥4	𝑥4	ADJ
cana-6282	77	10	)	)	PUNCT
cana-6282	78	1	+	+	CCONJ
cana-6282	78	2	𝐹(𝑥1	𝐹(𝑥1	ADJ
cana-6282	78	3	,	,	PUNCT
cana-6282	78	4	𝑥2	𝑥2	NOUN
cana-6282	78	5	,	,	PUNCT
cana-6282	78	6	𝑥3	𝑥3	NOUN
cana-6282	78	7	,	,	PUNCT
cana-6282	78	8	𝑥4	𝑥4	PROPN
cana-6282	78	9	)	)	PUNCT
cana-6282	78	10	,	,	PUNCT
cana-6282	78	11	where	where	SCONJ
cana-6282	78	12	𝐹	𝐹	PROPN
cana-6282	78	13	:	:	PUNCT
cana-6282	78	14	𝑆4	𝑆4	PROPN
cana-6282	78	15	→	→	SYM
cana-6282	78	16	𝑋	𝑋	PROPN
cana-6282	78	17	is	be	AUX
cana-6282	78	18	an	an	DET
cana-6282	78	19	arbitrary	arbitrary	ADJ
cana-6282	78	20	perturbation	perturbation	NOUN
cana-6282	78	21	function	function	NOUN
cana-6282	78	22	.	.	PUNCT
cana-6282	79	1	we	we	PRON
cana-6282	79	2	prove	prove	VERB
cana-6282	79	3	that	that	SCONJ
cana-6282	79	4	under	under	ADP
cana-6282	79	5	suitable	suitable	ADJ
cana-6282	79	6	assumptions	assumption	NOUN
cana-6282	79	7	,	,	PUNCT
cana-6282	79	8	any	any	DET
cana-6282	79	9	approximate	approximate	ADJ
cana-6282	79	10	solution	solution	NOUN
cana-6282	79	11	of	of	ADP
cana-6282	79	12	this	this	DET
cana-6282	79	13	perturbed	perturb	VERB
cana-6282	79	14	equation	equation	NOUN
cana-6282	79	15	must	must	AUX
cana-6282	79	16	be	be	AUX
cana-6282	79	17	an	an	DET
cana-6282	79	18	exact	exact	ADJ
cana-6282	79	19	solution	solution	NOUN
cana-6282	79	20	,	,	PUNCT
cana-6282	79	21	thereby	thereby	ADV
cana-6282	79	22	establishing	establish	VERB
cana-6282	79	23	its	its	PRON
cana-6282	79	24	hyperstability	hyperstability	NOUN
cana-6282	79	25	as	as	ADV
cana-6282	79	26	well	well	ADV
cana-6282	79	27	.	.	PUNCT
cana-6282	80	1	these	these	DET
cana-6282	80	2	results	result	NOUN
cana-6282	80	3	contribute	contribute	VERB
cana-6282	80	4	to	to	ADP
cana-6282	80	5	the	the	DET
cana-6282	80	6	ongoing	ongoing	ADJ
cana-6282	80	7	development	development	NOUN
cana-6282	80	8	of	of	ADP
cana-6282	80	9	the	the	DET
cana-6282	80	10	theory	theory	NOUN
cana-6282	80	11	of	of	ADP
cana-6282	80	12	stability	stability	NOUN
cana-6282	80	13	in	in	ADP
cana-6282	80	14	functional	functional	ADJ
cana-6282	80	15	equations	equation	NOUN
cana-6282	80	16	by	by	ADP
cana-6282	80	17	broadening	broaden	VERB
cana-6282	80	18	the	the	DET
cana-6282	80	19	class	class	NOUN
cana-6282	80	20	of	of	ADP
cana-6282	80	21	equations	equation	NOUN
cana-6282	80	22	and	and	CCONJ
cana-6282	80	23	algebraic	algebraic	ADJ
cana-6282	80	24	structures	structure	NOUN
cana-6282	80	25	for	for	ADP
cana-6282	80	26	which	which	DET
cana-6282	80	27	hyperstability	hyperstability	NOUN
cana-6282	80	28	can	can	AUX
cana-6282	80	29	be	be	AUX
cana-6282	80	30	rigorously	rigorously	ADV
cana-6282	80	31	validated	validate	VERB
cana-6282	80	32	.	.	PUNCT
cana-6282	81	1	one	one	NUM
cana-6282	81	2	of	of	ADP
cana-6282	81	3	the	the	DET
cana-6282	81	4	main	main	ADJ
cana-6282	81	5	goals	goal	NOUN
cana-6282	81	6	of	of	ADP
cana-6282	81	7	this	this	DET
cana-6282	81	8	paper	paper	NOUN
cana-6282	81	9	is	be	AUX
cana-6282	81	10	to	to	PART
cana-6282	81	11	study	study	VERB
cana-6282	81	12	the	the	DET
cana-6282	81	13	hyperstability	hyperstability	NOUN
cana-6282	81	14	of	of	ADP
cana-6282	81	15	a	a	DET
cana-6282	81	16	general	general	ADJ
cana-6282	81	17	functional	functional	ADJ
cana-6282	81	18	equation	equation	NOUN
cana-6282	81	19	involving	involve	VERB
cana-6282	81	20	involutions	involution	NOUN
cana-6282	81	21	on	on	ADP
cana-6282	81	22	semigroups	semigroup	NOUN
cana-6282	81	23	,	,	PUNCT
cana-6282	81	24	using	use	VERB
cana-6282	81	25	an	an	DET
cana-6282	81	26	asymptotic	asymptotic	ADJ
cana-6282	81	27	approach	approach	NOUN
cana-6282	81	28	inspired	inspire	VERB
cana-6282	81	29	by	by	ADP
cana-6282	81	30	the	the	DET
cana-6282	81	31	method	method	NOUN
cana-6282	81	32	of	of	ADP
cana-6282	81	33	maksa	maksa	ADJ
cana-6282	81	34	and	and	CCONJ
cana-6282	81	35	páles[15	páles[15	NOUN
cana-6282	81	36	]	]	PUNCT
cana-6282	81	37	.	.	PUNCT
cana-6282	82	1	our	our	PRON
cana-6282	82	2	analysis	analysis	NOUN
cana-6282	82	3	focuses	focus	VERB
cana-6282	82	4	on	on	ADP
cana-6282	82	5	identifying	identify	VERB
cana-6282	82	6	sufficient	sufficient	ADJ
cana-6282	82	7	conditions	condition	NOUN
cana-6282	82	8	under	under	ADP
cana-6282	82	9	which	which	PRON
cana-6282	82	10	the	the	DET
cana-6282	82	11	approximate	approximate	ADJ
cana-6282	82	12	solutions	solution	NOUN
cana-6282	82	13	of	of	ADP
cana-6282	82	14	the	the	DET
cana-6282	82	15	functional	functional	ADJ
cana-6282	82	16	equation	equation	NOUN
cana-6282	82	17	converge	converge	VERB
cana-6282	82	18	to	to	ADP
cana-6282	82	19	exact	exact	ADJ
cana-6282	82	20	solutions	solution	NOUN
cana-6282	82	21	,	,	PUNCT
cana-6282	82	22	thereby	thereby	ADV
cana-6282	82	23	establishing	establish	VERB
cana-6282	82	24	its	its	PRON
cana-6282	82	25	hyperstability	hyperstability	NOUN
cana-6282	82	26	in	in	ADP
cana-6282	82	27	the	the	DET
cana-6282	82	28	sense	sense	NOUN
cana-6282	82	29	of	of	ADP
cana-6282	82	30	ulam	ulam	NOUN
cana-6282	82	31	.	.	PUNCT
cana-6282	83	1	in	in	ADP
cana-6282	83	2	addition	addition	NOUN
cana-6282	83	3	to	to	ADP
cana-6282	83	4	analyzing	analyze	VERB
cana-6282	83	5	the	the	DET
cana-6282	83	6	hyperstability	hyperstability	NOUN
cana-6282	83	7	of	of	ADP
cana-6282	83	8	the	the	DET
cana-6282	83	9	original	original	ADJ
cana-6282	83	10	functional	functional	ADJ
cana-6282	83	11	equation	equation	NOUN
cana-6282	83	12	in	in	ADP
cana-6282	83	13	its	its	PRON
cana-6282	83	14	standard	standard	ADJ
cana-6282	83	15	form	form	NOUN
cana-6282	83	16	,	,	PUNCT
cana-6282	83	17	this	this	DET
cana-6282	83	18	work	work	NOUN
cana-6282	83	19	also	also	ADV
cana-6282	83	20	aims	aim	VERB
cana-6282	83	21	to	to	PART
cana-6282	83	22	recast	recast	VERB
cana-6282	83	23	the	the	DET
cana-6282	83	24	equation	equation	NOUN
cana-6282	83	25	within	within	ADP
cana-6282	83	26	a	a	DET
cana-6282	83	27	product	product	NOUN
cana-6282	83	28	semigroup	semigroup	NOUN
cana-6282	83	29	framework	framework	NOUN
cana-6282	83	30	.	.	PUNCT
cana-6282	84	1	by	by	ADP
cana-6282	84	2	identifying	identify	VERB
cana-6282	84	3	the	the	DET
cana-6282	84	4	underlying	underlie	VERB
cana-6282	84	5	structure	structure	NOUN
cana-6282	84	6	in	in	ADP
cana-6282	84	7	terms	term	NOUN
cana-6282	84	8	of	of	ADP
cana-6282	84	9	the	the	DET
cana-6282	84	10	product	product	NOUN
cana-6282	84	11	semigroup	semigroup	NOUN
cana-6282	84	12	𝒮	𝒮	PROPN
cana-6282	84	13	=	=	SYM
cana-6282	84	14	𝑆	𝑆	PROPN
cana-6282	84	15	×	×	PROPN
cana-6282	84	16	𝑆	𝑆	PROPN
cana-6282	84	17	and	and	CCONJ
cana-6282	84	18	an	an	DET
cana-6282	84	19	involutive	involutive	ADJ
cana-6282	84	20	automorphism	automorphism	NOUN
cana-6282	84	21	𝜑	𝜑	NOUN
cana-6282	84	22	:	:	PUNCT
cana-6282	84	23	𝒮	𝒮	PROPN
cana-6282	84	24	→	→	SYM
cana-6282	84	25	𝒮	𝒮	PROPN
cana-6282	84	26	,	,	PUNCT
cana-6282	84	27	we	we	PRON
cana-6282	84	28	reformulate	reformulate	VERB
cana-6282	84	29	the	the	DET
cana-6282	84	30	original	original	ADJ
cana-6282	84	31	equation	equation	NOUN
cana-6282	84	32	as	as	ADP
cana-6282	84	33	𝑓(𝜉휁	𝑓(𝜉휁	PROPN
cana-6282	84	34	)	)	PUNCT
cana-6282	85	1	+	+	CCONJ
cana-6282	86	1	𝑓(𝜉𝜑(휁	𝑓(𝜉𝜑(휁	NOUN
cana-6282	86	2	)	)	PUNCT
cana-6282	86	3	)	)	PUNCT
cana-6282	86	4	=	=	SYM
cana-6282	86	5	2𝑓(𝜉	2𝑓(𝜉	NUM
cana-6282	86	6	)	)	PUNCT
cana-6282	87	1	+	+	CCONJ
cana-6282	87	2	2𝑓(휁	2𝑓(휁	NUM
cana-6282	87	3	)	)	PUNCT
cana-6282	87	4	,	,	PUNCT
cana-6282	87	5	𝜉	𝜉	PROPN
cana-6282	87	6	,	,	PUNCT
cana-6282	87	7	휁	휁	PROPN
cana-6282	87	8	∈	∈	PROPN
cana-6282	87	9	𝒮	𝒮	NOUN
cana-6282	87	10	this	this	DET
cana-6282	87	11	alternative	alternative	ADJ
cana-6282	87	12	formulation	formulation	NOUN
cana-6282	87	13	,	,	PUNCT
cana-6282	87	14	which	which	PRON
cana-6282	87	15	reinterprets	reinterpret	VERB
cana-6282	87	16	the	the	DET
cana-6282	87	17	functional	functional	ADJ
cana-6282	87	18	equation	equation	NOUN
cana-6282	87	19	within	within	ADP
cana-6282	87	20	the	the	DET
cana-6282	87	21	cartesian	cartesian	ADJ
cana-6282	87	22	product	product	NOUN
cana-6282	87	23	semigroup	semigroup	NOUN
cana-6282	87	24	𝑆	𝑆	PROPN
cana-6282	87	25	×	×	PROPN
cana-6282	87	26	𝑆	𝑆	PROPN
cana-6282	87	27	,	,	PUNCT
cana-6282	87	28	serves	serve	VERB
cana-6282	87	29	a	a	DET
cana-6282	87	30	dual	dual	ADJ
cana-6282	87	31	purpose	purpose	NOUN
cana-6282	87	32	.	.	PUNCT
cana-6282	88	1	first	first	ADV
cana-6282	88	2	,	,	PUNCT
cana-6282	88	3	it	it	PRON
cana-6282	88	4	exposes	expose	VERB
cana-6282	88	5	the	the	DET
cana-6282	88	6	underlying	underlie	VERB
cana-6282	88	7	algebraic	algebraic	ADJ
cana-6282	88	8	symmetry	symmetry	NOUN
cana-6282	88	9	of	of	ADP
cana-6282	88	10	the	the	DET
cana-6282	88	11	equation	equation	NOUN
cana-6282	88	12	by	by	ADP
cana-6282	88	13	aligning	align	VERB
cana-6282	88	14	it	it	PRON
cana-6282	88	15	with	with	ADP
cana-6282	88	16	a	a	DET
cana-6282	88	17	structured	structured	ADJ
cana-6282	88	18	binary	binary	ADJ
cana-6282	88	19	operation	operation	NOUN
cana-6282	88	20	on	on	ADP
cana-6282	88	21	paired	pair	VERB
cana-6282	88	22	elements	element	NOUN
cana-6282	88	23	,	,	PUNCT
cana-6282	88	24	thus	thus	ADV
cana-6282	88	25	clarifying	clarify	VERB
cana-6282	88	26	the	the	DET
cana-6282	88	27	involutive	involutive	ADJ
cana-6282	88	28	behavior	behavior	NOUN
cana-6282	88	29	induced	induce	VERB
cana-6282	88	30	by	by	ADP
cana-6282	88	31	𝜎	𝜎	PROPN
cana-6282	88	32	and	and	CCONJ
cana-6282	88	33	𝜏.	𝜏.	PROPN
cana-6282	88	34	second	second	ADJ
cana-6282	88	35	,	,	PUNCT
cana-6282	88	36	this	this	DET
cana-6282	88	37	reformulation	reformulation	NOUN
cana-6282	88	38	enables	enable	VERB
cana-6282	88	39	the	the	DET
cana-6282	88	40	application	application	NOUN
cana-6282	88	41	of	of	ADP
cana-6282	88	42	general	general	ADJ
cana-6282	88	43	semigroup	semigroup	PROPN
cana-6282	88	44	techniques	technique	NOUN
cana-6282	88	45	to	to	ADP
cana-6282	88	46	a	a	DET
cana-6282	88	47	broader	broad	ADJ
cana-6282	88	48	class	class	NOUN
cana-6282	88	49	of	of	ADP
cana-6282	88	50	functional	functional	ADJ
cana-6282	88	51	equations	equation	NOUN
cana-6282	88	52	by	by	ADP
cana-6282	88	53	transforming	transform	VERB
cana-6282	88	54	the	the	DET
cana-6282	88	55	problem	problem	NOUN
cana-6282	88	56	into	into	ADP
cana-6282	88	57	an	an	DET
cana-6282	88	58	equivalent	equivalent	ADJ
cana-6282	88	59	one	one	NOUN
cana-6282	88	60	on	on	ADP
cana-6282	88	61	the	the	DET
cana-6282	88	62	product	product	NOUN
cana-6282	88	63	structure	structure	NOUN
cana-6282	88	64	.	.	PUNCT
cana-6282	89	1	such	such	DET
cana-6282	89	2	a	a	DET
cana-6282	89	3	representation	representation	NOUN
cana-6282	89	4	is	be	AUX
cana-6282	89	5	particularly	particularly	ADV
cana-6282	89	6	advantageous	advantageous	ADJ
cana-6282	89	7	when	when	SCONJ
cana-6282	89	8	exploring	explore	VERB
cana-6282	89	9	stability	stability	NOUN
cana-6282	89	10	properties	property	NOUN
cana-6282	89	11	,	,	PUNCT
cana-6282	89	12	as	as	SCONJ
cana-6282	89	13	it	it	PRON
cana-6282	89	14	allows	allow	VERB
cana-6282	89	15	the	the	DET
cana-6282	89	16	equation	equation	NOUN
cana-6282	89	17	to	to	PART
cana-6282	89	18	be	be	AUX
cana-6282	89	19	treated	treat	VERB
cana-6282	89	20	in	in	ADP
cana-6282	89	21	terms	term	NOUN
cana-6282	89	22	of	of	ADP
cana-6282	89	23	single	single	ADJ
cana-6282	89	24	-	-	PUNCT
cana-6282	89	25	variable	variable	ADJ
cana-6282	89	26	operations	operation	NOUN
cana-6282	89	27	on	on	ADP
cana-6282	89	28	pairs	pair	NOUN
cana-6282	89	29	,	,	PUNCT
cana-6282	89	30	thereby	thereby	ADV
cana-6282	89	31	facilitating	facilitate	VERB
cana-6282	89	32	the	the	DET
cana-6282	89	33	use	use	NOUN
cana-6282	89	34	of	of	ADP
cana-6282	89	35	asymptotic	asymptotic	ADJ
cana-6282	89	36	and	and	CCONJ
cana-6282	89	37	fixed	fix	VERB
cana-6282	89	38	-	-	PUNCT
cana-6282	89	39	point	point	NOUN
cana-6282	89	40	methods	method	NOUN
cana-6282	89	41	.	.	PUNCT
cana-6282	90	1	this	this	DET
cana-6282	90	2	perspective	perspective	NOUN
cana-6282	90	3	not	not	PART
cana-6282	90	4	only	only	ADV
cana-6282	90	5	streamlines	streamline	VERB
cana-6282	90	6	the	the	DET
cana-6282	90	7	analytical	analytical	ADJ
cana-6282	90	8	process	process	NOUN
cana-6282	90	9	but	but	CCONJ
cana-6282	90	10	also	also	ADV
cana-6282	90	11	opens	open	VERB
cana-6282	90	12	pathways	pathway	NOUN
cana-6282	90	13	for	for	ADP
cana-6282	90	14	further	further	ADJ
cana-6282	90	15	generalizations	generalization	NOUN
cana-6282	90	16	to	to	AUX
cana-6282	90	17	multi	multi	ADJ
cana-6282	90	18	-	-	ADJ
cana-6282	90	19	variable	variable	ADJ
cana-6282	90	20	or	or	CCONJ
cana-6282	90	21	higher	higher	ADV
cana-6282	90	22	-	-	PUNCT
cana-6282	90	23	dimensional	dimensional	ADJ
cana-6282	90	24	functional	functional	ADJ
cana-6282	90	25	equations	equation	NOUN
cana-6282	90	26	on	on	ADP
cana-6282	90	27	composite	composite	ADJ
cana-6282	90	28	algebraic	algebraic	ADJ
cana-6282	90	29	systems	system	NOUN
cana-6282	90	30	.	.	PUNCT
cana-6282	91	1	2	2	X
cana-6282	91	2	.	.	X
cana-6282	91	3	main	main	ADJ
cana-6282	91	4	results	result	NOUN
cana-6282	91	5	in	in	ADP
cana-6282	91	6	what	what	PRON
cana-6282	91	7	follows	follow	VERB
cana-6282	91	8	,	,	PUNCT
cana-6282	91	9	we	we	PRON
cana-6282	91	10	denote	denote	VERB
cana-6282	91	11	by	by	ADP
cana-6282	91	12	ℝ+the	ℝ+the	DET
cana-6282	91	13	set	set	NOUN
cana-6282	91	14	of	of	ADP
cana-6282	91	15	nonnegative	nonnegative	ADJ
cana-6282	91	16	real	real	ADJ
cana-6282	91	17	numbers	number	NOUN
cana-6282	91	18	and	and	CCONJ
cana-6282	91	19	by	by	ADP
cana-6282	91	20	ℕ	ℕ	PROPN
cana-6282	91	21	the	the	DET
cana-6282	91	22	set	set	NOUN
cana-6282	91	23	of	of	ADP
cana-6282	91	24	positive	positive	ADJ
cana-6282	91	25	integers	integer	NOUN
cana-6282	91	26	.	.	PUNCT
cana-6282	92	1	let	let	VERB
cana-6282	92	2	𝑋	𝑋	NOUN
cana-6282	92	3	be	be	AUX
cana-6282	92	4	a	a	DET
cana-6282	92	5	real	real	ADV
cana-6282	92	6	normed	normed	ADJ
cana-6282	92	7	space	space	NOUN
cana-6282	92	8	,	,	PUNCT
cana-6282	92	9	and	and	CCONJ
cana-6282	92	10	let	let	VERB
cana-6282	92	11	(	(	PUNCT
cana-6282	92	12	𝑆,⋅	𝑆,⋅	PROPN
cana-6282	92	13	)	)	PUNCT
cana-6282	92	14	be	be	AUX
cana-6282	92	15	a	a	DET
cana-6282	92	16	semigroup	semigroup	NOUN
cana-6282	92	17	.	.	PUNCT
cana-6282	93	1	throughout	throughout	ADP
cana-6282	93	2	this	this	DET
cana-6282	93	3	section	section	NOUN
cana-6282	93	4	,	,	PUNCT
cana-6282	93	5	we	we	PRON
cana-6282	93	6	consider	consider	VERB
cana-6282	93	7	two	two	NUM
cana-6282	93	8	mappings	mapping	NOUN
cana-6282	93	9	𝜎	𝜎	SYM
cana-6282	93	10	,	,	PUNCT
cana-6282	93	11	𝜏	𝜏	NOUN
cana-6282	93	12	:	:	PUNCT
cana-6282	93	13	𝑆	𝑆	PROPN
cana-6282	93	14	→	→	SYM
cana-6282	93	15	𝑆	𝑆	PROPN
cana-6282	93	16	that	that	PRON
cana-6282	93	17	are	be	AUX
cana-6282	93	18	assumed	assume	VERB
cana-6282	93	19	to	to	PART
cana-6282	93	20	be	be	AUX
cana-6282	93	21	involutive	involutive	ADJ
cana-6282	93	22	endomorphisms	endomorphism	NOUN
cana-6282	93	23	unless	unless	SCONJ
cana-6282	93	24	specified	specify	VERB
cana-6282	93	25	otherwise	otherwise	ADV
cana-6282	93	26	.	.	PUNCT
cana-6282	94	1	the	the	DET
cana-6282	94	2	analysis	analysis	NOUN
cana-6282	94	3	presented	present	VERB
cana-6282	94	4	in	in	ADP
cana-6282	94	5	this	this	DET
cana-6282	94	6	section	section	NOUN
cana-6282	94	7	is	be	AUX
cana-6282	94	8	motivated	motivate	VERB
cana-6282	94	9	by	by	ADP
cana-6282	94	10	the	the	DET
cana-6282	94	11	asymptotic	asymptotic	ADJ
cana-6282	94	12	technique	technique	NOUN
cana-6282	94	13	introduced	introduce	VERB
cana-6282	94	14	by	by	ADP
cana-6282	94	15	maksa	maksa	ADJ
cana-6282	94	16	and	and	CCONJ
cana-6282	94	17	páles	pále	NOUN
cana-6282	94	18	[	[	X
cana-6282	94	19	15	15	NUM
cana-6282	94	20	]	]	X
cana-6282	94	21	,	,	PUNCT
cana-6282	94	22	which	which	PRON
cana-6282	94	23	has	have	AUX
cana-6282	94	24	proven	prove	VERB
cana-6282	94	25	to	to	PART
cana-6282	94	26	be	be	AUX
cana-6282	94	27	a	a	DET
cana-6282	94	28	powerful	powerful	ADJ
cana-6282	94	29	tool	tool	NOUN
cana-6282	94	30	in	in	ADP
cana-6282	94	31	establishing	establish	VERB
cana-6282	94	32	hyperstability	hyperstability	NOUN
cana-6282	94	33	results	result	NOUN
cana-6282	94	34	.	.	PUNCT
cana-6282	95	1	our	our	PRON
cana-6282	95	2	approach	approach	NOUN
cana-6282	95	3	begins	begin	VERB
cana-6282	95	4	with	with	ADP
cana-6282	95	5	a	a	DET
cana-6282	95	6	key	key	ADJ
cana-6282	95	7	lemma	lemma	PROPN
cana-6282	95	8	that	that	PRON
cana-6282	95	9	forms	form	VERB
cana-6282	95	10	the	the	DET
cana-6282	95	11	basis	basis	NOUN
cana-6282	95	12	for	for	ADP
cana-6282	95	13	the	the	DET
cana-6282	95	14	subsequent	subsequent	ADJ
cana-6282	95	15	stability	stability	NOUN
cana-6282	95	16	results	result	VERB
cana-6282	95	17	.	.	PUNCT
cana-6282	96	1	lemma	lemma	PROPN
cana-6282	96	2	2.1	2.1	NUM
cana-6282	96	3	.	.	PUNCT
cana-6282	97	1	let	let	VERB
cana-6282	97	2	𝑓	𝑓	PRON
cana-6282	97	3	:	:	PUNCT
cana-6282	97	4	𝑆2	𝑆2	PROPN
cana-6282	97	5	⟶	⟶	NOUN
cana-6282	97	6	𝑋	𝑋	PROPN
cana-6282	97	7	be	be	VERB
cana-6282	97	8	an	an	DET
cana-6282	97	9	arbitrary	arbitrary	ADJ
cana-6282	97	10	function	function	NOUN
cana-6282	97	11	.	.	PUNCT
cana-6282	98	1	then	then	ADV
cana-6282	98	2	the	the	DET
cana-6282	98	3	function	function	NOUN
cana-6282	98	4	𝐷𝑓	𝐷𝑓	PROPN
cana-6282	98	5	:	:	PUNCT
cana-6282	98	6	𝑆4	𝑆4	PROPN
cana-6282	98	7	⟶	⟶	NOUN
cana-6282	98	8	𝑋	𝑋	PROPN
cana-6282	98	9	that	that	PRON
cana-6282	98	10	is	be	AUX
cana-6282	98	11	defined	define	VERB
cana-6282	98	12	by	by	ADP
cana-6282	98	13	𝐷𝑓(𝑥1	𝐷𝑓(𝑥1	PROPN
cana-6282	98	14	,	,	PUNCT
cana-6282	98	15	𝑥2	𝑥2	NOUN
cana-6282	98	16	,	,	PUNCT
cana-6282	98	17	𝑥3	𝑥3	NOUN
cana-6282	98	18	,	,	PUNCT
cana-6282	98	19	𝑥4	𝑥4	NOUN
cana-6282	98	20	)	)	PUNCT
cana-6282	98	21	=	=	PUNCT
cana-6282	98	22	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	98	23	⋅	⋅	ADJ
cana-6282	98	24	𝑥2	𝑥2	NOUN
cana-6282	98	25	,	,	PUNCT
cana-6282	98	26	𝑥3	𝑥3	ADJ
cana-6282	98	27	⋅	⋅	PROPN
cana-6282	98	28	𝑥4	𝑥4	NOUN
cana-6282	98	29	)	)	PUNCT
cana-6282	98	30	+	+	CCONJ
cana-6282	98	31	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	98	32	⋅	⋅	PROPN
cana-6282	98	33	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	98	34	)	)	PUNCT
cana-6282	98	35	,	,	PUNCT
cana-6282	98	36	𝑥3	𝑥3	ADJ
cana-6282	98	37	⋅	⋅	PROPN
cana-6282	98	38	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	98	39	)	)	PUNCT
cana-6282	98	40	)	)	PUNCT
cana-6282	99	1	−	−	ADP
cana-6282	99	2	2𝑓(𝑥1	2𝑓(𝑥1	NUM
cana-6282	99	3	,	,	PUNCT
cana-6282	99	4	𝑥3	𝑥3	NOUN
cana-6282	99	5	)	)	PUNCT
cana-6282	99	6	−	−	PROPN
cana-6282	99	7	2𝑓(𝑥2	2𝑓(𝑥2	NUM
cana-6282	99	8	,	,	PUNCT
cana-6282	99	9	𝑥4	𝑥4	PROPN
cana-6282	99	10	)	)	PUNCT
cana-6282	99	11	,	,	PUNCT
cana-6282	99	12	(	(	PUNCT
cana-6282	99	13	2.1	2.1	NUM
cana-6282	99	14	)	)	PUNCT
cana-6282	99	15	for	for	ADP
cana-6282	99	16	all	all	DET
cana-6282	99	17	𝑥1	𝑥1	NOUN
cana-6282	99	18	,	,	PUNCT
cana-6282	99	19	𝑥2	𝑥2	NOUN
cana-6282	99	20	,	,	PUNCT
cana-6282	99	21	𝑥3	𝑥3	NOUN
cana-6282	99	22	,	,	PUNCT
cana-6282	99	23	𝑥4	𝑥4	NOUN
cana-6282	99	24	∈	∈	PROPN
cana-6282	99	25	𝑆	𝑆	PROPN
cana-6282	99	26	,	,	PUNCT
cana-6282	99	27	satisfies	satisfy	VERB
cana-6282	99	28	the	the	DET
cana-6282	99	29	following	follow	VERB
cana-6282	99	30	functional	functional	ADJ
cana-6282	99	31	equation	equation	NOUN
cana-6282	99	32	communications	communication	NOUN
cana-6282	99	33	on	on	ADP
cana-6282	99	34	applied	apply	VERB
cana-6282	99	35	nonlinear	nonlinear	ADJ
cana-6282	99	36	analysis	analysis	NOUN
cana-6282	99	37	issn	issn	NOUN
cana-6282	99	38	:	:	PUNCT
cana-6282	99	39	1074	1074	NUM
cana-6282	99	40	-	-	PUNCT
cana-6282	99	41	133x	133x	NUM
cana-6282	99	42	vol	vol	VERB
cana-6282	99	43	32	32	NUM
cana-6282	99	44	no	no	NOUN
cana-6282	99	45	.	.	PUNCT
cana-6282	100	1	10s	10	NOUN
cana-6282	100	2	(	(	PUNCT
cana-6282	100	3	2025	2025	NUM
cana-6282	100	4	)	)	PUNCT
cana-6282	100	5	3689	3689	NUM
cana-6282	100	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-6282	100	7	2𝐷𝑓(𝑥1	2𝐷𝑓(𝑥1	NUM
cana-6282	100	8	,	,	PUNCT
cana-6282	100	9	𝑥2	𝑥2	NOUN
cana-6282	100	10	,	,	PUNCT
cana-6282	100	11	𝑥3	𝑥3	NOUN
cana-6282	100	12	,	,	PUNCT
cana-6282	100	13	𝑥4	𝑥4	NOUN
cana-6282	100	14	)	)	PUNCT
cana-6282	101	1	+	+	ADP
cana-6282	101	2	𝐷𝑓(𝑥1	𝐷𝑓(𝑥1	ADJ
cana-6282	101	3	⋅	⋅	ADJ
cana-6282	101	4	𝑥2	𝑥2	NOUN
cana-6282	101	5	,	,	PUNCT
cana-6282	101	6	𝑎	𝑎	NOUN
cana-6282	101	7	,	,	PUNCT
cana-6282	101	8	𝑥3	𝑥3	ADJ
cana-6282	101	9	⋅	⋅	PROPN
cana-6282	101	10	𝑥4	𝑥4	NOUN
cana-6282	101	11	,	,	PUNCT
cana-6282	101	12	𝑏	𝑏	NOUN
cana-6282	101	13	)	)	PUNCT
cana-6282	101	14	+	+	PROPN
cana-6282	101	15	𝐷𝑓(𝑥1	𝐷𝑓(𝑥1	ADJ
cana-6282	101	16	⋅	⋅	PROPN
cana-6282	101	17	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	101	18	)	)	PUNCT
cana-6282	101	19	,	,	PUNCT
cana-6282	101	20	𝑎	𝑎	X
cana-6282	101	21	,	,	PUNCT
cana-6282	101	22	𝑥3	𝑥3	ADJ
cana-6282	101	23	⋅	⋅	PROPN
cana-6282	101	24	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	101	25	)	)	PUNCT
cana-6282	101	26	,	,	PUNCT
cana-6282	101	27	𝑏	𝑏	NOUN
cana-6282	101	28	)	)	PUNCT
cana-6282	101	29	=	=	NOUN
cana-6282	101	30	𝐷𝑓(𝑥1	𝐷𝑓(𝑥1	ADJ
cana-6282	101	31	,	,	PUNCT
cana-6282	101	32	𝑥2	𝑥2	PROPN
cana-6282	101	33	⋅	⋅	PROPN
cana-6282	101	34	𝑎	𝑎	PROPN
cana-6282	101	35	,	,	PUNCT
cana-6282	101	36	𝑥3	𝑥3	NOUN
cana-6282	101	37	,	,	PUNCT
cana-6282	101	38	𝑥4	𝑥4	VERB
cana-6282	101	39	⋅	⋅	PROPN
cana-6282	101	40	𝑏	𝑏	PROPN
cana-6282	101	41	)	)	PUNCT
cana-6282	101	42	+	+	NOUN
cana-6282	101	43	𝐷𝑓(𝑥1	𝐷𝑓(𝑥1	ADJ
cana-6282	101	44	,	,	PUNCT
cana-6282	101	45	𝑥2	𝑥2	PROPN
cana-6282	101	46	⋅	⋅	PROPN
cana-6282	101	47	𝜎(𝑎	𝜎(𝑎	PROPN
cana-6282	101	48	)	)	PUNCT
cana-6282	101	49	,	,	PUNCT
cana-6282	101	50	𝑥3	𝑥3	NOUN
cana-6282	101	51	,	,	PUNCT
cana-6282	101	52	𝑥4	𝑥4	PROPN
cana-6282	101	53	⋅	⋅	PROPN
cana-6282	101	54	𝜏(𝑏	𝜏(𝑏	PROPN
cana-6282	101	55	)	)	PUNCT
cana-6282	101	56	)	)	PUNCT
cana-6282	102	1	+2𝐷𝑓(𝑥2	+2𝐷𝑓(𝑥2	PROPN
cana-6282	102	2	,	,	PUNCT
cana-6282	102	3	𝑎	𝑎	NOUN
cana-6282	102	4	,	,	PUNCT
cana-6282	102	5	𝑥4	𝑥4	NOUN
cana-6282	102	6	,	,	PUNCT
cana-6282	102	7	𝑏	𝑏	NOUN
cana-6282	102	8	)	)	PUNCT
cana-6282	102	9	(	(	PUNCT
cana-6282	102	10	2.2	2.2	NUM
cana-6282	102	11	)	)	PUNCT
cana-6282	102	12	for	for	ADP
cana-6282	102	13	all	all	DET
cana-6282	102	14	𝑥1	𝑥1	NOUN
cana-6282	102	15	,	,	PUNCT
cana-6282	102	16	𝑥2	𝑥2	NOUN
cana-6282	102	17	,	,	PUNCT
cana-6282	102	18	𝑥3	𝑥3	NOUN
cana-6282	102	19	,	,	PUNCT
cana-6282	102	20	𝑥4	𝑥4	NOUN
cana-6282	102	21	,	,	PUNCT
cana-6282	102	22	𝑎	𝑎	NOUN
cana-6282	102	23	,	,	PUNCT
cana-6282	102	24	𝑏	𝑏	PRON
cana-6282	102	25	∈	∈	ADJ
cana-6282	102	26	𝑆.	𝑆.	NOUN
cana-6282	102	27	proof	proof	NOUN
cana-6282	102	28	.	.	PUNCT
cana-6282	103	1	when	when	SCONJ
cana-6282	103	2	we	we	PRON
cana-6282	103	3	evaluate	evaluate	VERB
cana-6282	103	4	the	the	DET
cana-6282	103	5	left	left	ADJ
cana-6282	103	6	and	and	CCONJ
cana-6282	103	7	right	right	ADJ
cana-6282	103	8	sides	side	NOUN
cana-6282	103	9	of	of	ADP
cana-6282	103	10	(	(	PUNCT
cana-6282	103	11	2.2	2.2	NUM
cana-6282	103	12	)	)	PUNCT
cana-6282	103	13	,	,	PUNCT
cana-6282	103	14	we	we	PRON
cana-6282	103	15	obtain	obtain	VERB
cana-6282	103	16	2𝐷𝑓(𝑥1	2𝐷𝑓(𝑥1	PRON
cana-6282	103	17	,	,	PUNCT
cana-6282	103	18	𝑥2	𝑥2	NOUN
cana-6282	103	19	,	,	PUNCT
cana-6282	103	20	𝑥3	𝑥3	NOUN
cana-6282	103	21	,	,	PUNCT
cana-6282	103	22	𝑥4	𝑥4	ADJ
cana-6282	103	23	)	)	PUNCT
cana-6282	104	1	+	+	CCONJ
cana-6282	104	2	𝐷𝑓(𝑥1	𝐷𝑓(𝑥1	ADJ
cana-6282	104	3	⋅	⋅	PROPN
cana-6282	104	4	𝑥2	𝑥2	NOUN
cana-6282	104	5	,	,	PUNCT
cana-6282	104	6	𝑎	𝑎	NOUN
cana-6282	104	7	,	,	PUNCT
cana-6282	104	8	𝑥3	𝑥3	ADJ
cana-6282	104	9	⋅	⋅	PROPN
cana-6282	104	10	𝑥4	𝑥4	NOUN
cana-6282	104	11	,	,	PUNCT
cana-6282	104	12	𝑏	𝑏	NOUN
cana-6282	104	13	)	)	PUNCT
cana-6282	104	14	+	+	NUM
cana-6282	104	15	𝐷𝑓(𝑥1	𝐷𝑓(𝑥1	PROPN
cana-6282	104	16	⋅	⋅	PROPN
cana-6282	104	17	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	104	18	)	)	PUNCT
cana-6282	104	19	,	,	PUNCT
cana-6282	104	20	𝑎	𝑎	X
cana-6282	104	21	,	,	PUNCT
cana-6282	104	22	𝑥3	𝑥3	ADJ
cana-6282	104	23	⋅	⋅	PROPN
cana-6282	104	24	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	104	25	)	)	PUNCT
cana-6282	104	26	,	,	PUNCT
cana-6282	104	27	𝑏	𝑏	NOUN
cana-6282	104	28	)	)	PUNCT
cana-6282	104	29	=	=	SYM
cana-6282	104	30	2𝑓(𝑥1	2𝑓(𝑥1	NUM
cana-6282	104	31	⋅	⋅	PROPN
cana-6282	104	32	𝑥2	𝑥2	NOUN
cana-6282	104	33	,	,	PUNCT
cana-6282	104	34	𝑥3	𝑥3	ADJ
cana-6282	104	35	⋅	⋅	PROPN
cana-6282	104	36	𝑥4	𝑥4	NOUN
cana-6282	104	37	)	)	PUNCT
cana-6282	104	38	+	+	CCONJ
cana-6282	104	39	2𝑓(𝑥1	2𝑓(𝑥1	NUM
cana-6282	104	40	⋅	⋅	PROPN
cana-6282	104	41	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	104	42	)	)	PUNCT
cana-6282	104	43	,	,	PUNCT
cana-6282	104	44	𝑥3	𝑥3	ADJ
cana-6282	104	45	⋅	⋅	PROPN
cana-6282	104	46	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	104	47	)	)	PUNCT
cana-6282	104	48	)	)	PUNCT
cana-6282	105	1	−	−	PROPN
cana-6282	105	2	4𝑓(𝑥1	4𝑓(𝑥1	X
cana-6282	105	3	,	,	PUNCT
cana-6282	105	4	𝑥3	𝑥3	NOUN
cana-6282	105	5	)	)	PUNCT
cana-6282	105	6	−	−	PROPN
cana-6282	105	7	4𝑓(𝑥2	4𝑓(𝑥2	NUM
cana-6282	105	8	,	,	PUNCT
cana-6282	105	9	𝑥4	𝑥4	PROPN
cana-6282	105	10	)	)	PUNCT
cana-6282	106	1	+	+	ADP
cana-6282	106	2	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	106	3	⋅	⋅	ADJ
cana-6282	106	4	𝑥2	𝑥2	PROPN
cana-6282	106	5	⋅	⋅	PROPN
cana-6282	106	6	𝑎	𝑎	PROPN
cana-6282	106	7	,	,	PUNCT
cana-6282	106	8	𝑥3	𝑥3	ADJ
cana-6282	106	9	⋅	⋅	PROPN
cana-6282	106	10	𝑥4	𝑥4	PROPN
cana-6282	106	11	⋅	⋅	PROPN
cana-6282	106	12	𝑏	𝑏	PROPN
cana-6282	106	13	)	)	PUNCT
cana-6282	106	14	+	+	CCONJ
cana-6282	106	15	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	106	16	⋅	⋅	ADJ
cana-6282	106	17	𝑥2	𝑥2	PROPN
cana-6282	106	18	⋅	⋅	PROPN
cana-6282	106	19	𝜎(𝑎	𝜎(𝑎	PROPN
cana-6282	106	20	)	)	PUNCT
cana-6282	106	21	,	,	PUNCT
cana-6282	106	22	𝑥3	𝑥3	ADJ
cana-6282	106	23	⋅	⋅	PROPN
cana-6282	106	24	𝑥4	𝑥4	PROPN
cana-6282	106	25	⋅	⋅	PROPN
cana-6282	106	26	𝜏(𝑏	𝜏(𝑏	PROPN
cana-6282	106	27	)	)	PUNCT
cana-6282	106	28	)	)	PUNCT
cana-6282	107	1	−2𝑓(𝑥1	−2𝑓(𝑥1	PROPN
cana-6282	107	2	⋅	⋅	PROPN
cana-6282	107	3	𝑥2	𝑥2	PROPN
cana-6282	107	4	,	,	PUNCT
cana-6282	107	5	𝑥3	𝑥3	ADJ
cana-6282	107	6	⋅	⋅	PROPN
cana-6282	107	7	𝑥4	𝑥4	NOUN
cana-6282	107	8	)	)	PUNCT
cana-6282	107	9	−	−	PROPN
cana-6282	107	10	2𝑓(𝑎	2𝑓(𝑎	NUM
cana-6282	107	11	,	,	PUNCT
cana-6282	107	12	𝑏	𝑏	NOUN
cana-6282	107	13	)	)	PUNCT
cana-6282	107	14	+	+	CCONJ
cana-6282	107	15	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	107	16	⋅	⋅	PROPN
cana-6282	107	17	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	107	18	)	)	PUNCT
cana-6282	107	19	⋅	⋅	PROPN
cana-6282	107	20	𝑎	𝑎	NOUN
cana-6282	107	21	,	,	PUNCT
cana-6282	107	22	𝑥3	𝑥3	ADJ
cana-6282	107	23	⋅	⋅	PROPN
cana-6282	107	24	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	107	25	)	)	PUNCT
cana-6282	107	26	⋅	⋅	PROPN
cana-6282	107	27	𝑏	𝑏	NOUN
cana-6282	107	28	)	)	PUNCT
cana-6282	107	29	+	+	VERB
cana-6282	107	30	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	107	31	⋅	⋅	ADJ
cana-6282	107	32	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	107	33	)	)	PUNCT
cana-6282	107	34	⋅	⋅	PROPN
cana-6282	107	35	𝜎(𝑎	𝜎(𝑎	PROPN
cana-6282	107	36	)	)	PUNCT
cana-6282	107	37	,	,	PUNCT
cana-6282	107	38	𝑥3	𝑥3	ADJ
cana-6282	107	39	⋅	⋅	PROPN
cana-6282	107	40	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	107	41	)	)	PUNCT
cana-6282	107	42	⋅	⋅	PROPN
cana-6282	107	43	𝜏(𝑏	𝜏(𝑏	PROPN
cana-6282	107	44	)	)	PUNCT
cana-6282	107	45	)	)	PUNCT
cana-6282	107	46	−	−	PROPN
cana-6282	107	47	2𝑓(𝑥1	2𝑓(𝑥1	NUM
cana-6282	107	48	⋅	⋅	PROPN
cana-6282	107	49	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	107	50	)	)	PUNCT
cana-6282	107	51	,	,	PUNCT
cana-6282	107	52	𝑥3	𝑥3	ADJ
cana-6282	107	53	⋅	⋅	PROPN
cana-6282	107	54	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	107	55	)	)	PUNCT
cana-6282	107	56	)	)	PUNCT
cana-6282	108	1	−	−	PROPN
cana-6282	108	2	2𝑓(𝑎	2𝑓(𝑎	NUM
cana-6282	108	3	,	,	PUNCT
cana-6282	108	4	𝑏	𝑏	NOUN
cana-6282	108	5	)	)	PUNCT
cana-6282	108	6	=	=	SYM
cana-6282	109	1	−4𝑓(𝑥1	−4𝑓(𝑥1	PROPN
cana-6282	109	2	,	,	PUNCT
cana-6282	109	3	𝑥3	𝑥3	NOUN
cana-6282	109	4	)	)	PUNCT
cana-6282	109	5	−	−	PROPN
cana-6282	109	6	4𝑓(𝑥2	4𝑓(𝑥2	NUM
cana-6282	109	7	,	,	PUNCT
cana-6282	109	8	𝑥4	𝑥4	PROPN
cana-6282	109	9	)	)	PUNCT
cana-6282	109	10	+	+	CCONJ
cana-6282	109	11	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	109	12	⋅	⋅	ADJ
cana-6282	109	13	𝑥2	𝑥2	PROPN
cana-6282	109	14	⋅	⋅	PROPN
cana-6282	109	15	𝑎	𝑎	PROPN
cana-6282	109	16	,	,	PUNCT
cana-6282	109	17	𝑥3	𝑥3	ADJ
cana-6282	109	18	⋅	⋅	PROPN
cana-6282	109	19	𝑥4	𝑥4	PROPN
cana-6282	109	20	⋅	⋅	PROPN
cana-6282	109	21	𝑏	𝑏	PROPN
cana-6282	109	22	)	)	PUNCT
cana-6282	109	23	+	+	CCONJ
cana-6282	109	24	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	109	25	⋅	⋅	ADJ
cana-6282	109	26	𝑥2	𝑥2	PROPN
cana-6282	109	27	⋅	⋅	PROPN
cana-6282	109	28	𝜎(𝑎	𝜎(𝑎	PROPN
cana-6282	109	29	)	)	PUNCT
cana-6282	109	30	,	,	PUNCT
cana-6282	109	31	𝑥3	𝑥3	ADJ
cana-6282	109	32	⋅	⋅	PROPN
cana-6282	109	33	𝑥4	𝑥4	PROPN
cana-6282	109	34	⋅	⋅	PROPN
cana-6282	109	35	𝜏(𝑏	𝜏(𝑏	PROPN
cana-6282	109	36	)	)	PUNCT
cana-6282	109	37	)	)	PUNCT
cana-6282	109	38	−4𝑓(𝑎	−4𝑓(𝑎	NOUN
cana-6282	109	39	,	,	PUNCT
cana-6282	109	40	𝑏	𝑏	NOUN
cana-6282	109	41	)	)	PUNCT
cana-6282	109	42	+	+	CCONJ
cana-6282	109	43	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	109	44	⋅	⋅	PROPN
cana-6282	109	45	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	109	46	)	)	PUNCT
cana-6282	109	47	⋅	⋅	PROPN
cana-6282	109	48	𝑎	𝑎	NOUN
cana-6282	109	49	,	,	PUNCT
cana-6282	109	50	𝑥3	𝑥3	ADJ
cana-6282	109	51	⋅	⋅	PROPN
cana-6282	109	52	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	109	53	)	)	PUNCT
cana-6282	109	54	⋅	⋅	PROPN
cana-6282	109	55	𝑏	𝑏	NOUN
cana-6282	109	56	)	)	PUNCT
cana-6282	109	57	+	+	CCONJ
cana-6282	109	58	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	109	59	⋅	⋅	PROPN
cana-6282	109	60	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	109	61	)	)	PUNCT
cana-6282	109	62	⋅	⋅	PROPN
cana-6282	109	63	𝜎(𝑎	𝜎(𝑎	PROPN
cana-6282	109	64	)	)	PUNCT
cana-6282	109	65	,	,	PUNCT
cana-6282	109	66	𝑥3	𝑥3	ADJ
cana-6282	109	67	⋅	⋅	PROPN
cana-6282	109	68	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	109	69	)	)	PUNCT
cana-6282	109	70	⋅	⋅	PROPN
cana-6282	109	71	𝜏(𝑏	𝜏(𝑏	PROPN
cana-6282	109	72	)	)	PUNCT
cana-6282	109	73	)	)	PUNCT
cana-6282	109	74	,	,	PUNCT
cana-6282	109	75	for	for	ADP
cana-6282	109	76	all	all	DET
cana-6282	109	77	𝑥1	𝑥1	NOUN
cana-6282	109	78	,	,	PUNCT
cana-6282	109	79	𝑥2	𝑥2	NOUN
cana-6282	109	80	,	,	PUNCT
cana-6282	109	81	𝑥3	𝑥3	NOUN
cana-6282	109	82	,	,	PUNCT
cana-6282	109	83	𝑥4	𝑥4	NOUN
cana-6282	109	84	,	,	PUNCT
cana-6282	109	85	𝑎	𝑎	NOUN
cana-6282	109	86	,	,	PUNCT
cana-6282	109	87	𝑏	𝑏	PROPN
cana-6282	109	88	∈	∈	PROPN
cana-6282	109	89	𝑆	𝑆	PROPN
cana-6282	109	90	,	,	PUNCT
cana-6282	109	91	and	and	CCONJ
cana-6282	109	92	𝐷𝑓(𝑥1	𝐷𝑓(𝑥1	PROPN
cana-6282	109	93	,	,	PUNCT
cana-6282	109	94	𝑥2	𝑥2	PROPN
cana-6282	109	95	⋅	⋅	PROPN
cana-6282	109	96	𝑎	𝑎	PROPN
cana-6282	109	97	,	,	PUNCT
cana-6282	109	98	𝑥3	𝑥3	NOUN
cana-6282	109	99	,	,	PUNCT
cana-6282	109	100	𝑥4	𝑥4	VERB
cana-6282	109	101	⋅	⋅	PROPN
cana-6282	109	102	𝑏	𝑏	PROPN
cana-6282	109	103	)	)	PUNCT
cana-6282	109	104	+	+	X
cana-6282	109	105	𝐷𝑓(𝑥1	𝐷𝑓(𝑥1	ADJ
cana-6282	109	106	,	,	PUNCT
cana-6282	109	107	𝑥2	𝑥2	PROPN
cana-6282	109	108	⋅	⋅	PROPN
cana-6282	109	109	𝜎(𝑎	𝜎(𝑎	PROPN
cana-6282	109	110	)	)	PUNCT
cana-6282	109	111	,	,	PUNCT
cana-6282	109	112	𝑥3	𝑥3	NOUN
cana-6282	109	113	,	,	PUNCT
cana-6282	109	114	𝑥4	𝑥4	PROPN
cana-6282	109	115	⋅	⋅	PROPN
cana-6282	109	116	𝜏(𝑏	𝜏(𝑏	PROPN
cana-6282	109	117	)	)	PUNCT
cana-6282	109	118	)	)	PUNCT
cana-6282	110	1	+	+	CCONJ
cana-6282	110	2	2𝐷𝑓(𝑥2	2𝐷𝑓(𝑥2	NUM
cana-6282	110	3	,	,	PUNCT
cana-6282	110	4	𝑎	𝑎	NOUN
cana-6282	110	5	,	,	PUNCT
cana-6282	110	6	𝑥4	𝑥4	NOUN
cana-6282	110	7	,	,	PUNCT
cana-6282	110	8	𝑏	𝑏	NOUN
cana-6282	110	9	)	)	PUNCT
cana-6282	110	10	=	=	PUNCT
cana-6282	110	11	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	110	12	⋅	⋅	ADJ
cana-6282	110	13	𝑥2	𝑥2	PROPN
cana-6282	110	14	⋅	⋅	PROPN
cana-6282	110	15	𝑎	𝑎	PROPN
cana-6282	110	16	,	,	PUNCT
cana-6282	110	17	𝑥3	𝑥3	ADJ
cana-6282	110	18	⋅	⋅	PROPN
cana-6282	110	19	𝑥4	𝑥4	PROPN
cana-6282	110	20	⋅	⋅	PROPN
cana-6282	110	21	𝑏	𝑏	PROPN
cana-6282	110	22	)	)	PUNCT
cana-6282	111	1	+	+	VERB
cana-6282	111	2	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	111	3	⋅	⋅	ADJ
cana-6282	111	4	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	111	5	)	)	PUNCT
cana-6282	111	6	⋅	⋅	PROPN
cana-6282	111	7	𝜎(𝑎	𝜎(𝑎	PROPN
cana-6282	111	8	)	)	PUNCT
cana-6282	111	9	,	,	PUNCT
cana-6282	111	10	𝑥3	𝑥3	ADJ
cana-6282	111	11	⋅	⋅	PROPN
cana-6282	111	12	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	111	13	)	)	PUNCT
cana-6282	111	14	⋅	⋅	PROPN
cana-6282	111	15	𝜏(𝑏	𝜏(𝑏	PROPN
cana-6282	111	16	)	)	PUNCT
cana-6282	111	17	)	)	PUNCT
cana-6282	112	1	−	−	ADP
cana-6282	112	2	2𝑓(𝑥1	2𝑓(𝑥1	NUM
cana-6282	112	3	,	,	PUNCT
cana-6282	112	4	𝑥3	𝑥3	NOUN
cana-6282	112	5	)	)	PUNCT
cana-6282	112	6	−	−	PROPN
cana-6282	112	7	2𝑓(𝑥2	2𝑓(𝑥2	NUM
cana-6282	112	8	⋅	⋅	NUM
cana-6282	112	9	𝑎	𝑎	NOUN
cana-6282	112	10	,	,	PUNCT
cana-6282	112	11	𝑥4	𝑥4	ADJ
cana-6282	112	12	⋅	⋅	PROPN
cana-6282	112	13	𝑏	𝑏	NOUN
cana-6282	112	14	)	)	PUNCT
cana-6282	112	15	+	+	ADP
cana-6282	112	16	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	112	17	⋅	⋅	ADJ
cana-6282	112	18	𝑥2	𝑥2	PROPN
cana-6282	112	19	⋅	⋅	PROPN
cana-6282	112	20	𝜎(𝑎	𝜎(𝑎	PROPN
cana-6282	112	21	)	)	PUNCT
cana-6282	112	22	,	,	PUNCT
cana-6282	112	23	𝑥3	𝑥3	ADJ
cana-6282	112	24	⋅	⋅	PROPN
cana-6282	112	25	𝑥4	𝑥4	PROPN
cana-6282	112	26	⋅	⋅	PROPN
cana-6282	112	27	𝜏(𝑏	𝜏(𝑏	PROPN
cana-6282	112	28	)	)	PUNCT
cana-6282	112	29	)	)	PUNCT
cana-6282	113	1	+	+	CCONJ
cana-6282	113	2	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	113	3	⋅	⋅	PROPN
cana-6282	113	4	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	113	5	)	)	PUNCT
cana-6282	113	6	⋅	⋅	PROPN
cana-6282	113	7	𝑎	𝑎	NOUN
cana-6282	113	8	,	,	PUNCT
cana-6282	113	9	𝑥3	𝑥3	ADJ
cana-6282	113	10	⋅	⋅	PROPN
cana-6282	113	11	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	113	12	)	)	PUNCT
cana-6282	113	13	⋅	⋅	PROPN
cana-6282	113	14	𝑏	𝑏	NOUN
cana-6282	113	15	)	)	PUNCT
cana-6282	113	16	−	−	PROPN
cana-6282	113	17	2𝑓(𝑥1	2𝑓(𝑥1	NUM
cana-6282	113	18	,	,	PUNCT
cana-6282	113	19	𝑥3	𝑥3	NOUN
cana-6282	113	20	)	)	PUNCT
cana-6282	113	21	−2𝑓(𝑥2	−2𝑓(𝑥2	PROPN
cana-6282	113	22	⋅	⋅	PROPN
cana-6282	113	23	𝜎(𝑎	𝜎(𝑎	PROPN
cana-6282	113	24	)	)	PUNCT
cana-6282	113	25	,	,	PUNCT
cana-6282	113	26	𝑥4𝜏(𝑏	𝑥4𝜏(𝑏	PROPN
cana-6282	113	27	)	)	PUNCT
cana-6282	113	28	)	)	PUNCT
cana-6282	114	1	+	+	CCONJ
cana-6282	114	2	2𝑓(𝑥2	2𝑓(𝑥2	NUM
cana-6282	114	3	⋅	⋅	NUM
cana-6282	114	4	𝑎	𝑎	NOUN
cana-6282	114	5	,	,	PUNCT
cana-6282	114	6	𝑥4	𝑥4	ADJ
cana-6282	114	7	⋅	⋅	PROPN
cana-6282	114	8	𝑏	𝑏	NOUN
cana-6282	114	9	)	)	PUNCT
cana-6282	114	10	+	+	CCONJ
cana-6282	114	11	2𝑓(𝑥2	2𝑓(𝑥2	NUM
cana-6282	114	12	⋅	⋅	PROPN
cana-6282	114	13	𝜎(𝑎	𝜎(𝑎	NOUN
cana-6282	114	14	)	)	PUNCT
cana-6282	114	15	,	,	PUNCT
cana-6282	114	16	𝑥4𝜏(𝑏	𝑥4𝜏(𝑏	PROPN
cana-6282	114	17	)	)	PUNCT
cana-6282	114	18	)	)	PUNCT
cana-6282	115	1	−	−	PROPN
cana-6282	115	2	4(𝑥2	4(𝑥2	NUM
cana-6282	115	3	,	,	PUNCT
cana-6282	115	4	𝑥4	𝑥4	ADJ
cana-6282	115	5	)	)	PUNCT
cana-6282	115	6	−	−	PROPN
cana-6282	115	7	4𝑓(𝑎	4𝑓(𝑎	NUM
cana-6282	115	8	,	,	PUNCT
cana-6282	115	9	𝑏	𝑏	NOUN
cana-6282	115	10	)	)	PUNCT
cana-6282	115	11	=	=	SYM
cana-6282	116	1	−4𝑓(𝑥1	−4𝑓(𝑥1	PROPN
cana-6282	116	2	,	,	PUNCT
cana-6282	116	3	𝑥3	𝑥3	NOUN
cana-6282	116	4	)	)	PUNCT
cana-6282	116	5	−	−	PROPN
cana-6282	116	6	4𝑓(𝑥2	4𝑓(𝑥2	NUM
cana-6282	116	7	,	,	PUNCT
cana-6282	116	8	𝑥4	𝑥4	PROPN
cana-6282	116	9	)	)	PUNCT
cana-6282	116	10	+	+	CCONJ
cana-6282	116	11	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	116	12	⋅	⋅	ADJ
cana-6282	116	13	𝑥2	𝑥2	PROPN
cana-6282	116	14	⋅	⋅	PROPN
cana-6282	116	15	𝑎	𝑎	PROPN
cana-6282	116	16	,	,	PUNCT
cana-6282	116	17	𝑥3	𝑥3	ADJ
cana-6282	116	18	⋅	⋅	PROPN
cana-6282	116	19	𝑥4	𝑥4	PROPN
cana-6282	116	20	⋅	⋅	PROPN
cana-6282	116	21	𝑏	𝑏	PROPN
cana-6282	116	22	)	)	PUNCT
cana-6282	116	23	+	+	CCONJ
cana-6282	116	24	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	116	25	⋅	⋅	ADJ
cana-6282	116	26	𝑥2	𝑥2	PROPN
cana-6282	116	27	⋅	⋅	PROPN
cana-6282	116	28	𝜎(𝑎	𝜎(𝑎	PROPN
cana-6282	116	29	)	)	PUNCT
cana-6282	116	30	,	,	PUNCT
cana-6282	116	31	𝑥3	𝑥3	ADJ
cana-6282	116	32	⋅	⋅	PROPN
cana-6282	116	33	𝑥4	𝑥4	PROPN
cana-6282	116	34	⋅	⋅	PROPN
cana-6282	116	35	𝜏(𝑏	𝜏(𝑏	PROPN
cana-6282	116	36	)	)	PUNCT
cana-6282	116	37	)	)	PUNCT
cana-6282	116	38	−4𝑓(𝑎	−4𝑓(𝑎	NOUN
cana-6282	116	39	,	,	PUNCT
cana-6282	116	40	𝑏	𝑏	NOUN
cana-6282	116	41	)	)	PUNCT
cana-6282	116	42	+	+	CCONJ
cana-6282	116	43	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	116	44	⋅	⋅	PROPN
cana-6282	116	45	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	116	46	)	)	PUNCT
cana-6282	116	47	⋅	⋅	PROPN
cana-6282	116	48	𝑎	𝑎	NOUN
cana-6282	116	49	,	,	PUNCT
cana-6282	116	50	𝑥3	𝑥3	ADJ
cana-6282	116	51	⋅	⋅	PROPN
cana-6282	116	52	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	116	53	)	)	PUNCT
cana-6282	116	54	⋅	⋅	PROPN
cana-6282	116	55	𝑏	𝑏	NOUN
cana-6282	116	56	)	)	PUNCT
cana-6282	116	57	+	+	CCONJ
cana-6282	116	58	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	116	59	⋅	⋅	PROPN
cana-6282	116	60	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	116	61	)	)	PUNCT
cana-6282	116	62	⋅	⋅	PROPN
cana-6282	116	63	𝜎(𝑎	𝜎(𝑎	PROPN
cana-6282	116	64	)	)	PUNCT
cana-6282	116	65	,	,	PUNCT
cana-6282	116	66	𝑥3	𝑥3	ADJ
cana-6282	116	67	⋅	⋅	PROPN
cana-6282	116	68	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	116	69	)	)	PUNCT
cana-6282	116	70	⋅	⋅	PROPN
cana-6282	116	71	𝜏(𝑏	𝜏(𝑏	PROPN
cana-6282	116	72	)	)	PUNCT
cana-6282	116	73	)	)	PUNCT
cana-6282	116	74	,	,	PUNCT
cana-6282	116	75	for	for	ADP
cana-6282	116	76	all	all	DET
cana-6282	116	77	𝑥1	𝑥1	NOUN
cana-6282	116	78	,	,	PUNCT
cana-6282	116	79	𝑥2	𝑥2	NOUN
cana-6282	116	80	,	,	PUNCT
cana-6282	116	81	𝑥3	𝑥3	NOUN
cana-6282	116	82	,	,	PUNCT
cana-6282	116	83	𝑥4	𝑥4	NOUN
cana-6282	116	84	,	,	PUNCT
cana-6282	116	85	𝑎	𝑎	NOUN
cana-6282	116	86	,	,	PUNCT
cana-6282	116	87	𝑏	𝑏	PROPN
cana-6282	116	88	∈	∈	PROPN
cana-6282	116	89	𝑆.	𝑆.	PROPN
cana-6282	116	90	therefore	therefore	ADV
cana-6282	116	91	,	,	PUNCT
cana-6282	116	92	(	(	PUNCT
cana-6282	116	93	2.2	2.2	NUM
cana-6282	116	94	)	)	PUNCT
cana-6282	116	95	holds	hold	VERB
cana-6282	116	96	.	.	PUNCT
cana-6282	117	1	in	in	ADP
cana-6282	117	2	the	the	DET
cana-6282	117	3	next	next	ADJ
cana-6282	117	4	theorem	theorem	NOUN
cana-6282	117	5	,	,	PUNCT
cana-6282	117	6	we	we	PRON
cana-6282	117	7	study	study	VERB
cana-6282	117	8	the	the	DET
cana-6282	117	9	hyperstability	hyperstability	NOUN
cana-6282	117	10	of	of	ADP
cana-6282	117	11	the	the	DET
cana-6282	117	12	equation	equation	NOUN
cana-6282	117	13	(	(	PUNCT
cana-6282	117	14	1.5	1.5	NUM
cana-6282	117	15	)	)	PUNCT
cana-6282	117	16	.	.	PUNCT
cana-6282	118	1	specifically	specifically	ADV
cana-6282	118	2	,	,	PUNCT
cana-6282	118	3	we	we	PRON
cana-6282	118	4	prove	prove	VERB
cana-6282	118	5	that	that	SCONJ
cana-6282	118	6	the	the	DET
cana-6282	118	7	functional	functional	ADJ
cana-6282	118	8	equation	equation	NOUN
cana-6282	118	9	(	(	PUNCT
cana-6282	118	10	1.5	1.5	NUM
cana-6282	118	11	)	)	PUNCT
cana-6282	118	12	is	be	AUX
cana-6282	118	13	hyperstable	hyperstable	ADJ
cana-6282	118	14	in	in	ADP
cana-6282	118	15	the	the	DET
cana-6282	118	16	class	class	NOUN
cana-6282	118	17	of	of	ADP
cana-6282	118	18	functions	function	NOUN
cana-6282	118	19	𝑓	𝑓	X
cana-6282	118	20	:	:	PUNCT
cana-6282	118	21	𝑆	𝑆	PROPN
cana-6282	118	22	→	→	SYM
cana-6282	118	23	𝑋	𝑋	PROPN
cana-6282	118	24	for	for	ADP
cana-6282	118	25	certain	certain	ADJ
cana-6282	118	26	asymptotic	asymptotic	ADJ
cana-6282	118	27	properties	property	NOUN
cana-6282	118	28	of	of	ADP
cana-6282	118	29	the	the	DET
cana-6282	118	30	control	control	NOUN
cana-6282	118	31	function	function	VERB
cana-6282	118	32	휀	휀	NOUN
cana-6282	118	33	:	:	PUNCT
cana-6282	118	34	𝑆4	𝑆4	PROPN
cana-6282	118	35	→	→	SYM
cana-6282	118	36	ℝ+	ℝ+	PROPN
cana-6282	118	37	.	.	X
cana-6282	118	38	theorem	theorem	PROPN
cana-6282	118	39	2.2	2.2	NUM
cana-6282	118	40	.	.	PUNCT
cana-6282	119	1	let	let	VERB
cana-6282	119	2	휀	휀	NOUN
cana-6282	119	3	:	:	PUNCT
cana-6282	119	4	𝑆4	𝑆4	PROPN
cana-6282	119	5	→	→	SYM
cana-6282	119	6	ℝ+be	ℝ+be	NUM
cana-6282	119	7	a	a	DET
cana-6282	119	8	function	function	NOUN
cana-6282	119	9	,	,	PUNCT
cana-6282	119	10	and	and	CCONJ
cana-6282	119	11	consider	consider	VERB
cana-6282	119	12	that	that	SCONJ
cana-6282	119	13	there	there	PRON
cana-6282	119	14	exists	exist	VERB
cana-6282	119	15	a	a	DET
cana-6282	119	16	sequence	sequence	NOUN
cana-6282	119	17	{	{	PUNCT
cana-6282	119	18	𝑢𝑛}𝑛∈ℕ	𝑢𝑛}𝑛∈ℕ	NOUN
cana-6282	119	19	in	in	ADP
cana-6282	119	20	𝑆	𝑆	PROPN
cana-6282	119	21	that	that	PRON
cana-6282	119	22	satisfies	satisfy	VERB
cana-6282	119	23	the	the	DET
cana-6282	119	24	following	follow	VERB
cana-6282	119	25	two	two	NUM
cana-6282	119	26	conditions	condition	NOUN
cana-6282	119	27	lim	lim	NOUN
cana-6282	119	28	𝑛→∞	𝑛→∞	NUM
cana-6282	119	29	 	 	SPACE
cana-6282	119	30	inf휀(𝑥1	inf휀(𝑥1	ADJ
cana-6282	119	31	,	,	PUNCT
cana-6282	119	32	𝑥2	𝑥2	PROPN
cana-6282	119	33	⋅	⋅	PROPN
cana-6282	119	34	𝑢𝑛	𝑢𝑛	PROPN
cana-6282	119	35	,	,	PUNCT
cana-6282	119	36	𝑥3	𝑥3	NOUN
cana-6282	119	37	,	,	PUNCT
cana-6282	119	38	𝑥4	𝑥4	PROPN
cana-6282	119	39	⋅	⋅	PROPN
cana-6282	119	40	𝑢𝑛	𝑢𝑛	NUM
cana-6282	119	41	)	)	PUNCT
cana-6282	119	42	=	=	SYM
cana-6282	119	43	0	0	PUNCT
cana-6282	120	1	(	(	PUNCT
cana-6282	120	2	2.3	2.3	NUM
cana-6282	120	3	)	)	PUNCT
cana-6282	120	4	and	and	CCONJ
cana-6282	120	5	lim	lim	PROPN
cana-6282	120	6	𝑛→∞	𝑛→∞	NUM
cana-6282	120	7	 	 	SPACE
cana-6282	120	8	inf휀(𝑥1	inf휀(𝑥1	ADJ
cana-6282	120	9	,	,	PUNCT
cana-6282	120	10	𝑥2	𝑥2	PROPN
cana-6282	120	11	⋅	⋅	PROPN
cana-6282	120	12	𝜎(𝑢𝑛	𝜎(𝑢𝑛	NUM
cana-6282	120	13	)	)	PUNCT
cana-6282	120	14	,	,	PUNCT
cana-6282	120	15	𝑥3	𝑥3	NOUN
cana-6282	120	16	,	,	PUNCT
cana-6282	120	17	𝑥4	𝑥4	ADJ
cana-6282	120	18	⋅	⋅	X
cana-6282	120	19	𝜏(𝑢𝑛	𝜏(𝑢𝑛	NOUN
cana-6282	120	20	)	)	PUNCT
cana-6282	120	21	)	)	PUNCT
cana-6282	121	1	=	=	SYM
cana-6282	121	2	0	0	NUM
cana-6282	121	3	,	,	PUNCT
cana-6282	121	4	(	(	PUNCT
cana-6282	121	5	2.4	2.4	NUM
cana-6282	121	6	)	)	PUNCT
cana-6282	121	7	for	for	ADP
cana-6282	121	8	all	all	DET
cana-6282	121	9	𝑥1	𝑥1	NOUN
cana-6282	121	10	,	,	PUNCT
cana-6282	121	11	𝑥2	𝑥2	NOUN
cana-6282	121	12	,	,	PUNCT
cana-6282	121	13	𝑥3	𝑥3	NOUN
cana-6282	121	14	,	,	PUNCT
cana-6282	121	15	𝑥4	𝑥4	NOUN
cana-6282	121	16	∈	∈	PROPN
cana-6282	121	17	𝑆.	𝑆.	PROPN
cana-6282	121	18	assume	assume	VERB
cana-6282	121	19	that	that	SCONJ
cana-6282	121	20	𝑓	𝑓	X
cana-6282	121	21	:	:	PUNCT
cana-6282	121	22	𝑆2	𝑆2	PROPN
cana-6282	121	23	→	→	PUNCT
cana-6282	121	24	𝑋	𝑋	NOUN
cana-6282	121	25	satisfies	satisfy	VERB
cana-6282	121	26	the	the	DET
cana-6282	121	27	inequality	inequality	NOUN
cana-6282	121	28	‖𝑓(𝑥1	‖𝑓(𝑥1	ADJ
cana-6282	121	29	⋅	⋅	PROPN
cana-6282	121	30	𝑥2	𝑥2	NOUN
cana-6282	121	31	,	,	PUNCT
cana-6282	121	32	𝑥3	𝑥3	ADJ
cana-6282	121	33	⋅	⋅	PROPN
cana-6282	121	34	𝑥4	𝑥4	NOUN
cana-6282	121	35	)	)	PUNCT
cana-6282	121	36	+	+	CCONJ
cana-6282	121	37	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	121	38	⋅	⋅	PROPN
cana-6282	121	39	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	121	40	)	)	PUNCT
cana-6282	121	41	,	,	PUNCT
cana-6282	121	42	𝑥3	𝑥3	ADJ
cana-6282	121	43	⋅	⋅	PROPN
cana-6282	121	44	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	121	45	)	)	PUNCT
cana-6282	121	46	)	)	PUNCT
cana-6282	122	1	−	−	ADP
cana-6282	122	2	2𝑓(𝑥1	2𝑓(𝑥1	NUM
cana-6282	122	3	,	,	PUNCT
cana-6282	122	4	𝑥3	𝑥3	NOUN
cana-6282	122	5	)	)	PUNCT
cana-6282	122	6	−	−	PROPN
cana-6282	123	1	2𝑓(𝑥2	2𝑓(𝑥2	NUM
cana-6282	123	2	,	,	PUNCT
cana-6282	123	3	𝑥4)‖	𝑥4)‖	NOUN
cana-6282	123	4	≤	≤	NUM
cana-6282	123	5	휀(𝑥1	휀(𝑥1	NOUN
cana-6282	123	6	,	,	PUNCT
cana-6282	123	7	𝑥2	𝑥2	NOUN
cana-6282	123	8	,	,	PUNCT
cana-6282	123	9	𝑥3	𝑥3	NOUN
cana-6282	123	10	,	,	PUNCT
cana-6282	123	11	𝑥4	𝑥4	PROPN
cana-6282	123	12	)	)	PUNCT
cana-6282	123	13	(	(	PUNCT
cana-6282	123	14	2.5	2.5	NUM
cana-6282	123	15	)	)	PUNCT
cana-6282	123	16	for	for	ADP
cana-6282	123	17	all	all	DET
cana-6282	123	18	𝑥1	𝑥1	NOUN
cana-6282	123	19	,	,	PUNCT
cana-6282	123	20	𝑥2	𝑥2	NOUN
cana-6282	123	21	,	,	PUNCT
cana-6282	123	22	𝑥3	𝑥3	NOUN
cana-6282	123	23	,	,	PUNCT
cana-6282	123	24	𝑥4	𝑥4	NOUN
cana-6282	123	25	∈	∈	PROPN
cana-6282	123	26	𝑆.	𝑆.	PROPN
cana-6282	123	27	then	then	ADV
cana-6282	123	28	the	the	DET
cana-6282	123	29	functional	functional	ADJ
cana-6282	123	30	equation	equation	NOUN
cana-6282	123	31	(	(	PUNCT
cana-6282	123	32	1.5	1.5	NUM
cana-6282	123	33	)	)	PUNCT
cana-6282	123	34	is	be	AUX
cana-6282	123	35	hyperstable	hyperstable	ADJ
cana-6282	123	36	on	on	ADP
cana-6282	123	37	𝑆.	𝑆.	ADJ
cana-6282	123	38	proof	proof	NOUN
cana-6282	123	39	.	.	PUNCT
cana-6282	124	1	in	in	ADP
cana-6282	124	2	view	view	NOUN
cana-6282	124	3	of	of	ADP
cana-6282	124	4	the	the	DET
cana-6282	124	5	function	function	NOUN
cana-6282	124	6	𝐷𝑓	𝐷𝑓	PROPN
cana-6282	124	7	defined	define	VERB
cana-6282	124	8	by	by	ADP
cana-6282	124	9	(	(	PUNCT
cana-6282	124	10	2.1	2.1	NUM
cana-6282	124	11	)	)	PUNCT
cana-6282	124	12	,	,	PUNCT
cana-6282	124	13	the	the	DET
cana-6282	124	14	inequality	inequality	NOUN
cana-6282	124	15	(	(	PUNCT
cana-6282	124	16	2.5	2.5	NUM
cana-6282	124	17	)	)	PUNCT
cana-6282	124	18	becomes	become	VERB
cana-6282	124	19	𝐷𝑓(𝑥1	𝐷𝑓(𝑥1	PROPN
cana-6282	124	20	,	,	PUNCT
cana-6282	124	21	𝑥2	𝑥2	NOUN
cana-6282	124	22	,	,	PUNCT
cana-6282	124	23	𝑥3	𝑥3	NOUN
cana-6282	124	24	,	,	PUNCT
cana-6282	124	25	𝑥4	𝑥4	NOUN
cana-6282	124	26	)	)	PUNCT
cana-6282	124	27	≤	≤	NOUN
cana-6282	124	28	휀(𝑥1	휀(𝑥1	NOUN
cana-6282	124	29	,	,	PUNCT
cana-6282	124	30	𝑥2	𝑥2	NOUN
cana-6282	124	31	,	,	PUNCT
cana-6282	124	32	𝑥3	𝑥3	NOUN
cana-6282	124	33	,	,	PUNCT
cana-6282	124	34	𝑥4	𝑥4	PROPN
cana-6282	124	35	)	)	PUNCT
cana-6282	124	36	,	,	PUNCT
cana-6282	124	37	𝑥1	𝑥1	NOUN
cana-6282	124	38	,	,	PUNCT
cana-6282	124	39	𝑥2	𝑥2	NOUN
cana-6282	124	40	,	,	PUNCT
cana-6282	124	41	𝑥3	𝑥3	NOUN
cana-6282	124	42	,	,	PUNCT
cana-6282	124	43	𝑥4	𝑥4	NOUN
cana-6282	124	44	∈	∈	PROPN
cana-6282	124	45	𝑆	𝑆	PROPN
cana-6282	124	46	(	(	PUNCT
cana-6282	124	47	2.6	2.6	NUM
cana-6282	124	48	)	)	PUNCT
cana-6282	124	49	using	use	VERB
cana-6282	124	50	lemma	lemma	PROPN
cana-6282	124	51	2.1	2.1	NUM
cana-6282	124	52	,	,	PUNCT
cana-6282	124	53	we	we	PRON
cana-6282	124	54	get	get	VERB
cana-6282	124	55	communications	communication	NOUN
cana-6282	124	56	on	on	ADP
cana-6282	124	57	applied	apply	VERB
cana-6282	124	58	nonlinear	nonlinear	ADJ
cana-6282	124	59	analysis	analysis	NOUN
cana-6282	124	60	issn	issn	NOUN
cana-6282	124	61	:	:	PUNCT
cana-6282	124	62	1074	1074	NUM
cana-6282	124	63	-	-	PUNCT
cana-6282	124	64	133x	133x	NUM
cana-6282	124	65	vol	vol	VERB
cana-6282	124	66	32	32	NUM
cana-6282	124	67	no	no	NOUN
cana-6282	124	68	.	.	PUNCT
cana-6282	125	1	10s	10	NOUN
cana-6282	125	2	(	(	PUNCT
cana-6282	125	3	2025	2025	NUM
cana-6282	125	4	)	)	PUNCT
cana-6282	125	5	3690	3690	NUM
cana-6282	125	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-6282	125	7	2𝐷𝑓(𝑥1	2𝐷𝑓(𝑥1	NUM
cana-6282	125	8	,	,	PUNCT
cana-6282	125	9	𝑥2	𝑥2	NOUN
cana-6282	125	10	,	,	PUNCT
cana-6282	125	11	𝑥3	𝑥3	NOUN
cana-6282	125	12	,	,	PUNCT
cana-6282	125	13	𝑥4	𝑥4	ADJ
cana-6282	125	14	)	)	PUNCT
cana-6282	126	1	+	+	CCONJ
cana-6282	126	2	𝐷𝑓(𝑥1	𝐷𝑓(𝑥1	ADJ
cana-6282	126	3	⋅	⋅	PROPN
cana-6282	126	4	𝑥2	𝑥2	NOUN
cana-6282	126	5	,	,	PUNCT
cana-6282	126	6	𝑎	𝑎	NOUN
cana-6282	126	7	,	,	PUNCT
cana-6282	126	8	𝑥3	𝑥3	ADJ
cana-6282	126	9	⋅	⋅	PROPN
cana-6282	126	10	𝑥4	𝑥4	NOUN
cana-6282	126	11	,	,	PUNCT
cana-6282	126	12	𝑏	𝑏	NOUN
cana-6282	126	13	)	)	PUNCT
cana-6282	126	14	+	+	NUM
cana-6282	126	15	𝐷𝑓(𝑥1	𝐷𝑓(𝑥1	PROPN
cana-6282	126	16	⋅	⋅	PROPN
cana-6282	126	17	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	126	18	)	)	PUNCT
cana-6282	126	19	,	,	PUNCT
cana-6282	126	20	𝑎	𝑎	X
cana-6282	126	21	,	,	PUNCT
cana-6282	126	22	𝑥3	𝑥3	ADJ
cana-6282	126	23	⋅	⋅	PROPN
cana-6282	126	24	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	126	25	)	)	PUNCT
cana-6282	126	26	,	,	PUNCT
cana-6282	126	27	𝑏	𝑏	NOUN
cana-6282	126	28	)	)	PUNCT
cana-6282	126	29	=	=	SYM
cana-6282	126	30	𝐷𝑓(𝑥1	𝐷𝑓(𝑥1	ADJ
cana-6282	126	31	,	,	PUNCT
cana-6282	126	32	𝑥2	𝑥2	PROPN
cana-6282	126	33	⋅	⋅	PROPN
cana-6282	126	34	𝑎	𝑎	PROPN
cana-6282	126	35	,	,	PUNCT
cana-6282	126	36	𝑥3	𝑥3	NOUN
cana-6282	126	37	,	,	PUNCT
cana-6282	126	38	𝑥4	𝑥4	VERB
cana-6282	126	39	⋅	⋅	PROPN
cana-6282	126	40	𝑏	𝑏	PROPN
cana-6282	126	41	)	)	PUNCT
cana-6282	126	42	+	+	X
cana-6282	126	43	𝐷𝑓(𝑥1	𝐷𝑓(𝑥1	ADJ
cana-6282	126	44	,	,	PUNCT
cana-6282	126	45	𝑥2	𝑥2	PROPN
cana-6282	126	46	⋅	⋅	PROPN
cana-6282	126	47	𝜎(𝑎	𝜎(𝑎	PROPN
cana-6282	126	48	)	)	PUNCT
cana-6282	126	49	,	,	PUNCT
cana-6282	126	50	𝑥3	𝑥3	NOUN
cana-6282	126	51	,	,	PUNCT
cana-6282	126	52	𝑥4	𝑥4	PROPN
cana-6282	126	53	⋅	⋅	PROPN
cana-6282	126	54	𝜏(𝑏	𝜏(𝑏	PROPN
cana-6282	126	55	)	)	PUNCT
cana-6282	126	56	)	)	PUNCT
cana-6282	127	1	+	+	CCONJ
cana-6282	127	2	2𝐷𝑓(𝑥2	2𝐷𝑓(𝑥2	NUM
cana-6282	127	3	,	,	PUNCT
cana-6282	127	4	𝑎	𝑎	NOUN
cana-6282	127	5	,	,	PUNCT
cana-6282	127	6	𝑥4	𝑥4	NOUN
cana-6282	127	7	,	,	PUNCT
cana-6282	127	8	𝑏	𝑏	NOUN
cana-6282	127	9	)	)	PUNCT
cana-6282	127	10	(	(	PUNCT
cana-6282	127	11	2.7	2.7	NUM
cana-6282	127	12	)	)	PUNCT
cana-6282	127	13	for	for	ADP
cana-6282	127	14	all	all	DET
cana-6282	127	15	𝑥1	𝑥1	NOUN
cana-6282	127	16	,	,	PUNCT
cana-6282	127	17	𝑥2	𝑥2	NOUN
cana-6282	127	18	,	,	PUNCT
cana-6282	127	19	𝑥3	𝑥3	NOUN
cana-6282	127	20	,	,	PUNCT
cana-6282	127	21	𝑥4	𝑥4	NOUN
cana-6282	127	22	,	,	PUNCT
cana-6282	127	23	𝑎	𝑎	NOUN
cana-6282	127	24	,	,	PUNCT
cana-6282	127	25	𝑏	𝑏	PROPN
cana-6282	127	26	∈	∈	PROPN
cana-6282	127	27	𝑆.	𝑆.	PROPN
cana-6282	127	28	assume	assume	VERB
cana-6282	127	29	that	that	SCONJ
cana-6282	127	30	there	there	PRON
cana-6282	127	31	is	be	VERB
cana-6282	127	32	a	a	DET
cana-6282	127	33	sequence	sequence	NOUN
cana-6282	127	34	{	{	PUNCT
cana-6282	127	35	𝑢𝑛}𝑛∈ℕ	𝑢𝑛}𝑛∈ℕ	NOUN
cana-6282	127	36	of	of	ADP
cana-6282	127	37	elements	element	NOUN
cana-6282	127	38	of	of	ADP
cana-6282	127	39	𝑆	𝑆	PROPN
cana-6282	127	40	that	that	PRON
cana-6282	127	41	satisfying	satisfy	VERB
cana-6282	127	42	the	the	DET
cana-6282	127	43	conditions	condition	NOUN
cana-6282	127	44	(	(	PUNCT
cana-6282	127	45	2.3	2.3	NUM
cana-6282	127	46	)	)	PUNCT
cana-6282	127	47	and	and	CCONJ
cana-6282	127	48	(	(	PUNCT
cana-6282	127	49	2.4	2.4	NUM
cana-6282	127	50	)	)	PUNCT
cana-6282	127	51	.	.	PUNCT
cana-6282	128	1	replacing	replace	VERB
cana-6282	128	2	𝑥2	𝑥2	NOUN
cana-6282	128	3	by	by	ADP
cana-6282	128	4	𝑥2	𝑥2	PROPN
cana-6282	128	5	⋅	⋅	PROPN
cana-6282	128	6	𝑢𝑛	𝑢𝑛	NOUN
cana-6282	128	7	and	and	CCONJ
cana-6282	128	8	𝑥4	𝑥4	PROPN
cana-6282	128	9	by	by	ADP
cana-6282	128	10	𝑥4	𝑥4	PROPN
cana-6282	128	11	⋅	⋅	PROPN
cana-6282	128	12	𝑢𝑛	𝑢𝑛	PROPN
cana-6282	128	13	in	in	ADP
cana-6282	128	14	(	(	PUNCT
cana-6282	128	15	2.5	2.5	NUM
cana-6282	128	16	)	)	PUNCT
cana-6282	128	17	,	,	PUNCT
cana-6282	128	18	where	where	SCONJ
cana-6282	128	19	𝑛	𝑛	DET
cana-6282	128	20	∈	∈	PROPN
cana-6282	128	21	ℕ	ℕ	PROPN
cana-6282	128	22	,	,	PUNCT
cana-6282	128	23	we	we	PRON
cana-6282	128	24	get	get	VERB
cana-6282	128	25	‖𝐷𝑓(𝑥1	‖𝐷𝑓(𝑥1	ADV
cana-6282	128	26	,	,	PUNCT
cana-6282	128	27	𝑥2	𝑥2	PROPN
cana-6282	128	28	⋅	⋅	PROPN
cana-6282	128	29	𝑢𝑛	𝑢𝑛	PROPN
cana-6282	128	30	,	,	PUNCT
cana-6282	128	31	𝑥3	𝑥3	NOUN
cana-6282	128	32	,	,	PUNCT
cana-6282	128	33	𝑥4	𝑥4	ADJ
cana-6282	128	34	⋅	⋅	PROPN
cana-6282	128	35	𝑢𝑛)‖	𝑢𝑛)‖	PROPN
cana-6282	128	36	≤	≤	PROPN
cana-6282	128	37	휀(𝑥1	휀(𝑥1	NOUN
cana-6282	128	38	,	,	PUNCT
cana-6282	128	39	𝑥2	𝑥2	PROPN
cana-6282	128	40	⋅	⋅	PROPN
cana-6282	128	41	𝑢𝑛	𝑢𝑛	PROPN
cana-6282	128	42	,	,	PUNCT
cana-6282	128	43	𝑥3	𝑥3	NOUN
cana-6282	128	44	,	,	PUNCT
cana-6282	128	45	𝑥4	𝑥4	PROPN
cana-6282	128	46	⋅	⋅	PROPN
cana-6282	128	47	𝑢𝑛	𝑢𝑛	NUM
cana-6282	128	48	)	)	PUNCT
cana-6282	128	49	,	,	PUNCT
cana-6282	128	50	𝑥1	𝑥1	NOUN
cana-6282	128	51	,	,	PUNCT
cana-6282	128	52	𝑥2	𝑥2	NOUN
cana-6282	128	53	,	,	PUNCT
cana-6282	128	54	𝑥3	𝑥3	NOUN
cana-6282	128	55	,	,	PUNCT
cana-6282	128	56	𝑥4	𝑥4	NOUN
cana-6282	128	57	∈	∈	PROPN
cana-6282	128	58	𝑆	𝑆	PROPN
cana-6282	128	59	(	(	PUNCT
cana-6282	128	60	2.8	2.8	NUM
cana-6282	128	61	)	)	PUNCT
cana-6282	128	62	in	in	ADP
cana-6282	128	63	view	view	NOUN
cana-6282	128	64	of	of	ADP
cana-6282	128	65	(	(	PUNCT
cana-6282	128	66	2.3	2.3	NUM
cana-6282	128	67	)	)	PUNCT
cana-6282	128	68	,	,	PUNCT
cana-6282	128	69	we	we	PRON
cana-6282	128	70	obtain	obtain	VERB
cana-6282	128	71	lim	lim	PROPN
cana-6282	128	72	𝑛→∞	𝑛→∞	NUM
cana-6282	128	73	 	 	SPACE
cana-6282	128	74	inf	inf	PROPN
cana-6282	128	75	𝐷𝑓(𝑥1	𝐷𝑓(𝑥1	PROPN
cana-6282	128	76	,	,	PUNCT
cana-6282	128	77	𝑥2	𝑥2	PROPN
cana-6282	128	78	⋅	⋅	PROPN
cana-6282	128	79	𝑢𝑛	𝑢𝑛	PROPN
cana-6282	128	80	,	,	PUNCT
cana-6282	128	81	𝑥3	𝑥3	NOUN
cana-6282	128	82	,	,	PUNCT
cana-6282	128	83	𝑥4	𝑥4	PROPN
cana-6282	128	84	⋅	⋅	PROPN
cana-6282	128	85	𝑢𝑛	𝑢𝑛	NOUN
cana-6282	128	86	)	)	PUNCT
cana-6282	128	87	=	=	SYM
cana-6282	128	88	0	0	NUM
cana-6282	128	89	,	,	PUNCT
cana-6282	128	90	𝑥1	𝑥1	NOUN
cana-6282	128	91	,	,	PUNCT
cana-6282	128	92	𝑥2	𝑥2	NOUN
cana-6282	128	93	,	,	PUNCT
cana-6282	128	94	𝑥3	𝑥3	NOUN
cana-6282	128	95	,	,	PUNCT
cana-6282	128	96	𝑥4	𝑥4	NOUN
cana-6282	128	97	∈	∈	PROPN
cana-6282	128	98	𝑆	𝑆	PROPN
cana-6282	128	99	(	(	PUNCT
cana-6282	128	100	2.9	2.9	NUM
cana-6282	128	101	)	)	PUNCT
cana-6282	128	102	replacing	replace	VERB
cana-6282	128	103	𝑥2	𝑥2	NOUN
cana-6282	128	104	by	by	ADP
cana-6282	128	105	𝑥2	𝑥2	PROPN
cana-6282	128	106	⋅	⋅	PROPN
cana-6282	128	107	𝜎(𝑢𝑛	𝜎(𝑢𝑛	NUM
cana-6282	128	108	)	)	PUNCT
cana-6282	128	109	and	and	CCONJ
cana-6282	128	110	𝑥4	𝑥4	VERB
cana-6282	128	111	by	by	ADP
cana-6282	128	112	𝑥4	𝑥4	PROPN
cana-6282	128	113	⋅	⋅	PROPN
cana-6282	128	114	𝜏(𝑢𝑛	𝜏(𝑢𝑛	NOUN
cana-6282	128	115	)	)	PUNCT
cana-6282	128	116	in	in	ADP
cana-6282	128	117	(	(	PUNCT
cana-6282	128	118	2.5	2.5	NUM
cana-6282	128	119	)	)	PUNCT
cana-6282	128	120	,	,	PUNCT
cana-6282	128	121	where	where	SCONJ
cana-6282	128	122	𝑛	𝑛	DET
cana-6282	128	123	∈	∈	PROPN
cana-6282	128	124	ℕ	ℕ	PROPN
cana-6282	128	125	,	,	PUNCT
cana-6282	128	126	we	we	PRON
cana-6282	128	127	find	find	VERB
cana-6282	128	128	‖𝐷𝑓(𝑥1	‖𝐷𝑓(𝑥1	ADJ
cana-6282	128	129	,	,	PUNCT
cana-6282	129	1	𝑥2	𝑥2	PROPN
cana-6282	129	2	⋅	⋅	PROPN
cana-6282	129	3	𝜎(𝑢𝑛	𝜎(𝑢𝑛	NUM
cana-6282	129	4	)	)	PUNCT
cana-6282	129	5	,	,	PUNCT
cana-6282	129	6	𝑥3	𝑥3	NOUN
cana-6282	129	7	,	,	PUNCT
cana-6282	129	8	𝑥4	𝑥4	PROPN
cana-6282	129	9	⋅	⋅	PROPN
cana-6282	129	10	𝜏(𝑢𝑛))‖	𝜏(𝑢𝑛))‖	ADJ
cana-6282	129	11	≤	≤	ADJ
cana-6282	129	12	휀(𝑥1	휀(𝑥1	NOUN
cana-6282	129	13	,	,	PUNCT
cana-6282	129	14	𝑥2	𝑥2	PROPN
cana-6282	129	15	⋅	⋅	PROPN
cana-6282	129	16	𝜎(𝑢𝑛	𝜎(𝑢𝑛	NUM
cana-6282	129	17	)	)	PUNCT
cana-6282	129	18	,	,	PUNCT
cana-6282	129	19	𝑥3	𝑥3	NOUN
cana-6282	129	20	,	,	PUNCT
cana-6282	129	21	𝑥4	𝑥4	ADJ
cana-6282	129	22	⋅	⋅	X
cana-6282	129	23	𝜏(𝑢𝑛	𝜏(𝑢𝑛	NOUN
cana-6282	129	24	)	)	PUNCT
cana-6282	129	25	)	)	PUNCT
cana-6282	129	26	,	,	PUNCT
cana-6282	129	27	𝑥1	𝑥1	NOUN
cana-6282	129	28	,	,	PUNCT
cana-6282	129	29	𝑥2	𝑥2	NOUN
cana-6282	129	30	,	,	PUNCT
cana-6282	129	31	𝑥3	𝑥3	NOUN
cana-6282	129	32	,	,	PUNCT
cana-6282	129	33	𝑥4	𝑥4	NOUN
cana-6282	129	34	∈	∈	PROPN
cana-6282	129	35	𝑆	𝑆	PROPN
cana-6282	129	36	(	(	PUNCT
cana-6282	129	37	2.10	2.10	NUM
cana-6282	129	38	)	)	PUNCT
cana-6282	129	39	by	by	ADP
cana-6282	129	40	applying	apply	VERB
cana-6282	129	41	the	the	DET
cana-6282	129	42	condition	condition	NOUN
cana-6282	129	43	(	(	PUNCT
cana-6282	129	44	2.4	2.4	NUM
cana-6282	129	45	)	)	PUNCT
cana-6282	129	46	,	,	PUNCT
cana-6282	129	47	we	we	PRON
cana-6282	129	48	get	get	VERB
cana-6282	129	49	lim	lim	PROPN
cana-6282	129	50	𝑛→∞	𝑛→∞	NUM
cana-6282	129	51	 	 	SPACE
cana-6282	129	52	inf	inf	PROPN
cana-6282	129	53	𝐷𝑓(𝑥1	𝐷𝑓(𝑥1	PROPN
cana-6282	129	54	,	,	PUNCT
cana-6282	129	55	𝑥2	𝑥2	PROPN
cana-6282	129	56	⋅	⋅	PROPN
cana-6282	129	57	𝜎(𝑢𝑛	𝜎(𝑢𝑛	NUM
cana-6282	129	58	)	)	PUNCT
cana-6282	129	59	,	,	PUNCT
cana-6282	129	60	𝑥3	𝑥3	NOUN
cana-6282	129	61	,	,	PUNCT
cana-6282	129	62	𝑥4	𝑥4	ADJ
cana-6282	129	63	⋅	⋅	X
cana-6282	129	64	𝜏(𝑢𝑛	𝜏(𝑢𝑛	NOUN
cana-6282	129	65	)	)	PUNCT
cana-6282	129	66	)	)	PUNCT
cana-6282	130	1	=	=	SYM
cana-6282	130	2	0	0	NUM
cana-6282	130	3	,	,	PUNCT
cana-6282	130	4	𝑥1	𝑥1	NOUN
cana-6282	130	5	,	,	PUNCT
cana-6282	130	6	𝑥2	𝑥2	NOUN
cana-6282	130	7	,	,	PUNCT
cana-6282	130	8	𝑥3	𝑥3	NOUN
cana-6282	130	9	,	,	PUNCT
cana-6282	130	10	𝑥4	𝑥4	NOUN
cana-6282	130	11	∈	∈	PROPN
cana-6282	130	12	𝑆	𝑆	PROPN
cana-6282	130	13	(	(	PUNCT
cana-6282	130	14	2.11	2.11	NUM
cana-6282	130	15	)	)	PUNCT
cana-6282	130	16	let	let	VERB
cana-6282	130	17	𝑥1	𝑥1	NOUN
cana-6282	130	18	,	,	PUNCT
cana-6282	130	19	𝑥2	𝑥2	NOUN
cana-6282	130	20	,	,	PUNCT
cana-6282	130	21	𝑥3	𝑥3	NOUN
cana-6282	130	22	,	,	PUNCT
cana-6282	130	23	𝑥4	𝑥4	NOUN
cana-6282	130	24	,	,	PUNCT
cana-6282	130	25	𝛼	𝛼	X
cana-6282	130	26	,	,	PUNCT
cana-6282	130	27	𝛽	𝛽	PROPN
cana-6282	130	28	∈	∈	PROPN
cana-6282	130	29	𝑆	𝑆	PROPN
cana-6282	130	30	be	be	AUX
cana-6282	130	31	fixe	fixe	NOUN
cana-6282	130	32	.	.	PUNCT
cana-6282	131	1	by	by	ADP
cana-6282	131	2	replacing	replace	VERB
cana-6282	131	3	𝑎	𝑎	NOUN
cana-6282	131	4	by	by	ADP
cana-6282	131	5	𝛼	𝛼	PROPN
cana-6282	131	6	⋅	⋅	PROPN
cana-6282	131	7	𝑢𝑛	𝑢𝑛	NOUN
cana-6282	131	8	and	and	CCONJ
cana-6282	131	9	𝑏	𝑏	PROPN
cana-6282	131	10	by	by	ADP
cana-6282	131	11	𝛽	𝛽	NOUN
cana-6282	131	12	⋅	⋅	PROPN
cana-6282	131	13	𝑢𝑛	𝑢𝑛	NOUN
cana-6282	131	14	in	in	ADP
cana-6282	131	15	(	(	PUNCT
cana-6282	131	16	2.7	2.7	NUM
cana-6282	131	17	)	)	PUNCT
cana-6282	131	18	,	,	PUNCT
cana-6282	131	19	we	we	PRON
cana-6282	131	20	conclude	conclude	VERB
cana-6282	131	21	the	the	DET
cana-6282	131	22	following	follow	VERB
cana-6282	131	23	equality	equality	NOUN
cana-6282	131	24	2𝐷𝑓(𝑥1	2𝐷𝑓(𝑥1	NUM
cana-6282	131	25	,	,	PUNCT
cana-6282	131	26	𝑥2	𝑥2	NOUN
cana-6282	131	27	,	,	PUNCT
cana-6282	131	28	𝑥3	𝑥3	NOUN
cana-6282	131	29	,	,	PUNCT
cana-6282	131	30	𝑥4	𝑥4	ADJ
cana-6282	131	31	)	)	PUNCT
cana-6282	131	32	+	+	CCONJ
cana-6282	131	33	𝐷𝑓(𝑥1	𝐷𝑓(𝑥1	ADJ
cana-6282	131	34	⋅	⋅	PROPN
cana-6282	131	35	𝑥2	𝑥2	NOUN
cana-6282	131	36	,	,	PUNCT
cana-6282	131	37	𝛼	𝛼	PROPN
cana-6282	131	38	⋅	⋅	PROPN
cana-6282	131	39	𝑢𝑛	𝑢𝑛	PROPN
cana-6282	131	40	,	,	PUNCT
cana-6282	131	41	𝑥3	𝑥3	ADJ
cana-6282	131	42	⋅	⋅	PROPN
cana-6282	131	43	𝑥4	𝑥4	NOUN
cana-6282	131	44	,	,	PUNCT
cana-6282	131	45	𝛽	𝛽	PROPN
cana-6282	131	46	⋅	⋅	PROPN
cana-6282	131	47	𝑢𝑛	𝑢𝑛	NUM
cana-6282	131	48	)	)	PUNCT
cana-6282	131	49	+	+	PROPN
cana-6282	131	50	𝐷𝑓(𝑥1	𝐷𝑓(𝑥1	ADJ
cana-6282	131	51	⋅	⋅	PROPN
cana-6282	131	52	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	131	53	)	)	PUNCT
cana-6282	131	54	,	,	PUNCT
cana-6282	131	55	𝛼	𝛼	PROPN
cana-6282	131	56	⋅	⋅	PROPN
cana-6282	131	57	𝑢𝑛	𝑢𝑛	NOUN
cana-6282	131	58	,	,	PUNCT
cana-6282	131	59	𝑥3	𝑥3	ADJ
cana-6282	131	60	⋅	⋅	PROPN
cana-6282	131	61	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	131	62	)	)	PUNCT
cana-6282	131	63	,	,	PUNCT
cana-6282	131	64	𝛽	𝛽	PROPN
cana-6282	131	65	⋅	⋅	PROPN
cana-6282	131	66	𝑢𝑛	𝑢𝑛	NOUN
cana-6282	131	67	)	)	PUNCT
cana-6282	131	68	=	=	SYM
cana-6282	131	69	𝐷𝑓(𝑥1	𝐷𝑓(𝑥1	ADJ
cana-6282	131	70	,	,	PUNCT
cana-6282	131	71	𝑥2	𝑥2	PROPN
cana-6282	131	72	⋅	⋅	PROPN
cana-6282	131	73	𝛼	𝛼	PROPN
cana-6282	131	74	⋅	⋅	PROPN
cana-6282	131	75	𝑢𝑛	𝑢𝑛	PROPN
cana-6282	131	76	,	,	PUNCT
cana-6282	131	77	𝑥3	𝑥3	NOUN
cana-6282	131	78	,	,	PUNCT
cana-6282	131	79	𝑥4	𝑥4	PROPN
cana-6282	131	80	⋅	⋅	PROPN
cana-6282	131	81	𝛽	𝛽	PROPN
cana-6282	131	82	⋅	⋅	PROPN
cana-6282	131	83	𝑢𝑛	𝑢𝑛	NUM
cana-6282	131	84	)	)	PUNCT
cana-6282	131	85	+	+	X
cana-6282	131	86	𝐷𝑓(𝑥1	𝐷𝑓(𝑥1	ADJ
cana-6282	131	87	,	,	PUNCT
cana-6282	131	88	𝑥2	𝑥2	PROPN
cana-6282	131	89	⋅	⋅	PROPN
cana-6282	131	90	𝜎(𝛼	𝜎(𝛼	PROPN
cana-6282	131	91	⋅	⋅	PROPN
cana-6282	131	92	𝑢𝑛	𝑢𝑛	NUM
cana-6282	131	93	)	)	PUNCT
cana-6282	131	94	,	,	PUNCT
cana-6282	131	95	𝑥3	𝑥3	NOUN
cana-6282	131	96	,	,	PUNCT
cana-6282	131	97	𝑥4	𝑥4	PROPN
cana-6282	131	98	⋅	⋅	PROPN
cana-6282	131	99	𝜏(𝛽	𝜏(𝛽	PROPN
cana-6282	131	100	⋅	⋅	PROPN
cana-6282	131	101	𝑢𝑛	𝑢𝑛	NUM
cana-6282	131	102	)	)	PUNCT
cana-6282	131	103	)	)	PUNCT
cana-6282	132	1	+2𝐷𝑓(𝑥2	+2𝐷𝑓(𝑥2	PROPN
cana-6282	132	2	,	,	PUNCT
cana-6282	132	3	𝛼	𝛼	PROPN
cana-6282	132	4	⋅	⋅	PROPN
cana-6282	132	5	𝑢𝑛	𝑢𝑛	NOUN
cana-6282	132	6	,	,	PUNCT
cana-6282	132	7	𝑥4	𝑥4	NOUN
cana-6282	132	8	,	,	PUNCT
cana-6282	132	9	𝛽	𝛽	PROPN
cana-6282	132	10	⋅	⋅	PROPN
cana-6282	132	11	𝑢𝑛	𝑢𝑛	NUM
cana-6282	132	12	)	)	PUNCT
cana-6282	132	13	(	(	PUNCT
cana-6282	132	14	2.12	2.12	NUM
cana-6282	132	15	)	)	PUNCT
cana-6282	132	16	letting	let	VERB
cana-6282	132	17	𝑛	𝑛	PRON
cana-6282	132	18	→	→	SYM
cana-6282	132	19	∞	∞	PROPN
cana-6282	132	20	with	with	ADP
cana-6282	132	21	using	use	VERB
cana-6282	132	22	(	(	PUNCT
cana-6282	132	23	2.9	2.9	NUM
cana-6282	132	24	)	)	PUNCT
cana-6282	132	25	and	and	CCONJ
cana-6282	132	26	(	(	PUNCT
cana-6282	132	27	2.11	2.11	NUM
cana-6282	132	28	)	)	PUNCT
cana-6282	132	29	,	,	PUNCT
cana-6282	132	30	we	we	PRON
cana-6282	132	31	derive	derive	VERB
cana-6282	132	32	from	from	ADP
cana-6282	132	33	(	(	PUNCT
cana-6282	132	34	2.12	2.12	NUM
cana-6282	132	35	)	)	PUNCT
cana-6282	132	36	that	that	SCONJ
cana-6282	132	37	𝐷𝑓(𝑥1	𝐷𝑓(𝑥1	PROPN
cana-6282	132	38	,	,	PUNCT
cana-6282	132	39	𝑥2	𝑥2	NOUN
cana-6282	132	40	,	,	PUNCT
cana-6282	132	41	𝑥3	𝑥3	NOUN
cana-6282	132	42	,	,	PUNCT
cana-6282	132	43	𝑥4	𝑥4	NOUN
cana-6282	132	44	)	)	PUNCT
cana-6282	132	45	=	=	SYM
cana-6282	132	46	0	0	NUM
cana-6282	132	47	,	,	PUNCT
cana-6282	132	48	𝑥1	𝑥1	NOUN
cana-6282	132	49	,	,	PUNCT
cana-6282	132	50	𝑥2	𝑥2	NOUN
cana-6282	132	51	,	,	PUNCT
cana-6282	132	52	𝑥3	𝑥3	NOUN
cana-6282	132	53	,	,	PUNCT
cana-6282	132	54	𝑥4	𝑥4	NOUN
cana-6282	132	55	∈	∈	PROPN
cana-6282	132	56	𝑆	𝑆	PROPN
cana-6282	132	57	that	that	PRON
cana-6282	132	58	is	be	AUX
cana-6282	132	59	,	,	PUNCT
cana-6282	132	60	𝑓	𝑓	PRON
cana-6282	132	61	is	be	AUX
cana-6282	132	62	a	a	DET
cana-6282	132	63	solution	solution	NOUN
cana-6282	132	64	of	of	ADP
cana-6282	132	65	equation	equation	NOUN
cana-6282	132	66	(	(	PUNCT
cana-6282	132	67	1.5	1.5	NUM
cana-6282	132	68	)	)	PUNCT
cana-6282	132	69	which	which	PRON
cana-6282	132	70	means	mean	VERB
cana-6282	132	71	that	that	SCONJ
cana-6282	132	72	(	(	PUNCT
cana-6282	132	73	1.5	1.5	NUM
cana-6282	132	74	)	)	PUNCT
cana-6282	132	75	is	be	AUX
cana-6282	132	76	hyperstable	hyperstable	ADJ
cana-6282	132	77	on	on	ADP
cana-6282	132	78	𝑆.	𝑆.	PROPN
cana-6282	132	79	we	we	PRON
cana-6282	132	80	arrive	arrive	VERB
cana-6282	132	81	at	at	ADP
cana-6282	132	82	the	the	DET
cana-6282	132	83	following	follow	VERB
cana-6282	132	84	corollary	corollary	NOUN
cana-6282	132	85	as	as	ADP
cana-6282	132	86	a	a	DET
cana-6282	132	87	direct	direct	ADJ
cana-6282	132	88	result	result	NOUN
cana-6282	132	89	of	of	ADP
cana-6282	132	90	the	the	DET
cana-6282	132	91	theorem	theorem	ADJ
cana-6282	132	92	2.2	2.2	NUM
cana-6282	132	93	.	.	PUNCT
cana-6282	133	1	corollary	corollary	ADJ
cana-6282	133	2	2.3	2.3	NUM
cana-6282	133	3	.	.	PUNCT
cana-6282	134	1	let	let	VERB
cana-6282	134	2	휀	휀	NOUN
cana-6282	134	3	:	:	PUNCT
cana-6282	134	4	𝑆4	𝑆4	PROPN
cana-6282	134	5	⟶	⟶	NOUN
cana-6282	134	6	ℝ+be	ℝ+be	NUM
cana-6282	134	7	a	a	DET
cana-6282	134	8	function	function	NOUN
cana-6282	134	9	and	and	CCONJ
cana-6282	134	10	we	we	PRON
cana-6282	134	11	consider	consider	VERB
cana-6282	134	12	that	that	SCONJ
cana-6282	134	13	there	there	PRON
cana-6282	134	14	exist	exist	VERB
cana-6282	134	15	𝛼	𝛼	PRON
cana-6282	134	16	∈	∈	PROPN
cana-6282	134	17	𝑆	𝑆	PROPN
cana-6282	134	18	,	,	PUNCT
cana-6282	134	19	0	0	NUM
cana-6282	134	20	≤	≤	NUM
cana-6282	135	1	𝑠	𝑠	X
cana-6282	135	2	<	<	X
cana-6282	135	3	1	1	NUM
cana-6282	135	4	and	and	CCONJ
cana-6282	135	5	0	0	NUM
cana-6282	135	6	≤	≤	NUM
cana-6282	135	7	𝑡	𝑡	X
cana-6282	135	8	<	<	X
cana-6282	135	9	1	1	NUM
cana-6282	135	10	such	such	ADJ
cana-6282	135	11	that	that	SCONJ
cana-6282	135	12	{	{	PUNCT
cana-6282	135	13	휀(𝑥1	휀(𝑥1	NOUN
cana-6282	135	14	,	,	PUNCT
cana-6282	135	15	𝑥2	𝑥2	PROPN
cana-6282	135	16	⋅	⋅	PROPN
cana-6282	135	17	𝛼	𝛼	PROPN
cana-6282	135	18	,	,	PUNCT
cana-6282	135	19	𝑥3	𝑥3	NOUN
cana-6282	135	20	,	,	PUNCT
cana-6282	135	21	𝑥4	𝑥4	PROPN
cana-6282	135	22	⋅	⋅	PROPN
cana-6282	135	23	𝛼	𝛼	NOUN
cana-6282	135	24	)	)	PUNCT
cana-6282	135	25	≤	≤	NOUN
cana-6282	135	26	𝑠휀(𝑥1	𝑠휀(𝑥1	NOUN
cana-6282	135	27	,	,	PUNCT
cana-6282	135	28	𝑥2	𝑥2	NOUN
cana-6282	135	29	,	,	PUNCT
cana-6282	135	30	𝑥3	𝑥3	NOUN
cana-6282	135	31	,	,	PUNCT
cana-6282	135	32	𝑥4	𝑥4	NOUN
cana-6282	135	33	)	)	PUNCT
cana-6282	135	34	휀(𝑥1	휀(𝑥1	NOUN
cana-6282	135	35	,	,	PUNCT
cana-6282	135	36	𝑥2	𝑥2	PROPN
cana-6282	135	37	⋅	⋅	PROPN
cana-6282	135	38	𝜎(𝛼	𝜎(𝛼	PROPN
cana-6282	135	39	)	)	PUNCT
cana-6282	135	40	,	,	PUNCT
cana-6282	135	41	𝑥3	𝑥3	NOUN
cana-6282	135	42	,	,	PUNCT
cana-6282	135	43	𝑥4	𝑥4	PROPN
cana-6282	135	44	⋅	⋅	PROPN
cana-6282	135	45	𝜏(𝛼	𝜏(𝛼	PROPN
cana-6282	135	46	)	)	PUNCT
cana-6282	135	47	)	)	PUNCT
cana-6282	135	48	≤	≤	NUM
cana-6282	135	49	𝑡휀(𝑥1	𝑡휀(𝑥1	NOUN
cana-6282	135	50	,	,	PUNCT
cana-6282	135	51	𝑥2	𝑥2	NOUN
cana-6282	135	52	,	,	PUNCT
cana-6282	135	53	𝑥3	𝑥3	NOUN
cana-6282	135	54	,	,	PUNCT
cana-6282	135	55	𝑥4	𝑥4	PROPN
cana-6282	135	56	)	)	PUNCT
cana-6282	135	57	(	(	PUNCT
cana-6282	135	58	2.13	2.13	NUM
cana-6282	135	59	)	)	PUNCT
cana-6282	135	60	for	for	ADP
cana-6282	135	61	all	all	DET
cana-6282	135	62	𝑥1	𝑥1	NOUN
cana-6282	135	63	,	,	PUNCT
cana-6282	135	64	𝑥2	𝑥2	NOUN
cana-6282	135	65	,	,	PUNCT
cana-6282	135	66	𝑥3	𝑥3	NOUN
cana-6282	135	67	,	,	PUNCT
cana-6282	135	68	𝑥4	𝑥4	NOUN
cana-6282	135	69	∈	∈	PROPN
cana-6282	135	70	𝑆	𝑆	PROPN
cana-6282	135	71	and	and	CCONJ
cana-6282	135	72	all	all	DET
cana-6282	135	73	𝑛	𝑛	DET
cana-6282	135	74	∈	∈	NOUN
cana-6282	135	75	ℕ.	ℕ.	PROPN
cana-6282	135	76	if	if	SCONJ
cana-6282	135	77	𝑓	𝑓	X
cana-6282	135	78	:	:	PUNCT
cana-6282	135	79	𝑆	𝑆	PROPN
cana-6282	135	80	⟶	⟶	NOUN
cana-6282	135	81	𝑋	𝑋	NOUN
cana-6282	135	82	satisfies	satisfy	VERB
cana-6282	135	83	the	the	DET
cana-6282	135	84	inequality	inequality	NOUN
cana-6282	135	85	(	(	PUNCT
cana-6282	135	86	2.5	2.5	NUM
cana-6282	135	87	)	)	PUNCT
cana-6282	135	88	,	,	PUNCT
cana-6282	135	89	then	then	ADV
cana-6282	135	90	it	it	PRON
cana-6282	135	91	is	be	AUX
cana-6282	135	92	a	a	DET
cana-6282	135	93	solution	solution	NOUN
cana-6282	135	94	to	to	ADP
cana-6282	135	95	the	the	DET
cana-6282	135	96	functional	functional	ADJ
cana-6282	135	97	equation	equation	NOUN
cana-6282	135	98	(	(	PUNCT
cana-6282	135	99	1.5	1.5	NUM
cana-6282	135	100	)	)	PUNCT
cana-6282	135	101	.	.	PUNCT
cana-6282	136	1	proof	proof	NOUN
cana-6282	136	2	.	.	PUNCT
cana-6282	137	1	it	it	PRON
cana-6282	137	2	is	be	AUX
cana-6282	137	3	not	not	PART
cana-6282	137	4	hard	hard	ADJ
cana-6282	137	5	to	to	PART
cana-6282	137	6	demonstrate	demonstrate	VERB
cana-6282	137	7	,	,	PUNCT
cana-6282	137	8	using	use	VERB
cana-6282	137	9	induction	induction	NOUN
cana-6282	137	10	on	on	ADP
cana-6282	137	11	𝑛	𝑛	DET
cana-6282	137	12	∈	∈	PROPN
cana-6282	137	13	ℕ	ℕ	PROPN
cana-6282	137	14	,	,	PUNCT
cana-6282	137	15	that	that	SCONJ
cana-6282	137	16	{	{	PUNCT
cana-6282	137	17	휀(𝑥1	휀(𝑥1	ADV
cana-6282	137	18	,	,	PUNCT
cana-6282	137	19	𝑥2	𝑥2	PROPN
cana-6282	137	20	⋅	⋅	PROPN
cana-6282	137	21	𝛼𝑛	𝛼𝑛	PROPN
cana-6282	137	22	,	,	PUNCT
cana-6282	137	23	𝑥3	𝑥3	NOUN
cana-6282	137	24	,	,	PUNCT
cana-6282	137	25	𝑥4	𝑥4	PROPN
cana-6282	137	26	⋅	⋅	PROPN
cana-6282	137	27	𝛼𝑛	𝛼𝑛	PROPN
cana-6282	137	28	)	)	PUNCT
cana-6282	137	29	≤	≤	NOUN
cana-6282	137	30	𝑠𝑛휀(𝑥1	𝑠𝑛휀(𝑥1	PROPN
cana-6282	137	31	,	,	PUNCT
cana-6282	137	32	𝑥2	𝑥2	NOUN
cana-6282	137	33	,	,	PUNCT
cana-6282	137	34	𝑥3	𝑥3	NOUN
cana-6282	137	35	,	,	PUNCT
cana-6282	137	36	𝑥4	𝑥4	NOUN
cana-6282	137	37	)	)	PUNCT
cana-6282	137	38	휀(𝑥1	휀(𝑥1	NOUN
cana-6282	137	39	,	,	PUNCT
cana-6282	137	40	𝑥2	𝑥2	PROPN
cana-6282	137	41	⋅	⋅	PROPN
cana-6282	137	42	𝜎(𝛼𝑛	𝜎(𝛼𝑛	NUM
cana-6282	137	43	)	)	PUNCT
cana-6282	137	44	,	,	PUNCT
cana-6282	137	45	𝑥3	𝑥3	NOUN
cana-6282	137	46	,	,	PUNCT
cana-6282	137	47	𝑥4	𝑥4	PROPN
cana-6282	137	48	⋅	⋅	PROPN
cana-6282	137	49	𝜏(𝛼𝑛	𝜏(𝛼𝑛	PROPN
cana-6282	137	50	)	)	PUNCT
cana-6282	137	51	)	)	PUNCT
cana-6282	137	52	≤	≤	NUM
cana-6282	137	53	𝑡𝑛휀(𝑥1	𝑡𝑛휀(𝑥1	ADJ
cana-6282	137	54	,	,	PUNCT
cana-6282	137	55	𝑥2	𝑥2	NOUN
cana-6282	137	56	,	,	PUNCT
cana-6282	137	57	𝑥3	𝑥3	NOUN
cana-6282	137	58	,	,	PUNCT
cana-6282	137	59	𝑥4	𝑥4	PROPN
cana-6282	137	60	)	)	PUNCT
cana-6282	137	61	(	(	PUNCT
cana-6282	137	62	2.14	2.14	NUM
cana-6282	137	63	)	)	PUNCT
cana-6282	137	64	for	for	ADP
cana-6282	137	65	all	all	DET
cana-6282	137	66	𝑥1	𝑥1	NOUN
cana-6282	137	67	,	,	PUNCT
cana-6282	137	68	𝑥2	𝑥2	NOUN
cana-6282	137	69	,	,	PUNCT
cana-6282	137	70	𝑥3	𝑥3	NOUN
cana-6282	137	71	,	,	PUNCT
cana-6282	137	72	𝑥4	𝑥4	NOUN
cana-6282	137	73	∈	∈	PROPN
cana-6282	137	74	𝑆.	𝑆.	PROPN
cana-6282	137	75	therefore	therefore	ADV
cana-6282	137	76	,	,	PUNCT
cana-6282	137	77	the	the	DET
cana-6282	137	78	conditions	condition	NOUN
cana-6282	137	79	(	(	PUNCT
cana-6282	137	80	2.3	2.3	NUM
cana-6282	137	81	)	)	PUNCT
cana-6282	137	82	and	and	CCONJ
cana-6282	137	83	(	(	PUNCT
cana-6282	137	84	2.4	2.4	X
cana-6282	137	85	)	)	PUNCT
cana-6282	137	86	hold	hold	VERB
cana-6282	137	87	with	with	ADP
cana-6282	137	88	𝑢𝑛	𝑢𝑛	NOUN
cana-6282	137	89	=	=	SYM
cana-6282	137	90	𝛼𝑛	𝛼𝑛	PROPN
cana-6282	137	91	for	for	ADP
cana-6282	137	92	all	all	DET
cana-6282	137	93	𝑛	𝑛	DET
cana-6282	137	94	∈	∈	NOUN
cana-6282	137	95	ℕ.	ℕ.	PROPN
cana-6282	137	96	we	we	PRON
cana-6282	137	97	conclude	conclude	VERB
cana-6282	137	98	that	that	SCONJ
cana-6282	137	99	the	the	DET
cana-6282	137	100	functional	functional	ADJ
cana-6282	137	101	equation	equation	NOUN
cana-6282	137	102	(	(	PUNCT
cana-6282	137	103	1.5	1.5	NUM
cana-6282	137	104	)	)	PUNCT
cana-6282	137	105	is	be	AUX
cana-6282	137	106	hyperstable	hyperstable	ADJ
cana-6282	137	107	on	on	ADP
cana-6282	137	108	𝑆	𝑆	PROPN
cana-6282	137	109	based	base	VERB
cana-6282	137	110	on	on	ADP
cana-6282	137	111	theorem	theorem	ADJ
cana-6282	137	112	2.2	2.2	NUM
cana-6282	137	113	.	.	PUNCT
cana-6282	138	1	communications	communication	NOUN
cana-6282	138	2	on	on	ADP
cana-6282	138	3	applied	apply	VERB
cana-6282	138	4	nonlinear	nonlinear	ADJ
cana-6282	138	5	analysis	analysis	NOUN
cana-6282	138	6	issn	issn	NOUN
cana-6282	138	7	:	:	PUNCT
cana-6282	138	8	1074	1074	NUM
cana-6282	138	9	-	-	PUNCT
cana-6282	138	10	133x	133x	NUM
cana-6282	138	11	vol	vol	VERB
cana-6282	138	12	32	32	NUM
cana-6282	138	13	no	no	NOUN
cana-6282	138	14	.	.	PUNCT
cana-6282	139	1	10s	10	NOUN
cana-6282	139	2	(	(	PUNCT
cana-6282	139	3	2025	2025	NUM
cana-6282	139	4	)	)	PUNCT
cana-6282	139	5	3691	3691	NUM
cana-6282	139	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-6282	139	7	in	in	ADP
cana-6282	139	8	the	the	DET
cana-6282	139	9	next	next	ADJ
cana-6282	139	10	theorem	theorem	NOUN
cana-6282	139	11	,	,	PUNCT
cana-6282	139	12	we	we	PRON
cana-6282	139	13	discuss	discuss	VERB
cana-6282	139	14	the	the	DET
cana-6282	139	15	hyperstability	hyperstability	NOUN
cana-6282	139	16	of	of	ADP
cana-6282	139	17	the	the	DET
cana-6282	139	18	following	follow	VERB
cana-6282	139	19	inhomogeneous	inhomogeneous	ADJ
cana-6282	139	20	functional	functional	ADJ
cana-6282	139	21	equation	equation	NOUN
cana-6282	139	22	:	:	PUNCT
cana-6282	139	23	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	139	24	⋅	⋅	ADJ
cana-6282	139	25	𝑥2	𝑥2	NOUN
cana-6282	139	26	,	,	PUNCT
cana-6282	139	27	𝑥3	𝑥3	ADJ
cana-6282	139	28	⋅	⋅	PROPN
cana-6282	139	29	𝑥4	𝑥4	NOUN
cana-6282	139	30	)	)	PUNCT
cana-6282	139	31	+	+	CCONJ
cana-6282	139	32	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	139	33	⋅	⋅	PROPN
cana-6282	139	34	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	139	35	)	)	PUNCT
cana-6282	139	36	,	,	PUNCT
cana-6282	139	37	𝑥3	𝑥3	ADJ
cana-6282	139	38	⋅	⋅	PROPN
cana-6282	139	39	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	139	40	)	)	PUNCT
cana-6282	139	41	)	)	PUNCT
cana-6282	140	1	=	=	SYM
cana-6282	140	2	2𝑓(𝑥1	2𝑓(𝑥1	NUM
cana-6282	140	3	,	,	PUNCT
cana-6282	140	4	𝑥3	𝑥3	NOUN
cana-6282	140	5	)	)	PUNCT
cana-6282	140	6	+	+	CCONJ
cana-6282	140	7	2𝑓(𝑥2	2𝑓(𝑥2	NUM
cana-6282	140	8	,	,	PUNCT
cana-6282	140	9	𝑥4	𝑥4	ADJ
cana-6282	140	10	)	)	PUNCT
cana-6282	141	1	+	+	CCONJ
cana-6282	141	2	𝐹(𝑥1	𝐹(𝑥1	ADJ
cana-6282	141	3	,	,	PUNCT
cana-6282	141	4	𝑥2	𝑥2	NOUN
cana-6282	141	5	,	,	PUNCT
cana-6282	141	6	𝑥3	𝑥3	NOUN
cana-6282	141	7	,	,	PUNCT
cana-6282	141	8	𝑥4	𝑥4	PROPN
cana-6282	141	9	)	)	PUNCT
cana-6282	141	10	(	(	PUNCT
cana-6282	141	11	2.15	2.15	NUM
cana-6282	141	12	)	)	PUNCT
cana-6282	141	13	for	for	ADP
cana-6282	141	14	all	all	DET
cana-6282	141	15	𝑥1	𝑥1	NOUN
cana-6282	141	16	,	,	PUNCT
cana-6282	141	17	𝑥2	𝑥2	NOUN
cana-6282	141	18	,	,	PUNCT
cana-6282	141	19	𝑥3	𝑥3	NOUN
cana-6282	141	20	,	,	PUNCT
cana-6282	141	21	𝑥4	𝑥4	NOUN
cana-6282	141	22	∈	∈	PROPN
cana-6282	141	23	𝑆	𝑆	PROPN
cana-6282	141	24	,	,	PUNCT
cana-6282	141	25	where	where	SCONJ
cana-6282	141	26	𝐹	𝐹	PROPN
cana-6282	141	27	:	:	PUNCT
cana-6282	141	28	𝑆4	𝑆4	PROPN
cana-6282	141	29	→	→	SYM
cana-6282	141	30	𝑋.	𝑋.	PROPN
cana-6282	141	31	theorem	theorem	VERB
cana-6282	141	32	2.4	2.4	NUM
cana-6282	141	33	.	.	PUNCT
cana-6282	142	1	let	let	VERB
cana-6282	142	2	𝑋	𝑋	PROPN
cana-6282	142	3	,	,	PUNCT
cana-6282	142	4	𝑆	𝑆	PROPN
cana-6282	142	5	,	,	PUNCT
cana-6282	142	6	𝜎	𝜎	PROPN
cana-6282	142	7	,	,	PUNCT
cana-6282	142	8	and	and	CCONJ
cana-6282	142	9	𝜏	𝜏	NOUN
cana-6282	142	10	remain	remain	VERB
cana-6282	142	11	unchanged	unchanged	ADJ
cana-6282	142	12	from	from	ADP
cana-6282	142	13	theorem	theorem	ADJ
cana-6282	142	14	2.2	2.2	NUM
cana-6282	142	15	.	.	PUNCT
cana-6282	143	1	let	let	VERB
cana-6282	143	2	𝑓	𝑓	PRON
cana-6282	143	3	:	:	PUNCT
cana-6282	143	4	𝑆2	𝑆2	PROPN
cana-6282	143	5	→	→	SYM
cana-6282	143	6	𝑋	𝑋	PROPN
cana-6282	143	7	and	and	CCONJ
cana-6282	143	8	𝐹	𝐹	PROPN
cana-6282	143	9	:	:	PUNCT
cana-6282	143	10	𝑆4	𝑆4	PROPN
cana-6282	143	11	→	→	SYM
cana-6282	143	12	𝑋	𝑋	PROPN
cana-6282	143	13	be	be	VERB
cana-6282	143	14	two	two	NUM
cana-6282	143	15	functions	function	NOUN
cana-6282	143	16	satisfy	satisfy	VERB
cana-6282	143	17	the	the	DET
cana-6282	143	18	inequality	inequality	NOUN
cana-6282	143	19	‖𝑓(𝑥1	‖𝑓(𝑥1	ADJ
cana-6282	143	20	⋅	⋅	PROPN
cana-6282	143	21	𝑥2	𝑥2	NOUN
cana-6282	143	22	,	,	PUNCT
cana-6282	143	23	𝑥3	𝑥3	ADJ
cana-6282	143	24	⋅	⋅	PROPN
cana-6282	143	25	𝑥4	𝑥4	NOUN
cana-6282	143	26	)	)	PUNCT
cana-6282	143	27	+	+	CCONJ
cana-6282	143	28	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	143	29	⋅	⋅	PROPN
cana-6282	143	30	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	143	31	)	)	PUNCT
cana-6282	143	32	,	,	PUNCT
cana-6282	143	33	𝑥3	𝑥3	ADJ
cana-6282	143	34	⋅	⋅	PROPN
cana-6282	143	35	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	143	36	)	)	PUNCT
cana-6282	143	37	)	)	PUNCT
cana-6282	144	1	−2𝑓(𝑥1	−2𝑓(𝑥1	PROPN
cana-6282	144	2	,	,	PUNCT
cana-6282	144	3	𝑥3	𝑥3	NOUN
cana-6282	144	4	)	)	PUNCT
cana-6282	145	1	−	−	PROPN
cana-6282	145	2	2𝑓(𝑥2	2𝑓(𝑥2	NUM
cana-6282	145	3	,	,	PUNCT
cana-6282	145	4	𝑥4	𝑥4	ADJ
cana-6282	145	5	)	)	PUNCT
cana-6282	145	6	−	−	PROPN
cana-6282	145	7	𝐹(𝑥1	𝐹(𝑥1	PROPN
cana-6282	145	8	,	,	PUNCT
cana-6282	145	9	𝑥2	𝑥2	NOUN
cana-6282	145	10	,	,	PUNCT
cana-6282	145	11	𝑥3	𝑥3	NOUN
cana-6282	145	12	,	,	PUNCT
cana-6282	145	13	𝑥4)‖	𝑥4)‖	NOUN
cana-6282	145	14	≤	≤	NUM
cana-6282	145	15	휀(𝑥1	휀(𝑥1	NOUN
cana-6282	145	16	,	,	PUNCT
cana-6282	145	17	𝑥2	𝑥2	NOUN
cana-6282	145	18	,	,	PUNCT
cana-6282	145	19	𝑥3	𝑥3	NOUN
cana-6282	145	20	,	,	PUNCT
cana-6282	145	21	𝑥4	𝑥4	PROPN
cana-6282	145	22	)	)	PUNCT
cana-6282	145	23	(	(	PUNCT
cana-6282	145	24	2.16	2.16	NUM
cana-6282	145	25	)	)	PUNCT
cana-6282	145	26	for	for	ADP
cana-6282	145	27	all	all	DET
cana-6282	145	28	𝑥1	𝑥1	NOUN
cana-6282	145	29	,	,	PUNCT
cana-6282	145	30	𝑥2	𝑥2	NOUN
cana-6282	145	31	,	,	PUNCT
cana-6282	145	32	𝑥3	𝑥3	NOUN
cana-6282	145	33	,	,	PUNCT
cana-6282	145	34	𝑥4	𝑥4	NOUN
cana-6282	145	35	∈	∈	PROPN
cana-6282	145	36	𝑆.	𝑆.	PROPN
cana-6282	145	37	if	if	SCONJ
cana-6282	145	38	the	the	DET
cana-6282	145	39	functional	functional	ADJ
cana-6282	145	40	equation	equation	NOUN
cana-6282	145	41	(	(	PUNCT
cana-6282	145	42	2.15	2.15	NUM
cana-6282	145	43	)	)	PUNCT
cana-6282	145	44	admits	admit	VERB
cana-6282	145	45	a	a	DET
cana-6282	145	46	solution	solution	NOUN
cana-6282	145	47	𝑓0	𝑓0	NOUN
cana-6282	145	48	:	:	PUNCT
cana-6282	145	49	𝑆2	𝑆2	PROPN
cana-6282	145	50	→	→	SYM
cana-6282	145	51	𝑋	𝑋	PROPN
cana-6282	145	52	,	,	PUNCT
cana-6282	145	53	then	then	ADV
cana-6282	145	54	(	(	PUNCT
cana-6282	145	55	2.15	2.15	NUM
cana-6282	145	56	)	)	PUNCT
cana-6282	145	57	is	be	AUX
cana-6282	145	58	hyperstable	hyperstable	ADJ
cana-6282	145	59	on	on	ADP
cana-6282	145	60	𝑆.	𝑆.	ADJ
cana-6282	145	61	proof	proof	NOUN
cana-6282	145	62	.	.	PUNCT
cana-6282	146	1	let	let	VERB
cana-6282	146	2	𝑔	𝑔	NOUN
cana-6282	146	3	:	:	PUNCT
cana-6282	146	4	𝑆2	𝑆2	PROPN
cana-6282	146	5	→	→	PUNCT
cana-6282	146	6	𝑋	𝑋	NOUN
cana-6282	146	7	be	be	VERB
cana-6282	146	8	a	a	DET
cana-6282	146	9	function	function	NOUN
cana-6282	146	10	defined	define	VERB
cana-6282	146	11	by	by	ADP
cana-6282	146	12	𝑔(𝑥1	𝑔(𝑥1	NOUN
cana-6282	146	13	,	,	PUNCT
cana-6282	146	14	𝑥2	𝑥2	NOUN
cana-6282	146	15	):	):	PUNCT
cana-6282	146	16	=	=	SYM
cana-6282	146	17	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	146	18	,	,	PUNCT
cana-6282	146	19	𝑥2	𝑥2	NOUN
cana-6282	146	20	)	)	PUNCT
cana-6282	146	21	−	−	PROPN
cana-6282	146	22	𝑓0(𝑥1	𝑓0(𝑥1	ADP
cana-6282	146	23	,	,	PUNCT
cana-6282	146	24	𝑥2	𝑥2	NOUN
cana-6282	146	25	)	)	PUNCT
cana-6282	146	26	for	for	ADP
cana-6282	146	27	all	all	DET
cana-6282	146	28	𝑥1	𝑥1	NOUN
cana-6282	146	29	,	,	PUNCT
cana-6282	146	30	𝑥2	𝑥2	PROPN
cana-6282	146	31	∈	∈	NOUN
cana-6282	146	32	𝑆.	𝑆.	PROPN
cana-6282	146	33	then	then	ADV
cana-6282	146	34	‖𝑔(𝑥1	‖𝑔(𝑥1	ADJ
cana-6282	146	35	⋅	⋅	PROPN
cana-6282	146	36	𝑥2	𝑥2	NOUN
cana-6282	146	37	,	,	PUNCT
cana-6282	146	38	𝑥3	𝑥3	ADJ
cana-6282	146	39	⋅	⋅	PROPN
cana-6282	146	40	𝑥4	𝑥4	NOUN
cana-6282	146	41	)	)	PUNCT
cana-6282	147	1	+	+	NUM
cana-6282	147	2	𝑔(𝑥1	𝑔(𝑥1	PROPN
cana-6282	147	3	⋅	⋅	PROPN
cana-6282	147	4	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	147	5	)	)	PUNCT
cana-6282	147	6	,	,	PUNCT
cana-6282	147	7	𝑥3	𝑥3	ADJ
cana-6282	147	8	⋅	⋅	PROPN
cana-6282	147	9	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	147	10	)	)	PUNCT
cana-6282	147	11	)	)	PUNCT
cana-6282	148	1	−	−	ADP
cana-6282	148	2	2𝑔(𝑥1	2𝑔(𝑥1	NUM
cana-6282	148	3	,	,	PUNCT
cana-6282	148	4	𝑥3	𝑥3	NOUN
cana-6282	148	5	)	)	PUNCT
cana-6282	148	6	−	−	PROPN
cana-6282	148	7	2𝑔(𝑥2	2𝑔(𝑥2	NUM
cana-6282	148	8	,	,	PUNCT
cana-6282	148	9	𝑥4)‖	𝑥4)‖	NOUN
cana-6282	148	10	=	=	SYM
cana-6282	148	11	‖𝑓(𝑥1	‖𝑓(𝑥1	ADJ
cana-6282	148	12	⋅	⋅	PROPN
cana-6282	148	13	𝑥2	𝑥2	NOUN
cana-6282	148	14	,	,	PUNCT
cana-6282	148	15	𝑥3	𝑥3	ADJ
cana-6282	148	16	⋅	⋅	PROPN
cana-6282	148	17	𝑥4	𝑥4	NOUN
cana-6282	148	18	)	)	PUNCT
cana-6282	148	19	+	+	CCONJ
cana-6282	148	20	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	148	21	⋅	⋅	PROPN
cana-6282	148	22	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	148	23	)	)	PUNCT
cana-6282	148	24	,	,	PUNCT
cana-6282	148	25	𝑥3	𝑥3	ADJ
cana-6282	148	26	⋅	⋅	PROPN
cana-6282	148	27	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	148	28	)	)	PUNCT
cana-6282	148	29	)	)	PUNCT
cana-6282	149	1	−	−	PROPN
cana-6282	149	2	(	(	PUNCT
cana-6282	149	3	𝑓0(𝑥1	𝑓0(𝑥1	ADP
cana-6282	149	4	⋅	⋅	PROPN
cana-6282	149	5	𝑥2	𝑥2	NOUN
cana-6282	149	6	,	,	PUNCT
cana-6282	149	7	𝑥3	𝑥3	ADJ
cana-6282	149	8	⋅	⋅	PROPN
cana-6282	149	9	𝑥4	𝑥4	NOUN
cana-6282	149	10	)	)	PUNCT
cana-6282	150	1	+	+	ADP
cana-6282	150	2	𝑓0(𝑥1	𝑓0(𝑥1	ADP
cana-6282	150	3	⋅	⋅	PROPN
cana-6282	150	4	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	150	5	)	)	PUNCT
cana-6282	150	6	,	,	PUNCT
cana-6282	150	7	𝑥3	𝑥3	ADJ
cana-6282	150	8	⋅	⋅	PROPN
cana-6282	150	9	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	150	10	)	)	PUNCT
cana-6282	150	11	)	)	PUNCT
cana-6282	150	12	)	)	PUNCT
cana-6282	151	1	−	−	ADP
cana-6282	151	2	2𝑓(𝑥1	2𝑓(𝑥1	NUM
cana-6282	151	3	,	,	PUNCT
cana-6282	151	4	𝑥3	𝑥3	NOUN
cana-6282	151	5	)	)	PUNCT
cana-6282	151	6	−	−	PROPN
cana-6282	151	7	2𝑓(𝑥2	2𝑓(𝑥2	NUM
cana-6282	151	8	,	,	PUNCT
cana-6282	151	9	𝑥4	𝑥4	ADJ
cana-6282	151	10	)	)	PUNCT
cana-6282	151	11	−	−	PROPN
cana-6282	151	12	𝐹(𝑥1	𝐹(𝑥1	PROPN
cana-6282	151	13	,	,	PUNCT
cana-6282	151	14	𝑥2	𝑥2	NOUN
cana-6282	151	15	,	,	PUNCT
cana-6282	151	16	𝑥3	𝑥3	NOUN
cana-6282	151	17	,	,	PUNCT
cana-6282	151	18	𝑥4	𝑥4	NOUN
cana-6282	151	19	)	)	PUNCT
cana-6282	151	20	−(2𝑓0(𝑥1	−(2𝑓0(𝑥1	PROPN
cana-6282	151	21	,	,	PUNCT
cana-6282	151	22	𝑥3	𝑥3	NOUN
cana-6282	151	23	)	)	PUNCT
cana-6282	151	24	+	+	NUM
cana-6282	151	25	2𝑓0(𝑥2	2𝑓0(𝑥2	NUM
cana-6282	151	26	,	,	PUNCT
cana-6282	151	27	𝑥4	𝑥4	ADJ
cana-6282	151	28	)	)	PUNCT
cana-6282	151	29	+	+	CCONJ
cana-6282	151	30	𝐹(𝑥1	𝐹(𝑥1	ADJ
cana-6282	151	31	,	,	PUNCT
cana-6282	151	32	𝑥2	𝑥2	NOUN
cana-6282	151	33	,	,	PUNCT
cana-6282	151	34	𝑥3	𝑥3	NOUN
cana-6282	151	35	,	,	PUNCT
cana-6282	151	36	𝑥4))‖	𝑥4))‖	PROPN
cana-6282	151	37	≤	≤	NUM
cana-6282	151	38	‖𝑓(𝑥1	‖𝑓(𝑥1	ADJ
cana-6282	151	39	⋅	⋅	PROPN
cana-6282	151	40	𝑥2	𝑥2	NOUN
cana-6282	151	41	,	,	PUNCT
cana-6282	151	42	𝑥3	𝑥3	ADJ
cana-6282	151	43	⋅	⋅	PROPN
cana-6282	151	44	𝑥4	𝑥4	NOUN
cana-6282	151	45	)	)	PUNCT
cana-6282	151	46	+	+	CCONJ
cana-6282	151	47	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	151	48	⋅	⋅	PROPN
cana-6282	151	49	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	151	50	)	)	PUNCT
cana-6282	151	51	,	,	PUNCT
cana-6282	151	52	𝑥3	𝑥3	ADJ
cana-6282	151	53	⋅	⋅	PROPN
cana-6282	151	54	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	151	55	)	)	PUNCT
cana-6282	151	56	)	)	PUNCT
cana-6282	152	1	−	−	ADP
cana-6282	152	2	2𝑓(𝑥1	2𝑓(𝑥1	NUM
cana-6282	152	3	,	,	PUNCT
cana-6282	152	4	𝑥3	𝑥3	NOUN
cana-6282	152	5	)	)	PUNCT
cana-6282	153	1	−	−	PROPN
cana-6282	153	2	2𝑓(𝑥2	2𝑓(𝑥2	NUM
cana-6282	153	3	,	,	PUNCT
cana-6282	153	4	𝑥4	𝑥4	ADJ
cana-6282	153	5	)	)	PUNCT
cana-6282	153	6	−𝐹(𝑥1	−𝐹(𝑥1	NOUN
cana-6282	153	7	,	,	PUNCT
cana-6282	153	8	𝑥2	𝑥2	NOUN
cana-6282	153	9	,	,	PUNCT
cana-6282	153	10	𝑥3	𝑥3	NOUN
cana-6282	153	11	,	,	PUNCT
cana-6282	153	12	𝑥4)‖	𝑥4)‖	DET
cana-6282	153	13	+	+	ADJ
cana-6282	153	14	‖𝑓0(𝑥1	‖𝑓0(𝑥1	ADJ
cana-6282	153	15	⋅	⋅	ADJ
cana-6282	153	16	𝑥2	𝑥2	NOUN
cana-6282	153	17	,	,	PUNCT
cana-6282	153	18	𝑥3	𝑥3	ADJ
cana-6282	153	19	⋅	⋅	PROPN
cana-6282	153	20	𝑥4	𝑥4	NOUN
cana-6282	153	21	)	)	PUNCT
cana-6282	153	22	+	+	CCONJ
cana-6282	154	1	𝑓0(𝑥1	𝑓0(𝑥1	ADP
cana-6282	154	2	⋅	⋅	PROPN
cana-6282	154	3	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	154	4	)	)	PUNCT
cana-6282	154	5	,	,	PUNCT
cana-6282	154	6	𝑥3	𝑥3	ADJ
cana-6282	154	7	⋅	⋅	PROPN
cana-6282	154	8	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	154	9	)	)	PUNCT
cana-6282	154	10	)	)	PUNCT
cana-6282	155	1	−	−	ADP
cana-6282	155	2	2𝑓0(𝑥1	2𝑓0(𝑥1	NUM
cana-6282	155	3	,	,	PUNCT
cana-6282	155	4	𝑥3	𝑥3	NOUN
cana-6282	155	5	)	)	PUNCT
cana-6282	155	6	−	−	PROPN
cana-6282	156	1	2𝑓0(𝑥2	2𝑓0(𝑥2	NUM
cana-6282	156	2	,	,	PUNCT
cana-6282	156	3	𝑥4	𝑥4	ADJ
cana-6282	156	4	)	)	PUNCT
cana-6282	156	5	−𝐹(𝑥1	−𝐹(𝑥1	NOUN
cana-6282	156	6	,	,	PUNCT
cana-6282	156	7	𝑥2	𝑥2	NOUN
cana-6282	156	8	,	,	PUNCT
cana-6282	156	9	𝑥3	𝑥3	NOUN
cana-6282	156	10	,	,	PUNCT
cana-6282	156	11	𝑥4)‖	𝑥4)‖	NOUN
cana-6282	156	12	=	=	SYM
cana-6282	156	13	‖𝑓(𝑥1	‖𝑓(𝑥1	ADJ
cana-6282	156	14	⋅	⋅	PROPN
cana-6282	156	15	𝑥2	𝑥2	NOUN
cana-6282	156	16	,	,	PUNCT
cana-6282	156	17	𝑥3	𝑥3	ADJ
cana-6282	156	18	⋅	⋅	PROPN
cana-6282	156	19	𝑥4	𝑥4	NOUN
cana-6282	156	20	)	)	PUNCT
cana-6282	156	21	+	+	CCONJ
cana-6282	156	22	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	156	23	⋅	⋅	PROPN
cana-6282	156	24	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	156	25	)	)	PUNCT
cana-6282	156	26	,	,	PUNCT
cana-6282	156	27	𝑥3	𝑥3	ADJ
cana-6282	156	28	⋅	⋅	PROPN
cana-6282	156	29	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	156	30	)	)	PUNCT
cana-6282	156	31	)	)	PUNCT
cana-6282	157	1	−	−	ADP
cana-6282	157	2	2𝑓(𝑥1	2𝑓(𝑥1	NUM
cana-6282	157	3	,	,	PUNCT
cana-6282	157	4	𝑥3	𝑥3	NOUN
cana-6282	157	5	)	)	PUNCT
cana-6282	158	1	−	−	PROPN
cana-6282	158	2	2𝑓(𝑥2	2𝑓(𝑥2	NUM
cana-6282	158	3	,	,	PUNCT
cana-6282	158	4	𝑥4	𝑥4	ADJ
cana-6282	158	5	)	)	PUNCT
cana-6282	158	6	−𝐹(𝑥1	−𝐹(𝑥1	NOUN
cana-6282	158	7	,	,	PUNCT
cana-6282	158	8	𝑥2	𝑥2	NOUN
cana-6282	158	9	,	,	PUNCT
cana-6282	158	10	𝑥3	𝑥3	NOUN
cana-6282	158	11	,	,	PUNCT
cana-6282	158	12	𝑥4)‖	𝑥4)‖	NOUN
cana-6282	158	13	≤	≤	NUM
cana-6282	158	14	휀(𝑥1	휀(𝑥1	NOUN
cana-6282	158	15	,	,	PUNCT
cana-6282	158	16	𝑥2	𝑥2	NOUN
cana-6282	158	17	,	,	PUNCT
cana-6282	158	18	𝑥3	𝑥3	NOUN
cana-6282	158	19	,	,	PUNCT
cana-6282	158	20	𝑥4	𝑥4	PROPN
cana-6282	158	21	)	)	PUNCT
cana-6282	158	22	,	,	PUNCT
cana-6282	158	23	𝑥1	𝑥1	NOUN
cana-6282	158	24	,	,	PUNCT
cana-6282	158	25	𝑥2	𝑥2	NOUN
cana-6282	158	26	,	,	PUNCT
cana-6282	158	27	𝑥3	𝑥3	NOUN
cana-6282	158	28	,	,	PUNCT
cana-6282	158	29	𝑥4	𝑥4	NOUN
cana-6282	158	30	∈	∈	PROPN
cana-6282	158	31	𝑆.	𝑆.	PROPN
cana-6282	158	32	according	accord	VERB
cana-6282	158	33	to	to	ADP
cana-6282	158	34	theorem	theorem	ADJ
cana-6282	158	35	2.2	2.2	NUM
cana-6282	158	36	,	,	PUNCT
cana-6282	158	37	the	the	DET
cana-6282	158	38	function	function	NOUN
cana-6282	158	39	𝑔	𝑔	PROPN
cana-6282	158	40	=	=	PUNCT
cana-6282	158	41	𝑓	𝑓	PRON
cana-6282	158	42	−	−	PROPN
cana-6282	158	43	𝑓0	𝑓0	PROPN
cana-6282	158	44	satisfies	satisfy	VERB
cana-6282	158	45	the	the	DET
cana-6282	158	46	homogeneous	homogeneous	ADJ
cana-6282	158	47	equation	equation	NOUN
cana-6282	158	48	(	(	PUNCT
cana-6282	158	49	1.5	1.5	NUM
cana-6282	158	50	)	)	PUNCT
cana-6282	158	51	.	.	PUNCT
cana-6282	159	1	therefore	therefore	ADV
cana-6282	159	2	,	,	PUNCT
cana-6282	159	3	by	by	ADP
cana-6282	159	4	linearity	linearity	NOUN
cana-6282	159	5	,	,	PUNCT
cana-6282	159	6	we	we	PRON
cana-6282	159	7	have	have	VERB
cana-6282	159	8	:	:	PUNCT
cana-6282	159	9	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	159	10	⋅	⋅	ADJ
cana-6282	159	11	𝑥2	𝑥2	NOUN
cana-6282	159	12	,	,	PUNCT
cana-6282	159	13	𝑥3	𝑥3	ADJ
cana-6282	159	14	⋅	⋅	PROPN
cana-6282	159	15	𝑥4	𝑥4	NOUN
cana-6282	159	16	)	)	PUNCT
cana-6282	159	17	+	+	CCONJ
cana-6282	159	18	𝑓(𝑥1	𝑓(𝑥1	ADJ
cana-6282	159	19	⋅	⋅	PROPN
cana-6282	159	20	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	159	21	)	)	PUNCT
cana-6282	159	22	,	,	PUNCT
cana-6282	159	23	𝑥3	𝑥3	ADJ
cana-6282	159	24	⋅	⋅	PROPN
cana-6282	159	25	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	159	26	)	)	PUNCT
cana-6282	159	27	)	)	PUNCT
cana-6282	160	1	−	−	ADP
cana-6282	160	2	2𝑓(𝑥1	2𝑓(𝑥1	NUM
cana-6282	160	3	,	,	PUNCT
cana-6282	160	4	𝑥3	𝑥3	NOUN
cana-6282	160	5	)	)	PUNCT
cana-6282	161	1	−	−	PROPN
cana-6282	161	2	2𝑓(𝑥2	2𝑓(𝑥2	NUM
cana-6282	161	3	,	,	PUNCT
cana-6282	161	4	𝑥4	𝑥4	ADJ
cana-6282	161	5	)	)	PUNCT
cana-6282	161	6	−	−	PROPN
cana-6282	161	7	𝐹(𝑥1	𝐹(𝑥1	PROPN
cana-6282	161	8	,	,	PUNCT
cana-6282	161	9	𝑥2	𝑥2	NOUN
cana-6282	161	10	,	,	PUNCT
cana-6282	161	11	𝑥3	𝑥3	NOUN
cana-6282	161	12	,	,	PUNCT
cana-6282	161	13	𝑥4	𝑥4	ADJ
cana-6282	161	14	)	)	PUNCT
cana-6282	161	15	=	=	SYM
cana-6282	161	16	𝑔(𝑥1	𝑔(𝑥1	ADJ
cana-6282	161	17	⋅	⋅	PROPN
cana-6282	161	18	𝑥2	𝑥2	PROPN
cana-6282	161	19	,	,	PUNCT
cana-6282	161	20	𝑥3	𝑥3	ADJ
cana-6282	161	21	⋅	⋅	PROPN
cana-6282	161	22	𝑥4	𝑥4	NOUN
cana-6282	161	23	)	)	PUNCT
cana-6282	161	24	+	+	NUM
cana-6282	161	25	𝑔(𝑥1	𝑔(𝑥1	PROPN
cana-6282	161	26	⋅	⋅	PROPN
cana-6282	161	27	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	161	28	)	)	PUNCT
cana-6282	161	29	,	,	PUNCT
cana-6282	161	30	𝑥3	𝑥3	ADJ
cana-6282	161	31	⋅	⋅	PROPN
cana-6282	161	32	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	161	33	)	)	PUNCT
cana-6282	161	34	)	)	PUNCT
cana-6282	162	1	−	−	ADP
cana-6282	162	2	2𝑔(𝑥1	2𝑔(𝑥1	NUM
cana-6282	162	3	,	,	PUNCT
cana-6282	162	4	𝑥3	𝑥3	NOUN
cana-6282	162	5	)	)	PUNCT
cana-6282	162	6	−	−	PROPN
cana-6282	162	7	2𝑔(𝑥2	2𝑔(𝑥2	NUM
cana-6282	162	8	,	,	PUNCT
cana-6282	162	9	𝑥4	𝑥4	NOUN
cana-6282	162	10	)	)	PUNCT
cana-6282	163	1	+	+	ADJ
cana-6282	163	2	𝑓0(𝑥1	𝑓0(𝑥1	ADP
cana-6282	163	3	⋅	⋅	ADJ
cana-6282	163	4	𝑥2	𝑥2	NOUN
cana-6282	163	5	,	,	PUNCT
cana-6282	163	6	𝑥3	𝑥3	ADJ
cana-6282	163	7	⋅	⋅	PROPN
cana-6282	163	8	𝑥4	𝑥4	NOUN
cana-6282	163	9	)	)	PUNCT
cana-6282	164	1	+	+	CCONJ
cana-6282	164	2	𝑓0(𝑥1	𝑓0(𝑥1	ADP
cana-6282	164	3	⋅	⋅	PROPN
cana-6282	164	4	𝜎(𝑥2	𝜎(𝑥2	NUM
cana-6282	164	5	)	)	PUNCT
cana-6282	164	6	,	,	PUNCT
cana-6282	164	7	𝑥3	𝑥3	ADJ
cana-6282	164	8	⋅	⋅	PROPN
cana-6282	164	9	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	164	10	)	)	PUNCT
cana-6282	164	11	)	)	PUNCT
cana-6282	165	1	−	−	ADP
cana-6282	165	2	2𝑓0(𝑥1	2𝑓0(𝑥1	NUM
cana-6282	165	3	,	,	PUNCT
cana-6282	165	4	𝑥3	𝑥3	NOUN
cana-6282	165	5	)	)	PUNCT
cana-6282	165	6	−	−	PROPN
cana-6282	165	7	2𝑓0(𝑥2	2𝑓0(𝑥2	NUM
cana-6282	165	8	,	,	PUNCT
cana-6282	165	9	𝑥4	𝑥4	ADJ
cana-6282	165	10	)	)	PUNCT
cana-6282	165	11	−	−	PROPN
cana-6282	165	12	𝐹(𝑥1	𝐹(𝑥1	PROPN
cana-6282	165	13	,	,	PUNCT
cana-6282	165	14	𝑥2	𝑥2	NOUN
cana-6282	165	15	,	,	PUNCT
cana-6282	165	16	𝑥3	𝑥3	NOUN
cana-6282	165	17	,	,	PUNCT
cana-6282	165	18	𝑥4	𝑥4	NOUN
cana-6282	165	19	)	)	PUNCT
cana-6282	165	20	=	=	SYM
cana-6282	165	21	0	0	NUM
cana-6282	165	22	for	for	ADP
cana-6282	165	23	all	all	DET
cana-6282	165	24	𝑥1	𝑥1	NOUN
cana-6282	165	25	,	,	PUNCT
cana-6282	165	26	𝑥2	𝑥2	NOUN
cana-6282	165	27	,	,	PUNCT
cana-6282	165	28	𝑥3	𝑥3	NOUN
cana-6282	165	29	,	,	PUNCT
cana-6282	165	30	𝑥4	𝑥4	NOUN
cana-6282	165	31	∈	∈	PROPN
cana-6282	165	32	𝑆.	𝑆.	PROPN
cana-6282	165	33	since	since	SCONJ
cana-6282	165	34	both	both	PRON
cana-6282	165	35	𝑔	𝑔	PROPN
cana-6282	165	36	and	and	CCONJ
cana-6282	165	37	𝑓0	𝑓0	PROPN
cana-6282	165	38	satisfy	satisfy	VERB
cana-6282	165	39	their	their	PRON
cana-6282	165	40	respective	respective	ADJ
cana-6282	165	41	equations	equation	NOUN
cana-6282	165	42	,	,	PUNCT
cana-6282	165	43	each	each	DET
cana-6282	165	44	component	component	NOUN
cana-6282	165	45	vanishes	vanish	VERB
cana-6282	165	46	,	,	PUNCT
cana-6282	165	47	and	and	CCONJ
cana-6282	165	48	we	we	PRON
cana-6282	165	49	conclude	conclude	VERB
cana-6282	165	50	that	that	SCONJ
cana-6282	165	51	𝑓	𝑓	PRON
cana-6282	165	52	is	be	AUX
cana-6282	165	53	also	also	ADV
cana-6282	165	54	a	a	DET
cana-6282	165	55	solution	solution	NOUN
cana-6282	165	56	of	of	ADP
cana-6282	165	57	(	(	PUNCT
cana-6282	165	58	2.15	2.15	NUM
cana-6282	165	59	)	)	PUNCT
cana-6282	165	60	.	.	PUNCT
cana-6282	166	1	before	before	ADP
cana-6282	166	2	closing	close	VERB
cana-6282	166	3	this	this	DET
cana-6282	166	4	section	section	NOUN
cana-6282	166	5	,	,	PUNCT
cana-6282	166	6	we	we	PRON
cana-6282	166	7	reformulate	reformulate	VERB
cana-6282	166	8	the	the	DET
cana-6282	166	9	main	main	ADJ
cana-6282	166	10	functional	functional	ADJ
cana-6282	166	11	equation	equation	NOUN
cana-6282	166	12	within	within	ADP
cana-6282	166	13	the	the	DET
cana-6282	166	14	framework	framework	NOUN
cana-6282	166	15	of	of	ADP
cana-6282	166	16	the	the	DET
cana-6282	166	17	cartesian	cartesian	ADJ
cana-6282	166	18	product	product	NOUN
cana-6282	166	19	semigroup	semigroup	NOUN
cana-6282	166	20	𝑆	𝑆	PROPN
cana-6282	166	21	×	×	NOUN
cana-6282	166	22	𝑆.	𝑆.	NOUN
cana-6282	166	23	this	this	DET
cana-6282	166	24	structural	structural	ADJ
cana-6282	166	25	reinterpretation	reinterpretation	NOUN
cana-6282	166	26	allows	allow	VERB
cana-6282	166	27	us	we	PRON
cana-6282	166	28	to	to	PART
cana-6282	166	29	express	express	VERB
cana-6282	166	30	the	the	DET
cana-6282	166	31	equation	equation	NOUN
cana-6282	166	32	in	in	ADP
cana-6282	166	33	a	a	DET
cana-6282	166	34	more	more	ADV
cana-6282	166	35	symmetric	symmetric	ADJ
cana-6282	166	36	and	and	CCONJ
cana-6282	166	37	compact	compact	ADJ
cana-6282	166	38	form	form	NOUN
cana-6282	166	39	,	,	PUNCT
cana-6282	166	40	which	which	PRON
cana-6282	166	41	not	not	PART
cana-6282	166	42	only	only	ADV
cana-6282	166	43	reflects	reflect	VERB
cana-6282	166	44	the	the	DET
cana-6282	166	45	underlying	underlie	VERB
cana-6282	166	46	algebraic	algebraic	ADJ
cana-6282	166	47	properties	property	NOUN
cana-6282	166	48	induced	induce	VERB
cana-6282	166	49	by	by	ADP
cana-6282	166	50	the	the	DET
cana-6282	166	51	involutions	involution	NOUN
cana-6282	166	52	𝜎	𝜎	PROPN
cana-6282	166	53	and	and	CCONJ
cana-6282	166	54	𝜏	𝜏	NOUN
cana-6282	166	55	,	,	PUNCT
cana-6282	166	56	but	but	CCONJ
cana-6282	166	57	also	also	ADV
cana-6282	166	58	enables	enable	VERB
cana-6282	166	59	broader	broad	ADJ
cana-6282	166	60	generalizations	generalization	NOUN
cana-6282	166	61	.	.	PUNCT
cana-6282	167	1	by	by	ADP
cana-6282	167	2	identifying	identify	VERB
cana-6282	167	3	elements	element	NOUN
cana-6282	167	4	of	of	ADP
cana-6282	167	5	𝑆2	𝑆2	PROPN
cana-6282	167	6	with	with	ADP
cana-6282	167	7	ordered	order	VERB
cana-6282	167	8	pairs	pair	NOUN
cana-6282	167	9	and	and	CCONJ
cana-6282	167	10	defining	define	VERB
cana-6282	167	11	the	the	DET
cana-6282	167	12	multiplication	multiplication	NOUN
cana-6282	167	13	and	and	CCONJ
cana-6282	167	14	involutive	involutive	ADJ
cana-6282	167	15	mappings	mapping	NOUN
cana-6282	167	16	accordingly	accordingly	ADV
cana-6282	167	17	,	,	PUNCT
cana-6282	167	18	we	we	PRON
cana-6282	167	19	communications	communication	VERB
cana-6282	167	20	on	on	ADP
cana-6282	167	21	applied	apply	VERB
cana-6282	167	22	nonlinear	nonlinear	ADJ
cana-6282	167	23	analysis	analysis	NOUN
cana-6282	167	24	issn	issn	NOUN
cana-6282	167	25	:	:	PUNCT
cana-6282	167	26	1074	1074	NUM
cana-6282	167	27	-	-	PUNCT
cana-6282	167	28	133x	133x	NUM
cana-6282	167	29	vol	vol	VERB
cana-6282	167	30	32	32	NUM
cana-6282	167	31	no	no	NOUN
cana-6282	167	32	.	.	PUNCT
cana-6282	168	1	10s	10	NOUN
cana-6282	168	2	(	(	PUNCT
cana-6282	168	3	2025	2025	NUM
cana-6282	168	4	)	)	PUNCT
cana-6282	168	5	3692	3692	NUM
cana-6282	168	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-6282	168	7	derive	derive	VERB
cana-6282	168	8	a	a	DET
cana-6282	168	9	natural	natural	ADJ
cana-6282	168	10	equivalent	equivalent	NOUN
cana-6282	168	11	of	of	ADP
cana-6282	168	12	the	the	DET
cana-6282	168	13	original	original	ADJ
cana-6282	168	14	equation	equation	NOUN
cana-6282	168	15	on	on	ADP
cana-6282	168	16	the	the	DET
cana-6282	168	17	product	product	NOUN
cana-6282	168	18	semigroup	semigroup	NOUN
cana-6282	168	19	.	.	PUNCT
cana-6282	169	1	the	the	DET
cana-6282	169	2	following	follow	VERB
cana-6282	169	3	corollary	corollary	NOUN
cana-6282	169	4	confirms	confirm	VERB
cana-6282	169	5	that	that	SCONJ
cana-6282	169	6	the	the	DET
cana-6282	169	7	hyperstability	hyperstability	NOUN
cana-6282	169	8	property	property	NOUN
cana-6282	169	9	persists	persist	VERB
cana-6282	169	10	in	in	ADP
cana-6282	169	11	this	this	DET
cana-6282	169	12	extended	extended	ADJ
cana-6282	169	13	setting	setting	NOUN
cana-6282	169	14	.	.	PUNCT
cana-6282	170	1	corollary	corollary	ADJ
cana-6282	170	2	2.5	2.5	NUM
cana-6282	170	3	.	.	PUNCT
cana-6282	171	1	let	let	VERB
cana-6282	171	2	𝒮	𝒮	NOUN
cana-6282	171	3	:	:	PUNCT
cana-6282	171	4	=	=	SYM
cana-6282	171	5	𝑆	𝑆	PROPN
cana-6282	171	6	×	×	NOUN
cana-6282	171	7	𝑆	𝑆	PROPN
cana-6282	171	8	be	be	AUX
cana-6282	171	9	the	the	DET
cana-6282	171	10	direct	direct	ADJ
cana-6282	171	11	product	product	NOUN
cana-6282	171	12	of	of	ADP
cana-6282	171	13	a	a	DET
cana-6282	171	14	semigroup	semigroup	ADJ
cana-6282	171	15	𝑆	𝑆	PROPN
cana-6282	171	16	with	with	ADP
cana-6282	171	17	itself	itself	PRON
cana-6282	171	18	,	,	PUNCT
cana-6282	171	19	equipped	equip	VERB
cana-6282	171	20	with	with	ADP
cana-6282	171	21	the	the	DET
cana-6282	171	22	product	product	NOUN
cana-6282	171	23	operation	operation	NOUN
cana-6282	171	24	(	(	PUNCT
cana-6282	171	25	𝑥1	𝑥1	NOUN
cana-6282	171	26	,	,	PUNCT
cana-6282	171	27	𝑥3)(𝑥2	𝑥3)(𝑥2	NUM
cana-6282	171	28	,	,	PUNCT
cana-6282	171	29	𝑥4	𝑥4	NUM
cana-6282	171	30	):	):	PUNCT
cana-6282	172	1	=	=	SYM
cana-6282	172	2	(	(	PUNCT
cana-6282	172	3	𝑥1𝑥2	𝑥1𝑥2	NOUN
cana-6282	172	4	,	,	PUNCT
cana-6282	172	5	𝑥3𝑥4	𝑥3𝑥4	NOUN
cana-6282	172	6	)	)	PUNCT
cana-6282	172	7	and	and	CCONJ
cana-6282	172	8	let	let	VERB
cana-6282	172	9	𝜑	𝜑	PRON
cana-6282	172	10	:	:	PUNCT
cana-6282	172	11	𝒮	𝒮	PROPN
cana-6282	172	12	→	→	SYM
cana-6282	172	13	𝒮	𝒮	NOUN
cana-6282	172	14	be	be	AUX
cana-6282	172	15	defined	define	VERB
cana-6282	172	16	by	by	ADP
cana-6282	172	17	𝜑(𝑥2	𝜑(𝑥2	NOUN
cana-6282	172	18	,	,	PUNCT
cana-6282	172	19	𝑥4	𝑥4	NUM
cana-6282	172	20	):	):	PUNCT
cana-6282	172	21	=	=	SYM
cana-6282	172	22	(	(	PUNCT
cana-6282	172	23	𝜎(𝑥2	𝜎(𝑥2	NOUN
cana-6282	172	24	)	)	PUNCT
cana-6282	172	25	,	,	PUNCT
cana-6282	172	26	𝜏(𝑥4	𝜏(𝑥4	NOUN
cana-6282	172	27	)	)	PUNCT
cana-6282	172	28	)	)	PUNCT
cana-6282	172	29	,	,	PUNCT
cana-6282	172	30	where	where	SCONJ
cana-6282	172	31	𝜎	𝜎	PROPN
cana-6282	172	32	and	and	CCONJ
cana-6282	172	33	𝜏	𝜏	NOUN
cana-6282	172	34	are	be	AUX
cana-6282	172	35	involutions	involution	NOUN
cana-6282	172	36	on	on	ADP
cana-6282	172	37	𝑆.	𝑆.	PROPN
cana-6282	172	38	suppose	suppose	VERB
cana-6282	172	39	that	that	SCONJ
cana-6282	172	40	𝑓	𝑓	X
cana-6282	172	41	:	:	PUNCT
cana-6282	172	42	𝒮	𝒮	NOUN
cana-6282	172	43	→	→	PUNCT
cana-6282	172	44	𝑋	𝑋	PROPN
cana-6282	172	45	satisfies	satisfy	VERB
cana-6282	172	46	the	the	DET
cana-6282	172	47	inequality	inequality	NOUN
cana-6282	172	48	‖𝑓(𝜉휁	‖𝑓(𝜉휁	NOUN
cana-6282	172	49	)	)	PUNCT
cana-6282	172	50	+	+	CCONJ
cana-6282	172	51	𝑓(𝜉𝜑(휁	𝑓(𝜉𝜑(휁	NOUN
cana-6282	172	52	)	)	PUNCT
cana-6282	172	53	)	)	PUNCT
cana-6282	172	54	−	−	PROPN
cana-6282	172	55	2𝑓(𝜉	2𝑓(𝜉	NUM
cana-6282	172	56	)	)	PUNCT
cana-6282	172	57	−	−	ADP
cana-6282	172	58	2𝑓(휁)‖	2𝑓(휁)‖	NUM
cana-6282	172	59	≤	≤	NUM
cana-6282	172	60	휀(𝜉	휀(𝜉	NOUN
cana-6282	172	61	,	,	PUNCT
cana-6282	172	62	휁	휁	NOUN
cana-6282	172	63	)	)	PUNCT
cana-6282	172	64	for	for	ADP
cana-6282	172	65	all	all	DET
cana-6282	172	66	𝜉	𝜉	NOUN
cana-6282	172	67	,	,	PUNCT
cana-6282	172	68	휁	휁	PROPN
cana-6282	172	69	∈	∈	PROPN
cana-6282	172	70	𝒮	𝒮	PROPN
cana-6282	172	71	,	,	PUNCT
cana-6282	172	72	where	where	SCONJ
cana-6282	172	73	휀	휀	X
cana-6282	172	74	:	:	PUNCT
cana-6282	172	75	𝒮2	𝒮2	PROPN
cana-6282	172	76	→	→	PUNCT
cana-6282	172	77	ℝ+satisfies	ℝ+satisfie	NOUN
cana-6282	172	78	the	the	DET
cana-6282	172	79	asymptotic	asymptotic	ADJ
cana-6282	172	80	conditions	condition	NOUN
cana-6282	172	81	analogous	analogous	ADJ
cana-6282	172	82	to	to	ADP
cana-6282	172	83	those	those	PRON
cana-6282	172	84	in	in	ADP
cana-6282	172	85	theorem	theorem	ADJ
cana-6282	172	86	2.2	2.2	NUM
cana-6282	172	87	.	.	PUNCT
cana-6282	173	1	then	then	ADV
cana-6282	173	2	the	the	DET
cana-6282	173	3	functional	functional	ADJ
cana-6282	173	4	equation	equation	NOUN
cana-6282	173	5	𝑓(𝜉휁	𝑓(𝜉휁	PROPN
cana-6282	173	6	)	)	PUNCT
cana-6282	174	1	+	+	CCONJ
cana-6282	175	1	𝑓(𝜉𝜑(휁	𝑓(𝜉𝜑(휁	NOUN
cana-6282	175	2	)	)	PUNCT
cana-6282	175	3	)	)	PUNCT
cana-6282	175	4	=	=	SYM
cana-6282	175	5	2𝑓(𝜉	2𝑓(𝜉	NUM
cana-6282	175	6	)	)	PUNCT
cana-6282	176	1	+	+	CCONJ
cana-6282	176	2	2𝑓(휁	2𝑓(휁	NUM
cana-6282	176	3	)	)	PUNCT
cana-6282	176	4	,	,	PUNCT
cana-6282	176	5	∀𝜉	∀𝜉	PROPN
cana-6282	176	6	,	,	PUNCT
cana-6282	176	7	휁	휁	PROPN
cana-6282	176	8	∈	∈	PROPN
cana-6282	176	9	𝒮	𝒮	PROPN
cana-6282	176	10	(	(	PUNCT
cana-6282	176	11	2.17	2.17	NUM
cana-6282	176	12	)	)	PUNCT
cana-6282	176	13	is	be	AUX
cana-6282	176	14	hyperstable	hyperstable	ADJ
cana-6282	176	15	on	on	ADP
cana-6282	176	16	𝒮.	𝒮.	PROPN
cana-6282	176	17	proof	proof	NOUN
cana-6282	176	18	.	.	PUNCT
cana-6282	177	1	let	let	VERB
cana-6282	177	2	𝜉	𝜉	X
cana-6282	177	3	=	=	PUNCT
cana-6282	177	4	(	(	PUNCT
cana-6282	177	5	𝑥1	𝑥1	NOUN
cana-6282	177	6	,	,	PUNCT
cana-6282	177	7	𝑥3	𝑥3	NOUN
cana-6282	177	8	)	)	PUNCT
cana-6282	177	9	and	and	CCONJ
cana-6282	177	10	휁	휁	X
cana-6282	177	11	=	=	PUNCT
cana-6282	177	12	(	(	PUNCT
cana-6282	177	13	𝑥2	𝑥2	NOUN
cana-6282	177	14	,	,	PUNCT
cana-6282	177	15	𝑥4	𝑥4	NOUN
cana-6282	177	16	)	)	PUNCT
cana-6282	177	17	be	be	AUX
cana-6282	177	18	arbitrary	arbitrary	ADJ
cana-6282	177	19	elements	element	NOUN
cana-6282	177	20	in	in	ADP
cana-6282	177	21	𝒮.	𝒮.	PROPN
cana-6282	177	22	then	then	ADV
cana-6282	177	23	:	:	PUNCT
cana-6282	178	1	𝜉휁	𝜉휁	ADP
cana-6282	178	2	=	=	SYM
cana-6282	178	3	(	(	PUNCT
cana-6282	178	4	𝑥1𝑥2	𝑥1𝑥2	NOUN
cana-6282	178	5	,	,	PUNCT
cana-6282	178	6	𝑥3𝑥4	𝑥3𝑥4	NOUN
cana-6282	178	7	)	)	PUNCT
cana-6282	178	8	𝜉𝜑(휁	𝜉𝜑(휁	NUM
cana-6282	178	9	)	)	PUNCT
cana-6282	178	10	=	=	SYM
cana-6282	178	11	(	(	PUNCT
cana-6282	178	12	𝑥1𝜎(𝑥2	𝑥1𝜎(𝑥2	PROPN
cana-6282	178	13	)	)	PUNCT
cana-6282	178	14	,	,	PUNCT
cana-6282	178	15	𝑥3𝜏(𝑥4	𝑥3𝜏(𝑥4	PROPN
cana-6282	178	16	)	)	PUNCT
cana-6282	178	17	)	)	PUNCT
cana-6282	179	1	hence	hence	ADV
cana-6282	179	2	,	,	PUNCT
cana-6282	179	3	the	the	DET
cana-6282	179	4	equation	equation	NOUN
cana-6282	179	5	𝑓(𝜉휁	𝑓(𝜉휁	PROPN
cana-6282	179	6	)	)	PUNCT
cana-6282	180	1	+	+	CCONJ
cana-6282	181	1	𝑓(𝜉𝜑(휁	𝑓(𝜉𝜑(휁	NOUN
cana-6282	181	2	)	)	PUNCT
cana-6282	181	3	)	)	PUNCT
cana-6282	181	4	=	=	SYM
cana-6282	181	5	2𝑓(𝜉	2𝑓(𝜉	NUM
cana-6282	181	6	)	)	PUNCT
cana-6282	182	1	+	+	CCONJ
cana-6282	183	1	2𝑓(휁	2𝑓(휁	NOUN
cana-6282	183	2	)	)	PUNCT
cana-6282	183	3	is	be	AUX
cana-6282	183	4	equivalent	equivalent	ADJ
cana-6282	183	5	to	to	ADP
cana-6282	183	6	the	the	DET
cana-6282	183	7	original	original	ADJ
cana-6282	183	8	functional	functional	ADJ
cana-6282	183	9	equation	equation	NOUN
cana-6282	183	10	:	:	PUNCT
cana-6282	183	11	𝑓(𝑥1𝑥2	𝑓(𝑥1𝑥2	NOUN
cana-6282	183	12	,	,	PUNCT
cana-6282	183	13	𝑥3𝑥4	𝑥3𝑥4	NOUN
cana-6282	183	14	)	)	PUNCT
cana-6282	183	15	+	+	NUM
cana-6282	183	16	𝑓(𝑥1𝜎(𝑥2	𝑓(𝑥1𝜎(𝑥2	ADV
cana-6282	183	17	)	)	PUNCT
cana-6282	183	18	,	,	PUNCT
cana-6282	183	19	𝑥3𝜏(𝑥4	𝑥3𝜏(𝑥4	PROPN
cana-6282	183	20	)	)	PUNCT
cana-6282	183	21	)	)	PUNCT
cana-6282	184	1	=	=	SYM
cana-6282	184	2	2𝑓(𝑥1	2𝑓(𝑥1	NUM
cana-6282	184	3	,	,	PUNCT
cana-6282	184	4	𝑥3	𝑥3	NOUN
cana-6282	184	5	)	)	PUNCT
cana-6282	184	6	+	+	CCONJ
cana-6282	184	7	2𝑓(𝑥2	2𝑓(𝑥2	NUM
cana-6282	184	8	,	,	PUNCT
cana-6282	184	9	𝑥4	𝑥4	PROPN
cana-6282	184	10	)	)	PUNCT
cana-6282	184	11	which	which	PRON
cana-6282	184	12	was	be	AUX
cana-6282	184	13	shown	show	VERB
cana-6282	184	14	in	in	ADP
cana-6282	184	15	theorem	theorem	ADJ
cana-6282	184	16	2.2	2.2	NUM
cana-6282	184	17	to	to	PART
cana-6282	184	18	be	be	AUX
cana-6282	184	19	hyperstable	hyperstable	ADJ
cana-6282	184	20	under	under	ADP
cana-6282	184	21	the	the	DET
cana-6282	184	22	given	give	VERB
cana-6282	184	23	asymptotic	asymptotic	ADJ
cana-6282	184	24	condition	condition	NOUN
cana-6282	184	25	on	on	ADP
cana-6282	184	26	휀	휀	PROPN
cana-6282	184	27	.	.	PUNCT
cana-6282	185	1	moreover	moreover	ADV
cana-6282	185	2	,	,	PUNCT
cana-6282	185	3	we	we	PRON
cana-6282	185	4	can	can	AUX
cana-6282	185	5	define	define	VERB
cana-6282	185	6	a	a	DET
cana-6282	185	7	sequence	sequence	NOUN
cana-6282	185	8	{	{	PUNCT
cana-6282	185	9	𝑈𝑛}𝑛∈ℕ	𝑈𝑛}𝑛∈ℕ	X
cana-6282	185	10	⊂	⊂	PROPN
cana-6282	185	11	𝒮	𝒮	NOUN
cana-6282	185	12	by	by	ADP
cana-6282	185	13	𝑈𝑛	𝑈𝑛	PROPN
cana-6282	185	14	=	=	SYM
cana-6282	185	15	(	(	PUNCT
cana-6282	185	16	𝛼𝑛	𝛼𝑛	PROPN
cana-6282	185	17	,	,	PUNCT
cana-6282	185	18	𝛼𝑛	𝛼𝑛	PROPN
cana-6282	185	19	)	)	PUNCT
cana-6282	185	20	for	for	ADP
cana-6282	185	21	some	some	DET
cana-6282	185	22	fixed	fix	VERB
cana-6282	185	23	𝛼	𝛼	PROPN
cana-6282	185	24	∈	∈	PROPN
cana-6282	185	25	𝑆	𝑆	PROPN
cana-6282	185	26	satisfying	satisfy	VERB
cana-6282	185	27	the	the	DET
cana-6282	185	28	decay	decay	NOUN
cana-6282	185	29	condition	condition	NOUN
cana-6282	185	30	:	:	PUNCT
cana-6282	185	31	휀(𝜉	휀(𝜉	PROPN
cana-6282	185	32	⋅	⋅	PROPN
cana-6282	185	33	𝑈𝑛	𝑈𝑛	PROPN
cana-6282	185	34	,	,	PUNCT
cana-6282	185	35	휁	휁	PRON
cana-6282	185	36	⋅	⋅	PROPN
cana-6282	185	37	𝑈𝑛	𝑈𝑛	PROPN
cana-6282	185	38	)	)	PUNCT
cana-6282	185	39	→	→	SYM
cana-6282	185	40	0	0	NUM
cana-6282	185	41	and	and	CCONJ
cana-6282	185	42	휀(𝜉	휀(𝜉	PROPN
cana-6282	185	43	⋅	⋅	PROPN
cana-6282	185	44	𝜑(𝑈𝑛	𝜑(𝑈𝑛	NUM
cana-6282	185	45	)	)	PUNCT
cana-6282	185	46	,	,	PUNCT
cana-6282	185	47	휁	휁	X
cana-6282	185	48	⋅	⋅	PROPN
cana-6282	185	49	𝜑(𝑈𝑛	𝜑(𝑈𝑛	NUM
cana-6282	185	50	)	)	PUNCT
cana-6282	185	51	)	)	PUNCT
cana-6282	186	1	→	→	SYM
cana-6282	186	2	0	0	PUNCT
cana-6282	186	3	as	as	ADP
cana-6282	186	4	𝑛	𝑛	PROPN
cana-6282	186	5	→	→	SYM
cana-6282	186	6	∞	∞	PROPN
cana-6282	186	7	,	,	PUNCT
cana-6282	186	8	for	for	ADP
cana-6282	186	9	all	all	DET
cana-6282	186	10	𝜉	𝜉	NOUN
cana-6282	186	11	,	,	PUNCT
cana-6282	186	12	휁	휁	PROPN
cana-6282	186	13	∈	∈	PROPN
cana-6282	186	14	𝒮.	𝒮.	PROPN
cana-6282	186	15	therefore	therefore	ADV
cana-6282	186	16	,	,	PUNCT
cana-6282	186	17	by	by	ADP
cana-6282	186	18	direct	direct	ADJ
cana-6282	186	19	application	application	NOUN
cana-6282	186	20	of	of	ADP
cana-6282	186	21	the	the	DET
cana-6282	186	22	same	same	ADJ
cana-6282	186	23	reasoning	reasoning	NOUN
cana-6282	186	24	in	in	ADP
cana-6282	186	25	theorem	theorem	ADJ
cana-6282	186	26	2.2	2.2	NUM
cana-6282	186	27	,	,	PUNCT
cana-6282	186	28	we	we	PRON
cana-6282	186	29	conclude	conclude	VERB
cana-6282	186	30	that	that	SCONJ
cana-6282	186	31	𝑓(𝜉휁	𝑓(𝜉휁	NOUN
cana-6282	186	32	)	)	PUNCT
cana-6282	187	1	+	+	CCONJ
cana-6282	188	1	𝑓(𝜉𝜑(휁	𝑓(𝜉𝜑(휁	NOUN
cana-6282	188	2	)	)	PUNCT
cana-6282	188	3	)	)	PUNCT
cana-6282	188	4	=	=	SYM
cana-6282	188	5	2𝑓(𝜉	2𝑓(𝜉	NUM
cana-6282	188	6	)	)	PUNCT
cana-6282	189	1	+	+	CCONJ
cana-6282	189	2	2𝑓(휁	2𝑓(휁	NUM
cana-6282	189	3	)	)	PUNCT
cana-6282	189	4	holds	hold	VERB
cana-6282	189	5	for	for	ADP
cana-6282	189	6	all	all	DET
cana-6282	189	7	𝜉	𝜉	NOUN
cana-6282	189	8	,	,	PUNCT
cana-6282	189	9	휁	휁	PROPN
cana-6282	189	10	∈	∈	PROPN
cana-6282	189	11	𝒮	𝒮	PROPN
cana-6282	189	12	,	,	PUNCT
cana-6282	189	13	i.e.	i.e.	X
cana-6282	189	14	,	,	PUNCT
cana-6282	189	15	the	the	DET
cana-6282	189	16	equation	equation	NOUN
cana-6282	189	17	(	(	PUNCT
cana-6282	189	18	2.17	2.17	NUM
cana-6282	189	19	)	)	PUNCT
cana-6282	189	20	is	be	AUX
cana-6282	189	21	hyperstable	hyperstable	ADJ
cana-6282	189	22	on	on	ADP
cana-6282	189	23	𝒮.	𝒮.	PROPN
cana-6282	189	24	conclusion	conclusion	NOUN
cana-6282	189	25	in	in	ADP
cana-6282	189	26	this	this	DET
cana-6282	189	27	paper	paper	NOUN
cana-6282	189	28	,	,	PUNCT
cana-6282	189	29	we	we	PRON
cana-6282	189	30	have	have	AUX
cana-6282	189	31	investigated	investigate	VERB
cana-6282	189	32	the	the	DET
cana-6282	189	33	hyperstability	hyperstability	NOUN
cana-6282	189	34	of	of	ADP
cana-6282	189	35	a	a	DET
cana-6282	189	36	general	general	ADJ
cana-6282	189	37	functional	functional	ADJ
cana-6282	189	38	equation	equation	NOUN
cana-6282	189	39	involving	involve	VERB
cana-6282	189	40	two	two	NUM
cana-6282	189	41	involutive	involutive	ADJ
cana-6282	189	42	mappings	mapping	NOUN
cana-6282	189	43	defined	define	VERB
cana-6282	189	44	on	on	ADP
cana-6282	189	45	an	an	DET
cana-6282	189	46	arbitrary	arbitrary	ADJ
cana-6282	189	47	semigroup	semigroup	NOUN
cana-6282	189	48	.	.	PUNCT
cana-6282	190	1	using	use	VERB
cana-6282	190	2	an	an	DET
cana-6282	190	3	asymptotic	asymptotic	ADJ
cana-6282	190	4	approach	approach	NOUN
cana-6282	190	5	based	base	VERB
cana-6282	190	6	on	on	ADP
cana-6282	190	7	the	the	DET
cana-6282	190	8	method	method	NOUN
cana-6282	190	9	of	of	ADP
cana-6282	190	10	maksa	maksa	ADJ
cana-6282	190	11	and	and	CCONJ
cana-6282	190	12	páles	pále	NOUN
cana-6282	190	13	[	[	X
cana-6282	190	14	15	15	NUM
cana-6282	190	15	]	]	PUNCT
cana-6282	190	16	,	,	PUNCT
cana-6282	190	17	we	we	PRON
cana-6282	190	18	established	establish	VERB
cana-6282	190	19	sufficient	sufficient	ADJ
cana-6282	190	20	conditions	condition	NOUN
cana-6282	190	21	under	under	ADP
cana-6282	190	22	which	which	PRON
cana-6282	190	23	approximate	approximate	ADJ
cana-6282	190	24	solutions	solution	NOUN
cana-6282	190	25	to	to	ADP
cana-6282	190	26	the	the	DET
cana-6282	190	27	equation	equation	NOUN
cana-6282	190	28	𝑓(𝑥1𝑥2	𝑓(𝑥1𝑥2	NOUN
cana-6282	190	29	,	,	PUNCT
cana-6282	190	30	𝑥3𝑥4	𝑥3𝑥4	NOUN
cana-6282	190	31	)	)	PUNCT
cana-6282	190	32	+	+	NUM
cana-6282	190	33	𝑓(𝑥1𝜎(𝑥2	𝑓(𝑥1𝜎(𝑥2	ADV
cana-6282	190	34	)	)	PUNCT
cana-6282	190	35	,	,	PUNCT
cana-6282	190	36	𝑥3𝜏(𝑥4	𝑥3𝜏(𝑥4	PROPN
cana-6282	190	37	)	)	PUNCT
cana-6282	190	38	)	)	PUNCT
cana-6282	191	1	=	=	SYM
cana-6282	191	2	2𝑓(𝑥1	2𝑓(𝑥1	NUM
cana-6282	191	3	,	,	PUNCT
cana-6282	191	4	𝑥3	𝑥3	NOUN
cana-6282	191	5	)	)	PUNCT
cana-6282	191	6	+	+	CCONJ
cana-6282	191	7	2𝑓(𝑥2	2𝑓(𝑥2	NUM
cana-6282	191	8	,	,	PUNCT
cana-6282	191	9	𝑥4	𝑥4	ADJ
cana-6282	191	10	)	)	PUNCT
cana-6282	191	11	communications	communication	NOUN
cana-6282	191	12	on	on	ADP
cana-6282	191	13	applied	apply	VERB
cana-6282	191	14	nonlinear	nonlinear	ADJ
cana-6282	191	15	analysis	analysis	NOUN
cana-6282	191	16	issn	issn	NOUN
cana-6282	191	17	:	:	PUNCT
cana-6282	191	18	1074	1074	NUM
cana-6282	191	19	-	-	PUNCT
cana-6282	191	20	133x	133x	NUM
cana-6282	191	21	vol	vol	VERB
cana-6282	191	22	32	32	NUM
cana-6282	191	23	no	no	NOUN
cana-6282	191	24	.	.	PUNCT
cana-6282	192	1	10s	10	NOUN
cana-6282	192	2	(	(	PUNCT
cana-6282	192	3	2025	2025	NUM
cana-6282	192	4	)	)	PUNCT
cana-6282	192	5	3693	3693	NUM
cana-6282	192	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-6282	192	7	must	must	AUX
cana-6282	192	8	in	in	ADP
cana-6282	192	9	fact	fact	NOUN
cana-6282	192	10	be	be	AUX
cana-6282	192	11	exact	exact	ADJ
cana-6282	192	12	solutions	solution	NOUN
cana-6282	192	13	.	.	PUNCT
cana-6282	193	1	these	these	DET
cana-6282	193	2	results	result	NOUN
cana-6282	193	3	extend	extend	VERB
cana-6282	193	4	prior	prior	ADV
cana-6282	193	5	stability	stability	NOUN
cana-6282	193	6	theorems	theorem	NOUN
cana-6282	193	7	and	and	CCONJ
cana-6282	193	8	contribute	contribute	VERB
cana-6282	193	9	to	to	ADP
cana-6282	193	10	the	the	DET
cana-6282	193	11	ongoing	ongoing	ADJ
cana-6282	193	12	development	development	NOUN
cana-6282	193	13	of	of	ADP
cana-6282	193	14	the	the	DET
cana-6282	193	15	theory	theory	NOUN
cana-6282	193	16	of	of	ADP
cana-6282	193	17	functional	functional	ADJ
cana-6282	193	18	equations	equation	NOUN
cana-6282	193	19	in	in	ADP
cana-6282	193	20	algebraic	algebraic	ADJ
cana-6282	193	21	settings	setting	NOUN
cana-6282	193	22	involving	involve	VERB
cana-6282	193	23	involutions	involution	NOUN
cana-6282	193	24	.	.	PUNCT
cana-6282	194	1	furthermore	furthermore	ADV
cana-6282	194	2	,	,	PUNCT
cana-6282	194	3	we	we	PRON
cana-6282	194	4	provided	provide	VERB
cana-6282	194	5	a	a	DET
cana-6282	194	6	reformulation	reformulation	NOUN
cana-6282	194	7	of	of	ADP
cana-6282	194	8	the	the	DET
cana-6282	194	9	equation	equation	NOUN
cana-6282	194	10	over	over	ADP
cana-6282	194	11	the	the	DET
cana-6282	194	12	product	product	NOUN
cana-6282	194	13	semigroup	semigroup	NOUN
cana-6282	194	14	𝑆	𝑆	PROPN
cana-6282	194	15	=	=	SYM
cana-6282	194	16	𝑆	𝑆	PROPN
cana-6282	194	17	×	×	PROPN
cana-6282	194	18	𝑆	𝑆	PROPN
cana-6282	194	19	,	,	PUNCT
cana-6282	194	20	where	where	SCONJ
cana-6282	194	21	we	we	PRON
cana-6282	194	22	demonstrated	demonstrate	VERB
cana-6282	194	23	via	via	ADP
cana-6282	194	24	corollary	corollary	ADJ
cana-6282	194	25	2.5	2.5	NUM
cana-6282	194	26	that	that	PRON
cana-6282	194	27	the	the	DET
cana-6282	194	28	hyperstability	hyperstability	NOUN
cana-6282	194	29	property	property	NOUN
cana-6282	194	30	is	be	AUX
cana-6282	194	31	preserved	preserve	VERB
cana-6282	194	32	under	under	ADP
cana-6282	194	33	this	this	DET
cana-6282	194	34	transformation	transformation	NOUN
cana-6282	194	35	.	.	PUNCT
cana-6282	195	1	this	this	DET
cana-6282	195	2	reformulation	reformulation	NOUN
cana-6282	195	3	not	not	PART
cana-6282	195	4	only	only	ADV
cana-6282	195	5	highlights	highlight	VERB
cana-6282	195	6	the	the	DET
cana-6282	195	7	underlying	underlie	VERB
cana-6282	195	8	algebraic	algebraic	ADJ
cana-6282	195	9	symmetry	symmetry	NOUN
cana-6282	195	10	of	of	ADP
cana-6282	195	11	the	the	DET
cana-6282	195	12	problem	problem	NOUN
cana-6282	195	13	but	but	CCONJ
cana-6282	195	14	also	also	ADV
cana-6282	195	15	broadens	broaden	VERB
cana-6282	195	16	the	the	DET
cana-6282	195	17	structural	structural	ADJ
cana-6282	195	18	scope	scope	NOUN
cana-6282	195	19	of	of	ADP
cana-6282	195	20	the	the	DET
cana-6282	195	21	stability	stability	NOUN
cana-6282	195	22	analysis	analysis	NOUN
cana-6282	195	23	.	.	PUNCT
cana-6282	196	1	future	future	ADJ
cana-6282	196	2	work	work	NOUN
cana-6282	196	3	may	may	AUX
cana-6282	196	4	explore	explore	VERB
cana-6282	196	5	analogous	analogous	ADJ
cana-6282	196	6	phenomena	phenomenon	NOUN
cana-6282	196	7	in	in	ADP
cana-6282	196	8	more	more	ADV
cana-6282	196	9	complex	complex	ADJ
cana-6282	196	10	algebraic	algebraic	ADJ
cana-6282	196	11	systems	system	NOUN
cana-6282	196	12	,	,	PUNCT
cana-6282	196	13	including	include	VERB
cana-6282	196	14	inverse	inverse	NOUN
cana-6282	196	15	and	and	CCONJ
cana-6282	196	16	topological	topological	ADJ
cana-6282	196	17	semigroups	semigroup	NOUN
cana-6282	196	18	or	or	CCONJ
cana-6282	196	19	frameworks	framework	NOUN
cana-6282	196	20	involving	involve	VERB
cana-6282	196	21	multiple	multiple	ADJ
cana-6282	196	22	interacting	interacting	ADJ
cana-6282	196	23	involutions	involution	NOUN
cana-6282	196	24	.	.	PUNCT
cana-6282	197	1	additional	additional	ADJ
cana-6282	197	2	investigations	investigation	NOUN
cana-6282	197	3	may	may	AUX
cana-6282	197	4	also	also	ADV
cana-6282	197	5	target	target	VERB
cana-6282	197	6	non	non	ADJ
cana-6282	197	7	-	-	ADJ
cana-6282	197	8	quadratic	quadratic	ADJ
cana-6282	197	9	functional	functional	ADJ
cana-6282	197	10	equations	equation	NOUN
cana-6282	197	11	and	and	CCONJ
cana-6282	197	12	their	their	PRON
cana-6282	197	13	inhomogeneous	inhomogeneous	ADJ
cana-6282	197	14	counterparts	counterpart	NOUN
cana-6282	197	15	under	under	ADP
cana-6282	197	16	similar	similar	ADJ
cana-6282	197	17	asymptotic	asymptotic	ADJ
cana-6282	197	18	stability	stability	NOUN
cana-6282	197	19	conditions	condition	NOUN
cana-6282	197	20	.	.	PUNCT
cana-6282	198	1	refrences	refrence	VERB
cana-6282	199	1	[	[	X
cana-6282	199	2	1	1	X
cana-6282	199	3	]	]	X
cana-6282	199	4	y.	y.	PROPN
cana-6282	199	5	aissi	aissi	PROPN
cana-6282	199	6	,	,	PUNCT
cana-6282	199	7	d.	d.	PROPN
cana-6282	199	8	zeglami	zeglami	PROPN
cana-6282	199	9	and	and	CCONJ
cana-6282	199	10	a.	a.	PROPN
cana-6282	199	11	mouzoun	mouzoun	PROPN
cana-6282	199	12	,	,	PUNCT
cana-6282	199	13	a	a	DET
cana-6282	199	14	quadratic	quadratic	ADJ
cana-6282	199	15	functional	functional	ADJ
cana-6282	199	16	equation	equation	NOUN
cana-6282	199	17	with	with	ADP
cana-6282	199	18	involutive	involutive	ADJ
cana-6282	199	19	automorphisms	automorphism	NOUN
cana-6282	199	20	on	on	ADP
cana-6282	199	21	semigroups	semigroup	NOUN
cana-6282	199	22	,	,	PUNCT
cana-6282	199	23	bol	bol	NOUN
cana-6282	199	24	.	.	PUNCT
cana-6282	200	1	soc	soc	PROPN
cana-6282	200	2	.	.	PUNCT
cana-6282	201	1	mat	mat	PROPN
cana-6282	201	2	.	.	PUNCT
cana-6282	201	3	mex	mex	PROPN
cana-6282	201	4	.	.	PUNCT
cana-6282	202	1	28	28	NUM
cana-6282	202	2	,	,	PUNCT
cana-6282	202	3	19	19	NUM
cana-6282	202	4	(	(	PUNCT
cana-6282	202	5	2022	2022	NUM
cana-6282	202	6	)	)	PUNCT
cana-6282	202	7	.	.	PUNCT
cana-6282	203	1	[	[	X
cana-6282	203	2	2	2	NUM
cana-6282	203	3	]	]	PUNCT
cana-6282	203	4	a.	a.	NOUN
cana-6282	203	5	akkaoui	akkaoui	PROPN
cana-6282	203	6	and	and	CCONJ
cana-6282	203	7	b.	b.	PROPN
cana-6282	203	8	fadli	fadli	PROPN
cana-6282	203	9	,	,	PUNCT
cana-6282	203	10	new	new	ADJ
cana-6282	203	11	results	result	NOUN
cana-6282	203	12	about	about	ADP
cana-6282	203	13	quadratic	quadratic	ADJ
cana-6282	203	14	functional	functional	ADJ
cana-6282	203	15	equation	equation	NOUN
cana-6282	203	16	on	on	ADP
cana-6282	203	17	semigroups	semigroup	NOUN
cana-6282	203	18	,	,	PUNCT
cana-6282	203	19	annales	annale	VERB
cana-6282	203	20	mathematicae	mathematicae	PROPN
cana-6282	203	21	silesianae	silesianae	NOUN
cana-6282	203	22	,	,	PUNCT
cana-6282	203	23	39	39	NUM
cana-6282	203	24	(	(	PUNCT
cana-6282	203	25	2025	2025	NUM
cana-6282	203	26	)	)	PUNCT
cana-6282	203	27	,	,	PUNCT
cana-6282	203	28	no	no	INTJ
cana-6282	203	29	.	.	NOUN
cana-6282	203	30	2	2	NUM
cana-6282	203	31	,	,	PUNCT
cana-6282	203	32	209–222	209–222	NUM
cana-6282	203	33	.	.	PUNCT
cana-6282	204	1	[	[	X
cana-6282	204	2	3	3	X
cana-6282	204	3	]	]	PUNCT
cana-6282	204	4	m.	m.	NOUN
cana-6282	204	5	almahalebi	almahalebi	NOUN
cana-6282	204	6	,	,	PUNCT
cana-6282	204	7	on	on	ADP
cana-6282	204	8	the	the	DET
cana-6282	204	9	hyperstability	hyperstability	NOUN
cana-6282	204	10	of	of	ADP
cana-6282	204	11	σ	σ	PROPN
cana-6282	204	12	-	-	PUNCT
cana-6282	204	13	drygas	drygas	ADJ
cana-6282	204	14	functional	functional	ADJ
cana-6282	204	15	equation	equation	NOUN
cana-6282	204	16	on	on	ADP
cana-6282	204	17	semigroups	semigroup	NOUN
cana-6282	204	18	,	,	PUNCT
cana-6282	204	19	aequat	aequat	PROPN
cana-6282	204	20	.	.	PUNCT
cana-6282	205	1	math	math	NOUN
cana-6282	205	2	.	.	PUNCT
cana-6282	206	1	,90	,90	PROPN
cana-6282	206	2	(	(	PUNCT
cana-6282	206	3	2016	2016	NUM
cana-6282	206	4	)	)	PUNCT
cana-6282	206	5	,	,	PUNCT
cana-6282	206	6	849	849	NUM
cana-6282	206	7	-	-	SYM
cana-6282	206	8	857	857	NUM
cana-6282	206	9	.	.	PUNCT
cana-6282	207	1	[	[	X
cana-6282	207	2	4	4	X
cana-6282	207	3	]	]	PUNCT
cana-6282	207	4	t.	t.	PROPN
cana-6282	207	5	aoki	aoki	PROPN
cana-6282	207	6	,	,	PUNCT
cana-6282	207	7	on	on	ADP
cana-6282	207	8	the	the	DET
cana-6282	207	9	stability	stability	NOUN
cana-6282	207	10	of	of	ADP
cana-6282	207	11	the	the	DET
cana-6282	207	12	linear	linear	ADJ
cana-6282	207	13	transformation	transformation	NOUN
cana-6282	207	14	in	in	ADP
cana-6282	207	15	banach	banach	NOUN
cana-6282	207	16	spaces	space	NOUN
cana-6282	207	17	,	,	PUNCT
cana-6282	207	18	j.	j.	PROPN
cana-6282	207	19	math	math	PROPN
cana-6282	207	20	.	.	PUNCT
cana-6282	208	1	soc	soc	PROPN
cana-6282	208	2	.	.	PUNCT
cana-6282	209	1	japan	japan	PROPN
cana-6282	209	2	,	,	PUNCT
cana-6282	209	3	2	2	NUM
cana-6282	209	4	(	(	PUNCT
cana-6282	209	5	1950	1950	NUM
cana-6282	209	6	)	)	PUNCT
cana-6282	209	7	,	,	PUNCT
cana-6282	209	8	64	64	NUM
cana-6282	209	9	-	-	SYM
cana-6282	209	10	66	66	NUM
cana-6282	209	11	.	.	PUNCT
cana-6282	210	1	[	[	X
cana-6282	210	2	5	5	NUM
cana-6282	210	3	]	]	X
cana-6282	210	4	j.-h	j.-h	NOUN
cana-6282	210	5	.	.	PUNCT
cana-6282	211	1	bae	bae	NOUN
cana-6282	211	2	and	and	CCONJ
cana-6282	211	3	w.-g	w.-g	PROPN
cana-6282	211	4	.	.	PUNCT
cana-6282	212	1	park	park	NOUN
cana-6282	212	2	,	,	PUNCT
cana-6282	212	3	a	a	DET
cana-6282	212	4	functional	functional	ADJ
cana-6282	212	5	equation	equation	NOUN
cana-6282	212	6	originating	originate	VERB
cana-6282	212	7	from	from	ADP
cana-6282	212	8	quadratic	quadratic	ADJ
cana-6282	212	9	forms	form	NOUN
cana-6282	212	10	,	,	PUNCT
cana-6282	212	11	j.	j.	PROPN
cana-6282	212	12	math	math	PROPN
cana-6282	212	13	.	.	PUNCT
cana-6282	213	1	anal	anal	PROPN
cana-6282	213	2	.	.	PUNCT
cana-6282	213	3	appl	appl	PROPN
cana-6282	213	4	.	.	PROPN
cana-6282	213	5	,	,	PUNCT
cana-6282	213	6	326	326	NUM
cana-6282	213	7	(	(	PUNCT
cana-6282	213	8	2	2	NUM
cana-6282	213	9	)	)	PUNCT
cana-6282	213	10	,	,	PUNCT
cana-6282	213	11	(	(	PUNCT
cana-6282	213	12	2007	2007	NUM
cana-6282	213	13	)	)	PUNCT
cana-6282	213	14	,	,	PUNCT
cana-6282	213	15	1142	1142	NUM
cana-6282	213	16	-	-	SYM
cana-6282	213	17	1148	1148	NUM
cana-6282	213	18	.	.	PUNCT
cana-6282	214	1	[	[	X
cana-6282	214	2	6	6	NUM
cana-6282	214	3	]	]	SYM
cana-6282	214	4	j.-h	j.-h	NOUN
cana-6282	214	5	.	.	PUNCT
cana-6282	215	1	bae	bae	NOUN
cana-6282	215	2	and	and	CCONJ
cana-6282	215	3	w.-g	w.-g	PROPN
cana-6282	215	4	.	.	PUNCT
cana-6282	216	1	park	park	NOUN
cana-6282	216	2	,	,	PUNCT
cana-6282	216	3	a	a	DET
cana-6282	216	4	fixed	fix	VERB
cana-6282	216	5	-	-	PUNCT
cana-6282	216	6	point	point	NOUN
cana-6282	216	7	approach	approach	NOUN
cana-6282	216	8	to	to	ADP
cana-6282	216	9	the	the	DET
cana-6282	216	10	stability	stability	NOUN
cana-6282	216	11	of	of	ADP
cana-6282	216	12	a	a	DET
cana-6282	216	13	functional	functional	ADJ
cana-6282	216	14	equation	equation	NOUN
cana-6282	216	15	on	on	ADP
cana-6282	216	16	quadratic	quadratic	ADJ
cana-6282	216	17	forms	form	NOUN
cana-6282	216	18	j.	j.	PROPN
cana-6282	216	19	inequal	inequal	PROPN
cana-6282	216	20	.	.	PUNCT
cana-6282	217	1	appl	appl	PROPN
cana-6282	217	2	.	.	PROPN
cana-6282	217	3	,	,	PUNCT
cana-6282	217	4	2011	2011	NUM
cana-6282	217	5	,	,	PUNCT
cana-6282	217	6	82	82	NUM
cana-6282	217	7	,	,	PUNCT
cana-6282	217	8	(	(	PUNCT
cana-6282	217	9	2011	2011	NUM
cana-6282	217	10	)	)	PUNCT
cana-6282	217	11	,	,	PUNCT
cana-6282	217	12	https://doi.org/10.1186/1029242x-2011-82	https://doi.org/10.1186/1029242x-2011-82	NOUN
cana-6282	217	13	[	[	X
cana-6282	217	14	7	7	NUM
cana-6282	217	15	]	]	X
cana-6282	217	16	j.-h	j.-h	NOUN
cana-6282	217	17	.	.	PUNCT
cana-6282	218	1	bae	bae	NOUN
cana-6282	218	2	and	and	CCONJ
cana-6282	218	3	w.-g	w.-g	PROPN
cana-6282	218	4	.	.	PUNCT
cana-6282	219	1	park	park	NOUN
cana-6282	219	2	,	,	PUNCT
cana-6282	219	3	approximate	approximate	ADJ
cana-6282	219	4	property	property	NOUN
cana-6282	219	5	of	of	ADP
cana-6282	219	6	a	a	DET
cana-6282	219	7	functional	functional	ADJ
cana-6282	219	8	equation	equation	NOUN
cana-6282	219	9	with	with	ADP
cana-6282	219	10	a	a	DET
cana-6282	219	11	general	general	ADJ
cana-6282	219	12	involution	involution	NOUN
cana-6282	219	13	demonstratio	demonstratio	PROPN
cana-6282	219	14	mathematica	mathematica	PROPN
cana-6282	219	15	,	,	PUNCT
cana-6282	219	16	51	51	NUM
cana-6282	219	17	(	(	PUNCT
cana-6282	219	18	1	1	NUM
cana-6282	219	19	)	)	PUNCT
cana-6282	219	20	,	,	PUNCT
cana-6282	219	21	(	(	PUNCT
cana-6282	219	22	2018	2018	NUM
cana-6282	219	23	)	)	PUNCT
cana-6282	219	24	,	,	PUNCT
cana-6282	219	25	304	304	NUM
cana-6282	219	26	-	-	SYM
cana-6282	219	27	308	308	NUM
cana-6282	219	28	.	.	PUNCT
cana-6282	220	1	[	[	X
cana-6282	220	2	8	8	NUM
cana-6282	220	3	]	]	X
cana-6282	220	4	d.	d.	PROPN
cana-6282	220	5	g.	g.	PROPN
cana-6282	220	6	bourgin	bourgin	PROPN
cana-6282	220	7	,	,	PUNCT
cana-6282	220	8	approximately	approximately	ADV
cana-6282	220	9	isometric	isometric	ADJ
cana-6282	220	10	and	and	CCONJ
cana-6282	220	11	multiplicative	multiplicative	ADJ
cana-6282	220	12	transformations	transformation	NOUN
cana-6282	220	13	on	on	ADP
cana-6282	220	14	continuous	continuous	ADJ
cana-6282	220	15	function	function	NOUN
cana-6282	220	16	rings	ring	NOUN
cana-6282	220	17	,	,	PUNCT
cana-6282	220	18	duke	duke	PROPN
cana-6282	220	19	math	math	PROPN
cana-6282	220	20	.	.	PUNCT
cana-6282	221	1	j.	j.	PROPN
cana-6282	221	2	,	,	PUNCT
cana-6282	221	3	16	16	NUM
cana-6282	221	4	(	(	PUNCT
cana-6282	221	5	1949	1949	NUM
cana-6282	221	6	)	)	PUNCT
cana-6282	221	7	,	,	PUNCT
cana-6282	221	8	385	385	NUM
cana-6282	221	9	-	-	SYM
cana-6282	221	10	397	397	NUM
cana-6282	221	11	.	.	PUNCT
cana-6282	222	1	[	[	X
cana-6282	222	2	9	9	NUM
cana-6282	222	3	]	]	PUNCT
cana-6282	222	4	d.	d.	PROPN
cana-6282	222	5	g.	g.	PROPN
cana-6282	222	6	bourgin	bourgin	PROPN
cana-6282	222	7	,	,	PUNCT
cana-6282	222	8	classes	class	NOUN
cana-6282	222	9	of	of	ADP
cana-6282	222	10	transformations	transformation	NOUN
cana-6282	222	11	and	and	CCONJ
cana-6282	222	12	bordering	border	VERB
cana-6282	222	13	transformations	transformation	NOUN
cana-6282	222	14	,	,	PUNCT
cana-6282	222	15	bull	bull	NOUN
cana-6282	222	16	.	.	PUNCT
cana-6282	223	1	amer	amer	PROPN
cana-6282	223	2	.	.	PUNCT
cana-6282	223	3	math	math	PROPN
cana-6282	223	4	.	.	PUNCT
cana-6282	224	1	soc	soc	PROPN
cana-6282	224	2	.	.	PUNCT
cana-6282	224	3	,	,	PUNCT
cana-6282	224	4	57	57	NUM
cana-6282	224	5	(	(	PUNCT
cana-6282	224	6	1951	1951	NUM
cana-6282	224	7	)	)	PUNCT
cana-6282	224	8	,	,	PUNCT
cana-6282	224	9	223	223	NUM
cana-6282	224	10	-	-	SYM
cana-6282	224	11	237	237	NUM
cana-6282	224	12	.	.	PUNCT
cana-6282	225	1	[	[	X
cana-6282	225	2	10	10	NUM
cana-6282	225	3	]	]	X
cana-6282	225	4	j.	j.	PROPN
cana-6282	225	5	brzd¸ek	brzd¸ek	PROPN
cana-6282	225	6	and	and	CCONJ
cana-6282	225	7	k.	k.	PROPN
cana-6282	225	8	ciepliński	ciepliński	PROPN
cana-6282	225	9	,	,	PUNCT
cana-6282	225	10	hyperstability	hyperstability	NOUN
cana-6282	225	11	and	and	CCONJ
cana-6282	225	12	superstability	superstability	NOUN
cana-6282	225	13	,	,	PUNCT
cana-6282	225	14	abs	ab	NOUN
cana-6282	225	15	.	.	PUNCT
cana-6282	225	16	appl	appl	PROPN
cana-6282	225	17	.	.	PUNCT
cana-6282	226	1	anal	anal	PROPN
cana-6282	226	2	.	.	PROPN
cana-6282	226	3	,	,	PUNCT
cana-6282	226	4	2013	2013	NUM
cana-6282	226	5	(	(	PUNCT
cana-6282	226	6	2013	2013	NUM
cana-6282	226	7	)	)	PUNCT
cana-6282	226	8	,	,	PUNCT
cana-6282	226	9	article	article	NOUN
cana-6282	226	10	i	i	PROPN
cana-6282	226	11	d	d	PROPN
cana-6282	226	12	401756	401756	NUM
cana-6282	226	13	,	,	PUNCT
cana-6282	226	14	13	13	NUM
cana-6282	226	15	pp	pp	NOUN
cana-6282	226	16	.	.	PUNCT
cana-6282	227	1	[	[	X
cana-6282	227	2	11	11	NUM
cana-6282	227	3	]	]	PUNCT
cana-6282	227	4	b.	b.	PROPN
cana-6282	227	5	fadli	fadli	PROPN
cana-6282	227	6	,	,	PUNCT
cana-6282	227	7	d.	d.	PROPN
cana-6282	227	8	zeglami	zeglami	PROPN
cana-6282	227	9	and	and	CCONJ
cana-6282	227	10	s.	s.	PROPN
cana-6282	227	11	kabbaj	kabbaj	PROPN
cana-6282	227	12	,	,	PUNCT
cana-6282	227	13	a	a	DET
cana-6282	227	14	variant	variant	NOUN
cana-6282	227	15	of	of	ADP
cana-6282	227	16	the	the	DET
cana-6282	227	17	quadratic	quadratic	ADJ
cana-6282	227	18	functional	functional	ADJ
cana-6282	227	19	equation	equation	NOUN
cana-6282	227	20	on	on	ADP
cana-6282	227	21	semigroups	semigroup	NOUN
cana-6282	227	22	,	,	PUNCT
cana-6282	227	23	proyecciones	proyeccione	NOUN
cana-6282	227	24	(	(	PUNCT
cana-6282	227	25	antofagasta	antofagasta	PROPN
cana-6282	227	26	)	)	PUNCT
cana-6282	227	27	,	,	PUNCT
cana-6282	227	28	37	37	NUM
cana-6282	227	29	(	(	PUNCT
cana-6282	227	30	1	1	NUM
cana-6282	227	31	)	)	PUNCT
cana-6282	227	32	,	,	PUNCT
cana-6282	227	33	(	(	PUNCT
cana-6282	227	34	2018	2018	NUM
cana-6282	227	35	)	)	PUNCT
cana-6282	227	36	,	,	PUNCT
cana-6282	227	37	45–55	45–55	X
cana-6282	227	38	.	.	PUNCT
cana-6282	228	1	[	[	X
cana-6282	228	2	12	12	NUM
cana-6282	228	3	]	]	X
cana-6282	228	4	g.	g.	PROPN
cana-6282	228	5	l.	l.	PROPN
cana-6282	228	6	forti	forti	PROPN
cana-6282	228	7	,	,	PUNCT
cana-6282	228	8	an	an	DET
cana-6282	228	9	existence	existence	NOUN
cana-6282	228	10	and	and	CCONJ
cana-6282	228	11	stability	stability	NOUN
cana-6282	228	12	theorem	theorem	VERB
cana-6282	228	13	for	for	ADP
cana-6282	228	14	a	a	DET
cana-6282	228	15	class	class	NOUN
cana-6282	228	16	of	of	ADP
cana-6282	228	17	functional	functional	ADJ
cana-6282	228	18	equations	equation	NOUN
cana-6282	228	19	,	,	PUNCT
cana-6282	228	20	stochastica	stochastica	NOUN
cana-6282	228	21	4(1980	4(1980	NUM
cana-6282	228	22	)	)	PUNCT
cana-6282	228	23	,	,	PUNCT
cana-6282	228	24	23	23	NUM
cana-6282	228	25	-	-	SYM
cana-6282	228	26	30	30	NUM
cana-6282	228	27	.	.	PUNCT
cana-6282	229	1	communications	communication	NOUN
cana-6282	229	2	on	on	ADP
cana-6282	229	3	applied	apply	VERB
cana-6282	229	4	nonlinear	nonlinear	ADJ
cana-6282	229	5	analysis	analysis	NOUN
cana-6282	229	6	issn	issn	NOUN
cana-6282	229	7	:	:	PUNCT
cana-6282	229	8	1074	1074	NUM
cana-6282	229	9	-	-	PUNCT
cana-6282	229	10	133x	133x	NUM
cana-6282	229	11	vol	vol	VERB
cana-6282	229	12	32	32	NUM
cana-6282	229	13	no	no	NOUN
cana-6282	229	14	.	.	PUNCT
cana-6282	230	1	10s	10	NOUN
cana-6282	230	2	(	(	PUNCT
cana-6282	230	3	2025	2025	NUM
cana-6282	230	4	)	)	PUNCT
cana-6282	230	5	3694	3694	NUM
cana-6282	230	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-6282	231	1	[	[	X
cana-6282	231	2	13	13	NUM
cana-6282	231	3	]	]	PUNCT
cana-6282	231	4	p.	p.	NOUN
cana-6282	231	5	găvruţă	găvruţă	NOUN
cana-6282	231	6	,	,	PUNCT
cana-6282	231	7	a	a	DET
cana-6282	231	8	generalization	generalization	NOUN
cana-6282	231	9	of	of	ADP
cana-6282	231	10	the	the	DET
cana-6282	231	11	hyers	hyers	PROPN
cana-6282	231	12	-	-	PUNCT
cana-6282	231	13	ulam	ulam	ADJ
cana-6282	231	14	-	-	PUNCT
cana-6282	231	15	rassias	rassias	PROPN
cana-6282	231	16	stability	stability	NOUN
cana-6282	231	17	of	of	ADP
cana-6282	231	18	approximately	approximately	ADV
cana-6282	231	19	additive	additive	ADJ
cana-6282	231	20	mappings	mapping	NOUN
cana-6282	231	21	,	,	PUNCT
cana-6282	231	22	j.	j.	PROPN
cana-6282	231	23	math	math	PROPN
cana-6282	231	24	.	.	PUNCT
cana-6282	232	1	anal	anal	PROPN
cana-6282	232	2	.	.	PUNCT
cana-6282	232	3	appl	appl	PROPN
cana-6282	232	4	.	.	PUNCT
cana-6282	233	1	184	184	NUM
cana-6282	233	2	(	(	PUNCT
cana-6282	233	3	1994	1994	NUM
cana-6282	233	4	)	)	PUNCT
cana-6282	233	5	,	,	PUNCT
cana-6282	233	6	431	431	NUM
cana-6282	233	7	-	-	SYM
cana-6282	233	8	436	436	NUM
cana-6282	233	9	.	.	PUNCT
cana-6282	234	1	[	[	X
cana-6282	234	2	14	14	NUM
cana-6282	234	3	]	]	X
cana-6282	234	4	d.	d.	PROPN
cana-6282	234	5	h.	h.	PROPN
cana-6282	234	6	hyers	hyers	PROPN
cana-6282	234	7	,	,	PUNCT
cana-6282	234	8	on	on	ADP
cana-6282	234	9	the	the	DET
cana-6282	234	10	stability	stability	NOUN
cana-6282	234	11	of	of	ADP
cana-6282	234	12	the	the	DET
cana-6282	234	13	linear	linear	ADJ
cana-6282	234	14	functional	functional	ADJ
cana-6282	234	15	equation	equation	NOUN
cana-6282	234	16	,	,	PUNCT
cana-6282	234	17	proc	proc	NOUN
cana-6282	234	18	.	.	PUNCT
cana-6282	235	1	nat	nat	PROPN
cana-6282	235	2	.	.	PUNCT
cana-6282	236	1	acad	acad	PROPN
cana-6282	236	2	.	.	PUNCT
cana-6282	237	1	sci	sci	PROPN
cana-6282	237	2	.	.	PUNCT
cana-6282	237	3	u.	u.	PROPN
cana-6282	237	4	s.	s.	PROPN
cana-6282	237	5	a.	a.	PROPN
cana-6282	237	6	27	27	NUM
cana-6282	237	7	(	(	PUNCT
cana-6282	237	8	1941	1941	NUM
cana-6282	237	9	)	)	PUNCT
cana-6282	237	10	,	,	PUNCT
cana-6282	237	11	222	222	NUM
cana-6282	237	12	-	-	SYM
cana-6282	237	13	224	224	NUM
cana-6282	237	14	.	.	PUNCT
cana-6282	238	1	[	[	X
cana-6282	238	2	15	15	NUM
cana-6282	238	3	]	]	X
cana-6282	238	4	gy	gy	NOUN
cana-6282	238	5	.	.	PROPN
cana-6282	238	6	maksa	maksa	PROPN
cana-6282	238	7	and	and	CCONJ
cana-6282	238	8	zs	zs	PROPN
cana-6282	238	9	.	.	PUNCT
cana-6282	238	10	páles	pále	NOUN
cana-6282	238	11	,	,	PUNCT
cana-6282	238	12	hyperstability	hyperstability	NOUN
cana-6282	238	13	of	of	ADP
cana-6282	238	14	a	a	DET
cana-6282	238	15	class	class	NOUN
cana-6282	238	16	of	of	ADP
cana-6282	238	17	linear	linear	ADJ
cana-6282	238	18	functional	functional	ADJ
cana-6282	238	19	equations	equation	NOUN
cana-6282	238	20	,	,	PUNCT
cana-6282	238	21	acta	acta	PROPN
cana-6282	238	22	math	math	PROPN
cana-6282	238	23	.	.	PUNCT
cana-6282	239	1	acad	acad	PROPN
cana-6282	239	2	.	.	PUNCT
cana-6282	240	1	paedag	paedag	PROPN
cana-6282	240	2	.	.	PUNCT
cana-6282	241	1	nyíregyháziensis	nyíregyháziensis	PROPN
cana-6282	241	2	,	,	PUNCT
cana-6282	241	3	17	17	NUM
cana-6282	241	4	(	(	PUNCT
cana-6282	241	5	2001	2001	NUM
cana-6282	241	6	)	)	PUNCT
cana-6282	241	7	,	,	PUNCT
cana-6282	241	8	107	107	NUM
cana-6282	241	9	-	-	SYM
cana-6282	241	10	112	112	NUM
cana-6282	241	11	.	.	PUNCT
cana-6282	242	1	[	[	X
cana-6282	242	2	16	16	NUM
cana-6282	242	3	]	]	X
cana-6282	242	4	th	th	X
cana-6282	242	5	.	.	PUNCT
cana-6282	242	6	m.	m.	NOUN
cana-6282	242	7	rassias	rassias	PROPN
cana-6282	242	8	,	,	PUNCT
cana-6282	242	9	on	on	ADP
cana-6282	242	10	the	the	DET
cana-6282	242	11	stability	stability	NOUN
cana-6282	242	12	of	of	ADP
cana-6282	242	13	linear	linear	PROPN
cana-6282	242	14	mapping	mapping	NOUN
cana-6282	242	15	in	in	ADP
cana-6282	242	16	banach	banach	NOUN
cana-6282	242	17	spaces	space	NOUN
cana-6282	242	18	,	,	PUNCT
cana-6282	242	19	proc	proc	NOUN
cana-6282	242	20	.	.	PUNCT
cana-6282	243	1	amer	amer	PROPN
cana-6282	243	2	.	.	PUNCT
cana-6282	243	3	math	math	PROPN
cana-6282	243	4	.	.	PUNCT
cana-6282	244	1	soc	soc	PROPN
cana-6282	244	2	.	.	PUNCT
cana-6282	244	3	,	,	PUNCT
cana-6282	244	4	72	72	NUM
cana-6282	244	5	(	(	PUNCT
cana-6282	244	6	1978	1978	NUM
cana-6282	244	7	)	)	PUNCT
cana-6282	244	8	,	,	PUNCT
cana-6282	244	9	297	297	NUM
cana-6282	244	10	-	-	SYM
cana-6282	244	11	300	300	NUM
cana-6282	244	12	.	.	PUNCT
cana-6282	245	1	[	[	X
cana-6282	245	2	17	17	NUM
cana-6282	245	3	]	]	PUNCT
cana-6282	245	4	s.	s.	PROPN
cana-6282	245	5	m.	m.	PROPN
cana-6282	245	6	ulam	ulam	PROPN
cana-6282	245	7	.	.	PUNCT
cana-6282	245	8	problems	problem	NOUN
cana-6282	245	9	in	in	ADP
cana-6282	245	10	modern	modern	ADJ
cana-6282	245	11	mathematics	mathematic	NOUN
cana-6282	245	12	,	,	PUNCT
cana-6282	245	13	science	science	NOUN
cana-6282	245	14	editions	edition	NOUN
cana-6282	245	15	john	john	PROPN
cana-6282	245	16	wiley	wiley	PROPN
cana-6282	245	17	sons	sons	PROPN
cana-6282	245	18	,	,	PUNCT
cana-6282	245	19	inc	inc	PROPN
cana-6282	245	20	.	.	PROPN
cana-6282	245	21	,	,	PUNCT
cana-6282	245	22	new	new	ADJ
cana-6282	245	23	york,(1964	york,(1964	NOUN
cana-6282	245	24	)	)	PUNCT
cana-6282	245	25	.	.	PUNCT