id	sid	tid	token	lemma	pos
cana-1961	1	1	communications	communication	NOUN
cana-1961	1	2	on	on	ADP
cana-1961	1	3	applied	apply	VERB
cana-1961	1	4	nonlinear	nonlinear	ADJ
cana-1961	1	5	analysis	analysis	NOUN
cana-1961	1	6	issn	issn	NOUN
cana-1961	1	7	:	:	PUNCT
cana-1961	1	8	1074	1074	NUM
cana-1961	1	9	-	-	PUNCT
cana-1961	1	10	133x	133x	NUM
cana-1961	1	11	vol	vol	NOUN
cana-1961	1	12	32	32	NUM
cana-1961	1	13	no	no	NOUN
cana-1961	1	14	.	.	NOUN
cana-1961	1	15	3	3	NUM
cana-1961	1	16	(	(	PUNCT
cana-1961	1	17	2025	2025	NUM
cana-1961	1	18	)	)	PUNCT
cana-1961	1	19	301	301	NUM
cana-1961	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1961	1	21	stability	stability	NOUN
cana-1961	1	22	of	of	ADP
cana-1961	1	23	a	a	DET
cana-1961	1	24	general	general	ADJ
cana-1961	1	25	quadratic	quadratic	ADJ
cana-1961	1	26	-	-	PUNCT
cana-1961	1	27	cubic	cubic	ADJ
cana-1961	1	28	functional	functional	ADJ
cana-1961	1	29	equation	equation	NOUN
cana-1961	1	30	in	in	ADP
cana-1961	1	31	nonarchimedean	nonarchimedean	ADJ
cana-1961	1	32	2	2	NUM
cana-1961	1	33	-	-	PUNCT
cana-1961	1	34	normed	norme	VERB
cana-1961	1	35	spaces	space	NOUN
cana-1961	1	36	elumalai	elumalai	PROPN
cana-1961	1	37	p1	p1	PROPN
cana-1961	1	38	,	,	PUNCT
cana-1961	1	39	sangeetha	sangeetha	PROPN
cana-1961	1	40	s2	s2	PROPN
cana-1961	1	41	*	*	PROPN
cana-1961	1	42	1,2	1,2	NUM
cana-1961	1	43	department	department	NOUN
cana-1961	1	44	of	of	ADP
cana-1961	1	45	mathematics	mathematic	NOUN
cana-1961	1	46	,	,	PUNCT
cana-1961	1	47	college	college	NOUN
cana-1961	1	48	of	of	ADP
cana-1961	1	49	engineering	engineering	NOUN
cana-1961	1	50	and	and	CCONJ
cana-1961	1	51	technology	technology	NOUN
cana-1961	1	52	,	,	PUNCT
cana-1961	1	53	srm	srm	PROPN
cana-1961	1	54	institute	institute	PROPN
cana-1961	1	55	of	of	ADP
cana-1961	1	56	science	science	NOUN
cana-1961	1	57	and	and	CCONJ
cana-1961	1	58	technology	technology	NOUN
cana-1961	1	59	,	,	PUNCT
cana-1961	1	60	kattankulathur	kattankulathur	PROPN
cana-1961	1	61	,	,	PUNCT
cana-1961	1	62	chengalpattu	chengalpattu	ADV
cana-1961	1	63	,	,	PUNCT
cana-1961	1	64	tamil	tamil	PROPN
cana-1961	1	65	nadu-603	nadu-603	NOUN
cana-1961	1	66	203	203	NUM
cana-1961	1	67	,	,	PUNCT
cana-1961	1	68	india	india	PROPN
cana-1961	1	69	.	.	PUNCT
cana-1961	2	1	ep5583@srmist.edu.in	ep5583@srmist.edu.in	PROPN
cana-1961	2	2	corresponding	correspond	VERB
cana-1961	2	3	author	author	NOUN
cana-1961	2	4	:	:	PUNCT
cana-1961	2	5	sangeets@srmist.edu.in	sangeets@srmist.edu.in	PROPN
cana-1961	2	6	article	article	NOUN
cana-1961	2	7	history	history	NOUN
cana-1961	2	8	:	:	PUNCT
cana-1961	2	9	received	receive	VERB
cana-1961	2	10	:	:	PUNCT
cana-1961	2	11	31	31	NUM
cana-1961	2	12	-	-	SYM
cana-1961	2	13	07	07	NUM
cana-1961	2	14	-	-	PUNCT
cana-1961	2	15	2024	2024	NUM
cana-1961	2	16	revised	revise	VERB
cana-1961	2	17	:	:	PUNCT
cana-1961	2	18	21	21	NUM
cana-1961	2	19	-	-	SYM
cana-1961	2	20	09	09	NUM
cana-1961	2	21	-	-	PUNCT
cana-1961	2	22	2024	2024	NUM
cana-1961	2	23	accepted	accept	VERB
cana-1961	2	24	:	:	PUNCT
cana-1961	2	25	02	02	NUM
cana-1961	2	26	-	-	PUNCT
cana-1961	2	27	10	10	NUM
cana-1961	2	28	-	-	PUNCT
cana-1961	2	29	2024	2024	NUM
cana-1961	2	30	abstract	abstract	NOUN
cana-1961	2	31	this	this	DET
cana-1961	2	32	research	research	NOUN
cana-1961	2	33	aims	aim	VERB
cana-1961	2	34	to	to	PART
cana-1961	2	35	investigate	investigate	VERB
cana-1961	2	36	the	the	DET
cana-1961	2	37	hyers	hyer	NOUN
cana-1961	2	38	-	-	PUNCT
cana-1961	2	39	ulam	ulam	ADJ
cana-1961	2	40	stability	stability	NOUN
cana-1961	2	41	of	of	ADP
cana-1961	2	42	the	the	DET
cana-1961	2	43	mixed	mixed	ADJ
cana-1961	2	44	-	-	PUNCT
cana-1961	2	45	type	type	NOUN
cana-1961	2	46	quadraticcubic	quadraticcubic	ADJ
cana-1961	2	47	functional	functional	ADJ
cana-1961	2	48	equation	equation	NOUN
cana-1961	2	49	in	in	ADP
cana-1961	2	50	non	non	ADJ
cana-1961	2	51	-	-	ADJ
cana-1961	2	52	archimedean	archimedean	ADJ
cana-1961	2	53	2	2	NUM
cana-1961	2	54	-	-	PUNCT
cana-1961	2	55	normed	norme	VERB
cana-1961	2	56	spaces	space	NOUN
cana-1961	2	57	using	use	VERB
cana-1961	2	58	the	the	DET
cana-1961	2	59	fixed	fix	VERB
cana-1961	2	60	-	-	PUNCT
cana-1961	2	61	point	point	NOUN
cana-1961	2	62	method	method	NOUN
cana-1961	2	63	.	.	PUNCT
cana-1961	3	1	additionally	additionally	ADV
cana-1961	3	2	,	,	PUNCT
cana-1961	3	3	some	some	DET
cana-1961	3	4	counter	counter	NOUN
cana-1961	3	5	-	-	NOUN
cana-1961	3	6	examples	example	NOUN
cana-1961	3	7	are	be	AUX
cana-1961	3	8	illustrated	illustrate	VERB
cana-1961	3	9	for	for	ADP
cana-1961	3	10	instability	instability	NOUN
cana-1961	3	11	.	.	PUNCT
cana-1961	4	1	the	the	DET
cana-1961	4	2	exciting	exciting	ADJ
cana-1961	4	3	possibilities	possibility	NOUN
cana-1961	4	4	of	of	ADP
cana-1961	4	5	this	this	DET
cana-1961	4	6	cutting	cut	VERB
cana-1961	4	7	-	-	PUNCT
cana-1961	4	8	edge	edge	NOUN
cana-1961	4	9	research	research	NOUN
cana-1961	4	10	and	and	CCONJ
cana-1961	4	11	unlocking	unlock	VERB
cana-1961	4	12	new	new	ADJ
cana-1961	4	13	frontiers	frontier	NOUN
cana-1961	4	14	in	in	ADP
cana-1961	4	15	mathematical	mathematical	ADJ
cana-1961	4	16	analysis	analysis	NOUN
cana-1961	4	17	are	be	AUX
cana-1961	4	18	explored	explore	VERB
cana-1961	4	19	.	.	PUNCT
cana-1961	5	1	keywords	keyword	NOUN
cana-1961	5	2	:	:	PUNCT
cana-1961	5	3	fixed	fix	VERB
cana-1961	5	4	point	point	NOUN
cana-1961	5	5	method	method	NOUN
cana-1961	5	6	,	,	PUNCT
cana-1961	5	7	hyers	hyers	PROPN
cana-1961	5	8	-	-	PUNCT
cana-1961	5	9	ulam	ulam	PROPN
cana-1961	5	10	stability	stability	PROPN
cana-1961	5	11	,	,	PUNCT
cana-1961	5	12	non	non	ADJ
cana-1961	5	13	-	-	ADJ
cana-1961	5	14	archimedean	archimedean	ADJ
cana-1961	5	15	2	2	NUM
cana-1961	5	16	-	-	PUNCT
cana-1961	5	17	normed	norme	VERB
cana-1961	5	18	spaces	space	NOUN
cana-1961	5	19	,	,	PUNCT
cana-1961	5	20	p	p	ADJ
cana-1961	5	21	-	-	PUNCT
cana-1961	5	22	adic	adic	ADJ
cana-1961	5	23	field	field	NOUN
cana-1961	5	24	,	,	PUNCT
cana-1961	5	25	quadratic	quadratic	ADJ
cana-1961	5	26	-	-	PUNCT
cana-1961	5	27	cubic	cubic	ADJ
cana-1961	5	28	functional	functional	ADJ
cana-1961	5	29	equation	equation	NOUN
cana-1961	5	30	.	.	PUNCT
cana-1961	6	1	1	1	X
cana-1961	6	2	.	.	X
cana-1961	6	3	introduction	introduction	NOUN
cana-1961	6	4	functional	functional	ADJ
cana-1961	6	5	equations	equation	NOUN
cana-1961	6	6	play	play	VERB
cana-1961	6	7	an	an	DET
cana-1961	6	8	essential	essential	ADJ
cana-1961	6	9	and	and	CCONJ
cana-1961	6	10	fascinating	fascinating	ADJ
cana-1961	6	11	role	role	NOUN
cana-1961	6	12	in	in	ADP
cana-1961	6	13	mathematics	mathematic	NOUN
cana-1961	6	14	,	,	PUNCT
cana-1961	6	15	employing	employ	VERB
cana-1961	6	16	simple	simple	ADJ
cana-1961	6	17	algebraic	algebraic	ADJ
cana-1961	6	18	procedures	procedure	NOUN
cana-1961	6	19	that	that	PRON
cana-1961	6	20	lead	lead	VERB
cana-1961	6	21	to	to	ADP
cana-1961	6	22	intriguing	intriguing	ADJ
cana-1961	6	23	solutions	solution	NOUN
cana-1961	6	24	.	.	PUNCT
cana-1961	7	1	the	the	DET
cana-1961	7	2	theory	theory	NOUN
cana-1961	7	3	of	of	ADP
cana-1961	7	4	functional	functional	ADJ
cana-1961	7	5	equations	equation	NOUN
cana-1961	7	6	is	be	AUX
cana-1961	7	7	also	also	ADV
cana-1961	7	8	applied	apply	VERB
cana-1961	7	9	in	in	ADP
cana-1961	7	10	developing	develop	VERB
cana-1961	7	11	other	other	ADJ
cana-1961	7	12	domains	domain	NOUN
cana-1961	7	13	such	such	ADJ
cana-1961	7	14	as	as	ADP
cana-1961	7	15	analysis	analysis	NOUN
cana-1961	7	16	,	,	PUNCT
cana-1961	7	17	algebra	algebra	NOUN
cana-1961	7	18	,	,	PUNCT
cana-1961	7	19	geometry	geometry	NOUN
cana-1961	7	20	,	,	PUNCT
cana-1961	7	21	and	and	CCONJ
cana-1961	7	22	more	more	ADJ
cana-1961	7	23	.	.	PUNCT
cana-1961	8	1	new	new	ADJ
cana-1961	8	2	approaches	approach	NOUN
cana-1961	8	3	and	and	CCONJ
cana-1961	8	4	techniques	technique	NOUN
cana-1961	8	5	are	be	AUX
cana-1961	8	6	utilized	utilize	VERB
cana-1961	8	7	in	in	ADP
cana-1961	8	8	problem	problem	NOUN
cana-1961	8	9	-	-	PUNCT
cana-1961	8	10	solving	solving	NOUN
cana-1961	8	11	across	across	ADP
cana-1961	8	12	fields	field	NOUN
cana-1961	8	13	like	like	ADP
cana-1961	8	14	it	it	PRON
cana-1961	8	15	,	,	PUNCT
cana-1961	8	16	finance	finance	NOUN
cana-1961	8	17	,	,	PUNCT
cana-1961	8	18	geometry	geometry	NOUN
cana-1961	8	19	,	,	PUNCT
cana-1961	8	20	wireless	wireless	ADJ
cana-1961	8	21	sensor	sensor	NOUN
cana-1961	8	22	networks	network	NOUN
cana-1961	8	23	,	,	PUNCT
cana-1961	8	24	and	and	CCONJ
cana-1961	8	25	beyond	beyond	ADP
cana-1961	8	26	.	.	PUNCT
cana-1961	9	1	ulam	ulam	PROPN
cana-1961	9	2	stability	stability	PROPN
cana-1961	9	3	is	be	AUX
cana-1961	9	4	a	a	DET
cana-1961	9	5	crucial	crucial	ADJ
cana-1961	9	6	concept	concept	NOUN
cana-1961	9	7	in	in	ADP
cana-1961	9	8	studying	study	VERB
cana-1961	9	9	functional	functional	ADJ
cana-1961	9	10	equations	equation	NOUN
cana-1961	9	11	and	and	CCONJ
cana-1961	9	12	their	their	PRON
cana-1961	9	13	solutions	solution	NOUN
cana-1961	9	14	.	.	PUNCT
cana-1961	10	1	this	this	DET
cana-1961	10	2	theory	theory	NOUN
cana-1961	10	3	examines	examine	VERB
cana-1961	10	4	whether	whether	SCONJ
cana-1961	10	5	a	a	DET
cana-1961	10	6	function	function	NOUN
cana-1961	10	7	that	that	PRON
cana-1961	10	8	approximately	approximately	ADV
cana-1961	10	9	satisfies	satisfy	VERB
cana-1961	10	10	a	a	DET
cana-1961	10	11	certain	certain	ADJ
cana-1961	10	12	functional	functional	ADJ
cana-1961	10	13	equation	equation	NOUN
cana-1961	10	14	is	be	AUX
cana-1961	10	15	close	close	ADJ
cana-1961	10	16	to	to	ADP
cana-1961	10	17	a	a	DET
cana-1961	10	18	function	function	NOUN
cana-1961	10	19	that	that	PRON
cana-1961	10	20	exactly	exactly	ADV
cana-1961	10	21	satisfies	satisfy	VERB
cana-1961	10	22	the	the	DET
cana-1961	10	23	equation	equation	NOUN
cana-1961	10	24	.	.	PUNCT
cana-1961	11	1	numerous	numerous	ADJ
cana-1961	11	2	researchers	researcher	NOUN
cana-1961	11	3	across	across	ADP
cana-1961	11	4	various	various	ADJ
cana-1961	11	5	fields	field	NOUN
cana-1961	11	6	have	have	AUX
cana-1961	11	7	explored	explore	VERB
cana-1961	11	8	different	different	ADJ
cana-1961	11	9	types	type	NOUN
cana-1961	11	10	of	of	ADP
cana-1961	11	11	functional	functional	ADJ
cana-1961	11	12	equation	equation	NOUN
cana-1961	11	13	stability	stability	NOUN
cana-1961	11	14	,	,	PUNCT
cana-1961	11	15	such	such	ADJ
cana-1961	11	16	as	as	ADP
cana-1961	11	17	hyers	hyer	NOUN
cana-1961	11	18	-	-	PUNCT
cana-1961	11	19	ulam	ulam	INTJ
cana-1961	11	20	(	(	PUNCT
cana-1961	11	21	h	h	NOUN
cana-1961	11	22	-	-	PUNCT
cana-1961	11	23	u	u	NOUN
cana-1961	11	24	)	)	PUNCT
cana-1961	11	25	stability	stability	NOUN
cana-1961	11	26	,	,	PUNCT
cana-1961	11	27	hyers	hyers	PROPN
cana-1961	11	28	-	-	PUNCT
cana-1961	11	29	ulam	ulam	INTJ
cana-1961	11	30	-	-	PUNCT
cana-1961	11	31	rassias	rassias	PROPN
cana-1961	11	32	(	(	PUNCT
cana-1961	11	33	h	h	NOUN
cana-1961	11	34	-	-	PUNCT
cana-1961	11	35	u	u	NOUN
cana-1961	11	36	-	-	PUNCT
cana-1961	11	37	r	r	NOUN
cana-1961	11	38	)	)	PUNCT
cana-1961	11	39	stability	stability	NOUN
cana-1961	11	40	,	,	PUNCT
cana-1961	11	41	and	and	CCONJ
cana-1961	11	42	generalized	generalized	ADJ
cana-1961	11	43	hyers	hyer	NOUN
cana-1961	11	44	-	-	PUNCT
cana-1961	11	45	ulam	ulam	PROPN
cana-1961	11	46	stability	stability	NOUN
cana-1961	11	47	,	,	PUNCT
cana-1961	11	48	particularly	particularly	ADV
cana-1961	11	49	in	in	ADP
cana-1961	11	50	the	the	DET
cana-1961	11	51	context	context	NOUN
cana-1961	11	52	of	of	ADP
cana-1961	11	53	different	different	ADJ
cana-1961	11	54	functional	functional	ADJ
cana-1961	11	55	equations	equation	NOUN
cana-1961	11	56	and	and	CCONJ
cana-1961	11	57	mixed	mixed	ADJ
cana-1961	11	58	types	type	NOUN
cana-1961	11	59	over	over	ADP
cana-1961	11	60	recent	recent	ADJ
cana-1961	11	61	decades	decade	NOUN
cana-1961	11	62	.	.	PUNCT
cana-1961	12	1	additionally	additionally	ADV
cana-1961	12	2	,	,	PUNCT
cana-1961	12	3	many	many	ADJ
cana-1961	12	4	authors	author	NOUN
cana-1961	12	5	have	have	AUX
cana-1961	12	6	investigated	investigate	VERB
cana-1961	12	7	the	the	DET
cana-1961	12	8	stability	stability	NOUN
cana-1961	12	9	of	of	ADP
cana-1961	12	10	various	various	ADJ
cana-1961	12	11	functional	functional	ADJ
cana-1961	12	12	equations	equation	NOUN
cana-1961	12	13	,	,	PUNCT
cana-1961	12	14	yielding	yield	VERB
cana-1961	12	15	fascinating	fascinating	ADJ
cana-1961	12	16	results	result	NOUN
cana-1961	12	17	in	in	ADP
cana-1961	12	18	the	the	DET
cana-1961	12	19	classical	classical	ADJ
cana-1961	12	20	(	(	PUNCT
cana-1961	12	21	archimedean	archimedean	ADJ
cana-1961	12	22	)	)	PUNCT
cana-1961	12	23	case	case	NOUN
cana-1961	12	24	.	.	PUNCT
cana-1961	13	1	in	in	ADP
cana-1961	13	2	recent	recent	ADJ
cana-1961	13	3	years	year	NOUN
cana-1961	13	4	,	,	PUNCT
cana-1961	13	5	the	the	DET
cana-1961	13	6	stability	stability	NOUN
cana-1961	13	7	problems	problem	NOUN
cana-1961	13	8	of	of	ADP
cana-1961	13	9	these	these	DET
cana-1961	13	10	functional	functional	ADJ
cana-1961	13	11	equations	equation	NOUN
cana-1961	13	12	in	in	ADP
cana-1961	13	13	non	non	ADJ
cana-1961	13	14	-	-	ADJ
cana-1961	13	15	archimedean	archimedean	ADJ
cana-1961	13	16	(	(	PUNCT
cana-1961	13	17	na	na	NOUN
cana-1961	13	18	)	)	PUNCT
cana-1961	13	19	spaces	space	NOUN
cana-1961	13	20	have	have	AUX
cana-1961	13	21	also	also	ADV
cana-1961	13	22	been	be	AUX
cana-1961	13	23	examined	examine	VERB
cana-1961	13	24	.	.	PUNCT
cana-1961	14	1	in	in	ADP
cana-1961	14	2	the	the	DET
cana-1961	14	3	last	last	ADJ
cana-1961	14	4	few	few	ADJ
cana-1961	14	5	decades	decade	NOUN
cana-1961	14	6	,	,	PUNCT
cana-1961	14	7	researchers	researcher	NOUN
cana-1961	14	8	have	have	AUX
cana-1961	14	9	become	become	VERB
cana-1961	14	10	much	much	ADV
cana-1961	14	11	more	more	ADV
cana-1961	14	12	interested	interested	ADJ
cana-1961	14	13	in	in	ADP
cana-1961	14	14	the	the	DET
cana-1961	14	15	study	study	NOUN
cana-1961	14	16	of	of	ADP
cana-1961	14	17	hyers	hyers	PROPN
cana-1961	14	18	-	-	PUNCT
cana-1961	14	19	ulam	ulam	PROPN
cana-1961	14	20	stability	stability	NOUN
cana-1961	14	21	of	of	ADP
cana-1961	14	22	functional	functional	ADJ
cana-1961	14	23	equations	equation	NOUN
cana-1961	14	24	(	(	PUNCT
cana-1961	14	25	fes	fes	NOUN
cana-1961	14	26	)	)	PUNCT
cana-1961	14	27	.	.	PUNCT
cana-1961	15	1	numerous	numerous	ADJ
cana-1961	15	2	high	high	ADJ
cana-1961	15	3	-	-	PUNCT
cana-1961	15	4	quality	quality	NOUN
cana-1961	15	5	research	research	NOUN
cana-1961	15	6	papers	paper	NOUN
cana-1961	15	7	have	have	AUX
cana-1961	15	8	been	be	AUX
cana-1961	15	9	published	publish	VERB
cana-1961	15	10	on	on	ADP
cana-1961	15	11	this	this	DET
cana-1961	15	12	topic	topic	NOUN
cana-1961	15	13	,	,	PUNCT
cana-1961	15	14	as	as	SCONJ
cana-1961	15	15	evidenced	evidence	VERB
cana-1961	15	16	by	by	ADP
cana-1961	15	17	references	reference	NOUN
cana-1961	15	18	[	[	X
cana-1961	15	19	1	1	NUM
cana-1961	15	20	-	-	SYM
cana-1961	15	21	3	3	NUM
cana-1961	15	22	]	]	PUNCT
cana-1961	15	23	and	and	CCONJ
cana-1961	15	24	the	the	DET
cana-1961	15	25	associated	associated	ADJ
cana-1961	15	26	citations	citation	NOUN
cana-1961	15	27	in	in	ADP
cana-1961	15	28	references	reference	NOUN
cana-1961	15	29	[	[	X
cana-1961	15	30	4	4	NUM
cana-1961	15	31	-	-	SYM
cana-1961	15	32	6	6	NUM
cana-1961	15	33	]	]	PUNCT
cana-1961	15	34	.	.	PUNCT
cana-1961	16	1	as	as	SCONJ
cana-1961	16	2	the	the	DET
cana-1961	16	3	field	field	NOUN
cana-1961	16	4	continues	continue	VERB
cana-1961	16	5	to	to	PART
cana-1961	16	6	grow	grow	VERB
cana-1961	16	7	,	,	PUNCT
cana-1961	16	8	various	various	ADJ
cana-1961	16	9	approaches	approach	NOUN
cana-1961	16	10	,	,	PUNCT
cana-1961	16	11	including	include	VERB
cana-1961	16	12	direct	direct	ADJ
cana-1961	16	13	methods	method	NOUN
cana-1961	16	14	,	,	PUNCT
cana-1961	16	15	fixed	fix	VERB
cana-1961	16	16	-	-	PUNCT
cana-1961	16	17	point	point	NOUN
cana-1961	16	18	techniques	technique	NOUN
cana-1961	16	19	,	,	PUNCT
cana-1961	16	20	and	and	CCONJ
cana-1961	16	21	others	other	NOUN
cana-1961	16	22	,	,	PUNCT
cana-1961	16	23	have	have	AUX
cana-1961	16	24	been	be	AUX
cana-1961	16	25	developed	develop	VERB
cana-1961	16	26	and	and	CCONJ
cana-1961	16	27	applied	apply	VERB
cana-1961	16	28	to	to	PART
cana-1961	16	29	solve	solve	VERB
cana-1961	16	30	different	different	ADJ
cana-1961	16	31	types	type	NOUN
cana-1961	16	32	of	of	ADP
cana-1961	16	33	fes	fes	NOUN
cana-1961	16	34	[	[	X
cana-1961	16	35	7	7	NUM
cana-1961	16	36	-	-	SYM
cana-1961	16	37	9	9	NUM
cana-1961	16	38	]	]	PUNCT
cana-1961	16	39	.	.	PUNCT
cana-1961	17	1	typically	typically	ADV
cana-1961	17	2	,	,	PUNCT
cana-1961	17	3	when	when	SCONJ
cana-1961	17	4	employing	employ	VERB
cana-1961	17	5	the	the	DET
cana-1961	17	6	direct	direct	ADJ
cana-1961	17	7	method	method	NOUN
cana-1961	17	8	to	to	PART
cana-1961	17	9	establish	establish	VERB
cana-1961	17	10	stability	stability	NOUN
cana-1961	17	11	outcomes	outcome	NOUN
cana-1961	17	12	for	for	ADP
cana-1961	17	13	fes	fes	NOUN
cana-1961	17	14	,	,	PUNCT
cana-1961	17	15	one	one	PRON
cana-1961	17	16	must	must	AUX
cana-1961	17	17	satisfy	satisfy	VERB
cana-1961	17	18	either	either	PRON
cana-1961	17	19	of	of	ADP
cana-1961	17	20	the	the	DET
cana-1961	17	21	two	two	NUM
cana-1961	17	22	conditions	condition	NOUN
cana-1961	17	23	:	:	PUNCT
cana-1961	17	24	‖𝔣(ω1	‖𝔣(ω1	NUM
cana-1961	17	25	)	)	PUNCT
cana-1961	17	26	−	−	NOUN
cana-1961	17	27	1	1	NUM
cana-1961	17	28	α	α	NOUN
cana-1961	17	29	𝔣(αω1)‖	𝔣(αω1)‖	NOUN
cana-1961	17	30	≤	≤	NUM
cana-1961	17	31	𝔲(ω1	𝔲(ω1	ADV
cana-1961	17	32	)	)	PUNCT
cana-1961	17	33	or	or	CCONJ
cana-1961	17	34	‖𝔣(ω1	‖𝔣(ω1	NUM
cana-1961	17	35	)	)	PUNCT
cana-1961	17	36	−	−	PROPN
cana-1961	17	37	α𝔣	α𝔣	ADP
cana-1961	17	38	(	(	PUNCT
cana-1961	17	39	ω1	ω1	PROPN
cana-1961	17	40	α	α	PROPN
cana-1961	17	41	)	)	PUNCT
cana-1961	17	42	‖	‖	PROPN
cana-1961	17	43	≤	≤	ADJ
cana-1961	17	44	𝔲	𝔲	PROPN
cana-1961	17	45	(	(	PUNCT
cana-1961	17	46	ω1	ω1	PROPN
cana-1961	17	47	α	α	PROPN
cana-1961	17	48	)	)	PUNCT
cana-1961	17	49	.	.	PUNCT
cana-1961	18	1	the	the	DET
cana-1961	18	2	choice	choice	NOUN
cana-1961	18	3	between	between	ADP
cana-1961	18	4	these	these	DET
cana-1961	18	5	conditions	condition	NOUN
cana-1961	18	6	depends	depend	VERB
cana-1961	18	7	on	on	ADP
cana-1961	18	8	specific	specific	ADJ
cana-1961	18	9	assumptions	assumption	NOUN
cana-1961	18	10	,	,	PUNCT
cana-1961	18	11	necessitating	necessitate	VERB
cana-1961	18	12	distinctions	distinction	NOUN
cana-1961	18	13	to	to	PART
cana-1961	18	14	apply	apply	VERB
cana-1961	18	15	an	an	DET
cana-1961	18	16	appropriate	appropriate	ADJ
cana-1961	18	17	approach	approach	NOUN
cana-1961	18	18	to	to	ADP
cana-1961	18	19	solving	solve	VERB
cana-1961	18	20	distinct	distinct	ADJ
cana-1961	18	21	problems	problem	NOUN
cana-1961	18	22	[	[	X
cana-1961	18	23	10	10	NUM
cana-1961	18	24	-	-	SYM
cana-1961	18	25	12	12	NUM
cana-1961	18	26	]	]	PUNCT
cana-1961	18	27	.	.	PUNCT
cana-1961	19	1	it	it	PRON
cana-1961	19	2	was	be	AUX
cana-1961	19	3	ulam	ulam	NOUN
cana-1961	19	4	[	[	X
cana-1961	19	5	13	13	NUM
cana-1961	19	6	]	]	PUNCT
cana-1961	19	7	who	who	PRON
cana-1961	19	8	first	first	ADV
cana-1961	19	9	raised	raise	VERB
cana-1961	19	10	the	the	DET
cana-1961	19	11	stability	stability	NOUN
cana-1961	19	12	problem	problem	NOUN
cana-1961	19	13	in	in	ADP
cana-1961	19	14	1940	1940	NUM
cana-1961	19	15	,	,	PUNCT
cana-1961	19	16	and	and	CCONJ
cana-1961	19	17	it	it	PRON
cana-1961	19	18	was	be	AUX
cana-1961	19	19	hyers	hyer	NOUN
cana-1961	19	20	[	[	X
cana-1961	19	21	14	14	NUM
cana-1961	19	22	]	]	PUNCT
cana-1961	19	23	who	who	PRON
cana-1961	19	24	responded	respond	VERB
cana-1961	19	25	affirmatively	affirmatively	ADV
cana-1961	19	26	in	in	ADP
cana-1961	19	27	1941	1941	NUM
cana-1961	19	28	for	for	ADP
cana-1961	19	29	a	a	DET
cana-1961	19	30	banach	banach	NOUN
cana-1961	19	31	mailto:ep5583@srmist.edu.in	mailto:ep5583@srmist.edu.in	INTJ
cana-1961	19	32	mailto:sangeets@srmist.edu.in	mailto:sangeets@srmist.edu.in	NOUN
cana-1961	19	33	communications	communication	NOUN
cana-1961	19	34	on	on	ADP
cana-1961	19	35	applied	apply	VERB
cana-1961	19	36	nonlinear	nonlinear	ADJ
cana-1961	19	37	analysis	analysis	NOUN
cana-1961	19	38	issn	issn	NOUN
cana-1961	19	39	:	:	PUNCT
cana-1961	19	40	1074	1074	NUM
cana-1961	19	41	-	-	PUNCT
cana-1961	19	42	133x	133x	NUM
cana-1961	19	43	vol	vol	NOUN
cana-1961	19	44	32	32	NUM
cana-1961	19	45	no	no	NOUN
cana-1961	19	46	.	.	NOUN
cana-1961	19	47	3	3	NUM
cana-1961	19	48	(	(	PUNCT
cana-1961	19	49	2025	2025	NUM
cana-1961	19	50	)	)	PUNCT
cana-1961	19	51	302	302	NUM
cana-1961	19	52	https://internationalpubls.com	https://internationalpubls.com	X
cana-1961	19	53	space	space	NOUN
cana-1961	19	54	.	.	PUNCT
cana-1961	20	1	according	accord	VERB
cana-1961	20	2	to	to	ADP
cana-1961	20	3	rassias	rassias	PROPN
cana-1961	20	4	,	,	PUNCT
cana-1961	20	5	the	the	DET
cana-1961	20	6	bound	bind	VERB
cana-1961	20	7	for	for	ADP
cana-1961	20	8	the	the	DET
cana-1961	20	9	norm	norm	NOUN
cana-1961	20	10	of	of	ADP
cana-1961	20	11	the	the	DET
cana-1961	20	12	cauchy	cauchy	ADJ
cana-1961	20	13	difference	difference	NOUN
cana-1961	20	14	is	be	AUX
cana-1961	20	15	weakened	weaken	VERB
cana-1961	20	16	after	after	ADP
cana-1961	20	17	the	the	DET
cana-1961	20	18	fe	fe	X
cana-1961	20	19	∥	∥	PUNCT
cana-1961	20	20	𝔤(ω1	𝔤(ω1	NOUN
cana-1961	20	21	+	+	CCONJ
cana-1961	20	22	ω2	ω2	ADJ
cana-1961	20	23	)	)	PUNCT
cana-1961	20	24	−	−	NOUN
cana-1961	20	25	𝔤(ω1	𝔤(ω1	NOUN
cana-1961	20	26	)	)	PUNCT
cana-1961	20	27	−	−	ADP
cana-1961	20	28	𝔤(ω2	𝔤(ω2	NOUN
cana-1961	20	29	)	)	PUNCT
cana-1961	20	30	∥	∥	PUNCT
cana-1961	20	31	≤	≤	PUNCT
cana-1961	21	1	ε(∥	ε(∥	ADJ
cana-1961	21	2	ω1	ω1	PROPN
cana-1961	21	3	∥p	∥p	PROPN
cana-1961	21	4	+	+	NOUN
cana-1961	21	5	∥	∥	X
cana-1961	21	6	ω2	ω2	ADJ
cana-1961	21	7	∥p	∥p	NOUN
cana-1961	21	8	)	)	PUNCT
cana-1961	21	9	which	which	PRON
cana-1961	21	10	leads	lead	VERB
cana-1961	21	11	to	to	ADP
cana-1961	21	12	the	the	DET
cana-1961	21	13	generalized	generalized	ADJ
cana-1961	21	14	h	h	NOUN
cana-1961	21	15	-	-	PUNCT
cana-1961	21	16	u	u	NOUN
cana-1961	21	17	stability	stability	NOUN
cana-1961	21	18	theorem	theorem	VERB
cana-1961	21	19	for	for	ADP
cana-1961	21	20	additive	additive	ADJ
cana-1961	21	21	mapping	mapping	NOUN
cana-1961	21	22	.	.	PUNCT
cana-1961	22	1	concerning	concern	VERB
cana-1961	22	2	hyers	hyer	NOUN
cana-1961	22	3	’	'	PUNCT
cana-1961	22	4	theorem	theorem	NOUN
cana-1961	22	5	of	of	ADP
cana-1961	22	6	additive	additive	ADJ
cana-1961	22	7	mappings	mapping	NOUN
cana-1961	22	8	,	,	PUNCT
cana-1961	22	9	papers	paper	NOUN
cana-1961	22	10	were	be	AUX
cana-1961	22	11	published	publish	VERB
cana-1961	22	12	by	by	ADP
cana-1961	22	13	aoki	aoki	PROPN
cana-1961	23	1	[	[	X
cana-1961	23	2	15	15	NUM
cana-1961	23	3	]	]	PUNCT
cana-1961	23	4	and	and	CCONJ
cana-1961	23	5	rassias	rassia	VERB
cana-1961	24	1	[	[	X
cana-1961	24	2	16	16	NUM
cana-1961	24	3	]	]	PUNCT
cana-1961	24	4	.	.	PUNCT
cana-1961	25	1	gavruta	gavruta	PROPN
cana-1961	25	2	presented	present	VERB
cana-1961	25	3	a	a	DET
cana-1961	25	4	generalization	generalization	NOUN
cana-1961	25	5	of	of	ADP
cana-1961	25	6	the	the	DET
cana-1961	25	7	rassias	rassias	PROPN
cana-1961	25	8	theorem	theorem	VERB
cana-1961	25	9	in	in	ADP
cana-1961	25	10	1994	1994	NUM
cana-1961	25	11	[	[	X
cana-1961	25	12	17	17	NUM
cana-1961	25	13	]	]	PUNCT
cana-1961	25	14	.	.	PUNCT
cana-1961	26	1	the	the	DET
cana-1961	26	2	fe	fe	NOUN
cana-1961	26	3	𝔤(ω1	𝔤(ω1	NOUN
cana-1961	26	4	+	+	CCONJ
cana-1961	26	5	ω2	ω2	ADJ
cana-1961	26	6	)	)	PUNCT
cana-1961	26	7	=	=	SYM
cana-1961	26	8	𝔤(ω1	𝔤(ω1	NOUN
cana-1961	26	9	)	)	PUNCT
cana-1961	26	10	+	+	NUM
cana-1961	26	11	𝔤(ω2	𝔤(ω2	NOUN
cana-1961	26	12	)	)	PUNCT
cana-1961	26	13	1	1	NUM
cana-1961	26	14	is	be	AUX
cana-1961	26	15	referred	refer	VERB
cana-1961	26	16	to	to	ADP
cana-1961	26	17	as	as	ADP
cana-1961	26	18	an	an	DET
cana-1961	26	19	additive	additive	ADJ
cana-1961	26	20	fe	fe	NOUN
cana-1961	26	21	.	.	PUNCT
cana-1961	27	1	specifically	specifically	ADV
cana-1961	27	2	,	,	PUNCT
cana-1961	27	3	every	every	DET
cana-1961	27	4	solution	solution	NOUN
cana-1961	27	5	of	of	ADP
cana-1961	27	6	the	the	DET
cana-1961	27	7	additive	additive	ADJ
cana-1961	27	8	fe	fe	NOUN
cana-1961	27	9	is	be	AUX
cana-1961	27	10	termed	term	VERB
cana-1961	27	11	an	an	DET
cana-1961	27	12	additive	additive	ADJ
cana-1961	27	13	mapping	mapping	NOUN
cana-1961	27	14	.	.	PUNCT
cana-1961	28	1	in	in	ADP
cana-1961	28	2	the	the	DET
cana-1961	28	3	work	work	NOUN
cana-1961	28	4	by	by	ADP
cana-1961	28	5	gahler	gahler	NOUN
cana-1961	28	6	[	[	X
cana-1961	28	7	18	18	NUM
cana-1961	28	8	]	]	PUNCT
cana-1961	28	9	,	,	PUNCT
cana-1961	28	10	the	the	DET
cana-1961	28	11	theory	theory	NOUN
cana-1961	28	12	of	of	ADP
cana-1961	28	13	2	2	NUM
cana-1961	28	14	-	-	PUNCT
cana-1961	28	15	norms	norm	NOUN
cana-1961	28	16	and	and	CCONJ
cana-1961	28	17	n	n	CCONJ
cana-1961	28	18	-	-	PUNCT
cana-1961	28	19	norms	norm	NOUN
cana-1961	28	20	within	within	ADP
cana-1961	28	21	a	a	DET
cana-1961	28	22	linear	linear	ADJ
cana-1961	28	23	space	space	NOUN
cana-1961	28	24	was	be	AUX
cana-1961	28	25	introduced	introduce	VERB
cana-1961	28	26	.	.	PUNCT
cana-1961	29	1	subsequently	subsequently	ADV
cana-1961	29	2	,	,	PUNCT
cana-1961	29	3	gahler	gahler	NOUN
cana-1961	29	4	and	and	CCONJ
cana-1961	29	5	white	white	PROPN
cana-1961	29	6	proposed	propose	VERB
cana-1961	29	7	the	the	DET
cana-1961	29	8	idea	idea	NOUN
cana-1961	29	9	of	of	ADP
cana-1961	29	10	2	2	NUM
cana-1961	29	11	-	-	PUNCT
cana-1961	29	12	banach	banach	NOUN
cana-1961	29	13	spaces	space	NOUN
cana-1961	29	14	[	[	X
cana-1961	29	15	19	19	NUM
cana-1961	29	16	]	]	PUNCT
cana-1961	29	17	.	.	PUNCT
cana-1961	30	1	kim	kim	PROPN
cana-1961	30	2	and	and	CCONJ
cana-1961	30	3	park	park	NOUN
cana-1961	30	4	[	[	X
cana-1961	30	5	20	20	NUM
cana-1961	30	6	]	]	PUNCT
cana-1961	30	7	investigated	investigate	VERB
cana-1961	30	8	the	the	DET
cana-1961	30	9	generalized	generalized	ADJ
cana-1961	30	10	h	h	NOUN
cana-1961	30	11	-	-	PUNCT
cana-1961	30	12	u	u	NOUN
cana-1961	30	13	stability	stability	NOUN
cana-1961	30	14	of	of	ADP
cana-1961	30	15	additive	additive	ADJ
cana-1961	30	16	fes	fe	NOUN
cana-1961	30	17	in	in	ADP
cana-1961	30	18	na	na	PART
cana-1961	30	19	2	2	NUM
cana-1961	30	20	-	-	PUNCT
cana-1961	30	21	normed	norme	VERB
cana-1961	30	22	space	space	NOUN
cana-1961	30	23	in	in	ADP
cana-1961	30	24	2014	2014	NUM
cana-1961	30	25	.	.	PUNCT
cana-1961	31	1	in	in	ADP
cana-1961	31	2	2020	2020	NUM
cana-1961	31	3	,	,	PUNCT
cana-1961	31	4	wand	wand	NOUN
cana-1961	31	5	,	,	PUNCT
cana-1961	31	6	park	park	NOUN
cana-1961	31	7	,	,	PUNCT
cana-1961	31	8	and	and	CCONJ
cana-1961	31	9	shin	shin	NOUN
cana-1961	31	10	[	[	X
cana-1961	31	11	21	21	NUM
cana-1961	31	12	]	]	PUNCT
cana-1961	31	13	examined	examine	VERB
cana-1961	31	14	the	the	DET
cana-1961	31	15	h	h	NOUN
cana-1961	31	16	-	-	PUNCT
cana-1961	31	17	u	u	NOUN
cana-1961	31	18	stability	stability	NOUN
cana-1961	31	19	of	of	ADP
cana-1961	31	20	additive	additive	ADJ
cana-1961	31	21	ρ	ρ	ADJ
cana-1961	31	22	-	-	ADJ
cana-1961	31	23	functional	functional	ADJ
cana-1961	31	24	equations	equation	NOUN
cana-1961	31	25	in	in	ADP
cana-1961	31	26	na	na	PRON
cana-1961	31	27	2normed	2normed	NUM
cana-1961	31	28	space	space	NOUN
cana-1961	31	29	.	.	PUNCT
cana-1961	32	1	recently	recently	ADV
cana-1961	32	2	,	,	PUNCT
cana-1961	32	3	ghali	ghali	NOUN
cana-1961	32	4	and	and	CCONJ
cana-1961	32	5	kabbaj	kabbaj	NOUN
cana-1961	33	1	[	[	X
cana-1961	33	2	22	22	NUM
cana-1961	33	3	]	]	PUNCT
cana-1961	33	4	investigated	investigate	VERB
cana-1961	33	5	the	the	DET
cana-1961	33	6	hyperstability	hyperstability	NOUN
cana-1961	33	7	of	of	ADP
cana-1961	33	8	the	the	DET
cana-1961	33	9	cauchy	cauchy	PROPN
cana-1961	33	10	-	-	PUNCT
cana-1961	33	11	jensen	jensen	PROPN
cana-1961	33	12	fe	fe	PROPN
cana-1961	33	13	in	in	ADP
cana-1961	33	14	na	na	DET
cana-1961	33	15	2	2	NUM
cana-1961	33	16	-	-	PUNCT
cana-1961	33	17	banach	banach	NOUN
cana-1961	33	18	spaces	space	NOUN
cana-1961	33	19	and	and	CCONJ
cana-1961	33	20	some	some	PRON
cana-1961	33	21	of	of	ADP
cana-1961	33	22	its	its	PRON
cana-1961	33	23	applications	application	NOUN
cana-1961	33	24	.	.	PUNCT
cana-1961	34	1	in	in	ADP
cana-1961	34	2	2020	2020	NUM
cana-1961	34	3	,	,	PUNCT
cana-1961	34	4	cho	cho	PROPN
cana-1961	34	5	,	,	PUNCT
cana-1961	34	6	gordji	gordji	NOUN
cana-1961	34	7	,	,	PUNCT
cana-1961	34	8	and	and	CCONJ
cana-1961	34	9	zolfaghari	zolfaghari	PROPN
cana-1961	34	10	[	[	X
cana-1961	34	11	23	23	NUM
cana-1961	34	12	]	]	PUNCT
cana-1961	34	13	investigated	investigate	VERB
cana-1961	34	14	the	the	DET
cana-1961	34	15	solution	solution	NOUN
cana-1961	34	16	and	and	CCONJ
cana-1961	34	17	stability	stability	NOUN
cana-1961	34	18	of	of	ADP
cana-1961	34	19	a	a	DET
cana-1961	34	20	generalized	generalize	VERB
cana-1961	34	21	mixed	mixed	ADJ
cana-1961	34	22	-	-	PUNCT
cana-1961	34	23	type	type	NOUN
cana-1961	34	24	quadratic	quadratic	ADJ
cana-1961	34	25	-	-	PUNCT
cana-1961	34	26	cubic(𝑄2	cubic(𝑄2	ADJ
cana-1961	34	27	́	́	PROPN
cana-1961	34	28	−	−	PROPN
cana-1961	34	29	𝒞3	𝒞3	NOUN
cana-1961	34	30	́	́	PUNCT
cana-1961	34	31	)	)	PUNCT
cana-1961	34	32	functional	functional	ADJ
cana-1961	34	33	equation	equation	NOUN
cana-1961	34	34	in	in	ADP
cana-1961	34	35	random	random	ADJ
cana-1961	34	36	normed	normed	ADJ
cana-1961	34	37	spaces	space	NOUN
cana-1961	34	38	.	.	PUNCT
cana-1961	35	1	in	in	ADP
cana-1961	35	2	2022	2022	NUM
cana-1961	35	3	,	,	PUNCT
cana-1961	35	4	mohiuddine	mohiuddine	NOUN
cana-1961	35	5	,	,	PUNCT
cana-1961	35	6	tamilvannan	tamilvannan	PROPN
cana-1961	35	7	,	,	PUNCT
cana-1961	35	8	mursaleen	mursaleen	NOUN
cana-1961	35	9	,	,	PUNCT
cana-1961	35	10	and	and	CCONJ
cana-1961	35	11	alotaibi	alotaibi	NOUN
cana-1961	35	12	[	[	X
cana-1961	35	13	24	24	NUM
cana-1961	35	14	]	]	PUNCT
cana-1961	35	15	examined	examine	VERB
cana-1961	35	16	the	the	DET
cana-1961	35	17	stability	stability	NOUN
cana-1961	35	18	of	of	ADP
cana-1961	35	19	a	a	DET
cana-1961	35	20	quartic	quartic	ADJ
cana-1961	35	21	functional	functional	ADJ
cana-1961	35	22	equation	equation	NOUN
cana-1961	35	23	in	in	ADP
cana-1961	35	24	modular	modular	ADJ
cana-1961	35	25	spaces	space	NOUN
cana-1961	35	26	using	use	VERB
cana-1961	35	27	hyers	hyer	NOUN
cana-1961	35	28	and	and	CCONJ
cana-1961	35	29	fixed	fix	VERB
cana-1961	35	30	-	-	PUNCT
cana-1961	35	31	point	point	NOUN
cana-1961	35	32	methods	method	NOUN
cana-1961	35	33	.	.	PUNCT
cana-1961	36	1	for	for	ADP
cana-1961	36	2	example	example	NOUN
cana-1961	36	3	,	,	PUNCT
cana-1961	36	4	hensel	hensel	PROPN
cana-1961	36	5	[	[	X
cana-1961	36	6	25	25	NUM
cana-1961	36	7	]	]	PUNCT
cana-1961	36	8	discovered	discover	VERB
cana-1961	36	9	the	the	DET
cana-1961	36	10	p	p	ADJ
cana-1961	36	11	-	-	PUNCT
cana-1961	36	12	adic	adic	ADJ
cana-1961	36	13	numbers	number	NOUN
cana-1961	36	14	in	in	ADP
cana-1961	36	15	1897	1897	NUM
cana-1961	36	16	as	as	ADP
cana-1961	36	17	a	a	DET
cana-1961	36	18	number	number	NOUN
cana-1961	36	19	theory	theory	NOUN
cana-1961	36	20	equivalent	equivalent	ADJ
cana-1961	36	21	to	to	ADP
cana-1961	36	22	power	power	NOUN
cana-1961	36	23	series	series	NOUN
cana-1961	36	24	in	in	ADP
cana-1961	36	25	complex	complex	ADJ
cana-1961	36	26	analysis	analysis	NOUN
cana-1961	36	27	.	.	PUNCT
cana-1961	37	1	he	he	PRON
cana-1961	37	2	created	create	VERB
cana-1961	37	3	a	a	DET
cana-1961	37	4	field	field	NOUN
cana-1961	37	5	with	with	ADP
cana-1961	37	6	a	a	DET
cana-1961	37	7	valuation	valuation	NOUN
cana-1961	37	8	standard	standard	NOUN
cana-1961	37	9	that	that	PRON
cana-1961	37	10	lacks	lack	VERB
cana-1961	37	11	the	the	DET
cana-1961	37	12	archimedean	archimedean	ADJ
cana-1961	37	13	property	property	NOUN
cana-1961	37	14	.	.	PUNCT
cana-1961	38	1	numbers	number	NOUN
cana-1961	38	2	with	with	ADP
cana-1961	38	3	p	p	NOUN
cana-1961	38	4	-	-	PUNCT
cana-1961	38	5	adic	adic	ADJ
cana-1961	38	6	numbers	number	NOUN
cana-1961	38	7	are	be	AUX
cana-1961	38	8	the	the	DET
cana-1961	38	9	best	good	ADJ
cana-1961	38	10	examples	example	NOUN
cana-1961	38	11	of	of	ADP
cana-1961	38	12	na	na	ADP
cana-1961	38	13	spaces	space	NOUN
cana-1961	38	14	.	.	PUNCT
cana-1961	39	1	those	those	PRON
cana-1961	39	2	who	who	PRON
cana-1961	39	3	work	work	VERB
cana-1961	39	4	with	with	ADP
cana-1961	39	5	p	p	NOUN
cana-1961	39	6	-	-	PUNCT
cana-1961	39	7	adic	adic	ADJ
cana-1961	39	8	numbers	number	NOUN
cana-1961	39	9	understand	understand	VERB
cana-1961	39	10	that	that	SCONJ
cana-1961	39	11	they	they	PRON
cana-1961	39	12	do	do	AUX
cana-1961	39	13	not	not	PART
cana-1961	39	14	adhere	adhere	VERB
cana-1961	39	15	to	to	ADP
cana-1961	39	16	archimedean	archimedean	ADJ
cana-1961	39	17	properties	property	NOUN
cana-1961	39	18	,	,	PUNCT
cana-1961	39	19	which	which	PRON
cana-1961	39	20	state	state	VERB
cana-1961	39	21	that	that	SCONJ
cana-1961	39	22	for	for	ADP
cana-1961	39	23	any	any	DET
cana-1961	39	24	positive	positive	ADJ
cana-1961	39	25	number	number	NOUN
cana-1961	39	26	n	n	CCONJ
cana-1961	39	27	,	,	PUNCT
cana-1961	39	28	there	there	PRON
cana-1961	39	29	exists	exist	VERB
cana-1961	39	30	an	an	DET
cana-1961	39	31	integer	integer	NOUN
cana-1961	39	32	x	x	PUNCT
cana-1961	39	33	such	such	ADJ
cana-1961	39	34	that	that	DET
cana-1961	39	35	𝑛𝑥	𝑛𝑥	PROPN
cana-1961	39	36	>	>	X
cana-1961	39	37	𝑦	𝑦	NOUN
cana-1961	39	38	for	for	ADP
cana-1961	39	39	another	another	DET
cana-1961	39	40	positive	positive	ADJ
cana-1961	39	41	integer	integer	NOUN
cana-1961	39	42	y.	y.	NOUN
cana-1961	39	43	in	in	ADP
cana-1961	39	44	the	the	DET
cana-1961	39	45	past	past	ADJ
cana-1961	39	46	30	30	NUM
cana-1961	39	47	years	year	NOUN
cana-1961	39	48	,	,	PUNCT
cana-1961	39	49	physicists	physicist	NOUN
cana-1961	39	50	have	have	AUX
cana-1961	39	51	shown	show	VERB
cana-1961	39	52	increased	increased	ADJ
cana-1961	39	53	interest	interest	NOUN
cana-1961	39	54	in	in	ADP
cana-1961	39	55	the	the	DET
cana-1961	39	56	theory	theory	NOUN
cana-1961	39	57	of	of	ADP
cana-1961	39	58	na	na	ADP
cana-1961	39	59	spaces	space	NOUN
cana-1961	39	60	,	,	PUNCT
cana-1961	39	61	especially	especially	ADV
cana-1961	39	62	with	with	ADP
cana-1961	39	63	problems	problem	NOUN
cana-1961	39	64	in	in	ADP
cana-1961	39	65	quantum	quantum	ADJ
cana-1961	39	66	physics	physics	NOUN
cana-1961	39	67	,	,	PUNCT
cana-1961	39	68	p	p	ADJ
cana-1961	39	69	-	-	PUNCT
cana-1961	39	70	adic	adic	ADJ
cana-1961	39	71	physics	physics	NOUN
cana-1961	39	72	,	,	PUNCT
cana-1961	39	73	and	and	CCONJ
cana-1961	39	74	superstring	superstring	NOUN
cana-1961	39	75	theory	theory	NOUN
cana-1961	39	76	.	.	PUNCT
cana-1961	40	1	the	the	DET
cana-1961	40	2	na	na	PROPN
cana-1961	40	3	version	version	NOUN
cana-1961	40	4	of	of	ADP
cana-1961	40	5	several	several	ADJ
cana-1961	40	6	results	result	NOUN
cana-1961	40	7	in	in	ADP
cana-1961	40	8	the	the	DET
cana-1961	40	9	usual	usual	ADJ
cana-1961	40	10	normed	norme	VERB
cana-1961	40	11	spaces	space	NOUN
cana-1961	40	12	theory	theory	NOUN
cana-1961	40	13	differs	differ	VERB
cana-1961	40	14	significantly	significantly	ADV
cana-1961	40	15	in	in	ADP
cana-1961	40	16	their	their	PRON
cana-1961	40	17	proofs	proof	NOUN
cana-1961	40	18	,	,	PUNCT
cana-1961	40	19	requiring	require	VERB
cana-1961	40	20	a	a	DET
cana-1961	40	21	new	new	ADJ
cana-1961	40	22	level	level	NOUN
cana-1961	40	23	of	of	ADP
cana-1961	40	24	understanding	understanding	NOUN
cana-1961	40	25	.	.	PUNCT
cana-1961	41	1	notably	notably	ADV
cana-1961	41	2	,	,	PUNCT
cana-1961	41	3	in	in	ADP
cana-1961	41	4	every	every	DET
cana-1961	41	5	valuation	valuation	NOUN
cana-1961	41	6	field	field	NOUN
cana-1961	41	7	where	where	SCONJ
cana-1961	41	8	|n|	|n|	VERB
cana-1961	41	9	≤	≤	NUM
cana-1961	41	10	1	1	NUM
cana-1961	41	11	,	,	PUNCT
cana-1961	41	12	every	every	DET
cana-1961	41	13	triangle	triangle	NOUN
cana-1961	41	14	is	be	AUX
cana-1961	41	15	isosceles	isoscele	NOUN
cana-1961	41	16	,	,	PUNCT
cana-1961	41	17	and	and	CCONJ
cana-1961	41	18	there	there	PRON
cana-1961	41	19	may	may	AUX
cana-1961	41	20	not	not	PART
cana-1961	41	21	be	be	AUX
cana-1961	41	22	a	a	DET
cana-1961	41	23	unit	unit	NOUN
cana-1961	41	24	vector	vector	NOUN
cana-1961	41	25	in	in	ADP
cana-1961	41	26	a	a	DET
cana-1961	41	27	non	non	ADJ
cana-1961	41	28	-	-	ADJ
cana-1961	41	29	archimedean	archimedean	ADJ
cana-1961	41	30	space	space	NOUN
cana-1961	41	31	.	.	PUNCT
cana-1961	42	1	these	these	DET
cana-1961	42	2	details	detail	NOUN
cana-1961	42	3	highlight	highlight	VERB
cana-1961	42	4	the	the	DET
cana-1961	42	5	remarkable	remarkable	ADJ
cana-1961	42	6	structure	structure	NOUN
cana-1961	42	7	of	of	ADP
cana-1961	42	8	na	na	ADP
cana-1961	42	9	spaces	space	NOUN
cana-1961	42	10	.	.	PUNCT
cana-1961	43	1	in	in	ADP
cana-1961	43	2	this	this	DET
cana-1961	43	3	present	present	ADJ
cana-1961	43	4	article	article	NOUN
cana-1961	43	5	,	,	PUNCT
cana-1961	43	6	we	we	PRON
cana-1961	43	7	will	will	AUX
cana-1961	43	8	use	use	VERB
cana-1961	43	9	the	the	DET
cana-1961	43	10	alternative	alternative	ADJ
cana-1961	43	11	fixed	fix	VERB
cana-1961	43	12	point	point	NOUN
cana-1961	43	13	method	method	NOUN
cana-1961	43	14	to	to	PART
cana-1961	43	15	investigate	investigate	VERB
cana-1961	43	16	the	the	DET
cana-1961	43	17	h	h	NOUN
cana-1961	43	18	-	-	PUNCT
cana-1961	43	19	u	u	NOUN
cana-1961	43	20	stability	stability	NOUN
cana-1961	43	21	of	of	ADP
cana-1961	43	22	the	the	DET
cana-1961	43	23	general	general	ADJ
cana-1961	43	24	quadratic	quadratic	ADJ
cana-1961	43	25	-	-	PUNCT
cana-1961	43	26	cubic	cubic	ADJ
cana-1961	43	27	fe	fe	NOUN
cana-1961	43	28	over	over	ADP
cana-1961	43	29	na	na	PART
cana-1961	43	30	2	2	NUM
cana-1961	43	31	-	-	PUNCT
cana-1961	43	32	normed	norme	VERB
cana-1961	43	33	spaces	space	NOUN
cana-1961	43	34	.	.	PUNCT
cana-1961	44	1	2	2	X
cana-1961	44	2	.	.	X
cana-1961	44	3	preliminaries	preliminary	NOUN
cana-1961	44	4	some	some	DET
cana-1961	44	5	basic	basic	ADJ
cana-1961	44	6	definitions	definition	NOUN
cana-1961	44	7	and	and	CCONJ
cana-1961	44	8	theorems	theorem	NOUN
cana-1961	44	9	are	be	AUX
cana-1961	44	10	presented	present	VERB
cana-1961	44	11	in	in	ADP
cana-1961	44	12	this	this	DET
cana-1961	44	13	section	section	NOUN
cana-1961	44	14	as	as	ADP
cana-1961	44	15	a	a	DET
cana-1961	44	16	reminder	reminder	NOUN
cana-1961	44	17	.	.	PUNCT
cana-1961	45	1	definition	definition	NOUN
cana-1961	45	2	1	1	NUM
cana-1961	45	3	:	:	PUNCT
cana-1961	46	1	[	[	X
cana-1961	46	2	21	21	NUM
cana-1961	46	3	]	]	X
cana-1961	46	4	let	let	VERB
cana-1961	46	5	𝒲	𝒲	NOUN
cana-1961	46	6	be	be	AUX
cana-1961	46	7	a	a	DET
cana-1961	46	8	vector	vector	NOUN
cana-1961	46	9	space	space	NOUN
cana-1961	46	10	over	over	ADP
cana-1961	46	11	a	a	DET
cana-1961	46	12	scalar	scalar	ADJ
cana-1961	46	13	field	field	NOUN
cana-1961	46	14	𝕂	𝕂	NOUN
cana-1961	46	15	with	with	ADP
cana-1961	46	16	an	an	DET
cana-1961	46	17	na	na	PRON
cana-1961	46	18	non	non	ADJ
cana-1961	46	19	-	-	ADJ
cana-1961	46	20	trivial	trivial	ADJ
cana-1961	46	21	valuation	valuation	NOUN
cana-1961	46	22	|	|	NOUN
cana-1961	46	23	.	.	PUNCT
cana-1961	47	1	|	|	ADV
cana-1961	47	2	.	.	PUNCT
cana-1961	48	1	a	a	DET
cana-1961	48	2	function	function	NOUN
cana-1961	48	3	∥.	∥.	X
cana-1961	48	4	∥	∥	PUNCT
cana-1961	48	5	from	from	ADP
cana-1961	48	6	𝒲	𝒲	NOUN
cana-1961	48	7	to	to	ADP
cana-1961	48	8	ℝ	ℝ	PROPN
cana-1961	48	9	is	be	AUX
cana-1961	48	10	called	call	VERB
cana-1961	48	11	an	an	DET
cana-1961	48	12	na	na	NOUN
cana-1961	48	13	norm	norm	NOUN
cana-1961	48	14	(	(	PUNCT
cana-1961	48	15	valuation	valuation	NOUN
cana-1961	48	16	)	)	PUNCT
cana-1961	48	17	if	if	SCONJ
cana-1961	48	18	it	it	PRON
cana-1961	48	19	satisfies	satisfy	VERB
cana-1961	48	20	the	the	DET
cana-1961	48	21	following	follow	VERB
cana-1961	48	22	conditions	condition	NOUN
cana-1961	48	23	:	:	PUNCT
cana-1961	48	24	(	(	PUNCT
cana-1961	48	25	1	1	X
cana-1961	48	26	)	)	PUNCT
cana-1961	48	27	∥	∥	NOUN
cana-1961	48	28	ω1	ω1	PROPN
cana-1961	48	29	∥=	∥=	NOUN
cana-1961	48	30	0	0	PUNCT
cana-1961	49	1	if	if	SCONJ
cana-1961	49	2	and	and	CCONJ
cana-1961	49	3	only	only	ADV
cana-1961	49	4	if	if	SCONJ
cana-1961	49	5	ω1	ω1	PROPN
cana-1961	49	6	=	=	PROPN
cana-1961	49	7	0	0	NUM
cana-1961	49	8	;	;	PUNCT
cana-1961	49	9	(	(	PUNCT
cana-1961	49	10	2	2	X
cana-1961	49	11	)	)	PUNCT
cana-1961	49	12	∥	∥	NOUN
cana-1961	49	13	rω1	rω1	NOUN
cana-1961	49	14	∥=	∥=	ADJ
cana-1961	49	15	|r|	|r|	NOUN
cana-1961	49	16	∥	∥	PUNCT
cana-1961	49	17	ω1	ω1	PROPN
cana-1961	49	18	∥	∥	PUNCT
cana-1961	49	19	for	for	ADP
cana-1961	49	20	all	all	DET
cana-1961	49	21	r	r	NOUN
cana-1961	49	22	∈	∈	NOUN
cana-1961	49	23	𝕂	𝕂	PROPN
cana-1961	49	24	,	,	PUNCT
cana-1961	49	25	ω1	ω1	PROPN
cana-1961	49	26	∈	∈	PROPN
cana-1961	49	27	𝒲	𝒲	PROPN
cana-1961	49	28	;	;	PUNCT
cana-1961	49	29	(	(	PUNCT
cana-1961	49	30	3	3	X
cana-1961	49	31	)	)	PUNCT
cana-1961	49	32	∥	∥	NUM
cana-1961	49	33	ω1	ω1	PROPN
cana-1961	49	34	,	,	PUNCT
cana-1961	49	35	ω2	ω2	PROPN
cana-1961	49	36	∥≤	∥≤	PROPN
cana-1961	49	37	max{∥	max{∥	PROPN
cana-1961	49	38	ω1	ω1	PROPN
cana-1961	49	39	∥	∥	PROPN
cana-1961	49	40	,	,	PUNCT
cana-1961	49	41	∥	∥	X
cana-1961	49	42	ω2	ω2	CCONJ
cana-1961	49	43	∥	∥	NUM
cana-1961	49	44	}	}	PUNCT
cana-1961	49	45	(	(	PUNCT
cana-1961	49	46	satisfying	satisfy	VERB
cana-1961	49	47	the	the	DET
cana-1961	49	48	strong	strong	ADJ
cana-1961	49	49	triangle	triangle	NOUN
cana-1961	49	50	inequality	inequality	NOUN
cana-1961	49	51	or	or	CCONJ
cana-1961	49	52	ultra	ultra	ADJ
cana-1961	49	53	-	-	ADJ
cana-1961	49	54	metric	metric	ADJ
cana-1961	49	55	property	property	NOUN
cana-1961	49	56	)	)	PUNCT
cana-1961	49	57	for	for	ADP
cana-1961	49	58	all	all	DET
cana-1961	49	59	ω1	ω1	PROPN
cana-1961	49	60	,	,	PUNCT
cana-1961	49	61	ω2	ω2	NOUN
cana-1961	49	62	∈	∈	PROPN
cana-1961	49	63	𝒲.	𝒲.	PROPN
cana-1961	49	64	then	then	ADV
cana-1961	49	65	(	(	PUNCT
cana-1961	49	66	𝒲	𝒲	NOUN
cana-1961	49	67	,	,	PUNCT
cana-1961	49	68	∥.	∥.	X
cana-1961	49	69	∥	∥	X
cana-1961	49	70	)	)	PUNCT
cana-1961	49	71	is	be	AUX
cana-1961	49	72	called	call	VERB
cana-1961	49	73	an	an	DET
cana-1961	49	74	na	na	NOUN
cana-1961	49	75	-	-	PUNCT
cana-1961	49	76	normed	normed	ADJ
cana-1961	49	77	space	space	NOUN
cana-1961	49	78	.	.	PUNCT
cana-1961	49	79	example	example	NOUN
cana-1961	50	1	1	1	NUM
cana-1961	50	2	:	:	PUNCT
cana-1961	50	3	let	let	VERB
cana-1961	50	4	ρ	ρ	NOUN
cana-1961	50	5	be	be	AUX
cana-1961	50	6	a	a	DET
cana-1961	50	7	fixed	fix	VERB
cana-1961	50	8	prime	prime	ADJ
cana-1961	50	9	number	number	NOUN
cana-1961	50	10	.	.	PUNCT
cana-1961	51	1	for	for	ADP
cana-1961	51	2	any	any	DET
cana-1961	51	3	non	non	ADJ
cana-1961	51	4	-	-	ADJ
cana-1961	51	5	zero	zero	ADJ
cana-1961	51	6	rational	rational	ADJ
cana-1961	51	7	number	number	NOUN
cana-1961	51	8	ω1	ω1	PROPN
cana-1961	51	9	,	,	PUNCT
cana-1961	51	10	there	there	PRON
cana-1961	51	11	is	be	VERB
cana-1961	51	12	a	a	DET
cana-1961	51	13	unique	unique	ADJ
cana-1961	51	14	integer	integer	NOUN
cana-1961	51	15	nω1	nω1	NOUN
cana-1961	51	16	∈	∈	PROPN
cana-1961	51	17	ℤ	ℤ	PROPN
cana-1961	51	18	such	such	DET
cana-1961	51	19	that	that	SCONJ
cana-1961	51	20	communications	communication	NOUN
cana-1961	51	21	on	on	ADP
cana-1961	51	22	applied	apply	VERB
cana-1961	51	23	nonlinear	nonlinear	ADJ
cana-1961	51	24	analysis	analysis	NOUN
cana-1961	51	25	issn	issn	NOUN
cana-1961	51	26	:	:	PUNCT
cana-1961	51	27	1074	1074	NUM
cana-1961	51	28	-	-	PUNCT
cana-1961	51	29	133x	133x	NUM
cana-1961	51	30	vol	vol	NOUN
cana-1961	51	31	32	32	NUM
cana-1961	51	32	no	no	NOUN
cana-1961	51	33	.	.	NOUN
cana-1961	51	34	3	3	NUM
cana-1961	51	35	(	(	PUNCT
cana-1961	51	36	2025	2025	NUM
cana-1961	51	37	)	)	PUNCT
cana-1961	51	38	303	303	NUM
cana-1961	51	39	https://internationalpubls.com	https://internationalpubls.com	X
cana-1961	51	40	ω1	ω1	PROPN
cana-1961	51	41	=	=	PUNCT
cana-1961	51	42	α	α	PROPN
cana-1961	51	43	β	β	PROPN
cana-1961	51	44	ρnω1	ρnω1	PROPN
cana-1961	51	45	,	,	PUNCT
cana-1961	51	46	where	where	SCONJ
cana-1961	51	47	α	α	NOUN
cana-1961	51	48	and	and	CCONJ
cana-1961	51	49	β	β	PROPN
cana-1961	51	50	are	be	AUX
cana-1961	51	51	integers	integer	NOUN
cana-1961	51	52	not	not	PART
cana-1961	51	53	divisible	divisible	ADJ
cana-1961	51	54	by	by	ADP
cana-1961	51	55	ρ	ρ	PROPN
cana-1961	51	56	.	.	PUNCT
cana-1961	52	1	then	then	ADV
cana-1961	52	2	,	,	PUNCT
cana-1961	52	3	the	the	DET
cana-1961	52	4	function	function	NOUN
cana-1961	52	5	|	|	NOUN
cana-1961	52	6	.	.	PUNCT
cana-1961	53	1	|ρ	|ρ	NOUN
cana-1961	53	2	:	:	PUNCT
cana-1961	53	3	ℚ	ℚ	PROPN
cana-1961	53	4	ρ	ρ	PROPN
cana-1961	53	5	→	→	SYM
cana-1961	54	1	[	[	X
cana-1961	54	2	0	0	NUM
cana-1961	54	3	,	,	PUNCT
cana-1961	54	4	+	+	NOUN
cana-1961	54	5	∞	∞	NOUN
cana-1961	54	6	)	)	PUNCT
cana-1961	54	7	defined	define	VERB
cana-1961	54	8	by	by	ADP
cana-1961	54	9	|ω1|	|ω1|	NOUN
cana-1961	54	10	=	=	SYM
cana-1961	54	11	{	{	PUNCT
cana-1961	54	12	0	0	NUM
cana-1961	54	13	,	,	PUNCT
cana-1961	54	14	ω1	ω1	PROPN
cana-1961	54	15	=	=	SYM
cana-1961	54	16	0	0	NUM
cana-1961	54	17	,	,	PUNCT
cana-1961	54	18	ρ−nω1	ρ−nω1	NOUN
cana-1961	54	19	,	,	PUNCT
cana-1961	54	20	ω1	ω1	PROPN
cana-1961	54	21	≠	≠	PROPN
cana-1961	54	22	0	0	NUM
cana-1961	54	23	is	be	AUX
cana-1961	54	24	an	an	DET
cana-1961	54	25	na	na	NOUN
cana-1961	54	26	valuation	valuation	NOUN
cana-1961	54	27	on	on	ADP
cana-1961	54	28	ℚ	ℚ	PROPN
cana-1961	54	29	ρ	ρ	PROPN
cana-1961	54	30	.	.	PUNCT
cana-1961	54	31	example	example	NOUN
cana-1961	55	1	2	2	NUM
cana-1961	55	2	:	:	PUNCT
cana-1961	55	3	let	let	VERB
cana-1961	55	4	ω1	ω1	PROPN
cana-1961	55	5	=	=	PROPN
cana-1961	55	6	70	70	NUM
cana-1961	55	7	13	13	NUM
cana-1961	55	8	.	.	PUNCT
cana-1961	56	1	in	in	ADP
cana-1961	56	2	this	this	DET
cana-1961	56	3	case	case	NOUN
cana-1961	56	4	,	,	PUNCT
cana-1961	56	5	let	let	VERB
cana-1961	56	6	us	we	PRON
cana-1961	56	7	find	find	VERB
cana-1961	56	8	its	its	PRON
cana-1961	56	9	5	5	NUM
cana-1961	56	10	-	-	PUNCT
cana-1961	56	11	adic	adic	ADJ
cana-1961	56	12	absolute	absolute	ADJ
cana-1961	56	13	value	value	NOUN
cana-1961	56	14	(	(	PUNCT
cana-1961	56	15	ρ	ρ	NOUN
cana-1961	56	16	=	=	SYM
cana-1961	56	17	5	5	NUM
cana-1961	56	18	)	)	PUNCT
cana-1961	56	19	as	as	SCONJ
cana-1961	56	20	given	give	VERB
cana-1961	56	21	below	below	ADP
cana-1961	56	22	ω1	ω1	PROPN
cana-1961	56	23	=	=	PROPN
cana-1961	56	24	70	70	NUM
cana-1961	56	25	13	13	NUM
cana-1961	56	26	=	=	SYM
cana-1961	56	27	51	51	NUM
cana-1961	56	28	.	.	PUNCT
cana-1961	56	29	14	14	NUM
cana-1961	56	30	13	13	NUM
cana-1961	56	31	which	which	PRON
cana-1961	56	32	means	mean	VERB
cana-1961	56	33	|ω1|5	|ω1|5	PROPN
cana-1961	56	34	=	=	SYM
cana-1961	56	35	1	1	NUM
cana-1961	56	36	5	5	NUM
cana-1961	56	37	or	or	CCONJ
cana-1961	56	38	5−1	5−1	NUM
cana-1961	56	39	.	.	PUNCT
cana-1961	57	1	it	it	PRON
cana-1961	57	2	will	will	AUX
cana-1961	57	3	be	be	AUX
cana-1961	57	4	simple	simple	ADJ
cana-1961	57	5	to	to	ADP
cana-1961	57	6	|ω1|13	|ω1|13	PROPN
cana-1961	57	7	=	=	SYM
cana-1961	57	8	13	13	NUM
cana-1961	57	9	,	,	PUNCT
cana-1961	57	10	because	because	SCONJ
cana-1961	57	11	ω1	ω1	PROPN
cana-1961	57	12	=	=	SYM
cana-1961	57	13	13	13	NUM
cana-1961	57	14	−1	−1	NOUN
cana-1961	57	15	.	.	PUNCT
cana-1961	58	1	75	75	NUM
cana-1961	58	2	|ω1|13	|ω1|13	NOUN
cana-1961	58	3	=	=	SYM
cana-1961	58	4	1	1	NUM
cana-1961	58	5	13	13	NUM
cana-1961	58	6	−1	−1	NOUN
cana-1961	58	7	=	=	NOUN
cana-1961	58	8	13	13	NUM
cana-1961	58	9	.	.	NOUN
cana-1961	58	10	which	which	PRON
cana-1961	58	11	means	mean	VERB
cana-1961	58	12	|ω1|13	|ω1|13	PROPN
cana-1961	58	13	=	=	SYM
cana-1961	58	14	13	13	NUM
cana-1961	58	15	.	.	PUNCT
cana-1961	59	1	definition	definition	NOUN
cana-1961	59	2	2	2	NUM
cana-1961	59	3	:	:	PUNCT
cana-1961	60	1	[	[	X
cana-1961	60	2	18	18	NUM
cana-1961	60	3	]	]	PUNCT
cana-1961	60	4	let	let	VERB
cana-1961	60	5	𝒲	𝒲	NOUN
cana-1961	60	6	be	be	AUX
cana-1961	60	7	a	a	DET
cana-1961	60	8	vector	vector	NOUN
cana-1961	60	9	space	space	NOUN
cana-1961	60	10	over	over	ADP
cana-1961	60	11	a	a	DET
cana-1961	60	12	scalar	scalar	ADJ
cana-1961	60	13	field	field	NOUN
cana-1961	60	14	𝕂	𝕂	NOUN
cana-1961	60	15	with	with	ADP
cana-1961	60	16	an	an	DET
cana-1961	60	17	na	na	PRON
cana-1961	60	18	non	non	ADJ
cana-1961	60	19	-	-	ADJ
cana-1961	60	20	trivial	trivial	ADJ
cana-1961	60	21	valuation	valuation	NOUN
cana-1961	60	22	|	|	NOUN
cana-1961	60	23	.	.	PUNCT
cana-1961	61	1	|	|	ADV
cana-1961	61	2	with	with	ADP
cana-1961	61	3	dim	dim	ADJ
cana-1961	61	4	𝒲	𝒲	NOUN
cana-1961	61	5	>	>	X
cana-1961	61	6	1	1	NUM
cana-1961	61	7	.	.	PUNCT
cana-1961	62	1	a	a	DET
cana-1961	62	2	function	function	NOUN
cana-1961	62	3	∥.	∥.	X
cana-1961	62	4	,	,	PUNCT
cana-1961	62	5	.	.	PUNCT
cana-1961	63	1	∥	∥	X
cana-1961	63	2	from	from	ADP
cana-1961	63	3	𝒲	𝒲	NOUN
cana-1961	63	4	to	to	ADP
cana-1961	63	5	ℝ	ℝ	PROPN
cana-1961	63	6	is	be	AUX
cana-1961	63	7	called	call	VERB
cana-1961	63	8	an	an	DET
cana-1961	63	9	na	na	NOUN
cana-1961	63	10	2	2	NUM
cana-1961	63	11	-	-	PUNCT
cana-1961	63	12	norm	norm	NOUN
cana-1961	63	13	(	(	PUNCT
cana-1961	63	14	valuation	valuation	NOUN
cana-1961	63	15	)	)	PUNCT
cana-1961	63	16	if	if	SCONJ
cana-1961	63	17	it	it	PRON
cana-1961	63	18	satisfies	satisfy	VERB
cana-1961	63	19	the	the	DET
cana-1961	63	20	following	follow	VERB
cana-1961	63	21	conditions	condition	NOUN
cana-1961	63	22	:	:	PUNCT
cana-1961	63	23	(	(	PUNCT
cana-1961	63	24	1	1	X
cana-1961	63	25	)	)	PUNCT
cana-1961	63	26	∥	∥	NUM
cana-1961	63	27	ω1	ω1	PROPN
cana-1961	63	28	,	,	PUNCT
cana-1961	63	29	ω2	ω2	ADJ
cana-1961	63	30	∥=	∥=	NOUN
cana-1961	63	31	0	0	PUNCT
cana-1961	64	1	if	if	SCONJ
cana-1961	64	2	and	and	CCONJ
cana-1961	64	3	only	only	ADV
cana-1961	64	4	if	if	SCONJ
cana-1961	64	5	ω1	ω1	PROPN
cana-1961	64	6	,	,	PUNCT
cana-1961	64	7	ω2	ω2	NUM
cana-1961	64	8	are	be	AUX
cana-1961	64	9	linearly	linearly	ADV
cana-1961	64	10	dependent	dependent	ADJ
cana-1961	64	11	;	;	PUNCT
cana-1961	64	12	(	(	PUNCT
cana-1961	64	13	2	2	X
cana-1961	64	14	)	)	PUNCT
cana-1961	64	15	∥	∥	NUM
cana-1961	64	16	ω1	ω1	PROPN
cana-1961	64	17	,	,	PUNCT
cana-1961	64	18	ω2	ω2	ADJ
cana-1961	64	19	∥=∥	∥=∥	X
cana-1961	64	20	ω2	ω2	ADJ
cana-1961	64	21	,	,	PUNCT
cana-1961	64	22	ω1	ω1	PROPN
cana-1961	64	23	∥	∥	NUM
cana-1961	64	24	;	;	PUNCT
cana-1961	64	25	(	(	PUNCT
cana-1961	64	26	3	3	X
cana-1961	64	27	)	)	PUNCT
cana-1961	64	28	∥	∥	NUM
cana-1961	64	29	r	r	NOUN
cana-1961	64	30	ω1	ω1	PROPN
cana-1961	64	31	,	,	PUNCT
cana-1961	64	32	ω2	ω2	ADJ
cana-1961	64	33	∥=	∥=	ADJ
cana-1961	64	34	|r|	|r|	PROPN
cana-1961	64	35	∥	∥	NOUN
cana-1961	64	36	ω1	ω1	PROPN
cana-1961	64	37	,	,	PUNCT
cana-1961	64	38	ω2	ω2	ADV
cana-1961	64	39	∥	∥	NUM
cana-1961	64	40	for	for	ADP
cana-1961	64	41	all	all	DET
cana-1961	64	42	r	r	NOUN
cana-1961	64	43	∈	∈	PROPN
cana-1961	64	44	𝕂	𝕂	PROPN
cana-1961	64	45	,	,	PUNCT
cana-1961	64	46	ω1	ω1	PROPN
cana-1961	64	47	,	,	PUNCT
cana-1961	64	48	ω2	ω2	NOUN
cana-1961	64	49	∈	∈	PROPN
cana-1961	64	50	𝒲	𝒲	PROPN
cana-1961	64	51	;	;	PUNCT
cana-1961	64	52	(	(	PUNCT
cana-1961	64	53	4	4	X
cana-1961	64	54	)	)	PUNCT
cana-1961	64	55	∥	∥	NUM
cana-1961	64	56	ω1	ω1	PROPN
cana-1961	64	57	,	,	PUNCT
cana-1961	64	58	ω2	ω2	NOUN
cana-1961	64	59	+	+	CCONJ
cana-1961	64	60	υ	υ	PROPN
cana-1961	64	61	∥≤	∥≤	PROPN
cana-1961	64	62	max{∥	max{∥	PROPN
cana-1961	64	63	ω1	ω1	PROPN
cana-1961	64	64	,	,	PUNCT
cana-1961	64	65	ω2	ω2	ADJ
cana-1961	64	66	∥	∥	PROPN
cana-1961	64	67	,	,	PUNCT
cana-1961	64	68	∥	∥	PROPN
cana-1961	64	69	ω1	ω1	PROPN
cana-1961	64	70	,	,	PUNCT
cana-1961	64	71	υ	υ	NOUN
cana-1961	64	72	∥	∥	X
cana-1961	64	73	}	}	PUNCT
cana-1961	64	74	for	for	ADP
cana-1961	64	75	all	all	DET
cana-1961	64	76	ω1	ω1	PROPN
cana-1961	64	77	,	,	PUNCT
cana-1961	64	78	ω2	ω2	ADJ
cana-1961	64	79	,	,	PUNCT
cana-1961	64	80	υ	υ	PROPN
cana-1961	64	81	∈	∈	PROPN
cana-1961	64	82	𝒲	𝒲	PROPN
cana-1961	64	83	then	then	ADV
cana-1961	64	84	(	(	PUNCT
cana-1961	64	85	𝒲	𝒲	NOUN
cana-1961	64	86	,	,	PUNCT
cana-1961	64	87	∥.	∥.	X
cana-1961	64	88	,	,	PUNCT
cana-1961	64	89	.	.	PUNCT
cana-1961	65	1	∥	∥	X
cana-1961	65	2	)	)	PUNCT
cana-1961	66	1	is	be	AUX
cana-1961	66	2	called	call	VERB
cana-1961	66	3	an	an	DET
cana-1961	66	4	na	na	NOUN
cana-1961	66	5	2	2	NUM
cana-1961	66	6	-	-	PUNCT
cana-1961	66	7	normed	norme	VERB
cana-1961	66	8	space	space	NOUN
cana-1961	66	9	.	.	PUNCT
cana-1961	67	1	the	the	DET
cana-1961	67	2	following	follow	VERB
cana-1961	67	3	lemma	lemma	PROPN
cana-1961	67	4	follows	follow	VERB
cana-1961	67	5	from	from	ADP
cana-1961	67	6	definition	definition	NOUN
cana-1961	67	7	2	2	NUM
cana-1961	67	8	.	.	PUNCT
cana-1961	68	1	lemma	lemma	PROPN
cana-1961	68	2	1	1	NUM
cana-1961	68	3	:	:	PUNCT
cana-1961	68	4	[	[	X
cana-1961	68	5	18	18	NUM
cana-1961	68	6	]	]	X
cana-1961	68	7	let	let	NOUN
cana-1961	68	8	(	(	PUNCT
cana-1961	68	9	𝒲	𝒲	NOUN
cana-1961	68	10	,	,	PUNCT
cana-1961	68	11	∥.	∥.	X
cana-1961	68	12	,	,	PUNCT
cana-1961	68	13	.	.	PUNCT
cana-1961	69	1	∥	∥	X
cana-1961	69	2	)	)	PUNCT
cana-1961	69	3	be	be	VERB
cana-1961	69	4	an	an	DET
cana-1961	69	5	na	na	SYM
cana-1961	69	6	2	2	NUM
cana-1961	69	7	-	-	PUNCT
cana-1961	69	8	normed	norme	VERB
cana-1961	69	9	space	space	NOUN
cana-1961	69	10	.	.	PUNCT
cana-1961	70	1	if	if	SCONJ
cana-1961	70	2	ω1	ω1	PROPN
cana-1961	70	3	∈	∈	PROPN
cana-1961	70	4	𝒲	𝒲	PROPN
cana-1961	70	5	and	and	CCONJ
cana-1961	70	6	∥	∥	NUM
cana-1961	70	7	ω1	ω1	PROPN
cana-1961	70	8	,	,	PUNCT
cana-1961	70	9	ω2	ω2	ADJ
cana-1961	70	10	∥=	∥=	NOUN
cana-1961	70	11	0	0	PUNCT
cana-1961	70	12	for	for	ADP
cana-1961	70	13	all	all	DET
cana-1961	70	14	ω2	ω2	ADJ
cana-1961	70	15	∈	∈	PROPN
cana-1961	70	16	𝒲	𝒲	PROPN
cana-1961	70	17	,	,	PUNCT
cana-1961	70	18	then	then	ADV
cana-1961	70	19	ω1	ω1	PROPN
cana-1961	70	20	=	=	SYM
cana-1961	70	21	0	0	PROPN
cana-1961	70	22	.	.	PUNCT
cana-1961	71	1	definition	definition	NOUN
cana-1961	71	2	3	3	NUM
cana-1961	71	3	:	:	PUNCT
cana-1961	72	1	[	[	X
cana-1961	72	2	18	18	NUM
cana-1961	72	3	]	]	PUNCT
cana-1961	72	4	a	a	DET
cana-1961	72	5	sequence	sequence	NOUN
cana-1961	72	6	{	{	PUNCT
cana-1961	72	7	ω1𝔫	ω1𝔫	PROPN
cana-1961	72	8	}	}	PUNCT
cana-1961	72	9	in	in	ADP
cana-1961	72	10	an	an	DET
cana-1961	72	11	na	na	NOUN
cana-1961	72	12	2	2	NUM
cana-1961	72	13	-	-	PUNCT
cana-1961	72	14	normed	norme	VERB
cana-1961	72	15	space	space	NOUN
cana-1961	72	16	(	(	PUNCT
cana-1961	72	17	𝒲	𝒲	NOUN
cana-1961	72	18	,	,	PUNCT
cana-1961	72	19	∥.	∥.	X
cana-1961	72	20	,	,	PUNCT
cana-1961	72	21	.	.	PUNCT
cana-1961	72	22	∥	∥	X
cana-1961	72	23	)	)	PUNCT
cana-1961	72	24	is	be	AUX
cana-1961	72	25	called	call	VERB
cana-1961	72	26	a	a	DET
cana-1961	72	27	cauchy	cauchy	ADJ
cana-1961	72	28	sequence	sequence	NOUN
cana-1961	72	29	if	if	SCONJ
cana-1961	72	30	there	there	PRON
cana-1961	72	31	are	be	VERB
cana-1961	72	32	two	two	NUM
cana-1961	72	33	linearly	linearly	ADV
cana-1961	72	34	independent	independent	ADJ
cana-1961	72	35	points	point	NOUN
cana-1961	72	36	ω1	ω1	PROPN
cana-1961	72	37	,	,	PUNCT
cana-1961	72	38	ω2	ω2	NOUN
cana-1961	72	39	∈	∈	PROPN
cana-1961	72	40	𝒲	𝒲	NOUN
cana-1961	72	41	such	such	ADJ
cana-1961	72	42	that	that	SCONJ
cana-1961	72	43	lim	lim	PROPN
cana-1961	72	44	𝔪,𝔫	𝔪,𝔫	NOUN
cana-1961	72	45	∥	∥	X
cana-1961	72	46	ω1𝔫	ω1𝔫	ADP
cana-1961	72	47	−	−	X
cana-1961	72	48	ω1𝔪	ω1𝔪	X
cana-1961	72	49	,	,	PUNCT
cana-1961	72	50	ω2	ω2	ADJ
cana-1961	72	51	∥=	∥=	NOUN
cana-1961	72	52	0	0	PUNCT
cana-1961	73	1	and	and	CCONJ
cana-1961	73	2	lim	lim	PROPN
cana-1961	73	3	𝔪,𝔫	𝔪,𝔫	VERB
cana-1961	73	4	∥	∥	X
cana-1961	73	5	ω1𝔫	ω1𝔫	ADP
cana-1961	73	6	−	−	X
cana-1961	73	7	ω1𝔪	ω1𝔪	X
cana-1961	73	8	,	,	PUNCT
cana-1961	73	9	ω2	ω2	ADJ
cana-1961	73	10	∥=	∥=	NOUN
cana-1961	73	11	0	0	NUM
cana-1961	73	12	.	.	PUNCT
cana-1961	74	1	definition	definition	NOUN
cana-1961	74	2	4	4	NUM
cana-1961	74	3	:	:	PUNCT
cana-1961	75	1	[	[	X
cana-1961	75	2	18	18	NUM
cana-1961	75	3	]	]	PUNCT
cana-1961	75	4	a	a	DET
cana-1961	75	5	sequence	sequence	NOUN
cana-1961	75	6	{	{	PUNCT
cana-1961	75	7	ω1𝔫	ω1𝔫	PROPN
cana-1961	75	8	}	}	PUNCT
cana-1961	75	9	in	in	ADP
cana-1961	75	10	an	an	DET
cana-1961	75	11	na	na	NOUN
cana-1961	75	12	2	2	NUM
cana-1961	75	13	-	-	PUNCT
cana-1961	75	14	normed	norme	VERB
cana-1961	75	15	space	space	NOUN
cana-1961	75	16	(	(	PUNCT
cana-1961	75	17	𝒲	𝒲	NOUN
cana-1961	75	18	,	,	PUNCT
cana-1961	75	19	∥.	∥.	X
cana-1961	75	20	,	,	PUNCT
cana-1961	75	21	.	.	PUNCT
cana-1961	75	22	∥	∥	X
cana-1961	75	23	)	)	PUNCT
cana-1961	75	24	is	be	AUX
cana-1961	75	25	called	call	VERB
cana-1961	75	26	a	a	DET
cana-1961	75	27	convergent	convergent	NOUN
cana-1961	75	28	sequence	sequence	NOUN
cana-1961	75	29	if	if	SCONJ
cana-1961	75	30	there	there	PRON
cana-1961	75	31	exists	exist	VERB
cana-1961	75	32	an	an	DET
cana-1961	75	33	ω1	ω1	PROPN
cana-1961	75	34	∈	∈	PROPN
cana-1961	75	35	𝒲	𝒲	NOUN
cana-1961	75	36	such	such	ADJ
cana-1961	75	37	that	that	SCONJ
cana-1961	75	38	lim	lim	PROPN
cana-1961	75	39	𝔫	𝔫	PROPN
cana-1961	75	40	∥	∥	X
cana-1961	75	41	ω1𝔫	ω1𝔫	ADP
cana-1961	75	42	−	−	PROPN
cana-1961	75	43	ω1	ω1	PROPN
cana-1961	75	44	,	,	PUNCT
cana-1961	75	45	ω2	ω2	ADJ
cana-1961	75	46	∥=	∥=	NOUN
cana-1961	75	47	0	0	PUNCT
cana-1961	75	48	for	for	ADP
cana-1961	75	49	all	all	DET
cana-1961	75	50	ω1	ω1	PROPN
cana-1961	75	51	,	,	PUNCT
cana-1961	75	52	ω2	ω2	NOUN
cana-1961	75	53	∈	∈	PROPN
cana-1961	75	54	𝒲.	𝒲.	PROPN
cana-1961	75	55	in	in	ADP
cana-1961	75	56	this	this	DET
cana-1961	75	57	case	case	NOUN
cana-1961	75	58	,	,	PUNCT
cana-1961	75	59	recall	recall	VERB
cana-1961	75	60	that	that	SCONJ
cana-1961	75	61	{	{	PUNCT
cana-1961	75	62	ω1𝔫	ω1𝔫	ADP
cana-1961	75	63	}	}	PUNCT
cana-1961	75	64	converges	converge	NOUN
cana-1961	75	65	to	to	ADP
cana-1961	75	66	ω1	ω1	PROPN
cana-1961	75	67	or	or	CCONJ
cana-1961	75	68	that	that	SCONJ
cana-1961	75	69	ω1	ω1	PROPN
cana-1961	75	70	is	be	AUX
cana-1961	75	71	the	the	DET
cana-1961	75	72	limit	limit	NOUN
cana-1961	75	73	of	of	ADP
cana-1961	75	74	{	{	PUNCT
cana-1961	75	75	ω1𝔫	ω1𝔫	PROPN
cana-1961	75	76	}	}	PUNCT
cana-1961	75	77	,	,	PUNCT
cana-1961	75	78	write	write	VERB
cana-1961	75	79	{	{	PUNCT
cana-1961	75	80	ω1𝔫	ω1𝔫	PROPN
cana-1961	75	81	}	}	PUNCT
cana-1961	75	82	→	→	SYM
cana-1961	75	83	ω1	ω1	PROPN
cana-1961	75	84	as	as	ADP
cana-1961	75	85	𝔫	𝔫	PROPN
cana-1961	75	86	→	→	SYM
cana-1961	75	87	∞	∞	PROPN
cana-1961	75	88	or	or	CCONJ
cana-1961	75	89	lim	lim	PROPN
cana-1961	75	90	𝔫→∞	𝔫→∞	NUM
cana-1961	75	91	ω1𝔫	ω1𝔫	PROPN
cana-1961	75	92	=	=	SYM
cana-1961	75	93	ω1	ω1	PROPN
cana-1961	75	94	.	.	PUNCT
cana-1961	75	95	communications	communication	NOUN
cana-1961	75	96	on	on	ADP
cana-1961	75	97	applied	apply	VERB
cana-1961	75	98	nonlinear	nonlinear	ADJ
cana-1961	75	99	analysis	analysis	NOUN
cana-1961	75	100	issn	issn	NOUN
cana-1961	75	101	:	:	PUNCT
cana-1961	75	102	1074	1074	NUM
cana-1961	75	103	-	-	PUNCT
cana-1961	75	104	133x	133x	NUM
cana-1961	75	105	vol	vol	NOUN
cana-1961	75	106	32	32	NUM
cana-1961	75	107	no	no	NOUN
cana-1961	75	108	.	.	NOUN
cana-1961	75	109	3	3	NUM
cana-1961	75	110	(	(	PUNCT
cana-1961	75	111	2025	2025	NUM
cana-1961	75	112	)	)	PUNCT
cana-1961	75	113	304	304	NUM
cana-1961	75	114	https://internationalpubls.com	https://internationalpubls.com	X
cana-1961	75	115	by	by	ADP
cana-1961	75	116	definition	definition	NOUN
cana-1961	75	117	2	2	NUM
cana-1961	75	118	(	(	PUNCT
cana-1961	75	119	4	4	NUM
cana-1961	75	120	)	)	PUNCT
cana-1961	75	121	,	,	PUNCT
cana-1961	75	122	we	we	PRON
cana-1961	75	123	have	have	VERB
cana-1961	75	124	∥	∥	NUM
cana-1961	75	125	ω1𝔫	ω1𝔫	ADP
cana-1961	75	126	−	−	X
cana-1961	75	127	ω1𝔪	ω1𝔪	X
cana-1961	75	128	,	,	PUNCT
cana-1961	75	129	ω2	ω2	PROPN
cana-1961	75	130	∥≤	∥≤	PROPN
cana-1961	75	131	max{∥	max{∥	PROPN
cana-1961	75	132	ω1ȷ+1	ω1ȷ+1	PROPN
cana-1961	75	133	−	−	PROPN
cana-1961	75	134	ω1ȷ	ω1ȷ	PROPN
cana-1961	75	135	,	,	PUNCT
cana-1961	75	136	ω2	ω2	ADV
cana-1961	75	137	∥	∥	NUM
cana-1961	75	138	:	:	PUNCT
cana-1961	75	139	𝔪	𝔪	X
cana-1961	75	140	≤	≤	NUM
cana-1961	75	141	ȷ	ȷ	ADP
cana-1961	75	142	≤	≤	X
cana-1961	75	143	𝔫	𝔫	NOUN
cana-1961	75	144	−	−	PROPN
cana-1961	75	145	1	1	NUM
cana-1961	75	146	}	}	PUNCT
cana-1961	75	147	,	,	PUNCT
cana-1961	75	148	(	(	PUNCT
cana-1961	75	149	𝔫	𝔫	X
cana-1961	75	150	>	>	X
cana-1961	75	151	𝔪	𝔪	NOUN
cana-1961	75	152	)	)	PUNCT
cana-1961	75	153	,	,	PUNCT
cana-1961	75	154	for	for	ADP
cana-1961	75	155	all	all	DET
cana-1961	75	156	𝜔1	𝜔1	ADJ
cana-1961	75	157	,	,	PUNCT
cana-1961	75	158	ω2	ω2	NOUN
cana-1961	75	159	∈	∈	PROPN
cana-1961	75	160	𝒲.	𝒲.	PROPN
cana-1961	75	161	hence	hence	ADV
cana-1961	75	162	,	,	PUNCT
cana-1961	75	163	a	a	DET
cana-1961	75	164	sequence	sequence	NOUN
cana-1961	75	165	{	{	PUNCT
cana-1961	75	166	ω1𝔫	ω1𝔫	PROPN
cana-1961	75	167	}	}	PUNCT
cana-1961	75	168	in	in	ADP
cana-1961	75	169	cauchy	cauchy	NOUN
cana-1961	75	170	in	in	ADP
cana-1961	75	171	(	(	PUNCT
cana-1961	75	172	𝒲	𝒲	NOUN
cana-1961	75	173	,	,	PUNCT
cana-1961	75	174	∥.	∥.	X
cana-1961	75	175	,	,	PUNCT
cana-1961	75	176	.	.	PUNCT
cana-1961	76	1	∥	∥	X
cana-1961	76	2	)	)	PUNCT
cana-1961	77	1	if	if	SCONJ
cana-1961	77	2	and	and	CCONJ
cana-1961	77	3	only	only	ADV
cana-1961	77	4	if	if	SCONJ
cana-1961	77	5	{	{	PUNCT
cana-1961	77	6	ω1𝔫+1	ω1𝔫+1	ADJ
cana-1961	77	7	−	−	NOUN
cana-1961	77	8	ω1𝔫	ω1𝔫	ADP
cana-1961	77	9	}	}	PUNCT
cana-1961	77	10	converges	converge	VERB
cana-1961	77	11	to	to	ADP
cana-1961	77	12	0	0	NUM
cana-1961	77	13	in	in	ADP
cana-1961	77	14	an	an	DET
cana-1961	77	15	na	na	NOUN
cana-1961	77	16	2	2	NUM
cana-1961	77	17	-	-	PUNCT
cana-1961	77	18	normed	norme	VERB
cana-1961	77	19	space	space	NOUN
cana-1961	77	20	(	(	PUNCT
cana-1961	77	21	𝒲	𝒲	NOUN
cana-1961	77	22	,	,	PUNCT
cana-1961	77	23	∥.	∥.	X
cana-1961	77	24	,	,	PUNCT
cana-1961	77	25	.	.	PUNCT
cana-1961	78	1	∥	∥	NUM
cana-1961	78	2	)	)	PUNCT
cana-1961	78	3	.	.	PUNCT
cana-1961	79	1	remark	remark	VERB
cana-1961	79	2	1	1	NUM
cana-1961	79	3	:	:	PUNCT
cana-1961	80	1	[	[	X
cana-1961	80	2	18	18	NUM
cana-1961	80	3	]	]	X
cana-1961	80	4	let	let	NOUN
cana-1961	80	5	(	(	PUNCT
cana-1961	80	6	𝒲	𝒲	NOUN
cana-1961	80	7	,	,	PUNCT
cana-1961	80	8	∥.	∥.	X
cana-1961	80	9	,	,	PUNCT
cana-1961	80	10	.	.	PUNCT
cana-1961	80	11	∥	∥	X
cana-1961	80	12	)	)	PUNCT
cana-1961	80	13	be	be	VERB
cana-1961	80	14	an	an	DET
cana-1961	80	15	na	na	SYM
cana-1961	80	16	2	2	NUM
cana-1961	80	17	-	-	PUNCT
cana-1961	80	18	normed	norme	VERB
cana-1961	80	19	space	space	NOUN
cana-1961	80	20	.	.	PUNCT
cana-1961	81	1	one	one	PRON
cana-1961	81	2	can	can	AUX
cana-1961	81	3	show	show	VERB
cana-1961	81	4	that	that	SCONJ
cana-1961	81	5	conditions	condition	NOUN
cana-1961	81	6	(	(	PUNCT
cana-1961	81	7	2	2	NUM
cana-1961	81	8	)	)	PUNCT
cana-1961	81	9	and	and	CCONJ
cana-1961	81	10	(	(	PUNCT
cana-1961	81	11	4	4	X
cana-1961	81	12	)	)	PUNCT
cana-1961	81	13	in	in	ADP
cana-1961	81	14	definition	definition	NOUN
cana-1961	81	15	2	2	NUM
cana-1961	81	16	imply	imply	VERB
cana-1961	81	17	that	that	SCONJ
cana-1961	81	18	∥	∥	PROPN
cana-1961	81	19	ω1	ω1	PROPN
cana-1961	81	20	+	+	CCONJ
cana-1961	81	21	ω2	ω2	ADJ
cana-1961	81	22	,	,	PUNCT
cana-1961	81	23	υ	υ	PROPN
cana-1961	81	24	∥≤∥	∥≤∥	PROPN
cana-1961	81	25	ω1	ω1	PROPN
cana-1961	81	26	,	,	PUNCT
cana-1961	81	27	υ	υ	NOUN
cana-1961	81	28	∥	∥	PUNCT
cana-1961	81	29	+	+	NOUN
cana-1961	81	30	∥	∥	NOUN
cana-1961	81	31	ω2	ω2	ADJ
cana-1961	81	32	,	,	PUNCT
cana-1961	81	33	υ	υ	NOUN
cana-1961	81	34	∥	∥	NOUN
cana-1961	81	35	and	and	CCONJ
cana-1961	81	36	|	|	ADV
cana-1961	81	37	∥	∥	PUNCT
cana-1961	81	38	ω1	ω1	PROPN
cana-1961	81	39	−	−	PROPN
cana-1961	81	40	υ	υ	PROPN
cana-1961	81	41	∥	∥	PUNCT
cana-1961	81	42	−∥	−∥	X
cana-1961	81	43	ω2	ω2	ADJ
cana-1961	81	44	,	,	PUNCT
cana-1961	81	45	υ	υ	NOUN
cana-1961	81	46	∥	∥	PUNCT
cana-1961	81	47	|	|	ADV
cana-1961	81	48	≤∥	≤∥	VERB
cana-1961	81	49	ω1	ω1	PROPN
cana-1961	81	50	−	−	PROPN
cana-1961	81	51	ω2	ω2	PROPN
cana-1961	81	52	,	,	PUNCT
cana-1961	81	53	υ	υ	NOUN
cana-1961	81	54	∥	∥	NOUN
cana-1961	81	55	for	for	ADP
cana-1961	81	56	all	all	DET
cana-1961	81	57	ω1	ω1	PROPN
cana-1961	81	58	,	,	PUNCT
cana-1961	81	59	ω2	ω2	ADJ
cana-1961	81	60	,	,	PUNCT
cana-1961	81	61	υ	υ	PROPN
cana-1961	81	62	∈	∈	PROPN
cana-1961	81	63	𝒲.	𝒲.	PROPN
cana-1961	81	64	it	it	PRON
cana-1961	81	65	is	be	AUX
cana-1961	81	66	effortless	effortless	ADJ
cana-1961	81	67	to	to	PART
cana-1961	81	68	get	get	VERB
cana-1961	81	69	the	the	DET
cana-1961	81	70	following	follow	VERB
cana-1961	81	71	lemma	lemma	PROPN
cana-1961	81	72	by	by	ADP
cana-1961	81	73	using	use	VERB
cana-1961	81	74	remark	remark	NOUN
cana-1961	81	75	1	1	NUM
cana-1961	81	76	.	.	PUNCT
cana-1961	82	1	lemma	lemma	PROPN
cana-1961	82	2	2	2	NUM
cana-1961	82	3	:	:	PUNCT
cana-1961	82	4	[	[	X
cana-1961	82	5	18	18	NUM
cana-1961	82	6	]	]	PUNCT
cana-1961	82	7	for	for	ADP
cana-1961	82	8	a	a	DET
cana-1961	82	9	convergent	convergent	NOUN
cana-1961	82	10	sequence	sequence	NOUN
cana-1961	82	11	{	{	PUNCT
cana-1961	82	12	ω1𝔫	ω1𝔫	PROPN
cana-1961	82	13	}	}	PUNCT
cana-1961	82	14	in	in	ADP
cana-1961	82	15	an	an	DET
cana-1961	82	16	na	na	NOUN
cana-1961	82	17	2	2	NUM
cana-1961	82	18	-	-	PUNCT
cana-1961	82	19	normed	norme	VERB
cana-1961	82	20	space	space	NOUN
cana-1961	82	21	(	(	PUNCT
cana-1961	82	22	𝒲	𝒲	NOUN
cana-1961	82	23	,	,	PUNCT
cana-1961	82	24	∥.	∥.	X
cana-1961	82	25	,	,	PUNCT
cana-1961	82	26	.	.	PUNCT
cana-1961	83	1	∥	∥	NUM
cana-1961	83	2	)	)	PUNCT
cana-1961	84	1	,	,	PUNCT
cana-1961	84	2	lim	lim	NOUN
cana-1961	84	3	𝔫→∞	𝔫→∞	NUM
cana-1961	84	4	∥	∥	X
cana-1961	84	5	ω1𝔫	ω1𝔫	PUNCT
cana-1961	84	6	,	,	PUNCT
cana-1961	84	7	ω2	ω2	ADJ
cana-1961	84	8	∥=∥	∥=∥	CCONJ
cana-1961	84	9	lim	lim	PROPN
cana-1961	84	10	𝔫→∞	𝔫→∞	NUM
cana-1961	84	11	ω1𝔫	ω1𝔫	PROPN
cana-1961	84	12	,	,	PUNCT
cana-1961	84	13	ω2	ω2	ADV
cana-1961	84	14	∥	∥	NUM
cana-1961	84	15	for	for	ADP
cana-1961	84	16	all	all	DET
cana-1961	84	17	ω2	ω2	NOUN
cana-1961	84	18	∈	∈	PROPN
cana-1961	84	19	𝒲.	𝒲.	PROPN
cana-1961	84	20	definition	definition	NOUN
cana-1961	84	21	5	5	NUM
cana-1961	84	22	:	:	PUNCT
cana-1961	84	23	[	[	X
cana-1961	84	24	18	18	NUM
cana-1961	84	25	]	]	PUNCT
cana-1961	84	26	if	if	SCONJ
cana-1961	84	27	every	every	DET
cana-1961	84	28	cauchy	cauchy	ADJ
cana-1961	84	29	sequence	sequence	NOUN
cana-1961	84	30	in	in	ADP
cana-1961	84	31	𝒲	𝒲	NOUN
cana-1961	84	32	converges	converge	VERB
cana-1961	84	33	,	,	PUNCT
cana-1961	84	34	then	then	ADV
cana-1961	84	35	the	the	DET
cana-1961	84	36	na	na	PROPN
cana-1961	84	37	2	2	NUM
cana-1961	84	38	-	-	PUNCT
cana-1961	84	39	normed	norme	VERB
cana-1961	84	40	space	space	NOUN
cana-1961	84	41	𝒲	𝒲	PROPN
cana-1961	84	42	is	be	AUX
cana-1961	84	43	called	call	VERB
cana-1961	84	44	an	an	DET
cana-1961	84	45	na	na	NOUN
cana-1961	84	46	2	2	NUM
cana-1961	84	47	-	-	PUNCT
cana-1961	84	48	banach	banach	NOUN
cana-1961	84	49	space	space	NOUN
cana-1961	84	50	or	or	CCONJ
cana-1961	84	51	an	an	DET
cana-1961	84	52	ultrametric	ultrametric	ADJ
cana-1961	84	53	2	2	NUM
cana-1961	84	54	-	-	PUNCT
cana-1961	84	55	banach	banach	NOUN
cana-1961	84	56	space	space	NOUN
cana-1961	84	57	.	.	PUNCT
cana-1961	85	1	definition	definition	NOUN
cana-1961	85	2	6	6	NUM
cana-1961	85	3	:	:	PUNCT
cana-1961	86	1	[	[	X
cana-1961	86	2	26	26	NUM
cana-1961	86	3	]	]	X
cana-1961	86	4	let	let	VERB
cana-1961	86	5	𝒲	𝒲	NOUN
cana-1961	86	6	be	be	AUX
cana-1961	86	7	a	a	DET
cana-1961	86	8	set	set	NOUN
cana-1961	86	9	.	.	PUNCT
cana-1961	87	1	a	a	DET
cana-1961	87	2	function	function	NOUN
cana-1961	87	3	ð	ð	NUM
cana-1961	87	4	:	:	PUNCT
cana-1961	87	5	𝒲	𝒲	NOUN
cana-1961	87	6	×	×	NOUN
cana-1961	87	7	𝒲	𝒲	NOUN
cana-1961	87	8	→	→	SYM
cana-1961	87	9	[	[	X
cana-1961	87	10	0	0	NUM
cana-1961	87	11	,	,	PUNCT
cana-1961	87	12	∞	∞	PROPN
cana-1961	87	13	]	]	PUNCT
cana-1961	87	14	is	be	AUX
cana-1961	87	15	called	call	VERB
cana-1961	87	16	a	a	DET
cana-1961	87	17	generalized	generalize	VERB
cana-1961	87	18	metric(gm	metric(gm	NOUN
cana-1961	87	19	)	)	PUNCT
cana-1961	87	20	on	on	ADP
cana-1961	87	21	𝒲	𝒲	NOUN
cana-1961	87	22	if	if	SCONJ
cana-1961	87	23	ð	ð	PROPN
cana-1961	87	24	satisfies	satisfy	VERB
cana-1961	87	25	the	the	DET
cana-1961	87	26	following	follow	VERB
cana-1961	87	27	conditions	condition	NOUN
cana-1961	87	28	:	:	PUNCT
cana-1961	87	29	(	(	PUNCT
cana-1961	87	30	1	1	X
cana-1961	87	31	)	)	PUNCT
cana-1961	87	32	ð(ω1	ð(ω1	NOUN
cana-1961	87	33	,	,	PUNCT
cana-1961	87	34	ω2	ω2	ADJ
cana-1961	87	35	)	)	PUNCT
cana-1961	87	36	=	=	SYM
cana-1961	87	37	0	0	PUNCT
cana-1961	88	1	if	if	SCONJ
cana-1961	88	2	and	and	CCONJ
cana-1961	88	3	only	only	ADV
cana-1961	88	4	if	if	SCONJ
cana-1961	88	5	ω1	ω1	PROPN
cana-1961	88	6	=	=	SYM
cana-1961	88	7	ω2	ω2	PROPN
cana-1961	88	8	;	;	PUNCT
cana-1961	88	9	(	(	PUNCT
cana-1961	88	10	2	2	X
cana-1961	88	11	)	)	PUNCT
cana-1961	88	12	ð(ω1	ð(ω1	NOUN
cana-1961	88	13	,	,	PUNCT
cana-1961	88	14	ω2	ω2	NUM
cana-1961	88	15	)	)	PUNCT
cana-1961	88	16	=	=	SYM
cana-1961	88	17	ð(ω2	ð(ω2	X
cana-1961	88	18	,	,	PUNCT
cana-1961	88	19	ω1	ω1	PROPN
cana-1961	88	20	)	)	PUNCT
cana-1961	88	21	for	for	ADP
cana-1961	88	22	all	all	DET
cana-1961	88	23	ω1	ω1	PROPN
cana-1961	88	24	,	,	PUNCT
cana-1961	88	25	ω2	ω2	NOUN
cana-1961	88	26	∈	∈	PROPN
cana-1961	88	27	𝒲	𝒲	PROPN
cana-1961	88	28	;	;	PUNCT
cana-1961	88	29	(	(	PUNCT
cana-1961	88	30	3	3	X
cana-1961	88	31	)	)	PUNCT
cana-1961	88	32	ð(ω1	ð(ω1	NOUN
cana-1961	88	33	,	,	PUNCT
cana-1961	88	34	υ	υ	NOUN
cana-1961	88	35	)	)	PUNCT
cana-1961	88	36	≤	≤	NOUN
cana-1961	88	37	ð(ω1	ð(ω1	NOUN
cana-1961	88	38	,	,	PUNCT
cana-1961	88	39	ω2	ω2	ADJ
cana-1961	88	40	)	)	PUNCT
cana-1961	88	41	+	+	CCONJ
cana-1961	88	42	ð(ω2	ð(ω2	X
cana-1961	88	43	,	,	PUNCT
cana-1961	88	44	υ	υ	NOUN
cana-1961	88	45	)	)	PUNCT
cana-1961	88	46	for	for	ADP
cana-1961	88	47	all	all	DET
cana-1961	88	48	ω1	ω1	PROPN
cana-1961	88	49	,	,	PUNCT
cana-1961	88	50	ω2	ω2	ADJ
cana-1961	88	51	,	,	PUNCT
cana-1961	88	52	υ	υ	PROPN
cana-1961	88	53	∈	∈	PROPN
cana-1961	88	54	𝒲.	𝒲.	PROPN
cana-1961	88	55	theorem	theorem	VERB
cana-1961	88	56	1	1	NUM
cana-1961	88	57	:	:	PUNCT
cana-1961	88	58	[	[	X
cana-1961	88	59	26	26	NUM
cana-1961	88	60	]	]	X
cana-1961	88	61	let	let	VERB
cana-1961	88	62	(	(	PUNCT
cana-1961	88	63	𝒲	𝒲	PROPN
cana-1961	88	64	,	,	PUNCT
cana-1961	88	65	ð	ð	NUM
cana-1961	88	66	)	)	PUNCT
cana-1961	88	67	be	be	AUX
cana-1961	88	68	complete	complete	ADJ
cana-1961	88	69	and	and	CCONJ
cana-1961	88	70	λ	λ	NOUN
cana-1961	88	71	:	:	PUNCT
cana-1961	88	72	𝒲	𝒲	NOUN
cana-1961	88	73	→	→	SYM
cana-1961	88	74	𝒲	𝒲	NOUN
cana-1961	88	75	be	be	VERB
cana-1961	88	76	strictly	strictly	ADV
cana-1961	88	77	contractive	contractive	ADJ
cana-1961	88	78	mapping	mapping	NOUN
cana-1961	88	79	with	with	ADP
cana-1961	88	80	0	0	NUM
cana-1961	88	81	<	<	X
cana-1961	88	82	£	£	SYM
cana-1961	88	83	<	<	X
cana-1961	88	84	1	1	NUM
cana-1961	88	85	.	.	PUNCT
cana-1961	89	1	then	then	ADV
cana-1961	89	2	for	for	ADP
cana-1961	89	3	each	each	DET
cana-1961	89	4	given	give	VERB
cana-1961	89	5	element	element	NOUN
cana-1961	89	6	ω1	ω1	PROPN
cana-1961	89	7	∈	∈	PROPN
cana-1961	89	8	𝒲	𝒲	PROPN
cana-1961	89	9	,	,	PUNCT
cana-1961	89	10	either	either	CCONJ
cana-1961	89	11	ð(λ	ð(λ	PROPN
cana-1961	89	12	n	n	PRON
cana-1961	89	13	ω1	ω1	PROPN
cana-1961	89	14	,	,	PUNCT
cana-1961	89	15	λn+1	λn+1	X
cana-1961	89	16	ω1	ω1	PROPN
cana-1961	89	17	)	)	PUNCT
cana-1961	89	18	=	=	SYM
cana-1961	89	19	∞	∞	PROPN
cana-1961	89	20	for	for	ADP
cana-1961	89	21	all	all	DET
cana-1961	89	22	n	n	DET
cana-1961	89	23	≥	≥	NOUN
cana-1961	89	24	0	0	NUM
cana-1961	89	25	or	or	CCONJ
cana-1961	89	26	there	there	PRON
cana-1961	89	27	exists	exist	VERB
cana-1961	89	28	a	a	DET
cana-1961	89	29	natural	natural	ADJ
cana-1961	89	30	number	number	NOUN
cana-1961	89	31	n0	n0	NOUN
cana-1961	90	1	such	such	ADJ
cana-1961	90	2	that	that	SCONJ
cana-1961	90	3	(	(	PUNCT
cana-1961	90	4	1	1	X
cana-1961	90	5	)	)	PUNCT
cana-1961	90	6	ð(λ	ð(λ	PROPN
cana-1961	90	7	n	n	SYM
cana-1961	90	8	ω1	ω1	PROPN
cana-1961	90	9	,	,	PUNCT
cana-1961	90	10	λ	λ	PROPN
cana-1961	90	11	n+1	n+1	PROPN
cana-1961	90	12	ω1	ω1	PROPN
cana-1961	90	13	)	)	PUNCT
cana-1961	90	14	<	<	X
cana-1961	90	15	∞	∞	PROPN
cana-1961	90	16	,	,	PUNCT
cana-1961	90	17	for	for	ADP
cana-1961	90	18	all	all	DET
cana-1961	90	19	n	n	PRON
cana-1961	90	20	≥	≥	NOUN
cana-1961	90	21	n0	n0	NUM
cana-1961	90	22	;	;	PUNCT
cana-1961	90	23	(	(	PUNCT
cana-1961	90	24	2	2	X
cana-1961	90	25	)	)	PUNCT
cana-1961	90	26	the	the	DET
cana-1961	90	27	sequence	sequence	NOUN
cana-1961	90	28	{	{	PUNCT
cana-1961	90	29	λ	λ	PROPN
cana-1961	90	30	n	n	CCONJ
cana-1961	90	31	ω1	ω1	PROPN
cana-1961	90	32	}	}	PUNCT
cana-1961	90	33	converges	converge	VERB
cana-1961	90	34	to	to	ADP
cana-1961	90	35	a	a	DET
cana-1961	90	36	fixed	fix	VERB
cana-1961	90	37	point	point	NOUN
cana-1961	90	38	ω2	ω2	ADP
cana-1961	90	39	∗	∗	NOUN
cana-1961	90	40	of	of	ADP
cana-1961	90	41	λ	λ	NOUN
cana-1961	90	42	;	;	PUNCT
cana-1961	90	43	(	(	PUNCT
cana-1961	90	44	3	3	X
cana-1961	90	45	)	)	PUNCT
cana-1961	90	46	ω2	ω2	ADJ
cana-1961	90	47	∗	∗	NOUN
cana-1961	90	48	is	be	AUX
cana-1961	90	49	the	the	DET
cana-1961	90	50	unique	unique	ADJ
cana-1961	90	51	fixed	fix	VERB
cana-1961	90	52	point	point	NOUN
cana-1961	90	53	of	of	ADP
cana-1961	90	54	λ	λ	PROPN
cana-1961	90	55	in	in	ADP
cana-1961	90	56	the	the	DET
cana-1961	90	57	set	set	NOUN
cana-1961	90	58	ψ	ψ	X
cana-1961	90	59	=	=	PUNCT
cana-1961	90	60	{	{	PUNCT
cana-1961	90	61	ω2	ω2	NOUN
cana-1961	90	62	∈	∈	PROPN
cana-1961	90	63	𝒲	𝒲	PROPN
cana-1961	90	64	∶	∶	ADV
cana-1961	90	65	ð(λ	ð(λ	PROPN
cana-1961	90	66	n0ω1	n0ω1	SYM
cana-1961	90	67	,	,	PUNCT
cana-1961	90	68	ω2	ω2	ADJ
cana-1961	90	69	)	)	PUNCT
cana-1961	90	70	<	<	X
cana-1961	90	71	∞	∞	PROPN
cana-1961	90	72	}	}	PUNCT
cana-1961	90	73	;	;	PUNCT
cana-1961	90	74	(	(	PUNCT
cana-1961	90	75	4	4	X
cana-1961	90	76	)	)	PUNCT
cana-1961	90	77	ð(ω2	ð(ω2	NOUN
cana-1961	90	78	,	,	PUNCT
cana-1961	90	79	ω2	ω2	ADJ
cana-1961	90	80	∗	∗	NOUN
cana-1961	90	81	)	)	PUNCT
cana-1961	90	82	≤	≤	NUM
cana-1961	90	83	1	1	NUM
cana-1961	90	84	1−£	1−£	NUM
cana-1961	90	85	ð	ð	X
cana-1961	90	86	(	(	PUNCT
cana-1961	90	87	ω2	ω2	ADJ
cana-1961	90	88	,	,	PUNCT
cana-1961	90	89	λω2	λω2	PROPN
cana-1961	90	90	)	)	PUNCT
cana-1961	90	91	,	,	PUNCT
cana-1961	90	92	for	for	ADP
cana-1961	90	93	all	all	DET
cana-1961	90	94	ω2	ω2	NOUN
cana-1961	90	95	∈	∈	PROPN
cana-1961	90	96	ψ	ψ	NOUN
cana-1961	90	97	.	.	PUNCT
cana-1961	90	98	”	"	PUNCT
cana-1961	91	1	in	in	ADP
cana-1961	91	2	this	this	DET
cana-1961	91	3	article	article	NOUN
cana-1961	91	4	,	,	PUNCT
cana-1961	91	5	let	let	VERB
cana-1961	91	6	𝒲	𝒲	NOUN
cana-1961	91	7	be	be	AUX
cana-1961	91	8	an	an	DET
cana-1961	91	9	na	na	NOUN
cana-1961	91	10	2	2	NUM
cana-1961	91	11	-	-	PUNCT
cana-1961	91	12	normed	norme	VERB
cana-1961	91	13	space	space	NOUN
cana-1961	91	14	with	with	ADP
cana-1961	91	15	dim	dim	ADJ
cana-1961	91	16	𝒲	𝒲	NOUN
cana-1961	91	17	>	>	SYM
cana-1961	91	18	1	1	NUM
cana-1961	91	19	and	and	CCONJ
cana-1961	91	20	𝒵	𝒵	PRON
cana-1961	91	21	be	be	VERB
cana-1961	91	22	an	an	DET
cana-1961	91	23	na	na	NOUN
cana-1961	91	24	2	2	NUM
cana-1961	91	25	-	-	PUNCT
cana-1961	91	26	banach	banach	NOUN
cana-1961	91	27	space	space	NOUN
cana-1961	91	28	with	with	ADP
cana-1961	91	29	dim	dim	ADJ
cana-1961	91	30	𝒵	𝒵	PROPN
cana-1961	91	31	>	>	X
cana-1961	91	32	1	1	X
cana-1961	91	33	.	.	PUNCT
cana-1961	91	34	suppose	suppose	VERB
cana-1961	91	35	for	for	ADP
cana-1961	91	36	a	a	DET
cana-1961	91	37	mapping	mapping	NOUN
cana-1961	91	38	𝔤	𝔤	NOUN
cana-1961	91	39	:	:	PUNCT
cana-1961	91	40	𝒲	𝒲	NOUN
cana-1961	91	41	→	→	SYM
cana-1961	91	42	𝒵	𝒵	PROPN
cana-1961	91	43	,	,	PUNCT
cana-1961	91	44	d𝔤(ω1	d𝔤(ω1	NOUN
cana-1961	91	45	,	,	PUNCT
cana-1961	91	46	ω2	ω2	ADJ
cana-1961	91	47	)	)	PUNCT
cana-1961	91	48	∶=	∶=	NUM
cana-1961	91	49	𝔤(ω1	𝔤(ω1	VERB
cana-1961	91	50	+	+	CCONJ
cana-1961	91	51	𝔪ω2	𝔪ω2	NOUN
cana-1961	91	52	)	)	PUNCT
cana-1961	92	1	+	+	CCONJ
cana-1961	92	2	𝔤(ω1	𝔤(ω1	NOUN
cana-1961	92	3	−	−	NOUN
cana-1961	92	4	𝔪ω2	𝔪ω2	NOUN
cana-1961	92	5	)	)	PUNCT
cana-1961	92	6	−	−	PROPN
cana-1961	93	1	𝔪2[𝔤(ω1	𝔪2[𝔤(ω1	PROPN
cana-1961	93	2	+	+	CCONJ
cana-1961	93	3	ω2	ω2	NUM
cana-1961	93	4	)	)	PUNCT
cana-1961	93	5	+	+	CCONJ
cana-1961	93	6	𝔤(ω1	𝔤(ω1	NOUN
cana-1961	93	7	−	−	PROPN
cana-1961	93	8	ω2	ω2	NUM
cana-1961	93	9	)	)	PUNCT
cana-1961	93	10	]	]	PUNCT
cana-1961	94	1	−	−	PROPN
cana-1961	94	2	2(𝔪2−1	2(𝔪2−1	NUM
cana-1961	94	3	)	)	PUNCT
cana-1961	94	4	𝔪2(𝔪−2	𝔪2(𝔪−2	NOUN
cana-1961	94	5	)	)	PUNCT
cana-1961	94	6	𝔤(𝔪ω1	𝔤(𝔪ω1	PROPN
cana-1961	94	7	)	)	PUNCT
cana-1961	95	1	+	+	CCONJ
cana-1961	95	2	𝔪3−𝔪2−𝔪+1	𝔪3−𝔪2−𝔪+1	NOUN
cana-1961	95	3	2(𝔪−2	2(𝔪−2	NOUN
cana-1961	95	4	)	)	PUNCT
cana-1961	95	5	𝔤(2ω1	𝔤(2ω1	NOUN
cana-1961	95	6	)	)	PUNCT
cana-1961	96	1	−	−	PROPN
cana-1961	96	2	(	(	PUNCT
cana-1961	96	3	𝔤(2ω2	𝔤(2ω2	NOUN
cana-1961	96	4	)	)	PUNCT
cana-1961	96	5	−	−	NOUN
cana-1961	96	6	𝔤(−2ω2	𝔤(−2ω2	NOUN
cana-1961	96	7	)	)	PUNCT
cana-1961	96	8	)	)	PUNCT
cana-1961	97	1	+	+	CCONJ
cana-1961	97	2	8(𝔤(ω2	8(𝔤(ω2	NUM
cana-1961	97	3	)	)	PUNCT
cana-1961	97	4	−	−	PROPN
cana-1961	97	5	𝔤(−ω2))2	𝔤(−ω2))2	PRON
cana-1961	97	6	for	for	ADP
cana-1961	97	7	all	all	DET
cana-1961	97	8	ω1	ω1	PROPN
cana-1961	97	9	,	,	PUNCT
cana-1961	97	10	ω2	ω2	NOUN
cana-1961	97	11	∈	∈	PROPN
cana-1961	97	12	𝒲.	𝒲.	PROPN
cana-1961	97	13	communications	communication	NOUN
cana-1961	97	14	on	on	ADP
cana-1961	97	15	applied	apply	VERB
cana-1961	97	16	nonlinear	nonlinear	ADJ
cana-1961	97	17	analysis	analysis	NOUN
cana-1961	97	18	issn	issn	NOUN
cana-1961	97	19	:	:	PUNCT
cana-1961	97	20	1074	1074	NUM
cana-1961	97	21	-	-	PUNCT
cana-1961	97	22	133x	133x	NUM
cana-1961	97	23	vol	vol	NOUN
cana-1961	97	24	32	32	NUM
cana-1961	97	25	no	no	NOUN
cana-1961	97	26	.	.	NOUN
cana-1961	97	27	3	3	NUM
cana-1961	97	28	(	(	PUNCT
cana-1961	97	29	2025	2025	NUM
cana-1961	97	30	)	)	PUNCT
cana-1961	97	31	305	305	NUM
cana-1961	97	32	https://internationalpubls.com	https://internationalpubls.com	X
cana-1961	97	33	3	3	X
cana-1961	97	34	.	.	NOUN
cana-1961	97	35	results	result	NOUN
cana-1961	97	36	and	and	CCONJ
cana-1961	97	37	discussion	discussion	NOUN
cana-1961	97	38	stability	stability	NOUN
cana-1961	97	39	of	of	ADP
cana-1961	97	40	the	the	DET
cana-1961	97	41	fe	fe	NOUN
cana-1961	97	42	–	–	PUNCT
cana-1961	97	43	even	even	ADV
cana-1961	97	44	case	case	NOUN
cana-1961	97	45	one	one	PRON
cana-1961	97	46	can	can	AUX
cana-1961	97	47	easily	easily	ADV
cana-1961	97	48	demonstrate	demonstrate	VERB
cana-1961	97	49	this	this	PRON
cana-1961	97	50	by	by	ADP
cana-1961	97	51	applying	apply	VERB
cana-1961	97	52	the	the	DET
cana-1961	97	53	conditons	conditon	NOUN
cana-1961	97	54	for	for	ADP
cana-1961	97	55	even	even	ADV
cana-1961	97	56	and	and	CCONJ
cana-1961	97	57	odd	odd	ADJ
cana-1961	97	58	functions	function	NOUN
cana-1961	97	59	in	in	ADP
cana-1961	97	60	this	this	DET
cana-1961	97	61	section	section	NOUN
cana-1961	97	62	.	.	PUNCT
cana-1961	98	1	the	the	DET
cana-1961	98	2	condition	condition	NOUN
cana-1961	98	3	for	for	ADP
cana-1961	98	4	an	an	DET
cana-1961	98	5	even	even	ADJ
cana-1961	98	6	function	function	NOUN
cana-1961	98	7	is	be	AUX
cana-1961	98	8	𝔤(−a	𝔤(−a	NOUN
cana-1961	98	9	)	)	PUNCT
cana-1961	98	10	=	=	SYM
cana-1961	98	11	𝔤(a	𝔤(a	PROPN
cana-1961	98	12	)	)	PUNCT
cana-1961	98	13	,	,	PUNCT
cana-1961	98	14	and	and	CCONJ
cana-1961	98	15	for	for	ADP
cana-1961	98	16	and	and	CCONJ
cana-1961	98	17	odd	odd	ADJ
cana-1961	98	18	function	function	NOUN
cana-1961	98	19	,	,	PUNCT
cana-1961	98	20	it	it	PRON
cana-1961	98	21	is	be	AUX
cana-1961	98	22	𝔤(−a	𝔤(−a	ADV
cana-1961	98	23	)	)	PUNCT
cana-1961	98	24	=	=	SYM
cana-1961	98	25	−𝔤(a	−𝔤(a	PROPN
cana-1961	98	26	)	)	PUNCT
cana-1961	98	27	.	.	PUNCT
cana-1961	99	1	an	an	DET
cana-1961	99	2	even	even	ADV
cana-1961	99	3	mapping	mapping	NOUN
cana-1961	99	4	𝔤	𝔤	NOUN
cana-1961	99	5	:	:	PUNCT
cana-1961	99	6	𝒲	𝒲	NOUN
cana-1961	99	7	→	→	SYM
cana-1961	99	8	𝒵	𝒵	PROPN
cana-1961	99	9	with	with	ADP
cana-1961	99	10	𝔤(0	𝔤(0	PROPN
cana-1961	99	11	)	)	PUNCT
cana-1961	99	12	=	=	SYM
cana-1961	100	1	0	0	NUM
cana-1961	100	2	satisfies	satisfie	NOUN
cana-1961	100	3	eq	eq	ADP
cana-1961	100	4	2	2	NUM
cana-1961	100	5	,	,	PUNCT
cana-1961	100	6	if	if	SCONJ
cana-1961	100	7	and	and	CCONJ
cana-1961	100	8	only	only	ADV
cana-1961	100	9	if	if	SCONJ
cana-1961	100	10	the	the	DET
cana-1961	100	11	even	even	ADV
cana-1961	100	12	mapping	mapping	NOUN
cana-1961	100	13	𝔤	𝔤	NOUN
cana-1961	100	14	:	:	PUNCT
cana-1961	100	15	𝒲	𝒲	NOUN
cana-1961	100	16	→	→	SYM
cana-1961	100	17	𝒵	𝒵	PROPN
cana-1961	100	18	is	be	AUX
cana-1961	100	19	a	a	DET
cana-1961	100	20	quadratic	quadratic	ADJ
cana-1961	100	21	(	(	PUNCT
cana-1961	100	22	𝑄2	𝑄2	PROPN
cana-1961	100	23	́	́	PROPN
cana-1961	100	24	)	)	PUNCT
cana-1961	101	1	mapping	mapping	NOUN
cana-1961	101	2	,	,	PUNCT
cana-1961	101	3	that	that	ADV
cana-1961	101	4	is	is	ADV
cana-1961	101	5	,	,	PUNCT
cana-1961	101	6	𝔤(ω1	𝔤(ω1	ADJ
cana-1961	101	7	+	+	CCONJ
cana-1961	101	8	ω2	ω2	NUM
cana-1961	101	9	)	)	PUNCT
cana-1961	101	10	+	+	CCONJ
cana-1961	101	11	𝔤(ω1	𝔤(ω1	NOUN
cana-1961	101	12	+	+	CCONJ
cana-1961	101	13	ω2	ω2	ADJ
cana-1961	101	14	)	)	PUNCT
cana-1961	101	15	=	=	SYM
cana-1961	101	16	2𝔤(ω1	2𝔤(ω1	NUM
cana-1961	101	17	)	)	PUNCT
cana-1961	101	18	+	+	NUM
cana-1961	101	19	2𝔤(ω2	2𝔤(ω2	NUM
cana-1961	101	20	)	)	PUNCT
cana-1961	101	21	and	and	CCONJ
cana-1961	101	22	an	an	DET
cana-1961	101	23	odd	odd	ADJ
cana-1961	101	24	mapping	mapping	NOUN
cana-1961	101	25	𝔤	𝔤	NOUN
cana-1961	101	26	:	:	PUNCT
cana-1961	101	27	𝒲	𝒲	NOUN
cana-1961	101	28	→	→	SYM
cana-1961	101	29	𝒵	𝒵	PROPN
cana-1961	101	30	satisfies	satisfy	VERB
cana-1961	101	31	eq	eq	ADP
cana-1961	101	32	2	2	NUM
cana-1961	101	33	if	if	SCONJ
cana-1961	101	34	and	and	CCONJ
cana-1961	101	35	only	only	ADV
cana-1961	101	36	if	if	SCONJ
cana-1961	101	37	the	the	DET
cana-1961	101	38	odd	odd	ADJ
cana-1961	101	39	mapping	mapping	NOUN
cana-1961	101	40	𝔤	𝔤	NOUN
cana-1961	101	41	:	:	PUNCT
cana-1961	101	42	𝒲	𝒲	NOUN
cana-1961	101	43	→	→	SYM
cana-1961	101	44	𝒵	𝒵	PROPN
cana-1961	101	45	is	be	AUX
cana-1961	101	46	a	a	DET
cana-1961	101	47	cubic	cubic	ADJ
cana-1961	101	48	(	(	PUNCT
cana-1961	101	49	𝐶3	𝐶3	PROPN
cana-1961	101	50	́	́	PROPN
cana-1961	101	51	)	)	PUNCT
cana-1961	101	52	mapping	mapping	NOUN
cana-1961	101	53	,	,	PUNCT
cana-1961	101	54	that	that	ADV
cana-1961	101	55	is	is	ADV
cana-1961	101	56	,	,	PUNCT
cana-1961	101	57	𝔤(2ω1	𝔤(2ω1	NOUN
cana-1961	101	58	+	+	CCONJ
cana-1961	101	59	ω2	ω2	ADJ
cana-1961	101	60	)	)	PUNCT
cana-1961	101	61	+	+	NUM
cana-1961	101	62	𝔤(2ω1	𝔤(2ω1	NOUN
cana-1961	101	63	+	+	CCONJ
cana-1961	101	64	ω2	ω2	ADJ
cana-1961	101	65	)	)	PUNCT
cana-1961	101	66	=	=	SYM
cana-1961	102	1	2𝔤(ω1	2𝔤(ω1	NUM
cana-1961	102	2	+	+	CCONJ
cana-1961	102	3	ω2	ω2	NUM
cana-1961	102	4	)	)	PUNCT
cana-1961	102	5	+	+	CCONJ
cana-1961	102	6	2𝔤(ω1	2𝔤(ω1	NUM
cana-1961	102	7	+	+	CCONJ
cana-1961	102	8	ω2	ω2	NUM
cana-1961	102	9	)	)	PUNCT
cana-1961	102	10	+	+	SYM
cana-1961	102	11	12𝔤(ω1	12𝔤(ω1	X
cana-1961	102	12	)	)	PUNCT
cana-1961	103	1	it	it	PRON
cana-1961	103	2	was	be	AUX
cana-1961	103	3	proved	prove	VERB
cana-1961	103	4	in	in	ADP
cana-1961	103	5	[	[	X
cana-1961	103	6	23	23	NUM
cana-1961	103	7	]	]	PUNCT
cana-1961	103	8	,	,	PUNCT
cana-1961	103	9	that	that	DET
cana-1961	103	10	𝔤(ω1	𝔤(ω1	NOUN
cana-1961	103	11	)	)	PUNCT
cana-1961	103	12	=	=	SYM
cana-1961	103	13	f(2ω1	f(2ω1	PROPN
cana-1961	103	14	)	)	PUNCT
cana-1961	103	15	−	−	NOUN
cana-1961	103	16	4f(ω1	4f(ω1	NOUN
cana-1961	103	17	)	)	PUNCT
cana-1961	103	18	and	and	CCONJ
cana-1961	103	19	𝔤(ω1	𝔤(ω1	VERB
cana-1961	103	20	)	)	PUNCT
cana-1961	103	21	=	=	SYM
cana-1961	103	22	f(2ω1	f(2ω1	PROPN
cana-1961	103	23	)	)	PUNCT
cana-1961	103	24	−	−	PROPN
cana-1961	103	25	8f(ω1	8f(ω1	NUM
cana-1961	103	26	)	)	PUNCT
cana-1961	103	27	are	be	AUX
cana-1961	103	28	𝑄2	𝑄2	PROPN
cana-1961	103	29	́	́	PROPN
cana-1961	103	30	and	and	CCONJ
cana-1961	103	31	𝐶3	𝐶3	PROPN
cana-1961	103	32	́	́	PUNCT
cana-1961	103	33	mappings	mapping	NOUN
cana-1961	103	34	respectively	respectively	ADV
cana-1961	103	35	.	.	PUNCT
cana-1961	104	1	in	in	ADP
cana-1961	104	2	this	this	DET
cana-1961	104	3	section	section	NOUN
cana-1961	104	4	,	,	PUNCT
cana-1961	104	5	to	to	PART
cana-1961	104	6	prove	prove	VERB
cana-1961	104	7	the	the	DET
cana-1961	104	8	generalized	generalized	ADJ
cana-1961	104	9	h	h	NOUN
cana-1961	104	10	-	-	PUNCT
cana-1961	104	11	u	u	NOUN
cana-1961	104	12	stability	stability	NOUN
cana-1961	104	13	of	of	ADP
cana-1961	104	14	the	the	DET
cana-1961	104	15	fe	fe	NOUN
cana-1961	104	16	d𝔤(ω1	d𝔤(ω1	ADP
cana-1961	104	17	,	,	PUNCT
cana-1961	104	18	ω2	ω2	ADJ
cana-1961	104	19	)	)	PUNCT
cana-1961	104	20	=	=	SYM
cana-1961	105	1	0	0	NUM
cana-1961	105	2	in	in	ADP
cana-1961	105	3	na	na	DET
cana-1961	105	4	2	2	NUM
cana-1961	105	5	-	-	PUNCT
cana-1961	105	6	normed	norme	VERB
cana-1961	105	7	space	space	NOUN
cana-1961	105	8	is	be	AUX
cana-1961	105	9	discussed	discuss	VERB
cana-1961	105	10	for	for	ADP
cana-1961	105	11	even	even	ADV
cana-1961	105	12	case	case	NOUN
cana-1961	105	13	.	.	PUNCT
cana-1961	106	1	theorem	theorem	NOUN
cana-1961	106	2	2	2	NUM
cana-1961	106	3	:	:	PUNCT
cana-1961	106	4	let	let	VERB
cana-1961	106	5	φ	φ	NUM
cana-1961	106	6	:	:	PUNCT
cana-1961	106	7	𝒲	𝒲	NOUN
cana-1961	106	8	×	×	NOUN
cana-1961	106	9	𝒲	𝒲	NOUN
cana-1961	106	10	→	→	SYM
cana-1961	107	1	[	[	X
cana-1961	107	2	0	0	NUM
cana-1961	107	3	,	,	PUNCT
cana-1961	107	4	∞	∞	PROPN
cana-1961	107	5	)	)	PUNCT
cana-1961	107	6	be	be	VERB
cana-1961	107	7	an	an	DET
cana-1961	107	8	even	even	ADV
cana-1961	107	9	function	function	NOUN
cana-1961	107	10	such	such	ADJ
cana-1961	107	11	that	that	SCONJ
cana-1961	107	12	there	there	PRON
cana-1961	107	13	exists	exist	VERB
cana-1961	107	14	a	a	DET
cana-1961	107	15	constant	constant	ADJ
cana-1961	107	16	0	0	NUM
cana-1961	107	17	<	<	X
cana-1961	107	18	£	£	SYM
cana-1961	107	19	<	<	X
cana-1961	107	20	1	1	NUM
cana-1961	107	21	with	with	ADP
cana-1961	107	22	φ(𝔪ω1	φ(𝔪ω1	NOUN
cana-1961	107	23	,	,	PUNCT
cana-1961	107	24	𝔪ω2	𝔪ω2	NOUN
cana-1961	107	25	)	)	PUNCT
cana-1961	107	26	≤	≤	NOUN
cana-1961	107	27	|𝔪|2	|𝔪|2	PROPN
cana-1961	107	28	£	£	NOUN
cana-1961	107	29	φ(ω1	φ(ω1	NOUN
cana-1961	107	30	,	,	PUNCT
cana-1961	107	31	ω2	ω2	NUM
cana-1961	107	32	)	)	PUNCT
cana-1961	107	33	3	3	NUM
cana-1961	107	34	for	for	ADP
cana-1961	107	35	all	all	DET
cana-1961	107	36	ω1	ω1	PROPN
cana-1961	107	37	,	,	PUNCT
cana-1961	107	38	ω2	ω2	NOUN
cana-1961	107	39	∈	∈	PROPN
cana-1961	107	40	𝒲.	𝒲.	PROPN
cana-1961	107	41	let	let	VERB
cana-1961	107	42	𝔤	𝔤	PRON
cana-1961	107	43	:	:	PUNCT
cana-1961	107	44	𝒲	𝒲	NOUN
cana-1961	107	45	→	→	PUNCT
cana-1961	107	46	𝒵	𝒵	PRON
cana-1961	107	47	be	be	VERB
cana-1961	107	48	an	an	DET
cana-1961	107	49	even	even	ADV
cana-1961	107	50	mapping	mapping	NOUN
cana-1961	107	51	satisfying	satisfy	VERB
cana-1961	107	52	‖d𝔤(ω1	‖d𝔤(ω1	NOUN
cana-1961	107	53	,	,	PUNCT
cana-1961	107	54	ω2	ω2	NUM
cana-1961	107	55	)	)	PUNCT
cana-1961	107	56	,	,	PUNCT
cana-1961	107	57	υ‖	υ‖	VERB
cana-1961	107	58	≤	≤	NOUN
cana-1961	107	59	φ(ω1	φ(ω1	X
cana-1961	107	60	,	,	PUNCT
cana-1961	107	61	ω2	ω2	NUM
cana-1961	107	62	)	)	PUNCT
cana-1961	107	63	4	4	NUM
cana-1961	107	64	for	for	ADP
cana-1961	107	65	all	all	DET
cana-1961	107	66	ω1	ω1	PROPN
cana-1961	107	67	,	,	PUNCT
cana-1961	107	68	ω2	ω2	NOUN
cana-1961	107	69	∈	∈	PROPN
cana-1961	107	70	𝒲	𝒲	PROPN
cana-1961	107	71	,	,	PUNCT
cana-1961	107	72	υ	υ	PRON
cana-1961	107	73	∈	∈	PROPN
cana-1961	107	74	𝒵.	𝒵.	PROPN
cana-1961	107	75	then	then	ADV
cana-1961	107	76	there	there	PRON
cana-1961	107	77	is	be	VERB
cana-1961	107	78	a	a	DET
cana-1961	107	79	unique	unique	ADJ
cana-1961	107	80	quadratic	quadratic	ADJ
cana-1961	107	81	mapping	mapping	NOUN
cana-1961	107	82	𝑄2	𝑄2	PROPN
cana-1961	108	1	́	́	NOUN
cana-1961	108	2	:	:	PUNCT
cana-1961	108	3	𝒲	𝒲	NOUN
cana-1961	108	4	→	→	SYM
cana-1961	108	5	𝒵	𝒵	NOUN
cana-1961	108	6	such	such	ADJ
cana-1961	108	7	that	that	DET
cana-1961	108	8	‖𝔤(ω1	‖𝔤(ω1	NOUN
cana-1961	108	9	)	)	PUNCT
cana-1961	108	10	−	−	PROPN
cana-1961	108	11	𝑄2	𝑄2	PROPN
cana-1961	109	1	́	́	PROPN
cana-1961	109	2	(	(	PUNCT
cana-1961	109	3	ω1	ω1	PROPN
cana-1961	109	4	)	)	PUNCT
cana-1961	109	5	,	,	PUNCT
cana-1961	109	6	υ‖	υ‖	VERB
cana-1961	109	7	≤	≤	NOUN
cana-1961	109	8	1	1	NUM
cana-1961	109	9	|2|	|2|	PROPN
cana-1961	109	10	|𝔪|2	|𝔪|2	PROPN
cana-1961	109	11	(	(	PUNCT
cana-1961	109	12	1−£	1−£	NUM
cana-1961	109	13	)	)	PUNCT
cana-1961	109	14	φ(0	φ(0	ADJ
cana-1961	109	15	,	,	PUNCT
cana-1961	109	16	ω1	ω1	PROPN
cana-1961	109	17	)	)	PUNCT
cana-1961	109	18	5	5	NUM
cana-1961	109	19	for	for	ADP
cana-1961	109	20	all	all	DET
cana-1961	109	21	ω1	ω1	PROPN
cana-1961	109	22	∈	∈	PROPN
cana-1961	109	23	𝒲	𝒲	PROPN
cana-1961	109	24	,	,	PUNCT
cana-1961	109	25	υ	υ	DET
cana-1961	109	26	∈	∈	NOUN
cana-1961	109	27	𝒵.	𝒵.	NOUN
cana-1961	109	28	proof	proof	NOUN
cana-1961	109	29	:	:	PUNCT
cana-1961	109	30	putting	put	VERB
cana-1961	109	31	ω1	ω1	PROPN
cana-1961	109	32	=	=	PUNCT
cana-1961	109	33	0	0	NUM
cana-1961	109	34	in	in	ADP
cana-1961	109	35	eq	eq	NOUN
cana-1961	109	36	2	2	NUM
cana-1961	109	37	,	,	PUNCT
cana-1961	109	38	implies	imply	VERB
cana-1961	109	39	‖2𝔤(𝔪ω2	‖2𝔤(𝔪ω2	NOUN
cana-1961	109	40	)	)	PUNCT
cana-1961	110	1	−	−	PROPN
cana-1961	110	2	2𝔪2𝔤(ω2	2𝔪2𝔤(ω2	NUM
cana-1961	110	3	)	)	PUNCT
cana-1961	110	4	,	,	PUNCT
cana-1961	110	5	υ‖	υ‖	VERB
cana-1961	110	6	≤	≤	ADJ
cana-1961	110	7	φ(0	φ(0	ADJ
cana-1961	110	8	,	,	PUNCT
cana-1961	110	9	ω2	ω2	NUM
cana-1961	110	10	)	)	PUNCT
cana-1961	110	11	6	6	NUM
cana-1961	110	12	for	for	ADP
cana-1961	110	13	all	all	DET
cana-1961	110	14	ω2	ω2	ADJ
cana-1961	110	15	∈	∈	PROPN
cana-1961	110	16	𝒲	𝒲	PROPN
cana-1961	110	17	,	,	PUNCT
cana-1961	110	18	υ	υ	PRON
cana-1961	110	19	∈	∈	PROPN
cana-1961	110	20	𝒵.	𝒵.	PROPN
cana-1961	110	21	interchanging	interchange	VERB
cana-1961	110	22	ω2	ω2	NOUN
cana-1961	110	23	by	by	ADP
cana-1961	110	24	ω1	ω1	PROPN
cana-1961	110	25	in	in	ADP
cana-1961	110	26	eq	eq	ADP
cana-1961	110	27	6	6	NUM
cana-1961	110	28	and	and	CCONJ
cana-1961	110	29	dividing	divide	VERB
cana-1961	110	30	on	on	ADP
cana-1961	110	31	both	both	DET
cana-1961	110	32	sides	side	NOUN
cana-1961	110	33	of	of	ADP
cana-1961	110	34	eq	eq	NOUN
cana-1961	110	35	6	6	NUM
cana-1961	110	36	by	by	ADP
cana-1961	110	37	2	2	NUM
cana-1961	110	38	,	,	PUNCT
cana-1961	110	39	gives	give	VERB
cana-1961	110	40	‖𝔤(𝔪ω1	‖𝔤(𝔪ω1	NOUN
cana-1961	110	41	)	)	PUNCT
cana-1961	110	42	−	−	ADP
cana-1961	111	1	𝔪2𝔤(ω1	𝔪2𝔤(ω1	PROPN
cana-1961	111	2	)	)	PUNCT
cana-1961	111	3	,	,	PUNCT
cana-1961	111	4	υ‖	υ‖	VERB
cana-1961	111	5	≤	≤	NOUN
cana-1961	111	6	1	1	NUM
cana-1961	111	7	|2|	|2|	PROPN
cana-1961	111	8	φ(0	φ(0	PROPN
cana-1961	111	9	,	,	PUNCT
cana-1961	111	10	ω1	ω1	PROPN
cana-1961	111	11	)	)	PUNCT
cana-1961	111	12	‖	‖	PROPN
cana-1961	111	13	𝔤(𝔪ω1	𝔤(𝔪ω1	PROPN
cana-1961	111	14	)	)	PUNCT
cana-1961	111	15	𝔪2	𝔪2	NOUN
cana-1961	111	16	−	−	PROPN
cana-1961	111	17	𝔤(ω1	𝔤(ω1	NOUN
cana-1961	111	18	)	)	PUNCT
cana-1961	111	19	,	,	PUNCT
cana-1961	111	20	υ‖	υ‖	VERB
cana-1961	111	21	≤	≤	NUM
cana-1961	111	22	1	1	NUM
cana-1961	111	23	|2|	|2|	PROPN
cana-1961	111	24	1	1	NUM
cana-1961	111	25	|𝔪|2	|𝔪|2	PROPN
cana-1961	111	26	φ(0	φ(0	PROPN
cana-1961	111	27	,	,	PUNCT
cana-1961	111	28	ω1	ω1	PROPN
cana-1961	111	29	)	)	PUNCT
cana-1961	111	30	7	7	NUM
cana-1961	111	31	for	for	ADP
cana-1961	111	32	all	all	DET
cana-1961	111	33	ω1	ω1	PROPN
cana-1961	111	34	∈	∈	PROPN
cana-1961	111	35	𝒲	𝒲	PROPN
cana-1961	111	36	,	,	PUNCT
cana-1961	111	37	υ	υ	PRON
cana-1961	111	38	∈	∈	NOUN
cana-1961	111	39	𝒵.	𝒵.	PROPN
cana-1961	111	40	consider	consider	VERB
cana-1961	111	41	the	the	DET
cana-1961	111	42	set	set	NOUN
cana-1961	111	43	γ	γ	X
cana-1961	111	44	=	=	PRON
cana-1961	111	45	{	{	PUNCT
cana-1961	111	46	𝔣	𝔣	NOUN
cana-1961	111	47	:	:	PUNCT
cana-1961	111	48	𝒲	𝒲	NOUN
cana-1961	111	49	→	→	SYM
cana-1961	111	50	𝒵	𝒵	PROPN
cana-1961	111	51	}	}	PUNCT
cana-1961	111	52	8	8	NUM
cana-1961	111	53	and	and	CCONJ
cana-1961	111	54	define	define	VERB
cana-1961	111	55	the	the	DET
cana-1961	111	56	generalized	generalized	ADJ
cana-1961	111	57	metric	metric	NOUN
cana-1961	111	58	ð	ð	PROPN
cana-1961	111	59	in	in	ADP
cana-1961	111	60	γ	γ	X
cana-1961	111	61	by	by	ADP
cana-1961	111	62	ð(𝔣	ð(𝔣	PROPN
cana-1961	111	63	,	,	PUNCT
cana-1961	111	64	𝔥	𝔥	NOUN
cana-1961	111	65	)	)	PUNCT
cana-1961	111	66	=	=	PUNCT
cana-1961	111	67	inf{σ	inf{σ	PROPN
cana-1961	111	68	∈	∈	NOUN
cana-1961	111	69	(	(	PUNCT
cana-1961	111	70	0	0	NUM
cana-1961	111	71	,	,	PUNCT
cana-1961	111	72	∞	∞	NUM
cana-1961	111	73	):	):	PUNCT
cana-1961	111	74	‖𝔣(ω1	‖𝔣(ω1	NUM
cana-1961	111	75	)	)	PUNCT
cana-1961	111	76	−	−	NOUN
cana-1961	111	77	𝔥(ω1	𝔥(ω1	ADV
cana-1961	111	78	)	)	PUNCT
cana-1961	111	79	,	,	PUNCT
cana-1961	111	80	υ‖	υ‖	VERB
cana-1961	111	81	≤	≤	NOUN
cana-1961	112	1	σ	σ	NUM
cana-1961	112	2	φ(0	φ(0	PROPN
cana-1961	112	3	,	,	PUNCT
cana-1961	112	4	ω1	ω1	PROPN
cana-1961	112	5	)	)	PUNCT
cana-1961	112	6	,	,	PUNCT
cana-1961	112	7	∀	∀	X
cana-1961	112	8	ω1	ω1	PROPN
cana-1961	112	9	∈	∈	PROPN
cana-1961	112	10	𝒲	𝒲	PROPN
cana-1961	112	11	}	}	PUNCT
cana-1961	112	12	.	.	PUNCT
cana-1961	113	1	9	9	NUM
cana-1961	113	2	it	it	PRON
cana-1961	113	3	is	be	AUX
cana-1961	113	4	simple	simple	ADJ
cana-1961	113	5	to	to	PART
cana-1961	113	6	prove	prove	VERB
cana-1961	113	7	that	that	SCONJ
cana-1961	113	8	(	(	PUNCT
cana-1961	113	9	γ	γ	X
cana-1961	113	10	,	,	PUNCT
cana-1961	113	11	ð	ð	NUM
cana-1961	113	12	)	)	PUNCT
cana-1961	113	13	is	be	AUX
cana-1961	113	14	complete	complete	ADJ
cana-1961	113	15	[	[	X
cana-1961	113	16	26	26	NUM
cana-1961	113	17	]	]	PUNCT
cana-1961	113	18	.	.	PUNCT
cana-1961	114	1	now	now	ADV
cana-1961	114	2	,	,	PUNCT
cana-1961	114	3	define	define	VERB
cana-1961	114	4	the	the	DET
cana-1961	114	5	function	function	NOUN
cana-1961	114	6	λ	λ	PROPN
cana-1961	114	7	∶	∶	NOUN
cana-1961	114	8	γ	γ	X
cana-1961	114	9	→	→	SYM
cana-1961	114	10	γ	γ	X
cana-1961	114	11	such	such	ADJ
cana-1961	114	12	that	that	SCONJ
cana-1961	114	13	communications	communication	NOUN
cana-1961	114	14	on	on	ADP
cana-1961	114	15	applied	apply	VERB
cana-1961	114	16	nonlinear	nonlinear	ADJ
cana-1961	114	17	analysis	analysis	NOUN
cana-1961	114	18	issn	issn	NOUN
cana-1961	114	19	:	:	PUNCT
cana-1961	114	20	1074	1074	NUM
cana-1961	114	21	-	-	PUNCT
cana-1961	114	22	133x	133x	NUM
cana-1961	114	23	vol	vol	NOUN
cana-1961	114	24	32	32	NUM
cana-1961	114	25	no	no	NOUN
cana-1961	114	26	.	.	NOUN
cana-1961	114	27	3	3	NUM
cana-1961	114	28	(	(	PUNCT
cana-1961	114	29	2025	2025	NUM
cana-1961	114	30	)	)	PUNCT
cana-1961	114	31	306	306	NUM
cana-1961	114	32	https://internationalpubls.com	https://internationalpubls.com	X
cana-1961	114	33	λ𝔣(ω1	λ𝔣(ω1	NOUN
cana-1961	114	34	)	)	PUNCT
cana-1961	114	35	=	=	SYM
cana-1961	114	36	1	1	NUM
cana-1961	114	37	𝔪2	𝔪2	NOUN
cana-1961	114	38	𝔣(𝔪ω1	𝔣(𝔪ω1	NOUN
cana-1961	114	39	)	)	PUNCT
cana-1961	114	40	10	10	NUM
cana-1961	114	41	for	for	ADP
cana-1961	114	42	all	all	DET
cana-1961	114	43	ω1	ω1	PROPN
cana-1961	114	44	∈	∈	PROPN
cana-1961	114	45	𝒲.	𝒲.	PROPN
cana-1961	114	46	let	let	VERB
cana-1961	114	47	𝔣	𝔣	ADP
cana-1961	114	48	,	,	PUNCT
cana-1961	114	49	𝔥	𝔥	PROPN
cana-1961	114	50	∈	∈	PROPN
cana-1961	114	51	γ	γ	NOUN
cana-1961	114	52	be	be	AUX
cana-1961	114	53	given	give	VERB
cana-1961	114	54	such	such	ADJ
cana-1961	114	55	that	that	DET
cana-1961	114	56	ð(𝔣	ð(𝔣	PROPN
cana-1961	114	57	,	,	PUNCT
cana-1961	114	58	𝔥	𝔥	NOUN
cana-1961	114	59	)	)	PUNCT
cana-1961	114	60	=	=	VERB
cana-1961	115	1	ϵ.	ϵ.	NOUN
cana-1961	115	2	then	then	ADV
cana-1961	115	3	‖𝔣(ω1	‖𝔣(ω1	NUM
cana-1961	115	4	)	)	PUNCT
cana-1961	115	5	−	−	NOUN
cana-1961	115	6	𝔥(ω1	𝔥(ω1	ADV
cana-1961	115	7	)	)	PUNCT
cana-1961	115	8	,	,	PUNCT
cana-1961	115	9	υ‖	υ‖	VERB
cana-1961	115	10	≤	≤	NOUN
cana-1961	115	11	ϵ	ϵ	ADP
cana-1961	115	12	φ(0	φ(0	ADJ
cana-1961	115	13	,	,	PUNCT
cana-1961	115	14	ω1	ω1	PROPN
cana-1961	115	15	)	)	PUNCT
cana-1961	115	16	11	11	NUM
cana-1961	115	17	for	for	ADP
cana-1961	115	18	all	all	DET
cana-1961	115	19	ω1	ω1	PROPN
cana-1961	115	20	∈	∈	PROPN
cana-1961	115	21	𝒲	𝒲	PROPN
cana-1961	115	22	,	,	PUNCT
cana-1961	115	23	υ	υ	PRON
cana-1961	115	24	∈	∈	NOUN
cana-1961	115	25	𝒵.	𝒵.	PROPN
cana-1961	115	26	hence	hence	ADV
cana-1961	115	27	‖λ𝔣(𝔪ω1	‖λ𝔣(𝔪ω1	NOUN
cana-1961	115	28	)	)	PUNCT
cana-1961	115	29	−	−	PROPN
cana-1961	115	30	λ𝔥(𝔪ω1	λ𝔥(𝔪ω1	NOUN
cana-1961	115	31	)	)	PUNCT
cana-1961	115	32	,	,	PUNCT
cana-1961	115	33	υ‖	υ‖	X
cana-1961	115	34	=	=	PUNCT
cana-1961	116	1	‖	‖	PROPN
cana-1961	116	2	1	1	NUM
cana-1961	116	3	𝔪2	𝔪2	NOUN
cana-1961	116	4	𝔣(𝔪ω1	𝔣(𝔪ω1	NOUN
cana-1961	116	5	)	)	PUNCT
cana-1961	116	6	−	−	PROPN
cana-1961	116	7	1	1	NUM
cana-1961	116	8	𝔪2	𝔪2	NOUN
cana-1961	116	9	𝔥(𝔪ω1	𝔥(𝔪ω1	NOUN
cana-1961	116	10	)	)	PUNCT
cana-1961	116	11	,	,	PUNCT
cana-1961	116	12	υ‖	υ‖	VERB
cana-1961	116	13	≤	≤	NOUN
cana-1961	116	14	1	1	NUM
cana-1961	116	15	|𝔪|2	|𝔪|2	PROPN
cana-1961	116	16	.	.	PUNCT
cana-1961	117	1	ϵ.	ϵ.	NOUN
cana-1961	117	2	φ(0	φ(0	ADJ
cana-1961	117	3	,	,	PUNCT
cana-1961	117	4	𝔪ω1	𝔪ω1	NOUN
cana-1961	117	5	)	)	PUNCT
cana-1961	117	6	≤	≤	NOUN
cana-1961	117	7	1	1	NUM
cana-1961	117	8	|𝔪|2	|𝔪|2	PROPN
cana-1961	117	9	.	.	PUNCT
cana-1961	118	1	ϵ.	ϵ.	NOUN
cana-1961	118	2	|𝔪|2	|𝔪|2	PROPN
cana-1961	118	3	.	.	PUNCT
cana-1961	119	1	£	£	SYM
cana-1961	119	2	.	.	PUNCT
cana-1961	120	1	φ(0	φ(0	ADJ
cana-1961	120	2	,	,	PUNCT
cana-1961	120	3	ω1	ω1	PROPN
cana-1961	120	4	)	)	PUNCT
cana-1961	120	5	≤	≤	NOUN
cana-1961	121	1	ϵ.	ϵ.	NOUN
cana-1961	121	2	£	£	NOUN
cana-1961	121	3	.	.	PUNCT
cana-1961	122	1	φ(0	φ(0	ADJ
cana-1961	122	2	,	,	PUNCT
cana-1961	122	3	ω1	ω1	PROPN
cana-1961	122	4	)	)	PUNCT
cana-1961	122	5	for	for	ADP
cana-1961	122	6	all	all	DET
cana-1961	122	7	ω1	ω1	PROPN
cana-1961	122	8	∈	∈	PROPN
cana-1961	122	9	𝒲	𝒲	PROPN
cana-1961	122	10	,	,	PUNCT
cana-1961	122	11	υ	υ	PRON
cana-1961	122	12	∈	∈	PROPN
cana-1961	122	13	𝒵	𝒵	PROPN
cana-1961	122	14	,	,	PUNCT
cana-1961	122	15	that	that	PRON
cana-1961	122	16	is	be	AUX
cana-1961	122	17	ð(λ𝔣	ð(λ𝔣	NOUN
cana-1961	122	18	,	,	PUNCT
cana-1961	122	19	λh	λh	VERB
cana-1961	122	20	)	)	PUNCT
cana-1961	122	21	≤	≤	NOUN
cana-1961	122	22	£	£	SYM
cana-1961	122	23	ε	ε	PROPN
cana-1961	122	24	.	.	PUNCT
cana-1961	123	1	therefore	therefore	PROPN
cana-1961	123	2	ð(λ𝔣	ð(λ𝔣	NOUN
cana-1961	123	3	,	,	PUNCT
cana-1961	123	4	λh	λh	VERB
cana-1961	123	5	)	)	PUNCT
cana-1961	123	6	≤	≤	NOUN
cana-1961	123	7	£	£	SYM
cana-1961	123	8	ð(𝔣	ð(𝔣	PROPN
cana-1961	123	9	,	,	PUNCT
cana-1961	123	10	h	h	NOUN
cana-1961	123	11	)	)	PUNCT
cana-1961	123	12	for	for	ADP
cana-1961	123	13	all	all	DET
cana-1961	123	14	𝔣	𝔣	ADJ
cana-1961	123	15	,	,	PUNCT
cana-1961	123	16	𝔥	𝔥	PROPN
cana-1961	123	17	∈	∈	PROPN
cana-1961	123	18	γ	γ	X
cana-1961	123	19	.	.	PUNCT
cana-1961	124	1	according	accord	VERB
cana-1961	124	2	to	to	ADP
cana-1961	124	3	eq	eq	NOUN
cana-1961	124	4	7	7	NUM
cana-1961	124	5	ð(𝔤	ð(𝔤	NOUN
cana-1961	124	6	,	,	PUNCT
cana-1961	124	7	λ𝔤	λ𝔤	NOUN
cana-1961	124	8	)	)	PUNCT
cana-1961	124	9	≤	≤	NOUN
cana-1961	124	10	1	1	NUM
cana-1961	124	11	|2|	|2|	PROPN
cana-1961	124	12	.	.	PUNCT
cana-1961	124	13	1	1	NUM
cana-1961	124	14	|𝔪|2	|𝔪|2	NOUN
cana-1961	124	15	<	<	X
cana-1961	124	16	+	+	PROPN
cana-1961	124	17	∞.	∞.	PROPN
cana-1961	124	18	12	12	NUM
cana-1961	124	19	by	by	ADP
cana-1961	124	20	theorem	theorem	NOUN
cana-1961	124	21	1	1	NUM
cana-1961	124	22	,	,	PUNCT
cana-1961	124	23	there	there	PRON
cana-1961	124	24	is	be	VERB
cana-1961	124	25	a	a	DET
cana-1961	124	26	mapping	mapping	NOUN
cana-1961	124	27	𝑄2	𝑄2	PROPN
cana-1961	124	28	́	́	PROPN
cana-1961	124	29	:	:	PUNCT
cana-1961	124	30	𝒲	𝒲	NOUN
cana-1961	124	31	→	→	SYM
cana-1961	124	32	𝒵	𝒵	PROPN
cana-1961	124	33	satisfying	satisfy	VERB
cana-1961	124	34	the	the	DET
cana-1961	124	35	following	follow	VERB
cana-1961	124	36	conditions	condition	NOUN
cana-1961	124	37	:	:	PUNCT
cana-1961	124	38	(	(	PUNCT
cana-1961	124	39	1	1	X
cana-1961	124	40	)	)	PUNCT
cana-1961	124	41	𝑄2	𝑄2	NOUN
cana-1961	125	1	́	́	PROPN
cana-1961	125	2	is	be	AUX
cana-1961	125	3	a	a	DET
cana-1961	125	4	fixed	fix	VERB
cana-1961	125	5	point	point	NOUN
cana-1961	125	6	of	of	ADP
cana-1961	125	7	λ	λ	PROPN
cana-1961	125	8	,	,	PUNCT
cana-1961	125	9	that	that	ADV
cana-1961	125	10	is	is	ADV
cana-1961	125	11	,	,	PUNCT
cana-1961	125	12	𝑄2	𝑄2	PROPN
cana-1961	125	13	́	́	PROPN
cana-1961	125	14	(	(	PUNCT
cana-1961	125	15	𝔪ω1	𝔪ω1	NOUN
cana-1961	125	16	)	)	PUNCT
cana-1961	125	17	=	=	PUNCT
cana-1961	125	18	𝔪2	𝔪2	NOUN
cana-1961	125	19	𝑄2	𝑄2	X
cana-1961	125	20	́	́	PROPN
cana-1961	125	21	(	(	PUNCT
cana-1961	125	22	ω1	ω1	PROPN
cana-1961	125	23	)	)	PUNCT
cana-1961	125	24	13	13	NUM
cana-1961	125	25	𝑄2	𝑄2	NOUN
cana-1961	125	26	́	́	PROPN
cana-1961	125	27	is	be	AUX
cana-1961	125	28	a	a	DET
cana-1961	125	29	unique	unique	ADJ
cana-1961	125	30	fixed	fix	VERB
cana-1961	125	31	point	point	NOUN
cana-1961	125	32	of	of	ADP
cana-1961	125	33	the	the	DET
cana-1961	125	34	set	set	NOUN
cana-1961	125	35	denoted	denote	VERB
cana-1961	125	36	by	by	ADP
cana-1961	125	37	λ	λ	NOUN
cana-1961	125	38	.	.	PROPN
cana-1961	125	39	s	s	PART
cana-1961	125	40	=	=	X
cana-1961	125	41	{	{	PUNCT
cana-1961	125	42	𝔥	𝔥	NOUN
cana-1961	125	43	∈	∈	PROPN
cana-1961	125	44	γ	γ	PROPN
cana-1961	125	45	∶	∶	NOUN
cana-1961	125	46	ð(𝔣	ð(𝔣	PROPN
cana-1961	125	47	,	,	PUNCT
cana-1961	125	48	𝔥	𝔥	NOUN
cana-1961	125	49	)	)	PUNCT
cana-1961	125	50	<	<	X
cana-1961	125	51	∞	∞	PROPN
cana-1961	125	52	}	}	PUNCT
cana-1961	125	53	.	.	PUNCT
cana-1961	126	1	this	this	PRON
cana-1961	126	2	indicates	indicate	VERB
cana-1961	126	3	that	that	SCONJ
cana-1961	126	4	𝑄2	𝑄2	PROPN
cana-1961	126	5	́	́	PROPN
cana-1961	126	6	is	be	AUX
cana-1961	126	7	a	a	DET
cana-1961	126	8	unique	unique	ADJ
cana-1961	126	9	mapping	mapping	NOUN
cana-1961	126	10	satisfying	satisfy	VERB
cana-1961	126	11	eq	eq	ADP
cana-1961	126	12	13	13	NUM
cana-1961	126	13	such	such	ADJ
cana-1961	126	14	that	that	SCONJ
cana-1961	126	15	there	there	PRON
cana-1961	126	16	is	be	VERB
cana-1961	126	17	a	a	DET
cana-1961	126	18	σ	σ	NUM
cana-1961	126	19	∈	∈	PROPN
cana-1961	126	20	(	(	PUNCT
cana-1961	126	21	0	0	NUM
cana-1961	126	22	,	,	PUNCT
cana-1961	126	23	∞	∞	NUM
cana-1961	126	24	)	)	PUNCT
cana-1961	126	25	satisfying	satisfy	VERB
cana-1961	126	26	‖𝔤(ω1	‖𝔤(ω1	NOUN
cana-1961	126	27	)	)	PUNCT
cana-1961	127	1	−	−	PROPN
cana-1961	127	2	𝑄2	𝑄2	PROPN
cana-1961	127	3	́	́	PROPN
cana-1961	127	4	(	(	PUNCT
cana-1961	127	5	ω1	ω1	PROPN
cana-1961	127	6	)	)	PUNCT
cana-1961	127	7	,	,	PUNCT
cana-1961	127	8	υ‖	υ‖	VERB
cana-1961	127	9	≤	≤	PROPN
cana-1961	127	10	σ	σ	NOUN
cana-1961	127	11	.	.	PUNCT
cana-1961	128	1	φ(0	φ(0	ADJ
cana-1961	128	2	,	,	PUNCT
cana-1961	128	3	ω1	ω1	PROPN
cana-1961	128	4	)	)	PUNCT
cana-1961	128	5	,	,	PUNCT
cana-1961	128	6	∀	∀	X
cana-1961	128	7	ω1	ω1	PROPN
cana-1961	128	8	∈	∈	PROPN
cana-1961	128	9	𝒲	𝒲	PROPN
cana-1961	128	10	,	,	PUNCT
cana-1961	128	11	υ	υ	PRON
cana-1961	128	12	∈	∈	PROPN
cana-1961	128	13	𝒵.	𝒵.	PROPN
cana-1961	128	14	(	(	PUNCT
cana-1961	128	15	2	2	NUM
cana-1961	128	16	)	)	PUNCT
cana-1961	128	17	ð(λ	ð(λ	PROPN
cana-1961	128	18	n𝔤	n𝔤	PROPN
cana-1961	128	19	,	,	PUNCT
cana-1961	128	20	𝑄2	𝑄2	PROPN
cana-1961	128	21	́	́	PROPN
cana-1961	128	22	)	)	PUNCT
cana-1961	129	1	→	→	SYM
cana-1961	129	2	0	0	NUM
cana-1961	129	3	as	as	ADP
cana-1961	129	4	n	n	NOUN
cana-1961	129	5	→	→	SYM
cana-1961	129	6	∞.	∞.	PROPN
cana-1961	129	7	this	this	PRON
cana-1961	129	8	gives	give	VERB
cana-1961	129	9	that	that	PRON
cana-1961	129	10	,	,	PUNCT
cana-1961	129	11	lim	lim	PROPN
cana-1961	129	12	n→∞	n→∞	X
cana-1961	129	13	(	(	PUNCT
cana-1961	129	14	λ	λ	NOUN
cana-1961	129	15	n𝔤)(ω1	n𝔤)(ω1	NOUN
cana-1961	129	16	)	)	PUNCT
cana-1961	129	17	=	=	SYM
cana-1961	129	18	lim	lim	PROPN
cana-1961	129	19	n→∞	n→∞	X
cana-1961	129	20	𝔤(𝔪nω1	𝔤(𝔪nω1	NUM
cana-1961	129	21	)	)	PUNCT
cana-1961	129	22	𝔪2n	𝔪2n	PUNCT
cana-1961	130	1	=	=	SYM
cana-1961	130	2	𝑄2	𝑄2	PROPN
cana-1961	131	1	́	́	PUNCT
cana-1961	131	2	(	(	PUNCT
cana-1961	131	3	ω1	ω1	PROPN
cana-1961	131	4	)	)	PUNCT
cana-1961	131	5	,	,	PUNCT
cana-1961	131	6	∀	∀	X
cana-1961	131	7	ω1	ω1	PROPN
cana-1961	131	8	∈	∈	PROPN
cana-1961	131	9	𝒲.	𝒲.	PROPN
cana-1961	131	10	(	(	PUNCT
cana-1961	131	11	3	3	X
cana-1961	131	12	)	)	PUNCT
cana-1961	131	13	ð(𝔤	ð(𝔤	PROPN
cana-1961	131	14	,	,	PUNCT
cana-1961	131	15	𝑄2	𝑄2	PROPN
cana-1961	131	16	́	́	PROPN
cana-1961	131	17	)	)	PUNCT
cana-1961	131	18	≤	≤	ADV
cana-1961	131	19	1	1	NUM
cana-1961	131	20	1−£	1−£	NUM
cana-1961	131	21	ð(𝔤	ð(𝔤	NOUN
cana-1961	131	22	,	,	PUNCT
cana-1961	131	23	λ	λ	PROPN
cana-1961	131	24	n𝔤	n𝔤	NUM
cana-1961	131	25	)	)	PUNCT
cana-1961	131	26	,	,	PUNCT
cana-1961	131	27	which	which	PRON
cana-1961	131	28	gives	give	VERB
cana-1961	131	29	the	the	DET
cana-1961	131	30	inequality	inequality	NOUN
cana-1961	131	31	ð(𝔤	ð(𝔤	PROPN
cana-1961	131	32	,	,	PUNCT
cana-1961	131	33	𝑄2	𝑄2	PROPN
cana-1961	131	34	́	́	PROPN
cana-1961	131	35	)	)	PUNCT
cana-1961	132	1	≤	≤	ADV
cana-1961	132	2	1	1	NUM
cana-1961	132	3	1	1	NUM
cana-1961	132	4	−	−	NOUN
cana-1961	132	5	£	£	PROPN
cana-1961	132	6	ð(𝔤	ð(𝔤	NOUN
cana-1961	132	7	,	,	PUNCT
cana-1961	132	8	λ𝔤	λ𝔤	NOUN
cana-1961	132	9	)	)	PUNCT
cana-1961	132	10	≤	≤	NOUN
cana-1961	132	11	1	1	NUM
cana-1961	132	12	|2|	|2|	PROPN
cana-1961	132	13	1	1	NUM
cana-1961	132	14	|𝔪|2	|𝔪|2	NOUN
cana-1961	132	15	1	1	NUM
cana-1961	132	16	(	(	PUNCT
cana-1961	132	17	1	1	NUM
cana-1961	132	18	−	−	NOUN
cana-1961	132	19	£	£	NUM
cana-1961	132	20	)	)	PUNCT
cana-1961	132	21	.	.	PUNCT
cana-1961	133	1	this	this	PRON
cana-1961	133	2	indicates	indicate	VERB
cana-1961	133	3	that	that	SCONJ
cana-1961	133	4	the	the	DET
cana-1961	133	5	inequality	inequality	NOUN
cana-1961	133	6	eq	eq	ADP
cana-1961	133	7	5	5	NUM
cana-1961	133	8	remains	remain	VERB
cana-1961	133	9	valid	valid	ADJ
cana-1961	133	10	.	.	PUNCT
cana-1961	134	1	according	accord	VERB
cana-1961	134	2	to	to	ADP
cana-1961	134	3	eq	eq	NOUN
cana-1961	134	4	3	3	NUM
cana-1961	134	5	and	and	CCONJ
cana-1961	134	6	eq	eq	ADP
cana-1961	134	7	4	4	NUM
cana-1961	134	8	∥	∥	NUM
cana-1961	134	9	d𝑄2	d𝑄2	PROPN
cana-1961	134	10	́	́	PROPN
cana-1961	134	11	(	(	PUNCT
cana-1961	134	12	ω1	ω1	PROPN
cana-1961	134	13	,	,	PUNCT
cana-1961	134	14	ω2	ω2	NUM
cana-1961	134	15	)	)	PUNCT
cana-1961	134	16	,	,	PUNCT
cana-1961	134	17	υ	υ	PRON
cana-1961	134	18	∥=	∥=	ADJ
cana-1961	134	19	lim	lim	PROPN
cana-1961	134	20	n→∞	n→∞	X
cana-1961	134	21	∥	∥	PROPN
cana-1961	134	22	𝔪−2nd𝔤(𝔪nω1	𝔪−2nd𝔤(𝔪nω1	PROPN
cana-1961	134	23	,	,	PUNCT
cana-1961	134	24	𝔪nω2	𝔪nω2	PROPN
cana-1961	134	25	)	)	PUNCT
cana-1961	134	26	,	,	PUNCT
cana-1961	134	27	υ	υ	NOUN
cana-1961	134	28	∥	∥	PUNCT
cana-1961	134	29	≤	≤	NUM
cana-1961	134	30	lim	lim	PROPN
cana-1961	134	31	n→∞	n→∞	NUM
cana-1961	134	32	1	1	NUM
cana-1961	134	33	|𝔪|2n	|𝔪|2n	NOUN
cana-1961	134	34	φ(𝔪nω1	φ(𝔪nω1	NOUN
cana-1961	134	35	,	,	PUNCT
cana-1961	134	36	𝔪nω2	𝔪nω2	NOUN
cana-1961	134	37	)	)	PUNCT
cana-1961	134	38	communications	communication	NOUN
cana-1961	134	39	on	on	ADP
cana-1961	134	40	applied	apply	VERB
cana-1961	134	41	nonlinear	nonlinear	ADJ
cana-1961	134	42	analysis	analysis	NOUN
cana-1961	134	43	issn	issn	NOUN
cana-1961	134	44	:	:	PUNCT
cana-1961	134	45	1074	1074	NUM
cana-1961	134	46	-	-	PUNCT
cana-1961	134	47	133x	133x	NUM
cana-1961	134	48	vol	vol	NOUN
cana-1961	134	49	32	32	NUM
cana-1961	134	50	no	no	NOUN
cana-1961	134	51	.	.	NOUN
cana-1961	134	52	3	3	NUM
cana-1961	134	53	(	(	PUNCT
cana-1961	134	54	2025	2025	NUM
cana-1961	134	55	)	)	PUNCT
cana-1961	134	56	307	307	NUM
cana-1961	134	57	https://internationalpubls.com	https://internationalpubls.com	X
cana-1961	134	58	≤	≤	NUM
cana-1961	134	59	lim	lim	PROPN
cana-1961	134	60	n→∞	n→∞	NUM
cana-1961	134	61	1	1	NUM
cana-1961	134	62	|𝔪|2n	|𝔪|2n	NOUN
cana-1961	134	63	£	£	SYM
cana-1961	134	64	n|𝔪|2nφ(ω1	n|𝔪|2nφ(ω1	NOUN
cana-1961	134	65	,	,	PUNCT
cana-1961	134	66	ω2	ω2	ADJ
cana-1961	134	67	)	)	PUNCT
cana-1961	134	68	≤	≤	NOUN
cana-1961	134	69	lim	lim	PROPN
cana-1961	134	70	n→∞	n→∞	PRON
cana-1961	134	71	£	£	SYM
cana-1961	134	72	n	n	PRON
cana-1961	134	73	φ(ω1	φ(ω1	NOUN
cana-1961	134	74	,	,	PUNCT
cana-1961	134	75	ω2	ω2	NUM
cana-1961	134	76	)	)	PUNCT
cana-1961	134	77	=	=	SYM
cana-1961	134	78	0	0	NUM
cana-1961	134	79	for	for	ADP
cana-1961	134	80	all	all	DET
cana-1961	134	81	ω1	ω1	PROPN
cana-1961	134	82	,	,	PUNCT
cana-1961	134	83	ω2	ω2	NOUN
cana-1961	134	84	∈	∈	PROPN
cana-1961	134	85	𝒲	𝒲	PROPN
cana-1961	134	86	,	,	PUNCT
cana-1961	134	87	υ	υ	PRON
cana-1961	134	88	∈	∈	PROPN
cana-1961	134	89	𝒵	𝒵	PROPN
cana-1961	134	90	,	,	PUNCT
cana-1961	134	91	and	and	CCONJ
cana-1961	134	92	n	n	DET
cana-1961	134	93	∈	∈	PROPN
cana-1961	134	94	ℕ.	ℕ.	PROPN
cana-1961	134	95	so,∥	so,∥	PROPN
cana-1961	134	96	d𝑄2	d𝑄2	PROPN
cana-1961	134	97	́	́	PROPN
cana-1961	134	98	(	(	PUNCT
cana-1961	134	99	ω1	ω1	PROPN
cana-1961	134	100	,	,	PUNCT
cana-1961	134	101	ω2	ω2	NUM
cana-1961	134	102	)	)	PUNCT
cana-1961	134	103	,	,	PUNCT
cana-1961	134	104	υ	υ	PRON
cana-1961	134	105	∥=	∥=	NOUN
cana-1961	134	106	0	0	NUM
cana-1961	134	107	.	.	PUNCT
cana-1961	135	1	thus	thus	ADV
cana-1961	135	2	the	the	DET
cana-1961	135	3	mapping	mapping	NOUN
cana-1961	135	4	𝑄2	𝑄2	PROPN
cana-1961	135	5	́	́	PROPN
cana-1961	135	6	:	:	PUNCT
cana-1961	135	7	𝒲	𝒲	NOUN
cana-1961	135	8	→	→	SYM
cana-1961	135	9	𝒵	𝒵	PROPN
cana-1961	135	10	is	be	AUX
cana-1961	135	11	𝑄2	𝑄2	PROPN
cana-1961	135	12	́	́	PROPN
cana-1961	135	13	as	as	SCONJ
cana-1961	135	14	desired	desire	VERB
cana-1961	135	15	.	.	PUNCT
cana-1961	136	1	corollary	corollary	ADJ
cana-1961	136	2	1	1	NUM
cana-1961	136	3	:	:	PUNCT
cana-1961	136	4	let	let	VERB
cana-1961	136	5	θ	θ	PROPN
cana-1961	136	6	≥	≥	X
cana-1961	136	7	0	0	NUM
cana-1961	136	8	and	and	CCONJ
cana-1961	136	9	τ	τ	X
cana-1961	136	10	=	=	SYM
cana-1961	136	11	s	s	PART
cana-1961	137	1	+	+	NUM
cana-1961	137	2	t	t	AUX
cana-1961	137	3	be	be	VERB
cana-1961	137	4	a	a	DET
cana-1961	137	5	positive	positive	ADJ
cana-1961	137	6	real	real	ADJ
cana-1961	137	7	number	number	NOUN
cana-1961	137	8	with	with	ADP
cana-1961	137	9	τ	τ	PROPN
cana-1961	137	10	<	<	X
cana-1961	137	11	2	2	X
cana-1961	137	12	.	.	PUNCT
cana-1961	138	1	let	let	VERB
cana-1961	138	2	𝔤	𝔤	PRON
cana-1961	138	3	:	:	PUNCT
cana-1961	138	4	𝒲	𝒲	NOUN
cana-1961	138	5	→	→	PUNCT
cana-1961	138	6	𝒵	𝒵	PRON
cana-1961	138	7	be	be	VERB
cana-1961	138	8	an	an	DET
cana-1961	138	9	even	even	ADV
cana-1961	138	10	mapping	mapping	NOUN
cana-1961	138	11	with	with	ADP
cana-1961	138	12	𝔤(0	𝔤(0	PROPN
cana-1961	138	13	)	)	PUNCT
cana-1961	139	1	=	=	SYM
cana-1961	139	2	0	0	PUNCT
cana-1961	139	3	satisfying	satisfy	VERB
cana-1961	139	4	∥	∥	PUNCT
cana-1961	139	5	d𝔤(ω1	d𝔤(ω1	NOUN
cana-1961	139	6	,	,	PUNCT
cana-1961	139	7	ω2	ω2	NUM
cana-1961	139	8	)	)	PUNCT
cana-1961	139	9	,	,	PUNCT
cana-1961	139	10	υ	υ	PROPN
cana-1961	139	11	∥≤	∥≤	PROPN
cana-1961	139	12	θ(∥	θ(∥	PROPN
cana-1961	139	13	ω1	ω1	PROPN
cana-1961	139	14	∥τ	∥τ	PROPN
cana-1961	140	1	+	+	PROPN
cana-1961	140	2	∥	∥	PROPN
cana-1961	140	3	ω2	ω2	NOUN
cana-1961	140	4	∥τ	∥τ	PROPN
cana-1961	141	1	+	+	PROPN
cana-1961	141	2	∥	∥	PROPN
cana-1961	141	3	ω1	ω1	PROPN
cana-1961	141	4	∥s	∥s	PROPN
cana-1961	141	5	.	.	PUNCT
cana-1961	142	1	∥	∥	NOUN
cana-1961	142	2	ω2	ω2	NUM
cana-1961	142	3	∥t	∥t	NOUN
cana-1961	142	4	)	)	PUNCT
cana-1961	142	5	for	for	ADP
cana-1961	142	6	all	all	DET
cana-1961	142	7	ω1	ω1	PROPN
cana-1961	142	8	,	,	PUNCT
cana-1961	142	9	ω2	ω2	NOUN
cana-1961	142	10	∈	∈	PROPN
cana-1961	142	11	𝒲	𝒲	PROPN
cana-1961	142	12	,	,	PUNCT
cana-1961	142	13	υ	υ	PRON
cana-1961	142	14	∈	∈	PROPN
cana-1961	142	15	𝒵	𝒵	PROPN
cana-1961	142	16	,	,	PUNCT
cana-1961	142	17	and	and	CCONJ
cana-1961	142	18	𝑄2	𝑄2	PROPN
cana-1961	143	1	́	́	PROPN
cana-1961	143	2	:	:	PUNCT
cana-1961	143	3	𝒲	𝒲	NOUN
cana-1961	143	4	→	→	SYM
cana-1961	143	5	𝒵	𝒵	PROPN
cana-1961	143	6	is	be	AUX
cana-1961	143	7	a	a	DET
cana-1961	143	8	quadratic	quadratic	ADJ
cana-1961	143	9	mapping	mapping	NOUN
cana-1961	143	10	such	such	ADJ
cana-1961	143	11	that	that	SCONJ
cana-1961	143	12	∥	∥	NUM
cana-1961	143	13	𝔤(ω1	𝔤(ω1	NOUN
cana-1961	143	14	)	)	PUNCT
cana-1961	143	15	−	−	PROPN
cana-1961	143	16	𝑄2	𝑄2	PROPN
cana-1961	144	1	́	́	PROPN
cana-1961	144	2	(	(	PUNCT
cana-1961	144	3	ω1	ω1	PROPN
cana-1961	144	4	)	)	PUNCT
cana-1961	144	5	∥≤	∥≤	PROPN
cana-1961	144	6	|𝔪|τ	|𝔪|τ	PROPN
cana-1961	144	7	|𝔪|2(|𝔪|τ	|𝔪|2(|𝔪|τ	PROPN
cana-1961	144	8	−	−	PROPN
cana-1961	144	9	|𝔪|2	|𝔪|2	PROPN
cana-1961	144	10	)	)	PUNCT
cana-1961	144	11	θ	θ	PROPN
cana-1961	144	12	∥	∥	PUNCT
cana-1961	144	13	ω1	ω1	PROPN
cana-1961	144	14	∥τ	∥τ	PROPN
cana-1961	144	15	|2|	|2|	PROPN
cana-1961	144	16	for	for	ADP
cana-1961	144	17	all	all	DET
cana-1961	144	18	ω1	ω1	PROPN
cana-1961	144	19	∈	∈	PROPN
cana-1961	144	20	𝒲.	𝒲.	PROPN
cana-1961	144	21	proof	proof	NOUN
cana-1961	144	22	:	:	PUNCT
cana-1961	144	23	assuming	assume	VERB
cana-1961	144	24	φ(ω1	φ(ω1	NOUN
cana-1961	144	25	,	,	PUNCT
cana-1961	144	26	ω2	ω2	ADJ
cana-1961	144	27	)	)	PUNCT
cana-1961	144	28	∶=	∶=	NUM
cana-1961	144	29	θ	θ	NOUN
cana-1961	144	30	(	(	PUNCT
cana-1961	144	31	‖ω1‖τ	‖ω1‖τ	X
cana-1961	144	32	+	+	CCONJ
cana-1961	144	33	‖ω2‖τ+∥	‖ω2‖τ+∥	PROPN
cana-1961	144	34	ω1	ω1	PROPN
cana-1961	144	35	∥s	∥s	PROPN
cana-1961	144	36	.	.	PUNCT
cana-1961	145	1	∥	∥	NOUN
cana-1961	145	2	ω2	ω2	NUM
cana-1961	145	3	∥t	∥t	NOUN
cana-1961	145	4	)	)	PUNCT
cana-1961	145	5	for	for	ADP
cana-1961	145	6	all	all	DET
cana-1961	145	7	ω1	ω1	PROPN
cana-1961	145	8	,	,	PUNCT
cana-1961	145	9	ω2	ω2	NOUN
cana-1961	145	10	∈	∈	PROPN
cana-1961	145	11	𝒲	𝒲	PROPN
cana-1961	145	12	,	,	PUNCT
cana-1961	145	13	and	and	CCONJ
cana-1961	145	14	by	by	ADP
cana-1961	145	15	choosing	choose	VERB
cana-1961	145	16	£	£	PROPN
cana-1961	145	17	=	=	SYM
cana-1961	145	18	|𝔪|2−τ	|𝔪|2−τ	PROPN
cana-1961	145	19	,	,	PUNCT
cana-1961	145	20	the	the	DET
cana-1961	145	21	expected	expect	VERB
cana-1961	145	22	result	result	NOUN
cana-1961	145	23	can	can	AUX
cana-1961	145	24	be	be	AUX
cana-1961	145	25	obtained	obtain	VERB
cana-1961	145	26	by	by	ADP
cana-1961	145	27	theorem	theorem	ADJ
cana-1961	145	28	2	2	NUM
cana-1961	145	29	.	.	PUNCT
cana-1961	145	30	theorem	theorem	NOUN
cana-1961	145	31	3	3	NUM
cana-1961	145	32	:	:	PUNCT
cana-1961	145	33	let	let	VERB
cana-1961	145	34	φ	φ	NUM
cana-1961	145	35	:	:	PUNCT
cana-1961	145	36	𝒲	𝒲	NOUN
cana-1961	145	37	×	×	NOUN
cana-1961	145	38	𝒲	𝒲	NOUN
cana-1961	145	39	→	→	SYM
cana-1961	145	40	[	[	X
cana-1961	145	41	0	0	NUM
cana-1961	145	42	,	,	PUNCT
cana-1961	145	43	∞	∞	PROPN
cana-1961	145	44	)	)	PUNCT
cana-1961	145	45	be	be	VERB
cana-1961	145	46	an	an	DET
cana-1961	145	47	even	even	ADV
cana-1961	145	48	function	function	NOUN
cana-1961	145	49	such	such	ADJ
cana-1961	145	50	that	that	SCONJ
cana-1961	145	51	there	there	PRON
cana-1961	145	52	is	be	VERB
cana-1961	145	53	a	a	DET
cana-1961	145	54	constant	constant	ADJ
cana-1961	145	55	0	0	NUM
cana-1961	145	56	<	<	X
cana-1961	145	57	£	£	X
cana-1961	145	58	<	<	X
cana-1961	145	59	1	1	NUM
cana-1961	145	60	with	with	ADP
cana-1961	145	61	φ	φ	PROPN
cana-1961	145	62	(	(	PUNCT
cana-1961	145	63	ω1	ω1	PROPN
cana-1961	145	64	𝔪	𝔪	NOUN
cana-1961	145	65	,	,	PUNCT
cana-1961	145	66	ω2	ω2	ADJ
cana-1961	145	67	𝔪	𝔪	NOUN
cana-1961	145	68	)	)	PUNCT
cana-1961	146	1	≤	≤	NUM
cana-1961	147	1	£	£	SYM
cana-1961	147	2	|𝔪|2	|𝔪|2	NOUN
cana-1961	147	3	φ(ω1	φ(ω1	NOUN
cana-1961	147	4	,	,	PUNCT
cana-1961	147	5	ω2	ω2	NUM
cana-1961	147	6	)	)	PUNCT
cana-1961	147	7	for	for	ADP
cana-1961	147	8	all	all	DET
cana-1961	147	9	ω1	ω1	PROPN
cana-1961	147	10	,	,	PUNCT
cana-1961	147	11	ω2	ω2	NOUN
cana-1961	147	12	∈	∈	PROPN
cana-1961	147	13	𝒲.	𝒲.	PROPN
cana-1961	147	14	let	let	VERB
cana-1961	147	15	𝔤	𝔤	PRON
cana-1961	147	16	:	:	PUNCT
cana-1961	147	17	𝒲	𝒲	NOUN
cana-1961	147	18	→	→	PUNCT
cana-1961	147	19	𝒵	𝒵	PRON
cana-1961	147	20	be	be	VERB
cana-1961	147	21	an	an	DET
cana-1961	147	22	even	even	ADV
cana-1961	147	23	mapping	mapping	NOUN
cana-1961	147	24	satisfying	satisfying	ADJ
cana-1961	147	25	eq	eq	ADP
cana-1961	147	26	2	2	NUM
cana-1961	147	27	.	.	PUNCT
cana-1961	148	1	then	then	ADV
cana-1961	148	2	there	there	PRON
cana-1961	148	3	is	be	VERB
cana-1961	148	4	a	a	DET
cana-1961	148	5	unique	unique	ADJ
cana-1961	148	6	quadratic	quadratic	ADJ
cana-1961	148	7	mapping	mapping	NOUN
cana-1961	148	8	𝑄2	𝑄2	PROPN
cana-1961	149	1	́	́	NOUN
cana-1961	149	2	:	:	PUNCT
cana-1961	149	3	𝒲	𝒲	NOUN
cana-1961	149	4	→	→	SYM
cana-1961	149	5	𝒵	𝒵	NOUN
cana-1961	149	6	such	such	ADJ
cana-1961	149	7	that	that	DET
cana-1961	149	8	‖𝔤(ω1	‖𝔤(ω1	NOUN
cana-1961	149	9	)	)	PUNCT
cana-1961	149	10	−	−	PROPN
cana-1961	149	11	𝑄2	𝑄2	PROPN
cana-1961	150	1	́	́	PROPN
cana-1961	150	2	(	(	PUNCT
cana-1961	150	3	ω1	ω1	PROPN
cana-1961	150	4	)	)	PUNCT
cana-1961	150	5	,	,	PUNCT
cana-1961	150	6	υ‖	υ‖	VERB
cana-1961	150	7	≤	≤	NUM
cana-1961	150	8	1	1	NUM
cana-1961	150	9	|2|	|2|	PROPN
cana-1961	150	10	1	1	NUM
cana-1961	150	11	|𝔪|2	|𝔪|2	PROPN
cana-1961	150	12	£	£	PROPN
cana-1961	150	13	(	(	PUNCT
cana-1961	150	14	1−£	1−£	NUM
cana-1961	150	15	)	)	PUNCT
cana-1961	150	16	φ(0	φ(0	ADJ
cana-1961	150	17	,	,	PUNCT
cana-1961	150	18	ω1	ω1	PROPN
cana-1961	150	19	)	)	PUNCT
cana-1961	150	20	14	14	NUM
cana-1961	150	21	for	for	ADP
cana-1961	150	22	all	all	DET
cana-1961	150	23	ω1	ω1	PROPN
cana-1961	150	24	∈	∈	PROPN
cana-1961	150	25	𝒲	𝒲	PROPN
cana-1961	150	26	,	,	PUNCT
cana-1961	150	27	υ	υ	DET
cana-1961	150	28	∈	∈	NOUN
cana-1961	150	29	𝒵.	𝒵.	NOUN
cana-1961	150	30	proof	proof	NOUN
cana-1961	150	31	:	:	PUNCT
cana-1961	150	32	putting	put	VERB
cana-1961	150	33	ω1	ω1	PROPN
cana-1961	150	34	=	=	PUNCT
cana-1961	150	35	0	0	NUM
cana-1961	150	36	in	in	ADP
cana-1961	150	37	eq	eq	NOUN
cana-1961	150	38	2	2	NUM
cana-1961	150	39	implies	imply	VERB
cana-1961	150	40	‖2𝔤(𝔪ω2	‖2𝔤(𝔪ω2	NUM
cana-1961	150	41	)	)	PUNCT
cana-1961	150	42	−	−	PROPN
cana-1961	151	1	2𝔪2𝔤(ω2	2𝔪2𝔤(ω2	NUM
cana-1961	151	2	)	)	PUNCT
cana-1961	151	3	,	,	PUNCT
cana-1961	151	4	υ‖	υ‖	VERB
cana-1961	151	5	≤	≤	ADJ
cana-1961	151	6	φ(0	φ(0	ADJ
cana-1961	151	7	,	,	PUNCT
cana-1961	151	8	ω2	ω2	NUM
cana-1961	151	9	)	)	PUNCT
cana-1961	151	10	,	,	PUNCT
cana-1961	151	11	15	15	NUM
cana-1961	151	12	for	for	ADP
cana-1961	151	13	all	all	DET
cana-1961	151	14	ω2	ω2	ADJ
cana-1961	151	15	∈	∈	PROPN
cana-1961	151	16	𝒲	𝒲	PROPN
cana-1961	151	17	,	,	PUNCT
cana-1961	151	18	υ	υ	DET
cana-1961	151	19	∈	∈	PROPN
cana-1961	151	20	𝒵.	𝒵.	PROPN
cana-1961	151	21	replacing	replacing	NOUN
cana-1961	151	22	ω2	ω2	ADV
cana-1961	151	23	by	by	ADP
cana-1961	151	24	(	(	PUNCT
cana-1961	151	25	ω1	ω1	PROPN
cana-1961	151	26	𝔪	𝔪	NOUN
cana-1961	151	27	)	)	PUNCT
cana-1961	151	28	in	in	ADP
cana-1961	151	29	eq	eq	ADP
cana-1961	151	30	15	15	NUM
cana-1961	151	31	and	and	CCONJ
cana-1961	151	32	dividing	divide	VERB
cana-1961	151	33	on	on	ADP
cana-1961	151	34	both	both	DET
cana-1961	151	35	sides	side	NOUN
cana-1961	151	36	of	of	ADP
cana-1961	151	37	eq	eq	NOUN
cana-1961	151	38	6	6	NUM
cana-1961	151	39	by	by	ADP
cana-1961	151	40	2	2	NUM
cana-1961	151	41	,	,	PUNCT
cana-1961	151	42	it	it	PRON
cana-1961	151	43	gives	give	VERB
cana-1961	151	44	‖𝔤(ω1	‖𝔤(ω1	NOUN
cana-1961	151	45	)	)	PUNCT
cana-1961	151	46	−	−	PROPN
cana-1961	151	47	𝔪2𝔤	𝔪2𝔤	PUNCT
cana-1961	151	48	(	(	PUNCT
cana-1961	151	49	ω1	ω1	PROPN
cana-1961	151	50	𝔪	𝔪	NOUN
cana-1961	151	51	)	)	PUNCT
cana-1961	151	52	,	,	PUNCT
cana-1961	151	53	υ‖	υ‖	VERB
cana-1961	151	54	≤	≤	NOUN
cana-1961	151	55	1	1	NUM
cana-1961	151	56	|2|	|2|	PROPN
cana-1961	151	57	φ	φ	X
cana-1961	151	58	(	(	PUNCT
cana-1961	151	59	0	0	NUM
cana-1961	151	60	,	,	PUNCT
cana-1961	151	61	ω1	ω1	ADJ
cana-1961	151	62	𝔪	𝔪	NOUN
cana-1961	151	63	)	)	PUNCT
cana-1961	151	64	,	,	PUNCT
cana-1961	151	65	16	16	NUM
cana-1961	151	66	for	for	ADP
cana-1961	151	67	all	all	DET
cana-1961	151	68	ω1	ω1	PROPN
cana-1961	151	69	∈	∈	PROPN
cana-1961	151	70	𝒲	𝒲	PROPN
cana-1961	151	71	,	,	PUNCT
cana-1961	151	72	υ	υ	PRON
cana-1961	151	73	∈	∈	PROPN
cana-1961	151	74	𝒵.	𝒵.	PROPN
cana-1961	151	75	let	let	VERB
cana-1961	151	76	(	(	PUNCT
cana-1961	151	77	γ	γ	X
cana-1961	151	78	,	,	PUNCT
cana-1961	151	79	ð	ð	NUM
cana-1961	151	80	)	)	PUNCT
cana-1961	151	81	be	be	VERB
cana-1961	151	82	the	the	DET
cana-1961	151	83	gms	gms	NOUN
cana-1961	151	84	as	as	SCONJ
cana-1961	151	85	defined	define	VERB
cana-1961	151	86	by	by	ADP
cana-1961	151	87	theorem	theorem	NOUN
cana-1961	151	88	2	2	NUM
cana-1961	151	89	.	.	PUNCT
cana-1961	152	1	now	now	ADV
cana-1961	152	2	,	,	PUNCT
cana-1961	152	3	define	define	VERB
cana-1961	152	4	the	the	DET
cana-1961	152	5	function	function	NOUN
cana-1961	152	6	λ	λ	PROPN
cana-1961	152	7	:	:	PUNCT
cana-1961	152	8	γ	γ	X
cana-1961	152	9	→	→	SYM
cana-1961	152	10	γ	γ	X
cana-1961	152	11	such	such	ADJ
cana-1961	152	12	that	that	PRON
cana-1961	152	13	λ𝔣(ω1	λ𝔣(ω1	ADP
cana-1961	152	14	)	)	PUNCT
cana-1961	152	15	=	=	SYM
cana-1961	152	16	𝔪2𝔣	𝔪2𝔣	PROPN
cana-1961	152	17	(	(	PUNCT
cana-1961	152	18	ω1	ω1	PROPN
cana-1961	152	19	𝔪	𝔪	NOUN
cana-1961	152	20	)	)	PUNCT
cana-1961	152	21	17	17	NUM
cana-1961	152	22	for	for	ADP
cana-1961	152	23	all	all	DET
cana-1961	152	24	ω1	ω1	PROPN
cana-1961	152	25	∈	∈	PROPN
cana-1961	152	26	𝒲.	𝒲.	PROPN
cana-1961	153	1	so	so	SCONJ
cana-1961	153	2	ð(𝔤	ð(𝔤	PROPN
cana-1961	153	3	,	,	PUNCT
cana-1961	153	4	λ𝔤	λ𝔤	NOUN
cana-1961	153	5	)	)	PUNCT
cana-1961	153	6	≤	≤	NOUN
cana-1961	153	7	£	£	NUM
cana-1961	153	8	|2||𝔪|2	|2||𝔪|2	NOUN
cana-1961	153	9	.	.	PUNCT
cana-1961	154	1	according	accord	VERB
cana-1961	154	2	to	to	ADP
cana-1961	154	3	eq	eq	PROPN
cana-1961	154	4	16	16	NUM
cana-1961	154	5	ð(𝔤	ð(𝔤	PROPN
cana-1961	154	6	,	,	PUNCT
cana-1961	154	7	𝑄2	𝑄2	PROPN
cana-1961	154	8	́	́	PROPN
cana-1961	154	9	)	)	PUNCT
cana-1961	154	10	≤	≤	ADV
cana-1961	154	11	1	1	NUM
cana-1961	154	12	|2|	|2|	PROPN
cana-1961	154	13	.	.	PUNCT
cana-1961	155	1	£	£	SYM
cana-1961	155	2	|𝔪|2(1	|𝔪|2(1	NOUN
cana-1961	155	3	−	−	ADP
cana-1961	155	4	£	£	NOUN
cana-1961	155	5	)	)	PUNCT
cana-1961	155	6	communications	communication	NOUN
cana-1961	155	7	on	on	ADP
cana-1961	155	8	applied	apply	VERB
cana-1961	155	9	nonlinear	nonlinear	ADJ
cana-1961	155	10	analysis	analysis	NOUN
cana-1961	155	11	issn	issn	NOUN
cana-1961	155	12	:	:	PUNCT
cana-1961	155	13	1074	1074	NUM
cana-1961	155	14	-	-	PUNCT
cana-1961	155	15	133x	133x	NUM
cana-1961	155	16	vol	vol	NOUN
cana-1961	155	17	32	32	NUM
cana-1961	155	18	no	no	NOUN
cana-1961	155	19	.	.	NOUN
cana-1961	155	20	3	3	NUM
cana-1961	155	21	(	(	PUNCT
cana-1961	155	22	2025	2025	NUM
cana-1961	155	23	)	)	PUNCT
cana-1961	155	24	308	308	NUM
cana-1961	155	25	https://internationalpubls.com	https://internationalpubls.com	X
cana-1961	155	26	for	for	ADP
cana-1961	155	27	all	all	DET
cana-1961	155	28	ω1	ω1	PROPN
cana-1961	155	29	∈	∈	PROPN
cana-1961	155	30	𝒲.	𝒲.	PROPN
cana-1961	155	31	this	this	DET
cana-1961	155	32	proof	proof	NOUN
cana-1961	155	33	follows	follow	VERB
cana-1961	155	34	the	the	DET
cana-1961	155	35	same	same	ADJ
cana-1961	155	36	pattern	pattern	NOUN
cana-1961	155	37	as	as	ADP
cana-1961	155	38	theorem	theorem	ADJ
cana-1961	155	39	2	2	NUM
cana-1961	155	40	.	.	PUNCT
cana-1961	155	41	corollary	corollary	ADJ
cana-1961	155	42	2	2	NUM
cana-1961	155	43	:	:	PUNCT
cana-1961	155	44	let	let	VERB
cana-1961	155	45	θ	θ	PROPN
cana-1961	155	46	≥	≥	X
cana-1961	155	47	0	0	NUM
cana-1961	155	48	and	and	CCONJ
cana-1961	155	49	τ	τ	X
cana-1961	155	50	=	=	SYM
cana-1961	155	51	s	s	PART
cana-1961	156	1	+	+	NUM
cana-1961	156	2	t	t	AUX
cana-1961	156	3	be	be	VERB
cana-1961	156	4	a	a	DET
cana-1961	156	5	positive	positive	ADJ
cana-1961	156	6	real	real	ADJ
cana-1961	156	7	number	number	NOUN
cana-1961	156	8	with	with	ADP
cana-1961	156	9	τ	τ	PROPN
cana-1961	156	10	>	>	X
cana-1961	156	11	2	2	X
cana-1961	156	12	.	.	PUNCT
cana-1961	157	1	let	let	VERB
cana-1961	157	2	𝔤	𝔤	PRON
cana-1961	157	3	:	:	PUNCT
cana-1961	157	4	𝒲	𝒲	NOUN
cana-1961	157	5	→	→	PUNCT
cana-1961	157	6	𝒵	𝒵	PRON
cana-1961	157	7	be	be	VERB
cana-1961	157	8	an	an	DET
cana-1961	157	9	even	even	ADV
cana-1961	157	10	mapping	mapping	NOUN
cana-1961	157	11	with	with	ADP
cana-1961	157	12	𝔤(0	𝔤(0	PROPN
cana-1961	157	13	)	)	PUNCT
cana-1961	158	1	=	=	SYM
cana-1961	158	2	0	0	PUNCT
cana-1961	158	3	satisfying	satisfy	VERB
cana-1961	158	4	∥	∥	PUNCT
cana-1961	158	5	d𝔤(ω1	d𝔤(ω1	NOUN
cana-1961	158	6	,	,	PUNCT
cana-1961	158	7	ω2	ω2	NUM
cana-1961	158	8	)	)	PUNCT
cana-1961	158	9	,	,	PUNCT
cana-1961	158	10	υ	υ	PROPN
cana-1961	158	11	∥≤	∥≤	PROPN
cana-1961	158	12	θ(∥	θ(∥	PROPN
cana-1961	158	13	ω1	ω1	PROPN
cana-1961	158	14	∥τ	∥τ	PROPN
cana-1961	159	1	+	+	PROPN
cana-1961	159	2	∥	∥	PROPN
cana-1961	159	3	ω2	ω2	NOUN
cana-1961	159	4	∥τ	∥τ	PROPN
cana-1961	160	1	+	+	PROPN
cana-1961	160	2	∥	∥	PROPN
cana-1961	160	3	ω1	ω1	PROPN
cana-1961	160	4	∥s	∥s	PROPN
cana-1961	160	5	.	.	PUNCT
cana-1961	161	1	∥	∥	NOUN
cana-1961	161	2	ω2	ω2	NUM
cana-1961	161	3	∥t	∥t	NOUN
cana-1961	161	4	)	)	PUNCT
cana-1961	161	5	,	,	PUNCT
cana-1961	161	6	for	for	ADP
cana-1961	161	7	all	all	DET
cana-1961	161	8	ω1	ω1	PROPN
cana-1961	161	9	∈	∈	PROPN
cana-1961	161	10	𝒲	𝒲	PROPN
cana-1961	161	11	,	,	PUNCT
cana-1961	161	12	υ	υ	PRON
cana-1961	161	13	∈	∈	PROPN
cana-1961	161	14	𝒵.	𝒵.	PROPN
cana-1961	161	15	then	then	ADV
cana-1961	161	16	there	there	PRON
cana-1961	161	17	is	be	VERB
cana-1961	161	18	a	a	DET
cana-1961	161	19	unique	unique	ADJ
cana-1961	161	20	𝑄2	𝑄2	NOUN
cana-1961	161	21	́	́	PUNCT
cana-1961	162	1	mapping	map	VERB
cana-1961	162	2	𝑄2	𝑄2	PROPN
cana-1961	162	3	́	́	PROPN
cana-1961	162	4	:	:	PUNCT
cana-1961	162	5	𝒲	𝒲	NOUN
cana-1961	162	6	→	→	SYM
cana-1961	162	7	𝒵	𝒵	NOUN
cana-1961	162	8	such	such	ADJ
cana-1961	162	9	that	that	SCONJ
cana-1961	162	10	∥	∥	NUM
cana-1961	162	11	𝔤(ω1	𝔤(ω1	NOUN
cana-1961	162	12	)	)	PUNCT
cana-1961	162	13	−	−	PROPN
cana-1961	162	14	𝑄2	𝑄2	PROPN
cana-1961	163	1	́	́	PROPN
cana-1961	163	2	(	(	PUNCT
cana-1961	163	3	ω1	ω1	PROPN
cana-1961	163	4	)	)	PUNCT
cana-1961	163	5	,	,	PUNCT
cana-1961	163	6	υ	υ	PROPN
cana-1961	163	7	∥≤	∥≤	PROPN
cana-1961	163	8	|𝔪|τ	|𝔪|τ	ADV
cana-1961	163	9	|𝔪|2(|𝔪|2	|𝔪|2(|𝔪|2	VERB
cana-1961	163	10	−	−	PROPN
cana-1961	163	11	|𝔪|τ	|𝔪|τ	NUM
cana-1961	163	12	)	)	PUNCT
cana-1961	163	13	θ	θ	PROPN
cana-1961	163	14	∥	∥	PUNCT
cana-1961	163	15	ω1	ω1	PROPN
cana-1961	163	16	∥τ	∥τ	PROPN
cana-1961	163	17	|2|	|2|	PROPN
cana-1961	163	18	for	for	ADP
cana-1961	163	19	all	all	DET
cana-1961	163	20	ω1	ω1	PROPN
cana-1961	163	21	∈	∈	PROPN
cana-1961	163	22	𝒲	𝒲	PROPN
cana-1961	163	23	,	,	PUNCT
cana-1961	163	24	υ	υ	DET
cana-1961	163	25	∈	∈	NOUN
cana-1961	163	26	𝒵.	𝒵.	PROPN
cana-1961	163	27	proof	proof	NOUN
cana-1961	163	28	:	:	PUNCT
cana-1961	163	29	assuming	assume	VERB
cana-1961	163	30	φ(ω1	φ(ω1	NOUN
cana-1961	163	31	,	,	PUNCT
cana-1961	163	32	ω2	ω2	ADJ
cana-1961	163	33	)	)	PUNCT
cana-1961	163	34	∶=	∶=	NUM
cana-1961	163	35	θ	θ	NOUN
cana-1961	163	36	(	(	PUNCT
cana-1961	163	37	‖ω1‖τ	‖ω1‖τ	X
cana-1961	163	38	+	+	CCONJ
cana-1961	163	39	‖ω2‖τ+∥	‖ω2‖τ+∥	PROPN
cana-1961	163	40	ω1	ω1	PROPN
cana-1961	163	41	∥s	∥s	PROPN
cana-1961	163	42	.	.	PUNCT
cana-1961	164	1	∥	∥	NOUN
cana-1961	164	2	ω2	ω2	NUM
cana-1961	164	3	∥t	∥t	NOUN
cana-1961	164	4	)	)	PUNCT
cana-1961	164	5	for	for	ADP
cana-1961	164	6	all	all	DET
cana-1961	164	7	ω1	ω1	PROPN
cana-1961	164	8	,	,	PUNCT
cana-1961	164	9	ω2	ω2	NOUN
cana-1961	164	10	∈	∈	PROPN
cana-1961	164	11	𝒲	𝒲	PROPN
cana-1961	164	12	,	,	PUNCT
cana-1961	164	13	and	and	CCONJ
cana-1961	164	14	by	by	ADP
cana-1961	164	15	choosing	choose	VERB
cana-1961	164	16	£	£	PROPN
cana-1961	164	17	=	=	PUNCT
cana-1961	164	18	|𝔪|τ−2	|𝔪|τ−2	NOUN
cana-1961	164	19	,	,	PUNCT
cana-1961	164	20	the	the	DET
cana-1961	164	21	expected	expect	VERB
cana-1961	164	22	result	result	NOUN
cana-1961	164	23	can	can	AUX
cana-1961	164	24	be	be	AUX
cana-1961	164	25	obtained	obtain	VERB
cana-1961	164	26	by	by	ADP
cana-1961	164	27	theorem	theorem	ADJ
cana-1961	164	28	3	3	NUM
cana-1961	164	29	.	.	NOUN
cana-1961	164	30	example	example	NOUN
cana-1961	165	1	3	3	NUM
cana-1961	165	2	:	:	PUNCT
cana-1961	165	3	let	let	VERB
cana-1961	165	4	ρ	ρ	PROPN
cana-1961	165	5	>	>	X
cana-1961	165	6	2	2	NUM
cana-1961	165	7	be	be	AUX
cana-1961	165	8	a	a	DET
cana-1961	165	9	prime	prime	ADJ
cana-1961	165	10	number	number	NOUN
cana-1961	165	11	and	and	CCONJ
cana-1961	165	12	𝒲	𝒲	NOUN
cana-1961	165	13	=	=	SYM
cana-1961	165	14	𝒵	𝒵	NOUN
cana-1961	165	15	=	=	SYM
cana-1961	165	16	ℚ	ℚ	PROPN
cana-1961	165	17	𝔭	𝔭	NOUN
cana-1961	165	18	.	.	PUNCT
cana-1961	166	1	define	define	VERB
cana-1961	166	2	𝔤	𝔤	NOUN
cana-1961	166	3	:	:	PUNCT
cana-1961	166	4	𝒲	𝒲	NOUN
cana-1961	166	5	→	→	SYM
cana-1961	166	6	𝒵	𝒵	PROPN
cana-1961	166	7	by	by	ADP
cana-1961	166	8	𝔤(ω_1	𝔤(ω_1	NOUN
cana-1961	166	9	)	)	PUNCT
cana-1961	166	10	=	=	SYM
cana-1961	166	11	ω1	ω1	PROPN
cana-1961	166	12	2	2	NUM
cana-1961	166	13	+	+	CCONJ
cana-1961	166	14	1	1	NUM
cana-1961	166	15	for	for	ADP
cana-1961	166	16	all	all	DET
cana-1961	166	17	ω1	ω1	PROPN
cana-1961	166	18	∈	∈	PROPN
cana-1961	166	19	𝒲.	𝒲.	PROPN
cana-1961	166	20	since	since	SCONJ
cana-1961	166	21	|2n|ρ	|2n|ρ	NOUN
cana-1961	166	22	=	=	SYM
cana-1961	166	23	1	1	NUM
cana-1961	166	24	.	.	PUNCT
cana-1961	166	25	|d𝔤(ω1	|d𝔤(ω1	NOUN
cana-1961	166	26	,	,	PUNCT
cana-1961	166	27	ω2)|	ω2)|	NOUN
cana-1961	166	28	=	=	SYM
cana-1961	167	1	|	|	ADV
cana-1961	167	2	88	88	NUM
cana-1961	167	3	9	9	NUM
cana-1961	167	4	|	|	ADV
cana-1961	167	5	≤	≤	PUNCT
cana-1961	168	1	θ(‖ω1‖τ	θ(‖ω1‖τ	PROPN
cana-1961	168	2	+	+	CCONJ
cana-1961	168	3	‖ω2‖τ+∥	‖ω2‖τ+∥	PROPN
cana-1961	168	4	ω1	ω1	PROPN
cana-1961	168	5	∥s	∥s	PROPN
cana-1961	168	6	.	.	PUNCT
cana-1961	169	1	∥	∥	NOUN
cana-1961	169	2	ω2	ω2	NUM
cana-1961	169	3	∥t	∥t	NOUN
cana-1961	169	4	)	)	PUNCT
cana-1961	169	5	(	(	PUNCT
cana-1961	169	6	∀	∀	X
cana-1961	169	7	ω1	ω1	PROPN
cana-1961	169	8	,	,	PUNCT
cana-1961	169	9	ω2	ω2	NOUN
cana-1961	169	10	∈	∈	PROPN
cana-1961	169	11	𝒲	𝒲	PROPN
cana-1961	169	12	)	)	PUNCT
cana-1961	169	13	and	and	CCONJ
cana-1961	169	14	‖	‖	PROPN
cana-1961	169	15	h(2	h(2	PROPN
cana-1961	169	16	n	n	SYM
cana-1961	169	17	ω1	ω1	NOUN
cana-1961	169	18	)	)	PUNCT
cana-1961	169	19	2	2	NUM
cana-1961	169	20	2n	2n	NUM
cana-1961	169	21	−	−	PROPN
cana-1961	169	22	h(2	h(2	PROPN
cana-1961	169	23	n−1	n−1	PROPN
cana-1961	169	24	ω1	ω1	PROPN
cana-1961	169	25	)	)	PUNCT
cana-1961	169	26	2	2	NUM
cana-1961	169	27	2(n−1	2(n−1	NOUN
cana-1961	169	28	)	)	PUNCT
cana-1961	169	29	‖	‖	PROPN
cana-1961	169	30	=	=	SYM
cana-1961	170	1	|9|	|9|	NOUN
cana-1961	170	2	≠	≠	PROPN
cana-1961	170	3	0	0	X
cana-1961	170	4	.	.	PUNCT
cana-1961	171	1	hence	hence	ADV
cana-1961	171	2	{	{	PUNCT
cana-1961	171	3	2	2	NUM
cana-1961	171	4	−2n	−2n	PROPN
cana-1961	171	5	h(2	h(2	PROPN
cana-1961	171	6	n	n	SYM
cana-1961	171	7	ω1	ω1	NOUN
cana-1961	171	8	)	)	PUNCT
cana-1961	171	9	}	}	PUNCT
cana-1961	171	10	is	be	AUX
cana-1961	171	11	not	not	PART
cana-1961	171	12	a	a	DET
cana-1961	171	13	cauchy	cauchy	ADJ
cana-1961	171	14	sequence	sequence	NOUN
cana-1961	171	15	.	.	PUNCT
cana-1961	172	1	where	where	SCONJ
cana-1961	172	2	h(ω1)=𝔤(2ω1	h(ω1)=𝔤(2ω1	NOUN
cana-1961	172	3	)	)	PUNCT
cana-1961	172	4	−	−	NOUN
cana-1961	172	5	4𝔤(ω1	4𝔤(ω1	NUM
cana-1961	172	6	)	)	PUNCT
cana-1961	172	7	.	.	PUNCT
cana-1961	173	1	stability	stability	NOUN
cana-1961	173	2	of	of	ADP
cana-1961	173	3	the	the	DET
cana-1961	173	4	fe	fe	NOUN
cana-1961	173	5	eq	eq	ADP
cana-1961	173	6	2	2	NUM
cana-1961	173	7	:	:	PUNCT
cana-1961	173	8	odd	odd	ADJ
cana-1961	173	9	case	case	NOUN
cana-1961	173	10	theorem	theorem	VERB
cana-1961	173	11	4	4	NUM
cana-1961	173	12	:	:	PUNCT
cana-1961	173	13	let	let	VERB
cana-1961	173	14	φ	φ	NUM
cana-1961	173	15	:	:	PUNCT
cana-1961	173	16	𝒲	𝒲	NOUN
cana-1961	173	17	×	×	NOUN
cana-1961	173	18	𝒲	𝒲	NOUN
cana-1961	173	19	→	→	SYM
cana-1961	173	20	[	[	X
cana-1961	173	21	0	0	NUM
cana-1961	173	22	,	,	PUNCT
cana-1961	173	23	∞	∞	PROPN
cana-1961	173	24	)	)	PUNCT
cana-1961	173	25	be	be	VERB
cana-1961	173	26	an	an	DET
cana-1961	173	27	odd	odd	ADJ
cana-1961	173	28	function	function	NOUN
cana-1961	173	29	such	such	ADJ
cana-1961	173	30	that	that	SCONJ
cana-1961	173	31	there	there	PRON
cana-1961	173	32	is	be	VERB
cana-1961	173	33	a	a	DET
cana-1961	173	34	constant	constant	ADJ
cana-1961	173	35	0	0	NUM
cana-1961	173	36	<	<	X
cana-1961	173	37	£	£	SYM
cana-1961	173	38	<	<	X
cana-1961	173	39	1	1	NUM
cana-1961	173	40	with	with	ADP
cana-1961	173	41	φ(𝔪ω1	φ(𝔪ω1	NOUN
cana-1961	173	42	,	,	PUNCT
cana-1961	173	43	𝔪ω2	𝔪ω2	NOUN
cana-1961	173	44	)	)	PUNCT
cana-1961	173	45	≤	≤	NOUN
cana-1961	174	1	|𝔪|3£φ(ω1	|𝔪|3£φ(ω1	ADV
cana-1961	174	2	,	,	PUNCT
cana-1961	174	3	ω2	ω2	NUM
cana-1961	174	4	)	)	PUNCT
cana-1961	174	5	18	18	NUM
cana-1961	174	6	for	for	ADP
cana-1961	174	7	all	all	DET
cana-1961	174	8	ω1	ω1	PROPN
cana-1961	174	9	,	,	PUNCT
cana-1961	174	10	ω2	ω2	NOUN
cana-1961	174	11	∈	∈	PROPN
cana-1961	174	12	𝒲.	𝒲.	PROPN
cana-1961	174	13	let	let	VERB
cana-1961	174	14	𝔤	𝔤	PRON
cana-1961	174	15	:	:	PUNCT
cana-1961	174	16	𝒲	𝒲	NOUN
cana-1961	174	17	→	→	PUNCT
cana-1961	174	18	𝒵	𝒵	PRON
cana-1961	174	19	be	be	VERB
cana-1961	174	20	an	an	DET
cana-1961	174	21	odd	odd	ADJ
cana-1961	174	22	mapping	mapping	NOUN
cana-1961	174	23	satisfying	satisfy	VERB
cana-1961	174	24	‖d𝔤(ω1	‖d𝔤(ω1	NOUN
cana-1961	174	25	,	,	PUNCT
cana-1961	174	26	ω2	ω2	NUM
cana-1961	174	27	)	)	PUNCT
cana-1961	174	28	,	,	PUNCT
cana-1961	174	29	υ‖	υ‖	VERB
cana-1961	174	30	≤	≤	NOUN
cana-1961	174	31	φ(ω1	φ(ω1	X
cana-1961	174	32	,	,	PUNCT
cana-1961	174	33	ω2	ω2	NUM
cana-1961	174	34	)	)	PUNCT
cana-1961	174	35	19	19	NUM
cana-1961	174	36	for	for	ADP
cana-1961	174	37	all	all	DET
cana-1961	174	38	ω1	ω1	PROPN
cana-1961	174	39	,	,	PUNCT
cana-1961	174	40	ω2	ω2	NOUN
cana-1961	174	41	∈	∈	PROPN
cana-1961	174	42	𝒲	𝒲	PROPN
cana-1961	174	43	,	,	PUNCT
cana-1961	174	44	υ	υ	PRON
cana-1961	174	45	∈	∈	PROPN
cana-1961	174	46	𝒵.	𝒵.	PROPN
cana-1961	174	47	then	then	ADV
cana-1961	174	48	there	there	PRON
cana-1961	174	49	is	be	VERB
cana-1961	174	50	a	a	DET
cana-1961	174	51	unique	unique	ADJ
cana-1961	174	52	c3	c3	NOUN
cana-1961	174	53	́	́	PUNCT
cana-1961	174	54	mapping	map	VERB
cana-1961	174	55	c3	c3	NOUN
cana-1961	174	56	́	́	PROPN
cana-1961	174	57	:	:	PUNCT
cana-1961	174	58	𝒲	𝒲	NOUN
cana-1961	174	59	→	→	SYM
cana-1961	174	60	𝒵	𝒵	NOUN
cana-1961	174	61	such	such	ADJ
cana-1961	174	62	that	that	DET
cana-1961	174	63	‖𝔤(ω1	‖𝔤(ω1	NOUN
cana-1961	174	64	)	)	PUNCT
cana-1961	175	1	−	−	PROPN
cana-1961	176	1	c3	c3	PROPN
cana-1961	176	2	́	́	PROPN
cana-1961	176	3	(	(	PUNCT
cana-1961	176	4	ω1	ω1	PROPN
cana-1961	176	5	)	)	PUNCT
cana-1961	176	6	,	,	PUNCT
cana-1961	176	7	υ‖	υ‖	VERB
cana-1961	176	8	≤	≤	NOUN
cana-1961	176	9	1	1	NUM
cana-1961	176	10	|𝔪|3(1−£	|𝔪|3(1−£	PROPN
cana-1961	176	11	)	)	PUNCT
cana-1961	176	12	φ(ω1	φ(ω1	NOUN
cana-1961	176	13	)	)	PUNCT
cana-1961	176	14	,	,	PUNCT
cana-1961	176	15	20	20	NUM
cana-1961	176	16	where	where	SCONJ
cana-1961	176	17	φ(ω1	φ(ω1	NOUN
cana-1961	176	18	)	)	PUNCT
cana-1961	176	19	=	=	SYM
cana-1961	176	20	max	max	PROPN
cana-1961	176	21	{	{	PUNCT
cana-1961	176	22	|𝔪2(𝔪	|𝔪2(𝔪	PROPN
cana-1961	176	23	−	−	PROPN
cana-1961	176	24	1)|	1)|	NUM
cana-1961	176	25	|2|	|2|	PROPN
cana-1961	176	26	.	.	PUNCT
cana-1961	177	1	|4|	|4|	PROPN
cana-1961	177	2	φ(0	φ(0	PROPN
cana-1961	177	3	,	,	PUNCT
cana-1961	177	4	ω1	ω1	PROPN
cana-1961	177	5	)	)	PUNCT
cana-1961	177	6	,	,	PUNCT
cana-1961	177	7	|𝔪2(𝔪	|𝔪2(𝔪	PROPN
cana-1961	177	8	−	−	PROPN
cana-1961	177	9	2)|	2)|	NUM
cana-1961	178	1	|2(1	|2(1	ADP
cana-1961	178	2	−	−	PROPN
cana-1961	178	3	𝔪2)|	𝔪2)|	PROPN
cana-1961	178	4	φ(ω1	φ(ω1	NOUN
cana-1961	178	5	,	,	PUNCT
cana-1961	178	6	0	0	NUM
cana-1961	178	7	)	)	PUNCT
cana-1961	178	8	}	}	PUNCT
cana-1961	178	9	for	for	ADP
cana-1961	178	10	all	all	DET
cana-1961	178	11	ω1	ω1	PROPN
cana-1961	178	12	∈	∈	PROPN
cana-1961	178	13	𝒲	𝒲	PROPN
cana-1961	178	14	,	,	PUNCT
cana-1961	178	15	υ	υ	PRON
cana-1961	178	16	∈	∈	NOUN
cana-1961	178	17	𝒵.	𝒵.	NOUN
cana-1961	178	18	proof	proof	NOUN
cana-1961	178	19	:	:	PUNCT
cana-1961	178	20	replacing	replace	VERB
cana-1961	178	21	ω1	ω1	PROPN
cana-1961	178	22	=	=	SYM
cana-1961	178	23	0	0	NUM
cana-1961	179	1	in	in	ADP
cana-1961	179	2	eq	eq	NOUN
cana-1961	179	3	2	2	NUM
cana-1961	179	4	,	,	PUNCT
cana-1961	179	5	obtains	obtain	VERB
cana-1961	179	6	‖2𝔤(2ω2	‖2𝔤(2ω2	ADJ
cana-1961	179	7	)	)	PUNCT
cana-1961	180	1	−	−	ADP
cana-1961	180	2	16𝔤(ω2	16𝔤(ω2	NUM
cana-1961	180	3	)	)	PUNCT
cana-1961	180	4	,	,	PUNCT
cana-1961	180	5	υ‖	υ‖	VERB
cana-1961	180	6	≤	≤	ADJ
cana-1961	180	7	φ(0	φ(0	ADJ
cana-1961	180	8	,	,	PUNCT
cana-1961	180	9	ω2	ω2	NUM
cana-1961	180	10	)	)	PUNCT
cana-1961	180	11	21	21	NUM
cana-1961	180	12	for	for	ADP
cana-1961	180	13	all	all	DET
cana-1961	180	14	ω2	ω2	ADJ
cana-1961	180	15	∈	∈	PROPN
cana-1961	180	16	𝒲	𝒲	PROPN
cana-1961	180	17	,	,	PUNCT
cana-1961	180	18	υ	υ	PRON
cana-1961	180	19	∈	∈	PROPN
cana-1961	180	20	𝒵.	𝒵.	PROPN
cana-1961	180	21	interchanging	interchange	VERB
cana-1961	180	22	ω2	ω2	NOUN
cana-1961	180	23	by	by	ADP
cana-1961	180	24	ω1	ω1	PROPN
cana-1961	180	25	in	in	ADP
cana-1961	180	26	eq	eq	NOUN
cana-1961	180	27	21	21	NUM
cana-1961	180	28	and	and	CCONJ
cana-1961	180	29	dividing	divide	VERB
cana-1961	180	30	on	on	ADP
cana-1961	180	31	both	both	DET
cana-1961	180	32	sides	side	NOUN
cana-1961	180	33	of	of	ADP
cana-1961	180	34	eq	eq	NOUN
cana-1961	180	35	21	21	NUM
cana-1961	180	36	by	by	ADP
cana-1961	180	37	2	2	NUM
cana-1961	180	38	,	,	PUNCT
cana-1961	180	39	which	which	PRON
cana-1961	180	40	gives	give	VERB
cana-1961	180	41	communications	communication	NOUN
cana-1961	180	42	on	on	ADP
cana-1961	180	43	applied	apply	VERB
cana-1961	180	44	nonlinear	nonlinear	ADJ
cana-1961	180	45	analysis	analysis	NOUN
cana-1961	180	46	issn	issn	NOUN
cana-1961	180	47	:	:	PUNCT
cana-1961	180	48	1074	1074	NUM
cana-1961	180	49	-	-	PUNCT
cana-1961	180	50	133x	133x	NUM
cana-1961	180	51	vol	vol	NOUN
cana-1961	180	52	32	32	NUM
cana-1961	180	53	no	no	NOUN
cana-1961	180	54	.	.	NOUN
cana-1961	180	55	3	3	NUM
cana-1961	180	56	(	(	PUNCT
cana-1961	180	57	2025	2025	NUM
cana-1961	180	58	)	)	PUNCT
cana-1961	180	59	309	309	NUM
cana-1961	180	60	https://internationalpubls.com	https://internationalpubls.com	X
cana-1961	180	61	‖𝔤(2ω1	‖𝔤(2ω1	NOUN
cana-1961	180	62	)	)	PUNCT
cana-1961	180	63	−	−	NOUN
cana-1961	180	64	8𝔤(ω1	8𝔤(ω1	NUM
cana-1961	180	65	)	)	PUNCT
cana-1961	180	66	,	,	PUNCT
cana-1961	180	67	υ‖	υ‖	VERB
cana-1961	180	68	≤	≤	NOUN
cana-1961	180	69	1	1	NUM
cana-1961	180	70	|2|	|2|	PROPN
cana-1961	180	71	φ(0	φ(0	PROPN
cana-1961	180	72	,	,	PUNCT
cana-1961	180	73	ω1	ω1	PROPN
cana-1961	180	74	)	)	PUNCT
cana-1961	180	75	22	22	NUM
cana-1961	180	76	for	for	ADP
cana-1961	180	77	all	all	DET
cana-1961	180	78	ω1	ω1	PROPN
cana-1961	180	79	∈	∈	PROPN
cana-1961	180	80	𝒲	𝒲	PROPN
cana-1961	180	81	,	,	PUNCT
cana-1961	180	82	υ	υ	PRON
cana-1961	180	83	∈	∈	PROPN
cana-1961	180	84	𝒵.	𝒵.	PROPN
cana-1961	180	85	letting	let	VERB
cana-1961	180	86	ω2	ω2	NOUN
cana-1961	181	1	=	=	SYM
cana-1961	181	2	0	0	NUM
cana-1961	181	3	in	in	ADP
cana-1961	181	4	eq	eq	NOUN
cana-1961	181	5	2	2	NUM
cana-1961	181	6	,	,	PUNCT
cana-1961	181	7	it	it	PRON
cana-1961	181	8	gives	give	VERB
cana-1961	181	9	‖2(1	‖2(1	NOUN
cana-1961	181	10	−	−	ADP
cana-1961	181	11	𝔪2)𝔤(ω1	𝔪2)𝔤(ω1	NOUN
cana-1961	181	12	)	)	PUNCT
cana-1961	181	13	+	+	CCONJ
cana-1961	181	14	2(1−𝔪2	2(1−𝔪2	NOUN
cana-1961	181	15	)	)	PUNCT
cana-1961	181	16	𝔪2(𝔪−2	𝔪2(𝔪−2	NOUN
cana-1961	181	17	)	)	PUNCT
cana-1961	181	18	𝔤(𝔪ω1	𝔤(𝔪ω1	PROPN
cana-1961	181	19	)	)	PUNCT
cana-1961	182	1	+	+	CCONJ
cana-1961	182	2	𝔪3−𝔪2−𝔪+1	𝔪3−𝔪2−𝔪+1	NOUN
cana-1961	182	3	2(𝔪−2	2(𝔪−2	NOUN
cana-1961	182	4	)	)	PUNCT
cana-1961	182	5	𝔤(2ω1	𝔤(2ω1	NOUN
cana-1961	182	6	)	)	PUNCT
cana-1961	182	7	,	,	PUNCT
cana-1961	182	8	υ‖	υ‖	VERB
cana-1961	182	9	≤	≤	NOUN
cana-1961	182	10	φ(ω1	φ(ω1	X
cana-1961	182	11	,	,	PUNCT
cana-1961	182	12	0	0	NUM
cana-1961	182	13	)	)	PUNCT
cana-1961	182	14	23	23	NUM
cana-1961	182	15	for	for	ADP
cana-1961	182	16	all	all	DET
cana-1961	182	17	ω1	ω1	PROPN
cana-1961	182	18	∈	∈	PROPN
cana-1961	182	19	𝒲	𝒲	PROPN
cana-1961	182	20	,	,	PUNCT
cana-1961	182	21	υ	υ	DET
cana-1961	182	22	∈	∈	PROPN
cana-1961	182	23	𝒵.	𝒵.	PROPN
cana-1961	182	24	therefore	therefore	ADV
cana-1961	182	25	‖𝔪2(𝔪	‖𝔪2(𝔪	NOUN
cana-1961	182	26	−	−	PROPN
cana-1961	182	27	2)𝔤(ω1	2)𝔤(ω1	NUM
cana-1961	182	28	)	)	PUNCT
cana-1961	183	1	+	+	CCONJ
cana-1961	183	2	𝔤(𝔪ω1	𝔤(𝔪ω1	PROPN
cana-1961	183	3	)	)	PUNCT
cana-1961	183	4	−	−	PROPN
cana-1961	183	5	𝔪2(𝔪−1	𝔪2(𝔪−1	SYM
cana-1961	183	6	)	)	PUNCT
cana-1961	183	7	4	4	NUM
cana-1961	183	8	𝔤(2ω1	𝔤(2ω1	NOUN
cana-1961	183	9	)	)	PUNCT
cana-1961	183	10	,	,	PUNCT
cana-1961	183	11	υ‖	υ‖	VERB
cana-1961	183	12	≤	≤	NUM
cana-1961	183	13	|𝔪2(𝔪−2)|	|𝔪2(𝔪−2)|	NOUN
cana-1961	184	1	|2(1−𝔪2)|	|2(1−𝔪2)|	PROPN
cana-1961	184	2	φ(ω1	φ(ω1	NOUN
cana-1961	184	3	,	,	PUNCT
cana-1961	184	4	0	0	NUM
cana-1961	184	5	)	)	PUNCT
cana-1961	184	6	24	24	NUM
cana-1961	184	7	for	for	ADP
cana-1961	184	8	all	all	DET
cana-1961	184	9	ω1	ω1	PROPN
cana-1961	184	10	∈	∈	PROPN
cana-1961	184	11	𝒲	𝒲	PROPN
cana-1961	184	12	,	,	PUNCT
cana-1961	184	13	υ	υ	PRON
cana-1961	184	14	∈	∈	PROPN
cana-1961	184	15	𝒵.	𝒵.	PROPN
cana-1961	184	16	according	accord	VERB
cana-1961	184	17	to	to	ADP
cana-1961	184	18	eq	eq	NOUN
cana-1961	184	19	22	22	NUM
cana-1961	184	20	and	and	CCONJ
cana-1961	184	21	eq	eq	ADP
cana-1961	184	22	24	24	NUM
cana-1961	184	23	∥	∥	NUM
cana-1961	184	24	𝔤(𝔪ω1	𝔤(𝔪ω1	PROPN
cana-1961	184	25	)	)	PUNCT
cana-1961	185	1	−	−	ADP
cana-1961	185	2	𝔪3𝔤(ω1	𝔪3𝔤(ω1	PROPN
cana-1961	185	3	)	)	PUNCT
cana-1961	185	4	,	,	PUNCT
cana-1961	185	5	υ	υ	PROPN
cana-1961	185	6	∥≤	∥≤	PROPN
cana-1961	185	7	max	max	PROPN
cana-1961	185	8	{	{	PUNCT
cana-1961	185	9	|𝔪2(𝔪−1)|	|𝔪2(𝔪−1)|	X
cana-1961	185	10	|2|.|4|	|2|.|4|	NOUN
cana-1961	185	11	φ(0	φ(0	ADJ
cana-1961	185	12	,	,	PUNCT
cana-1961	185	13	ω1	ω1	PROPN
cana-1961	185	14	)	)	PUNCT
cana-1961	185	15	,	,	PUNCT
cana-1961	185	16	|𝔪2(𝔪−2)|	|𝔪2(𝔪−2)|	NOUN
cana-1961	185	17	|2(1−𝔪2)|	|2(1−𝔪2)|	PROPN
cana-1961	185	18	φ(ω1	φ(ω1	NOUN
cana-1961	185	19	,	,	PUNCT
cana-1961	185	20	0	0	NUM
cana-1961	185	21	)	)	PUNCT
cana-1961	185	22	}	}	PUNCT
cana-1961	185	23	25	25	NUM
cana-1961	185	24	where	where	SCONJ
cana-1961	185	25	φ(ω1	φ(ω1	NOUN
cana-1961	185	26	)	)	PUNCT
cana-1961	185	27	=	=	SYM
cana-1961	185	28	max	max	PROPN
cana-1961	185	29	{	{	PUNCT
cana-1961	185	30	|𝔪2(𝔪	|𝔪2(𝔪	PROPN
cana-1961	185	31	−	−	PROPN
cana-1961	185	32	1)|	1)|	NUM
cana-1961	185	33	|2|	|2|	PROPN
cana-1961	185	34	.	.	PUNCT
cana-1961	185	35	|4|	|4|	PROPN
cana-1961	185	36	φ(0	φ(0	PROPN
cana-1961	185	37	,	,	PUNCT
cana-1961	185	38	ω1	ω1	PROPN
cana-1961	185	39	)	)	PUNCT
cana-1961	185	40	,	,	PUNCT
cana-1961	185	41	|𝔪2(𝔪	|𝔪2(𝔪	PROPN
cana-1961	185	42	−	−	PROPN
cana-1961	185	43	2)|	2)|	NUM
cana-1961	185	44	|2(1	|2(1	ADP
cana-1961	185	45	−	−	PROPN
cana-1961	185	46	𝔪2)|	𝔪2)|	PROPN
cana-1961	185	47	φ(ω1	φ(ω1	NOUN
cana-1961	185	48	,	,	PUNCT
cana-1961	185	49	0	0	NUM
cana-1961	185	50	)	)	PUNCT
cana-1961	185	51	}	}	PUNCT
cana-1961	185	52	.	.	PUNCT
cana-1961	186	1	then	then	ADV
cana-1961	186	2	,	,	PUNCT
cana-1961	186	3	∥	∥	PROPN
cana-1961	186	4	𝔤(𝔪ω1	𝔤(𝔪ω1	PROPN
cana-1961	186	5	)	)	PUNCT
cana-1961	186	6	−	−	ADP
cana-1961	186	7	𝔪3𝔤(ω1	𝔪3𝔤(ω1	PROPN
cana-1961	186	8	)	)	PUNCT
cana-1961	186	9	,	,	PUNCT
cana-1961	186	10	υ	υ	PROPN
cana-1961	186	11	∥≤	∥≤	PROPN
cana-1961	186	12	φ(ω1	φ(ω1	NOUN
cana-1961	186	13	)	)	PUNCT
cana-1961	186	14	26	26	NUM
cana-1961	186	15	for	for	ADP
cana-1961	186	16	all	all	DET
cana-1961	186	17	ω1	ω1	PROPN
cana-1961	186	18	∈	∈	PROPN
cana-1961	186	19	𝒲	𝒲	PROPN
cana-1961	186	20	,	,	PUNCT
cana-1961	186	21	υ	υ	PRON
cana-1961	186	22	∈	∈	PROPN
cana-1961	186	23	𝒵.	𝒵.	PROPN
cana-1961	186	24	according	accord	VERB
cana-1961	186	25	to	to	ADP
cana-1961	186	26	eq	eq	NOUN
cana-1961	186	27	26	26	NUM
cana-1961	186	28	,	,	PUNCT
cana-1961	186	29	obtains	obtain	VERB
cana-1961	186	30	‖	‖	PROPN
cana-1961	186	31	𝔤(𝔪ω1	𝔤(𝔪ω1	PROPN
cana-1961	186	32	)	)	PUNCT
cana-1961	186	33	𝔪3	𝔪3	NOUN
cana-1961	186	34	−	−	NOUN
cana-1961	186	35	𝔤(ω1	𝔤(ω1	NOUN
cana-1961	186	36	)	)	PUNCT
cana-1961	186	37	,	,	PUNCT
cana-1961	186	38	υ‖	υ‖	VERB
cana-1961	186	39	≤	≤	ADJ
cana-1961	186	40	1	1	NUM
cana-1961	186	41	|𝔪|3	|𝔪|3	NOUN
cana-1961	186	42	φ(ω1	φ(ω1	NOUN
cana-1961	186	43	)	)	PUNCT
cana-1961	186	44	27	27	NUM
cana-1961	186	45	for	for	ADP
cana-1961	186	46	all	all	DET
cana-1961	186	47	ω1	ω1	PROPN
cana-1961	186	48	∈	∈	PROPN
cana-1961	186	49	𝒲	𝒲	PROPN
cana-1961	186	50	,	,	PUNCT
cana-1961	186	51	υ	υ	PRON
cana-1961	186	52	∈	∈	NOUN
cana-1961	186	53	𝒵.	𝒵.	PROPN
cana-1961	186	54	consider	consider	VERB
cana-1961	186	55	the	the	DET
cana-1961	186	56	set	set	NOUN
cana-1961	186	57	γ	γ	X
cana-1961	186	58	=	=	PRON
cana-1961	186	59	{	{	PUNCT
cana-1961	186	60	𝔣	𝔣	NOUN
cana-1961	186	61	:	:	PUNCT
cana-1961	186	62	𝒲	𝒲	NOUN
cana-1961	186	63	→	→	SYM
cana-1961	186	64	𝒵	𝒵	PROPN
cana-1961	186	65	}	}	PUNCT
cana-1961	186	66	28	28	NUM
cana-1961	186	67	and	and	CCONJ
cana-1961	186	68	define	define	VERB
cana-1961	186	69	the	the	DET
cana-1961	186	70	generalized	generalized	ADJ
cana-1961	186	71	metric	metric	NOUN
cana-1961	186	72	ð	ð	PROPN
cana-1961	186	73	in	in	ADP
cana-1961	186	74	γ	γ	X
cana-1961	186	75	by	by	ADP
cana-1961	186	76	ð(𝔣	ð(𝔣	PROPN
cana-1961	186	77	,	,	PUNCT
cana-1961	186	78	𝔥	𝔥	NOUN
cana-1961	186	79	)	)	PUNCT
cana-1961	186	80	=	=	PUNCT
cana-1961	186	81	inf{σ	inf{σ	PROPN
cana-1961	186	82	∈	∈	NOUN
cana-1961	186	83	(	(	PUNCT
cana-1961	186	84	0	0	NUM
cana-1961	186	85	,	,	PUNCT
cana-1961	186	86	∞	∞	NUM
cana-1961	186	87	):	):	PUNCT
cana-1961	186	88	‖𝔣(ω1	‖𝔣(ω1	NUM
cana-1961	186	89	)	)	PUNCT
cana-1961	186	90	−	−	NOUN
cana-1961	186	91	𝔥(ω1	𝔥(ω1	ADV
cana-1961	186	92	)	)	PUNCT
cana-1961	186	93	,	,	PUNCT
cana-1961	186	94	υ‖	υ‖	VERB
cana-1961	186	95	≤	≤	NOUN
cana-1961	186	96	σ	σ	NUM
cana-1961	186	97	φ(ω1	φ(ω1	NOUN
cana-1961	186	98	)	)	PUNCT
cana-1961	186	99	,	,	PUNCT
cana-1961	186	100	∀	∀	X
cana-1961	186	101	ω1	ω1	PROPN
cana-1961	186	102	∈	∈	PROPN
cana-1961	186	103	𝒲	𝒲	PROPN
cana-1961	186	104	,	,	PUNCT
cana-1961	186	105	υ	υ	PRON
cana-1961	186	106	∈	∈	PROPN
cana-1961	186	107	𝒵	𝒵	PROPN
cana-1961	186	108	}	}	PUNCT
cana-1961	186	109	.	.	PUNCT
cana-1961	187	1	29	29	NUM
cana-1961	188	1	it	it	PRON
cana-1961	188	2	is	be	AUX
cana-1961	188	3	simple	simple	ADJ
cana-1961	188	4	to	to	PART
cana-1961	188	5	prove	prove	VERB
cana-1961	188	6	that	that	SCONJ
cana-1961	188	7	(	(	PUNCT
cana-1961	188	8	γ	γ	X
cana-1961	188	9	,	,	PUNCT
cana-1961	188	10	ð	ð	NUM
cana-1961	188	11	)	)	PUNCT
cana-1961	188	12	is	be	AUX
cana-1961	188	13	complete	complete	ADJ
cana-1961	188	14	[	[	X
cana-1961	188	15	26	26	NUM
cana-1961	188	16	]	]	PUNCT
cana-1961	188	17	.	.	PUNCT
cana-1961	189	1	now	now	ADV
cana-1961	189	2	,	,	PUNCT
cana-1961	189	3	define	define	VERB
cana-1961	189	4	the	the	DET
cana-1961	189	5	function	function	NOUN
cana-1961	189	6	λ	λ	PROPN
cana-1961	189	7	:	:	PUNCT
cana-1961	189	8	γ	γ	X
cana-1961	189	9	→	→	SYM
cana-1961	189	10	γ	γ	X
cana-1961	189	11	such	such	ADJ
cana-1961	189	12	that	that	PRON
cana-1961	189	13	λ𝔣(ω1	λ𝔣(ω1	ADP
cana-1961	189	14	)	)	PUNCT
cana-1961	189	15	=	=	SYM
cana-1961	189	16	1	1	NUM
cana-1961	189	17	𝔪3	𝔪3	NOUN
cana-1961	189	18	𝔣(𝔪ω1	𝔣(𝔪ω1	NOUN
cana-1961	189	19	)	)	PUNCT
cana-1961	189	20	30	30	NUM
cana-1961	189	21	for	for	ADP
cana-1961	189	22	all	all	DET
cana-1961	189	23	ω1	ω1	PROPN
cana-1961	189	24	∈	∈	PROPN
cana-1961	189	25	𝒲.	𝒲.	PROPN
cana-1961	189	26	let	let	VERB
cana-1961	189	27	𝔣	𝔣	ADP
cana-1961	189	28	,	,	PUNCT
cana-1961	189	29	𝔥	𝔥	PROPN
cana-1961	189	30	∈	∈	PROPN
cana-1961	189	31	γ	γ	NOUN
cana-1961	189	32	be	be	AUX
cana-1961	189	33	given	give	VERB
cana-1961	189	34	such	such	ADJ
cana-1961	189	35	that	that	DET
cana-1961	189	36	ð(𝔣	ð(𝔣	PROPN
cana-1961	189	37	,	,	PUNCT
cana-1961	189	38	𝔥	𝔥	NOUN
cana-1961	189	39	)	)	PUNCT
cana-1961	189	40	=	=	VERB
cana-1961	190	1	ϵ.	ϵ.	NOUN
cana-1961	190	2	then	then	ADV
cana-1961	190	3	‖𝔣(ω1	‖𝔣(ω1	NUM
cana-1961	190	4	)	)	PUNCT
cana-1961	190	5	−	−	NOUN
cana-1961	190	6	𝔥(ω1	𝔥(ω1	ADV
cana-1961	190	7	)	)	PUNCT
cana-1961	190	8	,	,	PUNCT
cana-1961	190	9	υ‖	υ‖	VERB
cana-1961	190	10	≤	≤	X
cana-1961	190	11	ϵ	ϵ	X
cana-1961	190	12	φ(ω1	φ(ω1	NOUN
cana-1961	190	13	)	)	PUNCT
cana-1961	190	14	31	31	NUM
cana-1961	190	15	for	for	ADP
cana-1961	190	16	all	all	DET
cana-1961	190	17	ω1	ω1	PROPN
cana-1961	190	18	∈	∈	PROPN
cana-1961	190	19	𝒲	𝒲	PROPN
cana-1961	190	20	,	,	PUNCT
cana-1961	190	21	υ	υ	PRON
cana-1961	190	22	∈	∈	NOUN
cana-1961	190	23	𝒵.	𝒵.	PROPN
cana-1961	190	24	hence	hence	ADV
cana-1961	190	25	‖λ𝔣(𝔪ω1	‖λ𝔣(𝔪ω1	NOUN
cana-1961	190	26	)	)	PUNCT
cana-1961	190	27	−	−	PROPN
cana-1961	190	28	λ𝔥(𝔪ω1	λ𝔥(𝔪ω1	NOUN
cana-1961	190	29	)	)	PUNCT
cana-1961	190	30	,	,	PUNCT
cana-1961	190	31	υ‖	υ‖	X
cana-1961	190	32	=	=	PUNCT
cana-1961	191	1	‖	‖	PROPN
cana-1961	191	2	1	1	NUM
cana-1961	191	3	𝔪3	𝔪3	NOUN
cana-1961	191	4	𝔣(𝔪ω1	𝔣(𝔪ω1	NOUN
cana-1961	191	5	)	)	PUNCT
cana-1961	192	1	−	−	PROPN
cana-1961	192	2	1	1	NUM
cana-1961	192	3	𝔪3	𝔪3	NOUN
cana-1961	192	4	𝔥(𝔪ω1	𝔥(𝔪ω1	NOUN
cana-1961	192	5	)	)	PUNCT
cana-1961	192	6	,	,	PUNCT
cana-1961	192	7	υ‖	υ‖	VERB
cana-1961	192	8	≤	≤	NOUN
cana-1961	192	9	1	1	NUM
cana-1961	192	10	|𝔪|3	|𝔪|3	NOUN
cana-1961	192	11	ϵ	ϵ	ADP
cana-1961	192	12	φ(𝔪ω1	φ(𝔪ω1	NOUN
cana-1961	192	13	)	)	PUNCT
cana-1961	192	14	≤	≤	NUM
cana-1961	192	15	1	1	NUM
cana-1961	192	16	|𝔪|3	|𝔪|3	NOUN
cana-1961	192	17	ϵ	ϵ	ADP
cana-1961	192	18	|𝔪|3	|𝔪|3	NOUN
cana-1961	192	19	£	£	NOUN
cana-1961	192	20	φ(ω1	φ(ω1	NOUN
cana-1961	192	21	)	)	PUNCT
cana-1961	192	22	≤	≤	NOUN
cana-1961	192	23	ϵ	ϵ	SCONJ
cana-1961	192	24	£	£	SYM
cana-1961	192	25	φ(ω1	φ(ω1	NOUN
cana-1961	192	26	)	)	PUNCT
cana-1961	192	27	communications	communication	NOUN
cana-1961	192	28	on	on	ADP
cana-1961	192	29	applied	apply	VERB
cana-1961	192	30	nonlinear	nonlinear	ADJ
cana-1961	192	31	analysis	analysis	NOUN
cana-1961	192	32	issn	issn	NOUN
cana-1961	192	33	:	:	PUNCT
cana-1961	192	34	1074	1074	NUM
cana-1961	192	35	-	-	PUNCT
cana-1961	192	36	133x	133x	NUM
cana-1961	192	37	vol	vol	NOUN
cana-1961	192	38	32	32	NUM
cana-1961	192	39	no	no	NOUN
cana-1961	192	40	.	.	NOUN
cana-1961	192	41	3	3	NUM
cana-1961	192	42	(	(	PUNCT
cana-1961	192	43	2025	2025	NUM
cana-1961	192	44	)	)	PUNCT
cana-1961	192	45	310	310	NUM
cana-1961	192	46	https://internationalpubls.com	https://internationalpubls.com	X
cana-1961	192	47	for	for	ADP
cana-1961	192	48	all	all	DET
cana-1961	192	49	ω1	ω1	PROPN
cana-1961	192	50	∈	∈	PROPN
cana-1961	192	51	𝒲	𝒲	PROPN
cana-1961	192	52	,	,	PUNCT
cana-1961	192	53	υ	υ	PRON
cana-1961	192	54	∈	∈	PROPN
cana-1961	192	55	𝒵	𝒵	PROPN
cana-1961	192	56	,	,	PUNCT
cana-1961	192	57	that	that	PRON
cana-1961	192	58	is	be	AUX
cana-1961	192	59	ð(λ𝔣	ð(λ𝔣	NOUN
cana-1961	192	60	,	,	PUNCT
cana-1961	192	61	λ𝔥	λ𝔥	ADJ
cana-1961	192	62	)	)	PUNCT
cana-1961	192	63	≤	≤	NOUN
cana-1961	192	64	£	£	SYM
cana-1961	192	65	ε	ε	PROPN
cana-1961	192	66	.	.	PUNCT
cana-1961	193	1	therefore	therefore	PROPN
cana-1961	193	2	ð(λ𝔣	ð(λ𝔣	PROPN
cana-1961	193	3	,	,	PUNCT
cana-1961	193	4	λ𝔥	λ𝔥	NOUN
cana-1961	193	5	)	)	PUNCT
cana-1961	193	6	≤	≤	NOUN
cana-1961	193	7	£	£	SYM
cana-1961	193	8	ð(𝔣	ð(𝔣	PROPN
cana-1961	193	9	,	,	PUNCT
cana-1961	193	10	𝔥	𝔥	NOUN
cana-1961	193	11	)	)	PUNCT
cana-1961	193	12	for	for	ADP
cana-1961	193	13	all	all	DET
cana-1961	193	14	𝔣	𝔣	ADJ
cana-1961	193	15	,	,	PUNCT
cana-1961	193	16	𝔥	𝔥	PROPN
cana-1961	193	17	∈	∈	PROPN
cana-1961	193	18	γ	γ	X
cana-1961	193	19	.	.	PUNCT
cana-1961	194	1	according	accord	VERB
cana-1961	194	2	to	to	ADP
cana-1961	194	3	eq	eq	PROPN
cana-1961	194	4	27	27	NUM
cana-1961	194	5	ð(𝔤	ð(𝔤	NOUN
cana-1961	194	6	,	,	PUNCT
cana-1961	194	7	λ𝔤	λ𝔤	NOUN
cana-1961	194	8	)	)	PUNCT
cana-1961	194	9	≤	≤	NOUN
cana-1961	194	10	1	1	NUM
cana-1961	194	11	|𝔪|3	|𝔪|3	NOUN
cana-1961	194	12	<	<	X
cana-1961	194	13	+	+	NOUN
cana-1961	194	14	∞.	∞.	PROPN
cana-1961	194	15	32	32	NUM
cana-1961	194	16	by	by	ADP
cana-1961	194	17	theorem	theorem	NOUN
cana-1961	194	18	1	1	NUM
cana-1961	194	19	,	,	PUNCT
cana-1961	194	20	there	there	PRON
cana-1961	194	21	is	be	VERB
cana-1961	194	22	a	a	DET
cana-1961	194	23	function	function	NOUN
cana-1961	194	24	c3	c3	NOUN
cana-1961	194	25	́	́	PROPN
cana-1961	194	26	:	:	PUNCT
cana-1961	194	27	𝒲	𝒲	NOUN
cana-1961	194	28	→	→	SYM
cana-1961	194	29	𝒵	𝒵	PROPN
cana-1961	194	30	satisfying	satisfy	VERB
cana-1961	194	31	the	the	DET
cana-1961	194	32	following	follow	VERB
cana-1961	194	33	conditions	condition	NOUN
cana-1961	194	34	:	:	PUNCT
cana-1961	194	35	(	(	PUNCT
cana-1961	194	36	1	1	X
cana-1961	194	37	)	)	PUNCT
cana-1961	194	38	c3	c3	NOUN
cana-1961	194	39	́	́	PROPN
cana-1961	194	40	is	be	AUX
cana-1961	194	41	a	a	DET
cana-1961	194	42	fixed	fix	VERB
cana-1961	194	43	point	point	NOUN
cana-1961	194	44	of	of	ADP
cana-1961	194	45	λ	λ	PROPN
cana-1961	194	46	,	,	PUNCT
cana-1961	194	47	that	that	ADV
cana-1961	194	48	is	is	ADV
cana-1961	194	49	,	,	PUNCT
cana-1961	194	50	c3	c3	PROPN
cana-1961	194	51	́	́	PROPN
cana-1961	194	52	(	(	PUNCT
cana-1961	194	53	𝔪ω1	𝔪ω1	NOUN
cana-1961	194	54	)	)	PUNCT
cana-1961	194	55	=	=	PUNCT
cana-1961	194	56	𝔪3	𝔪3	NOUN
cana-1961	194	57	c3	c3	NOUN
cana-1961	194	58	́	́	PROPN
cana-1961	194	59	(	(	PUNCT
cana-1961	194	60	ω1	ω1	PROPN
cana-1961	194	61	)	)	PUNCT
cana-1961	194	62	33	33	NUM
cana-1961	194	63	for	for	ADP
cana-1961	194	64	all	all	DET
cana-1961	194	65	ω1	ω1	PROPN
cana-1961	194	66	∈	∈	PROPN
cana-1961	194	67	𝒲.	𝒲.	PROPN
cana-1961	194	68	c3	c3	PROPN
cana-1961	194	69	́	́	PROPN
cana-1961	194	70	is	be	AUX
cana-1961	194	71	a	a	DET
cana-1961	194	72	unique	unique	ADJ
cana-1961	194	73	fixed	fix	VERB
cana-1961	194	74	point	point	NOUN
cana-1961	194	75	of	of	ADP
cana-1961	194	76	the	the	DET
cana-1961	194	77	set	set	NOUN
cana-1961	194	78	denoted	denote	VERB
cana-1961	194	79	by	by	ADP
cana-1961	194	80	λ	λ	PROPN
cana-1961	194	81	s	s	PART
cana-1961	194	82	=	=	X
cana-1961	194	83	{	{	PUNCT
cana-1961	194	84	𝔥	𝔥	NOUN
cana-1961	194	85	∈	∈	PROPN
cana-1961	194	86	γ	γ	PROPN
cana-1961	194	87	∶	∶	NOUN
cana-1961	194	88	ð(𝔣	ð(𝔣	PROPN
cana-1961	194	89	,	,	PUNCT
cana-1961	194	90	𝔥	𝔥	NOUN
cana-1961	194	91	)	)	PUNCT
cana-1961	194	92	<	<	X
cana-1961	194	93	∞	∞	PROPN
cana-1961	194	94	}	}	PUNCT
cana-1961	194	95	.	.	PUNCT
cana-1961	195	1	this	this	PRON
cana-1961	195	2	indicates	indicate	VERB
cana-1961	195	3	that	that	SCONJ
cana-1961	195	4	c3	c3	PROPN
cana-1961	195	5	́	́	PROPN
cana-1961	195	6	is	be	AUX
cana-1961	195	7	a	a	DET
cana-1961	195	8	unique	unique	ADJ
cana-1961	195	9	mapping	mapping	NOUN
cana-1961	195	10	satisfying	satisfying	NOUN
cana-1961	195	11	eq	eq	ADP
cana-1961	195	12	33	33	NUM
cana-1961	195	13	such	such	ADJ
cana-1961	195	14	that	that	SCONJ
cana-1961	195	15	there	there	PRON
cana-1961	195	16	is	be	VERB
cana-1961	195	17	a	a	DET
cana-1961	195	18	σ	σ	NUM
cana-1961	195	19	∈	∈	PROPN
cana-1961	195	20	(	(	PUNCT
cana-1961	195	21	0	0	NUM
cana-1961	195	22	,	,	PUNCT
cana-1961	195	23	∞	∞	NUM
cana-1961	195	24	)	)	PUNCT
cana-1961	195	25	satisfying	satisfy	VERB
cana-1961	195	26	‖𝔤(ω1	‖𝔤(ω1	NOUN
cana-1961	195	27	)	)	PUNCT
cana-1961	195	28	−	−	PROPN
cana-1961	196	1	c3	c3	PROPN
cana-1961	196	2	́	́	PROPN
cana-1961	196	3	(	(	PUNCT
cana-1961	196	4	ω1	ω1	PROPN
cana-1961	196	5	)	)	PUNCT
cana-1961	196	6	,	,	PUNCT
cana-1961	196	7	υ‖	υ‖	VERB
cana-1961	196	8	≤	≤	NOUN
cana-1961	196	9	σ	σ	NUM
cana-1961	196	10	φ(ω1	φ(ω1	NOUN
cana-1961	196	11	)	)	PUNCT
cana-1961	196	12	∀	∀	PUNCT
cana-1961	196	13	ω1	ω1	PROPN
cana-1961	196	14	∈	∈	PROPN
cana-1961	196	15	𝒲	𝒲	PROPN
cana-1961	196	16	,	,	PUNCT
cana-1961	196	17	υ	υ	PRON
cana-1961	196	18	∈	∈	PROPN
cana-1961	196	19	𝒵.	𝒵.	PROPN
cana-1961	196	20	(	(	PUNCT
cana-1961	196	21	2	2	NUM
cana-1961	196	22	)	)	PUNCT
cana-1961	196	23	ð(λ	ð(λ	PROPN
cana-1961	196	24	n𝔤	n𝔤	PROPN
cana-1961	196	25	,	,	PUNCT
cana-1961	196	26	c3	c3	PROPN
cana-1961	196	27	́	́	PROPN
cana-1961	196	28	)	)	PUNCT
cana-1961	196	29	→	→	SYM
cana-1961	196	30	0	0	NUM
cana-1961	196	31	as	as	ADP
cana-1961	196	32	n	n	NOUN
cana-1961	196	33	→	→	SYM
cana-1961	196	34	∞.	∞.	PROPN
cana-1961	196	35	this	this	PRON
cana-1961	196	36	indicates	indicate	VERB
cana-1961	196	37	the	the	DET
cana-1961	196	38	equality	equality	NOUN
cana-1961	196	39	,	,	PUNCT
cana-1961	196	40	lim	lim	PROPN
cana-1961	196	41	n→∞	n→∞	X
cana-1961	196	42	(	(	PUNCT
cana-1961	196	43	λ	λ	NOUN
cana-1961	196	44	n𝔤)(ω1	n𝔤)(ω1	NOUN
cana-1961	196	45	)	)	PUNCT
cana-1961	196	46	=	=	SYM
cana-1961	196	47	lim	lim	PROPN
cana-1961	196	48	n→∞	n→∞	NUM
cana-1961	196	49	𝔤(𝔪nω1	𝔤(𝔪nω1	NUM
cana-1961	196	50	)	)	PUNCT
cana-1961	196	51	𝔪3n	𝔪3n	PROPN
cana-1961	196	52	=	=	PUNCT
cana-1961	196	53	c3	c3	PROPN
cana-1961	196	54	́	́	PROPN
cana-1961	196	55	(	(	PUNCT
cana-1961	196	56	ω1	ω1	PROPN
cana-1961	196	57	)	)	PUNCT
cana-1961	196	58	,	,	PUNCT
cana-1961	196	59	∀	∀	X
cana-1961	196	60	ω1	ω1	PROPN
cana-1961	196	61	∈	∈	PROPN
cana-1961	196	62	𝒲.	𝒲.	PROPN
cana-1961	196	63	(	(	PUNCT
cana-1961	196	64	3	3	X
cana-1961	196	65	)	)	PUNCT
cana-1961	196	66	ð(𝔤	ð(𝔤	PROPN
cana-1961	196	67	,	,	PUNCT
cana-1961	196	68	c3	c3	PROPN
cana-1961	196	69	́	́	PROPN
cana-1961	196	70	)	)	PUNCT
cana-1961	196	71	≤	≤	ADV
cana-1961	196	72	1	1	NUM
cana-1961	196	73	1−£	1−£	NUM
cana-1961	196	74	ð(𝔤	ð(𝔤	NOUN
cana-1961	196	75	,	,	PUNCT
cana-1961	196	76	λn𝔤	λn𝔤	PROPN
cana-1961	196	77	)	)	PUNCT
cana-1961	196	78	,	,	PUNCT
cana-1961	196	79	which	which	PRON
cana-1961	196	80	implies	imply	VERB
cana-1961	196	81	ð(𝔤	ð(𝔤	PROPN
cana-1961	196	82	,	,	PUNCT
cana-1961	196	83	c3	c3	PROPN
cana-1961	196	84	́	́	PROPN
cana-1961	196	85	)	)	PUNCT
cana-1961	197	1	≤	≤	ADV
cana-1961	197	2	1	1	NUM
cana-1961	197	3	1	1	NUM
cana-1961	197	4	−	−	NOUN
cana-1961	197	5	£	£	PROPN
cana-1961	197	6	ð(𝔤	ð(𝔤	NOUN
cana-1961	197	7	,	,	PUNCT
cana-1961	197	8	λ𝔤	λ𝔤	NOUN
cana-1961	197	9	)	)	PUNCT
cana-1961	197	10	≤	≤	NOUN
cana-1961	197	11	1	1	NUM
cana-1961	197	12	|𝔪|3	|𝔪|3	NOUN
cana-1961	197	13	1	1	NUM
cana-1961	197	14	(	(	PUNCT
cana-1961	197	15	1	1	NUM
cana-1961	197	16	−	−	NOUN
cana-1961	197	17	£	£	NUM
cana-1961	197	18	)	)	PUNCT
cana-1961	197	19	.	.	PUNCT
cana-1961	198	1	this	this	PRON
cana-1961	198	2	indicates	indicate	VERB
cana-1961	198	3	that	that	SCONJ
cana-1961	198	4	the	the	DET
cana-1961	198	5	inequality	inequality	NOUN
cana-1961	198	6	eq	eq	ADP
cana-1961	198	7	20	20	NUM
cana-1961	198	8	remains	remain	VERB
cana-1961	198	9	valid	valid	ADJ
cana-1961	198	10	.	.	PUNCT
cana-1961	199	1	according	accord	VERB
cana-1961	199	2	to	to	ADP
cana-1961	199	3	eq	eq	NOUN
cana-1961	199	4	18	18	NUM
cana-1961	199	5	and	and	CCONJ
cana-1961	199	6	eq	eq	ADP
cana-1961	199	7	19	19	NUM
cana-1961	199	8	∥	∥	PROPN
cana-1961	199	9	dc3	dc3	NOUN
cana-1961	199	10	́	́	PROPN
cana-1961	199	11	(	(	PUNCT
cana-1961	199	12	ω1	ω1	PROPN
cana-1961	199	13	,	,	PUNCT
cana-1961	199	14	ω2	ω2	NUM
cana-1961	199	15	)	)	PUNCT
cana-1961	199	16	,	,	PUNCT
cana-1961	199	17	υ	υ	PRON
cana-1961	199	18	∥=	∥=	ADJ
cana-1961	199	19	lim	lim	PROPN
cana-1961	199	20	n→∞	n→∞	NUM
cana-1961	199	21	∥	∥	PROPN
cana-1961	199	22	𝔪−3nd𝔤(𝔪nω1	𝔪−3nd𝔤(𝔪nω1	PROPN
cana-1961	199	23	,	,	PUNCT
cana-1961	199	24	𝔪nω2	𝔪nω2	PROPN
cana-1961	199	25	)	)	PUNCT
cana-1961	199	26	,	,	PUNCT
cana-1961	199	27	υ	υ	NOUN
cana-1961	199	28	∥	∥	PUNCT
cana-1961	199	29	≤	≤	NUM
cana-1961	199	30	lim	lim	PROPN
cana-1961	199	31	n→∞	n→∞	NUM
cana-1961	199	32	1	1	NUM
cana-1961	199	33	|𝔪|3n	|𝔪|3n	NOUN
cana-1961	199	34	φ(𝔪nω1	φ(𝔪nω1	NOUN
cana-1961	199	35	,	,	PUNCT
cana-1961	199	36	𝔪nω2	𝔪nω2	NOUN
cana-1961	199	37	)	)	PUNCT
cana-1961	199	38	≤	≤	PROPN
cana-1961	199	39	lim	lim	PROPN
cana-1961	199	40	n→∞	n→∞	NUM
cana-1961	199	41	1	1	NUM
cana-1961	199	42	|𝔪|3n	|𝔪|3n	NOUN
cana-1961	199	43	£	£	SYM
cana-1961	199	44	n|𝔪|3nφ(ω1	n|𝔪|3nφ(ω1	NUM
cana-1961	199	45	,	,	PUNCT
cana-1961	199	46	ω2	ω2	ADJ
cana-1961	199	47	)	)	PUNCT
cana-1961	199	48	≤	≤	NOUN
cana-1961	199	49	lim	lim	PROPN
cana-1961	199	50	n→∞	n→∞	PRON
cana-1961	199	51	£	£	SYM
cana-1961	199	52	n	n	PRON
cana-1961	199	53	φ(ω1	φ(ω1	NOUN
cana-1961	199	54	,	,	PUNCT
cana-1961	199	55	ω2	ω2	NUM
cana-1961	199	56	)	)	PUNCT
cana-1961	199	57	=	=	SYM
cana-1961	199	58	0	0	X
cana-1961	199	59	.	.	PUNCT
cana-1961	200	1	for	for	ADP
cana-1961	200	2	all	all	DET
cana-1961	200	3	ω1	ω1	PROPN
cana-1961	200	4	,	,	PUNCT
cana-1961	200	5	ω2	ω2	NOUN
cana-1961	200	6	∈	∈	PROPN
cana-1961	200	7	𝒲	𝒲	PROPN
cana-1961	200	8	,	,	PUNCT
cana-1961	200	9	υ	υ	PROPN
cana-1961	200	10	∈	∈	PROPN
cana-1961	200	11	𝒵	𝒵	PROPN
cana-1961	200	12	and	and	CCONJ
cana-1961	200	13	n	n	CCONJ
cana-1961	200	14	∈	∈	NOUN
cana-1961	200	15	ℕ.	ℕ.	PROPN
cana-1961	200	16	so	so	ADV
cana-1961	200	17	∥	∥	X
cana-1961	200	18	dc3	dc3	NOUN
cana-1961	200	19	́	́	PROPN
cana-1961	200	20	(	(	PUNCT
cana-1961	200	21	ω1	ω1	PROPN
cana-1961	200	22	,	,	PUNCT
cana-1961	200	23	ω2	ω2	NUM
cana-1961	200	24	)	)	PUNCT
cana-1961	200	25	,	,	PUNCT
cana-1961	200	26	υ	υ	PRON
cana-1961	200	27	∥=	∥=	NOUN
cana-1961	200	28	0	0	NUM
cana-1961	200	29	.	.	PUNCT
cana-1961	201	1	thus	thus	ADV
cana-1961	201	2	the	the	DET
cana-1961	201	3	mapping	mapping	NOUN
cana-1961	201	4	c3	c3	X
cana-1961	201	5	́	́	PROPN
cana-1961	201	6	:	:	PUNCT
cana-1961	201	7	𝒲	𝒲	NOUN
cana-1961	201	8	→	→	SYM
cana-1961	201	9	𝒵	𝒵	PROPN
cana-1961	201	10	is	be	AUX
cana-1961	201	11	c3	c3	NOUN
cana-1961	201	12	́	́	PROPN
cana-1961	201	13	as	as	SCONJ
cana-1961	201	14	desired	desire	VERB
cana-1961	201	15	.	.	PUNCT
cana-1961	202	1	corollary	corollary	ADJ
cana-1961	202	2	3	3	NUM
cana-1961	202	3	:	:	PUNCT
cana-1961	202	4	let	let	VERB
cana-1961	202	5	θ	θ	PROPN
cana-1961	202	6	≥	≥	X
cana-1961	202	7	0	0	NUM
cana-1961	202	8	and	and	CCONJ
cana-1961	202	9	τ	τ	X
cana-1961	202	10	=	=	SYM
cana-1961	202	11	s	s	PART
cana-1961	203	1	+	+	NUM
cana-1961	203	2	t	t	AUX
cana-1961	203	3	be	be	VERB
cana-1961	203	4	a	a	DET
cana-1961	203	5	positive	positive	ADJ
cana-1961	203	6	real	real	ADJ
cana-1961	203	7	number	number	NOUN
cana-1961	203	8	with	with	ADP
cana-1961	203	9	τ	τ	PROPN
cana-1961	203	10	<	<	X
cana-1961	203	11	3	3	X
cana-1961	203	12	.	.	PUNCT
cana-1961	204	1	let	let	VERB
cana-1961	204	2	𝔤	𝔤	PRON
cana-1961	204	3	:	:	PUNCT
cana-1961	204	4	𝒲	𝒲	NOUN
cana-1961	204	5	→	→	PUNCT
cana-1961	204	6	𝒵	𝒵	PRON
cana-1961	204	7	be	be	VERB
cana-1961	204	8	an	an	DET
cana-1961	204	9	odd	odd	ADJ
cana-1961	204	10	mapping	mapping	NOUN
cana-1961	204	11	with	with	ADP
cana-1961	204	12	𝔤(0	𝔤(0	PROPN
cana-1961	204	13	)	)	PUNCT
cana-1961	205	1	=	=	SYM
cana-1961	205	2	0	0	PUNCT
cana-1961	205	3	satisfying	satisfy	VERB
cana-1961	205	4	∥	∥	PUNCT
cana-1961	205	5	d𝔤(ω1	d𝔤(ω1	NOUN
cana-1961	205	6	,	,	PUNCT
cana-1961	205	7	ω2	ω2	NUM
cana-1961	205	8	)	)	PUNCT
cana-1961	205	9	,	,	PUNCT
cana-1961	205	10	υ	υ	PROPN
cana-1961	205	11	∥≤	∥≤	PROPN
cana-1961	205	12	θ(∥	θ(∥	PROPN
cana-1961	205	13	ω1	ω1	PROPN
cana-1961	205	14	∥τ	∥τ	PROPN
cana-1961	206	1	+	+	PROPN
cana-1961	206	2	∥	∥	PROPN
cana-1961	206	3	ω2	ω2	NOUN
cana-1961	206	4	∥τ	∥τ	PROPN
cana-1961	207	1	+	+	PROPN
cana-1961	207	2	∥	∥	PROPN
cana-1961	207	3	ω1	ω1	PROPN
cana-1961	207	4	∥s	∥s	PROPN
cana-1961	207	5	.	.	PUNCT
cana-1961	208	1	∥	∥	NOUN
cana-1961	208	2	ω2	ω2	NUM
cana-1961	208	3	∥t	∥t	NOUN
cana-1961	208	4	)	)	PUNCT
cana-1961	208	5	for	for	ADP
cana-1961	208	6	all	all	DET
cana-1961	208	7	ω1	ω1	PROPN
cana-1961	208	8	,	,	PUNCT
cana-1961	208	9	ω2	ω2	NOUN
cana-1961	208	10	∈	∈	PROPN
cana-1961	208	11	𝒲	𝒲	PROPN
cana-1961	208	12	,	,	PUNCT
cana-1961	208	13	υ	υ	PRON
cana-1961	208	14	∈	∈	PROPN
cana-1961	208	15	𝒵.	𝒵.	PROPN
cana-1961	208	16	then	then	ADV
cana-1961	208	17	there	there	PRON
cana-1961	208	18	is	be	VERB
cana-1961	208	19	a	a	DET
cana-1961	208	20	unique	unique	ADJ
cana-1961	208	21	c3	c3	NOUN
cana-1961	208	22	́	́	PUNCT
cana-1961	208	23	mapping	map	VERB
cana-1961	208	24	c3	c3	NOUN
cana-1961	208	25	́	́	PROPN
cana-1961	208	26	:	:	PUNCT
cana-1961	209	1	𝒲	𝒲	NOUN
cana-1961	209	2	→	→	SYM
cana-1961	209	3	𝒵	𝒵	NOUN
cana-1961	209	4	such	such	ADJ
cana-1961	209	5	that	that	SCONJ
cana-1961	209	6	∥	∥	NUM
cana-1961	209	7	𝔤(ω1	𝔤(ω1	NOUN
cana-1961	209	8	)	)	PUNCT
cana-1961	209	9	−	−	PROPN
cana-1961	209	10	c3	c3	PROPN
cana-1961	209	11	́	́	PROPN
cana-1961	209	12	(	(	PUNCT
cana-1961	209	13	ω1	ω1	PROPN
cana-1961	209	14	)	)	PUNCT
cana-1961	209	15	,	,	PUNCT
cana-1961	209	16	υ	υ	PROPN
cana-1961	209	17	∥≤	∥≤	PROPN
cana-1961	209	18	|𝔪|τ	|𝔪|τ	PROPN
cana-1961	209	19	|𝔪|3(|𝔪|τ−|𝔪|3	|𝔪|3(|𝔪|τ−|𝔪|3	PROPN
cana-1961	209	20	)	)	PUNCT
cana-1961	209	21	max	max	PROPN
cana-1961	209	22	{	{	PUNCT
cana-1961	209	23	|𝔪2(𝔪−1)|	|𝔪2(𝔪−1)|	PROPN
cana-1961	209	24	|2|.|4|	|2|.|4|	NOUN
cana-1961	209	25	θ	θ	PART
cana-1961	209	26	∥	∥	PUNCT
cana-1961	209	27	ω1	ω1	PROPN
cana-1961	209	28	∥τ	∥τ	PROPN
cana-1961	209	29	,	,	PUNCT
cana-1961	209	30	|𝔪2(𝔪−2)|	|𝔪2(𝔪−2)|	NOUN
cana-1961	209	31	|2(1−𝔪2)|	|2(1−𝔪2)|	PROPN
cana-1961	209	32	θ	θ	PROPN
cana-1961	209	33	∥	∥	PUNCT
cana-1961	209	34	ω1	ω1	PROPN
cana-1961	209	35	∥τ	∥τ	PROPN
cana-1961	209	36	}	}	PUNCT
cana-1961	209	37	for	for	ADP
cana-1961	209	38	all	all	DET
cana-1961	209	39	ω1	ω1	PROPN
cana-1961	209	40	∈	∈	PROPN
cana-1961	209	41	𝒲	𝒲	PROPN
cana-1961	209	42	,	,	PUNCT
cana-1961	209	43	υ	υ	DET
cana-1961	209	44	∈	∈	PROPN
cana-1961	209	45	𝒵.	𝒵.	PROPN
cana-1961	209	46	communications	communication	NOUN
cana-1961	209	47	on	on	ADP
cana-1961	209	48	applied	apply	VERB
cana-1961	209	49	nonlinear	nonlinear	ADJ
cana-1961	209	50	analysis	analysis	NOUN
cana-1961	209	51	issn	issn	NOUN
cana-1961	209	52	:	:	PUNCT
cana-1961	209	53	1074	1074	NUM
cana-1961	209	54	-	-	PUNCT
cana-1961	209	55	133x	133x	NUM
cana-1961	209	56	vol	vol	NOUN
cana-1961	209	57	32	32	NUM
cana-1961	209	58	no	no	NOUN
cana-1961	209	59	.	.	NOUN
cana-1961	209	60	3	3	NUM
cana-1961	209	61	(	(	PUNCT
cana-1961	209	62	2025	2025	NUM
cana-1961	209	63	)	)	PUNCT
cana-1961	210	1	311	311	NUM
cana-1961	210	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-1961	210	3	proof	proof	NOUN
cana-1961	210	4	:	:	PUNCT
cana-1961	210	5	assuming	assume	VERB
cana-1961	210	6	φ(ω1	φ(ω1	NOUN
cana-1961	210	7	,	,	PUNCT
cana-1961	210	8	ω2	ω2	ADJ
cana-1961	210	9	)	)	PUNCT
cana-1961	210	10	∶=	∶=	NUM
cana-1961	210	11	θ	θ	NOUN
cana-1961	210	12	(	(	PUNCT
cana-1961	210	13	‖ω1‖τ	‖ω1‖τ	X
cana-1961	210	14	+	+	CCONJ
cana-1961	210	15	‖ω2‖τ+∥	‖ω2‖τ+∥	PROPN
cana-1961	210	16	ω1	ω1	PROPN
cana-1961	210	17	∥s	∥s	PROPN
cana-1961	210	18	.	.	PUNCT
cana-1961	211	1	∥	∥	NOUN
cana-1961	211	2	ω2	ω2	NUM
cana-1961	211	3	∥t	∥t	NOUN
cana-1961	211	4	)	)	PUNCT
cana-1961	211	5	for	for	ADP
cana-1961	211	6	all	all	DET
cana-1961	211	7	ω1	ω1	PROPN
cana-1961	211	8	,	,	PUNCT
cana-1961	211	9	ω2	ω2	NOUN
cana-1961	211	10	∈	∈	PROPN
cana-1961	211	11	𝒲	𝒲	PROPN
cana-1961	211	12	,	,	PUNCT
cana-1961	211	13	and	and	CCONJ
cana-1961	211	14	by	by	ADP
cana-1961	211	15	choosing	choose	VERB
cana-1961	211	16	£	£	PROPN
cana-1961	211	17	=	=	SYM
cana-1961	211	18	|𝔪|3−τ	|𝔪|3−τ	PROPN
cana-1961	211	19	,	,	PUNCT
cana-1961	211	20	the	the	DET
cana-1961	211	21	expected	expect	VERB
cana-1961	211	22	result	result	NOUN
cana-1961	211	23	can	can	AUX
cana-1961	211	24	be	be	AUX
cana-1961	211	25	obtained	obtain	VERB
cana-1961	211	26	by	by	ADP
cana-1961	211	27	theorem	theorem	ADJ
cana-1961	211	28	4	4	NUM
cana-1961	211	29	.	.	PUNCT
cana-1961	212	1	theorem	theorem	NOUN
cana-1961	212	2	5	5	NUM
cana-1961	212	3	:	:	PUNCT
cana-1961	212	4	let	let	VERB
cana-1961	212	5	φ	φ	NUM
cana-1961	212	6	:	:	PUNCT
cana-1961	212	7	𝒲	𝒲	NOUN
cana-1961	212	8	×	×	NOUN
cana-1961	212	9	𝒲	𝒲	NOUN
cana-1961	212	10	→	→	SYM
cana-1961	212	11	[	[	X
cana-1961	212	12	0	0	NUM
cana-1961	212	13	,	,	PUNCT
cana-1961	212	14	∞	∞	PROPN
cana-1961	212	15	)	)	PUNCT
cana-1961	212	16	be	be	VERB
cana-1961	212	17	an	an	DET
cana-1961	212	18	odd	odd	ADJ
cana-1961	212	19	function	function	NOUN
cana-1961	212	20	such	such	ADJ
cana-1961	212	21	that	that	SCONJ
cana-1961	212	22	there	there	PRON
cana-1961	212	23	is	be	VERB
cana-1961	212	24	a	a	DET
cana-1961	212	25	constant	constant	ADJ
cana-1961	212	26	0	0	NUM
cana-1961	212	27	<	<	X
cana-1961	212	28	£	£	X
cana-1961	212	29	<	<	X
cana-1961	212	30	1	1	NUM
cana-1961	212	31	with	with	ADP
cana-1961	212	32	φ	φ	PROPN
cana-1961	212	33	(	(	PUNCT
cana-1961	212	34	ω1	ω1	PROPN
cana-1961	212	35	𝔪	𝔪	NOUN
cana-1961	212	36	,	,	PUNCT
cana-1961	212	37	ω2	ω2	ADJ
cana-1961	212	38	𝔪	𝔪	NOUN
cana-1961	212	39	)	)	PUNCT
cana-1961	212	40	≤	≤	NUM
cana-1961	212	41	£	£	SYM
cana-1961	212	42	|𝔪|3	|𝔪|3	NOUN
cana-1961	212	43	φ(ω1	φ(ω1	NOUN
cana-1961	212	44	,	,	PUNCT
cana-1961	212	45	ω2	ω2	NUM
cana-1961	212	46	)	)	PUNCT
cana-1961	212	47	34	34	NUM
cana-1961	212	48	for	for	ADP
cana-1961	212	49	all	all	DET
cana-1961	212	50	ω1	ω1	PROPN
cana-1961	212	51	,	,	PUNCT
cana-1961	212	52	ω2	ω2	NOUN
cana-1961	212	53	∈	∈	PROPN
cana-1961	212	54	𝒲.	𝒲.	PROPN
cana-1961	212	55	let	let	VERB
cana-1961	212	56	𝔤	𝔤	PRON
cana-1961	212	57	:	:	PUNCT
cana-1961	212	58	𝒲	𝒲	NOUN
cana-1961	212	59	→	→	PUNCT
cana-1961	212	60	𝒵	𝒵	PRON
cana-1961	212	61	be	be	VERB
cana-1961	212	62	an	an	DET
cana-1961	212	63	odd	odd	ADJ
cana-1961	212	64	mapping	mapping	NOUN
cana-1961	212	65	satisfying	satisfying	NOUN
cana-1961	212	66	eq	eq	ADP
cana-1961	212	67	19	19	NUM
cana-1961	212	68	.	.	PUNCT
cana-1961	213	1	then	then	ADV
cana-1961	213	2	there	there	PRON
cana-1961	213	3	is	be	VERB
cana-1961	213	4	a	a	DET
cana-1961	213	5	unique	unique	ADJ
cana-1961	213	6	c3	c3	NOUN
cana-1961	213	7	́	́	PUNCT
cana-1961	213	8	mapping	map	VERB
cana-1961	213	9	c3	c3	NOUN
cana-1961	213	10	́	́	PROPN
cana-1961	213	11	:	:	PUNCT
cana-1961	213	12	𝒲	𝒲	NOUN
cana-1961	213	13	→	→	SYM
cana-1961	213	14	𝒵	𝒵	NOUN
cana-1961	213	15	such	such	ADJ
cana-1961	213	16	that	that	DET
cana-1961	213	17	‖𝔤(ω1	‖𝔤(ω1	NOUN
cana-1961	213	18	)	)	PUNCT
cana-1961	213	19	−	−	PROPN
cana-1961	214	1	c3	c3	PROPN
cana-1961	214	2	́	́	PROPN
cana-1961	214	3	(	(	PUNCT
cana-1961	214	4	ω1	ω1	PROPN
cana-1961	214	5	)	)	PUNCT
cana-1961	214	6	,	,	PUNCT
cana-1961	214	7	υ‖	υ‖	VERB
cana-1961	214	8	≤	≤	ADJ
cana-1961	214	9	£	£	SYM
cana-1961	214	10	|𝔪|3(1−£	|𝔪|3(1−£	PROPN
cana-1961	214	11	)	)	PUNCT
cana-1961	214	12	φ(ω1	φ(ω1	NOUN
cana-1961	214	13	)	)	PUNCT
cana-1961	214	14	35	35	NUM
cana-1961	214	15	where	where	SCONJ
cana-1961	214	16	φ(ω1	φ(ω1	NOUN
cana-1961	214	17	)	)	PUNCT
cana-1961	214	18	=	=	SYM
cana-1961	214	19	max	max	PROPN
cana-1961	214	20	{	{	PUNCT
cana-1961	214	21	|𝔪2(𝔪	|𝔪2(𝔪	PROPN
cana-1961	214	22	−	−	PROPN
cana-1961	214	23	1)|	1)|	NUM
cana-1961	214	24	|2|	|2|	PROPN
cana-1961	214	25	.	.	PUNCT
cana-1961	215	1	|4|	|4|	PROPN
cana-1961	215	2	φ(0	φ(0	PROPN
cana-1961	215	3	,	,	PUNCT
cana-1961	215	4	ω1	ω1	PROPN
cana-1961	215	5	)	)	PUNCT
cana-1961	215	6	,	,	PUNCT
cana-1961	215	7	|𝔪2(𝔪	|𝔪2(𝔪	PROPN
cana-1961	215	8	−	−	PROPN
cana-1961	215	9	2)|	2)|	NUM
cana-1961	216	1	|2(1	|2(1	ADP
cana-1961	216	2	−	−	PROPN
cana-1961	216	3	𝔪2)|	𝔪2)|	PROPN
cana-1961	216	4	φ(ω1	φ(ω1	NOUN
cana-1961	216	5	,	,	PUNCT
cana-1961	216	6	0	0	NUM
cana-1961	216	7	)	)	PUNCT
cana-1961	216	8	}	}	PUNCT
cana-1961	216	9	for	for	ADP
cana-1961	216	10	all	all	DET
cana-1961	216	11	ω1	ω1	PROPN
cana-1961	216	12	∈	∈	PROPN
cana-1961	216	13	𝒲	𝒲	PROPN
cana-1961	216	14	,	,	PUNCT
cana-1961	216	15	υ	υ	DET
cana-1961	216	16	∈	∈	NOUN
cana-1961	216	17	𝒵.	𝒵.	PROPN
cana-1961	216	18	proof	proof	NOUN
cana-1961	216	19	:	:	PUNCT
cana-1961	216	20	according	accord	VERB
cana-1961	216	21	to	to	ADP
cana-1961	216	22	eq	eq	ADP
cana-1961	216	23	26	26	NUM
cana-1961	216	24	,	,	PUNCT
cana-1961	216	25	‖𝔤(ω1	‖𝔤(ω1	NOUN
cana-1961	216	26	)	)	PUNCT
cana-1961	216	27	−	−	NOUN
cana-1961	217	1	𝔪3𝔤	𝔪3𝔤	PROPN
cana-1961	217	2	(	(	PUNCT
cana-1961	217	3	ω1	ω1	PROPN
cana-1961	217	4	𝔪	𝔪	NOUN
cana-1961	217	5	)	)	PUNCT
cana-1961	217	6	,	,	PUNCT
cana-1961	217	7	υ‖	υ‖	VERB
cana-1961	217	8	≤	≤	PROPN
cana-1961	217	9	φ	φ	PROPN
cana-1961	217	10	(	(	PUNCT
cana-1961	217	11	ω1	ω1	PROPN
cana-1961	217	12	𝔪	𝔪	NOUN
cana-1961	217	13	)	)	PUNCT
cana-1961	217	14	36	36	NUM
cana-1961	217	15	for	for	ADP
cana-1961	217	16	all	all	DET
cana-1961	217	17	ω1	ω1	PROPN
cana-1961	217	18	∈	∈	PROPN
cana-1961	217	19	𝒲	𝒲	PROPN
cana-1961	217	20	,	,	PUNCT
cana-1961	217	21	υ	υ	PRON
cana-1961	217	22	∈	∈	PROPN
cana-1961	217	23	𝒵.	𝒵.	PROPN
cana-1961	217	24	let	let	VERB
cana-1961	217	25	(	(	PUNCT
cana-1961	217	26	γ	γ	X
cana-1961	217	27	,	,	PUNCT
cana-1961	217	28	ð	ð	NUM
cana-1961	217	29	)	)	PUNCT
cana-1961	217	30	be	be	VERB
cana-1961	217	31	the	the	DET
cana-1961	217	32	gms	gms	NOUN
cana-1961	217	33	as	as	SCONJ
cana-1961	217	34	defined	define	VERB
cana-1961	217	35	by	by	ADP
cana-1961	217	36	theorem	theorem	NOUN
cana-1961	217	37	2	2	NUM
cana-1961	217	38	.	.	PUNCT
cana-1961	217	39	now	now	ADV
cana-1961	217	40	,	,	PUNCT
cana-1961	217	41	define	define	VERB
cana-1961	217	42	the	the	DET
cana-1961	217	43	function	function	NOUN
cana-1961	217	44	λ	λ	PROPN
cana-1961	217	45	:	:	PUNCT
cana-1961	217	46	γ	γ	X
cana-1961	217	47	→	→	SYM
cana-1961	217	48	γ	γ	X
cana-1961	217	49	such	such	ADJ
cana-1961	217	50	that	that	PRON
cana-1961	217	51	λ𝔣(ω1	λ𝔣(ω1	ADP
cana-1961	217	52	)	)	PUNCT
cana-1961	217	53	=	=	PUNCT
cana-1961	217	54	𝔪3𝔣	𝔪3𝔣	X
cana-1961	217	55	(	(	PUNCT
cana-1961	217	56	ω1	ω1	PROPN
cana-1961	217	57	𝔪	𝔪	NOUN
cana-1961	217	58	)	)	PUNCT
cana-1961	217	59	37	37	NUM
cana-1961	217	60	for	for	ADP
cana-1961	217	61	all	all	PRON
cana-1961	217	62	ω1	ω1	PROPN
cana-1961	217	63	∈	∈	PROPN
cana-1961	217	64	𝒲.	𝒲.	PROPN
cana-1961	217	65	let	let	VERB
cana-1961	217	66	𝔣	𝔣	ADP
cana-1961	217	67	,	,	PUNCT
cana-1961	217	68	𝔥	𝔥	PROPN
cana-1961	217	69	∈	∈	PROPN
cana-1961	217	70	γ	γ	PROPN
cana-1961	217	71	,	,	PUNCT
cana-1961	217	72	be	be	AUX
cana-1961	217	73	given	give	VERB
cana-1961	217	74	such	such	ADJ
cana-1961	217	75	that	that	DET
cana-1961	217	76	ð(𝔣	ð(𝔣	PROPN
cana-1961	217	77	,	,	PUNCT
cana-1961	217	78	𝔥	𝔥	NOUN
cana-1961	217	79	)	)	PUNCT
cana-1961	217	80	=	=	VERB
cana-1961	218	1	ϵ.	ϵ.	NOUN
cana-1961	218	2	then	then	ADV
cana-1961	218	3	∥	∥	NUM
cana-1961	218	4	𝔣(ω1	𝔣(ω1	NOUN
cana-1961	218	5	)	)	PUNCT
cana-1961	218	6	−	−	ADP
cana-1961	218	7	𝔥(ω1	𝔥(ω1	ADJ
cana-1961	218	8	)	)	PUNCT
cana-1961	218	9	,	,	PUNCT
cana-1961	218	10	υ	υ	PROPN
cana-1961	218	11	∥≤	∥≤	PROPN
cana-1961	218	12	ϵ	ϵ	X
cana-1961	218	13	φ	φ	PROPN
cana-1961	218	14	(	(	PUNCT
cana-1961	218	15	ω1	ω1	PROPN
cana-1961	218	16	𝔪	𝔪	NOUN
cana-1961	218	17	)	)	PUNCT
cana-1961	218	18	38	38	NUM
cana-1961	218	19	for	for	ADP
cana-1961	218	20	all	all	DET
cana-1961	218	21	ω1	ω1	PROPN
cana-1961	218	22	∈	∈	PROPN
cana-1961	218	23	𝒲	𝒲	PROPN
cana-1961	218	24	,	,	PUNCT
cana-1961	218	25	υ	υ	PRON
cana-1961	218	26	∈	∈	NOUN
cana-1961	218	27	𝒵.	𝒵.	PROPN
cana-1961	218	28	hence	hence	ADV
cana-1961	218	29	∥	∥	PUNCT
cana-1961	218	30	λ𝔣(ω1	λ𝔣(ω1	ADP
cana-1961	218	31	)	)	PUNCT
cana-1961	218	32	−	−	PROPN
cana-1961	218	33	λ𝔥(ω1	λ𝔥(ω1	NOUN
cana-1961	218	34	)	)	PUNCT
cana-1961	218	35	,	,	PUNCT
cana-1961	218	36	υ	υ	PRON
cana-1961	218	37	∥=	∥=	ADJ
cana-1961	218	38	‖𝔪3𝔣	‖𝔪3𝔣	PROPN
cana-1961	218	39	(	(	PUNCT
cana-1961	218	40	ω1	ω1	PROPN
cana-1961	218	41	𝔪	𝔪	NOUN
cana-1961	218	42	)	)	PUNCT
cana-1961	218	43	−	−	PROPN
cana-1961	219	1	𝔪3𝔥	𝔪3𝔥	PROPN
cana-1961	219	2	(	(	PUNCT
cana-1961	219	3	ω1	ω1	PROPN
cana-1961	219	4	𝔪	𝔪	NOUN
cana-1961	219	5	)	)	PUNCT
cana-1961	219	6	,	,	PUNCT
cana-1961	219	7	υ‖	υ‖	VERB
cana-1961	219	8	≤	≤	PROPN
cana-1961	219	9	|𝔪|3ϵ	|𝔪|3ϵ	PROPN
cana-1961	219	10	φ	φ	PROPN
cana-1961	219	11	(	(	PUNCT
cana-1961	219	12	ω1	ω1	PROPN
cana-1961	219	13	𝔪	𝔪	NOUN
cana-1961	219	14	)	)	PUNCT
cana-1961	219	15	≤	≤	NUM
cana-1961	219	16	|𝔪|3ϵ	|𝔪|3ϵ	X
cana-1961	219	17	£	£	SYM
cana-1961	219	18	|𝔪|3	|𝔪|3	NOUN
cana-1961	219	19	.	.	PUNCT
cana-1961	220	1	φ(ω1	φ(ω1	NOUN
cana-1961	220	2	)	)	PUNCT
cana-1961	220	3	for	for	ADP
cana-1961	220	4	all	all	DET
cana-1961	220	5	ω1	ω1	PROPN
cana-1961	220	6	∈	∈	PROPN
cana-1961	220	7	𝒲	𝒲	PROPN
cana-1961	220	8	,	,	PUNCT
cana-1961	220	9	υ	υ	PRON
cana-1961	220	10	∈	∈	PROPN
cana-1961	220	11	𝒵	𝒵	PROPN
cana-1961	220	12	,	,	PUNCT
cana-1961	220	13	that	that	PRON
cana-1961	220	14	is	be	AUX
cana-1961	220	15	ð(λ𝔣	ð(λ𝔣	NOUN
cana-1961	220	16	,	,	PUNCT
cana-1961	220	17	λ𝔥	λ𝔥	NOUN
cana-1961	220	18	)	)	PUNCT
cana-1961	220	19	≤	≤	NOUN
cana-1961	220	20	£	£	SYM
cana-1961	220	21	ϵ.	ϵ.	NOUN
cana-1961	220	22	therefore	therefore	ADV
cana-1961	220	23	ð(λ𝔣	ð(λ𝔣	NOUN
cana-1961	220	24	,	,	PUNCT
cana-1961	220	25	λ𝔥	λ𝔥	NOUN
cana-1961	220	26	)	)	PUNCT
cana-1961	220	27	≤	≤	NOUN
cana-1961	220	28	£	£	SYM
cana-1961	220	29	ð(𝔣	ð(𝔣	PROPN
cana-1961	220	30	,	,	PUNCT
cana-1961	220	31	𝔥	𝔥	NOUN
cana-1961	220	32	)	)	PUNCT
cana-1961	220	33	39	39	NUM
cana-1961	220	34	for	for	ADP
cana-1961	220	35	all	all	DET
cana-1961	220	36	𝔣	𝔣	ADJ
cana-1961	220	37	,	,	PUNCT
cana-1961	220	38	𝔥	𝔥	PROPN
cana-1961	220	39	∈	∈	PROPN
cana-1961	220	40	γ	γ	X
cana-1961	220	41	.	.	PUNCT
cana-1961	221	1	according	accord	VERB
cana-1961	221	2	to	to	ADP
cana-1961	221	3	eq	eq	ADP
cana-1961	221	4	36	36	NUM
cana-1961	221	5	,	,	PUNCT
cana-1961	221	6	ð(𝔤	ð(𝔤	PROPN
cana-1961	221	7	,	,	PUNCT
cana-1961	221	8	λ𝔤	λ𝔤	NOUN
cana-1961	221	9	)	)	PUNCT
cana-1961	221	10	≤	≤	NUM
cana-1961	221	11	£	£	NUM
cana-1961	221	12	|𝔪|3	|𝔪|3	NOUN
cana-1961	221	13	<	<	X
cana-1961	221	14	∞.	∞.	PROPN
cana-1961	221	15	40	40	NUM
cana-1961	221	16	so	so	ADV
cana-1961	221	17	communications	communication	NOUN
cana-1961	221	18	on	on	ADP
cana-1961	221	19	applied	apply	VERB
cana-1961	221	20	nonlinear	nonlinear	ADJ
cana-1961	221	21	analysis	analysis	NOUN
cana-1961	221	22	issn	issn	NOUN
cana-1961	221	23	:	:	PUNCT
cana-1961	221	24	1074	1074	NUM
cana-1961	221	25	-	-	PUNCT
cana-1961	221	26	133x	133x	NUM
cana-1961	221	27	vol	vol	NOUN
cana-1961	221	28	32	32	NUM
cana-1961	221	29	no	no	NOUN
cana-1961	221	30	.	.	NOUN
cana-1961	221	31	3	3	NUM
cana-1961	221	32	(	(	PUNCT
cana-1961	221	33	2025	2025	NUM
cana-1961	221	34	)	)	PUNCT
cana-1961	221	35	312	312	NUM
cana-1961	221	36	https://internationalpubls.com	https://internationalpubls.com	X
cana-1961	221	37	ð(𝔤	ð(𝔤	PROPN
cana-1961	221	38	,	,	PUNCT
cana-1961	221	39	c3	c3	PROPN
cana-1961	221	40	́	́	PROPN
cana-1961	221	41	)	)	PUNCT
cana-1961	221	42	≤	≤	PROPN
cana-1961	221	43	£	£	SYM
cana-1961	221	44	|𝔪|3(1−£	|𝔪|3(1−£	PROPN
cana-1961	221	45	)	)	PUNCT
cana-1961	221	46	.	.	PUNCT
cana-1961	222	1	this	this	PRON
cana-1961	222	2	gives	give	VERB
cana-1961	222	3	us	we	PRON
cana-1961	222	4	the	the	DET
cana-1961	222	5	inequality	inequality	NOUN
cana-1961	222	6	eq	eq	ADP
cana-1961	222	7	35	35	NUM
cana-1961	222	8	.	.	PUNCT
cana-1961	223	1	this	this	DET
cana-1961	223	2	proof	proof	NOUN
cana-1961	223	3	follows	follow	VERB
cana-1961	223	4	the	the	DET
cana-1961	223	5	same	same	ADJ
cana-1961	223	6	pattern	pattern	NOUN
cana-1961	223	7	as	as	ADP
cana-1961	223	8	theorem	theorem	ADJ
cana-1961	223	9	4	4	NUM
cana-1961	223	10	.	.	PUNCT
cana-1961	223	11	corollary	corollary	ADJ
cana-1961	223	12	4	4	NUM
cana-1961	223	13	:	:	PUNCT
cana-1961	223	14	let	let	VERB
cana-1961	223	15	θ	θ	PROPN
cana-1961	223	16	≥	≥	X
cana-1961	223	17	0	0	NUM
cana-1961	223	18	and	and	CCONJ
cana-1961	223	19	τ	τ	X
cana-1961	223	20	=	=	SYM
cana-1961	223	21	s	s	PART
cana-1961	224	1	+	+	NUM
cana-1961	224	2	t	t	AUX
cana-1961	224	3	be	be	VERB
cana-1961	224	4	a	a	DET
cana-1961	224	5	positive	positive	ADJ
cana-1961	224	6	real	real	ADJ
cana-1961	224	7	number	number	NOUN
cana-1961	224	8	with	with	ADP
cana-1961	224	9	τ	τ	PROPN
cana-1961	224	10	>	>	X
cana-1961	224	11	3	3	X
cana-1961	224	12	.	.	PUNCT
cana-1961	225	1	let	let	VERB
cana-1961	225	2	𝔤	𝔤	PRON
cana-1961	225	3	:	:	PUNCT
cana-1961	225	4	𝒲	𝒲	NOUN
cana-1961	225	5	→	→	PUNCT
cana-1961	225	6	𝒵	𝒵	PRON
cana-1961	225	7	be	be	VERB
cana-1961	225	8	an	an	DET
cana-1961	225	9	odd	odd	ADJ
cana-1961	225	10	mapping	mapping	NOUN
cana-1961	225	11	with	with	ADP
cana-1961	225	12	𝔤(0	𝔤(0	PROPN
cana-1961	225	13	)	)	PUNCT
cana-1961	226	1	=	=	SYM
cana-1961	226	2	0	0	PUNCT
cana-1961	226	3	satisfying	satisfy	VERB
cana-1961	226	4	∥	∥	PUNCT
cana-1961	226	5	d𝔤(ω1	d𝔤(ω1	NOUN
cana-1961	226	6	,	,	PUNCT
cana-1961	226	7	ω2	ω2	NUM
cana-1961	226	8	)	)	PUNCT
cana-1961	226	9	,	,	PUNCT
cana-1961	226	10	υ	υ	PROPN
cana-1961	226	11	∥≤	∥≤	PROPN
cana-1961	226	12	θ(∥	θ(∥	PROPN
cana-1961	226	13	ω1	ω1	PROPN
cana-1961	226	14	∥τ	∥τ	PROPN
cana-1961	227	1	+	+	PROPN
cana-1961	227	2	∥	∥	PROPN
cana-1961	227	3	ω2	ω2	NOUN
cana-1961	227	4	∥τ	∥τ	PROPN
cana-1961	228	1	+	+	PROPN
cana-1961	228	2	∥	∥	PROPN
cana-1961	228	3	ω1	ω1	PROPN
cana-1961	228	4	∥s	∥s	PROPN
cana-1961	228	5	.	.	PUNCT
cana-1961	229	1	∥	∥	NOUN
cana-1961	229	2	ω2	ω2	NUM
cana-1961	229	3	∥t	∥t	NOUN
cana-1961	229	4	)	)	PUNCT
cana-1961	229	5	for	for	ADP
cana-1961	229	6	all	all	DET
cana-1961	229	7	ω1	ω1	PROPN
cana-1961	229	8	,	,	PUNCT
cana-1961	229	9	ω2	ω2	NOUN
cana-1961	229	10	∈	∈	PROPN
cana-1961	229	11	𝒲	𝒲	PROPN
cana-1961	229	12	,	,	PUNCT
cana-1961	229	13	υ	υ	PRON
cana-1961	229	14	∈	∈	PROPN
cana-1961	229	15	𝒵.	𝒵.	PROPN
cana-1961	229	16	then	then	ADV
cana-1961	229	17	there	there	PRON
cana-1961	229	18	is	be	VERB
cana-1961	229	19	a	a	DET
cana-1961	229	20	unique	unique	ADJ
cana-1961	229	21	c3	c3	NOUN
cana-1961	229	22	́	́	PUNCT
cana-1961	229	23	mapping	map	VERB
cana-1961	229	24	c3	c3	NOUN
cana-1961	229	25	́	́	PROPN
cana-1961	229	26	:	:	PUNCT
cana-1961	230	1	𝒲	𝒲	NOUN
cana-1961	230	2	→	→	SYM
cana-1961	230	3	𝒵	𝒵	NOUN
cana-1961	230	4	such	such	ADJ
cana-1961	230	5	that	that	SCONJ
cana-1961	230	6	∥	∥	NUM
cana-1961	230	7	𝔤(ω1	𝔤(ω1	NOUN
cana-1961	230	8	)	)	PUNCT
cana-1961	230	9	−	−	PROPN
cana-1961	230	10	c3	c3	PROPN
cana-1961	230	11	́	́	PROPN
cana-1961	230	12	(	(	PUNCT
cana-1961	230	13	ω1	ω1	PROPN
cana-1961	230	14	)	)	PUNCT
cana-1961	230	15	,	,	PUNCT
cana-1961	230	16	υ	υ	PROPN
cana-1961	230	17	∥≤	∥≤	PROPN
cana-1961	230	18	|𝔪|τ	|𝔪|τ	PROPN
cana-1961	230	19	|𝔪|3(|𝔪|3−|𝔪|τ	|𝔪|3(|𝔪|3−|𝔪|τ	NUM
cana-1961	230	20	)	)	PUNCT
cana-1961	230	21	max	max	PROPN
cana-1961	230	22	{	{	PUNCT
cana-1961	230	23	|𝔪2(𝔪−1)|	|𝔪2(𝔪−1)|	PROPN
cana-1961	230	24	|2|.|4|	|2|.|4|	NOUN
cana-1961	230	25	θ	θ	PART
cana-1961	230	26	∥	∥	PUNCT
cana-1961	230	27	ω1	ω1	PROPN
cana-1961	230	28	∥τ	∥τ	PROPN
cana-1961	230	29	,	,	PUNCT
cana-1961	230	30	|𝔪2(𝔪−2)|	|𝔪2(𝔪−2)|	NOUN
cana-1961	230	31	|2(1−𝔪2)|	|2(1−𝔪2)|	PROPN
cana-1961	230	32	θ	θ	PROPN
cana-1961	230	33	∥	∥	PUNCT
cana-1961	230	34	ω1	ω1	PROPN
cana-1961	230	35	∥τ	∥τ	PROPN
cana-1961	230	36	}	}	PUNCT
cana-1961	230	37	for	for	ADP
cana-1961	230	38	all	all	DET
cana-1961	230	39	ω1	ω1	PROPN
cana-1961	230	40	∈	∈	PROPN
cana-1961	230	41	𝒲	𝒲	PROPN
cana-1961	230	42	,	,	PUNCT
cana-1961	230	43	υ	υ	DET
cana-1961	230	44	∈	∈	NOUN
cana-1961	230	45	𝒵.	𝒵.	PROPN
cana-1961	230	46	proof	proof	NOUN
cana-1961	230	47	:	:	PUNCT
cana-1961	230	48	assuming	assume	VERB
cana-1961	230	49	φ(ω1	φ(ω1	NOUN
cana-1961	230	50	,	,	PUNCT
cana-1961	230	51	ω2	ω2	ADJ
cana-1961	230	52	):	):	PUNCT
cana-1961	230	53	=	=	SYM
cana-1961	230	54	θ	θ	PROPN
cana-1961	230	55	(	(	PUNCT
cana-1961	230	56	‖ω1‖τ	‖ω1‖τ	X
cana-1961	230	57	+	+	CCONJ
cana-1961	230	58	‖ω2‖τ+∥	‖ω2‖τ+∥	PROPN
cana-1961	230	59	ω1	ω1	PROPN
cana-1961	230	60	∥s	∥s	PROPN
cana-1961	230	61	.	.	PUNCT
cana-1961	231	1	∥	∥	NOUN
cana-1961	231	2	ω2	ω2	NUM
cana-1961	231	3	∥t	∥t	NOUN
cana-1961	231	4	)	)	PUNCT
cana-1961	231	5	for	for	ADP
cana-1961	231	6	all	all	DET
cana-1961	231	7	ω1	ω1	PROPN
cana-1961	231	8	,	,	PUNCT
cana-1961	231	9	ω2	ω2	NOUN
cana-1961	231	10	∈	∈	PROPN
cana-1961	231	11	𝒲	𝒲	PROPN
cana-1961	231	12	,	,	PUNCT
cana-1961	231	13	and	and	CCONJ
cana-1961	231	14	by	by	ADP
cana-1961	231	15	choosing	choose	VERB
cana-1961	231	16	£	£	PROPN
cana-1961	231	17	=	=	SYM
cana-1961	231	18	|𝔪|τ−3	|𝔪|τ−3	NOUN
cana-1961	231	19	,	,	PUNCT
cana-1961	231	20	the	the	DET
cana-1961	231	21	expected	expect	VERB
cana-1961	231	22	result	result	NOUN
cana-1961	231	23	can	can	AUX
cana-1961	231	24	be	be	AUX
cana-1961	231	25	obtained	obtain	VERB
cana-1961	231	26	by	by	ADP
cana-1961	231	27	theorem	theorem	NOUN
cana-1961	231	28	5	5	NUM
cana-1961	231	29	.	.	NOUN
cana-1961	231	30	example	example	NOUN
cana-1961	231	31	4	4	NUM
cana-1961	231	32	:	:	PUNCT
cana-1961	231	33	let	let	VERB
cana-1961	231	34	ρ	ρ	PROPN
cana-1961	231	35	>	>	X
cana-1961	231	36	2	2	NUM
cana-1961	231	37	be	be	AUX
cana-1961	231	38	a	a	DET
cana-1961	231	39	prime	prime	ADJ
cana-1961	231	40	number	number	NOUN
cana-1961	231	41	and	and	CCONJ
cana-1961	231	42	𝒲	𝒲	NOUN
cana-1961	231	43	=	=	SYM
cana-1961	231	44	𝒵	𝒵	NOUN
cana-1961	231	45	=	=	SYM
cana-1961	231	46	ℚ	ℚ	PROPN
cana-1961	231	47	𝔭	𝔭	NOUN
cana-1961	231	48	.	.	PUNCT
cana-1961	232	1	define	define	VERB
cana-1961	232	2	𝔤	𝔤	NOUN
cana-1961	232	3	:	:	PUNCT
cana-1961	232	4	𝒲	𝒲	NOUN
cana-1961	232	5	→	→	SYM
cana-1961	232	6	𝒵	𝒵	PROPN
cana-1961	232	7	by	by	ADP
cana-1961	232	8	𝔤(ω1	𝔤(ω1	NOUN
cana-1961	232	9	)	)	PUNCT
cana-1961	232	10	=	=	SYM
cana-1961	232	11	ω1	ω1	X
cana-1961	232	12	3	3	NUM
cana-1961	232	13	+	+	SYM
cana-1961	232	14	1	1	NUM
cana-1961	232	15	for	for	ADP
cana-1961	232	16	all	all	DET
cana-1961	232	17	ω1	ω1	PROPN
cana-1961	232	18	∈	∈	PROPN
cana-1961	232	19	𝒲.	𝒲.	PROPN
cana-1961	232	20	since	since	SCONJ
cana-1961	232	21	|2n|ρ	|2n|ρ	NOUN
cana-1961	232	22	=	=	SYM
cana-1961	232	23	1	1	NUM
cana-1961	232	24	.	.	PUNCT
cana-1961	232	25	|d𝔤(ω1	|d𝔤(ω1	NOUN
cana-1961	232	26	,	,	PUNCT
cana-1961	232	27	ω2)|	ω2)|	NOUN
cana-1961	232	28	=	=	SYM
cana-1961	233	1	|	|	ADV
cana-1961	233	2	88	88	NUM
cana-1961	233	3	9	9	NUM
cana-1961	233	4	|	|	ADV
cana-1961	233	5	≤	≤	PUNCT
cana-1961	234	1	θ(‖ω1‖τ	θ(‖ω1‖τ	PROPN
cana-1961	234	2	+	+	CCONJ
cana-1961	234	3	‖ω2‖τ+∥	‖ω2‖τ+∥	PROPN
cana-1961	234	4	ω1	ω1	PROPN
cana-1961	234	5	∥s	∥s	PROPN
cana-1961	234	6	.	.	PUNCT
cana-1961	235	1	∥	∥	NOUN
cana-1961	235	2	ω2	ω2	NUM
cana-1961	235	3	∥t	∥t	NOUN
cana-1961	235	4	)	)	PUNCT
cana-1961	235	5	(	(	PUNCT
cana-1961	235	6	∀	∀	X
cana-1961	235	7	ω1	ω1	PROPN
cana-1961	235	8	,	,	PUNCT
cana-1961	235	9	ω2	ω2	NOUN
cana-1961	235	10	∈	∈	PROPN
cana-1961	235	11	𝒲	𝒲	PROPN
cana-1961	235	12	)	)	PUNCT
cana-1961	235	13	,	,	PUNCT
cana-1961	235	14	and	and	CCONJ
cana-1961	235	15	‖	‖	ADJ
cana-1961	235	16	h(2nω1	h(2nω1	ADJ
cana-1961	235	17	)	)	PUNCT
cana-1961	235	18	23n	23n	NOUN
cana-1961	235	19	−	−	PROPN
cana-1961	235	20	h(2n−1ω1	h(2n−1ω1	SYM
cana-1961	235	21	)	)	PUNCT
cana-1961	235	22	23(n−1	23(n−1	NUM
cana-1961	235	23	)	)	PUNCT
cana-1961	236	1	‖	‖	ADJ
cana-1961	236	2	=	=	PUNCT
cana-1961	236	3	|49|	|49|	NOUN
cana-1961	236	4	≠	≠	PROPN
cana-1961	236	5	0	0	NUM
cana-1961	236	6	.	.	PUNCT
cana-1961	237	1	hence	hence	ADV
cana-1961	237	2	{	{	PUNCT
cana-1961	237	3	2	2	NUM
cana-1961	237	4	−3n	−3n	NUM
cana-1961	237	5	h(2	h(2	NOUN
cana-1961	237	6	n	n	SYM
cana-1961	237	7	ω1	ω1	NOUN
cana-1961	237	8	)	)	PUNCT
cana-1961	237	9	}	}	PUNCT
cana-1961	237	10	is	be	AUX
cana-1961	237	11	not	not	PART
cana-1961	237	12	a	a	DET
cana-1961	237	13	cauchy	cauchy	ADJ
cana-1961	237	14	sequence	sequence	NOUN
cana-1961	237	15	.	.	PUNCT
cana-1961	238	1	where	where	SCONJ
cana-1961	238	2	h(ω1)=𝔤(2ω1	h(ω1)=𝔤(2ω1	NOUN
cana-1961	238	3	)	)	PUNCT
cana-1961	238	4	−	−	NOUN
cana-1961	238	5	8𝔤(ω1	8𝔤(ω1	NUM
cana-1961	238	6	)	)	PUNCT
cana-1961	238	7	.	.	PUNCT
cana-1961	239	1	stability	stability	NOUN
cana-1961	239	2	of	of	ADP
cana-1961	239	3	the	the	DET
cana-1961	239	4	fe	fe	NOUN
cana-1961	239	5	eq	eq	ADP
cana-1961	239	6	2	2	NUM
cana-1961	239	7	:	:	PUNCT
cana-1961	239	8	mixed	mixed	ADJ
cana-1961	239	9	case	case	NOUN
cana-1961	239	10	our	our	PRON
cana-1961	239	11	goal	goal	NOUN
cana-1961	239	12	in	in	ADP
cana-1961	239	13	this	this	DET
cana-1961	239	14	section	section	NOUN
cana-1961	239	15	will	will	AUX
cana-1961	239	16	be	be	AUX
cana-1961	239	17	to	to	PART
cana-1961	239	18	establish	establish	VERB
cana-1961	239	19	the	the	DET
cana-1961	239	20	generalized	generalized	ADJ
cana-1961	239	21	h	h	NOUN
cana-1961	239	22	-	-	PUNCT
cana-1961	239	23	u	u	NOUN
cana-1961	239	24	stability	stability	NOUN
cana-1961	239	25	of	of	ADP
cana-1961	239	26	the	the	DET
cana-1961	239	27	𝑄2	𝑄2	PROPN
cana-1961	239	28	́	́	PUNCT
cana-1961	240	1	−	−	PROPN
cana-1961	240	2	c3	c3	PROPN
cana-1961	240	3	́	́	PROPN
cana-1961	240	4	fe	fe	X
cana-1961	240	5	eq	eq	ADP
cana-1961	240	6	2	2	NUM
cana-1961	240	7	,	,	PUNCT
cana-1961	240	8	in	in	ADP
cana-1961	240	9	na	na	DET
cana-1961	240	10	2	2	NUM
cana-1961	240	11	-	-	PUNCT
cana-1961	240	12	normed	norme	VERB
cana-1961	240	13	spaces	space	NOUN
cana-1961	240	14	.	.	PUNCT
cana-1961	241	1	for	for	ADP
cana-1961	241	2	a	a	DET
cana-1961	241	3	given	give	VERB
cana-1961	241	4	mapping	mapping	NOUN
cana-1961	241	5	𝔤	𝔤	NOUN
cana-1961	241	6	:	:	PUNCT
cana-1961	241	7	𝒲	𝒲	NOUN
cana-1961	241	8	→	→	SYM
cana-1961	241	9	𝒵	𝒵	PROPN
cana-1961	241	10	,	,	PUNCT
cana-1961	241	11	let	let	VERB
cana-1961	241	12	𝔤o(ω1	𝔤o(ω1	PRON
cana-1961	241	13	)	)	PUNCT
cana-1961	241	14	=	=	SYM
cana-1961	241	15	𝔤(ω1)−𝔤(−ω1	𝔤(ω1)−𝔤(−ω1	NOUN
cana-1961	241	16	)	)	PUNCT
cana-1961	241	17	2	2	NUM
cana-1961	241	18	and	and	CCONJ
cana-1961	241	19	𝔤e(ω1	𝔤e(ω1	NOUN
cana-1961	241	20	)	)	PUNCT
cana-1961	241	21	=	=	SYM
cana-1961	241	22	𝔤(ω1)+𝔤(−ω1	𝔤(ω1)+𝔤(−ω1	NOUN
cana-1961	241	23	)	)	PUNCT
cana-1961	241	24	2	2	NUM
cana-1961	241	25	.	.	PUNCT
cana-1961	242	1	then	then	ADV
cana-1961	242	2	𝔤o	𝔤o	INTJ
cana-1961	242	3	is	be	AUX
cana-1961	242	4	odd	odd	ADJ
cana-1961	242	5	and	and	CCONJ
cana-1961	242	6	𝔤e	𝔤e	PROPN
cana-1961	242	7	is	be	AUX
cana-1961	242	8	even	even	ADV
cana-1961	242	9	.	.	PUNCT
cana-1961	243	1	theorem	theorem	ADJ
cana-1961	243	2	6	6	NUM
cana-1961	243	3	:	:	PUNCT
cana-1961	243	4	let	let	VERB
cana-1961	243	5	φ	φ	NUM
cana-1961	243	6	:	:	PUNCT
cana-1961	243	7	𝒲	𝒲	NOUN
cana-1961	243	8	×	×	NOUN
cana-1961	243	9	𝒲	𝒲	NOUN
cana-1961	243	10	→	→	SYM
cana-1961	244	1	[	[	X
cana-1961	244	2	0	0	NUM
cana-1961	244	3	,	,	PUNCT
cana-1961	244	4	∞	∞	PROPN
cana-1961	244	5	)	)	PUNCT
cana-1961	244	6	be	be	VERB
cana-1961	244	7	a	a	DET
cana-1961	244	8	function	function	NOUN
cana-1961	244	9	such	such	ADJ
cana-1961	244	10	that	that	SCONJ
cana-1961	244	11	there	there	PRON
cana-1961	244	12	is	be	VERB
cana-1961	244	13	a	a	DET
cana-1961	244	14	constant	constant	ADJ
cana-1961	244	15	0	0	NUM
cana-1961	244	16	<	<	X
cana-1961	244	17	£	£	SYM
cana-1961	244	18	<	<	X
cana-1961	244	19	1	1	NUM
cana-1961	244	20	with	with	ADP
cana-1961	244	21	φ(𝔪ω1	φ(𝔪ω1	NOUN
cana-1961	244	22	,	,	PUNCT
cana-1961	244	23	𝔪ω2	𝔪ω2	NOUN
cana-1961	244	24	)	)	PUNCT
cana-1961	244	25	≤	≤	NUM
cana-1961	244	26	|𝔪|3	|𝔪|3	PROPN
cana-1961	244	27	£	£	NOUN
cana-1961	244	28	φ(ω1	φ(ω1	NOUN
cana-1961	244	29	,	,	PUNCT
cana-1961	244	30	ω2	ω2	NUM
cana-1961	244	31	)	)	PUNCT
cana-1961	244	32	41	41	NUM
cana-1961	244	33	for	for	ADP
cana-1961	244	34	all	all	DET
cana-1961	244	35	ω1	ω1	PROPN
cana-1961	244	36	,	,	PUNCT
cana-1961	244	37	ω2	ω2	NOUN
cana-1961	244	38	∈	∈	PROPN
cana-1961	244	39	𝒲.	𝒲.	PROPN
cana-1961	244	40	suppose	suppose	VERB
cana-1961	244	41	𝔤	𝔤	NOUN
cana-1961	244	42	:	:	PUNCT
cana-1961	244	43	𝒲	𝒲	NOUN
cana-1961	244	44	→	→	SYM
cana-1961	244	45	𝒵	𝒵	PROPN
cana-1961	244	46	is	be	AUX
cana-1961	244	47	a	a	DET
cana-1961	244	48	mapping	mapping	NOUN
cana-1961	244	49	satisfying	satisfy	VERB
cana-1961	244	50	the	the	DET
cana-1961	244	51	inequality	inequality	NOUN
cana-1961	244	52	‖d𝔤(ω1	‖d𝔤(ω1	NOUN
cana-1961	244	53	,	,	PUNCT
cana-1961	244	54	ω2	ω2	NUM
cana-1961	244	55	)	)	PUNCT
cana-1961	244	56	,	,	PUNCT
cana-1961	244	57	υ‖	υ‖	VERB
cana-1961	244	58	≤	≤	NOUN
cana-1961	244	59	φ(ω1	φ(ω1	X
cana-1961	244	60	,	,	PUNCT
cana-1961	244	61	ω2	ω2	NUM
cana-1961	244	62	)	)	PUNCT
cana-1961	244	63	42	42	NUM
cana-1961	244	64	for	for	ADP
cana-1961	244	65	all	all	DET
cana-1961	244	66	ω1	ω1	PROPN
cana-1961	244	67	,	,	PUNCT
cana-1961	244	68	ω2	ω2	NOUN
cana-1961	244	69	∈	∈	PROPN
cana-1961	244	70	𝒲	𝒲	PROPN
cana-1961	244	71	,	,	PUNCT
cana-1961	244	72	υ	υ	PRON
cana-1961	244	73	∈	∈	PROPN
cana-1961	244	74	𝒵.	𝒵.	PROPN
cana-1961	245	1	then	then	ADV
cana-1961	245	2	there	there	PRON
cana-1961	245	3	are	be	VERB
cana-1961	245	4	a	a	DET
cana-1961	245	5	unique	unique	ADJ
cana-1961	245	6	𝑄2	𝑄2	NOUN
cana-1961	245	7	́	́	PUNCT
cana-1961	245	8	mapping	map	VERB
cana-1961	245	9	𝑄2	𝑄2	PROPN
cana-1961	245	10	́	́	PROPN
cana-1961	245	11	:	:	PUNCT
cana-1961	245	12	𝒲	𝒲	NOUN
cana-1961	245	13	→	→	SYM
cana-1961	245	14	𝒲	𝒲	NOUN
cana-1961	245	15	and	and	CCONJ
cana-1961	245	16	a	a	DET
cana-1961	245	17	unique	unique	ADJ
cana-1961	245	18	c3	c3	NOUN
cana-1961	245	19	́	́	PUNCT
cana-1961	245	20	mapping	map	VERB
cana-1961	245	21	c3	c3	NOUN
cana-1961	245	22	́	́	PROPN
cana-1961	245	23	:	:	PUNCT
cana-1961	245	24	𝒲	𝒲	NOUN
cana-1961	245	25	→	→	SYM
cana-1961	245	26	𝒵	𝒵	NOUN
cana-1961	245	27	such	such	ADJ
cana-1961	245	28	that	that	SCONJ
cana-1961	245	29	∥	∥	NUM
cana-1961	245	30	𝔤(ω1	𝔤(ω1	NOUN
cana-1961	245	31	)	)	PUNCT
cana-1961	245	32	−	−	PROPN
cana-1961	245	33	𝑄2	𝑄2	PROPN
cana-1961	246	1	́	́	PUNCT
cana-1961	246	2	(	(	PUNCT
cana-1961	246	3	ω1	ω1	PROPN
cana-1961	246	4	)	)	PUNCT
cana-1961	246	5	−	−	PROPN
cana-1961	246	6	c3	c3	PROPN
cana-1961	246	7	́	́	PROPN
cana-1961	246	8	(	(	PUNCT
cana-1961	246	9	ω1	ω1	PROPN
cana-1961	246	10	)	)	PUNCT
cana-1961	246	11	,	,	PUNCT
cana-1961	246	12	υ	υ	NOUN
cana-1961	246	13	∥	∥	PUNCT
cana-1961	246	14	≤	≤	NUM
cana-1961	246	15	max	max	NOUN
cana-1961	246	16	{	{	PUNCT
cana-1961	246	17	1	1	NUM
cana-1961	246	18	|2|	|2|	PROPN
cana-1961	246	19	max	max	PROPN
cana-1961	246	20	{	{	PUNCT
cana-1961	246	21	1	1	NUM
cana-1961	246	22	|2|.|𝔪|2.(1−£	|2|.|𝔪|2.(1−£	PROPN
cana-1961	246	23	)	)	PUNCT
cana-1961	246	24	φ(0	φ(0	PROPN
cana-1961	246	25	,	,	PUNCT
cana-1961	246	26	ω1	ω1	PROPN
cana-1961	246	27	)	)	PUNCT
cana-1961	246	28	φ(0	φ(0	ADJ
cana-1961	246	29	,	,	PUNCT
cana-1961	246	30	−ω1	−ω1	NOUN
cana-1961	246	31	)	)	PUNCT
cana-1961	246	32	}	}	PUNCT
cana-1961	246	33	,	,	PUNCT
cana-1961	246	34	1	1	NUM
cana-1961	246	35	|2|	|2|	PROPN
cana-1961	246	36	max	max	X
cana-1961	246	37	{	{	PUNCT
cana-1961	246	38	1	1	NUM
cana-1961	246	39	|𝔪|3(1−£	|𝔪|3(1−£	PROPN
cana-1961	246	40	)	)	PUNCT
cana-1961	246	41	φ(ω1	φ(ω1	NOUN
cana-1961	246	42	)	)	PUNCT
cana-1961	246	43	,	,	PUNCT
cana-1961	246	44	1	1	NUM
cana-1961	246	45	|𝔪|3(1−£	|𝔪|3(1−£	PROPN
cana-1961	246	46	)	)	PUNCT
cana-1961	246	47	φ(−ω1	φ(−ω1	NOUN
cana-1961	246	48	)	)	PUNCT
cana-1961	246	49	}	}	PUNCT
cana-1961	246	50	}	}	PUNCT
cana-1961	246	51	43	43	NUM
cana-1961	246	52	communications	communication	NOUN
cana-1961	246	53	on	on	ADP
cana-1961	246	54	applied	apply	VERB
cana-1961	246	55	nonlinear	nonlinear	ADJ
cana-1961	246	56	analysis	analysis	NOUN
cana-1961	246	57	issn	issn	NOUN
cana-1961	246	58	:	:	PUNCT
cana-1961	246	59	1074	1074	NUM
cana-1961	246	60	-	-	PUNCT
cana-1961	246	61	133x	133x	NUM
cana-1961	246	62	vol	vol	NOUN
cana-1961	246	63	32	32	NUM
cana-1961	246	64	no	no	NOUN
cana-1961	246	65	.	.	NOUN
cana-1961	246	66	3	3	NUM
cana-1961	246	67	(	(	PUNCT
cana-1961	246	68	2025	2025	NUM
cana-1961	246	69	)	)	PUNCT
cana-1961	246	70	313	313	NUM
cana-1961	246	71	https://internationalpubls.com	https://internationalpubls.com	X
cana-1961	246	72	where	where	SCONJ
cana-1961	246	73	φ(ω1	φ(ω1	NOUN
cana-1961	246	74	)	)	PUNCT
cana-1961	246	75	=	=	SYM
cana-1961	246	76	max	max	X
cana-1961	246	77	{	{	PUNCT
cana-1961	246	78	|𝔪2(𝔪−1)|	|𝔪2(𝔪−1)|	X
cana-1961	246	79	|2|.|4|	|2|.|4|	NOUN
cana-1961	246	80	φ(0	φ(0	ADJ
cana-1961	246	81	,	,	PUNCT
cana-1961	246	82	ω1	ω1	PROPN
cana-1961	246	83	)	)	PUNCT
cana-1961	246	84	,	,	PUNCT
cana-1961	246	85	|𝔪2(𝔪−2)|	|𝔪2(𝔪−2)|	NOUN
cana-1961	247	1	|2(1−𝔪2)|	|2(1−𝔪2)|	PROPN
cana-1961	247	2	φ(ω1	φ(ω1	NOUN
cana-1961	247	3	,	,	PUNCT
cana-1961	247	4	0	0	NUM
cana-1961	247	5	)	)	PUNCT
cana-1961	247	6	}	}	PUNCT
cana-1961	247	7	for	for	ADP
cana-1961	247	8	all	all	DET
cana-1961	247	9	ω1	ω1	PROPN
cana-1961	247	10	∈	∈	PROPN
cana-1961	247	11	𝒲	𝒲	PROPN
cana-1961	247	12	,	,	PUNCT
cana-1961	247	13	υ	υ	PRON
cana-1961	247	14	∈	∈	NOUN
cana-1961	247	15	𝒵.	𝒵.	PROPN
cana-1961	247	16	proof	proof	NOUN
cana-1961	247	17	:	:	PUNCT
cana-1961	247	18	assume	assume	VERB
cana-1961	247	19	that	that	SCONJ
cana-1961	247	20	𝔤(ω1	𝔤(ω1	NOUN
cana-1961	247	21	)	)	PUNCT
cana-1961	247	22	=	=	SYM
cana-1961	247	23	𝔤e(ω1	𝔤e(ω1	NOUN
cana-1961	247	24	)	)	PUNCT
cana-1961	247	25	+	+	PUNCT
cana-1961	247	26	𝔤o(ω1	𝔤o(ω1	NOUN
cana-1961	247	27	)	)	PUNCT
cana-1961	247	28	.	.	PUNCT
cana-1961	248	1	let	let	VERB
cana-1961	248	2	φ(ω1	φ(ω1	NOUN
cana-1961	248	3	,	,	PUNCT
cana-1961	248	4	ω2	ω2	ADJ
cana-1961	248	5	)	)	PUNCT
cana-1961	248	6	=	=	SYM
cana-1961	249	1	1	1	NUM
cana-1961	249	2	|2|	|2|	PROPN
cana-1961	249	3	max{φ(ω1	max{φ(ω1	NOUN
cana-1961	249	4	,	,	PUNCT
cana-1961	249	5	ω2	ω2	NUM
cana-1961	249	6	)	)	PUNCT
cana-1961	249	7	,	,	PUNCT
cana-1961	249	8	ϕ(−ω1	ϕ(−ω1	PROPN
cana-1961	249	9	,	,	PUNCT
cana-1961	249	10	−ω2	−ω2	NOUN
cana-1961	249	11	)	)	PUNCT
cana-1961	249	12	}	}	PUNCT
cana-1961	249	13	then	then	ADV
cana-1961	249	14	by	by	ADP
cana-1961	249	15	eq	eq	NOUN
cana-1961	249	16	41	41	NUM
cana-1961	249	17	,	,	PUNCT
cana-1961	249	18	and	and	CCONJ
cana-1961	249	19	eq	eq	ADP
cana-1961	249	20	42	42	NUM
cana-1961	249	21	,	,	PUNCT
cana-1961	249	22	which	which	PRON
cana-1961	249	23	gives	give	VERB
cana-1961	249	24	φ(𝔪ω1	φ(𝔪ω1	NOUN
cana-1961	249	25	,	,	PUNCT
cana-1961	249	26	𝔪ω2	𝔪ω2	NOUN
cana-1961	249	27	)	)	PUNCT
cana-1961	249	28	≤	≤	NUM
cana-1961	249	29	|𝔪|3	|𝔪|3	PROPN
cana-1961	249	30	£	£	NOUN
cana-1961	249	31	φ(ω1	φ(ω1	NOUN
cana-1961	249	32	,	,	PUNCT
cana-1961	249	33	ω2	ω2	ADJ
cana-1961	249	34	)	)	PUNCT
cana-1961	249	35	≤	≤	NOUN
cana-1961	249	36	|𝔪|2	|𝔪|2	PROPN
cana-1961	249	37	£	£	NOUN
cana-1961	249	38	φ(ω1	φ(ω1	NOUN
cana-1961	249	39	,	,	PUNCT
cana-1961	249	40	ω2	ω2	NUM
cana-1961	249	41	)	)	PUNCT
cana-1961	249	42	∥	∥	PUNCT
cana-1961	249	43	d𝔤o(ω1	d𝔤o(ω1	ADJ
cana-1961	249	44	,	,	PUNCT
cana-1961	249	45	ω2	ω2	ADJ
cana-1961	249	46	)	)	PUNCT
cana-1961	249	47	,	,	PUNCT
cana-1961	249	48	υ	υ	PROPN
cana-1961	249	49	∥≤	∥≤	PROPN
cana-1961	249	50	φ(ω1	φ(ω1	X
cana-1961	249	51	,	,	PUNCT
cana-1961	249	52	ω2	ω2	NUM
cana-1961	249	53	)	)	PUNCT
cana-1961	249	54	,	,	PUNCT
cana-1961	249	55	∥	∥	PUNCT
cana-1961	249	56	d𝔤e(ω1	d𝔤e(ω1	NOUN
cana-1961	249	57	,	,	PUNCT
cana-1961	249	58	ω2	ω2	NUM
cana-1961	249	59	)	)	PUNCT
cana-1961	249	60	,	,	PUNCT
cana-1961	249	61	υ	υ	PROPN
cana-1961	249	62	∥≤	∥≤	PROPN
cana-1961	249	63	φ(ω1	φ(ω1	NOUN
cana-1961	249	64	,	,	PUNCT
cana-1961	249	65	ω2	ω2	NUM
cana-1961	249	66	)	)	PUNCT
cana-1961	249	67	,	,	PUNCT
cana-1961	249	68	hence	hence	ADV
cana-1961	249	69	by	by	ADP
cana-1961	249	70	theorem	theorem	ADJ
cana-1961	249	71	2	2	NUM
cana-1961	249	72	and	and	CCONJ
cana-1961	249	73	theorem	theorem	VERB
cana-1961	249	74	4	4	NUM
cana-1961	249	75	,	,	PUNCT
cana-1961	249	76	there	there	PRON
cana-1961	249	77	are	be	VERB
cana-1961	249	78	a	a	DET
cana-1961	249	79	unique	unique	ADJ
cana-1961	249	80	𝑄2	𝑄2	NOUN
cana-1961	249	81	́	́	PUNCT
cana-1961	249	82	mapping	map	VERB
cana-1961	249	83	𝑄2	𝑄2	PROPN
cana-1961	249	84	́	́	PROPN
cana-1961	249	85	:	:	PUNCT
cana-1961	249	86	𝒲	𝒲	NOUN
cana-1961	249	87	→	→	SYM
cana-1961	249	88	𝒵	𝒵	PROPN
cana-1961	249	89	and	and	CCONJ
cana-1961	249	90	a	a	DET
cana-1961	249	91	unique	unique	ADJ
cana-1961	249	92	c3	c3	NOUN
cana-1961	249	93	́	́	PUNCT
cana-1961	249	94	mapping	map	VERB
cana-1961	249	95	c3	c3	NOUN
cana-1961	249	96	́	́	PROPN
cana-1961	249	97	:	:	PUNCT
cana-1961	249	98	𝒲	𝒲	NOUN
cana-1961	249	99	→	→	SYM
cana-1961	249	100	𝒵	𝒵	NOUN
cana-1961	249	101	such	such	ADJ
cana-1961	249	102	that	that	SCONJ
cana-1961	249	103	∥	∥	PROPN
cana-1961	249	104	𝔤o(ω1	𝔤o(ω1	NOUN
cana-1961	249	105	)	)	PUNCT
cana-1961	249	106	−	−	PROPN
cana-1961	249	107	𝑄2	𝑄2	PROPN
cana-1961	250	1	́	́	PUNCT
cana-1961	250	2	(	(	PUNCT
cana-1961	250	3	ω1	ω1	PROPN
cana-1961	250	4	)	)	PUNCT
cana-1961	250	5	,	,	PUNCT
cana-1961	250	6	υ	υ	PROPN
cana-1961	250	7	∥≤	∥≤	PROPN
cana-1961	250	8	1	1	NUM
cana-1961	250	9	|2|	|2|	PROPN
cana-1961	250	10	.	.	PUNCT
cana-1961	250	11	|𝔪|2	|𝔪|2	PROPN
cana-1961	250	12	.	.	PUNCT
cana-1961	251	1	(	(	PUNCT
cana-1961	251	2	1	1	NUM
cana-1961	251	3	−	−	NOUN
cana-1961	251	4	£	£	NOUN
cana-1961	251	5	)	)	PUNCT
cana-1961	251	6	φ(0	φ(0	PROPN
cana-1961	251	7	,	,	PUNCT
cana-1961	251	8	ω1	ω1	PROPN
cana-1961	251	9	)	)	PUNCT
cana-1961	251	10	and	and	CCONJ
cana-1961	251	11	∥	∥	NUM
cana-1961	251	12	𝔤e(ω1	𝔤e(ω1	NOUN
cana-1961	251	13	)	)	PUNCT
cana-1961	252	1	−	−	PROPN
cana-1961	252	2	c3	c3	PROPN
cana-1961	252	3	́	́	PROPN
cana-1961	252	4	(	(	PUNCT
cana-1961	252	5	ω1	ω1	PROPN
cana-1961	252	6	)	)	PUNCT
cana-1961	252	7	,	,	PUNCT
cana-1961	252	8	υ	υ	PROPN
cana-1961	252	9	∥≤	∥≤	PROPN
cana-1961	252	10	1	1	NUM
cana-1961	252	11	|𝔪|3(1	|𝔪|3(1	ADV
cana-1961	252	12	−	−	PRON
cana-1961	252	13	£	£	NOUN
cana-1961	252	14	)	)	PUNCT
cana-1961	252	15	φ(ω1	φ(ω1	NOUN
cana-1961	252	16	)	)	PUNCT
cana-1961	252	17	for	for	ADP
cana-1961	252	18	all	all	DET
cana-1961	252	19	ω1	ω1	PROPN
cana-1961	252	20	∈	∈	PROPN
cana-1961	252	21	𝒲	𝒲	PROPN
cana-1961	252	22	,	,	PUNCT
cana-1961	252	23	υ	υ	PRON
cana-1961	252	24	∈	∈	PROPN
cana-1961	252	25	𝒵.	𝒵.	PROPN
cana-1961	252	26	therefore	therefore	ADV
cana-1961	252	27	∥	∥	NUM
cana-1961	252	28	𝔤(ω1	𝔤(ω1	NOUN
cana-1961	252	29	)	)	PUNCT
cana-1961	252	30	−	−	PROPN
cana-1961	252	31	c3	c3	PROPN
cana-1961	252	32	́	́	PROPN
cana-1961	252	33	(	(	PUNCT
cana-1961	252	34	ω1	ω1	PROPN
cana-1961	252	35	)	)	PUNCT
cana-1961	252	36	−	−	PROPN
cana-1961	252	37	𝑄2	𝑄2	PROPN
cana-1961	253	1	́	́	PROPN
cana-1961	253	2	(	(	PUNCT
cana-1961	253	3	ω1	ω1	PROPN
cana-1961	253	4	)	)	PUNCT
cana-1961	253	5	,	,	PUNCT
cana-1961	253	6	υ	υ	PRON
cana-1961	253	7	∥=	∥=	ADJ
cana-1961	253	8	‖𝔤o(ω1	‖𝔤o(ω1	ADV
cana-1961	253	9	)	)	PUNCT
cana-1961	253	10	+	+	CCONJ
cana-1961	253	11	𝔤e(ω1	𝔤e(ω1	NOUN
cana-1961	253	12	)	)	PUNCT
cana-1961	253	13	−	−	PROPN
cana-1961	253	14	c3	c3	PROPN
cana-1961	253	15	́	́	PROPN
cana-1961	253	16	(	(	PUNCT
cana-1961	253	17	ω1	ω1	PROPN
cana-1961	253	18	)	)	PUNCT
cana-1961	253	19	−	−	PROPN
cana-1961	253	20	𝑄2	𝑄2	PROPN
cana-1961	254	1	́	́	PROPN
cana-1961	254	2	(	(	PUNCT
cana-1961	254	3	ω1	ω1	PROPN
cana-1961	254	4	)	)	PUNCT
cana-1961	254	5	,	,	PUNCT
cana-1961	254	6	υ‖	υ‖	VERB
cana-1961	254	7	≤	≤	X
cana-1961	254	8	max{∥	max{∥	NUM
cana-1961	254	9	𝔤o(ω_1	𝔤o(ω_1	NOUN
cana-1961	254	10	)	)	PUNCT
cana-1961	255	1	−	−	PROPN
cana-1961	255	2	c3	c3	PROPN
cana-1961	255	3	́	́	PUNCT
cana-1961	255	4	(	(	PUNCT
cana-1961	255	5	ω_1	ω_1	NOUN
cana-1961	255	6	)	)	PUNCT
cana-1961	255	7	,	,	PUNCT
cana-1961	255	8	υ	υ	NOUN
cana-1961	255	9	∥	∥	NOUN
cana-1961	255	10	,	,	PUNCT
cana-1961	255	11	∥	∥	X
cana-1961	255	12	𝔤e(ω_1	𝔤e(ω_1	NOUN
cana-1961	255	13	)	)	PUNCT
cana-1961	255	14	−	−	PROPN
cana-1961	255	15	𝑄2	𝑄2	PROPN
cana-1961	256	1	́	́	PUNCT
cana-1961	256	2	(	(	PUNCT
cana-1961	256	3	ω_1	ω_1	NOUN
cana-1961	256	4	)	)	PUNCT
cana-1961	256	5	,	,	PUNCT
cana-1961	256	6	υ	υ	NOUN
cana-1961	256	7	∥	∥	X
cana-1961	256	8	}	}	PUNCT
cana-1961	256	9	≤	≤	NUM
cana-1961	256	10	max	max	NOUN
cana-1961	256	11	{	{	PUNCT
cana-1961	256	12	1	1	NUM
cana-1961	256	13	|2|	|2|	PROPN
cana-1961	256	14	max	max	PROPN
cana-1961	256	15	{	{	PUNCT
cana-1961	256	16	1	1	NUM
cana-1961	256	17	|2|.|𝔪|2.(1−£	|2|.|𝔪|2.(1−£	PROPN
cana-1961	256	18	)	)	PUNCT
cana-1961	256	19	φ(0	φ(0	PROPN
cana-1961	256	20	,	,	PUNCT
cana-1961	256	21	ω1	ω1	PROPN
cana-1961	256	22	)	)	PUNCT
cana-1961	256	23	,	,	PUNCT
cana-1961	256	24	1	1	NUM
cana-1961	256	25	|2|.|𝔪|2.(1−£	|2|.|𝔪|2.(1−£	NOUN
cana-1961	256	26	)	)	PUNCT
cana-1961	256	27	φ(0	φ(0	PROPN
cana-1961	256	28	,	,	PUNCT
cana-1961	256	29	−ω1	−ω1	NOUN
cana-1961	256	30	)	)	PUNCT
cana-1961	256	31	}	}	PUNCT
cana-1961	256	32	,	,	PUNCT
cana-1961	256	33	1	1	NUM
cana-1961	256	34	|2|	|2|	PROPN
cana-1961	256	35	max	max	X
cana-1961	256	36	{	{	PUNCT
cana-1961	256	37	1	1	NUM
cana-1961	256	38	|𝔪|3(1−£	|𝔪|3(1−£	PROPN
cana-1961	256	39	)	)	PUNCT
cana-1961	256	40	φ(ω1	φ(ω1	NOUN
cana-1961	256	41	)	)	PUNCT
cana-1961	256	42	,	,	PUNCT
cana-1961	256	43	1	1	NUM
cana-1961	256	44	|𝔪|3(1−£	|𝔪|3(1−£	PROPN
cana-1961	256	45	)	)	PUNCT
cana-1961	256	46	φ(−ω1	φ(−ω1	NOUN
cana-1961	256	47	)	)	PUNCT
cana-1961	256	48	}	}	PUNCT
cana-1961	256	49	}	}	PUNCT
cana-1961	256	50	for	for	ADP
cana-1961	256	51	all	all	DET
cana-1961	256	52	ω1	ω1	PROPN
cana-1961	256	53	∈	∈	PROPN
cana-1961	256	54	𝒲	𝒲	PROPN
cana-1961	256	55	,	,	PUNCT
cana-1961	256	56	υ	υ	PRON
cana-1961	256	57	∈	∈	PROPN
cana-1961	256	58	𝒵.	𝒵.	PROPN
cana-1961	256	59	this	this	PRON
cana-1961	256	60	completes	complete	VERB
cana-1961	256	61	the	the	DET
cana-1961	256	62	proof	proof	NOUN
cana-1961	256	63	.	.	PUNCT
cana-1961	257	1	4	4	X
cana-1961	257	2	.	.	X
cana-1961	257	3	conclusion	conclusion	NOUN
cana-1961	257	4	lorem	lorem	VERB
cana-1961	257	5	in	in	ADP
cana-1961	257	6	this	this	DET
cana-1961	257	7	article	article	NOUN
cana-1961	257	8	,	,	PUNCT
cana-1961	257	9	we	we	PRON
cana-1961	257	10	investigated	investigate	VERB
cana-1961	257	11	the	the	DET
cana-1961	257	12	h	h	NOUN
cana-1961	257	13	-	-	PUNCT
cana-1961	257	14	u	u	NOUN
cana-1961	257	15	stability	stability	NOUN
cana-1961	257	16	of	of	ADP
cana-1961	257	17	the	the	DET
cana-1961	257	18	quadratic	quadratic	ADJ
cana-1961	257	19	-	-	PUNCT
cana-1961	257	20	cubic	cubic	ADJ
cana-1961	257	21	functional	functional	ADJ
cana-1961	257	22	equation	equation	NOUN
cana-1961	257	23	(	(	PUNCT
cana-1961	257	24	eq	eq	NOUN
cana-1961	257	25	2	2	NUM
cana-1961	257	26	)	)	PUNCT
cana-1961	257	27	in	in	ADP
cana-1961	257	28	na	na	DET
cana-1961	257	29	2	2	NUM
cana-1961	257	30	-	-	PUNCT
cana-1961	257	31	normed	norme	VERB
cana-1961	257	32	spaces	space	NOUN
cana-1961	257	33	using	use	VERB
cana-1961	257	34	fixed	fix	VERB
cana-1961	257	35	-	-	PUNCT
cana-1961	257	36	point	point	NOUN
cana-1961	257	37	methods	method	NOUN
cana-1961	257	38	and	and	CCONJ
cana-1961	257	39	provided	provide	VERB
cana-1961	257	40	a	a	DET
cana-1961	257	41	suitable	suitable	ADJ
cana-1961	257	42	counterexample	counterexample	NOUN
cana-1961	257	43	.	.	PUNCT
cana-1961	258	1	in	in	ADP
cana-1961	258	2	light	light	NOUN
cana-1961	258	3	of	of	ADP
cana-1961	258	4	this	this	DET
cana-1961	258	5	work	work	NOUN
cana-1961	258	6	,	,	PUNCT
cana-1961	258	7	it	it	PRON
cana-1961	258	8	has	have	AUX
cana-1961	258	9	become	become	VERB
cana-1961	258	10	possible	possible	ADJ
cana-1961	258	11	to	to	PART
cana-1961	258	12	gain	gain	VERB
cana-1961	258	13	a	a	DET
cana-1961	258	14	more	more	ADV
cana-1961	258	15	comprehensive	comprehensive	ADJ
cana-1961	258	16	understanding	understanding	NOUN
cana-1961	258	17	of	of	ADP
cana-1961	258	18	the	the	DET
cana-1961	258	19	stability	stability	NOUN
cana-1961	258	20	problems	problem	NOUN
cana-1961	258	21	of	of	ADP
cana-1961	258	22	functional	functional	ADJ
cana-1961	258	23	equations	equation	NOUN
cana-1961	258	24	within	within	ADP
cana-1961	258	25	the	the	DET
cana-1961	258	26	context	context	NOUN
cana-1961	258	27	of	of	ADP
cana-1961	258	28	na	na	ADP
cana-1961	258	29	2	2	NUM
cana-1961	258	30	-	-	PUNCT
cana-1961	258	31	normed	norme	VERB
cana-1961	258	32	spaces	space	NOUN
cana-1961	258	33	.	.	PUNCT
cana-1961	259	1	conflict	conflict	NOUN
cana-1961	259	2	of	of	ADP
cana-1961	259	3	interest	interest	NOUN
cana-1961	259	4	the	the	DET
cana-1961	259	5	authors	author	NOUN
cana-1961	259	6	declare	declare	VERB
cana-1961	259	7	that	that	SCONJ
cana-1961	259	8	they	they	PRON
cana-1961	259	9	have	have	VERB
cana-1961	259	10	no	no	DET
cana-1961	259	11	competing	compete	VERB
cana-1961	259	12	interests	interest	NOUN
cana-1961	259	13	.	.	PUNCT
cana-1961	260	1	references	reference	NOUN
cana-1961	260	2	[	[	X
cana-1961	260	3	1	1	NUM
cana-1961	260	4	]	]	X
cana-1961	260	5	elghali	elghali	PROPN
cana-1961	260	6	r	r	PROPN
cana-1961	260	7	,	,	PUNCT
cana-1961	260	8	kabbaji	kabbaji	PROPN
cana-1961	260	9	s.	s.	PROPN
cana-1961	260	10	some	some	DET
cana-1961	260	11	hyperstability	hyperstability	NOUN
cana-1961	260	12	results	result	VERB
cana-1961	260	13	in	in	ADP
cana-1961	260	14	non	non	ADJ
cana-1961	260	15	-	-	ADJ
cana-1961	260	16	archimedean	archimedean	ADJ
cana-1961	260	17	2	2	NUM
cana-1961	260	18	-	-	PUNCT
cana-1961	260	19	banach	banach	NOUN
cana-1961	260	20	space	space	NOUN
cana-1961	260	21	for	for	ADP
cana-1961	260	22	a	a	DET
cana-1961	260	23	𝜎-jensen	𝜎-jensen	NOUN
cana-1961	260	24	functional	functional	ADJ
cana-1961	260	25	equations	equation	NOUN
cana-1961	260	26	.	.	PUNCT
cana-1961	261	1	approx	approx	PROPN
cana-1961	261	2	comput	comput	PROPN
cana-1961	261	3	sci	sci	PROPN
cana-1961	261	4	engin	engin	PROPN
cana-1961	261	5	germany	germany	PROPN
cana-1961	261	6	,	,	PUNCT
cana-1961	261	7	2022	2022	NUM
cana-1961	261	8	;	;	PUNCT
cana-1961	261	9	180	180	NUM
cana-1961	261	10	;	;	PUNCT
cana-1961	261	11	349	349	NUM
cana-1961	261	12	-	-	SYM
cana-1961	261	13	367	367	NUM
cana-1961	261	14	.	.	PUNCT
cana-1961	262	1	https://doi.org/10.1007/978-3-030-84122-5_19	https://doi.org/10.1007/978-3-030-84122-5_19	ADJ
cana-1961	263	1	[	[	X
cana-1961	263	2	2	2	X
cana-1961	263	3	]	]	X
cana-1961	263	4	kim	kim	PROPN
cana-1961	263	5	c	c	PROPN
cana-1961	263	6	,	,	PUNCT
cana-1961	263	7	han	han	PROPN
cana-1961	263	8	g.	g.	PROPN
cana-1961	263	9	a	a	DET
cana-1961	263	10	fixed	fix	VERB
cana-1961	263	11	point	point	NOUN
cana-1961	263	12	theorem	theorem	ADJ
cana-1961	263	13	and	and	CCONJ
cana-1961	263	14	generalized	generalize	VERB
cana-1961	263	15	non	non	ADJ
cana-1961	263	16	-	-	ADJ
cana-1961	263	17	archimedean	archimedean	ADJ
cana-1961	263	18	quasi	quasi	NOUN
cana-1961	263	19	-	-	ADJ
cana-1961	263	20	ordered	order	VERB
cana-1961	263	21	metric	metric	ADJ
cana-1961	263	22	spaces	space	NOUN
cana-1961	263	23	and	and	CCONJ
cana-1961	263	24	its	its	PRON
cana-1961	263	25	applications	application	NOUN
cana-1961	263	26	.	.	PUNCT
cana-1961	264	1	thai	thai	PROPN
cana-1961	264	2	j	j	PROPN
cana-1961	264	3	math	math	PROPN
cana-1961	264	4	.	.	PUNCT
cana-1961	265	1	2023	2023	NUM
cana-1961	265	2	;	;	PUNCT
cana-1961	265	3	21(3	21(3	NUM
cana-1961	265	4	):	):	PUNCT
cana-1961	265	5	481	481	NUM
cana-1961	265	6	-	-	SYM
cana-1961	265	7	90	90	NUM
cana-1961	265	8	.	.	PUNCT
cana-1961	266	1	available	available	ADJ
cana-1961	266	2	from	from	ADP
cana-1961	266	3	:	:	PUNCT
cana-1961	266	4	https://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1521	https://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1521	PROPN
cana-1961	266	5	[	[	X
cana-1961	266	6	3	3	NUM
cana-1961	266	7	]	]	X
cana-1961	266	8	tamilvanan	tamilvanan	ADJ
cana-1961	266	9	k	k	NOUN
cana-1961	266	10	,	,	PUNCT
cana-1961	266	11	alanazi	alanazi	PROPN
cana-1961	266	12	am	be	AUX
cana-1961	266	13	,	,	PUNCT
cana-1961	266	14	alshehri	alshehri	VERB
cana-1961	266	15	mg	mg	PROPN
cana-1961	266	16	,	,	PUNCT
cana-1961	266	17	kafle	kafle	PROPN
cana-1961	266	18	j.	j.	PROPN
cana-1961	266	19	hyers	hyers	PROPN
cana-1961	266	20	-	-	PUNCT
cana-1961	266	21	ulam	ulam	PROPN
cana-1961	266	22	stability	stability	NOUN
cana-1961	266	23	of	of	ADP
cana-1961	266	24	quadratic	quadratic	ADJ
cana-1961	266	25	functional	functional	ADJ
cana-1961	266	26	equation	equation	NOUN
cana-1961	266	27	based	base	VERB
cana-1961	266	28	on	on	ADP
cana-1961	266	29	fixed	fix	VERB
cana-1961	266	30	point	point	NOUN
cana-1961	266	31	technique	technique	NOUN
cana-1961	266	32	in	in	ADP
cana-1961	266	33	banach	banach	NOUN
cana-1961	266	34	spaces	space	NOUN
cana-1961	266	35	and	and	CCONJ
cana-1961	266	36	non	non	ADJ
cana-1961	266	37	-	-	ADJ
cana-1961	266	38	archimedean	archimedean	ADJ
cana-1961	266	39	banach	banach	NOUN
cana-1961	266	40	spaces	space	VERB
cana-1961	266	41	.	.	PUNCT
cana-1961	267	1	mathematics	mathematic	NOUN
cana-1961	267	2	.	.	PUNCT
cana-1961	267	3	2021	2021	NUM
cana-1961	267	4	;	;	PUNCT
cana-1961	267	5	9(20	9(20	NUM
cana-1961	267	6	):	):	PUNCT
cana-1961	267	7	2575	2575	NUM
cana-1961	267	8	.	.	PUNCT
cana-1961	268	1	doi	doi	NOUN
cana-1961	268	2	:	:	PUNCT
cana-1961	268	3	https://doi.org/10.3390/math9202575	https://doi.org/10.3390/math9202575	PROPN
cana-1961	268	4	https://doi.org/10.1007/978-3-030-84122-5_19	https://doi.org/10.1007/978-3-030-84122-5_19	PROPN
cana-1961	268	5	https://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1521	https://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/1521	PROPN
cana-1961	268	6	https://doi.org/10.3390/math9202575	https://doi.org/10.3390/math9202575	NOUN
cana-1961	268	7	communications	communication	NOUN
cana-1961	268	8	on	on	ADP
cana-1961	268	9	applied	apply	VERB
cana-1961	268	10	nonlinear	nonlinear	ADJ
cana-1961	268	11	analysis	analysis	NOUN
cana-1961	268	12	issn	issn	NOUN
cana-1961	268	13	:	:	PUNCT
cana-1961	268	14	1074	1074	NUM
cana-1961	268	15	-	-	PUNCT
cana-1961	268	16	133x	133x	NUM
cana-1961	268	17	vol	vol	NOUN
cana-1961	268	18	32	32	NUM
cana-1961	268	19	no	no	NOUN
cana-1961	268	20	.	.	NOUN
cana-1961	268	21	3	3	NUM
cana-1961	268	22	(	(	PUNCT
cana-1961	268	23	2025	2025	NUM
cana-1961	268	24	)	)	PUNCT
cana-1961	268	25	314	314	NUM
cana-1961	268	26	https://internationalpubls.com	https://internationalpubls.com	X
cana-1961	269	1	[	[	X
cana-1961	269	2	4	4	X
cana-1961	269	3	]	]	X
cana-1961	269	4	mohiuddine	mohiuddine	PROPN
cana-1961	269	5	sa	sa	PROPN
cana-1961	269	6	,	,	PUNCT
cana-1961	269	7	tamilvanan	tamilvanan	ADJ
cana-1961	269	8	k	k	NOUN
cana-1961	269	9	,	,	PUNCT
cana-1961	269	10	mursaleen	mursaleen	PROPN
cana-1961	269	11	m	m	PROPN
cana-1961	269	12	,	,	PUNCT
cana-1961	269	13	alotaibi	alotaibi	NOUN
cana-1961	269	14	t.	t.	NOUN
cana-1961	269	15	stability	stability	NOUN
cana-1961	269	16	of	of	ADP
cana-1961	269	17	the	the	DET
cana-1961	269	18	quartic	quartic	ADJ
cana-1961	269	19	functional	functional	ADJ
cana-1961	269	20	equation	equation	NOUN
cana-1961	269	21	in	in	ADP
cana-1961	269	22	modular	modular	ADJ
cana-1961	269	23	spaces	space	NOUN
cana-1961	269	24	via	via	ADP
cana-1961	269	25	hyers	hyer	NOUN
cana-1961	269	26	and	and	CCONJ
cana-1961	269	27	fixed	fix	VERB
cana-1961	269	28	-	-	PUNCT
cana-1961	269	29	point	point	NOUN
cana-1961	269	30	methods	method	NOUN
cana-1961	269	31	.	.	PUNCT
cana-1961	270	1	mathematics	mathematic	NOUN
cana-1961	270	2	.	.	PUNCT
cana-1961	271	1	2022	2022	NUM
cana-1961	271	2	;	;	PUNCT
cana-1961	271	3	10	10	NUM
cana-1961	271	4	:	:	SYM
cana-1961	271	5	19	19	NUM
cana-1961	271	6	-	-	SYM
cana-1961	271	7	38	38	NUM
cana-1961	271	8	.	.	PUNCT
cana-1961	272	1	doi	doi	NOUN
cana-1961	272	2	:	:	PUNCT
cana-1961	272	3	https://doi.org/10.3390/math10111938	https://doi.org/10.3390/math10111938	PROPN
cana-1961	272	4	[	[	X
cana-1961	272	5	5	5	NUM
cana-1961	272	6	]	]	X
cana-1961	272	7	aribou	aribou	PROPN
cana-1961	272	8	y	y	PROPN
cana-1961	272	9	,	,	PUNCT
cana-1961	272	10	kabbaj	kabbaj	PROPN
cana-1961	272	11	s.	s.	PROPN
cana-1961	272	12	the	the	DET
cana-1961	272	13	stability	stability	NOUN
cana-1961	272	14	of	of	ADP
cana-1961	272	15	n	n	CCONJ
cana-1961	272	16	-	-	PUNCT
cana-1961	272	17	dimensional	dimensional	ADJ
cana-1961	272	18	quadratic	quadratic	ADJ
cana-1961	272	19	functional	functional	ADJ
cana-1961	272	20	inequality	inequality	NOUN
cana-1961	272	21	in	in	ADP
cana-1961	272	22	non	non	ADJ
cana-1961	272	23	-	-	ADJ
cana-1961	272	24	archimedean	archimedean	ADJ
cana-1961	272	25	banach	banach	NOUN
cana-1961	272	26	spaces	space	VERB
cana-1961	272	27	.	.	PUNCT
cana-1961	273	1	são	são	PROPN
cana-1961	273	2	paulo	paulo	PROPN
cana-1961	273	3	j.	j.	PROPN
cana-1961	273	4	math	math	PROPN
cana-1961	273	5	.	.	PUNCT
cana-1961	274	1	sci	sci	PROPN
cana-1961	274	2	.	.	PROPN
cana-1961	274	3	2021	2021	NUM
cana-1961	274	4	;	;	PUNCT
cana-1961	274	5	16	16	NUM
cana-1961	274	6	:	:	SYM
cana-1961	274	7	1382	1382	NUM
cana-1961	274	8	-	-	SYM
cana-1961	274	9	1400	1400	NUM
cana-1961	274	10	.	.	PUNCT
cana-1961	275	1	doi	doi	NOUN
cana-1961	275	2	:	:	PUNCT
cana-1961	275	3	http://dx.doi.org/10.1007/s40863-021-00220-9	http://dx.doi.org/10.1007/s40863-021-00220-9	PUNCT
cana-1961	276	1	[	[	X
cana-1961	276	2	6	6	NUM
cana-1961	276	3	]	]	X
cana-1961	276	4	jabbar	jabbar	PROPN
cana-1961	276	5	ha	ha	INTJ
cana-1961	276	6	,	,	PUNCT
cana-1961	276	7	kadhim	kadhim	PROPN
cana-1961	276	8	sn	sn	PROPN
cana-1961	276	9	,	,	PUNCT
cana-1961	276	10	abed	abe	VERB
cana-1961	276	11	ss	ss	PROPN
cana-1961	276	12	.	.	PUNCT
cana-1961	277	1	best	good	ADJ
cana-1961	277	2	approximation	approximation	NOUN
cana-1961	277	3	in	in	ADP
cana-1961	277	4	b	b	NOUN
cana-1961	277	5	-	-	PUNCT
cana-1961	277	6	modular	modular	ADJ
cana-1961	277	7	spaces	space	NOUN
cana-1961	277	8	.	.	PUNCT
cana-1961	278	1	baghdad	baghdad	PROPN
cana-1961	278	2	sci	sci	PROPN
cana-1961	278	3	.	.	PUNCT
cana-1961	279	1	j.	j.	PROPN
cana-1961	279	2	2024	2024	NUM
cana-1961	279	3	;	;	PUNCT
cana-1961	279	4	21(3	21(3	NUM
cana-1961	279	5	):	):	PUNCT
cana-1961	279	6	1080	1080	NUM
cana-1961	279	7	-	-	SYM
cana-1961	279	8	1085	1085	NUM
cana-1961	279	9	.	.	PUNCT
cana-1961	280	1	doi	doi	NOUN
cana-1961	280	2	:	:	PUNCT
cana-1961	280	3	https://dx.doi.org/10.21123/bsj.2023.8230	https://dx.doi.org/10.21123/bsj.2023.8230	X
cana-1961	281	1	[	[	X
cana-1961	281	2	7	7	X
cana-1961	281	3	]	]	X
cana-1961	281	4	van	van	PROPN
cana-1961	281	5	an	an	DET
cana-1961	281	6	l.	l.	PROPN
cana-1961	281	7	generalized	generalized	ADJ
cana-1961	281	8	stability	stability	NOUN
cana-1961	281	9	of	of	ADP
cana-1961	281	10	the	the	DET
cana-1961	281	11	quadratic	quadratic	ADJ
cana-1961	281	12	type	type	NOUN
cana-1961	281	13	λ	λ	ADJ
cana-1961	281	14	-	-	ADJ
cana-1961	281	15	functional	functional	ADJ
cana-1961	281	16	equation	equation	NOUN
cana-1961	281	17	with	with	ADP
cana-1961	281	18	3k	3k	NOUN
cana-1961	281	19	-	-	PUNCT
cana-1961	281	20	variables	variable	NOUN
cana-1961	281	21	in	in	ADP
cana-1961	281	22	non	non	ADJ
cana-1961	281	23	-	-	ADJ
cana-1961	281	24	archimedean	archimedean	ADJ
cana-1961	281	25	banach	banach	NOUN
cana-1961	281	26	space	space	NOUN
cana-1961	281	27	and	and	CCONJ
cana-1961	281	28	non	non	ADJ
cana-1961	281	29	-	-	ADJ
cana-1961	281	30	archimedean	archimedean	ADJ
cana-1961	281	31	random	random	ADJ
cana-1961	281	32	normed	normed	ADJ
cana-1961	281	33	space	space	NOUN
cana-1961	281	34	.	.	PUNCT
cana-1961	282	1	open	open	ADJ
cana-1961	282	2	access	access	NOUN
cana-1961	282	3	library	library	NOUN
cana-1961	282	4	j.	j.	PROPN
cana-1961	282	5	2023	2023	NUM
cana-1961	282	6	;	;	PUNCT
cana-1961	282	7	10(2	10(2	NUM
cana-1961	282	8	):	):	PUNCT
cana-1961	282	9	1	1	NUM
cana-1961	282	10	-	-	SYM
cana-1961	282	11	21	21	NUM
cana-1961	282	12	.	.	PUNCT
cana-1961	283	1	doi	doi	NOUN
cana-1961	283	2	:	:	PUNCT
cana-1961	283	3	https://doi.org/10.4236/oalib.1109821	https://doi.org/10.4236/oalib.1109821	NOUN
cana-1961	283	4	[	[	X
cana-1961	283	5	8	8	NUM
cana-1961	283	6	]	]	PUNCT
cana-1961	283	7	hussein	hussein	PROPN
cana-1961	283	8	luaibi	luaibi	PROPN
cana-1961	283	9	h	h	NOUN
cana-1961	283	10	,	,	PUNCT
cana-1961	283	11	abed	abed	PROPN
cana-1961	283	12	s	s	PART
cana-1961	283	13	s.	s.	PROPN
cana-1961	283	14	fixed	fix	VERB
cana-1961	283	15	point	point	NOUN
cana-1961	283	16	theorems	theorem	NOUN
cana-1961	283	17	in	in	ADP
cana-1961	283	18	general	general	ADJ
cana-1961	283	19	metric	metric	ADJ
cana-1961	283	20	space	space	NOUN
cana-1961	283	21	with	with	ADP
cana-1961	283	22	an	an	DET
cana-1961	283	23	application	application	NOUN
cana-1961	283	24	.	.	PUNCT
cana-1961	284	1	baghdad	baghdad	PROPN
cana-1961	284	2	sci	sci	PROPN
cana-1961	284	3	.	.	PUNCT
cana-1961	285	1	j.	j.	PROPN
cana-1961	285	2	2021	2021	NUM
cana-1961	285	3	;	;	PUNCT
cana-1961	285	4	18(1(suppl	18(1(suppl	PROPN
cana-1961	285	5	.	.	PUNCT
cana-1961	285	6	)	)	PUNCT
cana-1961	285	7	):	):	PUNCT
cana-1961	285	8	0812	0812	NUM
cana-1961	285	9	-	-	SYM
cana-1961	285	10	0815	0815	NUM
cana-1961	285	11	.	.	PUNCT
cana-1961	286	1	doi	doi	NOUN
cana-1961	286	2	:	:	PUNCT
cana-1961	286	3	http://dx.doi.org/10.21123/bsj.2021.18.1(suppl.).0812	http://dx.doi.org/10.21123/bsj.2021.18.1(suppl.).0812	PROPN
cana-1961	287	1	[	[	X
cana-1961	287	2	9	9	NUM
cana-1961	287	3	]	]	PUNCT
cana-1961	287	4	thanyacharoen	thanyacharoen	NOUN
cana-1961	287	5	a	a	NOUN
cana-1961	287	6	,	,	PUNCT
cana-1961	287	7	sintunavarat	sintunavarat	PROPN
cana-1961	287	8	w.	w.	PROPN
cana-1961	287	9	the	the	DET
cana-1961	287	10	new	new	ADJ
cana-1961	287	11	investigation	investigation	NOUN
cana-1961	287	12	of	of	ADP
cana-1961	287	13	the	the	DET
cana-1961	287	14	stability	stability	NOUN
cana-1961	287	15	of	of	ADP
cana-1961	287	16	mixed	mixed	ADJ
cana-1961	287	17	-	-	PUNCT
cana-1961	287	18	type	type	NOUN
cana-1961	287	19	additive	additive	NOUN
cana-1961	287	20	-	-	PUNCT
cana-1961	287	21	quartic	quartic	ADJ
cana-1961	287	22	functional	functional	ADJ
cana-1961	287	23	equations	equation	NOUN
cana-1961	287	24	in	in	ADP
cana-1961	287	25	non	non	ADJ
cana-1961	287	26	-	-	ADJ
cana-1961	287	27	archimedean	archimedean	ADJ
cana-1961	287	28	spaces	space	NOUN
cana-1961	287	29	.	.	PUNCT
cana-1961	288	1	demonstr	demonstr	NOUN
cana-1961	288	2	math	math	NOUN
cana-1961	288	3	.	.	PUNCT
cana-1961	289	1	2020	2020	NUM
cana-1961	289	2	;	;	PUNCT
cana-1961	289	3	53	53	NUM
cana-1961	289	4	:	:	PUNCT
cana-1961	289	5	174–192	174–192	NUM
cana-1961	289	6	.	.	PUNCT
cana-1961	290	1	doi	doi	NOUN
cana-1961	290	2	:	:	PUNCT
cana-1961	290	3	https://doi.org/10.1515/dema-20200009	https://doi.org/10.1515/dema-20200009	PROPN
cana-1961	290	4	[	[	X
cana-1961	290	5	10	10	NUM
cana-1961	290	6	]	]	X
cana-1961	290	7	ramachandran	ramachandran	PROPN
cana-1961	290	8	a	a	PROPN
cana-1961	290	9	,	,	PUNCT
cana-1961	290	10	sangeetha	sangeetha	PROPN
cana-1961	290	11	s.	s.	PROPN
cana-1961	290	12	on	on	ADP
cana-1961	290	13	the	the	DET
cana-1961	290	14	generalized	generalized	ADJ
cana-1961	290	15	quadratic	quadratic	ADJ
cana-1961	290	16	-	-	PUNCT
cana-1961	290	17	quartic	quartic	ADJ
cana-1961	290	18	cauchy	cauchy	ADJ
cana-1961	290	19	functional	functional	ADJ
cana-1961	290	20	equation	equation	NOUN
cana-1961	290	21	and	and	CCONJ
cana-1961	290	22	its	its	PRON
cana-1961	290	23	stability	stability	NOUN
cana-1961	290	24	over	over	ADP
cana-1961	290	25	non	non	ADJ
cana-1961	290	26	-	-	ADJ
cana-1961	290	27	archimedean	archimedean	ADJ
cana-1961	290	28	normed	normed	ADJ
cana-1961	290	29	space	space	NOUN
cana-1961	290	30	.	.	PUNCT
cana-1961	291	1	math	math	NOUN
cana-1961	291	2	.	.	PUNCT
cana-1961	292	1	stat	stat	PROPN
cana-1961	292	2	.	.	PUNCT
cana-1961	293	1	2022	2022	NUM
cana-1961	293	2	;	;	PUNCT
cana-1961	293	3	10(6	10(6	NUM
cana-1961	293	4	):	):	PUNCT
cana-1961	293	5	1210	1210	NUM
cana-1961	293	6	-	-	SYM
cana-1961	293	7	7	7	NUM
cana-1961	293	8	.	.	PUNCT
cana-1961	293	9	available	available	ADJ
cana-1961	293	10	from	from	ADP
cana-1961	293	11	:	:	PUNCT
cana-1961	293	12	https://www.hrpub.org/download/20221030/ms7-13428783.pdf	https://www.hrpub.org/download/20221030/ms7-13428783.pdf	PROPN
cana-1961	294	1	[	[	X
cana-1961	294	2	11	11	NUM
cana-1961	294	3	]	]	X
cana-1961	294	4	majani	majani	PROPN
cana-1961	294	5	h.	h.	PROPN
cana-1961	294	6	stability	stability	NOUN
cana-1961	294	7	of	of	ADP
cana-1961	294	8	a	a	DET
cana-1961	294	9	system	system	NOUN
cana-1961	294	10	of	of	ADP
cana-1961	294	11	euler	euler	NOUN
cana-1961	294	12	-	-	PUNCT
cana-1961	294	13	lagrange	lagrange	NOUN
cana-1961	294	14	type	type	NOUN
cana-1961	294	15	cubic	cubic	ADJ
cana-1961	294	16	functional	functional	ADJ
cana-1961	294	17	equations	equation	NOUN
cana-1961	294	18	in	in	ADP
cana-1961	294	19	non	non	ADJ
cana-1961	294	20	-	-	ADJ
cana-1961	294	21	archimedean	archimedean	ADJ
cana-1961	294	22	2	2	NUM
cana-1961	294	23	-	-	PUNCT
cana-1961	294	24	normed	norme	VERB
cana-1961	294	25	spaces	space	NOUN
cana-1961	294	26	.	.	PUNCT
cana-1961	295	1	j	j	PROPN
cana-1961	295	2	adv	adv	PROPN
cana-1961	295	3	math	math	PROPN
cana-1961	295	4	model	model	PROPN
cana-1961	295	5	.	.	PUNCT
cana-1961	295	6	2021	2021	NUM
cana-1961	295	7	;	;	PUNCT
cana-1961	295	8	11(1	11(1	NUM
cana-1961	295	9	):	):	PUNCT
cana-1961	295	10	11	11	NUM
cana-1961	295	11	-	-	SYM
cana-1961	295	12	24	24	NUM
cana-1961	295	13	.	.	PUNCT
cana-1961	296	1	doi	doi	NOUN
cana-1961	296	2	:	:	PUNCT
cana-1961	296	3	https://doi.org/10.22055/jamm.2020.28513.1685	https://doi.org/10.22055/jamm.2020.28513.1685	PROPN
cana-1961	296	4	[	[	SYM
cana-1961	296	5	12	12	NUM
cana-1961	296	6	]	]	X
cana-1961	296	7	ulam	ulam	PROPN
cana-1961	296	8	s	s	PROPN
cana-1961	296	9	m.	m.	NOUN
cana-1961	296	10	problem	problem	NOUN
cana-1961	296	11	in	in	ADP
cana-1961	296	12	modern	modern	ADJ
cana-1961	296	13	mathematics	mathematic	NOUN
cana-1961	296	14	(	(	PUNCT
cana-1961	296	15	science	science	NOUN
cana-1961	296	16	edition	edition	PROPN
cana-1961	296	17	)	)	PUNCT
cana-1961	296	18	.	.	PUNCT
cana-1961	297	1	john	john	PROPN
cana-1961	297	2	wiley	wiley	PROPN
cana-1961	297	3	and	and	CCONJ
cana-1961	297	4	sons	son	NOUN
cana-1961	297	5	,	,	PUNCT
cana-1961	297	6	inc	inc	PROPN
cana-1961	297	7	.	.	PROPN
cana-1961	297	8	new	new	PROPN
cana-1961	297	9	york	york	PROPN
cana-1961	297	10	;	;	PUNCT
cana-1961	297	11	1964	1964	NUM
cana-1961	297	12	.	.	PUNCT
cana-1961	298	1	[	[	X
cana-1961	298	2	13	13	NUM
cana-1961	298	3	]	]	X
cana-1961	298	4	hyers	hyer	NOUN
cana-1961	298	5	d	d	X
cana-1961	298	6	h.	h.	PROPN
cana-1961	298	7	on	on	ADP
cana-1961	298	8	the	the	DET
cana-1961	298	9	stability	stability	NOUN
cana-1961	298	10	of	of	ADP
cana-1961	298	11	the	the	DET
cana-1961	298	12	linear	linear	ADJ
cana-1961	298	13	functional	functional	ADJ
cana-1961	298	14	equation	equation	NOUN
cana-1961	298	15	.	.	PUNCT
cana-1961	299	1	proc	proc	PROPN
cana-1961	299	2	nat	nat	PROPN
cana-1961	299	3	acad	acad	PROPN
cana-1961	299	4	sci	sci	PROPN
cana-1961	299	5	.	.	PUNCT
cana-1961	299	6	usa	usa	PROPN
cana-1961	299	7	.	.	PROPN
cana-1961	299	8	1941	1941	NUM
cana-1961	299	9	;	;	PUNCT
cana-1961	299	10	27	27	NUM
cana-1961	299	11	:	:	PUNCT
cana-1961	300	1	222–224	222–224	NUM
cana-1961	300	2	.	.	PUNCT
cana-1961	301	1	doi	doi	NOUN
cana-1961	301	2	:	:	PUNCT
cana-1961	301	3	https://doi.org/10.1073/pnas.27.4.222	https://doi.org/10.1073/pnas.27.4.222	PROPN
cana-1961	302	1	[	[	X
cana-1961	302	2	14	14	NUM
cana-1961	302	3	]	]	X
cana-1961	302	4	aoki	aoki	PROPN
cana-1961	302	5	t.	t.	PROPN
cana-1961	302	6	on	on	ADP
cana-1961	302	7	the	the	DET
cana-1961	302	8	stability	stability	NOUN
cana-1961	302	9	of	of	ADP
cana-1961	302	10	the	the	DET
cana-1961	302	11	linear	linear	ADJ
cana-1961	302	12	transformation	transformation	NOUN
cana-1961	302	13	in	in	ADP
cana-1961	302	14	banach	banach	NOUN
cana-1961	302	15	spaces	space	NOUN
cana-1961	302	16	.	.	PUNCT
cana-1961	303	1	j	j	PROPN
cana-1961	303	2	math	math	PROPN
cana-1961	303	3	soc	soc	PROPN
cana-1961	303	4	.	.	PUNCT
cana-1961	303	5	1950	1950	NUM
cana-1961	303	6	;	;	PUNCT
cana-1961	303	7	2	2	NUM
cana-1961	303	8	:	:	SYM
cana-1961	303	9	64–66	64–66	NUM
cana-1961	303	10	.	.	PUNCT
cana-1961	303	11	available	available	ADJ
cana-1961	303	12	from	from	ADP
cana-1961	303	13	:	:	PUNCT
cana-1961	303	14	https://projecteuclid.org/journalarticle/download?urlid=10.2969%2fjmsj%2f00210064	https://projecteuclid.org/journalarticle/download?urlid=10.2969%2fjmsj%2f00210064	PROPN
cana-1961	304	1	[	[	X
cana-1961	304	2	15	15	NUM
cana-1961	304	3	]	]	PUNCT
cana-1961	304	4	rassias	rassia	NOUN
cana-1961	304	5	t	t	PROPN
cana-1961	304	6	m.	m.	NOUN
cana-1961	304	7	on	on	ADP
cana-1961	304	8	the	the	DET
cana-1961	304	9	stability	stability	NOUN
cana-1961	304	10	of	of	ADP
cana-1961	304	11	the	the	DET
cana-1961	304	12	linear	linear	ADJ
cana-1961	304	13	mapping	mapping	NOUN
cana-1961	304	14	in	in	ADP
cana-1961	304	15	banach	banach	NOUN
cana-1961	304	16	spaces	space	NOUN
cana-1961	304	17	.	.	PUNCT
cana-1961	305	1	proc	proc	NOUN
cana-1961	305	2	am	be	AUX
cana-1961	305	3	math	math	NOUN
cana-1961	305	4	soc	soc	NOUN
cana-1961	305	5	.	.	PUNCT
cana-1961	306	1	1978	1978	NUM
cana-1961	306	2	;	;	PUNCT
cana-1961	306	3	72	72	NUM
cana-1961	306	4	:	:	SYM
cana-1961	307	1	297–300	297–300	NUM
cana-1961	307	2	.	.	PUNCT
cana-1961	307	3	doi	doi	NOUN
cana-1961	307	4	:	:	PUNCT
cana-1961	307	5	http://dx.doi.org/10.1090/s0002-9939-1978-0507327-1	http://dx.doi.org/10.1090/s0002-9939-1978-0507327-1	PROPN
cana-1961	308	1	[	[	X
cana-1961	308	2	16	16	NUM
cana-1961	308	3	]	]	PUNCT
cana-1961	308	4	gavruta	gavruta	PROPN
cana-1961	308	5	p.	p.	PROPN
cana-1961	308	6	a	a	DET
cana-1961	308	7	generalization	generalization	NOUN
cana-1961	308	8	of	of	ADP
cana-1961	308	9	the	the	DET
cana-1961	308	10	hyers	hyers	PROPN
cana-1961	308	11	-	-	PUNCT
cana-1961	308	12	ulam	ulam	NOUN
cana-1961	308	13	-	-	PUNCT
cana-1961	308	14	rassias	rassia	NOUN
cana-1961	308	15	of	of	ADP
cana-1961	308	16	approximate	approximate	ADJ
cana-1961	308	17	additive	additive	ADJ
cana-1961	308	18	mappings	mapping	NOUN
cana-1961	308	19	.	.	PUNCT
cana-1961	309	1	j	j	PROPN
cana-1961	309	2	math	math	PROPN
cana-1961	309	3	anal	anal	PROPN
cana-1961	309	4	appl	appl	PROPN
cana-1961	309	5	.	.	PROPN
cana-1961	309	6	1994	1994	NUM
cana-1961	309	7	;	;	PUNCT
cana-1961	309	8	184	184	NUM
cana-1961	309	9	:	:	SYM
cana-1961	309	10	431–436	431–436	NUM
cana-1961	309	11	.	.	PUNCT
cana-1961	310	1	doi	doi	NOUN
cana-1961	310	2	:	:	PUNCT
cana-1961	310	3	https://doi.org/10.1006/jmaa.1994.1211	https://doi.org/10.1006/jmaa.1994.1211	PROPN
cana-1961	311	1	[	[	X
cana-1961	311	2	17	17	NUM
cana-1961	311	3	]	]	PUNCT
cana-1961	311	4	gahler	gahler	NOUN
cana-1961	311	5	s.	s.	PROPN
cana-1961	311	6	lineare	lineare	PROPN
cana-1961	311	7	2	2	NUM
cana-1961	311	8	-	-	PUNCT
cana-1961	311	9	normierte	normierte	NOUN
cana-1961	311	10	rumen	ruman	NOUN
cana-1961	311	11	.	.	PUNCT
cana-1961	312	1	math	math	PROPN
cana-1961	312	2	nachr	nachr	PROPN
cana-1961	312	3	.	.	PUNCT
cana-1961	313	1	1964	1964	NUM
cana-1961	313	2	;	;	PUNCT
cana-1961	313	3	28	28	NUM
cana-1961	313	4	:	:	PUNCT
cana-1961	313	5	1–43	1–43	NUM
cana-1961	313	6	.	.	PUNCT
cana-1961	314	1	doi	doi	PROPN
cana-1961	314	2	:	:	PUNCT
cana-1961	314	3	http://dx.doi.org/10.1002/mana.19640280102	http://dx.doi.org/10.1002/mana.19640280102	X
cana-1961	314	4	[	[	X
cana-1961	314	5	18	18	NUM
cana-1961	314	6	]	]	X
cana-1961	314	7	white	white	ADJ
cana-1961	314	8	a.	a.	NOUN
cana-1961	314	9	2	2	NUM
cana-1961	314	10	-	-	PUNCT
cana-1961	314	11	banach	banach	NOUN
cana-1961	314	12	spaces	space	NOUN
cana-1961	314	13	.	.	PUNCT
cana-1961	315	1	math	math	PROPN
cana-1961	315	2	nachr	nachr	PROPN
cana-1961	315	3	.	.	PUNCT
cana-1961	316	1	1969	1969	NUM
cana-1961	316	2	;	;	PUNCT
cana-1961	316	3	42	42	NUM
cana-1961	316	4	:	:	PUNCT
cana-1961	316	5	43–60	43–60	NUM
cana-1961	316	6	.	.	PUNCT
cana-1961	317	1	doi	doi	NOUN
cana-1961	317	2	:	:	PUNCT
cana-1961	317	3	https://doi.org/10.1002/mana.19690420104	https://doi.org/10.1002/mana.19690420104	X
cana-1961	318	1	[	[	X
cana-1961	318	2	19	19	NUM
cana-1961	318	3	]	]	PUNCT
cana-1961	318	4	park	park	NOUN
cana-1961	318	5	s	s	PROPN
cana-1961	318	6	,	,	PUNCT
cana-1961	318	7	kim	kim	PROPN
cana-1961	318	8	c.	c.	PROPN
cana-1961	318	9	the	the	DET
cana-1961	318	10	generalized	generalize	VERB
cana-1961	318	11	hyers	hyers	PROPN
cana-1961	318	12	-	-	PUNCT
cana-1961	318	13	ulam	ulam	ADJ
cana-1961	318	14	stability	stability	NOUN
cana-1961	318	15	of	of	ADP
cana-1961	318	16	additive	additive	ADJ
cana-1961	318	17	functional	functional	ADJ
cana-1961	318	18	inequalities	inequality	NOUN
cana-1961	318	19	in	in	ADP
cana-1961	318	20	non	non	ADJ
cana-1961	318	21	-	-	ADJ
cana-1961	318	22	archimedean	archimedean	ADJ
cana-1961	318	23	2normed	2normed	NUM
cana-1961	318	24	space	space	NOUN
cana-1961	318	25	.	.	PUNCT
cana-1961	319	1	korean	korean	PROPN
cana-1961	319	2	j	j	PROPN
cana-1961	319	3	math	math	PROPN
cana-1961	319	4	.	.	PUNCT
cana-1961	320	1	2014	2014	NUM
cana-1961	320	2	;	;	PUNCT
cana-1961	321	1	22(2	22(2	NUM
cana-1961	321	2	):	):	PUNCT
cana-1961	321	3	339	339	NUM
cana-1961	321	4	-	-	SYM
cana-1961	321	5	348	348	NUM
cana-1961	321	6	.	.	PUNCT
cana-1961	322	1	doi	doi	NOUN
cana-1961	322	2	:	:	PUNCT
cana-1961	322	3	https://doi.org/10.11568/kjm.2014.22.2.339	https://doi.org/10.11568/kjm.2014.22.2.339	PROPN
cana-1961	322	4	[	[	PUNCT
cana-1961	322	5	20	20	NUM
cana-1961	322	6	]	]	X
cana-1961	322	7	wang	wang	PROPN
cana-1961	322	8	z	z	PROPN
cana-1961	322	9	,	,	PUNCT
cana-1961	322	10	park	park	NOUN
cana-1961	322	11	c	c	PROPN
cana-1961	322	12	,	,	PUNCT
cana-1961	322	13	shin	shin	PROPN
cana-1961	322	14	dy	dy	X
cana-1961	322	15	.	.	PROPN
cana-1961	322	16	additive	additive	PROPN
cana-1961	322	17	ρ	ρ	ADJ
cana-1961	322	18	-	-	ADJ
cana-1961	322	19	functional	functional	ADJ
cana-1961	322	20	inequalities	inequality	NOUN
cana-1961	322	21	in	in	ADP
cana-1961	322	22	non	non	ADJ
cana-1961	322	23	-	-	ADJ
cana-1961	322	24	archimedean	archimedean	ADJ
cana-1961	322	25	2	2	NUM
cana-1961	322	26	-	-	PUNCT
cana-1961	322	27	normed	norme	VERB
cana-1961	322	28	spaces	space	NOUN
cana-1961	322	29	.	.	PUNCT
cana-1961	323	1	aims	aim	VERB
cana-1961	323	2	math	math	NOUN
cana-1961	323	3	.	.	PUNCT
cana-1961	323	4	2021	2021	NUM
cana-1961	323	5	;	;	PUNCT
cana-1961	323	6	6(2	6(2	NUM
cana-1961	323	7	)	)	PUNCT
cana-1961	323	8	.	.	PUNCT
cana-1961	324	1	available	available	ADJ
cana-1961	324	2	from	from	ADP
cana-1961	324	3	:	:	PUNCT
cana-1961	324	4	http://www.aimspress.com/article/doi/10.3934/math.2021116	http://www.aimspress.com/article/doi/10.3934/math.2021116	NOUN
cana-1961	325	1	[	[	X
cana-1961	325	2	21	21	NUM
cana-1961	325	3	]	]	X
cana-1961	325	4	ghali	ghali	NOUN
cana-1961	325	5	r	r	NOUN
cana-1961	325	6	,	,	PUNCT
cana-1961	325	7	kabbaj	kabbaj	PROPN
cana-1961	325	8	s.	s.	PROPN
cana-1961	325	9	a	a	DET
cana-1961	325	10	new	new	ADJ
cana-1961	325	11	approach	approach	NOUN
cana-1961	325	12	to	to	ADP
cana-1961	325	13	fixed	fix	VERB
cana-1961	325	14	point	point	NOUN
cana-1961	325	15	result	result	NOUN
cana-1961	325	16	in	in	ADP
cana-1961	325	17	non	non	ADJ
cana-1961	325	18	-	-	ADJ
cana-1961	325	19	archimedean	archimedean	ADJ
cana-1961	325	20	2	2	NUM
cana-1961	325	21	-	-	PUNCT
cana-1961	325	22	banach	banach	NOUN
cana-1961	325	23	space	space	NOUN
cana-1961	325	24	and	and	CCONJ
cana-1961	325	25	some	some	PRON
cana-1961	325	26	of	of	ADP
cana-1961	325	27	its	its	PRON
cana-1961	325	28	applications	application	NOUN
cana-1961	325	29	.	.	PUNCT
cana-1961	326	1	palestine	palestine	PROPN
cana-1961	326	2	j	j	PROPN
cana-1961	326	3	math	math	PROPN
cana-1961	326	4	.	.	PUNCT
cana-1961	327	1	2022	2022	NUM
cana-1961	327	2	;	;	PUNCT
cana-1961	327	3	11(3	11(3	NUM
cana-1961	327	4	):	):	PUNCT
cana-1961	327	5	586	586	NUM
cana-1961	327	6	-	-	SYM
cana-1961	327	7	597	597	NUM
cana-1961	327	8	.	.	PUNCT
cana-1961	328	1	available	available	ADJ
cana-1961	328	2	from	from	ADP
cana-1961	328	3	:	:	PUNCT
cana-1961	328	4	https://pjm.ppu.edu/sites/default/files/papers/pjm_may_%283%292022_586_to_597.pdf	https://pjm.ppu.edu/sites/default/files/papers/pjm_may_%283%292022_586_to_597.pdf	NOUN
cana-1961	328	5	[	[	X
cana-1961	328	6	22	22	NUM
cana-1961	328	7	]	]	X
cana-1961	328	8	cho	cho	PROPN
cana-1961	328	9	y	y	PROPN
cana-1961	328	10	,	,	PUNCT
cana-1961	328	11	gordji	gordji	PROPN
cana-1961	328	12	m	m	PROPN
cana-1961	328	13	,	,	PUNCT
cana-1961	328	14	zolfaghari	zolfaghari	PROPN
cana-1961	328	15	s.	s.	PROPN
cana-1961	328	16	solutions	solutions	PROPN
cana-1961	328	17	and	and	CCONJ
cana-1961	328	18	stability	stability	NOUN
cana-1961	328	19	of	of	ADP
cana-1961	328	20	generalized	generalized	ADJ
cana-1961	328	21	mixed	mixed	ADJ
cana-1961	328	22	type	type	NOUN
cana-1961	328	23	qc	qc	PROPN
cana-1961	328	24	functional	functional	ADJ
cana-1961	328	25	equations	equation	NOUN
cana-1961	328	26	in	in	ADP
cana-1961	328	27	random	random	ADJ
cana-1961	328	28	normed	normed	ADJ
cana-1961	328	29	spaces	space	NOUN
cana-1961	328	30	.	.	PUNCT
cana-1961	329	1	j	j	PROPN
cana-1961	329	2	inequal	inequal	PROPN
cana-1961	329	3	appl	appl	PROPN
cana-1961	329	4	.	.	PUNCT
cana-1961	330	1	2020	2020	NUM
cana-1961	330	2	;	;	PUNCT
cana-1961	330	3	2020	2020	NUM
cana-1961	330	4	:	:	PUNCT
cana-1961	331	1	403101	403101	NUM
cana-1961	331	2	.	.	PUNCT
cana-1961	332	1	available	available	ADJ
cana-1961	332	2	from	from	ADP
cana-1961	332	3	:	:	PUNCT
cana-1961	332	4	https://journalofinequalitiesandapplications.springeropen.com/counter/pdf/10.1155/2010/403101.pdf	https://journalofinequalitiesandapplications.springeropen.com/counter/pdf/10.1155/2010/403101.pdf	PROPN
cana-1961	332	5	.	.	PUNCT
cana-1961	333	1	doi	doi	PROPN
cana-1961	333	2	:	:	PUNCT
cana-1961	333	3	https://doi.org/10.1155/2010/403101	https://doi.org/10.1155/2010/403101	PROPN
cana-1961	333	4	.	.	PUNCT
cana-1961	334	1	[	[	X
cana-1961	334	2	23	23	NUM
cana-1961	334	3	]	]	X
cana-1961	334	4	mohiuddine	mohiuddine	PROPN
cana-1961	334	5	sa	sa	PROPN
cana-1961	334	6	,	,	PUNCT
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cana-1961	334	8	k	k	NOUN
cana-1961	334	9	,	,	PUNCT
cana-1961	334	10	mursaleen	mursaleen	PROPN
cana-1961	334	11	m	m	PROPN
cana-1961	334	12	,	,	PUNCT
cana-1961	334	13	alotaibi	alotaibi	NOUN
cana-1961	334	14	t.	t.	NOUN
cana-1961	334	15	stability	stability	NOUN
cana-1961	334	16	of	of	ADP
cana-1961	334	17	quartic	quartic	ADJ
cana-1961	334	18	functional	functional	ADJ
cana-1961	334	19	equation	equation	NOUN
cana-1961	334	20	in	in	ADP
cana-1961	334	21	modular	modular	ADJ
cana-1961	334	22	spaces	space	NOUN
cana-1961	334	23	via	via	ADP
cana-1961	334	24	hyers	hyer	NOUN
cana-1961	334	25	and	and	CCONJ
cana-1961	334	26	fixed	fix	VERB
cana-1961	334	27	-	-	PUNCT
cana-1961	334	28	point	point	NOUN
cana-1961	334	29	methods	method	NOUN
cana-1961	334	30	.	.	PUNCT
cana-1961	335	1	mathematics	mathematic	NOUN
cana-1961	335	2	.	.	PUNCT
cana-1961	336	1	2022	2022	NUM
cana-1961	336	2	;	;	PUNCT
cana-1961	336	3	10(11	10(11	NUM
cana-1961	336	4	):	):	PUNCT
cana-1961	336	5	1938	1938	NUM
cana-1961	336	6	.	.	PUNCT
cana-1961	337	1	doi	doi	NOUN
cana-1961	337	2	:	:	PUNCT
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cana-1961	337	4	[	[	X
cana-1961	337	5	24	24	NUM
cana-1961	337	6	]	]	X
cana-1961	337	7	hensel	hensel	PROPN
cana-1961	337	8	k.	k.	PROPN
cana-1961	337	9	ubereine	ubereine	PROPN
cana-1961	337	10	neue	neue	PROPN
cana-1961	337	11	begrndung	begrndung	PROPN
cana-1961	337	12	der	der	NOUN
cana-1961	337	13	theorie	theorie	PROPN
cana-1961	337	14	der	der	PROPN
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cana-1961	337	16	zahlen	zahlen	PROPN
cana-1961	337	17	.	.	PUNCT
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cana-1961	338	2	dtsch	dtsch	PROPN
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cana-1961	338	5	.	.	PUNCT
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cana-1961	339	2	;	;	PUNCT
cana-1961	339	3	6	6	NUM
cana-1961	339	4	:	:	SYM
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cana-1961	339	6	.	.	PUNCT
cana-1961	339	7	available	available	ADJ
cana-1961	339	8	from	from	ADP
cana-1961	339	9	:	:	PUNCT
cana-1961	339	10	http://resolver.sub.uni-goettingen.de/purl?ppn37721857x_0006	http://resolver.sub.uni-goettingen.de/purl?ppn37721857x_0006	NOUN
cana-1961	340	1	[	[	X
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cana-1961	340	3	]	]	X
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cana-1961	340	10	points	point	NOUN
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cana-1961	340	12	the	the	DET
cana-1961	340	13	stability	stability	NOUN
cana-1961	340	14	of	of	ADP
cana-1961	340	15	jensen	jensen	PROPN
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cana-1961	340	17	functional	functional	ADJ
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cana-1961	341	1	j	j	PROPN
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cana-1961	341	3	pure	pure	ADJ
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cana-1961	342	8	.	.	PUNCT
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cana-1961	342	10	from	from	ADP
cana-1961	342	11	:	:	PUNCT
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cana-1961	342	13	.	.	PUNCT
cana-1961	343	1	[	[	X
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cana-1961	343	3	]	]	X
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cana-1961	344	1	a	a	DET
cana-1961	344	2	fixed	fix	VERB
cana-1961	344	3	point	point	NOUN
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cana-1961	344	12	generalized	generalized	ADJ
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cana-1961	350	7	https://journalofinequalitiesandapplications.springeropen.com/counter/pdf/10.1155/2010/403101.pdf	https://journalofinequalitiesandapplications.springeropen.com/counter/pdf/10.1155/2010/403101.pdf	PROPN
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cana-1961	350	11	http://www.emis.de/journals/jipam/images/075_02_jipam/075_02.pdf	http://www.emis.de/journals/jipam/images/075_02_jipam/075_02.pdf	NOUN
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