id	sid	tid	token	lemma	pos
cana-525	1	1	communications	communication	NOUN
cana-525	1	2	on	on	ADP
cana-525	1	3	applied	apply	VERB
cana-525	1	4	nonlinear	nonlinear	ADJ
cana-525	1	5	analysis	analysis	NOUN
cana-525	1	6	issn	issn	NOUN
cana-525	1	7	:	:	PUNCT
cana-525	1	8	1074	1074	NUM
cana-525	1	9	-	-	PUNCT
cana-525	1	10	133x	133x	NUM
cana-525	1	11	vol	vol	NOUN
cana-525	1	12	31	31	NUM
cana-525	1	13	no	no	NOUN
cana-525	1	14	.	.	NOUN
cana-525	1	15	2	2	NUM
cana-525	1	16	(	(	PUNCT
cana-525	1	17	2024	2024	NUM
cana-525	1	18	)	)	PUNCT
cana-525	1	19	129	129	NUM
cana-525	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-525	1	21	fuzzy	fuzzy	ADJ
cana-525	1	22	soft	soft	ADJ
cana-525	1	23	paranormal	paranormal	ADJ
cana-525	1	24	operator	operator	NOUN
cana-525	1	25	in	in	ADP
cana-525	1	26	fuzzy	fuzzy	ADJ
cana-525	1	27	soft	soft	ADJ
cana-525	1	28	hilbert	hilbert	NOUN
cana-525	1	29	space	space	NOUN
cana-525	1	30	dr	dr	PROPN
cana-525	1	31	a	a	DET
cana-525	1	32	radharamani1	radharamani1	PROPN
cana-525	1	33	,	,	PUNCT
cana-525	1	34	t	t	PROPN
cana-525	1	35	nagajothi2	nagajothi2	NOUN
cana-525	2	1	1assistant	1assistant	NUM
cana-525	2	2	professor	professor	NOUN
cana-525	2	3	department	department	NOUN
cana-525	2	4	of	of	ADP
cana-525	2	5	mathematics	mathematics	PROPN
cana-525	2	6	chikkanna	chikkanna	NOUN
cana-525	2	7	government	government	NOUN
cana-525	2	8	arts	arts	PROPN
cana-525	2	9	college	college	PROPN
cana-525	2	10	,	,	PUNCT
cana-525	2	11	tirupur	tirupur	PROPN
cana-525	2	12	–	–	PUNCT
cana-525	2	13	641602	641602	NUM
cana-525	2	14	mail	mail	NOUN
cana-525	2	15	i	i	NOUN
cana-525	2	16	d	d	PROPN
cana-525	2	17	:	:	PUNCT
cana-525	2	18	radhabtk@gmail.com	radhabtk@gmail.com	X
cana-525	3	1	2assistant	2assistant	NUM
cana-525	3	2	professor	professor	NOUN
cana-525	3	3	department	department	PROPN
cana-525	3	4	of	of	ADP
cana-525	3	5	mathematics	mathematics	PROPN
cana-525	3	6	psg	psg	PROPN
cana-525	3	7	college	college	PROPN
cana-525	3	8	of	of	ADP
cana-525	3	9	arts	arts	PROPN
cana-525	3	10	&	&	CCONJ
cana-525	3	11	science	science	PROPN
cana-525	3	12	,	,	PUNCT
cana-525	3	13	coimbatore	coimbatore	PROPN
cana-525	3	14	–	–	PUNCT
cana-525	3	15	641014	641014	NUM
cana-525	3	16	mail	mail	NOUN
cana-525	3	17	i	i	NOUN
cana-525	3	18	d	d	PROPN
cana-525	3	19	:	:	PUNCT
cana-525	3	20	cnpp1977@gmail.com	cnpp1977@gmail.com	X
cana-525	3	21	article	article	NOUN
cana-525	3	22	history	history	NOUN
cana-525	3	23	:	:	PUNCT
cana-525	3	24	received	receive	VERB
cana-525	3	25	:	:	PUNCT
cana-525	3	26	23	23	NUM
cana-525	3	27	-	-	SYM
cana-525	3	28	01	01	NUM
cana-525	3	29	-	-	PUNCT
cana-525	3	30	2024	2024	NUM
cana-525	3	31	revised	revise	VERB
cana-525	3	32	:	:	PUNCT
cana-525	3	33	08	08	NUM
cana-525	3	34	-	-	PUNCT
cana-525	3	35	04	04	NUM
cana-525	3	36	-	-	PUNCT
cana-525	3	37	2024	2024	NUM
cana-525	3	38	accepted	accept	VERB
cana-525	3	39	:	:	PUNCT
cana-525	3	40	26	26	NUM
cana-525	3	41	-	-	PUNCT
cana-525	3	42	04	04	NUM
cana-525	3	43	-	-	PUNCT
cana-525	3	44	2024	2024	NUM
cana-525	3	45	abstract	abstract	NOUN
cana-525	3	46	:	:	PUNCT
cana-525	3	47	this	this	DET
cana-525	3	48	paper	paper	NOUN
cana-525	3	49	defines	define	VERB
cana-525	3	50	the	the	DET
cana-525	3	51	fuzzy	fuzzy	ADJ
cana-525	3	52	soft	soft	ADJ
cana-525	3	53	paranormal	paranormal	ADJ
cana-525	3	54	operator	operator	NOUN
cana-525	3	55	and	and	CCONJ
cana-525	3	56	discusses	discuss	VERB
cana-525	3	57	several	several	ADJ
cana-525	3	58	fundamental	fundamental	ADJ
cana-525	3	59	fuzzy	fuzzy	ADJ
cana-525	3	60	soft	soft	ADJ
cana-525	3	61	paranormal	paranormal	ADJ
cana-525	3	62	operator	operator	NOUN
cana-525	3	63	properties	property	NOUN
cana-525	3	64	in	in	ADP
cana-525	3	65	fuzzy	fuzzy	ADJ
cana-525	3	66	soft	soft	ADJ
cana-525	3	67	hilbert	hilbert	NOUN
cana-525	3	68	space	space	NOUN
cana-525	3	69	.	.	PUNCT
cana-525	4	1	some	some	DET
cana-525	4	2	concepts	concept	NOUN
cana-525	4	3	relevant	relevant	ADJ
cana-525	4	4	to	to	ADP
cana-525	4	5	the	the	DET
cana-525	4	6	fuzzy	fuzzy	ADJ
cana-525	4	7	soft	soft	ADJ
cana-525	4	8	paranormal	paranormal	ADJ
cana-525	4	9	operator	operator	NOUN
cana-525	4	10	have	have	AUX
cana-525	4	11	been	be	AUX
cana-525	4	12	defined	define	VERB
cana-525	4	13	in	in	ADP
cana-525	4	14	fuzzy	fuzzy	ADJ
cana-525	4	15	soft	soft	ADJ
cana-525	4	16	hilbert	hilbert	NOUN
cana-525	4	17	space	space	NOUN
cana-525	4	18	.	.	PUNCT
cana-525	5	1	keywords	keyword	NOUN
cana-525	5	2	:	:	PUNCT
cana-525	5	3	fuzzy	fuzzy	ADJ
cana-525	5	4	soft	soft	ADJ
cana-525	5	5	normal	normal	ADJ
cana-525	5	6	operator	operator	NOUN
cana-525	5	7	,	,	PUNCT
cana-525	5	8	fuzzy	fuzzy	ADJ
cana-525	5	9	soft	soft	ADJ
cana-525	5	10	hilbert	hilbert	NOUN
cana-525	5	11	space	space	NOUN
cana-525	5	12	,	,	PUNCT
cana-525	5	13	fuzzy	fuzzy	ADJ
cana-525	5	14	soft	soft	ADJ
cana-525	5	15	hyponormal	hyponormal	ADJ
cana-525	5	16	operator	operator	NOUN
cana-525	5	17	,	,	PUNCT
cana-525	5	18	fuzzy	fuzzy	ADJ
cana-525	5	19	soft	soft	ADJ
cana-525	5	20	paranormal	paranormal	ADJ
cana-525	5	21	operator	operator	NOUN
cana-525	5	22	i	i	PRON
cana-525	5	23	introduction	introduction	VERB
cana-525	5	24	more	more	ADJ
cana-525	5	25	than	than	ADP
cana-525	5	26	a	a	DET
cana-525	5	27	century	century	NOUN
cana-525	5	28	ago	ago	ADV
cana-525	5	29	,	,	PUNCT
cana-525	5	30	the	the	DET
cana-525	5	31	field	field	NOUN
cana-525	5	32	of	of	ADP
cana-525	5	33	functional	functional	ADJ
cana-525	5	34	analysis	analysis	NOUN
cana-525	5	35	was	be	AUX
cana-525	5	36	established	establish	VERB
cana-525	5	37	to	to	PART
cana-525	5	38	address	address	VERB
cana-525	5	39	a	a	DET
cana-525	5	40	number	number	NOUN
cana-525	5	41	of	of	ADP
cana-525	5	42	problems	problem	NOUN
cana-525	5	43	in	in	ADP
cana-525	5	44	pure	pure	ADJ
cana-525	5	45	mathematics	mathematic	NOUN
cana-525	5	46	.	.	PUNCT
cana-525	6	1	in	in	ADP
cana-525	6	2	addition	addition	NOUN
cana-525	6	3	to	to	PART
cana-525	6	4	regularly	regularly	ADV
cana-525	6	5	presenting	present	VERB
cana-525	6	6	us	we	PRON
cana-525	6	7	with	with	ADP
cana-525	6	8	uncertainty	uncertainty	NOUN
cana-525	6	9	,	,	PUNCT
cana-525	6	10	the	the	DET
cana-525	6	11	phenomena	phenomenon	NOUN
cana-525	6	12	under	under	ADP
cana-525	6	13	study	study	NOUN
cana-525	6	14	's	's	PART
cana-525	6	15	ambiguity	ambiguity	NOUN
cana-525	6	16	also	also	ADV
cana-525	6	17	provides	provide	VERB
cana-525	6	18	us	we	PRON
cana-525	6	19	with	with	ADP
cana-525	6	20	instruments	instrument	NOUN
cana-525	6	21	for	for	ADP
cana-525	6	22	assessing	assess	VERB
cana-525	6	23	faults	fault	NOUN
cana-525	6	24	in	in	ADP
cana-525	6	25	solutions	solution	NOUN
cana-525	6	26	to	to	ADP
cana-525	6	27	issues	issue	NOUN
cana-525	6	28	with	with	ADP
cana-525	6	29	both	both	CCONJ
cana-525	6	30	infinite	infinite	ADJ
cana-525	6	31	and	and	CCONJ
cana-525	6	32	limited	limited	ADJ
cana-525	6	33	dimensions	dimension	NOUN
cana-525	6	34	.	.	PUNCT
cana-525	7	1	in	in	ADP
cana-525	7	2	a	a	DET
cana-525	7	3	variety	variety	NOUN
cana-525	7	4	of	of	ADP
cana-525	7	5	fields	field	NOUN
cana-525	7	6	,	,	PUNCT
cana-525	7	7	including	include	VERB
cana-525	7	8	engineering	engineering	NOUN
cana-525	7	9	,	,	PUNCT
cana-525	7	10	business	business	NOUN
cana-525	7	11	,	,	PUNCT
cana-525	7	12	medicine	medicine	NOUN
cana-525	7	13	,	,	PUNCT
cana-525	7	14	and	and	CCONJ
cana-525	7	15	economics	economic	NOUN
cana-525	7	16	,	,	PUNCT
cana-525	7	17	this	this	DET
cana-525	7	18	kind	kind	NOUN
cana-525	7	19	of	of	ADP
cana-525	7	20	problem	problem	NOUN
cana-525	7	21	might	might	AUX
cana-525	7	22	be	be	AUX
cana-525	7	23	encountered	encounter	VERB
cana-525	7	24	.	.	PUNCT
cana-525	8	1	our	our	PRON
cana-525	8	2	conventional	conventional	ADJ
cana-525	8	3	mathematical	mathematical	ADJ
cana-525	8	4	methods	method	NOUN
cana-525	8	5	frequently	frequently	ADV
cana-525	8	6	fall	fall	VERB
cana-525	8	7	short	short	ADJ
cana-525	8	8	in	in	ADP
cana-525	8	9	addressing	address	VERB
cana-525	8	10	such	such	ADJ
cana-525	8	11	problems	problem	NOUN
cana-525	8	12	.	.	PUNCT
cana-525	9	1	thus	thus	ADV
cana-525	9	2	,	,	PUNCT
cana-525	9	3	l.	l.	PROPN
cana-525	9	4	zadeh[3	zadeh[3	PROPN
cana-525	9	5	]	]	PUNCT
cana-525	9	6	provided	provide	VERB
cana-525	9	7	an	an	DET
cana-525	9	8	extension	extension	NOUN
cana-525	9	9	of	of	ADP
cana-525	9	10	set	set	NOUN
cana-525	9	11	theory	theory	NOUN
cana-525	9	12	in	in	ADP
cana-525	9	13	1965	1965	NUM
cana-525	9	14	.	.	PUNCT
cana-525	10	1	fuzzy	fuzzy	ADJ
cana-525	10	2	set	set	NOUN
cana-525	10	3	theory	theory	NOUN
cana-525	10	4	was	be	AUX
cana-525	10	5	the	the	DET
cana-525	10	6	term	term	NOUN
cana-525	10	7	given	give	VERB
cana-525	10	8	to	to	ADP
cana-525	10	9	the	the	DET
cana-525	10	10	resulting	result	VERB
cana-525	10	11	theory	theory	NOUN
cana-525	10	12	.	.	PUNCT
cana-525	11	1	fuzzy	fuzzy	ADJ
cana-525	11	2	set	set	PROPN
cana-525	11	3	theory	theory	NOUN
cana-525	11	4	quickly	quickly	ADV
cana-525	11	5	established	establish	VERB
cana-525	11	6	itself	itself	PRON
cana-525	11	7	as	as	ADP
cana-525	11	8	an	an	DET
cana-525	11	9	effective	effective	ADJ
cana-525	11	10	method	method	NOUN
cana-525	11	11	for	for	ADP
cana-525	11	12	dealing	deal	VERB
cana-525	11	13	with	with	ADP
cana-525	11	14	ambiguous	ambiguous	ADJ
cana-525	11	15	circumstances	circumstance	NOUN
cana-525	11	16	.	.	PUNCT
cana-525	12	1	the	the	DET
cana-525	12	2	basis	basis	NOUN
cana-525	12	3	function	function	NOUN
cana-525	12	4	from	from	ADP
cana-525	12	5	a	a	DET
cana-525	12	6	set	set	NOUN
cana-525	12	7	x	x	PUNCT
cana-525	12	8	to	to	ADP
cana-525	12	9	a	a	DET
cana-525	12	10	set	set	NOUN
cana-525	12	11	[	[	X
cana-525	12	12	0,1	0,1	NUM
cana-525	12	13	]	]	PUNCT
cana-525	12	14	defines	define	VERB
cana-525	12	15	the	the	DET
cana-525	12	16	set	set	NOUN
cana-525	12	17	x	x	PUNCT
cana-525	12	18	in	in	ADP
cana-525	12	19	classical	classical	ADJ
cana-525	12	20	set	set	NOUN
cana-525	12	21	theory	theory	NOUN
cana-525	12	22	.	.	PUNCT
cana-525	13	1	in	in	ADP
cana-525	13	2	contrast	contrast	NOUN
cana-525	13	3	,	,	PUNCT
cana-525	13	4	a	a	DET
cana-525	13	5	set	set	NOUN
cana-525	13	6	in	in	ADP
cana-525	13	7	fuzzy	fuzzy	ADJ
cana-525	13	8	set	set	NOUN
cana-525	13	9	theory	theory	NOUN
cana-525	13	10	is	be	AUX
cana-525	13	11	described	describe	VERB
cana-525	13	12	by	by	ADP
cana-525	13	13	its	its	PRON
cana-525	13	14	membership	membership	NOUN
cana-525	13	15	function	function	NOUN
cana-525	13	16	,	,	PUNCT
cana-525	13	17	which	which	PRON
cana-525	13	18	ranges	range	VERB
cana-525	13	19	from	from	ADP
cana-525	13	20	x	x	PRON
cana-525	13	21	to	to	ADP
cana-525	13	22	the	the	DET
cana-525	13	23	closed	closed	ADJ
cana-525	13	24	range	range	NOUN
cana-525	13	25	between	between	ADP
cana-525	13	26	0	0	NUM
cana-525	13	27	and	and	CCONJ
cana-525	13	28	1	1	NUM
cana-525	13	29	.	.	PUNCT
cana-525	14	1	in	in	ADP
cana-525	14	2	1999	1999	NUM
cana-525	14	3	,	,	PUNCT
cana-525	14	4	molodtsov[4	molodtsov[4	PROPN
cana-525	14	5	]	]	PUNCT
cana-525	14	6	also	also	ADV
cana-525	14	7	developed	develop	VERB
cana-525	14	8	a	a	DET
cana-525	14	9	fresh	fresh	ADJ
cana-525	14	10	generalisation	generalisation	NOUN
cana-525	14	11	for	for	ADP
cana-525	14	12	dealing	deal	VERB
cana-525	14	13	with	with	ADP
cana-525	14	14	uncertainty	uncertainty	NOUN
cana-525	14	15	.	.	PUNCT
cana-525	15	1	soft	soft	ADJ
cana-525	15	2	set	set	NOUN
cana-525	15	3	theory	theory	NOUN
cana-525	15	4	was	be	AUX
cana-525	15	5	created	create	VERB
cana-525	15	6	as	as	ADP
cana-525	15	7	a	a	DET
cana-525	15	8	result	result	NOUN
cana-525	15	9	of	of	ADP
cana-525	15	10	this	this	DET
cana-525	15	11	research	research	NOUN
cana-525	15	12	.	.	PUNCT
cana-525	16	1	since	since	SCONJ
cana-525	16	2	then	then	ADV
cana-525	16	3	,	,	PUNCT
cana-525	16	4	it	it	PRON
cana-525	16	5	has	have	AUX
cana-525	16	6	been	be	AUX
cana-525	16	7	applied	apply	VERB
cana-525	16	8	to	to	PART
cana-525	16	9	tackle	tackle	VERB
cana-525	16	10	difficult	difficult	ADJ
cana-525	16	11	issues	issue	NOUN
cana-525	16	12	in	in	ADP
cana-525	16	13	a	a	DET
cana-525	16	14	number	number	NOUN
cana-525	16	15	of	of	ADP
cana-525	16	16	fields	field	NOUN
cana-525	16	17	,	,	PUNCT
cana-525	16	18	including	include	VERB
cana-525	16	19	computer	computer	NOUN
cana-525	16	20	science	science	NOUN
cana-525	16	21	,	,	PUNCT
cana-525	16	22	engineering	engineering	NOUN
cana-525	16	23	,	,	PUNCT
cana-525	16	24	medicine	medicine	NOUN
cana-525	16	25	,	,	PUNCT
cana-525	16	26	and	and	CCONJ
cana-525	16	27	others	other	NOUN
cana-525	16	28	.	.	PUNCT
cana-525	17	1	a	a	DET
cana-525	17	2	soft	soft	ADJ
cana-525	17	3	set	set	NOUN
cana-525	17	4	is	be	AUX
cana-525	17	5	a	a	DET
cana-525	17	6	collection	collection	NOUN
cana-525	17	7	of	of	ADP
cana-525	17	8	universal	universal	ADJ
cana-525	17	9	sets	set	NOUN
cana-525	17	10	that	that	PRON
cana-525	17	11	has	have	AUX
cana-525	17	12	been	be	AUX
cana-525	17	13	parametrized	parametrize	VERB
cana-525	17	14	.	.	PUNCT
cana-525	18	1	soft	soft	ADJ
cana-525	18	2	set	set	NOUN
cana-525	18	3	gave	give	VERB
cana-525	18	4	rise	rise	NOUN
cana-525	18	5	to	to	ADP
cana-525	18	6	the	the	DET
cana-525	18	7	ideas	idea	NOUN
cana-525	18	8	of	of	ADP
cana-525	18	9	soft	soft	ADJ
cana-525	18	10	point	point	NOUN
cana-525	18	11	,	,	PUNCT
cana-525	18	12	soft	soft	ADJ
cana-525	18	13	normed	normed	ADJ
cana-525	18	14	space	space	NOUN
cana-525	18	15	,	,	PUNCT
cana-525	18	16	soft	soft	ADJ
cana-525	18	17	inner	inner	ADJ
cana-525	18	18	product	product	NOUN
cana-525	18	19	space	space	NOUN
cana-525	18	20	,	,	PUNCT
cana-525	18	21	and	and	CCONJ
cana-525	18	22	soft	soft	ADJ
cana-525	18	23	hilbert	hilbert	NOUN
cana-525	18	24	space	space	NOUN
cana-525	18	25	,	,	PUNCT
cana-525	18	26	which	which	PRON
cana-525	18	27	were	be	AUX
cana-525	18	28	later	later	ADV
cana-525	18	29	applied	apply	VERB
cana-525	18	30	in	in	ADP
cana-525	18	31	functional	functional	ADJ
cana-525	18	32	analysis	analysis	NOUN
cana-525	18	33	to	to	PART
cana-525	18	34	tackle	tackle	VERB
cana-525	18	35	a	a	DET
cana-525	18	36	number	number	NOUN
cana-525	18	37	of	of	ADP
cana-525	18	38	different	different	ADJ
cana-525	18	39	mathematical	mathematical	ADJ
cana-525	18	40	topics	topic	NOUN
cana-525	18	41	.	.	PUNCT
cana-525	19	1	the	the	DET
cana-525	19	2	concept	concept	NOUN
cana-525	19	3	of	of	ADP
cana-525	19	4	a	a	DET
cana-525	19	5	fuzzy	fuzzy	ADJ
cana-525	19	6	soft	soft	ADJ
cana-525	19	7	set	set	NOUN
cana-525	19	8	was	be	AUX
cana-525	19	9	initially	initially	ADV
cana-525	19	10	introduced	introduce	VERB
cana-525	19	11	in	in	ADP
cana-525	19	12	2001	2001	NUM
cana-525	19	13	by	by	ADP
cana-525	19	14	maji[5	maji[5	SYM
cana-525	19	15	]	]	X
cana-525	19	16	et	et	PROPN
cana-525	19	17	al	al	PROPN
cana-525	19	18	.	.	PUNCT
cana-525	20	1	the	the	DET
cana-525	20	2	idea	idea	NOUN
cana-525	20	3	was	be	AUX
cana-525	20	4	created	create	VERB
cana-525	20	5	by	by	ADP
cana-525	20	6	using	use	VERB
cana-525	20	7	a	a	DET
cana-525	20	8	soft	soft	ADJ
cana-525	20	9	set	set	NOUN
cana-525	20	10	and	and	CCONJ
cana-525	20	11	a	a	DET
cana-525	20	12	fuzzy	fuzzy	ADJ
cana-525	20	13	set	set	NOUN
cana-525	20	14	.	.	PUNCT
cana-525	21	1	to	to	PART
cana-525	21	2	provide	provide	VERB
cana-525	21	3	more	more	ADV
cana-525	21	4	precise	precise	ADJ
cana-525	21	5	and	and	CCONJ
cana-525	21	6	thorough	thorough	ADJ
cana-525	21	7	findings	finding	NOUN
cana-525	21	8	,	,	PUNCT
cana-525	21	9	it	it	PRON
cana-525	21	10	was	be	AUX
cana-525	21	11	necessary	necessary	ADJ
cana-525	21	12	to	to	PART
cana-525	21	13	merge	merge	VERB
cana-525	21	14	the	the	DET
cana-525	21	15	two	two	NUM
cana-525	21	16	concepts	concept	NOUN
cana-525	21	17	.	.	PUNCT
cana-525	22	1	fuzzy	fuzzy	ADJ
cana-525	22	2	soft	soft	ADJ
cana-525	22	3	point[6	point[6	NOUN
cana-525	22	4	]	]	PUNCT
cana-525	22	5	and	and	CCONJ
cana-525	22	6	fuzzy	fuzzy	ADJ
cana-525	22	7	soft	soft	ADJ
cana-525	22	8	normed	norme	VERB
cana-525	22	9	space[7	space[7	X
cana-525	22	10	]	]	PUNCT
cana-525	22	11	were	be	AUX
cana-525	22	12	created	create	VERB
cana-525	22	13	as	as	ADP
cana-525	22	14	a	a	DET
cana-525	22	15	result	result	NOUN
cana-525	22	16	of	of	ADP
cana-525	22	17	the	the	DET
cana-525	22	18	framework	framework	NOUN
cana-525	22	19	's	's	PART
cana-525	22	20	expansion	expansion	NOUN
cana-525	22	21	to	to	PART
cana-525	22	22	include	include	VERB
cana-525	22	23	these	these	DET
cana-525	22	24	new	new	ADJ
cana-525	22	25	concepts	concept	NOUN
cana-525	22	26	.	.	PUNCT
cana-525	23	1	faried	faried	ADJ
cana-525	23	2	communications	communication	NOUN
cana-525	23	3	on	on	ADP
cana-525	23	4	applied	apply	VERB
cana-525	23	5	nonlinear	nonlinear	ADJ
cana-525	23	6	analysis	analysis	NOUN
cana-525	23	7	issn	issn	NOUN
cana-525	23	8	:	:	PUNCT
cana-525	23	9	1074	1074	NUM
cana-525	23	10	-	-	PUNCT
cana-525	23	11	133x	133x	NUM
cana-525	23	12	vol	vol	NOUN
cana-525	23	13	31	31	NUM
cana-525	23	14	no	no	NOUN
cana-525	23	15	.	.	NOUN
cana-525	23	16	2	2	NUM
cana-525	23	17	(	(	PUNCT
cana-525	23	18	2024	2024	NUM
cana-525	23	19	)	)	PUNCT
cana-525	23	20	130	130	NUM
cana-525	23	21	https://internationalpubls.com	https://internationalpubls.com	X
cana-525	24	1	[	[	X
cana-525	24	2	10]et	10]et	NUM
cana-525	24	3	al	al	PROPN
cana-525	24	4	.	.	PROPN
cana-525	24	5	presented	present	VERB
cana-525	24	6	fuzzy	fuzzy	ADJ
cana-525	24	7	soft	soft	ADJ
cana-525	24	8	hilbert	hilbert	NOUN
cana-525	24	9	spaces	space	NOUN
cana-525	24	10	in	in	ADP
cana-525	24	11	2020	2020	NUM
cana-525	24	12	.	.	PUNCT
cana-525	25	1	the	the	DET
cana-525	25	2	fuzzy	fuzzy	ADJ
cana-525	25	3	soft	soft	ADJ
cana-525	25	4	linear	linear	NOUN
cana-525	25	5	operators	operator	NOUN
cana-525	25	6	are	be	AUX
cana-525	25	7	also	also	ADV
cana-525	25	8	included	include	VERB
cana-525	25	9	.	.	PUNCT
cana-525	26	1	we	we	PRON
cana-525	26	2	introduce	introduce	VERB
cana-525	26	3	a	a	DET
cana-525	26	4	brand	brand	NOUN
cana-525	26	5	-	-	PUNCT
cana-525	26	6	new	new	ADJ
cana-525	26	7	class	class	NOUN
cana-525	26	8	of	of	ADP
cana-525	26	9	fuzzy	fuzzy	ADJ
cana-525	26	10	soft	soft	ADJ
cana-525	26	11	paranormal	paranormal	ADJ
cana-525	26	12	operator	operator	NOUN
cana-525	26	13	and	and	CCONJ
cana-525	26	14	establish	establish	VERB
cana-525	26	15	a	a	DET
cana-525	26	16	number	number	NOUN
cana-525	26	17	of	of	ADP
cana-525	26	18	associated	associate	VERB
cana-525	26	19	theorems	theorem	NOUN
cana-525	26	20	in	in	ADP
cana-525	26	21	this	this	DET
cana-525	26	22	article	article	NOUN
cana-525	26	23	.	.	PUNCT
cana-525	27	1	ii	ii	PROPN
cana-525	27	2	preliminaries	preliminary	NOUN
cana-525	27	3	this	this	DET
cana-525	27	4	section	section	NOUN
cana-525	27	5	serves	serve	VERB
cana-525	27	6	as	as	ADP
cana-525	27	7	a	a	DET
cana-525	27	8	preface	preface	NOUN
cana-525	27	9	to	to	ADP
cana-525	27	10	the	the	DET
cana-525	27	11	topic	topic	NOUN
cana-525	27	12	that	that	PRON
cana-525	27	13	follows	follow	VERB
cana-525	27	14	by	by	ADP
cana-525	27	15	providing	provide	VERB
cana-525	27	16	specific	specific	ADJ
cana-525	27	17	notations	notation	NOUN
cana-525	27	18	,	,	PUNCT
cana-525	27	19	definitions	definition	NOUN
cana-525	27	20	,	,	PUNCT
cana-525	27	21	and	and	CCONJ
cana-525	27	22	preliminaries	preliminary	NOUN
cana-525	27	23	for	for	ADP
cana-525	27	24	fuzzy	fuzzy	ADJ
cana-525	27	25	set	set	NOUN
cana-525	27	26	,	,	PUNCT
cana-525	27	27	soft	soft	ADJ
cana-525	27	28	set	set	NOUN
cana-525	27	29	,	,	PUNCT
cana-525	27	30	and	and	CCONJ
cana-525	27	31	fuzzy	fuzzy	ADJ
cana-525	27	32	soft	soft	ADJ
cana-525	27	33	set	set	NOUN
cana-525	27	34	.	.	PUNCT
cana-525	28	1	definition	definition	NOUN
cana-525	28	2	2.1	2.1	NUM
cana-525	28	3	:	:	PUNCT
cana-525	29	1	[	[	X
cana-525	29	2	3	3	NUM
cana-525	29	3	]	]	X
cana-525	29	4	fuzzy	fuzzy	ADJ
cana-525	29	5	set	set	NOUN
cana-525	29	6	let	let	VERB
cana-525	29	7	ԏ	ԏ	PART
cana-525	29	8	be	be	AUX
cana-525	29	9	a	a	DET
cana-525	29	10	universal	universal	ADJ
cana-525	29	11	set	set	NOUN
cana-525	29	12	.	.	PUNCT
cana-525	30	1	a	a	DET
cana-525	30	2	fuzzy	fuzzy	ADJ
cana-525	30	3	set	set	NOUN
cana-525	30	4	₳	₳	PROPN
cana-525	30	5	̌	̌	PUNCT
cana-525	30	6	over	over	ADP
cana-525	30	7	ԏ	ԏ	PROPN
cana-525	30	8	is	be	AUX
cana-525	30	9	a	a	DET
cana-525	30	10	set	set	NOUN
cana-525	30	11	characterized	characterize	VERB
cana-525	30	12	by	by	ADP
cana-525	30	13	a	a	DET
cana-525	30	14	function	function	NOUN
cana-525	30	15	𝜂₳	𝜂₳	NOUN
cana-525	30	16	̌	̌	PRON
cana-525	30	17	:	:	PUNCT
cana-525	30	18	ԏ	ԏ	X
cana-525	30	19	→	→	X
cana-525	30	20	[	[	X
cana-525	30	21	0,1	0,1	NUM
cana-525	30	22	]	]	PUNCT
cana-525	30	23	.	.	PUNCT
cana-525	31	1	𝜂₳	𝜂₳	NOUN
cana-525	31	2	̌	̌	NUM
cana-525	31	3	is	be	AUX
cana-525	31	4	called	call	VERB
cana-525	31	5	the	the	DET
cana-525	31	6	membership	membership	NOUN
cana-525	31	7	,	,	PUNCT
cana-525	31	8	characteristic	characteristic	ADJ
cana-525	31	9	or	or	CCONJ
cana-525	31	10	indicator	indicator	NOUN
cana-525	31	11	function	function	NOUN
cana-525	31	12	of	of	ADP
cana-525	31	13	the	the	DET
cana-525	31	14	fuzzy	fuzzy	ADJ
cana-525	31	15	set	set	NOUN
cana-525	31	16	₳	₳	PROPN
cana-525	31	17	̌	̌	PROPN
cana-525	31	18	and	and	CCONJ
cana-525	31	19	the	the	DET
cana-525	31	20	value	value	NOUN
cana-525	31	21	𝜂₳	𝜂₳	PROPN
cana-525	31	22	̌(𝔵	̌(𝔵	PART
cana-525	31	23	)	)	PUNCT
cana-525	31	24	is	be	AUX
cana-525	31	25	termed	term	VERB
cana-525	31	26	the	the	DET
cana-525	31	27	grade	grade	NOUN
cana-525	31	28	of	of	ADP
cana-525	31	29	membership	membership	NOUN
cana-525	31	30	of	of	ADP
cana-525	31	31	𝔵	𝔵	DET
cana-525	31	32	∈	∈	NOUN
cana-525	31	33	ԏ	ԏ	X
cana-525	31	34	in	in	ADP
cana-525	31	35	₳	₳	PROPN
cana-525	31	36	̌.	̌.	PROPN
cana-525	31	37	definition	definition	NOUN
cana-525	31	38	2.2	2.2	NUM
cana-525	31	39	:	:	PUNCT
cana-525	32	1	[	[	X
cana-525	32	2	4	4	NUM
cana-525	32	3	,	,	PUNCT
cana-525	32	4	10	10	NUM
cana-525	32	5	]	]	X
cana-525	32	6	soft	soft	ADJ
cana-525	32	7	set	set	NOUN
cana-525	32	8	assume	assume	VERB
cana-525	32	9	that	that	SCONJ
cana-525	32	10	𝒫(ԏ	𝒫(ԏ	NOUN
cana-525	32	11	)	)	PUNCT
cana-525	32	12	the	the	DET
cana-525	32	13	power	power	NOUN
cana-525	32	14	set	set	NOUN
cana-525	32	15	of	of	ADP
cana-525	32	16	ԏ	ԏ	PROPN
cana-525	32	17	and	and	CCONJ
cana-525	32	18	e	e	AUX
cana-525	32	19	be	be	AUX
cana-525	32	20	the	the	DET
cana-525	32	21	collection	collection	NOUN
cana-525	32	22	of	of	ADP
cana-525	32	23	parameters	parameter	NOUN
cana-525	32	24	and	and	CCONJ
cana-525	32	25	⊆	⊆	NUM
cana-525	32	26	𝐸.	𝐸.	NOUN
cana-525	32	27	the	the	DET
cana-525	32	28	mapping	mapping	NOUN
cana-525	32	29	ɠ	ɠ	NOUN
cana-525	32	30	:	:	PUNCT
cana-525	32	31	₳	₳	PROPN
cana-525	32	32	̌	̌	PROPN
cana-525	32	33	→	→	SYM
cana-525	32	34	բ(ԏ	բ(ԏ	X
cana-525	32	35	)	)	PUNCT
cana-525	32	36	,	,	PUNCT
cana-525	32	37	where	where	SCONJ
cana-525	32	38	(	(	PUNCT
cana-525	32	39	ɠ	ɠ	NOUN
cana-525	32	40	,	,	PUNCT
cana-525	32	41	₳	₳	PROPN
cana-525	32	42	̌	̌	NUM
cana-525	32	43	)	)	PUNCT
cana-525	32	44	=	=	PRON
cana-525	32	45	{	{	PUNCT
cana-525	32	46	ɠ(𝑙	ɠ(𝑙	PROPN
cana-525	32	47	)	)	PUNCT
cana-525	32	48	𝜖	𝜖	PROPN
cana-525	32	49	բ(ԏ	բ(ԏ	NOUN
cana-525	32	50	):	):	PUNCT
cana-525	32	51	𝑙	𝑙	PROPN
cana-525	32	52	∈	∈	PROPN
cana-525	32	53	₳	₳	PROPN
cana-525	32	54	̌	̌	ADV
cana-525	32	55	}	}	PUNCT
cana-525	32	56	.	.	PUNCT
cana-525	33	1	as	as	ADP
cana-525	33	2	a	a	DET
cana-525	33	3	result	result	NOUN
cana-525	33	4	(	(	PUNCT
cana-525	33	5	ɠ	ɠ	NOUN
cana-525	33	6	,	,	PUNCT
cana-525	33	7	₳	₳	PROPN
cana-525	33	8	̌	̌	NUM
cana-525	33	9	)	)	PUNCT
cana-525	33	10	is	be	AUX
cana-525	33	11	called	call	VERB
cana-525	33	12	the	the	DET
cana-525	33	13	soft	soft	ADJ
cana-525	33	14	set	set	NOUN
cana-525	33	15	.	.	PUNCT
cana-525	34	1	definition	definition	NOUN
cana-525	34	2	2.3	2.3	NUM
cana-525	34	3	:	:	PUNCT
cana-525	35	1	[	[	X
cana-525	35	2	5	5	NUM
cana-525	35	3	]	]	X
cana-525	35	4	fuzzy	fuzzy	ADJ
cana-525	35	5	soft	soft	ADJ
cana-525	35	6	set	set	NOUN
cana-525	35	7	letԏ	letԏ	NOUN
cana-525	35	8	be	be	AUX
cana-525	35	9	a	a	DET
cana-525	35	10	universal	universal	ADJ
cana-525	35	11	set	set	NOUN
cana-525	35	12	,	,	PUNCT
cana-525	35	13	e	e	X
cana-525	35	14	be	be	AUX
cana-525	35	15	a	a	DET
cana-525	35	16	set	set	NOUN
cana-525	35	17	of	of	ADP
cana-525	35	18	parameters	parameter	NOUN
cana-525	35	19	and	and	CCONJ
cana-525	35	20	₳	₳	PROPN
cana-525	35	21	̌	̌	PROPN
cana-525	36	1	⊆	⊆	NUM
cana-525	36	2	e.	e.	PROPN
cana-525	36	3	a	a	DET
cana-525	36	4	pair	pair	NOUN
cana-525	36	5	(	(	PUNCT
cana-525	36	6	ɠ	ɠ	NOUN
cana-525	36	7	,	,	PUNCT
cana-525	36	8	₳	₳	PROPN
cana-525	36	9	)	)	PUNCT
cana-525	36	10	is	be	AUX
cana-525	36	11	called	call	VERB
cana-525	36	12	a	a	DET
cana-525	36	13	fuzzy	fuzzy	ADJ
cana-525	36	14	soft	soft	ADJ
cana-525	36	15	set	set	NOUN
cana-525	36	16	over	over	ADP
cana-525	36	17	ԏ	ԏ	X
cana-525	36	18	,	,	PUNCT
cana-525	36	19	where	where	SCONJ
cana-525	36	20	ɠ	ɠ	PROPN
cana-525	36	21	is	be	AUX
cana-525	36	22	a	a	DET
cana-525	36	23	mapping	mapping	NOUN
cana-525	36	24	given	give	VERB
cana-525	36	25	by	by	ADP
cana-525	36	26	ɠ	ɠ	NOUN
cana-525	36	27	:	:	PUNCT
cana-525	36	28	₳	₳	PROPN
cana-525	36	29	→	→	SYM
cana-525	36	30	ℱ(ԏ	ℱ(ԏ	NUM
cana-525	36	31	)	)	PUNCT
cana-525	36	32	,	,	PUNCT
cana-525	36	33	ℱ(ԏ	ℱ(ԏ	CCONJ
cana-525	36	34	)	)	PUNCT
cana-525	36	35	is	be	AUX
cana-525	36	36	the	the	DET
cana-525	36	37	family	family	NOUN
cana-525	36	38	of	of	ADP
cana-525	36	39	all	all	DET
cana-525	36	40	fuzzy	fuzzy	ADJ
cana-525	36	41	subsets	subset	NOUN
cana-525	36	42	of	of	ADP
cana-525	36	43	ԏ	ԏ	PROPN
cana-525	36	44	and	and	CCONJ
cana-525	36	45	the	the	DET
cana-525	36	46	fuzzy	fuzzy	ADJ
cana-525	36	47	subset	subset	NOUN
cana-525	36	48	of	of	ADP
cana-525	36	49	ԏ	ԏ	PROPN
cana-525	36	50	is	be	AUX
cana-525	36	51	defined	define	VERB
cana-525	36	52	as	as	ADP
cana-525	36	53	a	a	DET
cana-525	36	54	map	map	NOUN
cana-525	36	55	𝜂	𝜂	NOUN
cana-525	36	56	from	from	ADP
cana-525	36	57	ԏ	ԏ	PRON
cana-525	36	58	to	to	ADP
cana-525	36	59	[	[	X
cana-525	36	60	0,1	0,1	NUM
cana-525	36	61	]	]	PUNCT
cana-525	36	62	.	.	PUNCT
cana-525	37	1	the	the	DET
cana-525	37	2	family	family	NOUN
cana-525	37	3	of	of	ADP
cana-525	37	4	all	all	DET
cana-525	37	5	fuzzy	fuzzy	ADJ
cana-525	37	6	soft	soft	ADJ
cana-525	37	7	sets	set	NOUN
cana-525	37	8	(	(	PUNCT
cana-525	37	9	ɠ	ɠ	NOUN
cana-525	37	10	,	,	PUNCT
cana-525	37	11	₳	₳	NOUN
cana-525	37	12	)	)	PUNCT
cana-525	37	13	over	over	ADP
cana-525	37	14	a	a	DET
cana-525	37	15	universal	universal	ADJ
cana-525	37	16	set	set	NOUN
cana-525	37	17	ԏ	ԏ	X
cana-525	37	18	,	,	PUNCT
cana-525	37	19	in	in	ADP
cana-525	37	20	which	which	PRON
cana-525	37	21	all	all	DET
cana-525	37	22	the	the	DET
cana-525	37	23	parameter	parameter	NOUN
cana-525	37	24	sets	set	VERB
cana-525	37	25	₳	₳	PROPN
cana-525	37	26	̌	̌	NUM
cana-525	37	27	are	be	AUX
cana-525	37	28	the	the	DET
cana-525	37	29	same	same	ADJ
cana-525	37	30	,	,	PUNCT
cana-525	37	31	is	be	AUX
cana-525	37	32	denoted	denote	VERB
cana-525	37	33	by	by	ADP
cana-525	37	34	𝐹𝑆𝑆(ԏ	𝐹𝑆𝑆(ԏ	NUM
cana-525	37	35	)	)	PUNCT
cana-525	38	1	₳	₳	ADJ
cana-525	38	2	̌	̌	PUNCT
cana-525	39	1	=	=	NUM
cana-525	39	2	𝐹𝑆𝑆(ԏ	𝐹𝑆𝑆(ԏ	NUM
cana-525	39	3	)	)	PUNCT
cana-525	39	4	definition	definition	NOUN
cana-525	39	5	2.4	2.4	NUM
cana-525	39	6	:	:	PUNCT
cana-525	40	1	[	[	X
cana-525	40	2	9	9	NUM
cana-525	40	3	]	]	X
cana-525	40	4	fuzzy	fuzzy	ADJ
cana-525	40	5	soft	soft	ADJ
cana-525	40	6	hilbert	hilbert	NOUN
cana-525	40	7	space	space	NOUN
cana-525	40	8	a	a	DET
cana-525	40	9	fuzzy	fuzzy	ADJ
cana-525	40	10	soft	soft	ADJ
cana-525	40	11	inner	inner	ADJ
cana-525	40	12	product	product	NOUN
cana-525	40	13	space	space	NOUN
cana-525	40	14	is	be	AUX
cana-525	40	15	defined	define	VERB
cana-525	40	16	as(ԏ	as(ԏ	PUNCT
cana-525	40	17	̃	̃	PROPN
cana-525	40	18	,	,	PUNCT
cana-525	40	19	〈	〈	PROPN
cana-525	40	20	.	.	PROPN
cana-525	40	21	,	,	PUNCT
cana-525	40	22	.	.	PUNCT
cana-525	41	1	〉	〉	NOUN
cana-525	41	2	̃	̃	PROPN
cana-525	41	3	)	)	PUNCT
cana-525	41	4	.	.	PUNCT
cana-525	42	1	this	this	DET
cana-525	42	2	space	space	NOUN
cana-525	42	3	,	,	PUNCT
cana-525	42	4	which	which	PRON
cana-525	42	5	is	be	AUX
cana-525	42	6	fuzzy	fuzzy	ADJ
cana-525	42	7	soft	soft	ADJ
cana-525	42	8	complete	complete	ADJ
cana-525	42	9	in	in	ADP
cana-525	42	10	the	the	DET
cana-525	42	11	induced	induce	VERB
cana-525	42	12	fuzzy	fuzzy	ADJ
cana-525	42	13	soft	soft	ADJ
cana-525	42	14	normed	normed	ADJ
cana-525	42	15	space	space	NOUN
cana-525	42	16	called	call	VERB
cana-525	42	17	as	as	ADP
cana-525	42	18	a	a	DET
cana-525	42	19	fuzzy	fuzzy	ADJ
cana-525	42	20	soft	soft	ADJ
cana-525	42	21	hilbert	hilbert	NOUN
cana-525	42	22	space	space	NOUN
cana-525	42	23	and	and	CCONJ
cana-525	42	24	denoted	denote	VERB
cana-525	42	25	by	by	ADP
cana-525	42	26	(	(	PUNCT
cana-525	42	27	�	�	PROPN
cana-525	42	28	̃	̃	PROPN
cana-525	42	29	�	�	PROPN
cana-525	42	30	,	,	PUNCT
cana-525	42	31	〈	〈	PROPN
cana-525	42	32	.	.	PROPN
cana-525	42	33	,	,	PUNCT
cana-525	42	34	.	.	PUNCT
cana-525	43	1	〉	〉	NOUN
cana-525	43	2	̃	̃	PROPN
cana-525	43	3	)	)	PUNCT
cana-525	43	4	.	.	PUNCT
cana-525	44	1	every	every	DET
cana-525	44	2	fuzzy	fuzzy	ADJ
cana-525	44	3	soft	soft	ADJ
cana-525	44	4	hilbert	hilbert	NOUN
cana-525	44	5	space	space	NOUN
cana-525	44	6	is	be	AUX
cana-525	44	7	obviously	obviously	ADV
cana-525	44	8	a	a	DET
cana-525	44	9	fuzzy	fuzzy	ADJ
cana-525	44	10	soft	soft	ADJ
cana-525	44	11	banach	banach	NOUN
cana-525	44	12	space	space	NOUN
cana-525	44	13	.	.	PUNCT
cana-525	45	1	definition	definition	NOUN
cana-525	45	2	2.5	2.5	NUM
cana-525	45	3	:	:	PUNCT
cana-525	46	1	[	[	X
cana-525	46	2	2	2	NUM
cana-525	46	3	]	]	X
cana-525	46	4	fuzzy	fuzzy	ADJ
cana-525	46	5	soft	soft	ADJ
cana-525	46	6	linear	linear	NOUN
cana-525	46	7	operator	operator	NOUN
cana-525	46	8	in	in	ADP
cana-525	46	9	�	�	PROPN
cana-525	46	10	̃	̃	PROPN
cana-525	46	11	�	�	PROPN
cana-525	46	12	consider	consider	VERB
cana-525	46	13	�	�	PROPN
cana-525	46	14	̃	̃	PROPN
cana-525	46	15	�	�	PROPN
cana-525	46	16	to	to	PART
cana-525	46	17	be	be	AUX
cana-525	46	18	a	a	DET
cana-525	46	19	fuzzy	fuzzy	ADJ
cana-525	46	20	soft	soft	ADJ
cana-525	46	21	hilbert	hilbert	NOUN
cana-525	46	22	space	space	NOUN
cana-525	46	23	.	.	PUNCT
cana-525	47	1	a	a	DET
cana-525	47	2	fuzzy	fuzzy	ADJ
cana-525	47	3	soft	soft	ADJ
cana-525	47	4	linear	linear	NOUN
cana-525	47	5	operator	operator	NOUN
cana-525	47	6	₮	₮	ADP
cana-525	47	7	̃	̃	PROPN
cana-525	47	8	:	:	PUNCT
cana-525	47	9	�	�	PROPN
cana-525	47	10	̃	̃	PROPN
cana-525	47	11	�	�	PROPN
cana-525	47	12	→	→	SYM
cana-525	47	13	�	�	PROPN
cana-525	47	14	̃	̃	PROPN
cana-525	47	15	�	�	PROPN
cana-525	47	16	is	be	AUX
cana-525	47	17	called	call	VERB
cana-525	47	18	a	a	DET
cana-525	47	19	fuzzy	fuzzy	ADJ
cana-525	47	20	soft	soft	ADJ
cana-525	47	21	linear	linear	NOUN
cana-525	47	22	operator	operator	NOUN
cana-525	47	23	in	in	ADP
cana-525	47	24	�	�	PROPN
cana-525	47	25	̃	̃	PROPN
cana-525	47	26	�	�	PROPN
cana-525	47	27	,	,	PUNCT
cana-525	47	28	then	then	ADV
cana-525	47	29	₮	₮	ADP
cana-525	47	30	̃	̃	PROPN
cana-525	47	31	is	be	AUX
cana-525	47	32	a	a	DET
cana-525	47	33	fuzzy	fuzzy	ADJ
cana-525	47	34	soft	soft	ADJ
cana-525	47	35	linear	linear	NOUN
cana-525	47	36	operator	operator	NOUN
cana-525	47	37	on	on	ADP
cana-525	47	38	�	�	PROPN
cana-525	47	39	̃	̃	PROPN
cana-525	47	40	�	�	PROPN
cana-525	47	41	which	which	PRON
cana-525	47	42	is	be	AUX
cana-525	47	43	denoted	denote	VERB
cana-525	47	44	as	as	ADP
cana-525	47	45	₮	₮	ADP
cana-525	47	46	̃	̃	PROPN
cana-525	47	47	∈̃	∈̃	PROPN
cana-525	47	48	�	�	PROPN
cana-525	47	49	̃	̃	PROPN
cana-525	47	50	�	�	PROPN
cana-525	47	51	(	(	PUNCT
cana-525	47	52	�	�	PROPN
cana-525	47	53	̃	̃	NOUN
cana-525	47	54	�	�	PROPN
cana-525	47	55	)	)	PUNCT
cana-525	47	56	.	.	PUNCT
cana-525	48	1	₮	₮	PUNCT
cana-525	48	2	̃	̃	NOUN
cana-525	48	3	is	be	AUX
cana-525	48	4	fuzzy	fuzzy	ADJ
cana-525	48	5	soft	soft	ADJ
cana-525	48	6	bounded	bound	VERB
cana-525	48	7	if	if	SCONJ
cana-525	48	8	there	there	PRON
cana-525	48	9	exists	exist	VERB
cana-525	48	10	�	�	PROPN
cana-525	48	11	̃	̃	PROPN
cana-525	48	12	�	�	PROPN
cana-525	48	13	∈̃	∈̃	NOUN
cana-525	48	14	ℜ(₳	ℜ(₳	PROPN
cana-525	48	15	)	)	PUNCT
cana-525	48	16	:	:	PUNCT
cana-525	49	1	‖₮̃	‖₮̃	NOUN
cana-525	49	2	(	(	PUNCT
cana-525	49	3	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	49	4	)	)	PUNCT
cana-525	49	5	)	)	PUNCT
cana-525	50	1	̃	̃	PROPN
cana-525	50	2	‖	‖	PROPN
cana-525	50	3	≤̃	≤̃	PROPN
cana-525	50	4	�	�	PROPN
cana-525	50	5	̃	̃	PROPN
cana-525	50	6	�	�	PROPN
cana-525	50	7	‖𝑙𝜂ɠ(𝑒	‖𝑙𝜂ɠ(𝑒	NOUN
cana-525	50	8	)	)	PUNCT
cana-525	50	9	‖	‖	NUM
cana-525	50	10	∀	∀	X
cana-525	50	11	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	NOUN
cana-525	50	12	)	)	PUNCT
cana-525	50	13	∈̃	∈̃	PROPN
cana-525	50	14	�	�	PROPN
cana-525	50	15	̃	̃	PROPN
cana-525	50	16	�	�	PROPN
cana-525	50	17	,	,	PUNCT
cana-525	50	18	then	then	ADV
cana-525	50	19	₮	₮	ADP
cana-525	50	20	̃	̃	PROPN
cana-525	50	21	∈̃	∈̃	PROPN
cana-525	50	22	�	�	PROPN
cana-525	50	23	̃	̃	PROPN
cana-525	50	24	�	�	PROPN
cana-525	50	25	(	(	PUNCT
cana-525	50	26	�	�	PROPN
cana-525	50	27	̃	̃	PROPN
cana-525	50	28	�	�	PROPN
cana-525	50	29	)	)	PUNCT
cana-525	50	30	definition	definition	NOUN
cana-525	50	31	2.6	2.6	NUM
cana-525	50	32	:	:	PUNCT
cana-525	51	1	[	[	X
cana-525	51	2	2	2	NUM
cana-525	51	3	]	]	X
cana-525	51	4	fuzzy	fuzzy	ADJ
cana-525	51	5	soft	soft	ADJ
cana-525	51	6	adjoint	adjoint	NOUN
cana-525	51	7	operator	operator	NOUN
cana-525	51	8	in	in	ADP
cana-525	51	9	�	�	PROPN
cana-525	51	10	̃	̃	PROPN
cana-525	51	11	�	�	PROPN
cana-525	51	12	the	the	DET
cana-525	51	13	fuzzy	fuzzy	ADJ
cana-525	51	14	soft	soft	ADJ
cana-525	51	15	adjoint	adjoint	NOUN
cana-525	51	16	operator	operator	NOUN
cana-525	51	17	₮	₮	ADP
cana-525	51	18	̃∗̃of	̃∗̃of	VERB
cana-525	51	19	a	a	DET
cana-525	51	20	fuzzy	fuzzy	ADJ
cana-525	51	21	soft	soft	ADJ
cana-525	51	22	linear	linear	NOUN
cana-525	51	23	operator	operator	NOUN
cana-525	51	24	₮	₮	ADP
cana-525	51	25	̃	̃	PROPN
cana-525	51	26	is	be	AUX
cana-525	51	27	defined	define	VERB
cana-525	51	28	by	by	ADP
cana-525	51	29	〈	〈	NOUN
cana-525	51	30	₮	₮	PROPN
cana-525	51	31	̃𝑙1	̃𝑙1	NOUN
cana-525	51	32	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	NOUN
cana-525	51	33	)	)	PUNCT
cana-525	51	34	,	,	PUNCT
cana-525	51	35	𝑙2	𝑙2	PROPN
cana-525	51	36	𝜂2ɠ(2	𝜂2ɠ(2	PROPN
cana-525	51	37	)	)	PUNCT
cana-525	51	38	̃	̃	NOUN
cana-525	51	39	〉	〉	NOUN
cana-525	51	40	=	=	SYM
cana-525	51	41	̃	̃	NOUN
cana-525	51	42	〈	〈	NOUN
cana-525	51	43	𝑙1	𝑙1	PROPN
cana-525	51	44	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	NOUN
cana-525	51	45	)	)	PUNCT
cana-525	51	46	,	,	PUNCT
cana-525	51	47	₮	₮	PROPN
cana-525	51	48	̃∗̃𝑙2	̃∗̃𝑙2	PROPN
cana-525	51	49	𝜂2ɠ(2	𝜂2ɠ(2	NOUN
cana-525	51	50	)	)	PUNCT
cana-525	51	51	̃	̃	NOUN
cana-525	51	52	〉	〉	NOUN
cana-525	51	53	for	for	ADP
cana-525	51	54	all	all	DET
cana-525	51	55	𝑙1	𝑙1	PROPN
cana-525	51	56	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	51	57	)	)	PUNCT
cana-525	51	58	,	,	PUNCT
cana-525	51	59	𝑙2	𝑙2	PROPN
cana-525	51	60	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	51	61	)	)	PUNCT
cana-525	51	62	∈̃	∈̃	PROPN
cana-525	51	63	�	�	PROPN
cana-525	51	64	̃	̃	PROPN
cana-525	51	65	�	�	PROPN
cana-525	51	66	communications	communication	NOUN
cana-525	51	67	on	on	ADP
cana-525	51	68	applied	apply	VERB
cana-525	51	69	nonlinear	nonlinear	ADJ
cana-525	51	70	analysis	analysis	NOUN
cana-525	51	71	issn	issn	NOUN
cana-525	51	72	:	:	PUNCT
cana-525	51	73	1074	1074	NUM
cana-525	51	74	-	-	PUNCT
cana-525	51	75	133x	133x	NUM
cana-525	51	76	vol	vol	NOUN
cana-525	51	77	31	31	NUM
cana-525	51	78	no	no	NOUN
cana-525	51	79	.	.	NOUN
cana-525	51	80	2	2	NUM
cana-525	51	81	(	(	PUNCT
cana-525	51	82	2024	2024	NUM
cana-525	51	83	)	)	PUNCT
cana-525	51	84	131	131	NUM
cana-525	51	85	https://internationalpubls.com	https://internationalpubls.com	X
cana-525	51	86	definition	definition	NOUN
cana-525	51	87	2.7:[11	2.7:[11	NUM
cana-525	51	88	]	]	X
cana-525	51	89	fuzzy	fuzzy	ADJ
cana-525	51	90	soft	soft	ADJ
cana-525	51	91	normal	normal	ADJ
cana-525	51	92	operator	operator	NOUN
cana-525	51	93	let	let	VERB
cana-525	51	94	�	�	PROPN
cana-525	51	95	̃	̃	PROPN
cana-525	51	96	�	�	PROPN
cana-525	51	97	be	be	AUX
cana-525	51	98	an	an	DET
cana-525	51	99	fs	fs	X
cana-525	51	100	hilbert	hilbert	NOUN
cana-525	51	101	space	space	NOUN
cana-525	51	102	and	and	CCONJ
cana-525	51	103	₮	₮	ADP
cana-525	51	104	̃	̃	PROPN
cana-525	51	105	∈̃	∈̃	PROPN
cana-525	51	106	�	�	PROPN
cana-525	51	107	̃	̃	PROPN
cana-525	51	108	�	�	PROPN
cana-525	51	109	(	(	PUNCT
cana-525	51	110	�	�	PROPN
cana-525	51	111	̃	̃	PROPN
cana-525	51	112	�	�	PROPN
cana-525	51	113	)	)	PUNCT
cana-525	51	114	.	.	PUNCT
cana-525	52	1	then	then	ADV
cana-525	52	2	,	,	PUNCT
cana-525	52	3	₮	₮	ADP
cana-525	52	4	̃	̃	NOUN
cana-525	52	5	is	be	AUX
cana-525	52	6	said	say	VERB
cana-525	52	7	to	to	PART
cana-525	52	8	be	be	AUX
cana-525	52	9	an	an	DET
cana-525	52	10	fs	fs	X
cana-525	52	11	normal	normal	ADJ
cana-525	52	12	operator	operator	NOUN
cana-525	52	13	if	if	SCONJ
cana-525	52	14	₮	₮	NOUN
cana-525	52	15	̃₮̃∗̃	̃₮̃∗̃	NOUN
cana-525	52	16	=	=	SYM
cana-525	52	17	̃	̃	NOUN
cana-525	52	18	₮	₮	ADP
cana-525	52	19	̃∗̃₮̃	̃∗̃₮̃	ADJ
cana-525	52	20	definition	definition	NOUN
cana-525	52	21	2.8	2.8	NUM
cana-525	52	22	:	:	PUNCT
cana-525	53	1	[	[	X
cana-525	53	2	11	11	NUM
cana-525	53	3	]	]	X
cana-525	53	4	fuzzy	fuzzy	ADJ
cana-525	53	5	soft	soft	ADJ
cana-525	53	6	self	self	NOUN
cana-525	53	7	adjoint	adjoint	NOUN
cana-525	53	8	operator	operator	NOUN
cana-525	53	9	the	the	DET
cana-525	53	10	fs	fs	NOUN
cana-525	53	11	-	-	NOUN
cana-525	53	12	operator	operator	NOUN
cana-525	53	13	₮	₮	ADP
cana-525	53	14	̃	̃	NOUN
cana-525	53	15	of	of	ADP
cana-525	53	16	fsh	fsh	ADJ
cana-525	53	17	-	-	PUNCT
cana-525	53	18	space	space	NOUN
cana-525	53	19	�	�	PROPN
cana-525	53	20	̃	̃	PROPN
cana-525	53	21	�	�	PROPN
cana-525	53	22	is	be	AUX
cana-525	53	23	called	call	VERB
cana-525	53	24	fuzzy	fuzzy	ADJ
cana-525	53	25	soft	soft	ADJ
cana-525	53	26	self	self	NOUN
cana-525	53	27	adjoint	adjoint	NOUN
cana-525	53	28	(	(	PUNCT
cana-525	53	29	fs	fs	ADJ
cana-525	53	30	-	-	PUNCT
cana-525	53	31	self	self	NOUN
cana-525	53	32	adjoint	adjoint	NOUN
cana-525	53	33	operator	operator	NOUN
cana-525	53	34	)	)	PUNCT
cana-525	53	35	if	if	SCONJ
cana-525	53	36	₮	₮	ADP
cana-525	53	37	̃	̃	NOUN
cana-525	53	38	=	=	SYM
cana-525	53	39	̃	̃	NOUN
cana-525	53	40	₮	₮	NOUN
cana-525	53	41	̃∗̃	̃∗̃	PROPN
cana-525	53	42	definition	definition	NOUN
cana-525	53	43	2.9	2.9	NUM
cana-525	53	44	:	:	PUNCT
cana-525	54	1	[	[	X
cana-525	54	2	14	14	NUM
cana-525	54	3	]	]	X
cana-525	54	4	fuzzy	fuzzy	ADJ
cana-525	54	5	soft	soft	ADJ
cana-525	54	6	isometry	isometry	NOUN
cana-525	54	7	operator	operator	NOUN
cana-525	54	8	let	let	VERB
cana-525	54	9	�	�	PROPN
cana-525	54	10	̃	̃	PROPN
cana-525	54	11	�	�	PROPN
cana-525	54	12	be	be	AUX
cana-525	54	13	an	an	DET
cana-525	54	14	fs	fs	X
cana-525	54	15	hilbert	hilbert	NOUN
cana-525	54	16	space	space	NOUN
cana-525	54	17	and	and	CCONJ
cana-525	54	18	₮	₮	ADP
cana-525	54	19	̃	̃	PROPN
cana-525	54	20	∈̃	∈̃	PROPN
cana-525	54	21	�	�	PROPN
cana-525	54	22	̃	̃	PROPN
cana-525	54	23	�	�	PROPN
cana-525	54	24	(	(	PUNCT
cana-525	54	25	�	�	PROPN
cana-525	54	26	̃	̃	NOUN
cana-525	54	27	�	�	PROPN
cana-525	54	28	)	)	PUNCT
cana-525	54	29	.	.	PUNCT
cana-525	55	1	then	then	ADV
cana-525	55	2	,	,	PUNCT
cana-525	55	3	₮	₮	ADP
cana-525	55	4	̃	̃	NOUN
cana-525	55	5	is	be	AUX
cana-525	55	6	said	say	VERB
cana-525	55	7	to	to	PART
cana-525	55	8	be	be	AUX
cana-525	55	9	an	an	DET
cana-525	55	10	fs	fs	X
cana-525	55	11	isometry	isometry	NOUN
cana-525	55	12	operator	operator	NOUN
cana-525	55	13	if	if	SCONJ
cana-525	55	14	〈	〈	NOUN
cana-525	55	15	₮	₮	NOUN
cana-525	55	16	̃𝑙1	̃𝑙1	NOUN
cana-525	55	17	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	NOUN
cana-525	55	18	)	)	PUNCT
cana-525	55	19	,	,	PUNCT
cana-525	55	20	₮	₮	ADP
cana-525	55	21	̃̃	̃̃	NOUN
cana-525	55	22	𝑙2	𝑙2	NOUN
cana-525	55	23	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	55	24	)	)	PUNCT
cana-525	55	25	〉	〉	NOUN
cana-525	55	26	=	=	SYM
cana-525	55	27	̃	̃	NOUN
cana-525	55	28	〈	〈	NOUN
cana-525	55	29	𝑙1	𝑙1	PROPN
cana-525	55	30	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	NOUN
cana-525	55	31	)	)	PUNCT
cana-525	55	32	,	,	PUNCT
cana-525	55	33	𝑙2	𝑙2	PROPN
cana-525	55	34	𝜂2ɠ(2	𝜂2ɠ(2	PROPN
cana-525	55	35	)	)	PUNCT
cana-525	55	36	̃	̃	NOUN
cana-525	55	37	〉	〉	NOUN
cana-525	55	38	for	for	ADP
cana-525	55	39	all	all	DET
cana-525	55	40	𝑙1	𝑙1	PROPN
cana-525	55	41	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	55	42	)	)	PUNCT
cana-525	55	43	,	,	PUNCT
cana-525	55	44	𝑙2	𝑙2	PROPN
cana-525	55	45	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	55	46	)	)	PUNCT
cana-525	55	47	∈̃	∈̃	PROPN
cana-525	55	48	�	�	PROPN
cana-525	55	49	̃	̃	PROPN
cana-525	55	50	�	�	PROPN
cana-525	55	51	definition	definition	NOUN
cana-525	55	52	2.10	2.10	NUM
cana-525	55	53	:	:	PUNCT
cana-525	56	1	[	[	X
cana-525	56	2	13	13	NUM
cana-525	56	3	]	]	X
cana-525	56	4	fuzzy	fuzzy	ADJ
cana-525	56	5	soft	soft	ADJ
cana-525	56	6	projection	projection	NOUN
cana-525	56	7	operator	operator	NOUN
cana-525	56	8	consider	consider	VERB
cana-525	56	9	�	�	PROPN
cana-525	56	10	̃	̃	PROPN
cana-525	56	11	�	�	PROPN
cana-525	56	12	to	to	PART
cana-525	56	13	be	be	AUX
cana-525	56	14	a	a	DET
cana-525	56	15	fuzzy	fuzzy	ADJ
cana-525	56	16	soft	soft	ADJ
cana-525	56	17	hilbert	hilbert	NOUN
cana-525	56	18	space	space	NOUN
cana-525	56	19	.	.	PUNCT
cana-525	57	1	a	a	DET
cana-525	57	2	fuzzy	fuzzy	ADJ
cana-525	57	3	soft	soft	ADJ
cana-525	57	4	linear	linear	NOUN
cana-525	57	5	operator	operator	NOUN
cana-525	57	6	₮	₮	ADP
cana-525	57	7	̃	̃	PROPN
cana-525	57	8	:	:	PUNCT
cana-525	57	9	�	�	PROPN
cana-525	57	10	̃	̃	PROPN
cana-525	57	11	�	�	PROPN
cana-525	57	12	→	→	SYM
cana-525	57	13	�	�	PROPN
cana-525	57	14	̃	̃	PROPN
cana-525	57	15	�	�	PROPN
cana-525	57	16	is	be	AUX
cana-525	57	17	called	call	VERB
cana-525	57	18	a	a	DET
cana-525	57	19	fuzzy	fuzzy	ADJ
cana-525	57	20	soft	soft	ADJ
cana-525	57	21	projection	projection	NOUN
cana-525	57	22	operator	operator	NOUN
cana-525	57	23	in	in	ADP
cana-525	57	24	�	�	PROPN
cana-525	57	25	̃	̃	PROPN
cana-525	57	26	�	�	PROPN
cana-525	57	27	if	if	SCONJ
cana-525	57	28	₮	₮	NOUN
cana-525	57	29	̃2	̃2	PROPN
cana-525	57	30	=	=	SYM
cana-525	57	31	̃	̃	NOUN
cana-525	57	32	₮	₮	ADP
cana-525	57	33	̃	̃	PROPN
cana-525	57	34	ie	ie	X
cana-525	57	35	,	,	PUNCT
cana-525	57	36	₮	₮	ADP
cana-525	57	37	̃is	̃is	NOUN
cana-525	57	38	an	an	DET
cana-525	57	39	idempotent	idempotent	NOUN
cana-525	57	40	.	.	PUNCT
cana-525	58	1	definition	definition	NOUN
cana-525	58	2	2.11	2.11	NUM
cana-525	58	3	:	:	PUNCT
cana-525	59	1	[	[	X
cana-525	59	2	15	15	NUM
cana-525	59	3	]	]	X
cana-525	59	4	fuzzy	fuzzy	ADJ
cana-525	59	5	soft	soft	ADJ
cana-525	59	6	hyponormal	hyponormal	ADJ
cana-525	59	7	operator	operator	NOUN
cana-525	59	8	consider	consider	VERB
cana-525	59	9	�	�	PROPN
cana-525	59	10	̃	̃	PROPN
cana-525	59	11	�	�	PROPN
cana-525	59	12	to	to	PART
cana-525	59	13	be	be	AUX
cana-525	59	14	a	a	DET
cana-525	59	15	fuzzy	fuzzy	ADJ
cana-525	59	16	soft	soft	ADJ
cana-525	59	17	hilbert	hilbert	NOUN
cana-525	59	18	space	space	NOUN
cana-525	59	19	.	.	PUNCT
cana-525	60	1	₮	₮	ADP
cana-525	60	2	̃	̃	PROPN
cana-525	60	3	∈	∈	PROPN
cana-525	60	4	�	�	PROPN
cana-525	60	5	̃	̃	PROPN
cana-525	60	6	�	�	PROPN
cana-525	60	7	(	(	PUNCT
cana-525	60	8	�	�	PROPN
cana-525	60	9	̃	̃	PROPN
cana-525	60	10	�	�	PROPN
cana-525	60	11	)	)	PUNCT
cana-525	60	12	is	be	AUX
cana-525	60	13	called	call	VERB
cana-525	60	14	fuzzy	fuzzy	ADJ
cana-525	60	15	soft	soft	ADJ
cana-525	60	16	hyponormal	hyponormal	ADJ
cana-525	60	17	operator	operator	NOUN
cana-525	60	18	if	if	SCONJ
cana-525	60	19	‖₮̃∗̃𝑙𝜂ɠ(𝑒	‖₮̃∗̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	60	20	)	)	PUNCT
cana-525	60	21	‖	‖	PROPN
cana-525	60	22	≤	≤	PROPN
cana-525	60	23	‖₮̃𝑙𝜂ɠ(𝑒	‖₮̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	60	24	)	)	PUNCT
cana-525	60	25	‖	‖	PROPN
cana-525	60	26	for	for	ADP
cana-525	60	27	all	all	DET
cana-525	60	28	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	60	29	)	)	PUNCT
cana-525	60	30	∈	∈	PROPN
cana-525	60	31	̃	̃	PROPN
cana-525	60	32	�	�	PROPN
cana-525	60	33	̃	̃	NOUN
cana-525	60	34	�	�	PROPN
cana-525	60	35	or	or	CCONJ
cana-525	60	36	equivalently	equivalently	ADV
cana-525	60	37	₮	₮	ADP
cana-525	60	38	̃∗̃₮̃	̃∗̃₮̃	PROPN
cana-525	60	39	≥	≥	NUM
cana-525	60	40	₮	₮	ADP
cana-525	60	41	̃₮̃∗̃	̃₮̃∗̃	NOUN
cana-525	60	42	definition	definition	NOUN
cana-525	60	43	2.12	2.12	NUM
cana-525	60	44	:	:	PUNCT
cana-525	61	1	[	[	X
cana-525	61	2	16	16	NUM
cana-525	61	3	]	]	X
cana-525	61	4	m	m	ADJ
cana-525	61	5	-	-	ADJ
cana-525	61	6	fuzzy	fuzzy	ADJ
cana-525	61	7	soft	soft	ADJ
cana-525	61	8	hyponormal	hyponormal	ADJ
cana-525	61	9	operator	operator	NOUN
cana-525	61	10	let	let	VERB
cana-525	61	11	�	�	PROPN
cana-525	61	12	̃	̃	PROPN
cana-525	61	13	�	�	PROPN
cana-525	61	14	be	be	AUX
cana-525	61	15	an	an	DET
cana-525	61	16	fs	fs	X
cana-525	61	17	hilbert	hilbert	NOUN
cana-525	61	18	space	space	NOUN
cana-525	61	19	and	and	CCONJ
cana-525	61	20	let	let	VERB
cana-525	61	21	₮	₮	PART
cana-525	61	22	̃	̃	PROPN
cana-525	61	23	∈	∈	PROPN
cana-525	61	24	�	�	PROPN
cana-525	61	25	̃	̃	PROPN
cana-525	61	26	�	�	PROPN
cana-525	61	27	(	(	PUNCT
cana-525	61	28	�	�	PROPN
cana-525	61	29	̃	̃	PROPN
cana-525	61	30	�	�	PROPN
cana-525	61	31	)	)	PUNCT
cana-525	61	32	is	be	AUX
cana-525	61	33	called	call	VERB
cana-525	61	34	m	m	PROPN
cana-525	61	35	–	–	PUNCT
cana-525	61	36	fuzzy	fuzzy	ADJ
cana-525	61	37	soft	soft	ADJ
cana-525	61	38	hyponormal	hyponormal	ADJ
cana-525	61	39	operator	operator	NOUN
cana-525	61	40	if	if	SCONJ
cana-525	61	41	there	there	PRON
cana-525	61	42	exist	exist	VERB
cana-525	61	43	a	a	DET
cana-525	61	44	real	real	ADJ
cana-525	61	45	number	number	NOUN
cana-525	61	46	ℳ	ℳ	NOUN
cana-525	61	47	,	,	PUNCT
cana-525	61	48	such	such	ADJ
cana-525	61	49	that	that	SCONJ
cana-525	61	50	‖(₮̃−̃₴̃𝐼	‖(₮̃−̃₴̃𝐼	PROPN
cana-525	61	51	)	)	PUNCT
cana-525	61	52	∗̃	∗̃	NUM
cana-525	61	53	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	61	54	)	)	PUNCT
cana-525	61	55	̃	̃	PROPN
cana-525	62	1	‖	‖	PROPN
cana-525	62	2	≤̃	≤̃	NOUN
cana-525	63	1	ℳ̃	ℳ̃	NUM
cana-525	63	2	‖(₮̃−̃₴̃𝐼)𝑙𝜂ɠ(𝑒	‖(₮̃−̃₴̃𝐼)𝑙𝜂ɠ(𝑒	ADV
cana-525	63	3	)	)	PUNCT
cana-525	63	4	̃	̃	NOUN
cana-525	63	5	‖	‖	ADJ
cana-525	63	6	for	for	ADP
cana-525	63	7	all	all	DET
cana-525	63	8	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	NOUN
cana-525	63	9	)	)	PUNCT
cana-525	63	10	∈̃	∈̃	PROPN
cana-525	63	11	�	�	PROPN
cana-525	63	12	̃	̃	PROPN
cana-525	63	13	�	�	PROPN
cana-525	63	14	and	and	CCONJ
cana-525	63	15	for	for	ADP
cana-525	63	16	all	all	DET
cana-525	63	17	₴	₴	NOUN
cana-525	63	18	̃	̃	PROPN
cana-525	63	19	∈̃	∈̃	PROPN
cana-525	63	20	ℂ̃(₳	ℂ̃(₳	NUM
cana-525	63	21	)	)	PUNCT
cana-525	63	22	iii	iii	NUM
cana-525	63	23	main	main	ADJ
cana-525	63	24	results	result	NOUN
cana-525	63	25	the	the	DET
cana-525	63	26	definition	definition	NOUN
cana-525	63	27	of	of	ADP
cana-525	63	28	the	the	DET
cana-525	63	29	fuzzy	fuzzy	ADJ
cana-525	63	30	soft	soft	ADJ
cana-525	63	31	paranormal	paranormal	ADJ
cana-525	63	32	operator	operator	NOUN
cana-525	63	33	in	in	ADP
cana-525	63	34	fuzzy	fuzzy	ADJ
cana-525	63	35	soft	soft	ADJ
cana-525	63	36	hilbert	hilbert	NOUN
cana-525	63	37	space	space	NOUN
cana-525	63	38	is	be	AUX
cana-525	63	39	provided	provide	VERB
cana-525	63	40	in	in	ADP
cana-525	63	41	this	this	DET
cana-525	63	42	section	section	NOUN
cana-525	63	43	.	.	PUNCT
cana-525	64	1	definition	definition	NOUN
cana-525	64	2	3.1	3.1	NUM
cana-525	64	3	:	:	PUNCT
cana-525	64	4	fuzzy	fuzzy	ADJ
cana-525	64	5	soft	soft	ADJ
cana-525	64	6	paranormal	paranormal	ADJ
cana-525	64	7	operator	operator	NOUN
cana-525	64	8	(	(	PUNCT
cana-525	64	9	fspn	fspn	NOUN
cana-525	64	10	)	)	PUNCT
cana-525	64	11	let	let	VERB
cana-525	64	12	�	�	PROPN
cana-525	64	13	̃	̃	PROPN
cana-525	64	14	�	�	PROPN
cana-525	64	15	be	be	AUX
cana-525	64	16	an	an	DET
cana-525	64	17	fs	fs	X
cana-525	64	18	hilbert	hilbert	NOUN
cana-525	64	19	space	space	NOUN
cana-525	64	20	and	and	CCONJ
cana-525	64	21	let	let	VERB
cana-525	64	22	ʈ̃	ʈ̃	PROPN
cana-525	64	23	∈	∈	PROPN
cana-525	64	24	�	�	PROPN
cana-525	64	25	̃	̃	PROPN
cana-525	64	26	�	�	PROPN
cana-525	64	27	(	(	PUNCT
cana-525	64	28	�	�	PROPN
cana-525	64	29	̃	̃	PROPN
cana-525	64	30	�	�	PROPN
cana-525	64	31	)	)	PUNCT
cana-525	64	32	then	then	ADV
cana-525	64	33	ʈ̃	ʈ̃	PROPN
cana-525	64	34	is	be	AUX
cana-525	64	35	a	a	DET
cana-525	64	36	fspn	fspn	ADJ
cana-525	64	37	operator	operator	NOUN
cana-525	64	38	if	if	SCONJ
cana-525	64	39	‖ʈ̃2𝑙𝜂ɠ(𝑒	‖ʈ̃2𝑙𝜂ɠ(𝑒	ADV
cana-525	64	40	)	)	PUNCT
cana-525	64	41	̃	̃	PROPN
cana-525	64	42	‖	‖	ADJ
cana-525	64	43	‖𝑙𝜂ɠ(𝑒	‖𝑙𝜂ɠ(𝑒	ADJ
cana-525	64	44	)	)	PUNCT
cana-525	64	45	̃‖	̃‖	VERB
cana-525	64	46	≥̃	≥̃	PUNCT
cana-525	64	47	‖ʈ̃𝑙𝜂ɠ(𝑒	‖ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	64	48	)	)	PUNCT
cana-525	64	49	̃	̃	NOUN
cana-525	64	50	‖	‖	ADJ
cana-525	64	51	2	2	NUM
cana-525	64	52	for	for	ADP
cana-525	64	53	all	all	DET
cana-525	64	54	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	64	55	)	)	PUNCT
cana-525	64	56	∈̃	∈̃	PROPN
cana-525	64	57	�	�	PROPN
cana-525	64	58	̃	̃	PROPN
cana-525	64	59	�	�	PROPN
cana-525	64	60	note	note	NOUN
cana-525	64	61	:	:	PUNCT
cana-525	64	62	an	an	DET
cana-525	64	63	operator	operator	NOUN
cana-525	64	64	ʈ̃	ʈ̃	PROPN
cana-525	64	65	∈	∈	PROPN
cana-525	64	66	�	�	PROPN
cana-525	64	67	̃	̃	PROPN
cana-525	64	68	�	�	PROPN
cana-525	64	69	(	(	PUNCT
cana-525	64	70	�	�	PROPN
cana-525	64	71	̃	̃	PROPN
cana-525	64	72	�	�	PROPN
cana-525	64	73	)	)	PUNCT
cana-525	64	74	and	and	CCONJ
cana-525	64	75	�	�	PROPN
cana-525	64	76	̃	̃	PROPN
cana-525	64	77	�	�	PROPN
cana-525	64	78	be	be	VERB
cana-525	64	79	a	a	DET
cana-525	64	80	fshs	fshs	NOUN
cana-525	64	81	then	then	ADV
cana-525	64	82	ʈ̃	ʈ̃	PROPN
cana-525	64	83	is	be	AUX
cana-525	64	84	said	say	VERB
cana-525	64	85	to	to	PART
cana-525	64	86	be	be	AUX
cana-525	64	87	an	an	DET
cana-525	64	88	fspn	fspn	ADJ
cana-525	64	89	operator	operator	NOUN
cana-525	64	90	if	if	SCONJ
cana-525	64	91	‖ʈ̃𝑙𝜂ɠ(𝑒	‖ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	64	92	)	)	PUNCT
cana-525	64	93	̃	̃	NOUN
cana-525	64	94	‖	‖	ADJ
cana-525	64	95	2	2	NUM
cana-525	64	96	≤̃	≤̃	NOUN
cana-525	64	97	‖ʈ̃2𝑙𝜂ɠ(𝑒	‖ʈ̃2𝑙𝜂ɠ(𝑒	ADV
cana-525	64	98	)	)	PUNCT
cana-525	64	99	̃	̃	NOUN
cana-525	64	100	‖	‖	ADJ
cana-525	64	101	,	,	PUNCT
cana-525	64	102	for	for	ADP
cana-525	64	103	every	every	DET
cana-525	64	104	unit	unit	NOUN
cana-525	64	105	vector	vector	NOUN
cana-525	64	106	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	64	107	)	)	PUNCT
cana-525	64	108	in	in	ADP
cana-525	64	109	�	�	PROPN
cana-525	64	110	̃	̃	PROPN
cana-525	64	111	�	�	PROPN
cana-525	64	112	.	.	PUNCT
cana-525	65	1	communications	communication	NOUN
cana-525	65	2	on	on	ADP
cana-525	65	3	applied	apply	VERB
cana-525	65	4	nonlinear	nonlinear	ADJ
cana-525	65	5	analysis	analysis	NOUN
cana-525	65	6	issn	issn	NOUN
cana-525	65	7	:	:	PUNCT
cana-525	65	8	1074	1074	NUM
cana-525	65	9	-	-	PUNCT
cana-525	65	10	133x	133x	NUM
cana-525	65	11	vol	vol	NOUN
cana-525	65	12	31	31	NUM
cana-525	65	13	no	no	NOUN
cana-525	65	14	.	.	NOUN
cana-525	65	15	2	2	NUM
cana-525	65	16	(	(	PUNCT
cana-525	65	17	2024	2024	NUM
cana-525	65	18	)	)	PUNCT
cana-525	65	19	132	132	NUM
cana-525	65	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-525	65	21	remark	remark	NOUN
cana-525	65	22	:	:	PUNCT
cana-525	65	23	let	let	VERB
cana-525	65	24	ʈ̃	ʈ̃	PROPN
cana-525	65	25	∈	∈	PROPN
cana-525	65	26	�	�	PROPN
cana-525	65	27	̃	̃	PROPN
cana-525	65	28	�	�	PROPN
cana-525	65	29	(	(	PUNCT
cana-525	65	30	�	�	PROPN
cana-525	65	31	̃	̃	PROPN
cana-525	65	32	�	�	PROPN
cana-525	65	33	)	)	PUNCT
cana-525	65	34	,	,	PUNCT
cana-525	65	35	�	�	PROPN
cana-525	65	36	̃	̃	PROPN
cana-525	65	37	�	�	NOUN
cana-525	65	38	=	=	SYM
cana-525	65	39	̃	̃	PROPN
cana-525	65	40	𝑙2(	𝑙2(	NOUN
cana-525	65	41	�	�	PROPN
cana-525	65	42	̃	̃	NOUN
cana-525	65	43	�	�	PROPN
cana-525	65	44	)	)	PUNCT
cana-525	65	45	ie	ie	ADJ
cana-525	65	46	)	)	PUNCT
cana-525	65	47	𝑙2(	𝑙2(	NOUN
cana-525	65	48	�	�	PROPN
cana-525	65	49	̃	̃	NOUN
cana-525	65	50	�	�	NOUN
cana-525	65	51	)	)	PUNCT
cana-525	66	1	=	=	SYM
cana-525	66	2	̃	̃	PROPN
cana-525	66	3	{	{	PUNCT
cana-525	66	4	𝑙𝜂𝔾(𝑒	𝑙𝜂𝔾(𝑒	NOUN
cana-525	66	5	)	)	PUNCT
cana-525	67	1	=	=	SYM
cana-525	67	2	̃	̃	PROPN
cana-525	67	3	(	(	PUNCT
cana-525	67	4	𝑙1	𝑙1	PROPN
cana-525	67	5	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	67	6	)	)	PUNCT
cana-525	67	7	,	,	PUNCT
cana-525	67	8	𝑙2	𝑙2	PROPN
cana-525	67	9	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	67	10	)	)	PUNCT
cana-525	67	11	…	…	PUNCT
cana-525	67	12	)	)	PUNCT
cana-525	68	1	̃	̃	ADV
cana-525	68	2	:	:	PUNCT
cana-525	68	3	∑	∑	PUNCT
cana-525	68	4	|𝑙𝑖𝜂𝑖ɠ(𝑒𝑖	|𝑙𝑖𝜂𝑖ɠ(𝑒𝑖	PROPN
cana-525	68	5	)	)	PUNCT
cana-525	68	6	̃	̃	PROPN
cana-525	68	7	|	|	ADV
cana-525	68	8	2	2	NUM
cana-525	68	9	<	<	X
cana-525	68	10	∞,∞	∞,∞	PROPN
cana-525	68	11	𝑖=1	𝑖=1	PROPN
cana-525	68	12	𝑙𝑖𝜂𝑖ɠ(𝑒𝑖	𝑙𝑖𝜂𝑖ɠ(𝑒𝑖	PROPN
cana-525	68	13	)	)	PUNCT
cana-525	68	14	∈	∈	PROPN
cana-525	68	15	𝒞𝑛(𝒜)̃	𝒞𝑛(𝒜)̃	PROPN
cana-525	68	16	}	}	PUNCT
cana-525	68	17	for	for	ADP
cana-525	68	18	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	NOUN
cana-525	68	19	)	)	PUNCT
cana-525	68	20	∈̃	∈̃	PROPN
cana-525	68	21	𝑙2(	𝑙2(	X
cana-525	68	22	�	�	PROPN
cana-525	68	23	̃	̃	NOUN
cana-525	68	24	�	�	PROPN
cana-525	68	25	)	)	PUNCT
cana-525	68	26	,	,	PUNCT
cana-525	68	27	defined	define	VERB
cana-525	68	28	‖𝑙𝜂ɠ(𝑒	‖𝑙𝜂ɠ(𝑒	ADV
cana-525	68	29	)	)	PUNCT
cana-525	68	30	‖	‖	PROPN
cana-525	69	1	̃	̃	PROPN
cana-525	69	2	=	=	SYM
cana-525	69	3	̃	̃	NOUN
cana-525	69	4	〈	〈	NOUN
cana-525	69	5	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	NOUN
cana-525	69	6	)	)	PUNCT
cana-525	69	7	,	,	PUNCT
cana-525	69	8	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	69	9	)	)	PUNCT
cana-525	69	10	〉	〉	NOUN
cana-525	69	11	1	1	NUM
cana-525	69	12	2⁄̃	2⁄̃	NUM
cana-525	69	13	=	=	SYM
cana-525	69	14	̃	̃	PROPN
cana-525	69	15	(	(	PUNCT
cana-525	69	16	∑	∑	INTJ
cana-525	69	17	|𝑙𝑖𝜂𝑖ɠ(𝑒𝑖	|𝑙𝑖𝜂𝑖ɠ(𝑒𝑖	PROPN
cana-525	69	18	)	)	PUNCT
cana-525	70	1	|	|	ADV
cana-525	70	2	2	2	NUM
cana-525	70	3	∞	∞	NUM
cana-525	70	4	𝑖=1	𝑖=1	PUNCT
cana-525	70	5	)	)	PUNCT
cana-525	70	6	1	1	NUM
cana-525	70	7	2⁄̃	2⁄̃	NUM
cana-525	70	8	let	let	VERB
cana-525	70	9	ʈ̃	ʈ̃	PROPN
cana-525	70	10	:	:	PUNCT
cana-525	70	11	�	�	PROPN
cana-525	70	12	̃	̃	PROPN
cana-525	70	13	�	�	PROPN
cana-525	70	14	→	→	SYM
cana-525	70	15	�	�	PROPN
cana-525	70	16	̃	̃	PROPN
cana-525	70	17	�	�	PROPN
cana-525	70	18	defined	define	VERB
cana-525	70	19	by	by	ADP
cana-525	70	20	ʈ̃	ʈ̃	PROPN
cana-525	70	21	(	(	PUNCT
cana-525	70	22	𝑙1	𝑙1	PROPN
cana-525	70	23	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	70	24	)	)	PUNCT
cana-525	70	25	,	,	PUNCT
cana-525	70	26	𝑙2	𝑙2	PROPN
cana-525	70	27	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	70	28	)	)	PUNCT
cana-525	70	29	…	…	PUNCT
cana-525	70	30	)	)	PUNCT
cana-525	71	1	̃	̃	NOUN
cana-525	71	2	=	=	SYM
cana-525	71	3	̃	̃	PROPN
cana-525	71	4	(	(	PUNCT
cana-525	71	5	𝜃	𝜃	NOUN
cana-525	71	6	,	,	PUNCT
cana-525	71	7	𝑙1	𝑙1	PROPN
cana-525	71	8	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	71	9	)	)	PUNCT
cana-525	71	10	,	,	PUNCT
cana-525	71	11	𝑙2	𝑙2	PROPN
cana-525	71	12	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	71	13	)	)	PUNCT
cana-525	71	14	…	…	PUNCT
cana-525	71	15	)	)	PUNCT
cana-525	72	1	̃	̃	NOUN
cana-525	72	2	∀	∀	X
cana-525	72	3	(	(	PUNCT
cana-525	72	4	𝑙1	𝑙1	PROPN
cana-525	72	5	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	72	6	)	)	PUNCT
cana-525	72	7	,	,	PUNCT
cana-525	72	8	𝑙2	𝑙2	PROPN
cana-525	72	9	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	72	10	)	)	PUNCT
cana-525	72	11	…	…	PUNCT
cana-525	72	12	)	)	PUNCT
cana-525	72	13	∈̃	∈̃	PROPN
cana-525	72	14	̃	̃	PROPN
cana-525	72	15	𝑙2(	𝑙2(	X
cana-525	72	16	�	�	PROPN
cana-525	72	17	̃	̃	NOUN
cana-525	72	18	�	�	PROPN
cana-525	72	19	)	)	PUNCT
cana-525	72	20	a	a	NOUN
cana-525	72	21	)	)	PUNCT
cana-525	72	22	to	to	PART
cana-525	72	23	find	find	VERB
cana-525	72	24	ʈ̃	ʈ̃	PROPN
cana-525	72	25	is	be	AUX
cana-525	72	26	linear	linear	ADJ
cana-525	72	27	take	take	VERB
cana-525	72	28	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	ADV
cana-525	72	29	)	)	PUNCT
cana-525	72	30	=	=	SYM
cana-525	72	31	̃	̃	PROPN
cana-525	72	32	(	(	PUNCT
cana-525	72	33	𝑙1	𝑙1	PROPN
cana-525	72	34	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	72	35	)	)	PUNCT
cana-525	72	36	,	,	PUNCT
cana-525	72	37	𝑙2	𝑙2	PROPN
cana-525	72	38	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	72	39	)	)	PUNCT
cana-525	72	40	…	…	PUNCT
cana-525	72	41	)	)	PUNCT
cana-525	72	42	̃	̃	PROPN
cana-525	72	43	�	�	PROPN
cana-525	72	44	̃	̃	PROPN
cana-525	72	45	�	�	NOUN
cana-525	72	46	𝛾ɠ(𝑎	𝛾ɠ(𝑎	NUM
cana-525	72	47	)	)	PUNCT
cana-525	72	48	=	=	SYM
cana-525	72	49	̃	̃	PROPN
cana-525	72	50	(	(	PUNCT
cana-525	72	51	�	�	PROPN
cana-525	72	52	̃	̃	NOUN
cana-525	72	53	�	�	PROPN
cana-525	72	54	1	1	NUM
cana-525	72	55	𝛾1ɠ(𝑎1	𝛾1ɠ(𝑎1	PROPN
cana-525	72	56	)	)	PUNCT
cana-525	72	57	,	,	PUNCT
cana-525	72	58	𝑚2	𝑚2	NOUN
cana-525	72	59	𝛾2ɠ(𝑎2	𝛾2ɠ(𝑎2	NOUN
cana-525	72	60	)	)	PUNCT
cana-525	72	61	…	…	PUNCT
cana-525	72	62	)	)	PUNCT
cana-525	73	1	̃	̃	ADV
cana-525	73	2	∈̃	∈̃	PROPN
cana-525	73	3	𝑙2(	𝑙2(	X
cana-525	73	4	�	�	PROPN
cana-525	73	5	̃	̃	PROPN
cana-525	73	6	�	�	PROPN
cana-525	73	7	)	)	PUNCT
cana-525	73	8	ʈ̃	ʈ̃	PROPN
cana-525	73	9	(	(	PUNCT
cana-525	73	10	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	73	11	)	)	PUNCT
cana-525	73	12	+	+	CCONJ
cana-525	73	13	�	�	PROPN
cana-525	73	14	̃	̃	PROPN
cana-525	73	15	�	�	NOUN
cana-525	73	16	𝛾ɠ(𝑎	𝛾ɠ(𝑎	NUM
cana-525	73	17	)	)	PUNCT
cana-525	73	18	)	)	PUNCT
cana-525	74	1	̃	̃	PROPN
cana-525	74	2	=	=	SYM
cana-525	74	3	̃	̃	NOUN
cana-525	74	4	ʈ̃	ʈ̃	PROPN
cana-525	74	5	(	(	PUNCT
cana-525	74	6	𝑙1	𝑙1	PROPN
cana-525	74	7	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	74	8	)	)	PUNCT
cana-525	75	1	+	+	CCONJ
cana-525	75	2	�	�	PROPN
cana-525	75	3	̃	̃	NOUN
cana-525	75	4	�	�	PROPN
cana-525	75	5	1	1	NUM
cana-525	75	6	𝛾1ɠ(𝑎1	𝛾1ɠ(𝑎1	PROPN
cana-525	75	7	)	)	PUNCT
cana-525	75	8	,	,	PUNCT
cana-525	75	9	𝑙2	𝑙2	PROPN
cana-525	75	10	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	75	11	)	)	PUNCT
cana-525	76	1	+	+	NUM
cana-525	76	2	𝑚2	𝑚2	NOUN
cana-525	76	3	𝛾2ɠ(𝑎2	𝛾2ɠ(𝑎2	NOUN
cana-525	76	4	)	)	PUNCT
cana-525	76	5	,	,	PUNCT
cana-525	76	6	…	…	PUNCT
cana-525	76	7	.	.	PUNCT
cana-525	76	8	)	)	PUNCT
cana-525	77	1	̃	̃	NOUN
cana-525	77	2	=	=	SYM
cana-525	77	3	̃	̃	PROPN
cana-525	77	4	(	(	PUNCT
cana-525	77	5	𝜃	𝜃	NOUN
cana-525	77	6	,	,	PUNCT
cana-525	77	7	𝑙1	𝑙1	PROPN
cana-525	77	8	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	77	9	)	)	PUNCT
cana-525	77	10	+	+	CCONJ
cana-525	77	11	�	�	PROPN
cana-525	77	12	̃	̃	NOUN
cana-525	77	13	�	�	PROPN
cana-525	77	14	1	1	NUM
cana-525	77	15	𝛾1ɠ(𝑎1	𝛾1ɠ(𝑎1	PROPN
cana-525	77	16	)	)	PUNCT
cana-525	77	17	,	,	PUNCT
cana-525	77	18	𝑙2	𝑙2	PROPN
cana-525	77	19	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	77	20	)	)	PUNCT
cana-525	78	1	+	+	NUM
cana-525	78	2	𝑚2	𝑚2	NOUN
cana-525	78	3	𝛾2ɠ(𝑎2	𝛾2ɠ(𝑎2	NOUN
cana-525	78	4	)	)	PUNCT
cana-525	78	5	,	,	PUNCT
cana-525	78	6	…	…	PUNCT
cana-525	78	7	.	.	PUNCT
cana-525	78	8	)	)	PUNCT
cana-525	79	1	̃	̃	NOUN
cana-525	79	2	=	=	SYM
cana-525	79	3	̃	̃	PROPN
cana-525	79	4	(	(	PUNCT
cana-525	79	5	𝜃	𝜃	NOUN
cana-525	79	6	,	,	PUNCT
cana-525	79	7	𝑙1	𝑙1	PROPN
cana-525	79	8	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	79	9	)	)	PUNCT
cana-525	79	10	,	,	PUNCT
cana-525	79	11	𝑙2	𝑙2	PROPN
cana-525	79	12	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	79	13	)	)	PUNCT
cana-525	79	14	…	…	PUNCT
cana-525	79	15	)	)	PUNCT
cana-525	80	1	̃	̃	PROPN
cana-525	80	2	+	+	PUNCT
cana-525	80	3	̃	̃	PROPN
cana-525	80	4	(	(	PUNCT
cana-525	80	5	�	�	PROPN
cana-525	80	6	̃	̃	NOUN
cana-525	80	7	�	�	PROPN
cana-525	80	8	1	1	NUM
cana-525	80	9	𝛾1ɠ(𝑎1	𝛾1ɠ(𝑎1	PROPN
cana-525	80	10	)	)	PUNCT
cana-525	80	11	,	,	PUNCT
cana-525	80	12	𝑚2	𝑚2	NOUN
cana-525	80	13	𝛾2ɠ(𝑎2	𝛾2ɠ(𝑎2	NOUN
cana-525	80	14	)	)	PUNCT
cana-525	80	15	…	…	PUNCT
cana-525	80	16	)	)	PUNCT
cana-525	81	1	̃	̃	PROPN
cana-525	81	2	ʈ̃	ʈ̃	PROPN
cana-525	81	3	(	(	PUNCT
cana-525	81	4	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	81	5	)	)	PUNCT
cana-525	81	6	+	+	CCONJ
cana-525	81	7	�	�	PROPN
cana-525	81	8	̃	̃	PROPN
cana-525	81	9	�	�	NOUN
cana-525	81	10	𝛾ɠ(𝑎	𝛾ɠ(𝑎	NUM
cana-525	81	11	)	)	PUNCT
cana-525	81	12	)	)	PUNCT
cana-525	82	1	̃	̃	PROPN
cana-525	82	2	=	=	SYM
cana-525	82	3	̃	̃	NOUN
cana-525	82	4	ʈ̃	ʈ̃	PROPN
cana-525	82	5	(	(	PUNCT
cana-525	82	6	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	82	7	)	)	PUNCT
cana-525	82	8	)	)	PUNCT
cana-525	83	1	̃	̃	PROPN
cana-525	83	2	+	+	PUNCT
cana-525	83	3	̃	̃	NOUN
cana-525	83	4	ʈ̃	ʈ̃	PROPN
cana-525	83	5	(	(	PUNCT
cana-525	83	6	�	�	PROPN
cana-525	83	7	̃	̃	PROPN
cana-525	83	8	�	�	NOUN
cana-525	83	9	𝛾ɠ(𝑎	𝛾ɠ(𝑎	NUM
cana-525	83	10	)	)	PUNCT
cana-525	83	11	)	)	PUNCT
cana-525	84	1	̃	̃	PROPN
cana-525	84	2	ʈ̃	ʈ̃	PROPN
cana-525	84	3	(	(	PUNCT
cana-525	84	4	𝛼	𝛼	NOUN
cana-525	84	5	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	84	6	)	)	PUNCT
cana-525	84	7	)	)	PUNCT
cana-525	85	1	̃	̃	NOUN
cana-525	85	2	=	=	SYM
cana-525	85	3	̃	̃	PROPN
cana-525	85	4	(	(	PUNCT
cana-525	85	5	𝜃	𝜃	NOUN
cana-525	85	6	,	,	PUNCT
cana-525	85	7	𝛼𝑙1	𝛼𝑙1	NOUN
cana-525	85	8	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	NOUN
cana-525	85	9	)	)	PUNCT
cana-525	85	10	,	,	PUNCT
cana-525	85	11	𝛼	𝛼	PROPN
cana-525	85	12	�	�	PROPN
cana-525	85	13	̃	̃	NOUN
cana-525	85	14	�	�	NOUN
cana-525	85	15	2	2	NUM
cana-525	85	16	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	85	17	)	)	PUNCT
cana-525	85	18	…	…	PUNCT
cana-525	85	19	)	)	PUNCT
cana-525	86	1	̃	̃	NOUN
cana-525	86	2	=	=	SYM
cana-525	86	3	̃	̃	PROPN
cana-525	86	4	�	�	PROPN
cana-525	86	5	̃	̃	PROPN
cana-525	86	6	�	�	PROPN
cana-525	86	7	(	(	PUNCT
cana-525	86	8	𝜃	𝜃	PROPN
cana-525	86	9	,	,	PUNCT
cana-525	86	10	𝑙1	𝑙1	PROPN
cana-525	86	11	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	86	12	)	)	PUNCT
cana-525	86	13	,	,	PUNCT
cana-525	86	14	𝑙2	𝑙2	PROPN
cana-525	86	15	𝜂2𝑣(𝑒2	𝜂2𝑣(𝑒2	PROPN
cana-525	86	16	)	)	PUNCT
cana-525	86	17	…	…	PUNCT
cana-525	86	18	)	)	PUNCT
cana-525	87	1	̃	̃	NOUN
cana-525	87	2	=	=	SYM
cana-525	87	3	̃	̃	PROPN
cana-525	87	4	�	�	NOUN
cana-525	87	5	̃	̃	PROPN
cana-525	87	6	�	�	PROPN
cana-525	87	7	ʈ̃	ʈ̃	PROPN
cana-525	87	8	(	(	PUNCT
cana-525	87	9	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	87	10	)	)	PUNCT
cana-525	87	11	)	)	PUNCT
cana-525	88	1	̃	̃	PROPN
cana-525	88	2	b	b	X
cana-525	88	3	)	)	PUNCT
cana-525	88	4	to	to	PART
cana-525	88	5	find	find	VERB
cana-525	88	6	ʈ̃	ʈ̃	PROPN
cana-525	88	7	is	be	AUX
cana-525	88	8	finite	finite	ADJ
cana-525	88	9	take	take	NOUN
cana-525	88	10	(	(	PUNCT
cana-525	88	11	𝑙1	𝑙1	PROPN
cana-525	88	12	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	88	13	)	)	PUNCT
cana-525	88	14	,	,	PUNCT
cana-525	88	15	𝑙2	𝑙2	PROPN
cana-525	88	16	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	88	17	)	)	PUNCT
cana-525	88	18	…	…	PUNCT
cana-525	88	19	)	)	PUNCT
cana-525	89	1	̃	̃	ADV
cana-525	89	2	∈̃	∈̃	PROPN
cana-525	89	3	𝑙2(	𝑙2(	X
cana-525	89	4	�	�	PROPN
cana-525	89	5	̃	̃	NOUN
cana-525	89	6	�	�	NOUN
cana-525	89	7	)	)	PUNCT
cana-525	89	8	‖ʈ̃	‖ʈ̃	PUNCT
cana-525	90	1	(	(	PUNCT
cana-525	90	2	𝑙1	𝑙1	PROPN
cana-525	90	3	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	90	4	)	)	PUNCT
cana-525	90	5	,	,	PUNCT
cana-525	90	6	𝑙2	𝑙2	PROPN
cana-525	90	7	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	90	8	)	)	PUNCT
cana-525	90	9	…	…	PUNCT
cana-525	90	10	)	)	PUNCT
cana-525	91	1	‖	‖	PROPN
cana-525	91	2	̃	̃	ADJ
cana-525	91	3	2	2	NUM
cana-525	91	4	=	=	SYM
cana-525	91	5	̃	̃	NOUN
cana-525	91	6	‖(𝜃	‖(𝜃	NOUN
cana-525	91	7	,	,	PUNCT
cana-525	91	8	𝑙1	𝑙1	PROPN
cana-525	91	9	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	91	10	)	)	PUNCT
cana-525	91	11	,	,	PUNCT
cana-525	91	12	𝑙2	𝑙2	PROPN
cana-525	91	13	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	91	14	)	)	PUNCT
cana-525	91	15	…	…	PUNCT
cana-525	91	16	)	)	PUNCT
cana-525	92	1	‖	‖	PROPN
cana-525	93	1	2̃	2̃	NOUN
cana-525	93	2	=	=	X
cana-525	93	3	̃	̃	ADV
cana-525	93	4	∑	∑	PUNCT
cana-525	93	5	|𝑙𝑖𝜂𝑖ɠ(𝑒𝑖	|𝑙𝑖𝜂𝑖ɠ(𝑒𝑖	PROPN
cana-525	93	6	)	)	PUNCT
cana-525	93	7	̃	̃	PROPN
cana-525	93	8	|	|	ADV
cana-525	93	9	2	2	NUM
cana-525	93	10	∞	∞	NUM
cana-525	93	11	𝑖=1	𝑖=1	PROPN
cana-525	93	12	communications	communication	NOUN
cana-525	93	13	on	on	ADP
cana-525	93	14	applied	apply	VERB
cana-525	93	15	nonlinear	nonlinear	ADJ
cana-525	93	16	analysis	analysis	NOUN
cana-525	93	17	issn	issn	NOUN
cana-525	93	18	:	:	PUNCT
cana-525	93	19	1074	1074	NUM
cana-525	93	20	-	-	PUNCT
cana-525	93	21	133x	133x	NUM
cana-525	93	22	vol	vol	NOUN
cana-525	93	23	31	31	NUM
cana-525	93	24	no	no	NOUN
cana-525	93	25	.	.	NOUN
cana-525	93	26	2	2	NUM
cana-525	93	27	(	(	PUNCT
cana-525	93	28	2024	2024	NUM
cana-525	93	29	)	)	PUNCT
cana-525	94	1	133	133	NUM
cana-525	94	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-525	94	3	=	=	SYM
cana-525	94	4	̃	̃	NOUN
cana-525	94	5	‖𝑙𝜂ɠ(𝑒	‖𝑙𝜂ɠ(𝑒	NOUN
cana-525	94	6	)	)	PUNCT
cana-525	94	7	‖	‖	PROPN
cana-525	94	8	̃	̃	PROPN
cana-525	94	9	2	2	NUM
cana-525	94	10	ie	ie	ADJ
cana-525	94	11	)	)	PUNCT
cana-525	94	12	‖ʈ̃	‖ʈ̃	PUNCT
cana-525	94	13	(	(	PUNCT
cana-525	94	14	𝑙1	𝑙1	PROPN
cana-525	94	15	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	94	16	)	)	PUNCT
cana-525	94	17	,	,	PUNCT
cana-525	94	18	𝑙2	𝑙2	PROPN
cana-525	94	19	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	94	20	)	)	PUNCT
cana-525	94	21	…	…	PUNCT
cana-525	94	22	)	)	PUNCT
cana-525	95	1	‖	‖	PROPN
cana-525	95	2	̃	̃	ADJ
cana-525	95	3	2	2	NUM
cana-525	95	4	=	=	SYM
cana-525	95	5	̃	̃	NOUN
cana-525	95	6	‖𝑙𝜂ɠ(𝑒	‖𝑙𝜂ɠ(𝑒	NOUN
cana-525	95	7	)	)	PUNCT
cana-525	95	8	‖	‖	PROPN
cana-525	95	9	̃	̃	PROPN
cana-525	95	10	2	2	NUM
cana-525	95	11	‖ʈ̃𝑙𝜂ɠ(𝑒	‖ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	95	12	)	)	PUNCT
cana-525	96	1	‖	‖	PROPN
cana-525	96	2	̃	̃	ADJ
cana-525	96	3	2	2	NUM
cana-525	96	4	=	=	SYM
cana-525	96	5	̃	̃	NOUN
cana-525	96	6	‖𝑙𝜂ɠ(𝑒	‖𝑙𝜂ɠ(𝑒	NOUN
cana-525	96	7	)	)	PUNCT
cana-525	96	8	‖	‖	PROPN
cana-525	96	9	̃	̃	PROPN
cana-525	96	10	2	2	NUM
cana-525	96	11	iff	iff	PROPN
cana-525	96	12	‖ʈ̃𝑙𝜂ɠ(𝑒	‖ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	96	13	)	)	PUNCT
cana-525	96	14	‖	‖	PROPN
cana-525	96	15	̃	̃	PROPN
cana-525	96	16	=	=	SYM
cana-525	96	17	̃	̃	NOUN
cana-525	96	18	‖𝑙𝜂ɠ(𝑒	‖𝑙𝜂ɠ(𝑒	NOUN
cana-525	96	19	)	)	PUNCT
cana-525	96	20	‖	‖	PROPN
cana-525	96	21	̃	̃	PROPN
cana-525	96	22	which	which	PRON
cana-525	96	23	implies	imply	VERB
cana-525	96	24	ʈ̃	ʈ̃	PROPN
cana-525	96	25	is	be	AUX
cana-525	96	26	finite	finite	ADJ
cana-525	96	27	therefore	therefore	ADV
cana-525	96	28	,	,	PUNCT
cana-525	96	29	ʈ̃	ʈ̃	PROPN
cana-525	96	30	∈	∈	PROPN
cana-525	96	31	�	�	PROPN
cana-525	96	32	̃	̃	PROPN
cana-525	96	33	�	�	PROPN
cana-525	96	34	(	(	PUNCT
cana-525	96	35	�	�	PROPN
cana-525	96	36	̃	̃	NOUN
cana-525	96	37	�	�	PROPN
cana-525	96	38	)	)	PUNCT
cana-525	96	39	c	c	X
cana-525	96	40	)	)	PUNCT
cana-525	96	41	to	to	PART
cana-525	96	42	find	find	VERB
cana-525	96	43	ʈ̃	ʈ̃	PROPN
cana-525	96	44	is	be	AUX
cana-525	96	45	fspn	fspn	ADJ
cana-525	96	46	take	take	NOUN
cana-525	96	47	(	(	PUNCT
cana-525	96	48	𝑙1	𝑙1	PROPN
cana-525	96	49	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	96	50	)	)	PUNCT
cana-525	96	51	,	,	PUNCT
cana-525	96	52	𝑙2	𝑙2	PROPN
cana-525	96	53	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	96	54	)	)	PUNCT
cana-525	96	55	…	…	PUNCT
cana-525	96	56	)	)	PUNCT
cana-525	97	1	̃	̃	ADV
cana-525	97	2	∈̃	∈̃	PROPN
cana-525	97	3	𝑙2(	𝑙2(	X
cana-525	97	4	�	�	PROPN
cana-525	97	5	̃	̃	NOUN
cana-525	97	6	�	�	NOUN
cana-525	97	7	)	)	PUNCT
cana-525	97	8	‖ʈ̃	‖ʈ̃	PUNCT
cana-525	98	1	(	(	PUNCT
cana-525	98	2	𝑙1	𝑙1	PROPN
cana-525	98	3	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	98	4	)	)	PUNCT
cana-525	98	5	,	,	PUNCT
cana-525	98	6	𝑙2	𝑙2	PROPN
cana-525	98	7	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	98	8	)	)	PUNCT
cana-525	98	9	…	…	PUNCT
cana-525	98	10	)	)	PUNCT
cana-525	99	1	‖	‖	PROPN
cana-525	99	2	̃	̃	ADJ
cana-525	99	3	2	2	NUM
cana-525	99	4	=	=	SYM
cana-525	99	5	̃	̃	NOUN
cana-525	99	6	‖(𝜃	‖(𝜃	NOUN
cana-525	99	7	,	,	PUNCT
cana-525	99	8	𝑙1	𝑙1	PROPN
cana-525	99	9	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	99	10	)	)	PUNCT
cana-525	99	11	,	,	PUNCT
cana-525	99	12	𝑙2	𝑙2	PROPN
cana-525	99	13	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	99	14	)	)	PUNCT
cana-525	99	15	…	…	PUNCT
cana-525	99	16	)	)	PUNCT
cana-525	100	1	‖	‖	PROPN
cana-525	101	1	2̃	2̃	NOUN
cana-525	101	2	=	=	X
cana-525	101	3	̃	̃	ADV
cana-525	101	4	∑	∑	PUNCT
cana-525	101	5	|𝑙𝑖𝜂𝑖ɠ(𝑒𝑖	|𝑙𝑖𝜂𝑖ɠ(𝑒𝑖	PROPN
cana-525	101	6	)	)	PUNCT
cana-525	101	7	̃	̃	PROPN
cana-525	101	8	|	|	ADV
cana-525	101	9	2	2	NUM
cana-525	101	10	∞	∞	NUM
cana-525	101	11	𝑖=1	𝑖=1	PUNCT
cana-525	102	1	=	=	PUNCT
cana-525	102	2	̃	̃	NOUN
cana-525	102	3	‖	‖	ADJ
cana-525	102	4	(	(	PUNCT
cana-525	102	5	𝑙1	𝑙1	PROPN
cana-525	102	6	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	102	7	)	)	PUNCT
cana-525	102	8	,	,	PUNCT
cana-525	102	9	𝑙2	𝑙2	PROPN
cana-525	102	10	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	102	11	)	)	PUNCT
cana-525	102	12	…	…	PUNCT
cana-525	102	13	)	)	PUNCT
cana-525	103	1	‖	‖	PROPN
cana-525	103	2	̃	̃	PROPN
cana-525	103	3	2	2	NUM
cana-525	103	4	⇔	⇔	NOUN
cana-525	103	5	‖ʈ̃	‖ʈ̃	PUNCT
cana-525	103	6	(	(	PUNCT
cana-525	103	7	𝑙1	𝑙1	PROPN
cana-525	103	8	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	103	9	)	)	PUNCT
cana-525	103	10	,	,	PUNCT
cana-525	103	11	𝑙2	𝑙2	PROPN
cana-525	103	12	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	103	13	)	)	PUNCT
cana-525	103	14	…	…	PUNCT
cana-525	103	15	)	)	PUNCT
cana-525	104	1	‖	‖	PROPN
cana-525	104	2	=	=	X
cana-525	104	3	̃	̃	ADP
cana-525	104	4	̃	̃	NOUN
cana-525	104	5	‖	‖	PROPN
cana-525	104	6	(	(	PUNCT
cana-525	104	7	𝑙1	𝑙1	PROPN
cana-525	104	8	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	104	9	)	)	PUNCT
cana-525	104	10	,	,	PUNCT
cana-525	104	11	𝑙2	𝑙2	PROPN
cana-525	104	12	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	104	13	)	)	PUNCT
cana-525	104	14	…	…	PUNCT
cana-525	104	15	)	)	PUNCT
cana-525	105	1	‖	‖	PROPN
cana-525	105	2	̃	̃	ADJ
cana-525	105	3	d	d	NOUN
cana-525	105	4	)	)	PUNCT
cana-525	105	5	take	take	NOUN
cana-525	105	6	(	(	PUNCT
cana-525	105	7	𝑙1	𝑙1	PROPN
cana-525	105	8	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	105	9	)	)	PUNCT
cana-525	105	10	,	,	PUNCT
cana-525	105	11	𝑙2	𝑙2	PROPN
cana-525	105	12	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	105	13	)	)	PUNCT
cana-525	105	14	…	…	PUNCT
cana-525	105	15	)	)	PUNCT
cana-525	106	1	̃	̃	ADV
cana-525	106	2	∈̃	∈̃	PROPN
cana-525	106	3	𝑙2(	𝑙2(	X
cana-525	106	4	�	�	PROPN
cana-525	106	5	̃	̃	NOUN
cana-525	106	6	�	�	NOUN
cana-525	106	7	)	)	PUNCT
cana-525	106	8	‖ʈ̃	‖ʈ̃	PUNCT
cana-525	107	1	(	(	PUNCT
cana-525	107	2	𝑙1	𝑙1	PROPN
cana-525	107	3	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	107	4	)	)	PUNCT
cana-525	107	5	,	,	PUNCT
cana-525	107	6	𝑙2	𝑙2	PROPN
cana-525	107	7	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	107	8	)	)	PUNCT
cana-525	107	9	…	…	PUNCT
cana-525	107	10	)	)	PUNCT
cana-525	108	1	‖	‖	PROPN
cana-525	108	2	=	=	NUM
cana-525	108	3	̃	̃	PROPN
cana-525	108	4	̃	̃	PROPN
cana-525	108	5	‖(𝜃	‖(𝜃	PROPN
cana-525	108	6	,	,	PUNCT
cana-525	108	7	𝑙1	𝑙1	PROPN
cana-525	108	8	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	108	9	)	)	PUNCT
cana-525	108	10	,	,	PUNCT
cana-525	108	11	𝑙2	𝑙2	PROPN
cana-525	108	12	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	108	13	)	)	PUNCT
cana-525	108	14	…	…	PUNCT
cana-525	108	15	)	)	PUNCT
cana-525	109	1	̃	̃	NOUN
cana-525	109	2	‖	‖	ADJ
cana-525	109	3	let	let	VERB
cana-525	109	4	ʈ̃2	ʈ̃2	PROPN
cana-525	109	5	(	(	PUNCT
cana-525	109	6	𝑙1	𝑙1	PROPN
cana-525	109	7	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	109	8	)	)	PUNCT
cana-525	109	9	,	,	PUNCT
cana-525	109	10	𝑙2	𝑙2	PROPN
cana-525	109	11	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	109	12	)	)	PUNCT
cana-525	109	13	…	…	PUNCT
cana-525	109	14	)	)	PUNCT
cana-525	110	1	̃	̃	NOUN
cana-525	110	2	=	=	SYM
cana-525	110	3	̃	̃	NOUN
cana-525	110	4	ʈ̃	ʈ̃	PROPN
cana-525	110	5	(	(	PUNCT
cana-525	110	6	ʈ̃	ʈ̃	PROPN
cana-525	110	7	(	(	PUNCT
cana-525	110	8	𝑙1	𝑙1	PROPN
cana-525	110	9	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	110	10	)	)	PUNCT
cana-525	110	11	,	,	PUNCT
cana-525	110	12	𝑙2	𝑙2	PROPN
cana-525	110	13	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	110	14	)	)	PUNCT
cana-525	110	15	…	…	PUNCT
cana-525	110	16	)	)	PUNCT
cana-525	111	1	̃	̃	ADV
cana-525	111	2	)	)	PUNCT
cana-525	112	1	=	=	X
cana-525	112	2	̃	̃	NOUN
cana-525	112	3	ʈ̃	ʈ̃	PROPN
cana-525	112	4	(	(	PUNCT
cana-525	112	5	𝜃	𝜃	PROPN
cana-525	112	6	,	,	PUNCT
cana-525	112	7	𝑙1	𝑙1	PROPN
cana-525	112	8	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	112	9	)	)	PUNCT
cana-525	112	10	,	,	PUNCT
cana-525	112	11	𝑙2	𝑙2	PROPN
cana-525	112	12	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	112	13	)	)	PUNCT
cana-525	112	14	…	…	PUNCT
cana-525	112	15	)	)	PUNCT
cana-525	113	1	̃	̃	PROPN
cana-525	113	2	ʈ̃2	ʈ̃2	PROPN
cana-525	113	3	(	(	PUNCT
cana-525	113	4	𝑙1	𝑙1	PROPN
cana-525	113	5	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	113	6	)	)	PUNCT
cana-525	113	7	,	,	PUNCT
cana-525	113	8	𝑙2	𝑙2	PROPN
cana-525	113	9	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	113	10	)	)	PUNCT
cana-525	113	11	…	…	PUNCT
cana-525	113	12	)	)	PUNCT
cana-525	114	1	̃	̃	NOUN
cana-525	114	2	=	=	SYM
cana-525	114	3	̃	̃	PROPN
cana-525	114	4	(	(	PUNCT
cana-525	114	5	𝜃	𝜃	NOUN
cana-525	114	6	,	,	PUNCT
cana-525	114	7	𝜃𝑙1	𝜃𝑙1	ADJ
cana-525	114	8	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	NOUN
cana-525	114	9	)	)	PUNCT
cana-525	114	10	,	,	PUNCT
cana-525	114	11	𝑙2	𝑙2	PROPN
cana-525	114	12	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	114	13	)	)	PUNCT
cana-525	114	14	…	…	PUNCT
cana-525	114	15	)	)	PUNCT
cana-525	115	1	̃	̃	PROPN
cana-525	115	2	‖	‖	PROPN
cana-525	115	3	ʈ̃2	ʈ̃2	PROPN
cana-525	115	4	(	(	PUNCT
cana-525	115	5	𝑙1	𝑙1	PROPN
cana-525	115	6	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	115	7	)	)	PUNCT
cana-525	115	8	,	,	PUNCT
cana-525	115	9	𝑙2	𝑙2	PROPN
cana-525	115	10	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	115	11	)	)	PUNCT
cana-525	115	12	…	…	PUNCT
cana-525	115	13	)	)	PUNCT
cana-525	116	1	̃	̃	ADP
cana-525	116	2	‖	‖	ADJ
cana-525	116	3	=	=	SYM
cana-525	116	4	̃	̃	PROPN
cana-525	116	5	‖(𝜃	‖(𝜃	PROPN
cana-525	116	6	,	,	PUNCT
cana-525	116	7	𝜃𝑙1	𝜃𝑙1	ADJ
cana-525	116	8	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	NOUN
cana-525	116	9	)	)	PUNCT
cana-525	116	10	,	,	PUNCT
cana-525	116	11	𝑙2	𝑙2	PROPN
cana-525	116	12	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	116	13	)	)	PUNCT
cana-525	116	14	…	…	PUNCT
cana-525	116	15	)	)	PUNCT
cana-525	117	1	̃	̃	PROPN
cana-525	117	2	‖	‖	PROPN
cana-525	117	3	‖	‖	PROPN
cana-525	117	4	ʈ̃2	ʈ̃2	PROPN
cana-525	117	5	(	(	PUNCT
cana-525	117	6	𝑙1	𝑙1	PROPN
cana-525	117	7	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	117	8	)	)	PUNCT
cana-525	117	9	,	,	PUNCT
cana-525	117	10	𝑙2	𝑙2	PROPN
cana-525	117	11	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	117	12	)	)	PUNCT
cana-525	117	13	…	…	PUNCT
cana-525	117	14	)	)	PUNCT
cana-525	118	1	̃	̃	ADP
cana-525	118	2	‖	‖	ADJ
cana-525	118	3	=	=	SYM
cana-525	118	4	̃	̃	ADJ
cana-525	118	5	∑	∑	PUNCT
cana-525	118	6	|𝑙𝑖𝜂𝑖ɠ(𝑒𝑖	|𝑙𝑖𝜂𝑖ɠ(𝑒𝑖	PROPN
cana-525	118	7	)	)	PUNCT
cana-525	118	8	|	|	ADV
cana-525	119	1	̃	̃	NOUN
cana-525	119	2	∞	∞	PUNCT
cana-525	119	3	𝑖=1	𝑖=1	PUNCT
cana-525	120	1	e	e	X
cana-525	120	2	)	)	PUNCT
cana-525	120	3	taken	take	VERB
cana-525	120	4	any	any	DET
cana-525	120	5	(	(	PUNCT
cana-525	120	6	𝑙1	𝑙1	PROPN
cana-525	120	7	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	120	8	)	)	PUNCT
cana-525	120	9	,	,	PUNCT
cana-525	120	10	𝑙2	𝑙2	PROPN
cana-525	120	11	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	120	12	)	)	PUNCT
cana-525	120	13	…	…	PUNCT
cana-525	120	14	)	)	PUNCT
cana-525	121	1	̃	̃	ADV
cana-525	121	2	∈̃	∈̃	PROPN
cana-525	121	3	𝑙2(	𝑙2(	X
cana-525	121	4	�	�	PROPN
cana-525	121	5	̃	̃	PROPN
cana-525	121	6	�	�	PROPN
cana-525	121	7	)	)	PUNCT
cana-525	121	8	ʈ̃	ʈ̃	PROPN
cana-525	121	9	(	(	PUNCT
cana-525	121	10	𝑙1	𝑙1	PROPN
cana-525	121	11	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	121	12	)	)	PUNCT
cana-525	121	13	,	,	PUNCT
cana-525	121	14	𝑙2	𝑙2	PROPN
cana-525	121	15	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	121	16	)	)	PUNCT
cana-525	121	17	…	…	PUNCT
cana-525	121	18	)	)	PUNCT
cana-525	122	1	̃	̃	NOUN
cana-525	122	2	=	=	SYM
cana-525	122	3	̃	̃	PROPN
cana-525	122	4	(	(	PUNCT
cana-525	122	5	𝜃	𝜃	NOUN
cana-525	122	6	,	,	PUNCT
cana-525	122	7	𝑙1	𝑙1	PROPN
cana-525	122	8	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	122	9	)	)	PUNCT
cana-525	122	10	,	,	PUNCT
cana-525	122	11	𝑙2	𝑙2	PROPN
cana-525	122	12	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	122	13	)	)	PUNCT
cana-525	122	14	…	…	PUNCT
cana-525	122	15	)	)	PUNCT
cana-525	123	1	̃	̃	NOUN
cana-525	123	2	communications	communication	NOUN
cana-525	123	3	on	on	ADP
cana-525	123	4	applied	apply	VERB
cana-525	123	5	nonlinear	nonlinear	ADJ
cana-525	123	6	analysis	analysis	NOUN
cana-525	123	7	issn	issn	NOUN
cana-525	123	8	:	:	PUNCT
cana-525	123	9	1074	1074	NUM
cana-525	123	10	-	-	PUNCT
cana-525	123	11	133x	133x	NUM
cana-525	123	12	vol	vol	NOUN
cana-525	123	13	31	31	NUM
cana-525	123	14	no	no	NOUN
cana-525	123	15	.	.	NOUN
cana-525	123	16	2	2	NUM
cana-525	123	17	(	(	PUNCT
cana-525	123	18	2024	2024	NUM
cana-525	123	19	)	)	PUNCT
cana-525	123	20	134	134	NUM
cana-525	123	21	https://internationalpubls.com	https://internationalpubls.com	X
cana-525	123	22	‖ʈ̃	‖ʈ̃	PUNCT
cana-525	123	23	(	(	PUNCT
cana-525	123	24	𝑙1	𝑙1	PROPN
cana-525	123	25	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	123	26	)	)	PUNCT
cana-525	123	27	,	,	PUNCT
cana-525	123	28	𝑙2	𝑙2	PROPN
cana-525	123	29	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	123	30	)	)	PUNCT
cana-525	123	31	…	…	PUNCT
cana-525	123	32	)	)	PUNCT
cana-525	124	1	‖	‖	PROPN
cana-525	124	2	̃	̃	ADJ
cana-525	124	3	2	2	NUM
cana-525	124	4	=	=	SYM
cana-525	124	5	̃	̃	NOUN
cana-525	124	6	‖(𝜃	‖(𝜃	NOUN
cana-525	124	7	,	,	PUNCT
cana-525	124	8	𝑙1	𝑙1	PROPN
cana-525	124	9	𝜂1ɠ(𝑒1	𝜂1ɠ(𝑒1	PROPN
cana-525	124	10	)	)	PUNCT
cana-525	124	11	,	,	PUNCT
cana-525	124	12	𝑙2	𝑙2	PROPN
cana-525	124	13	𝜂2ɠ(𝑒2	𝜂2ɠ(𝑒2	NOUN
cana-525	124	14	)	)	PUNCT
cana-525	124	15	…	…	PUNCT
cana-525	124	16	)	)	PUNCT
cana-525	125	1	‖	‖	PROPN
cana-525	126	1	2̃	2̃	NOUN
cana-525	126	2	=	=	X
cana-525	126	3	̃	̃	ADV
cana-525	126	4	∑	∑	PUNCT
cana-525	126	5	|𝑙𝑖𝜂𝑖ɠ(𝑒𝑖	|𝑙𝑖𝜂𝑖ɠ(𝑒𝑖	PROPN
cana-525	126	6	)	)	PUNCT
cana-525	126	7	̃	̃	PROPN
cana-525	126	8	|	|	ADV
cana-525	126	9	2	2	NUM
cana-525	126	10	∞	∞	NUM
cana-525	126	11	𝑖=1	𝑖=1	PUNCT
cana-525	126	12	from	from	ADP
cana-525	126	13	d	d	PROPN
cana-525	126	14	)	)	PUNCT
cana-525	126	15	and	and	CCONJ
cana-525	126	16	e	e	X
cana-525	126	17	)	)	PUNCT
cana-525	126	18	,	,	PUNCT
cana-525	126	19	we	we	PRON
cana-525	126	20	get	get	VERB
cana-525	126	21	‖ʈ̃(	‖ʈ̃(	NOUN
cana-525	126	22	�	�	PROPN
cana-525	126	23	̃	̃	NOUN
cana-525	126	24	�	�	NOUN
cana-525	126	25	𝜂ɠ(𝑒	𝜂ɠ(𝑒	NUM
cana-525	126	26	)	)	PUNCT
cana-525	126	27	)	)	PUNCT
cana-525	127	1	̃	̃	NOUN
cana-525	127	2	‖	‖	ADJ
cana-525	127	3	2	2	NUM
cana-525	127	4	≤̃	≤̃	NOUN
cana-525	127	5	‖ʈ̃2𝑙𝜂ɠ(𝑒	‖ʈ̃2𝑙𝜂ɠ(𝑒	ADV
cana-525	127	6	)	)	PUNCT
cana-525	127	7	̃	̃	PROPN
cana-525	127	8	‖	‖	ADJ
cana-525	127	9	therefore	therefore	ADV
cana-525	127	10	,	,	PUNCT
cana-525	127	11	ʈ̃	ʈ̃	PROPN
cana-525	127	12	is	be	AUX
cana-525	127	13	fspn	fspn	ADJ
cana-525	127	14	operator	operator	NOUN
cana-525	127	15	theorem	theorem	VERB
cana-525	127	16	3.3	3.3	NUM
cana-525	127	17	:	:	PUNCT
cana-525	127	18	‖ʈ̃3𝑙𝜂ɠ(𝑒	‖ʈ̃3𝑙𝜂ɠ(𝑒	ADV
cana-525	127	19	)	)	PUNCT
cana-525	128	1	‖	‖	PROPN
cana-525	128	2	̃	̃	PROPN
cana-525	128	3	≥̃	≥̃	X
cana-525	128	4	‖ʈ̃2𝑙𝜂ɠ(𝑒	‖ʈ̃2𝑙𝜂ɠ(𝑒	ADJ
cana-525	128	5	)	)	PUNCT
cana-525	128	6	̃	̃	ADV
cana-525	128	7	‖	‖	ADJ
cana-525	128	8	‖ʈ̃	‖ʈ̃	PUNCT
cana-525	128	9	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PRON
cana-525	128	10	)	)	PUNCT
cana-525	129	1	‖	‖	PROPN
cana-525	129	2	̃	̃	PROPN
cana-525	129	3	for	for	ADP
cana-525	129	4	every	every	DET
cana-525	129	5	unit	unit	NOUN
cana-525	129	6	vector	vector	NOUN
cana-525	129	7	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	129	8	)	)	PUNCT
cana-525	129	9	in	in	ADP
cana-525	129	10	�	�	PROPN
cana-525	129	11	̃	̃	NOUN
cana-525	129	12	�	�	NOUN
cana-525	129	13	proof	proof	NOUN
cana-525	129	14	:	:	PUNCT
cana-525	129	15	for	for	SCONJ
cana-525	129	16	every	every	DET
cana-525	129	17	unit	unit	NOUN
cana-525	129	18	vector	vector	NOUN
cana-525	129	19	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	129	20	)	)	PUNCT
cana-525	129	21	∈̃	∈̃	PROPN
cana-525	129	22	�	�	PROPN
cana-525	129	23	̃	̃	PROPN
cana-525	129	24	�	�	PROPN
cana-525	129	25	let	let	VERB
cana-525	129	26	‖ʈ̃3𝑙𝜂ɠ(𝑒	‖ʈ̃3𝑙𝜂ɠ(𝑒	ADV
cana-525	129	27	)	)	PUNCT
cana-525	130	1	‖	‖	PROPN
cana-525	131	1	̃	̃	ADJ
cana-525	131	2	2	2	NUM
cana-525	131	3	=	=	SYM
cana-525	131	4	̃	̃	NOUN
cana-525	131	5	〈	〈	NOUN
cana-525	131	6	ʈ̃3𝑙𝜂ɠ(𝑒	ʈ̃3𝑙𝜂ɠ(𝑒	NOUN
cana-525	131	7	)	)	PUNCT
cana-525	131	8	,	,	PUNCT
cana-525	131	9	ʈ̃3𝑙𝜂ɠ(𝑒	ʈ̃3𝑙𝜂ɠ(𝑒	NUM
cana-525	131	10	)	)	PUNCT
cana-525	131	11	〉	〉	NOUN
cana-525	132	1	̃	̃	NOUN
cana-525	132	2	=	=	SYM
cana-525	132	3	̃	̃	NOUN
cana-525	132	4	〈	〈	NOUN
cana-525	132	5	ʈ̃	ʈ̃	PROPN
cana-525	132	6	ʈ̃2𝑙𝜂ɠ(𝑒	ʈ̃2𝑙𝜂ɠ(𝑒	NUM
cana-525	132	7	)	)	PUNCT
cana-525	132	8	,	,	PUNCT
cana-525	132	9	ʈ̃ʈ̃2𝑙𝜂ɠ(𝑒	ʈ̃ʈ̃2𝑙𝜂ɠ(𝑒	NOUN
cana-525	132	10	)	)	PUNCT
cana-525	132	11	〉	〉	NOUN
cana-525	132	12	̃	̃	NOUN
cana-525	132	13	=	=	SYM
cana-525	132	14	̃	̃	NOUN
cana-525	132	15	〈	〈	NOUN
cana-525	132	16	ʈ̃∗̃ʈ̃	ʈ̃∗̃ʈ̃	NOUN
cana-525	132	17	ʈ̃2𝑙𝜂ɠ(𝑒	ʈ̃2𝑙𝜂ɠ(𝑒	NUM
cana-525	132	18	)	)	PUNCT
cana-525	132	19	,	,	PUNCT
cana-525	132	20	ʈ̃2𝑙𝜂ɠ(𝑒	ʈ̃2𝑙𝜂ɠ(𝑒	X
cana-525	132	21	)	)	PUNCT
cana-525	132	22	〉	〉	NOUN
cana-525	133	1	̃	̃	NOUN
cana-525	133	2	=	=	SYM
cana-525	133	3	̃	̃	ADP
cana-525	133	4	〈	〈	PROPN
cana-525	133	5	ʈ̃2	ʈ̃2	PROPN
cana-525	133	6	ʈ̃2𝑙𝜂ɠ(𝑒	ʈ̃2𝑙𝜂ɠ(𝑒	NUM
cana-525	133	7	)	)	PUNCT
cana-525	133	8	,	,	PUNCT
cana-525	133	9	ʈ̃2𝑙𝜂ɠ(𝑒	ʈ̃2𝑙𝜂ɠ(𝑒	X
cana-525	133	10	)	)	PUNCT
cana-525	133	11	〉	〉	NOUN
cana-525	133	12	̃	̃	NOUN
cana-525	133	13	=	=	SYM
cana-525	133	14	̃	̃	NOUN
cana-525	133	15	〈	〈	NOUN
cana-525	133	16	ʈ̃4𝑙𝜂ɠ(𝑒	ʈ̃4𝑙𝜂ɠ(𝑒	NUM
cana-525	133	17	)	)	PUNCT
cana-525	133	18	,	,	PUNCT
cana-525	133	19	ʈ̃2𝑙𝜂ɠ(𝑒	ʈ̃2𝑙𝜂ɠ(𝑒	X
cana-525	133	20	)	)	PUNCT
cana-525	133	21	〉	〉	NOUN
cana-525	133	22	̃	̃	NOUN
cana-525	133	23	≤̃	≤̃	NOUN
cana-525	133	24	‖ʈ̃4𝑙𝜂ɠ(𝑒	‖ʈ̃4𝑙𝜂ɠ(𝑒	NUM
cana-525	133	25	)	)	PUNCT
cana-525	133	26	‖	‖	PROPN
cana-525	133	27	̃	̃	PROPN
cana-525	133	28	‖	‖	ADJ
cana-525	133	29	ʈ̃2𝑙𝜂ɠ(𝑒	ʈ̃2𝑙𝜂ɠ(𝑒	NUM
cana-525	133	30	)	)	PUNCT
cana-525	133	31	‖	‖	PROPN
cana-525	133	32	̃	̃	PROPN
cana-525	133	33	‖ʈ̃3𝑙𝜂ɠ(𝑒	‖ʈ̃3𝑙𝜂ɠ(𝑒	NOUN
cana-525	133	34	)	)	PUNCT
cana-525	133	35	‖	‖	PROPN
cana-525	133	36	̃	̃	PROPN
cana-525	133	37	2	2	NUM
cana-525	133	38	≥̃	≥̃	X
cana-525	133	39	‖ʈ̃𝑙𝜂ɠ(𝑒	‖ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	133	40	)	)	PUNCT
cana-525	133	41	̃	̃	NOUN
cana-525	133	42	‖	‖	ADJ
cana-525	133	43	4	4	NUM
cana-525	133	44	‖ʈ̃𝑙𝜂ɠ(𝑒	‖ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	133	45	)	)	PUNCT
cana-525	133	46	̃	̃	NOUN
cana-525	133	47	‖	‖	ADJ
cana-525	133	48	2	2	NUM
cana-525	133	49	(	(	PUNCT
cana-525	133	50	since	since	SCONJ
cana-525	133	51	ʈ̃	ʈ̃	PROPN
cana-525	133	52	is	be	AUX
cana-525	133	53	fspn	fspn	ADJ
cana-525	133	54	operator	operator	NOUN
cana-525	133	55	)	)	PUNCT
cana-525	133	56	⇒	⇒	NOUN
cana-525	133	57	‖ʈ̃3𝑙𝜂ɠ(𝑒	‖ʈ̃3𝑙𝜂ɠ(𝑒	ADV
cana-525	133	58	)	)	PUNCT
cana-525	133	59	‖	‖	PROPN
cana-525	133	60	̃	̃	PROPN
cana-525	133	61	≥̃	≥̃	X
cana-525	133	62	‖ʈ̃𝑙𝜂ɠ(𝑒	‖ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	133	63	)	)	PUNCT
cana-525	134	1	̃	̃	NOUN
cana-525	134	2	‖	‖	ADJ
cana-525	134	3	2	2	NUM
cana-525	134	4	‖ʈ̃	‖ʈ̃	NUM
cana-525	134	5	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PRON
cana-525	134	6	)	)	PUNCT
cana-525	135	1	‖	‖	PROPN
cana-525	135	2	̃	̃	PROPN
cana-525	135	3	hence	hence	ADV
cana-525	135	4	‖ʈ̃3𝑙𝜂ɠ(𝑒	‖ʈ̃3𝑙𝜂ɠ(𝑒	ADV
cana-525	135	5	)	)	PUNCT
cana-525	136	1	‖	‖	PROPN
cana-525	136	2	̃	̃	PROPN
cana-525	136	3	≥̃	≥̃	X
cana-525	136	4	‖ʈ̃2𝑙𝜂ɠ(𝑒	‖ʈ̃2𝑙𝜂ɠ(𝑒	ADJ
cana-525	136	5	)	)	PUNCT
cana-525	136	6	̃	̃	ADV
cana-525	136	7	‖	‖	ADJ
cana-525	136	8	‖ʈ̃	‖ʈ̃	PUNCT
cana-525	136	9	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PRON
cana-525	136	10	)	)	PUNCT
cana-525	136	11	‖	‖	PROPN
cana-525	136	12	̃	̃	PROPN
cana-525	136	13	theorem	theorem	ADJ
cana-525	136	14	3.4	3.4	NUM
cana-525	136	15	:	:	PUNCT
cana-525	136	16	let	let	VERB
cana-525	136	17	�	�	PROPN
cana-525	136	18	̃	̃	PROPN
cana-525	136	19	�	�	PROPN
cana-525	136	20	be	be	AUX
cana-525	136	21	a	a	DET
cana-525	136	22	fs	fs	ADP
cana-525	136	23	hilbert	hilbert	NOUN
cana-525	136	24	space	space	NOUN
cana-525	136	25	and	and	CCONJ
cana-525	136	26	let	let	VERB
cana-525	136	27	ʈ̃	ʈ̃	PROPN
cana-525	136	28	∈	∈	PROPN
cana-525	136	29	�	�	PROPN
cana-525	136	30	̃	̃	PROPN
cana-525	136	31	�	�	PROPN
cana-525	136	32	(	(	PUNCT
cana-525	136	33	�	�	PROPN
cana-525	136	34	̃	̃	PROPN
cana-525	136	35	�	�	PROPN
cana-525	136	36	)	)	PUNCT
cana-525	136	37	be	be	AUX
cana-525	136	38	a	a	DET
cana-525	136	39	fspn	fspn	ADJ
cana-525	136	40	operator	operator	NOUN
cana-525	136	41	.	.	PUNCT
cana-525	137	1	then	then	ADV
cana-525	137	2	‖ʈ̃𝑘+1𝑙𝜂ɠ(𝑒	‖ʈ̃𝑘+1𝑙𝜂ɠ(𝑒	NOUN
cana-525	137	3	)	)	PUNCT
cana-525	138	1	‖	‖	PROPN
cana-525	138	2	̃	̃	PROPN
cana-525	138	3	2	2	NUM
cana-525	138	4	≥̃	≥̃	X
cana-525	138	5	‖ʈ̃𝑘𝑙𝜂ɠ(𝑒	‖ʈ̃𝑘𝑙𝜂ɠ(𝑒	PUNCT
cana-525	138	6	)	)	PUNCT
cana-525	138	7	‖	‖	PROPN
cana-525	139	1	̃	̃	PROPN
cana-525	139	2	2	2	NUM
cana-525	139	3	‖ʈ̃2𝑙𝜂ɠ(𝑒	‖ʈ̃2𝑙𝜂ɠ(𝑒	ADJ
cana-525	139	4	)	)	PUNCT
cana-525	139	5	̃	̃	NOUN
cana-525	139	6	‖	‖	ADJ
cana-525	139	7	for	for	ADP
cana-525	139	8	every	every	DET
cana-525	139	9	positive	positive	ADJ
cana-525	139	10	integer	integer	NOUN
cana-525	139	11	𝑘	𝑘	PRON
cana-525	139	12	≥	≥	NOUN
cana-525	139	13	1	1	NUM
cana-525	139	14	and	and	CCONJ
cana-525	139	15	for	for	ADP
cana-525	139	16	every	every	DET
cana-525	139	17	unit	unit	NOUN
cana-525	139	18	vector	vector	NOUN
cana-525	139	19	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	139	20	)	)	PUNCT
cana-525	139	21	in	in	ADP
cana-525	139	22	�	�	PROPN
cana-525	139	23	̃	̃	PROPN
cana-525	139	24	�	�	PROPN
cana-525	139	25	.	.	PUNCT
cana-525	140	1	proof	proof	NOUN
cana-525	140	2	:	:	PUNCT
cana-525	140	3	let	let	VERB
cana-525	140	4	ʈ̃	ʈ̃	PROPN
cana-525	140	5	∈	∈	PROPN
cana-525	140	6	�	�	PROPN
cana-525	140	7	̃	̃	PROPN
cana-525	140	8	�	�	PROPN
cana-525	140	9	(	(	PUNCT
cana-525	140	10	�	�	PROPN
cana-525	140	11	̃	̃	PROPN
cana-525	140	12	�	�	PROPN
cana-525	140	13	)	)	PUNCT
cana-525	140	14	be	be	AUX
cana-525	140	15	a	a	DET
cana-525	140	16	fspn	fspn	ADJ
cana-525	140	17	operator	operator	NOUN
cana-525	140	18	by	by	ADP
cana-525	140	19	using	use	VERB
cana-525	140	20	the	the	DET
cana-525	140	21	induction	induction	NOUN
cana-525	140	22	hypothesis	hypothesis	NOUN
cana-525	140	23	,	,	PUNCT
cana-525	140	24	we	we	PRON
cana-525	140	25	will	will	AUX
cana-525	140	26	prove	prove	VERB
cana-525	140	27	the	the	DET
cana-525	140	28	theorem	theorem	NOUN
cana-525	140	29	.	.	PROPN
cana-525	141	1	for	for	ADP
cana-525	141	2	the	the	DET
cana-525	141	3	case	case	NOUN
cana-525	141	4	𝑘	𝑘	X
cana-525	141	5	=	=	SYM
cana-525	141	6	1	1	NUM
cana-525	141	7	,	,	PUNCT
cana-525	141	8	communications	communication	NOUN
cana-525	141	9	on	on	ADP
cana-525	141	10	applied	apply	VERB
cana-525	141	11	nonlinear	nonlinear	ADJ
cana-525	141	12	analysis	analysis	NOUN
cana-525	141	13	issn	issn	NOUN
cana-525	141	14	:	:	PUNCT
cana-525	141	15	1074	1074	NUM
cana-525	141	16	-	-	PUNCT
cana-525	141	17	133x	133x	NUM
cana-525	141	18	vol	vol	NOUN
cana-525	141	19	31	31	NUM
cana-525	141	20	no	no	NOUN
cana-525	141	21	.	.	NOUN
cana-525	141	22	2	2	NUM
cana-525	141	23	(	(	PUNCT
cana-525	141	24	2024	2024	NUM
cana-525	141	25	)	)	PUNCT
cana-525	142	1	135	135	NUM
cana-525	142	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-525	142	3	‖ʈ̃2𝑙𝜂ɠ(𝑒	‖ʈ̃2𝑙𝜂ɠ(𝑒	PROPN
cana-525	142	4	)	)	PUNCT
cana-525	142	5	‖	‖	PROPN
cana-525	142	6	̃	̃	PROPN
cana-525	142	7	2	2	NUM
cana-525	142	8	≥̃	≥̃	X
cana-525	142	9	‖ʈ̃𝑙𝜂ɠ(𝑒	‖ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	142	10	)	)	PUNCT
cana-525	142	11	̃	̃	NOUN
cana-525	142	12	‖	‖	ADJ
cana-525	142	13	2	2	NUM
cana-525	142	14	‖ʈ̃2𝑙𝜂ɠ(𝑒	‖ʈ̃2𝑙𝜂ɠ(𝑒	ADJ
cana-525	142	15	)	)	PUNCT
cana-525	142	16	̃	̃	NOUN
cana-525	142	17	‖	‖	ADJ
cana-525	142	18	now	now	ADV
cana-525	142	19	suppose	suppose	VERB
cana-525	142	20	that	that	SCONJ
cana-525	142	21	‖ʈ̃𝑘+1𝑙𝜂ɠ(𝑒	‖ʈ̃𝑘+1𝑙𝜂ɠ(𝑒	NOUN
cana-525	142	22	)	)	PUNCT
cana-525	143	1	‖	‖	PROPN
cana-525	143	2	̃	̃	PROPN
cana-525	143	3	2	2	NUM
cana-525	143	4	≥̃	≥̃	X
cana-525	143	5	‖ʈ̃𝑘𝑙𝜂ɠ(𝑒	‖ʈ̃𝑘𝑙𝜂ɠ(𝑒	PUNCT
cana-525	143	6	)	)	PUNCT
cana-525	143	7	‖	‖	PROPN
cana-525	144	1	̃	̃	PROPN
cana-525	144	2	2	2	NUM
cana-525	144	3	‖ʈ̃2𝑙𝜂ɠ(𝑒	‖ʈ̃2𝑙𝜂ɠ(𝑒	ADJ
cana-525	144	4	)	)	PUNCT
cana-525	145	1	̃	̃	NOUN
cana-525	145	2	‖	‖	ADJ
cana-525	145	3	is	be	AUX
cana-525	145	4	valid	valid	ADJ
cana-525	145	5	for	for	ADP
cana-525	145	6	k.	k.	PROPN
cana-525	145	7	then	then	ADV
cana-525	145	8	𝑘	𝑘	PROPN
cana-525	145	9	=	=	PUNCT
cana-525	145	10	𝑘	𝑘	PROPN
cana-525	145	11	+	+	ADJ
cana-525	145	12	1	1	NUM
cana-525	145	13	‖ʈ̃𝑘+2𝑙𝜂ɠ(𝑒	‖ʈ̃𝑘+2𝑙𝜂ɠ(𝑒	NOUN
cana-525	145	14	)	)	PUNCT
cana-525	146	1	‖	‖	PROPN
cana-525	146	2	̃	̃	ADJ
cana-525	146	3	2	2	NUM
cana-525	146	4	=	=	SYM
cana-525	146	5	̃	̃	NOUN
cana-525	146	6	〈	〈	NOUN
cana-525	146	7	ʈ̃𝑘+2𝑙𝜂ɠ(𝑒	ʈ̃𝑘+2𝑙𝜂ɠ(𝑒	NOUN
cana-525	146	8	)	)	PUNCT
cana-525	146	9	,	,	PUNCT
cana-525	146	10	ʈ̃𝑘+2𝑙𝜂ɠ(𝑒	ʈ̃𝑘+2𝑙𝜂ɠ(𝑒	X
cana-525	146	11	)	)	PUNCT
cana-525	146	12	〉	〉	NOUN
cana-525	146	13	̃	̃	NOUN
cana-525	146	14	=	=	SYM
cana-525	146	15	̃	̃	NOUN
cana-525	146	16	〈	〈	NOUN
cana-525	146	17	(	(	PUNCT
cana-525	146	18	ʈ̃𝑘	ʈ̃𝑘	NOUN
cana-525	146	19	)	)	PUNCT
cana-525	146	20	∗̃	∗̃	X
cana-525	146	21	ʈ̃𝑘+2𝑙𝜂ɠ(𝑒	ʈ̃𝑘+2𝑙𝜂ɠ(𝑒	NOUN
cana-525	146	22	)	)	PUNCT
cana-525	146	23	,	,	PUNCT
cana-525	146	24	ʈ̃2𝑙𝜂ɠ(𝑒	ʈ̃2𝑙𝜂ɠ(𝑒	NUM
cana-525	146	25	)	)	PUNCT
cana-525	146	26	〉	〉	NOUN
cana-525	147	1	̃	̃	PROPN
cana-525	147	2	=	=	SYM
cana-525	147	3	̃	̃	NOUN
cana-525	147	4	〈	〈	NOUN
cana-525	147	5	(	(	PUNCT
cana-525	147	6	ʈ̃∗̃	ʈ̃∗̃	NOUN
cana-525	147	7	)	)	PUNCT
cana-525	147	8	𝑘	𝑘	PRON
cana-525	147	9	ʈ̃𝑘+2𝑙𝜂ɠ(𝑒	ʈ̃𝑘+2𝑙𝜂ɠ(𝑒	NOUN
cana-525	147	10	)	)	PUNCT
cana-525	147	11	,	,	PUNCT
cana-525	147	12	ʈ̃2𝑙𝜂ɠ(𝑒	ʈ̃2𝑙𝜂ɠ(𝑒	NUM
cana-525	147	13	)	)	PUNCT
cana-525	147	14	〉	〉	NOUN
cana-525	148	1	̃	̃	PROPN
cana-525	148	2	=	=	SYM
cana-525	148	3	̃	̃	NOUN
cana-525	148	4	〈	〈	NOUN
cana-525	148	5	ʈ̃2𝑘+2𝑙𝜂ɠ(𝑒	ʈ̃2𝑘+2𝑙𝜂ɠ(𝑒	NOUN
cana-525	148	6	)	)	PUNCT
cana-525	148	7	,	,	PUNCT
cana-525	148	8	ʈ̃2𝑙𝜂ɠ(𝑒	ʈ̃2𝑙𝜂ɠ(𝑒	X
cana-525	148	9	)	)	PUNCT
cana-525	148	10	〉	〉	NOUN
cana-525	148	11	̃	̃	NOUN
cana-525	148	12	=	=	SYM
cana-525	148	13	̃	̃	NOUN
cana-525	148	14	〈	〈	NOUN
cana-525	148	15	ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒	ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒	NOUN
cana-525	148	16	)	)	PUNCT
cana-525	148	17	,	,	PUNCT
cana-525	148	18	ʈ̃2𝑙𝜂ɠ(𝑒	ʈ̃2𝑙𝜂ɠ(𝑒	X
cana-525	148	19	)	)	PUNCT
cana-525	148	20	〉	〉	NOUN
cana-525	148	21	̃	̃	NOUN
cana-525	148	22	≤̃	≤̃	ADJ
cana-525	148	23	‖ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒	‖ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒	NOUN
cana-525	148	24	)	)	PUNCT
cana-525	148	25	‖	‖	PROPN
cana-525	148	26	̃	̃	PROPN
cana-525	148	27	‖ʈ̃2𝑙𝜂ɠ(𝑒	‖ʈ̃2𝑙𝜂ɠ(𝑒	PROPN
cana-525	148	28	)	)	PUNCT
cana-525	149	1	‖	‖	PROPN
cana-525	149	2	̃	̃	NOUN
cana-525	149	3	since	since	SCONJ
cana-525	149	4	‖ʈ2𝑙𝜂ɠ(𝑒	‖ʈ2𝑙𝜂ɠ(𝑒	NOUN
cana-525	149	5	)	)	PUNCT
cana-525	149	6	̃	̃	PROPN
cana-525	149	7	‖	‖	ADJ
cana-525	149	8	≥̃	≥̃	X
cana-525	149	9	‖ʈ̃𝑙𝜂ɠ(𝑒	‖ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	149	10	)	)	PUNCT
cana-525	149	11	̃	̃	NOUN
cana-525	149	12	‖	‖	ADJ
cana-525	149	13	2	2	NUM
cana-525	149	14	‖𝑙𝜂ɠ(𝑒	‖𝑙𝜂ɠ(𝑒	PRON
cana-525	149	15	)	)	PUNCT
cana-525	149	16	̃‖	̃‖	X
cana-525	149	17	∀	∀	PUNCT
cana-525	149	18	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	NOUN
cana-525	149	19	)	)	PUNCT
cana-525	150	1	∈̃	∈̃	PROPN
cana-525	150	2	�	�	PROPN
cana-525	150	3	̃	̃	PROPN
cana-525	150	4	�	�	PROPN
cana-525	150	5	,	,	PUNCT
cana-525	150	6	‖ʈ̃𝑘+2𝑙𝜂ɠ(𝑒	‖ʈ̃𝑘+2𝑙𝜂ɠ(𝑒	X
cana-525	150	7	)	)	PUNCT
cana-525	150	8	‖	‖	PROPN
cana-525	150	9	̃	̃	PROPN
cana-525	150	10	2	2	NUM
cana-525	150	11	≥̃	≥̃	X
cana-525	150	12	‖ʈ̃𝑘+1𝑙𝜂ɠ(𝑒	‖ʈ̃𝑘+1𝑙𝜂ɠ(𝑒	NOUN
cana-525	150	13	)	)	PUNCT
cana-525	150	14	‖	‖	PROPN
cana-525	150	15	̃	̃	PROPN
cana-525	150	16	2	2	NUM
cana-525	150	17	‖ʈ̃2𝑙𝜂ɠ(𝑒	‖ʈ̃2𝑙𝜂ɠ(𝑒	PART
cana-525	150	18	)	)	PUNCT
cana-525	151	1	‖	‖	PROPN
cana-525	151	2	so	so	ADV
cana-525	151	3	𝑘	𝑘	ADP
cana-525	152	1	=	=	PUNCT
cana-525	152	2	𝑘	𝑘	PROPN
cana-525	153	1	+	+	NOUN
cana-525	153	2	1	1	NUM
cana-525	153	3	is	be	AUX
cana-525	153	4	valid	valid	ADJ
cana-525	153	5	and	and	CCONJ
cana-525	153	6	the	the	DET
cana-525	153	7	proof	proof	NOUN
cana-525	153	8	is	be	AUX
cana-525	153	9	complete	complete	ADJ
cana-525	153	10	by	by	ADP
cana-525	153	11	the	the	DET
cana-525	153	12	mathematical	mathematical	ADJ
cana-525	153	13	induction	induction	NOUN
cana-525	153	14	.	.	PUNCT
cana-525	154	1	lemma	lemma	PROPN
cana-525	154	2	3.5	3.5	NUM
cana-525	154	3	:	:	PUNCT
cana-525	154	4	let	let	VERB
cana-525	154	5	ʈ̃	ʈ̃	PROPN
cana-525	154	6	∈	∈	PROPN
cana-525	154	7	�	�	PROPN
cana-525	154	8	̃	̃	PROPN
cana-525	154	9	�	�	PROPN
cana-525	154	10	(	(	PUNCT
cana-525	154	11	�	�	PROPN
cana-525	154	12	̃	̃	PROPN
cana-525	154	13	�	�	PROPN
cana-525	154	14	)	)	PUNCT
cana-525	154	15	be	be	AUX
cana-525	154	16	a	a	DET
cana-525	154	17	fspn	fspn	ADJ
cana-525	154	18	operator	operator	NOUN
cana-525	154	19	.	.	PUNCT
cana-525	155	1	then	then	ADV
cana-525	155	2	ʈ̃𝑛	ʈ̃𝑛	VERB
cana-525	155	3	is	be	AUX
cana-525	155	4	also	also	ADV
cana-525	155	5	fspn	fspn	ADJ
cana-525	155	6	for	for	ADP
cana-525	155	7	every	every	DET
cana-525	155	8	integer	integer	NOUN
cana-525	155	9	𝑛	𝑛	PRON
cana-525	155	10	≥	≥	NUM
cana-525	155	11	1	1	NUM
cana-525	155	12	proof	proof	NOUN
cana-525	155	13	:	:	PUNCT
cana-525	155	14	it	it	PRON
cana-525	155	15	is	be	AUX
cana-525	155	16	sufficient	sufficient	ADJ
cana-525	155	17	to	to	PART
cana-525	155	18	show	show	VERB
cana-525	155	19	that	that	SCONJ
cana-525	155	20	if	if	SCONJ
cana-525	155	21	ʈ̃	ʈ̃	PROPN
cana-525	155	22	and	and	CCONJ
cana-525	155	23	ʈ̃𝑘	ʈ̃𝑘	NOUN
cana-525	155	24	is	be	AUX
cana-525	155	25	a	a	DET
cana-525	155	26	fspn	fspn	NOUN
cana-525	155	27	then	then	ADV
cana-525	155	28	ʈ̃𝑘+1	ʈ̃𝑘+1	PROPN
cana-525	155	29	is	be	AUX
cana-525	155	30	also	also	ADV
cana-525	155	31	fspn	fspn	ADJ
cana-525	155	32	for	for	ADP
cana-525	155	33	every	every	DET
cana-525	155	34	unit	unit	NOUN
cana-525	155	35	vector	vector	NOUN
cana-525	155	36	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	155	37	)	)	PUNCT
cana-525	155	38	in	in	ADP
cana-525	155	39	�	�	PROPN
cana-525	155	40	̃	̃	ADP
cana-525	155	41	�	�	PROPN
cana-525	155	42	let	let	VERB
cana-525	155	43	‖ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒	‖ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒	NOUN
cana-525	155	44	)	)	PUNCT
cana-525	156	1	‖	‖	PROPN
cana-525	157	1	̃	̃	ADJ
cana-525	157	2	2	2	NUM
cana-525	157	3	=	=	SYM
cana-525	157	4	̃	̃	NOUN
cana-525	157	5	〈	〈	NOUN
cana-525	157	6	ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒	ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒	NOUN
cana-525	157	7	)	)	PUNCT
cana-525	157	8	,	,	PUNCT
cana-525	157	9	ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒	ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒	NOUN
cana-525	157	10	)	)	PUNCT
cana-525	157	11	〉	〉	NOUN
cana-525	157	12	̃	̃	NOUN
cana-525	157	13	=	=	SYM
cana-525	157	14	̃	̃	NOUN
cana-525	157	15	〈	〈	NOUN
cana-525	157	16	(	(	PUNCT
cana-525	157	17	ʈ̃2(𝑘+1	ʈ̃2(𝑘+1	NOUN
cana-525	157	18	)	)	PUNCT
cana-525	157	19	)	)	PUNCT
cana-525	158	1	∗̃	∗̃	NUM
cana-525	158	2	ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒	ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒	NOUN
cana-525	158	3	)	)	PUNCT
cana-525	158	4	,	,	PUNCT
cana-525	158	5	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	158	6	)	)	PUNCT
cana-525	158	7	〉	〉	NOUN
cana-525	159	1	̃	̃	PROPN
cana-525	159	2	=	=	SYM
cana-525	159	3	̃	̃	NOUN
cana-525	159	4	〈	〈	NOUN
cana-525	159	5	(	(	PUNCT
cana-525	159	6	ʈ̃∗̃	ʈ̃∗̃	NOUN
cana-525	159	7	)	)	PUNCT
cana-525	159	8	2(𝑘+1	2(𝑘+1	NUM
cana-525	159	9	)	)	PUNCT
cana-525	159	10	ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒	ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒	NOUN
cana-525	159	11	)	)	PUNCT
cana-525	159	12	,	,	PUNCT
cana-525	159	13	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PRON
cana-525	159	14	)	)	PUNCT
cana-525	159	15	〉	〉	NOUN
cana-525	160	1	̃	̃	PROPN
cana-525	160	2	=	=	SYM
cana-525	160	3	̃	̃	NOUN
cana-525	160	4	〈	〈	NOUN
cana-525	160	5	ʈ̃4𝑘+4𝑙𝜂ɠ(𝑒	ʈ̃4𝑘+4𝑙𝜂ɠ(𝑒	NOUN
cana-525	160	6	)	)	PUNCT
cana-525	160	7	,	,	PUNCT
cana-525	160	8	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	160	9	)	)	PUNCT
cana-525	160	10	〉	〉	NOUN
cana-525	160	11	̃	̃	NOUN
cana-525	160	12	=	=	SYM
cana-525	160	13	̃	̃	NOUN
cana-525	160	14	〈	〈	NOUN
cana-525	160	15	ʈ̃4(𝑘+1)𝑙𝜂ɠ(𝑒	ʈ̃4(𝑘+1)𝑙𝜂ɠ(𝑒	X
cana-525	160	16	)	)	PUNCT
cana-525	160	17	,	,	PUNCT
cana-525	160	18	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	160	19	)	)	PUNCT
cana-525	160	20	〉	〉	NOUN
cana-525	160	21	̃	̃	NOUN
cana-525	160	22	communications	communication	NOUN
cana-525	160	23	on	on	ADP
cana-525	160	24	applied	apply	VERB
cana-525	160	25	nonlinear	nonlinear	ADJ
cana-525	160	26	analysis	analysis	NOUN
cana-525	160	27	issn	issn	NOUN
cana-525	160	28	:	:	PUNCT
cana-525	160	29	1074	1074	NUM
cana-525	160	30	-	-	PUNCT
cana-525	160	31	133x	133x	NUM
cana-525	160	32	vol	vol	NOUN
cana-525	160	33	31	31	NUM
cana-525	160	34	no	no	NOUN
cana-525	160	35	.	.	NOUN
cana-525	160	36	2	2	NUM
cana-525	160	37	(	(	PUNCT
cana-525	160	38	2024	2024	NUM
cana-525	160	39	)	)	PUNCT
cana-525	160	40	136	136	NUM
cana-525	160	41	https://internationalpubls.com	https://internationalpubls.com	X
cana-525	160	42	≤̃	≤̃	NOUN
cana-525	160	43	‖ʈ̃4(𝑘+1)𝑙𝜂ɠ(𝑒	‖ʈ̃4(𝑘+1)𝑙𝜂ɠ(𝑒	X
cana-525	160	44	)	)	PUNCT
cana-525	160	45	‖	‖	PROPN
cana-525	160	46	̃	̃	PROPN
cana-525	160	47	‖𝑙𝜂ɠ(𝑒	‖𝑙𝜂ɠ(𝑒	NOUN
cana-525	160	48	)	)	PUNCT
cana-525	160	49	‖	‖	PROPN
cana-525	160	50	̃	̃	PROPN
cana-525	160	51	≤̃	≤̃	ADJ
cana-525	160	52	‖ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒	‖ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒	NOUN
cana-525	160	53	)	)	PUNCT
cana-525	160	54	‖	‖	PROPN
cana-525	160	55	̃	̃	NOUN
cana-525	160	56	‖ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒	‖ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒	NOUN
cana-525	160	57	)	)	PUNCT
cana-525	160	58	‖	‖	PROPN
cana-525	160	59	̃	̃	PROPN
cana-525	160	60	‖𝑙𝜂ɠ(𝑒	‖𝑙𝜂ɠ(𝑒	NOUN
cana-525	160	61	)	)	PUNCT
cana-525	160	62	̃‖	̃‖	VERB
cana-525	160	63	ie	ie	NOUN
cana-525	160	64	)	)	PUNCT
cana-525	160	65	‖ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒	‖ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒	NOUN
cana-525	160	66	)	)	PUNCT
cana-525	161	1	‖	‖	PROPN
cana-525	162	1	̃	̃	PROPN
cana-525	162	2	2	2	NUM
cana-525	162	3	≥̃	≥̃	X
cana-525	162	4	‖ʈ̃𝑘+1𝑙𝜂ɠ(𝑒	‖ʈ̃𝑘+1𝑙𝜂ɠ(𝑒	NOUN
cana-525	162	5	)	)	PUNCT
cana-525	162	6	‖	‖	PROPN
cana-525	163	1	̃	̃	PROPN
cana-525	163	2	4	4	NUM
cana-525	163	3	‖𝑙𝜂ɠ(𝑒	‖𝑙𝜂ɠ(𝑒	PRON
cana-525	163	4	)	)	PUNCT
cana-525	164	1	‖	‖	PROPN
cana-525	164	2	implies	imply	VERB
cana-525	164	3	that	that	SCONJ
cana-525	164	4	‖ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒	‖ʈ̃2(𝑘+1)𝑙𝜂ɠ(𝑒	NOUN
cana-525	164	5	)	)	PUNCT
cana-525	164	6	̃	̃	ADP
cana-525	164	7	‖	‖	ADJ
cana-525	164	8	≥̃	≥̃	X
cana-525	164	9	‖ʈ̃𝑘+1𝑙𝜂ɠ(𝑒	‖ʈ̃𝑘+1𝑙𝜂ɠ(𝑒	NOUN
cana-525	164	10	)	)	PUNCT
cana-525	164	11	‖	‖	PROPN
cana-525	164	12	̃	̃	PROPN
cana-525	164	13	2	2	NUM
cana-525	164	14	by	by	ADP
cana-525	164	15	the	the	DET
cana-525	164	16	above	above	ADJ
cana-525	164	17	lemma	lemma	PROPN
cana-525	164	18	,	,	PUNCT
cana-525	164	19	so	so	SCONJ
cana-525	164	20	ʈ̃(𝑘+1	ʈ̃(𝑘+1	NOUN
cana-525	164	21	)	)	PUNCT
cana-525	164	22	is	be	AUX
cana-525	164	23	also	also	ADV
cana-525	164	24	fspn	fspn	ADJ
cana-525	164	25	.	.	PUNCT
cana-525	165	1	theorem	theorem	VERB
cana-525	165	2	3.6	3.6	NUM
cana-525	165	3	:	:	PUNCT
cana-525	165	4	let	let	VERB
cana-525	165	5	ʈ̃	ʈ̃	PROPN
cana-525	165	6	∈	∈	PROPN
cana-525	165	7	�	�	PROPN
cana-525	165	8	̃	̃	PROPN
cana-525	165	9	�	�	PROPN
cana-525	165	10	(	(	PUNCT
cana-525	165	11	�	�	PROPN
cana-525	165	12	̃	̃	PROPN
cana-525	165	13	�	�	PROPN
cana-525	165	14	)	)	PUNCT
cana-525	165	15	is	be	AUX
cana-525	165	16	a	a	DET
cana-525	165	17	self	self	NOUN
cana-525	165	18	-	-	PUNCT
cana-525	165	19	adjoint	adjoint	NOUN
cana-525	165	20	fuzzy	fuzzy	ADJ
cana-525	165	21	soft	soft	ADJ
cana-525	165	22	operator	operator	NOUN
cana-525	165	23	then	then	ADV
cana-525	165	24	ʈ̃	ʈ̃	PROPN
cana-525	165	25	is	be	AUX
cana-525	165	26	fspn	fspn	ADJ
cana-525	165	27	.	.	PUNCT
cana-525	166	1	proof	proof	NOUN
cana-525	166	2	:	:	PUNCT
cana-525	166	3	for	for	ADP
cana-525	166	4	any	any	DET
cana-525	166	5	𝑙𝜂𝔾(𝑒	𝑙𝜂𝔾(𝑒	NOUN
cana-525	166	6	)	)	PUNCT
cana-525	166	7	in	in	ADP
cana-525	166	8	�	�	PROPN
cana-525	166	9	̃	̃	PROPN
cana-525	166	10	�	�	PROPN
cana-525	166	11	with	with	ADP
cana-525	166	12	‖𝑙𝜂ɠ(𝑒	‖𝑙𝜂ɠ(𝑒	NOUN
cana-525	166	13	)	)	PUNCT
cana-525	167	1	̃‖	̃‖	NOUN
cana-525	167	2	=	=	SYM
cana-525	167	3	̃	̃	NOUN
cana-525	167	4	1	1	NUM
cana-525	167	5	,	,	PUNCT
cana-525	167	6	we	we	PRON
cana-525	167	7	know	know	VERB
cana-525	167	8	that	that	SCONJ
cana-525	167	9	ʈ̃	ʈ̃	PROPN
cana-525	167	10	is	be	AUX
cana-525	167	11	a	a	DET
cana-525	167	12	self	self	NOUN
cana-525	167	13	-	-	PUNCT
cana-525	167	14	adjoint	adjoint	NOUN
cana-525	167	15	fuzzy	fuzzy	ADJ
cana-525	167	16	soft	soft	ADJ
cana-525	167	17	operator	operator	NOUN
cana-525	167	18	ie	ie	NOUN
cana-525	167	19	)	)	PUNCT
cana-525	167	20	ʈ̃	ʈ̃	PROPN
cana-525	167	21	=	=	PUNCT
cana-525	167	22	̃	̃	PROPN
cana-525	167	23	ʈ̃∗̃	ʈ̃∗̃	NOUN
cana-525	167	24	let	let	VERB
cana-525	167	25	‖ʈ̃𝑙𝜂ɠ(𝑒	‖ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	167	26	)	)	PUNCT
cana-525	167	27	̃	̃	NOUN
cana-525	167	28	‖	‖	ADJ
cana-525	167	29	2	2	NUM
cana-525	167	30	=	=	SYM
cana-525	167	31	̃	̃	NOUN
cana-525	167	32	〈	〈	NOUN
cana-525	167	33	ʈ̃𝑙𝜂ɠ(𝑒	ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	167	34	)	)	PUNCT
cana-525	167	35	,	,	PUNCT
cana-525	167	36	ʈ̃𝑙𝜂ɠ(𝑒	ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	167	37	)	)	PUNCT
cana-525	167	38	̃	̃	NOUN
cana-525	167	39	〉	〉	NOUN
cana-525	167	40	=	=	SYM
cana-525	167	41	̃	̃	NOUN
cana-525	167	42	〈	〈	NOUN
cana-525	167	43	ʈ̃∗̃ʈ̃𝑙𝜂ɠ(𝑒	ʈ̃∗̃ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	167	44	)	)	PUNCT
cana-525	167	45	,	,	PUNCT
cana-525	167	46	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	167	47	)	)	PUNCT
cana-525	167	48	̃	̃	NOUN
cana-525	167	49	〉	〉	NOUN
cana-525	167	50	=	=	SYM
cana-525	167	51	̃	̃	NOUN
cana-525	167	52	〈	〈	NOUN
cana-525	167	53	ʈ̃	ʈ̃	PROPN
cana-525	167	54	ʈ̃𝑙𝜂ɠ(𝑒	ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	167	55	)	)	PUNCT
cana-525	167	56	,	,	PUNCT
cana-525	167	57	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	167	58	)	)	PUNCT
cana-525	167	59	̃	̃	NOUN
cana-525	167	60	〉	〉	NOUN
cana-525	167	61	=	=	SYM
cana-525	167	62	̃	̃	NOUN
cana-525	167	63	〈	〈	NOUN
cana-525	167	64	ʈ̃2𝑙𝜂ɠ(𝑒	ʈ̃2𝑙𝜂ɠ(𝑒	NUM
cana-525	167	65	)	)	PUNCT
cana-525	167	66	,	,	PUNCT
cana-525	167	67	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	167	68	)	)	PUNCT
cana-525	167	69	̃	̃	PROPN
cana-525	167	70	〉	〉	NOUN
cana-525	167	71	≤̃	≤̃	NOUN
cana-525	167	72	‖ʈ̃2𝑙𝜂ɠ(𝑒	‖ʈ̃2𝑙𝜂ɠ(𝑒	ADV
cana-525	167	73	)	)	PUNCT
cana-525	168	1	‖	‖	PROPN
cana-525	168	2	̃	̃	PROPN
cana-525	168	3	‖𝑙𝜂ɠ(𝑒	‖𝑙𝜂ɠ(𝑒	ADJ
cana-525	168	4	)	)	PUNCT
cana-525	168	5	̃‖	̃‖	NOUN
cana-525	168	6	‖ʈ̃𝑙𝜂ɠ(𝑒	‖ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	168	7	)	)	PUNCT
cana-525	168	8	̃	̃	NOUN
cana-525	168	9	‖	‖	ADJ
cana-525	168	10	2	2	NUM
cana-525	168	11	≤̃	≤̃	NOUN
cana-525	168	12	‖ʈ̃2𝑙𝜂ɠ(𝑒	‖ʈ̃2𝑙𝜂ɠ(𝑒	ADV
cana-525	168	13	)	)	PUNCT
cana-525	168	14	‖	‖	PROPN
cana-525	168	15	̃	̃	PROPN
cana-525	168	16	‖𝑙𝜂ɠ(𝑒	‖𝑙𝜂ɠ(𝑒	NOUN
cana-525	168	17	)	)	PUNCT
cana-525	169	1	̃‖	̃‖	NOUN
cana-525	169	2	implies	imply	VERB
cana-525	169	3	that	that	SCONJ
cana-525	169	4	‖ʈ̃𝑙𝜂ɠ(𝑒	‖ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	169	5	)	)	PUNCT
cana-525	169	6	̃	̃	NOUN
cana-525	169	7	‖	‖	ADJ
cana-525	169	8	2	2	NUM
cana-525	169	9	≤̃	≤̃	NOUN
cana-525	169	10	‖ʈ̃2𝑙𝜂ɠ(𝑒	‖ʈ̃2𝑙𝜂ɠ(𝑒	ADV
cana-525	169	11	)	)	PUNCT
cana-525	169	12	‖	‖	PROPN
cana-525	170	1	̃	̃	PROPN
cana-525	170	2	so	so	SCONJ
cana-525	170	3	ʈ̃	ʈ̃	PROPN
cana-525	170	4	is	be	AUX
cana-525	170	5	fspn	fspn	ADJ
cana-525	170	6	.	.	PUNCT
cana-525	171	1	theorem	theorem	VERB
cana-525	171	2	3.7	3.7	NUM
cana-525	171	3	:	:	PUNCT
cana-525	171	4	let	let	VERB
cana-525	171	5	ʈ̃	ʈ̃	PROPN
cana-525	171	6	∈	∈	PROPN
cana-525	171	7	�	�	PROPN
cana-525	171	8	̃	̃	PROPN
cana-525	171	9	�	�	PROPN
cana-525	171	10	(	(	PUNCT
cana-525	171	11	�	�	PROPN
cana-525	171	12	̃	̃	PROPN
cana-525	171	13	�	�	PROPN
cana-525	171	14	)	)	PUNCT
cana-525	171	15	be	be	AUX
cana-525	171	16	fspn	fspn	ADJ
cana-525	171	17	and	and	CCONJ
cana-525	171	18	fuzzy	fuzzy	ADJ
cana-525	171	19	soft	soft	ADJ
cana-525	171	20	self	self	NOUN
cana-525	171	21	adjoint	adjoint	NOUN
cana-525	171	22	operator	operator	NOUN
cana-525	171	23	then	then	ADV
cana-525	171	24	ʈ̃∗̃	ʈ̃∗̃	NOUN
cana-525	171	25	is	be	AUX
cana-525	171	26	fspn	fspn	ADJ
cana-525	171	27	.	.	PUNCT
cana-525	172	1	proof	proof	NOUN
cana-525	172	2	:	:	PUNCT
cana-525	172	3	for	for	ADP
cana-525	172	4	any	any	DET
cana-525	172	5	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	NOUN
cana-525	172	6	)	)	PUNCT
cana-525	172	7	∈̃	∈̃	PROPN
cana-525	172	8	�	�	PROPN
cana-525	172	9	̃	̃	PROPN
cana-525	172	10	�	�	PROPN
cana-525	172	11	,	,	PUNCT
cana-525	172	12	‖𝑙𝜂ɠ(𝑒	‖𝑙𝜂ɠ(𝑒	ADJ
cana-525	172	13	)	)	PUNCT
cana-525	172	14	̃‖	̃‖	NOUN
cana-525	172	15	=	=	SYM
cana-525	172	16	̃	̃	PROPN
cana-525	172	17	1	1	NUM
cana-525	172	18	let	let	VERB
cana-525	172	19	‖ʈ̃∗̃𝑙𝜂ɠ(𝑒	‖ʈ̃∗̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	172	20	)	)	PUNCT
cana-525	173	1	̃	̃	ADV
cana-525	173	2	‖	‖	ADJ
cana-525	173	3	2	2	NUM
cana-525	173	4	=	=	SYM
cana-525	173	5	̃	̃	NOUN
cana-525	173	6	〈	〈	NOUN
cana-525	173	7	ʈ̃∗̃𝑙𝜂ɠ(𝑒	ʈ̃∗̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	173	8	)	)	PUNCT
cana-525	173	9	,	,	PUNCT
cana-525	173	10	ʈ̃∗̃𝑙𝜂ɠ(𝑒	ʈ̃∗̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	173	11	)	)	PUNCT
cana-525	173	12	̃	̃	NOUN
cana-525	173	13	〉	〉	NOUN
cana-525	173	14	=	=	SYM
cana-525	173	15	̃	̃	NOUN
cana-525	173	16	〈	〈	NOUN
cana-525	173	17	ʈ̃ʈ̃∗̃𝑙𝜂ɠ(𝑒	ʈ̃ʈ̃∗̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	173	18	)	)	PUNCT
cana-525	173	19	,	,	PUNCT
cana-525	173	20	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	173	21	)	)	PUNCT
cana-525	173	22	̃	̃	PROPN
cana-525	173	23	〉	〉	NOUN
cana-525	173	24	communications	communication	NOUN
cana-525	173	25	on	on	ADP
cana-525	173	26	applied	apply	VERB
cana-525	173	27	nonlinear	nonlinear	ADJ
cana-525	173	28	analysis	analysis	NOUN
cana-525	173	29	issn	issn	NOUN
cana-525	173	30	:	:	PUNCT
cana-525	173	31	1074	1074	NUM
cana-525	173	32	-	-	PUNCT
cana-525	173	33	133x	133x	NUM
cana-525	173	34	vol	vol	NOUN
cana-525	173	35	31	31	NUM
cana-525	173	36	no	no	NOUN
cana-525	173	37	.	.	NOUN
cana-525	173	38	2	2	NUM
cana-525	173	39	(	(	PUNCT
cana-525	173	40	2024	2024	NUM
cana-525	173	41	)	)	PUNCT
cana-525	173	42	137	137	NUM
cana-525	174	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-525	174	2	=	=	SYM
cana-525	174	3	̃	̃	NOUN
cana-525	174	4	〈	〈	NOUN
cana-525	174	5	(	(	PUNCT
cana-525	174	6	ʈ̃∗̃	ʈ̃∗̃	NOUN
cana-525	174	7	)	)	PUNCT
cana-525	174	8	2	2	NUM
cana-525	174	9	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	NOUN
cana-525	174	10	)	)	PUNCT
cana-525	174	11	,	,	PUNCT
cana-525	174	12	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	174	13	)	)	PUNCT
cana-525	174	14	̃	̃	PROPN
cana-525	174	15	〉	〉	NOUN
cana-525	174	16	≤̃	≤̃	NOUN
cana-525	174	17	‖(ʈ̃∗̃	‖(ʈ̃∗̃	NOUN
cana-525	174	18	)	)	PUNCT
cana-525	174	19	2	2	NUM
cana-525	174	20	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PRON
cana-525	174	21	)	)	PUNCT
cana-525	174	22	‖	‖	PROPN
cana-525	174	23	̃	̃	PROPN
cana-525	174	24	‖𝑙𝜂ɠ(𝑒	‖𝑙𝜂ɠ(𝑒	ADJ
cana-525	174	25	)	)	PUNCT
cana-525	174	26	̃‖	̃‖	VERB
cana-525	174	27	‖ʈ̃∗̃𝑙𝜂ɠ(𝑒	‖ʈ̃∗̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	174	28	)	)	PUNCT
cana-525	174	29	̃	̃	ADV
cana-525	174	30	‖	‖	ADJ
cana-525	174	31	2	2	NUM
cana-525	174	32	≤̃	≤̃	NOUN
cana-525	174	33	‖(ʈ̃∗̃	‖(ʈ̃∗̃	NOUN
cana-525	174	34	)	)	PUNCT
cana-525	174	35	2	2	NUM
cana-525	174	36	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PRON
cana-525	174	37	)	)	PUNCT
cana-525	174	38	‖	‖	PROPN
cana-525	174	39	̃	̃	PROPN
cana-525	174	40	‖𝑙𝜂ɠ(𝑒	‖𝑙𝜂ɠ(𝑒	NOUN
cana-525	174	41	)	)	PUNCT
cana-525	175	1	̃‖	̃‖	NOUN
cana-525	175	2	implies	imply	VERB
cana-525	175	3	that	that	SCONJ
cana-525	175	4	‖ʈ̃∗̃𝑙𝜂ɠ(𝑒	‖ʈ̃∗̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	175	5	)	)	PUNCT
cana-525	175	6	̃	̃	ADV
cana-525	175	7	‖	‖	ADJ
cana-525	175	8	2	2	NUM
cana-525	175	9	≤̃	≤̃	NOUN
cana-525	175	10	‖(ʈ̃∗̃	‖(ʈ̃∗̃	NOUN
cana-525	175	11	)	)	PUNCT
cana-525	175	12	2	2	NUM
cana-525	175	13	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PRON
cana-525	175	14	)	)	PUNCT
cana-525	175	15	‖	‖	PROPN
cana-525	175	16	̃	̃	PROPN
cana-525	175	17	ie	ie	X
cana-525	175	18	)	)	PUNCT
cana-525	175	19	‖(ʈ̃∗̃	‖(ʈ̃∗̃	NOUN
cana-525	175	20	)	)	PUNCT
cana-525	175	21	2	2	NUM
cana-525	175	22	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PRON
cana-525	175	23	)	)	PUNCT
cana-525	175	24	‖	‖	PROPN
cana-525	176	1	̃	̃	PROPN
cana-525	176	2	≥̃	≥̃	PUNCT
cana-525	176	3	‖ʈ̃∗̃𝑙𝜂ɠ(𝑒	‖ʈ̃∗̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	176	4	)	)	PUNCT
cana-525	177	1	̃	̃	NOUN
cana-525	177	2	‖	‖	ADJ
cana-525	177	3	2	2	NUM
cana-525	177	4	therefore	therefore	ADV
cana-525	177	5	,	,	PUNCT
cana-525	177	6	ʈ̃	ʈ̃	PROPN
cana-525	177	7	is	be	AUX
cana-525	177	8	fspn	fspn	ADJ
cana-525	177	9	.	.	PUNCT
cana-525	178	1	theorem	theorem	VERB
cana-525	178	2	3.8	3.8	NUM
cana-525	178	3	:	:	PUNCT
cana-525	178	4	let	let	VERB
cana-525	178	5	₷	₷	NOUN
cana-525	178	6	̃	̃	ADJ
cana-525	178	7	and	and	CCONJ
cana-525	178	8	ʈ̃	ʈ̃	PROPN
cana-525	178	9	∈̃	∈̃	PROPN
cana-525	178	10	�	�	PROPN
cana-525	178	11	̃	̃	PROPN
cana-525	178	12	�	�	PROPN
cana-525	178	13	is	be	AUX
cana-525	178	14	a	a	DET
cana-525	178	15	fspn	fspn	ADJ
cana-525	178	16	operator	operator	NOUN
cana-525	178	17	and	and	CCONJ
cana-525	178	18	fuzzy	fuzzy	ADJ
cana-525	178	19	soft	soft	ADJ
cana-525	178	20	self	self	NOUN
cana-525	178	21	adjoint	adjoint	NOUN
cana-525	178	22	operator	operator	NOUN
cana-525	178	23	.	.	PUNCT
cana-525	179	1	then	then	ADV
cana-525	179	2	a	a	X
cana-525	179	3	)	)	PUNCT
cana-525	179	4	₷	₷	NOUN
cana-525	179	5	̃	̃	ADJ
cana-525	179	6	+	+	ADJ
cana-525	179	7	̃	̃	ADJ
cana-525	179	8	ʈ̃	ʈ̃	PROPN
cana-525	179	9	b	b	PROPN
cana-525	179	10	)	)	PUNCT
cana-525	179	11	₷	₷	NOUN
cana-525	179	12	̃	̃	NOUN
cana-525	179	13	.ʈ̃	.ʈ̃	X
cana-525	179	14	are	be	AUX
cana-525	179	15	also	also	ADV
cana-525	179	16	as	as	ADP
cana-525	179	17	fspn	fspn	ADJ
cana-525	179	18	.	.	PUNCT
cana-525	180	1	proof	proof	NOUN
cana-525	180	2	:	:	PUNCT
cana-525	180	3	for	for	ADP
cana-525	180	4	every	every	DET
cana-525	180	5	unit	unit	NOUN
cana-525	180	6	vector	vector	NOUN
cana-525	180	7	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	180	8	)	)	PUNCT
cana-525	180	9	∈̃	∈̃	PROPN
cana-525	180	10	�	�	PROPN
cana-525	180	11	̃	̃	PROPN
cana-525	180	12	�	�	PROPN
cana-525	180	13	we	we	PRON
cana-525	180	14	know	know	VERB
cana-525	180	15	that	that	SCONJ
cana-525	180	16	‖ʈ̃2𝑙𝜂ɠ(𝑒	‖ʈ̃2𝑙𝜂ɠ(𝑒	ADV
cana-525	180	17	)	)	PUNCT
cana-525	180	18	‖	‖	PROPN
cana-525	180	19	̃	̃	PROPN
cana-525	180	20	≥̃	≥̃	X
cana-525	180	21	‖ʈ̃𝑙𝜂ɠ(𝑒	‖ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	180	22	)	)	PUNCT
cana-525	180	23	̃	̃	NOUN
cana-525	180	24	‖	‖	ADJ
cana-525	180	25	2	2	NUM
cana-525	180	26	‖₷̃	‖₷̃	PROPN
cana-525	180	27	2	2	NUM
cana-525	180	28	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PRON
cana-525	180	29	)	)	PUNCT
cana-525	180	30	‖	‖	PROPN
cana-525	181	1	̃	̃	PROPN
cana-525	181	2	≥̃	≥̃	X
cana-525	181	3	‖₷̃𝑙𝜂ɠ(𝑒	‖₷̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	181	4	)	)	PUNCT
cana-525	181	5	̃	̃	NOUN
cana-525	181	6	‖	‖	ADJ
cana-525	181	7	2	2	NUM
cana-525	181	8	and	and	CCONJ
cana-525	181	9	₷	₷	NOUN
cana-525	181	10	̃	̃	ADJ
cana-525	181	11	=	=	SYM
cana-525	181	12	̃	̃	NOUN
cana-525	181	13	₷	₷	NOUN
cana-525	181	14	̃	̃	NOUN
cana-525	181	15	∗̃	∗̃	X
cana-525	181	16	,	,	PUNCT
cana-525	181	17	ʈ̃	ʈ̃	PROPN
cana-525	181	18	=	=	SYM
cana-525	181	19	̃	̃	PROPN
cana-525	181	20	ʈ̃∗̃	ʈ̃∗̃	NOUN
cana-525	181	21	a	a	NOUN
cana-525	181	22	)	)	PUNCT
cana-525	181	23	to	to	PART
cana-525	181	24	prove	prove	VERB
cana-525	181	25	that	that	SCONJ
cana-525	181	26	₷	₷	X
cana-525	181	27	̃	̃	ADJ
cana-525	181	28	+	+	ADJ
cana-525	181	29	̃	̃	NOUN
cana-525	181	30	ʈ̃	ʈ̃	PROPN
cana-525	181	31	is	be	AUX
cana-525	181	32	a	a	DET
cana-525	181	33	fspn	fspn	ADJ
cana-525	181	34	operator	operator	NOUN
cana-525	181	35	let	let	VERB
cana-525	181	36	‖(₷̃	‖(₷̃	NOUN
cana-525	181	37	+	+	ADJ
cana-525	181	38	̃	̃	ADJ
cana-525	181	39	ʈ̃)𝑙𝜂ɠ(𝑒	ʈ̃)𝑙𝜂ɠ(𝑒	NOUN
cana-525	181	40	)	)	PUNCT
cana-525	181	41	̃	̃	NOUN
cana-525	181	42	‖	‖	ADJ
cana-525	181	43	2	2	NUM
cana-525	181	44	=	=	SYM
cana-525	181	45	̃	̃	NOUN
cana-525	181	46	〈	〈	NOUN
cana-525	181	47	(	(	PUNCT
cana-525	181	48	₷	₷	NOUN
cana-525	181	49	̃	̃	ADJ
cana-525	181	50	+	+	ADJ
cana-525	181	51	̃	̃	NOUN
cana-525	181	52	ʈ̃)𝑙𝜂ɠ(𝑒	ʈ̃)𝑙𝜂ɠ(𝑒	NOUN
cana-525	181	53	)	)	PUNCT
cana-525	181	54	,	,	PUNCT
cana-525	181	55	(	(	PUNCT
cana-525	181	56	₷	₷	X
cana-525	181	57	̃	̃	ADJ
cana-525	181	58	+	+	ADJ
cana-525	181	59	̃	̃	ADJ
cana-525	181	60	ʈ̃)𝑙𝜂ɠ(𝑒	ʈ̃)𝑙𝜂ɠ(𝑒	NOUN
cana-525	181	61	)	)	PUNCT
cana-525	181	62	〉	〉	NOUN
cana-525	182	1	̃	̃	NOUN
cana-525	182	2	=	=	SYM
cana-525	182	3	̃	̃	NOUN
cana-525	182	4	〈	〈	NOUN
cana-525	182	5	(	(	PUNCT
cana-525	182	6	₷	₷	NOUN
cana-525	182	7	̃	̃	ADJ
cana-525	182	8	+	+	ADJ
cana-525	182	9	̃	̃	ADJ
cana-525	182	10	ʈ̃	ʈ̃	NOUN
cana-525	182	11	)	)	PUNCT
cana-525	182	12	∗̃	∗̃	X
cana-525	182	13	(	(	PUNCT
cana-525	182	14	₷	₷	X
cana-525	182	15	̃	̃	ADJ
cana-525	182	16	+	+	ADJ
cana-525	182	17	̃	̃	ADJ
cana-525	182	18	ʈ̃	ʈ̃	PROPN
cana-525	182	19	)	)	PUNCT
cana-525	182	20	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	182	21	)	)	PUNCT
cana-525	182	22	,	,	PUNCT
cana-525	182	23	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	182	24	)	)	PUNCT
cana-525	182	25	̃	̃	NOUN
cana-525	182	26	〉	〉	NOUN
cana-525	182	27	=	=	SYM
cana-525	182	28	̃	̃	NOUN
cana-525	182	29	〈	〈	NOUN
cana-525	182	30	(	(	PUNCT
cana-525	182	31	₷	₷	NOUN
cana-525	182	32	̃	̃	ADJ
cana-525	182	33	∗̃	∗̃	X
cana-525	182	34	+	+	ADJ
cana-525	182	35	̃	̃	NOUN
cana-525	182	36	ʈ̃∗̃	ʈ̃∗̃	NOUN
cana-525	182	37	)	)	PUNCT
cana-525	182	38	(	(	PUNCT
cana-525	182	39	₷	₷	X
cana-525	182	40	̃	̃	ADJ
cana-525	182	41	+	+	ADJ
cana-525	182	42	̃	̃	ADJ
cana-525	182	43	ʈ̃	ʈ̃	PROPN
cana-525	182	44	)	)	PUNCT
cana-525	182	45	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	182	46	)	)	PUNCT
cana-525	182	47	,	,	PUNCT
cana-525	182	48	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	182	49	)	)	PUNCT
cana-525	182	50	̃	̃	NOUN
cana-525	182	51	〉	〉	NOUN
cana-525	182	52	=	=	SYM
cana-525	182	53	̃	̃	NOUN
cana-525	182	54	〈	〈	NOUN
cana-525	182	55	(	(	PUNCT
cana-525	182	56	₷	₷	NOUN
cana-525	182	57	̃	̃	ADJ
cana-525	182	58	+	+	ADJ
cana-525	182	59	̃	̃	NOUN
cana-525	182	60	ʈ̃	ʈ̃	PROPN
cana-525	182	61	)	)	PUNCT
cana-525	182	62	(	(	PUNCT
cana-525	182	63	₷	₷	X
cana-525	182	64	̃	̃	ADJ
cana-525	182	65	+	+	ADJ
cana-525	182	66	̃	̃	ADJ
cana-525	182	67	ʈ̃	ʈ̃	PROPN
cana-525	182	68	)	)	PUNCT
cana-525	182	69	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	182	70	)	)	PUNCT
cana-525	182	71	,	,	PUNCT
cana-525	182	72	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	182	73	)	)	PUNCT
cana-525	182	74	̃	̃	PROPN
cana-525	182	75	〉	〉	NOUN
cana-525	182	76	≤̃	≤̃	VERB
cana-525	182	77	‖(₷̃	‖(₷̃	NUM
cana-525	182	78	+	+	ADJ
cana-525	182	79	̃	̃	ADJ
cana-525	182	80	ʈ̃	ʈ̃	PROPN
cana-525	182	81	)	)	PUNCT
cana-525	182	82	2	2	NUM
cana-525	182	83	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	NOUN
cana-525	182	84	)	)	PUNCT
cana-525	182	85	‖	‖	PROPN
cana-525	183	1	̃	̃	PROPN
cana-525	183	2	‖𝑙𝜂ɠ(𝑒	‖𝑙𝜂ɠ(𝑒	NOUN
cana-525	183	3	)	)	PUNCT
cana-525	184	1	̃‖	̃‖	NOUN
cana-525	184	2	implies	imply	VERB
cana-525	184	3	that	that	SCONJ
cana-525	184	4	‖(₷̃	‖(₷̃	PROPN
cana-525	184	5	+	+	ADJ
cana-525	184	6	̃	̃	ADJ
cana-525	184	7	ʈ̃)𝑙𝜂ɠ(𝑒	ʈ̃)𝑙𝜂ɠ(𝑒	NOUN
cana-525	184	8	)	)	PUNCT
cana-525	184	9	̃	̃	NOUN
cana-525	184	10	‖	‖	ADJ
cana-525	184	11	2	2	NUM
cana-525	184	12	≤̃	≤̃	NOUN
cana-525	184	13	‖(₷̃	‖(₷̃	PROPN
cana-525	184	14	+	+	PROPN
cana-525	184	15	̃	̃	ADJ
cana-525	184	16	ʈ̃	ʈ̃	PROPN
cana-525	184	17	)	)	PUNCT
cana-525	184	18	2	2	NUM
cana-525	184	19	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	NOUN
cana-525	184	20	)	)	PUNCT
cana-525	184	21	‖	‖	PROPN
cana-525	184	22	̃	̃	PROPN
cana-525	184	23	therefore	therefore	ADV
cana-525	184	24	,	,	PUNCT
cana-525	184	25	₷	₷	X
cana-525	184	26	̃	̃	ADJ
cana-525	184	27	+	+	ADJ
cana-525	184	28	̃	̃	NOUN
cana-525	184	29	ʈ̃	ʈ̃	PROPN
cana-525	184	30	is	be	AUX
cana-525	184	31	a	a	DET
cana-525	184	32	fspn	fspn	ADJ
cana-525	184	33	operator	operator	NOUN
cana-525	184	34	.	.	PUNCT
cana-525	185	1	b	b	X
cana-525	185	2	)	)	PUNCT
cana-525	185	3	to	to	PART
cana-525	185	4	prove	prove	VERB
cana-525	185	5	that	that	SCONJ
cana-525	185	6	₷	₷	NOUN
cana-525	185	7	̃.ʈ̃	̃.ʈ̃	NOUN
cana-525	185	8	is	be	AUX
cana-525	185	9	a	a	DET
cana-525	185	10	fspn	fspn	ADJ
cana-525	185	11	operator	operator	NOUN
cana-525	185	12	let	let	VERB
cana-525	185	13	‖(₷̃.	‖(₷̃.	PROPN
cana-525	185	14	ʈ̃)𝑙𝜂ɠ(𝑒	ʈ̃)𝑙𝜂ɠ(𝑒	ADJ
cana-525	185	15	)	)	PUNCT
cana-525	185	16	̃	̃	NOUN
cana-525	185	17	‖	‖	ADJ
cana-525	185	18	2	2	NUM
cana-525	185	19	=	=	SYM
cana-525	185	20	̃	̃	NOUN
cana-525	185	21	〈	〈	NOUN
cana-525	185	22	(	(	PUNCT
cana-525	185	23	₷	₷	NOUN
cana-525	185	24	̃.	̃.	NOUN
cana-525	185	25	ʈ̃)𝑙𝜂ɠ(𝑒	ʈ̃)𝑙𝜂ɠ(𝑒	NOUN
cana-525	185	26	)	)	PUNCT
cana-525	185	27	,	,	PUNCT
cana-525	185	28	(	(	PUNCT
cana-525	185	29	₷	₷	ADJ
cana-525	185	30	̃.	̃.	ADJ
cana-525	185	31	ʈ̃)𝑙𝜂ɠ(𝑒	ʈ̃)𝑙𝜂ɠ(𝑒	NOUN
cana-525	185	32	)	)	PUNCT
cana-525	185	33	〉	〉	NOUN
cana-525	186	1	̃	̃	NOUN
cana-525	186	2	=	=	SYM
cana-525	186	3	̃	̃	NOUN
cana-525	186	4	〈	〈	NOUN
cana-525	186	5	(	(	PUNCT
cana-525	186	6	₷	₷	NOUN
cana-525	186	7	̃.	̃.	PROPN
cana-525	186	8	ʈ̃	ʈ̃	PROPN
cana-525	186	9	)	)	PUNCT
cana-525	186	10	∗̃	∗̃	X
cana-525	186	11	(	(	PUNCT
cana-525	186	12	₷	₷	NOUN
cana-525	186	13	̃.	̃.	PROPN
cana-525	186	14	ʈ̃	ʈ̃	PROPN
cana-525	186	15	)	)	PUNCT
cana-525	186	16	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	186	17	)	)	PUNCT
cana-525	186	18	,	,	PUNCT
cana-525	186	19	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	186	20	)	)	PUNCT
cana-525	186	21	̃	̃	PROPN
cana-525	186	22	〉	〉	NOUN
cana-525	186	23	communications	communication	NOUN
cana-525	186	24	on	on	ADP
cana-525	186	25	applied	apply	VERB
cana-525	186	26	nonlinear	nonlinear	ADJ
cana-525	186	27	analysis	analysis	NOUN
cana-525	186	28	issn	issn	NOUN
cana-525	186	29	:	:	PUNCT
cana-525	186	30	1074	1074	NUM
cana-525	186	31	-	-	PUNCT
cana-525	186	32	133x	133x	NUM
cana-525	186	33	vol	vol	NOUN
cana-525	186	34	31	31	NUM
cana-525	186	35	no	no	NOUN
cana-525	186	36	.	.	NOUN
cana-525	186	37	2	2	NUM
cana-525	186	38	(	(	PUNCT
cana-525	186	39	2024	2024	NUM
cana-525	186	40	)	)	PUNCT
cana-525	186	41	138	138	NUM
cana-525	186	42	https://internationalpubls.com	https://internationalpubls.com	X
cana-525	186	43	=	=	SYM
cana-525	186	44	̃	̃	NOUN
cana-525	186	45	〈	〈	NOUN
cana-525	186	46	(	(	PUNCT
cana-525	186	47	ʈ̃∗̃	ʈ̃∗̃	VERB
cana-525	186	48	₷	₷	NOUN
cana-525	186	49	̃	̃	NOUN
cana-525	186	50	∗̃	∗̃	X
cana-525	186	51	)	)	PUNCT
cana-525	186	52	(	(	PUNCT
cana-525	186	53	₷	₷	X
cana-525	186	54	̃.	̃.	PROPN
cana-525	186	55	ʈ̃	ʈ̃	PROPN
cana-525	186	56	)	)	PUNCT
cana-525	186	57	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	186	58	)	)	PUNCT
cana-525	186	59	,	,	PUNCT
cana-525	186	60	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	186	61	)	)	PUNCT
cana-525	186	62	̃	̃	NOUN
cana-525	186	63	〉	〉	NOUN
cana-525	186	64	=	=	SYM
cana-525	186	65	̃	̃	NOUN
cana-525	186	66	〈	〈	PROPN
cana-525	186	67	(	(	PUNCT
cana-525	186	68	ʈ̃	ʈ̃	PROPN
cana-525	186	69	₷	₷	NOUN
cana-525	186	70	̃	̃	NOUN
cana-525	186	71	)	)	PUNCT
cana-525	186	72	(	(	PUNCT
cana-525	186	73	₷	₷	NOUN
cana-525	186	74	̃.	̃.	PROPN
cana-525	186	75	ʈ̃	ʈ̃	PROPN
cana-525	186	76	)	)	PUNCT
cana-525	186	77	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	186	78	)	)	PUNCT
cana-525	186	79	,	,	PUNCT
cana-525	186	80	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	186	81	)	)	PUNCT
cana-525	186	82	̃	̃	NOUN
cana-525	186	83	〉	〉	NOUN
cana-525	186	84	=	=	SYM
cana-525	186	85	̃	̃	NOUN
cana-525	186	86	〈	〈	NOUN
cana-525	186	87	(	(	PUNCT
cana-525	186	88	₷	₷	NOUN
cana-525	186	89	̃.	̃.	PROPN
cana-525	186	90	ʈ̃	ʈ̃	PROPN
cana-525	186	91	)	)	PUNCT
cana-525	186	92	(	(	PUNCT
cana-525	186	93	₷	₷	NOUN
cana-525	186	94	̃.	̃.	PROPN
cana-525	186	95	ʈ̃	ʈ̃	PROPN
cana-525	186	96	)	)	PUNCT
cana-525	186	97	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	186	98	)	)	PUNCT
cana-525	186	99	,	,	PUNCT
cana-525	186	100	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	186	101	)	)	PUNCT
cana-525	186	102	̃	̃	PROPN
cana-525	186	103	〉	〉	NOUN
cana-525	186	104	≤̃	≤̃	VERB
cana-525	186	105	‖(₷̃.	‖(₷̃.	PROPN
cana-525	186	106	ʈ̃	ʈ̃	PROPN
cana-525	186	107	)	)	PUNCT
cana-525	186	108	2	2	NUM
cana-525	186	109	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	NOUN
cana-525	186	110	)	)	PUNCT
cana-525	186	111	‖	‖	PROPN
cana-525	186	112	̃	̃	PROPN
cana-525	186	113	‖𝑙𝜂ɠ(𝑒	‖𝑙𝜂ɠ(𝑒	NOUN
cana-525	186	114	)	)	PUNCT
cana-525	186	115	̃‖	̃‖	VERB
cana-525	186	116	‖(₷̃.	‖(₷̃.	PROPN
cana-525	186	117	ʈ̃)𝑙𝜂ɠ(𝑒	ʈ̃)𝑙𝜂ɠ(𝑒	PROPN
cana-525	186	118	)	)	PUNCT
cana-525	186	119	̃	̃	NOUN
cana-525	186	120	‖	‖	ADJ
cana-525	186	121	2	2	NUM
cana-525	186	122	≤̃	≤̃	NOUN
cana-525	186	123	‖(₷̃.	‖(₷̃.	PROPN
cana-525	186	124	ʈ̃	ʈ̃	PROPN
cana-525	186	125	)	)	PUNCT
cana-525	186	126	2	2	NUM
cana-525	186	127	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	NOUN
cana-525	186	128	)	)	PUNCT
cana-525	186	129	‖	‖	PROPN
cana-525	186	130	̃	̃	PROPN
cana-525	186	131	‖𝑙𝜂ɠ(𝑒	‖𝑙𝜂ɠ(𝑒	NOUN
cana-525	186	132	)	)	PUNCT
cana-525	186	133	̃‖	̃‖	NOUN
cana-525	186	134	implies	imply	VERB
cana-525	186	135	that	that	SCONJ
cana-525	186	136	‖(₷̃.	‖(₷̃.	PROPN
cana-525	186	137	ʈ̃)𝑙𝜂ɠ(𝑒	ʈ̃)𝑙𝜂ɠ(𝑒	PROPN
cana-525	186	138	)	)	PUNCT
cana-525	186	139	̃	̃	NOUN
cana-525	186	140	‖	‖	ADJ
cana-525	186	141	2	2	NUM
cana-525	186	142	≤̃	≤̃	NOUN
cana-525	186	143	‖(₷̃.	‖(₷̃.	PROPN
cana-525	186	144	ʈ̃	ʈ̃	PROPN
cana-525	186	145	)	)	PUNCT
cana-525	186	146	2	2	NUM
cana-525	186	147	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PRON
cana-525	186	148	)	)	PUNCT
cana-525	187	1	‖	‖	PROPN
cana-525	187	2	̃	̃	PROPN
cana-525	187	3	therefore	therefore	ADV
cana-525	187	4	,	,	PUNCT
cana-525	187	5	₷	₷	NOUN
cana-525	187	6	̃.ʈ̃	̃.ʈ̃	NOUN
cana-525	187	7	is	be	AUX
cana-525	187	8	a	a	DET
cana-525	187	9	fspn	fspn	ADJ
cana-525	187	10	operator	operator	NOUN
cana-525	187	11	.	.	PUNCT
cana-525	188	1	theorem	theorem	VERB
cana-525	188	2	3.9	3.9	NUM
cana-525	188	3	:	:	PUNCT
cana-525	188	4	let	let	VERB
cana-525	188	5	ʈ̃	ʈ̃	PROPN
cana-525	188	6	∈	∈	PROPN
cana-525	188	7	�	�	PROPN
cana-525	188	8	̃	̃	PROPN
cana-525	188	9	�	�	PROPN
cana-525	188	10	(	(	PUNCT
cana-525	188	11	�	�	PROPN
cana-525	188	12	̃	̃	PROPN
cana-525	188	13	�	�	PROPN
cana-525	188	14	)	)	PUNCT
cana-525	188	15	is	be	AUX
cana-525	188	16	a	a	DET
cana-525	188	17	fsn	fsn	NOUN
cana-525	188	18	operator	operator	NOUN
cana-525	188	19	then	then	ADV
cana-525	188	20	ʈ̃	ʈ̃	PROPN
cana-525	188	21	is	be	AUX
cana-525	188	22	a	a	DET
cana-525	188	23	fspn	fspn	ADJ
cana-525	188	24	operator	operator	NOUN
cana-525	188	25	proof	proof	NOUN
cana-525	188	26	:	:	PUNCT
cana-525	188	27	for	for	SCONJ
cana-525	188	28	every	every	DET
cana-525	188	29	unit	unit	NOUN
cana-525	188	30	vector	vector	NOUN
cana-525	188	31	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	188	32	)	)	PUNCT
cana-525	188	33	∈̃	∈̃	PROPN
cana-525	188	34	�	�	PROPN
cana-525	188	35	̃	̃	PROPN
cana-525	188	36	�	�	PROPN
cana-525	188	37	let	let	VERB
cana-525	188	38	‖ʈ̃𝑙𝜂ɠ(𝑒	‖ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	188	39	)	)	PUNCT
cana-525	188	40	̃	̃	NOUN
cana-525	188	41	‖	‖	ADJ
cana-525	188	42	2	2	NUM
cana-525	188	43	=	=	SYM
cana-525	188	44	̃	̃	NOUN
cana-525	188	45	〈	〈	NOUN
cana-525	188	46	ʈ̃𝑙𝜂ɠ(𝑒	ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	188	47	)	)	PUNCT
cana-525	188	48	,	,	PUNCT
cana-525	189	1	ʈ̃𝑙𝜂ɠ(𝑒	ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	189	2	)	)	PUNCT
cana-525	189	3	̃	̃	NOUN
cana-525	189	4	〉	〉	NOUN
cana-525	189	5	=	=	SYM
cana-525	189	6	̃	̃	NOUN
cana-525	189	7	〈	〈	NOUN
cana-525	189	8	ʈ̃∗̃	ʈ̃∗̃	NOUN
cana-525	189	9	ʈ̃𝑙𝜂ɠ(𝑒	ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	189	10	)	)	PUNCT
cana-525	189	11	,	,	PUNCT
cana-525	189	12	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	189	13	)	)	PUNCT
cana-525	189	14	̃	̃	NOUN
cana-525	189	15	〉	〉	NOUN
cana-525	189	16	=	=	SYM
cana-525	189	17	̃	̃	NOUN
cana-525	189	18	〈	〈	NOUN
cana-525	189	19	(	(	PUNCT
cana-525	189	20	ʈ̃ʈ̃∗̃	ʈ̃ʈ̃∗̃	NOUN
cana-525	189	21	)	)	PUNCT
cana-525	189	22	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	189	23	)	)	PUNCT
cana-525	189	24	,	,	PUNCT
cana-525	189	25	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	189	26	)	)	PUNCT
cana-525	189	27	̃	̃	NOUN
cana-525	189	28	〉	〉	NOUN
cana-525	189	29	=	=	SYM
cana-525	189	30	̃	̃	NOUN
cana-525	189	31	〈	〈	NOUN
cana-525	189	32	ʈ̃2𝑙𝜂ɠ(𝑒	ʈ̃2𝑙𝜂ɠ(𝑒	NUM
cana-525	189	33	)	)	PUNCT
cana-525	189	34	,	,	PUNCT
cana-525	189	35	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	189	36	)	)	PUNCT
cana-525	189	37	̃	̃	PROPN
cana-525	189	38	〉	〉	NOUN
cana-525	189	39	≤̃	≤̃	NOUN
cana-525	189	40	‖ʈ̃2𝑙𝜂ɠ(𝑒	‖ʈ̃2𝑙𝜂ɠ(𝑒	ADV
cana-525	189	41	)	)	PUNCT
cana-525	190	1	‖	‖	PROPN
cana-525	190	2	̃	̃	PROPN
cana-525	190	3	‖𝑙𝜂ɠ(𝑒	‖𝑙𝜂ɠ(𝑒	NOUN
cana-525	190	4	)	)	PUNCT
cana-525	190	5	̃‖	̃‖	VERB
cana-525	190	6	ie	ie	ADJ
cana-525	190	7	)	)	PUNCT
cana-525	190	8	‖ʈ̃𝑙𝜂ɠ(𝑒	‖ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	190	9	)	)	PUNCT
cana-525	191	1	̃	̃	NOUN
cana-525	191	2	‖	‖	ADJ
cana-525	191	3	2	2	NUM
cana-525	191	4	≤̃	≤̃	NOUN
cana-525	191	5	‖ʈ̃2𝑙𝜂ɠ(𝑒	‖ʈ̃2𝑙𝜂ɠ(𝑒	ADV
cana-525	191	6	)	)	PUNCT
cana-525	191	7	‖	‖	PROPN
cana-525	191	8	̃	̃	PROPN
cana-525	191	9	‖𝑙𝜂ɠ(𝑒	‖𝑙𝜂ɠ(𝑒	NOUN
cana-525	191	10	)	)	PUNCT
cana-525	191	11	̃‖	̃‖	NOUN
cana-525	191	12	implies	imply	VERB
cana-525	191	13	that	that	SCONJ
cana-525	191	14	‖ʈ̃𝑙𝜂ɠ(𝑒	‖ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	191	15	)	)	PUNCT
cana-525	191	16	̃	̃	NOUN
cana-525	191	17	‖	‖	ADJ
cana-525	191	18	2	2	NUM
cana-525	191	19	≤̃	≤̃	NOUN
cana-525	191	20	‖ʈ̃2𝑙𝜂ɠ(𝑒	‖ʈ̃2𝑙𝜂ɠ(𝑒	ADV
cana-525	191	21	)	)	PUNCT
cana-525	191	22	‖	‖	PROPN
cana-525	192	1	̃	̃	PROPN
cana-525	192	2	therefore	therefore	ADV
cana-525	192	3	,	,	PUNCT
cana-525	192	4	ʈ̃	ʈ̃	PROPN
cana-525	192	5	is	be	AUX
cana-525	192	6	fspn	fspn	ADJ
cana-525	192	7	.	.	PUNCT
cana-525	193	1	theorem	theorem	VERB
cana-525	193	2	3.10	3.10	NUM
cana-525	193	3	:	:	PUNCT
cana-525	193	4	let	let	VERB
cana-525	193	5	ʈ̃	ʈ̃	PROPN
cana-525	193	6	∈	∈	PROPN
cana-525	193	7	�	�	PROPN
cana-525	193	8	̃	̃	PROPN
cana-525	193	9	�	�	PROPN
cana-525	193	10	(	(	PUNCT
cana-525	193	11	�	�	PROPN
cana-525	193	12	̃	̃	PROPN
cana-525	193	13	�	�	PROPN
cana-525	193	14	)	)	PUNCT
cana-525	193	15	is	be	AUX
cana-525	193	16	a	a	DET
cana-525	193	17	fspn	fspn	ADJ
cana-525	193	18	operator	operator	NOUN
cana-525	193	19	and	and	CCONJ
cana-525	193	20	fshn	fshn	NOUN
cana-525	193	21	.	.	PUNCT
cana-525	194	1	then	then	ADV
cana-525	194	2	‖ʈ̃	‖ʈ̃	PUNCT
cana-525	195	1	‖	‖	PROPN
cana-525	195	2	≥̃	≥̃	PUNCT
cana-525	195	3	‖ʈ̃∗̃‖	‖ʈ̃∗̃‖	X
cana-525	195	4	is	be	AUX
cana-525	195	5	a	a	DET
cana-525	195	6	fspn	fspn	ADJ
cana-525	195	7	operator	operator	NOUN
cana-525	195	8	.	.	PUNCT
cana-525	196	1	proof	proof	NOUN
cana-525	196	2	:	:	PUNCT
cana-525	196	3	for	for	SCONJ
cana-525	196	4	every	every	DET
cana-525	196	5	unit	unit	NOUN
cana-525	196	6	vector	vector	NOUN
cana-525	196	7	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	196	8	)	)	PUNCT
cana-525	196	9	∈̃	∈̃	PROPN
cana-525	196	10	�	�	PROPN
cana-525	196	11	̃	̃	PROPN
cana-525	196	12	�	�	PROPN
cana-525	196	13	let	let	VERB
cana-525	196	14	‖ʈ̃𝑙𝜂ɠ(𝑒	‖ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	196	15	)	)	PUNCT
cana-525	196	16	̃	̃	NOUN
cana-525	196	17	‖	‖	ADJ
cana-525	196	18	2	2	NUM
cana-525	196	19	=	=	SYM
cana-525	196	20	̃	̃	NOUN
cana-525	196	21	〈	〈	NOUN
cana-525	196	22	ʈ̃𝑙𝜂ɠ(𝑒	ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	196	23	)	)	PUNCT
cana-525	196	24	,	,	PUNCT
cana-525	197	1	ʈ̃𝑙𝜂ɠ(𝑒	ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	197	2	)	)	PUNCT
cana-525	197	3	̃	̃	NOUN
cana-525	197	4	〉	〉	NOUN
cana-525	197	5	=	=	SYM
cana-525	197	6	̃	̃	NOUN
cana-525	197	7	〈	〈	NOUN
cana-525	197	8	ʈ̃∗̃	ʈ̃∗̃	NOUN
cana-525	197	9	ʈ̃𝑙𝜂ɠ(𝑒	ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	197	10	)	)	PUNCT
cana-525	197	11	,	,	PUNCT
cana-525	197	12	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	197	13	)	)	PUNCT
cana-525	197	14	̃	̃	NOUN
cana-525	197	15	〉	〉	NOUN
cana-525	197	16	≥̃	≥̃	X
cana-525	197	17	〈	〈	NOUN
cana-525	197	18	(	(	PUNCT
cana-525	197	19	ʈ̃ʈ̃∗̃	ʈ̃ʈ̃∗̃	NOUN
cana-525	197	20	)	)	PUNCT
cana-525	197	21	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	197	22	)	)	PUNCT
cana-525	197	23	,	,	PUNCT
cana-525	197	24	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	197	25	)	)	PUNCT
cana-525	197	26	̃	̃	NOUN
cana-525	197	27	〉	〉	NOUN
cana-525	197	28	≥̃	≥̃	X
cana-525	197	29	〈	〈	NOUN
cana-525	197	30	ʈ̃∗̃	ʈ̃∗̃	NOUN
cana-525	197	31	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	NOUN
cana-525	197	32	)	)	PUNCT
cana-525	197	33	,	,	PUNCT
cana-525	197	34	ʈ̃∗̃	ʈ̃∗̃	ADJ
cana-525	197	35	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	NOUN
cana-525	197	36	)	)	PUNCT
cana-525	197	37	̃	̃	PROPN
cana-525	197	38	〉	〉	NOUN
cana-525	197	39	since	since	SCONJ
cana-525	197	40	‖ʈ̃2𝑙𝜂ɠ(𝑒	‖ʈ̃2𝑙𝜂ɠ(𝑒	ADV
cana-525	197	41	)	)	PUNCT
cana-525	197	42	‖	‖	PROPN
cana-525	197	43	̃	̃	PROPN
cana-525	197	44	≥̃	≥̃	X
cana-525	197	45	‖ʈ̃𝑙𝜂ɠ(𝑒	‖ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	197	46	)	)	PUNCT
cana-525	197	47	̃	̃	NOUN
cana-525	197	48	‖	‖	ADJ
cana-525	197	49	2	2	NUM
cana-525	197	50	and	and	CCONJ
cana-525	197	51	ʈ̃∗̃ʈ̃	ʈ̃∗̃ʈ̃	NOUN
cana-525	197	52	−	−	VERB
cana-525	197	53	ʈ̃ʈ̃∗̃	ʈ̃ʈ̃∗̃	X
cana-525	197	54	≥̃	≥̃	PUNCT
cana-525	197	55	0̃	0̃	NOUN
cana-525	197	56	∀	∀	NUM
cana-525	197	57	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	NOUN
cana-525	197	58	)	)	PUNCT
cana-525	197	59	∈̃	∈̃	PROPN
cana-525	197	60	�	�	PROPN
cana-525	197	61	̃	̃	PROPN
cana-525	197	62	�	�	PROPN
cana-525	197	63	communications	communication	NOUN
cana-525	197	64	on	on	ADP
cana-525	197	65	applied	apply	VERB
cana-525	197	66	nonlinear	nonlinear	ADJ
cana-525	197	67	analysis	analysis	NOUN
cana-525	197	68	issn	issn	NOUN
cana-525	197	69	:	:	PUNCT
cana-525	197	70	1074	1074	NUM
cana-525	197	71	-	-	PUNCT
cana-525	197	72	133x	133x	NUM
cana-525	197	73	vol	vol	NOUN
cana-525	197	74	31	31	NUM
cana-525	197	75	no	no	NOUN
cana-525	197	76	.	.	NOUN
cana-525	197	77	2	2	NUM
cana-525	197	78	(	(	PUNCT
cana-525	197	79	2024	2024	NUM
cana-525	197	80	)	)	PUNCT
cana-525	197	81	139	139	NUM
cana-525	197	82	https://internationalpubls.com	https://internationalpubls.com	X
cana-525	197	83	‖ʈ̃𝑙𝜂ɠ(𝑒	‖ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	197	84	)	)	PUNCT
cana-525	197	85	̃	̃	NOUN
cana-525	197	86	‖	‖	ADJ
cana-525	197	87	2	2	NUM
cana-525	197	88	≥̃	≥̃	X
cana-525	197	89	‖ʈ̃∗̃𝑙𝜂ɠ(𝑒	‖ʈ̃∗̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	197	90	)	)	PUNCT
cana-525	197	91	̃	̃	NOUN
cana-525	197	92	‖	‖	ADJ
cana-525	197	93	2	2	NUM
cana-525	197	94	⇒	⇒	NOUN
cana-525	197	95	‖ʈ̃	‖ʈ̃	PUNCT
cana-525	197	96	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PRON
cana-525	197	97	)	)	PUNCT
cana-525	198	1	‖	‖	PROPN
cana-525	198	2	̃	̃	PROPN
cana-525	198	3	≥̃	≥̃	X
cana-525	198	4	‖ʈ̃∗̃	‖ʈ̃∗̃	X
cana-525	198	5	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	198	6	)	)	PUNCT
cana-525	198	7	‖	‖	PROPN
cana-525	198	8	̃	̃	PROPN
cana-525	198	9	implies	imply	VERB
cana-525	198	10	that	that	SCONJ
cana-525	198	11	‖ʈ̃	‖ʈ̃	PUNCT
cana-525	198	12	‖	‖	PROPN
cana-525	198	13	≥̃	≥̃	X
cana-525	198	14	‖ʈ̃∗̃‖	‖ʈ̃∗̃‖	X
cana-525	198	15	theorem	theorem	VERB
cana-525	198	16	3.11	3.11	NUM
cana-525	198	17	:	:	PUNCT
cana-525	198	18	let	let	VERB
cana-525	198	19	ʈ̃	ʈ̃	PROPN
cana-525	198	20	∈	∈	PROPN
cana-525	198	21	�	�	PROPN
cana-525	198	22	̃	̃	PROPN
cana-525	198	23	�	�	PROPN
cana-525	198	24	(	(	PUNCT
cana-525	198	25	�	�	PROPN
cana-525	198	26	̃	̃	PROPN
cana-525	198	27	�	�	PROPN
cana-525	198	28	)	)	PUNCT
cana-525	198	29	is	be	AUX
cana-525	198	30	a	a	DET
cana-525	198	31	fspn	fspn	ADJ
cana-525	198	32	operator	operator	NOUN
cana-525	198	33	and	and	CCONJ
cana-525	198	34	₷	₷	NOUN
cana-525	198	35	̃	̃	NOUN
cana-525	198	36	is	be	AUX
cana-525	198	37	unitarily	unitarily	ADV
cana-525	198	38	equivalent	equivalent	ADJ
cana-525	198	39	to	to	ADP
cana-525	198	40	ʈ̃	ʈ̃	PROPN
cana-525	198	41	then	then	ADV
cana-525	198	42	₷	₷	NOUN
cana-525	198	43	̃	̃	PROPN
cana-525	198	44	is	be	AUX
cana-525	198	45	a	a	DET
cana-525	198	46	fspn	fspn	NOUN
cana-525	198	47	.	.	PUNCT
cana-525	199	1	proof	proof	NOUN
cana-525	199	2	:	:	PUNCT
cana-525	199	3	for	for	ADP
cana-525	199	4	₷	₷	NOUN
cana-525	199	5	̃	̃	PROPN
cana-525	199	6	is	be	AUX
cana-525	199	7	unitarily	unitarily	ADV
cana-525	199	8	equivalent	equivalent	ADJ
cana-525	199	9	to	to	ADP
cana-525	199	10	ʈ̃	ʈ̃	PROPN
cana-525	199	11	,	,	PUNCT
cana-525	199	12	we	we	PRON
cana-525	199	13	have	have	VERB
cana-525	199	14	₷	₷	NUM
cana-525	199	15	̃	̃	PROPN
cana-525	199	16	=	=	SYM
cana-525	199	17	̃	̃	PROPN
cana-525	199	18	�	�	PROPN
cana-525	199	19	̃	̃	PROPN
cana-525	199	20	�	�	PROPN
cana-525	199	21	ʈ̃	ʈ̃	PROPN
cana-525	199	22	�	�	PROPN
cana-525	199	23	̃	̃	PROPN
cana-525	199	24	�	�	PROPN
cana-525	199	25	∗̃	∗̃	NUM
cana-525	199	26	for	for	ADP
cana-525	199	27	some	some	DET
cana-525	199	28	unitarily	unitarily	ADV
cana-525	199	29	equivalent	equivalent	ADJ
cana-525	199	30	to	to	ADP
cana-525	199	31	₷	₷	NOUN
cana-525	199	32	̃	̃	ADJ
cana-525	199	33	2	2	NUM
cana-525	199	34	=	=	SYM
cana-525	199	35	̃	̃	PROPN
cana-525	199	36	�	�	NOUN
cana-525	199	37	̃	̃	PROPN
cana-525	199	38	�	�	PROPN
cana-525	199	39	ʈ̃2	ʈ̃2	PROPN
cana-525	199	40	�	�	PROPN
cana-525	199	41	̃	̃	PROPN
cana-525	199	42	�	�	PROPN
cana-525	199	43	∗̃	∗̃	X
cana-525	199	44	⇒	⇒	NOUN
cana-525	199	45	‖₷̃	‖₷̃	PROPN
cana-525	199	46	2	2	NUM
cana-525	199	47	𝑙𝜂𝔾(𝑒	𝑙𝜂𝔾(𝑒	NOUN
cana-525	199	48	)	)	PUNCT
cana-525	199	49	‖	‖	PROPN
cana-525	200	1	̃	̃	PROPN
cana-525	200	2	=	=	SYM
cana-525	200	3	̃	̃	PROPN
cana-525	200	4	‖	‖	ADJ
cana-525	200	5	�	�	PROPN
cana-525	200	6	̃	̃	PROPN
cana-525	200	7	�	�	PROPN
cana-525	200	8	ʈ̃2	ʈ̃2	PROPN
cana-525	200	9	�	�	PROPN
cana-525	200	10	̃	̃	PROPN
cana-525	200	11	�	�	PROPN
cana-525	200	12	∗̃	∗̃	PUNCT
cana-525	200	13	𝑙𝜂𝔾(𝑒	𝑙𝜂𝔾(𝑒	PROPN
cana-525	200	14	)	)	PUNCT
cana-525	201	1	‖	‖	PROPN
cana-525	201	2	let	let	VERB
cana-525	201	3	‖₷̃𝑙𝜂ɠ(𝑒	‖₷̃𝑙𝜂ɠ(𝑒	X
cana-525	201	4	)	)	PUNCT
cana-525	201	5	̃	̃	NOUN
cana-525	201	6	‖	‖	ADJ
cana-525	201	7	2	2	NUM
cana-525	201	8	=	=	SYM
cana-525	201	9	̃	̃	NOUN
cana-525	201	10	‖(	‖(	NOUN
cana-525	201	11	�	�	PROPN
cana-525	201	12	̃	̃	PROPN
cana-525	201	13	�	�	PROPN
cana-525	201	14	ʈ̃	ʈ̃	PROPN
cana-525	201	15	�	�	PROPN
cana-525	201	16	̃	̃	PROPN
cana-525	201	17	�	�	NOUN
cana-525	201	18	∗̃)𝑙𝜂ɠ(𝑒	∗̃)𝑙𝜂ɠ(𝑒	PART
cana-525	201	19	)	)	PUNCT
cana-525	202	1	̃‖	̃‖	PROPN
cana-525	202	2	2	2	NUM
cana-525	202	3	〈	〈	PRON
cana-525	202	4	₷	₷	NOUN
cana-525	202	5	̃𝑙𝜂ɠ(𝑒	̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	202	6	)	)	PUNCT
cana-525	202	7	,	,	PUNCT
cana-525	202	8	₷	₷	NOUN
cana-525	202	9	̃𝑙𝜂ɠ(𝑒	̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	202	10	)	)	PUNCT
cana-525	202	11	〉	〉	NOUN
cana-525	202	12	̃	̃	NOUN
cana-525	202	13	=	=	SYM
cana-525	202	14	̃	̃	NOUN
cana-525	202	15	〈	〈	PROPN
cana-525	202	16	(	(	PUNCT
cana-525	202	17	�	�	PROPN
cana-525	202	18	̃	̃	PROPN
cana-525	202	19	�	�	PROPN
cana-525	202	20	ʈ̃	ʈ̃	PROPN
cana-525	202	21	�	�	PROPN
cana-525	202	22	̃	̃	PROPN
cana-525	202	23	�	�	NOUN
cana-525	202	24	∗̃)𝑙𝜂ɠ(𝑒	∗̃)𝑙𝜂ɠ(𝑒	NUM
cana-525	202	25	)	)	PUNCT
cana-525	202	26	,	,	PUNCT
cana-525	202	27	(	(	PUNCT
cana-525	202	28	�	�	PROPN
cana-525	202	29	̃	̃	PROPN
cana-525	202	30	�	�	PROPN
cana-525	202	31	ʈ̃	ʈ̃	PROPN
cana-525	202	32	�	�	PROPN
cana-525	202	33	̃	̃	PROPN
cana-525	202	34	�	�	NOUN
cana-525	202	35	∗̃)𝑙𝜂ɠ(𝑒	∗̃)𝑙𝜂ɠ(𝑒	PART
cana-525	202	36	)	)	PUNCT
cana-525	202	37	〉	〉	NOUN
cana-525	203	1	̃	̃	NOUN
cana-525	203	2	=	=	SYM
cana-525	203	3	̃	̃	NOUN
cana-525	203	4	〈	〈	PROPN
cana-525	203	5	(	(	PUNCT
cana-525	203	6	ʈ̃	ʈ̃	PROPN
cana-525	203	7	�	�	PROPN
cana-525	203	8	̃	̃	PROPN
cana-525	203	9	�	�	NOUN
cana-525	203	10	∗̃)𝑙𝜂ɠ(𝑒	∗̃)𝑙𝜂ɠ(𝑒	NUM
cana-525	203	11	)	)	PUNCT
cana-525	203	12	,	,	PUNCT
cana-525	203	13	�	�	PROPN
cana-525	203	14	̃	̃	PROPN
cana-525	203	15	�	�	PROPN
cana-525	203	16	∗̃	∗̃	SYM
cana-525	203	17	�	�	PROPN
cana-525	203	18	̃	̃	PROPN
cana-525	203	19	�	�	PROPN
cana-525	203	20	(	(	PUNCT
cana-525	203	21	ʈ̃	ʈ̃	PROPN
cana-525	203	22	�	�	PROPN
cana-525	203	23	̃	̃	PROPN
cana-525	203	24	�	�	NOUN
cana-525	203	25	∗̃)𝑙𝜂ɠ(𝑒	∗̃)𝑙𝜂ɠ(𝑒	PART
cana-525	203	26	)	)	PUNCT
cana-525	203	27	〉	〉	NOUN
cana-525	203	28	̃	̃	NOUN
cana-525	203	29	=	=	SYM
cana-525	203	30	̃	̃	NOUN
cana-525	203	31	〈	〈	PROPN
cana-525	203	32	(	(	PUNCT
cana-525	203	33	ʈ̃	ʈ̃	PROPN
cana-525	203	34	�	�	PROPN
cana-525	203	35	̃	̃	PROPN
cana-525	203	36	�	�	NOUN
cana-525	203	37	∗̃)𝑙𝜂ɠ(𝑒	∗̃)𝑙𝜂ɠ(𝑒	NUM
cana-525	203	38	)	)	PUNCT
cana-525	203	39	,	,	PUNCT
cana-525	203	40	(	(	PUNCT
cana-525	203	41	ʈ̃	ʈ̃	PROPN
cana-525	203	42	�	�	PROPN
cana-525	203	43	̃	̃	PROPN
cana-525	203	44	�	�	NOUN
cana-525	203	45	∗̃)𝑙𝜂ɠ(𝑒	∗̃)𝑙𝜂ɠ(𝑒	PART
cana-525	203	46	)	)	PUNCT
cana-525	203	47	〉	〉	NOUN
cana-525	203	48	̃	̃	PROPN
cana-525	203	49	[	[	PUNCT
cana-525	203	50	since	since	SCONJ
cana-525	203	51	�	�	PROPN
cana-525	203	52	̃	̃	PROPN
cana-525	203	53	�	�	PROPN
cana-525	203	54	is	be	AUX
cana-525	203	55	fs	fs	ADP
cana-525	203	56	isometry	isometry	NOUN
cana-525	203	57	]	]	X
cana-525	203	58	=	=	SYM
cana-525	203	59	̃	̃	NOUN
cana-525	203	60	〈	〈	PROPN
cana-525	203	61	(	(	PUNCT
cana-525	203	62	ʈ̃	ʈ̃	PROPN
cana-525	203	63	�	�	PROPN
cana-525	203	64	̃	̃	PROPN
cana-525	203	65	�	�	NOUN
cana-525	203	66	∗̃	∗̃	NUM
cana-525	203	67	)	)	PUNCT
cana-525	203	68	∗̃	∗̃	X
cana-525	203	69	(	(	PUNCT
cana-525	203	70	ʈ̃	ʈ̃	PROPN
cana-525	203	71	�	�	PROPN
cana-525	203	72	̃	̃	PROPN
cana-525	203	73	�	�	NOUN
cana-525	203	74	∗̃)𝑙𝜂ɠ(𝑒	∗̃)𝑙𝜂ɠ(𝑒	NUM
cana-525	203	75	)	)	PUNCT
cana-525	203	76	,	,	PUNCT
cana-525	203	77	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	203	78	)	)	PUNCT
cana-525	203	79	〉	〉	NOUN
cana-525	204	1	̃	̃	PROPN
cana-525	204	2	=	=	SYM
cana-525	204	3	̃	̃	ADP
cana-525	204	4	〈	〈	NOUN
cana-525	204	5	�	�	PROPN
cana-525	204	6	̃	̃	PROPN
cana-525	204	7	�	�	PROPN
cana-525	204	8	ʈ̃∗̃(ʈ̃	ʈ̃∗̃(ʈ̃	PROPN
cana-525	204	9	�	�	PROPN
cana-525	204	10	̃	̃	PROPN
cana-525	204	11	�	�	NOUN
cana-525	204	12	∗̃)𝑙𝜂ɠ(𝑒	∗̃)𝑙𝜂ɠ(𝑒	NUM
cana-525	204	13	)	)	PUNCT
cana-525	204	14	,	,	PUNCT
cana-525	204	15	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	204	16	)	)	PUNCT
cana-525	204	17	〉	〉	NOUN
cana-525	205	1	̃	̃	NOUN
cana-525	205	2	=	=	SYM
cana-525	205	3	̃	̃	ADP
cana-525	205	4	〈	〈	NOUN
cana-525	205	5	�	�	PROPN
cana-525	205	6	̃	̃	PROPN
cana-525	205	7	�	�	PROPN
cana-525	205	8	ʈ̃2	ʈ̃2	PROPN
cana-525	205	9	�	�	PROPN
cana-525	205	10	̃	̃	PROPN
cana-525	205	11	�	�	PROPN
cana-525	205	12	∗̃	∗̃	NUM
cana-525	205	13	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	205	14	)	)	PUNCT
cana-525	205	15	,	,	PUNCT
cana-525	205	16	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	205	17	)	)	PUNCT
cana-525	205	18	〉	〉	NOUN
cana-525	205	19	̃	̃	ADP
cana-525	205	20	≤̃	≤̃	NOUN
cana-525	205	21	‖	‖	PROPN
cana-525	205	22	�	�	PROPN
cana-525	205	23	̃	̃	PROPN
cana-525	205	24	�	�	PROPN
cana-525	205	25	ʈ̃2	ʈ̃2	PROPN
cana-525	205	26	�	�	PROPN
cana-525	205	27	̃	̃	PROPN
cana-525	205	28	�	�	PROPN
cana-525	205	29	∗̃	∗̃	NUM
cana-525	205	30	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	205	31	)	)	PUNCT
cana-525	205	32	‖	‖	PROPN
cana-525	205	33	‖	‖	PROPN
cana-525	205	34	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	205	35	)	)	PUNCT
cana-525	205	36	‖	‖	PROPN
cana-525	205	37	‖₷̃𝑙𝜂ɠ(𝑒	‖₷̃𝑙𝜂ɠ(𝑒	SYM
cana-525	205	38	)	)	PUNCT
cana-525	205	39	̃	̃	NOUN
cana-525	205	40	‖	‖	ADJ
cana-525	205	41	2	2	NUM
cana-525	205	42	≤̃	≤̃	PROPN
cana-525	205	43	‖	‖	PROPN
cana-525	205	44	�	�	PROPN
cana-525	205	45	̃	̃	PROPN
cana-525	205	46	�	�	PROPN
cana-525	205	47	ʈ̃2	ʈ̃2	PROPN
cana-525	205	48	�	�	PROPN
cana-525	205	49	̃	̃	PROPN
cana-525	205	50	�	�	PROPN
cana-525	205	51	∗̃	∗̃	NUM
cana-525	205	52	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	205	53	)	)	PUNCT
cana-525	205	54	‖	‖	PROPN
cana-525	205	55	‖	‖	PROPN
cana-525	205	56	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	205	57	)	)	PUNCT
cana-525	206	1	‖	‖	PROPN
cana-525	206	2	implies	imply	VERB
cana-525	206	3	that	that	SCONJ
cana-525	206	4	‖₷̃𝑙𝜂ɠ(𝑒	‖₷̃𝑙𝜂ɠ(𝑒	X
cana-525	206	5	)	)	PUNCT
cana-525	206	6	̃	̃	NOUN
cana-525	206	7	‖	‖	ADJ
cana-525	206	8	2	2	NUM
cana-525	206	9	≤̃	≤̃	NOUN
cana-525	206	10	‖₷̃	‖₷̃	PROPN
cana-525	206	11	2	2	NUM
cana-525	206	12	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	NOUN
cana-525	206	13	)	)	PUNCT
cana-525	206	14	‖	‖	PROPN
cana-525	206	15	‖	‖	PROPN
cana-525	206	16	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	206	17	)	)	PUNCT
cana-525	206	18	‖	‖	ADJ
cana-525	206	19	hence	hence	ADV
cana-525	206	20	₷	₷	NOUN
cana-525	206	21	̃	̃	NOUN
cana-525	206	22	is	be	AUX
cana-525	206	23	a	a	DET
cana-525	206	24	fspn	fspn	NOUN
cana-525	206	25	theorem	theorem	ADJ
cana-525	206	26	3.12	3.12	NUM
cana-525	206	27	:	:	PUNCT
cana-525	206	28	let	let	VERB
cana-525	206	29	ʈ̃	ʈ̃	PROPN
cana-525	206	30	∈	∈	PROPN
cana-525	206	31	�	�	PROPN
cana-525	206	32	̃	̃	PROPN
cana-525	206	33	�	�	PROPN
cana-525	206	34	(	(	PUNCT
cana-525	206	35	�	�	PROPN
cana-525	206	36	̃	̃	PROPN
cana-525	206	37	�	�	PROPN
cana-525	206	38	)	)	PUNCT
cana-525	206	39	is	be	AUX
cana-525	206	40	an	an	DET
cana-525	206	41	invertible	invertible	ADJ
cana-525	206	42	and	and	CCONJ
cana-525	206	43	fspn	fspn	ADJ
cana-525	206	44	operator	operator	NOUN
cana-525	206	45	.	.	PUNCT
cana-525	207	1	then	then	ADV
cana-525	207	2	ʈ̃−1	ʈ̃−1	PROPN
cana-525	207	3	is	be	AUX
cana-525	207	4	also	also	ADV
cana-525	207	5	a	a	DET
cana-525	207	6	fspn	fspn	NOUN
cana-525	207	7	.	.	PUNCT
cana-525	208	1	proof	proof	NOUN
cana-525	208	2	:	:	PUNCT
cana-525	208	3	for	for	ADP
cana-525	208	4	every	every	DET
cana-525	208	5	unit	unit	NOUN
cana-525	208	6	vector	vector	NOUN
cana-525	208	7	𝑙𝜂𝔾(𝑒	𝑙𝜂𝔾(𝑒	PROPN
cana-525	208	8	)	)	PUNCT
cana-525	208	9	∈̃	∈̃	PROPN
cana-525	208	10	�	�	PROPN
cana-525	208	11	̃	̃	PROPN
cana-525	208	12	�	�	PROPN
cana-525	208	13	let	let	VERB
cana-525	208	14	‖ʈ̃𝑙𝜂ɠ(𝑒	‖ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	208	15	)	)	PUNCT
cana-525	208	16	̃	̃	NOUN
cana-525	208	17	‖	‖	ADJ
cana-525	208	18	2	2	NUM
cana-525	208	19	=	=	SYM
cana-525	208	20	̃	̃	NOUN
cana-525	208	21	〈	〈	NOUN
cana-525	208	22	ʈ̃𝑙𝜂ɠ(𝑒	ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	208	23	)	)	PUNCT
cana-525	208	24	,	,	PUNCT
cana-525	209	1	ʈ̃𝑙𝜂ɠ(𝑒	ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	209	2	)	)	PUNCT
cana-525	209	3	̃	̃	NOUN
cana-525	209	4	〉	〉	NOUN
cana-525	209	5	=	=	SYM
cana-525	209	6	̃	̃	NOUN
cana-525	209	7	〈	〈	NOUN
cana-525	209	8	ʈ̃∗̃	ʈ̃∗̃	NOUN
cana-525	209	9	ʈ̃𝑙𝜂ɠ(𝑒	ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	209	10	)	)	PUNCT
cana-525	209	11	,	,	PUNCT
cana-525	209	12	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	209	13	)	)	PUNCT
cana-525	209	14	̃	̃	NOUN
cana-525	209	15	〉	〉	NOUN
cana-525	209	16	=	=	SYM
cana-525	209	17	̃	̃	NOUN
cana-525	209	18	〈	〈	NOUN
cana-525	209	19	ʈ̃	ʈ̃	PROPN
cana-525	209	20	ʈ̃∗̃	ʈ̃∗̃	NOUN
cana-525	209	21	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	209	22	)	)	PUNCT
cana-525	209	23	,	,	PUNCT
cana-525	209	24	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	209	25	)	)	PUNCT
cana-525	209	26	̃	̃	NOUN
cana-525	209	27	〉	〉	NOUN
cana-525	209	28	=	=	SYM
cana-525	209	29	̃	̃	ADP
cana-525	209	30	〈	〈	NOUN
cana-525	209	31	ʈ̃2	ʈ̃2	PROPN
cana-525	209	32	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	209	33	)	)	PUNCT
cana-525	209	34	,	,	PUNCT
cana-525	209	35	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	209	36	)	)	PUNCT
cana-525	209	37	̃	̃	PROPN
cana-525	209	38	〉	〉	NOUN
cana-525	209	39	≤̃	≤̃	VERB
cana-525	209	40	‖ʈ̃2	‖ʈ̃2	PROPN
cana-525	209	41	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	209	42	)	)	PUNCT
cana-525	210	1	‖	‖	PROPN
cana-525	210	2	‖	‖	PROPN
cana-525	210	3	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	210	4	)	)	PUNCT
cana-525	210	5	‖	‖	PROPN
cana-525	211	1	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	211	2	)	)	PUNCT
cana-525	211	3	is	be	AUX
cana-525	211	4	replaced	replace	VERB
cana-525	211	5	by	by	ADP
cana-525	211	6	ʈ̃−2	ʈ̃−2	PROPN
cana-525	211	7	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	211	8	)	)	PUNCT
cana-525	211	9	communications	communication	NOUN
cana-525	211	10	on	on	ADP
cana-525	211	11	applied	apply	VERB
cana-525	211	12	nonlinear	nonlinear	ADJ
cana-525	211	13	analysis	analysis	NOUN
cana-525	211	14	issn	issn	NOUN
cana-525	211	15	:	:	PUNCT
cana-525	211	16	1074	1074	NUM
cana-525	211	17	-	-	PUNCT
cana-525	211	18	133x	133x	NUM
cana-525	211	19	vol	vol	NOUN
cana-525	211	20	31	31	NUM
cana-525	211	21	no	no	NOUN
cana-525	211	22	.	.	NOUN
cana-525	211	23	2	2	NUM
cana-525	211	24	(	(	PUNCT
cana-525	211	25	2024	2024	NUM
cana-525	211	26	)	)	PUNCT
cana-525	211	27	140	140	NUM
cana-525	211	28	https://internationalpubls.com	https://internationalpubls.com	X
cana-525	211	29	‖ʈ̃ʈ̃−2	‖ʈ̃ʈ̃−2	PROPN
cana-525	211	30	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	211	31	)	)	PUNCT
cana-525	212	1	̃	̃	PROPN
cana-525	212	2	‖	‖	ADJ
cana-525	212	3	2	2	NUM
cana-525	212	4	≤̃	≤̃	NOUN
cana-525	212	5	‖ʈ̃2	‖ʈ̃2	PROPN
cana-525	212	6	ʈ̃−2	ʈ̃−2	PROPN
cana-525	212	7	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	NOUN
cana-525	212	8	)	)	PUNCT
cana-525	212	9	‖	‖	PROPN
cana-525	212	10	‖	‖	PROPN
cana-525	212	11	ʈ̃−2	ʈ̃−2	PROPN
cana-525	212	12	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	NOUN
cana-525	212	13	)	)	PUNCT
cana-525	212	14	‖	‖	PROPN
cana-525	212	15	‖ʈ̃−1𝑙𝜂ɠ(𝑒	‖ʈ̃−1𝑙𝜂ɠ(𝑒	PROPN
cana-525	212	16	)	)	PUNCT
cana-525	212	17	̃	̃	PROPN
cana-525	212	18	‖	‖	ADJ
cana-525	212	19	2	2	NUM
cana-525	212	20	≤̃	≤̃	NOUN
cana-525	212	21	‖	‖	PROPN
cana-525	212	22	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	212	23	)	)	PUNCT
cana-525	212	24	‖	‖	PROPN
cana-525	212	25	‖	‖	PROPN
cana-525	212	26	ʈ̃−2	ʈ̃−2	PROPN
cana-525	212	27	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	NOUN
cana-525	212	28	)	)	PUNCT
cana-525	212	29	‖	‖	PROPN
cana-525	212	30	‖ʈ̃−1𝑙𝜂ɠ(𝑒	‖ʈ̃−1𝑙𝜂ɠ(𝑒	PROPN
cana-525	212	31	)	)	PUNCT
cana-525	212	32	̃	̃	PROPN
cana-525	212	33	‖	‖	ADJ
cana-525	212	34	2	2	NUM
cana-525	212	35	≤̃	≤̃	NOUN
cana-525	212	36	‖	‖	PROPN
cana-525	212	37	ʈ̃−2	ʈ̃−2	PROPN
cana-525	212	38	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	NOUN
cana-525	212	39	)	)	PUNCT
cana-525	212	40	‖	‖	PROPN
cana-525	212	41	‖	‖	PROPN
cana-525	212	42	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	212	43	)	)	PUNCT
cana-525	212	44	‖	‖	PROPN
cana-525	212	45	implies	imply	VERB
cana-525	212	46	that	that	SCONJ
cana-525	212	47	‖	‖	PROPN
cana-525	212	48	ʈ̃−2	ʈ̃−2	PROPN
cana-525	212	49	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	NOUN
cana-525	212	50	)	)	PUNCT
cana-525	212	51	‖	‖	PROPN
cana-525	212	52	‖	‖	PROPN
cana-525	212	53	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	212	54	)	)	PUNCT
cana-525	212	55	‖	‖	PROPN
cana-525	212	56	≥̃	≥̃	X
cana-525	212	57	‖ʈ̃−1𝑙𝜂ɠ(𝑒	‖ʈ̃−1𝑙𝜂ɠ(𝑒	NOUN
cana-525	212	58	)	)	PUNCT
cana-525	212	59	̃	̃	NOUN
cana-525	212	60	‖	‖	ADJ
cana-525	212	61	2	2	NUM
cana-525	212	62	ie	ie	X
cana-525	212	63	)	)	PUNCT
cana-525	212	64	‖	‖	PROPN
cana-525	212	65	(	(	PUNCT
cana-525	212	66	ʈ̃−1	ʈ̃−1	ADJ
cana-525	212	67	)	)	PUNCT
cana-525	212	68	2	2	NUM
cana-525	212	69	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	NOUN
cana-525	212	70	)	)	PUNCT
cana-525	212	71	‖	‖	PROPN
cana-525	212	72	‖	‖	PROPN
cana-525	212	73	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	212	74	)	)	PUNCT
cana-525	212	75	‖	‖	PROPN
cana-525	212	76	≥̃	≥̃	X
cana-525	212	77	‖ʈ̃−1𝑙𝜂ɠ(𝑒	‖ʈ̃−1𝑙𝜂ɠ(𝑒	NOUN
cana-525	212	78	)	)	PUNCT
cana-525	212	79	̃	̃	NOUN
cana-525	212	80	‖	‖	ADJ
cana-525	212	81	2	2	NUM
cana-525	212	82	hence	hence	ADV
cana-525	212	83	ʈ̃−1	ʈ̃−1	PROPN
cana-525	212	84	is	be	AUX
cana-525	212	85	also	also	ADV
cana-525	212	86	a	a	DET
cana-525	212	87	fspn	fspn	NOUN
cana-525	212	88	theorem	theorem	NOUN
cana-525	212	89	3.13	3.13	NUM
cana-525	212	90	:	:	PUNCT
cana-525	212	91	if	if	SCONJ
cana-525	212	92	ʈ̃∗̃𝟐	ʈ̃∗̃𝟐	NUM
cana-525	212	93	ʈ̃2	ʈ̃2	PROPN
cana-525	212	94	≥̃	≥̃	X
cana-525	212	95	(	(	PUNCT
cana-525	212	96	ʈ̃∗̃ʈ̃	ʈ̃∗̃ʈ̃	X
cana-525	212	97	)	)	PUNCT
cana-525	212	98	2	2	NUM
cana-525	212	99	,	,	PUNCT
cana-525	212	100	then	then	ADV
cana-525	212	101	ʈ̃	ʈ̃	PROPN
cana-525	212	102	is	be	AUX
cana-525	212	103	fspn	fspn	ADJ
cana-525	212	104	operator	operator	NOUN
cana-525	212	105	proof	proof	NOUN
cana-525	212	106	:	:	PUNCT
cana-525	212	107	for	for	SCONJ
cana-525	212	108	every	every	DET
cana-525	212	109	unit	unit	NOUN
cana-525	212	110	vector	vector	NOUN
cana-525	212	111	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	212	112	)	)	PUNCT
cana-525	212	113	∈̃	∈̃	PROPN
cana-525	212	114	�	�	PROPN
cana-525	212	115	̃	̃	PROPN
cana-525	212	116	�	�	PROPN
cana-525	212	117	let	let	VERB
cana-525	212	118	ʈ̃∗̃𝟐	ʈ̃∗̃𝟐	NUM
cana-525	212	119	ʈ̃2	ʈ̃2	PROPN
cana-525	212	120	≥̃	≥̃	X
cana-525	212	121	(	(	PUNCT
cana-525	212	122	ʈ̃∗̃ʈ̃	ʈ̃∗̃ʈ̃	X
cana-525	212	123	)	)	PUNCT
cana-525	212	124	2	2	NUM
cana-525	212	125	ʈ̃∗̃𝟐	ʈ̃∗̃𝟐	NUM
cana-525	212	126	ʈ̃2	ʈ̃2	NUM
cana-525	212	127	−	−	PROPN
cana-525	212	128	(	(	PUNCT
cana-525	212	129	ʈ̃∗̃ʈ̃	ʈ̃∗̃ʈ̃	X
cana-525	212	130	)	)	PUNCT
cana-525	212	131	2	2	NUM
cana-525	212	132	≥̃	≥̃	X
cana-525	212	133	0̃	0̃	NOUN
cana-525	212	134	〈	〈	PROPN
cana-525	212	135	(	(	PUNCT
cana-525	212	136	ʈ̃∗̃𝟐	ʈ̃∗̃𝟐	PROPN
cana-525	212	137	ʈ̃2	ʈ̃2	NUM
cana-525	212	138	−	−	PROPN
cana-525	212	139	(	(	PUNCT
cana-525	212	140	ʈ̃∗̃ʈ̃	ʈ̃∗̃ʈ̃	X
cana-525	212	141	)	)	PUNCT
cana-525	212	142	2	2	NUM
cana-525	212	143	)	)	PUNCT
cana-525	212	144	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	212	145	)	)	PUNCT
cana-525	212	146	,	,	PUNCT
cana-525	212	147	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	212	148	)	)	PUNCT
cana-525	212	149	〉	〉	NOUN
cana-525	212	150	̃	̃	ADP
cana-525	212	151	≥̃	≥̃	X
cana-525	212	152	0̃	0̃	NOUN
cana-525	212	153	〈	〈	PROPN
cana-525	212	154	(	(	PUNCT
cana-525	212	155	ʈ̃∗̃𝟐	ʈ̃∗̃𝟐	NUM
cana-525	212	156	ʈ̃2	ʈ̃2	PROPN
cana-525	212	157	)	)	PUNCT
cana-525	212	158	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	212	159	)	)	PUNCT
cana-525	212	160	,	,	PUNCT
cana-525	212	161	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	212	162	)	)	PUNCT
cana-525	212	163	〉	〉	NOUN
cana-525	212	164	−	−	NOUN
cana-525	212	165	〈	〈	PROPN
cana-525	212	166	(	(	PUNCT
cana-525	212	167	(	(	PUNCT
cana-525	212	168	ʈ̃∗̃ʈ̃	ʈ̃∗̃ʈ̃	X
cana-525	212	169	)	)	PUNCT
cana-525	212	170	2	2	NUM
cana-525	212	171	)	)	PUNCT
cana-525	212	172	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	212	173	)	)	PUNCT
cana-525	212	174	,	,	PUNCT
cana-525	212	175	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	212	176	)	)	PUNCT
cana-525	212	177	〉	〉	NOUN
cana-525	212	178	̃	̃	ADP
cana-525	212	179	≥̃	≥̃	X
cana-525	212	180	0̃	0̃	NOUN
cana-525	212	181	〈	〈	PROPN
cana-525	212	182	(	(	PUNCT
cana-525	212	183	ʈ̃∗̃𝟐	ʈ̃∗̃𝟐	NUM
cana-525	212	184	ʈ̃2	ʈ̃2	PROPN
cana-525	212	185	)	)	PUNCT
cana-525	212	186	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	212	187	)	)	PUNCT
cana-525	212	188	,	,	PUNCT
cana-525	212	189	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	212	190	)	)	PUNCT
cana-525	212	191	〉	〉	NOUN
cana-525	212	192	≥̃	≥̃	X
cana-525	212	193	〈	〈	NOUN
cana-525	212	194	(	(	PUNCT
cana-525	212	195	(	(	PUNCT
cana-525	212	196	ʈ̃∗̃ʈ̃	ʈ̃∗̃ʈ̃	X
cana-525	212	197	)	)	PUNCT
cana-525	212	198	2	2	NUM
cana-525	212	199	)	)	PUNCT
cana-525	212	200	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	212	201	)	)	PUNCT
cana-525	212	202	,	,	PUNCT
cana-525	212	203	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	212	204	)	)	PUNCT
cana-525	212	205	〉	〉	NOUN
cana-525	212	206	̃	̃	ADP
cana-525	212	207	〈	〈	PROPN
cana-525	212	208	ʈ̃2	ʈ̃2	PROPN
cana-525	212	209	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	212	210	)	)	PUNCT
cana-525	212	211	,	,	PUNCT
cana-525	212	212	ʈ̃2𝑙𝜂ɠ(𝑒	ʈ̃2𝑙𝜂ɠ(𝑒	NUM
cana-525	212	213	)	)	PUNCT
cana-525	212	214	〉	〉	NOUN
cana-525	212	215	≥̃	≥̃	X
cana-525	212	216	〈	〈	NOUN
cana-525	212	217	ʈ̃∗̃ʈ̃𝑙𝜂ɠ(𝑒	ʈ̃∗̃ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	212	218	)	)	PUNCT
cana-525	212	219	,	,	PUNCT
cana-525	212	220	ʈ̃∗̃ʈ̃	ʈ̃∗̃ʈ̃	NOUN
cana-525	212	221	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	NOUN
cana-525	212	222	)	)	PUNCT
cana-525	212	223	〉	〉	NOUN
cana-525	213	1	̃	̃	PROPN
cana-525	213	2	since	since	SCONJ
cana-525	213	3	‖ʈ̃∗̃ʈ̃̃‖	‖ʈ̃∗̃ʈ̃̃‖	PROPN
cana-525	213	4	=	=	SYM
cana-525	213	5	̃	̃	PROPN
cana-525	213	6	‖ʈ̃‖	‖ʈ̃‖	NOUN
cana-525	213	7	2	2	NUM
cana-525	213	8	‖ʈ̃2𝑙𝜂ɠ(𝑒	‖ʈ̃2𝑙𝜂ɠ(𝑒	ADJ
cana-525	214	1	)	)	PUNCT
cana-525	214	2	‖	‖	PROPN
cana-525	214	3	2	2	NUM
cana-525	214	4	≥̃	≥̃	X
cana-525	214	5	‖ʈ̃𝑙𝜂ɠ(𝑒	‖ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	214	6	)	)	PUNCT
cana-525	214	7	̃	̃	NOUN
cana-525	214	8	‖	‖	ADJ
cana-525	214	9	4	4	NUM
cana-525	214	10	⇒	⇒	NOUN
cana-525	214	11	‖ʈ̃2𝑙𝜂ɠ(𝑒	‖ʈ̃2𝑙𝜂ɠ(𝑒	ADV
cana-525	214	12	)	)	PUNCT
cana-525	214	13	‖	‖	ADJ
cana-525	214	14	≥̃	≥̃	PUNCT
cana-525	214	15	‖ʈ̃𝑙𝜂ɠ(𝑒	‖ʈ̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	214	16	)	)	PUNCT
cana-525	214	17	̃	̃	NOUN
cana-525	214	18	‖	‖	ADJ
cana-525	214	19	2	2	NUM
cana-525	214	20	hence	hence	ADV
cana-525	214	21	ʈ̃	ʈ̃	PROPN
cana-525	214	22	is	be	AUX
cana-525	214	23	fspn	fspn	ADJ
cana-525	214	24	operator	operator	NOUN
cana-525	214	25	theorem	theorem	VERB
cana-525	214	26	3.14	3.14	NUM
cana-525	214	27	:	:	PUNCT
cana-525	214	28	let	let	VERB
cana-525	214	29	ʈ̃	ʈ̃	PROPN
cana-525	214	30	∈	∈	PROPN
cana-525	214	31	�	�	PROPN
cana-525	214	32	̃	̃	PROPN
cana-525	214	33	�	�	PROPN
cana-525	214	34	(	(	PUNCT
cana-525	214	35	�	�	PROPN
cana-525	214	36	̃	̃	PROPN
cana-525	214	37	�	�	PROPN
cana-525	214	38	)	)	PUNCT
cana-525	214	39	is	be	AUX
cana-525	214	40	a	a	DET
cana-525	214	41	fsn	fsn	NOUN
cana-525	214	42	then	then	ADV
cana-525	214	43	ʈ̃∗̃	ʈ̃∗̃	NOUN
cana-525	214	44	is	be	AUX
cana-525	214	45	a	a	DET
cana-525	214	46	fspn	fspn	ADJ
cana-525	214	47	proof	proof	NOUN
cana-525	214	48	:	:	PUNCT
cana-525	214	49	since	since	SCONJ
cana-525	214	50	ʈ̃	ʈ̃	PROPN
cana-525	214	51	is	be	AUX
cana-525	214	52	fuzzy	fuzzy	ADJ
cana-525	214	53	soft	soft	ADJ
cana-525	214	54	normal	normal	ADJ
cana-525	214	55	operator	operator	NOUN
cana-525	214	56	we	we	PRON
cana-525	214	57	know	know	VERB
cana-525	214	58	that	that	SCONJ
cana-525	214	59	ʈ̃∗̃ʈ	ʈ̃∗̃ʈ	VERB
cana-525	215	1	̃	̃	PROPN
cana-525	215	2	=	=	SYM
cana-525	215	3	̃	̃	NOUN
cana-525	215	4	ʈ̃	ʈ̃	PROPN
cana-525	215	5	ʈ̃∗̃	ʈ̃∗̃	NOUN
cana-525	215	6	if	if	SCONJ
cana-525	215	7	and	and	CCONJ
cana-525	215	8	only	only	ADV
cana-525	215	9	if	if	SCONJ
cana-525	215	10	‖ʈ̃∗̃	‖ʈ̃∗̃	ADJ
cana-525	215	11	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	215	12	)	)	PUNCT
cana-525	215	13	‖	‖	PROPN
cana-525	216	1	̃	̃	PROPN
cana-525	216	2	=	=	SYM
cana-525	216	3	̃	̃	PROPN
cana-525	216	4	‖ʈ̃	‖ʈ̃	PUNCT
cana-525	216	5	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PRON
cana-525	216	6	)	)	PUNCT
cana-525	217	1	‖	‖	PROPN
cana-525	217	2	̃	̃	PROPN
cana-525	217	3	for	for	ADP
cana-525	217	4	every	every	DET
cana-525	217	5	unit	unit	NOUN
cana-525	217	6	vector	vector	NOUN
cana-525	217	7	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	217	8	)	)	PUNCT
cana-525	217	9	∈̃	∈̃	PROPN
cana-525	217	10	�	�	PROPN
cana-525	217	11	̃	̃	PROPN
cana-525	217	12	�	�	PROPN
cana-525	217	13	let	let	VERB
cana-525	217	14	‖ʈ̃∗̃	‖ʈ̃∗̃	PROPN
cana-525	217	15	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	ADV
cana-525	217	16	)	)	PUNCT
cana-525	217	17	‖	‖	PROPN
cana-525	218	1	̃	̃	PROPN
cana-525	218	2	=	=	SYM
cana-525	218	3	̃	̃	NOUN
cana-525	218	4	〈	〈	NOUN
cana-525	218	5	ʈ̃∗̃	ʈ̃∗̃	NOUN
cana-525	218	6	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	NOUN
cana-525	218	7	)	)	PUNCT
cana-525	218	8	,	,	PUNCT
cana-525	218	9	ʈ̃∗̃	ʈ̃∗̃	ADJ
cana-525	218	10	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	NOUN
cana-525	218	11	)	)	PUNCT
cana-525	218	12	̃	̃	NOUN
cana-525	218	13	〉	〉	NOUN
cana-525	218	14	=	=	SYM
cana-525	218	15	̃	̃	NOUN
cana-525	218	16	〈	〈	NOUN
cana-525	218	17	ʈ̃∗̃	ʈ̃∗̃	NOUN
cana-525	218	18	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	NOUN
cana-525	218	19	)	)	PUNCT
cana-525	218	20	,	,	PUNCT
cana-525	218	21	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	218	22	)	)	PUNCT
cana-525	218	23	̃	̃	NOUN
cana-525	218	24	〉	〉	NOUN
cana-525	218	25	=	=	SYM
cana-525	218	26	̃	̃	NOUN
cana-525	218	27	〈	〈	NOUN
cana-525	218	28	ʈ̃	ʈ̃	PROPN
cana-525	218	29	ʈ̃∗̃	ʈ̃∗̃	NOUN
cana-525	218	30	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	218	31	)	)	PUNCT
cana-525	218	32	,	,	PUNCT
cana-525	218	33	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	218	34	)	)	PUNCT
cana-525	218	35	̃	̃	PROPN
cana-525	218	36	〉	〉	NOUN
cana-525	218	37	communications	communication	NOUN
cana-525	218	38	on	on	ADP
cana-525	218	39	applied	apply	VERB
cana-525	218	40	nonlinear	nonlinear	ADJ
cana-525	218	41	analysis	analysis	NOUN
cana-525	218	42	issn	issn	NOUN
cana-525	218	43	:	:	PUNCT
cana-525	218	44	1074	1074	NUM
cana-525	218	45	-	-	PUNCT
cana-525	218	46	133x	133x	NUM
cana-525	218	47	vol	vol	NOUN
cana-525	218	48	31	31	NUM
cana-525	218	49	no	no	NOUN
cana-525	218	50	.	.	NOUN
cana-525	218	51	2	2	NUM
cana-525	218	52	(	(	PUNCT
cana-525	218	53	2024	2024	NUM
cana-525	218	54	)	)	PUNCT
cana-525	219	1	141	141	NUM
cana-525	219	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-525	219	3	=	=	SYM
cana-525	219	4	̃	̃	NOUN
cana-525	219	5	〈	〈	ADJ
cana-525	219	6	(	(	PUNCT
cana-525	219	7	ʈ̃∗̃ʈ	ʈ̃∗̃ʈ	NOUN
cana-525	219	8	̃)𝑙𝜂ɠ(𝑒	̃)𝑙𝜂ɠ(𝑒	NOUN
cana-525	219	9	)	)	PUNCT
cana-525	219	10	,	,	PUNCT
cana-525	219	11	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	X
cana-525	219	12	)	)	PUNCT
cana-525	219	13	̃	̃	NOUN
cana-525	219	14	〉	〉	NOUN
cana-525	219	15	=	=	SYM
cana-525	219	16	̃	̃	ADP
cana-525	219	17	〈	〈	NOUN
cana-525	219	18	ʈ̃2	ʈ̃2	PROPN
cana-525	219	19	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	219	20	)	)	PUNCT
cana-525	219	21	,	,	PUNCT
cana-525	219	22	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PROPN
cana-525	219	23	)	)	PUNCT
cana-525	219	24	〉	〉	NOUN
cana-525	219	25	≤̃	≤̃	NOUN
cana-525	219	26	‖(ʈ̃∗̃	‖(ʈ̃∗̃	NOUN
cana-525	219	27	)	)	PUNCT
cana-525	219	28	2	2	NUM
cana-525	219	29	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PRON
cana-525	219	30	)	)	PUNCT
cana-525	219	31	‖	‖	PROPN
cana-525	219	32	̃	̃	PROPN
cana-525	219	33	‖𝑙𝜂ɠ(𝑒	‖𝑙𝜂ɠ(𝑒	ADJ
cana-525	219	34	)	)	PUNCT
cana-525	219	35	̃‖	̃‖	VERB
cana-525	219	36	‖ʈ̃∗̃𝑙𝜂ɠ(𝑒	‖ʈ̃∗̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	219	37	)	)	PUNCT
cana-525	219	38	̃	̃	ADV
cana-525	219	39	‖	‖	ADJ
cana-525	219	40	2	2	NUM
cana-525	219	41	≤̃	≤̃	NOUN
cana-525	219	42	‖(ʈ̃∗̃	‖(ʈ̃∗̃	NOUN
cana-525	219	43	)	)	PUNCT
cana-525	219	44	2	2	NUM
cana-525	219	45	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PRON
cana-525	219	46	)	)	PUNCT
cana-525	219	47	‖	‖	PROPN
cana-525	219	48	̃	̃	PROPN
cana-525	219	49	‖𝑙𝜂ɠ(𝑒	‖𝑙𝜂ɠ(𝑒	NOUN
cana-525	219	50	)	)	PUNCT
cana-525	219	51	̃‖	̃‖	NOUN
cana-525	219	52	implies	imply	VERB
cana-525	219	53	that	that	SCONJ
cana-525	219	54	‖ʈ̃∗̃𝑙𝜂ɠ(𝑒	‖ʈ̃∗̃𝑙𝜂ɠ(𝑒	NOUN
cana-525	219	55	)	)	PUNCT
cana-525	219	56	̃	̃	ADV
cana-525	219	57	‖	‖	ADJ
cana-525	219	58	2	2	NUM
cana-525	219	59	≤̃	≤̃	NOUN
cana-525	219	60	‖(ʈ̃∗̃	‖(ʈ̃∗̃	NOUN
cana-525	219	61	)	)	PUNCT
cana-525	219	62	2	2	NUM
cana-525	219	63	𝑙𝜂ɠ(𝑒	𝑙𝜂ɠ(𝑒	PRON
cana-525	219	64	)	)	PUNCT
cana-525	219	65	‖	‖	PROPN
cana-525	220	1	̃	̃	PROPN
cana-525	220	2	therefore	therefore	ADV
cana-525	220	3	,	,	PUNCT
cana-525	220	4	ʈ̃∗̃	ʈ̃∗̃	NOUN
cana-525	220	5	is	be	AUX
cana-525	220	6	fspn	fspn	ADJ
cana-525	220	7	iv	iv	NUM
cana-525	220	8	conclusion	conclusion	NOUN
cana-525	220	9	the	the	DET
cana-525	220	10	ideas	idea	NOUN
cana-525	220	11	of	of	ADP
cana-525	220	12	normed	normed	ADJ
cana-525	220	13	space	space	NOUN
cana-525	220	14	,	,	PUNCT
cana-525	220	15	metric	metric	ADJ
cana-525	220	16	space	space	NOUN
cana-525	220	17	,	,	PUNCT
cana-525	220	18	and	and	CCONJ
cana-525	220	19	hilbert	hilbert	NOUN
cana-525	220	20	space	space	NOUN
cana-525	220	21	provide	provide	VERB
cana-525	220	22	the	the	DET
cana-525	220	23	soft	soft	ADJ
cana-525	220	24	and	and	CCONJ
cana-525	220	25	fuzzy	fuzzy	ADJ
cana-525	220	26	updates	update	NOUN
cana-525	220	27	.	.	PUNCT
cana-525	221	1	there	there	PRON
cana-525	221	2	are	be	VERB
cana-525	221	3	many	many	ADJ
cana-525	221	4	uses	use	NOUN
cana-525	221	5	for	for	ADP
cana-525	221	6	combining	combine	VERB
cana-525	221	7	fuzzy	fuzzy	ADJ
cana-525	221	8	and	and	CCONJ
cana-525	221	9	soft	soft	ADJ
cana-525	221	10	ideas	idea	NOUN
cana-525	221	11	.	.	PUNCT
cana-525	222	1	the	the	DET
cana-525	222	2	fuzzy	fuzzy	ADJ
cana-525	222	3	soft	soft	ADJ
cana-525	222	4	paranormal	paranormal	ADJ
cana-525	222	5	operator	operator	NOUN
cana-525	222	6	has	have	AUX
cana-525	222	7	been	be	AUX
cana-525	222	8	defined	define	VERB
cana-525	222	9	and	and	CCONJ
cana-525	222	10	explained	explain	VERB
cana-525	222	11	in	in	ADP
cana-525	222	12	this	this	DET
cana-525	222	13	article	article	NOUN
cana-525	222	14	.	.	PUNCT
cana-525	223	1	acknowledgement	acknowledgement	NOUN
cana-525	223	2	the	the	DET
cana-525	223	3	authors	author	NOUN
cana-525	223	4	want	want	VERB
cana-525	223	5	to	to	PART
cana-525	223	6	express	express	VERB
cana-525	223	7	their	their	PRON
cana-525	223	8	heartfelt	heartfelt	ADJ
cana-525	223	9	appreciation	appreciation	NOUN
cana-525	223	10	to	to	ADP
cana-525	223	11	the	the	DET
cana-525	223	12	editor	editor	NOUN
cana-525	223	13	and	and	CCONJ
cana-525	223	14	reviewers	reviewer	NOUN
cana-525	223	15	for	for	ADP
cana-525	223	16	their	their	PRON
cana-525	223	17	insightful	insightful	ADJ
cana-525	223	18	criticism	criticism	NOUN
cana-525	223	19	and	and	CCONJ
cana-525	223	20	recommendations	recommendation	NOUN
cana-525	223	21	,	,	PUNCT
cana-525	223	22	which	which	PRON
cana-525	223	23	helped	help	VERB
cana-525	223	24	to	to	PART
cana-525	223	25	shape	shape	VERB
cana-525	223	26	the	the	DET
cana-525	223	27	manuscript	manuscript	NOUN
cana-525	223	28	.	.	PUNCT
cana-525	224	1	references	reference	NOUN
cana-525	224	2	[	[	X
cana-525	224	3	1	1	NUM
cana-525	224	4	]	]	X
cana-525	224	5	radharamani	radharamani	X
cana-525	224	6	.	.	PUNCT
cana-525	225	1	a	a	DET
cana-525	225	2	etal	etal	NOUN
cana-525	225	3	.	.	PUNCT
cana-525	225	4	,	,	PUNCT
cana-525	225	5	fuzzy	fuzzy	ADJ
cana-525	225	6	unitary	unitary	ADJ
cana-525	225	7	operator	operator	NOUN
cana-525	225	8	in	in	ADP
cana-525	225	9	fuzzy	fuzzy	ADJ
cana-525	225	10	hilbert	hilbert	NOUN
cana-525	225	11	space	space	NOUN
cana-525	225	12	and	and	CCONJ
cana-525	225	13	its	its	PRON
cana-525	225	14	properties	property	NOUN
cana-525	225	15	,	,	PUNCT
cana-525	225	16	international	international	ADJ
cana-525	225	17	journal	journal	NOUN
cana-525	225	18	of	of	ADP
cana-525	225	19	research	research	NOUN
cana-525	225	20	and	and	CCONJ
cana-525	225	21	analytic	analytic	ADJ
cana-525	225	22	reviews(ijrar	reviews(ijrar	NOUN
cana-525	225	23	)	)	PUNCT
cana-525	225	24	,	,	PUNCT
cana-525	225	25	2018	2018	NUM
cana-525	225	26	,	,	PUNCT
cana-525	225	27	5(4	5(4	NUM
cana-525	225	28	)	)	PUNCT
cana-525	225	29	,	,	PUNCT
cana-525	225	30	258	258	NUM
cana-525	225	31	-	-	SYM
cana-525	225	32	261	261	NUM
cana-525	225	33	.	.	PUNCT
cana-525	226	1	[	[	X
cana-525	226	2	2	2	NUM
cana-525	226	3	]	]	X
cana-525	226	4	n.	n.	NOUN
cana-525	226	5	faried	farie	VERB
cana-525	226	6	,	,	PUNCT
cana-525	226	7	m.s.s	m.s.s	PROPN
cana-525	226	8	.	.	PROPN
cana-525	226	9	ali	ali	PROPN
cana-525	226	10	and	and	CCONJ
cana-525	226	11	h.h	h.h	PROPN
cana-525	226	12	.	.	PROPN
cana-525	226	13	sakr	sakr	PROPN
cana-525	226	14	,	,	PUNCT
cana-525	226	15	on	on	ADP
cana-525	226	16	fuzzy	fuzzy	ADJ
cana-525	226	17	soft	soft	ADJ
cana-525	226	18	linear	linear	NOUN
cana-525	226	19	operators	operator	NOUN
cana-525	226	20	in	in	ADP
cana-525	226	21	fuzzy	fuzzy	ADJ
cana-525	226	22	soft	soft	ADJ
cana-525	226	23	hilbert	hilbert	NOUN
cana-525	226	24	spaces	space	NOUN
cana-525	226	25	,	,	PUNCT
cana-525	226	26	abst	abst	PROPN
cana-525	226	27	.	.	PUNCT
cana-525	226	28	appl	appl	PROPN
cana-525	226	29	.	.	PUNCT
cana-525	227	1	anal.2020	anal.2020	X
cana-525	227	2	.	.	PUNCT
cana-525	228	1	[	[	X
cana-525	228	2	3	3	NUM
cana-525	228	3	]	]	X
cana-525	228	4	la	la	PROPN
cana-525	228	5	.	.	PUNCT
cana-525	228	6	zadeh	zadeh	PROPN
cana-525	228	7	,	,	PUNCT
cana-525	228	8	fuzzy	fuzzy	ADJ
cana-525	228	9	sets	set	NOUN
cana-525	228	10	,	,	PUNCT
cana-525	228	11	inf.control	inf.control	PROPN
cana-525	228	12	,	,	PUNCT
cana-525	228	13	vol.8	vol.8	PROPN
cana-525	228	14	,	,	PUNCT
cana-525	228	15	no.3,pp	no.3,pp	NOUN
cana-525	228	16	.	.	PUNCT
cana-525	228	17	338	338	NUM
cana-525	228	18	-	-	SYM
cana-525	228	19	353	353	NUM
cana-525	228	20	,	,	PUNCT
cana-525	228	21	1965	1965	NUM
cana-525	228	22	.	.	PUNCT
cana-525	229	1	[	[	X
cana-525	229	2	4	4	X
cana-525	229	3	]	]	X
cana-525	229	4	d.	d.	PROPN
cana-525	229	5	molodtsov	molodtsov	PROPN
cana-525	229	6	,	,	PUNCT
cana-525	229	7	soft	soft	ADJ
cana-525	229	8	set	set	NOUN
cana-525	229	9	theory	theory	NOUN
cana-525	229	10	-	-	PUNCT
cana-525	229	11	first	first	ADJ
cana-525	229	12	results	result	NOUN
cana-525	229	13	,	,	PUNCT
cana-525	229	14	comput	comput	NOUN
cana-525	229	15	.	.	PUNCT
cana-525	230	1	math.appl	math.appl	NOUN
cana-525	230	2	.	.	PUNCT
cana-525	231	1	37	37	NUM
cana-525	231	2	.	.	X
cana-525	231	3	19	19	NUM
cana-525	231	4	-	-	SYM
cana-525	231	5	31	31	NUM
cana-525	231	6	(	(	PUNCT
cana-525	231	7	1999	1999	NUM
cana-525	231	8	)	)	PUNCT
cana-525	231	9	.	.	PUNCT
cana-525	232	1	[	[	X
cana-525	232	2	5	5	X
cana-525	232	3	]	]	X
cana-525	232	4	p.k	p.k	PROPN
cana-525	232	5	.	.	PROPN
cana-525	232	6	maji	maji	PROPN
cana-525	232	7	,	,	PUNCT
cana-525	232	8	r.	r.	PROPN
cana-525	232	9	biswas	biswas	PROPN
cana-525	232	10	and	and	CCONJ
cana-525	232	11	a.r	a.r	PROPN
cana-525	232	12	.	.	PROPN
cana-525	232	13	roy	roy	PROPN
cana-525	232	14	,	,	PUNCT
cana-525	232	15	fuzzy	fuzzy	ADJ
cana-525	232	16	soft	soft	ADJ
cana-525	232	17	set	set	NOUN
cana-525	232	18	,	,	PUNCT
cana-525	232	19	j.	j.	PROPN
cana-525	232	20	fuzzy	fuzzy	PROPN
cana-525	232	21	math	math	PROPN
cana-525	232	22	..	..	X
cana-525	232	23	9(3	9(3	NUM
cana-525	232	24	)	)	PUNCT
cana-525	232	25	,	,	PUNCT
cana-525	232	26	677	677	NUM
cana-525	232	27	-	-	SYM
cana-525	232	28	692	692	NUM
cana-525	232	29	(	(	PUNCT
cana-525	232	30	2001	2001	NUM
cana-525	232	31	)	)	PUNCT
cana-525	232	32	.	.	PUNCT
cana-525	233	1	[	[	X
cana-525	233	2	6	6	NUM
cana-525	233	3	]	]	PUNCT
cana-525	233	4	t.	t.	NOUN
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cana-525	233	7	m.m	m.m	PROPN
cana-525	233	8	.	.	PROPN
cana-525	233	9	priyanga	priyanga	PROPN
cana-525	233	10	,	,	PUNCT
cana-525	233	11	a	a	DET
cana-525	233	12	new	new	ADJ
cana-525	233	13	notion	notion	NOUN
cana-525	233	14	for	for	ADP
cana-525	233	15	fuzzy	fuzzy	ADJ
cana-525	233	16	soft	soft	ADJ
cana-525	233	17	normed	norme	VERB
cana-525	233	18	linear	linear	ADJ
cana-525	233	19	space	space	NOUN
cana-525	233	20	,	,	PUNCT
cana-525	233	21	int	int	NOUN
cana-525	233	22	.	.	PUNCT
cana-525	234	1	j.	j.	PROPN
cana-525	234	2	fuzzy	fuzzy	PROPN
cana-525	234	3	math	math	PROPN
cana-525	234	4	.	.	PUNCT
cana-525	235	1	arch	arch	NOUN
cana-525	235	2	.	.	PUNCT
cana-525	236	1	9(1	9(1	NUM
cana-525	236	2	)	)	PUNCT
cana-525	236	3	,	,	PUNCT
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cana-525	236	5	-	-	SYM
cana-525	236	6	90	90	NUM
cana-525	236	7	(	(	PUNCT
cana-525	236	8	2015	2015	NUM
cana-525	236	9	)	)	PUNCT
cana-525	236	10	.	.	PUNCT
cana-525	237	1	[	[	X
cana-525	237	2	7	7	X
cana-525	237	3	]	]	X
cana-525	237	4	t.	t.	NOUN
cana-525	237	5	beaula	beaula	NOUN
cana-525	237	6	and	and	CCONJ
cana-525	237	7	c.	c.	PROPN
cana-525	237	8	gunaseeli	gunaseeli	PROPN
cana-525	237	9	,	,	PUNCT
cana-525	237	10	on	on	ADP
cana-525	237	11	fuzzy	fuzzy	ADJ
cana-525	237	12	soft	soft	ADJ
cana-525	237	13	metric	metric	ADJ
cana-525	237	14	spaces	space	NOUN
cana-525	237	15	,	,	PUNCT
cana-525	237	16	malaya	malaya	PROPN
cana-525	237	17	j.	j.	PROPN
cana-525	237	18	mat	mat	PROPN
cana-525	237	19	.	.	PROPN
cana-525	237	20	2(3	2(3	NUM
cana-525	237	21	)	)	PUNCT
cana-525	237	22	,	,	PUNCT
cana-525	237	23	197	197	NUM
cana-525	237	24	-	-	SYM
cana-525	237	25	202	202	NUM
cana-525	237	26	(	(	PUNCT
cana-525	237	27	2014	2014	NUM
cana-525	237	28	)	)	PUNCT
cana-525	237	29	.	.	PUNCT
cana-525	238	1	[	[	X
cana-525	238	2	8	8	NUM
cana-525	238	3	]	]	X
cana-525	238	4	n.	n.	NOUN
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cana-525	238	6	,	,	PUNCT
cana-525	238	7	m.s.s	m.s.s	PROPN
cana-525	238	8	.	.	PROPN
cana-525	238	9	ali	ali	PROPN
cana-525	238	10	and	and	CCONJ
cana-525	238	11	h.h	h.h	PROPN
cana-525	238	12	.	.	PROPN
cana-525	238	13	sakr	sakr	PROPN
cana-525	238	14	.	.	PUNCT
cana-525	239	1	fuzzy	fuzzy	ADJ
cana-525	239	2	soft	soft	ADJ
cana-525	239	3	inner	inner	ADJ
cana-525	239	4	product	product	NOUN
cana-525	239	5	spaces	space	NOUN
cana-525	239	6	,	,	PUNCT
cana-525	239	7	appl.math	appl.math	PROPN
cana-525	239	8	.	.	PUNCT
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cana-525	239	10	.	.	PUNCT
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cana-525	240	4	)	)	PUNCT
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cana-525	240	7	)	)	PUNCT
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cana-525	241	2	9	9	NUM
cana-525	241	3	]	]	X
cana-525	241	4	n.	n.	NOUN
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cana-525	241	6	,	,	PUNCT
cana-525	241	7	m.s.s	m.s.s	PROPN
cana-525	241	8	.	.	PROPN
cana-525	241	9	ali	ali	PROPN
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cana-525	241	11	h.h	h.h	PROPN
cana-525	241	12	.	.	PROPN
cana-525	241	13	sakr	sakr	PROPN
cana-525	241	14	.	.	PUNCT
cana-525	242	1	fuzzy	fuzzy	ADJ
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cana-525	242	5	,	,	PUNCT
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cana-525	242	8	.	.	PUNCT
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cana-525	243	2	.	.	PUNCT
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cana-525	243	4	)	)	PUNCT
cana-525	243	5	,	,	PUNCT
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cana-525	243	7	-	-	SYM
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cana-525	243	11	)	)	PUNCT
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cana-525	244	1	[	[	X
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cana-525	244	3	]	]	X
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cana-525	244	5	.	.	PROPN
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cana-525	244	8	r.	r.	PROPN
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cana-525	244	12	a.r	a.r	PROPN
cana-525	244	13	.	.	PROPN
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cana-525	244	16	“	"	PUNCT
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cana-525	244	18	set	set	NOUN
cana-525	244	19	theory	theory	NOUN
cana-525	244	20	”	"	PUNCT
cana-525	244	21	,	,	PUNCT
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cana-525	244	23	&	&	CCONJ
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cana-525	244	33	-	-	SYM
cana-525	244	34	5	5	NUM
cana-525	244	35	,	,	PUNCT
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cana-525	244	37	.	.	PUNCT
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cana-525	245	2	-	-	SYM
cana-525	245	3	562	562	NUM
cana-525	245	4	,	,	PUNCT
cana-525	245	5	2003	2003	NUM
cana-525	245	6	.	.	PUNCT
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cana-525	246	2	11	11	NUM
cana-525	246	3	]	]	X
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cana-525	246	5	.	.	PROPN
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cana-525	246	9	ali	ali	PROPN
cana-525	246	10	qassim	qassim	PROPN
cana-525	246	11	jabur	jabur	PROPN
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cana-525	246	13	on	on	ADP
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cana-525	246	18	,	,	PUNCT
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cana-525	246	22	:	:	PUNCT
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cana-525	246	26	)	)	PUNCT
cana-525	246	27	032002	032002	NUM
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cana-525	247	3	]	]	X
cana-525	247	4	radharamani	radharamani	X
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cana-525	248	1	a	a	DET
cana-525	248	2	etal	etal	NOUN
cana-525	248	3	.	.	PUNCT
cana-525	248	4	,	,	PUNCT
cana-525	248	5	fuzzy	fuzzy	ADJ
cana-525	248	6	partial	partial	ADJ
cana-525	248	7	isometry	isometry	NOUN
cana-525	248	8	operator	operator	NOUN
cana-525	248	9	and	and	CCONJ
cana-525	248	10	its	its	PRON
cana-525	248	11	characteristics	characteristic	NOUN
cana-525	248	12	,	,	PUNCT
cana-525	248	13	iosr	iosr	ADJ
cana-525	248	14	journal	journal	NOUN
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cana-525	248	17	(	(	PUNCT
cana-525	248	18	iosrjen	iosrjen	NOUN
cana-525	248	19	)	)	PUNCT
cana-525	248	20	,	,	PUNCT
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cana-525	248	22	,	,	PUNCT
cana-525	248	23	2019	2019	NUM
cana-525	248	24	,	,	PUNCT
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cana-525	248	26	-	-	PUNCT
cana-525	248	27	58	58	NUM
cana-525	248	28	.	.	PUNCT
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cana-525	249	2	13	13	NUM
cana-525	249	3	]	]	X
cana-525	249	4	dr	dr	PROPN
cana-525	249	5	.	.	PROPN
cana-525	249	6	salim	salim	PROPN
cana-525	249	7	dawood	dawood	PROPN
cana-525	249	8	,	,	PUNCT
cana-525	249	9	ali	ali	PROPN
cana-525	249	10	qassim	qassim	PROPN
cana-525	249	11	jabur	jabur	PROPN
cana-525	249	12	,	,	PUNCT
cana-525	249	13	on	on	ADP
cana-525	249	14	fuzzy	fuzzy	ADJ
cana-525	249	15	soft	soft	ADJ
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cana-525	249	17	operators	operator	NOUN
cana-525	249	18	in	in	ADP
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cana-525	249	21	,	,	PUNCT
cana-525	249	22	al	al	PROPN
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cana-525	249	29	vol.(26	vol.(26	PROPN
cana-525	249	30	)	)	PUNCT
cana-525	249	31	issue	issue	NOUN
cana-525	249	32	(	(	PUNCT
cana-525	249	33	1)(2021)pp	1)(2021)pp	NUM
cana-525	249	34	math.112	math.112	PROPN
cana-525	249	35	-	-	PUNCT
cana-525	249	36	123	123	NUM
cana-525	250	1	[	[	X
cana-525	250	2	14	14	NUM
cana-525	250	3	]	]	X
cana-525	250	4	n.	n.	PROPN
cana-525	250	5	faried	faried	PROPN
cana-525	250	6	,	,	PUNCT
cana-525	250	7	m.s.s	m.s.s	PROPN
cana-525	250	8	.	.	PROPN
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cana-525	250	10	and	and	CCONJ
cana-525	250	11	h.h	h.h	PROPN
cana-525	250	12	.	.	PROPN
cana-525	250	13	sakr	sakr	PROPN
cana-525	250	14	.	.	PUNCT
cana-525	251	1	a	a	DET
cana-525	251	2	note	note	NOUN
cana-525	251	3	on	on	ADP
cana-525	251	4	fuzzy	fuzzy	ADJ
cana-525	251	5	soft	soft	ADJ
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cana-525	251	7	operators	operator	NOUN
cana-525	251	8	,	,	PUNCT
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cana-525	251	10	,	,	PUNCT
cana-525	251	11	no.1	no.1	NUM
cana-525	251	12	,	,	PUNCT
cana-525	251	13	13(2021	13(2021	NUM
cana-525	251	14	)	)	PUNCT
cana-525	252	1	[	[	X
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cana-525	252	3	]	]	X
cana-525	252	4	radharamani	radharamani	X
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cana-525	253	1	a	a	PRON
cana-525	253	2	,	,	PUNCT
cana-525	253	3	t	t	PROPN
cana-525	253	4	nagajothi	nagajothi	ADV
cana-525	253	5	,	,	PUNCT
cana-525	253	6	fuzzy	fuzzy	ADJ
cana-525	253	7	soft	soft	ADJ
cana-525	253	8	hyponormal	hyponormal	ADJ
cana-525	253	9	operator	operator	NOUN
cana-525	253	10	in	in	ADP
cana-525	253	11	fuzzy	fuzzy	ADJ
cana-525	253	12	soft	soft	ADJ
cana-525	253	13	hilbert	hilbert	NOUN
cana-525	253	14	space	space	NOUN
cana-525	253	15	,	,	PUNCT
cana-525	253	16	strad	strad	PROPN
cana-525	253	17	research	research	PROPN
cana-525	253	18	,	,	PUNCT
cana-525	253	19	vol.9	vol.9	PROPN
cana-525	253	20	,	,	PUNCT
cana-525	253	21	issue	issue	VERB
cana-525	253	22	3	3	NUM
cana-525	253	23	-	-	SYM
cana-525	253	24	2022	2022	NUM
cana-525	253	25	[	[	X
cana-525	253	26	16	16	NUM
cana-525	253	27	]	]	X
cana-525	253	28	radharamani	radharamani	X
cana-525	253	29	.	.	PUNCT
cana-525	254	1	a	a	DET
cana-525	254	2	,	,	PUNCT
cana-525	254	3	t	t	PROPN
cana-525	254	4	nagajothi	nagajothi	ADV
cana-525	254	5	,	,	PUNCT
cana-525	254	6	m	m	NOUN
cana-525	254	7	-	-	ADJ
cana-525	254	8	fuzzy	fuzzy	ADJ
cana-525	254	9	soft	soft	ADJ
cana-525	254	10	hyponormal	hyponormal	ADJ
cana-525	254	11	operator	operator	NOUN
cana-525	254	12	in	in	ADP
cana-525	254	13	fuzzy	fuzzy	ADJ
cana-525	254	14	soft	soft	ADJ
cana-525	254	15	hilbert	hilbert	NOUN
cana-525	254	16	space	space	NOUN
cana-525	254	17	,	,	PUNCT
cana-525	254	18	journal	journal	NOUN
cana-525	254	19	of	of	ADP
cana-525	254	20	data	datum	NOUN
cana-525	254	21	acquisition	acquisition	NOUN
cana-525	254	22	and	and	CCONJ
cana-525	254	23	processing	processing	NOUN
cana-525	254	24	,	,	PUNCT
cana-525	254	25	vol.38	vol.38	NOUN
cana-525	254	26	,	,	PUNCT
cana-525	254	27	issue	issue	NOUN
cana-525	254	28	(	(	PUNCT
cana-525	254	29	1)2023	1)2023	NUM