id	sid	tid	token	lemma	pos
cana-3950	1	1	communications	communication	NOUN
cana-3950	1	2	on	on	ADP
cana-3950	1	3	applied	apply	VERB
cana-3950	1	4	nonlinear	nonlinear	ADJ
cana-3950	1	5	analysis	analysis	NOUN
cana-3950	1	6	issn	issn	NOUN
cana-3950	1	7	:	:	PUNCT
cana-3950	1	8	1074	1074	NUM
cana-3950	1	9	-	-	PUNCT
cana-3950	1	10	133x	133x	NUM
cana-3950	1	11	vol	vol	NOUN
cana-3950	1	12	32	32	NUM
cana-3950	1	13	no	no	NOUN
cana-3950	1	14	.	.	PUNCT
cana-3950	2	1	9s	9s	NUM
cana-3950	2	2	(	(	PUNCT
cana-3950	2	3	2025	2025	NUM
cana-3950	2	4	)	)	PUNCT
cana-3950	2	5	386	386	NUM
cana-3950	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3950	2	7	application	application	NOUN
cana-3950	2	8	of	of	ADP
cana-3950	2	9	beal	beal	NOUN
cana-3950	2	10	’s	’s	PART
cana-3950	2	11	fuzzy	fuzzy	ADJ
cana-3950	2	12	sets	set	NOUN
cana-3950	2	13	in	in	ADP
cana-3950	2	14	pattern	pattern	NOUN
cana-3950	2	15	recognitions	recognition	NOUN
cana-3950	2	16	under	under	ADP
cana-3950	2	17	similarity	similarity	NOUN
cana-3950	2	18	measures	measure	NOUN
cana-3950	2	19	dr.k.balasubramaniyan1	dr.k.balasubramaniyan1	PROPN
cana-3950	2	20	c.dilly	c.dilly	ADV
cana-3950	2	21	rani	rani	VERB
cana-3950	2	22	2	2	NUM
cana-3950	2	23	1	1	NUM
cana-3950	2	24	assistant	assistant	NOUN
cana-3950	2	25	professor	professor	NOUN
cana-3950	2	26	,	,	PUNCT
cana-3950	2	27	department	department	NOUN
cana-3950	2	28	of	of	ADP
cana-3950	2	29	mathematics	mathematic	NOUN
cana-3950	2	30	,	,	PUNCT
cana-3950	2	31	(	(	PUNCT
cana-3950	2	32	deputed	depute	VERB
cana-3950	2	33	to	to	ADP
cana-3950	2	34	annamalai	annamalai	PROPN
cana-3950	2	35	university	university	PROPN
cana-3950	2	36	,	,	PUNCT
cana-3950	2	37	annamalai	annamalai	PROPN
cana-3950	2	38	nagar	nagar	PROPN
cana-3950	2	39	,	,	PUNCT
cana-3950	2	40	chidambaram	chidambaram	PROPN
cana-3950	2	41	)	)	PUNCT
cana-3950	2	42	arignar	arignar	PROPN
cana-3950	2	43	anna	anna	PROPN
cana-3950	2	44	government	government	PROPN
cana-3950	2	45	arts	arts	PROPN
cana-3950	2	46	college	college	NOUN
cana-3950	2	47	vadachennimalai-636121	vadachennimalai-636121	NOUN
cana-3950	2	48	,	,	PUNCT
cana-3950	2	49	attur	attur	NOUN
cana-3950	2	50	,	,	PUNCT
cana-3950	2	51	tamilnadu	tamilnadu	NOUN
cana-3950	2	52	,	,	PUNCT
cana-3950	2	53	india	india	PROPN
cana-3950	2	54	.	.	PUNCT
cana-3950	3	1	email	email	NOUN
cana-3950	3	2	id	id	ADP
cana-3950	3	3	:	:	PUNCT
cana-3950	3	4	kgbalumaths@gmail	kgbalumaths@gmail	PROPN
cana-3950	3	5	.	.	NOUN
cana-3950	3	6	2	2	NUM
cana-3950	3	7	assistant	assistant	NOUN
cana-3950	3	8	professor	professor	NOUN
cana-3950	3	9	department	department	PROPN
cana-3950	3	10	of	of	ADP
cana-3950	3	11	mathematics	mathematics	PROPN
cana-3950	3	12	,	,	PUNCT
cana-3950	3	13	j.j.college	j.j.college	NOUN
cana-3950	3	14	of	of	ADP
cana-3950	3	15	engineering	engineering	NOUN
cana-3950	3	16	and	and	CCONJ
cana-3950	3	17	technology	technology	NOUN
cana-3950	3	18	(	(	PUNCT
cana-3950	3	19	sowdambikaa	sowdambikaa	ADJ
cana-3950	3	20	groups	group	NOUN
cana-3950	3	21	of	of	ADP
cana-3950	3	22	institutions	institution	NOUN
cana-3950	3	23	)	)	PUNCT
cana-3950	3	24	,	,	PUNCT
cana-3950	3	25	tiruchirappalli-600	tiruchirappalli-600	NOUN
cana-3950	3	26	009	009	NUM
cana-3950	3	27	,	,	PUNCT
cana-3950	3	28	tamilnadu	tamilnadu	ADJ
cana-3950	3	29	,	,	PUNCT
cana-3950	3	30	india	india	PROPN
cana-3950	3	31	.	.	PUNCT
cana-3950	4	1	email	email	NOUN
cana-3950	5	1	i	i	PROPN
cana-3950	5	2	d	d	PROPN
cana-3950	5	3	:	:	PUNCT
cana-3950	5	4	kani21bala@gmail.com	kani21bala@gmail.com	X
cana-3950	5	5	article	article	NOUN
cana-3950	5	6	history	history	NOUN
cana-3950	5	7	:	:	PUNCT
cana-3950	5	8	received	receive	VERB
cana-3950	5	9	:	:	PUNCT
cana-3950	5	10	10	10	NUM
cana-3950	5	11	-	-	SYM
cana-3950	5	12	11	11	NUM
cana-3950	5	13	-	-	PUNCT
cana-3950	5	14	2024	2024	NUM
cana-3950	5	15	revised	revise	VERB
cana-3950	5	16	:	:	PUNCT
cana-3950	5	17	25	25	NUM
cana-3950	5	18	-	-	SYM
cana-3950	5	19	12	12	NUM
cana-3950	5	20	-	-	PUNCT
cana-3950	5	21	2024	2024	NUM
cana-3950	5	22	accepted	accept	VERB
cana-3950	5	23	:	:	PUNCT
cana-3950	5	24	12	12	NUM
cana-3950	5	25	-	-	SYM
cana-3950	5	26	01	01	NUM
cana-3950	5	27	-	-	PUNCT
cana-3950	5	28	2025	2025	NUM
cana-3950	5	29	abstract	abstract	NOUN
cana-3950	5	30	:	:	PUNCT
cana-3950	5	31	in	in	ADP
cana-3950	5	32	this	this	DET
cana-3950	5	33	article	article	NOUN
cana-3950	5	34	,	,	PUNCT
cana-3950	5	35	we	we	PRON
cana-3950	5	36	established	establish	VERB
cana-3950	5	37	to	to	ADP
cana-3950	5	38	basic	basic	ADJ
cana-3950	5	39	collection	collection	NOUN
cana-3950	5	40	of	of	ADP
cana-3950	5	41	operations	operation	NOUN
cana-3950	5	42	and	and	CCONJ
cana-3950	5	43	construct	construct	VERB
cana-3950	5	44	the	the	DET
cana-3950	5	45	abstract	abstract	ADJ
cana-3950	5	46	properties	property	NOUN
cana-3950	5	47	that	that	PRON
cana-3950	5	48	can	can	AUX
cana-3950	5	49	be	be	AUX
cana-3950	5	50	transmitted	transmit	VERB
cana-3950	5	51	to	to	ADP
cana-3950	5	52	the	the	DET
cana-3950	5	53	different	different	ADJ
cana-3950	5	54	models	model	NOUN
cana-3950	5	55	they	they	PRON
cana-3950	5	56	are	be	AUX
cana-3950	5	57	in	in	ADP
cana-3950	5	58	joined	join	VERB
cana-3950	5	59	with	with	ADP
cana-3950	5	60	then	then	ADV
cana-3950	5	61	to	to	PART
cana-3950	5	62	rank	rank	VERB
cana-3950	5	63	beal	beal	PROPN
cana-3950	5	64	’s	’s	PART
cana-3950	5	65	fuzzy	fuzzy	ADJ
cana-3950	5	66	set	set	NOUN
cana-3950	5	67	.	.	PUNCT
cana-3950	6	1	we	we	PRON
cana-3950	6	2	obtain	obtain	VERB
cana-3950	6	3	the	the	DET
cana-3950	6	4	functions	function	NOUN
cana-3950	6	5	of	of	ADP
cana-3950	6	6	score	score	NOUN
cana-3950	6	7	and	and	CCONJ
cana-3950	6	8	accuracy	accuracy	NOUN
cana-3950	6	9	,	,	PUNCT
cana-3950	6	10	and	and	CCONJ
cana-3950	6	11	formulate	formulate	VERB
cana-3950	6	12	aggregate	aggregate	ADJ
cana-3950	6	13	operators	operator	NOUN
cana-3950	6	14	to	to	PART
cana-3950	6	15	be	be	AUX
cana-3950	6	16	used	use	VERB
cana-3950	6	17	with	with	ADP
cana-3950	6	18	beal	beal	NOUN
cana-3950	6	19	’s	’s	PART
cana-3950	6	20	fuzzy	fuzzy	ADJ
cana-3950	6	21	sets	set	NOUN
cana-3950	6	22	.	.	PUNCT
cana-3950	7	1	alternatively	alternatively	ADV
cana-3950	7	2	,	,	PUNCT
cana-3950	7	3	we	we	PRON
cana-3950	7	4	develop	develop	VERB
cana-3950	7	5	the	the	DET
cana-3950	7	6	successful	successful	ADJ
cana-3950	7	7	techniques	technique	NOUN
cana-3950	7	8	“	"	PUNCT
cana-3950	7	9	aggregative	aggregative	ADJ
cana-3950	7	10	operators	operator	NOUN
cana-3950	7	11	”	"	PUNCT
cana-3950	7	12	to	to	PART
cana-3950	7	13	handle	handle	VERB
cana-3950	7	14	multi	multi	ADJ
cana-3950	7	15	-	-	ADJ
cana-3950	7	16	criteria	criteria	ADJ
cana-3950	7	17	decision	decision	NOUN
cana-3950	7	18	making	make	VERB
cana-3950	7	19	problems	problem	NOUN
cana-3950	7	20	in	in	ADP
cana-3950	7	21	the	the	DET
cana-3950	7	22	beal	beal	NOUN
cana-3950	7	23	’s	’s	PART
cana-3950	7	24	fuzzy	fuzzy	ADJ
cana-3950	7	25	set	set	VERB
cana-3950	7	26	environment	environment	NOUN
cana-3950	7	27	.	.	PUNCT
cana-3950	8	1	the	the	DET
cana-3950	8	2	proposed	propose	VERB
cana-3950	8	3	techniques	technique	NOUN
cana-3950	8	4	has	have	AUX
cana-3950	8	5	been	be	AUX
cana-3950	8	6	illustrated	illustrate	VERB
cana-3950	8	7	and	and	CCONJ
cana-3950	8	8	analysed	analyse	VERB
cana-3950	8	9	through	through	ADP
cana-3950	8	10	suitable	suitable	ADJ
cana-3950	8	11	example	example	NOUN
cana-3950	8	12	.	.	PUNCT
cana-3950	9	1	keywords	keyword	NOUN
cana-3950	9	2	:	:	PUNCT
cana-3950	9	3	fuzzy	fuzzy	ADJ
cana-3950	9	4	set	set	NOUN
cana-3950	9	5	,	,	PUNCT
cana-3950	9	6	beal	beal	NOUN
cana-3950	9	7	’s	’s	PART
cana-3950	9	8	fuzzy	fuzzy	ADJ
cana-3950	9	9	set	set	NOUN
cana-3950	9	10	,	,	PUNCT
cana-3950	9	11	score	score	NOUN
cana-3950	9	12	function	function	NOUN
cana-3950	9	13	,	,	PUNCT
cana-3950	9	14	accuracy	accuracy	NOUN
cana-3950	9	15	function	function	NOUN
cana-3950	9	16	,	,	PUNCT
cana-3950	9	17	aggregate	aggregate	ADJ
cana-3950	9	18	operators	operator	NOUN
cana-3950	9	19	,	,	PUNCT
cana-3950	9	20	decision	decision	NOUN
cana-3950	9	21	making	making	NOUN
cana-3950	9	22	.	.	PUNCT
cana-3950	10	1	ams	am	NOUN
cana-3950	10	2	subject	subject	ADJ
cana-3950	10	3	classification	classification	NOUN
cana-3950	10	4	(	(	PUNCT
cana-3950	10	5	2010	2010	NUM
cana-3950	10	6	):	):	PUNCT
cana-3950	10	7	20n25	20n25	NUM
cana-3950	10	8	,	,	PUNCT
cana-3950	10	9	03f72	03f72	NUM
cana-3950	10	10	,	,	PUNCT
cana-3950	10	11	20n99	20n99	NUM
cana-3950	10	12	1	1	NUM
cana-3950	10	13	.	.	PUNCT
cana-3950	10	14	introduction	introduction	NOUN
cana-3950	10	15	the	the	DET
cana-3950	10	16	problems	problem	NOUN
cana-3950	10	17	in	in	ADP
cana-3950	10	18	the	the	DET
cana-3950	10	19	real	real	ADJ
cana-3950	10	20	world	world	NOUN
cana-3950	10	21	are	be	AUX
cana-3950	10	22	too	too	ADV
cana-3950	10	23	complicated	complicated	ADJ
cana-3950	10	24	since	since	SCONJ
cana-3950	10	25	it	it	PRON
cana-3950	10	26	includes	include	VERB
cana-3950	10	27	ambiguity	ambiguity	NOUN
cana-3950	10	28	,	,	PUNCT
cana-3950	10	29	uncertainty	uncertainty	NOUN
cana-3950	10	30	,	,	PUNCT
cana-3950	10	31	or	or	CCONJ
cana-3950	10	32	insufficient	insufficient	ADJ
cana-3950	10	33	knowledge	knowledge	NOUN
cana-3950	10	34	.	.	PUNCT
cana-3950	11	1	so	so	ADV
cana-3950	11	2	,	,	PUNCT
cana-3950	11	3	decision	decision	NOUN
cana-3950	11	4	-	-	PUNCT
cana-3950	11	5	makers	maker	NOUN
cana-3950	11	6	treat	treat	VERB
cana-3950	11	7	these	these	DET
cana-3950	11	8	problems	problem	NOUN
cana-3950	11	9	using	use	VERB
cana-3950	11	10	the	the	DET
cana-3950	11	11	methodology	methodology	NOUN
cana-3950	11	12	of	of	ADP
cana-3950	11	13	fuzzification	fuzzification	NOUN
cana-3950	11	14	which	which	PRON
cana-3950	11	15	is	be	AUX
cana-3950	11	16	a	a	DET
cana-3950	11	17	vital	vital	ADJ
cana-3950	11	18	method	method	NOUN
cana-3950	11	19	to	to	PART
cana-3950	11	20	address	address	VERB
cana-3950	11	21	humanistic	humanistic	ADJ
cana-3950	11	22	systems	system	NOUN
cana-3950	11	23	existing	exist	VERB
cana-3950	11	24	in	in	ADP
cana-3950	11	25	real	real	ADJ
cana-3950	11	26	-	-	PUNCT
cana-3950	11	27	world	world	NOUN
cana-3950	11	28	problems	problem	NOUN
cana-3950	11	29	.	.	PUNCT
cana-3950	12	1	l.a.zadeh	l.a.zadeh	NOUN
cana-3950	12	2	[	[	X
cana-3950	12	3	19	19	NUM
cana-3950	12	4	]	]	PUNCT
cana-3950	12	5	in	in	ADP
cana-3950	12	6	1965	1965	NUM
cana-3950	12	7	,	,	PUNCT
cana-3950	12	8	created	create	VERB
cana-3950	12	9	fuzzy	fuzzy	ADJ
cana-3950	12	10	set	set	NOUN
cana-3950	12	11	(	(	PUNCT
cana-3950	12	12	fss	fss	PROPN
cana-3950	12	13	)	)	PUNCT
cana-3950	12	14	as	as	ADP
cana-3950	12	15	a	a	DET
cana-3950	12	16	generalization	generalization	NOUN
cana-3950	12	17	of	of	ADP
cana-3950	12	18	classical	classical	ADJ
cana-3950	12	19	collection	collection	NOUN
cana-3950	12	20	which	which	PRON
cana-3950	12	21	are	be	AUX
cana-3950	12	22	characterized	characterize	VERB
cana-3950	12	23	by	by	ADP
cana-3950	12	24	membership	membership	NOUN
cana-3950	12	25	functions	function	NOUN
cana-3950	12	26	from	from	ADP
cana-3950	12	27	the	the	DET
cana-3950	12	28	universe	universe	NOUN
cana-3950	12	29	of	of	ADP
cana-3950	12	30	discourse	discourse	NOUN
cana-3950	12	31	to	to	ADP
cana-3950	12	32	the	the	DET
cana-3950	12	33	closed	closed	ADJ
cana-3950	12	34	interval	interval	NOUN
cana-3950	12	35	[	[	X
cana-3950	12	36	0,1	0,1	NUM
cana-3950	12	37	]	]	PUNCT
cana-3950	12	38	.	.	PUNCT
cana-3950	13	1	fs	fs	PROPN
cana-3950	13	2	theory	theory	NOUN
cana-3950	13	3	is	be	AUX
cana-3950	13	4	applicable	applicable	ADJ
cana-3950	13	5	in	in	ADP
cana-3950	13	6	various	various	ADJ
cana-3950	13	7	areas	area	NOUN
cana-3950	13	8	such	such	ADJ
cana-3950	13	9	as	as	ADP
cana-3950	13	10	control	control	NOUN
cana-3950	13	11	theory	theory	NOUN
cana-3950	13	12	,	,	PUNCT
cana-3950	13	13	artificial	artificial	ADJ
cana-3950	13	14	intelligence	intelligence	NOUN
cana-3950	13	15	,	,	PUNCT
cana-3950	13	16	pattern	pattern	NOUN
cana-3950	13	17	recognition	recognition	NOUN
cana-3950	13	18	,	,	PUNCT
cana-3950	13	19	database	database	NOUN
cana-3950	13	20	systems	system	NOUN
cana-3950	13	21	,	,	PUNCT
cana-3950	13	22	and	and	CCONJ
cana-3950	13	23	medical	medical	ADJ
cana-3950	13	24	diagnosis	diagnosis	NOUN
cana-3950	13	25	.	.	PUNCT
cana-3950	14	1	the	the	DET
cana-3950	14	2	fusion	fusion	NOUN
cana-3950	14	3	of	of	ADP
cana-3950	14	4	technology	technology	NOUN
cana-3950	14	5	and	and	CCONJ
cana-3950	14	6	generalized	generalized	ADJ
cana-3950	14	7	forms	form	NOUN
cana-3950	14	8	of	of	ADP
cana-3950	14	9	classical	classical	ADJ
cana-3950	14	10	sets	set	NOUN
cana-3950	14	11	is	be	AUX
cana-3950	14	12	very	very	ADV
cana-3950	14	13	useful	useful	ADJ
cana-3950	14	14	to	to	PART
cana-3950	14	15	solve	solve	VERB
cana-3950	14	16	many	many	ADJ
cana-3950	14	17	real	real	ADJ
cana-3950	14	18	-	-	PUNCT
cana-3950	14	19	world	world	NOUN
cana-3950	14	20	complex	complex	ADJ
cana-3950	14	21	problems	problem	NOUN
cana-3950	14	22	that	that	PRON
cana-3950	14	23	involve	involve	VERB
cana-3950	14	24	vague	vague	ADJ
cana-3950	14	25	and	and	CCONJ
cana-3950	14	26	uncertain	uncertain	ADJ
cana-3950	14	27	information	information	NOUN
cana-3950	14	28	.	.	PUNCT
cana-3950	15	1	a	a	DET
cana-3950	15	2	applicable	applicable	ADJ
cana-3950	15	3	set	set	NOUN
cana-3950	15	4	is	be	AUX
cana-3950	15	5	defined	define	VERB
cana-3950	15	6	by	by	ADP
cana-3950	15	7	its	its	PRON
cana-3950	15	8	members	member	NOUN
cana-3950	15	9	function	function	VERB
cana-3950	15	10	from	from	ADP
cana-3950	15	11	the	the	DET
cana-3950	15	12	universe	universe	NOUN
cana-3950	15	13	of	of	ADP
cana-3950	15	14	discourse	discourse	NOUN
cana-3950	15	15	to	to	ADP
cana-3950	15	16	the	the	DET
cana-3950	15	17	two	two	NUM
cana-3950	15	18	-	-	PUNCT
cana-3950	15	19	point	point	NOUN
cana-3950	15	20	set	set	NOUN
cana-3950	15	21	{	{	PUNCT
cana-3950	15	22	0	0	NUM
cana-3950	15	23	,	,	PUNCT
cana-3950	15	24	1	1	NUM
cana-3950	15	25	}	}	PUNCT
cana-3950	15	26	.	.	PUNCT
cana-3950	16	1	classical	classical	ADJ
cana-3950	16	2	set	set	NOUN
cana-3950	16	3	theory	theory	NOUN
cana-3950	16	4	is	be	AUX
cana-3950	16	5	insufficient	insufficient	ADJ
cana-3950	16	6	to	to	PART
cana-3950	16	7	take	take	VERB
cana-3950	16	8	the	the	DET
cana-3950	16	9	complex	complex	ADJ
cana-3950	16	10	problems	problem	NOUN
cana-3950	16	11	involving	involve	VERB
cana-3950	16	12	vague	vague	ADJ
cana-3950	16	13	and	and	CCONJ
cana-3950	16	14	fuzzy	fuzzy	ADJ
cana-3950	16	15	information	information	NOUN
cana-3950	16	16	.	.	PUNCT
cana-3950	17	1	to	to	PART
cana-3950	17	2	handle	handle	VERB
cana-3950	17	3	the	the	DET
cana-3950	17	4	vagueness	vagueness	NOUN
cana-3950	17	5	and	and	CCONJ
cana-3950	17	6	uncertainty	uncertainty	NOUN
cana-3950	17	7	fuzzy	fuzzy	ADJ
cana-3950	17	8	sets	set	NOUN
cana-3950	17	9	(	(	PUNCT
cana-3950	17	10	ifs	ifs	PROPN
cana-3950	17	11	)	)	PUNCT
cana-3950	17	12	.	.	PUNCT
cana-3950	18	1	ifss	ifss	PROPN
cana-3950	18	2	are	be	AUX
cana-3950	18	3	widely	widely	ADV
cana-3950	18	4	used	use	VERB
cana-3950	18	5	in	in	ADP
cana-3950	18	6	many	many	ADJ
cana-3950	18	7	fields	field	NOUN
cana-3950	18	8	of	of	ADP
cana-3950	18	9	mathematics	mathematic	NOUN
cana-3950	18	10	,	,	PUNCT
cana-3950	18	11	computer	computer	NOUN
cana-3950	18	12	science	science	NOUN
cana-3950	18	13	,	,	PUNCT
cana-3950	18	14	management	management	NOUN
cana-3950	18	15	and	and	CCONJ
cana-3950	18	16	medical	medical	ADJ
cana-3950	18	17	sciences	science	NOUN
cana-3950	18	18	.	.	PUNCT
cana-3950	19	1	szmidt	szmidt	PROPN
cana-3950	20	1	[	[	X
cana-3950	20	2	15	15	NUM
cana-3950	20	3	]	]	PUNCT
cana-3950	20	4	and	and	CCONJ
cana-3950	20	5	kacprzyk	kacprzyk	VERB
cana-3950	21	1	[	[	X
cana-3950	21	2	16	16	NUM
cana-3950	21	3	]	]	PUNCT
cana-3950	21	4	and	and	CCONJ
cana-3950	21	5	wang	wang	PROPN
cana-3950	21	6	and	and	CCONJ
cana-3950	21	7	xin	xin	PROPN
cana-3950	22	1	[	[	X
cana-3950	22	2	17	17	NUM
cana-3950	22	3	]	]	PUNCT
cana-3950	22	4	developed	develop	VERB
cana-3950	22	5	various	various	ADJ
cana-3950	22	6	distances	distance	NOUN
cana-3950	22	7	and	and	CCONJ
cana-3950	22	8	similarity	similarity	NOUN
cana-3950	22	9	measures	measure	NOUN
cana-3950	22	10	between	between	ADP
cana-3950	22	11	ifss	ifss	ADJ
cana-3950	22	12	and	and	CCONJ
cana-3950	22	13	studied	studied	ADJ
cana-3950	22	14	applications	application	NOUN
cana-3950	22	15	of	of	ADP
cana-3950	22	16	distance	distance	NOUN
cana-3950	22	17	and	and	CCONJ
cana-3950	22	18	of	of	ADP
cana-3950	22	19	atanassov	atanassov	NOUN
cana-3950	22	20	[	[	X
cana-3950	22	21	2	2	NUM
cana-3950	22	22	]	]	X
cana-3950	22	23	paper	paper	NOUN
cana-3950	22	24	,	,	PUNCT
cana-3950	22	25	several	several	ADJ
cana-3950	22	26	generalizations	generalization	NOUN
cana-3950	22	27	of	of	ADP
cana-3950	22	28	ifss	ifss	NOUN
cana-3950	22	29	have	have	AUX
cana-3950	22	30	appeared	appear	VERB
cana-3950	22	31	in	in	ADP
cana-3950	22	32	the	the	DET
cana-3950	22	33	literature	literature	NOUN
cana-3950	22	34	.	.	PUNCT
cana-3950	23	1	in	in	ADP
cana-3950	23	2	2020	2020	NUM
cana-3950	23	3	,	,	PUNCT
cana-3950	23	4	a	a	DET
cana-3950	23	5	new	new	ADJ
cana-3950	23	6	notion	notion	NOUN
cana-3950	23	7	called	call	VERB
cana-3950	23	8	npythagorean	npythagorean	ADJ
cana-3950	23	9	fuzzy	fuzzy	ADJ
cana-3950	23	10	sets	set	NOUN
cana-3950	23	11	(	(	PUNCT
cana-3950	23	12	n	n	CCONJ
cana-3950	23	13	-	-	PUNCT
cana-3950	23	14	pfs	pf	VERB
cana-3950	23	15	)	)	PUNCT
cana-3950	23	16	was	be	AUX
cana-3950	23	17	created	create	VERB
cana-3950	23	18	by	by	ADP
cana-3950	23	19	bryniarska	bryniarska	NOUN
cana-3950	23	20	[	[	X
cana-3950	23	21	3	3	X
cana-3950	23	22	]	]	PUNCT
cana-3950	23	23	as	as	ADP
cana-3950	23	24	a	a	DET
cana-3950	23	25	super	super	ADJ
cana-3950	23	26	class	class	NOUN
cana-3950	23	27	of	of	ADP
cana-3950	23	28	ffss	ffss	NOUN
cana-3950	23	29	and	and	CCONJ
cana-3950	23	30	studied	study	VERB
cana-3950	23	31	yager	yager	NOUN
cana-3950	23	32	’s	’s	PART
cana-3950	23	33	aggregation	aggregation	NOUN
cana-3950	23	34	operations	operation	NOUN
cana-3950	23	35	for	for	ADP
cana-3950	23	36	n	n	X
cana-3950	23	37	-	-	PUNCT
cana-3950	23	38	pfss	pfss	NOUN
cana-3950	23	39	.	.	PUNCT
cana-3950	24	1	the	the	DET
cana-3950	24	2	distance	distance	NOUN
cana-3950	24	3	and	and	CCONJ
cana-3950	24	4	similarity	similarity	NOUN
cana-3950	24	5	measure	measure	NOUN
cana-3950	24	6	on	on	ADP
cana-3950	24	7	n	n	CCONJ
cana-3950	24	8	-	-	PUNCT
cana-3950	24	9	pfss	pfss	NOUN
cana-3950	24	10	and	and	CCONJ
cana-3950	24	11	their	their	PRON
cana-3950	24	12	applications	application	NOUN
cana-3950	24	13	in	in	ADP
cana-3950	24	14	mcdm	mcdm	ADJ
cana-3950	24	15	problems	problem	NOUN
cana-3950	24	16	were	be	AUX
cana-3950	24	17	studied	study	VERB
cana-3950	24	18	by	by	ADP
cana-3950	24	19	liu	liu	PROPN
cana-3950	24	20	,	,	PUNCT
cana-3950	24	21	chen	chen	PROPN
cana-3950	24	22	,	,	PUNCT
cana-3950	24	23	and	and	CCONJ
cana-3950	24	24	peng	peng	PROPN
cana-3950	25	1	[	[	X
cana-3950	25	2	10	10	NUM
cana-3950	25	3	]	]	PUNCT
cana-3950	25	4	and	and	CCONJ
cana-3950	25	5	peng	peng	PROPN
cana-3950	25	6	and	and	CCONJ
cana-3950	25	7	liu	liu	PROPN
cana-3950	26	1	[	[	X
cana-3950	26	2	11	11	NUM
cana-3950	26	3	]	]	PUNCT
cana-3950	26	4	communications	communication	NOUN
cana-3950	26	5	on	on	ADP
cana-3950	26	6	applied	apply	VERB
cana-3950	26	7	nonlinear	nonlinear	ADJ
cana-3950	26	8	analysis	analysis	NOUN
cana-3950	26	9	issn	issn	NOUN
cana-3950	26	10	:	:	PUNCT
cana-3950	26	11	1074	1074	NUM
cana-3950	26	12	-	-	PUNCT
cana-3950	26	13	133x	133x	NUM
cana-3950	26	14	vol	vol	NOUN
cana-3950	26	15	32	32	NUM
cana-3950	26	16	no	no	NOUN
cana-3950	26	17	.	.	PUNCT
cana-3950	27	1	9s	9s	NUM
cana-3950	27	2	(	(	PUNCT
cana-3950	27	3	2025	2025	NUM
cana-3950	27	4	)	)	PUNCT
cana-3950	27	5	387	387	NUM
cana-3950	27	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3950	27	7	ibrahim	ibrahim	PROPN
cana-3950	27	8	and	and	CCONJ
cana-3950	27	9	his	his	PRON
cana-3950	27	10	coworkers	coworker	NOUN
cana-3950	28	1	[	[	X
cana-3950	28	2	6,7	6,7	NUM
cana-3950	28	3	]	]	PUNCT
cana-3950	28	4	initiated	initiate	VERB
cana-3950	28	5	the	the	DET
cana-3950	28	6	study	study	NOUN
cana-3950	28	7	of	of	ADP
cana-3950	28	8	(	(	PUNCT
cana-3950	28	9	3,2)-fuzzy	3,2)-fuzzy	NUM
cana-3950	28	10	sets	set	NOUN
cana-3950	28	11	was	be	AUX
cana-3950	28	12	created	create	VERB
cana-3950	28	13	by	by	ADP
cana-3950	28	14	alshami	alshami	NOUN
cana-3950	29	1	[	[	X
cana-3950	29	2	1	1	NUM
cana-3950	29	3	]	]	PUNCT
cana-3950	30	1	and	and	CCONJ
cana-3950	30	2	presented	present	VERB
cana-3950	30	3	their	their	PRON
cana-3950	30	4	applications	application	NOUN
cana-3950	30	5	to	to	ADP
cana-3950	30	6	mcdm	mcdm	ADJ
cana-3950	30	7	methods	method	NOUN
cana-3950	30	8	.	.	PUNCT
cana-3950	31	1	recently	recently	ADV
cana-3950	31	2	,	,	PUNCT
cana-3950	31	3	jun	jun	PROPN
cana-3950	31	4	and	and	CCONJ
cana-3950	31	5	his	his	PRON
cana-3950	31	6	co	co	NOUN
cana-3950	31	7	-	-	NOUN
cana-3950	31	8	workers	worker	NOUN
cana-3950	31	9	[	[	X
cana-3950	31	10	8	8	NUM
cana-3950	31	11	]	]	PUNCT
cana-3950	31	12	created	create	VERB
cana-3950	31	13	the	the	DET
cana-3950	31	14	class	class	NOUN
cana-3950	31	15	of	of	ADP
cana-3950	31	16	beal	beal	NOUN
cana-3950	31	17	’s	’s	PART
cana-3950	31	18	fuzzy	fuzzy	ADJ
cana-3950	31	19	sets	set	NOUN
cana-3950	31	20	(	(	PUNCT
cana-3950	31	21	bfss	bfss	NOUN
cana-3950	31	22	)	)	PUNCT
cana-3950	31	23	as	as	ADP
cana-3950	31	24	a	a	DET
cana-3950	31	25	super	super	ADJ
cana-3950	31	26	class	class	NOUN
cana-3950	31	27	of	of	ADP
cana-3950	31	28	n	n	CCONJ
cana-3950	31	29	-	-	PUNCT
cana-3950	31	30	pfss	pfss	NOUN
cana-3950	31	31	.	.	PUNCT
cana-3950	32	1	in	in	ADP
cana-3950	32	2	this	this	DET
cana-3950	32	3	paper	paper	NOUN
cana-3950	32	4	,	,	PUNCT
cana-3950	32	5	we	we	PRON
cana-3950	32	6	establish	establish	VERB
cana-3950	32	7	to	to	ADP
cana-3950	32	8	basic	basic	ADJ
cana-3950	32	9	set	set	NOUN
cana-3950	32	10	of	of	ADP
cana-3950	32	11	operations	operation	NOUN
cana-3950	32	12	and	and	CCONJ
cana-3950	32	13	investigate	investigate	VERB
cana-3950	32	14	the	the	DET
cana-3950	32	15	abstract	abstract	ADJ
cana-3950	32	16	properties	property	NOUN
cana-3950	32	17	that	that	PRON
cana-3950	32	18	can	can	AUX
cana-3950	32	19	be	be	AUX
cana-3950	32	20	transmitted	transmit	VERB
cana-3950	32	21	to	to	ADP
cana-3950	32	22	the	the	DET
cana-3950	32	23	different	different	ADJ
cana-3950	32	24	models	model	NOUN
cana-3950	32	25	they	they	PRON
cana-3950	32	26	are	be	AUX
cana-3950	32	27	in	in	ADP
cana-3950	32	28	connection	connection	NOUN
cana-3950	32	29	with	with	ADP
cana-3950	32	30	then	then	ADV
cana-3950	32	31	to	to	PART
cana-3950	32	32	rank	rank	VERB
cana-3950	32	33	beal	beal	PROPN
cana-3950	32	34	’s	’s	PART
cana-3950	32	35	fuzzy	fuzzy	ADJ
cana-3950	32	36	sets	set	NOUN
cana-3950	32	37	,	,	PUNCT
cana-3950	32	38	we	we	PRON
cana-3950	32	39	obtain	obtain	VERB
cana-3950	32	40	the	the	DET
cana-3950	32	41	mapping	mapping	NOUN
cana-3950	32	42	of	of	ADP
cana-3950	32	43	score	score	NOUN
cana-3950	32	44	and	and	CCONJ
cana-3950	32	45	accuracy	accuracy	NOUN
cana-3950	32	46	,	,	PUNCT
cana-3950	32	47	and	and	CCONJ
cana-3950	32	48	formulate	formulate	VERB
cana-3950	32	49	aggregate	aggregate	ADJ
cana-3950	32	50	operators	operator	NOUN
cana-3950	32	51	to	to	PART
cana-3950	32	52	be	be	AUX
cana-3950	32	53	applied	apply	VERB
cana-3950	32	54	with	with	ADP
cana-3950	32	55	beal	beal	NOUN
cana-3950	32	56	’s	’s	PART
cana-3950	32	57	fuzzy	fuzzy	ADJ
cana-3950	32	58	sets	set	NOUN
cana-3950	32	59	.	.	PUNCT
cana-3950	33	1	ultimately	ultimately	ADV
cana-3950	33	2	,	,	PUNCT
cana-3950	33	3	we	we	PRON
cana-3950	33	4	develop	develop	VERB
cana-3950	33	5	the	the	DET
cana-3950	33	6	successful	successful	ADJ
cana-3950	33	7	techniques	technique	NOUN
cana-3950	33	8	“	"	PUNCT
cana-3950	33	9	aggregative	aggregative	ADJ
cana-3950	33	10	operators	operator	NOUN
cana-3950	33	11	”	"	PUNCT
cana-3950	33	12	to	to	PART
cana-3950	33	13	handle	handle	VERB
cana-3950	33	14	multi	multi	ADJ
cana-3950	33	15	-	-	ADJ
cana-3950	33	16	criteria	criteria	ADJ
cana-3950	33	17	decision	decision	NOUN
cana-3950	33	18	making	make	VERB
cana-3950	33	19	problems	problem	NOUN
cana-3950	33	20	in	in	ADP
cana-3950	33	21	the	the	DET
cana-3950	33	22	beal	beal	NOUN
cana-3950	33	23	’s	’s	PART
cana-3950	33	24	fuzzy	fuzzy	ADJ
cana-3950	33	25	sets	set	NOUN
cana-3950	33	26	environment	environment	NOUN
cana-3950	33	27	.	.	PUNCT
cana-3950	34	1	the	the	DET
cana-3950	34	2	proposed	propose	VERB
cana-3950	34	3	technique	technique	NOUN
cana-3950	34	4	has	have	AUX
cana-3950	34	5	been	be	AUX
cana-3950	34	6	explained	explain	VERB
cana-3950	34	7	and	and	CCONJ
cana-3950	34	8	analysed	analyse	VERB
cana-3950	34	9	through	through	ADP
cana-3950	34	10	suitable	suitable	ADJ
cana-3950	34	11	example	example	NOUN
cana-3950	34	12	.	.	PUNCT
cana-3950	35	1	1.1	1.1	NUM
cana-3950	35	2	.	.	PUNCT
cana-3950	35	3	extension	extension	NOUN
cana-3950	35	4	structure	structure	NOUN
cana-3950	35	5	of	of	ADP
cana-3950	35	6	fuzzy	fuzzy	ADJ
cana-3950	35	7	sets	set	NOUN
cana-3950	35	8	2	2	NUM
cana-3950	35	9	.	.	PUNCT
cana-3950	35	10	preliminaries	preliminary	NOUN
cana-3950	35	11	throughout	throughout	ADP
cana-3950	35	12	this	this	DET
cana-3950	35	13	article	article	NOUN
cana-3950	35	14	x	x	PART
cana-3950	35	15	be	be	AUX
cana-3950	35	16	a	a	DET
cana-3950	35	17	universe	universe	NOUN
cana-3950	35	18	of	of	ADP
cana-3950	35	19	discourse	discourse	NOUN
cana-3950	35	20	n	n	PROPN
cana-3950	35	21	referred	refer	VERB
cana-3950	35	22	to	to	ADP
cana-3950	35	23	the	the	DET
cana-3950	35	24	collection	collection	NOUN
cana-3950	35	25	all	all	DET
cana-3950	35	26	natural	natural	ADJ
cana-3950	35	27	numbers	number	NOUN
cana-3950	35	28	and	and	CCONJ
cana-3950	35	29	𝑚,𝑛	𝑚,𝑛	VERB
cana-3950	35	30	∈	∈	PROPN
cana-3950	35	31	𝑁.	𝑁.	PROPN
cana-3950	35	32	definition	definition	NOUN
cana-3950	35	33	2.1	2.1	NUM
cana-3950	35	34	[	[	X
cana-3950	35	35	fuzzy	fuzzy	ADJ
cana-3950	35	36	set	set	NOUN
cana-3950	35	37	]	]	PUNCT
cana-3950	35	38	let	let	VERB
cana-3950	35	39	x	x	PRON
cana-3950	35	40	be	be	AUX
cana-3950	35	41	a	a	DET
cana-3950	35	42	nonempty	nonempty	ADJ
cana-3950	35	43	set	set	VERB
cana-3950	35	44	.	.	PUNCT
cana-3950	36	1	then	then	ADV
cana-3950	36	2	a	a	DET
cana-3950	36	3	fuzzy	fuzzy	ADJ
cana-3950	36	4	set	set	NOUN
cana-3950	36	5	on	on	ADP
cana-3950	36	6	x	x	PUNCT
cana-3950	36	7	is	be	AUX
cana-3950	36	8	defined	define	VERB
cana-3950	36	9	by	by	ADP
cana-3950	36	10	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	36	11	:	:	PUNCT
cana-3950	36	12	𝑥	𝑥	X
cana-3950	36	13	→	→	SYM
cana-3950	37	1	[	[	X
cana-3950	37	2	0,1	0,1	NUM
cana-3950	37	3	]	]	PUNCT
cana-3950	37	4	.	.	PUNCT
cana-3950	38	1	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	38	2	is	be	AUX
cana-3950	38	3	called	call	VERB
cana-3950	38	4	the	the	DET
cana-3950	38	5	membership	membership	NOUN
cana-3950	38	6	function	function	NOUN
cana-3950	38	7	.	.	PUNCT
cana-3950	39	1	𝐽𝐴(𝑥	𝐽𝐴(𝑥	NOUN
cana-3950	39	2	)	)	PUNCT
cana-3950	39	3	is	be	AUX
cana-3950	39	4	called	call	VERB
cana-3950	39	5	the	the	DET
cana-3950	39	6	membership	membership	NOUN
cana-3950	39	7	grade	grade	NOUN
cana-3950	39	8	of	of	ADP
cana-3950	39	9	x	x	PUNCT
cana-3950	39	10	in	in	ADP
cana-3950	39	11	𝐽𝐴.	𝐽𝐴.	NOUN
cana-3950	39	12	we	we	PRON
cana-3950	39	13	also	also	ADV
cana-3950	39	14	write	write	VERB
cana-3950	39	15	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	39	16	=	=	PRON
cana-3950	39	17	{	{	PUNCT
cana-3950	39	18	(	(	PUNCT
cana-3950	39	19	𝑥	𝑥	NOUN
cana-3950	39	20	,	,	PUNCT
cana-3950	39	21	𝐽𝐴(𝑥))/𝑥	𝐽𝐴(𝑥))/𝑥	PROPN
cana-3950	39	22	∈	∈	PROPN
cana-3950	39	23	𝑋	𝑋	PROPN
cana-3950	39	24	}	}	PUNCT
cana-3950	39	25	.	.	PUNCT
cana-3950	39	26	example	example	NOUN
cana-3950	39	27	2.2	2.2	NUM
cana-3950	39	28	let	let	VERB
cana-3950	39	29	𝑥	𝑥	PRON
cana-3950	39	30	=	=	PUNCT
cana-3950	39	31	{	{	PUNCT
cana-3950	39	32	𝑎	𝑎	NOUN
cana-3950	39	33	,	,	PUNCT
cana-3950	39	34	𝑏	𝑏	NOUN
cana-3950	39	35	,	,	PUNCT
cana-3950	39	36	𝑐	𝑐	NOUN
cana-3950	39	37	,	,	PUNCT
cana-3950	39	38	𝑑	𝑑	NOUN
cana-3950	39	39	}	}	PUNCT
cana-3950	39	40	and	and	CCONJ
cana-3950	39	41	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	39	42	:	:	PUNCT
cana-3950	39	43	𝑥	𝑥	X
cana-3950	39	44	→	→	SYM
cana-3950	40	1	[	[	X
cana-3950	40	2	0,1	0,1	NUM
cana-3950	40	3	]	]	PUNCT
cana-3950	40	4	defined	define	VERB
cana-3950	40	5	by	by	ADP
cana-3950	40	6	𝐽𝐴(𝑎	𝐽𝐴(𝑎	NOUN
cana-3950	40	7	)	)	PUNCT
cana-3950	40	8	=	=	SYM
cana-3950	40	9	0	0	NUM
cana-3950	40	10	,	,	PUNCT
cana-3950	40	11	𝐽𝐴(𝑏	𝐽𝐴(𝑏	PUNCT
cana-3950	40	12	)	)	PUNCT
cana-3950	40	13	=	=	SYM
cana-3950	40	14	0.6	0.6	NUM
cana-3950	40	15	,	,	PUNCT
cana-3950	40	16	𝐽𝐴(𝑐	𝐽𝐴(𝑐	PROPN
cana-3950	40	17	)	)	PUNCT
cana-3950	40	18	=	=	SYM
cana-3950	40	19	0.3	0.3	NUM
cana-3950	40	20	,	,	PUNCT
cana-3950	40	21	𝐽𝐴(𝑑	𝐽𝐴(𝑑	NOUN
cana-3950	40	22	)	)	PUNCT
cana-3950	40	23	=	=	SYM
cana-3950	41	1	1	1	X
cana-3950	41	2	.	.	PUNCT
cana-3950	41	3	definition	definition	NOUN
cana-3950	41	4	2.3	2.3	NUM
cana-3950	41	5	[	[	X
cana-3950	41	6	intuitionistic	intuitionistic	ADJ
cana-3950	41	7	fuzzy	fuzzy	ADJ
cana-3950	41	8	set	set	NOUN
cana-3950	41	9	]	]	X
cana-3950	41	10	:	:	PUNCT
cana-3950	41	11	an	an	DET
cana-3950	41	12	ifs	ifs	PROPN
cana-3950	41	13	is	be	AUX
cana-3950	41	14	an	an	DET
cana-3950	41	15	extension	extension	NOUN
cana-3950	41	16	of	of	ADP
cana-3950	41	17	a	a	DET
cana-3950	41	18	fuzzy	fuzzy	ADJ
cana-3950	41	19	set	set	NOUN
cana-3950	41	20	introduced	introduce	VERB
cana-3950	41	21	by	by	ADP
cana-3950	41	22	k.atanassov	k.atanassov	NOUN
cana-3950	41	23	in	in	ADP
cana-3950	41	24	1983	1983	NUM
cana-3950	41	25	.	.	PUNCT
cana-3950	42	1	an	an	DET
cana-3950	42	2	intuitionistic	intuitionistic	ADJ
cana-3950	42	3	fuzzy	fuzzy	NOUN
cana-3950	42	4	set	set	VERB
cana-3950	42	5	a	a	DET
cana-3950	42	6	in	in	ADP
cana-3950	42	7	x	x	SYM
cana-3950	42	8	is	be	AUX
cana-3950	42	9	defined	define	VERB
cana-3950	42	10	as	as	ADP
cana-3950	42	11	𝐴	𝐴	PROPN
cana-3950	42	12	=	=	SYM
cana-3950	42	13	{	{	PUNCT
cana-3950	42	14	〈	〈	NOUN
cana-3950	42	15	𝑥	𝑥	NOUN
cana-3950	42	16	,	,	PUNCT
cana-3950	42	17	𝐽𝐴(𝑥	𝐽𝐴(𝑥	NOUN
cana-3950	42	18	)	)	PUNCT
cana-3950	42	19	,	,	PUNCT
cana-3950	42	20	𝐾𝐴(𝑥)〉/	𝐾𝐴(𝑥)〉/	PROPN
cana-3950	42	21	𝑥	𝑥	PROPN
cana-3950	42	22	∈	∈	PROPN
cana-3950	42	23	𝑋	𝑋	PROPN
cana-3950	42	24	}	}	PUNCT
cana-3950	42	25	,	,	PUNCT
cana-3950	42	26	where	where	SCONJ
cana-3950	42	27	𝐽𝐴	𝐽𝐴	NOUN
cana-3950	42	28	:	:	PUNCT
cana-3950	42	29	𝑋	𝑋	NOUN
cana-3950	42	30	→	→	SYM
cana-3950	42	31	[	[	X
cana-3950	42	32	0,1	0,1	NUM
cana-3950	42	33	]	]	PUNCT
cana-3950	42	34	and	and	CCONJ
cana-3950	42	35	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	42	36	:	:	PUNCT
cana-3950	42	37	𝑋	𝑋	NOUN
cana-3950	42	38	→	→	SYM
cana-3950	42	39	[	[	X
cana-3950	42	40	0,1	0,1	NUM
cana-3950	42	41	]	]	PUNCT
cana-3950	42	42	are	be	AUX
cana-3950	42	43	respectively	respectively	ADV
cana-3950	42	44	degree	degree	NOUN
cana-3950	42	45	of	of	ADP
cana-3950	42	46	membership	membership	NOUN
cana-3950	42	47	and	and	CCONJ
cana-3950	42	48	degree	degree	NOUN
cana-3950	42	49	of	of	ADP
cana-3950	42	50	noncommunications	noncommunication	NOUN
cana-3950	42	51	on	on	ADP
cana-3950	42	52	applied	apply	VERB
cana-3950	42	53	nonlinear	nonlinear	ADJ
cana-3950	42	54	analysis	analysis	NOUN
cana-3950	42	55	issn	issn	NOUN
cana-3950	42	56	:	:	PUNCT
cana-3950	42	57	1074	1074	NUM
cana-3950	42	58	-	-	PUNCT
cana-3950	42	59	133x	133x	NUM
cana-3950	42	60	vol	vol	NOUN
cana-3950	42	61	32	32	NUM
cana-3950	42	62	no	no	NOUN
cana-3950	42	63	.	.	PUNCT
cana-3950	43	1	9s	9s	NUM
cana-3950	43	2	(	(	PUNCT
cana-3950	43	3	2025	2025	NUM
cana-3950	43	4	)	)	PUNCT
cana-3950	43	5	388	388	NUM
cana-3950	43	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3950	43	7	membership	membership	NOUN
cana-3950	43	8	for	for	ADP
cana-3950	43	9	every	every	DET
cana-3950	43	10	𝑥	𝑥	DET
cana-3950	43	11	∈	∈	PROPN
cana-3950	43	12	𝑋	𝑋	NOUN
cana-3950	43	13	with	with	ADP
cana-3950	43	14	0	0	NUM
cana-3950	43	15	≤	≤	NUM
cana-3950	43	16	𝐽𝐴(𝑥	𝐽𝐴(𝑥	NOUN
cana-3950	43	17	)	)	PUNCT
cana-3950	44	1	+	+	CCONJ
cana-3950	44	2	𝐾𝐴(𝑥	𝐾𝐴(𝑥	X
cana-3950	44	3	)	)	PUNCT
cana-3950	44	4	≤	≤	NUM
cana-3950	44	5	1	1	NUM
cana-3950	44	6	and	and	CCONJ
cana-3950	44	7	𝜋𝐴(𝑥	𝜋𝐴(𝑥	NUM
cana-3950	44	8	)	)	PUNCT
cana-3950	44	9	=	=	SYM
cana-3950	44	10	1	1	NUM
cana-3950	44	11	−	−	PROPN
cana-3950	44	12	(	(	PUNCT
cana-3950	44	13	𝐽𝐴(𝑥	𝐽𝐴(𝑥	NOUN
cana-3950	44	14	)	)	PUNCT
cana-3950	44	15	+	+	CCONJ
cana-3950	44	16	𝐾𝐴(𝑥	𝐾𝐴(𝑥	ADJ
cana-3950	44	17	)	)	PUNCT
cana-3950	44	18	)	)	PUNCT
cana-3950	44	19	is	be	AUX
cana-3950	44	20	the	the	DET
cana-3950	44	21	degree	degree	NOUN
cana-3950	44	22	of	of	ADP
cana-3950	44	23	indeterminacy	indeterminacy	NOUN
cana-3950	44	24	of	of	ADP
cana-3950	44	25	𝑥	𝑥	DET
cana-3950	44	26	∈	∈	PROPN
cana-3950	44	27	𝑋.	𝑋.	PROPN
cana-3950	44	28	sometimes	sometimes	ADV
cana-3950	44	29	ifs	ifs	PROPN
cana-3950	44	30	is	be	AUX
cana-3950	44	31	obviously	obviously	ADV
cana-3950	44	32	called	call	VERB
cana-3950	44	33	bifuzzy	bifuzzy	PROPN
cana-3950	44	34	set	set	NOUN
cana-3950	44	35	.	.	PUNCT
cana-3950	45	1	definition	definition	NOUN
cana-3950	45	2	2.4	2.4	NUM
cana-3950	45	3	a	a	DET
cana-3950	45	4	structure	structure	NOUN
cana-3950	45	5	𝐴	𝐴	NOUN
cana-3950	45	6	=	=	SYM
cana-3950	45	7	{	{	PUNCT
cana-3950	45	8	〈	〈	NOUN
cana-3950	45	9	𝑥	𝑥	NOUN
cana-3950	45	10	,	,	PUNCT
cana-3950	45	11	𝐽𝐴(𝑥	𝐽𝐴(𝑥	NOUN
cana-3950	45	12	)	)	PUNCT
cana-3950	45	13	,	,	PUNCT
cana-3950	45	14	𝐾𝐴(𝑥)〉/	𝐾𝐴(𝑥)〉/	PROPN
cana-3950	45	15	𝑥	𝑥	PROPN
cana-3950	45	16	∈	∈	PROPN
cana-3950	45	17	𝑋	𝑋	PROPN
cana-3950	45	18	}	}	PUNCT
cana-3950	45	19	,	,	PUNCT
cana-3950	45	20	where	where	SCONJ
cana-3950	45	21	𝐽𝐴	𝐽𝐴	NOUN
cana-3950	45	22	:	:	PUNCT
cana-3950	45	23	𝑋	𝑋	NOUN
cana-3950	45	24	→	→	SYM
cana-3950	45	25	[	[	X
cana-3950	45	26	0,1	0,1	NUM
cana-3950	45	27	]	]	PUNCT
cana-3950	45	28	and	and	CCONJ
cana-3950	45	29	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	45	30	:	:	PUNCT
cana-3950	45	31	𝑋	𝑋	NOUN
cana-3950	45	32	→	→	SYM
cana-3950	45	33	[	[	X
cana-3950	45	34	0,1	0,1	NUM
cana-3950	45	35	]	]	PUNCT
cana-3950	45	36	are	be	AUX
cana-3950	45	37	respectively	respectively	ADV
cana-3950	45	38	degree	degree	NOUN
cana-3950	45	39	of	of	ADP
cana-3950	45	40	membership	membership	NOUN
cana-3950	45	41	and	and	CCONJ
cana-3950	45	42	degree	degree	NOUN
cana-3950	45	43	of	of	ADP
cana-3950	45	44	non	non	ADJ
cana-3950	45	45	-	-	NOUN
cana-3950	45	46	membership	membership	NOUN
cana-3950	45	47	for	for	ADP
cana-3950	45	48	every	every	DET
cana-3950	45	49	𝑥	𝑥	PRON
cana-3950	45	50	∈	∈	PROPN
cana-3950	45	51	𝑋	𝑋	NOUN
cana-3950	45	52	to	to	ADP
cana-3950	45	53	a	a	PRON
cana-3950	45	54	is	be	AUX
cana-3950	45	55	called	call	VERB
cana-3950	45	56	(	(	PUNCT
cana-3950	45	57	i	i	NOUN
cana-3950	45	58	)	)	PUNCT
cana-3950	45	59	intuitionistic	intuitionistic	ADJ
cana-3950	45	60	fuzzy	fuzzy	ADJ
cana-3950	45	61	set	set	NOUN
cana-3950	45	62	in	in	ADP
cana-3950	45	63	x	x	PUNCT
cana-3950	45	64	if	if	SCONJ
cana-3950	45	65	0	0	NUM
cana-3950	45	66	≤	≤	NUM
cana-3950	45	67	𝐽𝐴(𝑥	𝐽𝐴(𝑥	NOUN
cana-3950	45	68	)	)	PUNCT
cana-3950	46	1	+	+	CCONJ
cana-3950	46	2	𝐾𝐴(𝑥	𝐾𝐴(𝑥	X
cana-3950	46	3	)	)	PUNCT
cana-3950	46	4	≤	≤	NUM
cana-3950	46	5	1	1	NUM
cana-3950	46	6	for	for	SCONJ
cana-3950	46	7	all	all	PRON
cana-3950	46	8	𝑥	𝑥	DET
cana-3950	46	9	∈	∈	PROPN
cana-3950	46	10	𝑋.	𝑋.	PROPN
cana-3950	46	11	(	(	PUNCT
cana-3950	46	12	ii	ii	NOUN
cana-3950	46	13	)	)	PUNCT
cana-3950	46	14	pythagorean	pythagorean	PROPN
cana-3950	46	15	fuzzy	fuzzy	ADJ
cana-3950	46	16	set	set	VERB
cana-3950	46	17	in	in	ADP
cana-3950	46	18	x	x	PUNCT
cana-3950	46	19	if	if	SCONJ
cana-3950	46	20	0	0	NUM
cana-3950	46	21	≤	≤	NUM
cana-3950	46	22	𝐽𝐴	𝐽𝐴	NOUN
cana-3950	46	23	2(𝑥	2(𝑥	NUM
cana-3950	46	24	)	)	PUNCT
cana-3950	47	1	+	+	CCONJ
cana-3950	47	2	𝐾𝐴	𝐾𝐴	NOUN
cana-3950	47	3	2(𝑥	2(𝑥	NUM
cana-3950	47	4	)	)	PUNCT
cana-3950	47	5	≤	≤	NUM
cana-3950	47	6	1	1	NUM
cana-3950	47	7	for	for	ADP
cana-3950	47	8	all	all	PRON
cana-3950	47	9	𝑥	𝑥	DET
cana-3950	47	10	∈	∈	PROPN
cana-3950	47	11	𝑋.	𝑋.	PROPN
cana-3950	47	12	(	(	PUNCT
cana-3950	47	13	iii	iii	NOUN
cana-3950	47	14	)	)	PUNCT
cana-3950	47	15	(	(	PUNCT
cana-3950	47	16	2	2	NUM
cana-3950	47	17	,	,	PUNCT
cana-3950	47	18	1	1	NUM
cana-3950	47	19	)	)	PUNCT
cana-3950	47	20	fuzzy	fuzzy	ADJ
cana-3950	47	21	set	set	VERB
cana-3950	47	22	in	in	ADP
cana-3950	47	23	x	x	PUNCT
cana-3950	47	24	if	if	SCONJ
cana-3950	47	25	0	0	NUM
cana-3950	47	26	≤	≤	NUM
cana-3950	47	27	𝐽𝐴	𝐽𝐴	NOUN
cana-3950	47	28	2(𝑥	2(𝑥	NUM
cana-3950	47	29	)	)	PUNCT
cana-3950	47	30	+	+	CCONJ
cana-3950	48	1	𝐾𝐴(𝑥	𝐾𝐴(𝑥	X
cana-3950	48	2	)	)	PUNCT
cana-3950	48	3	≤	≤	NUM
cana-3950	48	4	1	1	NUM
cana-3950	48	5	for	for	ADP
cana-3950	48	6	all	all	PRON
cana-3950	48	7	𝑥	𝑥	DET
cana-3950	48	8	∈	∈	PROPN
cana-3950	48	9	𝑋.	𝑋.	PROPN
cana-3950	48	10	(	(	PUNCT
cana-3950	48	11	iv	iv	NOUN
cana-3950	48	12	)	)	PUNCT
cana-3950	48	13	(	(	PUNCT
cana-3950	48	14	3	3	NUM
cana-3950	48	15	,	,	PUNCT
cana-3950	48	16	2	2	NUM
cana-3950	48	17	)	)	PUNCT
cana-3950	48	18	fuzzy	fuzzy	ADJ
cana-3950	48	19	set	set	VERB
cana-3950	48	20	in	in	ADP
cana-3950	48	21	x	x	X
cana-3950	48	22	of	of	ADP
cana-3950	48	23	0	0	NUM
cana-3950	48	24	≤	≤	NUM
cana-3950	48	25	𝐽𝐴	𝐽𝐴	NOUN
cana-3950	48	26	3(𝑥	3(𝑥	NUM
cana-3950	48	27	)	)	PUNCT
cana-3950	48	28	+	+	CCONJ
cana-3950	48	29	𝐾𝐴	𝐾𝐴	NOUN
cana-3950	48	30	2(𝑥	2(𝑥	NUM
cana-3950	48	31	)	)	PUNCT
cana-3950	48	32	≤	≤	NUM
cana-3950	48	33	1	1	NUM
cana-3950	48	34	for	for	SCONJ
cana-3950	48	35	all	all	PRON
cana-3950	48	36	𝑥	𝑥	DET
cana-3950	48	37	∈	∈	PROPN
cana-3950	48	38	𝑋.	𝑋.	PROPN
cana-3950	48	39	(	(	PUNCT
cana-3950	48	40	v	v	NOUN
cana-3950	48	41	)	)	PUNCT
cana-3950	48	42	fermatean	fermatean	ADJ
cana-3950	48	43	fuzzy	fuzzy	NOUN
cana-3950	48	44	set	set	VERB
cana-3950	48	45	in	in	ADP
cana-3950	48	46	x	x	PUNCT
cana-3950	48	47	if	if	SCONJ
cana-3950	48	48	0	0	NUM
cana-3950	48	49	≤	≤	NUM
cana-3950	48	50	𝐽𝐴	𝐽𝐴	NOUN
cana-3950	48	51	3(𝑥	3(𝑥	NUM
cana-3950	48	52	)	)	PUNCT
cana-3950	48	53	+	+	CCONJ
cana-3950	48	54	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	48	55	3(𝑥	3(𝑥	NUM
cana-3950	48	56	)	)	PUNCT
cana-3950	48	57	≤	≤	NUM
cana-3950	48	58	1	1	NUM
cana-3950	48	59	for	for	ADP
cana-3950	48	60	all	all	PRON
cana-3950	48	61	𝑥	𝑥	DET
cana-3950	48	62	∈	∈	PROPN
cana-3950	48	63	𝑋.	𝑋.	PROPN
cana-3950	48	64	(	(	PUNCT
cana-3950	48	65	vi	vi	NOUN
cana-3950	48	66	)	)	PUNCT
cana-3950	48	67	(	(	PUNCT
cana-3950	48	68	4	4	NUM
cana-3950	48	69	,	,	PUNCT
cana-3950	48	70	2	2	NUM
cana-3950	48	71	)	)	PUNCT
cana-3950	48	72	–	–	PUNCT
cana-3950	48	73	fuzzy	fuzzy	ADJ
cana-3950	48	74	set	set	VERB
cana-3950	48	75	in	in	ADP
cana-3950	48	76	x	x	SYM
cana-3950	48	77	if	if	SCONJ
cana-3950	48	78	0	0	NUM
cana-3950	48	79	≤	≤	NUM
cana-3950	48	80	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	48	81	4(𝑥	4(𝑥	NUM
cana-3950	48	82	)	)	PUNCT
cana-3950	48	83	+	+	CCONJ
cana-3950	48	84	𝐾𝐴	𝐾𝐴	NOUN
cana-3950	48	85	2(𝑥	2(𝑥	NUM
cana-3950	48	86	)	)	PUNCT
cana-3950	48	87	≤	≤	NUM
cana-3950	48	88	1	1	NUM
cana-3950	48	89	for	for	ADP
cana-3950	48	90	all	all	PRON
cana-3950	48	91	𝑥	𝑥	DET
cana-3950	48	92	∈	∈	PROPN
cana-3950	48	93	𝑋.	𝑋.	PROPN
cana-3950	48	94	(	(	PUNCT
cana-3950	48	95	vii	vii	PROPN
cana-3950	48	96	)	)	PUNCT
cana-3950	48	97	n	n	CCONJ
cana-3950	48	98	–	–	PUNCT
cana-3950	48	99	fuzzy	fuzzy	ADJ
cana-3950	48	100	set	set	NOUN
cana-3950	48	101	when	when	SCONJ
cana-3950	48	102	𝑚,𝑛	𝑚,𝑛	PROPN
cana-3950	48	103	∈	∈	NOUN
cana-3950	48	104	𝑁	𝑁	PROPN
cana-3950	48	105	in	in	ADP
cana-3950	48	106	x	x	SYM
cana-3950	48	107	if	if	SCONJ
cana-3950	48	108	0	0	NUM
cana-3950	48	109	≤	≤	NUM
cana-3950	48	110	𝐽𝐴	𝐽𝐴	NOUN
cana-3950	48	111	𝑚(𝑥	𝑚(𝑥	NOUN
cana-3950	48	112	)	)	PUNCT
cana-3950	49	1	+	+	CCONJ
cana-3950	49	2	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	49	3	𝑛(𝑥	𝑛(𝑥	PROPN
cana-3950	49	4	)	)	PUNCT
cana-3950	49	5	≤	≤	NOUN
cana-3950	49	6	1	1	NUM
cana-3950	49	7	for	for	ADP
cana-3950	49	8	all	all	DET
cana-3950	49	9	𝑥	𝑥	DET
cana-3950	49	10	∈	∈	PROPN
cana-3950	49	11	𝑋	𝑋	NOUN
cana-3950	49	12	and	and	CCONJ
cana-3950	49	13	𝑚	𝑚	ADP
cana-3950	49	14	≥	≥	NUM
cana-3950	49	15	4	4	NUM
cana-3950	49	16	and	and	CCONJ
cana-3950	49	17	𝑛	𝑛	PRON
cana-3950	49	18	≥	≥	NUM
cana-3950	49	19	4	4	NUM
cana-3950	49	20	.	.	PUNCT
cana-3950	49	21	(	(	PUNCT
cana-3950	49	22	ix	ix	PROPN
cana-3950	49	23	)	)	PUNCT
cana-3950	49	24	q	q	ADJ
cana-3950	49	25	-	-	PUNCT
cana-3950	49	26	runk	runk	ADJ
cana-3950	49	27	fuzzy	fuzzy	ADJ
cana-3950	49	28	set	set	VERB
cana-3950	49	29	when	when	SCONJ
cana-3950	49	30	𝑚	𝑚	PROPN
cana-3950	49	31	=	=	SYM
cana-3950	49	32	𝑛	𝑛	NOUN
cana-3950	49	33	=	=	PUNCT
cana-3950	49	34	𝑞	𝑞	X
cana-3950	49	35	i.e.	i.e.	X
cana-3950	49	36	,	,	PUNCT
cana-3950	49	37	0	0	NUM
cana-3950	49	38	≤	≤	NUM
cana-3950	49	39	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	49	40	𝑞(𝑥	𝑞(𝑥	PROPN
cana-3950	49	41	)	)	PUNCT
cana-3950	50	1	+	+	CCONJ
cana-3950	50	2	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	50	3	𝑞(𝑥	𝑞(𝑥	PROPN
cana-3950	50	4	)	)	PUNCT
cana-3950	50	5	≤	≤	NOUN
cana-3950	50	6	1	1	NUM
cana-3950	50	7	for	for	ADP
cana-3950	50	8	all	all	PRON
cana-3950	50	9	𝑥	𝑥	DET
cana-3950	50	10	∈	∈	NOUN
cana-3950	50	11	𝑋.	𝑋.	PROPN
cana-3950	50	12	in	in	ADP
cana-3950	50	13	what	what	PRON
cana-3950	50	14	follows	follow	VERB
cana-3950	50	15	,	,	PUNCT
cana-3950	50	16	we	we	PRON
cana-3950	50	17	compare	compare	VERB
cana-3950	50	18	beal	beal	NOUN
cana-3950	50	19	’s	’s	PART
cana-3950	50	20	fuzzy	fuzzy	ADJ
cana-3950	50	21	set	set	VERB
cana-3950	50	22	with	with	ADP
cana-3950	50	23	the	the	DET
cana-3950	50	24	previous	previous	ADJ
cana-3950	50	25	generalization	generalization	NOUN
cana-3950	50	26	of	of	ADP
cana-3950	50	27	intuitionstic	intuitionstic	ADJ
cana-3950	50	28	fuzzy	fuzzy	ADJ
cana-3950	50	29	sets	set	NOUN
cana-3950	50	30	.	.	PUNCT
cana-3950	51	1	proposition	proposition	NOUN
cana-3950	51	2	2.5	2.5	NUM
cana-3950	51	3	(	(	PUNCT
cana-3950	51	4	i	i	NOUN
cana-3950	51	5	)	)	PUNCT
cana-3950	51	6	every	every	DET
cana-3950	51	7	bifuzzy	bifuzzy	NOUN
cana-3950	51	8	set	set	VERB
cana-3950	51	9	is	be	AUX
cana-3950	51	10	a	a	DET
cana-3950	51	11	beal	beal	NOUN
cana-3950	51	12	’s	’s	PART
cana-3950	51	13	fuzzy	fuzzy	ADJ
cana-3950	51	14	set	set	NOUN
cana-3950	51	15	.	.	PUNCT
cana-3950	52	1	(	(	PUNCT
cana-3950	52	2	ii	ii	NOUN
cana-3950	52	3	)	)	PUNCT
cana-3950	52	4	if	if	SCONJ
cana-3950	52	5	𝑚	𝑚	PROPN
cana-3950	52	6	≥	≥	NUM
cana-3950	52	7	2	2	NUM
cana-3950	52	8	and	and	CCONJ
cana-3950	52	9	𝑛	𝑛	PRON
cana-3950	52	10	≥	≥	NUM
cana-3950	52	11	2	2	NUM
cana-3950	52	12	,	,	PUNCT
cana-3950	52	13	then	then	ADV
cana-3950	52	14	a	a	DET
cana-3950	52	15	pythagorean	pythagorean	ADJ
cana-3950	52	16	fuzzy	fuzzy	ADJ
cana-3950	52	17	set	set	NOUN
cana-3950	52	18	is	be	AUX
cana-3950	52	19	a	a	DET
cana-3950	52	20	beal	beal	NOUN
cana-3950	52	21	’s	’s	PART
cana-3950	52	22	fuzzy	fuzzy	ADJ
cana-3950	52	23	set	set	NOUN
cana-3950	52	24	.	.	PUNCT
cana-3950	53	1	(	(	PUNCT
cana-3950	53	2	iii	iii	X
cana-3950	53	3	)	)	PUNCT
cana-3950	53	4	if	if	SCONJ
cana-3950	53	5	𝑚	𝑚	PROPN
cana-3950	53	6	≥	≥	NUM
cana-3950	53	7	3	3	NUM
cana-3950	53	8	and	and	CCONJ
cana-3950	53	9	𝑛	𝑛	PRON
cana-3950	53	10	≥	≥	NUM
cana-3950	53	11	3	3	NUM
cana-3950	53	12	,	,	PUNCT
cana-3950	53	13	then	then	ADV
cana-3950	53	14	a	a	DET
cana-3950	53	15	fermatean	fermatean	ADJ
cana-3950	53	16	fuzzy	fuzzy	ADJ
cana-3950	53	17	set	set	NOUN
cana-3950	53	18	is	be	AUX
cana-3950	53	19	a	a	DET
cana-3950	53	20	beal	beal	NOUN
cana-3950	53	21	’s	’s	PART
cana-3950	53	22	fuzzy	fuzzy	ADJ
cana-3950	53	23	set	set	NOUN
cana-3950	53	24	.	.	PUNCT
cana-3950	54	1	(	(	PUNCT
cana-3950	54	2	iv	iv	X
cana-3950	54	3	)	)	PUNCT
cana-3950	54	4	if	if	SCONJ
cana-3950	54	5	𝑚	𝑚	PROPN
cana-3950	54	6	≥	≥	NOUN
cana-3950	54	7	𝑞	𝑞	X
cana-3950	54	8	and	and	CCONJ
cana-3950	54	9	𝑛	𝑛	DET
cana-3950	54	10	≥	≥	NUM
cana-3950	54	11	𝑞	𝑞	NOUN
cana-3950	54	12	,	,	PUNCT
cana-3950	54	13	then	then	ADV
cana-3950	54	14	a	a	DET
cana-3950	54	15	q	q	NOUN
cana-3950	54	16	-	-	PUNCT
cana-3950	54	17	runk	runk	ADJ
cana-3950	54	18	ortho	ortho	PROPN
cana-3950	54	19	pair	pair	PROPN
cana-3950	54	20	fuzzy	fuzzy	ADJ
cana-3950	54	21	set	set	NOUN
cana-3950	54	22	is	be	AUX
cana-3950	54	23	a	a	DET
cana-3950	54	24	beal	beal	NOUN
cana-3950	54	25	’s	’s	PART
cana-3950	54	26	fuzzy	fuzzy	ADJ
cana-3950	54	27	set	set	NOUN
cana-3950	54	28	.	.	PUNCT
cana-3950	55	1	(	(	PUNCT
cana-3950	55	2	v	v	NOUN
cana-3950	55	3	)	)	PUNCT
cana-3950	55	4	if	if	SCONJ
cana-3950	55	5	𝑚	𝑚	PROPN
cana-3950	55	6	≥	≥	NUM
cana-3950	55	7	2	2	NUM
cana-3950	55	8	and	and	CCONJ
cana-3950	55	9	𝑛	𝑛	PRON
cana-3950	55	10	≥	≥	NUM
cana-3950	55	11	3	3	NUM
cana-3950	55	12	,	,	PUNCT
cana-3950	55	13	then	then	ADV
cana-3950	55	14	a	a	DET
cana-3950	55	15	(	(	PUNCT
cana-3950	55	16	2	2	NUM
cana-3950	55	17	,	,	PUNCT
cana-3950	55	18	3	3	NUM
cana-3950	55	19	)	)	PUNCT
cana-3950	55	20	–	–	PUNCT
cana-3950	55	21	fuzzy	fuzzy	ADJ
cana-3950	55	22	set	set	NOUN
cana-3950	55	23	is	be	AUX
cana-3950	55	24	a	a	DET
cana-3950	55	25	beal	beal	NOUN
cana-3950	55	26	’s	’s	PART
cana-3950	55	27	fuzzy	fuzzy	ADJ
cana-3950	55	28	set	set	NOUN
cana-3950	55	29	.	.	PUNCT
cana-3950	56	1	proof	proof	NOUN
cana-3950	56	2	:	:	PUNCT
cana-3950	56	3	the	the	DET
cana-3950	56	4	proof	proof	NOUN
cana-3950	56	5	is	be	AUX
cana-3950	56	6	straight	straight	ADV
cana-3950	56	7	forward	forward	ADV
cana-3950	56	8	.	.	PUNCT
cana-3950	57	1	remark	remark	NOUN
cana-3950	57	2	:	:	PUNCT
cana-3950	57	3	for	for	ADP
cana-3950	57	4	all	all	DET
cana-3950	57	5	𝑎	𝑎	NOUN
cana-3950	57	6	,	,	PUNCT
cana-3950	57	7	𝑏	𝑏	PROPN
cana-3950	57	8	∈	∈	PROPN
cana-3950	58	1	[	[	X
cana-3950	58	2	0,1	0,1	NUM
cana-3950	58	3	]	]	PUNCT
cana-3950	58	4	,	,	PUNCT
cana-3950	58	5	we	we	PRON
cana-3950	58	6	have	have	VERB
cana-3950	58	7	𝑎	𝑎	NOUN
cana-3950	58	8	+	+	NOUN
cana-3950	58	9	𝑏	𝑏	ADJ
cana-3950	58	10	≤	≤	NUM
cana-3950	58	11	1	1	NUM
cana-3950	58	12	⟹	⟹	NOUN
cana-3950	58	13	𝑎2	𝑎2	PROPN
cana-3950	58	14	+	+	CCONJ
cana-3950	58	15	𝑏2	𝑏2	PROPN
cana-3950	58	16	≤	≤	PROPN
cana-3950	58	17	1	1	NUM
cana-3950	58	18	⟹	⟹	NUM
cana-3950	58	19	𝑎3	𝑎3	PROPN
cana-3950	58	20	+	+	CCONJ
cana-3950	58	21	𝑏3	𝑏3	NOUN
cana-3950	58	22	≤	≤	NUM
cana-3950	58	23	1	1	NUM
cana-3950	58	24	⟹	⟹	NUM
cana-3950	58	25	𝑎4	𝑎4	PROPN
cana-3950	58	26	+	+	CCONJ
cana-3950	58	27	𝑏4	𝑏4	PROPN
cana-3950	58	28	≤	≤	NUM
cana-3950	58	29	1⟹	1⟹	NUM
cana-3950	58	30	𝑎𝑛	𝑎𝑛	NOUN
cana-3950	58	31	+	+	PROPN
cana-3950	58	32	𝑏𝑛	𝑏𝑛	ADP
cana-3950	58	33	≤	≤	NUM
cana-3950	58	34	1	1	NUM
cana-3950	58	35	⟹	⟹	NOUN
cana-3950	58	36	𝑎𝑚	𝑎𝑚	NOUN
cana-3950	59	1	+	+	ADP
cana-3950	59	2	𝑏𝑛	𝑏𝑛	ADP
cana-3950	59	3	≤	≤	ADJ
cana-3950	59	4	1	1	NUM
cana-3950	59	5	,	,	PUNCT
cana-3950	59	6	𝑚	𝑚	X
cana-3950	59	7	≥	≥	NUM
cana-3950	59	8	4	4	NUM
cana-3950	59	9	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-3950	59	10	𝑛	𝑛	PRON
cana-3950	59	11	≥	≥	NOUN
cana-3950	59	12	4	4	NUM
cana-3950	59	13	from	from	ADP
cana-3950	59	14	definition	definition	NOUN
cana-3950	59	15	2.1	2.1	NUM
cana-3950	59	16	.	.	PUNCT
cana-3950	60	1	definition	definition	NOUN
cana-3950	60	2	2.6	2.6	NUM
cana-3950	60	3	let	let	VERB
cana-3950	60	4	𝐴	𝐴	PROPN
cana-3950	60	5	=	=	SYM
cana-3950	60	6	(	(	PUNCT
cana-3950	60	7	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	60	8	,	,	PUNCT
cana-3950	60	9	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	60	10	)	)	PUNCT
cana-3950	60	11	∈	∈	PROPN
cana-3950	60	12	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	60	13	𝑛	𝑛	PROPN
cana-3950	60	14	(	(	PUNCT
cana-3950	60	15	𝑋	𝑋	NOUN
cana-3950	60	16	)	)	PUNCT
cana-3950	60	17	𝑎𝑛𝑑	𝑎𝑛𝑑	VERB
cana-3950	60	18	𝑥	𝑥	PROPN
cana-3950	60	19	∈	∈	PROPN
cana-3950	60	20	𝑋.	𝑋.	PROPN
cana-3950	60	21	then	then	ADV
cana-3950	60	22	the	the	DET
cana-3950	60	23	expression	expression	NOUN
cana-3950	60	24	∏	∏	PROPN
cana-3950	60	25	(	(	PUNCT
cana-3950	60	26	𝑥	𝑥	NOUN
cana-3950	60	27	)	)	PUNCT
cana-3950	60	28	=	=	SYM
cana-3950	61	1	(	(	PUNCT
cana-3950	61	2	1−	1−	NUM
cana-3950	61	3	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	61	4	𝑚(𝑥	𝑚(𝑥	NOUN
cana-3950	61	5	)	)	PUNCT
cana-3950	62	1	−	−	NUM
cana-3950	62	2	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	62	3	𝑛(𝑥	𝑛(𝑥	PROPN
cana-3950	62	4	)	)	PUNCT
cana-3950	62	5	)	)	PUNCT
cana-3950	62	6	2	2	NUM
cana-3950	62	7	𝑚+𝑛	𝑚+𝑛	NUM
cana-3950	62	8	𝐴	𝐴	PROPN
cana-3950	62	9	is	be	AUX
cana-3950	62	10	said	say	VERB
cana-3950	62	11	to	to	PART
cana-3950	62	12	be	be	AUX
cana-3950	62	13	the	the	DET
cana-3950	62	14	degree	degree	NOUN
cana-3950	62	15	of	of	ADP
cana-3950	62	16	indeterminacy	indeterminacy	NOUN
cana-3950	62	17	of	of	ADP
cana-3950	62	18	x	x	PUNCT
cana-3950	62	19	to	to	ADP
cana-3950	62	20	a.	a.	NOUN
cana-3950	62	21	remark	remark	PROPN
cana-3950	62	22	:	:	PUNCT
cana-3950	62	23	clearly	clearly	ADV
cana-3950	62	24	,	,	PUNCT
cana-3950	62	25	∏	∏	PROPN
cana-3950	62	26	(	(	PUNCT
cana-3950	62	27	𝑥	𝑥	NOUN
cana-3950	62	28	)	)	PUNCT
cana-3950	62	29	(	(	PUNCT
cana-3950	62	30	𝑚+𝑛	𝑚+𝑛	NUM
cana-3950	62	31	)	)	PUNCT
cana-3950	62	32	2	2	NUM
cana-3950	62	33	𝐴	𝐴	PROPN
cana-3950	62	34	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-3950	62	35	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
cana-3950	62	36	𝑥	𝑥	DET
cana-3950	62	37	∈	∈	PROPN
cana-3950	62	38	𝑋.	𝑋.	PROPN
cana-3950	62	39	definition	definition	NOUN
cana-3950	62	40	2.7	2.7	NUM
cana-3950	62	41	let	let	VERB
cana-3950	62	42	𝐴	𝐴	PROPN
cana-3950	62	43	=	=	SYM
cana-3950	62	44	(	(	PUNCT
cana-3950	62	45	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	62	46	,	,	PUNCT
cana-3950	62	47	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	62	48	)	)	PUNCT
cana-3950	62	49	∈	∈	PROPN
cana-3950	62	50	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	62	51	𝑛	𝑛	PROPN
cana-3950	62	52	(	(	PUNCT
cana-3950	62	53	𝑋	𝑋	PROPN
cana-3950	62	54	)	)	PUNCT
cana-3950	62	55	.	.	PUNCT
cana-3950	63	1	then	then	ADV
cana-3950	63	2	the	the	DET
cana-3950	63	3	complement	complement	NOUN
cana-3950	63	4	of	of	ADP
cana-3950	63	5	a	a	PRON
cana-3950	63	6	,	,	PUNCT
cana-3950	63	7	denoted	denote	VERB
cana-3950	63	8	by	by	ADP
cana-3950	63	9	𝐴𝑐	𝐴𝑐	PROPN
cana-3950	63	10	,	,	PUNCT
cana-3950	63	11	is	be	AUX
cana-3950	63	12	defined	define	VERB
cana-3950	63	13	as	as	SCONJ
cana-3950	63	14	follows	follow	VERB
cana-3950	63	15	:	:	PUNCT
cana-3950	63	16	𝐴𝑐	𝐴𝑐	PROPN
cana-3950	63	17	=	=	SYM
cana-3950	63	18	(	(	PUNCT
cana-3950	63	19	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	63	20	𝑐	𝑐	PROPN
cana-3950	63	21	,	,	PUNCT
cana-3950	63	22	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	63	23	𝑐	𝑐	NOUN
cana-3950	63	24	)	)	PUNCT
cana-3950	63	25	=	=	SYM
cana-3950	63	26	(	(	PUNCT
cana-3950	63	27	𝐾𝐴	𝐾𝐴	NOUN
cana-3950	63	28	𝑛	𝑛	ADP
cana-3950	63	29	𝑚	𝑚	NOUN
cana-3950	63	30	,	,	PUNCT
cana-3950	63	31	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	63	32	𝑛	𝑛	PRON
cana-3950	63	33	𝑚	𝑚	NOUN
cana-3950	63	34	)	)	PUNCT
cana-3950	63	35	.	.	PUNCT
cana-3950	64	1	theorem	theorem	ADJ
cana-3950	64	2	2.8	2.8	NUM
cana-3950	64	3	let	let	VERB
cana-3950	64	4	𝐴	𝐴	PROPN
cana-3950	64	5	=	=	SYM
cana-3950	64	6	(	(	PUNCT
cana-3950	64	7	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	64	8	,	,	PUNCT
cana-3950	64	9	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	64	10	)	)	PUNCT
cana-3950	64	11	,	,	PUNCT
cana-3950	64	12	𝐴1	𝐴1	PROPN
cana-3950	65	1	=	=	SYM
cana-3950	65	2	(	(	PUNCT
cana-3950	65	3	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	65	4	,	,	PUNCT
cana-3950	65	5	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	65	6	)	)	PUNCT
cana-3950	65	7	,	,	PUNCT
cana-3950	65	8	𝐴2	𝐴2	PROPN
cana-3950	65	9	=	=	SYM
cana-3950	65	10	(	(	PUNCT
cana-3950	65	11	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	65	12	,	,	PUNCT
cana-3950	65	13	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	65	14	)	)	PUNCT
cana-3950	65	15	∈	∈	PROPN
cana-3950	65	16	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	65	17	𝑛	𝑛	PROPN
cana-3950	65	18	(	(	PUNCT
cana-3950	65	19	𝑋	𝑋	PROPN
cana-3950	65	20	)	)	PUNCT
cana-3950	65	21	.	.	PUNCT
cana-3950	66	1	then	then	ADV
cana-3950	66	2	communications	communication	NOUN
cana-3950	66	3	on	on	ADP
cana-3950	66	4	applied	apply	VERB
cana-3950	66	5	nonlinear	nonlinear	ADJ
cana-3950	66	6	analysis	analysis	NOUN
cana-3950	66	7	issn	issn	NOUN
cana-3950	66	8	:	:	PUNCT
cana-3950	66	9	1074	1074	NUM
cana-3950	66	10	-	-	PUNCT
cana-3950	66	11	133x	133x	NUM
cana-3950	66	12	vol	vol	NOUN
cana-3950	66	13	32	32	NUM
cana-3950	66	14	no	no	NOUN
cana-3950	66	15	.	.	PUNCT
cana-3950	67	1	9s	9s	NUM
cana-3950	67	2	(	(	PUNCT
cana-3950	67	3	2025	2025	NUM
cana-3950	67	4	)	)	PUNCT
cana-3950	67	5	389	389	NUM
cana-3950	67	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3950	67	7	(	(	PUNCT
cana-3950	67	8	𝑖	𝑖	X
cana-3950	67	9	)	)	PUNCT
cana-3950	67	10	𝐴𝑐	𝐴𝑐	PROPN
cana-3950	67	11	∈	∈	NOUN
cana-3950	67	12	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	67	13	𝑛	𝑛	PROPN
cana-3950	67	14	(	(	PUNCT
cana-3950	67	15	𝑋	𝑋	PROPN
cana-3950	67	16	)	)	PUNCT
cana-3950	67	17	(	(	PUNCT
cana-3950	67	18	𝑖𝑖)(𝐴𝑐)𝑐	𝑖𝑖)(𝐴𝑐)𝑐	NUM
cana-3950	67	19	=	=	SYM
cana-3950	67	20	𝐴	𝐴	PROPN
cana-3950	67	21	(	(	PUNCT
cana-3950	67	22	𝑖𝑖𝑖)(𝐴1	𝑖𝑖𝑖)(𝐴1	PROPN
cana-3950	67	23	∪	∪	PROPN
cana-3950	67	24	𝐴2	𝐴2	PROPN
cana-3950	67	25	)	)	PUNCT
cana-3950	67	26	𝑐	𝑐	PROPN
cana-3950	67	27	=	=	SYM
cana-3950	67	28	𝐴1	𝐴1	PROPN
cana-3950	67	29	𝑐	𝑐	PROPN
cana-3950	67	30	∩	∩	PROPN
cana-3950	67	31	𝐴2	𝐴2	PROPN
cana-3950	67	32	𝑐	𝑐	PROPN
cana-3950	67	33	(	(	PUNCT
cana-3950	67	34	𝑖𝑣)(𝐴1	𝑖𝑣)(𝐴1	PROPN
cana-3950	67	35	∩	∩	X
cana-3950	67	36	𝐴2	𝐴2	PROPN
cana-3950	67	37	)	)	PUNCT
cana-3950	67	38	𝑐	𝑐	PROPN
cana-3950	67	39	=	=	SYM
cana-3950	67	40	𝐴1	𝐴1	PROPN
cana-3950	67	41	𝑐	𝑐	PROPN
cana-3950	67	42	∪	∪	VERB
cana-3950	67	43	𝐴2	𝐴2	PROPN
cana-3950	67	44	𝑐.	𝑐.	NOUN
cana-3950	67	45	proof	proof	NOUN
cana-3950	67	46	:	:	PUNCT
cana-3950	67	47	(	(	PUNCT
cana-3950	67	48	i	i	NOUN
cana-3950	67	49	)	)	PUNCT
cana-3950	67	50	since	since	ADV
cana-3950	67	51	,	,	PUNCT
cana-3950	67	52	𝐴𝑐	𝐴𝑐	PROPN
cana-3950	67	53	=	=	SYM
cana-3950	67	54	(	(	PUNCT
cana-3950	67	55	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	67	56	𝑐	𝑐	PROPN
cana-3950	67	57	,	,	PUNCT
cana-3950	67	58	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	67	59	𝑐	𝑐	NOUN
cana-3950	67	60	)	)	PUNCT
cana-3950	67	61	=	=	SYM
cana-3950	67	62	(	(	PUNCT
cana-3950	67	63	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	67	64	𝑚	𝑚	ADP
cana-3950	67	65	𝑛	𝑛	PROPN
cana-3950	67	66	,	,	PUNCT
cana-3950	67	67	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	67	68	𝑚	𝑚	ADP
cana-3950	67	69	𝑛	𝑛	PROPN
cana-3950	67	70	)	)	PUNCT
cana-3950	67	71	,	,	PUNCT
cana-3950	67	72	we	we	PRON
cana-3950	67	73	have	have	VERB
cana-3950	67	74	0	0	NUM
cana-3950	67	75	≤	≤	NUM
cana-3950	67	76	𝐽𝐴𝑐	𝐽𝐴𝑐	NOUN
cana-3950	67	77	𝑚	𝑚	PROPN
cana-3950	68	1	+	+	NUM
cana-3950	68	2	𝐾𝐴𝑐	𝐾𝐴𝑐	NOUN
cana-3950	68	3	𝑛	𝑛	NOUN
cana-3950	68	4	=	=	SYM
cana-3950	68	5	(	(	PUNCT
cana-3950	68	6	(	(	PUNCT
cana-3950	68	7	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	68	8	)	)	PUNCT
cana-3950	68	9	𝑛	𝑛	DET
cana-3950	68	10	𝑚	𝑚	NOUN
cana-3950	68	11	)	)	PUNCT
cana-3950	68	12	𝑚	𝑚	NOUN
cana-3950	68	13	+	+	CCONJ
cana-3950	68	14	(	(	PUNCT
cana-3950	68	15	(	(	PUNCT
cana-3950	68	16	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	68	17	)	)	PUNCT
cana-3950	68	18	𝑚	𝑚	ADP
cana-3950	68	19	𝑛	𝑛	PROPN
cana-3950	68	20	)	)	PUNCT
cana-3950	68	21	𝑛	𝑛	NOUN
cana-3950	68	22	=	=	SYM
cana-3950	68	23	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	68	24	𝑛	𝑛	NOUN
cana-3950	68	25	+	+	NUM
cana-3950	68	26	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	68	27	𝑚	𝑚	NOUN
cana-3950	68	28	=	=	SYM
cana-3950	68	29	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	68	30	𝑚	𝑚	NOUN
cana-3950	68	31	+	+	CCONJ
cana-3950	68	32	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	68	33	𝑛	𝑛	ADJ
cana-3950	68	34	≤	≤	NUM
cana-3950	68	35	1	1	NUM
cana-3950	68	36	therefore	therefore	ADV
cana-3950	68	37	,	,	PUNCT
cana-3950	68	38	𝐴𝑐	𝐴𝑐	PROPN
cana-3950	68	39	∈	∈	PROPN
cana-3950	68	40	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	68	41	𝑛	𝑛	PROPN
cana-3950	68	42	(	(	PUNCT
cana-3950	68	43	𝑋	𝑋	PROPN
cana-3950	68	44	)	)	PUNCT
cana-3950	68	45	.	.	PUNCT
cana-3950	69	1	(	(	PUNCT
cana-3950	69	2	ii	ii	NOUN
cana-3950	69	3	)	)	PUNCT
cana-3950	69	4	easy	easy	ADJ
cana-3950	69	5	and	and	CCONJ
cana-3950	69	6	left	leave	VERB
cana-3950	69	7	to	to	ADP
cana-3950	69	8	the	the	DET
cana-3950	69	9	reader	reader	NOUN
cana-3950	69	10	.	.	PUNCT
cana-3950	70	1	(	(	PUNCT
cana-3950	70	2	iii	iii	X
cana-3950	70	3	)	)	PUNCT
cana-3950	70	4	we	we	PRON
cana-3950	70	5	have	have	VERB
cana-3950	70	6	(	(	PUNCT
cana-3950	70	7	𝐴1	𝐴1	PROPN
cana-3950	70	8	∪	∪	ADP
cana-3950	70	9	𝐴2	𝐴2	PROPN
cana-3950	70	10	)	)	PUNCT
cana-3950	70	11	𝑐	𝑐	PROPN
cana-3950	70	12	=	=	SYM
cana-3950	70	13	(	(	PUNCT
cana-3950	70	14	max	max	PROPN
cana-3950	70	15	{	{	PUNCT
cana-3950	70	16	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	70	17	,	,	PUNCT
cana-3950	70	18	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	70	19	)	)	PUNCT
cana-3950	70	20	,	,	PUNCT
cana-3950	70	21	min	min	PROPN
cana-3950	70	22	{	{	PUNCT
cana-3950	70	23	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	70	24	,	,	PUNCT
cana-3950	70	25	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	70	26	}	}	PUNCT
cana-3950	70	27	)	)	PUNCT
cana-3950	70	28	𝑐	𝑐	PROPN
cana-3950	70	29	=	=	SYM
cana-3950	70	30	(	(	PUNCT
cana-3950	70	31	min	min	X
cana-3950	70	32	{	{	PUNCT
cana-3950	70	33	𝐾𝐴1	𝐾𝐴1	NOUN
cana-3950	70	34	𝑛	𝑛	ADP
cana-3950	70	35	𝑚	𝑚	NOUN
cana-3950	70	36	,	,	PUNCT
cana-3950	70	37	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	70	38	𝑛	𝑛	ADP
cana-3950	70	39	𝑚	𝑚	PROPN
cana-3950	70	40	}	}	PUNCT
cana-3950	70	41	,	,	PUNCT
cana-3950	70	42	max	max	PROPN
cana-3950	70	43	{	{	PUNCT
cana-3950	70	44	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	70	45	𝑚	𝑚	PROPN
cana-3950	70	46	𝑛	𝑛	PROPN
cana-3950	70	47	,	,	PUNCT
cana-3950	71	1	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	71	2	(	(	PUNCT
cana-3950	71	3	𝑚	𝑚	PROPN
cana-3950	71	4	𝑛	𝑛	PROPN
cana-3950	71	5	)	)	PUNCT
cana-3950	71	6	}	}	PUNCT
cana-3950	71	7	)	)	PUNCT
cana-3950	72	1	=	=	SYM
cana-3950	72	2	(	(	PUNCT
cana-3950	72	3	𝐾𝐴1	𝐾𝐴1	NOUN
cana-3950	72	4	𝑛	𝑛	ADP
cana-3950	72	5	𝑚	𝑚	NOUN
cana-3950	72	6	,	,	PUNCT
cana-3950	72	7	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	72	8	𝑚	𝑚	PROPN
cana-3950	72	9	𝑛	𝑛	PROPN
cana-3950	72	10	)	)	PUNCT
cana-3950	72	11	∩	∩	NOUN
cana-3950	72	12	(	(	PUNCT
cana-3950	72	13	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	72	14	𝑛	𝑛	ADP
cana-3950	72	15	𝑚	𝑚	X
cana-3950	72	16	,	,	PUNCT
cana-3950	72	17	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	72	18	(	(	PUNCT
cana-3950	72	19	𝑚	𝑚	PROPN
cana-3950	72	20	𝑛	𝑛	PROPN
cana-3950	72	21	)	)	PUNCT
cana-3950	72	22	)	)	PUNCT
cana-3950	73	1	=	=	SYM
cana-3950	73	2	𝐴1	𝐴1	PROPN
cana-3950	73	3	𝑐	𝑐	PROPN
cana-3950	73	4	∩	∩	PROPN
cana-3950	73	5	𝐴2	𝐴2	PROPN
cana-3950	73	6	𝑐.	𝑐.	NOUN
cana-3950	73	7	(	(	PUNCT
cana-3950	73	8	i	i	NOUN
cana-3950	73	9	)	)	PUNCT
cana-3950	73	10	follows	follow	VERB
cana-3950	73	11	by	by	ADP
cana-3950	73	12	a	a	DET
cana-3950	73	13	similar	similar	ADJ
cana-3950	73	14	process	process	NOUN
cana-3950	73	15	to	to	ADP
cana-3950	73	16	(	(	PUNCT
cana-3950	73	17	iii	iii	NOUN
cana-3950	73	18	)	)	PUNCT
cana-3950	73	19	.	.	PUNCT
cana-3950	74	1	definition	definition	NOUN
cana-3950	74	2	2.9	2.9	NUM
cana-3950	74	3	let	let	VERB
cana-3950	74	4	𝐴	𝐴	PROPN
cana-3950	74	5	=	=	SYM
cana-3950	74	6	(	(	PUNCT
cana-3950	74	7	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	74	8	,	,	PUNCT
cana-3950	74	9	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	74	10	)	)	PUNCT
cana-3950	74	11	∈	∈	PROPN
cana-3950	74	12	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	74	13	𝑛	𝑛	PROPN
cana-3950	74	14	(	(	PUNCT
cana-3950	74	15	𝑋	𝑋	PROPN
cana-3950	74	16	)	)	PUNCT
cana-3950	74	17	.	.	PUNCT
cana-3950	75	1	then	then	ADV
cana-3950	75	2	we	we	PRON
cana-3950	75	3	need	need	VERB
cana-3950	75	4	of	of	ADP
cana-3950	75	5	terms	term	NOUN
cana-3950	75	6	and	and	CCONJ
cana-3950	75	7	the	the	DET
cana-3950	75	8	possibility	possibility	NOUN
cana-3950	75	9	measure	measure	NOUN
cana-3950	75	10	on	on	ADP
cana-3950	75	11	a	a	PRON
cana-3950	75	12	are	be	AUX
cana-3950	75	13	called	call	VERB
cana-3950	75	14	as	as	SCONJ
cana-3950	75	15	follows	follow	VERB
cana-3950	75	16	:	:	PUNCT
cana-3950	75	17	(	(	PUNCT
cana-3950	75	18	𝑖	𝑖	X
cana-3950	75	19	)	)	PUNCT
cana-3950	75	20	□	□	PUNCT
cana-3950	75	21	a	a	X
cana-3950	75	22	=	=	X
cana-3950	75	23	(	(	PUNCT
cana-3950	75	24	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	75	25	,	,	PUNCT
cana-3950	75	26	(	(	PUNCT
cana-3950	75	27	1−	1−	NUM
cana-3950	75	28	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	75	29	𝑚	𝑚	NOUN
cana-3950	75	30	)	)	PUNCT
cana-3950	75	31	1	1	NUM
cana-3950	75	32	𝑛	𝑛	NOUN
cana-3950	75	33	)	)	PUNCT
cana-3950	75	34	.	.	PUNCT
cana-3950	76	1	(	(	PUNCT
cana-3950	76	2	𝑖𝑖)	𝑖𝑖)	PUNCT
cana-3950	76	3	◊	◊	NOUN
cana-3950	76	4	a	a	X
cana-3950	76	5	=	=	PUNCT
cana-3950	76	6	(	(	PUNCT
cana-3950	76	7	(	(	PUNCT
cana-3950	76	8	1−	1−	NUM
cana-3950	76	9	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	76	10	𝑛	𝑛	NOUN
cana-3950	76	11	)	)	PUNCT
cana-3950	76	12	1	1	NUM
cana-3950	76	13	𝑚	𝑚	NOUN
cana-3950	76	14	,	,	PUNCT
cana-3950	76	15	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	76	16	)	)	PUNCT
cana-3950	76	17	.	.	PUNCT
cana-3950	77	1	example	example	NOUN
cana-3950	77	2	2.10	2.10	NUM
cana-3950	77	3	let	let	VERB
cana-3950	77	4	𝑋	𝑋	NOUN
cana-3950	77	5	=	=	SYM
cana-3950	77	6	{	{	PUNCT
cana-3950	77	7	𝑥	𝑥	X
cana-3950	77	8	}	}	PUNCT
cana-3950	77	9	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-3950	77	10	𝐴	𝐴	NOUN
cana-3950	77	11	=	=	SYM
cana-3950	77	12	{	{	PUNCT
cana-3950	77	13	〈	〈	NOUN
cana-3950	77	14	0.75,0.8	0.75,0.8	ADJ
cana-3950	77	15	〉	〉	NOUN
cana-3950	77	16	}	}	PUNCT
cana-3950	77	17	∈	∈	PROPN
cana-3950	77	18	𝐵6	𝐵6	NOUN
cana-3950	77	19	5(𝑥	5(𝑥	NUM
cana-3950	77	20	)	)	PUNCT
cana-3950	77	21	.	.	PUNCT
cana-3950	78	1	then	then	ADV
cana-3950	78	2	∏	∏	NUM
cana-3950	78	3	(	(	PUNCT
cana-3950	78	4	𝑥	𝑥	NOUN
cana-3950	78	5	)	)	PUNCT
cana-3950	78	6	=	=	PUNCT
cana-3950	78	7	0.6321432,𝐴	0.6321432,𝐴	X
cana-3950	78	8	□	□	PUNCT
cana-3950	78	9	a	a	DET
cana-3950	78	10	=	=	NOUN
cana-3950	78	11	{	{	PUNCT
cana-3950	78	12	{	{	PUNCT
cana-3950	78	13	〈	〈	NOUN
cana-3950	78	14	0.75,0.8	0.75,0.8	ADJ
cana-3950	78	15	〉	〉	NOUN
cana-3950	78	16	}	}	PUNCT
cana-3950	78	17	,	,	PUNCT
cana-3950	78	18	◊	◊	PROPN
cana-3950	78	19	a	a	X
cana-3950	78	20	=	=	PUNCT
cana-3950	78	21	〈	〈	NOUN
cana-3950	78	22	𝑥	𝑥	PROPN
cana-3950	78	23	,	,	PUNCT
cana-3950	78	24	0.75932	0.75932	NUM
cana-3950	78	25	,	,	PUNCT
cana-3950	78	26	0.8	0.8	NUM
cana-3950	78	27	〉	〉	NOUN
cana-3950	78	28	}	}	PUNCT
cana-3950	78	29	,	,	PUNCT
cana-3950	78	30	𝐴𝑐	𝐴𝑐	PROPN
cana-3950	78	31	=	=	PUNCT
cana-3950	78	32	{	{	PUNCT
cana-3950	78	33	〈	〈	NOUN
cana-3950	78	34	𝑥	𝑥	NOUN
cana-3950	78	35	,	,	PUNCT
cana-3950	78	36	0.8,0.723	0.8,0.723	NUM
cana-3950	78	37	〉	〉	NOUN
cana-3950	78	38	}	}	PUNCT
cana-3950	78	39	.	.	PUNCT
cana-3950	79	1	theorem	theorem	VERB
cana-3950	79	2	2.11	2.11	NUM
cana-3950	79	3	𝐼𝑓	𝐼𝑓	PROPN
cana-3950	79	4	𝐴	𝐴	NOUN
cana-3950	79	5	=	=	SYM
cana-3950	79	6	(	(	PUNCT
cana-3950	79	7	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	79	8	,	,	PUNCT
cana-3950	79	9	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	79	10	)	)	PUNCT
cana-3950	79	11	∈	∈	PROPN
cana-3950	79	12	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	79	13	𝑛	𝑛	PROPN
cana-3950	79	14	(	(	PUNCT
cana-3950	79	15	𝑋	𝑋	PROPN
cana-3950	79	16	)	)	PUNCT
cana-3950	79	17	,	,	PUNCT
cana-3950	79	18	then	then	ADV
cana-3950	79	19	(	(	PUNCT
cana-3950	79	20	𝑖	𝑖	X
cana-3950	79	21	)	)	PUNCT
cana-3950	79	22	□	□	PUNCT
cana-3950	79	23	a	a	DET
cana-3950	79	24	∈	∈	PROPN
cana-3950	79	25	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	79	26	𝑛	𝑛	PROPN
cana-3950	79	27	(	(	PUNCT
cana-3950	79	28	𝑋	𝑋	PROPN
cana-3950	79	29	)	)	PUNCT
cana-3950	79	30	,	,	PUNCT
cana-3950	79	31	(	(	PUNCT
cana-3950	79	32	𝑖𝑖)	𝑖𝑖)	X
cana-3950	79	33	◊	◊	PROPN
cana-3950	79	34	a	a	DET
cana-3950	79	35	∈	∈	NOUN
cana-3950	79	36	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	79	37	𝑛	𝑛	PROPN
cana-3950	79	38	(	(	PUNCT
cana-3950	79	39	𝑋	𝑋	PROPN
cana-3950	79	40	)	)	PUNCT
cana-3950	79	41	,	,	PUNCT
cana-3950	80	1	proof	proof	NOUN
cana-3950	80	2	:	:	PUNCT
cana-3950	80	3	(	(	PUNCT
cana-3950	80	4	i	i	NOUN
cana-3950	80	5	)	)	PUNCT
cana-3950	80	6	follows	follow	VERB
cana-3950	80	7	on	on	ADP
cana-3950	80	8	noting	note	VERB
cana-3950	80	9	that	that	SCONJ
cana-3950	80	10	:	:	PUNCT
cana-3950	80	11	𝐽	𝐽	PROPN
cana-3950	80	12	□	□	PROPN
cana-3950	80	13	a	a	PRON
cana-3950	80	14	𝑚	𝑚	X
cana-3950	80	15	+	+	PROPN
cana-3950	80	16	𝐾	𝐾	PROPN
cana-3950	80	17	□	□	PROPN
cana-3950	80	18	a	a	DET
cana-3950	80	19	𝑚	𝑚	NOUN
cana-3950	80	20	=	=	SYM
cana-3950	80	21	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	80	22	𝑚	𝑚	NOUN
cana-3950	80	23	+	+	CCONJ
cana-3950	80	24	(	(	PUNCT
cana-3950	80	25	(	(	PUNCT
cana-3950	80	26	1−	1−	NUM
cana-3950	80	27	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	80	28	𝑚	𝑚	NOUN
cana-3950	80	29	)	)	PUNCT
cana-3950	80	30	1	1	NUM
cana-3950	80	31	𝑛	𝑛	NOUN
cana-3950	80	32	)	)	PUNCT
cana-3950	80	33	𝑛	𝑛	NOUN
cana-3950	80	34	=	=	SYM
cana-3950	80	35	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	80	36	𝑚	𝑚	NOUN
cana-3950	80	37	+	+	CCONJ
cana-3950	80	38	(	(	PUNCT
cana-3950	80	39	1	1	NUM
cana-3950	80	40	−	−	PROPN
cana-3950	80	41	𝐽𝐴	𝐽𝐴	NOUN
cana-3950	80	42	𝑚	𝑚	NOUN
cana-3950	80	43	)	)	PUNCT
cana-3950	80	44	communications	communication	NOUN
cana-3950	80	45	on	on	ADP
cana-3950	80	46	applied	apply	VERB
cana-3950	80	47	nonlinear	nonlinear	ADJ
cana-3950	80	48	analysis	analysis	NOUN
cana-3950	80	49	issn	issn	NOUN
cana-3950	80	50	:	:	PUNCT
cana-3950	80	51	1074	1074	NUM
cana-3950	80	52	-	-	PUNCT
cana-3950	80	53	133x	133x	NUM
cana-3950	80	54	vol	vol	NOUN
cana-3950	80	55	32	32	NUM
cana-3950	80	56	no	no	NOUN
cana-3950	80	57	.	.	PUNCT
cana-3950	81	1	9s	9s	NUM
cana-3950	81	2	(	(	PUNCT
cana-3950	81	3	2025	2025	NUM
cana-3950	81	4	)	)	PUNCT
cana-3950	81	5	390	390	NUM
cana-3950	81	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3950	81	7	=	=	SYM
cana-3950	81	8	1	1	X
cana-3950	81	9	.	.	PUNCT
cana-3950	82	1	(	(	PUNCT
cana-3950	82	2	ii)follows	ii)follow	VERB
cana-3950	82	3	on	on	ADP
cana-3950	82	4	noting	note	VERB
cana-3950	82	5	that	that	SCONJ
cana-3950	82	6	𝐽	𝐽	PROPN
cana-3950	82	7	◊	◊	PROPN
cana-3950	82	8	a	a	PRON
cana-3950	82	9	𝑚	𝑚	NOUN
cana-3950	82	10	+	+	CCONJ
cana-3950	82	11	𝐾	𝐾	PROPN
cana-3950	82	12	◊	◊	PROPN
cana-3950	82	13	a	a	DET
cana-3950	82	14	𝑛	𝑛	NOUN
cana-3950	82	15	=	=	PUNCT
cana-3950	82	16	(	(	PUNCT
cana-3950	82	17	(	(	PUNCT
cana-3950	82	18	1−	1−	NUM
cana-3950	82	19	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	82	20	𝑛	𝑛	NOUN
cana-3950	82	21	)	)	PUNCT
cana-3950	82	22	1	1	NUM
cana-3950	82	23	𝑚	𝑚	NOUN
cana-3950	82	24	)	)	PUNCT
cana-3950	82	25	𝑚	𝑚	PROPN
cana-3950	83	1	+	+	CCONJ
cana-3950	83	2	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	83	3	𝑛	𝑛	NOUN
cana-3950	83	4	=	=	PUNCT
cana-3950	83	5	(	(	PUNCT
cana-3950	83	6	1−	1−	NUM
cana-3950	83	7	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	83	8	𝑛	𝑛	NOUN
cana-3950	83	9	)	)	PUNCT
cana-3950	83	10	+	+	CCONJ
cana-3950	83	11	𝐾𝐴	𝐾𝐴	NOUN
cana-3950	83	12	𝑛	𝑛	NOUN
cana-3950	83	13	=	=	SYM
cana-3950	83	14	1	1	X
cana-3950	83	15	.	.	PUNCT
cana-3950	83	16	definition	definition	NOUN
cana-3950	83	17	2.12	2.12	NUM
cana-3950	83	18	let	let	VERB
cana-3950	83	19	𝐴	𝐴	PROPN
cana-3950	83	20	=	=	SYM
cana-3950	83	21	(	(	PUNCT
cana-3950	83	22	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	83	23	,	,	PUNCT
cana-3950	83	24	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	83	25	)	)	PUNCT
cana-3950	83	26	,	,	PUNCT
cana-3950	83	27	𝐴1	𝐴1	PROPN
cana-3950	84	1	=	=	SYM
cana-3950	84	2	(	(	PUNCT
cana-3950	84	3	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	84	4	,	,	PUNCT
cana-3950	84	5	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	84	6	)	)	PUNCT
cana-3950	84	7	,	,	PUNCT
cana-3950	84	8	𝐴2	𝐴2	PROPN
cana-3950	84	9	=	=	SYM
cana-3950	84	10	(	(	PUNCT
cana-3950	84	11	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	84	12	,	,	PUNCT
cana-3950	84	13	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	84	14	)	)	PUNCT
cana-3950	84	15	∈	∈	PROPN
cana-3950	84	16	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	84	17	𝑛	𝑛	PROPN
cana-3950	84	18	(	(	PUNCT
cana-3950	84	19	𝑋	𝑋	PROPN
cana-3950	84	20	)	)	PUNCT
cana-3950	84	21	and	and	CCONJ
cana-3950	84	22	𝑞	𝑞	PROPN
cana-3950	84	23	∈	∈	PROPN
cana-3950	84	24	𝑁.	𝑁.	PROPN
cana-3950	84	25	then	then	ADV
cana-3950	84	26	the	the	DET
cana-3950	84	27	operations	operation	NOUN
cana-3950	84	28	𝐴1⊕𝐴2	𝐴1⊕𝐴2	PROPN
cana-3950	84	29	,	,	PUNCT
cana-3950	84	30	𝐴1	𝐴1	PROPN
cana-3950	84	31	⨂𝐴2	⨂𝐴2	PROPN
cana-3950	84	32	,	,	PUNCT
cana-3950	84	33	𝑞𝐴	𝑞𝐴	VERB
cana-3950	84	34	,	,	PUNCT
cana-3950	84	35	and	and	CCONJ
cana-3950	84	36	𝐴𝑞are	𝐴𝑞are	PROPN
cana-3950	84	37	defined	define	VERB
cana-3950	84	38	as	as	ADP
cana-3950	84	39	follows	follow	VERB
cana-3950	84	40	(	(	PUNCT
cana-3950	84	41	i	i	NOUN
cana-3950	84	42	)	)	PUNCT
cana-3950	85	1	𝐴1⊕𝐴2	𝐴1⊕𝐴2	PROPN
cana-3950	85	2	=	=	SYM
cana-3950	85	3	(	(	PUNCT
cana-3950	85	4	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	85	5	𝑚	𝑚	PROPN
cana-3950	86	1	+	+	CCONJ
cana-3950	86	2	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	86	3	𝑚	𝑚	NOUN
cana-3950	86	4	−	−	PROPN
cana-3950	86	5	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	86	6	𝑚𝐽𝐴2	𝑚𝐽𝐴2	NOUN
cana-3950	86	7	𝑚	𝑚	X
cana-3950	86	8	,	,	PUNCT
cana-3950	86	9	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	86	10	𝑛𝐾𝐴2	𝑛𝐾𝐴2	VERB
cana-3950	86	11	𝑛	𝑛	PROPN
cana-3950	86	12	)	)	PUNCT
cana-3950	86	13	,	,	PUNCT
cana-3950	86	14	(	(	PUNCT
cana-3950	86	15	ii	ii	X
cana-3950	86	16	)	)	PUNCT
cana-3950	86	17	𝐴1	𝐴1	PROPN
cana-3950	86	18	⨂𝐴2	⨂𝐴2	PROPN
cana-3950	87	1	=	=	SYM
cana-3950	87	2	(	(	PUNCT
cana-3950	87	3	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	87	4	𝑚𝐽𝐴2	𝑚𝐽𝐴2	VERB
cana-3950	87	5	𝑚	𝑚	X
cana-3950	87	6	,	,	PUNCT
cana-3950	87	7	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	87	8	𝑛	𝑛	PROPN
cana-3950	87	9	+	+	CCONJ
cana-3950	87	10	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	87	11	𝑛	𝑛	DET
cana-3950	87	12	−	−	PROPN
cana-3950	87	13	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	87	14	𝑛𝐾𝐴2	𝑛𝐾𝐴2	VERB
cana-3950	87	15	𝑛	𝑛	NOUN
cana-3950	87	16	)	)	PUNCT
cana-3950	87	17	(	(	PUNCT
cana-3950	87	18	iii	iii	X
cana-3950	87	19	)	)	PUNCT
cana-3950	87	20	𝑞𝐴	𝑞𝐴	NOUN
cana-3950	87	21	=	=	SYM
cana-3950	87	22	(	(	PUNCT
cana-3950	87	23	1−	1−	NUM
cana-3950	87	24	(	(	PUNCT
cana-3950	87	25	1	1	NUM
cana-3950	87	26	−	−	PROPN
cana-3950	87	27	𝐽𝐴	𝐽𝐴	NOUN
cana-3950	87	28	𝑚)𝑞	𝑚)𝑞	NOUN
cana-3950	87	29	,	,	PUNCT
cana-3950	87	30	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	87	31	𝑛	𝑛	NOUN
cana-3950	87	32	)	)	PUNCT
cana-3950	87	33	,	,	PUNCT
cana-3950	87	34	(	(	PUNCT
cana-3950	87	35	iv	iv	X
cana-3950	87	36	)	)	PUNCT
cana-3950	87	37	𝐴𝑞	𝐴𝑞	X
cana-3950	87	38	=	=	SYM
cana-3950	87	39	(	(	PUNCT
cana-3950	87	40	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	87	41	𝑚𝑞	𝑚𝑞	ADP
cana-3950	87	42	,	,	PUNCT
cana-3950	87	43	1−	1−	NUM
cana-3950	87	44	(	(	PUNCT
cana-3950	87	45	1−	1−	NUM
cana-3950	87	46	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	87	47	𝑛)𝑞	𝑛)𝑞	NOUN
cana-3950	87	48	)	)	PUNCT
cana-3950	87	49	.	.	PUNCT
cana-3950	88	1	theorem	theorem	NOUN
cana-3950	88	2	2.13	2.13	NUM
cana-3950	88	3	let	let	VERB
cana-3950	88	4	𝐴	𝐴	PROPN
cana-3950	88	5	=	=	SYM
cana-3950	88	6	(	(	PUNCT
cana-3950	88	7	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	88	8	,	,	PUNCT
cana-3950	88	9	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	88	10	)	)	PUNCT
cana-3950	88	11	,	,	PUNCT
cana-3950	88	12	𝐴1	𝐴1	PROPN
cana-3950	89	1	=	=	SYM
cana-3950	89	2	(	(	PUNCT
cana-3950	89	3	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	89	4	,	,	PUNCT
cana-3950	89	5	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	89	6	)	)	PUNCT
cana-3950	89	7	,	,	PUNCT
cana-3950	89	8	𝐴2	𝐴2	PROPN
cana-3950	89	9	=	=	SYM
cana-3950	89	10	(	(	PUNCT
cana-3950	89	11	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	89	12	,	,	PUNCT
cana-3950	89	13	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	89	14	)	)	PUNCT
cana-3950	89	15	∈	∈	PROPN
cana-3950	89	16	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	89	17	𝑛	𝑛	PROPN
cana-3950	89	18	(	(	PUNCT
cana-3950	89	19	𝑋	𝑋	PROPN
cana-3950	89	20	)	)	PUNCT
cana-3950	89	21	and	and	CCONJ
cana-3950	89	22	𝑞	𝑞	X
cana-3950	89	23	∈	∈	PROPN
cana-3950	89	24	𝑁.	𝑁.	PROPN
cana-3950	89	25	then	then	ADV
cana-3950	89	26	(	(	PUNCT
cana-3950	89	27	i	i	NOUN
cana-3950	89	28	)	)	PUNCT
cana-3950	89	29	𝐴1⊕𝐴2	𝐴1⊕𝐴2	PROPN
cana-3950	89	30	∈	∈	PROPN
cana-3950	89	31	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	89	32	𝑛	𝑛	PROPN
cana-3950	89	33	(	(	PUNCT
cana-3950	89	34	𝑋	𝑋	PROPN
cana-3950	89	35	)	)	PUNCT
cana-3950	89	36	,	,	PUNCT
cana-3950	89	37	(	(	PUNCT
cana-3950	89	38	ii	ii	X
cana-3950	89	39	)	)	PUNCT
cana-3950	89	40	𝐴1	𝐴1	PROPN
cana-3950	89	41	⨂𝐴2	⨂𝐴2	PROPN
cana-3950	90	1	∈	∈	PROPN
cana-3950	90	2	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	90	3	𝑛	𝑛	PROPN
cana-3950	90	4	(	(	PUNCT
cana-3950	90	5	𝑋	𝑋	PROPN
cana-3950	90	6	)	)	PUNCT
cana-3950	90	7	,	,	PUNCT
cana-3950	90	8	(	(	PUNCT
cana-3950	90	9	iii	iii	X
cana-3950	90	10	)	)	PUNCT
cana-3950	90	11	𝐴1	𝐴1	PROPN
cana-3950	90	12	∪	∪	VERB
cana-3950	90	13	𝐴2	𝐴2	PROPN
cana-3950	90	14	∈	∈	PROPN
cana-3950	90	15	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	90	16	𝑛	𝑛	PROPN
cana-3950	90	17	(	(	PUNCT
cana-3950	90	18	𝑋	𝑋	PROPN
cana-3950	90	19	)	)	PUNCT
cana-3950	90	20	,	,	PUNCT
cana-3950	90	21	(	(	PUNCT
cana-3950	90	22	iv	iv	X
cana-3950	90	23	)	)	PUNCT
cana-3950	90	24	𝐴1	𝐴1	PROPN
cana-3950	90	25	∩	∩	SCONJ
cana-3950	90	26	𝐴2	𝐴2	PROPN
cana-3950	90	27	∈	∈	PROPN
cana-3950	90	28	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	90	29	𝑛	𝑛	PROPN
cana-3950	90	30	(	(	PUNCT
cana-3950	90	31	𝑋	𝑋	PROPN
cana-3950	90	32	)	)	PUNCT
cana-3950	90	33	,	,	PUNCT
cana-3950	90	34	(	(	PUNCT
cana-3950	90	35	v	v	NOUN
cana-3950	90	36	)	)	PUNCT
cana-3950	90	37	𝑞𝐴	𝑞𝐴	NOUN
cana-3950	90	38	∈	∈	PROPN
cana-3950	91	1	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	91	2	𝑛	𝑛	PROPN
cana-3950	91	3	(	(	PUNCT
cana-3950	91	4	𝑋	𝑋	PROPN
cana-3950	91	5	)	)	PUNCT
cana-3950	91	6	,	,	PUNCT
cana-3950	91	7	(	(	PUNCT
cana-3950	91	8	vi	vi	X
cana-3950	91	9	)	)	PUNCT
cana-3950	91	10	𝐴𝑞	𝐴𝑞	VERB
cana-3950	91	11	∈	∈	NOUN
cana-3950	91	12	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	91	13	𝑛	𝑛	PROPN
cana-3950	91	14	(	(	PUNCT
cana-3950	91	15	𝑋	𝑋	PROPN
cana-3950	91	16	)	)	PUNCT
cana-3950	91	17	.	.	PUNCT
cana-3950	92	1	proof	proof	NOUN
cana-3950	92	2	:	:	PUNCT
cana-3950	92	3	(	(	PUNCT
cana-3950	92	4	i	i	NOUN
cana-3950	92	5	)	)	PUNCT
cana-3950	92	6	since	since	SCONJ
cana-3950	92	7	,	,	PUNCT
cana-3950	92	8	𝐴1⊕𝐴2	𝐴1⊕𝐴2	PROPN
cana-3950	92	9	=	=	SYM
cana-3950	92	10	(	(	PUNCT
cana-3950	92	11	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	92	12	𝑚	𝑚	PROPN
cana-3950	93	1	+	+	CCONJ
cana-3950	93	2	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	93	3	𝑚	𝑚	NOUN
cana-3950	93	4	−	−	PROPN
cana-3950	93	5	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	93	6	𝑚𝐽𝐴2	𝑚𝐽𝐴2	NOUN
cana-3950	93	7	𝑚	𝑚	X
cana-3950	93	8	,	,	PUNCT
cana-3950	93	9	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	93	10	𝑛𝐾𝐴2	𝑛𝐾𝐴2	VERB
cana-3950	93	11	𝑛	𝑛	PROPN
cana-3950	93	12	)	)	PUNCT
cana-3950	93	13	,	,	PUNCT
cana-3950	93	14	we	we	PRON
cana-3950	93	15	have	have	VERB
cana-3950	93	16	,	,	PUNCT
cana-3950	93	17	𝐽𝐴1⊕𝐴2	𝐽𝐴1⊕𝐴2	PROPN
cana-3950	93	18	𝑚	𝑚	ADP
cana-3950	93	19	+	+	CCONJ
cana-3950	93	20	𝐾𝐴1⊕𝐴2	𝐾𝐴1⊕𝐴2	ADJ
cana-3950	93	21	𝑛	𝑛	X
cana-3950	93	22	=	=	PUNCT
cana-3950	93	23	(	(	PUNCT
cana-3950	93	24	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	93	25	𝑚	𝑚	PROPN
cana-3950	94	1	+	+	CCONJ
cana-3950	94	2	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	94	3	𝑚	𝑚	NOUN
cana-3950	94	4	−	−	NOUN
cana-3950	94	5	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	94	6	𝑚𝐽𝐴2	𝑚𝐽𝐴2	NOUN
cana-3950	94	7	𝑚	𝑚	NOUN
cana-3950	94	8	)	)	PUNCT
cana-3950	94	9	𝑚	𝑚	NOUN
cana-3950	95	1	+	+	CCONJ
cana-3950	95	2	(	(	PUNCT
cana-3950	95	3	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	95	4	𝑛𝐾𝐴2	𝑛𝐾𝐴2	VERB
cana-3950	95	5	𝑛	𝑛	NOUN
cana-3950	95	6	)	)	PUNCT
cana-3950	95	7	𝑛	𝑛	NOUN
cana-3950	95	8	=	=	SYM
cana-3950	95	9	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	95	10	𝑚(1−	𝑚(1−	NOUN
cana-3950	95	11	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	95	12	𝑚	𝑚	NOUN
cana-3950	95	13	)	)	PUNCT
cana-3950	95	14	+	+	CCONJ
cana-3950	95	15	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	95	16	𝑚	𝑚	X
cana-3950	95	17	+	+	CCONJ
cana-3950	95	18	(	(	PUNCT
cana-3950	95	19	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	95	20	𝑛𝐾𝐴2	𝑛𝐾𝐴2	VERB
cana-3950	95	21	𝑛	𝑛	PRON
cana-3950	95	22	)	)	PUNCT
cana-3950	95	23	𝑛	𝑛	PRON
cana-3950	95	24	≥	≥	NOUN
cana-3950	95	25	0	0	NUM
cana-3950	95	26	and	and	CCONJ
cana-3950	95	27	𝐽𝐴1⊕𝐴2	𝐽𝐴1⊕𝐴2	NUM
cana-3950	95	28	𝑚	𝑚	ADP
cana-3950	95	29	+	+	NOUN
cana-3950	95	30	𝐾𝐴1⊕𝐴2	𝐾𝐴1⊕𝐴2	ADJ
cana-3950	95	31	𝑛	𝑛	X
cana-3950	95	32	=	=	PUNCT
cana-3950	95	33	(	(	PUNCT
cana-3950	95	34	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	95	35	𝑚	𝑚	PROPN
cana-3950	96	1	+	+	CCONJ
cana-3950	96	2	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	96	3	𝑚	𝑚	NOUN
cana-3950	96	4	−	−	NOUN
cana-3950	96	5	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	96	6	𝑚𝐽𝐴2	𝑚𝐽𝐴2	NOUN
cana-3950	96	7	𝑚	𝑚	NOUN
cana-3950	96	8	)	)	PUNCT
cana-3950	96	9	𝑚	𝑚	NOUN
cana-3950	97	1	+	+	CCONJ
cana-3950	97	2	(	(	PUNCT
cana-3950	97	3	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	97	4	𝑛𝐾𝐴2	𝑛𝐾𝐴2	VERB
cana-3950	97	5	𝑛	𝑛	NOUN
cana-3950	97	6	)	)	PUNCT
cana-3950	97	7	𝑛	𝑛	PRON
cana-3950	97	8	≤	≤	NUM
cana-3950	97	9	(	(	PUNCT
cana-3950	97	10	(	(	PUNCT
cana-3950	97	11	1	1	NUM
cana-3950	97	12	−	−	NOUN
cana-3950	97	13	𝐾𝐴1	𝐾𝐴1	NOUN
cana-3950	97	14	𝑛	𝑛	PROPN
cana-3950	97	15	)	)	PUNCT
cana-3950	98	1	+	+	CCONJ
cana-3950	98	2	(	(	PUNCT
cana-3950	98	3	1	1	NUM
cana-3950	98	4	−	−	PROPN
cana-3950	98	5	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	98	6	𝑛	𝑛	PROPN
cana-3950	98	7	)	)	PUNCT
cana-3950	98	8	−	−	PROPN
cana-3950	98	9	(	(	PUNCT
cana-3950	98	10	1−	1−	NUM
cana-3950	98	11	𝐾𝐴1	𝐾𝐴1	NOUN
cana-3950	98	12	𝑛	𝑛	PROPN
cana-3950	98	13	)	)	PUNCT
cana-3950	98	14	(	(	PUNCT
cana-3950	98	15	1−	1−	NUM
cana-3950	98	16	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	98	17	𝑛	𝑛	PROPN
cana-3950	98	18	)	)	PUNCT
cana-3950	98	19	)	)	PUNCT
cana-3950	99	1	𝑚	𝑚	X
cana-3950	99	2	+	+	CCONJ
cana-3950	99	3	(	(	PUNCT
cana-3950	99	4	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	99	5	𝑛𝐾𝐴2	𝑛𝐾𝐴2	VERB
cana-3950	99	6	𝑛	𝑛	NOUN
cana-3950	99	7	)	)	PUNCT
cana-3950	99	8	𝑛	𝑛	NOUN
cana-3950	99	9	=	=	PUNCT
cana-3950	99	10	(	(	PUNCT
cana-3950	99	11	1−	1−	NUM
cana-3950	99	12	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	99	13	𝑛𝐾𝐴1	𝑛𝐾𝐴1	NOUN
cana-3950	99	14	𝑛	𝑛	PROPN
cana-3950	99	15	)	)	PUNCT
cana-3950	100	1	𝑚	𝑚	PROPN
cana-3950	100	2	+	+	CCONJ
cana-3950	100	3	(	(	PUNCT
cana-3950	100	4	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	100	5	𝑛𝐾𝐴2	𝑛𝐾𝐴2	VERB
cana-3950	100	6	𝑛	𝑛	PRON
cana-3950	100	7	)	)	PUNCT
cana-3950	100	8	𝑛	𝑛	PRON
cana-3950	100	9	≤	≤	NUM
cana-3950	100	10	1	1	NUM
cana-3950	100	11	because	because	SCONJ
cana-3950	100	12	0	0	NUM
cana-3950	100	13	≤	≤	NUM
cana-3950	100	14	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	100	15	𝑛𝐾𝐴2	𝑛𝐾𝐴2	VERB
cana-3950	100	16	𝑛	𝑛	DET
cana-3950	100	17	≤	≤	NUM
cana-3950	100	18	1	1	NUM
cana-3950	100	19	and	and	CCONJ
cana-3950	100	20	𝑚,𝑛	𝑚,𝑛	VERB
cana-3950	100	21	≥	≥	NUM
cana-3950	100	22	1	1	NUM
cana-3950	100	23	.	.	PUNCT
cana-3950	100	24	communications	communication	NOUN
cana-3950	100	25	on	on	ADP
cana-3950	100	26	applied	apply	VERB
cana-3950	100	27	nonlinear	nonlinear	ADJ
cana-3950	100	28	analysis	analysis	NOUN
cana-3950	100	29	issn	issn	NOUN
cana-3950	100	30	:	:	PUNCT
cana-3950	100	31	1074	1074	NUM
cana-3950	100	32	-	-	PUNCT
cana-3950	100	33	133x	133x	NUM
cana-3950	100	34	vol	vol	NOUN
cana-3950	100	35	32	32	NUM
cana-3950	100	36	no	no	NOUN
cana-3950	100	37	.	.	PUNCT
cana-3950	101	1	9s	9s	NUM
cana-3950	101	2	(	(	PUNCT
cana-3950	101	3	2025	2025	NUM
cana-3950	101	4	)	)	PUNCT
cana-3950	101	5	391	391	NUM
cana-3950	101	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3950	101	7	hence	hence	ADV
cana-3950	101	8	,	,	PUNCT
cana-3950	101	9	𝐴1⊕𝐴2	𝐴1⊕𝐴2	PROPN
cana-3950	101	10	∈	∈	PROPN
cana-3950	101	11	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	101	12	𝑛	𝑛	PROPN
cana-3950	101	13	(	(	PUNCT
cana-3950	101	14	𝑋	𝑋	PROPN
cana-3950	101	15	)	)	PUNCT
cana-3950	101	16	,	,	PUNCT
cana-3950	101	17	(	(	PUNCT
cana-3950	101	18	ii	ii	NOUN
cana-3950	101	19	)	)	PUNCT
cana-3950	101	20	similar	similar	ADJ
cana-3950	101	21	to	to	ADP
cana-3950	101	22	(	(	PUNCT
cana-3950	101	23	i	i	NOUN
cana-3950	101	24	)	)	PUNCT
cana-3950	101	25	.	.	PUNCT
cana-3950	102	1	(	(	PUNCT
cana-3950	102	2	iii)suppose	iii)suppose	PROPN
cana-3950	102	3	max	max	PROPN
cana-3950	102	4	{	{	PUNCT
cana-3950	102	5	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	102	6	,	,	PUNCT
cana-3950	102	7	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	102	8	}	}	PUNCT
cana-3950	102	9	≤	≤	NOUN
cana-3950	102	10	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	102	11	,	,	PUNCT
cana-3950	102	12	we	we	PRON
cana-3950	102	13	have	have	VERB
cana-3950	102	14	0	0	NUM
cana-3950	102	15	≤	≤	NUM
cana-3950	102	16	𝐽𝐴1∪𝐴2	𝐽𝐴1∪𝐴2	X
cana-3950	102	17	𝑚	𝑚	X
cana-3950	103	1	+	+	NOUN
cana-3950	103	2	𝐾𝐴1∪𝐴2	𝐾𝐴1∪𝐴2	NOUN
cana-3950	104	1	=	=	X
cana-3950	104	2	(	(	PUNCT
cana-3950	104	3	max{𝐽𝐴1	max{𝐽𝐴1	PROPN
cana-3950	104	4	,	,	PUNCT
cana-3950	104	5	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	104	6	}	}	PUNCT
cana-3950	104	7	)	)	PUNCT
cana-3950	104	8	𝑚	𝑚	PROPN
cana-3950	104	9	+	+	PUNCT
cana-3950	104	10	(	(	PUNCT
cana-3950	104	11	(	(	PUNCT
cana-3950	104	12	min{𝐾𝐴1	min{𝐾𝐴1	NOUN
cana-3950	104	13	,	,	PUNCT
cana-3950	104	14	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	104	15	}	}	PUNCT
cana-3950	104	16	)	)	PUNCT
cana-3950	105	1	𝑛	𝑛	PRON
cana-3950	105	2	≤	≤	NUM
cana-3950	105	3	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	105	4	𝑚	𝑚	PROPN
cana-3950	106	1	+	+	CCONJ
cana-3950	106	2	𝐾𝐴1	𝐾𝐴1	NOUN
cana-3950	106	3	𝑛	𝑛	PRON
cana-3950	106	4	≤	≤	NUM
cana-3950	106	5	1	1	NUM
cana-3950	106	6	suppose	suppose	VERB
cana-3950	106	7	now	now	ADV
cana-3950	106	8	max	max	PROPN
cana-3950	106	9	{	{	PUNCT
cana-3950	106	10	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	106	11	,	,	PUNCT
cana-3950	106	12	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	106	13	}	}	PUNCT
cana-3950	106	14	=	=	SYM
cana-3950	106	15	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	106	16	.	.	PUNCT
cana-3950	107	1	since	since	SCONJ
cana-3950	107	2	min{𝐾𝐴1	min{𝐾𝐴1	NOUN
cana-3950	107	3	,	,	PUNCT
cana-3950	107	4	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	107	5	}	}	PUNCT
cana-3950	107	6	≤	≤	PROPN
cana-3950	107	7	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	107	8	,	,	PUNCT
cana-3950	107	9	we	we	PRON
cana-3950	107	10	have	have	VERB
cana-3950	107	11	0	0	NUM
cana-3950	107	12	≤	≤	NOUN
cana-3950	107	13	𝐽𝐴1∪𝐴2	𝐽𝐴1∪𝐴2	NOUN
cana-3950	107	14	𝑚	𝑚	X
cana-3950	107	15	+	+	X
cana-3950	107	16	𝐾𝑛𝐴1∪𝐴2	𝐾𝑛𝐴1∪𝐴2	ADJ
cana-3950	107	17	,	,	PUNCT
cana-3950	107	18	=	=	SYM
cana-3950	107	19	(	(	PUNCT
cana-3950	107	20	max{𝐽𝐴1	max{𝐽𝐴1	PROPN
cana-3950	107	21	,	,	PUNCT
cana-3950	107	22	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	107	23	}	}	PUNCT
cana-3950	107	24	)	)	PUNCT
cana-3950	107	25	𝑚	𝑚	PROPN
cana-3950	108	1	+	+	PUNCT
cana-3950	108	2	(	(	PUNCT
cana-3950	108	3	(	(	PUNCT
cana-3950	108	4	min{𝐾𝐴1	min{𝐾𝐴1	NOUN
cana-3950	108	5	,	,	PUNCT
cana-3950	108	6	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	108	7	}	}	PUNCT
cana-3950	108	8	)	)	PUNCT
cana-3950	109	1	𝑛	𝑛	DET
cana-3950	109	2	≤	≤	NUM
cana-3950	109	3	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	109	4	𝑚	𝑚	PROPN
cana-3950	109	5	+	+	CCONJ
cana-3950	109	6	𝐾𝐴1	𝐾𝐴1	NOUN
cana-3950	109	7	𝑛	𝑛	DET
cana-3950	109	8	≤	≤	NUM
cana-3950	109	9	1	1	NUM
cana-3950	109	10	thus	thus	ADV
cana-3950	109	11	,	,	PUNCT
cana-3950	109	12	the	the	DET
cana-3950	109	13	proof	proof	NOUN
cana-3950	109	14	of	of	ADP
cana-3950	109	15	(	(	PUNCT
cana-3950	109	16	iii	iii	NOUN
cana-3950	109	17	)	)	PUNCT
cana-3950	109	18	is	be	AUX
cana-3950	109	19	completed	complete	VERB
cana-3950	109	20	.	.	PUNCT
cana-3950	110	1	(	(	PUNCT
cana-3950	110	2	iv	iv	X
cana-3950	110	3	)	)	PUNCT
cana-3950	110	4	similar	similar	ADJ
cana-3950	110	5	to	to	ADP
cana-3950	110	6	(	(	PUNCT
cana-3950	110	7	iii	iii	NOUN
cana-3950	110	8	)	)	PUNCT
cana-3950	110	9	(	(	PUNCT
cana-3950	110	10	v	v	NOUN
cana-3950	110	11	)	)	PUNCT
cana-3950	110	12	since	since	SCONJ
cana-3950	110	13	𝐴	𝐴	PROPN
cana-3950	110	14	∈	∈	PROPN
cana-3950	110	15	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	110	16	𝑛	𝑛	PROPN
cana-3950	110	17	(	(	PUNCT
cana-3950	110	18	𝑋	𝑋	PROPN
cana-3950	110	19	)	)	PUNCT
cana-3950	110	20	,	,	PUNCT
cana-3950	110	21	we	we	PRON
cana-3950	110	22	have	have	VERB
cana-3950	111	1	0	0	NUM
cana-3950	111	2	≤	≤	NUM
cana-3950	111	3	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	111	4	𝑚	𝑚	ADP
cana-3950	111	5	≤	≤	NUM
cana-3950	111	6	1	1	NUM
cana-3950	111	7	and	and	CCONJ
cana-3950	111	8	0	0	NUM
cana-3950	111	9	≤	≤	NUM
cana-3950	111	10	𝐾𝐴	𝐾𝐴	NOUN
cana-3950	111	11	𝑛	𝑛	ADJ
cana-3950	111	12	≤	≤	NUM
cana-3950	111	13	1	1	NUM
cana-3950	111	14	.	.	PUNCT
cana-3950	112	1	𝑞𝐴	𝑞𝐴	PROPN
cana-3950	112	2	=	=	SYM
cana-3950	113	1	(	(	PUNCT
cana-3950	113	2	1−	1−	NUM
cana-3950	113	3	(	(	PUNCT
cana-3950	113	4	1	1	NUM
cana-3950	113	5	−	−	PROPN
cana-3950	113	6	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	113	7	𝑚)𝑞	𝑚)𝑞	NOUN
cana-3950	113	8	,	,	PUNCT
cana-3950	113	9	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	113	10	𝑛𝑞	𝑛𝑞	NOUN
cana-3950	113	11	)	)	PUNCT
cana-3950	113	12	,	,	PUNCT
cana-3950	113	13	we	we	PRON
cana-3950	113	14	have	have	VERB
cana-3950	113	15	≤	≤	NOUN
cana-3950	113	16	𝐽𝑞𝐴	𝐽𝑞𝐴	ADP
cana-3950	113	17	𝑚	𝑚	ADP
cana-3950	113	18	+	+	NOUN
cana-3950	113	19	𝐾𝑞𝐴	𝐾𝑞𝐴	X
cana-3950	113	20	𝑛	𝑛	NOUN
cana-3950	113	21	=	=	SYM
cana-3950	113	22	(	(	PUNCT
cana-3950	113	23	1−	1−	NUM
cana-3950	113	24	(	(	PUNCT
cana-3950	113	25	1	1	NUM
cana-3950	113	26	−	−	PROPN
cana-3950	113	27	𝐽𝐴	𝐽𝐴	NOUN
cana-3950	113	28	𝑚)𝑞)𝑚	𝑚)𝑞)𝑚	NOUN
cana-3950	114	1	+	+	CCONJ
cana-3950	114	2	(	(	PUNCT
cana-3950	114	3	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	114	4	𝑛𝑞	𝑛𝑞	NOUN
cana-3950	114	5	)	)	PUNCT
cana-3950	114	6	𝑛	𝑛	DET
cana-3950	114	7	≤	≤	NUM
cana-3950	114	8	(	(	PUNCT
cana-3950	114	9	1−	1−	NUM
cana-3950	114	10	(	(	PUNCT
cana-3950	114	11	1−	1−	NUM
cana-3950	114	12	𝐽𝐴	𝐽𝐴	NOUN
cana-3950	114	13	𝑚)𝑞)𝑚	𝑚)𝑞)𝑚	NOUN
cana-3950	115	1	+	+	CCONJ
cana-3950	115	2	(	(	PUNCT
cana-3950	115	3	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	115	4	𝑛𝑞	𝑛𝑞	NOUN
cana-3950	115	5	)	)	PUNCT
cana-3950	115	6	𝑛	𝑛	DET
cana-3950	115	7	≤	≤	NUM
cana-3950	115	8	(	(	PUNCT
cana-3950	115	9	1−	1−	NUM
cana-3950	115	10	(	(	PUNCT
cana-3950	115	11	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	115	12	𝑛𝑞	𝑛𝑞	PROPN
cana-3950	115	13	)	)	PUNCT
cana-3950	115	14	𝑚	𝑚	NOUN
cana-3950	115	15	)	)	PUNCT
cana-3950	116	1	+	+	CCONJ
cana-3950	116	2	(	(	PUNCT
cana-3950	116	3	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	116	4	𝑛𝑞	𝑛𝑞	NOUN
cana-3950	116	5	)	)	PUNCT
cana-3950	116	6	𝑛	𝑛	DET
cana-3950	116	7	≤	≤	NUM
cana-3950	116	8	1	1	NUM
cana-3950	116	9	theorem	theorem	VERB
cana-3950	116	10	2.14	2.14	NUM
cana-3950	116	11	let	let	VERB
cana-3950	116	12	𝐴	𝐴	PROPN
cana-3950	116	13	=	=	SYM
cana-3950	116	14	(	(	PUNCT
cana-3950	116	15	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	116	16	,	,	PUNCT
cana-3950	116	17	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	116	18	)	)	PUNCT
cana-3950	116	19	,	,	PUNCT
cana-3950	116	20	𝐴1	𝐴1	PROPN
cana-3950	117	1	=	=	SYM
cana-3950	117	2	(	(	PUNCT
cana-3950	117	3	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	117	4	,	,	PUNCT
cana-3950	117	5	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	117	6	)	)	PUNCT
cana-3950	117	7	,	,	PUNCT
cana-3950	117	8	𝐴2	𝐴2	PROPN
cana-3950	117	9	=	=	SYM
cana-3950	117	10	(	(	PUNCT
cana-3950	117	11	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	117	12	,	,	PUNCT
cana-3950	117	13	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	117	14	)	)	PUNCT
cana-3950	117	15	∈	∈	PROPN
cana-3950	117	16	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	117	17	𝑛	𝑛	PROPN
cana-3950	117	18	(	(	PUNCT
cana-3950	117	19	𝑋	𝑋	PROPN
cana-3950	117	20	)	)	PUNCT
cana-3950	117	21	and	and	CCONJ
cana-3950	117	22	𝑞	𝑞	PROPN
cana-3950	117	23	,	,	PUNCT
cana-3950	117	24	𝑞1	𝑞1	PROPN
cana-3950	117	25	,	,	PUNCT
cana-3950	117	26	𝑞2	𝑞2	PROPN
cana-3950	117	27	∈	∈	PROPN
cana-3950	117	28	𝑁.	𝑁.	PROPN
cana-3950	117	29	then	then	ADV
cana-3950	117	30	(	(	PUNCT
cana-3950	117	31	𝑖	𝑖	X
cana-3950	117	32	)	)	PUNCT
cana-3950	117	33	𝐴1⊕𝐴2	𝐴1⊕𝐴2	PROPN
cana-3950	117	34	=	=	SYM
cana-3950	117	35	𝐴2⊕𝐴1	𝐴2⊕𝐴1	PROPN
cana-3950	117	36	(	(	PUNCT
cana-3950	117	37	𝑖𝑖	𝑖𝑖	NOUN
cana-3950	117	38	)	)	PUNCT
cana-3950	117	39	𝐴1	𝐴1	PROPN
cana-3950	117	40	⨂𝐴2	⨂𝐴2	PROPN
cana-3950	118	1	=	=	PUNCT
cana-3950	118	2	𝐴2	𝐴2	PROPN
cana-3950	118	3	⨂𝐴1	⨂𝐴1	PROPN
cana-3950	118	4	(	(	PUNCT
cana-3950	118	5	𝑖𝑖𝑖	𝑖𝑖𝑖	NOUN
cana-3950	118	6	)	)	PUNCT
cana-3950	118	7	𝑞(𝐴1⊕𝐴2	𝑞(𝐴1⊕𝐴2	NOUN
cana-3950	118	8	)	)	PUNCT
cana-3950	119	1	=	=	SYM
cana-3950	119	2	𝑞𝐴1⊕𝑞𝐴2	𝑞𝐴1⊕𝑞𝐴2	NOUN
cana-3950	120	1	(	(	PUNCT
cana-3950	120	2	𝑖𝑣)(𝑞1	𝑖𝑣)(𝑞1	X
cana-3950	120	3	+	+	NUM
cana-3950	120	4	𝑞2)𝐴	𝑞2)𝐴	NOUN
cana-3950	120	5	=	=	SYM
cana-3950	120	6	𝑞1𝐴	𝑞1𝐴	SYM
cana-3950	120	7	⊕	⊕	PROPN
cana-3950	120	8	𝑞2𝐴	𝑞2𝐴	PROPN
cana-3950	120	9	,	,	PUNCT
cana-3950	120	10	communications	communication	NOUN
cana-3950	120	11	on	on	ADP
cana-3950	120	12	applied	apply	VERB
cana-3950	120	13	nonlinear	nonlinear	ADJ
cana-3950	120	14	analysis	analysis	NOUN
cana-3950	120	15	issn	issn	NOUN
cana-3950	120	16	:	:	PUNCT
cana-3950	120	17	1074	1074	NUM
cana-3950	120	18	-	-	PUNCT
cana-3950	120	19	133x	133x	NUM
cana-3950	120	20	vol	vol	NOUN
cana-3950	120	21	32	32	NUM
cana-3950	120	22	no	no	NOUN
cana-3950	120	23	.	.	PUNCT
cana-3950	121	1	9s	9s	NUM
cana-3950	121	2	(	(	PUNCT
cana-3950	121	3	2025	2025	NUM
cana-3950	121	4	)	)	PUNCT
cana-3950	121	5	392	392	NUM
cana-3950	121	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3950	121	7	(	(	PUNCT
cana-3950	121	8	𝑣)(𝐴1⊗𝐴2	𝑣)(𝐴1⊗𝐴2	NUM
cana-3950	121	9	)	)	PUNCT
cana-3950	121	10	𝑞	𝑞	X
cana-3950	121	11	=	=	SYM
cana-3950	121	12	𝐴1	𝐴1	PROPN
cana-3950	122	1	𝑞	𝑞	X
cana-3950	122	2	⊗	⊗	PROPN
cana-3950	122	3	𝐴2	𝐴2	PROPN
cana-3950	122	4	𝑞	𝑞	PROPN
cana-3950	122	5	(	(	PUNCT
cana-3950	122	6	𝑣𝑖	𝑣𝑖	PROPN
cana-3950	122	7	)	)	PUNCT
cana-3950	122	8	𝐴𝑞1	𝐴𝑞1	NOUN
cana-3950	122	9	⊗𝐴𝑞2	⊗𝐴𝑞2	NOUN
cana-3950	122	10	=	=	PUNCT
cana-3950	122	11	𝐴𝑞1+𝑞2	𝐴𝑞1+𝑞2	PROPN
cana-3950	122	12	.	.	PUNCT
cana-3950	123	1	proof	proof	NOUN
cana-3950	123	2	:	:	PUNCT
cana-3950	123	3	the	the	DET
cana-3950	123	4	proof	proof	NOUN
cana-3950	123	5	of	of	ADP
cana-3950	123	6	above	above	ADV
cana-3950	123	7	can	can	AUX
cana-3950	123	8	easily	easily	ADV
cana-3950	123	9	understand	understand	VERB
cana-3950	123	10	.	.	PUNCT
cana-3950	124	1	theorem	theorem	VERB
cana-3950	124	2	2.15	2.15	NUM
cana-3950	124	3	let	let	VERB
cana-3950	124	4	𝐴	𝐴	PROPN
cana-3950	124	5	=	=	SYM
cana-3950	124	6	(	(	PUNCT
cana-3950	124	7	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	124	8	,	,	PUNCT
cana-3950	124	9	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	124	10	)	)	PUNCT
cana-3950	124	11	∈	∈	PROPN
cana-3950	124	12	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	124	13	𝑛	𝑛	PROPN
cana-3950	124	14	(	(	PUNCT
cana-3950	124	15	𝑋	𝑋	NOUN
cana-3950	124	16	)	)	PUNCT
cana-3950	124	17	and	and	CCONJ
cana-3950	124	18	𝑥	𝑥	DET
cana-3950	124	19	∈	∈	PROPN
cana-3950	124	20	𝑋.	𝑋.	PROPN
cana-3950	125	1	if	if	SCONJ
cana-3950	125	2	∏	∏	PROPN
cana-3950	125	3	(	(	PUNCT
cana-3950	125	4	𝑥	𝑥	NOUN
cana-3950	125	5	)	)	PUNCT
cana-3950	125	6	=	=	SYM
cana-3950	125	7	0,𝐴	0,𝐴	NUM
cana-3950	126	1	then	then	ADV
cana-3950	126	2	∏	∏	PROPN
cana-3950	126	3	(	(	PUNCT
cana-3950	126	4	𝑥	𝑥	NOUN
cana-3950	126	5	)	)	PUNCT
cana-3950	126	6	=	=	SYM
cana-3950	126	7	0𝐴𝑞	0𝐴𝑞	NOUN
cana-3950	126	8	for	for	ADP
cana-3950	126	9	all	all	PRON
cana-3950	126	10	𝑞	𝑞	PROPN
cana-3950	126	11	∈	∈	PROPN
cana-3950	126	12	𝑁.	𝑁.	PROPN
cana-3950	126	13	proof	proof	NOUN
cana-3950	126	14	:	:	PUNCT
cana-3950	126	15	since	since	SCONJ
cana-3950	126	16	∏	∏	PROPN
cana-3950	126	17	(	(	PUNCT
cana-3950	126	18	𝑥	𝑥	NOUN
cana-3950	126	19	)	)	PUNCT
cana-3950	127	1	=	=	SYM
cana-3950	127	2	(	(	PUNCT
cana-3950	127	3	1−	1−	NUM
cana-3950	127	4	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	127	5	𝑚(𝑥	𝑚(𝑥	NOUN
cana-3950	127	6	)	)	PUNCT
cana-3950	127	7	−	−	NUM
cana-3950	127	8	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	127	9	𝑛(𝑥	𝑛(𝑥	PROPN
cana-3950	127	10	)	)	PUNCT
cana-3950	127	11	)	)	PUNCT
cana-3950	128	1	2	2	NUM
cana-3950	128	2	𝑚+𝑛	𝑚+𝑛	NUM
cana-3950	129	1	𝐴	𝐴	NOUN
cana-3950	129	2	we	we	PRON
cana-3950	129	3	have	have	VERB
cana-3950	129	4	∏	∏	NUM
cana-3950	129	5	(	(	PUNCT
cana-3950	129	6	𝑥	𝑥	NOUN
cana-3950	129	7	)	)	PUNCT
cana-3950	130	1	=	=	SYM
cana-3950	130	2	0	0	NUM
cana-3950	130	3	⟹	⟹	NUM
cana-3950	130	4	(	(	PUNCT
cana-3950	130	5	1−	1−	NUM
cana-3950	130	6	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	130	7	𝑚(𝑥	𝑚(𝑥	NOUN
cana-3950	130	8	)	)	PUNCT
cana-3950	130	9	−	−	NUM
cana-3950	130	10	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	130	11	𝑛(𝑥	𝑛(𝑥	PROPN
cana-3950	130	12	)	)	PUNCT
cana-3950	130	13	)	)	PUNCT
cana-3950	131	1	2	2	NUM
cana-3950	131	2	𝑚+𝑛	𝑚+𝑛	PUNCT
cana-3950	131	3	𝐴	𝐴	PROPN
cana-3950	131	4	⟹	⟹	NUM
cana-3950	131	5	𝐽𝐴(𝑥	𝐽𝐴(𝑥	NOUN
cana-3950	131	6	)	)	PUNCT
cana-3950	132	1	+	+	CCONJ
cana-3950	132	2	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	132	3	𝑛(𝑥	𝑛(𝑥	PROPN
cana-3950	132	4	)	)	PUNCT
cana-3950	132	5	=	=	SYM
cana-3950	132	6	1	1	NUM
cana-3950	132	7	⟹	⟹	NUM
cana-3950	132	8	𝐽𝐴(𝑥	𝐽𝐴(𝑥	NOUN
cana-3950	132	9	)	)	PUNCT
cana-3950	132	10	=	=	SYM
cana-3950	132	11	1	1	NUM
cana-3950	132	12	−	−	NOUN
cana-3950	132	13	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	132	14	𝑛(𝑥	𝑛(𝑥	PROPN
cana-3950	132	15	)	)	PUNCT
cana-3950	132	16	.	.	PUNCT
cana-3950	133	1	by	by	ADP
cana-3950	133	2	using	use	VERB
cana-3950	133	3	this	this	DET
cana-3950	133	4	result	result	NOUN
cana-3950	133	5	,	,	PUNCT
cana-3950	133	6	we	we	PRON
cana-3950	133	7	have	have	VERB
cana-3950	133	8	𝐴𝑞	𝐴𝑞	ADJ
cana-3950	133	9	=	=	SYM
cana-3950	133	10	(	(	PUNCT
cana-3950	133	11	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	133	12	𝑚𝑞	𝑚𝑞	ADP
cana-3950	133	13	,	,	PUNCT
cana-3950	133	14	1−	1−	NUM
cana-3950	133	15	(	(	PUNCT
cana-3950	133	16	1−	1−	NUM
cana-3950	133	17	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	133	18	𝑛)𝑞	𝑛)𝑞	NOUN
cana-3950	133	19	)	)	PUNCT
cana-3950	133	20	=	=	PRON
cana-3950	134	1	(	(	PUNCT
cana-3950	134	2	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	134	3	𝑚𝑞	𝑚𝑞	ADP
cana-3950	134	4	,	,	PUNCT
cana-3950	134	5	1−	1−	NUM
cana-3950	134	6	(	(	PUNCT
cana-3950	134	7	1−	1−	NUM
cana-3950	134	8	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	134	9	𝑚)𝑞	𝑚)𝑞	NOUN
cana-3950	134	10	)	)	PUNCT
cana-3950	135	1	=	=	SYM
cana-3950	135	2	(	(	PUNCT
cana-3950	135	3	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	135	4	𝑚𝑞	𝑚𝑞	ADP
cana-3950	135	5	,	,	PUNCT
cana-3950	135	6	1−	1−	NUM
cana-3950	135	7	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	135	8	𝑚𝑞	𝑚𝑞	NOUN
cana-3950	135	9	)	)	PUNCT
cana-3950	135	10	hence	hence	ADV
cana-3950	135	11	,	,	PUNCT
cana-3950	135	12	∏	∏	PROPN
cana-3950	135	13	(	(	PUNCT
cana-3950	135	14	𝑥	𝑥	NOUN
cana-3950	135	15	)	)	PUNCT
cana-3950	135	16	=	=	SYM
cana-3950	135	17	(	(	PUNCT
cana-3950	135	18	1	1	NUM
cana-3950	135	19	−	−	PROPN
cana-3950	135	20	(	(	PUNCT
cana-3950	135	21	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	135	22	𝑚𝑞	𝑚𝑞	NOUN
cana-3950	135	23	(	(	PUNCT
cana-3950	135	24	𝑥	𝑥	NOUN
cana-3950	135	25	)	)	PUNCT
cana-3950	135	26	)	)	PUNCT
cana-3950	136	1	𝑚	𝑚	ADP
cana-3950	136	2	−	−	PROPN
cana-3950	136	3	(	(	PUNCT
cana-3950	136	4	1	1	NUM
cana-3950	136	5	−	−	NOUN
cana-3950	136	6	𝐽𝐴	𝐽𝐴	NOUN
cana-3950	136	7	𝑚𝑞	𝑚𝑞	NOUN
cana-3950	136	8	(	(	PUNCT
cana-3950	136	9	𝑥	𝑥	NOUN
cana-3950	136	10	)	)	PUNCT
cana-3950	136	11	)	)	PUNCT
cana-3950	136	12	𝑛	𝑛	X
cana-3950	136	13	)	)	PUNCT
cana-3950	136	14	2	2	NUM
cana-3950	136	15	𝑚+𝑛	𝑚+𝑛	PUNCT
cana-3950	137	1	𝐴𝑞	𝐴𝑞	VERB
cana-3950	137	2	⟹	⟹	NUM
cana-3950	137	3	∏	∏	PROPN
cana-3950	137	4	(	(	PUNCT
cana-3950	137	5	𝑥	𝑥	NOUN
cana-3950	137	6	)	)	PUNCT
cana-3950	137	7	=	=	SYM
cana-3950	137	8	(	(	PUNCT
cana-3950	137	9	1−	1−	NUM
cana-3950	137	10	(	(	PUNCT
cana-3950	137	11	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	137	12	𝑚𝑞	𝑚𝑞	PART
cana-3950	137	13	(	(	PUNCT
cana-3950	137	14	𝑥	𝑥	NOUN
cana-3950	137	15	)	)	PUNCT
cana-3950	137	16	)	)	PUNCT
cana-3950	138	1	𝑚	𝑚	ADP
cana-3950	138	2	−	−	PROPN
cana-3950	138	3	(	(	PUNCT
cana-3950	138	4	1	1	NUM
cana-3950	138	5	−	−	NOUN
cana-3950	138	6	𝐽𝐴	𝐽𝐴	NOUN
cana-3950	138	7	𝑚𝑞	𝑚𝑞	NOUN
cana-3950	138	8	(	(	PUNCT
cana-3950	138	9	𝑥	𝑥	NOUN
cana-3950	138	10	)	)	PUNCT
cana-3950	138	11	)	)	PUNCT
cana-3950	138	12	𝑛	𝑛	X
cana-3950	138	13	)	)	PUNCT
cana-3950	138	14	2	2	NUM
cana-3950	138	15	𝑚+𝑛	𝑚+𝑛	PUNCT
cana-3950	139	1	𝐴𝑞	𝐴𝑞	VERB
cana-3950	139	2	⟹	⟹	NUM
cana-3950	139	3	∏	∏	PROPN
cana-3950	139	4	(	(	PUNCT
cana-3950	139	5	𝑥	𝑥	NOUN
cana-3950	139	6	)	)	PUNCT
cana-3950	139	7	=	=	SYM
cana-3950	139	8	0𝐴𝑞	0𝐴𝑞	NOUN
cana-3950	139	9	theorem	theorem	VERB
cana-3950	139	10	2.16	2.16	NUM
cana-3950	139	11	let	let	VERB
cana-3950	139	12	𝐴	𝐴	PROPN
cana-3950	139	13	=	=	SYM
cana-3950	139	14	(	(	PUNCT
cana-3950	139	15	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	139	16	,	,	PUNCT
cana-3950	139	17	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	139	18	)	)	PUNCT
cana-3950	139	19	∈	∈	PROPN
cana-3950	140	1	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	140	2	𝑛	𝑛	PROPN
cana-3950	140	3	(	(	PUNCT
cana-3950	140	4	𝑥	𝑥	NOUN
cana-3950	140	5	)	)	PUNCT
cana-3950	140	6	and	and	CCONJ
cana-3950	140	7	𝑥	𝑥	PRON
cana-3950	140	8	∈	∈	PROPN
cana-3950	140	9	𝑋	𝑋	PROPN
cana-3950	140	10	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-3950	140	11	𝑞	𝑞	PROPN
cana-3950	140	12	,	,	PUNCT
cana-3950	140	13	𝑞1	𝑞1	PROPN
cana-3950	140	14	,	,	PUNCT
cana-3950	140	15	𝑞2	𝑞2	PROPN
cana-3950	140	16	∈	∈	PROPN
cana-3950	140	17	𝑁.	𝑁.	PROPN
cana-3950	140	18	then	then	ADV
cana-3950	140	19	(	(	PUNCT
cana-3950	140	20	𝑖)𝑞1	𝑖)𝑞1	PROPN
cana-3950	140	21	≥	≥	NOUN
cana-3950	140	22	𝑞2	𝑞2	PROPN
cana-3950	140	23	⟹	⟹	NUM
cana-3950	140	24	𝐴𝑞1	𝐴𝑞1	NOUN
cana-3950	140	25	⊂	⊂	PROPN
cana-3950	140	26	𝐴𝑞2	𝐴𝑞2	PROPN
cana-3950	140	27	(	(	PUNCT
cana-3950	140	28	𝑖𝑖	𝑖𝑖	NOUN
cana-3950	140	29	)	)	PUNCT
cana-3950	140	30	𝑞1	𝑞1	PROPN
cana-3950	140	31	≥	≥	NUM
cana-3950	140	32	𝑞2	𝑞2	PROPN
cana-3950	140	33	⟹	⟹	PROPN
cana-3950	140	34	𝑞2𝐴	𝑞2𝐴	ADP
cana-3950	140	35	⊂	⊂	PROPN
cana-3950	140	36	𝑞1𝐴	𝑞1𝐴	X
cana-3950	140	37	1−	1−	NUM
cana-3950	140	38	𝐽𝐴2	𝐽𝐴2	VERB
cana-3950	140	39	𝑚	𝑚	VERB
cana-3950	140	40	≤	≤	ADJ
cana-3950	140	41	1	1	NUM
cana-3950	140	42	−	−	NOUN
cana-3950	140	43	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	140	44	𝑚	𝑚	PROPN
cana-3950	140	45	⟹	⟹	NUM
cana-3950	140	46	(	(	PUNCT
cana-3950	140	47	1	1	NUM
cana-3950	140	48	−	−	PROPN
cana-3950	140	49	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	140	50	𝑚	𝑚	NOUN
cana-3950	140	51	)	)	PUNCT
cana-3950	140	52	𝑞	𝑞	PROPN
cana-3950	140	53	≤	≤	X
cana-3950	140	54	(	(	PUNCT
cana-3950	140	55	1−	1−	NUM
cana-3950	140	56	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	140	57	𝑚	𝑚	NOUN
cana-3950	140	58	)	)	PUNCT
cana-3950	140	59	𝑞	𝑞	X
cana-3950	140	60	⟹	⟹	X
cana-3950	140	61	(	(	PUNCT
cana-3950	140	62	1−	1−	NUM
cana-3950	140	63	(	(	PUNCT
cana-3950	140	64	1	1	NUM
cana-3950	140	65	−	−	NOUN
cana-3950	140	66	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	140	67	𝑚	𝑚	NOUN
cana-3950	140	68	)	)	PUNCT
cana-3950	140	69	𝑞	𝑞	NOUN
cana-3950	140	70	)	)	PUNCT
cana-3950	140	71	≤	≤	NOUN
cana-3950	140	72	(	(	PUNCT
cana-3950	140	73	1−	1−	NUM
cana-3950	140	74	(	(	PUNCT
cana-3950	140	75	1−	1−	NUM
cana-3950	140	76	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	140	77	𝑚	𝑚	NOUN
cana-3950	140	78	)	)	PUNCT
cana-3950	140	79	𝑞	𝑞	NOUN
cana-3950	140	80	)	)	PUNCT
cana-3950	140	81	⟹	⟹	PROPN
cana-3950	140	82	𝐽𝑞𝐴1	𝐽𝑞𝐴1	VERB
cana-3950	140	83	≤	≤	ADJ
cana-3950	140	84	𝐽𝑞𝐴2	𝐽𝑞𝐴2	NOUN
cana-3950	140	85	and	and	CCONJ
cana-3950	140	86	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	140	87	≥	≥	PRON
cana-3950	141	1	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	141	2	⟹𝐾𝐴1	⟹𝐾𝐴1	VERB
cana-3950	141	3	𝑛	𝑛	PRON
cana-3950	141	4	≥	≥	NOUN
cana-3950	141	5	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	141	6	𝑛	𝑛	PROPN
cana-3950	141	7	⟹	⟹	NUM
cana-3950	141	8	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	141	9	𝑛𝑞	𝑛𝑞	ADP
cana-3950	141	10	≥	≥	NOUN
cana-3950	141	11	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	141	12	𝑛𝑞	𝑛𝑞	SCONJ
cana-3950	141	13	communications	communication	NOUN
cana-3950	141	14	on	on	ADP
cana-3950	141	15	applied	apply	VERB
cana-3950	141	16	nonlinear	nonlinear	ADJ
cana-3950	141	17	analysis	analysis	NOUN
cana-3950	141	18	issn	issn	NOUN
cana-3950	141	19	:	:	PUNCT
cana-3950	141	20	1074	1074	NUM
cana-3950	141	21	-	-	PUNCT
cana-3950	141	22	133x	133x	NUM
cana-3950	141	23	vol	vol	NOUN
cana-3950	141	24	32	32	NUM
cana-3950	141	25	no	no	NOUN
cana-3950	141	26	.	.	PUNCT
cana-3950	142	1	9s	9s	NUM
cana-3950	142	2	(	(	PUNCT
cana-3950	142	3	2025	2025	NUM
cana-3950	142	4	)	)	PUNCT
cana-3950	142	5	393	393	NUM
cana-3950	142	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3950	142	7	⟹	⟹	NUM
cana-3950	142	8	𝐾𝑞𝐴1	𝐾𝑞𝐴1	NUM
cana-3950	142	9	≥	≥	X
cana-3950	142	10	𝐾𝑞𝐴2	𝐾𝑞𝐴2	NOUN
cana-3950	142	11	hence	hence	ADV
cana-3950	142	12	,	,	PUNCT
cana-3950	142	13	𝑞𝐴1	𝑞𝐴1	PROPN
cana-3950	142	14	⊂	⊂	PROPN
cana-3950	142	15	𝑞𝐴2	𝑞𝐴2	PROPN
cana-3950	142	16	.	.	PUNCT
cana-3950	143	1	(	(	PUNCT
cana-3950	143	2	i)similar	i)similar	ADJ
cana-3950	143	3	to	to	ADP
cana-3950	143	4	that	that	PRON
cana-3950	143	5	of	of	ADP
cana-3950	143	6	(	(	PUNCT
cana-3950	143	7	i	i	NOUN
cana-3950	143	8	)	)	PUNCT
cana-3950	143	9	(	(	PUNCT
cana-3950	143	10	ii)follow	ii)follow	VERB
cana-3950	143	11	since	since	ADV
cana-3950	143	12	:	:	PUNCT
cana-3950	143	13	𝐴1	𝐴1	PROPN
cana-3950	143	14	∪	∪	VERB
cana-3950	143	15	𝐴2	𝐴2	PROPN
cana-3950	143	16	=	=	SYM
cana-3950	143	17	(	(	PUNCT
cana-3950	143	18	max	max	PROPN
cana-3950	143	19	{	{	PUNCT
cana-3950	143	20	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	143	21	,	,	PUNCT
cana-3950	143	22	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	143	23	}	}	PUNCT
cana-3950	143	24	,	,	PUNCT
cana-3950	143	25	min{𝐾𝐴1	min{𝐾𝐴1	PROPN
cana-3950	143	26	,	,	PUNCT
cana-3950	143	27	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	143	28	}	}	PUNCT
cana-3950	143	29	)	)	PUNCT
cana-3950	143	30	and	and	CCONJ
cana-3950	143	31	(	(	PUNCT
cana-3950	143	32	𝐴1	𝐴1	PROPN
cana-3950	143	33	∪	∪	ADP
cana-3950	143	34	𝐴2	𝐴2	PROPN
cana-3950	143	35	)	)	PUNCT
cana-3950	143	36	𝑞	𝑞	X
cana-3950	143	37	=	=	PUNCT
cana-3950	143	38	(	(	PUNCT
cana-3950	143	39	𝐽𝐴1∪𝐴2	𝐽𝐴1∪𝐴2	PROPN
cana-3950	143	40	,	,	PUNCT
cana-3950	143	41	𝐾𝐴1∪𝐴2	𝐾𝐴1∪𝐴2	NOUN
cana-3950	143	42	)	)	PUNCT
cana-3950	144	1	=	=	PUNCT
cana-3950	144	2	(	(	PUNCT
cana-3950	144	3	(	(	PUNCT
cana-3950	144	4	max	max	PROPN
cana-3950	144	5	{	{	PUNCT
cana-3950	144	6	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	144	7	,	,	PUNCT
cana-3950	144	8	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	144	9	}	}	PUNCT
cana-3950	144	10	)	)	PUNCT
cana-3950	144	11	𝑚𝑞	𝑚𝑞	ADP
cana-3950	144	12	,	,	PUNCT
cana-3950	144	13	1−	1−	NUM
cana-3950	144	14	(	(	PUNCT
cana-3950	144	15	1	1	NUM
cana-3950	144	16	−	−	NOUN
cana-3950	144	17	(	(	PUNCT
cana-3950	144	18	min{𝐾𝐴1	min{𝐾𝐴1	PROPN
cana-3950	144	19	,	,	PUNCT
cana-3950	144	20	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	144	21	}	}	PUNCT
cana-3950	144	22	)	)	PUNCT
cana-3950	144	23	𝑛	𝑛	PROPN
cana-3950	144	24	)	)	PUNCT
cana-3950	144	25	𝑞	𝑞	NOUN
cana-3950	144	26	)	)	PUNCT
cana-3950	144	27	=	=	SYM
cana-3950	144	28	(	(	PUNCT
cana-3950	144	29	max	max	PROPN
cana-3950	144	30	{	{	PUNCT
cana-3950	144	31	{	{	PUNCT
cana-3950	144	32	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	144	33	𝑚𝑞	𝑚𝑞	NOUN
cana-3950	144	34	,	,	PUNCT
cana-3950	144	35	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	144	36	𝑚𝑞	𝑚𝑞	ADP
cana-3950	144	37	}	}	PUNCT
cana-3950	144	38	,	,	PUNCT
cana-3950	144	39	1−	1−	NUM
cana-3950	144	40	(	(	PUNCT
cana-3950	144	41	1−min{𝐾𝐴1	1−min{𝐾𝐴1	NUM
cana-3950	144	42	𝑛	𝑛	PROPN
cana-3950	144	43	,	,	PUNCT
cana-3950	144	44	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	144	45	𝑛	𝑛	PROPN
cana-3950	144	46	}	}	PUNCT
cana-3950	144	47	)	)	PUNCT
cana-3950	144	48	𝑞	𝑞	NOUN
cana-3950	144	49	)	)	PUNCT
cana-3950	144	50	=	=	SYM
cana-3950	144	51	(	(	PUNCT
cana-3950	144	52	max	max	PROPN
cana-3950	144	53	{	{	PUNCT
cana-3950	144	54	{	{	PUNCT
cana-3950	144	55	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	144	56	𝑚𝑞	𝑚𝑞	NOUN
cana-3950	144	57	,	,	PUNCT
cana-3950	144	58	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	144	59	𝑚𝑞	𝑚𝑞	ADP
cana-3950	144	60	}	}	PUNCT
cana-3950	144	61	,	,	PUNCT
cana-3950	144	62	1−	1−	NUM
cana-3950	144	63	max{1−	max{1−	VERB
cana-3950	144	64	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	144	65	𝑛	𝑛	PROPN
cana-3950	144	66	,	,	PUNCT
cana-3950	144	67	1−	1−	NUM
cana-3950	144	68	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	144	69	𝑛	𝑛	PROPN
cana-3950	144	70	}	}	PUNCT
cana-3950	144	71	)	)	PUNCT
cana-3950	144	72	𝑞	𝑞	NOUN
cana-3950	144	73	)	)	PUNCT
cana-3950	144	74	=	=	SYM
cana-3950	144	75	(	(	PUNCT
cana-3950	144	76	max	max	PROPN
cana-3950	144	77	{	{	PUNCT
cana-3950	144	78	{	{	PUNCT
cana-3950	144	79	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	144	80	𝑚𝑞	𝑚𝑞	NOUN
cana-3950	144	81	,	,	PUNCT
cana-3950	144	82	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	144	83	𝑚𝑞	𝑚𝑞	ADP
cana-3950	144	84	}	}	PUNCT
cana-3950	144	85	,	,	PUNCT
cana-3950	144	86	1−	1−	NUM
cana-3950	144	87	max{1−	max{1−	VERB
cana-3950	144	88	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	144	89	𝑛	𝑛	PROPN
cana-3950	144	90	,	,	PUNCT
cana-3950	144	91	1−	1−	NUM
cana-3950	144	92	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	144	93	𝑛	𝑛	PROPN
cana-3950	144	94	}	}	PUNCT
cana-3950	144	95	)	)	PUNCT
cana-3950	144	96	𝑞	𝑞	NOUN
cana-3950	144	97	)	)	PUNCT
cana-3950	144	98	=	=	SYM
cana-3950	144	99	(	(	PUNCT
cana-3950	144	100	max	max	PROPN
cana-3950	144	101	{	{	PUNCT
cana-3950	144	102	{	{	PUNCT
cana-3950	144	103	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	144	104	𝑚𝑞	𝑚𝑞	NOUN
cana-3950	144	105	,	,	PUNCT
cana-3950	144	106	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	144	107	𝑚𝑞	𝑚𝑞	ADP
cana-3950	144	108	}	}	PUNCT
cana-3950	144	109	,	,	PUNCT
cana-3950	144	110	1	1	NUM
cana-3950	144	111	−	−	PROPN
cana-3950	144	112	(	(	PUNCT
cana-3950	144	113	max{1	max{1	NOUN
cana-3950	144	114	−	−	PROPN
cana-3950	144	115	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	144	116	𝑛	𝑛	PROPN
cana-3950	144	117	)	)	PUNCT
cana-3950	144	118	𝑞	𝑞	NOUN
cana-3950	144	119	,	,	PUNCT
cana-3950	144	120	(	(	PUNCT
cana-3950	144	121	1	1	NUM
cana-3950	144	122	−	−	PROPN
cana-3950	144	123	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	144	124	𝑛	𝑛	PROPN
cana-3950	144	125	)	)	PUNCT
cana-3950	144	126	𝑞	𝑞	NOUN
cana-3950	144	127	}	}	PUNCT
cana-3950	144	128	)	)	PUNCT
cana-3950	144	129	.	.	PUNCT
cana-3950	145	1	proof	proof	NOUN
cana-3950	145	2	:	:	PUNCT
cana-3950	145	3	(	(	PUNCT
cana-3950	145	4	i	i	NOUN
cana-3950	145	5	)	)	PUNCT
cana-3950	145	6	since	since	SCONJ
cana-3950	145	7	𝐴𝑞1	𝐴𝑞1	NOUN
cana-3950	145	8	=	=	SYM
cana-3950	145	9	(	(	PUNCT
cana-3950	145	10	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	145	11	𝑚𝑞1	𝑚𝑞1	PROPN
cana-3950	145	12	,	,	PUNCT
cana-3950	145	13	1−	1−	NUM
cana-3950	145	14	(	(	PUNCT
cana-3950	145	15	1−	1−	NUM
cana-3950	145	16	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	145	17	𝑛)𝑞1	𝑛)𝑞1	PROPN
cana-3950	145	18	)	)	PUNCT
cana-3950	145	19	𝐴𝑞2	𝐴𝑞2	NOUN
cana-3950	146	1	=	=	PUNCT
cana-3950	147	1	(	(	PUNCT
cana-3950	147	2	𝐽𝐴	𝐽𝐴	NOUN
cana-3950	147	3	𝑚𝑞2	𝑚𝑞2	PROPN
cana-3950	147	4	,	,	PUNCT
cana-3950	147	5	1−	1−	NUM
cana-3950	147	6	(	(	PUNCT
cana-3950	147	7	1−	1−	NUM
cana-3950	147	8	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	147	9	𝑛)𝑞2	𝑛)𝑞2	PROPN
cana-3950	147	10	)	)	PUNCT
cana-3950	147	11	we	we	PRON
cana-3950	147	12	have	have	VERB
cana-3950	147	13	,	,	PUNCT
cana-3950	147	14	𝑞1	𝑞1	PROPN
cana-3950	147	15	≥	≥	NUM
cana-3950	147	16	𝑞2	𝑞2	PROPN
cana-3950	147	17	⟹	⟹	NUM
cana-3950	147	18	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	147	19	𝑞2	𝑞2	PROPN
cana-3950	147	20	≥	≥	PROPN
cana-3950	147	21	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	147	22	𝑞1	𝑞1	PROPN
cana-3950	147	23	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-3950	147	24	(	(	PUNCT
cana-3950	147	25	1	1	NUM
cana-3950	147	26	−	−	NOUN
cana-3950	147	27	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	147	28	𝑛)𝑞1	𝑛)𝑞1	PROPN
cana-3950	147	29	≤	≤	NUM
cana-3950	147	30	(	(	PUNCT
cana-3950	147	31	1−	1−	NUM
cana-3950	147	32	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	147	33	𝑛)𝑞2	𝑛)𝑞2	PROPN
cana-3950	147	34	⟹	⟹	PUNCT
cana-3950	147	35	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	147	36	𝑚𝑞2	𝑚𝑞2	PROPN
cana-3950	147	37	≥	≥	NOUN
cana-3950	147	38	𝐽𝑚𝐴	𝐽𝑚𝐴	ADV
cana-3950	147	39	𝑞1	𝑞1	PROPN
cana-3950	147	40	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-3950	147	41	1	1	NUM
cana-3950	147	42	−	−	PROPN
cana-3950	147	43	(	(	PUNCT
cana-3950	147	44	1−	1−	NUM
cana-3950	147	45	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	147	46	𝑛)𝑞2	𝑛)𝑞2	PROPN
cana-3950	147	47	≤	≤	NOUN
cana-3950	147	48	(	(	PUNCT
cana-3950	147	49	1−	1−	NUM
cana-3950	147	50	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	147	51	𝑛)𝑞1	𝑛)𝑞1	PROPN
cana-3950	147	52	⟹	⟹	NUM
cana-3950	147	53	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	147	54	𝑞2	𝑞2	PROPN
cana-3950	147	55	≥	≥	PROPN
cana-3950	147	56	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	147	57	𝑞1	𝑞1	PROPN
cana-3950	147	58	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-3950	147	59	𝐾𝐴	𝐾𝐴	NOUN
cana-3950	147	60	𝑞2	𝑞2	NOUN
cana-3950	147	61	≤	≤	NUM
cana-3950	147	62	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	147	63	𝑞1	𝑞1	NOUN
cana-3950	147	64	.	.	PUNCT
cana-3950	148	1	hence	hence	ADV
cana-3950	148	2	𝐴𝑞1	𝐴𝑞1	VERB
cana-3950	148	3	⊂	⊂	PROPN
cana-3950	148	4	𝐴𝑞2	𝐴𝑞2	PROPN
cana-3950	148	5	.	.	PUNCT
cana-3950	149	1	(	(	PUNCT
cana-3950	149	2	ii	ii	NOUN
cana-3950	149	3	)	)	PUNCT
cana-3950	149	4	similar	similar	ADJ
cana-3950	149	5	to	to	ADP
cana-3950	149	6	that	that	PRON
cana-3950	149	7	of	of	ADP
cana-3950	149	8	(	(	PUNCT
cana-3950	149	9	i	i	NOUN
cana-3950	149	10	)	)	PUNCT
cana-3950	149	11	.	.	PUNCT
cana-3950	150	1	theorem	theorem	NOUN
cana-3950	150	2	2.17	2.17	NUM
cana-3950	150	3	let	let	VERB
cana-3950	150	4	𝐴1	𝐴1	PROPN
cana-3950	150	5	=	=	SYM
cana-3950	150	6	(	(	PUNCT
cana-3950	150	7	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	150	8	,	,	PUNCT
cana-3950	150	9	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	150	10	)	)	PUNCT
cana-3950	150	11	,	,	PUNCT
cana-3950	150	12	𝐴2	𝐴2	PROPN
cana-3950	150	13	=	=	SYM
cana-3950	150	14	(	(	PUNCT
cana-3950	150	15	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	150	16	,	,	PUNCT
cana-3950	150	17	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	150	18	)	)	PUNCT
cana-3950	150	19	∈	∈	PROPN
cana-3950	150	20	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	150	21	𝑛	𝑛	PROPN
cana-3950	150	22	(	(	PUNCT
cana-3950	150	23	𝑋	𝑋	PROPN
cana-3950	150	24	)	)	PUNCT
cana-3950	150	25	𝑎𝑛𝑑	𝑎𝑛𝑑	NOUN
cana-3950	150	26	𝑞	𝑞	X
cana-3950	150	27	∈	∈	PROPN
cana-3950	150	28	𝑁.	𝑁.	PROPN
cana-3950	150	29	then	then	ADV
cana-3950	150	30	(	(	PUNCT
cana-3950	151	1	𝑖	𝑖	X
cana-3950	151	2	)	)	PUNCT
cana-3950	151	3	𝐴1	𝐴1	PROPN
cana-3950	152	1	⊂	⊂	PROPN
cana-3950	152	2	𝐴2	𝐴2	PROPN
cana-3950	152	3	⟹	⟹	PUNCT
cana-3950	152	4	𝑞𝐴1	𝑞𝐴1	PROPN
cana-3950	152	5	⊂	⊂	PROPN
cana-3950	152	6	𝑞𝐴2	𝑞𝐴2	PROPN
cana-3950	152	7	(	(	PUNCT
cana-3950	152	8	𝑖𝑖	𝑖𝑖	NOUN
cana-3950	152	9	)	)	PUNCT
cana-3950	152	10	𝐴1	𝐴1	PROPN
cana-3950	153	1	⊂	⊂	PROPN
cana-3950	153	2	𝐴2	𝐴2	PROPN
cana-3950	153	3	⟹	⟹	X
cana-3950	153	4	𝐴1	𝐴1	PROPN
cana-3950	153	5	𝑞	𝑞	PROPN
cana-3950	153	6	⊂	⊂	PROPN
cana-3950	153	7	𝐴2	𝐴2	PROPN
cana-3950	153	8	𝑞	𝑞	PROPN
cana-3950	153	9	(	(	PUNCT
cana-3950	153	10	𝑖𝑖𝑖)(𝐴1	𝑖𝑖𝑖)(𝐴1	PROPN
cana-3950	153	11	∪	∪	PROPN
cana-3950	153	12	𝐴2	𝐴2	PROPN
cana-3950	153	13	)	)	PUNCT
cana-3950	153	14	𝑞	𝑞	X
cana-3950	153	15	⟹	⟹	X
cana-3950	153	16	𝐴1	𝐴1	PROPN
cana-3950	153	17	𝑞	𝑞	AUX
cana-3950	153	18	∪	∪	VERB
cana-3950	153	19	𝐴2	𝐴2	PROPN
cana-3950	153	20	𝑞	𝑞	PROPN
cana-3950	153	21	(	(	PUNCT
cana-3950	153	22	𝑖𝑣	𝑖𝑣	ADP
cana-3950	153	23	)	)	PUNCT
cana-3950	153	24	𝑞(𝐴1	𝑞(𝐴1	X
cana-3950	153	25	∪	∪	PROPN
cana-3950	153	26	𝐴2	𝐴2	PROPN
cana-3950	153	27	)	)	PUNCT
cana-3950	153	28	⟹	⟹	VERB
cana-3950	154	1	𝑞𝐴1	𝑞𝐴1	PROPN
cana-3950	154	2	∪	∪	PROPN
cana-3950	154	3	𝑞𝐴2	𝑞𝐴2	PROPN
cana-3950	154	4	(	(	PUNCT
cana-3950	154	5	𝑣)(𝐴1	𝑣)(𝐴1	PROPN
cana-3950	154	6	∩	∩	PROPN
cana-3950	154	7	𝐴2	𝐴2	PROPN
cana-3950	154	8	)	)	PUNCT
cana-3950	154	9	⟹	⟹	PUNCT
cana-3950	155	1	𝐴1	𝐴1	PROPN
cana-3950	155	2	𝑞	𝑞	PROPN
cana-3950	155	3	∩	∩	VERB
cana-3950	155	4	𝐴2	𝐴2	PROPN
cana-3950	155	5	𝑞	𝑞	PROPN
cana-3950	155	6	(	(	PUNCT
cana-3950	155	7	𝑣𝑖)𝑞(𝐴1	𝑣𝑖)𝑞(𝐴1	X
cana-3950	155	8	∩	∩	X
cana-3950	155	9	𝐴2	𝐴2	PROPN
cana-3950	155	10	)	)	PUNCT
cana-3950	155	11	⟹	⟹	PUNCT
cana-3950	156	1	𝑞𝐴1	𝑞𝐴1	PROPN
cana-3950	156	2	∩	∩	PROPN
cana-3950	156	3	𝑞𝐴2	𝑞𝐴2	PROPN
cana-3950	156	4	.	.	PUNCT
cana-3950	156	5	communications	communication	NOUN
cana-3950	156	6	on	on	ADP
cana-3950	156	7	applied	apply	VERB
cana-3950	156	8	nonlinear	nonlinear	ADJ
cana-3950	156	9	analysis	analysis	NOUN
cana-3950	156	10	issn	issn	NOUN
cana-3950	156	11	:	:	PUNCT
cana-3950	156	12	1074	1074	NUM
cana-3950	156	13	-	-	PUNCT
cana-3950	156	14	133x	133x	NUM
cana-3950	156	15	vol	vol	NOUN
cana-3950	156	16	32	32	NUM
cana-3950	156	17	no	no	NOUN
cana-3950	156	18	.	.	PUNCT
cana-3950	157	1	9s	9s	NUM
cana-3950	157	2	(	(	PUNCT
cana-3950	157	3	2025	2025	NUM
cana-3950	157	4	)	)	PUNCT
cana-3950	157	5	394	394	NUM
cana-3950	158	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3950	158	2	proof	proof	NOUN
cana-3950	158	3	:	:	PUNCT
cana-3950	158	4	since	since	SCONJ
cana-3950	158	5	𝐴1	𝐴1	PROPN
cana-3950	158	6	⊂	⊂	PROPN
cana-3950	158	7	𝐴2	𝐴2	PROPN
cana-3950	158	8	,	,	PUNCT
cana-3950	158	9	we	we	PRON
cana-3950	158	10	have	have	VERB
cana-3950	158	11	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	158	12	≤	≤	PUNCT
cana-3950	158	13	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	159	1	→	→	SYM
cana-3950	159	2	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	159	3	𝑚	𝑚	PROPN
cana-3950	159	4	≤	≤	NUM
cana-3950	159	5	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	159	6	𝑚	𝑚	PROPN
cana-3950	159	7	=	=	SYM
cana-3950	159	8	max	max	PROPN
cana-3950	159	9	{	{	PUNCT
cana-3950	159	10	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	159	11	𝑚𝑞	𝑚𝑞	INTJ
cana-3950	159	12	,	,	PUNCT
cana-3950	159	13	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	159	14	𝑚𝑞	𝑚𝑞	ADP
cana-3950	159	15	}	}	PUNCT
cana-3950	159	16	,	,	PUNCT
cana-3950	159	17	min	min	NOUN
cana-3950	159	18	{	{	PUNCT
cana-3950	159	19	(	(	PUNCT
cana-3950	159	20	1−	1−	NUM
cana-3950	159	21	(	(	PUNCT
cana-3950	159	22	1	1	NUM
cana-3950	159	23	−	−	NOUN
cana-3950	159	24	𝐾𝐴1	𝐾𝐴1	NOUN
cana-3950	159	25	𝑛	𝑛	NOUN
cana-3950	159	26	)	)	PUNCT
cana-3950	159	27	𝑞	𝑞	NOUN
cana-3950	159	28	)	)	PUNCT
cana-3950	159	29	}	}	PUNCT
cana-3950	159	30	=	=	PUNCT
cana-3950	160	1	𝐴1	𝐴1	PROPN
cana-3950	160	2	𝑞	𝑞	X
cana-3950	160	3	∪	∪	VERB
cana-3950	160	4	𝐴2	𝐴2	PROPN
cana-3950	160	5	𝑞	𝑞	PROPN
cana-3950	160	6	(	(	PUNCT
cana-3950	160	7	iv	iv	X
cana-3950	160	8	)	)	PUNCT
cana-3950	160	9	follows	follow	VERB
cana-3950	160	10	since	since	SCONJ
cana-3950	160	11	𝐴1	𝐴1	PROPN
cana-3950	160	12	∩	∩	NOUN
cana-3950	160	13	𝐴2	𝐴2	PROPN
cana-3950	160	14	=	=	SYM
cana-3950	160	15	(	(	PUNCT
cana-3950	160	16	min{𝐽𝐴1	min{𝐽𝐴1	PROPN
cana-3950	160	17	,	,	PUNCT
cana-3950	160	18	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	160	19	}	}	PUNCT
cana-3950	160	20	,	,	PUNCT
cana-3950	160	21	max{𝐾𝐴1	max{𝐾𝐴1	NOUN
cana-3950	160	22	,	,	PUNCT
cana-3950	160	23	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	160	24	}	}	PUNCT
cana-3950	160	25	)	)	PUNCT
cana-3950	160	26	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
cana-3950	160	27	(	(	PUNCT
cana-3950	160	28	𝐴1	𝐴1	PROPN
cana-3950	160	29	∩	∩	PROPN
cana-3950	160	30	𝐴2	𝐴2	PROPN
cana-3950	160	31	)	)	PUNCT
cana-3950	160	32	𝑞	𝑞	X
cana-3950	160	33	=	=	SYM
cana-3950	160	34	(	(	PUNCT
cana-3950	160	35	𝐽𝐴1∩𝐴2	𝐽𝐴1∩𝐴2	PROPN
cana-3950	160	36	,	,	PUNCT
cana-3950	160	37	𝐾𝐴1∩𝐴2	𝐾𝐴1∩𝐴2	NUM
cana-3950	160	38	)	)	PUNCT
cana-3950	160	39	=	=	SYM
cana-3950	160	40	(	(	PUNCT
cana-3950	160	41	(	(	PUNCT
cana-3950	160	42	min{𝐽𝐴1	min{𝐽𝐴1	PROPN
cana-3950	160	43	,	,	PUNCT
cana-3950	160	44	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	160	45	}	}	PUNCT
cana-3950	160	46	)	)	PUNCT
cana-3950	160	47	𝑚𝑞	𝑚𝑞	ADP
cana-3950	160	48	,	,	PUNCT
cana-3950	160	49	1−	1−	NUM
cana-3950	160	50	(	(	PUNCT
cana-3950	160	51	1−	1−	NUM
cana-3950	160	52	{	{	PUNCT
cana-3950	160	53	max{𝐾𝐴1	max{𝐾𝐴1	NOUN
cana-3950	160	54	,	,	PUNCT
cana-3950	160	55	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	160	56	}	}	PUNCT
cana-3950	160	57	)	)	PUNCT
cana-3950	161	1	𝑛	𝑛	X
cana-3950	161	2	)	)	PUNCT
cana-3950	161	3	𝑞	𝑞	NOUN
cana-3950	161	4	}	}	PUNCT
cana-3950	161	5	=	=	SYM
cana-3950	161	6	(	(	PUNCT
cana-3950	161	7	(	(	PUNCT
cana-3950	161	8	min{𝐽𝐴1	min{𝐽𝐴1	PROPN
cana-3950	161	9	𝑚𝑞	𝑚𝑞	ADP
cana-3950	161	10	,	,	PUNCT
cana-3950	161	11	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	161	12	𝑚𝑞	𝑚𝑞	NOUN
cana-3950	161	13	}	}	PUNCT
cana-3950	161	14	)	)	PUNCT
cana-3950	161	15	,	,	PUNCT
cana-3950	161	16	1	1	NUM
cana-3950	161	17	−	−	PROPN
cana-3950	161	18	(	(	PUNCT
cana-3950	161	19	1−max{𝐾𝐴1	1−max{𝐾𝐴1	NOUN
cana-3950	161	20	𝑛	𝑛	PROPN
cana-3950	161	21	,	,	PUNCT
cana-3950	161	22	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	161	23	𝑛	𝑛	PROPN
cana-3950	161	24	}	}	PUNCT
cana-3950	161	25	)	)	PUNCT
cana-3950	161	26	𝑘	𝑘	X
cana-3950	161	27	)	)	PUNCT
cana-3950	161	28	=	=	SYM
cana-3950	162	1	(	(	PUNCT
cana-3950	162	2	min{𝐽𝐴1	min{𝐽𝐴1	PROPN
cana-3950	162	3	𝑚𝑞	𝑚𝑞	ADP
cana-3950	162	4	,	,	PUNCT
cana-3950	162	5	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	162	6	𝑚𝑞	𝑚𝑞	X
cana-3950	162	7	}	}	PUNCT
cana-3950	162	8	,	,	PUNCT
cana-3950	162	9	1	1	NUM
cana-3950	162	10	−	−	PROPN
cana-3950	162	11	(	(	PUNCT
cana-3950	162	12	min(1−	min(1−	PROPN
cana-3950	162	13	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	162	14	𝑛	𝑛	PROPN
cana-3950	162	15	,	,	PUNCT
cana-3950	162	16	1−	1−	NUM
cana-3950	162	17	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	162	18	𝑛	𝑛	PROPN
cana-3950	162	19	)	)	PUNCT
cana-3950	162	20	𝑞	𝑞	NOUN
cana-3950	162	21	)	)	PUNCT
cana-3950	162	22	)	)	PUNCT
cana-3950	163	1	=	=	PRON
cana-3950	163	2	(	(	PUNCT
cana-3950	163	3	min{𝐽𝐴1	min{𝐽𝐴1	PROPN
cana-3950	163	4	𝑚𝑞	𝑚𝑞	ADP
cana-3950	163	5	,	,	PUNCT
cana-3950	163	6	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	163	7	𝑚𝑞	𝑚𝑞	ADP
cana-3950	163	8	}	}	PUNCT
cana-3950	163	9	,	,	PUNCT
cana-3950	163	10	max{1−	max{1−	X
cana-3950	163	11	(	(	PUNCT
cana-3950	163	12	1	1	NUM
cana-3950	163	13	−	−	NOUN
cana-3950	163	14	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	163	15	𝑛)𝑞	𝑛)𝑞	NOUN
cana-3950	163	16	,	,	PUNCT
cana-3950	163	17	(	(	PUNCT
cana-3950	163	18	1−	1−	NUM
cana-3950	163	19	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	163	20	𝑛	𝑛	X
cana-3950	163	21	)	)	PUNCT
cana-3950	163	22	𝑞	𝑞	NOUN
cana-3950	163	23	}	}	PUNCT
cana-3950	163	24	)	)	PUNCT
cana-3950	163	25	=	=	SYM
cana-3950	164	1	(	(	PUNCT
cana-3950	164	2	𝑚𝑖𝑛{𝐽𝐴1	𝑚𝑖𝑛{𝐽𝐴1	NOUN
cana-3950	164	3	𝑚𝑞	𝑚𝑞	ADP
cana-3950	164	4	,	,	PUNCT
cana-3950	164	5	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	164	6	𝑚𝑞	𝑚𝑞	ADV
cana-3950	164	7	}	}	PUNCT
cana-3950	164	8	,	,	PUNCT
cana-3950	164	9	max{1−	max{1−	X
cana-3950	164	10	(	(	PUNCT
cana-3950	164	11	1	1	NUM
cana-3950	164	12	−	−	NOUN
cana-3950	164	13	𝐾𝐴1	𝐾𝐴1	NOUN
cana-3950	164	14	𝑛	𝑛	PROPN
cana-3950	164	15	)	)	PUNCT
cana-3950	164	16	𝑞	𝑞	PROPN
cana-3950	164	17	(	(	PUNCT
cana-3950	164	18	1−	1−	NUM
cana-3950	164	19	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	164	20	𝑛	𝑛	PROPN
cana-3950	164	21	)	)	PUNCT
cana-3950	164	22	𝑞	𝑞	NOUN
cana-3950	164	23	}	}	PUNCT
cana-3950	164	24	)	)	PUNCT
cana-3950	165	1	=	=	PUNCT
cana-3950	165	2	𝐴1	𝐴1	PROPN
cana-3950	165	3	𝑘	𝑘	X
cana-3950	165	4	∩	∩	NOUN
cana-3950	165	5	𝐴2	𝐴2	PROPN
cana-3950	165	6	𝑘	𝑘	PROPN
cana-3950	165	7	(	(	PUNCT
cana-3950	165	8	v	v	NOUN
cana-3950	165	9	)	)	PUNCT
cana-3950	165	10	the	the	DET
cana-3950	165	11	same	same	ADJ
cana-3950	165	12	that	that	PRON
cana-3950	165	13	of	of	ADP
cana-3950	165	14	result	result	NOUN
cana-3950	165	15	(	(	PUNCT
cana-3950	165	16	iii	iii	NOUN
cana-3950	165	17	)	)	PUNCT
cana-3950	165	18	(	(	PUNCT
cana-3950	165	19	vi	vi	NOUN
cana-3950	165	20	)	)	PUNCT
cana-3950	165	21	some	some	DET
cana-3950	165	22	way	way	NOUN
cana-3950	165	23	that	that	PRON
cana-3950	165	24	𝑓(𝑣	𝑓(𝑣	VERB
cana-3950	165	25	)	)	PUNCT
cana-3950	165	26	definition	definition	NOUN
cana-3950	165	27	2.18	2.18	NUM
cana-3950	165	28	let	let	VERB
cana-3950	165	29	𝐴	𝐴	PROPN
cana-3950	165	30	=	=	SYM
cana-3950	165	31	(	(	PUNCT
cana-3950	165	32	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	165	33	,	,	PUNCT
cana-3950	165	34	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	165	35	)	)	PUNCT
cana-3950	165	36	∈	∈	PROPN
cana-3950	166	1	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	166	2	𝑛	𝑛	PROPN
cana-3950	166	3	(	(	PUNCT
cana-3950	166	4	𝑥	𝑥	NOUN
cana-3950	166	5	)	)	PUNCT
cana-3950	166	6	and	and	CCONJ
cana-3950	166	7	𝛼	𝛼	PRON
cana-3950	166	8	∈	∈	PROPN
cana-3950	167	1	[	[	X
cana-3950	167	2	0,1	0,1	NUM
cana-3950	167	3	]	]	PUNCT
cana-3950	167	4	.	.	PUNCT
cana-3950	168	1	then	then	ADV
cana-3950	168	2	the	the	DET
cana-3950	168	3	operator	operator	NOUN
cana-3950	168	4	𝐺𝛼(𝐴	𝐺𝛼(𝐴	ADV
cana-3950	168	5	)	)	PUNCT
cana-3950	168	6	is	be	AUX
cana-3950	168	7	expressed	express	VERB
cana-3950	168	8	as	as	SCONJ
cana-3950	168	9	follows	follow	VERB
cana-3950	168	10	:	:	PUNCT
cana-3950	168	11	𝐺𝛼(𝐴	𝐺𝛼(𝐴	NUM
cana-3950	168	12	)	)	PUNCT
cana-3950	168	13	=	=	SYM
cana-3950	169	1	(	(	PUNCT
cana-3950	169	2	(	(	PUNCT
cana-3950	169	3	𝐽𝐴	𝐽𝐴	NOUN
cana-3950	169	4	𝑚	𝑚	NOUN
cana-3950	169	5	+	+	CCONJ
cana-3950	169	6	𝛼∏	𝛼∏	PROPN
cana-3950	169	7	𝑚+𝑛	𝑚+𝑛	NUM
cana-3950	169	8	2	2	NUM
cana-3950	169	9	)	)	PUNCT
cana-3950	169	10	1	1	NUM
cana-3950	169	11	𝑚	𝑚	NOUN
cana-3950	169	12	,	,	PUNCT
cana-3950	169	13	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	169	14	𝑛	𝑛	PROPN
cana-3950	169	15	+	+	X
cana-3950	169	16	(	(	PUNCT
cana-3950	169	17	(	(	PUNCT
cana-3950	169	18	1−	1−	NUM
cana-3950	169	19	𝛼)∏	𝛼)∏	NUM
cana-3950	169	20	𝑚+𝑛	𝑚+𝑛	NUM
cana-3950	169	21	2	2	NUM
cana-3950	169	22	)	)	PUNCT
cana-3950	169	23	1	1	NUM
cana-3950	169	24	𝑛	𝑛	NOUN
cana-3950	169	25	)	)	PUNCT
cana-3950	169	26	theorem	theorem	VERB
cana-3950	169	27	2.19	2.19	NUM
cana-3950	169	28	let	let	VERB
cana-3950	170	1	𝐴	𝐴	PROPN
cana-3950	170	2	=	=	SYM
cana-3950	170	3	(	(	PUNCT
cana-3950	170	4	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	170	5	,	,	PUNCT
cana-3950	170	6	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	170	7	)	)	PUNCT
cana-3950	170	8	∈	∈	PROPN
cana-3950	171	1	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	171	2	𝑛	𝑛	PROPN
cana-3950	171	3	(	(	PUNCT
cana-3950	171	4	𝑥	𝑥	NOUN
cana-3950	171	5	)	)	PUNCT
cana-3950	171	6	and	and	CCONJ
cana-3950	171	7	𝛼	𝛼	X
cana-3950	171	8	,	,	PUNCT
cana-3950	171	9	𝛽	𝛽	PROPN
cana-3950	171	10	∈	∈	NOUN
cana-3950	172	1	[	[	X
cana-3950	172	2	0,1	0,1	NUM
cana-3950	172	3	]	]	PUNCT
cana-3950	172	4	.	.	PUNCT
cana-3950	173	1	then	then	ADV
cana-3950	173	2	(	(	PUNCT
cana-3950	173	3	𝑖)𝛼	𝑖)𝛼	ADJ
cana-3950	173	4	≤	≤	NUM
cana-3950	173	5	𝛽	𝛽	NOUN
cana-3950	173	6	⟹	⟹	NUM
cana-3950	173	7	𝐺𝛼(𝐴	𝐺𝛼(𝐴	NUM
cana-3950	173	8	)	)	PUNCT
cana-3950	173	9	⊂	⊂	PROPN
cana-3950	173	10	𝐺𝛽1(𝐴	𝐺𝛽1(𝐴	PROPN
cana-3950	173	11	)	)	PUNCT
cana-3950	173	12	(	(	PUNCT
cana-3950	173	13	𝑖𝑖)𝐺0(𝐴	𝑖𝑖)𝐺0(𝐴	NOUN
cana-3950	173	14	)	)	PUNCT
cana-3950	173	15	=	=	SYM
cana-3950	173	16	□	□	PUNCT
cana-3950	173	17	(	(	PUNCT
cana-3950	173	18	𝑖𝑖𝑖)𝐺1(𝐴	𝑖𝑖𝑖)𝐺1(𝐴	NUM
cana-3950	173	19	)	)	PUNCT
cana-3950	173	20	=	=	SYM
cana-3950	174	1	◊	◊	ADJ
cana-3950	174	2	a	a	DET
cana-3950	174	3	proof	proof	NOUN
cana-3950	174	4	:	:	PUNCT
cana-3950	174	5	(	(	PUNCT
cana-3950	174	6	i	i	NOUN
cana-3950	174	7	)	)	PUNCT
cana-3950	174	8	immediate	immediate	ADJ
cana-3950	174	9	result	result	NOUN
cana-3950	174	10	to	to	PART
cana-3950	174	11	obtained	obtain	VERB
cana-3950	174	12	(	(	PUNCT
cana-3950	174	13	ii	ii	NOUN
cana-3950	174	14	)	)	PUNCT
cana-3950	174	15	since	since	SCONJ
cana-3950	174	16	,	,	PUNCT
cana-3950	174	17	𝐺0(𝐴	𝐺0(𝐴	NOUN
cana-3950	174	18	)	)	PUNCT
cana-3950	174	19	=	=	SYM
cana-3950	174	20	(	(	PUNCT
cana-3950	174	21	(	(	PUNCT
cana-3950	174	22	𝐽𝐴	𝐽𝐴	NOUN
cana-3950	174	23	𝑚	𝑚	NOUN
cana-3950	174	24	+	+	CCONJ
cana-3950	174	25	𝛼∏	𝛼∏	PROPN
cana-3950	174	26	𝑚+𝑛	𝑚+𝑛	NUM
cana-3950	174	27	2	2	NUM
cana-3950	174	28	)	)	PUNCT
cana-3950	174	29	1	1	NUM
cana-3950	174	30	𝑚	𝑚	X
cana-3950	174	31	(	(	PUNCT
cana-3950	174	32	𝐾𝐴	𝐾𝐴	NOUN
cana-3950	174	33	𝑛	𝑛	PROPN
cana-3950	174	34	+	+	X
cana-3950	174	35	(	(	PUNCT
cana-3950	174	36	1	1	NUM
cana-3950	174	37	−	−	PROPN
cana-3950	174	38	𝛼)∏	𝛼)∏	NUM
cana-3950	174	39	𝑚+𝑛	𝑚+𝑛	NUM
cana-3950	174	40	2	2	NUM
cana-3950	174	41	)	)	PUNCT
cana-3950	174	42	(	(	PUNCT
cana-3950	174	43	1	1	NUM
cana-3950	174	44	𝑛	𝑛	NOUN
cana-3950	174	45	)	)	PUNCT
cana-3950	174	46	)	)	PUNCT
cana-3950	174	47	communications	communication	NOUN
cana-3950	174	48	on	on	ADP
cana-3950	174	49	applied	apply	VERB
cana-3950	174	50	nonlinear	nonlinear	ADJ
cana-3950	174	51	analysis	analysis	NOUN
cana-3950	174	52	issn	issn	NOUN
cana-3950	174	53	:	:	PUNCT
cana-3950	174	54	1074	1074	NUM
cana-3950	174	55	-	-	PUNCT
cana-3950	174	56	133x	133x	NUM
cana-3950	174	57	vol	vol	NOUN
cana-3950	174	58	32	32	NUM
cana-3950	174	59	no	no	NOUN
cana-3950	174	60	.	.	PUNCT
cana-3950	175	1	9s	9s	NUM
cana-3950	175	2	(	(	PUNCT
cana-3950	175	3	2025	2025	NUM
cana-3950	175	4	)	)	PUNCT
cana-3950	175	5	395	395	NUM
cana-3950	175	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3950	175	7	=	=	SYM
cana-3950	175	8	(	(	PUNCT
cana-3950	175	9	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	175	10	,	,	PUNCT
cana-3950	175	11	(	(	PUNCT
cana-3950	175	12	𝐾𝐴	𝐾𝐴	NOUN
cana-3950	175	13	𝑛	𝑛	PROPN
cana-3950	175	14	+	+	CCONJ
cana-3950	175	15	∏	∏	PROPN
cana-3950	175	16	𝑚+𝑛	𝑚+𝑛	X
cana-3950	175	17	2	2	NUM
cana-3950	175	18	)	)	PUNCT
cana-3950	175	19	1	1	NUM
cana-3950	175	20	𝑛	𝑛	NOUN
cana-3950	175	21	)	)	PUNCT
cana-3950	175	22	=	=	SYM
cana-3950	175	23	(	(	PUNCT
cana-3950	175	24	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	175	25	,	,	PUNCT
cana-3950	175	26	(	(	PUNCT
cana-3950	175	27	𝐾𝐴	𝐾𝐴	NOUN
cana-3950	175	28	+	+	CCONJ
cana-3950	175	29	1−	1−	NUM
cana-3950	175	30	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	175	31	𝑚	𝑚	ADP
cana-3950	175	32	−	−	PROPN
cana-3950	175	33	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	175	34	𝑛	𝑛	NOUN
cana-3950	175	35	)	)	PUNCT
cana-3950	175	36	1	1	NUM
cana-3950	175	37	𝑛	𝑛	NOUN
cana-3950	175	38	)	)	PUNCT
cana-3950	175	39	=	=	SYM
cana-3950	175	40	(	(	PUNCT
cana-3950	175	41	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	175	42	,	,	PUNCT
cana-3950	175	43	(	(	PUNCT
cana-3950	175	44	1−	1−	NUM
cana-3950	175	45	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	175	46	𝑚	𝑚	NOUN
cana-3950	175	47	)	)	PUNCT
cana-3950	175	48	1	1	NUM
cana-3950	175	49	𝑛	𝑛	NOUN
cana-3950	175	50	)	)	PUNCT
cana-3950	175	51	=	=	SYM
cana-3950	176	1	□	□	PUNCT
cana-3950	176	2	a	a	DET
cana-3950	176	3	the	the	DET
cana-3950	176	4	proof	proof	NOUN
cana-3950	176	5	is	be	AUX
cana-3950	176	6	completed	complete	VERB
cana-3950	176	7	.	.	PUNCT
cana-3950	177	1	it	it	PRON
cana-3950	177	2	follows	follow	VERB
cana-3950	177	3	on	on	ADP
cana-3950	177	4	nothing	nothing	PRON
cana-3950	177	5	that	that	PRON
cana-3950	177	6	,	,	PUNCT
cana-3950	177	7	𝐺1(𝐴	𝐺1(𝐴	NOUN
cana-3950	177	8	)	)	PUNCT
cana-3950	177	9	=	=	SYM
cana-3950	178	1	(	(	PUNCT
cana-3950	178	2	(	(	PUNCT
cana-3950	178	3	𝐽𝐴	𝐽𝐴	NOUN
cana-3950	178	4	𝑚	𝑚	NOUN
cana-3950	178	5	+	+	CCONJ
cana-3950	178	6	(	(	PUNCT
cana-3950	178	7	1)∏	1)∏	NUM
cana-3950	178	8	𝑚+𝑛	𝑚+𝑛	NUM
cana-3950	178	9	2	2	NUM
cana-3950	178	10	)	)	PUNCT
cana-3950	178	11	1	1	NUM
cana-3950	178	12	𝑚	𝑚	NOUN
cana-3950	178	13	,	,	PUNCT
cana-3950	178	14	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	178	15	+	+	CCONJ
cana-3950	178	16	(	(	PUNCT
cana-3950	178	17	1−	1−	NUM
cana-3950	178	18	1)∏	1)∏	NUM
cana-3950	178	19	𝑚+𝑛	𝑚+𝑛	SYM
cana-3950	178	20	2	2	NUM
cana-3950	178	21	)	)	PUNCT
cana-3950	178	22	1	1	NUM
cana-3950	178	23	𝑛	𝑛	NOUN
cana-3950	178	24	)	)	PUNCT
cana-3950	178	25	=	=	SYM
cana-3950	179	1	(	(	PUNCT
cana-3950	179	2	(	(	PUNCT
cana-3950	179	3	𝐽𝐴	𝐽𝐴	NOUN
cana-3950	179	4	𝑚	𝑚	NOUN
cana-3950	179	5	+	+	CCONJ
cana-3950	179	6	∏	∏	X
cana-3950	179	7	𝑚+𝑛	𝑚+𝑛	X
cana-3950	179	8	2	2	NUM
cana-3950	179	9	)	)	PUNCT
cana-3950	179	10	1	1	NUM
cana-3950	179	11	𝑚	𝑚	NOUN
cana-3950	179	12	,	,	PUNCT
cana-3950	179	13	(	(	PUNCT
cana-3950	179	14	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	179	15	𝑛	𝑛	PROPN
cana-3950	179	16	)	)	PUNCT
cana-3950	179	17	1	1	NUM
cana-3950	179	18	𝑛	𝑛	NOUN
cana-3950	179	19	)	)	PUNCT
cana-3950	179	20	=	=	SYM
cana-3950	179	21	(	(	PUNCT
cana-3950	179	22	(	(	PUNCT
cana-3950	179	23	𝐽𝐴	𝐽𝐴	NOUN
cana-3950	179	24	𝑚	𝑚	NOUN
cana-3950	179	25	+	+	CCONJ
cana-3950	179	26	(	(	PUNCT
cana-3950	179	27	1	1	NUM
cana-3950	179	28	−	−	PROPN
cana-3950	179	29	𝐽𝐴	𝐽𝐴	NOUN
cana-3950	179	30	𝑚	𝑚	ADP
cana-3950	179	31	−	−	PROPN
cana-3950	179	32	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	179	33	𝑛	𝑛	NOUN
cana-3950	179	34	)	)	PUNCT
cana-3950	179	35	1	1	NUM
cana-3950	179	36	𝑚	𝑚	NOUN
cana-3950	179	37	,	,	PUNCT
cana-3950	179	38	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	179	39	)	)	PUNCT
cana-3950	179	40	)	)	PUNCT
cana-3950	180	1	=	=	PUNCT
cana-3950	180	2	(	(	PUNCT
cana-3950	180	3	(	(	PUNCT
cana-3950	180	4	1−	1−	NUM
cana-3950	180	5	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	180	6	𝑛	𝑛	NOUN
cana-3950	180	7	)	)	PUNCT
cana-3950	180	8	1	1	NUM
cana-3950	180	9	𝑚	𝑚	NOUN
cana-3950	180	10	,	,	PUNCT
cana-3950	180	11	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	180	12	)	)	PUNCT
cana-3950	180	13	=	=	PUNCT
cana-3950	181	1	◊	◊	VERB
cana-3950	181	2	a	a	DET
cana-3950	181	3	definition	definition	NOUN
cana-3950	181	4	2.20	2.20	NUM
cana-3950	181	5	let	let	VERB
cana-3950	181	6	𝐴	𝐴	PROPN
cana-3950	181	7	=	=	SYM
cana-3950	181	8	(	(	PUNCT
cana-3950	181	9	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	181	10	,	,	PUNCT
cana-3950	181	11	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	181	12	)	)	PUNCT
cana-3950	181	13	∈	∈	PROPN
cana-3950	181	14	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	181	15	𝑛	𝑛	PROPN
cana-3950	181	16	(	(	PUNCT
cana-3950	181	17	𝑥	𝑥	NOUN
cana-3950	181	18	)	)	PUNCT
cana-3950	181	19	and	and	CCONJ
cana-3950	181	20	𝛼	𝛼	X
cana-3950	181	21	,	,	PUNCT
cana-3950	181	22	𝛽	𝛽	PROPN
cana-3950	181	23	∈	∈	NOUN
cana-3950	181	24	[	[	X
cana-3950	181	25	0,1	0,1	NUM
cana-3950	181	26	]	]	PUNCT
cana-3950	181	27	where	where	SCONJ
cana-3950	181	28	𝛼	𝛼	X
cana-3950	181	29	+	+	X
cana-3950	181	30	𝛽	𝛽	NOUN
cana-3950	181	31	≤	≤	NOUN
cana-3950	181	32	1	1	NUM
cana-3950	181	33	.	.	PUNCT
cana-3950	182	1	we	we	PRON
cana-3950	182	2	define	define	VERB
cana-3950	182	3	the	the	DET
cana-3950	182	4	operator	operator	NOUN
cana-3950	182	5	𝐻𝛼,𝛽(𝐴	𝐻𝛼,𝛽(𝐴	NOUN
cana-3950	182	6	)	)	PUNCT
cana-3950	182	7	as	as	ADP
cana-3950	182	8	𝐻𝛼,𝛽(𝐴	𝐻𝛼,𝛽(𝐴	NOUN
cana-3950	182	9	)	)	PUNCT
cana-3950	182	10	=	=	SYM
cana-3950	182	11	(	(	PUNCT
cana-3950	182	12	(	(	PUNCT
cana-3950	182	13	𝐽𝐴	𝐽𝐴	NOUN
cana-3950	182	14	𝑚	𝑚	NOUN
cana-3950	182	15	+	+	CCONJ
cana-3950	182	16	𝛼∏	𝛼∏	PROPN
cana-3950	182	17	𝑚+𝑛	𝑚+𝑛	NUM
cana-3950	182	18	2	2	NUM
cana-3950	182	19	)	)	PUNCT
cana-3950	182	20	1	1	NUM
cana-3950	182	21	𝑚	𝑚	NOUN
cana-3950	182	22	,	,	PUNCT
cana-3950	182	23	(	(	PUNCT
cana-3950	182	24	𝐾𝐴	𝐾𝐴	NOUN
cana-3950	182	25	𝑛	𝑛	PROPN
cana-3950	182	26	+	+	PROPN
cana-3950	182	27	𝛽∏	𝛽∏	PROPN
cana-3950	182	28	𝑚+𝑛	𝑚+𝑛	PROPN
cana-3950	182	29	2	2	X
cana-3950	182	30	)	)	SYM
cana-3950	182	31	1	1	NUM
cana-3950	182	32	𝑛	𝑛	NOUN
cana-3950	182	33	)	)	PUNCT
cana-3950	182	34	theorem	theorem	VERB
cana-3950	182	35	2.21	2.21	NUM
cana-3950	182	36	for	for	ADP
cana-3950	182	37	any	any	DET
cana-3950	182	38	𝐴	𝐴	PROPN
cana-3950	182	39	=	=	SYM
cana-3950	182	40	(	(	PUNCT
cana-3950	182	41	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	182	42	,	,	PUNCT
cana-3950	182	43	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	182	44	)	)	PUNCT
cana-3950	182	45	∈	∈	PROPN
cana-3950	183	1	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	183	2	𝑛	𝑛	PROPN
cana-3950	183	3	(	(	PUNCT
cana-3950	183	4	𝑥	𝑥	NOUN
cana-3950	183	5	)	)	PUNCT
cana-3950	183	6	and	and	CCONJ
cana-3950	183	7	𝛼	𝛼	X
cana-3950	183	8	,	,	PUNCT
cana-3950	183	9	𝛽	𝛽	PROPN
cana-3950	183	10	∈	∈	NOUN
cana-3950	184	1	[	[	X
cana-3950	184	2	0,1	0,1	NUM
cana-3950	184	3	]	]	PUNCT
cana-3950	184	4	where	where	SCONJ
cana-3950	184	5	𝛼	𝛼	X
cana-3950	184	6	+	+	X
cana-3950	184	7	𝛽	𝛽	NOUN
cana-3950	184	8	≤	≤	NOUN
cana-3950	184	9	1	1	NUM
cana-3950	184	10	.	.	PUNCT
cana-3950	185	1	we	we	PRON
cana-3950	185	2	have	have	VERB
cana-3950	185	3	(	(	PUNCT
cana-3950	185	4	𝑖)𝐻𝛼,𝛽(𝐴	𝑖)𝐻𝛼,𝛽(𝐴	ADJ
cana-3950	185	5	)	)	PUNCT
cana-3950	185	6	∈	∈	PROPN
cana-3950	186	1	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	186	2	𝑛	𝑛	PROPN
cana-3950	186	3	(	(	PUNCT
cana-3950	186	4	𝑥	𝑥	NOUN
cana-3950	186	5	)	)	PUNCT
cana-3950	186	6	(	(	PUNCT
cana-3950	186	7	𝑖𝑖	𝑖𝑖	NOUN
cana-3950	186	8	)	)	PUNCT
cana-3950	186	9	0	0	NUM
cana-3950	186	10	≤	≤	NUM
cana-3950	186	11	𝑡	𝑡	PROPN
cana-3950	186	12	≤	≤	NUM
cana-3950	186	13	𝛼	𝛼	NUM
cana-3950	186	14	⟹	⟹	NUM
cana-3950	186	15	𝐻𝛼,𝛽(𝐴	𝐻𝛼,𝛽(𝐴	NOUN
cana-3950	186	16	)	)	PUNCT
cana-3950	187	1	⊂	⊂	NOUN
cana-3950	187	2	𝐻𝛼,𝛽(𝐴	𝐻𝛼,𝛽(𝐴	NOUN
cana-3950	187	3	)	)	PUNCT
cana-3950	187	4	(	(	PUNCT
cana-3950	187	5	𝑖𝑖𝑖	𝑖𝑖𝑖	NOUN
cana-3950	187	6	)	)	PUNCT
cana-3950	187	7	0	0	NUM
cana-3950	187	8	≤	≤	NUM
cana-3950	187	9	𝑡	𝑡	PROPN
cana-3950	187	10	≤	≤	PROPN
cana-3950	187	11	𝛽	𝛽	NOUN
cana-3950	187	12	⟹	⟹	NUM
cana-3950	187	13	𝐻𝛼,𝛽(𝐴	𝐻𝛼,𝛽(𝐴	NOUN
cana-3950	187	14	)	)	PUNCT
cana-3950	187	15	⊂	⊂	X
cana-3950	187	16	𝐻𝛼,𝑡(𝐴	𝐻𝛼,𝑡(𝐴	NUM
cana-3950	187	17	)	)	PUNCT
cana-3950	187	18	(	(	PUNCT
cana-3950	187	19	𝑖𝑣)𝐺𝛼(𝐴	𝑖𝑣)𝐺𝛼(𝐴	ADV
cana-3950	187	20	)	)	PUNCT
cana-3950	187	21	=	=	SYM
cana-3950	188	1	𝐻𝛼	𝐻𝛼	NOUN
cana-3950	188	2	,	,	PUNCT
cana-3950	188	3	1−𝛼	1−𝛼	NUM
cana-3950	188	4	(	(	PUNCT
cana-3950	188	5	𝐴	𝐴	PROPN
cana-3950	188	6	)	)	PUNCT
cana-3950	188	7	(	(	PUNCT
cana-3950	188	8	𝑣)	𝑣)	NOUN
cana-3950	188	9	□	□	NUM
cana-3950	188	10	a	a	DET
cana-3950	188	11	=	=	SYM
cana-3950	188	12	𝐻0,1	𝐻0,1	PROPN
cana-3950	188	13	(	(	PUNCT
cana-3950	188	14	𝐴	𝐴	PROPN
cana-3950	188	15	)	)	PUNCT
cana-3950	188	16	(	(	PUNCT
cana-3950	188	17	𝑣𝑖	𝑣𝑖	NOUN
cana-3950	188	18	)	)	PUNCT
cana-3950	188	19	◊	◊	NOUN
cana-3950	188	20	a	a	NOUN
cana-3950	188	21	=	=	SYM
cana-3950	188	22	h1,0	h1,0	NOUN
cana-3950	188	23	(	(	PUNCT
cana-3950	188	24	a	a	NOUN
cana-3950	188	25	)	)	PUNCT
cana-3950	188	26	(	(	PUNCT
cana-3950	188	27	𝑣𝑖𝑖)𝐻𝛼,3	𝑣𝑖𝑖)𝐻𝛼,3	PROPN
cana-3950	188	28	𝑐	𝑐	PROPN
cana-3950	188	29	(	(	PUNCT
cana-3950	188	30	𝑀𝑐	𝑀𝑐	PROPN
cana-3950	188	31	)	)	PUNCT
cana-3950	188	32	=	=	SYM
cana-3950	188	33	𝐻3,𝛼(𝐴	𝐻3,𝛼(𝐴	NOUN
cana-3950	188	34	)	)	PUNCT
cana-3950	188	35	proof	proof	NOUN
cana-3950	188	36	:	:	PUNCT
cana-3950	188	37	(	(	PUNCT
cana-3950	188	38	i	i	NOUN
cana-3950	188	39	)	)	PUNCT
cana-3950	188	40	follows	follow	VERB
cana-3950	188	41	since	since	ADV
cana-3950	188	42	,	,	PUNCT
cana-3950	188	43	𝐽1+𝛼,3	𝐽1+𝛼,3	PROPN
cana-3950	188	44	𝑚	𝑚	PROPN
cana-3950	188	45	(	(	PUNCT
cana-3950	188	46	𝐴	𝐴	PROPN
cana-3950	188	47	)	)	PUNCT
cana-3950	188	48	+	+	CCONJ
cana-3950	188	49	𝐾1+𝛼,3(𝐴	𝐾1+𝛼,3(𝐴	NOUN
cana-3950	188	50	)	)	PUNCT
cana-3950	188	51	=	=	SYM
cana-3950	188	52	(	(	PUNCT
cana-3950	188	53	(	(	PUNCT
cana-3950	188	54	𝐽𝐴	𝐽𝐴	NOUN
cana-3950	188	55	𝑚	𝑚	NOUN
cana-3950	188	56	+	+	CCONJ
cana-3950	188	57	𝛼∏	𝛼∏	PROPN
cana-3950	188	58	𝑚+𝑛	𝑚+𝑛	NUM
cana-3950	188	59	2	2	NUM
cana-3950	188	60	)	)	PUNCT
cana-3950	188	61	1	1	NUM
cana-3950	188	62	𝑚	𝑚	NOUN
cana-3950	188	63	)	)	PUNCT
cana-3950	188	64	𝑚	𝑚	PROPN
cana-3950	189	1	+	+	CCONJ
cana-3950	189	2	(	(	PUNCT
cana-3950	189	3	(	(	PUNCT
cana-3950	189	4	𝐾𝐴	𝐾𝐴	NOUN
cana-3950	189	5	𝑛	𝑛	PROPN
cana-3950	189	6	+	+	PROPN
cana-3950	189	7	𝛽∏	𝛽∏	PROPN
cana-3950	189	8	𝑚+𝑛	𝑚+𝑛	PROPN
cana-3950	189	9	2	2	X
cana-3950	189	10	)	)	PUNCT
cana-3950	189	11	1	1	NUM
cana-3950	189	12	𝑛	𝑛	NOUN
cana-3950	189	13	)	)	PUNCT
cana-3950	189	14	𝑛	𝑛	PRON
cana-3950	189	15	communications	communication	NOUN
cana-3950	189	16	on	on	ADP
cana-3950	189	17	applied	apply	VERB
cana-3950	189	18	nonlinear	nonlinear	ADJ
cana-3950	189	19	analysis	analysis	NOUN
cana-3950	189	20	issn	issn	NOUN
cana-3950	189	21	:	:	PUNCT
cana-3950	189	22	1074	1074	NUM
cana-3950	189	23	-	-	PUNCT
cana-3950	189	24	133x	133x	NUM
cana-3950	189	25	vol	vol	NOUN
cana-3950	189	26	32	32	NUM
cana-3950	189	27	no	no	NOUN
cana-3950	189	28	.	.	PUNCT
cana-3950	190	1	9s	9s	NUM
cana-3950	190	2	(	(	PUNCT
cana-3950	190	3	2025	2025	NUM
cana-3950	190	4	)	)	PUNCT
cana-3950	190	5	396	396	NUM
cana-3950	190	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3950	190	7	=	=	PUNCT
cana-3950	190	8	𝐺𝛼(𝐴	𝐺𝛼(𝐴	PROPN
cana-3950	190	9	)	)	PUNCT
cana-3950	190	10	(	(	PUNCT
cana-3950	190	11	v	v	NOUN
cana-3950	190	12	)	)	PUNCT
cana-3950	190	13	follows	follow	VERB
cana-3950	190	14	since	since	SCONJ
cana-3950	190	15	𝐻𝛼,1−𝛼(𝑥	𝐻𝛼,1−𝛼(𝑥	NOUN
cana-3950	190	16	)	)	PUNCT
cana-3950	190	17	=	=	PUNCT
cana-3950	190	18	𝐺𝛼(𝐴	𝐺𝛼(𝐴	X
cana-3950	190	19	)	)	PUNCT
cana-3950	190	20	⟹	⟹	PUNCT
cana-3950	191	1	𝐻0,1(𝐴	𝐻0,1(𝐴	NOUN
cana-3950	191	2	)	)	PUNCT
cana-3950	191	3	=	=	SYM
cana-3950	191	4	𝐺0(𝐴	𝐺0(𝐴	NOUN
cana-3950	191	5	)	)	PUNCT
cana-3950	191	6	.	.	PUNCT
cana-3950	192	1	⟹𝐻0,1(𝐴	⟹𝐻0,1(𝐴	X
cana-3950	192	2	)	)	PUNCT
cana-3950	193	1	=	=	PUNCT
cana-3950	193	2	□	□	PUNCT
cana-3950	193	3	a	a	DET
cana-3950	193	4	(	(	PUNCT
cana-3950	193	5	vi	vi	NOUN
cana-3950	193	6	)	)	PUNCT
cana-3950	193	7	follows	follow	VERB
cana-3950	193	8	on	on	ADP
cana-3950	193	9	nothing	nothing	PRON
cana-3950	193	10	that	that	PRON
cana-3950	193	11	,	,	PUNCT
cana-3950	193	12	𝐻𝛼,1−2(𝐴	𝐻𝛼,1−2(𝐴	NOUN
cana-3950	193	13	)	)	PUNCT
cana-3950	193	14	=	=	SYM
cana-3950	193	15	𝐺𝛼(𝐴	𝐺𝛼(𝐴	X
cana-3950	193	16	)	)	PUNCT
cana-3950	193	17	⟹	⟹	PUNCT
cana-3950	194	1	𝐻1,0	𝐻1,0	NOUN
cana-3950	194	2	(	(	PUNCT
cana-3950	194	3	𝐴	𝐴	PROPN
cana-3950	194	4	)	)	PUNCT
cana-3950	194	5	=	=	SYM
cana-3950	194	6	𝐺1	𝐺1	NOUN
cana-3950	194	7	(	(	PUNCT
cana-3950	194	8	𝐴	𝐴	PROPN
cana-3950	194	9	)	)	PUNCT
cana-3950	194	10	⟹𝐻1,0(𝐴	⟹𝐻1,0(𝐴	NUM
cana-3950	194	11	)	)	PUNCT
cana-3950	194	12	=	=	PUNCT
cana-3950	195	1	◊	◊	VERB
cana-3950	195	2	a	a	DET
cana-3950	195	3	(	(	PUNCT
cana-3950	195	4	vii)since	vii)since	PROPN
cana-3950	195	5	𝑀𝑐	𝑀𝑐	PROPN
cana-3950	195	6	=	=	SYM
cana-3950	195	7	(	(	PUNCT
cana-3950	195	8	𝐾𝐴	𝐾𝐴	NOUN
cana-3950	195	9	𝑛	𝑛	ADP
cana-3950	195	10	𝑚	𝑚	NOUN
cana-3950	195	11	,	,	PUNCT
cana-3950	195	12	𝐽𝑛	𝐽𝑛	PROPN
cana-3950	195	13	𝑚	𝑚	PROPN
cana-3950	195	14	𝑛	𝑛	NOUN
cana-3950	195	15	)	)	PUNCT
cana-3950	195	16	we	we	PRON
cana-3950	195	17	have	have	VERB
cana-3950	195	18	𝐻𝛼,𝛽(𝑀	𝐻𝛼,𝛽(𝑀	NUM
cana-3950	195	19	𝑐	𝑐	NOUN
cana-3950	195	20	)	)	PUNCT
cana-3950	195	21	=	=	SYM
cana-3950	196	1	(	(	PUNCT
cana-3950	196	2	(	(	PUNCT
cana-3950	196	3	𝐾𝐴	𝐾𝐴	NOUN
cana-3950	196	4	𝑛	𝑛	NOUN
cana-3950	196	5	𝑚	𝑚	NOUN
cana-3950	196	6	)	)	PUNCT
cana-3950	196	7	𝑚	𝑚	PROPN
cana-3950	196	8	+	+	CCONJ
cana-3950	197	1	𝛼∏	𝛼∏	PROPN
cana-3950	197	2	𝑚+𝑛	𝑚+𝑛	NUM
cana-3950	197	3	2	2	NUM
cana-3950	197	4	)	)	PUNCT
cana-3950	197	5	1	1	NUM
cana-3950	197	6	𝑚	𝑚	NOUN
cana-3950	197	7	,	,	PUNCT
cana-3950	197	8	(	(	PUNCT
cana-3950	197	9	(	(	PUNCT
cana-3950	197	10	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	197	11	𝑚	𝑚	ADP
cana-3950	197	12	𝑛	𝑛	NOUN
cana-3950	197	13	)	)	PUNCT
cana-3950	197	14	𝑛	𝑛	PRON
cana-3950	198	1	+	+	NUM
cana-3950	198	2	𝛽∏	𝛽∏	PROPN
cana-3950	198	3	𝑚+𝑛	𝑚+𝑛	PROPN
cana-3950	198	4	2	2	X
cana-3950	198	5	)	)	PUNCT
cana-3950	198	6	𝟏	𝟏	NUM
cana-3950	199	1	𝒏	𝒏	PROPN
cana-3950	199	2	=	=	PUNCT
cana-3950	199	3	(	(	PUNCT
cana-3950	199	4	(	(	PUNCT
cana-3950	199	5	𝐾𝐴	𝐾𝐴	NOUN
cana-3950	199	6	𝑛	𝑛	PROPN
cana-3950	199	7	+	+	CCONJ
cana-3950	199	8	𝛼∏	𝛼∏	PROPN
cana-3950	199	9	𝑚+𝑛	𝑚+𝑛	NUM
cana-3950	199	10	2	2	NUM
cana-3950	199	11	)	)	PUNCT
cana-3950	199	12	1	1	NUM
cana-3950	199	13	𝑚	𝑚	X
cana-3950	199	14	(	(	PUNCT
cana-3950	199	15	𝐽𝐴	𝐽𝐴	NOUN
cana-3950	199	16	𝑚	𝑚	PROPN
cana-3950	199	17	+	+	PROPN
cana-3950	199	18	𝛽∏	𝛽∏	PROPN
cana-3950	199	19	𝑚+𝑛	𝑚+𝑛	PROPN
cana-3950	199	20	2	2	X
cana-3950	199	21	)	)	PUNCT
cana-3950	199	22	1	1	NUM
cana-3950	199	23	𝑛	𝑛	NOUN
cana-3950	199	24	)	)	PUNCT
cana-3950	199	25	𝑑𝐵𝑚𝑛	𝑑𝐵𝑚𝑛	PROPN
cana-3950	199	26	𝐻	𝐻	PROPN
cana-3950	199	27	(	(	PUNCT
cana-3950	199	28	𝐴1	𝐴1	PROPN
cana-3950	199	29	,	,	PUNCT
cana-3950	199	30	𝐴2	𝐴2	PROPN
cana-3950	199	31	)	)	PUNCT
cana-3950	199	32	=	=	SYM
cana-3950	199	33	1	1	NUM
cana-3950	199	34	2	2	NUM
cana-3950	199	35	∑{|𝐽𝐴1	∑{|𝐽𝐴1	NOUN
cana-3950	199	36	(	(	PUNCT
cana-3950	199	37	𝑥𝑖	𝑥𝑖	NOUN
cana-3950	199	38	)	)	PUNCT
cana-3950	199	39	−	−	PROPN
cana-3950	200	1	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	200	2	(	(	PUNCT
cana-3950	200	3	𝑥𝑖)|	𝑥𝑖)|	PROPN
cana-3950	200	4	𝑛	𝑛	PRON
cana-3950	200	5	𝑖=1	𝑖=1	PROPN
cana-3950	200	6	+	+	NUM
cana-3950	200	7	|𝐾𝐴1	|𝐾𝐴1	X
cana-3950	200	8	(	(	PUNCT
cana-3950	200	9	𝑥𝑖	𝑥𝑖	NOUN
cana-3950	200	10	)	)	PUNCT
cana-3950	200	11	−	−	PROPN
cana-3950	200	12	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	200	13	(	(	PUNCT
cana-3950	200	14	𝑥𝑖)|	𝑥𝑖)|	PROPN
cana-3950	200	15	+	+	NUM
cana-3950	200	16	|∏	|∏	PROPN
cana-3950	200	17	(	(	PUNCT
cana-3950	200	18	𝑥𝑖	𝑥𝑖	PROPN
cana-3950	200	19	)	)	PUNCT
cana-3950	200	20	−	−	PROPN
cana-3950	200	21	∏	∏	PROPN
cana-3950	200	22	(	(	PUNCT
cana-3950	200	23	𝑥𝑖)|	𝑥𝑖)|	PROPN
cana-3950	200	24	}	}	PUNCT
cana-3950	200	25	𝐴2𝐴1	𝐴2𝐴1	NOUN
cana-3950	200	26	the	the	DET
cana-3950	200	27	normalize	normalize	NOUN
cana-3950	200	28	hamming	hamming	NOUN
cana-3950	200	29	distance	distance	NOUN
cana-3950	200	30	is	be	AUX
cana-3950	200	31	explained	explain	VERB
cana-3950	200	32	as	as	ADP
cana-3950	200	33	𝑑𝐵𝑚𝑛	𝑑𝐵𝑚𝑛	PROPN
cana-3950	200	34	𝐻	𝐻	PROPN
cana-3950	200	35	(	(	PUNCT
cana-3950	200	36	𝐴1	𝐴1	PROPN
cana-3950	200	37	,	,	PUNCT
cana-3950	200	38	𝐴2	𝐴2	PROPN
cana-3950	200	39	)	)	PUNCT
cana-3950	200	40	=	=	PUNCT
cana-3950	200	41	1	1	NUM
cana-3950	200	42	2𝑟	2𝑟	NUM
cana-3950	200	43	∑	∑	PUNCT
cana-3950	200	44	{	{	PUNCT
cana-3950	200	45	|𝐽𝐴1(𝑥𝑖	|𝐽𝐴1(𝑥𝑖	PROPN
cana-3950	200	46	)	)	PUNCT
cana-3950	200	47	−	−	ADP
cana-3950	200	48	𝐽𝐴2(𝑥𝑖)|	𝐽𝐴2(𝑥𝑖)|	NOUN
cana-3950	201	1	𝑛	𝑛	PRON
cana-3950	201	2	𝑖=1	𝑖=1	PROPN
cana-3950	201	3	+	+	CCONJ
cana-3950	201	4	|𝐾𝐴1(𝑥𝑖	|𝐾𝐴1(𝑥𝑖	PROPN
cana-3950	201	5	)	)	PUNCT
cana-3950	201	6	−	−	ADP
cana-3950	201	7	𝐾𝐴2(𝑥𝑖)|	𝐾𝐴2(𝑥𝑖)|	PUNCT
cana-3950	202	1	+	+	NUM
cana-3950	202	2	|∏	|∏	PROPN
cana-3950	202	3	(	(	PUNCT
cana-3950	202	4	𝑥𝑖	𝑥𝑖	PROPN
cana-3950	202	5	)	)	PUNCT
cana-3950	202	6	−	−	PROPN
cana-3950	202	7	∏	∏	PROPN
cana-3950	202	8	(	(	PUNCT
cana-3950	202	9	𝑥𝑖𝐴2𝐴1	𝑥𝑖𝐴2𝐴1	PROPN
cana-3950	202	10	)	)	PUNCT
cana-3950	202	11	|	|	ADV
cana-3950	202	12	the	the	DET
cana-3950	202	13	euclidean	euclidean	ADJ
cana-3950	202	14	distance	distance	NOUN
cana-3950	202	15	is	be	AUX
cana-3950	202	16	defined	define	VERB
cana-3950	202	17	as	as	ADP
cana-3950	202	18	𝑑𝐵𝑚𝑛	𝑑𝐵𝑚𝑛	PROPN
cana-3950	202	19	𝐻	𝐻	PROPN
cana-3950	202	20	(	(	PUNCT
cana-3950	202	21	𝐴1	𝐴1	PROPN
cana-3950	202	22	,	,	PUNCT
cana-3950	202	23	𝐴2	𝐴2	PROPN
cana-3950	202	24	)	)	PUNCT
cana-3950	202	25	=	=	SYM
cana-3950	203	1	√	√	NUM
cana-3950	203	2	1	1	NUM
cana-3950	203	3	2	2	NUM
cana-3950	203	4	∑	∑	PUNCT
cana-3950	203	5	{	{	PUNCT
cana-3950	203	6	(	(	PUNCT
cana-3950	203	7	𝐽𝐴1(𝑥𝑖	𝐽𝐴1(𝑥𝑖	PROPN
cana-3950	203	8	)	)	PUNCT
cana-3950	203	9	−	−	PROPN
cana-3950	203	10	𝐽𝐴2(𝑥2	𝐽𝐴2(𝑥2	NUM
cana-3950	203	11	)	)	PUNCT
cana-3950	203	12	)	)	PUNCT
cana-3950	203	13	2	2	NUM
cana-3950	204	1	+	+	CCONJ
cana-3950	204	2	(	(	PUNCT
cana-3950	204	3	𝐾𝐴1(𝑥𝑖	𝐾𝐴1(𝑥𝑖	NUM
cana-3950	204	4	)	)	PUNCT
cana-3950	204	5	−	−	PROPN
cana-3950	204	6	𝐾𝐴2(𝑥𝑖	𝐾𝐴2(𝑥𝑖	NUM
cana-3950	204	7	)	)	PUNCT
cana-3950	204	8	)	)	PUNCT
cana-3950	204	9	2	2	NUM
cana-3950	205	1	+	+	CCONJ
cana-3950	205	2	(	(	PUNCT
cana-3950	205	3	∏	∏	PROPN
cana-3950	205	4	(	(	PUNCT
cana-3950	205	5	𝑥𝑖	𝑥𝑖	NOUN
cana-3950	205	6	)	)	PUNCT
cana-3950	205	7	−	−	PROPN
cana-3950	205	8	∏	∏	PROPN
cana-3950	205	9	(	(	PUNCT
cana-3950	205	10	𝑥𝑖	𝑥𝑖	PROPN
cana-3950	205	11	)	)	PUNCT
cana-3950	205	12	)	)	PUNCT
cana-3950	206	1	2𝐴2𝐴1	2𝐴2𝐴1	NUM
cana-3950	206	2	𝑟	𝑟	NOUN
cana-3950	206	3	𝑖=0	𝑖=0	SYM
cana-3950	206	4	}	}	PUNCT
cana-3950	206	5	the	the	DET
cana-3950	206	6	normalized	normalize	VERB
cana-3950	206	7	euclidean	euclidean	ADJ
cana-3950	206	8	distance	distance	NOUN
cana-3950	206	9	is	be	AUX
cana-3950	206	10	defined	define	VERB
cana-3950	206	11	as	as	ADP
cana-3950	206	12	𝑑𝐵𝑚𝑛	𝑑𝐵𝑚𝑛	PROPN
cana-3950	206	13	𝐻	𝐻	PROPN
cana-3950	206	14	(	(	PUNCT
cana-3950	206	15	𝐴1	𝐴1	PROPN
cana-3950	206	16	,	,	PUNCT
cana-3950	206	17	𝐴2	𝐴2	PROPN
cana-3950	206	18	)	)	PUNCT
cana-3950	207	1	=	=	SYM
cana-3950	208	1	√	√	NUM
cana-3950	208	2	1	1	NUM
cana-3950	208	3	2𝑟	2𝑟	NUM
cana-3950	208	4	∑	∑	PUNCT
cana-3950	208	5	{	{	PUNCT
cana-3950	208	6	(	(	PUNCT
cana-3950	208	7	𝐽𝐴1(𝑥𝑖	𝐽𝐴1(𝑥𝑖	PROPN
cana-3950	208	8	)	)	PUNCT
cana-3950	208	9	−	−	PROPN
cana-3950	208	10	𝐽𝐴2(𝑥𝑖	𝐽𝐴2(𝑥𝑖	NUM
cana-3950	208	11	)	)	PUNCT
cana-3950	208	12	)	)	PUNCT
cana-3950	208	13	2	2	NUM
cana-3950	209	1	+	+	CCONJ
cana-3950	209	2	(	(	PUNCT
cana-3950	209	3	𝐾𝐴1(𝑥𝑖	𝐾𝐴1(𝑥𝑖	NUM
cana-3950	209	4	)	)	PUNCT
cana-3950	209	5	−	−	PROPN
cana-3950	209	6	𝐾𝐴2(𝑥𝑖	𝐾𝐴2(𝑥𝑖	NUM
cana-3950	209	7	)	)	PUNCT
cana-3950	209	8	)	)	PUNCT
cana-3950	209	9	2	2	NUM
cana-3950	210	1	+	+	CCONJ
cana-3950	210	2	(	(	PUNCT
cana-3950	210	3	∏	∏	PROPN
cana-3950	210	4	(	(	PUNCT
cana-3950	210	5	𝑥𝑖	𝑥𝑖	NOUN
cana-3950	210	6	)	)	PUNCT
cana-3950	210	7	−	−	PROPN
cana-3950	210	8	∏	∏	PROPN
cana-3950	210	9	(	(	PUNCT
cana-3950	210	10	𝑥𝑖	𝑥𝑖	PROPN
cana-3950	210	11	)	)	PUNCT
cana-3950	210	12	)	)	PUNCT
cana-3950	210	13	2	2	NUM
cana-3950	210	14	𝐴2𝐴1	𝐴2𝐴1	NOUN
cana-3950	210	15	𝑟	𝑟	NOUN
cana-3950	210	16	𝑖=0	𝑖=0	PUNCT
cana-3950	210	17	}	}	PUNCT
cana-3950	210	18	and	and	CCONJ
cana-3950	210	19	so	so	ADV
cana-3950	210	20	,	,	PUNCT
cana-3950	210	21	(	(	PUNCT
cana-3950	210	22	𝐻𝛼,3(𝑀	𝐻𝛼,3(𝑀	NOUN
cana-3950	210	23	𝑐))𝑐	𝑐))𝑐	NOUN
cana-3950	210	24	=	=	SYM
cana-3950	210	25	(	(	PUNCT
cana-3950	210	26	(	(	PUNCT
cana-3950	210	27	(	(	PUNCT
cana-3950	210	28	𝐽𝐴	𝐽𝐴	NOUN
cana-3950	210	29	𝑚	𝑚	PROPN
cana-3950	210	30	+	+	PROPN
cana-3950	210	31	𝛽∏	𝛽∏	PROPN
cana-3950	210	32	𝑚+𝑛	𝑚+𝑛	PROPN
cana-3950	210	33	2	2	X
cana-3950	210	34	)	)	PUNCT
cana-3950	210	35	1	1	NUM
cana-3950	210	36	𝑛	𝑛	NOUN
cana-3950	210	37	)	)	PUNCT
cana-3950	210	38	𝑚	𝑚	X
cana-3950	210	39	𝑛	𝑛	PROPN
cana-3950	210	40	+	+	PUNCT
cana-3950	210	41	(	(	PUNCT
cana-3950	210	42	(	(	PUNCT
cana-3950	210	43	𝐾𝐴	𝐾𝐴	NOUN
cana-3950	210	44	𝑛	𝑛	PROPN
cana-3950	210	45	+	+	CCONJ
cana-3950	210	46	𝛼∏	𝛼∏	PROPN
cana-3950	210	47	𝑚+𝑛	𝑚+𝑛	NUM
cana-3950	210	48	2	2	NUM
cana-3950	210	49	)	)	PUNCT
cana-3950	210	50	1	1	NUM
cana-3950	210	51	𝑚	𝑚	NOUN
cana-3950	210	52	)	)	PUNCT
cana-3950	210	53	𝑚	𝑚	PROPN
cana-3950	210	54	𝑛	𝑛	PROPN
cana-3950	210	55	)	)	PUNCT
cana-3950	210	56	=	=	SYM
cana-3950	210	57	(	(	PUNCT
cana-3950	210	58	(	(	PUNCT
cana-3950	210	59	𝐽𝑎	𝐽𝑎	NOUN
cana-3950	210	60	𝑚	𝑚	PROPN
cana-3950	210	61	+	+	PROPN
cana-3950	210	62	𝛽∏	𝛽∏	PROPN
cana-3950	210	63	𝑚+𝑛	𝑚+𝑛	PROPN
cana-3950	210	64	2	2	X
cana-3950	210	65	)	)	PUNCT
cana-3950	210	66	1	1	NUM
cana-3950	210	67	𝑚	𝑚	NOUN
cana-3950	210	68	,	,	PUNCT
cana-3950	210	69	(	(	PUNCT
cana-3950	210	70	𝐾𝐴	𝐾𝐴	NOUN
cana-3950	210	71	𝑛	𝑛	PROPN
cana-3950	210	72	+	+	CCONJ
cana-3950	210	73	𝛼∏	𝛼∏	PROPN
cana-3950	210	74	𝑚+𝑛	𝑚+𝑛	NUM
cana-3950	210	75	2	2	NUM
cana-3950	210	76	)	)	PUNCT
cana-3950	210	77	1	1	NUM
cana-3950	210	78	𝑛	𝑛	NOUN
cana-3950	210	79	)	)	PUNCT
cana-3950	210	80	=	=	SYM
cana-3950	211	1	𝐻𝛽,𝛼	𝐻𝛽,𝛼	PROPN
cana-3950	211	2	(	(	PUNCT
cana-3950	211	3	𝐴	𝐴	PROPN
cana-3950	211	4	)	)	PUNCT
cana-3950	211	5	.	.	PUNCT
cana-3950	212	1	communications	communication	NOUN
cana-3950	212	2	on	on	ADP
cana-3950	212	3	applied	apply	VERB
cana-3950	212	4	nonlinear	nonlinear	ADJ
cana-3950	212	5	analysis	analysis	NOUN
cana-3950	212	6	issn	issn	NOUN
cana-3950	212	7	:	:	PUNCT
cana-3950	212	8	1074	1074	NUM
cana-3950	212	9	-	-	PUNCT
cana-3950	212	10	133x	133x	NUM
cana-3950	212	11	vol	vol	NOUN
cana-3950	212	12	32	32	NUM
cana-3950	212	13	no	no	NOUN
cana-3950	212	14	.	.	PUNCT
cana-3950	213	1	9s	9s	NUM
cana-3950	213	2	(	(	PUNCT
cana-3950	213	3	2025	2025	NUM
cana-3950	213	4	)	)	PUNCT
cana-3950	213	5	397	397	NUM
cana-3950	213	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3950	213	7	hence	hence	ADV
cana-3950	213	8	the	the	DET
cana-3950	213	9	proof	proof	NOUN
cana-3950	213	10	.	.	PUNCT
cana-3950	214	1	3	3	X
cana-3950	214	2	.	.	X
cana-3950	214	3	various	various	ADJ
cana-3950	214	4	types	type	NOUN
cana-3950	214	5	of	of	ADP
cana-3950	214	6	distance	distance	NOUN
cana-3950	214	7	function	function	NOUN
cana-3950	214	8	over	over	ADP
cana-3950	214	9	beal	beal	NOUN
cana-3950	214	10	’s	’s	PART
cana-3950	214	11	fuzzy	fuzzy	ADJ
cana-3950	214	12	set	set	VERB
cana-3950	214	13	definition	definition	NOUN
cana-3950	214	14	3.1	3.1	NUM
cana-3950	214	15	let	let	VERB
cana-3950	214	16	𝑋	𝑋	PROPN
cana-3950	214	17	=	=	SYM
cana-3950	214	18	{	{	PUNCT
cana-3950	214	19	𝑥1	𝑥1	NOUN
cana-3950	214	20	,	,	PUNCT
cana-3950	214	21	𝑥2	𝑥2	NOUN
cana-3950	214	22	,	,	PUNCT
cana-3950	214	23	𝑥3	𝑥3	NOUN
cana-3950	214	24	…	…	PUNCT
cana-3950	214	25	,	,	PUNCT
cana-3950	214	26	𝑥𝑟	𝑥𝑟	AUX
cana-3950	214	27	}	}	PUNCT
cana-3950	214	28	be	be	AUX
cana-3950	214	29	a	a	DET
cana-3950	214	30	whole	whole	NOUN
cana-3950	214	31	of	of	ADP
cana-3950	214	32	discourse	discourse	NOUN
cana-3950	214	33	and	and	CCONJ
cana-3950	214	34	𝐴1	𝐴1	PROPN
cana-3950	214	35	=	=	SYM
cana-3950	214	36	(	(	PUNCT
cana-3950	214	37	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	214	38	,	,	PUNCT
cana-3950	214	39	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	214	40	)	)	PUNCT
cana-3950	214	41	,	,	PUNCT
cana-3950	214	42	𝐴2	𝐴2	PROPN
cana-3950	214	43	=	=	SYM
cana-3950	214	44	(	(	PUNCT
cana-3950	214	45	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	214	46	,	,	PUNCT
cana-3950	214	47	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	214	48	)	)	PUNCT
cana-3950	214	49	,	,	PUNCT
cana-3950	214	50	𝐴3	𝐴3	PROPN
cana-3950	214	51	=	=	SYM
cana-3950	214	52	(	(	PUNCT
cana-3950	214	53	𝐽𝐴3	𝐽𝐴3	PROPN
cana-3950	214	54	,	,	PUNCT
cana-3950	214	55	𝐾𝐴3	𝐾𝐴3	PROPN
cana-3950	214	56	)	)	PUNCT
cana-3950	214	57	∈	∈	PROPN
cana-3950	214	58	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	214	59	𝑛	𝑛	PROPN
cana-3950	214	60	(	(	PUNCT
cana-3950	214	61	𝑋	𝑋	PROPN
cana-3950	214	62	)	)	PUNCT
cana-3950	214	63	,	,	PUNCT
cana-3950	214	64	the	the	DET
cana-3950	214	65	distance	distance	NOUN
cana-3950	214	66	function	function	NOUN
cana-3950	214	67	𝑑	𝑑	NOUN
cana-3950	214	68	:	:	PUNCT
cana-3950	214	69	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	214	70	𝑛	𝑛	PROPN
cana-3950	214	71	(	(	PUNCT
cana-3950	214	72	𝑋	𝑋	NOUN
cana-3950	214	73	)	)	PUNCT
cana-3950	214	74	×	×	NOUN
cana-3950	214	75	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	214	76	𝑛	𝑛	PROPN
cana-3950	214	77	(	(	PUNCT
cana-3950	214	78	𝑋	𝑋	NOUN
cana-3950	214	79	)	)	PUNCT
cana-3950	214	80	→	→	PUNCT
cana-3950	215	1	[	[	X
cana-3950	215	2	0,1	0,1	NUM
cana-3950	215	3	]	]	PUNCT
cana-3950	215	4	is	be	AUX
cana-3950	215	5	defined	define	VERB
cana-3950	215	6	as	as	ADP
cana-3950	215	7	(	(	PUNCT
cana-3950	215	8	𝑖)0	𝑖)0	X
cana-3950	215	9	≤	≤	PROPN
cana-3950	215	10	𝑑(𝐴1	𝑑(𝐴1	PROPN
cana-3950	215	11	,	,	PUNCT
cana-3950	215	12	𝐴2	𝐴2	PROPN
cana-3950	215	13	)	)	PUNCT
cana-3950	215	14	≤	≤	NOUN
cana-3950	215	15	1	1	NUM
cana-3950	215	16	(	(	PUNCT
cana-3950	215	17	boundness	boundness	NOUN
cana-3950	215	18	)	)	PUNCT
cana-3950	215	19	(	(	PUNCT
cana-3950	215	20	𝑖𝑖)𝑑(𝐴1	𝑖𝑖)𝑑(𝐴1	PROPN
cana-3950	215	21	,	,	PUNCT
cana-3950	215	22	𝐴2	𝐴2	PROPN
cana-3950	215	23	)	)	PUNCT
cana-3950	216	1	=	=	SYM
cana-3950	216	2	0	0	PUNCT
cana-3950	216	3	⟹	⟹	NUM
cana-3950	216	4	𝐴1	𝐴1	PROPN
cana-3950	216	5	=	=	PUNCT
cana-3950	216	6	𝐴2	𝐴2	PROPN
cana-3950	216	7	(	(	PUNCT
cana-3950	216	8	separable	separable	PROPN
cana-3950	216	9	)	)	PUNCT
cana-3950	216	10	(	(	PUNCT
cana-3950	216	11	𝑖𝑖𝑖)𝑑(𝐴1	𝑖𝑖𝑖)𝑑(𝐴1	X
cana-3950	216	12	,	,	PUNCT
cana-3950	216	13	𝐴2	𝐴2	PROPN
cana-3950	216	14	)	)	PUNCT
cana-3950	216	15	=	=	SYM
cana-3950	216	16	𝑑(𝐴2	𝑑(𝐴2	PROPN
cana-3950	216	17	,	,	PUNCT
cana-3950	216	18	𝐴1	𝐴1	PROPN
cana-3950	216	19	)	)	PUNCT
cana-3950	216	20	(	(	PUNCT
cana-3950	216	21	symmetric	symmetric	ADJ
cana-3950	216	22	)	)	PUNCT
cana-3950	216	23	(	(	PUNCT
cana-3950	216	24	𝑖𝑣)𝑑(𝐴1	𝑖𝑣)𝑑(𝐴1	PROPN
cana-3950	216	25	,	,	PUNCT
cana-3950	216	26	𝐴3	𝐴3	PROPN
cana-3950	216	27	)	)	PUNCT
cana-3950	216	28	+	+	PROPN
cana-3950	216	29	𝑑(𝐴2	𝑑(𝐴2	PROPN
cana-3950	216	30	,	,	PUNCT
cana-3950	216	31	𝐴3	𝐴3	PROPN
cana-3950	216	32	)	)	PUNCT
cana-3950	216	33	≥	≥	NOUN
cana-3950	216	34	𝑑(𝐴1	𝑑(𝐴1	PROPN
cana-3950	216	35	,	,	PUNCT
cana-3950	216	36	𝐴2	𝐴2	PROPN
cana-3950	216	37	)	)	PUNCT
cana-3950	216	38	(	(	PUNCT
cana-3950	216	39	triangle	triangle	NOUN
cana-3950	216	40	inequality	inequality	NOUN
cana-3950	216	41	)	)	PUNCT
cana-3950	216	42	definition	definition	NOUN
cana-3950	216	43	3.2	3.2	NUM
cana-3950	216	44	let	let	VERB
cana-3950	216	45	𝑋	𝑋	PROPN
cana-3950	216	46	=	=	SYM
cana-3950	216	47	(	(	PUNCT
cana-3950	216	48	𝑥1	𝑥1	NOUN
cana-3950	216	49	,	,	PUNCT
cana-3950	216	50	𝑥2	𝑥2	NOUN
cana-3950	216	51	}	}	PUNCT
cana-3950	216	52	be	be	AUX
cana-3950	216	53	a	a	DET
cana-3950	216	54	whole	whole	NOUN
cana-3950	216	55	of	of	ADP
cana-3950	216	56	discourse	discourse	NOUN
cana-3950	216	57	and	and	CCONJ
cana-3950	216	58	𝐴1	𝐴1	PROPN
cana-3950	216	59	,	,	PUNCT
cana-3950	216	60	𝐴2	𝐴2	PROPN
cana-3950	216	61	,	,	PUNCT
cana-3950	216	62	𝐴3	𝐴3	PROPN
cana-3950	216	63	∈	∈	PROPN
cana-3950	216	64	𝐵7	𝐵7	PROPN
cana-3950	216	65	5(𝑋	5(𝑋	PROPN
cana-3950	216	66	)	)	PUNCT
cana-3950	216	67	,	,	PUNCT
cana-3950	216	68	where	where	SCONJ
cana-3950	216	69	𝐴1	𝐴1	PROPN
cana-3950	216	70	=	=	SYM
cana-3950	216	71	{	{	PUNCT
cana-3950	216	72	〈	〈	PROPN
cana-3950	216	73	𝑥1	𝑥1	PROPN
cana-3950	216	74	,	,	PUNCT
cana-3950	216	75	0.5,0.6	0.5,0.6	NOUN
cana-3950	216	76	〉	〉	NOUN
cana-3950	216	77	,	,	PUNCT
cana-3950	216	78	〈	〈	NOUN
cana-3950	216	79	𝑥2	𝑥2	NOUN
cana-3950	216	80	,	,	PUNCT
cana-3950	216	81	0.4	0.4	NUM
cana-3950	216	82	,	,	PUNCT
cana-3950	216	83	0.5	0.5	NUM
cana-3950	216	84	〉	〉	NOUN
cana-3950	216	85	}	}	PUNCT
cana-3950	216	86	,	,	PUNCT
cana-3950	216	87	𝐴2	𝐴2	PROPN
cana-3950	216	88	=	=	SYM
cana-3950	216	89	{	{	PUNCT
cana-3950	216	90	〈	〈	PROPN
cana-3950	216	91	𝑥1	𝑥1	PROPN
cana-3950	216	92	,	,	PUNCT
cana-3950	216	93	0.8,0.3	0.8,0.3	PROPN
cana-3950	216	94	〉	〉	NOUN
cana-3950	216	95	,	,	PUNCT
cana-3950	216	96	〈	〈	NOUN
cana-3950	216	97	𝑥2	𝑥2	NOUN
cana-3950	216	98	,	,	PUNCT
cana-3950	216	99	0.9	0.9	NUM
cana-3950	216	100	,	,	PUNCT
cana-3950	216	101	0.5	0.5	NUM
cana-3950	216	102	〉	〉	NOUN
cana-3950	216	103	}	}	PUNCT
cana-3950	216	104	,	,	PUNCT
cana-3950	216	105	𝐴3	𝐴3	PROPN
cana-3950	216	106	=	=	SYM
cana-3950	216	107	{	{	PUNCT
cana-3950	216	108	〈	〈	PROPN
cana-3950	216	109	𝑥1	𝑥1	PROPN
cana-3950	216	110	,	,	PUNCT
cana-3950	216	111	0.6,0.8	0.6,0.8	NUM
cana-3950	216	112	〉	〉	NOUN
cana-3950	216	113	,	,	PUNCT
cana-3950	216	114	〈	〈	NOUN
cana-3950	216	115	𝑥2	𝑥2	NOUN
cana-3950	216	116	,	,	PUNCT
cana-3950	216	117	0.8	0.8	NUM
cana-3950	216	118	,	,	PUNCT
cana-3950	216	119	0.5	0.5	NUM
cana-3950	216	120	〉	〉	NOUN
cana-3950	216	121	}	}	PUNCT
cana-3950	216	122	,	,	PUNCT
cana-3950	216	123	find	find	VERB
cana-3950	216	124	the	the	DET
cana-3950	216	125	following	following	NOUN
cana-3950	216	126	:	:	PUNCT
cana-3950	216	127	(	(	PUNCT
cana-3950	216	128	1	1	X
cana-3950	216	129	)	)	PUNCT
cana-3950	216	130	hamming	hamming	NOUN
cana-3950	216	131	distance	distance	NOUN
cana-3950	216	132	(	(	PUNCT
cana-3950	216	133	2	2	NUM
cana-3950	216	134	)	)	PUNCT
cana-3950	216	135	normalized	normalize	VERB
cana-3950	216	136	hamming	hamming	NOUN
cana-3950	216	137	distance	distance	NOUN
cana-3950	216	138	(	(	PUNCT
cana-3950	216	139	3	3	X
cana-3950	216	140	)	)	PUNCT
cana-3950	216	141	euclidean	euclidean	ADJ
cana-3950	216	142	distance	distance	NOUN
cana-3950	216	143	(	(	PUNCT
cana-3950	216	144	4	4	X
cana-3950	216	145	)	)	PUNCT
cana-3950	216	146	normalized	normalize	VERB
cana-3950	216	147	euclidean	euclidean	ADJ
cana-3950	216	148	distance	distance	NOUN
cana-3950	216	149	𝑑𝐵7	𝑑𝐵7	PROPN
cana-3950	216	150	5(𝐴1	5(𝐴1	NUM
cana-3950	216	151	,	,	PUNCT
cana-3950	216	152	𝐴2	𝐴2	PROPN
cana-3950	216	153	)	)	PUNCT
cana-3950	216	154	=	=	PUNCT
cana-3950	216	155	0.6	0.6	NUM
cana-3950	216	156	𝑑𝐵7	𝑑𝐵7	PROPN
cana-3950	216	157	5	5	NUM
cana-3950	216	158	(	(	PUNCT
cana-3950	216	159	𝐴3	𝐴3	PROPN
cana-3950	216	160	,	,	PUNCT
cana-3950	216	161	𝐴1	𝐴1	PROPN
cana-3950	216	162	)	)	PUNCT
cana-3950	216	163	=	=	SYM
cana-3950	216	164	0.5	0.5	NUM
cana-3950	216	165	𝑑𝐸𝐵7	𝑑𝐸𝐵7	NOUN
cana-3950	216	166	5	5	NUM
cana-3950	216	167	(	(	PUNCT
cana-3950	216	168	𝐴1	𝐴1	PROPN
cana-3950	216	169	,	,	PUNCT
cana-3950	216	170	𝐴2	𝐴2	PROPN
cana-3950	216	171	)	)	PUNCT
cana-3950	216	172	=	=	NUM
cana-3950	216	173	0.3	0.3	NUM
cana-3950	216	174	𝑑𝐵7	𝑑𝐵7	PROPN
cana-3950	216	175	5	5	NUM
cana-3950	216	176	(	(	PUNCT
cana-3950	216	177	𝐴3	𝐴3	PROPN
cana-3950	216	178	,	,	PUNCT
cana-3950	216	179	𝐴1	𝐴1	PROPN
cana-3950	216	180	)	)	PUNCT
cana-3950	216	181	=	=	PUNCT
cana-3950	216	182	0.2	0.2	NUM
cana-3950	216	183	𝑑𝑛𝐵𝑚𝑛	𝑑𝑛𝐵𝑚𝑛	NOUN
cana-3950	216	184	5	5	NUM
cana-3950	216	185	(	(	PUNCT
cana-3950	216	186	𝐴1	𝐴1	PROPN
cana-3950	216	187	,	,	PUNCT
cana-3950	216	188	𝐴2	𝐴2	PROPN
cana-3950	216	189	)	)	PUNCT
cana-3950	216	190	=	=	PUNCT
cana-3950	216	191	0.6	0.6	NUM
cana-3950	216	192	𝑑𝐵7	𝑑𝐵7	PROPN
cana-3950	216	193	5	5	NUM
cana-3950	216	194	(	(	PUNCT
cana-3950	216	195	𝐴3	𝐴3	PROPN
cana-3950	216	196	,	,	PUNCT
cana-3950	216	197	𝐴1	𝐴1	PROPN
cana-3950	216	198	)	)	PUNCT
cana-3950	216	199	=	=	SYM
cana-3950	217	1	0.5	0.5	NUM
cana-3950	217	2	𝑑𝐻𝐵7	𝑑𝐻𝐵7	ADJ
cana-3950	217	3	5	5	NUM
cana-3950	217	4	(	(	PUNCT
cana-3950	217	5	𝐴2	𝐴2	PROPN
cana-3950	217	6	,	,	PUNCT
cana-3950	217	7	𝐴3	𝐴3	PROPN
cana-3950	217	8	)	)	PUNCT
cana-3950	217	9	=	=	NOUN
cana-3950	217	10	0.5	0.5	NUM
cana-3950	217	11	𝑑𝑛𝐵7	𝑑𝑛𝐵7	NOUN
cana-3950	217	12	5	5	NUM
cana-3950	217	13	(	(	PUNCT
cana-3950	217	14	𝐴3	𝐴3	PROPN
cana-3950	217	15	,	,	PUNCT
cana-3950	217	16	𝐴1	𝐴1	PROPN
cana-3950	217	17	)	)	PUNCT
cana-3950	217	18	=	=	SYM
cana-3950	218	1	0.3	0.3	NUM
cana-3950	218	2	𝑑𝐻𝐵7	𝑑𝐻𝐵7	VERB
cana-3950	218	3	5	5	NUM
cana-3950	218	4	(	(	PUNCT
cana-3950	218	5	𝐴2	𝐴2	PROPN
cana-3950	218	6	,	,	PUNCT
cana-3950	218	7	𝐴3	𝐴3	PROPN
cana-3950	218	8	)	)	PUNCT
cana-3950	218	9	=	=	PUNCT
cana-3950	218	10	0.2	0.2	NUM
cana-3950	218	11	𝑑𝐸𝐵7	𝑑𝐸𝐵7	NOUN
cana-3950	218	12	5	5	NUM
cana-3950	218	13	=	=	SYM
cana-3950	218	14	0.5	0.5	NUM
cana-3950	218	15	𝑑𝑛𝐵7	𝑑𝑛𝐵7	NOUN
cana-3950	218	16	5	5	NUM
cana-3950	218	17	=	=	SYM
cana-3950	218	18	0.4	0.4	NUM
cana-3950	218	19	example	example	NOUN
cana-3950	218	20	3.3	3.3	NUM
cana-3950	218	21	let	let	VERB
cana-3950	218	22	𝑋	𝑋	PROPN
cana-3950	218	23	=	=	SYM
cana-3950	218	24	{	{	PUNCT
cana-3950	218	25	𝑥1	𝑥1	NOUN
cana-3950	218	26	,	,	PUNCT
cana-3950	218	27	𝑥2	𝑥2	NOUN
cana-3950	218	28	,	,	PUNCT
cana-3950	218	29	𝑥3	𝑥3	NOUN
cana-3950	218	30	,	,	PUNCT
cana-3950	218	31	𝑥4	𝑥4	NOUN
cana-3950	218	32	}	}	PUNCT
cana-3950	218	33	and	and	CCONJ
cana-3950	218	34	𝐴1	𝐴1	PROPN
cana-3950	218	35	,	,	PUNCT
cana-3950	218	36	𝐴2	𝐴2	PROPN
cana-3950	218	37	,	,	PUNCT
cana-3950	218	38	𝐴3	𝐴3	PROPN
cana-3950	218	39	∈	∈	PROPN
cana-3950	218	40	𝐵8	𝐵8	PROPN
cana-3950	218	41	6(𝑋	6(𝑋	NOUN
cana-3950	218	42	)	)	PUNCT
cana-3950	218	43	where	where	SCONJ
cana-3950	218	44	𝐴1	𝐴1	PROPN
cana-3950	218	45	=	=	SYM
cana-3950	218	46	{	{	PUNCT
cana-3950	218	47	〈	〈	PROPN
cana-3950	218	48	𝑥1	𝑥1	NOUN
cana-3950	218	49	,	,	PUNCT
cana-3950	218	50	0.2,0.7	0.2,0.7	PROPN
cana-3950	218	51	〉	〉	NOUN
cana-3950	218	52	,	,	PUNCT
cana-3950	218	53	〈	〈	NOUN
cana-3950	218	54	𝑥2	𝑥2	NOUN
cana-3950	218	55	,	,	PUNCT
cana-3950	218	56	0.9	0.9	NUM
cana-3950	218	57	,	,	PUNCT
cana-3950	218	58	0.7	0.7	NUM
cana-3950	218	59	〉	〉	NOUN
cana-3950	218	60	}	}	PUNCT
cana-3950	218	61	,	,	PUNCT
cana-3950	218	62	𝐴2	𝐴2	PROPN
cana-3950	218	63	=	=	SYM
cana-3950	218	64	{	{	PUNCT
cana-3950	218	65	〈	〈	PROPN
cana-3950	218	66	𝑥1	𝑥1	PROPN
cana-3950	218	67	,	,	PUNCT
cana-3950	218	68	0.9,0.3	0.9,0.3	PROPN
cana-3950	218	69	〉	〉	NOUN
cana-3950	218	70	,	,	PUNCT
cana-3950	218	71	〈	〈	NOUN
cana-3950	218	72	𝑥2	𝑥2	NOUN
cana-3950	218	73	,	,	PUNCT
cana-3950	218	74	0.5	0.5	NUM
cana-3950	218	75	,	,	PUNCT
cana-3950	218	76	0.5	0.5	NUM
cana-3950	218	77	〉	〉	NOUN
cana-3950	218	78	}	}	PUNCT
cana-3950	218	79	,	,	PUNCT
cana-3950	218	80	𝐴3	𝐴3	PROPN
cana-3950	218	81	=	=	SYM
cana-3950	218	82	{	{	PUNCT
cana-3950	218	83	〈	〈	PROPN
cana-3950	218	84	𝑥1	𝑥1	NOUN
cana-3950	218	85	,	,	PUNCT
cana-3950	218	86	0.6,0.7	0.6,0.7	PROPN
cana-3950	218	87	〉	〉	NOUN
cana-3950	218	88	,	,	PUNCT
cana-3950	218	89	〈	〈	NOUN
cana-3950	218	90	𝑥2	𝑥2	NOUN
cana-3950	218	91	,	,	PUNCT
cana-3950	218	92	0.8	0.8	NUM
cana-3950	218	93	,	,	PUNCT
cana-3950	218	94	0.5	0.5	NUM
cana-3950	218	95	〉	〉	NOUN
cana-3950	218	96	}	}	PUNCT
cana-3950	218	97	,	,	PUNCT
cana-3950	218	98	𝑑𝐵8	𝑑𝐵8	PROPN
cana-3950	218	99	5(𝐴1	5(𝐴1	NUM
cana-3950	218	100	,	,	PUNCT
cana-3950	218	101	𝐴2	𝐴2	PROPN
cana-3950	218	102	)	)	PUNCT
cana-3950	218	103	=	=	SYM
cana-3950	218	104	0.9	0.9	NUM
cana-3950	218	105	𝑑𝐻𝐵8	𝑑𝐻𝐵8	NOUN
cana-3950	218	106	5(𝐴1	5(𝐴1	NUM
cana-3950	218	107	,	,	PUNCT
cana-3950	218	108	𝐴2	𝐴2	PROPN
cana-3950	218	109	)	)	PUNCT
cana-3950	218	110	=	=	SYM
cana-3950	218	111	0.3	0.3	NUM
cana-3950	218	112	𝑑𝑛	𝑑𝑛	X
cana-3950	218	113	𝐵8	𝐵8	PROPN
cana-3950	218	114	5(𝐴1	5(𝐴1	NUM
cana-3950	218	115	,	,	PUNCT
cana-3950	218	116	𝐴2	𝐴2	PROPN
cana-3950	218	117	)	)	PUNCT
cana-3950	218	118	=	=	SYM
cana-3950	218	119	0.6	0.6	NUM
cana-3950	218	120	𝑑𝐸𝐵8	𝑑𝐸𝐵8	NOUN
cana-3950	218	121	5(𝐴1	5(𝐴1	NUM
cana-3950	218	122	,	,	PUNCT
cana-3950	218	123	𝐴2	𝐴2	PROPN
cana-3950	218	124	)	)	PUNCT
cana-3950	218	125	=	=	SYM
cana-3950	218	126	0.8	0.8	NUM
cana-3950	218	127	definition	definition	NOUN
cana-3950	218	128	3.4	3.4	NUM
cana-3950	218	129	let	let	VERB
cana-3950	218	130	𝑋	𝑋	PROPN
cana-3950	218	131	=	=	SYM
cana-3950	218	132	{	{	PUNCT
cana-3950	218	133	𝑥1	𝑥1	NOUN
cana-3950	218	134	,	,	PUNCT
cana-3950	218	135	𝑥2	𝑥2	NOUN
cana-3950	218	136	,	,	PUNCT
cana-3950	218	137	𝑥3	𝑥3	NOUN
cana-3950	218	138	…	…	PUNCT
cana-3950	218	139	,	,	PUNCT
cana-3950	218	140	𝑥𝑟	𝑥𝑟	AUX
cana-3950	218	141	}	}	PUNCT
cana-3950	218	142	be	be	AUX
cana-3950	218	143	a	a	DET
cana-3950	218	144	whole	whole	NOUN
cana-3950	218	145	of	of	ADP
cana-3950	218	146	discourse	discourse	NOUN
cana-3950	218	147	and	and	CCONJ
cana-3950	218	148	𝐴1	𝐴1	PROPN
cana-3950	218	149	=	=	SYM
cana-3950	218	150	(	(	PUNCT
cana-3950	218	151	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	218	152	,	,	PUNCT
cana-3950	218	153	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	218	154	)	)	PUNCT
cana-3950	218	155	,	,	PUNCT
cana-3950	218	156	𝐴2	𝐴2	PROPN
cana-3950	218	157	=	=	SYM
cana-3950	218	158	(	(	PUNCT
cana-3950	218	159	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	218	160	,	,	PUNCT
cana-3950	218	161	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	218	162	)	)	PUNCT
cana-3950	218	163	,	,	PUNCT
cana-3950	218	164	𝐴3	𝐴3	PROPN
cana-3950	218	165	=	=	SYM
cana-3950	218	166	(	(	PUNCT
cana-3950	218	167	𝐽𝐴3	𝐽𝐴3	PROPN
cana-3950	218	168	,	,	PUNCT
cana-3950	218	169	𝐾𝐴3	𝐾𝐴3	PROPN
cana-3950	218	170	)	)	PUNCT
cana-3950	218	171	∈	∈	PROPN
cana-3950	218	172	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	218	173	𝑛	𝑛	PROPN
cana-3950	218	174	(	(	PUNCT
cana-3950	218	175	𝑋	𝑋	PROPN
cana-3950	218	176	)	)	PUNCT
cana-3950	218	177	.	.	PUNCT
cana-3950	219	1	the	the	DET
cana-3950	219	2	similarity	similarity	NOUN
cana-3950	219	3	measure	measure	NOUN
cana-3950	219	4	𝑆	𝑆	PROPN
cana-3950	219	5	:	:	PUNCT
cana-3950	219	6	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	219	7	𝑛	𝑛	PROPN
cana-3950	219	8	(	(	PUNCT
cana-3950	219	9	𝑋	𝑋	NOUN
cana-3950	219	10	)	)	PUNCT
cana-3950	219	11	×	×	NOUN
cana-3950	219	12	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	219	13	𝑛	𝑛	PROPN
cana-3950	219	14	(	(	PUNCT
cana-3950	219	15	𝑋	𝑋	NOUN
cana-3950	219	16	)	)	PUNCT
cana-3950	219	17	→	→	PUNCT
cana-3950	220	1	[	[	X
cana-3950	220	2	0,1	0,1	NUM
cana-3950	220	3	]	]	PUNCT
cana-3950	220	4	is	be	AUX
cana-3950	220	5	expressed	express	VERB
cana-3950	220	6	as	as	SCONJ
cana-3950	220	7	communications	communication	NOUN
cana-3950	220	8	on	on	ADP
cana-3950	220	9	applied	apply	VERB
cana-3950	220	10	nonlinear	nonlinear	ADJ
cana-3950	220	11	analysis	analysis	NOUN
cana-3950	220	12	issn	issn	NOUN
cana-3950	220	13	:	:	PUNCT
cana-3950	220	14	1074	1074	NUM
cana-3950	220	15	-	-	PUNCT
cana-3950	220	16	133x	133x	NUM
cana-3950	220	17	vol	vol	NOUN
cana-3950	220	18	32	32	NUM
cana-3950	220	19	no	no	NOUN
cana-3950	220	20	.	.	PUNCT
cana-3950	221	1	9s	9s	NUM
cana-3950	221	2	(	(	PUNCT
cana-3950	221	3	2025	2025	NUM
cana-3950	221	4	)	)	PUNCT
cana-3950	221	5	398	398	NUM
cana-3950	221	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3950	221	7	(	(	PUNCT
cana-3950	221	8	𝑖)0	𝑖)0	PROPN
cana-3950	221	9	≤	≤	PROPN
cana-3950	221	10	𝑆(𝐴1	𝑆(𝐴1	NOUN
cana-3950	221	11	,	,	PUNCT
cana-3950	221	12	𝐴2	𝐴2	PROPN
cana-3950	221	13	)	)	PUNCT
cana-3950	221	14	≤	≤	NOUN
cana-3950	221	15	1	1	NUM
cana-3950	221	16	(	(	PUNCT
cana-3950	221	17	boundedness	boundedness	NOUN
cana-3950	221	18	)	)	PUNCT
cana-3950	221	19	(	(	PUNCT
cana-3950	221	20	𝑖𝑖)𝑆(𝐴1	𝑖𝑖)𝑆(𝐴1	ADJ
cana-3950	221	21	,	,	PUNCT
cana-3950	221	22	𝐴2	𝐴2	PROPN
cana-3950	221	23	)	)	PUNCT
cana-3950	221	24	=	=	SYM
cana-3950	222	1	1	1	NUM
cana-3950	222	2	⟺	⟺	NOUN
cana-3950	222	3	𝐴1	𝐴1	PROPN
cana-3950	222	4	=	=	PUNCT
cana-3950	222	5	𝐴2	𝐴2	PROPN
cana-3950	222	6	(	(	PUNCT
cana-3950	222	7	separability	separability	PROPN
cana-3950	222	8	)	)	PUNCT
cana-3950	222	9	(	(	PUNCT
cana-3950	222	10	𝑖𝑖𝑖)𝑆(𝐴1	𝑖𝑖𝑖)𝑆(𝐴1	X
cana-3950	222	11	,	,	PUNCT
cana-3950	222	12	𝐴2	𝐴2	PROPN
cana-3950	222	13	)	)	PUNCT
cana-3950	223	1	=	=	PUNCT
cana-3950	223	2	𝑆(𝐴2	𝑆(𝐴2	PROPN
cana-3950	223	3	,	,	PUNCT
cana-3950	223	4	𝐴1	𝐴1	PROPN
cana-3950	223	5	)	)	PUNCT
cana-3950	223	6	(	(	PUNCT
cana-3950	223	7	symmetric	symmetric	ADJ
cana-3950	223	8	)	)	PUNCT
cana-3950	223	9	(	(	PUNCT
cana-3950	223	10	𝑖𝑣)𝑆(𝐴1	𝑖𝑣)𝑆(𝐴1	PROPN
cana-3950	223	11	,	,	PUNCT
cana-3950	223	12	𝐴3	𝐴3	PROPN
cana-3950	223	13	)	)	PUNCT
cana-3950	223	14	+	+	SYM
cana-3950	224	1	𝑆(𝐴2	𝑆(𝐴2	ADJ
cana-3950	224	2	,	,	PUNCT
cana-3950	224	3	𝐴1	𝐴1	PROPN
cana-3950	224	4	)	)	PUNCT
cana-3950	224	5	≥	≥	PROPN
cana-3950	224	6	𝑆(𝐴1	𝑆(𝐴1	NOUN
cana-3950	224	7	,	,	PUNCT
cana-3950	224	8	𝐴2	𝐴2	PROPN
cana-3950	224	9	)	)	PUNCT
cana-3950	224	10	(	(	PUNCT
cana-3950	224	11	triangle	triangle	NOUN
cana-3950	224	12	inequality	inequality	NOUN
cana-3950	224	13	)	)	PUNCT
cana-3950	224	14	using	use	VERB
cana-3950	224	15	distance	distance	NOUN
cana-3950	224	16	functions	function	NOUN
cana-3950	224	17	and	and	CCONJ
cana-3950	224	18	similarity	similarity	NOUN
cana-3950	224	19	measures	measure	NOUN
cana-3950	224	20	,	,	PUNCT
cana-3950	224	21	the	the	DET
cana-3950	224	22	below	below	NOUN
cana-3950	224	23	of	of	ADP
cana-3950	224	24	result	result	NOUN
cana-3950	224	25	proved	prove	VERB
cana-3950	224	26	.	.	PUNCT
cana-3950	225	1	theorem	theorem	VERB
cana-3950	225	2	3.5	3.5	NUM
cana-3950	225	3	let𝐴1	let𝐴1	NOUN
cana-3950	225	4	=	=	SYM
cana-3950	225	5	(	(	PUNCT
cana-3950	225	6	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	225	7	,	,	PUNCT
cana-3950	225	8	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	225	9	)	)	PUNCT
cana-3950	225	10	,	,	PUNCT
cana-3950	226	1	𝐴2	𝐴2	PROPN
cana-3950	226	2	=	=	SYM
cana-3950	226	3	(	(	PUNCT
cana-3950	226	4	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	226	5	,	,	PUNCT
cana-3950	226	6	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	226	7	)	)	PUNCT
cana-3950	226	8	∈	∈	PROPN
cana-3950	226	9	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	226	10	𝑛	𝑛	PROPN
cana-3950	226	11	(	(	PUNCT
cana-3950	226	12	𝑋	𝑋	PROPN
cana-3950	226	13	)	)	PUNCT
cana-3950	226	14	.	.	PUNCT
cana-3950	227	1	if	if	SCONJ
cana-3950	227	2	𝑑(𝐴1	𝑑(𝐴1	PROPN
cana-3950	227	3	,	,	PUNCT
cana-3950	227	4	𝐴2	𝐴2	PROPN
cana-3950	227	5	)	)	PUNCT
cana-3950	227	6	is	be	AUX
cana-3950	227	7	a	a	DET
cana-3950	227	8	distance	distance	NOUN
cana-3950	227	9	measure	measure	NOUN
cana-3950	227	10	between	between	ADP
cana-3950	227	11	beal	beal	NOUN
cana-3950	227	12	’s	’s	PART
cana-3950	227	13	fuzzy	fuzzy	ADJ
cana-3950	227	14	sets	set	VERB
cana-3950	227	15	𝐴1	𝐴1	PROPN
cana-3950	227	16	and	and	CCONJ
cana-3950	227	17	𝐴2	𝐴2	PROPN
cana-3950	227	18	,	,	PUNCT
cana-3950	227	19	then	then	ADV
cana-3950	227	20	𝑆(𝐴1	𝑆(𝐴1	PROPN
cana-3950	227	21	,	,	PUNCT
cana-3950	227	22	𝐴2	𝐴2	PROPN
cana-3950	227	23	)	)	PUNCT
cana-3950	227	24	=	=	SYM
cana-3950	227	25	1−	1−	NUM
cana-3950	227	26	𝑑(𝐴1	𝑑(𝐴1	PROPN
cana-3950	227	27	,	,	PUNCT
cana-3950	227	28	𝐴2	𝐴2	PROPN
cana-3950	227	29	)	)	PUNCT
cana-3950	227	30	is	be	AUX
cana-3950	227	31	a	a	DET
cana-3950	227	32	similarity	similarity	NOUN
cana-3950	227	33	measure	measure	NOUN
cana-3950	227	34	of	of	ADP
cana-3950	227	35	𝐴1	𝐴1	PROPN
cana-3950	227	36	and	and	CCONJ
cana-3950	227	37	𝐴2	𝐴2	PROPN
cana-3950	227	38	.	.	PUNCT
cana-3950	228	1	theorem	theorem	VERB
cana-3950	228	2	3.6	3.6	NUM
cana-3950	228	3	let	let	VERB
cana-3950	228	4	𝐴1	𝐴1	PROPN
cana-3950	228	5	=	=	SYM
cana-3950	228	6	(	(	PUNCT
cana-3950	228	7	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	228	8	,	,	PUNCT
cana-3950	228	9	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	228	10	)	)	PUNCT
cana-3950	228	11	,	,	PUNCT
cana-3950	228	12	𝐴2	𝐴2	PROPN
cana-3950	228	13	=	=	SYM
cana-3950	228	14	(	(	PUNCT
cana-3950	228	15	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	228	16	,	,	PUNCT
cana-3950	228	17	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	228	18	)	)	PUNCT
cana-3950	228	19	∈	∈	PROPN
cana-3950	228	20	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	228	21	𝑛	𝑛	PROPN
cana-3950	228	22	(	(	PUNCT
cana-3950	228	23	𝑋	𝑋	PROPN
cana-3950	228	24	)	)	PUNCT
cana-3950	228	25	.	.	PUNCT
cana-3950	229	1	suppose	suppose	VERB
cana-3950	230	1	𝐴1	𝐴1	PROPN
cana-3950	230	2	⊂	⊂	PROPN
cana-3950	230	3	𝐴2	𝐴2	PROPN
cana-3950	230	4	⊂	⊂	PROPN
cana-3950	230	5	𝐴3	𝐴3	PROPN
cana-3950	230	6	,	,	PUNCT
cana-3950	230	7	then	then	ADV
cana-3950	230	8	(	(	PUNCT
cana-3950	230	9	𝑖)𝑑(𝐴1	𝑖)𝑑(𝐴1	ADJ
cana-3950	230	10	,	,	PUNCT
cana-3950	230	11	𝐴2	𝐴2	PROPN
cana-3950	230	12	)	)	PUNCT
cana-3950	230	13	≥	≥	NOUN
cana-3950	230	14	𝑑(𝐴1	𝑑(𝐴1	PROPN
cana-3950	230	15	,	,	PUNCT
cana-3950	230	16	𝐴2)𝑎𝑛𝑑	𝐴2)𝑎𝑛𝑑	PROPN
cana-3950	230	17	(	(	PUNCT
cana-3950	230	18	𝑖𝑖)𝑑(𝐴1	𝑖𝑖)𝑑(𝐴1	NOUN
cana-3950	230	19	,	,	PUNCT
cana-3950	230	20	𝐴3	𝐴3	PROPN
cana-3950	230	21	)	)	PUNCT
cana-3950	230	22	≥	≥	PROPN
cana-3950	230	23	𝑑(𝐴2	𝑑(𝐴2	PROPN
cana-3950	230	24	,	,	PUNCT
cana-3950	230	25	𝐴3	𝐴3	PROPN
cana-3950	230	26	)	)	PUNCT
cana-3950	230	27	(	(	PUNCT
cana-3950	230	28	𝑖𝑖𝑖)𝑆(𝐴1	𝑖𝑖𝑖)𝑆(𝐴1	X
cana-3950	230	29	,	,	PUNCT
cana-3950	230	30	𝐴3	𝐴3	PROPN
cana-3950	230	31	)	)	PUNCT
cana-3950	230	32	≤	≤	NOUN
cana-3950	230	33	𝑆(𝐴1	𝑆(𝐴1	NOUN
cana-3950	230	34	,	,	PUNCT
cana-3950	230	35	𝐴2)𝑎𝑛𝑑	𝐴2)𝑎𝑛𝑑	PROPN
cana-3950	230	36	(	(	PUNCT
cana-3950	230	37	𝑖𝑣)𝑆(𝑀1	𝑖𝑣)𝑆(𝑀1	NOUN
cana-3950	230	38	,	,	PUNCT
cana-3950	230	39	𝑀3	𝑀3	PROPN
cana-3950	230	40	)	)	PUNCT
cana-3950	230	41	≤	≤	NOUN
cana-3950	230	42	𝑆(𝑀2	𝑆(𝑀2	PROPN
cana-3950	230	43	,	,	PUNCT
cana-3950	230	44	𝑀3	𝑀3	NOUN
cana-3950	230	45	)	)	PUNCT
cana-3950	230	46	.	.	PUNCT
cana-3950	231	1	theorem	theorem	VERB
cana-3950	231	2	3.7	3.7	NUM
cana-3950	231	3	let	let	VERB
cana-3950	231	4	𝐴1	𝐴1	PROPN
cana-3950	231	5	=	=	SYM
cana-3950	231	6	(	(	PUNCT
cana-3950	231	7	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	231	8	,	,	PUNCT
cana-3950	231	9	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	231	10	)	)	PUNCT
cana-3950	231	11	,	,	PUNCT
cana-3950	231	12	𝐴2	𝐴2	PROPN
cana-3950	231	13	=	=	SYM
cana-3950	231	14	(	(	PUNCT
cana-3950	231	15	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	231	16	,	,	PUNCT
cana-3950	231	17	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	231	18	)	)	PUNCT
cana-3950	231	19	∈	∈	PROPN
cana-3950	231	20	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	231	21	𝑛	𝑛	PROPN
cana-3950	231	22	(	(	PUNCT
cana-3950	231	23	𝑋	𝑋	PROPN
cana-3950	231	24	)	)	PUNCT
cana-3950	231	25	.	.	PUNCT
cana-3950	232	1	then	then	ADV
cana-3950	232	2	(	(	PUNCT
cana-3950	232	3	𝑖)𝑑(𝐴1	𝑖)𝑑(𝐴1	ADJ
cana-3950	232	4	,	,	PUNCT
cana-3950	232	5	𝐴2	𝐴2	PROPN
cana-3950	232	6	)	)	PUNCT
cana-3950	232	7	=	=	PUNCT
cana-3950	232	8	𝑑(𝐴1	𝑑(𝐴1	PROPN
cana-3950	232	9	𝑐	𝑐	PROPN
cana-3950	232	10	,	,	PUNCT
cana-3950	232	11	𝐴2	𝐴2	PROPN
cana-3950	232	12	2	2	NUM
cana-3950	232	13	)	)	PUNCT
cana-3950	232	14	(	(	PUNCT
cana-3950	232	15	𝑖𝑖)𝑆(𝐴1	𝑖𝑖)𝑆(𝐴1	ADJ
cana-3950	232	16	,	,	PUNCT
cana-3950	232	17	𝐴2	𝐴2	PROPN
cana-3950	232	18	)	)	PUNCT
cana-3950	232	19	=	=	PUNCT
cana-3950	233	1	𝑆(𝐴1	𝑆(𝐴1	PROPN
cana-3950	233	2	𝑐	𝑐	PROPN
cana-3950	233	3	,	,	PUNCT
cana-3950	233	4	𝐴2	𝐴2	PROPN
cana-3950	233	5	𝑐	𝑐	PROPN
cana-3950	233	6	)	)	PUNCT
cana-3950	233	7	.	.	PUNCT
cana-3950	234	1	note	note	VERB
cana-3950	234	2	.	.	PUNCT
cana-3950	235	1	the	the	DET
cana-3950	235	2	two	two	NUM
cana-3950	235	3	similarity	similarity	NOUN
cana-3950	235	4	measures	measure	NOUN
cana-3950	235	5	are	be	AUX
cana-3950	235	6	defined	define	VERB
cana-3950	235	7	are	be	AUX
cana-3950	235	8	defined	define	VERB
cana-3950	235	9	as	as	SCONJ
cana-3950	235	10	follows	follow	VERB
cana-3950	235	11	𝑆1	𝑆1	NOUN
cana-3950	235	12	=	=	SYM
cana-3950	235	13	1−	1−	NUM
cana-3950	235	14	𝑑𝐵𝑚𝑛	𝑑𝐵𝑚𝑛	PROPN
cana-3950	235	15	𝑛	𝑛	PRON
cana-3950	235	16	𝐻(𝐴1	𝐻(𝐴1	NOUN
cana-3950	235	17	,	,	PUNCT
cana-3950	235	18	𝐴2	𝐴2	PROPN
cana-3950	235	19	)	)	PUNCT
cana-3950	235	20	,	,	PUNCT
cana-3950	235	21	𝑆2	𝑆2	PROPN
cana-3950	235	22	=	=	SYM
cana-3950	235	23	1−	1−	NUM
cana-3950	235	24	𝑑𝐵𝑚𝑛	𝑑𝐵𝑚𝑛	PROPN
cana-3950	235	25	𝑛	𝑛	PRON
cana-3950	235	26	𝐸(𝐴1	𝐸(𝐴1	NOUN
cana-3950	235	27	,	,	PUNCT
cana-3950	235	28	𝐴2	𝐴2	PROPN
cana-3950	235	29	)	)	PUNCT
cana-3950	235	30	note	note	NOUN
cana-3950	235	31	:	:	PUNCT
cana-3950	235	32	𝐴1	𝐴1	PROPN
cana-3950	235	33	,	,	PUNCT
cana-3950	235	34	𝐴2	𝐴2	PROPN
cana-3950	235	35	∈	∈	PROPN
cana-3950	235	36	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	235	37	𝑛	𝑛	PROPN
cana-3950	235	38	(	(	PUNCT
cana-3950	235	39	𝑋	𝑋	PROPN
cana-3950	235	40	)	)	PUNCT
cana-3950	235	41	,	,	PUNCT
cana-3950	235	42	𝑋	𝑋	PROPN
cana-3950	235	43	=	=	SYM
cana-3950	235	44	{	{	PUNCT
cana-3950	235	45	𝑥1	𝑥1	NOUN
cana-3950	235	46	,	,	PUNCT
cana-3950	235	47	𝑥2	𝑥2	NOUN
cana-3950	235	48	}	}	PUNCT
cana-3950	235	49	,	,	PUNCT
cana-3950	235	50	𝑆1	𝑆1	NOUN
cana-3950	235	51	∈	∈	PROPN
cana-3950	235	52	(	(	PUNCT
cana-3950	235	53	𝐴1	𝐴1	PROPN
cana-3950	235	54	,	,	PUNCT
cana-3950	235	55	𝐴2	𝐴2	PROPN
cana-3950	235	56	)	)	PUNCT
cana-3950	235	57	,	,	PUNCT
cana-3950	235	58	𝑆2	𝑆2	PROPN
cana-3950	235	59	∈	∈	PROPN
cana-3950	235	60	(	(	PUNCT
cana-3950	235	61	𝐴1	𝐴1	PROPN
cana-3950	235	62	,	,	PUNCT
cana-3950	235	63	𝐴2	𝐴2	PROPN
cana-3950	235	64	)	)	PUNCT
cana-3950	235	65	,	,	PUNCT
cana-3950	235	66	𝐴1	𝐴1	PROPN
cana-3950	235	67	,	,	PUNCT
cana-3950	235	68	𝐴2	𝐴2	PROPN
cana-3950	235	69	,	,	PUNCT
cana-3950	235	70	𝐴3	𝐴3	PROPN
cana-3950	235	71	∈	∈	PROPN
cana-3950	235	72	𝐵𝑚	𝐵𝑚	PROPN
cana-3950	235	73	𝑛	𝑛	PROPN
cana-3950	235	74	(	(	PUNCT
cana-3950	235	75	𝑋	𝑋	NOUN
cana-3950	235	76	)	)	PUNCT
cana-3950	235	77	𝑆1(𝐴1	𝑆1(𝐴1	PROPN
cana-3950	235	78	,	,	PUNCT
cana-3950	235	79	𝐴2	𝐴2	PROPN
cana-3950	235	80	)	)	PUNCT
cana-3950	235	81	=	=	SYM
cana-3950	235	82	0.6	0.6	NUM
cana-3950	235	83	,	,	PUNCT
cana-3950	235	84	𝑆2(𝐴1	𝑆2(𝐴1	PROPN
cana-3950	235	85	,	,	PUNCT
cana-3950	235	86	𝐴2	𝐴2	PROPN
cana-3950	235	87	)	)	PUNCT
cana-3950	235	88	=	=	SYM
cana-3950	235	89	0.8	0.8	NUM
cana-3950	235	90	,	,	PUNCT
cana-3950	235	91	𝑆1(𝐴1	𝑆1(𝐴1	ADV
cana-3950	235	92	,	,	PUNCT
cana-3950	235	93	𝐴3	𝐴3	PROPN
cana-3950	235	94	)	)	PUNCT
cana-3950	235	95	=	=	SYM
cana-3950	235	96	0.6	0.6	NUM
cana-3950	235	97	,	,	PUNCT
cana-3950	235	98	𝑆2(𝐴1	𝑆2(𝐴1	PROPN
cana-3950	235	99	,	,	PUNCT
cana-3950	235	100	𝐴3	𝐴3	PROPN
cana-3950	235	101	)	)	PUNCT
cana-3950	235	102	=	=	PUNCT
cana-3950	236	1	0.8	0.8	NUM
cana-3950	236	2	.	.	PUNCT
cana-3950	237	1	in	in	ADP
cana-3950	237	2	order	order	NOUN
cana-3950	237	3	to	to	PART
cana-3950	237	4	rank	rank	VERB
cana-3950	237	5	beal	beal	PROPN
cana-3950	237	6	’s	’s	PART
cana-3950	237	7	fuzzy	fuzzy	ADJ
cana-3950	237	8	sets	set	NOUN
cana-3950	237	9	,	,	PUNCT
cana-3950	237	10	we	we	PRON
cana-3950	237	11	give	give	VERB
cana-3950	237	12	the	the	DET
cana-3950	237	13	score	score	NOUN
cana-3950	237	14	mapping	mapping	NOUN
cana-3950	237	15	and	and	CCONJ
cana-3950	237	16	accuracy	accuracy	NOUN
cana-3950	237	17	,	,	PUNCT
cana-3950	237	18	mapping	mapping	NOUN
cana-3950	237	19	of	of	ADP
cana-3950	237	20	the	the	DET
cana-3950	237	21	beal	beal	NOUN
cana-3950	237	22	’s	’s	PART
cana-3950	237	23	fuzzy	fuzzy	ADJ
cana-3950	237	24	set	set	NOUN
cana-3950	237	25	.	.	PUNCT
cana-3950	238	1	definition	definition	NOUN
cana-3950	238	2	3.8	3.8	NUM
cana-3950	238	3	the	the	DET
cana-3950	238	4	score	score	NOUN
cana-3950	238	5	mapping	mapping	NOUN
cana-3950	238	6	function	function	NOUN
cana-3950	238	7	of	of	ADP
cana-3950	238	8	a	a	DET
cana-3950	238	9	beal	beal	NOUN
cana-3950	238	10	’s	’s	PART
cana-3950	238	11	fuzzy	fuzzy	ADJ
cana-3950	238	12	set	set	VERB
cana-3950	238	13	𝐴	𝐴	NOUN
cana-3950	238	14	=	=	SYM
cana-3950	238	15	(	(	PUNCT
cana-3950	238	16	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	238	17	,	,	PUNCT
cana-3950	238	18	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	238	19	)	)	PUNCT
cana-3950	238	20	can	can	AUX
cana-3950	238	21	be	be	AUX
cana-3950	238	22	given	give	VERB
cana-3950	238	23	as	as	ADP
cana-3950	238	24	𝑆(𝐴	𝑆(𝐴	NOUN
cana-3950	238	25	)	)	PUNCT
cana-3950	238	26	=	=	SYM
cana-3950	238	27	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	238	28	𝑛	𝑛	DET
cana-3950	238	29	−	−	PROPN
cana-3950	238	30	𝐾𝐴	𝐾𝐴	NOUN
cana-3950	238	31	𝑚.	𝑚.	NOUN
cana-3950	238	32	the	the	DET
cana-3950	238	33	accuracy	accuracy	NOUN
cana-3950	238	34	function	function	NOUN
cana-3950	238	35	of	of	ADP
cana-3950	238	36	a	a	DET
cana-3950	238	37	beal	beal	NOUN
cana-3950	238	38	’s	’s	PART
cana-3950	238	39	fuzzy	fuzzy	ADJ
cana-3950	238	40	set	set	VERB
cana-3950	238	41	𝐴	𝐴	NOUN
cana-3950	238	42	=	=	SYM
cana-3950	238	43	(	(	PUNCT
cana-3950	238	44	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	238	45	,	,	PUNCT
cana-3950	238	46	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	238	47	)	)	PUNCT
cana-3950	238	48	can	can	AUX
cana-3950	238	49	be	be	AUX
cana-3950	238	50	given	give	VERB
cana-3950	238	51	as	as	ADP
cana-3950	238	52	𝛼	𝛼	X
cana-3950	238	53	(	(	PUNCT
cana-3950	238	54	a	a	NOUN
cana-3950	238	55	)	)	PUNCT
cana-3950	238	56	=	=	SYM
cana-3950	238	57	𝐽𝐴	𝐽𝐴	NOUN
cana-3950	238	58	𝑛	𝑛	PROPN
cana-3950	238	59	+	+	CCONJ
cana-3950	238	60	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	238	61	𝑚.	𝑚.	ADJ
cana-3950	238	62	example	example	NOUN
cana-3950	238	63	3.9	3.9	NUM
cana-3950	238	64	consider	consider	VERB
cana-3950	238	65	the	the	DET
cana-3950	238	66	beal	beal	NOUN
cana-3950	238	67	’s	’s	PART
cana-3950	238	68	fuzzy	fuzzy	ADJ
cana-3950	238	69	set	set	VERB
cana-3950	238	70	𝐴	𝐴	NOUN
cana-3950	238	71	=	=	SYM
cana-3950	238	72	(	(	PUNCT
cana-3950	238	73	0.62	0.62	NUM
cana-3950	238	74	,	,	PUNCT
cana-3950	238	75	0.54	0.54	NUM
cana-3950	238	76	)	)	PUNCT
cana-3950	238	77	then	then	ADV
cana-3950	238	78	the	the	DET
cana-3950	238	79	score	score	NOUN
cana-3950	238	80	function	function	NOUN
cana-3950	238	81	is	be	AUX
cana-3950	238	82	communications	communication	NOUN
cana-3950	238	83	on	on	ADP
cana-3950	238	84	applied	apply	VERB
cana-3950	238	85	nonlinear	nonlinear	ADJ
cana-3950	238	86	analysis	analysis	NOUN
cana-3950	238	87	issn	issn	NOUN
cana-3950	238	88	:	:	PUNCT
cana-3950	238	89	1074	1074	NUM
cana-3950	238	90	-	-	PUNCT
cana-3950	238	91	133x	133x	NUM
cana-3950	238	92	vol	vol	NOUN
cana-3950	238	93	32	32	NUM
cana-3950	238	94	no	no	NOUN
cana-3950	238	95	.	.	PUNCT
cana-3950	239	1	9s	9s	NUM
cana-3950	239	2	(	(	PUNCT
cana-3950	239	3	2025	2025	NUM
cana-3950	239	4	)	)	PUNCT
cana-3950	239	5	399	399	NUM
cana-3950	239	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3950	239	7	𝑆(𝐴	𝑆(𝐴	NOUN
cana-3950	239	8	)	)	PUNCT
cana-3950	239	9	=	=	PRON
cana-3950	239	10	{	{	PUNCT
cana-3950	240	1	−0.14	−0.14	INTJ
cana-3950	240	2	𝑖𝑓	𝑖𝑓	NUM
cana-3950	240	3	𝑛	𝑛	NOUN
cana-3950	240	4	=	=	SYM
cana-3950	240	5	10	10	NUM
cana-3950	240	6	,	,	PUNCT
cana-3950	240	7	𝑚	𝑚	X
cana-3950	240	8	=	=	SYM
cana-3950	240	9	3	3	NUM
cana-3950	240	10	−0.00	−0.00	PART
cana-3950	240	11	𝑖𝑓	𝑖𝑓	NUM
cana-3950	240	12	𝑛	𝑛	NOUN
cana-3950	240	13	=	=	SYM
cana-3950	240	14	8	8	NUM
cana-3950	240	15	,	,	PUNCT
cana-3950	240	16	𝑚	𝑚	NOUN
cana-3950	240	17	=	=	SYM
cana-3950	240	18	6	6	NUM
cana-3950	241	1	−0.01	−0.01	ADV
cana-3950	241	2	𝑖𝑓	𝑖𝑓	ADP
cana-3950	241	3	𝑛	𝑛	PROPN
cana-3950	241	4	=	=	SYM
cana-3950	241	5	7	7	NUM
cana-3950	241	6	,	,	PUNCT
cana-3950	241	7	𝑚	𝑚	NOUN
cana-3950	241	8	=	=	SYM
cana-3950	241	9	5	5	NUM
cana-3950	241	10	0.06	0.06	NUM
cana-3950	241	11	𝑖𝑓	𝑖𝑓	ADP
cana-3950	241	12	𝑛	𝑛	PRON
cana-3950	241	13	=	=	SYM
cana-3950	241	14	5	5	NUM
cana-3950	241	15	,	,	PUNCT
cana-3950	241	16	𝑚	𝑚	NOUN
cana-3950	241	17	=	=	SYM
cana-3950	241	18	6	6	NUM
cana-3950	241	19	and	and	CCONJ
cana-3950	241	20	the	the	DET
cana-3950	241	21	accuracy	accuracy	NOUN
cana-3950	241	22	function	function	NOUN
cana-3950	241	23	is	be	AUX
cana-3950	241	24	𝛼(𝐴	𝛼(𝐴	NUM
cana-3950	241	25	)	)	PUNCT
cana-3950	242	1	=	=	PRON
cana-3950	242	2	{	{	PUNCT
cana-3950	242	3	0.16	0.16	NUM
cana-3950	242	4	𝑖𝑓	𝑖𝑓	NOUN
cana-3950	242	5	𝑛	𝑛	PRON
cana-3950	242	6	=	=	SYM
cana-3950	242	7	10	10	NUM
cana-3950	242	8	,	,	PUNCT
cana-3950	242	9	𝑚	𝑚	X
cana-3950	242	10	=	=	SYM
cana-3950	242	11	3	3	NUM
cana-3950	242	12	0.05	0.05	NUM
cana-3950	242	13	𝑖𝑓	𝑖𝑓	NOUN
cana-3950	242	14	𝑛	𝑛	PRON
cana-3950	242	15	=	=	SYM
cana-3950	242	16	8	8	NUM
cana-3950	242	17	,	,	PUNCT
cana-3950	242	18	𝑚	𝑚	X
cana-3950	242	19	=	=	SYM
cana-3950	242	20	6	6	NUM
cana-3950	242	21	0.08	0.08	NUM
cana-3950	242	22	𝑖𝑓	𝑖𝑓	ADP
cana-3950	242	23	𝑛	𝑛	PRON
cana-3950	242	24	=	=	SYM
cana-3950	242	25	7	7	NUM
cana-3950	242	26	,	,	PUNCT
cana-3950	242	27	𝑚	𝑚	NOUN
cana-3950	242	28	=	=	SYM
cana-3950	242	29	5	5	NUM
cana-3950	242	30	0.11	0.11	NUM
cana-3950	242	31	𝑖𝑓	𝑖𝑓	NOUN
cana-3950	242	32	𝑛	𝑛	PRON
cana-3950	242	33	=	=	SYM
cana-3950	242	34	5	5	NUM
cana-3950	242	35	,	,	PUNCT
cana-3950	242	36	𝑚	𝑚	NOUN
cana-3950	242	37	=	=	SYM
cana-3950	242	38	6	6	NUM
cana-3950	242	39	theorem	theorem	VERB
cana-3950	242	40	3.10	3.10	NUM
cana-3950	242	41	let	let	VERB
cana-3950	242	42	𝐴	𝐴	PROPN
cana-3950	242	43	=	=	SYM
cana-3950	242	44	(	(	PUNCT
cana-3950	242	45	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	242	46	,	,	PUNCT
cana-3950	242	47	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	242	48	)	)	PUNCT
cana-3950	242	49	be	be	VERB
cana-3950	242	50	a	a	DET
cana-3950	242	51	beal	beal	NOUN
cana-3950	242	52	’s	’s	PART
cana-3950	242	53	fuzzy	fuzzy	ADJ
cana-3950	242	54	set	set	NOUN
cana-3950	242	55	.	.	PUNCT
cana-3950	243	1	then	then	ADV
cana-3950	243	2	the	the	DET
cana-3950	243	3	suggested	suggest	VERB
cana-3950	243	4	score	score	NOUN
cana-3950	243	5	fuzzy	fuzzy	ADJ
cana-3950	243	6	mapping	mapping	NOUN
cana-3950	243	7	𝑆(𝐴	𝑆(𝐴	NOUN
cana-3950	243	8	)	)	PUNCT
cana-3950	243	9	∈	∈	NOUN
cana-3950	244	1	[	[	X
cana-3950	244	2	−1,1	−1,1	X
cana-3950	244	3	]	]	PUNCT
cana-3950	244	4	.	.	PUNCT
cana-3950	245	1	proof	proof	NOUN
cana-3950	245	2	:	:	PUNCT
cana-3950	245	3	since	since	SCONJ
cana-3950	245	4	for	for	ADP
cana-3950	245	5	any	any	DET
cana-3950	245	6	beal	beal	NOUN
cana-3950	245	7	’s	’s	PART
cana-3950	245	8	fuzzy	fuzzy	ADJ
cana-3950	245	9	set	set	VERB
cana-3950	245	10	a.	a.	NOUN
cana-3950	245	11	we	we	PRON
cana-3950	245	12	have	have	VERB
cana-3950	245	13	𝐽𝐴	𝐽𝐴	NOUN
cana-3950	245	14	𝑛	𝑛	PROPN
cana-3950	245	15	+	+	CCONJ
cana-3950	245	16	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	245	17	𝑛	𝑛	ADJ
cana-3950	245	18	≤	≤	NUM
cana-3950	245	19	1	1	NUM
cana-3950	245	20	hence	hence	ADV
cana-3950	245	21	,	,	PUNCT
cana-3950	245	22	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	245	23	𝑛	𝑛	DET
cana-3950	245	24	−	−	PROPN
cana-3950	245	25	𝐾𝐴	𝐾𝐴	NOUN
cana-3950	245	26	𝑛	𝑛	DET
cana-3950	245	27	≤	≤	ADJ
cana-3950	245	28	𝐽𝐴	𝐽𝐴	NOUN
cana-3950	245	29	𝑛	𝑛	ADP
cana-3950	245	30	≤	≤	NUM
cana-3950	245	31	1	1	NUM
cana-3950	245	32	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-3950	245	33	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	245	34	𝑛	𝑛	PRON
cana-3950	245	35	−	−	PROPN
cana-3950	245	36	𝐾𝐴	𝐾𝐴	NOUN
cana-3950	245	37	𝑛	𝑛	PROPN
cana-3950	245	38	≥	≥	NOUN
cana-3950	245	39	−𝐾𝐴	−𝐾𝐴	VERB
cana-3950	245	40	𝑚	𝑚	PROPN
cana-3950	245	41	≥	≥	NUM
cana-3950	245	42	−1	−1	NOUN
cana-3950	245	43	thus	thus	ADV
cana-3950	245	44	−1	−1	NOUN
cana-3950	245	45	≤	≤	ADJ
cana-3950	245	46	𝐽𝐴	𝐽𝐴	NOUN
cana-3950	245	47	𝑛	𝑛	DET
cana-3950	245	48	−	−	PROPN
cana-3950	246	1	𝐾𝐴	𝐾𝐴	NOUN
cana-3950	246	2	𝑛	𝑛	ADJ
cana-3950	246	3	≤	≤	NUM
cana-3950	246	4	1	1	NUM
cana-3950	246	5	,	,	PUNCT
cana-3950	246	6	namely	namely	ADV
cana-3950	246	7	𝑆(𝐴	𝑆(𝐴	NUM
cana-3950	246	8	)	)	PUNCT
cana-3950	246	9	∈	∈	NOUN
cana-3950	247	1	[	[	X
cana-3950	247	2	−1,1	−1,1	NOUN
cana-3950	247	3	]	]	X
cana-3950	247	4	.	.	PUNCT
cana-3950	248	1	in	in	ADP
cana-3950	248	2	such	such	ADJ
cana-3950	248	3	way	way	NOUN
cana-3950	248	4	that	that	SCONJ
cana-3950	248	5	,	,	PUNCT
cana-3950	248	6	if	if	SCONJ
cana-3950	248	7	𝐴	𝐴	PROPN
cana-3950	248	8	=	=	SYM
cana-3950	248	9	(	(	PUNCT
cana-3950	248	10	0,1	0,1	NUM
cana-3950	248	11	)	)	PUNCT
cana-3950	248	12	then	then	ADV
cana-3950	248	13	𝑆(𝐴	𝑆(𝐴	NUM
cana-3950	248	14	)	)	PUNCT
cana-3950	248	15	=	=	SYM
cana-3950	248	16	−1	−1	NOUN
cana-3950	248	17	and	and	CCONJ
cana-3950	248	18	if	if	SCONJ
cana-3950	248	19	𝐴	𝐴	PROPN
cana-3950	248	20	=	=	SYM
cana-3950	248	21	(	(	PUNCT
cana-3950	248	22	0,1	0,1	NUM
cana-3950	248	23	)	)	PUNCT
cana-3950	248	24	then	then	ADV
cana-3950	248	25	𝑆(𝐴	𝑆(𝐴	NUM
cana-3950	248	26	)	)	PUNCT
cana-3950	248	27	=	=	SYM
cana-3950	249	1	1	1	X
cana-3950	249	2	.	.	X
cana-3950	249	3	remark	remark	NOUN
cana-3950	249	4	:	:	PUNCT
cana-3950	249	5	for	for	SCONJ
cana-3950	249	6	any	any	DET
cana-3950	249	7	beal	beal	NOUN
cana-3950	249	8	’s	’s	PART
cana-3950	249	9	fuzzy	fuzzy	ADJ
cana-3950	249	10	set	set	VERB
cana-3950	249	11	𝐴1	𝐴1	PROPN
cana-3950	249	12	=	=	SYM
cana-3950	249	13	(	(	PUNCT
cana-3950	249	14	𝐽𝐴1	𝐽𝐴1	PROPN
cana-3950	249	15	,	,	PUNCT
cana-3950	249	16	𝐾𝐴1	𝐾𝐴1	PROPN
cana-3950	249	17	)	)	PUNCT
cana-3950	249	18	and	and	CCONJ
cana-3950	249	19	𝐴2	𝐴2	PROPN
cana-3950	249	20	=	=	SYM
cana-3950	249	21	(	(	PUNCT
cana-3950	249	22	𝐽𝐴2	𝐽𝐴2	PROPN
cana-3950	249	23	,	,	PUNCT
cana-3950	249	24	𝐾𝐴2	𝐾𝐴2	PROPN
cana-3950	249	25	)	)	PUNCT
cana-3950	249	26	,	,	PUNCT
cana-3950	249	27	the	the	DET
cana-3950	249	28	comparison	comparison	NOUN
cana-3950	249	29	technique	technique	NOUN
cana-3950	249	30	is	be	AUX
cana-3950	249	31	supposed	suppose	VERB
cana-3950	249	32	as	as	ADP
cana-3950	249	33	(	(	PUNCT
cana-3950	249	34	𝑖)𝐼𝑓	𝑖)𝐼𝑓	NOUN
cana-3950	249	35	𝑆(𝐴1	𝑆(𝐴1	NOUN
cana-3950	249	36	)	)	PUNCT
cana-3950	249	37	<	<	X
cana-3950	249	38	𝑆(𝐴2	𝑆(𝐴2	PROPN
cana-3950	249	39	)	)	PUNCT
cana-3950	249	40	𝑡ℎ𝑒𝑛	𝑡ℎ𝑒𝑛	VERB
cana-3950	249	41	𝐴1	𝐴1	PROPN
cana-3950	249	42	<	<	X
cana-3950	249	43	𝐴2	𝐴2	PROPN
cana-3950	249	44	(	(	PUNCT
cana-3950	249	45	𝑖𝑖)𝐼𝑓	𝑖𝑖)𝐼𝑓	PROPN
cana-3950	249	46	𝑆(𝐴1	𝑆(𝐴1	NOUN
cana-3950	249	47	)	)	PUNCT
cana-3950	249	48	>	>	X
cana-3950	249	49	𝑆(𝐴2	𝑆(𝐴2	PROPN
cana-3950	249	50	)	)	PUNCT
cana-3950	249	51	𝑡ℎ𝑒𝑛	𝑡ℎ𝑒𝑛	VERB
cana-3950	249	52	𝐴1	𝐴1	PROPN
cana-3950	249	53	>	>	X
cana-3950	249	54	𝐴2	𝐴2	PROPN
cana-3950	249	55	(	(	PUNCT
cana-3950	249	56	𝑖𝑖𝑖)𝐼𝑓	𝑖𝑖𝑖)𝐼𝑓	PROPN
cana-3950	249	57	𝑆(𝐴1	𝑆(𝐴1	NOUN
cana-3950	249	58	)	)	PUNCT
cana-3950	250	1	=	=	SYM
cana-3950	250	2	𝑆(𝐴2	𝑆(𝐴2	NOUN
cana-3950	250	3	)	)	PUNCT
cana-3950	250	4	𝑡ℎ𝑒𝑛	𝑡ℎ𝑒𝑛	VERB
cana-3950	250	5	𝐴1	𝐴1	PROPN
cana-3950	251	1	=	=	PUNCT
cana-3950	251	2	𝐴2	𝐴2	PROPN
cana-3950	251	3	(	(	PUNCT
cana-3950	251	4	𝑖𝑣	𝑖𝑣	PROPN
cana-3950	251	5	)	)	PUNCT
cana-3950	251	6	𝛼	𝛼	NOUN
cana-3950	251	7	̅(𝐴1	̅(𝐴1	NOUN
cana-3950	251	8	)	)	PUNCT
cana-3950	251	9	<	<	X
cana-3950	251	10	𝛼	𝛼	X
cana-3950	251	11	̅(𝐴2	̅(𝐴2	NOUN
cana-3950	251	12	)	)	PUNCT
cana-3950	251	13	𝑡ℎ𝑒𝑛	𝑡ℎ𝑒𝑛	VERB
cana-3950	251	14	𝐴1	𝐴1	PROPN
cana-3950	251	15	<	<	X
cana-3950	251	16	𝐴2	𝐴2	PROPN
cana-3950	251	17	(	(	PUNCT
cana-3950	251	18	𝑣)𝐼𝑓	𝑣)𝐼𝑓	NOUN
cana-3950	251	19	𝛼	𝛼	NOUN
cana-3950	251	20	̅(𝐴1	̅(𝐴1	NOUN
cana-3950	251	21	)	)	PUNCT
cana-3950	251	22	>	>	X
cana-3950	252	1	𝛼	𝛼	X
cana-3950	252	2	̅(𝐴2	̅(𝐴2	NOUN
cana-3950	252	3	)	)	PUNCT
cana-3950	252	4	𝑡ℎ𝑒𝑛	𝑡ℎ𝑒𝑛	VERB
cana-3950	252	5	𝐴1	𝐴1	PROPN
cana-3950	252	6	>	>	X
cana-3950	252	7	𝐴2	𝐴2	PROPN
cana-3950	252	8	(	(	PUNCT
cana-3950	252	9	𝑣𝑖)𝐼𝑓	𝑣𝑖)𝐼𝑓	PROPN
cana-3950	252	10	𝛼	𝛼	PRON
cana-3950	252	11	̅(𝐴1	̅(𝐴1	NOUN
cana-3950	252	12	)	)	PUNCT
cana-3950	253	1	=	=	SYM
cana-3950	253	2	𝛼	𝛼	NOUN
cana-3950	253	3	̅(𝐴2	̅(𝐴2	NOUN
cana-3950	253	4	)	)	PUNCT
cana-3950	253	5	𝑡ℎ𝑒𝑛	𝑡ℎ𝑒𝑛	VERB
cana-3950	253	6	𝐴1	𝐴1	PROPN
cana-3950	254	1	=	=	SYM
cana-3950	254	2	𝐴2	𝐴2	PROPN
cana-3950	254	3	.	.	PUNCT
cana-3950	254	4	beal	beal	PROPN
cana-3950	254	5	’s	’s	PART
cana-3950	254	6	fuzzy	fuzzy	ADJ
cana-3950	254	7	weighted	weight	VERB
cana-3950	254	8	power	power	NOUN
cana-3950	254	9	average	average	NOUN
cana-3950	254	10	in	in	ADP
cana-3950	254	11	this	this	DET
cana-3950	254	12	section	section	NOUN
cana-3950	254	13	,	,	PUNCT
cana-3950	254	14	we	we	PRON
cana-3950	254	15	study	study	VERB
cana-3950	254	16	the	the	DET
cana-3950	254	17	various	various	ADJ
cana-3950	254	18	operation	operation	NOUN
cana-3950	254	19	on	on	ADP
cana-3950	254	20	beal	beal	NOUN
cana-3950	254	21	’s	’s	PART
cana-3950	254	22	fuzzy	fuzzy	ADJ
cana-3950	254	23	sets	set	NOUN
cana-3950	254	24	,	,	PUNCT
cana-3950	254	25	and	and	CCONJ
cana-3950	254	26	some	some	DET
cana-3950	254	27	intellectual	intellectual	ADJ
cana-3950	254	28	structure	structure	NOUN
cana-3950	254	29	are	be	AUX
cana-3950	254	30	indicated	indicate	VERB
cana-3950	254	31	in	in	ADP
cana-3950	254	32	detail	detail	NOUN
cana-3950	254	33	.	.	PUNCT
cana-3950	255	1	definition	definition	NOUN
cana-3950	255	2	4.1	4.1	NUM
cana-3950	255	3	.	.	PUNCT
cana-3950	256	1	let	let	VERB
cana-3950	256	2	𝐴𝑖	𝐴𝑖	PROPN
cana-3950	256	3	=	=	PUNCT
cana-3950	256	4	(	(	PUNCT
cana-3950	256	5	𝐽𝐴𝑖	𝐽𝐴𝑖	NOUN
cana-3950	256	6	,	,	PUNCT
cana-3950	256	7	𝐾𝐴𝑖	𝐾𝐴𝑖	PROPN
cana-3950	256	8	)	)	PUNCT
cana-3950	256	9	(	(	PUNCT
cana-3950	256	10	𝑖	𝑖	SYM
cana-3950	256	11	=	=	SYM
cana-3950	256	12	1,2	1,2	NUM
cana-3950	256	13	,	,	PUNCT
cana-3950	256	14	…	…	PUNCT
cana-3950	256	15	.	.	PUNCT
cana-3950	257	1	𝑘	𝑘	X
cana-3950	257	2	)	)	PUNCT
cana-3950	257	3	be	be	VERB
cana-3950	257	4	a	a	DET
cana-3950	257	5	value	value	NOUN
cana-3950	257	6	of	of	ADP
cana-3950	257	7	beal	beal	NOUN
cana-3950	257	8	’s	’s	PART
cana-3950	257	9	fuzzy	fuzzy	ADJ
cana-3950	257	10	sets	set	NOUN
cana-3950	257	11	and	and	CCONJ
cana-3950	257	12	∈=	∈=	PUNCT
cana-3950	257	13	(	(	PUNCT
cana-3950	257	14	∈1	∈1	NOUN
cana-3950	257	15	,	,	PUNCT
cana-3950	257	16	∈2	∈2	ADJ
cana-3950	257	17	,	,	PUNCT
cana-3950	257	18	…	…	PUNCT
cana-3950	257	19	∈𝑘	∈𝑘	NUM
cana-3950	257	20	)	)	PUNCT
cana-3950	257	21	𝑇	𝑇	PROPN
cana-3950	257	22	be	be	AUX
cana-3950	257	23	a	a	DET
cana-3950	257	24	weighted	weighted	ADJ
cana-3950	257	25	vector	vector	NOUN
cana-3950	257	26	of	of	ADP
cana-3950	257	27	𝐴𝑖	𝐴𝑖	PROPN
cana-3950	257	28	with	with	ADP
cana-3950	257	29	∈𝑖	∈𝑖	NOUN
cana-3950	257	30	>	>	X
cana-3950	257	31	0	0	NUM
cana-3950	257	32	,	,	PUNCT
cana-3950	257	33	∑	∑	PROPN
cana-3950	257	34	∈𝑖=	∈𝑖=	NOUN
cana-3950	257	35	1.𝑘	1.𝑘	NUM
cana-3950	257	36	𝑖=1	𝑖=1	PUNCT
cana-3950	258	1	then	then	ADV
cana-3950	258	2	a	a	DET
cana-3950	258	3	beal	beal	NOUN
cana-3950	258	4	’s	’s	PART
cana-3950	258	5	fuzzy	fuzzy	ADJ
cana-3950	258	6	weighted	weight	VERB
cana-3950	258	7	power	power	NOUN
cana-3950	258	8	mean	mean	NOUN
cana-3950	258	9	operator	operator	NOUN
cana-3950	258	10	is	be	AUX
cana-3950	258	11	a	a	DET
cana-3950	258	12	function	function	NOUN
cana-3950	258	13	𝐵𝐹𝑊𝑃𝐴	𝐵𝐹𝑊𝑃𝐴	PROPN
cana-3950	258	14	:	:	PUNCT
cana-3950	259	1	𝐴𝑘	𝐴𝑘	PROPN
cana-3950	259	2	→	→	SYM
cana-3950	259	3	𝐴	𝐴	PROPN
cana-3950	259	4	where	where	SCONJ
cana-3950	259	5	communications	communication	NOUN
cana-3950	259	6	on	on	ADP
cana-3950	259	7	applied	apply	VERB
cana-3950	259	8	nonlinear	nonlinear	ADJ
cana-3950	259	9	analysis	analysis	NOUN
cana-3950	259	10	issn	issn	NOUN
cana-3950	259	11	:	:	PUNCT
cana-3950	259	12	1074	1074	NUM
cana-3950	259	13	-	-	PUNCT
cana-3950	259	14	133x	133x	NUM
cana-3950	259	15	vol	vol	NOUN
cana-3950	259	16	32	32	NUM
cana-3950	259	17	no	no	NOUN
cana-3950	259	18	.	.	PUNCT
cana-3950	260	1	9s	9s	NUM
cana-3950	260	2	(	(	PUNCT
cana-3950	260	3	2025	2025	NUM
cana-3950	260	4	)	)	PUNCT
cana-3950	260	5	400	400	NUM
cana-3950	260	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3950	260	7	𝐵𝐹𝑊𝑃𝐴(𝐴1	𝐵𝐹𝑊𝑃𝐴(𝐴1	PROPN
cana-3950	260	8	,	,	PUNCT
cana-3950	260	9	𝐴2	𝐴2	PROPN
cana-3950	260	10	,	,	PUNCT
cana-3950	260	11	…	…	PUNCT
cana-3950	260	12	𝐴𝑘	𝐴𝑘	PROPN
cana-3950	260	13	)	)	PUNCT
cana-3950	260	14	=	=	SYM
cana-3950	260	15	(	(	PUNCT
cana-3950	260	16	(	(	PUNCT
cana-3950	260	17	∑	∑	X
cana-3950	260	18	∈𝑖	∈𝑖	VERB
cana-3950	260	19	𝐽𝐴𝑖	𝐽𝐴𝑖	NOUN
cana-3950	260	20	𝑛𝑘	𝑛𝑘	X
cana-3950	260	21	𝑖=1	𝑖=1	PROPN
cana-3950	260	22	)	)	PUNCT
cana-3950	260	23	1	1	NUM
cana-3950	260	24	𝑛	𝑛	PROPN
cana-3950	260	25	,	,	PUNCT
cana-3950	260	26	(	(	PUNCT
cana-3950	260	27	∑	∑	PUNCT
cana-3950	260	28	∈𝑖	∈𝑖	VERB
cana-3950	260	29	𝐾𝐴𝑖	𝐾𝐴𝑖	NOUN
cana-3950	260	30	𝑚𝑘	𝑚𝑘	PRON
cana-3950	260	31	𝑖=1	𝑖=1	PROPN
cana-3950	260	32	)	)	PUNCT
cana-3950	260	33	1	1	NUM
cana-3950	260	34	𝑚	𝑚	NOUN
cana-3950	260	35	)	)	PUNCT
cana-3950	260	36	.	.	PUNCT
cana-3950	261	1	example	example	NOUN
cana-3950	261	2	4.2	4.2	NUM
cana-3950	261	3	.	.	PUNCT
cana-3950	262	1	consider	consider	VERB
cana-3950	262	2	𝐴1	𝐴1	PROPN
cana-3950	262	3	=	=	SYM
cana-3950	262	4	(	(	PUNCT
cana-3950	262	5	0.8	0.8	NUM
cana-3950	262	6	,	,	PUNCT
cana-3950	262	7	0.4	0.4	NUM
cana-3950	262	8	)	)	PUNCT
cana-3950	262	9	,	,	PUNCT
cana-3950	262	10	𝐴2	𝐴2	PROPN
cana-3950	262	11	=	=	PUNCT
cana-3950	262	12	(	(	PUNCT
cana-3950	262	13	0.3	0.3	NUM
cana-3950	262	14	,	,	PUNCT
cana-3950	262	15	0.9	0.9	NUM
cana-3950	262	16	)	)	PUNCT
cana-3950	262	17	,	,	PUNCT
cana-3950	262	18	𝐴3	𝐴3	PROPN
cana-3950	262	19	=	=	PUNCT
cana-3950	262	20	(	(	PUNCT
cana-3950	262	21	0.6	0.6	NUM
cana-3950	262	22	,	,	PUNCT
cana-3950	262	23	0.7	0.7	NUM
cana-3950	262	24	)	)	PUNCT
cana-3950	262	25	,	,	PUNCT
cana-3950	262	26	𝐴4	𝐴4	PROPN
cana-3950	262	27	=	=	PUNCT
cana-3950	262	28	(	(	PUNCT
cana-3950	262	29	0.8	0.8	NUM
cana-3950	262	30	,	,	PUNCT
cana-3950	262	31	0.7	0.7	NUM
cana-3950	262	32	)	)	PUNCT
cana-3950	262	33	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-3950	262	34	𝐴5	𝐴5	NOUN
cana-3950	262	35	=	=	PUNCT
cana-3950	262	36	(	(	PUNCT
cana-3950	262	37	0.9	0.9	NUM
cana-3950	262	38	,	,	PUNCT
cana-3950	262	39	0.5	0.5	NUM
cana-3950	262	40	)	)	PUNCT
cana-3950	262	41	as	as	ADP
cana-3950	262	42	five	five	NUM
cana-3950	262	43	beal	beal	NOUN
cana-3950	262	44	’s	’s	PART
cana-3950	262	45	fuzzy	fuzzy	ADJ
cana-3950	262	46	sets	set	NOUN
cana-3950	262	47	and	and	CCONJ
cana-3950	262	48	let	let	VERB
cana-3950	262	49	∈=	∈=	NUM
cana-3950	262	50	(	(	PUNCT
cana-3950	262	51	0.18	0.18	NUM
cana-3950	262	52	,	,	PUNCT
cana-3950	262	53	0.27	0.27	NUM
cana-3950	262	54	,	,	PUNCT
cana-3950	262	55	0.25	0.25	NUM
cana-3950	262	56	,	,	PUNCT
cana-3950	262	57	0.16	0.16	NUM
cana-3950	262	58	,	,	PUNCT
cana-3950	262	59	0.14)𝑇	0.14)𝑇	PROPN
cana-3950	262	60	be	be	AUX
cana-3950	262	61	a	a	DET
cana-3950	262	62	weighted	weighted	ADJ
cana-3950	262	63	vector	vector	NOUN
cana-3950	262	64	of	of	ADP
cana-3950	262	65	𝐴𝑖(𝑖	𝐴𝑖(𝑖	PROPN
cana-3950	262	66	=	=	SYM
cana-3950	262	67	1,2,3,4,5	1,2,3,4,5	NUM
cana-3950	262	68	)	)	PUNCT
cana-3950	262	69	.	.	PUNCT
cana-3950	263	1	then	then	ADV
cana-3950	263	2	bfwpa	bfwpa	PROPN
cana-3950	263	3	(	(	PUNCT
cana-3950	263	4	𝐴1,𝐴2	𝐴1,𝐴2	PROPN
cana-3950	263	5	,	,	PUNCT
cana-3950	263	6	…	…	PUNCT
cana-3950	263	7	.	.	PUNCT
cana-3950	264	1	𝐴5	𝐴5	NOUN
cana-3950	264	2	)	)	PUNCT
cana-3950	264	3	=	=	PUNCT
cana-3950	265	1	(	(	PUNCT
cana-3950	265	2	(	(	PUNCT
cana-3950	265	3	0.8𝑛	0.8𝑛	NOUN
cana-3950	265	4	×	×	NOUN
cana-3950	265	5	0.18	0.18	NUM
cana-3950	265	6	+	+	CCONJ
cana-3950	265	7	0.3𝑛	0.3𝑛	PROPN
cana-3950	265	8	×	×	NOUN
cana-3950	265	9	0.27	0.27	NUM
cana-3950	265	10	+	+	NUM
cana-3950	265	11	0.6𝑛	0.6𝑛	PRON
cana-3950	265	12	×	×	NOUN
cana-3950	265	13	0.25	0.25	NUM
cana-3950	265	14	+	+	NUM
cana-3950	265	15	0.8𝑛	0.8𝑛	NOUN
cana-3950	265	16	×	×	NOUN
cana-3950	265	17	0.16	0.16	NUM
cana-3950	265	18	+	+	CCONJ
cana-3950	265	19	0.9𝑛	0.9𝑛	ADJ
cana-3950	265	20	×	×	NOUN
cana-3950	265	21	0.14	0.14	NUM
cana-3950	265	22	)	)	PUNCT
cana-3950	265	23	1	1	NUM
cana-3950	265	24	𝑛	𝑛	NOUN
cana-3950	265	25	,	,	PUNCT
cana-3950	265	26	(	(	PUNCT
cana-3950	265	27	0.4𝑛	0.4𝑛	X
cana-3950	265	28	×	×	NOUN
cana-3950	265	29	0.18	0.18	NUM
cana-3950	265	30	+	+	NOUN
cana-3950	265	31	0.9𝑛	0.9𝑛	ADJ
cana-3950	265	32	×	×	NOUN
cana-3950	265	33	0.27	0.27	NUM
cana-3950	265	34	+	+	CCONJ
cana-3950	265	35	0.7𝑛	0.7𝑛	PROPN
cana-3950	265	36	×	×	NOUN
cana-3950	265	37	0.25	0.25	NUM
cana-3950	265	38	+	+	CCONJ
cana-3950	265	39	0.7𝑛	0.7𝑛	NUM
cana-3950	265	40	×	×	NOUN
cana-3950	265	41	0.16	0.16	NUM
cana-3950	265	42	+	+	CCONJ
cana-3950	265	43	0.5𝑛	0.5𝑛	NUM
cana-3950	265	44	×	×	NOUN
cana-3950	265	45	0.14	0.14	NUM
cana-3950	265	46	)	)	PUNCT
cana-3950	265	47	1	1	NUM
cana-3950	265	48	𝑚	𝑚	NOUN
cana-3950	265	49	)	)	PUNCT
cana-3950	265	50	=	=	NOUN
cana-3950	265	51	{	{	PUNCT
cana-3950	265	52	0.667	0.667	NUM
cana-3950	265	53	,	,	PUNCT
cana-3950	265	54	0.818	0.818	NUM
cana-3950	265	55	𝑖𝑓	𝑖𝑓	NOUN
cana-3950	265	56	𝑛	𝑛	PRON
cana-3950	265	57	=	=	SYM
cana-3950	265	58	2	2	NUM
cana-3950	265	59	,	,	PUNCT
cana-3950	265	60	𝑚	𝑚	X
cana-3950	265	61	=	=	SYM
cana-3950	265	62	3	3	NUM
cana-3950	265	63	0.696,0.832	0.696,0.832	NUM
cana-3950	265	64	𝑖𝑓	𝑖𝑓	NUM
cana-3950	265	65	𝑛	𝑛	PROPN
cana-3950	265	66	=	=	SYM
cana-3950	265	67	3	3	NUM
cana-3950	265	68	,	,	PUNCT
cana-3950	265	69	𝑚	𝑚	X
cana-3950	265	70	=	=	SYM
cana-3950	265	71	5	5	NUM
cana-3950	265	72	0.717,0.862	0.717,0.862	NUM
cana-3950	265	73	𝑖𝑓	𝑖𝑓	NOUN
cana-3950	265	74	𝑛	𝑛	NOUN
cana-3950	265	75	=	=	SYM
cana-3950	265	76	4	4	NUM
cana-3950	265	77	,	,	PUNCT
cana-3950	265	78	𝑚	𝑚	NOUN
cana-3950	265	79	=	=	SYM
cana-3950	265	80	8	8	NUM
cana-3950	265	81	0.783,0.851	0.783,0.851	NUM
cana-3950	265	82	𝑖𝑓	𝑖𝑓	NUM
cana-3950	265	83	𝑛	𝑛	NOUN
cana-3950	265	84	=	=	SYM
cana-3950	265	85	10	10	NUM
cana-3950	265	86	,	,	PUNCT
cana-3950	265	87	𝑚	𝑚	X
cana-3950	265	88	=	=	SYM
cana-3950	265	89	14	14	NUM
cana-3950	265	90	0.824,0.89	0.824,0.89	NUM
cana-3950	265	91	𝑖𝑓	𝑖𝑓	ADP
cana-3950	265	92	𝑛	𝑛	PRON
cana-3950	265	93	=	=	SYM
cana-3950	265	94	20	20	NUM
cana-3950	265	95	,	,	PUNCT
cana-3950	265	96	𝑚	𝑚	X
cana-3950	265	97	=	=	SYM
cana-3950	265	98	30	30	NUM
cana-3950	265	99	0.875,0.84	0.875,0.84	NUM
cana-3950	265	100	𝑖𝑓	𝑖𝑓	NUM
cana-3950	265	101	𝑛	𝑛	PRON
cana-3950	265	102	=	=	SYM
cana-3950	265	103	70	70	NUM
cana-3950	265	104	,	,	PUNCT
cana-3950	265	105	𝑚	𝑚	X
cana-3950	265	106	=	=	SYM
cana-3950	265	107	50	50	NUM
cana-3950	265	108	.	.	PUNCT
cana-3950	265	109	theorem	theorem	VERB
cana-3950	265	110	4.3	4.3	NUM
cana-3950	265	111	.	.	PUNCT
cana-3950	266	1	let	let	VERB
cana-3950	266	2	𝐴𝑖	𝐴𝑖	PROPN
cana-3950	266	3	=	=	PUNCT
cana-3950	266	4	(	(	PUNCT
cana-3950	266	5	𝐽𝐴𝑖	𝐽𝐴𝑖	PROPN
cana-3950	266	6	,	,	PUNCT
cana-3950	266	7	𝐾𝐴𝑖)(𝑖	𝐾𝐴𝑖)(𝑖	NOUN
cana-3950	266	8	=	=	SYM
cana-3950	266	9	1,2	1,2	NUM
cana-3950	266	10	,	,	PUNCT
cana-3950	266	11	…	…	PUNCT
cana-3950	266	12	𝑘	𝑘	X
cana-3950	266	13	)	)	PUNCT
cana-3950	266	14	be	be	VERB
cana-3950	266	15	a	a	DET
cana-3950	266	16	family	family	NOUN
cana-3950	266	17	of	of	ADP
cana-3950	266	18	beal	beal	NOUN
cana-3950	266	19	’s	’s	PART
cana-3950	266	20	fuzzy	fuzzy	ADJ
cana-3950	266	21	sets	set	NOUN
cana-3950	266	22	and	and	CCONJ
cana-3950	266	23	let	let	VERB
cana-3950	266	24	∈=	∈=	NUM
cana-3950	266	25	(	(	PUNCT
cana-3950	266	26	∈1	∈1	NOUN
cana-3950	266	27	,	,	PUNCT
cana-3950	266	28	∈2	∈2	NOUN
cana-3950	266	29	.	.	PUNCT
cana-3950	266	30	.	.	PUNCT
cana-3950	267	1	∈𝑘	∈𝑘	X
cana-3950	267	2	)	)	PUNCT
cana-3950	267	3	𝑇	𝑇	PROPN
cana-3950	267	4	be	be	AUX
cana-3950	267	5	a	a	DET
cana-3950	267	6	weighted	weighted	ADJ
cana-3950	267	7	vector	vector	NOUN
cana-3950	267	8	of	of	ADP
cana-3950	267	9	𝐴𝑖	𝐴𝑖	PROPN
cana-3950	267	10	with	with	ADP
cana-3950	267	11	∈𝑖	∈𝑖	VERB
cana-3950	267	12	>	>	X
cana-3950	267	13	0	0	NUM
cana-3950	267	14	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-3950	267	15	∑	∑	PROPN
cana-3950	267	16	∈𝑖=	∈𝑖=	PROPN
cana-3950	267	17	1𝑘	1𝑘	NUM
cana-3950	267	18	𝑖=1	𝑖=1	PUNCT
cana-3950	267	19	.	.	PUNCT
cana-3950	268	1	then	then	ADV
cana-3950	268	2	beal	beal	PROPN
cana-3950	268	3	’s	’s	PART
cana-3950	268	4	fuzzy	fuzzy	ADJ
cana-3950	268	5	weighted	weight	VERB
cana-3950	268	6	power	power	NOUN
cana-3950	268	7	mean	mean	NOUN
cana-3950	268	8	(	(	PUNCT
cana-3950	268	9	𝐴1	𝐴1	PROPN
cana-3950	268	10	,	,	PUNCT
cana-3950	268	11	𝐴2	𝐴2	PROPN
cana-3950	268	12	,	,	PUNCT
cana-3950	268	13	…	…	PUNCT
cana-3950	268	14	𝐴𝑘	𝐴𝑘	PROPN
cana-3950	268	15	)	)	PUNCT
cana-3950	268	16	is	be	AUX
cana-3950	268	17	a	a	DET
cana-3950	268	18	beal	beal	NOUN
cana-3950	268	19	’s	’s	PART
cana-3950	268	20	fuzzy	fuzzy	ADJ
cana-3950	268	21	set	set	NOUN
cana-3950	268	22	.	.	PUNCT
cana-3950	269	1	theorem	theorem	VERB
cana-3950	269	2	4.4	4.4	NUM
cana-3950	269	3	.	.	PUNCT
cana-3950	270	1	let	let	VERB
cana-3950	270	2	𝐴𝑖	𝐴𝑖	PROPN
cana-3950	270	3	=	=	PUNCT
cana-3950	270	4	(	(	PUNCT
cana-3950	270	5	𝐽𝐴𝑖	𝐽𝐴𝑖	NOUN
cana-3950	270	6	,	,	PUNCT
cana-3950	270	7	𝐾𝐴𝑖)(𝑖	𝐾𝐴𝑖)(𝑖	NOUN
cana-3950	270	8	=	=	SYM
cana-3950	270	9	1,2	1,2	NUM
cana-3950	270	10	,	,	PUNCT
cana-3950	270	11	.	.	PUNCT
cana-3950	270	12	.	.	PUNCT
cana-3950	271	1	𝑘	𝑘	X
cana-3950	271	2	)	)	PUNCT
cana-3950	271	3	be	be	VERB
cana-3950	271	4	a	a	DET
cana-3950	271	5	value	value	NOUN
cana-3950	271	6	of	of	ADP
cana-3950	271	7	beal	beal	NOUN
cana-3950	271	8	’s	’s	PART
cana-3950	271	9	fuzzy	fuzzy	ADJ
cana-3950	271	10	sets	set	NOUN
cana-3950	271	11	𝐴	𝐴	PROPN
cana-3950	271	12	=	=	SYM
cana-3950	271	13	(	(	PUNCT
cana-3950	271	14	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	271	15	,	,	PUNCT
cana-3950	271	16	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	271	17	)	)	PUNCT
cana-3950	271	18	be	be	VERB
cana-3950	271	19	a	a	DET
cana-3950	271	20	beal	beal	NOUN
cana-3950	271	21	’s	’s	PART
cana-3950	271	22	fuzzy	fuzzy	ADJ
cana-3950	271	23	set	set	NOUN
cana-3950	271	24	and	and	CCONJ
cana-3950	271	25	∈=	∈=	ADP
cana-3950	271	26	(	(	PUNCT
cana-3950	271	27	∈1	∈1	NOUN
cana-3950	271	28	,	,	PUNCT
cana-3950	271	29	∈2	∈2	ADJ
cana-3950	271	30	,	,	PUNCT
cana-3950	271	31	…	…	PUNCT
cana-3950	271	32	∈𝑘	∈𝑘	NUM
cana-3950	271	33	)	)	PUNCT
cana-3950	271	34	𝑇	𝑇	PROPN
cana-3950	271	35	be	be	AUX
cana-3950	271	36	a	a	DET
cana-3950	271	37	weighted	weighted	ADJ
cana-3950	271	38	vector	vector	NOUN
cana-3950	271	39	of	of	ADP
cana-3950	271	40	𝐴𝑖with	𝐴𝑖with	PROPN
cana-3950	271	41	∑	∑	PROPN
cana-3950	271	42	∈𝑖=	∈𝑖=	PROPN
cana-3950	271	43	1𝑘	1𝑘	NUM
cana-3950	271	44	𝑖=1	𝑖=1	PUNCT
cana-3950	271	45	.	.	PUNCT
cana-3950	272	1	then	then	ADV
cana-3950	272	2	𝐵𝐹𝑊𝑃𝐴(𝐴1⊕𝐴2⊕	𝐵𝐹𝑊𝑃𝐴(𝐴1⊕𝐴2⊕	PROPN
cana-3950	272	3	…	…	SYM
cana-3950	272	4	.𝐴𝑘⊕𝐴	.𝐴𝑘⊕𝐴	PUNCT
cana-3950	272	5	)	)	PUNCT
cana-3950	272	6	≥	≥	NOUN
cana-3950	272	7	𝐵𝐹𝑊𝑃𝐴(𝐴1⨂𝐴2⨂	𝐵𝐹𝑊𝑃𝐴(𝐴1⨂𝐴2⨂	NOUN
cana-3950	272	8	…	…	SYM
cana-3950	272	9	𝐴𝑘⨂𝐴	𝐴𝑘⨂𝐴	NUM
cana-3950	272	10	)	)	PUNCT
cana-3950	272	11	.	.	PUNCT
cana-3950	273	1	theorem	theorem	VERB
cana-3950	273	2	4.5	4.5	NUM
cana-3950	273	3	.	.	PUNCT
cana-3950	274	1	let	let	VERB
cana-3950	274	2	𝐴𝑖	𝐴𝑖	PROPN
cana-3950	274	3	=	=	PUNCT
cana-3950	274	4	(	(	PUNCT
cana-3950	274	5	𝐽𝐴𝑖	𝐽𝐴𝑖	PROPN
cana-3950	274	6	,	,	PUNCT
cana-3950	274	7	𝐾𝐴𝑖)(𝑖	𝐾𝐴𝑖)(𝑖	NOUN
cana-3950	274	8	=	=	SYM
cana-3950	274	9	1,2	1,2	NUM
cana-3950	274	10	,	,	PUNCT
cana-3950	274	11	…	…	PUNCT
cana-3950	274	12	,	,	PUNCT
cana-3950	274	13	𝑘	𝑘	X
cana-3950	274	14	)	)	PUNCT
cana-3950	274	15	be	be	VERB
cana-3950	274	16	a	a	DET
cana-3950	274	17	value	value	NOUN
cana-3950	274	18	of	of	ADP
cana-3950	274	19	beal	beal	NOUN
cana-3950	274	20	’s	’s	PART
cana-3950	274	21	fuzzy	fuzzy	ADJ
cana-3950	274	22	sets	set	NOUN
cana-3950	274	23	,	,	PUNCT
cana-3950	274	24	𝐴	𝐴	PROPN
cana-3950	274	25	=	=	SYM
cana-3950	274	26	(	(	PUNCT
cana-3950	274	27	𝐽𝐴	𝐽𝐴	PROPN
cana-3950	274	28	,	,	PUNCT
cana-3950	274	29	𝐾𝐴	𝐾𝐴	PROPN
cana-3950	274	30	)	)	PUNCT
cana-3950	274	31	be	be	VERB
cana-3950	274	32	a	a	DET
cana-3950	274	33	beal	beal	NOUN
cana-3950	274	34	’s	’s	PART
cana-3950	274	35	fuzzy	fuzzy	ADJ
cana-3950	274	36	set	set	NOUN
cana-3950	274	37	,	,	PUNCT
cana-3950	274	38	and	and	CCONJ
cana-3950	274	39	∈=	∈=	ADP
cana-3950	274	40	(	(	PUNCT
cana-3950	274	41	∈1	∈1	NOUN
cana-3950	274	42	,	,	PUNCT
cana-3950	274	43	∈2	∈2	ADJ
cana-3950	274	44	,	,	PUNCT
cana-3950	274	45	…	…	PUNCT
cana-3950	274	46	∈𝑘	∈𝑘	NUM
cana-3950	274	47	)	)	PUNCT
cana-3950	274	48	𝑇	𝑇	PROPN
cana-3950	274	49	be	be	AUX
cana-3950	274	50	a	a	DET
cana-3950	274	51	weighted	weighted	ADJ
cana-3950	274	52	vector	vector	NOUN
cana-3950	274	53	of	of	ADP
cana-3950	274	54	𝐴𝑖with	𝐴𝑖with	PROPN
cana-3950	274	55	∑	∑	PROPN
cana-3950	274	56	∈𝑖=	∈𝑖=	PROPN
cana-3950	274	57	1.𝑘	1.𝑘	NUM
cana-3950	274	58	𝑖=1	𝑖=1	PUNCT
cana-3950	275	1	then	then	ADV
cana-3950	275	2	(	(	PUNCT
cana-3950	275	3	𝑖)𝐵𝐹𝑊𝑃𝐴	𝑖)𝐵𝐹𝑊𝑃𝐴	PROPN
cana-3950	275	4	(	(	PUNCT
cana-3950	275	5	𝐴1⊕𝐴2⊕	𝐴1⊕𝐴2⊕	PROPN
cana-3950	275	6	…	…	PUNCT
cana-3950	275	7	.𝐴𝑘⊕𝐴	.𝐴𝑘⊕𝐴	PUNCT
cana-3950	275	8	)	)	PUNCT
cana-3950	275	9	≥	≥	PROPN
cana-3950	275	10	𝐵𝐹𝑊𝑃𝐴(𝐴1	𝐵𝐹𝑊𝑃𝐴(𝐴1	PROPN
cana-3950	275	11	,	,	PUNCT
cana-3950	275	12	𝐴2	𝐴2	PROPN
cana-3950	275	13	,	,	PUNCT
cana-3950	275	14	…	…	PUNCT
cana-3950	275	15	𝐴𝐾	𝐴𝐾	PROPN
cana-3950	275	16	)	)	PUNCT
cana-3950	275	17	⊗	⊗	PROPN
cana-3950	275	18	𝐴.	𝐴.	PROPN
cana-3950	275	19	(	(	PUNCT
cana-3950	275	20	𝑖𝑖)𝐵𝐹𝑊𝑃𝐴	𝑖𝑖)𝐵𝐹𝑊𝑃𝐴	PROPN
cana-3950	275	21	(	(	PUNCT
cana-3950	275	22	𝐴1	𝐴1	PROPN
cana-3950	275	23	,	,	PUNCT
cana-3950	275	24	𝐴2	𝐴2	PROPN
cana-3950	275	25	,	,	PUNCT
cana-3950	275	26	…	…	PUNCT
cana-3950	275	27	𝐴𝑘	𝐴𝑘	PROPN
cana-3950	275	28	)	)	PUNCT
cana-3950	275	29	⊕	⊕	PROPN
cana-3950	275	30	𝐴	𝐴	PROPN
cana-3950	275	31	≥	≥	NUM
cana-3950	275	32	𝐵𝐹𝑊𝑃𝐴(𝐴1	𝐵𝐹𝑊𝑃𝐴(𝐴1	PROPN
cana-3950	275	33	,	,	PUNCT
cana-3950	275	34	𝐴2	𝐴2	PROPN
cana-3950	275	35	,	,	PUNCT
cana-3950	275	36	…	…	PUNCT
cana-3950	275	37	,	,	PUNCT
cana-3950	275	38	𝐴𝑘)⨂𝐴.	𝐴𝑘)⨂𝐴.	PROPN
cana-3950	275	39	theorem	theorem	VERB
cana-3950	275	40	4.6	4.6	NUM
cana-3950	275	41	.	.	PUNCT
cana-3950	276	1	let	let	VERB
cana-3950	276	2	𝐴𝑖	𝐴𝑖	PROPN
cana-3950	276	3	=	=	PUNCT
cana-3950	276	4	(	(	PUNCT
cana-3950	276	5	𝐽𝐴𝑖	𝐽𝐴𝑖	NOUN
cana-3950	276	6	,	,	PUNCT
cana-3950	276	7	𝐾𝐴𝑖	𝐾𝐴𝑖	NOUN
cana-3950	276	8	)	)	PUNCT
cana-3950	276	9	and	and	CCONJ
cana-3950	276	10	𝐵𝑖	𝐵𝑖	PROPN
cana-3950	276	11	=	=	PUNCT
cana-3950	276	12	(	(	PUNCT
cana-3950	276	13	𝐽𝐵𝑖	𝐽𝐵𝑖	PROPN
cana-3950	276	14	,	,	PUNCT
cana-3950	276	15	𝐾𝐵𝑖)(𝑖	𝐾𝐵𝑖)(𝑖	NOUN
cana-3950	276	16	=	=	SYM
cana-3950	276	17	1,2	1,2	NUM
cana-3950	276	18	,	,	PUNCT
cana-3950	276	19	…	…	PUNCT
cana-3950	276	20	,	,	PUNCT
cana-3950	276	21	𝑘	𝑘	X
cana-3950	276	22	)	)	PUNCT
cana-3950	276	23	be	be	VERB
cana-3950	276	24	two	two	NUM
cana-3950	276	25	values	value	NOUN
cana-3950	276	26	of	of	ADP
cana-3950	276	27	beal	beal	NOUN
cana-3950	276	28	’s	’s	PART
cana-3950	276	29	fuzzy	fuzzy	ADJ
cana-3950	276	30	sets	set	NOUN
cana-3950	276	31	and	and	CCONJ
cana-3950	276	32	∈=	∈=	PUNCT
cana-3950	276	33	(	(	PUNCT
cana-3950	276	34	∈1	∈1	NOUN
cana-3950	276	35	,	,	PUNCT
cana-3950	276	36	∈2	∈2	ADJ
cana-3950	276	37	,	,	PUNCT
cana-3950	276	38	…	…	PUNCT
cana-3950	276	39	∈𝑘	∈𝑘	NUM
cana-3950	276	40	)	)	PUNCT
cana-3950	276	41	𝑇	𝑇	PROPN
cana-3950	276	42	be	be	AUX
cana-3950	276	43	a	a	DET
cana-3950	276	44	weighted	weighted	ADJ
cana-3950	276	45	vector	vector	NOUN
cana-3950	276	46	of	of	ADP
cana-3950	276	47	them	they	PRON
cana-3950	276	48	with	with	ADP
cana-3950	276	49	∑	∑	PROPN
cana-3950	276	50	∈𝑖=	∈𝑖=	PROPN
cana-3950	276	51	1𝑘	1𝑘	NUM
cana-3950	276	52	𝑖=1	𝑖=1	PUNCT
cana-3950	276	53	.	.	PUNCT
cana-3950	277	1	then	then	ADV
cana-3950	277	2	(	(	PUNCT
cana-3950	277	3	𝑖)𝐵𝐹𝑊𝑃𝐴(𝐴1⨁𝐵1	𝑖)𝐵𝐹𝑊𝑃𝐴(𝐴1⨁𝐵1	NOUN
cana-3950	277	4	,	,	PUNCT
cana-3950	277	5	𝐴2⨁𝐵2	𝐴2⨁𝐵2	NOUN
cana-3950	277	6	,	,	PUNCT
cana-3950	277	7	…	…	PUNCT
cana-3950	277	8	,	,	PUNCT
cana-3950	277	9	𝐴𝑘⊕𝐵𝑘	𝐴𝑘⊕𝐵𝑘	NOUN
cana-3950	277	10	)	)	PUNCT
cana-3950	277	11	≥	≥	NOUN
cana-3950	277	12	(	(	PUNCT
cana-3950	277	13	𝐴1⨂𝐵1	𝐴1⨂𝐵1	PROPN
cana-3950	277	14	,	,	PUNCT
cana-3950	277	15	𝐴2⨂𝐵2	𝐴2⨂𝐵2	PROPN
cana-3950	277	16	,	,	PUNCT
cana-3950	277	17	…	…	PUNCT
cana-3950	277	18	,	,	PUNCT
cana-3950	277	19	𝐴𝑘⨂𝐵𝑘	𝐴𝑘⨂𝐵𝑘	PROPN
cana-3950	277	20	)	)	PUNCT
cana-3950	277	21	.	.	PUNCT
cana-3950	278	1	(	(	PUNCT
cana-3950	278	2	𝑖𝑖)𝐵𝐹𝑊𝑃𝐴	𝑖𝑖)𝐵𝐹𝑊𝑃𝐴	PROPN
cana-3950	278	3	(	(	PUNCT
cana-3950	278	4	𝐴1	𝐴1	PROPN
cana-3950	278	5	,	,	PUNCT
cana-3950	278	6	𝐴2	𝐴2	PROPN
cana-3950	278	7	,	,	PUNCT
cana-3950	278	8	…	…	PUNCT
cana-3950	278	9	,	,	PUNCT
cana-3950	278	10	𝐴𝑘)⨁𝐵𝐹𝑊𝑃𝐴	𝐴𝑘)⨁𝐵𝐹𝑊𝑃𝐴	PROPN
cana-3950	278	11	(	(	PUNCT
cana-3950	278	12	𝐵1	𝐵1	PROPN
cana-3950	278	13	,	,	PUNCT
cana-3950	278	14	𝐵2	𝐵2	NOUN
cana-3950	278	15	,	,	PUNCT
cana-3950	278	16	…	…	PUNCT
cana-3950	278	17	,	,	PUNCT
cana-3950	278	18	𝐵𝑘	𝐵𝑘	PROPN
cana-3950	278	19	)	)	PUNCT
cana-3950	278	20	(	(	PUNCT
cana-3950	278	21	𝑖𝑖𝑖)𝐵𝐹𝑊𝑃𝐴	𝑖𝑖𝑖)𝐵𝐹𝑊𝑃𝐴	X
cana-3950	278	22	(	(	PUNCT
cana-3950	278	23	𝐴1	𝐴1	PROPN
cana-3950	278	24	,	,	PUNCT
cana-3950	278	25	𝐴2	𝐴2	PROPN
cana-3950	278	26	,	,	PUNCT
cana-3950	278	27	…	…	PUNCT
cana-3950	278	28	,	,	PUNCT
cana-3950	278	29	𝐴𝑘)⨂𝐵𝐹𝑊𝑃𝐴	𝐴𝑘)⨂𝐵𝐹𝑊𝑃𝐴	PRON
cana-3950	278	30	(	(	PUNCT
cana-3950	278	31	𝐵1	𝐵1	PROPN
cana-3950	278	32	,	,	PUNCT
cana-3950	278	33	𝐵2	𝐵2	NOUN
cana-3950	278	34	,	,	PUNCT
cana-3950	278	35	…	…	PUNCT
cana-3950	278	36	,	,	PUNCT
cana-3950	278	37	𝐵𝑘	𝐵𝑘	PROPN
cana-3950	278	38	)	)	PUNCT
cana-3950	278	39	=	=	SYM
cana-3950	278	40	(	(	PUNCT
cana-3950	278	41	(	(	PUNCT
cana-3950	278	42	∑	∑	X
cana-3950	278	43	∈𝑖	∈𝑖	VERB
cana-3950	278	44	𝐽𝐴𝑖	𝐽𝐴𝑖	NOUN
cana-3950	278	45	𝑛𝑘	𝑛𝑘	X
cana-3950	278	46	𝑖=1	𝑖=1	PROPN
cana-3950	278	47	)	)	PUNCT
cana-3950	278	48	1	1	NUM
cana-3950	278	49	𝑛	𝑛	PROPN
cana-3950	278	50	(	(	PUNCT
cana-3950	278	51	∑	∑	X
cana-3950	278	52	∈𝑖	∈𝑖	VERB
cana-3950	278	53	𝐽𝐵𝑖	𝐽𝐵𝑖	NOUN
cana-3950	278	54	𝑛𝑘	𝑛𝑘	X
cana-3950	278	55	𝑖=1	𝑖=1	PROPN
cana-3950	278	56	)	)	PUNCT
cana-3950	278	57	1	1	NUM
cana-3950	278	58	𝑛	𝑛	PROPN
cana-3950	278	59	√∑	√∑	ADV
cana-3950	278	60	∈𝑖	∈𝑖	VERB
cana-3950	278	61	𝐾𝐴𝑖	𝐾𝐴𝑖	NOUN
cana-3950	278	62	𝑚	𝑚	X
cana-3950	278	63	+	+	CCONJ
cana-3950	278	64	∑	∑	ADV
cana-3950	278	65	∈𝑖	∈𝑖	VERB
cana-3950	278	66	𝐾𝐴𝑖	𝐾𝐴𝑖	NOUN
cana-3950	278	67	𝑚	𝑚	ADP
cana-3950	278	68	−	−	PROPN
cana-3950	278	69	∑	∑	PUNCT
cana-3950	278	70	∈𝑖	∈𝑖	VERB
cana-3950	278	71	𝐾𝐴𝑖	𝐾𝐴𝑖	PROPN
cana-3950	278	72	𝑚	𝑚	PROPN
cana-3950	278	73	∑	∑	PUNCT
cana-3950	278	74	∈𝑖	∈𝑖	VERB
cana-3950	278	75	𝐾𝐴𝐵𝑖	𝐾𝐴𝐵𝑖	PROPN
cana-3950	278	76	𝑚	𝑚	NOUN
cana-3950	278	77	)	)	PUNCT
cana-3950	278	78	𝑛	𝑛	PROPN
cana-3950	278	79	𝑖=1	𝑖=1	PROPN
cana-3950	279	1	𝑘	𝑘	X
cana-3950	279	2	𝑖=1	𝑖=1	PUNCT
cana-3950	280	1	𝑘	𝑘	X
cana-3950	280	2	𝑖=1	𝑖=1	PUNCT
cana-3950	280	3	𝑘	𝑘	DET
cana-3950	280	4	𝑖=1	𝑖=1	PROPN
cana-3950	280	5	application	application	NOUN
cana-3950	280	6	of	of	ADP
cana-3950	280	7	beal	beal	NOUN
cana-3950	280	8	’s	’s	PART
cana-3950	280	9	fuzzy	fuzzy	ADJ
cana-3950	280	10	set	set	VERB
cana-3950	280	11	in	in	ADP
cana-3950	280	12	pattern	pattern	NOUN
cana-3950	280	13	recognition	recognition	NOUN
cana-3950	280	14	in	in	ADP
cana-3950	280	15	this	this	DET
cana-3950	280	16	section	section	NOUN
cana-3950	280	17	,	,	PUNCT
cana-3950	280	18	we	we	PRON
cana-3950	280	19	propose	propose	VERB
cana-3950	280	20	similarity	similarity	NOUN
cana-3950	280	21	measures	measure	NOUN
cana-3950	280	22	that	that	PRON
cana-3950	280	23	can	can	AUX
cana-3950	280	24	be	be	AUX
cana-3950	280	25	dignity	dignity	NOUN
cana-3950	280	26	in	in	ADP
cana-3950	280	27	commander	commander	NOUN
cana-3950	280	28	’s	’s	PART
cana-3950	280	29	chosen	choose	VERB
cana-3950	280	30	in	in	ADP
cana-3950	280	31	military	military	ADJ
cana-3950	280	32	office	office	NOUN
cana-3950	280	33	,	,	PUNCT
cana-3950	280	34	medical	medical	ADJ
cana-3950	280	35	diagnosis	diagnosis	NOUN
cana-3950	280	36	of	of	ADP
cana-3950	280	37	disease	disease	NOUN
cana-3950	280	38	.	.	PUNCT
cana-3950	281	1	plant	plant	NOUN
cana-3950	281	2	leaf	leaf	NOUN
cana-3950	281	3	diseases	disease	NOUN
cana-3950	281	4	classifications	classification	NOUN
cana-3950	281	5	,	,	PUNCT
cana-3950	281	6	construction	construction	NOUN
cana-3950	281	7	material	material	NOUN
cana-3950	281	8	selections	selection	NOUN
cana-3950	281	9	and	and	CCONJ
cana-3950	281	10	other	other	ADJ
cana-3950	281	11	multi	multi	ADJ
cana-3950	281	12	-	-	ADJ
cana-3950	281	13	attribute	attribute	NOUN
cana-3950	281	14	decision	decision	NOUN
cana-3950	281	15	-	-	PUNCT
cana-3950	281	16	making	make	VERB
cana-3950	281	17	problems	problem	NOUN
cana-3950	281	18	.	.	PUNCT
cana-3950	282	1	communications	communication	NOUN
cana-3950	282	2	on	on	ADP
cana-3950	282	3	applied	apply	VERB
cana-3950	282	4	nonlinear	nonlinear	ADJ
cana-3950	282	5	analysis	analysis	NOUN
cana-3950	282	6	issn	issn	NOUN
cana-3950	282	7	:	:	PUNCT
cana-3950	282	8	1074	1074	NUM
cana-3950	282	9	-	-	PUNCT
cana-3950	282	10	133x	133x	NUM
cana-3950	282	11	vol	vol	NOUN
cana-3950	282	12	32	32	NUM
cana-3950	282	13	no	no	NOUN
cana-3950	282	14	.	.	PUNCT
cana-3950	283	1	9s	9s	NUM
cana-3950	283	2	(	(	PUNCT
cana-3950	283	3	2025	2025	NUM
cana-3950	283	4	)	)	PUNCT
cana-3950	283	5	401	401	NUM
cana-3950	283	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3950	284	1	the	the	DET
cana-3950	284	2	following	follow	VERB
cana-3950	284	3	contribution	contribution	NOUN
cana-3950	284	4	showed	show	VERB
cana-3950	284	5	to	to	PART
cana-3950	284	6	accept	accept	VERB
cana-3950	284	7	on	on	ADP
cana-3950	284	8	unknown	unknown	ADJ
cana-3950	284	9	pattern	pattern	NOUN
cana-3950	284	10	using	use	VERB
cana-3950	284	11	the	the	DET
cana-3950	284	12	new	new	ADJ
cana-3950	284	13	idea	idea	NOUN
cana-3950	284	14	of	of	ADP
cana-3950	284	15	similarity	similarity	NOUN
cana-3950	284	16	theorem	theorem	VERB
cana-3950	284	17	.	.	PROPN
cana-3950	284	18	example	example	NOUN
cana-3950	284	19	4.7	4.7	NUM
cana-3950	284	20	let	let	VERB
cana-3950	284	21	us	we	PRON
cana-3950	284	22	taken	take	VERB
cana-3950	284	23	three	three	NUM
cana-3950	284	24	unknown	unknown	ADJ
cana-3950	284	25	patterns	pattern	NOUN
cana-3950	285	1	𝑃𝑖(𝑖	𝑃𝑖(𝑖	ADP
cana-3950	285	2	=	=	SYM
cana-3950	285	3	1,2,3	1,2,3	NUM
cana-3950	285	4	)	)	PUNCT
cana-3950	285	5	which	which	PRON
cana-3950	285	6	are	be	AUX
cana-3950	285	7	represented	represent	VERB
cana-3950	285	8	by	by	ADP
cana-3950	285	9	the	the	DET
cana-3950	285	10	beal	beal	NOUN
cana-3950	285	11	’s	’s	PART
cana-3950	285	12	fuzzy	fuzzy	ADJ
cana-3950	285	13	set	set	NOUN
cana-3950	285	14	(	(	PUNCT
cana-3950	285	15	𝑚	𝑚	NOUN
cana-3950	285	16	=	=	SYM
cana-3950	285	17	5	5	NUM
cana-3950	285	18	,	,	PUNCT
cana-3950	285	19	𝑛	𝑛	PRON
cana-3950	285	20	=	=	SYM
cana-3950	285	21	5)𝑃𝑖(𝑖	5)𝑃𝑖(𝑖	PROPN
cana-3950	285	22	=	=	SYM
cana-3950	285	23	1,2,3	1,2,3	NUM
cana-3950	285	24	)	)	PUNCT
cana-3950	285	25	in	in	ADP
cana-3950	285	26	the	the	DET
cana-3950	285	27	characteristic	characteristic	NOUN
cana-3950	285	28	as	as	ADP
cana-3950	285	29	:	:	PUNCT
cana-3950	285	30	𝑅	𝑅	PROPN
cana-3950	285	31	=	=	SYM
cana-3950	285	32	{	{	PUNCT
cana-3950	285	33	𝑟1	𝑟1	NOUN
cana-3950	285	34	,	,	PUNCT
cana-3950	285	35	𝑟2	𝑟2	NOUN
cana-3950	285	36	,	,	PUNCT
cana-3950	285	37	𝑟3	𝑟3	NOUN
cana-3950	285	38	}	}	PUNCT
cana-3950	285	39	.	.	PUNCT
cana-3950	286	1	𝑃1	𝑃1	NOUN
cana-3950	286	2	=	=	SYM
cana-3950	286	3	〈	〈	PROPN
cana-3950	286	4	𝑟1	𝑟1	PROPN
cana-3950	286	5	,	,	PUNCT
cana-3950	286	6	0.5,0.6	0.5,0.6	PROPN
cana-3950	286	7	〉	〉	NOUN
cana-3950	286	8	,	,	PUNCT
cana-3950	286	9	〈	〈	PROPN
cana-3950	286	10	𝑟2,0.1,0.7	𝑟2,0.1,0.7	NOUN
cana-3950	286	11	〉	〉	NOUN
cana-3950	286	12	,	,	PUNCT
cana-3950	286	13	〈	〈	NOUN
cana-3950	286	14	𝑟3	𝑟3	NOUN
cana-3950	286	15	,	,	PUNCT
cana-3950	286	16	0.4	0.4	NUM
cana-3950	286	17	,	,	PUNCT
cana-3950	286	18	0.7	0.7	NUM
cana-3950	286	19	〉	〉	NOUN
cana-3950	286	20	𝑃2	𝑃2	NOUN
cana-3950	286	21	=	=	SYM
cana-3950	286	22	〈	〈	PROPN
cana-3950	286	23	𝑟1,0.1,0.8	𝑟1,0.1,0.8	ADJ
cana-3950	286	24	〉	〉	NOUN
cana-3950	286	25	,	,	PUNCT
cana-3950	286	26	〈	〈	NOUN
cana-3950	286	27	𝑟2	𝑟2	NOUN
cana-3950	286	28	,	,	PUNCT
cana-3950	286	29	0.8,0.8	0.8,0.8	NUM
cana-3950	286	30	〉	〉	NOUN
cana-3950	286	31	,	,	PUNCT
cana-3950	286	32	〈	〈	NOUN
cana-3950	286	33	𝑟3	𝑟3	NOUN
cana-3950	286	34	,	,	PUNCT
cana-3950	286	35	0.7	0.7	NUM
cana-3950	286	36	,	,	PUNCT
cana-3950	286	37	0.6	0.6	NUM
cana-3950	286	38	〉	〉	NOUN
cana-3950	286	39	𝑃3	𝑃3	NOUN
cana-3950	286	40	=	=	PUNCT
cana-3950	286	41	〈	〈	PROPN
cana-3950	286	42	𝑟1,0.5,0.7	𝑟1,0.5,0.7	NOUN
cana-3950	286	43	〉	〉	NOUN
cana-3950	286	44	,	,	PUNCT
cana-3950	286	45	〈	〈	NOUN
cana-3950	286	46	𝑟2	𝑟2	NOUN
cana-3950	286	47	,	,	PUNCT
cana-3950	286	48	0.6,0.6	0.6,0.6	NOUN
cana-3950	286	49	〉	〉	NOUN
cana-3950	286	50	,	,	PUNCT
cana-3950	286	51	〈	〈	NOUN
cana-3950	286	52	𝑟3	𝑟3	NOUN
cana-3950	286	53	,	,	PUNCT
cana-3950	286	54	0.7	0.7	NUM
cana-3950	286	55	,	,	PUNCT
cana-3950	286	56	0.5	0.5	NUM
cana-3950	286	57	〉	〉	NOUN
cana-3950	286	58	consider	consider	VERB
cana-3950	286	59	,	,	PUNCT
cana-3950	286	60	as	as	ADP
cana-3950	286	61	unknown	unknown	ADJ
cana-3950	286	62	pattern	pattern	NOUN
cana-3950	286	63	𝑃	𝑃	PROPN
cana-3950	286	64	∈	∈	PROPN
cana-3950	286	65	𝐵4	𝐵4	NOUN
cana-3950	286	66	5	5	NUM
cana-3950	286	67	(	(	PUNCT
cana-3950	286	68	𝑅	𝑅	PROPN
cana-3950	286	69	)	)	PUNCT
cana-3950	286	70	that	that	PRON
cana-3950	286	71	will	will	AUX
cana-3950	286	72	be	be	AUX
cana-3950	286	73	recognized	recognize	VERB
cana-3950	286	74	,	,	PUNCT
cana-3950	286	75	where	where	SCONJ
cana-3950	286	76	:	:	PUNCT
cana-3950	286	77	𝑃	𝑃	NOUN
cana-3950	286	78	=	=	PUNCT
cana-3950	286	79	〈	〈	NOUN
cana-3950	286	80	𝑟1,0.9,0.7	𝑟1,0.9,0.7	NOUN
cana-3950	286	81	〉	〉	NOUN
cana-3950	286	82	,	,	PUNCT
cana-3950	286	83	〈	〈	NOUN
cana-3950	286	84	𝑟2	𝑟2	NOUN
cana-3950	286	85	,	,	PUNCT
cana-3950	286	86	0.8,0.7	0.8,0.7	PROPN
cana-3950	286	87	〉	〉	NOUN
cana-3950	286	88	,	,	PUNCT
cana-3950	286	89	〈	〈	NOUN
cana-3950	286	90	𝑟1	𝑟1	NOUN
cana-3950	286	91	,	,	PUNCT
cana-3950	286	92	0.6,0.8	0.6,0.8	NUM
cana-3950	286	93	〉	〉	NOUN
cana-3950	286	94	.	.	PUNCT
cana-3950	287	1	then	then	ADV
cana-3950	287	2	the	the	DET
cana-3950	287	3	proposed	propose	VERB
cana-3950	287	4	similarity	similarity	NOUN
cana-3950	287	5	techniques	technique	NOUN
cana-3950	287	6	𝑆1	𝑆1	VERB
cana-3950	287	7	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
cana-3950	287	8	𝑆2	𝑆2	PROPN
cana-3950	287	9	which	which	PRON
cana-3950	287	10	have	have	AUX
cana-3950	287	11	been	be	AUX
cana-3950	287	12	evaluated	evaluate	VERB
cana-3950	287	13	from	from	ADP
cana-3950	287	14	p	p	NOUN
cana-3950	287	15	to	to	ADP
cana-3950	287	16	𝑃𝑖(𝑖	𝑃𝑖(𝑖	NOUN
cana-3950	287	17	=	=	PUNCT
cana-3950	287	18	1,2,3	1,2,3	NUM
cana-3950	287	19	)	)	PUNCT
cana-3950	287	20	are	be	AUX
cana-3950	287	21	given	give	VERB
cana-3950	287	22	in	in	ADP
cana-3950	287	23	the	the	DET
cana-3950	287	24	following	follow	VERB
cana-3950	287	25	table	table	NOUN
cana-3950	287	26	.	.	PUNCT
cana-3950	288	1	from	from	ADP
cana-3950	288	2	the	the	DET
cana-3950	288	3	number	number	NOUN
cana-3950	288	4	point	point	NOUN
cana-3950	288	5	of	of	ADP
cana-3950	288	6	given	give	VERB
cana-3950	288	7	presented	present	VERB
cana-3950	288	8	in	in	ADP
cana-3950	288	9	the	the	DET
cana-3950	288	10	previous	previous	ADJ
cana-3950	288	11	data	datum	NOUN
cana-3950	288	12	,	,	PUNCT
cana-3950	288	13	we	we	PRON
cana-3950	288	14	know	know	VERB
cana-3950	288	15	the	the	DET
cana-3950	288	16	similarity	similarity	NOUN
cana-3950	288	17	measures	measure	NOUN
cana-3950	288	18	between	between	ADP
cana-3950	288	19	𝑃2	𝑃2	NOUN
cana-3950	288	20	and	and	CCONJ
cana-3950	288	21	𝑃	𝑃	PROPN
cana-3950	288	22	are	be	AUX
cana-3950	288	23	the	the	DET
cana-3950	288	24	biggest	big	ADJ
cana-3950	288	25	or	or	CCONJ
cana-3950	288	26	highest	high	ADJ
cana-3950	288	27	one	one	NUM
cana-3950	288	28	.	.	PUNCT
cana-3950	289	1	similarity	similarity	NOUN
cana-3950	289	2	measure	measure	NOUN
cana-3950	289	3	between	between	ADP
cana-3950	289	4	𝑃𝑖(𝑖	𝑃𝑖(𝑖	NOUN
cana-3950	289	5	=	=	NOUN
cana-3950	289	6	1,2,3	1,2,3	NUM
cana-3950	289	7	)	)	PUNCT
cana-3950	289	8	and	and	CCONJ
cana-3950	289	9	p	p	NOUN
cana-3950	289	10	similarity	similarity	NOUN
cana-3950	289	11	measure	measure	NOUN
cana-3950	289	12	(	(	PUNCT
cana-3950	289	13	𝑃1	𝑃1	NOUN
cana-3950	289	14	,	,	PUNCT
cana-3950	289	15	𝑃	𝑃	PROPN
cana-3950	289	16	)	)	PUNCT
cana-3950	289	17	(	(	PUNCT
cana-3950	289	18	𝑃2	𝑃2	PROPN
cana-3950	289	19	,	,	PUNCT
cana-3950	289	20	𝑃	𝑃	PROPN
cana-3950	289	21	)	)	PUNCT
cana-3950	289	22	(	(	PUNCT
cana-3950	289	23	𝑃3	𝑃3	NOUN
cana-3950	289	24	,	,	PUNCT
cana-3950	289	25	𝑃	𝑃	NOUN
cana-3950	289	26	)	)	PUNCT
cana-3950	289	27	𝑆1(𝑃𝑖	𝑆1(𝑃𝑖	PROPN
cana-3950	289	28	,	,	PUNCT
cana-3950	289	29	𝑃	𝑃	PROPN
cana-3950	289	30	)	)	PUNCT
cana-3950	289	31	0.7180	0.7180	NUM
cana-3950	289	32	0.8487	0.8487	NUM
cana-3950	289	33	0.7589	0.7589	NUM
cana-3950	289	34	𝑆2(𝑃𝑖	𝑆2(𝑃𝑖	NOUN
cana-3950	289	35	,	,	PUNCT
cana-3950	289	36	𝑃	𝑃	NOUN
cana-3950	289	37	)	)	PUNCT
cana-3950	289	38	0.7679	0.7679	NUM
cana-3950	289	39	0.8382	0.8382	NUM
cana-3950	289	40	0.7625	0.7625	NUM
cana-3950	289	41	among	among	ADP
cana-3950	289	42	the	the	DET
cana-3950	289	43	calculated	calculate	VERB
cana-3950	289	44	measures	measure	NOUN
cana-3950	289	45	,	,	PUNCT
cana-3950	289	46	(	(	PUNCT
cana-3950	289	47	𝑃2	𝑃2	NOUN
cana-3950	289	48	,	,	PUNCT
cana-3950	289	49	𝑃	𝑃	PROPN
cana-3950	289	50	)	)	PUNCT
cana-3950	289	51	is	be	AUX
cana-3950	289	52	the	the	DET
cana-3950	289	53	largest	large	ADJ
cana-3950	289	54	one	one	NUM
cana-3950	289	55	.	.	PUNCT
cana-3950	290	1	5	5	NUM
cana-3950	290	2	.	.	X
cana-3950	290	3	conclusion	conclusion	NOUN
cana-3950	290	4	in	in	ADP
cana-3950	290	5	the	the	DET
cana-3950	290	6	beal	beal	NOUN
cana-3950	290	7	’s	’s	PART
cana-3950	290	8	fuzzy	fuzzy	ADJ
cana-3950	290	9	set	set	NOUN
cana-3950	290	10	,	,	PUNCT
cana-3950	290	11	the	the	DET
cana-3950	290	12	complement	complement	NOUN
cana-3950	290	13	,	,	PUNCT
cana-3950	290	14	necessity	necessity	NOUN
cana-3950	290	15	,	,	PUNCT
cana-3950	290	16	possibility	possibility	NOUN
cana-3950	290	17	and	and	CCONJ
cana-3950	290	18	various	various	ADJ
cana-3950	290	19	arithmetic	arithmetic	ADJ
cana-3950	290	20	operators	operator	NOUN
cana-3950	290	21	are	be	AUX
cana-3950	290	22	explained	explain	VERB
cana-3950	290	23	and	and	CCONJ
cana-3950	290	24	various	various	ADJ
cana-3950	290	25	results	result	NOUN
cana-3950	290	26	related	relate	VERB
cana-3950	290	27	to	to	ADP
cana-3950	290	28	properties	property	NOUN
cana-3950	290	29	of	of	ADP
cana-3950	290	30	these	these	DET
cana-3950	290	31	operations	operation	NOUN
cana-3950	290	32	have	have	AUX
cana-3950	290	33	been	be	AUX
cana-3950	290	34	demonstrated	demonstrate	VERB
cana-3950	290	35	in	in	ADP
cana-3950	290	36	this	this	DET
cana-3950	290	37	article	article	NOUN
cana-3950	290	38	.	.	PUNCT
cana-3950	291	1	furthermore	furthermore	ADV
cana-3950	291	2	,	,	PUNCT
cana-3950	291	3	some	some	DET
cana-3950	291	4	distance	distance	NOUN
cana-3950	291	5	and	and	CCONJ
cana-3950	291	6	similarity	similarity	NOUN
cana-3950	291	7	measures	measure	NOUN
cana-3950	291	8	over	over	ADP
cana-3950	291	9	beal	beal	NOUN
cana-3950	291	10	’s	’s	PART
cana-3950	291	11	fuzzy	fuzzy	ADJ
cana-3950	291	12	set	set	VERB
cana-3950	291	13	as	as	SCONJ
cana-3950	291	14	developed	develop	VERB
cana-3950	291	15	and	and	CCONJ
cana-3950	291	16	their	their	PRON
cana-3950	291	17	validity	validity	NOUN
cana-3950	291	18	are	be	AUX
cana-3950	291	19	verified	verify	VERB
cana-3950	291	20	by	by	ADP
cana-3950	291	21	suitable	suitable	ADJ
cana-3950	291	22	examples	example	NOUN
cana-3950	291	23	.	.	PUNCT
cana-3950	292	1	an	an	DET
cana-3950	292	2	example	example	NOUN
cana-3950	292	3	of	of	ADP
cana-3950	292	4	the	the	DET
cana-3950	292	5	application	application	NOUN
cana-3950	292	6	of	of	ADP
cana-3950	292	7	similarity	similarity	NOUN
cana-3950	292	8	measures	measure	NOUN
cana-3950	292	9	exposed	expose	VERB
cana-3950	292	10	in	in	ADP
cana-3950	292	11	pattern	pattern	NOUN
cana-3950	292	12	recognition	recognition	NOUN
cana-3950	292	13	is	be	AUX
cana-3950	292	14	investigated	investigate	VERB
cana-3950	292	15	.	.	PUNCT
cana-3950	293	1	6	6	X
cana-3950	293	2	.	.	X
cana-3950	293	3	future	future	ADJ
cana-3950	293	4	work	work	NOUN
cana-3950	293	5	1	1	NUM
cana-3950	293	6	.	.	PUNCT
cana-3950	294	1	researchers	researcher	NOUN
cana-3950	294	2	may	may	AUX
cana-3950	294	3	study	study	VERB
cana-3950	294	4	the	the	DET
cana-3950	294	5	various	various	ADJ
cana-3950	294	6	hybrid	hybrid	ADJ
cana-3950	294	7	structure	structure	NOUN
cana-3950	294	8	of	of	ADP
cana-3950	294	9	beal	beal	NOUN
cana-3950	294	10	’s	’s	PART
cana-3950	294	11	fuzzy	fuzzy	ADJ
cana-3950	294	12	sets	set	NOUN
cana-3950	294	13	with	with	ADP
cana-3950	294	14	soft	soft	ADJ
cana-3950	294	15	sets	set	NOUN
cana-3950	294	16	and	and	CCONJ
cana-3950	294	17	rough	rough	ADJ
cana-3950	294	18	sets	set	NOUN
cana-3950	294	19	.	.	PUNCT
cana-3950	295	1	these	these	PRON
cana-3950	295	2	can	can	AUX
cana-3950	295	3	be	be	AUX
cana-3950	295	4	identified	identify	VERB
cana-3950	295	5	and	and	CCONJ
cana-3950	295	6	their	their	PRON
cana-3950	295	7	uses	use	NOUN
cana-3950	295	8	in	in	ADP
cana-3950	295	9	different	different	ADJ
cana-3950	295	10	fields	field	NOUN
cana-3950	295	11	of	of	ADP
cana-3950	295	12	mathematics	mathematic	NOUN
cana-3950	295	13	can	can	AUX
cana-3950	295	14	be	be	AUX
cana-3950	295	15	applied	apply	VERB
cana-3950	295	16	.	.	PUNCT
cana-3950	296	1	2	2	X
cana-3950	296	2	.	.	X
cana-3950	296	3	also	also	ADV
cana-3950	296	4	,	,	PUNCT
cana-3950	296	5	the	the	DET
cana-3950	296	6	dice	dice	NOUN
cana-3950	296	7	,	,	PUNCT
cana-3950	296	8	cosine	cosine	NOUN
cana-3950	296	9	and	and	CCONJ
cana-3950	296	10	cotangent	cotangent	NOUN
cana-3950	296	11	similarity	similarity	NOUN
cana-3950	296	12	measures	measure	NOUN
cana-3950	296	13	and	and	CCONJ
cana-3950	296	14	weighted	weight	VERB
cana-3950	296	15	aggregation	aggregation	NOUN
cana-3950	296	16	operators	operator	NOUN
cana-3950	296	17	of	of	ADP
cana-3950	296	18	beal	beal	NOUN
cana-3950	296	19	’s	’s	PART
cana-3950	296	20	fuzzy	fuzzy	ADJ
cana-3950	296	21	set	set	NOUN
cana-3950	296	22	can	can	AUX
cana-3950	296	23	be	be	AUX
cana-3950	296	24	studied	study	VERB
cana-3950	296	25	and	and	CCONJ
cana-3950	296	26	their	their	PRON
cana-3950	296	27	applications	application	NOUN
cana-3950	296	28	in	in	ADP
cana-3950	296	29	multicriterion	multicriterion	NOUN
cana-3950	296	30	decision	decision	NOUN
cana-3950	296	31	making	making	NOUN
cana-3950	296	32	and	and	CCONJ
cana-3950	296	33	multi	multi	ADJ
cana-3950	296	34	attribute	attribute	NOUN
cana-3950	296	35	decision	decision	NOUN
cana-3950	296	36	making	make	VERB
cana-3950	296	37	problems	problem	NOUN
cana-3950	296	38	can	can	AUX
cana-3950	296	39	be	be	AUX
cana-3950	296	40	survived	survive	VERB
cana-3950	296	41	.	.	PUNCT
cana-3950	297	1	references	reference	NOUN
cana-3950	297	2	[	[	X
cana-3950	297	3	1	1	NUM
cana-3950	297	4	]	]	PUNCT
cana-3950	297	5	alshami	alshami	NOUN
cana-3950	297	6	t.	t.	NOUN
cana-3950	297	7	m.	m.	NOUN
cana-3950	297	8	,	,	PUNCT
cana-3950	297	9	(	(	PUNCT
cana-3950	297	10	2022	2022	NUM
cana-3950	297	11	)	)	PUNCT
cana-3950	297	12	,	,	PUNCT
cana-3950	297	13	(	(	PUNCT
cana-3950	297	14	2,1)fuzzy	2,1)fuzzy	NUM
cana-3950	297	15	sets	set	VERB
cana-3950	297	16	:	:	PUNCT
cana-3950	297	17	properties	property	NOUN
cana-3950	297	18	,	,	PUNCT
cana-3950	297	19	weighted	weight	VERB
cana-3950	297	20	aggregated	aggregate	VERB
cana-3950	297	21	operators	operator	NOUN
cana-3950	297	22	and	and	CCONJ
cana-3950	297	23	their	their	PRON
cana-3950	297	24	applications	application	NOUN
cana-3950	297	25	to	to	ADP
cana-3950	297	26	multi	multi	ADJ
cana-3950	297	27	-	-	NOUN
cana-3950	297	28	criteria	criterion	NOUN
cana-3950	297	29	decision	decision	NOUN
cana-3950	297	30	–	–	PUNCT
cana-3950	297	31	making	make	VERB
cana-3950	297	32	methods	method	NOUN
cana-3950	297	33	.	.	PUNCT
cana-3950	298	1	complex	complex	ADJ
cana-3950	298	2	and	and	CCONJ
cana-3950	298	3	intelligent	intelligent	ADJ
cana-3950	298	4	systems	system	NOUN
cana-3950	298	5	.	.	PUNCT
cana-3950	299	1	[	[	X
cana-3950	299	2	2	2	NUM
cana-3950	299	3	]	]	PUNCT
cana-3950	299	4	atanassov	atanassov	NOUN
cana-3950	299	5	.k	.k	PROPN
cana-3950	299	6	,	,	PUNCT
cana-3950	299	7	(	(	PUNCT
cana-3950	299	8	1986	1986	NUM
cana-3950	299	9	)	)	PUNCT
cana-3950	299	10	,	,	PUNCT
cana-3950	299	11	intuitionistic	intuitionistic	ADJ
cana-3950	299	12	fuzzy	fuzzy	ADJ
cana-3950	299	13	sets	set	NOUN
cana-3950	299	14	.	.	PUNCT
cana-3950	300	1	fuzzy	fuzzy	ADJ
cana-3950	300	2	sets	set	NOUN
cana-3950	300	3	and	and	CCONJ
cana-3950	300	4	systems	system	NOUN
cana-3950	300	5	,	,	PUNCT
cana-3950	300	6	(	(	PUNCT
cana-3950	300	7	20	20	NUM
cana-3950	300	8	)	)	PUNCT
cana-3950	300	9	87	87	NUM
cana-3950	300	10	-	-	SYM
cana-3950	300	11	96	96	NUM
cana-3950	300	12	.	.	PUNCT
cana-3950	301	1	communications	communication	NOUN
cana-3950	301	2	on	on	ADP
cana-3950	301	3	applied	apply	VERB
cana-3950	301	4	nonlinear	nonlinear	ADJ
cana-3950	301	5	analysis	analysis	NOUN
cana-3950	301	6	issn	issn	NOUN
cana-3950	301	7	:	:	PUNCT
cana-3950	301	8	1074	1074	NUM
cana-3950	301	9	-	-	PUNCT
cana-3950	301	10	133x	133x	NUM
cana-3950	301	11	vol	vol	NOUN
cana-3950	301	12	32	32	NUM
cana-3950	301	13	no	no	NOUN
cana-3950	301	14	.	.	PUNCT
cana-3950	302	1	9s	9s	NUM
cana-3950	302	2	(	(	PUNCT
cana-3950	302	3	2025	2025	NUM
cana-3950	302	4	)	)	PUNCT
cana-3950	302	5	402	402	NUM
cana-3950	302	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3950	303	1	[	[	X
cana-3950	303	2	3	3	NUM
cana-3950	303	3	]	]	X
cana-3950	303	4	bryniarska	bryniarska	NOUN
cana-3950	303	5	.	.	PUNCT
cana-3950	304	1	a	a	DET
cana-3950	304	2	,	,	PUNCT
cana-3950	304	3	(	(	PUNCT
cana-3950	304	4	2020	2020	NUM
cana-3950	304	5	)	)	PUNCT
cana-3950	304	6	,	,	PUNCT
cana-3950	304	7	the	the	DET
cana-3950	304	8	n	n	CCONJ
cana-3950	304	9	-	-	PUNCT
cana-3950	304	10	pythagorean	pythagorean	ADJ
cana-3950	304	11	fuzzy	fuzzy	ADJ
cana-3950	304	12	sets	set	NOUN
cana-3950	304	13	.	.	PUNCT
cana-3950	305	1	symmetry	symmetry	NOUN
cana-3950	305	2	,	,	PUNCT
cana-3950	305	3	12	12	NUM
cana-3950	305	4	(	(	PUNCT
cana-3950	305	5	11	11	NUM
cana-3950	305	6	)	)	PUNCT
cana-3950	305	7	.	.	PUNCT
cana-3950	306	1	[	[	X
cana-3950	306	2	4	4	NUM
cana-3950	306	3	]	]	PUNCT
cana-3950	306	4	ejegwa	ejegwa	NOUN
cana-3950	306	5	.	.	PUNCT
cana-3950	307	1	p.	p.	NOUN
cana-3950	307	2	a.	a.	NOUN
cana-3950	308	1	and	and	CCONJ
cana-3950	308	2	onyeke	onyeke	ADJ
cana-3950	308	3	.	.	PUNCT
cana-3950	309	1	i.c	i.c	PROPN
cana-3950	309	2	.	.	PROPN
cana-3950	309	3	,	,	PUNCT
cana-3950	309	4	(	(	PUNCT
cana-3950	309	5	2021	2021	NUM
cana-3950	309	6	)	)	PUNCT
cana-3950	309	7	,	,	PUNCT
cana-3950	309	8	fermatean	fermatean	ADJ
cana-3950	309	9	fuzzy	fuzzy	ADJ
cana-3950	309	10	similarity	similarity	NOUN
cana-3950	309	11	measure	measure	NOUN
cana-3950	309	12	algorithm	algorithm	NOUN
cana-3950	309	13	,	,	PUNCT
cana-3950	309	14	and	and	CCONJ
cana-3950	309	15	its	its	PRON
cana-3950	309	16	application	application	NOUN
cana-3950	309	17	in	in	ADP
cana-3950	309	18	students	student	NOUN
cana-3950	309	19	admission	admission	NOUN
cana-3950	309	20	process	process	NOUN
cana-3950	309	21	.	.	PUNCT
cana-3950	310	1	international	international	ADJ
cana-3950	310	2	journal	journal	NOUN
cana-3950	310	3	of	of	ADP
cana-3950	310	4	fuzzy	fuzzy	ADJ
cana-3950	310	5	computation	computation	NOUN
cana-3950	310	6	and	and	CCONJ
cana-3950	310	7	modelling	modelling	NOUN
cana-3950	310	8	,	,	PUNCT
cana-3950	310	9	4	4	NUM
cana-3950	310	10	(	(	PUNCT
cana-3950	310	11	1	1	NUM
cana-3950	310	12	)	)	PUNCT
cana-3950	310	13	:	:	PUNCT
cana-3950	310	14	34	34	NUM
cana-3950	310	15	-	-	SYM
cana-3950	310	16	50	50	NUM
cana-3950	310	17	.	.	PUNCT
cana-3950	311	1	[	[	X
cana-3950	311	2	5	5	X
cana-3950	311	3	]	]	X
cana-3950	311	4	hussian	hussian	NOUN
cana-3950	311	5	.	.	PUNCT
cana-3950	312	1	z.	z.	PROPN
cana-3950	312	2	and	and	CCONJ
cana-3950	312	3	yang	yang	PROPN
cana-3950	312	4	.	.	PUNCT
cana-3950	313	1	m.s	m.s	PROPN
cana-3950	313	2	.	.	PROPN
cana-3950	313	3	,	,	PUNCT
cana-3950	313	4	(	(	PUNCT
cana-3950	313	5	2019	2019	NUM
cana-3950	313	6	)	)	PUNCT
cana-3950	313	7	,	,	PUNCT
cana-3950	313	8	distance	distance	NOUN
cana-3950	313	9	and	and	CCONJ
cana-3950	313	10	similarity	similarity	NOUN
cana-3950	313	11	measures	measure	NOUN
cana-3950	313	12	of	of	ADP
cana-3950	313	13	pythagorean	pythagorean	ADJ
cana-3950	313	14	fuzzy	fuzzy	ADJ
cana-3950	313	15	sets	set	NOUN
cana-3950	313	16	based	base	VERB
cana-3950	313	17	on	on	ADP
cana-3950	313	18	hausdorff	hausdorff	NOUN
cana-3950	313	19	metric	metric	ADJ
cana-3950	313	20	with	with	ADP
cana-3950	313	21	application	application	NOUN
cana-3950	313	22	to	to	ADP
cana-3950	313	23	fuzzy	fuzzy	ADJ
cana-3950	313	24	topics	topic	NOUN
cana-3950	313	25	.	.	PUNCT
cana-3950	314	1	journal	journal	NOUN
cana-3950	314	2	of	of	ADP
cana-3950	314	3	intelligent	intelligent	ADJ
cana-3950	314	4	systems	system	NOUN
cana-3950	314	5	,	,	PUNCT
cana-3950	314	6	1	1	NUM
cana-3950	314	7	-	-	SYM
cana-3950	314	8	22	22	NUM
cana-3950	314	9	.	.	PUNCT
cana-3950	315	1	[	[	X
cana-3950	315	2	6	6	NUM
cana-3950	315	3	]	]	X
cana-3950	315	4	ibrahim	ibrahim	PROPN
cana-3950	315	5	,	,	PUNCT
cana-3950	315	6	h.	h.	PROPN
cana-3950	315	7	al	al	PROPN
cana-3950	315	8	-	-	PUNCT
cana-3950	315	9	shami	shami	PROPN
cana-3950	315	10	,	,	PUNCT
cana-3950	315	11	t.	t.	NOUN
cana-3950	315	12	m.	m.	NOUN
cana-3950	315	13	and	and	CCONJ
cana-3950	315	14	elbarbary	elbarbary	PROPN
cana-3950	315	15	.	.	PUNCT
cana-3950	316	1	o.	o.	PROPN
cana-3950	316	2	g.	g.	PROPN
cana-3950	316	3	,	,	PUNCT
cana-3950	316	4	(	(	PUNCT
cana-3950	316	5	2021	2021	NUM
cana-3950	316	6	):	):	PUNCT
cana-3950	316	7	(	(	PUNCT
cana-3950	316	8	3,2)fuzzy	3,2)fuzzy	NUM
cana-3950	316	9	sets	set	NOUN
cana-3950	316	10	and	and	CCONJ
cana-3950	316	11	their	their	PRON
cana-3950	316	12	applications	application	NOUN
cana-3950	316	13	to	to	ADP
cana-3950	316	14	topology	topology	NOUN
cana-3950	316	15	and	and	CCONJ
cana-3950	316	16	optimal	optimal	ADJ
cana-3950	316	17	choice	choice	NOUN
cana-3950	316	18	.	.	PUNCT
cana-3950	317	1	computational	computational	ADJ
cana-3950	317	2	intelligence	intelligence	NOUN
cana-3950	317	3	and	and	CCONJ
cana-3950	317	4	neuroscience	neuroscience	NOUN
cana-3950	317	5	.	.	PUNCT
cana-3950	318	1	[	[	X
cana-3950	318	2	7	7	X
cana-3950	318	3	]	]	X
cana-3950	318	4	ibrahim	ibrahim	PROPN
cana-3950	318	5	.	.	PUNCT
cana-3950	319	1	h.	h.	PROPN
cana-3950	319	2	and	and	CCONJ
cana-3950	319	3	murad	murad	PROPN
cana-3950	319	4	.	.	PUNCT
cana-3950	320	1	k.	k.	PROPN
cana-3950	320	2	k.	k.	PROPN
cana-3950	320	3	,	,	PUNCT
cana-3950	320	4	(	(	PUNCT
cana-3950	320	5	2020	2020	NUM
cana-3950	320	6	):	):	PUNCT
cana-3950	320	7	(	(	PUNCT
cana-3950	320	8	3,4)fuzzy	3,4)fuzzy	NUM
cana-3950	320	9	sets	set	NOUN
cana-3950	320	10	and	and	CCONJ
cana-3950	320	11	their	their	PRON
cana-3950	320	12	topological	topological	ADJ
cana-3950	320	13	spaces	space	NOUN
cana-3950	320	14	.	.	PUNCT
cana-3950	321	1	journal	journal	NOUN
cana-3950	321	2	of	of	ADP
cana-3950	321	3	mathematics	mathematic	NOUN
cana-3950	321	4	and	and	CCONJ
cana-3950	321	5	computer	computer	NOUN
cana-3950	321	6	science	science	NOUN
cana-3950	321	7	,	,	PUNCT
cana-3950	321	8	28	28	NUM
cana-3950	321	9	(	(	PUNCT
cana-3950	321	10	2	2	NUM
cana-3950	321	11	)	)	PUNCT
cana-3950	321	12	:	:	PUNCT
cana-3950	321	13	158	158	NUM
cana-3950	321	14	-	-	SYM
cana-3950	321	15	170	170	NUM
cana-3950	321	16	.	.	PUNCT
cana-3950	322	1	[	[	X
cana-3950	322	2	8	8	NUM
cana-3950	322	3	]	]	X
cana-3950	322	4	jun	jun	PROPN
cana-3950	322	5	.	.	PROPN
cana-3950	322	6	y.	y.	PROPN
cana-3950	322	7	b.	b.	PROPN
cana-3950	322	8	and	and	CCONJ
cana-3950	322	9	hur	hur	PROPN
cana-3950	322	10	kul	kul	PROPN
cana-3950	322	11	.	.	PUNCT
cana-3950	323	1	(	(	PUNCT
cana-3950	323	2	2022	2022	NUM
cana-3950	323	3	)	)	PUNCT
cana-3950	323	4	,	,	PUNCT
cana-3950	323	5	the	the	DET
cana-3950	323	6	(	(	PUNCT
cana-3950	323	7	m	m	PROPN
cana-3950	323	8	,	,	PUNCT
cana-3950	323	9	n)fuzzy	n)fuzzy	ADV
cana-3950	323	10	set	set	VERB
cana-3950	323	11	and	and	CCONJ
cana-3950	323	12	its	its	PRON
cana-3950	323	13	application	application	NOUN
cana-3950	323	14	in	in	ADP
cana-3950	323	15	bck	bck	PROPN
cana-3950	323	16	-	-	PUNCT
cana-3950	323	17	algebras	algebras	PROPN
cana-3950	323	18	.	.	PUNCT
cana-3950	324	1	annals	annal	NOUN
cana-3950	324	2	of	of	ADP
cana-3950	324	3	fuzzy	fuzzy	ADJ
cana-3950	324	4	mathematics	mathematic	NOUN
cana-3950	324	5	and	and	CCONJ
cana-3950	324	6	informatics	informatic	NOUN
cana-3950	324	7	,	,	PUNCT
cana-3950	324	8	24	24	NUM
cana-3950	324	9	(	(	PUNCT
cana-3950	324	10	1	1	NUM
cana-3950	324	11	):	):	PUNCT
cana-3950	324	12	17	17	NUM
cana-3950	324	13	-	-	SYM
cana-3950	324	14	29	29	NUM
cana-3950	324	15	.	.	PUNCT
cana-3950	325	1	[	[	X
cana-3950	325	2	9	9	NUM
cana-3950	325	3	]	]	SYM
cana-3950	325	4	kirisci	kirisci	NOUN
cana-3950	325	5	.	.	PUNCT
cana-3950	325	6	m.	m.	NOUN
cana-3950	325	7	(	(	PUNCT
cana-3950	325	8	2023	2023	NUM
cana-3950	325	9	)	)	PUNCT
cana-3950	325	10	,	,	PUNCT
cana-3950	325	11	new	new	ADJ
cana-3950	325	12	cosine	cosine	NOUN
cana-3950	325	13	similarity	similarity	NOUN
cana-3950	325	14	,	,	PUNCT
cana-3950	325	15	and	and	CCONJ
cana-3950	325	16	distance	distance	NOUN
cana-3950	325	17	measures	measure	NOUN
cana-3950	325	18	for	for	ADP
cana-3950	325	19	fermatean	fermatean	ADJ
cana-3950	325	20	fuzzy	fuzzy	ADJ
cana-3950	325	21	sets	set	NOUN
cana-3950	325	22	and	and	CCONJ
cana-3950	325	23	topics	topic	NOUN
cana-3950	325	24	approach	approach	NOUN
cana-3950	325	25	.	.	PUNCT
cana-3950	326	1	knowledge	knowledge	NOUN
cana-3950	326	2	and	and	CCONJ
cana-3950	326	3	information	information	NOUN
cana-3950	326	4	systems	system	NOUN
cana-3950	326	5	,	,	PUNCT
cana-3950	326	6	65	65	NUM
cana-3950	326	7	:	:	SYM
cana-3950	326	8	855	855	NUM
cana-3950	326	9	-	-	SYM
cana-3950	326	10	868	868	NUM
cana-3950	326	11	.	.	PUNCT
cana-3950	327	1	[	[	X
cana-3950	327	2	10	10	NUM
cana-3950	327	3	]	]	X
cana-3950	327	4	liu	liu	PROPN
cana-3950	327	5	.	.	PUNCT
cana-3950	327	6	d.	d.	PROPN
cana-3950	327	7	,	,	PUNCT
cana-3950	327	8	chen	chen	PROPN
cana-3950	327	9	.	.	PUNCT
cana-3950	328	1	x.	x.	PROPN
cana-3950	328	2	and	and	CCONJ
cana-3950	328	3	peng	peng	PROPN
cana-3950	328	4	.	.	PUNCT
cana-3950	329	1	d.	d.	PROPN
cana-3950	329	2	(	(	PUNCT
cana-3950	329	3	2019	2019	NUM
cana-3950	329	4	)	)	PUNCT
cana-3950	329	5	,	,	PUNCT
cana-3950	329	6	some	some	DET
cana-3950	329	7	cosine	cosine	NOUN
cana-3950	329	8	similarity	similarity	NOUN
cana-3950	329	9	measures	measure	NOUN
cana-3950	329	10	,	,	PUNCT
cana-3950	329	11	and	and	CCONJ
cana-3950	329	12	distance	distance	NOUN
cana-3950	329	13	measures	measure	NOUN
cana-3950	329	14	between	between	ADP
cana-3950	329	15	q	q	NOUN
cana-3950	329	16	-	-	PUNCT
cana-3950	329	17	rung	rung	ADJ
cana-3950	329	18	orthopair	orthopair	ADJ
cana-3950	329	19	fuzzy	fuzzy	ADJ
cana-3950	329	20	sets	set	NOUN
cana-3950	329	21	.	.	PUNCT
cana-3950	330	1	international	international	ADJ
cana-3950	330	2	journal	journal	NOUN
cana-3950	330	3	of	of	ADP
cana-3950	330	4	intelligent	intelligent	ADJ
cana-3950	330	5	systems	system	NOUN
cana-3950	330	6	,	,	PUNCT
cana-3950	330	7	34	34	NUM
cana-3950	330	8	:	:	SYM
cana-3950	330	9	2454	2454	NUM
cana-3950	330	10	-	-	SYM
cana-3950	330	11	2469	2469	NUM
cana-3950	330	12	.	.	PUNCT
cana-3950	331	1	[	[	X
cana-3950	331	2	11	11	NUM
cana-3950	331	3	]	]	X
cana-3950	331	4	peng	peng	PROPN
cana-3950	331	5	.	.	PUNCT
cana-3950	332	1	x	x	PUNCT
cana-3950	332	2	and	and	CCONJ
cana-3950	332	3	liu	liu	PROPN
cana-3950	332	4	.l	.l	PROPN
cana-3950	332	5	.	.	PUNCT
cana-3950	332	6	,	,	PUNCT
cana-3950	332	7	(	(	PUNCT
cana-3950	332	8	2019	2019	NUM
cana-3950	332	9	)	)	PUNCT
cana-3950	332	10	,	,	PUNCT
cana-3950	332	11	information	information	NOUN
cana-3950	332	12	measures	measure	NOUN
cana-3950	332	13	for	for	ADP
cana-3950	332	14	q	q	NOUN
cana-3950	332	15	-	-	PUNCT
cana-3950	332	16	rung	rung	ADJ
cana-3950	332	17	orthopair	orthopair	ADJ
cana-3950	332	18	fuzzy	fuzzy	ADJ
cana-3950	332	19	sets	set	NOUN
cana-3950	332	20	.	.	PUNCT
cana-3950	333	1	international	international	ADJ
cana-3950	333	2	journal	journal	PROPN
cana-3950	333	3	of	of	ADP
cana-3950	333	4	intelligents	intelligent	NOUN
cana-3950	333	5	systems	system	NOUN
cana-3950	333	6	,	,	PUNCT
cana-3950	333	7	34	34	NUM
cana-3950	333	8	:	:	SYM
cana-3950	333	9	1795	1795	NUM
cana-3950	333	10	-	-	SYM
cana-3950	333	11	1834	1834	NUM
cana-3950	333	12	.	.	PUNCT
cana-3950	334	1	[	[	X
cana-3950	334	2	12	12	NUM
cana-3950	334	3	]	]	X
cana-3950	334	4	sahoo	sahoo	PROPN
cana-3950	334	5	.	.	PUNCT
cana-3950	334	6	l.	l.	PROPN
cana-3950	334	7	(	(	PUNCT
cana-3950	334	8	2022	2022	NUM
cana-3950	334	9	)	)	PUNCT
cana-3950	334	10	,	,	PUNCT
cana-3950	334	11	similarity	similarity	NOUN
cana-3950	334	12	measures	measure	NOUN
cana-3950	334	13	for	for	ADP
cana-3950	334	14	fermatean	fermatean	ADJ
cana-3950	334	15	fuzzy	fuzzy	ADJ
cana-3950	334	16	sets	set	NOUN
cana-3950	334	17	,	,	PUNCT
cana-3950	334	18	and	and	CCONJ
cana-3950	334	19	applications	application	NOUN
cana-3950	334	20	in	in	ADP
cana-3950	334	21	group	group	NOUN
cana-3950	334	22	decision	decision	NOUN
cana-3950	334	23	making	making	NOUN
cana-3950	334	24	.	.	PUNCT
cana-3950	335	1	decision	decision	NOUN
cana-3950	335	2	science	science	NOUN
cana-3950	335	3	letters	letter	NOUN
cana-3950	335	4	.	.	PUNCT
cana-3950	336	1	11	11	NUM
cana-3950	336	2	:	:	PUNCT
cana-3950	336	3	167	167	NUM
cana-3950	336	4	-	-	SYM
cana-3950	336	5	180	180	NUM
cana-3950	336	6	.	.	PUNCT
cana-3950	337	1	[	[	X
cana-3950	337	2	13	13	NUM
cana-3950	337	3	]	]	PUNCT
cana-3950	337	4	senapati	senapati	NOUN
cana-3950	337	5	.	.	PUNCT
cana-3950	338	1	t	t	NOUN
cana-3950	338	2	and	and	CCONJ
cana-3950	338	3	yager	yager	NOUN
cana-3950	338	4	.	.	PUNCT
cana-3950	339	1	r.	r.	PROPN
cana-3950	339	2	r.	r.	PROPN
cana-3950	339	3	,	,	PUNCT
cana-3950	339	4	(	(	PUNCT
cana-3950	339	5	2019	2019	NUM
cana-3950	339	6	)	)	PUNCT
cana-3950	339	7	,	,	PUNCT
cana-3950	339	8	some	some	DET
cana-3950	339	9	new	new	ADJ
cana-3950	339	10	operations	operation	NOUN
cana-3950	339	11	over	over	ADP
cana-3950	339	12	fermatean	fermatean	ADJ
cana-3950	339	13	fuzzy	fuzzy	ADJ
cana-3950	339	14	numbers	number	NOUN
cana-3950	339	15	and	and	CCONJ
cana-3950	339	16	application	application	NOUN
cana-3950	339	17	of	of	ADP
cana-3950	339	18	fermatean	fermatean	NOUN
cana-3950	339	19	fuzzy	fuzzy	ADJ
cana-3950	339	20	in	in	ADP
cana-3950	339	21	multicriteria	multicriteria	PROPN
cana-3950	339	22	decision	decision	NOUN
cana-3950	339	23	making	making	NOUN
cana-3950	339	24	.	.	PUNCT
cana-3950	340	1	informatics	informatic	NOUN
cana-3950	340	2	,	,	PUNCT
cana-3950	340	3	30	30	NUM
cana-3950	340	4	(	(	PUNCT
cana-3950	340	5	2	2	NUM
cana-3950	340	6	):	):	PUNCT
cana-3950	340	7	391	391	NUM
cana-3950	340	8	-	-	SYM
cana-3950	340	9	411	411	NUM
cana-3950	340	10	.	.	PUNCT
cana-3950	341	1	[	[	X
cana-3950	341	2	14	14	NUM
cana-3950	341	3	]	]	PUNCT
cana-3950	341	4	senapati	senapati	PROPN
cana-3950	341	5	.	.	PUNCT
cana-3950	342	1	t	t	NOUN
cana-3950	342	2	and	and	CCONJ
cana-3950	342	3	yager	yager	NOUN
cana-3950	342	4	.	.	PUNCT
cana-3950	343	1	r.	r.	PROPN
cana-3950	343	2	r.	r.	PROPN
cana-3950	343	3	,	,	PUNCT
cana-3950	343	4	(	(	PUNCT
cana-3950	343	5	2020	2020	NUM
cana-3950	343	6	)	)	PUNCT
cana-3950	343	7	,	,	PUNCT
cana-3950	343	8	fermatean	fermatean	ADJ
cana-3950	343	9	fuzzy	fuzzy	ADJ
cana-3950	343	10	sets	set	NOUN
cana-3950	343	11	.	.	PUNCT
cana-3950	344	1	journal	journal	PROPN
cana-3950	344	2	of	of	ADP
cana-3950	344	3	ambient	ambient	ADJ
cana-3950	344	4	intelligence	intelligence	NOUN
cana-3950	344	5	and	and	CCONJ
cana-3950	344	6	humanized	humanize	VERB
cana-3950	344	7	computing	computing	NOUN
cana-3950	344	8	.	.	PUNCT
cana-3950	345	1	11	11	NUM
cana-3950	345	2	:	:	PUNCT
cana-3950	345	3	663	663	NUM
cana-3950	345	4	-	-	SYM
cana-3950	345	5	674	674	NUM
cana-3950	345	6	.	.	PUNCT
cana-3950	346	1	[	[	X
cana-3950	346	2	15	15	NUM
cana-3950	346	3	]	]	X
cana-3950	346	4	szmidt	szmidt	PROPN
cana-3950	346	5	.	.	PUNCT
cana-3950	347	1	e.	e.	PROPN
cana-3950	347	2	(	(	PUNCT
cana-3950	347	3	2014	2014	NUM
cana-3950	347	4	)	)	PUNCT
cana-3950	347	5	.	.	PUNCT
cana-3950	348	1	distances	distance	NOUN
cana-3950	348	2	and	and	CCONJ
cana-3950	348	3	similarities	similarity	NOUN
cana-3950	348	4	in	in	ADP
cana-3950	348	5	intuitionistic	intuitionistic	ADJ
cana-3950	348	6	fuzzy	fuzzy	ADJ
cana-3950	348	7	sets	set	NOUN
cana-3950	348	8	.	.	PUNCT
cana-3950	349	1	springer	springer	NOUN
cana-3950	349	2	international	international	ADJ
cana-3950	349	3	publishing	publishing	PROPN
cana-3950	349	4	,	,	PUNCT
cana-3950	349	5	switzerland	switzerland	PROPN
cana-3950	349	6	.	.	PUNCT
cana-3950	350	1	[	[	X
cana-3950	350	2	16	16	NUM
cana-3950	350	3	]	]	PUNCT
cana-3950	350	4	szmidt	szmidt	PROPN
cana-3950	350	5	.	.	PUNCT
cana-3950	351	1	e.	e.	PROPN
cana-3950	351	2	and	and	CCONJ
cana-3950	351	3	kacprzyk	kacprzyk	PROPN
cana-3950	351	4	.j	.j	PROPN
cana-3950	351	5	(	(	PUNCT
cana-3950	351	6	2000	2000	NUM
cana-3950	351	7	)	)	PUNCT
cana-3950	351	8	,	,	PUNCT
cana-3950	351	9	distances	distance	NOUN
cana-3950	351	10	between	between	ADP
cana-3950	351	11	intuitionistic	intuitionistic	ADJ
cana-3950	351	12	fuzzy	fuzzy	ADJ
cana-3950	351	13	sets	set	NOUN
cana-3950	351	14	.	.	PUNCT
cana-3950	352	1	fuzzy	fuzzy	ADJ
cana-3950	352	2	sets	set	NOUN
cana-3950	352	3	and	and	CCONJ
cana-3950	352	4	systems	system	NOUN
cana-3950	352	5	.	.	PUNCT
cana-3950	353	1	114	114	NUM
cana-3950	353	2	(	(	PUNCT
cana-3950	353	3	3):505	3):505	PROPN
cana-3950	353	4	-	-	NOUN
cana-3950	353	5	518	518	NUM
cana-3950	353	6	.	.	PUNCT
cana-3950	354	1	[	[	X
cana-3950	354	2	17	17	NUM
cana-3950	354	3	]	]	X
cana-3950	354	4	wang	wang	PROPN
cana-3950	354	5	.	.	PUNCT
cana-3950	355	1	w	w	PROPN
cana-3950	355	2	and	and	CCONJ
cana-3950	355	3	xin	xin	PROPN
cana-3950	355	4	.	.	PUNCT
cana-3950	356	1	x.	x.	NOUN
cana-3950	356	2	(	(	PUNCT
cana-3950	356	3	2009	2009	NUM
cana-3950	356	4	)	)	PUNCT
cana-3950	356	5	,	,	PUNCT
cana-3950	356	6	distance	distance	NOUN
cana-3950	356	7	measure	measure	NOUN
cana-3950	356	8	between	between	ADP
cana-3950	356	9	intuitionistic	intuitionistic	ADJ
cana-3950	356	10	fuzzy	fuzzy	ADJ
cana-3950	356	11	sets	set	NOUN
cana-3950	356	12	.	.	PUNCT
cana-3950	357	1	pattern	pattern	NOUN
cana-3950	357	2	recognition	recognition	NOUN
cana-3950	357	3	letters	letter	NOUN
cana-3950	357	4	.	.	PUNCT
cana-3950	358	1	26	26	NUM
cana-3950	358	2	:	:	PUNCT
cana-3950	358	3	2063	2063	NUM
cana-3950	358	4	-	-	SYM
cana-3950	358	5	2069	2069	NUM
cana-3950	358	6	.	.	PUNCT
cana-3950	359	1	[	[	X
cana-3950	359	2	18	18	NUM
cana-3950	359	3	]	]	X
cana-3950	359	4	yager	yager	NOUN
cana-3950	359	5	.	.	PUNCT
cana-3950	359	6	r.	r.	PROPN
cana-3950	359	7	r.	r.	PROPN
cana-3950	359	8	(	(	PUNCT
cana-3950	359	9	2013	2013	NUM
cana-3950	359	10	)	)	PUNCT
cana-3950	359	11	,	,	PUNCT
cana-3950	359	12	pythagorean	pythagorean	PROPN
cana-3950	359	13	fuzzy	fuzzy	PROPN
cana-3950	359	14	subset	subset	NOUN
cana-3950	359	15	.	.	PUNCT
cana-3950	360	1	in	in	ADP
cana-3950	360	2	pedrycz	pedrycz	NOUN
cana-3950	360	3	.	.	PUNCT
cana-3950	361	1	w.	w.	NOUN
cana-3950	361	2	reformat	reformat	NOUN
cana-3950	361	3	.	.	PUNCT
cana-3950	362	1	m.	m.	NOUN
cana-3950	362	2	and	and	CCONJ
cana-3950	362	3	marek	marek	PROPN
cana-3950	362	4	.	.	PUNCT
cana-3950	363	1	z	z	PROPN
cana-3950	363	2	.	.	PUNCT
cana-3950	364	1	,	,	PUNCT
cana-3950	364	2	editors	editor	NOUN
cana-3950	364	3	,	,	PUNCT
cana-3950	364	4	proceedings	proceeding	NOUN
cana-3950	364	5	of	of	ADP
cana-3950	364	6	the	the	DET
cana-3950	364	7	2013	2013	NUM
cana-3950	364	8	joint	joint	ADJ
cana-3950	364	9	ifsa	ifsa	PROPN
cana-3950	364	10	world	world	PROPN
cana-3950	364	11	congress	congress	PROPN
cana-3950	364	12	and	and	CCONJ
cana-3950	364	13	nafips	nafip	NOUN
cana-3950	364	14	annual	annual	ADJ
cana-3950	364	15	meeting	meeting	NOUN
cana-3950	364	16	(	(	PUNCT
cana-3950	364	17	ifsa	ifsa	PROPN
cana-3950	364	18	/	/	SYM
cana-3950	364	19	nafips	nafip	NOUN
cana-3950	364	20	)	)	PUNCT
cana-3950	364	21	,	,	PUNCT
cana-3950	364	22	57	57	NUM
cana-3950	364	23	-	-	SYM
cana-3950	364	24	61	61	NUM
cana-3950	364	25	,	,	PUNCT
cana-3950	364	26	edmonton	edmonton	PROPN
cana-3950	364	27	,	,	PUNCT
cana-3950	364	28	canada	canada	PROPN
cana-3950	364	29	,	,	PUNCT
cana-3950	364	30	ieee	ieee	NOUN
cana-3950	364	31	.	.	PUNCT
cana-3950	365	1	[	[	X
cana-3950	365	2	19	19	NUM
cana-3950	365	3	]	]	SYM
cana-3950	365	4	zadeh	zadeh	PROPN
cana-3950	365	5	.	.	PUNCT
cana-3950	365	6	l.	l.	PROPN
cana-3950	365	7	a.	a.	PROPN
cana-3950	365	8	(	(	PUNCT
cana-3950	365	9	1965	1965	NUM
cana-3950	365	10	)	)	PUNCT
cana-3950	365	11	,	,	PUNCT
cana-3950	365	12	information	information	NOUN
cana-3950	365	13	and	and	CCONJ
cana-3950	365	14	control	control	NOUN
cana-3950	365	15	,	,	PUNCT
cana-3950	365	16	8	8	NUM
cana-3950	365	17	:	:	SYM
cana-3950	365	18	338	338	NUM
cana-3950	365	19	-	-	SYM
cana-3950	365	20	353	353	NUM
cana-3950	365	21	.	.	PUNCT
cana-3950	366	1	[	[	X
cana-3950	366	2	20	20	NUM
cana-3950	366	3	]	]	X
cana-3950	366	4	zeng	zeng	PROPN
cana-3950	366	5	.	.	PUNCT
cana-3950	367	1	w.	w.	PROPN
cana-3950	367	2	,	,	PUNCT
cana-3950	367	3	li	li	PROPN
cana-3950	367	4	.	.	PROPN
cana-3950	367	5	d.	d.	PROPN
cana-3950	367	6	and	and	CCONJ
cana-3950	367	7	yin	yin	PROPN
cana-3950	367	8	.	.	PUNCT
cana-3950	368	1	q.	q.	PROPN
cana-3950	368	2	(	(	PUNCT
cana-3950	368	3	2018	2018	NUM
cana-3950	368	4	)	)	PUNCT
cana-3950	368	5	,	,	PUNCT
cana-3950	368	6	distance	distance	NOUN
cana-3950	368	7	and	and	CCONJ
cana-3950	368	8	similarity	similarity	NOUN
cana-3950	368	9	measures	measure	NOUN
cana-3950	368	10	of	of	ADP
cana-3950	368	11	pythagorean	pythagorean	ADJ
cana-3950	368	12	fuzzy	fuzzy	ADJ
cana-3950	368	13	sets	set	NOUN
cana-3950	368	14	and	and	CCONJ
cana-3950	368	15	their	their	PRON
cana-3950	368	16	applications	application	NOUN
cana-3950	368	17	to	to	ADP
cana-3950	368	18	multiple	multiple	ADJ
cana-3950	368	19	criteria	criterion	NOUN
cana-3950	368	20	group	group	NOUN
cana-3950	368	21	decision	decision	NOUN
cana-3950	368	22	making	making	NOUN
cana-3950	368	23	.	.	PUNCT
cana-3950	369	1	international	international	ADJ
cana-3950	369	2	journal	journal	NOUN
cana-3950	369	3	of	of	ADP
cana-3950	369	4	intelligent	intelligent	ADJ
cana-3950	369	5	system	system	NOUN
cana-3950	369	6	,	,	PUNCT
cana-3950	369	7	33:22362254	33:22362254	NUM
cana-3950	369	8	.	.	PUNCT