id	sid	tid	token	lemma	pos
cana-2695	1	1	communications	communication	NOUN
cana-2695	1	2	on	on	ADP
cana-2695	1	3	applied	apply	VERB
cana-2695	1	4	nonlinear	nonlinear	ADJ
cana-2695	1	5	analysis	analysis	NOUN
cana-2695	1	6	issn	issn	NOUN
cana-2695	1	7	:	:	PUNCT
cana-2695	1	8	1074	1074	NUM
cana-2695	1	9	-	-	PUNCT
cana-2695	1	10	133x	133x	NUM
cana-2695	1	11	vol	vol	NOUN
cana-2695	1	12	32	32	NUM
cana-2695	1	13	no	no	NOUN
cana-2695	1	14	.	.	PUNCT
cana-2695	2	1	3s	3s	NUM
cana-2695	2	2	(	(	PUNCT
cana-2695	2	3	2025	2025	NUM
cana-2695	2	4	)	)	PUNCT
cana-2695	2	5	547	547	NUM
cana-2695	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2695	2	7	a	a	DET
cana-2695	2	8	novel	novel	ADJ
cana-2695	2	9	case	case	NOUN
cana-2695	2	10	on	on	ADP
cana-2695	2	11	fuzzy	fuzzy	ADJ
cana-2695	2	12	resolving	resolving	NOUN
cana-2695	2	13	sets	set	NOUN
cana-2695	2	14	on	on	ADP
cana-2695	2	15	interval	interval	NOUN
cana-2695	2	16	-	-	PUNCT
cana-2695	2	17	valued	value	VERB
cana-2695	2	18	fuzzy	fuzzy	ADJ
cana-2695	2	19	graph	graph	NOUN
cana-2695	2	20	and	and	CCONJ
cana-2695	2	21	its	its	PRON
cana-2695	2	22	application	application	NOUN
cana-2695	2	23	on	on	ADP
cana-2695	2	24	signal	signal	NOUN
cana-2695	2	25	processing	processing	NOUN
cana-2695	2	26	unit	unit	NOUN
cana-2695	2	27	r.	r.	PROPN
cana-2695	2	28	shanmugapriya1	shanmugapriya1	PROPN
cana-2695	2	29	&	&	CCONJ
cana-2695	2	30	vasuki	vasuki	PROPN
cana-2695	2	31	m1	m1	PROPN
cana-2695	2	32	1,1department	1,1department	NUM
cana-2695	2	33	of	of	ADP
cana-2695	2	34	mathematics	mathematic	NOUN
cana-2695	2	35	,	,	PUNCT
cana-2695	2	36	vel	vel	ADJ
cana-2695	2	37	tech	tech	NOUN
cana-2695	2	38	rangarajan	rangarajan	NOUN
cana-2695	2	39	dr.sagunthala	dr.sagunthala	NOUN
cana-2695	2	40	r&d	r&d	PROPN
cana-2695	2	41	institute	institute	PROPN
cana-2695	2	42	of	of	ADP
cana-2695	2	43	science	science	NOUN
cana-2695	2	44	and	and	CCONJ
cana-2695	2	45	technology	technology	NOUN
cana-2695	2	46	,	,	PUNCT
cana-2695	2	47	chennai	chennai	PROPN
cana-2695	2	48	.	.	PUNCT
cana-2695	3	1	email	email	NOUN
cana-2695	4	1	i	i	PROPN
cana-2695	4	2	d	d	PROPN
cana-2695	4	3	:	:	PUNCT
cana-2695	4	4	spriyasathish11@gmail.com1	spriyasathish11@gmail.com1	PROPN
cana-2695	4	5	&	&	CCONJ
cana-2695	4	6	vasukimani1997@gmail.com1	vasukimani1997@gmail.com1	PROPN
cana-2695	4	7	article	article	NOUN
cana-2695	4	8	history	history	NOUN
cana-2695	4	9	:	:	PUNCT
cana-2695	4	10	received	receive	VERB
cana-2695	4	11	:	:	PUNCT
cana-2695	4	12	23	23	NUM
cana-2695	4	13	-	-	SYM
cana-2695	4	14	09	09	NUM
cana-2695	4	15	-	-	PUNCT
cana-2695	4	16	2024	2024	NUM
cana-2695	4	17	revised	revise	VERB
cana-2695	4	18	:	:	PUNCT
cana-2695	4	19	09	09	NUM
cana-2695	4	20	-	-	SYM
cana-2695	4	21	11	11	NUM
cana-2695	4	22	-	-	PUNCT
cana-2695	4	23	2024	2024	NUM
cana-2695	4	24	accepted	accept	VERB
cana-2695	4	25	:	:	PUNCT
cana-2695	4	26	27	27	NUM
cana-2695	4	27	-	-	SYM
cana-2695	4	28	11	11	NUM
cana-2695	4	29	-	-	PUNCT
cana-2695	4	30	2024	2024	NUM
cana-2695	4	31	abstract	abstract	NOUN
cana-2695	4	32	:	:	PUNCT
cana-2695	4	33	a	a	DET
cana-2695	4	34	fuzzy	fuzzy	ADJ
cana-2695	4	35	graph	graph	NOUN
cana-2695	4	36	generalisation	generalisation	NOUN
cana-2695	4	37	known	know	VERB
cana-2695	4	38	as	as	ADP
cana-2695	4	39	an	an	DET
cana-2695	4	40	interval	interval	NOUN
cana-2695	4	41	-	-	PUNCT
cana-2695	4	42	valued	value	VERB
cana-2695	4	43	fuzzy	fuzzy	ADJ
cana-2695	4	44	graph	graph	NOUN
cana-2695	4	45	(	(	PUNCT
cana-2695	4	46	ivfg	ivfg	NOUN
cana-2695	4	47	)	)	PUNCT
cana-2695	4	48	reflects	reflect	VERB
cana-2695	4	49	uncertainty	uncertainty	NOUN
cana-2695	4	50	more	more	ADV
cana-2695	4	51	flexibly	flexibly	ADV
cana-2695	4	52	by	by	ADP
cana-2695	4	53	representing	represent	VERB
cana-2695	4	54	the	the	DET
cana-2695	4	55	membership	membership	NOUN
cana-2695	4	56	values	value	NOUN
cana-2695	4	57	of	of	ADP
cana-2695	4	58	vertices	vertex	NOUN
cana-2695	4	59	and	and	CCONJ
cana-2695	4	60	edges	edge	NOUN
cana-2695	4	61	as	as	ADP
cana-2695	4	62	intervals	interval	NOUN
cana-2695	4	63	rather	rather	ADV
cana-2695	4	64	than	than	ADP
cana-2695	4	65	fixed	fix	VERB
cana-2695	4	66	numbers	number	NOUN
cana-2695	4	67	.	.	PUNCT
cana-2695	5	1	we	we	PRON
cana-2695	5	2	have	have	AUX
cana-2695	5	3	defined	define	VERB
cana-2695	5	4	fuzzy	fuzzy	ADJ
cana-2695	5	5	resolving	resolving	NOUN
cana-2695	5	6	set	set	VERB
cana-2695	5	7	to	to	ADP
cana-2695	5	8	intervalvalued	intervalvalue	VERB
cana-2695	5	9	fuzzy	fuzzy	ADJ
cana-2695	5	10	graphs	graph	NOUN
cana-2695	5	11	such	such	ADJ
cana-2695	5	12	as	as	ADP
cana-2695	5	13	interval	interval	NOUN
cana-2695	5	14	-	-	PUNCT
cana-2695	5	15	valued	value	VERB
cana-2695	5	16	fuzzy	fuzzy	ADJ
cana-2695	5	17	resolving	resolving	NOUN
cana-2695	5	18	set	set	VERB
cana-2695	5	19	and	and	CCONJ
cana-2695	5	20	interval	interval	NOUN
cana-2695	5	21	-	-	PUNCT
cana-2695	5	22	valued	value	VERB
cana-2695	5	23	fuzzy	fuzzy	ADJ
cana-2695	5	24	super	super	ADJ
cana-2695	5	25	resolving	resolve	VERB
cana-2695	5	26	set	set	NOUN
cana-2695	5	27	.	.	PUNCT
cana-2695	6	1	we	we	PRON
cana-2695	6	2	also	also	ADV
cana-2695	6	3	derived	derive	VERB
cana-2695	6	4	the	the	DET
cana-2695	6	5	properties	property	NOUN
cana-2695	6	6	of	of	ADP
cana-2695	6	7	isomorphism	isomorphism	NOUN
cana-2695	6	8	in	in	ADP
cana-2695	6	9	this	this	DET
cana-2695	6	10	topic	topic	NOUN
cana-2695	6	11	and	and	CCONJ
cana-2695	6	12	explained	explain	VERB
cana-2695	6	13	an	an	DET
cana-2695	6	14	application	application	NOUN
cana-2695	6	15	based	base	VERB
cana-2695	6	16	on	on	ADP
cana-2695	6	17	them	they	PRON
cana-2695	6	18	.	.	PUNCT
cana-2695	7	1	keywords	keyword	NOUN
cana-2695	7	2	:	:	PUNCT
cana-2695	7	3	fuzzy	fuzzy	ADJ
cana-2695	7	4	graph	graph	NOUN
cana-2695	7	5	,	,	PUNCT
cana-2695	7	6	fuzzy	fuzzy	ADJ
cana-2695	7	7	resolving	resolving	NOUN
cana-2695	7	8	set	set	NOUN
cana-2695	7	9	,	,	PUNCT
cana-2695	7	10	interval	interval	NOUN
cana-2695	7	11	-	-	PUNCT
cana-2695	7	12	valued	value	VERB
cana-2695	7	13	fuzzy	fuzzy	ADJ
cana-2695	7	14	graph	graph	NOUN
cana-2695	7	15	,	,	PUNCT
cana-2695	7	16	interval	interval	NOUN
cana-2695	7	17	valued	value	VERB
cana-2695	7	18	fuzzy	fuzzy	ADJ
cana-2695	7	19	resolving	resolving	NOUN
cana-2695	7	20	set	set	VERB
cana-2695	7	21	1	1	NUM
cana-2695	7	22	.	.	PUNCT
cana-2695	8	1	introduction	introduction	NOUN
cana-2695	8	2	the	the	DET
cana-2695	8	3	degree	degree	NOUN
cana-2695	8	4	of	of	ADP
cana-2695	8	5	certainty	certainty	NOUN
cana-2695	8	6	or	or	CCONJ
cana-2695	8	7	uncertainty	uncertainty	NOUN
cana-2695	8	8	can	can	AUX
cana-2695	8	9	be	be	AUX
cana-2695	8	10	expressed	express	VERB
cana-2695	8	11	using	use	VERB
cana-2695	8	12	degrees	degree	NOUN
cana-2695	8	13	of	of	ADP
cana-2695	8	14	membership	membership	NOUN
cana-2695	8	15	in	in	ADP
cana-2695	8	16	fuzzy	fuzzy	ADJ
cana-2695	8	17	sets	set	NOUN
cana-2695	8	18	.	.	PUNCT
cana-2695	9	1	instead	instead	ADV
cana-2695	9	2	of	of	ADP
cana-2695	9	3	dealing	deal	VERB
cana-2695	9	4	with	with	ADP
cana-2695	9	5	precise	precise	ADJ
cana-2695	9	6	distances	distance	NOUN
cana-2695	9	7	,	,	PUNCT
cana-2695	9	8	we	we	PRON
cana-2695	9	9	deal	deal	VERB
cana-2695	9	10	with	with	ADP
cana-2695	9	11	fuzzy	fuzzy	ADJ
cana-2695	9	12	distances	distance	NOUN
cana-2695	9	13	or	or	CCONJ
cana-2695	9	14	fuzzy	fuzzy	ADJ
cana-2695	9	15	sets	set	NOUN
cana-2695	9	16	of	of	ADP
cana-2695	9	17	distances	distance	NOUN
cana-2695	9	18	in	in	ADP
cana-2695	9	19	an	an	DET
cana-2695	9	20	frs	frs	NOUN
cana-2695	9	21	.	.	PUNCT
cana-2695	10	1	in	in	ADP
cana-2695	10	2	summary	summary	NOUN
cana-2695	10	3	,	,	PUNCT
cana-2695	10	4	an	an	DET
cana-2695	10	5	frs	frs	NOUN
cana-2695	10	6	provides	provide	VERB
cana-2695	10	7	a	a	DET
cana-2695	10	8	more	more	ADV
cana-2695	10	9	flexible	flexible	ADJ
cana-2695	10	10	framework	framework	NOUN
cana-2695	10	11	for	for	ADP
cana-2695	10	12	resolving	resolve	VERB
cana-2695	10	13	network	network	NOUN
cana-2695	10	14	vertices	vertex	NOUN
cana-2695	10	15	when	when	SCONJ
cana-2695	10	16	precise	precise	ADJ
cana-2695	10	17	information	information	NOUN
cana-2695	10	18	is	be	AUX
cana-2695	10	19	not	not	PART
cana-2695	10	20	easily	easily	ADV
cana-2695	10	21	accessible	accessible	ADJ
cana-2695	10	22	.	.	PUNCT
cana-2695	11	1	it	it	PRON
cana-2695	11	2	is	be	AUX
cana-2695	11	3	an	an	DET
cana-2695	11	4	extension	extension	NOUN
cana-2695	11	5	of	of	ADP
cana-2695	11	6	traditional	traditional	ADJ
cana-2695	11	7	resolving	resolving	NOUN
cana-2695	11	8	sets	set	NOUN
cana-2695	11	9	that	that	PRON
cana-2695	11	10	accounts	account	VERB
cana-2695	11	11	for	for	ADP
cana-2695	11	12	measurement	measurement	NOUN
cana-2695	11	13	uncertainty	uncertainty	NOUN
cana-2695	11	14	in	in	ADP
cana-2695	11	15	the	the	DET
cana-2695	11	16	vertex	vertex	NOUN
cana-2695	11	17	-	-	PUNCT
cana-2695	11	18	to	to	ADP
cana-2695	11	19	-	-	PUNCT
cana-2695	11	20	vertex	vertex	NOUN
cana-2695	11	21	distance	distance	NOUN
cana-2695	11	22	.	.	PUNCT
cana-2695	12	1	frs	frs	PROPN
cana-2695	12	2	can	can	AUX
cana-2695	12	3	be	be	AUX
cana-2695	12	4	used	use	VERB
cana-2695	12	5	to	to	PART
cana-2695	12	6	model	model	VERB
cana-2695	12	7	the	the	DET
cana-2695	12	8	transmission	transmission	NOUN
cana-2695	12	9	of	of	ADP
cana-2695	12	10	illnesses	illness	NOUN
cana-2695	12	11	when	when	SCONJ
cana-2695	12	12	precise	precise	ADJ
cana-2695	12	13	distance	distance	NOUN
cana-2695	12	14	measurements	measurement	NOUN
cana-2695	12	15	between	between	ADP
cana-2695	12	16	sites	site	NOUN
cana-2695	12	17	or	or	CCONJ
cana-2695	12	18	between	between	ADP
cana-2695	12	19	individuals	individual	NOUN
cana-2695	12	20	are	be	AUX
cana-2695	12	21	difficult	difficult	ADJ
cana-2695	12	22	to	to	PART
cana-2695	12	23	obtain	obtain	VERB
cana-2695	12	24	.	.	PUNCT
cana-2695	13	1	an	an	DET
cana-2695	13	2	interval	interval	NOUN
cana-2695	13	3	-	-	PUNCT
cana-2695	13	4	valued	value	VERB
cana-2695	13	5	fuzzy	fuzzy	ADJ
cana-2695	13	6	graph	graph	NOUN
cana-2695	13	7	(	(	PUNCT
cana-2695	13	8	ivfg	ivfg	NOUN
cana-2695	13	9	)	)	PUNCT
cana-2695	13	10	is	be	AUX
cana-2695	13	11	a	a	DET
cana-2695	13	12	sophisticated	sophisticated	ADJ
cana-2695	13	13	mathematical	mathematical	ADJ
cana-2695	13	14	framework	framework	NOUN
cana-2695	13	15	that	that	PRON
cana-2695	13	16	models	model	VERB
cana-2695	13	17	uncertainty	uncertainty	NOUN
cana-2695	13	18	in	in	ADP
cana-2695	13	19	real	real	ADJ
cana-2695	13	20	-	-	PUNCT
cana-2695	13	21	world	world	NOUN
cana-2695	13	22	systems	system	NOUN
cana-2695	13	23	by	by	ADP
cana-2695	13	24	fusing	fuse	VERB
cana-2695	13	25	fuzzy	fuzzy	ADJ
cana-2695	13	26	logic	logic	NOUN
cana-2695	13	27	and	and	CCONJ
cana-2695	13	28	graph	graph	NOUN
cana-2695	13	29	theory	theory	NOUN
cana-2695	13	30	.	.	PUNCT
cana-2695	14	1	ivfgs	ivfgs	PROPN
cana-2695	14	2	represent	represent	VERB
cana-2695	14	3	uncertainty	uncertainty	NOUN
cana-2695	14	4	in	in	ADP
cana-2695	14	5	the	the	DET
cana-2695	14	6	degree	degree	NOUN
cana-2695	14	7	of	of	ADP
cana-2695	14	8	membership	membership	NOUN
cana-2695	14	9	of	of	ADP
cana-2695	14	10	vertices	vertex	NOUN
cana-2695	14	11	and	and	CCONJ
cana-2695	14	12	edges	edge	NOUN
cana-2695	14	13	using	use	VERB
cana-2695	14	14	intervals	interval	NOUN
cana-2695	14	15	,	,	PUNCT
cana-2695	14	16	as	as	SCONJ
cana-2695	14	17	opposed	oppose	VERB
cana-2695	14	18	to	to	ADP
cana-2695	14	19	standard	standard	ADJ
cana-2695	14	20	graphs	graph	NOUN
cana-2695	14	21	or	or	CCONJ
cana-2695	14	22	fuzzy	fuzzy	ADJ
cana-2695	14	23	graphs	graph	NOUN
cana-2695	14	24	,	,	PUNCT
cana-2695	14	25	where	where	SCONJ
cana-2695	14	26	memberships	membership	NOUN
cana-2695	14	27	are	be	AUX
cana-2695	14	28	discrete	discrete	ADJ
cana-2695	14	29	or	or	CCONJ
cana-2695	14	30	single	single	ADJ
cana-2695	14	31	values	value	NOUN
cana-2695	14	32	.	.	PUNCT
cana-2695	15	1	while	while	SCONJ
cana-2695	15	2	euler	euler	PROPN
cana-2695	15	3	announced	announce	VERB
cana-2695	15	4	graph	graph	NOUN
cana-2695	15	5	theory	theory	NOUN
cana-2695	15	6	,	,	PUNCT
cana-2695	15	7	zadeh[13	zadeh[13	NOUN
cana-2695	15	8	]	]	PUNCT
cana-2695	15	9	evaluated	evaluate	VERB
cana-2695	15	10	in	in	ADP
cana-2695	15	11	1965	1965	NUM
cana-2695	15	12	.	.	PUNCT
cana-2695	16	1	kauffman	kauffman	PROPN
cana-2695	16	2	developed	develop	VERB
cana-2695	16	3	the	the	DET
cana-2695	16	4	fuzzy	fuzzy	ADJ
cana-2695	16	5	set	set	NOUN
cana-2695	16	6	,	,	PUNCT
cana-2695	16	7	which	which	PRON
cana-2695	16	8	served	serve	VERB
cana-2695	16	9	as	as	ADP
cana-2695	16	10	the	the	DET
cana-2695	16	11	foundation	foundation	NOUN
cana-2695	16	12	for	for	ADP
cana-2695	16	13	the	the	DET
cana-2695	16	14	fuzzy	fuzzy	ADJ
cana-2695	16	15	graph	graph	NOUN
cana-2695	16	16	,	,	PUNCT
cana-2695	16	17	in	in	ADP
cana-2695	16	18	1973	1973	NUM
cana-2695	16	19	.	.	PUNCT
cana-2695	17	1	moderson	moderson	NOUN
cana-2695	17	2	and	and	CCONJ
cana-2695	17	3	associates	associate	NOUN
cana-2695	18	1	[	[	X
cana-2695	18	2	6	6	NUM
cana-2695	18	3	]	]	PUNCT
cana-2695	18	4	created	create	VERB
cana-2695	18	5	a	a	DET
cana-2695	18	6	fuzzy	fuzzy	ADJ
cana-2695	18	7	graph	graph	NOUN
cana-2695	18	8	that	that	PRON
cana-2695	18	9	has	have	VERB
cana-2695	18	10	numerous	numerous	ADJ
cana-2695	18	11	uses	use	NOUN
cana-2695	18	12	.	.	PUNCT
cana-2695	19	1	resolvability	resolvability	NOUN
cana-2695	19	2	and	and	CCONJ
cana-2695	19	3	an	an	DET
cana-2695	19	4	upper	upper	ADJ
cana-2695	19	5	dimension	dimension	NOUN
cana-2695	19	6	of	of	ADP
cana-2695	19	7	graphs	graph	NOUN
cana-2695	19	8	were	be	AUX
cana-2695	19	9	first	first	ADV
cana-2695	19	10	suggested	suggest	VERB
cana-2695	19	11	by	by	ADP
cana-2695	19	12	chartrand[2	chartrand[2	PROPN
cana-2695	19	13	]	]	X
cana-2695	19	14	in	in	ADP
cana-2695	19	15	2000	2000	NUM
cana-2695	19	16	.	.	PUNCT
cana-2695	20	1	shanmugapriya	shanmugapriya	PROPN
cana-2695	20	2	and	and	CCONJ
cana-2695	20	3	mary	mary	PROPN
cana-2695	20	4	jiny	jiny	PROPN
cana-2695	21	1	[	[	X
cana-2695	21	2	9	9	NUM
cana-2695	21	3	]	]	PUNCT
cana-2695	21	4	subsequently	subsequently	ADV
cana-2695	21	5	extended	extend	VERB
cana-2695	21	6	this	this	DET
cana-2695	21	7	concept	concept	NOUN
cana-2695	21	8	to	to	ADP
cana-2695	21	9	resolvability	resolvability	NOUN
cana-2695	21	10	in	in	ADP
cana-2695	21	11	fuzzy	fuzzy	ADJ
cana-2695	21	12	graphs	graph	NOUN
cana-2695	21	13	.	.	PUNCT
cana-2695	22	1	additionally	additionally	ADV
cana-2695	22	2	,	,	PUNCT
cana-2695	22	3	they	they	PRON
cana-2695	22	4	both	both	PRON
cana-2695	22	5	introduced	introduce	VERB
cana-2695	22	6	modified	modify	VERB
cana-2695	22	7	frn	frn	PROPN
cana-2695	22	8	and	and	CCONJ
cana-2695	22	9	frs	frs	PROPN
cana-2695	22	10	characteristics	characteristic	NOUN
cana-2695	22	11	.	.	PUNCT
cana-2695	23	1	furthermore	furthermore	ADV
cana-2695	23	2	,	,	PUNCT
cana-2695	23	3	atanassov[1	atanassov[1	PROPN
cana-2695	23	4	]	]	PUNCT
cana-2695	23	5	extended	extend	VERB
cana-2695	23	6	the	the	DET
cana-2695	23	7	fg	fg	PROPN
cana-2695	23	8	to	to	PART
cana-2695	23	9	create	create	VERB
cana-2695	23	10	an	an	DET
cana-2695	23	11	intuitive	intuitive	ADJ
cana-2695	23	12	fuzzy	fuzzy	ADJ
cana-2695	23	13	graph	graph	NOUN
cana-2695	23	14	(	(	PUNCT
cana-2695	23	15	ifg	ifg	PROPN
cana-2695	23	16	)	)	PUNCT
cana-2695	23	17	.	.	PUNCT
cana-2695	24	1	an	an	DET
cana-2695	24	2	interval	interval	NOUN
cana-2695	24	3	-	-	PUNCT
cana-2695	24	4	valued	value	VERB
cana-2695	24	5	fuzzy	fuzzy	ADJ
cana-2695	24	6	graph	graph	NOUN
cana-2695	24	7	has	have	AUX
cana-2695	24	8	been	be	AUX
cana-2695	24	9	found	find	VERB
cana-2695	24	10	by	by	ADP
cana-2695	24	11	muhammad	muhammad	PROPN
cana-2695	24	12	akram	akram	PROPN
cana-2695	24	13	and	and	CCONJ
cana-2695	24	14	a.	a.	NOUN
cana-2695	24	15	dudek	dudek	PROPN
cana-2695	24	16	in	in	ADP
cana-2695	24	17	2011[7	2011[7	NUM
cana-2695	24	18	]	]	PUNCT
cana-2695	24	19	.	.	PUNCT
cana-2695	25	1	it	it	PRON
cana-2695	25	2	has	have	AUX
cana-2695	25	3	been	be	AUX
cana-2695	25	4	further	far	ADV
cana-2695	25	5	elaborated	elaborate	VERB
cana-2695	25	6	by	by	ADP
cana-2695	25	7	pramanik	pramanik	NOUN
cana-2695	25	8	and	and	CCONJ
cana-2695	25	9	others	other	NOUN
cana-2695	25	10	in	in	ADP
cana-2695	25	11	2020[10	2020[10	NUM
cana-2695	25	12	]	]	PUNCT
cana-2695	25	13	.	.	PUNCT
cana-2695	26	1	also	also	ADV
cana-2695	26	2	,	,	PUNCT
cana-2695	26	3	the	the	DET
cana-2695	26	4	properties	property	NOUN
cana-2695	26	5	of	of	ADP
cana-2695	26	6	interval	interval	NOUN
cana-2695	26	7	-	-	PUNCT
cana-2695	26	8	valued	value	VERB
cana-2695	26	9	fuzzy	fuzzy	ADJ
cana-2695	26	10	graph	graph	NOUN
cana-2695	26	11	has	have	AUX
cana-2695	26	12	been	be	AUX
cana-2695	26	13	developed	develop	VERB
cana-2695	26	14	by	by	ADP
cana-2695	26	15	xiaoli	xiaoli	PROPN
cana-2695	26	16	qiang	qiang	PROPN
cana-2695	26	17	and	and	CCONJ
cana-2695	26	18	others	other	NOUN
cana-2695	26	19	in	in	ADP
cana-2695	26	20	the	the	DET
cana-2695	26	21	year	year	NOUN
cana-2695	26	22	2022[12	2022[12	NUM
cana-2695	26	23	]	]	PUNCT
cana-2695	26	24	.	.	PUNCT
cana-2695	27	1	they	they	PRON
cana-2695	27	2	also	also	ADV
cana-2695	27	3	collaborated	collaborate	VERB
cana-2695	27	4	with	with	ADP
cana-2695	27	5	numerous	numerous	ADJ
cana-2695	27	6	others	other	NOUN
cana-2695	27	7	to	to	PART
cana-2695	27	8	create	create	VERB
cana-2695	27	9	a	a	DET
cana-2695	27	10	number	number	NOUN
cana-2695	27	11	of	of	ADP
cana-2695	27	12	research	research	NOUN
cana-2695	27	13	works	work	VERB
cana-2695	27	14	on	on	ADP
cana-2695	27	15	fuzzy	fuzzy	ADJ
cana-2695	27	16	graphs	graph	NOUN
cana-2695	27	17	,	,	PUNCT
cana-2695	27	18	including	include	VERB
cana-2695	27	19	studies	study	NOUN
cana-2695	27	20	on	on	ADP
cana-2695	27	21	the	the	DET
cana-2695	27	22	size	size	NOUN
cana-2695	27	23	,	,	PUNCT
cana-2695	27	24	order	order	NOUN
cana-2695	27	25	,	,	PUNCT
cana-2695	27	26	and	and	CCONJ
cana-2695	27	27	strong	strong	ADJ
cana-2695	27	28	and	and	CCONJ
cana-2695	27	29	weak	weak	ADJ
cana-2695	27	30	domination	domination	NOUN
cana-2695	27	31	of	of	ADP
cana-2695	27	32	fuzzy	fuzzy	ADJ
cana-2695	27	33	graphs	graph	NOUN
cana-2695	27	34	.	.	PUNCT
cana-2695	28	1	the	the	DET
cana-2695	28	2	parameter	parameter	PROPN
cana-2695	28	3	mailto	mailto	PROPN
cana-2695	28	4	:	:	PUNCT
cana-2695	28	5	spriyasathish@gmail.com1	spriyasathish@gmail.com1	PROPN
cana-2695	28	6	mailto:vasukimani1997@gmail.com	mailto:vasukimani1997@gmail.com	PROPN
cana-2695	28	7	communications	communication	NOUN
cana-2695	28	8	on	on	ADP
cana-2695	28	9	applied	apply	VERB
cana-2695	28	10	nonlinear	nonlinear	ADJ
cana-2695	28	11	analysis	analysis	NOUN
cana-2695	28	12	issn	issn	NOUN
cana-2695	28	13	:	:	PUNCT
cana-2695	28	14	1074	1074	NUM
cana-2695	28	15	-	-	PUNCT
cana-2695	28	16	133x	133x	NUM
cana-2695	28	17	vol	vol	NOUN
cana-2695	28	18	32	32	NUM
cana-2695	28	19	no	no	NOUN
cana-2695	28	20	.	.	PUNCT
cana-2695	29	1	3s	3s	NUM
cana-2695	29	2	(	(	PUNCT
cana-2695	29	3	2025	2025	NUM
cana-2695	29	4	)	)	PUNCT
cana-2695	29	5	548	548	NUM
cana-2695	29	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2695	29	7	of	of	ADP
cana-2695	29	8	clustering	cluster	VERB
cana-2695	29	9	the	the	DET
cana-2695	29	10	sets	set	NOUN
cana-2695	29	11	is	be	AUX
cana-2695	29	12	used	use	VERB
cana-2695	29	13	in	in	ADP
cana-2695	29	14	the	the	DET
cana-2695	29	15	development	development	NOUN
cana-2695	29	16	of	of	ADP
cana-2695	29	17	numerous	numerous	ADJ
cana-2695	29	18	applications	application	NOUN
cana-2695	29	19	.	.	PUNCT
cana-2695	30	1	in	in	ADP
cana-2695	30	2	2023	2023	NUM
cana-2695	30	3	,	,	PUNCT
cana-2695	30	4	fuzzy	fuzzy	ADJ
cana-2695	30	5	resolving	resolving	NOUN
cana-2695	30	6	domination	domination	NOUN
cana-2695	30	7	set	set	NOUN
cana-2695	30	8	has	have	AUX
cana-2695	30	9	been	be	AUX
cana-2695	30	10	introduced	introduce	VERB
cana-2695	30	11	by	by	ADP
cana-2695	30	12	shanmugapriya	shanmugapriya	PROPN
cana-2695	30	13	and	and	CCONJ
cana-2695	30	14	co	co	VERB
cana-2695	31	1	[	[	X
cana-2695	31	2	8][11	8][11	NOUN
cana-2695	31	3	]	]	X
cana-2695	31	4	.	.	PUNCT
cana-2695	32	1	let	let	VERB
cana-2695	32	2	𝐺	𝐺	PRON
cana-2695	32	3	be	be	AUX
cana-2695	32	4	a	a	DET
cana-2695	32	5	connected	connected	ADJ
cana-2695	32	6	simple	simple	ADJ
cana-2695	32	7	finite	finite	NOUN
cana-2695	32	8	fg	fg	PROPN
cana-2695	32	9	with	with	ADP
cana-2695	32	10	the	the	DET
cana-2695	32	11	number	number	NOUN
cana-2695	32	12	of	of	ADP
cana-2695	32	13	vertices	vertex	NOUN
cana-2695	32	14	greater	great	ADJ
cana-2695	32	15	than	than	ADP
cana-2695	32	16	3	3	NUM
cana-2695	32	17	.	.	NOUN
cana-2695	32	18	standard	standard	ADJ
cana-2695	32	19	fuzzy	fuzzy	ADJ
cana-2695	32	20	graph	graph	NOUN
cana-2695	32	21	and	and	CCONJ
cana-2695	32	22	interval	interval	NOUN
cana-2695	32	23	-	-	PUNCT
cana-2695	32	24	valued	value	VERB
cana-2695	32	25	fuzzy	fuzzy	ADJ
cana-2695	32	26	graph	graph	NOUN
cana-2695	32	27	definitions	definition	NOUN
cana-2695	32	28	were	be	AUX
cana-2695	32	29	covered	cover	VERB
cana-2695	32	30	in	in	ADP
cana-2695	32	31	section	section	NOUN
cana-2695	32	32	2	2	NUM
cana-2695	32	33	,	,	PUNCT
cana-2695	32	34	whereas	whereas	SCONJ
cana-2695	32	35	ivfrs	ivfrs	NOUN
cana-2695	32	36	,	,	PUNCT
cana-2695	32	37	ivfsrs	ivfsrs	ADJ
cana-2695	32	38	and	and	CCONJ
cana-2695	32	39	its	its	PRON
cana-2695	32	40	theorems	theorem	NOUN
cana-2695	32	41	were	be	AUX
cana-2695	32	42	presented	present	VERB
cana-2695	32	43	in	in	ADP
cana-2695	32	44	section	section	NOUN
cana-2695	32	45	3	3	NUM
cana-2695	32	46	and	and	CCONJ
cana-2695	32	47	4	4	NUM
cana-2695	32	48	.	.	X
cana-2695	33	1	we	we	PRON
cana-2695	33	2	have	have	AUX
cana-2695	33	3	created	create	VERB
cana-2695	33	4	an	an	DET
cana-2695	33	5	application	application	NOUN
cana-2695	33	6	based	base	VERB
cana-2695	33	7	on	on	ADP
cana-2695	33	8	the	the	DET
cana-2695	33	9	idea	idea	NOUN
cana-2695	33	10	in	in	ADP
cana-2695	33	11	section	section	NOUN
cana-2695	33	12	5	5	NUM
cana-2695	33	13	.	.	PUNCT
cana-2695	34	1	we	we	PRON
cana-2695	34	2	are	be	AUX
cana-2695	34	3	keen	keen	ADJ
cana-2695	34	4	to	to	PART
cana-2695	34	5	explore	explore	VERB
cana-2695	34	6	into	into	ADP
cana-2695	34	7	further	further	ADJ
cana-2695	34	8	fuzzy	fuzzy	ADJ
cana-2695	34	9	resolving	resolve	VERB
cana-2695	34	10	set	set	NOUN
cana-2695	34	11	-	-	PUNCT
cana-2695	34	12	related	relate	VERB
cana-2695	34	13	subjects	subject	NOUN
cana-2695	34	14	in	in	ADP
cana-2695	34	15	the	the	DET
cana-2695	34	16	future	future	ADJ
cana-2695	34	17	studies	study	NOUN
cana-2695	34	18	.	.	PUNCT
cana-2695	35	1	names	name	NOUN
cana-2695	35	2	notations	notation	NOUN
cana-2695	35	3	fuzzy	fuzzy	ADJ
cana-2695	35	4	resolving	resolving	NOUN
cana-2695	35	5	set	set	VERB
cana-2695	35	6	frs	frs	ADJ
cana-2695	35	7	interval	interval	NOUN
cana-2695	35	8	-	-	PUNCT
cana-2695	35	9	valued	value	VERB
cana-2695	35	10	fuzzy	fuzzy	ADJ
cana-2695	35	11	graph	graph	NOUN
cana-2695	35	12	ivfg	ivfg	ADJ
cana-2695	35	13	interval	interval	NOUN
cana-2695	35	14	-	-	PUNCT
cana-2695	35	15	valued	value	VERB
cana-2695	35	16	fuzzy	fuzzy	ADJ
cana-2695	35	17	resolving	resolving	NOUN
cana-2695	35	18	set	set	VERB
cana-2695	35	19	ivfrs	ivfrs	NOUN
cana-2695	35	20	interval	interval	NOUN
cana-2695	35	21	-	-	PUNCT
cana-2695	35	22	valued	value	VERB
cana-2695	35	23	fuzzy	fuzzy	ADJ
cana-2695	35	24	super	super	ADJ
cana-2695	35	25	resolving	resolving	NOUN
cana-2695	35	26	set	set	VERB
cana-2695	35	27	ivfsrs	ivfsrs	ADJ
cana-2695	35	28	interval	interval	NOUN
cana-2695	35	29	-	-	PUNCT
cana-2695	35	30	valued	value	VERB
cana-2695	35	31	fuzzy	fuzzy	ADJ
cana-2695	35	32	resolving	resolving	NOUN
cana-2695	35	33	matrix	matrix	NOUN
cana-2695	35	34	ivfrm	ivfrm	NOUN
cana-2695	35	35	interval	interval	NOUN
cana-2695	35	36	-	-	PUNCT
cana-2695	35	37	valued	value	VERB
cana-2695	35	38	fuzzy	fuzzy	ADJ
cana-2695	35	39	super	super	ADJ
cana-2695	35	40	resolving	resolving	NOUN
cana-2695	35	41	matrix	matrix	NOUN
cana-2695	35	42	ivfsrm	ivfsrm	NOUN
cana-2695	35	43	table	table	NOUN
cana-2695	35	44	1	1	NUM
cana-2695	35	45	2	2	NUM
cana-2695	35	46	.	.	PUNCT
cana-2695	35	47	preliminaries	preliminary	NOUN
cana-2695	35	48	in	in	ADP
cana-2695	35	49	this	this	DET
cana-2695	35	50	section	section	NOUN
cana-2695	35	51	,	,	PUNCT
cana-2695	35	52	we	we	PRON
cana-2695	35	53	have	have	AUX
cana-2695	35	54	defined	define	VERB
cana-2695	35	55	a	a	DET
cana-2695	35	56	basic	basic	ADJ
cana-2695	35	57	definition	definition	NOUN
cana-2695	35	58	to	to	PART
cana-2695	35	59	deal	deal	VERB
cana-2695	35	60	with	with	ADP
cana-2695	35	61	the	the	DET
cana-2695	35	62	ivfrs	ivfrs	NOUN
cana-2695	35	63	.	.	PUNCT
cana-2695	36	1	definition	definition	NOUN
cana-2695	36	2	2.1	2.1	NUM
cana-2695	36	3	.	.	PUNCT
cana-2695	37	1	a	a	DET
cana-2695	37	2	fg	fg	PROPN
cana-2695	37	3	g(v	g(v	PROPN
cana-2695	37	4	,	,	PUNCT
cana-2695	37	5	σ	σ	PROPN
cana-2695	37	6	,	,	PUNCT
cana-2695	37	7	µ	µ	NOUN
cana-2695	37	8	)	)	PUNCT
cana-2695	37	9	where	where	SCONJ
cana-2695	37	10	µ	µ	NOUN
cana-2695	37	11	is	be	AUX
cana-2695	37	12	a	a	DET
cana-2695	37	13	symmetric	symmetric	ADJ
cana-2695	37	14	fuzzy	fuzzy	ADJ
cana-2695	37	15	relation	relation	NOUN
cana-2695	37	16	on	on	ADP
cana-2695	37	17	σ	σ	PROPN
cana-2695	37	18	:	:	PUNCT
cana-2695	37	19	v	v	NOUN
cana-2695	37	20	→	→	SYM
cana-2695	37	21	[	[	X
cana-2695	37	22	0,1	0,1	NUM
cana-2695	37	23	]	]	PUNCT
cana-2695	37	24	and	and	CCONJ
cana-2695	37	25	µ	µ	X
cana-2695	37	26	:	:	PUNCT
cana-2695	37	27	v	v	NUM
cana-2695	37	28	×	×	NOUN
cana-2695	37	29	v	v	NOUN
cana-2695	37	30	→	→	SYM
cana-2695	37	31	[	[	X
cana-2695	37	32	0,1	0,1	NUM
cana-2695	37	33	]	]	PUNCT
cana-2695	37	34	such	such	ADJ
cana-2695	37	35	that	that	SCONJ
cana-2695	37	36	µ(x	µ(x	VERB
cana-2695	37	37	,	,	PUNCT
cana-2695	37	38	z	z	NOUN
cana-2695	37	39	)	)	PUNCT
cana-2695	37	40	≤	≤	NUM
cana-2695	37	41	σ(x)ʌσ(z	σ(x)ʌσ(z	NOUN
cana-2695	37	42	)	)	PUNCT
cana-2695	37	43	,	,	PUNCT
cana-2695	37	44	∀	∀	X
cana-2695	37	45	x	x	NOUN
cana-2695	37	46	,	,	PUNCT
cana-2695	37	47	z	z	PROPN
cana-2695	37	48	∈	∈	PROPN
cana-2695	37	49	v.	v.	ADP
cana-2695	37	50	definition	definition	NOUN
cana-2695	37	51	2.2	2.2	NUM
cana-2695	37	52	.	.	PUNCT
cana-2695	37	53	consider	consider	VERB
cana-2695	37	54	an	an	DET
cana-2695	37	55	ordered	order	VERB
cana-2695	37	56	fuzzy	fuzzy	ADJ
cana-2695	37	57	subset	subset	NOUN
cana-2695	37	58	h	h	NOUN
cana-2695	37	59	=	=	PRON
cana-2695	37	60	{	{	PUNCT
cana-2695	37	61	(	(	PUNCT
cana-2695	37	62	u1	u1	NOUN
cana-2695	37	63	,	,	PUNCT
cana-2695	37	64	σ(u1	σ(u1	NOUN
cana-2695	37	65	)	)	PUNCT
cana-2695	37	66	)	)	PUNCT
cana-2695	37	67	,	,	PUNCT
cana-2695	37	68	(	(	PUNCT
cana-2695	37	69	u2	u2	NOUN
cana-2695	37	70	,	,	PUNCT
cana-2695	37	71	σ(u2	σ(u2	NOUN
cana-2695	37	72	)	)	PUNCT
cana-2695	37	73	)	)	PUNCT
cana-2695	37	74	,	,	PUNCT
cana-2695	37	75	…	…	PUNCT
cana-2695	37	76	(	(	PUNCT
cana-2695	37	77	uk	uk	PROPN
cana-2695	37	78	,	,	PUNCT
cana-2695	37	79	σ(uk	σ(uk	NOUN
cana-2695	37	80	)	)	PUNCT
cana-2695	37	81	)	)	PUNCT
cana-2695	37	82	}	}	PUNCT
cana-2695	37	83	,	,	PUNCT
cana-2695	37	84	|h|	|h|	PROPN
cana-2695	37	85	≥	≥	NOUN
cana-2695	37	86	2	2	NUM
cana-2695	37	87	,	,	PUNCT
cana-2695	37	88	the	the	DET
cana-2695	37	89	representation	representation	NOUN
cana-2695	37	90	of	of	ADP
cana-2695	37	91	(	(	PUNCT
cana-2695	37	92	z	z	NOUN
cana-2695	37	93	,	,	PUNCT
cana-2695	37	94	σ(z))ϵσ	σ(z))ϵσ	NOUN
cana-2695	37	95	−	−	NOUN
cana-2695	37	96	h	h	NOUN
cana-2695	37	97	=	=	PRON
cana-2695	37	98	{	{	PUNCT
cana-2695	37	99	(	(	PUNCT
cana-2695	37	100	uk+1	uk+1	NOUN
cana-2695	37	101	,	,	PUNCT
cana-2695	37	102	σ(uk+1	σ(uk+1	NUM
cana-2695	37	103	)	)	PUNCT
cana-2695	37	104	)	)	PUNCT
cana-2695	37	105	,	,	PUNCT
cana-2695	37	106	(	(	PUNCT
cana-2695	37	107	uk+2	uk+2	NOUN
cana-2695	37	108	,	,	PUNCT
cana-2695	37	109	σ(uk+2	σ(uk+2	NOUN
cana-2695	37	110	)	)	PUNCT
cana-2695	37	111	)	)	PUNCT
cana-2695	37	112	,	,	PUNCT
cana-2695	37	113	…	…	PUNCT
cana-2695	37	114	(	(	PUNCT
cana-2695	37	115	un	un	PROPN
cana-2695	37	116	,	,	PUNCT
cana-2695	37	117	σ(un	σ(un	NOUN
cana-2695	37	118	)	)	PUNCT
cana-2695	37	119	)	)	PUNCT
cana-2695	37	120	}	}	PUNCT
cana-2695	37	121	and	and	CCONJ
cana-2695	37	122	{	{	PUNCT
cana-2695	37	123	w(z	w(z	PROPN
cana-2695	37	124	,	,	PUNCT
cana-2695	37	125	u1	u1	NOUN
cana-2695	37	126	)	)	PUNCT
cana-2695	37	127	,	,	PUNCT
cana-2695	37	128	w(z	w(z	PROPN
cana-2695	37	129	,	,	PUNCT
cana-2695	37	130	u2	u2	NOUN
cana-2695	37	131	)	)	PUNCT
cana-2695	37	132	,	,	PUNCT
cana-2695	37	133	…	…	PUNCT
cana-2695	38	1	w(z	w(z	X
cana-2695	38	2	,	,	PUNCT
cana-2695	38	3	uk	uk	PROPN
cana-2695	38	4	)	)	PUNCT
cana-2695	38	5	}	}	PUNCT
cana-2695	38	6	,	,	PUNCT
cana-2695	38	7	where	where	SCONJ
cana-2695	38	8	the	the	DET
cana-2695	38	9	significance	significance	NOUN
cana-2695	38	10	of	of	ADP
cana-2695	38	11	the	the	DET
cana-2695	38	12	link	link	NOUN
cana-2695	38	13	between	between	ADP
cana-2695	38	14	z	z	PROPN
cana-2695	38	15	and	and	CCONJ
cana-2695	38	16	y	y	PROPN
cana-2695	38	17	is	be	AUX
cana-2695	38	18	denoted	denote	VERB
cana-2695	38	19	by	by	ADP
cana-2695	38	20	w(z	w(z	PROPN
cana-2695	38	21	,	,	PUNCT
cana-2695	38	22	y	y	PROPN
cana-2695	38	23	)	)	PUNCT
cana-2695	38	24	.	.	PUNCT
cana-2695	39	1	the	the	DET
cana-2695	39	2	set	set	NOUN
cana-2695	39	3	is	be	AUX
cana-2695	39	4	called	call	VERB
cana-2695	39	5	a	a	DET
cana-2695	39	6	frs	frs	NOUN
cana-2695	39	7	if	if	SCONJ
cana-2695	39	8	every	every	DET
cana-2695	39	9	pair	pair	NOUN
cana-2695	39	10	of	of	ADP
cana-2695	39	11	items	item	NOUN
cana-2695	39	12	in	in	ADP
cana-2695	39	13	the	the	DET
cana-2695	39	14	fuzzy	fuzzy	ADJ
cana-2695	39	15	subset	subset	NOUN
cana-2695	39	16	h	h	NOUN
cana-2695	39	17	has	have	VERB
cana-2695	39	18	a	a	DET
cana-2695	39	19	different	different	ADJ
cana-2695	39	20	representation	representation	NOUN
cana-2695	39	21	with	with	ADP
cana-2695	39	22	respect	respect	NOUN
cana-2695	39	23	to	to	ADP
cana-2695	39	24	h	h	PROPN
cana-2695	39	25	of	of	ADP
cana-2695	39	26	g.	g.	PROPN
cana-2695	39	27	the	the	DET
cana-2695	39	28	frn	frn	PROPN
cana-2695	39	29	is	be	AUX
cana-2695	39	30	represented	represent	VERB
cana-2695	39	31	as	as	ADP
cana-2695	39	32	fr(g	fr(g	NUM
cana-2695	39	33	)	)	PUNCT
cana-2695	39	34	,	,	PUNCT
cana-2695	39	35	the	the	DET
cana-2695	39	36	minimal	minimal	ADJ
cana-2695	39	37	cardinality	cardinality	NOUN
cana-2695	39	38	of	of	ADP
cana-2695	39	39	frs	frs	PROPN
cana-2695	39	40	.	.	PUNCT
cana-2695	40	1	definition	definition	NOUN
cana-2695	40	2	2.3	2.3	NUM
cana-2695	40	3	.	.	PUNCT
cana-2695	41	1	a	a	DET
cana-2695	41	2	fuzzy	fuzzy	ADJ
cana-2695	41	3	graph	graph	NOUN
cana-2695	41	4	g	g	PROPN
cana-2695	41	5	's	's	PART
cana-2695	41	6	frs	frs	NOUN
cana-2695	41	7	is	be	AUX
cana-2695	41	8	considered	consider	VERB
cana-2695	41	9	a	a	DET
cana-2695	41	10	fuzzy	fuzzy	ADJ
cana-2695	41	11	super	super	ADJ
cana-2695	41	12	resolving	resolve	VERB
cana-2695	41	13	set	set	VERB
cana-2695	41	14	(	(	PUNCT
cana-2695	41	15	fsrs	fsr	NOUN
cana-2695	41	16	)	)	PUNCT
cana-2695	41	17	if	if	SCONJ
cana-2695	41	18	any	any	DET
cana-2695	41	19	two	two	NUM
cana-2695	41	20	of	of	ADP
cana-2695	41	21	its	its	PRON
cana-2695	41	22	elements	element	NOUN
cana-2695	41	23	have	have	VERB
cana-2695	41	24	unique	unique	ADJ
cana-2695	41	25	representations	representation	NOUN
cana-2695	41	26	with	with	ADP
cana-2695	41	27	respect	respect	NOUN
cana-2695	41	28	to	to	ADP
cana-2695	41	29	h.	h.	PROPN
cana-2695	41	30	we	we	PRON
cana-2695	41	31	also	also	ADV
cana-2695	41	32	write	write	VERB
cana-2695	41	33	the	the	DET
cana-2695	41	34	fsrn	fsrn	NOUN
cana-2695	41	35	,	,	PUNCT
cana-2695	41	36	denoted	denote	VERB
cana-2695	41	37	by	by	ADP
cana-2695	41	38	sr(g	sr(g	NUM
cana-2695	41	39	)	)	PUNCT
cana-2695	41	40	,	,	PUNCT
cana-2695	41	41	and	and	CCONJ
cana-2695	41	42	the	the	DET
cana-2695	41	43	super	super	ADJ
cana-2695	41	44	resolving	resolving	NOUN
cana-2695	41	45	matrix	matrix	NOUN
cana-2695	41	46	,	,	PUNCT
cana-2695	41	47	denoted	denote	VERB
cana-2695	41	48	by	by	ADP
cana-2695	41	49	sn×k	sn×k	PROPN
cana-2695	41	50	,	,	PUNCT
cana-2695	41	51	which	which	PRON
cana-2695	41	52	is	be	AUX
cana-2695	41	53	the	the	DET
cana-2695	41	54	lowest	low	ADJ
cana-2695	41	55	cardinality	cardinality	NOUN
cana-2695	41	56	of	of	ADP
cana-2695	41	57	a	a	DET
cana-2695	41	58	fg	fg	PROPN
cana-2695	41	59	g.	g.	PROPN
cana-2695	41	60	definition	definition	NOUN
cana-2695	41	61	2.4	2.4	NUM
cana-2695	41	62	.	.	PUNCT
cana-2695	42	1	a	a	DET
cana-2695	42	2	set	set	NOUN
cana-2695	42	3	c	c	NOUN
cana-2695	42	4	of	of	ADP
cana-2695	42	5	vertices	vertex	NOUN
cana-2695	42	6	v	v	NOUN
cana-2695	42	7	is	be	AUX
cana-2695	42	8	a	a	DET
cana-2695	42	9	dominating	dominating	NOUN
cana-2695	42	10	set	set	NOUN
cana-2695	42	11	if	if	SCONJ
cana-2695	42	12	every	every	DET
cana-2695	42	13	vertex	vertex	NOUN
cana-2695	42	14	of	of	ADP
cana-2695	42	15	v\c	v\c	NOUN
cana-2695	42	16	is	be	AUX
cana-2695	42	17	adjacent	adjacent	ADJ
cana-2695	42	18	to	to	ADP
cana-2695	42	19	any	any	DET
cana-2695	42	20	vertex	vertex	NOUN
cana-2695	42	21	of	of	ADP
cana-2695	42	22	𝐶.	𝐶.	PROPN
cana-2695	42	23	the	the	DET
cana-2695	42	24	smallest	small	ADJ
cana-2695	42	25	cardinality	cardinality	NOUN
cana-2695	42	26	of	of	ADP
cana-2695	42	27	this	this	DET
cana-2695	42	28	set	set	NOUN
cana-2695	42	29	is	be	AUX
cana-2695	42	30	called	call	VERB
cana-2695	42	31	the	the	DET
cana-2695	42	32	fuzzy	fuzzy	ADJ
cana-2695	42	33	domination	domination	NOUN
cana-2695	42	34	number(fdn	number(fdn	NOUN
cana-2695	42	35	)	)	PUNCT
cana-2695	42	36	and	and	CCONJ
cana-2695	42	37	it	it	PRON
cana-2695	42	38	is	be	AUX
cana-2695	42	39	denoted	denote	VERB
cana-2695	42	40	by	by	ADP
cana-2695	42	41	fγ(g	fγ(g	NUM
cana-2695	42	42	)	)	PUNCT
cana-2695	42	43	.	.	PUNCT
cana-2695	43	1	communications	communication	NOUN
cana-2695	43	2	on	on	ADP
cana-2695	43	3	applied	apply	VERB
cana-2695	43	4	nonlinear	nonlinear	ADJ
cana-2695	43	5	analysis	analysis	NOUN
cana-2695	43	6	issn	issn	NOUN
cana-2695	43	7	:	:	PUNCT
cana-2695	43	8	1074	1074	NUM
cana-2695	43	9	-	-	PUNCT
cana-2695	43	10	133x	133x	NUM
cana-2695	43	11	vol	vol	NOUN
cana-2695	43	12	32	32	NUM
cana-2695	43	13	no	no	NOUN
cana-2695	43	14	.	.	PUNCT
cana-2695	44	1	3s	3s	NUM
cana-2695	44	2	(	(	PUNCT
cana-2695	44	3	2025	2025	NUM
cana-2695	44	4	)	)	PUNCT
cana-2695	44	5	549	549	NUM
cana-2695	44	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2695	44	7	definition	definition	NOUN
cana-2695	44	8	2.5	2.5	NUM
cana-2695	44	9	an	an	DET
cana-2695	44	10	interval	interval	NOUN
cana-2695	44	11	-	-	PUNCT
cana-2695	44	12	valued	value	VERB
cana-2695	44	13	fuzzy	fuzzy	ADJ
cana-2695	44	14	relation	relation	NOUN
cana-2695	44	15	q	q	NOUN
cana-2695	44	16	is	be	AUX
cana-2695	44	17	a	a	DET
cana-2695	44	18	mapping	mapping	NOUN
cana-2695	44	19	,	,	PUNCT
cana-2695	44	20	q	q	NOUN
cana-2695	44	21	:	:	PUNCT
cana-2695	44	22	xxy	xxy	PROPN
cana-2695	44	23	→	→	PROPN
cana-2695	44	24	d	d	X
cana-2695	45	1	[	[	X
cana-2695	45	2	0,1	0,1	NUM
cana-2695	45	3	]	]	PUNCT
cana-2695	46	1	such	such	ADJ
cana-2695	46	2	that	that	PRON
cana-2695	46	3	q	q	X
cana-2695	46	4	(	(	PUNCT
cana-2695	46	5	x	x	NOUN
cana-2695	46	6	,	,	PUNCT
cana-2695	46	7	y	y	NOUN
cana-2695	46	8	)	)	PUNCT
cana-2695	46	9	=	=	PUNCT
cana-2695	47	1	[	[	X
cana-2695	47	2	q−	q−	PROPN
cana-2695	47	3	(	(	PUNCT
cana-2695	47	4	x	x	NOUN
cana-2695	47	5	,	,	PUNCT
cana-2695	47	6	y	y	PROPN
cana-2695	47	7	)	)	PUNCT
cana-2695	47	8	,	,	PUNCT
cana-2695	47	9	q+	q+	X
cana-2695	47	10	(	(	PUNCT
cana-2695	47	11	x	x	NOUN
cana-2695	47	12	,	,	PUNCT
cana-2695	47	13	y	y	PROPN
cana-2695	47	14	)	)	PUNCT
cana-2695	47	15	]	]	PUNCT
cana-2695	47	16	∈	∈	PROPN
cana-2695	48	1	d	d	X
cana-2695	48	2	[	[	X
cana-2695	48	3	0,1	0,1	NUM
cana-2695	48	4	]	]	PUNCT
cana-2695	48	5	̬for	̬for	ADP
cana-2695	48	6	all	all	DET
cana-2695	48	7	pairs	pair	NOUN
cana-2695	48	8	(	(	PUNCT
cana-2695	48	9	x	x	NOUN
cana-2695	48	10	,	,	PUNCT
cana-2695	48	11	y	y	PROPN
cana-2695	48	12	)	)	PUNCT
cana-2695	48	13	ϵ	ϵ	PROPN
cana-2695	48	14	xxy	xxy	PROPN
cana-2695	48	15	.	.	PUNCT
cana-2695	49	1	definition	definition	NOUN
cana-2695	49	2	2.6	2.6	NUM
cana-2695	49	3	we	we	PRON
cana-2695	49	4	define	define	VERB
cana-2695	49	5	g	g	PROPN
cana-2695	49	6	=	=	PUNCT
cana-2695	50	1	[	[	X
cana-2695	50	2	e	e	NOUN
cana-2695	50	3	,	,	PUNCT
cana-2695	50	4	f	f	X
cana-2695	50	5	]	]	X
cana-2695	50	6	,	,	PUNCT
cana-2695	50	7	where	where	SCONJ
cana-2695	50	8	e	e	X
cana-2695	50	9	=	=	PUNCT
cana-2695	51	1	[	[	X
cana-2695	51	2	μ	μ	NUM
cana-2695	51	3	e	e	NOUN
cana-2695	51	4	−	−	PROPN
cana-2695	51	5	,	,	PUNCT
cana-2695	51	6	μ	μ	NOUN
cana-2695	51	7	e	e	X
cana-2695	51	8	+	+	NOUN
cana-2695	51	9	]	]	X
cana-2695	51	10	and	and	CCONJ
cana-2695	51	11	f	f	X
cana-2695	52	1	=	=	PUNCT
cana-2695	53	1	[	[	X
cana-2695	53	2	μ	μ	NUM
cana-2695	53	3	f	f	PROPN
cana-2695	53	4	−	−	PROPN
cana-2695	53	5	,	,	PUNCT
cana-2695	53	6	μ	μ	PROPN
cana-2695	53	7	f	f	PROPN
cana-2695	54	1	+	+	X
cana-2695	54	2	]	]	X
cana-2695	54	3	is	be	AUX
cana-2695	54	4	called	call	VERB
cana-2695	54	5	an	an	DET
cana-2695	54	6	interval	interval	NOUN
cana-2695	54	7	-	-	PUNCT
cana-2695	54	8	valued	value	VERB
cana-2695	54	9	fuzzy	fuzzy	ADJ
cana-2695	54	10	graph	graph	NOUN
cana-2695	54	11	(	(	PUNCT
cana-2695	54	12	ivfg	ivfg	NOUN
cana-2695	54	13	)	)	PUNCT
cana-2695	54	14	,	,	PUNCT
cana-2695	54	15	if	if	SCONJ
cana-2695	54	16	e	e	NOUN
cana-2695	54	17	is	be	AUX
cana-2695	54	18	an	an	DET
cana-2695	54	19	interval	interval	NOUN
cana-2695	54	20	-	-	PUNCT
cana-2695	54	21	valued	value	VERB
cana-2695	54	22	fuzzy	fuzzy	ADJ
cana-2695	54	23	set	set	VERB
cana-2695	54	24	on	on	ADP
cana-2695	54	25	v	v	NOUN
cana-2695	54	26	and	and	CCONJ
cana-2695	54	27	f	f	PROPN
cana-2695	54	28	is	be	AUX
cana-2695	54	29	an	an	DET
cana-2695	54	30	interval	interval	NOUN
cana-2695	54	31	-	-	PUNCT
cana-2695	54	32	valued	value	VERB
cana-2695	54	33	fuzzy	fuzzy	ADJ
cana-2695	54	34	relation	relation	NOUN
cana-2695	54	35	such	such	ADJ
cana-2695	54	36	that	that	SCONJ
cana-2695	54	37	μ	μ	PROPN
cana-2695	54	38	f	f	PROPN
cana-2695	54	39	−	−	PROPN
cana-2695	54	40	(	(	PUNCT
cana-2695	54	41	ef	ef	PROPN
cana-2695	54	42	)	)	PUNCT
cana-2695	54	43	≤	≤	NUM
cana-2695	54	44	min	min	NOUN
cana-2695	54	45	(	(	PUNCT
cana-2695	54	46	μ	μ	NOUN
cana-2695	54	47	e	e	NOUN
cana-2695	54	48	−	−	PROPN
cana-2695	54	49	(	(	PUNCT
cana-2695	54	50	e	e	NOUN
cana-2695	54	51	)	)	PUNCT
cana-2695	54	52	,	,	PUNCT
cana-2695	54	53	μ	μ	PROPN
cana-2695	54	54	e	e	X
cana-2695	54	55	−	−	PROPN
cana-2695	54	56	(	(	PUNCT
cana-2695	54	57	f	f	NOUN
cana-2695	54	58	)	)	PUNCT
cana-2695	54	59	)	)	PUNCT
cana-2695	54	60	,	,	PUNCT
cana-2695	54	61	μ	μ	PROPN
cana-2695	54	62	f	f	PROPN
cana-2695	54	63	+	+	CCONJ
cana-2695	54	64	(	(	PUNCT
cana-2695	54	65	ef	ef	PROPN
cana-2695	54	66	)	)	PUNCT
cana-2695	54	67	≤	≤	NUM
cana-2695	54	68	min	min	NOUN
cana-2695	54	69	(	(	PUNCT
cana-2695	54	70	μ	μ	NOUN
cana-2695	54	71	e	e	X
cana-2695	54	72	+	+	CCONJ
cana-2695	54	73	(	(	PUNCT
cana-2695	54	74	e	e	NOUN
cana-2695	54	75	)	)	PUNCT
cana-2695	54	76	,	,	PUNCT
cana-2695	54	77	μ	μ	PROPN
cana-2695	54	78	e	e	X
cana-2695	54	79	+	+	CCONJ
cana-2695	54	80	(	(	PUNCT
cana-2695	54	81	f	f	NOUN
cana-2695	54	82	)	)	PUNCT
cana-2695	54	83	)	)	PUNCT
cana-2695	54	84	,	,	PUNCT
cana-2695	54	85	∀	∀	NOUN
cana-2695	54	86	ef	ef	PROPN
cana-2695	54	87	∈	∈	PROPN
cana-2695	54	88	e.	e.	PROPN
cana-2695	54	89	definition	definition	NOUN
cana-2695	54	90	2.7	2.7	NUM
cana-2695	54	91	the	the	DET
cana-2695	54	92	order	order	NOUN
cana-2695	54	93	and	and	CCONJ
cana-2695	54	94	size	size	NOUN
cana-2695	54	95	of	of	ADP
cana-2695	54	96	ivfg	ivfg	NOUN
cana-2695	54	97	is	be	AUX
cana-2695	54	98	defined	define	VERB
cana-2695	54	99	by	by	ADP
cana-2695	54	100	ο(g	ο(g	NOUN
cana-2695	54	101	)	)	PUNCT
cana-2695	55	1	=	=	PUNCT
cana-2695	55	2	∑	∑	PUNCT
cana-2695	55	3	1	1	NUM
cana-2695	56	1	+	+	NUM
cana-2695	56	2	μ	μ	NUM
cana-2695	56	3	e	e	NOUN
cana-2695	56	4	+	+	CCONJ
cana-2695	56	5	(	(	PUNCT
cana-2695	56	6	r	r	NOUN
cana-2695	56	7	)	)	PUNCT
cana-2695	56	8	−	−	PROPN
cana-2695	56	9	μ	μ	NUM
cana-2695	56	10	e	e	X
cana-2695	56	11	−	−	PROPN
cana-2695	56	12	(	(	PUNCT
cana-2695	56	13	r	r	NOUN
cana-2695	56	14	)	)	PUNCT
cana-2695	56	15	2	2	NUM
cana-2695	56	16	r∈v	r∈v	NOUN
cana-2695	56	17	𝒮(g	𝒮(g	NOUN
cana-2695	56	18	)	)	PUNCT
cana-2695	56	19	=	=	PUNCT
cana-2695	56	20	∑	∑	PUNCT
cana-2695	56	21	1	1	NUM
cana-2695	56	22	+	+	SYM
cana-2695	56	23	μ	μ	PROPN
cana-2695	56	24	f	f	PROPN
cana-2695	56	25	+	+	CCONJ
cana-2695	56	26	(	(	PUNCT
cana-2695	56	27	rq	rq	INTJ
cana-2695	56	28	)	)	PUNCT
cana-2695	56	29	−	−	PROPN
cana-2695	56	30	μ	μ	PROPN
cana-2695	56	31	f	f	PROPN
cana-2695	57	1	−	−	PROPN
cana-2695	57	2	(	(	PUNCT
cana-2695	57	3	rq	rq	NOUN
cana-2695	57	4	)	)	PUNCT
cana-2695	57	5	2	2	NUM
cana-2695	57	6	r∈v	r∈v	NOUN
cana-2695	57	7	definition	definition	NOUN
cana-2695	57	8	2.8	2.8	NUM
cana-2695	57	9	an	an	DET
cana-2695	57	10	ivfg	ivfg	NOUN
cana-2695	57	11	is	be	AUX
cana-2695	57	12	called	call	VERB
cana-2695	57	13	complete	complete	ADJ
cana-2695	57	14	if	if	SCONJ
cana-2695	57	15	μ	μ	PROPN
cana-2695	57	16	f	f	PROPN
cana-2695	57	17	−(ef	−(ef	PROPN
cana-2695	57	18	)	)	PUNCT
cana-2695	57	19	=	=	SYM
cana-2695	57	20	min	min	PROPN
cana-2695	57	21	(	(	PUNCT
cana-2695	57	22	μ	μ	NOUN
cana-2695	57	23	e	e	NOUN
cana-2695	57	24	−	−	PROPN
cana-2695	57	25	(	(	PUNCT
cana-2695	57	26	e	e	NOUN
cana-2695	57	27	)	)	PUNCT
cana-2695	57	28	,	,	PUNCT
cana-2695	57	29	μ	μ	PROPN
cana-2695	57	30	e	e	X
cana-2695	57	31	−	−	PROPN
cana-2695	57	32	(	(	PUNCT
cana-2695	57	33	f	f	NOUN
cana-2695	57	34	)	)	PUNCT
cana-2695	57	35	)	)	PUNCT
cana-2695	57	36	,	,	PUNCT
cana-2695	57	37	μ	μ	PROPN
cana-2695	57	38	f	f	PROPN
cana-2695	57	39	+	+	CCONJ
cana-2695	57	40	(	(	PUNCT
cana-2695	57	41	ef	ef	PROPN
cana-2695	57	42	)	)	PUNCT
cana-2695	57	43	=	=	SYM
cana-2695	57	44	max	max	PROPN
cana-2695	57	45	(	(	PUNCT
cana-2695	57	46	μ	μ	NOUN
cana-2695	57	47	e	e	X
cana-2695	57	48	+	+	CCONJ
cana-2695	57	49	(	(	PUNCT
cana-2695	57	50	e	e	NOUN
cana-2695	57	51	)	)	PUNCT
cana-2695	57	52	,	,	PUNCT
cana-2695	57	53	μ	μ	PROPN
cana-2695	57	54	e	e	X
cana-2695	57	55	+	+	CCONJ
cana-2695	57	56	(	(	PUNCT
cana-2695	57	57	f	f	NOUN
cana-2695	57	58	)	)	PUNCT
cana-2695	57	59	)	)	PUNCT
cana-2695	57	60	,	,	PUNCT
cana-2695	57	61	∀	∀	NOUN
cana-2695	57	62	ef	ef	PROPN
cana-2695	57	63	∈	∈	PROPN
cana-2695	57	64	e.	e.	PROPN
cana-2695	57	65	definition	definition	NOUN
cana-2695	57	66	2.9	2.9	NUM
cana-2695	57	67	an	an	DET
cana-2695	57	68	isomorphism	isomorphism	NOUN
cana-2695	57	69	φ	φ	NOUN
cana-2695	57	70	:	:	PUNCT
cana-2695	57	71	g1	g1	PROPN
cana-2695	57	72	→	→	PUNCT
cana-2695	57	73	g2	g2	PROPN
cana-2695	57	74	is	be	AUX
cana-2695	57	75	a	a	DET
cana-2695	57	76	bijective	bijective	ADJ
cana-2695	57	77	mapping	mapping	NOUN
cana-2695	57	78	of	of	ADP
cana-2695	57	79	two	two	NUM
cana-2695	57	80	ivfgs	ivfg	VERB
cana-2695	57	81	,	,	PUNCT
cana-2695	57	82	φ	φ	NOUN
cana-2695	57	83	:	:	PUNCT
cana-2695	57	84	x1	x1	PROPN
cana-2695	57	85	→	→	SYM
cana-2695	57	86	x2	x2	PROPN
cana-2695	57	87	,	,	PUNCT
cana-2695	57	88	∋	∋	NOUN
cana-2695	57	89	μ	μ	PROPN
cana-2695	57	90	e	e	PROPN
cana-2695	57	91	−	−	PROPN
cana-2695	58	1	(	(	PUNCT
cana-2695	58	2	r	r	NOUN
cana-2695	58	3	)	)	PUNCT
cana-2695	58	4	=	=	SYM
cana-2695	58	5	μ	μ	NOUN
cana-2695	58	6	e	e	NOUN
cana-2695	58	7	−′(φ	−′(φ	ADJ
cana-2695	58	8	(	(	PUNCT
cana-2695	58	9	r	r	NOUN
cana-2695	58	10	)	)	PUNCT
cana-2695	58	11	)	)	PUNCT
cana-2695	58	12	,	,	PUNCT
cana-2695	58	13	μ	μ	PROPN
cana-2695	58	14	e	e	X
cana-2695	58	15	+	+	CCONJ
cana-2695	58	16	(	(	PUNCT
cana-2695	58	17	r	r	NOUN
cana-2695	58	18	)	)	PUNCT
cana-2695	58	19	=	=	SYM
cana-2695	59	1	μ	μ	NUM
cana-2695	59	2	e	e	X
cana-2695	59	3	+	+	NOUN
cana-2695	59	4	′	′	NUM
cana-2695	59	5	(	(	PUNCT
cana-2695	59	6	φ(r	φ(r	NOUN
cana-2695	59	7	)	)	PUNCT
cana-2695	59	8	)	)	PUNCT
cana-2695	59	9	,	,	PUNCT
cana-2695	59	10	μ	μ	PROPN
cana-2695	59	11	f	f	X
cana-2695	60	1	−	−	PROPN
cana-2695	60	2	(	(	PUNCT
cana-2695	60	3	r	r	NOUN
cana-2695	60	4	,	,	PUNCT
cana-2695	60	5	u	u	NOUN
cana-2695	60	6	)	)	PUNCT
cana-2695	60	7	=	=	SYM
cana-2695	61	1	μ	μ	PROPN
cana-2695	61	2	f	f	X
cana-2695	61	3	−′	−′	X
cana-2695	61	4	(	(	PUNCT
cana-2695	61	5	φ(r	φ(r	NOUN
cana-2695	61	6	)	)	PUNCT
cana-2695	61	7	,	,	PUNCT
cana-2695	61	8	φ(u	φ(u	NOUN
cana-2695	61	9	)	)	PUNCT
cana-2695	61	10	)	)	PUNCT
cana-2695	61	11	,	,	PUNCT
cana-2695	61	12	μ	μ	PROPN
cana-2695	61	13	f	f	PROPN
cana-2695	62	1	+	+	CCONJ
cana-2695	62	2	(	(	PUNCT
cana-2695	62	3	r	r	NOUN
cana-2695	62	4	,	,	PUNCT
cana-2695	62	5	u	u	NOUN
cana-2695	62	6	)	)	PUNCT
cana-2695	62	7	=	=	SYM
cana-2695	62	8	μ	μ	NOUN
cana-2695	62	9	f	f	X
cana-2695	63	1	+	+	NOUN
cana-2695	63	2	′	′	NUM
cana-2695	63	3	(	(	PUNCT
cana-2695	63	4	φ(r	φ(r	NOUN
cana-2695	63	5	)	)	PUNCT
cana-2695	63	6	,	,	PUNCT
cana-2695	63	7	φ(u	φ(u	NOUN
cana-2695	63	8	)	)	PUNCT
cana-2695	63	9	)	)	PUNCT
cana-2695	64	1	∀r	∀r	NOUN
cana-2695	64	2	,	,	PUNCT
cana-2695	64	3	u	u	PROPN
cana-2695	64	4	∈	∈	PROPN
cana-2695	64	5	e.	e.	PROPN
cana-2695	64	6	definition	definition	NOUN
cana-2695	64	7	2.10	2.10	NUM
cana-2695	64	8	a	a	DET
cana-2695	64	9	homomorphism	homomorphism	NOUN
cana-2695	64	10	φ	φ	X
cana-2695	64	11	:	:	PUNCT
cana-2695	64	12	g1	g1	PROPN
cana-2695	64	13	→	→	PUNCT
cana-2695	64	14	g2	g2	PROPN
cana-2695	64	15	is	be	AUX
cana-2695	64	16	a	a	DET
cana-2695	64	17	mapping	mapping	NOUN
cana-2695	64	18	φ	φ	NOUN
cana-2695	64	19	:	:	PUNCT
cana-2695	64	20	x1	x1	PROPN
cana-2695	64	21	→	→	SYM
cana-2695	64	22	x2	x2	PROPN
cana-2695	64	23	of	of	ADP
cana-2695	64	24	two	two	NUM
cana-2695	64	25	ivfgs	ivfg	VERB
cana-2695	64	26	such	such	ADJ
cana-2695	64	27	that	that	SCONJ
cana-2695	64	28	μ	μ	PROPN
cana-2695	64	29	e	e	NOUN
cana-2695	64	30	−	−	PROPN
cana-2695	64	31	(	(	PUNCT
cana-2695	64	32	r	r	NOUN
cana-2695	64	33	)	)	PUNCT
cana-2695	64	34	≤	≤	NOUN
cana-2695	64	35	μ	μ	PROPN
cana-2695	64	36	e	e	NOUN
cana-2695	64	37	−′(φ	−′(φ	ADJ
cana-2695	64	38	(	(	PUNCT
cana-2695	64	39	r	r	NOUN
cana-2695	64	40	)	)	PUNCT
cana-2695	64	41	)	)	PUNCT
cana-2695	64	42	,	,	PUNCT
cana-2695	64	43	μ	μ	PROPN
cana-2695	64	44	e	e	X
cana-2695	64	45	+	+	CCONJ
cana-2695	64	46	(	(	PUNCT
cana-2695	64	47	r	r	NOUN
cana-2695	64	48	)	)	PUNCT
cana-2695	64	49	≤	≤	NOUN
cana-2695	64	50	μ	μ	NOUN
cana-2695	64	51	e	e	X
cana-2695	64	52	+	+	NOUN
cana-2695	64	53	′	′	NUM
cana-2695	64	54	(	(	PUNCT
cana-2695	64	55	φ(r	φ(r	NOUN
cana-2695	64	56	)	)	PUNCT
cana-2695	64	57	)	)	PUNCT
cana-2695	64	58	,	,	PUNCT
cana-2695	64	59	μ	μ	PROPN
cana-2695	64	60	f	f	X
cana-2695	65	1	−	−	PROPN
cana-2695	65	2	(	(	PUNCT
cana-2695	65	3	r	r	NOUN
cana-2695	65	4	,	,	PUNCT
cana-2695	65	5	u	u	NOUN
cana-2695	65	6	)	)	PUNCT
cana-2695	65	7	≤	≤	NOUN
cana-2695	66	1	μ	μ	PROPN
cana-2695	66	2	f	f	PROPN
cana-2695	67	1	−′	−′	PROPN
cana-2695	67	2	(	(	PUNCT
cana-2695	67	3	φ(r	φ(r	NOUN
cana-2695	67	4	)	)	PUNCT
cana-2695	67	5	,	,	PUNCT
cana-2695	67	6	φ(u	φ(u	NOUN
cana-2695	67	7	)	)	PUNCT
cana-2695	67	8	)	)	PUNCT
cana-2695	67	9	,	,	PUNCT
cana-2695	67	10	μ	μ	PROPN
cana-2695	67	11	f	f	PROPN
cana-2695	68	1	+	+	CCONJ
cana-2695	68	2	(	(	PUNCT
cana-2695	68	3	r	r	NOUN
cana-2695	68	4	,	,	PUNCT
cana-2695	68	5	u	u	NOUN
cana-2695	68	6	)	)	PUNCT
cana-2695	68	7	≤	≤	NOUN
cana-2695	68	8	μ	μ	NUM
cana-2695	68	9	f	f	PROPN
cana-2695	69	1	+	+	NOUN
cana-2695	69	2	′	′	NUM
cana-2695	69	3	(	(	PUNCT
cana-2695	69	4	φ(r	φ(r	NOUN
cana-2695	69	5	)	)	PUNCT
cana-2695	69	6	,	,	PUNCT
cana-2695	69	7	φ(u	φ(u	NOUN
cana-2695	69	8	)	)	PUNCT
cana-2695	69	9	)	)	PUNCT
cana-2695	70	1	∀r	∀r	NOUN
cana-2695	70	2	,	,	PUNCT
cana-2695	70	3	u	u	PROPN
cana-2695	70	4	∈	∈	PROPN
cana-2695	70	5	e.	e.	PROPN
cana-2695	70	6	definition	definition	NOUN
cana-2695	70	7	2.11	2.11	NUM
cana-2695	70	8	a	a	DET
cana-2695	70	9	co	co	ADJ
cana-2695	70	10	-	-	ADJ
cana-2695	70	11	weak	weak	ADJ
cana-2695	70	12	isomorphism	isomorphism	NOUN
cana-2695	70	13	φ	φ	NOUN
cana-2695	70	14	:	:	PUNCT
cana-2695	70	15	g1	g1	PROPN
cana-2695	70	16	→	→	PUNCT
cana-2695	70	17	g2	g2	PROPN
cana-2695	70	18	is	be	AUX
cana-2695	70	19	a	a	DET
cana-2695	70	20	bijective	bijective	ADJ
cana-2695	70	21	mapping	mapping	NOUN
cana-2695	70	22	,	,	PUNCT
cana-2695	70	23	φ	φ	PROPN
cana-2695	70	24	:	:	PUNCT
cana-2695	70	25	x1	x1	PROPN
cana-2695	70	26	→	→	SYM
cana-2695	70	27	x2	x2	PROPN
cana-2695	70	28	,	,	PUNCT
cana-2695	70	29	such	such	ADJ
cana-2695	70	30	that	that	SCONJ
cana-2695	70	31	(	(	PUNCT
cana-2695	70	32	i	i	NOUN
cana-2695	70	33	)	)	PUNCT
cana-2695	70	34	φ	φ	PROPN
cana-2695	70	35	is	be	AUX
cana-2695	70	36	homomorphism	homomorphism	PROPN
cana-2695	70	37	(	(	PUNCT
cana-2695	70	38	ii	ii	NOUN
cana-2695	70	39	)	)	PUNCT
cana-2695	70	40	μ	μ	PROPN
cana-2695	70	41	f	f	PROPN
cana-2695	71	1	−	−	PROPN
cana-2695	71	2	(	(	PUNCT
cana-2695	71	3	r	r	NOUN
cana-2695	71	4	,	,	PUNCT
cana-2695	71	5	u	u	NOUN
cana-2695	71	6	)	)	PUNCT
cana-2695	71	7	=	=	SYM
cana-2695	72	1	μ	μ	PROPN
cana-2695	72	2	f	f	X
cana-2695	72	3	−′	−′	X
cana-2695	72	4	(	(	PUNCT
cana-2695	72	5	φ(r	φ(r	NOUN
cana-2695	72	6	)	)	PUNCT
cana-2695	72	7	,	,	PUNCT
cana-2695	72	8	φ(u	φ(u	NOUN
cana-2695	72	9	)	)	PUNCT
cana-2695	72	10	)	)	PUNCT
cana-2695	72	11	,	,	PUNCT
cana-2695	72	12	μ	μ	PROPN
cana-2695	72	13	f	f	PROPN
cana-2695	73	1	+	+	CCONJ
cana-2695	73	2	(	(	PUNCT
cana-2695	73	3	r	r	NOUN
cana-2695	73	4	,	,	PUNCT
cana-2695	73	5	u	u	NOUN
cana-2695	73	6	)	)	PUNCT
cana-2695	73	7	=	=	SYM
cana-2695	73	8	μ	μ	NOUN
cana-2695	73	9	f	f	X
cana-2695	74	1	+	+	NOUN
cana-2695	74	2	′	′	NUM
cana-2695	74	3	(	(	PUNCT
cana-2695	74	4	φ(r	φ(r	NOUN
cana-2695	74	5	)	)	PUNCT
cana-2695	74	6	,	,	PUNCT
cana-2695	74	7	φ(u	φ(u	NOUN
cana-2695	74	8	)	)	PUNCT
cana-2695	74	9	)	)	PUNCT
cana-2695	75	1	∀r	∀r	NOUN
cana-2695	75	2	,	,	PUNCT
cana-2695	75	3	u	u	PROPN
cana-2695	75	4	∈	∈	PROPN
cana-2695	75	5	e.	e.	PROPN
cana-2695	75	6	3	3	X
cana-2695	75	7	.	.	PUNCT
cana-2695	75	8	fuzzy	fuzzy	ADJ
cana-2695	75	9	resolving	resolving	NOUN
cana-2695	75	10	sets	set	NOUN
cana-2695	75	11	of	of	ADP
cana-2695	75	12	interval	interval	NOUN
cana-2695	75	13	-	-	PUNCT
cana-2695	75	14	valued	value	VERB
cana-2695	75	15	fuzzy	fuzzy	ADJ
cana-2695	75	16	graphs	graph	NOUN
cana-2695	75	17	consider	consider	VERB
cana-2695	75	18	a	a	DET
cana-2695	75	19	fuzzy	fuzzy	ADJ
cana-2695	75	20	subset	subset	NOUN
cana-2695	75	21	ℋ	ℋ	PROPN
cana-2695	75	22	,	,	PUNCT
cana-2695	75	23	then	then	ADV
cana-2695	75	24	the	the	DET
cana-2695	75	25	representation	representation	NOUN
cana-2695	75	26	of	of	ADP
cana-2695	75	27	ℋ	ℋ	PROPN
cana-2695	75	28	and	and	CCONJ
cana-2695	75	29	(	(	PUNCT
cana-2695	75	30	u	u	NOUN
cana-2695	75	31	,	,	PUNCT
cana-2695	75	32	σ−(u	σ−(u	PROPN
cana-2695	75	33	)	)	PUNCT
cana-2695	75	34	,	,	PUNCT
cana-2695	75	35	σ+(u	σ+(u	ADJ
cana-2695	75	36	)	)	PUNCT
cana-2695	75	37	)	)	PUNCT
cana-2695	76	1	∈	∈	PROPN
cana-2695	76	2	σ	σ	NOUN
cana-2695	76	3	−	−	PROPN
cana-2695	76	4	ℋ	ℋ	PROPN
cana-2695	76	5	are	be	AUX
cana-2695	76	6	all	all	ADV
cana-2695	76	7	distinct	distinct	ADJ
cana-2695	76	8	,	,	PUNCT
cana-2695	76	9	where	where	SCONJ
cana-2695	76	10	ℋ	ℋ	PROPN
cana-2695	76	11	=	=	SYM
cana-2695	76	12	{	{	PUNCT
cana-2695	76	13	(	(	PUNCT
cana-2695	76	14	x1	x1	PROPN
cana-2695	76	15	,	,	PUNCT
cana-2695	76	16	σ	σ	PROPN
cana-2695	76	17	−(x1	−(x1	PROPN
cana-2695	76	18	)	)	PUNCT
cana-2695	76	19	,	,	PUNCT
cana-2695	76	20	σ+(x1	σ+(x1	NOUN
cana-2695	76	21	)	)	PUNCT
cana-2695	76	22	)	)	PUNCT
cana-2695	76	23	,	,	PUNCT
cana-2695	76	24	(	(	PUNCT
cana-2695	76	25	x2	x2	PROPN
cana-2695	76	26	,	,	PUNCT
cana-2695	76	27	σ	σ	PROPN
cana-2695	76	28	−(x2	−(x2	NOUN
cana-2695	76	29	)	)	PUNCT
cana-2695	76	30	,	,	PUNCT
cana-2695	76	31	σ+(x2	σ+(x2	NOUN
cana-2695	76	32	)	)	PUNCT
cana-2695	76	33	)	)	PUNCT
cana-2695	76	34	,	,	PUNCT
cana-2695	76	35	.	.	PUNCT
cana-2695	76	36	.	.	PUNCT
cana-2695	76	37	.	.	PUNCT
cana-2695	77	1	(	(	PUNCT
cana-2695	77	2	xk	xk	PROPN
cana-2695	77	3	,	,	PUNCT
cana-2695	77	4	σ	σ	PROPN
cana-2695	77	5	−(x2	−(x2	NOUN
cana-2695	77	6	)	)	PUNCT
cana-2695	77	7	,	,	PUNCT
cana-2695	77	8	σ	σ	PROPN
cana-2695	78	1	+	+	PROPN
cana-2695	78	2	(	(	PUNCT
cana-2695	78	3	x2	x2	PROPN
cana-2695	78	4	)	)	PUNCT
cana-2695	78	5	)	)	PUNCT
cana-2695	78	6	}	}	PUNCT
cana-2695	78	7	and	and	CCONJ
cana-2695	78	8	σ	σ	NUM
cana-2695	78	9	−	−	PROPN
cana-2695	78	10	ℋ	ℋ	PROPN
cana-2695	78	11	=	=	SYM
cana-2695	78	12	{	{	PUNCT
cana-2695	78	13	(	(	PUNCT
cana-2695	78	14	xk+1	xk+1	PROPN
cana-2695	78	15	,	,	PUNCT
cana-2695	78	16	σ	σ	PROPN
cana-2695	78	17	−(xk+1	−(xk+1	NOUN
cana-2695	78	18	)	)	PUNCT
cana-2695	78	19	,	,	PUNCT
cana-2695	78	20	σ+(xk+1	σ+(xk+1	PROPN
cana-2695	78	21	)	)	PUNCT
cana-2695	78	22	)	)	PUNCT
cana-2695	78	23	,	,	PUNCT
cana-2695	78	24	(	(	PUNCT
cana-2695	78	25	xk+2	xk+2	PROPN
cana-2695	78	26	,	,	PUNCT
cana-2695	78	27	σ	σ	PROPN
cana-2695	78	28	−(xk+2	−(xk+2	NOUN
cana-2695	78	29	)	)	PUNCT
cana-2695	78	30	,	,	PUNCT
cana-2695	78	31	σ+(xk+2	σ+(xk+2	PROPN
cana-2695	78	32	)	)	PUNCT
cana-2695	78	33	)	)	PUNCT
cana-2695	78	34	,	,	PUNCT
cana-2695	78	35	.	.	PUNCT
cana-2695	78	36	.	.	PUNCT
cana-2695	78	37	.	.	PUNCT
cana-2695	79	1	(	(	PUNCT
cana-2695	79	2	xn	xn	PROPN
cana-2695	79	3	,	,	PUNCT
cana-2695	79	4	σ	σ	PROPN
cana-2695	79	5	−(xn	−(xn	PROPN
cana-2695	79	6	)	)	PUNCT
cana-2695	79	7	,	,	PUNCT
cana-2695	79	8	σ	σ	PROPN
cana-2695	79	9	+	+	PROPN
cana-2695	79	10	(	(	PUNCT
cana-2695	79	11	xn	xn	PROPN
cana-2695	79	12	)	)	PUNCT
cana-2695	79	13	)	)	PUNCT
cana-2695	79	14	}	}	PUNCT
cana-2695	79	15	,	,	PUNCT
cana-2695	79	16	then	then	ADV
cana-2695	79	17	ℋ	ℋ	PROPN
cana-2695	79	18	is	be	AUX
cana-2695	79	19	communications	communication	NOUN
cana-2695	79	20	on	on	ADP
cana-2695	79	21	applied	apply	VERB
cana-2695	79	22	nonlinear	nonlinear	ADJ
cana-2695	79	23	analysis	analysis	NOUN
cana-2695	79	24	issn	issn	NOUN
cana-2695	79	25	:	:	PUNCT
cana-2695	79	26	1074	1074	NUM
cana-2695	79	27	-	-	PUNCT
cana-2695	79	28	133x	133x	NUM
cana-2695	79	29	vol	vol	NOUN
cana-2695	79	30	32	32	NUM
cana-2695	79	31	no	no	NOUN
cana-2695	79	32	.	.	PUNCT
cana-2695	80	1	3s	3s	NUM
cana-2695	80	2	(	(	PUNCT
cana-2695	80	3	2025	2025	NUM
cana-2695	80	4	)	)	PUNCT
cana-2695	80	5	550	550	NUM
cana-2695	80	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2695	80	7	known	know	VERB
cana-2695	80	8	as	as	ADP
cana-2695	80	9	an	an	DET
cana-2695	80	10	interval	interval	NOUN
cana-2695	80	11	-	-	PUNCT
cana-2695	80	12	valued	value	VERB
cana-2695	80	13	fuzzy	fuzzy	ADJ
cana-2695	80	14	resolving	resolving	NOUN
cana-2695	80	15	set	set	NOUN
cana-2695	80	16	(	(	PUNCT
cana-2695	80	17	ivfrs	ivfrs	NOUN
cana-2695	80	18	)	)	PUNCT
cana-2695	80	19	.	.	PUNCT
cana-2695	81	1	the	the	DET
cana-2695	81	2	minimum	minimum	ADJ
cana-2695	81	3	cardinality	cardinality	NOUN
cana-2695	81	4	of	of	ADP
cana-2695	81	5	this	this	DET
cana-2695	81	6	concerned	concerned	ADJ
cana-2695	81	7	set	set	NOUN
cana-2695	81	8	is	be	AUX
cana-2695	81	9	called	call	VERB
cana-2695	81	10	as	as	ADP
cana-2695	81	11	interval	interval	NOUN
cana-2695	81	12	-	-	PUNCT
cana-2695	81	13	valued	value	VERB
cana-2695	81	14	resolving	resolving	NOUN
cana-2695	81	15	number	number	NOUN
cana-2695	81	16	(	(	PUNCT
cana-2695	81	17	ivrn	ivrn	PROPN
cana-2695	81	18	)	)	PUNCT
cana-2695	81	19	,	,	PUNCT
cana-2695	81	20	denoted	denote	VERB
cana-2695	81	21	by	by	ADP
cana-2695	81	22	ℐr(g	ℐr(g	NOUN
cana-2695	81	23	)	)	PUNCT
cana-2695	81	24	.	.	PUNCT
cana-2695	82	1	3.1	3.1	NUM
cana-2695	82	2	definition	definition	NOUN
cana-2695	82	3	the	the	DET
cana-2695	82	4	representation	representation	NOUN
cana-2695	82	5	of	of	ADP
cana-2695	82	6	σ	σ	PROPN
cana-2695	82	7	−	−	PROPN
cana-2695	82	8	ℋ	ℋ	PROPN
cana-2695	82	9	with	with	ADP
cana-2695	82	10	respect	respect	NOUN
cana-2695	82	11	to	to	ADP
cana-2695	82	12	ℋ	ℋ	PROPN
cana-2695	82	13	is	be	AUX
cana-2695	82	14	written	write	VERB
cana-2695	82	15	as	as	ADP
cana-2695	82	16	[	[	X
cana-2695	82	17	aij	aij	PROPN
cana-2695	82	18	−	−	PROPN
cana-2695	82	19	,	,	PUNCT
cana-2695	82	20	aij	aij	PROPN
cana-2695	82	21	+	+	PROPN
cana-2695	82	22	]	]	X
cana-2695	82	23	,	,	PUNCT
cana-2695	82	24	where	where	SCONJ
cana-2695	82	25	aij	aij	PROPN
cana-2695	82	26	−	−	PROPN
cana-2695	82	27	=	=	PUNCT
cana-2695	83	1	[	[	X
cana-2695	83	2	w−(uj	w−(uj	NOUN
cana-2695	83	3	,	,	PUNCT
cana-2695	83	4	x1	x1	PROPN
cana-2695	83	5	)	)	PUNCT
cana-2695	83	6	,	,	PUNCT
cana-2695	83	7	w	w	PROPN
cana-2695	83	8	−(uj	−(uj	PROPN
cana-2695	83	9	,	,	PUNCT
cana-2695	83	10	x2	x2	PROPN
cana-2695	83	11	)	)	PUNCT
cana-2695	83	12	,	,	PUNCT
cana-2695	83	13	.	.	PUNCT
cana-2695	83	14	.	.	PUNCT
cana-2695	83	15	.	.	PUNCT
cana-2695	84	1	w	w	PROPN
cana-2695	84	2	−(uj	−(uj	PROPN
cana-2695	84	3	,	,	PUNCT
cana-2695	84	4	xk	xk	PROPN
cana-2695	84	5	)	)	PUNCT
cana-2695	84	6	]	]	PUNCT
cana-2695	84	7	,	,	PUNCT
cana-2695	84	8	aij	aij	PROPN
cana-2695	84	9	+	+	SYM
cana-2695	84	10	=	=	SYM
cana-2695	85	1	[	[	X
cana-2695	85	2	w+(uj	w+(uj	PROPN
cana-2695	85	3	,	,	PUNCT
cana-2695	85	4	x1	x1	PROPN
cana-2695	85	5	)	)	PUNCT
cana-2695	85	6	,	,	PUNCT
cana-2695	85	7	w	w	PROPN
cana-2695	85	8	+	+	PROPN
cana-2695	85	9	(	(	PUNCT
cana-2695	85	10	uj	uj	PROPN
cana-2695	85	11	,	,	PUNCT
cana-2695	85	12	x2	x2	PROPN
cana-2695	85	13	)	)	PUNCT
cana-2695	85	14	,	,	PUNCT
cana-2695	85	15	.	.	PUNCT
cana-2695	85	16	.	.	PUNCT
cana-2695	85	17	.	.	PUNCT
cana-2695	86	1	w	w	PROPN
cana-2695	87	1	+	+	PROPN
cana-2695	87	2	(	(	PUNCT
cana-2695	87	3	uj	uj	PROPN
cana-2695	87	4	,	,	PUNCT
cana-2695	87	5	xk	xk	PROPN
cana-2695	87	6	)	)	PUNCT
cana-2695	87	7	]	]	PUNCT
cana-2695	87	8	for	for	ADP
cana-2695	87	9	j	j	PROPN
cana-2695	87	10	=	=	SYM
cana-2695	87	11	k	k	PROPN
cana-2695	88	1	+	+	PROPN
cana-2695	88	2	1	1	NUM
cana-2695	88	3	,	,	PUNCT
cana-2695	88	4	k	k	PROPN
cana-2695	88	5	+	+	PROPN
cana-2695	88	6	2	2	NUM
cana-2695	88	7	,	,	PUNCT
cana-2695	88	8	…	…	PUNCT
cana-2695	88	9	n	n	X
cana-2695	88	10	are	be	AUX
cana-2695	88	11	written	write	VERB
cana-2695	88	12	in	in	ADP
cana-2695	88	13	a	a	DET
cana-2695	88	14	form	form	NOUN
cana-2695	88	15	𝐴𝑛−𝑘×𝑘	𝐴𝑛−𝑘×𝑘	NOUN
cana-2695	88	16	.	.	PUNCT
cana-2695	89	1	this	this	DET
cana-2695	89	2	matrix	matrix	NOUN
cana-2695	89	3	is	be	AUX
cana-2695	89	4	called	call	VERB
cana-2695	89	5	interval	interval	NOUN
cana-2695	89	6	-	-	PUNCT
cana-2695	89	7	valued	value	VERB
cana-2695	89	8	fuzzy	fuzzy	ADJ
cana-2695	89	9	resolving	resolve	VERB
cana-2695	89	10	matrix(ivfrm	matrix(ivfrm	PROPN
cana-2695	89	11	)	)	PUNCT
cana-2695	89	12	.	.	PUNCT
cana-2695	90	1	3.2	3.2	NUM
cana-2695	90	2	illustration	illustration	NOUN
cana-2695	90	3	fig	fig	NOUN
cana-2695	90	4	1	1	NUM
cana-2695	90	5	:	:	PUNCT
cana-2695	90	6	ivfg	ivfg	NOUN
cana-2695	90	7	𝝁𝑬	𝝁𝑬	NUM
cana-2695	90	8	𝑸𝟏	𝑸𝟏	PROPN
cana-2695	90	9	𝑸𝟐	𝑸𝟐	PROPN
cana-2695	90	10	𝑸𝟑	𝑸𝟑	PROPN
cana-2695	90	11	𝑸𝟒	𝑸𝟒	PROPN
cana-2695	90	12	𝝁𝑬	𝝁𝑬	PUNCT
cana-2695	91	1	−	−	NOUN
cana-2695	91	2	𝝁𝑬	𝝁𝑬	PUNCT
cana-2695	92	1	+	+	CCONJ
cana-2695	92	2	0.4	0.4	NUM
cana-2695	92	3	0.5	0.5	NUM
cana-2695	92	4	0.6	0.6	NUM
cana-2695	92	5	0.6	0.6	NUM
cana-2695	92	6	0.8	0.8	NUM
cana-2695	92	7	0.7	0.7	NUM
cana-2695	92	8	0.8	0.8	NUM
cana-2695	92	9	0.7	0.7	NUM
cana-2695	92	10	𝝁𝑭	𝝁𝑭	NOUN
cana-2695	92	11	𝑸𝟏𝑸𝟐	𝑸𝟏𝑸𝟐	PROPN
cana-2695	92	12	𝑸𝟐𝑸𝟑	𝑸𝟐𝑸𝟑	PROPN
cana-2695	92	13	𝑸𝟏𝑸𝟑	𝑸𝟏𝑸𝟑	PROPN
cana-2695	92	14	𝑸𝟑𝑸𝟒	𝑸𝟑𝑸𝟒	PROPN
cana-2695	92	15	𝝁𝑭	𝝁𝑭	PUNCT
cana-2695	92	16	−	−	PROPN
cana-2695	92	17	𝝁𝑭	𝝁𝑭	PUNCT
cana-2695	93	1	+	+	CCONJ
cana-2695	93	2	0.4	0.4	NUM
cana-2695	93	3	0.4	0.4	NUM
cana-2695	93	4	0.5	0.5	NUM
cana-2695	93	5	0.3	0.3	NUM
cana-2695	93	6	0.3	0.3	NUM
cana-2695	93	7	0.5	0.5	NUM
cana-2695	93	8	0.7	0.7	NUM
cana-2695	93	9	0.6	0.6	NUM
cana-2695	93	10	let	let	VERB
cana-2695	93	11	ℋ	ℋ	VERB
cana-2695	93	12	=	=	SYM
cana-2695	93	13	{	{	PUNCT
cana-2695	93	14	q	q	NOUN
cana-2695	93	15	1	1	NUM
cana-2695	93	16	,	,	PUNCT
cana-2695	93	17	q	q	NOUN
cana-2695	93	18	3	3	NUM
cana-2695	93	19	}	}	PUNCT
cana-2695	93	20	,	,	PUNCT
cana-2695	93	21	then	then	ADV
cana-2695	93	22	σ	σ	PROPN
cana-2695	93	23	−	−	PROPN
cana-2695	93	24	ℋ	ℋ	PROPN
cana-2695	93	25	=	=	SYM
cana-2695	93	26	{	{	PUNCT
cana-2695	93	27	q	q	NOUN
cana-2695	93	28	2	2	NUM
cana-2695	93	29	,	,	PUNCT
cana-2695	93	30	q	q	NOUN
cana-2695	93	31	4	4	NUM
cana-2695	93	32	}	}	PUNCT
cana-2695	93	33	where	where	SCONJ
cana-2695	93	34	𝑄1	𝑄1	NOUN
cana-2695	93	35	=	=	PUNCT
cana-2695	93	36	{	{	PUNCT
cana-2695	93	37	𝜎1	𝜎1	PROPN
cana-2695	93	38	−	−	PROPN
cana-2695	93	39	,	,	PUNCT
cana-2695	93	40	𝜎1	𝜎1	PROPN
cana-2695	93	41	+	+	NOUN
cana-2695	93	42	}	}	PUNCT
cana-2695	93	43	,	,	PUNCT
cana-2695	93	44	𝑄2	𝑄2	X
cana-2695	94	1	=	=	PRON
cana-2695	95	1	{	{	PUNCT
cana-2695	95	2	𝜎2	𝜎2	PROPN
cana-2695	95	3	−	−	PROPN
cana-2695	95	4	,	,	PUNCT
cana-2695	95	5	𝜎2	𝜎2	NOUN
cana-2695	95	6	+	+	NOUN
cana-2695	95	7	}	}	PUNCT
cana-2695	95	8	,	,	PUNCT
cana-2695	95	9	𝑄3	𝑄3	X
cana-2695	95	10	=	=	SYM
cana-2695	95	11	{	{	PUNCT
cana-2695	95	12	𝜎3	𝜎3	NOUN
cana-2695	95	13	−	−	NOUN
cana-2695	95	14	,	,	PUNCT
cana-2695	95	15	𝜎3	𝜎3	VERB
cana-2695	95	16	+	+	NOUN
cana-2695	95	17	}	}	PUNCT
cana-2695	95	18	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-2695	95	19	𝑄4	𝑄4	NOUN
cana-2695	95	20	=	=	PUNCT
cana-2695	95	21	{	{	PUNCT
cana-2695	95	22	𝜎4	𝜎4	PROPN
cana-2695	95	23	−	−	PROPN
cana-2695	95	24	,	,	PUNCT
cana-2695	95	25	𝜎4	𝜎4	VERB
cana-2695	95	26	+	+	ADP
cana-2695	95	27	}	}	PUNCT
cana-2695	95	28	then	then	ADV
cana-2695	95	29	,	,	PUNCT
cana-2695	95	30	𝜎2	𝜎2	PROPN
cana-2695	95	31	−/ℋ	−/ℋ	NOUN
cana-2695	95	32	=	=	SYM
cana-2695	95	33	{	{	PUNCT
cana-2695	95	34	μ∞(𝜎2	μ∞(𝜎2	NUM
cana-2695	95	35	−	−	NOUN
cana-2695	95	36	,	,	PUNCT
cana-2695	95	37	𝜎1	𝜎1	ADJ
cana-2695	95	38	−	−	NOUN
cana-2695	95	39	)	)	PUNCT
cana-2695	95	40	,	,	PUNCT
cana-2695	95	41	μ∞(𝜎2	μ∞(𝜎2	NUM
cana-2695	95	42	−	−	NOUN
cana-2695	95	43	,	,	PUNCT
cana-2695	95	44	𝜎3	𝜎3	ADJ
cana-2695	95	45	−	−	NOUN
cana-2695	95	46	)	)	PUNCT
cana-2695	95	47	}	}	PUNCT
cana-2695	95	48	=	=	SYM
cana-2695	95	49	(	(	PUNCT
cana-2695	95	50	0.4	0.4	NUM
cana-2695	95	51	,	,	PUNCT
cana-2695	95	52	0.5	0.5	NUM
cana-2695	95	53	)	)	PUNCT
cana-2695	95	54	𝜎2	𝜎2	NOUN
cana-2695	95	55	+	+	NOUN
cana-2695	95	56	/ℋ	/ℋ	PUNCT
cana-2695	95	57	=	=	PUNCT
cana-2695	95	58	{	{	PUNCT
cana-2695	95	59	μ∞(𝜎2	μ∞(𝜎2	NUM
cana-2695	95	60	+	+	PROPN
cana-2695	95	61	,	,	PUNCT
cana-2695	95	62	𝜎1	𝜎1	PROPN
cana-2695	95	63	+	+	NOUN
cana-2695	95	64	)	)	PUNCT
cana-2695	95	65	,	,	PUNCT
cana-2695	95	66	μ∞(𝜎2	μ∞(𝜎2	X
cana-2695	95	67	+	+	ADJ
cana-2695	95	68	,	,	PUNCT
cana-2695	95	69	𝜎3	𝜎3	ADJ
cana-2695	95	70	+	+	NOUN
cana-2695	95	71	)	)	PUNCT
cana-2695	95	72	}	}	PUNCT
cana-2695	95	73	=	=	SYM
cana-2695	95	74	(	(	PUNCT
cana-2695	95	75	0.4	0.4	NUM
cana-2695	95	76	,	,	PUNCT
cana-2695	95	77	0.4	0.4	NUM
cana-2695	95	78	)	)	PUNCT
cana-2695	95	79	𝜎4	𝜎4	VERB
cana-2695	95	80	−/ℋ	−/ℋ	NOUN
cana-2695	95	81	=	=	SYM
cana-2695	95	82	{	{	PUNCT
cana-2695	95	83	μ∞(𝜎4	μ∞(𝜎4	PROPN
cana-2695	95	84	−	−	PROPN
cana-2695	95	85	,	,	PUNCT
cana-2695	95	86	𝜎1	𝜎1	ADJ
cana-2695	95	87	−	−	NUM
cana-2695	95	88	)	)	PUNCT
cana-2695	95	89	,	,	PUNCT
cana-2695	95	90	μ∞(𝜎4	μ∞(𝜎4	PROPN
cana-2695	95	91	−	−	NUM
cana-2695	95	92	,	,	PUNCT
cana-2695	95	93	𝜎3	𝜎3	VERB
cana-2695	95	94	−	−	PROPN
cana-2695	95	95	)	)	PUNCT
cana-2695	95	96	}	}	PUNCT
cana-2695	96	1	=	=	SYM
cana-2695	96	2	(	(	PUNCT
cana-2695	96	3	0.4	0.4	NUM
cana-2695	96	4	,	,	PUNCT
cana-2695	96	5	0.7	0.7	NUM
cana-2695	96	6	)	)	PUNCT
cana-2695	96	7	𝜎4	𝜎4	VERB
cana-2695	96	8	+	+	PROPN
cana-2695	96	9	/ℋ	/ℋ	NOUN
cana-2695	96	10	=	=	X
cana-2695	96	11	{	{	PUNCT
cana-2695	96	12	μ∞(𝜎4	μ∞(𝜎4	PROPN
cana-2695	96	13	+	+	PROPN
cana-2695	96	14	,	,	PUNCT
cana-2695	96	15	𝜎1	𝜎1	PROPN
cana-2695	96	16	+	+	NOUN
cana-2695	96	17	)	)	PUNCT
cana-2695	96	18	,	,	PUNCT
cana-2695	96	19	μ∞(𝜎4	μ∞(𝜎4	PROPN
cana-2695	96	20	+	+	NOUN
cana-2695	96	21	,	,	PUNCT
cana-2695	96	22	𝜎3	𝜎3	ADJ
cana-2695	96	23	+	+	NOUN
cana-2695	96	24	)	)	PUNCT
cana-2695	96	25	}	}	PUNCT
cana-2695	96	26	=	=	SYM
cana-2695	96	27	(	(	PUNCT
cana-2695	96	28	0.5	0.5	NUM
cana-2695	96	29	,	,	PUNCT
cana-2695	96	30	0.6	0.6	NUM
cana-2695	96	31	)	)	PUNCT
cana-2695	96	32	therefore	therefore	ADV
cana-2695	96	33	,	,	PUNCT
cana-2695	96	34	aij	aij	PROPN
cana-2695	96	35	−	−	PROPN
cana-2695	96	36	=	=	SYM
cana-2695	96	37	[	[	PUNCT
cana-2695	96	38	0.4	0.4	NUM
cana-2695	96	39	0.5	0.5	NUM
cana-2695	96	40	0.4	0.4	NUM
cana-2695	96	41	0.7	0.7	NUM
cana-2695	96	42	]	]	PUNCT
cana-2695	96	43	,	,	PUNCT
cana-2695	96	44	aij	aij	PROPN
cana-2695	96	45	+	+	SYM
cana-2695	96	46	=	=	X
cana-2695	96	47	[	[	PUNCT
cana-2695	96	48	0.4	0.4	NUM
cana-2695	96	49	0.4	0.4	NUM
cana-2695	96	50	0.5	0.5	NUM
cana-2695	96	51	0.6	0.6	NUM
cana-2695	96	52	]	]	PUNCT
cana-2695	96	53	communications	communication	NOUN
cana-2695	96	54	on	on	ADP
cana-2695	96	55	applied	apply	VERB
cana-2695	96	56	nonlinear	nonlinear	ADJ
cana-2695	96	57	analysis	analysis	NOUN
cana-2695	96	58	issn	issn	NOUN
cana-2695	96	59	:	:	PUNCT
cana-2695	96	60	1074	1074	NUM
cana-2695	96	61	-	-	PUNCT
cana-2695	96	62	133x	133x	NUM
cana-2695	96	63	vol	vol	NOUN
cana-2695	96	64	32	32	NUM
cana-2695	96	65	no	no	NOUN
cana-2695	96	66	.	.	PUNCT
cana-2695	97	1	3s	3s	NUM
cana-2695	97	2	(	(	PUNCT
cana-2695	97	3	2025	2025	NUM
cana-2695	97	4	)	)	PUNCT
cana-2695	97	5	551	551	NUM
cana-2695	97	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2695	97	7	hence	hence	ADV
cana-2695	97	8	,	,	PUNCT
cana-2695	97	9	the	the	DET
cana-2695	97	10	values	value	NOUN
cana-2695	97	11	are	be	AUX
cana-2695	97	12	all	all	ADV
cana-2695	97	13	distinct	distinct	ADJ
cana-2695	97	14	therefore	therefore	ADV
cana-2695	97	15	,	,	PUNCT
cana-2695	97	16	ℐr(g	ℐr(g	ADV
cana-2695	97	17	)	)	PUNCT
cana-2695	97	18	=	=	SYM
cana-2695	97	19	2	2	NUM
cana-2695	97	20	3.3	3.3	NUM
cana-2695	97	21	theorem	theorem	NOUN
cana-2695	97	22	if	if	SCONJ
cana-2695	97	23	g	g	PROPN
cana-2695	97	24	and	and	CCONJ
cana-2695	97	25	g′	g′	NOUN
cana-2695	97	26	are	be	AUX
cana-2695	97	27	isomorphic	isomorphic	ADJ
cana-2695	97	28	,	,	PUNCT
cana-2695	97	29	then	then	ADV
cana-2695	97	30	ℐr(g	ℐr(g	ADV
cana-2695	97	31	)	)	PUNCT
cana-2695	97	32	=	=	SYM
cana-2695	97	33	ℐr(g′	ℐr(g′	X
cana-2695	97	34	)	)	PUNCT
cana-2695	97	35	proof	proof	NOUN
cana-2695	97	36	:	:	PUNCT
cana-2695	97	37	let	let	VERB
cana-2695	97	38	us	we	PRON
cana-2695	97	39	assume	assume	VERB
cana-2695	97	40	that	that	SCONJ
cana-2695	97	41	v	v	X
cana-2695	97	42	=	=	SYM
cana-2695	97	43	{	{	PUNCT
cana-2695	97	44	r1	r1	PROPN
cana-2695	97	45	,	,	PUNCT
cana-2695	97	46	r2	r2	PROPN
cana-2695	97	47	,	,	PUNCT
cana-2695	97	48	.	.	PUNCT
cana-2695	97	49	.	.	PUNCT
cana-2695	97	50	.	.	PUNCT
cana-2695	98	1	,	,	PUNCT
cana-2695	98	2	rn	rn	PROPN
cana-2695	98	3	}	}	PUNCT
cana-2695	98	4	,	,	PUNCT
cana-2695	98	5	ℐr(g)=	ℐr(g)=	NOUN
cana-2695	98	6	k	k	NOUN
cana-2695	98	7	,	,	PUNCT
cana-2695	98	8	and	and	CCONJ
cana-2695	98	9	let	let	VERB
cana-2695	98	10	ℋ	ℋ	PROPN
cana-2695	98	11	=	=	SYM
cana-2695	98	12	{	{	PUNCT
cana-2695	98	13	(	(	PUNCT
cana-2695	98	14	σ1	σ1	PROPN
cana-2695	98	15	−	−	PROPN
cana-2695	98	16	,	,	PUNCT
cana-2695	98	17	σ1	σ1	PROPN
cana-2695	98	18	+	+	PROPN
cana-2695	98	19	)	)	PUNCT
cana-2695	98	20	,	,	PUNCT
cana-2695	98	21	(	(	PUNCT
cana-2695	98	22	σ2	σ2	NOUN
cana-2695	98	23	−	−	PROPN
cana-2695	98	24	,	,	PUNCT
cana-2695	98	25	σ2	σ2	NOUN
cana-2695	98	26	+	+	PROPN
cana-2695	98	27	)	)	PUNCT
cana-2695	98	28	,	,	PUNCT
cana-2695	98	29	.	.	PUNCT
cana-2695	98	30	.	.	PUNCT
cana-2695	99	1	.	.	PUNCT
cana-2695	100	1	,	,	PUNCT
cana-2695	100	2	(	(	PUNCT
cana-2695	100	3	σk	σk	ADP
cana-2695	100	4	−	−	NOUN
cana-2695	100	5	,	,	PUNCT
cana-2695	100	6	σk	σk	X
cana-2695	100	7	+	+	PUNCT
cana-2695	100	8	)	)	PUNCT
cana-2695	100	9	is	be	AUX
cana-2695	100	10	the	the	DET
cana-2695	100	11	corresponding	corresponding	ADJ
cana-2695	100	12	frs	frs	PROPN
cana-2695	100	13	.	.	PUNCT
cana-2695	101	1	we	we	PRON
cana-2695	101	2	denote	denote	VERB
cana-2695	101	3	(	(	PUNCT
cana-2695	101	4	r1	r1	PROPN
cana-2695	101	5	,	,	PUNCT
cana-2695	101	6	σ	σ	PROPN
cana-2695	101	7	−(r1	−(r1	NUM
cana-2695	101	8	)	)	PUNCT
cana-2695	101	9	)	)	PUNCT
cana-2695	102	1	=	=	SYM
cana-2695	102	2	σ1	σ1	PROPN
cana-2695	102	3	−	−	PROPN
cana-2695	102	4	,	,	PUNCT
cana-2695	102	5	(	(	PUNCT
cana-2695	102	6	r1	r1	PROPN
cana-2695	102	7	,	,	PUNCT
cana-2695	102	8	σ	σ	PROPN
cana-2695	102	9	+	+	PROPN
cana-2695	102	10	(	(	PUNCT
cana-2695	102	11	r1	r1	NOUN
cana-2695	102	12	)	)	PUNCT
cana-2695	102	13	)	)	PUNCT
cana-2695	103	1	=	=	SYM
cana-2695	103	2	σ1	σ1	PROPN
cana-2695	103	3	+	+	X
cana-2695	103	4	,	,	PUNCT
cana-2695	103	5	(	(	PUNCT
cana-2695	103	6	φ(r1	φ(r1	ADV
cana-2695	103	7	)	)	PUNCT
cana-2695	103	8	,	,	PUNCT
cana-2695	103	9	σ	σ	PROPN
cana-2695	103	10	−(φ(r	−(φ(r	PROPN
cana-2695	103	11	1	1	NUM
cana-2695	103	12	)	)	PUNCT
cana-2695	103	13	)	)	PUNCT
cana-2695	104	1	=	=	SYM
cana-2695	104	2	σ1	σ1	NOUN
cana-2695	104	3	−′	−′	NOUN
cana-2695	104	4	,	,	PUNCT
cana-2695	104	5	(	(	PUNCT
cana-2695	104	6	φ(r1	φ(r1	ADV
cana-2695	104	7	)	)	PUNCT
cana-2695	104	8	,	,	PUNCT
cana-2695	104	9	σ	σ	PROPN
cana-2695	105	1	+	+	PROPN
cana-2695	105	2	(	(	PUNCT
cana-2695	105	3	φ(r	φ(r	NOUN
cana-2695	105	4	1	1	NUM
cana-2695	105	5	)	)	PUNCT
cana-2695	105	6	)	)	PUNCT
cana-2695	106	1	=	=	SYM
cana-2695	106	2	σ1	σ1	NOUN
cana-2695	106	3	+	+	ADP
cana-2695	106	4	′.	′.	NOUN
cana-2695	106	5	the	the	DET
cana-2695	106	6	representation	representation	NOUN
cana-2695	106	7	,	,	PUNCT
cana-2695	106	8	𝜎𝑘+𝑖	𝜎𝑘+𝑖	NUM
cana-2695	106	9	−	−	NOUN
cana-2695	106	10	/ℋ	/ℋ	PUNCT
cana-2695	106	11	=(	=(	ADJ
cana-2695	106	12	w−	w−	PROPN
cana-2695	106	13	(	(	PUNCT
cana-2695	106	14	rk+i	rk+i	PROPN
cana-2695	106	15	,	,	PUNCT
cana-2695	106	16	r1	r1	PROPN
cana-2695	106	17	)	)	PUNCT
cana-2695	106	18	,	,	PUNCT
cana-2695	106	19	w	w	PROPN
cana-2695	106	20	−	−	PROPN
cana-2695	106	21	(	(	PUNCT
cana-2695	106	22	rk+i	rk+i	PROPN
cana-2695	106	23	,	,	PUNCT
cana-2695	106	24	r2	r2	PROPN
cana-2695	106	25	)	)	PUNCT
cana-2695	106	26	,	,	PUNCT
cana-2695	106	27	.	.	PUNCT
cana-2695	106	28	.	.	PUNCT
cana-2695	106	29	.	.	PUNCT
cana-2695	107	1	,	,	PUNCT
cana-2695	107	2	w	w	ADP
cana-2695	107	3	−	−	PROPN
cana-2695	107	4	(	(	PUNCT
cana-2695	107	5	rk+i	rk+i	PROPN
cana-2695	107	6	,	,	PUNCT
cana-2695	107	7	rk	rk	NOUN
cana-2695	107	8	)	)	PUNCT
cana-2695	107	9	,	,	PUNCT
cana-2695	107	10	𝜎𝑘+𝑖	𝜎𝑘+𝑖	X
cana-2695	108	1	+	+	CCONJ
cana-2695	108	2	/ℋ	/ℋ	SYM
cana-2695	108	3	=(	=(	NOUN
cana-2695	108	4	w+	w+	X
cana-2695	108	5	(	(	PUNCT
cana-2695	108	6	rk+i	rk+i	PROPN
cana-2695	108	7	,	,	PUNCT
cana-2695	108	8	r1	r1	PROPN
cana-2695	108	9	)	)	PUNCT
cana-2695	108	10	,	,	PUNCT
cana-2695	108	11	w	w	ADP
cana-2695	108	12	+	+	CCONJ
cana-2695	108	13	(	(	PUNCT
cana-2695	108	14	rk+i	rk+i	PROPN
cana-2695	108	15	,	,	PUNCT
cana-2695	108	16	r2	r2	PROPN
cana-2695	108	17	)	)	PUNCT
cana-2695	108	18	,	,	PUNCT
cana-2695	108	19	.	.	PUNCT
cana-2695	108	20	.	.	PUNCT
cana-2695	108	21	.	.	PUNCT
cana-2695	109	1	,	,	PUNCT
cana-2695	109	2	w	w	X
cana-2695	109	3	+	+	CCONJ
cana-2695	109	4	(	(	PUNCT
cana-2695	109	5	rk+i	rk+i	PROPN
cana-2695	109	6	,	,	PUNCT
cana-2695	109	7	rk	rk	NOUN
cana-2695	109	8	)	)	PUNCT
cana-2695	109	9	all	all	PRON
cana-2695	109	10	are	be	AUX
cana-2695	109	11	distinct	distinct	ADJ
cana-2695	109	12	,	,	PUNCT
cana-2695	109	13	∀	∀	VERB
cana-2695	109	14	i	i	NOUN
cana-2695	109	15	=	=	NOUN
cana-2695	109	16	1,2	1,2	NUM
cana-2695	109	17	,	,	PUNCT
cana-2695	109	18	.	.	PUNCT
cana-2695	109	19	.	.	PUNCT
cana-2695	110	1	.	.	PUNCT
cana-2695	111	1	n	n	PRON
cana-2695	111	2	−	−	PROPN
cana-2695	111	3	k.	k.	NOUN
cana-2695	112	1	if	if	SCONJ
cana-2695	112	2	g	g	PROPN
cana-2695	112	3	and	and	CCONJ
cana-2695	112	4	g′	g′	NOUN
cana-2695	112	5	are	be	AUX
cana-2695	112	6	isomorphic	isomorphic	ADJ
cana-2695	112	7	,	,	PUNCT
cana-2695	112	8	then	then	ADV
cana-2695	112	9	∃	∃	PROPN
cana-2695	112	10	a	a	DET
cana-2695	112	11	bijection	bijection	NOUN
cana-2695	112	12	φ	φ	NOUN
cana-2695	112	13	:	:	PUNCT
cana-2695	112	14	v	v	PROPN
cana-2695	112	15	→	→	SYM
cana-2695	112	16	v′	v′	NUM
cana-2695	112	17	which	which	PRON
cana-2695	112	18	satisfies	satisfy	VERB
cana-2695	112	19	,	,	PUNCT
cana-2695	112	20	μ	μ	PROPN
cana-2695	112	21	e	e	NOUN
cana-2695	112	22	−	−	PROPN
cana-2695	112	23	(	(	PUNCT
cana-2695	112	24	r	r	NOUN
cana-2695	112	25	)	)	PUNCT
cana-2695	112	26	=	=	SYM
cana-2695	112	27	μ	μ	NOUN
cana-2695	112	28	e	e	NOUN
cana-2695	112	29	−′(φ	−′(φ	ADJ
cana-2695	112	30	(	(	PUNCT
cana-2695	112	31	r	r	NOUN
cana-2695	112	32	)	)	PUNCT
cana-2695	112	33	)	)	PUNCT
cana-2695	112	34	,	,	PUNCT
cana-2695	112	35	μ	μ	PROPN
cana-2695	112	36	e	e	X
cana-2695	112	37	+	+	CCONJ
cana-2695	112	38	(	(	PUNCT
cana-2695	112	39	r	r	NOUN
cana-2695	112	40	)	)	PUNCT
cana-2695	112	41	=	=	SYM
cana-2695	112	42	μ	μ	NUM
cana-2695	112	43	e	e	X
cana-2695	112	44	+	+	NOUN
cana-2695	112	45	′	′	NUM
cana-2695	112	46	(	(	PUNCT
cana-2695	112	47	φ(r	φ(r	NOUN
cana-2695	112	48	)	)	PUNCT
cana-2695	112	49	)	)	PUNCT
cana-2695	112	50	,	,	PUNCT
cana-2695	112	51	μ	μ	PROPN
cana-2695	112	52	f	f	X
cana-2695	113	1	−	−	PROPN
cana-2695	113	2	(	(	PUNCT
cana-2695	113	3	r	r	NOUN
cana-2695	113	4	,	,	PUNCT
cana-2695	113	5	u	u	NOUN
cana-2695	113	6	)	)	PUNCT
cana-2695	113	7	=	=	SYM
cana-2695	114	1	μ	μ	PROPN
cana-2695	114	2	f	f	X
cana-2695	114	3	−′	−′	X
cana-2695	114	4	(	(	PUNCT
cana-2695	114	5	φ(r	φ(r	NOUN
cana-2695	114	6	)	)	PUNCT
cana-2695	114	7	,	,	PUNCT
cana-2695	114	8	φ(u	φ(u	NOUN
cana-2695	114	9	)	)	PUNCT
cana-2695	114	10	)	)	PUNCT
cana-2695	114	11	and	and	CCONJ
cana-2695	114	12	μ	μ	NOUN
cana-2695	114	13	f	f	PROPN
cana-2695	115	1	+	+	CCONJ
cana-2695	115	2	(	(	PUNCT
cana-2695	115	3	r	r	NOUN
cana-2695	115	4	,	,	PUNCT
cana-2695	115	5	u	u	NOUN
cana-2695	115	6	)	)	PUNCT
cana-2695	115	7	=	=	SYM
cana-2695	115	8	μ	μ	NOUN
cana-2695	115	9	f	f	X
cana-2695	116	1	+	+	NOUN
cana-2695	116	2	′	′	NUM
cana-2695	116	3	(	(	PUNCT
cana-2695	116	4	φ(r	φ(r	NOUN
cana-2695	116	5	)	)	PUNCT
cana-2695	116	6	,	,	PUNCT
cana-2695	116	7	φ(u	φ(u	NOUN
cana-2695	116	8	)	)	PUNCT
cana-2695	116	9	)	)	PUNCT
cana-2695	117	1	∀r	∀r	NOUN
cana-2695	117	2	,	,	PUNCT
cana-2695	117	3	u	u	PROPN
cana-2695	117	4	∈	∈	PROPN
cana-2695	117	5	e.	e.	PROPN
cana-2695	117	6	now	now	ADV
cana-2695	117	7	,	,	PUNCT
cana-2695	117	8	we	we	PRON
cana-2695	117	9	define	define	VERB
cana-2695	117	10	ℋ1	ℋ1	NOUN
cana-2695	117	11	=	=	PUNCT
cana-2695	117	12	{	{	PUNCT
cana-2695	117	13	(	(	PUNCT
cana-2695	117	14	σ1	σ1	PROPN
cana-2695	117	15	−′	−′	PROPN
cana-2695	117	16	,	,	PUNCT
cana-2695	117	17	σ1	σ1	PROPN
cana-2695	117	18	+	+	PROPN
cana-2695	117	19	′	′	NUM
cana-2695	117	20	)	)	PUNCT
cana-2695	117	21	,	,	PUNCT
cana-2695	117	22	(	(	PUNCT
cana-2695	117	23	σ2	σ2	PROPN
cana-2695	117	24	−′	−′	PROPN
cana-2695	117	25	,	,	PUNCT
cana-2695	117	26	σ2	σ2	PROPN
cana-2695	117	27	+	+	PROPN
cana-2695	117	28	′	′	NOUN
cana-2695	117	29	)	)	PUNCT
cana-2695	117	30	,	,	PUNCT
cana-2695	117	31	.	.	PUNCT
cana-2695	117	32	.	.	PUNCT
cana-2695	117	33	.	.	PUNCT
cana-2695	118	1	,	,	PUNCT
cana-2695	118	2	(	(	PUNCT
cana-2695	118	3	σk	σk	ADP
cana-2695	118	4	−′	−′	NUM
cana-2695	118	5	,	,	PUNCT
cana-2695	118	6	σk	σk	ADV
cana-2695	118	7	+	+	NOUN
cana-2695	118	8	′	′	NUM
cana-2695	118	9	)	)	PUNCT
cana-2695	118	10	}	}	PUNCT
cana-2695	118	11	for	for	ADP
cana-2695	118	12	i	i	PROPN
cana-2695	118	13	=	=	SYM
cana-2695	118	14	1,2	1,2	NUM
cana-2695	118	15	,	,	PUNCT
cana-2695	118	16	.	.	PUNCT
cana-2695	118	17	.	.	PUNCT
cana-2695	118	18	.	.	PUNCT
cana-2695	119	1	n	n	CCONJ
cana-2695	120	1	−	−	PROPN
cana-2695	120	2	k	k	PROPN
cana-2695	120	3	,	,	PUNCT
cana-2695	120	4	𝜎𝑘+𝑖	𝜎𝑘+𝑖	X
cana-2695	121	1	−′	−′	X
cana-2695	121	2	/ℋ	/ℋ	PUNCT
cana-2695	122	1	=	=	PUNCT
cana-2695	122	2	{	{	PUNCT
cana-2695	122	3	w−	w−	NOUN
cana-2695	122	4	(	(	PUNCT
cana-2695	122	5	φ(r	φ(r	PROPN
cana-2695	122	6	k+i	k+i	NUM
cana-2695	122	7	)	)	PUNCT
cana-2695	122	8	,	,	PUNCT
cana-2695	122	9	φ(r	φ(r	NOUN
cana-2695	122	10	1	1	NUM
cana-2695	122	11	)	)	PUNCT
cana-2695	122	12	)	)	PUNCT
cana-2695	122	13	,	,	PUNCT
cana-2695	122	14	w−	w−	X
cana-2695	122	15	(	(	PUNCT
cana-2695	122	16	φ(r	φ(r	PROPN
cana-2695	122	17	k+i	k+i	NUM
cana-2695	122	18	)	)	PUNCT
cana-2695	122	19	,	,	PUNCT
cana-2695	122	20	φ(r	φ(r	PROPN
cana-2695	122	21	2	2	NUM
cana-2695	122	22	)	)	PUNCT
cana-2695	122	23	)	)	PUNCT
cana-2695	122	24	,	,	PUNCT
cana-2695	122	25	.	.	PUNCT
cana-2695	122	26	.	.	PUNCT
cana-2695	122	27	.	.	PUNCT
cana-2695	123	1	,	,	PUNCT
cana-2695	123	2	w−	w−	X
cana-2695	123	3	(	(	PUNCT
cana-2695	123	4	φ(r	φ(r	PROPN
cana-2695	123	5	k+i	k+i	NUM
cana-2695	123	6	)	)	PUNCT
cana-2695	123	7	,	,	PUNCT
cana-2695	123	8	φ(r	φ(r	PROPN
cana-2695	123	9	k	k	NOUN
cana-2695	123	10	)	)	PUNCT
cana-2695	123	11	}	}	PUNCT
cana-2695	123	12	=	=	SYM
cana-2695	123	13	{	{	PUNCT
cana-2695	123	14	(	(	PUNCT
cana-2695	123	15	(	(	PUNCT
cana-2695	123	16	μ−′)∞(φ(r	μ−′)∞(φ(r	NOUN
cana-2695	123	17	k+i	k+i	NUM
cana-2695	123	18	)	)	PUNCT
cana-2695	123	19	,	,	PUNCT
cana-2695	123	20	(	(	PUNCT
cana-2695	123	21	μ−′)∞(φ(r	μ−′)∞(φ(r	NOUN
cana-2695	123	22	1	1	NUM
cana-2695	123	23	)	)	PUNCT
cana-2695	123	24	)	)	PUNCT
cana-2695	123	25	,	,	PUNCT
cana-2695	123	26	(	(	PUNCT
cana-2695	123	27	(	(	PUNCT
cana-2695	123	28	μ−′)∞(φ(r	μ−′)∞(φ(r	NOUN
cana-2695	123	29	k+i	k+i	NUM
cana-2695	123	30	)	)	PUNCT
cana-2695	123	31	,	,	PUNCT
cana-2695	123	32	(	(	PUNCT
cana-2695	123	33	μ−′)∞(φ(r	μ−′)∞(φ(r	NOUN
cana-2695	123	34	2	2	NUM
cana-2695	123	35	)	)	PUNCT
cana-2695	123	36	)	)	PUNCT
cana-2695	123	37	,	,	PUNCT
cana-2695	123	38	.	.	PUNCT
cana-2695	124	1	..	..	PUNCT
cana-2695	125	1	(	(	PUNCT
cana-2695	125	2	(	(	PUNCT
cana-2695	125	3	μ−′)∞(φ(r	μ−′)∞(φ(r	NOUN
cana-2695	125	4	k+i	k+i	NUM
cana-2695	125	5	)	)	PUNCT
cana-2695	125	6	,	,	PUNCT
cana-2695	125	7	(	(	PUNCT
cana-2695	125	8	μ−′)∞(φ(r	μ−′)∞(φ(r	NOUN
cana-2695	125	9	k	k	NOUN
cana-2695	125	10	)	)	PUNCT
cana-2695	125	11	)	)	PUNCT
cana-2695	126	1	=	=	SYM
cana-2695	126	2	μ−(rk+i	μ−(rk+i	NOUN
cana-2695	126	3	,	,	PUNCT
cana-2695	126	4	r1	r1	PROPN
cana-2695	126	5	)	)	PUNCT
cana-2695	126	6	,	,	PUNCT
cana-2695	126	7	μ	μ	PROPN
cana-2695	126	8	−(rk+i	−(rk+i	NOUN
cana-2695	126	9	,	,	PUNCT
cana-2695	126	10	r2	r2	PROPN
cana-2695	126	11	)	)	PUNCT
cana-2695	126	12	,	,	PUNCT
cana-2695	126	13	.	.	PUNCT
cana-2695	126	14	.	.	PUNCT
cana-2695	126	15	.	.	PUNCT
cana-2695	127	1	,	,	PUNCT
cana-2695	127	2	μ	μ	PROPN
cana-2695	127	3	−(rk+i	−(rk+i	NOUN
cana-2695	127	4	,	,	PUNCT
cana-2695	127	5	rk	rk	NOUN
cana-2695	127	6	)	)	PUNCT
cana-2695	127	7	=	=	SYM
cana-2695	128	1	w−	w−	NOUN
cana-2695	128	2	(	(	PUNCT
cana-2695	128	3	rk+i	rk+i	PROPN
cana-2695	128	4	,	,	PUNCT
cana-2695	128	5	r1	r1	PROPN
cana-2695	128	6	)	)	PUNCT
cana-2695	128	7	,	,	PUNCT
cana-2695	128	8	w	w	PROPN
cana-2695	128	9	−	−	PROPN
cana-2695	128	10	(	(	PUNCT
cana-2695	128	11	rk+i	rk+i	PROPN
cana-2695	128	12	,	,	PUNCT
cana-2695	128	13	r2	r2	PROPN
cana-2695	128	14	)	)	PUNCT
cana-2695	128	15	,	,	PUNCT
cana-2695	128	16	.	.	PUNCT
cana-2695	128	17	.	.	PUNCT
cana-2695	128	18	.	.	PUNCT
cana-2695	129	1	,	,	PUNCT
cana-2695	129	2	w	w	ADP
cana-2695	129	3	−	−	PROPN
cana-2695	129	4	(	(	PUNCT
cana-2695	129	5	rk+i	rk+i	PROPN
cana-2695	129	6	,	,	PUNCT
cana-2695	129	7	rk	rk	NOUN
cana-2695	129	8	)	)	PUNCT
cana-2695	129	9	𝜎𝑘+𝑖	𝜎𝑘+𝑖	PUNCT
cana-2695	130	1	+	+	NOUN
cana-2695	130	2	′	′	NUM
cana-2695	130	3	/ℋ	/ℋ	PUNCT
cana-2695	131	1	=	=	PUNCT
cana-2695	131	2	{	{	PUNCT
cana-2695	131	3	w+	w+	X
cana-2695	131	4	(	(	PUNCT
cana-2695	131	5	φ(r	φ(r	PROPN
cana-2695	131	6	k+i	k+i	NUM
cana-2695	131	7	)	)	PUNCT
cana-2695	131	8	,	,	PUNCT
cana-2695	131	9	φ(r	φ(r	NOUN
cana-2695	131	10	1	1	NUM
cana-2695	131	11	)	)	PUNCT
cana-2695	131	12	)	)	PUNCT
cana-2695	131	13	,	,	PUNCT
cana-2695	131	14	w+(φ(r	w+(φ(r	NOUN
cana-2695	131	15	k+i	k+i	PROPN
cana-2695	131	16	)	)	PUNCT
cana-2695	131	17	,	,	PUNCT
cana-2695	131	18	φ(r	φ(r	PROPN
cana-2695	131	19	2	2	NUM
cana-2695	131	20	)	)	PUNCT
cana-2695	131	21	)	)	PUNCT
cana-2695	131	22	,	,	PUNCT
cana-2695	131	23	.	.	PUNCT
cana-2695	131	24	.	.	PUNCT
cana-2695	131	25	.	.	PUNCT
cana-2695	132	1	,	,	PUNCT
cana-2695	132	2	w−+(φ(r	w−+(φ(r	ADJ
cana-2695	132	3	k+i	k+i	NUM
cana-2695	132	4	)	)	PUNCT
cana-2695	132	5	,	,	PUNCT
cana-2695	132	6	φ(r	φ(r	PROPN
cana-2695	132	7	k	k	NOUN
cana-2695	132	8	)	)	PUNCT
cana-2695	132	9	}	}	PUNCT
cana-2695	133	1	=	=	SYM
cana-2695	133	2	{	{	PUNCT
cana-2695	133	3	(	(	PUNCT
cana-2695	133	4	(	(	PUNCT
cana-2695	133	5	μ+′)∞(φ(r	μ+′)∞(φ(r	NOUN
cana-2695	133	6	k+i	k+i	NUM
cana-2695	133	7	)	)	PUNCT
cana-2695	133	8	,	,	PUNCT
cana-2695	133	9	(	(	PUNCT
cana-2695	133	10	μ+′)∞(φ(r	μ+′)∞(φ(r	NOUN
cana-2695	133	11	1	1	NUM
cana-2695	133	12	)	)	PUNCT
cana-2695	133	13	)	)	PUNCT
cana-2695	133	14	,	,	PUNCT
cana-2695	133	15	(	(	PUNCT
cana-2695	133	16	(	(	PUNCT
cana-2695	133	17	μ+′)∞(φ(r	μ+′)∞(φ(r	NOUN
cana-2695	133	18	k+i	k+i	NUM
cana-2695	133	19	)	)	PUNCT
cana-2695	133	20	,	,	PUNCT
cana-2695	133	21	(	(	PUNCT
cana-2695	133	22	μ+′)∞(φ(r	μ+′)∞(φ(r	NOUN
cana-2695	133	23	2	2	NUM
cana-2695	133	24	)	)	PUNCT
cana-2695	133	25	)	)	PUNCT
cana-2695	133	26	,	,	PUNCT
cana-2695	133	27	.	.	PUNCT
cana-2695	133	28	..	..	PUNCT
cana-2695	134	1	(	(	PUNCT
cana-2695	134	2	(	(	PUNCT
cana-2695	134	3	μ+′)∞(φ(r	μ+′)∞(φ(r	NOUN
cana-2695	134	4	k+i	k+i	NUM
cana-2695	134	5	)	)	PUNCT
cana-2695	134	6	,	,	PUNCT
cana-2695	134	7	(	(	PUNCT
cana-2695	134	8	μ+′)∞(φ(r	μ+′)∞(φ(r	NOUN
cana-2695	134	9	k	k	PROPN
cana-2695	134	10	)	)	PUNCT
cana-2695	134	11	)	)	PUNCT
cana-2695	135	1	=	=	SYM
cana-2695	135	2	μ+(rk+i	μ+(rk+i	NOUN
cana-2695	135	3	,	,	PUNCT
cana-2695	135	4	r1	r1	PROPN
cana-2695	135	5	)	)	PUNCT
cana-2695	135	6	,	,	PUNCT
cana-2695	135	7	μ	μ	PROPN
cana-2695	135	8	+	+	PROPN
cana-2695	135	9	(	(	PUNCT
cana-2695	135	10	rk+i	rk+i	PROPN
cana-2695	135	11	,	,	PUNCT
cana-2695	135	12	r2	r2	PROPN
cana-2695	135	13	)	)	PUNCT
cana-2695	135	14	,	,	PUNCT
cana-2695	135	15	.	.	PUNCT
cana-2695	135	16	.	.	PUNCT
cana-2695	135	17	.	.	PUNCT
cana-2695	136	1	,	,	PUNCT
cana-2695	136	2	μ	μ	PROPN
cana-2695	136	3	+	+	PROPN
cana-2695	136	4	(	(	PUNCT
cana-2695	136	5	rk+i	rk+i	PROPN
cana-2695	136	6	,	,	PUNCT
cana-2695	136	7	rk	rk	NOUN
cana-2695	136	8	)	)	PUNCT
cana-2695	136	9	=	=	SYM
cana-2695	136	10	w+(rk+i	w+(rk+i	NOUN
cana-2695	136	11	,	,	PUNCT
cana-2695	136	12	r1	r1	PROPN
cana-2695	136	13	)	)	PUNCT
cana-2695	136	14	,	,	PUNCT
cana-2695	136	15	w	w	ADP
cana-2695	136	16	+	+	CCONJ
cana-2695	136	17	(	(	PUNCT
cana-2695	136	18	rk+i	rk+i	PROPN
cana-2695	136	19	,	,	PUNCT
cana-2695	136	20	r2	r2	PROPN
cana-2695	136	21	)	)	PUNCT
cana-2695	136	22	,	,	PUNCT
cana-2695	136	23	.	.	PUNCT
cana-2695	136	24	.	.	PUNCT
cana-2695	137	1	.	.	PUNCT
cana-2695	138	1	,	,	PUNCT
cana-2695	138	2	w	w	X
cana-2695	138	3	+	+	CCONJ
cana-2695	138	4	(	(	PUNCT
cana-2695	138	5	rk+i	rk+i	PROPN
cana-2695	138	6	,	,	PUNCT
cana-2695	138	7	rk	rk	NOUN
cana-2695	138	8	)	)	PUNCT
cana-2695	138	9	which	which	PRON
cana-2695	138	10	are	be	AUX
cana-2695	138	11	all	all	ADV
cana-2695	138	12	distinct	distinct	ADJ
cana-2695	138	13	for	for	ADP
cana-2695	138	14	i	i	PRON
cana-2695	138	15	=	=	NOUN
cana-2695	138	16	1	1	NUM
cana-2695	138	17	to	to	ADP
cana-2695	138	18	n	n	PROPN
cana-2695	138	19	=	=	SYM
cana-2695	138	20	k.	k.	PROPN
cana-2695	138	21	therefore	therefore	ADV
cana-2695	138	22	,	,	PUNCT
cana-2695	138	23	ℋ1	ℋ1	PROPN
cana-2695	138	24	is	be	AUX
cana-2695	138	25	the	the	DET
cana-2695	138	26	ivfrs	ivfrs	NOUN
cana-2695	138	27	of	of	ADP
cana-2695	138	28	g′	g′	NOUN
cana-2695	138	29	and	and	CCONJ
cana-2695	138	30	│	│	NOUN
cana-2695	138	31	ℋ1	ℋ1	NOUN
cana-2695	138	32	│	│	X
cana-2695	138	33	=	=	PUNCT
cana-2695	138	34	k.	k.	PROPN
cana-2695	138	35	now	now	ADV
cana-2695	138	36	to	to	PART
cana-2695	138	37	prove	prove	VERB
cana-2695	138	38	that	that	SCONJ
cana-2695	138	39	ℋ1	ℋ1	PROPN
cana-2695	138	40	is	be	AUX
cana-2695	138	41	the	the	DET
cana-2695	138	42	minimum	minimum	ADJ
cana-2695	138	43	ivfrs	ivfrs	NOUN
cana-2695	138	44	of	of	ADP
cana-2695	138	45	g	g	NOUN
cana-2695	138	46	’	'	PUNCT
cana-2695	138	47	.	.	PUNCT
cana-2695	139	1	assume	assume	VERB
cana-2695	139	2	that	that	SCONJ
cana-2695	139	3	there	there	PRON
cana-2695	139	4	exists	exist	VERB
cana-2695	139	5	a	a	DET
cana-2695	139	6	frs	frs	ADJ
cana-2695	139	7	r	r	NOUN
cana-2695	139	8	of	of	ADP
cana-2695	139	9	g	g	NOUN
cana-2695	139	10	’	'	PUNCT
cana-2695	139	11	such	such	ADJ
cana-2695	139	12	that	that	DET
cana-2695	139	13	r	r	NOUN
cana-2695	139	14	=	=	SYM
cana-2695	139	15	{	{	PUNCT
cana-2695	139	16	(	(	PUNCT
cana-2695	139	17	σ1	σ1	PROPN
cana-2695	139	18	−	−	PROPN
cana-2695	139	19	,	,	PUNCT
cana-2695	139	20	σ1	σ1	PROPN
cana-2695	139	21	+	+	PROPN
cana-2695	139	22	)	)	PUNCT
cana-2695	139	23	,	,	PUNCT
cana-2695	139	24	(	(	PUNCT
cana-2695	139	25	σ2	σ2	NOUN
cana-2695	139	26	−	−	PROPN
cana-2695	139	27	,	,	PUNCT
cana-2695	139	28	σ2	σ2	NOUN
cana-2695	139	29	+	+	PROPN
cana-2695	139	30	)	)	PUNCT
cana-2695	139	31	,	,	PUNCT
cana-2695	139	32	.	.	PUNCT
cana-2695	139	33	.	.	PUNCT
cana-2695	139	34	.	.	PUNCT
cana-2695	140	1	,	,	PUNCT
cana-2695	140	2	(	(	PUNCT
cana-2695	140	3	σk1	σk1	NOUN
cana-2695	140	4	−	−	PROPN
cana-2695	140	5	,	,	PUNCT
cana-2695	140	6	σk1	σk1	NOUN
cana-2695	140	7	+	+	CCONJ
cana-2695	140	8	)	)	PUNCT
cana-2695	140	9	}	}	PUNCT
cana-2695	140	10	and	and	CCONJ
cana-2695	140	11	│	│	NOUN
cana-2695	140	12	ℋ1	ℋ1	NOUN
cana-2695	140	13	│	│	X
cana-2695	140	14	=	=	SYM
cana-2695	140	15	k	k	X
cana-2695	140	16	>	>	PUNCT
cana-2695	140	17	│	│	PUNCT
cana-2695	140	18	r	r	NOUN
cana-2695	140	19	│	│	NOUN
cana-2695	140	20	=	=	SYM
cana-2695	140	21	k1	k1	NOUN
cana-2695	140	22	,	,	PUNCT
cana-2695	140	23	then	then	ADV
cana-2695	140	24	𝜎𝑘+𝑖	𝜎𝑘+𝑖	NUM
cana-2695	140	25	−	−	NOUN
cana-2695	140	26	/𝑅	/𝑅	PUNCT
cana-2695	140	27	and	and	CCONJ
cana-2695	140	28	𝜎𝑘+𝑖	𝜎𝑘+𝑖	X
cana-2695	141	1	+	+	CCONJ
cana-2695	141	2	/r	/r	VERB
cana-2695	141	3	are	be	AUX
cana-2695	141	4	all	all	ADV
cana-2695	141	5	distinct	distinct	ADJ
cana-2695	141	6	for	for	ADP
cana-2695	141	7	n	n	NOUN
cana-2695	141	8	=	=	SYM
cana-2695	141	9	1	1	NUM
cana-2695	141	10	,	,	PUNCT
cana-2695	141	11	2	2	NUM
cana-2695	141	12	,	,	PUNCT
cana-2695	141	13	.	.	PUNCT
cana-2695	141	14	.	.	PUNCT
cana-2695	142	1	.	.	PUNCT
cana-2695	143	1	,	,	PUNCT
cana-2695	143	2	n	n	CCONJ
cana-2695	143	3	−	−	PROPN
cana-2695	143	4	k1	k1	NOUN
cana-2695	143	5	.	.	PUNCT
cana-2695	144	1	let	let	VERB
cana-2695	144	2	r1	r1	PROPN
cana-2695	144	3	=	=	SYM
cana-2695	144	4	{	{	PUNCT
cana-2695	144	5	(	(	PUNCT
cana-2695	144	6	σ1	σ1	PROPN
cana-2695	144	7	−	−	PROPN
cana-2695	144	8	,	,	PUNCT
cana-2695	144	9	σ1	σ1	PROPN
cana-2695	144	10	+	+	PROPN
cana-2695	144	11	)	)	PUNCT
cana-2695	144	12	,	,	PUNCT
cana-2695	144	13	(	(	PUNCT
cana-2695	144	14	σ2	σ2	NOUN
cana-2695	144	15	−	−	PROPN
cana-2695	144	16	,	,	PUNCT
cana-2695	144	17	σ2	σ2	NOUN
cana-2695	144	18	+	+	PROPN
cana-2695	144	19	)	)	PUNCT
cana-2695	144	20	,	,	PUNCT
cana-2695	144	21	.	.	PUNCT
cana-2695	144	22	.	.	PUNCT
cana-2695	145	1	.	.	PUNCT
cana-2695	146	1	,	,	PUNCT
cana-2695	146	2	(	(	PUNCT
cana-2695	146	3	σk1	σk1	NOUN
cana-2695	146	4	−	−	PROPN
cana-2695	146	5	,	,	PUNCT
cana-2695	146	6	σk1	σk1	NOUN
cana-2695	146	7	+	+	CCONJ
cana-2695	146	8	)	)	PUNCT
cana-2695	146	9	}	}	PUNCT
cana-2695	146	10	,	,	PUNCT
cana-2695	146	11	𝜎𝑘+𝑖	𝜎𝑘+𝑖	X
cana-2695	147	1	−	−	NOUN
cana-2695	147	2	/r1	/r1	PUNCT
cana-2695	148	1	=	=	PUNCT
cana-2695	148	2	w−	w−	NOUN
cana-2695	148	3	(	(	PUNCT
cana-2695	148	4	rk+i	rk+i	PROPN
cana-2695	148	5	,	,	PUNCT
cana-2695	148	6	r1	r1	PROPN
cana-2695	148	7	)	)	PUNCT
cana-2695	148	8	,	,	PUNCT
cana-2695	148	9	w	w	PROPN
cana-2695	148	10	−	−	PROPN
cana-2695	148	11	(	(	PUNCT
cana-2695	148	12	rk+i	rk+i	PROPN
cana-2695	148	13	,	,	PUNCT
cana-2695	148	14	r2	r2	PROPN
cana-2695	148	15	)	)	PUNCT
cana-2695	148	16	,	,	PUNCT
cana-2695	148	17	.	.	PUNCT
cana-2695	148	18	.	.	PUNCT
cana-2695	148	19	.	.	PUNCT
cana-2695	149	1	,	,	PUNCT
cana-2695	149	2	w	w	ADP
cana-2695	149	3	−	−	PROPN
cana-2695	149	4	(	(	PUNCT
cana-2695	149	5	rk+i	rk+i	PROPN
cana-2695	149	6	,	,	PUNCT
cana-2695	149	7	rk1	rk1	NOUN
cana-2695	149	8	)	)	PUNCT
cana-2695	150	1	=	=	PRON
cana-2695	150	2	{	{	PUNCT
cana-2695	150	3	(	(	PUNCT
cana-2695	150	4	μ−)∞(rk+i	μ−)∞(rk+i	NUM
cana-2695	150	5	,	,	PUNCT
cana-2695	150	6	r1	r1	PROPN
cana-2695	150	7	)	)	PUNCT
cana-2695	150	8	,	,	PUNCT
cana-2695	150	9	(	(	PUNCT
cana-2695	150	10	μ−)∞(rk+i	μ−)∞(rk+i	NOUN
cana-2695	150	11	,	,	PUNCT
cana-2695	150	12	r2	r2	PROPN
cana-2695	150	13	)	)	PUNCT
cana-2695	150	14	,	,	PUNCT
cana-2695	150	15	.	.	PUNCT
cana-2695	150	16	..	..	PUNCT
cana-2695	151	1	(	(	PUNCT
cana-2695	151	2	μ−)∞(rk+i	μ−)∞(rk+i	NUM
cana-2695	151	3	,	,	PUNCT
cana-2695	151	4	rk1	rk1	NOUN
cana-2695	151	5	)	)	PUNCT
cana-2695	151	6	}	}	PUNCT
cana-2695	151	7	=	=	PRON
cana-2695	151	8	{	{	PUNCT
cana-2695	151	9	(	(	PUNCT
cana-2695	151	10	(	(	PUNCT
cana-2695	151	11	μ−′)∞(φ(r	μ−′)∞(φ(r	NOUN
cana-2695	151	12	k+i	k+i	NUM
cana-2695	151	13	)	)	PUNCT
cana-2695	151	14	,	,	PUNCT
cana-2695	151	15	(	(	PUNCT
cana-2695	151	16	μ−′)∞(φ(r	μ−′)∞(φ(r	NOUN
cana-2695	151	17	1	1	NUM
cana-2695	151	18	)	)	PUNCT
cana-2695	151	19	)	)	PUNCT
cana-2695	151	20	,	,	PUNCT
cana-2695	151	21	(	(	PUNCT
cana-2695	151	22	(	(	PUNCT
cana-2695	151	23	μ−′)∞(φ(r	μ−′)∞(φ(r	NOUN
cana-2695	151	24	k+i	k+i	NUM
cana-2695	151	25	)	)	PUNCT
cana-2695	151	26	,	,	PUNCT
cana-2695	151	27	(	(	PUNCT
cana-2695	151	28	μ−′)∞(φ(r	μ−′)∞(φ(r	NOUN
cana-2695	151	29	2	2	NUM
cana-2695	151	30	)	)	PUNCT
cana-2695	151	31	)	)	PUNCT
cana-2695	151	32	,	,	PUNCT
cana-2695	151	33	.	.	PUNCT
cana-2695	151	34	..	..	PUNCT
cana-2695	152	1	(	(	PUNCT
cana-2695	152	2	(	(	PUNCT
cana-2695	152	3	μ−′)∞(φ(r	μ−′)∞(φ(r	NOUN
cana-2695	152	4	k+i	k+i	NUM
cana-2695	152	5	)	)	PUNCT
cana-2695	152	6	,	,	PUNCT
cana-2695	152	7	(	(	PUNCT
cana-2695	152	8	μ−′)∞(φ(r	μ−′)∞(φ(r	ADJ
cana-2695	152	9	k1	k1	NOUN
cana-2695	152	10	)	)	PUNCT
cana-2695	152	11	)	)	PUNCT
cana-2695	152	12	similarly	similarly	ADV
cana-2695	152	13	,	,	PUNCT
cana-2695	152	14	communications	communication	NOUN
cana-2695	152	15	on	on	ADP
cana-2695	152	16	applied	apply	VERB
cana-2695	152	17	nonlinear	nonlinear	ADJ
cana-2695	152	18	analysis	analysis	NOUN
cana-2695	152	19	issn	issn	NOUN
cana-2695	152	20	:	:	PUNCT
cana-2695	152	21	1074	1074	NUM
cana-2695	152	22	-	-	PUNCT
cana-2695	152	23	133x	133x	NUM
cana-2695	152	24	vol	vol	NOUN
cana-2695	152	25	32	32	NUM
cana-2695	152	26	no	no	NOUN
cana-2695	152	27	.	.	PUNCT
cana-2695	153	1	3s	3s	NUM
cana-2695	153	2	(	(	PUNCT
cana-2695	153	3	2025	2025	NUM
cana-2695	153	4	)	)	PUNCT
cana-2695	153	5	552	552	NUM
cana-2695	153	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2695	153	7	𝜎𝑘+𝑖	𝜎𝑘+𝑖	PUNCT
cana-2695	154	1	+	+	NUM
cana-2695	154	2	/r1	/r1	SYM
cana-2695	154	3	=	=	SYM
cana-2695	154	4	w+	w+	X
cana-2695	154	5	(	(	PUNCT
cana-2695	154	6	rk+i	rk+i	PROPN
cana-2695	154	7	,	,	PUNCT
cana-2695	154	8	r1	r1	PROPN
cana-2695	154	9	)	)	PUNCT
cana-2695	154	10	,	,	PUNCT
cana-2695	154	11	w	w	ADP
cana-2695	154	12	+	+	CCONJ
cana-2695	154	13	(	(	PUNCT
cana-2695	154	14	rk+i	rk+i	PROPN
cana-2695	154	15	,	,	PUNCT
cana-2695	154	16	r2	r2	PROPN
cana-2695	154	17	)	)	PUNCT
cana-2695	154	18	,	,	PUNCT
cana-2695	154	19	.	.	PUNCT
cana-2695	154	20	.	.	PUNCT
cana-2695	154	21	.	.	PUNCT
cana-2695	155	1	,	,	PUNCT
cana-2695	155	2	w	w	X
cana-2695	155	3	+	+	CCONJ
cana-2695	155	4	(	(	PUNCT
cana-2695	155	5	rk+i	rk+i	PROPN
cana-2695	155	6	,	,	PUNCT
cana-2695	155	7	rk1	rk1	NOUN
cana-2695	155	8	)	)	PUNCT
cana-2695	156	1	=	=	PRON
cana-2695	156	2	{	{	PUNCT
cana-2695	156	3	(	(	PUNCT
cana-2695	156	4	μ+)∞(rk+i	μ+)∞(rk+i	NOUN
cana-2695	156	5	,	,	PUNCT
cana-2695	156	6	r1	r1	PROPN
cana-2695	156	7	)	)	PUNCT
cana-2695	156	8	,	,	PUNCT
cana-2695	156	9	(	(	PUNCT
cana-2695	156	10	μ+)∞(rk+i	μ+)∞(rk+i	NOUN
cana-2695	156	11	,	,	PUNCT
cana-2695	156	12	r2	r2	PROPN
cana-2695	156	13	)	)	PUNCT
cana-2695	156	14	,	,	PUNCT
cana-2695	156	15	.	.	PUNCT
cana-2695	157	1	..	..	PUNCT
cana-2695	157	2	(	(	PUNCT
cana-2695	157	3	μ+)∞(rk+i	μ+)∞(rk+i	NOUN
cana-2695	157	4	,	,	PUNCT
cana-2695	157	5	rk1	rk1	NOUN
cana-2695	157	6	)	)	PUNCT
cana-2695	157	7	}	}	PUNCT
cana-2695	157	8	=	=	PRON
cana-2695	157	9	{	{	PUNCT
cana-2695	157	10	(	(	PUNCT
cana-2695	157	11	(	(	PUNCT
cana-2695	157	12	μ+′)∞(φ(r	μ+′)∞(φ(r	NOUN
cana-2695	157	13	k+i	k+i	NUM
cana-2695	157	14	)	)	PUNCT
cana-2695	157	15	,	,	PUNCT
cana-2695	157	16	(	(	PUNCT
cana-2695	157	17	μ+′)∞(φ(r	μ+′)∞(φ(r	NOUN
cana-2695	157	18	1	1	NUM
cana-2695	157	19	)	)	PUNCT
cana-2695	157	20	)	)	PUNCT
cana-2695	157	21	,	,	PUNCT
cana-2695	157	22	(	(	PUNCT
cana-2695	157	23	(	(	PUNCT
cana-2695	157	24	μ+′)∞(φ(r	μ+′)∞(φ(r	NOUN
cana-2695	157	25	k+i	k+i	NUM
cana-2695	157	26	)	)	PUNCT
cana-2695	157	27	,	,	PUNCT
cana-2695	157	28	(	(	PUNCT
cana-2695	157	29	μ+′)∞(φ(r	μ+′)∞(φ(r	NOUN
cana-2695	157	30	2	2	NUM
cana-2695	157	31	)	)	PUNCT
cana-2695	157	32	)	)	PUNCT
cana-2695	157	33	,	,	PUNCT
cana-2695	157	34	.	.	PUNCT
cana-2695	157	35	..	..	PUNCT
cana-2695	158	1	(	(	PUNCT
cana-2695	158	2	(	(	PUNCT
cana-2695	158	3	μ+′)∞(φ(r	μ+′)∞(φ(r	NOUN
cana-2695	158	4	k+i	k+i	NUM
cana-2695	158	5	)	)	PUNCT
cana-2695	158	6	,	,	PUNCT
cana-2695	158	7	(	(	PUNCT
cana-2695	158	8	μ+′)∞(φ(r	μ+′)∞(φ(r	NOUN
cana-2695	158	9	k1	k1	NOUN
cana-2695	158	10	)	)	PUNCT
cana-2695	158	11	)	)	PUNCT
cana-2695	158	12	which	which	PRON
cana-2695	158	13	are	be	AUX
cana-2695	158	14	all	all	ADV
cana-2695	158	15	distinct	distinct	ADJ
cana-2695	158	16	for	for	ADP
cana-2695	158	17	i	i	X
cana-2695	158	18	=	=	NOUN
cana-2695	158	19	1,2	1,2	NUM
cana-2695	158	20	,	,	PUNCT
cana-2695	158	21	.	.	PUNCT
cana-2695	158	22	.	.	PUNCT
cana-2695	158	23	.	.	PUNCT
cana-2695	159	1	n	n	PRON
cana-2695	159	2	−	−	PROPN
cana-2695	159	3	k.	k.	PROPN
cana-2695	160	1	therefore	therefore	ADV
cana-2695	160	2	,	,	PUNCT
cana-2695	160	3	r1	r1	PROPN
cana-2695	160	4	is	be	AUX
cana-2695	160	5	the	the	DET
cana-2695	160	6	ivfrs	ivfrs	NOUN
cana-2695	160	7	of	of	ADP
cana-2695	160	8	g	g	PROPN
cana-2695	160	9	and	and	CCONJ
cana-2695	160	10	ℐr(g)=𝑘1	ℐr(g)=𝑘1	PROPN
cana-2695	161	1	[	[	X
cana-2695	161	2	𝑘1	𝑘1	NOUN
cana-2695	161	3	<	<	X
cana-2695	161	4	k	k	X
cana-2695	161	5	]	]	X
cana-2695	161	6	,	,	PUNCT
cana-2695	161	7	which	which	PRON
cana-2695	161	8	is	be	AUX
cana-2695	161	9	a	a	DET
cana-2695	161	10	contradiction	contradiction	NOUN
cana-2695	161	11	to	to	ADP
cana-2695	161	12	our	our	PRON
cana-2695	161	13	assumption	assumption	NOUN
cana-2695	161	14	that	that	SCONJ
cana-2695	161	15	ℐr(g	ℐr(g	ADV
cana-2695	161	16	)	)	PUNCT
cana-2695	161	17	=	=	SYM
cana-2695	161	18	k.	k.	PROPN
cana-2695	161	19	hence	hence	ADV
cana-2695	161	20	,	,	PUNCT
cana-2695	161	21	ℋ1	ℋ1	PROPN
cana-2695	161	22	is	be	AUX
cana-2695	161	23	the	the	DET
cana-2695	161	24	minimum	minimum	ADJ
cana-2695	161	25	ivrs	ivrs	NOUN
cana-2695	161	26	of	of	ADP
cana-2695	161	27	g′.	g′.	PROPN
cana-2695	161	28	therefore,ℐr(g	therefore,ℐr(g	PROPN
cana-2695	161	29	)	)	PUNCT
cana-2695	161	30	≅	≅	PROPN
cana-2695	161	31	ℐr(g′	ℐr(g′	X
cana-2695	161	32	)	)	PUNCT
cana-2695	161	33	.	.	PUNCT
cana-2695	162	1	3.4	3.4	NUM
cana-2695	162	2	theorem	theorem	VERB
cana-2695	162	3	if	if	SCONJ
cana-2695	162	4	g	g	PROPN
cana-2695	162	5	and	and	CCONJ
cana-2695	162	6	g′	g′	NOUN
cana-2695	162	7	are	be	AUX
cana-2695	162	8	homomorphic	homomorphic	ADJ
cana-2695	162	9	functions	function	NOUN
cana-2695	162	10	,	,	PUNCT
cana-2695	162	11	then	then	ADV
cana-2695	162	12	ℐr(g	ℐr(g	ADV
cana-2695	162	13	)	)	PUNCT
cana-2695	162	14	and	and	CCONJ
cana-2695	162	15	ℐr(g′	ℐr(g′	PRON
cana-2695	162	16	)	)	PUNCT
cana-2695	162	17	are	be	AUX
cana-2695	162	18	homomorphic	homomorphic	ADJ
cana-2695	162	19	.	.	PUNCT
cana-2695	163	1	proof	proof	NOUN
cana-2695	163	2	:	:	PUNCT
cana-2695	163	3	let	let	VERB
cana-2695	163	4	us	we	PRON
cana-2695	163	5	assume	assume	VERB
cana-2695	163	6	that	that	SCONJ
cana-2695	163	7	v	v	X
cana-2695	163	8	=	=	SYM
cana-2695	163	9	{	{	PUNCT
cana-2695	163	10	r1	r1	PROPN
cana-2695	163	11	,	,	PUNCT
cana-2695	163	12	r2	r2	PROPN
cana-2695	163	13	,	,	PUNCT
cana-2695	163	14	.	.	PUNCT
cana-2695	163	15	.	.	PUNCT
cana-2695	163	16	.	.	PUNCT
cana-2695	164	1	,	,	PUNCT
cana-2695	164	2	rn	rn	PROPN
cana-2695	164	3	}	}	PUNCT
cana-2695	164	4	,	,	PUNCT
cana-2695	164	5	ℐr(g)=	ℐr(g)=	NOUN
cana-2695	164	6	k	k	NOUN
cana-2695	164	7	,	,	PUNCT
cana-2695	164	8	and	and	CCONJ
cana-2695	164	9	let	let	VERB
cana-2695	164	10	ℋ	ℋ	PROPN
cana-2695	164	11	=	=	SYM
cana-2695	164	12	{	{	PUNCT
cana-2695	164	13	(	(	PUNCT
cana-2695	164	14	σ1	σ1	PROPN
cana-2695	164	15	−	−	PROPN
cana-2695	164	16	,	,	PUNCT
cana-2695	164	17	σ1	σ1	PROPN
cana-2695	164	18	+	+	PROPN
cana-2695	164	19	)	)	PUNCT
cana-2695	164	20	,	,	PUNCT
cana-2695	164	21	(	(	PUNCT
cana-2695	164	22	σ2	σ2	NOUN
cana-2695	164	23	−	−	PROPN
cana-2695	164	24	,	,	PUNCT
cana-2695	164	25	σ2	σ2	NOUN
cana-2695	164	26	+	+	PROPN
cana-2695	164	27	)	)	PUNCT
cana-2695	164	28	,	,	PUNCT
cana-2695	164	29	.	.	PUNCT
cana-2695	164	30	.	.	PUNCT
cana-2695	165	1	.	.	PUNCT
cana-2695	166	1	,	,	PUNCT
cana-2695	166	2	(	(	PUNCT
cana-2695	166	3	σk	σk	ADP
cana-2695	166	4	−	−	NOUN
cana-2695	166	5	,	,	PUNCT
cana-2695	166	6	σk	σk	X
cana-2695	166	7	+	+	PUNCT
cana-2695	166	8	)	)	PUNCT
cana-2695	166	9	is	be	AUX
cana-2695	166	10	the	the	DET
cana-2695	166	11	corresponding	corresponding	ADJ
cana-2695	166	12	frs	frs	PROPN
cana-2695	166	13	.	.	PUNCT
cana-2695	167	1	we	we	PRON
cana-2695	167	2	denote	denote	VERB
cana-2695	167	3	(	(	PUNCT
cana-2695	167	4	r1	r1	PROPN
cana-2695	167	5	,	,	PUNCT
cana-2695	167	6	σ	σ	PROPN
cana-2695	167	7	−(r1	−(r1	NUM
cana-2695	167	8	)	)	PUNCT
cana-2695	167	9	)	)	PUNCT
cana-2695	168	1	=	=	SYM
cana-2695	168	2	σ1	σ1	PROPN
cana-2695	168	3	−	−	PROPN
cana-2695	168	4	,	,	PUNCT
cana-2695	168	5	(	(	PUNCT
cana-2695	168	6	r1	r1	PROPN
cana-2695	168	7	,	,	PUNCT
cana-2695	168	8	σ	σ	PROPN
cana-2695	168	9	+	+	PROPN
cana-2695	168	10	(	(	PUNCT
cana-2695	168	11	r1	r1	NOUN
cana-2695	168	12	)	)	PUNCT
cana-2695	168	13	)	)	PUNCT
cana-2695	169	1	=	=	SYM
cana-2695	169	2	σ1	σ1	PROPN
cana-2695	169	3	+	+	X
cana-2695	169	4	,	,	PUNCT
cana-2695	169	5	(	(	PUNCT
cana-2695	169	6	φ(r1	φ(r1	ADV
cana-2695	169	7	)	)	PUNCT
cana-2695	169	8	,	,	PUNCT
cana-2695	169	9	σ	σ	PROPN
cana-2695	169	10	−(φ(r	−(φ(r	PROPN
cana-2695	169	11	1	1	NUM
cana-2695	169	12	)	)	PUNCT
cana-2695	169	13	)	)	PUNCT
cana-2695	170	1	=	=	SYM
cana-2695	170	2	σ1	σ1	NOUN
cana-2695	170	3	−′	−′	NOUN
cana-2695	170	4	,	,	PUNCT
cana-2695	170	5	(	(	PUNCT
cana-2695	170	6	φ(r1	φ(r1	ADV
cana-2695	170	7	)	)	PUNCT
cana-2695	170	8	,	,	PUNCT
cana-2695	170	9	σ	σ	PROPN
cana-2695	171	1	+	+	PROPN
cana-2695	171	2	(	(	PUNCT
cana-2695	171	3	φ(r	φ(r	NOUN
cana-2695	171	4	1	1	NUM
cana-2695	171	5	)	)	PUNCT
cana-2695	171	6	)	)	PUNCT
cana-2695	172	1	=	=	SYM
cana-2695	172	2	σ1	σ1	NOUN
cana-2695	172	3	+	+	ADP
cana-2695	172	4	′.	′.	NOUN
cana-2695	172	5	the	the	DET
cana-2695	172	6	representation	representation	NOUN
cana-2695	172	7	,	,	PUNCT
cana-2695	172	8	𝜎𝑘+𝑖	𝜎𝑘+𝑖	NUM
cana-2695	172	9	−	−	NOUN
cana-2695	172	10	/ℋ	/ℋ	PUNCT
cana-2695	172	11	=(	=(	ADJ
cana-2695	172	12	w−	w−	PROPN
cana-2695	172	13	(	(	PUNCT
cana-2695	172	14	rk+i	rk+i	PROPN
cana-2695	172	15	,	,	PUNCT
cana-2695	172	16	r1	r1	PROPN
cana-2695	172	17	)	)	PUNCT
cana-2695	172	18	,	,	PUNCT
cana-2695	172	19	w	w	PROPN
cana-2695	172	20	−	−	PROPN
cana-2695	172	21	(	(	PUNCT
cana-2695	172	22	rk+i	rk+i	PROPN
cana-2695	172	23	,	,	PUNCT
cana-2695	172	24	r2	r2	PROPN
cana-2695	172	25	)	)	PUNCT
cana-2695	172	26	,	,	PUNCT
cana-2695	172	27	.	.	PUNCT
cana-2695	172	28	.	.	PUNCT
cana-2695	172	29	.	.	PUNCT
cana-2695	173	1	,	,	PUNCT
cana-2695	173	2	w	w	ADP
cana-2695	173	3	−	−	PROPN
cana-2695	173	4	(	(	PUNCT
cana-2695	173	5	rk+i	rk+i	PROPN
cana-2695	173	6	,	,	PUNCT
cana-2695	173	7	rk	rk	NOUN
cana-2695	173	8	)	)	PUNCT
cana-2695	173	9	,	,	PUNCT
cana-2695	173	10	𝜎𝑘+𝑖	𝜎𝑘+𝑖	X
cana-2695	174	1	+	+	CCONJ
cana-2695	174	2	/ℋ	/ℋ	SYM
cana-2695	174	3	=(	=(	NOUN
cana-2695	174	4	w+	w+	X
cana-2695	174	5	(	(	PUNCT
cana-2695	174	6	rk+i	rk+i	PROPN
cana-2695	174	7	,	,	PUNCT
cana-2695	174	8	r1	r1	PROPN
cana-2695	174	9	)	)	PUNCT
cana-2695	174	10	,	,	PUNCT
cana-2695	174	11	w	w	ADP
cana-2695	174	12	+	+	CCONJ
cana-2695	174	13	(	(	PUNCT
cana-2695	174	14	rk+i	rk+i	PROPN
cana-2695	174	15	,	,	PUNCT
cana-2695	174	16	r2	r2	PROPN
cana-2695	174	17	)	)	PUNCT
cana-2695	174	18	,	,	PUNCT
cana-2695	174	19	.	.	PUNCT
cana-2695	174	20	.	.	PUNCT
cana-2695	174	21	.	.	PUNCT
cana-2695	175	1	,	,	PUNCT
cana-2695	175	2	w	w	X
cana-2695	175	3	+	+	CCONJ
cana-2695	175	4	(	(	PUNCT
cana-2695	175	5	rk+i	rk+i	PROPN
cana-2695	175	6	,	,	PUNCT
cana-2695	175	7	rk	rk	NOUN
cana-2695	175	8	)	)	PUNCT
cana-2695	175	9	all	all	PRON
cana-2695	175	10	are	be	AUX
cana-2695	175	11	distinct	distinct	ADJ
cana-2695	175	12	,	,	PUNCT
cana-2695	175	13	∀	∀	VERB
cana-2695	175	14	i	i	NOUN
cana-2695	175	15	=	=	NOUN
cana-2695	175	16	1,2	1,2	NUM
cana-2695	175	17	,	,	PUNCT
cana-2695	175	18	.	.	PUNCT
cana-2695	175	19	.	.	PUNCT
cana-2695	176	1	.	.	PUNCT
cana-2695	177	1	n	n	PRON
cana-2695	177	2	−	−	PROPN
cana-2695	177	3	k.	k.	NOUN
cana-2695	178	1	if	if	SCONJ
cana-2695	178	2	g	g	PROPN
cana-2695	178	3	and	and	CCONJ
cana-2695	178	4	g′	g′	NOUN
cana-2695	178	5	are	be	AUX
cana-2695	178	6	homomorphic	homomorphic	ADJ
cana-2695	178	7	,	,	PUNCT
cana-2695	178	8	then	then	ADV
cana-2695	178	9	∃	∃	PROPN
cana-2695	178	10	a	a	DET
cana-2695	178	11	bijection	bijection	NOUN
cana-2695	178	12	φ	φ	NOUN
cana-2695	178	13	:	:	PUNCT
cana-2695	178	14	v	v	PROPN
cana-2695	178	15	→	→	SYM
cana-2695	178	16	v′	v′	NUM
cana-2695	178	17	which	which	PRON
cana-2695	178	18	satisfies	satisfy	VERB
cana-2695	178	19	,	,	PUNCT
cana-2695	178	20	μ	μ	PROPN
cana-2695	178	21	e	e	NOUN
cana-2695	178	22	−	−	PROPN
cana-2695	178	23	(	(	PUNCT
cana-2695	178	24	r	r	NOUN
cana-2695	178	25	)	)	PUNCT
cana-2695	178	26	≤	≤	NOUN
cana-2695	178	27	μ	μ	PROPN
cana-2695	178	28	e	e	NOUN
cana-2695	178	29	−′(φ	−′(φ	ADJ
cana-2695	178	30	(	(	PUNCT
cana-2695	178	31	r	r	NOUN
cana-2695	178	32	)	)	PUNCT
cana-2695	178	33	)	)	PUNCT
cana-2695	178	34	,	,	PUNCT
cana-2695	178	35	μ	μ	PROPN
cana-2695	178	36	e	e	X
cana-2695	178	37	+	+	CCONJ
cana-2695	178	38	(	(	PUNCT
cana-2695	178	39	r	r	NOUN
cana-2695	178	40	)	)	PUNCT
cana-2695	178	41	≤	≤	NOUN
cana-2695	178	42	μ	μ	NOUN
cana-2695	178	43	e	e	X
cana-2695	178	44	+	+	NOUN
cana-2695	178	45	′	′	NUM
cana-2695	178	46	(	(	PUNCT
cana-2695	178	47	φ(r	φ(r	NOUN
cana-2695	178	48	)	)	PUNCT
cana-2695	178	49	)	)	PUNCT
cana-2695	178	50	,	,	PUNCT
cana-2695	178	51	μ	μ	PROPN
cana-2695	178	52	f	f	X
cana-2695	179	1	−	−	PROPN
cana-2695	179	2	(	(	PUNCT
cana-2695	179	3	r	r	NOUN
cana-2695	179	4	,	,	PUNCT
cana-2695	179	5	u	u	NOUN
cana-2695	179	6	)	)	PUNCT
cana-2695	179	7	≤	≤	NOUN
cana-2695	180	1	μ	μ	PROPN
cana-2695	180	2	f	f	PROPN
cana-2695	181	1	−′	−′	PROPN
cana-2695	181	2	(	(	PUNCT
cana-2695	181	3	φ(r	φ(r	NOUN
cana-2695	181	4	)	)	PUNCT
cana-2695	181	5	,	,	PUNCT
cana-2695	181	6	φ(u	φ(u	NOUN
cana-2695	181	7	)	)	PUNCT
cana-2695	181	8	)	)	PUNCT
cana-2695	181	9	and	and	CCONJ
cana-2695	181	10	μ	μ	NOUN
cana-2695	181	11	f	f	PROPN
cana-2695	182	1	+	+	CCONJ
cana-2695	182	2	(	(	PUNCT
cana-2695	182	3	r	r	NOUN
cana-2695	182	4	,	,	PUNCT
cana-2695	182	5	u	u	NOUN
cana-2695	182	6	)	)	PUNCT
cana-2695	182	7	≤	≤	NOUN
cana-2695	182	8	μ	μ	NUM
cana-2695	182	9	f	f	PROPN
cana-2695	183	1	+	+	NOUN
cana-2695	183	2	′	′	NUM
cana-2695	183	3	(	(	PUNCT
cana-2695	183	4	φ(r	φ(r	NOUN
cana-2695	183	5	)	)	PUNCT
cana-2695	183	6	,	,	PUNCT
cana-2695	183	7	φ(u	φ(u	NOUN
cana-2695	183	8	)	)	PUNCT
cana-2695	183	9	)	)	PUNCT
cana-2695	184	1	∀r	∀r	NOUN
cana-2695	184	2	,	,	PUNCT
cana-2695	184	3	u	u	PROPN
cana-2695	184	4	∈	∈	PROPN
cana-2695	184	5	e.	e.	PROPN
cana-2695	184	6	now	now	ADV
cana-2695	184	7	,	,	PUNCT
cana-2695	184	8	we	we	PRON
cana-2695	184	9	define	define	VERB
cana-2695	184	10	ℋ1	ℋ1	NOUN
cana-2695	184	11	=	=	PUNCT
cana-2695	184	12	{	{	PUNCT
cana-2695	184	13	(	(	PUNCT
cana-2695	184	14	σ1	σ1	PROPN
cana-2695	184	15	−′	−′	PROPN
cana-2695	184	16	,	,	PUNCT
cana-2695	184	17	σ1	σ1	PROPN
cana-2695	184	18	+	+	PROPN
cana-2695	184	19	′	′	NUM
cana-2695	184	20	)	)	PUNCT
cana-2695	184	21	,	,	PUNCT
cana-2695	184	22	(	(	PUNCT
cana-2695	184	23	σ2	σ2	PROPN
cana-2695	184	24	−′	−′	PROPN
cana-2695	184	25	,	,	PUNCT
cana-2695	184	26	σ2	σ2	PROPN
cana-2695	184	27	+	+	PROPN
cana-2695	184	28	′	′	NOUN
cana-2695	184	29	)	)	PUNCT
cana-2695	184	30	,	,	PUNCT
cana-2695	184	31	.	.	PUNCT
cana-2695	184	32	.	.	PUNCT
cana-2695	184	33	.	.	PUNCT
cana-2695	185	1	,	,	PUNCT
cana-2695	185	2	(	(	PUNCT
cana-2695	185	3	σk	σk	ADP
cana-2695	185	4	−′	−′	NUM
cana-2695	185	5	,	,	PUNCT
cana-2695	185	6	σk	σk	ADV
cana-2695	185	7	+	+	NOUN
cana-2695	185	8	′	′	NUM
cana-2695	185	9	)	)	PUNCT
cana-2695	185	10	}	}	PUNCT
cana-2695	185	11	for	for	ADP
cana-2695	185	12	i	i	PROPN
cana-2695	185	13	=	=	SYM
cana-2695	185	14	1,2	1,2	NUM
cana-2695	185	15	,	,	PUNCT
cana-2695	185	16	.	.	PUNCT
cana-2695	185	17	.	.	PUNCT
cana-2695	185	18	.	.	PUNCT
cana-2695	186	1	n	n	CCONJ
cana-2695	187	1	−	−	PROPN
cana-2695	187	2	k	k	PROPN
cana-2695	187	3	,	,	PUNCT
cana-2695	187	4	𝜎𝑘+𝑖	𝜎𝑘+𝑖	X
cana-2695	188	1	−′	−′	X
cana-2695	188	2	/ℋ	/ℋ	PUNCT
cana-2695	189	1	=	=	PUNCT
cana-2695	189	2	{	{	PUNCT
cana-2695	189	3	w−	w−	NOUN
cana-2695	189	4	(	(	PUNCT
cana-2695	189	5	φ(r	φ(r	PROPN
cana-2695	189	6	k+i	k+i	NUM
cana-2695	189	7	)	)	PUNCT
cana-2695	189	8	,	,	PUNCT
cana-2695	189	9	φ(r	φ(r	NOUN
cana-2695	189	10	1	1	NUM
cana-2695	189	11	)	)	PUNCT
cana-2695	189	12	)	)	PUNCT
cana-2695	189	13	,	,	PUNCT
cana-2695	189	14	w−	w−	X
cana-2695	189	15	(	(	PUNCT
cana-2695	189	16	φ(r	φ(r	PROPN
cana-2695	189	17	k+i	k+i	NUM
cana-2695	189	18	)	)	PUNCT
cana-2695	189	19	,	,	PUNCT
cana-2695	189	20	φ(r	φ(r	PROPN
cana-2695	189	21	2	2	NUM
cana-2695	189	22	)	)	PUNCT
cana-2695	189	23	)	)	PUNCT
cana-2695	189	24	,	,	PUNCT
cana-2695	189	25	.	.	PUNCT
cana-2695	189	26	.	.	PUNCT
cana-2695	189	27	.	.	PUNCT
cana-2695	190	1	,	,	PUNCT
cana-2695	190	2	w−	w−	X
cana-2695	190	3	(	(	PUNCT
cana-2695	190	4	φ(r	φ(r	PROPN
cana-2695	190	5	k+i	k+i	NUM
cana-2695	190	6	)	)	PUNCT
cana-2695	190	7	,	,	PUNCT
cana-2695	190	8	φ(r	φ(r	PROPN
cana-2695	190	9	k	k	NOUN
cana-2695	190	10	)	)	PUNCT
cana-2695	190	11	}	}	PUNCT
cana-2695	190	12	=	=	SYM
cana-2695	190	13	{	{	PUNCT
cana-2695	190	14	(	(	PUNCT
cana-2695	190	15	(	(	PUNCT
cana-2695	190	16	μ−′)∞(φ(r	μ−′)∞(φ(r	NOUN
cana-2695	190	17	k+i	k+i	NUM
cana-2695	190	18	)	)	PUNCT
cana-2695	190	19	,	,	PUNCT
cana-2695	190	20	(	(	PUNCT
cana-2695	190	21	μ−′)∞(φ(r	μ−′)∞(φ(r	NOUN
cana-2695	190	22	1	1	NUM
cana-2695	190	23	)	)	PUNCT
cana-2695	190	24	)	)	PUNCT
cana-2695	190	25	,	,	PUNCT
cana-2695	190	26	(	(	PUNCT
cana-2695	190	27	(	(	PUNCT
cana-2695	190	28	μ−′)∞(φ(r	μ−′)∞(φ(r	NOUN
cana-2695	190	29	k+i	k+i	NUM
cana-2695	190	30	)	)	PUNCT
cana-2695	190	31	,	,	PUNCT
cana-2695	190	32	(	(	PUNCT
cana-2695	190	33	μ−′)∞(φ(r	μ−′)∞(φ(r	NOUN
cana-2695	190	34	2	2	NUM
cana-2695	190	35	)	)	PUNCT
cana-2695	190	36	)	)	PUNCT
cana-2695	190	37	,	,	PUNCT
cana-2695	190	38	.	.	PUNCT
cana-2695	191	1	..	..	PUNCT
cana-2695	192	1	(	(	PUNCT
cana-2695	192	2	(	(	PUNCT
cana-2695	192	3	μ−′)∞(φ(r	μ−′)∞(φ(r	NOUN
cana-2695	192	4	k+i	k+i	NUM
cana-2695	192	5	)	)	PUNCT
cana-2695	192	6	,	,	PUNCT
cana-2695	192	7	(	(	PUNCT
cana-2695	192	8	μ−′)∞(φ(r	μ−′)∞(φ(r	NOUN
cana-2695	192	9	k	k	NOUN
cana-2695	192	10	)	)	PUNCT
cana-2695	192	11	)	)	PUNCT
cana-2695	192	12	≥	≥	NOUN
cana-2695	192	13	μ−(rk+i	μ−(rk+i	NOUN
cana-2695	192	14	,	,	PUNCT
cana-2695	192	15	r1	r1	PROPN
cana-2695	192	16	)	)	PUNCT
cana-2695	192	17	,	,	PUNCT
cana-2695	192	18	μ	μ	PROPN
cana-2695	192	19	−(rk+i	−(rk+i	NOUN
cana-2695	192	20	,	,	PUNCT
cana-2695	192	21	r2	r2	PROPN
cana-2695	192	22	)	)	PUNCT
cana-2695	192	23	,	,	PUNCT
cana-2695	192	24	.	.	PUNCT
cana-2695	192	25	.	.	PUNCT
cana-2695	192	26	.	.	PUNCT
cana-2695	193	1	,	,	PUNCT
cana-2695	193	2	μ	μ	PROPN
cana-2695	193	3	−(rk+i	−(rk+i	NOUN
cana-2695	193	4	,	,	PUNCT
cana-2695	193	5	rk	rk	NOUN
cana-2695	193	6	)	)	PUNCT
cana-2695	193	7	=	=	SYM
cana-2695	194	1	w−	w−	NOUN
cana-2695	194	2	(	(	PUNCT
cana-2695	194	3	rk+i	rk+i	PROPN
cana-2695	194	4	,	,	PUNCT
cana-2695	194	5	r1	r1	PROPN
cana-2695	194	6	)	)	PUNCT
cana-2695	194	7	,	,	PUNCT
cana-2695	194	8	w	w	PROPN
cana-2695	194	9	−	−	PROPN
cana-2695	194	10	(	(	PUNCT
cana-2695	194	11	rk+i	rk+i	PROPN
cana-2695	194	12	,	,	PUNCT
cana-2695	194	13	r2	r2	PROPN
cana-2695	194	14	)	)	PUNCT
cana-2695	194	15	,	,	PUNCT
cana-2695	194	16	.	.	PUNCT
cana-2695	194	17	.	.	PUNCT
cana-2695	194	18	.	.	PUNCT
cana-2695	195	1	,	,	PUNCT
cana-2695	195	2	w	w	ADP
cana-2695	195	3	−	−	PROPN
cana-2695	195	4	(	(	PUNCT
cana-2695	195	5	rk+i	rk+i	PROPN
cana-2695	195	6	,	,	PUNCT
cana-2695	195	7	rk	rk	NOUN
cana-2695	195	8	)	)	PUNCT
cana-2695	195	9	𝜎𝑘+𝑖	𝜎𝑘+𝑖	PUNCT
cana-2695	196	1	+	+	NOUN
cana-2695	196	2	′	′	NUM
cana-2695	196	3	/ℋ	/ℋ	PUNCT
cana-2695	197	1	=	=	PUNCT
cana-2695	197	2	{	{	PUNCT
cana-2695	197	3	w+	w+	X
cana-2695	197	4	(	(	PUNCT
cana-2695	197	5	φ(r	φ(r	PROPN
cana-2695	197	6	k+i	k+i	NUM
cana-2695	197	7	)	)	PUNCT
cana-2695	197	8	,	,	PUNCT
cana-2695	197	9	φ(r	φ(r	NOUN
cana-2695	197	10	1	1	NUM
cana-2695	197	11	)	)	PUNCT
cana-2695	197	12	)	)	PUNCT
cana-2695	197	13	,	,	PUNCT
cana-2695	197	14	w+(φ(r	w+(φ(r	NOUN
cana-2695	197	15	k+i	k+i	PROPN
cana-2695	197	16	)	)	PUNCT
cana-2695	197	17	,	,	PUNCT
cana-2695	197	18	φ(r	φ(r	PROPN
cana-2695	197	19	2	2	NUM
cana-2695	197	20	)	)	PUNCT
cana-2695	197	21	)	)	PUNCT
cana-2695	197	22	,	,	PUNCT
cana-2695	197	23	.	.	PUNCT
cana-2695	197	24	.	.	PUNCT
cana-2695	197	25	.	.	PUNCT
cana-2695	198	1	,	,	PUNCT
cana-2695	198	2	w−+(φ(r	w−+(φ(r	ADJ
cana-2695	198	3	k+i	k+i	NUM
cana-2695	198	4	)	)	PUNCT
cana-2695	198	5	,	,	PUNCT
cana-2695	198	6	φ(r	φ(r	PROPN
cana-2695	198	7	k	k	NOUN
cana-2695	198	8	)	)	PUNCT
cana-2695	198	9	}	}	PUNCT
cana-2695	199	1	=	=	SYM
cana-2695	199	2	{	{	PUNCT
cana-2695	199	3	(	(	PUNCT
cana-2695	199	4	(	(	PUNCT
cana-2695	199	5	μ+′)∞(φ(r	μ+′)∞(φ(r	NOUN
cana-2695	199	6	k+i	k+i	NUM
cana-2695	199	7	)	)	PUNCT
cana-2695	199	8	,	,	PUNCT
cana-2695	199	9	(	(	PUNCT
cana-2695	199	10	μ+′)∞(φ(r	μ+′)∞(φ(r	NOUN
cana-2695	199	11	1	1	NUM
cana-2695	199	12	)	)	PUNCT
cana-2695	199	13	)	)	PUNCT
cana-2695	199	14	,	,	PUNCT
cana-2695	199	15	(	(	PUNCT
cana-2695	199	16	(	(	PUNCT
cana-2695	199	17	μ+′)∞(φ(r	μ+′)∞(φ(r	NOUN
cana-2695	199	18	k+i	k+i	NUM
cana-2695	199	19	)	)	PUNCT
cana-2695	199	20	,	,	PUNCT
cana-2695	199	21	(	(	PUNCT
cana-2695	199	22	μ+′)∞(φ(r	μ+′)∞(φ(r	NOUN
cana-2695	199	23	2	2	NUM
cana-2695	199	24	)	)	PUNCT
cana-2695	199	25	)	)	PUNCT
cana-2695	199	26	,	,	PUNCT
cana-2695	199	27	.	.	PUNCT
cana-2695	199	28	..	..	PUNCT
cana-2695	200	1	(	(	PUNCT
cana-2695	200	2	(	(	PUNCT
cana-2695	200	3	μ+′)∞(φ(r	μ+′)∞(φ(r	NOUN
cana-2695	200	4	k+i	k+i	NUM
cana-2695	200	5	)	)	PUNCT
cana-2695	200	6	,	,	PUNCT
cana-2695	200	7	(	(	PUNCT
cana-2695	200	8	μ+′)∞(φ(r	μ+′)∞(φ(r	NOUN
cana-2695	200	9	k	k	PROPN
cana-2695	200	10	)	)	PUNCT
cana-2695	200	11	)	)	PUNCT
cana-2695	200	12	≥	≥	PROPN
cana-2695	200	13	μ+(rk+i	μ+(rk+i	NOUN
cana-2695	200	14	,	,	PUNCT
cana-2695	200	15	r1	r1	PROPN
cana-2695	200	16	)	)	PUNCT
cana-2695	200	17	,	,	PUNCT
cana-2695	200	18	μ	μ	PROPN
cana-2695	200	19	+	+	PROPN
cana-2695	200	20	(	(	PUNCT
cana-2695	200	21	rk+i	rk+i	PROPN
cana-2695	200	22	,	,	PUNCT
cana-2695	200	23	r2	r2	PROPN
cana-2695	200	24	)	)	PUNCT
cana-2695	200	25	,	,	PUNCT
cana-2695	200	26	.	.	PUNCT
cana-2695	200	27	.	.	PUNCT
cana-2695	201	1	.	.	PUNCT
cana-2695	202	1	,	,	PUNCT
cana-2695	202	2	μ	μ	PROPN
cana-2695	202	3	+	+	PROPN
cana-2695	202	4	(	(	PUNCT
cana-2695	202	5	rk+i	rk+i	PROPN
cana-2695	202	6	,	,	PUNCT
cana-2695	202	7	rk	rk	NOUN
cana-2695	202	8	)	)	PUNCT
cana-2695	202	9	=	=	SYM
cana-2695	202	10	w+(rk+i	w+(rk+i	NOUN
cana-2695	202	11	,	,	PUNCT
cana-2695	202	12	r1	r1	PROPN
cana-2695	202	13	)	)	PUNCT
cana-2695	202	14	,	,	PUNCT
cana-2695	202	15	w	w	ADP
cana-2695	202	16	+	+	CCONJ
cana-2695	202	17	(	(	PUNCT
cana-2695	202	18	rk+i	rk+i	PROPN
cana-2695	202	19	,	,	PUNCT
cana-2695	202	20	r2	r2	PROPN
cana-2695	202	21	)	)	PUNCT
cana-2695	202	22	,	,	PUNCT
cana-2695	202	23	.	.	PUNCT
cana-2695	202	24	.	.	PUNCT
cana-2695	203	1	.	.	PUNCT
cana-2695	204	1	,	,	PUNCT
cana-2695	204	2	w	w	X
cana-2695	204	3	+	+	CCONJ
cana-2695	204	4	(	(	PUNCT
cana-2695	204	5	rk+i	rk+i	PROPN
cana-2695	204	6	,	,	PUNCT
cana-2695	204	7	rk	rk	NOUN
cana-2695	204	8	)	)	PUNCT
cana-2695	204	9	which	which	PRON
cana-2695	204	10	are	be	AUX
cana-2695	204	11	all	all	ADV
cana-2695	204	12	distinct	distinct	ADJ
cana-2695	204	13	for	for	ADP
cana-2695	204	14	i	i	PRON
cana-2695	204	15	=	=	NOUN
cana-2695	204	16	1	1	NUM
cana-2695	204	17	to	to	ADP
cana-2695	204	18	n	n	PROPN
cana-2695	204	19	=	=	SYM
cana-2695	204	20	k.	k.	PROPN
cana-2695	204	21	therefore	therefore	ADV
cana-2695	204	22	,	,	PUNCT
cana-2695	204	23	ℋ1	ℋ1	PROPN
cana-2695	204	24	is	be	AUX
cana-2695	204	25	the	the	DET
cana-2695	204	26	ivfrs	ivfrs	NOUN
cana-2695	204	27	of	of	ADP
cana-2695	204	28	g′	g′	NOUN
cana-2695	204	29	and	and	CCONJ
cana-2695	204	30	│	│	NOUN
cana-2695	204	31	ℋ1	ℋ1	NOUN
cana-2695	204	32	│	│	X
cana-2695	204	33	=	=	PUNCT
cana-2695	204	34	k.	k.	PROPN
cana-2695	204	35	now	now	ADV
cana-2695	204	36	to	to	PART
cana-2695	204	37	prove	prove	VERB
cana-2695	204	38	that	that	SCONJ
cana-2695	204	39	ℋ1	ℋ1	PROPN
cana-2695	204	40	is	be	AUX
cana-2695	204	41	the	the	DET
cana-2695	204	42	minimum	minimum	ADJ
cana-2695	204	43	ivfrs	ivfrs	NOUN
cana-2695	204	44	of	of	ADP
cana-2695	204	45	g	g	NOUN
cana-2695	204	46	’	'	PUNCT
cana-2695	204	47	.	.	PUNCT
cana-2695	205	1	assume	assume	VERB
cana-2695	205	2	that	that	SCONJ
cana-2695	205	3	there	there	PRON
cana-2695	205	4	exists	exist	VERB
cana-2695	205	5	a	a	DET
cana-2695	205	6	frs	frs	ADJ
cana-2695	205	7	r	r	NOUN
cana-2695	205	8	of	of	ADP
cana-2695	205	9	g	g	NOUN
cana-2695	205	10	’	'	PUNCT
cana-2695	205	11	such	such	ADJ
cana-2695	205	12	that	that	DET
cana-2695	205	13	r	r	NOUN
cana-2695	205	14	=	=	SYM
cana-2695	205	15	{	{	PUNCT
cana-2695	205	16	(	(	PUNCT
cana-2695	205	17	σ1	σ1	PROPN
cana-2695	205	18	−	−	PROPN
cana-2695	205	19	,	,	PUNCT
cana-2695	205	20	σ1	σ1	PROPN
cana-2695	205	21	+	+	PROPN
cana-2695	205	22	)	)	PUNCT
cana-2695	205	23	,	,	PUNCT
cana-2695	205	24	(	(	PUNCT
cana-2695	205	25	σ2	σ2	NOUN
cana-2695	205	26	−	−	PROPN
cana-2695	205	27	,	,	PUNCT
cana-2695	205	28	σ2	σ2	NOUN
cana-2695	205	29	+	+	PROPN
cana-2695	205	30	)	)	PUNCT
cana-2695	205	31	,	,	PUNCT
cana-2695	205	32	.	.	PUNCT
cana-2695	205	33	.	.	PUNCT
cana-2695	205	34	.	.	PUNCT
cana-2695	206	1	,	,	PUNCT
cana-2695	206	2	(	(	PUNCT
cana-2695	206	3	σk1	σk1	NOUN
cana-2695	206	4	−	−	PROPN
cana-2695	206	5	,	,	PUNCT
cana-2695	206	6	σk1	σk1	NOUN
cana-2695	206	7	+	+	CCONJ
cana-2695	206	8	)	)	PUNCT
cana-2695	206	9	}	}	PUNCT
cana-2695	206	10	and	and	CCONJ
cana-2695	206	11	│	│	NOUN
cana-2695	206	12	ℋ1	ℋ1	NOUN
cana-2695	206	13	│	│	X
cana-2695	206	14	=	=	SYM
cana-2695	206	15	k	k	X
cana-2695	206	16	>	>	PUNCT
cana-2695	206	17	│	│	PUNCT
cana-2695	206	18	r	r	NOUN
cana-2695	206	19	│	│	NOUN
cana-2695	206	20	=	=	SYM
cana-2695	206	21	k1	k1	NOUN
cana-2695	206	22	,	,	PUNCT
cana-2695	206	23	then	then	ADV
cana-2695	206	24	𝜎𝑘+𝑖	𝜎𝑘+𝑖	NUM
cana-2695	206	25	−	−	NOUN
cana-2695	206	26	/𝑅	/𝑅	PUNCT
cana-2695	206	27	and	and	CCONJ
cana-2695	206	28	𝜎𝑘+𝑖	𝜎𝑘+𝑖	X
cana-2695	207	1	+	+	CCONJ
cana-2695	207	2	/r	/r	VERB
cana-2695	207	3	are	be	AUX
cana-2695	207	4	all	all	ADV
cana-2695	207	5	distinct	distinct	ADJ
cana-2695	207	6	for	for	ADP
cana-2695	207	7	n	n	NOUN
cana-2695	207	8	=	=	SYM
cana-2695	207	9	1	1	NUM
cana-2695	207	10	,	,	PUNCT
cana-2695	207	11	2	2	NUM
cana-2695	207	12	,	,	PUNCT
cana-2695	207	13	.	.	PUNCT
cana-2695	207	14	.	.	PUNCT
cana-2695	208	1	.	.	PUNCT
cana-2695	209	1	,	,	PUNCT
cana-2695	209	2	n	n	CCONJ
cana-2695	209	3	−	−	PROPN
cana-2695	209	4	k1	k1	NOUN
cana-2695	209	5	.	.	PUNCT
cana-2695	210	1	let	let	VERB
cana-2695	210	2	r1	r1	PROPN
cana-2695	210	3	=	=	SYM
cana-2695	210	4	{	{	PUNCT
cana-2695	210	5	(	(	PUNCT
cana-2695	210	6	σ1	σ1	PROPN
cana-2695	210	7	−	−	PROPN
cana-2695	210	8	,	,	PUNCT
cana-2695	210	9	σ1	σ1	PROPN
cana-2695	210	10	+	+	PROPN
cana-2695	210	11	)	)	PUNCT
cana-2695	210	12	,	,	PUNCT
cana-2695	210	13	(	(	PUNCT
cana-2695	210	14	σ2	σ2	NOUN
cana-2695	210	15	−	−	PROPN
cana-2695	210	16	,	,	PUNCT
cana-2695	210	17	σ2	σ2	NOUN
cana-2695	210	18	+	+	PROPN
cana-2695	210	19	)	)	PUNCT
cana-2695	210	20	,	,	PUNCT
cana-2695	210	21	.	.	PUNCT
cana-2695	210	22	.	.	PUNCT
cana-2695	211	1	.	.	PUNCT
cana-2695	212	1	,	,	PUNCT
cana-2695	212	2	(	(	PUNCT
cana-2695	212	3	σk1	σk1	NOUN
cana-2695	212	4	−	−	PROPN
cana-2695	212	5	,	,	PUNCT
cana-2695	212	6	σk1	σk1	NOUN
cana-2695	212	7	+	+	CCONJ
cana-2695	212	8	)	)	PUNCT
cana-2695	212	9	}	}	PUNCT
cana-2695	212	10	,	,	PUNCT
cana-2695	212	11	𝜎𝑘+𝑖	𝜎𝑘+𝑖	X
cana-2695	213	1	−	−	NOUN
cana-2695	213	2	/r1	/r1	PUNCT
cana-2695	214	1	=	=	PUNCT
cana-2695	214	2	w−	w−	NOUN
cana-2695	214	3	(	(	PUNCT
cana-2695	214	4	rk+i	rk+i	PROPN
cana-2695	214	5	,	,	PUNCT
cana-2695	214	6	r1	r1	PROPN
cana-2695	214	7	)	)	PUNCT
cana-2695	214	8	,	,	PUNCT
cana-2695	214	9	w	w	PROPN
cana-2695	214	10	−	−	PROPN
cana-2695	214	11	(	(	PUNCT
cana-2695	214	12	rk+i	rk+i	PROPN
cana-2695	214	13	,	,	PUNCT
cana-2695	214	14	r2	r2	PROPN
cana-2695	214	15	)	)	PUNCT
cana-2695	214	16	,	,	PUNCT
cana-2695	214	17	.	.	PUNCT
cana-2695	214	18	.	.	PUNCT
cana-2695	214	19	.	.	PUNCT
cana-2695	215	1	,	,	PUNCT
cana-2695	215	2	w	w	ADP
cana-2695	215	3	−	−	PROPN
cana-2695	215	4	(	(	PUNCT
cana-2695	215	5	rk+i	rk+i	PROPN
cana-2695	215	6	,	,	PUNCT
cana-2695	215	7	rk1	rk1	NOUN
cana-2695	215	8	)	)	PUNCT
cana-2695	215	9	communications	communication	NOUN
cana-2695	215	10	on	on	ADP
cana-2695	215	11	applied	apply	VERB
cana-2695	215	12	nonlinear	nonlinear	ADJ
cana-2695	215	13	analysis	analysis	NOUN
cana-2695	215	14	issn	issn	NOUN
cana-2695	215	15	:	:	PUNCT
cana-2695	215	16	1074	1074	NUM
cana-2695	215	17	-	-	PUNCT
cana-2695	215	18	133x	133x	NUM
cana-2695	215	19	vol	vol	NOUN
cana-2695	215	20	32	32	NUM
cana-2695	215	21	no	no	NOUN
cana-2695	215	22	.	.	PUNCT
cana-2695	216	1	3s	3s	NUM
cana-2695	216	2	(	(	PUNCT
cana-2695	216	3	2025	2025	NUM
cana-2695	216	4	)	)	PUNCT
cana-2695	216	5	553	553	NUM
cana-2695	216	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2695	216	7	=	=	SYM
cana-2695	216	8	{	{	PUNCT
cana-2695	216	9	(	(	PUNCT
cana-2695	216	10	μ−)∞(rk+i	μ−)∞(rk+i	NUM
cana-2695	216	11	,	,	PUNCT
cana-2695	216	12	r1	r1	PROPN
cana-2695	216	13	)	)	PUNCT
cana-2695	216	14	,	,	PUNCT
cana-2695	216	15	(	(	PUNCT
cana-2695	216	16	μ−)∞(rk+i	μ−)∞(rk+i	NOUN
cana-2695	216	17	,	,	PUNCT
cana-2695	216	18	r2	r2	PROPN
cana-2695	216	19	)	)	PUNCT
cana-2695	216	20	,	,	PUNCT
cana-2695	216	21	.	.	PUNCT
cana-2695	216	22	..	..	PUNCT
cana-2695	217	1	(	(	PUNCT
cana-2695	217	2	μ−)∞(rk+i	μ−)∞(rk+i	NUM
cana-2695	217	3	,	,	PUNCT
cana-2695	217	4	rk1	rk1	NOUN
cana-2695	217	5	)	)	PUNCT
cana-2695	217	6	}	}	PUNCT
cana-2695	217	7	≤	≤	NOUN
cana-2695	217	8	{	{	PUNCT
cana-2695	217	9	(	(	PUNCT
cana-2695	217	10	(	(	PUNCT
cana-2695	217	11	μ−′)∞(φ(r	μ−′)∞(φ(r	NOUN
cana-2695	217	12	k+i	k+i	NUM
cana-2695	217	13	)	)	PUNCT
cana-2695	217	14	,	,	PUNCT
cana-2695	217	15	(	(	PUNCT
cana-2695	217	16	μ−′)∞(φ(r	μ−′)∞(φ(r	NOUN
cana-2695	217	17	1	1	NUM
cana-2695	217	18	)	)	PUNCT
cana-2695	217	19	)	)	PUNCT
cana-2695	217	20	,	,	PUNCT
cana-2695	217	21	(	(	PUNCT
cana-2695	217	22	(	(	PUNCT
cana-2695	217	23	μ−′)∞(φ(r	μ−′)∞(φ(r	NOUN
cana-2695	217	24	k+i	k+i	NUM
cana-2695	217	25	)	)	PUNCT
cana-2695	217	26	,	,	PUNCT
cana-2695	217	27	(	(	PUNCT
cana-2695	217	28	μ−′)∞(φ(r	μ−′)∞(φ(r	NOUN
cana-2695	217	29	2	2	NUM
cana-2695	217	30	)	)	PUNCT
cana-2695	217	31	)	)	PUNCT
cana-2695	217	32	,	,	PUNCT
cana-2695	217	33	.	.	PUNCT
cana-2695	217	34	..	..	PUNCT
cana-2695	218	1	(	(	PUNCT
cana-2695	218	2	(	(	PUNCT
cana-2695	218	3	μ−′)∞(φ(r	μ−′)∞(φ(r	NOUN
cana-2695	218	4	k+i	k+i	NUM
cana-2695	218	5	)	)	PUNCT
cana-2695	218	6	,	,	PUNCT
cana-2695	218	7	(	(	PUNCT
cana-2695	218	8	μ−′)∞(φ(r	μ−′)∞(φ(r	ADJ
cana-2695	218	9	k1	k1	NOUN
cana-2695	218	10	)	)	PUNCT
cana-2695	218	11	)	)	PUNCT
cana-2695	218	12	similarly	similarly	ADV
cana-2695	218	13	,	,	PUNCT
cana-2695	218	14	𝜎𝑘+𝑖	𝜎𝑘+𝑖	X
cana-2695	219	1	+	+	NUM
cana-2695	219	2	/r1	/r1	SYM
cana-2695	219	3	=	=	SYM
cana-2695	219	4	w+	w+	X
cana-2695	219	5	(	(	PUNCT
cana-2695	219	6	rk+i	rk+i	PROPN
cana-2695	219	7	,	,	PUNCT
cana-2695	219	8	r1	r1	PROPN
cana-2695	219	9	)	)	PUNCT
cana-2695	219	10	,	,	PUNCT
cana-2695	219	11	w	w	ADP
cana-2695	219	12	+	+	CCONJ
cana-2695	219	13	(	(	PUNCT
cana-2695	219	14	rk+i	rk+i	PROPN
cana-2695	219	15	,	,	PUNCT
cana-2695	219	16	r2	r2	PROPN
cana-2695	219	17	)	)	PUNCT
cana-2695	219	18	,	,	PUNCT
cana-2695	219	19	.	.	PUNCT
cana-2695	219	20	.	.	PUNCT
cana-2695	219	21	.	.	PUNCT
cana-2695	220	1	,	,	PUNCT
cana-2695	220	2	w	w	X
cana-2695	220	3	+	+	CCONJ
cana-2695	220	4	(	(	PUNCT
cana-2695	220	5	rk+i	rk+i	PROPN
cana-2695	220	6	,	,	PUNCT
cana-2695	220	7	rk1	rk1	NOUN
cana-2695	220	8	)	)	PUNCT
cana-2695	221	1	=	=	PRON
cana-2695	221	2	{	{	PUNCT
cana-2695	221	3	(	(	PUNCT
cana-2695	221	4	μ+)∞(rk+i	μ+)∞(rk+i	NOUN
cana-2695	221	5	,	,	PUNCT
cana-2695	221	6	r1	r1	PROPN
cana-2695	221	7	)	)	PUNCT
cana-2695	221	8	,	,	PUNCT
cana-2695	221	9	(	(	PUNCT
cana-2695	221	10	μ+)∞(rk+i	μ+)∞(rk+i	NOUN
cana-2695	221	11	,	,	PUNCT
cana-2695	221	12	r2	r2	PROPN
cana-2695	221	13	)	)	PUNCT
cana-2695	221	14	,	,	PUNCT
cana-2695	221	15	.	.	PUNCT
cana-2695	222	1	..	..	PUNCT
cana-2695	222	2	(	(	PUNCT
cana-2695	222	3	μ+)∞(rk+i	μ+)∞(rk+i	NOUN
cana-2695	222	4	,	,	PUNCT
cana-2695	222	5	rk1	rk1	NOUN
cana-2695	222	6	)	)	PUNCT
cana-2695	222	7	}	}	PUNCT
cana-2695	222	8	≤	≤	NOUN
cana-2695	222	9	{	{	PUNCT
cana-2695	222	10	(	(	PUNCT
cana-2695	222	11	(	(	PUNCT
cana-2695	222	12	μ+′)∞(φ(r	μ+′)∞(φ(r	NOUN
cana-2695	222	13	k+i	k+i	NUM
cana-2695	222	14	)	)	PUNCT
cana-2695	222	15	,	,	PUNCT
cana-2695	222	16	(	(	PUNCT
cana-2695	222	17	μ+′)∞(φ(r	μ+′)∞(φ(r	NOUN
cana-2695	222	18	1	1	NUM
cana-2695	222	19	)	)	PUNCT
cana-2695	222	20	)	)	PUNCT
cana-2695	222	21	,	,	PUNCT
cana-2695	222	22	(	(	PUNCT
cana-2695	222	23	(	(	PUNCT
cana-2695	222	24	μ+′)∞(φ(r	μ+′)∞(φ(r	NOUN
cana-2695	222	25	k+i	k+i	NUM
cana-2695	222	26	)	)	PUNCT
cana-2695	222	27	,	,	PUNCT
cana-2695	222	28	(	(	PUNCT
cana-2695	222	29	μ+′)∞(φ(r	μ+′)∞(φ(r	NOUN
cana-2695	222	30	2	2	NUM
cana-2695	222	31	)	)	PUNCT
cana-2695	222	32	)	)	PUNCT
cana-2695	222	33	,	,	PUNCT
cana-2695	222	34	.	.	PUNCT
cana-2695	222	35	..	..	PUNCT
cana-2695	223	1	(	(	PUNCT
cana-2695	223	2	(	(	PUNCT
cana-2695	223	3	μ+′)∞(φ(r	μ+′)∞(φ(r	NOUN
cana-2695	223	4	k+i	k+i	NUM
cana-2695	223	5	)	)	PUNCT
cana-2695	223	6	,	,	PUNCT
cana-2695	223	7	(	(	PUNCT
cana-2695	223	8	μ+′)∞(φ(r	μ+′)∞(φ(r	NOUN
cana-2695	223	9	k1	k1	NOUN
cana-2695	223	10	)	)	PUNCT
cana-2695	223	11	)	)	PUNCT
cana-2695	223	12	which	which	PRON
cana-2695	223	13	are	be	AUX
cana-2695	223	14	all	all	ADV
cana-2695	223	15	distinct	distinct	ADJ
cana-2695	223	16	for	for	ADP
cana-2695	223	17	i	i	X
cana-2695	223	18	=	=	NOUN
cana-2695	223	19	1,2	1,2	NUM
cana-2695	223	20	,	,	PUNCT
cana-2695	223	21	.	.	PUNCT
cana-2695	223	22	.	.	PUNCT
cana-2695	223	23	.	.	PUNCT
cana-2695	224	1	n	n	PRON
cana-2695	224	2	−	−	PROPN
cana-2695	224	3	k.	k.	PROPN
cana-2695	225	1	therefore	therefore	ADV
cana-2695	225	2	,	,	PUNCT
cana-2695	225	3	r1	r1	PROPN
cana-2695	225	4	is	be	AUX
cana-2695	225	5	the	the	DET
cana-2695	225	6	ivfrs	ivfrs	NOUN
cana-2695	225	7	of	of	ADP
cana-2695	225	8	g	g	PROPN
cana-2695	225	9	and	and	CCONJ
cana-2695	225	10	ℐr(g)=𝑘1	ℐr(g)=𝑘1	PROPN
cana-2695	226	1	[	[	X
cana-2695	226	2	𝑘1	𝑘1	NOUN
cana-2695	226	3	<	<	X
cana-2695	226	4	k	k	X
cana-2695	226	5	]	]	X
cana-2695	226	6	,	,	PUNCT
cana-2695	226	7	which	which	PRON
cana-2695	226	8	is	be	AUX
cana-2695	226	9	a	a	DET
cana-2695	226	10	contradiction	contradiction	NOUN
cana-2695	226	11	to	to	ADP
cana-2695	226	12	our	our	PRON
cana-2695	226	13	assumption	assumption	NOUN
cana-2695	226	14	that	that	SCONJ
cana-2695	226	15	ℐr(g	ℐr(g	ADV
cana-2695	226	16	)	)	PUNCT
cana-2695	226	17	=	=	SYM
cana-2695	226	18	k.	k.	PROPN
cana-2695	226	19	hence	hence	ADV
cana-2695	226	20	,	,	PUNCT
cana-2695	226	21	ℋ1	ℋ1	PROPN
cana-2695	226	22	is	be	AUX
cana-2695	226	23	the	the	DET
cana-2695	226	24	minimum	minimum	ADJ
cana-2695	226	25	ivrs	ivrs	NOUN
cana-2695	226	26	of	of	ADP
cana-2695	226	27	g′.	g′.	PROPN
cana-2695	226	28	therefore,ℐr(g	therefore,ℐr(g	PROPN
cana-2695	226	29	)	)	PUNCT
cana-2695	226	30	and	and	CCONJ
cana-2695	226	31	ℐr(g′	ℐr(g′	PRON
cana-2695	226	32	)	)	PUNCT
cana-2695	226	33	are	be	AUX
cana-2695	226	34	homomorphic	homomorphic	ADJ
cana-2695	226	35	functions	function	NOUN
cana-2695	226	36	.	.	PUNCT
cana-2695	227	1	3.5	3.5	NUM
cana-2695	227	2	corollary	corollary	NOUN
cana-2695	227	3	if	if	SCONJ
cana-2695	227	4	g	g	PROPN
cana-2695	227	5	and	and	CCONJ
cana-2695	227	6	g′	g′	NOUN
cana-2695	227	7	are	be	AUX
cana-2695	227	8	co	co	ADJ
cana-2695	227	9	-	-	ADJ
cana-2695	227	10	weak	weak	ADJ
cana-2695	227	11	isomorphic	isomorphic	NOUN
cana-2695	227	12	,	,	PUNCT
cana-2695	227	13	then	then	ADV
cana-2695	227	14	ℐr(g	ℐr(g	ADV
cana-2695	227	15	)	)	PUNCT
cana-2695	227	16	and	and	CCONJ
cana-2695	227	17	ℐr(g′	ℐr(g′	PRON
cana-2695	227	18	)	)	PUNCT
cana-2695	227	19	also	also	ADV
cana-2695	227	20	have	have	VERB
cana-2695	227	21	a	a	DET
cana-2695	227	22	co	co	ADJ
cana-2695	227	23	-	-	ADJ
cana-2695	227	24	weak	weak	ADJ
cana-2695	227	25	isomorphism	isomorphism	NOUN
cana-2695	227	26	.	.	PUNCT
cana-2695	228	1	3.6	3.6	NUM
cana-2695	228	2	theorem	theorem	VERB
cana-2695	228	3	:	:	PUNCT
cana-2695	228	4	for	for	ADP
cana-2695	228	5	a	a	DET
cana-2695	228	6	connected	connected	ADJ
cana-2695	228	7	ivfg	ivfg	NOUN
cana-2695	228	8	with	with	ADP
cana-2695	228	9	atleast	atleast	ADJ
cana-2695	228	10	three	three	NUM
cana-2695	228	11	different	different	ADJ
cana-2695	228	12	membership	membership	NOUN
cana-2695	228	13	value	value	NOUN
cana-2695	228	14	with	with	ADP
cana-2695	228	15	both	both	CCONJ
cana-2695	228	16	𝜎−	𝜎−	PROPN
cana-2695	228	17	and	and	CCONJ
cana-2695	228	18	𝜎+	𝜎+	NUM
cana-2695	228	19	,	,	PUNCT
cana-2695	228	20	there	there	PRON
cana-2695	228	21	exists	exist	VERB
cana-2695	228	22	an	an	DET
cana-2695	228	23	ivfrss	ivfrss	NOUN
cana-2695	228	24	,	,	PUNCT
cana-2695	228	25	such	such	ADJ
cana-2695	228	26	that	that	SCONJ
cana-2695	228	27	2	2	NUM
cana-2695	228	28	≤	≤	NOUN
cana-2695	228	29	ℐr(g	ℐr(g	ADV
cana-2695	228	30	)	)	PUNCT
cana-2695	228	31	≤	≤	NOUN
cana-2695	228	32	n	n	CCONJ
cana-2695	228	33	−	−	PROPN
cana-2695	228	34	1	1	NUM
cana-2695	228	35	.	.	PUNCT
cana-2695	229	1	[	[	X
cana-2695	229	2	n	n	CCONJ
cana-2695	229	3	≥	≥	NOUN
cana-2695	229	4	3	3	NUM
cana-2695	229	5	]	]	X
cana-2695	229	6	proof	proof	NOUN
cana-2695	229	7	:	:	PUNCT
cana-2695	229	8	let	let	VERB
cana-2695	229	9	us	we	PRON
cana-2695	229	10	take	take	VERB
cana-2695	229	11	n	n	NOUN
cana-2695	229	12	=	=	SYM
cana-2695	229	13	3	3	NUM
cana-2695	229	14	,	,	PUNCT
cana-2695	229	15	then	then	ADV
cana-2695	229	16	the	the	DET
cana-2695	229	17	defined	define	VERB
cana-2695	229	18	three	three	NUM
cana-2695	229	19	vertices	vertex	NOUN
cana-2695	229	20	are	be	AUX
cana-2695	229	21	a	a	DET
cana-2695	229	22	,	,	PUNCT
cana-2695	229	23	b	b	NOUN
cana-2695	229	24	,	,	PUNCT
cana-2695	229	25	c.	c.	PROPN
cana-2695	229	26	this	this	PRON
cana-2695	229	27	implies	imply	VERB
cana-2695	229	28	that	that	SCONJ
cana-2695	229	29	,	,	PUNCT
cana-2695	229	30	𝜎−(a	𝜎−(a	ADV
cana-2695	229	31	)	)	PUNCT
cana-2695	229	32	=	=	SYM
cana-2695	229	33	α1	α1	PROPN
cana-2695	229	34	,	,	PUNCT
cana-2695	229	35	𝜎	𝜎	PROPN
cana-2695	230	1	+	+	ADJ
cana-2695	230	2	(	(	PUNCT
cana-2695	230	3	a	a	NOUN
cana-2695	230	4	)	)	PUNCT
cana-2695	230	5	=	=	SYM
cana-2695	230	6	β	β	X
cana-2695	230	7	1	1	NUM
cana-2695	230	8	,	,	PUNCT
cana-2695	230	9	𝜎−(b	𝜎−(b	PROPN
cana-2695	230	10	)	)	PUNCT
cana-2695	230	11	=	=	SYM
cana-2695	230	12	α2	α2	ADJ
cana-2695	230	13	,	,	PUNCT
cana-2695	230	14	𝜎	𝜎	PROPN
cana-2695	231	1	+	+	ADJ
cana-2695	231	2	(	(	PUNCT
cana-2695	231	3	b	b	NOUN
cana-2695	231	4	)	)	PUNCT
cana-2695	231	5	=	=	SYM
cana-2695	231	6	β	β	X
cana-2695	231	7	2	2	NUM
cana-2695	231	8	,	,	PUNCT
cana-2695	231	9	𝜎−(c	𝜎−(c	PROPN
cana-2695	231	10	)	)	PUNCT
cana-2695	231	11	=	=	SYM
cana-2695	231	12	α3	α3	PROPN
cana-2695	231	13	,	,	PUNCT
cana-2695	231	14	𝜎	𝜎	PROPN
cana-2695	232	1	+	+	ADJ
cana-2695	232	2	(	(	PUNCT
cana-2695	232	3	c	c	NOUN
cana-2695	232	4	)	)	PUNCT
cana-2695	232	5	=	=	NOUN
cana-2695	232	6	β	β	X
cana-2695	232	7	3	3	NUM
cana-2695	232	8	,	,	PUNCT
cana-2695	232	9	as	as	SCONJ
cana-2695	232	10	they	they	PRON
cana-2695	232	11	exist	exist	VERB
cana-2695	232	12	three	three	NUM
cana-2695	232	13	different	different	ADJ
cana-2695	232	14	membership	membership	NOUN
cana-2695	232	15	values	value	NOUN
cana-2695	232	16	,	,	PUNCT
cana-2695	232	17	then	then	ADV
cana-2695	232	18	the	the	DET
cana-2695	232	19	representation	representation	NOUN
cana-2695	232	20	of	of	ADP
cana-2695	232	21	any	any	DET
cana-2695	232	22	two	two	NUM
cana-2695	232	23	vertices	vertex	NOUN
cana-2695	232	24	in	in	ADP
cana-2695	232	25	h	h	NOUN
cana-2695	232	26	will	will	AUX
cana-2695	232	27	be	be	AUX
cana-2695	232	28	different	different	ADJ
cana-2695	232	29	.	.	PUNCT
cana-2695	233	1	hence	hence	ADV
cana-2695	233	2	ℐr(g	ℐr(g	ADV
cana-2695	233	3	)	)	PUNCT
cana-2695	233	4	=	=	SYM
cana-2695	233	5	2	2	NUM
cana-2695	233	6	,	,	PUNCT
cana-2695	233	7	for	for	ADP
cana-2695	233	8	n	n	NOUN
cana-2695	233	9	=	=	SYM
cana-2695	233	10	3	3	X
cana-2695	233	11	.	.	X
cana-2695	233	12	similarly	similarly	ADV
cana-2695	233	13	,	,	PUNCT
cana-2695	233	14	if	if	SCONJ
cana-2695	233	15	n	n	PROPN
cana-2695	234	1	=	=	SYM
cana-2695	234	2	k	k	NOUN
cana-2695	234	3	,	,	PUNCT
cana-2695	234	4	then	then	ADV
cana-2695	234	5	∃	∃	PROPN
cana-2695	234	6	atleast	atleast	NOUN
cana-2695	234	7	three	three	NUM
cana-2695	234	8	different	different	ADJ
cana-2695	234	9	membership	membership	NOUN
cana-2695	234	10	values	value	NOUN
cana-2695	234	11	of	of	ADP
cana-2695	234	12	both	both	DET
cana-2695	234	13	𝜎−	𝜎−	PROPN
cana-2695	234	14	and	and	CCONJ
cana-2695	234	15	𝜎+	𝜎+	NUM
cana-2695	234	16	.	.	PUNCT
cana-2695	235	1	hence	hence	ADV
cana-2695	235	2	the	the	DET
cana-2695	235	3	representation	representation	NOUN
cana-2695	235	4	will	will	AUX
cana-2695	235	5	be	be	AUX
cana-2695	235	6	unique	unique	ADJ
cana-2695	235	7	with	with	ADP
cana-2695	235	8	atleast	atleast	ADJ
cana-2695	235	9	(	(	PUNCT
cana-2695	235	10	n	n	CCONJ
cana-2695	235	11	−	−	PROPN
cana-2695	235	12	1	1	NUM
cana-2695	235	13	)	)	PUNCT
cana-2695	235	14	vertices	vertex	NOUN
cana-2695	235	15	.	.	PUNCT
cana-2695	236	1	that	that	PRON
cana-2695	236	2	is	is	ADV
cana-2695	236	3	,	,	PUNCT
cana-2695	236	4	ℐr(g	ℐr(g	ADV
cana-2695	236	5	)	)	PUNCT
cana-2695	236	6	≤	≤	NOUN
cana-2695	237	1	n	n	CCONJ
cana-2695	238	1	−	−	PROPN
cana-2695	238	2	1	1	NUM
cana-2695	238	3	.	.	PUNCT
cana-2695	238	4	hence	hence	ADV
cana-2695	238	5	2	2	NUM
cana-2695	238	6	≤	≤	NUM
cana-2695	238	7	ℐr(g	ℐr(g	ADV
cana-2695	238	8	)	)	PUNCT
cana-2695	238	9	≤	≤	NOUN
cana-2695	238	10	n	n	CCONJ
cana-2695	238	11	−	−	PROPN
cana-2695	238	12	1	1	NUM
cana-2695	238	13	.	.	X
cana-2695	238	14	4	4	NUM
cana-2695	238	15	.	.	NOUN
cana-2695	238	16	fuzzy	fuzzy	ADJ
cana-2695	238	17	super	super	ADJ
cana-2695	238	18	resolving	resolve	VERB
cana-2695	238	19	sets	set	NOUN
cana-2695	238	20	of	of	ADP
cana-2695	238	21	interval	interval	NOUN
cana-2695	238	22	-	-	PUNCT
cana-2695	238	23	valued	value	VERB
cana-2695	238	24	fuzzy	fuzzy	ADJ
cana-2695	238	25	graphs	graph	NOUN
cana-2695	238	26	an	an	DET
cana-2695	238	27	ivfrs	ivfrs	NOUN
cana-2695	238	28	is	be	AUX
cana-2695	238	29	called	call	VERB
cana-2695	238	30	interval	interval	NOUN
cana-2695	238	31	-	-	PUNCT
cana-2695	238	32	valued	value	VERB
cana-2695	238	33	fuzzy	fuzzy	ADJ
cana-2695	238	34	super	super	ADJ
cana-2695	238	35	resolving	resolve	VERB
cana-2695	238	36	set	set	VERB
cana-2695	238	37	(	(	PUNCT
cana-2695	238	38	ivfsrs	ivfsrs	PROPN
cana-2695	238	39	)	)	PUNCT
cana-2695	238	40	if	if	SCONJ
cana-2695	238	41	any	any	DET
cana-2695	238	42	two	two	NUM
cana-2695	238	43	elements	element	NOUN
cana-2695	238	44	of	of	ADP
cana-2695	238	45	[	[	X
cana-2695	238	46	𝜎−	𝜎−	NOUN
cana-2695	238	47	,	,	PUNCT
cana-2695	238	48	𝜎+	𝜎+	X
cana-2695	238	49	]	]	PUNCT
cana-2695	238	50	have	have	VERB
cana-2695	238	51	different	different	ADJ
cana-2695	238	52	representation	representation	NOUN
cana-2695	238	53	with	with	ADP
cana-2695	238	54	regards	regard	NOUN
cana-2695	238	55	to	to	ADP
cana-2695	238	56	ℋ	ℋ	PROPN
cana-2695	238	57	.	.	PUNCT
cana-2695	239	1	here	here	ADV
cana-2695	239	2	we	we	PRON
cana-2695	239	3	take	take	VERB
cana-2695	239	4	w−	w−	NOUN
cana-2695	239	5	(	(	PUNCT
cana-2695	239	6	r	r	NOUN
cana-2695	239	7	,	,	PUNCT
cana-2695	239	8	r	r	NOUN
cana-2695	239	9	)	)	PUNCT
cana-2695	239	10	=	=	SYM
cana-2695	239	11	σ−(r	σ−(r	PROPN
cana-2695	239	12	)	)	PUNCT
cana-2695	239	13	and	and	CCONJ
cana-2695	239	14	w+(r	w+(r	PRON
cana-2695	239	15	,	,	PUNCT
cana-2695	239	16	r	r	NOUN
cana-2695	239	17	)	)	PUNCT
cana-2695	239	18	=	=	SYM
cana-2695	239	19	σ+(r	σ+(r	NOUN
cana-2695	239	20	)	)	PUNCT
cana-2695	239	21	,	,	PUNCT
cana-2695	239	22	and	and	CCONJ
cana-2695	239	23	the	the	DET
cana-2695	239	24	least	least	ADJ
cana-2695	239	25	cardinality	cardinality	NOUN
cana-2695	239	26	of	of	ADP
cana-2695	239	27	this	this	DET
cana-2695	239	28	set	set	NOUN
cana-2695	239	29	is	be	AUX
cana-2695	239	30	known	know	VERB
cana-2695	239	31	as	as	ADP
cana-2695	239	32	interval	interval	NOUN
cana-2695	239	33	-	-	PUNCT
cana-2695	239	34	valued	value	VERB
cana-2695	239	35	super	super	ADJ
cana-2695	239	36	resolving	resolve	VERB
cana-2695	239	37	number	number	NOUN
cana-2695	239	38	(	(	PUNCT
cana-2695	239	39	ivsrn	ivsrn	PROPN
cana-2695	239	40	)	)	PUNCT
cana-2695	239	41	which	which	PRON
cana-2695	239	42	is	be	AUX
cana-2695	239	43	denoted	denote	VERB
cana-2695	239	44	by	by	ADP
cana-2695	239	45	ℐsr(g	ℐsr(g	PROPN
cana-2695	239	46	)	)	PUNCT
cana-2695	239	47	,	,	PUNCT
cana-2695	239	48	if	if	SCONJ
cana-2695	239	49	we	we	PRON
cana-2695	239	50	arrange	arrange	VERB
cana-2695	239	51	the	the	DET
cana-2695	239	52	obtained	obtain	VERB
cana-2695	239	53	set	set	VERB
cana-2695	239	54	in	in	ADP
cana-2695	239	55	a	a	DET
cana-2695	239	56	row	row	NOUN
cana-2695	239	57	form	form	NOUN
cana-2695	239	58	,	,	PUNCT
cana-2695	239	59	then	then	ADV
cana-2695	239	60	it	it	PRON
cana-2695	239	61	is	be	AUX
cana-2695	239	62	named	name	VERB
cana-2695	239	63	as	as	ADP
cana-2695	239	64	interval	interval	NOUN
cana-2695	239	65	-	-	PUNCT
cana-2695	239	66	valued	value	VERB
cana-2695	239	67	fuzzy	fuzzy	ADJ
cana-2695	239	68	super	super	ADJ
cana-2695	239	69	resolving	resolve	VERB
cana-2695	239	70	matrix	matrix	NOUN
cana-2695	239	71	denoted	denote	VERB
cana-2695	239	72	as	as	ADP
cana-2695	239	73	[	[	X
cana-2695	239	74	𝑆𝑛×𝑘	𝑆𝑛×𝑘	NOUN
cana-2695	239	75	−	−	PROPN
cana-2695	239	76	,	,	PUNCT
cana-2695	239	77	𝑆𝑛×𝑘	𝑆𝑛×𝑘	PROPN
cana-2695	240	1	+	+	X
cana-2695	240	2	]	]	PUNCT
cana-2695	240	3	.	.	PUNCT
cana-2695	241	1	communications	communication	NOUN
cana-2695	241	2	on	on	ADP
cana-2695	241	3	applied	apply	VERB
cana-2695	241	4	nonlinear	nonlinear	ADJ
cana-2695	241	5	analysis	analysis	NOUN
cana-2695	241	6	issn	issn	NOUN
cana-2695	241	7	:	:	PUNCT
cana-2695	241	8	1074	1074	NUM
cana-2695	241	9	-	-	PUNCT
cana-2695	241	10	133x	133x	NUM
cana-2695	241	11	vol	vol	NOUN
cana-2695	241	12	32	32	NUM
cana-2695	241	13	no	no	NOUN
cana-2695	241	14	.	.	PUNCT
cana-2695	242	1	3s	3s	NUM
cana-2695	242	2	(	(	PUNCT
cana-2695	242	3	2025	2025	NUM
cana-2695	242	4	)	)	PUNCT
cana-2695	242	5	554	554	NUM
cana-2695	242	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2695	242	7	4.1	4.1	NUM
cana-2695	242	8	illustration	illustration	NOUN
cana-2695	242	9	fig	fig	NOUN
cana-2695	242	10	2	2	NUM
cana-2695	242	11	:	:	PUNCT
cana-2695	242	12	ivfg	ivfg	ADJ
cana-2695	242	13	𝑴𝑬	𝑴𝑬	PROPN
cana-2695	242	14	𝑺𝟏	𝑺𝟏	PROPN
cana-2695	242	15	𝑺𝟐	𝑺𝟐	PROPN
cana-2695	242	16	𝑺𝟑	𝑺𝟑	PROPN
cana-2695	242	17	𝑺𝟒	𝑺𝟒	PROPN
cana-2695	243	1	𝑺𝟓	𝑺𝟓	PROPN
cana-2695	243	2	𝑴𝑬	𝑴𝑬	PROPN
cana-2695	243	3	−	−	PROPN
cana-2695	243	4	𝑴𝑬	𝑴𝑬	PROPN
cana-2695	243	5	+	+	CCONJ
cana-2695	243	6	0.7	0.7	NUM
cana-2695	243	7	0.4	0.4	NUM
cana-2695	243	8	0.8	0.8	NUM
cana-2695	243	9	0.5	0.5	NUM
cana-2695	243	10	0.6	0.6	NUM
cana-2695	243	11	0.6	0.6	NUM
cana-2695	243	12	0.5	0.5	NUM
cana-2695	243	13	0.7	0.7	NUM
cana-2695	243	14	0.4	0.4	NUM
cana-2695	243	15	0.8	0.8	NUM
cana-2695	243	16	𝑴𝑭	𝑴𝑭	PROPN
cana-2695	243	17	𝑺𝟏𝑺𝟐	𝑺𝟏𝑺𝟐	PROPN
cana-2695	243	18	𝑺𝟏𝑺𝟑	𝑺𝟏𝑺𝟑	X
cana-2695	243	19	𝑺𝟒𝑺𝟐	𝑺𝟒𝑺𝟐	X
cana-2695	243	20	𝑺𝟑𝑺𝟒	𝑺𝟑𝑺𝟒	PROPN
cana-2695	243	21	𝑺𝟒𝑺𝟓	𝑺𝟒𝑺𝟓	NOUN
cana-2695	243	22	𝑴𝑭	𝑴𝑭	PROPN
cana-2695	243	23	−	−	PROPN
cana-2695	243	24	𝑴𝑭	𝑴𝑭	PROPN
cana-2695	244	1	+	+	CCONJ
cana-2695	244	2	0.6	0.6	NUM
cana-2695	244	3	0.4	0.4	NUM
cana-2695	244	4	0.5	0.5	NUM
cana-2695	244	5	0.3	0.3	NUM
cana-2695	244	6	0.5	0.5	NUM
cana-2695	244	7	0.4	0.4	NUM
cana-2695	244	8	0.3	0.3	NUM
cana-2695	244	9	0.6	0.6	NUM
cana-2695	244	10	0.4	0.4	NUM
cana-2695	244	11	0.7	0.7	NUM
cana-2695	244	12	the	the	DET
cana-2695	244	13	strength	strength	NOUN
cana-2695	244	14	of	of	ADP
cana-2695	244	15	connectedness	connectedness	NOUN
cana-2695	244	16	matrix	matrix	NOUN
cana-2695	244	17	is	be	AUX
cana-2695	244	18	s1	s1	PROPN
cana-2695	244	19	s2	s2	PROPN
cana-2695	244	20	s3	s3	PROPN
cana-2695	244	21	s4	s4	PROPN
cana-2695	244	22	s5	s5	PROPN
cana-2695	244	23	s1	s1	PROPN
cana-2695	244	24	0.7	0.7	NUM
cana-2695	244	25	0.6	0.6	NUM
cana-2695	244	26	0.5	0.5	NUM
cana-2695	244	27	0.5	0.5	NUM
cana-2695	244	28	0.4	0.4	NUM
cana-2695	244	29	s2	s2	NOUN
cana-2695	244	30	0.6	0.6	NUM
cana-2695	244	31	0.8	0.8	NUM
cana-2695	244	32	0.5	0.5	NUM
cana-2695	244	33	0.5	0.5	NUM
cana-2695	244	34	0.4	0.4	NUM
cana-2695	244	35	𝐶𝑖𝑗	𝐶𝑖𝑗	PROPN
cana-2695	244	36	−	−	NOUN
cana-2695	245	1	=	=	SYM
cana-2695	245	2	s3	s3	PROPN
cana-2695	245	3	0.5	0.5	NUM
cana-2695	245	4	0.5	0.5	NUM
cana-2695	245	5	0.6	0.6	NUM
cana-2695	245	6	0.5	0.5	NUM
cana-2695	245	7	0.4	0.4	NUM
cana-2695	245	8	s4	s4	PROPN
cana-2695	245	9	0.5	0.5	NUM
cana-2695	245	10	0.5	0.5	NUM
cana-2695	245	11	0.5	0.5	NUM
cana-2695	245	12	0.5	0.5	NUM
cana-2695	245	13	0.4	0.4	NUM
cana-2695	245	14	s5	s5	PROPN
cana-2695	245	15	0.4	0.4	NUM
cana-2695	245	16	0.4	0.4	NUM
cana-2695	245	17	0.4	0.4	NUM
cana-2695	245	18	0.4	0.4	NUM
cana-2695	245	19	0.4	0.4	NUM
cana-2695	245	20	s1	s1	PROPN
cana-2695	245	21	s2	s2	PROPN
cana-2695	245	22	s3	s3	PROPN
cana-2695	245	23	s4	s4	PROPN
cana-2695	245	24	s5	s5	PROPN
cana-2695	245	25	s1	s1	PROPN
cana-2695	245	26	0.4	0.4	NUM
cana-2695	245	27	0.4	0.4	NUM
cana-2695	245	28	0.4	0.4	NUM
cana-2695	245	29	0.4	0.4	NUM
cana-2695	245	30	0.4	0.4	NUM
cana-2695	245	31	s2	s2	NOUN
cana-2695	245	32	0.4	0.4	NUM
cana-2695	245	33	0.5	0.5	NUM
cana-2695	245	34	0.4	0.4	NUM
cana-2695	245	35	0.4	0.4	NUM
cana-2695	245	36	0.4	0.4	NUM
cana-2695	246	1	𝐶𝑖𝑗	𝐶𝑖𝑗	PROPN
cana-2695	246	2	−	−	NOUN
cana-2695	246	3	=	=	SYM
cana-2695	246	4	s3	s3	PROPN
cana-2695	246	5	0.4	0.4	NUM
cana-2695	246	6	0.4	0.4	NUM
cana-2695	246	7	0.6	0.6	NUM
cana-2695	246	8	0.6	0.6	NUM
cana-2695	246	9	0.6	0.6	NUM
cana-2695	246	10	s4	s4	PROPN
cana-2695	246	11	0.4	0.4	NUM
cana-2695	246	12	0.4	0.4	NUM
cana-2695	246	13	0.6	0.6	NUM
cana-2695	246	14	0.7	0.7	NUM
cana-2695	246	15	0.7	0.7	NUM
cana-2695	246	16	s5	s5	NOUN
cana-2695	246	17	0.4	0.4	NUM
cana-2695	246	18	0.4	0.4	NUM
cana-2695	246	19	0.6	0.6	NUM
cana-2695	246	20	0.7	0.7	NUM
cana-2695	246	21	0.8	0.8	NUM
cana-2695	246	22	communications	communication	NOUN
cana-2695	246	23	on	on	ADP
cana-2695	246	24	applied	apply	VERB
cana-2695	246	25	nonlinear	nonlinear	ADJ
cana-2695	246	26	analysis	analysis	NOUN
cana-2695	246	27	issn	issn	NOUN
cana-2695	246	28	:	:	PUNCT
cana-2695	246	29	1074	1074	NUM
cana-2695	246	30	-	-	PUNCT
cana-2695	246	31	133x	133x	NUM
cana-2695	246	32	vol	vol	NOUN
cana-2695	246	33	32	32	NUM
cana-2695	246	34	no	no	NOUN
cana-2695	246	35	.	.	PUNCT
cana-2695	247	1	3s	3s	NUM
cana-2695	247	2	(	(	PUNCT
cana-2695	247	3	2025	2025	NUM
cana-2695	247	4	)	)	PUNCT
cana-2695	247	5	555	555	NUM
cana-2695	247	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2695	247	7	the	the	DET
cana-2695	247	8	ivfsrs	ivfsrs	NOUN
cana-2695	247	9	is	be	AUX
cana-2695	247	10	{	{	PUNCT
cana-2695	247	11	𝑆2	𝑆2	PROPN
cana-2695	247	12	,	,	PUNCT
cana-2695	247	13	𝑆4	𝑆4	PROPN
cana-2695	247	14	,	,	PUNCT
cana-2695	247	15	𝑆5	𝑆5	PROPN
cana-2695	247	16	}	}	PUNCT
cana-2695	247	17	and	and	CCONJ
cana-2695	247	18	ivfsrm	ivfsrm	PROPN
cana-2695	247	19	is	be	AUX
cana-2695	247	20	written	write	VERB
cana-2695	247	21	as	as	ADP
cana-2695	247	22	,	,	PUNCT
cana-2695	247	23	[	[	PUNCT
cana-2695	247	24	0.6	0.6	NUM
cana-2695	247	25	0.5	0.5	NUM
cana-2695	247	26	0.4	0.4	NUM
cana-2695	247	27	0.8	0.8	NUM
cana-2695	247	28	0.5	0.5	NUM
cana-2695	247	29	0.4	0.4	NUM
cana-2695	247	30	0.5	0.5	NUM
cana-2695	247	31	0.6	0.6	NUM
cana-2695	247	32	0.4	0.4	NUM
cana-2695	247	33	0.5	0.5	NUM
cana-2695	247	34	0.5	0.5	NUM
cana-2695	247	35	0.4	0.4	NUM
cana-2695	247	36	0.4	0.4	NUM
cana-2695	247	37	0.4	0.4	NUM
cana-2695	247	38	0.4	0.4	NUM
cana-2695	247	39	]	]	PUNCT
cana-2695	247	40	and	and	CCONJ
cana-2695	247	41	[	[	PUNCT
cana-2695	247	42	0.4	0.4	NUM
cana-2695	247	43	0.4	0.4	NUM
cana-2695	247	44	0.4	0.4	NUM
cana-2695	247	45	0.5	0.5	NUM
cana-2695	247	46	0.4	0.4	NUM
cana-2695	247	47	0.4	0.4	NUM
cana-2695	247	48	0.4	0.4	NUM
cana-2695	247	49	0.6	0.6	NUM
cana-2695	247	50	0.6	0.6	NUM
cana-2695	247	51	0.4	0.4	NUM
cana-2695	247	52	0.6	0.6	NUM
cana-2695	247	53	0.7	0.7	NUM
cana-2695	247	54	0.4	0.4	NUM
cana-2695	247	55	0.6	0.6	NUM
cana-2695	247	56	0.8	0.8	NUM
cana-2695	247	57	]	]	PUNCT
cana-2695	247	58	hence	hence	ADV
cana-2695	247	59	ℐsr(g	ℐsr(g	PUNCT
cana-2695	247	60	)	)	PUNCT
cana-2695	248	1	=	=	SYM
cana-2695	249	1	3	3	X
cana-2695	249	2	.	.	X
cana-2695	249	3	4.2	4.2	NUM
cana-2695	249	4	corollary	corollary	NOUN
cana-2695	249	5	if	if	SCONJ
cana-2695	249	6	g	g	PROPN
cana-2695	249	7	and	and	CCONJ
cana-2695	249	8	g′	g′	NOUN
cana-2695	249	9	are	be	AUX
cana-2695	249	10	isomorphic	isomorphic	ADJ
cana-2695	249	11	,	,	PUNCT
cana-2695	249	12	then	then	ADV
cana-2695	249	13	ℐsr(g	ℐsr(g	PUNCT
cana-2695	249	14	)	)	PUNCT
cana-2695	250	1	≅	≅	PROPN
cana-2695	250	2	ℐsr(g′	ℐsr(g′	PROPN
cana-2695	250	3	)	)	PUNCT
cana-2695	250	4	.	.	PUNCT
cana-2695	251	1	4.3	4.3	NUM
cana-2695	251	2	theorem	theorem	VERB
cana-2695	251	3	an	an	DET
cana-2695	251	4	ivers	iver	NOUN
cana-2695	251	5	is	be	AUX
cana-2695	251	6	also	also	ADV
cana-2695	251	7	an	an	DET
cana-2695	251	8	ivfrs	ivfrs	NOUN
cana-2695	251	9	but	but	CCONJ
cana-2695	251	10	the	the	DET
cana-2695	251	11	converse	converse	NOUN
cana-2695	251	12	may	may	AUX
cana-2695	251	13	or	or	CCONJ
cana-2695	251	14	may	may	AUX
cana-2695	251	15	not	not	PART
cana-2695	251	16	be	be	AUX
cana-2695	251	17	true	true	ADJ
cana-2695	251	18	.	.	PUNCT
cana-2695	252	1	proof	proof	NOUN
cana-2695	252	2	:	:	PUNCT
cana-2695	252	3	let	let	VERB
cana-2695	252	4	g	g	PRON
cana-2695	252	5	be	be	AUX
cana-2695	252	6	an	an	DET
cana-2695	252	7	ivfg	ivfg	NOUN
cana-2695	252	8	with	with	ADP
cana-2695	252	9	‘	'	PUNCT
cana-2695	252	10	n	n	CCONJ
cana-2695	252	11	’	'	PUNCT
cana-2695	252	12	vertices	vertex	NOUN
cana-2695	252	13	.	.	PUNCT
cana-2695	253	1	let	let	VERB
cana-2695	253	2	ℋ	ℋ	PROPN
cana-2695	253	3	=	=	SYM
cana-2695	253	4	{	{	PUNCT
cana-2695	253	5	(	(	PUNCT
cana-2695	253	6	σ1	σ1	PROPN
cana-2695	253	7	−	−	PROPN
cana-2695	253	8	,	,	PUNCT
cana-2695	253	9	σ1	σ1	PROPN
cana-2695	253	10	+	+	PROPN
cana-2695	253	11	)	)	PUNCT
cana-2695	253	12	,	,	PUNCT
cana-2695	253	13	(	(	PUNCT
cana-2695	253	14	σ2	σ2	NOUN
cana-2695	253	15	−	−	PROPN
cana-2695	253	16	,	,	PUNCT
cana-2695	253	17	σ2	σ2	NOUN
cana-2695	253	18	+	+	PROPN
cana-2695	253	19	)	)	PUNCT
cana-2695	253	20	,	,	PUNCT
cana-2695	253	21	.	.	PUNCT
cana-2695	253	22	.	.	PUNCT
cana-2695	254	1	.	.	PUNCT
cana-2695	255	1	,	,	PUNCT
cana-2695	255	2	(	(	PUNCT
cana-2695	255	3	σm	σm	INTJ
cana-2695	255	4	−	−	PROPN
cana-2695	255	5	,	,	PUNCT
cana-2695	255	6	σm	σm	X
cana-2695	255	7	+	+	ADJ
cana-2695	255	8	)	)	PUNCT
cana-2695	255	9	}	}	PUNCT
cana-2695	255	10	be	be	AUX
cana-2695	255	11	an	an	DET
cana-2695	255	12	ivfrs	ivfrs	NOUN
cana-2695	255	13	of	of	ADP
cana-2695	255	14	g	g	NOUN
cana-2695	255	15	,	,	PUNCT
cana-2695	255	16	then	then	ADV
cana-2695	255	17	the	the	DET
cana-2695	255	18	representation	representation	NOUN
cana-2695	255	19	of	of	ADP
cana-2695	255	20	𝜎𝑖	𝜎𝑖	DET
cana-2695	255	21	−/ℋ	−/ℋ	NOUN
cana-2695	255	22	and	and	CCONJ
cana-2695	255	23	𝜎𝑖	𝜎𝑖	DET
cana-2695	255	24	+	+	NOUN
cana-2695	255	25	/ℋ	/ℋ	PROPN
cana-2695	255	26	,	,	PUNCT
cana-2695	255	27	for	for	ADP
cana-2695	255	28	i	i	PRON
cana-2695	255	29	=	=	VERB
cana-2695	255	30	m	m	VERB
cana-2695	255	31	+	+	NOUN
cana-2695	255	32	1	1	NUM
cana-2695	255	33	,	,	PUNCT
cana-2695	255	34	m	m	VERB
cana-2695	255	35	+	+	ADJ
cana-2695	255	36	2	2	NUM
cana-2695	255	37	,	,	PUNCT
cana-2695	255	38	.	.	PUNCT
cana-2695	255	39	.	.	PUNCT
cana-2695	255	40	.	.	PUNCT
cana-2695	256	1	n	n	PRON
cana-2695	256	2	are	be	AUX
cana-2695	256	3	all	all	ADV
cana-2695	256	4	distinct	distinct	ADJ
cana-2695	256	5	.	.	PUNCT
cana-2695	257	1	also	also	ADV
cana-2695	257	2	,	,	PUNCT
cana-2695	257	3	𝜎𝑖	𝜎𝑖	X
cana-2695	257	4	−/ℋ	−/ℋ	NOUN
cana-2695	257	5	and	and	CCONJ
cana-2695	257	6	𝜎𝑖	𝜎𝑖	DET
cana-2695	257	7	+	+	ADJ
cana-2695	257	8	/ℋ	/ℋ	NOUN
cana-2695	257	9	need	need	VERB
cana-2695	257	10	not	not	PART
cana-2695	257	11	to	to	PART
cana-2695	257	12	be	be	AUX
cana-2695	257	13	distinct	distinct	ADJ
cana-2695	257	14	for	for	ADP
cana-2695	257	15	i	i	X
cana-2695	257	16	=	=	NOUN
cana-2695	257	17	1,2	1,2	NUM
cana-2695	257	18	,	,	PUNCT
cana-2695	257	19	.	.	PUNCT
cana-2695	257	20	.	.	PUNCT
cana-2695	257	21	.	.	PUNCT
cana-2695	258	1	n.	n.	NOUN
cana-2695	258	2	(	(	PUNCT
cana-2695	258	3	i.	i.	NOUN
cana-2695	258	4	,e	,e	PUNCT
cana-2695	258	5	)	)	PUNCT
cana-2695	258	6	ℋ	ℋ	PROPN
cana-2695	258	7	does	do	AUX
cana-2695	258	8	not	not	PART
cana-2695	258	9	need	need	VERB
cana-2695	258	10	to	to	PART
cana-2695	258	11	be	be	AUX
cana-2695	258	12	an	an	DET
cana-2695	258	13	ivfsrs	ivfsrs	NOUN
cana-2695	258	14	.	.	PUNCT
cana-2695	259	1	conversely	conversely	ADV
cana-2695	259	2	,	,	PUNCT
cana-2695	259	3	let	let	VERB
cana-2695	259	4	ℋ	ℋ	PROPN
cana-2695	259	5	=	=	SYM
cana-2695	259	6	{	{	PUNCT
cana-2695	259	7	(	(	PUNCT
cana-2695	259	8	σ1	σ1	PROPN
cana-2695	259	9	−	−	PROPN
cana-2695	259	10	,	,	PUNCT
cana-2695	259	11	σ1	σ1	PROPN
cana-2695	259	12	+	+	PROPN
cana-2695	259	13	)	)	PUNCT
cana-2695	259	14	,	,	PUNCT
cana-2695	259	15	(	(	PUNCT
cana-2695	259	16	σ2	σ2	NOUN
cana-2695	259	17	−	−	PROPN
cana-2695	259	18	,	,	PUNCT
cana-2695	259	19	σ2	σ2	NOUN
cana-2695	259	20	+	+	PROPN
cana-2695	259	21	)	)	PUNCT
cana-2695	259	22	,	,	PUNCT
cana-2695	259	23	.	.	PUNCT
cana-2695	259	24	.	.	PUNCT
cana-2695	259	25	.	.	PUNCT
cana-2695	260	1	,	,	PUNCT
cana-2695	260	2	(	(	PUNCT
cana-2695	260	3	σm	σm	INTJ
cana-2695	260	4	−	−	PROPN
cana-2695	260	5	,	,	PUNCT
cana-2695	260	6	σm	σm	X
cana-2695	261	1	+	+	ADJ
cana-2695	261	2	)	)	PUNCT
cana-2695	261	3	}	}	PUNCT
cana-2695	261	4	is	be	AUX
cana-2695	261	5	an	an	DET
cana-2695	261	6	ivfsrs	ivfsrs	NOUN
cana-2695	261	7	of	of	ADP
cana-2695	261	8	g	g	NOUN
cana-2695	261	9	,	,	PUNCT
cana-2695	261	10	then	then	ADV
cana-2695	261	11	𝜎𝑖	𝜎𝑖	NUM
cana-2695	261	12	−/ℋ	−/ℋ	NOUN
cana-2695	261	13	and	and	CCONJ
cana-2695	261	14	𝜎𝑖	𝜎𝑖	DET
cana-2695	261	15	+	+	NOUN
cana-2695	261	16	/ℋ	/ℋ	PROPN
cana-2695	261	17	,	,	PUNCT
cana-2695	261	18	for	for	ADP
cana-2695	261	19	i	i	PROPN
cana-2695	261	20	=	=	SYM
cana-2695	261	21	1,2	1,2	NUM
cana-2695	261	22	,	,	PUNCT
cana-2695	261	23	.	.	PUNCT
cana-2695	261	24	.	.	PUNCT
cana-2695	261	25	.	.	PUNCT
cana-2695	262	1	m	m	VERB
cana-2695	263	1	+	+	ADJ
cana-2695	263	2	1	1	NUM
cana-2695	263	3	,	,	PUNCT
cana-2695	263	4	.	.	PUNCT
cana-2695	263	5	.	.	PUNCT
cana-2695	263	6	.	.	PUNCT
cana-2695	264	1	n	n	CCONJ
cana-2695	264	2	,	,	PUNCT
cana-2695	264	3	are	be	AUX
cana-2695	264	4	all	all	ADV
cana-2695	264	5	distinct	distinct	ADJ
cana-2695	264	6	which	which	PRON
cana-2695	264	7	means	mean	VERB
cana-2695	264	8	that	that	SCONJ
cana-2695	264	9	the	the	DET
cana-2695	264	10	representation	representation	NOUN
cana-2695	264	11	of	of	ADP
cana-2695	264	12	𝜎𝑖	𝜎𝑖	DET
cana-2695	264	13	−/ℋ	−/ℋ	NOUN
cana-2695	264	14	and	and	CCONJ
cana-2695	264	15	𝜎𝑖	𝜎𝑖	DET
cana-2695	264	16	+	+	NOUN
cana-2695	264	17	/ℋ	/ℋ	PROPN
cana-2695	264	18	,	,	PUNCT
cana-2695	264	19	for	for	ADP
cana-2695	264	20	i	i	PRON
cana-2695	264	21	=	=	VERB
cana-2695	264	22	m	m	VERB
cana-2695	264	23	+	+	NOUN
cana-2695	264	24	1	1	NUM
cana-2695	264	25	,	,	PUNCT
cana-2695	264	26	m	m	VERB
cana-2695	264	27	+	+	ADJ
cana-2695	264	28	2	2	NUM
cana-2695	264	29	,	,	PUNCT
cana-2695	264	30	.	.	PUNCT
cana-2695	264	31	.	.	PUNCT
cana-2695	264	32	.	.	PUNCT
cana-2695	265	1	n	n	CCONJ
cana-2695	265	2	,	,	PUNCT
cana-2695	265	3	are	be	AUX
cana-2695	265	4	all	all	ADV
cana-2695	265	5	distinct	distinct	ADJ
cana-2695	265	6	,	,	PUNCT
cana-2695	265	7	hence	hence	ADV
cana-2695	265	8	,	,	PUNCT
cana-2695	265	9	ℋ	ℋ	PROPN
cana-2695	265	10	is	be	AUX
cana-2695	265	11	also	also	ADV
cana-2695	265	12	an	an	DET
cana-2695	265	13	ivfrs	ivfrs	NOUN
cana-2695	265	14	.	.	PUNCT
cana-2695	266	1	4.4	4.4	NUM
cana-2695	266	2	theorem	theorem	VERB
cana-2695	266	3	for	for	ADP
cana-2695	266	4	an	an	DET
cana-2695	266	5	interval	interval	NOUN
cana-2695	266	6	-	-	PUNCT
cana-2695	266	7	valued	value	VERB
cana-2695	266	8	fuzzy	fuzzy	ADJ
cana-2695	266	9	star	star	NOUN
cana-2695	266	10	graph	graph	NOUN
cana-2695	266	11	(	(	PUNCT
cana-2695	266	12	ivfsg	ivfsg	ADJ
cana-2695	266	13	)	)	PUNCT
cana-2695	266	14	with	with	ADP
cana-2695	266	15	distinct	distinct	ADJ
cana-2695	266	16	membership	membership	NOUN
cana-2695	266	17	values	value	NOUN
cana-2695	266	18	,	,	PUNCT
cana-2695	266	19	∃	∃	PROPN
cana-2695	266	20	any	any	DET
cana-2695	266	21	two	two	NUM
cana-2695	266	22	vertices	vertex	NOUN
cana-2695	266	23	∋	∋	NOUN
cana-2695	266	24	r	r	NOUN
cana-2695	266	25	,	,	PUNCT
cana-2695	266	26	u	u	NOUN
cana-2695	266	27	∈	∈	PROPN
cana-2695	266	28	e	e	NOUN
cana-2695	266	29	,	,	PUNCT
cana-2695	266	30	ℐsr(g	ℐsr(g	NOUN
cana-2695	266	31	)	)	PUNCT
cana-2695	266	32	=	=	PUNCT
cana-2695	267	1	ℐr(g	ℐr(g	ADV
cana-2695	267	2	)	)	PUNCT
cana-2695	267	3	=	=	SYM
cana-2695	267	4	2	2	NUM
cana-2695	267	5	proof	proof	NOUN
cana-2695	267	6	:	:	PUNCT
cana-2695	267	7	by	by	ADP
cana-2695	267	8	the	the	DET
cana-2695	267	9	definition	definition	NOUN
cana-2695	267	10	of	of	ADP
cana-2695	267	11	ivfsg	ivfsg	ADJ
cana-2695	267	12	,	,	PUNCT
cana-2695	267	13	𝜇𝐸	𝜇𝐸	PROPN
cana-2695	267	14	−(𝑟	−(𝑟	PROPN
cana-2695	267	15	,	,	PUNCT
cana-2695	267	16	𝑢𝑖	𝑢𝑖	PROPN
cana-2695	267	17	)	)	PUNCT
cana-2695	267	18	>	>	X
cana-2695	267	19	0	0	NUM
cana-2695	267	20	,	,	PUNCT
cana-2695	267	21	𝜇𝐸	𝜇𝐸	PROPN
cana-2695	267	22	+	+	PROPN
cana-2695	267	23	(	(	PUNCT
cana-2695	267	24	𝑟	𝑟	NOUN
cana-2695	267	25	,	,	PUNCT
cana-2695	267	26	𝑢𝑖	𝑢𝑖	ADP
cana-2695	267	27	)	)	PUNCT
cana-2695	267	28	>	>	X
cana-2695	267	29	0	0	NUM
cana-2695	267	30	,	,	PUNCT
cana-2695	267	31	𝜇𝐸	𝜇𝐸	PROPN
cana-2695	267	32	−(𝑢𝑖	−(𝑢𝑖	PROPN
cana-2695	267	33	,	,	PUNCT
cana-2695	267	34	𝑢𝑖+1	𝑢𝑖+1	PROPN
cana-2695	267	35	)	)	PUNCT
cana-2695	267	36	=	=	SYM
cana-2695	267	37	0	0	NUM
cana-2695	267	38	,	,	PUNCT
cana-2695	267	39	𝜇𝐸	𝜇𝐸	PROPN
cana-2695	267	40	−(𝑢𝑖	−(𝑢𝑖	PROPN
cana-2695	267	41	,	,	PUNCT
cana-2695	267	42	𝑢𝑖+1	𝑢𝑖+1	PROPN
cana-2695	267	43	)	)	PUNCT
cana-2695	267	44	=	=	SYM
cana-2695	267	45	0	0	PUNCT
cana-2695	268	1	for	for	ADP
cana-2695	268	2	i	i	PRON
cana-2695	268	3	=	=	SYM
cana-2695	268	4	1,2	1,2	NUM
cana-2695	268	5	,	,	PUNCT
cana-2695	268	6	.	.	PUNCT
cana-2695	268	7	.	.	PUNCT
cana-2695	268	8	.	.	PUNCT
cana-2695	269	1	n.	n.	INTJ
cana-2695	269	2	it	it	PRON
cana-2695	269	3	is	be	AUX
cana-2695	269	4	evident	evident	ADJ
cana-2695	269	5	that	that	SCONJ
cana-2695	269	6	we	we	PRON
cana-2695	269	7	can	can	AUX
cana-2695	269	8	find	find	VERB
cana-2695	269	9	any	any	DET
cana-2695	269	10	two	two	NUM
cana-2695	269	11	vertices	vertex	NOUN
cana-2695	269	12	with	with	ADP
cana-2695	269	13	distinct	distinct	ADJ
cana-2695	269	14	membership	membership	NOUN
cana-2695	269	15	values	value	NOUN
cana-2695	269	16	in	in	ADP
cana-2695	269	17	the	the	DET
cana-2695	269	18	ivfsrm	ivfsrm	NOUN
cana-2695	269	19	as	as	ADV
cana-2695	269	20	well	well	ADV
cana-2695	269	21	as	as	ADP
cana-2695	269	22	ivfrm	ivfrm	NOUN
cana-2695	269	23	.	.	PUNCT
cana-2695	270	1	hence	hence	ADV
cana-2695	270	2	,	,	PUNCT
cana-2695	270	3	ℐsr(g	ℐsr(g	NOUN
cana-2695	270	4	)	)	PUNCT
cana-2695	270	5	=	=	PUNCT
cana-2695	271	1	ℐr(g	ℐr(g	ADV
cana-2695	271	2	)	)	PUNCT
cana-2695	271	3	=	=	SYM
cana-2695	271	4	2	2	NUM
cana-2695	271	5	.	.	NOUN
cana-2695	271	6	5	5	NUM
cana-2695	271	7	.	.	X
cana-2695	271	8	application	application	NOUN
cana-2695	271	9	in	in	ADP
cana-2695	271	10	several	several	ADJ
cana-2695	271	11	domains	domain	NOUN
cana-2695	271	12	,	,	PUNCT
cana-2695	271	13	interval	interval	NOUN
cana-2695	271	14	-	-	PUNCT
cana-2695	271	15	valued	value	VERB
cana-2695	271	16	fuzzy	fuzzy	ADJ
cana-2695	271	17	graphs	graph	NOUN
cana-2695	271	18	(	(	PUNCT
cana-2695	271	19	ivfgs	ivfg	VERB
cana-2695	271	20	)	)	PUNCT
cana-2695	271	21	are	be	AUX
cana-2695	271	22	employed	employ	VERB
cana-2695	271	23	to	to	PART
cana-2695	271	24	manage	manage	VERB
cana-2695	271	25	imprecision	imprecision	NOUN
cana-2695	271	26	and	and	CCONJ
cana-2695	271	27	uncertainty.compared	uncertainty.compare	VERB
cana-2695	271	28	to	to	ADP
cana-2695	271	29	crisp	crisp	ADJ
cana-2695	271	30	or	or	CCONJ
cana-2695	271	31	normal	normal	ADJ
cana-2695	271	32	fuzzy	fuzzy	ADJ
cana-2695	271	33	graphs	graph	NOUN
cana-2695	271	34	,	,	PUNCT
cana-2695	271	35	interval	interval	NOUN
cana-2695	271	36	-	-	PUNCT
cana-2695	271	37	valued	value	VERB
cana-2695	271	38	fuzzy	fuzzy	ADJ
cana-2695	271	39	graphs	graph	NOUN
cana-2695	271	40	(	(	PUNCT
cana-2695	271	41	ivfgs	ivfg	VERB
cana-2695	271	42	)	)	PUNCT
cana-2695	271	43	are	be	AUX
cana-2695	271	44	better	well	ADJ
cana-2695	271	45	at	at	ADP
cana-2695	271	46	capturing	capture	VERB
cana-2695	271	47	imprecision	imprecision	NOUN
cana-2695	271	48	because	because	SCONJ
cana-2695	271	49	they	they	PRON
cana-2695	271	50	use	use	VERB
cana-2695	271	51	intervals	interval	NOUN
cana-2695	271	52	for	for	ADP
cana-2695	271	53	edge	edge	NOUN
cana-2695	271	54	and	and	CCONJ
cana-2695	271	55	vertex	vertex	NOUN
cana-2695	271	56	values	value	NOUN
cana-2695	271	57	to	to	PART
cana-2695	271	58	depict	depict	VERB
cana-2695	271	59	interactions	interaction	NOUN
cana-2695	271	60	with	with	ADP
cana-2695	271	61	uncertainty	uncertainty	NOUN
cana-2695	271	62	.	.	PUNCT
cana-2695	272	1	they	they	PRON
cana-2695	272	2	are	be	AUX
cana-2695	272	3	used	use	VERB
cana-2695	272	4	to	to	PART
cana-2695	272	5	efficiently	efficiently	ADV
cana-2695	272	6	handle	handle	VERB
cana-2695	272	7	varying	vary	VERB
cana-2695	272	8	or	or	CCONJ
cana-2695	272	9	imprecise	imprecise	ADJ
cana-2695	272	10	data	datum	NOUN
cana-2695	272	11	in	in	ADP
cana-2695	272	12	uncertain	uncertain	ADJ
cana-2695	272	13	systems	system	NOUN
cana-2695	272	14	such	such	ADJ
cana-2695	272	15	as	as	ADP
cana-2695	272	16	networks	network	NOUN
cana-2695	272	17	,	,	PUNCT
cana-2695	272	18	decision	decision	NOUN
cana-2695	272	19	-	-	PUNCT
cana-2695	272	20	making	making	NOUN
cana-2695	272	21	,	,	PUNCT
cana-2695	272	22	biology	biology	NOUN
cana-2695	272	23	,	,	PUNCT
cana-2695	272	24	and	and	CCONJ
cana-2695	272	25	transportation	transportation	NOUN
cana-2695	272	26	.	.	PUNCT
cana-2695	273	1	ivfgs	ivfgs	PROPN
cana-2695	273	2	use	use	VERB
cana-2695	273	3	interval	interval	NOUN
cana-2695	273	4	values	value	NOUN
cana-2695	273	5	to	to	PART
cana-2695	273	6	represent	represent	VERB
cana-2695	273	7	relationships	relationship	NOUN
cana-2695	273	8	in	in	ADP
cana-2695	273	9	social	social	ADJ
cana-2695	273	10	networks	network	NOUN
cana-2695	273	11	that	that	PRON
cana-2695	273	12	have	have	VERB
cana-2695	273	13	unclear	unclear	ADJ
cana-2695	273	14	strengths	strength	NOUN
cana-2695	273	15	,	,	PUNCT
cana-2695	273	16	such	such	ADJ
cana-2695	273	17	as	as	ADP
cana-2695	273	18	differing	differ	VERB
cana-2695	273	19	degrees	degree	NOUN
cana-2695	273	20	of	of	ADP
cana-2695	273	21	influence	influence	NOUN
cana-2695	273	22	,	,	PUNCT
cana-2695	273	23	friendship	friendship	NOUN
cana-2695	273	24	,	,	PUNCT
cana-2695	273	25	or	or	CCONJ
cana-2695	273	26	trust	trust	NOUN
cana-2695	273	27	.	.	PUNCT
cana-2695	274	1	this	this	DET
cana-2695	274	2	aids	aid	NOUN
cana-2695	274	3	in	in	ADP
cana-2695	274	4	identifying	identify	VERB
cana-2695	274	5	important	important	ADJ
cana-2695	274	6	influencers	influencer	NOUN
cana-2695	274	7	or	or	CCONJ
cana-2695	274	8	clusters	cluster	NOUN
cana-2695	274	9	and	and	CCONJ
cana-2695	274	10	analysing	analyse	VERB
cana-2695	274	11	dynamic	dynamic	ADJ
cana-2695	274	12	,	,	PUNCT
cana-2695	274	13	imprecise	imprecise	ADJ
cana-2695	274	14	interactions	interaction	NOUN
cana-2695	274	15	.	.	PUNCT
cana-2695	275	1	ivfgs	ivfgs	PROPN
cana-2695	275	2	are	be	AUX
cana-2695	275	3	used	use	VERB
cana-2695	275	4	in	in	ADP
cana-2695	275	5	signal	signal	ADJ
cana-2695	275	6	processing	processing	NOUN
cana-2695	275	7	to	to	PART
cana-2695	275	8	describe	describe	VERB
cana-2695	275	9	uncertain	uncertain	ADJ
cana-2695	275	10	relationships	relationship	NOUN
cana-2695	275	11	between	between	ADP
cana-2695	275	12	elements	element	NOUN
cana-2695	275	13	,	,	PUNCT
cana-2695	275	14	such	such	ADJ
cana-2695	275	15	as	as	ADP
cana-2695	275	16	varying	vary	VERB
cana-2695	275	17	noise	noise	NOUN
cana-2695	275	18	levels	level	NOUN
cana-2695	275	19	or	or	CCONJ
cana-2695	275	20	signal	signal	ADJ
cana-2695	275	21	intensities	intensity	NOUN
cana-2695	275	22	.	.	PUNCT
cana-2695	276	1	they	they	PRON
cana-2695	276	2	help	help	VERB
cana-2695	276	3	with	with	ADP
cana-2695	276	4	data	datum	NOUN
cana-2695	276	5	transmission	transmission	NOUN
cana-2695	276	6	uncertainty	uncertainty	NOUN
cana-2695	276	7	management	management	NOUN
cana-2695	276	8	,	,	PUNCT
cana-2695	276	9	pathway	pathway	NOUN
cana-2695	276	10	optimisation	optimisation	NOUN
cana-2695	276	11	,	,	PUNCT
cana-2695	276	12	and	and	CCONJ
cana-2695	276	13	robust	robust	ADJ
cana-2695	276	14	system	system	NOUN
cana-2695	276	15	design	design	NOUN
cana-2695	276	16	.	.	PUNCT
cana-2695	277	1	communications	communication	NOUN
cana-2695	277	2	on	on	ADP
cana-2695	277	3	applied	apply	VERB
cana-2695	277	4	nonlinear	nonlinear	ADJ
cana-2695	277	5	analysis	analysis	NOUN
cana-2695	277	6	issn	issn	NOUN
cana-2695	277	7	:	:	PUNCT
cana-2695	277	8	1074	1074	NUM
cana-2695	277	9	-	-	PUNCT
cana-2695	277	10	133x	133x	NUM
cana-2695	277	11	vol	vol	NOUN
cana-2695	277	12	32	32	NUM
cana-2695	277	13	no	no	NOUN
cana-2695	277	14	.	.	PUNCT
cana-2695	278	1	3s	3s	NUM
cana-2695	278	2	(	(	PUNCT
cana-2695	278	3	2025	2025	NUM
cana-2695	278	4	)	)	PUNCT
cana-2695	278	5	556	556	NUM
cana-2695	278	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2695	278	7	transportation	transportation	NOUN
cana-2695	278	8	networks	network	NOUN
cana-2695	278	9	,	,	PUNCT
cana-2695	278	10	such	such	ADJ
cana-2695	278	11	as	as	ADP
cana-2695	278	12	airline	airline	NOUN
cana-2695	278	13	or	or	CCONJ
cana-2695	278	14	urban	urban	ADJ
cana-2695	278	15	transit	transit	NOUN
cana-2695	278	16	systems	system	NOUN
cana-2695	278	17	,	,	PUNCT
cana-2695	278	18	are	be	AUX
cana-2695	278	19	a	a	DET
cana-2695	278	20	real	real	ADJ
cana-2695	278	21	-	-	PUNCT
cana-2695	278	22	world	world	NOUN
cana-2695	278	23	example	example	NOUN
cana-2695	278	24	of	of	ADP
cana-2695	278	25	an	an	DET
cana-2695	278	26	interval	interval	NOUN
cana-2695	278	27	-	-	PUNCT
cana-2695	278	28	valued	value	VERB
cana-2695	278	29	fuzzy	fuzzy	ADJ
cana-2695	278	30	graph	graph	NOUN
cana-2695	278	31	(	(	PUNCT
cana-2695	278	32	ivfg	ivfg	NOUN
cana-2695	278	33	)	)	PUNCT
cana-2695	278	34	with	with	ADP
cana-2695	278	35	a	a	DET
cana-2695	278	36	significant	significant	ADJ
cana-2695	278	37	number	number	NOUN
cana-2695	278	38	of	of	ADP
cana-2695	278	39	vertices	vertex	NOUN
cana-2695	278	40	.	.	PUNCT
cana-2695	279	1	we	we	PRON
cana-2695	279	2	simulate	simulate	VERB
cana-2695	279	3	the	the	DET
cana-2695	279	4	unpredictability	unpredictability	NOUN
cana-2695	279	5	of	of	ADP
cana-2695	279	6	passenger	passenger	NOUN
cana-2695	279	7	demand	demand	NOUN
cana-2695	279	8	and	and	CCONJ
cana-2695	279	9	flight	flight	NOUN
cana-2695	279	10	connectivity	connectivity	NOUN
cana-2695	279	11	between	between	ADP
cana-2695	279	12	cities	city	NOUN
cana-2695	279	13	.	.	PUNCT
cana-2695	280	1	every	every	DET
cana-2695	280	2	vertex	vertex	NOUN
cana-2695	280	3	denotes	denote	VERB
cana-2695	280	4	a	a	DET
cana-2695	280	5	city	city	NOUN
cana-2695	280	6	,	,	PUNCT
cana-2695	280	7	such	such	ADJ
cana-2695	280	8	as	as	ADP
cana-2695	280	9	tokyo	tokyo	PROPN
cana-2695	280	10	,	,	PUNCT
cana-2695	280	11	london	london	PROPN
cana-2695	280	12	,	,	PUNCT
cana-2695	280	13	or	or	CCONJ
cana-2695	280	14	new	new	PROPN
cana-2695	280	15	york	york	PROPN
cana-2695	280	16	.	.	PUNCT
cana-2695	281	1	this	this	PRON
cana-2695	281	2	might	might	AUX
cana-2695	281	3	include	include	VERB
cana-2695	281	4	hundreds	hundred	NOUN
cana-2695	281	5	or	or	CCONJ
cana-2695	281	6	thousands	thousand	NOUN
cana-2695	281	7	of	of	ADP
cana-2695	281	8	cities	city	NOUN
cana-2695	281	9	for	for	ADP
cana-2695	281	10	a	a	DET
cana-2695	281	11	global	global	ADJ
cana-2695	281	12	airline	airline	NOUN
cana-2695	281	13	network	network	NOUN
cana-2695	281	14	.	.	PUNCT
cana-2695	282	1	every	every	DET
cana-2695	282	2	edge	edge	NOUN
cana-2695	282	3	depicts	depict	VERB
cana-2695	282	4	a	a	DET
cana-2695	282	5	flight	flight	NOUN
cana-2695	282	6	path	path	NOUN
cana-2695	282	7	between	between	ADP
cana-2695	282	8	two	two	NUM
cana-2695	282	9	cities	city	NOUN
cana-2695	282	10	,	,	PUNCT
cana-2695	282	11	with	with	ADP
cana-2695	282	12	an	an	DET
cana-2695	282	13	interval	interval	NOUN
cana-2695	282	14	signifying	signify	VERB
cana-2695	282	15	the	the	DET
cana-2695	282	16	degree	degree	NOUN
cana-2695	282	17	of	of	ADP
cana-2695	282	18	popularity	popularity	NOUN
cana-2695	282	19	or	or	CCONJ
cana-2695	282	20	dependability	dependability	NOUN
cana-2695	282	21	of	of	ADP
cana-2695	282	22	the	the	DET
cana-2695	282	23	route	route	NOUN
cana-2695	282	24	.	.	PUNCT
cana-2695	283	1	the	the	DET
cana-2695	283	2	uncertainty	uncertainty	NOUN
cana-2695	283	3	of	of	ADP
cana-2695	283	4	a	a	DET
cana-2695	283	5	city	city	NOUN
cana-2695	283	6	's	's	PART
cana-2695	283	7	traffic	traffic	NOUN
cana-2695	283	8	capacity	capacity	NOUN
cana-2695	283	9	or	or	CCONJ
cana-2695	283	10	significance	significance	NOUN
cana-2695	283	11	in	in	ADP
cana-2695	283	12	the	the	DET
cana-2695	283	13	network	network	NOUN
cana-2695	283	14	is	be	AUX
cana-2695	283	15	indicated	indicate	VERB
cana-2695	283	16	by	by	ADP
cana-2695	283	17	the	the	DET
cana-2695	283	18	vertex	vertex	NOUN
cana-2695	283	19	membership	membership	NOUN
cana-2695	283	20	value	value	NOUN
cana-2695	283	21	.	.	PUNCT
cana-2695	284	1	uncertainty	uncertainty	NOUN
cana-2695	284	2	in	in	ADP
cana-2695	284	3	passenger	passenger	NOUN
cana-2695	284	4	demand	demand	NOUN
cana-2695	284	5	or	or	CCONJ
cana-2695	284	6	flight	flight	NOUN
cana-2695	284	7	dependability	dependability	NOUN
cana-2695	284	8	is	be	AUX
cana-2695	284	9	represented	represent	VERB
cana-2695	284	10	by	by	ADP
cana-2695	284	11	the	the	DET
cana-2695	284	12	edge	edge	NOUN
cana-2695	284	13	membership	membership	NOUN
cana-2695	284	14	value	value	NOUN
cana-2695	284	15	.	.	PUNCT
cana-2695	285	1	the	the	DET
cana-2695	285	2	benefits	benefit	NOUN
cana-2695	285	3	includes	include	VERB
cana-2695	285	4	assisting	assist	VERB
cana-2695	285	5	airlines	airline	NOUN
cana-2695	285	6	in	in	ADP
cana-2695	285	7	optimising	optimise	VERB
cana-2695	285	8	their	their	PRON
cana-2695	285	9	schedules	schedule	NOUN
cana-2695	285	10	in	in	ADP
cana-2695	285	11	spite	spite	NOUN
cana-2695	285	12	of	of	ADP
cana-2695	285	13	demand	demand	NOUN
cana-2695	285	14	fluctuations	fluctuation	NOUN
cana-2695	285	15	,	,	PUNCT
cana-2695	285	16	helps	helps	AUX
cana-2695	285	17	identify	identify	VERB
cana-2695	285	18	important	important	ADJ
cana-2695	285	19	cities	city	NOUN
cana-2695	285	20	or	or	CCONJ
cana-2695	285	21	routes	route	NOUN
cana-2695	285	22	for	for	ADP
cana-2695	285	23	investment	investment	NOUN
cana-2695	285	24	,	,	PUNCT
cana-2695	285	25	beneficial	beneficial	ADJ
cana-2695	285	26	for	for	ADP
cana-2695	285	27	managing	manage	VERB
cana-2695	285	28	disruptions	disruption	NOUN
cana-2695	285	29	(	(	PUNCT
cana-2695	285	30	such	such	ADJ
cana-2695	285	31	as	as	ADP
cana-2695	285	32	delays	delay	NOUN
cana-2695	285	33	and	and	CCONJ
cana-2695	285	34	bad	bad	ADJ
cana-2695	285	35	weather	weather	NOUN
cana-2695	285	36	)	)	PUNCT
cana-2695	285	37	.	.	PUNCT
cana-2695	286	1	these	these	DET
cana-2695	286	2	models	model	NOUN
cana-2695	286	3	effectively	effectively	ADV
cana-2695	286	4	handle	handle	VERB
cana-2695	286	5	big	big	ADJ
cana-2695	286	6	,	,	PUNCT
cana-2695	286	7	complicated	complicated	ADJ
cana-2695	286	8	systems	system	NOUN
cana-2695	286	9	with	with	ADP
cana-2695	286	10	a	a	DET
cana-2695	286	11	lot	lot	NOUN
cana-2695	286	12	of	of	ADP
cana-2695	286	13	uncertainty	uncertainty	NOUN
cana-2695	286	14	.	.	PUNCT
cana-2695	287	1	for	for	ADP
cana-2695	287	2	example	example	NOUN
cana-2695	287	3	,	,	PUNCT
cana-2695	287	4	𝑀𝐸	𝑀𝐸	PROPN
cana-2695	287	5	(	(	PUNCT
cana-2695	287	6	𝐴1	𝐴1	PROPN
cana-2695	287	7	:	:	PUNCT
cana-2695	287	8	london	london	PROPN
cana-2695	287	9	)	)	PUNCT
cana-2695	287	10	=	=	PUNCT
cana-2695	287	11	(	(	PUNCT
cana-2695	287	12	0.5	0.5	NUM
cana-2695	287	13	,	,	PUNCT
cana-2695	287	14	0.4	0.4	NUM
cana-2695	287	15	)	)	PUNCT
cana-2695	287	16	,	,	PUNCT
cana-2695	287	17	𝑀𝐸	𝑀𝐸	PROPN
cana-2695	287	18	(	(	PUNCT
cana-2695	287	19	𝐴2	𝐴2	PROPN
cana-2695	287	20	:	:	PUNCT
cana-2695	287	21	new	new	PROPN
cana-2695	287	22	york	york	PROPN
cana-2695	287	23	)	)	PUNCT
cana-2695	288	1	=	=	PRON
cana-2695	288	2	(	(	PUNCT
cana-2695	288	3	0.7	0.7	NUM
cana-2695	288	4	,	,	PUNCT
cana-2695	288	5	0.6	0.6	NUM
cana-2695	288	6	)	)	PUNCT
cana-2695	288	7	,	,	PUNCT
cana-2695	288	8	𝑀𝐸	𝑀𝐸	PROPN
cana-2695	288	9	(	(	PUNCT
cana-2695	288	10	𝐴3	𝐴3	PROPN
cana-2695	288	11	:	:	PUNCT
cana-2695	288	12	malaysia	malaysia	PROPN
cana-2695	288	13	)	)	PUNCT
cana-2695	288	14	=	=	PUNCT
cana-2695	289	1	(	(	PUNCT
cana-2695	289	2	0.8	0.8	NUM
cana-2695	289	3	,	,	PUNCT
cana-2695	289	4	0.6	0.6	NUM
cana-2695	289	5	)	)	PUNCT
cana-2695	290	1	and	and	CCONJ
cana-2695	290	2	so	so	ADV
cana-2695	290	3	on	on	ADV
cana-2695	290	4	.	.	PUNCT
cana-2695	291	1	also	also	ADV
cana-2695	291	2	,	,	PUNCT
cana-2695	291	3	𝑀𝐹	𝑀𝐹	PROPN
cana-2695	291	4	(	(	PUNCT
cana-2695	291	5	𝐴1𝐴2	𝐴1𝐴2	NOUN
cana-2695	291	6	)	)	PUNCT
cana-2695	291	7	=	=	PUNCT
cana-2695	291	8	(	(	PUNCT
cana-2695	291	9	0.4	0.4	NUM
cana-2695	291	10	,	,	PUNCT
cana-2695	291	11	0.4	0.4	NUM
cana-2695	291	12	)	)	PUNCT
cana-2695	291	13	,	,	PUNCT
cana-2695	291	14	𝑀𝐹	𝑀𝐹	PROPN
cana-2695	291	15	(	(	PUNCT
cana-2695	291	16	𝐴1𝐴3	𝐴1𝐴3	PROPN
cana-2695	291	17	)	)	PUNCT
cana-2695	291	18	=	=	PUNCT
cana-2695	292	1	(	(	PUNCT
cana-2695	292	2	0.4	0.4	NUM
cana-2695	292	3	,	,	PUNCT
cana-2695	292	4	0.3	0.3	NUM
cana-2695	292	5	)	)	PUNCT
cana-2695	292	6	and	and	CCONJ
cana-2695	292	7	so	so	ADV
cana-2695	292	8	on	on	ADV
cana-2695	292	9	.	.	PUNCT
cana-2695	293	1	using	use	VERB
cana-2695	293	2	ivfrs	ivfrs	NOUN
cana-2695	293	3	,	,	PUNCT
cana-2695	293	4	we	we	PRON
cana-2695	293	5	can	can	AUX
cana-2695	293	6	obtain	obtain	VERB
cana-2695	293	7	the	the	DET
cana-2695	293	8	few	few	ADJ
cana-2695	293	9	set	set	NOUN
cana-2695	293	10	of	of	ADP
cana-2695	293	11	vertices	vertex	NOUN
cana-2695	293	12	such	such	ADJ
cana-2695	293	13	that	that	SCONJ
cana-2695	293	14	the	the	DET
cana-2695	293	15	resulted	resulted	ADJ
cana-2695	293	16	set	set	NOUN
cana-2695	293	17	will	will	AUX
cana-2695	293	18	have	have	VERB
cana-2695	293	19	a	a	DET
cana-2695	293	20	shortest	short	ADJ
cana-2695	293	21	route	route	NOUN
cana-2695	293	22	between	between	ADP
cana-2695	293	23	them	they	PRON
cana-2695	293	24	considering	consider	VERB
cana-2695	293	25	all	all	DET
cana-2695	293	26	the	the	DET
cana-2695	293	27	traffic	traffic	NOUN
cana-2695	293	28	routes	route	NOUN
cana-2695	293	29	and	and	CCONJ
cana-2695	293	30	the	the	DET
cana-2695	293	31	passenger	passenger	NOUN
cana-2695	293	32	dependancy	dependancy	NOUN
cana-2695	293	33	.	.	PUNCT
cana-2695	294	1	5	5	X
cana-2695	294	2	.	.	X
cana-2695	294	3	conclusion	conclusion	NOUN
cana-2695	294	4	one	one	NUM
cana-2695	294	5	effective	effective	ADJ
cana-2695	294	6	paradigm	paradigm	NOUN
cana-2695	294	7	for	for	ADP
cana-2695	294	8	simulating	simulate	VERB
cana-2695	294	9	uncertainty	uncertainty	NOUN
cana-2695	294	10	in	in	ADP
cana-2695	294	11	complex	complex	ADJ
cana-2695	294	12	systems	system	NOUN
cana-2695	294	13	is	be	AUX
cana-2695	294	14	offered	offer	VERB
cana-2695	294	15	by	by	ADP
cana-2695	294	16	ivfgs	ivfgs	PROPN
cana-2695	294	17	.	.	PUNCT
cana-2695	295	1	ivfgs	ivfgs	PROPN
cana-2695	295	2	are	be	AUX
cana-2695	295	3	well	well	ADV
cana-2695	295	4	suited	suited	ADJ
cana-2695	295	5	for	for	ADP
cana-2695	295	6	real	real	ADJ
cana-2695	295	7	-	-	PUNCT
cana-2695	295	8	world	world	NOUN
cana-2695	295	9	applications	application	NOUN
cana-2695	295	10	like	like	ADP
cana-2695	295	11	as	as	ADP
cana-2695	295	12	social	social	ADJ
cana-2695	295	13	systems	system	NOUN
cana-2695	295	14	,	,	PUNCT
cana-2695	295	15	biological	biological	ADJ
cana-2695	295	16	interactions	interaction	NOUN
cana-2695	295	17	,	,	PUNCT
cana-2695	295	18	and	and	CCONJ
cana-2695	295	19	transportation	transportation	NOUN
cana-2695	295	20	networks	network	NOUN
cana-2695	295	21	because	because	SCONJ
cana-2695	295	22	they	they	PRON
cana-2695	295	23	incorporate	incorporate	VERB
cana-2695	295	24	interval	interval	NOUN
cana-2695	295	25	-	-	PUNCT
cana-2695	295	26	based	base	VERB
cana-2695	295	27	membership	membership	NOUN
cana-2695	295	28	values	value	NOUN
cana-2695	295	29	for	for	ADP
cana-2695	295	30	vertices	vertex	NOUN
cana-2695	295	31	and	and	CCONJ
cana-2695	295	32	edges	edge	NOUN
cana-2695	295	33	,	,	PUNCT
cana-2695	295	34	which	which	PRON
cana-2695	295	35	effectively	effectively	ADV
cana-2695	295	36	express	express	VERB
cana-2695	295	37	imprecise	imprecise	ADJ
cana-2695	295	38	relationships	relationship	NOUN
cana-2695	295	39	.	.	PUNCT
cana-2695	296	1	in	in	ADP
cana-2695	296	2	this	this	DET
cana-2695	296	3	article	article	NOUN
cana-2695	296	4	,	,	PUNCT
cana-2695	296	5	we	we	PRON
cana-2695	296	6	have	have	AUX
cana-2695	296	7	defined	define	VERB
cana-2695	296	8	intervalvalued	intervalvalue	VERB
cana-2695	296	9	fuzzy	fuzzy	ADJ
cana-2695	296	10	resolving	resolving	NOUN
cana-2695	296	11	sets	set	NOUN
cana-2695	296	12	,	,	PUNCT
cana-2695	296	13	interval	interval	NOUN
cana-2695	296	14	-	-	PUNCT
cana-2695	296	15	valued	value	VERB
cana-2695	296	16	fuzzy	fuzzy	ADJ
cana-2695	296	17	super	super	ADJ
cana-2695	296	18	resolving	resolve	VERB
cana-2695	296	19	sets	set	NOUN
cana-2695	296	20	and	and	CCONJ
cana-2695	296	21	its	its	PRON
cana-2695	296	22	properties	property	NOUN
cana-2695	296	23	.	.	PUNCT
cana-2695	297	1	also	also	ADV
cana-2695	297	2	,	,	PUNCT
cana-2695	297	3	we	we	PRON
cana-2695	297	4	have	have	AUX
cana-2695	297	5	defined	define	VERB
cana-2695	297	6	an	an	DET
cana-2695	297	7	application	application	NOUN
cana-2695	297	8	based	base	VERB
cana-2695	297	9	on	on	ADP
cana-2695	297	10	interval	interval	NOUN
cana-2695	297	11	-	-	PUNCT
cana-2695	297	12	valued	value	VERB
cana-2695	297	13	fuzzy	fuzzy	ADJ
cana-2695	297	14	resolving	resolving	NOUN
cana-2695	297	15	set	set	NOUN
cana-2695	297	16	.	.	PUNCT
cana-2695	298	1	in	in	ADP
cana-2695	298	2	future	future	NOUN
cana-2695	298	3	,	,	PUNCT
cana-2695	298	4	we	we	PRON
cana-2695	298	5	may	may	AUX
cana-2695	298	6	look	look	VERB
cana-2695	298	7	into	into	ADP
cana-2695	298	8	more	more	ADJ
cana-2695	298	9	problems	problem	NOUN
cana-2695	298	10	related	relate	VERB
cana-2695	298	11	to	to	ADP
cana-2695	298	12	interval	interval	NOUN
cana-2695	298	13	-	-	PUNCT
cana-2695	298	14	valued	value	VERB
cana-2695	298	15	fuzzy	fuzzy	ADJ
cana-2695	298	16	resolving	resolving	NOUN
cana-2695	298	17	set	set	NOUN
cana-2695	298	18	.	.	PUNCT
cana-2695	299	1	reference	reference	NOUN
cana-2695	299	2	[	[	X
cana-2695	299	3	1	1	NUM
cana-2695	299	4	]	]	X
cana-2695	299	5	atanassov	atanassov	PROPN
cana-2695	299	6	,	,	PUNCT
cana-2695	299	7	k.t	k.t	PROPN
cana-2695	299	8	.	.	PROPN
cana-2695	299	9	(	(	PUNCT
cana-2695	299	10	1986	1986	NUM
cana-2695	299	11	)	)	PUNCT
cana-2695	299	12	.	.	PUNCT
cana-2695	300	1	intuitionistic	intuitionistic	ADJ
cana-2695	300	2	fuzzy	fuzzy	ADJ
cana-2695	300	3	sets	set	NOUN
cana-2695	300	4	,	,	PUNCT
cana-2695	300	5	fuzzy	fuzzy	ADJ
cana-2695	300	6	sets	set	NOUN
cana-2695	300	7	and	and	CCONJ
cana-2695	300	8	systems	system	NOUN
cana-2695	300	9	,	,	PUNCT
cana-2695	300	10	20(1	20(1	NUM
cana-2695	300	11	)	)	PUNCT
cana-2695	300	12	,	,	PUNCT
cana-2695	300	13	87	87	NUM
cana-2695	300	14	-	-	SYM
cana-2695	300	15	96	96	NUM
cana-2695	300	16	.	.	PUNCT
cana-2695	301	1	[	[	X
cana-2695	301	2	2	2	NUM
cana-2695	301	3	]	]	SYM
cana-2695	301	4	chartrand	chartrand	NOUN
cana-2695	301	5	,	,	PUNCT
cana-2695	301	6	g.	g.	PROPN
cana-2695	301	7	poisson	poisson	PROPN
cana-2695	301	8	,	,	PUNCT
cana-2695	301	9	c.	c.	PROPN
cana-2695	301	10	zhang	zhang	PROPN
cana-2695	301	11	,	,	PUNCT
cana-2695	301	12	p.	p.	PROPN
cana-2695	301	13	(	(	PUNCT
cana-2695	301	14	2000	2000	NUM
cana-2695	301	15	)	)	PUNCT
cana-2695	301	16	.	.	PUNCT
cana-2695	302	1	resolvability	resolvability	NOUN
cana-2695	302	2	and	and	CCONJ
cana-2695	302	3	the	the	DET
cana-2695	302	4	upper	upper	ADJ
cana-2695	302	5	dimension	dimension	NOUN
cana-2695	302	6	of	of	ADP
cana-2695	302	7	graphs	graph	NOUN
cana-2695	302	8	,	,	PUNCT
cana-2695	302	9	comput	comput	NOUN
cana-2695	302	10	.	.	PUNCT
cana-2695	303	1	mathematical	mathematical	ADJ
cana-2695	303	2	applications	application	NOUN
cana-2695	303	3	,	,	PUNCT
cana-2695	303	4	39	39	NUM
cana-2695	303	5	,	,	PUNCT
cana-2695	303	6	19	19	NUM
cana-2695	303	7	-	-	SYM
cana-2695	303	8	28	28	NUM
cana-2695	303	9	.	.	PUNCT
cana-2695	304	1	[	[	X
cana-2695	304	2	3	3	NUM
cana-2695	304	3	]	]	X
cana-2695	304	4	gani	gani	X
cana-2695	304	5	,	,	PUNCT
cana-2695	304	6	a.	a.	NOUN
cana-2695	304	7	n.	n.	PROPN
cana-2695	304	8	and	and	CCONJ
cana-2695	304	9	basheer	basheer	PROPN
cana-2695	304	10	ahamed	ahamed	PROPN
cana-2695	304	11	,	,	PUNCT
cana-2695	304	12	m.	m.	NOUN
cana-2695	304	13	,	,	PUNCT
cana-2695	304	14	(	(	PUNCT
cana-2695	304	15	2003	2003	NUM
cana-2695	304	16	)	)	PUNCT
cana-2695	304	17	,	,	PUNCT
cana-2695	304	18	order	order	NOUN
cana-2695	304	19	and	and	CCONJ
cana-2695	304	20	size	size	NOUN
cana-2695	304	21	in	in	ADP
cana-2695	304	22	fuzzy	fuzzy	ADJ
cana-2695	304	23	graphs	graph	NOUN
cana-2695	304	24	,	,	PUNCT
cana-2695	304	25	bull	bull	NOUN
cana-2695	304	26	.	.	PUNCT
cana-2695	305	1	pure	pure	ADJ
cana-2695	305	2	appl	appl	PROPN
cana-2695	305	3	.	.	PUNCT
cana-2695	306	1	sci	sci	PROPN
cana-2695	306	2	.	.	PROPN
cana-2695	306	3	,	,	PUNCT
cana-2695	306	4	22e	22e	NUM
cana-2695	306	5	,	,	PUNCT
cana-2695	306	6	145	145	NUM
cana-2695	306	7	-	-	SYM
cana-2695	306	8	148	148	NUM
cana-2695	306	9	.	.	PUNCT
cana-2695	307	1	[	[	X
cana-2695	307	2	6	6	NUM
cana-2695	307	3	]	]	SYM
cana-2695	307	4	mordeson	mordeson	NOUN
cana-2695	307	5	,	,	PUNCT
cana-2695	307	6	j.	j.	PROPN
cana-2695	307	7	n.	n.	PROPN
cana-2695	307	8	mathew	mathew	PROPN
cana-2695	307	9	,	,	PUNCT
cana-2695	307	10	s.	s.	PROPN
cana-2695	307	11	,	,	PUNCT
cana-2695	307	12	(	(	PUNCT
cana-2695	307	13	2019	2019	NUM
cana-2695	307	14	)	)	PUNCT
cana-2695	307	15	,	,	PUNCT
cana-2695	307	16	advanced	advanced	ADJ
cana-2695	307	17	topics	topic	NOUN
cana-2695	307	18	in	in	ADP
cana-2695	307	19	fuzzy	fuzzy	ADJ
cana-2695	307	20	graph	graph	NOUN
cana-2695	307	21	theory	theory	NOUN
cana-2695	307	22	,	,	PUNCT
cana-2695	307	23	springer	springer	NOUN
cana-2695	307	24	science	science	NOUN
cana-2695	307	25	and	and	CCONJ
cana-2695	307	26	business	business	NOUN
cana-2695	307	27	media	medium	NOUN
cana-2695	307	28	.	.	PUNCT
cana-2695	308	1	[	[	X
cana-2695	308	2	7	7	X
cana-2695	308	3	]	]	X
cana-2695	308	4	muhammad	muhammad	PROPN
cana-2695	308	5	akram	akram	PROPN
cana-2695	308	6	.	.	PROPN
cana-2695	308	7	wieslaw	wieslaw	PROPN
cana-2695	308	8	a.	a.	PROPN
cana-2695	308	9	dudek	dudek	PROPN
cana-2695	308	10	.	.	PUNCT
cana-2695	309	1	(	(	PUNCT
cana-2695	309	2	2011	2011	NUM
cana-2695	309	3	)	)	PUNCT
cana-2695	309	4	,	,	PUNCT
cana-2695	309	5	interval	interval	NOUN
cana-2695	309	6	-	-	PUNCT
cana-2695	309	7	valued	value	VERB
cana-2695	309	8	fuzzy	fuzzy	ADJ
cana-2695	309	9	graphs	graph	NOUN
cana-2695	309	10	,	,	PUNCT
cana-2695	309	11	computers	computer	NOUN
cana-2695	309	12	and	and	CCONJ
cana-2695	309	13	mathematics	mathematic	NOUN
cana-2695	309	14	with	with	ADP
cana-2695	309	15	applications	application	NOUN
cana-2695	309	16	,	,	PUNCT
cana-2695	309	17	289	289	NUM
cana-2695	309	18	-	-	SYM
cana-2695	309	19	299	299	NUM
cana-2695	309	20	.	.	PUNCT
cana-2695	310	1	[	[	X
cana-2695	310	2	8	8	NUM
cana-2695	310	3	]	]	SYM
cana-2695	310	4	shanmugapriya	shanmugapriya	PROPN
cana-2695	310	5	,	,	PUNCT
cana-2695	310	6	r.	r.	PROPN
cana-2695	310	7	and	and	CCONJ
cana-2695	310	8	hemalatha	hemalatha	NOUN
cana-2695	310	9	,	,	PUNCT
cana-2695	310	10	p.	p.	PROPN
cana-2695	310	11	k.	k.	PROPN
cana-2695	310	12	,	,	PUNCT
cana-2695	310	13	(	(	PUNCT
cana-2695	310	14	2023	2023	NUM
cana-2695	310	15	)	)	PUNCT
cana-2695	310	16	,	,	PUNCT
cana-2695	310	17	a	a	DET
cana-2695	310	18	study	study	NOUN
cana-2695	310	19	of	of	ADP
cana-2695	310	20	independency	independency	NOUN
cana-2695	310	21	on	on	ADP
cana-2695	310	22	fuzzy	fuzzy	ADJ
cana-2695	310	23	resolving	resolving	NOUN
cana-2695	310	24	sets	set	NOUN
cana-2695	310	25	of	of	ADP
cana-2695	310	26	labelling	labelling	NOUN
cana-2695	310	27	graphs	graph	NOUN
cana-2695	310	28	,	,	PUNCT
cana-2695	310	29	mathematics,11	mathematics,11	NOUN
cana-2695	310	30	,	,	PUNCT
cana-2695	310	31	3440	3440	NUM
cana-2695	310	32	.	.	PUNCT
cana-2695	311	1	[	[	X
cana-2695	311	2	9	9	NUM
cana-2695	311	3	]	]	SYM
cana-2695	311	4	shanmugapriya	shanmugapriya	NOUN
cana-2695	311	5	,	,	PUNCT
cana-2695	311	6	r.	r.	PROPN
cana-2695	311	7	and	and	CCONJ
cana-2695	311	8	mary	mary	PROPN
cana-2695	311	9	jiny	jiny	PROPN
cana-2695	311	10	,	,	PUNCT
cana-2695	311	11	d.	d.	PROPN
cana-2695	311	12	(	(	PUNCT
cana-2695	311	13	2021	2021	NUM
cana-2695	311	14	)	)	PUNCT
cana-2695	311	15	,	,	PUNCT
cana-2695	311	16	fuzzy	fuzzy	ADJ
cana-2695	311	17	super	super	ADJ
cana-2695	311	18	resolving	resolve	VERB
cana-2695	311	19	number	number	NOUN
cana-2695	311	20	and	and	CCONJ
cana-2695	311	21	resolving	resolve	VERB
cana-2695	311	22	number	number	NOUN
cana-2695	311	23	of	of	ADP
cana-2695	311	24	some	some	DET
cana-2695	311	25	special	special	ADJ
cana-2695	311	26	graphs	graph	NOUN
cana-2695	311	27	,	,	PUNCT
cana-2695	311	28	twms	twms	PROPN
cana-2695	311	29	j.app	j.app	PROPN
cana-2695	311	30	.	.	PUNCT
cana-2695	312	1	and	and	CCONJ
cana-2695	312	2	eng	eng	PROPN
cana-2695	312	3	math	math	NOUN
cana-2695	312	4	,	,	PUNCT
cana-2695	312	5	11	11	NUM
cana-2695	312	6	,	,	PUNCT
cana-2695	312	7	459	459	NUM
cana-2695	312	8	-	-	SYM
cana-2695	312	9	468	468	NUM
cana-2695	312	10	.	.	PUNCT
cana-2695	313	1	[	[	X
cana-2695	313	2	10	10	NUM
cana-2695	313	3	]	]	X
cana-2695	313	4	tarasankar	tarasankar	NOUN
cana-2695	313	5	pramanik	pramanik	AUX
cana-2695	313	6	.	.	PUNCT
cana-2695	314	1	sovan	sovan	PROPN
cana-2695	314	2	samanta	samanta	PROPN
cana-2695	314	3	.	.	PUNCT
cana-2695	315	1	and	and	CCONJ
cana-2695	315	2	madhumangal	madhumangal	ADJ
cana-2695	315	3	pal	pal	NOUN
cana-2695	315	4	.	.	PUNCT
cana-2695	316	1	(	(	PUNCT
cana-2695	316	2	2020	2020	NUM
cana-2695	316	3	)	)	PUNCT
cana-2695	316	4	,	,	PUNCT
cana-2695	316	5	interval	interval	NOUN
cana-2695	316	6	-	-	PUNCT
cana-2695	316	7	valued	value	VERB
cana-2695	316	8	fuzzy	fuzzy	ADJ
cana-2695	316	9	graphs	graph	NOUN
cana-2695	316	10	,	,	PUNCT
cana-2695	316	11	international	international	ADJ
cana-2695	316	12	journal	journal	NOUN
cana-2695	316	13	of	of	ADP
cana-2695	316	14	fuzzy	fuzzy	ADJ
cana-2695	316	15	logic	logic	NOUN
cana-2695	316	16	and	and	CCONJ
cana-2695	316	17	intelligent	intelligent	ADJ
cana-2695	316	18	systems	system	NOUN
cana-2695	316	19	,	,	PUNCT
cana-2695	316	20	316	316	NUM
cana-2695	316	21	-	-	SYM
cana-2695	316	22	323	323	NUM
cana-2695	316	23	.	.	PUNCT
cana-2695	317	1	[	[	X
cana-2695	317	2	11	11	NUM
cana-2695	317	3	]	]	SYM
cana-2695	317	4	vasuki	vasuki	NOUN
cana-2695	317	5	,	,	PUNCT
cana-2695	317	6	m.	m.	NOUN
cana-2695	317	7	shanmugapriya	shanmugapriya	PROPN
cana-2695	317	8	,	,	PUNCT
cana-2695	317	9	r.	r.	PROPN
cana-2695	317	10	mahdal	mahdal	PROPN
cana-2695	317	11	,	,	PUNCT
cana-2695	317	12	m.	m.	NOUN
cana-2695	317	13	and	and	CCONJ
cana-2695	317	14	robert	robert	PROPN
cana-2695	317	15	cep	cep	PROPN
cana-2695	317	16	.	.	PUNCT
cana-2695	318	1	(	(	PUNCT
cana-2695	318	2	2023	2023	NUM
cana-2695	318	3	)	)	PUNCT
cana-2695	318	4	,	,	PUNCT
cana-2695	318	5	a	a	DET
cana-2695	318	6	study	study	NOUN
cana-2695	318	7	on	on	ADP
cana-2695	318	8	fuzzy	fuzzy	ADJ
cana-2695	318	9	resolving	resolving	NOUN
cana-2695	318	10	domination	domination	NOUN
cana-2695	318	11	sets	set	NOUN
cana-2695	318	12	and	and	CCONJ
cana-2695	318	13	their	their	PRON
cana-2695	318	14	application	application	NOUN
cana-2695	318	15	in	in	ADP
cana-2695	318	16	network	network	NOUN
cana-2695	318	17	theory	theory	NOUN
cana-2695	318	18	,	,	PUNCT
cana-2695	318	19	mathematics	mathematic	NOUN
cana-2695	318	20	,	,	PUNCT
cana-2695	318	21	11	11	NUM
cana-2695	318	22	,	,	PUNCT
cana-2695	318	23	317	317	NUM
cana-2695	318	24	.	.	PUNCT
cana-2695	319	1	[	[	X
cana-2695	319	2	12	12	NUM
cana-2695	319	3	]	]	X
cana-2695	319	4	xiaoli	xiaoli	PROPN
cana-2695	319	5	qiang	qiang	PROPN
cana-2695	319	6	.	.	PUNCT
cana-2695	320	1	qian	qian	PROPN
cana-2695	320	2	-	-	PUNCT
cana-2695	320	3	ru	ru	PROPN
cana-2695	320	4	xiao	xiao	PROPN
cana-2695	320	5	.	.	PROPN
cana-2695	320	6	aysha	aysha	PROPN
cana-2695	320	7	khan	khan	PROPN
cana-2695	320	8	,	,	PUNCT
cana-2695	320	9	a.	a.	PROPN
cana-2695	320	10	talebi	talebi	PROPN
cana-2695	320	11	,	,	PUNCT
cana-2695	320	12	a.	a.	NOUN
cana-2695	320	13	arun	arun	PROPN
cana-2695	320	14	kumar	kumar	PROPN
cana-2695	320	15	sivaraman	sivaraman	PROPN
cana-2695	320	16	.	.	PUNCT
cana-2695	321	1	mojahedfar	mojahedfar	PROPN
cana-2695	321	2	,	,	PUNCT
cana-2695	321	3	m.	m.	NOUN
cana-2695	321	4	(	(	PUNCT
cana-2695	321	5	2022	2022	NUM
cana-2695	321	6	)	)	PUNCT
cana-2695	321	7	,	,	PUNCT
cana-2695	321	8	astudy	astudy	VERB
cana-2695	321	9	on	on	ADP
cana-2695	321	10	interval	interval	NOUN
cana-2695	321	11	-	-	PUNCT
cana-2695	321	12	valued	value	VERB
cana-2695	321	13	fuzzy	fuzzy	ADJ
cana-2695	321	14	graph	graph	NOUN
cana-2695	321	15	with	with	ADP
cana-2695	321	16	application	application	NOUN
cana-2695	321	17	in	in	ADP
cana-2695	321	18	energy	energy	NOUN
cana-2695	321	19	industry	industry	NOUN
cana-2695	321	20	management	management	NOUN
cana-2695	321	21	,	,	PUNCT
cana-2695	321	22	discrete	discrete	ADJ
cana-2695	321	23	dynamics	dynamic	NOUN
cana-2695	321	24	in	in	ADP
cana-2695	321	25	nature	nature	NOUN
cana-2695	321	26	and	and	CCONJ
cana-2695	321	27	society	society	NOUN
cana-2695	321	28	,	,	PUNCT
cana-2695	321	29	9	9	NUM
cana-2695	321	30	.	.	PUNCT
cana-2695	322	1	[	[	X
cana-2695	322	2	13	13	NUM
cana-2695	322	3	]	]	SYM
cana-2695	322	4	zadeh	zadeh	PROPN
cana-2695	322	5	,	,	PUNCT
cana-2695	322	6	l.a	l.a	PROPN
cana-2695	322	7	.	.	PROPN
cana-2695	322	8	(	(	PUNCT
cana-2695	322	9	1965	1965	NUM
cana-2695	322	10	)	)	PUNCT
cana-2695	322	11	,	,	PUNCT
cana-2695	322	12	fuzzy	fuzzy	ADJ
cana-2695	322	13	sets	set	NOUN
cana-2695	322	14	,	,	PUNCT
cana-2695	322	15	information	information	NOUN
cana-2695	322	16	and	and	CCONJ
cana-2695	322	17	control	control	NOUN
cana-2695	322	18	,	,	PUNCT
cana-2695	322	19	8	8	NUM
cana-2695	322	20	,	,	PUNCT
cana-2695	322	21	338	338	NUM
cana-2695	322	22	-	-	SYM
cana-2695	322	23	353	353	NUM
cana-2695	322	24	.	.	PUNCT
cana-2695	322	25	communications	communication	NOUN
cana-2695	322	26	on	on	ADP
cana-2695	322	27	applied	apply	VERB
cana-2695	322	28	nonlinear	nonlinear	ADJ
cana-2695	322	29	analysis	analysis	NOUN
cana-2695	322	30	issn	issn	NOUN
cana-2695	322	31	:	:	PUNCT
cana-2695	322	32	1074	1074	NUM
cana-2695	322	33	-	-	PUNCT
cana-2695	322	34	133x	133x	NUM
cana-2695	322	35	vol	vol	NOUN
cana-2695	322	36	32	32	NUM
cana-2695	322	37	no	no	NOUN
cana-2695	322	38	.	.	PUNCT
cana-2695	323	1	3s	3s	NUM
cana-2695	323	2	(	(	PUNCT
cana-2695	323	3	2025	2025	NUM
cana-2695	323	4	)	)	PUNCT
cana-2695	323	5	557	557	NUM
cana-2695	323	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2695	324	1	[	[	X
cana-2695	324	2	14	14	NUM
cana-2695	324	3	]	]	SYM
cana-2695	324	4	renukadevi	renukadevi	NOUN
cana-2695	324	5	,	,	PUNCT
cana-2695	324	6	r.	r.	PROPN
cana-2695	324	7	et	et	PROPN
cana-2695	324	8	al	al	PROPN
cana-2695	324	9	.	.	PUNCT
cana-2695	325	1	“	"	PUNCT
cana-2695	325	2	an	an	DET
cana-2695	325	3	improved	improved	ADJ
cana-2695	325	4	collaborative	collaborative	ADJ
cana-2695	325	5	user	user	NOUN
cana-2695	325	6	product	product	NOUN
cana-2695	325	7	recommendation	recommendation	NOUN
cana-2695	325	8	system	system	NOUN
cana-2695	325	9	using	use	VERB
cana-2695	325	10	computational	computational	ADJ
cana-2695	325	11	intelligence	intelligence	NOUN
cana-2695	325	12	with	with	ADP
cana-2695	325	13	association	association	NOUN
cana-2695	325	14	rules	rule	NOUN
cana-2695	325	15	.	.	PUNCT
cana-2695	325	16	”	"	PUNCT
cana-2695	326	1	communications	communication	NOUN
cana-2695	326	2	on	on	ADP
cana-2695	326	3	applied	apply	VERB
cana-2695	326	4	nonlinear	nonlinear	ADJ
cana-2695	326	5	analysis	analysis	NOUN
cana-2695	326	6	(	(	PUNCT
cana-2695	326	7	2024	2024	NUM
cana-2695	326	8	):	):	PUNCT
cana-2695	326	9	n.	n.	PROPN
cana-2695	326	10	pag	pag	PROPN
cana-2695	326	11	.	.	PUNCT
cana-2695	327	1	https://doi.org/10.52783/cana.v31.1243	https://doi.org/10.52783/cana.v31.1243	PROPN
cana-2695	328	1	[	[	X
cana-2695	328	2	15	15	NUM
cana-2695	328	3	]	]	X
cana-2695	328	4	kumar	kumar	PROPN
cana-2695	328	5	,	,	PUNCT
cana-2695	328	6	e.	e.	PROPN
cana-2695	328	7	boopathi	boopathi	PROPN
cana-2695	328	8	,	,	PUNCT
cana-2695	328	9	and	and	CCONJ
cana-2695	328	10	m.	m.	PROPN
cana-2695	328	11	sundaresan	sundaresan	PROPN
cana-2695	328	12	.	.	PUNCT
cana-2695	329	1	"	"	PUNCT
cana-2695	329	2	edge	edge	NOUN
cana-2695	329	3	detection	detection	NOUN
cana-2695	329	4	using	use	VERB
cana-2695	329	5	trapezoidal	trapezoidal	ADJ
cana-2695	329	6	membership	membership	NOUN
cana-2695	329	7	function	function	NOUN
cana-2695	329	8	based	base	VERB
cana-2695	329	9	on	on	ADP
cana-2695	329	10	fuzzy	fuzzy	PROPN
cana-2695	329	11	's	's	PART
cana-2695	329	12	mamdani	mamdani	PROPN
cana-2695	329	13	inference	inference	NOUN
cana-2695	329	14	system	system	NOUN
cana-2695	329	15	.	.	PUNCT
cana-2695	329	16	"	"	PUNCT
cana-2695	330	1	2014	2014	NUM
cana-2695	330	2	international	international	ADJ
cana-2695	330	3	conference	conference	NOUN
cana-2695	330	4	on	on	ADP
cana-2695	330	5	computing	compute	VERB
cana-2695	330	6	for	for	ADP
cana-2695	330	7	sustainable	sustainable	ADJ
cana-2695	330	8	global	global	ADJ
cana-2695	330	9	development	development	NOUN
cana-2695	330	10	(	(	PUNCT
cana-2695	330	11	indiacom	indiacom	NOUN
cana-2695	330	12	)	)	PUNCT
cana-2695	330	13	.	.	PUNCT
cana-2695	331	1	ieee	ieee	PROPN
cana-2695	331	2	,	,	PUNCT
cana-2695	331	3	2014	2014	NUM
cana-2695	331	4	.	.	PUNCT
cana-2695	332	1	[	[	X
cana-2695	332	2	16	16	NUM
cana-2695	332	3	]	]	X
cana-2695	332	4	yookesh	yookesh	ADJ
cana-2695	332	5	,	,	PUNCT
cana-2695	332	6	t.	t.	PROPN
cana-2695	332	7	l.	l.	PROPN
cana-2695	332	8	,	,	PUNCT
cana-2695	332	9	et	et	PROPN
cana-2695	332	10	al	al	PROPN
cana-2695	332	11	.	.	PUNCT
cana-2695	333	1	"	"	PUNCT
cana-2695	333	2	efficiency	efficiency	NOUN
cana-2695	333	3	of	of	ADP
cana-2695	333	4	iterative	iterative	ADJ
cana-2695	333	5	filtering	filtering	NOUN
cana-2695	333	6	method	method	NOUN
cana-2695	333	7	for	for	ADP
cana-2695	333	8	solving	solve	VERB
cana-2695	333	9	volterra	volterra	NOUN
cana-2695	333	10	fuzzy	fuzzy	ADJ
cana-2695	333	11	integral	integral	ADJ
cana-2695	333	12	equations	equation	NOUN
cana-2695	333	13	with	with	ADP
cana-2695	333	14	a	a	DET
cana-2695	333	15	delay	delay	NOUN
cana-2695	333	16	and	and	CCONJ
cana-2695	333	17	material	material	NOUN
cana-2695	333	18	investigation	investigation	NOUN
cana-2695	333	19	.	.	PUNCT
cana-2695	333	20	"	"	PUNCT
cana-2695	334	1	materials	material	NOUN
cana-2695	334	2	today	today	NOUN
cana-2695	334	3	:	:	PUNCT
cana-2695	334	4	proceedings	proceeding	NOUN
cana-2695	334	5	47	47	NUM
cana-2695	334	6	(	(	PUNCT
cana-2695	334	7	2021	2021	NUM
cana-2695	334	8	):	):	PUNCT
cana-2695	334	9	6101	6101	NUM
cana-2695	334	10	-	-	SYM
cana-2695	334	11	6104	6104	NUM
cana-2695	334	12	.	.	PUNCT
cana-2695	335	1	https://doi.org/10.52783/cana.v31.1243	https://doi.org/10.52783/cana.v31.1243	PROPN