id	sid	tid	token	lemma	pos
cana-1322	1	1	communications	communication	NOUN
cana-1322	1	2	on	on	ADP
cana-1322	1	3	applied	apply	VERB
cana-1322	1	4	nonlinear	nonlinear	ADJ
cana-1322	1	5	analysis	analysis	NOUN
cana-1322	1	6	issn	issn	NOUN
cana-1322	1	7	:	:	PUNCT
cana-1322	1	8	1074	1074	NUM
cana-1322	1	9	-	-	PUNCT
cana-1322	1	10	133x	133x	NUM
cana-1322	1	11	vol	vol	NOUN
cana-1322	1	12	31	31	NUM
cana-1322	1	13	no	no	NOUN
cana-1322	1	14	.	.	PUNCT
cana-1322	2	1	7s	7	NOUN
cana-1322	2	2	(	(	PUNCT
cana-1322	2	3	2024	2024	NUM
cana-1322	2	4	)	)	PUNCT
cana-1322	2	5	437	437	NUM
cana-1322	3	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1322	3	2	a	a	DET
cana-1322	3	3	research	research	NOUN
cana-1322	3	4	on	on	ADP
cana-1322	3	5	bipolar	bipolar	ADJ
cana-1322	3	6	valued	value	VERB
cana-1322	3	7	vague	vague	ADJ
cana-1322	3	8	normal	normal	ADJ
cana-1322	3	9	subrings	subring	NOUN
cana-1322	3	10	of	of	ADP
cana-1322	3	11	a	a	DET
cana-1322	3	12	ring	ring	NOUN
cana-1322	3	13	1	1	NUM
cana-1322	3	14	b.deeba	b.deeba	NOUN
cana-1322	3	15	,	,	PUNCT
cana-1322	3	16	2	2	NUM
cana-1322	3	17	s.	s.	PROPN
cana-1322	3	18	naganathan	naganathan	PROPN
cana-1322	3	19	&	&	CCONJ
cana-1322	3	20	3	3	NUM
cana-1322	3	21	k.arjunan	k.arjunan	NOUN
cana-1322	3	22	1	1	NUM
cana-1322	3	23	.	.	PUNCT
cana-1322	4	1	department	department	NOUN
cana-1322	4	2	of	of	ADP
cana-1322	4	3	mathematics	mathematics	PROPN
cana-1322	4	4	,	,	PUNCT
cana-1322	4	5	idhaya	idhaya	VERB
cana-1322	4	6	college	college	NOUN
cana-1322	4	7	for	for	ADP
cana-1322	4	8	women	woman	NOUN
cana-1322	4	9	(	(	PUNCT
cana-1322	4	10	affiliated	affiliate	VERB
cana-1322	4	11	to	to	PART
cana-1322	4	12	alagappa	alagappa	VERB
cana-1322	4	13	university	university	PROPN
cana-1322	4	14	,	,	PUNCT
cana-1322	4	15	karaikudi	karaikudi	PROPN
cana-1322	4	16	)	)	PUNCT
cana-1322	4	17	,	,	PUNCT
cana-1322	4	18	sarugani	sarugani	PROPN
cana-1322	4	19	–	–	PUNCT
cana-1322	4	20	630411	630411	NUM
cana-1322	4	21	,	,	PUNCT
cana-1322	4	22	tamilnadu	tamilnadu	NOUN
cana-1322	4	23	,	,	PUNCT
cana-1322	4	24	india	india	PROPN
cana-1322	4	25	.	.	PUNCT
cana-1322	5	1	email:bdeepa85@gmail.com	email:bdeepa85@gmail.com	PROPN
cana-1322	5	2	2	2	NUM
cana-1322	5	3	.	.	PUNCT
cana-1322	5	4	department	department	NOUN
cana-1322	5	5	of	of	ADP
cana-1322	5	6	mathematics	mathematic	NOUN
cana-1322	5	7	,	,	PUNCT
cana-1322	5	8	sethupathy	sethupathy	ADJ
cana-1322	5	9	government	government	NOUN
cana-1322	5	10	arts	arts	PROPN
cana-1322	5	11	college	college	PROPN
cana-1322	5	12	(	(	PUNCT
cana-1322	5	13	affiliated	affiliate	VERB
cana-1322	5	14	to	to	PART
cana-1322	5	15	alagappa	alagappa	VERB
cana-1322	5	16	university	university	PROPN
cana-1322	5	17	,	,	PUNCT
cana-1322	5	18	karaikudi	karaikudi	PROPN
cana-1322	5	19	)	)	PUNCT
cana-1322	5	20	,	,	PUNCT
cana-1322	5	21	ramanathapuram	ramanathapuram	NOUN
cana-1322	5	22	-623	-623	PROPN
cana-1322	5	23	502	502	NUM
cana-1322	5	24	,	,	PUNCT
cana-1322	5	25	tamilnadu	tamilnadu	ADJ
cana-1322	5	26	,	,	PUNCT
cana-1322	5	27	india	india	PROPN
cana-1322	5	28	.	.	PUNCT
cana-1322	5	29	email	email	NOUN
cana-1322	5	30	:	:	PUNCT
cana-1322	5	31	nathanaga	nathanaga	X
cana-1322	5	32	@yahoo.com	@yahoo.com	X
cana-1322	5	33	3	3	X
cana-1322	5	34	.	.	PUNCT
cana-1322	5	35	department	department	NOUN
cana-1322	5	36	of	of	ADP
cana-1322	5	37	mathematics	mathematic	NOUN
cana-1322	5	38	,	,	PUNCT
cana-1322	5	39	alagappa	alagappa	VERB
cana-1322	5	40	government	government	NOUN
cana-1322	5	41	arts	arts	PROPN
cana-1322	5	42	college	college	PROPN
cana-1322	5	43	(	(	PUNCT
cana-1322	5	44	affiliated	affiliate	VERB
cana-1322	5	45	to	to	PART
cana-1322	5	46	alagappa	alagappa	VERB
cana-1322	5	47	university	university	PROPN
cana-1322	5	48	,	,	PUNCT
cana-1322	5	49	karaikudi	karaikudi	PROPN
cana-1322	5	50	)	)	PUNCT
cana-1322	5	51	,	,	PUNCT
cana-1322	5	52	karaikudi	karaikudi	PROPN
cana-1322	5	53	–	–	PUNCT
cana-1322	5	54	630003	630003	NUM
cana-1322	5	55	,	,	PUNCT
cana-1322	5	56	tamilnadu	tamilnadu	NOUN
cana-1322	5	57	,	,	PUNCT
cana-1322	5	58	india	india	PROPN
cana-1322	5	59	.	.	PUNCT
cana-1322	5	60	email	email	NOUN
cana-1322	5	61	:	:	PUNCT
cana-1322	5	62	arjunan.karmegam@gmail.com	arjunan.karmegam@gmail.com	X
cana-1322	6	1	article	article	NOUN
cana-1322	6	2	history	history	NOUN
cana-1322	6	3	:	:	PUNCT
cana-1322	6	4	received	receive	VERB
cana-1322	6	5	:	:	PUNCT
cana-1322	6	6	01	01	NUM
cana-1322	6	7	-	-	PUNCT
cana-1322	6	8	06	06	NUM
cana-1322	6	9	-	-	PUNCT
cana-1322	6	10	2024	2024	NUM
cana-1322	6	11	revised	revise	VERB
cana-1322	6	12	:	:	PUNCT
cana-1322	6	13	03	03	NUM
cana-1322	6	14	-	-	PUNCT
cana-1322	6	15	07	07	NUM
cana-1322	6	16	-	-	PUNCT
cana-1322	6	17	2024	2024	NUM
cana-1322	6	18	accepted	accept	VERB
cana-1322	6	19	:	:	PUNCT
cana-1322	6	20	29	29	NUM
cana-1322	6	21	-	-	SYM
cana-1322	6	22	07	07	NUM
cana-1322	6	23	-	-	PUNCT
cana-1322	6	24	2024	2024	NUM
cana-1322	6	25	abstract	abstract	NOUN
cana-1322	6	26	:	:	PUNCT
cana-1322	6	27	this	this	DET
cana-1322	6	28	paper	paper	NOUN
cana-1322	6	29	introduces	introduce	NOUN
cana-1322	6	30	and	and	CCONJ
cana-1322	6	31	discusses	discuss	VERB
cana-1322	6	32	certain	certain	ADJ
cana-1322	6	33	properties	property	NOUN
cana-1322	6	34	of	of	ADP
cana-1322	6	35	bipolar	bipolar	ADJ
cana-1322	6	36	valued	value	VERB
cana-1322	6	37	vague	vague	ADJ
cana-1322	6	38	normal	normal	ADJ
cana-1322	6	39	subring	subring	NOUN
cana-1322	6	40	of	of	ADP
cana-1322	6	41	a	a	DET
cana-1322	6	42	ring	ring	NOUN
cana-1322	6	43	.	.	PUNCT
cana-1322	7	1	keywords	keyword	NOUN
cana-1322	7	2	:	:	PUNCT
cana-1322	7	3	introduction	introduction	NOUN
cana-1322	7	4	.	.	PUNCT
cana-1322	8	1	[	[	PUNCT
cana-1322	8	2	]	]	X
cana-1322	8	3	succeeding	succeed	VERB
cana-1322	8	4	years	year	NOUN
cana-1322	8	5	,	,	PUNCT
cana-1322	8	6	fuzzy	fuzzy	ADJ
cana-1322	8	7	set	set	NOUN
cana-1322	8	8	was	be	AUX
cana-1322	8	9	grown	grow	VERB
cana-1322	8	10	in	in	ADP
cana-1322	8	11	different	different	ADJ
cana-1322	8	12	ways	way	NOUN
cana-1322	8	13	.	.	PUNCT
cana-1322	9	1	the	the	DET
cana-1322	9	2	following	follow	VERB
cana-1322	9	3	are	be	AUX
cana-1322	9	4	extension	extension	NOUN
cana-1322	9	5	of	of	ADP
cana-1322	9	6	fuzzy	fuzzy	ADJ
cana-1322	9	7	set	set	NOUN
cana-1322	9	8	,	,	PUNCT
cana-1322	9	9	they	they	PRON
cana-1322	9	10	are	be	AUX
cana-1322	9	11	vague	vague	ADJ
cana-1322	9	12	set	set	NOUN
cana-1322	9	13	,	,	PUNCT
cana-1322	9	14	intuitionistic	intuitionistic	ADJ
cana-1322	9	15	fuzzy	fuzzy	ADJ
cana-1322	9	16	set	set	NOUN
cana-1322	9	17	,	,	PUNCT
cana-1322	9	18	bipolar	bipolar	ADJ
cana-1322	9	19	valued	value	VERB
cana-1322	9	20	fuzzy	fuzzy	ADJ
cana-1322	9	21	set	set	NOUN
cana-1322	9	22	and	and	CCONJ
cana-1322	9	23	etc	etc	X
cana-1322	9	24	.	.	X
cana-1322	10	1	v	v	X
cana-1322	10	2	[	[	PUNCT
cana-1322	10	3	]	]	PUNCT
cana-1322	10	4	rosenfeld	rosenfeld	PROPN
cana-1322	11	1	[	[	X
cana-1322	11	2	2	2	NUM
cana-1322	11	3	]	]	PUNCT
cana-1322	11	4	;	;	PUNCT
cana-1322	11	5	bipolar	bipolar	ADJ
cana-1322	11	6	valued	value	VERB
cana-1322	11	7	fuzzy	fuzzy	ADJ
cana-1322	11	8	subset	subset	VERB
cana-1322	11	9	by	by	ADP
cana-1322	11	10	w.r.zhang[16	w.r.zhang[16	PROPN
cana-1322	11	11	]	]	X
cana-1322	11	12	;	;	PUNCT
cana-1322	11	13	vague	vague	ADJ
cana-1322	11	14	group	group	NOUN
cana-1322	11	15	by	by	ADP
cana-1322	11	16	ranjitbiswas	ranjitbiswas	PROPN
cana-1322	12	1	[	[	X
cana-1322	12	2	12	12	NUM
cana-1322	12	3	]	]	X
cana-1322	12	4	;	;	PUNCT
cana-1322	12	5	bipolar	bipolar	ADJ
cana-1322	12	6	vague	vague	NOUN
cana-1322	12	7	set	set	VERB
cana-1322	12	8	by	by	ADP
cana-1322	12	9	cicily	cicily	ADV
cana-1322	12	10	flora	flora	NOUN
cana-1322	12	11	.	.	PUNCT
cana-1322	13	1	s	s	VERB
cana-1322	13	2	and	and	CCONJ
cana-1322	13	3	arockiarani.i	arockiarani.i	NOUN
cana-1322	14	1	[	[	X
cana-1322	14	2	4	4	NUM
cana-1322	14	3	]	]	X
cana-1322	14	4	;	;	PUNCT
cana-1322	14	5	bipolar	bipolar	ADJ
cana-1322	14	6	valued	value	VERB
cana-1322	14	7	fuzzy	fuzzy	ADJ
cana-1322	14	8	subgroup	subgroup	NOUN
cana-1322	14	9	by	by	ADP
cana-1322	14	10	anitha.m.s	anitha.m.	NOUN
cana-1322	14	11	.	.	PUNCT
cana-1322	14	12	,	,	PUNCT
cana-1322	15	1	et.al.[1	et.al.[1	PROPN
cana-1322	15	2	]	]	X
cana-1322	15	3	;	;	PUNCT
cana-1322	15	4	in	in	ADP
cana-1322	15	5	similar	similar	ADJ
cana-1322	15	6	way	way	NOUN
cana-1322	15	7	,	,	PUNCT
cana-1322	15	8	[	[	X
cana-1322	15	9	3	3	NUM
cana-1322	15	10	]	]	PUNCT
cana-1322	15	11	,	,	PUNCT
cana-1322	15	12	[	[	X
cana-1322	15	13	11	11	NUM
cana-1322	15	14	]	]	PUNCT
cana-1322	15	15	,	,	PUNCT
cana-1322	15	16	[	[	X
cana-1322	15	17	13	13	NUM
cana-1322	15	18	]	]	PUNCT
cana-1322	15	19	,	,	PUNCT
cana-1322	15	20	[	[	X
cana-1322	15	21	14	14	NUM
cana-1322	15	22	]	]	PUNCT
cana-1322	15	23	,	,	PUNCT
cana-1322	15	24	[	[	X
cana-1322	15	25	5	5	NUM
cana-1322	15	26	]	]	PUNCT
cana-1322	15	27	,	,	PUNCT
cana-1322	15	28	[	[	X
cana-1322	15	29	6	6	NUM
cana-1322	15	30	]	]	PUNCT
cana-1322	15	31	,	,	PUNCT
cana-1322	15	32	[	[	X
cana-1322	15	33	8	8	NUM
cana-1322	15	34	]	]	PUNCT
cana-1322	15	35	,	,	PUNCT
cana-1322	15	36	[	[	X
cana-1322	15	37	9	9	NUM
cana-1322	15	38	]	]	PUNCT
cana-1322	15	39	and	and	CCONJ
cana-1322	15	40	[	[	X
cana-1322	15	41	10	10	NUM
cana-1322	15	42	]	]	PUNCT
cana-1322	15	43	were	be	AUX
cana-1322	15	44	useful	useful	ADJ
cana-1322	15	45	to	to	PART
cana-1322	15	46	write	write	VERB
cana-1322	15	47	this	this	DET
cana-1322	15	48	paper	paper	NOUN
cana-1322	15	49	.	.	PUNCT
cana-1322	16	1	1.preliminaries	1.preliminaries	NUM
cana-1322	16	2	.	.	PUNCT
cana-1322	17	1	definition	definition	NOUN
cana-1322	17	2	1.1	1.1	NUM
cana-1322	17	3	[	[	X
cana-1322	17	4	15	15	NUM
cana-1322	17	5	]	]	X
cana-1322	17	6	[	[	PUNCT
cana-1322	17	7	]	]	PUNCT
cana-1322	17	8	definition	definition	NOUN
cana-1322	17	9	1.2	1.2	NUM
cana-1322	18	1	[	[	X
cana-1322	18	2	7	7	X
cana-1322	18	3	]	]	X
cana-1322	18	4	{	{	PUNCT
cana-1322	18	5	[	[	PUNCT
cana-1322	18	6	]	]	X
cana-1322	18	7	}	}	PUNCT
cana-1322	18	8	a	a	PRON
cana-1322	18	9	[	[	PUNCT
cana-1322	18	10	]	]	X
cana-1322	18	11	map	map	NOUN
cana-1322	18	12	and	and	CCONJ
cana-1322	18	13	[	[	PUNCT
cana-1322	18	14	]	]	X
cana-1322	18	15	is	be	AUX
cana-1322	18	16	a	a	DET
cana-1322	18	17	false	false	ADJ
cana-1322	18	18	membership	membership	NOUN
cana-1322	18	19	map	map	NOUN
cana-1322	18	20	,	,	PUNCT
cana-1322	18	21	such	such	ADJ
cana-1322	18	22	that	that	PRON
cana-1322	18	23	.	.	PUNCT
cana-1322	19	1	definition	definition	NOUN
cana-1322	19	2	1.3	1.3	NUM
cana-1322	20	1	[	[	X
cana-1322	20	2	7	7	NUM
cana-1322	20	3	]	]	PUNCT
cana-1322	20	4	[	[	PUNCT
cana-1322	20	5	]	]	X
cana-1322	20	6	[	[	PUNCT
cana-1322	20	7	]	]	X
cana-1322	20	8	example	example	NOUN
cana-1322	20	9	1.4	1.4	NUM
cana-1322	20	10	.	.	PUNCT
cana-1322	21	1	=	=	PRON
cana-1322	21	2	{	{	PUNCT
cana-1322	21	3	<	<	X
cana-1322	21	4	,	,	PUNCT
cana-1322	21	5	[	[	X
cana-1322	21	6	0.5	0.5	NUM
cana-1322	21	7	,	,	PUNCT
cana-1322	21	8	0.6	0.6	NUM
cana-1322	21	9	]	]	PUNCT
cana-1322	21	10	>	>	X
cana-1322	21	11	,	,	PUNCT
cana-1322	21	12	<	<	X
cana-1322	21	13	,	,	PUNCT
cana-1322	21	14	[	[	X
cana-1322	21	15	0.7	0.7	NUM
cana-1322	21	16	,	,	PUNCT
cana-1322	21	17	0.8	0.8	NUM
cana-1322	21	18	]	]	PUNCT
cana-1322	21	19	>	>	X
cana-1322	21	20	,	,	PUNCT
cana-1322	21	21	<	<	X
cana-1322	21	22	,	,	PUNCT
cana-1322	21	23	[	[	X
cana-1322	21	24	0.4	0.4	NUM
cana-1322	21	25	,	,	PUNCT
cana-1322	21	26	0.9	0.9	NUM
cana-1322	21	27	]	]	PUNCT
cana-1322	21	28	>	>	X
cana-1322	21	29	}	}	PUNCT
cana-1322	21	30	is	be	AUX
cana-1322	21	31	a	a	DET
cana-1322	21	32	vague	vague	ADJ
cana-1322	21	33	set	set	NOUN
cana-1322	21	34	of	of	ADP
cana-1322	21	35	{	{	PUNCT
cana-1322	21	36	}	}	PUNCT
cana-1322	21	37	definition	definition	NOUN
cana-1322	21	38	1.5	1.5	NUM
cana-1322	21	39	[	[	X
cana-1322	21	40	16	16	NUM
cana-1322	21	41	]	]	PUNCT
cana-1322	21	42	{	{	PUNCT
cana-1322	21	43	(	(	PUNCT
cana-1322	21	44	)	)	PUNCT
cana-1322	21	45	}	}	PUNCT
cana-1322	21	46	a	a	DET
cana-1322	21	47	bipolar	bipolar	ADJ
cana-1322	21	48	[	[	PUNCT
cana-1322	21	49	]	]	X
cana-1322	21	50	map	map	NOUN
cana-1322	21	51	and	and	CCONJ
cana-1322	21	52	[	[	PUNCT
cana-1322	21	53	]	]	X
cana-1322	21	54	is	be	AUX
cana-1322	21	55	a	a	DET
cana-1322	21	56	negative	negative	ADJ
cana-1322	21	57	membership	membership	NOUN
cana-1322	21	58	map	map	NOUN
cana-1322	21	59	.	.	PUNCT
cana-1322	22	1	communications	communication	NOUN
cana-1322	22	2	on	on	ADP
cana-1322	22	3	applied	apply	VERB
cana-1322	22	4	nonlinear	nonlinear	ADJ
cana-1322	22	5	analysis	analysis	NOUN
cana-1322	22	6	issn	issn	NOUN
cana-1322	22	7	:	:	PUNCT
cana-1322	22	8	1074	1074	NUM
cana-1322	22	9	-	-	PUNCT
cana-1322	22	10	133x	133x	NUM
cana-1322	22	11	vol	vol	NOUN
cana-1322	22	12	31	31	NUM
cana-1322	22	13	no	no	NOUN
cana-1322	22	14	.	.	PUNCT
cana-1322	23	1	7s	7	NOUN
cana-1322	23	2	(	(	PUNCT
cana-1322	23	3	2024	2024	NUM
cana-1322	23	4	)	)	PUNCT
cana-1322	23	5	438	438	NUM
cana-1322	23	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1322	23	7	definition	definition	NOUN
cana-1322	23	8	1.6	1.6	NUM
cana-1322	24	1	[	[	X
cana-1322	24	2	4	4	NUM
cana-1322	24	3	]	]	X
cana-1322	24	4	{	{	PUNCT
cana-1322	24	5	[	[	PUNCT
cana-1322	24	6	]	]	X
cana-1322	24	7	[	[	PUNCT
cana-1322	24	8	]	]	X
cana-1322	24	9	}	}	PUNCT
cana-1322	24	10	[	[	PUNCT
cana-1322	24	11	]	]	X
cana-1322	24	12	[	[	PUNCT
cana-1322	24	13	]	]	X
cana-1322	24	14	[	[	PUNCT
cana-1322	24	15	]	]	X
cana-1322	24	16	[	[	PUNCT
cana-1322	24	17	]	]	X
cana-1322	24	18	{	{	PUNCT
cana-1322	24	19	(	(	PUNCT
cana-1322	24	20	)	)	PUNCT
cana-1322	24	21	}	}	PUNCT
cana-1322	24	22	,	,	PUNCT
cana-1322	24	23	where	where	SCONJ
cana-1322	24	24	=[	=[	NOUN
cana-1322	24	25	]	]	PUNCT
cana-1322	24	26	and	and	CCONJ
cana-1322	24	27	=	=	SYM
cana-1322	24	28	[	[	PUNCT
cana-1322	24	29	]	]	X
cana-1322	24	30	it	it	PRON
cana-1322	24	31	is	be	AUX
cana-1322	24	32	denoted	denote	VERB
cana-1322	24	33	as	as	ADP
cana-1322	24	34	example	example	NOUN
cana-1322	24	35	1.7	1.7	NUM
cana-1322	24	36	.	.	PUNCT
cana-1322	25	1	=	=	PRON
cana-1322	25	2	{	{	PUNCT
cana-1322	25	3	<	<	X
cana-1322	25	4	,	,	PUNCT
cana-1322	25	5	[	[	X
cana-1322	25	6	0.5	0.5	NUM
cana-1322	25	7	,	,	PUNCT
cana-1322	25	8	0.75	0.75	NUM
cana-1322	25	9	]	]	PUNCT
cana-1322	25	10	,	,	PUNCT
cana-1322	25	11	[	[	X
cana-1322	25	12	0.55	0.55	NOUN
cana-1322	25	13	,	,	PUNCT
cana-1322	25	14	0.32	0.32	ADV
cana-1322	25	15	]	]	PUNCT
cana-1322	25	16	>	>	X
cana-1322	25	17	,	,	PUNCT
cana-1322	25	18	<	<	X
cana-1322	25	19	,	,	PUNCT
cana-1322	25	20	[	[	X
cana-1322	25	21	0.7	0.7	NUM
cana-1322	25	22	,	,	PUNCT
cana-1322	25	23	0.8	0.8	NUM
cana-1322	25	24	]	]	PUNCT
cana-1322	25	25	,	,	PUNCT
cana-1322	25	26	[	[	X
cana-1322	25	27	0.45	0.45	ADV
cana-1322	25	28	,	,	PUNCT
cana-1322	25	29	0.23	0.23	PROPN
cana-1322	25	30	]	]	PUNCT
cana-1322	25	31	>	>	X
cana-1322	25	32	,	,	PUNCT
cana-1322	25	33	<	<	X
cana-1322	25	34	,	,	PUNCT
cana-1322	25	35	[	[	X
cana-1322	25	36	0.4	0.4	NUM
cana-1322	25	37	,	,	PUNCT
cana-1322	25	38	0.9	0.9	NUM
cana-1322	25	39	]	]	PUNCT
cana-1322	25	40	,	,	PUNCT
cana-1322	25	41	[	[	PUNCT
cana-1322	25	42	0.005	0.005	ADJ
cana-1322	25	43	,	,	PUNCT
cana-1322	25	44	0.002	0.002	NOUN
cana-1322	25	45	]	]	PUNCT
cana-1322	25	46	>	>	X
cana-1322	25	47	}	}	PUNCT
cana-1322	25	48	is	be	AUX
cana-1322	25	49	a	a	DET
cana-1322	25	50	of	of	ADP
cana-1322	25	51	{	{	PUNCT
cana-1322	25	52	}	}	PUNCT
cana-1322	25	53	definition	definition	NOUN
cana-1322	25	54	1.8	1.8	NUM
cana-1322	25	55	[	[	X
cana-1322	25	56	4	4	NUM
cana-1322	25	57	]	]	X
cana-1322	25	58	let	let	VERB
cana-1322	25	59	=	=	PRON
cana-1322	25	60			X
cana-1322	25	61	,	,	PUNCT
cana-1322	25	62			PROPN
cana-1322	25	63	and	and	CCONJ
cana-1322	25	64	=	=	NOUN
cana-1322	25	65			X
cana-1322	25	66	,	,	PUNCT
cana-1322	25	67			PROPN
cana-1322	25	68	be	be	VERB
cana-1322	25	69	.	.	PUNCT
cana-1322	26	1	(	(	PUNCT
cana-1322	26	2	i	i	NOUN
cana-1322	26	3	)	)	PUNCT
cana-1322	26	4	and	and	CCONJ
cana-1322	26	5	(	(	PUNCT
cana-1322	26	6	ii	ii	NOUN
cana-1322	26	7	)	)	PUNCT
cana-1322	26	8	=	=	PRON
cana-1322	26	9	{	{	PUNCT
cana-1322	26	10			X
cana-1322	26	11	rmin	rmin	NOUN
cana-1322	26	12	(	(	PUNCT
cana-1322	26	13	,	,	PUNCT
cana-1322	26	14	,	,	PUNCT
cana-1322	26	15	rmax	rmax	X
cana-1322	26	16	(	(	PUNCT
cana-1322	26	17	,	,	PUNCT
cana-1322	26	18	)	)	PUNCT
cana-1322	26	19			PROPN
cana-1322	26	20	/	/	SYM
cana-1322	26	21	}	}	PUNCT
cana-1322	26	22	.	.	PUNCT
cana-1322	27	1	definition	definition	NOUN
cana-1322	27	2	1.9	1.9	NUM
cana-1322	28	1	[	[	X
cana-1322	28	2	5	5	NUM
cana-1322	28	3	]	]	PUNCT
cana-1322	28	4			X
cana-1322	28	5	,	,	PUNCT
cana-1322	28	6			PROPN
cana-1322	28	7	valued	value	VERB
cana-1322	28	8	(	(	PUNCT
cana-1322	28	9	i	i	NOUN
cana-1322	28	10	)	)	PUNCT
cana-1322	28	11	{	{	PUNCT
cana-1322	28	12	}	}	PUNCT
cana-1322	28	13	(	(	PUNCT
cana-1322	28	14	ii	ii	NOUN
cana-1322	28	15	)	)	PUNCT
cana-1322	28	16	{	{	PUNCT
cana-1322	28	17	}	}	PUNCT
cana-1322	28	18	(	(	PUNCT
cana-1322	28	19	iii	iii	NOUN
cana-1322	28	20	)	)	PUNCT
cana-1322	28	21	{	{	PUNCT
cana-1322	28	22	}	}	PUNCT
cana-1322	28	23	(	(	PUNCT
cana-1322	28	24	iv	iv	X
cana-1322	28	25	)	)	PUNCT
cana-1322	28	26	{	{	PUNCT
cana-1322	28	27	}	}	PUNCT
cana-1322	28	28	where	where	SCONJ
cana-1322	28	29	{	{	PUNCT
cana-1322	28	30	[	[	PUNCT
cana-1322	28	31	]	]	X
cana-1322	28	32	[	[	PUNCT
cana-1322	28	33	]	]	X
cana-1322	28	34	}	}	PUNCT
cana-1322	28	35	[	[	PUNCT
cana-1322	28	36	{	{	PUNCT
cana-1322	28	37	}	}	PUNCT
cana-1322	28	38	{	{	PUNCT
cana-1322	28	39	}	}	PUNCT
cana-1322	28	40	]	]	PUNCT
cana-1322	28	41	and	and	CCONJ
cana-1322	28	42	{	{	PUNCT
cana-1322	28	43	[	[	PUNCT
cana-1322	28	44	]	]	X
cana-1322	28	45	[	[	PUNCT
cana-1322	28	46	]	]	X
cana-1322	28	47	}	}	PUNCT
cana-1322	28	48	[	[	PUNCT
cana-1322	28	49	{	{	PUNCT
cana-1322	28	50	}	}	PUNCT
cana-1322	28	51	{	{	PUNCT
cana-1322	28	52	}	}	PUNCT
cana-1322	28	53	]	]	PUNCT
cana-1322	28	54	example	example	NOUN
cana-1322	28	55	1.10	1.10	NUM
cana-1322	28	56	.	.	PUNCT
cana-1322	28	57	{	{	PUNCT
cana-1322	29	1	[	[	PUNCT
cana-1322	29	2	]	]	X
cana-1322	29	3	[	[	X
cana-1322	29	4			NOUN
cana-1322	29	5			PROPN
cana-1322	29	6	]	]	PUNCT
cana-1322	29	7	[	[	PUNCT
cana-1322	29	8	]	]	X
cana-1322	29	9	[	[	X
cana-1322	29	10			NOUN
cana-1322	29	11			PROPN
cana-1322	29	12	]	]	PUNCT
cana-1322	29	13	[	[	PUNCT
cana-1322	29	14	]	]	X
cana-1322	29	15	[	[	X
cana-1322	29	16			NOUN
cana-1322	29	17			PROPN
cana-1322	29	18	]	]	PUNCT
cana-1322	29	19	}	}	PUNCT
cana-1322	29	20	{	{	PUNCT
cana-1322	29	21	}	}	PUNCT
cana-1322	29	22	definition	definition	NOUN
cana-1322	29	23	1.11	1.11	NUM
cana-1322	29	24			NUM
cana-1322	29	25	,	,	PUNCT
cana-1322	29	26			PROPN
cana-1322	29	27	valued	value	VERB
cana-1322	29	28	(	(	PUNCT
cana-1322	29	29	i	i	NOUN
cana-1322	29	30	)	)	PUNCT
cana-1322	29	31	(	(	PUNCT
cana-1322	29	32	ii	ii	NOUN
cana-1322	29	33	)	)	PUNCT
cana-1322	29	34	definition	definition	NOUN
cana-1322	29	35	1.12	1.12	NUM
cana-1322	29	36	.	.	PUNCT
cana-1322	30	1	[	[	X
cana-1322	30	2	5	5	NUM
cana-1322	30	3	]	]	PUNCT
cana-1322	30	4			X
cana-1322	30	5	,	,	PUNCT
cana-1322	30	6			PROPN
cana-1322	30	7	the	the	DET
cana-1322	30	8	strongest	strong	ADJ
cana-1322	30	9	that	that	PRON
cana-1322	30	10	is	be	AUX
cana-1322	30	11	a	a	DET
cana-1322	30	12	on	on	ADP
cana-1322	30	13	{	{	PUNCT
cana-1322	30	14			X
cana-1322	30	15	(	(	PUNCT
cana-1322	30	16	,	,	PUNCT
cana-1322	30	17	)	)	PUNCT
cana-1322	30	18	,	,	PUNCT
cana-1322	30	19	(	(	PUNCT
cana-1322	30	20	,	,	PUNCT
cana-1322	30	21	)	)	PUNCT
cana-1322	30	22	,	,	PUNCT
cana-1322	30	23	(	(	PUNCT
cana-1322	30	24	,	,	PUNCT
cana-1322	30	25	)	)	PUNCT
cana-1322	30	26			PROPN
cana-1322	30	27	/	/	PUNCT
cana-1322	30	28	for	for	ADP
cana-1322	30	29	all	all	PRON
cana-1322	30	30	,	,	PUNCT
cana-1322	30	31			NOUN
cana-1322	30	32	}	}	PUNCT
cana-1322	30	33	,	,	PUNCT
cana-1322	30	34	where	where	SCONJ
cana-1322	30	35	(	(	PUNCT
cana-1322	30	36	,	,	PUNCT
cana-1322	30	37	)	)	PUNCT
cana-1322	30	38	=	=	SYM
cana-1322	30	39	rmin	rmin	NOUN
cana-1322	30	40	{	{	PUNCT
cana-1322	30	41	(	(	PUNCT
cana-1322	30	42	)	)	PUNCT
cana-1322	30	43	,	,	PUNCT
cana-1322	30	44	(	(	PUNCT
cana-1322	30	45	)	)	PUNCT
cana-1322	30	46	}	}	PUNCT
cana-1322	30	47	and	and	CCONJ
cana-1322	30	48	(	(	PUNCT
cana-1322	30	49	,	,	PUNCT
cana-1322	30	50	)	)	PUNCT
cana-1322	30	51	=	=	SYM
cana-1322	30	52	rmax	rmax	ADJ
cana-1322	30	53	{	{	PUNCT
cana-1322	30	54	(	(	PUNCT
cana-1322	30	55	)	)	PUNCT
cana-1322	30	56	,	,	PUNCT
cana-1322	30	57	(	(	PUNCT
cana-1322	30	58	)	)	PUNCT
cana-1322	30	59	}	}	PUNCT
cana-1322	30	60	,	,	PUNCT
cana-1322	30	61	for	for	ADP
cana-1322	30	62	all	all	PRON
cana-1322	30	63	,	,	PUNCT
cana-1322	30	64			NOUN
cana-1322	30	65	.	.	PUNCT
cana-1322	31	1	definition	definition	NOUN
cana-1322	31	2	1.13	1.13	NUM
cana-1322	31	3	.	.	PUNCT
cana-1322	32	1	[	[	X
cana-1322	32	2	5	5	NUM
cana-1322	32	3	]	]	PUNCT
cana-1322	32	4			X
cana-1322	32	5	,	,	PUNCT
cana-1322	32	6			PROPN
cana-1322	32	7	and	and	CCONJ
cana-1322	32	8			PRON
cana-1322	32	9			PROPN
cana-1322	32	10	and	and	CCONJ
cana-1322	32	11	,	,	PUNCT
cana-1322	32	12	denoted	denote	VERB
cana-1322	32	13	by	by	ADP
cana-1322	32	14	,	,	PUNCT
cana-1322	32	15	is	be	AUX
cana-1322	32	16	defined	define	VERB
cana-1322	32	17	as	as	ADP
cana-1322	32	18	=	=	VERB
cana-1322	32	19	{	{	PUNCT
cana-1322	32	20			X
cana-1322	32	21	(	(	PUNCT
cana-1322	32	22	,	,	PUNCT
cana-1322	32	23	)	)	PUNCT
cana-1322	32	24	,	,	PUNCT
cana-1322	32	25	(	(	PUNCT
cana-1322	32	26	×	×	NOUN
cana-1322	32	27	)	)	PUNCT
cana-1322	33	1	+	+	CCONJ
cana-1322	33	2	(	(	PUNCT
cana-1322	33	3	,	,	PUNCT
cana-1322	33	4	)	)	PUNCT
cana-1322	33	5	,	,	PUNCT
cana-1322	33	6	(	(	PUNCT
cana-1322	33	7	×	×	NOUN
cana-1322	33	8	)	)	PUNCT
cana-1322	33	9			PROPN
cana-1322	33	10	(	(	PUNCT
cana-1322	33	11	,	,	PUNCT
cana-1322	33	12	)	)	PUNCT
cana-1322	33	13			PROPN
cana-1322	33	14	/	/	PUNCT
cana-1322	33	15	for	for	ADP
cana-1322	33	16	all	all	PRON
cana-1322	33	17	(	(	PUNCT
cana-1322	33	18	,	,	PUNCT
cana-1322	33	19	)	)	PUNCT
cana-1322	33	20			NOUN
cana-1322	33	21	}	}	PUNCT
cana-1322	33	22	,	,	PUNCT
cana-1322	33	23	where	where	SCONJ
cana-1322	33	24	(	(	PUNCT
cana-1322	33	25	×	×	NOUN
cana-1322	33	26	)	)	PUNCT
cana-1322	33	27	+	+	CCONJ
cana-1322	33	28	(	(	PUNCT
cana-1322	33	29	,	,	PUNCT
cana-1322	33	30	)	)	PUNCT
cana-1322	33	31	=	=	SYM
cana-1322	33	32	rmin	rmin	NOUN
cana-1322	33	33	{	{	PUNCT
cana-1322	33	34	+	+	X
cana-1322	33	35	(	(	PUNCT
cana-1322	33	36	)	)	PUNCT
cana-1322	33	37	,	,	PUNCT
cana-1322	33	38	+	+	CCONJ
cana-1322	33	39	(	(	PUNCT
cana-1322	33	40	)	)	PUNCT
cana-1322	33	41	}	}	PUNCT
cana-1322	33	42	and	and	CCONJ
cana-1322	33	43	(	(	PUNCT
cana-1322	33	44	×	×	NOUN
cana-1322	33	45	)	)	PUNCT
cana-1322	33	46			PROPN
cana-1322	33	47	(	(	PUNCT
cana-1322	33	48	,	,	PUNCT
cana-1322	33	49	)	)	PUNCT
cana-1322	33	50	=	=	SYM
cana-1322	33	51	rmax	rmax	ADJ
cana-1322	33	52	{	{	PUNCT
cana-1322	33	53			NOUN
cana-1322	33	54	(	(	PUNCT
cana-1322	33	55	)	)	PUNCT
cana-1322	33	56	,	,	PUNCT
cana-1322	33	57			NOUN
cana-1322	33	58	(	(	PUNCT
cana-1322	33	59	)	)	PUNCT
cana-1322	33	60	}	}	PUNCT
cana-1322	33	61	.	.	PUNCT
cana-1322	34	1	communications	communication	NOUN
cana-1322	34	2	on	on	ADP
cana-1322	34	3	applied	apply	VERB
cana-1322	34	4	nonlinear	nonlinear	ADJ
cana-1322	34	5	analysis	analysis	NOUN
cana-1322	34	6	issn	issn	NOUN
cana-1322	34	7	:	:	PUNCT
cana-1322	34	8	1074	1074	NUM
cana-1322	34	9	-	-	PUNCT
cana-1322	34	10	133x	133x	NUM
cana-1322	34	11	vol	vol	NOUN
cana-1322	34	12	31	31	NUM
cana-1322	34	13	no	no	NOUN
cana-1322	34	14	.	.	PUNCT
cana-1322	35	1	7s	7	NOUN
cana-1322	35	2	(	(	PUNCT
cana-1322	35	3	2024	2024	NUM
cana-1322	35	4	)	)	PUNCT
cana-1322	35	5	439	439	NUM
cana-1322	35	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1322	35	7	definition	definition	NOUN
cana-1322	35	8	1.14	1.14	NUM
cana-1322	35	9	.	.	PUNCT
cana-1322	36	1			X
cana-1322	36	2			PROPN
cana-1322	36	3	be	be	VERB
cana-1322	36	4	a	a	PRON
cana-1322	36	5	of	of	ADP
cana-1322	36	6	a	a	DET
cana-1322	36	7	set	set	NOUN
cana-1322	36	8			X
cana-1322	36	9			NOUN
cana-1322	36	10	which	which	PRON
cana-1322	36	11	is	be	AUX
cana-1322	36	12	defined	define	VERB
cana-1322	36	13	as	as	ADP
cana-1322	36	14	and	and	CCONJ
cana-1322	36	15	for	for	ADP
cana-1322	36	16	all	all	PRON
cana-1322	36	17	.	.	PUNCT
cana-1322	37	1	definition	definition	NOUN
cana-1322	37	2	1.15	1.15	NUM
cana-1322	37	3	.	.	PUNCT
cana-1322	38	1			X
cana-1322	38	2			PROPN
cana-1322	38	3	be	be	AUX
cana-1322	38	4	a	a	PRON
cana-1322	38	5	of	of	ADP
cana-1322	38	6	a	a	DET
cana-1322	38	7	set	set	NOUN
cana-1322	38	8			X
cana-1322	38	9			NOUN
cana-1322	38	10	is	be	AUX
cana-1322	38	11	defined	define	VERB
cana-1322	38	12	as	as	ADP
cana-1322	38	13	[	[	PUNCT
cana-1322	38	14	]	]	X
cana-1322	38	15	and	and	CCONJ
cana-1322	38	16	[	[	PUNCT
cana-1322	38	17	]	]	X
cana-1322	38	18	for	for	ADP
cana-1322	38	19	all	all	PRON
cana-1322	38	20	.	.	PUNCT
cana-1322	39	1	2	2	NUM
cana-1322	39	2	–	–	PUNCT
cana-1322	39	3	theorems	theorem	NOUN
cana-1322	39	4	.	.	PUNCT
cana-1322	39	5	theorem	theorem	VERB
cana-1322	39	6	2.1	2.1	NUM
cana-1322	39	7	.	.	PUNCT
cana-1322	40	1	[	[	X
cana-1322	40	2	5	5	NUM
cana-1322	40	3	]	]	PUNCT
cana-1322	40	4			X
cana-1322	40	5	,	,	PUNCT
cana-1322	40	6			PROPN
cana-1322	40	7	is	be	AUX
cana-1322	40	8	a	a	PRON
cana-1322	40	9	where	where	SCONJ
cana-1322	40	10	is	be	AUX
cana-1322	40	11	an	an	DET
cana-1322	40	12	first	first	ADJ
cana-1322	40	13	operation	operation	NOUN
cana-1322	40	14	identity	identity	NOUN
cana-1322	40	15	element	element	NOUN
cana-1322	40	16	of	of	ADP
cana-1322	40	17	theorem	theorem	ADJ
cana-1322	40	18	2.2	2.2	NUM
cana-1322	40	19	.	.	PUNCT
cana-1322	41	1			PUNCT
cana-1322	41	2	,	,	PUNCT
cana-1322	41	3			PROPN
cana-1322	41	4	is	be	AUX
cana-1322	41	5	a	a	PRON
cana-1322	41	6	where	where	SCONJ
cana-1322	41	7	is	be	AUX
cana-1322	41	8	an	an	DET
cana-1322	41	9	first	first	ADJ
cana-1322	41	10	operation	operation	NOUN
cana-1322	41	11	identity	identity	NOUN
cana-1322	41	12	element	element	NOUN
cana-1322	41	13	of	of	ADP
cana-1322	41	14	proof	proof	NOUN
cana-1322	41	15	.	.	PUNCT
cana-1322	42	1	by	by	ADP
cana-1322	42	2	the	the	DET
cana-1322	42	3	theorem	theorem	NOUN
cana-1322	42	4	2.1	2.1	NUM
cana-1322	42	5	,	,	PUNCT
cana-1322	42	6	it	it	PRON
cana-1322	42	7	can	can	AUX
cana-1322	42	8	be	be	AUX
cana-1322	42	9	easily	easily	ADV
cana-1322	42	10	shown	show	VERB
cana-1322	42	11	.	.	PUNCT
cana-1322	43	1	theorem	theorem	VERB
cana-1322	43	2	2.3	2.3	NUM
cana-1322	43	3	.	.	PUNCT
cana-1322	44	1	[	[	X
cana-1322	44	2	5	5	NUM
cana-1322	44	3	]	]	PUNCT
cana-1322	44	4			X
cana-1322	44	5	,	,	PUNCT
cana-1322	44	6			PROPN
cana-1322	44	7	be	be	VERB
cana-1322	44	8	a	a	DET
cana-1322	44	9	(	(	PUNCT
cana-1322	44	10			X
cana-1322	44	11	[	[	PUNCT
cana-1322	44	12	]	]	X
cana-1322	44	13	(	(	PUNCT
cana-1322	44	14	[	[	PUNCT
cana-1322	44	15	]	]	X
cana-1322	44	16	(	(	PUNCT
cana-1322	44	17	[	[	PUNCT
cana-1322	44	18	]	]	X
cana-1322	44	19			NOUN
cana-1322	44	20	(	(	PUNCT
cana-1322	44	21	[	[	PUNCT
cana-1322	44	22	]	]	X
cana-1322	44	23	(	(	PUNCT
cana-1322	44	24	[	[	PUNCT
cana-1322	44	25	]	]	X
cana-1322	44	26	(	(	PUNCT
cana-1322	44	27	[	[	PUNCT
cana-1322	44	28	]	]	X
cana-1322	44	29			NOUN
cana-1322	44	30	(	(	PUNCT
cana-1322	44	31			PROPN
cana-1322	44	32	[	[	PUNCT
cana-1322	44	33	]	]	X
cana-1322	44	34	(	(	PUNCT
cana-1322	44	35	[	[	PUNCT
cana-1322	44	36	]	]	X
cana-1322	44	37	(	(	PUNCT
cana-1322	44	38	[	[	PUNCT
cana-1322	44	39	]	]	X
cana-1322	44	40			NOUN
cana-1322	44	41	(	(	PUNCT
cana-1322	44	42	[	[	PUNCT
cana-1322	44	43	]	]	X
cana-1322	44	44	(	(	PUNCT
cana-1322	44	45	[	[	PUNCT
cana-1322	44	46	]	]	X
cana-1322	44	47	(	(	PUNCT
cana-1322	44	48	[	[	PUNCT
cana-1322	44	49	]	]	X
cana-1322	44	50			NOUN
cana-1322	44	51	theorem	theorem	VERB
cana-1322	44	52	2.4	2.4	NUM
cana-1322	44	53	.	.	PUNCT
cana-1322	45	1			PUNCT
cana-1322	45	2	,	,	PUNCT
cana-1322	45	3			PROPN
cana-1322	45	4	be	be	VERB
cana-1322	45	5	a	a	DET
cana-1322	45	6	(	(	PUNCT
cana-1322	45	7			X
cana-1322	45	8	[	[	PUNCT
cana-1322	45	9	]	]	X
cana-1322	45	10	(	(	PUNCT
cana-1322	45	11	[	[	PUNCT
cana-1322	45	12	]	]	X
cana-1322	45	13	(	(	PUNCT
cana-1322	45	14	[	[	PUNCT
cana-1322	45	15	]	]	X
cana-1322	45	16			NOUN
cana-1322	45	17	(	(	PUNCT
cana-1322	45	18	[	[	PUNCT
cana-1322	45	19	]	]	X
cana-1322	45	20	(	(	PUNCT
cana-1322	45	21	[	[	PUNCT
cana-1322	45	22	]	]	X
cana-1322	45	23	(	(	PUNCT
cana-1322	45	24	[	[	PUNCT
cana-1322	45	25	]	]	X
cana-1322	45	26			NOUN
cana-1322	45	27	(	(	PUNCT
cana-1322	45	28			PROPN
cana-1322	45	29	[	[	PUNCT
cana-1322	45	30	]	]	X
cana-1322	45	31	(	(	PUNCT
cana-1322	45	32	[	[	PUNCT
cana-1322	45	33	]	]	X
cana-1322	45	34	(	(	PUNCT
cana-1322	45	35	[	[	PUNCT
cana-1322	45	36	]	]	X
cana-1322	45	37			NOUN
cana-1322	45	38	(	(	PUNCT
cana-1322	45	39	[	[	PUNCT
cana-1322	45	40	]	]	X
cana-1322	45	41	(	(	PUNCT
cana-1322	45	42	[	[	PUNCT
cana-1322	45	43	]	]	X
cana-1322	45	44	(	(	PUNCT
cana-1322	45	45	[	[	PUNCT
cana-1322	45	46	]	]	PUNCT
cana-1322	45	47			NOUN
cana-1322	45	48	proof	proof	NOUN
cana-1322	45	49	.	.	PUNCT
cana-1322	46	1	by	by	ADP
cana-1322	46	2	the	the	DET
cana-1322	46	3	theorem	theorem	NOUN
cana-1322	46	4	2.3	2.3	NUM
cana-1322	46	5	,	,	PUNCT
cana-1322	46	6	it	it	PRON
cana-1322	46	7	can	can	AUX
cana-1322	46	8	be	be	AUX
cana-1322	46	9	easily	easily	ADV
cana-1322	46	10	shown	show	VERB
cana-1322	46	11	.	.	PUNCT
cana-1322	47	1	theorem	theorem	VERB
cana-1322	47	2	2.5	2.5	NUM
cana-1322	47	3	.	.	PUNCT
cana-1322	48	1	[	[	X
cana-1322	48	2	5	5	NUM
cana-1322	48	3	]	]	PUNCT
cana-1322	48	4			X
cana-1322	48	5	,	,	PUNCT
cana-1322	48	6			PROPN
cana-1322	48	7	is	be	AUX
cana-1322	48	8	a	a	DET
cana-1322	48	9	{	{	PUNCT
cana-1322	48	10	[	[	PUNCT
cana-1322	48	11	]	]	X
cana-1322	48	12	[	[	PUNCT
cana-1322	48	13	]	]	X
cana-1322	48	14	}	}	PUNCT
cana-1322	48	15	is	be	AUX
cana-1322	48	16	either	either	CCONJ
cana-1322	48	17	empty	empty	ADJ
cana-1322	48	18	or	or	CCONJ
cana-1322	48	19	theorem	theorem	VERB
cana-1322	48	20	2.6	2.6	NUM
cana-1322	48	21	.	.	PUNCT
cana-1322	48	22			PUNCT
cana-1322	48	23	,	,	PUNCT
cana-1322	48	24			PROPN
cana-1322	48	25	is	be	AUX
cana-1322	48	26	a	a	DET
cana-1322	48	27	{	{	PUNCT
cana-1322	48	28	[	[	PUNCT
cana-1322	48	29	]	]	X
cana-1322	48	30	[	[	PUNCT
cana-1322	48	31	]	]	X
cana-1322	48	32	}	}	PUNCT
cana-1322	48	33	is	be	AUX
cana-1322	48	34	either	either	CCONJ
cana-1322	48	35	empty	empty	ADJ
cana-1322	48	36	or	or	CCONJ
cana-1322	48	37	proof	proof	ADJ
cana-1322	48	38	.	.	PUNCT
cana-1322	49	1	by	by	ADP
cana-1322	49	2	the	the	DET
cana-1322	49	3	theorem	theorem	ADJ
cana-1322	49	4	2.5	2.5	NUM
cana-1322	49	5	,	,	PUNCT
cana-1322	49	6	it	it	PRON
cana-1322	49	7	can	can	AUX
cana-1322	49	8	be	be	AUX
cana-1322	49	9	easily	easily	ADV
cana-1322	49	10	shown	show	VERB
cana-1322	49	11	.	.	PUNCT
cana-1322	50	1	theorem	theorem	ADJ
cana-1322	50	2	2.7	2.7	NUM
cana-1322	50	3	.	.	PUNCT
cana-1322	51	1	[	[	X
cana-1322	51	2	5	5	NUM
cana-1322	51	3	]	]	PUNCT
cana-1322	51	4			CCONJ
cana-1322	51	5			NOUN
cana-1322	51	6			NUM
cana-1322	51	7			PROPN
cana-1322	51	8	communications	communication	NOUN
cana-1322	51	9	on	on	ADP
cana-1322	51	10	applied	apply	VERB
cana-1322	51	11	nonlinear	nonlinear	ADJ
cana-1322	51	12	analysis	analysis	NOUN
cana-1322	51	13	issn	issn	NOUN
cana-1322	51	14	:	:	PUNCT
cana-1322	51	15	1074	1074	NUM
cana-1322	51	16	-	-	PUNCT
cana-1322	51	17	133x	133x	NUM
cana-1322	51	18	vol	vol	NOUN
cana-1322	51	19	31	31	NUM
cana-1322	51	20	no	no	NOUN
cana-1322	51	21	.	.	PUNCT
cana-1322	52	1	7s	7	NOUN
cana-1322	52	2	(	(	PUNCT
cana-1322	52	3	2024	2024	NUM
cana-1322	52	4	)	)	PUNCT
cana-1322	52	5	440	440	NUM
cana-1322	52	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1322	52	7	theorem	theorem	VERB
cana-1322	52	8	2.8	2.8	NUM
cana-1322	52	9	.	.	PUNCT
cana-1322	53	1			X
cana-1322	53	2			NOUN
cana-1322	53	3			NUM
cana-1322	53	4			PROPN
cana-1322	53	5	proof	proof	NOUN
cana-1322	53	6	.	.	PUNCT
cana-1322	54	1	let	let	AUX
cana-1322	54	2	be	be	AUX
cana-1322	54	3	in	in	ADP
cana-1322	54	4	.	.	PUNCT
cana-1322	55	1	let	let	VERB
cana-1322	55	2	then	then	ADV
cana-1322	55	3	(	(	PUNCT
cana-1322	55	4	)	)	PUNCT
cana-1322	55	5	=	=	SYM
cana-1322	55	6	rmin	rmin	NOUN
cana-1322	55	7	{	{	PUNCT
cana-1322	55	8	(	(	PUNCT
cana-1322	55	9	)	)	PUNCT
cana-1322	55	10	,	,	PUNCT
cana-1322	55	11	(	(	PUNCT
cana-1322	55	12	)	)	PUNCT
cana-1322	55	13	}	}	PUNCT
cana-1322	55	14	=	=	SYM
cana-1322	55	15	rmin	rmin	NOUN
cana-1322	55	16	{	{	PUNCT
cana-1322	55	17	(	(	PUNCT
cana-1322	55	18	)	)	PUNCT
cana-1322	55	19	,	,	PUNCT
cana-1322	55	20	(	(	PUNCT
cana-1322	55	21	)	)	PUNCT
cana-1322	55	22	}	}	PUNCT
cana-1322	55	23	=	=	SYM
cana-1322	55	24	(	(	PUNCT
cana-1322	55	25	)	)	PUNCT
cana-1322	55	26	,	,	PUNCT
cana-1322	55	27			VERB
cana-1322	55	28	in	in	ADP
cana-1322	55	29	.	.	PUNCT
cana-1322	56	1	and	and	CCONJ
cana-1322	56	2	(	(	PUNCT
cana-1322	56	3	)	)	PUNCT
cana-1322	56	4	=	=	SYM
cana-1322	56	5	rmax	rmax	ADJ
cana-1322	56	6	{	{	PUNCT
cana-1322	56	7	(	(	PUNCT
cana-1322	56	8	)	)	PUNCT
cana-1322	56	9	,	,	PUNCT
cana-1322	56	10	(	(	PUNCT
cana-1322	56	11	)	)	PUNCT
cana-1322	56	12	}	}	PUNCT
cana-1322	56	13	=	=	SYM
cana-1322	56	14	rmax	rmax	ADJ
cana-1322	56	15	{	{	PUNCT
cana-1322	56	16	(	(	PUNCT
cana-1322	56	17	)	)	PUNCT
cana-1322	56	18	,	,	PUNCT
cana-1322	56	19	(	(	PUNCT
cana-1322	56	20	)	)	PUNCT
cana-1322	56	21	}	}	PUNCT
cana-1322	56	22	=	=	SYM
cana-1322	56	23	(	(	PUNCT
cana-1322	56	24	)	)	PUNCT
cana-1322	56	25	,	,	PUNCT
cana-1322	56	26			VERB
cana-1322	56	27	in	in	ADP
cana-1322	56	28	.	.	PUNCT
cana-1322	57	1	hence	hence	ADV
cana-1322	57	2	theorem	theorem	VERB
cana-1322	57	3	2.9	2.9	NUM
cana-1322	57	4	.	.	PUNCT
cana-1322	58	1	[	[	X
cana-1322	58	2	5	5	NUM
cana-1322	58	3	]	]	PUNCT
cana-1322	58	4	,	,	PUNCT
cana-1322	58	5	,	,	PUNCT
cana-1322	58	6	…	…	PUNCT
cana-1322	58	7	and	and	CCONJ
cana-1322	58	8	…	…	PUNCT
cana-1322	58	9	theorem	theorem	VERB
cana-1322	58	10	2.10	2.10	NUM
cana-1322	58	11	.	.	PUNCT
cana-1322	58	12	,	,	PUNCT
cana-1322	58	13	,	,	PUNCT
cana-1322	58	14	…	…	PUNCT
cana-1322	58	15	and	and	CCONJ
cana-1322	58	16	…	…	PUNCT
cana-1322	58	17	proof	proof	NOUN
cana-1322	58	18	.	.	PUNCT
cana-1322	59	1	by	by	ADP
cana-1322	59	2	the	the	DET
cana-1322	59	3	theorem	theorem	NOUN
cana-1322	59	4	2.9	2.9	NUM
cana-1322	59	5	,	,	PUNCT
cana-1322	59	6	it	it	PRON
cana-1322	59	7	can	can	AUX
cana-1322	59	8	be	be	AUX
cana-1322	59	9	easily	easily	ADV
cana-1322	59	10	shown	show	VERB
cana-1322	59	11	.	.	PUNCT
cana-1322	60	1	theorem	theorem	VERB
cana-1322	60	2	2.11	2.11	NUM
cana-1322	60	3	.	.	PUNCT
cana-1322	61	1	[	[	X
cana-1322	61	2	5	5	NUM
cana-1322	61	3	]	]	PUNCT
cana-1322	61	4	,	,	PUNCT
cana-1322	61	5	,	,	PUNCT
cana-1322	61	6	…	…	PUNCT
cana-1322	61	7	intersection	intersection	NOUN
cana-1322	61	8	…	…	PUNCT
cana-1322	61	9	theorem	theorem	VERB
cana-1322	61	10	2.12	2.12	NUM
cana-1322	61	11	.	.	PUNCT
cana-1322	61	12	,	,	PUNCT
cana-1322	61	13	,	,	PUNCT
cana-1322	61	14	…	…	PUNCT
cana-1322	61	15	…	…	PUNCT
cana-1322	61	16	proof	proof	NOUN
cana-1322	61	17	.	.	PUNCT
cana-1322	62	1	by	by	ADP
cana-1322	62	2	the	the	DET
cana-1322	62	3	theorem	theorem	NOUN
cana-1322	62	4	2.11	2.11	NUM
cana-1322	62	5	,	,	PUNCT
cana-1322	62	6	it	it	PRON
cana-1322	62	7	can	can	AUX
cana-1322	62	8	be	be	AUX
cana-1322	62	9	easily	easily	ADV
cana-1322	62	10	shown	show	VERB
cana-1322	62	11	.	.	PUNCT
cana-1322	63	1	theorem	theorem	NOUN
cana-1322	63	2	2.13	2.13	NUM
cana-1322	63	3	.	.	PUNCT
cana-1322	64	1	[	[	X
cana-1322	64	2	5	5	NUM
cana-1322	64	3	]	]	PUNCT
cana-1322	64	4	and	and	CCONJ
cana-1322	64	5	be	be	AUX
cana-1322	64	6	the	the	DET
cana-1322	64	7	stronget	stronget	NOUN
cana-1322	64	8	relation	relation	NOUN
cana-1322	64	9	of	of	ADP
cana-1322	64	10	.	.	PUNCT
cana-1322	65	1	then	then	ADV
cana-1322	65	2	is	be	AUX
cana-1322	65	3	a	a	PRON
cana-1322	65	4	of	of	ADP
cana-1322	65	5	if	if	SCONJ
cana-1322	65	6	and	and	CCONJ
cana-1322	65	7	only	only	ADV
cana-1322	65	8	if	if	SCONJ
cana-1322	65	9	is	be	AUX
cana-1322	65	10	a	a	PRON
cana-1322	65	11	of	of	ADP
cana-1322	65	12	×	×	NOUN
cana-1322	65	13	theorem	theorem	ADJ
cana-1322	65	14	2.14	2.14	NUM
cana-1322	65	15	.	.	PUNCT
cana-1322	66	1	and	and	CCONJ
cana-1322	66	2	be	be	AUX
cana-1322	66	3	the	the	DET
cana-1322	66	4	stronget	stronget	NOUN
cana-1322	66	5	relation	relation	NOUN
cana-1322	66	6	of	of	ADP
cana-1322	66	7	.	.	PUNCT
cana-1322	67	1	then	then	ADV
cana-1322	67	2	is	be	AUX
cana-1322	67	3	a	a	PRON
cana-1322	67	4	of	of	ADP
cana-1322	67	5	if	if	SCONJ
cana-1322	67	6	and	and	CCONJ
cana-1322	67	7	only	only	ADV
cana-1322	67	8	if	if	SCONJ
cana-1322	67	9	is	be	AUX
cana-1322	67	10	a	a	PRON
cana-1322	67	11	of	of	ADP
cana-1322	67	12	×	×	NOUN
cana-1322	67	13	proof	proof	NOUN
cana-1322	67	14	.	.	PUNCT
cana-1322	68	1	let	let	AUX
cana-1322	68	2	be	be	AUX
cana-1322	68	3	in	in	ADP
cana-1322	68	4	.	.	PUNCT
cana-1322	69	1	then	then	ADV
cana-1322	69	2	(	(	PUNCT
cana-1322	69	3	,	,	PUNCT
cana-1322	69	4	)	)	PUNCT
cana-1322	69	5	and	and	CCONJ
cana-1322	69	6	(	(	PUNCT
cana-1322	69	7	,	,	PUNCT
cana-1322	69	8	)	)	PUNCT
cana-1322	69	9	are	be	AUX
cana-1322	69	10	in	in	ADP
cana-1322	69	11	×	×	PROPN
cana-1322	69	12	.	.	PUNCT
cana-1322	70	1	by	by	ADP
cana-1322	70	2	theorem	theorem	NOUN
cana-1322	70	3	2.13	2.13	NUM
cana-1322	70	4	,	,	PUNCT
cana-1322	70	5	is	be	AUX
cana-1322	70	6	a	a	PRON
cana-1322	70	7	of	of	ADP
cana-1322	70	8	×	×	NOUN
cana-1322	70	9	then	then	ADV
cana-1322	70	10	[	[	X
cana-1322	70	11	(	(	PUNCT
cana-1322	70	12	,	,	PUNCT
cana-1322	70	13	)	)	PUNCT
cana-1322	70	14	(	(	PUNCT
cana-1322	70	15	,	,	PUNCT
cana-1322	70	16	)	)	PUNCT
cana-1322	70	17	]	]	PUNCT
cana-1322	71	1	=	=	PUNCT
cana-1322	71	2	(	(	PUNCT
cana-1322	71	3	,	,	PUNCT
cana-1322	71	4	)	)	PUNCT
cana-1322	71	5	=	=	SYM
cana-1322	71	6	rmin	rmin	NOUN
cana-1322	71	7	{	{	PUNCT
cana-1322	71	8	+	+	X
cana-1322	71	9	(	(	PUNCT
cana-1322	71	10	)	)	PUNCT
cana-1322	71	11	,	,	PUNCT
cana-1322	71	12	+	+	CCONJ
cana-1322	71	13	(	(	PUNCT
cana-1322	71	14	)	)	PUNCT
cana-1322	71	15	}	}	PUNCT
cana-1322	71	16	=	=	SYM
cana-1322	71	17	rmin	rmin	NOUN
cana-1322	71	18	{	{	PUNCT
cana-1322	71	19	+	+	X
cana-1322	71	20	(	(	PUNCT
cana-1322	71	21	)	)	PUNCT
cana-1322	71	22	,	,	PUNCT
cana-1322	71	23	+	+	CCONJ
cana-1322	71	24	(	(	PUNCT
cana-1322	71	25	)	)	PUNCT
cana-1322	71	26	}	}	PUNCT
cana-1322	71	27	=	=	SYM
cana-1322	71	28	(	(	PUNCT
cana-1322	71	29	,	,	PUNCT
cana-1322	71	30	)	)	PUNCT
cana-1322	71	31	=	=	PUNCT
cana-1322	72	1	[	[	X
cana-1322	72	2	(	(	PUNCT
cana-1322	72	3	,	,	PUNCT
cana-1322	72	4	)	)	PUNCT
cana-1322	72	5	(	(	PUNCT
cana-1322	72	6	,	,	PUNCT
cana-1322	72	7	)	)	PUNCT
cana-1322	72	8	]	]	PUNCT
cana-1322	72	9	,	,	PUNCT
cana-1322	72	10			NOUN
cana-1322	72	11	(	(	PUNCT
cana-1322	72	12	,	,	PUNCT
cana-1322	72	13	)	)	PUNCT
cana-1322	72	14	,	,	PUNCT
cana-1322	72	15	(	(	PUNCT
cana-1322	72	16	,	,	PUNCT
cana-1322	72	17	)	)	PUNCT
cana-1322	72	18			NOUN
cana-1322	72	19	×	×	NOUN
cana-1322	72	20	.	.	PUNCT
cana-1322	73	1	and	and	CCONJ
cana-1322	74	1	[	[	X
cana-1322	74	2	(	(	PUNCT
cana-1322	74	3	,	,	PUNCT
cana-1322	74	4	)	)	PUNCT
cana-1322	74	5	(	(	PUNCT
cana-1322	74	6	,	,	PUNCT
cana-1322	74	7	)	)	PUNCT
cana-1322	74	8	]	]	PUNCT
cana-1322	75	1	=	=	PUNCT
cana-1322	75	2	(	(	PUNCT
cana-1322	75	3	,	,	PUNCT
cana-1322	75	4	)	)	PUNCT
cana-1322	75	5	=	=	SYM
cana-1322	75	6	rmax	rmax	ADJ
cana-1322	75	7	{	{	PUNCT
cana-1322	75	8	(	(	PUNCT
cana-1322	75	9	)	)	PUNCT
cana-1322	75	10	,	,	PUNCT
cana-1322	75	11	(	(	PUNCT
cana-1322	75	12	)	)	PUNCT
cana-1322	75	13	}	}	PUNCT
cana-1322	75	14	=	=	SYM
cana-1322	75	15	rmax	rmax	ADJ
cana-1322	75	16	{	{	PUNCT
cana-1322	75	17	(	(	PUNCT
cana-1322	75	18	)	)	PUNCT
cana-1322	75	19	,	,	PUNCT
cana-1322	75	20	(	(	PUNCT
cana-1322	75	21	)	)	PUNCT
cana-1322	75	22	}	}	PUNCT
cana-1322	75	23	=	=	SYM
cana-1322	75	24	(	(	PUNCT
cana-1322	75	25	,	,	PUNCT
cana-1322	75	26	)	)	PUNCT
cana-1322	75	27	=	=	PUNCT
cana-1322	76	1	[	[	X
cana-1322	76	2	(	(	PUNCT
cana-1322	76	3	,	,	PUNCT
cana-1322	76	4	)	)	PUNCT
cana-1322	76	5	(	(	PUNCT
cana-1322	76	6	,	,	PUNCT
cana-1322	76	7	)	)	PUNCT
cana-1322	76	8	]	]	PUNCT
cana-1322	76	9	,	,	PUNCT
cana-1322	76	10			NOUN
cana-1322	76	11	(	(	PUNCT
cana-1322	76	12	,	,	PUNCT
cana-1322	76	13	)	)	PUNCT
cana-1322	76	14	,	,	PUNCT
cana-1322	76	15	(	(	PUNCT
cana-1322	76	16	,	,	PUNCT
cana-1322	76	17	)	)	PUNCT
cana-1322	76	18			NOUN
cana-1322	76	19	×	×	NOUN
cana-1322	76	20	.	.	PUNCT
cana-1322	77	1	hence	hence	ADV
cana-1322	77	2	is	be	AUX
cana-1322	77	3	a	a	PRON
cana-1322	77	4	of	of	ADP
cana-1322	77	5	×	×	NOUN
cana-1322	77	6	.	.	PUNCT
cana-1322	78	1	conversely	conversely	ADV
cana-1322	78	2	,	,	PUNCT
cana-1322	78	3	assume	assume	VERB
cana-1322	78	4	that	that	PRON
cana-1322	78	5	is	be	AUX
cana-1322	78	6	a	a	PRON
cana-1322	78	7	of	of	ADP
cana-1322	78	8	×	×	NOUN
cana-1322	78	9	.	.	PUNCT
cana-1322	79	1	by	by	ADP
cana-1322	79	2	theorem	theorem	NOUN
cana-1322	79	3	2.13	2.13	NUM
cana-1322	79	4	,	,	PUNCT
cana-1322	79	5	is	be	AUX
cana-1322	79	6	a	a	PRON
cana-1322	79	7	of	of	ADP
cana-1322	79	8	rmin	rmin	NOUN
cana-1322	79	9	{	{	PUNCT
cana-1322	79	10	+	+	X
cana-1322	79	11	(	(	PUNCT
cana-1322	79	12	)	)	PUNCT
cana-1322	79	13	,	,	PUNCT
cana-1322	79	14	+	+	CCONJ
cana-1322	79	15	(	(	PUNCT
cana-1322	79	16	)	)	PUNCT
cana-1322	79	17	}	}	PUNCT
cana-1322	79	18	=	=	SYM
cana-1322	79	19	(	(	PUNCT
cana-1322	79	20	,	,	PUNCT
cana-1322	79	21	)	)	PUNCT
cana-1322	79	22	=	=	PUNCT
cana-1322	80	1	[	[	X
cana-1322	80	2	(	(	PUNCT
cana-1322	80	3	,	,	PUNCT
cana-1322	80	4	)	)	PUNCT
cana-1322	80	5	(	(	PUNCT
cana-1322	80	6	,	,	PUNCT
cana-1322	80	7	)	)	PUNCT
cana-1322	80	8	]	]	PUNCT
cana-1322	81	1	=	=	PUNCT
cana-1322	82	1	[	[	X
cana-1322	82	2	(	(	PUNCT
cana-1322	82	3	,	,	PUNCT
cana-1322	82	4	)	)	PUNCT
cana-1322	82	5	(	(	PUNCT
cana-1322	82	6	,	,	PUNCT
cana-1322	82	7	)	)	PUNCT
cana-1322	82	8	]	]	PUNCT
cana-1322	83	1	=	=	PUNCT
cana-1322	83	2	(	(	PUNCT
cana-1322	83	3	,	,	PUNCT
cana-1322	83	4	)	)	PUNCT
cana-1322	83	5	=	=	SYM
cana-1322	83	6	rmin	rmin	NOUN
cana-1322	83	7	{	{	PUNCT
cana-1322	83	8	+	+	X
cana-1322	83	9	(	(	PUNCT
cana-1322	83	10	)	)	PUNCT
cana-1322	83	11	,	,	PUNCT
cana-1322	83	12	+	+	CCONJ
cana-1322	83	13	(	(	PUNCT
cana-1322	83	14	)	)	PUNCT
cana-1322	83	15	}	}	PUNCT
cana-1322	83	16	,	,	PUNCT
cana-1322	83	17	put	put	VERB
cana-1322	83	18	and	and	CCONJ
cana-1322	83	19	,	,	PUNCT
cana-1322	83	20	where	where	SCONJ
cana-1322	83	21	is	be	AUX
cana-1322	83	22	an	an	DET
cana-1322	83	23	first	first	ADJ
cana-1322	83	24	operation	operation	NOUN
cana-1322	83	25	identity	identity	NOUN
cana-1322	83	26	element	element	NOUN
cana-1322	83	27	of	of	ADP
cana-1322	83	28	,	,	PUNCT
cana-1322	83	29	then	then	ADV
cana-1322	83	30	+	+	CCONJ
cana-1322	83	31	(	(	PUNCT
cana-1322	83	32	)	)	PUNCT
cana-1322	83	33	=	=	PUNCT
cana-1322	84	1	+	+	PUNCT
cana-1322	84	2	(	(	PUNCT
cana-1322	84	3	)	)	PUNCT
cana-1322	84	4	,	,	PUNCT
cana-1322	84	5			NOUN
cana-1322	84	6	,	,	PUNCT
cana-1322	84	7			PROPN
cana-1322	84	8	.	.	PUNCT
cana-1322	85	1	and	and	CCONJ
cana-1322	85	2	rmax	rmax	ADJ
cana-1322	85	3	{	{	PUNCT
cana-1322	85	4			NOUN
cana-1322	85	5	(	(	PUNCT
cana-1322	85	6	)	)	PUNCT
cana-1322	85	7	,	,	PUNCT
cana-1322	85	8			NOUN
cana-1322	85	9	(	(	PUNCT
cana-1322	85	10	)	)	PUNCT
cana-1322	85	11	}	}	PUNCT
cana-1322	85	12	=	=	SYM
cana-1322	85	13	(	(	PUNCT
cana-1322	85	14	,	,	PUNCT
cana-1322	85	15	)	)	PUNCT
cana-1322	85	16	=	=	PUNCT
cana-1322	86	1	[	[	X
cana-1322	86	2	(	(	PUNCT
cana-1322	86	3	,	,	PUNCT
cana-1322	86	4	)	)	PUNCT
cana-1322	86	5	(	(	PUNCT
cana-1322	86	6	,	,	PUNCT
cana-1322	86	7	)	)	PUNCT
cana-1322	86	8	]	]	PUNCT
cana-1322	87	1	=	=	PUNCT
cana-1322	88	1	[	[	X
cana-1322	88	2	(	(	PUNCT
cana-1322	88	3	,	,	PUNCT
cana-1322	88	4	)	)	PUNCT
cana-1322	88	5	(	(	PUNCT
cana-1322	88	6	,	,	PUNCT
cana-1322	88	7	)	)	PUNCT
cana-1322	88	8	]	]	PUNCT
cana-1322	89	1	=	=	PUNCT
cana-1322	89	2	(	(	PUNCT
cana-1322	89	3	,	,	PUNCT
cana-1322	89	4	)	)	PUNCT
cana-1322	89	5	=	=	SYM
cana-1322	89	6	rmax	rmax	ADJ
cana-1322	89	7	{	{	PUNCT
cana-1322	89	8			NOUN
cana-1322	89	9	(	(	PUNCT
cana-1322	89	10	)	)	PUNCT
cana-1322	89	11	,	,	PUNCT
cana-1322	89	12			NOUN
cana-1322	89	13	(	(	PUNCT
cana-1322	89	14	)	)	PUNCT
cana-1322	89	15	}	}	PUNCT
cana-1322	89	16	,	,	PUNCT
cana-1322	89	17	put	put	VERB
cana-1322	89	18	and	and	CCONJ
cana-1322	89	19	,	,	PUNCT
cana-1322	89	20	where	where	SCONJ
cana-1322	89	21	is	be	AUX
cana-1322	89	22	an	an	DET
cana-1322	89	23	first	first	ADJ
cana-1322	89	24	operation	operation	NOUN
cana-1322	89	25	identity	identity	NOUN
cana-1322	89	26	element	element	NOUN
cana-1322	89	27	of	of	ADP
cana-1322	89	28	,	,	PUNCT
cana-1322	89	29	then	then	ADV
cana-1322	89	30			PROPN
cana-1322	89	31	(	(	PUNCT
cana-1322	89	32	)	)	PUNCT
cana-1322	89	33	=	=	SYM
cana-1322	89	34			NOUN
cana-1322	89	35	(	(	PUNCT
cana-1322	89	36	)	)	PUNCT
cana-1322	89	37	,	,	PUNCT
cana-1322	89	38			NOUN
cana-1322	89	39	,	,	PUNCT
cana-1322	89	40			NOUN
cana-1322	89	41	.	.	PUNCT
cana-1322	90	1	hence	hence	ADV
cana-1322	90	2	is	be	AUX
cana-1322	90	3	a	a	PRON
cana-1322	90	4	of	of	ADP
cana-1322	90	5	.	.	PUNCT
cana-1322	91	1	communications	communication	NOUN
cana-1322	91	2	on	on	ADP
cana-1322	91	3	applied	apply	VERB
cana-1322	91	4	nonlinear	nonlinear	ADJ
cana-1322	91	5	analysis	analysis	NOUN
cana-1322	91	6	issn	issn	NOUN
cana-1322	91	7	:	:	PUNCT
cana-1322	91	8	1074	1074	NUM
cana-1322	91	9	-	-	PUNCT
cana-1322	91	10	133x	133x	NUM
cana-1322	91	11	vol	vol	NOUN
cana-1322	91	12	31	31	NUM
cana-1322	91	13	no	no	NOUN
cana-1322	91	14	.	.	PUNCT
cana-1322	92	1	7s	7	NOUN
cana-1322	92	2	(	(	PUNCT
cana-1322	92	3	2024	2024	NUM
cana-1322	92	4	)	)	PUNCT
cana-1322	92	5	441	441	NUM
cana-1322	92	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1322	92	7	theorem	theorem	VERB
cana-1322	92	8	2.15	2.15	NUM
cana-1322	92	9	.	.	PUNCT
cana-1322	93	1	[	[	X
cana-1322	93	2	5	5	NUM
cana-1322	93	3	]	]	PUNCT
cana-1322	93	4	,	,	PUNCT
cana-1322	93	5	,	,	PUNCT
cana-1322	93	6	…	…	PUNCT
cana-1322	93	7	,	,	PUNCT
cana-1322	93	8	be	be	AUX
cana-1322	93	9	and	and	CCONJ
cana-1322	93	10	be	be	AUX
cana-1322	93	11	the	the	DET
cana-1322	93	12	strongest	strong	ADJ
cana-1322	93	13	ndimensional	ndimensional	ADJ
cana-1322	93	14	relation	relation	NOUN
cana-1322	93	15	of	of	ADP
cana-1322	93	16	.	.	PUNCT
cana-1322	94	1	then	then	ADV
cana-1322	94	2	,	,	PUNCT
cana-1322	94	3	,	,	PUNCT
cana-1322	94	4	…	…	PUNCT
cana-1322	94	5	,	,	PUNCT
cana-1322	94	6	are	be	AUX
cana-1322	94	7	×	×	NOUN
cana-1322	94	8	…	…	SYM
cana-1322	94	9	×	×	NOUN
cana-1322	94	10	(	(	PUNCT
cana-1322	94	11	m	m	NOUN
cana-1322	94	12	times	time	NOUN
cana-1322	94	13	)	)	PUNCT
cana-1322	94	14	.	.	PUNCT
cana-1322	95	1	theorem	theorem	VERB
cana-1322	95	2	2.16	2.16	NUM
cana-1322	95	3	.	.	PUNCT
cana-1322	96	1	,	,	PUNCT
cana-1322	96	2	,	,	PUNCT
cana-1322	96	3	…	…	PUNCT
cana-1322	96	4	,	,	PUNCT
cana-1322	96	5	be	be	AUX
cana-1322	96	6	and	and	CCONJ
cana-1322	96	7	be	be	AUX
cana-1322	96	8	the	the	DET
cana-1322	96	9	strongest	strong	ADJ
cana-1322	96	10	n	n	CCONJ
cana-1322	96	11	-	-	PUNCT
cana-1322	96	12	dimensional	dimensional	ADJ
cana-1322	96	13	relation	relation	NOUN
cana-1322	96	14	of	of	ADP
cana-1322	96	15	.	.	PUNCT
cana-1322	97	1	then	then	ADV
cana-1322	97	2	,	,	PUNCT
cana-1322	97	3	,	,	PUNCT
cana-1322	97	4	…	…	PUNCT
cana-1322	97	5	,	,	PUNCT
cana-1322	97	6	are	be	AUX
cana-1322	97	7	×	×	NOUN
cana-1322	97	8	…	…	SYM
cana-1322	97	9	×	×	NOUN
cana-1322	97	10	(	(	PUNCT
cana-1322	97	11	m	m	NOUN
cana-1322	97	12	times	time	NOUN
cana-1322	97	13	)	)	PUNCT
cana-1322	97	14	.	.	PUNCT
cana-1322	98	1	proof	proof	NOUN
cana-1322	98	2	.	.	PUNCT
cana-1322	99	1	by	by	ADP
cana-1322	99	2	the	the	DET
cana-1322	99	3	theorem	theorem	NOUN
cana-1322	99	4	2.15	2.15	NUM
cana-1322	99	5	,	,	PUNCT
cana-1322	99	6	it	it	PRON
cana-1322	99	7	can	can	AUX
cana-1322	99	8	be	be	AUX
cana-1322	99	9	easily	easily	ADV
cana-1322	99	10	shown	show	VERB
cana-1322	99	11	.	.	PUNCT
cana-1322	100	1	theorem	theorem	VERB
cana-1322	100	2	2.17	2.17	NUM
cana-1322	100	3	.	.	PUNCT
cana-1322	101	1	[	[	X
cana-1322	101	2	5	5	NUM
cana-1322	101	3	]	]	PUNCT
cana-1322	101	4	and	and	CCONJ
cana-1322	101	5	is	be	AUX
cana-1322	101	6	a	a	DET
cana-1322	101	7	theorem	theorem	NOUN
cana-1322	101	8	2.18	2.18	NUM
cana-1322	101	9	.	.	PUNCT
cana-1322	102	1	and	and	CCONJ
cana-1322	102	2	is	be	AUX
cana-1322	102	3	a	a	DET
cana-1322	102	4	proof	proof	NOUN
cana-1322	102	5	.	.	PUNCT
cana-1322	103	1	let	let	AUX
cana-1322	103	2	be	be	AUX
cana-1322	103	3	in	in	ADP
cana-1322	103	4	1	1	NUM
cana-1322	103	5	and	and	CCONJ
cana-1322	103	6	be	be	AUX
cana-1322	103	7	in	in	ADP
cana-1322	103	8	2	2	NUM
cana-1322	103	9	.	.	PUNCT
cana-1322	104	1	then	then	ADV
cana-1322	104	2	(	(	PUNCT
cana-1322	104	3	,	,	PUNCT
cana-1322	104	4	)	)	PUNCT
cana-1322	104	5	,	,	PUNCT
cana-1322	104	6	(	(	PUNCT
cana-1322	104	7	,	,	PUNCT
cana-1322	104	8	)	)	PUNCT
cana-1322	104	9			NOUN
cana-1322	104	10	1×	1×	NUM
cana-1322	104	11	2	2	NUM
cana-1322	104	12	.	.	PUNCT
cana-1322	104	13	by	by	ADP
cana-1322	104	14	theorem	theorem	NOUN
cana-1322	104	15	2.17	2.17	NUM
cana-1322	104	16	,	,	PUNCT
cana-1322	104	17	is	be	AUX
cana-1322	104	18	a	a	DET
cana-1322	104	19	then	then	ADV
cana-1322	104	20	(	(	PUNCT
cana-1322	104	21	×	×	NOUN
cana-1322	104	22	)	)	PUNCT
cana-1322	104	23	+	+	PUNCT
cana-1322	105	1	[	[	X
cana-1322	105	2	(	(	PUNCT
cana-1322	105	3	,	,	PUNCT
cana-1322	105	4	)	)	PUNCT
cana-1322	105	5	(	(	PUNCT
cana-1322	105	6	,	,	PUNCT
cana-1322	105	7	)	)	PUNCT
cana-1322	105	8	]	]	PUNCT
cana-1322	106	1	=	=	PUNCT
cana-1322	106	2	(	(	PUNCT
cana-1322	106	3	×	×	PROPN
cana-1322	106	4	)	)	PUNCT
cana-1322	106	5	+	+	CCONJ
cana-1322	106	6	(	(	PUNCT
cana-1322	106	7	,	,	PUNCT
cana-1322	106	8	)	)	PUNCT
cana-1322	106	9	=	=	SYM
cana-1322	106	10	rmin	rmin	NOUN
cana-1322	106	11	{	{	PUNCT
cana-1322	106	12	+	+	X
cana-1322	106	13	(	(	PUNCT
cana-1322	106	14	)	)	PUNCT
cana-1322	106	15	,	,	PUNCT
cana-1322	106	16	+	+	CCONJ
cana-1322	106	17	(	(	PUNCT
cana-1322	106	18	)	)	PUNCT
cana-1322	106	19	}	}	PUNCT
cana-1322	106	20	=	=	SYM
cana-1322	106	21	rmin	rmin	NOUN
cana-1322	106	22	{	{	PUNCT
cana-1322	106	23	+	+	X
cana-1322	106	24	(	(	PUNCT
cana-1322	106	25	)	)	PUNCT
cana-1322	106	26	,	,	PUNCT
cana-1322	106	27	+	+	CCONJ
cana-1322	106	28	(	(	PUNCT
cana-1322	106	29	)	)	PUNCT
cana-1322	106	30	}	}	PUNCT
cana-1322	106	31	=	=	SYM
cana-1322	106	32	(	(	PUNCT
cana-1322	106	33	×	×	NOUN
cana-1322	106	34	)	)	PUNCT
cana-1322	106	35	+	+	CCONJ
cana-1322	106	36	(	(	PUNCT
cana-1322	106	37	,	,	PUNCT
cana-1322	106	38	)	)	PUNCT
cana-1322	106	39	=	=	SYM
cana-1322	106	40	(	(	PUNCT
cana-1322	106	41	×	×	NOUN
cana-1322	106	42	)	)	PUNCT
cana-1322	106	43	+	+	PUNCT
cana-1322	107	1	[	[	X
cana-1322	107	2	(	(	PUNCT
cana-1322	107	3	,	,	PUNCT
cana-1322	107	4	)	)	PUNCT
cana-1322	107	5	(	(	PUNCT
cana-1322	107	6	,	,	PUNCT
cana-1322	107	7	)	)	PUNCT
cana-1322	107	8	]	]	PUNCT
cana-1322	107	9	,	,	PUNCT
cana-1322	107	10			NOUN
cana-1322	107	11	(	(	PUNCT
cana-1322	107	12	,	,	PUNCT
cana-1322	107	13	)	)	PUNCT
cana-1322	107	14	,	,	PUNCT
cana-1322	107	15	(	(	PUNCT
cana-1322	107	16	,	,	PUNCT
cana-1322	107	17	)	)	PUNCT
cana-1322	107	18			NOUN
cana-1322	107	19	1×	1×	NUM
cana-1322	107	20	2	2	NUM
cana-1322	107	21	.	.	PUNCT
cana-1322	107	22	and	and	CCONJ
cana-1322	107	23	(	(	PUNCT
cana-1322	107	24	×	×	NOUN
cana-1322	107	25	)	)	PUNCT
cana-1322	107	26			NOUN
cana-1322	107	27	[	[	X
cana-1322	107	28	(	(	PUNCT
cana-1322	107	29	,	,	PUNCT
cana-1322	107	30	)	)	PUNCT
cana-1322	107	31	(	(	PUNCT
cana-1322	107	32	,	,	PUNCT
cana-1322	107	33	)	)	PUNCT
cana-1322	107	34	]	]	PUNCT
cana-1322	108	1	=	=	PUNCT
cana-1322	108	2	(	(	PUNCT
cana-1322	108	3	×	×	NOUN
cana-1322	108	4	)	)	PUNCT
cana-1322	108	5			PROPN
cana-1322	108	6	(	(	PUNCT
cana-1322	108	7	,	,	PUNCT
cana-1322	108	8	)	)	PUNCT
cana-1322	108	9	=	=	SYM
cana-1322	108	10	rmax	rmax	ADJ
cana-1322	108	11	{	{	PUNCT
cana-1322	108	12			NOUN
cana-1322	108	13	(	(	PUNCT
cana-1322	108	14	)	)	PUNCT
cana-1322	108	15	,	,	PUNCT
cana-1322	108	16			NOUN
cana-1322	108	17	(	(	PUNCT
cana-1322	108	18	)	)	PUNCT
cana-1322	108	19	}	}	PUNCT
cana-1322	108	20	=	=	SYM
cana-1322	108	21	rmax	rmax	ADJ
cana-1322	108	22	{	{	PUNCT
cana-1322	108	23			NOUN
cana-1322	108	24	(	(	PUNCT
cana-1322	108	25	)	)	PUNCT
cana-1322	108	26	,	,	PUNCT
cana-1322	108	27			NOUN
cana-1322	108	28	(	(	PUNCT
cana-1322	108	29	)	)	PUNCT
cana-1322	108	30	}	}	PUNCT
cana-1322	108	31	=	=	SYM
cana-1322	108	32	(	(	PUNCT
cana-1322	108	33	×	×	NOUN
cana-1322	108	34	)	)	PUNCT
cana-1322	108	35			PROPN
cana-1322	108	36	(	(	PUNCT
cana-1322	108	37	,	,	PUNCT
cana-1322	108	38	)	)	PUNCT
cana-1322	108	39	=	=	SYM
cana-1322	108	40	(	(	PUNCT
cana-1322	108	41	×	×	NOUN
cana-1322	108	42	)	)	PUNCT
cana-1322	108	43			NOUN
cana-1322	108	44	[	[	X
cana-1322	108	45	(	(	PUNCT
cana-1322	108	46	,	,	PUNCT
cana-1322	108	47	)	)	PUNCT
cana-1322	108	48	(	(	PUNCT
cana-1322	108	49	,	,	PUNCT
cana-1322	108	50	)	)	PUNCT
cana-1322	108	51	]	]	PUNCT
cana-1322	108	52	,	,	PUNCT
cana-1322	108	53			NOUN
cana-1322	108	54	(	(	PUNCT
cana-1322	108	55	,	,	PUNCT
cana-1322	108	56	)	)	PUNCT
cana-1322	108	57	,	,	PUNCT
cana-1322	108	58	(	(	PUNCT
cana-1322	108	59	,	,	PUNCT
cana-1322	108	60	)	)	PUNCT
cana-1322	108	61			NOUN
cana-1322	108	62	1×	1×	NUM
cana-1322	108	63	2	2	NUM
cana-1322	108	64	.	.	PUNCT
cana-1322	109	1	hence	hence	ADV
cana-1322	109	2	×	×	NOUN
cana-1322	109	3	is	be	AUX
cana-1322	109	4	a	a	PRON
cana-1322	109	5	of	of	ADP
cana-1322	109	6	1×	1×	NUM
cana-1322	109	7	2	2	NUM
cana-1322	109	8	.	.	PUNCT
cana-1322	109	9	theorem	theorem	VERB
cana-1322	109	10	2.19	2.19	NUM
cana-1322	109	11	.	.	PUNCT
cana-1322	110	1	[	[	X
cana-1322	110	2	5	5	NUM
cana-1322	110	3	]	]	PUNCT
cana-1322	110	4	,	,	PUNCT
cana-1322	110	5	,	,	PUNCT
cana-1322	110	6	…	…	PUNCT
cana-1322	110	7	,	,	PUNCT
cana-1322	110	8	is	be	AUX
cana-1322	110	9	a	a	DET
cana-1322	110	10	theorem	theorem	ADJ
cana-1322	110	11	2.20	2.20	NUM
cana-1322	110	12	.	.	PUNCT
cana-1322	110	13	,	,	PUNCT
cana-1322	110	14	,	,	PUNCT
cana-1322	110	15	…	…	PUNCT
cana-1322	110	16	,	,	PUNCT
cana-1322	110	17	is	be	AUX
cana-1322	110	18	a	a	DET
cana-1322	110	19	proof	proof	NOUN
cana-1322	110	20	.	.	PUNCT
cana-1322	111	1	by	by	ADP
cana-1322	111	2	the	the	DET
cana-1322	111	3	theorem	theorem	NOUN
cana-1322	111	4	2.19	2.19	NUM
cana-1322	111	5	,	,	PUNCT
cana-1322	111	6	it	it	PRON
cana-1322	111	7	can	can	AUX
cana-1322	111	8	be	be	AUX
cana-1322	111	9	easily	easily	ADV
cana-1322	111	10	shown	show	VERB
cana-1322	111	11	.	.	PUNCT
cana-1322	112	1	theorem	theorem	VERB
cana-1322	112	2	2.21	2.21	NUM
cana-1322	112	3	.	.	PUNCT
cana-1322	113	1	[	[	X
cana-1322	113	2	5	5	X
cana-1322	113	3	]	]	X
cana-1322	113	4	if	if	SCONJ
cana-1322	113	5	is	be	AUX
cana-1322	113	6	a	a	DET
cana-1322	113	7	,	,	PUNCT
cana-1322	113	8	then	then	ADV
cana-1322	113	9	(	(	PUNCT
cana-1322	113	10	)	)	PUNCT
cana-1322	113	11	is	be	AUX
cana-1322	113	12	a	a	PRON
cana-1322	113	13	of	of	ADP
cana-1322	113	14	.	.	PUNCT
cana-1322	114	1	theorem	theorem	ADJ
cana-1322	114	2	2.22.if	2.22.if	NUM
cana-1322	114	3	is	be	AUX
cana-1322	114	4	a	a	DET
cana-1322	114	5	,	,	PUNCT
cana-1322	114	6	then	then	ADV
cana-1322	114	7	(	(	PUNCT
cana-1322	114	8	)	)	PUNCT
cana-1322	114	9	is	be	AUX
cana-1322	114	10	a	a	PRON
cana-1322	114	11	of	of	ADP
cana-1322	114	12	.	.	PUNCT
cana-1322	115	1	proof	proof	NOUN
cana-1322	115	2	.	.	PUNCT
cana-1322	116	1	let	let	AUX
cana-1322	116	2	be	be	AUX
cana-1322	116	3	in	in	ADP
cana-1322	116	4	1	1	NUM
cana-1322	116	5	.	.	PUNCT
cana-1322	117	1	by	by	ADP
cana-1322	117	2	theorem	theorem	NOUN
cana-1322	117	3	2.21	2.21	NUM
cana-1322	117	4	,	,	PUNCT
cana-1322	117	5	(	(	PUNCT
cana-1322	117	6	)	)	PUNCT
cana-1322	117	7	is	be	AUX
cana-1322	117	8	a	a	PRON
cana-1322	117	9	of	of	ADP
cana-1322	117	10	,	,	PUNCT
cana-1322	117	11	(	(	PUNCT
cana-1322	117	12	)	)	PUNCT
cana-1322	117	13	=	=	PUNCT
cana-1322	117	14	)	)	PUNCT
cana-1322	118	1	+	+	PUNCT
cana-1322	118	2	[	[	X
cana-1322	118	3	1	1	NUM
cana-1322	118	4	]	]	PUNCT
cana-1322	118	5	)	)	PUNCT
cana-1322	118	6	=	=	PUNCT
cana-1322	119	1	)	)	PUNCT
cana-1322	120	1	+	+	PUNCT
cana-1322	120	2	[	[	X
cana-1322	120	3	1	1	NUM
cana-1322	120	4	]	]	PUNCT
cana-1322	120	5	)	)	PUNCT
cana-1322	120	6	=	=	SYM
cana-1322	120	7	(	(	PUNCT
cana-1322	120	8	)	)	PUNCT
cana-1322	120	9	,	,	PUNCT
cana-1322	120	10			NOUN
cana-1322	120	11	,	,	PUNCT
cana-1322	120	12			NOUN
cana-1322	120	13	1	1	NUM
cana-1322	120	14	.	.	PUNCT
cana-1322	121	1	and	and	CCONJ
cana-1322	121	2	(	(	PUNCT
cana-1322	121	3	)	)	PUNCT
cana-1322	121	4	=	=	PUNCT
cana-1322	121	5	)	)	PUNCT
cana-1322	122	1	+	+	PUNCT
cana-1322	122	2	[	[	X
cana-1322	122	3	1	1	NUM
cana-1322	122	4	]	]	PUNCT
cana-1322	122	5	)	)	PUNCT
cana-1322	122	6	=	=	PUNCT
cana-1322	123	1	)	)	PUNCT
cana-1322	124	1	+	+	PUNCT
cana-1322	124	2	[	[	X
cana-1322	124	3	1	1	NUM
cana-1322	124	4	]	]	PUNCT
cana-1322	124	5	)	)	PUNCT
cana-1322	124	6	=	=	SYM
cana-1322	124	7	(	(	PUNCT
cana-1322	124	8	)	)	PUNCT
cana-1322	124	9	,	,	PUNCT
cana-1322	124	10			NOUN
cana-1322	124	11	,	,	PUNCT
cana-1322	124	12			NOUN
cana-1322	124	13	1	1	NUM
cana-1322	124	14	.	.	PUNCT
cana-1322	124	15	hence	hence	ADV
cana-1322	124	16	(	(	PUNCT
cana-1322	124	17	)	)	PUNCT
cana-1322	124	18	is	be	AUX
cana-1322	124	19	a	a	PRON
cana-1322	124	20	of	of	ADP
cana-1322	124	21	.	.	PUNCT
cana-1322	125	1	theorem	theorem	VERB
cana-1322	125	2	2.23	2.23	NUM
cana-1322	125	3	.	.	PUNCT
cana-1322	126	1	[	[	X
cana-1322	126	2	5	5	NUM
cana-1322	126	3	]	]	PUNCT
cana-1322	126	4	let	let	NOUN
cana-1322	126	5	is	be	AUX
cana-1322	126	6	a	a	PRON
cana-1322	126	7	.	.	PUNCT
cana-1322	127	1	(	(	PUNCT
cana-1322	127	2	i	i	NOUN
cana-1322	127	3	)	)	PUNCT
cana-1322	127	4	then	then	ADV
cana-1322	127	5	(	(	PUNCT
cana-1322	127	6	[	[	PUNCT
cana-1322	127	7	]	]	X
cana-1322	127	8	(	(	PUNCT
cana-1322	127	9	[	[	PUNCT
cana-1322	127	10	]	]	X
cana-1322	127	11	.	.	PUNCT
cana-1322	128	1	(	(	PUNCT
cana-1322	128	2	ii	ii	NOUN
cana-1322	128	3	)	)	PUNCT
cana-1322	128	4	[	[	PUNCT
cana-1322	128	5	]	]	X
cana-1322	128	6	[	[	PUNCT
cana-1322	128	7	]	]	X
cana-1322	128	8	(	(	PUNCT
cana-1322	128	9	.	.	PUNCT
cana-1322	128	10	(	(	PUNCT
cana-1322	128	11	iii	iii	X
cana-1322	128	12	)	)	PUNCT
cana-1322	128	13	communications	communication	NOUN
cana-1322	128	14	on	on	ADP
cana-1322	128	15	applied	apply	VERB
cana-1322	128	16	nonlinear	nonlinear	ADJ
cana-1322	128	17	analysis	analysis	NOUN
cana-1322	128	18	issn	issn	NOUN
cana-1322	128	19	:	:	PUNCT
cana-1322	128	20	1074	1074	NUM
cana-1322	128	21	-	-	PUNCT
cana-1322	128	22	133x	133x	NUM
cana-1322	128	23	vol	vol	NOUN
cana-1322	128	24	31	31	NUM
cana-1322	128	25	no	no	NOUN
cana-1322	128	26	.	.	PUNCT
cana-1322	129	1	7s	7	NOUN
cana-1322	129	2	(	(	PUNCT
cana-1322	129	3	2024	2024	NUM
cana-1322	129	4	)	)	PUNCT
cana-1322	129	5	442	442	NUM
cana-1322	129	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1322	129	7	(	(	PUNCT
cana-1322	129	8	[	[	PUNCT
cana-1322	129	9	]	]	X
cana-1322	129	10	(	(	PUNCT
cana-1322	129	11	[	[	PUNCT
cana-1322	129	12	]	]	X
cana-1322	129	13	.	.	PUNCT
cana-1322	130	1	(	(	PUNCT
cana-1322	130	2	iv	iv	X
cana-1322	130	3	)	)	PUNCT
cana-1322	130	4	[	[	PUNCT
cana-1322	130	5	]	]	X
cana-1322	130	6	[	[	PUNCT
cana-1322	130	7	]	]	X
cana-1322	130	8	(	(	PUNCT
cana-1322	130	9	[	[	PUNCT
cana-1322	130	10	]	]	X
cana-1322	130	11	(	(	PUNCT
cana-1322	130	12	[	[	PUNCT
cana-1322	130	13	]	]	X
cana-1322	130	14	.	.	PUNCT
cana-1322	131	1	(	(	PUNCT
cana-1322	131	2	v	v	NOUN
cana-1322	131	3	)	)	PUNCT
cana-1322	131	4	(	(	PUNCT
cana-1322	131	5	.	.	PUNCT
cana-1322	132	1	(	(	PUNCT
cana-1322	132	2	vi	vi	NOUN
cana-1322	132	3	)	)	PUNCT
cana-1322	132	4	(	(	PUNCT
cana-1322	132	5	vii	vii	PROPN
cana-1322	132	6	)	)	PUNCT
cana-1322	132	7	(	(	PUNCT
cana-1322	132	8	viii	viii	NOUN
cana-1322	132	9	)	)	PUNCT
cana-1322	132	10	theorem	theorem	VERB
cana-1322	132	11	2.24	2.24	NUM
cana-1322	132	12	.	.	PUNCT
cana-1322	133	1	let	let	VERB
cana-1322	133	2	is	be	AUX
cana-1322	133	3	a	a	PRON
cana-1322	133	4	.	.	PUNCT
cana-1322	134	1	(	(	PUNCT
cana-1322	134	2	i	i	NOUN
cana-1322	134	3	)	)	PUNCT
cana-1322	134	4	then	then	ADV
cana-1322	134	5	(	(	PUNCT
cana-1322	134	6	[	[	PUNCT
cana-1322	134	7	]	]	X
cana-1322	134	8	(	(	PUNCT
cana-1322	134	9	[	[	PUNCT
cana-1322	134	10	]	]	X
cana-1322	134	11	.	.	PUNCT
cana-1322	135	1	(	(	PUNCT
cana-1322	135	2	ii	ii	NOUN
cana-1322	135	3	)	)	PUNCT
cana-1322	135	4	[	[	PUNCT
cana-1322	135	5	]	]	X
cana-1322	135	6	[	[	PUNCT
cana-1322	135	7	]	]	X
cana-1322	135	8	(	(	PUNCT
cana-1322	135	9	.	.	PUNCT
cana-1322	136	1	(	(	PUNCT
cana-1322	136	2	iii	iii	X
cana-1322	136	3	)	)	PUNCT
cana-1322	136	4	(	(	PUNCT
cana-1322	136	5	[	[	PUNCT
cana-1322	136	6	]	]	X
cana-1322	136	7	(	(	PUNCT
cana-1322	136	8	[	[	PUNCT
cana-1322	136	9	]	]	X
cana-1322	136	10	.	.	PUNCT
cana-1322	137	1	(	(	PUNCT
cana-1322	137	2	iv	iv	X
cana-1322	137	3	)	)	PUNCT
cana-1322	137	4	[	[	PUNCT
cana-1322	137	5	]	]	X
cana-1322	137	6	[	[	PUNCT
cana-1322	137	7	]	]	X
cana-1322	137	8	(	(	PUNCT
cana-1322	137	9	[	[	PUNCT
cana-1322	137	10	]	]	X
cana-1322	137	11	(	(	PUNCT
cana-1322	137	12	[	[	PUNCT
cana-1322	137	13	]	]	X
cana-1322	137	14	.	.	PUNCT
cana-1322	138	1	(	(	PUNCT
cana-1322	138	2	v	v	NOUN
cana-1322	138	3	)	)	PUNCT
cana-1322	138	4	(	(	PUNCT
cana-1322	138	5	.	.	PUNCT
cana-1322	139	1	(	(	PUNCT
cana-1322	139	2	vi	vi	NOUN
cana-1322	139	3	)	)	PUNCT
cana-1322	139	4	(	(	PUNCT
cana-1322	139	5	vii	vii	PROPN
cana-1322	139	6	)	)	PUNCT
cana-1322	139	7	(	(	PUNCT
cana-1322	139	8	viii	viii	NOUN
cana-1322	139	9	)	)	PUNCT
cana-1322	139	10	proof	proof	NOUN
cana-1322	139	11	.	.	PUNCT
cana-1322	140	1	by	by	ADP
cana-1322	140	2	the	the	DET
cana-1322	140	3	theorem	theorem	NOUN
cana-1322	140	4	2.23	2.23	NUM
cana-1322	140	5	,	,	PUNCT
cana-1322	140	6	it	it	PRON
cana-1322	140	7	can	can	AUX
cana-1322	140	8	be	be	AUX
cana-1322	140	9	easily	easily	ADV
cana-1322	140	10	shown	show	VERB
cana-1322	140	11	.	.	PUNCT
cana-1322	141	1	conclusion	conclusion	NOUN
cana-1322	141	2	using	use	VERB
cana-1322	141	3	the	the	DET
cana-1322	141	4	above	above	ADJ
cana-1322	141	5	theorems	theorem	NOUN
cana-1322	141	6	,	,	PUNCT
cana-1322	141	7	we	we	PRON
cana-1322	141	8	can	can	AUX
cana-1322	141	9	find	find	VERB
cana-1322	141	10	more	more	ADJ
cana-1322	141	11	results	result	NOUN
cana-1322	141	12	.	.	PUNCT
cana-1322	142	1	it	it	PRON
cana-1322	142	2	can	can	AUX
cana-1322	142	3	be	be	AUX
cana-1322	142	4	extended	extend	VERB
cana-1322	142	5	into	into	ADP
cana-1322	142	6	different	different	ADJ
cana-1322	142	7	types	type	NOUN
cana-1322	142	8	of	of	ADP
cana-1322	142	9	algebra	algebra	NOUN
cana-1322	142	10	.	.	PUNCT
cana-1322	143	1	references	reference	NOUN
cana-1322	143	2	[	[	X
cana-1322	143	3	1	1	NUM
cana-1322	143	4	]	]	PUNCT
cana-1322	143	5	anitha.m.s	anitha.m.s	ADV
cana-1322	143	6	.	.	PUNCT
cana-1322	143	7	,	,	PUNCT
cana-1322	143	8	muruganantha	muruganantha	PROPN
cana-1322	143	9	prasad	prasad	PROPN
cana-1322	143	10	&	&	CCONJ
cana-1322	143	11	k.arjunan	k.arjunan	PROPN
cana-1322	143	12	,	,	PUNCT
cana-1322	143	13	notes	note	NOUN
cana-1322	143	14	on	on	ADP
cana-1322	143	15	bipolar	bipolar	ADJ
cana-1322	143	16	valued	value	VERB
cana-1322	143	17	fuzzy	fuzzy	ADJ
cana-1322	143	18	subgroups	subgroup	NOUN
cana-1322	143	19	of	of	ADP
cana-1322	143	20	a	a	DET
cana-1322	143	21	group	group	NOUN
cana-1322	143	22	,	,	PUNCT
cana-1322	143	23	bulletin	bulletin	NOUN
cana-1322	143	24	of	of	ADP
cana-1322	143	25	society	society	NOUN
cana-1322	143	26	for	for	ADP
cana-1322	143	27	mathematical	mathematical	ADJ
cana-1322	143	28	services	service	NOUN
cana-1322	143	29	and	and	CCONJ
cana-1322	143	30	standards	standard	NOUN
cana-1322	143	31	,	,	PUNCT
cana-1322	143	32	vol	vol	NOUN
cana-1322	143	33	.	.	NOUN
cana-1322	143	34	2	2	NUM
cana-1322	144	1	no	no	NOUN
cana-1322	144	2	.	.	NOUN
cana-1322	144	3	3	3	NUM
cana-1322	144	4	(	(	PUNCT
cana-1322	144	5	2013	2013	NUM
cana-1322	144	6	)	)	PUNCT
cana-1322	144	7	,	,	PUNCT
cana-1322	144	8	pp.52	pp.52	NOUN
cana-1322	144	9	-	-	SYM
cana-1322	144	10	59	59	NUM
cana-1322	144	11	.	.	PUNCT
cana-1322	145	1	[	[	X
cana-1322	145	2	2	2	X
cana-1322	145	3	]	]	PUNCT
cana-1322	145	4	azriel	azriel	PROPN
cana-1322	145	5	rosenfeld	rosenfeld	PROPN
cana-1322	145	6	,	,	PUNCT
cana-1322	145	7	fuzzy	fuzzy	ADJ
cana-1322	145	8	groups	group	NOUN
cana-1322	145	9	,	,	PUNCT
cana-1322	145	10	journal	journal	NOUN
cana-1322	145	11	of	of	ADP
cana-1322	145	12	mathematical	mathematical	ADJ
cana-1322	145	13	analysis	analysis	NOUN
cana-1322	145	14	and	and	CCONJ
cana-1322	145	15	applications	application	NOUN
cana-1322	145	16	35(1971	35(1971	NUM
cana-1322	145	17	)	)	PUNCT
cana-1322	145	18	,	,	PUNCT
cana-1322	145	19	512	512	NUM
cana-1322	145	20	-	-	SYM
cana-1322	145	21	517	517	NUM
cana-1322	145	22	.	.	PUNCT
cana-1322	146	1	[	[	X
cana-1322	146	2	3	3	NUM
cana-1322	146	3	]	]	SYM
cana-1322	146	4	balasubramanian.a	balasubramanian.a	PROPN
cana-1322	146	5	,	,	PUNCT
cana-1322	146	6	k.l.muruganantha	k.l.muruganantha	PROPN
cana-1322	146	7	prasad	prasad	PROPN
cana-1322	146	8	&	&	CCONJ
cana-1322	146	9	k.arjunan	k.arjunan	PROPN
cana-1322	146	10	,	,	PUNCT
cana-1322	146	11	“	"	PUNCT
cana-1322	146	12	properties	property	NOUN
cana-1322	146	13	of	of	ADP
cana-1322	146	14	bipolar	bipolar	ADJ
cana-1322	146	15	interval	interval	NOUN
cana-1322	146	16	valued	value	VERB
cana-1322	146	17	fuzzy	fuzzy	ADJ
cana-1322	146	18	subgroups	subgroup	NOUN
cana-1322	146	19	of	of	ADP
cana-1322	146	20	a	a	DET
cana-1322	146	21	group	group	NOUN
cana-1322	146	22	”	"	PUNCT
cana-1322	146	23	,	,	PUNCT
cana-1322	146	24	international	international	ADJ
cana-1322	146	25	journal	journal	NOUN
cana-1322	146	26	of	of	ADP
cana-1322	146	27	scientific	scientific	ADJ
cana-1322	146	28	research	research	NOUN
cana-1322	146	29	,	,	PUNCT
cana-1322	146	30	vol	vol	NOUN
cana-1322	146	31	.	.	PROPN
cana-1322	146	32	4	4	NUM
cana-1322	146	33	,	,	PUNCT
cana-1322	146	34	iss	iss	PROPN
cana-1322	146	35	.	.	PROPN
cana-1322	146	36	4	4	NUM
cana-1322	146	37	(	(	PUNCT
cana-1322	146	38	2015	2015	NUM
cana-1322	146	39	)	)	PUNCT
cana-1322	146	40	,	,	PUNCT
cana-1322	146	41	262	262	NUM
cana-1322	146	42	268	268	NUM
cana-1322	146	43	.	.	PUNCT
cana-1322	147	1	[	[	X
cana-1322	147	2	4	4	X
cana-1322	147	3	]	]	X
cana-1322	147	4	cicily	cicily	ADV
cana-1322	147	5	flora	flora	NOUN
cana-1322	147	6	.	.	PUNCT
cana-1322	148	1	s	s	PART
cana-1322	148	2	and	and	CCONJ
cana-1322	148	3	arockiarani.i	arockiarani.i	PROPN
cana-1322	148	4	,	,	PUNCT
cana-1322	148	5	a	a	DET
cana-1322	148	6	new	new	ADJ
cana-1322	148	7	class	class	NOUN
cana-1322	148	8	of	of	ADP
cana-1322	148	9	generalized	generalized	ADJ
cana-1322	148	10	bipolar	bipolar	ADJ
cana-1322	148	11	vague	vague	ADJ
cana-1322	148	12	sets	set	NOUN
cana-1322	148	13	,	,	PUNCT
cana-1322	148	14	international	international	ADJ
cana-1322	148	15	journal	journal	NOUN
cana-1322	148	16	of	of	ADP
cana-1322	148	17	[	[	X
cana-1322	148	18	5	5	NUM
cana-1322	148	19	]	]	SYM
cana-1322	148	20	deeba.b	deeba.b	NUM
cana-1322	148	21	,	,	PUNCT
cana-1322	148	22	s.naganathan	s.naganathan	NOUN
cana-1322	148	23	&	&	CCONJ
cana-1322	148	24	k.arjunan	k.arjunan	PROPN
cana-1322	148	25	,	,	PUNCT
cana-1322	148	26	“	"	PUNCT
cana-1322	148	27	a	a	DET
cana-1322	148	28	study	study	NOUN
cana-1322	148	29	on	on	ADP
cana-1322	148	30	bipolar	bipolar	ADJ
cana-1322	148	31	valued	value	VERB
cana-1322	148	32	vague	vague	ADJ
cana-1322	148	33	subrings	subring	NOUN
cana-1322	148	34	of	of	ADP
cana-1322	148	35	a	a	DET
cana-1322	148	36	ring	ring	NOUN
cana-1322	148	37	”	"	PUNCT
cana-1322	148	38	,	,	PUNCT
cana-1322	148	39	journal	journal	NOUN
cana-1322	148	40	of	of	ADP
cana-1322	148	41	shanghai	shanghai	PROPN
cana-1322	148	42	jiaotong	jiaotong	PROPN
cana-1322	148	43	university	university	PROPN
cana-1322	148	44	,	,	PUNCT
cana-1322	148	45	vol	vol	NOUN
cana-1322	148	46	.	.	PROPN
cana-1322	148	47	16	16	NUM
cana-1322	148	48	,	,	PUNCT
cana-1322	148	49	issue	issue	NOUN
cana-1322	148	50	10	10	NUM
cana-1322	148	51	(	(	PUNCT
cana-1322	148	52	2020),512	2020),512	NUM
cana-1322	148	53	–	–	PUNCT
cana-1322	148	54	518	518	NUM
cana-1322	148	55	.	.	PUNCT
cana-1322	149	1	[	[	X
cana-1322	149	2	6	6	NUM
cana-1322	149	3	]	]	SYM
cana-1322	149	4	deepa.b	deepa.b	NOUN
cana-1322	149	5	,	,	PUNCT
cana-1322	149	6	s.naganathan	s.naganathan	NOUN
cana-1322	149	7	&	&	CCONJ
cana-1322	149	8	k.arjunan	k.arjunan	PROPN
cana-1322	149	9	,	,	PUNCT
cana-1322	149	10	“	"	PUNCT
cana-1322	149	11	homomorphism	homomorphism	NOUN
cana-1322	149	12	and	and	CCONJ
cana-1322	149	13	anti	anti	ADJ
cana-1322	149	14	homomorphism	homomorphism	NOUN
cana-1322	149	15	functions	function	NOUN
cana-1322	149	16	in	in	ADP
cana-1322	149	17	bipolar	bipolar	ADJ
cana-1322	149	18	valued	value	VERB
cana-1322	149	19	vague	vague	ADJ
cana-1322	149	20	subrngs	subrng	NOUN
cana-1322	149	21	of	of	ADP
cana-1322	149	22	a	a	DET
cana-1322	149	23	ring	ring	NOUN
cana-1322	149	24	”	"	PUNCT
cana-1322	149	25	international	international	ADJ
cana-1322	149	26	journal	journal	NOUN
cana-1322	149	27	of	of	ADP
cana-1322	149	28	mathematical	mathematical	ADJ
cana-1322	149	29	archive	archive	NOUN
cana-1322	149	30	,	,	PUNCT
cana-1322	149	31	12(7	12(7	NUM
cana-1322	149	32	)	)	PUNCT
cana-1322	149	33	,	,	PUNCT
cana-1322	149	34	(	(	PUNCT
cana-1322	149	35	2021	2021	NUM
cana-1322	149	36	)	)	PUNCT
cana-1322	149	37	,	,	PUNCT
cana-1322	149	38	1	1	NUM
cana-1322	149	39	-	-	SYM
cana-1322	149	40	5	5	NUM
cana-1322	149	41	.	.	PUNCT
cana-1322	150	1	[	[	X
cana-1322	150	2	7	7	X
cana-1322	150	3	]	]	X
cana-1322	150	4	[	[	X
cana-1322	150	5	8	8	NUM
cana-1322	150	6	]	]	X
cana-1322	150	7	grattan	grattan	PROPN
cana-1322	150	8	-	-	PUNCT
cana-1322	150	9	guiness	guiness	PROPN
cana-1322	150	10	,	,	PUNCT
cana-1322	150	11	“	"	PUNCT
cana-1322	150	12	fuzzy	fuzzy	ADJ
cana-1322	150	13	membership	membership	NOUN
cana-1322	150	14	mapped	map	VERB
cana-1322	150	15	onto	onto	ADP
cana-1322	150	16	interval	interval	NOUN
cana-1322	150	17	and	and	CCONJ
cana-1322	150	18	many	many	ADJ
cana-1322	150	19	valued	value	VERB
cana-1322	150	20	quantities	quantity	NOUN
cana-1322	150	21	”	"	PUNCT
cana-1322	150	22	,	,	PUNCT
cana-1322	150	23	z.math.logik	z.math.logik	PROPN
cana-1322	150	24	.	.	PUNCT
cana-1322	151	1	[	[	X
cana-1322	151	2	9	9	NUM
cana-1322	151	3	]	]	X
cana-1322	151	4	k.m.lee	k.m.lee	PROPN
cana-1322	151	5	,	,	PUNCT
cana-1322	151	6	bipolar	bipolar	ADJ
cana-1322	151	7	valued	value	VERB
cana-1322	151	8	fuzzy	fuzzy	ADJ
cana-1322	151	9	sets	set	NOUN
cana-1322	151	10	and	and	CCONJ
cana-1322	151	11	their	their	PRON
cana-1322	151	12	operations	operation	NOUN
cana-1322	151	13	.	.	PUNCT
cana-1322	152	1	proc	proc	NOUN
cana-1322	152	2	.	.	PUNCT
cana-1322	153	1	int	int	NOUN
cana-1322	153	2	.	.	PUNCT
cana-1322	153	3	conf	conf	PROPN
cana-1322	153	4	.	.	PUNCT
cana-1322	154	1	on	on	ADP
cana-1322	154	2	intelligent	intelligent	ADJ
cana-1322	154	3	technologies	technology	NOUN
cana-1322	154	4	,	,	PUNCT
cana-1322	154	5	bangkok	bangkok	PROPN
cana-1322	154	6	,	,	PUNCT
cana-1322	154	7	thailand	thailand	PROPN
cana-1322	154	8	(	(	PUNCT
cana-1322	154	9	2000	2000	NUM
cana-1322	154	10	)	)	PUNCT
cana-1322	154	11	,	,	PUNCT
cana-1322	154	12	307	307	NUM
cana-1322	154	13	-	-	SYM
cana-1322	154	14	312	312	NUM
cana-1322	154	15	.	.	PUNCT
cana-1322	154	16	communications	communication	NOUN
cana-1322	154	17	on	on	ADP
cana-1322	154	18	applied	apply	VERB
cana-1322	154	19	nonlinear	nonlinear	ADJ
cana-1322	154	20	analysis	analysis	NOUN
cana-1322	154	21	issn	issn	NOUN
cana-1322	154	22	:	:	PUNCT
cana-1322	154	23	1074	1074	NUM
cana-1322	154	24	-	-	PUNCT
cana-1322	154	25	133x	133x	NUM
cana-1322	154	26	vol	vol	NOUN
cana-1322	154	27	31	31	NUM
cana-1322	154	28	no	no	NOUN
cana-1322	154	29	.	.	PUNCT
cana-1322	155	1	7s	7	NOUN
cana-1322	155	2	(	(	PUNCT
cana-1322	155	3	2024	2024	NUM
cana-1322	155	4	)	)	PUNCT
cana-1322	155	5	443	443	NUM
cana-1322	155	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1322	156	1	[	[	X
cana-1322	156	2	10	10	NUM
cana-1322	156	3	]	]	X
cana-1322	156	4	k.m.lee	k.m.lee	PROPN
cana-1322	156	5	,	,	PUNCT
cana-1322	156	6	comparison	comparison	NOUN
cana-1322	156	7	of	of	ADP
cana-1322	156	8	interval	interval	NOUN
cana-1322	156	9	valued	value	VERB
cana-1322	156	10	fuzzy	fuzzy	ADJ
cana-1322	156	11	sets	set	NOUN
cana-1322	156	12	,	,	PUNCT
cana-1322	156	13	intuitionistic	intuitionistic	ADJ
cana-1322	156	14	fuzzy	fuzzy	ADJ
cana-1322	156	15	sets	set	NOUN
cana-1322	156	16	and	and	CCONJ
cana-1322	156	17	bipolar	bipolar	ADJ
cana-1322	156	18	valued	value	VERB
cana-1322	156	19	fuzzy	fuzzy	ADJ
cana-1322	156	20	sets	set	NOUN
cana-1322	156	21	.	.	PUNCT
cana-1322	157	1	j.	j.	PROPN
cana-1322	157	2	fuzzy	fuzzy	ADJ
cana-1322	157	3	logic	logic	NOUN
cana-1322	157	4	intelligent	intelligent	ADJ
cana-1322	157	5	systems	system	NOUN
cana-1322	157	6	,	,	PUNCT
cana-1322	157	7	14	14	NUM
cana-1322	157	8	(	(	PUNCT
cana-1322	157	9	2	2	NUM
cana-1322	157	10	)	)	PUNCT
cana-1322	157	11	(	(	PUNCT
cana-1322	157	12	2004	2004	NUM
cana-1322	157	13	)	)	PUNCT
cana-1322	157	14	,	,	PUNCT
cana-1322	157	15	125	125	NUM
cana-1322	157	16	-	-	SYM
cana-1322	157	17	129	129	NUM
cana-1322	157	18	.	.	PUNCT
cana-1322	158	1	[	[	X
cana-1322	158	2	11	11	NUM
cana-1322	158	3	]	]	X
cana-1322	158	4	k.murugalingam	k.murugalingam	PROPN
cana-1322	158	5	&	&	CCONJ
cana-1322	158	6	k.arjunan	k.arjunan	PROPN
cana-1322	158	7	,	,	PUNCT
cana-1322	158	8	a	a	DET
cana-1322	158	9	study	study	NOUN
cana-1322	158	10	on	on	ADP
cana-1322	158	11	interval	interval	NOUN
cana-1322	158	12	valued	value	VERB
cana-1322	158	13	fuzzy	fuzzy	ADJ
cana-1322	158	14	subsemiring	subsemiring	NOUN
cana-1322	158	15	of	of	ADP
cana-1322	158	16	a	a	DET
cana-1322	158	17	semiring	semiring	NOUN
cana-1322	158	18	,	,	PUNCT
cana-1322	158	19	international	international	ADJ
cana-1322	158	20	journal	journal	NOUN
cana-1322	158	21	of	of	ADP
cana-1322	158	22	applied	apply	VERB
cana-1322	158	23	mathematics	mathematic	NOUN
cana-1322	158	24	modeling	modeling	NOUN
cana-1322	158	25	,	,	PUNCT
cana-1322	158	26	vol.1	vol.1	PROPN
cana-1322	158	27	,	,	PUNCT
cana-1322	158	28	no.5	no.5	PROPN
cana-1322	158	29	,	,	PUNCT
cana-1322	158	30	1	1	NUM
cana-1322	158	31	-	-	SYM
cana-1322	158	32	6	6	NUM
cana-1322	158	33	,	,	PUNCT
cana-1322	158	34	(	(	PUNCT
cana-1322	158	35	2013	2013	NUM
cana-1322	158	36	)	)	PUNCT
cana-1322	158	37	.	.	PUNCT
cana-1322	159	1	[	[	X
cana-1322	159	2	12	12	NUM
cana-1322	159	3	]	]	X
cana-1322	159	4	[	[	X
cana-1322	159	5	13	13	NUM
cana-1322	159	6	]	]	PUNCT
cana-1322	159	7	g.	g.	PROPN
cana-1322	159	8	srinivasa	srinivasa	PROPN
cana-1322	159	9	rao	rao	PROPN
cana-1322	159	10	,	,	PUNCT
cana-1322	159	11	d.	d.	PROPN
cana-1322	159	12	madhusudhanarao	madhusudhanarao	PROPN
cana-1322	159	13	and	and	CCONJ
cana-1322	159	14	p.	p.	PROPN
cana-1322	159	15	siva	siva	PROPN
cana-1322	159	16	prasad	prasad	PROPN
cana-1322	159	17	,	,	PUNCT
cana-1322	159	18	simple	simple	ADJ
cana-1322	159	19	ternary	ternary	ADJ
cana-1322	159	20	semi	semi	NOUN
cana-1322	159	21	-	-	NOUN
cana-1322	159	22	rings	ring	NOUN
cana-1322	159	23	,	,	PUNCT
cana-1322	159	24	the	the	DET
cana-1322	159	25	global	global	ADJ
cana-1322	159	26	journal	journal	NOUN
cana-1322	159	27	of	of	ADP
cana-1322	159	28	mathematics	mathematics	PROPN
cana-1322	159	29	&	&	CCONJ
cana-1322	159	30	mathematical	mathematical	PROPN
cana-1322	159	31	sciences	sciences	PROPN
cana-1322	159	32	,	,	PUNCT
cana-1322	159	33	9(2	9(2	NUM
cana-1322	159	34	)	)	PUNCT
cana-1322	159	35	(	(	PUNCT
cana-1322	159	36	2016	2016	NUM
cana-1322	159	37	)	)	PUNCT
cana-1322	159	38	,	,	PUNCT
cana-1322	159	39	185	185	NUM
cana-1322	159	40	-	-	SYM
cana-1322	159	41	196	196	NUM
cana-1322	159	42	.	.	PUNCT
cana-1322	160	1	[	[	X
cana-1322	160	2	14	14	NUM
cana-1322	160	3	]	]	X
cana-1322	160	4	d.	d.	PROPN
cana-1322	160	5	madhusudhana	madhusudhana	PROPN
cana-1322	160	6	rao	rao	PROPN
cana-1322	160	7	,	,	PUNCT
cana-1322	160	8	g.	g.	PROPN
cana-1322	160	9	srinivasa	srinivasa	PROPN
cana-1322	160	10	rao	rao	PROPN
cana-1322	160	11	,	,	PUNCT
cana-1322	160	12	special	special	ADJ
cana-1322	160	13	elements	element	NOUN
cana-1322	160	14	in	in	ADP
cana-1322	160	15	ternary	ternary	ADJ
cana-1322	160	16	semi	semi	ADJ
cana-1322	160	17	rings	ring	NOUN
cana-1322	160	18	,	,	PUNCT
cana-1322	160	19	international	international	ADJ
cana-1322	160	20	journal	journal	NOUN
cana-1322	160	21	of	of	ADP
cana-1322	160	22	engineering	engineering	NOUN
cana-1322	160	23	research	research	NOUN
cana-1322	160	24	and	and	CCONJ
cana-1322	160	25	applications	application	NOUN
cana-1322	160	26	,	,	PUNCT
cana-1322	160	27	4(11	4(11	NUM
cana-1322	160	28	)	)	PUNCT
cana-1322	160	29	(	(	PUNCT
cana-1322	160	30	2014	2014	NUM
cana-1322	160	31	)	)	PUNCT
cana-1322	160	32	,	,	PUNCT
cana-1322	160	33	123	123	NUM
cana-1322	160	34	-	-	SYM
cana-1322	160	35	130	130	NUM
cana-1322	160	36	.	.	PUNCT
cana-1322	161	1	[	[	X
cana-1322	161	2	15	15	NUM
cana-1322	161	3	]	]	X
cana-1322	161	4	g.	g.	PROPN
cana-1322	161	5	srinivasa	srinivasa	PROPN
cana-1322	161	6	rao	rao	PROPN
cana-1322	161	7	,	,	PUNCT
cana-1322	161	8	d.	d.	PROPN
cana-1322	161	9	madhusudhana	madhusudhana	PROPN
cana-1322	161	10	rao	rao	PROPN
cana-1322	161	11	,	,	PUNCT
cana-1322	161	12	structure	structure	NOUN
cana-1322	161	13	of	of	ADP
cana-1322	161	14	certain	certain	ADJ
cana-1322	161	15	ideals	ideal	NOUN
cana-1322	161	16	in	in	ADP
cana-1322	161	17	ternary	ternary	ADJ
cana-1322	161	18	semi	semi	ADJ
cana-1322	161	19	rings	ring	NOUN
cana-1322	161	20	,	,	PUNCT
cana-1322	161	21	int	int	NOUN
cana-1322	161	22	.	.	PUNCT
cana-1322	162	1	j.	j.	PROPN
cana-1322	162	2	of	of	ADP
cana-1322	162	3	innovative	innovative	ADJ
cana-1322	162	4	science	science	NOUN
cana-1322	162	5	and	and	CCONJ
cana-1322	162	6	modern	modern	ADJ
cana-1322	162	7	engg	engg	PROPN
cana-1322	162	8	.	.	PUNCT
cana-1322	162	9	,	,	PUNCT
cana-1322	162	10	3(3	3(3	NUM
cana-1322	162	11	)	)	PUNCT
cana-1322	162	12	(	(	PUNCT
cana-1322	162	13	2015	2015	NUM
cana-1322	162	14	)	)	PUNCT
cana-1322	162	15	,	,	PUNCT
cana-1322	162	16	49	49	NUM
cana-1322	162	17	-	-	SYM
cana-1322	162	18	56	56	NUM
cana-1322	162	19	.	.	PUNCT
cana-1322	163	1	[	[	X
cana-1322	163	2	16	16	NUM
cana-1322	163	3	]	]	X
cana-1322	163	4	g.	g.	PROPN
cana-1322	163	5	srinivasa	srinivasa	PROPN
cana-1322	163	6	rao	rao	PROPN
cana-1322	163	7	,	,	PUNCT
cana-1322	163	8	d.	d.	PROPN
cana-1322	163	9	madhusudhana	madhusudhana	PROPN
cana-1322	163	10	rao	rao	PROPN
cana-1322	163	11	,	,	PUNCT
cana-1322	163	12	a	a	DET
cana-1322	163	13	study	study	NOUN
cana-1322	163	14	on	on	ADP
cana-1322	163	15	ternary	ternary	ADJ
cana-1322	163	16	semi	semi	ADJ
cana-1322	163	17	rings	ring	NOUN
cana-1322	163	18	,	,	PUNCT
cana-1322	163	19	int	int	NOUN
cana-1322	163	20	.	.	PUNCT
cana-1322	164	1	j.	j.	PROPN
cana-1322	164	2	of	of	ADP
cana-1322	164	3	math	math	PROPN
cana-1322	164	4	.	.	PUNCT
cana-1322	165	1	archive	archive	NOUN
cana-1322	165	2	,	,	PUNCT
cana-1322	165	3	5(12	5(12	NUM
cana-1322	165	4	)	)	PUNCT
cana-1322	165	5	(	(	PUNCT
cana-1322	165	6	2014	2014	NUM
cana-1322	165	7	)	)	PUNCT
cana-1322	165	8	,	,	PUNCT
cana-1322	165	9	24	24	NUM
cana-1322	165	10	-	-	SYM
cana-1322	165	11	30	30	NUM
cana-1322	165	12	.	.	PUNCT
cana-1322	166	1	[	[	X
cana-1322	166	2	17	17	NUM
cana-1322	166	3	]	]	X
cana-1322	166	4	g.	g.	PROPN
cana-1322	166	5	srinivasa	srinivasa	PROPN
cana-1322	166	6	rao	rao	PROPN
cana-1322	166	7	,	,	PUNCT
cana-1322	166	8	d.	d.	PROPN
cana-1322	166	9	madhusudhana	madhusudhana	PROPN
cana-1322	166	10	rao	rao	PROPN
cana-1322	166	11	,	,	PUNCT
cana-1322	166	12	characteristics	characteristic	NOUN
cana-1322	166	13	of	of	ADP
cana-1322	166	14	ternary	ternary	ADJ
cana-1322	166	15	semi	semi	ADJ
cana-1322	166	16	rings	ring	NOUN
cana-1322	166	17	,	,	PUNCT
cana-1322	166	18	int.j	int.j	PROPN
cana-1322	166	19	.	.	PROPN
cana-1322	166	20	of	of	ADP
cana-1322	166	21	engg	engg	PROPN
cana-1322	166	22	.	.	PUNCT
cana-1322	167	1	res	re	NOUN
cana-1322	167	2	.	.	PUNCT
cana-1322	167	3	and	and	CCONJ
cana-1322	167	4	mgt	mgt	PROPN
cana-1322	167	5	.	.	PUNCT
cana-1322	167	6	,	,	PUNCT
cana-1322	167	7	2(1	2(1	NUM
cana-1322	167	8	)	)	PUNCT
cana-1322	167	9	(	(	PUNCT
cana-1322	167	10	2015	2015	NUM
cana-1322	167	11	)	)	PUNCT
cana-1322	167	12	,	,	PUNCT
cana-1322	167	13	3	3	NUM
cana-1322	167	14	-	-	SYM
cana-1322	167	15	6	6	NUM
cana-1322	167	16	.	.	PUNCT
cana-1322	168	1	[	[	X
cana-1322	168	2	18	18	NUM
cana-1322	168	3	]	]	X
cana-1322	168	4	g.	g.	PROPN
cana-1322	168	5	srinivasa	srinivasa	PROPN
cana-1322	168	6	rao	rao	PROPN
cana-1322	168	7	,	,	PUNCT
cana-1322	168	8	a.	a.	PROPN
cana-1322	168	9	nagamalleswara	nagamalleswara	PROPN
cana-1322	168	10	rao	rao	PROPN
cana-1322	168	11	,	,	PUNCT
cana-1322	168	12	p.l.n	p.l.n	PROPN
cana-1322	168	13	.	.	PROPN
cana-1322	168	14	varma	varma	PROPN
cana-1322	168	15	,	,	PUNCT
cana-1322	168	16	d.madhusudhana	d.madhusudhana	PROPN
cana-1322	168	17	rao	rao	PROPN
cana-1322	168	18	,	,	PUNCT
cana-1322	168	19	ch	ch	NOUN
cana-1322	168	20	.	.	PROPN
cana-1322	168	21	ramprasad	ramprasad	ADJ
cana-1322	168	22	,	,	PUNCT
cana-1322	168	23	prime	prime	ADJ
cana-1322	168	24	biinterior	biinterior	PROPN
cana-1322	168	25	ideals	ideal	NOUN
cana-1322	168	26	in	in	ADP
cana-1322	168	27	tgsr	tgsr	ADJ
cana-1322	168	28	,	,	PUNCT
cana-1322	168	29	malaya	malaya	PROPN
cana-1322	168	30	journal	journal	PROPN
cana-1322	168	31	of	of	ADP
cana-1322	168	32	mathematika	mathematika	NOUN
cana-1322	168	33	,	,	PUNCT
cana-1322	168	34	vol.9	vol.9	PROPN
cana-1322	168	35	,	,	PUNCT
cana-1322	168	36	no.1	no.1	NUM
cana-1322	168	37	,	,	PUNCT
cana-1322	168	38	pp:542	pp:542	ADV
cana-1322	168	39	-	-	PUNCT
cana-1322	168	40	546	546	NUM
cana-1322	168	41	,	,	PUNCT
cana-1322	168	42	2021	2021	NUM
cana-1322	168	43	.	.	PUNCT
cana-1322	169	1	[	[	X
cana-1322	169	2	19	19	NUM
cana-1322	169	3	]	]	PUNCT
cana-1322	169	4	g.	g.	PROPN
cana-1322	169	5	srinivasa	srinivasa	PROPN
cana-1322	169	6	rao	rao	PROPN
cana-1322	169	7	,	,	PUNCT
cana-1322	169	8	a.	a.	PROPN
cana-1322	169	9	nagamalleswara	nagamalleswara	PROPN
cana-1322	169	10	rao	rao	PROPN
cana-1322	169	11	,	,	PUNCT
cana-1322	169	12	p.l.n	p.l.n	PROPN
cana-1322	169	13	.	.	PROPN
cana-1322	169	14	varma	varma	PROPN
cana-1322	169	15	,	,	PUNCT
cana-1322	169	16	d.	d.	PROPN
cana-1322	169	17	madhusudhana	madhusudhana	PROPN
cana-1322	169	18	rao	rao	PROPN
cana-1322	169	19	,	,	PUNCT
cana-1322	169	20	ch	ch	NOUN
cana-1322	169	21	.	.	PROPN
cana-1322	169	22	ramprasad	ramprasad	ADJ
cana-1322	169	23	,	,	PUNCT
cana-1322	169	24	bi	bi	ADJ
cana-1322	169	25	-	-	ADJ
cana-1322	169	26	interior	interior	ADJ
cana-1322	169	27	ideals	ideal	NOUN
cana-1322	169	28	in	in	ADP
cana-1322	169	29	tgsr	tgsr	ADJ
cana-1322	169	30	,	,	PUNCT
cana-1322	169	31	advances	advance	NOUN
cana-1322	169	32	in	in	ADP
cana-1322	169	33	mathematics	mathematics	NOUN
cana-1322	169	34	scientific	scientific	ADJ
cana-1322	169	35	journal	journal	NOUN
cana-1322	169	36	,	,	PUNCT
cana-1322	169	37	10	10	NUM
cana-1322	169	38	(	(	PUNCT
cana-1322	169	39	2021	2021	NUM
cana-1322	169	40	)	)	PUNCT
cana-1322	169	41	,	,	PUNCT
cana-1322	169	42	no.3	no.3	VERB
cana-1322	169	43	,	,	PUNCT
cana-1322	169	44	pp	pp	CCONJ
cana-1322	169	45	:	:	PUNCT
cana-1322	169	46	1183	1183	NUM
cana-1322	169	47	-	-	SYM
cana-1322	169	48	1195	1195	NUM
cana-1322	169	49	.	.	PUNCT
cana-1322	170	1	[	[	X
cana-1322	170	2	20	20	NUM
cana-1322	170	3	]	]	PUNCT
cana-1322	170	4	somasundra	somasundra	NOUN
cana-1322	170	5	moorthy.m.g	moorthy.m.g	NUM
cana-1322	170	6	.	.	PUNCT
cana-1322	170	7	,	,	PUNCT
cana-1322	170	8	“	"	PUNCT
cana-1322	170	9	a	a	DET
cana-1322	170	10	study	study	NOUN
cana-1322	170	11	on	on	ADP
cana-1322	170	12	interval	interval	NOUN
cana-1322	170	13	valued	value	VERB
cana-1322	170	14	fuzzy	fuzzy	ADJ
cana-1322	170	15	,	,	PUNCT
cana-1322	170	16	anti	anti	X
cana-1322	170	17	fuzzy	fuzzy	ADJ
cana-1322	170	18	,	,	PUNCT
cana-1322	170	19	intuitionistic	intuitionistic	ADJ
cana-1322	170	20	fuzzy	fuzzy	ADJ
cana-1322	170	21	subrings	subring	NOUN
cana-1322	170	22	of	of	ADP
cana-1322	170	23	a	a	DET
cana-1322	170	24	ring	ring	NOUN
cana-1322	170	25	”	"	PUNCT
cana-1322	170	26	,	,	PUNCT
cana-1322	170	27	ph.d	ph.d	PROPN
cana-1322	170	28	thesis	thesis	NOUN
cana-1322	170	29	,	,	PUNCT
cana-1322	170	30	bharathidasan	bharathidasan	ADJ
cana-1322	170	31	university	university	NOUN
cana-1322	170	32	,	,	PUNCT
cana-1322	170	33	trichy	trichy	NOUN
cana-1322	170	34	,	,	PUNCT
cana-1322	170	35	tamilnadu	tamilnadu	NOUN
cana-1322	170	36	,	,	PUNCT
cana-1322	170	37	india	india	PROPN
cana-1322	170	38	(	(	PUNCT
cana-1322	170	39	2014	2014	NUM
cana-1322	170	40	)	)	PUNCT
cana-1322	170	41	.	.	PUNCT
cana-1322	171	1	[	[	X
cana-1322	171	2	21	21	NUM
cana-1322	171	3	]	]	SYM
cana-1322	171	4	yasodara.b	yasodara.b	PUNCT
cana-1322	171	5	and	and	CCONJ
cana-1322	171	6	ke.sathappan	ke.sathappan	NOUN
cana-1322	171	7	,	,	PUNCT
cana-1322	171	8	“	"	PUNCT
cana-1322	171	9	bipolar	bipolar	ADJ
cana-1322	171	10	-	-	PUNCT
cana-1322	171	11	valued	value	VERB
cana-1322	171	12	multi	multi	ADJ
cana-1322	171	13	fuzzy	fuzzy	ADJ
cana-1322	171	14	subsemirings	subsemiring	NOUN
cana-1322	171	15	of	of	ADP
cana-1322	171	16	a	a	DET
cana-1322	171	17	semiring	semiring	NOUN
cana-1322	171	18	”	"	PUNCT
cana-1322	171	19	,	,	PUNCT
cana-1322	171	20	international	international	ADJ
cana-1322	171	21	journal	journal	NOUN
cana-1322	171	22	of	of	ADP
cana-1322	171	23	[	[	X
cana-1322	171	24	22	22	NUM
cana-1322	171	25	]	]	PUNCT
cana-1322	171	26	l.a.zadeh	l.a.zadeh	NOUN
cana-1322	171	27	,	,	PUNCT
cana-1322	171	28	fuzzy	fuzzy	ADJ
cana-1322	171	29	sets	set	NOUN
cana-1322	171	30	,	,	PUNCT
cana-1322	171	31	inform	inform	NOUN
cana-1322	171	32	.	.	PUNCT
cana-1322	172	1	and	and	CCONJ
cana-1322	172	2	control	control	NOUN
cana-1322	172	3	,	,	PUNCT
cana-1322	172	4	8(1965	8(1965	NUM
cana-1322	172	5	)	)	PUNCT
cana-1322	172	6	,	,	PUNCT
cana-1322	172	7	338	338	NUM
cana-1322	172	8	-	-	SYM
cana-1322	172	9	353	353	NUM
cana-1322	172	10	.	.	PUNCT
cana-1322	173	1	[	[	X
cana-1322	173	2	23	23	NUM
cana-1322	173	3	]	]	X
cana-1322	173	4	w.r.zhang	w.r.zhang	PROPN
cana-1322	173	5	,	,	PUNCT
cana-1322	173	6	bipolar	bipolar	ADJ
cana-1322	173	7	fuzzy	fuzzy	ADJ
cana-1322	173	8	sets	set	NOUN
cana-1322	173	9	and	and	CCONJ
cana-1322	173	10	relations	relation	NOUN
cana-1322	173	11	,	,	PUNCT
cana-1322	173	12	a	a	DET
cana-1322	173	13	computational	computational	ADJ
cana-1322	173	14	frame	frame	NOUN
cana-1322	173	15	work	work	NOUN
cana-1322	173	16	for	for	ADP
cana-1322	173	17	cognitive	cognitive	ADJ
cana-1322	173	18	modeling	modeling	NOUN
cana-1322	173	19	and	and	CCONJ
cana-1322	173	20	multiple	multiple	ADJ