id	sid	tid	token	lemma	pos
cana-5869	1	1	communications	communication	NOUN
cana-5869	1	2	on	on	ADP
cana-5869	1	3	applied	apply	VERB
cana-5869	1	4	nonlinear	nonlinear	ADJ
cana-5869	1	5	analysis	analysis	NOUN
cana-5869	1	6	issn	issn	NOUN
cana-5869	1	7	:	:	PUNCT
cana-5869	1	8	1074	1074	NUM
cana-5869	1	9	-	-	PUNCT
cana-5869	1	10	133x	133x	NUM
cana-5869	1	11	vol	vol	NOUN
cana-5869	1	12	31	31	NUM
cana-5869	1	13	no	no	NOUN
cana-5869	1	14	.	.	PUNCT
cana-5869	2	1	2s	2s	NUM
cana-5869	2	2	(	(	PUNCT
cana-5869	2	3	2024	2024	NUM
cana-5869	2	4	)	)	PUNCT
cana-5869	2	5	754	754	NUM
cana-5869	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5869	2	7	bicomplex	bicomplex	NOUN
cana-5869	2	8	sequence	sequence	NOUN
cana-5869	2	9	spaces	space	VERB
cana-5869	2	10	:	:	PUNCT
cana-5869	2	11	duality	duality	NOUN
cana-5869	2	12	via	via	ADP
cana-5869	2	13	idempotent	idempotent	ADJ
cana-5869	2	14	decomposition	decomposition	NOUN
cana-5869	2	15	mamta	mamta	PROPN
cana-5869	2	16	amol	amol	PROPN
cana-5869	2	17	wagh	wagh	PROPN
cana-5869	2	18	department	department	PROPN
cana-5869	2	19	of	of	ADP
cana-5869	2	20	mathematics	mathematics	PROPN
cana-5869	2	21	,	,	PUNCT
cana-5869	2	22	deen	deen	PROPN
cana-5869	2	23	dayal	dayal	PROPN
cana-5869	2	24	upadhyaya	upadhyaya	PROPN
cana-5869	2	25	college	college	PROPN
cana-5869	2	26	,	,	PUNCT
cana-5869	2	27	university	university	PROPN
cana-5869	2	28	of	of	ADP
cana-5869	2	29	delhi	delhi	PROPN
cana-5869	2	30	email	email	NOUN
cana-5869	2	31	–	–	PUNCT
cana-5869	2	32	mamtanigam@ddu.du.ac.in	mamtanigam@ddu.du.ac.in	PROPN
cana-5869	2	33	article	article	NOUN
cana-5869	2	34	history	history	NOUN
cana-5869	2	35	:	:	PUNCT
cana-5869	2	36	received	receive	VERB
cana-5869	2	37	:	:	PUNCT
cana-5869	2	38	14	14	NUM
cana-5869	2	39	-	-	SYM
cana-5869	2	40	02	02	NUM
cana-5869	2	41	-	-	PUNCT
cana-5869	2	42	2024	2024	NUM
cana-5869	2	43	revised	revise	VERB
cana-5869	2	44	:	:	PUNCT
cana-5869	2	45	15	15	NUM
cana-5869	2	46	-	-	SYM
cana-5869	2	47	03	03	NUM
cana-5869	2	48	-	-	PUNCT
cana-5869	2	49	2024	2024	NUM
cana-5869	2	50	accepted	accept	VERB
cana-5869	2	51	:	:	PUNCT
cana-5869	2	52	21	21	NUM
cana-5869	2	53	-	-	PUNCT
cana-5869	2	54	04	04	NUM
cana-5869	2	55	-	-	PUNCT
cana-5869	2	56	2024	2024	NUM
cana-5869	2	57	abstract	abstract	NOUN
cana-5869	2	58	:	:	PUNCT
cana-5869	2	59	this	this	DET
cana-5869	2	60	paper	paper	NOUN
cana-5869	2	61	investigates	investigate	VERB
cana-5869	2	62	the	the	DET
cana-5869	2	63	duals	dual	NOUN
cana-5869	2	64	of	of	ADP
cana-5869	2	65	some	some	DET
cana-5869	2	66	bicomplex	bicomplex	NOUN
cana-5869	2	67	sequence	sequence	NOUN
cana-5869	2	68	spaces	space	VERB
cana-5869	2	69	corresponding	correspond	VERB
cana-5869	2	70	to	to	ADP
cana-5869	2	71	bicomplex	bicomplex	NOUN
cana-5869	2	72	functions	function	NOUN
cana-5869	2	73	that	that	PRON
cana-5869	2	74	are	be	AUX
cana-5869	2	75	holomorphic	holomorphic	ADJ
cana-5869	2	76	in	in	ADP
cana-5869	2	77	the	the	DET
cana-5869	2	78	bicomplex	bicomplex	NOUN
cana-5869	2	79	space	space	NOUN
cana-5869	2	80	ℂ2	ℂ2	NOUN
cana-5869	2	81	,	,	PUNCT
cana-5869	2	82	or	or	CCONJ
cana-5869	2	83	entire	entire	ADJ
cana-5869	2	84	bicomplex	bicomplex	NOUN
cana-5869	2	85	sequence	sequence	NOUN
cana-5869	2	86	spaces	space	VERB
cana-5869	2	87	.	.	PUNCT
cana-5869	3	1	we	we	PRON
cana-5869	3	2	investigate	investigate	VERB
cana-5869	3	3	these	these	DET
cana-5869	3	4	spaces	space	NOUN
cana-5869	3	5	through	through	ADP
cana-5869	3	6	their	their	PRON
cana-5869	3	7	idempotent	idempotent	ADJ
cana-5869	3	8	decompositions	decomposition	NOUN
cana-5869	3	9	and	and	CCONJ
cana-5869	3	10	examine	examine	VERB
cana-5869	3	11	the	the	DET
cana-5869	3	12	β	β	NOUN
cana-5869	3	13	-	-	ADJ
cana-5869	3	14	dual	dual	ADJ
cana-5869	3	15	,	,	PUNCT
cana-5869	3	16	γ	γ	NOUN
cana-5869	3	17	-	-	ADJ
cana-5869	3	18	dual	dual	ADJ
cana-5869	3	19	,	,	PUNCT
cana-5869	3	20	and	and	CCONJ
cana-5869	3	21	δ	δ	PROPN
cana-5869	3	22	-	-	PUNCT
cana-5869	3	23	dual	dual	ADJ
cana-5869	3	24	.	.	PUNCT
cana-5869	4	1	precise	precise	ADJ
cana-5869	4	2	definitions	definition	NOUN
cana-5869	4	3	and	and	CCONJ
cana-5869	4	4	analyses	analysis	NOUN
cana-5869	4	5	of	of	ADP
cana-5869	4	6	these	these	DET
cana-5869	4	7	duals	dual	NOUN
cana-5869	4	8	are	be	AUX
cana-5869	4	9	presented	present	VERB
cana-5869	4	10	.	.	PUNCT
cana-5869	5	1	our	our	PRON
cana-5869	5	2	results	result	NOUN
cana-5869	5	3	demonstrate	demonstrate	VERB
cana-5869	5	4	that	that	SCONJ
cana-5869	5	5	the	the	DET
cana-5869	5	6	duals	dual	NOUN
cana-5869	5	7	of	of	ADP
cana-5869	5	8	the	the	DET
cana-5869	5	9	original	original	ADJ
cana-5869	5	10	sequence	sequence	NOUN
cana-5869	5	11	spaces	space	NOUN
cana-5869	5	12	are	be	AUX
cana-5869	5	13	strictly	strictly	ADV
cana-5869	5	14	contained	contain	VERB
cana-5869	5	15	within	within	ADP
cana-5869	5	16	the	the	DET
cana-5869	5	17	duals	dual	NOUN
cana-5869	5	18	of	of	ADP
cana-5869	5	19	their	their	PRON
cana-5869	5	20	corresponding	corresponding	ADJ
cana-5869	5	21	idempotent	idempotent	ADJ
cana-5869	5	22	subclasses	subclass	NOUN
cana-5869	5	23	.	.	PUNCT
cana-5869	6	1	these	these	DET
cana-5869	6	2	findings	finding	NOUN
cana-5869	6	3	are	be	AUX
cana-5869	6	4	also	also	ADV
cana-5869	6	5	discussed	discuss	VERB
cana-5869	6	6	in	in	ADP
cana-5869	6	7	the	the	DET
cana-5869	6	8	context	context	NOUN
cana-5869	6	9	of	of	ADP
cana-5869	6	10	algebra	algebra	NOUN
cana-5869	6	11	homomorphisms	homomorphism	NOUN
cana-5869	6	12	between	between	ADP
cana-5869	6	13	the	the	DET
cana-5869	6	14	original	original	ADJ
cana-5869	6	15	sequence	sequence	NOUN
cana-5869	6	16	space	space	NOUN
cana-5869	6	17	ℵ	ℵ	NOUN
cana-5869	6	18	and	and	CCONJ
cana-5869	6	19	its	its	PRON
cana-5869	6	20	idempotent	idempotent	ADJ
cana-5869	6	21	subclasses	subclass	NOUN
cana-5869	6	22	1	1	NUM
cana-5869	6	23	ℵ	ℵ	NOUN
cana-5869	6	24	and	and	CCONJ
cana-5869	6	25	2	2	NUM
cana-5869	6	26	ℵ.	ℵ.	NOUN
cana-5869	6	27	keywords	keyword	NOUN
cana-5869	6	28	:	:	PUNCT
cana-5869	7	1	bicomplex	bicomplex	NOUN
cana-5869	7	2	numbers	number	NOUN
cana-5869	7	3	,	,	PUNCT
cana-5869	7	4	entire	entire	ADJ
cana-5869	7	5	bicomplex	bicomplex	NOUN
cana-5869	7	6	sequence	sequence	NOUN
cana-5869	7	7	spaces	space	NOUN
cana-5869	7	8	,	,	PUNCT
cana-5869	7	9	idempotent	idempotent	ADJ
cana-5869	7	10	sequence	sequence	NOUN
cana-5869	7	11	spaces	space	NOUN
cana-5869	7	12	,	,	PUNCT
cana-5869	7	13	köthe	köthe	ADJ
cana-5869	7	14	–	–	PUNCT
cana-5869	7	15	toeplitz	toeplitz	NOUN
cana-5869	7	16	duals	dual	NOUN
cana-5869	7	17	2020	2020	NUM
cana-5869	7	18	mathematics	mathematic	NOUN
cana-5869	7	19	subject	subject	ADJ
cana-5869	7	20	classification	classification	NOUN
cana-5869	7	21	:	:	PUNCT
cana-5869	7	22	46e10	46e10	NUM
cana-5869	7	23	,	,	PUNCT
cana-5869	7	24	46e15	46e15	NUM
cana-5869	7	25	,	,	PUNCT
cana-5869	7	26	46e25	46e25	NUM
cana-5869	7	27	1	1	NUM
cana-5869	7	28	.	.	PUNCT
cana-5869	7	29	introduction	introduction	NOUN
cana-5869	7	30	1.1	1.1	NUM
cana-5869	7	31	duals	dual	NOUN
cana-5869	7	32	of	of	ADP
cana-5869	7	33	sequence	sequence	NOUN
cana-5869	7	34	spaces	space	NOUN
cana-5869	7	35	there	there	PRON
cana-5869	7	36	are	be	VERB
cana-5869	7	37	two	two	NUM
cana-5869	7	38	primary	primary	ADJ
cana-5869	7	39	types	type	NOUN
cana-5869	7	40	of	of	ADP
cana-5869	7	41	duals	dual	NOUN
cana-5869	7	42	associated	associate	VERB
cana-5869	7	43	with	with	ADP
cana-5869	7	44	a	a	DET
cana-5869	7	45	sequence	sequence	NOUN
cana-5869	7	46	space	space	NOUN
cana-5869	7	47	:	:	PUNCT
cana-5869	7	48	the	the	DET
cana-5869	7	49	algebraic	algebraic	PROPN
cana-5869	7	50	dual	dual	ADJ
cana-5869	7	51	and	and	CCONJ
cana-5869	7	52	the	the	DET
cana-5869	7	53	topological	topological	ADJ
cana-5869	7	54	dual	dual	NOUN
cana-5869	7	55	.	.	PUNCT
cana-5869	8	1	the	the	DET
cana-5869	8	2	algebraic	algebraic	PROPN
cana-5869	8	3	dual	dual	ADJ
cana-5869	8	4	of	of	ADP
cana-5869	8	5	a	a	DET
cana-5869	8	6	linear	linear	ADJ
cana-5869	8	7	space	space	NOUN
cana-5869	8	8	v	v	NOUN
cana-5869	8	9	is	be	AUX
cana-5869	8	10	the	the	DET
cana-5869	8	11	set	set	NOUN
cana-5869	8	12	of	of	ADP
cana-5869	8	13	all	all	DET
cana-5869	8	14	linear	linear	ADJ
cana-5869	8	15	functionals	functional	NOUN
cana-5869	8	16	from	from	ADP
cana-5869	8	17	v	v	NUM
cana-5869	8	18	to	to	ADP
cana-5869	8	19	a	a	DET
cana-5869	8	20	scalar	scalar	ADJ
cana-5869	8	21	field	field	NOUN
cana-5869	8	22	k	k	NOUN
cana-5869	8	23	,	,	PUNCT
cana-5869	8	24	and	and	CCONJ
cana-5869	8	25	is	be	AUX
cana-5869	8	26	denoted	denote	VERB
cana-5869	8	27	by	by	ADP
cana-5869	8	28	l(v	l(v	PROPN
cana-5869	8	29	,	,	PUNCT
cana-5869	8	30	k	k	NOUN
cana-5869	8	31	)	)	PUNCT
cana-5869	8	32	=	=	SYM
cana-5869	8	33	v	v	NOUN
cana-5869	8	34	#	#	NOUN
cana-5869	8	35	.	.	PUNCT
cana-5869	9	1	on	on	ADP
cana-5869	9	2	the	the	DET
cana-5869	9	3	other	other	ADJ
cana-5869	9	4	hand	hand	NOUN
cana-5869	9	5	,	,	PUNCT
cana-5869	9	6	the	the	DET
cana-5869	9	7	topological	topological	ADJ
cana-5869	9	8	dual	dual	ADJ
cana-5869	9	9	consists	consist	NOUN
cana-5869	9	10	of	of	ADP
cana-5869	9	11	all	all	DET
cana-5869	9	12	continuous	continuous	ADJ
cana-5869	9	13	linear	linear	ADJ
cana-5869	9	14	functionals	functional	NOUN
cana-5869	9	15	on	on	ADP
cana-5869	9	16	v	v	NOUN
cana-5869	9	17	and	and	CCONJ
cana-5869	9	18	is	be	AUX
cana-5869	9	19	denoted	denote	VERB
cana-5869	9	20	by	by	ADP
cana-5869	9	21	v	v	NOUN
cana-5869	9	22	*	*	NOUN
cana-5869	9	23	.	.	PUNCT
cana-5869	10	1	the	the	DET
cana-5869	10	2	only	only	ADJ
cana-5869	10	3	sequence	sequence	NOUN
cana-5869	10	4	space	space	NOUN
cana-5869	10	5	with	with	ADP
cana-5869	10	6	a	a	DET
cana-5869	10	7	well	well	ADV
cana-5869	10	8	-	-	PUNCT
cana-5869	10	9	behaved	behave	VERB
cana-5869	10	10	algebraic	algebraic	ADJ
cana-5869	10	11	dual	dual	ADJ
cana-5869	10	12	consisting	consist	VERB
cana-5869	10	13	of	of	ADP
cana-5869	10	14	sequences	sequence	NOUN
cana-5869	10	15	is	be	AUX
cana-5869	10	16	ϕ	ϕ	NOUN
cana-5869	10	17	,	,	PUNCT
cana-5869	10	18	whose	whose	DET
cana-5869	10	19	dual	dual	ADV
cana-5869	10	20	is	be	AUX
cana-5869	10	21	ω	ω	PROPN
cana-5869	10	22	.	.	PUNCT
cana-5869	11	1	therefore	therefore	ADV
cana-5869	11	2	,	,	PUNCT
cana-5869	11	3	in	in	ADP
cana-5869	11	4	duality	duality	NOUN
cana-5869	11	5	theory	theory	NOUN
cana-5869	11	6	,	,	PUNCT
cana-5869	11	7	it	it	PRON
cana-5869	11	8	is	be	AUX
cana-5869	11	9	more	more	ADV
cana-5869	11	10	effective	effective	ADJ
cana-5869	11	11	to	to	PART
cana-5869	11	12	study	study	VERB
cana-5869	11	13	sequence	sequence	NOUN
cana-5869	11	14	spaces	space	NOUN
cana-5869	11	15	with	with	ADP
cana-5869	11	16	linear	linear	PROPN
cana-5869	11	17	topologies	topology	NOUN
cana-5869	11	18	,	,	PUNCT
cana-5869	11	19	though	though	SCONJ
cana-5869	11	20	finding	find	VERB
cana-5869	11	21	their	their	PRON
cana-5869	11	22	topological	topological	ADJ
cana-5869	11	23	duals	dual	NOUN
cana-5869	11	24	is	be	AUX
cana-5869	11	25	often	often	ADV
cana-5869	11	26	challenging	challenge	VERB
cana-5869	11	27	.	.	PUNCT
cana-5869	12	1	to	to	PART
cana-5869	12	2	address	address	VERB
cana-5869	12	3	this	this	PRON
cana-5869	12	4	,	,	PUNCT
cana-5869	12	5	köthe	köthe	NOUN
cana-5869	12	6	and	and	CCONJ
cana-5869	12	7	toeplitz	toeplitz	NOUN
cana-5869	13	1	[	[	X
cana-5869	13	2	3	3	X
cana-5869	13	3	]	]	PUNCT
cana-5869	13	4	introduced	introduce	VERB
cana-5869	13	5	the	the	DET
cana-5869	13	6	α	α	NOUN
cana-5869	13	7	-	-	ADJ
cana-5869	13	8	dual	dual	ADJ
cana-5869	13	9	and	and	CCONJ
cana-5869	13	10	β	β	NOUN
cana-5869	13	11	-	-	ADJ
cana-5869	13	12	dual	dual	ADJ
cana-5869	13	13	,	,	PUNCT
cana-5869	13	14	which	which	PRON
cana-5869	13	15	facilitate	facilitate	VERB
cana-5869	13	16	a	a	DET
cana-5869	13	17	more	more	ADV
cana-5869	13	18	practical	practical	ADJ
cana-5869	13	19	dual	dual	ADJ
cana-5869	13	20	system	system	NOUN
cana-5869	13	21	.	.	PUNCT
cana-5869	14	1	later	later	ADV
cana-5869	14	2	,	,	PUNCT
cana-5869	14	3	garling	garle	VERB
cana-5869	14	4	[	[	X
cana-5869	14	5	1	1	NUM
cana-5869	14	6	]	]	PUNCT
cana-5869	14	7	proposed	propose	VERB
cana-5869	14	8	the	the	DET
cana-5869	14	9	more	more	ADV
cana-5869	14	10	general	general	ADJ
cana-5869	14	11	γ	γ	X
cana-5869	14	12	-	-	ADJ
cana-5869	14	13	dual	dual	ADJ
cana-5869	14	14	,	,	PUNCT
cana-5869	14	15	and	and	CCONJ
cana-5869	14	16	for	for	ADP
cana-5869	14	17	symmetric	symmetric	ADJ
cana-5869	14	18	sequence	sequence	NOUN
cana-5869	14	19	spaces	space	NOUN
cana-5869	14	20	,	,	PUNCT
cana-5869	14	21	the	the	DET
cana-5869	14	22	δ	δ	PROPN
cana-5869	14	23	-	-	PROPN
cana-5869	14	24	dual	dual	ADV
cana-5869	14	25	was	be	AUX
cana-5869	14	26	introduced	introduce	VERB
cana-5869	14	27	by	by	ADP
cana-5869	14	28	garling	garle	VERB
cana-5869	14	29	[	[	X
cana-5869	14	30	2	2	NUM
cana-5869	14	31	]	]	PUNCT
cana-5869	14	32	and	and	CCONJ
cana-5869	14	33	ruckle	ruckle	VERB
cana-5869	14	34	[	[	X
cana-5869	14	35	5	5	NUM
cana-5869	14	36	]	]	PUNCT
cana-5869	14	37	.	.	PUNCT
cana-5869	15	1	1.2	1.2	NUM
cana-5869	15	2	bicomplex	bicomplex	NOUN
cana-5869	15	3	space	space	NOUN
cana-5869	15	4	ℂ2	ℂ2	NOUN
cana-5869	15	5	bicomplex	bicomplex	NOUN
cana-5869	15	6	numbers	number	NOUN
cana-5869	15	7	were	be	AUX
cana-5869	15	8	defined	define	VERB
cana-5869	15	9	by	by	ADP
cana-5869	15	10	corrado	corrado	PROPN
cana-5869	15	11	segre	segre	PROPN
cana-5869	15	12	(	(	PUNCT
cana-5869	15	13	1860	1860	NUM
cana-5869	15	14	–	–	PUNCT
cana-5869	15	15	1924	1924	NUM
cana-5869	15	16	)	)	PUNCT
cana-5869	15	17	in	in	ADP
cana-5869	15	18	1892	1892	NUM
cana-5869	15	19	.	.	PUNCT
cana-5869	16	1	infinite	infinite	ADJ
cana-5869	16	2	set	set	NOUN
cana-5869	16	3	of	of	ADP
cana-5869	16	4	algebras	algebra	NOUN
cana-5869	16	5	and	and	CCONJ
cana-5869	16	6	the	the	DET
cana-5869	16	7	concept	concept	NOUN
cana-5869	16	8	of	of	ADP
cana-5869	16	9	multicomplex	multicomplex	ADJ
cana-5869	16	10	numbers	number	NOUN
cana-5869	16	11	was	be	AUX
cana-5869	16	12	given	give	VERB
cana-5869	16	13	in	in	ADP
cana-5869	16	14	[	[	X
cana-5869	16	15	6	6	NUM
cana-5869	16	16	]	]	PUNCT
cana-5869	16	17	.	.	PUNCT
cana-5869	17	1	the	the	DET
cana-5869	17	2	set	set	NOUN
cana-5869	17	3	of	of	ADP
cana-5869	17	4	bicomplex	bicomplex	NOUN
cana-5869	17	5	numbers	number	NOUN
cana-5869	17	6	is	be	AUX
cana-5869	17	7	given	give	VERB
cana-5869	17	8	by	by	ADP
cana-5869	17	9	ℂ2	ℂ2	NOUN
cana-5869	17	10	=	=	SYM
cana-5869	17	11	{	{	PUNCT
cana-5869	17	12	𝜇1	𝜇1	PROPN
cana-5869	17	13	+	+	CCONJ
cana-5869	17	14	𝑖1𝜇2	𝑖1𝜇2	PUNCT
cana-5869	18	1	+	+	NUM
cana-5869	18	2	𝑖2𝜇3	𝑖2𝜇3	X
cana-5869	18	3	+	+	CCONJ
cana-5869	18	4	𝑖1𝑖2𝜇4	𝑖1𝑖2𝜇4	ADJ
cana-5869	18	5	:	:	PUNCT
cana-5869	18	6	𝜇1	𝜇1	ADJ
cana-5869	18	7	,	,	PUNCT
cana-5869	18	8	𝜇2	𝜇2	PROPN
cana-5869	18	9	,	,	PUNCT
cana-5869	18	10	𝜇3	𝜇3	PROPN
cana-5869	18	11	,	,	PUNCT
cana-5869	18	12	𝜇4	𝜇4	NOUN
cana-5869	18	13	∈	∈	PROPN
cana-5869	18	14	ℂ0	ℂ0	PROPN
cana-5869	18	15	}	}	PUNCT
cana-5869	18	16	,	,	PUNCT
cana-5869	18	17	where	where	SCONJ
cana-5869	18	18	𝑖1	𝑖1	PROPN
cana-5869	18	19	2	2	NUM
cana-5869	18	20	=	=	SYM
cana-5869	18	21	𝑖2	𝑖2	PROPN
cana-5869	18	22	2	2	NUM
cana-5869	18	23	=	=	SYM
cana-5869	18	24	−1	−1	NOUN
cana-5869	18	25	,	,	PUNCT
cana-5869	18	26	𝑖1𝑖2	𝑖1𝑖2	X
cana-5869	18	27	=	=	ADJ
cana-5869	18	28	𝑖2𝑖1	𝑖2𝑖1	PROPN
cana-5869	18	29	.	.	PUNCT
cana-5869	19	1	communications	communication	NOUN
cana-5869	19	2	on	on	ADP
cana-5869	19	3	applied	apply	VERB
cana-5869	19	4	nonlinear	nonlinear	ADJ
cana-5869	19	5	analysis	analysis	NOUN
cana-5869	19	6	issn	issn	NOUN
cana-5869	19	7	:	:	PUNCT
cana-5869	19	8	1074	1074	NUM
cana-5869	19	9	-	-	PUNCT
cana-5869	19	10	133x	133x	NUM
cana-5869	19	11	vol	vol	NOUN
cana-5869	19	12	31	31	NUM
cana-5869	19	13	no	no	NOUN
cana-5869	19	14	.	.	PUNCT
cana-5869	20	1	2s	2s	NUM
cana-5869	20	2	(	(	PUNCT
cana-5869	20	3	2024	2024	NUM
cana-5869	20	4	)	)	PUNCT
cana-5869	20	5	755	755	NUM
cana-5869	20	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5869	20	7	the	the	DET
cana-5869	20	8	binary	binary	ADJ
cana-5869	20	9	operations	operation	NOUN
cana-5869	20	10	of	of	ADP
cana-5869	20	11	addition	addition	NOUN
cana-5869	20	12	and	and	CCONJ
cana-5869	20	13	scalar	scalar	ADJ
cana-5869	20	14	multiplication	multiplication	NOUN
cana-5869	20	15	on	on	ADP
cana-5869	20	16	ℂ2	ℂ2	NOUN
cana-5869	20	17	are	be	AUX
cana-5869	20	18	defined	define	VERB
cana-5869	20	19	coordinate	coordinate	NOUN
cana-5869	20	20	-	-	PUNCT
cana-5869	20	21	wise	wise	ADJ
cana-5869	20	22	,	,	PUNCT
cana-5869	20	23	and	and	CCONJ
cana-5869	20	24	multiplication	multiplication	NOUN
cana-5869	20	25	is	be	AUX
cana-5869	20	26	defined	define	VERB
cana-5869	20	27	component	component	NOUN
cana-5869	20	28	-	-	PUNCT
cana-5869	20	29	wise	wise	ADJ
cana-5869	20	30	(	(	PUNCT
cana-5869	20	31	i.e.	i.e.	X
cana-5869	20	32	,	,	PUNCT
cana-5869	20	33	term	term	NOUN
cana-5869	20	34	by	by	ADP
cana-5869	20	35	term	term	NOUN
cana-5869	20	36	)	)	PUNCT
cana-5869	20	37	.	.	PUNCT
cana-5869	21	1	with	with	ADP
cana-5869	21	2	these	these	DET
cana-5869	21	3	operations	operation	NOUN
cana-5869	21	4	,	,	PUNCT
cana-5869	21	5	ℂ2	ℂ2	NOUN
cana-5869	21	6	forms	form	VERB
cana-5869	21	7	a	a	DET
cana-5869	21	8	commutative	commutative	ADJ
cana-5869	21	9	algebra	algebra	NOUN
cana-5869	21	10	with	with	ADP
cana-5869	21	11	identity	identity	NOUN
cana-5869	21	12	.	.	PUNCT
cana-5869	22	1	there	there	PRON
cana-5869	22	2	are	be	VERB
cana-5869	22	3	several	several	ADJ
cana-5869	22	4	notable	notable	ADJ
cana-5869	22	5	differences	difference	NOUN
cana-5869	22	6	between	between	ADP
cana-5869	22	7	the	the	DET
cana-5869	22	8	algebraic	algebraic	ADJ
cana-5869	22	9	structures	structure	NOUN
cana-5869	22	10	of	of	ADP
cana-5869	22	11	ℂ2	ℂ2	NOUN
cana-5869	22	12	and	and	CCONJ
cana-5869	22	13	ℂ1	ℂ1	PROPN
cana-5869	22	14	,	,	PUNCT
cana-5869	22	15	as	as	SCONJ
cana-5869	22	16	outlined	outline	VERB
cana-5869	22	17	by	by	ADP
cana-5869	22	18	price	price	NOUN
cana-5869	22	19	in	in	ADP
cana-5869	22	20	[	[	X
cana-5869	22	21	4	4	NUM
cana-5869	22	22	]	]	PUNCT
cana-5869	22	23	.	.	PUNCT
cana-5869	23	1	although	although	SCONJ
cana-5869	23	2	bicomplex	bicomplex	NOUN
cana-5869	23	3	numbers	number	NOUN
cana-5869	23	4	,	,	PUNCT
cana-5869	23	5	like	like	ADP
cana-5869	23	6	quaternions	quaternion	NOUN
cana-5869	23	7	,	,	PUNCT
cana-5869	23	8	form	form	VERB
cana-5869	23	9	a	a	DET
cana-5869	23	10	four	four	NUM
cana-5869	23	11	-	-	PUNCT
cana-5869	23	12	dimensional	dimensional	ADJ
cana-5869	23	13	algebra	algebra	NOUN
cana-5869	23	14	,	,	PUNCT
cana-5869	23	15	they	they	PRON
cana-5869	23	16	differ	differ	VERB
cana-5869	23	17	in	in	ADP
cana-5869	23	18	that	that	DET
cana-5869	23	19	bicomplex	bicomplex	NOUN
cana-5869	23	20	numbers	number	NOUN
cana-5869	23	21	are	be	AUX
cana-5869	23	22	commutative	commutative	ADJ
cana-5869	23	23	,	,	PUNCT
cana-5869	23	24	whereas	whereas	SCONJ
cana-5869	23	25	quaternions	quaternion	NOUN
cana-5869	23	26	are	be	AUX
cana-5869	23	27	not	not	PART
cana-5869	23	28	.	.	PUNCT
cana-5869	24	1	idempotent	idempotent	ADJ
cana-5869	24	2	elements	element	NOUN
cana-5869	24	3	–	–	PUNCT
cana-5869	24	4	apart	apart	ADV
cana-5869	24	5	from	from	ADP
cana-5869	24	6	0	0	NUM
cana-5869	24	7	and	and	CCONJ
cana-5869	24	8	1	1	NUM
cana-5869	24	9	,	,	PUNCT
cana-5869	24	10	the	the	DET
cana-5869	24	11	structure	structure	NOUN
cana-5869	24	12	contains	contain	VERB
cana-5869	24	13	two	two	NUM
cana-5869	24	14	distinct	distinct	ADJ
cana-5869	24	15	nontrivial	nontrivial	ADJ
cana-5869	24	16	idempotent	idempotent	ADJ
cana-5869	24	17	elements	element	NOUN
cana-5869	24	18	given	give	VERB
cana-5869	24	19	by	by	ADP
cana-5869	24	20	𝑒1	𝑒1	NOUN
cana-5869	24	21	=	=	SYM
cana-5869	24	22	1+𝑖1𝑖2	1+𝑖1𝑖2	NUM
cana-5869	24	23	2	2	NUM
cana-5869	24	24	and	and	CCONJ
cana-5869	24	25	𝑒2	𝑒2	NOUN
cana-5869	24	26	=	=	NOUN
cana-5869	25	1	1−𝑖1𝑖2	1−𝑖1𝑖2	NOUN
cana-5869	25	2	2	2	NUM
cana-5869	25	3	.	.	PUNCT
cana-5869	26	1	the	the	DET
cana-5869	26	2	addition	addition	NOUN
cana-5869	26	3	of	of	ADP
cana-5869	26	4	these	these	DET
cana-5869	26	5	two	two	NUM
cana-5869	26	6	idempotent	idempotent	ADJ
cana-5869	26	7	elements	element	NOUN
cana-5869	26	8	is	be	AUX
cana-5869	26	9	1	1	NUM
cana-5869	26	10	and	and	CCONJ
cana-5869	26	11	their	their	PRON
cana-5869	26	12	product	product	NOUN
cana-5869	26	13	is	be	AUX
cana-5869	26	14	zero	zero	NUM
cana-5869	26	15	.	.	PUNCT
cana-5869	27	1	there	there	PRON
cana-5869	27	2	are	be	VERB
cana-5869	27	3	two	two	NUM
cana-5869	27	4	principal	principal	ADJ
cana-5869	27	5	ideals	ideal	NOUN
cana-5869	27	6	generated	generate	VERB
cana-5869	27	7	by	by	ADP
cana-5869	27	8	these	these	DET
cana-5869	27	9	idempotent	idempotent	ADJ
cana-5869	27	10	elements	element	NOUN
cana-5869	27	11	.	.	PUNCT
cana-5869	28	1	intersection	intersection	NOUN
cana-5869	28	2	of	of	ADP
cana-5869	28	3	these	these	DET
cana-5869	28	4	ideals	ideal	NOUN
cana-5869	28	5	is	be	AUX
cana-5869	28	6	zero	zero	NUM
cana-5869	28	7	and	and	CCONJ
cana-5869	28	8	their	their	PRON
cana-5869	28	9	union	union	NOUN
cana-5869	28	10	is	be	AUX
cana-5869	28	11	the	the	DET
cana-5869	28	12	set	set	NOUN
cana-5869	28	13	of	of	ADP
cana-5869	28	14	all	all	DET
cana-5869	28	15	singular	singular	ADJ
cana-5869	28	16	elements	element	NOUN
cana-5869	28	17	of	of	ADP
cana-5869	28	18	ℂ2	ℂ2	NOUN
cana-5869	28	19	.	.	PUNCT
cana-5869	29	1	two	two	NUM
cana-5869	29	2	bicomplex	bicomplex	NOUN
cana-5869	29	3	numbers	number	NOUN
cana-5869	29	4	are	be	AUX
cana-5869	29	5	zero	zero	NUM
cana-5869	29	6	divisors	divisor	NOUN
cana-5869	29	7	precisely	precisely	ADV
cana-5869	29	8	when	when	SCONJ
cana-5869	29	9	one	one	PRON
cana-5869	29	10	is	be	AUX
cana-5869	29	11	a	a	DET
cana-5869	29	12	complex	complex	ADJ
cana-5869	29	13	multiple	multiple	NOUN
cana-5869	29	14	of	of	ADP
cana-5869	29	15	one	one	NUM
cana-5869	29	16	idempotent	idempotent	ADJ
cana-5869	29	17	element	element	NOUN
cana-5869	29	18	and	and	CCONJ
cana-5869	29	19	the	the	DET
cana-5869	29	20	other	other	ADJ
cana-5869	29	21	is	be	AUX
cana-5869	29	22	a	a	DET
cana-5869	29	23	complex	complex	ADJ
cana-5869	29	24	multiple	multiple	NOUN
cana-5869	29	25	of	of	ADP
cana-5869	29	26	the	the	DET
cana-5869	29	27	other	other	ADJ
cana-5869	29	28	idempotent	idempotent	ADJ
cana-5869	29	29	element	element	NOUN
cana-5869	29	30	.	.	PUNCT
cana-5869	30	1	the	the	DET
cana-5869	30	2	detailed	detailed	ADJ
cana-5869	30	3	study	study	NOUN
cana-5869	30	4	of	of	ADP
cana-5869	30	5	ℂ2	ℂ2	NOUN
cana-5869	30	6	is	be	AUX
cana-5869	30	7	provided	provide	VERB
cana-5869	30	8	in	in	ADP
cana-5869	30	9	[	[	NOUN
cana-5869	30	10	13	13	NUM
cana-5869	30	11	]	]	SYM
cana-5869	30	12	.	.	PUNCT
cana-5869	31	1	2	2	X
cana-5869	31	2	.	.	X
cana-5869	31	3	objectives	objective	VERB
cana-5869	31	4	the	the	DET
cana-5869	31	5	objective	objective	NOUN
cana-5869	31	6	of	of	ADP
cana-5869	31	7	this	this	DET
cana-5869	31	8	paper	paper	NOUN
cana-5869	31	9	is	be	AUX
cana-5869	31	10	to	to	PART
cana-5869	31	11	investigate	investigate	VERB
cana-5869	31	12	the	the	DET
cana-5869	31	13	β-	β-	X
cana-5869	31	14	,	,	PUNCT
cana-5869	31	15	γ-	γ-	X
cana-5869	31	16	,	,	PUNCT
cana-5869	31	17	and	and	CCONJ
cana-5869	31	18	δ	δ	NOUN
cana-5869	31	19	-	-	NOUN
cana-5869	31	20	duals	dual	NOUN
cana-5869	31	21	of	of	ADP
cana-5869	31	22	certain	certain	ADJ
cana-5869	31	23	classes	class	NOUN
cana-5869	31	24	of	of	ADP
cana-5869	31	25	bicomplex	bicomplex	NOUN
cana-5869	31	26	sequences	sequence	NOUN
cana-5869	31	27	,	,	PUNCT
cana-5869	31	28	and	and	CCONJ
cana-5869	31	29	to	to	PART
cana-5869	31	30	explore	explore	VERB
cana-5869	31	31	their	their	PRON
cana-5869	31	32	relationships	relationship	NOUN
cana-5869	31	33	with	with	ADP
cana-5869	31	34	the	the	DET
cana-5869	31	35	duals	dual	NOUN
cana-5869	31	36	of	of	ADP
cana-5869	31	37	corresponding	correspond	VERB
cana-5869	31	38	idempotent	idempotent	ADJ
cana-5869	31	39	subclasses	subclass	NOUN
cana-5869	31	40	,	,	PUNCT
cana-5869	31	41	supported	support	VERB
cana-5869	31	42	by	by	ADP
cana-5869	31	43	illustrative	illustrative	ADJ
cana-5869	31	44	examples	example	NOUN
cana-5869	31	45	and	and	CCONJ
cana-5869	31	46	counterexamples	counterexample	NOUN
cana-5869	31	47	3	3	X
cana-5869	32	1	.	.	PUNCT
cana-5869	32	2	methods	method	NOUN
cana-5869	32	3	idempotent	idempotent	ADJ
cana-5869	32	4	technique	technique	NOUN
cana-5869	32	5	has	have	AUX
cana-5869	32	6	been	be	AUX
cana-5869	32	7	used	use	VERB
cana-5869	32	8	to	to	PART
cana-5869	32	9	investigate	investigate	VERB
cana-5869	32	10	the	the	DET
cana-5869	32	11	duals	dual	NOUN
cana-5869	32	12	of	of	ADP
cana-5869	32	13	bicomplex	bicomplex	NOUN
cana-5869	32	14	sequence	sequence	NOUN
cana-5869	32	15	spaces	space	NOUN
cana-5869	32	16	and	and	CCONJ
cana-5869	32	17	their	their	PRON
cana-5869	32	18	subclasses	subclass	NOUN
cana-5869	32	19	.	.	PUNCT
cana-5869	33	1	4	4	X
cana-5869	33	2	.	.	NOUN
cana-5869	33	3	results	result	VERB
cana-5869	33	4	bicomplex	bicomplex	PROPN
cana-5869	33	5	köthe	köthe	ADJ
cana-5869	33	6	–	–	PUNCT
cana-5869	33	7	toeplitz	toeplitz	NOUN
cana-5869	33	8	duals	dual	NOUN
cana-5869	33	9	if	if	SCONJ
cana-5869	33	10	ω	ω	NOUN
cana-5869	33	11	’	'	PUNCT
cana-5869	33	12	is	be	AUX
cana-5869	33	13	the	the	DET
cana-5869	33	14	family	family	NOUN
cana-5869	33	15	of	of	ADP
cana-5869	33	16	all	all	DET
cana-5869	33	17	bicomplex	bicomplex	NOUN
cana-5869	33	18	sequences	sequence	NOUN
cana-5869	33	19	𝜉	𝜉	PART
cana-5869	33	20	=	=	SYM
cana-5869	33	21	(	(	PUNCT
cana-5869	33	22	𝜉𝑘	𝜉𝑘	PROPN
cana-5869	33	23	)	)	PUNCT
cana-5869	33	24	with	with	ADP
cana-5869	33	25	𝜉𝑘	𝜉𝑘	PROPN
cana-5869	33	26	∈	∈	PROPN
cana-5869	33	27	ℂ2	ℂ2	NOUN
cana-5869	33	28	,	,	PUNCT
cana-5869	33	29	k	k	PROPN
cana-5869	33	30	≥	≥	NUM
cana-5869	33	31	1	1	NUM
cana-5869	33	32	,	,	PUNCT
cana-5869	33	33	where	where	SCONJ
cana-5869	33	34	ℂ2	ℂ2	NOUN
cana-5869	33	35	is	be	AUX
cana-5869	33	36	the	the	DET
cana-5869	33	37	space	space	NOUN
cana-5869	33	38	of	of	ADP
cana-5869	33	39	all	all	DET
cana-5869	33	40	bicomplex	bicomplex	NOUN
cana-5869	33	41	numbers	number	NOUN
cana-5869	33	42	.	.	PUNCT
cana-5869	34	1	if	if	SCONJ
cana-5869	34	2	𝜓	𝜓	NOUN
cana-5869	34	3	be	be	VERB
cana-5869	34	4	a	a	DET
cana-5869	34	5	bicomplex	bicomplex	NOUN
cana-5869	34	6	sequence	sequence	NOUN
cana-5869	34	7	space	space	NOUN
cana-5869	34	8	,	,	PUNCT
cana-5869	34	9	then	then	ADV
cana-5869	34	10	we	we	PRON
cana-5869	34	11	denote	denote	VERB
cana-5869	34	12	α	α	PROPN
cana-5869	34	13	–	–	PUNCT
cana-5869	34	14	,	,	PUNCT
cana-5869	34	15	β	β	X
cana-5869	34	16	–	–	PUNCT
cana-5869	34	17	,	,	PUNCT
cana-5869	34	18	γ	γ	X
cana-5869	34	19	–	–	PUNCT
cana-5869	34	20	,	,	PUNCT
cana-5869	34	21	and	and	CCONJ
cana-5869	34	22	δ	δ	PROPN
cana-5869	34	23	–	–	PUNCT
cana-5869	34	24	duals	dual	NOUN
cana-5869	34	25	of	of	ADP
cana-5869	34	26	𝜓	𝜓	NOUN
cana-5869	34	27	by	by	ADP
cana-5869	34	28	𝜓𝛼	𝜓𝛼	ADP
cana-5869	34	29	,	,	PUNCT
cana-5869	34	30	𝜓𝛽	𝜓𝛽	INTJ
cana-5869	34	31	,	,	PUNCT
cana-5869	34	32	𝜓𝛾	𝜓𝛾	PROPN
cana-5869	34	33	and	and	CCONJ
cana-5869	34	34	𝜓𝛿	𝜓𝛿	ADP
cana-5869	34	35	respectively	respectively	ADV
cana-5869	34	36	.	.	PUNCT
cana-5869	35	1	these	these	DET
cana-5869	35	2	duals	dual	NOUN
cana-5869	35	3	have	have	AUX
cana-5869	35	4	been	be	AUX
cana-5869	35	5	defined	define	VERB
cana-5869	35	6	in	in	ADP
cana-5869	35	7	[	[	X
cana-5869	35	8	10	10	NUM
cana-5869	35	9	]	]	PUNCT
cana-5869	35	10	.	.	PUNCT
cana-5869	36	1	let	let	VERB
cana-5869	36	2	us	we	PRON
cana-5869	36	3	see	see	VERB
cana-5869	36	4	these	these	DET
cana-5869	36	5	definitions	definition	NOUN
cana-5869	36	6	for	for	ADP
cana-5869	36	7	our	our	PRON
cana-5869	36	8	ready	ready	ADJ
cana-5869	36	9	reference	reference	NOUN
cana-5869	36	10	.	.	PUNCT
cana-5869	37	1	𝜓𝛼	𝜓𝛼	VERB
cana-5869	37	2	=	=	SYM
cana-5869	37	3	{	{	PUNCT
cana-5869	37	4	𝜉	𝜉	X
cana-5869	37	5	:	:	PUNCT
cana-5869	37	6	𝜉	𝜉	PROPN
cana-5869	37	7	∈	∈	PROPN
cana-5869	37	8	𝜔′	𝜔′	PROPN
cana-5869	37	9	,	,	PUNCT
cana-5869	37	10	∑‖𝜉𝑖𝜂𝑖‖	∑‖𝜉𝑖𝜂𝑖‖	PRON
cana-5869	37	11	<	<	X
cana-5869	37	12	∞	∞	PROPN
cana-5869	37	13	,	,	PUNCT
cana-5869	37	14	∀	∀	PUNCT
cana-5869	37	15	𝜂𝑖	𝜂𝑖	ADP
cana-5869	37	16	∈	∈	PROPN
cana-5869	37	17	𝜆	𝜆	DET
cana-5869	37	18	𝑖≥1	𝑖≥1	NOUN
cana-5869	37	19	}	}	PUNCT
cana-5869	37	20	𝜓𝛽	𝜓𝛽	ADP
cana-5869	37	21	=	=	PUNCT
cana-5869	37	22	{	{	PUNCT
cana-5869	37	23	𝜉	𝜉	X
cana-5869	37	24	:	:	PUNCT
cana-5869	37	25	𝜉	𝜉	PROPN
cana-5869	37	26	∈	∈	PROPN
cana-5869	37	27	𝜔′	𝜔′	NOUN
cana-5869	37	28	,	,	PUNCT
cana-5869	37	29	‖∑	‖∑	ADP
cana-5869	37	30	𝜉𝑖𝜂𝑖𝑖≥1	𝜉𝑖𝜂𝑖𝑖≥1	SYM
cana-5869	37	31	‖	‖	PROPN
cana-5869	37	32	<	<	X
cana-5869	37	33	∞	∞	PROPN
cana-5869	37	34	,	,	PUNCT
cana-5869	37	35	∀	∀	PUNCT
cana-5869	37	36	𝜂𝑖	𝜂𝑖	ADP
cana-5869	37	37	∈	∈	PROPN
cana-5869	37	38	𝜆	𝜆	X
cana-5869	37	39	}	}	PUNCT
cana-5869	37	40	𝜓𝛾	𝜓𝛾	NOUN
cana-5869	37	41	=	=	PUNCT
cana-5869	37	42	{	{	PUNCT
cana-5869	37	43	𝜉	𝜉	X
cana-5869	37	44	:	:	PUNCT
cana-5869	37	45	𝜉	𝜉	PROPN
cana-5869	37	46	∈	∈	PROPN
cana-5869	37	47	𝜔′	𝜔′	NOUN
cana-5869	37	48	,	,	PUNCT
cana-5869	37	49	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-5869	37	50	𝑛	𝑛	DET
cana-5869	37	51	‖∑	‖∑	ADV
cana-5869	37	52	𝜉𝑖𝜂𝑖	𝜉𝑖𝜂𝑖	NOUN
cana-5869	37	53	𝑛	𝑛	X
cana-5869	37	54	𝑖=1	𝑖=1	PUNCT
cana-5869	37	55	‖	‖	PROPN
cana-5869	37	56	<	<	X
cana-5869	37	57	∞	∞	PROPN
cana-5869	37	58	,	,	PUNCT
cana-5869	37	59	∀	∀	PUNCT
cana-5869	37	60	𝜂𝑖	𝜂𝑖	ADP
cana-5869	37	61	∈	∈	PROPN
cana-5869	37	62	𝜆	𝜆	X
cana-5869	37	63	}	}	PUNCT
cana-5869	37	64	𝜆𝛿	𝜆𝛿	NOUN
cana-5869	37	65	=	=	NOUN
cana-5869	37	66	{	{	PUNCT
cana-5869	37	67	𝜉	𝜉	X
cana-5869	37	68	:	:	PUNCT
cana-5869	37	69	𝜉	𝜉	PROPN
cana-5869	37	70	∈	∈	PROPN
cana-5869	37	71	𝜔′	𝜔′	NOUN
cana-5869	37	72	,	,	PUNCT
cana-5869	37	73	∑	∑	ADV
cana-5869	37	74	‖𝜉𝑖𝜂𝜌(𝑖)‖	‖𝜉𝑖𝜂𝜌(𝑖)‖	X
cana-5869	37	75	<	<	X
cana-5869	37	76	∞	∞	PROPN
cana-5869	37	77	,	,	PUNCT
cana-5869	38	1	∀𝜂𝑖	∀𝜂𝑖	PROPN
cana-5869	38	2	∈	∈	PROPN
cana-5869	38	3	𝜆𝑖≥1	𝜆𝑖≥1	NOUN
cana-5869	38	4	𝑎𝑛𝑑𝜌	𝑎𝑛𝑑𝜌	NOUN
cana-5869	38	5	∈	∈	PROPN
cana-5869	38	6	𝜋	𝜋	NOUN
cana-5869	38	7	}	}	PUNCT
cana-5869	38	8	where	where	SCONJ
cana-5869	38	9	𝜋	𝜋	NOUN
cana-5869	38	10	is	be	AUX
cana-5869	38	11	the	the	DET
cana-5869	38	12	set	set	NOUN
cana-5869	38	13	of	of	ADP
cana-5869	38	14	all	all	DET
cana-5869	38	15	permutations	permutation	NOUN
cana-5869	38	16	of	of	ADP
cana-5869	38	17	ℕ.	ℕ.	PROPN
cana-5869	38	18	communications	communication	NOUN
cana-5869	38	19	on	on	ADP
cana-5869	38	20	applied	apply	VERB
cana-5869	38	21	nonlinear	nonlinear	ADJ
cana-5869	38	22	analysis	analysis	NOUN
cana-5869	38	23	issn	issn	NOUN
cana-5869	38	24	:	:	PUNCT
cana-5869	38	25	1074	1074	NUM
cana-5869	38	26	-	-	PUNCT
cana-5869	38	27	133x	133x	NUM
cana-5869	38	28	vol	vol	NOUN
cana-5869	38	29	31	31	NUM
cana-5869	38	30	no	no	NOUN
cana-5869	38	31	.	.	PUNCT
cana-5869	39	1	2s	2s	NUM
cana-5869	39	2	(	(	PUNCT
cana-5869	39	3	2024	2024	NUM
cana-5869	39	4	)	)	PUNCT
cana-5869	39	5	756	756	NUM
cana-5869	39	6	https://internationalpubls.com	https://internationalpubls.com	SYM
cana-5869	39	7	4.1	4.1	NUM
cana-5869	39	8	bicomplex	bicomplex	NOUN
cana-5869	39	9	sequenc	sequenc	NOUN
cana-5869	39	10	spaces	space	NOUN
cana-5869	39	11	now	now	ADV
cana-5869	39	12	,	,	PUNCT
cana-5869	39	13	let	let	VERB
cana-5869	39	14	us	we	PRON
cana-5869	39	15	first	first	ADV
cana-5869	39	16	describe	describe	VERB
cana-5869	39	17	the	the	DET
cana-5869	39	18	classes	class	NOUN
cana-5869	39	19	of	of	ADP
cana-5869	39	20	sequences	sequence	NOUN
cana-5869	39	21	whose	whose	DET
cana-5869	39	22	duals	dual	NOUN
cana-5869	39	23	we	we	PRON
cana-5869	39	24	aim	aim	VERB
cana-5869	39	25	to	to	PART
cana-5869	39	26	study	study	VERB
cana-5869	39	27	:	:	PUNCT
cana-5869	39	28	(	(	PUNCT
cana-5869	39	29	i	i	NOUN
cana-5869	39	30	)	)	PUNCT
cana-5869	39	31	ℵ	ℵ	PROPN
cana-5869	40	1	=	=	X
cana-5869	40	2	{	{	PUNCT
cana-5869	40	3	𝒻	𝒻	NOUN
cana-5869	40	4	:	:	PUNCT
cana-5869	40	5	𝒻	𝒻	PROPN
cana-5869	40	6	=	=	PRON
cana-5869	40	7	{	{	PUNCT
cana-5869	40	8	𝒳𝒿	𝒳𝒿	PROPN
cana-5869	40	9	}	}	PUNCT
cana-5869	40	10	=	=	SYM
cana-5869	40	11	{	{	PUNCT
cana-5869	40	12	1𝒳𝒿.	1𝒳𝒿.	NOUN
cana-5869	40	13	𝑒1	𝑒1	NOUN
cana-5869	40	14	+	+	CCONJ
cana-5869	40	15	2𝒳𝒿.	2𝒳𝒿.	PROPN
cana-5869	40	16	𝑒2	𝑒2	NOUN
cana-5869	40	17	}	}	PUNCT
cana-5869	40	18	:	:	PUNCT
cana-5869	40	19	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-5869	40	20	𝒿≥1	𝒿≥1	PROPN
cana-5869	40	21	𝒿𝒿	𝒿𝒿	PRON
cana-5869	40	22	|1𝒳𝒿	|1𝒳𝒿	VERB
cana-5869	40	23	|	|	ADV
cana-5869	40	24	ℂ1	ℂ1	NOUN
cana-5869	40	25	<	<	X
cana-5869	40	26	∞	∞	PROPN
cana-5869	40	27	,	,	PUNCT
cana-5869	40	28	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-5869	40	29	𝒿≥1	𝒿≥1	PROPN
cana-5869	40	30	𝒿𝒿	𝒿𝒿	PRON
cana-5869	40	31	|2𝒳𝒿	|2𝒳𝒿	AUX
cana-5869	40	32	|	|	ADV
cana-5869	40	33	ℂ1	ℂ1	VERB
cana-5869	40	34	<	<	X
cana-5869	40	35	∞	∞	PROPN
cana-5869	40	36	}	}	PUNCT
cana-5869	40	37	(	(	PUNCT
cana-5869	40	38	ii	ii	NOUN
cana-5869	40	39	)	)	PUNCT
cana-5869	40	40	1ℵ	1ℵ	NOUN
cana-5869	40	41	=	=	SYM
cana-5869	40	42	{	{	PUNCT
cana-5869	40	43	𝒻	𝒻	NOUN
cana-5869	40	44	:	:	PUNCT
cana-5869	40	45	𝒻	𝒻	PROPN
cana-5869	40	46	=	=	PRON
cana-5869	40	47	{	{	PUNCT
cana-5869	40	48	1𝒳𝒿.	1𝒳𝒿.	PROPN
cana-5869	40	49	𝑒1	𝑒1	NOUN
cana-5869	40	50	}	}	PUNCT
cana-5869	40	51	:	:	PUNCT
cana-5869	40	52	𝑠𝑢𝑝	𝑠𝑢𝑝	VERB
cana-5869	40	53	𝒿≥1	𝒿≥1	PROPN
cana-5869	40	54	𝒿𝒿	𝒿𝒿	PRON
cana-5869	40	55	|1𝒳𝒿	|1𝒳𝒿	VERB
cana-5869	40	56	|	|	ADV
cana-5869	40	57	ℂ1	ℂ1	NOUN
cana-5869	40	58	<	<	X
cana-5869	40	59	∞	∞	PROPN
cana-5869	40	60	}	}	PUNCT
cana-5869	40	61	(	(	PUNCT
cana-5869	40	62	iii	iii	X
cana-5869	40	63	)	)	PUNCT
cana-5869	40	64	2ℵ	2ℵ	NOUN
cana-5869	40	65	=	=	PUNCT
cana-5869	40	66	{	{	PUNCT
cana-5869	40	67	𝒻	𝒻	NOUN
cana-5869	40	68	:	:	PUNCT
cana-5869	40	69	𝒻	𝒻	PROPN
cana-5869	40	70	=	=	PUNCT
cana-5869	40	71	{	{	PUNCT
cana-5869	40	72	2𝒳𝒿.	2𝒳𝒿.	PROPN
cana-5869	40	73	𝑒1	𝑒1	NOUN
cana-5869	40	74	}	}	PUNCT
cana-5869	40	75	:	:	PUNCT
cana-5869	40	76	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-5869	40	77	𝒿≥1	𝒿≥1	PROPN
cana-5869	40	78	𝒿𝒿	𝒿𝒿	PRON
cana-5869	40	79	|2𝒳𝒿	|2𝒳𝒿	AUX
cana-5869	40	80	|	|	ADV
cana-5869	40	81	ℂ1	ℂ1	VERB
cana-5869	40	82	<	<	X
cana-5869	40	83	∞	∞	PROPN
cana-5869	40	84	}	}	PUNCT
cana-5869	40	85	the	the	DET
cana-5869	40	86	class	class	NOUN
cana-5869	40	87	ℵ	ℵ	NOUN
cana-5869	40	88	given	give	VERB
cana-5869	40	89	in	in	ADP
cana-5869	40	90	(	(	PUNCT
cana-5869	40	91	i	i	NOUN
cana-5869	40	92	)	)	PUNCT
cana-5869	40	93	has	have	AUX
cana-5869	40	94	been	be	AUX
cana-5869	40	95	studied	study	VERB
cana-5869	40	96	in	in	ADP
cana-5869	40	97	[	[	X
cana-5869	40	98	8	8	NUM
cana-5869	40	99	]	]	PUNCT
cana-5869	40	100	by	by	ADP
cana-5869	40	101	srivastava	srivastava	PROPN
cana-5869	40	102	&	&	CCONJ
cana-5869	40	103	srivastava	srivastava	PROPN
cana-5869	40	104	and	and	CCONJ
cana-5869	40	105	the	the	DET
cana-5869	40	106	subspaces	subspace	NOUN
cana-5869	40	107	given	give	VERB
cana-5869	40	108	in	in	ADP
cana-5869	40	109	(	(	PUNCT
cana-5869	40	110	ii	ii	NOUN
cana-5869	40	111	)	)	PUNCT
cana-5869	40	112	and	and	CCONJ
cana-5869	40	113	(	(	PUNCT
cana-5869	40	114	iii	iii	X
cana-5869	40	115	)	)	PUNCT
cana-5869	40	116	have	have	AUX
cana-5869	40	117	been	be	AUX
cana-5869	40	118	studied	study	VERB
cana-5869	40	119	in	in	ADP
cana-5869	40	120	[	[	X
cana-5869	40	121	9	9	NUM
cana-5869	40	122	]	]	PUNCT
cana-5869	40	123	by	by	ADP
cana-5869	40	124	wagh	wagh	PROPN
cana-5869	40	125	.	.	PUNCT
cana-5869	41	1	the	the	DET
cana-5869	41	2	subclass	subclass	NOUN
cana-5869	41	3	in	in	ADP
cana-5869	41	4	(	(	PUNCT
cana-5869	41	5	ii	ii	NOUN
cana-5869	41	6	)	)	PUNCT
cana-5869	41	7	have	have	AUX
cana-5869	41	8	been	be	AUX
cana-5869	41	9	studied	study	VERB
cana-5869	41	10	with	with	ADP
cana-5869	41	11	a	a	DET
cana-5869	41	12	functional	functional	ADJ
cana-5869	41	13	analytic	analytic	ADJ
cana-5869	41	14	viewpoint	viewpoint	NOUN
cana-5869	41	15	in	in	ADP
cana-5869	41	16	[	[	X
cana-5869	41	17	11	11	NUM
cana-5869	41	18	]	]	PUNCT
cana-5869	41	19	by	by	ADP
cana-5869	41	20	wagh	wagh	PROPN
cana-5869	41	21	.	.	PUNCT
cana-5869	42	1	these	these	DET
cana-5869	42	2	subclasses	subclass	NOUN
cana-5869	42	3	are	be	AUX
cana-5869	42	4	the	the	DET
cana-5869	42	5	subspaces	subspace	NOUN
cana-5869	42	6	of	of	ADP
cana-5869	42	7	our	our	PRON
cana-5869	42	8	space	space	NOUN
cana-5869	42	9	ℵ	ℵ	NOUN
cana-5869	42	10	in	in	ADP
cana-5869	42	11	the	the	DET
cana-5869	42	12	sense	sense	NOUN
cana-5869	42	13	that	that	SCONJ
cana-5869	42	14	they	they	PRON
cana-5869	42	15	are	be	AUX
cana-5869	42	16	formed	form	VERB
cana-5869	42	17	by	by	ADP
cana-5869	42	18	idempotent	idempotent	ADJ
cana-5869	42	19	sequences	sequence	NOUN
cana-5869	42	20	of	of	ADP
cana-5869	42	21	ℵ.	ℵ.	PROPN
cana-5869	42	22	in	in	ADP
cana-5869	42	23	the	the	DET
cana-5869	42	24	above	above	ADJ
cana-5869	42	25	spaces	space	NOUN
cana-5869	42	26	,	,	PUNCT
cana-5869	42	27	the	the	DET
cana-5869	42	28	notation	notation	NOUN
cana-5869	42	29	|	|	NOUN
cana-5869	42	30	.	.	PUNCT
cana-5869	43	1	|ℂ1	|ℂ1	PUNCT
cana-5869	43	2	represents	represent	VERB
cana-5869	43	3	the	the	DET
cana-5869	43	4	complex	complex	ADJ
cana-5869	43	5	norm	norm	NOUN
cana-5869	43	6	.	.	PUNCT
cana-5869	44	1	for	for	ADP
cana-5869	44	2	any	any	DET
cana-5869	44	3	bicomplex	bicomplex	NOUN
cana-5869	44	4	sequence	sequence	NOUN
cana-5869	44	5	{	{	PUNCT
cana-5869	44	6	𝒳𝒿	𝒳𝒿	PROPN
cana-5869	44	7	}	}	PUNCT
cana-5869	44	8	=	=	SYM
cana-5869	44	9	{	{	PUNCT
cana-5869	44	10	1𝒳𝒿.	1𝒳𝒿.	NOUN
cana-5869	44	11	𝑒1	𝑒1	NOUN
cana-5869	44	12	+	+	CCONJ
cana-5869	44	13	2𝒳𝒿.	2𝒳𝒿.	PROPN
cana-5869	44	14	𝑒2	𝑒2	NOUN
cana-5869	44	15	}	}	PUNCT
cana-5869	44	16	,	,	PUNCT
cana-5869	44	17	the	the	DET
cana-5869	44	18	following	follow	VERB
cana-5869	44	19	two	two	NUM
cana-5869	44	20	conditions	condition	NOUN
cana-5869	44	21	(	(	PUNCT
cana-5869	44	22	iv	iv	X
cana-5869	44	23	)	)	PUNCT
cana-5869	44	24	and	and	CCONJ
cana-5869	44	25	(	(	PUNCT
cana-5869	44	26	v	v	NOUN
cana-5869	44	27	)	)	PUNCT
cana-5869	44	28	are	be	AUX
cana-5869	44	29	equivalent	equivalent	ADJ
cana-5869	44	30	:	:	PUNCT
cana-5869	44	31	(	(	PUNCT
cana-5869	44	32	iv	iv	X
cana-5869	44	33	)	)	PUNCT
cana-5869	44	34	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-5869	44	35	𝒿≥1	𝒿≥1	PROPN
cana-5869	44	36	𝒿𝒿	𝒿𝒿	PRON
cana-5869	44	37	|1𝒳𝒿	|1𝒳𝒿	VERB
cana-5869	44	38	|	|	ADV
cana-5869	44	39	ℂ1	ℂ1	NOUN
cana-5869	44	40	<	<	X
cana-5869	44	41	∞	∞	PROPN
cana-5869	44	42	and	and	CCONJ
cana-5869	44	43	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-5869	44	44	𝒿≥1	𝒿≥1	PROPN
cana-5869	44	45	𝒿𝒿	𝒿𝒿	PRON
cana-5869	44	46	|2𝒳𝒿	|2𝒳𝒿	AUX
cana-5869	44	47	|	|	ADV
cana-5869	44	48	ℂ1	ℂ1	VERB
cana-5869	44	49	<	<	X
cana-5869	44	50	∞	∞	PROPN
cana-5869	44	51	(	(	PUNCT
cana-5869	44	52	v	v	NOUN
cana-5869	44	53	)	)	PUNCT
cana-5869	44	54	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-5869	44	55	𝒿≥1	𝒿≥1	PROPN
cana-5869	44	56	𝒿𝒿‖𝒳𝒿‖	𝒿𝒿‖𝒳𝒿‖	PROPN
cana-5869	44	57	ℂ2	ℂ2	X
cana-5869	44	58	<	<	X
cana-5869	44	59	∞	∞	NUM
cana-5869	44	60	where	where	SCONJ
cana-5869	44	61	‖	‖	PROPN
cana-5869	44	62	‖ℂ2	‖ℂ2	PROPN
cana-5869	44	63	represents	represent	VERB
cana-5869	44	64	the	the	DET
cana-5869	44	65	bicomplex	bicomplex	NOUN
cana-5869	44	66	norm	norm	NOUN
cana-5869	44	67	,	,	PUNCT
cana-5869	44	68	given	give	VERB
cana-5869	44	69	by	by	ADP
cana-5869	44	70	‖𝜍‖ℂ2	‖𝜍‖ℂ2	PROPN
cana-5869	44	71	=	=	NOUN
cana-5869	44	72	{	{	PUNCT
cana-5869	44	73	|1𝜍|	|1𝜍|	PROPN
cana-5869	44	74	ℂ1	ℂ1	NOUN
cana-5869	44	75	2	2	NUM
cana-5869	45	1	+	+	NOUN
cana-5869	45	2	|2𝜍|	|2𝜍|	PROPN
cana-5869	45	3	ℂ1	ℂ1	NOUN
cana-5869	45	4	2	2	NUM
cana-5869	45	5	2	2	NUM
cana-5869	45	6	}	}	PUNCT
cana-5869	45	7	1/2	1/2	NUM
cana-5869	45	8	,	,	PUNCT
cana-5869	45	9	where	where	SCONJ
cana-5869	45	10	(	(	PUNCT
cana-5869	45	11	vi	vi	NOUN
cana-5869	45	12	)	)	PUNCT
cana-5869	45	13	𝜍	𝜍	X
cana-5869	45	14	=	=	PUNCT
cana-5869	45	15	𝑢1	𝑢1	PROPN
cana-5869	45	16	+	+	CCONJ
cana-5869	45	17	𝑖2𝑢2	𝑖2𝑢2	X
cana-5869	45	18	=	=	SYM
cana-5869	45	19	(	(	PUNCT
cana-5869	45	20	𝑢1	𝑢1	PROPN
cana-5869	45	21	−	−	PROPN
cana-5869	45	22	𝑖1𝑢2)𝑒1	𝑖1𝑢2)𝑒1	X
cana-5869	45	23	+	+	CCONJ
cana-5869	45	24	(	(	PUNCT
cana-5869	45	25	𝑢1	𝑢1	PROPN
cana-5869	45	26	+	+	CCONJ
cana-5869	45	27	𝑖1𝑢2)𝑒2	𝑖1𝑢2)𝑒2	NOUN
cana-5869	45	28	=	=	NOUN
cana-5869	45	29	1𝜍𝑒1	1𝜍𝑒1	NUM
cana-5869	45	30	+	+	CCONJ
cana-5869	45	31	2𝜍𝑒2	2𝜍𝑒2	NUM
cana-5869	45	32	∈	∈	PROPN
cana-5869	45	33	ℂ2	ℂ2	NOUN
cana-5869	45	34	,	,	PUNCT
cana-5869	45	35	1𝜍	1𝜍	NUM
cana-5869	45	36	,	,	PUNCT
cana-5869	45	37	2𝜍	2𝜍	PROPN
cana-5869	45	38	∈	∈	PROPN
cana-5869	45	39	ℂ1	ℂ1	NOUN
cana-5869	45	40	communications	communication	NOUN
cana-5869	45	41	on	on	ADP
cana-5869	45	42	applied	apply	VERB
cana-5869	45	43	nonlinear	nonlinear	ADJ
cana-5869	45	44	analysis	analysis	NOUN
cana-5869	45	45	issn	issn	NOUN
cana-5869	45	46	:	:	PUNCT
cana-5869	45	47	1074	1074	NUM
cana-5869	45	48	-	-	PUNCT
cana-5869	45	49	133x	133x	NUM
cana-5869	45	50	vol	vol	NOUN
cana-5869	45	51	31	31	NUM
cana-5869	45	52	no	no	NOUN
cana-5869	45	53	.	.	PUNCT
cana-5869	46	1	2s	2s	NUM
cana-5869	46	2	(	(	PUNCT
cana-5869	46	3	2024	2024	NUM
cana-5869	46	4	)	)	PUNCT
cana-5869	46	5	757	757	NUM
cana-5869	46	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5869	46	7	1𝜍	1𝜍	NUM
cana-5869	46	8	and	and	CCONJ
cana-5869	46	9	2𝜍	2𝜍	NUM
cana-5869	46	10	are	be	AUX
cana-5869	46	11	first	first	ADV
cana-5869	46	12	idempotent	idempotent	ADJ
cana-5869	46	13	component	component	NOUN
cana-5869	46	14	and	and	CCONJ
cana-5869	46	15	second	second	ADJ
cana-5869	46	16	idempotent	idempotent	ADJ
cana-5869	46	17	component	component	NOUN
cana-5869	46	18	of	of	ADP
cana-5869	46	19	𝜍	𝜍	PRON
cana-5869	46	20	respectively	respectively	ADV
cana-5869	46	21	.	.	PUNCT
cana-5869	47	1	the	the	DET
cana-5869	47	2	idempotent	idempotent	ADJ
cana-5869	47	3	representation	representation	NOUN
cana-5869	47	4	given	give	VERB
cana-5869	47	5	in	in	ADP
cana-5869	47	6	(	(	PUNCT
cana-5869	47	7	vi	vi	NOUN
cana-5869	47	8	)	)	PUNCT
cana-5869	47	9	is	be	AUX
cana-5869	47	10	unique	unique	ADJ
cana-5869	47	11	and	and	CCONJ
cana-5869	47	12	was	be	AUX
cana-5869	47	13	given	give	VERB
cana-5869	47	14	by	by	ADP
cana-5869	47	15	srivastava	srivastava	PROPN
cana-5869	47	16	in	in	ADP
cana-5869	47	17	[	[	X
cana-5869	47	18	7	7	NUM
cana-5869	47	19	]	]	PUNCT
cana-5869	47	20	.	.	PUNCT
cana-5869	48	1	thus	thus	ADV
cana-5869	48	2	,	,	PUNCT
cana-5869	48	3	the	the	DET
cana-5869	48	4	class	class	NOUN
cana-5869	48	5	in	in	ADP
cana-5869	48	6	(	(	PUNCT
cana-5869	48	7	i	i	NOUN
cana-5869	48	8	)	)	PUNCT
cana-5869	48	9	has	have	VERB
cana-5869	48	10	an	an	DET
cana-5869	48	11	equivalent	equivalent	ADJ
cana-5869	48	12	representation	representation	NOUN
cana-5869	48	13	given	give	VERB
cana-5869	48	14	by	by	ADP
cana-5869	48	15	ℵ	ℵ	NOUN
cana-5869	48	16	=	=	SYM
cana-5869	48	17	{	{	PUNCT
cana-5869	48	18	𝒻	𝒻	NOUN
cana-5869	48	19	:	:	PUNCT
cana-5869	48	20	𝒻	𝒻	PROPN
cana-5869	49	1	=	=	PRON
cana-5869	49	2	{	{	PUNCT
cana-5869	49	3	𝒳𝒿	𝒳𝒿	PROPN
cana-5869	49	4	}	}	PUNCT
cana-5869	49	5	:	:	PUNCT
cana-5869	49	6	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-5869	49	7	𝒿≥1	𝒿≥1	PROPN
cana-5869	49	8	𝒿𝒿‖𝒳𝒿‖	𝒿𝒿‖𝒳𝒿‖	PROPN
cana-5869	49	9	ℂ2	ℂ2	X
cana-5869	49	10	<	<	X
cana-5869	49	11	∞	∞	PROPN
cana-5869	49	12	}	}	PUNCT
cana-5869	49	13	(	(	PUNCT
cana-5869	49	14	𝜉𝑗	𝜉𝑗	NOUN
cana-5869	49	15	)	)	PUNCT
cana-5869	49	16	∈	∈	PROPN
cana-5869	49	17	ℵ	ℵ	NOUN
cana-5869	49	18	can	can	AUX
cana-5869	49	19	be	be	AUX
cana-5869	49	20	written	write	VERB
cana-5869	49	21	as	as	ADP
cana-5869	49	22	𝜉𝑗	𝜉𝑗	NOUN
cana-5869	49	23	=	=	PUNCT
cana-5869	49	24	𝑧1𝑗	𝑧1𝑗	PROPN
cana-5869	50	1	+	+	CCONJ
cana-5869	50	2	𝑖2𝑧2𝑗	𝑖2𝑧2𝑗	SYM
cana-5869	50	3	=	=	NOUN
cana-5869	50	4	𝛽1𝑗𝑒1	𝛽1𝑗𝑒1	NOUN
cana-5869	50	5	+	+	CCONJ
cana-5869	50	6	𝛽2𝑗𝑒2	𝛽2𝑗𝑒2	NOUN
cana-5869	50	7	,	,	PUNCT
cana-5869	50	8	where	where	SCONJ
cana-5869	50	9	,	,	PUNCT
cana-5869	50	10	𝛽1𝑗	𝛽1𝑗	PROPN
cana-5869	50	11	=	=	PUNCT
cana-5869	50	12	𝑧1𝑗	𝑧1𝑗	PROPN
cana-5869	50	13	−	−	NOUN
cana-5869	50	14	𝑖1𝑧2𝑗	𝑖1𝑧2𝑗	NUM
cana-5869	50	15	,	,	PUNCT
cana-5869	50	16	𝛽2𝑗	𝛽2𝑗	X
cana-5869	50	17	=	=	PUNCT
cana-5869	50	18	𝑧1𝑗	𝑧1𝑗	PROPN
cana-5869	50	19	+	+	CCONJ
cana-5869	50	20	𝑖1𝑧2𝑗	𝑖1𝑧2𝑗	NUM
cana-5869	50	21	(	(	PUNCT
cana-5869	50	22	𝛽1𝑗	𝛽1𝑗	PROPN
cana-5869	50	23	)	)	PUNCT
cana-5869	50	24	and	and	CCONJ
cana-5869	50	25	(	(	PUNCT
cana-5869	50	26	𝛽2𝑗	𝛽2𝑗	NOUN
cana-5869	50	27	)	)	PUNCT
cana-5869	50	28	are	be	AUX
cana-5869	50	29	complex	complex	ADJ
cana-5869	50	30	sequences	sequence	NOUN
cana-5869	50	31	i.e.	i.e.	X
cana-5869	50	32	,	,	PUNCT
cana-5869	50	33	𝛽1𝑗	𝛽1𝑗	PROPN
cana-5869	50	34	,	,	PUNCT
cana-5869	50	35	𝛽2𝑗	𝛽2𝑗	PROPN
cana-5869	50	36	∈	∈	PROPN
cana-5869	50	37	ℂ1(𝑖1	ℂ1(𝑖1	NOUN
cana-5869	50	38	)	)	PUNCT
cana-5869	50	39	.	.	PUNCT
cana-5869	51	1	4.2	4.2	NUM
cana-5869	51	2	algebra	algebra	NOUN
cana-5869	51	3	homomorphism	homomorphism	NOUN
cana-5869	51	4	between	between	ADP
cana-5869	51	5	ℵ	ℵ	NOUN
cana-5869	51	6	and	and	CCONJ
cana-5869	51	7	its	its	PRON
cana-5869	51	8	subclasses	subclass	NOUN
cana-5869	51	9	algebra	algebra	NOUN
cana-5869	51	10	homomorphism	homomorphism	NOUN
cana-5869	51	11	(	(	PUNCT
cana-5869	51	12	denoted	denote	VERB
cana-5869	51	13	by	by	ADP
cana-5869	51	14	t1	t1	NOUN
cana-5869	51	15	and	and	CCONJ
cana-5869	51	16	t2	t2	NOUN
cana-5869	51	17	)	)	PUNCT
cana-5869	51	18	between	between	ADP
cana-5869	51	19	ℵ	ℵ	NOUN
cana-5869	51	20	and	and	CCONJ
cana-5869	51	21	its	its	PRON
cana-5869	51	22	subclasses	subclass	NOUN
cana-5869	51	23	1ℵ	1ℵ	NOUN
cana-5869	51	24	and	and	CCONJ
cana-5869	51	25	2ℵ	2ℵ	NOUN
cana-5869	51	26	have	have	AUX
cana-5869	51	27	been	be	AUX
cana-5869	51	28	investigated	investigate	VERB
cana-5869	51	29	in	in	ADP
cana-5869	51	30	[	[	X
cana-5869	51	31	10	10	NUM
cana-5869	51	32	]	]	PUNCT
cana-5869	51	33	given	give	VERB
cana-5869	51	34	by	by	ADP
cana-5869	51	35	:	:	PUNCT
cana-5869	51	36	𝑇1	𝑇1	NOUN
cana-5869	51	37	:	:	PUNCT
cana-5869	51	38	ℵ	ℵ	PROPN
cana-5869	51	39	→	→	SYM
cana-5869	51	40	1ℵ	1ℵ	NOUN
cana-5869	51	41	as	as	ADP
cana-5869	51	42	𝑇1(𝑓	𝑇1(𝑓	NUM
cana-5869	51	43	)	)	PUNCT
cana-5869	51	44	=	=	PUNCT
cana-5869	52	1	𝑇1({𝜉𝑗	𝑇1({𝜉𝑗	NOUN
cana-5869	52	2	}	}	PUNCT
cana-5869	52	3	)	)	PUNCT
cana-5869	53	1	=	=	PRON
cana-5869	53	2	{	{	PUNCT
cana-5869	53	3	1𝜉𝑗	1𝜉𝑗	ADJ
cana-5869	53	4	𝑒1	𝑒1	NOUN
cana-5869	53	5	}	}	PUNCT
cana-5869	53	6	∈	∈	PROPN
cana-5869	53	7	1ℵ	1ℵ	NOUN
cana-5869	53	8	and	and	CCONJ
cana-5869	53	9	𝑇2	𝑇2	NOUN
cana-5869	53	10	:	:	PUNCT
cana-5869	53	11	ℵ	ℵ	X
cana-5869	53	12	→	→	SYM
cana-5869	53	13	2ℵ	2ℵ	NOUN
cana-5869	53	14	as	as	ADP
cana-5869	53	15	𝑇2(𝑓	𝑇2(𝑓	NOUN
cana-5869	53	16	)	)	PUNCT
cana-5869	53	17	=	=	PUNCT
cana-5869	53	18	𝑇2({𝜉𝑗	𝑇2({𝜉𝑗	NOUN
cana-5869	53	19	}	}	PUNCT
cana-5869	53	20	)	)	PUNCT
cana-5869	54	1	=	=	PRON
cana-5869	54	2	{	{	PUNCT
cana-5869	54	3	2𝜉𝑗	2𝜉𝑗	ADJ
cana-5869	54	4	𝑒2	𝑒2	NOUN
cana-5869	54	5	}	}	PUNCT
cana-5869	54	6	∈	∈	PROPN
cana-5869	54	7	2ℵ	2ℵ	NOUN
cana-5869	54	8	,	,	PUNCT
cana-5869	54	9	{	{	PUNCT
cana-5869	54	10	𝜉𝑘	𝜉𝑘	NOUN
cana-5869	54	11	}	}	PUNCT
cana-5869	54	12	∈	∈	PROPN
cana-5869	54	13	𝐵	𝐵	NOUN
cana-5869	54	14	its	its	PRON
cana-5869	54	15	pictorial	pictorial	ADJ
cana-5869	54	16	representation	representation	NOUN
cana-5869	54	17	is	be	AUX
cana-5869	54	18	given	give	VERB
cana-5869	54	19	below	below	ADP
cana-5869	54	20	thus	thus	ADV
cana-5869	54	21	we	we	PRON
cana-5869	54	22	can	can	AUX
cana-5869	54	23	say	say	VERB
cana-5869	54	24	that	that	PRON
cana-5869	54	25	(	(	PUNCT
cana-5869	54	26	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	54	27	)	)	PUNCT
cana-5869	54	28	∈	∈	NOUN
cana-5869	54	29	ℵ𝛼	ℵ𝛼	ADV
cana-5869	54	30	if	if	SCONJ
cana-5869	54	31	and	and	CCONJ
cana-5869	54	32	only	only	ADV
cana-5869	54	33	if	if	SCONJ
cana-5869	54	34	𝑇1(𝜂𝑗	𝑇1(𝜂𝑗	NUM
cana-5869	54	35	)	)	PUNCT
cana-5869	54	36	∈	∈	PROPN
cana-5869	54	37	(	(	PUNCT
cana-5869	54	38	1ℵ)𝛼	1ℵ)𝛼	NUM
cana-5869	54	39	,	,	PUNCT
cana-5869	54	40	𝑇2(𝜂𝑗	𝑇2(𝜂𝑗	PROPN
cana-5869	54	41	)	)	PUNCT
cana-5869	54	42	∈	∈	PROPN
cana-5869	54	43	(	(	PUNCT
cana-5869	54	44	2ℵ)𝛼	2ℵ)𝛼	NUM
cana-5869	54	45	t1	t1	NOUN
cana-5869	54	46	ℵ	ℵ	ADP
cana-5869	54	47	2ℵ	2ℵ	NUM
cana-5869	54	48	1ℵ	1ℵ	ADJ
cana-5869	54	49	communications	communication	NOUN
cana-5869	54	50	on	on	ADP
cana-5869	54	51	applied	apply	VERB
cana-5869	54	52	nonlinear	nonlinear	ADJ
cana-5869	54	53	analysis	analysis	NOUN
cana-5869	54	54	issn	issn	NOUN
cana-5869	54	55	:	:	PUNCT
cana-5869	54	56	1074	1074	NUM
cana-5869	54	57	-	-	PUNCT
cana-5869	54	58	133x	133x	NUM
cana-5869	54	59	vol	vol	NOUN
cana-5869	54	60	31	31	NUM
cana-5869	54	61	no	no	NOUN
cana-5869	54	62	.	.	PUNCT
cana-5869	55	1	2s	2s	NUM
cana-5869	55	2	(	(	PUNCT
cana-5869	55	3	2024	2024	NUM
cana-5869	55	4	)	)	PUNCT
cana-5869	55	5	758	758	NUM
cana-5869	55	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5869	55	7	or	or	CCONJ
cana-5869	55	8	in	in	ADP
cana-5869	55	9	words	word	NOUN
cana-5869	55	10	we	we	PRON
cana-5869	55	11	can	can	AUX
cana-5869	55	12	also	also	ADV
cana-5869	55	13	say	say	VERB
cana-5869	55	14	that	that	SCONJ
cana-5869	55	15	a	a	DET
cana-5869	55	16	sequence	sequence	NOUN
cana-5869	55	17	belongs	belong	VERB
cana-5869	55	18	to	to	ADP
cana-5869	55	19	𝛼	𝛼	PRON
cana-5869	55	20	dual	dual	ADJ
cana-5869	55	21	of	of	ADP
cana-5869	55	22	the	the	DET
cana-5869	55	23	class	class	NOUN
cana-5869	55	24	ℵ	ℵ	NOUN
cana-5869	55	25	if	if	SCONJ
cana-5869	55	26	and	and	CCONJ
cana-5869	55	27	only	only	ADV
cana-5869	55	28	if	if	SCONJ
cana-5869	55	29	its	its	PRON
cana-5869	55	30	t1	t1	NOUN
cana-5869	55	31	–	–	PUNCT
cana-5869	55	32	image	image	NOUN
cana-5869	55	33	belongs	belong	VERB
cana-5869	55	34	to	to	ADP
cana-5869	55	35	𝛼	𝛼	PRON
cana-5869	55	36	dual	dual	ADJ
cana-5869	55	37	of	of	ADP
cana-5869	55	38	the	the	DET
cana-5869	55	39	first	first	ADJ
cana-5869	55	40	idempotent	idempotent	ADJ
cana-5869	55	41	component	component	NOUN
cana-5869	55	42	of	of	ADP
cana-5869	55	43	ℵ	ℵ	NOUN
cana-5869	55	44	or	or	CCONJ
cana-5869	55	45	its	its	PRON
cana-5869	55	46	first	first	ADJ
cana-5869	55	47	subclass	subclass	NOUN
cana-5869	55	48	and	and	CCONJ
cana-5869	55	49	the	the	DET
cana-5869	55	50	t2	t2	NOUN
cana-5869	55	51	–	–	PUNCT
cana-5869	55	52	image	image	NOUN
cana-5869	55	53	belongs	belong	VERB
cana-5869	55	54	to	to	ADP
cana-5869	55	55	the	the	DET
cana-5869	55	56	𝛼	𝛼	NOUN
cana-5869	55	57	dual	dual	ADJ
cana-5869	55	58	of	of	ADP
cana-5869	55	59	its	its	PRON
cana-5869	55	60	second	second	ADJ
cana-5869	55	61	idempotent	idempotent	ADJ
cana-5869	55	62	component	component	NOUN
cana-5869	55	63	or	or	CCONJ
cana-5869	55	64	its	its	PRON
cana-5869	55	65	second	second	ADJ
cana-5869	55	66	subclass	subclass	NOUN
cana-5869	55	67	.	.	PUNCT
cana-5869	56	1	this	this	PRON
cana-5869	56	2	is	be	AUX
cana-5869	56	3	to	to	PART
cana-5869	56	4	be	be	AUX
cana-5869	56	5	noted	note	VERB
cana-5869	56	6	that	that	SCONJ
cana-5869	56	7	the	the	DET
cana-5869	56	8	𝛼	𝛼	NOUN
cana-5869	56	9	–	–	PUNCT
cana-5869	56	10	dual	dual	ADJ
cana-5869	56	11	of	of	ADP
cana-5869	56	12	the	the	DET
cana-5869	56	13	class	class	NOUN
cana-5869	56	14	ℵ	ℵ	NOUN
cana-5869	56	15	has	have	AUX
cana-5869	56	16	been	be	AUX
cana-5869	56	17	studied	study	VERB
cana-5869	56	18	in	in	ADP
cana-5869	56	19	[	[	X
cana-5869	56	20	12	12	NUM
cana-5869	56	21	]	]	PUNCT
cana-5869	56	22	.	.	PUNCT
cana-5869	57	1	in	in	ADP
cana-5869	57	2	this	this	DET
cana-5869	57	3	paper	paper	NOUN
cana-5869	57	4	,	,	PUNCT
cana-5869	57	5	we	we	PRON
cana-5869	57	6	are	be	AUX
cana-5869	57	7	going	go	VERB
cana-5869	57	8	to	to	PART
cana-5869	57	9	analyze	analyze	VERB
cana-5869	57	10	other	other	ADJ
cana-5869	57	11	duals	dual	NOUN
cana-5869	57	12	of	of	ADP
cana-5869	57	13	these	these	DET
cana-5869	57	14	spaces	space	NOUN
cana-5869	57	15	.	.	PUNCT
cana-5869	58	1	β	β	X
cana-5869	58	2	–	–	PUNCT
cana-5869	58	3	dual	dual	ADJ
cana-5869	58	4	of	of	ADP
cana-5869	58	5	the	the	DET
cana-5869	58	6	class	class	NOUN
cana-5869	58	7	ℵ	ℵ	NOUN
cana-5869	58	8	and	and	CCONJ
cana-5869	58	9	its	its	PRON
cana-5869	58	10	subclasses	subclass	NOUN
cana-5869	58	11	1ℵ	1ℵ	NOUN
cana-5869	58	12	and	and	CCONJ
cana-5869	58	13	2ℵ	2ℵ	NOUN
cana-5869	58	14	are	be	AUX
cana-5869	58	15	given	give	VERB
cana-5869	58	16	by	by	ADP
cana-5869	58	17	ℵ𝛽	ℵ𝛽	PROPN
cana-5869	58	18	=	=	SYM
cana-5869	58	19	{	{	PUNCT
cana-5869	58	20	(	(	PUNCT
cana-5869	58	21	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	58	22	)	)	PUNCT
cana-5869	58	23	∈	∈	NOUN
cana-5869	58	24	𝜔′	𝜔′	NUM
cana-5869	58	25	:	:	PUNCT
cana-5869	58	26	‖∑	‖∑	ADP
cana-5869	58	27	𝜂𝑗𝜉𝑗	𝜂𝑗𝜉𝑗	NOUN
cana-5869	58	28	𝑗≥1	𝑗≥1	NOUN
cana-5869	58	29	‖	‖	X
cana-5869	58	30	<	<	X
cana-5869	58	31	∞	∞	PROPN
cana-5869	58	32	,	,	PUNCT
cana-5869	58	33	∀(𝜉𝑗	∀(𝜉𝑗	X
cana-5869	58	34	)	)	PUNCT
cana-5869	58	35	∈	∈	PROPN
cana-5869	58	36	ℵ	ℵ	NOUN
cana-5869	58	37	}	}	PUNCT
cana-5869	58	38	or	or	CCONJ
cana-5869	58	39	{	{	PUNCT
cana-5869	58	40	(	(	PUNCT
cana-5869	58	41	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	58	42	)	)	PUNCT
cana-5869	58	43	∈	∈	NOUN
cana-5869	58	44	𝜔′	𝜔′	NUM
cana-5869	58	45	:	:	PUNCT
cana-5869	58	46	|∑	|∑	NUM
cana-5869	58	47	1𝜂𝑗	1𝜂𝑗	ADJ
cana-5869	58	48	1𝜉𝑗	1𝜉𝑗	ADJ
cana-5869	58	49	𝑗≥1	𝑗≥1	NOUN
cana-5869	59	1	|	|	ADV
cana-5869	59	2	<	<	X
cana-5869	59	3	∞	∞	PROPN
cana-5869	59	4	,	,	PUNCT
cana-5869	59	5	|∑	|∑	VERB
cana-5869	59	6	2𝜂𝑗	2𝜂𝑗	ADJ
cana-5869	59	7	2𝜉𝑗	2𝜉𝑗	ADJ
cana-5869	59	8	𝑗≥1	𝑗≥1	NOUN
cana-5869	59	9	|	|	ADV
cana-5869	59	10	<	<	X
cana-5869	59	11	∞	∞	PROPN
cana-5869	59	12	,	,	PUNCT
cana-5869	59	13	∀(𝜉𝑗	∀(𝜉𝑗	X
cana-5869	59	14	)	)	PUNCT
cana-5869	59	15	∈	∈	PROPN
cana-5869	59	16	ℵ	ℵ	NOUN
cana-5869	59	17	}	}	PUNCT
cana-5869	59	18	(	(	PUNCT
cana-5869	59	19	1ℵ)𝛽	1ℵ)𝛽	NUM
cana-5869	59	20	=	=	SYM
cana-5869	59	21	{	{	PUNCT
cana-5869	59	22	(	(	PUNCT
cana-5869	59	23	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	59	24	)	)	PUNCT
cana-5869	59	25	∈	∈	NOUN
cana-5869	59	26	𝜔′	𝜔′	NUM
cana-5869	59	27	:	:	PUNCT
cana-5869	59	28	|∑	|∑	NUM
cana-5869	59	29	1𝜂𝑗	1𝜂𝑗	ADJ
cana-5869	59	30	1𝜉𝑗	1𝜉𝑗	ADJ
cana-5869	59	31	𝑗≥1	𝑗≥1	NOUN
cana-5869	59	32	|	|	ADV
cana-5869	59	33	<	<	X
cana-5869	59	34	∞	∞	PROPN
cana-5869	59	35	,	,	PUNCT
cana-5869	59	36	∀(𝜉𝑗	∀(𝜉𝑗	X
cana-5869	59	37	)	)	PUNCT
cana-5869	59	38	∈	∈	PROPN
cana-5869	59	39	ℵ	ℵ	PROPN
cana-5869	59	40	}	}	PUNCT
cana-5869	59	41	(	(	PUNCT
cana-5869	59	42	2ℵ)𝛽	2ℵ)𝛽	NUM
cana-5869	59	43	=	=	SYM
cana-5869	59	44	{	{	PUNCT
cana-5869	59	45	(	(	PUNCT
cana-5869	59	46	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	59	47	)	)	PUNCT
cana-5869	59	48	∈	∈	NOUN
cana-5869	59	49	𝜔′	𝜔′	NUM
cana-5869	59	50	:	:	PUNCT
cana-5869	59	51	|∑	|∑	PROPN
cana-5869	59	52	2𝜂𝑗	2𝜂𝑗	ADJ
cana-5869	59	53	2𝜉𝑗	2𝜉𝑗	ADJ
cana-5869	59	54	𝑗≥1	𝑗≥1	NOUN
cana-5869	59	55	|	|	ADV
cana-5869	59	56	<	<	X
cana-5869	59	57	∞	∞	PROPN
cana-5869	59	58	,	,	PUNCT
cana-5869	59	59	∀(𝜉𝑗	∀(𝜉𝑗	X
cana-5869	59	60	)	)	PUNCT
cana-5869	59	61	∈	∈	PROPN
cana-5869	59	62	ℵ	ℵ	NOUN
cana-5869	59	63	}	}	PUNCT
cana-5869	59	64	theorem	theorem	VERB
cana-5869	59	65	4.1	4.1	NUM
cana-5869	59	66	:	:	PUNCT
cana-5869	59	67	(	(	PUNCT
cana-5869	59	68	i	i	NOUN
cana-5869	59	69	)	)	PUNCT
cana-5869	59	70	ℵ𝛽	ℵ𝛽	PROPN
cana-5869	60	1	⊂	⊂	PROPN
cana-5869	60	2	(	(	PUNCT
cana-5869	60	3	1ℵ)𝛽.	1ℵ)𝛽.	NUM
cana-5869	60	4	(	(	PUNCT
cana-5869	60	5	ii	ii	NOUN
cana-5869	60	6	)	)	PUNCT
cana-5869	60	7	ℵ𝛽	ℵ𝛽	PROPN
cana-5869	61	1	⊂	⊂	PROPN
cana-5869	61	2	(	(	PUNCT
cana-5869	61	3	2ℵ)𝛽	2ℵ)𝛽	NUM
cana-5869	61	4	proof	proof	NOUN
cana-5869	61	5	:	:	PUNCT
cana-5869	61	6	(	(	PUNCT
cana-5869	61	7	i	i	NOUN
cana-5869	61	8	)	)	PUNCT
cana-5869	61	9	we	we	PRON
cana-5869	61	10	know	know	VERB
cana-5869	61	11	1ℵ	1ℵ	ADJ
cana-5869	61	12	⊆	⊆	NUM
cana-5869	61	13	ℵ	ℵ	NOUN
cana-5869	61	14	and	and	CCONJ
cana-5869	61	15	ℵ𝛽	ℵ𝛽	NOUN
cana-5869	61	16	⊆	⊆	NUM
cana-5869	61	17	(	(	PUNCT
cana-5869	61	18	1ℵ)𝛽	1ℵ)𝛽	NUM
cana-5869	61	19	(	(	PUNCT
cana-5869	61	20	since	since	SCONJ
cana-5869	61	21	dual	dual	ADJ
cana-5869	61	22	of	of	ADP
cana-5869	61	23	a	a	DET
cana-5869	61	24	set	set	NOUN
cana-5869	61	25	is	be	AUX
cana-5869	61	26	contained	contain	VERB
cana-5869	61	27	in	in	ADP
cana-5869	61	28	the	the	DET
cana-5869	61	29	dual	dual	ADJ
cana-5869	61	30	of	of	ADP
cana-5869	61	31	its	its	PRON
cana-5869	61	32	subset	subset	NOUN
cana-5869	61	33	)	)	PUNCT
cana-5869	61	34	to	to	PART
cana-5869	61	35	establish	establish	VERB
cana-5869	61	36	that	that	SCONJ
cana-5869	61	37	the	the	DET
cana-5869	61	38	inclusion	inclusion	NOUN
cana-5869	61	39	is	be	AUX
cana-5869	61	40	proper	proper	ADJ
cana-5869	61	41	,	,	PUNCT
cana-5869	61	42	we	we	PRON
cana-5869	61	43	construct	construct	VERB
cana-5869	61	44	a	a	DET
cana-5869	61	45	sequence	sequence	NOUN
cana-5869	61	46	contained	contain	VERB
cana-5869	61	47	in	in	ADP
cana-5869	61	48	the	the	DET
cana-5869	61	49	β	β	X
cana-5869	61	50	–	–	PUNCT
cana-5869	61	51	dual	dual	ADJ
cana-5869	61	52	of	of	ADP
cana-5869	61	53	1ℵ	1ℵ	NOUN
cana-5869	61	54	which	which	PRON
cana-5869	61	55	does	do	AUX
cana-5869	61	56	not	not	PART
cana-5869	61	57	belong	belong	VERB
cana-5869	61	58	to	to	ADP
cana-5869	61	59	β	β	X
cana-5869	61	60	–	–	PUNCT
cana-5869	61	61	dual	dual	ADJ
cana-5869	61	62	of	of	ADP
cana-5869	61	63	ℵ.	ℵ.	PROPN
cana-5869	61	64	communications	communication	NOUN
cana-5869	61	65	on	on	ADP
cana-5869	61	66	applied	apply	VERB
cana-5869	61	67	nonlinear	nonlinear	ADJ
cana-5869	61	68	analysis	analysis	NOUN
cana-5869	61	69	issn	issn	NOUN
cana-5869	61	70	:	:	PUNCT
cana-5869	61	71	1074	1074	NUM
cana-5869	61	72	-	-	PUNCT
cana-5869	61	73	133x	133x	NUM
cana-5869	61	74	vol	vol	NOUN
cana-5869	61	75	31	31	NUM
cana-5869	61	76	no	no	NOUN
cana-5869	61	77	.	.	PUNCT
cana-5869	62	1	2s	2s	NUM
cana-5869	62	2	(	(	PUNCT
cana-5869	62	3	2024	2024	NUM
cana-5869	62	4	)	)	PUNCT
cana-5869	62	5	759	759	NUM
cana-5869	62	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5869	62	7	consider	consider	VERB
cana-5869	62	8	the	the	DET
cana-5869	62	9	sequence	sequence	NOUN
cana-5869	62	10	,	,	PUNCT
cana-5869	62	11	(	(	PUNCT
cana-5869	62	12	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	62	13	)	)	PUNCT
cana-5869	62	14	=	=	SYM
cana-5869	62	15	(	(	PUNCT
cana-5869	62	16	1	1	NUM
cana-5869	62	17	𝑗3	𝑗3	NOUN
cana-5869	62	18	𝑒1	𝑒1	NOUN
cana-5869	62	19	+	+	CCONJ
cana-5869	62	20	𝑗3+𝑗𝑒2	𝑗3+𝑗𝑒2	NUM
cana-5869	62	21	)	)	PUNCT
cana-5869	62	22	.	.	PUNCT
cana-5869	63	1	first	first	ADV
cana-5869	63	2	we	we	PRON
cana-5869	63	3	show	show	VERB
cana-5869	63	4	that(𝜂𝑗	that(𝜂𝑗	NOUN
cana-5869	63	5	)	)	PUNCT
cana-5869	63	6	∈	∈	NOUN
cana-5869	63	7	(	(	PUNCT
cana-5869	63	8	1ℵ)𝛽.	1ℵ)𝛽.	NUM
cana-5869	63	9	let	let	VERB
cana-5869	63	10	(	(	PUNCT
cana-5869	63	11	1𝜉𝑗	1𝜉𝑗	ADJ
cana-5869	63	12	𝑒1	𝑒1	NOUN
cana-5869	63	13	)	)	PUNCT
cana-5869	63	14	∈	∈	PROPN
cana-5869	63	15	1ℵ	1ℵ	ADJ
cana-5869	63	16	⇒	⇒	NOUN
cana-5869	63	17	𝑠𝑢𝑝	𝑠𝑢𝑝	VERB
cana-5869	63	18	𝑗	𝑗	INTJ
cana-5869	63	19	𝑗𝑗	𝑗𝑗	ADP
cana-5869	63	20	|1𝜉𝑗	|1𝜉𝑗	NOUN
cana-5869	63	21	|	|	ADV
cana-5869	63	22	<	<	X
cana-5869	63	23	∞	∞	PROPN
cana-5869	63	24	⇒	⇒	NOUN
cana-5869	63	25	𝑗𝑗	𝑗𝑗	ADP
cana-5869	63	26	|1𝜉𝑗	|1𝜉𝑗	PROPN
cana-5869	63	27	|	|	ADV
cana-5869	63	28	<	<	X
cana-5869	63	29	𝑀	𝑀	PROPN
cana-5869	63	30	,	,	PUNCT
cana-5869	63	31	∀𝑗	∀𝑗	NOUN
cana-5869	63	32	≥	≥	NOUN
cana-5869	63	33	1	1	NUM
cana-5869	63	34	for	for	ADP
cana-5869	63	35	some	some	DET
cana-5869	63	36	m.	m.	NOUN
cana-5869	63	37	⇒	⇒	PROPN
cana-5869	63	38	|1𝜉𝑗	|1𝜉𝑗	PROPN
cana-5869	64	1	|	|	ADV
cana-5869	64	2	<	<	X
cana-5869	64	3	𝑀	𝑀	PROPN
cana-5869	64	4	𝑗𝑗	𝑗𝑗	NOUN
cana-5869	64	5	,	,	PUNCT
cana-5869	64	6	∀𝑗	∀𝑗	NOUN
cana-5869	64	7	≥	≥	NOUN
cana-5869	64	8	1	1	NUM
cana-5869	64	9	(	(	PUNCT
cana-5869	64	10	i	i	NOUN
cana-5869	64	11	)	)	PUNCT
cana-5869	64	12	𝑗𝑗	𝑗𝑗	ADP
cana-5869	64	13	>	>	X
cana-5869	64	14	𝑗3	𝑗3	PROPN
cana-5869	64	15	,	,	PUNCT
cana-5869	64	16	∀𝑗	∀𝑗	NOUN
cana-5869	64	17	≥	≥	NOUN
cana-5869	64	18	3	3	NUM
cana-5869	64	19	⇒	⇒	NOUN
cana-5869	64	20	1	1	NUM
cana-5869	64	21	𝑗𝑗	𝑗𝑗	ADP
cana-5869	64	22	<	<	X
cana-5869	64	23	1	1	NUM
cana-5869	64	24	𝑗3	𝑗3	NOUN
cana-5869	64	25	,	,	PUNCT
cana-5869	64	26	∀𝑗	∀𝑗	NOUN
cana-5869	64	27	≥	≥	NOUN
cana-5869	64	28	3	3	NUM
cana-5869	64	29	and	and	CCONJ
cana-5869	64	30	∑	∑	PROPN
cana-5869	64	31	1	1	NUM
cana-5869	64	32	𝑗3	𝑗3	PROPN
cana-5869	64	33	is	be	AUX
cana-5869	64	34	convergent	convergent	ADJ
cana-5869	64	35	⇒	⇒	NOUN
cana-5869	64	36	∑	∑	PROPN
cana-5869	64	37	1	1	NUM
cana-5869	64	38	𝑗𝑗	𝑗𝑗	NOUN
cana-5869	64	39	is	be	AUX
cana-5869	64	40	convergent	convergent	ADJ
cana-5869	64	41	,	,	PUNCT
cana-5869	64	42	by	by	ADP
cana-5869	64	43	comparison	comparison	NOUN
cana-5869	64	44	test	test	NOUN
cana-5869	64	45	.	.	PUNCT
cana-5869	65	1	∑	∑	PUNCT
cana-5869	65	2	𝑀	𝑀	PROPN
cana-5869	65	3	𝑗𝑗	𝑗𝑗	NOUN
cana-5869	65	4	is	be	AUX
cana-5869	65	5	a	a	DET
cana-5869	65	6	convergent	convergent	NOUN
cana-5869	65	7	series	series	NOUN
cana-5869	65	8	.	.	PUNCT
cana-5869	66	1	(	(	PUNCT
cana-5869	66	2	ii	ii	NOUN
cana-5869	66	3	)	)	PUNCT
cana-5869	66	4	∴	∴	NOUN
cana-5869	66	5	from	from	ADP
cana-5869	66	6	(	(	PUNCT
cana-5869	66	7	i	i	NOUN
cana-5869	66	8	)	)	PUNCT
cana-5869	66	9	and	and	CCONJ
cana-5869	66	10	(	(	PUNCT
cana-5869	66	11	ii	ii	NOUN
cana-5869	66	12	)	)	PUNCT
cana-5869	66	13	∑	∑	PUNCT
cana-5869	66	14	|1𝜉𝑗	|1𝜉𝑗	NOUN
cana-5869	66	15	|is	|is	DET
cana-5869	66	16	convergent	convergent	NOUN
cana-5869	66	17	.	.	PUNCT
cana-5869	67	1	and	and	CCONJ
cana-5869	67	2	|∑	|∑	VERB
cana-5869	67	3	1𝜉𝑗	1𝜉𝑗	NOUN
cana-5869	67	4	|	|	ADV
cana-5869	67	5	≤	≤	NUM
cana-5869	67	6	∑	∑	PUNCT
cana-5869	67	7	|1𝜉𝑗	|1𝜉𝑗	PROPN
cana-5869	67	8	|	|	ADV
cana-5869	67	9	<	<	X
cana-5869	67	10	∞	∞	PROPN
cana-5869	67	11	in	in	ADP
cana-5869	67	12	(	(	PUNCT
cana-5869	67	13	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	67	14	)	)	PUNCT
cana-5869	67	15	=	=	SYM
cana-5869	67	16	(	(	PUNCT
cana-5869	67	17	1	1	NUM
cana-5869	67	18	𝑗3	𝑗3	NOUN
cana-5869	67	19	𝑒1	𝑒1	NOUN
cana-5869	67	20	+	+	CCONJ
cana-5869	67	21	𝑗3+𝑗𝑒2	𝑗3+𝑗𝑒2	NUM
cana-5869	67	22	)	)	PUNCT
cana-5869	67	23	,	,	PUNCT
cana-5869	67	24	|∑	|∑	X
cana-5869	67	25	1𝜂𝑗	1𝜂𝑗	NOUN
cana-5869	68	1	|	|	ADV
cana-5869	68	2	≤	≤	NUM
cana-5869	68	3	∑	∑	PUNCT
cana-5869	68	4	|1𝜂𝑗	|1𝜂𝑗	VERB
cana-5869	68	5	|	|	NOUN
cana-5869	68	6	=	=	SYM
cana-5869	68	7	∑	∑	PROPN
cana-5869	68	8	1	1	NUM
cana-5869	68	9	𝑗3	𝑗3	PROPN
cana-5869	68	10	<	<	X
cana-5869	68	11	∞	∞	PROPN
cana-5869	68	12	and|∑	and|∑	NOUN
cana-5869	68	13	1𝜉𝑗	1𝜉𝑗	NOUN
cana-5869	69	1	|	|	ADV
cana-5869	69	2	<	<	X
cana-5869	69	3	∞	∞	PROPN
cana-5869	69	4	,	,	PUNCT
cana-5869	69	5	the	the	DET
cana-5869	69	6	term	term	NOUN
cana-5869	69	7	-	-	PUNCT
cana-5869	69	8	by	by	ADP
cana-5869	69	9	-	-	PUNCT
cana-5869	69	10	term	term	NOUN
cana-5869	69	11	product	product	NOUN
cana-5869	69	12	of	of	ADP
cana-5869	69	13	two	two	NUM
cana-5869	69	14	convergent	convergent	NOUN
cana-5869	69	15	series	series	NOUN
cana-5869	69	16	is	be	AUX
cana-5869	69	17	itself	itself	PRON
cana-5869	69	18	convergent	convergent	NOUN
cana-5869	69	19	,	,	PUNCT
cana-5869	69	20	therefore	therefore	ADV
cana-5869	69	21	|∑	|∑	X
cana-5869	69	22	1	1	NUM
cana-5869	69	23	𝑗3	𝑗3	PROPN
cana-5869	69	24	1	1	NUM
cana-5869	69	25	𝜉𝑗|	𝜉𝑗|	NOUN
cana-5869	69	26	<	<	X
cana-5869	69	27	∞	∞	PROPN
cana-5869	69	28	,	,	PUNCT
cana-5869	69	29	∀	∀	X
cana-5869	69	30	(	(	PUNCT
cana-5869	69	31	1𝜉𝑗	1𝜉𝑗	ADJ
cana-5869	69	32	𝑒1	𝑒1	NOUN
cana-5869	69	33	)	)	PUNCT
cana-5869	69	34	∈	∈	PROPN
cana-5869	69	35	1ℵ.	1ℵ.	NUM
cana-5869	69	36	thus(𝜂𝑗	thus(𝜂𝑗	NOUN
cana-5869	69	37	)	)	PUNCT
cana-5869	69	38	∈	∈	PROPN
cana-5869	69	39	(	(	PUNCT
cana-5869	69	40	1ℵ)𝛽.	1ℵ)𝛽.	NUM
cana-5869	69	41	(	(	PUNCT
cana-5869	69	42	a	a	NOUN
cana-5869	69	43	)	)	PUNCT
cana-5869	69	44	to	to	PART
cana-5869	69	45	show	show	VERB
cana-5869	69	46	that(𝜂𝑗	that(𝜂𝑗	PROPN
cana-5869	69	47	)	)	PUNCT
cana-5869	69	48	∉	∉	PROPN
cana-5869	69	49	ℵ𝛽	ℵ𝛽	PROPN
cana-5869	69	50	,	,	PUNCT
cana-5869	69	51	we	we	PRON
cana-5869	69	52	must	must	AUX
cana-5869	69	53	show	show	VERB
cana-5869	69	54	that	that	SCONJ
cana-5869	69	55	for	for	ADP
cana-5869	69	56	some	some	DET
cana-5869	69	57	element(𝜉𝑗	element(𝜉𝑗	PROPN
cana-5869	69	58	)	)	PUNCT
cana-5869	69	59	∈	∈	PROPN
cana-5869	69	60	ℵ	ℵ	NOUN
cana-5869	69	61	,	,	PUNCT
cana-5869	69	62	∑	∑	PUNCT
cana-5869	69	63	𝜉𝑗𝜂𝑗	𝜉𝑗𝜂𝑗	VERB
cana-5869	69	64	is	be	AUX
cana-5869	69	65	not	not	PART
cana-5869	69	66	convergent	convergent	ADJ
cana-5869	69	67	.	.	PUNCT
cana-5869	70	1	consider	consider	VERB
cana-5869	70	2	the	the	DET
cana-5869	70	3	sequence(𝜉𝑗	sequence(𝜉𝑗	NOUN
cana-5869	70	4	)	)	PUNCT
cana-5869	71	1	=	=	SYM
cana-5869	71	2	𝑗−𝑗	𝑗−𝑗	PROPN
cana-5869	71	3	(	(	PUNCT
cana-5869	71	4	1	1	NUM
cana-5869	71	5	𝑗2	𝑗2	NOUN
cana-5869	71	6	𝑒1	𝑒1	NOUN
cana-5869	71	7	+	+	CCONJ
cana-5869	71	8	1	1	NUM
cana-5869	71	9	𝑗3	𝑗3	PROPN
cana-5869	71	10	𝑒2	𝑒2	PROPN
cana-5869	71	11	)	)	PUNCT
cana-5869	71	12	=	=	PUNCT
cana-5869	72	1	(	(	PUNCT
cana-5869	72	2	𝑗−𝑗−2𝑒1	𝑗−𝑗−2𝑒1	NOUN
cana-5869	72	3	+	+	CCONJ
cana-5869	72	4	𝑗−𝑗−3𝑒2	𝑗−𝑗−3𝑒2	NOUN
cana-5869	72	5	)	)	PUNCT
cana-5869	72	6	.	.	PUNCT
cana-5869	73	1	note	note	VERB
cana-5869	73	2	first	first	ADV
cana-5869	73	3	that	that	SCONJ
cana-5869	73	4	,	,	PUNCT
cana-5869	73	5	communications	communication	NOUN
cana-5869	73	6	on	on	ADP
cana-5869	73	7	applied	apply	VERB
cana-5869	73	8	nonlinear	nonlinear	ADJ
cana-5869	73	9	analysis	analysis	NOUN
cana-5869	73	10	issn	issn	NOUN
cana-5869	73	11	:	:	PUNCT
cana-5869	73	12	1074	1074	NUM
cana-5869	73	13	-	-	PUNCT
cana-5869	73	14	133x	133x	NUM
cana-5869	73	15	vol	vol	NOUN
cana-5869	73	16	31	31	NUM
cana-5869	73	17	no	no	NOUN
cana-5869	73	18	.	.	PUNCT
cana-5869	74	1	2s	2s	NUM
cana-5869	74	2	(	(	PUNCT
cana-5869	74	3	2024	2024	NUM
cana-5869	74	4	)	)	PUNCT
cana-5869	74	5	760	760	NUM
cana-5869	74	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5869	74	7	𝑠𝑢𝑝	𝑠𝑢𝑝	VERB
cana-5869	74	8	𝑗	𝑗	PRON
cana-5869	74	9	𝑗𝑗‖𝜉𝑗‖	𝑗𝑗‖𝜉𝑗‖	NOUN
cana-5869	74	10	=	=	PUNCT
cana-5869	74	11	𝑠𝑢𝑝	𝑠𝑢𝑝	VERB
cana-5869	74	12	𝑗	𝑗	INTJ
cana-5869	74	13	𝑗𝑗	𝑗𝑗	NOUN
cana-5869	75	1	(	(	PUNCT
cana-5869	75	2	|1𝜉𝑗	|1𝜉𝑗	NOUN
cana-5869	75	3	|	|	ADV
cana-5869	75	4	2	2	NUM
cana-5869	76	1	+	+	CCONJ
cana-5869	77	1	|2𝜉𝑗	|2𝜉𝑗	NOUN
cana-5869	77	2	|	|	ADV
cana-5869	77	3	2	2	NUM
cana-5869	77	4	2	2	NUM
cana-5869	77	5	)	)	PUNCT
cana-5869	77	6	1/2	1/2	NUM
cana-5869	77	7	=	=	PUNCT
cana-5869	77	8	𝑠𝑢𝑝	𝑠𝑢𝑝	VERB
cana-5869	77	9	𝑗	𝑗	INTJ
cana-5869	77	10	𝑗𝑗	𝑗𝑗	NOUN
cana-5869	77	11	(	(	PUNCT
cana-5869	77	12	|𝑗−2𝑗−4|+|𝑗−2𝑗−6|	|𝑗−2𝑗−4|+|𝑗−2𝑗−6|	NOUN
cana-5869	77	13	2	2	NUM
cana-5869	77	14	)	)	PUNCT
cana-5869	77	15	1/2	1/2	NUM
cana-5869	77	16	=	=	PUNCT
cana-5869	77	17	𝑠𝑢𝑝	𝑠𝑢𝑝	VERB
cana-5869	77	18	𝑗	𝑗	INTJ
cana-5869	77	19	{	{	PUNCT
cana-5869	77	20	1	1	NUM
cana-5869	77	21	2	2	NUM
cana-5869	77	22	(	(	PUNCT
cana-5869	77	23	1	1	NUM
cana-5869	77	24	𝑗4	𝑗4	NOUN
cana-5869	77	25	+	+	CCONJ
cana-5869	77	26	1	1	NUM
cana-5869	77	27	𝑗6	𝑗6	NOUN
cana-5869	77	28	)	)	PUNCT
cana-5869	77	29	}	}	PUNCT
cana-5869	78	1	1/2	1/2	NUM
cana-5869	78	2	<	<	NOUN
cana-5869	78	3	∞	∞	NUM
cana-5869	78	4	so	so	ADV
cana-5869	78	5	that(𝜉𝑗	that(𝜉𝑗	ADJ
cana-5869	78	6	)	)	PUNCT
cana-5869	78	7	∈	∈	PROPN
cana-5869	78	8	ℵ.	ℵ.	NOUN
cana-5869	79	1	now	now	ADV
cana-5869	79	2	,	,	PUNCT
cana-5869	79	3	‖∑	‖∑	ADP
cana-5869	79	4	𝜉𝑗𝜂𝑗𝑗≥1	𝜉𝑗𝜂𝑗𝑗≥1	NOUN
cana-5869	79	5	‖	‖	PROPN
cana-5869	79	6	≤	≤	PROPN
cana-5869	79	7	∑	∑	PUNCT
cana-5869	79	8	‖𝜉𝑗𝜂𝑗‖	‖𝜉𝑗𝜂𝑗‖	PROPN
cana-5869	79	9	=	=	PUNCT
cana-5869	79	10	∑	∑	PUNCT
cana-5869	79	11	‖	‖	PROPN
cana-5869	79	12	(	(	PUNCT
cana-5869	79	13	1	1	NUM
cana-5869	79	14	𝑗2+𝑗	𝑗2+𝑗	NOUN
cana-5869	79	15	𝑒1	𝑒1	NOUN
cana-5869	79	16	+	+	CCONJ
cana-5869	79	17	1	1	NUM
cana-5869	79	18	𝑗3+𝑗	𝑗3+𝑗	PROPN
cana-5869	79	19	𝑒2	𝑒2	PROPN
cana-5869	79	20	)	)	PUNCT
cana-5869	79	21	.	.	PUNCT
cana-5869	80	1	(	(	PUNCT
cana-5869	80	2	1	1	NUM
cana-5869	80	3	𝑗2	𝑗2	NOUN
cana-5869	80	4	𝑒1	𝑒1	NOUN
cana-5869	80	5	+	+	CCONJ
cana-5869	80	6	𝑗2+𝑗𝑒2)‖𝑗≥1𝑗≥1	𝑗2+𝑗𝑒2)‖𝑗≥1𝑗≥1	PROPN
cana-5869	80	7	=	=	SYM
cana-5869	80	8	∑	∑	PUNCT
cana-5869	80	9	‖	‖	PROPN
cana-5869	80	10	(	(	PUNCT
cana-5869	80	11	1	1	NUM
cana-5869	80	12	𝑗4+𝑗	𝑗4+𝑗	NOUN
cana-5869	80	13	𝑒1	𝑒1	NOUN
cana-5869	80	14	+	+	CCONJ
cana-5869	80	15	1	1	NUM
cana-5869	80	16	𝑗	𝑗	PROPN
cana-5869	80	17	𝑒2)‖𝑗≥1	𝑒2)‖𝑗≥1	PROPN
cana-5869	80	18	𝑗4+𝑗	𝑗4+𝑗	PROPN
cana-5869	80	19	>	>	PUNCT
cana-5869	80	20	𝑗	𝑗	PROPN
cana-5869	80	21	⇒	⇒	NOUN
cana-5869	80	22	1	1	NUM
cana-5869	80	23	𝑗4+𝑗	𝑗4+𝑗	NOUN
cana-5869	80	24	<	<	X
cana-5869	80	25	1	1	NUM
cana-5869	80	26	𝑗4	𝑗4	NOUN
cana-5869	80	27	and	and	CCONJ
cana-5869	80	28	∑	∑	ADV
cana-5869	80	29	|	|	ADV
cana-5869	80	30	1	1	NUM
cana-5869	80	31	𝑗4	𝑗4	PROPN
cana-5869	80	32	|𝑗≥1	|𝑗≥1	NOUN
cana-5869	80	33	is	be	AUX
cana-5869	80	34	convergent	convergent	NOUN
cana-5869	80	35	therefore	therefore	ADV
cana-5869	80	36	by	by	ADP
cana-5869	80	37	comparison	comparison	NOUN
cana-5869	80	38	test	test	NOUN
cana-5869	80	39	∑	∑	PUNCT
cana-5869	80	40	|	|	ADV
cana-5869	80	41	1	1	NUM
cana-5869	80	42	𝑗4+𝑗	𝑗4+𝑗	NOUN
cana-5869	80	43	|𝑗≥1	|𝑗≥1	NOUN
cana-5869	80	44	is	be	AUX
cana-5869	80	45	also	also	ADV
cana-5869	80	46	convergent	convergent	ADJ
cana-5869	80	47	,	,	PUNCT
cana-5869	80	48	but	but	CCONJ
cana-5869	80	49	∑	∑	ADV
cana-5869	80	50	|	|	ADV
cana-5869	80	51	1	1	NUM
cana-5869	80	52	𝑗4	𝑗4	PROPN
cana-5869	80	53	|𝑗≥1	|𝑗≥1	NOUN
cana-5869	80	54	is	be	AUX
cana-5869	80	55	not	not	PART
cana-5869	80	56	convergent	convergent	ADJ
cana-5869	80	57	.	.	PUNCT
cana-5869	81	1	therefore	therefore	ADV
cana-5869	81	2	,	,	PUNCT
cana-5869	81	3	(	(	PUNCT
cana-5869	81	4	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	81	5	)	)	PUNCT
cana-5869	81	6	∉	∉	PROPN
cana-5869	81	7	ℵ𝛽	ℵ𝛽	PROPN
cana-5869	81	8	(	(	PUNCT
cana-5869	81	9	b	b	NOUN
cana-5869	81	10	)	)	PUNCT
cana-5869	81	11	hence	hence	ADV
cana-5869	81	12	from	from	ADP
cana-5869	81	13	(	(	PUNCT
cana-5869	81	14	a	a	X
cana-5869	81	15	)	)	PUNCT
cana-5869	81	16	and	and	CCONJ
cana-5869	81	17	(	(	PUNCT
cana-5869	81	18	b	b	X
cana-5869	81	19	)	)	PUNCT
cana-5869	81	20	we	we	PRON
cana-5869	81	21	get	get	VERB
cana-5869	81	22	(	(	PUNCT
cana-5869	81	23	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	81	24	)	)	PUNCT
cana-5869	81	25	∈	∈	PROPN
cana-5869	81	26	(	(	PUNCT
cana-5869	81	27	1ℵ)𝛽	1ℵ)𝛽	NUM
cana-5869	81	28	but(𝜂𝑗	but(𝜂𝑗	NOUN
cana-5869	81	29	)	)	PUNCT
cana-5869	81	30	∉	∉	PROPN
cana-5869	82	1	ℵ𝛽.	ℵ𝛽.	AUX
cana-5869	82	2	note	note	VERB
cana-5869	82	3	1	1	NUM
cana-5869	82	4	:	:	PUNCT
cana-5869	82	5	beta	beta	VERB
cana-5869	82	6	dual	dual	NOUN
cana-5869	82	7	of	of	ADP
cana-5869	82	8	the	the	DET
cana-5869	82	9	class	class	NOUN
cana-5869	82	10	ℵ	ℵ	NOUN
cana-5869	82	11	is	be	AUX
cana-5869	82	12	properly	properly	ADV
cana-5869	82	13	contained	contain	VERB
cana-5869	82	14	in	in	ADP
cana-5869	82	15	the	the	DET
cana-5869	82	16	beta	beta	NOUN
cana-5869	82	17	dual	dual	NOUN
cana-5869	82	18	of	of	ADP
cana-5869	82	19	its	its	PRON
cana-5869	82	20	t1	t1	NOUN
cana-5869	82	21	image	image	NOUN
cana-5869	82	22	.	.	PUNCT
cana-5869	83	1	next	next	ADV
cana-5869	83	2	,	,	PUNCT
cana-5869	83	3	gamma	gamma	PROPN
cana-5869	83	4	dual	dual	ADJ
cana-5869	83	5	of	of	ADP
cana-5869	83	6	the	the	DET
cana-5869	83	7	class	class	NOUN
cana-5869	83	8	ℵ	ℵ	NOUN
cana-5869	83	9	and	and	CCONJ
cana-5869	83	10	its	its	PRON
cana-5869	83	11	subclasses	subclass	NOUN
cana-5869	83	12	are	be	AUX
cana-5869	83	13	given	give	VERB
cana-5869	83	14	by	by	ADP
cana-5869	83	15	ℵ𝛾	ℵ𝛾	NOUN
cana-5869	83	16	=	=	SYM
cana-5869	83	17	{	{	PUNCT
cana-5869	83	18	(	(	PUNCT
cana-5869	83	19	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	83	20	)	)	PUNCT
cana-5869	83	21	∈	∈	NOUN
cana-5869	83	22	𝜔′	𝜔′	NUM
cana-5869	83	23	:	:	PUNCT
cana-5869	83	24	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-5869	83	25	𝑛	𝑛	PRON
cana-5869	83	26	‖∑	‖∑	ADJ
cana-5869	83	27	𝜂𝑗𝜉𝑗	𝜂𝑗𝜉𝑗	X
cana-5869	83	28	𝑛	𝑛	PRON
cana-5869	83	29	𝑗=1	𝑗=1	SYM
cana-5869	83	30	‖	‖	PROPN
cana-5869	83	31	<	<	X
cana-5869	83	32	∞	∞	PROPN
cana-5869	83	33	,	,	PUNCT
cana-5869	83	34	∀(𝜉𝑗	∀(𝜉𝑗	X
cana-5869	83	35	)	)	PUNCT
cana-5869	83	36	∈	∈	PROPN
cana-5869	83	37	ℵ	ℵ	PROPN
cana-5869	83	38	}	}	PUNCT
cana-5869	83	39	(	(	PUNCT
cana-5869	83	40	iii	iii	NOUN
cana-5869	83	41	)	)	PUNCT
cana-5869	83	42	or	or	CCONJ
cana-5869	83	43	{	{	PUNCT
cana-5869	83	44	(	(	PUNCT
cana-5869	83	45	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	83	46	)	)	PUNCT
cana-5869	83	47	∈	∈	NOUN
cana-5869	83	48	𝜔′	𝜔′	NUM
cana-5869	83	49	:	:	PUNCT
cana-5869	83	50	𝑠𝑢𝑝	𝑠𝑢𝑝	VERB
cana-5869	84	1	𝑛	𝑛	ADP
cana-5869	84	2	|∑	|∑	NUM
cana-5869	84	3	1𝜂𝑗	1𝜂𝑗	ADJ
cana-5869	84	4	1𝜉𝑗	1𝜉𝑗	NOUN
cana-5869	84	5	𝑛	𝑛	VERB
cana-5869	84	6	𝑗=1	𝑗=1	PROPN
cana-5869	84	7	|	|	ADV
cana-5869	84	8	<	<	X
cana-5869	84	9	∞	∞	PROPN
cana-5869	84	10	,	,	PUNCT
cana-5869	84	11	𝑠𝑢𝑝	𝑠𝑢𝑝	VERB
cana-5869	84	12	𝑛	𝑛	ADP
cana-5869	84	13	|∑	|∑	VERB
cana-5869	84	14	1𝜂𝑗	1𝜂𝑗	ADJ
cana-5869	84	15	2𝜉𝑗	2𝜉𝑗	ADJ
cana-5869	84	16	𝑛	𝑛	PRON
cana-5869	84	17	𝑗=1	𝑗=1	PROPN
cana-5869	85	1	|	|	ADV
cana-5869	85	2	<	<	X
cana-5869	85	3	∞	∞	PROPN
cana-5869	85	4	,	,	PUNCT
cana-5869	85	5	∀(𝜉𝑗	∀(𝜉𝑗	X
cana-5869	85	6	)	)	PUNCT
cana-5869	85	7	∈	∈	PROPN
cana-5869	85	8	ℵ	ℵ	NOUN
cana-5869	85	9	}	}	PUNCT
cana-5869	85	10	.	.	PUNCT
cana-5869	86	1	communications	communication	NOUN
cana-5869	86	2	on	on	ADP
cana-5869	86	3	applied	apply	VERB
cana-5869	86	4	nonlinear	nonlinear	ADJ
cana-5869	86	5	analysis	analysis	NOUN
cana-5869	86	6	issn	issn	NOUN
cana-5869	86	7	:	:	PUNCT
cana-5869	86	8	1074	1074	NUM
cana-5869	86	9	-	-	PUNCT
cana-5869	86	10	133x	133x	NUM
cana-5869	86	11	vol	vol	NOUN
cana-5869	86	12	31	31	NUM
cana-5869	86	13	no	no	NOUN
cana-5869	86	14	.	.	PUNCT
cana-5869	87	1	2s	2s	NUM
cana-5869	87	2	(	(	PUNCT
cana-5869	87	3	2024	2024	NUM
cana-5869	87	4	)	)	PUNCT
cana-5869	87	5	761	761	NUM
cana-5869	87	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5869	87	7	(	(	PUNCT
cana-5869	87	8	1ℵ)𝛾	1ℵ)𝛾	NUM
cana-5869	87	9	=	=	SYM
cana-5869	87	10	{	{	PUNCT
cana-5869	87	11	(	(	PUNCT
cana-5869	87	12	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	87	13	)	)	PUNCT
cana-5869	87	14	∈	∈	NOUN
cana-5869	87	15	𝜔′	𝜔′	NUM
cana-5869	87	16	:	:	PUNCT
cana-5869	87	17	𝑠𝑢𝑝	𝑠𝑢𝑝	VERB
cana-5869	87	18	𝑛	𝑛	ADP
cana-5869	87	19	|∑	|∑	NUM
cana-5869	87	20	1𝜂𝑗	1𝜂𝑗	ADJ
cana-5869	87	21	1𝜉𝑗	1𝜉𝑗	NOUN
cana-5869	88	1	𝑛	𝑛	VERB
cana-5869	88	2	𝑗=1	𝑗=1	PROPN
cana-5869	89	1	|	|	ADV
cana-5869	89	2	<	<	X
cana-5869	89	3	∞	∞	PROPN
cana-5869	89	4	,	,	PUNCT
cana-5869	89	5	∀(𝜉𝑗	∀(𝜉𝑗	X
cana-5869	89	6	)	)	PUNCT
cana-5869	89	7	∈	∈	PROPN
cana-5869	89	8	ℵ	ℵ	NOUN
cana-5869	89	9	}	}	PUNCT
cana-5869	89	10	(	(	PUNCT
cana-5869	89	11	iv	iv	X
cana-5869	89	12	)	)	PUNCT
cana-5869	89	13	(	(	PUNCT
cana-5869	89	14	2ℵ)𝛾	2ℵ)𝛾	NUM
cana-5869	89	15	=	=	SYM
cana-5869	89	16	{	{	PUNCT
cana-5869	89	17	(	(	PUNCT
cana-5869	89	18	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	89	19	)	)	PUNCT
cana-5869	89	20	∈	∈	NOUN
cana-5869	89	21	𝜔′	𝜔′	NUM
cana-5869	89	22	:	:	PUNCT
cana-5869	89	23	𝑠𝑢𝑝	𝑠𝑢𝑝	VERB
cana-5869	89	24	𝑛	𝑛	PRON
cana-5869	89	25	|∑	|∑	PROPN
cana-5869	89	26	2𝜂𝑗	2𝜂𝑗	ADJ
cana-5869	89	27	2𝜉𝑗	2𝜉𝑗	NOUN
cana-5869	89	28	𝑛	𝑛	PRON
cana-5869	89	29	𝑗=1	𝑗=1	PROPN
cana-5869	90	1	|	|	ADV
cana-5869	90	2	<	<	X
cana-5869	90	3	∞	∞	PROPN
cana-5869	90	4	,	,	PUNCT
cana-5869	90	5	∀(𝜉𝑗	∀(𝜉𝑗	X
cana-5869	90	6	)	)	PUNCT
cana-5869	90	7	∈	∈	PROPN
cana-5869	90	8	ℵ	ℵ	NOUN
cana-5869	90	9	}	}	PUNCT
cana-5869	90	10	(	(	PUNCT
cana-5869	90	11	v	v	NOUN
cana-5869	90	12	)	)	PUNCT
cana-5869	90	13	theorem	theorem	VERB
cana-5869	90	14	4.2	4.2	NUM
cana-5869	90	15	:	:	PUNCT
cana-5869	90	16	a	a	DET
cana-5869	90	17	sequence	sequence	NOUN
cana-5869	90	18	(	(	PUNCT
cana-5869	90	19	𝜂𝑘	𝜂𝑘	NOUN
cana-5869	90	20	)	)	PUNCT
cana-5869	90	21	belongs	belong	VERB
cana-5869	90	22	to	to	ADP
cana-5869	90	23	𝛾	𝛾	ADP
cana-5869	90	24	dual	dual	ADJ
cana-5869	90	25	of	of	ADP
cana-5869	90	26	the	the	DET
cana-5869	90	27	class	class	NOUN
cana-5869	90	28	ℵ	ℵ	NOUN
cana-5869	90	29	if	if	SCONJ
cana-5869	91	1	and	and	CCONJ
cana-5869	91	2	only	only	ADV
cana-5869	91	3	if	if	SCONJ
cana-5869	91	4	its	its	PRON
cana-5869	91	5	first	first	ADJ
cana-5869	91	6	idempotent	idempotent	ADJ
cana-5869	91	7	sequence	sequence	NOUN
cana-5869	91	8	belongs	belong	VERB
cana-5869	91	9	to	to	ADP
cana-5869	91	10	𝛾	𝛾	ADP
cana-5869	91	11	dual	dual	ADJ
cana-5869	91	12	of	of	ADP
cana-5869	91	13	the	the	DET
cana-5869	91	14	class	class	NOUN
cana-5869	91	15	1ℵ	1ℵ	NOUN
cana-5869	91	16	and	and	CCONJ
cana-5869	91	17	the	the	DET
cana-5869	91	18	second	second	ADJ
cana-5869	91	19	idempotent	idempotent	ADJ
cana-5869	91	20	sequence	sequence	NOUN
cana-5869	91	21	belongs	belong	VERB
cana-5869	91	22	to	to	ADP
cana-5869	91	23	𝛾	𝛾	ADP
cana-5869	91	24	dual	dual	ADJ
cana-5869	91	25	of	of	ADP
cana-5869	91	26	the	the	DET
cana-5869	91	27	class	class	NOUN
cana-5869	91	28	2ℵ	2ℵ	NOUN
cana-5869	91	29	that	that	PRON
cana-5869	91	30	is	be	AUX
cana-5869	91	31	(	(	PUNCT
cana-5869	91	32	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	91	33	)	)	PUNCT
cana-5869	91	34	∈	∈	PROPN
cana-5869	91	35	𝐵𝛾	𝐵𝛾	PROPN
cana-5869	91	36	⇔	⇔	X
cana-5869	91	37	(	(	PUNCT
cana-5869	91	38	1𝜂𝑗	1𝜂𝑗	ADJ
cana-5869	91	39	𝑒1	𝑒1	NOUN
cana-5869	91	40	)	)	PUNCT
cana-5869	91	41	∈	∈	PROPN
cana-5869	91	42	(	(	PUNCT
cana-5869	91	43	1ℵ)𝛾	1ℵ)𝛾	NUM
cana-5869	91	44	𝑎𝑛𝑑	𝑎𝑛𝑑	NOUN
cana-5869	91	45	(	(	PUNCT
cana-5869	91	46	2𝜂𝑗	2𝜂𝑗	ADJ
cana-5869	91	47	𝑒2	𝑒2	PROPN
cana-5869	91	48	)	)	PUNCT
cana-5869	91	49	∈	∈	PROPN
cana-5869	91	50	(	(	PUNCT
cana-5869	91	51	2ℵ)𝛾.	2ℵ)𝛾.	NUM
cana-5869	91	52	proof	proof	NOUN
cana-5869	91	53	:	:	PUNCT
cana-5869	91	54	let	let	VERB
cana-5869	91	55	(	(	PUNCT
cana-5869	91	56	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	91	57	)	)	PUNCT
cana-5869	91	58	∈	∈	NOUN
cana-5869	91	59	ℵ𝛾	ℵ𝛾	NOUN
cana-5869	91	60	be	be	AUX
cana-5869	91	61	any	any	DET
cana-5869	91	62	sequence	sequence	NOUN
cana-5869	91	63	.	.	PUNCT
cana-5869	92	1	by	by	ADP
cana-5869	92	2	(	(	PUNCT
cana-5869	92	3	iii	iii	NOUN
cana-5869	92	4	)	)	PUNCT
cana-5869	92	5	,	,	PUNCT
cana-5869	92	6	(	(	PUNCT
cana-5869	92	7	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	92	8	)	)	PUNCT
cana-5869	92	9	∈	∈	PROPN
cana-5869	93	1	𝐵𝛾	𝐵𝛾	PROPN
cana-5869	93	2	⇔	⇔	PROPN
cana-5869	93	3	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-5869	93	4	𝑛	𝑛	PRON
cana-5869	93	5	‖∑	‖∑	ADJ
cana-5869	93	6	𝜂𝑗𝜉𝑗	𝜂𝑗𝜉𝑗	X
cana-5869	93	7	𝑛	𝑛	PRON
cana-5869	93	8	𝑗=1	𝑗=1	SYM
cana-5869	93	9	‖	‖	PROPN
cana-5869	93	10	<	<	X
cana-5869	93	11	∞	∞	PROPN
cana-5869	93	12	,	,	PUNCT
cana-5869	93	13	∀(𝜉𝑗	∀(𝜉𝑗	X
cana-5869	93	14	)	)	PUNCT
cana-5869	93	15	∈	∈	PROPN
cana-5869	93	16	ℵ	ℵ	X
cana-5869	93	17	⇔	⇔	X
cana-5869	93	18	{	{	PUNCT
cana-5869	93	19	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-5869	93	20	𝑛	𝑛	PROPN
cana-5869	93	21	|∑	|∑	NUM
cana-5869	93	22	1𝜂𝑗	1𝜂𝑗	ADJ
cana-5869	93	23	1𝜉𝑗	1𝜉𝑗	NOUN
cana-5869	93	24	𝑛	𝑛	VERB
cana-5869	93	25	𝑗=1	𝑗=1	PUNCT
cana-5869	93	26	|	|	ADV
cana-5869	93	27	}	}	PUNCT
cana-5869	93	28	𝑒1	𝑒1	NOUN
cana-5869	93	29	+	+	CCONJ
cana-5869	93	30	{	{	PUNCT
cana-5869	93	31	𝑠𝑢𝑝	𝑠𝑢𝑝	VERB
cana-5869	93	32	𝑛	𝑛	PRON
cana-5869	93	33	|∑	|∑	NUM
cana-5869	93	34	2𝜂𝑗	2𝜂𝑗	ADJ
cana-5869	93	35	2𝜉𝑗	2𝜉𝑗	ADJ
cana-5869	93	36	𝑛	𝑛	PRON
cana-5869	93	37	𝑗=1	𝑗=1	X
cana-5869	93	38	|	|	ADJ
cana-5869	93	39	}	}	PUNCT
cana-5869	93	40	𝑒2	𝑒2	PROPN
cana-5869	93	41	<	<	X
cana-5869	93	42	∞	∞	PROPN
cana-5869	93	43	,	,	PUNCT
cana-5869	93	44	∀(𝜉𝑗	∀(𝜉𝑗	X
cana-5869	93	45	)	)	PUNCT
cana-5869	93	46	∈	∈	PROPN
cana-5869	93	47	ℵ	ℵ	PROPN
cana-5869	93	48	⇔	⇔	PROPN
cana-5869	93	49	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
cana-5869	93	50	𝑛	𝑛	PROPN
cana-5869	93	51	|∑	|∑	X
cana-5869	93	52	1𝑗1𝜉𝑗	1𝑗1𝜉𝑗	NUM
cana-5869	93	53	𝑛	𝑛	PRON
cana-5869	93	54	𝑗=1	𝑗=1	NOUN
cana-5869	94	1	|	|	ADV
cana-5869	94	2	<	<	X
cana-5869	94	3	∞	∞	PROPN
cana-5869	94	4	and	and	CCONJ
cana-5869	94	5	𝑠𝑢𝑝	𝑠𝑢𝑝	VERB
cana-5869	95	1	𝑛	𝑛	PRON
cana-5869	95	2	|∑	|∑	PROPN
cana-5869	95	3	2𝜂𝑗	2𝜂𝑗	ADJ
cana-5869	95	4	2𝜉𝑗	2𝜉𝑗	NOUN
cana-5869	95	5	𝑛	𝑛	PRON
cana-5869	95	6	𝑗=1	𝑗=1	PROPN
cana-5869	96	1	|	|	ADV
cana-5869	96	2	<	<	X
cana-5869	96	3	∞	∞	PROPN
cana-5869	96	4	,	,	PUNCT
cana-5869	96	5	∀(𝜉𝑗	∀(𝜉𝑗	X
cana-5869	96	6	)	)	PUNCT
cana-5869	96	7	∈	∈	PROPN
cana-5869	96	8	ℵ	ℵ	NOUN
cana-5869	96	9	hence	hence	ADV
cana-5869	96	10	,	,	PUNCT
cana-5869	96	11	(	(	PUNCT
cana-5869	96	12	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	96	13	)	)	PUNCT
cana-5869	96	14	∈	∈	PROPN
cana-5869	96	15	ℵ𝛾	ℵ𝛾	ADP
cana-5869	96	16	⇔	⇔	X
cana-5869	96	17	(	(	PUNCT
cana-5869	96	18	1𝜂𝑗𝑒1	1𝜂𝑗𝑒1	PROPN
cana-5869	96	19	)	)	PUNCT
cana-5869	96	20	∈	∈	PROPN
cana-5869	96	21	(	(	PUNCT
cana-5869	96	22	1ℵ)𝛾	1ℵ)𝛾	NUM
cana-5869	96	23	and	and	CCONJ
cana-5869	96	24	(	(	PUNCT
cana-5869	96	25	2𝜂𝑗𝑒2	2𝜂𝑗𝑒2	X
cana-5869	96	26	)	)	PUNCT
cana-5869	96	27	∈	∈	PROPN
cana-5869	96	28	(	(	PUNCT
cana-5869	96	29	2ℵ)𝛾	2ℵ)𝛾	NUM
cana-5869	96	30	,	,	PUNCT
cana-5869	96	31	by	by	ADP
cana-5869	96	32	(	(	PUNCT
cana-5869	96	33	iii	iii	NOUN
cana-5869	96	34	)	)	PUNCT
cana-5869	96	35	and	and	CCONJ
cana-5869	96	36	(	(	PUNCT
cana-5869	96	37	iv	iv	X
cana-5869	96	38	)	)	PUNCT
cana-5869	96	39	.	.	PUNCT
cana-5869	97	1	(	(	PUNCT
cana-5869	97	2	c	c	X
cana-5869	97	3	)	)	PUNCT
cana-5869	97	4	corollary	corollary	NOUN
cana-5869	97	5	1	1	NUM
cana-5869	97	6	:	:	PUNCT
cana-5869	97	7	thus	thus	ADV
cana-5869	97	8	from	from	ADP
cana-5869	97	9	(	(	PUNCT
cana-5869	97	10	b	b	NOUN
cana-5869	97	11	)	)	PUNCT
cana-5869	97	12	and	and	CCONJ
cana-5869	97	13	(	(	PUNCT
cana-5869	97	14	c	c	X
cana-5869	97	15	)	)	PUNCT
cana-5869	97	16	we	we	PRON
cana-5869	97	17	can	can	AUX
cana-5869	97	18	say	say	VERB
cana-5869	97	19	that	that	DET
cana-5869	97	20	(	(	PUNCT
cana-5869	97	21	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	97	22	)	)	PUNCT
cana-5869	97	23	∈	∈	PROPN
cana-5869	97	24	ℵ𝛾if	ℵ𝛾if	NOUN
cana-5869	97	25	and	and	CCONJ
cana-5869	97	26	only	only	ADV
cana-5869	97	27	if	if	SCONJ
cana-5869	97	28	𝑇1(𝜂𝑗	𝑇1(𝜂𝑗	NUM
cana-5869	97	29	)	)	PUNCT
cana-5869	97	30	∈	∈	PROPN
cana-5869	97	31	(	(	PUNCT
cana-5869	97	32	1ℵ	1ℵ	NOUN
cana-5869	97	33	)	)	PUNCT
cana-5869	97	34	,	,	PUNCT
cana-5869	97	35	𝑇2(𝜂𝑗	𝑇2(𝜂𝑗	PROPN
cana-5869	97	36	)	)	PUNCT
cana-5869	97	37	∈	∈	PROPN
cana-5869	97	38	(	(	PUNCT
cana-5869	97	39	2ℵ)𝛾	2ℵ)𝛾	NUM
cana-5869	97	40	or	or	CCONJ
cana-5869	97	41	in	in	ADP
cana-5869	97	42	words	word	NOUN
cana-5869	97	43	we	we	PRON
cana-5869	97	44	can	can	AUX
cana-5869	97	45	also	also	ADV
cana-5869	97	46	say	say	VERB
cana-5869	97	47	that	that	SCONJ
cana-5869	97	48	a	a	DET
cana-5869	97	49	sequence	sequence	NOUN
cana-5869	97	50	belongs	belong	VERB
cana-5869	97	51	to	to	ADP
cana-5869	97	52	𝛾	𝛾	ADP
cana-5869	97	53	dual	dual	ADJ
cana-5869	97	54	of	of	ADP
cana-5869	97	55	the	the	DET
cana-5869	97	56	class	class	NOUN
cana-5869	97	57	ℵ	ℵ	NOUN
cana-5869	97	58	if	if	SCONJ
cana-5869	97	59	and	and	CCONJ
cana-5869	97	60	only	only	ADV
cana-5869	97	61	if	if	SCONJ
cana-5869	97	62	its	its	PRON
cana-5869	97	63	t1	t1	NOUN
cana-5869	97	64	–	–	PUNCT
cana-5869	97	65	image	image	NOUN
cana-5869	97	66	belongs	belong	VERB
cana-5869	97	67	to	to	ADP
cana-5869	97	68	𝛾	𝛾	ADP
cana-5869	97	69	dual	dual	ADJ
cana-5869	97	70	of	of	ADP
cana-5869	97	71	the	the	DET
cana-5869	97	72	first	first	ADJ
cana-5869	97	73	idempotent	idempotent	ADJ
cana-5869	97	74	component	component	NOUN
cana-5869	97	75	of	of	ADP
cana-5869	97	76	ℵ	ℵ	NOUN
cana-5869	97	77	or	or	CCONJ
cana-5869	97	78	its	its	PRON
cana-5869	97	79	first	first	ADJ
cana-5869	97	80	subclass	subclass	NOUN
cana-5869	97	81	and	and	CCONJ
cana-5869	97	82	the	the	DET
cana-5869	97	83	t2	t2	NOUN
cana-5869	97	84	–	–	PUNCT
cana-5869	97	85	image	image	NOUN
cana-5869	97	86	belongs	belong	VERB
cana-5869	97	87	to	to	ADP
cana-5869	97	88	the	the	DET
cana-5869	97	89	𝛾	𝛾	NOUN
cana-5869	97	90	dual	dual	ADJ
cana-5869	97	91	of	of	ADP
cana-5869	97	92	its	its	PRON
cana-5869	97	93	second	second	ADJ
cana-5869	97	94	idempotent	idempotent	ADJ
cana-5869	97	95	component	component	NOUN
cana-5869	97	96	or	or	CCONJ
cana-5869	97	97	its	its	PRON
cana-5869	97	98	second	second	ADJ
cana-5869	97	99	subclass	subclass	NOUN
cana-5869	97	100	.	.	PUNCT
cana-5869	98	1	communications	communication	NOUN
cana-5869	98	2	on	on	ADP
cana-5869	98	3	applied	apply	VERB
cana-5869	98	4	nonlinear	nonlinear	ADJ
cana-5869	98	5	analysis	analysis	NOUN
cana-5869	98	6	issn	issn	NOUN
cana-5869	98	7	:	:	PUNCT
cana-5869	98	8	1074	1074	NUM
cana-5869	98	9	-	-	PUNCT
cana-5869	98	10	133x	133x	NUM
cana-5869	98	11	vol	vol	NOUN
cana-5869	98	12	31	31	NUM
cana-5869	98	13	no	no	NOUN
cana-5869	98	14	.	.	PUNCT
cana-5869	99	1	2s	2s	NUM
cana-5869	99	2	(	(	PUNCT
cana-5869	99	3	2024	2024	NUM
cana-5869	99	4	)	)	PUNCT
cana-5869	99	5	762	762	NUM
cana-5869	99	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5869	99	7	theorem	theorem	VERB
cana-5869	99	8	4.3	4.3	NUM
cana-5869	99	9	:	:	PUNCT
cana-5869	99	10	(	(	PUNCT
cana-5869	99	11	i)ℵ𝛾	i)ℵ𝛾	PROPN
cana-5869	99	12	⊂	⊂	X
cana-5869	99	13	(	(	PUNCT
cana-5869	99	14	1ℵ)𝛾.	1ℵ)𝛾.	PROPN
cana-5869	99	15	(	(	PUNCT
cana-5869	99	16	ii	ii	NOUN
cana-5869	99	17	)	)	PUNCT
cana-5869	99	18	ℵ𝛾	ℵ𝛾	NOUN
cana-5869	99	19	⊂	⊂	PROPN
cana-5869	99	20	(	(	PUNCT
cana-5869	99	21	2ℵ)𝛾	2ℵ)𝛾	NUM
cana-5869	99	22	proof	proof	NOUN
cana-5869	99	23	:	:	PUNCT
cana-5869	99	24	similar	similar	ADJ
cana-5869	99	25	example	example	NOUN
cana-5869	99	26	may	may	AUX
cana-5869	99	27	be	be	AUX
cana-5869	99	28	considered	consider	VERB
cana-5869	99	29	.	.	PUNCT
cana-5869	100	1	note	note	VERB
cana-5869	100	2	2	2	NUM
cana-5869	100	3	:	:	PUNCT
cana-5869	100	4	gamma	gamma	NOUN
cana-5869	100	5	dual	dual	ADJ
cana-5869	100	6	of	of	ADP
cana-5869	100	7	the	the	DET
cana-5869	100	8	class	class	NOUN
cana-5869	100	9	ℵ	ℵ	NOUN
cana-5869	100	10	is	be	AUX
cana-5869	100	11	properly	properly	ADV
cana-5869	100	12	contained	contain	VERB
cana-5869	100	13	in	in	ADP
cana-5869	100	14	the	the	DET
cana-5869	100	15	gamma	gamma	NOUN
cana-5869	100	16	dual	dual	ADJ
cana-5869	100	17	of	of	ADP
cana-5869	100	18	its	its	PRON
cana-5869	100	19	idempotent	idempotent	NOUN
cana-5869	100	20	parts/	parts/	PROPN
cana-5869	100	21	t1\	t1\	PROPN
cana-5869	100	22	]	]	PUNCT
cana-5869	100	23	,	,	PUNCT
cana-5869	100	24	and	and	CCONJ
cana-5869	100	25	t2	t2	NOUN
cana-5869	100	26	–	–	PUNCT
cana-5869	100	27	images	image	NOUN
cana-5869	100	28	.	.	PUNCT
cana-5869	101	1	δ	δ	PROPN
cana-5869	101	2	–	–	PUNCT
cana-5869	101	3	dual	dual	ADJ
cana-5869	101	4	of	of	ADP
cana-5869	101	5	the	the	DET
cana-5869	101	6	class	class	NOUN
cana-5869	101	7	ℵ	ℵ	NOUN
cana-5869	101	8	and	and	CCONJ
cana-5869	101	9	its	its	PRON
cana-5869	101	10	subclasses	subclass	NOUN
cana-5869	101	11	ℵ𝛿	ℵ𝛿	ADV
cana-5869	101	12	=	=	PRON
cana-5869	101	13	{	{	PUNCT
cana-5869	101	14	(	(	PUNCT
cana-5869	101	15	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	101	16	)	)	PUNCT
cana-5869	101	17	∈	∈	NOUN
cana-5869	101	18	𝜔′	𝜔′	NUM
cana-5869	101	19	:	:	PUNCT
cana-5869	101	20	∑	∑	PUNCT
cana-5869	101	21	‖𝜂𝑗𝜉𝜌(𝑗)‖	‖𝜂𝑗𝜉𝜌(𝑗)‖	X
cana-5869	101	22	<	<	X
cana-5869	101	23	∞	∞	PROPN
cana-5869	101	24	,	,	PUNCT
cana-5869	101	25	∀(𝜉𝑗	∀(𝜉𝑗	X
cana-5869	101	26	)	)	PUNCT
cana-5869	101	27	∈	∈	PROPN
cana-5869	101	28	ℵ	ℵ	NOUN
cana-5869	101	29	𝑗≥1	𝑗≥1	NOUN
cana-5869	101	30	,	,	PUNCT
cana-5869	101	31	𝜌	𝜌	X
cana-5869	101	32	∈	∈	PROPN
cana-5869	101	33	𝜋	𝜋	NOUN
cana-5869	101	34	}	}	PUNCT
cana-5869	101	35	(	(	PUNCT
cana-5869	101	36	vi	vi	NOUN
cana-5869	101	37	)	)	PUNCT
cana-5869	101	38	=	=	PRON
cana-5869	101	39	{	{	PUNCT
cana-5869	101	40	(	(	PUNCT
cana-5869	101	41	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	101	42	)	)	PUNCT
cana-5869	101	43	∈	∈	NOUN
cana-5869	101	44	𝜔′	𝜔′	NUM
cana-5869	101	45	:	:	PUNCT
cana-5869	101	46	∑	∑	PUNCT
cana-5869	101	47	|1𝜂𝑗	|1𝜂𝑗	NOUN
cana-5869	101	48	1𝜉𝜌(𝑗	1𝜉𝜌(𝑗	NUM
cana-5869	101	49	)	)	PUNCT
cana-5869	101	50	|	|	ADV
cana-5869	101	51	<	<	X
cana-5869	101	52	∞	∞	PROPN
cana-5869	101	53	,	,	PUNCT
cana-5869	101	54	𝑗≥1	𝑗≥1	NOUN
cana-5869	101	55	∑	∑	NOUN
cana-5869	101	56	|2𝜂𝑗	|2𝜂𝑗	PROPN
cana-5869	101	57	2𝜉𝜌(𝑗	2𝜉𝜌(𝑗	NUM
cana-5869	101	58	)	)	PUNCT
cana-5869	101	59	|	|	ADV
cana-5869	101	60	<	<	X
cana-5869	101	61	∞	∞	PROPN
cana-5869	101	62	,	,	PUNCT
cana-5869	101	63	𝑗≥1	𝑗≥1	NOUN
cana-5869	101	64	∀(𝜉𝑗	∀(𝜉𝑗	X
cana-5869	101	65	)	)	PUNCT
cana-5869	101	66	∈	∈	PROPN
cana-5869	101	67	ℵ	ℵ	NOUN
cana-5869	101	68	,	,	PUNCT
cana-5869	101	69	𝜌	𝜌	X
cana-5869	101	70	∈	∈	PROPN
cana-5869	101	71	𝜋	𝜋	NOUN
cana-5869	101	72	}	}	PUNCT
cana-5869	101	73	(	(	PUNCT
cana-5869	101	74	1ℵ)𝛿	1ℵ)𝛿	NUM
cana-5869	101	75	=	=	SYM
cana-5869	101	76	{	{	PUNCT
cana-5869	101	77	(	(	PUNCT
cana-5869	101	78	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	101	79	)	)	PUNCT
cana-5869	101	80	∈	∈	NOUN
cana-5869	101	81	𝜔′	𝜔′	NUM
cana-5869	101	82	:	:	PUNCT
cana-5869	101	83	∑	∑	PUNCT
cana-5869	101	84	|1𝜂𝑗	|1𝜂𝑗	VERB
cana-5869	101	85	1𝜉𝜌(𝑗)|	1𝜉𝜌(𝑗)|	NUM
cana-5869	101	86	<	<	X
cana-5869	101	87	∞	∞	PROPN
cana-5869	101	88	,	,	PUNCT
cana-5869	101	89	∀(𝜉𝑗	∀(𝜉𝑗	X
cana-5869	101	90	)	)	PUNCT
cana-5869	101	91	∈	∈	PROPN
cana-5869	101	92	ℵ𝑗≥1	ℵ𝑗≥1	NOUN
cana-5869	101	93	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-5869	101	94	𝜌	𝜌	ADP
cana-5869	101	95	∈	∈	PROPN
cana-5869	101	96	𝜋	𝜋	NOUN
cana-5869	101	97	}	}	PUNCT
cana-5869	101	98	(	(	PUNCT
cana-5869	101	99	vii	vii	PROPN
cana-5869	101	100	)	)	PUNCT
cana-5869	101	101	(	(	PUNCT
cana-5869	101	102	2ℵ)𝛿	2ℵ)𝛿	NUM
cana-5869	101	103	=	=	SYM
cana-5869	101	104	{	{	PUNCT
cana-5869	101	105	(	(	PUNCT
cana-5869	101	106	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	101	107	)	)	PUNCT
cana-5869	101	108	∈	∈	NOUN
cana-5869	101	109	𝜔′	𝜔′	NUM
cana-5869	101	110	:	:	PUNCT
cana-5869	101	111	∑	∑	PUNCT
cana-5869	101	112	|2𝜂𝑗	|2𝜂𝑗	NOUN
cana-5869	101	113	2𝜉𝜌(𝑗)|	2𝜉𝜌(𝑗)|	NUM
cana-5869	101	114	<	<	X
cana-5869	101	115	∞	∞	PROPN
cana-5869	101	116	,	,	PUNCT
cana-5869	101	117	∀(𝜉𝑗	∀(𝜉𝑗	X
cana-5869	101	118	)	)	PUNCT
cana-5869	101	119	∈	∈	PROPN
cana-5869	101	120	ℵ𝑗≥1	ℵ𝑗≥1	NOUN
cana-5869	101	121	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-5869	101	122	𝜌	𝜌	ADP
cana-5869	101	123	∈	∈	PROPN
cana-5869	101	124	𝜋	𝜋	NOUN
cana-5869	101	125	}	}	PUNCT
cana-5869	101	126	(	(	PUNCT
cana-5869	101	127	viii	viii	NOUN
cana-5869	101	128	)	)	PUNCT
cana-5869	101	129	theorem	theorem	VERB
cana-5869	101	130	4.4	4.4	NUM
cana-5869	101	131	:	:	PUNCT
cana-5869	101	132	a	a	DET
cana-5869	101	133	sequence	sequence	NOUN
cana-5869	101	134	(	(	PUNCT
cana-5869	101	135	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	101	136	)	)	PUNCT
cana-5869	101	137	belongs	belong	VERB
cana-5869	101	138	to	to	ADP
cana-5869	101	139	𝛿	𝛿	PROPN
cana-5869	101	140	dual	dual	ADJ
cana-5869	101	141	of	of	ADP
cana-5869	101	142	the	the	DET
cana-5869	101	143	class	class	NOUN
cana-5869	101	144	ℵ	ℵ	NOUN
cana-5869	101	145	if	if	SCONJ
cana-5869	102	1	and	and	CCONJ
cana-5869	102	2	only	only	ADV
cana-5869	102	3	if	if	SCONJ
cana-5869	102	4	its	its	PRON
cana-5869	102	5	first	first	ADJ
cana-5869	102	6	idempotent	idempotent	ADJ
cana-5869	102	7	sequence	sequence	NOUN
cana-5869	102	8	belongs	belong	VERB
cana-5869	102	9	to	to	ADP
cana-5869	102	10	𝛿	𝛿	PROPN
cana-5869	102	11	dual	dual	ADJ
cana-5869	102	12	of	of	ADP
cana-5869	102	13	the	the	DET
cana-5869	102	14	class	class	NOUN
cana-5869	102	15	1ℵ	1ℵ	NOUN
cana-5869	102	16	and	and	CCONJ
cana-5869	102	17	the	the	DET
cana-5869	102	18	second	second	ADJ
cana-5869	102	19	idempotent	idempotent	ADJ
cana-5869	102	20	sequence	sequence	NOUN
cana-5869	102	21	belongs	belong	VERB
cana-5869	102	22	to	to	ADP
cana-5869	102	23	𝛿	𝛿	PROPN
cana-5869	102	24	dual	dual	ADJ
cana-5869	102	25	of	of	ADP
cana-5869	102	26	the	the	DET
cana-5869	102	27	class	class	NOUN
cana-5869	102	28	2ℵ	2ℵ	NOUN
cana-5869	102	29	,	,	PUNCT
cana-5869	103	1	that	that	ADV
cana-5869	103	2	is	is	ADV
cana-5869	103	3	(	(	PUNCT
cana-5869	103	4	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	103	5	)	)	PUNCT
cana-5869	103	6	∈	∈	PROPN
cana-5869	103	7	𝐵𝛿	𝐵𝛿	PROPN
cana-5869	103	8	⇔	⇔	X
cana-5869	103	9	(	(	PUNCT
cana-5869	103	10	1𝜂𝑗𝑒1	1𝜂𝑗𝑒1	PROPN
cana-5869	103	11	)	)	PUNCT
cana-5869	103	12	∈	∈	PROPN
cana-5869	103	13	(	(	PUNCT
cana-5869	103	14	1ℵ)𝛿	1ℵ)𝛿	NUM
cana-5869	103	15	and	and	CCONJ
cana-5869	103	16	(	(	PUNCT
cana-5869	103	17	2𝜂𝑗𝑒2	2𝜂𝑗𝑒2	X
cana-5869	103	18	)	)	PUNCT
cana-5869	103	19	∈	∈	PROPN
cana-5869	103	20	(	(	PUNCT
cana-5869	103	21	2ℵ)𝛿	2ℵ)𝛿	NUM
cana-5869	103	22	proof	proof	NOUN
cana-5869	103	23	:	:	PUNCT
cana-5869	103	24	let	let	VERB
cana-5869	103	25	(	(	PUNCT
cana-5869	103	26	𝜂𝑘	𝜂𝑘	NOUN
cana-5869	103	27	)	)	PUNCT
cana-5869	103	28	∈	∈	PROPN
cana-5869	103	29	𝐵𝛿be	𝐵𝛿be	PROPN
cana-5869	103	30	any	any	DET
cana-5869	103	31	sequence	sequence	NOUN
cana-5869	103	32	.	.	PUNCT
cana-5869	104	1	from	from	ADP
cana-5869	104	2	(	(	PUNCT
cana-5869	104	3	vi	vi	NOUN
cana-5869	104	4	)	)	PUNCT
cana-5869	104	5	,	,	PUNCT
cana-5869	104	6	(	(	PUNCT
cana-5869	104	7	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	104	8	)	)	PUNCT
cana-5869	104	9	∈	∈	PROPN
cana-5869	104	10	ℵ𝛿	ℵ𝛿	ADP
cana-5869	104	11	⇔	⇔	PROPN
cana-5869	104	12	∑‖𝜂𝑗𝜉𝜌(𝑗)‖	∑‖𝜂𝑗𝜉𝜌(𝑗)‖	PROPN
cana-5869	104	13	<	<	X
cana-5869	104	14	∞	∞	PROPN
cana-5869	104	15	,	,	PUNCT
cana-5869	104	16	∀(𝜉𝑗	∀(𝜉𝑗	X
cana-5869	104	17	)	)	PUNCT
cana-5869	104	18	∈	∈	PROPN
cana-5869	104	19	ℵ	ℵ	NOUN
cana-5869	104	20	𝑗≥1	𝑗≥1	NOUN
cana-5869	104	21	and	and	CCONJ
cana-5869	104	22	𝜌	𝜌	ADP
cana-5869	104	23	∈	∈	PROPN
cana-5869	104	24	𝜋	𝜋	X
cana-5869	104	25	⇔	⇔	X
cana-5869	104	26	{	{	PUNCT
cana-5869	104	27	{	{	PUNCT
cana-5869	104	28	∑	∑	PROPN
cana-5869	104	29	|1𝜂𝑗	|1𝜂𝑗	NOUN
cana-5869	104	30	1𝜉𝜌(𝑗)|𝑗≥1	1𝜉𝜌(𝑗)|𝑗≥1	NUM
cana-5869	104	31	}	}	PUNCT
cana-5869	104	32	𝑒1	𝑒1	NOUN
cana-5869	104	33	+	+	CCONJ
cana-5869	104	34	{	{	PUNCT
cana-5869	104	35	∑	∑	PROPN
cana-5869	104	36	|2𝜂𝑗	|2𝜂𝑗	NOUN
cana-5869	104	37	2𝜉𝜌(𝑗)|𝑗≥1	2𝜉𝜌(𝑗)|𝑗≥1	NUM
cana-5869	104	38	}	}	PUNCT
cana-5869	104	39	𝑒2	𝑒2	NOUN
cana-5869	104	40	}	}	PUNCT
cana-5869	104	41	<	<	X
cana-5869	104	42	∞	∞	PROPN
cana-5869	104	43	,	,	PUNCT
cana-5869	104	44	∀(𝜉𝑗	∀(𝜉𝑗	X
cana-5869	104	45	)	)	PUNCT
cana-5869	104	46	∈	∈	PROPN
cana-5869	104	47	ℵ	ℵ	X
cana-5869	104	48	⇔	⇔	X
cana-5869	104	49	∑	∑	PROPN
cana-5869	104	50	|1𝜂𝑗	|1𝜂𝑗	NOUN
cana-5869	104	51	1𝜉𝜌(𝑗)|	1𝜉𝜌(𝑗)|	NUM
cana-5869	104	52	<	<	X
cana-5869	104	53	∞𝑗≥1	∞𝑗≥1	PROPN
cana-5869	104	54	and	and	CCONJ
cana-5869	104	55	∑	∑	ADP
cana-5869	104	56	|2𝜂𝑗	|2𝜂𝑗	PROPN
cana-5869	104	57	2𝜉𝜌(𝑗)|	2𝜉𝜌(𝑗)|	NUM
cana-5869	104	58	<	<	X
cana-5869	104	59	∞𝑗≥1	∞𝑗≥1	PROPN
cana-5869	104	60	,	,	PUNCT
cana-5869	104	61	∀(𝜉𝑗	∀(𝜉𝑗	PROPN
cana-5869	104	62	)	)	PUNCT
cana-5869	104	63	∈	∈	PROPN
cana-5869	104	64	ℵ	ℵ	NOUN
cana-5869	104	65	hence	hence	ADV
cana-5869	104	66	,	,	PUNCT
cana-5869	104	67	(	(	PUNCT
cana-5869	104	68	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	104	69	)	)	PUNCT
cana-5869	104	70	∈	∈	PROPN
cana-5869	104	71	ℵ𝛿	ℵ𝛿	ADP
cana-5869	104	72	⇔	⇔	PROPN
cana-5869	104	73	(	(	PUNCT
cana-5869	104	74	1𝜂𝑗𝑒1	1𝜂𝑗𝑒1	PROPN
cana-5869	104	75	)	)	PUNCT
cana-5869	104	76	∈	∈	PROPN
cana-5869	104	77	(	(	PUNCT
cana-5869	104	78	1ℵ)𝛿	1ℵ)𝛿	NUM
cana-5869	104	79	and	and	CCONJ
cana-5869	104	80	(	(	PUNCT
cana-5869	104	81	2𝜂𝑗𝑒2	2𝜂𝑗𝑒2	X
cana-5869	104	82	)	)	PUNCT
cana-5869	104	83	∈	∈	PROPN
cana-5869	104	84	(	(	PUNCT
cana-5869	104	85	2ℵ)𝛿	2ℵ)𝛿	NUM
cana-5869	104	86	,	,	PUNCT
cana-5869	104	87	by	by	ADP
cana-5869	104	88	(	(	PUNCT
cana-5869	104	89	vii	vii	PROPN
cana-5869	104	90	)	)	PUNCT
cana-5869	104	91	and	and	CCONJ
cana-5869	104	92	(	(	PUNCT
cana-5869	104	93	viii	viii	NOUN
cana-5869	104	94	)	)	PUNCT
cana-5869	104	95	.	.	PUNCT
cana-5869	105	1	(	(	PUNCT
cana-5869	105	2	d	d	X
cana-5869	105	3	)	)	PUNCT
cana-5869	105	4	communications	communication	NOUN
cana-5869	105	5	on	on	ADP
cana-5869	105	6	applied	apply	VERB
cana-5869	105	7	nonlinear	nonlinear	ADJ
cana-5869	105	8	analysis	analysis	NOUN
cana-5869	105	9	issn	issn	NOUN
cana-5869	105	10	:	:	PUNCT
cana-5869	105	11	1074	1074	NUM
cana-5869	105	12	-	-	PUNCT
cana-5869	105	13	133x	133x	NUM
cana-5869	105	14	vol	vol	NOUN
cana-5869	105	15	31	31	NUM
cana-5869	105	16	no	no	NOUN
cana-5869	105	17	.	.	PUNCT
cana-5869	106	1	2s	2s	NUM
cana-5869	106	2	(	(	PUNCT
cana-5869	106	3	2024	2024	NUM
cana-5869	106	4	)	)	PUNCT
cana-5869	106	5	763	763	NUM
cana-5869	106	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5869	106	7	corollary	corollary	NOUN
cana-5869	106	8	2	2	NUM
cana-5869	106	9	:	:	PUNCT
cana-5869	106	10	from	from	ADP
cana-5869	106	11	(	(	PUNCT
cana-5869	106	12	d	d	NOUN
cana-5869	106	13	)	)	PUNCT
cana-5869	106	14	and	and	CCONJ
cana-5869	106	15	the	the	DET
cana-5869	106	16	definitions	definition	NOUN
cana-5869	106	17	of	of	ADP
cana-5869	106	18	t1	t1	NOUN
cana-5869	106	19	and	and	CCONJ
cana-5869	106	20	t2	t2	NOUN
cana-5869	106	21	we	we	PRON
cana-5869	106	22	can	can	AUX
cana-5869	106	23	say	say	VERB
cana-5869	106	24	that	that	PRON
cana-5869	106	25	(	(	PUNCT
cana-5869	106	26	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	106	27	)	)	PUNCT
cana-5869	106	28	∈	∈	NOUN
cana-5869	106	29	ℵ𝛿	ℵ𝛿	ADP
cana-5869	106	30	if	if	SCONJ
cana-5869	106	31	and	and	CCONJ
cana-5869	106	32	only	only	ADV
cana-5869	106	33	if	if	SCONJ
cana-5869	106	34	𝑇1(𝜂𝑗	𝑇1(𝜂𝑗	NUM
cana-5869	106	35	)	)	PUNCT
cana-5869	106	36	∈	∈	PROPN
cana-5869	106	37	(	(	PUNCT
cana-5869	106	38	1ℵ	1ℵ	NOUN
cana-5869	106	39	)	)	PUNCT
cana-5869	106	40	,	,	PUNCT
cana-5869	106	41	𝑇2(𝜂𝑗	𝑇2(𝜂𝑗	PROPN
cana-5869	106	42	)	)	PUNCT
cana-5869	106	43	∈	∈	PROPN
cana-5869	106	44	(	(	PUNCT
cana-5869	106	45	2ℵ	2ℵ	NUM
cana-5869	106	46	)	)	PUNCT
cana-5869	106	47	or	or	CCONJ
cana-5869	106	48	in	in	ADP
cana-5869	106	49	words	word	NOUN
cana-5869	106	50	we	we	PRON
cana-5869	106	51	can	can	AUX
cana-5869	106	52	also	also	ADV
cana-5869	106	53	say	say	VERB
cana-5869	106	54	that	that	SCONJ
cana-5869	106	55	a	a	DET
cana-5869	106	56	sequence	sequence	NOUN
cana-5869	106	57	belongs	belong	VERB
cana-5869	106	58	to	to	ADP
cana-5869	106	59	𝛿	𝛿	PROPN
cana-5869	106	60	dual	dual	ADJ
cana-5869	106	61	of	of	ADP
cana-5869	106	62	the	the	DET
cana-5869	106	63	class	class	NOUN
cana-5869	106	64	ℵ	ℵ	NOUN
cana-5869	106	65	if	if	SCONJ
cana-5869	106	66	and	and	CCONJ
cana-5869	106	67	only	only	ADV
cana-5869	106	68	if	if	SCONJ
cana-5869	106	69	its	its	PRON
cana-5869	106	70	t1	t1	NOUN
cana-5869	106	71	–	–	PUNCT
cana-5869	106	72	image	image	NOUN
cana-5869	106	73	belongs	belong	VERB
cana-5869	106	74	to	to	ADP
cana-5869	106	75	𝛿	𝛿	PROPN
cana-5869	106	76	dual	dual	ADJ
cana-5869	106	77	of	of	ADP
cana-5869	106	78	the	the	DET
cana-5869	106	79	first	first	ADJ
cana-5869	106	80	idempotent	idempotent	ADJ
cana-5869	106	81	component	component	NOUN
cana-5869	106	82	of	of	ADP
cana-5869	106	83	ℵ	ℵ	NOUN
cana-5869	106	84	or	or	CCONJ
cana-5869	106	85	its	its	PRON
cana-5869	106	86	first	first	ADJ
cana-5869	106	87	subclass	subclass	NOUN
cana-5869	106	88	and	and	CCONJ
cana-5869	106	89	the	the	DET
cana-5869	106	90	t2	t2	NOUN
cana-5869	106	91	–	–	PUNCT
cana-5869	106	92	image	image	NOUN
cana-5869	106	93	belongs	belong	VERB
cana-5869	106	94	to	to	ADP
cana-5869	106	95	the	the	DET
cana-5869	106	96	𝛿	𝛿	ADJ
cana-5869	106	97	dual	dual	ADJ
cana-5869	106	98	of	of	ADP
cana-5869	106	99	its	its	PRON
cana-5869	106	100	second	second	ADJ
cana-5869	106	101	idempotent	idempotent	ADJ
cana-5869	106	102	component	component	NOUN
cana-5869	106	103	or	or	CCONJ
cana-5869	106	104	its	its	PRON
cana-5869	106	105	second	second	ADJ
cana-5869	106	106	subclass	subclass	NOUN
cana-5869	106	107	.	.	PUNCT
cana-5869	107	1	theorem	theorem	VERB
cana-5869	107	2	4.5	4.5	NUM
cana-5869	107	3	:	:	PUNCT
cana-5869	107	4	(	(	PUNCT
cana-5869	107	5	i)ℵ𝛿	i)ℵ𝛿	PROPN
cana-5869	107	6	⊂	⊂	PROPN
cana-5869	107	7	(	(	PUNCT
cana-5869	107	8	1ℵ)𝛿.	1ℵ)𝛿.	NUM
cana-5869	107	9	(	(	PUNCT
cana-5869	107	10	ii	ii	NOUN
cana-5869	107	11	)	)	PUNCT
cana-5869	107	12	ℵ𝛿	ℵ𝛿	VERB
cana-5869	108	1	⊂	⊂	PROPN
cana-5869	108	2	(	(	PUNCT
cana-5869	108	3	2ℵ)𝛿	2ℵ)𝛿	NUM
cana-5869	108	4	proof	proof	NOUN
cana-5869	108	5	:	:	PUNCT
cana-5869	108	6	can	can	AUX
cana-5869	108	7	be	be	AUX
cana-5869	108	8	shown	show	VERB
cana-5869	108	9	easily	easily	ADV
cana-5869	108	10	with	with	ADP
cana-5869	108	11	the	the	DET
cana-5869	108	12	help	help	NOUN
cana-5869	108	13	of	of	ADP
cana-5869	108	14	similar	similar	ADJ
cana-5869	108	15	example	example	NOUN
cana-5869	108	16	.	.	PUNCT
cana-5869	109	1	note	note	VERB
cana-5869	109	2	3	3	NUM
cana-5869	109	3	:	:	PUNCT
cana-5869	109	4	delta	delta	NOUN
cana-5869	109	5	dual	dual	ADV
cana-5869	109	6	of	of	ADP
cana-5869	109	7	the	the	DET
cana-5869	109	8	class	class	NOUN
cana-5869	109	9	ℵ	ℵ	NOUN
cana-5869	109	10	is	be	AUX
cana-5869	109	11	properly	properly	ADV
cana-5869	109	12	contained	contain	VERB
cana-5869	109	13	in	in	ADP
cana-5869	109	14	the	the	DET
cana-5869	109	15	delta	delta	NOUN
cana-5869	109	16	dual	dual	ADV
cana-5869	109	17	of	of	ADP
cana-5869	109	18	its	its	PRON
cana-5869	109	19	idempotent	idempotent	ADJ
cana-5869	109	20	parts	part	NOUN
cana-5869	109	21	/	/	SYM
cana-5869	109	22	t1	t1	NOUN
cana-5869	109	23	and	and	CCONJ
cana-5869	109	24	t2	t2	NOUN
cana-5869	109	25	images	image	NOUN
cana-5869	109	26	.	.	PUNCT
cana-5869	110	1	constructing	construct	VERB
cana-5869	110	2	other	other	ADJ
cana-5869	110	3	counter	counter	NOUN
cana-5869	110	4	–	–	PUNCT
cana-5869	110	5	examples	example	NOUN
cana-5869	110	6	we	we	PRON
cana-5869	110	7	can	can	AUX
cana-5869	110	8	also	also	ADV
cana-5869	110	9	construct	construct	VERB
cana-5869	110	10	other	other	ADJ
cana-5869	110	11	counter	counter	NOUN
cana-5869	110	12	–	–	PUNCT
cana-5869	110	13	examples	example	NOUN
cana-5869	110	14	.	.	PUNCT
cana-5869	111	1	for	for	ADP
cana-5869	111	2	instance	instance	NOUN
cana-5869	111	3	,	,	PUNCT
cana-5869	111	4	for	for	ADP
cana-5869	111	5	part	part	NOUN
cana-5869	111	6	(	(	PUNCT
cana-5869	111	7	i	i	NOUN
cana-5869	111	8	)	)	PUNCT
cana-5869	111	9	of	of	ADP
cana-5869	111	10	the	the	DET
cana-5869	111	11	theorems	theorem	NOUN
cana-5869	111	12	4.1	4.1	NUM
cana-5869	111	13	,	,	PUNCT
cana-5869	111	14	4.3	4.3	NUM
cana-5869	111	15	and	and	CCONJ
cana-5869	111	16	4.5	4.5	NUM
cana-5869	111	17	,	,	PUNCT
cana-5869	111	18	we	we	PRON
cana-5869	111	19	can	can	AUX
cana-5869	111	20	take	take	VERB
cana-5869	111	21	(	(	PUNCT
cana-5869	111	22	𝜉𝑗	𝜉𝑗	NOUN
cana-5869	111	23	)	)	PUNCT
cana-5869	111	24	=	=	SYM
cana-5869	111	25	𝑗−𝑗	𝑗−𝑗	PROPN
cana-5869	111	26	(	(	PUNCT
cana-5869	111	27	1	1	NUM
cana-5869	111	28	𝑗3	𝑗3	NOUN
cana-5869	111	29	𝑒1	𝑒1	NOUN
cana-5869	111	30	+	+	CCONJ
cana-5869	111	31	1	1	NUM
cana-5869	111	32	𝑗4	𝑗4	PROPN
cana-5869	111	33	𝑒2	𝑒2	NOUN
cana-5869	111	34	)	)	PUNCT
cana-5869	111	35	=	=	SYM
cana-5869	111	36	1	1	NUM
cana-5869	111	37	𝑗3+𝑗	𝑗3+𝑗	PROPN
cana-5869	111	38	𝑒1	𝑒1	NOUN
cana-5869	111	39	+	+	CCONJ
cana-5869	111	40	1	1	NUM
cana-5869	111	41	𝑗4+𝑗	𝑗4+𝑗	NOUN
cana-5869	111	42	𝑒2	𝑒2	NOUN
cana-5869	111	43	,	,	PUNCT
cana-5869	111	44	(	(	PUNCT
cana-5869	111	45	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	111	46	)	)	PUNCT
cana-5869	111	47	=	=	SYM
cana-5869	111	48	(	(	PUNCT
cana-5869	111	49	1	1	NUM
cana-5869	111	50	𝑗2	𝑗2	PROPN
cana-5869	111	51	𝑒1	𝑒1	NOUN
cana-5869	111	52	+	+	CCONJ
cana-5869	111	53	𝑗𝑗+3𝑒2	𝑗𝑗+3𝑒2	NOUN
cana-5869	111	54	)	)	PUNCT
cana-5869	111	55	for	for	ADP
cana-5869	111	56	part	part	NOUN
cana-5869	111	57	(	(	PUNCT
cana-5869	111	58	ii	ii	NOUN
cana-5869	111	59	)	)	PUNCT
cana-5869	111	60	,	,	PUNCT
cana-5869	111	61	take	take	VERB
cana-5869	111	62	(	(	PUNCT
cana-5869	111	63	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	111	64	)	)	PUNCT
cana-5869	111	65	=	=	SYM
cana-5869	112	1	(	(	PUNCT
cana-5869	112	2	𝑗𝑗+2𝑒1	𝑗𝑗+2𝑒1	PUNCT
cana-5869	112	3	+	+	CCONJ
cana-5869	112	4	1	1	NUM
cana-5869	112	5	𝑗2	𝑗2	PROPN
cana-5869	112	6	𝑒2	𝑒2	PROPN
cana-5869	112	7	)	)	PUNCT
cana-5869	112	8	or	or	CCONJ
cana-5869	112	9	(	(	PUNCT
cana-5869	112	10	𝑗𝑗+2𝑒1	𝑗𝑗+2𝑒1	PUNCT
cana-5869	112	11	+	+	CCONJ
cana-5869	112	12	1	1	NUM
cana-5869	112	13	𝑗3	𝑗3	PROPN
cana-5869	112	14	𝑒2	𝑒2	PROPN
cana-5869	112	15	)	)	PUNCT
cana-5869	112	16	generalized	generalize	VERB
cana-5869	112	17	counter	counter	NOUN
cana-5869	112	18	–	–	PUNCT
cana-5869	112	19	example	example	NOUN
cana-5869	112	20	,	,	PUNCT
cana-5869	112	21	for	for	ADP
cana-5869	112	22	part	part	NOUN
cana-5869	112	23	(	(	PUNCT
cana-5869	112	24	i	i	NOUN
cana-5869	112	25	)	)	PUNCT
cana-5869	112	26	of	of	ADP
cana-5869	112	27	the	the	DET
cana-5869	112	28	theorems	theorem	NOUN
cana-5869	112	29	4.1	4.1	NUM
cana-5869	112	30	,	,	PUNCT
cana-5869	112	31	4.3	4.3	NUM
cana-5869	112	32	and	and	CCONJ
cana-5869	112	33	4.5	4.5	NUM
cana-5869	112	34	,	,	PUNCT
cana-5869	112	35	take	take	VERB
cana-5869	112	36	(	(	PUNCT
cana-5869	112	37	𝜉𝑗	𝜉𝑗	NOUN
cana-5869	112	38	)	)	PUNCT
cana-5869	112	39	=	=	SYM
cana-5869	112	40	𝑗−𝑗	𝑗−𝑗	PROPN
cana-5869	112	41	(	(	PUNCT
cana-5869	112	42	1	1	NUM
cana-5869	112	43	𝑗𝑛	𝑗𝑛	ADP
cana-5869	112	44	𝑒1	𝑒1	NOUN
cana-5869	113	1	+	+	CCONJ
cana-5869	113	2	1	1	NUM
cana-5869	113	3	𝑗𝑛+1	𝑗𝑛+1	NUM
cana-5869	113	4	𝑒2	𝑒2	NOUN
cana-5869	113	5	)	)	PUNCT
cana-5869	114	1	=	=	SYM
cana-5869	114	2	1	1	NUM
cana-5869	114	3	𝑗𝑛+𝑗	𝑗𝑛+𝑗	NOUN
cana-5869	114	4	𝑒1	𝑒1	NOUN
cana-5869	114	5	+	+	CCONJ
cana-5869	114	6	1	1	NUM
cana-5869	114	7	𝑗𝑛+1+𝑗	𝑗𝑛+1+𝑗	NUM
cana-5869	114	8	𝑒2	𝑒2	PROPN
cana-5869	114	9	,	,	PUNCT
cana-5869	114	10	𝑛	𝑛	DET
cana-5869	114	11	≥	≥	NUM
cana-5869	114	12	2	2	NUM
cana-5869	114	13	(	(	PUNCT
cana-5869	114	14	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	114	15	)	)	PUNCT
cana-5869	114	16	=	=	SYM
cana-5869	114	17	(	(	PUNCT
cana-5869	114	18	1	1	NUM
cana-5869	114	19	𝑗𝑛−1	𝑗𝑛−1	PROPN
cana-5869	114	20	𝑒1	𝑒1	NOUN
cana-5869	114	21	+	+	CCONJ
cana-5869	114	22	𝑗𝑗+𝑛𝑒2	𝑗𝑗+𝑛𝑒2	PROPN
cana-5869	114	23	)	)	PUNCT
cana-5869	114	24	for	for	ADP
cana-5869	114	25	part	part	NOUN
cana-5869	114	26	(	(	PUNCT
cana-5869	114	27	ii	ii	NOUN
cana-5869	114	28	)	)	PUNCT
cana-5869	114	29	,	,	PUNCT
cana-5869	114	30	take	take	VERB
cana-5869	114	31	(	(	PUNCT
cana-5869	114	32	𝜉𝑗	𝜉𝑗	NOUN
cana-5869	114	33	)	)	PUNCT
cana-5869	114	34	=	=	SYM
cana-5869	114	35	𝑗−𝑗	𝑗−𝑗	PROPN
cana-5869	114	36	(	(	PUNCT
cana-5869	114	37	1	1	NUM
cana-5869	114	38	𝑗𝑛	𝑗𝑛	ADP
cana-5869	114	39	𝑒1	𝑒1	NOUN
cana-5869	114	40	+	+	CCONJ
cana-5869	114	41	1	1	NUM
cana-5869	114	42	𝑗𝑛+1	𝑗𝑛+1	NUM
cana-5869	114	43	𝑒2	𝑒2	NOUN
cana-5869	114	44	)	)	PUNCT
cana-5869	114	45	=	=	SYM
cana-5869	114	46	1	1	NUM
cana-5869	114	47	𝑗𝑛+𝑗	𝑗𝑛+𝑗	NOUN
cana-5869	114	48	𝑒1	𝑒1	NOUN
cana-5869	114	49	+	+	CCONJ
cana-5869	114	50	1	1	NUM
cana-5869	114	51	𝑗𝑛+1+𝑗	𝑗𝑛+1+𝑗	NUM
cana-5869	114	52	𝑒2	𝑒2	PROPN
cana-5869	114	53	,	,	PUNCT
cana-5869	114	54	𝑛	𝑛	PRON
cana-5869	114	55	≥	≥	NUM
cana-5869	114	56	2	2	NUM
cana-5869	114	57	(	(	PUNCT
cana-5869	114	58	𝜂𝑗	𝜂𝑗	NOUN
cana-5869	114	59	)	)	PUNCT
cana-5869	114	60	=	=	SYM
cana-5869	114	61	(	(	PUNCT
cana-5869	114	62	𝑗𝑗+𝑛−1𝑒1	𝑗𝑗+𝑛−1𝑒1	NOUN
cana-5869	114	63	+	+	CCONJ
cana-5869	114	64	1	1	NUM
cana-5869	114	65	𝑗𝑛−2	𝑗𝑛−2	PROPN
cana-5869	114	66	𝑒2	𝑒2	NOUN
cana-5869	114	67	)	)	PUNCT
cana-5869	114	68	.	.	PUNCT
cana-5869	115	1	communications	communication	NOUN
cana-5869	115	2	on	on	ADP
cana-5869	115	3	applied	apply	VERB
cana-5869	115	4	nonlinear	nonlinear	ADJ
cana-5869	115	5	analysis	analysis	NOUN
cana-5869	115	6	issn	issn	NOUN
cana-5869	115	7	:	:	PUNCT
cana-5869	115	8	1074	1074	NUM
cana-5869	115	9	-	-	PUNCT
cana-5869	115	10	133x	133x	NUM
cana-5869	115	11	vol	vol	NOUN
cana-5869	115	12	31	31	NUM
cana-5869	115	13	no	no	NOUN
cana-5869	115	14	.	.	PUNCT
cana-5869	116	1	2s	2s	NUM
cana-5869	116	2	(	(	PUNCT
cana-5869	116	3	2024	2024	NUM
cana-5869	116	4	)	)	PUNCT
cana-5869	116	5	764	764	NUM
cana-5869	116	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5869	116	7	conclusion	conclusion	NOUN
cana-5869	116	8	:	:	PUNCT
cana-5869	116	9	we	we	PRON
cana-5869	116	10	have	have	AUX
cana-5869	116	11	studied	study	VERB
cana-5869	116	12	beta	beta	NOUN
cana-5869	116	13	,	,	PUNCT
cana-5869	116	14	gamma	gamma	NOUN
cana-5869	116	15	and	and	CCONJ
cana-5869	116	16	delta	delta	NOUN
cana-5869	116	17	duals	dual	NOUN
cana-5869	116	18	of	of	ADP
cana-5869	116	19	the	the	DET
cana-5869	116	20	idempotent	idempotent	ADJ
cana-5869	116	21	subspaces	subspace	NOUN
cana-5869	116	22	of	of	ADP
cana-5869	116	23	an	an	DET
cana-5869	116	24	entire	entire	ADJ
cana-5869	116	25	bicomplex	bicomplex	NOUN
cana-5869	116	26	sequence	sequence	NOUN
cana-5869	116	27	space	space	NOUN
cana-5869	116	28	and	and	CCONJ
cana-5869	116	29	shown	show	VERB
cana-5869	116	30	that	that	SCONJ
cana-5869	116	31	these	these	DET
cana-5869	116	32	duals	dual	NOUN
cana-5869	116	33	of	of	ADP
cana-5869	116	34	idempotent	idempotent	ADJ
cana-5869	116	35	sequences	sequence	NOUN
cana-5869	116	36	are	be	AUX
cana-5869	116	37	properly	properly	ADV
cana-5869	116	38	contained	contain	VERB
cana-5869	116	39	in	in	ADP
cana-5869	116	40	the	the	DET
cana-5869	116	41	dual	dual	ADJ
cana-5869	116	42	of	of	ADP
cana-5869	116	43	our	our	PRON
cana-5869	116	44	original	original	ADJ
cana-5869	116	45	space	space	NOUN
cana-5869	116	46	.	.	PUNCT
cana-5869	117	1	we	we	PRON
cana-5869	117	2	have	have	AUX
cana-5869	117	3	also	also	ADV
cana-5869	117	4	given	give	VERB
cana-5869	117	5	the	the	DET
cana-5869	117	6	generalized	generalized	ADJ
cana-5869	117	7	example	example	NOUN
cana-5869	117	8	for	for	ADP
cana-5869	117	9	this	this	DET
cana-5869	117	10	proper	proper	ADJ
cana-5869	117	11	containment	containment	NOUN
cana-5869	117	12	.	.	PUNCT
cana-5869	118	1	conflict	conflict	NOUN
cana-5869	118	2	of	of	ADP
cana-5869	118	3	interest	interest	NOUN
cana-5869	118	4	:	:	PUNCT
cana-5869	118	5	there	there	PRON
cana-5869	118	6	is	be	VERB
cana-5869	118	7	no	no	DET
cana-5869	118	8	conflict	conflict	NOUN
cana-5869	118	9	of	of	ADP
cana-5869	118	10	interest	interest	NOUN
cana-5869	118	11	.	.	PUNCT
cana-5869	119	1	references	reference	NOUN
cana-5869	119	2	[	[	X
cana-5869	119	3	1	1	NUM
cana-5869	119	4	]	]	X
cana-5869	119	5	garling	garling	NOUN
cana-5869	119	6	,	,	PUNCT
cana-5869	119	7	d.g.h	d.g.h	PROPN
cana-5869	119	8	.	.	PUNCT
cana-5869	119	9	,	,	PUNCT
cana-5869	119	10	“	"	PUNCT
cana-5869	119	11	the	the	DET
cana-5869	119	12	β	β	X
cana-5869	119	13	and	and	CCONJ
cana-5869	119	14	γ	γ	PROPN
cana-5869	119	15	duality	duality	NOUN
cana-5869	119	16	”	"	PUNCT
cana-5869	119	17	,	,	PUNCT
cana-5869	119	18	proc	proc	NOUN
cana-5869	119	19	.	.	PUNCT
cana-5869	120	1	cambridge	cambridge	PROPN
cana-5869	120	2	philos	philos	PROPN
cana-5869	120	3	.	.	PUNCT
cana-5869	120	4	soc	soc	PROPN
cana-5869	120	5	.	.	PUNCT
cana-5869	121	1	,	,	PUNCT
cana-5869	121	2	1967	1967	NUM
cana-5869	121	3	,	,	PUNCT
cana-5869	121	4	963	963	NUM
cana-5869	121	5	–	–	PUNCT
cana-5869	121	6	981	981	NUM
cana-5869	121	7	.	.	PUNCT
cana-5869	122	1	[	[	X
cana-5869	122	2	2	2	NUM
cana-5869	122	3	]	]	X
cana-5869	122	4	garling	garling	NOUN
cana-5869	122	5	,	,	PUNCT
cana-5869	122	6	d.g.h	d.g.h	NOUN
cana-5869	122	7	,	,	PUNCT
cana-5869	122	8	“	"	PUNCT
cana-5869	122	9	on	on	ADP
cana-5869	122	10	symmetric	symmetric	ADJ
cana-5869	122	11	sequence	sequence	NOUN
cana-5869	122	12	spaces	space	NOUN
cana-5869	122	13	”	"	PUNCT
cana-5869	122	14	,	,	PUNCT
cana-5869	122	15	proc	proc	NOUN
cana-5869	122	16	.	.	PUNCT
cana-5869	123	1	london	london	PROPN
cana-5869	123	2	math	math	PROPN
cana-5869	123	3	.	.	PUNCT
cana-5869	124	1	soc	soc	PROPN
cana-5869	124	2	.	.	PUNCT
cana-5869	125	1	16	16	NUM
cana-5869	125	2	(	(	PUNCT
cana-5869	125	3	3	3	NUM
cana-5869	125	4	)	)	PUNCT
cana-5869	125	5	,	,	PUNCT
cana-5869	125	6	1966	1966	NUM
cana-5869	125	7	,	,	PUNCT
cana-5869	125	8	85	85	NUM
cana-5869	125	9	–	–	PUNCT
cana-5869	125	10	106	106	NUM
cana-5869	125	11	.	.	PUNCT
cana-5869	126	1	[	[	X
cana-5869	126	2	3	3	NUM
cana-5869	126	3	]	]	X
cana-5869	126	4	köthe	köthe	NOUN
cana-5869	126	5	,	,	PUNCT
cana-5869	126	6	g	g	PROPN
cana-5869	126	7	&	&	CCONJ
cana-5869	126	8	toeplitz	toeplitz	PROPN
cana-5869	126	9	,	,	PUNCT
cana-5869	126	10	o.	o.	INTJ
cana-5869	126	11	,	,	PUNCT
cana-5869	126	12	“	"	PUNCT
cana-5869	126	13	lineare	lineare	ADJ
cana-5869	126	14	raume	raume	NOUN
cana-5869	126	15	mit	mit	PROPN
cana-5869	126	16	unendlich	unendlich	PROPN
cana-5869	126	17	vielen	vielen	VERB
cana-5869	126	18	koordinaten	koordinaten	VERB
cana-5869	126	19	und	und	NOUN
cana-5869	126	20	ringe	ringe	PROPN
cana-5869	126	21	unendlicher	unendlicher	ADJ
cana-5869	126	22	matrizen	matrizen	PROPN
cana-5869	126	23	”	"	PUNCT
cana-5869	126	24	,	,	PUNCT
cana-5869	126	25	jour	jour	X
cana-5869	126	26	.	.	PROPN
cana-5869	126	27	reine	reine	PROPN
cana-5869	126	28	angew	angew	PROPN
cana-5869	126	29	.	.	PUNCT
cana-5869	127	1	math	math	NOUN
cana-5869	127	2	.	.	PUNCT
cana-5869	128	1	171	171	NUM
cana-5869	128	2	,	,	PUNCT
cana-5869	128	3	1934	1934	NUM
cana-5869	128	4	,	,	PUNCT
cana-5869	128	5	193	193	NUM
cana-5869	128	6	–	–	SYM
cana-5869	128	7	226	226	NUM
cana-5869	128	8	.	.	PUNCT
cana-5869	129	1	[	[	X
cana-5869	129	2	4	4	NUM
cana-5869	129	3	]	]	X
cana-5869	129	4	price	price	NOUN
cana-5869	129	5	,	,	PUNCT
cana-5869	129	6	g.	g.	PROPN
cana-5869	129	7	baley	baley	PROPN
cana-5869	129	8	,	,	PUNCT
cana-5869	129	9	“	"	PUNCT
cana-5869	129	10	an	an	DET
cana-5869	129	11	introduction	introduction	NOUN
cana-5869	129	12	to	to	ADP
cana-5869	129	13	multicomplex	multicomplex	ADJ
cana-5869	129	14	spaces	space	NOUN
cana-5869	129	15	and	and	CCONJ
cana-5869	129	16	functions	function	NOUN
cana-5869	129	17	”	"	PUNCT
cana-5869	129	18	,	,	PUNCT
cana-5869	129	19	marcel	marcel	PROPN
cana-5869	129	20	dekker	dekker	PROPN
cana-5869	129	21	,	,	PUNCT
cana-5869	129	22	inc	inc	PROPN
cana-5869	129	23	.	.	PROPN
cana-5869	129	24	,	,	PUNCT
cana-5869	129	25	1991	1991	NUM
cana-5869	129	26	.	.	PUNCT
cana-5869	130	1	[	[	X
cana-5869	130	2	5	5	NUM
cana-5869	130	3	]	]	PUNCT
cana-5869	130	4	ruckle	ruckle	NOUN
cana-5869	130	5	,	,	PUNCT
cana-5869	130	6	w	w	PROPN
cana-5869	130	7	,	,	PUNCT
cana-5869	130	8	“	"	PUNCT
cana-5869	130	9	on	on	ADP
cana-5869	130	10	the	the	DET
cana-5869	130	11	characterization	characterization	NOUN
cana-5869	130	12	of	of	ADP
cana-5869	130	13	sequence	sequence	NOUN
cana-5869	130	14	spaces	space	NOUN
cana-5869	130	15	associated	associate	VERB
cana-5869	130	16	with	with	ADP
cana-5869	130	17	schauder	schauder	NOUN
cana-5869	130	18	bases	basis	NOUN
cana-5869	130	19	”	"	PUNCT
cana-5869	130	20	,	,	PUNCT
cana-5869	130	21	studia	studia	PROPN
cana-5869	130	22	math	math	PROPN
cana-5869	130	23	.	.	PUNCT
cana-5869	130	24	,	,	PUNCT
cana-5869	130	25	28	28	NUM
cana-5869	130	26	,	,	PUNCT
cana-5869	130	27	1967	1967	NUM
cana-5869	130	28	,	,	PUNCT
cana-5869	130	29	279	279	NUM
cana-5869	130	30	–	–	PUNCT
cana-5869	130	31	288	288	NUM
cana-5869	130	32	.	.	PUNCT
cana-5869	131	1	[	[	X
cana-5869	131	2	6	6	NUM
cana-5869	131	3	]	]	PUNCT
cana-5869	131	4	segre	segre	PROPN
cana-5869	131	5	c.	c.	PROPN
cana-5869	131	6	,“le	,“le	PUNCT
cana-5869	131	7	rappresentazioni	rappresentazioni	PROPN
cana-5869	131	8	reali	reali	PROPN
cana-5869	131	9	delle	delle	PROPN
cana-5869	131	10	forme	forme	PROPN
cana-5869	131	11	complesse	complesse	PROPN
cana-5869	131	12	e	e	PROPN
cana-5869	131	13	gli	gli	NOUN
cana-5869	131	14	enti	enti	X
cana-5869	131	15	iperalgebrici	iperalgebrici	NOUN
cana-5869	131	16	”	"	PUNCT
cana-5869	131	17	,	,	PUNCT
cana-5869	131	18	math	math	NOUN
cana-5869	131	19	.	.	PUNCT
cana-5869	132	1	ann	ann	PROPN
cana-5869	132	2	.	.	PROPN
cana-5869	132	3	,	,	PUNCT
cana-5869	132	4	40	40	NUM
cana-5869	132	5	,	,	PUNCT
cana-5869	132	6	1892	1892	NUM
cana-5869	132	7	,	,	PUNCT
cana-5869	132	8	413	413	NUM
cana-5869	132	9	-	-	SYM
cana-5869	132	10	467	467	NUM
cana-5869	132	11	.	.	PUNCT
cana-5869	133	1	[	[	X
cana-5869	133	2	7	7	X
cana-5869	133	3	]	]	X
cana-5869	133	4	srivastava	srivastava	PROPN
cana-5869	133	5	,	,	PUNCT
cana-5869	133	6	rajiv	rajiv	PROPN
cana-5869	133	7	k.	k.	PROPN
cana-5869	133	8	,	,	PUNCT
cana-5869	133	9	“	"	PUNCT
cana-5869	133	10	certain	certain	ADJ
cana-5869	133	11	topological	topological	ADJ
cana-5869	133	12	aspects	aspect	NOUN
cana-5869	133	13	of	of	ADP
cana-5869	133	14	bicomplex	bicomplex	NOUN
cana-5869	133	15	space	space	NOUN
cana-5869	133	16	”	"	PUNCT
cana-5869	133	17	,	,	PUNCT
cana-5869	133	18	bull	bull	NOUN
cana-5869	133	19	.	.	PUNCT
cana-5869	134	1	pure	pure	PROPN
cana-5869	134	2	&	&	CCONJ
cana-5869	134	3	appl	appl	PROPN
cana-5869	134	4	.	.	PROPN
cana-5869	134	5	math	math	PROPN
cana-5869	134	6	.	.	PUNCT
cana-5869	135	1	dec	dec	PROPN
cana-5869	135	2	.	.	PROPN
cana-5869	135	3	,	,	PUNCT
cana-5869	135	4	2(2	2(2	NUM
cana-5869	135	5	)	)	PUNCT
cana-5869	135	6	,	,	PUNCT
cana-5869	135	7	2008	2008	NUM
cana-5869	135	8	,	,	PUNCT
cana-5869	135	9	222	222	NUM
cana-5869	135	10	–	–	SYM
cana-5869	135	11	234	234	NUM
cana-5869	135	12	.	.	PUNCT
cana-5869	136	1	[	[	X
cana-5869	136	2	8	8	NUM
cana-5869	136	3	]	]	X
cana-5869	136	4	srivastava	srivastava	PROPN
cana-5869	136	5	,	,	PUNCT
cana-5869	136	6	rajiv	rajiv	PROPN
cana-5869	136	7	k.	k.	PROPN
cana-5869	136	8	&	&	CCONJ
cana-5869	136	9	srivastava	srivastava	PROPN
cana-5869	136	10	,	,	PUNCT
cana-5869	136	11	naveen	naveen	PROPN
cana-5869	136	12	k.	k.	PROPN
cana-5869	136	13	,	,	PUNCT
cana-5869	136	14	“	"	PUNCT
cana-5869	136	15	on	on	ADP
cana-5869	136	16	a	a	DET
cana-5869	136	17	class	class	NOUN
cana-5869	136	18	of	of	ADP
cana-5869	136	19	entire	entire	ADJ
cana-5869	136	20	bicomplex	bicomplex	NOUN
cana-5869	136	21	sequences	sequence	NOUN
cana-5869	136	22	”	"	PUNCT
cana-5869	136	23	,	,	PUNCT
cana-5869	136	24	south	south	PROPN
cana-5869	136	25	east	east	PROPN
cana-5869	136	26	.	.	PUNCT
cana-5869	137	1	asian	asian	PROPN
cana-5869	137	2	j.	j.	PROPN
cana-5869	137	3	math	math	PROPN
cana-5869	137	4	&	&	CCONJ
cana-5869	137	5	math	math	PROPN
cana-5869	137	6	sc	sc	PROPN
cana-5869	137	7	.	.	PUNCT
cana-5869	137	8	5(3	5(3	NUM
cana-5869	137	9	)	)	PUNCT
cana-5869	137	10	,	,	PUNCT
cana-5869	137	11	2007	2007	NUM
cana-5869	137	12	,	,	PUNCT
cana-5869	137	13	47	47	NUM
cana-5869	137	14	-	-	SYM
cana-5869	137	15	68	68	NUM
cana-5869	137	16	.	.	PUNCT
cana-5869	138	1	[	[	X
cana-5869	138	2	9	9	NUM
cana-5869	138	3	]	]	PUNCT
cana-5869	138	4	wagh	wagh	NOUN
cana-5869	138	5	,	,	PUNCT
cana-5869	138	6	m.	m.	NOUN
cana-5869	138	7	a.	a.	NOUN
cana-5869	138	8	,	,	PUNCT
cana-5869	138	9	“	"	PUNCT
cana-5869	138	10	on	on	ADP
cana-5869	138	11	certain	certain	ADJ
cana-5869	138	12	spaces	space	NOUN
cana-5869	138	13	of	of	ADP
cana-5869	138	14	bicomplex	bicomplex	NOUN
cana-5869	138	15	sequences	sequence	NOUN
cana-5869	138	16	”	"	PUNCT
cana-5869	138	17	,	,	PUNCT
cana-5869	138	18	inter	inter	PROPN
cana-5869	138	19	.	.	PUNCT
cana-5869	139	1	j.	j.	PROPN
cana-5869	139	2	phy	phy	PROPN
cana-5869	139	3	.	.	PROPN
cana-5869	139	4	chem	chem	PROPN
cana-5869	139	5	.	.	PUNCT
cana-5869	140	1	math	math	NOUN
cana-5869	140	2	.	.	PUNCT
cana-5869	141	1	fundam	fundam	ADJ
cana-5869	141	2	,	,	PUNCT
cana-5869	141	3	2014	2014	NUM
cana-5869	141	4	,	,	PUNCT
cana-5869	141	5	7(1	7(1	NUM
cana-5869	141	6	)	)	PUNCT
cana-5869	141	7	,	,	PUNCT
cana-5869	141	8	1	1	NUM
cana-5869	141	9	-	-	SYM
cana-5869	141	10	6	6	NUM
cana-5869	141	11	.	.	PUNCT
cana-5869	142	1	[	[	X
cana-5869	142	2	10	10	NUM
cana-5869	142	3	]	]	X
cana-5869	142	4	wagh	wagh	PROPN
cana-5869	142	5	,	,	PUNCT
cana-5869	142	6	m.	m.	NOUN
cana-5869	142	7	a.	a.	PROPN
cana-5869	142	8	,	,	PUNCT
cana-5869	142	9	&	&	CCONJ
cana-5869	142	10	kumar	kumar	PROPN
cana-5869	142	11	,	,	PUNCT
cana-5869	142	12	s.	s.	PROPN
cana-5869	142	13	,	,	PUNCT
cana-5869	142	14	“	"	PUNCT
cana-5869	142	15	on	on	ADP
cana-5869	142	16	certain	certain	ADJ
cana-5869	142	17	bicomplex	bicomplex	NOUN
cana-5869	142	18	duals	dual	NOUN
cana-5869	142	19	”	"	PUNCT
cana-5869	142	20	,	,	PUNCT
cana-5869	142	21	global	global	PROPN
cana-5869	142	22	j.	j.	PROPN
cana-5869	142	23	sc	sc	PROPN
cana-5869	142	24	.	.	PROPN
cana-5869	142	25	&	&	CCONJ
cana-5869	142	26	frontier	frontier	PROPN
cana-5869	142	27	research	research	NOUN
cana-5869	142	28	(	(	PUNCT
cana-5869	142	29	gjsfr	gjsfr	NOUN
cana-5869	142	30	)	)	PUNCT
cana-5869	142	31	,	,	PUNCT
cana-5869	142	32	2014	2014	NUM
cana-5869	142	33	,	,	PUNCT
cana-5869	142	34	14(6	14(6	NOUN
cana-5869	142	35	)	)	PUNCT
cana-5869	142	36	,	,	PUNCT
cana-5869	142	37	17	17	NUM
cana-5869	142	38	–	–	SYM
cana-5869	142	39	23	23	NUM
cana-5869	142	40	.	.	PUNCT
cana-5869	143	1	[	[	X
cana-5869	143	2	11	11	NUM
cana-5869	143	3	]	]	PUNCT
cana-5869	143	4	wagh	wagh	PROPN
cana-5869	143	5	,	,	PUNCT
cana-5869	143	6	m.a	m.a	PROPN
cana-5869	143	7	.	.	PROPN
cana-5869	143	8	,	,	PUNCT
cana-5869	143	9	“	"	PUNCT
cana-5869	143	10	on	on	ADP
cana-5869	143	11	a	a	DET
cana-5869	143	12	class	class	NOUN
cana-5869	143	13	of	of	ADP
cana-5869	143	14	bicomplex	bicomplex	NOUN
cana-5869	143	15	sequences	sequence	NOUN
cana-5869	143	16	”	"	PUNCT
cana-5869	143	17	,	,	PUNCT
cana-5869	143	18	international	international	ADJ
cana-5869	143	19	journal	journal	NOUN
cana-5869	143	20	of	of	ADP
cana-5869	143	21	trends	trend	NOUN
cana-5869	143	22	in	in	ADP
cana-5869	143	23	mathematics	mathematic	NOUN
cana-5869	143	24	and	and	CCONJ
cana-5869	143	25	statisitics	statisitic	NOUN
cana-5869	143	26	,	,	PUNCT
cana-5869	143	27	2014	2014	NUM
cana-5869	143	28	,	,	PUNCT
cana-5869	143	29	3(5	3(5	NUM
cana-5869	143	30	)	)	PUNCT
cana-5869	143	31	,	,	PUNCT
cana-5869	143	32	158	158	NUM
cana-5869	143	33	–	–	SYM
cana-5869	143	34	171	171	NUM
cana-5869	143	35	.	.	PUNCT
cana-5869	144	1	[	[	X
cana-5869	144	2	12	12	NUM
cana-5869	144	3	]	]	PUNCT
cana-5869	144	4	wagh	wagh	PROPN
cana-5869	144	5	,	,	PUNCT
cana-5869	144	6	m.	m.	NOUN
cana-5869	144	7	a.	a.	PROPN
cana-5869	144	8	,	,	PUNCT
cana-5869	144	9	&	&	CCONJ
cana-5869	144	10	kumar	kumar	PROPN
cana-5869	144	11	,	,	PUNCT
cana-5869	144	12	s.	s.	PROPN
cana-5869	144	13	,	,	PUNCT
cana-5869	144	14	“	"	PUNCT
cana-5869	144	15	köthe	köthe	ADJ
cana-5869	144	16	toeplitz	toeplitz	NOUN
cana-5869	144	17	duals	dual	NOUN
cana-5869	144	18	of	of	ADP
cana-5869	144	19	certain	certain	ADJ
cana-5869	144	20	bicomplex	bicomplex	NOUN
cana-5869	144	21	sequence	sequence	NOUN
cana-5869	144	22	spaces	space	VERB
cana-5869	144	23	”	"	PUNCT
cana-5869	144	24	,	,	PUNCT
cana-5869	144	25	2014	2014	NUM
cana-5869	144	26	,	,	PUNCT
cana-5869	144	27	inter	inter	PROPN
cana-5869	144	28	.	.	PUNCT
cana-5869	145	1	j.	j.	PROPN
cana-5869	145	2	math	math	PROPN
cana-5869	145	3	.	.	PUNCT
cana-5869	145	4	&	&	CCONJ
cana-5869	145	5	comp	comp	PROPN
cana-5869	145	6	.	.	PUNCT
cana-5869	146	1	app	app	PROPN
cana-5869	146	2	.	.	PUNCT
cana-5869	147	1	research	research	NOUN
cana-5869	147	2	(	(	PUNCT
cana-5869	147	3	ijmcar	ijmcar	NOUN
cana-5869	147	4	)	)	PUNCT
cana-5869	147	5	,	,	PUNCT
cana-5869	147	6	4(3	4(3	NUM
cana-5869	147	7	)	)	PUNCT
cana-5869	147	8	,	,	PUNCT
cana-5869	147	9	87	87	NUM
cana-5869	147	10	–	–	SYM
cana-5869	147	11	98	98	NUM