id	sid	tid	token	lemma	pos
cana-1064	1	1	communications	communication	NOUN
cana-1064	1	2	on	on	ADP
cana-1064	1	3	applied	apply	VERB
cana-1064	1	4	nonlinear	nonlinear	ADJ
cana-1064	1	5	analysis	analysis	NOUN
cana-1064	1	6	issn	issn	NOUN
cana-1064	1	7	:	:	PUNCT
cana-1064	1	8	1074	1074	NUM
cana-1064	1	9	-	-	PUNCT
cana-1064	1	10	133x	133x	NUM
cana-1064	1	11	vol	vol	NOUN
cana-1064	1	12	31	31	NUM
cana-1064	1	13	no	no	NOUN
cana-1064	1	14	.	.	PUNCT
cana-1064	2	1	5s	5s	NUM
cana-1064	2	2	(	(	PUNCT
cana-1064	2	3	2024	2024	NUM
cana-1064	2	4	)	)	PUNCT
cana-1064	2	5	449	449	NUM
cana-1064	2	6	https://internationalpubls.com	https://internationalpubls.com	PROPN
cana-1064	2	7	matthews	matthews	PROPN
cana-1064	2	8	partial	partial	ADJ
cana-1064	2	9	metric	metric	ADJ
cana-1064	2	10	space	space	NOUN
cana-1064	2	11	using	use	VERB
cana-1064	2	12	𝓕-contraction	𝓕-contraction	PROPN
cana-1064	2	13	bonuga	bonuga	NOUN
cana-1064	2	14	vijayabaskerreddy1	vijayabaskerreddy1	PROPN
cana-1064	2	15	,	,	PUNCT
cana-1064	2	16	veladi	veladi	PROPN
cana-1064	2	17	srinivas	srinivas	PROPN
cana-1064	2	18	2	2	NUM
cana-1064	2	19	1	1	NUM
cana-1064	2	20	department	department	NOUN
cana-1064	2	21	of	of	ADP
cana-1064	2	22	mathematics	mathematic	NOUN
cana-1064	2	23	,	,	PUNCT
cana-1064	2	24	sreenidhi	sreenidhi	PROPN
cana-1064	2	25	institute	institute	PROPN
cana-1064	2	26	of	of	ADP
cana-1064	2	27	science	science	NOUN
cana-1064	2	28	and	and	CCONJ
cana-1064	2	29	technology	technology	NOUN
cana-1064	2	30	(	(	PUNCT
cana-1064	2	31	snist	snist	NOUN
cana-1064	2	32	)	)	PUNCT
cana-1064	2	33	,	,	PUNCT
cana-1064	2	34	hyderabad	hyderabad	PROPN
cana-1064	2	35	,	,	PUNCT
cana-1064	2	36	india	india	PROPN
cana-1064	2	37	.	.	PUNCT
cana-1064	3	1	email	email	NOUN
cana-1064	3	2	:	:	PUNCT
cana-1064	3	3	basker.bonuga@gmail.com	basker.bonuga@gmail.com	X
cana-1064	3	4	(	(	PUNCT
cana-1064	3	5	orcid	orcid	NOUN
cana-1064	3	6	i	i	NOUN
cana-1064	3	7	d	d	NOUN
cana-1064	3	8	:	:	PUNCT
cana-1064	3	9	0000	0000	NUM
cana-1064	3	10	-	-	PUNCT
cana-1064	3	11	0001	0001	NUM
cana-1064	3	12	-	-	PUNCT
cana-1064	3	13	6254	6254	NUM
cana-1064	3	14	-	-	PUNCT
cana-1064	3	15	3973	3973	NUM
cana-1064	3	16	)	)	PUNCT
cana-1064	3	17	2	2	NUM
cana-1064	3	18	department	department	NOUN
cana-1064	3	19	of	of	ADP
cana-1064	3	20	mathematics	mathematic	NOUN
cana-1064	3	21	,	,	PUNCT
cana-1064	3	22	university	university	NOUN
cana-1064	3	23	college	college	NOUN
cana-1064	3	24	of	of	ADP
cana-1064	3	25	science	science	PROPN
cana-1064	3	26	,	,	PUNCT
cana-1064	3	27	osmania	osmania	PROPN
cana-1064	3	28	university	university	PROPN
cana-1064	3	29	,	,	PUNCT
cana-1064	3	30	hyderabad	hyderabad	PROPN
cana-1064	3	31	,	,	PUNCT
cana-1064	3	32	india	india	PROPN
cana-1064	3	33	.	.	PUNCT
cana-1064	3	34	email	email	NOUN
cana-1064	3	35	:	:	PUNCT
cana-1064	3	36	srinivasmaths4141@gmail.com	srinivasmaths4141@gmail.com	X
cana-1064	4	1	(	(	PUNCT
cana-1064	4	2	orcid	orcid	NOUN
cana-1064	4	3	id:0000	id:0000	NUM
cana-1064	4	4	-	-	PUNCT
cana-1064	4	5	0003	0003	NUM
cana-1064	4	6	-	-	PUNCT
cana-1064	4	7	1991	1991	NUM
cana-1064	4	8	-	-	PUNCT
cana-1064	4	9	4569	4569	NUM
cana-1064	4	10	)	)	PUNCT
cana-1064	4	11	article	article	NOUN
cana-1064	4	12	history	history	NOUN
cana-1064	4	13	:	:	PUNCT
cana-1064	4	14	received	receive	VERB
cana-1064	4	15	:	:	PUNCT
cana-1064	4	16	12	12	NUM
cana-1064	4	17	-	-	PUNCT
cana-1064	4	18	05	05	NUM
cana-1064	4	19	-	-	PUNCT
cana-1064	4	20	2024	2024	NUM
cana-1064	4	21	revised	revise	VERB
cana-1064	4	22	:	:	PUNCT
cana-1064	4	23	22	22	NUM
cana-1064	4	24	-	-	SYM
cana-1064	4	25	06	06	NUM
cana-1064	4	26	-	-	PUNCT
cana-1064	4	27	2024	2024	NUM
cana-1064	4	28	accepted	accept	VERB
cana-1064	4	29	:	:	PUNCT
cana-1064	4	30	09	09	NUM
cana-1064	4	31	-	-	SYM
cana-1064	4	32	07	07	NUM
cana-1064	4	33	-	-	PUNCT
cana-1064	4	34	2024	2024	NUM
cana-1064	4	35	abstract	abstract	NOUN
cana-1064	4	36	:	:	PUNCT
cana-1064	4	37	following	follow	VERB
cana-1064	4	38	the	the	DET
cana-1064	4	39	notion	notion	NOUN
cana-1064	4	40	of	of	ADP
cana-1064	4	41	partial	partial	ADJ
cana-1064	4	42	metric	metric	ADJ
cana-1064	4	43	space	space	NOUN
cana-1064	4	44	(	(	PUNCT
cana-1064	4	45	briefly	briefly	NOUN
cana-1064	4	46	pms	pm	NOUN
cana-1064	4	47	)	)	PUNCT
cana-1064	4	48	established	establish	VERB
cana-1064	4	49	by	by	ADP
cana-1064	4	50	matthaws	matthaw	NOUN
cana-1064	4	51	[	[	X
cana-1064	4	52	1	1	NUM
cana-1064	4	53	]	]	PUNCT
cana-1064	4	54	,	,	PUNCT
cana-1064	4	55	in	in	ADP
cana-1064	4	56	this	this	DET
cana-1064	4	57	present	present	ADJ
cana-1064	4	58	research	research	NOUN
cana-1064	4	59	article	article	NOUN
cana-1064	4	60	,	,	PUNCT
cana-1064	4	61	we	we	PRON
cana-1064	4	62	proved	prove	VERB
cana-1064	4	63	common	common	ADJ
cana-1064	4	64	fixed	fix	VERB
cana-1064	4	65	point	point	NOUN
cana-1064	4	66	theorem	theorem	VERB
cana-1064	4	67	for	for	ADP
cana-1064	4	68	two	two	NUM
cana-1064	4	69	pair	pair	NOUN
cana-1064	4	70	of	of	ADP
cana-1064	4	71	self	self	NOUN
cana-1064	4	72	maps	map	NOUN
cana-1064	4	73	using	use	VERB
cana-1064	4	74	the	the	DET
cana-1064	4	75	weakly	weakly	ADJ
cana-1064	4	76	compatible	compatible	ADJ
cana-1064	4	77	mappings	mapping	NOUN
cana-1064	4	78	through	through	ADP
cana-1064	4	79	ℱ	ℱ	PROPN
cana-1064	4	80	-contraction	-contraction	NOUN
cana-1064	4	81	.	.	PUNCT
cana-1064	5	1	in	in	ADP
cana-1064	5	2	addition	addition	NOUN
cana-1064	5	3	,	,	PUNCT
cana-1064	5	4	we	we	PRON
cana-1064	5	5	give	give	VERB
cana-1064	5	6	an	an	DET
cana-1064	5	7	illustrative	illustrative	ADJ
cana-1064	5	8	example	example	NOUN
cana-1064	5	9	.	.	PUNCT
cana-1064	6	1	keywords	keyword	NOUN
cana-1064	6	2	:	:	PUNCT
cana-1064	6	3	partial	partial	ADJ
cana-1064	6	4	metric	metric	ADJ
cana-1064	6	5	space	space	NOUN
cana-1064	6	6	(	(	PUNCT
cana-1064	6	7	pms	pms	PROPN
cana-1064	6	8	)	)	PUNCT
cana-1064	6	9	,	,	PUNCT
cana-1064	6	10	ℱ-contraction	ℱ-contraction	PROPN
cana-1064	6	11	,	,	PUNCT
cana-1064	6	12	weakly	weakly	ADV
cana-1064	6	13	compatible	compatible	ADJ
cana-1064	6	14	(	(	PUNCT
cana-1064	6	15	wc	wc	PROPN
cana-1064	6	16	)	)	PUNCT
cana-1064	6	17	.	.	PUNCT
cana-1064	7	1	msc	msc	PROPN
cana-1064	7	2	(	(	PUNCT
cana-1064	7	3	2000	2000	NUM
cana-1064	7	4	):	):	PUNCT
cana-1064	7	5	54h25;47h10	54h25;47h10	NUM
cana-1064	7	6	.	.	NOUN
cana-1064	7	7	1	1	NUM
cana-1064	7	8	.	.	X
cana-1064	7	9	introduction	introduction	NOUN
cana-1064	7	10	as	as	ADP
cana-1064	7	11	a	a	DET
cana-1064	7	12	generalization	generalization	NOUN
cana-1064	7	13	of	of	ADP
cana-1064	7	14	metric	metric	ADJ
cana-1064	7	15	space	space	NOUN
cana-1064	7	16	,	,	PUNCT
cana-1064	7	17	the	the	DET
cana-1064	7	18	concept	concept	NOUN
cana-1064	7	19	of	of	ADP
cana-1064	7	20	partial	partial	ADJ
cana-1064	7	21	metric	metric	ADJ
cana-1064	7	22	space	space	NOUN
cana-1064	7	23	emphasized	emphasize	VERB
cana-1064	7	24	by	by	ADP
cana-1064	7	25	matthews	matthews	PROPN
cana-1064	7	26	[	[	X
cana-1064	7	27	1	1	X
cana-1064	7	28	]	]	PUNCT
cana-1064	7	29	in	in	ADP
cana-1064	7	30	the	the	DET
cana-1064	7	31	year	year	NOUN
cana-1064	7	32	1994.recently	1994.recently	ADV
cana-1064	7	33	,	,	PUNCT
cana-1064	7	34	many	many	ADJ
cana-1064	7	35	fixed	fix	VERB
cana-1064	7	36	point	point	NOUN
cana-1064	7	37	theory	theory	NOUN
cana-1064	7	38	researchers	researcher	NOUN
cana-1064	7	39	established	establish	VERB
cana-1064	7	40	fixed	fix	VERB
cana-1064	7	41	point	point	NOUN
cana-1064	7	42	theorems	theorem	NOUN
cana-1064	7	43	using	use	VERB
cana-1064	7	44	different	different	ADJ
cana-1064	7	45	contractions	contraction	NOUN
cana-1064	7	46	with	with	ADP
cana-1064	7	47	different	different	ADJ
cana-1064	7	48	weaker	weak	ADJ
cana-1064	7	49	conditions	condition	NOUN
cana-1064	7	50	.	.	PUNCT
cana-1064	8	1	the	the	DET
cana-1064	8	2	pme	pme	PROPN
cana-1064	8	3	play	play	VERB
cana-1064	8	4	an	an	DET
cana-1064	8	5	vital	vital	ADJ
cana-1064	8	6	role	role	NOUN
cana-1064	8	7	in	in	ADP
cana-1064	8	8	study	study	NOUN
cana-1064	8	9	of	of	ADP
cana-1064	8	10	data	datum	NOUN
cana-1064	8	11	flows	flow	VERB
cana-1064	8	12	network	network	NOUN
cana-1064	8	13	and	and	CCONJ
cana-1064	8	14	also	also	ADV
cana-1064	8	15	theory	theory	NOUN
cana-1064	8	16	of	of	ADP
cana-1064	8	17	computation	computation	NOUN
cana-1064	8	18	in	in	ADP
cana-1064	8	19	the	the	DET
cana-1064	8	20	computer	computer	NOUN
cana-1064	8	21	science	science	NOUN
cana-1064	8	22	.	.	PUNCT
cana-1064	9	1	some	some	DET
cana-1064	9	2	authors	author	NOUN
cana-1064	9	3	prove	prove	VERB
cana-1064	9	4	fixed	fixed	ADJ
cana-1064	9	5	point	point	NOUN
cana-1064	9	6	theorems	theorem	NOUN
cana-1064	9	7	in	in	ADP
cana-1064	9	8	pms	pm	NOUN
cana-1064	9	9	like	like	ADP
cana-1064	9	10	,	,	PUNCT
cana-1064	9	11	[	[	X
cana-1064	9	12	2],[3],[4],[6	2],[3],[4],[6	X
cana-1064	9	13	]	]	X
cana-1064	9	14	,	,	PUNCT
cana-1064	9	15	and	and	CCONJ
cana-1064	9	16	[	[	X
cana-1064	9	17	8	8	NUM
cana-1064	9	18	]	]	PUNCT
cana-1064	9	19	.	.	PUNCT
cana-1064	10	1	in	in	ADP
cana-1064	10	2	the	the	DET
cana-1064	10	3	metric	metric	ADJ
cana-1064	10	4	space	space	NOUN
cana-1064	10	5	the	the	DET
cana-1064	10	6	notion	notion	NOUN
cana-1064	10	7	of	of	ADP
cana-1064	10	8	f	f	PROPN
cana-1064	10	9	-	-	PUNCT
cana-1064	10	10	contraction	contraction	NOUN
cana-1064	10	11	proposed	propose	VERB
cana-1064	10	12	by	by	ADP
cana-1064	10	13	wardowski	wardowski	PROPN
cana-1064	10	14	[	[	X
cana-1064	10	15	5	5	NUM
cana-1064	10	16	]	]	PUNCT
cana-1064	10	17	,	,	PUNCT
cana-1064	10	18	which	which	PRON
cana-1064	10	19	is	be	AUX
cana-1064	10	20	generalization	generalization	NOUN
cana-1064	10	21	of	of	ADP
cana-1064	10	22	well	well	ADV
cana-1064	10	23	known	know	VERB
cana-1064	10	24	banach	banach	NOUN
cana-1064	10	25	contraction	contraction	NOUN
cana-1064	10	26	principle	principle	NOUN
cana-1064	10	27	.	.	PUNCT
cana-1064	11	1	recently	recently	ADV
cana-1064	11	2	,	,	PUNCT
cana-1064	11	3	nazam	nazam	PROPN
cana-1064	11	4	m	m	VERB
cana-1064	11	5	et.al	et.al	ADJ
cana-1064	11	6	,	,	PUNCT
cana-1064	11	7	proved	prove	VERB
cana-1064	11	8	common	common	ADJ
cana-1064	11	9	fixed	fix	VERB
cana-1064	11	10	point	point	NOUN
cana-1064	11	11	theorems	theorem	NOUN
cana-1064	11	12	using	use	VERB
cana-1064	11	13	one	one	NUM
cana-1064	11	14	and	and	CCONJ
cana-1064	11	15	two	two	NUM
cana-1064	11	16	self	self	NOUN
cana-1064	11	17	mappings	mapping	NOUN
cana-1064	11	18	concerning	concern	VERB
cana-1064	11	19	f	f	NOUN
cana-1064	11	20	-	-	PUNCT
cana-1064	11	21	contraction	contraction	NOUN
cana-1064	11	22	in	in	ADP
cana-1064	11	23	pms	pms	PROPN
cana-1064	11	24	[	[	X
cana-1064	11	25	7	7	NUM
cana-1064	11	26	]	]	PUNCT
cana-1064	11	27	.	.	PUNCT
cana-1064	12	1	on	on	ADP
cana-1064	12	2	the	the	DET
cana-1064	12	3	other	other	ADJ
cana-1064	12	4	hand	hand	NOUN
cana-1064	12	5	,	,	PUNCT
cana-1064	12	6	sessa	sessa	PROPN
cana-1064	12	7	initiated	initiate	VERB
cana-1064	12	8	the	the	DET
cana-1064	12	9	notation	notation	NOUN
cana-1064	12	10	of	of	ADP
cana-1064	12	11	weakly	weakly	ADJ
cana-1064	12	12	commuting	commuting	NOUN
cana-1064	12	13	maps	map	NOUN
cana-1064	12	14	which	which	PRON
cana-1064	12	15	generalized	generalize	VERB
cana-1064	12	16	the	the	DET
cana-1064	12	17	concept	concept	NOUN
cana-1064	12	18	of	of	ADP
cana-1064	12	19	commuting	commute	VERB
cana-1064	12	20	mappings	mapping	NOUN
cana-1064	12	21	consequently	consequently	ADV
cana-1064	12	22	jungck	jungck	VERB
cana-1064	12	23	g	g	PROPN
cana-1064	12	24	,	,	PUNCT
cana-1064	12	25	rhoades	rhoade	VERB
cana-1064	12	26	b	b	PROPN
cana-1064	12	27	e	e	PROPN
cana-1064	13	1	[	[	X
cana-1064	13	2	9	9	NUM
cana-1064	13	3	]	]	PUNCT
cana-1064	13	4	generalized	generalize	VERB
cana-1064	13	5	this	this	DET
cana-1064	13	6	idea	idea	NOUN
cana-1064	13	7	first	first	ADV
cana-1064	13	8	to	to	ADP
cana-1064	13	9	compatible	compatible	ADJ
cana-1064	13	10	mappings	mapping	NOUN
cana-1064	13	11	and	and	CCONJ
cana-1064	13	12	later	later	ADV
cana-1064	13	13	to	to	PART
cana-1064	13	14	weakly	weakly	ADV
cana-1064	13	15	compatible	compatible	ADJ
cana-1064	13	16	(	(	PUNCT
cana-1064	13	17	shortly	shortly	ADV
cana-1064	13	18	wc	wc	NOUN
cana-1064	13	19	)	)	PUNCT
cana-1064	13	20	mappings	mapping	NOUN
cana-1064	13	21	.	.	PUNCT
cana-1064	14	1	the	the	DET
cana-1064	14	2	aim	aim	NOUN
cana-1064	14	3	of	of	ADP
cana-1064	14	4	the	the	DET
cana-1064	14	5	research	research	NOUN
cana-1064	14	6	article	article	NOUN
cana-1064	14	7	is	be	AUX
cana-1064	14	8	to	to	PART
cana-1064	14	9	establish	establish	VERB
cana-1064	14	10	existence	existence	NOUN
cana-1064	14	11	of	of	ADP
cana-1064	14	12	unique	unique	ADJ
cana-1064	14	13	common	common	ADJ
cana-1064	14	14	fixed	fix	VERB
cana-1064	14	15	point	point	NOUN
cana-1064	14	16	theorem	theorem	VERB
cana-1064	14	17	for	for	ADP
cana-1064	14	18	four	four	NUM
cana-1064	14	19	self	self	NOUN
cana-1064	14	20	mappings	mapping	NOUN
cana-1064	14	21	through	through	ADP
cana-1064	14	22	f	f	PROPN
cana-1064	14	23	-contraction	-contraction	PROPN
cana-1064	14	24	using	use	VERB
cana-1064	14	25	the	the	DET
cana-1064	14	26	idea	idea	NOUN
cana-1064	14	27	of	of	ADP
cana-1064	14	28	wc	wc	PROPN
cana-1064	14	29	mappings	mapping	NOUN
cana-1064	14	30	in	in	ADP
cana-1064	14	31	pms	pms	PROPN
cana-1064	14	32	.	.	PUNCT
cana-1064	15	1	now	now	ADV
cana-1064	15	2	we	we	PRON
cana-1064	15	3	recall	recall	VERB
cana-1064	15	4	useful	useful	ADJ
cana-1064	15	5	fundamental	fundamental	ADJ
cana-1064	15	6	definitions	definition	NOUN
cana-1064	15	7	,	,	PUNCT
cana-1064	15	8	lemmas	lemma	NOUN
cana-1064	15	9	of	of	ADP
cana-1064	15	10	pms	pms	PROPN
cana-1064	15	11	.	.	PUNCT
cana-1064	16	1	definition	definition	NOUN
cana-1064	16	2	1.1[1	1.1[1	NUM
cana-1064	16	3	]	]	X
cana-1064	16	4	:	:	PUNCT
cana-1064	16	5	a	a	DET
cana-1064	16	6	partial	partial	ADJ
cana-1064	16	7	metric	metric	NOUN
cana-1064	16	8	on	on	ADP
cana-1064	16	9	non	non	ADJ
cana-1064	16	10	-	-	ADJ
cana-1064	16	11	empty	empty	ADJ
cana-1064	16	12	set	set	NOUN
cana-1064	16	13	𝔛	𝔛	PROPN
cana-1064	16	14	is	be	AUX
cana-1064	16	15	a	a	DET
cana-1064	16	16	function	function	NOUN
cana-1064	16	17	𝒫:𝔛	𝒫:𝔛	PROPN
cana-1064	16	18	×	×	NOUN
cana-1064	16	19	𝔛	𝔛	PROPN
cana-1064	16	20	→	→	PUNCT
cana-1064	16	21	ℝ+	ℝ+	PUNCT
cana-1064	16	22	such	such	ADJ
cana-1064	16	23	that	that	PRON
cana-1064	16	24	for	for	SCONJ
cana-1064	16	25	all	all	PRON
cana-1064	16	26	𝜆	𝜆	NOUN
cana-1064	16	27	,	,	PUNCT
cana-1064	16	28	𝜇	𝜇	ADP
cana-1064	16	29	,	,	PUNCT
cana-1064	16	30	𝜉	𝜉	ADP
cana-1064	16	31	𝑖𝑛	𝑖𝑛	NOUN
cana-1064	16	32	𝔛	𝔛	NOUN
cana-1064	16	33	:	:	PUNCT
cana-1064	16	34	(	(	PUNCT
cana-1064	16	35	𝒫ℳ𝒮1	𝒫ℳ𝒮1	ADJ
cana-1064	16	36	):	):	PUNCT
cana-1064	16	37	𝒫	𝒫	PROPN
cana-1064	16	38	(	(	PUNCT
cana-1064	16	39	𝜆	𝜆	NOUN
cana-1064	16	40	,	,	PUNCT
cana-1064	16	41	𝜆	𝜆	X
cana-1064	16	42	)	)	PUNCT
cana-1064	16	43	=	=	SYM
cana-1064	16	44	𝒫(𝜆	𝒫(𝜆	NOUN
cana-1064	16	45	,	,	PUNCT
cana-1064	16	46	𝜇	𝜇	ADP
cana-1064	16	47	)	)	PUNCT
cana-1064	16	48	=	=	SYM
cana-1064	16	49	𝒫(𝜇	𝒫(𝜇	PROPN
cana-1064	16	50	,	,	PUNCT
cana-1064	16	51	𝜇	𝜇	NOUN
cana-1064	16	52	)	)	PUNCT
cana-1064	16	53	⇔	⇔	X
cana-1064	16	54	𝜆	𝜆	NOUN
cana-1064	16	55	=	=	X
cana-1064	16	56	𝜇	𝜇	X
cana-1064	16	57	(	(	PUNCT
cana-1064	16	58	𝒫ℳ𝒮2	𝒫ℳ𝒮2	PROPN
cana-1064	16	59	):	):	PUNCT
cana-1064	16	60	𝒫(𝜆	𝒫(𝜆	NUM
cana-1064	16	61	,	,	PUNCT
cana-1064	16	62	𝜆	𝜆	NOUN
cana-1064	16	63	)	)	PUNCT
cana-1064	16	64	≤	≤	NUM
cana-1064	16	65	𝒫(𝜆	𝒫(𝜆	NOUN
cana-1064	16	66	,	,	PUNCT
cana-1064	16	67	𝜇	𝜇	NOUN
cana-1064	16	68	)	)	PUNCT
cana-1064	16	69	mailto:basker.bonuga@gmail.com	mailto:basker.bonuga@gmail.com	X
cana-1064	16	70	communications	communication	NOUN
cana-1064	16	71	on	on	ADP
cana-1064	16	72	applied	apply	VERB
cana-1064	16	73	nonlinear	nonlinear	ADJ
cana-1064	16	74	analysis	analysis	NOUN
cana-1064	16	75	issn	issn	NOUN
cana-1064	16	76	:	:	PUNCT
cana-1064	16	77	1074	1074	NUM
cana-1064	16	78	-	-	PUNCT
cana-1064	16	79	133x	133x	NUM
cana-1064	16	80	vol	vol	NOUN
cana-1064	16	81	31	31	NUM
cana-1064	16	82	no	no	NOUN
cana-1064	16	83	.	.	PUNCT
cana-1064	17	1	5s	5s	NUM
cana-1064	17	2	(	(	PUNCT
cana-1064	17	3	2024	2024	NUM
cana-1064	17	4	)	)	PUNCT
cana-1064	17	5	450	450	NUM
cana-1064	17	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1064	17	7	(	(	PUNCT
cana-1064	17	8	𝒫ℳ𝒮3	𝒫ℳ𝒮3	NUM
cana-1064	17	9	):	):	PUNCT
cana-1064	17	10	𝒫(𝜆	𝒫(𝜆	NUM
cana-1064	17	11	,	,	PUNCT
cana-1064	17	12	𝜇	𝜇	ADP
cana-1064	17	13	)	)	PUNCT
cana-1064	17	14	=	=	SYM
cana-1064	17	15	𝒫(𝜇	𝒫(𝜇	PROPN
cana-1064	17	16	,	,	PUNCT
cana-1064	17	17	𝜆	𝜆	NOUN
cana-1064	17	18	)	)	PUNCT
cana-1064	17	19	(	(	PUNCT
cana-1064	17	20	𝒫ℳ𝒮4	𝒫ℳ𝒮4	NUM
cana-1064	17	21	):	):	PUNCT
cana-1064	17	22	𝒫	𝒫	NOUN
cana-1064	17	23	(	(	PUNCT
cana-1064	17	24	𝜆	𝜆	NOUN
cana-1064	17	25	,	,	PUNCT
cana-1064	17	26	𝜇	𝜇	NOUN
cana-1064	17	27	)	)	PUNCT
cana-1064	17	28	≤	≤	NUM
cana-1064	17	29	𝒫(𝜆	𝒫(𝜆	NOUN
cana-1064	17	30	,	,	PUNCT
cana-1064	17	31	𝜉	𝜉	X
cana-1064	17	32	)	)	PUNCT
cana-1064	17	33	+	+	CCONJ
cana-1064	17	34	𝒫(𝜉	𝒫(𝜉	PROPN
cana-1064	17	35	,	,	PUNCT
cana-1064	17	36	𝜇	𝜇	ADP
cana-1064	17	37	)	)	PUNCT
cana-1064	17	38	−	−	PROPN
cana-1064	18	1	𝒫(𝜉	𝒫(𝜉	PROPN
cana-1064	18	2	,	,	PUNCT
cana-1064	18	3	𝜉	𝜉	PROPN
cana-1064	18	4	)	)	PUNCT
cana-1064	18	5	.	.	PUNCT
cana-1064	19	1	the	the	DET
cana-1064	19	2	pair	pair	NOUN
cana-1064	19	3	(	(	PUNCT
cana-1064	19	4	𝔛	𝔛	PROPN
cana-1064	19	5	,	,	PUNCT
cana-1064	19	6	𝒫	𝒫	NOUN
cana-1064	19	7	)	)	PUNCT
cana-1064	19	8	is	be	AUX
cana-1064	19	9	called	call	VERB
cana-1064	19	10	partial	partial	ADJ
cana-1064	19	11	metric	metric	ADJ
cana-1064	19	12	space	space	NOUN
cana-1064	19	13	(	(	PUNCT
cana-1064	19	14	briefly	briefly	NOUN
cana-1064	19	15	pms	pm	NOUN
cana-1064	19	16	)	)	PUNCT
cana-1064	19	17	and	and	CCONJ
cana-1064	19	18	𝒫	𝒫	NOUN
cana-1064	19	19	is	be	AUX
cana-1064	19	20	a	a	DET
cana-1064	19	21	partial	partial	ADJ
cana-1064	19	22	metric	metric	NOUN
cana-1064	19	23	on	on	ADP
cana-1064	19	24	𝔛.	𝔛.	PROPN
cana-1064	19	25	a	a	DET
cana-1064	19	26	mapping	mapping	NOUN
cana-1064	19	27	𝒫𝑠	𝒫𝑠	NOUN
cana-1064	19	28	:	:	PUNCT
cana-1064	19	29	:	:	PUNCT
cana-1064	19	30	𝔛	𝔛	NOUN
cana-1064	19	31	×	×	NOUN
cana-1064	19	32	𝔛	𝔛	PROPN
cana-1064	19	33	→	→	PUNCT
cana-1064	19	34	ℝ+	ℝ+	PUNCT
cana-1064	19	35	is	be	AUX
cana-1064	19	36	defined	define	VERB
cana-1064	19	37	by	by	ADP
cana-1064	19	38	𝒫𝑠(𝜆	𝒫𝑠(𝜆	PROPN
cana-1064	19	39	,	,	PUNCT
cana-1064	19	40	𝜇	𝜇	ADP
cana-1064	19	41	)	)	PUNCT
cana-1064	19	42	=	=	SYM
cana-1064	19	43	2𝒫(𝜆	2𝒫(𝜆	NUM
cana-1064	19	44	,	,	PUNCT
cana-1064	19	45	𝜇	𝜇	ADP
cana-1064	19	46	)	)	PUNCT
cana-1064	19	47	−	−	PROPN
cana-1064	20	1	𝒫(𝜆	𝒫(𝜆	NOUN
cana-1064	20	2	,	,	PUNCT
cana-1064	20	3	𝜆	𝜆	NOUN
cana-1064	20	4	)	)	PUNCT
cana-1064	20	5	−	−	PROPN
cana-1064	21	1	𝒫(𝜇	𝒫(𝜇	SYM
cana-1064	21	2	,	,	PUNCT
cana-1064	21	3	𝜇	𝜇	NOUN
cana-1064	21	4	)	)	PUNCT
cana-1064	21	5	is	be	AUX
cana-1064	21	6	usual	usual	ADJ
cana-1064	21	7	metric	metric	NOUN
cana-1064	21	8	,	,	PUNCT
cana-1064	21	9	where	where	SCONJ
cana-1064	21	10	𝒫	𝒫	NOUN
cana-1064	21	11	is	be	AUX
cana-1064	21	12	partial	partial	ADJ
cana-1064	21	13	metric	metric	NOUN
cana-1064	21	14	on	on	ADP
cana-1064	21	15	𝔛.	𝔛.	PROPN
cana-1064	21	16	example	example	NOUN
cana-1064	22	1	1.2[2,3	1.2[2,3	ADJ
cana-1064	22	2	]	]	X
cana-1064	22	3	:	:	PUNCT
cana-1064	22	4	suppose	suppose	VERB
cana-1064	22	5	that	that	SCONJ
cana-1064	22	6	𝔛	𝔛	PROPN
cana-1064	22	7	=	=	SYM
cana-1064	22	8	ℝ+⋃	ℝ+⋃	PROPN
cana-1064	22	9	{	{	PUNCT
cana-1064	22	10	0	0	NUM
cana-1064	22	11	}	}	PUNCT
cana-1064	22	12	and	and	CCONJ
cana-1064	22	13	we	we	PRON
cana-1064	22	14	defined	define	VERB
cana-1064	22	15	𝒫(𝜆	𝒫(𝜆	NOUN
cana-1064	22	16	,	,	PUNCT
cana-1064	22	17	𝜇	𝜇	ADP
cana-1064	22	18	)	)	PUNCT
cana-1064	22	19	=	=	SYM
cana-1064	22	20	max	max	X
cana-1064	22	21	{	{	PUNCT
cana-1064	22	22	𝜆	𝜆	NOUN
cana-1064	22	23	,	,	PUNCT
cana-1064	22	24	𝜇	𝜇	ADP
cana-1064	22	25	}	}	PUNCT
cana-1064	22	26	∀𝜆	∀𝜆	NUM
cana-1064	22	27	,	,	PUNCT
cana-1064	22	28	𝜇	𝜇	ADP
cana-1064	22	29	∈	∈	PROPN
cana-1064	22	30	𝔛.then	𝔛.then	ADP
cana-1064	22	31	(	(	PUNCT
cana-1064	22	32	𝔛	𝔛	PROPN
cana-1064	22	33	,	,	PUNCT
cana-1064	22	34	𝒫	𝒫	NOUN
cana-1064	22	35	)	)	PUNCT
cana-1064	22	36	is	be	AUX
cana-1064	22	37	pms	pm	NOUN
cana-1064	22	38	but	but	CCONJ
cana-1064	22	39	not	not	PART
cana-1064	22	40	(	(	PUNCT
cana-1064	22	41	usual	usual	ADJ
cana-1064	22	42	)	)	PUNCT
cana-1064	22	43	metric	metric	ADJ
cana-1064	22	44	space	space	NOUN
cana-1064	22	45	.	.	PUNCT
cana-1064	23	1	definition	definition	NOUN
cana-1064	23	2	1.3[1	1.3[1	NUM
cana-1064	23	3	]	]	X
cana-1064	23	4	:	:	PUNCT
cana-1064	23	5	let	let	VERB
cana-1064	23	6	(	(	PUNCT
cana-1064	23	7	𝔛	𝔛	NOUN
cana-1064	23	8	,	,	PUNCT
cana-1064	23	9	𝒫	𝒫	NOUN
cana-1064	23	10	)	)	PUNCT
cana-1064	23	11	be	be	VERB
cana-1064	23	12	a	a	DET
cana-1064	23	13	pms	pm	NOUN
cana-1064	23	14	.	.	PUNCT
cana-1064	24	1	a	a	DET
cana-1064	24	2	sequence	sequence	NOUN
cana-1064	24	3	{	{	PUNCT
cana-1064	24	4	𝜆𝓃	𝜆𝓃	ADP
cana-1064	24	5	}	}	PUNCT
cana-1064	24	6	⊆	⊆	NUM
cana-1064	24	7	𝒳	𝒳	PROPN
cana-1064	24	8	converges	converge	VERB
cana-1064	24	9	to	to	ADP
cana-1064	24	10	𝜆	𝜆	DET
cana-1064	24	11	∈	∈	X
cana-1064	24	12	𝔛	𝔛	NOUN
cana-1064	24	13	if	if	SCONJ
cana-1064	24	14	and	and	CCONJ
cana-1064	24	15	only	only	ADV
cana-1064	24	16	if	if	SCONJ
cana-1064	24	17	𝒫(𝜆	𝒫(𝜆	NOUN
cana-1064	24	18	,	,	PUNCT
cana-1064	24	19	𝜆	𝜆	NOUN
cana-1064	24	20	)	)	PUNCT
cana-1064	24	21	=	=	SYM
cana-1064	24	22	lim	lim	PROPN
cana-1064	24	23	𝜂→∞	𝜂→∞	NUM
cana-1064	24	24	𝒫(𝜆	𝒫(𝜆	PROPN
cana-1064	24	25	,	,	PUNCT
cana-1064	24	26	𝜆𝜂	𝜆𝜂	PRON
cana-1064	24	27	)	)	PUNCT
cana-1064	24	28	.	.	PUNCT
cana-1064	25	1	also	also	ADV
cana-1064	25	2	cauchy	cauchy	VERB
cana-1064	25	3	sequence	sequence	NOUN
cana-1064	25	4	⇔	⇔	PROPN
cana-1064	25	5	lim	lim	PROPN
cana-1064	25	6	𝜂,𝜁→∞	𝜂,𝜁→∞	NOUN
cana-1064	25	7	𝒫(𝜆𝜂	𝒫(𝜆𝜂	PROPN
cana-1064	25	8	,	,	PUNCT
cana-1064	25	9	𝜆𝜁	𝜆𝜁	CCONJ
cana-1064	25	10	)	)	PUNCT
cana-1064	25	11	exists	exist	VERB
cana-1064	25	12	finitely	finitely	ADV
cana-1064	25	13	.	.	PUNCT
cana-1064	26	1	lemma	lemma	PROPN
cana-1064	26	2	1.4	1.4	NUM
cana-1064	27	1	[	[	X
cana-1064	27	2	1	1	NUM
cana-1064	27	3	]	]	PUNCT
cana-1064	27	4	:	:	PUNCT
cana-1064	27	5	let	let	VERB
cana-1064	27	6	(	(	PUNCT
cana-1064	27	7	𝔛	𝔛	NOUN
cana-1064	27	8	,	,	PUNCT
cana-1064	27	9	𝒫	𝒫	NOUN
cana-1064	27	10	)	)	PUNCT
cana-1064	27	11	be	be	VERB
cana-1064	27	12	a	a	DET
cana-1064	27	13	pms	pm	NOUN
cana-1064	27	14	and	and	CCONJ
cana-1064	27	15	then	then	ADV
cana-1064	27	16	,	,	PUNCT
cana-1064	27	17	(	(	PUNCT
cana-1064	27	18	i	i	NOUN
cana-1064	27	19	)	)	PUNCT
cana-1064	27	20	{	{	PUNCT
cana-1064	27	21	𝜆𝜂	𝜆𝜂	CCONJ
cana-1064	27	22	}	}	PUNCT
cana-1064	27	23	is	be	AUX
cana-1064	27	24	cauchy	cauchy	ADJ
cana-1064	27	25	sequence	sequence	NOUN
cana-1064	27	26	in	in	ADP
cana-1064	27	27	(	(	PUNCT
cana-1064	27	28	𝔛	𝔛	PROPN
cana-1064	27	29	,	,	PUNCT
cana-1064	27	30	𝒫	𝒫	NOUN
cana-1064	27	31	)	)	PUNCT
cana-1064	27	32	if	if	SCONJ
cana-1064	27	33	and	and	CCONJ
cana-1064	27	34	only	only	ADV
cana-1064	27	35	if	if	SCONJ
cana-1064	27	36	it	it	PRON
cana-1064	27	37	is	be	AUX
cana-1064	27	38	a	a	DET
cana-1064	27	39	cauchy	cauchy	NOUN
cana-1064	27	40	in	in	ADP
cana-1064	27	41	metric	metric	ADJ
cana-1064	27	42	space	space	NOUN
cana-1064	27	43	(	(	PUNCT
cana-1064	27	44	𝔛	𝔛	PROPN
cana-1064	27	45	,	,	PUNCT
cana-1064	27	46	𝒫𝑠	𝒫𝑠	PROPN
cana-1064	27	47	)	)	PUNCT
cana-1064	27	48	.	.	PUNCT
cana-1064	28	1	(	(	PUNCT
cana-1064	28	2	ii	ii	NOUN
cana-1064	28	3	)	)	PUNCT
cana-1064	28	4	(	(	PUNCT
cana-1064	28	5	𝔛	𝔛	PROPN
cana-1064	28	6	,	,	PUNCT
cana-1064	28	7	𝒫	𝒫	NOUN
cana-1064	28	8	)	)	PUNCT
cana-1064	28	9	is	be	AUX
cana-1064	28	10	complete	complete	ADJ
cana-1064	28	11	pms	pm	NOUN
cana-1064	28	12	if	if	SCONJ
cana-1064	28	13	and	and	CCONJ
cana-1064	28	14	only	only	ADV
cana-1064	28	15	if	if	SCONJ
cana-1064	28	16	(	(	PUNCT
cana-1064	28	17	𝔛	𝔛	PROPN
cana-1064	28	18	,	,	PUNCT
cana-1064	28	19	𝒫𝑠	𝒫𝑠	PROPN
cana-1064	28	20	)	)	PUNCT
cana-1064	28	21	is	be	AUX
cana-1064	28	22	complete	complete	ADJ
cana-1064	28	23	.	.	PUNCT
cana-1064	29	1	moreover	moreover	ADV
cana-1064	29	2	,	,	PUNCT
cana-1064	29	3	lim	lim	PROPN
cana-1064	29	4	𝜂→∞	𝜂→∞	PROPN
cana-1064	29	5	𝒫𝑠	𝒫𝑠	PROPN
cana-1064	29	6	(	(	PUNCT
cana-1064	29	7	𝜆	𝜆	NOUN
cana-1064	29	8	,	,	PUNCT
cana-1064	29	9	𝜆𝜂	𝜆𝜂	PRON
cana-1064	29	10	)	)	PUNCT
cana-1064	29	11	=	=	SYM
cana-1064	29	12	0	0	NUM
cana-1064	29	13	⇔	⇔	PROPN
cana-1064	29	14	𝒫	𝒫	PROPN
cana-1064	29	15	(	(	PUNCT
cana-1064	29	16	𝜆	𝜆	NOUN
cana-1064	29	17	,	,	PUNCT
cana-1064	29	18	𝜆	𝜆	NOUN
cana-1064	29	19	)	)	PUNCT
cana-1064	29	20	=	=	SYM
cana-1064	29	21	lim	lim	PROPN
cana-1064	29	22	𝜂→∞	𝜂→∞	PROPN
cana-1064	29	23	𝒫	𝒫	PROPN
cana-1064	29	24	(	(	PUNCT
cana-1064	29	25	𝜆	𝜆	NOUN
cana-1064	29	26	,	,	PUNCT
cana-1064	29	27	𝜆𝜂	𝜆𝜂	PRON
cana-1064	29	28	)	)	PUNCT
cana-1064	30	1	=	=	VERB
cana-1064	30	2	lim	lim	NOUN
cana-1064	30	3	𝜂,𝜁→∞	𝜂,𝜁→∞	NOUN
cana-1064	31	1	𝒫	𝒫	NOUN
cana-1064	31	2	(	(	PUNCT
cana-1064	31	3	𝜆𝜂	𝜆𝜂	PART
cana-1064	31	4	,	,	PUNCT
cana-1064	31	5	𝜆𝜁	𝜆𝜁	PROPN
cana-1064	31	6	)	)	PUNCT
cana-1064	31	7	.	.	PUNCT
cana-1064	32	1	lemma1.5	lemma1.5	PUNCT
cana-1064	32	2	[	[	X
cana-1064	32	3	4	4	NUM
cana-1064	32	4	]	]	PUNCT
cana-1064	32	5	:	:	PUNCT
cana-1064	32	6	assume	assume	VERB
cana-1064	32	7	that	that	SCONJ
cana-1064	32	8	𝜆𝓃	𝜆𝓃	VERB
cana-1064	32	9	⟶	⟶	NOUN
cana-1064	32	10	𝜉	𝜉	NOUN
cana-1064	32	11	as	as	ADP
cana-1064	32	12	𝜂	𝜂	PROPN
cana-1064	32	13	→	→	SYM
cana-1064	32	14	∞	∞	PROPN
cana-1064	32	15	in	in	ADP
cana-1064	32	16	(	(	PUNCT
cana-1064	32	17	𝔛	𝔛	PROPN
cana-1064	32	18	,	,	PUNCT
cana-1064	32	19	𝒫	𝒫	NOUN
cana-1064	32	20	)	)	PUNCT
cana-1064	32	21	with	with	ADP
cana-1064	32	22	𝒫(𝜉	𝒫(𝜉	PROPN
cana-1064	32	23	,	,	PUNCT
cana-1064	32	24	𝜉	𝜉	NOUN
cana-1064	32	25	)	)	PUNCT
cana-1064	32	26	=	=	SYM
cana-1064	32	27	0	0	PUNCT
cana-1064	33	1	then	then	ADV
cana-1064	33	2	lim	lim	PROPN
cana-1064	33	3	𝜂→∞	𝜂→∞	PROPN
cana-1064	33	4	𝒫(𝜆𝓃	𝒫(𝜆𝓃	PROPN
cana-1064	33	5	,	,	PUNCT
cana-1064	33	6	𝜇	𝜇	ADP
cana-1064	33	7	)	)	PUNCT
cana-1064	33	8	=	=	SYM
cana-1064	33	9	𝒫(𝜉	𝒫(𝜉	PROPN
cana-1064	33	10	,	,	PUNCT
cana-1064	33	11	𝜇	𝜇	NOUN
cana-1064	33	12	)	)	PUNCT
cana-1064	33	13	∀	∀	X
cana-1064	33	14	𝜇	𝜇	ADP
cana-1064	33	15	∈	∈	PROPN
cana-1064	33	16	𝔛.	𝔛.	X
cana-1064	33	17	lemma	lemma	PROPN
cana-1064	33	18	1.6[4	1.6[4	NUM
cana-1064	33	19	]	]	X
cana-1064	33	20	:	:	PUNCT
cana-1064	33	21	let	let	VERB
cana-1064	33	22	(	(	PUNCT
cana-1064	33	23	𝔛	𝔛	NOUN
cana-1064	33	24	,	,	PUNCT
cana-1064	33	25	𝒫	𝒫	NOUN
cana-1064	33	26	)	)	PUNCT
cana-1064	33	27	be	be	VERB
cana-1064	33	28	a	a	DET
cana-1064	33	29	pms	pm	NOUN
cana-1064	33	30	.	.	PUNCT
cana-1064	34	1	(	(	PUNCT
cana-1064	34	2	i	i	NOUN
cana-1064	34	3	)	)	PUNCT
cana-1064	34	4	if	if	SCONJ
cana-1064	34	5	𝒫	𝒫	NOUN
cana-1064	34	6	(	(	PUNCT
cana-1064	34	7	𝜆	𝜆	NOUN
cana-1064	34	8	,	,	PUNCT
cana-1064	34	9	𝜇	𝜇	NOUN
cana-1064	34	10	)	)	PUNCT
cana-1064	34	11	=	=	SYM
cana-1064	34	12	0	0	PUNCT
cana-1064	34	13	then	then	ADV
cana-1064	34	14	𝜆	𝜆	X
cana-1064	34	15	=	=	X
cana-1064	34	16	𝜇.	𝜇.	X
cana-1064	34	17	(	(	PUNCT
cana-1064	34	18	ii	ii	NOUN
cana-1064	34	19	)	)	PUNCT
cana-1064	34	20	if	if	SCONJ
cana-1064	34	21	𝜆	𝜆	DET
cana-1064	34	22	≠	≠	PROPN
cana-1064	34	23	𝜇	𝜇	ADP
cana-1064	34	24	then	then	ADV
cana-1064	34	25	𝒫	𝒫	NOUN
cana-1064	34	26	(	(	PUNCT
cana-1064	34	27	𝜆	𝜆	NOUN
cana-1064	34	28	,	,	PUNCT
cana-1064	34	29	𝜇	𝜇	ADP
cana-1064	34	30	)	)	PUNCT
cana-1064	34	31	>	>	X
cana-1064	34	32	0	0	X
cana-1064	34	33	.	.	PUNCT
cana-1064	35	1	definition	definition	NOUN
cana-1064	35	2	1.7[5	1.7[5	NUM
cana-1064	35	3	]	]	X
cana-1064	35	4	:	:	PUNCT
cana-1064	35	5	a	a	DET
cana-1064	35	6	mapping	mapping	NOUN
cana-1064	35	7	ℱ	ℱ	NOUN
cana-1064	35	8	:	:	PUNCT
cana-1064	35	9	ℝ+→ℝ	ℝ+→ℝ	NOUN
cana-1064	35	10	is	be	AUX
cana-1064	35	11	said	say	VERB
cana-1064	35	12	to	to	PART
cana-1064	35	13	be	be	AUX
cana-1064	35	14	ℱ	ℱ	PROPN
cana-1064	35	15	-contraction	-contraction	NOUN
cana-1064	35	16	if	if	SCONJ
cana-1064	35	17	it	it	PRON
cana-1064	35	18	satisfying	satisfy	VERB
cana-1064	35	19	following	follow	VERB
cana-1064	35	20	conditions	condition	NOUN
cana-1064	35	21	(	(	PUNCT
cana-1064	35	22	ℱ1	ℱ1	NUM
cana-1064	35	23	):	):	PUNCT
cana-1064	35	24	if	if	SCONJ
cana-1064	35	25	𝜆	𝜆	NOUN
cana-1064	35	26	,	,	PUNCT
cana-1064	35	27	𝜇	𝜇	ADP
cana-1064	35	28	∈	∈	PROPN
cana-1064	35	29	ℝ+	ℝ+	PUNCT
cana-1064	35	30	such	such	ADJ
cana-1064	35	31	that	that	SCONJ
cana-1064	35	32	𝜆	𝜆	DET
cana-1064	35	33	<	<	X
cana-1064	35	34	𝜇	𝜇	ADP
cana-1064	35	35	⇒	⇒	NOUN
cana-1064	35	36	ℱ(𝜆	ℱ(𝜆	NUM
cana-1064	35	37	)	)	PUNCT
cana-1064	35	38	<	<	X
cana-1064	35	39	ℱ(𝜇	ℱ(𝜇	PROPN
cana-1064	35	40	)	)	PUNCT
cana-1064	35	41	(	(	PUNCT
cana-1064	35	42	ℱ2	ℱ2	NUM
cana-1064	35	43	):	):	PUNCT
cana-1064	35	44	for	for	ADP
cana-1064	35	45	each	each	DET
cana-1064	35	46	{	{	PUNCT
cana-1064	35	47	𝛼𝜂}𝜂∈ℕ	𝛼𝜂}𝜂∈ℕ	NOUN
cana-1064	35	48	∈	∈	PROPN
cana-1064	35	49	ℝ+	ℝ+	PROPN
cana-1064	35	50	,	,	PUNCT
cana-1064	35	51	lim	lim	NOUN
cana-1064	35	52	𝜂→∞	𝜂→∞	PROPN
cana-1064	35	53	𝛼𝜂	𝛼𝜂	PROPN
cana-1064	35	54	=	=	NOUN
cana-1064	35	55	0	0	PUNCT
cana-1064	36	1	if	if	SCONJ
cana-1064	36	2	and	and	CCONJ
cana-1064	36	3	only	only	ADV
cana-1064	36	4	if	if	SCONJ
cana-1064	36	5	lim	lim	PROPN
cana-1064	36	6	𝜂→∞	𝜂→∞	NUM
cana-1064	36	7	ℱ(𝛼𝜂	ℱ(𝛼𝜂	NOUN
cana-1064	36	8	)	)	PUNCT
cana-1064	37	1	=	=	SYM
cana-1064	37	2	−∞.	−∞.	NOUN
cana-1064	37	3	(	(	PUNCT
cana-1064	37	4	ℱ3	ℱ3	NOUN
cana-1064	37	5	)	)	PUNCT
cana-1064	37	6	∃	∃	PROPN
cana-1064	37	7	real	real	ADJ
cana-1064	37	8	number	number	NOUN
cana-1064	37	9	𝜃	𝜃	X
cana-1064	37	10	∈	∈	PROPN
cana-1064	37	11	(	(	PUNCT
cana-1064	37	12	0,1	0,1	NOUN
cana-1064	37	13	)	)	PUNCT
cana-1064	37	14	such	such	ADJ
cana-1064	37	15	that	that	SCONJ
cana-1064	37	16	lim	lim	PROPN
cana-1064	37	17	𝛼→0	𝛼→0	PROPN
cana-1064	37	18	+	+	CCONJ
cana-1064	37	19	𝛼𝜃ℱ(𝛼	𝛼𝜃ℱ(𝛼	NUM
cana-1064	37	20	)	)	PUNCT
cana-1064	37	21	=	=	SYM
cana-1064	37	22	0	0	X
cana-1064	37	23	.	.	PUNCT
cana-1064	37	24	notation	notation	NOUN
cana-1064	37	25	1.8[5	1.8[5	NUM
cana-1064	37	26	]	]	PUNCT
cana-1064	37	27	:	:	PUNCT
cana-1064	37	28	we	we	PRON
cana-1064	37	29	symbolize	symbolize	VERB
cana-1064	37	30	the	the	DET
cana-1064	37	31	collection	collection	NOUN
cana-1064	37	32	of	of	ADP
cana-1064	37	33	all	all	DET
cana-1064	37	34	functions	function	NOUN
cana-1064	37	35	which	which	PRON
cana-1064	37	36	satisfy	satisfy	VERB
cana-1064	37	37	the	the	DET
cana-1064	37	38	above	above	ADJ
cana-1064	37	39	specified	specified	ADJ
cana-1064	37	40	constraints	constraint	NOUN
cana-1064	37	41	ℱ1	ℱ1	ADP
cana-1064	37	42	to	to	ADP
cana-1064	37	43	ℱ3	ℱ3	NOUN
cana-1064	37	44	by	by	ADP
cana-1064	37	45	∆ℱ	∆ℱ	PROPN
cana-1064	37	46	.	.	PUNCT
cana-1064	38	1	definition	definition	NOUN
cana-1064	38	2	1.9[5	1.9[5	NUM
cana-1064	38	3	]	]	X
cana-1064	38	4	:	:	PUNCT
cana-1064	38	5	a	a	DET
cana-1064	38	6	self	self	NOUN
cana-1064	38	7	mapping	map	VERB
cana-1064	38	8	𝔄	𝔄	NOUN
cana-1064	38	9	:	:	PUNCT
cana-1064	38	10	𝔛→𝔛	𝔛→𝔛	PROPN
cana-1064	38	11	is	be	AUX
cana-1064	38	12	said	say	VERB
cana-1064	38	13	to	to	ADP
cana-1064	38	14	ℱ	ℱ	PROPN
cana-1064	38	15	contraction	contraction	NOUN
cana-1064	38	16	if	if	SCONJ
cana-1064	38	17	there	there	PRON
cana-1064	38	18	exists	exist	VERB
cana-1064	38	19	a	a	DET
cana-1064	38	20	𝜏	𝜏	NOUN
cana-1064	38	21	>	>	X
cana-1064	38	22	0	0	NUM
cana-1064	38	23	such	such	ADJ
cana-1064	38	24	that	that	PRON
cana-1064	38	25	for	for	ADP
cana-1064	38	26	all	all	DET
cana-1064	38	27	𝜆	𝜆	NOUN
cana-1064	38	28	,	,	PUNCT
cana-1064	38	29	𝜇	𝜇	ADP
cana-1064	38	30	∈	∈	PROPN
cana-1064	38	31	𝔛	𝔛	NOUN
cana-1064	38	32	,	,	PUNCT
cana-1064	38	33	𝒹(𝔄𝜆	𝒹(𝔄𝜆	X
cana-1064	38	34	,	,	PUNCT
cana-1064	38	35	𝔄𝜇	𝔄𝜇	PROPN
cana-1064	38	36	)	)	PUNCT
cana-1064	38	37	>	>	X
cana-1064	38	38	0	0	PUNCT
cana-1064	39	1	and	and	CCONJ
cana-1064	39	2	we	we	PRON
cana-1064	39	3	have	have	VERB
cana-1064	39	4	𝜏	𝜏	NOUN
cana-1064	39	5	+	+	X
cana-1064	39	6	ℱ(𝒹(𝔄𝜆	ℱ(𝒹(𝔄𝜆	PROPN
cana-1064	39	7	,	,	PUNCT
cana-1064	39	8	𝔄𝜇	𝔄𝜇	PROPN
cana-1064	39	9	)	)	PUNCT
cana-1064	39	10	)	)	PUNCT
cana-1064	40	1	≤	≤	NOUN
cana-1064	40	2	ℱ(𝒹(𝜆	ℱ(𝒹(𝜆	NUM
cana-1064	40	3	,	,	PUNCT
cana-1064	40	4	𝜇	𝜇	NOUN
cana-1064	40	5	)	)	PUNCT
cana-1064	40	6	)	)	PUNCT
cana-1064	40	7	.	.	PUNCT
cana-1064	41	1	example	example	NOUN
cana-1064	42	1	1.10	1.10	NUM
cana-1064	42	2	:	:	PUNCT
cana-1064	42	3	let	let	VERB
cana-1064	42	4	ℱ	ℱ	PROPN
cana-1064	42	5	:	:	PUNCT
cana-1064	42	6	ℝ+	ℝ+	ADP
cana-1064	42	7	→	→	PUNCT
cana-1064	42	8	ℝ	ℝ	PROPN
cana-1064	42	9	be	be	AUX
cana-1064	42	10	given	give	VERB
cana-1064	42	11	by	by	ADP
cana-1064	42	12	ℱ(𝛼	ℱ(𝛼	PUNCT
cana-1064	42	13	)	)	PUNCT
cana-1064	42	14	=	=	SYM
cana-1064	42	15	log𝑒	log𝑒	PROPN
cana-1064	42	16	𝛼	𝛼	NOUN
cana-1064	42	17	and	and	CCONJ
cana-1064	42	18	satisfies	satisfy	VERB
cana-1064	42	19	ℱ1	ℱ1	VERB
cana-1064	42	20	to	to	PART
cana-1064	42	21	ℱ3.each	ℱ3.each	VERB
cana-1064	42	22	mapping	map	VERB
cana-1064	42	23	𝔄	𝔄	NOUN
cana-1064	42	24	:	:	PUNCT
cana-1064	42	25	𝔛→𝔛	𝔛→𝔛	PROPN
cana-1064	42	26	is	be	AUX
cana-1064	42	27	an	an	DET
cana-1064	42	28	ℱ	ℱ	PROPN
cana-1064	42	29	-contraction	-contraction	NOUN
cana-1064	42	30	such	such	ADJ
cana-1064	42	31	that	that	PRON
cana-1064	42	32	for	for	SCONJ
cana-1064	42	33	all	all	DET
cana-1064	42	34	𝜆	𝜆	NOUN
cana-1064	42	35	,	,	PUNCT
cana-1064	42	36	𝜇	𝜇	ADP
cana-1064	42	37	∈	∈	PROPN
cana-1064	42	38	𝔛	𝔛	NOUN
cana-1064	42	39	,	,	PUNCT
cana-1064	42	40	𝔄𝜆	𝔄𝜆	PROPN
cana-1064	42	41	≠	≠	PROPN
cana-1064	42	42	𝔄𝜇,𝒹(𝔄𝜆	𝔄𝜇,𝒹(𝔄𝜆	NOUN
cana-1064	42	43	,	,	PUNCT
cana-1064	42	44	𝔄𝜇	𝔄𝜇	PROPN
cana-1064	42	45	)	)	PUNCT
cana-1064	42	46	≤	≤	NOUN
cana-1064	42	47	𝑒−𝜏𝒹(𝜆	𝑒−𝜏𝒹(𝜆	VERB
cana-1064	42	48	,	,	PUNCT
cana-1064	42	49	𝜇	𝜇	ADP
cana-1064	42	50	)	)	PUNCT
cana-1064	42	51	.	.	PUNCT
cana-1064	43	1	it	it	PRON
cana-1064	43	2	is	be	AUX
cana-1064	43	3	clear	clear	ADJ
cana-1064	43	4	that	that	SCONJ
cana-1064	43	5	𝜆	𝜆	X
cana-1064	43	6	,	,	PUNCT
cana-1064	43	7	𝜇	𝜇	ADP
cana-1064	43	8	∈	∈	X
cana-1064	43	9	𝔛	𝔛	NOUN
cana-1064	43	10	such	such	ADJ
cana-1064	43	11	that	that	SCONJ
cana-1064	44	1	𝔄𝜆	𝔄𝜆	PROPN
cana-1064	44	2	=	=	PUNCT
cana-1064	44	3	𝔄𝜇	𝔄𝜇	PROPN
cana-1064	44	4	then	then	ADV
cana-1064	44	5	𝒹(𝔄𝜆	𝒹(𝔄𝜆	NUM
cana-1064	44	6	,	,	PUNCT
cana-1064	44	7	𝔄𝜇	𝔄𝜇	PROPN
cana-1064	44	8	)	)	PUNCT
cana-1064	44	9	≤	≤	NOUN
cana-1064	44	10	𝑒−𝜏𝒹(𝜆	𝑒−𝜏𝒹(𝜆	VERB
cana-1064	44	11	,	,	PUNCT
cana-1064	44	12	𝜇	𝜇	X
cana-1064	44	13	)	)	PUNCT
cana-1064	44	14	also	also	ADV
cana-1064	44	15	satisfying	satisfy	VERB
cana-1064	44	16	i.e.	i.e.	X
cana-1064	44	17	𝔄	𝔄	PROPN
cana-1064	44	18	is	be	AUX
cana-1064	44	19	banach	banach	NOUN
cana-1064	44	20	contraction	contraction	NOUN
cana-1064	44	21	.	.	PUNCT
cana-1064	45	1	communications	communication	NOUN
cana-1064	45	2	on	on	ADP
cana-1064	45	3	applied	apply	VERB
cana-1064	45	4	nonlinear	nonlinear	ADJ
cana-1064	45	5	analysis	analysis	NOUN
cana-1064	45	6	issn	issn	NOUN
cana-1064	45	7	:	:	PUNCT
cana-1064	45	8	1074	1074	NUM
cana-1064	45	9	-	-	PUNCT
cana-1064	45	10	133x	133x	NUM
cana-1064	45	11	vol	vol	NOUN
cana-1064	45	12	31	31	NUM
cana-1064	45	13	no	no	NOUN
cana-1064	45	14	.	.	PUNCT
cana-1064	46	1	5s	5s	NUM
cana-1064	46	2	(	(	PUNCT
cana-1064	46	3	2024	2024	NUM
cana-1064	46	4	)	)	PUNCT
cana-1064	46	5	451	451	NUM
cana-1064	47	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1064	47	2	example	example	NOUN
cana-1064	47	3	1.11	1.11	NUM
cana-1064	47	4	:	:	PUNCT
cana-1064	47	5	defined	define	VERB
cana-1064	47	6	the	the	DET
cana-1064	47	7	complete	complete	ADJ
cana-1064	47	8	pms	pm	NOUN
cana-1064	47	9	(	(	PUNCT
cana-1064	47	10	𝔛	𝔛	PROPN
cana-1064	47	11	,	,	PUNCT
cana-1064	47	12	𝒫	𝒫	NOUN
cana-1064	47	13	)	)	PUNCT
cana-1064	47	14	by	by	ADP
cana-1064	47	15	𝒫	𝒫	PROPN
cana-1064	47	16	(	(	PUNCT
cana-1064	47	17	𝜆	𝜆	NOUN
cana-1064	47	18	,	,	PUNCT
cana-1064	47	19	𝜇	𝜇	NOUN
cana-1064	47	20	)	)	PUNCT
cana-1064	47	21	=	=	SYM
cana-1064	48	1	𝑚𝑎𝑥{𝜆	𝑚𝑎𝑥{𝜆	ADJ
cana-1064	48	2	,	,	PUNCT
cana-1064	48	3	𝜇	𝜇	ADP
cana-1064	48	4	}	}	PUNCT
cana-1064	48	5	and	and	CCONJ
cana-1064	48	6	also	also	ADV
cana-1064	48	7	complete	complete	VERB
cana-1064	48	8	metric	metric	ADJ
cana-1064	48	9	space	space	NOUN
cana-1064	48	10	(	(	PUNCT
cana-1064	48	11	𝔛	𝔛	PROPN
cana-1064	48	12	,	,	PUNCT
cana-1064	48	13	𝒹	𝒹	PROPN
cana-1064	48	14	)	)	PUNCT
cana-1064	48	15	by	by	ADP
cana-1064	48	16	𝒹(𝜆	𝒹(𝜆	NOUN
cana-1064	48	17	,	,	PUNCT
cana-1064	48	18	𝜇	𝜇	ADP
cana-1064	48	19	)	)	PUNCT
cana-1064	48	20	=	=	SYM
cana-1064	48	21	|𝜆	|𝜆	PROPN
cana-1064	48	22	−	−	PROPN
cana-1064	48	23	𝜇|	𝜇|	PROPN
cana-1064	48	24	for	for	ADP
cana-1064	48	25	all	all	PRON
cana-1064	48	26	𝜆	𝜆	NOUN
cana-1064	48	27	,	,	PUNCT
cana-1064	48	28	𝜇	𝜇	ADP
cana-1064	48	29	∈	∈	NOUN
cana-1064	48	30	𝔛.	𝔛.	NOUN
cana-1064	48	31	define	define	VERB
cana-1064	48	32	mappings	mapping	NOUN
cana-1064	48	33	ℱ	ℱ	PROPN
cana-1064	48	34	∶	∶	NOUN
cana-1064	48	35	ℝ+	ℝ+	PUNCT
cana-1064	48	36	⟶	⟶	NOUN
cana-1064	48	37	ℝ	ℝ	PROPN
cana-1064	48	38	and	and	CCONJ
cana-1064	48	39	ℱ(𝛼	ℱ(𝛼	NUM
cana-1064	48	40	)	)	PUNCT
cana-1064	49	1	=	=	SYM
cana-1064	49	2	log𝑒	log𝑒	PROPN
cana-1064	49	3	𝛼	𝛼	PROPN
cana-1064	49	4	and	and	CCONJ
cana-1064	49	5	𝔄	𝔄	PROPN
cana-1064	49	6	by	by	ADP
cana-1064	49	7	𝔄(𝜆	𝔄(𝜆	PROPN
cana-1064	49	8	)	)	PUNCT
cana-1064	49	9	=	=	PRON
cana-1064	49	10	{	{	PUNCT
cana-1064	49	11	𝜆	𝜆	PROPN
cana-1064	49	12	2	2	NUM
cana-1064	49	13	if	if	SCONJ
cana-1064	49	14	𝜆	𝜆	PRON
cana-1064	49	15	∈	∈	PROPN
cana-1064	50	1	[	[	X
cana-1064	50	2	0,1	0,1	NUM
cana-1064	50	3	)	)	PUNCT
cana-1064	50	4	3	3	NUM
cana-1064	50	5	4	4	NUM
cana-1064	50	6	if	if	SCONJ
cana-1064	50	7	𝜆	𝜆	NOUN
cana-1064	50	8	=	=	SYM
cana-1064	50	9	1	1	NUM
cana-1064	50	10	then	then	ADV
cana-1064	50	11	𝔄	𝔄	PROPN
cana-1064	50	12	is	be	AUX
cana-1064	50	13	not	not	PART
cana-1064	50	14	a	a	DET
cana-1064	50	15	ℱ	ℱ	PROPN
cana-1064	50	16	-contraction	-contraction	NOUN
cana-1064	50	17	in	in	ADP
cana-1064	50	18	metric	metric	ADJ
cana-1064	50	19	space	space	NOUN
cana-1064	50	20	certainly	certainly	ADV
cana-1064	50	21	for	for	ADP
cana-1064	50	22	𝜆	𝜆	DET
cana-1064	50	23	=	=	NOUN
cana-1064	50	24	1	1	NUM
cana-1064	50	25	and	and	CCONJ
cana-1064	50	26	𝜇	𝜇	X
cana-1064	50	27	=	=	NOUN
cana-1064	50	28	1/2	1/2	NUM
cana-1064	50	29	,	,	PUNCT
cana-1064	50	30	𝒹(𝔄𝜆	𝒹(𝔄𝜆	ADJ
cana-1064	50	31	,	,	PUNCT
cana-1064	50	32	𝔄𝜇	𝔄𝜇	PROPN
cana-1064	50	33	)	)	PUNCT
cana-1064	50	34	>	>	PUNCT
cana-1064	50	35	0	0	PUNCT
cana-1064	51	1	and	and	CCONJ
cana-1064	51	2	we	we	PRON
cana-1064	51	3	have	have	VERB
cana-1064	51	4	𝜏	𝜏	NOUN
cana-1064	51	5	+	+	X
cana-1064	51	6	ℱ(𝒹(𝔄𝜆	ℱ(𝒹(𝔄𝜆	PROPN
cana-1064	51	7	,	,	PUNCT
cana-1064	51	8	𝔄𝜇	𝔄𝜇	PROPN
cana-1064	51	9	)	)	PUNCT
cana-1064	51	10	)	)	PUNCT
cana-1064	52	1	≤	≤	NOUN
cana-1064	52	2	ℱ(𝒹(𝜆	ℱ(𝒹(𝜆	NUM
cana-1064	52	3	,	,	PUNCT
cana-1064	52	4	𝜇	𝜇	NOUN
cana-1064	52	5	)	)	PUNCT
cana-1064	52	6	)	)	PUNCT
cana-1064	52	7	⇒	⇒	NOUN
cana-1064	52	8	𝜏	𝜏	X
cana-1064	52	9	+	+	CCONJ
cana-1064	52	10	ℱ	ℱ	PROPN
cana-1064	52	11	(	(	PUNCT
cana-1064	52	12	𝒹	𝒹	X
cana-1064	52	13	(	(	PUNCT
cana-1064	52	14	𝔄(1	𝔄(1	PROPN
cana-1064	52	15	)	)	PUNCT
cana-1064	52	16	,	,	PUNCT
cana-1064	52	17	𝔄	𝔄	PROPN
cana-1064	52	18	(	(	PUNCT
cana-1064	52	19	1	1	NUM
cana-1064	52	20	2	2	NUM
cana-1064	52	21	)	)	PUNCT
cana-1064	52	22	)	)	PUNCT
cana-1064	52	23	)	)	PUNCT
cana-1064	52	24	≤	≤	NUM
cana-1064	52	25	ℱ	ℱ	PROPN
cana-1064	52	26	(	(	PUNCT
cana-1064	52	27	𝒹	𝒹	X
cana-1064	52	28	(	(	PUNCT
cana-1064	52	29	1	1	NUM
cana-1064	52	30	,	,	PUNCT
cana-1064	52	31	1	1	NUM
cana-1064	52	32	2	2	NUM
cana-1064	52	33	)	)	PUNCT
cana-1064	52	34	)	)	PUNCT
cana-1064	52	35	⇒	⇒	VERB
cana-1064	52	36	𝜏	𝜏	X
cana-1064	53	1	+	+	CCONJ
cana-1064	53	2	|	|	ADV
cana-1064	53	3	3	3	NUM
cana-1064	53	4	4	4	NUM
cana-1064	53	5	−	−	NOUN
cana-1064	53	6	1	1	NUM
cana-1064	53	7	4	4	NUM
cana-1064	53	8	|	|	ADV
cana-1064	53	9	≤	≤	PUNCT
cana-1064	53	10	|1	|1	PRON
cana-1064	54	1	−	−	NUM
cana-1064	54	2	1	1	NUM
cana-1064	54	3	2	2	NUM
cana-1064	54	4	|	|	ADV
cana-1064	54	5	⇒	⇒	VERB
cana-1064	54	6	𝜏	𝜏	X
cana-1064	54	7	+	+	CCONJ
cana-1064	54	8	1	1	NUM
cana-1064	54	9	2	2	NUM
cana-1064	54	10	≤	≤	NUM
cana-1064	54	11	1	1	NUM
cana-1064	54	12	2	2	NUM
cana-1064	54	13	which	which	PRON
cana-1064	54	14	is	be	AUX
cana-1064	54	15	a	a	DET
cana-1064	54	16	contradiction	contradiction	NOUN
cana-1064	54	17	for	for	ADP
cana-1064	54	18	all	all	PRON
cana-1064	54	19	𝜏	𝜏	PART
cana-1064	54	20	>	>	X
cana-1064	54	21	0	0	X
cana-1064	54	22	.	.	PUNCT
cana-1064	55	1	now	now	ADV
cana-1064	55	2	if	if	SCONJ
cana-1064	55	3	we	we	PRON
cana-1064	55	4	studying	study	VERB
cana-1064	55	5	in	in	ADP
cana-1064	55	6	pms	pms	PROPN
cana-1064	55	7	(	(	PUNCT
cana-1064	55	8	𝔛	𝔛	PROPN
cana-1064	55	9	,	,	PUNCT
cana-1064	55	10	𝒫	𝒫	NOUN
cana-1064	55	11	)	)	PUNCT
cana-1064	55	12	we	we	PRON
cana-1064	55	13	get	get	VERB
cana-1064	55	14	,	,	PUNCT
cana-1064	55	15	𝜏	𝜏	X
cana-1064	55	16	+	+	X
cana-1064	55	17	ℱ(𝒫(𝔄𝜆	ℱ(𝒫(𝔄𝜆	PROPN
cana-1064	55	18	,	,	PUNCT
cana-1064	55	19	𝔄𝜇	𝔄𝜇	PROPN
cana-1064	55	20	)	)	PUNCT
cana-1064	55	21	)	)	PUNCT
cana-1064	56	1	≤	≤	NUM
cana-1064	56	2	ℱ(𝒫(𝜆	ℱ(𝒫(𝜆	NOUN
cana-1064	56	3	,	,	PUNCT
cana-1064	56	4	𝜇	𝜇	NOUN
cana-1064	56	5	)	)	PUNCT
cana-1064	56	6	)	)	PUNCT
cana-1064	56	7	⇒	⇒	NOUN
cana-1064	56	8	𝜏	𝜏	X
cana-1064	56	9	+	+	CCONJ
cana-1064	56	10	ℱ	ℱ	PROPN
cana-1064	56	11	(	(	PUNCT
cana-1064	56	12	𝒫	𝒫	NOUN
cana-1064	56	13	(	(	PUNCT
cana-1064	56	14	𝔄(1	𝔄(1	NUM
cana-1064	56	15	)	)	PUNCT
cana-1064	56	16	,	,	PUNCT
cana-1064	56	17	𝔄	𝔄	PROPN
cana-1064	56	18	(	(	PUNCT
cana-1064	56	19	1	1	NUM
cana-1064	56	20	2	2	NUM
cana-1064	56	21	)	)	PUNCT
cana-1064	56	22	)	)	PUNCT
cana-1064	56	23	)	)	PUNCT
cana-1064	56	24	≤	≤	PUNCT
cana-1064	56	25	ℱ	ℱ	PROPN
cana-1064	56	26	(	(	PUNCT
cana-1064	56	27	𝒫	𝒫	NOUN
cana-1064	56	28	(	(	PUNCT
cana-1064	56	29	1	1	NUM
cana-1064	56	30	,	,	PUNCT
cana-1064	56	31	1	1	NUM
cana-1064	56	32	2	2	NUM
cana-1064	56	33	)	)	PUNCT
cana-1064	56	34	)	)	PUNCT
cana-1064	56	35	⇒	⇒	VERB
cana-1064	56	36	𝜏	𝜏	X
cana-1064	57	1	+	+	CCONJ
cana-1064	57	2	ℱ	ℱ	PROPN
cana-1064	57	3	(	(	PUNCT
cana-1064	57	4	max	max	PROPN
cana-1064	57	5	{	{	PUNCT
cana-1064	57	6	3	3	NUM
cana-1064	57	7	4	4	NUM
cana-1064	57	8	,	,	PUNCT
cana-1064	57	9	1	1	NUM
cana-1064	57	10	4	4	NUM
cana-1064	57	11	}	}	PUNCT
cana-1064	57	12	)	)	PUNCT
cana-1064	57	13	≤	≤	PUNCT
cana-1064	57	14	ℱ	ℱ	PROPN
cana-1064	57	15	(	(	PUNCT
cana-1064	57	16	max	max	PROPN
cana-1064	57	17	{	{	PUNCT
cana-1064	57	18	1	1	NUM
cana-1064	57	19	,	,	PUNCT
cana-1064	57	20	1	1	NUM
cana-1064	57	21	2	2	NUM
cana-1064	57	22	}	}	PUNCT
cana-1064	57	23	)	)	PUNCT
cana-1064	57	24	⇒	⇒	NOUN
cana-1064	57	25	𝜏	𝜏	X
cana-1064	57	26	+	+	CCONJ
cana-1064	57	27	ℱ	ℱ	PROPN
cana-1064	57	28	(	(	PUNCT
cana-1064	57	29	3	3	NUM
cana-1064	57	30	4	4	NUM
cana-1064	57	31	)	)	PUNCT
cana-1064	57	32	≤	≤	NOUN
cana-1064	57	33	ℱ(1	ℱ(1	NUM
cana-1064	57	34	)	)	PUNCT
cana-1064	57	35	which	which	PRON
cana-1064	57	36	is	be	AUX
cana-1064	57	37	true	true	ADJ
cana-1064	57	38	.	.	PUNCT
cana-1064	58	1	in	in	ADP
cana-1064	58	2	similar	similar	ADJ
cana-1064	58	3	manner	manner	NOUN
cana-1064	58	4	our	our	PRON
cana-1064	58	5	assertion	assertion	NOUN
cana-1064	58	6	is	be	AUX
cana-1064	58	7	true	true	ADJ
cana-1064	58	8	for	for	ADP
cana-1064	58	9	every	every	DET
cana-1064	58	10	other	other	ADJ
cana-1064	58	11	points	point	NOUN
cana-1064	58	12	in	in	ADP
cana-1064	58	13	𝔛.	𝔛.	PROPN
cana-1064	58	14	definition	definition	NOUN
cana-1064	58	15	1.12	1.12	NUM
cana-1064	58	16	:	:	PUNCT
cana-1064	58	17	let	let	VERB
cana-1064	58	18	a	a	DET
cana-1064	58	19	pair	pair	NOUN
cana-1064	58	20	if	if	SCONJ
cana-1064	58	21	self	self	NOUN
cana-1064	58	22	mappings	mapping	VERB
cana-1064	58	23	𝔣	𝔣	ADJ
cana-1064	58	24	and	and	CCONJ
cana-1064	58	25	𝔤	𝔤	NOUN
cana-1064	58	26	are	be	AUX
cana-1064	58	27	defined	define	VERB
cana-1064	58	28	on	on	ADP
cana-1064	58	29	a	a	DET
cana-1064	58	30	set	set	NOUN
cana-1064	58	31	𝔛	𝔛	PROPN
cana-1064	58	32	is	be	AUX
cana-1064	58	33	weakly	weakly	ADV
cana-1064	58	34	compatible	compatible	ADJ
cana-1064	58	35	(	(	PUNCT
cana-1064	58	36	wc	wc	NOUN
cana-1064	58	37	)	)	PUNCT
cana-1064	58	38	if	if	SCONJ
cana-1064	58	39	a	a	DET
cana-1064	58	40	point	point	NOUN
cana-1064	58	41	𝜆	𝜆	ADP
cana-1064	58	42	∈	∈	NOUN
cana-1064	59	1	𝔛	𝔛	PROPN
cana-1064	59	2	is	be	AUX
cana-1064	59	3	such	such	ADJ
cana-1064	59	4	that	that	SCONJ
cana-1064	59	5	𝔣𝜆	𝔣𝜆	ADV
cana-1064	59	6	=	=	PRON
cana-1064	59	7	𝔤𝜆	𝔤𝜆	NOUN
cana-1064	59	8	implies	imply	VERB
cana-1064	59	9	𝔣𝔤𝜆	𝔣𝔤𝜆	NOUN
cana-1064	59	10	=	=	SYM
cana-1064	59	11	𝔤𝔣𝜆.	𝔤𝔣𝜆.	NOUN
cana-1064	59	12	the	the	DET
cana-1064	59	13	aim	aim	NOUN
cana-1064	59	14	of	of	ADP
cana-1064	59	15	this	this	DET
cana-1064	59	16	paper	paper	NOUN
cana-1064	59	17	is	be	AUX
cana-1064	59	18	to	to	PART
cana-1064	59	19	develop	develop	VERB
cana-1064	59	20	a	a	DET
cana-1064	59	21	fixed	fix	VERB
cana-1064	59	22	point	point	NOUN
cana-1064	59	23	theorem	theorem	NOUN
cana-1064	59	24	for	for	ADP
cana-1064	59	25	ℱ	ℱ	PROPN
cana-1064	59	26	−	−	PROPN
cana-1064	59	27	contraction	contraction	NOUN
cana-1064	59	28	in	in	ADP
cana-1064	59	29	pms	pms	PROPN
cana-1064	59	30	(	(	PUNCT
cana-1064	59	31	𝔛	𝔛	PROPN
cana-1064	59	32	,	,	PUNCT
cana-1064	59	33	𝒫	𝒫	NOUN
cana-1064	59	34	)	)	PUNCT
cana-1064	59	35	using	use	VERB
cana-1064	59	36	the	the	DET
cana-1064	59	37	notation	notation	NOUN
cana-1064	59	38	of	of	ADP
cana-1064	59	39	weakly	weakly	ADJ
cana-1064	59	40	compatibility	compatibility	NOUN
cana-1064	59	41	.	.	PUNCT
cana-1064	60	1	in	in	ADP
cana-1064	60	2	the	the	DET
cana-1064	60	3	next	next	ADJ
cana-1064	60	4	section	section	NOUN
cana-1064	60	5	we	we	PRON
cana-1064	60	6	present	present	VERB
cana-1064	60	7	our	our	PRON
cana-1064	60	8	main	main	ADJ
cana-1064	60	9	result	result	NOUN
cana-1064	60	10	.	.	PUNCT
cana-1064	61	1	3.main	3.main	NUM
cana-1064	61	2	results	result	NOUN
cana-1064	61	3	theorem	theorem	VERB
cana-1064	61	4	3.1	3.1	NUM
cana-1064	61	5	:	:	PUNCT
cana-1064	61	6	let	let	VERB
cana-1064	61	7	(	(	PUNCT
cana-1064	61	8	𝔛	𝔛	NOUN
cana-1064	61	9	,	,	PUNCT
cana-1064	61	10	𝒫	𝒫	NOUN
cana-1064	61	11	)	)	PUNCT
cana-1064	61	12	be	be	VERB
cana-1064	61	13	a	a	DET
cana-1064	61	14	complete	complete	ADJ
cana-1064	61	15	pms	pm	NOUN
cana-1064	61	16	.	.	PUNCT
cana-1064	62	1	suppose	suppose	VERB
cana-1064	62	2	that	that	SCONJ
cana-1064	62	3	𝔣	𝔣	NOUN
cana-1064	62	4	,	,	PUNCT
cana-1064	62	5	𝔤	𝔤	PROPN
cana-1064	62	6	,	,	PUNCT
cana-1064	62	7	𝔖	𝔖	PROPN
cana-1064	62	8	and	and	CCONJ
cana-1064	62	9	𝔗	𝔗	PROPN
cana-1064	62	10	:	:	PUNCT
cana-1064	62	11	𝔛→𝔛	𝔛→𝔛	PROPN
cana-1064	62	12	are	be	AUX
cana-1064	62	13	four	four	NUM
cana-1064	62	14	self	self	NOUN
cana-1064	62	15	mappings	mapping	NOUN
cana-1064	62	16	satisfying	satisfy	VERB
cana-1064	62	17	(	(	PUNCT
cana-1064	62	18	i	i	NOUN
cana-1064	62	19	)	)	PUNCT
cana-1064	62	20	𝔣(𝔛)	𝔣(𝔛)	NOUN
cana-1064	62	21	𝔗(𝔛	𝔗(𝔛	NOUN
cana-1064	62	22	)	)	PUNCT
cana-1064	62	23	and	and	CCONJ
cana-1064	62	24	𝔤	𝔤	PROPN
cana-1064	62	25	(	(	PUNCT
cana-1064	62	26	𝔛	𝔛	NOUN
cana-1064	62	27	)	)	PUNCT
cana-1064	62	28			PROPN
cana-1064	62	29	𝔖(𝔛	𝔖(𝔛	NUM
cana-1064	62	30	)	)	PUNCT
cana-1064	62	31	(	(	PUNCT
cana-1064	62	32	ii	ii	NOUN
cana-1064	62	33	)	)	PUNCT
cana-1064	62	34	two	two	NUM
cana-1064	62	35	pairs	pair	NOUN
cana-1064	62	36	(	(	PUNCT
cana-1064	62	37	𝔣	𝔣	NOUN
cana-1064	62	38	,	,	PUNCT
cana-1064	62	39	𝔖	𝔖	PROPN
cana-1064	62	40	)	)	PUNCT
cana-1064	62	41	and	and	CCONJ
cana-1064	62	42	(	(	PUNCT
cana-1064	62	43	𝔤	𝔤	PROPN
cana-1064	62	44	,	,	PUNCT
cana-1064	62	45	𝔗	𝔗	PROPN
cana-1064	62	46	)	)	PUNCT
cana-1064	62	47	are	be	AUX
cana-1064	62	48	wc	wc	PROPN
cana-1064	62	49	mappings	mapping	NOUN
cana-1064	62	50	(	(	PUNCT
cana-1064	62	51	iii	iii	NOUN
cana-1064	62	52	)	)	PUNCT
cana-1064	62	53	𝔣(𝔛	𝔣(𝔛	NOUN
cana-1064	62	54	)	)	PUNCT
cana-1064	62	55	or	or	CCONJ
cana-1064	62	56	𝔗(𝔛	𝔗(𝔛	NOUN
cana-1064	62	57	)	)	PUNCT
cana-1064	62	58	or	or	CCONJ
cana-1064	62	59	𝔤	𝔤	PROPN
cana-1064	62	60	(	(	PUNCT
cana-1064	62	61	𝔛	𝔛	NOUN
cana-1064	62	62	)	)	PUNCT
cana-1064	62	63	or	or	CCONJ
cana-1064	62	64	𝔖(𝔛	𝔖(𝔛	NUM
cana-1064	62	65	)	)	PUNCT
cana-1064	62	66	is	be	AUX
cana-1064	62	67	closed	close	VERB
cana-1064	62	68	subset	subset	NOUN
cana-1064	62	69	of	of	ADP
cana-1064	62	70	(	(	PUNCT
cana-1064	62	71	𝔛	𝔛	PROPN
cana-1064	62	72	,	,	PUNCT
cana-1064	62	73	𝒫	𝒫	NOUN
cana-1064	62	74	)	)	PUNCT
cana-1064	62	75	(	(	PUNCT
cana-1064	62	76	iv	iv	X
cana-1064	62	77	)	)	PUNCT
cana-1064	62	78	assume	assume	VERB
cana-1064	62	79	that	that	SCONJ
cana-1064	62	80	there	there	PRON
cana-1064	62	81	exists	exist	VERB
cana-1064	62	82	ℱ	ℱ	PROPN
cana-1064	62	83	∆ℱ	∆ℱ	PROPN
cana-1064	62	84	and	and	CCONJ
cana-1064	62	85	𝜏	𝜏	X
cana-1064	62	86	>	>	X
cana-1064	62	87	0	0	NUM
cana-1064	62	88	for	for	ADP
cana-1064	62	89	𝜆	𝜆	NOUN
cana-1064	62	90	,	,	PUNCT
cana-1064	62	91	𝜇	𝜇	ADP
cana-1064	62	92	∈	∈	X
cana-1064	62	93	𝔛	𝔛	NOUN
cana-1064	62	94	such	such	ADJ
cana-1064	62	95	that	that	DET
cana-1064	62	96	𝒫(𝔣𝜆	𝒫(𝔣𝜆	NOUN
cana-1064	62	97	,	,	PUNCT
cana-1064	62	98	𝔤𝜇	𝔤𝜇	NOUN
cana-1064	62	99	)	)	PUNCT
cana-1064	62	100	>	>	X
cana-1064	62	101	0	0	NUM
cana-1064	62	102	⟹	⟹	NUM
cana-1064	62	103	𝜏	𝜏	PROPN
cana-1064	62	104	+	+	SYM
cana-1064	62	105	ℱ	ℱ	PROPN
cana-1064	62	106	(	(	PUNCT
cana-1064	62	107	𝒫(𝔣𝜆	𝒫(𝔣𝜆	NOUN
cana-1064	62	108	,	,	PUNCT
cana-1064	62	109	𝔤𝜇	𝔤𝜇	NOUN
cana-1064	62	110	)	)	PUNCT
cana-1064	62	111	)	)	PUNCT
cana-1064	62	112	≤	≤	NUM
cana-1064	62	113	ℱ(ℳ(𝜆	ℱ(ℳ(𝜆	NOUN
cana-1064	62	114	,	,	PUNCT
cana-1064	62	115	𝜇))	𝜇))	NOUN
cana-1064	62	116	…	…	PUNCT
cana-1064	62	117	…	…	SYM
cana-1064	62	118	…	…	SYM
cana-1064	62	119	…	…	PUNCT
cana-1064	62	120	.(3.1.1	.(3.1.1	NOUN
cana-1064	62	121	)	)	PUNCT
cana-1064	62	122	where	where	SCONJ
cana-1064	62	123	ℳ(𝜆	ℳ(𝜆	NOUN
cana-1064	62	124	,	,	PUNCT
cana-1064	62	125	𝜇	𝜇	NOUN
cana-1064	62	126	)	)	PUNCT
cana-1064	62	127	=	=	SYM
cana-1064	62	128	𝑚𝑎𝑥{𝒫(𝔖𝜆	𝑚𝑎𝑥{𝒫(𝔖𝜆	NOUN
cana-1064	62	129	,	,	PUNCT
cana-1064	62	130	𝔗𝜇	𝔗𝜇	NOUN
cana-1064	62	131	)	)	PUNCT
cana-1064	62	132	,	,	PUNCT
cana-1064	62	133	𝒫(𝔣𝜆	𝒫(𝔣𝜆	NOUN
cana-1064	62	134	,	,	PUNCT
cana-1064	62	135	𝔖𝜆	𝔖𝜆	PROPN
cana-1064	62	136	)	)	PUNCT
cana-1064	62	137	,	,	PUNCT
cana-1064	62	138	𝒫(𝔤𝜇	𝒫(𝔤𝜇	NOUN
cana-1064	62	139	,	,	PUNCT
cana-1064	62	140	𝔗𝜇	𝔗𝜇	NOUN
cana-1064	62	141	)	)	PUNCT
cana-1064	62	142	,	,	PUNCT
cana-1064	62	143	1	1	NUM
cana-1064	62	144	2	2	NUM
cana-1064	62	145	[	[	X
cana-1064	62	146	𝒫(𝔣𝜆	𝒫(𝔣𝜆	NOUN
cana-1064	62	147	,	,	PUNCT
cana-1064	62	148	𝔗𝜇	𝔗𝜇	NOUN
cana-1064	62	149	)	)	PUNCT
cana-1064	62	150	+	+	NUM
cana-1064	62	151	𝒫(𝔤𝜇	𝒫(𝔤𝜇	NOUN
cana-1064	62	152	,	,	PUNCT
cana-1064	62	153	𝔖𝜆	𝔖𝜆	PROPN
cana-1064	62	154	)	)	PUNCT
cana-1064	62	155	]	]	PUNCT
cana-1064	62	156	.	.	PUNCT
cana-1064	63	1	communications	communication	NOUN
cana-1064	63	2	on	on	ADP
cana-1064	63	3	applied	apply	VERB
cana-1064	63	4	nonlinear	nonlinear	ADJ
cana-1064	63	5	analysis	analysis	NOUN
cana-1064	63	6	issn	issn	NOUN
cana-1064	63	7	:	:	PUNCT
cana-1064	63	8	1074	1074	NUM
cana-1064	63	9	-	-	PUNCT
cana-1064	63	10	133x	133x	NUM
cana-1064	63	11	vol	vol	NOUN
cana-1064	63	12	31	31	NUM
cana-1064	63	13	no	no	NOUN
cana-1064	63	14	.	.	PUNCT
cana-1064	64	1	5s	5s	NUM
cana-1064	64	2	(	(	PUNCT
cana-1064	64	3	2024	2024	NUM
cana-1064	64	4	)	)	PUNCT
cana-1064	64	5	452	452	NUM
cana-1064	65	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1064	65	2	then	then	ADV
cana-1064	65	3	𝔖	𝔖	PROPN
cana-1064	65	4	,	,	PUNCT
cana-1064	65	5	𝔗	𝔗	PROPN
cana-1064	65	6	,	,	PUNCT
cana-1064	65	7	𝔣	𝔣	ADJ
cana-1064	65	8	and	and	CCONJ
cana-1064	65	9	𝔤	𝔤	PRON
cana-1064	65	10	have	have	VERB
cana-1064	65	11	a	a	DET
cana-1064	65	12	unique	unique	ADJ
cana-1064	65	13	common	common	ADJ
cana-1064	65	14	fixed	fix	VERB
cana-1064	65	15	point	point	NOUN
cana-1064	65	16	in	in	ADP
cana-1064	65	17	𝔛	𝔛	PROPN
cana-1064	65	18	.	.	PUNCT
cana-1064	66	1	proof	proof	NOUN
cana-1064	66	2	:	:	PUNCT
cana-1064	66	3	let	let	VERB
cana-1064	66	4	λ0𝔛	λ0𝔛	PRON
cana-1064	66	5	be	be	AUX
cana-1064	66	6	any	any	DET
cana-1064	66	7	point	point	NOUN
cana-1064	66	8	.	.	PUNCT
cana-1064	67	1	by	by	ADP
cana-1064	67	2	the	the	DET
cana-1064	67	3	(	(	PUNCT
cana-1064	67	4	i	i	NOUN
cana-1064	67	5	)	)	PUNCT
cana-1064	67	6	of	of	ADP
cana-1064	67	7	(	(	PUNCT
cana-1064	67	8	3.1	3.1	NUM
cana-1064	67	9	)	)	PUNCT
cana-1064	67	10	,	,	PUNCT
cana-1064	67	11	we	we	PRON
cana-1064	67	12	construct	construct	VERB
cana-1064	67	13	sequences	sequence	NOUN
cana-1064	67	14	{	{	PUNCT
cana-1064	67	15	𝜆𝜂	𝜆𝜂	SYM
cana-1064	67	16	}	}	PUNCT
cana-1064	67	17	𝑎𝑛𝑑	𝑎𝑛𝑑	X
cana-1064	67	18	{	{	PUNCT
cana-1064	67	19	𝜇𝜂	𝜇𝜂	NOUN
cana-1064	67	20	}	}	PUNCT
cana-1064	67	21	𝑖𝑛	𝑖𝑛	NOUN
cana-1064	68	1	𝔛	𝔛	NOUN
cana-1064	68	2	satisfying	satisfy	VERB
cana-1064	68	3	𝔗𝜆2𝜂+1	𝔗𝜆2𝜂+1	NOUN
cana-1064	68	4	=	=	SYM
cana-1064	68	5	𝔣𝜆2𝜂	𝔣𝜆2𝜂	PROPN
cana-1064	68	6	=	=	PUNCT
cana-1064	68	7	𝜇2𝜂+1	𝜇2𝜂+1	PROPN
cana-1064	68	8	and	and	CCONJ
cana-1064	68	9	𝔖𝜆2𝜂+2	𝔖𝜆2𝜂+2	NUM
cana-1064	68	10	=	=	SYM
cana-1064	68	11	𝔤𝜆2𝜂+1	𝔤𝜆2𝜂+1	PROPN
cana-1064	68	12	=	=	SYM
cana-1064	68	13	𝜇2𝜂+2	𝜇2𝜂+2	X
cana-1064	68	14	…	…	PUNCT
cana-1064	68	15	…	…	PUNCT
cana-1064	68	16	…	…	PUNCT
cana-1064	68	17	……	……	NOUN
cana-1064	68	18	……	……	NOUN
cana-1064	68	19	.(3.1.2	.(3.1.2	NOUN
cana-1064	68	20	)	)	PUNCT
cana-1064	68	21	for	for	ADP
cana-1064	68	22	𝜂	𝜂	NOUN
cana-1064	68	23	=	=	SYM
cana-1064	68	24	0,1	0,1	NUM
cana-1064	68	25	,	,	PUNCT
cana-1064	68	26	2	2	NUM
cana-1064	68	27	,	,	PUNCT
cana-1064	68	28	…	…	PUNCT
cana-1064	68	29	…	…	PUNCT
cana-1064	68	30	.	.	PUNCT
cana-1064	69	1	step	step	NOUN
cana-1064	69	2	-	-	PUNCT
cana-1064	69	3	i	i	NOUN
cana-1064	69	4	:	:	PUNCT
cana-1064	69	5	to	to	PART
cana-1064	69	6	prove	prove	VERB
cana-1064	69	7	that	that	DET
cana-1064	69	8	𝒫(𝜇𝜂	𝒫(𝜇𝜂	NOUN
cana-1064	69	9	,	,	PUNCT
cana-1064	69	10	𝜇𝜂+1	𝜇𝜂+1	NUM
cana-1064	69	11	)	)	PUNCT
cana-1064	69	12	⟶	⟶	NOUN
cana-1064	69	13	0	0	NUM
cana-1064	69	14	as	as	ADP
cana-1064	69	15	𝜂	𝜂	NOUN
cana-1064	69	16	⟶	⟶	NOUN
cana-1064	69	17	0	0	NUM
cana-1064	69	18	.	.	PUNCT
cana-1064	70	1	𝜏	𝜏	PRON
cana-1064	70	2	+	+	NUM
cana-1064	70	3	ℱ(𝒫(𝜇2𝜂+1	ℱ(𝒫(𝜇2𝜂+1	PROPN
cana-1064	70	4	,	,	PUNCT
cana-1064	70	5	𝜇2𝜂+2	𝜇2𝜂+2	NOUN
cana-1064	70	6	)	)	PUNCT
cana-1064	70	7	)	)	PUNCT
cana-1064	70	8	≤	≤	PROPN
cana-1064	70	9	ℱ(ℳ(𝜆2𝜂	ℱ(ℳ(𝜆2𝜂	NOUN
cana-1064	70	10	,	,	PUNCT
cana-1064	70	11	𝜆2𝜂+1	𝜆2𝜂+1	PROPN
cana-1064	70	12	)	)	PUNCT
cana-1064	70	13	)	)	PUNCT
cana-1064	71	1	…	…	PUNCT
cana-1064	71	2	……	……	NOUN
cana-1064	71	3	……	……	NOUN
cana-1064	71	4	……	……	NOUN
cana-1064	71	5	……	……	NOUN
cana-1064	71	6	……	……	NOUN
cana-1064	71	7	..	..	PUNCT
cana-1064	71	8	.(3.1.3	.(3.1.3	PUNCT
cana-1064	71	9	)	)	PUNCT
cana-1064	72	1	it	it	PRON
cana-1064	72	2	follows	follow	VERB
cana-1064	72	3	from	from	ADP
cana-1064	72	4	(	(	PUNCT
cana-1064	72	5	𝒫ℳ𝒮2)and	𝒫ℳ𝒮2)and	X
cana-1064	72	6	(	(	PUNCT
cana-1064	72	7	𝒫ℳ𝒮4	𝒫ℳ𝒮4	PROPN
cana-1064	72	8	)	)	PUNCT
cana-1064	72	9	that	that	DET
cana-1064	72	10	ℳ(𝜆2𝜂	ℳ(𝜆2𝜂	NOUN
cana-1064	72	11	,	,	PUNCT
cana-1064	72	12	𝜆2𝜂+1	𝜆2𝜂+1	PROPN
cana-1064	72	13	)	)	PUNCT
cana-1064	73	1	=	=	PUNCT
cana-1064	74	1	ℳ(𝜆	ℳ(𝜆	NOUN
cana-1064	74	2	,	,	PUNCT
cana-1064	74	3	𝜇	𝜇	NOUN
cana-1064	74	4	)	)	PUNCT
cana-1064	74	5	=	=	SYM
cana-1064	74	6	max	max	X
cana-1064	74	7	{	{	PUNCT
cana-1064	74	8	𝒫	𝒫	PROPN
cana-1064	74	9	(	(	PUNCT
cana-1064	74	10	𝔖𝜆2𝜂	𝔖𝜆2𝜂	PROPN
cana-1064	74	11	,	,	PUNCT
cana-1064	74	12	𝔗𝜆2𝜂+1	𝔗𝜆2𝜂+1	PROPN
cana-1064	74	13	)	)	PUNCT
cana-1064	74	14	,	,	PUNCT
cana-1064	74	15	𝒫(𝔣𝜆2𝜂	𝒫(𝔣𝜆2𝜂	PROPN
cana-1064	74	16	,	,	PUNCT
cana-1064	74	17	𝔖𝜆2𝜂	𝔖𝜆2𝜂	PROPN
cana-1064	74	18	)	)	PUNCT
cana-1064	74	19	,	,	PUNCT
cana-1064	74	20	𝒫(𝔤𝜆2𝜂+1	𝒫(𝔤𝜆2𝜂+1	PROPN
cana-1064	74	21	,	,	PUNCT
cana-1064	74	22	𝔗𝜆2𝜂+1	𝔗𝜆2𝜂+1	PROPN
cana-1064	74	23	)	)	PUNCT
cana-1064	74	24	,	,	PUNCT
cana-1064	74	25	1	1	NUM
cana-1064	74	26	2	2	NUM
cana-1064	75	1	[	[	X
cana-1064	75	2	𝒫(𝔣𝜆2𝜂	𝒫(𝔣𝜆2𝜂	PROPN
cana-1064	75	3	,	,	PUNCT
cana-1064	75	4	𝔗𝜆2𝜂+1	𝔗𝜆2𝜂+1	PROPN
cana-1064	75	5	)	)	PUNCT
cana-1064	75	6	+	+	CCONJ
cana-1064	75	7	𝒫(𝔤𝜆2𝜂+1	𝒫(𝔤𝜆2𝜂+1	PROPN
cana-1064	75	8	,	,	PUNCT
cana-1064	75	9	𝔖𝜆2𝜂	𝔖𝜆2𝜂	NOUN
cana-1064	75	10	)	)	PUNCT
cana-1064	75	11	]	]	PUNCT
cana-1064	76	1	=	=	PUNCT
cana-1064	76	2	max	max	PROPN
cana-1064	76	3	{	{	PUNCT
cana-1064	76	4	𝒫(𝜇2𝜂𝓃	𝒫(𝜇2𝜂𝓃	NOUN
cana-1064	76	5	,	,	PUNCT
cana-1064	76	6	𝜇2𝓃𝜂+1	𝜇2𝓃𝜂+1	NOUN
cana-1064	76	7	)	)	PUNCT
cana-1064	76	8	,	,	PUNCT
cana-1064	76	9	𝒫(𝜇2𝓃𝜂+1	𝒫(𝜇2𝓃𝜂+1	NOUN
cana-1064	76	10	,	,	PUNCT
cana-1064	76	11	𝜇2𝜂𝓃	𝜇2𝜂𝓃	NUM
cana-1064	76	12	)	)	PUNCT
cana-1064	76	13	,	,	PUNCT
cana-1064	76	14	𝒫(𝜇2𝑛𝜂+2	𝒫(𝜇2𝑛𝜂+2	NOUN
cana-1064	76	15	,	,	PUNCT
cana-1064	76	16	𝜇2𝓃𝜂+1	𝜇2𝓃𝜂+1	NOUN
cana-1064	76	17	)	)	PUNCT
cana-1064	76	18	,	,	PUNCT
cana-1064	76	19	1	1	NUM
cana-1064	76	20	2	2	NUM
cana-1064	76	21	[	[	X
cana-1064	76	22	𝒫(𝜇2𝜂𝓃	𝒫(𝜇2𝜂𝓃	NUM
cana-1064	76	23	,	,	PUNCT
cana-1064	76	24	𝜇2𝜂+2	𝜇2𝜂+2	PROPN
cana-1064	76	25	)	)	PUNCT
cana-1064	76	26	+	+	CCONJ
cana-1064	77	1	𝒫(𝜇2𝜂+1	𝒫(𝜇2𝜂+1	PROPN
cana-1064	77	2	,	,	PUNCT
cana-1064	77	3	𝜇2𝜂+1	𝜇2𝜂+1	NOUN
cana-1064	77	4	)	)	PUNCT
cana-1064	77	5	]	]	PUNCT
cana-1064	77	6	}	}	PUNCT
cana-1064	77	7	≤	≤	NUM
cana-1064	77	8	𝑚𝑎𝑥	𝑚𝑎𝑥	X
cana-1064	77	9	{	{	PUNCT
cana-1064	77	10	𝒫(𝜇2𝜂	𝒫(𝜇2𝜂	NOUN
cana-1064	77	11	,	,	PUNCT
cana-1064	77	12	𝜇2𝜂+1	𝜇2𝜂+1	PROPN
cana-1064	77	13	)	)	PUNCT
cana-1064	77	14	,	,	PUNCT
cana-1064	77	15	𝒫(𝜇2𝜂+2	𝒫(𝜇2𝜂+2	PROPN
cana-1064	77	16	,	,	PUNCT
cana-1064	77	17	𝜇2𝜂+1	𝜇2𝜂+1	NOUN
cana-1064	77	18	)	)	PUNCT
cana-1064	77	19	,	,	PUNCT
cana-1064	77	20	1	1	NUM
cana-1064	77	21	2	2	NUM
cana-1064	77	22	[	[	X
cana-1064	77	23	𝒫(𝜇2𝜂+1	𝒫(𝜇2𝜂+1	PROPN
cana-1064	77	24	,	,	PUNCT
cana-1064	77	25	𝜇2𝜂+1	𝜇2𝜂+1	ADJ
cana-1064	77	26	)	)	PUNCT
cana-1064	77	27	+	+	CCONJ
cana-1064	77	28	𝒫(𝜇2𝜂	𝒫(𝜇2𝜂	PROPN
cana-1064	77	29	,	,	PUNCT
cana-1064	77	30	𝜇2𝜂+1	𝜇2𝜂+1	NOUN
cana-1064	77	31	)	)	PUNCT
cana-1064	77	32	+	+	CCONJ
cana-1064	78	1	𝒫(𝜇2𝜂+1	𝒫(𝜇2𝜂+1	PROPN
cana-1064	78	2	,	,	PUNCT
cana-1064	78	3	𝜇2𝜂+2	𝜇2𝜂+2	PROPN
cana-1064	78	4	)	)	PUNCT
cana-1064	78	5	−	−	PROPN
cana-1064	79	1	𝒫(𝜇2𝜂+1	𝒫(𝜇2𝜂+1	PROPN
cana-1064	79	2	,	,	PUNCT
cana-1064	79	3	𝜇2𝜂+1	𝜇2𝜂+1	PROPN
cana-1064	79	4	)	)	PUNCT
cana-1064	79	5	]	]	PUNCT
cana-1064	79	6	}	}	PUNCT
cana-1064	79	7	≤	≤	NUM
cana-1064	79	8	𝑚𝑎𝑥{𝒫(𝜇2𝜂	𝑚𝑎𝑥{𝒫(𝜇2𝜂	NOUN
cana-1064	79	9	,	,	PUNCT
cana-1064	79	10	𝜇2𝜂+1	𝜇2𝜂+1	NOUN
cana-1064	79	11	)	)	PUNCT
cana-1064	79	12	,	,	PUNCT
cana-1064	79	13	𝒫(𝜇2𝜂+2	𝒫(𝜇2𝜂+2	PROPN
cana-1064	79	14	,	,	PUNCT
cana-1064	79	15	𝜇2𝜂+1	𝜇2𝜂+1	NOUN
cana-1064	79	16	)	)	PUNCT
cana-1064	79	17	}	}	PUNCT
cana-1064	79	18	if	if	SCONJ
cana-1064	79	19	𝑚𝑎𝑥{𝒫(𝜇2𝜂	𝑚𝑎𝑥{𝒫(𝜇2𝜂	NOUN
cana-1064	79	20	,	,	PUNCT
cana-1064	79	21	𝜇2𝜂+1	𝜇2𝜂+1	NOUN
cana-1064	79	22	)	)	PUNCT
cana-1064	79	23	,	,	PUNCT
cana-1064	79	24	𝒫(𝜇2𝜂+2	𝒫(𝜇2𝜂+2	PROPN
cana-1064	79	25	,	,	PUNCT
cana-1064	79	26	𝜇2𝜂+1	𝜇2𝜂+1	NOUN
cana-1064	79	27	)	)	PUNCT
cana-1064	79	28	}	}	PUNCT
cana-1064	79	29	=	=	SYM
cana-1064	79	30	𝒫(𝜇2𝜂+1	𝒫(𝜇2𝜂+1	PROPN
cana-1064	79	31	,	,	PUNCT
cana-1064	79	32	𝜇2𝜂+2	𝜇2𝜂+2	PROPN
cana-1064	79	33	)	)	PUNCT
cana-1064	79	34	then	then	ADV
cana-1064	79	35	𝜏	𝜏	X
cana-1064	79	36	+	+	SYM
cana-1064	79	37	ℱ(𝒫(𝜇2𝜂+1	ℱ(𝒫(𝜇2𝜂+1	PROPN
cana-1064	79	38	,	,	PUNCT
cana-1064	79	39	𝜇2𝜂+2	𝜇2𝜂+2	NOUN
cana-1064	79	40	)	)	PUNCT
cana-1064	79	41	)	)	PUNCT
cana-1064	80	1	≤	≤	PUNCT
cana-1064	81	1	ℱ(𝒫(𝜇2𝜂+1	ℱ(𝒫(𝜇2𝜂+1	PROPN
cana-1064	81	2	,	,	PUNCT
cana-1064	81	3	𝜇2𝜂+2	𝜇2𝜂+2	NOUN
cana-1064	81	4	)	)	PUNCT
cana-1064	81	5	)	)	PUNCT
cana-1064	82	1	this	this	PRON
cana-1064	82	2	implies	imply	VERB
cana-1064	82	3	ℱ(𝒫(𝜇2𝜂+1	ℱ(𝒫(𝜇2𝜂+1	PROPN
cana-1064	82	4	,	,	PUNCT
cana-1064	82	5	𝜇2𝜂+2	𝜇2𝜂+2	NOUN
cana-1064	82	6	)	)	PUNCT
cana-1064	82	7	)	)	PUNCT
cana-1064	82	8	≤	≤	PUNCT
cana-1064	83	1	ℱ(𝒫(𝜇2𝜂+1	ℱ(𝒫(𝜇2𝜂+1	PROPN
cana-1064	83	2	,	,	PUNCT
cana-1064	83	3	𝜇2𝜂+2	𝜇2𝜂+2	NOUN
cana-1064	83	4	)	)	PUNCT
cana-1064	83	5	)	)	PUNCT
cana-1064	84	1	−	−	ADP
cana-1064	84	2	𝜏	𝜏	NUM
cana-1064	84	3	which	which	PRON
cana-1064	84	4	is	be	AUX
cana-1064	84	5	congtradiction	congtradiction	NOUN
cana-1064	84	6	to	to	ADP
cana-1064	84	7	(	(	PUNCT
cana-1064	84	8	ℱ	ℱ	PROPN
cana-1064	84	9	-1	-1	NOUN
cana-1064	84	10	)	)	PUNCT
cana-1064	84	11	.	.	PUNCT
cana-1064	85	1	thus	thus	ADV
cana-1064	85	2	𝑚𝑎𝑥{𝒫(𝜇2𝜂	𝑚𝑎𝑥{𝒫(𝜇2𝜂	X
cana-1064	85	3	,	,	PUNCT
cana-1064	85	4	𝜇2𝜂+1	𝜇2𝜂+1	NOUN
cana-1064	85	5	)	)	PUNCT
cana-1064	85	6	,	,	PUNCT
cana-1064	85	7	𝒫(𝜇2𝜂+2	𝒫(𝜇2𝜂+2	PROPN
cana-1064	85	8	,	,	PUNCT
cana-1064	85	9	𝜇2𝜂+1	𝜇2𝜂+1	NOUN
cana-1064	85	10	)	)	PUNCT
cana-1064	85	11	}	}	PUNCT
cana-1064	85	12	=	=	SYM
cana-1064	85	13	𝒫(𝜇2𝜂	𝒫(𝜇2𝜂	PROPN
cana-1064	85	14	,	,	PUNCT
cana-1064	85	15	𝜇2𝜂+1	𝜇2𝜂+1	PROPN
cana-1064	85	16	)	)	PUNCT
cana-1064	85	17	for	for	ADP
cana-1064	85	18	all	all	PRON
cana-1064	85	19	𝜂	𝜂	PRON
cana-1064	85	20	∈	∈	NOUN
cana-1064	85	21	𝑁.	𝑁.	PROPN
cana-1064	85	22	from	from	ADP
cana-1064	85	23	(	(	PUNCT
cana-1064	85	24	10	10	NUM
cana-1064	85	25	)	)	PUNCT
cana-1064	85	26	,	,	PUNCT
cana-1064	85	27	𝜏	𝜏	X
cana-1064	85	28	+	+	SYM
cana-1064	86	1	ℱ(𝒫(𝜇2𝜂+1	ℱ(𝒫(𝜇2𝜂+1	PROPN
cana-1064	86	2	,	,	PUNCT
cana-1064	86	3	𝜇2𝜂+2	𝜇2𝜂+2	NOUN
cana-1064	86	4	)	)	PUNCT
cana-1064	86	5	)	)	PUNCT
cana-1064	87	1	≤	≤	NUM
cana-1064	87	2	ℱ(𝒫(𝜇2𝜂	ℱ(𝒫(𝜇2𝜂	X
cana-1064	87	3	,	,	PUNCT
cana-1064	87	4	𝜇2𝜂+1	𝜇2𝜂+1	NOUN
cana-1064	87	5	)	)	PUNCT
cana-1064	87	6	)	)	PUNCT
cana-1064	87	7	…	…	PUNCT
cana-1064	87	8	(	(	PUNCT
cana-1064	87	9	3.1.4	3.1.4	NUM
cana-1064	87	10	)	)	PUNCT
cana-1064	87	11	constituting	constitute	VERB
cana-1064	87	12	this	this	DET
cana-1064	87	13	way	way	NOUN
cana-1064	87	14	,	,	PUNCT
cana-1064	87	15	it	it	PRON
cana-1064	87	16	follows	follow	VERB
cana-1064	87	17	that	that	SCONJ
cana-1064	87	18	ℱ(𝒫(𝜇2𝜂	ℱ(𝒫(𝜇2𝜂	PROPN
cana-1064	87	19	,	,	PUNCT
cana-1064	87	20	𝜇2𝜂+1	𝜇2𝜂+1	NOUN
cana-1064	87	21	)	)	PUNCT
cana-1064	87	22	)	)	PUNCT
cana-1064	87	23	≤	≤	NUM
cana-1064	88	1	𝐹	𝐹	PROPN
cana-1064	88	2	(	(	PUNCT
cana-1064	88	3	𝒫(𝜇2𝜂−1	𝒫(𝜇2𝜂−1	PROPN
cana-1064	88	4	,	,	PUNCT
cana-1064	88	5	𝜇2𝜂	𝜇2𝜂	NOUN
cana-1064	88	6	)	)	PUNCT
cana-1064	88	7	)	)	PUNCT
cana-1064	89	1	−	−	PROPN
cana-1064	89	2	𝜏	𝜏	X
cana-1064	89	3	…	…	PUNCT
cana-1064	89	4	…	…	PUNCT
cana-1064	89	5	(	(	PUNCT
cana-1064	89	6	3.1.5	3.1.5	NOUN
cana-1064	89	7	)	)	PUNCT
cana-1064	89	8	.	.	PUNCT
cana-1064	90	1	using	use	VERB
cana-1064	90	2	(	(	PUNCT
cana-1064	90	3	3.1.4	3.1.4	NUM
cana-1064	90	4	)	)	PUNCT
cana-1064	90	5	and	and	CCONJ
cana-1064	90	6	(	(	PUNCT
cana-1064	90	7	3.1.5	3.1.5	X
cana-1064	90	8	)	)	PUNCT
cana-1064	90	9	ℱ(𝒫(𝜇2𝜂+1	ℱ(𝒫(𝜇2𝜂+1	PROPN
cana-1064	90	10	,	,	PUNCT
cana-1064	90	11	𝜇2𝜂+2	𝜇2𝜂+2	NOUN
cana-1064	90	12	)	)	PUNCT
cana-1064	90	13	)	)	PUNCT
cana-1064	91	1	≤	≤	NUM
cana-1064	91	2	ℱ(𝒫(𝜇2𝜂	ℱ(𝒫(𝜇2𝜂	X
cana-1064	91	3	,	,	PUNCT
cana-1064	91	4	𝜇2𝜂+1	𝜇2𝜂+1	NOUN
cana-1064	91	5	)	)	PUNCT
cana-1064	91	6	)	)	PUNCT
cana-1064	92	1	−	−	ADP
cana-1064	92	2	𝜏	𝜏	NOUN
cana-1064	92	3	≤	≤	NUM
cana-1064	92	4	ℱ(𝒫(𝜇2𝜂−1	ℱ(𝒫(𝜇2𝜂−1	NOUN
cana-1064	92	5	,	,	PUNCT
cana-1064	92	6	𝜇2𝜂	𝜇2𝜂	NOUN
cana-1064	92	7	)	)	PUNCT
cana-1064	92	8	)	)	PUNCT
cana-1064	93	1	−	−	PROPN
cana-1064	94	1	2𝜏	2𝜏	NOUN
cana-1064	94	2	≤	≤	PUNCT
cana-1064	94	3	ℱ(𝒫(𝜇2𝜂−2	ℱ(𝒫(𝜇2𝜂−2	PROPN
cana-1064	94	4	,	,	PUNCT
cana-1064	94	5	𝜇2𝜂−1	𝜇2𝜂−1	NOUN
cana-1064	94	6	)	)	PUNCT
cana-1064	94	7	)	)	PUNCT
cana-1064	95	1	−	−	PROPN
cana-1064	96	1	3𝜏	3𝜏	NOUN
cana-1064	96	2	≤	≤	NUM
cana-1064	96	3	ℱ(𝒫(𝜇0	ℱ(𝒫(𝜇0	ADJ
cana-1064	96	4	,	,	PUNCT
cana-1064	96	5	𝜇1	𝜇1	NOUN
cana-1064	96	6	)	)	PUNCT
cana-1064	96	7	)	)	PUNCT
cana-1064	97	1	−	−	PROPN
cana-1064	97	2	(	(	PUNCT
cana-1064	97	3	2𝜂	2𝜂	NOUN
cana-1064	97	4	+	+	CCONJ
cana-1064	97	5	1)𝜏	1)𝜏	NUM
cana-1064	97	6	…	…	SYM
cana-1064	97	7	…	…	PUNCT
cana-1064	97	8	.(3.1.6	.(3.1.6	NUM
cana-1064	97	9	)	)	PUNCT
cana-1064	97	10	and	and	CCONJ
cana-1064	97	11	ℱ(𝒫(𝜇2𝜂	ℱ(𝒫(𝜇2𝜂	NOUN
cana-1064	97	12	,	,	PUNCT
cana-1064	97	13	𝜇2𝜂+1	𝜇2𝜂+1	NOUN
cana-1064	97	14	)	)	PUNCT
cana-1064	97	15	)	)	PUNCT
cana-1064	97	16	≤	≤	NUM
cana-1064	97	17	ℱ(𝒫(𝜇2𝜂−1	ℱ(𝒫(𝜇2𝜂−1	NOUN
cana-1064	97	18	,	,	PUNCT
cana-1064	97	19	𝜇2𝜂	𝜇2𝜂	NOUN
cana-1064	97	20	)	)	PUNCT
cana-1064	97	21	)	)	PUNCT
cana-1064	98	1	−	−	ADP
cana-1064	98	2	𝜏	𝜏	NOUN
cana-1064	98	3	communications	communication	NOUN
cana-1064	98	4	on	on	ADP
cana-1064	98	5	applied	apply	VERB
cana-1064	98	6	nonlinear	nonlinear	ADJ
cana-1064	98	7	analysis	analysis	NOUN
cana-1064	98	8	issn	issn	NOUN
cana-1064	98	9	:	:	PUNCT
cana-1064	98	10	1074	1074	NUM
cana-1064	98	11	-	-	PUNCT
cana-1064	98	12	133x	133x	NUM
cana-1064	98	13	vol	vol	NOUN
cana-1064	98	14	31	31	NUM
cana-1064	98	15	no	no	NOUN
cana-1064	98	16	.	.	PUNCT
cana-1064	99	1	5s	5s	NUM
cana-1064	99	2	(	(	PUNCT
cana-1064	99	3	2024	2024	NUM
cana-1064	99	4	)	)	PUNCT
cana-1064	99	5	453	453	NUM
cana-1064	99	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1064	99	7	≤	≤	NUM
cana-1064	99	8	ℱ	ℱ	PROPN
cana-1064	99	9	(	(	PUNCT
cana-1064	99	10	𝒫((𝜇2𝜂−2	𝒫((𝜇2𝜂−2	PROPN
cana-1064	99	11	,	,	PUNCT
cana-1064	99	12	𝜇2𝜂−1	𝜇2𝜂−1	NOUN
cana-1064	99	13	)	)	PUNCT
cana-1064	99	14	)	)	PUNCT
cana-1064	100	1	−	−	PROPN
cana-1064	101	1	2𝜏	2𝜏	NUM
cana-1064	101	2	≤	≤	NOUN
cana-1064	101	3	ℱ(𝒫((𝜇0	ℱ(𝒫((𝜇0	PROPN
cana-1064	101	4	,	,	PUNCT
cana-1064	101	5	𝜇1	𝜇1	NOUN
cana-1064	101	6	)	)	PUNCT
cana-1064	101	7	)	)	PUNCT
cana-1064	102	1	−	−	PROPN
cana-1064	102	2	(	(	PUNCT
cana-1064	102	3	2𝜂)𝜏	2𝜂)𝜏	NUM
cana-1064	102	4	…	…	PUNCT
cana-1064	102	5	..	..	PUNCT
cana-1064	102	6	(3.1.7	(3.1.7	NOUN
cana-1064	102	7	)	)	PUNCT
cana-1064	102	8	repeating	repeat	VERB
cana-1064	102	9	,	,	PUNCT
cana-1064	102	10	ℱ(𝒫(𝜇𝜂	ℱ(𝒫(𝜇𝜂	X
cana-1064	102	11	,	,	PUNCT
cana-1064	102	12	𝜇𝜂+1	𝜇𝜂+1	NUM
cana-1064	102	13	)	)	PUNCT
cana-1064	102	14	)	)	PUNCT
cana-1064	102	15	≤	≤	PROPN
cana-1064	102	16	ℱ(𝒫((𝜇0	ℱ(𝒫((𝜇0	NUM
cana-1064	102	17	,	,	PUNCT
cana-1064	102	18	𝜇1	𝜇1	NOUN
cana-1064	102	19	)	)	PUNCT
cana-1064	102	20	)	)	PUNCT
cana-1064	103	1	−	−	PROPN
cana-1064	104	1	𝜂𝜏	𝜂𝜏	ADV
cana-1064	104	2	then	then	ADV
cana-1064	104	3	it	it	PRON
cana-1064	104	4	follows	follow	VERB
cana-1064	104	5	lim	lim	PROPN
cana-1064	104	6	𝓃𝜂→∞	𝓃𝜂→∞	PROPN
cana-1064	104	7	ℱ(𝒫(𝜇𝜂	ℱ(𝒫(𝜇𝜂	X
cana-1064	104	8	,	,	PUNCT
cana-1064	104	9	𝜇𝜂+1	𝜇𝜂+1	NUM
cana-1064	104	10	)	)	PUNCT
cana-1064	104	11	)	)	PUNCT
cana-1064	105	1	=	=	PUNCT
cana-1064	105	2	−∞	−∞	PUNCT
cana-1064	105	3	by	by	ADP
cana-1064	105	4	ℱ∆ℱ	ℱ∆ℱ	NOUN
cana-1064	105	5	and	and	CCONJ
cana-1064	105	6	(	(	PUNCT
cana-1064	105	7	ℱ	ℱ	PROPN
cana-1064	105	8	-2	-2	NOUN
cana-1064	105	9	)	)	PUNCT
cana-1064	105	10	we	we	PRON
cana-1064	105	11	have	have	VERB
cana-1064	105	12	lim	lim	NOUN
cana-1064	105	13	𝜂→∞	𝜂→∞	NUM
cana-1064	105	14	𝒫(𝜇𝜂	𝒫(𝜇𝜂	NOUN
cana-1064	105	15	,	,	PUNCT
cana-1064	105	16	𝜇𝜂+1	𝜇𝜂+1	NUM
cana-1064	105	17	)	)	PUNCT
cana-1064	105	18	=	=	SYM
cana-1064	105	19	0	0	NUM
cana-1064	105	20	..	..	PUNCT
cana-1064	105	21	…	…	PUNCT
cana-1064	105	22	..	..	PUNCT
cana-1064	105	23	(3.1.8	(3.1.8	PROPN
cana-1064	105	24	)	)	PUNCT
cana-1064	105	25	step	step	NOUN
cana-1064	105	26	-	-	PUNCT
cana-1064	105	27	ii	ii	NOUN
cana-1064	105	28	now	now	ADV
cana-1064	105	29	we	we	PRON
cana-1064	105	30	prove	prove	VERB
cana-1064	105	31	that	that	SCONJ
cana-1064	105	32	{	{	PUNCT
cana-1064	105	33	𝜇𝜂	𝜇𝜂	NOUN
cana-1064	105	34	}	}	PUNCT
cana-1064	105	35	is	be	AUX
cana-1064	105	36	𝒫	𝒫	NOUN
cana-1064	105	37	-cauchy	-cauchy	ADJ
cana-1064	105	38	sequence	sequence	NOUN
cana-1064	105	39	.	.	PUNCT
cana-1064	106	1	by	by	ADP
cana-1064	106	2	ℱ∆ℱ	ℱ∆ℱ	NOUN
cana-1064	106	3	and	and	CCONJ
cana-1064	106	4	(	(	PUNCT
cana-1064	106	5	ℱ	ℱ	PROPN
cana-1064	106	6	-3	-3	PUNCT
cana-1064	106	7	)	)	PUNCT
cana-1064	106	8	thereexists	thereexist	NOUN
cana-1064	106	9	𝓀	𝓀	PROPN
cana-1064	106	10	∈	∈	PROPN
cana-1064	106	11	(	(	PUNCT
cana-1064	106	12	0,1	0,1	NOUN
cana-1064	106	13	)	)	PUNCT
cana-1064	106	14	such	such	ADJ
cana-1064	106	15	that	that	SCONJ
cana-1064	106	16	lim	lim	PROPN
cana-1064	106	17	𝜂→∞	𝜂→∞	NUM
cana-1064	106	18	𝒫(𝜇𝜂	𝒫(𝜇𝜂	NOUN
cana-1064	106	19	,	,	PUNCT
cana-1064	106	20	𝜇𝜂+1)𝓀	𝜇𝜂+1)𝓀	X
cana-1064	106	21	ℱ(𝒫(𝜇𝜂	ℱ(𝒫(𝜇𝜂	NUM
cana-1064	106	22	,	,	PUNCT
cana-1064	106	23	𝜇𝜂+1	𝜇𝜂+1	NUM
cana-1064	106	24	)	)	PUNCT
cana-1064	106	25	)	)	PUNCT
cana-1064	106	26	=	=	PUNCT
cana-1064	106	27	0	0	NUM
cana-1064	106	28	…	…	NUM
cana-1064	106	29	.	.	PUNCT
cana-1064	107	1	(	(	PUNCT
cana-1064	107	2	3.1.9	3.1.9	NUM
cana-1064	107	3	)	)	PUNCT
cana-1064	107	4	by	by	ADP
cana-1064	107	5	(	(	PUNCT
cana-1064	107	6	3.1.6	3.1.6	NUM
cana-1064	107	7	)	)	PUNCT
cana-1064	107	8	and	and	CCONJ
cana-1064	107	9	(	(	PUNCT
cana-1064	107	10	3.1.7	3.1.7	NUM
cana-1064	107	11	)	)	PUNCT
cana-1064	107	12	lim	lim	NOUN
cana-1064	107	13	𝜂→∞	𝜂→∞	NUM
cana-1064	107	14	𝒫(𝜇𝜂	𝒫(𝜇𝜂	NOUN
cana-1064	107	15	,	,	PUNCT
cana-1064	107	16	𝜇𝜂+1)𝓀	𝜇𝜂+1)𝓀	X
cana-1064	107	17	ℱ(𝒫(𝜇𝜂	ℱ(𝒫(𝜇𝜂	NUM
cana-1064	107	18	,	,	PUNCT
cana-1064	107	19	𝜇𝜂+1	𝜇𝜂+1	NUM
cana-1064	107	20	)	)	PUNCT
cana-1064	107	21	)	)	PUNCT
cana-1064	108	1	=	=	SYM
cana-1064	108	2	0	0	NUM
cana-1064	109	1	lim	lim	PROPN
cana-1064	109	2	𝜂→∞	𝜂→∞	PROPN
cana-1064	109	3	𝒫(𝜇2𝜂+1	𝒫(𝜇2𝜂+1	PROPN
cana-1064	109	4	,	,	PUNCT
cana-1064	109	5	𝜇2𝜂+2)𝓀	𝜇2𝜂+2)𝓀	VERB
cana-1064	109	6	ℱ	ℱ	PROPN
cana-1064	109	7	(	(	PUNCT
cana-1064	109	8	𝒫(𝜇2𝜂+1	𝒫(𝜇2𝜂+1	PROPN
cana-1064	109	9	,	,	PUNCT
cana-1064	109	10	𝜇2𝜂+2	𝜇2𝜂+2	NOUN
cana-1064	109	11	)	)	PUNCT
cana-1064	109	12	)	)	PUNCT
cana-1064	110	1	−	−	PROPN
cana-1064	110	2	ℱ(𝒫(𝜇0	ℱ(𝒫(𝜇0	ADJ
cana-1064	110	3	,	,	PUNCT
cana-1064	110	4	𝜇1	𝜇1	NOUN
cana-1064	110	5	)	)	PUNCT
cana-1064	110	6	≤	≤	NOUN
cana-1064	111	1	𝒫(𝜇2𝜂+1	𝒫(𝜇2𝜂+1	PROPN
cana-1064	111	2	,	,	PUNCT
cana-1064	111	3	𝜇2𝜂+2	𝜇2𝜂+2	PROPN
cana-1064	111	4	)	)	PUNCT
cana-1064	111	5	𝓀	𝓀	PROPN
cana-1064	111	6	−	−	PROPN
cana-1064	111	7	(	(	PUNCT
cana-1064	111	8	2𝜂	2𝜂	NOUN
cana-1064	111	9	+	+	CCONJ
cana-1064	111	10	1)𝜏	1)𝜏	NUM
cana-1064	111	11	≤	≤	NUM
cana-1064	111	12	0	0	NUM
cana-1064	111	13	-	-	PUNCT
cana-1064	111	14	----(3.1.10	----(3.1.10	NOUN
cana-1064	111	15	)	)	PUNCT
cana-1064	111	16	and	and	CCONJ
cana-1064	111	17	lim	lim	PROPN
cana-1064	111	18	𝜂→∞	𝜂→∞	PROPN
cana-1064	111	19	𝒫(𝜇2𝜂	𝒫(𝜇2𝜂	PROPN
cana-1064	111	20	,	,	PUNCT
cana-1064	111	21	𝜇2𝜂+1)𝓀	𝜇2𝜂+1)𝓀	PROPN
cana-1064	111	22	ℱ	ℱ	PROPN
cana-1064	111	23	(	(	PUNCT
cana-1064	111	24	𝒫(𝜇2𝜂	𝒫(𝜇2𝜂	INTJ
cana-1064	111	25	,	,	PUNCT
cana-1064	111	26	𝜇2𝜂+1	𝜇2𝜂+1	NOUN
cana-1064	111	27	)	)	PUNCT
cana-1064	111	28	)	)	PUNCT
cana-1064	112	1	−	−	PROPN
cana-1064	112	2	ℱ(𝒫(𝜇0	ℱ(𝒫(𝜇0	ADJ
cana-1064	112	3	,	,	PUNCT
cana-1064	112	4	𝜇1	𝜇1	NOUN
cana-1064	112	5	)	)	PUNCT
cana-1064	112	6	≤	≤	NOUN
cana-1064	112	7	𝒫(𝜇2𝜂	𝒫(𝜇2𝜂	NOUN
cana-1064	112	8	,	,	PUNCT
cana-1064	112	9	𝜇2𝜂+1	𝜇2𝜂+1	PROPN
cana-1064	112	10	)	)	PUNCT
cana-1064	112	11	𝓀	𝓀	PROPN
cana-1064	112	12	−	−	PROPN
cana-1064	112	13	(	(	PUNCT
cana-1064	112	14	2𝜂)𝜏	2𝜂)𝜏	NUM
cana-1064	112	15	≤	≤	NUM
cana-1064	112	16	0	0	NUM
cana-1064	112	17	-	-	PUNCT
cana-1064	112	18	---(3.1.11	---(3.1.11	NOUN
cana-1064	112	19	)	)	PUNCT
cana-1064	112	20	using	use	VERB
cana-1064	112	21	the	the	DET
cana-1064	112	22	above	above	ADJ
cana-1064	112	23	inequality	inequality	NOUN
cana-1064	112	24	and	and	CCONJ
cana-1064	112	25	(	(	PUNCT
cana-1064	112	26	3.1.9	3.1.9	NUM
cana-1064	112	27	)	)	PUNCT
cana-1064	112	28	lim	lim	NOUN
cana-1064	112	29	𝜂→∞	𝜂→∞	NUM
cana-1064	112	30	𝜂	𝜂	PROPN
cana-1064	112	31	𝒫(𝜇2𝜂	𝒫(𝜇2𝜂	PROPN
cana-1064	112	32	,	,	PUNCT
cana-1064	112	33	𝜇2𝜂+1	𝜇2𝜂+1	NOUN
cana-1064	112	34	)	)	PUNCT
cana-1064	112	35	𝓀	𝓀	PROPN
cana-1064	112	36	=	=	PUNCT
cana-1064	112	37	0	0	X
cana-1064	112	38	.	.	PUNCT
cana-1064	113	1	therefore	therefore	ADV
cana-1064	113	2	,	,	PUNCT
cana-1064	113	3	there	there	PRON
cana-1064	113	4	exists	exist	VERB
cana-1064	113	5	𝜂	𝜂	PROPN
cana-1064	113	6	∈	∈	PROPN
cana-1064	113	7	ℕ	ℕ	PROPN
cana-1064	113	8	such	such	ADJ
cana-1064	113	9	that	that	DET
cana-1064	113	10	𝜂.	𝜂.	NOUN
cana-1064	113	11	𝒫(𝜇2𝜂	𝒫(𝜇2𝜂	PROPN
cana-1064	113	12	,	,	PUNCT
cana-1064	113	13	𝜇2𝜂+1	𝜇2𝜂+1	PROPN
cana-1064	113	14	)	)	PUNCT
cana-1064	113	15	𝓀	𝓀	PROPN
cana-1064	113	16	<	<	X
cana-1064	113	17	1	1	NUM
cana-1064	113	18	for	for	ADP
cana-1064	113	19	all	all	DET
cana-1064	113	20	𝜂	𝜂	PRON
cana-1064	113	21	≥	≥	NUM
cana-1064	113	22	𝜂1	𝜂1	NOUN
cana-1064	113	23	.	.	PUNCT
cana-1064	114	1	(	(	PUNCT
cana-1064	114	2	or	or	CCONJ
cana-1064	114	3	)	)	PUNCT
cana-1064	114	4	𝒫(𝜇𝓃	𝒫(𝜇𝓃	NOUN
cana-1064	114	5	,	,	PUNCT
cana-1064	114	6	𝜇𝓃+1	𝜇𝓃+1	NOUN
cana-1064	114	7	)	)	PUNCT
cana-1064	114	8	<	<	X
cana-1064	114	9	1	1	NUM
cana-1064	114	10	𝜂1/𝓀	𝜂1/𝓀	NUM
cana-1064	114	11	…	…	PUNCT
cana-1064	114	12	…	…	PUNCT
cana-1064	114	13	…	…	PUNCT
cana-1064	114	14	..	..	PUNCT
cana-1064	114	15	(3.1.12	(3.1.12	PUNCT
cana-1064	114	16	)	)	PUNCT
cana-1064	114	17	let	let	VERB
cana-1064	114	18	𝜁	𝜁	NOUN
cana-1064	114	19	,	,	PUNCT
cana-1064	114	20	𝜂	𝜂	PROPN
cana-1064	114	21	∈	∈	PROPN
cana-1064	114	22	ℕ	ℕ	PROPN
cana-1064	114	23	with	with	ADP
cana-1064	114	24	𝜁	𝜁	PROPN
cana-1064	114	25	>	>	X
cana-1064	114	26	𝜂	𝜂	X
cana-1064	114	27	>	>	X
cana-1064	114	28	𝜂1	𝜂1	PROPN
cana-1064	114	29	using	use	VERB
cana-1064	114	30	triangular	triangular	NOUN
cana-1064	114	31	inequality	inequality	NOUN
cana-1064	114	32	we	we	PRON
cana-1064	114	33	have	have	VERB
cana-1064	114	34	𝒫(𝜇𝜂	𝒫(𝜇𝜂	NOUN
cana-1064	114	35	,	,	PUNCT
cana-1064	114	36	𝜇𝜁	𝜇𝜁	NUM
cana-1064	114	37	)	)	PUNCT
cana-1064	114	38	=	=	SYM
cana-1064	114	39	𝒫(𝜇𝓃𝜂	𝒫(𝜇𝓃𝜂	PROPN
cana-1064	114	40	,	,	PUNCT
cana-1064	114	41	𝜇𝜂+1	𝜇𝜂+1	NUM
cana-1064	114	42	)	)	PUNCT
cana-1064	115	1	+	+	CCONJ
cana-1064	115	2	𝒫(𝜇𝜂+1	𝒫(𝜇𝜂+1	PROPN
cana-1064	115	3	,	,	PUNCT
cana-1064	115	4	𝜇𝜂+2	𝜇𝜂+2	NUM
cana-1064	115	5	)	)	PUNCT
cana-1064	115	6	…	…	PUNCT
cana-1064	115	7	.	.	PUNCT
cana-1064	116	1	+	+	PUNCT
cana-1064	116	2	𝒫(𝜇𝜁−1	𝒫(𝜇𝜁−1	NOUN
cana-1064	116	3	,	,	PUNCT
cana-1064	116	4	𝜇𝜁	𝜇𝜁	NUM
cana-1064	116	5	)	)	PUNCT
cana-1064	116	6	−[𝒫(𝜇𝜂+1	−[𝒫(𝜇𝜂+1	PROPN
cana-1064	116	7	,	,	PUNCT
cana-1064	116	8	𝜇𝜂+1	𝜇𝜂+1	NUM
cana-1064	116	9	)	)	PUNCT
cana-1064	116	10	+	+	CCONJ
cana-1064	116	11	𝒫(𝜇𝜂+2	𝒫(𝜇𝜂+2	NOUN
cana-1064	116	12	,	,	PUNCT
cana-1064	116	13	𝜇𝜂+2	𝜇𝜂+2	NUM
cana-1064	116	14	)	)	PUNCT
cana-1064	117	1	+	+	CCONJ
cana-1064	117	2	⋯	⋯	NOUN
cana-1064	117	3	.	.	PUNCT
cana-1064	118	1	𝒫(𝜇𝜁−1	𝒫(𝜇𝜁−1	PROPN
cana-1064	118	2	,	,	PUNCT
cana-1064	118	3	𝜇𝜁−1	𝜇𝜁−1	PROPN
cana-1064	118	4	)	)	PUNCT
cana-1064	118	5	]	]	PUNCT
cana-1064	118	6	≤	≤	PROPN
cana-1064	118	7	∑	∑	PUNCT
cana-1064	118	8	𝒫(𝜇𝒾	𝒫(𝜇𝒾	NOUN
cana-1064	118	9	,	,	PUNCT
cana-1064	118	10	𝜇𝒾+1	𝜇𝒾+1	NOUN
cana-1064	118	11	)	)	PUNCT
cana-1064	118	12	≤	≤	NOUN
cana-1064	118	13	∑	∑	PUNCT
cana-1064	118	14	𝒫(𝜇𝒾	𝒫(𝜇𝒾	NOUN
cana-1064	118	15	,	,	PUNCT
cana-1064	118	16	𝜇𝒾+1	𝜇𝒾+1	NOUN
cana-1064	118	17	)	)	PUNCT
cana-1064	118	18	≤	≤	NOUN
cana-1064	118	19	∑	∑	ADP
cana-1064	118	20	1	1	NUM
cana-1064	118	21	𝒾	𝒾	SYM
cana-1064	118	22	1	1	NUM
cana-1064	118	23	𝓀	𝓀	PROPN
cana-1064	118	24	.∞	.∞	NOUN
cana-1064	118	25	𝒾=1	𝒾=1	PUNCT
cana-1064	118	26	∞	∞	NUM
cana-1064	118	27	𝒾=1	𝒾=1	X
cana-1064	118	28	𝜁−1	𝜁−1	NUM
cana-1064	118	29	𝒾=1	𝒾=1	PUNCT
cana-1064	118	30	as	as	ADP
cana-1064	118	31	𝓀	𝓀	PROPN
cana-1064	118	32	∈	∈	PROPN
cana-1064	118	33	(	(	PUNCT
cana-1064	118	34	0,1	0,1	NUM
cana-1064	118	35	)	)	PUNCT
cana-1064	118	36	the	the	DET
cana-1064	118	37	infinite	infinite	ADJ
cana-1064	118	38	series	series	NOUN
cana-1064	118	39	∑	∑	PROPN
cana-1064	118	40	1	1	NUM
cana-1064	118	41	𝒾	𝒾	SYM
cana-1064	118	42	1	1	NUM
cana-1064	118	43	𝓀	𝓀	NUM
cana-1064	118	44	∞	∞	NUM
cana-1064	118	45	𝒾=1	𝒾=1	PUNCT
cana-1064	118	46	converges	converge	NOUN
cana-1064	118	47	,	,	PUNCT
cana-1064	118	48	consequently	consequently	ADV
cana-1064	118	49	we	we	PRON
cana-1064	118	50	get	get	VERB
cana-1064	118	51	lim	lim	NOUN
cana-1064	118	52	𝜁,𝜂→∞	𝜁,𝜂→∞	ADJ
cana-1064	118	53	𝒫(𝜇𝜂	𝒫(𝜇𝜂	NOUN
cana-1064	118	54	,	,	PUNCT
cana-1064	118	55	𝜇𝜁	𝜇𝜁	NUM
cana-1064	118	56	)	)	PUNCT
cana-1064	118	57	=	=	SYM
cana-1064	119	1	0	0	X
cana-1064	119	2	.	.	PUNCT
cana-1064	120	1	this	this	DET
cana-1064	120	2	communications	communication	NOUN
cana-1064	120	3	on	on	ADP
cana-1064	120	4	applied	apply	VERB
cana-1064	120	5	nonlinear	nonlinear	ADJ
cana-1064	120	6	analysis	analysis	NOUN
cana-1064	120	7	issn	issn	NOUN
cana-1064	120	8	:	:	PUNCT
cana-1064	120	9	1074	1074	NUM
cana-1064	120	10	-	-	PUNCT
cana-1064	120	11	133x	133x	NUM
cana-1064	120	12	vol	vol	NOUN
cana-1064	120	13	31	31	NUM
cana-1064	120	14	no	no	NOUN
cana-1064	120	15	.	.	PUNCT
cana-1064	121	1	5s	5s	NUM
cana-1064	121	2	(	(	PUNCT
cana-1064	121	3	2024	2024	NUM
cana-1064	121	4	)	)	PUNCT
cana-1064	121	5	454	454	NUM
cana-1064	121	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1064	121	7	proves	prove	VERB
cana-1064	121	8	that	that	SCONJ
cana-1064	121	9	{	{	PUNCT
cana-1064	121	10	𝜇𝜂	𝜇𝜂	NOUN
cana-1064	121	11	}	}	PUNCT
cana-1064	121	12	is	be	AUX
cana-1064	121	13	a	a	DET
cana-1064	121	14	cauchy	cauchy	ADJ
cana-1064	121	15	sequence	sequence	NOUN
cana-1064	121	16	in	in	ADP
cana-1064	121	17	(	(	PUNCT
cana-1064	121	18	𝔛	𝔛	PROPN
cana-1064	121	19	,	,	PUNCT
cana-1064	121	20	𝒫	𝒫	NOUN
cana-1064	121	21	)	)	PUNCT
cana-1064	121	22	and	and	CCONJ
cana-1064	121	23	consequently	consequently	ADV
cana-1064	121	24	we	we	PRON
cana-1064	121	25	get	get	VERB
cana-1064	121	26	{	{	PUNCT
cana-1064	121	27	𝜇𝜂	𝜇𝜂	NOUN
cana-1064	121	28	}	}	PUNCT
cana-1064	121	29	is	be	AUX
cana-1064	121	30	cauchy	cauchy	ADJ
cana-1064	121	31	in	in	ADP
cana-1064	121	32	(	(	PUNCT
cana-1064	121	33	𝔛	𝔛	PROPN
cana-1064	121	34	,	,	PUNCT
cana-1064	121	35	𝒹𝒫	𝒹𝒫	NOUN
cana-1064	121	36	)	)	PUNCT
cana-1064	121	37	.	.	PUNCT
cana-1064	122	1	since	since	SCONJ
cana-1064	122	2	(	(	PUNCT
cana-1064	122	3	𝔛	𝔛	PROPN
cana-1064	122	4	,	,	PUNCT
cana-1064	122	5	𝒫	𝒫	NOUN
cana-1064	122	6	)	)	PUNCT
cana-1064	122	7	is	be	AUX
cana-1064	122	8	complete	complete	ADJ
cana-1064	122	9	pms	pm	NOUN
cana-1064	122	10	,	,	PUNCT
cana-1064	122	11	this	this	PRON
cana-1064	122	12	ensures	ensure	VERB
cana-1064	122	13	that	that	SCONJ
cana-1064	122	14	(	(	PUNCT
cana-1064	122	15	𝔛	𝔛	PROPN
cana-1064	122	16	,	,	PUNCT
cana-1064	122	17	𝒹𝒫	𝒹𝒫	NOUN
cana-1064	122	18	)	)	PUNCT
cana-1064	122	19	is	be	AUX
cana-1064	122	20	complete	complete	ADJ
cana-1064	122	21	metric	metric	ADJ
cana-1064	122	22	space	space	NOUN
cana-1064	122	23	,	,	PUNCT
cana-1064	122	24	then	then	ADV
cana-1064	122	25	∃	∃	PROPN
cana-1064	122	26	𝓊𝓅	𝓊𝓅	NOUN
cana-1064	122	27	∈	∈	PROPN
cana-1064	122	28	𝔛	𝔛	PROPN
cana-1064	122	29	such	such	ADJ
cana-1064	122	30	that	that	SCONJ
cana-1064	122	31	lim	lim	PROPN
cana-1064	122	32	𝜂→∞	𝜂→∞	NUM
cana-1064	122	33	𝒹𝒫(𝜇𝜂	𝒹𝒫(𝜇𝜂	X
cana-1064	122	34	,	,	PUNCT
cana-1064	122	35	𝓊𝓅	𝓊𝓅	NOUN
cana-1064	122	36	)	)	PUNCT
cana-1064	122	37	=	=	SYM
cana-1064	122	38	0	0	X
cana-1064	122	39	.	.	PUNCT
cana-1064	123	1	moreover	moreover	ADV
cana-1064	123	2	𝒫(𝓊𝓅	𝒫(𝓊𝓅	NOUN
cana-1064	123	3	,	,	PUNCT
cana-1064	123	4	𝓊𝓅	𝓊𝓅	NOUN
cana-1064	123	5	)	)	PUNCT
cana-1064	123	6	=	=	SYM
cana-1064	123	7	lim	lim	PROPN
cana-1064	123	8	𝓃→∞	𝓃→∞	NUM
cana-1064	123	9	𝒫(𝜇𝜂	𝒫(𝜇𝜂	X
cana-1064	123	10	,	,	PUNCT
cana-1064	123	11	𝓊𝓅	𝓊𝓅	NOUN
cana-1064	123	12	)	)	PUNCT
cana-1064	124	1	=	=	SYM
cana-1064	124	2	lim	lim	PROPN
cana-1064	124	3	𝜁,,𝜂→∞	𝜁,,𝜂→∞	PROPN
cana-1064	124	4	𝒫(𝜇𝜂	𝒫(𝜇𝜂	NOUN
cana-1064	124	5	,	,	PUNCT
cana-1064	124	6	𝜇𝜁	𝜇𝜁	NUM
cana-1064	124	7	)	)	PUNCT
cana-1064	124	8	=	=	SYM
cana-1064	124	9	0	0	X
cana-1064	124	10	.......	.......	SYM
cana-1064	124	11	(3.1.13	(3.1.13	PROPN
cana-1064	124	12	)	)	PUNCT
cana-1064	124	13	since	since	SCONJ
cana-1064	124	14	𝜇𝜂	𝜇𝜂	PROPN
cana-1064	124	15	→	→	SYM
cana-1064	124	16	𝓊𝓅	𝓊𝓅	NOUN
cana-1064	124	17	then	then	ADV
cana-1064	124	18	𝔗𝜆2𝜂+1	𝔗𝜆2𝜂+1	AUX
cana-1064	124	19	,	,	PUNCT
cana-1064	124	20	𝔣𝜆2𝜂	𝔣𝜆2𝜂	PROPN
cana-1064	124	21	𝔖𝜆2𝜂+2	𝔖𝜆2𝜂+2	NUM
cana-1064	124	22	and	and	CCONJ
cana-1064	124	23	𝔤𝜆2𝜂+1	𝔤𝜆2𝜂+1	PROPN
cana-1064	124	24	converges	converge	NOUN
cana-1064	124	25	to	to	PART
cana-1064	124	26	𝓊𝓅.	𝓊𝓅.	VERB
cana-1064	124	27	step	step	NOUN
cana-1064	124	28	-	-	PUNCT
cana-1064	124	29	iii	iii	NOUN
cana-1064	124	30	now	now	ADV
cana-1064	124	31	we	we	PRON
cana-1064	124	32	claim	claim	VERB
cana-1064	124	33	that	that	SCONJ
cana-1064	124	34	the	the	DET
cana-1064	124	35	mappings	mapping	NOUN
cana-1064	124	36	𝔖	𝔖	PROPN
cana-1064	124	37	,	,	PUNCT
cana-1064	124	38	𝔗	𝔗	PROPN
cana-1064	124	39	,	,	PUNCT
cana-1064	124	40	𝔣	𝔣	ADJ
cana-1064	124	41	and	and	CCONJ
cana-1064	124	42	𝔤	𝔤	PROPN
cana-1064	124	43	have	have	VERB
cana-1064	124	44	a	a	DET
cana-1064	124	45	common	common	ADJ
cana-1064	124	46	fixed	fix	VERB
cana-1064	124	47	point	point	NOUN
cana-1064	124	48	.	.	PUNCT
cana-1064	124	49	suppose	suppose	VERB
cana-1064	124	50	that	that	SCONJ
cana-1064	124	51	the	the	DET
cana-1064	124	52	range	range	NOUN
cana-1064	124	53	𝔗(𝔛	𝔗(𝔛	NOUN
cana-1064	124	54	)	)	PUNCT
cana-1064	124	55	is	be	AUX
cana-1064	124	56	closed	close	VERB
cana-1064	124	57	,	,	PUNCT
cana-1064	124	58	then	then	ADV
cana-1064	124	59	∃	∃	PROPN
cana-1064	124	60	𝜈𝓅	𝜈𝓅	NOUN
cana-1064	124	61	∈	∈	PROPN
cana-1064	124	62	𝔛	𝔛	PROPN
cana-1064	124	63	such	such	ADJ
cana-1064	124	64	that	that	DET
cana-1064	124	65	𝔗𝜈𝓅	𝔗𝜈𝓅	PROPN
cana-1064	124	66	=	=	PUNCT
cana-1064	124	67	𝓊𝓅	𝓊𝓅	VERB
cana-1064	124	68	……	……	NOUN
cana-1064	124	69	.	.	PUNCT
cana-1064	125	1	(	(	PUNCT
cana-1064	125	2	3.1.14	3.1.14	NUM
cana-1064	125	3	)	)	PUNCT
cana-1064	125	4	now	now	ADV
cana-1064	125	5	assuming	assume	VERB
cana-1064	125	6	that	that	SCONJ
cana-1064	125	7	𝔤𝜈𝓅	𝔤𝜈𝓅	NOUN
cana-1064	125	8	≠	≠	PROPN
cana-1064	125	9	𝓊𝓅	𝓊𝓅	NOUN
cana-1064	125	10	and	and	CCONJ
cana-1064	125	11	put	put	VERB
cana-1064	125	12	𝜆	𝜆	PRON
cana-1064	125	13	=	=	X
cana-1064	125	14	𝜆2𝜂	𝜆2𝜂	NOUN
cana-1064	125	15	,	,	PUNCT
cana-1064	125	16	𝜇	𝜇	X
cana-1064	125	17	=	=	X
cana-1064	125	18	𝜈𝓅	𝜈𝓅	NOUN
cana-1064	125	19	in	in	ADP
cana-1064	125	20	(	(	PUNCT
cana-1064	125	21	3.1.1	3.1.1	NUM
cana-1064	125	22	)	)	PUNCT
cana-1064	125	23	then	then	ADV
cana-1064	125	24	(	(	PUNCT
cana-1064	125	25	𝔣𝜆2𝜂	𝔣𝜆2𝜂	PROPN
cana-1064	125	26	,	,	PUNCT
cana-1064	125	27	𝔤𝜈𝓅	𝔤𝜈𝓅	PROPN
cana-1064	125	28	)	)	PUNCT
cana-1064	125	29	>	>	X
cana-1064	125	30	0	0	NUM
cana-1064	125	31	⟹	⟹	NUM
cana-1064	126	1	𝜏	𝜏	PROPN
cana-1064	126	2	+	+	SYM
cana-1064	126	3	ℱ	ℱ	PROPN
cana-1064	126	4	(	(	PUNCT
cana-1064	126	5	𝒫(𝔣𝜆2𝜂	𝒫(𝔣𝜆2𝜂	PROPN
cana-1064	126	6	,	,	PUNCT
cana-1064	126	7	𝔤𝜈𝓅	𝔤𝜈𝓅	PROPN
cana-1064	126	8	)	)	PUNCT
cana-1064	126	9	)	)	PUNCT
cana-1064	126	10	≤	≤	PUNCT
cana-1064	126	11	ℱ	ℱ	PROPN
cana-1064	126	12	(	(	PUNCT
cana-1064	126	13	ℳ(𝜆2𝜂	ℳ(𝜆2𝜂	NOUN
cana-1064	126	14	,	,	PUNCT
cana-1064	126	15	𝜈𝓅))	𝜈𝓅))	PROPN
cana-1064	126	16	…	…	SYM
cana-1064	126	17	..	..	PUNCT
cana-1064	126	18	(3.1.15	(3.1.15	PROPN
cana-1064	126	19	)	)	PUNCT
cana-1064	126	20	where	where	SCONJ
cana-1064	126	21	ℳ(𝜆2𝜂	ℳ(𝜆2𝜂	NOUN
cana-1064	126	22	,	,	PUNCT
cana-1064	126	23	𝜈𝓅	𝜈𝓅	NOUN
cana-1064	126	24	)	)	PUNCT
cana-1064	126	25	=	=	SYM
cana-1064	127	1	𝑚𝑎𝑥{𝒫(𝔖𝜆2𝜂	𝑚𝑎𝑥{𝒫(𝔖𝜆2𝜂	PROPN
cana-1064	127	2	,	,	PUNCT
cana-1064	127	3	𝔗𝜈𝓅	𝔗𝜈𝓅	PROPN
cana-1064	127	4	)	)	PUNCT
cana-1064	127	5	,	,	PUNCT
cana-1064	127	6	𝒫(𝔣𝜆2𝜂	𝒫(𝔣𝜆2𝜂	PROPN
cana-1064	127	7	,	,	PUNCT
cana-1064	127	8	𝔖𝜆2𝜂	𝔖𝜆2𝜂	PROPN
cana-1064	127	9	)	)	PUNCT
cana-1064	127	10	,	,	PUNCT
cana-1064	127	11	𝒫(𝔤𝜈𝓅	𝒫(𝔤𝜈𝓅	PROPN
cana-1064	127	12	,	,	PUNCT
cana-1064	127	13	𝔗𝜈𝓅	𝔗𝜈𝓅	PROPN
cana-1064	127	14	)	)	PUNCT
cana-1064	127	15	,	,	PUNCT
cana-1064	127	16	1	1	NUM
cana-1064	127	17	2	2	NUM
cana-1064	128	1	[	[	X
cana-1064	128	2	𝒫(𝔣𝜆2𝜂	𝒫(𝔣𝜆2𝜂	PROPN
cana-1064	128	3	,	,	PUNCT
cana-1064	128	4	𝔗𝜈𝓅	𝔗𝜈𝓅	PROPN
cana-1064	128	5	)	)	PUNCT
cana-1064	129	1	+	+	CCONJ
cana-1064	129	2	𝒫(𝔤𝜈𝓅	𝒫(𝔤𝜈𝓅	PROPN
cana-1064	129	3	,	,	PUNCT
cana-1064	129	4	𝔖𝜆2𝜂	𝔖𝜆2𝜂	PROPN
cana-1064	129	5	)	)	PUNCT
cana-1064	129	6	]	]	PUNCT
cana-1064	129	7	}	}	PUNCT
cana-1064	129	8	=	=	SYM
cana-1064	129	9	max	max	X
cana-1064	129	10	{	{	PUNCT
cana-1064	129	11	𝒫(𝓊𝓅	𝒫(𝓊𝓅	PROPN
cana-1064	129	12	,	,	PUNCT
cana-1064	129	13	𝓊𝓅	𝓊𝓅	NOUN
cana-1064	129	14	)	)	PUNCT
cana-1064	129	15	,	,	PUNCT
cana-1064	129	16	𝒫(𝓊𝓅	𝒫(𝓊𝓅	NOUN
cana-1064	129	17	,	,	PUNCT
cana-1064	129	18	𝔤𝜈𝓅	𝔤𝜈𝓅	PROPN
cana-1064	129	19	)	)	PUNCT
cana-1064	129	20	,	,	PUNCT
cana-1064	129	21	𝒫(𝓊𝓅	𝒫(𝓊𝓅	NOUN
cana-1064	129	22	,	,	PUNCT
cana-1064	129	23	𝓊𝓅	𝓊𝓅	NOUN
cana-1064	129	24	)	)	PUNCT
cana-1064	129	25	,	,	PUNCT
cana-1064	129	26	1	1	NUM
cana-1064	129	27	2	2	NUM
cana-1064	129	28	[	[	X
cana-1064	129	29	𝒫(𝓊𝓅	𝒫(𝓊𝓅	NOUN
cana-1064	129	30	,	,	PUNCT
cana-1064	129	31	𝓊𝓅	𝓊𝓅	NOUN
cana-1064	129	32	)	)	PUNCT
cana-1064	129	33	+	+	NUM
cana-1064	129	34	𝒫(𝓊𝓅	𝒫(𝓊𝓅	NOUN
cana-1064	129	35	,	,	PUNCT
cana-1064	129	36	𝔤𝜈𝓅	𝔤𝜈𝓅	PROPN
cana-1064	129	37	)	)	PUNCT
cana-1064	129	38	]	]	PUNCT
cana-1064	129	39	}	}	PUNCT
cana-1064	129	40	𝜏	𝜏	X
cana-1064	129	41	+	+	CCONJ
cana-1064	129	42	ℱ(𝒫(𝓊𝓅	ℱ(𝒫(𝓊𝓅	NOUN
cana-1064	129	43	,	,	PUNCT
cana-1064	129	44	𝔤𝜈𝓅	𝔤𝜈𝓅	NOUN
cana-1064	129	45	)	)	PUNCT
cana-1064	129	46	)	)	PUNCT
cana-1064	130	1	≤	≤	PUNCT
cana-1064	130	2	ℱ	ℱ	PROPN
cana-1064	130	3	(	(	PUNCT
cana-1064	130	4	𝒫(𝓊𝓅	𝒫(𝓊𝓅	PROPN
cana-1064	130	5	,	,	PUNCT
cana-1064	130	6	𝔤𝜈𝓅	𝔤𝜈𝓅	PROPN
cana-1064	130	7	)	)	PUNCT
cana-1064	130	8	)	)	PUNCT
cana-1064	130	9	this	this	PRON
cana-1064	130	10	a	a	DET
cana-1064	130	11	contradiction	contradiction	NOUN
cana-1064	130	12	with	with	ADP
cana-1064	130	13	𝜏	𝜏	X
cana-1064	130	14	>	>	X
cana-1064	130	15	0	0	NUM
cana-1064	130	16	.	.	PUNCT
cana-1064	131	1	thus	thus	ADV
cana-1064	131	2	𝔤𝜈𝓅=𝓊𝓅.	𝔤𝜈𝓅=𝓊𝓅.	PROPN
cana-1064	131	3	…	…	PUNCT
cana-1064	131	4	…	…	PUNCT
cana-1064	131	5	(	(	PUNCT
cana-1064	131	6	3.1.15	3.1.15	NUM
cana-1064	131	7	)	)	PUNCT
cana-1064	131	8	therefore	therefore	ADV
cana-1064	131	9	using	use	VERB
cana-1064	131	10	(	(	PUNCT
cana-1064	131	11	3.1.14	3.1.14	NUM
cana-1064	131	12	)	)	PUNCT
cana-1064	131	13	and	and	CCONJ
cana-1064	131	14	(	(	PUNCT
cana-1064	131	15	3.1.15	3.1.15	X
cana-1064	131	16	)	)	PUNCT
cana-1064	131	17	we	we	PRON
cana-1064	131	18	obtain	obtain	VERB
cana-1064	131	19	𝔗𝜈𝓅	𝔗𝜈𝓅	PROPN
cana-1064	131	20	=	=	NOUN
cana-1064	131	21	𝔤𝜈𝓅	𝔤𝜈𝓅	NOUN
cana-1064	131	22	=	=	NOUN
cana-1064	131	23	𝓊𝓅	𝓊𝓅	NOUN
cana-1064	131	24	.since	.since	NOUN
cana-1064	131	25	𝔤	𝔤	NOUN
cana-1064	131	26	and	and	CCONJ
cana-1064	131	27	𝔗	𝔗	PROPN
cana-1064	131	28	are	be	AUX
cana-1064	131	29	wc	wc	PROPN
cana-1064	131	30	mappings	mapping	NOUN
cana-1064	131	31	then	then	ADV
cana-1064	131	32	,	,	PUNCT
cana-1064	132	1	𝔤𝓊𝓅	𝔤𝓊𝓅	NOUN
cana-1064	132	2	=	=	SYM
cana-1064	132	3	𝔤𝔗𝜈𝓅	𝔤𝔗𝜈𝓅	PROPN
cana-1064	132	4	=	=	PUNCT
cana-1064	133	1	𝔗𝔤𝜈𝓅	𝔗𝔤𝜈𝓅	PROPN
cana-1064	133	2	=	=	SYM
cana-1064	133	3	𝔗𝓊𝓅	𝔗𝓊𝓅	PROPN
cana-1064	133	4	.	.	PUNCT
cana-1064	134	1	…	…	PUNCT
cana-1064	134	2	…	…	PUNCT
cana-1064	134	3	…	…	SYM
cana-1064	134	4	…	…	PUNCT
cana-1064	134	5	.(3.1.16	.(3.1.16	NUM
cana-1064	134	6	)	)	PUNCT
cana-1064	134	7	now	now	ADV
cana-1064	134	8	we	we	PRON
cana-1064	134	9	show	show	VERB
cana-1064	134	10	that	that	DET
cana-1064	134	11	𝔤𝓊𝓅	𝔤𝓊𝓅	NOUN
cana-1064	134	12	=	=	SYM
cana-1064	134	13	𝜈𝓅	𝜈𝓅	NOUN
cana-1064	134	14	.	.	PUNCT
cana-1064	135	1	on	on	ADP
cana-1064	135	2	contrary	contrary	ADV
cana-1064	135	3	,	,	PUNCT
cana-1064	135	4	let	let	VERB
cana-1064	135	5	𝔤𝓊𝓅	𝔤𝓊𝓅	VERB
cana-1064	135	6	≠	≠	PROPN
cana-1064	135	7	𝜈𝓅	𝜈𝓅	NOUN
cana-1064	135	8	and	and	CCONJ
cana-1064	135	9	use	use	VERB
cana-1064	135	10	𝜆	𝜆	PRON
cana-1064	135	11	=	=	PUNCT
cana-1064	135	12	𝜆2𝜂	𝜆2𝜂	NOUN
cana-1064	135	13	,	,	PUNCT
cana-1064	135	14	𝜇	𝜇	X
cana-1064	135	15	=	=	X
cana-1064	135	16	𝜈𝓅	𝜈𝓅	NOUN
cana-1064	135	17	in	in	ADP
cana-1064	135	18	(	(	PUNCT
cana-1064	135	19	3.1.1	3.1.1	NUM
cana-1064	135	20	)	)	PUNCT
cana-1064	135	21	then	then	ADV
cana-1064	135	22	𝜏	𝜏	X
cana-1064	135	23	+	+	CCONJ
cana-1064	135	24	ℱ(𝒫(𝔣𝜆2𝜂	ℱ(𝒫(𝔣𝜆2𝜂	PROPN
cana-1064	135	25	,	,	PUNCT
cana-1064	135	26	𝔤𝜈𝓅	𝔤𝜈𝓅	NOUN
cana-1064	135	27	)	)	PUNCT
cana-1064	135	28	)	)	PUNCT
cana-1064	135	29	≤	≤	PUNCT
cana-1064	135	30	ℱ	ℱ	PROPN
cana-1064	135	31	(	(	PUNCT
cana-1064	135	32	ℳ(𝜆2𝜂𝓊𝓅	ℳ(𝜆2𝜂𝓊𝓅	NOUN
cana-1064	135	33	)	)	PUNCT
cana-1064	135	34	)	)	PUNCT
cana-1064	135	35	ℳ(𝜆2𝜂	ℳ(𝜆2𝜂	NOUN
cana-1064	135	36	,	,	PUNCT
cana-1064	135	37	𝓊𝓅	𝓊𝓅	NOUN
cana-1064	135	38	)	)	PUNCT
cana-1064	135	39	=	=	SYM
cana-1064	135	40	𝑚𝑎𝑥{𝒫(𝔖𝜆2𝜂	𝑚𝑎𝑥{𝒫(𝔖𝜆2𝜂	PROPN
cana-1064	135	41	,	,	PUNCT
cana-1064	135	42	𝔗𝓊𝓅	𝔗𝓊𝓅	PROPN
cana-1064	135	43	)	)	PUNCT
cana-1064	135	44	,	,	PUNCT
cana-1064	135	45	𝒫(𝔣𝓍𝜆2𝓃	𝒫(𝔣𝓍𝜆2𝓃	NOUN
cana-1064	135	46	,	,	PUNCT
cana-1064	135	47	𝔖𝜆2𝜂	𝔖𝜆2𝜂	NOUN
cana-1064	135	48	)	)	PUNCT
cana-1064	135	49	,	,	PUNCT
cana-1064	135	50	𝒫(𝔤𝓊𝓅	𝒫(𝔤𝓊𝓅	PROPN
cana-1064	135	51	,	,	PUNCT
cana-1064	135	52	𝔗𝓊𝓅	𝔗𝓊𝓅	PROPN
cana-1064	135	53	)	)	PUNCT
cana-1064	135	54	,	,	PUNCT
cana-1064	135	55	1	1	NUM
cana-1064	135	56	2	2	NUM
cana-1064	135	57	[	[	X
cana-1064	135	58	𝒫(𝔣𝜆2𝜂	𝒫(𝔣𝜆2𝜂	PROPN
cana-1064	135	59	,	,	PUNCT
cana-1064	135	60	𝔗𝓊𝓅	𝔗𝓊𝓅	PROPN
cana-1064	135	61	)	)	PUNCT
cana-1064	135	62	+	+	NUM
cana-1064	135	63	𝒫(𝔤𝓊𝓅	𝒫(𝔤𝓊𝓅	PROPN
cana-1064	135	64	,	,	PUNCT
cana-1064	135	65	𝔖𝜆2𝓃	𝔖𝜆2𝓃	PROPN
cana-1064	135	66	)	)	PUNCT
cana-1064	135	67	]	]	PUNCT
cana-1064	135	68	}	}	PUNCT
cana-1064	135	69	=	=	SYM
cana-1064	135	70	𝒫(𝓊𝓅	𝒫(𝓊𝓅	NOUN
cana-1064	135	71	,	,	PUNCT
cana-1064	135	72	𝔤𝓊𝓅	𝔤𝓊𝓅	NOUN
cana-1064	135	73	)	)	PUNCT
cana-1064	135	74	.	.	PUNCT
cana-1064	136	1	by	by	ADP
cana-1064	136	2	permitting	permit	VERB
cana-1064	136	3	,	,	PUNCT
cana-1064	136	4	continuity	continuity	NOUN
cana-1064	136	5	of	of	ADP
cana-1064	136	6	ℱ	ℱ	PROPN
cana-1064	136	7	and	and	CCONJ
cana-1064	136	8	applying	apply	VERB
cana-1064	136	9	the	the	DET
cana-1064	136	10	limit	limit	NOUN
cana-1064	136	11	as	as	ADP
cana-1064	136	12	𝜂→∞	𝜂→∞	NUM
cana-1064	136	13	,	,	PUNCT
cana-1064	136	14	we	we	PRON
cana-1064	136	15	have	have	VERB
cana-1064	136	16	𝜏	𝜏	NOUN
cana-1064	136	17	+	+	NUM
cana-1064	136	18	ℱ(𝒫(𝓊𝓅	ℱ(𝒫(𝓊𝓅	NOUN
cana-1064	136	19	,	,	PUNCT
cana-1064	136	20	𝔤𝓊𝓅	𝔤𝓊𝓅	NOUN
cana-1064	136	21	)	)	PUNCT
cana-1064	136	22	)	)	PUNCT
cana-1064	137	1	≤	≤	PUNCT
cana-1064	137	2	ℱ	ℱ	PROPN
cana-1064	137	3	(	(	PUNCT
cana-1064	137	4	ℳ(𝓊𝓅	ℳ(𝓊𝓅	NOUN
cana-1064	137	5	,	,	PUNCT
cana-1064	137	6	𝔤𝓊𝓅	𝔤𝓊𝓅	NOUN
cana-1064	137	7	)	)	PUNCT
cana-1064	137	8	)	)	PUNCT
cana-1064	137	9	which	which	PRON
cana-1064	137	10	is	be	AUX
cana-1064	137	11	contradiction	contradiction	NOUN
cana-1064	137	12	.	.	PUNCT
cana-1064	138	1	communications	communication	NOUN
cana-1064	138	2	on	on	ADP
cana-1064	138	3	applied	apply	VERB
cana-1064	138	4	nonlinear	nonlinear	ADJ
cana-1064	138	5	analysis	analysis	NOUN
cana-1064	138	6	issn	issn	NOUN
cana-1064	138	7	:	:	PUNCT
cana-1064	138	8	1074	1074	NUM
cana-1064	138	9	-	-	PUNCT
cana-1064	138	10	133x	133x	NUM
cana-1064	138	11	vol	vol	NOUN
cana-1064	138	12	31	31	NUM
cana-1064	138	13	no	no	NOUN
cana-1064	138	14	.	.	PUNCT
cana-1064	139	1	5s	5s	NUM
cana-1064	139	2	(	(	PUNCT
cana-1064	139	3	2024	2024	NUM
cana-1064	139	4	)	)	PUNCT
cana-1064	139	5	455	455	NUM
cana-1064	140	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1064	140	2	therefore	therefore	ADV
cana-1064	140	3	𝒫(𝓊𝓅	𝒫(𝓊𝓅	PROPN
cana-1064	140	4	,	,	PUNCT
cana-1064	140	5	𝔤𝓊𝓅	𝔤𝓊𝓅	NOUN
cana-1064	140	6	)	)	PUNCT
cana-1064	140	7	=	=	SYM
cana-1064	140	8	0	0	NUM
cana-1064	140	9	,	,	PUNCT
cana-1064	140	10	this	this	PRON
cana-1064	140	11	yields	yield	VERB
cana-1064	140	12	𝔗𝓊𝓅	𝔗𝓊𝓅	PROPN
cana-1064	140	13	=	=	SYM
cana-1064	140	14	𝔤𝓊𝓅	𝔤𝓊𝓅	NOUN
cana-1064	140	15	=	=	SYM
cana-1064	140	16	𝓊𝓅	𝓊𝓅	NOUN
cana-1064	140	17	…	…	PUNCT
cana-1064	140	18	…	…	PUNCT
cana-1064	140	19	.(3.1.17	.(3.1.17	NUM
cana-1064	140	20	)	)	PUNCT
cana-1064	141	1	now	now	ADV
cana-1064	141	2	we	we	PRON
cana-1064	141	3	show	show	VERB
cana-1064	141	4	that	that	DET
cana-1064	141	5	𝓊𝓅	𝓊𝓅	NOUN
cana-1064	141	6	is	be	AUX
cana-1064	141	7	a	a	DET
cana-1064	141	8	fixed	fix	VERB
cana-1064	141	9	pint	pint	NOUN
cana-1064	141	10	of	of	ADP
cana-1064	141	11	the	the	DET
cana-1064	141	12	mappings	mapping	NOUN
cana-1064	141	13	𝔣	𝔣	ADJ
cana-1064	141	14	and	and	CCONJ
cana-1064	141	15	𝔖	𝔖	PROPN
cana-1064	141	16	.	.	PUNCT
cana-1064	142	1	since	since	SCONJ
cana-1064	142	2	𝔤	𝔤	PROPN
cana-1064	142	3	(	(	PUNCT
cana-1064	142	4	𝒳	𝒳	PROPN
cana-1064	142	5	)	)	PUNCT
cana-1064	142	6			PROPN
cana-1064	142	7	𝔖(𝒳	𝔖(𝒳	NUM
cana-1064	142	8	)	)	PUNCT
cana-1064	142	9	∃	∃	PROPN
cana-1064	142	10	a	a	DET
cana-1064	142	11	point	point	NOUN
cana-1064	142	12	𝔷𝓅	𝔷𝓅	NOUN
cana-1064	142	13	∈	∈	PROPN
cana-1064	142	14	𝒳	𝒳	NOUN
cana-1064	142	15	such	such	ADJ
cana-1064	142	16	that	that	DET
cana-1064	142	17	𝔤𝓊𝓅	𝔤𝓊𝓅	NOUN
cana-1064	142	18	=	=	SYM
cana-1064	142	19	𝔖𝔷𝓅.	𝔖𝔷𝓅.	PROPN
cana-1064	142	20	suppose	suppose	VERB
cana-1064	142	21	that	that	SCONJ
cana-1064	142	22	𝔣𝔷𝓅	𝔣𝔷𝓅	PROPN
cana-1064	142	23	≠	≠	PROPN
cana-1064	142	24	𝔖𝔷𝓅	𝔖𝔷𝓅	PROPN
cana-1064	142	25	,	,	PUNCT
cana-1064	142	26	then	then	ADV
cana-1064	142	27	𝜏	𝜏	PROPN
cana-1064	142	28	+	+	CCONJ
cana-1064	142	29	ℱ(𝒫(𝔣𝔷𝓅.	ℱ(𝒫(𝔣𝔷𝓅.	PROPN
cana-1064	142	30	,	,	PUNCT
cana-1064	142	31	𝔤𝓊𝓅	𝔤𝓊𝓅	NOUN
cana-1064	142	32	)	)	PUNCT
cana-1064	142	33	)	)	PUNCT
cana-1064	142	34	≤	≤	PUNCT
cana-1064	142	35	ℱ	ℱ	PROPN
cana-1064	142	36	(	(	PUNCT
cana-1064	142	37	ℳ(𝔷𝓅.	ℳ(𝔷𝓅.	NOUN
cana-1064	142	38	,	,	PUNCT
cana-1064	142	39	𝓊𝓅	𝓊𝓅	NOUN
cana-1064	142	40	)	)	PUNCT
cana-1064	142	41	)	)	PUNCT
cana-1064	142	42	ℳ(𝔷𝓅.	ℳ(𝔷𝓅.	NOUN
cana-1064	142	43	,	,	PUNCT
cana-1064	142	44	𝓊𝓅	𝓊𝓅	NOUN
cana-1064	142	45	)	)	PUNCT
cana-1064	142	46	=	=	SYM
cana-1064	142	47	𝑚𝑎𝑥{𝒫(𝔖𝔷𝓅.	𝑚𝑎𝑥{𝒫(𝔖𝔷𝓅.	NOUN
cana-1064	142	48	,	,	PUNCT
cana-1064	142	49	𝔗𝓊𝓅	𝔗𝓊𝓅	PROPN
cana-1064	142	50	)	)	PUNCT
cana-1064	142	51	,	,	PUNCT
cana-1064	142	52	𝒫(𝔣𝔷𝓅.	𝒫(𝔣𝔷𝓅.	PROPN
cana-1064	142	53	,	,	PUNCT
cana-1064	142	54	𝔖𝔷𝓅.	𝔖𝔷𝓅.	PROPN
cana-1064	142	55	)	)	PUNCT
cana-1064	142	56	,	,	PUNCT
cana-1064	142	57	𝒫(𝔤𝓊𝓅	𝒫(𝔤𝓊𝓅	PROPN
cana-1064	142	58	,	,	PUNCT
cana-1064	142	59	𝔗𝓊𝓅	𝔗𝓊𝓅	PROPN
cana-1064	142	60	)	)	PUNCT
cana-1064	142	61	,	,	PUNCT
cana-1064	142	62	1	1	NUM
cana-1064	142	63	2	2	NUM
cana-1064	142	64	[	[	X
cana-1064	142	65	𝒫(𝔣𝔷𝓅.	𝒫(𝔣𝔷𝓅.	PROPN
cana-1064	142	66	,	,	PUNCT
cana-1064	142	67	𝔗𝓊𝓅	𝔗𝓊𝓅	PROPN
cana-1064	142	68	)	)	PUNCT
cana-1064	142	69	+	+	NUM
cana-1064	142	70	𝒫(𝔤𝓊𝓅	𝒫(𝔤𝓊𝓅	PROPN
cana-1064	142	71	,	,	PUNCT
cana-1064	142	72	𝔖𝔷𝓅.	𝔖𝔷𝓅.	PROPN
cana-1064	142	73	)	)	PUNCT
cana-1064	142	74	]	]	PUNCT
cana-1064	142	75	}	}	PUNCT
cana-1064	142	76	=	=	SYM
cana-1064	142	77	𝑚𝑎𝑥{𝒫(𝔤𝓊𝓅	𝑚𝑎𝑥{𝒫(𝔤𝓊𝓅	PROPN
cana-1064	142	78	,	,	PUNCT
cana-1064	142	79	𝔤𝓊𝓅	𝔤𝓊𝓅	PROPN
cana-1064	142	80	)	)	PUNCT
cana-1064	142	81	,	,	PUNCT
cana-1064	142	82	𝒫(𝔤𝓊𝓅	𝒫(𝔤𝓊𝓅	PROPN
cana-1064	142	83	,	,	PUNCT
cana-1064	142	84	𝔤𝓊𝓅	𝔤𝓊𝓅	NOUN
cana-1064	142	85	)	)	PUNCT
cana-1064	142	86	,	,	PUNCT
cana-1064	142	87	𝒫(𝔤𝓊𝓅	𝒫(𝔤𝓊𝓅	PROPN
cana-1064	142	88	,	,	PUNCT
cana-1064	142	89	𝔣𝔷𝓅.	𝔣𝔷𝓅.	NOUN
cana-1064	142	90	)	)	PUNCT
cana-1064	142	91	,	,	PUNCT
cana-1064	142	92	1	1	NUM
cana-1064	142	93	2	2	NUM
cana-1064	142	94	[	[	X
cana-1064	142	95	𝒫(𝔣𝔷𝓅.	𝒫(𝔣𝔷𝓅.	PROPN
cana-1064	142	96	,	,	PUNCT
cana-1064	142	97	𝔤𝓊𝓅	𝔤𝓊𝓅	NOUN
cana-1064	142	98	)	)	PUNCT
cana-1064	142	99	+	+	CCONJ
cana-1064	142	100	𝒫(𝔤𝓊𝓅	𝒫(𝔤𝓊𝓅	PROPN
cana-1064	142	101	,	,	PUNCT
cana-1064	142	102	𝔤𝓊𝓅	𝔤𝓊𝓅	NOUN
cana-1064	142	103	)	)	PUNCT
cana-1064	142	104	]	]	PUNCT
cana-1064	142	105	}	}	PUNCT
cana-1064	142	106	=	=	SYM
cana-1064	142	107	𝒫(𝔤𝓊𝓅	𝒫(𝔤𝓊𝓅	ADJ
cana-1064	142	108	,	,	PUNCT
cana-1064	142	109	𝔣𝔷𝓅.	𝔣𝔷𝓅.	NOUN
cana-1064	142	110	)	)	PUNCT
cana-1064	142	111	𝜏	𝜏	PROPN
cana-1064	142	112	+	+	NUM
cana-1064	142	113	ℱ(𝒫(𝔣𝔷𝓅.	ℱ(𝒫(𝔣𝔷𝓅.	PROPN
cana-1064	142	114	,	,	PUNCT
cana-1064	142	115	𝔤𝓊𝓅	𝔤𝓊𝓅	NOUN
cana-1064	142	116	)	)	PUNCT
cana-1064	142	117	)	)	PUNCT
cana-1064	142	118	≤	≤	PROPN
cana-1064	142	119	𝒫(𝔤𝓊𝓅	𝒫(𝔤𝓊𝓅	PROPN
cana-1064	142	120	,	,	PUNCT
cana-1064	142	121	𝔣𝔷𝓅	𝔣𝔷𝓅	PROPN
cana-1064	142	122	)	)	PUNCT
cana-1064	142	123	this	this	PRON
cana-1064	142	124	a	a	DET
cana-1064	142	125	contradiction	contradiction	NOUN
cana-1064	142	126	with	with	ADP
cana-1064	142	127	𝜏	𝜏	X
cana-1064	142	128	>	>	X
cana-1064	142	129	0	0	NUM
cana-1064	142	130	.	.	PUNCT
cana-1064	142	131	thus	thus	ADV
cana-1064	142	132	𝔤𝓊𝓅	𝔤𝓊𝓅	VERB
cana-1064	142	133	=	=	SYM
cana-1064	142	134	𝔣𝔷𝓅	𝔣𝔷𝓅	PROPN
cana-1064	142	135	=	=	SYM
cana-1064	142	136	𝔖𝔷𝓅	𝔖𝔷𝓅	PROPN
cana-1064	142	137	…	…	X
cana-1064	142	138	.	.	PUNCT
cana-1064	143	1	(	(	PUNCT
cana-1064	143	2	3.1.18	3.1.18	NUM
cana-1064	143	3	)	)	PUNCT
cana-1064	143	4	.	.	PUNCT
cana-1064	144	1	by	by	ADP
cana-1064	144	2	using	use	VERB
cana-1064	144	3	weakly	weakly	ADJ
cana-1064	144	4	compatible	compatible	ADJ
cana-1064	144	5	nature	nature	NOUN
cana-1064	144	6	of	of	ADP
cana-1064	144	7	𝔣	𝔣	PROPN
cana-1064	144	8	and	and	CCONJ
cana-1064	144	9	𝔖	𝔖	PROPN
cana-1064	144	10	,	,	PUNCT
cana-1064	144	11	we	we	PRON
cana-1064	144	12	get	get	VERB
cana-1064	144	13	𝔖𝓊𝓅	𝔖𝓊𝓅	PROPN
cana-1064	144	14	=	=	SYM
cana-1064	144	15	𝔣	𝔣	PROPN
cana-1064	144	16	𝔖𝔷𝓅	𝔖𝔷𝓅	PROPN
cana-1064	144	17	=	=	SYM
cana-1064	144	18	𝔖𝔣𝔷𝓅	𝔖𝔣𝔷𝓅	PROPN
cana-1064	144	19	=	=	SYM
cana-1064	144	20	𝔣𝓊𝓅.	𝔣𝓊𝓅.	X
cana-1064	144	21	…	…	PUNCT
cana-1064	144	22	…	…	PUNCT
cana-1064	144	23	…	…	PUNCT
cana-1064	144	24	…	…	PUNCT
cana-1064	144	25	(	(	PUNCT
cana-1064	144	26	3.1.19	3.1.19	NUM
cana-1064	144	27	)	)	PUNCT
cana-1064	144	28	.	.	PUNCT
cana-1064	145	1	finally	finally	ADV
cana-1064	145	2	we	we	PRON
cana-1064	145	3	show	show	VERB
cana-1064	145	4	that	that	SCONJ
cana-1064	145	5	𝔣𝓊𝓅	𝔣𝓊𝓅	NOUN
cana-1064	145	6	=	=	SYM
cana-1064	145	7	𝓊𝓅	𝓊𝓅	NOUN
cana-1064	145	8	.assume	.assume	X
cana-1064	145	9	𝔣𝓊𝓅	𝔣𝓊𝓅	ADJ
cana-1064	145	10	≠	≠	ADJ
cana-1064	145	11	𝓊𝓅	𝓊𝓅	NOUN
cana-1064	145	12	and	and	CCONJ
cana-1064	145	13	put	put	VERB
cana-1064	145	14	𝜆	𝜆	PRON
cana-1064	145	15	=	=	X
cana-1064	145	16	𝜇	𝜇	X
cana-1064	145	17	=	=	X
cana-1064	145	18	𝓊𝓅	𝓊𝓅	NOUN
cana-1064	145	19	in	in	ADP
cana-1064	145	20	(	(	PUNCT
cana-1064	145	21	3.1.1	3.1.1	X
cana-1064	145	22	)	)	PUNCT
cana-1064	145	23	we	we	PRON
cana-1064	145	24	get	get	VERB
cana-1064	145	25	𝜏	𝜏	NOUN
cana-1064	145	26	+	+	NOUN
cana-1064	145	27	ℱ(𝒫(𝔣𝓊𝓅	ℱ(𝒫(𝔣𝓊𝓅	NOUN
cana-1064	145	28	,	,	PUNCT
cana-1064	145	29	𝓊𝓅	𝓊𝓅	NOUN
cana-1064	145	30	)	)	PUNCT
cana-1064	145	31	)	)	PUNCT
cana-1064	145	32	≤	≤	NOUN
cana-1064	145	33	ℱ(𝒫(𝔣𝓊𝓅	ℱ(𝒫(𝔣𝓊𝓅	ADJ
cana-1064	145	34	,	,	PUNCT
cana-1064	145	35	𝔤𝓊𝓅	𝔤𝓊𝓅	NOUN
cana-1064	145	36	)	)	PUNCT
cana-1064	145	37	)	)	PUNCT
cana-1064	146	1	≤	≤	PUNCT
cana-1064	146	2	ℱ	ℱ	PROPN
cana-1064	146	3	(	(	PUNCT
cana-1064	146	4	ℳ(𝓊𝓅	ℳ(𝓊𝓅	NOUN
cana-1064	146	5	,	,	PUNCT
cana-1064	146	6	𝓊𝓅	𝓊𝓅	NOUN
cana-1064	146	7	)	)	PUNCT
cana-1064	146	8	)	)	PUNCT
cana-1064	146	9	ℳ(𝓊𝓅	ℳ(𝓊𝓅	NOUN
cana-1064	146	10	,	,	PUNCT
cana-1064	146	11	𝓊𝓅	𝓊𝓅	NOUN
cana-1064	146	12	)	)	PUNCT
cana-1064	146	13	=	=	SYM
cana-1064	146	14	𝑚𝑎𝑥{𝒫(𝔖𝓊𝓅	𝑚𝑎𝑥{𝒫(𝔖𝓊𝓅	PROPN
cana-1064	146	15	,	,	PUNCT
cana-1064	146	16	𝔗𝓊𝓅	𝔗𝓊𝓅	PROPN
cana-1064	146	17	)	)	PUNCT
cana-1064	146	18	,	,	PUNCT
cana-1064	146	19	𝒫(𝔣𝓊𝓅	𝒫(𝔣𝓊𝓅	NUM
cana-1064	146	20	,	,	PUNCT
cana-1064	146	21	𝔖𝓊𝓅	𝔖𝓊𝓅	PROPN
cana-1064	146	22	)	)	PUNCT
cana-1064	146	23	,	,	PUNCT
cana-1064	146	24	𝒫(𝔤𝓊𝓅	𝒫(𝔤𝓊𝓅	PROPN
cana-1064	146	25	,	,	PUNCT
cana-1064	146	26	𝔗𝓊𝓅	𝔗𝓊𝓅	PROPN
cana-1064	146	27	)	)	PUNCT
cana-1064	146	28	,	,	PUNCT
cana-1064	146	29	1	1	NUM
cana-1064	146	30	2	2	NUM
cana-1064	146	31	[	[	SYM
cana-1064	146	32	𝒫(𝔣𝓊𝓅	𝒫(𝔣𝓊𝓅	X
cana-1064	146	33	,	,	PUNCT
cana-1064	146	34	𝔗𝓊𝓅	𝔗𝓊𝓅	PROPN
cana-1064	146	35	)	)	PUNCT
cana-1064	146	36	+	+	NUM
cana-1064	146	37	𝒫(𝔤𝓊𝓅	𝒫(𝔤𝓊𝓅	PROPN
cana-1064	146	38	,	,	PUNCT
cana-1064	146	39	𝔖𝓊𝓅	𝔖𝓊𝓅	PROPN
cana-1064	146	40	)	)	PUNCT
cana-1064	146	41	]	]	PUNCT
cana-1064	146	42	}	}	PUNCT
cana-1064	146	43	=	=	SYM
cana-1064	146	44	𝒫(𝓊𝓅	𝒫(𝓊𝓅	NOUN
cana-1064	146	45	,	,	PUNCT
cana-1064	146	46	𝔣𝓊𝓅	𝔣𝓊𝓅	NOUN
cana-1064	146	47	)	)	PUNCT
cana-1064	146	48	𝜏	𝜏	X
cana-1064	146	49	+	+	SYM
cana-1064	146	50	ℱ(𝒫(𝔣𝓊𝓅	ℱ(𝒫(𝔣𝓊𝓅	NUM
cana-1064	146	51	,	,	PUNCT
cana-1064	146	52	𝓊𝓅	𝓊𝓅	NOUN
cana-1064	146	53	)	)	PUNCT
cana-1064	146	54	)	)	PUNCT
cana-1064	146	55	≤	≤	NUM
cana-1064	146	56	𝒫(𝔣𝓊𝓅	𝒫(𝔣𝓊𝓅	NUM
cana-1064	146	57	,	,	PUNCT
cana-1064	146	58	𝓊𝓅	𝓊𝓅	NOUN
cana-1064	146	59	)	)	PUNCT
cana-1064	146	60	this	this	PRON
cana-1064	146	61	a	a	DET
cana-1064	146	62	contradiction	contradiction	NOUN
cana-1064	146	63	with	with	ADP
cana-1064	146	64	𝜏	𝜏	X
cana-1064	146	65	>	>	X
cana-1064	146	66	0	0	NUM
cana-1064	146	67	.	.	PUNCT
cana-1064	146	68	thus	thus	ADV
cana-1064	146	69	𝔣𝓊𝓅	𝔣𝓊𝓅	ADJ
cana-1064	146	70	=	=	SYM
cana-1064	146	71	𝔖𝓊𝓅	𝔖𝓊𝓅	PROPN
cana-1064	146	72	=	=	SYM
cana-1064	146	73	𝓊𝓅	𝓊𝓅	NOUN
cana-1064	146	74	…	…	PUNCT
cana-1064	146	75	..	..	PUNCT
cana-1064	146	76	(3.1.20	(3.1.20	PROPN
cana-1064	146	77	)	)	PUNCT
cana-1064	146	78	.	.	PUNCT
cana-1064	147	1	using	use	VERB
cana-1064	147	2	(	(	PUNCT
cana-1064	147	3	3.1.17	3.1.17	NUM
cana-1064	147	4	)	)	PUNCT
cana-1064	147	5	and	and	CCONJ
cana-1064	147	6	(	(	PUNCT
cana-1064	147	7	3.1.20	3.1.20	NUM
cana-1064	147	8	)	)	PUNCT
cana-1064	147	9	𝓊𝓅	𝓊𝓅	NOUN
cana-1064	147	10	is	be	AUX
cana-1064	147	11	fixed	fix	VERB
cana-1064	147	12	point	point	NOUN
cana-1064	147	13	of	of	ADP
cana-1064	147	14	𝔣	𝔣	PROPN
cana-1064	147	15	,	,	PUNCT
cana-1064	147	16	𝔤	𝔤	PROPN
cana-1064	147	17	,	,	PUNCT
cana-1064	147	18	𝔖	𝔖	PROPN
cana-1064	147	19	and	and	CCONJ
cana-1064	147	20	𝔗	𝔗	PROPN
cana-1064	147	21	.	.	PUNCT
cana-1064	148	1	step	step	NOUN
cana-1064	148	2	-	-	PUNCT
cana-1064	148	3	iv	iv	NUM
cana-1064	148	4	:	:	PUNCT
cana-1064	148	5	let	let	AUX
cana-1064	148	6	𝜔𝓅(≠𝓊𝓅	𝜔𝓅(≠𝓊𝓅	NOUN
cana-1064	148	7	)	)	PUNCT
cana-1064	148	8	be	be	AUX
cana-1064	148	9	the	the	DET
cana-1064	148	10	another	another	DET
cana-1064	148	11	fixed	fix	VERB
cana-1064	148	12	point	point	NOUN
cana-1064	148	13	of	of	ADP
cana-1064	148	14	𝔣	𝔣	PROPN
cana-1064	148	15	,	,	PUNCT
cana-1064	148	16	𝔤	𝔤	PROPN
cana-1064	148	17	,	,	PUNCT
cana-1064	148	18	𝔖	𝔖	PROPN
cana-1064	148	19	and	and	CCONJ
cana-1064	148	20	𝔗.	𝔗.	PROPN
cana-1064	148	21	put	put	VERB
cana-1064	148	22	𝜆	𝜆	PRON
cana-1064	148	23	=	=	X
cana-1064	148	24	𝓊𝓅	𝓊𝓅	NOUN
cana-1064	148	25	,	,	PUNCT
cana-1064	148	26	𝜇	𝜇	X
cana-1064	148	27	=	=	X
cana-1064	148	28	𝜔𝓅	𝜔𝓅	ADP
cana-1064	148	29	in	in	ADP
cana-1064	148	30	the	the	DET
cana-1064	148	31	contraction	contraction	NOUN
cana-1064	148	32	(	(	PUNCT
cana-1064	148	33	3.1.1	3.1.1	NUM
cana-1064	148	34	)	)	PUNCT
cana-1064	148	35	,	,	PUNCT
cana-1064	148	36	we	we	PRON
cana-1064	148	37	get	get	VERB
cana-1064	148	38	𝜏	𝜏	PRON
cana-1064	148	39	+	+	X
cana-1064	148	40	ℱ	ℱ	PROPN
cana-1064	148	41	(	(	PUNCT
cana-1064	148	42	𝒫(𝔣𝓊𝓅	𝒫(𝔣𝓊𝓅	X
cana-1064	148	43	,	,	PUNCT
cana-1064	148	44	𝜔𝓅	𝜔𝓅	NOUN
cana-1064	148	45	)	)	PUNCT
cana-1064	148	46	)	)	PUNCT
cana-1064	149	1	=	=	SYM
cana-1064	150	1	𝜏	𝜏	PROPN
cana-1064	150	2	+	+	NUM
cana-1064	150	3	ℱ	ℱ	PROPN
cana-1064	150	4	(	(	PUNCT
cana-1064	150	5	𝒫(𝔣𝓊𝓅	𝒫(𝔣𝓊𝓅	X
cana-1064	150	6	,	,	PUNCT
cana-1064	150	7	𝔤𝜔𝓅	𝔤𝜔𝓅	NOUN
cana-1064	150	8	)	)	PUNCT
cana-1064	150	9	)	)	PUNCT
cana-1064	150	10	≤	≤	NUM
cana-1064	150	11	ℱ	ℱ	PROPN
cana-1064	150	12	(	(	PUNCT
cana-1064	150	13	ℳ(𝓊𝓅	ℳ(𝓊𝓅	NOUN
cana-1064	150	14	,	,	PUNCT
cana-1064	150	15	𝜔𝓅	𝜔𝓅	ADJ
cana-1064	150	16	)	)	PUNCT
cana-1064	150	17	)	)	PUNCT
cana-1064	150	18	communications	communication	NOUN
cana-1064	150	19	on	on	ADP
cana-1064	150	20	applied	apply	VERB
cana-1064	150	21	nonlinear	nonlinear	ADJ
cana-1064	150	22	analysis	analysis	NOUN
cana-1064	150	23	issn	issn	NOUN
cana-1064	150	24	:	:	PUNCT
cana-1064	150	25	1074	1074	NUM
cana-1064	150	26	-	-	PUNCT
cana-1064	150	27	133x	133x	NUM
cana-1064	150	28	vol	vol	NOUN
cana-1064	150	29	31	31	NUM
cana-1064	150	30	no	no	NOUN
cana-1064	150	31	.	.	PUNCT
cana-1064	151	1	5s	5s	NUM
cana-1064	151	2	(	(	PUNCT
cana-1064	151	3	2024	2024	NUM
cana-1064	151	4	)	)	PUNCT
cana-1064	151	5	456	456	NUM
cana-1064	152	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1064	152	2	ℳ(𝓊𝓅	ℳ(𝓊𝓅	NOUN
cana-1064	152	3	,	,	PUNCT
cana-1064	152	4	,	,	PUNCT
cana-1064	152	5	𝜔𝓅	𝜔𝓅	ADP
cana-1064	152	6	)	)	PUNCT
cana-1064	152	7	=	=	NOUN
cana-1064	152	8	𝑚𝑎𝑥{𝒫(𝔖𝓊𝓅	𝑚𝑎𝑥{𝒫(𝔖𝓊𝓅	NOUN
cana-1064	152	9	,	,	PUNCT
cana-1064	152	10	𝔗𝜔𝓅	𝔗𝜔𝓅	PROPN
cana-1064	152	11	)	)	PUNCT
cana-1064	152	12	,	,	PUNCT
cana-1064	152	13	𝒫(𝔣𝓊𝓅	𝒫(𝔣𝓊𝓅	X
cana-1064	152	14	,	,	PUNCT
cana-1064	152	15	𝔖𝜔𝓅	𝔖𝜔𝓅	PROPN
cana-1064	152	16	)	)	PUNCT
cana-1064	152	17	,	,	PUNCT
cana-1064	152	18	𝒫(𝔤𝜔𝓅	𝒫(𝔤𝜔𝓅	NUM
cana-1064	152	19	,	,	PUNCT
cana-1064	152	20	𝔗𝜔𝓅	𝔗𝜔𝓅	NOUN
cana-1064	152	21	)	)	PUNCT
cana-1064	152	22	,	,	PUNCT
cana-1064	152	23	1	1	NUM
cana-1064	152	24	2	2	NUM
cana-1064	152	25	[	[	PUNCT
cana-1064	152	26	𝒫(𝔣𝓊𝓅	𝒫(𝔣𝓊𝓅	NUM
cana-1064	152	27	,	,	PUNCT
cana-1064	152	28	𝔗𝜔𝓅	𝔗𝜔𝓅	NOUN
cana-1064	152	29	)	)	PUNCT
cana-1064	152	30	+	+	CCONJ
cana-1064	152	31	𝒫(𝔤𝜔𝓅	𝒫(𝔤𝜔𝓅	NUM
cana-1064	152	32	,	,	PUNCT
cana-1064	152	33	𝔖𝓊𝓅	𝔖𝓊𝓅	PROPN
cana-1064	152	34	)	)	PUNCT
cana-1064	152	35	]	]	PUNCT
cana-1064	152	36	}	}	PUNCT
cana-1064	152	37	=	=	SYM
cana-1064	152	38	𝒫(𝓊𝓅	𝒫(𝓊𝓅	NOUN
cana-1064	152	39	,	,	PUNCT
cana-1064	152	40	𝜔𝓅	𝜔𝓅	ADP
cana-1064	152	41	)	)	PUNCT
cana-1064	152	42	𝜏	𝜏	X
cana-1064	153	1	+	+	NUM
cana-1064	153	2	ℱ	ℱ	PROPN
cana-1064	153	3	(	(	PUNCT
cana-1064	153	4	𝒫(𝔣𝓊𝓅	𝒫(𝔣𝓊𝓅	NUM
cana-1064	153	5	,	,	PUNCT
cana-1064	153	6	𝜔𝓅	𝜔𝓅	NOUN
cana-1064	153	7	)	)	PUNCT
cana-1064	153	8	)	)	PUNCT
cana-1064	153	9	≤	≤	NUM
cana-1064	153	10	𝒫(𝓊𝓅	𝒫(𝓊𝓅	NOUN
cana-1064	153	11	,	,	PUNCT
cana-1064	153	12	𝜔𝓅	𝜔𝓅	ADP
cana-1064	153	13	)	)	PUNCT
cana-1064	153	14	this	this	PRON
cana-1064	153	15	a	a	DET
cana-1064	153	16	contradiction	contradiction	NOUN
cana-1064	153	17	with	with	ADP
cana-1064	153	18	𝜏	𝜏	X
cana-1064	153	19	>	>	X
cana-1064	153	20	0	0	NUM
cana-1064	153	21	.	.	PUNCT
cana-1064	153	22	thus	thus	ADV
cana-1064	153	23	𝓊𝓅	𝓊𝓅	VERB
cana-1064	153	24	=	=	SYM
cana-1064	153	25	𝜔𝓅.	𝜔𝓅.	NOUN
cana-1064	153	26	in	in	ADP
cana-1064	153	27	conclusion	conclusion	NOUN
cana-1064	153	28	,	,	PUNCT
cana-1064	153	29	these	these	DET
cana-1064	153	30	four	four	NUM
cana-1064	153	31	mappings	mapping	NOUN
cana-1064	153	32	have	have	VERB
cana-1064	153	33	a	a	DET
cana-1064	153	34	unique	unique	ADJ
cana-1064	153	35	common	common	ADJ
cana-1064	153	36	fixed	fix	VERB
cana-1064	153	37	point	point	NOUN
cana-1064	153	38	.	.	PUNCT
cana-1064	153	39	example	example	NOUN
cana-1064	153	40	3.2	3.2	NUM
cana-1064	153	41	:	:	PUNCT
cana-1064	153	42	let	let	VERB
cana-1064	153	43	(	(	PUNCT
cana-1064	153	44	𝔛	𝔛	NOUN
cana-1064	153	45	,	,	PUNCT
cana-1064	153	46	𝒫	𝒫	NOUN
cana-1064	153	47	)	)	PUNCT
cana-1064	153	48	with	with	ADP
cana-1064	153	49	𝔛	𝔛	PROPN
cana-1064	153	50	=	=	PUNCT
cana-1064	154	1	[	[	X
cana-1064	154	2	0,1	0,1	NUM
cana-1064	154	3	]	]	PUNCT
cana-1064	154	4	is	be	AUX
cana-1064	154	5	a	a	DET
cana-1064	154	6	complete	complete	ADJ
cana-1064	154	7	pms	pm	NOUN
cana-1064	154	8	and	and	CCONJ
cana-1064	154	9	𝒫	𝒫	PROPN
cana-1064	154	10	(	(	PUNCT
cana-1064	154	11	𝜆	𝜆	NOUN
cana-1064	154	12	,	,	PUNCT
cana-1064	154	13	𝜇)=max	𝜇)=max	PROPN
cana-1064	154	14	{	{	PUNCT
cana-1064	154	15	𝜆	𝜆	NOUN
cana-1064	154	16	,	,	PUNCT
cana-1064	154	17	𝜇	𝜇	X
cana-1064	154	18	}	}	PUNCT
cana-1064	154	19	∀	∀	X
cana-1064	154	20	𝜆	𝜆	NOUN
cana-1064	154	21	,	,	PUNCT
cana-1064	154	22	𝜇	𝜇	ADP
cana-1064	154	23	∈	∈	PROPN
cana-1064	154	24	𝔛.	𝔛.	NOUN
cana-1064	154	25	define	define	VERB
cana-1064	154	26	mappings	mapping	NOUN
cana-1064	154	27	ℱ	ℱ	PROPN
cana-1064	154	28	:	:	PUNCT
cana-1064	154	29	ℝ+	ℝ+	PUNCT
cana-1064	154	30	⟶	⟶	NOUN
cana-1064	154	31	ℝ	ℝ	PROPN
cana-1064	154	32	and	and	CCONJ
cana-1064	154	33	ℱ(𝛼	ℱ(𝛼	NUM
cana-1064	154	34	)	)	PUNCT
cana-1064	155	1	=	=	SYM
cana-1064	155	2	log𝑒	log𝑒	PROPN
cana-1064	155	3	𝛼.	𝛼.	PROPN
cana-1064	155	4	let	let	VERB
cana-1064	155	5	𝔣	𝔣	NOUN
cana-1064	155	6	,	,	PUNCT
cana-1064	155	7	𝔤	𝔤	PROPN
cana-1064	155	8	,	,	PUNCT
cana-1064	155	9	𝔖	𝔖	PROPN
cana-1064	155	10	and	and	CCONJ
cana-1064	155	11	𝔗	𝔗	PROPN
cana-1064	155	12	:	:	PUNCT
cana-1064	155	13	𝔛	𝔛	NOUN
cana-1064	155	14	→	→	SYM
cana-1064	155	15	𝔛	𝔛	PROPN
cana-1064	155	16	be	be	AUX
cana-1064	155	17	defined	define	VERB
cana-1064	155	18	as	as	ADP
cana-1064	155	19	𝔣(𝜆	𝔣(𝜆	NOUN
cana-1064	155	20	)	)	PUNCT
cana-1064	155	21	=	=	SYM
cana-1064	155	22	𝜆	𝜆	ADP
cana-1064	155	23	4	4	NUM
cana-1064	155	24	,	,	PUNCT
cana-1064	155	25	𝔤	𝔤	X
cana-1064	155	26	(	(	PUNCT
cana-1064	155	27	𝜆	𝜆	X
cana-1064	155	28	)	)	PUNCT
cana-1064	155	29	=	=	SYM
cana-1064	155	30	0	0	NUM
cana-1064	155	31	,	,	PUNCT
cana-1064	155	32	𝔖(𝜆	𝔖(𝜆	PRON
cana-1064	155	33	)	)	PUNCT
cana-1064	155	34	=	=	SYM
cana-1064	155	35	3𝜆	3𝜆	NUM
cana-1064	155	36	2	2	NUM
cana-1064	155	37	,	,	PUNCT
cana-1064	155	38	𝔗(𝜆	𝔗(𝜆	NOUN
cana-1064	155	39	)	)	PUNCT
cana-1064	155	40	=	=	SYM
cana-1064	156	1	𝜆	𝜆	X
cana-1064	156	2	,	,	PUNCT
cana-1064	156	3	then	then	ADV
cana-1064	156	4	𝔣(𝔛	𝔣(𝔛	ADP
cana-1064	156	5	)	)	PUNCT
cana-1064	156	6	=	=	PUNCT
cana-1064	157	1	[	[	X
cana-1064	157	2	0	0	NUM
cana-1064	157	3	,	,	PUNCT
cana-1064	157	4	1	1	NUM
cana-1064	157	5	4	4	NUM
cana-1064	157	6	]	]	PUNCT
cana-1064	157	7	,	,	PUNCT
cana-1064	157	8	𝔤	𝔤	PROPN
cana-1064	157	9	(	(	PUNCT
cana-1064	157	10	𝔛	𝔛	NOUN
cana-1064	157	11	)	)	PUNCT
cana-1064	157	12	=	=	PUNCT
cana-1064	157	13	{	{	PUNCT
cana-1064	157	14	0	0	NUM
cana-1064	157	15	}	}	PUNCT
cana-1064	157	16	,	,	PUNCT
cana-1064	157	17	𝔖(𝔛	𝔖(𝔛	NUM
cana-1064	157	18	)	)	PUNCT
cana-1064	157	19	=	=	PUNCT
cana-1064	158	1	[	[	X
cana-1064	158	2	0	0	NUM
cana-1064	158	3	,	,	PUNCT
cana-1064	158	4	3	3	NUM
cana-1064	158	5	4	4	NUM
cana-1064	158	6	]	]	PUNCT
cana-1064	158	7	,	,	PUNCT
cana-1064	158	8	and	and	CCONJ
cana-1064	158	9	𝔗(𝔛	𝔗(𝔛	NOUN
cana-1064	158	10	)	)	PUNCT
cana-1064	158	11	=	=	PUNCT
cana-1064	159	1	[	[	X
cana-1064	159	2	0,1	0,1	NUM
cana-1064	159	3	]	]	PUNCT
cana-1064	159	4	these	these	DET
cana-1064	159	5	ranges	range	NOUN
cana-1064	159	6	of	of	ADP
cana-1064	159	7	mappings	mapping	NOUN
cana-1064	159	8	satisfying	satisfy	VERB
cana-1064	159	9	the	the	DET
cana-1064	159	10	inclusion	inclusion	NOUN
cana-1064	159	11	inequalities	inequality	NOUN
cana-1064	159	12	(	(	PUNCT
cana-1064	159	13	i	i	NOUN
cana-1064	159	14	)	)	PUNCT
cana-1064	159	15	of	of	ADP
cana-1064	159	16	theorem	theorem	NOUN
cana-1064	159	17	3.1	3.1	NUM
cana-1064	159	18	.	.	PUNCT
cana-1064	160	1	now	now	ADV
cana-1064	160	2	clearly	clearly	ADV
cana-1064	160	3	𝔣(0	𝔣(0	VERB
cana-1064	160	4	)	)	PUNCT
cana-1064	160	5	=	=	SYM
cana-1064	160	6	𝔖(0	𝔖(0	NUM
cana-1064	160	7	)	)	PUNCT
cana-1064	160	8	=	=	SYM
cana-1064	160	9	0	0	PROPN
cana-1064	160	10	gives	give	VERB
cana-1064	160	11	𝔣𝔖(0	𝔣𝔖(0	NOUN
cana-1064	160	12	)	)	PUNCT
cana-1064	160	13	=	=	SYM
cana-1064	160	14	𝔖𝔣(0	𝔖𝔣(0	NOUN
cana-1064	160	15	)	)	PUNCT
cana-1064	160	16	which	which	PRON
cana-1064	160	17	gives	give	VERB
cana-1064	160	18	the	the	DET
cana-1064	160	19	pair	pair	NOUN
cana-1064	160	20	(	(	PUNCT
cana-1064	160	21	𝔣	𝔣	ADJ
cana-1064	160	22	,	,	PUNCT
cana-1064	160	23	𝔖	𝔖	NOUN
cana-1064	160	24	)	)	PUNCT
cana-1064	160	25	is	be	AUX
cana-1064	160	26	weakly	weakly	ADV
cana-1064	160	27	compatible	compatible	ADJ
cana-1064	160	28	and	and	CCONJ
cana-1064	161	1	𝔤(0	𝔤(0	PROPN
cana-1064	161	2	)	)	PUNCT
cana-1064	162	1	=	=	SYM
cana-1064	162	2	𝔗(0	𝔗(0	X
cana-1064	162	3	)	)	PUNCT
cana-1064	162	4	=	=	SYM
cana-1064	162	5	0	0	PUNCT
cana-1064	162	6	gives	give	VERB
cana-1064	162	7	𝔤𝔗(0	𝔤𝔗(0	NOUN
cana-1064	162	8	)	)	PUNCT
cana-1064	162	9	=	=	SYM
cana-1064	163	1	𝔗𝔤(0	𝔗𝔤(0	ADV
cana-1064	163	2	)	)	PUNCT
cana-1064	163	3	this	this	PRON
cana-1064	163	4	implies	imply	VERB
cana-1064	163	5	the	the	DET
cana-1064	163	6	pair	pair	NOUN
cana-1064	163	7	(	(	PUNCT
cana-1064	163	8	𝔤	𝔤	PROPN
cana-1064	163	9	,	,	PUNCT
cana-1064	163	10	𝔗	𝔗	PROPN
cana-1064	163	11	)	)	PUNCT
cana-1064	163	12	is	be	AUX
cana-1064	163	13	weakly	weakly	ADV
cana-1064	163	14	compatible	compatible	ADJ
cana-1064	163	15	at	at	ADP
cana-1064	163	16	the	the	DET
cana-1064	163	17	coincident	coincident	ADJ
cana-1064	163	18	point	point	NOUN
cana-1064	163	19	zero	zero	NUM
cana-1064	163	20	.	.	PUNCT
cana-1064	164	1	we	we	PRON
cana-1064	164	2	discuss	discuss	VERB
cana-1064	164	3	the	the	DET
cana-1064	164	4	existence	existence	NOUN
cana-1064	164	5	of	of	ADP
cana-1064	164	6	contraction	contraction	NOUN
cana-1064	164	7	condition	condition	NOUN
cana-1064	164	8	(	(	PUNCT
cana-1064	164	9	3.1.1	3.1.1	NUM
cana-1064	164	10	)	)	PUNCT
cana-1064	164	11	under	under	ADP
cana-1064	164	12	some	some	DET
cana-1064	164	13	conditions	condition	NOUN
cana-1064	164	14	now	now	ADV
cana-1064	164	15	𝒫(𝔣𝜆	𝒫(𝔣𝜆	VERB
cana-1064	164	16	,	,	PUNCT
cana-1064	164	17	𝔤𝜇	𝔤𝜇	NOUN
cana-1064	164	18	)	)	PUNCT
cana-1064	164	19	=	=	SYM
cana-1064	164	20	max	max	X
cana-1064	164	21	{	{	PUNCT
cana-1064	164	22	𝜆	𝜆	PROPN
cana-1064	164	23	4	4	NUM
cana-1064	164	24	,	,	PUNCT
cana-1064	164	25	0	0	NUM
cana-1064	164	26	}	}	PUNCT
cana-1064	164	27	,	,	PUNCT
cana-1064	164	28	𝒫(𝔖𝜆	𝒫(𝔖𝜆	NOUN
cana-1064	164	29	,	,	PUNCT
cana-1064	164	30	𝔗𝜇	𝔗𝜇	NOUN
cana-1064	164	31	)	)	PUNCT
cana-1064	164	32	=	=	PUNCT
cana-1064	164	33	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
cana-1064	164	34	{	{	PUNCT
cana-1064	164	35	3𝜆	3𝜆	NOUN
cana-1064	164	36	2	2	NUM
cana-1064	164	37	,	,	PUNCT
cana-1064	164	38	𝜇},𝒫(𝔣𝜆	𝜇},𝒫(𝔣𝜆	ADJ
cana-1064	164	39	,	,	PUNCT
cana-1064	164	40	𝔖𝜆	𝔖𝜆	PROPN
cana-1064	164	41	)	)	PUNCT
cana-1064	164	42	=	=	SYM
cana-1064	164	43	max	max	X
cana-1064	164	44	{	{	PUNCT
cana-1064	164	45	𝜆	𝜆	PROPN
cana-1064	164	46	4	4	NUM
cana-1064	164	47	,	,	PUNCT
cana-1064	164	48	3𝜆	3𝜆	ADJ
cana-1064	164	49	2	2	NUM
cana-1064	164	50	}	}	PUNCT
cana-1064	164	51	,	,	PUNCT
cana-1064	164	52	𝒫(𝔤𝜇	𝒫(𝔤𝜇	NOUN
cana-1064	164	53	,	,	PUNCT
cana-1064	164	54	𝔗𝜇	𝔗𝜇	NOUN
cana-1064	164	55	)	)	PUNCT
cana-1064	164	56	=	=	SYM
cana-1064	164	57	max{0	max{0	PROPN
cana-1064	164	58	,	,	PUNCT
cana-1064	164	59	𝜇	𝜇	ADP
cana-1064	164	60	}	}	PUNCT
cana-1064	164	61	,	,	PUNCT
cana-1064	164	62	𝒫(𝔣𝜆	𝒫(𝔣𝜆	NOUN
cana-1064	164	63	,	,	PUNCT
cana-1064	164	64	𝔗𝜇	𝔗𝜇	NOUN
cana-1064	164	65	)	)	PUNCT
cana-1064	164	66	=	=	SYM
cana-1064	164	67	max	max	X
cana-1064	164	68	{	{	PUNCT
cana-1064	164	69	𝜆	𝜆	PROPN
cana-1064	164	70	4	4	NUM
cana-1064	164	71	,	,	PUNCT
cana-1064	164	72	𝜇	𝜇	ADP
cana-1064	164	73	}	}	PUNCT
cana-1064	164	74	and	and	CCONJ
cana-1064	164	75	𝒫(𝔤𝜇	𝒫(𝔤𝜇	NOUN
cana-1064	164	76	,	,	PUNCT
cana-1064	164	77	𝔖𝜆	𝔖𝜆	NOUN
cana-1064	164	78	)	)	PUNCT
cana-1064	164	79	=	=	SYM
cana-1064	164	80	max	max	X
cana-1064	164	81	{	{	PUNCT
cana-1064	164	82	0	0	NUM
cana-1064	164	83	,	,	PUNCT
cana-1064	164	84	3𝜆	3𝜆	NOUN
cana-1064	164	85	2	2	NUM
cana-1064	164	86	}	}	PUNCT
cana-1064	164	87	.	.	PUNCT
cana-1064	165	1	case	case	NOUN
cana-1064	166	1	i	i	PRON
cana-1064	166	2	:	:	PUNCT
cana-1064	166	3	if	if	SCONJ
cana-1064	166	4	3𝜆	3𝜆	PROPN
cana-1064	166	5	2	2	NUM
cana-1064	166	6	>	>	X
cana-1064	166	7	𝜇	𝜇	ADP
cana-1064	166	8	then	then	ADV
cana-1064	166	9	ℳ(𝜆	ℳ(𝜆	VERB
cana-1064	166	10	,	,	PUNCT
cana-1064	166	11	𝜇	𝜇	NOUN
cana-1064	166	12	)	)	PUNCT
cana-1064	166	13	=	=	SYM
cana-1064	166	14	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
cana-1064	166	15	{	{	PUNCT
cana-1064	166	16	3𝜆	3𝜆	NOUN
cana-1064	166	17	2	2	NUM
cana-1064	166	18	,	,	PUNCT
cana-1064	166	19	3𝜆	3𝜆	NOUN
cana-1064	166	20	2	2	NUM
cana-1064	166	21	,	,	PUNCT
cana-1064	166	22	𝜇	𝜇	ADP
cana-1064	166	23	,	,	PUNCT
cana-1064	166	24	1	1	NUM
cana-1064	166	25	2	2	NUM
cana-1064	166	26	[	[	PUNCT
cana-1064	166	27	𝜆	𝜆	PROPN
cana-1064	166	28	4	4	NUM
cana-1064	166	29	+	+	CCONJ
cana-1064	166	30	3𝜆	3𝜆	ADJ
cana-1064	166	31	2	2	NUM
cana-1064	166	32	]	]	PUNCT
cana-1064	166	33	}	}	PUNCT
cana-1064	166	34	=	=	SYM
cana-1064	166	35	3𝜆	3𝜆	PROPN
cana-1064	166	36	2	2	NUM
cana-1064	166	37	𝒫(𝔣𝜆	𝒫(𝔣𝜆	NOUN
cana-1064	166	38	,	,	PUNCT
cana-1064	166	39	𝔤𝜇	𝔤𝜇	NOUN
cana-1064	166	40	)	)	PUNCT
cana-1064	166	41	=	=	PUNCT
cana-1064	166	42	𝑚𝑎𝑥	𝑚𝑎𝑥	NOUN
cana-1064	166	43	{	{	PUNCT
cana-1064	166	44	𝜆	𝜆	NOUN
cana-1064	166	45	4	4	NUM
cana-1064	166	46	,	,	PUNCT
cana-1064	166	47	0	0	NUM
cana-1064	166	48	}	}	PUNCT
cana-1064	166	49	=	=	SYM
cana-1064	166	50	𝜆	𝜆	ADP
cana-1064	166	51	4	4	NUM
cana-1064	166	52	>	>	SYM
cana-1064	166	53	0	0	NUM
cana-1064	166	54	⟹	⟹	NUM
cana-1064	166	55	𝜏	𝜏	NOUN
cana-1064	166	56	+	+	X
cana-1064	166	57	ℱ	ℱ	PROPN
cana-1064	166	58	(	(	PUNCT
cana-1064	166	59	𝜆	𝜆	PROPN
cana-1064	166	60	4	4	X
cana-1064	166	61	)	)	PUNCT
cana-1064	166	62	≤	≤	NOUN
cana-1064	166	63	ℱ	ℱ	PROPN
cana-1064	166	64	(	(	PUNCT
cana-1064	166	65	3𝜆	3𝜆	NOUN
cana-1064	166	66	2	2	NUM
cana-1064	166	67	)	)	PUNCT
cana-1064	166	68	⟹	⟹	PUNCT
cana-1064	167	1	𝜏	𝜏	PROPN
cana-1064	167	2	+	+	NUM
cana-1064	167	3	log𝑒	log𝑒	PROPN
cana-1064	167	4	(	(	PUNCT
cana-1064	167	5	𝜆	𝜆	PROPN
cana-1064	167	6	4	4	X
cana-1064	167	7	)	)	PUNCT
cana-1064	167	8	≤	≤	NOUN
cana-1064	167	9	log𝑒	log𝑒	PROPN
cana-1064	167	10	(	(	PUNCT
cana-1064	167	11	6𝜆	6𝜆	PROPN
cana-1064	167	12	4	4	NUM
cana-1064	167	13	)	)	PUNCT
cana-1064	167	14	.	.	PUNCT
cana-1064	168	1	case	case	NOUN
cana-1064	168	2	ii	ii	NOUN
cana-1064	168	3	:	:	PUNCT
cana-1064	168	4	if	if	SCONJ
cana-1064	168	5	3𝜆	3𝜆	ADJ
cana-1064	168	6	2	2	NUM
cana-1064	168	7	<	<	X
cana-1064	168	8	𝜇	𝜇	ADP
cana-1064	168	9	then	then	ADV
cana-1064	168	10	ℳ(𝜆	ℳ(𝜆	VERB
cana-1064	168	11	,	,	PUNCT
cana-1064	168	12	𝜇	𝜇	NOUN
cana-1064	168	13	)	)	PUNCT
cana-1064	168	14	=	=	SYM
cana-1064	168	15	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
cana-1064	168	16	{	{	PUNCT
cana-1064	168	17	𝜇	𝜇	ADP
cana-1064	168	18	,	,	PUNCT
cana-1064	168	19	3𝜆	3𝜆	ADJ
cana-1064	168	20	2	2	NUM
cana-1064	168	21	,	,	PUNCT
cana-1064	168	22	𝜇	𝜇	ADP
cana-1064	168	23	,	,	PUNCT
cana-1064	168	24	1	1	NUM
cana-1064	168	25	2	2	NUM
cana-1064	168	26	[	[	X
cana-1064	168	27	𝜇	𝜇	X
cana-1064	168	28	+	+	NOUN
cana-1064	168	29	3	3	NUM
cana-1064	168	30	4	4	NUM
cana-1064	168	31	]	]	PUNCT
cana-1064	168	32	}	}	PUNCT
cana-1064	168	33	=	=	PUNCT
cana-1064	168	34	𝜇	𝜇	ADP
cana-1064	168	35	𝒫(𝔣𝜆	𝒫(𝔣𝜆	NOUN
cana-1064	168	36	,	,	PUNCT
cana-1064	168	37	𝔤𝜇	𝔤𝜇	NOUN
cana-1064	168	38	)	)	PUNCT
cana-1064	168	39	=	=	PUNCT
cana-1064	168	40	𝑚𝑎𝑥	𝑚𝑎𝑥	NOUN
cana-1064	168	41	{	{	PUNCT
cana-1064	168	42	𝜆	𝜆	NOUN
cana-1064	168	43	4	4	NUM
cana-1064	168	44	,	,	PUNCT
cana-1064	168	45	0	0	NUM
cana-1064	168	46	}	}	PUNCT
cana-1064	168	47	=	=	SYM
cana-1064	168	48	𝜆	𝜆	ADP
cana-1064	168	49	4	4	NUM
cana-1064	168	50	>	>	SYM
cana-1064	168	51	0	0	NUM
cana-1064	168	52	⟹	⟹	NUM
cana-1064	168	53	𝜏	𝜏	PROPN
cana-1064	169	1	+	+	X
cana-1064	169	2	𝐹	𝐹	PROPN
cana-1064	169	3	(	(	PUNCT
cana-1064	169	4	𝜆	𝜆	ADP
cana-1064	169	5	4	4	X
cana-1064	169	6	)	)	PUNCT
cana-1064	169	7	≤	≤	NOUN
cana-1064	169	8	𝐹(𝜇	𝐹(𝜇	NUM
cana-1064	169	9	)	)	PUNCT
cana-1064	169	10	⟹	⟹	PUNCT
cana-1064	170	1	𝜏	𝜏	PROPN
cana-1064	170	2	+	+	NUM
cana-1064	170	3	log𝑒	log𝑒	PROPN
cana-1064	170	4	(	(	PUNCT
cana-1064	170	5	𝜆	𝜆	PROPN
cana-1064	170	6	4	4	X
cana-1064	170	7	)	)	PUNCT
cana-1064	170	8	≤	≤	NUM
cana-1064	170	9	log𝑒(𝜇	log𝑒(𝜇	PROPN
cana-1064	170	10	)	)	PUNCT
cana-1064	170	11	.	.	PUNCT
cana-1064	171	1	this	this	PRON
cana-1064	171	2	gives	give	VERB
cana-1064	171	3	𝜏	𝜏	PROPN
cana-1064	171	4	+	+	NUM
cana-1064	171	5	log𝑒	log𝑒	PROPN
cana-1064	171	6	1	1	NUM
cana-1064	171	7	6	6	NUM
cana-1064	171	8	(	(	PUNCT
cana-1064	171	9	3𝜆	3𝜆	NOUN
cana-1064	171	10	2	2	NUM
cana-1064	171	11	)	)	PUNCT
cana-1064	171	12	≤	≤	NUM
cana-1064	171	13	log𝑒(𝜇	log𝑒(𝜇	PROPN
cana-1064	171	14	)	)	PUNCT
cana-1064	171	15	.	.	PUNCT
cana-1064	172	1	in	in	ADP
cana-1064	172	2	both	both	CCONJ
cana-1064	172	3	the	the	DET
cana-1064	172	4	cases	case	NOUN
cana-1064	172	5	,	,	PUNCT
cana-1064	172	6	inequality	inequality	NOUN
cana-1064	172	7	(	(	PUNCT
cana-1064	172	8	3.1.1	3.1.1	NUM
cana-1064	172	9	)	)	PUNCT
cana-1064	172	10	is	be	AUX
cana-1064	172	11	satified	satifie	VERB
cana-1064	172	12	for	for	ADP
cana-1064	172	13	all	all	DET
cana-1064	172	14	λ	λ	PROPN
cana-1064	172	15	,	,	PUNCT
cana-1064	172	16	μ	μ	NOUN
cana-1064	172	17	.	.	PUNCT
cana-1064	173	1	in	in	ADP
cana-1064	173	2	this	this	DET
cana-1064	173	3	illustration	illustration	NOUN
cana-1064	173	4	,	,	PUNCT
cana-1064	173	5	it	it	PRON
cana-1064	173	6	is	be	AUX
cana-1064	173	7	observed	observe	VERB
cana-1064	173	8	that	that	SCONJ
cana-1064	173	9	zero	zero	NUM
cana-1064	173	10	is	be	AUX
cana-1064	173	11	the	the	DET
cana-1064	173	12	unique	unique	ADJ
cana-1064	173	13	common	common	ADJ
cana-1064	173	14	fixed	fix	VERB
cana-1064	173	15	point	point	NOUN
cana-1064	173	16	.	.	PUNCT
cana-1064	174	1	communications	communication	NOUN
cana-1064	174	2	on	on	ADP
cana-1064	174	3	applied	apply	VERB
cana-1064	174	4	nonlinear	nonlinear	ADJ
cana-1064	174	5	analysis	analysis	NOUN
cana-1064	174	6	issn	issn	NOUN
cana-1064	174	7	:	:	PUNCT
cana-1064	174	8	1074	1074	NUM
cana-1064	174	9	-	-	PUNCT
cana-1064	174	10	133x	133x	NUM
cana-1064	174	11	vol	vol	NOUN
cana-1064	174	12	31	31	NUM
cana-1064	174	13	no	no	NOUN
cana-1064	174	14	.	.	PUNCT
cana-1064	175	1	5s	5s	NUM
cana-1064	175	2	(	(	PUNCT
cana-1064	175	3	2024	2024	NUM
cana-1064	175	4	)	)	PUNCT
cana-1064	175	5	457	457	NUM
cana-1064	176	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1064	176	2	4.conclusion	4.conclusion	NUM
cana-1064	176	3	:	:	PUNCT
cana-1064	176	4	in	in	ADP
cana-1064	176	5	this	this	DET
cana-1064	176	6	research	research	NOUN
cana-1064	176	7	article	article	NOUN
cana-1064	176	8	,	,	PUNCT
cana-1064	176	9	we	we	PRON
cana-1064	176	10	have	have	AUX
cana-1064	176	11	proved	prove	VERB
cana-1064	176	12	a	a	DET
cana-1064	176	13	fixed	fix	VERB
cana-1064	176	14	point	point	NOUN
cana-1064	176	15	result	result	NOUN
cana-1064	176	16	for	for	ADP
cana-1064	176	17	ℱ	ℱ	PROPN
cana-1064	176	18	−	−	PROPN
cana-1064	176	19	contraction	contraction	NOUN
cana-1064	176	20	in	in	ADP
cana-1064	176	21	pms	pms	PROPN
cana-1064	176	22	via	via	ADP
cana-1064	176	23	weakly	weakly	ADJ
cana-1064	176	24	compatible	compatible	ADJ
cana-1064	176	25	mappings	mapping	NOUN
cana-1064	176	26	.	.	PUNCT
cana-1064	177	1	finally	finally	ADV
cana-1064	177	2	,	,	PUNCT
cana-1064	177	3	we	we	PRON
cana-1064	177	4	have	have	AUX
cana-1064	177	5	provided	provide	VERB
cana-1064	177	6	one	one	NUM
cana-1064	177	7	example	example	NOUN
cana-1064	177	8	to	to	PART
cana-1064	177	9	support	support	VERB
cana-1064	177	10	our	our	PRON
cana-1064	177	11	main	main	ADJ
cana-1064	177	12	result	result	NOUN
cana-1064	177	13	with	with	ADP
cana-1064	177	14	unique	unique	ADJ
cana-1064	177	15	common	common	ADJ
cana-1064	177	16	fixed	fix	VERB
cana-1064	177	17	point	point	NOUN
cana-1064	177	18	.	.	PUNCT
cana-1064	178	1	references	reference	NOUN
cana-1064	178	2	:	:	PUNCT
cana-1064	179	1	[	[	X
cana-1064	179	2	1	1	X
cana-1064	179	3	]	]	PUNCT
cana-1064	179	4	matthwes	matthwe	NOUN
cana-1064	179	5	s	s	PART
cana-1064	179	6	g	g	NOUN
cana-1064	179	7	,	,	PUNCT
cana-1064	179	8	partial	partial	ADJ
cana-1064	179	9	metric	metric	ADJ
cana-1064	179	10	topology	topology	NOUN
cana-1064	179	11	,	,	PUNCT
cana-1064	179	12	papers	paper	NOUN
cana-1064	179	13	on	on	ADP
cana-1064	179	14	general	general	ADJ
cana-1064	179	15	topology	topology	NOUN
cana-1064	179	16	and	and	CCONJ
cana-1064	179	17	applications	application	NOUN
cana-1064	179	18	(	(	PUNCT
cana-1064	179	19	flushing	flush	VERB
cana-1064	179	20	,	,	PUNCT
cana-1064	179	21	ny,1992	ny,1992	NOUN
cana-1064	179	22	)	)	PUNCT
cana-1064	179	23	,	,	PUNCT
cana-1064	179	24	ann	ann	PROPN
cana-1064	179	25	.	.	PUNCT
cana-1064	180	1	new	new	PROPN
cana-1064	180	2	york	york	PROPN
cana-1064	180	3	acad.sci	acad.sci	PROPN
cana-1064	180	4	.	.	PUNCT
cana-1064	180	5	,1994	,1994	PUNCT
cana-1064	180	6	,	,	PUNCT
cana-1064	180	7	vol	vol	NOUN
cana-1064	180	8	728	728	NUM
cana-1064	180	9	,	,	PUNCT
cana-1064	180	10	pp	pp	ADJ
cana-1064	180	11	.	.	PUNCT
cana-1064	181	1	183	183	NUM
cana-1064	181	2	-	-	PUNCT
cana-1064	181	3	197.proc.8th	197.proc.8th	NUM
cana-1064	181	4	summer	summer	NOUN
cana-1064	181	5	conference	conference	NOUN
cana-1064	181	6	on	on	ADP
cana-1064	181	7	general	general	ADJ
cana-1064	181	8	topology	topology	NOUN
cana-1064	181	9	and	and	CCONJ
cana-1064	181	10	applications	application	NOUN
cana-1064	181	11	.	.	PUNCT
cana-1064	182	1	https://doi.org/10.1111/j.1749-6632.1994.tb44144.x	https://doi.org/10.1111/j.1749-6632.1994.tb44144.x	X
cana-1064	182	2	.	.	PUNCT
cana-1064	183	1	[	[	X
cana-1064	183	2	2	2	NUM
cana-1064	183	3	]	]	PUNCT
cana-1064	183	4	ciri	ciri	PROPN
cana-1064	183	5	l	l	PROPN
cana-1064	183	6	samet	samet	PROPN
cana-1064	184	1	b	b	PROPN
cana-1064	184	2	,	,	PUNCT
cana-1064	184	3	aydi	aydi	VERB
cana-1064	184	4	h	h	NOUN
cana-1064	184	5	,	,	PUNCT
cana-1064	184	6	vetro	vetro	NOUN
cana-1064	184	7	c	c	NOUN
cana-1064	184	8	,	,	PUNCT
cana-1064	184	9	common	common	ADJ
cana-1064	184	10	fixed	fix	VERB
cana-1064	184	11	points	point	NOUN
cana-1064	184	12	of	of	ADP
cana-1064	184	13	generalized	generalized	ADJ
cana-1064	184	14	contractions	contraction	NOUN
cana-1064	184	15	on	on	ADP
cana-1064	184	16	partial	partial	ADJ
cana-1064	184	17	metric	metric	ADJ
cana-1064	184	18	space	space	NOUN
cana-1064	184	19	and	and	CCONJ
cana-1064	184	20	applications	application	NOUN
cana-1064	184	21	,	,	PUNCT
cana-1064	184	22	applied	apply	VERB
cana-1064	184	23	mathematics	mathematic	NOUN
cana-1064	184	24	and	and	CCONJ
cana-1064	184	25	computations	computation	NOUN
cana-1064	184	26	2011	2011	NUM
cana-1064	184	27	,	,	PUNCT
cana-1064	184	28	vol	vol	NOUN
cana-1064	184	29	218	218	NUM
cana-1064	184	30	no.6	no.6	NOUN
cana-1064	184	31	,	,	PUNCT
cana-1064	184	32	pp.2398	pp.2398	PROPN
cana-1064	184	33	-	-	PUNCT
cana-1064	184	34	2406	2406	NUM
cana-1064	184	35	.	.	PUNCT
cana-1064	185	1	https://doi.org/10.1016/j.amc.2011.07.005	https://doi.org/10.1016/j.amc.2011.07.005	X
cana-1064	185	2	.	.	PUNCT
cana-1064	186	1	[	[	X
cana-1064	186	2	3	3	X
cana-1064	186	3	]	]	PUNCT
cana-1064	186	4	aydi	aydi	VERB
cana-1064	186	5	h	h	NOUN
cana-1064	186	6	,	,	PUNCT
cana-1064	186	7	abbas	abbas	PROPN
cana-1064	186	8	m	m	PROPN
cana-1064	186	9	and	and	CCONJ
cana-1064	186	10	vetro	vetro	VERB
cana-1064	186	11	c	c	NOUN
cana-1064	186	12	,	,	PUNCT
cana-1064	186	13	partial	partial	ADJ
cana-1064	186	14	hausdorff	hausdorff	NOUN
cana-1064	186	15	metric	metric	ADJ
cana-1064	186	16	and	and	CCONJ
cana-1064	186	17	nadler	nadler	PROPN
cana-1064	186	18	’s	’s	PART
cana-1064	186	19	fixed	fix	VERB
cana-1064	186	20	point	point	NOUN
cana-1064	186	21	theorem	theorem	VERB
cana-1064	186	22	on	on	ADP
cana-1064	186	23	partial	partial	ADJ
cana-1064	186	24	metric	metric	ADJ
cana-1064	186	25	spaces	space	NOUN
cana-1064	186	26	,	,	PUNCT
cana-1064	186	27	topology	topology	NOUN
cana-1064	186	28	and	and	CCONJ
cana-1064	186	29	its	its	PRON
cana-1064	186	30	applications	application	NOUN
cana-1064	186	31	,	,	PUNCT
cana-1064	186	32	2012	2012	NUM
cana-1064	186	33	,	,	PUNCT
cana-1064	186	34	vol	vol	NOUN
cana-1064	186	35	159	159	NUM
cana-1064	186	36	,	,	PUNCT
cana-1064	186	37	no.14	no.14	PROPN
cana-1064	186	38	,	,	PUNCT
cana-1064	186	39	pp.3234	pp.3234	PROPN
cana-1064	186	40	-	-	PUNCT
cana-1064	186	41	3242	3242	NUM
cana-1064	186	42	.	.	PUNCT
cana-1064	187	1	http://dx.doi.org/10.1016/j.topol.2012.06.012	http://dx.doi.org/10.1016/j.topol.2012.06.012	NOUN
cana-1064	187	2	.	.	PUNCT
cana-1064	188	1	[	[	X
cana-1064	188	2	4	4	X
cana-1064	188	3	]	]	X
cana-1064	188	4	abdeljawad	abdeljawad	ADJ
cana-1064	188	5	thabet	thabet	NOUN
cana-1064	188	6	,	,	PUNCT
cana-1064	188	7	karapinar	karapinar	PROPN
cana-1064	188	8	erdal	erdal	X
cana-1064	188	9	,	,	PUNCT
cana-1064	188	10	tas	tas	PROPN
cana-1064	188	11	k	k	PROPN
cana-1064	188	12	,	,	PUNCT
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cana-1064	188	14	and	and	CCONJ
cana-1064	188	15	uniqueness	uniqueness	NOUN
cana-1064	188	16	of	of	ADP
cana-1064	188	17	a	a	DET
cana-1064	188	18	common	common	ADJ
cana-1064	188	19	fixed	fix	VERB
cana-1064	188	20	point	point	NOUN
cana-1064	188	21	on	on	ADP
cana-1064	188	22	partial	partial	ADJ
cana-1064	188	23	metric	metric	ADJ
cana-1064	188	24	spaces	space	NOUN
cana-1064	188	25	,	,	PUNCT
cana-1064	188	26	appl	appl	PROPN
cana-1064	188	27	.	.	PROPN
cana-1064	188	28	math	math	PROPN
cana-1064	188	29	.	.	PUNCT
cana-1064	189	1	lett	lett	PROPN
cana-1064	189	2	2011	2011	NUM
cana-1064	189	3	,	,	PUNCT
cana-1064	189	4	vol	vol	NOUN
cana-1064	189	5	24	24	NUM
cana-1064	189	6	,	,	PUNCT
cana-1064	189	7	pp.1900	pp.1900	NOUN
cana-1064	189	8	-	-	SYM
cana-1064	189	9	1904	1904	NUM
cana-1064	189	10	.	.	PUNCT
cana-1064	190	1	https://doi.org/10.1016/j.aml.2011.05.014	https://doi.org/10.1016/j.aml.2011.05.014	X
cana-1064	191	1	[	[	X
cana-1064	191	2	5	5	NUM
cana-1064	191	3	]	]	PUNCT
cana-1064	191	4	wardowski	wardowski	NOUN
cana-1064	191	5	,	,	PUNCT
cana-1064	191	6	fixed	fix	VERB
cana-1064	191	7	points	point	NOUN
cana-1064	191	8	of	of	ADP
cana-1064	191	9	a	a	DET
cana-1064	191	10	new	new	ADJ
cana-1064	191	11	type	type	NOUN
cana-1064	191	12	of	of	ADP
cana-1064	191	13	contractive	contractive	ADJ
cana-1064	191	14	mappings	mapping	NOUN
cana-1064	191	15	in	in	ADP
cana-1064	191	16	complete	complete	ADJ
cana-1064	191	17	metric	metric	ADJ
cana-1064	191	18	spaces	space	NOUN
cana-1064	191	19	,	,	PUNCT
cana-1064	191	20	fixed	fix	VERB
cana-1064	191	21	point	point	NOUN
cana-1064	191	22	theory	theory	NOUN
cana-1064	191	23	and	and	CCONJ
cana-1064	191	24	applications,2012	applications,2012	NOUN
cana-1064	191	25	,	,	PUNCT
cana-1064	191	26	vol	vol	VERB
cana-1064	191	27	2012(94	2012(94	NUM
cana-1064	191	28	)	)	PUNCT
cana-1064	191	29	.	.	PUNCT
cana-1064	192	1	https://doi.org/10.1186/1687-1812-2012-94	https://doi.org/10.1186/1687-1812-2012-94	PROPN
cana-1064	192	2	.	.	PUNCT
cana-1064	193	1	[	[	X
cana-1064	193	2	6	6	NUM
cana-1064	193	3	]	]	PUNCT
cana-1064	193	4	vildan	vildan	PROPN
cana-1064	193	5	o	o	NOUN
cana-1064	193	6	,	,	PUNCT
cana-1064	193	7	integral	integral	ADJ
cana-1064	193	8	type	type	NOUN
cana-1064	193	9	f	f	NOUN
cana-1064	193	10	-	-	PUNCT
cana-1064	193	11	contraction	contraction	NOUN
cana-1064	193	12	in	in	ADP
cana-1064	193	13	partial	partial	ADJ
cana-1064	193	14	metric	metric	ADJ
cana-1064	193	15	spaces	space	NOUN
cana-1064	193	16	,	,	PUNCT
cana-1064	193	17	journal	journal	NOUN
cana-1064	193	18	of	of	ADP
cana-1064	193	19	function	function	NOUN
cana-1064	193	20	spaces	space	NOUN
cana-1064	193	21	,	,	PUNCT
cana-1064	193	22	2019	2019	NUM
cana-1064	193	23	,	,	PUNCT
cana-1064	193	24	article	article	NOUN
cana-1064	193	25	i	i	PROPN
cana-1064	193	26	d	d	PROPN
cana-1064	193	27	5193862	5193862	NUM
cana-1064	193	28	,	,	PUNCT
cana-1064	193	29	vol	vol	NOUN
cana-1064	193	30	2019	2019	NUM
cana-1064	193	31	,	,	PUNCT
cana-1064	193	32	pp.1	pp.1	NOUN
cana-1064	193	33	-	-	PUNCT
cana-1064	193	34	8	8	NUM
cana-1064	193	35	.	.	PUNCT
cana-1064	194	1	https://doi.org/10.1155/2019/5193862	https://doi.org/10.1155/2019/5193862	NOUN
cana-1064	194	2	.	.	PUNCT
cana-1064	195	1	[	[	X
cana-1064	195	2	7	7	NUM
cana-1064	195	3	]	]	X
cana-1064	195	4	nazam	nazam	PROPN
cana-1064	195	5	m	m	PROPN
cana-1064	195	6	,	,	PUNCT
cana-1064	195	7	arshad	arshad	PROPN
cana-1064	195	8	m	m	PROPN
cana-1064	195	9	,	,	PUNCT
cana-1064	195	10	abbas	abbas	PROPN
cana-1064	195	11	m	m	PROPN
cana-1064	195	12	,	,	PUNCT
cana-1064	195	13	existence	existence	NOUN
cana-1064	195	14	of	of	ADP
cana-1064	195	15	common	common	ADJ
cana-1064	195	16	fixed	fix	VERB
cana-1064	195	17	points	point	NOUN
cana-1064	195	18	of	of	ADP
cana-1064	195	19	improved	improved	ADJ
cana-1064	195	20	f	f	NOUN
cana-1064	195	21	-	-	PUNCT
cana-1064	195	22	contraction	contraction	NOUN
cana-1064	195	23	on	on	ADP
cana-1064	195	24	partial	partial	ADJ
cana-1064	195	25	metric	metric	ADJ
cana-1064	195	26	spaces	space	NOUN
cana-1064	195	27	.	.	PUNCT
cana-1064	196	1	applied	apply	VERB
cana-1064	196	2	general	general	ADJ
cana-1064	196	3	topology	topology	NOUN
cana-1064	196	4	,	,	PUNCT
cana-1064	196	5	2017	2017	NUM
cana-1064	196	6	,	,	PUNCT
cana-1064	196	7	vol	vol	NOUN
cana-1064	196	8	18.no.2	18.no.2	NUM
cana-1064	196	9	,	,	PUNCT
cana-1064	196	10	pp.277	pp.277	PROPN
cana-1064	196	11	-	-	PUNCT
cana-1064	196	12	287	287	NUM
cana-1064	196	13	.	.	PUNCT
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cana-1064	197	2	.	.	PUNCT
cana-1064	198	1	[	[	X
cana-1064	198	2	8	8	NUM
cana-1064	198	3	]	]	PUNCT
cana-1064	198	4	v.	v.	ADP
cana-1064	198	5	m.	m.	PROPN
cana-1064	198	6	l.	l.	PROPN
cana-1064	198	7	hima	hima	PROPN
cana-1064	198	8	bindu	bindu	PROPN
cana-1064	198	9	,	,	PUNCT
cana-1064	198	10	g.	g.	PROPN
cana-1064	198	11	n.	n.	PROPN
cana-1064	198	12	v.	v.	PROPN
cana-1064	198	13	kishore	kishore	PROPN
cana-1064	198	14	,	,	PUNCT
cana-1064	198	15	k.	k.	PROPN
cana-1064	199	1	p.	p.	PROPN
cana-1064	199	2	r.	r.	PROPN
cana-1064	199	3	rao	rao	PROPN
cana-1064	199	4	,	,	PUNCT
cana-1064	199	5	y.	y.	PROPN
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cana-1064	199	7	,	,	PUNCT
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cana-1064	199	11	common	common	ADJ
cana-1064	199	12	fixed	fix	VERB
cana-1064	199	13	point	point	NOUN
cana-1064	199	14	theorem	theorem	VERB
cana-1064	199	15	in	in	ADP
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