id	sid	tid	token	lemma	pos
ejpam-5952	1	1	european	european	PROPN
ejpam-5952	1	2	journal	journal	PROPN
ejpam-5952	1	3	of	of	ADP
ejpam-5952	1	4	pure	pure	ADJ
ejpam-5952	1	5	and	and	CCONJ
ejpam-5952	1	6	applied	applied	ADJ
ejpam-5952	1	7	mathematics	mathematic	NOUN
ejpam-5952	1	8	2025	2025	NUM
ejpam-5952	1	9	,	,	PUNCT
ejpam-5952	1	10	vol	vol	NOUN
ejpam-5952	1	11	.	.	PROPN
ejpam-5952	1	12	18	18	NUM
ejpam-5952	1	13	,	,	PUNCT
ejpam-5952	1	14	issue	issue	NOUN
ejpam-5952	1	15	2	2	NUM
ejpam-5952	1	16	,	,	PUNCT
ejpam-5952	1	17	article	article	NOUN
ejpam-5952	1	18	number	number	NOUN
ejpam-5952	1	19	5952	5952	NUM
ejpam-5952	1	20	issn	issn	PROPN
ejpam-5952	1	21	1307	1307	NUM
ejpam-5952	1	22	-	-	SYM
ejpam-5952	1	23	5543	5543	NUM
ejpam-5952	1	24	–	–	PUNCT
ejpam-5952	1	25	ejpam.com	ejpam.com	X
ejpam-5952	1	26	published	publish	VERB
ejpam-5952	1	27	by	by	ADP
ejpam-5952	1	28	new	new	PROPN
ejpam-5952	1	29	york	york	PROPN
ejpam-5952	1	30	business	business	PROPN
ejpam-5952	1	31	global	global	ADJ
ejpam-5952	1	32	convergence	convergence	NOUN
ejpam-5952	1	33	and	and	CCONJ
ejpam-5952	1	34	fixed	fix	VERB
ejpam-5952	1	35	points	point	NOUN
ejpam-5952	1	36	of	of	ADP
ejpam-5952	1	37	self	self	NOUN
ejpam-5952	1	38	-	-	PUNCT
ejpam-5952	1	39	mappings	mapping	NOUN
ejpam-5952	1	40	in	in	ADP
ejpam-5952	1	41	mr	mr	PROPN
ejpam-5952	1	42	-	-	PUNCT
ejpam-5952	1	43	metric	metric	ADJ
ejpam-5952	1	44	spaces	space	NOUN
ejpam-5952	1	45	:	:	PUNCT
ejpam-5952	1	46	theory	theory	NOUN
ejpam-5952	1	47	and	and	CCONJ
ejpam-5952	1	48	applications	application	NOUN
ejpam-5952	1	49	abed	abe	VERB
ejpam-5952	1	50	al	al	PROPN
ejpam-5952	1	51	-	-	PUNCT
ejpam-5952	1	52	rahman	rahman	PROPN
ejpam-5952	1	53	m.	m.	NOUN
ejpam-5952	1	54	malkawi	malkawi	ADP
ejpam-5952	1	55	department	department	PROPN
ejpam-5952	1	56	of	of	ADP
ejpam-5952	1	57	mathematics	mathematic	NOUN
ejpam-5952	1	58	,	,	PUNCT
ejpam-5952	1	59	faculty	faculty	NOUN
ejpam-5952	1	60	of	of	ADP
ejpam-5952	1	61	arts	art	NOUN
ejpam-5952	1	62	and	and	CCONJ
ejpam-5952	1	63	science	science	NOUN
ejpam-5952	1	64	,	,	PUNCT
ejpam-5952	1	65	amman	amman	PROPN
ejpam-5952	1	66	arab	arab	PROPN
ejpam-5952	1	67	university	university	PROPN
ejpam-5952	1	68	,	,	PUNCT
ejpam-5952	1	69	amman	amman	PROPN
ejpam-5952	1	70	11953	11953	NUM
ejpam-5952	1	71	,	,	PUNCT
ejpam-5952	1	72	jordan	jordan	PROPN
ejpam-5952	1	73	abstract	abstract	PROPN
ejpam-5952	1	74	.	.	PUNCT
ejpam-5952	2	1	in	in	ADP
ejpam-5952	2	2	this	this	DET
ejpam-5952	2	3	paper	paper	NOUN
ejpam-5952	2	4	,	,	PUNCT
ejpam-5952	2	5	we	we	PRON
ejpam-5952	2	6	present	present	VERB
ejpam-5952	2	7	some	some	DET
ejpam-5952	2	8	important	important	ADJ
ejpam-5952	2	9	results	result	NOUN
ejpam-5952	2	10	for	for	ADP
ejpam-5952	2	11	self	self	NOUN
ejpam-5952	2	12	-	-	PUNCT
ejpam-5952	2	13	mappings	mapping	NOUN
ejpam-5952	2	14	in	in	ADP
ejpam-5952	2	15	mr	mr	PROPN
ejpam-5952	2	16	−	−	PROPN
ejpam-5952	2	17	metric	metric	ADJ
ejpam-5952	2	18	spaces	space	NOUN
ejpam-5952	2	19	,	,	PUNCT
ejpam-5952	2	20	ensuring	ensure	VERB
ejpam-5952	2	21	the	the	DET
ejpam-5952	2	22	existence	existence	NOUN
ejpam-5952	2	23	and	and	CCONJ
ejpam-5952	2	24	uniqueness	uniqueness	NOUN
ejpam-5952	2	25	of	of	ADP
ejpam-5952	2	26	fixed	fix	VERB
ejpam-5952	2	27	points	point	NOUN
ejpam-5952	2	28	in	in	ADP
ejpam-5952	2	29	contraction	contraction	NOUN
ejpam-5952	2	30	mappings	mapping	NOUN
ejpam-5952	2	31	.	.	PUNCT
ejpam-5952	3	1	the	the	DET
ejpam-5952	3	2	study	study	NOUN
ejpam-5952	3	3	reveals	reveal	VERB
ejpam-5952	3	4	the	the	DET
ejpam-5952	3	5	important	important	ADJ
ejpam-5952	3	6	role	role	NOUN
ejpam-5952	3	7	of	of	ADP
ejpam-5952	3	8	contraction	contraction	NOUN
ejpam-5952	3	9	properties	property	NOUN
ejpam-5952	3	10	in	in	ADP
ejpam-5952	3	11	achieving	achieve	VERB
ejpam-5952	3	12	convergence	convergence	NOUN
ejpam-5952	3	13	.	.	PUNCT
ejpam-5952	4	1	the	the	DET
ejpam-5952	4	2	paper	paper	NOUN
ejpam-5952	4	3	continues	continue	VERB
ejpam-5952	4	4	to	to	PART
ejpam-5952	4	5	develop	develop	VERB
ejpam-5952	4	6	these	these	DET
ejpam-5952	4	7	concepts	concept	NOUN
ejpam-5952	4	8	by	by	ADP
ejpam-5952	4	9	providing	provide	VERB
ejpam-5952	4	10	concrete	concrete	ADJ
ejpam-5952	4	11	examples	example	NOUN
ejpam-5952	4	12	that	that	PRON
ejpam-5952	4	13	demonstrate	demonstrate	VERB
ejpam-5952	4	14	their	their	PRON
ejpam-5952	4	15	importance	importance	NOUN
ejpam-5952	4	16	in	in	ADP
ejpam-5952	4	17	measurable	measurable	ADJ
ejpam-5952	4	18	spaces	space	NOUN
ejpam-5952	4	19	.	.	PUNCT
ejpam-5952	5	1	finally	finally	ADV
ejpam-5952	5	2	,	,	PUNCT
ejpam-5952	5	3	these	these	DET
ejpam-5952	5	4	results	result	NOUN
ejpam-5952	5	5	lay	lie	VERB
ejpam-5952	5	6	the	the	DET
ejpam-5952	5	7	foundation	foundation	NOUN
ejpam-5952	5	8	for	for	ADP
ejpam-5952	5	9	future	future	ADJ
ejpam-5952	5	10	investigations	investigation	NOUN
ejpam-5952	5	11	into	into	ADP
ejpam-5952	5	12	the	the	DET
ejpam-5952	5	13	stability	stability	NOUN
ejpam-5952	5	14	of	of	ADP
ejpam-5952	5	15	fixed	fix	VERB
ejpam-5952	5	16	points	point	NOUN
ejpam-5952	5	17	and	and	CCONJ
ejpam-5952	5	18	their	their	PRON
ejpam-5952	5	19	applications	application	NOUN
ejpam-5952	5	20	via	via	ADP
ejpam-5952	5	21	mathematical	mathematical	ADJ
ejpam-5952	5	22	frameworks	framework	NOUN
ejpam-5952	5	23	applicable	applicable	ADJ
ejpam-5952	5	24	to	to	ADP
ejpam-5952	5	25	real	real	ADJ
ejpam-5952	5	26	life	life	NOUN
ejpam-5952	5	27	.	.	PUNCT
ejpam-5952	6	1	2020	2020	NUM
ejpam-5952	6	2	mathematics	mathematic	NOUN
ejpam-5952	6	3	subject	subject	NOUN
ejpam-5952	6	4	classifications	classification	NOUN
ejpam-5952	6	5	:	:	PUNCT
ejpam-5952	6	6	47h10	47h10	NUM
ejpam-5952	6	7	,	,	PUNCT
ejpam-5952	6	8	54h25	54h25	NUM
ejpam-5952	6	9	,	,	PUNCT
ejpam-5952	6	10	46n10	46n10	NUM
ejpam-5952	6	11	,	,	PUNCT
ejpam-5952	6	12	54e50	54e50	NUM
ejpam-5952	6	13	,	,	PUNCT
ejpam-5952	6	14	28a33	28a33	NUM
ejpam-5952	6	15	key	key	ADJ
ejpam-5952	6	16	words	word	NOUN
ejpam-5952	6	17	and	and	CCONJ
ejpam-5952	6	18	phrases	phrase	NOUN
ejpam-5952	6	19	:	:	PUNCT
ejpam-5952	6	20	mr	mr	PROPN
ejpam-5952	6	21	−	−	PROPN
ejpam-5952	6	22	metric	metric	ADJ
ejpam-5952	6	23	space	space	NOUN
ejpam-5952	6	24	,	,	PUNCT
ejpam-5952	6	25	mr	mr	PROPN
ejpam-5952	6	26	-	-	PUNCT
ejpam-5952	6	27	cauchy	cauchy	PROPN
ejpam-5952	6	28	,	,	PUNCT
ejpam-5952	6	29	fixed	fix	VERB
ejpam-5952	6	30	point	point	NOUN
ejpam-5952	6	31	theorems	theorem	NOUN
ejpam-5952	6	32	,	,	PUNCT
ejpam-5952	6	33	mrconvergent	mrconvergent	ADJ
ejpam-5952	6	34	,	,	PUNCT
ejpam-5952	6	35	self	self	NOUN
ejpam-5952	6	36	-	-	PUNCT
ejpam-5952	6	37	mappings	mapping	NOUN
ejpam-5952	6	38	1	1	NUM
ejpam-5952	6	39	.	.	PUNCT
ejpam-5952	7	1	introduction	introduction	NOUN
ejpam-5952	7	2	this	this	DET
ejpam-5952	7	3	paper	paper	NOUN
ejpam-5952	7	4	presents	present	VERB
ejpam-5952	7	5	a	a	DET
ejpam-5952	7	6	comprehensive	comprehensive	ADJ
ejpam-5952	7	7	study	study	NOUN
ejpam-5952	7	8	on	on	ADP
ejpam-5952	7	9	the	the	DET
ejpam-5952	7	10	convergence	convergence	NOUN
ejpam-5952	7	11	and	and	CCONJ
ejpam-5952	7	12	fixed	fix	VERB
ejpam-5952	7	13	points	point	NOUN
ejpam-5952	7	14	of	of	ADP
ejpam-5952	7	15	self	self	NOUN
ejpam-5952	7	16	-	-	PUNCT
ejpam-5952	7	17	mappings	mapping	NOUN
ejpam-5952	7	18	in	in	ADP
ejpam-5952	7	19	mr	mr	PROPN
ejpam-5952	7	20	-	-	PUNCT
ejpam-5952	7	21	metric	metric	ADJ
ejpam-5952	7	22	spaces	space	NOUN
ejpam-5952	7	23	,	,	PUNCT
ejpam-5952	7	24	a	a	DET
ejpam-5952	7	25	recently	recently	ADV
ejpam-5952	7	26	introduced	introduce	VERB
ejpam-5952	7	27	generalization	generalization	NOUN
ejpam-5952	7	28	of	of	ADP
ejpam-5952	7	29	metric	metric	ADJ
ejpam-5952	7	30	spaces	space	NOUN
ejpam-5952	7	31	.	.	PUNCT
ejpam-5952	8	1	we	we	PRON
ejpam-5952	8	2	establish	establish	VERB
ejpam-5952	8	3	several	several	ADJ
ejpam-5952	8	4	key	key	ADJ
ejpam-5952	8	5	results	result	NOUN
ejpam-5952	8	6	,	,	PUNCT
ejpam-5952	8	7	including	include	VERB
ejpam-5952	8	8	the	the	DET
ejpam-5952	8	9	existence	existence	NOUN
ejpam-5952	8	10	and	and	CCONJ
ejpam-5952	8	11	uniqueness	uniqueness	NOUN
ejpam-5952	8	12	of	of	ADP
ejpam-5952	8	13	fixed	fix	VERB
ejpam-5952	8	14	points	point	NOUN
ejpam-5952	8	15	for	for	ADP
ejpam-5952	8	16	contraction	contraction	NOUN
ejpam-5952	8	17	mappings	mapping	NOUN
ejpam-5952	8	18	,	,	PUNCT
ejpam-5952	8	19	the	the	DET
ejpam-5952	8	20	convergence	convergence	NOUN
ejpam-5952	8	21	of	of	ADP
ejpam-5952	8	22	cauchy	cauchy	ADJ
ejpam-5952	8	23	sequences	sequence	NOUN
ejpam-5952	8	24	,	,	PUNCT
ejpam-5952	8	25	and	and	CCONJ
ejpam-5952	8	26	the	the	DET
ejpam-5952	8	27	convergence	convergence	NOUN
ejpam-5952	8	28	in	in	ADP
ejpam-5952	8	29	measure	measure	NOUN
ejpam-5952	8	30	of	of	ADP
ejpam-5952	8	31	iterates	iterate	NOUN
ejpam-5952	8	32	to	to	ADP
ejpam-5952	8	33	fixed	fix	VERB
ejpam-5952	8	34	points	point	NOUN
ejpam-5952	8	35	.	.	PUNCT
ejpam-5952	9	1	these	these	DET
ejpam-5952	9	2	findings	finding	NOUN
ejpam-5952	9	3	have	have	VERB
ejpam-5952	9	4	significant	significant	ADJ
ejpam-5952	9	5	implications	implication	NOUN
ejpam-5952	9	6	for	for	ADP
ejpam-5952	9	7	various	various	ADJ
ejpam-5952	9	8	fields	field	NOUN
ejpam-5952	9	9	,	,	PUNCT
ejpam-5952	9	10	including	include	VERB
ejpam-5952	9	11	optimization	optimization	NOUN
ejpam-5952	9	12	,	,	PUNCT
ejpam-5952	9	13	machine	machine	NOUN
ejpam-5952	9	14	learning	learning	NOUN
ejpam-5952	9	15	,	,	PUNCT
ejpam-5952	9	16	and	and	CCONJ
ejpam-5952	9	17	numerical	numerical	ADJ
ejpam-5952	9	18	analysis	analysis	NOUN
ejpam-5952	9	19	,	,	PUNCT
ejpam-5952	9	20	and	and	CCONJ
ejpam-5952	9	21	provide	provide	VERB
ejpam-5952	9	22	a	a	DET
ejpam-5952	9	23	solid	solid	ADJ
ejpam-5952	9	24	foundation	foundation	NOUN
ejpam-5952	9	25	for	for	ADP
ejpam-5952	9	26	further	further	ADJ
ejpam-5952	9	27	research	research	NOUN
ejpam-5952	9	28	and	and	CCONJ
ejpam-5952	9	29	applications	application	NOUN
ejpam-5952	9	30	in	in	ADP
ejpam-5952	9	31	mr	mr	PROPN
ejpam-5952	9	32	-	-	PUNCT
ejpam-5952	9	33	metric	metric	ADJ
ejpam-5952	9	34	spaces	space	NOUN
ejpam-5952	9	35	.	.	PUNCT
ejpam-5952	10	1	for	for	ADP
ejpam-5952	10	2	further	further	ADJ
ejpam-5952	10	3	details	detail	NOUN
ejpam-5952	10	4	,	,	PUNCT
ejpam-5952	10	5	we	we	PRON
ejpam-5952	10	6	refer	refer	VERB
ejpam-5952	10	7	readers	reader	NOUN
ejpam-5952	10	8	to	to	ADP
ejpam-5952	10	9	the	the	DET
ejpam-5952	10	10	works	work	NOUN
ejpam-5952	10	11	cited	cite	VERB
ejpam-5952	10	12	in	in	ADP
ejpam-5952	10	13	[	[	X
ejpam-5952	10	14	1–24	1–24	NOUN
ejpam-5952	10	15	]	]	PUNCT
ejpam-5952	10	16	.	.	PUNCT
ejpam-5952	11	1	definition	definition	NOUN
ejpam-5952	11	2	1	1	NUM
ejpam-5952	11	3	.	.	PUNCT
ejpam-5952	12	1	[	[	X
ejpam-5952	12	2	5	5	NUM
ejpam-5952	12	3	]	]	PUNCT
ejpam-5952	12	4	consider	consider	VERB
ejpam-5952	12	5	a	a	DET
ejpam-5952	12	6	non	non	ADJ
ejpam-5952	12	7	-	-	ADJ
ejpam-5952	12	8	empty	empty	ADJ
ejpam-5952	12	9	set	set	NOUN
ejpam-5952	12	10	x	x	PUNCT
ejpam-5952	12	11	̸=	̸=	PROPN
ejpam-5952	12	12	∅	∅	NOUN
ejpam-5952	12	13	and	and	CCONJ
ejpam-5952	12	14	a	a	DET
ejpam-5952	12	15	real	real	ADJ
ejpam-5952	12	16	number	number	NOUN
ejpam-5952	12	17	r	r	NOUN
ejpam-5952	12	18	>	>	X
ejpam-5952	12	19	1	1	NUM
ejpam-5952	12	20	.	.	PUNCT
ejpam-5952	13	1	a	a	DET
ejpam-5952	13	2	function	function	NOUN
ejpam-5952	13	3	m	m	VERB
ejpam-5952	13	4	:	:	PUNCT
ejpam-5952	13	5	x	x	X
ejpam-5952	13	6	×	×	NOUN
ejpam-5952	13	7	x	x	SYM
ejpam-5952	13	8	×	×	NOUN
ejpam-5952	13	9	x	x	INTJ
ejpam-5952	13	10	→	→	X
ejpam-5952	13	11	[	[	X
ejpam-5952	13	12	0,∞	0,∞	NOUN
ejpam-5952	13	13	)	)	PUNCT
ejpam-5952	13	14	is	be	AUX
ejpam-5952	13	15	termed	term	VERB
ejpam-5952	13	16	an	an	DET
ejpam-5952	13	17	mr	mr	PROPN
ejpam-5952	13	18	-	-	PUNCT
ejpam-5952	13	19	metric	metric	NOUN
ejpam-5952	13	20	if	if	SCONJ
ejpam-5952	13	21	it	it	PRON
ejpam-5952	13	22	satisfies	satisfy	VERB
ejpam-5952	13	23	the	the	DET
ejpam-5952	13	24	following	follow	VERB
ejpam-5952	13	25	conditions	condition	NOUN
ejpam-5952	13	26	for	for	ADP
ejpam-5952	13	27	all	all	DET
ejpam-5952	13	28	υ	υ	PROPN
ejpam-5952	13	29	,	,	PUNCT
ejpam-5952	13	30	ξ,ℑ	ξ,ℑ	PROPN
ejpam-5952	13	31	∈	∈	PROPN
ejpam-5952	14	1	x	x	X
ejpam-5952	14	2	:	:	PUNCT
ejpam-5952	14	3	•	•	PRON
ejpam-5952	14	4	(	(	PUNCT
ejpam-5952	14	5	m1	m1	NOUN
ejpam-5952	14	6	)	)	PUNCT
ejpam-5952	14	7	m(υ	m(υ	PROPN
ejpam-5952	14	8	,	,	PUNCT
ejpam-5952	14	9	ξ,ℑ	ξ,ℑ	PROPN
ejpam-5952	14	10	)	)	PUNCT
ejpam-5952	14	11	≥	≥	NOUN
ejpam-5952	14	12	0	0	NUM
ejpam-5952	14	13	.	.	NOUN
ejpam-5952	14	14	•	•	NUM
ejpam-5952	14	15	(	(	PUNCT
ejpam-5952	14	16	m2	m2	PROPN
ejpam-5952	14	17	)	)	PUNCT
ejpam-5952	14	18	m(υ	m(υ	PROPN
ejpam-5952	14	19	,	,	PUNCT
ejpam-5952	14	20	ξ,ℑ	ξ,ℑ	PROPN
ejpam-5952	14	21	)	)	PUNCT
ejpam-5952	15	1	=	=	SYM
ejpam-5952	15	2	0	0	PUNCT
ejpam-5952	16	1	if	if	SCONJ
ejpam-5952	16	2	and	and	CCONJ
ejpam-5952	16	3	only	only	ADV
ejpam-5952	16	4	if	if	SCONJ
ejpam-5952	16	5	υ	υ	PROPN
ejpam-5952	16	6	=	=	SYM
ejpam-5952	16	7	ξ	ξ	NOUN
ejpam-5952	16	8	=	=	SYM
ejpam-5952	16	9	ℑ.	ℑ.	NOUN
ejpam-5952	16	10	doi	doi	NOUN
ejpam-5952	16	11	:	:	PUNCT
ejpam-5952	16	12	https://doi.org/10.29020/nybg.ejpam.v18i2.5952	https://doi.org/10.29020/nybg.ejpam.v18i2.5952	NOUN
ejpam-5952	16	13	email	email	NOUN
ejpam-5952	16	14	addresses	address	NOUN
ejpam-5952	16	15	:	:	PUNCT
ejpam-5952	16	16	a.malkawi@aau.edu.jo	a.malkawi@aau.edu.jo	PROPN
ejpam-5952	16	17	and	and	CCONJ
ejpam-5952	16	18	math.malkawi@gmail.com	math.malkawi@gmail.com	X
ejpam-5952	16	19	(	(	PUNCT
ejpam-5952	16	20	a.	a.	NOUN
ejpam-5952	16	21	m.	m.	PROPN
ejpam-5952	16	22	m.	m.	PROPN
ejpam-5952	16	23	malkawi	malkawi	PROPN
ejpam-5952	16	24	)	)	PUNCT
ejpam-5952	16	25	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5952	16	26	1	1	NUM
ejpam-5952	16	27	copyright	copyright	NOUN
ejpam-5952	16	28	:	:	PUNCT
ejpam-5952	16	29	©	©	PROPN
ejpam-5952	16	30	2025	2025	NUM
ejpam-5952	16	31	the	the	DET
ejpam-5952	16	32	author(s	author(s	NOUN
ejpam-5952	16	33	)	)	PUNCT
ejpam-5952	16	34	.	.	PUNCT
ejpam-5952	17	1	(	(	PUNCT
ejpam-5952	17	2	cc	cc	NOUN
ejpam-5952	17	3	by	by	ADP
ejpam-5952	17	4	-	-	PUNCT
ejpam-5952	17	5	nc	nc	PROPN
ejpam-5952	17	6	4.0	4.0	NUM
ejpam-5952	17	7	)	)	PUNCT
ejpam-5952	17	8	a.	a.	NOUN
ejpam-5952	17	9	a.	a.	NOUN
ejpam-5952	17	10	m.	m.	NOUN
ejpam-5952	17	11	malkawi	malkawi	ADP
ejpam-5952	17	12	/	/	SYM
ejpam-5952	17	13	eur	eur	PROPN
ejpam-5952	17	14	.	.	PUNCT
ejpam-5952	18	1	j.	j.	PROPN
ejpam-5952	18	2	pure	pure	PROPN
ejpam-5952	18	3	appl	appl	PROPN
ejpam-5952	18	4	.	.	PROPN
ejpam-5952	18	5	math	math	PROPN
ejpam-5952	18	6	,	,	PUNCT
ejpam-5952	18	7	18	18	NUM
ejpam-5952	18	8	(	(	PUNCT
ejpam-5952	18	9	2	2	NUM
ejpam-5952	18	10	)	)	PUNCT
ejpam-5952	18	11	(	(	PUNCT
ejpam-5952	18	12	2025	2025	NUM
ejpam-5952	18	13	)	)	PUNCT
ejpam-5952	18	14	,	,	PUNCT
ejpam-5952	18	15	5952	5952	NUM
ejpam-5952	18	16	2	2	NUM
ejpam-5952	18	17	of	of	ADP
ejpam-5952	18	18	14	14	NUM
ejpam-5952	18	19	•	•	NOUN
ejpam-5952	18	20	(	(	PUNCT
ejpam-5952	18	21	m3	m3	PROPN
ejpam-5952	18	22	)	)	PUNCT
ejpam-5952	18	23	m(υ	m(υ	PROPN
ejpam-5952	18	24	,	,	PUNCT
ejpam-5952	18	25	ξ,ℑ	ξ,ℑ	PROPN
ejpam-5952	18	26	)	)	PUNCT
ejpam-5952	18	27	remains	remain	VERB
ejpam-5952	18	28	invariant	invariant	ADJ
ejpam-5952	18	29	under	under	ADP
ejpam-5952	18	30	any	any	DET
ejpam-5952	18	31	permutation	permutation	NOUN
ejpam-5952	18	32	p(υ	p(υ	NOUN
ejpam-5952	18	33	,	,	PUNCT
ejpam-5952	18	34	ξ,ℑ	ξ,ℑ	PROPN
ejpam-5952	18	35	)	)	PUNCT
ejpam-5952	18	36	,	,	PUNCT
ejpam-5952	18	37	i.e.	i.e.	X
ejpam-5952	18	38	,	,	PUNCT
ejpam-5952	18	39	m(υ	m(υ	PROPN
ejpam-5952	18	40	,	,	PUNCT
ejpam-5952	18	41	ξ,ℑ	ξ,ℑ	PROPN
ejpam-5952	18	42	)	)	PUNCT
ejpam-5952	19	1	=	=	SYM
ejpam-5952	20	1	m(p(υ	m(p(υ	PROPN
ejpam-5952	20	2	,	,	PUNCT
ejpam-5952	20	3	ξ,ℑ	ξ,ℑ	PROPN
ejpam-5952	20	4	)	)	PUNCT
ejpam-5952	20	5	)	)	PUNCT
ejpam-5952	20	6	.	.	PUNCT
ejpam-5952	21	1	•	•	NUM
ejpam-5952	21	2	(	(	PUNCT
ejpam-5952	21	3	m4	m4	PROPN
ejpam-5952	21	4	)	)	PUNCT
ejpam-5952	21	5	the	the	DET
ejpam-5952	21	6	following	follow	VERB
ejpam-5952	21	7	inequality	inequality	NOUN
ejpam-5952	21	8	holds	hold	VERB
ejpam-5952	21	9	:	:	PUNCT
ejpam-5952	22	1	m(υ	m(υ	PROPN
ejpam-5952	22	2	,	,	PUNCT
ejpam-5952	22	3	ξ,ℑ	ξ,ℑ	PROPN
ejpam-5952	22	4	)	)	PUNCT
ejpam-5952	22	5	≤	≤	NUM
ejpam-5952	22	6	r	r	NOUN
ejpam-5952	23	1	[	[	X
ejpam-5952	23	2	m(υ	m(υ	PROPN
ejpam-5952	23	3	,	,	PUNCT
ejpam-5952	23	4	ξ	ξ	PROPN
ejpam-5952	23	5	,	,	PUNCT
ejpam-5952	23	6	ℓ1	ℓ1	NOUN
ejpam-5952	23	7	)	)	PUNCT
ejpam-5952	24	1	+	+	SYM
ejpam-5952	24	2	m(υ	m(υ	PROPN
ejpam-5952	24	3	,	,	PUNCT
ejpam-5952	24	4	ℓ1,ℑ	ℓ1,ℑ	NOUN
ejpam-5952	24	5	)	)	PUNCT
ejpam-5952	25	1	+	+	ADJ
ejpam-5952	25	2	m(ℓ1	m(ℓ1	NOUN
ejpam-5952	25	3	,	,	PUNCT
ejpam-5952	25	4	ξ,ℑ	ξ,ℑ	PROPN
ejpam-5952	25	5	)	)	PUNCT
ejpam-5952	25	6	]	]	PUNCT
ejpam-5952	25	7	.	.	PUNCT
ejpam-5952	26	1	a	a	DET
ejpam-5952	26	2	structure	structure	NOUN
ejpam-5952	26	3	(	(	PUNCT
ejpam-5952	26	4	x	x	X
ejpam-5952	26	5	,	,	PUNCT
ejpam-5952	26	6	m	m	NOUN
ejpam-5952	26	7	)	)	PUNCT
ejpam-5952	26	8	that	that	PRON
ejpam-5952	26	9	adheres	adhere	VERB
ejpam-5952	26	10	to	to	ADP
ejpam-5952	26	11	these	these	DET
ejpam-5952	26	12	properties	property	NOUN
ejpam-5952	26	13	is	be	AUX
ejpam-5952	26	14	defined	define	VERB
ejpam-5952	26	15	as	as	ADP
ejpam-5952	26	16	an	an	DET
ejpam-5952	26	17	mr	mr	PROPN
ejpam-5952	26	18	-	-	PUNCT
ejpam-5952	26	19	metric	metric	ADJ
ejpam-5952	26	20	space	space	NOUN
ejpam-5952	26	21	.	.	PUNCT
ejpam-5952	27	1	definition	definition	NOUN
ejpam-5952	27	2	2	2	NUM
ejpam-5952	27	3	.	.	PUNCT
ejpam-5952	28	1	[	[	X
ejpam-5952	28	2	5	5	NUM
ejpam-5952	28	3	]	]	PUNCT
ejpam-5952	28	4	consider	consider	VERB
ejpam-5952	28	5	a	a	DET
ejpam-5952	28	6	sequence	sequence	NOUN
ejpam-5952	28	7	{	{	PUNCT
ejpam-5952	28	8	υin	υin	NOUN
ejpam-5952	28	9	}	}	PUNCT
ejpam-5952	28	10	in	in	ADP
ejpam-5952	28	11	an	an	DET
ejpam-5952	28	12	mr	mr	PROPN
ejpam-5952	28	13	-	-	PUNCT
ejpam-5952	28	14	metric	metric	ADJ
ejpam-5952	28	15	space	space	NOUN
ejpam-5952	28	16	(	(	PUNCT
ejpam-5952	28	17	x	x	X
ejpam-5952	28	18	,	,	PUNCT
ejpam-5952	28	19	m	m	NOUN
ejpam-5952	28	20	)	)	PUNCT
ejpam-5952	28	21	.	.	PUNCT
ejpam-5952	29	1	this	this	DET
ejpam-5952	29	2	sequence	sequence	NOUN
ejpam-5952	29	3	is	be	AUX
ejpam-5952	29	4	said	say	VERB
ejpam-5952	29	5	to	to	PART
ejpam-5952	29	6	be	be	AUX
ejpam-5952	29	7	mr	mr	NOUN
ejpam-5952	29	8	-	-	PUNCT
ejpam-5952	29	9	convergent	convergent	NOUN
ejpam-5952	29	10	if	if	SCONJ
ejpam-5952	29	11	there	there	PRON
ejpam-5952	29	12	exists	exist	VERB
ejpam-5952	29	13	an	an	DET
ejpam-5952	29	14	element	element	ADJ
ejpam-5952	29	15	υi1	υi1	NOUN
ejpam-5952	29	16	∈	∈	PROPN
ejpam-5952	29	17	x	x	PUNCT
ejpam-5952	29	18	such	such	ADJ
ejpam-5952	29	19	that	that	PRON
ejpam-5952	29	20	for	for	ADP
ejpam-5952	29	21	any	any	DET
ejpam-5952	29	22	ϵ	ϵ	X
ejpam-5952	29	23	>	>	X
ejpam-5952	29	24	0	0	NUM
ejpam-5952	29	25	,	,	PUNCT
ejpam-5952	29	26	there	there	PRON
ejpam-5952	29	27	exists	exist	VERB
ejpam-5952	29	28	a	a	DET
ejpam-5952	29	29	positive	positive	ADJ
ejpam-5952	29	30	integer	integer	NOUN
ejpam-5952	29	31	n	n	CCONJ
ejpam-5952	29	32	satisfying	satisfy	VERB
ejpam-5952	29	33	the	the	DET
ejpam-5952	29	34	condition	condition	NOUN
ejpam-5952	29	35	m(υin	m(υin	NOUN
ejpam-5952	29	36	,	,	PUNCT
ejpam-5952	29	37	υim	υim	PROPN
ejpam-5952	29	38	,	,	PUNCT
ejpam-5952	29	39	υi1	υi1	PROPN
ejpam-5952	29	40	)	)	PUNCT
ejpam-5952	29	41	<	<	X
ejpam-5952	30	1	ϵ	ϵ	X
ejpam-5952	30	2	,	,	PUNCT
ejpam-5952	30	3	for	for	ADP
ejpam-5952	30	4	all	all	DET
ejpam-5952	30	5	m	m	PROPN
ejpam-5952	30	6	,	,	PUNCT
ejpam-5952	30	7	n	n	PRON
ejpam-5952	30	8	≥	≥	NOUN
ejpam-5952	30	9	n.	n.	NOUN
ejpam-5952	30	10	in	in	ADP
ejpam-5952	30	11	this	this	DET
ejpam-5952	30	12	case	case	NOUN
ejpam-5952	30	13	,	,	PUNCT
ejpam-5952	30	14	we	we	PRON
ejpam-5952	30	15	say	say	VERB
ejpam-5952	30	16	that	that	SCONJ
ejpam-5952	30	17	{	{	PUNCT
ejpam-5952	30	18	υin	υin	NOUN
ejpam-5952	30	19	}	}	PUNCT
ejpam-5952	30	20	converges	converge	VERB
ejpam-5952	30	21	in	in	ADP
ejpam-5952	30	22	the	the	DET
ejpam-5952	30	23	mr	mr	PROPN
ejpam-5952	30	24	-	-	PUNCT
ejpam-5952	30	25	metric	metric	ADJ
ejpam-5952	30	26	sense	sense	NOUN
ejpam-5952	30	27	to	to	PART
ejpam-5952	30	28	υi1	υi1	VERB
ejpam-5952	30	29	,	,	PUNCT
ejpam-5952	30	30	and	and	CCONJ
ejpam-5952	30	31	we	we	PRON
ejpam-5952	30	32	refer	refer	VERB
ejpam-5952	30	33	to	to	ADP
ejpam-5952	30	34	υi1	υi1	NOUN
ejpam-5952	30	35	as	as	ADP
ejpam-5952	30	36	the	the	DET
ejpam-5952	30	37	limit	limit	NOUN
ejpam-5952	30	38	of	of	ADP
ejpam-5952	30	39	the	the	DET
ejpam-5952	30	40	sequence	sequence	NOUN
ejpam-5952	30	41	.	.	PUNCT
ejpam-5952	31	1	definition	definition	NOUN
ejpam-5952	31	2	3	3	NUM
ejpam-5952	31	3	.	.	PUNCT
ejpam-5952	32	1	[	[	X
ejpam-5952	32	2	5	5	NUM
ejpam-5952	32	3	]	]	PUNCT
ejpam-5952	32	4	a	a	DET
ejpam-5952	32	5	sequence	sequence	NOUN
ejpam-5952	32	6	{	{	PUNCT
ejpam-5952	32	7	υin	υin	NOUN
ejpam-5952	32	8	}	}	PUNCT
ejpam-5952	32	9	in	in	ADP
ejpam-5952	32	10	an	an	DET
ejpam-5952	32	11	mr	mr	PROPN
ejpam-5952	32	12	-	-	PUNCT
ejpam-5952	32	13	metric	metric	ADJ
ejpam-5952	32	14	space	space	NOUN
ejpam-5952	32	15	(	(	PUNCT
ejpam-5952	32	16	x	x	X
ejpam-5952	32	17	,	,	PUNCT
ejpam-5952	32	18	m	m	VERB
ejpam-5952	32	19	)	)	PUNCT
ejpam-5952	32	20	is	be	AUX
ejpam-5952	32	21	termed	term	VERB
ejpam-5952	32	22	mr	mr	PROPN
ejpam-5952	32	23	-	-	PUNCT
ejpam-5952	32	24	cauchy	cauchy	PROPN
ejpam-5952	32	25	if	if	SCONJ
ejpam-5952	32	26	for	for	ADP
ejpam-5952	32	27	every	every	DET
ejpam-5952	32	28	ϵ	ϵ	X
ejpam-5952	32	29	>	>	X
ejpam-5952	32	30	0	0	NUM
ejpam-5952	32	31	,	,	PUNCT
ejpam-5952	32	32	there	there	PRON
ejpam-5952	32	33	exists	exist	VERB
ejpam-5952	32	34	a	a	DET
ejpam-5952	32	35	positive	positive	ADJ
ejpam-5952	32	36	integer	integer	NOUN
ejpam-5952	32	37	n	n	CCONJ
ejpam-5952	32	38	such	such	ADJ
ejpam-5952	32	39	that	that	SCONJ
ejpam-5952	32	40	the	the	DET
ejpam-5952	32	41	inequality	inequality	NOUN
ejpam-5952	32	42	m(υin	m(υin	NOUN
ejpam-5952	32	43	,	,	PUNCT
ejpam-5952	32	44	υim	υim	PROPN
ejpam-5952	32	45	,	,	PUNCT
ejpam-5952	32	46	υip	υip	ADJ
ejpam-5952	32	47	)	)	PUNCT
ejpam-5952	32	48	<	<	X
ejpam-5952	33	1	ϵ	ϵ	X
ejpam-5952	33	2	holds	hold	VERB
ejpam-5952	33	3	for	for	ADP
ejpam-5952	33	4	all	all	DET
ejpam-5952	33	5	m	m	PROPN
ejpam-5952	33	6	,	,	PUNCT
ejpam-5952	33	7	n	n	CCONJ
ejpam-5952	33	8	,	,	PUNCT
ejpam-5952	33	9	p	p	PRON
ejpam-5952	33	10	≥	≥	NOUN
ejpam-5952	33	11	n.	n.	NOUN
ejpam-5952	33	12	definition	definition	NOUN
ejpam-5952	33	13	4	4	NUM
ejpam-5952	33	14	.	.	PUNCT
ejpam-5952	34	1	[	[	X
ejpam-5952	34	2	5	5	X
ejpam-5952	34	3	]	]	PUNCT
ejpam-5952	34	4	an	an	DET
ejpam-5952	34	5	mr	mr	PROPN
ejpam-5952	34	6	-	-	PUNCT
ejpam-5952	34	7	metric	metric	ADJ
ejpam-5952	34	8	space	space	NOUN
ejpam-5952	34	9	(	(	PUNCT
ejpam-5952	34	10	x	x	X
ejpam-5952	34	11	,	,	PUNCT
ejpam-5952	34	12	m	m	VERB
ejpam-5952	34	13	)	)	PUNCT
ejpam-5952	34	14	is	be	AUX
ejpam-5952	34	15	said	say	VERB
ejpam-5952	34	16	to	to	PART
ejpam-5952	34	17	be	be	AUX
ejpam-5952	34	18	bounded	bound	VERB
ejpam-5952	34	19	if	if	SCONJ
ejpam-5952	34	20	there	there	PRON
ejpam-5952	34	21	exists	exist	VERB
ejpam-5952	34	22	a	a	DET
ejpam-5952	34	23	constant	constant	ADJ
ejpam-5952	34	24	l	l	NOUN
ejpam-5952	34	25	>	>	X
ejpam-5952	34	26	0	0	NUM
ejpam-5952	34	27	such	such	ADJ
ejpam-5952	34	28	that	that	SCONJ
ejpam-5952	34	29	m(υ	m(υ	PROPN
ejpam-5952	34	30	,	,	PUNCT
ejpam-5952	34	31	ξ,ℑ	ξ,ℑ	PROPN
ejpam-5952	34	32	)	)	PUNCT
ejpam-5952	34	33	≤	≤	NUM
ejpam-5952	34	34	l	l	NOUN
ejpam-5952	34	35	for	for	ADP
ejpam-5952	34	36	all	all	DET
ejpam-5952	34	37	υ	υ	PROPN
ejpam-5952	34	38	,	,	PUNCT
ejpam-5952	34	39	ξ,ℑ	ξ,ℑ	PROPN
ejpam-5952	34	40	∈	∈	PROPN
ejpam-5952	34	41	x.	x.	NOUN
ejpam-5952	34	42	in	in	ADP
ejpam-5952	34	43	this	this	DET
ejpam-5952	34	44	case	case	NOUN
ejpam-5952	34	45	,	,	PUNCT
ejpam-5952	34	46	the	the	DET
ejpam-5952	34	47	function	function	NOUN
ejpam-5952	34	48	m	m	AUX
ejpam-5952	34	49	is	be	AUX
ejpam-5952	34	50	called	call	VERB
ejpam-5952	34	51	an	an	DET
ejpam-5952	34	52	mr	mr	PROPN
ejpam-5952	34	53	-	-	PUNCT
ejpam-5952	34	54	bound	bind	VERB
ejpam-5952	34	55	for	for	ADP
ejpam-5952	34	56	the	the	DET
ejpam-5952	34	57	metric	metric	NOUN
ejpam-5952	34	58	.	.	PUNCT
ejpam-5952	35	1	definition	definition	NOUN
ejpam-5952	35	2	5	5	NUM
ejpam-5952	35	3	(	(	PUNCT
ejpam-5952	35	4	[	[	X
ejpam-5952	35	5	25	25	NUM
ejpam-5952	35	6	]	]	PUNCT
ejpam-5952	35	7	,	,	PUNCT
ejpam-5952	35	8	definition	definition	NOUN
ejpam-5952	35	9	1.1	1.1	NUM
ejpam-5952	35	10	)	)	PUNCT
ejpam-5952	35	11	.	.	PUNCT
ejpam-5952	36	1	a	a	DET
ejpam-5952	36	2	measure	measure	NOUN
ejpam-5952	36	3	space	space	NOUN
ejpam-5952	36	4	is	be	AUX
ejpam-5952	36	5	a	a	DET
ejpam-5952	36	6	triplet	triplet	NOUN
ejpam-5952	36	7	(	(	PUNCT
ejpam-5952	36	8	x	x	NOUN
ejpam-5952	36	9	,	,	PUNCT
ejpam-5952	36	10	σ	σ	PROPN
ejpam-5952	36	11	,	,	PUNCT
ejpam-5952	36	12	µ	µ	NOUN
ejpam-5952	36	13	)	)	PUNCT
ejpam-5952	36	14	,	,	PUNCT
ejpam-5952	36	15	where	where	SCONJ
ejpam-5952	36	16	:	:	PUNCT
ejpam-5952	36	17	•	•	NOUN
ejpam-5952	37	1	x	x	X
ejpam-5952	37	2	is	be	AUX
ejpam-5952	37	3	a	a	DET
ejpam-5952	37	4	non	non	ADJ
ejpam-5952	37	5	-	-	ADJ
ejpam-5952	37	6	empty	empty	ADJ
ejpam-5952	37	7	set	set	NOUN
ejpam-5952	37	8	.	.	PUNCT
ejpam-5952	38	1	•	•	NUM
ejpam-5952	38	2	σ	σ	PROPN
ejpam-5952	38	3	is	be	AUX
ejpam-5952	38	4	a	a	DET
ejpam-5952	38	5	σ	σ	NOUN
ejpam-5952	38	6	-	-	PUNCT
ejpam-5952	38	7	algebra	algebra	NOUN
ejpam-5952	38	8	on	on	ADP
ejpam-5952	38	9	x	x	NOUN
ejpam-5952	38	10	,	,	PUNCT
ejpam-5952	38	11	which	which	PRON
ejpam-5952	38	12	satisfies	satisfy	VERB
ejpam-5952	38	13	the	the	DET
ejpam-5952	38	14	following	follow	VERB
ejpam-5952	38	15	properties	property	NOUN
ejpam-5952	38	16	:	:	PUNCT
ejpam-5952	38	17	(	(	PUNCT
ejpam-5952	38	18	i	i	NOUN
ejpam-5952	38	19	)	)	PUNCT
ejpam-5952	38	20	x	x	SYM
ejpam-5952	38	21	∈	∈	PROPN
ejpam-5952	38	22	σ	σ	PROPN
ejpam-5952	38	23	.	.	PUNCT
ejpam-5952	38	24	(	(	PUNCT
ejpam-5952	38	25	ii	ii	NOUN
ejpam-5952	38	26	)	)	PUNCT
ejpam-5952	38	27	if	if	SCONJ
ejpam-5952	38	28	a	a	DET
ejpam-5952	38	29	∈	∈	PROPN
ejpam-5952	38	30	σ	σ	PROPN
ejpam-5952	38	31	,	,	PUNCT
ejpam-5952	38	32	then	then	ADV
ejpam-5952	38	33	the	the	DET
ejpam-5952	38	34	complement	complement	NOUN
ejpam-5952	38	35	ac	ac	PROPN
ejpam-5952	38	36	∈	∈	PROPN
ejpam-5952	38	37	σ	σ	PROPN
ejpam-5952	38	38	.	.	PUNCT
ejpam-5952	39	1	(	(	PUNCT
ejpam-5952	39	2	iii	iii	X
ejpam-5952	39	3	)	)	PUNCT
ejpam-5952	39	4	if	if	SCONJ
ejpam-5952	39	5	{	{	PUNCT
ejpam-5952	39	6	an}∞n=1	an}∞n=1	X
ejpam-5952	39	7	⊆	⊆	NUM
ejpam-5952	39	8	σ	σ	NOUN
ejpam-5952	39	9	,	,	PUNCT
ejpam-5952	39	10	then	then	ADV
ejpam-5952	39	11	⋃∞	⋃∞	PUNCT
ejpam-5952	39	12	n=1an	n=1an	PROPN
ejpam-5952	40	1	∈	∈	PROPN
ejpam-5952	40	2	σ	σ	PROPN
ejpam-5952	40	3	.	.	PROPN
ejpam-5952	40	4	•	•	NUM
ejpam-5952	40	5	µ	µ	X
ejpam-5952	40	6	:	:	PUNCT
ejpam-5952	40	7	σ	σ	NOUN
ejpam-5952	40	8	→	→	PUNCT
ejpam-5952	40	9	[	[	X
ejpam-5952	40	10	0,∞	0,∞	X
ejpam-5952	40	11	]	]	PUNCT
ejpam-5952	40	12	is	be	AUX
ejpam-5952	40	13	a	a	DET
ejpam-5952	40	14	function	function	NOUN
ejpam-5952	40	15	satisfying	satisfy	VERB
ejpam-5952	40	16	:	:	PUNCT
ejpam-5952	40	17	(	(	PUNCT
ejpam-5952	40	18	i	i	NOUN
ejpam-5952	40	19	)	)	PUNCT
ejpam-5952	40	20	non	non	ADJ
ejpam-5952	40	21	-	-	NOUN
ejpam-5952	40	22	negativity	negativity	ADJ
ejpam-5952	40	23	:	:	PUNCT
ejpam-5952	40	24	µ(a	µ(a	PROPN
ejpam-5952	40	25	)	)	PUNCT
ejpam-5952	40	26	≥	≥	NOUN
ejpam-5952	40	27	0	0	NUM
ejpam-5952	40	28	for	for	ADP
ejpam-5952	40	29	all	all	DET
ejpam-5952	40	30	a	a	DET
ejpam-5952	40	31	∈	∈	PROPN
ejpam-5952	40	32	σ	σ	PROPN
ejpam-5952	40	33	.	.	PUNCT
ejpam-5952	40	34	(	(	PUNCT
ejpam-5952	40	35	ii	ii	NOUN
ejpam-5952	40	36	)	)	PUNCT
ejpam-5952	40	37	null	null	ADJ
ejpam-5952	40	38	empty	empty	ADJ
ejpam-5952	40	39	set	set	NOUN
ejpam-5952	40	40	:	:	PUNCT
ejpam-5952	40	41	µ(∅	µ(∅	X
ejpam-5952	40	42	)	)	PUNCT
ejpam-5952	41	1	=	=	SYM
ejpam-5952	41	2	0	0	X
ejpam-5952	41	3	.	.	PUNCT
ejpam-5952	42	1	(	(	PUNCT
ejpam-5952	42	2	iii	iii	NOUN
ejpam-5952	42	3	)	)	PUNCT
ejpam-5952	42	4	countable	countable	ADJ
ejpam-5952	42	5	additivity	additivity	NOUN
ejpam-5952	42	6	(	(	PUNCT
ejpam-5952	42	7	σ	σ	NOUN
ejpam-5952	42	8	-	-	NOUN
ejpam-5952	42	9	additivity	additivity	NOUN
ejpam-5952	42	10	):	):	PUNCT
ejpam-5952	42	11	if	if	SCONJ
ejpam-5952	42	12	{	{	PUNCT
ejpam-5952	42	13	an}∞n=1	an}∞n=1	X
ejpam-5952	42	14	is	be	AUX
ejpam-5952	42	15	a	a	DET
ejpam-5952	42	16	countable	countable	ADJ
ejpam-5952	42	17	collection	collection	NOUN
ejpam-5952	42	18	of	of	ADP
ejpam-5952	42	19	disjoint	disjoint	NOUN
ejpam-5952	42	20	sets	set	NOUN
ejpam-5952	42	21	in	in	ADP
ejpam-5952	42	22	σ	σ	PROPN
ejpam-5952	42	23	,	,	PUNCT
ejpam-5952	42	24	then	then	ADV
ejpam-5952	42	25	:	:	PUNCT
ejpam-5952	42	26	µ	µ	X
ejpam-5952	42	27	(	(	PUNCT
ejpam-5952	42	28	∞⋃	∞⋃	PROPN
ejpam-5952	42	29	n=1	n=1	PROPN
ejpam-5952	42	30	an	an	X
ejpam-5952	42	31	)	)	PUNCT
ejpam-5952	42	32	=	=	SYM
ejpam-5952	43	1	∞∑	∞∑	NUM
ejpam-5952	43	2	n=1	n=1	PROPN
ejpam-5952	43	3	µ(an	µ(an	PROPN
ejpam-5952	43	4	)	)	PUNCT
ejpam-5952	43	5	.	.	PUNCT
ejpam-5952	44	1	a.	a.	NOUN
ejpam-5952	44	2	a.	a.	PROPN
ejpam-5952	44	3	m.	m.	PROPN
ejpam-5952	44	4	malkawi	malkawi	ADP
ejpam-5952	44	5	/	/	SYM
ejpam-5952	44	6	eur	eur	PROPN
ejpam-5952	44	7	.	.	PUNCT
ejpam-5952	45	1	j.	j.	PROPN
ejpam-5952	45	2	pure	pure	PROPN
ejpam-5952	45	3	appl	appl	PROPN
ejpam-5952	45	4	.	.	PROPN
ejpam-5952	45	5	math	math	PROPN
ejpam-5952	45	6	,	,	PUNCT
ejpam-5952	45	7	18	18	NUM
ejpam-5952	45	8	(	(	PUNCT
ejpam-5952	45	9	2	2	NUM
ejpam-5952	45	10	)	)	PUNCT
ejpam-5952	45	11	(	(	PUNCT
ejpam-5952	45	12	2025	2025	NUM
ejpam-5952	45	13	)	)	PUNCT
ejpam-5952	45	14	,	,	PUNCT
ejpam-5952	45	15	5952	5952	NUM
ejpam-5952	45	16	3	3	NUM
ejpam-5952	45	17	of	of	ADP
ejpam-5952	45	18	14	14	NUM
ejpam-5952	45	19	the	the	DET
ejpam-5952	45	20	function	function	NOUN
ejpam-5952	45	21	µ	µ	NOUN
ejpam-5952	45	22	is	be	AUX
ejpam-5952	45	23	called	call	VERB
ejpam-5952	45	24	a	a	DET
ejpam-5952	45	25	measure	measure	NOUN
ejpam-5952	45	26	,	,	PUNCT
ejpam-5952	45	27	and	and	CCONJ
ejpam-5952	45	28	x	x	X
ejpam-5952	45	29	is	be	AUX
ejpam-5952	45	30	referred	refer	VERB
ejpam-5952	45	31	to	to	ADP
ejpam-5952	45	32	as	as	ADP
ejpam-5952	45	33	the	the	DET
ejpam-5952	45	34	measurable	measurable	ADJ
ejpam-5952	45	35	space	space	NOUN
ejpam-5952	45	36	.	.	PUNCT
ejpam-5952	46	1	definition	definition	NOUN
ejpam-5952	46	2	6	6	NUM
ejpam-5952	46	3	(	(	PUNCT
ejpam-5952	46	4	[	[	X
ejpam-5952	46	5	26	26	NUM
ejpam-5952	46	6	]	]	PUNCT
ejpam-5952	46	7	,	,	PUNCT
ejpam-5952	46	8	section	section	NOUN
ejpam-5952	46	9	2	2	NUM
ejpam-5952	46	10	)	)	PUNCT
ejpam-5952	46	11	.	.	PUNCT
ejpam-5952	47	1	a	a	DET
ejpam-5952	47	2	measure	measure	NOUN
ejpam-5952	47	3	µ	µ	NOUN
ejpam-5952	47	4	is	be	AUX
ejpam-5952	47	5	said	say	VERB
ejpam-5952	47	6	to	to	PART
ejpam-5952	47	7	be	be	AUX
ejpam-5952	47	8	σ	σ	NOUN
ejpam-5952	47	9	-	-	NOUN
ejpam-5952	47	10	finite	finite	NOUN
ejpam-5952	47	11	if	if	SCONJ
ejpam-5952	47	12	there	there	PRON
ejpam-5952	47	13	exists	exist	VERB
ejpam-5952	47	14	a	a	DET
ejpam-5952	47	15	countable	countable	ADJ
ejpam-5952	47	16	collection	collection	NOUN
ejpam-5952	47	17	of	of	ADP
ejpam-5952	47	18	measurable	measurable	ADJ
ejpam-5952	47	19	sets	set	NOUN
ejpam-5952	47	20	{	{	PUNCT
ejpam-5952	47	21	xn}∞n=1	xn}∞n=1	PROPN
ejpam-5952	47	22	⊆	⊆	NUM
ejpam-5952	47	23	σ	σ	NOUN
ejpam-5952	47	24	such	such	ADJ
ejpam-5952	47	25	that	that	PRON
ejpam-5952	47	26	:	:	PUNCT
ejpam-5952	47	27	x	x	SYM
ejpam-5952	47	28	=	=	SYM
ejpam-5952	47	29	∞⋃	∞⋃	PROPN
ejpam-5952	47	30	n=1	n=1	PUNCT
ejpam-5952	47	31	xn	xn	PROPN
ejpam-5952	47	32	and	and	CCONJ
ejpam-5952	47	33	µ(xn	µ(xn	PROPN
ejpam-5952	47	34	)	)	PUNCT
ejpam-5952	47	35	<	<	X
ejpam-5952	47	36	∞	∞	PROPN
ejpam-5952	47	37	for	for	ADP
ejpam-5952	47	38	every	every	DET
ejpam-5952	47	39	n.	n.	NOUN
ejpam-5952	47	40	definition	definition	NOUN
ejpam-5952	47	41	7	7	NUM
ejpam-5952	47	42	(	(	PUNCT
ejpam-5952	47	43	[	[	X
ejpam-5952	47	44	27	27	NUM
ejpam-5952	47	45	]	]	PUNCT
ejpam-5952	47	46	,	,	PUNCT
ejpam-5952	47	47	section	section	NOUN
ejpam-5952	47	48	3.1	3.1	NUM
ejpam-5952	47	49	)	)	PUNCT
ejpam-5952	47	50	.	.	PUNCT
ejpam-5952	48	1	a	a	DET
ejpam-5952	48	2	measure	measure	NOUN
ejpam-5952	48	3	µ	µ	NOUN
ejpam-5952	48	4	is	be	AUX
ejpam-5952	48	5	called	call	VERB
ejpam-5952	48	6	absolutely	absolutely	ADV
ejpam-5952	48	7	continuous	continuous	ADJ
ejpam-5952	48	8	with	with	ADP
ejpam-5952	48	9	respect	respect	NOUN
ejpam-5952	48	10	to	to	ADP
ejpam-5952	48	11	another	another	DET
ejpam-5952	48	12	measure	measure	NOUN
ejpam-5952	48	13	ν	ν	X
ejpam-5952	48	14	(	(	PUNCT
ejpam-5952	48	15	denoted	denote	VERB
ejpam-5952	48	16	µ	µ	PRON
ejpam-5952	48	17	≪	≪	ADJ
ejpam-5952	48	18	ν	ν	NOUN
ejpam-5952	48	19	)	)	PUNCT
ejpam-5952	48	20	if	if	SCONJ
ejpam-5952	48	21	for	for	SCONJ
ejpam-5952	48	22	every	every	DET
ejpam-5952	48	23	measurable	measurable	NOUN
ejpam-5952	48	24	set	set	VERB
ejpam-5952	48	25	a	a	DET
ejpam-5952	48	26	∈	∈	PROPN
ejpam-5952	48	27	σ	σ	PROPN
ejpam-5952	48	28	,	,	PUNCT
ejpam-5952	48	29	ν(a	ν(a	PROPN
ejpam-5952	48	30	)	)	PUNCT
ejpam-5952	48	31	=	=	SYM
ejpam-5952	48	32	0	0	NUM
ejpam-5952	48	33	⇒	⇒	NOUN
ejpam-5952	48	34	µ(a	µ(a	PROPN
ejpam-5952	48	35	)	)	PUNCT
ejpam-5952	49	1	=	=	PUNCT
ejpam-5952	49	2	0	0	NUM
ejpam-5952	49	3	.	.	NOUN
ejpam-5952	49	4	2	2	NUM
ejpam-5952	49	5	.	.	X
ejpam-5952	49	6	main	main	ADJ
ejpam-5952	49	7	result	result	NOUN
ejpam-5952	49	8	in	in	ADP
ejpam-5952	49	9	this	this	DET
ejpam-5952	49	10	section	section	NOUN
ejpam-5952	49	11	,	,	PUNCT
ejpam-5952	49	12	we	we	PRON
ejpam-5952	49	13	present	present	VERB
ejpam-5952	49	14	the	the	DET
ejpam-5952	49	15	main	main	ADJ
ejpam-5952	49	16	results	result	NOUN
ejpam-5952	49	17	for	for	ADP
ejpam-5952	49	18	a	a	DET
ejpam-5952	49	19	complete	complete	ADJ
ejpam-5952	49	20	mr	mr	ADJ
ejpam-5952	49	21	-	-	PUNCT
ejpam-5952	49	22	metric	metric	ADJ
ejpam-5952	49	23	space	space	NOUN
ejpam-5952	49	24	within	within	ADP
ejpam-5952	49	25	a	a	DET
ejpam-5952	49	26	convergent	convergent	NOUN
ejpam-5952	49	27	sequence	sequence	NOUN
ejpam-5952	49	28	,	,	PUNCT
ejpam-5952	49	29	and	and	CCONJ
ejpam-5952	49	30	we	we	PRON
ejpam-5952	49	31	have	have	AUX
ejpam-5952	49	32	given	give	VERB
ejpam-5952	49	33	some	some	DET
ejpam-5952	49	34	examples	example	NOUN
ejpam-5952	49	35	for	for	ADP
ejpam-5952	49	36	the	the	DET
ejpam-5952	49	37	main	main	ADJ
ejpam-5952	49	38	results	result	NOUN
ejpam-5952	49	39	.	.	PUNCT
ejpam-5952	50	1	theorem	theorem	NOUN
ejpam-5952	50	2	1	1	NUM
ejpam-5952	50	3	.	.	PUNCT
ejpam-5952	51	1	let	let	AUX
ejpam-5952	51	2	(	(	PUNCT
ejpam-5952	51	3	x	x	X
ejpam-5952	51	4	,	,	PUNCT
ejpam-5952	51	5	m	m	VERB
ejpam-5952	51	6	)	)	PUNCT
ejpam-5952	51	7	be	be	VERB
ejpam-5952	51	8	an	an	DET
ejpam-5952	51	9	mr	mr	ADJ
ejpam-5952	51	10	-	-	PUNCT
ejpam-5952	51	11	metric	metric	ADJ
ejpam-5952	51	12	space	space	NOUN
ejpam-5952	51	13	with	with	ADP
ejpam-5952	51	14	r	r	NOUN
ejpam-5952	51	15	>	>	X
ejpam-5952	51	16	1	1	NUM
ejpam-5952	51	17	,	,	PUNCT
ejpam-5952	51	18	and	and	CCONJ
ejpam-5952	51	19	let	let	VERB
ejpam-5952	51	20	t	t	NOUN
ejpam-5952	51	21	:	:	PUNCT
ejpam-5952	51	22	x	x	X
ejpam-5952	51	23	→	→	PUNCT
ejpam-5952	51	24	x	x	PUNCT
ejpam-5952	51	25	be	be	AUX
ejpam-5952	51	26	a	a	DET
ejpam-5952	51	27	self	self	NOUN
ejpam-5952	51	28	-	-	PUNCT
ejpam-5952	51	29	mapping	mapping	NOUN
ejpam-5952	51	30	satisfying	satisfy	VERB
ejpam-5952	51	31	the	the	DET
ejpam-5952	51	32	contraction	contraction	NOUN
ejpam-5952	51	33	condition	condition	NOUN
ejpam-5952	51	34	:	:	PUNCT
ejpam-5952	52	1	m(tυ	m(tυ	NOUN
ejpam-5952	52	2	,	,	PUNCT
ejpam-5952	52	3	tξ	tξ	VERB
ejpam-5952	52	4	,	,	PUNCT
ejpam-5952	52	5	tℑ	tℑ	NOUN
ejpam-5952	52	6	)	)	PUNCT
ejpam-5952	52	7	≤	≤	NOUN
ejpam-5952	52	8	αm(υ	αm(υ	NUM
ejpam-5952	52	9	,	,	PUNCT
ejpam-5952	52	10	ξ,ℑ	ξ,ℑ	PROPN
ejpam-5952	52	11	)	)	PUNCT
ejpam-5952	52	12	,	,	PUNCT
ejpam-5952	52	13	∀υ	∀υ	NOUN
ejpam-5952	52	14	,	,	PUNCT
ejpam-5952	52	15	ξ,ℑ	ξ,ℑ	PROPN
ejpam-5952	52	16	∈	∈	PROPN
ejpam-5952	52	17	x	x	X
ejpam-5952	52	18	,	,	PUNCT
ejpam-5952	52	19	where	where	SCONJ
ejpam-5952	52	20	0	0	NUM
ejpam-5952	52	21	≤	≤	NUM
ejpam-5952	52	22	α	α	NOUN
ejpam-5952	52	23	<	<	X
ejpam-5952	52	24	1	1	NUM
ejpam-5952	52	25	/	/	SYM
ejpam-5952	52	26	r.	r.	PROPN
ejpam-5952	52	27	then	then	ADV
ejpam-5952	52	28	,	,	PUNCT
ejpam-5952	52	29	t	t	PROPN
ejpam-5952	52	30	has	have	VERB
ejpam-5952	52	31	a	a	DET
ejpam-5952	52	32	unique	unique	ADJ
ejpam-5952	52	33	fixed	fix	VERB
ejpam-5952	52	34	point	point	NOUN
ejpam-5952	52	35	υ∗	υ∗	NOUN
ejpam-5952	52	36	∈	∈	PROPN
ejpam-5952	52	37	x	x	PUNCT
ejpam-5952	52	38	such	such	ADJ
ejpam-5952	52	39	that	that	DET
ejpam-5952	52	40	t	t	PROPN
ejpam-5952	52	41	(	(	PUNCT
ejpam-5952	52	42	υ∗	υ∗	PROPN
ejpam-5952	52	43	)	)	PUNCT
ejpam-5952	52	44	=	=	SYM
ejpam-5952	52	45	υ∗.	υ∗.	VERB
ejpam-5952	52	46	proof	proof	NOUN
ejpam-5952	52	47	.	.	PUNCT
ejpam-5952	53	1	consider	consider	VERB
ejpam-5952	53	2	the	the	DET
ejpam-5952	53	3	sequence	sequence	NOUN
ejpam-5952	53	4	{	{	PUNCT
ejpam-5952	53	5	υn	υn	NOUN
ejpam-5952	53	6	}	}	PUNCT
ejpam-5952	53	7	in	in	ADP
ejpam-5952	53	8	x	x	NOUN
ejpam-5952	53	9	,	,	PUNCT
ejpam-5952	53	10	which	which	PRON
ejpam-5952	53	11	is	be	AUX
ejpam-5952	53	12	generated	generate	VERB
ejpam-5952	53	13	recursively	recursively	ADV
ejpam-5952	53	14	by	by	ADP
ejpam-5952	53	15	the	the	DET
ejpam-5952	53	16	relation	relation	NOUN
ejpam-5952	53	17	υn+1	υn+1	PROPN
ejpam-5952	53	18	=	=	SYM
ejpam-5952	53	19	t	t	PROPN
ejpam-5952	53	20	(	(	PUNCT
ejpam-5952	53	21	υn	υn	NOUN
ejpam-5952	53	22	)	)	PUNCT
ejpam-5952	53	23	,	,	PUNCT
ejpam-5952	53	24	starting	start	VERB
ejpam-5952	53	25	from	from	ADP
ejpam-5952	53	26	an	an	DET
ejpam-5952	53	27	initial	initial	ADJ
ejpam-5952	53	28	element	element	NOUN
ejpam-5952	53	29	υ0	υ0	NOUN
ejpam-5952	53	30	∈	∈	PROPN
ejpam-5952	53	31	x.	x.	NOUN
ejpam-5952	54	1	our	our	PRON
ejpam-5952	54	2	objective	objective	NOUN
ejpam-5952	54	3	is	be	AUX
ejpam-5952	54	4	to	to	PART
ejpam-5952	54	5	establish	establish	VERB
ejpam-5952	54	6	that	that	SCONJ
ejpam-5952	54	7	this	this	DET
ejpam-5952	54	8	sequence	sequence	NOUN
ejpam-5952	54	9	converges	converge	VERB
ejpam-5952	54	10	to	to	ADP
ejpam-5952	54	11	a	a	DET
ejpam-5952	54	12	unique	unique	ADJ
ejpam-5952	54	13	fixed	fix	VERB
ejpam-5952	54	14	point	point	NOUN
ejpam-5952	54	15	of	of	ADP
ejpam-5952	54	16	the	the	DET
ejpam-5952	54	17	mapping	mapping	NOUN
ejpam-5952	54	18	t	t	NOUN
ejpam-5952	54	19	.	.	PUNCT
ejpam-5952	55	1	to	to	PART
ejpam-5952	55	2	prove	prove	VERB
ejpam-5952	55	3	convergence	convergence	NOUN
ejpam-5952	55	4	,	,	PUNCT
ejpam-5952	55	5	we	we	PRON
ejpam-5952	55	6	begin	begin	VERB
ejpam-5952	55	7	by	by	ADP
ejpam-5952	55	8	using	use	VERB
ejpam-5952	55	9	the	the	DET
ejpam-5952	55	10	contraction	contraction	NOUN
ejpam-5952	55	11	condition	condition	NOUN
ejpam-5952	55	12	.	.	PUNCT
ejpam-5952	56	1	the	the	DET
ejpam-5952	56	2	distance	distance	NOUN
ejpam-5952	56	3	between	between	ADP
ejpam-5952	56	4	consecutive	consecutive	ADJ
ejpam-5952	56	5	terms	term	NOUN
ejpam-5952	56	6	of	of	ADP
ejpam-5952	56	7	the	the	DET
ejpam-5952	56	8	sequence	sequence	NOUN
ejpam-5952	56	9	is	be	AUX
ejpam-5952	56	10	controlled	control	VERB
ejpam-5952	56	11	by	by	ADP
ejpam-5952	56	12	the	the	DET
ejpam-5952	56	13	contraction	contraction	NOUN
ejpam-5952	56	14	condition	condition	NOUN
ejpam-5952	56	15	,	,	PUNCT
ejpam-5952	56	16	which	which	PRON
ejpam-5952	56	17	ensures	ensure	VERB
ejpam-5952	56	18	that	that	SCONJ
ejpam-5952	56	19	the	the	DET
ejpam-5952	56	20	sequence	sequence	NOUN
ejpam-5952	56	21	{	{	PUNCT
ejpam-5952	56	22	υn	υn	NOUN
ejpam-5952	56	23	}	}	PUNCT
ejpam-5952	56	24	is	be	AUX
ejpam-5952	56	25	a	a	DET
ejpam-5952	56	26	cauchy	cauchy	ADJ
ejpam-5952	56	27	sequence	sequence	NOUN
ejpam-5952	56	28	.	.	PUNCT
ejpam-5952	57	1	since	since	SCONJ
ejpam-5952	57	2	the	the	DET
ejpam-5952	57	3	mr	mr	PROPN
ejpam-5952	57	4	-	-	PUNCT
ejpam-5952	57	5	metric	metric	ADJ
ejpam-5952	57	6	space	space	NOUN
ejpam-5952	57	7	(	(	PUNCT
ejpam-5952	57	8	x	x	X
ejpam-5952	57	9	,	,	PUNCT
ejpam-5952	57	10	m	m	VERB
ejpam-5952	57	11	)	)	PUNCT
ejpam-5952	57	12	is	be	AUX
ejpam-5952	57	13	complete	complete	ADJ
ejpam-5952	57	14	,	,	PUNCT
ejpam-5952	57	15	every	every	DET
ejpam-5952	57	16	cauchy	cauchy	ADJ
ejpam-5952	57	17	sequence	sequence	NOUN
ejpam-5952	57	18	in	in	ADP
ejpam-5952	57	19	x	x	PRON
ejpam-5952	57	20	must	must	AUX
ejpam-5952	57	21	converge	converge	VERB
ejpam-5952	57	22	to	to	ADP
ejpam-5952	57	23	some	some	DET
ejpam-5952	57	24	point	point	NOUN
ejpam-5952	57	25	.	.	PUNCT
ejpam-5952	58	1	therefore	therefore	ADV
ejpam-5952	58	2	,	,	PUNCT
ejpam-5952	58	3	there	there	PRON
ejpam-5952	58	4	exists	exist	VERB
ejpam-5952	58	5	a	a	DET
ejpam-5952	58	6	point	point	NOUN
ejpam-5952	58	7	υ∗	υ∗	NOUN
ejpam-5952	58	8	∈	∈	PROPN
ejpam-5952	58	9	x	x	PUNCT
ejpam-5952	58	10	such	such	ADJ
ejpam-5952	58	11	that	that	DET
ejpam-5952	58	12	υn	υn	NOUN
ejpam-5952	58	13	→	→	SYM
ejpam-5952	58	14	υ∗	υ∗	NOUN
ejpam-5952	58	15	as	as	ADP
ejpam-5952	58	16	n	n	PROPN
ejpam-5952	58	17	→	→	SYM
ejpam-5952	58	18	∞.	∞.	PROPN
ejpam-5952	58	19	next	next	ADV
ejpam-5952	58	20	,	,	PUNCT
ejpam-5952	58	21	we	we	PRON
ejpam-5952	58	22	show	show	VERB
ejpam-5952	58	23	that	that	SCONJ
ejpam-5952	58	24	υ∗	υ∗	NOUN
ejpam-5952	58	25	is	be	AUX
ejpam-5952	58	26	a	a	DET
ejpam-5952	58	27	fixed	fix	VERB
ejpam-5952	58	28	point	point	NOUN
ejpam-5952	58	29	of	of	ADP
ejpam-5952	58	30	t	t	PROPN
ejpam-5952	58	31	.	.	PUNCT
ejpam-5952	59	1	since	since	SCONJ
ejpam-5952	59	2	t	t	PROPN
ejpam-5952	59	3	is	be	AUX
ejpam-5952	59	4	continuous	continuous	ADJ
ejpam-5952	59	5	and	and	CCONJ
ejpam-5952	59	6	υn	υn	NOUN
ejpam-5952	59	7	→	→	SYM
ejpam-5952	59	8	υ∗	υ∗	NOUN
ejpam-5952	59	9	,	,	PUNCT
ejpam-5952	59	10	we	we	PRON
ejpam-5952	59	11	can	can	AUX
ejpam-5952	59	12	take	take	VERB
ejpam-5952	59	13	the	the	DET
ejpam-5952	59	14	limit	limit	NOUN
ejpam-5952	59	15	of	of	ADP
ejpam-5952	59	16	both	both	DET
ejpam-5952	59	17	sides	side	NOUN
ejpam-5952	59	18	of	of	ADP
ejpam-5952	59	19	the	the	DET
ejpam-5952	59	20	equation	equation	NOUN
ejpam-5952	59	21	υn+1	υn+1	NUM
ejpam-5952	59	22	=	=	SYM
ejpam-5952	59	23	t	t	PROPN
ejpam-5952	59	24	(	(	PUNCT
ejpam-5952	59	25	υn	υn	NOUN
ejpam-5952	59	26	)	)	PUNCT
ejpam-5952	59	27	.	.	PUNCT
ejpam-5952	60	1	as	as	ADP
ejpam-5952	60	2	n	n	NUM
ejpam-5952	60	3	→	→	SYM
ejpam-5952	60	4	∞	∞	PROPN
ejpam-5952	60	5	,	,	PUNCT
ejpam-5952	60	6	we	we	PRON
ejpam-5952	60	7	get	get	VERB
ejpam-5952	60	8	:	:	PUNCT
ejpam-5952	60	9	υ∗	υ∗	NOUN
ejpam-5952	60	10	=	=	SYM
ejpam-5952	60	11	t	t	PROPN
ejpam-5952	60	12	(	(	PUNCT
ejpam-5952	60	13	υ∗	υ∗	PROPN
ejpam-5952	60	14	)	)	PUNCT
ejpam-5952	60	15	.	.	PUNCT
ejpam-5952	61	1	thus	thus	ADV
ejpam-5952	61	2	,	,	PUNCT
ejpam-5952	61	3	υ∗	υ∗	NOUN
ejpam-5952	61	4	is	be	AUX
ejpam-5952	61	5	a	a	DET
ejpam-5952	61	6	fixed	fix	VERB
ejpam-5952	61	7	point	point	NOUN
ejpam-5952	61	8	of	of	ADP
ejpam-5952	61	9	t	t	PROPN
ejpam-5952	61	10	.	.	PUNCT
ejpam-5952	62	1	to	to	PART
ejpam-5952	62	2	prove	prove	VERB
ejpam-5952	62	3	uniqueness	uniqueness	NOUN
ejpam-5952	62	4	,	,	PUNCT
ejpam-5952	62	5	suppose	suppose	VERB
ejpam-5952	62	6	that	that	SCONJ
ejpam-5952	62	7	t	t	PROPN
ejpam-5952	62	8	has	have	VERB
ejpam-5952	62	9	two	two	NUM
ejpam-5952	62	10	fixed	fix	VERB
ejpam-5952	62	11	points	point	NOUN
ejpam-5952	62	12	υ∗	υ∗	NOUN
ejpam-5952	62	13	and	and	CCONJ
ejpam-5952	62	14	υ∗∗	υ∗∗	NOUN
ejpam-5952	62	15	,	,	PUNCT
ejpam-5952	62	16	meaning	meaning	NOUN
ejpam-5952	62	17	t	t	X
ejpam-5952	62	18	(	(	PUNCT
ejpam-5952	62	19	υ∗	υ∗	NOUN
ejpam-5952	62	20	)	)	PUNCT
ejpam-5952	62	21	=	=	SYM
ejpam-5952	63	1	υ∗	υ∗	NOUN
ejpam-5952	63	2	and	and	CCONJ
ejpam-5952	63	3	t	t	PROPN
ejpam-5952	63	4	(	(	PUNCT
ejpam-5952	63	5	υ∗∗	υ∗∗	PROPN
ejpam-5952	63	6	)	)	PUNCT
ejpam-5952	64	1	=	=	NOUN
ejpam-5952	64	2	υ∗∗.	υ∗∗.	ADP
ejpam-5952	64	3	using	use	VERB
ejpam-5952	64	4	the	the	DET
ejpam-5952	64	5	contraction	contraction	NOUN
ejpam-5952	64	6	condition	condition	NOUN
ejpam-5952	64	7	for	for	ADP
ejpam-5952	64	8	υ∗	υ∗	NOUN
ejpam-5952	64	9	and	and	CCONJ
ejpam-5952	64	10	υ∗∗	υ∗∗	NOUN
ejpam-5952	64	11	,	,	PUNCT
ejpam-5952	64	12	we	we	PRON
ejpam-5952	64	13	have	have	VERB
ejpam-5952	64	14	:	:	PUNCT
ejpam-5952	64	15	m(t	m(t	NOUN
ejpam-5952	64	16	(	(	PUNCT
ejpam-5952	64	17	υ∗	υ∗	PROPN
ejpam-5952	64	18	)	)	PUNCT
ejpam-5952	64	19	,	,	PUNCT
ejpam-5952	64	20	t	t	PROPN
ejpam-5952	64	21	(	(	PUNCT
ejpam-5952	64	22	υ∗∗	υ∗∗	PROPN
ejpam-5952	64	23	)	)	PUNCT
ejpam-5952	64	24	,	,	PUNCT
ejpam-5952	64	25	t	t	PROPN
ejpam-5952	64	26	(	(	PUNCT
ejpam-5952	64	27	υ∗∗	υ∗∗	PROPN
ejpam-5952	64	28	)	)	PUNCT
ejpam-5952	64	29	)	)	PUNCT
ejpam-5952	64	30	≤	≤	NUM
ejpam-5952	64	31	αm(υ∗	αm(υ∗	NUM
ejpam-5952	64	32	,	,	PUNCT
ejpam-5952	64	33	υ∗∗	υ∗∗	X
ejpam-5952	64	34	,	,	PUNCT
ejpam-5952	64	35	υ∗∗	υ∗∗	NOUN
ejpam-5952	64	36	)	)	PUNCT
ejpam-5952	64	37	.	.	PUNCT
ejpam-5952	65	1	since	since	SCONJ
ejpam-5952	65	2	t	t	PROPN
ejpam-5952	65	3	(	(	PUNCT
ejpam-5952	65	4	υ∗	υ∗	NOUN
ejpam-5952	65	5	)	)	PUNCT
ejpam-5952	65	6	=	=	SYM
ejpam-5952	65	7	υ∗	υ∗	NOUN
ejpam-5952	65	8	and	and	CCONJ
ejpam-5952	65	9	t	t	PROPN
ejpam-5952	65	10	(	(	PUNCT
ejpam-5952	65	11	υ∗∗	υ∗∗	PROPN
ejpam-5952	65	12	)	)	PUNCT
ejpam-5952	65	13	=	=	PUNCT
ejpam-5952	65	14	υ∗∗	υ∗∗	NOUN
ejpam-5952	65	15	,	,	PUNCT
ejpam-5952	65	16	this	this	DET
ejpam-5952	65	17	simplifies	simplifie	NOUN
ejpam-5952	65	18	to	to	PART
ejpam-5952	65	19	:	:	PUNCT
ejpam-5952	65	20	m(υ∗	m(υ∗	ADV
ejpam-5952	65	21	,	,	PUNCT
ejpam-5952	65	22	υ∗∗	υ∗∗	VERB
ejpam-5952	65	23	,	,	PUNCT
ejpam-5952	65	24	υ∗∗	υ∗∗	X
ejpam-5952	65	25	)	)	PUNCT
ejpam-5952	65	26	≤	≤	NOUN
ejpam-5952	65	27	αm(υ∗	αm(υ∗	NUM
ejpam-5952	65	28	,	,	PUNCT
ejpam-5952	65	29	υ∗∗	υ∗∗	X
ejpam-5952	65	30	,	,	PUNCT
ejpam-5952	65	31	υ∗∗	υ∗∗	NOUN
ejpam-5952	65	32	)	)	PUNCT
ejpam-5952	65	33	.	.	PUNCT
ejpam-5952	66	1	a.	a.	NOUN
ejpam-5952	66	2	a.	a.	PROPN
ejpam-5952	66	3	m.	m.	PROPN
ejpam-5952	66	4	malkawi	malkawi	ADP
ejpam-5952	66	5	/	/	SYM
ejpam-5952	66	6	eur	eur	PROPN
ejpam-5952	66	7	.	.	PUNCT
ejpam-5952	67	1	j.	j.	PROPN
ejpam-5952	67	2	pure	pure	PROPN
ejpam-5952	67	3	appl	appl	PROPN
ejpam-5952	67	4	.	.	PROPN
ejpam-5952	67	5	math	math	PROPN
ejpam-5952	67	6	,	,	PUNCT
ejpam-5952	67	7	18	18	NUM
ejpam-5952	67	8	(	(	PUNCT
ejpam-5952	67	9	2	2	NUM
ejpam-5952	67	10	)	)	PUNCT
ejpam-5952	67	11	(	(	PUNCT
ejpam-5952	67	12	2025	2025	NUM
ejpam-5952	67	13	)	)	PUNCT
ejpam-5952	67	14	,	,	PUNCT
ejpam-5952	67	15	5952	5952	NUM
ejpam-5952	67	16	4	4	NUM
ejpam-5952	67	17	of	of	ADP
ejpam-5952	67	18	14	14	NUM
ejpam-5952	67	19	because	because	SCONJ
ejpam-5952	67	20	α	α	PROPN
ejpam-5952	67	21	<	<	X
ejpam-5952	67	22	1	1	NUM
ejpam-5952	67	23	/	/	SYM
ejpam-5952	67	24	r	r	NOUN
ejpam-5952	67	25	,	,	PUNCT
ejpam-5952	67	26	it	it	PRON
ejpam-5952	67	27	follows	follow	VERB
ejpam-5952	67	28	that	that	SCONJ
ejpam-5952	67	29	the	the	DET
ejpam-5952	67	30	right	right	ADJ
ejpam-5952	67	31	-	-	PUNCT
ejpam-5952	67	32	hand	hand	NOUN
ejpam-5952	67	33	side	side	NOUN
ejpam-5952	67	34	is	be	AUX
ejpam-5952	67	35	strictly	strictly	ADV
ejpam-5952	67	36	smaller	small	ADJ
ejpam-5952	67	37	than	than	ADP
ejpam-5952	67	38	the	the	DET
ejpam-5952	67	39	left	left	ADJ
ejpam-5952	67	40	-	-	PUNCT
ejpam-5952	67	41	hand	hand	NOUN
ejpam-5952	67	42	side	side	NOUN
ejpam-5952	67	43	,	,	PUNCT
ejpam-5952	67	44	implying	imply	VERB
ejpam-5952	67	45	that	that	SCONJ
ejpam-5952	67	46	:	:	PUNCT
ejpam-5952	67	47	m(υ∗	m(υ∗	X
ejpam-5952	67	48	,	,	PUNCT
ejpam-5952	67	49	υ∗∗	υ∗∗	X
ejpam-5952	67	50	,	,	PUNCT
ejpam-5952	67	51	υ∗∗	υ∗∗	X
ejpam-5952	67	52	)	)	PUNCT
ejpam-5952	67	53	=	=	SYM
ejpam-5952	67	54	0	0	X
ejpam-5952	67	55	.	.	PUNCT
ejpam-5952	68	1	this	this	DET
ejpam-5952	68	2	equation	equation	NOUN
ejpam-5952	68	3	implies	imply	VERB
ejpam-5952	68	4	that	that	DET
ejpam-5952	68	5	υ∗	υ∗	NOUN
ejpam-5952	68	6	=	=	PUNCT
ejpam-5952	68	7	υ∗∗	υ∗∗	NOUN
ejpam-5952	68	8	,	,	PUNCT
ejpam-5952	68	9	establishing	establish	VERB
ejpam-5952	68	10	the	the	DET
ejpam-5952	68	11	uniqueness	uniqueness	NOUN
ejpam-5952	68	12	of	of	ADP
ejpam-5952	68	13	the	the	DET
ejpam-5952	68	14	fixed	fix	VERB
ejpam-5952	68	15	point	point	NOUN
ejpam-5952	68	16	.	.	PUNCT
ejpam-5952	69	1	therefore	therefore	ADV
ejpam-5952	69	2	,	,	PUNCT
ejpam-5952	69	3	we	we	PRON
ejpam-5952	69	4	conclude	conclude	VERB
ejpam-5952	69	5	that	that	SCONJ
ejpam-5952	69	6	t	t	PROPN
ejpam-5952	69	7	has	have	VERB
ejpam-5952	69	8	a	a	DET
ejpam-5952	69	9	unique	unique	ADJ
ejpam-5952	69	10	fixed	fix	VERB
ejpam-5952	69	11	point	point	NOUN
ejpam-5952	69	12	υ∗	υ∗	NOUN
ejpam-5952	69	13	,	,	PUNCT
ejpam-5952	69	14	and	and	CCONJ
ejpam-5952	69	15	υn	υn	X
ejpam-5952	69	16	→	→	SYM
ejpam-5952	69	17	υ∗	υ∗	NOUN
ejpam-5952	69	18	as	as	ADP
ejpam-5952	69	19	n	n	PROPN
ejpam-5952	69	20	→	→	SYM
ejpam-5952	69	21	∞.	∞.	PROPN
ejpam-5952	69	22	example	example	NOUN
ejpam-5952	69	23	1	1	X
ejpam-5952	69	24	.	.	PUNCT
ejpam-5952	69	25	consider	consider	VERB
ejpam-5952	69	26	the	the	DET
ejpam-5952	69	27	mr	mr	PROPN
ejpam-5952	69	28	-	-	PUNCT
ejpam-5952	69	29	metric	metric	ADJ
ejpam-5952	69	30	space	space	NOUN
ejpam-5952	69	31	(	(	PUNCT
ejpam-5952	69	32	r	r	NOUN
ejpam-5952	69	33	,	,	PUNCT
ejpam-5952	69	34	m	m	NOUN
ejpam-5952	69	35	)	)	PUNCT
ejpam-5952	69	36	,	,	PUNCT
ejpam-5952	69	37	where	where	SCONJ
ejpam-5952	69	38	the	the	DET
ejpam-5952	69	39	mr	mr	PROPN
ejpam-5952	69	40	-	-	PUNCT
ejpam-5952	69	41	metric	metric	ADJ
ejpam-5952	69	42	m	m	NOUN
ejpam-5952	69	43	is	be	AUX
ejpam-5952	69	44	defined	define	VERB
ejpam-5952	69	45	as	as	ADP
ejpam-5952	69	46	:	:	PUNCT
ejpam-5952	69	47	m(υ	m(υ	PROPN
ejpam-5952	69	48	,	,	PUNCT
ejpam-5952	69	49	ξ,ℑ	ξ,ℑ	PROPN
ejpam-5952	69	50	)	)	PUNCT
ejpam-5952	70	1	=	=	SYM
ejpam-5952	70	2	|υ	|υ	NOUN
ejpam-5952	70	3	−	−	PROPN
ejpam-5952	70	4	ξ|+	ξ|+	PROPN
ejpam-5952	70	5	|ξ	|ξ	VERB
ejpam-5952	70	6	−ℑ|+	−ℑ|+	NOUN
ejpam-5952	70	7	|ℑ	|ℑ	NOUN
ejpam-5952	70	8	−	−	PROPN
ejpam-5952	70	9	υ|	υ|	PROPN
ejpam-5952	70	10	.	.	PUNCT
ejpam-5952	71	1	(	(	PUNCT
ejpam-5952	71	2	1	1	X
ejpam-5952	71	3	)	)	PUNCT
ejpam-5952	71	4	this	this	DET
ejpam-5952	71	5	metric	metric	NOUN
ejpam-5952	71	6	provides	provide	VERB
ejpam-5952	71	7	a	a	DET
ejpam-5952	71	8	way	way	NOUN
ejpam-5952	71	9	to	to	PART
ejpam-5952	71	10	measure	measure	VERB
ejpam-5952	71	11	the	the	DET
ejpam-5952	71	12	distance	distance	NOUN
ejpam-5952	71	13	between	between	ADP
ejpam-5952	71	14	three	three	NUM
ejpam-5952	71	15	points	point	NOUN
ejpam-5952	71	16	in	in	ADP
ejpam-5952	71	17	r.	r.	PROPN
ejpam-5952	71	18	let	let	VERB
ejpam-5952	71	19	the	the	DET
ejpam-5952	71	20	self	self	NOUN
ejpam-5952	71	21	-	-	PUNCT
ejpam-5952	71	22	mapping	mapping	NOUN
ejpam-5952	71	23	t	t	NOUN
ejpam-5952	71	24	:	:	PUNCT
ejpam-5952	71	25	r	r	NOUN
ejpam-5952	71	26	→	→	SYM
ejpam-5952	71	27	r	r	NOUN
ejpam-5952	71	28	be	be	AUX
ejpam-5952	71	29	defined	define	VERB
ejpam-5952	71	30	by	by	ADP
ejpam-5952	71	31	:	:	PUNCT
ejpam-5952	71	32	t	t	PROPN
ejpam-5952	71	33	(	(	PUNCT
ejpam-5952	71	34	υ	υ	NOUN
ejpam-5952	71	35	)	)	PUNCT
ejpam-5952	71	36	=	=	SYM
ejpam-5952	71	37	υ	υ	DET
ejpam-5952	71	38	2	2	NUM
ejpam-5952	71	39	.	.	PUNCT
ejpam-5952	72	1	(	(	PUNCT
ejpam-5952	72	2	2	2	X
ejpam-5952	72	3	)	)	PUNCT
ejpam-5952	72	4	this	this	DET
ejpam-5952	72	5	mapping	mapping	NOUN
ejpam-5952	72	6	divides	divide	VERB
ejpam-5952	72	7	any	any	DET
ejpam-5952	72	8	point	point	NOUN
ejpam-5952	72	9	in	in	ADP
ejpam-5952	72	10	r	r	NOUN
ejpam-5952	72	11	by	by	ADP
ejpam-5952	72	12	two	two	NUM
ejpam-5952	72	13	,	,	PUNCT
ejpam-5952	72	14	which	which	PRON
ejpam-5952	72	15	suggests	suggest	VERB
ejpam-5952	72	16	that	that	SCONJ
ejpam-5952	72	17	it	it	PRON
ejpam-5952	72	18	is	be	AUX
ejpam-5952	72	19	a	a	DET
ejpam-5952	72	20	contraction	contraction	NOUN
ejpam-5952	72	21	mapping	mapping	NOUN
ejpam-5952	72	22	.	.	PUNCT
ejpam-5952	73	1	to	to	PART
ejpam-5952	73	2	verify	verify	VERB
ejpam-5952	73	3	that	that	SCONJ
ejpam-5952	73	4	t	t	PROPN
ejpam-5952	73	5	satisfies	satisfy	VERB
ejpam-5952	73	6	the	the	DET
ejpam-5952	73	7	contraction	contraction	NOUN
ejpam-5952	73	8	condition	condition	NOUN
ejpam-5952	73	9	,	,	PUNCT
ejpam-5952	73	10	we	we	PRON
ejpam-5952	73	11	compute	compute	VERB
ejpam-5952	73	12	the	the	DET
ejpam-5952	73	13	mr	mr	PROPN
ejpam-5952	73	14	-	-	PUNCT
ejpam-5952	73	15	metric	metric	NOUN
ejpam-5952	73	16	between	between	ADP
ejpam-5952	73	17	tυ	tυ	X
ejpam-5952	73	18	,	,	PUNCT
ejpam-5952	73	19	tξ	tξ	ADP
ejpam-5952	73	20	,	,	PUNCT
ejpam-5952	73	21	and	and	CCONJ
ejpam-5952	73	22	tℑ.	tℑ.	NOUN
ejpam-5952	73	23	for	for	ADP
ejpam-5952	73	24	any	any	DET
ejpam-5952	73	25	υ	υ	NOUN
ejpam-5952	73	26	,	,	PUNCT
ejpam-5952	73	27	ξ,ℑ	ξ,ℑ	PROPN
ejpam-5952	73	28	∈	∈	PROPN
ejpam-5952	73	29	r	r	NOUN
ejpam-5952	73	30	,	,	PUNCT
ejpam-5952	73	31	we	we	PRON
ejpam-5952	73	32	have	have	VERB
ejpam-5952	73	33	:	:	PUNCT
ejpam-5952	73	34	m(tυ	m(tυ	NOUN
ejpam-5952	73	35	,	,	PUNCT
ejpam-5952	73	36	tξ	tξ	VERB
ejpam-5952	73	37	,	,	PUNCT
ejpam-5952	73	38	tℑ	tℑ	NOUN
ejpam-5952	73	39	)	)	PUNCT
ejpam-5952	74	1	=	=	SYM
ejpam-5952	74	2	∣∣∣∣υ2	∣∣∣∣υ2	PUNCT
ejpam-5952	75	1	−	−	ADP
ejpam-5952	75	2	ξ	ξ	SYM
ejpam-5952	75	3	2	2	NUM
ejpam-5952	75	4	∣∣∣∣+	∣∣∣∣+	NOUN
ejpam-5952	75	5	∣∣∣∣ξ2	∣∣∣∣ξ2	NUM
ejpam-5952	75	6	−	−	NOUN
ejpam-5952	75	7	ℑ	ℑ	NOUN
ejpam-5952	75	8	2	2	NUM
ejpam-5952	75	9	∣∣∣∣+	∣∣∣∣+	NOUN
ejpam-5952	75	10	∣∣∣∣ℑ2	∣∣∣∣ℑ2	NUM
ejpam-5952	75	11	−	−	PROPN
ejpam-5952	75	12	υ	υ	PROPN
ejpam-5952	75	13	2	2	NUM
ejpam-5952	75	14	∣∣∣∣	∣∣∣∣	NOUN
ejpam-5952	75	15	.	.	PUNCT
ejpam-5952	76	1	(	(	PUNCT
ejpam-5952	76	2	3	3	X
ejpam-5952	76	3	)	)	PUNCT
ejpam-5952	76	4	using	use	VERB
ejpam-5952	76	5	the	the	DET
ejpam-5952	76	6	definition	definition	NOUN
ejpam-5952	76	7	of	of	ADP
ejpam-5952	76	8	the	the	DET
ejpam-5952	76	9	mr	mr	PROPN
ejpam-5952	76	10	-	-	PUNCT
ejpam-5952	76	11	metric	metric	NOUN
ejpam-5952	76	12	,	,	PUNCT
ejpam-5952	76	13	this	this	PRON
ejpam-5952	76	14	becomes	become	VERB
ejpam-5952	76	15	:	:	PUNCT
ejpam-5952	76	16	m(tυ	m(tυ	NOUN
ejpam-5952	76	17	,	,	PUNCT
ejpam-5952	76	18	tξ	tξ	VERB
ejpam-5952	76	19	,	,	PUNCT
ejpam-5952	76	20	tℑ	tℑ	NOUN
ejpam-5952	76	21	)	)	PUNCT
ejpam-5952	76	22	=	=	SYM
ejpam-5952	76	23	1	1	NUM
ejpam-5952	76	24	2	2	NUM
ejpam-5952	76	25	m(υ	m(υ	PROPN
ejpam-5952	76	26	,	,	PUNCT
ejpam-5952	76	27	ξ,ℑ	ξ,ℑ	PROPN
ejpam-5952	76	28	)	)	PUNCT
ejpam-5952	76	29	.	.	PUNCT
ejpam-5952	77	1	(	(	PUNCT
ejpam-5952	77	2	4	4	X
ejpam-5952	77	3	)	)	PUNCT
ejpam-5952	77	4	since	since	SCONJ
ejpam-5952	77	5	α	α	NOUN
ejpam-5952	77	6	=	=	SYM
ejpam-5952	77	7	1	1	NUM
ejpam-5952	77	8	2	2	NUM
ejpam-5952	77	9	,	,	PUNCT
ejpam-5952	77	10	and	and	CCONJ
ejpam-5952	77	11	we	we	PRON
ejpam-5952	77	12	can	can	AUX
ejpam-5952	77	13	choose	choose	VERB
ejpam-5952	77	14	r	r	NOUN
ejpam-5952	77	15	>	>	X
ejpam-5952	77	16	2	2	NUM
ejpam-5952	77	17	so	so	SCONJ
ejpam-5952	77	18	that	that	SCONJ
ejpam-5952	77	19	α	α	PRON
ejpam-5952	77	20	<	<	X
ejpam-5952	77	21	1	1	NUM
ejpam-5952	77	22	r	r	NOUN
ejpam-5952	77	23	,	,	PUNCT
ejpam-5952	77	24	we	we	PRON
ejpam-5952	77	25	conclude	conclude	VERB
ejpam-5952	77	26	that	that	SCONJ
ejpam-5952	77	27	the	the	DET
ejpam-5952	77	28	contraction	contraction	NOUN
ejpam-5952	77	29	condition	condition	NOUN
ejpam-5952	77	30	is	be	AUX
ejpam-5952	77	31	satisfied	satisfied	ADJ
ejpam-5952	77	32	.	.	PUNCT
ejpam-5952	78	1	by	by	ADP
ejpam-5952	78	2	theorem	theorem	NOUN
ejpam-5952	78	3	2.1	2.1	NUM
ejpam-5952	78	4	,	,	PUNCT
ejpam-5952	78	5	which	which	PRON
ejpam-5952	78	6	guarantees	guarantee	VERB
ejpam-5952	78	7	the	the	DET
ejpam-5952	78	8	existence	existence	NOUN
ejpam-5952	78	9	and	and	CCONJ
ejpam-5952	78	10	uniqueness	uniqueness	NOUN
ejpam-5952	78	11	of	of	ADP
ejpam-5952	78	12	a	a	DET
ejpam-5952	78	13	fixed	fix	VERB
ejpam-5952	78	14	point	point	NOUN
ejpam-5952	78	15	for	for	ADP
ejpam-5952	78	16	a	a	DET
ejpam-5952	78	17	contraction	contraction	NOUN
ejpam-5952	78	18	mapping	mapping	NOUN
ejpam-5952	78	19	on	on	ADP
ejpam-5952	78	20	a	a	DET
ejpam-5952	78	21	complete	complete	ADJ
ejpam-5952	78	22	mr	mr	ADJ
ejpam-5952	78	23	-	-	PUNCT
ejpam-5952	78	24	metric	metric	ADJ
ejpam-5952	78	25	space	space	NOUN
ejpam-5952	78	26	,	,	PUNCT
ejpam-5952	78	27	we	we	PRON
ejpam-5952	78	28	conclude	conclude	VERB
ejpam-5952	78	29	that	that	SCONJ
ejpam-5952	78	30	there	there	PRON
ejpam-5952	78	31	exists	exist	VERB
ejpam-5952	78	32	a	a	DET
ejpam-5952	78	33	unique	unique	ADJ
ejpam-5952	78	34	fixed	fix	VERB
ejpam-5952	78	35	point	point	NOUN
ejpam-5952	78	36	υ∗	υ∗	NOUN
ejpam-5952	78	37	∈	∈	PROPN
ejpam-5952	78	38	r	r	NOUN
ejpam-5952	78	39	such	such	ADJ
ejpam-5952	78	40	that	that	PRON
ejpam-5952	78	41	:	:	PUNCT
ejpam-5952	78	42	t	t	PROPN
ejpam-5952	78	43	(	(	PUNCT
ejpam-5952	78	44	υ∗	υ∗	PROPN
ejpam-5952	78	45	)	)	PUNCT
ejpam-5952	78	46	=	=	PRON
ejpam-5952	78	47	υ∗.	υ∗.	PROPN
ejpam-5952	78	48	(	(	PUNCT
ejpam-5952	78	49	5	5	X
ejpam-5952	78	50	)	)	PUNCT
ejpam-5952	78	51	solving	solve	VERB
ejpam-5952	78	52	the	the	DET
ejpam-5952	78	53	equation	equation	NOUN
ejpam-5952	78	54	υ∗	υ∗	NOUN
ejpam-5952	78	55	=	=	PUNCT
ejpam-5952	78	56	υ∗	υ∗	NOUN
ejpam-5952	78	57	2	2	NUM
ejpam-5952	78	58	,	,	PUNCT
ejpam-5952	78	59	we	we	PRON
ejpam-5952	78	60	obtain	obtain	VERB
ejpam-5952	78	61	:	:	PUNCT
ejpam-5952	78	62	υ∗	υ∗	NOUN
ejpam-5952	78	63	=	=	SYM
ejpam-5952	78	64	0	0	NUM
ejpam-5952	78	65	,	,	PUNCT
ejpam-5952	78	66	(	(	PUNCT
ejpam-5952	78	67	6	6	NUM
ejpam-5952	78	68	)	)	PUNCT
ejpam-5952	78	69	which	which	PRON
ejpam-5952	78	70	confirms	confirm	VERB
ejpam-5952	78	71	that	that	SCONJ
ejpam-5952	78	72	the	the	DET
ejpam-5952	78	73	unique	unique	ADJ
ejpam-5952	78	74	fixed	fix	VERB
ejpam-5952	78	75	point	point	NOUN
ejpam-5952	78	76	of	of	ADP
ejpam-5952	78	77	t	t	PROPN
ejpam-5952	78	78	is	be	AUX
ejpam-5952	78	79	υ∗	υ∗	NOUN
ejpam-5952	78	80	=	=	SYM
ejpam-5952	78	81	0	0	X
ejpam-5952	78	82	.	.	PUNCT
ejpam-5952	78	83	theorem	theorem	NOUN
ejpam-5952	78	84	2	2	NUM
ejpam-5952	78	85	.	.	X
ejpam-5952	79	1	let	let	AUX
ejpam-5952	79	2	(	(	PUNCT
ejpam-5952	79	3	x	x	X
ejpam-5952	79	4	,	,	PUNCT
ejpam-5952	79	5	m	m	VERB
ejpam-5952	79	6	)	)	PUNCT
ejpam-5952	79	7	be	be	VERB
ejpam-5952	79	8	an	an	DET
ejpam-5952	79	9	mr	mr	ADJ
ejpam-5952	79	10	-	-	PUNCT
ejpam-5952	79	11	metric	metric	ADJ
ejpam-5952	79	12	space	space	NOUN
ejpam-5952	79	13	,	,	PUNCT
ejpam-5952	79	14	and	and	CCONJ
ejpam-5952	79	15	let	let	VERB
ejpam-5952	79	16	{	{	PUNCT
ejpam-5952	79	17	υn	υn	AUX
ejpam-5952	79	18	}	}	PUNCT
ejpam-5952	79	19	be	be	AUX
ejpam-5952	79	20	a	a	DET
ejpam-5952	79	21	sequence	sequence	NOUN
ejpam-5952	79	22	in	in	ADP
ejpam-5952	79	23	x	x	PUNCT
ejpam-5952	79	24	satisfying	satisfy	VERB
ejpam-5952	79	25	:	:	PUNCT
ejpam-5952	79	26	m(υn+1	m(υn+1	NUM
ejpam-5952	79	27	,	,	PUNCT
ejpam-5952	79	28	υn	υn	NOUN
ejpam-5952	79	29	,	,	PUNCT
ejpam-5952	79	30	υn−1	υn−1	PROPN
ejpam-5952	79	31	)	)	PUNCT
ejpam-5952	79	32	≤	≤	NUM
ejpam-5952	79	33	1	1	NUM
ejpam-5952	79	34	2	2	NUM
ejpam-5952	79	35	m(υn	m(υn	PROPN
ejpam-5952	79	36	,	,	PUNCT
ejpam-5952	79	37	υn−1	υn−1	ADJ
ejpam-5952	79	38	,	,	PUNCT
ejpam-5952	79	39	υn−2	υn−2	PROPN
ejpam-5952	79	40	)	)	PUNCT
ejpam-5952	79	41	.	.	PUNCT
ejpam-5952	80	1	(	(	PUNCT
ejpam-5952	80	2	7	7	NUM
ejpam-5952	80	3	)	)	PUNCT
ejpam-5952	80	4	then	then	ADV
ejpam-5952	80	5	,	,	PUNCT
ejpam-5952	80	6	{	{	PUNCT
ejpam-5952	80	7	υn	υn	NOUN
ejpam-5952	80	8	}	}	PUNCT
ejpam-5952	80	9	is	be	AUX
ejpam-5952	80	10	a	a	DET
ejpam-5952	80	11	cauchy	cauchy	ADJ
ejpam-5952	80	12	sequence	sequence	NOUN
ejpam-5952	80	13	and	and	CCONJ
ejpam-5952	80	14	converges	converge	VERB
ejpam-5952	80	15	to	to	ADP
ejpam-5952	80	16	some	some	DET
ejpam-5952	80	17	limit	limit	NOUN
ejpam-5952	81	1	υ∗	υ∗	NOUN
ejpam-5952	81	2	∈	∈	PROPN
ejpam-5952	81	3	x.	x.	NOUN
ejpam-5952	81	4	a.	a.	NOUN
ejpam-5952	81	5	a.	a.	PROPN
ejpam-5952	81	6	m.	m.	PROPN
ejpam-5952	82	1	malkawi	malkawi	ADP
ejpam-5952	82	2	/	/	SYM
ejpam-5952	82	3	eur	eur	PROPN
ejpam-5952	82	4	.	.	PUNCT
ejpam-5952	83	1	j.	j.	PROPN
ejpam-5952	83	2	pure	pure	PROPN
ejpam-5952	83	3	appl	appl	PROPN
ejpam-5952	83	4	.	.	PROPN
ejpam-5952	83	5	math	math	PROPN
ejpam-5952	83	6	,	,	PUNCT
ejpam-5952	83	7	18	18	NUM
ejpam-5952	83	8	(	(	PUNCT
ejpam-5952	83	9	2	2	NUM
ejpam-5952	83	10	)	)	PUNCT
ejpam-5952	83	11	(	(	PUNCT
ejpam-5952	83	12	2025	2025	NUM
ejpam-5952	83	13	)	)	PUNCT
ejpam-5952	83	14	,	,	PUNCT
ejpam-5952	83	15	5952	5952	NUM
ejpam-5952	83	16	5	5	NUM
ejpam-5952	83	17	of	of	ADP
ejpam-5952	83	18	14	14	NUM
ejpam-5952	83	19	proof	proof	NOUN
ejpam-5952	83	20	.	.	PUNCT
ejpam-5952	84	1	we	we	PRON
ejpam-5952	84	2	begin	begin	VERB
ejpam-5952	84	3	by	by	ADP
ejpam-5952	84	4	recursively	recursively	ADV
ejpam-5952	84	5	applying	apply	VERB
ejpam-5952	84	6	the	the	DET
ejpam-5952	84	7	given	give	VERB
ejpam-5952	84	8	inequality	inequality	NOUN
ejpam-5952	84	9	to	to	PART
ejpam-5952	84	10	estimate	estimate	VERB
ejpam-5952	84	11	the	the	DET
ejpam-5952	84	12	mrmetric	mrmetric	NOUN
ejpam-5952	84	13	between	between	ADP
ejpam-5952	84	14	terms	term	NOUN
ejpam-5952	84	15	of	of	ADP
ejpam-5952	84	16	the	the	DET
ejpam-5952	84	17	sequence	sequence	NOUN
ejpam-5952	84	18	.	.	PUNCT
ejpam-5952	85	1	specifically	specifically	ADV
ejpam-5952	85	2	,	,	PUNCT
ejpam-5952	85	3	using	use	VERB
ejpam-5952	85	4	the	the	DET
ejpam-5952	85	5	given	give	VERB
ejpam-5952	85	6	inequality	inequality	NOUN
ejpam-5952	85	7	,	,	PUNCT
ejpam-5952	85	8	m(υn+1	m(υn+1	NUM
ejpam-5952	85	9	,	,	PUNCT
ejpam-5952	85	10	υn	υn	NOUN
ejpam-5952	85	11	,	,	PUNCT
ejpam-5952	85	12	υn−1	υn−1	PROPN
ejpam-5952	85	13	)	)	PUNCT
ejpam-5952	85	14	≤	≤	NUM
ejpam-5952	85	15	1	1	NUM
ejpam-5952	85	16	2	2	NUM
ejpam-5952	85	17	m(υn	m(υn	PROPN
ejpam-5952	85	18	,	,	PUNCT
ejpam-5952	85	19	υn−1	υn−1	ADJ
ejpam-5952	85	20	,	,	PUNCT
ejpam-5952	85	21	υn−2	υn−2	PROPN
ejpam-5952	85	22	)	)	PUNCT
ejpam-5952	85	23	,	,	PUNCT
ejpam-5952	85	24	we	we	PRON
ejpam-5952	85	25	can	can	AUX
ejpam-5952	85	26	derive	derive	VERB
ejpam-5952	85	27	an	an	DET
ejpam-5952	85	28	upper	upper	ADJ
ejpam-5952	85	29	bound	bind	VERB
ejpam-5952	85	30	for	for	ADP
ejpam-5952	85	31	the	the	DET
ejpam-5952	85	32	mr	mr	NOUN
ejpam-5952	85	33	-	-	PUNCT
ejpam-5952	85	34	metric	metric	NOUN
ejpam-5952	85	35	for	for	ADP
ejpam-5952	85	36	the	the	DET
ejpam-5952	85	37	terms	term	NOUN
ejpam-5952	85	38	υn+k	υn+k	PROPN
ejpam-5952	85	39	,	,	PUNCT
ejpam-5952	85	40	υn+k−1	υn+k−1	PROPN
ejpam-5952	85	41	,	,	PUNCT
ejpam-5952	85	42	and	and	CCONJ
ejpam-5952	85	43	υn+k−2	υn+k−2	VERB
ejpam-5952	85	44	by	by	ADP
ejpam-5952	85	45	applying	apply	VERB
ejpam-5952	85	46	the	the	DET
ejpam-5952	85	47	inequality	inequality	NOUN
ejpam-5952	85	48	recursively	recursively	ADV
ejpam-5952	85	49	:	:	PUNCT
ejpam-5952	85	50	m(υn+k	m(υn+k	PROPN
ejpam-5952	85	51	,	,	PUNCT
ejpam-5952	85	52	υn+k−1	υn+k−1	PROPN
ejpam-5952	85	53	,	,	PUNCT
ejpam-5952	85	54	υn+k−2	υn+k−2	NOUN
ejpam-5952	85	55	)	)	PUNCT
ejpam-5952	85	56	≤	≤	NOUN
ejpam-5952	85	57	(	(	PUNCT
ejpam-5952	85	58	1	1	NUM
ejpam-5952	85	59	2	2	NUM
ejpam-5952	85	60	)	)	PUNCT
ejpam-5952	85	61	k	k	NOUN
ejpam-5952	85	62	m(υn	m(υn	PROPN
ejpam-5952	85	63	,	,	PUNCT
ejpam-5952	85	64	υn−1	υn−1	ADJ
ejpam-5952	85	65	,	,	PUNCT
ejpam-5952	85	66	υn−2	υn−2	PROPN
ejpam-5952	85	67	)	)	PUNCT
ejpam-5952	85	68	.	.	PUNCT
ejpam-5952	86	1	this	this	DET
ejpam-5952	86	2	inequality	inequality	NOUN
ejpam-5952	86	3	holds	hold	VERB
ejpam-5952	86	4	for	for	ADP
ejpam-5952	86	5	any	any	DET
ejpam-5952	86	6	k	k	PROPN
ejpam-5952	86	7	∈	∈	PROPN
ejpam-5952	86	8	n	n	CCONJ
ejpam-5952	86	9	,	,	PUNCT
ejpam-5952	86	10	and	and	CCONJ
ejpam-5952	86	11	as	as	ADP
ejpam-5952	86	12	k	k	PROPN
ejpam-5952	86	13	→	→	SYM
ejpam-5952	86	14	∞	∞	PROPN
ejpam-5952	86	15	,	,	PUNCT
ejpam-5952	86	16	the	the	DET
ejpam-5952	86	17	right	right	ADJ
ejpam-5952	86	18	-	-	PUNCT
ejpam-5952	86	19	hand	hand	NOUN
ejpam-5952	86	20	side	side	NOUN
ejpam-5952	86	21	of	of	ADP
ejpam-5952	86	22	the	the	DET
ejpam-5952	86	23	inequality	inequality	NOUN
ejpam-5952	86	24	tends	tend	VERB
ejpam-5952	86	25	to	to	ADP
ejpam-5952	86	26	zero	zero	NUM
ejpam-5952	86	27	because	because	SCONJ
ejpam-5952	86	28	(	(	PUNCT
ejpam-5952	86	29	1	1	NUM
ejpam-5952	86	30	2	2	NUM
ejpam-5952	86	31	)	)	PUNCT
ejpam-5952	86	32	k	k	PROPN
ejpam-5952	86	33	→	→	SYM
ejpam-5952	86	34	0	0	X
ejpam-5952	86	35	.	.	PUNCT
ejpam-5952	87	1	more	more	ADV
ejpam-5952	87	2	formally	formally	ADV
ejpam-5952	87	3	:	:	PUNCT
ejpam-5952	87	4	lim	lim	PROPN
ejpam-5952	87	5	k→∞	k→∞	PROPN
ejpam-5952	87	6	(	(	PUNCT
ejpam-5952	87	7	1	1	NUM
ejpam-5952	87	8	2	2	NUM
ejpam-5952	87	9	)	)	PUNCT
ejpam-5952	87	10	k	k	NOUN
ejpam-5952	87	11	=	=	PUNCT
ejpam-5952	87	12	0	0	PROPN
ejpam-5952	87	13	.	.	PUNCT
ejpam-5952	88	1	thus	thus	ADV
ejpam-5952	88	2	,	,	PUNCT
ejpam-5952	88	3	for	for	ADP
ejpam-5952	88	4	any	any	DET
ejpam-5952	88	5	ϵ	ϵ	X
ejpam-5952	88	6	>	>	X
ejpam-5952	88	7	0	0	NUM
ejpam-5952	88	8	,	,	PUNCT
ejpam-5952	88	9	there	there	PRON
ejpam-5952	88	10	exists	exist	VERB
ejpam-5952	88	11	an	an	DET
ejpam-5952	88	12	integer	integer	NOUN
ejpam-5952	88	13	n	n	CCONJ
ejpam-5952	88	14	such	such	ADJ
ejpam-5952	88	15	that	that	PRON
ejpam-5952	88	16	for	for	ADP
ejpam-5952	88	17	all	all	DET
ejpam-5952	88	18	k	k	PROPN
ejpam-5952	88	19	≥	≥	PROPN
ejpam-5952	88	20	n	n	PROPN
ejpam-5952	88	21	,	,	PUNCT
ejpam-5952	88	22	m(υn+k	m(υn+k	PROPN
ejpam-5952	88	23	,	,	PUNCT
ejpam-5952	88	24	υn+k−1	υn+k−1	PROPN
ejpam-5952	88	25	,	,	PUNCT
ejpam-5952	88	26	υn+k−2	υn+k−2	NOUN
ejpam-5952	88	27	)	)	PUNCT
ejpam-5952	88	28	<	<	X
ejpam-5952	88	29	ϵ.	ϵ.	NOUN
ejpam-5952	89	1	this	this	PRON
ejpam-5952	89	2	implies	imply	VERB
ejpam-5952	89	3	that	that	SCONJ
ejpam-5952	89	4	the	the	DET
ejpam-5952	89	5	sequence	sequence	NOUN
ejpam-5952	89	6	{	{	PUNCT
ejpam-5952	89	7	υn	υn	NOUN
ejpam-5952	89	8	}	}	PUNCT
ejpam-5952	89	9	is	be	AUX
ejpam-5952	89	10	cauchy	cauchy	ADJ
ejpam-5952	89	11	in	in	ADP
ejpam-5952	89	12	the	the	DET
ejpam-5952	89	13	mr	mr	PROPN
ejpam-5952	89	14	-	-	PUNCT
ejpam-5952	89	15	metric	metric	ADJ
ejpam-5952	89	16	space	space	NOUN
ejpam-5952	89	17	(	(	PUNCT
ejpam-5952	89	18	x	x	X
ejpam-5952	89	19	,	,	PUNCT
ejpam-5952	89	20	m	m	NOUN
ejpam-5952	89	21	)	)	PUNCT
ejpam-5952	89	22	,	,	PUNCT
ejpam-5952	89	23	since	since	SCONJ
ejpam-5952	89	24	for	for	ADP
ejpam-5952	89	25	any	any	DET
ejpam-5952	89	26	m	m	NOUN
ejpam-5952	89	27	,	,	PUNCT
ejpam-5952	89	28	n	n	PROPN
ejpam-5952	89	29	∈	∈	PROPN
ejpam-5952	89	30	n	n	CCONJ
ejpam-5952	89	31	,	,	PUNCT
ejpam-5952	89	32	the	the	DET
ejpam-5952	89	33	distance	distance	NOUN
ejpam-5952	89	34	between	between	ADP
ejpam-5952	89	35	terms	term	NOUN
ejpam-5952	89	36	in	in	ADP
ejpam-5952	89	37	the	the	DET
ejpam-5952	89	38	sequence	sequence	NOUN
ejpam-5952	89	39	can	can	AUX
ejpam-5952	89	40	be	be	AUX
ejpam-5952	89	41	made	make	VERB
ejpam-5952	89	42	arbitrarily	arbitrarily	ADV
ejpam-5952	89	43	small	small	ADJ
ejpam-5952	89	44	as	as	ADP
ejpam-5952	89	45	n	n	NUM
ejpam-5952	89	46	,	,	PUNCT
ejpam-5952	89	47	m	m	NOUN
ejpam-5952	89	48	increase	increase	NOUN
ejpam-5952	89	49	.	.	PUNCT
ejpam-5952	90	1	step	step	NOUN
ejpam-5952	90	2	1	1	NUM
ejpam-5952	90	3	:	:	PUNCT
ejpam-5952	90	4	cauchy	cauchy	NOUN
ejpam-5952	90	5	property	property	NOUN
ejpam-5952	90	6	to	to	PART
ejpam-5952	90	7	rigorously	rigorously	ADV
ejpam-5952	90	8	justify	justify	VERB
ejpam-5952	90	9	that	that	SCONJ
ejpam-5952	90	10	{	{	PUNCT
ejpam-5952	90	11	υn	υn	NOUN
ejpam-5952	90	12	}	}	PUNCT
ejpam-5952	90	13	is	be	AUX
ejpam-5952	90	14	a	a	DET
ejpam-5952	90	15	cauchy	cauchy	ADJ
ejpam-5952	90	16	sequence	sequence	NOUN
ejpam-5952	90	17	,	,	PUNCT
ejpam-5952	90	18	consider	consider	VERB
ejpam-5952	90	19	the	the	DET
ejpam-5952	90	20	mr	mr	NOUN
ejpam-5952	90	21	-	-	PUNCT
ejpam-5952	90	22	metric	metric	NOUN
ejpam-5952	90	23	for	for	ADP
ejpam-5952	90	24	the	the	DET
ejpam-5952	90	25	terms	term	NOUN
ejpam-5952	90	26	υn	υn	VERB
ejpam-5952	90	27	and	and	CCONJ
ejpam-5952	90	28	υm	υm	PROPN
ejpam-5952	90	29	.	.	PUNCT
ejpam-5952	91	1	we	we	PRON
ejpam-5952	91	2	have	have	VERB
ejpam-5952	91	3	:	:	PUNCT
ejpam-5952	91	4	m(υn	m(υn	ADJ
ejpam-5952	91	5	,	,	PUNCT
ejpam-5952	91	6	υm	υm	NOUN
ejpam-5952	91	7	,	,	PUNCT
ejpam-5952	91	8	υm−1	υm−1	PROPN
ejpam-5952	91	9	)	)	PUNCT
ejpam-5952	91	10	≤	≤	NOUN
ejpam-5952	91	11	(	(	PUNCT
ejpam-5952	91	12	1	1	NUM
ejpam-5952	91	13	2	2	NUM
ejpam-5952	91	14	)	)	PUNCT
ejpam-5952	91	15	|n−m|	|n−m|	PUNCT
ejpam-5952	92	1	m(υm	m(υm	NOUN
ejpam-5952	92	2	,	,	PUNCT
ejpam-5952	92	3	υm−1	υm−1	PROPN
ejpam-5952	92	4	,	,	PUNCT
ejpam-5952	92	5	υm−2	υm−2	NOUN
ejpam-5952	92	6	)	)	PUNCT
ejpam-5952	92	7	.	.	PUNCT
ejpam-5952	93	1	since	since	SCONJ
ejpam-5952	93	2	(	(	PUNCT
ejpam-5952	93	3	1	1	NUM
ejpam-5952	93	4	2	2	NUM
ejpam-5952	93	5	)	)	PUNCT
ejpam-5952	93	6	|n−m|	|n−m|	PUNCT
ejpam-5952	93	7	→	→	SYM
ejpam-5952	93	8	0	0	PUNCT
ejpam-5952	93	9	as	as	ADP
ejpam-5952	93	10	|n−m|	|n−m|	PUNCT
ejpam-5952	93	11	→	→	SYM
ejpam-5952	93	12	∞	∞	PROPN
ejpam-5952	93	13	,	,	PUNCT
ejpam-5952	93	14	we	we	PRON
ejpam-5952	93	15	conclude	conclude	VERB
ejpam-5952	93	16	that	that	SCONJ
ejpam-5952	93	17	m(υn	m(υn	PROPN
ejpam-5952	93	18	,	,	PUNCT
ejpam-5952	93	19	υm	υm	NOUN
ejpam-5952	93	20	,	,	PUNCT
ejpam-5952	93	21	υm−1	υm−1	PROPN
ejpam-5952	93	22	)	)	PUNCT
ejpam-5952	93	23	→	→	SYM
ejpam-5952	93	24	0	0	NUM
ejpam-5952	93	25	as	as	SCONJ
ejpam-5952	93	26	|n−m|	|n−m|	PUNCT
ejpam-5952	93	27	→	→	PUNCT
ejpam-5952	93	28	∞.	∞.	PROPN
ejpam-5952	93	29	this	this	PRON
ejpam-5952	93	30	shows	show	VERB
ejpam-5952	93	31	that	that	SCONJ
ejpam-5952	93	32	{	{	PUNCT
ejpam-5952	93	33	υn	υn	NOUN
ejpam-5952	93	34	}	}	PUNCT
ejpam-5952	93	35	is	be	AUX
ejpam-5952	93	36	cauchy	cauchy	NOUN
ejpam-5952	93	37	,	,	PUNCT
ejpam-5952	93	38	meaning	mean	VERB
ejpam-5952	93	39	that	that	SCONJ
ejpam-5952	93	40	the	the	DET
ejpam-5952	93	41	terms	term	NOUN
ejpam-5952	93	42	of	of	ADP
ejpam-5952	93	43	the	the	DET
ejpam-5952	93	44	sequence	sequence	NOUN
ejpam-5952	93	45	become	become	VERB
ejpam-5952	93	46	arbitrarily	arbitrarily	ADV
ejpam-5952	93	47	close	close	ADJ
ejpam-5952	93	48	to	to	ADP
ejpam-5952	93	49	each	each	DET
ejpam-5952	93	50	other	other	ADJ
ejpam-5952	93	51	as	as	ADP
ejpam-5952	93	52	n	n	NUM
ejpam-5952	93	53	and	and	CCONJ
ejpam-5952	93	54	m	m	VERB
ejpam-5952	93	55	increase	increase	NOUN
ejpam-5952	93	56	.	.	PUNCT
ejpam-5952	94	1	step	step	NOUN
ejpam-5952	94	2	2	2	NUM
ejpam-5952	94	3	:	:	PUNCT
ejpam-5952	94	4	convergence	convergence	NOUN
ejpam-5952	94	5	to	to	ADP
ejpam-5952	94	6	a	a	DET
ejpam-5952	94	7	limit	limit	NOUN
ejpam-5952	94	8	since	since	SCONJ
ejpam-5952	94	9	(	(	PUNCT
ejpam-5952	94	10	x	x	X
ejpam-5952	94	11	,	,	PUNCT
ejpam-5952	94	12	m	m	VERB
ejpam-5952	94	13	)	)	PUNCT
ejpam-5952	94	14	is	be	AUX
ejpam-5952	94	15	a	a	DET
ejpam-5952	94	16	complete	complete	ADJ
ejpam-5952	94	17	mr	mr	ADJ
ejpam-5952	94	18	-	-	PUNCT
ejpam-5952	94	19	metric	metric	ADJ
ejpam-5952	94	20	space	space	NOUN
ejpam-5952	94	21	,	,	PUNCT
ejpam-5952	94	22	every	every	DET
ejpam-5952	94	23	cauchy	cauchy	ADJ
ejpam-5952	94	24	sequence	sequence	NOUN
ejpam-5952	94	25	converges	converge	VERB
ejpam-5952	94	26	to	to	ADP
ejpam-5952	94	27	a	a	DET
ejpam-5952	94	28	limit	limit	NOUN
ejpam-5952	94	29	.	.	PUNCT
ejpam-5952	95	1	therefore	therefore	ADV
ejpam-5952	95	2	,	,	PUNCT
ejpam-5952	95	3	the	the	DET
ejpam-5952	95	4	sequence	sequence	NOUN
ejpam-5952	95	5	{	{	PUNCT
ejpam-5952	95	6	υn	υn	NOUN
ejpam-5952	95	7	}	}	PUNCT
ejpam-5952	95	8	converges	converge	NOUN
ejpam-5952	95	9	to	to	ADP
ejpam-5952	95	10	some	some	DET
ejpam-5952	95	11	point	point	NOUN
ejpam-5952	95	12	υ∗	υ∗	NOUN
ejpam-5952	95	13	∈	∈	PROPN
ejpam-5952	95	14	x.	x.	NOUN
ejpam-5952	95	15	in	in	ADP
ejpam-5952	95	16	other	other	ADJ
ejpam-5952	95	17	words	word	NOUN
ejpam-5952	95	18	,	,	PUNCT
ejpam-5952	95	19	there	there	PRON
ejpam-5952	95	20	exists	exist	VERB
ejpam-5952	95	21	υ∗	υ∗	NOUN
ejpam-5952	95	22	∈	∈	PROPN
ejpam-5952	95	23	x	x	PUNCT
ejpam-5952	95	24	such	such	ADJ
ejpam-5952	95	25	that	that	PRON
ejpam-5952	95	26	:	:	PUNCT
ejpam-5952	95	27	υn	υn	X
ejpam-5952	95	28	→	→	SYM
ejpam-5952	95	29	υ∗	υ∗	NOUN
ejpam-5952	95	30	as	as	ADP
ejpam-5952	95	31	n	n	PROPN
ejpam-5952	95	32	→	→	SYM
ejpam-5952	95	33	∞.	∞.	PROPN
ejpam-5952	95	34	thus	thus	ADV
ejpam-5952	95	35	,	,	PUNCT
ejpam-5952	95	36	we	we	PRON
ejpam-5952	95	37	have	have	AUX
ejpam-5952	95	38	shown	show	VERB
ejpam-5952	95	39	that	that	SCONJ
ejpam-5952	95	40	the	the	DET
ejpam-5952	95	41	sequence	sequence	NOUN
ejpam-5952	95	42	{	{	PUNCT
ejpam-5952	95	43	υn	υn	NOUN
ejpam-5952	95	44	}	}	PUNCT
ejpam-5952	95	45	converges	converge	NOUN
ejpam-5952	95	46	to	to	ADP
ejpam-5952	95	47	a	a	DET
ejpam-5952	95	48	limit	limit	NOUN
ejpam-5952	95	49	υ∗.	υ∗.	NOUN
ejpam-5952	95	50	conclusion	conclusion	NOUN
ejpam-5952	95	51	:	:	PUNCT
ejpam-5952	95	52	the	the	DET
ejpam-5952	95	53	sequence	sequence	NOUN
ejpam-5952	95	54	{	{	PUNCT
ejpam-5952	95	55	υn	υn	NOUN
ejpam-5952	95	56	}	}	PUNCT
ejpam-5952	95	57	is	be	AUX
ejpam-5952	95	58	cauchy	cauchy	NOUN
ejpam-5952	95	59	,	,	PUNCT
ejpam-5952	95	60	and	and	CCONJ
ejpam-5952	95	61	by	by	ADP
ejpam-5952	95	62	the	the	DET
ejpam-5952	95	63	completeness	completeness	NOUN
ejpam-5952	95	64	of	of	ADP
ejpam-5952	95	65	the	the	DET
ejpam-5952	95	66	mr	mr	PROPN
ejpam-5952	95	67	-	-	PUNCT
ejpam-5952	95	68	metric	metric	ADJ
ejpam-5952	95	69	space	space	NOUN
ejpam-5952	95	70	(	(	PUNCT
ejpam-5952	95	71	x	x	X
ejpam-5952	95	72	,	,	PUNCT
ejpam-5952	95	73	m	m	PROPN
ejpam-5952	95	74	)	)	PUNCT
ejpam-5952	95	75	,	,	PUNCT
ejpam-5952	95	76	it	it	PRON
ejpam-5952	95	77	converges	converge	VERB
ejpam-5952	95	78	to	to	ADP
ejpam-5952	95	79	a	a	DET
ejpam-5952	95	80	limit	limit	NOUN
ejpam-5952	95	81	υ∗	υ∗	NOUN
ejpam-5952	95	82	∈	∈	PROPN
ejpam-5952	95	83	x.	x.	NOUN
ejpam-5952	95	84	therefore	therefore	ADV
ejpam-5952	95	85	,	,	PUNCT
ejpam-5952	95	86	the	the	DET
ejpam-5952	95	87	theorem	theorem	NOUN
ejpam-5952	95	88	is	be	AUX
ejpam-5952	95	89	proved	prove	VERB
ejpam-5952	95	90	.	.	PUNCT
ejpam-5952	96	1	a.	a.	NOUN
ejpam-5952	96	2	a.	a.	PROPN
ejpam-5952	96	3	m.	m.	PROPN
ejpam-5952	96	4	malkawi	malkawi	ADP
ejpam-5952	96	5	/	/	SYM
ejpam-5952	96	6	eur	eur	PROPN
ejpam-5952	96	7	.	.	PUNCT
ejpam-5952	97	1	j.	j.	PROPN
ejpam-5952	97	2	pure	pure	PROPN
ejpam-5952	97	3	appl	appl	PROPN
ejpam-5952	97	4	.	.	PROPN
ejpam-5952	97	5	math	math	PROPN
ejpam-5952	97	6	,	,	PUNCT
ejpam-5952	97	7	18	18	NUM
ejpam-5952	97	8	(	(	PUNCT
ejpam-5952	97	9	2	2	NUM
ejpam-5952	97	10	)	)	PUNCT
ejpam-5952	97	11	(	(	PUNCT
ejpam-5952	97	12	2025	2025	NUM
ejpam-5952	97	13	)	)	PUNCT
ejpam-5952	97	14	,	,	PUNCT
ejpam-5952	97	15	5952	5952	NUM
ejpam-5952	97	16	6	6	NUM
ejpam-5952	97	17	of	of	ADP
ejpam-5952	97	18	14	14	NUM
ejpam-5952	97	19	example	example	NOUN
ejpam-5952	97	20	2	2	NUM
ejpam-5952	97	21	.	.	X
ejpam-5952	97	22	consider	consider	VERB
ejpam-5952	97	23	the	the	DET
ejpam-5952	97	24	mr	mr	PROPN
ejpam-5952	97	25	-	-	PUNCT
ejpam-5952	97	26	metric	metric	ADJ
ejpam-5952	97	27	space	space	NOUN
ejpam-5952	97	28	(	(	PUNCT
ejpam-5952	97	29	r	r	NOUN
ejpam-5952	97	30	,	,	PUNCT
ejpam-5952	97	31	m	m	NOUN
ejpam-5952	97	32	)	)	PUNCT
ejpam-5952	97	33	,	,	PUNCT
ejpam-5952	97	34	where	where	SCONJ
ejpam-5952	97	35	the	the	DET
ejpam-5952	97	36	mr	mr	PROPN
ejpam-5952	97	37	-	-	PUNCT
ejpam-5952	97	38	metric	metric	ADJ
ejpam-5952	97	39	m	m	NOUN
ejpam-5952	97	40	is	be	AUX
ejpam-5952	97	41	defined	define	VERB
ejpam-5952	97	42	as	as	ADP
ejpam-5952	97	43	:	:	PUNCT
ejpam-5952	97	44	m(υ	m(υ	PROPN
ejpam-5952	97	45	,	,	PUNCT
ejpam-5952	97	46	ξ,ℑ	ξ,ℑ	PROPN
ejpam-5952	97	47	)	)	PUNCT
ejpam-5952	97	48	=	=	SYM
ejpam-5952	97	49	|υ	|υ	NOUN
ejpam-5952	97	50	−	−	PROPN
ejpam-5952	97	51	ξ|+	ξ|+	PROPN
ejpam-5952	97	52	|ξ	|ξ	VERB
ejpam-5952	97	53	−ℑ|+	−ℑ|+	NOUN
ejpam-5952	97	54	|ℑ	|ℑ	NOUN
ejpam-5952	97	55	−	−	PROPN
ejpam-5952	97	56	υ|	υ|	PROPN
ejpam-5952	97	57	.	.	PUNCT
ejpam-5952	98	1	(	(	PUNCT
ejpam-5952	98	2	8)	8)	NUM
ejpam-5952	98	3	define	define	VERB
ejpam-5952	98	4	the	the	DET
ejpam-5952	98	5	sequence	sequence	NOUN
ejpam-5952	98	6	{	{	PUNCT
ejpam-5952	98	7	υn	υn	NOUN
ejpam-5952	98	8	}	}	PUNCT
ejpam-5952	98	9	in	in	ADP
ejpam-5952	98	10	r	r	NOUN
ejpam-5952	98	11	by	by	ADP
ejpam-5952	98	12	:	:	PUNCT
ejpam-5952	98	13	υn	υn	NOUN
ejpam-5952	98	14	=	=	SYM
ejpam-5952	98	15	1	1	NUM
ejpam-5952	98	16	2n	2n	NUM
ejpam-5952	98	17	.	.	PUNCT
ejpam-5952	99	1	(	(	PUNCT
ejpam-5952	99	2	9	9	X
ejpam-5952	99	3	)	)	PUNCT
ejpam-5952	99	4	we	we	PRON
ejpam-5952	99	5	need	need	VERB
ejpam-5952	99	6	to	to	PART
ejpam-5952	99	7	verify	verify	VERB
ejpam-5952	99	8	that	that	SCONJ
ejpam-5952	99	9	the	the	DET
ejpam-5952	99	10	sequence	sequence	NOUN
ejpam-5952	99	11	satisfies	satisfy	VERB
ejpam-5952	99	12	the	the	DET
ejpam-5952	99	13	condition	condition	NOUN
ejpam-5952	99	14	of	of	ADP
ejpam-5952	99	15	the	the	DET
ejpam-5952	99	16	theorem	theorem	NOUN
ejpam-5952	99	17	.	.	PUNCT
ejpam-5952	100	1	first	first	ADV
ejpam-5952	100	2	,	,	PUNCT
ejpam-5952	100	3	we	we	PRON
ejpam-5952	100	4	compute	compute	VERB
ejpam-5952	100	5	the	the	DET
ejpam-5952	100	6	mr	mr	PROPN
ejpam-5952	100	7	-	-	PUNCT
ejpam-5952	100	8	metric	metric	ADJ
ejpam-5952	100	9	m(υn+1	m(υn+1	PROPN
ejpam-5952	100	10	,	,	PUNCT
ejpam-5952	100	11	υn	υn	NOUN
ejpam-5952	100	12	,	,	PUNCT
ejpam-5952	100	13	υn−1	υn−1	PROPN
ejpam-5952	100	14	):	):	PUNCT
ejpam-5952	100	15	m(υn+1	m(υn+1	NUM
ejpam-5952	100	16	,	,	PUNCT
ejpam-5952	100	17	υn	υn	NOUN
ejpam-5952	100	18	,	,	PUNCT
ejpam-5952	100	19	υn−1	υn−1	PROPN
ejpam-5952	100	20	)	)	PUNCT
ejpam-5952	100	21	=	=	SYM
ejpam-5952	100	22	∣∣∣∣	∣∣∣∣	NOUN
ejpam-5952	100	23	1	1	NUM
ejpam-5952	100	24	2n+1	2n+1	PROPN
ejpam-5952	100	25	−	−	NUM
ejpam-5952	100	26	1	1	NUM
ejpam-5952	100	27	2n	2n	NUM
ejpam-5952	100	28	∣∣∣∣+	∣∣∣∣+	PROPN
ejpam-5952	100	29	∣∣∣∣	∣∣∣∣	PROPN
ejpam-5952	100	30	12n	12n	NOUN
ejpam-5952	100	31	−	−	PROPN
ejpam-5952	100	32	1	1	NUM
ejpam-5952	100	33	2n−1	2n−1	NUM
ejpam-5952	100	34	∣∣∣∣+	∣∣∣∣+	NOUN
ejpam-5952	100	35	∣∣∣∣	∣∣∣∣	PROPN
ejpam-5952	100	36	1	1	NUM
ejpam-5952	100	37	2n−1	2n−1	NUM
ejpam-5952	100	38	−	−	NUM
ejpam-5952	100	39	1	1	NUM
ejpam-5952	100	40	2n+1	2n+1	PROPN
ejpam-5952	100	41	∣∣∣∣	∣∣∣∣	NOUN
ejpam-5952	100	42	.	.	PUNCT
ejpam-5952	101	1	(	(	PUNCT
ejpam-5952	101	2	10	10	NUM
ejpam-5952	101	3	)	)	PUNCT
ejpam-5952	101	4	since	since	SCONJ
ejpam-5952	101	5	∣∣∣∣	∣∣∣∣	NOUN
ejpam-5952	101	6	1	1	NUM
ejpam-5952	101	7	2n+1	2n+1	PROPN
ejpam-5952	101	8	−	−	NUM
ejpam-5952	101	9	1	1	NUM
ejpam-5952	101	10	2n	2n	NUM
ejpam-5952	101	11	∣∣∣∣	∣∣∣∣	NOUN
ejpam-5952	101	12	=	=	SYM
ejpam-5952	101	13	1	1	NUM
ejpam-5952	101	14	2n	2n	NUM
ejpam-5952	101	15	−	−	ADP
ejpam-5952	101	16	1	1	NUM
ejpam-5952	101	17	2n+1	2n+1	NOUN
ejpam-5952	101	18	=	=	SYM
ejpam-5952	101	19	1	1	NUM
ejpam-5952	101	20	2n+1	2n+1	PROPN
ejpam-5952	101	21	,	,	PUNCT
ejpam-5952	101	22	(	(	PUNCT
ejpam-5952	101	23	11)∣∣∣∣	11)∣∣∣∣	NUM
ejpam-5952	101	24	12n	12n	NOUN
ejpam-5952	101	25	−	−	PROPN
ejpam-5952	101	26	1	1	NUM
ejpam-5952	101	27	2n−1	2n−1	NUM
ejpam-5952	101	28	∣∣∣∣	∣∣∣∣	NOUN
ejpam-5952	101	29	=	=	SYM
ejpam-5952	101	30	1	1	NUM
ejpam-5952	101	31	2n−1	2n−1	NUM
ejpam-5952	101	32	−	−	NUM
ejpam-5952	101	33	1	1	NUM
ejpam-5952	101	34	2n	2n	NUM
ejpam-5952	101	35	=	=	SYM
ejpam-5952	101	36	1	1	NUM
ejpam-5952	101	37	2n	2n	NUM
ejpam-5952	101	38	,	,	PUNCT
ejpam-5952	101	39	(	(	PUNCT
ejpam-5952	101	40	12)∣∣∣∣	12)∣∣∣∣	PROPN
ejpam-5952	101	41	1	1	NUM
ejpam-5952	101	42	2n−1	2n−1	NUM
ejpam-5952	101	43	−	−	NUM
ejpam-5952	101	44	1	1	NUM
ejpam-5952	101	45	2n+1	2n+1	PROPN
ejpam-5952	101	46	∣∣∣∣	∣∣∣∣	NOUN
ejpam-5952	101	47	=	=	NOUN
ejpam-5952	101	48	1	1	NUM
ejpam-5952	101	49	2n−1	2n−1	NUM
ejpam-5952	101	50	−	−	NUM
ejpam-5952	101	51	1	1	NUM
ejpam-5952	101	52	2n+1	2n+1	NOUN
ejpam-5952	101	53	=	=	SYM
ejpam-5952	101	54	1	1	NUM
ejpam-5952	101	55	2n+1	2n+1	NOUN
ejpam-5952	101	56	+	+	CCONJ
ejpam-5952	101	57	1	1	NUM
ejpam-5952	101	58	2n	2n	NUM
ejpam-5952	101	59	.	.	PUNCT
ejpam-5952	102	1	(	(	PUNCT
ejpam-5952	102	2	13	13	NUM
ejpam-5952	102	3	)	)	PUNCT
ejpam-5952	102	4	summing	sum	VERB
ejpam-5952	102	5	these	these	DET
ejpam-5952	102	6	terms	term	NOUN
ejpam-5952	102	7	,	,	PUNCT
ejpam-5952	102	8	we	we	PRON
ejpam-5952	102	9	get	get	VERB
ejpam-5952	102	10	:	:	PUNCT
ejpam-5952	102	11	m(υn+1	m(υn+1	NUM
ejpam-5952	102	12	,	,	PUNCT
ejpam-5952	102	13	υn	υn	NOUN
ejpam-5952	102	14	,	,	PUNCT
ejpam-5952	102	15	υn−1	υn−1	PROPN
ejpam-5952	102	16	)	)	PUNCT
ejpam-5952	102	17	=	=	SYM
ejpam-5952	102	18	1	1	NUM
ejpam-5952	102	19	2n+1	2n+1	NOUN
ejpam-5952	102	20	+	+	CCONJ
ejpam-5952	102	21	1	1	NUM
ejpam-5952	102	22	2n	2n	NUM
ejpam-5952	102	23	+	+	CCONJ
ejpam-5952	102	24	1	1	NUM
ejpam-5952	102	25	2n+1	2n+1	NOUN
ejpam-5952	102	26	+	+	CCONJ
ejpam-5952	102	27	1	1	NUM
ejpam-5952	102	28	2n	2n	NUM
ejpam-5952	102	29	=	=	SYM
ejpam-5952	102	30	3	3	NUM
ejpam-5952	102	31	2n+1	2n+1	PROPN
ejpam-5952	102	32	.	.	PUNCT
ejpam-5952	103	1	(	(	PUNCT
ejpam-5952	103	2	14	14	NUM
ejpam-5952	103	3	)	)	PUNCT
ejpam-5952	103	4	next	next	ADV
ejpam-5952	103	5	,	,	PUNCT
ejpam-5952	103	6	we	we	PRON
ejpam-5952	103	7	compute	compute	VERB
ejpam-5952	103	8	m(υn	m(υn	NUM
ejpam-5952	103	9	,	,	PUNCT
ejpam-5952	103	10	υn−1	υn−1	PROPN
ejpam-5952	103	11	,	,	PUNCT
ejpam-5952	103	12	υn−2	υn−2	PROPN
ejpam-5952	103	13	):	):	PUNCT
ejpam-5952	103	14	m(υn	m(υn	ADJ
ejpam-5952	103	15	,	,	PUNCT
ejpam-5952	103	16	υn−1	υn−1	ADJ
ejpam-5952	103	17	,	,	PUNCT
ejpam-5952	103	18	υn−2	υn−2	PROPN
ejpam-5952	103	19	)	)	PUNCT
ejpam-5952	103	20	=	=	SYM
ejpam-5952	103	21	∣∣∣∣	∣∣∣∣	NOUN
ejpam-5952	103	22	12n	12n	NOUN
ejpam-5952	103	23	−	−	PROPN
ejpam-5952	103	24	1	1	NUM
ejpam-5952	103	25	2n−1	2n−1	NUM
ejpam-5952	103	26	∣∣∣∣+	∣∣∣∣+	NOUN
ejpam-5952	103	27	∣∣∣∣	∣∣∣∣	PROPN
ejpam-5952	103	28	1	1	NUM
ejpam-5952	103	29	2n−1	2n−1	NUM
ejpam-5952	103	30	−	−	NUM
ejpam-5952	103	31	1	1	NUM
ejpam-5952	103	32	2n−2	2n−2	NUM
ejpam-5952	103	33	∣∣∣∣+	∣∣∣∣+	NOUN
ejpam-5952	103	34	∣∣∣∣	∣∣∣∣	PROPN
ejpam-5952	103	35	1	1	NUM
ejpam-5952	103	36	2n−2	2n−2	NUM
ejpam-5952	103	37	−	−	NUM
ejpam-5952	103	38	1	1	NUM
ejpam-5952	103	39	2n	2n	NUM
ejpam-5952	103	40	∣∣∣∣	∣∣∣∣	NOUN
ejpam-5952	103	41	.	.	PUNCT
ejpam-5952	104	1	(	(	PUNCT
ejpam-5952	104	2	15	15	NUM
ejpam-5952	104	3	)	)	PUNCT
ejpam-5952	104	4	this	this	DET
ejpam-5952	104	5	simplifies	simplifie	NOUN
ejpam-5952	104	6	to	to	PART
ejpam-5952	104	7	:	:	PUNCT
ejpam-5952	104	8	m(υn	m(υn	ADJ
ejpam-5952	104	9	,	,	PUNCT
ejpam-5952	104	10	υn−1	υn−1	ADJ
ejpam-5952	104	11	,	,	PUNCT
ejpam-5952	104	12	υn−2	υn−2	PROPN
ejpam-5952	104	13	)	)	PUNCT
ejpam-5952	104	14	=	=	SYM
ejpam-5952	104	15	3	3	NUM
ejpam-5952	104	16	2n	2n	NUM
ejpam-5952	104	17	.	.	PUNCT
ejpam-5952	105	1	(	(	PUNCT
ejpam-5952	105	2	16	16	NUM
ejpam-5952	105	3	)	)	PUNCT
ejpam-5952	105	4	thus	thus	ADV
ejpam-5952	105	5	,	,	PUNCT
ejpam-5952	105	6	we	we	PRON
ejpam-5952	105	7	observe	observe	VERB
ejpam-5952	105	8	that	that	SCONJ
ejpam-5952	105	9	:	:	PUNCT
ejpam-5952	105	10	m(υn+1	m(υn+1	NUM
ejpam-5952	105	11	,	,	PUNCT
ejpam-5952	105	12	υn	υn	NOUN
ejpam-5952	105	13	,	,	PUNCT
ejpam-5952	105	14	υn−1	υn−1	PROPN
ejpam-5952	105	15	)	)	PUNCT
ejpam-5952	105	16	=	=	SYM
ejpam-5952	105	17	1	1	NUM
ejpam-5952	105	18	2	2	NUM
ejpam-5952	105	19	m(υn	m(υn	PROPN
ejpam-5952	105	20	,	,	PUNCT
ejpam-5952	105	21	υn−1	υn−1	ADJ
ejpam-5952	105	22	,	,	PUNCT
ejpam-5952	105	23	υn−2	υn−2	PROPN
ejpam-5952	105	24	)	)	PUNCT
ejpam-5952	105	25	,	,	PUNCT
ejpam-5952	105	26	(	(	PUNCT
ejpam-5952	105	27	17	17	NUM
ejpam-5952	105	28	)	)	PUNCT
ejpam-5952	105	29	which	which	PRON
ejpam-5952	105	30	satisfies	satisfy	VERB
ejpam-5952	105	31	the	the	DET
ejpam-5952	105	32	condition	condition	NOUN
ejpam-5952	105	33	of	of	ADP
ejpam-5952	105	34	the	the	DET
ejpam-5952	105	35	theorem	theorem	NOUN
ejpam-5952	105	36	.	.	PUNCT
ejpam-5952	106	1	since	since	SCONJ
ejpam-5952	106	2	the	the	DET
ejpam-5952	106	3	sequence	sequence	NOUN
ejpam-5952	106	4	m(υn+1	m(υn+1	NUM
ejpam-5952	106	5	,	,	PUNCT
ejpam-5952	106	6	υn	υn	NOUN
ejpam-5952	106	7	,	,	PUNCT
ejpam-5952	106	8	υn−1	υn−1	PROPN
ejpam-5952	106	9	)	)	PUNCT
ejpam-5952	106	10	=	=	SYM
ejpam-5952	106	11	3	3	NUM
ejpam-5952	106	12	2n+1	2n+1	PROPN
ejpam-5952	106	13	converges	converge	VERB
ejpam-5952	106	14	to	to	ADP
ejpam-5952	106	15	0	0	NUM
ejpam-5952	106	16	as	as	ADP
ejpam-5952	106	17	n	n	PROPN
ejpam-5952	106	18	→	→	SYM
ejpam-5952	106	19	∞	∞	PROPN
ejpam-5952	106	20	,	,	PUNCT
ejpam-5952	106	21	the	the	DET
ejpam-5952	106	22	sequence	sequence	NOUN
ejpam-5952	106	23	{	{	PUNCT
ejpam-5952	106	24	υn	υn	NOUN
ejpam-5952	106	25	}	}	PUNCT
ejpam-5952	106	26	is	be	AUX
ejpam-5952	106	27	cauchy	cauchy	PROPN
ejpam-5952	106	28	.	.	PUNCT
ejpam-5952	107	1	since	since	SCONJ
ejpam-5952	107	2	r	r	NOUN
ejpam-5952	107	3	is	be	AUX
ejpam-5952	107	4	complete	complete	ADJ
ejpam-5952	107	5	,	,	PUNCT
ejpam-5952	107	6	it	it	PRON
ejpam-5952	107	7	follows	follow	VERB
ejpam-5952	107	8	that	that	SCONJ
ejpam-5952	107	9	there	there	PRON
ejpam-5952	107	10	exists	exist	VERB
ejpam-5952	107	11	a	a	DET
ejpam-5952	107	12	limit	limit	NOUN
ejpam-5952	107	13	υ∗	υ∗	NOUN
ejpam-5952	107	14	∈	∈	NOUN
ejpam-5952	107	15	r	r	NOUN
ejpam-5952	107	16	such	such	ADJ
ejpam-5952	107	17	that	that	PRON
ejpam-5952	107	18	:	:	PUNCT
ejpam-5952	108	1	lim	lim	PROPN
ejpam-5952	108	2	n→∞	n→∞	NUM
ejpam-5952	108	3	υn	υn	PROPN
ejpam-5952	108	4	=	=	SYM
ejpam-5952	108	5	υ∗.	υ∗.	PROPN
ejpam-5952	108	6	(	(	PUNCT
ejpam-5952	108	7	18	18	NUM
ejpam-5952	108	8	)	)	PUNCT
ejpam-5952	108	9	since	since	SCONJ
ejpam-5952	108	10	υn	υn	NOUN
ejpam-5952	108	11	=	=	SYM
ejpam-5952	108	12	1	1	NUM
ejpam-5952	108	13	2n	2n	NUM
ejpam-5952	108	14	,	,	PUNCT
ejpam-5952	108	15	we	we	PRON
ejpam-5952	108	16	have	have	VERB
ejpam-5952	108	17	:	:	PUNCT
ejpam-5952	108	18	υ∗	υ∗	NOUN
ejpam-5952	108	19	=	=	SYM
ejpam-5952	108	20	0	0	PROPN
ejpam-5952	108	21	.	.	PUNCT
ejpam-5952	109	1	(	(	PUNCT
ejpam-5952	109	2	19	19	NUM
ejpam-5952	109	3	)	)	PUNCT
ejpam-5952	109	4	thus	thus	ADV
ejpam-5952	109	5	,	,	PUNCT
ejpam-5952	109	6	the	the	DET
ejpam-5952	109	7	sequence	sequence	NOUN
ejpam-5952	109	8	{	{	PUNCT
ejpam-5952	109	9	υn	υn	NOUN
ejpam-5952	109	10	}	}	PUNCT
ejpam-5952	109	11	converges	converge	NOUN
ejpam-5952	109	12	to	to	ADP
ejpam-5952	109	13	υ∗	υ∗	NOUN
ejpam-5952	109	14	=	=	SYM
ejpam-5952	109	15	0	0	NUM
ejpam-5952	109	16	,	,	PUNCT
ejpam-5952	109	17	and	and	CCONJ
ejpam-5952	109	18	we	we	PRON
ejpam-5952	109	19	have	have	AUX
ejpam-5952	109	20	verified	verify	VERB
ejpam-5952	109	21	that	that	SCONJ
ejpam-5952	109	22	the	the	DET
ejpam-5952	109	23	sequence	sequence	NOUN
ejpam-5952	109	24	satisfies	satisfy	VERB
ejpam-5952	109	25	the	the	DET
ejpam-5952	109	26	conditions	condition	NOUN
ejpam-5952	109	27	of	of	ADP
ejpam-5952	109	28	the	the	DET
ejpam-5952	109	29	theorem	theorem	ADJ
ejpam-5952	109	30	2.2	2.2	NUM
ejpam-5952	109	31	and	and	CCONJ
ejpam-5952	109	32	converges	converge	VERB
ejpam-5952	109	33	to	to	ADP
ejpam-5952	109	34	the	the	DET
ejpam-5952	109	35	limit	limit	NOUN
ejpam-5952	109	36	.	.	PUNCT
ejpam-5952	110	1	a.	a.	NOUN
ejpam-5952	110	2	a.	a.	PROPN
ejpam-5952	110	3	m.	m.	PROPN
ejpam-5952	110	4	malkawi	malkawi	ADP
ejpam-5952	110	5	/	/	SYM
ejpam-5952	110	6	eur	eur	PROPN
ejpam-5952	110	7	.	.	PUNCT
ejpam-5952	111	1	j.	j.	PROPN
ejpam-5952	111	2	pure	pure	PROPN
ejpam-5952	111	3	appl	appl	PROPN
ejpam-5952	111	4	.	.	PROPN
ejpam-5952	111	5	math	math	PROPN
ejpam-5952	111	6	,	,	PUNCT
ejpam-5952	111	7	18	18	NUM
ejpam-5952	111	8	(	(	PUNCT
ejpam-5952	111	9	2	2	NUM
ejpam-5952	111	10	)	)	PUNCT
ejpam-5952	111	11	(	(	PUNCT
ejpam-5952	111	12	2025	2025	NUM
ejpam-5952	111	13	)	)	PUNCT
ejpam-5952	111	14	,	,	PUNCT
ejpam-5952	111	15	5952	5952	NUM
ejpam-5952	111	16	7	7	NUM
ejpam-5952	111	17	of	of	ADP
ejpam-5952	111	18	14	14	NUM
ejpam-5952	111	19	theorem	theorem	NOUN
ejpam-5952	111	20	3	3	X
ejpam-5952	111	21	.	.	PUNCT
ejpam-5952	112	1	let	let	AUX
ejpam-5952	112	2	(	(	PUNCT
ejpam-5952	112	3	x	x	X
ejpam-5952	112	4	,	,	PUNCT
ejpam-5952	112	5	m	m	VERB
ejpam-5952	112	6	)	)	PUNCT
ejpam-5952	112	7	be	be	AUX
ejpam-5952	112	8	a	a	DET
ejpam-5952	112	9	complete	complete	ADJ
ejpam-5952	112	10	mr	mr	ADJ
ejpam-5952	112	11	-	-	PUNCT
ejpam-5952	112	12	metric	metric	ADJ
ejpam-5952	112	13	space	space	NOUN
ejpam-5952	112	14	with	with	ADP
ejpam-5952	112	15	a	a	DET
ejpam-5952	112	16	σ	σ	NOUN
ejpam-5952	112	17	-	-	PUNCT
ejpam-5952	112	18	finite	finite	ADJ
ejpam-5952	112	19	measure	measure	NOUN
ejpam-5952	112	20	µ.	µ.	PROPN
ejpam-5952	112	21	suppose	suppose	VERB
ejpam-5952	112	22	that	that	SCONJ
ejpam-5952	112	23	t	t	NOUN
ejpam-5952	112	24	:	:	PUNCT
ejpam-5952	112	25	x	x	X
ejpam-5952	112	26	→	→	PUNCT
ejpam-5952	112	27	x	x	X
ejpam-5952	112	28	is	be	AUX
ejpam-5952	112	29	a	a	DET
ejpam-5952	112	30	measurable	measurable	ADJ
ejpam-5952	112	31	self	self	NOUN
ejpam-5952	112	32	-	-	PUNCT
ejpam-5952	112	33	mapping	mapping	NOUN
ejpam-5952	112	34	satisfying	satisfy	VERB
ejpam-5952	112	35	the	the	DET
ejpam-5952	112	36	contraction	contraction	NOUN
ejpam-5952	112	37	condition	condition	NOUN
ejpam-5952	112	38	:	:	PUNCT
ejpam-5952	112	39	m(tυ	m(tυ	NOUN
ejpam-5952	112	40	,	,	PUNCT
ejpam-5952	112	41	tξ	tξ	VERB
ejpam-5952	112	42	,	,	PUNCT
ejpam-5952	112	43	tℑ	tℑ	NOUN
ejpam-5952	112	44	)	)	PUNCT
ejpam-5952	112	45	≤	≤	NOUN
ejpam-5952	112	46	αm(υ	αm(υ	NUM
ejpam-5952	112	47	,	,	PUNCT
ejpam-5952	112	48	ξ,ℑ	ξ,ℑ	PROPN
ejpam-5952	112	49	)	)	PUNCT
ejpam-5952	112	50	,	,	PUNCT
ejpam-5952	112	51	∀υ	∀υ	NOUN
ejpam-5952	112	52	,	,	PUNCT
ejpam-5952	112	53	ξ,ℑ	ξ,ℑ	PROPN
ejpam-5952	112	54	∈	∈	PROPN
ejpam-5952	112	55	x	x	X
ejpam-5952	112	56	,	,	PUNCT
ejpam-5952	112	57	(	(	PUNCT
ejpam-5952	112	58	20	20	NUM
ejpam-5952	112	59	)	)	PUNCT
ejpam-5952	112	60	where	where	SCONJ
ejpam-5952	112	61	0	0	NUM
ejpam-5952	112	62	≤	≤	NUM
ejpam-5952	112	63	α	α	NOUN
ejpam-5952	112	64	<	<	X
ejpam-5952	112	65	1	1	NUM
ejpam-5952	112	66	r	r	NOUN
ejpam-5952	112	67	.	.	PUNCT
ejpam-5952	113	1	then	then	ADV
ejpam-5952	113	2	,	,	PUNCT
ejpam-5952	113	3	there	there	PRON
ejpam-5952	113	4	exists	exist	VERB
ejpam-5952	113	5	a	a	DET
ejpam-5952	113	6	unique	unique	ADJ
ejpam-5952	113	7	fixed	fix	VERB
ejpam-5952	113	8	point	point	NOUN
ejpam-5952	113	9	υ∗	υ∗	NOUN
ejpam-5952	113	10	∈	∈	PROPN
ejpam-5952	113	11	x	x	PUNCT
ejpam-5952	113	12	such	such	ADJ
ejpam-5952	113	13	that	that	PRON
ejpam-5952	113	14	t	t	PROPN
ejpam-5952	113	15	(	(	PUNCT
ejpam-5952	113	16	υ∗	υ∗	NOUN
ejpam-5952	113	17	)	)	PUNCT
ejpam-5952	113	18	=	=	SYM
ejpam-5952	113	19	υ∗	υ∗	NOUN
ejpam-5952	113	20	,	,	PUNCT
ejpam-5952	113	21	and	and	CCONJ
ejpam-5952	113	22	υn	υn	X
ejpam-5952	113	23	→	→	SYM
ejpam-5952	113	24	υ∗	υ∗	NOUN
ejpam-5952	113	25	in	in	ADP
ejpam-5952	113	26	measure	measure	NOUN
ejpam-5952	113	27	.	.	PUNCT
ejpam-5952	114	1	proof	proof	NOUN
ejpam-5952	114	2	.	.	PUNCT
ejpam-5952	115	1	step	step	NOUN
ejpam-5952	115	2	1	1	NUM
ejpam-5952	115	3	:	:	PUNCT
ejpam-5952	115	4	defining	define	VERB
ejpam-5952	115	5	the	the	DET
ejpam-5952	115	6	recursive	recursive	ADJ
ejpam-5952	115	7	sequence	sequence	NOUN
ejpam-5952	115	8	we	we	PRON
ejpam-5952	115	9	define	define	VERB
ejpam-5952	115	10	a	a	DET
ejpam-5952	115	11	sequence	sequence	NOUN
ejpam-5952	115	12	{	{	PUNCT
ejpam-5952	115	13	υn	υn	NOUN
ejpam-5952	115	14	}	}	PUNCT
ejpam-5952	115	15	in	in	ADP
ejpam-5952	115	16	x	x	PUNCT
ejpam-5952	115	17	recursively	recursively	ADV
ejpam-5952	115	18	by	by	ADP
ejpam-5952	115	19	setting	set	VERB
ejpam-5952	115	20	:	:	PUNCT
ejpam-5952	115	21	υn+1	υn+1	NUM
ejpam-5952	115	22	=	=	SYM
ejpam-5952	115	23	tυn	tυn	NOUN
ejpam-5952	115	24	,	,	PUNCT
ejpam-5952	115	25	for	for	ADP
ejpam-5952	115	26	some	some	DET
ejpam-5952	115	27	initial	initial	ADJ
ejpam-5952	115	28	point	point	NOUN
ejpam-5952	115	29	υ0	υ0	PROPN
ejpam-5952	115	30	∈	∈	PROPN
ejpam-5952	115	31	x.	x.	NOUN
ejpam-5952	116	1	our	our	PRON
ejpam-5952	116	2	goal	goal	NOUN
ejpam-5952	116	3	is	be	AUX
ejpam-5952	116	4	to	to	PART
ejpam-5952	116	5	show	show	VERB
ejpam-5952	116	6	that	that	SCONJ
ejpam-5952	116	7	this	this	DET
ejpam-5952	116	8	sequence	sequence	NOUN
ejpam-5952	116	9	{	{	PUNCT
ejpam-5952	116	10	υn	υn	NOUN
ejpam-5952	116	11	}	}	PUNCT
ejpam-5952	116	12	converges	converge	NOUN
ejpam-5952	116	13	to	to	ADP
ejpam-5952	116	14	a	a	DET
ejpam-5952	116	15	fixed	fix	VERB
ejpam-5952	116	16	point	point	NOUN
ejpam-5952	116	17	of	of	ADP
ejpam-5952	116	18	t	t	PROPN
ejpam-5952	116	19	and	and	CCONJ
ejpam-5952	116	20	that	that	SCONJ
ejpam-5952	116	21	the	the	DET
ejpam-5952	116	22	convergence	convergence	NOUN
ejpam-5952	116	23	is	be	AUX
ejpam-5952	116	24	in	in	ADP
ejpam-5952	116	25	measure	measure	NOUN
ejpam-5952	116	26	.	.	PUNCT
ejpam-5952	117	1	step	step	NOUN
ejpam-5952	117	2	2	2	NUM
ejpam-5952	117	3	:	:	PUNCT
ejpam-5952	117	4	applying	apply	VERB
ejpam-5952	117	5	the	the	DET
ejpam-5952	117	6	contraction	contraction	NOUN
ejpam-5952	117	7	condition	condition	NOUN
ejpam-5952	117	8	using	use	VERB
ejpam-5952	117	9	the	the	DET
ejpam-5952	117	10	contraction	contraction	NOUN
ejpam-5952	117	11	condition	condition	NOUN
ejpam-5952	117	12	,	,	PUNCT
ejpam-5952	117	13	we	we	PRON
ejpam-5952	117	14	obtain	obtain	VERB
ejpam-5952	117	15	the	the	DET
ejpam-5952	117	16	following	follow	VERB
ejpam-5952	117	17	inequality	inequality	NOUN
ejpam-5952	117	18	for	for	ADP
ejpam-5952	117	19	all	all	DET
ejpam-5952	117	20	n	n	PRON
ejpam-5952	117	21	≥	≥	NOUN
ejpam-5952	117	22	1	1	NUM
ejpam-5952	117	23	:	:	SYM
ejpam-5952	117	24	m(υn+1	m(υn+1	NUM
ejpam-5952	117	25	,	,	PUNCT
ejpam-5952	117	26	υn	υn	NOUN
ejpam-5952	117	27	,	,	PUNCT
ejpam-5952	117	28	υn−1	υn−1	PROPN
ejpam-5952	117	29	)	)	PUNCT
ejpam-5952	117	30	≤	≤	NOUN
ejpam-5952	117	31	αm(υn	αm(υn	VERB
ejpam-5952	117	32	,	,	PUNCT
ejpam-5952	117	33	υn−1	υn−1	PROPN
ejpam-5952	117	34	,	,	PUNCT
ejpam-5952	117	35	υn−2	υn−2	PROPN
ejpam-5952	117	36	)	)	PUNCT
ejpam-5952	117	37	.	.	PUNCT
ejpam-5952	118	1	this	this	DET
ejpam-5952	118	2	contraction	contraction	NOUN
ejpam-5952	118	3	condition	condition	NOUN
ejpam-5952	118	4	,	,	PUNCT
ejpam-5952	118	5	when	when	SCONJ
ejpam-5952	118	6	applied	apply	VERB
ejpam-5952	118	7	repeatedly	repeatedly	ADV
ejpam-5952	118	8	,	,	PUNCT
ejpam-5952	118	9	leads	lead	VERB
ejpam-5952	118	10	to	to	ADP
ejpam-5952	118	11	:	:	PUNCT
ejpam-5952	118	12	m(υn+k	m(υn+k	PROPN
ejpam-5952	118	13	,	,	PUNCT
ejpam-5952	118	14	υn+k−1	υn+k−1	PROPN
ejpam-5952	118	15	,	,	PUNCT
ejpam-5952	118	16	υn+k−2	υn+k−2	NOUN
ejpam-5952	118	17	)	)	PUNCT
ejpam-5952	118	18	≤	≤	NOUN
ejpam-5952	118	19	αkm(υn	αkm(υn	X
ejpam-5952	118	20	,	,	PUNCT
ejpam-5952	118	21	υn−1	υn−1	ADJ
ejpam-5952	118	22	,	,	PUNCT
ejpam-5952	118	23	υn−2	υn−2	PROPN
ejpam-5952	118	24	)	)	PUNCT
ejpam-5952	118	25	.	.	PUNCT
ejpam-5952	119	1	by	by	ADP
ejpam-5952	119	2	induction	induction	NOUN
ejpam-5952	119	3	,	,	PUNCT
ejpam-5952	119	4	it	it	PRON
ejpam-5952	119	5	can	can	AUX
ejpam-5952	119	6	be	be	AUX
ejpam-5952	119	7	shown	show	VERB
ejpam-5952	119	8	that	that	SCONJ
ejpam-5952	119	9	this	this	DET
ejpam-5952	119	10	inequality	inequality	NOUN
ejpam-5952	119	11	holds	hold	VERB
ejpam-5952	119	12	for	for	ADP
ejpam-5952	119	13	all	all	DET
ejpam-5952	119	14	k	k	PROPN
ejpam-5952	119	15	≥	≥	NUM
ejpam-5952	119	16	1	1	NUM
ejpam-5952	119	17	,	,	PUNCT
ejpam-5952	119	18	and	and	CCONJ
ejpam-5952	119	19	since	since	SCONJ
ejpam-5952	119	20	0	0	NUM
ejpam-5952	119	21	≤	≤	NUM
ejpam-5952	119	22	α	α	NOUN
ejpam-5952	119	23	<	<	X
ejpam-5952	119	24	1	1	NUM
ejpam-5952	119	25	/	/	SYM
ejpam-5952	119	26	r	r	NOUN
ejpam-5952	119	27	,	,	PUNCT
ejpam-5952	119	28	the	the	DET
ejpam-5952	119	29	right	right	ADJ
ejpam-5952	119	30	-	-	PUNCT
ejpam-5952	119	31	hand	hand	NOUN
ejpam-5952	119	32	side	side	NOUN
ejpam-5952	119	33	tends	tend	VERB
ejpam-5952	119	34	to	to	ADP
ejpam-5952	119	35	zero	zero	NUM
ejpam-5952	119	36	as	as	ADP
ejpam-5952	119	37	k	k	PROPN
ejpam-5952	119	38	→	→	SYM
ejpam-5952	119	39	∞.	∞.	PROPN
ejpam-5952	119	40	hence	hence	ADV
ejpam-5952	119	41	,	,	PUNCT
ejpam-5952	119	42	the	the	DET
ejpam-5952	119	43	distances	distance	NOUN
ejpam-5952	119	44	between	between	ADP
ejpam-5952	119	45	the	the	DET
ejpam-5952	119	46	terms	term	NOUN
ejpam-5952	119	47	of	of	ADP
ejpam-5952	119	48	the	the	DET
ejpam-5952	119	49	sequence	sequence	NOUN
ejpam-5952	119	50	shrink	shrink	VERB
ejpam-5952	119	51	exponentially	exponentially	ADV
ejpam-5952	119	52	,	,	PUNCT
ejpam-5952	119	53	implying	imply	VERB
ejpam-5952	119	54	that	that	SCONJ
ejpam-5952	119	55	the	the	DET
ejpam-5952	119	56	sequence	sequence	NOUN
ejpam-5952	119	57	{	{	PUNCT
ejpam-5952	119	58	υn	υn	NOUN
ejpam-5952	119	59	}	}	PUNCT
ejpam-5952	119	60	is	be	AUX
ejpam-5952	119	61	a	a	DET
ejpam-5952	119	62	cauchy	cauchy	ADJ
ejpam-5952	119	63	sequence	sequence	NOUN
ejpam-5952	119	64	.	.	PUNCT
ejpam-5952	120	1	step	step	NOUN
ejpam-5952	120	2	3	3	NUM
ejpam-5952	120	3	:	:	PUNCT
ejpam-5952	120	4	convergence	convergence	NOUN
ejpam-5952	120	5	of	of	ADP
ejpam-5952	120	6	the	the	DET
ejpam-5952	120	7	sequence	sequence	NOUN
ejpam-5952	120	8	since	since	SCONJ
ejpam-5952	120	9	(	(	PUNCT
ejpam-5952	120	10	x	x	X
ejpam-5952	120	11	,	,	PUNCT
ejpam-5952	120	12	m	m	VERB
ejpam-5952	120	13	)	)	PUNCT
ejpam-5952	120	14	is	be	AUX
ejpam-5952	120	15	a	a	DET
ejpam-5952	120	16	complete	complete	ADJ
ejpam-5952	120	17	mr	mr	ADJ
ejpam-5952	120	18	-	-	PUNCT
ejpam-5952	120	19	metric	metric	ADJ
ejpam-5952	120	20	space	space	NOUN
ejpam-5952	120	21	,	,	PUNCT
ejpam-5952	120	22	we	we	PRON
ejpam-5952	120	23	can	can	AUX
ejpam-5952	120	24	conclude	conclude	VERB
ejpam-5952	120	25	that	that	SCONJ
ejpam-5952	120	26	the	the	DET
ejpam-5952	120	27	sequence	sequence	NOUN
ejpam-5952	120	28	{	{	PUNCT
ejpam-5952	120	29	υn	υn	NOUN
ejpam-5952	120	30	}	}	PUNCT
ejpam-5952	120	31	converges	converge	NOUN
ejpam-5952	120	32	to	to	ADP
ejpam-5952	120	33	some	some	DET
ejpam-5952	120	34	point	point	NOUN
ejpam-5952	120	35	υ∗	υ∗	NOUN
ejpam-5952	120	36	∈	∈	PROPN
ejpam-5952	120	37	x.	x.	NOUN
ejpam-5952	120	38	therefore	therefore	ADV
ejpam-5952	120	39	,	,	PUNCT
ejpam-5952	120	40	we	we	PRON
ejpam-5952	120	41	have	have	VERB
ejpam-5952	120	42	:	:	PUNCT
ejpam-5952	120	43	υn	υn	X
ejpam-5952	120	44	→	→	SYM
ejpam-5952	120	45	υ∗	υ∗	NOUN
ejpam-5952	120	46	as	as	ADP
ejpam-5952	120	47	n	n	PROPN
ejpam-5952	120	48	→	→	SYM
ejpam-5952	120	49	∞.	∞.	PROPN
ejpam-5952	120	50	step	step	NOUN
ejpam-5952	120	51	4	4	NUM
ejpam-5952	120	52	:	:	PUNCT
ejpam-5952	120	53	showing	show	VERB
ejpam-5952	120	54	that	that	DET
ejpam-5952	120	55	υ∗	υ∗	NOUN
ejpam-5952	120	56	is	be	AUX
ejpam-5952	120	57	a	a	DET
ejpam-5952	120	58	fixed	fix	VERB
ejpam-5952	120	59	point	point	NOUN
ejpam-5952	120	60	to	to	PART
ejpam-5952	120	61	prove	prove	VERB
ejpam-5952	120	62	that	that	DET
ejpam-5952	120	63	υ∗	υ∗	NOUN
ejpam-5952	120	64	is	be	AUX
ejpam-5952	120	65	a	a	DET
ejpam-5952	120	66	fixed	fix	VERB
ejpam-5952	120	67	point	point	NOUN
ejpam-5952	120	68	of	of	ADP
ejpam-5952	120	69	t	t	PROPN
ejpam-5952	120	70	,	,	PUNCT
ejpam-5952	120	71	we	we	PRON
ejpam-5952	120	72	take	take	VERB
ejpam-5952	120	73	the	the	DET
ejpam-5952	120	74	limit	limit	NOUN
ejpam-5952	120	75	of	of	ADP
ejpam-5952	120	76	the	the	DET
ejpam-5952	120	77	contraction	contraction	NOUN
ejpam-5952	120	78	condition	condition	NOUN
ejpam-5952	120	79	as	as	ADP
ejpam-5952	120	80	n	n	PROPN
ejpam-5952	120	81	→	→	SYM
ejpam-5952	120	82	∞.	∞.	PROPN
ejpam-5952	120	83	by	by	ADP
ejpam-5952	120	84	continuity	continuity	NOUN
ejpam-5952	120	85	of	of	ADP
ejpam-5952	120	86	t	t	PROPN
ejpam-5952	120	87	and	and	CCONJ
ejpam-5952	120	88	the	the	DET
ejpam-5952	120	89	fact	fact	NOUN
ejpam-5952	120	90	that	that	SCONJ
ejpam-5952	120	91	υn	υn	NOUN
ejpam-5952	120	92	→	→	SYM
ejpam-5952	120	93	υ∗	υ∗	NOUN
ejpam-5952	120	94	,	,	PUNCT
ejpam-5952	120	95	we	we	PRON
ejpam-5952	120	96	obtain	obtain	VERB
ejpam-5952	120	97	:	:	PUNCT
ejpam-5952	120	98	m(tυ∗	m(tυ∗	PROPN
ejpam-5952	120	99	,	,	PUNCT
ejpam-5952	120	100	tυ∗	tυ∗	NOUN
ejpam-5952	120	101	,	,	PUNCT
ejpam-5952	120	102	tυ∗	tυ∗	NOUN
ejpam-5952	120	103	)	)	PUNCT
ejpam-5952	121	1	=	=	VERB
ejpam-5952	121	2	lim	lim	PROPN
ejpam-5952	121	3	n→∞	n→∞	NUM
ejpam-5952	122	1	m(tυn	m(tυn	PROPN
ejpam-5952	122	2	,	,	PUNCT
ejpam-5952	122	3	tυn−1	tυn−1	PROPN
ejpam-5952	122	4	,	,	PUNCT
ejpam-5952	122	5	tυn−2	tυn−2	PROPN
ejpam-5952	122	6	)	)	PUNCT
ejpam-5952	122	7	.	.	PUNCT
ejpam-5952	123	1	since	since	SCONJ
ejpam-5952	123	2	υn	υn	NOUN
ejpam-5952	123	3	→	→	SYM
ejpam-5952	123	4	υ∗	υ∗	NOUN
ejpam-5952	123	5	,	,	PUNCT
ejpam-5952	123	6	it	it	PRON
ejpam-5952	123	7	follows	follow	VERB
ejpam-5952	123	8	that	that	SCONJ
ejpam-5952	123	9	:	:	PUNCT
ejpam-5952	123	10	m(tυ∗	m(tυ∗	PROPN
ejpam-5952	123	11	,	,	PUNCT
ejpam-5952	123	12	tυ∗	tυ∗	NOUN
ejpam-5952	123	13	,	,	PUNCT
ejpam-5952	123	14	tυ∗	tυ∗	NOUN
ejpam-5952	123	15	)	)	PUNCT
ejpam-5952	123	16	≤	≤	NUM
ejpam-5952	123	17	αm(υ∗	αm(υ∗	NUM
ejpam-5952	123	18	,	,	PUNCT
ejpam-5952	123	19	υ∗	υ∗	NOUN
ejpam-5952	123	20	,	,	PUNCT
ejpam-5952	123	21	υ∗	υ∗	NOUN
ejpam-5952	123	22	)	)	PUNCT
ejpam-5952	124	1	=	=	SYM
ejpam-5952	124	2	0	0	X
ejpam-5952	124	3	.	.	PUNCT
ejpam-5952	125	1	therefore	therefore	ADV
ejpam-5952	125	2	,	,	PUNCT
ejpam-5952	125	3	we	we	PRON
ejpam-5952	125	4	have	have	VERB
ejpam-5952	125	5	:	:	PUNCT
ejpam-5952	125	6	m(tυ∗	m(tυ∗	PROPN
ejpam-5952	125	7	,	,	PUNCT
ejpam-5952	125	8	tυ∗	tυ∗	NOUN
ejpam-5952	125	9	,	,	PUNCT
ejpam-5952	125	10	tυ∗	tυ∗	NOUN
ejpam-5952	125	11	)	)	PUNCT
ejpam-5952	125	12	=	=	SYM
ejpam-5952	126	1	0	0	NUM
ejpam-5952	126	2	,	,	PUNCT
ejpam-5952	126	3	which	which	PRON
ejpam-5952	126	4	implies	imply	VERB
ejpam-5952	126	5	that	that	SCONJ
ejpam-5952	126	6	tυ∗	tυ∗	PRON
ejpam-5952	126	7	=	=	PRON
ejpam-5952	126	8	υ∗.	υ∗.	VERB
ejpam-5952	126	9	thus	thus	ADV
ejpam-5952	126	10	,	,	PUNCT
ejpam-5952	126	11	υ∗	υ∗	NOUN
ejpam-5952	126	12	is	be	AUX
ejpam-5952	126	13	a	a	DET
ejpam-5952	126	14	fixed	fix	VERB
ejpam-5952	126	15	point	point	NOUN
ejpam-5952	126	16	of	of	ADP
ejpam-5952	126	17	t	t	PROPN
ejpam-5952	126	18	.	.	PUNCT
ejpam-5952	127	1	a.	a.	NOUN
ejpam-5952	127	2	a.	a.	PROPN
ejpam-5952	127	3	m.	m.	PROPN
ejpam-5952	127	4	malkawi	malkawi	ADP
ejpam-5952	127	5	/	/	SYM
ejpam-5952	127	6	eur	eur	PROPN
ejpam-5952	127	7	.	.	PUNCT
ejpam-5952	128	1	j.	j.	PROPN
ejpam-5952	128	2	pure	pure	PROPN
ejpam-5952	128	3	appl	appl	PROPN
ejpam-5952	128	4	.	.	PROPN
ejpam-5952	128	5	math	math	PROPN
ejpam-5952	128	6	,	,	PUNCT
ejpam-5952	128	7	18	18	NUM
ejpam-5952	128	8	(	(	PUNCT
ejpam-5952	128	9	2	2	NUM
ejpam-5952	128	10	)	)	PUNCT
ejpam-5952	128	11	(	(	PUNCT
ejpam-5952	128	12	2025	2025	NUM
ejpam-5952	128	13	)	)	PUNCT
ejpam-5952	128	14	,	,	PUNCT
ejpam-5952	128	15	5952	5952	NUM
ejpam-5952	128	16	8	8	NUM
ejpam-5952	128	17	of	of	ADP
ejpam-5952	128	18	14	14	NUM
ejpam-5952	128	19	step	step	NOUN
ejpam-5952	128	20	5	5	NUM
ejpam-5952	128	21	:	:	PUNCT
ejpam-5952	128	22	showing	show	VERB
ejpam-5952	128	23	convergence	convergence	NOUN
ejpam-5952	128	24	in	in	ADP
ejpam-5952	128	25	measure	measure	NOUN
ejpam-5952	128	26	now	now	ADV
ejpam-5952	128	27	,	,	PUNCT
ejpam-5952	128	28	we	we	PRON
ejpam-5952	128	29	establish	establish	VERB
ejpam-5952	128	30	that	that	SCONJ
ejpam-5952	128	31	υn	υn	NOUN
ejpam-5952	128	32	→	→	SYM
ejpam-5952	128	33	υ∗	υ∗	NOUN
ejpam-5952	128	34	in	in	ADP
ejpam-5952	128	35	measure	measure	NOUN
ejpam-5952	128	36	.	.	PUNCT
ejpam-5952	129	1	for	for	ADP
ejpam-5952	129	2	this	this	PRON
ejpam-5952	129	3	,	,	PUNCT
ejpam-5952	129	4	we	we	PRON
ejpam-5952	129	5	consider	consider	VERB
ejpam-5952	129	6	the	the	DET
ejpam-5952	129	7	sets	set	NOUN
ejpam-5952	129	8	:	:	PUNCT
ejpam-5952	129	9	an	an	PRON
ejpam-5952	129	10	=	=	X
ejpam-5952	129	11	{	{	PUNCT
ejpam-5952	129	12	υ	υ	NOUN
ejpam-5952	129	13	∈	∈	PROPN
ejpam-5952	129	14	x	x	X
ejpam-5952	129	15	:	:	PUNCT
ejpam-5952	129	16	m(υn	m(υn	PROPN
ejpam-5952	129	17	,	,	PUNCT
ejpam-5952	129	18	υ	υ	PRON
ejpam-5952	129	19	∗	∗	NOUN
ejpam-5952	129	20	,	,	PUNCT
ejpam-5952	129	21	υ∗	υ∗	NOUN
ejpam-5952	129	22	)	)	PUNCT
ejpam-5952	129	23	≥	≥	NOUN
ejpam-5952	129	24	ϵ	ϵ	NOUN
ejpam-5952	129	25	}	}	PUNCT
ejpam-5952	129	26	,	,	PUNCT
ejpam-5952	129	27	where	where	SCONJ
ejpam-5952	129	28	ϵ	ϵ	X
ejpam-5952	129	29	>	>	X
ejpam-5952	129	30	0	0	PROPN
ejpam-5952	129	31	.	.	PUNCT
ejpam-5952	130	1	since	since	SCONJ
ejpam-5952	130	2	m(υn	m(υn	NUM
ejpam-5952	130	3	,	,	PUNCT
ejpam-5952	130	4	υ	υ	PRON
ejpam-5952	130	5	∗	∗	NOUN
ejpam-5952	130	6	,	,	PUNCT
ejpam-5952	130	7	υ∗	υ∗	NOUN
ejpam-5952	130	8	)	)	PUNCT
ejpam-5952	130	9	→	→	SYM
ejpam-5952	130	10	0	0	NUM
ejpam-5952	130	11	as	as	ADP
ejpam-5952	130	12	n	n	NUM
ejpam-5952	130	13	→	→	SYM
ejpam-5952	130	14	∞	∞	PROPN
ejpam-5952	130	15	,	,	PUNCT
ejpam-5952	130	16	it	it	PRON
ejpam-5952	130	17	follows	follow	VERB
ejpam-5952	130	18	that	that	SCONJ
ejpam-5952	130	19	for	for	ADP
ejpam-5952	130	20	sufficiently	sufficiently	ADV
ejpam-5952	130	21	large	large	ADJ
ejpam-5952	130	22	n	n	CCONJ
ejpam-5952	130	23	,	,	PUNCT
ejpam-5952	130	24	the	the	DET
ejpam-5952	130	25	measure	measure	NOUN
ejpam-5952	130	26	of	of	ADP
ejpam-5952	130	27	an	an	DET
ejpam-5952	130	28	becomes	become	VERB
ejpam-5952	130	29	arbitrarily	arbitrarily	ADV
ejpam-5952	130	30	small	small	ADJ
ejpam-5952	130	31	.	.	PUNCT
ejpam-5952	131	1	specifically	specifically	ADV
ejpam-5952	131	2	:	:	PUNCT
ejpam-5952	131	3	µ(an	µ(an	X
ejpam-5952	131	4	)	)	PUNCT
ejpam-5952	131	5	→	→	SYM
ejpam-5952	131	6	0	0	NUM
ejpam-5952	131	7	as	as	ADP
ejpam-5952	131	8	n	n	PROPN
ejpam-5952	131	9	→	→	SYM
ejpam-5952	131	10	∞.	∞.	PROPN
ejpam-5952	131	11	thus	thus	ADV
ejpam-5952	131	12	,	,	PUNCT
ejpam-5952	131	13	υn	υn	NOUN
ejpam-5952	131	14	→	→	SYM
ejpam-5952	131	15	υ∗	υ∗	NOUN
ejpam-5952	131	16	in	in	ADP
ejpam-5952	131	17	measure	measure	NOUN
ejpam-5952	131	18	.	.	PUNCT
ejpam-5952	132	1	step	step	VERB
ejpam-5952	132	2	6	6	NUM
ejpam-5952	132	3	:	:	PUNCT
ejpam-5952	132	4	uniqueness	uniqueness	NOUN
ejpam-5952	132	5	of	of	ADP
ejpam-5952	132	6	the	the	DET
ejpam-5952	132	7	fixed	fix	VERB
ejpam-5952	132	8	point	point	NOUN
ejpam-5952	132	9	finally	finally	ADV
ejpam-5952	132	10	,	,	PUNCT
ejpam-5952	132	11	we	we	PRON
ejpam-5952	132	12	show	show	VERB
ejpam-5952	132	13	the	the	DET
ejpam-5952	132	14	uniqueness	uniqueness	NOUN
ejpam-5952	132	15	of	of	ADP
ejpam-5952	132	16	the	the	DET
ejpam-5952	132	17	fixed	fix	VERB
ejpam-5952	132	18	point	point	NOUN
ejpam-5952	132	19	.	.	PUNCT
ejpam-5952	133	1	suppose	suppose	VERB
ejpam-5952	133	2	that	that	SCONJ
ejpam-5952	133	3	there	there	PRON
ejpam-5952	133	4	are	be	VERB
ejpam-5952	133	5	two	two	NUM
ejpam-5952	133	6	fixed	fix	VERB
ejpam-5952	133	7	points	point	NOUN
ejpam-5952	133	8	υ∗	υ∗	NOUN
ejpam-5952	133	9	and	and	CCONJ
ejpam-5952	133	10	υ∗∗	υ∗∗	VERB
ejpam-5952	133	11	such	such	ADJ
ejpam-5952	133	12	that	that	DET
ejpam-5952	133	13	tυ∗	tυ∗	NOUN
ejpam-5952	133	14	=	=	X
ejpam-5952	133	15	υ∗	υ∗	NOUN
ejpam-5952	133	16	and	and	CCONJ
ejpam-5952	133	17	tυ∗∗	tυ∗∗	PRON
ejpam-5952	134	1	=	=	SYM
ejpam-5952	134	2	υ∗∗.	υ∗∗.	X
ejpam-5952	134	3	from	from	ADP
ejpam-5952	134	4	the	the	DET
ejpam-5952	134	5	contraction	contraction	NOUN
ejpam-5952	134	6	condition	condition	NOUN
ejpam-5952	134	7	,	,	PUNCT
ejpam-5952	134	8	we	we	PRON
ejpam-5952	134	9	have	have	VERB
ejpam-5952	134	10	:	:	PUNCT
ejpam-5952	134	11	m(tυ∗	m(tυ∗	PROPN
ejpam-5952	134	12	,	,	PUNCT
ejpam-5952	134	13	tυ∗∗	tυ∗∗	PROPN
ejpam-5952	134	14	,	,	PUNCT
ejpam-5952	134	15	tυ∗∗	tυ∗∗	NUM
ejpam-5952	134	16	)	)	PUNCT
ejpam-5952	134	17	≤	≤	NUM
ejpam-5952	134	18	αm(υ∗	αm(υ∗	NUM
ejpam-5952	134	19	,	,	PUNCT
ejpam-5952	134	20	υ∗∗	υ∗∗	X
ejpam-5952	134	21	,	,	PUNCT
ejpam-5952	134	22	υ∗∗	υ∗∗	NOUN
ejpam-5952	134	23	)	)	PUNCT
ejpam-5952	134	24	.	.	PUNCT
ejpam-5952	135	1	since	since	SCONJ
ejpam-5952	135	2	tυ∗	tυ∗	NOUN
ejpam-5952	135	3	=	=	SYM
ejpam-5952	135	4	υ∗	υ∗	NOUN
ejpam-5952	135	5	and	and	CCONJ
ejpam-5952	135	6	tυ∗∗	tυ∗∗	NOUN
ejpam-5952	135	7	=	=	PUNCT
ejpam-5952	135	8	υ∗∗	υ∗∗	PROPN
ejpam-5952	135	9	,	,	PUNCT
ejpam-5952	135	10	we	we	PRON
ejpam-5952	135	11	obtain	obtain	VERB
ejpam-5952	135	12	:	:	PUNCT
ejpam-5952	135	13	m(υ∗	m(υ∗	ADV
ejpam-5952	135	14	,	,	PUNCT
ejpam-5952	135	15	υ∗∗	υ∗∗	VERB
ejpam-5952	135	16	,	,	PUNCT
ejpam-5952	135	17	υ∗∗	υ∗∗	X
ejpam-5952	135	18	)	)	PUNCT
ejpam-5952	135	19	≤	≤	NOUN
ejpam-5952	135	20	αm(υ∗	αm(υ∗	NUM
ejpam-5952	135	21	,	,	PUNCT
ejpam-5952	135	22	υ∗∗	υ∗∗	X
ejpam-5952	135	23	,	,	PUNCT
ejpam-5952	135	24	υ∗∗	υ∗∗	NOUN
ejpam-5952	135	25	)	)	PUNCT
ejpam-5952	135	26	.	.	PUNCT
ejpam-5952	136	1	since	since	SCONJ
ejpam-5952	136	2	α	α	PRON
ejpam-5952	136	3	<	<	X
ejpam-5952	136	4	1	1	NUM
ejpam-5952	136	5	/	/	SYM
ejpam-5952	136	6	r	r	NOUN
ejpam-5952	136	7	,	,	PUNCT
ejpam-5952	136	8	this	this	PRON
ejpam-5952	136	9	forces	force	VERB
ejpam-5952	136	10	the	the	DET
ejpam-5952	136	11	right	right	ADJ
ejpam-5952	136	12	-	-	PUNCT
ejpam-5952	136	13	hand	hand	NOUN
ejpam-5952	136	14	side	side	NOUN
ejpam-5952	136	15	to	to	PART
ejpam-5952	136	16	be	be	AUX
ejpam-5952	136	17	strictly	strictly	ADV
ejpam-5952	136	18	smaller	small	ADJ
ejpam-5952	136	19	than	than	ADP
ejpam-5952	136	20	the	the	DET
ejpam-5952	136	21	left	left	ADJ
ejpam-5952	136	22	-	-	PUNCT
ejpam-5952	136	23	hand	hand	NOUN
ejpam-5952	136	24	side	side	NOUN
ejpam-5952	136	25	,	,	PUNCT
ejpam-5952	136	26	leading	lead	VERB
ejpam-5952	136	27	to	to	ADP
ejpam-5952	136	28	:	:	PUNCT
ejpam-5952	136	29	m(υ∗	m(υ∗	ADV
ejpam-5952	136	30	,	,	PUNCT
ejpam-5952	136	31	υ∗∗	υ∗∗	VERB
ejpam-5952	136	32	,	,	PUNCT
ejpam-5952	136	33	υ∗∗	υ∗∗	X
ejpam-5952	136	34	)	)	PUNCT
ejpam-5952	136	35	=	=	SYM
ejpam-5952	136	36	0	0	X
ejpam-5952	136	37	.	.	PUNCT
ejpam-5952	137	1	thus	thus	ADV
ejpam-5952	137	2	,	,	PUNCT
ejpam-5952	137	3	υ∗	υ∗	NOUN
ejpam-5952	137	4	=	=	PUNCT
ejpam-5952	137	5	υ∗∗	υ∗∗	NOUN
ejpam-5952	137	6	,	,	PUNCT
ejpam-5952	137	7	which	which	PRON
ejpam-5952	137	8	proves	prove	VERB
ejpam-5952	137	9	the	the	DET
ejpam-5952	137	10	uniqueness	uniqueness	NOUN
ejpam-5952	137	11	of	of	ADP
ejpam-5952	137	12	the	the	DET
ejpam-5952	137	13	fixed	fix	VERB
ejpam-5952	137	14	point	point	NOUN
ejpam-5952	137	15	.	.	PUNCT
ejpam-5952	138	1	conclusion	conclusion	NOUN
ejpam-5952	138	2	we	we	PRON
ejpam-5952	138	3	have	have	AUX
ejpam-5952	138	4	shown	show	VERB
ejpam-5952	138	5	that	that	SCONJ
ejpam-5952	138	6	there	there	PRON
ejpam-5952	138	7	exists	exist	VERB
ejpam-5952	138	8	a	a	DET
ejpam-5952	138	9	unique	unique	ADJ
ejpam-5952	138	10	fixed	fix	VERB
ejpam-5952	138	11	point	point	NOUN
ejpam-5952	138	12	υ∗	υ∗	NOUN
ejpam-5952	138	13	such	such	ADJ
ejpam-5952	138	14	that	that	DET
ejpam-5952	138	15	t	t	PROPN
ejpam-5952	138	16	(	(	PUNCT
ejpam-5952	138	17	υ∗	υ∗	NOUN
ejpam-5952	138	18	)	)	PUNCT
ejpam-5952	138	19	=	=	SYM
ejpam-5952	138	20	υ∗	υ∗	NOUN
ejpam-5952	138	21	,	,	PUNCT
ejpam-5952	138	22	and	and	CCONJ
ejpam-5952	138	23	that	that	SCONJ
ejpam-5952	138	24	the	the	DET
ejpam-5952	138	25	sequence	sequence	NOUN
ejpam-5952	138	26	υn	υn	NOUN
ejpam-5952	138	27	converges	converge	VERB
ejpam-5952	138	28	to	to	ADP
ejpam-5952	138	29	υ∗	υ∗	NOUN
ejpam-5952	138	30	in	in	ADP
ejpam-5952	138	31	measure	measure	NOUN
ejpam-5952	138	32	.	.	PUNCT
ejpam-5952	139	1	this	this	PRON
ejpam-5952	139	2	completes	complete	VERB
ejpam-5952	139	3	the	the	DET
ejpam-5952	139	4	proof	proof	NOUN
ejpam-5952	139	5	.	.	PUNCT
ejpam-5952	140	1	example	example	NOUN
ejpam-5952	141	1	3	3	X
ejpam-5952	141	2	.	.	PUNCT
ejpam-5952	141	3	let	let	VERB
ejpam-5952	141	4	x	x	PUNCT
ejpam-5952	141	5	=	=	PUNCT
ejpam-5952	142	1	[	[	X
ejpam-5952	142	2	0	0	NUM
ejpam-5952	142	3	,	,	PUNCT
ejpam-5952	142	4	1	1	NUM
ejpam-5952	142	5	]	]	PUNCT
ejpam-5952	142	6	be	be	AUX
ejpam-5952	142	7	a	a	DET
ejpam-5952	142	8	metric	metric	ADJ
ejpam-5952	142	9	space	space	NOUN
ejpam-5952	142	10	equipped	equip	VERB
ejpam-5952	142	11	with	with	ADP
ejpam-5952	142	12	the	the	DET
ejpam-5952	142	13	mr	mr	PROPN
ejpam-5952	142	14	-	-	PUNCT
ejpam-5952	142	15	metric	metric	NOUN
ejpam-5952	142	16	:	:	PUNCT
ejpam-5952	142	17	m(υ	m(υ	PROPN
ejpam-5952	142	18	,	,	PUNCT
ejpam-5952	142	19	ξ,ℑ	ξ,ℑ	PROPN
ejpam-5952	142	20	)	)	PUNCT
ejpam-5952	143	1	=	=	SYM
ejpam-5952	143	2	|υ	|υ	NOUN
ejpam-5952	143	3	−	−	PROPN
ejpam-5952	143	4	ξ|+	ξ|+	PROPN
ejpam-5952	143	5	|ξ	|ξ	VERB
ejpam-5952	143	6	−ℑ|+	−ℑ|+	NOUN
ejpam-5952	143	7	|ℑ	|ℑ	NOUN
ejpam-5952	143	8	−	−	PROPN
ejpam-5952	143	9	υ|	υ|	PROPN
ejpam-5952	143	10	.	.	PUNCT
ejpam-5952	144	1	(	(	PUNCT
ejpam-5952	144	2	21	21	NUM
ejpam-5952	144	3	)	)	PUNCT
ejpam-5952	144	4	this	this	DET
ejpam-5952	144	5	function	function	NOUN
ejpam-5952	144	6	satisfies	satisfy	VERB
ejpam-5952	144	7	the	the	DET
ejpam-5952	144	8	properties	property	NOUN
ejpam-5952	144	9	of	of	ADP
ejpam-5952	144	10	an	an	DET
ejpam-5952	144	11	mr	mr	NOUN
ejpam-5952	144	12	-	-	PUNCT
ejpam-5952	144	13	metric	metric	NOUN
ejpam-5952	144	14	and	and	CCONJ
ejpam-5952	144	15	provides	provide	VERB
ejpam-5952	144	16	a	a	DET
ejpam-5952	144	17	measure	measure	NOUN
ejpam-5952	144	18	of	of	ADP
ejpam-5952	144	19	the	the	DET
ejpam-5952	144	20	threepoint	threepoint	NOUN
ejpam-5952	144	21	distance	distance	NOUN
ejpam-5952	144	22	in	in	ADP
ejpam-5952	144	23	the	the	DET
ejpam-5952	144	24	space	space	NOUN
ejpam-5952	144	25	x.	x.	NOUN
ejpam-5952	145	1	now	now	ADV
ejpam-5952	145	2	,	,	PUNCT
ejpam-5952	145	3	consider	consider	VERB
ejpam-5952	145	4	the	the	DET
ejpam-5952	145	5	mapping	mapping	NOUN
ejpam-5952	145	6	t	t	NOUN
ejpam-5952	145	7	:	:	PUNCT
ejpam-5952	145	8	x	x	X
ejpam-5952	145	9	→	→	SYM
ejpam-5952	145	10	x	x	PUNCT
ejpam-5952	145	11	defined	define	VERB
ejpam-5952	145	12	by	by	ADP
ejpam-5952	145	13	:	:	PUNCT
ejpam-5952	145	14	tυ	tυ	X
ejpam-5952	145	15	=	=	PUNCT
ejpam-5952	145	16	υ	υ	PROPN
ejpam-5952	145	17	2	2	NUM
ejpam-5952	145	18	.	.	PUNCT
ejpam-5952	146	1	(	(	PUNCT
ejpam-5952	146	2	22	22	NUM
ejpam-5952	146	3	)	)	PUNCT
ejpam-5952	146	4	this	this	DET
ejpam-5952	146	5	function	function	NOUN
ejpam-5952	146	6	maps	map	VERB
ejpam-5952	146	7	each	each	DET
ejpam-5952	146	8	point	point	NOUN
ejpam-5952	146	9	in	in	ADP
ejpam-5952	146	10	x	x	PUNCT
ejpam-5952	146	11	to	to	ADP
ejpam-5952	146	12	half	half	DET
ejpam-5952	146	13	its	its	PRON
ejpam-5952	146	14	value	value	NOUN
ejpam-5952	146	15	,	,	PUNCT
ejpam-5952	146	16	ensuring	ensure	VERB
ejpam-5952	146	17	that	that	SCONJ
ejpam-5952	146	18	the	the	DET
ejpam-5952	146	19	sequence	sequence	NOUN
ejpam-5952	146	20	{	{	PUNCT
ejpam-5952	146	21	υn	υn	NOUN
ejpam-5952	146	22	}	}	PUNCT
ejpam-5952	146	23	is	be	AUX
ejpam-5952	146	24	non	non	ADJ
ejpam-5952	146	25	-	-	ADJ
ejpam-5952	146	26	increasing	increase	VERB
ejpam-5952	146	27	and	and	CCONJ
ejpam-5952	146	28	convergent	convergent	NOUN
ejpam-5952	146	29	.	.	PUNCT
ejpam-5952	147	1	to	to	PART
ejpam-5952	147	2	verify	verify	VERB
ejpam-5952	147	3	that	that	SCONJ
ejpam-5952	147	4	t	t	PROPN
ejpam-5952	147	5	satisfies	satisfy	VERB
ejpam-5952	147	6	the	the	DET
ejpam-5952	147	7	contraction	contraction	NOUN
ejpam-5952	147	8	condition	condition	NOUN
ejpam-5952	147	9	,	,	PUNCT
ejpam-5952	147	10	we	we	PRON
ejpam-5952	147	11	compute	compute	VERB
ejpam-5952	147	12	:	:	PUNCT
ejpam-5952	148	1	m(tυ	m(tυ	NOUN
ejpam-5952	148	2	,	,	PUNCT
ejpam-5952	148	3	tξ	tξ	VERB
ejpam-5952	148	4	,	,	PUNCT
ejpam-5952	148	5	tℑ	tℑ	NOUN
ejpam-5952	148	6	)	)	PUNCT
ejpam-5952	148	7	=	=	SYM
ejpam-5952	148	8	∣∣∣∣υ2	∣∣∣∣υ2	PUNCT
ejpam-5952	149	1	−	−	ADP
ejpam-5952	149	2	ξ	ξ	SYM
ejpam-5952	149	3	2	2	NUM
ejpam-5952	149	4	∣∣∣∣+	∣∣∣∣+	NOUN
ejpam-5952	149	5	∣∣∣∣ξ2	∣∣∣∣ξ2	NUM
ejpam-5952	149	6	−	−	NOUN
ejpam-5952	149	7	ℑ	ℑ	NOUN
ejpam-5952	149	8	2	2	NUM
ejpam-5952	149	9	∣∣∣∣+	∣∣∣∣+	NOUN
ejpam-5952	149	10	∣∣∣∣ℑ2	∣∣∣∣ℑ2	NUM
ejpam-5952	149	11	−	−	PROPN
ejpam-5952	149	12	υ	υ	PROPN
ejpam-5952	149	13	2	2	NUM
ejpam-5952	149	14	∣∣∣∣	∣∣∣∣	NOUN
ejpam-5952	149	15	(	(	PUNCT
ejpam-5952	149	16	23	23	NUM
ejpam-5952	149	17	)	)	PUNCT
ejpam-5952	149	18	=	=	SYM
ejpam-5952	149	19	1	1	NUM
ejpam-5952	149	20	2	2	NUM
ejpam-5952	149	21	(	(	PUNCT
ejpam-5952	149	22	|υ	|υ	NOUN
ejpam-5952	149	23	−	−	ADP
ejpam-5952	149	24	ξ|+	ξ|+	PROPN
ejpam-5952	149	25	|ξ	|ξ	VERB
ejpam-5952	149	26	−ℑ|+	−ℑ|+	NOUN
ejpam-5952	149	27	|ℑ	|ℑ	NOUN
ejpam-5952	149	28	−	−	PROPN
ejpam-5952	149	29	υ|	υ|	PROPN
ejpam-5952	149	30	)	)	PUNCT
ejpam-5952	149	31	(	(	PUNCT
ejpam-5952	149	32	24	24	NUM
ejpam-5952	149	33	)	)	PUNCT
ejpam-5952	149	34	a.	a.	NOUN
ejpam-5952	149	35	a.	a.	NOUN
ejpam-5952	149	36	m.	m.	NOUN
ejpam-5952	149	37	malkawi	malkawi	ADP
ejpam-5952	149	38	/	/	SYM
ejpam-5952	149	39	eur	eur	PROPN
ejpam-5952	149	40	.	.	PUNCT
ejpam-5952	150	1	j.	j.	PROPN
ejpam-5952	150	2	pure	pure	PROPN
ejpam-5952	150	3	appl	appl	PROPN
ejpam-5952	150	4	.	.	PROPN
ejpam-5952	150	5	math	math	PROPN
ejpam-5952	150	6	,	,	PUNCT
ejpam-5952	150	7	18	18	NUM
ejpam-5952	150	8	(	(	PUNCT
ejpam-5952	150	9	2	2	NUM
ejpam-5952	150	10	)	)	PUNCT
ejpam-5952	150	11	(	(	PUNCT
ejpam-5952	150	12	2025	2025	NUM
ejpam-5952	150	13	)	)	PUNCT
ejpam-5952	150	14	,	,	PUNCT
ejpam-5952	150	15	5952	5952	NUM
ejpam-5952	150	16	9	9	NUM
ejpam-5952	150	17	of	of	ADP
ejpam-5952	150	18	14	14	NUM
ejpam-5952	150	19	=	=	SYM
ejpam-5952	150	20	1	1	NUM
ejpam-5952	150	21	2	2	NUM
ejpam-5952	150	22	m(υ	m(υ	PROPN
ejpam-5952	150	23	,	,	PUNCT
ejpam-5952	150	24	ξ,ℑ	ξ,ℑ	PROPN
ejpam-5952	150	25	)	)	PUNCT
ejpam-5952	150	26	.	.	PUNCT
ejpam-5952	151	1	(	(	PUNCT
ejpam-5952	151	2	25	25	NUM
ejpam-5952	151	3	)	)	PUNCT
ejpam-5952	151	4	thus	thus	ADV
ejpam-5952	151	5	,	,	PUNCT
ejpam-5952	151	6	t	t	PROPN
ejpam-5952	151	7	is	be	AUX
ejpam-5952	151	8	a	a	DET
ejpam-5952	151	9	contraction	contraction	NOUN
ejpam-5952	151	10	with	with	ADP
ejpam-5952	151	11	contraction	contraction	NOUN
ejpam-5952	151	12	coefficient	coefficient	NOUN
ejpam-5952	151	13	α	α	NOUN
ejpam-5952	151	14	=	=	NOUN
ejpam-5952	151	15	1	1	NUM
ejpam-5952	151	16	2	2	NUM
ejpam-5952	151	17	.	.	PUNCT
ejpam-5952	152	1	assuming	assume	VERB
ejpam-5952	152	2	that	that	SCONJ
ejpam-5952	152	3	r	r	NOUN
ejpam-5952	152	4	>	>	X
ejpam-5952	152	5	2	2	NUM
ejpam-5952	152	6	,	,	PUNCT
ejpam-5952	152	7	we	we	PRON
ejpam-5952	152	8	satisfy	satisfy	VERB
ejpam-5952	152	9	the	the	DET
ejpam-5952	152	10	requirement	requirement	NOUN
ejpam-5952	152	11	that	that	SCONJ
ejpam-5952	152	12	α	α	PRON
ejpam-5952	152	13	<	<	X
ejpam-5952	152	14	1	1	NUM
ejpam-5952	152	15	r	r	NOUN
ejpam-5952	152	16	.	.	PUNCT
ejpam-5952	153	1	next	next	ADV
ejpam-5952	153	2	,	,	PUNCT
ejpam-5952	153	3	we	we	PRON
ejpam-5952	153	4	define	define	VERB
ejpam-5952	153	5	the	the	DET
ejpam-5952	153	6	sequence	sequence	NOUN
ejpam-5952	153	7	recursively	recursively	ADV
ejpam-5952	153	8	by	by	ADP
ejpam-5952	153	9	:	:	PUNCT
ejpam-5952	153	10	υn+1	υn+1	ADJ
ejpam-5952	153	11	=	=	SYM
ejpam-5952	153	12	tυn	tυn	NOUN
ejpam-5952	153	13	,	,	PUNCT
ejpam-5952	153	14	(	(	PUNCT
ejpam-5952	153	15	26	26	NUM
ejpam-5952	153	16	)	)	PUNCT
ejpam-5952	153	17	with	with	ADP
ejpam-5952	153	18	an	an	DET
ejpam-5952	153	19	initial	initial	ADJ
ejpam-5952	153	20	point	point	NOUN
ejpam-5952	153	21	υ0	υ0	NOUN
ejpam-5952	153	22	∈	∈	PROPN
ejpam-5952	153	23	x.	x.	NOUN
ejpam-5952	154	1	we	we	PRON
ejpam-5952	154	2	get	get	VERB
ejpam-5952	154	3	:	:	PUNCT
ejpam-5952	154	4	υn	υn	NOUN
ejpam-5952	154	5	=	=	SYM
ejpam-5952	154	6	υ0	υ0	PROPN
ejpam-5952	154	7	2n	2n	NUM
ejpam-5952	154	8	.	.	PUNCT
ejpam-5952	155	1	(	(	PUNCT
ejpam-5952	155	2	27	27	NUM
ejpam-5952	155	3	)	)	PUNCT
ejpam-5952	155	4	since	since	SCONJ
ejpam-5952	155	5	0	0	NUM
ejpam-5952	155	6	≤	≤	NUM
ejpam-5952	155	7	υ0	υ0	NOUN
ejpam-5952	155	8	≤	≤	ADJ
ejpam-5952	155	9	1	1	NUM
ejpam-5952	155	10	,	,	PUNCT
ejpam-5952	155	11	it	it	PRON
ejpam-5952	155	12	follows	follow	VERB
ejpam-5952	155	13	that	that	SCONJ
ejpam-5952	155	14	:	:	PUNCT
ejpam-5952	156	1	lim	lim	PROPN
ejpam-5952	156	2	n→∞	n→∞	X
ejpam-5952	156	3	υn	υn	NOUN
ejpam-5952	156	4	=	=	NOUN
ejpam-5952	156	5	0	0	PROPN
ejpam-5952	156	6	.	.	PUNCT
ejpam-5952	157	1	(	(	PUNCT
ejpam-5952	157	2	28	28	NUM
ejpam-5952	157	3	)	)	PUNCT
ejpam-5952	157	4	this	this	PRON
ejpam-5952	157	5	shows	show	VERB
ejpam-5952	157	6	that	that	SCONJ
ejpam-5952	157	7	the	the	DET
ejpam-5952	157	8	sequence	sequence	NOUN
ejpam-5952	157	9	{	{	PUNCT
ejpam-5952	157	10	υn	υn	NOUN
ejpam-5952	157	11	}	}	PUNCT
ejpam-5952	157	12	is	be	AUX
ejpam-5952	157	13	convergent	convergent	ADJ
ejpam-5952	157	14	.	.	PUNCT
ejpam-5952	158	1	to	to	PART
ejpam-5952	158	2	confirm	confirm	VERB
ejpam-5952	158	3	that	that	SCONJ
ejpam-5952	158	4	{	{	PUNCT
ejpam-5952	158	5	υn	υn	NOUN
ejpam-5952	158	6	}	}	PUNCT
ejpam-5952	158	7	is	be	AUX
ejpam-5952	158	8	cauchy	cauchy	PROPN
ejpam-5952	158	9	,	,	PUNCT
ejpam-5952	158	10	we	we	PRON
ejpam-5952	158	11	check	check	VERB
ejpam-5952	158	12	that	that	PRON
ejpam-5952	158	13	for	for	ADP
ejpam-5952	158	14	any	any	DET
ejpam-5952	158	15	ϵ	ϵ	X
ejpam-5952	158	16	>	>	X
ejpam-5952	158	17	0	0	NUM
ejpam-5952	158	18	,	,	PUNCT
ejpam-5952	158	19	there	there	PRON
ejpam-5952	158	20	exists	exist	VERB
ejpam-5952	158	21	an	an	DET
ejpam-5952	158	22	integer	integer	NOUN
ejpam-5952	158	23	n	n	CCONJ
ejpam-5952	158	24	such	such	ADJ
ejpam-5952	158	25	that	that	PRON
ejpam-5952	158	26	for	for	ADP
ejpam-5952	158	27	all	all	DET
ejpam-5952	158	28	m	m	PROPN
ejpam-5952	158	29	,	,	PUNCT
ejpam-5952	158	30	n	n	PRON
ejpam-5952	158	31	≥	≥	NOUN
ejpam-5952	158	32	n	n	CCONJ
ejpam-5952	158	33	:	:	PUNCT
ejpam-5952	158	34	m(υm	m(υm	NOUN
ejpam-5952	158	35	,	,	PUNCT
ejpam-5952	158	36	υn	υn	NOUN
ejpam-5952	158	37	,	,	PUNCT
ejpam-5952	158	38	υn−1	υn−1	PROPN
ejpam-5952	158	39	)	)	PUNCT
ejpam-5952	158	40	<	<	X
ejpam-5952	158	41	ϵ.	ϵ.	NOUN
ejpam-5952	158	42	(	(	PUNCT
ejpam-5952	158	43	29	29	NUM
ejpam-5952	158	44	)	)	PUNCT
ejpam-5952	158	45	since	since	SCONJ
ejpam-5952	158	46	:	:	PUNCT
ejpam-5952	158	47	m(υm	m(υm	NOUN
ejpam-5952	158	48	,	,	PUNCT
ejpam-5952	158	49	υn	υn	NOUN
ejpam-5952	158	50	,	,	PUNCT
ejpam-5952	158	51	υn−1	υn−1	PROPN
ejpam-5952	158	52	)	)	PUNCT
ejpam-5952	158	53	=	=	SYM
ejpam-5952	158	54	1	1	NUM
ejpam-5952	158	55	2n	2n	NUM
ejpam-5952	158	56	m(υ0	m(υ0	PROPN
ejpam-5952	158	57	,	,	PUNCT
ejpam-5952	158	58	υ0	υ0	NOUN
ejpam-5952	158	59	,	,	PUNCT
ejpam-5952	158	60	υ0	υ0	NOUN
ejpam-5952	158	61	)	)	PUNCT
ejpam-5952	158	62	,	,	PUNCT
ejpam-5952	158	63	(	(	PUNCT
ejpam-5952	158	64	30	30	NUM
ejpam-5952	158	65	)	)	PUNCT
ejpam-5952	158	66	and	and	CCONJ
ejpam-5952	158	67	m(υ0	m(υ0	PROPN
ejpam-5952	158	68	,	,	PUNCT
ejpam-5952	158	69	υ0	υ0	NOUN
ejpam-5952	158	70	,	,	PUNCT
ejpam-5952	158	71	υ0	υ0	PROPN
ejpam-5952	158	72	)	)	PUNCT
ejpam-5952	158	73	is	be	AUX
ejpam-5952	158	74	bounded	bound	VERB
ejpam-5952	158	75	,	,	PUNCT
ejpam-5952	158	76	the	the	DET
ejpam-5952	158	77	right	right	ADJ
ejpam-5952	158	78	-	-	PUNCT
ejpam-5952	158	79	hand	hand	NOUN
ejpam-5952	158	80	side	side	NOUN
ejpam-5952	158	81	goes	go	VERB
ejpam-5952	158	82	to	to	ADP
ejpam-5952	158	83	zero	zero	NUM
ejpam-5952	158	84	as	as	ADP
ejpam-5952	158	85	n	n	PROPN
ejpam-5952	158	86	→	→	SYM
ejpam-5952	158	87	∞.	∞.	PROPN
ejpam-5952	158	88	hence	hence	ADV
ejpam-5952	158	89	,	,	PUNCT
ejpam-5952	158	90	{	{	PUNCT
ejpam-5952	158	91	υn	υn	NOUN
ejpam-5952	158	92	}	}	PUNCT
ejpam-5952	158	93	is	be	AUX
ejpam-5952	158	94	a	a	DET
ejpam-5952	158	95	cauchy	cauchy	ADJ
ejpam-5952	158	96	sequence	sequence	NOUN
ejpam-5952	158	97	.	.	PUNCT
ejpam-5952	159	1	since	since	SCONJ
ejpam-5952	159	2	(	(	PUNCT
ejpam-5952	159	3	x	x	X
ejpam-5952	159	4	,	,	PUNCT
ejpam-5952	159	5	m	m	VERB
ejpam-5952	159	6	)	)	PUNCT
ejpam-5952	159	7	is	be	AUX
ejpam-5952	159	8	complete	complete	ADJ
ejpam-5952	159	9	,	,	PUNCT
ejpam-5952	159	10	there	there	PRON
ejpam-5952	159	11	exists	exist	VERB
ejpam-5952	159	12	a	a	DET
ejpam-5952	159	13	limit	limit	NOUN
ejpam-5952	159	14	υ∗	υ∗	NOUN
ejpam-5952	159	15	∈	∈	PROPN
ejpam-5952	159	16	x	x	PUNCT
ejpam-5952	159	17	such	such	ADJ
ejpam-5952	159	18	that	that	SCONJ
ejpam-5952	159	19	:	:	PUNCT
ejpam-5952	160	1	lim	lim	PROPN
ejpam-5952	160	2	n→∞	n→∞	NUM
ejpam-5952	160	3	υn	υn	PROPN
ejpam-5952	160	4	=	=	SYM
ejpam-5952	160	5	υ∗.	υ∗.	PROPN
ejpam-5952	160	6	(	(	PUNCT
ejpam-5952	160	7	31	31	NUM
ejpam-5952	160	8	)	)	PUNCT
ejpam-5952	160	9	taking	take	VERB
ejpam-5952	160	10	the	the	DET
ejpam-5952	160	11	limit	limit	NOUN
ejpam-5952	160	12	in	in	ADP
ejpam-5952	160	13	the	the	DET
ejpam-5952	160	14	recursive	recursive	ADJ
ejpam-5952	160	15	relation	relation	NOUN
ejpam-5952	160	16	υn+1	υn+1	NOUN
ejpam-5952	160	17	=	=	SYM
ejpam-5952	160	18	tυn	tυn	NOUN
ejpam-5952	160	19	,	,	PUNCT
ejpam-5952	160	20	we	we	PRON
ejpam-5952	160	21	obtain	obtain	VERB
ejpam-5952	160	22	:	:	PUNCT
ejpam-5952	160	23	tυ∗	tυ∗	PROPN
ejpam-5952	161	1	=	=	PRON
ejpam-5952	161	2	υ∗.	υ∗.	PROPN
ejpam-5952	161	3	(	(	PUNCT
ejpam-5952	161	4	32	32	NUM
ejpam-5952	161	5	)	)	PUNCT
ejpam-5952	161	6	thus	thus	ADV
ejpam-5952	161	7	,	,	PUNCT
ejpam-5952	161	8	υ∗	υ∗	NOUN
ejpam-5952	161	9	=	=	SYM
ejpam-5952	161	10	0	0	NUM
ejpam-5952	161	11	is	be	AUX
ejpam-5952	161	12	a	a	DET
ejpam-5952	161	13	fixed	fix	VERB
ejpam-5952	161	14	point	point	NOUN
ejpam-5952	161	15	of	of	ADP
ejpam-5952	161	16	t	t	PROPN
ejpam-5952	161	17	.	.	PUNCT
ejpam-5952	162	1	finally	finally	ADV
ejpam-5952	162	2	,	,	PUNCT
ejpam-5952	162	3	since	since	SCONJ
ejpam-5952	162	4	the	the	DET
ejpam-5952	162	5	measure	measure	NOUN
ejpam-5952	162	6	µ	µ	NOUN
ejpam-5952	162	7	is	be	AUX
ejpam-5952	162	8	σ	σ	NOUN
ejpam-5952	162	9	-	-	NOUN
ejpam-5952	162	10	finite	finite	NOUN
ejpam-5952	162	11	,	,	PUNCT
ejpam-5952	162	12	and	and	CCONJ
ejpam-5952	162	13	υn	υn	X
ejpam-5952	162	14	→	→	SYM
ejpam-5952	162	15	υ∗	υ∗	NOUN
ejpam-5952	162	16	in	in	ADP
ejpam-5952	162	17	the	the	DET
ejpam-5952	162	18	mr	mr	PROPN
ejpam-5952	162	19	-	-	PUNCT
ejpam-5952	162	20	metric	metric	NOUN
ejpam-5952	162	21	,	,	PUNCT
ejpam-5952	162	22	we	we	PRON
ejpam-5952	162	23	conclude	conclude	VERB
ejpam-5952	162	24	that	that	PRON
ejpam-5952	162	25	υn	υn	NOUN
ejpam-5952	162	26	→	→	SYM
ejpam-5952	162	27	υ∗	υ∗	NOUN
ejpam-5952	162	28	in	in	ADP
ejpam-5952	162	29	measure	measure	NOUN
ejpam-5952	162	30	.	.	PUNCT
ejpam-5952	163	1	3	3	X
ejpam-5952	163	2	.	.	X
ejpam-5952	163	3	applications	application	NOUN
ejpam-5952	163	4	in	in	ADP
ejpam-5952	163	5	this	this	DET
ejpam-5952	163	6	section	section	NOUN
ejpam-5952	163	7	we	we	PRON
ejpam-5952	163	8	present	present	VERB
ejpam-5952	163	9	some	some	DET
ejpam-5952	163	10	important	important	ADJ
ejpam-5952	163	11	applications	application	NOUN
ejpam-5952	163	12	of	of	ADP
ejpam-5952	163	13	our	our	PRON
ejpam-5952	163	14	main	main	ADJ
ejpam-5952	163	15	results	result	NOUN
ejpam-5952	163	16	.	.	PUNCT
ejpam-5952	164	1	example	example	NOUN
ejpam-5952	164	2	4	4	NUM
ejpam-5952	164	3	.	.	PUNCT
ejpam-5952	165	1	consider	consider	VERB
ejpam-5952	165	2	the	the	DET
ejpam-5952	165	3	mr	mr	PROPN
ejpam-5952	165	4	-	-	PUNCT
ejpam-5952	165	5	metric	metric	ADJ
ejpam-5952	165	6	space	space	NOUN
ejpam-5952	165	7	(	(	PUNCT
ejpam-5952	165	8	rn	rn	PROPN
ejpam-5952	165	9	,	,	PUNCT
ejpam-5952	165	10	m	m	NOUN
ejpam-5952	165	11	)	)	PUNCT
ejpam-5952	165	12	,	,	PUNCT
ejpam-5952	165	13	where	where	SCONJ
ejpam-5952	165	14	the	the	DET
ejpam-5952	165	15	mr	mr	PROPN
ejpam-5952	165	16	-	-	PUNCT
ejpam-5952	165	17	metric	metric	ADJ
ejpam-5952	165	18	m	m	VERB
ejpam-5952	165	19	is	be	AUX
ejpam-5952	165	20	defined	define	VERB
ejpam-5952	165	21	by	by	ADP
ejpam-5952	165	22	:	:	PUNCT
ejpam-5952	165	23	m(υ	m(υ	PROPN
ejpam-5952	165	24	,	,	PUNCT
ejpam-5952	165	25	ξ,ℑ	ξ,ℑ	PROPN
ejpam-5952	165	26	)	)	PUNCT
ejpam-5952	166	1	=	=	SYM
ejpam-5952	166	2	∥υ	∥υ	PROPN
ejpam-5952	166	3	−	−	PROPN
ejpam-5952	166	4	ξ∥+	ξ∥+	VERB
ejpam-5952	166	5	∥ξ	∥ξ	PROPN
ejpam-5952	166	6	−ℑ∥+	−ℑ∥+	X
ejpam-5952	166	7	∥ℑ	∥ℑ	ADJ
ejpam-5952	166	8	−	−	PROPN
ejpam-5952	166	9	υ∥.	υ∥.	PROPN
ejpam-5952	166	10	a.	a.	NOUN
ejpam-5952	166	11	a.	a.	NOUN
ejpam-5952	166	12	m.	m.	NOUN
ejpam-5952	166	13	malkawi	malkawi	ADP
ejpam-5952	166	14	/	/	SYM
ejpam-5952	166	15	eur	eur	PROPN
ejpam-5952	166	16	.	.	PUNCT
ejpam-5952	167	1	j.	j.	PROPN
ejpam-5952	167	2	pure	pure	PROPN
ejpam-5952	167	3	appl	appl	PROPN
ejpam-5952	167	4	.	.	PROPN
ejpam-5952	167	5	math	math	PROPN
ejpam-5952	167	6	,	,	PUNCT
ejpam-5952	167	7	18	18	NUM
ejpam-5952	167	8	(	(	PUNCT
ejpam-5952	167	9	2	2	NUM
ejpam-5952	167	10	)	)	PUNCT
ejpam-5952	167	11	(	(	PUNCT
ejpam-5952	167	12	2025	2025	NUM
ejpam-5952	167	13	)	)	PUNCT
ejpam-5952	167	14	,	,	PUNCT
ejpam-5952	167	15	5952	5952	NUM
ejpam-5952	167	16	10	10	NUM
ejpam-5952	167	17	of	of	ADP
ejpam-5952	167	18	14	14	NUM
ejpam-5952	167	19	here	here	ADV
ejpam-5952	167	20	,	,	PUNCT
ejpam-5952	167	21	∥	∥	PROPN
ejpam-5952	167	22	·	·	PUNCT
ejpam-5952	168	1	∥	∥	PRON
ejpam-5952	168	2	represents	represent	VERB
ejpam-5952	168	3	the	the	DET
ejpam-5952	168	4	euclidean	euclidean	ADJ
ejpam-5952	168	5	norm	norm	NOUN
ejpam-5952	168	6	in	in	ADP
ejpam-5952	168	7	rn	rn	PROPN
ejpam-5952	168	8	.	.	PUNCT
ejpam-5952	169	1	the	the	DET
ejpam-5952	169	2	mr	mr	PROPN
ejpam-5952	169	3	-	-	PUNCT
ejpam-5952	169	4	metric	metric	NOUN
ejpam-5952	169	5	is	be	AUX
ejpam-5952	169	6	useful	useful	ADJ
ejpam-5952	169	7	for	for	ADP
ejpam-5952	169	8	distance	distance	NOUN
ejpam-5952	169	9	measurement	measurement	NOUN
ejpam-5952	169	10	between	between	ADP
ejpam-5952	169	11	three	three	NUM
ejpam-5952	169	12	points	point	NOUN
ejpam-5952	169	13	,	,	PUNCT
ejpam-5952	169	14	which	which	PRON
ejpam-5952	169	15	is	be	AUX
ejpam-5952	169	16	relevant	relevant	ADJ
ejpam-5952	169	17	in	in	ADP
ejpam-5952	169	18	analyzing	analyze	VERB
ejpam-5952	169	19	iterative	iterative	NOUN
ejpam-5952	169	20	methods	method	NOUN
ejpam-5952	169	21	.	.	PUNCT
ejpam-5952	170	1	next	next	ADV
ejpam-5952	170	2	,	,	PUNCT
ejpam-5952	170	3	consider	consider	VERB
ejpam-5952	170	4	the	the	DET
ejpam-5952	170	5	self	self	NOUN
ejpam-5952	170	6	-	-	PUNCT
ejpam-5952	170	7	mapping	mapping	NOUN
ejpam-5952	170	8	t	t	NOUN
ejpam-5952	170	9	:	:	PUNCT
ejpam-5952	170	10	rn	rn	PROPN
ejpam-5952	170	11	→	→	SYM
ejpam-5952	170	12	rn	rn	PROPN
ejpam-5952	170	13	given	give	VERB
ejpam-5952	170	14	by	by	ADP
ejpam-5952	170	15	:	:	PUNCT
ejpam-5952	170	16	t	t	PROPN
ejpam-5952	170	17	(	(	PUNCT
ejpam-5952	170	18	υ	υ	NOUN
ejpam-5952	170	19	)	)	PUNCT
ejpam-5952	170	20	=	=	SYM
ejpam-5952	170	21	υ	υ	DET
ejpam-5952	170	22	2	2	NUM
ejpam-5952	170	23	.	.	PUNCT
ejpam-5952	171	1	this	this	PRON
ejpam-5952	171	2	is	be	AUX
ejpam-5952	171	3	a	a	DET
ejpam-5952	171	4	contraction	contraction	NOUN
ejpam-5952	171	5	mapping	mapping	NOUN
ejpam-5952	171	6	because	because	SCONJ
ejpam-5952	171	7	it	it	PRON
ejpam-5952	171	8	scales	scale	VERB
ejpam-5952	171	9	any	any	DET
ejpam-5952	171	10	point	point	NOUN
ejpam-5952	171	11	υ	υ	PRON
ejpam-5952	171	12	∈	∈	PROPN
ejpam-5952	171	13	rn	rn	NOUN
ejpam-5952	171	14	by	by	ADP
ejpam-5952	171	15	1	1	NUM
ejpam-5952	171	16	2	2	NUM
ejpam-5952	171	17	,	,	PUNCT
ejpam-5952	171	18	and	and	CCONJ
ejpam-5952	171	19	such	such	ADJ
ejpam-5952	171	20	transformations	transformation	NOUN
ejpam-5952	171	21	are	be	AUX
ejpam-5952	171	22	frequently	frequently	ADV
ejpam-5952	171	23	used	use	VERB
ejpam-5952	171	24	in	in	ADP
ejpam-5952	171	25	optimization	optimization	NOUN
ejpam-5952	171	26	algorithms	algorithm	NOUN
ejpam-5952	171	27	like	like	ADP
ejpam-5952	171	28	gradient	gradient	ADJ
ejpam-5952	171	29	descent	descent	NOUN
ejpam-5952	171	30	,	,	PUNCT
ejpam-5952	171	31	where	where	SCONJ
ejpam-5952	171	32	the	the	DET
ejpam-5952	171	33	goal	goal	NOUN
ejpam-5952	171	34	is	be	AUX
ejpam-5952	171	35	to	to	PART
ejpam-5952	171	36	iteratively	iteratively	ADV
ejpam-5952	171	37	improve	improve	VERB
ejpam-5952	171	38	estimates	estimate	NOUN
ejpam-5952	171	39	toward	toward	ADP
ejpam-5952	171	40	an	an	DET
ejpam-5952	171	41	optimal	optimal	ADJ
ejpam-5952	171	42	solution	solution	NOUN
ejpam-5952	171	43	.	.	PUNCT
ejpam-5952	172	1	we	we	PRON
ejpam-5952	172	2	now	now	ADV
ejpam-5952	172	3	check	check	VERB
ejpam-5952	172	4	if	if	SCONJ
ejpam-5952	172	5	t	t	PROPN
ejpam-5952	172	6	satisfies	satisfy	VERB
ejpam-5952	172	7	the	the	DET
ejpam-5952	172	8	contraction	contraction	NOUN
ejpam-5952	172	9	condition	condition	NOUN
ejpam-5952	172	10	from	from	ADP
ejpam-5952	172	11	theorem	theorem	ADJ
ejpam-5952	172	12	2.1	2.1	NUM
ejpam-5952	172	13	:	:	PUNCT
ejpam-5952	172	14	m(tυ	m(tυ	NOUN
ejpam-5952	172	15	,	,	PUNCT
ejpam-5952	172	16	tξ	tξ	VERB
ejpam-5952	172	17	,	,	PUNCT
ejpam-5952	172	18	tℑ	tℑ	NOUN
ejpam-5952	172	19	)	)	PUNCT
ejpam-5952	172	20	≤	≤	NOUN
ejpam-5952	172	21	αm(υ	αm(υ	NUM
ejpam-5952	172	22	,	,	PUNCT
ejpam-5952	172	23	ξ,ℑ	ξ,ℑ	PROPN
ejpam-5952	172	24	)	)	PUNCT
ejpam-5952	172	25	,	,	PUNCT
ejpam-5952	172	26	∀υ	∀υ	NOUN
ejpam-5952	172	27	,	,	PUNCT
ejpam-5952	172	28	ξ,ℑ	ξ,ℑ	PROPN
ejpam-5952	172	29	∈	∈	PROPN
ejpam-5952	172	30	rn	rn	PROPN
ejpam-5952	172	31	.	.	PUNCT
ejpam-5952	173	1	substituting	substitute	VERB
ejpam-5952	173	2	t	t	PROPN
ejpam-5952	173	3	(	(	PUNCT
ejpam-5952	173	4	υ	υ	NOUN
ejpam-5952	173	5	)	)	PUNCT
ejpam-5952	173	6	=	=	SYM
ejpam-5952	173	7	υ	υ	PROPN
ejpam-5952	173	8	2	2	NUM
ejpam-5952	173	9	,	,	PUNCT
ejpam-5952	173	10	we	we	PRON
ejpam-5952	173	11	get	get	VERB
ejpam-5952	173	12	:	:	PUNCT
ejpam-5952	173	13	m(tυ	m(tυ	NOUN
ejpam-5952	173	14	,	,	PUNCT
ejpam-5952	173	15	tξ	tξ	VERB
ejpam-5952	173	16	,	,	PUNCT
ejpam-5952	173	17	tℑ	tℑ	NOUN
ejpam-5952	173	18	)	)	PUNCT
ejpam-5952	173	19	=	=	PUNCT
ejpam-5952	174	1	∥∥∥∥υ2	∥∥∥∥υ2	PROPN
ejpam-5952	174	2	−	−	ADP
ejpam-5952	174	3	ξ	ξ	SYM
ejpam-5952	174	4	2	2	NUM
ejpam-5952	174	5	∥∥∥∥+	∥∥∥∥+	NOUN
ejpam-5952	174	6	∥∥∥∥ξ2	∥∥∥∥ξ2	NOUN
ejpam-5952	174	7	−	−	NOUN
ejpam-5952	174	8	ℑ	ℑ	PROPN
ejpam-5952	174	9	2	2	NUM
ejpam-5952	174	10	∥∥∥∥+	∥∥∥∥+	NOUN
ejpam-5952	174	11	∥∥∥∥ℑ2	∥∥∥∥ℑ2	PROPN
ejpam-5952	174	12	−	−	NOUN
ejpam-5952	174	13	υ	υ	PROPN
ejpam-5952	174	14	2	2	NUM
ejpam-5952	174	15	∥∥∥∥	∥∥∥∥	NUM
ejpam-5952	174	16	.	.	PUNCT
ejpam-5952	175	1	since	since	SCONJ
ejpam-5952	175	2	the	the	DET
ejpam-5952	175	3	euclidean	euclidean	ADJ
ejpam-5952	175	4	norm	norm	NOUN
ejpam-5952	175	5	scales	scale	VERB
ejpam-5952	175	6	linearly	linearly	ADV
ejpam-5952	175	7	,	,	PUNCT
ejpam-5952	175	8	this	this	DET
ejpam-5952	175	9	simplifies	simplifie	NOUN
ejpam-5952	175	10	to	to	PART
ejpam-5952	175	11	:	:	PUNCT
ejpam-5952	175	12	m(tυ	m(tυ	NOUN
ejpam-5952	175	13	,	,	PUNCT
ejpam-5952	175	14	tξ	tξ	VERB
ejpam-5952	175	15	,	,	PUNCT
ejpam-5952	175	16	tℑ	tℑ	NOUN
ejpam-5952	175	17	)	)	PUNCT
ejpam-5952	175	18	=	=	SYM
ejpam-5952	176	1	1	1	NUM
ejpam-5952	176	2	2	2	NUM
ejpam-5952	176	3	m(υ	m(υ	PROPN
ejpam-5952	176	4	,	,	PUNCT
ejpam-5952	176	5	ξ,ℑ	ξ,ℑ	PROPN
ejpam-5952	176	6	)	)	PUNCT
ejpam-5952	176	7	.	.	PUNCT
ejpam-5952	177	1	thus	thus	ADV
ejpam-5952	177	2	,	,	PUNCT
ejpam-5952	177	3	t	t	PROPN
ejpam-5952	177	4	is	be	AUX
ejpam-5952	177	5	a	a	DET
ejpam-5952	177	6	contraction	contraction	NOUN
ejpam-5952	177	7	with	with	ADP
ejpam-5952	177	8	contraction	contraction	NOUN
ejpam-5952	177	9	factor	factor	NOUN
ejpam-5952	177	10	α	α	NOUN
ejpam-5952	177	11	=	=	SYM
ejpam-5952	177	12	1	1	NUM
ejpam-5952	177	13	2	2	NUM
ejpam-5952	177	14	,	,	PUNCT
ejpam-5952	177	15	which	which	PRON
ejpam-5952	177	16	satisfies	satisfy	VERB
ejpam-5952	177	17	α	α	X
ejpam-5952	177	18	<	<	X
ejpam-5952	177	19	1	1	NUM
ejpam-5952	177	20	.	.	PUNCT
ejpam-5952	177	21	according	accord	VERB
ejpam-5952	177	22	to	to	ADP
ejpam-5952	177	23	theorem	theorem	ADJ
ejpam-5952	177	24	2.1	2.1	NUM
ejpam-5952	177	25	,	,	PUNCT
ejpam-5952	177	26	this	this	PRON
ejpam-5952	177	27	guarantees	guarantee	VERB
ejpam-5952	177	28	the	the	DET
ejpam-5952	177	29	existence	existence	NOUN
ejpam-5952	177	30	of	of	ADP
ejpam-5952	177	31	a	a	DET
ejpam-5952	177	32	unique	unique	ADJ
ejpam-5952	177	33	fixed	fix	VERB
ejpam-5952	177	34	point	point	NOUN
ejpam-5952	177	35	υ∗	υ∗	NOUN
ejpam-5952	177	36	such	such	ADJ
ejpam-5952	177	37	that	that	PRON
ejpam-5952	177	38	:	:	PUNCT
ejpam-5952	177	39	t	t	PROPN
ejpam-5952	177	40	(	(	PUNCT
ejpam-5952	177	41	υ∗	υ∗	PROPN
ejpam-5952	177	42	)	)	PUNCT
ejpam-5952	177	43	=	=	SYM
ejpam-5952	177	44	υ∗.	υ∗.	VERB
ejpam-5952	177	45	solving	solve	VERB
ejpam-5952	177	46	t	t	PROPN
ejpam-5952	177	47	(	(	PUNCT
ejpam-5952	177	48	υ∗	υ∗	NOUN
ejpam-5952	177	49	)	)	PUNCT
ejpam-5952	178	1	=	=	SYM
ejpam-5952	178	2	υ∗	υ∗	NOUN
ejpam-5952	178	3	2	2	NUM
ejpam-5952	178	4	=	=	SYM
ejpam-5952	178	5	υ∗	υ∗	NOUN
ejpam-5952	178	6	,	,	PUNCT
ejpam-5952	178	7	we	we	PRON
ejpam-5952	178	8	find	find	VERB
ejpam-5952	178	9	υ∗	υ∗	NOUN
ejpam-5952	178	10	=	=	SYM
ejpam-5952	178	11	0	0	X
ejpam-5952	178	12	.	.	PUNCT
ejpam-5952	179	1	therefore	therefore	ADV
ejpam-5952	179	2	,	,	PUNCT
ejpam-5952	179	3	the	the	DET
ejpam-5952	179	4	sequence	sequence	NOUN
ejpam-5952	179	5	generated	generate	VERB
ejpam-5952	179	6	by	by	ADP
ejpam-5952	179	7	iteratively	iteratively	ADV
ejpam-5952	179	8	applying	apply	VERB
ejpam-5952	179	9	t	t	NOUN
ejpam-5952	179	10	converges	converge	NOUN
ejpam-5952	179	11	to	to	ADP
ejpam-5952	179	12	the	the	DET
ejpam-5952	179	13	fixed	fixed	ADJ
ejpam-5952	179	14	point	point	NOUN
ejpam-5952	179	15	υ∗	υ∗	NOUN
ejpam-5952	179	16	=	=	SYM
ejpam-5952	179	17	0	0	X
ejpam-5952	179	18	.	.	NOUN
ejpam-5952	179	19	practical	practical	ADJ
ejpam-5952	179	20	significance	significance	NOUN
ejpam-5952	179	21	:	:	PUNCT
ejpam-5952	179	22	the	the	DET
ejpam-5952	179	23	mr	mr	PROPN
ejpam-5952	179	24	-	-	PUNCT
ejpam-5952	179	25	metric	metric	NOUN
ejpam-5952	179	26	provides	provide	VERB
ejpam-5952	179	27	a	a	DET
ejpam-5952	179	28	reliable	reliable	ADJ
ejpam-5952	179	29	tool	tool	NOUN
ejpam-5952	179	30	for	for	ADP
ejpam-5952	179	31	proving	prove	VERB
ejpam-5952	179	32	the	the	DET
ejpam-5952	179	33	convergence	convergence	NOUN
ejpam-5952	179	34	of	of	ADP
ejpam-5952	179	35	iterative	iterative	ADJ
ejpam-5952	179	36	algorithms	algorithm	NOUN
ejpam-5952	179	37	,	,	PUNCT
ejpam-5952	179	38	ensuring	ensure	VERB
ejpam-5952	179	39	that	that	SCONJ
ejpam-5952	179	40	they	they	PRON
ejpam-5952	179	41	reach	reach	VERB
ejpam-5952	179	42	a	a	DET
ejpam-5952	179	43	unique	unique	ADJ
ejpam-5952	179	44	fixed	fix	VERB
ejpam-5952	179	45	point	point	NOUN
ejpam-5952	179	46	.	.	PUNCT
ejpam-5952	180	1	this	this	PRON
ejpam-5952	180	2	is	be	AUX
ejpam-5952	180	3	particularly	particularly	ADV
ejpam-5952	180	4	important	important	ADJ
ejpam-5952	180	5	in	in	ADP
ejpam-5952	180	6	optimization	optimization	NOUN
ejpam-5952	180	7	,	,	PUNCT
ejpam-5952	180	8	as	as	SCONJ
ejpam-5952	180	9	it	it	PRON
ejpam-5952	180	10	supports	support	VERB
ejpam-5952	180	11	the	the	DET
ejpam-5952	180	12	stability	stability	NOUN
ejpam-5952	180	13	and	and	CCONJ
ejpam-5952	180	14	efficiency	efficiency	NOUN
ejpam-5952	180	15	of	of	ADP
ejpam-5952	180	16	algorithms	algorithm	NOUN
ejpam-5952	180	17	used	use	VERB
ejpam-5952	180	18	in	in	ADP
ejpam-5952	180	19	fields	field	NOUN
ejpam-5952	180	20	like	like	ADP
ejpam-5952	180	21	machine	machine	NOUN
ejpam-5952	180	22	learning	learning	NOUN
ejpam-5952	180	23	and	and	CCONJ
ejpam-5952	180	24	numerical	numerical	ADJ
ejpam-5952	180	25	analysis	analysis	NOUN
ejpam-5952	180	26	.	.	PUNCT
ejpam-5952	180	27	example	example	NOUN
ejpam-5952	181	1	5	5	NUM
ejpam-5952	181	2	.	.	PUNCT
ejpam-5952	182	1	in	in	ADP
ejpam-5952	182	2	the	the	DET
ejpam-5952	182	3	context	context	NOUN
ejpam-5952	182	4	of	of	ADP
ejpam-5952	182	5	signal	signal	NOUN
ejpam-5952	182	6	processing	processing	NOUN
ejpam-5952	182	7	,	,	PUNCT
ejpam-5952	182	8	let	let	VERB
ejpam-5952	182	9	x	x	PRON
ejpam-5952	182	10	represent	represent	VERB
ejpam-5952	182	11	a	a	DET
ejpam-5952	182	12	space	space	NOUN
ejpam-5952	182	13	consisting	consist	VERB
ejpam-5952	182	14	of	of	ADP
ejpam-5952	182	15	signals	signal	NOUN
ejpam-5952	182	16	defined	define	VERB
ejpam-5952	182	17	over	over	ADP
ejpam-5952	182	18	the	the	DET
ejpam-5952	182	19	time	time	NOUN
ejpam-5952	182	20	interval	interval	NOUN
ejpam-5952	182	21	[	[	X
ejpam-5952	182	22	0	0	NUM
ejpam-5952	182	23	,	,	PUNCT
ejpam-5952	182	24	t	t	X
ejpam-5952	182	25	]	]	PUNCT
ejpam-5952	182	26	.	.	PUNCT
ejpam-5952	183	1	to	to	PART
ejpam-5952	183	2	quantify	quantify	VERB
ejpam-5952	183	3	the	the	DET
ejpam-5952	183	4	variation	variation	NOUN
ejpam-5952	183	5	among	among	ADP
ejpam-5952	183	6	three	three	NUM
ejpam-5952	183	7	signals	signal	NOUN
ejpam-5952	183	8	,	,	PUNCT
ejpam-5952	183	9	we	we	PRON
ejpam-5952	183	10	define	define	VERB
ejpam-5952	183	11	the	the	DET
ejpam-5952	183	12	mr	mr	PROPN
ejpam-5952	183	13	-	-	PUNCT
ejpam-5952	183	14	metric	metric	NOUN
ejpam-5952	183	15	as	as	SCONJ
ejpam-5952	183	16	follows	follow	VERB
ejpam-5952	183	17	:	:	PUNCT
ejpam-5952	183	18	m(υ	m(υ	PROPN
ejpam-5952	183	19	,	,	PUNCT
ejpam-5952	183	20	ξ,ℑ	ξ,ℑ	PROPN
ejpam-5952	183	21	)	)	PUNCT
ejpam-5952	184	1	=	=	SYM
ejpam-5952	185	1	∫	∫	PROPN
ejpam-5952	185	2	t	t	PROPN
ejpam-5952	185	3	0	0	NUM
ejpam-5952	185	4	|υ(t)−	|υ(t)−	PROPN
ejpam-5952	185	5	ξ(t)|+	ξ(t)|+	PROPN
ejpam-5952	186	1	|ξ(t)−ℑ(t)|+	|ξ(t)−ℑ(t)|+	PROPN
ejpam-5952	186	2	|ℑ(t)−	|ℑ(t)−	PROPN
ejpam-5952	186	3	υ(t)|	υ(t)|	NOUN
ejpam-5952	187	1	dt	dt	PROPN
ejpam-5952	187	2	,	,	PUNCT
ejpam-5952	187	3	where	where	SCONJ
ejpam-5952	187	4	υ(t	υ(t	NOUN
ejpam-5952	187	5	)	)	PUNCT
ejpam-5952	187	6	,	,	PUNCT
ejpam-5952	187	7	ξ(t	ξ(t	NOUN
ejpam-5952	187	8	)	)	PUNCT
ejpam-5952	187	9	,	,	PUNCT
ejpam-5952	187	10	and	and	CCONJ
ejpam-5952	187	11	ℑ(t	ℑ(t	NOUN
ejpam-5952	187	12	)	)	PUNCT
ejpam-5952	187	13	are	be	AUX
ejpam-5952	187	14	real	real	ADV
ejpam-5952	187	15	-	-	PUNCT
ejpam-5952	187	16	valued	value	VERB
ejpam-5952	187	17	functions	function	NOUN
ejpam-5952	187	18	over	over	ADP
ejpam-5952	187	19	the	the	DET
ejpam-5952	187	20	interval	interval	NOUN
ejpam-5952	188	1	[	[	X
ejpam-5952	188	2	0	0	NUM
ejpam-5952	188	3	,	,	PUNCT
ejpam-5952	188	4	t	t	X
ejpam-5952	188	5	]	]	PUNCT
ejpam-5952	188	6	.	.	PUNCT
ejpam-5952	189	1	this	this	DET
ejpam-5952	189	2	metric	metric	ADJ
ejpam-5952	189	3	captures	capture	VERB
ejpam-5952	189	4	the	the	DET
ejpam-5952	189	5	total	total	ADJ
ejpam-5952	189	6	deviation	deviation	NOUN
ejpam-5952	189	7	among	among	ADP
ejpam-5952	189	8	the	the	DET
ejpam-5952	189	9	three	three	NUM
ejpam-5952	189	10	signals	signal	NOUN
ejpam-5952	189	11	throughout	throughout	ADP
ejpam-5952	189	12	the	the	DET
ejpam-5952	189	13	specified	specified	ADJ
ejpam-5952	189	14	time	time	NOUN
ejpam-5952	189	15	domain	domain	NOUN
ejpam-5952	189	16	,	,	PUNCT
ejpam-5952	189	17	making	make	VERB
ejpam-5952	189	18	it	it	PRON
ejpam-5952	189	19	a	a	DET
ejpam-5952	189	20	useful	useful	ADJ
ejpam-5952	189	21	tool	tool	NOUN
ejpam-5952	189	22	for	for	ADP
ejpam-5952	189	23	comparing	compare	VERB
ejpam-5952	189	24	signal	signal	NOUN
ejpam-5952	189	25	differences	difference	NOUN
ejpam-5952	189	26	.	.	PUNCT
ejpam-5952	190	1	a	a	DET
ejpam-5952	190	2	significant	significant	ADJ
ejpam-5952	190	3	application	application	NOUN
ejpam-5952	190	4	of	of	ADP
ejpam-5952	190	5	this	this	DET
ejpam-5952	190	6	mr	mr	PROPN
ejpam-5952	190	7	-	-	PUNCT
ejpam-5952	190	8	metric	metric	NOUN
ejpam-5952	190	9	is	be	AUX
ejpam-5952	190	10	in	in	ADP
ejpam-5952	190	11	various	various	ADJ
ejpam-5952	190	12	signal	signal	NOUN
ejpam-5952	190	13	processing	processing	NOUN
ejpam-5952	190	14	techniques	technique	NOUN
ejpam-5952	190	15	,	,	PUNCT
ejpam-5952	190	16	such	such	ADJ
ejpam-5952	190	17	as	as	ADP
ejpam-5952	190	18	denoising	denoising	NOUN
ejpam-5952	190	19	,	,	PUNCT
ejpam-5952	190	20	compression	compression	NOUN
ejpam-5952	190	21	,	,	PUNCT
ejpam-5952	190	22	and	and	CCONJ
ejpam-5952	190	23	energy	energy	NOUN
ejpam-5952	190	24	optimization	optimization	NOUN
ejpam-5952	190	25	.	.	PUNCT
ejpam-5952	191	1	these	these	DET
ejpam-5952	191	2	methods	method	NOUN
ejpam-5952	191	3	often	often	ADV
ejpam-5952	191	4	rely	rely	VERB
ejpam-5952	191	5	on	on	ADP
ejpam-5952	191	6	a.	a.	NOUN
ejpam-5952	191	7	a.	a.	NOUN
ejpam-5952	191	8	m.	m.	NOUN
ejpam-5952	191	9	malkawi	malkawi	ADP
ejpam-5952	191	10	/	/	SYM
ejpam-5952	191	11	eur	eur	PROPN
ejpam-5952	191	12	.	.	PUNCT
ejpam-5952	192	1	j.	j.	PROPN
ejpam-5952	192	2	pure	pure	PROPN
ejpam-5952	192	3	appl	appl	PROPN
ejpam-5952	192	4	.	.	PROPN
ejpam-5952	192	5	math	math	PROPN
ejpam-5952	192	6	,	,	PUNCT
ejpam-5952	192	7	18	18	NUM
ejpam-5952	192	8	(	(	PUNCT
ejpam-5952	192	9	2	2	NUM
ejpam-5952	192	10	)	)	PUNCT
ejpam-5952	192	11	(	(	PUNCT
ejpam-5952	192	12	2025	2025	NUM
ejpam-5952	192	13	)	)	PUNCT
ejpam-5952	192	14	,	,	PUNCT
ejpam-5952	192	15	5952	5952	NUM
ejpam-5952	192	16	11	11	NUM
ejpam-5952	192	17	of	of	ADP
ejpam-5952	192	18	14	14	NUM
ejpam-5952	192	19	iterative	iterative	NOUN
ejpam-5952	192	20	procedures	procedure	NOUN
ejpam-5952	192	21	aimed	aim	VERB
ejpam-5952	192	22	at	at	ADP
ejpam-5952	192	23	improving	improve	VERB
ejpam-5952	192	24	signal	signal	ADJ
ejpam-5952	192	25	quality	quality	NOUN
ejpam-5952	192	26	,	,	PUNCT
ejpam-5952	192	27	whether	whether	SCONJ
ejpam-5952	192	28	by	by	ADP
ejpam-5952	192	29	reducing	reduce	VERB
ejpam-5952	192	30	noise	noise	NOUN
ejpam-5952	192	31	,	,	PUNCT
ejpam-5952	192	32	enhancing	enhance	VERB
ejpam-5952	192	33	representation	representation	NOUN
ejpam-5952	192	34	,	,	PUNCT
ejpam-5952	192	35	or	or	CCONJ
ejpam-5952	192	36	minimizing	minimize	VERB
ejpam-5952	192	37	energy	energy	NOUN
ejpam-5952	192	38	usage	usage	NOUN
ejpam-5952	192	39	.	.	PUNCT
ejpam-5952	193	1	the	the	DET
ejpam-5952	193	2	iterative	iterative	NOUN
ejpam-5952	193	3	transformations	transformation	NOUN
ejpam-5952	193	4	involved	involve	VERB
ejpam-5952	193	5	in	in	ADP
ejpam-5952	193	6	these	these	DET
ejpam-5952	193	7	processes	process	NOUN
ejpam-5952	193	8	can	can	AUX
ejpam-5952	193	9	be	be	AUX
ejpam-5952	193	10	effectively	effectively	ADV
ejpam-5952	193	11	represented	represent	VERB
ejpam-5952	193	12	using	use	VERB
ejpam-5952	193	13	self	self	NOUN
ejpam-5952	193	14	-	-	PUNCT
ejpam-5952	193	15	mappings	mapping	NOUN
ejpam-5952	193	16	.	.	PUNCT
ejpam-5952	194	1	consider	consider	VERB
ejpam-5952	194	2	the	the	DET
ejpam-5952	194	3	self	self	NOUN
ejpam-5952	194	4	-	-	PUNCT
ejpam-5952	194	5	mapping	mapping	NOUN
ejpam-5952	194	6	t	t	NOUN
ejpam-5952	194	7	:	:	PUNCT
ejpam-5952	194	8	x	x	X
ejpam-5952	194	9	→	→	PUNCT
ejpam-5952	194	10	x	x	PUNCT
ejpam-5952	194	11	given	give	VERB
ejpam-5952	194	12	by	by	ADP
ejpam-5952	194	13	:	:	PUNCT
ejpam-5952	194	14	t	t	PROPN
ejpam-5952	194	15	(	(	PUNCT
ejpam-5952	194	16	υ	υ	NOUN
ejpam-5952	194	17	)	)	PUNCT
ejpam-5952	194	18	=	=	SYM
ejpam-5952	194	19	υ	υ	DET
ejpam-5952	194	20	2	2	NUM
ejpam-5952	194	21	.	.	PUNCT
ejpam-5952	195	1	(	(	PUNCT
ejpam-5952	195	2	33	33	NUM
ejpam-5952	195	3	)	)	PUNCT
ejpam-5952	195	4	this	this	DET
ejpam-5952	195	5	operator	operator	NOUN
ejpam-5952	195	6	scales	scale	VERB
ejpam-5952	195	7	the	the	DET
ejpam-5952	195	8	amplitude	amplitude	NOUN
ejpam-5952	195	9	of	of	ADP
ejpam-5952	195	10	a	a	DET
ejpam-5952	195	11	signal	signal	NOUN
ejpam-5952	195	12	by	by	ADP
ejpam-5952	195	13	a	a	DET
ejpam-5952	195	14	factor	factor	NOUN
ejpam-5952	195	15	of	of	ADP
ejpam-5952	195	16	1/2	1/2	NUM
ejpam-5952	195	17	.	.	PUNCT
ejpam-5952	196	1	such	such	DET
ejpam-5952	196	2	an	an	DET
ejpam-5952	196	3	operation	operation	NOUN
ejpam-5952	196	4	is	be	AUX
ejpam-5952	196	5	commonly	commonly	ADV
ejpam-5952	196	6	employed	employ	VERB
ejpam-5952	196	7	in	in	ADP
ejpam-5952	196	8	signal	signal	ADJ
ejpam-5952	196	9	processing	processing	NOUN
ejpam-5952	196	10	algorithms	algorithm	NOUN
ejpam-5952	196	11	where	where	SCONJ
ejpam-5952	196	12	gradual	gradual	ADJ
ejpam-5952	196	13	attenuation	attenuation	NOUN
ejpam-5952	196	14	of	of	ADP
ejpam-5952	196	15	a	a	DET
ejpam-5952	196	16	signal	signal	NOUN
ejpam-5952	196	17	is	be	AUX
ejpam-5952	196	18	required	require	VERB
ejpam-5952	196	19	,	,	PUNCT
ejpam-5952	196	20	for	for	ADP
ejpam-5952	196	21	instance	instance	NOUN
ejpam-5952	196	22	,	,	PUNCT
ejpam-5952	196	23	to	to	PART
ejpam-5952	196	24	suppress	suppress	VERB
ejpam-5952	196	25	noise	noise	NOUN
ejpam-5952	196	26	or	or	CCONJ
ejpam-5952	196	27	reduce	reduce	VERB
ejpam-5952	196	28	energy	energy	NOUN
ejpam-5952	196	29	levels	level	NOUN
ejpam-5952	196	30	.	.	PUNCT
ejpam-5952	197	1	a	a	DET
ejpam-5952	197	2	practical	practical	ADJ
ejpam-5952	197	3	interpretation	interpretation	NOUN
ejpam-5952	197	4	of	of	ADP
ejpam-5952	197	5	this	this	DET
ejpam-5952	197	6	transformation	transformation	NOUN
ejpam-5952	197	7	is	be	AUX
ejpam-5952	197	8	in	in	ADP
ejpam-5952	197	9	the	the	DET
ejpam-5952	197	10	context	context	NOUN
ejpam-5952	197	11	of	of	ADP
ejpam-5952	197	12	iterative	iterative	NOUN
ejpam-5952	197	13	noise	noise	NOUN
ejpam-5952	197	14	suppression	suppression	NOUN
ejpam-5952	197	15	.	.	PUNCT
ejpam-5952	198	1	in	in	ADP
ejpam-5952	198	2	such	such	ADJ
ejpam-5952	198	3	scenarios	scenario	NOUN
ejpam-5952	198	4	,	,	PUNCT
ejpam-5952	198	5	unwanted	unwanted	ADJ
ejpam-5952	198	6	signal	signal	NOUN
ejpam-5952	198	7	fluctuations	fluctuation	NOUN
ejpam-5952	198	8	(	(	PUNCT
ejpam-5952	198	9	noise	noise	NOUN
ejpam-5952	198	10	)	)	PUNCT
ejpam-5952	198	11	are	be	AUX
ejpam-5952	198	12	progressively	progressively	ADV
ejpam-5952	198	13	diminished	diminish	VERB
ejpam-5952	198	14	through	through	ADP
ejpam-5952	198	15	repeated	repeat	VERB
ejpam-5952	198	16	applications	application	NOUN
ejpam-5952	198	17	of	of	ADP
ejpam-5952	198	18	t	t	PROPN
ejpam-5952	198	19	.	.	PUNCT
ejpam-5952	199	1	mathematically	mathematically	ADV
ejpam-5952	199	2	,	,	PUNCT
ejpam-5952	199	3	applying	apply	VERB
ejpam-5952	199	4	t	t	NOUN
ejpam-5952	199	5	iteratively	iteratively	ADV
ejpam-5952	199	6	to	to	ADP
ejpam-5952	199	7	an	an	DET
ejpam-5952	199	8	initial	initial	ADJ
ejpam-5952	199	9	signal	signal	ADJ
ejpam-5952	199	10	υ0	υ0	NOUN
ejpam-5952	199	11	generates	generate	VERB
ejpam-5952	199	12	the	the	DET
ejpam-5952	199	13	sequence	sequence	NOUN
ejpam-5952	199	14	:	:	PUNCT
ejpam-5952	199	15	υn	υn	NOUN
ejpam-5952	199	16	=	=	SYM
ejpam-5952	199	17	υ0	υ0	PROPN
ejpam-5952	199	18	2n	2n	NUM
ejpam-5952	199	19	.	.	PUNCT
ejpam-5952	200	1	as	as	ADP
ejpam-5952	200	2	n	n	NUM
ejpam-5952	200	3	→	→	SYM
ejpam-5952	200	4	∞	∞	PROPN
ejpam-5952	200	5	,	,	PUNCT
ejpam-5952	200	6	this	this	DET
ejpam-5952	200	7	sequence	sequence	NOUN
ejpam-5952	200	8	converges	converge	VERB
ejpam-5952	200	9	to	to	ADP
ejpam-5952	200	10	:	:	PUNCT
ejpam-5952	200	11	υ∗	υ∗	NOUN
ejpam-5952	200	12	=	=	SYM
ejpam-5952	200	13	0	0	X
ejpam-5952	200	14	.	.	PUNCT
ejpam-5952	201	1	this	this	DET
ejpam-5952	201	2	result	result	NOUN
ejpam-5952	201	3	indicates	indicate	VERB
ejpam-5952	201	4	that	that	SCONJ
ejpam-5952	201	5	,	,	PUNCT
ejpam-5952	201	6	given	give	VERB
ejpam-5952	201	7	enough	enough	ADJ
ejpam-5952	201	8	iterations	iteration	NOUN
ejpam-5952	201	9	,	,	PUNCT
ejpam-5952	201	10	the	the	DET
ejpam-5952	201	11	signal	signal	NOUN
ejpam-5952	201	12	will	will	AUX
ejpam-5952	201	13	converge	converge	VERB
ejpam-5952	201	14	to	to	ADP
ejpam-5952	201	15	the	the	DET
ejpam-5952	201	16	zero	zero	NUM
ejpam-5952	201	17	signal	signal	NOUN
ejpam-5952	201	18	,	,	PUNCT
ejpam-5952	201	19	representing	represent	VERB
ejpam-5952	201	20	complete	complete	ADJ
ejpam-5952	201	21	noise	noise	NOUN
ejpam-5952	201	22	suppression	suppression	NOUN
ejpam-5952	201	23	or	or	CCONJ
ejpam-5952	201	24	a	a	DET
ejpam-5952	201	25	fully	fully	ADV
ejpam-5952	201	26	attenuated	attenuate	VERB
ejpam-5952	201	27	signal	signal	NOUN
ejpam-5952	201	28	.	.	PUNCT
ejpam-5952	202	1	from	from	ADP
ejpam-5952	202	2	a	a	DET
ejpam-5952	202	3	theoretical	theoretical	ADJ
ejpam-5952	202	4	standpoint	standpoint	NOUN
ejpam-5952	202	5	,	,	PUNCT
ejpam-5952	202	6	verifying	verify	VERB
ejpam-5952	202	7	the	the	DET
ejpam-5952	202	8	contraction	contraction	NOUN
ejpam-5952	202	9	condition	condition	NOUN
ejpam-5952	202	10	is	be	AUX
ejpam-5952	202	11	crucial	crucial	ADJ
ejpam-5952	202	12	.	.	PUNCT
ejpam-5952	203	1	we	we	PRON
ejpam-5952	203	2	check	check	VERB
ejpam-5952	203	3	whether	whether	SCONJ
ejpam-5952	203	4	t	t	PROPN
ejpam-5952	203	5	satisfies	satisfy	VERB
ejpam-5952	203	6	the	the	DET
ejpam-5952	203	7	mr	mr	PROPN
ejpam-5952	203	8	-	-	PUNCT
ejpam-5952	203	9	metric	metric	ADJ
ejpam-5952	203	10	contraction	contraction	NOUN
ejpam-5952	203	11	property	property	NOUN
ejpam-5952	203	12	:	:	PUNCT
ejpam-5952	203	13	m(tυ	m(tυ	NOUN
ejpam-5952	203	14	,	,	PUNCT
ejpam-5952	203	15	tξ	tξ	VERB
ejpam-5952	203	16	,	,	PUNCT
ejpam-5952	203	17	tℑ	tℑ	NOUN
ejpam-5952	203	18	)	)	PUNCT
ejpam-5952	203	19	≤	≤	NOUN
ejpam-5952	203	20	αm(υ	αm(υ	NUM
ejpam-5952	203	21	,	,	PUNCT
ejpam-5952	203	22	ξ,ℑ	ξ,ℑ	PROPN
ejpam-5952	203	23	)	)	PUNCT
ejpam-5952	203	24	,	,	PUNCT
ejpam-5952	203	25	∀υ	∀υ	NOUN
ejpam-5952	203	26	,	,	PUNCT
ejpam-5952	203	27	ξ,ℑ	ξ,ℑ	PROPN
ejpam-5952	203	28	∈	∈	PROPN
ejpam-5952	203	29	x.	x.	NOUN
ejpam-5952	203	30	substituting	substitute	VERB
ejpam-5952	203	31	t	t	PROPN
ejpam-5952	203	32	(	(	PUNCT
ejpam-5952	203	33	υ	υ	NOUN
ejpam-5952	203	34	)	)	PUNCT
ejpam-5952	203	35	=	=	PUNCT
ejpam-5952	203	36	υ/2	υ/2	VERB
ejpam-5952	203	37	into	into	ADP
ejpam-5952	203	38	the	the	DET
ejpam-5952	203	39	mr	mr	PROPN
ejpam-5952	203	40	-	-	PUNCT
ejpam-5952	203	41	metric	metric	NOUN
ejpam-5952	203	42	,	,	PUNCT
ejpam-5952	203	43	we	we	PRON
ejpam-5952	203	44	compute	compute	VERB
ejpam-5952	203	45	:	:	PUNCT
ejpam-5952	204	1	m(tυ	m(tυ	NOUN
ejpam-5952	204	2	,	,	PUNCT
ejpam-5952	204	3	tξ	tξ	VERB
ejpam-5952	204	4	,	,	PUNCT
ejpam-5952	204	5	tℑ	tℑ	NOUN
ejpam-5952	204	6	)	)	PUNCT
ejpam-5952	204	7	=	=	SYM
ejpam-5952	205	1	∫	∫	PROPN
ejpam-5952	205	2	t	t	PROPN
ejpam-5952	205	3	0	0	NUM
ejpam-5952	205	4	∣∣∣∣υ(t)2	∣∣∣∣υ(t)2	PART
ejpam-5952	205	5	−	−	PROPN
ejpam-5952	205	6	ξ(t	ξ(t	NOUN
ejpam-5952	205	7	)	)	PUNCT
ejpam-5952	205	8	2	2	NUM
ejpam-5952	205	9	∣∣∣∣+	∣∣∣∣+	NOUN
ejpam-5952	205	10	∣∣∣∣ξ(t)2	∣∣∣∣ξ(t)2	PROPN
ejpam-5952	205	11	−	−	PROPN
ejpam-5952	205	12	ℑ(t	ℑ(t	NOUN
ejpam-5952	205	13	)	)	PUNCT
ejpam-5952	205	14	2	2	NUM
ejpam-5952	205	15	∣∣∣∣+	∣∣∣∣+	NOUN
ejpam-5952	205	16	∣∣∣∣ℑ(t)2	∣∣∣∣ℑ(t)2	NOUN
ejpam-5952	205	17	−	−	PROPN
ejpam-5952	205	18	υ(t	υ(t	PROPN
ejpam-5952	205	19	)	)	PUNCT
ejpam-5952	205	20	2	2	NUM
ejpam-5952	205	21	∣∣∣∣	∣∣∣∣	NOUN
ejpam-5952	205	22	dt	dt	NOUN
ejpam-5952	205	23	.	.	PUNCT
ejpam-5952	206	1	factoring	factor	VERB
ejpam-5952	206	2	out	out	ADP
ejpam-5952	206	3	the	the	DET
ejpam-5952	206	4	common	common	ADJ
ejpam-5952	206	5	term	term	NOUN
ejpam-5952	206	6	1/2	1/2	NUM
ejpam-5952	206	7	,	,	PUNCT
ejpam-5952	206	8	we	we	PRON
ejpam-5952	206	9	obtain	obtain	VERB
ejpam-5952	206	10	:	:	PUNCT
ejpam-5952	206	11	m(tυ	m(tυ	NOUN
ejpam-5952	206	12	,	,	PUNCT
ejpam-5952	206	13	tξ	tξ	VERB
ejpam-5952	206	14	,	,	PUNCT
ejpam-5952	206	15	tℑ	tℑ	NOUN
ejpam-5952	206	16	)	)	PUNCT
ejpam-5952	206	17	=	=	SYM
ejpam-5952	206	18	1	1	NUM
ejpam-5952	206	19	2	2	NUM
ejpam-5952	206	20	m(υ	m(υ	PROPN
ejpam-5952	206	21	,	,	PUNCT
ejpam-5952	206	22	ξ,ℑ	ξ,ℑ	PROPN
ejpam-5952	206	23	)	)	PUNCT
ejpam-5952	206	24	.	.	PUNCT
ejpam-5952	207	1	since	since	SCONJ
ejpam-5952	207	2	1	1	NUM
ejpam-5952	207	3	2	2	NUM
ejpam-5952	207	4	<	<	SYM
ejpam-5952	207	5	1	1	NUM
ejpam-5952	207	6	/	/	SYM
ejpam-5952	207	7	r	r	NOUN
ejpam-5952	207	8	for	for	ADP
ejpam-5952	207	9	any	any	DET
ejpam-5952	207	10	r	r	NOUN
ejpam-5952	207	11	>	>	X
ejpam-5952	207	12	1	1	NUM
ejpam-5952	207	13	,	,	PUNCT
ejpam-5952	207	14	the	the	DET
ejpam-5952	207	15	contraction	contraction	NOUN
ejpam-5952	207	16	condition	condition	NOUN
ejpam-5952	207	17	is	be	AUX
ejpam-5952	207	18	satisfied	satisfied	ADJ
ejpam-5952	207	19	,	,	PUNCT
ejpam-5952	207	20	ensuring	ensure	VERB
ejpam-5952	207	21	that	that	SCONJ
ejpam-5952	207	22	t	t	PROPN
ejpam-5952	207	23	has	have	VERB
ejpam-5952	207	24	a	a	DET
ejpam-5952	207	25	unique	unique	ADJ
ejpam-5952	207	26	fixed	fix	VERB
ejpam-5952	207	27	point	point	NOUN
ejpam-5952	207	28	,	,	PUNCT
ejpam-5952	207	29	υ∗	υ∗	NOUN
ejpam-5952	207	30	=	=	SYM
ejpam-5952	207	31	0	0	PROPN
ejpam-5952	207	32	,	,	PUNCT
ejpam-5952	207	33	which	which	PRON
ejpam-5952	207	34	is	be	AUX
ejpam-5952	207	35	reached	reach	VERB
ejpam-5952	207	36	as	as	ADP
ejpam-5952	207	37	the	the	DET
ejpam-5952	207	38	number	number	NOUN
ejpam-5952	207	39	of	of	ADP
ejpam-5952	207	40	iterations	iteration	NOUN
ejpam-5952	207	41	increases	increase	NOUN
ejpam-5952	207	42	.	.	PUNCT
ejpam-5952	208	1	this	this	DET
ejpam-5952	208	2	result	result	NOUN
ejpam-5952	208	3	has	have	VERB
ejpam-5952	208	4	significant	significant	ADJ
ejpam-5952	208	5	implications	implication	NOUN
ejpam-5952	208	6	in	in	ADP
ejpam-5952	208	7	signal	signal	ADJ
ejpam-5952	208	8	processing	processing	NOUN
ejpam-5952	208	9	.	.	PUNCT
ejpam-5952	209	1	for	for	ADP
ejpam-5952	209	2	instance	instance	NOUN
ejpam-5952	209	3	,	,	PUNCT
ejpam-5952	209	4	in	in	ADP
ejpam-5952	209	5	denoising	denoising	NOUN
ejpam-5952	209	6	applications	application	NOUN
ejpam-5952	209	7	,	,	PUNCT
ejpam-5952	209	8	an	an	DET
ejpam-5952	209	9	iterative	iterative	NOUN
ejpam-5952	209	10	algorithm	algorithm	NOUN
ejpam-5952	209	11	that	that	PRON
ejpam-5952	209	12	successively	successively	ADV
ejpam-5952	209	13	applies	apply	VERB
ejpam-5952	209	14	t	t	PROPN
ejpam-5952	209	15	can	can	AUX
ejpam-5952	209	16	remove	remove	VERB
ejpam-5952	209	17	high	high	ADJ
ejpam-5952	209	18	-	-	PUNCT
ejpam-5952	209	19	frequency	frequency	NOUN
ejpam-5952	209	20	noise	noise	NOUN
ejpam-5952	209	21	components	component	NOUN
ejpam-5952	209	22	while	while	SCONJ
ejpam-5952	209	23	preserving	preserve	VERB
ejpam-5952	209	24	the	the	DET
ejpam-5952	209	25	main	main	ADJ
ejpam-5952	209	26	structure	structure	NOUN
ejpam-5952	209	27	of	of	ADP
ejpam-5952	209	28	the	the	DET
ejpam-5952	209	29	signal	signal	NOUN
ejpam-5952	209	30	.	.	PUNCT
ejpam-5952	210	1	similarly	similarly	ADV
ejpam-5952	210	2	,	,	PUNCT
ejpam-5952	210	3	in	in	ADP
ejpam-5952	210	4	compression	compression	NOUN
ejpam-5952	210	5	algorithms	algorithm	NOUN
ejpam-5952	210	6	,	,	PUNCT
ejpam-5952	210	7	gradual	gradual	ADJ
ejpam-5952	210	8	amplitude	amplitude	NOUN
ejpam-5952	210	9	reduction	reduction	NOUN
ejpam-5952	210	10	can	can	AUX
ejpam-5952	210	11	lead	lead	VERB
ejpam-5952	210	12	to	to	ADP
ejpam-5952	210	13	a	a	DET
ejpam-5952	210	14	more	more	ADV
ejpam-5952	210	15	compact	compact	ADJ
ejpam-5952	210	16	and	and	CCONJ
ejpam-5952	210	17	efficient	efficient	ADJ
ejpam-5952	210	18	signal	signal	NOUN
ejpam-5952	210	19	representation	representation	NOUN
ejpam-5952	210	20	.	.	PUNCT
ejpam-5952	211	1	a.	a.	NOUN
ejpam-5952	211	2	a.	a.	PROPN
ejpam-5952	211	3	m.	m.	PROPN
ejpam-5952	211	4	malkawi	malkawi	ADP
ejpam-5952	211	5	/	/	SYM
ejpam-5952	211	6	eur	eur	PROPN
ejpam-5952	211	7	.	.	PUNCT
ejpam-5952	212	1	j.	j.	PROPN
ejpam-5952	212	2	pure	pure	PROPN
ejpam-5952	212	3	appl	appl	PROPN
ejpam-5952	212	4	.	.	PROPN
ejpam-5952	212	5	math	math	PROPN
ejpam-5952	212	6	,	,	PUNCT
ejpam-5952	212	7	18	18	NUM
ejpam-5952	212	8	(	(	PUNCT
ejpam-5952	212	9	2	2	NUM
ejpam-5952	212	10	)	)	PUNCT
ejpam-5952	212	11	(	(	PUNCT
ejpam-5952	212	12	2025	2025	NUM
ejpam-5952	212	13	)	)	PUNCT
ejpam-5952	212	14	,	,	PUNCT
ejpam-5952	212	15	5952	5952	NUM
ejpam-5952	212	16	12	12	NUM
ejpam-5952	212	17	of	of	ADP
ejpam-5952	212	18	14	14	NUM
ejpam-5952	212	19	the	the	DET
ejpam-5952	212	20	role	role	NOUN
ejpam-5952	212	21	of	of	ADP
ejpam-5952	212	22	mr	mr	NOUN
ejpam-5952	212	23	-	-	PUNCT
ejpam-5952	212	24	metrics	metric	NOUN
ejpam-5952	212	25	in	in	ADP
ejpam-5952	212	26	this	this	DET
ejpam-5952	212	27	context	context	NOUN
ejpam-5952	212	28	is	be	AUX
ejpam-5952	212	29	to	to	PART
ejpam-5952	212	30	provide	provide	VERB
ejpam-5952	212	31	a	a	DET
ejpam-5952	212	32	rigorous	rigorous	ADJ
ejpam-5952	212	33	framework	framework	NOUN
ejpam-5952	212	34	for	for	ADP
ejpam-5952	212	35	analyzing	analyze	VERB
ejpam-5952	212	36	the	the	DET
ejpam-5952	212	37	convergence	convergence	NOUN
ejpam-5952	212	38	of	of	ADP
ejpam-5952	212	39	iterative	iterative	ADJ
ejpam-5952	212	40	signal	signal	NOUN
ejpam-5952	212	41	transformations	transformation	NOUN
ejpam-5952	212	42	.	.	PUNCT
ejpam-5952	213	1	by	by	ADP
ejpam-5952	213	2	ensuring	ensure	VERB
ejpam-5952	213	3	that	that	SCONJ
ejpam-5952	213	4	the	the	DET
ejpam-5952	213	5	sequence	sequence	NOUN
ejpam-5952	213	6	of	of	ADP
ejpam-5952	213	7	processed	process	VERB
ejpam-5952	213	8	signals	signal	NOUN
ejpam-5952	213	9	converges	converge	VERB
ejpam-5952	213	10	reliably	reliably	ADV
ejpam-5952	213	11	to	to	ADP
ejpam-5952	213	12	a	a	DET
ejpam-5952	213	13	well	well	ADV
ejpam-5952	213	14	-	-	PUNCT
ejpam-5952	213	15	defined	define	VERB
ejpam-5952	213	16	state	state	NOUN
ejpam-5952	213	17	,	,	PUNCT
ejpam-5952	213	18	mr	mr	PROPN
ejpam-5952	213	19	-	-	PUNCT
ejpam-5952	213	20	metrics	metric	NOUN
ejpam-5952	213	21	help	help	NOUN
ejpam-5952	213	22	in	in	ADP
ejpam-5952	213	23	designing	design	VERB
ejpam-5952	213	24	stable	stable	ADJ
ejpam-5952	213	25	and	and	CCONJ
ejpam-5952	213	26	effective	effective	ADJ
ejpam-5952	213	27	signal	signal	ADJ
ejpam-5952	213	28	processing	processing	NOUN
ejpam-5952	213	29	algorithms	algorithm	NOUN
ejpam-5952	213	30	.	.	PUNCT
ejpam-5952	214	1	whether	whether	SCONJ
ejpam-5952	214	2	in	in	ADP
ejpam-5952	214	3	noise	noise	NOUN
ejpam-5952	214	4	reduction	reduction	NOUN
ejpam-5952	214	5	,	,	PUNCT
ejpam-5952	214	6	compression	compression	NOUN
ejpam-5952	214	7	,	,	PUNCT
ejpam-5952	214	8	or	or	CCONJ
ejpam-5952	214	9	energy	energy	NOUN
ejpam-5952	214	10	optimization	optimization	NOUN
ejpam-5952	214	11	,	,	PUNCT
ejpam-5952	214	12	the	the	DET
ejpam-5952	214	13	application	application	NOUN
ejpam-5952	214	14	of	of	ADP
ejpam-5952	214	15	mr	mr	PROPN
ejpam-5952	214	16	-	-	PUNCT
ejpam-5952	214	17	metrics	metric	NOUN
ejpam-5952	214	18	guarantees	guarantee	VERB
ejpam-5952	214	19	that	that	SCONJ
ejpam-5952	214	20	iterative	iterative	NOUN
ejpam-5952	214	21	methods	method	NOUN
ejpam-5952	214	22	lead	lead	VERB
ejpam-5952	214	23	to	to	ADP
ejpam-5952	214	24	meaningful	meaningful	ADJ
ejpam-5952	214	25	and	and	CCONJ
ejpam-5952	214	26	predictable	predictable	ADJ
ejpam-5952	214	27	outcomes	outcome	NOUN
ejpam-5952	214	28	.	.	PUNCT
ejpam-5952	215	1	4	4	X
ejpam-5952	215	2	.	.	X
ejpam-5952	215	3	conclusion	conclusion	NOUN
ejpam-5952	215	4	in	in	ADP
ejpam-5952	215	5	this	this	DET
ejpam-5952	215	6	paper	paper	NOUN
ejpam-5952	215	7	,	,	PUNCT
ejpam-5952	215	8	we	we	PRON
ejpam-5952	215	9	have	have	AUX
ejpam-5952	215	10	explored	explore	VERB
ejpam-5952	215	11	the	the	DET
ejpam-5952	215	12	existence	existence	NOUN
ejpam-5952	215	13	and	and	CCONJ
ejpam-5952	215	14	uniqueness	uniqueness	NOUN
ejpam-5952	215	15	of	of	ADP
ejpam-5952	215	16	fixed	fix	VERB
ejpam-5952	215	17	points	point	NOUN
ejpam-5952	215	18	for	for	ADP
ejpam-5952	215	19	self	self	NOUN
ejpam-5952	215	20	-	-	PUNCT
ejpam-5952	215	21	mappings	mapping	NOUN
ejpam-5952	215	22	in	in	ADP
ejpam-5952	215	23	mr	mr	PROPN
ejpam-5952	215	24	-	-	PUNCT
ejpam-5952	215	25	metric	metric	ADJ
ejpam-5952	215	26	spaces	space	NOUN
ejpam-5952	215	27	,	,	PUNCT
ejpam-5952	215	28	a	a	DET
ejpam-5952	215	29	generalized	generalize	VERB
ejpam-5952	215	30	framework	framework	NOUN
ejpam-5952	215	31	extending	extend	VERB
ejpam-5952	215	32	traditional	traditional	ADJ
ejpam-5952	215	33	metric	metric	ADJ
ejpam-5952	215	34	spaces	space	NOUN
ejpam-5952	215	35	.	.	PUNCT
ejpam-5952	216	1	by	by	ADP
ejpam-5952	216	2	establishing	establish	VERB
ejpam-5952	216	3	key	key	ADJ
ejpam-5952	216	4	contraction	contraction	NOUN
ejpam-5952	216	5	principles	principle	NOUN
ejpam-5952	216	6	and	and	CCONJ
ejpam-5952	216	7	convergence	convergence	NOUN
ejpam-5952	216	8	criteria	criterion	NOUN
ejpam-5952	216	9	,	,	PUNCT
ejpam-5952	216	10	we	we	PRON
ejpam-5952	216	11	have	have	AUX
ejpam-5952	216	12	demonstrated	demonstrate	VERB
ejpam-5952	216	13	that	that	SCONJ
ejpam-5952	216	14	iterative	iterative	ADJ
ejpam-5952	216	15	sequences	sequence	NOUN
ejpam-5952	216	16	in	in	ADP
ejpam-5952	216	17	mr	mr	PROPN
ejpam-5952	216	18	-	-	PUNCT
ejpam-5952	216	19	metric	metric	ADJ
ejpam-5952	216	20	spaces	space	NOUN
ejpam-5952	216	21	exhibit	exhibit	VERB
ejpam-5952	216	22	stable	stable	ADJ
ejpam-5952	216	23	and	and	CCONJ
ejpam-5952	216	24	predictable	predictable	ADJ
ejpam-5952	216	25	behavior	behavior	NOUN
ejpam-5952	216	26	,	,	PUNCT
ejpam-5952	216	27	leading	lead	VERB
ejpam-5952	216	28	to	to	ADP
ejpam-5952	216	29	unique	unique	ADJ
ejpam-5952	216	30	fixed	fix	VERB
ejpam-5952	216	31	points	point	NOUN
ejpam-5952	216	32	under	under	ADP
ejpam-5952	216	33	well	well	ADV
ejpam-5952	216	34	-	-	PUNCT
ejpam-5952	216	35	defined	define	VERB
ejpam-5952	216	36	conditions	condition	NOUN
ejpam-5952	216	37	.	.	PUNCT
ejpam-5952	217	1	our	our	PRON
ejpam-5952	217	2	results	result	NOUN
ejpam-5952	217	3	provide	provide	VERB
ejpam-5952	217	4	a	a	DET
ejpam-5952	217	5	solid	solid	ADJ
ejpam-5952	217	6	foundation	foundation	NOUN
ejpam-5952	217	7	for	for	ADP
ejpam-5952	217	8	further	further	ADJ
ejpam-5952	217	9	studies	study	NOUN
ejpam-5952	217	10	on	on	ADP
ejpam-5952	217	11	the	the	DET
ejpam-5952	217	12	stability	stability	NOUN
ejpam-5952	217	13	and	and	CCONJ
ejpam-5952	217	14	applications	application	NOUN
ejpam-5952	217	15	of	of	ADP
ejpam-5952	217	16	mr	mr	PROPN
ejpam-5952	217	17	-	-	PUNCT
ejpam-5952	217	18	metric	metric	ADJ
ejpam-5952	217	19	spaces	space	NOUN
ejpam-5952	217	20	,	,	PUNCT
ejpam-5952	217	21	particularly	particularly	ADV
ejpam-5952	217	22	in	in	ADP
ejpam-5952	217	23	optimization	optimization	NOUN
ejpam-5952	217	24	,	,	PUNCT
ejpam-5952	217	25	numerical	numerical	ADJ
ejpam-5952	217	26	analysis	analysis	NOUN
ejpam-5952	217	27	,	,	PUNCT
ejpam-5952	217	28	and	and	CCONJ
ejpam-5952	217	29	machine	machine	NOUN
ejpam-5952	217	30	learning	learning	NOUN
ejpam-5952	217	31	.	.	PUNCT
ejpam-5952	218	1	additionally	additionally	ADV
ejpam-5952	218	2	,	,	PUNCT
ejpam-5952	218	3	the	the	DET
ejpam-5952	218	4	examples	example	NOUN
ejpam-5952	218	5	presented	present	VERB
ejpam-5952	218	6	validate	validate	VERB
ejpam-5952	218	7	the	the	DET
ejpam-5952	218	8	theoretical	theoretical	ADJ
ejpam-5952	218	9	findings	finding	NOUN
ejpam-5952	218	10	,	,	PUNCT
ejpam-5952	218	11	highlighting	highlight	VERB
ejpam-5952	218	12	their	their	PRON
ejpam-5952	218	13	practical	practical	ADJ
ejpam-5952	218	14	significance	significance	NOUN
ejpam-5952	218	15	.	.	PUNCT
ejpam-5952	219	1	future	future	ADJ
ejpam-5952	219	2	research	research	NOUN
ejpam-5952	219	3	can	can	AUX
ejpam-5952	219	4	explore	explore	VERB
ejpam-5952	219	5	the	the	DET
ejpam-5952	219	6	extension	extension	NOUN
ejpam-5952	219	7	of	of	ADP
ejpam-5952	219	8	these	these	DET
ejpam-5952	219	9	results	result	NOUN
ejpam-5952	219	10	to	to	ADP
ejpam-5952	219	11	more	more	ADJ
ejpam-5952	219	12	complex	complex	ADJ
ejpam-5952	219	13	structures	structure	NOUN
ejpam-5952	219	14	,	,	PUNCT
ejpam-5952	219	15	such	such	ADJ
ejpam-5952	219	16	as	as	ADP
ejpam-5952	219	17	probabilistic	probabilistic	ADJ
ejpam-5952	219	18	or	or	CCONJ
ejpam-5952	219	19	fuzzy	fuzzy	ADJ
ejpam-5952	219	20	mr	mr	ADJ
ejpam-5952	219	21	-	-	PUNCT
ejpam-5952	219	22	metric	metric	ADJ
ejpam-5952	219	23	spaces	space	NOUN
ejpam-5952	219	24	,	,	PUNCT
ejpam-5952	219	25	and	and	CCONJ
ejpam-5952	219	26	investigate	investigate	VERB
ejpam-5952	219	27	their	their	PRON
ejpam-5952	219	28	applications	application	NOUN
ejpam-5952	219	29	in	in	ADP
ejpam-5952	219	30	real	real	ADJ
ejpam-5952	219	31	-	-	PUNCT
ejpam-5952	219	32	world	world	NOUN
ejpam-5952	219	33	problems	problem	NOUN
ejpam-5952	219	34	.	.	PUNCT
ejpam-5952	220	1	the	the	DET
ejpam-5952	220	2	theoretical	theoretical	ADJ
ejpam-5952	220	3	advancements	advancement	NOUN
ejpam-5952	220	4	presented	present	VERB
ejpam-5952	220	5	in	in	ADP
ejpam-5952	220	6	this	this	DET
ejpam-5952	220	7	study	study	NOUN
ejpam-5952	220	8	contribute	contribute	VERB
ejpam-5952	220	9	to	to	ADP
ejpam-5952	220	10	the	the	DET
ejpam-5952	220	11	broader	broad	ADJ
ejpam-5952	220	12	field	field	NOUN
ejpam-5952	220	13	of	of	ADP
ejpam-5952	220	14	fixed	fix	VERB
ejpam-5952	220	15	point	point	NOUN
ejpam-5952	220	16	theory	theory	NOUN
ejpam-5952	220	17	and	and	CCONJ
ejpam-5952	220	18	its	its	PRON
ejpam-5952	220	19	applications	application	NOUN
ejpam-5952	220	20	,	,	PUNCT
ejpam-5952	220	21	offering	offer	VERB
ejpam-5952	220	22	new	new	ADJ
ejpam-5952	220	23	insights	insight	NOUN
ejpam-5952	220	24	into	into	ADP
ejpam-5952	220	25	the	the	DET
ejpam-5952	220	26	convergence	convergence	NOUN
ejpam-5952	220	27	behavior	behavior	NOUN
ejpam-5952	220	28	of	of	ADP
ejpam-5952	220	29	mappings	mapping	NOUN
ejpam-5952	220	30	in	in	ADP
ejpam-5952	220	31	generalized	generalized	ADJ
ejpam-5952	220	32	metric	metric	ADJ
ejpam-5952	220	33	frameworks	framework	NOUN
ejpam-5952	220	34	.	.	PUNCT
ejpam-5952	221	1	references	reference	NOUN
ejpam-5952	221	2	[	[	X
ejpam-5952	221	3	1	1	NUM
ejpam-5952	221	4	]	]	PUNCT
ejpam-5952	221	5	b.	b.	PROPN
ejpam-5952	221	6	e.	e.	PROPN
ejpam-5952	221	7	rhoades	rhoades	PROPN
ejpam-5952	221	8	.	.	PUNCT
ejpam-5952	222	1	a	a	DET
ejpam-5952	222	2	fixed	fix	VERB
ejpam-5952	222	3	point	point	NOUN
ejpam-5952	222	4	theorem	theorem	NOUN
ejpam-5952	222	5	for	for	ADP
ejpam-5952	222	6	generalized	generalized	ADJ
ejpam-5952	222	7	metric	metric	ADJ
ejpam-5952	222	8	spaces	space	NOUN
ejpam-5952	222	9	.	.	PUNCT
ejpam-5952	223	1	international	international	ADJ
ejpam-5952	223	2	journal	journal	PROPN
ejpam-5952	223	3	of	of	ADP
ejpam-5952	223	4	mathematics	mathematics	PROPN
ejpam-5952	223	5	and	and	CCONJ
ejpam-5952	223	6	mathematical	mathematical	ADJ
ejpam-5952	223	7	sciences	science	NOUN
ejpam-5952	223	8	,	,	PUNCT
ejpam-5952	223	9	19(1):145–153	19(1):145–153	NUM
ejpam-5952	223	10	,	,	PUNCT
ejpam-5952	223	11	1996	1996	NUM
ejpam-5952	223	12	.	.	PUNCT
ejpam-5952	224	1	[	[	X
ejpam-5952	224	2	2	2	NUM
ejpam-5952	224	3	]	]	PUNCT
ejpam-5952	224	4	a.	a.	NOUN
ejpam-5952	224	5	malkawi	malkawi	PROPN
ejpam-5952	224	6	,	,	PUNCT
ejpam-5952	224	7	a.	a.	NOUN
ejpam-5952	224	8	tallafha	tallafha	NOUN
ejpam-5952	224	9	,	,	PUNCT
ejpam-5952	224	10	and	and	CCONJ
ejpam-5952	224	11	w.	w.	PROPN
ejpam-5952	224	12	shatanawi	shatanawi	PROPN
ejpam-5952	224	13	.	.	PUNCT
ejpam-5952	225	1	coincidence	coincidence	NOUN
ejpam-5952	225	2	and	and	CCONJ
ejpam-5952	225	3	fixed	fix	VERB
ejpam-5952	225	4	point	point	NOUN
ejpam-5952	225	5	results	result	NOUN
ejpam-5952	225	6	for	for	ADP
ejpam-5952	225	7	(	(	PUNCT
ejpam-5952	225	8	ψ	ψ	NOUN
ejpam-5952	225	9	,	,	PUNCT
ejpam-5952	225	10	l)-m	l)-m	PROPN
ejpam-5952	225	11	-weak	-weak	PROPN
ejpam-5952	225	12	contraction	contraction	NOUN
ejpam-5952	225	13	mapping	mapping	NOUN
ejpam-5952	225	14	on	on	ADP
ejpam-5952	225	15	mb	mb	ADJ
ejpam-5952	225	16	-	-	ADJ
ejpam-5952	225	17	metric	metric	ADJ
ejpam-5952	225	18	spaces	space	NOUN
ejpam-5952	225	19	.	.	PUNCT
ejpam-5952	226	1	italian	italian	ADJ
ejpam-5952	226	2	journal	journal	NOUN
ejpam-5952	226	3	of	of	ADP
ejpam-5952	226	4	pure	pure	ADJ
ejpam-5952	226	5	and	and	CCONJ
ejpam-5952	226	6	applied	applied	ADJ
ejpam-5952	226	7	mathematics	mathematic	NOUN
ejpam-5952	226	8	,	,	PUNCT
ejpam-5952	226	9	47:751–768	47:751–768	NUM
ejpam-5952	226	10	,	,	PUNCT
ejpam-5952	226	11	2022	2022	NUM
ejpam-5952	226	12	.	.	PUNCT
ejpam-5952	227	1	[	[	X
ejpam-5952	227	2	3	3	NUM
ejpam-5952	227	3	]	]	X
ejpam-5952	227	4	a.	a.	NOUN
ejpam-5952	227	5	malkawi	malkawi	PROPN
ejpam-5952	227	6	,	,	PUNCT
ejpam-5952	227	7	a.	a.	NOUN
ejpam-5952	227	8	tallafha	tallafha	NOUN
ejpam-5952	227	9	,	,	PUNCT
ejpam-5952	227	10	and	and	CCONJ
ejpam-5952	227	11	w.	w.	PROPN
ejpam-5952	227	12	shatanawi	shatanawi	PROPN
ejpam-5952	227	13	.	.	PUNCT
ejpam-5952	228	1	coincidence	coincidence	NOUN
ejpam-5952	228	2	and	and	CCONJ
ejpam-5952	228	3	fixed	fix	VERB
ejpam-5952	228	4	point	point	NOUN
ejpam-5952	228	5	results	result	NOUN
ejpam-5952	228	6	for	for	ADP
ejpam-5952	228	7	generalized	generalized	ADJ
ejpam-5952	228	8	weak	weak	ADJ
ejpam-5952	228	9	contraction	contraction	NOUN
ejpam-5952	228	10	mapping	mapping	NOUN
ejpam-5952	228	11	on	on	ADP
ejpam-5952	228	12	b	b	NOUN
ejpam-5952	228	13	-	-	PUNCT
ejpam-5952	228	14	metric	metric	ADJ
ejpam-5952	228	15	spaces	space	NOUN
ejpam-5952	228	16	.	.	PUNCT
ejpam-5952	229	1	nonlinear	nonlinear	ADJ
ejpam-5952	229	2	functional	functional	ADJ
ejpam-5952	229	3	analysis	analysis	NOUN
ejpam-5952	229	4	and	and	CCONJ
ejpam-5952	229	5	applications	application	NOUN
ejpam-5952	229	6	,	,	PUNCT
ejpam-5952	229	7	26(1):177–195	26(1):177–195	NOUN
ejpam-5952	229	8	,	,	PUNCT
ejpam-5952	229	9	2021	2021	NUM
ejpam-5952	229	10	.	.	PUNCT
ejpam-5952	230	1	[	[	X
ejpam-5952	230	2	4	4	NUM
ejpam-5952	230	3	]	]	PUNCT
ejpam-5952	230	4	a.	a.	NOUN
ejpam-5952	230	5	rabaiah	rabaiah	PROPN
ejpam-5952	230	6	,	,	PUNCT
ejpam-5952	230	7	a.	a.	NOUN
ejpam-5952	230	8	tallafha	tallafha	NOUN
ejpam-5952	230	9	,	,	PUNCT
ejpam-5952	230	10	and	and	CCONJ
ejpam-5952	230	11	w.	w.	PROPN
ejpam-5952	230	12	shatanawi	shatanawi	PROPN
ejpam-5952	230	13	.	.	PUNCT
ejpam-5952	231	1	common	common	ADJ
ejpam-5952	231	2	fixed	fix	VERB
ejpam-5952	231	3	point	point	NOUN
ejpam-5952	231	4	results	result	NOUN
ejpam-5952	231	5	for	for	ADP
ejpam-5952	231	6	mappings	mapping	NOUN
ejpam-5952	231	7	under	under	ADP
ejpam-5952	231	8	nonlinear	nonlinear	ADJ
ejpam-5952	231	9	contraction	contraction	NOUN
ejpam-5952	231	10	of	of	ADP
ejpam-5952	231	11	cyclic	cyclic	ADJ
ejpam-5952	231	12	form	form	NOUN
ejpam-5952	231	13	in	in	ADP
ejpam-5952	231	14	b	b	NOUN
ejpam-5952	231	15	-	-	ADJ
ejpam-5952	231	16	metric	metric	ADJ
ejpam-5952	231	17	spaces	space	NOUN
ejpam-5952	231	18	.	.	PUNCT
ejpam-5952	232	1	advances	advance	NOUN
ejpam-5952	232	2	in	in	ADP
ejpam-5952	232	3	mathematics	mathematic	NOUN
ejpam-5952	232	4	:	:	PUNCT
ejpam-5952	232	5	scientific	scientific	ADJ
ejpam-5952	232	6	journal	journal	NOUN
ejpam-5952	232	7	,	,	PUNCT
ejpam-5952	232	8	10(2):289–301	10(2):289–301	PROPN
ejpam-5952	232	9	,	,	PUNCT
ejpam-5952	232	10	2021	2021	NUM
ejpam-5952	232	11	.	.	PUNCT
ejpam-5952	233	1	[	[	X
ejpam-5952	233	2	5	5	NUM
ejpam-5952	233	3	]	]	PUNCT
ejpam-5952	233	4	a.	a.	NOUN
ejpam-5952	233	5	malkawi	malkawi	PROPN
ejpam-5952	233	6	,	,	PUNCT
ejpam-5952	233	7	a.	a.	PROPN
ejpam-5952	233	8	rabaiah	rabaiah	PROPN
ejpam-5952	233	9	,	,	PUNCT
ejpam-5952	233	10	w.	w.	PROPN
ejpam-5952	233	11	shatanawi	shatanawi	PROPN
ejpam-5952	233	12	,	,	PUNCT
ejpam-5952	233	13	and	and	CCONJ
ejpam-5952	233	14	a.	a.	NOUN
ejpam-5952	233	15	tallafha	tallafha	NOUN
ejpam-5952	233	16	.	.	PUNCT
ejpam-5952	234	1	mr	mr	PROPN
ejpam-5952	234	2	-	-	PUNCT
ejpam-5952	234	3	metric	metric	ADJ
ejpam-5952	234	4	spaces	space	NOUN
ejpam-5952	234	5	and	and	CCONJ
ejpam-5952	234	6	an	an	DET
ejpam-5952	234	7	application	application	NOUN
ejpam-5952	234	8	.	.	PUNCT
ejpam-5952	235	1	preprint	preprint	NOUN
ejpam-5952	235	2	,	,	PUNCT
ejpam-5952	235	3	2021	2021	NUM
ejpam-5952	235	4	.	.	PUNCT
ejpam-5952	236	1	[	[	X
ejpam-5952	236	2	6	6	NUM
ejpam-5952	236	3	]	]	X
ejpam-5952	236	4	g.	g.	PROPN
ejpam-5952	236	5	gharib	gharib	PROPN
ejpam-5952	236	6	,	,	PUNCT
ejpam-5952	236	7	a.	a.	PROPN
ejpam-5952	236	8	malkawi	malkawi	PROPN
ejpam-5952	236	9	,	,	PUNCT
ejpam-5952	236	10	a.	a.	PROPN
ejpam-5952	236	11	rabaiah	rabaiah	PROPN
ejpam-5952	236	12	,	,	PUNCT
ejpam-5952	236	13	w.	w.	PROPN
ejpam-5952	236	14	shatanawi	shatanawi	PROPN
ejpam-5952	236	15	,	,	PUNCT
ejpam-5952	236	16	and	and	CCONJ
ejpam-5952	236	17	m.	m.	NOUN
ejpam-5952	236	18	alsauodi	alsauodi	PROPN
ejpam-5952	236	19	.	.	PUNCT
ejpam-5952	237	1	a	a	DET
ejpam-5952	237	2	common	common	ADJ
ejpam-5952	237	3	fixed	fix	VERB
ejpam-5952	237	4	point	point	NOUN
ejpam-5952	237	5	theorem	theorem	VERB
ejpam-5952	237	6	in	in	ADP
ejpam-5952	237	7	m∗-metric	m∗-metric	ADJ
ejpam-5952	237	8	space	space	NOUN
ejpam-5952	237	9	and	and	CCONJ
ejpam-5952	237	10	an	an	DET
ejpam-5952	237	11	application	application	NOUN
ejpam-5952	237	12	.	.	PUNCT
ejpam-5952	238	1	nonlinear	nonlinear	ADJ
ejpam-5952	238	2	functional	functional	ADJ
ejpam-5952	238	3	analysis	analysis	NOUN
ejpam-5952	238	4	and	and	CCONJ
ejpam-5952	238	5	applications	application	NOUN
ejpam-5952	238	6	,	,	PUNCT
ejpam-5952	238	7	27(2):289–308	27(2):289–308	NUM
ejpam-5952	238	8	,	,	PUNCT
ejpam-5952	238	9	2022	2022	NUM
ejpam-5952	238	10	.	.	PUNCT
ejpam-5952	239	1	a.	a.	NOUN
ejpam-5952	239	2	a.	a.	PROPN
ejpam-5952	239	3	m.	m.	PROPN
ejpam-5952	239	4	malkawi	malkawi	ADP
ejpam-5952	239	5	/	/	SYM
ejpam-5952	239	6	eur	eur	PROPN
ejpam-5952	239	7	.	.	PUNCT
ejpam-5952	240	1	j.	j.	PROPN
ejpam-5952	240	2	pure	pure	PROPN
ejpam-5952	240	3	appl	appl	PROPN
ejpam-5952	240	4	.	.	PROPN
ejpam-5952	240	5	math	math	PROPN
ejpam-5952	240	6	,	,	PUNCT
ejpam-5952	240	7	18	18	NUM
ejpam-5952	240	8	(	(	PUNCT
ejpam-5952	240	9	2	2	NUM
ejpam-5952	240	10	)	)	PUNCT
ejpam-5952	240	11	(	(	PUNCT
ejpam-5952	240	12	2025	2025	NUM
ejpam-5952	240	13	)	)	PUNCT
ejpam-5952	240	14	,	,	PUNCT
ejpam-5952	240	15	5952	5952	NUM
ejpam-5952	240	16	13	13	NUM
ejpam-5952	240	17	of	of	ADP
ejpam-5952	240	18	14	14	NUM
ejpam-5952	240	19	[	[	X
ejpam-5952	240	20	7	7	NUM
ejpam-5952	240	21	]	]	PUNCT
ejpam-5952	240	22	a.	a.	NOUN
ejpam-5952	240	23	rabaiah	rabaiah	PROPN
ejpam-5952	240	24	,	,	PUNCT
ejpam-5952	240	25	a.	a.	NOUN
ejpam-5952	240	26	malkawi	malkawi	PROPN
ejpam-5952	240	27	,	,	PUNCT
ejpam-5952	240	28	a.	a.	PROPN
ejpam-5952	240	29	al	al	PROPN
ejpam-5952	240	30	-	-	PUNCT
ejpam-5952	240	31	rawabdeh	rawabdeh	PROPN
ejpam-5952	240	32	,	,	PUNCT
ejpam-5952	240	33	d.	d.	PROPN
ejpam-5952	240	34	mahmoud	mahmoud	PROPN
ejpam-5952	240	35	,	,	PUNCT
ejpam-5952	240	36	and	and	CCONJ
ejpam-5952	240	37	m.	m.	NOUN
ejpam-5952	240	38	qousini	qousini	PROPN
ejpam-5952	240	39	.	.	PUNCT
ejpam-5952	241	1	fixed	fix	VERB
ejpam-5952	241	2	point	point	NOUN
ejpam-5952	241	3	theorems	theorem	NOUN
ejpam-5952	241	4	in	in	ADP
ejpam-5952	241	5	mr	mr	PROPN
ejpam-5952	241	6	-	-	PUNCT
ejpam-5952	241	7	metric	metric	ADJ
ejpam-5952	241	8	space	space	NOUN
ejpam-5952	241	9	through	through	ADP
ejpam-5952	241	10	semi	semi	NOUN
ejpam-5952	241	11	-	-	NOUN
ejpam-5952	241	12	compatibility	compatibility	NOUN
ejpam-5952	241	13	.	.	PUNCT
ejpam-5952	242	1	advances	advance	NOUN
ejpam-5952	242	2	in	in	ADP
ejpam-5952	242	3	mathematics	mathematic	NOUN
ejpam-5952	242	4	:	:	PUNCT
ejpam-5952	242	5	scientific	scientific	ADJ
ejpam-5952	242	6	journal	journal	NOUN
ejpam-5952	242	7	,	,	PUNCT
ejpam-5952	242	8	10(6):2831–2845	10(6):2831–2845	NUM
ejpam-5952	242	9	,	,	PUNCT
ejpam-5952	242	10	2021	2021	NUM
ejpam-5952	242	11	.	.	PUNCT
ejpam-5952	243	1	[	[	X
ejpam-5952	243	2	8	8	NUM
ejpam-5952	243	3	]	]	PUNCT
ejpam-5952	243	4	m.	m.	NOUN
ejpam-5952	243	5	s.	s.	PROPN
ejpam-5952	243	6	alsauodi	alsauodi	PROPN
ejpam-5952	243	7	,	,	PUNCT
ejpam-5952	243	8	g.	g.	PROPN
ejpam-5952	243	9	m.	m.	PROPN
ejpam-5952	243	10	gharib	gharib	PROPN
ejpam-5952	243	11	,	,	PUNCT
ejpam-5952	243	12	a.	a.	PROPN
ejpam-5952	243	13	malkawi	malkawi	PROPN
ejpam-5952	243	14	,	,	PUNCT
ejpam-5952	243	15	a.	a.	PROPN
ejpam-5952	243	16	m.	m.	PROPN
ejpam-5952	243	17	rabaiah	rabaiah	PROPN
ejpam-5952	243	18	,	,	PUNCT
ejpam-5952	243	19	and	and	CCONJ
ejpam-5952	243	20	w.	w.	PROPN
ejpam-5952	243	21	a.	a.	PROPN
ejpam-5952	243	22	shatanawi	shatanawi	PROPN
ejpam-5952	243	23	.	.	PUNCT
ejpam-5952	244	1	fixed	fix	VERB
ejpam-5952	244	2	point	point	NOUN
ejpam-5952	244	3	theorems	theorem	NOUN
ejpam-5952	244	4	for	for	ADP
ejpam-5952	244	5	monotone	monotone	ADJ
ejpam-5952	244	6	mappings	mapping	NOUN
ejpam-5952	244	7	on	on	ADP
ejpam-5952	244	8	partial	partial	ADJ
ejpam-5952	244	9	m∗-metric	m∗-metric	ADJ
ejpam-5952	244	10	spaces	space	NOUN
ejpam-5952	244	11	.	.	PUNCT
ejpam-5952	245	1	italian	italian	ADJ
ejpam-5952	245	2	journal	journal	NOUN
ejpam-5952	245	3	of	of	ADP
ejpam-5952	245	4	pure	pure	ADJ
ejpam-5952	245	5	and	and	CCONJ
ejpam-5952	245	6	applied	applied	ADJ
ejpam-5952	245	7	mathematics	mathematic	NOUN
ejpam-5952	245	8	,	,	PUNCT
ejpam-5952	245	9	44:154–172	44:154–172	PROPN
ejpam-5952	245	10	,	,	PUNCT
ejpam-5952	245	11	2023	2023	NUM
ejpam-5952	245	12	.	.	PUNCT
ejpam-5952	246	1	[	[	X
ejpam-5952	246	2	9	9	NUM
ejpam-5952	246	3	]	]	PUNCT
ejpam-5952	246	4	t.	t.	NOUN
ejpam-5952	246	5	qawasmeh	qawasmeh	NOUN
ejpam-5952	246	6	.	.	PUNCT
ejpam-5952	247	1	h	h	NOUN
ejpam-5952	247	2	-	-	PUNCT
ejpam-5952	247	3	simulation	simulation	NOUN
ejpam-5952	247	4	functions	function	NOUN
ejpam-5952	247	5	and	and	CCONJ
ejpam-5952	247	6	ωb	ωb	NOUN
ejpam-5952	247	7	-	-	PUNCT
ejpam-5952	247	8	distance	distance	NOUN
ejpam-5952	247	9	mappings	mapping	NOUN
ejpam-5952	247	10	in	in	ADP
ejpam-5952	247	11	the	the	DET
ejpam-5952	247	12	setting	setting	NOUN
ejpam-5952	247	13	of	of	ADP
ejpam-5952	247	14	gb	gb	ADV
ejpam-5952	247	15	-	-	PUNCT
ejpam-5952	247	16	metric	metric	ADJ
ejpam-5952	247	17	spaces	space	NOUN
ejpam-5952	247	18	and	and	CCONJ
ejpam-5952	247	19	application	application	NOUN
ejpam-5952	247	20	.	.	PUNCT
ejpam-5952	248	1	nonlinear	nonlinear	ADJ
ejpam-5952	248	2	functional	functional	ADJ
ejpam-5952	248	3	analysis	analysis	NOUN
ejpam-5952	248	4	and	and	CCONJ
ejpam-5952	248	5	applications	application	NOUN
ejpam-5952	248	6	,	,	PUNCT
ejpam-5952	248	7	28(2):557–570	28(2):557–570	NOUN
ejpam-5952	248	8	,	,	PUNCT
ejpam-5952	248	9	2023	2023	NUM
ejpam-5952	248	10	.	.	PUNCT
ejpam-5952	249	1	[	[	X
ejpam-5952	249	2	10	10	NUM
ejpam-5952	249	3	]	]	PUNCT
ejpam-5952	249	4	a.	a.	NOUN
ejpam-5952	249	5	bataihah	bataihah	PROPN
ejpam-5952	249	6	and	and	CCONJ
ejpam-5952	249	7	t.	t.	NOUN
ejpam-5952	249	8	qawasmeh	qawasmeh	NOUN
ejpam-5952	249	9	.	.	PUNCT
ejpam-5952	250	1	a	a	DET
ejpam-5952	250	2	new	new	ADJ
ejpam-5952	250	3	type	type	NOUN
ejpam-5952	250	4	of	of	ADP
ejpam-5952	250	5	distance	distance	NOUN
ejpam-5952	250	6	spaces	space	NOUN
ejpam-5952	250	7	and	and	CCONJ
ejpam-5952	250	8	fixed	fix	VERB
ejpam-5952	250	9	point	point	NOUN
ejpam-5952	250	10	results	result	NOUN
ejpam-5952	250	11	.	.	PUNCT
ejpam-5952	251	1	journal	journal	NOUN
ejpam-5952	251	2	of	of	ADP
ejpam-5952	251	3	mathematical	mathematical	ADJ
ejpam-5952	251	4	analysis	analysis	NOUN
ejpam-5952	251	5	,	,	PUNCT
ejpam-5952	251	6	15(4):81–90	15(4):81–90	NUM
ejpam-5952	251	7	,	,	PUNCT
ejpam-5952	251	8	2024	2024	NUM
ejpam-5952	251	9	.	.	PUNCT
ejpam-5952	252	1	[	[	X
ejpam-5952	252	2	11	11	NUM
ejpam-5952	252	3	]	]	X
ejpam-5952	252	4	w.	w.	PROPN
ejpam-5952	252	5	shatanawi	shatanawi	PROPN
ejpam-5952	252	6	,	,	PUNCT
ejpam-5952	252	7	t.	t.	NOUN
ejpam-5952	252	8	qawasmeh	qawasmeh	NOUN
ejpam-5952	252	9	,	,	PUNCT
ejpam-5952	252	10	a.	a.	NOUN
ejpam-5952	252	11	bataihah	bataihah	PROPN
ejpam-5952	252	12	,	,	PUNCT
ejpam-5952	252	13	and	and	CCONJ
ejpam-5952	252	14	a.	a.	NOUN
ejpam-5952	252	15	tallafha	tallafha	NOUN
ejpam-5952	252	16	.	.	PUNCT
ejpam-5952	253	1	new	new	ADJ
ejpam-5952	253	2	contractions	contraction	NOUN
ejpam-5952	253	3	and	and	CCONJ
ejpam-5952	253	4	some	some	DET
ejpam-5952	253	5	fixed	fix	VERB
ejpam-5952	253	6	point	point	NOUN
ejpam-5952	253	7	results	result	NOUN
ejpam-5952	253	8	with	with	ADP
ejpam-5952	253	9	application	application	NOUN
ejpam-5952	253	10	based	base	VERB
ejpam-5952	253	11	on	on	ADP
ejpam-5952	253	12	extended	extended	ADJ
ejpam-5952	253	13	quasi	quasi	ADJ
ejpam-5952	253	14	b	b	NOUN
ejpam-5952	253	15	-	-	ADJ
ejpam-5952	253	16	metric	metric	ADJ
ejpam-5952	253	17	spaces	space	NOUN
ejpam-5952	253	18	.	.	PUNCT
ejpam-5952	254	1	u.p.b	u.p.b	ADJ
ejpam-5952	254	2	.	.	PUNCT
ejpam-5952	255	1	scientific	scientific	ADJ
ejpam-5952	255	2	bulletin	bulletin	NOUN
ejpam-5952	255	3	,	,	PUNCT
ejpam-5952	255	4	series	series	NOUN
ejpam-5952	255	5	a	a	PRON
ejpam-5952	255	6	,	,	PUNCT
ejpam-5952	255	7	83(2):53–64	83(2):53–64	NUM
ejpam-5952	255	8	,	,	PUNCT
ejpam-5952	255	9	2021	2021	NUM
ejpam-5952	255	10	.	.	PUNCT
ejpam-5952	256	1	[	[	X
ejpam-5952	256	2	12	12	NUM
ejpam-5952	256	3	]	]	PUNCT
ejpam-5952	256	4	t.	t.	NOUN
ejpam-5952	256	5	qawasmeh	qawasmeh	NOUN
ejpam-5952	256	6	,	,	PUNCT
ejpam-5952	256	7	w.	w.	PROPN
ejpam-5952	256	8	shatanawi	shatanawi	PROPN
ejpam-5952	256	9	,	,	PUNCT
ejpam-5952	256	10	a.	a.	NOUN
ejpam-5952	256	11	bataihah	bataihah	PROPN
ejpam-5952	256	12	,	,	PUNCT
ejpam-5952	256	13	and	and	CCONJ
ejpam-5952	256	14	a.	a.	NOUN
ejpam-5952	256	15	tallafha	tallafha	NOUN
ejpam-5952	256	16	.	.	PUNCT
ejpam-5952	257	1	fixed	fix	VERB
ejpam-5952	257	2	point	point	NOUN
ejpam-5952	257	3	results	result	NOUN
ejpam-5952	257	4	and	and	CCONJ
ejpam-5952	257	5	(	(	PUNCT
ejpam-5952	257	6	α	α	NOUN
ejpam-5952	257	7	,	,	PUNCT
ejpam-5952	257	8	β)-triangular	β)-triangular	ADJ
ejpam-5952	257	9	admissibility	admissibility	NOUN
ejpam-5952	257	10	in	in	ADP
ejpam-5952	257	11	the	the	DET
ejpam-5952	257	12	frame	frame	NOUN
ejpam-5952	257	13	of	of	ADP
ejpam-5952	257	14	complete	complete	ADJ
ejpam-5952	257	15	extended	extended	ADJ
ejpam-5952	257	16	b	b	NOUN
ejpam-5952	257	17	-	-	PUNCT
ejpam-5952	257	18	metric	metric	ADJ
ejpam-5952	257	19	spaces	space	NOUN
ejpam-5952	257	20	and	and	CCONJ
ejpam-5952	257	21	application	application	NOUN
ejpam-5952	257	22	.	.	PUNCT
ejpam-5952	258	1	u.p.b	u.p.b	PROPN
ejpam-5952	258	2	.	.	PUNCT
ejpam-5952	259	1	scientific	scientific	ADJ
ejpam-5952	259	2	bulletin	bulletin	NOUN
ejpam-5952	259	3	,	,	PUNCT
ejpam-5952	259	4	series	series	PROPN
ejpam-5952	259	5	a	a	PROPN
ejpam-5952	259	6	,	,	PUNCT
ejpam-5952	259	7	83(1):113–124	83(1):113–124	PROPN
ejpam-5952	259	8	,	,	PUNCT
ejpam-5952	259	9	2021	2021	NUM
ejpam-5952	259	10	.	.	PUNCT
ejpam-5952	260	1	[	[	X
ejpam-5952	260	2	13	13	NUM
ejpam-5952	260	3	]	]	PUNCT
ejpam-5952	260	4	a.	a.	NOUN
ejpam-5952	260	5	bataihah	bataihah	PROPN
ejpam-5952	260	6	,	,	PUNCT
ejpam-5952	260	7	a.	a.	NOUN
ejpam-5952	260	8	tallafha	tallafha	NOUN
ejpam-5952	260	9	,	,	PUNCT
ejpam-5952	260	10	and	and	CCONJ
ejpam-5952	260	11	w.	w.	PROPN
ejpam-5952	260	12	shatanawi	shatanawi	PROPN
ejpam-5952	260	13	.	.	PUNCT
ejpam-5952	261	1	fixed	fix	VERB
ejpam-5952	261	2	point	point	NOUN
ejpam-5952	261	3	results	result	NOUN
ejpam-5952	261	4	with	with	ADP
ejpam-5952	261	5	simulation	simulation	NOUN
ejpam-5952	261	6	functions	function	NOUN
ejpam-5952	261	7	.	.	PUNCT
ejpam-5952	262	1	nonlinear	nonlinear	ADJ
ejpam-5952	262	2	functional	functional	ADJ
ejpam-5952	262	3	analysis	analysis	NOUN
ejpam-5952	262	4	and	and	CCONJ
ejpam-5952	262	5	applications	application	NOUN
ejpam-5952	262	6	,	,	PUNCT
ejpam-5952	262	7	25(1):13–23	25(1):13–23	NUM
ejpam-5952	262	8	,	,	PUNCT
ejpam-5952	262	9	2020	2020	NUM
ejpam-5952	262	10	.	.	PUNCT
ejpam-5952	263	1	[	[	X
ejpam-5952	263	2	14	14	NUM
ejpam-5952	263	3	]	]	PUNCT
ejpam-5952	263	4	k.	k.	PROPN
ejpam-5952	263	5	abodayeh	abodayeh	PROPN
ejpam-5952	263	6	,	,	PUNCT
ejpam-5952	263	7	w.	w.	PROPN
ejpam-5952	263	8	shatanawi	shatanawi	PROPN
ejpam-5952	263	9	,	,	PUNCT
ejpam-5952	263	10	a.	a.	NOUN
ejpam-5952	263	11	bataihah	bataihah	PROPN
ejpam-5952	263	12	,	,	PUNCT
ejpam-5952	263	13	and	and	CCONJ
ejpam-5952	263	14	a.	a.	PROPN
ejpam-5952	263	15	h.	h.	PROPN
ejpam-5952	263	16	ansari	ansari	PROPN
ejpam-5952	263	17	.	.	PUNCT
ejpam-5952	264	1	some	some	DET
ejpam-5952	264	2	fixed	fix	VERB
ejpam-5952	264	3	point	point	NOUN
ejpam-5952	264	4	and	and	CCONJ
ejpam-5952	264	5	common	common	ADJ
ejpam-5952	264	6	fixed	fix	VERB
ejpam-5952	264	7	point	point	NOUN
ejpam-5952	264	8	results	result	NOUN
ejpam-5952	264	9	through	through	ADP
ejpam-5952	264	10	ω	ω	NOUN
ejpam-5952	264	11	-	-	PUNCT
ejpam-5952	264	12	distance	distance	NOUN
ejpam-5952	264	13	under	under	ADP
ejpam-5952	264	14	nonlinear	nonlinear	ADJ
ejpam-5952	264	15	contractions	contraction	NOUN
ejpam-5952	264	16	.	.	PUNCT
ejpam-5952	265	1	gazi	gazi	PROPN
ejpam-5952	265	2	university	university	PROPN
ejpam-5952	265	3	journal	journal	PROPN
ejpam-5952	265	4	of	of	ADP
ejpam-5952	265	5	science	science	NOUN
ejpam-5952	265	6	,	,	PUNCT
ejpam-5952	265	7	30(1):293–302	30(1):293–302	NOUN
ejpam-5952	265	8	,	,	PUNCT
ejpam-5952	265	9	2017	2017	NUM
ejpam-5952	265	10	.	.	PUNCT
ejpam-5952	266	1	[	[	X
ejpam-5952	266	2	15	15	NUM
ejpam-5952	266	3	]	]	X
ejpam-5952	266	4	a.	a.	NOUN
ejpam-5952	266	5	bataihah	bataihah	PROPN
ejpam-5952	266	6	,	,	PUNCT
ejpam-5952	266	7	a.	a.	NOUN
ejpam-5952	266	8	tallafha	tallafha	NOUN
ejpam-5952	266	9	,	,	PUNCT
ejpam-5952	266	10	and	and	CCONJ
ejpam-5952	266	11	w.	w.	PROPN
ejpam-5952	266	12	shatanawi	shatanawi	PROPN
ejpam-5952	266	13	.	.	PUNCT
ejpam-5952	267	1	fixed	fix	VERB
ejpam-5952	267	2	point	point	NOUN
ejpam-5952	267	3	results	result	NOUN
ejpam-5952	267	4	with	with	ADP
ejpam-5952	267	5	ω	ω	NOUN
ejpam-5952	267	6	-	-	PUNCT
ejpam-5952	267	7	distance	distance	NOUN
ejpam-5952	267	8	by	by	ADP
ejpam-5952	267	9	utilizing	utilize	VERB
ejpam-5952	267	10	simulation	simulation	NOUN
ejpam-5952	267	11	functions	function	NOUN
ejpam-5952	267	12	.	.	PUNCT
ejpam-5952	268	1	italian	italian	ADJ
ejpam-5952	268	2	journal	journal	NOUN
ejpam-5952	268	3	of	of	ADP
ejpam-5952	268	4	pure	pure	ADJ
ejpam-5952	268	5	and	and	CCONJ
ejpam-5952	268	6	applied	applied	ADJ
ejpam-5952	268	7	mathematics	mathematic	NOUN
ejpam-5952	268	8	,	,	PUNCT
ejpam-5952	268	9	43:185–196	43:185–196	NOUN
ejpam-5952	268	10	,	,	PUNCT
ejpam-5952	268	11	2020	2020	NUM
ejpam-5952	268	12	.	.	PUNCT
ejpam-5952	269	1	[	[	X
ejpam-5952	269	2	16	16	NUM
ejpam-5952	269	3	]	]	PUNCT
ejpam-5952	269	4	k.	k.	PROPN
ejpam-5952	269	5	abodayeh	abodayeh	PROPN
ejpam-5952	269	6	,	,	PUNCT
ejpam-5952	269	7	a.	a.	PROPN
ejpam-5952	269	8	bataihah	bataihah	PROPN
ejpam-5952	269	9	,	,	PUNCT
ejpam-5952	269	10	and	and	CCONJ
ejpam-5952	269	11	w.	w.	PROPN
ejpam-5952	269	12	shatanawi	shatanawi	PROPN
ejpam-5952	269	13	.	.	PUNCT
ejpam-5952	270	1	generalized	generalize	VERB
ejpam-5952	270	2	ω	ω	NUM
ejpam-5952	270	3	-	-	PUNCT
ejpam-5952	270	4	distance	distance	NOUN
ejpam-5952	270	5	mappings	mapping	NOUN
ejpam-5952	270	6	and	and	CCONJ
ejpam-5952	270	7	some	some	DET
ejpam-5952	270	8	fixed	fix	VERB
ejpam-5952	270	9	point	point	NOUN
ejpam-5952	270	10	theorems	theorem	NOUN
ejpam-5952	270	11	.	.	PUNCT
ejpam-5952	271	1	u.p.b	u.p.b	PROPN
ejpam-5952	271	2	.	.	PUNCT
ejpam-5952	272	1	scientific	scientific	ADJ
ejpam-5952	272	2	bulletin	bulletin	NOUN
ejpam-5952	272	3	,	,	PUNCT
ejpam-5952	272	4	series	series	NOUN
ejpam-5952	272	5	a	a	NOUN
ejpam-5952	272	6	,	,	PUNCT
ejpam-5952	272	7	79(4):223–232	79(4):223–232	NOUN
ejpam-5952	272	8	,	,	PUNCT
ejpam-5952	272	9	2017	2017	NUM
ejpam-5952	272	10	.	.	PUNCT
ejpam-5952	273	1	[	[	X
ejpam-5952	273	2	17	17	NUM
ejpam-5952	273	3	]	]	PUNCT
ejpam-5952	273	4	t.	t.	NOUN
ejpam-5952	273	5	qawasmeh	qawasmeh	NOUN
ejpam-5952	273	6	,	,	PUNCT
ejpam-5952	273	7	w.	w.	PROPN
ejpam-5952	273	8	shatanawi	shatanawi	PROPN
ejpam-5952	273	9	,	,	PUNCT
ejpam-5952	273	10	and	and	CCONJ
ejpam-5952	273	11	a.	a.	NOUN
ejpam-5952	273	12	bataihah	bataihah	PROPN
ejpam-5952	273	13	.	.	PUNCT
ejpam-5952	274	1	common	common	ADJ
ejpam-5952	274	2	fixed	fix	VERB
ejpam-5952	274	3	point	point	NOUN
ejpam-5952	274	4	results	result	NOUN
ejpam-5952	274	5	for	for	ADP
ejpam-5952	274	6	rational	rational	ADJ
ejpam-5952	274	7	(	(	PUNCT
ejpam-5952	274	8	α	α	NOUN
ejpam-5952	274	9	,	,	PUNCT
ejpam-5952	274	10	β)ϕ-mω	β)ϕ-mω	NOUN
ejpam-5952	274	11	contractions	contraction	NOUN
ejpam-5952	274	12	in	in	ADP
ejpam-5952	274	13	complete	complete	ADJ
ejpam-5952	274	14	quasi	quasi	ADJ
ejpam-5952	274	15	metric	metric	ADJ
ejpam-5952	274	16	spaces	space	NOUN
ejpam-5952	274	17	.	.	PUNCT
ejpam-5952	275	1	mathematics	mathematic	NOUN
ejpam-5952	275	2	,	,	PUNCT
ejpam-5952	275	3	7(5):392	7(5):392	NUM
ejpam-5952	275	4	,	,	PUNCT
ejpam-5952	275	5	2019	2019	NUM
ejpam-5952	275	6	.	.	PUNCT
ejpam-5952	276	1	[	[	X
ejpam-5952	276	2	18	18	NUM
ejpam-5952	276	3	]	]	X
ejpam-5952	276	4	i.	i.	PROPN
ejpam-5952	276	5	a.	a.	PROPN
ejpam-5952	276	6	bakhtin	bakhtin	PROPN
ejpam-5952	276	7	.	.	PUNCT
ejpam-5952	277	1	the	the	DET
ejpam-5952	277	2	contraction	contraction	NOUN
ejpam-5952	277	3	mapping	map	VERB
ejpam-5952	277	4	principle	principle	NOUN
ejpam-5952	277	5	in	in	ADP
ejpam-5952	277	6	almost	almost	ADV
ejpam-5952	277	7	metric	metric	ADJ
ejpam-5952	277	8	spaces	space	NOUN
ejpam-5952	277	9	.	.	PUNCT
ejpam-5952	278	1	functional	functional	ADJ
ejpam-5952	278	2	analysis	analysis	NOUN
ejpam-5952	278	3	,	,	PUNCT
ejpam-5952	278	4	30:26–37	30:26–37	PROPN
ejpam-5952	278	5	,	,	PUNCT
ejpam-5952	278	6	1989	1989	NUM
ejpam-5952	278	7	.	.	PUNCT
ejpam-5952	279	1	[	[	X
ejpam-5952	279	2	19	19	NUM
ejpam-5952	279	3	]	]	X
ejpam-5952	279	4	s.	s.	PROPN
ejpam-5952	279	5	czerwik	czerwik	PROPN
ejpam-5952	279	6	.	.	PUNCT
ejpam-5952	280	1	contraction	contraction	NOUN
ejpam-5952	280	2	mappings	mapping	NOUN
ejpam-5952	280	3	in	in	ADP
ejpam-5952	280	4	b	b	NOUN
ejpam-5952	280	5	-	-	ADJ
ejpam-5952	280	6	metric	metric	ADJ
ejpam-5952	280	7	spaces	space	NOUN
ejpam-5952	280	8	.	.	PUNCT
ejpam-5952	281	1	acta	acta	PROPN
ejpam-5952	281	2	mathematica	mathematica	PROPN
ejpam-5952	281	3	et	et	PROPN
ejpam-5952	281	4	informatica	informatica	PROPN
ejpam-5952	281	5	universitatis	universitatis	PROPN
ejpam-5952	281	6	ostraviensis	ostraviensis	PROPN
ejpam-5952	281	7	,	,	PUNCT
ejpam-5952	281	8	1:5–11	1:5–11	NUM
ejpam-5952	281	9	,	,	PUNCT
ejpam-5952	281	10	1993	1993	NUM
ejpam-5952	281	11	.	.	PUNCT
ejpam-5952	282	1	[	[	X
ejpam-5952	282	2	20	20	NUM
ejpam-5952	282	3	]	]	X
ejpam-5952	282	4	y.	y.	PROPN
ejpam-5952	282	5	j.	j.	PROPN
ejpam-5952	282	6	cho	cho	PROPN
ejpam-5952	282	7	,	,	PUNCT
ejpam-5952	282	8	p.	p.	NOUN
ejpam-5952	282	9	p.	p.	PROPN
ejpam-5952	283	1	murthy	murthy	ADJ
ejpam-5952	283	2	,	,	PUNCT
ejpam-5952	283	3	and	and	CCONJ
ejpam-5952	283	4	g.	g.	PROPN
ejpam-5952	283	5	jungck	jungck	PROPN
ejpam-5952	283	6	.	.	PUNCT
ejpam-5952	284	1	a	a	DET
ejpam-5952	284	2	common	common	ADJ
ejpam-5952	284	3	fixed	fix	VERB
ejpam-5952	284	4	point	point	NOUN
ejpam-5952	284	5	theorem	theorem	NOUN
ejpam-5952	284	6	of	of	ADP
ejpam-5952	284	7	meir	meir	PROPN
ejpam-5952	284	8	and	and	CCONJ
ejpam-5952	284	9	keeler	keeler	PROPN
ejpam-5952	284	10	type	type	NOUN
ejpam-5952	284	11	.	.	PUNCT
ejpam-5952	285	1	international	international	ADJ
ejpam-5952	285	2	journal	journal	PROPN
ejpam-5952	285	3	of	of	ADP
ejpam-5952	285	4	mathematics	mathematics	PROPN
ejpam-5952	285	5	and	and	CCONJ
ejpam-5952	285	6	mathematical	mathematical	ADJ
ejpam-5952	285	7	sciences	science	NOUN
ejpam-5952	285	8	,	,	PUNCT
ejpam-5952	285	9	16(4):669–674	16(4):669–674	NUM
ejpam-5952	285	10	,	,	PUNCT
ejpam-5952	285	11	1993	1993	NUM
ejpam-5952	285	12	.	.	PUNCT
ejpam-5952	286	1	[	[	X
ejpam-5952	286	2	21	21	NUM
ejpam-5952	286	3	]	]	X
ejpam-5952	286	4	r.	r.	PROPN
ejpam-5952	286	5	o.	o.	PROPN
ejpam-5952	286	6	davies	davies	PROPN
ejpam-5952	286	7	and	and	CCONJ
ejpam-5952	286	8	s.	s.	PROPN
ejpam-5952	286	9	sessa	sessa	PROPN
ejpam-5952	286	10	.	.	PUNCT
ejpam-5952	287	1	a	a	DET
ejpam-5952	287	2	common	common	ADJ
ejpam-5952	287	3	fixed	fix	VERB
ejpam-5952	287	4	point	point	NOUN
ejpam-5952	287	5	theorem	theorem	NOUN
ejpam-5952	287	6	of	of	ADP
ejpam-5952	287	7	gregus	gregus	NOUN
ejpam-5952	287	8	type	type	NOUN
ejpam-5952	287	9	for	for	ADP
ejpam-5952	287	10	compatible	compatible	ADJ
ejpam-5952	287	11	mappings	mapping	NOUN
ejpam-5952	287	12	.	.	PUNCT
ejpam-5952	288	1	facta	facta	PROPN
ejpam-5952	288	2	universitatis	universitatis	PROPN
ejpam-5952	288	3	,	,	PUNCT
ejpam-5952	288	4	series	series	NOUN
ejpam-5952	288	5	:	:	PUNCT
ejpam-5952	288	6	mathematics	mathematic	NOUN
ejpam-5952	288	7	and	and	CCONJ
ejpam-5952	288	8	informatics	informatic	NOUN
ejpam-5952	288	9	,	,	PUNCT
ejpam-5952	288	10	7:51–58	7:51–58	NOUN
ejpam-5952	288	11	,	,	PUNCT
ejpam-5952	288	12	1992	1992	NUM
ejpam-5952	288	13	.	.	PUNCT
ejpam-5952	289	1	[	[	X
ejpam-5952	289	2	22	22	NUM
ejpam-5952	289	3	]	]	X
ejpam-5952	289	4	b.	b.	PROPN
ejpam-5952	289	5	c.	c.	PROPN
ejpam-5952	289	6	dhage	dhage	PROPN
ejpam-5952	289	7	.	.	PUNCT
ejpam-5952	290	1	generalized	generalize	VERB
ejpam-5952	290	2	metric	metric	ADJ
ejpam-5952	290	3	spaces	space	NOUN
ejpam-5952	290	4	and	and	CCONJ
ejpam-5952	290	5	mappings	mapping	NOUN
ejpam-5952	290	6	with	with	ADP
ejpam-5952	290	7	fixed	fix	VERB
ejpam-5952	290	8	points	point	NOUN
ejpam-5952	290	9	.	.	PUNCT
ejpam-5952	291	1	bulletin	bulletin	NOUN
ejpam-5952	291	2	of	of	ADP
ejpam-5952	291	3	the	the	DET
ejpam-5952	291	4	calcutta	calcutta	PROPN
ejpam-5952	291	5	mathematical	mathematical	ADJ
ejpam-5952	291	6	society	society	NOUN
ejpam-5952	291	7	,	,	PUNCT
ejpam-5952	291	8	84:329–336	84:329–336	NUM
ejpam-5952	291	9	,	,	PUNCT
ejpam-5952	291	10	1992	1992	NUM
ejpam-5952	291	11	.	.	PUNCT
ejpam-5952	292	1	a.	a.	NOUN
ejpam-5952	292	2	a.	a.	PROPN
ejpam-5952	292	3	m.	m.	PROPN
ejpam-5952	292	4	malkawi	malkawi	ADP
ejpam-5952	292	5	/	/	SYM
ejpam-5952	292	6	eur	eur	PROPN
ejpam-5952	292	7	.	.	PUNCT
ejpam-5952	293	1	j.	j.	PROPN
ejpam-5952	293	2	pure	pure	PROPN
ejpam-5952	293	3	appl	appl	PROPN
ejpam-5952	293	4	.	.	PROPN
ejpam-5952	293	5	math	math	PROPN
ejpam-5952	293	6	,	,	PUNCT
ejpam-5952	293	7	18	18	NUM
ejpam-5952	293	8	(	(	PUNCT
ejpam-5952	293	9	2	2	NUM
ejpam-5952	293	10	)	)	PUNCT
ejpam-5952	293	11	(	(	PUNCT
ejpam-5952	293	12	2025	2025	NUM
ejpam-5952	293	13	)	)	PUNCT
ejpam-5952	293	14	,	,	PUNCT
ejpam-5952	293	15	5952	5952	NUM
ejpam-5952	293	16	14	14	NUM
ejpam-5952	293	17	of	of	ADP
ejpam-5952	293	18	14	14	NUM
ejpam-5952	294	1	[	[	X
ejpam-5952	294	2	23	23	NUM
ejpam-5952	294	3	]	]	X
ejpam-5952	294	4	s.	s.	PROPN
ejpam-5952	294	5	sedghi	sedghi	PROPN
ejpam-5952	294	6	,	,	PUNCT
ejpam-5952	294	7	d.	d.	PROPN
ejpam-5952	294	8	turkoglu	turkoglu	PROPN
ejpam-5952	294	9	,	,	PUNCT
ejpam-5952	294	10	and	and	CCONJ
ejpam-5952	294	11	n.	n.	PROPN
ejpam-5952	294	12	shobe	shobe	PROPN
ejpam-5952	294	13	.	.	PUNCT
ejpam-5952	295	1	common	common	ADJ
ejpam-5952	295	2	fixed	fix	VERB
ejpam-5952	295	3	point	point	NOUN
ejpam-5952	295	4	theorems	theorem	NOUN
ejpam-5952	295	5	for	for	ADP
ejpam-5952	295	6	six	six	NUM
ejpam-5952	295	7	weakly	weakly	ADJ
ejpam-5952	295	8	compatible	compatible	ADJ
ejpam-5952	295	9	mappings	mapping	NOUN
ejpam-5952	295	10	in	in	ADP
ejpam-5952	295	11	d∗-metric	d∗-metric	ADJ
ejpam-5952	295	12	spaces	space	NOUN
ejpam-5952	295	13	.	.	PUNCT
ejpam-5952	296	1	thai	thai	PROPN
ejpam-5952	296	2	journal	journal	PROPN
ejpam-5952	296	3	of	of	ADP
ejpam-5952	296	4	mathematics	mathematic	NOUN
ejpam-5952	296	5	,	,	PUNCT
ejpam-5952	296	6	7(2):381	7(2):381	NUM
ejpam-5952	296	7	–	–	PUNCT
ejpam-5952	296	8	391	391	NUM
ejpam-5952	296	9	,	,	PUNCT
ejpam-5952	296	10	2009	2009	NUM
ejpam-5952	296	11	.	.	PUNCT
ejpam-5952	297	1	[	[	X
ejpam-5952	297	2	24	24	NUM
ejpam-5952	297	3	]	]	PUNCT
ejpam-5952	297	4	a.	a.	NOUN
ejpam-5952	297	5	branciari	branciari	PROPN
ejpam-5952	297	6	.	.	PUNCT
ejpam-5952	298	1	a	a	DET
ejpam-5952	298	2	fixed	fix	VERB
ejpam-5952	298	3	point	point	NOUN
ejpam-5952	298	4	theorem	theorem	NOUN
ejpam-5952	298	5	for	for	ADP
ejpam-5952	298	6	mappings	mapping	NOUN
ejpam-5952	298	7	satisfying	satisfy	VERB
ejpam-5952	298	8	a	a	DET
ejpam-5952	298	9	general	general	ADJ
ejpam-5952	298	10	contractive	contractive	ADJ
ejpam-5952	298	11	condition	condition	NOUN
ejpam-5952	298	12	of	of	ADP
ejpam-5952	298	13	integral	integral	ADJ
ejpam-5952	298	14	type	type	NOUN
ejpam-5952	298	15	.	.	PUNCT
ejpam-5952	299	1	international	international	ADJ
ejpam-5952	299	2	journal	journal	PROPN
ejpam-5952	299	3	of	of	ADP
ejpam-5952	299	4	mathematics	mathematics	PROPN
ejpam-5952	299	5	and	and	CCONJ
ejpam-5952	299	6	mathematical	mathematical	ADJ
ejpam-5952	299	7	sciences	science	NOUN
ejpam-5952	299	8	,	,	PUNCT
ejpam-5952	299	9	29(9):531–536	29(9):531–536	NUM
ejpam-5952	299	10	,	,	PUNCT
ejpam-5952	299	11	2002	2002	NUM
ejpam-5952	299	12	.	.	PUNCT
ejpam-5952	300	1	[	[	X
ejpam-5952	300	2	25	25	NUM
ejpam-5952	300	3	]	]	X
ejpam-5952	300	4	w.	w.	PROPN
ejpam-5952	300	5	rudin	rudin	PROPN
ejpam-5952	300	6	.	.	PUNCT
ejpam-5952	301	1	real	real	ADJ
ejpam-5952	301	2	and	and	CCONJ
ejpam-5952	301	3	complex	complex	ADJ
ejpam-5952	301	4	analysis	analysis	NOUN
ejpam-5952	301	5	.	.	PUNCT
ejpam-5952	302	1	mcgraw	mcgraw	PROPN
ejpam-5952	302	2	-	-	PUNCT
ejpam-5952	302	3	hill	hill	PROPN
ejpam-5952	302	4	,	,	PUNCT
ejpam-5952	302	5	new	new	PROPN
ejpam-5952	302	6	york	york	PROPN
ejpam-5952	302	7	,	,	PUNCT
ejpam-5952	302	8	3	3	NUM
ejpam-5952	302	9	edition	edition	NOUN
ejpam-5952	302	10	,	,	PUNCT
ejpam-5952	302	11	1987	1987	NUM
ejpam-5952	302	12	.	.	PUNCT
ejpam-5952	303	1	[	[	X
ejpam-5952	303	2	26	26	NUM
ejpam-5952	303	3	]	]	PUNCT
ejpam-5952	303	4	p.	p.	PROPN
ejpam-5952	303	5	r.	r.	PROPN
ejpam-5952	303	6	halmos	halmos	PROPN
ejpam-5952	303	7	.	.	PUNCT
ejpam-5952	304	1	measure	measure	NOUN
ejpam-5952	304	2	theory	theory	NOUN
ejpam-5952	304	3	.	.	PUNCT
ejpam-5952	305	1	springer	springer	NOUN
ejpam-5952	305	2	,	,	PUNCT
ejpam-5952	305	3	new	new	PROPN
ejpam-5952	305	4	york	york	PROPN
ejpam-5952	305	5	,	,	PUNCT
ejpam-5952	305	6	1974	1974	NUM
ejpam-5952	305	7	.	.	PUNCT
ejpam-5952	306	1	[	[	X
ejpam-5952	306	2	27	27	NUM
ejpam-5952	306	3	]	]	X
ejpam-5952	306	4	h.	h.	PROPN
ejpam-5952	306	5	l.	l.	PROPN
ejpam-5952	306	6	royden	royden	PROPN
ejpam-5952	306	7	and	and	CCONJ
ejpam-5952	306	8	p.	p.	PROPN
ejpam-5952	306	9	m.	m.	PROPN
ejpam-5952	306	10	fitzpatrick	fitzpatrick	PROPN
ejpam-5952	306	11	.	.	PUNCT
ejpam-5952	307	1	real	real	ADJ
ejpam-5952	307	2	analysis	analysis	NOUN
ejpam-5952	307	3	.	.	PUNCT
ejpam-5952	308	1	prentice	prentice	PROPN
ejpam-5952	308	2	hall	hall	PROPN
ejpam-5952	308	3	,	,	PUNCT
ejpam-5952	308	4	upper	upper	ADJ
ejpam-5952	308	5	saddle	saddle	NOUN
ejpam-5952	308	6	river	river	PROPN
ejpam-5952	308	7	,	,	PUNCT
ejpam-5952	308	8	nj	nj	PROPN
ejpam-5952	308	9	,	,	PUNCT
ejpam-5952	308	10	4	4	NUM
ejpam-5952	308	11	edition	edition	NOUN
ejpam-5952	308	12	,	,	PUNCT
ejpam-5952	308	13	2010	2010	NUM
ejpam-5952	308	14	.	.	PUNCT