id	sid	tid	token	lemma	pos
ejpam-6528	1	1	european	european	PROPN
ejpam-6528	1	2	journal	journal	PROPN
ejpam-6528	1	3	of	of	ADP
ejpam-6528	1	4	pure	pure	ADJ
ejpam-6528	1	5	and	and	CCONJ
ejpam-6528	1	6	applied	applied	ADJ
ejpam-6528	1	7	mathematics	mathematic	NOUN
ejpam-6528	1	8	2025	2025	NUM
ejpam-6528	1	9	,	,	PUNCT
ejpam-6528	1	10	vol	vol	NOUN
ejpam-6528	1	11	.	.	PROPN
ejpam-6528	1	12	18	18	NUM
ejpam-6528	1	13	,	,	PUNCT
ejpam-6528	1	14	issue	issue	NOUN
ejpam-6528	1	15	3	3	NUM
ejpam-6528	1	16	,	,	PUNCT
ejpam-6528	1	17	article	article	NOUN
ejpam-6528	1	18	number	number	NOUN
ejpam-6528	1	19	6528	6528	NUM
ejpam-6528	1	20	issn	issn	PROPN
ejpam-6528	1	21	1307	1307	NUM
ejpam-6528	1	22	-	-	SYM
ejpam-6528	1	23	5543	5543	NUM
ejpam-6528	1	24	–	–	PUNCT
ejpam-6528	1	25	ejpam.com	ejpam.com	X
ejpam-6528	1	26	published	publish	VERB
ejpam-6528	1	27	by	by	ADP
ejpam-6528	1	28	new	new	PROPN
ejpam-6528	1	29	york	york	PROPN
ejpam-6528	1	30	business	business	PROPN
ejpam-6528	1	31	global	global	ADJ
ejpam-6528	1	32	applications	application	NOUN
ejpam-6528	1	33	of	of	ADP
ejpam-6528	1	34	mr	mr	PROPN
ejpam-6528	1	35	-	-	PUNCT
ejpam-6528	1	36	metric	metric	ADJ
ejpam-6528	1	37	spaces	space	NOUN
ejpam-6528	1	38	in	in	ADP
ejpam-6528	1	39	measure	measure	NOUN
ejpam-6528	1	40	theory	theory	NOUN
ejpam-6528	1	41	and	and	CCONJ
ejpam-6528	1	42	convergence	convergence	NOUN
ejpam-6528	1	43	analysis	analysis	NOUN
ejpam-6528	1	44	abed	abe	VERB
ejpam-6528	1	45	al	al	PROPN
ejpam-6528	1	46	-	-	PUNCT
ejpam-6528	1	47	rahman	rahman	PROPN
ejpam-6528	1	48	m.	m.	PROPN
ejpam-6528	1	49	malkawi1,∗	malkawi1,∗	PROPN
ejpam-6528	1	50	,	,	PUNCT
ejpam-6528	1	51	ayat	ayat	PROPN
ejpam-6528	1	52	m.	m.	NOUN
ejpam-6528	1	53	rabaiah1	rabaiah1	PROPN
ejpam-6528	1	54	1	1	NUM
ejpam-6528	1	55	department	department	NOUN
ejpam-6528	1	56	of	of	ADP
ejpam-6528	1	57	mathematics	mathematic	NOUN
ejpam-6528	1	58	,	,	PUNCT
ejpam-6528	1	59	faculty	faculty	NOUN
ejpam-6528	1	60	of	of	ADP
ejpam-6528	1	61	arts	art	NOUN
ejpam-6528	1	62	and	and	CCONJ
ejpam-6528	1	63	science	science	NOUN
ejpam-6528	1	64	,	,	PUNCT
ejpam-6528	1	65	amman	amman	PROPN
ejpam-6528	1	66	arab	arab	PROPN
ejpam-6528	1	67	university	university	PROPN
ejpam-6528	1	68	,	,	PUNCT
ejpam-6528	1	69	amman	amman	PROPN
ejpam-6528	1	70	11953	11953	NUM
ejpam-6528	1	71	,	,	PUNCT
ejpam-6528	1	72	jordan	jordan	PROPN
ejpam-6528	1	73	abstract	abstract	PROPN
ejpam-6528	1	74	.	.	PUNCT
ejpam-6528	2	1	this	this	DET
ejpam-6528	2	2	paper	paper	NOUN
ejpam-6528	2	3	investigates	investigate	VERB
ejpam-6528	2	4	the	the	DET
ejpam-6528	2	5	role	role	NOUN
ejpam-6528	2	6	of	of	ADP
ejpam-6528	2	7	mr	mr	PROPN
ejpam-6528	2	8	-	-	PUNCT
ejpam-6528	2	9	metric	metric	ADJ
ejpam-6528	2	10	spaces	space	NOUN
ejpam-6528	2	11	in	in	ADP
ejpam-6528	2	12	the	the	DET
ejpam-6528	2	13	fields	field	NOUN
ejpam-6528	2	14	of	of	ADP
ejpam-6528	2	15	measure	measure	NOUN
ejpam-6528	2	16	theory	theory	NOUN
ejpam-6528	2	17	and	and	CCONJ
ejpam-6528	2	18	convergence	convergence	NOUN
ejpam-6528	2	19	analysis	analysis	NOUN
ejpam-6528	2	20	.	.	PUNCT
ejpam-6528	3	1	we	we	PRON
ejpam-6528	3	2	analyze	analyze	VERB
ejpam-6528	3	3	how	how	SCONJ
ejpam-6528	3	4	measures	measure	NOUN
ejpam-6528	3	5	defined	define	VERB
ejpam-6528	3	6	through	through	ADP
ejpam-6528	3	7	a	a	DET
ejpam-6528	3	8	triadic	triadic	ADJ
ejpam-6528	3	9	metric	metric	ADJ
ejpam-6528	3	10	function	function	NOUN
ejpam-6528	3	11	reveal	reveal	VERB
ejpam-6528	3	12	key	key	ADJ
ejpam-6528	3	13	characteristics	characteristic	NOUN
ejpam-6528	3	14	such	such	ADJ
ejpam-6528	3	15	as	as	ADP
ejpam-6528	3	16	σ	σ	NOUN
ejpam-6528	3	17	-	-	PUNCT
ejpam-6528	3	18	finiteness	finiteness	ADJ
ejpam-6528	3	19	and	and	CCONJ
ejpam-6528	3	20	absolute	absolute	ADJ
ejpam-6528	3	21	continuity	continuity	NOUN
ejpam-6528	3	22	.	.	PUNCT
ejpam-6528	4	1	furthermore	furthermore	ADV
ejpam-6528	4	2	,	,	PUNCT
ejpam-6528	4	3	we	we	PRON
ejpam-6528	4	4	explore	explore	VERB
ejpam-6528	4	5	the	the	DET
ejpam-6528	4	6	convergence	convergence	NOUN
ejpam-6528	4	7	behavior	behavior	NOUN
ejpam-6528	4	8	of	of	ADP
ejpam-6528	4	9	sequences	sequence	NOUN
ejpam-6528	4	10	within	within	ADP
ejpam-6528	4	11	the	the	DET
ejpam-6528	4	12	mr	mr	PROPN
ejpam-6528	4	13	-	-	PUNCT
ejpam-6528	4	14	metric	metric	ADJ
ejpam-6528	4	15	framework	framework	NOUN
ejpam-6528	4	16	,	,	PUNCT
ejpam-6528	4	17	highlighting	highlight	VERB
ejpam-6528	4	18	their	their	PRON
ejpam-6528	4	19	relevance	relevance	NOUN
ejpam-6528	4	20	in	in	ADP
ejpam-6528	4	21	areas	area	NOUN
ejpam-6528	4	22	like	like	ADP
ejpam-6528	4	23	stochastic	stochastic	ADJ
ejpam-6528	4	24	processes	process	NOUN
ejpam-6528	4	25	,	,	PUNCT
ejpam-6528	4	26	optimization	optimization	NOUN
ejpam-6528	4	27	,	,	PUNCT
ejpam-6528	4	28	and	and	CCONJ
ejpam-6528	4	29	data	datum	NOUN
ejpam-6528	4	30	science	science	NOUN
ejpam-6528	4	31	.	.	PUNCT
ejpam-6528	5	1	the	the	DET
ejpam-6528	5	2	findings	finding	NOUN
ejpam-6528	5	3	offer	offer	VERB
ejpam-6528	5	4	valuable	valuable	ADJ
ejpam-6528	5	5	perspectives	perspective	NOUN
ejpam-6528	5	6	on	on	ADP
ejpam-6528	5	7	probability	probability	NOUN
ejpam-6528	5	8	distributions	distribution	NOUN
ejpam-6528	5	9	that	that	PRON
ejpam-6528	5	10	incorporate	incorporate	VERB
ejpam-6528	5	11	triadic	triadic	ADJ
ejpam-6528	5	12	dependencies	dependency	NOUN
ejpam-6528	5	13	,	,	PUNCT
ejpam-6528	5	14	the	the	DET
ejpam-6528	5	15	conditions	condition	NOUN
ejpam-6528	5	16	necessary	necessary	ADJ
ejpam-6528	5	17	for	for	ADP
ejpam-6528	5	18	stability	stability	NOUN
ejpam-6528	5	19	in	in	ADP
ejpam-6528	5	20	machine	machine	NOUN
ejpam-6528	5	21	learning	learning	NOUN
ejpam-6528	5	22	,	,	PUNCT
ejpam-6528	5	23	and	and	CCONJ
ejpam-6528	5	24	the	the	DET
ejpam-6528	5	25	clustering	cluster	VERB
ejpam-6528	5	26	behavior	behavior	NOUN
ejpam-6528	5	27	within	within	ADP
ejpam-6528	5	28	complex	complex	ADJ
ejpam-6528	5	29	networks	network	NOUN
ejpam-6528	5	30	.	.	PUNCT
ejpam-6528	6	1	these	these	DET
ejpam-6528	6	2	discoveries	discovery	NOUN
ejpam-6528	6	3	pave	pave	VERB
ejpam-6528	6	4	the	the	DET
ejpam-6528	6	5	way	way	NOUN
ejpam-6528	6	6	for	for	ADP
ejpam-6528	6	7	extending	extend	VERB
ejpam-6528	6	8	traditional	traditional	ADJ
ejpam-6528	6	9	metric	metric	ADJ
ejpam-6528	6	10	space	space	NOUN
ejpam-6528	6	11	theory	theory	NOUN
ejpam-6528	6	12	into	into	ADP
ejpam-6528	6	13	more	more	ADV
ejpam-6528	6	14	advanced	advanced	ADJ
ejpam-6528	6	15	settings	setting	NOUN
ejpam-6528	6	16	,	,	PUNCT
ejpam-6528	6	17	including	include	VERB
ejpam-6528	6	18	those	those	PRON
ejpam-6528	6	19	involving	involve	VERB
ejpam-6528	6	20	multi	multi	ADJ
ejpam-6528	6	21	-	-	ADJ
ejpam-6528	6	22	agent	agent	ADJ
ejpam-6528	6	23	systems	system	NOUN
ejpam-6528	6	24	and	and	CCONJ
ejpam-6528	6	25	high	high	ADJ
ejpam-6528	6	26	-	-	PUNCT
ejpam-6528	6	27	dimensional	dimensional	ADJ
ejpam-6528	6	28	analyses	analysis	NOUN
ejpam-6528	6	29	.	.	PUNCT
ejpam-6528	7	1	2020	2020	NUM
ejpam-6528	7	2	mathematics	mathematic	NOUN
ejpam-6528	7	3	subject	subject	NOUN
ejpam-6528	7	4	classifications	classification	NOUN
ejpam-6528	7	5	:	:	PUNCT
ejpam-6528	7	6	54e50	54e50	NUM
ejpam-6528	7	7	,	,	PUNCT
ejpam-6528	7	8	28a12	28a12	NUM
ejpam-6528	7	9	,	,	PUNCT
ejpam-6528	7	10	60b10	60b10	NOUN
ejpam-6528	7	11	,	,	PUNCT
ejpam-6528	7	12	46e27	46e27	NUM
ejpam-6528	7	13	,	,	PUNCT
ejpam-6528	7	14	37a50	37a50	NUM
ejpam-6528	7	15	key	key	ADJ
ejpam-6528	7	16	words	word	NOUN
ejpam-6528	7	17	and	and	CCONJ
ejpam-6528	7	18	phrases	phrase	NOUN
ejpam-6528	7	19	:	:	PUNCT
ejpam-6528	7	20	mr−metric	mr−metric	ADJ
ejpam-6528	7	21	space	space	NOUN
ejpam-6528	7	22	,	,	PUNCT
ejpam-6528	7	23	absolute	absolute	ADJ
ejpam-6528	7	24	continuity	continuity	NOUN
ejpam-6528	7	25	,	,	PUNCT
ejpam-6528	7	26	measure	measure	NOUN
ejpam-6528	7	27	theory	theory	NOUN
ejpam-6528	7	28	,	,	PUNCT
ejpam-6528	7	29	σ	σ	NOUN
ejpam-6528	7	30	-	-	PUNCT
ejpam-6528	7	31	finiteness	finiteness	NOUN
ejpam-6528	7	32	,	,	PUNCT
ejpam-6528	7	33	fixed	fix	VERB
ejpam-6528	7	34	point	point	NOUN
ejpam-6528	7	35	theorems	theorem	NOUN
ejpam-6528	7	36	,	,	PUNCT
ejpam-6528	7	37	mr	mr	NOUN
ejpam-6528	7	38	-	-	PUNCT
ejpam-6528	7	39	convergent	convergent	NOUN
ejpam-6528	7	40	,	,	PUNCT
ejpam-6528	7	41	optimization	optimization	NOUN
ejpam-6528	7	42	1	1	NUM
ejpam-6528	7	43	.	.	PUNCT
ejpam-6528	8	1	introduction	introduction	NOUN
ejpam-6528	8	2	this	this	DET
ejpam-6528	8	3	paper	paper	NOUN
ejpam-6528	8	4	presents	present	VERB
ejpam-6528	8	5	an	an	DET
ejpam-6528	8	6	in	in	ADP
ejpam-6528	8	7	-	-	PUNCT
ejpam-6528	8	8	depth	depth	NOUN
ejpam-6528	8	9	exploration	exploration	NOUN
ejpam-6528	8	10	of	of	ADP
ejpam-6528	8	11	mr	mr	PROPN
ejpam-6528	8	12	-	-	PUNCT
ejpam-6528	8	13	metric	metric	ADJ
ejpam-6528	8	14	spaces	space	NOUN
ejpam-6528	8	15	,	,	PUNCT
ejpam-6528	8	16	particularly	particularly	ADV
ejpam-6528	8	17	their	their	PRON
ejpam-6528	8	18	application	application	NOUN
ejpam-6528	8	19	within	within	ADP
ejpam-6528	8	20	the	the	DET
ejpam-6528	8	21	contexts	contexts	NOUN
ejpam-6528	8	22	of	of	ADP
ejpam-6528	8	23	measure	measure	NOUN
ejpam-6528	8	24	theory	theory	NOUN
ejpam-6528	8	25	and	and	CCONJ
ejpam-6528	8	26	convergence	convergence	NOUN
ejpam-6528	8	27	analysis	analysis	NOUN
ejpam-6528	8	28	.	.	PUNCT
ejpam-6528	9	1	mr	mr	PROPN
ejpam-6528	9	2	-	-	PUNCT
ejpam-6528	9	3	metric	metric	ADJ
ejpam-6528	9	4	spaces	space	NOUN
ejpam-6528	9	5	,	,	PUNCT
ejpam-6528	9	6	characterized	characterize	VERB
ejpam-6528	9	7	by	by	ADP
ejpam-6528	9	8	a	a	DET
ejpam-6528	9	9	triadic	triadic	ADJ
ejpam-6528	9	10	metric	metric	ADJ
ejpam-6528	9	11	function	function	NOUN
ejpam-6528	9	12	,	,	PUNCT
ejpam-6528	9	13	provide	provide	VERB
ejpam-6528	9	14	a	a	DET
ejpam-6528	9	15	novel	novel	ADJ
ejpam-6528	9	16	approach	approach	NOUN
ejpam-6528	9	17	to	to	ADP
ejpam-6528	9	18	understanding	understand	VERB
ejpam-6528	9	19	complex	complex	ADJ
ejpam-6528	9	20	relationships	relationship	NOUN
ejpam-6528	9	21	in	in	ADP
ejpam-6528	9	22	various	various	ADJ
ejpam-6528	9	23	mathematical	mathematical	ADJ
ejpam-6528	9	24	domains	domain	NOUN
ejpam-6528	9	25	.	.	PUNCT
ejpam-6528	10	1	by	by	ADP
ejpam-6528	10	2	leveraging	leverage	VERB
ejpam-6528	10	3	this	this	DET
ejpam-6528	10	4	triadic	triadic	ADJ
ejpam-6528	10	5	structure	structure	NOUN
ejpam-6528	10	6	,	,	PUNCT
ejpam-6528	10	7	we	we	PRON
ejpam-6528	10	8	examine	examine	VERB
ejpam-6528	10	9	foundational	foundational	ADJ
ejpam-6528	10	10	properties	property	NOUN
ejpam-6528	10	11	such	such	ADJ
ejpam-6528	10	12	as	as	ADP
ejpam-6528	10	13	σ	σ	NOUN
ejpam-6528	10	14	-	-	PUNCT
ejpam-6528	10	15	finiteness	finiteness	ADJ
ejpam-6528	10	16	and	and	CCONJ
ejpam-6528	10	17	absolute	absolute	ADJ
ejpam-6528	10	18	continuity	continuity	NOUN
ejpam-6528	10	19	,	,	PUNCT
ejpam-6528	10	20	which	which	PRON
ejpam-6528	10	21	are	be	AUX
ejpam-6528	10	22	crucial	crucial	ADJ
ejpam-6528	10	23	for	for	ADP
ejpam-6528	10	24	the	the	DET
ejpam-6528	10	25	rigorous	rigorous	ADJ
ejpam-6528	10	26	analysis	analysis	NOUN
ejpam-6528	10	27	of	of	ADP
ejpam-6528	10	28	measures	measure	NOUN
ejpam-6528	10	29	.	.	PUNCT
ejpam-6528	11	1	these	these	DET
ejpam-6528	11	2	properties	property	NOUN
ejpam-6528	11	3	enable	enable	VERB
ejpam-6528	11	4	a	a	DET
ejpam-6528	11	5	more	more	ADV
ejpam-6528	11	6	robust	robust	ADJ
ejpam-6528	11	7	framework	framework	NOUN
ejpam-6528	11	8	for	for	ADP
ejpam-6528	11	9	investigating	investigate	VERB
ejpam-6528	11	10	diverse	diverse	ADJ
ejpam-6528	11	11	mathematical	mathematical	ADJ
ejpam-6528	11	12	phenomena	phenomenon	NOUN
ejpam-6528	11	13	,	,	PUNCT
ejpam-6528	11	14	including	include	VERB
ejpam-6528	11	15	the	the	DET
ejpam-6528	11	16	behavior	behavior	NOUN
ejpam-6528	11	17	of	of	ADP
ejpam-6528	11	18	stochastic	stochastic	ADJ
ejpam-6528	11	19	processes	process	NOUN
ejpam-6528	11	20	.	.	PUNCT
ejpam-6528	12	1	furthermore	furthermore	ADV
ejpam-6528	12	2	,	,	PUNCT
ejpam-6528	12	3	the	the	DET
ejpam-6528	12	4	study	study	NOUN
ejpam-6528	12	5	explores	explore	VERB
ejpam-6528	12	6	how	how	SCONJ
ejpam-6528	12	7	these	these	DET
ejpam-6528	12	8	metric	metric	ADJ
ejpam-6528	12	9	spaces	space	NOUN
ejpam-6528	12	10	contribute	contribute	VERB
ejpam-6528	12	11	to	to	ADP
ejpam-6528	12	12	the	the	DET
ejpam-6528	12	13	stability	stability	NOUN
ejpam-6528	12	14	of	of	ADP
ejpam-6528	12	15	machine	machine	NOUN
ejpam-6528	12	16	learning	learning	NOUN
ejpam-6528	12	17	models	model	NOUN
ejpam-6528	12	18	,	,	PUNCT
ejpam-6528	12	19	offering	offer	VERB
ejpam-6528	12	20	new	new	ADJ
ejpam-6528	12	21	insights	insight	NOUN
ejpam-6528	12	22	into	into	ADP
ejpam-6528	12	23	optimization	optimization	NOUN
ejpam-6528	12	24	problems	problem	NOUN
ejpam-6528	12	25	and	and	CCONJ
ejpam-6528	12	26	enhancing	enhance	VERB
ejpam-6528	12	27	∗corresponding	∗corresponde	VERB
ejpam-6528	12	28	author	author	NOUN
ejpam-6528	12	29	.	.	PUNCT
ejpam-6528	13	1	doi	doi	NOUN
ejpam-6528	13	2	:	:	PUNCT
ejpam-6528	13	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6528	https://doi.org/10.29020/nybg.ejpam.v18i3.6528	PROPN
ejpam-6528	13	4	email	email	NOUN
ejpam-6528	13	5	addresses	address	VERB
ejpam-6528	13	6	:	:	PUNCT
ejpam-6528	13	7	a.malkawi@aau.edu.jo	a.malkawi@aau.edu.jo	PROPN
ejpam-6528	13	8	and	and	CCONJ
ejpam-6528	13	9	math.malkawi@gmail.com	math.malkawi@gmail.com	X
ejpam-6528	13	10	(	(	PUNCT
ejpam-6528	13	11	a.	a.	NOUN
ejpam-6528	13	12	malkawi	malkawi	PROPN
ejpam-6528	13	13	)	)	PUNCT
ejpam-6528	13	14	,	,	PUNCT
ejpam-6528	13	15	a.rabaieha@aau.edu.jo	a.rabaieha@aau.edu.jo	PROPN
ejpam-6528	13	16	(	(	PUNCT
ejpam-6528	13	17	a.	a.	NOUN
ejpam-6528	13	18	rabaiah	rabaiah	PROPN
ejpam-6528	13	19	)	)	PUNCT
ejpam-6528	13	20	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6528	14	1	1	1	NUM
ejpam-6528	14	2	copyright	copyright	NOUN
ejpam-6528	14	3	:	:	PUNCT
ejpam-6528	14	4	©	©	PROPN
ejpam-6528	14	5	2025	2025	NUM
ejpam-6528	14	6	the	the	DET
ejpam-6528	14	7	author(s	author(s	NOUN
ejpam-6528	14	8	)	)	PUNCT
ejpam-6528	14	9	.	.	PUNCT
ejpam-6528	15	1	(	(	PUNCT
ejpam-6528	15	2	cc	cc	NOUN
ejpam-6528	15	3	by	by	ADP
ejpam-6528	15	4	-	-	PUNCT
ejpam-6528	15	5	nc	nc	PROPN
ejpam-6528	15	6	4.0	4.0	NUM
ejpam-6528	15	7	)	)	PUNCT
ejpam-6528	15	8	a.	a.	NOUN
ejpam-6528	15	9	malkawi	malkawi	PROPN
ejpam-6528	15	10	,	,	PUNCT
ejpam-6528	15	11	a.	a.	PROPN
ejpam-6528	15	12	rabaiah	rabaiah	PROPN
ejpam-6528	15	13	/	/	SYM
ejpam-6528	15	14	eur	eur	PROPN
ejpam-6528	15	15	.	.	PUNCT
ejpam-6528	16	1	j.	j.	PROPN
ejpam-6528	16	2	pure	pure	PROPN
ejpam-6528	16	3	appl	appl	PROPN
ejpam-6528	16	4	.	.	PROPN
ejpam-6528	16	5	math	math	PROPN
ejpam-6528	16	6	,	,	PUNCT
ejpam-6528	16	7	18	18	NUM
ejpam-6528	16	8	(	(	PUNCT
ejpam-6528	16	9	3	3	NUM
ejpam-6528	16	10	)	)	PUNCT
ejpam-6528	16	11	(	(	PUNCT
ejpam-6528	16	12	2025	2025	NUM
ejpam-6528	16	13	)	)	PUNCT
ejpam-6528	16	14	,	,	PUNCT
ejpam-6528	16	15	6528	6528	NUM
ejpam-6528	16	16	2	2	NUM
ejpam-6528	16	17	of	of	ADP
ejpam-6528	16	18	12	12	NUM
ejpam-6528	16	19	algorithmic	algorithmic	ADJ
ejpam-6528	16	20	performance	performance	NOUN
ejpam-6528	16	21	in	in	ADP
ejpam-6528	16	22	the	the	DET
ejpam-6528	16	23	broader	broad	ADJ
ejpam-6528	16	24	field	field	NOUN
ejpam-6528	16	25	of	of	ADP
ejpam-6528	16	26	data	datum	NOUN
ejpam-6528	16	27	science	science	NOUN
ejpam-6528	16	28	.	.	PUNCT
ejpam-6528	17	1	by	by	ADP
ejpam-6528	17	2	addressing	address	VERB
ejpam-6528	17	3	the	the	DET
ejpam-6528	17	4	intricate	intricate	ADJ
ejpam-6528	17	5	connections	connection	NOUN
ejpam-6528	17	6	between	between	ADP
ejpam-6528	17	7	mr	mr	PROPN
ejpam-6528	17	8	-	-	PUNCT
ejpam-6528	17	9	metric	metric	ADJ
ejpam-6528	17	10	spaces	space	NOUN
ejpam-6528	17	11	and	and	CCONJ
ejpam-6528	17	12	probability	probability	NOUN
ejpam-6528	17	13	distributions	distribution	NOUN
ejpam-6528	17	14	that	that	PRON
ejpam-6528	17	15	incorporate	incorporate	VERB
ejpam-6528	17	16	triadic	triadic	ADJ
ejpam-6528	17	17	dependencies	dependency	NOUN
ejpam-6528	17	18	,	,	PUNCT
ejpam-6528	17	19	this	this	DET
ejpam-6528	17	20	research	research	NOUN
ejpam-6528	17	21	provides	provide	VERB
ejpam-6528	17	22	a	a	DET
ejpam-6528	17	23	deeper	deep	ADJ
ejpam-6528	17	24	understanding	understanding	NOUN
ejpam-6528	17	25	of	of	ADP
ejpam-6528	17	26	the	the	DET
ejpam-6528	17	27	underlying	underlying	ADJ
ejpam-6528	17	28	mathematical	mathematical	ADJ
ejpam-6528	17	29	principles	principle	NOUN
ejpam-6528	17	30	.	.	PUNCT
ejpam-6528	18	1	the	the	DET
ejpam-6528	18	2	results	result	NOUN
ejpam-6528	18	3	also	also	ADV
ejpam-6528	18	4	open	open	VERB
ejpam-6528	18	5	up	up	ADP
ejpam-6528	18	6	promising	promise	VERB
ejpam-6528	18	7	new	new	ADJ
ejpam-6528	18	8	avenues	avenue	NOUN
ejpam-6528	18	9	for	for	ADP
ejpam-6528	18	10	extending	extend	VERB
ejpam-6528	18	11	the	the	DET
ejpam-6528	18	12	theory	theory	NOUN
ejpam-6528	18	13	of	of	ADP
ejpam-6528	18	14	metric	metric	ADJ
ejpam-6528	18	15	spaces	space	NOUN
ejpam-6528	18	16	,	,	PUNCT
ejpam-6528	18	17	particularly	particularly	ADV
ejpam-6528	18	18	in	in	ADP
ejpam-6528	18	19	high	high	ADJ
ejpam-6528	18	20	-	-	PUNCT
ejpam-6528	18	21	dimensional	dimensional	ADJ
ejpam-6528	18	22	settings	setting	NOUN
ejpam-6528	18	23	and	and	CCONJ
ejpam-6528	18	24	complex	complex	ADJ
ejpam-6528	18	25	systems	system	NOUN
ejpam-6528	18	26	involving	involve	VERB
ejpam-6528	18	27	multiple	multiple	ADJ
ejpam-6528	18	28	interacting	interact	VERB
ejpam-6528	18	29	agents	agent	NOUN
ejpam-6528	18	30	,	,	PUNCT
ejpam-6528	18	31	such	such	ADJ
ejpam-6528	18	32	as	as	ADP
ejpam-6528	18	33	those	those	PRON
ejpam-6528	18	34	found	find	VERB
ejpam-6528	18	35	in	in	ADP
ejpam-6528	18	36	network	network	NOUN
ejpam-6528	18	37	analysis	analysis	NOUN
ejpam-6528	18	38	and	and	CCONJ
ejpam-6528	18	39	multi	multi	ADJ
ejpam-6528	18	40	-	-	ADJ
ejpam-6528	18	41	agent	agent	ADJ
ejpam-6528	18	42	models	model	NOUN
ejpam-6528	18	43	.	.	PUNCT
ejpam-6528	19	1	for	for	ADP
ejpam-6528	19	2	further	further	ADJ
ejpam-6528	19	3	details	detail	NOUN
ejpam-6528	19	4	,	,	PUNCT
ejpam-6528	19	5	we	we	PRON
ejpam-6528	19	6	refer	refer	VERB
ejpam-6528	19	7	readers	reader	NOUN
ejpam-6528	19	8	to	to	ADP
ejpam-6528	19	9	the	the	DET
ejpam-6528	19	10	works	work	NOUN
ejpam-6528	19	11	cited	cite	VERB
ejpam-6528	19	12	in	in	ADP
ejpam-6528	19	13	[	[	X
ejpam-6528	19	14	1–21	1–21	NOUN
ejpam-6528	19	15	]	]	PUNCT
ejpam-6528	19	16	.	.	PUNCT
ejpam-6528	20	1	definition	definition	NOUN
ejpam-6528	20	2	1	1	NUM
ejpam-6528	20	3	.	.	PUNCT
ejpam-6528	21	1	[	[	X
ejpam-6528	21	2	22	22	NUM
ejpam-6528	21	3	]	]	PUNCT
ejpam-6528	21	4	consider	consider	VERB
ejpam-6528	21	5	a	a	DET
ejpam-6528	21	6	non	non	ADJ
ejpam-6528	21	7	-	-	ADJ
ejpam-6528	21	8	empty	empty	ADJ
ejpam-6528	21	9	set	set	NOUN
ejpam-6528	21	10	x	x	PUNCT
ejpam-6528	21	11	̸=	̸=	PROPN
ejpam-6528	21	12	∅	∅	NOUN
ejpam-6528	21	13	and	and	CCONJ
ejpam-6528	21	14	a	a	DET
ejpam-6528	21	15	real	real	ADJ
ejpam-6528	21	16	number	number	NOUN
ejpam-6528	21	17	r	r	NOUN
ejpam-6528	21	18	>	>	X
ejpam-6528	21	19	1	1	NUM
ejpam-6528	21	20	.	.	PUNCT
ejpam-6528	22	1	a	a	DET
ejpam-6528	22	2	function	function	NOUN
ejpam-6528	22	3	m	m	VERB
ejpam-6528	22	4	:	:	PUNCT
ejpam-6528	22	5	x	x	X
ejpam-6528	22	6	×	×	NOUN
ejpam-6528	22	7	x	x	SYM
ejpam-6528	22	8	×	×	NOUN
ejpam-6528	22	9	x	x	INTJ
ejpam-6528	22	10	→	→	X
ejpam-6528	22	11	[	[	X
ejpam-6528	22	12	0,∞	0,∞	NOUN
ejpam-6528	22	13	)	)	PUNCT
ejpam-6528	22	14	is	be	AUX
ejpam-6528	22	15	termed	term	VERB
ejpam-6528	22	16	an	an	DET
ejpam-6528	22	17	mr	mr	PROPN
ejpam-6528	22	18	-	-	PUNCT
ejpam-6528	22	19	metric	metric	NOUN
ejpam-6528	22	20	if	if	SCONJ
ejpam-6528	22	21	it	it	PRON
ejpam-6528	22	22	satisfies	satisfy	VERB
ejpam-6528	22	23	the	the	DET
ejpam-6528	22	24	following	follow	VERB
ejpam-6528	22	25	conditions	condition	NOUN
ejpam-6528	22	26	for	for	ADP
ejpam-6528	22	27	all	all	DET
ejpam-6528	22	28	υ	υ	PROPN
ejpam-6528	22	29	,	,	PUNCT
ejpam-6528	22	30	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	22	31	∈	∈	PROPN
ejpam-6528	23	1	x	x	X
ejpam-6528	23	2	:	:	PUNCT
ejpam-6528	23	3	•	•	PRON
ejpam-6528	23	4	(	(	PUNCT
ejpam-6528	23	5	m1	m1	NOUN
ejpam-6528	23	6	)	)	PUNCT
ejpam-6528	23	7	m(υ	m(υ	PROPN
ejpam-6528	23	8	,	,	PUNCT
ejpam-6528	23	9	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	23	10	)	)	PUNCT
ejpam-6528	23	11	≥	≥	NOUN
ejpam-6528	23	12	0	0	NUM
ejpam-6528	23	13	.	.	NOUN
ejpam-6528	23	14	•	•	NUM
ejpam-6528	23	15	(	(	PUNCT
ejpam-6528	23	16	m2	m2	PROPN
ejpam-6528	23	17	)	)	PUNCT
ejpam-6528	23	18	m(υ	m(υ	PROPN
ejpam-6528	23	19	,	,	PUNCT
ejpam-6528	23	20	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	23	21	)	)	PUNCT
ejpam-6528	24	1	=	=	SYM
ejpam-6528	24	2	0	0	PUNCT
ejpam-6528	25	1	if	if	SCONJ
ejpam-6528	25	2	and	and	CCONJ
ejpam-6528	25	3	only	only	ADV
ejpam-6528	25	4	if	if	SCONJ
ejpam-6528	25	5	υ	υ	PROPN
ejpam-6528	25	6	=	=	SYM
ejpam-6528	25	7	ξ	ξ	NOUN
ejpam-6528	25	8	=	=	PUNCT
ejpam-6528	25	9	ℑ.	ℑ.	NOUN
ejpam-6528	25	10	•	•	NUM
ejpam-6528	25	11	(	(	PUNCT
ejpam-6528	25	12	m3)m(υ	m3)m(υ	PROPN
ejpam-6528	25	13	,	,	PUNCT
ejpam-6528	25	14	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	25	15	)	)	PUNCT
ejpam-6528	25	16	remains	remain	VERB
ejpam-6528	25	17	invariant	invariant	ADJ
ejpam-6528	25	18	under	under	ADP
ejpam-6528	25	19	any	any	DET
ejpam-6528	25	20	permutation	permutation	NOUN
ejpam-6528	25	21	p(υ	p(υ	NOUN
ejpam-6528	25	22	,	,	PUNCT
ejpam-6528	25	23	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	25	24	)	)	PUNCT
ejpam-6528	25	25	,	,	PUNCT
ejpam-6528	25	26	i.e.	i.e.	X
ejpam-6528	25	27	,m(υ	,m(υ	PUNCT
ejpam-6528	25	28	,	,	PUNCT
ejpam-6528	25	29	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	25	30	)	)	PUNCT
ejpam-6528	26	1	=	=	SYM
ejpam-6528	26	2	m(p(υ	m(p(υ	PROPN
ejpam-6528	26	3	,	,	PUNCT
ejpam-6528	26	4	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	26	5	)	)	PUNCT
ejpam-6528	26	6	)	)	PUNCT
ejpam-6528	26	7	.	.	PUNCT
ejpam-6528	27	1	•	•	NUM
ejpam-6528	27	2	(	(	PUNCT
ejpam-6528	27	3	m4	m4	PROPN
ejpam-6528	27	4	)	)	PUNCT
ejpam-6528	27	5	the	the	DET
ejpam-6528	27	6	following	follow	VERB
ejpam-6528	27	7	inequality	inequality	NOUN
ejpam-6528	27	8	holds	hold	VERB
ejpam-6528	27	9	:	:	PUNCT
ejpam-6528	28	1	m(υ	m(υ	PROPN
ejpam-6528	28	2	,	,	PUNCT
ejpam-6528	28	3	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	28	4	)	)	PUNCT
ejpam-6528	28	5	≤	≤	NUM
ejpam-6528	28	6	r	r	NOUN
ejpam-6528	29	1	[	[	X
ejpam-6528	29	2	m(υ	m(υ	PROPN
ejpam-6528	29	3	,	,	PUNCT
ejpam-6528	29	4	ξ	ξ	PROPN
ejpam-6528	29	5	,	,	PUNCT
ejpam-6528	29	6	ℓ1	ℓ1	NOUN
ejpam-6528	29	7	)	)	PUNCT
ejpam-6528	30	1	+	+	SYM
ejpam-6528	30	2	m(υ	m(υ	PROPN
ejpam-6528	30	3	,	,	PUNCT
ejpam-6528	30	4	ℓ1,ℑ	ℓ1,ℑ	NOUN
ejpam-6528	30	5	)	)	PUNCT
ejpam-6528	31	1	+	+	ADJ
ejpam-6528	31	2	m(ℓ1	m(ℓ1	NOUN
ejpam-6528	31	3	,	,	PUNCT
ejpam-6528	31	4	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	31	5	)	)	PUNCT
ejpam-6528	31	6	]	]	PUNCT
ejpam-6528	31	7	.	.	PUNCT
ejpam-6528	32	1	a	a	DET
ejpam-6528	32	2	structure	structure	NOUN
ejpam-6528	32	3	(	(	PUNCT
ejpam-6528	32	4	x	x	X
ejpam-6528	32	5	,	,	PUNCT
ejpam-6528	32	6	m	m	NOUN
ejpam-6528	32	7	)	)	PUNCT
ejpam-6528	32	8	that	that	PRON
ejpam-6528	32	9	adheres	adhere	VERB
ejpam-6528	32	10	to	to	ADP
ejpam-6528	32	11	these	these	DET
ejpam-6528	32	12	properties	property	NOUN
ejpam-6528	32	13	is	be	AUX
ejpam-6528	32	14	defined	define	VERB
ejpam-6528	32	15	as	as	ADP
ejpam-6528	32	16	an	an	DET
ejpam-6528	32	17	mr	mr	PROPN
ejpam-6528	32	18	-	-	PUNCT
ejpam-6528	32	19	metric	metric	ADJ
ejpam-6528	32	20	space	space	NOUN
ejpam-6528	32	21	.	.	PUNCT
ejpam-6528	33	1	definition	definition	NOUN
ejpam-6528	33	2	2	2	NUM
ejpam-6528	33	3	.	.	PUNCT
ejpam-6528	34	1	[	[	X
ejpam-6528	34	2	22	22	NUM
ejpam-6528	34	3	]	]	PUNCT
ejpam-6528	34	4	consider	consider	VERB
ejpam-6528	34	5	a	a	DET
ejpam-6528	34	6	sequence	sequence	NOUN
ejpam-6528	34	7	{	{	PUNCT
ejpam-6528	34	8	υin	υin	NOUN
ejpam-6528	34	9	}	}	PUNCT
ejpam-6528	34	10	in	in	ADP
ejpam-6528	34	11	an	an	DET
ejpam-6528	34	12	mr	mr	PROPN
ejpam-6528	34	13	-	-	PUNCT
ejpam-6528	34	14	metric	metric	ADJ
ejpam-6528	34	15	space	space	NOUN
ejpam-6528	34	16	(	(	PUNCT
ejpam-6528	34	17	x	x	X
ejpam-6528	34	18	,	,	PUNCT
ejpam-6528	34	19	m	m	NOUN
ejpam-6528	34	20	)	)	PUNCT
ejpam-6528	34	21	.	.	PUNCT
ejpam-6528	35	1	this	this	DET
ejpam-6528	35	2	sequence	sequence	NOUN
ejpam-6528	35	3	is	be	AUX
ejpam-6528	35	4	said	say	VERB
ejpam-6528	35	5	to	to	PART
ejpam-6528	35	6	be	be	AUX
ejpam-6528	35	7	mr	mr	NOUN
ejpam-6528	35	8	-	-	PUNCT
ejpam-6528	35	9	convergent	convergent	NOUN
ejpam-6528	35	10	if	if	SCONJ
ejpam-6528	35	11	there	there	PRON
ejpam-6528	35	12	exists	exist	VERB
ejpam-6528	35	13	an	an	DET
ejpam-6528	35	14	element	element	ADJ
ejpam-6528	35	15	υi1	υi1	NOUN
ejpam-6528	35	16	∈	∈	PROPN
ejpam-6528	35	17	x	x	PUNCT
ejpam-6528	35	18	such	such	ADJ
ejpam-6528	35	19	that	that	PRON
ejpam-6528	35	20	for	for	ADP
ejpam-6528	35	21	any	any	DET
ejpam-6528	35	22	ϵ	ϵ	X
ejpam-6528	35	23	>	>	X
ejpam-6528	35	24	0	0	NUM
ejpam-6528	35	25	,	,	PUNCT
ejpam-6528	35	26	there	there	PRON
ejpam-6528	35	27	exists	exist	VERB
ejpam-6528	35	28	a	a	DET
ejpam-6528	35	29	positive	positive	ADJ
ejpam-6528	35	30	integer	integer	NOUN
ejpam-6528	35	31	n	n	CCONJ
ejpam-6528	35	32	satisfying	satisfy	VERB
ejpam-6528	35	33	the	the	DET
ejpam-6528	35	34	condition	condition	NOUN
ejpam-6528	35	35	m(υin	m(υin	NOUN
ejpam-6528	35	36	,	,	PUNCT
ejpam-6528	35	37	υim	υim	PROPN
ejpam-6528	35	38	,	,	PUNCT
ejpam-6528	35	39	υi1	υi1	PROPN
ejpam-6528	35	40	)	)	PUNCT
ejpam-6528	35	41	<	<	X
ejpam-6528	36	1	ϵ	ϵ	X
ejpam-6528	36	2	,	,	PUNCT
ejpam-6528	36	3	for	for	ADP
ejpam-6528	36	4	all	all	DET
ejpam-6528	36	5	m	m	PROPN
ejpam-6528	36	6	,	,	PUNCT
ejpam-6528	36	7	n	n	PRON
ejpam-6528	36	8	≥	≥	NOUN
ejpam-6528	36	9	n.	n.	NOUN
ejpam-6528	36	10	in	in	ADP
ejpam-6528	36	11	this	this	DET
ejpam-6528	36	12	case	case	NOUN
ejpam-6528	36	13	,	,	PUNCT
ejpam-6528	36	14	we	we	PRON
ejpam-6528	36	15	say	say	VERB
ejpam-6528	36	16	that	that	SCONJ
ejpam-6528	36	17	{	{	PUNCT
ejpam-6528	36	18	υin	υin	NOUN
ejpam-6528	36	19	}	}	PUNCT
ejpam-6528	36	20	converges	converge	VERB
ejpam-6528	36	21	in	in	ADP
ejpam-6528	36	22	the	the	DET
ejpam-6528	36	23	mr	mr	PROPN
ejpam-6528	36	24	-	-	PUNCT
ejpam-6528	36	25	metric	metric	ADJ
ejpam-6528	36	26	sense	sense	NOUN
ejpam-6528	36	27	to	to	PART
ejpam-6528	36	28	υi1	υi1	VERB
ejpam-6528	36	29	,	,	PUNCT
ejpam-6528	36	30	and	and	CCONJ
ejpam-6528	36	31	we	we	PRON
ejpam-6528	36	32	refer	refer	VERB
ejpam-6528	36	33	to	to	ADP
ejpam-6528	36	34	υi1	υi1	NOUN
ejpam-6528	36	35	as	as	ADP
ejpam-6528	36	36	the	the	DET
ejpam-6528	36	37	limit	limit	NOUN
ejpam-6528	36	38	of	of	ADP
ejpam-6528	36	39	the	the	DET
ejpam-6528	36	40	sequence	sequence	NOUN
ejpam-6528	36	41	.	.	PUNCT
ejpam-6528	37	1	definition	definition	NOUN
ejpam-6528	37	2	3	3	NUM
ejpam-6528	37	3	.	.	PUNCT
ejpam-6528	38	1	[	[	X
ejpam-6528	38	2	22	22	NUM
ejpam-6528	38	3	]	]	PUNCT
ejpam-6528	38	4	a	a	DET
ejpam-6528	38	5	sequence	sequence	NOUN
ejpam-6528	38	6	{	{	PUNCT
ejpam-6528	38	7	υin	υin	NOUN
ejpam-6528	38	8	}	}	PUNCT
ejpam-6528	38	9	in	in	ADP
ejpam-6528	38	10	an	an	DET
ejpam-6528	38	11	mr	mr	PROPN
ejpam-6528	38	12	-	-	PUNCT
ejpam-6528	38	13	metric	metric	ADJ
ejpam-6528	38	14	space	space	NOUN
ejpam-6528	38	15	(	(	PUNCT
ejpam-6528	38	16	x	x	X
ejpam-6528	38	17	,	,	PUNCT
ejpam-6528	38	18	m	m	VERB
ejpam-6528	38	19	)	)	PUNCT
ejpam-6528	38	20	is	be	AUX
ejpam-6528	38	21	termed	term	VERB
ejpam-6528	38	22	mrcauchy	mrcauchy	ADJ
ejpam-6528	38	23	if	if	SCONJ
ejpam-6528	38	24	for	for	ADP
ejpam-6528	38	25	every	every	DET
ejpam-6528	38	26	ϵ	ϵ	X
ejpam-6528	38	27	>	>	X
ejpam-6528	38	28	0	0	NUM
ejpam-6528	38	29	,	,	PUNCT
ejpam-6528	38	30	there	there	PRON
ejpam-6528	38	31	exists	exist	VERB
ejpam-6528	38	32	a	a	DET
ejpam-6528	38	33	positive	positive	ADJ
ejpam-6528	38	34	integer	integer	NOUN
ejpam-6528	38	35	n	n	CCONJ
ejpam-6528	38	36	such	such	ADJ
ejpam-6528	38	37	that	that	SCONJ
ejpam-6528	38	38	the	the	DET
ejpam-6528	38	39	inequality	inequality	NOUN
ejpam-6528	38	40	m(υin	m(υin	NOUN
ejpam-6528	38	41	,	,	PUNCT
ejpam-6528	38	42	υim	υim	PROPN
ejpam-6528	38	43	,	,	PUNCT
ejpam-6528	38	44	υip	υip	ADJ
ejpam-6528	38	45	)	)	PUNCT
ejpam-6528	38	46	<	<	X
ejpam-6528	39	1	ϵ	ϵ	X
ejpam-6528	39	2	holds	hold	VERB
ejpam-6528	39	3	for	for	ADP
ejpam-6528	39	4	all	all	DET
ejpam-6528	39	5	m	m	PROPN
ejpam-6528	39	6	,	,	PUNCT
ejpam-6528	39	7	n	n	CCONJ
ejpam-6528	39	8	,	,	PUNCT
ejpam-6528	39	9	p	p	PRON
ejpam-6528	39	10	≥	≥	NOUN
ejpam-6528	39	11	n.	n.	NOUN
ejpam-6528	39	12	definition	definition	NOUN
ejpam-6528	39	13	4	4	NUM
ejpam-6528	39	14	.	.	PUNCT
ejpam-6528	40	1	[	[	X
ejpam-6528	40	2	22	22	NUM
ejpam-6528	40	3	]	]	PUNCT
ejpam-6528	40	4	an	an	DET
ejpam-6528	40	5	mr	mr	PROPN
ejpam-6528	40	6	-	-	PUNCT
ejpam-6528	40	7	metric	metric	ADJ
ejpam-6528	40	8	space	space	NOUN
ejpam-6528	40	9	(	(	PUNCT
ejpam-6528	40	10	x	x	X
ejpam-6528	40	11	,	,	PUNCT
ejpam-6528	40	12	m	m	VERB
ejpam-6528	40	13	)	)	PUNCT
ejpam-6528	40	14	is	be	AUX
ejpam-6528	40	15	said	say	VERB
ejpam-6528	40	16	to	to	PART
ejpam-6528	40	17	be	be	AUX
ejpam-6528	40	18	bounded	bound	VERB
ejpam-6528	40	19	if	if	SCONJ
ejpam-6528	40	20	there	there	PRON
ejpam-6528	40	21	exists	exist	VERB
ejpam-6528	40	22	a	a	DET
ejpam-6528	40	23	constant	constant	ADJ
ejpam-6528	40	24	l	l	NOUN
ejpam-6528	40	25	>	>	X
ejpam-6528	40	26	0	0	NUM
ejpam-6528	40	27	such	such	ADJ
ejpam-6528	40	28	that	that	SCONJ
ejpam-6528	40	29	m(υ	m(υ	PROPN
ejpam-6528	40	30	,	,	PUNCT
ejpam-6528	40	31	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	40	32	)	)	PUNCT
ejpam-6528	40	33	≤	≤	NUM
ejpam-6528	40	34	l	l	NOUN
ejpam-6528	40	35	for	for	ADP
ejpam-6528	40	36	all	all	DET
ejpam-6528	40	37	υ	υ	PROPN
ejpam-6528	40	38	,	,	PUNCT
ejpam-6528	40	39	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	40	40	∈	∈	PROPN
ejpam-6528	40	41	x.	x.	NOUN
ejpam-6528	40	42	in	in	ADP
ejpam-6528	40	43	this	this	DET
ejpam-6528	40	44	case	case	NOUN
ejpam-6528	40	45	,	,	PUNCT
ejpam-6528	40	46	the	the	DET
ejpam-6528	40	47	function	function	NOUN
ejpam-6528	40	48	m	m	AUX
ejpam-6528	40	49	is	be	AUX
ejpam-6528	40	50	called	call	VERB
ejpam-6528	40	51	an	an	DET
ejpam-6528	40	52	mr	mr	PROPN
ejpam-6528	40	53	-	-	PUNCT
ejpam-6528	40	54	bound	bind	VERB
ejpam-6528	40	55	for	for	ADP
ejpam-6528	40	56	the	the	DET
ejpam-6528	40	57	metric	metric	NOUN
ejpam-6528	40	58	.	.	PUNCT
ejpam-6528	41	1	definition	definition	NOUN
ejpam-6528	41	2	5	5	NUM
ejpam-6528	41	3	(	(	PUNCT
ejpam-6528	41	4	[	[	X
ejpam-6528	41	5	23	23	NUM
ejpam-6528	41	6	]	]	PUNCT
ejpam-6528	41	7	,	,	PUNCT
ejpam-6528	41	8	definition	definition	NOUN
ejpam-6528	41	9	1.1	1.1	NUM
ejpam-6528	41	10	)	)	PUNCT
ejpam-6528	41	11	.	.	PUNCT
ejpam-6528	42	1	a	a	DET
ejpam-6528	42	2	measure	measure	NOUN
ejpam-6528	42	3	space	space	NOUN
ejpam-6528	42	4	is	be	AUX
ejpam-6528	42	5	a	a	DET
ejpam-6528	42	6	triplet	triplet	NOUN
ejpam-6528	42	7	(	(	PUNCT
ejpam-6528	42	8	x	x	NOUN
ejpam-6528	42	9	,	,	PUNCT
ejpam-6528	42	10	σ	σ	PROPN
ejpam-6528	42	11	,	,	PUNCT
ejpam-6528	42	12	µ	µ	NOUN
ejpam-6528	42	13	)	)	PUNCT
ejpam-6528	42	14	,	,	PUNCT
ejpam-6528	42	15	where	where	SCONJ
ejpam-6528	42	16	:	:	PUNCT
ejpam-6528	42	17	a.	a.	NOUN
ejpam-6528	42	18	malkawi	malkawi	PROPN
ejpam-6528	42	19	,	,	PUNCT
ejpam-6528	42	20	a.	a.	PROPN
ejpam-6528	42	21	rabaiah	rabaiah	PROPN
ejpam-6528	42	22	/	/	SYM
ejpam-6528	42	23	eur	eur	PROPN
ejpam-6528	42	24	.	.	PUNCT
ejpam-6528	43	1	j.	j.	PROPN
ejpam-6528	43	2	pure	pure	PROPN
ejpam-6528	43	3	appl	appl	PROPN
ejpam-6528	43	4	.	.	PROPN
ejpam-6528	43	5	math	math	PROPN
ejpam-6528	43	6	,	,	PUNCT
ejpam-6528	43	7	18	18	NUM
ejpam-6528	43	8	(	(	PUNCT
ejpam-6528	43	9	3	3	NUM
ejpam-6528	43	10	)	)	PUNCT
ejpam-6528	43	11	(	(	PUNCT
ejpam-6528	43	12	2025	2025	NUM
ejpam-6528	43	13	)	)	PUNCT
ejpam-6528	43	14	,	,	PUNCT
ejpam-6528	43	15	6528	6528	NUM
ejpam-6528	43	16	3	3	NUM
ejpam-6528	43	17	of	of	ADP
ejpam-6528	43	18	12	12	NUM
ejpam-6528	43	19	•	•	NOUN
ejpam-6528	43	20	x	x	X
ejpam-6528	43	21	is	be	AUX
ejpam-6528	43	22	a	a	DET
ejpam-6528	43	23	non	non	ADJ
ejpam-6528	43	24	-	-	ADJ
ejpam-6528	43	25	empty	empty	ADJ
ejpam-6528	43	26	set	set	NOUN
ejpam-6528	43	27	.	.	PUNCT
ejpam-6528	44	1	•	•	NUM
ejpam-6528	44	2	σ	σ	PROPN
ejpam-6528	44	3	is	be	AUX
ejpam-6528	44	4	a	a	DET
ejpam-6528	44	5	σ	σ	NOUN
ejpam-6528	44	6	-	-	PUNCT
ejpam-6528	44	7	algebra	algebra	NOUN
ejpam-6528	44	8	on	on	ADP
ejpam-6528	44	9	x	x	NOUN
ejpam-6528	44	10	,	,	PUNCT
ejpam-6528	44	11	which	which	PRON
ejpam-6528	44	12	satisfies	satisfy	VERB
ejpam-6528	44	13	the	the	DET
ejpam-6528	44	14	following	follow	VERB
ejpam-6528	44	15	properties	property	NOUN
ejpam-6528	44	16	:	:	PUNCT
ejpam-6528	44	17	(	(	PUNCT
ejpam-6528	44	18	i	i	NOUN
ejpam-6528	44	19	)	)	PUNCT
ejpam-6528	44	20	x	x	SYM
ejpam-6528	44	21	∈	∈	PROPN
ejpam-6528	44	22	σ	σ	PROPN
ejpam-6528	44	23	.	.	PUNCT
ejpam-6528	44	24	(	(	PUNCT
ejpam-6528	44	25	ii	ii	NOUN
ejpam-6528	44	26	)	)	PUNCT
ejpam-6528	44	27	if	if	SCONJ
ejpam-6528	44	28	a	a	DET
ejpam-6528	44	29	∈	∈	PROPN
ejpam-6528	44	30	σ	σ	PROPN
ejpam-6528	44	31	,	,	PUNCT
ejpam-6528	44	32	then	then	ADV
ejpam-6528	44	33	the	the	DET
ejpam-6528	44	34	complement	complement	NOUN
ejpam-6528	44	35	ac	ac	PROPN
ejpam-6528	44	36	∈	∈	PROPN
ejpam-6528	44	37	σ	σ	PROPN
ejpam-6528	44	38	.	.	PUNCT
ejpam-6528	45	1	(	(	PUNCT
ejpam-6528	45	2	iii	iii	X
ejpam-6528	45	3	)	)	PUNCT
ejpam-6528	45	4	if	if	SCONJ
ejpam-6528	45	5	{	{	PUNCT
ejpam-6528	45	6	an}∞n=1	an}∞n=1	X
ejpam-6528	45	7	⊆	⊆	NUM
ejpam-6528	45	8	σ	σ	NOUN
ejpam-6528	45	9	,	,	PUNCT
ejpam-6528	45	10	then	then	ADV
ejpam-6528	45	11	⋃∞	⋃∞	PUNCT
ejpam-6528	45	12	n=1an	n=1an	PROPN
ejpam-6528	46	1	∈	∈	PROPN
ejpam-6528	46	2	σ	σ	PROPN
ejpam-6528	46	3	.	.	PROPN
ejpam-6528	46	4	•	•	NUM
ejpam-6528	46	5	µ	µ	X
ejpam-6528	46	6	:	:	PUNCT
ejpam-6528	46	7	σ	σ	NOUN
ejpam-6528	46	8	→	→	PUNCT
ejpam-6528	46	9	[	[	X
ejpam-6528	46	10	0,∞	0,∞	X
ejpam-6528	46	11	]	]	PUNCT
ejpam-6528	46	12	is	be	AUX
ejpam-6528	46	13	a	a	DET
ejpam-6528	46	14	function	function	NOUN
ejpam-6528	46	15	satisfying	satisfy	VERB
ejpam-6528	46	16	:	:	PUNCT
ejpam-6528	46	17	(	(	PUNCT
ejpam-6528	46	18	i	i	NOUN
ejpam-6528	46	19	)	)	PUNCT
ejpam-6528	46	20	non	non	ADJ
ejpam-6528	46	21	-	-	NOUN
ejpam-6528	46	22	negativity	negativity	ADJ
ejpam-6528	46	23	:	:	PUNCT
ejpam-6528	46	24	µ(a	µ(a	PROPN
ejpam-6528	46	25	)	)	PUNCT
ejpam-6528	46	26	≥	≥	NOUN
ejpam-6528	46	27	0	0	NUM
ejpam-6528	46	28	for	for	ADP
ejpam-6528	46	29	all	all	DET
ejpam-6528	46	30	a	a	DET
ejpam-6528	46	31	∈	∈	PROPN
ejpam-6528	46	32	σ	σ	PROPN
ejpam-6528	46	33	.	.	PUNCT
ejpam-6528	46	34	(	(	PUNCT
ejpam-6528	46	35	ii	ii	NOUN
ejpam-6528	46	36	)	)	PUNCT
ejpam-6528	46	37	null	null	ADJ
ejpam-6528	46	38	empty	empty	ADJ
ejpam-6528	46	39	set	set	NOUN
ejpam-6528	46	40	:	:	PUNCT
ejpam-6528	46	41	µ(∅	µ(∅	X
ejpam-6528	46	42	)	)	PUNCT
ejpam-6528	47	1	=	=	SYM
ejpam-6528	47	2	0	0	X
ejpam-6528	47	3	.	.	PUNCT
ejpam-6528	48	1	(	(	PUNCT
ejpam-6528	48	2	iii	iii	NOUN
ejpam-6528	48	3	)	)	PUNCT
ejpam-6528	48	4	countable	countable	ADJ
ejpam-6528	48	5	additivity	additivity	NOUN
ejpam-6528	48	6	(	(	PUNCT
ejpam-6528	48	7	σ	σ	NOUN
ejpam-6528	48	8	-	-	NOUN
ejpam-6528	48	9	additivity	additivity	NOUN
ejpam-6528	48	10	):	):	PUNCT
ejpam-6528	48	11	if	if	SCONJ
ejpam-6528	48	12	{	{	PUNCT
ejpam-6528	48	13	an}∞n=1	an}∞n=1	X
ejpam-6528	48	14	is	be	AUX
ejpam-6528	48	15	a	a	DET
ejpam-6528	48	16	countable	countable	ADJ
ejpam-6528	48	17	collection	collection	NOUN
ejpam-6528	48	18	of	of	ADP
ejpam-6528	48	19	disjoint	disjoint	NOUN
ejpam-6528	48	20	sets	set	NOUN
ejpam-6528	48	21	in	in	ADP
ejpam-6528	48	22	σ	σ	PROPN
ejpam-6528	48	23	,	,	PUNCT
ejpam-6528	48	24	then	then	ADV
ejpam-6528	48	25	:	:	PUNCT
ejpam-6528	48	26	µ	µ	X
ejpam-6528	48	27	(	(	PUNCT
ejpam-6528	48	28	∞⋃	∞⋃	PROPN
ejpam-6528	48	29	n=1	n=1	PROPN
ejpam-6528	48	30	an	an	X
ejpam-6528	48	31	)	)	PUNCT
ejpam-6528	48	32	=	=	SYM
ejpam-6528	49	1	∞∑	∞∑	NUM
ejpam-6528	49	2	n=1	n=1	PROPN
ejpam-6528	49	3	µ(an	µ(an	PROPN
ejpam-6528	49	4	)	)	PUNCT
ejpam-6528	49	5	.	.	PUNCT
ejpam-6528	50	1	the	the	DET
ejpam-6528	50	2	function	function	NOUN
ejpam-6528	50	3	µ	µ	NOUN
ejpam-6528	50	4	is	be	AUX
ejpam-6528	50	5	called	call	VERB
ejpam-6528	50	6	a	a	DET
ejpam-6528	50	7	measure	measure	NOUN
ejpam-6528	50	8	,	,	PUNCT
ejpam-6528	50	9	and	and	CCONJ
ejpam-6528	50	10	x	x	X
ejpam-6528	50	11	is	be	AUX
ejpam-6528	50	12	referred	refer	VERB
ejpam-6528	50	13	to	to	ADP
ejpam-6528	50	14	as	as	ADP
ejpam-6528	50	15	the	the	DET
ejpam-6528	50	16	measurable	measurable	ADJ
ejpam-6528	50	17	space	space	NOUN
ejpam-6528	50	18	.	.	PUNCT
ejpam-6528	51	1	definition	definition	NOUN
ejpam-6528	51	2	6	6	NUM
ejpam-6528	51	3	(	(	PUNCT
ejpam-6528	51	4	[	[	X
ejpam-6528	51	5	24	24	NUM
ejpam-6528	51	6	]	]	PUNCT
ejpam-6528	51	7	,	,	PUNCT
ejpam-6528	51	8	section	section	NOUN
ejpam-6528	51	9	2	2	NUM
ejpam-6528	51	10	)	)	PUNCT
ejpam-6528	51	11	.	.	PUNCT
ejpam-6528	52	1	a	a	DET
ejpam-6528	52	2	measure	measure	NOUN
ejpam-6528	52	3	µ	µ	NOUN
ejpam-6528	52	4	is	be	AUX
ejpam-6528	52	5	said	say	VERB
ejpam-6528	52	6	to	to	PART
ejpam-6528	52	7	be	be	AUX
ejpam-6528	52	8	σ	σ	NOUN
ejpam-6528	52	9	-	-	NOUN
ejpam-6528	52	10	finite	finite	NOUN
ejpam-6528	52	11	if	if	SCONJ
ejpam-6528	52	12	there	there	PRON
ejpam-6528	52	13	exists	exist	VERB
ejpam-6528	52	14	a	a	DET
ejpam-6528	52	15	countable	countable	ADJ
ejpam-6528	52	16	collection	collection	NOUN
ejpam-6528	52	17	of	of	ADP
ejpam-6528	52	18	measurable	measurable	ADJ
ejpam-6528	52	19	sets	set	NOUN
ejpam-6528	52	20	{	{	PUNCT
ejpam-6528	52	21	xn}∞n=1	xn}∞n=1	PROPN
ejpam-6528	52	22	⊆	⊆	NUM
ejpam-6528	52	23	σ	σ	NOUN
ejpam-6528	52	24	such	such	ADJ
ejpam-6528	52	25	that	that	PRON
ejpam-6528	52	26	:	:	PUNCT
ejpam-6528	52	27	x	x	SYM
ejpam-6528	52	28	=	=	SYM
ejpam-6528	52	29	∞⋃	∞⋃	PROPN
ejpam-6528	52	30	n=1	n=1	PUNCT
ejpam-6528	52	31	xn	xn	PROPN
ejpam-6528	52	32	and	and	CCONJ
ejpam-6528	52	33	µ(xn	µ(xn	PROPN
ejpam-6528	52	34	)	)	PUNCT
ejpam-6528	52	35	<	<	X
ejpam-6528	52	36	∞	∞	NUM
ejpam-6528	52	37	for	for	ADP
ejpam-6528	52	38	every	every	DET
ejpam-6528	52	39	n.	n.	NOUN
ejpam-6528	52	40	definition	definition	NOUN
ejpam-6528	52	41	7	7	NUM
ejpam-6528	52	42	(	(	PUNCT
ejpam-6528	52	43	[	[	X
ejpam-6528	52	44	25	25	NUM
ejpam-6528	52	45	]	]	PUNCT
ejpam-6528	52	46	,	,	PUNCT
ejpam-6528	52	47	section	section	NOUN
ejpam-6528	52	48	3.1	3.1	NUM
ejpam-6528	52	49	)	)	PUNCT
ejpam-6528	52	50	.	.	PUNCT
ejpam-6528	53	1	a	a	DET
ejpam-6528	53	2	measure	measure	NOUN
ejpam-6528	53	3	µ	µ	NOUN
ejpam-6528	53	4	is	be	AUX
ejpam-6528	53	5	called	call	VERB
ejpam-6528	53	6	absolutely	absolutely	ADV
ejpam-6528	53	7	continuous	continuous	ADJ
ejpam-6528	53	8	with	with	ADP
ejpam-6528	53	9	respect	respect	NOUN
ejpam-6528	53	10	to	to	ADP
ejpam-6528	53	11	another	another	DET
ejpam-6528	53	12	measure	measure	NOUN
ejpam-6528	53	13	ν	ν	X
ejpam-6528	53	14	(	(	PUNCT
ejpam-6528	53	15	denoted	denote	VERB
ejpam-6528	53	16	µ≪	µ≪	NOUN
ejpam-6528	53	17	ν	ν	NOUN
ejpam-6528	53	18	)	)	PUNCT
ejpam-6528	53	19	if	if	SCONJ
ejpam-6528	53	20	for	for	SCONJ
ejpam-6528	53	21	every	every	DET
ejpam-6528	53	22	measurable	measurable	NOUN
ejpam-6528	53	23	set	set	VERB
ejpam-6528	53	24	a	a	DET
ejpam-6528	53	25	∈	∈	PROPN
ejpam-6528	53	26	σ	σ	PROPN
ejpam-6528	53	27	,	,	PUNCT
ejpam-6528	53	28	ν(a	ν(a	PROPN
ejpam-6528	53	29	)	)	PUNCT
ejpam-6528	53	30	=	=	SYM
ejpam-6528	53	31	0	0	NUM
ejpam-6528	53	32	⇒	⇒	NOUN
ejpam-6528	53	33	µ(a	µ(a	PROPN
ejpam-6528	53	34	)	)	PUNCT
ejpam-6528	54	1	=	=	PUNCT
ejpam-6528	54	2	0	0	NUM
ejpam-6528	54	3	.	.	NOUN
ejpam-6528	54	4	2	2	NUM
ejpam-6528	54	5	.	.	X
ejpam-6528	54	6	main	main	ADJ
ejpam-6528	54	7	result	result	NOUN
ejpam-6528	54	8	theorem	theorem	VERB
ejpam-6528	54	9	1	1	X
ejpam-6528	54	10	.	.	PUNCT
ejpam-6528	55	1	let	let	AUX
ejpam-6528	55	2	(	(	PUNCT
ejpam-6528	55	3	x	x	X
ejpam-6528	55	4	,	,	PUNCT
ejpam-6528	55	5	m	m	VERB
ejpam-6528	55	6	)	)	PUNCT
ejpam-6528	55	7	be	be	VERB
ejpam-6528	55	8	an	an	DET
ejpam-6528	55	9	mr	mr	ADJ
ejpam-6528	55	10	-	-	PUNCT
ejpam-6528	55	11	metric	metric	ADJ
ejpam-6528	55	12	space	space	NOUN
ejpam-6528	55	13	,	,	PUNCT
ejpam-6528	55	14	and	and	CCONJ
ejpam-6528	55	15	let	let	VERB
ejpam-6528	55	16	µ	µ	X
ejpam-6528	55	17	be	be	AUX
ejpam-6528	55	18	a	a	DET
ejpam-6528	55	19	measure	measure	NOUN
ejpam-6528	55	20	on	on	ADP
ejpam-6528	55	21	x	x	SYM
ejpam-6528	55	22	such	such	ADJ
ejpam-6528	55	23	that	that	SCONJ
ejpam-6528	55	24	:	:	PUNCT
ejpam-6528	55	25	µ(a	µ(a	PROPN
ejpam-6528	55	26	)	)	PUNCT
ejpam-6528	56	1	=	=	SYM
ejpam-6528	56	2	∫	∫	PROPN
ejpam-6528	56	3	a	a	DET
ejpam-6528	56	4	m(υ	m(υ	PROPN
ejpam-6528	56	5	,	,	PUNCT
ejpam-6528	56	6	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	56	7	)	)	PUNCT
ejpam-6528	56	8	dµ(υ	dµ(υ	NUM
ejpam-6528	56	9	)	)	PUNCT
ejpam-6528	56	10	dµ(ξ	dµ(ξ	NUM
ejpam-6528	56	11	)	)	PUNCT
ejpam-6528	57	1	dµ(ℑ	dµ(ℑ	ADV
ejpam-6528	57	2	)	)	PUNCT
ejpam-6528	57	3	,	,	PUNCT
ejpam-6528	57	4	(	(	PUNCT
ejpam-6528	57	5	1	1	X
ejpam-6528	57	6	)	)	PUNCT
ejpam-6528	57	7	for	for	ADP
ejpam-6528	57	8	measurable	measurable	ADJ
ejpam-6528	57	9	sets	set	NOUN
ejpam-6528	57	10	a	a	DET
ejpam-6528	57	11	⊆	⊆	NUM
ejpam-6528	57	12	x.	x.	NOUN
ejpam-6528	57	13	if	if	SCONJ
ejpam-6528	57	14	m	m	NOUN
ejpam-6528	57	15	satisfies	satisfie	NOUN
ejpam-6528	57	16	(	(	PUNCT
ejpam-6528	57	17	m4	m4	PROPN
ejpam-6528	57	18	)	)	PUNCT
ejpam-6528	57	19	with	with	ADP
ejpam-6528	57	20	a	a	DET
ejpam-6528	57	21	finite	finite	ADJ
ejpam-6528	57	22	measure	measure	NOUN
ejpam-6528	57	23	space	space	NOUN
ejpam-6528	57	24	,	,	PUNCT
ejpam-6528	57	25	then	then	ADV
ejpam-6528	57	26	µ	µ	X
ejpam-6528	57	27	is	be	AUX
ejpam-6528	57	28	σ	σ	NOUN
ejpam-6528	57	29	-	-	NOUN
ejpam-6528	57	30	finite	finite	ADJ
ejpam-6528	57	31	and	and	CCONJ
ejpam-6528	57	32	absolutely	absolutely	ADV
ejpam-6528	57	33	continuous	continuous	ADJ
ejpam-6528	57	34	.	.	PUNCT
ejpam-6528	58	1	proof	proof	NOUN
ejpam-6528	58	2	.	.	PUNCT
ejpam-6528	59	1	step	step	NOUN
ejpam-6528	59	2	1	1	NUM
ejpam-6528	59	3	:	:	PUNCT
ejpam-6528	59	4	proving	prove	VERB
ejpam-6528	59	5	σ	σ	NOUN
ejpam-6528	59	6	-	-	PUNCT
ejpam-6528	59	7	finiteness	finiteness	NOUN
ejpam-6528	59	8	to	to	PART
ejpam-6528	59	9	demonstrate	demonstrate	VERB
ejpam-6528	59	10	that	that	SCONJ
ejpam-6528	59	11	µ	µ	NOUN
ejpam-6528	59	12	is	be	AUX
ejpam-6528	59	13	σ	σ	NOUN
ejpam-6528	59	14	-	-	NOUN
ejpam-6528	59	15	finite	finite	PROPN
ejpam-6528	59	16	,	,	PUNCT
ejpam-6528	59	17	we	we	PRON
ejpam-6528	59	18	need	need	VERB
ejpam-6528	59	19	to	to	PART
ejpam-6528	59	20	decompose	decompose	VERB
ejpam-6528	59	21	the	the	DET
ejpam-6528	59	22	space	space	NOUN
ejpam-6528	59	23	x	x	PUNCT
ejpam-6528	59	24	into	into	ADP
ejpam-6528	59	25	a	a	DET
ejpam-6528	59	26	countable	countable	ADJ
ejpam-6528	59	27	collection	collection	NOUN
ejpam-6528	59	28	of	of	ADP
ejpam-6528	59	29	subsets	subset	NOUN
ejpam-6528	59	30	{	{	PUNCT
ejpam-6528	59	31	xn	xn	X
ejpam-6528	59	32	}	}	PUNCT
ejpam-6528	59	33	such	such	ADJ
ejpam-6528	59	34	that	that	SCONJ
ejpam-6528	59	35	µ(xn	µ(xn	NOUN
ejpam-6528	59	36	)	)	PUNCT
ejpam-6528	59	37	<	<	X
ejpam-6528	59	38	∞	∞	NUM
ejpam-6528	59	39	for	for	ADP
ejpam-6528	59	40	all	all	DET
ejpam-6528	59	41	n.	n.	NOUN
ejpam-6528	59	42	a.	a.	NOUN
ejpam-6528	59	43	malkawi	malkawi	PROPN
ejpam-6528	59	44	,	,	PUNCT
ejpam-6528	59	45	a.	a.	PROPN
ejpam-6528	59	46	rabaiah	rabaiah	PROPN
ejpam-6528	59	47	/	/	SYM
ejpam-6528	59	48	eur	eur	PROPN
ejpam-6528	59	49	.	.	PUNCT
ejpam-6528	60	1	j.	j.	PROPN
ejpam-6528	60	2	pure	pure	PROPN
ejpam-6528	60	3	appl	appl	PROPN
ejpam-6528	60	4	.	.	PROPN
ejpam-6528	60	5	math	math	PROPN
ejpam-6528	60	6	,	,	PUNCT
ejpam-6528	60	7	18	18	NUM
ejpam-6528	60	8	(	(	PUNCT
ejpam-6528	60	9	3	3	NUM
ejpam-6528	60	10	)	)	PUNCT
ejpam-6528	60	11	(	(	PUNCT
ejpam-6528	60	12	2025	2025	NUM
ejpam-6528	60	13	)	)	PUNCT
ejpam-6528	60	14	,	,	PUNCT
ejpam-6528	60	15	6528	6528	NUM
ejpam-6528	60	16	4	4	NUM
ejpam-6528	60	17	of	of	ADP
ejpam-6528	60	18	12	12	NUM
ejpam-6528	60	19	first	first	ADJ
ejpam-6528	60	20	,	,	PUNCT
ejpam-6528	60	21	since	since	SCONJ
ejpam-6528	60	22	m	m	PROPN
ejpam-6528	60	23	satisfies	satisfie	NOUN
ejpam-6528	60	24	(	(	PUNCT
ejpam-6528	60	25	m4	m4	PROPN
ejpam-6528	60	26	)	)	PUNCT
ejpam-6528	61	1	,	,	PUNCT
ejpam-6528	61	2	we	we	PRON
ejpam-6528	61	3	have	have	VERB
ejpam-6528	61	4	the	the	DET
ejpam-6528	61	5	following	follow	VERB
ejpam-6528	61	6	inequality	inequality	NOUN
ejpam-6528	61	7	:	:	PUNCT
ejpam-6528	61	8	m(υ	m(υ	PROPN
ejpam-6528	61	9	,	,	PUNCT
ejpam-6528	61	10	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	61	11	)	)	PUNCT
ejpam-6528	61	12	≤	≤	NUM
ejpam-6528	62	1	r	r	NOUN
ejpam-6528	62	2	[	[	X
ejpam-6528	62	3	m(υ	m(υ	PROPN
ejpam-6528	62	4	,	,	PUNCT
ejpam-6528	62	5	ξ	ξ	PROPN
ejpam-6528	62	6	,	,	PUNCT
ejpam-6528	62	7	ℓ1	ℓ1	NOUN
ejpam-6528	62	8	)	)	PUNCT
ejpam-6528	63	1	+	+	SYM
ejpam-6528	63	2	m(υ	m(υ	PROPN
ejpam-6528	63	3	,	,	PUNCT
ejpam-6528	63	4	ℓ1,ℑ	ℓ1,ℑ	NOUN
ejpam-6528	63	5	)	)	PUNCT
ejpam-6528	64	1	+	+	ADJ
ejpam-6528	64	2	m(ℓ1	m(ℓ1	NOUN
ejpam-6528	64	3	,	,	PUNCT
ejpam-6528	64	4	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	64	5	)	)	PUNCT
ejpam-6528	64	6	]	]	PUNCT
ejpam-6528	64	7	,	,	PUNCT
ejpam-6528	64	8	(	(	PUNCT
ejpam-6528	64	9	2	2	X
ejpam-6528	64	10	)	)	PUNCT
ejpam-6528	64	11	where	where	SCONJ
ejpam-6528	64	12	r	r	NOUN
ejpam-6528	64	13	is	be	AUX
ejpam-6528	64	14	a	a	DET
ejpam-6528	64	15	constant	constant	ADJ
ejpam-6528	64	16	greater	great	ADJ
ejpam-6528	64	17	than	than	ADP
ejpam-6528	64	18	1	1	NUM
ejpam-6528	64	19	.	.	PUNCT
ejpam-6528	65	1	this	this	DET
ejpam-6528	65	2	inequality	inequality	NOUN
ejpam-6528	65	3	provides	provide	VERB
ejpam-6528	65	4	a	a	DET
ejpam-6528	65	5	useful	useful	ADJ
ejpam-6528	65	6	upper	upper	ADJ
ejpam-6528	65	7	bound	bind	VERB
ejpam-6528	65	8	on	on	ADP
ejpam-6528	65	9	the	the	DET
ejpam-6528	65	10	mr	mr	PROPN
ejpam-6528	65	11	-	-	PUNCT
ejpam-6528	65	12	metric	metric	NOUN
ejpam-6528	65	13	.	.	PUNCT
ejpam-6528	66	1	next	next	ADV
ejpam-6528	66	2	,	,	PUNCT
ejpam-6528	66	3	for	for	ADP
ejpam-6528	66	4	any	any	DET
ejpam-6528	66	5	measurable	measurable	NOUN
ejpam-6528	66	6	set	set	VERB
ejpam-6528	66	7	a	a	DET
ejpam-6528	66	8	⊆	⊆	NUM
ejpam-6528	66	9	x	x	SYM
ejpam-6528	66	10	,	,	PUNCT
ejpam-6528	66	11	we	we	PRON
ejpam-6528	66	12	integrate	integrate	VERB
ejpam-6528	66	13	both	both	DET
ejpam-6528	66	14	sides	side	NOUN
ejpam-6528	66	15	of	of	ADP
ejpam-6528	66	16	the	the	DET
ejpam-6528	66	17	inequality:∫	inequality:∫	NOUN
ejpam-6528	66	18	a	a	DET
ejpam-6528	66	19	m(υ	m(υ	PROPN
ejpam-6528	66	20	,	,	PUNCT
ejpam-6528	66	21	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	66	22	)	)	PUNCT
ejpam-6528	66	23	dµ(υ	dµ(υ	NUM
ejpam-6528	66	24	)	)	PUNCT
ejpam-6528	66	25	dµ(ξ	dµ(ξ	NUM
ejpam-6528	66	26	)	)	PUNCT
ejpam-6528	67	1	dµ(ℑ	dµ(ℑ	X
ejpam-6528	67	2	)	)	PUNCT
ejpam-6528	67	3	≤	≤	NOUN
ejpam-6528	67	4	r	r	NOUN
ejpam-6528	67	5	[	[	PUNCT
ejpam-6528	67	6	∫	∫	PROPN
ejpam-6528	67	7	a	a	DET
ejpam-6528	67	8	m(υ	m(υ	PROPN
ejpam-6528	67	9	,	,	PUNCT
ejpam-6528	67	10	ξ	ξ	PROPN
ejpam-6528	67	11	,	,	PUNCT
ejpam-6528	67	12	ℓ1	ℓ1	NOUN
ejpam-6528	67	13	)	)	PUNCT
ejpam-6528	67	14	dµ(υ	dµ(υ	NUM
ejpam-6528	67	15	)	)	PUNCT
ejpam-6528	67	16	dµ(ξ	dµ(ξ	NUM
ejpam-6528	67	17	)	)	PUNCT
ejpam-6528	67	18	dµ(ℑ	dµ(ℑ	ADV
ejpam-6528	67	19	)	)	PUNCT
ejpam-6528	68	1	+	+	CCONJ
ejpam-6528	68	2	∫	∫	PROPN
ejpam-6528	68	3	a	a	DET
ejpam-6528	68	4	m(υ	m(υ	PROPN
ejpam-6528	68	5	,	,	PUNCT
ejpam-6528	68	6	ℓ1,ℑ	ℓ1,ℑ	NOUN
ejpam-6528	68	7	)	)	PUNCT
ejpam-6528	68	8	dµ(υ	dµ(υ	NUM
ejpam-6528	68	9	)	)	PUNCT
ejpam-6528	68	10	dµ(ξ	dµ(ξ	NUM
ejpam-6528	68	11	)	)	PUNCT
ejpam-6528	68	12	dµ(ℑ	dµ(ℑ	ADV
ejpam-6528	68	13	)	)	PUNCT
ejpam-6528	69	1	+	+	CCONJ
ejpam-6528	69	2	∫	∫	PROPN
ejpam-6528	69	3	a	a	DET
ejpam-6528	69	4	m(ℓ1	m(ℓ1	NOUN
ejpam-6528	69	5	,	,	PUNCT
ejpam-6528	69	6	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	69	7	)	)	PUNCT
ejpam-6528	69	8	dµ(υ	dµ(υ	NUM
ejpam-6528	69	9	)	)	PUNCT
ejpam-6528	69	10	dµ(ξ	dµ(ξ	NUM
ejpam-6528	69	11	)	)	PUNCT
ejpam-6528	69	12	dµ(ℑ	dµ(ℑ	ADV
ejpam-6528	69	13	)	)	PUNCT
ejpam-6528	69	14	]	]	PUNCT
ejpam-6528	69	15	.	.	PUNCT
ejpam-6528	70	1	(	(	PUNCT
ejpam-6528	70	2	3	3	X
ejpam-6528	70	3	)	)	PUNCT
ejpam-6528	70	4	this	this	PRON
ejpam-6528	70	5	shows	show	VERB
ejpam-6528	70	6	that	that	SCONJ
ejpam-6528	70	7	µ(a	µ(a	PROPN
ejpam-6528	70	8	)	)	PUNCT
ejpam-6528	70	9	is	be	AUX
ejpam-6528	70	10	controlled	control	VERB
ejpam-6528	70	11	by	by	ADP
ejpam-6528	70	12	the	the	DET
ejpam-6528	70	13	integrals	integral	NOUN
ejpam-6528	70	14	of	of	ADP
ejpam-6528	70	15	m	m	PROPN
ejpam-6528	70	16	on	on	ADP
ejpam-6528	70	17	smaller	small	ADJ
ejpam-6528	70	18	subsets	subset	NOUN
ejpam-6528	70	19	.	.	PUNCT
ejpam-6528	71	1	we	we	PRON
ejpam-6528	71	2	can	can	AUX
ejpam-6528	71	3	now	now	ADV
ejpam-6528	71	4	use	use	VERB
ejpam-6528	71	5	the	the	DET
ejpam-6528	71	6	fact	fact	NOUN
ejpam-6528	71	7	that	that	SCONJ
ejpam-6528	71	8	m	m	NOUN
ejpam-6528	71	9	is	be	AUX
ejpam-6528	71	10	bounded	bound	VERB
ejpam-6528	71	11	for	for	ADP
ejpam-6528	71	12	all	all	DET
ejpam-6528	71	13	sets	set	NOUN
ejpam-6528	71	14	and	and	CCONJ
ejpam-6528	71	15	the	the	DET
ejpam-6528	71	16	measure	measure	NOUN
ejpam-6528	71	17	µ	µ	NOUN
ejpam-6528	71	18	is	be	AUX
ejpam-6528	71	19	finite	finite	ADJ
ejpam-6528	71	20	on	on	ADP
ejpam-6528	71	21	these	these	DET
ejpam-6528	71	22	subsets	subset	NOUN
ejpam-6528	71	23	.	.	PUNCT
ejpam-6528	72	1	thus	thus	ADV
ejpam-6528	72	2	,	,	PUNCT
ejpam-6528	72	3	it	it	PRON
ejpam-6528	72	4	is	be	AUX
ejpam-6528	72	5	possible	possible	ADJ
ejpam-6528	72	6	to	to	PART
ejpam-6528	72	7	partition	partition	VERB
ejpam-6528	72	8	x	x	PUNCT
ejpam-6528	72	9	into	into	ADP
ejpam-6528	72	10	a	a	DET
ejpam-6528	72	11	countable	countable	ADJ
ejpam-6528	72	12	union	union	NOUN
ejpam-6528	72	13	of	of	ADP
ejpam-6528	72	14	subsets	subset	NOUN
ejpam-6528	72	15	{	{	PUNCT
ejpam-6528	72	16	xn	xn	X
ejpam-6528	72	17	}	}	PUNCT
ejpam-6528	72	18	such	such	ADJ
ejpam-6528	72	19	that	that	SCONJ
ejpam-6528	72	20	µ(xn	µ(xn	NOUN
ejpam-6528	72	21	)	)	PUNCT
ejpam-6528	72	22	<	<	X
ejpam-6528	72	23	∞	∞	NUM
ejpam-6528	72	24	for	for	ADP
ejpam-6528	72	25	all	all	DET
ejpam-6528	72	26	n	n	CCONJ
ejpam-6528	72	27	,	,	PUNCT
ejpam-6528	72	28	proving	prove	VERB
ejpam-6528	72	29	that	that	SCONJ
ejpam-6528	72	30	µ	µ	NOUN
ejpam-6528	72	31	is	be	AUX
ejpam-6528	72	32	σ	σ	NOUN
ejpam-6528	72	33	-	-	NOUN
ejpam-6528	72	34	finite	finite	NOUN
ejpam-6528	72	35	.	.	PUNCT
ejpam-6528	73	1	step	step	NOUN
ejpam-6528	73	2	2	2	NUM
ejpam-6528	73	3	:	:	PUNCT
ejpam-6528	73	4	proving	prove	VERB
ejpam-6528	73	5	absolute	absolute	ADJ
ejpam-6528	73	6	continuity	continuity	NOUN
ejpam-6528	73	7	now	now	ADV
ejpam-6528	73	8	,	,	PUNCT
ejpam-6528	73	9	we	we	PRON
ejpam-6528	73	10	will	will	AUX
ejpam-6528	73	11	prove	prove	VERB
ejpam-6528	73	12	that	that	SCONJ
ejpam-6528	73	13	µ	µ	NOUN
ejpam-6528	73	14	is	be	AUX
ejpam-6528	73	15	absolutely	absolutely	ADV
ejpam-6528	73	16	continuous	continuous	ADJ
ejpam-6528	73	17	with	with	ADP
ejpam-6528	73	18	respect	respect	NOUN
ejpam-6528	73	19	to	to	ADP
ejpam-6528	73	20	the	the	DET
ejpam-6528	73	21	measure	measure	NOUN
ejpam-6528	73	22	µ.	µ.	PROPN
ejpam-6528	73	23	suppose	suppose	VERB
ejpam-6528	73	24	that	that	SCONJ
ejpam-6528	73	25	for	for	ADP
ejpam-6528	73	26	some	some	DET
ejpam-6528	73	27	measurable	measurable	NOUN
ejpam-6528	73	28	set	set	VERB
ejpam-6528	73	29	a	a	DET
ejpam-6528	73	30	⊆	⊆	NUM
ejpam-6528	73	31	x	x	SYM
ejpam-6528	73	32	,	,	PUNCT
ejpam-6528	73	33	we	we	PRON
ejpam-6528	73	34	have	have	VERB
ejpam-6528	73	35	µ(a	µ(a	PROPN
ejpam-6528	73	36	)	)	PUNCT
ejpam-6528	74	1	=	=	SYM
ejpam-6528	74	2	0	0	X
ejpam-6528	74	3	.	.	PUNCT
ejpam-6528	75	1	we	we	PRON
ejpam-6528	75	2	aim	aim	VERB
ejpam-6528	75	3	to	to	PART
ejpam-6528	75	4	show	show	VERB
ejpam-6528	75	5	that	that	SCONJ
ejpam-6528	75	6	this	this	PRON
ejpam-6528	75	7	implies	imply	VERB
ejpam-6528	75	8	that	that	SCONJ
ejpam-6528	75	9	m(υ	m(υ	PROPN
ejpam-6528	75	10	,	,	PUNCT
ejpam-6528	75	11	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	75	12	)	)	PUNCT
ejpam-6528	75	13	=	=	SYM
ejpam-6528	76	1	0	0	NUM
ejpam-6528	76	2	almost	almost	ADV
ejpam-6528	76	3	everywhere	everywhere	ADV
ejpam-6528	76	4	on	on	ADP
ejpam-6528	76	5	a.	a.	NOUN
ejpam-6528	76	6	by	by	ADP
ejpam-6528	76	7	the	the	DET
ejpam-6528	76	8	definition	definition	NOUN
ejpam-6528	76	9	of	of	ADP
ejpam-6528	76	10	µ	µ	NUM
ejpam-6528	76	11	,	,	PUNCT
ejpam-6528	76	12	we	we	PRON
ejpam-6528	76	13	have	have	VERB
ejpam-6528	76	14	:	:	PUNCT
ejpam-6528	76	15	µ(a	µ(a	PROPN
ejpam-6528	76	16	)	)	PUNCT
ejpam-6528	77	1	=	=	PRON
ejpam-6528	77	2	∫	∫	PROPN
ejpam-6528	77	3	a	a	DET
ejpam-6528	77	4	m(υ	m(υ	PROPN
ejpam-6528	77	5	,	,	PUNCT
ejpam-6528	77	6	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	77	7	)	)	PUNCT
ejpam-6528	77	8	dµ(υ	dµ(υ	NUM
ejpam-6528	77	9	)	)	PUNCT
ejpam-6528	77	10	dµ(ξ	dµ(ξ	NUM
ejpam-6528	77	11	)	)	PUNCT
ejpam-6528	78	1	dµ(ℑ	dµ(ℑ	ADV
ejpam-6528	78	2	)	)	PUNCT
ejpam-6528	78	3	=	=	SYM
ejpam-6528	78	4	0	0	X
ejpam-6528	78	5	.	.	PUNCT
ejpam-6528	79	1	(	(	PUNCT
ejpam-6528	79	2	4	4	NUM
ejpam-6528	79	3	)	)	PUNCT
ejpam-6528	79	4	since	since	SCONJ
ejpam-6528	79	5	m	m	PROPN
ejpam-6528	79	6	is	be	AUX
ejpam-6528	79	7	non	non	ADJ
ejpam-6528	79	8	-	-	ADJ
ejpam-6528	79	9	negative	negative	ADJ
ejpam-6528	79	10	,	,	PUNCT
ejpam-6528	79	11	we	we	PRON
ejpam-6528	79	12	conclude	conclude	VERB
ejpam-6528	79	13	that	that	SCONJ
ejpam-6528	79	14	the	the	DET
ejpam-6528	79	15	integrand	integrand	NOUN
ejpam-6528	79	16	must	must	AUX
ejpam-6528	79	17	be	be	AUX
ejpam-6528	79	18	zero	zero	NUM
ejpam-6528	79	19	almost	almost	ADV
ejpam-6528	79	20	everywhere	everywhere	ADV
ejpam-6528	79	21	on	on	ADP
ejpam-6528	79	22	a.	a.	NOUN
ejpam-6528	79	23	in	in	ADP
ejpam-6528	79	24	other	other	ADJ
ejpam-6528	79	25	words	word	NOUN
ejpam-6528	79	26	,	,	PUNCT
ejpam-6528	79	27	m(υ	m(υ	PROPN
ejpam-6528	79	28	,	,	PUNCT
ejpam-6528	79	29	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	79	30	)	)	PUNCT
ejpam-6528	79	31	=	=	SYM
ejpam-6528	79	32	0	0	NUM
ejpam-6528	79	33	for	for	ADP
ejpam-6528	79	34	µ-almost	µ-almost	ADJ
ejpam-6528	79	35	all	all	PRON
ejpam-6528	79	36	(	(	PUNCT
ejpam-6528	79	37	υ	υ	NOUN
ejpam-6528	79	38	,	,	PUNCT
ejpam-6528	79	39	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	79	40	)	)	PUNCT
ejpam-6528	79	41	∈	∈	PROPN
ejpam-6528	79	42	a.	a.	NOUN
ejpam-6528	79	43	because	because	SCONJ
ejpam-6528	79	44	m(υ	m(υ	PROPN
ejpam-6528	79	45	,	,	PUNCT
ejpam-6528	79	46	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	79	47	)	)	PUNCT
ejpam-6528	80	1	=	=	SYM
ejpam-6528	80	2	0	0	NUM
ejpam-6528	81	1	almost	almost	ADV
ejpam-6528	81	2	everywhere	everywhere	ADV
ejpam-6528	81	3	,	,	PUNCT
ejpam-6528	81	4	the	the	DET
ejpam-6528	81	5	measure	measure	NOUN
ejpam-6528	81	6	µ	µ	X
ejpam-6528	81	7	of	of	ADP
ejpam-6528	81	8	a	a	PRON
ejpam-6528	81	9	must	must	AUX
ejpam-6528	81	10	be	be	AUX
ejpam-6528	81	11	zero	zero	NUM
ejpam-6528	81	12	,	,	PUNCT
ejpam-6528	81	13	which	which	PRON
ejpam-6528	81	14	shows	show	VERB
ejpam-6528	81	15	that	that	SCONJ
ejpam-6528	81	16	a	a	PRON
ejpam-6528	81	17	is	be	AUX
ejpam-6528	81	18	a	a	DET
ejpam-6528	81	19	null	null	ADJ
ejpam-6528	81	20	set	set	NOUN
ejpam-6528	81	21	in	in	ADP
ejpam-6528	81	22	terms	term	NOUN
ejpam-6528	81	23	of	of	ADP
ejpam-6528	81	24	µ.	µ.	NOUN
ejpam-6528	81	25	this	this	PRON
ejpam-6528	81	26	proves	prove	VERB
ejpam-6528	81	27	that	that	SCONJ
ejpam-6528	81	28	µ	µ	NOUN
ejpam-6528	81	29	is	be	AUX
ejpam-6528	81	30	absolutely	absolutely	ADV
ejpam-6528	81	31	continuous	continuous	ADJ
ejpam-6528	81	32	.	.	PUNCT
ejpam-6528	82	1	conclusion	conclusion	NOUN
ejpam-6528	82	2	we	we	PRON
ejpam-6528	82	3	have	have	AUX
ejpam-6528	82	4	shown	show	VERB
ejpam-6528	82	5	that	that	SCONJ
ejpam-6528	82	6	µ	µ	NOUN
ejpam-6528	82	7	is	be	AUX
ejpam-6528	82	8	both	both	PRON
ejpam-6528	82	9	σ	σ	NOUN
ejpam-6528	82	10	-	-	NOUN
ejpam-6528	82	11	finite	finite	ADJ
ejpam-6528	82	12	and	and	CCONJ
ejpam-6528	82	13	absolutely	absolutely	ADV
ejpam-6528	82	14	continuous	continuous	ADJ
ejpam-6528	82	15	.	.	PUNCT
ejpam-6528	83	1	therefore	therefore	ADV
ejpam-6528	83	2	,	,	PUNCT
ejpam-6528	83	3	µ	µ	X
ejpam-6528	83	4	is	be	AUX
ejpam-6528	83	5	absolutely	absolutely	ADV
ejpam-6528	83	6	continuous	continuous	ADJ
ejpam-6528	83	7	with	with	ADP
ejpam-6528	83	8	respect	respect	NOUN
ejpam-6528	83	9	to	to	ADP
ejpam-6528	83	10	the	the	DET
ejpam-6528	83	11	product	product	NOUN
ejpam-6528	83	12	measure	measure	NOUN
ejpam-6528	83	13	µ(υ)µ(ξ)µ(ℑ	µ(υ)µ(ξ)µ(ℑ	NOUN
ejpam-6528	83	14	)	)	PUNCT
ejpam-6528	83	15	,	,	PUNCT
ejpam-6528	83	16	completing	complete	VERB
ejpam-6528	83	17	the	the	DET
ejpam-6528	83	18	proof	proof	NOUN
ejpam-6528	83	19	.	.	PUNCT
ejpam-6528	84	1	example	example	NOUN
ejpam-6528	84	2	1	1	NUM
ejpam-6528	84	3	.	.	X
ejpam-6528	85	1	consider	consider	VERB
ejpam-6528	85	2	the	the	DET
ejpam-6528	85	3	space	space	NOUN
ejpam-6528	85	4	x	x	PUNCT
ejpam-6528	86	1	=	=	PUNCT
ejpam-6528	86	2	r	r	NOUN
ejpam-6528	86	3	with	with	ADP
ejpam-6528	86	4	the	the	DET
ejpam-6528	86	5	mr	mr	PROPN
ejpam-6528	86	6	-	-	PUNCT
ejpam-6528	86	7	metric	metric	NOUN
ejpam-6528	86	8	defined	define	VERB
ejpam-6528	86	9	as	as	ADP
ejpam-6528	86	10	:	:	PUNCT
ejpam-6528	86	11	m(υ	m(υ	PROPN
ejpam-6528	86	12	,	,	PUNCT
ejpam-6528	86	13	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	86	14	)	)	PUNCT
ejpam-6528	87	1	=	=	SYM
ejpam-6528	87	2	|υ	|υ	NOUN
ejpam-6528	87	3	−	−	PROPN
ejpam-6528	87	4	ξ|+	ξ|+	PROPN
ejpam-6528	87	5	|ξ	|ξ	VERB
ejpam-6528	87	6	−ℑ|+	−ℑ|+	NOUN
ejpam-6528	87	7	|ℑ	|ℑ	NOUN
ejpam-6528	87	8	−	−	PROPN
ejpam-6528	87	9	υ|	υ|	PROPN
ejpam-6528	87	10	.	.	PUNCT
ejpam-6528	88	1	(	(	PUNCT
ejpam-6528	88	2	5	5	X
ejpam-6528	88	3	)	)	PUNCT
ejpam-6528	88	4	define	define	VERB
ejpam-6528	88	5	the	the	DET
ejpam-6528	88	6	measure	measure	NOUN
ejpam-6528	88	7	µ	µ	X
ejpam-6528	88	8	on	on	ADP
ejpam-6528	88	9	x	x	PUNCT
ejpam-6528	88	10	as	as	ADP
ejpam-6528	88	11	the	the	DET
ejpam-6528	88	12	lebesgue	lebesgue	NOUN
ejpam-6528	88	13	measure	measure	NOUN
ejpam-6528	88	14	.	.	PUNCT
ejpam-6528	89	1	the	the	DET
ejpam-6528	89	2	measure	measure	NOUN
ejpam-6528	89	3	of	of	ADP
ejpam-6528	89	4	a	a	DET
ejpam-6528	89	5	set	set	NOUN
ejpam-6528	89	6	a	a	DET
ejpam-6528	89	7	⊆	⊆	NUM
ejpam-6528	89	8	x	x	NUM
ejpam-6528	89	9	is	be	AUX
ejpam-6528	89	10	given	give	VERB
ejpam-6528	89	11	by	by	ADP
ejpam-6528	89	12	:	:	PUNCT
ejpam-6528	89	13	µ(a	µ(a	PROPN
ejpam-6528	89	14	)	)	PUNCT
ejpam-6528	90	1	=	=	PRON
ejpam-6528	90	2	∫	∫	PROPN
ejpam-6528	90	3	a	a	DET
ejpam-6528	90	4	m(υ	m(υ	PROPN
ejpam-6528	90	5	,	,	PUNCT
ejpam-6528	90	6	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	90	7	)	)	PUNCT
ejpam-6528	90	8	dυ	dυ	ADP
ejpam-6528	90	9	dξ	dξ	PROPN
ejpam-6528	90	10	dℑ.	dℑ.	PROPN
ejpam-6528	90	11	(	(	PUNCT
ejpam-6528	90	12	6	6	NUM
ejpam-6528	90	13	)	)	PUNCT
ejpam-6528	90	14	we	we	PRON
ejpam-6528	90	15	now	now	ADV
ejpam-6528	90	16	analyze	analyze	VERB
ejpam-6528	90	17	its	its	PRON
ejpam-6528	90	18	properties	property	NOUN
ejpam-6528	90	19	in	in	ADP
ejpam-6528	90	20	detail	detail	NOUN
ejpam-6528	90	21	:	:	PUNCT
ejpam-6528	90	22	1	1	X
ejpam-6528	90	23	.	.	X
ejpam-6528	90	24	σ	σ	NOUN
ejpam-6528	90	25	-	-	PUNCT
ejpam-6528	90	26	finiteness	finiteness	NOUN
ejpam-6528	90	27	:	:	PUNCT
ejpam-6528	90	28	a.	a.	NOUN
ejpam-6528	90	29	malkawi	malkawi	PROPN
ejpam-6528	90	30	,	,	PUNCT
ejpam-6528	90	31	a.	a.	PROPN
ejpam-6528	90	32	rabaiah	rabaiah	PROPN
ejpam-6528	90	33	/	/	SYM
ejpam-6528	90	34	eur	eur	PROPN
ejpam-6528	90	35	.	.	PUNCT
ejpam-6528	91	1	j.	j.	PROPN
ejpam-6528	91	2	pure	pure	PROPN
ejpam-6528	91	3	appl	appl	PROPN
ejpam-6528	91	4	.	.	PROPN
ejpam-6528	91	5	math	math	PROPN
ejpam-6528	91	6	,	,	PUNCT
ejpam-6528	91	7	18	18	NUM
ejpam-6528	91	8	(	(	PUNCT
ejpam-6528	91	9	3	3	NUM
ejpam-6528	91	10	)	)	PUNCT
ejpam-6528	91	11	(	(	PUNCT
ejpam-6528	91	12	2025	2025	NUM
ejpam-6528	91	13	)	)	PUNCT
ejpam-6528	91	14	,	,	PUNCT
ejpam-6528	91	15	6528	6528	NUM
ejpam-6528	91	16	5	5	NUM
ejpam-6528	91	17	of	of	ADP
ejpam-6528	91	18	12	12	NUM
ejpam-6528	91	19	•	•	NOUN
ejpam-6528	91	20	a	a	DET
ejpam-6528	91	21	measure	measure	NOUN
ejpam-6528	91	22	µ	µ	NOUN
ejpam-6528	91	23	is	be	AUX
ejpam-6528	91	24	called	call	VERB
ejpam-6528	91	25	σ	σ	NOUN
ejpam-6528	91	26	-	-	NOUN
ejpam-6528	91	27	finite	finite	NOUN
ejpam-6528	91	28	if	if	SCONJ
ejpam-6528	91	29	there	there	PRON
ejpam-6528	91	30	exists	exist	VERB
ejpam-6528	91	31	a	a	DET
ejpam-6528	91	32	countable	countable	ADJ
ejpam-6528	91	33	collection	collection	NOUN
ejpam-6528	91	34	of	of	ADP
ejpam-6528	91	35	measurable	measurable	ADJ
ejpam-6528	91	36	sets	set	NOUN
ejpam-6528	91	37	{	{	PUNCT
ejpam-6528	91	38	xn	xn	X
ejpam-6528	91	39	}	}	PUNCT
ejpam-6528	91	40	such	such	ADJ
ejpam-6528	91	41	that	that	SCONJ
ejpam-6528	91	42	:	:	PUNCT
ejpam-6528	91	43	x	x	SYM
ejpam-6528	91	44	=	=	SYM
ejpam-6528	91	45	∞⋃	∞⋃	PROPN
ejpam-6528	91	46	n=1	n=1	PUNCT
ejpam-6528	91	47	xn	xn	PROPN
ejpam-6528	91	48	and	and	CCONJ
ejpam-6528	91	49	µ(xn	µ(xn	PROPN
ejpam-6528	91	50	)	)	PUNCT
ejpam-6528	92	1	<	<	X
ejpam-6528	92	2	∞	∞	NUM
ejpam-6528	92	3	for	for	ADP
ejpam-6528	92	4	all	all	DET
ejpam-6528	92	5	n.	n.	NOUN
ejpam-6528	92	6	(	(	PUNCT
ejpam-6528	92	7	7	7	NUM
ejpam-6528	92	8	)	)	PUNCT
ejpam-6528	92	9	•	•	NUM
ejpam-6528	92	10	consider	consider	VERB
ejpam-6528	92	11	the	the	DET
ejpam-6528	92	12	sequence	sequence	NOUN
ejpam-6528	92	13	of	of	ADP
ejpam-6528	92	14	bounded	bounded	ADJ
ejpam-6528	92	15	intervals	interval	NOUN
ejpam-6528	92	16	xn	xn	PUNCT
ejpam-6528	93	1	=	=	PUNCT
ejpam-6528	94	1	[	[	X
ejpam-6528	94	2	−n	−n	ADJ
ejpam-6528	94	3	,	,	PUNCT
ejpam-6528	94	4	n	n	CCONJ
ejpam-6528	94	5	]	]	X
ejpam-6528	94	6	⊂	⊂	PROPN
ejpam-6528	94	7	r.	r.	PROPN
ejpam-6528	94	8	clearly	clearly	ADV
ejpam-6528	94	9	,	,	PUNCT
ejpam-6528	94	10	⋃∞	⋃∞	PUNCT
ejpam-6528	94	11	n=1xn	n=1xn	X
ejpam-6528	95	1	=	=	SYM
ejpam-6528	96	1	r	r	NOUN
ejpam-6528	96	2	,	,	PUNCT
ejpam-6528	96	3	covering	cover	VERB
ejpam-6528	96	4	the	the	DET
ejpam-6528	96	5	entire	entire	ADJ
ejpam-6528	96	6	space	space	NOUN
ejpam-6528	96	7	.	.	PUNCT
ejpam-6528	97	1	•	•	NOUN
ejpam-6528	97	2	for	for	ADP
ejpam-6528	97	3	each	each	DET
ejpam-6528	97	4	xn	xn	PROPN
ejpam-6528	97	5	,	,	PUNCT
ejpam-6528	97	6	the	the	DET
ejpam-6528	97	7	integral	integral	ADJ
ejpam-6528	97	8	defining	define	VERB
ejpam-6528	97	9	µ(xn	µ(xn	NOUN
ejpam-6528	97	10	)	)	PUNCT
ejpam-6528	97	11	is	be	AUX
ejpam-6528	97	12	:	:	PUNCT
ejpam-6528	97	13	µ(xn	µ(xn	X
ejpam-6528	97	14	)	)	PUNCT
ejpam-6528	97	15	=	=	SYM
ejpam-6528	98	1	∫	∫	PROPN
ejpam-6528	99	1	xn	xn	PROPN
ejpam-6528	99	2	∫	∫	PROPN
ejpam-6528	100	1	xn	xn	PROPN
ejpam-6528	100	2	∫	∫	PROPN
ejpam-6528	100	3	xn	xn	PROPN
ejpam-6528	101	1	m(υ	m(υ	PROPN
ejpam-6528	101	2	,	,	PUNCT
ejpam-6528	101	3	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	101	4	)	)	PUNCT
ejpam-6528	101	5	dυ	dυ	ADP
ejpam-6528	101	6	dξ	dξ	PROPN
ejpam-6528	101	7	dℑ.	dℑ.	PROPN
ejpam-6528	101	8	(	(	PUNCT
ejpam-6528	101	9	8)	8)	NUM
ejpam-6528	101	10	•	•	NOUN
ejpam-6528	101	11	since	since	SCONJ
ejpam-6528	101	12	m(υ	m(υ	PROPN
ejpam-6528	101	13	,	,	PUNCT
ejpam-6528	101	14	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	101	15	)	)	PUNCT
ejpam-6528	101	16	is	be	AUX
ejpam-6528	101	17	finite	finite	ADJ
ejpam-6528	101	18	for	for	ADP
ejpam-6528	101	19	bounded	bounded	ADJ
ejpam-6528	101	20	sets	set	NOUN
ejpam-6528	101	21	,	,	PUNCT
ejpam-6528	101	22	this	this	DET
ejpam-6528	101	23	integral	integral	ADJ
ejpam-6528	101	24	converges	converge	NOUN
ejpam-6528	101	25	,	,	PUNCT
ejpam-6528	101	26	implying	imply	VERB
ejpam-6528	101	27	that	that	SCONJ
ejpam-6528	101	28	µ(xn	µ(xn	NOUN
ejpam-6528	101	29	)	)	PUNCT
ejpam-6528	101	30	<	<	X
ejpam-6528	101	31	∞	∞	NUM
ejpam-6528	101	32	for	for	ADP
ejpam-6528	101	33	each	each	DET
ejpam-6528	101	34	n.	n.	NOUN
ejpam-6528	101	35	•	•	ADP
ejpam-6528	101	36	thus	thus	ADV
ejpam-6528	101	37	,	,	PUNCT
ejpam-6528	101	38	µ	µ	X
ejpam-6528	101	39	is	be	AUX
ejpam-6528	101	40	σ	σ	NOUN
ejpam-6528	101	41	-	-	NOUN
ejpam-6528	101	42	finite	finite	NOUN
ejpam-6528	101	43	.	.	NOUN
ejpam-6528	102	1	2	2	NUM
ejpam-6528	102	2	.	.	X
ejpam-6528	102	3	absolute	absolute	ADJ
ejpam-6528	102	4	continuity	continuity	NOUN
ejpam-6528	102	5	:	:	PUNCT
ejpam-6528	102	6	•	•	ADP
ejpam-6528	102	7	a	a	DET
ejpam-6528	102	8	measure	measure	NOUN
ejpam-6528	102	9	µ	µ	NOUN
ejpam-6528	102	10	is	be	AUX
ejpam-6528	102	11	absolutely	absolutely	ADV
ejpam-6528	102	12	continuous	continuous	ADJ
ejpam-6528	102	13	with	with	ADP
ejpam-6528	102	14	respect	respect	NOUN
ejpam-6528	102	15	to	to	ADP
ejpam-6528	102	16	the	the	DET
ejpam-6528	102	17	lebesgue	lebesgue	ADJ
ejpam-6528	102	18	measure	measure	NOUN
ejpam-6528	102	19	λ	λ	X
ejpam-6528	102	20	if	if	SCONJ
ejpam-6528	102	21	for	for	ADP
ejpam-6528	102	22	every	every	DET
ejpam-6528	102	23	measurable	measurable	NOUN
ejpam-6528	102	24	set	set	VERB
ejpam-6528	102	25	a	a	DET
ejpam-6528	102	26	:	:	PUNCT
ejpam-6528	102	27	λ(a	λ(a	NOUN
ejpam-6528	102	28	)	)	PUNCT
ejpam-6528	102	29	=	=	SYM
ejpam-6528	102	30	0	0	PUNCT
ejpam-6528	103	1	=	=	NOUN
ejpam-6528	103	2	⇒	⇒	NOUN
ejpam-6528	103	3	µ(a	µ(a	PROPN
ejpam-6528	103	4	)	)	PUNCT
ejpam-6528	103	5	=	=	SYM
ejpam-6528	104	1	0	0	X
ejpam-6528	104	2	.	.	PUNCT
ejpam-6528	105	1	(	(	PUNCT
ejpam-6528	105	2	9	9	NUM
ejpam-6528	105	3	)	)	PUNCT
ejpam-6528	105	4	•	•	NOUN
ejpam-6528	105	5	suppose	suppose	VERB
ejpam-6528	105	6	a	a	DET
ejpam-6528	105	7	⊆	⊆	NUM
ejpam-6528	105	8	x	x	X
ejpam-6528	105	9	is	be	AUX
ejpam-6528	105	10	a	a	DET
ejpam-6528	105	11	measurable	measurable	ADJ
ejpam-6528	105	12	set	set	VERB
ejpam-6528	105	13	with	with	ADP
ejpam-6528	105	14	lebesgue	lebesgue	NOUN
ejpam-6528	105	15	measure	measure	NOUN
ejpam-6528	105	16	λ(a	λ(a	NOUN
ejpam-6528	105	17	)	)	PUNCT
ejpam-6528	105	18	=	=	SYM
ejpam-6528	106	1	0	0	X
ejpam-6528	106	2	.	.	PUNCT
ejpam-6528	107	1	this	this	PRON
ejpam-6528	107	2	means	mean	VERB
ejpam-6528	107	3	that	that	SCONJ
ejpam-6528	107	4	the	the	DET
ejpam-6528	107	5	set	set	NOUN
ejpam-6528	107	6	has	have	VERB
ejpam-6528	107	7	no	no	DET
ejpam-6528	107	8	volume	volume	NOUN
ejpam-6528	107	9	in	in	ADP
ejpam-6528	107	10	the	the	DET
ejpam-6528	107	11	standard	standard	ADJ
ejpam-6528	107	12	lebesgue	lebesgue	NOUN
ejpam-6528	107	13	sense	sense	NOUN
ejpam-6528	107	14	.	.	PUNCT
ejpam-6528	108	1	•	•	NUM
ejpam-6528	108	2	by	by	ADP
ejpam-6528	108	3	the	the	DET
ejpam-6528	108	4	definition	definition	NOUN
ejpam-6528	108	5	of	of	ADP
ejpam-6528	108	6	µ(a	µ(a	PROPN
ejpam-6528	108	7	):	):	PUNCT
ejpam-6528	108	8	µ(a	µ(a	PROPN
ejpam-6528	108	9	)	)	PUNCT
ejpam-6528	109	1	=	=	SYM
ejpam-6528	109	2	∫	∫	PROPN
ejpam-6528	109	3	a	a	DET
ejpam-6528	109	4	m(υ	m(υ	PROPN
ejpam-6528	109	5	,	,	PUNCT
ejpam-6528	109	6	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	109	7	)	)	PUNCT
ejpam-6528	109	8	dυ	dυ	ADP
ejpam-6528	109	9	dξ	dξ	PROPN
ejpam-6528	109	10	dℑ.	dℑ.	PROPN
ejpam-6528	109	11	(	(	PUNCT
ejpam-6528	109	12	10	10	NUM
ejpam-6528	109	13	)	)	PUNCT
ejpam-6528	109	14	•	•	NOUN
ejpam-6528	109	15	since	since	SCONJ
ejpam-6528	109	16	the	the	DET
ejpam-6528	109	17	lebesgue	lebesgue	ADJ
ejpam-6528	109	18	measure	measure	NOUN
ejpam-6528	109	19	of	of	ADP
ejpam-6528	109	20	a	a	DET
ejpam-6528	109	21	is	be	AUX
ejpam-6528	109	22	zero	zero	NUM
ejpam-6528	109	23	,	,	PUNCT
ejpam-6528	109	24	the	the	DET
ejpam-6528	109	25	triple	triple	ADJ
ejpam-6528	109	26	integral	integral	ADJ
ejpam-6528	109	27	above	above	ADV
ejpam-6528	109	28	must	must	AUX
ejpam-6528	109	29	also	also	ADV
ejpam-6528	109	30	be	be	AUX
ejpam-6528	109	31	zero	zero	NUM
ejpam-6528	109	32	because	because	SCONJ
ejpam-6528	109	33	integration	integration	NOUN
ejpam-6528	109	34	over	over	ADP
ejpam-6528	109	35	a	a	DET
ejpam-6528	109	36	null	null	ADJ
ejpam-6528	109	37	set	set	VERB
ejpam-6528	109	38	yields	yield	NOUN
ejpam-6528	109	39	zero	zero	NUM
ejpam-6528	109	40	measure	measure	NOUN
ejpam-6528	109	41	.	.	PUNCT
ejpam-6528	110	1	•	•	NUM
ejpam-6528	110	2	thus	thus	ADV
ejpam-6528	110	3	,	,	PUNCT
ejpam-6528	110	4	we	we	PRON
ejpam-6528	110	5	conclude	conclude	VERB
ejpam-6528	110	6	that	that	SCONJ
ejpam-6528	110	7	µ(a	µ(a	PROPN
ejpam-6528	110	8	)	)	PUNCT
ejpam-6528	110	9	=	=	SYM
ejpam-6528	110	10	0	0	NUM
ejpam-6528	110	11	,	,	PUNCT
ejpam-6528	110	12	establishing	establish	VERB
ejpam-6528	110	13	absolute	absolute	ADJ
ejpam-6528	110	14	continuity	continuity	NOUN
ejpam-6528	110	15	of	of	ADP
ejpam-6528	110	16	µ	µ	NOUN
ejpam-6528	110	17	with	with	ADP
ejpam-6528	110	18	respect	respect	NOUN
ejpam-6528	110	19	to	to	ADP
ejpam-6528	110	20	the	the	DET
ejpam-6528	110	21	lebesgue	lebesgue	NOUN
ejpam-6528	110	22	measure	measure	NOUN
ejpam-6528	110	23	.	.	PUNCT
ejpam-6528	111	1	hence	hence	ADV
ejpam-6528	111	2	,	,	PUNCT
ejpam-6528	111	3	we	we	PRON
ejpam-6528	111	4	rigorously	rigorously	ADV
ejpam-6528	111	5	confirm	confirm	VERB
ejpam-6528	111	6	that	that	SCONJ
ejpam-6528	111	7	µ	µ	NOUN
ejpam-6528	111	8	is	be	AUX
ejpam-6528	111	9	both	both	PRON
ejpam-6528	111	10	σ	σ	NOUN
ejpam-6528	111	11	-	-	NOUN
ejpam-6528	111	12	finite	finite	ADJ
ejpam-6528	111	13	and	and	CCONJ
ejpam-6528	111	14	absolutely	absolutely	ADV
ejpam-6528	111	15	continuous	continuous	ADJ
ejpam-6528	111	16	.	.	PUNCT
ejpam-6528	112	1	theorem	theorem	NOUN
ejpam-6528	112	2	2	2	NUM
ejpam-6528	112	3	.	.	X
ejpam-6528	113	1	let	let	AUX
ejpam-6528	113	2	(	(	PUNCT
ejpam-6528	113	3	x	x	X
ejpam-6528	113	4	,	,	PUNCT
ejpam-6528	113	5	m	m	VERB
ejpam-6528	113	6	)	)	PUNCT
ejpam-6528	113	7	be	be	VERB
ejpam-6528	113	8	an	an	DET
ejpam-6528	113	9	mr	mr	ADJ
ejpam-6528	113	10	-	-	PUNCT
ejpam-6528	113	11	metric	metric	ADJ
ejpam-6528	113	12	space	space	NOUN
ejpam-6528	113	13	with	with	ADP
ejpam-6528	113	14	a	a	DET
ejpam-6528	113	15	measure	measure	NOUN
ejpam-6528	113	16	µ	µ	PRON
ejpam-6528	113	17	such	such	ADJ
ejpam-6528	113	18	that	that	PRON
ejpam-6528	113	19	for	for	ADP
ejpam-6528	113	20	a	a	DET
ejpam-6528	113	21	sequence	sequence	NOUN
ejpam-6528	113	22	{	{	PUNCT
ejpam-6528	113	23	υn	υn	NOUN
ejpam-6528	113	24	}	}	PUNCT
ejpam-6528	113	25	in	in	ADP
ejpam-6528	113	26	x	x	PROPN
ejpam-6528	113	27	,	,	PUNCT
ejpam-6528	113	28	m(υn	m(υn	ADJ
ejpam-6528	113	29	,	,	PUNCT
ejpam-6528	113	30	υn−1	υn−1	ADJ
ejpam-6528	113	31	,	,	PUNCT
ejpam-6528	113	32	υn−2	υn−2	PROPN
ejpam-6528	113	33	)	)	PUNCT
ejpam-6528	113	34	→	→	SYM
ejpam-6528	113	35	0	0	NUM
ejpam-6528	113	36	as	as	ADP
ejpam-6528	113	37	n→	n→	PROPN
ejpam-6528	113	38	∞.	∞.	PROPN
ejpam-6528	113	39	(	(	PUNCT
ejpam-6528	113	40	11	11	NUM
ejpam-6528	113	41	)	)	PUNCT
ejpam-6528	113	42	then	then	ADV
ejpam-6528	113	43	,	,	PUNCT
ejpam-6528	113	44	υn	υn	PROPN
ejpam-6528	113	45	converges	converge	VERB
ejpam-6528	113	46	to	to	ADP
ejpam-6528	113	47	some	some	DET
ejpam-6528	113	48	υ∗	υ∗	NOUN
ejpam-6528	113	49	∈	∈	PROPN
ejpam-6528	113	50	x	x	PUNCT
ejpam-6528	113	51	in	in	ADP
ejpam-6528	113	52	measure	measure	NOUN
ejpam-6528	113	53	,	,	PUNCT
ejpam-6528	113	54	meaning	mean	VERB
ejpam-6528	113	55	that	that	SCONJ
ejpam-6528	113	56	for	for	ADP
ejpam-6528	113	57	every	every	DET
ejpam-6528	113	58	ϵ	ϵ	PROPN
ejpam-6528	113	59	>	>	X
ejpam-6528	113	60	0	0	NUM
ejpam-6528	113	61	,	,	PUNCT
ejpam-6528	113	62	µ	µ	X
ejpam-6528	113	63	(	(	PUNCT
ejpam-6528	113	64	{	{	PUNCT
ejpam-6528	113	65	υ	υ	PROPN
ejpam-6528	113	66	∈	∈	PROPN
ejpam-6528	113	67	x	x	X
ejpam-6528	113	68	:	:	PUNCT
ejpam-6528	113	69	m(υn	m(υn	PROPN
ejpam-6528	113	70	,	,	PUNCT
ejpam-6528	113	71	υ	υ	PRON
ejpam-6528	113	72	∗	∗	NOUN
ejpam-6528	113	73	,	,	PUNCT
ejpam-6528	113	74	υ∗	υ∗	NOUN
ejpam-6528	113	75	)	)	PUNCT
ejpam-6528	113	76	≥	≥	NOUN
ejpam-6528	113	77	ϵ	ϵ	NOUN
ejpam-6528	113	78	}	}	PUNCT
ejpam-6528	113	79	)	)	PUNCT
ejpam-6528	113	80	→	→	SYM
ejpam-6528	113	81	0	0	NUM
ejpam-6528	113	82	as	as	ADP
ejpam-6528	113	83	n→	n→	PROPN
ejpam-6528	113	84	∞.	∞.	PROPN
ejpam-6528	113	85	(	(	PUNCT
ejpam-6528	113	86	12	12	NUM
ejpam-6528	113	87	)	)	PUNCT
ejpam-6528	113	88	a.	a.	NOUN
ejpam-6528	113	89	malkawi	malkawi	PROPN
ejpam-6528	113	90	,	,	PUNCT
ejpam-6528	113	91	a.	a.	PROPN
ejpam-6528	113	92	rabaiah	rabaiah	PROPN
ejpam-6528	113	93	/	/	SYM
ejpam-6528	113	94	eur	eur	PROPN
ejpam-6528	113	95	.	.	PUNCT
ejpam-6528	114	1	j.	j.	PROPN
ejpam-6528	114	2	pure	pure	PROPN
ejpam-6528	114	3	appl	appl	PROPN
ejpam-6528	114	4	.	.	PROPN
ejpam-6528	114	5	math	math	PROPN
ejpam-6528	114	6	,	,	PUNCT
ejpam-6528	114	7	18	18	NUM
ejpam-6528	114	8	(	(	PUNCT
ejpam-6528	114	9	3	3	NUM
ejpam-6528	114	10	)	)	PUNCT
ejpam-6528	114	11	(	(	PUNCT
ejpam-6528	114	12	2025	2025	NUM
ejpam-6528	114	13	)	)	PUNCT
ejpam-6528	114	14	,	,	PUNCT
ejpam-6528	114	15	6528	6528	NUM
ejpam-6528	114	16	6	6	NUM
ejpam-6528	114	17	of	of	ADP
ejpam-6528	114	18	12	12	NUM
ejpam-6528	114	19	proof	proof	NOUN
ejpam-6528	114	20	.	.	PUNCT
ejpam-6528	115	1	we	we	PRON
ejpam-6528	115	2	are	be	AUX
ejpam-6528	115	3	given	give	VERB
ejpam-6528	115	4	that	that	DET
ejpam-6528	115	5	m(υn	m(υn	ADJ
ejpam-6528	115	6	,	,	PUNCT
ejpam-6528	115	7	υn−1	υn−1	ADJ
ejpam-6528	115	8	,	,	PUNCT
ejpam-6528	115	9	υn−2	υn−2	PROPN
ejpam-6528	115	10	)	)	PUNCT
ejpam-6528	115	11	→	→	SYM
ejpam-6528	115	12	0	0	NUM
ejpam-6528	116	1	as	as	ADP
ejpam-6528	116	2	n	n	NUM
ejpam-6528	116	3	→	→	SYM
ejpam-6528	116	4	∞	∞	PROPN
ejpam-6528	116	5	,	,	PUNCT
ejpam-6528	116	6	which	which	PRON
ejpam-6528	116	7	implies	imply	VERB
ejpam-6528	116	8	that	that	SCONJ
ejpam-6528	116	9	the	the	DET
ejpam-6528	116	10	sequence	sequence	NOUN
ejpam-6528	116	11	{	{	PUNCT
ejpam-6528	116	12	υn	υn	NOUN
ejpam-6528	116	13	}	}	PUNCT
ejpam-6528	116	14	is	be	AUX
ejpam-6528	116	15	cauchy	cauchy	ADJ
ejpam-6528	116	16	in	in	ADP
ejpam-6528	116	17	the	the	DET
ejpam-6528	116	18	mr	mr	PROPN
ejpam-6528	116	19	-	-	PUNCT
ejpam-6528	116	20	metric	metric	ADJ
ejpam-6528	116	21	space	space	NOUN
ejpam-6528	116	22	(	(	PUNCT
ejpam-6528	116	23	x	x	X
ejpam-6528	116	24	,	,	PUNCT
ejpam-6528	116	25	m	m	NOUN
ejpam-6528	116	26	)	)	PUNCT
ejpam-6528	116	27	.	.	PUNCT
ejpam-6528	117	1	step	step	NOUN
ejpam-6528	117	2	1	1	NUM
ejpam-6528	117	3	:	:	PUNCT
ejpam-6528	117	4	verifying	verify	VERB
ejpam-6528	117	5	the	the	DET
ejpam-6528	117	6	cauchy	cauchy	ADJ
ejpam-6528	117	7	property	property	NOUN
ejpam-6528	117	8	the	the	DET
ejpam-6528	117	9	given	give	VERB
ejpam-6528	117	10	conditionm(υn	conditionm(υn	ADJ
ejpam-6528	117	11	,	,	PUNCT
ejpam-6528	117	12	υn−1	υn−1	PROPN
ejpam-6528	117	13	,	,	PUNCT
ejpam-6528	117	14	υn−2	υn−2	PROPN
ejpam-6528	117	15	)	)	PUNCT
ejpam-6528	117	16	→	→	SYM
ejpam-6528	117	17	0	0	NUM
ejpam-6528	117	18	as	as	ADP
ejpam-6528	117	19	n→	n→	NOUN
ejpam-6528	117	20	∞	∞	PROPN
ejpam-6528	117	21	implies	imply	VERB
ejpam-6528	117	22	that	that	SCONJ
ejpam-6528	117	23	for	for	ADP
ejpam-6528	117	24	every	every	DET
ejpam-6528	117	25	ϵ	ϵ	PROPN
ejpam-6528	117	26	>	>	X
ejpam-6528	117	27	0	0	NUM
ejpam-6528	117	28	,	,	PUNCT
ejpam-6528	117	29	there	there	PRON
ejpam-6528	117	30	exists	exist	VERB
ejpam-6528	117	31	an	an	DET
ejpam-6528	117	32	integer	integer	NOUN
ejpam-6528	117	33	n	n	CCONJ
ejpam-6528	117	34	such	such	ADJ
ejpam-6528	117	35	that	that	PRON
ejpam-6528	117	36	for	for	ADP
ejpam-6528	117	37	all	all	DET
ejpam-6528	117	38	n	n	PRON
ejpam-6528	117	39	≥	≥	NUM
ejpam-6528	117	40	n	n	CCONJ
ejpam-6528	117	41	,	,	PUNCT
ejpam-6528	117	42	m(υn	m(υn	PROPN
ejpam-6528	117	43	,	,	PUNCT
ejpam-6528	117	44	υn−1	υn−1	ADJ
ejpam-6528	117	45	,	,	PUNCT
ejpam-6528	117	46	υn−2	υn−2	PROPN
ejpam-6528	117	47	)	)	PUNCT
ejpam-6528	117	48	<	<	X
ejpam-6528	117	49	ϵ	ϵ	X
ejpam-6528	117	50	3	3	NUM
ejpam-6528	117	51	.	.	PUNCT
ejpam-6528	118	1	by	by	ADP
ejpam-6528	118	2	the	the	DET
ejpam-6528	118	3	mr	mr	PROPN
ejpam-6528	118	4	-	-	PUNCT
ejpam-6528	118	5	metric	metric	ADJ
ejpam-6528	118	6	inequality	inequality	NOUN
ejpam-6528	118	7	,	,	PUNCT
ejpam-6528	118	8	m(υn	m(υn	PROPN
ejpam-6528	118	9	,	,	PUNCT
ejpam-6528	118	10	υm	υm	NOUN
ejpam-6528	118	11	,	,	PUNCT
ejpam-6528	118	12	υk	υk	NOUN
ejpam-6528	118	13	)	)	PUNCT
ejpam-6528	118	14	≤	≤	NUM
ejpam-6528	118	15	r	r	NOUN
ejpam-6528	118	16	[	[	X
ejpam-6528	118	17	m(υn	m(υn	ADJ
ejpam-6528	118	18	,	,	PUNCT
ejpam-6528	118	19	υn−1	υn−1	ADJ
ejpam-6528	118	20	,	,	PUNCT
ejpam-6528	118	21	υn−2	υn−2	PROPN
ejpam-6528	118	22	)	)	PUNCT
ejpam-6528	118	23	+	+	NOUN
ejpam-6528	118	24	m(υn−1	m(υn−1	NOUN
ejpam-6528	118	25	,	,	PUNCT
ejpam-6528	118	26	υm	υm	NOUN
ejpam-6528	118	27	,	,	PUNCT
ejpam-6528	118	28	υk	υk	NOUN
ejpam-6528	118	29	)	)	PUNCT
ejpam-6528	118	30	+	+	NOUN
ejpam-6528	118	31	m(υm	m(υm	NOUN
ejpam-6528	118	32	,	,	PUNCT
ejpam-6528	118	33	υk	υk	NOUN
ejpam-6528	118	34	,	,	PUNCT
ejpam-6528	118	35	υn	υn	NOUN
ejpam-6528	118	36	)	)	PUNCT
ejpam-6528	118	37	]	]	PUNCT
ejpam-6528	118	38	.	.	PUNCT
ejpam-6528	119	1	since	since	SCONJ
ejpam-6528	119	2	each	each	DET
ejpam-6528	119	3	term	term	NOUN
ejpam-6528	119	4	on	on	ADP
ejpam-6528	119	5	the	the	DET
ejpam-6528	119	6	right	right	ADJ
ejpam-6528	119	7	-	-	PUNCT
ejpam-6528	119	8	hand	hand	NOUN
ejpam-6528	119	9	side	side	NOUN
ejpam-6528	119	10	approaches	approach	NOUN
ejpam-6528	119	11	zero	zero	NUM
ejpam-6528	119	12	,	,	PUNCT
ejpam-6528	119	13	it	it	PRON
ejpam-6528	119	14	follows	follow	VERB
ejpam-6528	119	15	that	that	SCONJ
ejpam-6528	119	16	m(υn	m(υn	ADJ
ejpam-6528	119	17	,	,	PUNCT
ejpam-6528	119	18	υm	υm	NOUN
ejpam-6528	119	19	,	,	PUNCT
ejpam-6528	119	20	υk	υk	NOUN
ejpam-6528	119	21	)	)	PUNCT
ejpam-6528	119	22	→	→	SYM
ejpam-6528	119	23	0	0	NUM
ejpam-6528	119	24	as	as	ADP
ejpam-6528	119	25	n	n	CCONJ
ejpam-6528	119	26	,	,	PUNCT
ejpam-6528	119	27	m	m	PROPN
ejpam-6528	119	28	,	,	PUNCT
ejpam-6528	119	29	k	k	PROPN
ejpam-6528	119	30	→	→	SYM
ejpam-6528	119	31	∞	∞	PROPN
ejpam-6528	119	32	,	,	PUNCT
ejpam-6528	119	33	establishing	establish	VERB
ejpam-6528	119	34	the	the	DET
ejpam-6528	119	35	cauchy	cauchy	ADJ
ejpam-6528	119	36	property	property	NOUN
ejpam-6528	119	37	.	.	PUNCT
ejpam-6528	120	1	step	step	NOUN
ejpam-6528	120	2	2	2	NUM
ejpam-6528	120	3	:	:	PUNCT
ejpam-6528	120	4	existence	existence	NOUN
ejpam-6528	120	5	of	of	ADP
ejpam-6528	120	6	limit	limit	NOUN
ejpam-6528	120	7	since	since	SCONJ
ejpam-6528	120	8	(	(	PUNCT
ejpam-6528	120	9	x	x	X
ejpam-6528	120	10	,	,	PUNCT
ejpam-6528	120	11	m	m	VERB
ejpam-6528	120	12	)	)	PUNCT
ejpam-6528	120	13	is	be	AUX
ejpam-6528	120	14	a	a	DET
ejpam-6528	120	15	complete	complete	ADJ
ejpam-6528	120	16	mr	mr	ADJ
ejpam-6528	120	17	-	-	PUNCT
ejpam-6528	120	18	metric	metric	ADJ
ejpam-6528	120	19	space	space	NOUN
ejpam-6528	120	20	,	,	PUNCT
ejpam-6528	120	21	every	every	DET
ejpam-6528	120	22	cauchy	cauchy	ADJ
ejpam-6528	120	23	sequence	sequence	NOUN
ejpam-6528	120	24	converges	converge	VERB
ejpam-6528	120	25	to	to	ADP
ejpam-6528	120	26	a	a	DET
ejpam-6528	120	27	unique	unique	ADJ
ejpam-6528	120	28	limit	limit	NOUN
ejpam-6528	120	29	υ∗	υ∗	NOUN
ejpam-6528	120	30	∈	∈	PROPN
ejpam-6528	120	31	x.	x.	NOUN
ejpam-6528	121	1	thus	thus	ADV
ejpam-6528	121	2	,	,	PUNCT
ejpam-6528	121	3	there	there	PRON
ejpam-6528	121	4	exists	exist	VERB
ejpam-6528	121	5	υ∗	υ∗	NOUN
ejpam-6528	121	6	∈	∈	PROPN
ejpam-6528	121	7	x	x	PUNCT
ejpam-6528	121	8	such	such	ADJ
ejpam-6528	121	9	that	that	SCONJ
ejpam-6528	121	10	lim	lim	PROPN
ejpam-6528	121	11	n→∞	n→∞	PRON
ejpam-6528	121	12	υn	υn	PROPN
ejpam-6528	122	1	=	=	PUNCT
ejpam-6528	122	2	υ∗.	υ∗.	VERB
ejpam-6528	122	3	this	this	PRON
ejpam-6528	122	4	ensures	ensure	VERB
ejpam-6528	122	5	that	that	SCONJ
ejpam-6528	122	6	υn	υn	NOUN
ejpam-6528	122	7	is	be	AUX
ejpam-6528	122	8	converging	converge	VERB
ejpam-6528	122	9	in	in	ADP
ejpam-6528	122	10	the	the	DET
ejpam-6528	122	11	mr	mr	PROPN
ejpam-6528	122	12	-	-	PUNCT
ejpam-6528	122	13	metric	metric	ADJ
ejpam-6528	122	14	space	space	NOUN
ejpam-6528	122	15	.	.	PUNCT
ejpam-6528	123	1	step	step	NOUN
ejpam-6528	123	2	3	3	NUM
ejpam-6528	123	3	:	:	PUNCT
ejpam-6528	123	4	convergence	convergence	NOUN
ejpam-6528	123	5	in	in	ADP
ejpam-6528	123	6	measure	measure	NOUN
ejpam-6528	123	7	to	to	PART
ejpam-6528	123	8	prove	prove	VERB
ejpam-6528	123	9	convergence	convergence	NOUN
ejpam-6528	123	10	in	in	ADP
ejpam-6528	123	11	measure	measure	NOUN
ejpam-6528	123	12	,	,	PUNCT
ejpam-6528	123	13	we	we	PRON
ejpam-6528	123	14	analyze	analyze	VERB
ejpam-6528	123	15	the	the	DET
ejpam-6528	123	16	set	set	NOUN
ejpam-6528	123	17	an	an	PRON
ejpam-6528	123	18	=	=	X
ejpam-6528	123	19	{	{	PUNCT
ejpam-6528	123	20	υ	υ	NOUN
ejpam-6528	123	21	∈	∈	PROPN
ejpam-6528	123	22	x	x	X
ejpam-6528	123	23	:	:	PUNCT
ejpam-6528	123	24	m(υn	m(υn	PROPN
ejpam-6528	123	25	,	,	PUNCT
ejpam-6528	123	26	υ	υ	PRON
ejpam-6528	123	27	∗	∗	NOUN
ejpam-6528	123	28	,	,	PUNCT
ejpam-6528	123	29	υ∗	υ∗	NOUN
ejpam-6528	123	30	)	)	PUNCT
ejpam-6528	123	31	≥	≥	NOUN
ejpam-6528	123	32	ϵ	ϵ	NOUN
ejpam-6528	123	33	}	}	PUNCT
ejpam-6528	123	34	.	.	PUNCT
ejpam-6528	124	1	for	for	ADP
ejpam-6528	124	2	large	large	ADJ
ejpam-6528	124	3	n	n	CCONJ
ejpam-6528	124	4	,	,	PUNCT
ejpam-6528	124	5	we	we	PRON
ejpam-6528	124	6	havem(υn	havem(υn	VERB
ejpam-6528	124	7	,	,	PUNCT
ejpam-6528	124	8	υ	υ	NOUN
ejpam-6528	124	9	∗	∗	NOUN
ejpam-6528	124	10	,	,	PUNCT
ejpam-6528	124	11	υ∗	υ∗	NOUN
ejpam-6528	124	12	)	)	PUNCT
ejpam-6528	124	13	<	<	X
ejpam-6528	125	1	ϵ	ϵ	X
ejpam-6528	125	2	,	,	PUNCT
ejpam-6528	125	3	meaning	mean	VERB
ejpam-6528	125	4	an	an	DET
ejpam-6528	125	5	shrinks	shrink	NOUN
ejpam-6528	125	6	as	as	ADP
ejpam-6528	125	7	n	n	DET
ejpam-6528	125	8	increases	increase	NOUN
ejpam-6528	125	9	.	.	PUNCT
ejpam-6528	126	1	the	the	DET
ejpam-6528	126	2	measure	measure	NOUN
ejpam-6528	126	3	µ	µ	NOUN
ejpam-6528	126	4	is	be	AUX
ejpam-6528	126	5	σ	σ	NOUN
ejpam-6528	126	6	-	-	NOUN
ejpam-6528	126	7	finite	finite	ADJ
ejpam-6528	126	8	,	,	PUNCT
ejpam-6528	126	9	meaning	mean	VERB
ejpam-6528	126	10	there	there	PRON
ejpam-6528	126	11	exist	exist	VERB
ejpam-6528	126	12	countable	countable	ADJ
ejpam-6528	126	13	subsets	subset	NOUN
ejpam-6528	126	14	of	of	ADP
ejpam-6528	126	15	finite	finite	ADJ
ejpam-6528	126	16	measure	measure	NOUN
ejpam-6528	126	17	covering	cover	VERB
ejpam-6528	126	18	x.	x.	NOUN
ejpam-6528	126	19	by	by	ADP
ejpam-6528	126	20	absolute	absolute	ADJ
ejpam-6528	126	21	continuity	continuity	NOUN
ejpam-6528	126	22	,	,	PUNCT
ejpam-6528	126	23	since	since	SCONJ
ejpam-6528	126	24	m(υn	m(υn	NUM
ejpam-6528	126	25	,	,	PUNCT
ejpam-6528	126	26	υ	υ	PRON
ejpam-6528	126	27	∗	∗	NOUN
ejpam-6528	126	28	,	,	PUNCT
ejpam-6528	126	29	υ∗	υ∗	NOUN
ejpam-6528	126	30	)	)	PUNCT
ejpam-6528	126	31	→	→	SYM
ejpam-6528	126	32	0	0	NUM
ejpam-6528	126	33	,	,	PUNCT
ejpam-6528	126	34	the	the	DET
ejpam-6528	126	35	measure	measure	NOUN
ejpam-6528	126	36	µ(an	µ(an	PROPN
ejpam-6528	126	37	)	)	PUNCT
ejpam-6528	126	38	→	→	SYM
ejpam-6528	126	39	0	0	NUM
ejpam-6528	126	40	as	as	ADV
ejpam-6528	126	41	well	well	ADV
ejpam-6528	126	42	.	.	PUNCT
ejpam-6528	127	1	conclusion	conclusion	NOUN
ejpam-6528	127	2	thus	thus	ADV
ejpam-6528	127	3	,	,	PUNCT
ejpam-6528	127	4	we	we	PRON
ejpam-6528	127	5	conclude	conclude	VERB
ejpam-6528	127	6	that	that	SCONJ
ejpam-6528	127	7	µ	µ	X
ejpam-6528	127	8	(	(	PUNCT
ejpam-6528	127	9	{	{	PUNCT
ejpam-6528	127	10	υ	υ	PROPN
ejpam-6528	127	11	∈	∈	PROPN
ejpam-6528	127	12	x	x	X
ejpam-6528	127	13	:	:	PUNCT
ejpam-6528	127	14	m(υn	m(υn	PROPN
ejpam-6528	127	15	,	,	PUNCT
ejpam-6528	127	16	υ	υ	PRON
ejpam-6528	127	17	∗	∗	NOUN
ejpam-6528	127	18	,	,	PUNCT
ejpam-6528	127	19	υ∗	υ∗	NOUN
ejpam-6528	127	20	)	)	PUNCT
ejpam-6528	127	21	≥	≥	NOUN
ejpam-6528	127	22	ϵ	ϵ	NOUN
ejpam-6528	127	23	}	}	PUNCT
ejpam-6528	127	24	)	)	PUNCT
ejpam-6528	127	25	→	→	SYM
ejpam-6528	127	26	0	0	NUM
ejpam-6528	127	27	as	as	ADP
ejpam-6528	127	28	n→	n→	PUNCT
ejpam-6528	127	29	∞.	∞.	PROPN
ejpam-6528	127	30	this	this	PRON
ejpam-6528	127	31	completes	complete	VERB
ejpam-6528	127	32	the	the	DET
ejpam-6528	127	33	proof	proof	NOUN
ejpam-6528	127	34	.	.	PUNCT
ejpam-6528	128	1	example	example	NOUN
ejpam-6528	128	2	2	2	NUM
ejpam-6528	128	3	.	.	X
ejpam-6528	128	4	consider	consider	VERB
ejpam-6528	128	5	the	the	DET
ejpam-6528	128	6	mr	mr	PROPN
ejpam-6528	128	7	-	-	PUNCT
ejpam-6528	128	8	metric	metric	ADJ
ejpam-6528	128	9	space	space	NOUN
ejpam-6528	128	10	(	(	PUNCT
ejpam-6528	128	11	r	r	NOUN
ejpam-6528	128	12	,	,	PUNCT
ejpam-6528	128	13	m	m	NOUN
ejpam-6528	128	14	)	)	PUNCT
ejpam-6528	128	15	where	where	SCONJ
ejpam-6528	128	16	the	the	DET
ejpam-6528	128	17	metric	metric	NOUN
ejpam-6528	128	18	is	be	AUX
ejpam-6528	128	19	given	give	VERB
ejpam-6528	128	20	by	by	ADP
ejpam-6528	128	21	m(υ	m(υ	PROPN
ejpam-6528	128	22	,	,	PUNCT
ejpam-6528	128	23	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	128	24	)	)	PUNCT
ejpam-6528	128	25	=	=	SYM
ejpam-6528	128	26	|υ	|υ	NOUN
ejpam-6528	128	27	−	−	PROPN
ejpam-6528	128	28	ξ|+	ξ|+	PROPN
ejpam-6528	128	29	|ξ	|ξ	VERB
ejpam-6528	128	30	−ℑ|+	−ℑ|+	NOUN
ejpam-6528	128	31	|ℑ	|ℑ	NOUN
ejpam-6528	128	32	−	−	PROPN
ejpam-6528	128	33	υ|	υ|	PROPN
ejpam-6528	128	34	.	.	PUNCT
ejpam-6528	129	1	(	(	PUNCT
ejpam-6528	129	2	13	13	NUM
ejpam-6528	129	3	)	)	PUNCT
ejpam-6528	129	4	let	let	VERB
ejpam-6528	129	5	the	the	DET
ejpam-6528	129	6	sequence	sequence	NOUN
ejpam-6528	129	7	υn	υn	NOUN
ejpam-6528	129	8	=	=	SYM
ejpam-6528	129	9	1	1	NUM
ejpam-6528	129	10	n	n	NOUN
ejpam-6528	129	11	for	for	ADP
ejpam-6528	129	12	n	n	PRON
ejpam-6528	129	13	∈	∈	PROPN
ejpam-6528	129	14	n.	n.	NOUN
ejpam-6528	129	15	we	we	PRON
ejpam-6528	129	16	verify	verify	VERB
ejpam-6528	129	17	that	that	SCONJ
ejpam-6528	129	18	:	:	PUNCT
ejpam-6528	129	19	m(υn	m(υn	ADJ
ejpam-6528	129	20	,	,	PUNCT
ejpam-6528	129	21	υn−1	υn−1	ADJ
ejpam-6528	129	22	,	,	PUNCT
ejpam-6528	129	23	υn−2	υn−2	PROPN
ejpam-6528	129	24	)	)	PUNCT
ejpam-6528	129	25	=	=	PUNCT
ejpam-6528	130	1	∣∣∣∣	∣∣∣∣	NOUN
ejpam-6528	130	2	1n	1n	NUM
ejpam-6528	131	1	−	−	NOUN
ejpam-6528	131	2	1	1	NUM
ejpam-6528	131	3	n−	n−	NOUN
ejpam-6528	131	4	1	1	NUM
ejpam-6528	131	5	∣∣∣∣+	∣∣∣∣+	NOUN
ejpam-6528	131	6	∣∣∣∣	∣∣∣∣	NOUN
ejpam-6528	131	7	1	1	NUM
ejpam-6528	131	8	n−	n−	NOUN
ejpam-6528	131	9	1	1	NUM
ejpam-6528	131	10	−	−	NUM
ejpam-6528	131	11	1	1	NUM
ejpam-6528	131	12	n−	n−	NOUN
ejpam-6528	131	13	2	2	NUM
ejpam-6528	131	14	∣∣∣∣+	∣∣∣∣+	NOUN
ejpam-6528	131	15	∣∣∣∣	∣∣∣∣	NOUN
ejpam-6528	131	16	1	1	NUM
ejpam-6528	131	17	n−	n−	NOUN
ejpam-6528	131	18	2	2	NUM
ejpam-6528	131	19	−	−	NOUN
ejpam-6528	131	20	1	1	NUM
ejpam-6528	131	21	n	n	PRON
ejpam-6528	131	22	∣∣∣∣	∣∣∣∣	NOUN
ejpam-6528	131	23	.	.	PUNCT
ejpam-6528	132	1	as	as	ADP
ejpam-6528	132	2	n	n	PROPN
ejpam-6528	132	3	→	→	SYM
ejpam-6528	132	4	∞	∞	PROPN
ejpam-6528	132	5	,	,	PUNCT
ejpam-6528	132	6	each	each	DET
ejpam-6528	132	7	term	term	NOUN
ejpam-6528	132	8	tends	tend	VERB
ejpam-6528	132	9	to	to	ADP
ejpam-6528	132	10	zero	zero	NUM
ejpam-6528	132	11	,	,	PUNCT
ejpam-6528	132	12	so	so	ADV
ejpam-6528	132	13	m(υn	m(υn	ADJ
ejpam-6528	132	14	,	,	PUNCT
ejpam-6528	132	15	υn−1	υn−1	ADJ
ejpam-6528	132	16	,	,	PUNCT
ejpam-6528	132	17	υn−2	υn−2	PROPN
ejpam-6528	132	18	)	)	PUNCT
ejpam-6528	132	19	→	→	SYM
ejpam-6528	132	20	0	0	NUM
ejpam-6528	132	21	,	,	PUNCT
ejpam-6528	132	22	satisfying	satisfy	VERB
ejpam-6528	132	23	the	the	DET
ejpam-6528	132	24	hypothesis	hypothesis	NOUN
ejpam-6528	132	25	of	of	ADP
ejpam-6528	132	26	the	the	DET
ejpam-6528	132	27	theorem	theorem	NOUN
ejpam-6528	132	28	.	.	PUNCT
ejpam-6528	133	1	the	the	DET
ejpam-6528	133	2	limit	limit	NOUN
ejpam-6528	133	3	of	of	ADP
ejpam-6528	133	4	υn	υn	NOUN
ejpam-6528	133	5	is	be	AUX
ejpam-6528	133	6	clearly	clearly	ADV
ejpam-6528	133	7	υ∗	υ∗	NOUN
ejpam-6528	133	8	=	=	SYM
ejpam-6528	133	9	0	0	NUM
ejpam-6528	133	10	,	,	PUNCT
ejpam-6528	133	11	and	and	CCONJ
ejpam-6528	133	12	we	we	PRON
ejpam-6528	133	13	check	check	VERB
ejpam-6528	133	14	convergence	convergence	NOUN
ejpam-6528	133	15	in	in	ADP
ejpam-6528	133	16	measure	measure	NOUN
ejpam-6528	133	17	.	.	PUNCT
ejpam-6528	134	1	for	for	ADP
ejpam-6528	134	2	any	any	PRON
ejpam-6528	134	3	ϵ	ϵ	PROPN
ejpam-6528	134	4	>	>	X
ejpam-6528	134	5	0	0	NUM
ejpam-6528	134	6	,	,	PUNCT
ejpam-6528	134	7	define	define	VERB
ejpam-6528	134	8	an	an	DET
ejpam-6528	134	9	=	=	PUNCT
ejpam-6528	134	10	{	{	PUNCT
ejpam-6528	134	11	υ	υ	NOUN
ejpam-6528	134	12	∈	∈	PROPN
ejpam-6528	134	13	r	r	NOUN
ejpam-6528	134	14	:	:	PUNCT
ejpam-6528	134	15	m(υn	m(υn	ADJ
ejpam-6528	134	16	,	,	PUNCT
ejpam-6528	134	17	0	0	NUM
ejpam-6528	134	18	,	,	PUNCT
ejpam-6528	134	19	0	0	NUM
ejpam-6528	134	20	)	)	PUNCT
ejpam-6528	134	21	≥	≥	NOUN
ejpam-6528	134	22	ϵ	ϵ	NOUN
ejpam-6528	134	23	}	}	PUNCT
ejpam-6528	134	24	.	.	PUNCT
ejpam-6528	135	1	for	for	ADP
ejpam-6528	135	2	large	large	ADJ
ejpam-6528	135	3	enough	enough	ADJ
ejpam-6528	135	4	n	n	CCONJ
ejpam-6528	135	5	,	,	PUNCT
ejpam-6528	135	6	m(υn	m(υn	X
ejpam-6528	135	7	,	,	PUNCT
ejpam-6528	135	8	0	0	NUM
ejpam-6528	135	9	,	,	PUNCT
ejpam-6528	135	10	0	0	NUM
ejpam-6528	135	11	)	)	PUNCT
ejpam-6528	135	12	=	=	SYM
ejpam-6528	135	13	3|υn|	3|υn|	PROPN
ejpam-6528	135	14	<	<	X
ejpam-6528	135	15	ϵ	ϵ	X
ejpam-6528	135	16	,	,	PUNCT
ejpam-6528	135	17	so	so	SCONJ
ejpam-6528	135	18	an	an	DET
ejpam-6528	135	19	eventually	eventually	ADV
ejpam-6528	135	20	becomes	become	VERB
ejpam-6528	135	21	empty	empty	ADJ
ejpam-6528	135	22	,	,	PUNCT
ejpam-6528	135	23	implying	imply	VERB
ejpam-6528	135	24	µ(an	µ(an	PROPN
ejpam-6528	135	25	)	)	PUNCT
ejpam-6528	135	26	→	→	SYM
ejpam-6528	135	27	0	0	NUM
ejpam-6528	135	28	.	.	PUNCT
ejpam-6528	136	1	thus	thus	ADV
ejpam-6528	136	2	,	,	PUNCT
ejpam-6528	136	3	υn	υn	X
ejpam-6528	136	4	→	→	SYM
ejpam-6528	136	5	0	0	NUM
ejpam-6528	136	6	in	in	ADP
ejpam-6528	136	7	measure	measure	NOUN
ejpam-6528	136	8	.	.	PUNCT
ejpam-6528	137	1	a.	a.	NOUN
ejpam-6528	137	2	malkawi	malkawi	PROPN
ejpam-6528	137	3	,	,	PUNCT
ejpam-6528	137	4	a.	a.	PROPN
ejpam-6528	137	5	rabaiah	rabaiah	PROPN
ejpam-6528	137	6	/	/	SYM
ejpam-6528	137	7	eur	eur	PROPN
ejpam-6528	137	8	.	.	PUNCT
ejpam-6528	138	1	j.	j.	PROPN
ejpam-6528	138	2	pure	pure	PROPN
ejpam-6528	138	3	appl	appl	PROPN
ejpam-6528	138	4	.	.	PROPN
ejpam-6528	138	5	math	math	PROPN
ejpam-6528	138	6	,	,	PUNCT
ejpam-6528	138	7	18	18	NUM
ejpam-6528	138	8	(	(	PUNCT
ejpam-6528	138	9	3	3	NUM
ejpam-6528	138	10	)	)	PUNCT
ejpam-6528	138	11	(	(	PUNCT
ejpam-6528	138	12	2025	2025	NUM
ejpam-6528	138	13	)	)	PUNCT
ejpam-6528	138	14	,	,	PUNCT
ejpam-6528	138	15	6528	6528	NUM
ejpam-6528	138	16	7	7	NUM
ejpam-6528	138	17	of	of	ADP
ejpam-6528	138	18	12	12	NUM
ejpam-6528	138	19	3	3	NUM
ejpam-6528	138	20	.	.	PUNCT
ejpam-6528	139	1	applications	application	NOUN
ejpam-6528	139	2	of	of	ADP
ejpam-6528	139	3	mr	mr	PROPN
ejpam-6528	139	4	-	-	PUNCT
ejpam-6528	139	5	metric	metric	ADJ
ejpam-6528	139	6	space	space	NOUN
ejpam-6528	139	7	theorems	theorem	VERB
ejpam-6528	139	8	3.1	3.1	NUM
ejpam-6528	139	9	.	.	PUNCT
ejpam-6528	140	1	applications	application	NOUN
ejpam-6528	140	2	of	of	ADP
ejpam-6528	140	3	the	the	DET
ejpam-6528	140	4	first	first	ADJ
ejpam-6528	140	5	theorem	theorem	NOUN
ejpam-6528	140	6	the	the	DET
ejpam-6528	140	7	first	first	ADJ
ejpam-6528	140	8	theorem	theorem	ADJ
ejpam-6528	140	9	concerns	concern	NOUN
ejpam-6528	140	10	measure	measure	NOUN
ejpam-6528	140	11	theory	theory	NOUN
ejpam-6528	140	12	in	in	ADP
ejpam-6528	140	13	mr	mr	PROPN
ejpam-6528	140	14	-	-	PUNCT
ejpam-6528	140	15	metric	metric	ADJ
ejpam-6528	140	16	spaces	space	NOUN
ejpam-6528	140	17	,	,	PUNCT
ejpam-6528	140	18	establishing	establish	VERB
ejpam-6528	140	19	that	that	SCONJ
ejpam-6528	140	20	any	any	DET
ejpam-6528	140	21	measure	measure	NOUN
ejpam-6528	140	22	satisfying	satisfy	VERB
ejpam-6528	140	23	the	the	DET
ejpam-6528	140	24	given	give	VERB
ejpam-6528	140	25	integral	integral	ADJ
ejpam-6528	140	26	condition	condition	NOUN
ejpam-6528	140	27	is	be	AUX
ejpam-6528	140	28	σ	σ	NOUN
ejpam-6528	140	29	-	-	NOUN
ejpam-6528	140	30	finite	finite	ADJ
ejpam-6528	140	31	and	and	CCONJ
ejpam-6528	140	32	absolutely	absolutely	ADV
ejpam-6528	140	33	continuous	continuous	ADJ
ejpam-6528	140	34	,	,	PUNCT
ejpam-6528	140	35	provided	provide	VERB
ejpam-6528	140	36	that	that	SCONJ
ejpam-6528	140	37	m	m	NOUN
ejpam-6528	140	38	satisfies	satisfie	NOUN
ejpam-6528	140	39	(	(	PUNCT
ejpam-6528	140	40	m4	m4	PROPN
ejpam-6528	140	41	)	)	PUNCT
ejpam-6528	140	42	with	with	ADP
ejpam-6528	140	43	a	a	DET
ejpam-6528	140	44	finite	finite	ADJ
ejpam-6528	140	45	measure	measure	NOUN
ejpam-6528	140	46	space	space	NOUN
ejpam-6528	140	47	.	.	PUNCT
ejpam-6528	141	1	possible	possible	ADJ
ejpam-6528	141	2	applications	application	NOUN
ejpam-6528	141	3	include	include	VERB
ejpam-6528	141	4	:	:	PUNCT
ejpam-6528	141	5	3.1.1	3.1.1	X
ejpam-6528	141	6	.	.	PUNCT
ejpam-6528	141	7	measure	measure	NOUN
ejpam-6528	141	8	theory	theory	NOUN
ejpam-6528	141	9	and	and	CCONJ
ejpam-6528	141	10	integration	integration	NOUN
ejpam-6528	141	11	the	the	DET
ejpam-6528	141	12	given	give	VERB
ejpam-6528	141	13	theorem	theorem	NOUN
ejpam-6528	141	14	plays	play	VERB
ejpam-6528	141	15	a	a	DET
ejpam-6528	141	16	crucial	crucial	ADJ
ejpam-6528	141	17	role	role	NOUN
ejpam-6528	141	18	in	in	ADP
ejpam-6528	141	19	measure	measure	NOUN
ejpam-6528	141	20	theory	theory	NOUN
ejpam-6528	141	21	and	and	CCONJ
ejpam-6528	141	22	integration	integration	NOUN
ejpam-6528	141	23	in	in	ADP
ejpam-6528	141	24	mr	mr	PROPN
ejpam-6528	141	25	-	-	PUNCT
ejpam-6528	141	26	metric	metric	ADJ
ejpam-6528	141	27	spaces	space	NOUN
ejpam-6528	141	28	,	,	PUNCT
ejpam-6528	141	29	where	where	SCONJ
ejpam-6528	141	30	distances	distance	NOUN
ejpam-6528	141	31	between	between	ADP
ejpam-6528	141	32	points	point	NOUN
ejpam-6528	141	33	are	be	AUX
ejpam-6528	141	34	determined	determine	VERB
ejpam-6528	141	35	by	by	ADP
ejpam-6528	141	36	a	a	DET
ejpam-6528	141	37	triple	triple	ADJ
ejpam-6528	141	38	metric	metric	ADJ
ejpam-6528	141	39	function	function	NOUN
ejpam-6528	141	40	m	m	VERB
ejpam-6528	141	41	instead	instead	ADV
ejpam-6528	141	42	of	of	ADP
ejpam-6528	141	43	the	the	DET
ejpam-6528	141	44	conventional	conventional	ADJ
ejpam-6528	141	45	pairwise	pairwise	NOUN
ejpam-6528	141	46	metric	metric	NOUN
ejpam-6528	141	47	.	.	PUNCT
ejpam-6528	142	1	below	below	ADV
ejpam-6528	142	2	,	,	PUNCT
ejpam-6528	142	3	we	we	PRON
ejpam-6528	142	4	discuss	discuss	VERB
ejpam-6528	142	5	its	its	PRON
ejpam-6528	142	6	implications	implication	NOUN
ejpam-6528	142	7	in	in	ADP
ejpam-6528	142	8	absolute	absolute	ADJ
ejpam-6528	142	9	continuity	continuity	NOUN
ejpam-6528	142	10	,	,	PUNCT
ejpam-6528	142	11	finiteness	finiteness	NOUN
ejpam-6528	142	12	of	of	ADP
ejpam-6528	142	13	measures	measure	NOUN
ejpam-6528	142	14	,	,	PUNCT
ejpam-6528	142	15	and	and	CCONJ
ejpam-6528	142	16	probability	probability	NOUN
ejpam-6528	142	17	density	density	NOUN
ejpam-6528	142	18	functions	function	NOUN
ejpam-6528	142	19	(	(	PUNCT
ejpam-6528	142	20	pdfs	pdfs	PROPN
ejpam-6528	142	21	)	)	PUNCT
ejpam-6528	142	22	.	.	PUNCT
ejpam-6528	143	1	•	•	NUM
ejpam-6528	143	2	preservation	preservation	NOUN
ejpam-6528	143	3	of	of	ADP
ejpam-6528	143	4	absolute	absolute	ADJ
ejpam-6528	143	5	continuity	continuity	NOUN
ejpam-6528	143	6	and	and	CCONJ
ejpam-6528	143	7	finiteness	finiteness	NOUN
ejpam-6528	143	8	of	of	ADP
ejpam-6528	143	9	measures	measure	NOUN
ejpam-6528	143	10	:	:	PUNCT
ejpam-6528	143	11	–	–	PUNCT
ejpam-6528	143	12	in	in	ADP
ejpam-6528	143	13	classical	classical	ADJ
ejpam-6528	143	14	measure	measure	NOUN
ejpam-6528	143	15	theory	theory	NOUN
ejpam-6528	143	16	,	,	PUNCT
ejpam-6528	143	17	a	a	DET
ejpam-6528	143	18	measure	measure	NOUN
ejpam-6528	143	19	µ	µ	NOUN
ejpam-6528	143	20	is	be	AUX
ejpam-6528	143	21	absolutely	absolutely	ADV
ejpam-6528	143	22	continuous	continuous	ADJ
ejpam-6528	143	23	with	with	ADP
ejpam-6528	143	24	respect	respect	NOUN
ejpam-6528	143	25	to	to	ADP
ejpam-6528	143	26	another	another	DET
ejpam-6528	143	27	measure	measure	NOUN
ejpam-6528	143	28	ν	ν	NOUN
ejpam-6528	143	29	if	if	SCONJ
ejpam-6528	143	30	for	for	ADP
ejpam-6528	143	31	every	every	DET
ejpam-6528	143	32	measurable	measurable	NOUN
ejpam-6528	143	33	set	set	VERB
ejpam-6528	143	34	a	a	PRON
ejpam-6528	143	35	,	,	PUNCT
ejpam-6528	143	36	ν(a	ν(a	PROPN
ejpam-6528	143	37	)	)	PUNCT
ejpam-6528	143	38	=	=	SYM
ejpam-6528	143	39	0	0	NUM
ejpam-6528	143	40	implies	imply	VERB
ejpam-6528	143	41	µ(a	µ(a	PROPN
ejpam-6528	143	42	)	)	PUNCT
ejpam-6528	143	43	=	=	SYM
ejpam-6528	144	1	0	0	X
ejpam-6528	144	2	.	.	X
ejpam-6528	144	3	–	–	PUNCT
ejpam-6528	144	4	the	the	DET
ejpam-6528	144	5	theorem	theorem	NOUN
ejpam-6528	144	6	ensures	ensure	VERB
ejpam-6528	144	7	that	that	SCONJ
ejpam-6528	144	8	measures	measure	NOUN
ejpam-6528	144	9	defined	define	VERB
ejpam-6528	144	10	using	use	VERB
ejpam-6528	144	11	m(υ	m(υ	PROPN
ejpam-6528	144	12	,	,	PUNCT
ejpam-6528	144	13	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	144	14	)	)	PUNCT
ejpam-6528	144	15	satisfy	satisfy	VERB
ejpam-6528	144	16	absolute	absolute	ADJ
ejpam-6528	144	17	continuity	continuity	NOUN
ejpam-6528	144	18	,	,	PUNCT
ejpam-6528	144	19	meaning	mean	VERB
ejpam-6528	144	20	that	that	SCONJ
ejpam-6528	144	21	regions	region	NOUN
ejpam-6528	144	22	of	of	ADP
ejpam-6528	144	23	measure	measure	NOUN
ejpam-6528	144	24	zero	zero	NUM
ejpam-6528	144	25	remain	remain	VERB
ejpam-6528	144	26	negligible	negligible	ADJ
ejpam-6528	144	27	under	under	ADP
ejpam-6528	144	28	the	the	DET
ejpam-6528	144	29	transformation	transformation	NOUN
ejpam-6528	144	30	imposed	impose	VERB
ejpam-6528	144	31	by	by	ADP
ejpam-6528	144	32	m	m	PROPN
ejpam-6528	144	33	.	.	PUNCT
ejpam-6528	145	1	–	–	PUNCT
ejpam-6528	145	2	additionally	additionally	ADV
ejpam-6528	145	3	,	,	PUNCT
ejpam-6528	145	4	the	the	DET
ejpam-6528	145	5	theorem	theorem	NOUN
ejpam-6528	145	6	establishes	establish	VERB
ejpam-6528	145	7	that	that	SCONJ
ejpam-6528	145	8	µ	µ	NOUN
ejpam-6528	145	9	is	be	AUX
ejpam-6528	145	10	σ	σ	NOUN
ejpam-6528	145	11	-	-	NOUN
ejpam-6528	145	12	finite	finite	NOUN
ejpam-6528	145	13	,	,	PUNCT
ejpam-6528	145	14	guaranteeing	guarantee	VERB
ejpam-6528	145	15	that	that	DET
ejpam-6528	145	16	space	space	NOUN
ejpam-6528	145	17	x	x	PUNCT
ejpam-6528	145	18	can	can	AUX
ejpam-6528	145	19	be	be	AUX
ejpam-6528	145	20	decomposed	decompose	VERB
ejpam-6528	145	21	into	into	ADP
ejpam-6528	145	22	countable	countable	ADJ
ejpam-6528	145	23	measurable	measurable	ADJ
ejpam-6528	145	24	subsets	subset	NOUN
ejpam-6528	145	25	of	of	ADP
ejpam-6528	145	26	finite	finite	ADJ
ejpam-6528	145	27	measure	measure	NOUN
ejpam-6528	145	28	,	,	PUNCT
ejpam-6528	145	29	making	make	VERB
ejpam-6528	145	30	integration	integration	NOUN
ejpam-6528	145	31	well	well	ADV
ejpam-6528	145	32	-	-	PUNCT
ejpam-6528	145	33	defined	define	VERB
ejpam-6528	145	34	.	.	PUNCT
ejpam-6528	146	1	•	•	NUM
ejpam-6528	146	2	existence	existence	NOUN
ejpam-6528	146	3	of	of	ADP
ejpam-6528	146	4	a	a	DET
ejpam-6528	146	5	probability	probability	NOUN
ejpam-6528	146	6	density	density	NOUN
ejpam-6528	146	7	function	function	NOUN
ejpam-6528	146	8	(	(	PUNCT
ejpam-6528	146	9	pdf	pdf	NOUN
ejpam-6528	146	10	)	)	PUNCT
ejpam-6528	146	11	in	in	ADP
ejpam-6528	146	12	mr	mr	PROPN
ejpam-6528	146	13	-	-	PUNCT
ejpam-6528	146	14	metric	metric	ADJ
ejpam-6528	146	15	spaces	space	NOUN
ejpam-6528	146	16	:	:	PUNCT
ejpam-6528	146	17	–	–	PUNCT
ejpam-6528	146	18	in	in	ADP
ejpam-6528	146	19	probability	probability	NOUN
ejpam-6528	146	20	theory	theory	NOUN
ejpam-6528	146	21	,	,	PUNCT
ejpam-6528	146	22	an	an	DET
ejpam-6528	146	23	absolutely	absolutely	ADV
ejpam-6528	146	24	continuous	continuous	ADJ
ejpam-6528	146	25	probability	probability	NOUN
ejpam-6528	146	26	measure	measure	NOUN
ejpam-6528	146	27	admits	admit	VERB
ejpam-6528	146	28	a	a	DET
ejpam-6528	146	29	density	density	NOUN
ejpam-6528	146	30	function	function	NOUN
ejpam-6528	146	31	f(υ	f(υ	PROPN
ejpam-6528	146	32	)	)	PUNCT
ejpam-6528	146	33	such	such	ADJ
ejpam-6528	146	34	that	that	SCONJ
ejpam-6528	146	35	:	:	PUNCT
ejpam-6528	146	36	µ(a	µ(a	PROPN
ejpam-6528	146	37	)	)	PUNCT
ejpam-6528	146	38	=	=	PRON
ejpam-6528	147	1	∫	∫	PROPN
ejpam-6528	147	2	a	a	DET
ejpam-6528	147	3	f(υ	f(υ	PROPN
ejpam-6528	147	4	)	)	PUNCT
ejpam-6528	147	5	dυ	dυ	NOUN
ejpam-6528	147	6	.	.	PUNCT
ejpam-6528	147	7	–	–	PUNCT
ejpam-6528	147	8	the	the	DET
ejpam-6528	147	9	theorem	theorem	NOUN
ejpam-6528	147	10	ensures	ensure	VERB
ejpam-6528	147	11	the	the	DET
ejpam-6528	147	12	existence	existence	NOUN
ejpam-6528	147	13	of	of	ADP
ejpam-6528	147	14	such	such	DET
ejpam-6528	147	15	a	a	DET
ejpam-6528	147	16	function	function	NOUN
ejpam-6528	147	17	when	when	SCONJ
ejpam-6528	147	18	the	the	DET
ejpam-6528	147	19	probability	probability	NOUN
ejpam-6528	147	20	distribution	distribution	NOUN
ejpam-6528	147	21	depends	depend	VERB
ejpam-6528	147	22	on	on	ADP
ejpam-6528	147	23	triadic	triadic	ADJ
ejpam-6528	147	24	interactions	interaction	NOUN
ejpam-6528	147	25	rather	rather	ADV
ejpam-6528	147	26	than	than	ADP
ejpam-6528	147	27	pairwise	pairwise	NOUN
ejpam-6528	147	28	distances	distance	NOUN
ejpam-6528	147	29	.	.	PUNCT
ejpam-6528	148	1	–	–	PUNCT
ejpam-6528	148	2	for	for	ADP
ejpam-6528	148	3	example	example	NOUN
ejpam-6528	148	4	,	,	PUNCT
ejpam-6528	148	5	consider	consider	VERB
ejpam-6528	148	6	a	a	DET
ejpam-6528	148	7	probability	probability	NOUN
ejpam-6528	148	8	distribution	distribution	NOUN
ejpam-6528	148	9	over	over	ADP
ejpam-6528	148	10	a	a	DET
ejpam-6528	148	11	space	space	NOUN
ejpam-6528	148	12	x	x	PUNCT
ejpam-6528	148	13	where	where	SCONJ
ejpam-6528	148	14	the	the	DET
ejpam-6528	148	15	likelihood	likelihood	NOUN
ejpam-6528	148	16	of	of	ADP
ejpam-6528	148	17	a	a	DET
ejpam-6528	148	18	point	point	NOUN
ejpam-6528	148	19	υ	υ	NOUN
ejpam-6528	148	20	is	be	AUX
ejpam-6528	148	21	influenced	influence	VERB
ejpam-6528	148	22	by	by	ADP
ejpam-6528	148	23	two	two	NUM
ejpam-6528	148	24	additional	additional	ADJ
ejpam-6528	148	25	reference	reference	NOUN
ejpam-6528	148	26	points	point	VERB
ejpam-6528	148	27	ξ	ξ	PROPN
ejpam-6528	148	28	and	and	CCONJ
ejpam-6528	148	29	ℑ.	ℑ.	PROPN
ejpam-6528	148	30	the	the	DET
ejpam-6528	148	31	measure	measure	NOUN
ejpam-6528	148	32	may	may	AUX
ejpam-6528	148	33	be	be	AUX
ejpam-6528	148	34	defined	define	VERB
ejpam-6528	148	35	as	as	ADP
ejpam-6528	148	36	:	:	PUNCT
ejpam-6528	148	37	µ(a	µ(a	PROPN
ejpam-6528	148	38	)	)	PUNCT
ejpam-6528	149	1	=	=	SYM
ejpam-6528	149	2	∫	∫	PROPN
ejpam-6528	149	3	a	a	DET
ejpam-6528	149	4	m(υ	m(υ	PROPN
ejpam-6528	149	5	,	,	PUNCT
ejpam-6528	149	6	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	149	7	)	)	PUNCT
ejpam-6528	149	8	dµ(υ	dµ(υ	NUM
ejpam-6528	149	9	)	)	PUNCT
ejpam-6528	149	10	dµ(ξ	dµ(ξ	NUM
ejpam-6528	149	11	)	)	PUNCT
ejpam-6528	150	1	dµ(ℑ	dµ(ℑ	ADV
ejpam-6528	150	2	)	)	PUNCT
ejpam-6528	150	3	,	,	PUNCT
ejpam-6528	150	4	where	where	SCONJ
ejpam-6528	150	5	m(υ	m(υ	PROPN
ejpam-6528	150	6	,	,	PUNCT
ejpam-6528	150	7	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	150	8	)	)	PUNCT
ejpam-6528	150	9	models	model	NOUN
ejpam-6528	150	10	the	the	DET
ejpam-6528	150	11	joint	joint	ADJ
ejpam-6528	150	12	influence	influence	NOUN
ejpam-6528	150	13	of	of	ADP
ejpam-6528	150	14	three	three	NUM
ejpam-6528	150	15	points	point	NOUN
ejpam-6528	150	16	rather	rather	ADV
ejpam-6528	150	17	than	than	ADP
ejpam-6528	150	18	traditional	traditional	ADJ
ejpam-6528	150	19	pairwise	pairwise	NOUN
ejpam-6528	150	20	interactions	interaction	NOUN
ejpam-6528	150	21	.	.	PUNCT
ejpam-6528	151	1	a.	a.	NOUN
ejpam-6528	151	2	malkawi	malkawi	PROPN
ejpam-6528	151	3	,	,	PUNCT
ejpam-6528	151	4	a.	a.	PROPN
ejpam-6528	151	5	rabaiah	rabaiah	PROPN
ejpam-6528	151	6	/	/	SYM
ejpam-6528	151	7	eur	eur	PROPN
ejpam-6528	151	8	.	.	PUNCT
ejpam-6528	152	1	j.	j.	PROPN
ejpam-6528	152	2	pure	pure	PROPN
ejpam-6528	152	3	appl	appl	PROPN
ejpam-6528	152	4	.	.	PROPN
ejpam-6528	152	5	math	math	PROPN
ejpam-6528	152	6	,	,	PUNCT
ejpam-6528	152	7	18	18	NUM
ejpam-6528	152	8	(	(	PUNCT
ejpam-6528	152	9	3	3	NUM
ejpam-6528	152	10	)	)	PUNCT
ejpam-6528	152	11	(	(	PUNCT
ejpam-6528	152	12	2025	2025	NUM
ejpam-6528	152	13	)	)	PUNCT
ejpam-6528	152	14	,	,	PUNCT
ejpam-6528	152	15	6528	6528	NUM
ejpam-6528	152	16	8	8	NUM
ejpam-6528	152	17	of	of	ADP
ejpam-6528	152	18	12	12	NUM
ejpam-6528	152	19	–	–	PUNCT
ejpam-6528	152	20	a	a	DET
ejpam-6528	152	21	practical	practical	ADJ
ejpam-6528	152	22	application	application	NOUN
ejpam-6528	152	23	arises	arise	VERB
ejpam-6528	152	24	in	in	ADP
ejpam-6528	152	25	spatial	spatial	ADJ
ejpam-6528	152	26	statistics	statistic	NOUN
ejpam-6528	152	27	,	,	PUNCT
ejpam-6528	152	28	where	where	SCONJ
ejpam-6528	152	29	the	the	DET
ejpam-6528	152	30	probability	probability	NOUN
ejpam-6528	152	31	of	of	ADP
ejpam-6528	152	32	an	an	DET
ejpam-6528	152	33	event	event	NOUN
ejpam-6528	152	34	depends	depend	VERB
ejpam-6528	152	35	on	on	ADP
ejpam-6528	152	36	the	the	DET
ejpam-6528	152	37	location	location	NOUN
ejpam-6528	152	38	of	of	ADP
ejpam-6528	152	39	three	three	NUM
ejpam-6528	152	40	interacting	interact	VERB
ejpam-6528	152	41	sites	site	NOUN
ejpam-6528	152	42	(	(	PUNCT
ejpam-6528	152	43	e.g.	e.g.	ADV
ejpam-6528	152	44	,	,	PUNCT
ejpam-6528	152	45	modeling	model	VERB
ejpam-6528	152	46	geological	geological	ADJ
ejpam-6528	152	47	formations	formation	NOUN
ejpam-6528	152	48	,	,	PUNCT
ejpam-6528	152	49	epidemic	epidemic	NOUN
ejpam-6528	152	50	spread	spread	NOUN
ejpam-6528	152	51	,	,	PUNCT
ejpam-6528	152	52	or	or	CCONJ
ejpam-6528	152	53	network	network	NOUN
ejpam-6528	152	54	communications	communication	NOUN
ejpam-6528	152	55	)	)	PUNCT
ejpam-6528	152	56	.	.	PUNCT
ejpam-6528	153	1	–	–	PUNCT
ejpam-6528	153	2	another	another	DET
ejpam-6528	153	3	example	example	NOUN
ejpam-6528	153	4	is	be	AUX
ejpam-6528	153	5	in	in	ADP
ejpam-6528	153	6	machine	machine	NOUN
ejpam-6528	153	7	learning	learning	NOUN
ejpam-6528	153	8	,	,	PUNCT
ejpam-6528	153	9	where	where	SCONJ
ejpam-6528	153	10	triplet	triplet	NOUN
ejpam-6528	153	11	-	-	PUNCT
ejpam-6528	153	12	based	base	VERB
ejpam-6528	153	13	distance	distance	NOUN
ejpam-6528	153	14	functions	function	NOUN
ejpam-6528	153	15	(	(	PUNCT
ejpam-6528	153	16	such	such	ADJ
ejpam-6528	153	17	as	as	ADP
ejpam-6528	153	18	in	in	ADP
ejpam-6528	153	19	triplet	triplet	NOUN
ejpam-6528	153	20	loss	loss	NOUN
ejpam-6528	153	21	functions	function	NOUN
ejpam-6528	153	22	)	)	PUNCT
ejpam-6528	153	23	are	be	AUX
ejpam-6528	153	24	used	use	VERB
ejpam-6528	153	25	to	to	PART
ejpam-6528	153	26	define	define	VERB
ejpam-6528	153	27	similarity	similarity	NOUN
ejpam-6528	153	28	measures	measure	NOUN
ejpam-6528	153	29	based	base	VERB
ejpam-6528	153	30	on	on	ADP
ejpam-6528	153	31	the	the	DET
ejpam-6528	153	32	interaction	interaction	NOUN
ejpam-6528	153	33	of	of	ADP
ejpam-6528	153	34	three	three	NUM
ejpam-6528	153	35	feature	feature	NOUN
ejpam-6528	153	36	vectors	vector	NOUN
ejpam-6528	153	37	.	.	PUNCT
ejpam-6528	154	1	3.1.2	3.1.2	X
ejpam-6528	154	2	.	.	PUNCT
ejpam-6528	155	1	applications	application	NOUN
ejpam-6528	155	2	in	in	ADP
ejpam-6528	155	3	dynamical	dynamical	ADJ
ejpam-6528	155	4	systems	system	NOUN
ejpam-6528	155	5	and	and	CCONJ
ejpam-6528	155	6	ergodic	ergodic	ADJ
ejpam-6528	155	7	theory	theory	NOUN
ejpam-6528	155	8	dynamical	dynamical	ADJ
ejpam-6528	155	9	systems	system	NOUN
ejpam-6528	155	10	and	and	CCONJ
ejpam-6528	155	11	ergodic	ergodic	ADJ
ejpam-6528	155	12	theory	theory	NOUN
ejpam-6528	155	13	study	study	VERB
ejpam-6528	155	14	the	the	DET
ejpam-6528	155	15	long	long	ADJ
ejpam-6528	155	16	-	-	PUNCT
ejpam-6528	155	17	term	term	NOUN
ejpam-6528	155	18	behavior	behavior	NOUN
ejpam-6528	155	19	of	of	ADP
ejpam-6528	155	20	systems	system	NOUN
ejpam-6528	155	21	evolving	evolve	VERB
ejpam-6528	155	22	over	over	ADP
ejpam-6528	155	23	time	time	NOUN
ejpam-6528	155	24	,	,	PUNCT
ejpam-6528	155	25	particularly	particularly	ADV
ejpam-6528	155	26	those	those	PRON
ejpam-6528	155	27	governed	govern	VERB
ejpam-6528	155	28	by	by	ADP
ejpam-6528	155	29	measure	measure	NOUN
ejpam-6528	155	30	-	-	PUNCT
ejpam-6528	155	31	preserving	preserve	VERB
ejpam-6528	155	32	transformations	transformation	NOUN
ejpam-6528	155	33	.	.	PUNCT
ejpam-6528	156	1	in	in	ADP
ejpam-6528	156	2	traditional	traditional	ADJ
ejpam-6528	156	3	ergodic	ergodic	ADJ
ejpam-6528	156	4	theory	theory	NOUN
ejpam-6528	156	5	,	,	PUNCT
ejpam-6528	156	6	measures	measure	NOUN
ejpam-6528	156	7	are	be	AUX
ejpam-6528	156	8	often	often	ADV
ejpam-6528	156	9	defined	define	VERB
ejpam-6528	156	10	in	in	ADP
ejpam-6528	156	11	terms	term	NOUN
ejpam-6528	156	12	of	of	ADP
ejpam-6528	156	13	pairwise	pairwise	NOUN
ejpam-6528	156	14	dependencies	dependency	NOUN
ejpam-6528	156	15	.	.	PUNCT
ejpam-6528	157	1	however	however	ADV
ejpam-6528	157	2	,	,	PUNCT
ejpam-6528	157	3	in	in	ADP
ejpam-6528	157	4	systems	system	NOUN
ejpam-6528	157	5	where	where	SCONJ
ejpam-6528	157	6	interactions	interaction	NOUN
ejpam-6528	157	7	are	be	AUX
ejpam-6528	157	8	inherently	inherently	ADV
ejpam-6528	157	9	tripartite	tripartite	ADJ
ejpam-6528	157	10	,	,	PUNCT
ejpam-6528	157	11	the	the	DET
ejpam-6528	157	12	theorem	theorem	NOUN
ejpam-6528	157	13	ensures	ensure	VERB
ejpam-6528	157	14	that	that	SCONJ
ejpam-6528	157	15	any	any	DET
ejpam-6528	157	16	ergodic	ergodic	ADJ
ejpam-6528	157	17	measure	measure	NOUN
ejpam-6528	157	18	defined	define	VERB
ejpam-6528	157	19	via	via	ADP
ejpam-6528	157	20	the	the	DET
ejpam-6528	157	21	three	three	NUM
ejpam-6528	157	22	-	-	PUNCT
ejpam-6528	157	23	point	point	NOUN
ejpam-6528	157	24	metric	metric	ADJ
ejpam-6528	157	25	function	function	NOUN
ejpam-6528	157	26	m(υ	m(υ	PROPN
ejpam-6528	157	27	,	,	PUNCT
ejpam-6528	157	28	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	157	29	)	)	PUNCT
ejpam-6528	157	30	remains	remain	VERB
ejpam-6528	157	31	finite	finite	ADJ
ejpam-6528	157	32	and	and	CCONJ
ejpam-6528	157	33	absolutely	absolutely	ADV
ejpam-6528	157	34	continuous	continuous	ADJ
ejpam-6528	157	35	,	,	PUNCT
ejpam-6528	157	36	making	make	VERB
ejpam-6528	157	37	it	it	PRON
ejpam-6528	157	38	suitable	suitable	ADJ
ejpam-6528	157	39	for	for	ADP
ejpam-6528	157	40	rigorous	rigorous	ADJ
ejpam-6528	157	41	probabilistic	probabilistic	ADJ
ejpam-6528	157	42	analysis	analysis	NOUN
ejpam-6528	157	43	.	.	PUNCT
ejpam-6528	158	1	•	•	NOUN
ejpam-6528	158	2	ergodic	ergodic	ADJ
ejpam-6528	158	3	measures	measure	NOUN
ejpam-6528	158	4	and	and	CCONJ
ejpam-6528	158	5	higher	high	ADJ
ejpam-6528	158	6	-	-	PUNCT
ejpam-6528	158	7	order	order	NOUN
ejpam-6528	158	8	dependencies	dependency	NOUN
ejpam-6528	158	9	:	:	PUNCT
ejpam-6528	158	10	–	–	PUNCT
ejpam-6528	158	11	in	in	ADP
ejpam-6528	158	12	classical	classical	ADJ
ejpam-6528	158	13	ergodic	ergodic	ADJ
ejpam-6528	158	14	theory	theory	NOUN
ejpam-6528	158	15	,	,	PUNCT
ejpam-6528	158	16	a	a	DET
ejpam-6528	158	17	measure	measure	NOUN
ejpam-6528	158	18	µ	µ	NOUN
ejpam-6528	158	19	is	be	AUX
ejpam-6528	158	20	called	call	VERB
ejpam-6528	158	21	ergodic	ergodic	ADJ
ejpam-6528	158	22	if	if	SCONJ
ejpam-6528	158	23	every	every	DET
ejpam-6528	158	24	invariant	invariant	ADJ
ejpam-6528	158	25	set	set	NOUN
ejpam-6528	158	26	under	under	ADP
ejpam-6528	158	27	the	the	DET
ejpam-6528	158	28	transformation	transformation	NOUN
ejpam-6528	158	29	t	t	PROPN
ejpam-6528	158	30	is	be	AUX
ejpam-6528	158	31	either	either	PRON
ejpam-6528	158	32	of	of	ADP
ejpam-6528	158	33	full	full	ADJ
ejpam-6528	158	34	or	or	CCONJ
ejpam-6528	158	35	zero	zero	NUM
ejpam-6528	158	36	measure	measure	NOUN
ejpam-6528	158	37	.	.	PUNCT
ejpam-6528	159	1	–	–	PUNCT
ejpam-6528	159	2	the	the	DET
ejpam-6528	159	3	theorem	theorem	NOUN
ejpam-6528	159	4	ensures	ensure	VERB
ejpam-6528	159	5	that	that	SCONJ
ejpam-6528	159	6	when	when	SCONJ
ejpam-6528	159	7	a	a	DET
ejpam-6528	159	8	measure	measure	NOUN
ejpam-6528	159	9	is	be	AUX
ejpam-6528	159	10	defined	define	VERB
ejpam-6528	159	11	through	through	ADP
ejpam-6528	159	12	the	the	DET
ejpam-6528	159	13	three	three	NUM
ejpam-6528	159	14	-	-	PUNCT
ejpam-6528	159	15	point	point	NOUN
ejpam-6528	159	16	metric	metric	ADJ
ejpam-6528	159	17	function	function	NOUN
ejpam-6528	159	18	m(υ	m(υ	PROPN
ejpam-6528	159	19	,	,	PUNCT
ejpam-6528	159	20	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	159	21	)	)	PUNCT
ejpam-6528	159	22	,	,	PUNCT
ejpam-6528	159	23	it	it	PRON
ejpam-6528	159	24	retains	retain	VERB
ejpam-6528	159	25	its	its	PRON
ejpam-6528	159	26	ergodic	ergodic	ADJ
ejpam-6528	159	27	properties	property	NOUN
ejpam-6528	159	28	while	while	SCONJ
ejpam-6528	159	29	remaining	remain	VERB
ejpam-6528	159	30	finite	finite	NOUN
ejpam-6528	159	31	and	and	CCONJ
ejpam-6528	159	32	absolutely	absolutely	ADV
ejpam-6528	159	33	continuous	continuous	ADJ
ejpam-6528	159	34	.	.	PUNCT
ejpam-6528	160	1	–	–	PUNCT
ejpam-6528	160	2	this	this	PRON
ejpam-6528	160	3	is	be	AUX
ejpam-6528	160	4	particularly	particularly	ADV
ejpam-6528	160	5	important	important	ADJ
ejpam-6528	160	6	in	in	ADP
ejpam-6528	160	7	stochastic	stochastic	ADJ
ejpam-6528	160	8	systems	system	NOUN
ejpam-6528	160	9	with	with	ADP
ejpam-6528	160	10	memory	memory	NOUN
ejpam-6528	160	11	,	,	PUNCT
ejpam-6528	160	12	where	where	SCONJ
ejpam-6528	160	13	the	the	DET
ejpam-6528	160	14	state	state	NOUN
ejpam-6528	160	15	of	of	ADP
ejpam-6528	160	16	a	a	DET
ejpam-6528	160	17	system	system	NOUN
ejpam-6528	160	18	at	at	ADP
ejpam-6528	160	19	time	time	NOUN
ejpam-6528	160	20	n	n	NUM
ejpam-6528	160	21	depends	depend	VERB
ejpam-6528	160	22	not	not	PART
ejpam-6528	160	23	just	just	ADV
ejpam-6528	160	24	on	on	ADP
ejpam-6528	160	25	the	the	DET
ejpam-6528	160	26	previous	previous	ADJ
ejpam-6528	160	27	state	state	NOUN
ejpam-6528	160	28	but	but	CCONJ
ejpam-6528	160	29	on	on	ADP
ejpam-6528	160	30	two	two	NUM
ejpam-6528	160	31	or	or	CCONJ
ejpam-6528	160	32	more	more	ADJ
ejpam-6528	160	33	past	past	ADJ
ejpam-6528	160	34	states	state	NOUN
ejpam-6528	160	35	.	.	PUNCT
ejpam-6528	161	1	•	•	NUM
ejpam-6528	161	2	tripartite	tripartite	ADJ
ejpam-6528	161	3	markov	markov	NOUN
ejpam-6528	161	4	processes	process	NOUN
ejpam-6528	161	5	and	and	CCONJ
ejpam-6528	161	6	measure	measure	VERB
ejpam-6528	161	7	evolution	evolution	NOUN
ejpam-6528	161	8	:	:	PUNCT
ejpam-6528	161	9	–	–	PUNCT
ejpam-6528	161	10	many	many	ADJ
ejpam-6528	161	11	real	real	ADJ
ejpam-6528	161	12	-	-	PUNCT
ejpam-6528	161	13	world	world	NOUN
ejpam-6528	161	14	stochastic	stochastic	NOUN
ejpam-6528	161	15	systems	system	NOUN
ejpam-6528	161	16	do	do	AUX
ejpam-6528	161	17	not	not	PART
ejpam-6528	161	18	follow	follow	VERB
ejpam-6528	161	19	a	a	DET
ejpam-6528	161	20	simple	simple	ADJ
ejpam-6528	161	21	first	first	ADJ
ejpam-6528	161	22	-	-	PUNCT
ejpam-6528	161	23	order	order	NOUN
ejpam-6528	161	24	markov	markov	NOUN
ejpam-6528	161	25	property	property	NOUN
ejpam-6528	161	26	but	but	CCONJ
ejpam-6528	161	27	instead	instead	ADV
ejpam-6528	161	28	involve	involve	VERB
ejpam-6528	161	29	higher	high	ADJ
ejpam-6528	161	30	-	-	PUNCT
ejpam-6528	161	31	order	order	NOUN
ejpam-6528	161	32	dependencies	dependency	NOUN
ejpam-6528	161	33	.	.	PUNCT
ejpam-6528	162	1	–	–	PUNCT
ejpam-6528	162	2	consider	consider	VERB
ejpam-6528	162	3	a	a	DET
ejpam-6528	162	4	tripartite	tripartite	ADJ
ejpam-6528	162	5	markov	markov	NOUN
ejpam-6528	162	6	process	process	NOUN
ejpam-6528	162	7	,	,	PUNCT
ejpam-6528	162	8	where	where	SCONJ
ejpam-6528	162	9	the	the	DET
ejpam-6528	162	10	transition	transition	NOUN
ejpam-6528	162	11	probabilities	probability	NOUN
ejpam-6528	162	12	depend	depend	VERB
ejpam-6528	162	13	on	on	ADP
ejpam-6528	162	14	the	the	DET
ejpam-6528	162	15	current	current	ADJ
ejpam-6528	162	16	state	state	NOUN
ejpam-6528	162	17	and	and	CCONJ
ejpam-6528	162	18	two	two	NUM
ejpam-6528	162	19	prior	prior	ADJ
ejpam-6528	162	20	states	state	NOUN
ejpam-6528	162	21	:	:	PUNCT
ejpam-6528	162	22	p	p	X
ejpam-6528	162	23	(	(	PUNCT
ejpam-6528	162	24	xn+1	xn+1	PROPN
ejpam-6528	162	25	|	|	ADV
ejpam-6528	162	26	xn	xn	PROPN
ejpam-6528	162	27	,	,	PUNCT
ejpam-6528	162	28	xn−1	xn−1	PROPN
ejpam-6528	162	29	,	,	PUNCT
ejpam-6528	162	30	xn−2	xn−2	PROPN
ejpam-6528	162	31	)	)	PUNCT
ejpam-6528	162	32	.	.	PUNCT
ejpam-6528	163	1	–	–	PUNCT
ejpam-6528	163	2	the	the	DET
ejpam-6528	163	3	theorem	theorem	NOUN
ejpam-6528	163	4	ensures	ensure	VERB
ejpam-6528	163	5	that	that	SCONJ
ejpam-6528	163	6	when	when	SCONJ
ejpam-6528	163	7	such	such	DET
ejpam-6528	163	8	a	a	DET
ejpam-6528	163	9	system	system	NOUN
ejpam-6528	163	10	is	be	AUX
ejpam-6528	163	11	described	describe	VERB
ejpam-6528	163	12	using	use	VERB
ejpam-6528	163	13	an	an	DET
ejpam-6528	163	14	mr	mr	PROPN
ejpam-6528	163	15	-	-	PUNCT
ejpam-6528	163	16	metric	metric	ADJ
ejpam-6528	163	17	space	space	NOUN
ejpam-6528	163	18	,	,	PUNCT
ejpam-6528	163	19	the	the	DET
ejpam-6528	163	20	induced	induced	ADJ
ejpam-6528	163	21	measure	measure	NOUN
ejpam-6528	163	22	remains	remain	VERB
ejpam-6528	163	23	well	well	ADV
ejpam-6528	163	24	-	-	PUNCT
ejpam-6528	163	25	behaved	behave	VERB
ejpam-6528	163	26	,	,	PUNCT
ejpam-6528	163	27	allowing	allow	VERB
ejpam-6528	163	28	for	for	ADP
ejpam-6528	163	29	meaningful	meaningful	ADJ
ejpam-6528	163	30	long	long	ADJ
ejpam-6528	163	31	-	-	PUNCT
ejpam-6528	163	32	term	term	NOUN
ejpam-6528	163	33	statistical	statistical	ADJ
ejpam-6528	163	34	analysis	analysis	NOUN
ejpam-6528	163	35	.	.	PUNCT
ejpam-6528	164	1	–	–	PUNCT
ejpam-6528	164	2	applications	application	NOUN
ejpam-6528	164	3	include	include	VERB
ejpam-6528	164	4	:	:	PUNCT
ejpam-6528	164	5	∗	∗	NOUN
ejpam-6528	164	6	stock	stock	NOUN
ejpam-6528	164	7	market	market	NOUN
ejpam-6528	164	8	modeling	modeling	NOUN
ejpam-6528	164	9	,	,	PUNCT
ejpam-6528	164	10	where	where	SCONJ
ejpam-6528	164	11	an	an	DET
ejpam-6528	164	12	asset	asset	NOUN
ejpam-6528	164	13	’s	’s	PART
ejpam-6528	164	14	price	price	NOUN
ejpam-6528	164	15	evolution	evolution	NOUN
ejpam-6528	164	16	may	may	AUX
ejpam-6528	164	17	depend	depend	VERB
ejpam-6528	164	18	on	on	ADP
ejpam-6528	164	19	its	its	PRON
ejpam-6528	164	20	value	value	NOUN
ejpam-6528	164	21	over	over	ADP
ejpam-6528	164	22	multiple	multiple	ADJ
ejpam-6528	164	23	previous	previous	ADJ
ejpam-6528	164	24	time	time	NOUN
ejpam-6528	164	25	steps	step	NOUN
ejpam-6528	164	26	.	.	PUNCT
ejpam-6528	165	1	∗	∗	NOUN
ejpam-6528	165	2	genetic	genetic	ADJ
ejpam-6528	165	3	sequence	sequence	NOUN
ejpam-6528	165	4	evolution	evolution	NOUN
ejpam-6528	165	5	,	,	PUNCT
ejpam-6528	165	6	where	where	SCONJ
ejpam-6528	165	7	the	the	DET
ejpam-6528	165	8	probability	probability	NOUN
ejpam-6528	165	9	of	of	ADP
ejpam-6528	165	10	a	a	DET
ejpam-6528	165	11	mutation	mutation	NOUN
ejpam-6528	165	12	may	may	AUX
ejpam-6528	165	13	be	be	AUX
ejpam-6528	165	14	influenced	influence	VERB
ejpam-6528	165	15	by	by	ADP
ejpam-6528	165	16	a	a	DET
ejpam-6528	165	17	combination	combination	NOUN
ejpam-6528	165	18	of	of	ADP
ejpam-6528	165	19	prior	prior	ADJ
ejpam-6528	165	20	genetic	genetic	ADJ
ejpam-6528	165	21	states	state	NOUN
ejpam-6528	165	22	.	.	PUNCT
ejpam-6528	166	1	a.	a.	PROPN
ejpam-6528	166	2	malkawi	malkawi	PROPN
ejpam-6528	166	3	,	,	PUNCT
ejpam-6528	166	4	a.	a.	PROPN
ejpam-6528	166	5	rabaiah	rabaiah	PROPN
ejpam-6528	166	6	/	/	SYM
ejpam-6528	166	7	eur	eur	PROPN
ejpam-6528	166	8	.	.	PUNCT
ejpam-6528	167	1	j.	j.	PROPN
ejpam-6528	167	2	pure	pure	PROPN
ejpam-6528	167	3	appl	appl	PROPN
ejpam-6528	167	4	.	.	PROPN
ejpam-6528	167	5	math	math	PROPN
ejpam-6528	167	6	,	,	PUNCT
ejpam-6528	167	7	18	18	NUM
ejpam-6528	167	8	(	(	PUNCT
ejpam-6528	167	9	3	3	NUM
ejpam-6528	167	10	)	)	PUNCT
ejpam-6528	167	11	(	(	PUNCT
ejpam-6528	167	12	2025	2025	NUM
ejpam-6528	167	13	)	)	PUNCT
ejpam-6528	167	14	,	,	PUNCT
ejpam-6528	167	15	6528	6528	NUM
ejpam-6528	167	16	9	9	NUM
ejpam-6528	167	17	of	of	ADP
ejpam-6528	167	18	12	12	NUM
ejpam-6528	167	19	∗	∗	NOUN
ejpam-6528	167	20	neural	neural	ADJ
ejpam-6528	167	21	activity	activity	NOUN
ejpam-6528	167	22	models	model	NOUN
ejpam-6528	167	23	,	,	PUNCT
ejpam-6528	167	24	where	where	SCONJ
ejpam-6528	167	25	the	the	DET
ejpam-6528	167	26	activation	activation	NOUN
ejpam-6528	167	27	of	of	ADP
ejpam-6528	167	28	a	a	DET
ejpam-6528	167	29	neuron	neuron	NOUN
ejpam-6528	167	30	may	may	AUX
ejpam-6528	167	31	depend	depend	VERB
ejpam-6528	167	32	on	on	ADP
ejpam-6528	167	33	the	the	DET
ejpam-6528	167	34	signals	signal	NOUN
ejpam-6528	167	35	from	from	ADP
ejpam-6528	167	36	a	a	DET
ejpam-6528	167	37	combination	combination	NOUN
ejpam-6528	167	38	of	of	ADP
ejpam-6528	167	39	previous	previous	ADJ
ejpam-6528	167	40	stimuli	stimulus	NOUN
ejpam-6528	167	41	rather	rather	ADV
ejpam-6528	167	42	than	than	ADP
ejpam-6528	167	43	just	just	ADV
ejpam-6528	167	44	the	the	DET
ejpam-6528	167	45	last	last	ADJ
ejpam-6528	167	46	input	input	NOUN
ejpam-6528	167	47	.	.	PUNCT
ejpam-6528	168	1	•	•	NUM
ejpam-6528	168	2	applications	application	NOUN
ejpam-6528	168	3	in	in	ADP
ejpam-6528	168	4	physics	physics	NOUN
ejpam-6528	168	5	and	and	CCONJ
ejpam-6528	168	6	ergodic	ergodic	ADJ
ejpam-6528	168	7	theory	theory	NOUN
ejpam-6528	168	8	:	:	PUNCT
ejpam-6528	168	9	–	–	PUNCT
ejpam-6528	168	10	in	in	ADP
ejpam-6528	168	11	statistical	statistical	ADJ
ejpam-6528	168	12	mechanics	mechanic	NOUN
ejpam-6528	168	13	,	,	PUNCT
ejpam-6528	168	14	tripartite	tripartite	ADJ
ejpam-6528	168	15	interactions	interaction	NOUN
ejpam-6528	168	16	arise	arise	VERB
ejpam-6528	168	17	naturally	naturally	ADV
ejpam-6528	168	18	in	in	ADP
ejpam-6528	168	19	models	model	NOUN
ejpam-6528	168	20	of	of	ADP
ejpam-6528	168	21	interacting	interact	VERB
ejpam-6528	168	22	particles	particle	NOUN
ejpam-6528	168	23	,	,	PUNCT
ejpam-6528	168	24	such	such	ADJ
ejpam-6528	168	25	as	as	ADP
ejpam-6528	168	26	in	in	ADP
ejpam-6528	168	27	spin	spin	NOUN
ejpam-6528	168	28	-	-	PUNCT
ejpam-6528	168	29	glass	glass	NOUN
ejpam-6528	168	30	systems	system	NOUN
ejpam-6528	168	31	,	,	PUNCT
ejpam-6528	168	32	where	where	SCONJ
ejpam-6528	168	33	the	the	DET
ejpam-6528	168	34	behavior	behavior	NOUN
ejpam-6528	168	35	of	of	ADP
ejpam-6528	168	36	a	a	DET
ejpam-6528	168	37	particle	particle	NOUN
ejpam-6528	168	38	depends	depend	VERB
ejpam-6528	168	39	on	on	ADP
ejpam-6528	168	40	multiple	multiple	ADJ
ejpam-6528	168	41	neighboring	neighboring	NOUN
ejpam-6528	168	42	interactions	interaction	NOUN
ejpam-6528	168	43	.	.	PUNCT
ejpam-6528	169	1	–	–	PUNCT
ejpam-6528	169	2	in	in	ADP
ejpam-6528	169	3	thermodynamics	thermodynamic	NOUN
ejpam-6528	169	4	,	,	PUNCT
ejpam-6528	169	5	the	the	DET
ejpam-6528	169	6	evolution	evolution	NOUN
ejpam-6528	169	7	of	of	ADP
ejpam-6528	169	8	macrostates	macrostate	NOUN
ejpam-6528	169	9	in	in	ADP
ejpam-6528	169	10	a	a	DET
ejpam-6528	169	11	system	system	NOUN
ejpam-6528	169	12	with	with	ADP
ejpam-6528	169	13	long	long	ADJ
ejpam-6528	169	14	-	-	PUNCT
ejpam-6528	169	15	range	range	NOUN
ejpam-6528	169	16	dependencies	dependency	NOUN
ejpam-6528	169	17	can	can	AUX
ejpam-6528	169	18	be	be	AUX
ejpam-6528	169	19	analyzed	analyze	VERB
ejpam-6528	169	20	using	use	VERB
ejpam-6528	169	21	a	a	DET
ejpam-6528	169	22	measure	measure	NOUN
ejpam-6528	169	23	induced	induce	VERB
ejpam-6528	169	24	by	by	ADP
ejpam-6528	169	25	m(υ	m(υ	PROPN
ejpam-6528	169	26	,	,	PUNCT
ejpam-6528	169	27	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	169	28	)	)	PUNCT
ejpam-6528	169	29	.	.	PUNCT
ejpam-6528	170	1	–	–	PUNCT
ejpam-6528	170	2	in	in	ADP
ejpam-6528	170	3	dynamical	dynamical	ADJ
ejpam-6528	170	4	astronomy	astronomy	NOUN
ejpam-6528	170	5	,	,	PUNCT
ejpam-6528	170	6	the	the	DET
ejpam-6528	170	7	behavior	behavior	NOUN
ejpam-6528	170	8	of	of	ADP
ejpam-6528	170	9	a	a	DET
ejpam-6528	170	10	celestial	celestial	ADJ
ejpam-6528	170	11	body	body	NOUN
ejpam-6528	170	12	in	in	ADP
ejpam-6528	170	13	an	an	DET
ejpam-6528	170	14	n	n	CCONJ
ejpam-6528	170	15	-	-	PUNCT
ejpam-6528	170	16	body	body	NOUN
ejpam-6528	170	17	problem	problem	NOUN
ejpam-6528	170	18	(	(	PUNCT
ejpam-6528	170	19	such	such	ADJ
ejpam-6528	170	20	as	as	ADP
ejpam-6528	170	21	a	a	DET
ejpam-6528	170	22	planetary	planetary	ADJ
ejpam-6528	170	23	system	system	NOUN
ejpam-6528	170	24	)	)	PUNCT
ejpam-6528	170	25	often	often	ADV
ejpam-6528	170	26	requires	require	VERB
ejpam-6528	170	27	a	a	DET
ejpam-6528	170	28	metric	metric	NOUN
ejpam-6528	170	29	that	that	PRON
ejpam-6528	170	30	accounts	account	VERB
ejpam-6528	170	31	for	for	ADP
ejpam-6528	170	32	more	more	ADJ
ejpam-6528	170	33	than	than	ADP
ejpam-6528	170	34	just	just	ADV
ejpam-6528	170	35	pairwise	pairwise	NOUN
ejpam-6528	170	36	gravitational	gravitational	ADJ
ejpam-6528	170	37	effects	effect	NOUN
ejpam-6528	170	38	.	.	PUNCT
ejpam-6528	171	1	–	–	PUNCT
ejpam-6528	171	2	the	the	DET
ejpam-6528	171	3	theorem	theorem	NOUN
ejpam-6528	171	4	ensures	ensure	VERB
ejpam-6528	171	5	that	that	SCONJ
ejpam-6528	171	6	such	such	ADJ
ejpam-6528	171	7	measure	measure	NOUN
ejpam-6528	171	8	-	-	PUNCT
ejpam-6528	171	9	preserving	preserve	VERB
ejpam-6528	171	10	systems	system	NOUN
ejpam-6528	171	11	remain	remain	VERB
ejpam-6528	171	12	statistically	statistically	ADV
ejpam-6528	171	13	predictable	predictable	ADJ
ejpam-6528	171	14	and	and	CCONJ
ejpam-6528	171	15	analyzable	analyzable	ADJ
ejpam-6528	171	16	,	,	PUNCT
ejpam-6528	171	17	supporting	support	VERB
ejpam-6528	171	18	long	long	ADJ
ejpam-6528	171	19	-	-	PUNCT
ejpam-6528	171	20	term	term	NOUN
ejpam-6528	171	21	stability	stability	NOUN
ejpam-6528	171	22	analyses	analyse	VERB
ejpam-6528	171	23	in	in	ADP
ejpam-6528	171	24	ergodic	ergodic	ADJ
ejpam-6528	171	25	theory	theory	NOUN
ejpam-6528	171	26	.	.	PUNCT
ejpam-6528	172	1	3.2	3.2	NUM
ejpam-6528	172	2	.	.	PUNCT
ejpam-6528	172	3	applications	application	NOUN
ejpam-6528	172	4	of	of	ADP
ejpam-6528	172	5	the	the	DET
ejpam-6528	172	6	second	second	ADJ
ejpam-6528	172	7	theorem	theorem	NOUN
ejpam-6528	172	8	the	the	DET
ejpam-6528	172	9	second	second	ADJ
ejpam-6528	172	10	theorem	theorem	ADJ
ejpam-6528	172	11	states	state	NOUN
ejpam-6528	172	12	that	that	SCONJ
ejpam-6528	172	13	if	if	SCONJ
ejpam-6528	172	14	a	a	DET
ejpam-6528	172	15	sequence	sequence	NOUN
ejpam-6528	172	16	{	{	PUNCT
ejpam-6528	172	17	υn	υn	NOUN
ejpam-6528	172	18	}	}	PUNCT
ejpam-6528	172	19	in	in	ADP
ejpam-6528	172	20	an	an	DET
ejpam-6528	172	21	mr	mr	PROPN
ejpam-6528	172	22	-	-	PUNCT
ejpam-6528	172	23	metric	metric	ADJ
ejpam-6528	172	24	space	space	NOUN
ejpam-6528	172	25	satisfies	satisfie	NOUN
ejpam-6528	172	26	m(υn	m(υn	X
ejpam-6528	172	27	,	,	PUNCT
ejpam-6528	172	28	υn−1	υn−1	ADJ
ejpam-6528	172	29	,	,	PUNCT
ejpam-6528	172	30	υn−2	υn−2	PROPN
ejpam-6528	172	31	)	)	PUNCT
ejpam-6528	172	32	→	→	SYM
ejpam-6528	172	33	0	0	NUM
ejpam-6528	172	34	as	as	ADP
ejpam-6528	172	35	n→	n→	ADV
ejpam-6528	172	36	∞	∞	PROPN
ejpam-6528	172	37	,	,	PUNCT
ejpam-6528	172	38	then	then	ADV
ejpam-6528	172	39	υn	υn	PROPN
ejpam-6528	172	40	converges	converge	NOUN
ejpam-6528	172	41	to	to	ADP
ejpam-6528	172	42	some	some	DET
ejpam-6528	172	43	υ∗	υ∗	NOUN
ejpam-6528	172	44	∈	∈	PROPN
ejpam-6528	172	45	x	x	PUNCT
ejpam-6528	172	46	in	in	ADP
ejpam-6528	172	47	measure	measure	NOUN
ejpam-6528	172	48	.	.	PUNCT
ejpam-6528	173	1	applications	application	NOUN
ejpam-6528	173	2	include	include	VERB
ejpam-6528	173	3	:	:	PUNCT
ejpam-6528	173	4	3.2.1	3.2.1	X
ejpam-6528	173	5	.	.	PUNCT
ejpam-6528	173	6	convergence	convergence	NOUN
ejpam-6528	173	7	in	in	ADP
ejpam-6528	173	8	probability	probability	NOUN
ejpam-6528	173	9	and	and	CCONJ
ejpam-6528	173	10	stochastic	stochastic	ADJ
ejpam-6528	173	11	analysis	analysis	NOUN
ejpam-6528	173	12	•	•	NOUN
ejpam-6528	173	13	this	this	DET
ejpam-6528	173	14	result	result	NOUN
ejpam-6528	173	15	provides	provide	VERB
ejpam-6528	173	16	a	a	DET
ejpam-6528	173	17	generalization	generalization	NOUN
ejpam-6528	173	18	of	of	ADP
ejpam-6528	173	19	cauchy	cauchy	ADJ
ejpam-6528	173	20	sequences	sequence	NOUN
ejpam-6528	173	21	in	in	ADP
ejpam-6528	173	22	mr	mr	PROPN
ejpam-6528	173	23	-	-	PUNCT
ejpam-6528	173	24	metric	metric	ADJ
ejpam-6528	173	25	spaces	space	NOUN
ejpam-6528	173	26	,	,	PUNCT
ejpam-6528	173	27	ensuring	ensure	VERB
ejpam-6528	173	28	that	that	SCONJ
ejpam-6528	173	29	under	under	ADP
ejpam-6528	173	30	the	the	DET
ejpam-6528	173	31	given	give	VERB
ejpam-6528	173	32	condition	condition	NOUN
ejpam-6528	173	33	,	,	PUNCT
ejpam-6528	173	34	sequences	sequence	NOUN
ejpam-6528	173	35	exhibit	exhibit	VERB
ejpam-6528	173	36	a	a	DET
ejpam-6528	173	37	form	form	NOUN
ejpam-6528	173	38	of	of	ADP
ejpam-6528	173	39	probabilistic	probabilistic	ADJ
ejpam-6528	173	40	convergence	convergence	NOUN
ejpam-6528	173	41	.	.	PUNCT
ejpam-6528	174	1	•	•	NOUN
ejpam-6528	174	2	it	it	PRON
ejpam-6528	174	3	can	can	AUX
ejpam-6528	174	4	be	be	AUX
ejpam-6528	174	5	used	use	VERB
ejpam-6528	174	6	to	to	PART
ejpam-6528	174	7	study	study	VERB
ejpam-6528	174	8	stochastic	stochastic	NOUN
ejpam-6528	174	9	processes	process	NOUN
ejpam-6528	174	10	where	where	SCONJ
ejpam-6528	174	11	transitions	transition	NOUN
ejpam-6528	174	12	depend	depend	VERB
ejpam-6528	174	13	on	on	ADP
ejpam-6528	174	14	three	three	NUM
ejpam-6528	174	15	previous	previous	ADJ
ejpam-6528	174	16	states	state	NOUN
ejpam-6528	174	17	rather	rather	ADV
ejpam-6528	174	18	than	than	ADP
ejpam-6528	174	19	just	just	ADV
ejpam-6528	174	20	one	one	NUM
ejpam-6528	174	21	(	(	PUNCT
ejpam-6528	174	22	e.g.	e.g.	ADV
ejpam-6528	174	23	,	,	PUNCT
ejpam-6528	174	24	higher	high	ADJ
ejpam-6528	174	25	-	-	PUNCT
ejpam-6528	174	26	order	order	NOUN
ejpam-6528	174	27	markov	markov	NOUN
ejpam-6528	174	28	chains	chain	NOUN
ejpam-6528	174	29	)	)	PUNCT
ejpam-6528	174	30	.	.	PUNCT
ejpam-6528	175	1	3.2.2	3.2.2	X
ejpam-6528	175	2	.	.	PUNCT
ejpam-6528	175	3	machine	machine	NOUN
ejpam-6528	175	4	learning	learning	NOUN
ejpam-6528	175	5	and	and	CCONJ
ejpam-6528	175	6	optimization	optimization	NOUN
ejpam-6528	175	7	•	•	NOUN
ejpam-6528	175	8	in	in	ADP
ejpam-6528	175	9	algorithms	algorithm	NOUN
ejpam-6528	175	10	that	that	PRON
ejpam-6528	175	11	rely	rely	VERB
ejpam-6528	175	12	on	on	ADP
ejpam-6528	175	13	a	a	DET
ejpam-6528	175	14	triplet	triplet	NOUN
ejpam-6528	175	15	-	-	PUNCT
ejpam-6528	175	16	based	base	VERB
ejpam-6528	175	17	distance	distance	NOUN
ejpam-6528	175	18	function	function	NOUN
ejpam-6528	175	19	,	,	PUNCT
ejpam-6528	175	20	such	such	ADJ
ejpam-6528	175	21	as	as	ADP
ejpam-6528	175	22	triplet	triplet	NOUN
ejpam-6528	175	23	loss	loss	NOUN
ejpam-6528	175	24	in	in	ADP
ejpam-6528	175	25	deep	deep	ADJ
ejpam-6528	175	26	learning	learning	NOUN
ejpam-6528	175	27	,	,	PUNCT
ejpam-6528	175	28	this	this	PRON
ejpam-6528	175	29	theorem	theorem	VERB
ejpam-6528	175	30	guarantees	guarantee	NOUN
ejpam-6528	175	31	that	that	SCONJ
ejpam-6528	175	32	optimization	optimization	NOUN
ejpam-6528	175	33	processes	process	NOUN
ejpam-6528	175	34	involving	involve	VERB
ejpam-6528	175	35	triple	triple	ADJ
ejpam-6528	175	36	relations	relation	NOUN
ejpam-6528	175	37	lead	lead	VERB
ejpam-6528	175	38	to	to	ADP
ejpam-6528	175	39	a	a	DET
ejpam-6528	175	40	well	well	ADV
ejpam-6528	175	41	-	-	PUNCT
ejpam-6528	175	42	defined	define	VERB
ejpam-6528	175	43	limit	limit	NOUN
ejpam-6528	175	44	.	.	PUNCT
ejpam-6528	176	1	•	•	INTJ
ejpam-6528	176	2	it	it	PRON
ejpam-6528	176	3	can	can	AUX
ejpam-6528	176	4	be	be	AUX
ejpam-6528	176	5	useful	useful	ADJ
ejpam-6528	176	6	in	in	ADP
ejpam-6528	176	7	metric	metric	ADJ
ejpam-6528	176	8	learning	learning	NOUN
ejpam-6528	176	9	applications	application	NOUN
ejpam-6528	176	10	where	where	SCONJ
ejpam-6528	176	11	distances	distance	NOUN
ejpam-6528	176	12	between	between	ADP
ejpam-6528	176	13	three	three	NUM
ejpam-6528	176	14	points	point	NOUN
ejpam-6528	176	15	determine	determine	NOUN
ejpam-6528	176	16	learning	learn	VERB
ejpam-6528	176	17	trajectories	trajectory	NOUN
ejpam-6528	176	18	.	.	PUNCT
ejpam-6528	177	1	a.	a.	PROPN
ejpam-6528	177	2	malkawi	malkawi	PROPN
ejpam-6528	177	3	,	,	PUNCT
ejpam-6528	177	4	a.	a.	PROPN
ejpam-6528	177	5	rabaiah	rabaiah	PROPN
ejpam-6528	177	6	/	/	SYM
ejpam-6528	177	7	eur	eur	PROPN
ejpam-6528	177	8	.	.	PUNCT
ejpam-6528	178	1	j.	j.	PROPN
ejpam-6528	178	2	pure	pure	PROPN
ejpam-6528	178	3	appl	appl	PROPN
ejpam-6528	178	4	.	.	PROPN
ejpam-6528	178	5	math	math	PROPN
ejpam-6528	178	6	,	,	PUNCT
ejpam-6528	178	7	18	18	NUM
ejpam-6528	178	8	(	(	PUNCT
ejpam-6528	178	9	3	3	NUM
ejpam-6528	178	10	)	)	PUNCT
ejpam-6528	178	11	(	(	PUNCT
ejpam-6528	178	12	2025	2025	NUM
ejpam-6528	178	13	)	)	PUNCT
ejpam-6528	178	14	,	,	PUNCT
ejpam-6528	178	15	6528	6528	NUM
ejpam-6528	178	16	10	10	NUM
ejpam-6528	178	17	of	of	ADP
ejpam-6528	178	18	12	12	NUM
ejpam-6528	178	19	3.2.3	3.2.3	NUM
ejpam-6528	178	20	.	.	PUNCT
ejpam-6528	179	1	geometric	geometric	ADJ
ejpam-6528	179	2	analysis	analysis	NOUN
ejpam-6528	179	3	and	and	CCONJ
ejpam-6528	179	4	fixed	fix	VERB
ejpam-6528	179	5	point	point	NOUN
ejpam-6528	179	6	theory	theory	NOUN
ejpam-6528	179	7	•	•	ADP
ejpam-6528	179	8	if	if	SCONJ
ejpam-6528	179	9	a	a	DET
ejpam-6528	179	10	dynamical	dynamical	ADJ
ejpam-6528	179	11	system	system	NOUN
ejpam-6528	179	12	is	be	AUX
ejpam-6528	179	13	governed	govern	VERB
ejpam-6528	179	14	by	by	ADP
ejpam-6528	179	15	a	a	DET
ejpam-6528	179	16	three	three	NUM
ejpam-6528	179	17	-	-	PUNCT
ejpam-6528	179	18	point	point	NOUN
ejpam-6528	179	19	metric	metric	ADJ
ejpam-6528	179	20	function	function	NOUN
ejpam-6528	179	21	,	,	PUNCT
ejpam-6528	179	22	this	this	PRON
ejpam-6528	179	23	theorem	theorem	VERB
ejpam-6528	179	24	ensures	ensure	VERB
ejpam-6528	179	25	that	that	SCONJ
ejpam-6528	179	26	iterative	iterative	NOUN
ejpam-6528	179	27	processes	process	NOUN
ejpam-6528	179	28	converge	converge	VERB
ejpam-6528	179	29	in	in	ADP
ejpam-6528	179	30	measure	measure	NOUN
ejpam-6528	179	31	,	,	PUNCT
ejpam-6528	179	32	which	which	PRON
ejpam-6528	179	33	is	be	AUX
ejpam-6528	179	34	useful	useful	ADJ
ejpam-6528	179	35	in	in	ADP
ejpam-6528	179	36	studying	study	VERB
ejpam-6528	179	37	generalized	generalized	ADJ
ejpam-6528	179	38	contraction	contraction	NOUN
ejpam-6528	179	39	mappings	mapping	NOUN
ejpam-6528	179	40	.	.	PUNCT
ejpam-6528	180	1	•	•	NOUN
ejpam-6528	180	2	it	it	PRON
ejpam-6528	180	3	can	can	AUX
ejpam-6528	180	4	be	be	AUX
ejpam-6528	180	5	applied	apply	VERB
ejpam-6528	180	6	in	in	ADP
ejpam-6528	180	7	establishing	establish	VERB
ejpam-6528	180	8	fixed	fix	VERB
ejpam-6528	180	9	points	point	NOUN
ejpam-6528	180	10	in	in	ADP
ejpam-6528	180	11	non	non	ADJ
ejpam-6528	180	12	-	-	ADJ
ejpam-6528	180	13	euclidean	euclidean	ADJ
ejpam-6528	180	14	geometries	geometry	NOUN
ejpam-6528	180	15	or	or	CCONJ
ejpam-6528	180	16	spaces	space	NOUN
ejpam-6528	180	17	where	where	SCONJ
ejpam-6528	180	18	standard	standard	ADJ
ejpam-6528	180	19	metric	metric	ADJ
ejpam-6528	180	20	properties	property	NOUN
ejpam-6528	180	21	do	do	AUX
ejpam-6528	180	22	not	not	PART
ejpam-6528	180	23	hold	hold	VERB
ejpam-6528	180	24	.	.	PUNCT
ejpam-6528	181	1	3.3	3.3	NUM
ejpam-6528	181	2	.	.	PUNCT
ejpam-6528	182	1	computational	computational	ADJ
ejpam-6528	182	2	implications	implication	NOUN
ejpam-6528	182	3	the	the	DET
ejpam-6528	182	4	theoretical	theoretical	ADJ
ejpam-6528	182	5	framework	framework	NOUN
ejpam-6528	182	6	of	of	ADP
ejpam-6528	182	7	mr	mr	PROPN
ejpam-6528	182	8	-	-	PUNCT
ejpam-6528	182	9	metric	metric	ADJ
ejpam-6528	182	10	spaces	space	NOUN
ejpam-6528	182	11	and	and	CCONJ
ejpam-6528	182	12	their	their	PRON
ejpam-6528	182	13	associated	associated	ADJ
ejpam-6528	182	14	measures	measure	NOUN
ejpam-6528	182	15	opens	open	VERB
ejpam-6528	182	16	avenues	avenue	NOUN
ejpam-6528	182	17	for	for	ADP
ejpam-6528	182	18	computational	computational	ADJ
ejpam-6528	182	19	exploration	exploration	NOUN
ejpam-6528	182	20	.	.	PUNCT
ejpam-6528	183	1	below	below	ADV
ejpam-6528	183	2	,	,	PUNCT
ejpam-6528	183	3	we	we	PRON
ejpam-6528	183	4	highlight	highlight	VERB
ejpam-6528	183	5	potential	potential	ADJ
ejpam-6528	183	6	implications	implication	NOUN
ejpam-6528	183	7	:	:	PUNCT
ejpam-6528	183	8	•	•	NUM
ejpam-6528	183	9	algorithmic	algorithmic	ADJ
ejpam-6528	183	10	complexity	complexity	NOUN
ejpam-6528	183	11	:	:	PUNCT
ejpam-6528	183	12	the	the	DET
ejpam-6528	183	13	triadic	triadic	ADJ
ejpam-6528	183	14	nature	nature	NOUN
ejpam-6528	183	15	of	of	ADP
ejpam-6528	183	16	the	the	DET
ejpam-6528	183	17	mr	mr	PROPN
ejpam-6528	183	18	-	-	PUNCT
ejpam-6528	183	19	metric	metric	ADJ
ejpam-6528	183	20	m(v	m(v	NOUN
ejpam-6528	183	21	,	,	PUNCT
ejpam-6528	183	22	ξ,ℑ	ξ,ℑ	PROPN
ejpam-6528	183	23	)	)	PUNCT
ejpam-6528	183	24	may	may	AUX
ejpam-6528	183	25	increase	increase	VERB
ejpam-6528	183	26	computational	computational	ADJ
ejpam-6528	183	27	overhead	overhead	NOUN
ejpam-6528	183	28	compared	compare	VERB
ejpam-6528	183	29	to	to	ADP
ejpam-6528	183	30	pairwise	pairwise	NOUN
ejpam-6528	183	31	metrics	metric	NOUN
ejpam-6528	183	32	,	,	PUNCT
ejpam-6528	183	33	as	as	SCONJ
ejpam-6528	183	34	evaluating	evaluate	VERB
ejpam-6528	183	35	m	m	NOUN
ejpam-6528	183	36	requires	require	VERB
ejpam-6528	183	37	o(n3	o(n3	NOUN
ejpam-6528	183	38	)	)	PUNCT
ejpam-6528	183	39	operations	operation	NOUN
ejpam-6528	183	40	for	for	ADP
ejpam-6528	183	41	n	n	NUM
ejpam-6528	183	42	points	point	NOUN
ejpam-6528	183	43	.	.	PUNCT
ejpam-6528	184	1	however	however	ADV
ejpam-6528	184	2	,	,	PUNCT
ejpam-6528	184	3	symmetry	symmetry	NOUN
ejpam-6528	184	4	(	(	PUNCT
ejpam-6528	184	5	m3	m3	PROPN
ejpam-6528	184	6	)	)	PUNCT
ejpam-6528	184	7	and	and	CCONJ
ejpam-6528	184	8	boundedness	boundedness	NOUN
ejpam-6528	184	9	(	(	PUNCT
ejpam-6528	184	10	m4	m4	PROPN
ejpam-6528	184	11	)	)	PUNCT
ejpam-6528	184	12	could	could	AUX
ejpam-6528	184	13	be	be	AUX
ejpam-6528	184	14	exploited	exploit	VERB
ejpam-6528	184	15	to	to	PART
ejpam-6528	184	16	optimize	optimize	VERB
ejpam-6528	184	17	calculations	calculation	NOUN
ejpam-6528	184	18	,	,	PUNCT
ejpam-6528	184	19	e.g.	e.g.	ADV
ejpam-6528	184	20	,	,	PUNCT
ejpam-6528	184	21	by	by	ADP
ejpam-6528	184	22	caching	cache	VERB
ejpam-6528	184	23	repeated	repeat	VERB
ejpam-6528	184	24	terms	term	NOUN
ejpam-6528	184	25	or	or	CCONJ
ejpam-6528	184	26	pruning	prune	VERB
ejpam-6528	184	27	negligible	negligible	ADJ
ejpam-6528	184	28	contributions	contribution	NOUN
ejpam-6528	184	29	.	.	PUNCT
ejpam-6528	185	1	•	•	NUM
ejpam-6528	185	2	high	high	ADV
ejpam-6528	185	3	-	-	PUNCT
ejpam-6528	185	4	dimensional	dimensional	ADJ
ejpam-6528	185	5	data	datum	NOUN
ejpam-6528	185	6	:	:	PUNCT
ejpam-6528	185	7	in	in	ADP
ejpam-6528	185	8	machine	machine	NOUN
ejpam-6528	185	9	learning	learning	NOUN
ejpam-6528	185	10	,	,	PUNCT
ejpam-6528	185	11	mr	mr	PROPN
ejpam-6528	185	12	-	-	PUNCT
ejpam-6528	185	13	metrics	metric	NOUN
ejpam-6528	185	14	could	could	AUX
ejpam-6528	185	15	model	model	VERB
ejpam-6528	185	16	higherorder	higherorder	NOUN
ejpam-6528	185	17	dependencies	dependency	NOUN
ejpam-6528	185	18	in	in	ADP
ejpam-6528	185	19	data	datum	NOUN
ejpam-6528	185	20	(	(	PUNCT
ejpam-6528	185	21	e.g.	e.g.	ADV
ejpam-6528	185	22	,	,	PUNCT
ejpam-6528	185	23	triplet	triplet	NOUN
ejpam-6528	185	24	interactions	interaction	NOUN
ejpam-6528	185	25	in	in	ADP
ejpam-6528	185	26	graph	graph	NOUN
ejpam-6528	185	27	embeddings	embedding	NOUN
ejpam-6528	185	28	)	)	PUNCT
ejpam-6528	185	29	.	.	PUNCT
ejpam-6528	186	1	while	while	SCONJ
ejpam-6528	186	2	this	this	PRON
ejpam-6528	186	3	enriches	enrich	VERB
ejpam-6528	186	4	representation	representation	NOUN
ejpam-6528	186	5	,	,	PUNCT
ejpam-6528	186	6	scalability	scalability	NOUN
ejpam-6528	186	7	challenges	challenge	NOUN
ejpam-6528	186	8	arise	arise	VERB
ejpam-6528	186	9	.	.	PUNCT
ejpam-6528	187	1	approximation	approximation	NOUN
ejpam-6528	187	2	techniques	technique	NOUN
ejpam-6528	187	3	,	,	PUNCT
ejpam-6528	187	4	such	such	ADJ
ejpam-6528	187	5	as	as	ADP
ejpam-6528	187	6	sampling	sample	VERB
ejpam-6528	187	7	or	or	CCONJ
ejpam-6528	187	8	low	low	ADJ
ejpam-6528	187	9	-	-	PUNCT
ejpam-6528	187	10	rank	rank	NOUN
ejpam-6528	187	11	tensor	tensor	NOUN
ejpam-6528	187	12	decompositions	decomposition	NOUN
ejpam-6528	187	13	,	,	PUNCT
ejpam-6528	187	14	might	might	AUX
ejpam-6528	187	15	mitigate	mitigate	VERB
ejpam-6528	187	16	these	these	DET
ejpam-6528	187	17	costs	cost	NOUN
ejpam-6528	187	18	.	.	PUNCT
ejpam-6528	188	1	•	•	NUM
ejpam-6528	188	2	convergence	convergence	NOUN
ejpam-6528	188	3	verification	verification	NOUN
ejpam-6528	188	4	:	:	PUNCT
ejpam-6528	188	5	theorem	theorem	VERB
ejpam-6528	188	6	2	2	NUM
ejpam-6528	188	7	’s	’s	PART
ejpam-6528	188	8	condition	condition	NOUN
ejpam-6528	188	9	m(vn	m(vn	PROPN
ejpam-6528	188	10	,	,	PUNCT
ejpam-6528	188	11	vn−1	vn−1	ADJ
ejpam-6528	188	12	,	,	PUNCT
ejpam-6528	188	13	vn−2	vn−2	PROPN
ejpam-6528	188	14	)	)	PUNCT
ejpam-6528	188	15	→	→	SYM
ejpam-6528	188	16	0	0	NUM
ejpam-6528	188	17	suggests	suggest	VERB
ejpam-6528	188	18	iterative	iterative	NOUN
ejpam-6528	188	19	algorithms	algorithm	NOUN
ejpam-6528	188	20	could	could	AUX
ejpam-6528	188	21	leverage	leverage	VERB
ejpam-6528	188	22	mr	mr	PROPN
ejpam-6528	188	23	-	-	PUNCT
ejpam-6528	188	24	metrics	metric	NOUN
ejpam-6528	188	25	to	to	PART
ejpam-6528	188	26	monitor	monitor	VERB
ejpam-6528	188	27	convergence	convergence	NOUN
ejpam-6528	188	28	in	in	ADP
ejpam-6528	188	29	multi	multi	ADJ
ejpam-6528	188	30	-	-	ADJ
ejpam-6528	188	31	agent	agent	ADJ
ejpam-6528	188	32	systems	system	NOUN
ejpam-6528	188	33	or	or	CCONJ
ejpam-6528	188	34	gradient	gradient	NOUN
ejpam-6528	188	35	-	-	PUNCT
ejpam-6528	188	36	based	base	VERB
ejpam-6528	188	37	optimization	optimization	NOUN
ejpam-6528	188	38	.	.	PUNCT
ejpam-6528	189	1	early	early	ADJ
ejpam-6528	189	2	stopping	stopping	NOUN
ejpam-6528	189	3	criteria	criterion	NOUN
ejpam-6528	189	4	could	could	AUX
ejpam-6528	189	5	adapt	adapt	VERB
ejpam-6528	189	6	this	this	DET
ejpam-6528	189	7	condition	condition	NOUN
ejpam-6528	189	8	numerically	numerically	ADV
ejpam-6528	189	9	.	.	PUNCT
ejpam-6528	190	1	•	•	NUM
ejpam-6528	190	2	speculative	speculative	ADJ
ejpam-6528	190	3	applications	application	NOUN
ejpam-6528	190	4	:	:	PUNCT
ejpam-6528	190	5	quantum	quantum	NOUN
ejpam-6528	190	6	computing	computing	NOUN
ejpam-6528	190	7	might	might	AUX
ejpam-6528	190	8	benefit	benefit	VERB
ejpam-6528	190	9	from	from	ADP
ejpam-6528	190	10	mr	mr	PROPN
ejpam-6528	190	11	-	-	PUNCT
ejpam-6528	190	12	metrics	metric	NOUN
ejpam-6528	190	13	’	'	PUNCT
ejpam-6528	190	14	triadic	triadic	ADJ
ejpam-6528	190	15	structure	structure	NOUN
ejpam-6528	190	16	,	,	PUNCT
ejpam-6528	190	17	as	as	SCONJ
ejpam-6528	190	18	quantum	quantum	NOUN
ejpam-6528	190	19	states	state	NOUN
ejpam-6528	190	20	naturally	naturally	ADV
ejpam-6528	190	21	encode	encode	VERB
ejpam-6528	190	22	multi	multi	ADJ
ejpam-6528	190	23	-	-	ADJ
ejpam-6528	190	24	particle	particle	ADJ
ejpam-6528	190	25	correlations	correlation	NOUN
ejpam-6528	190	26	.	.	PUNCT
ejpam-6528	191	1	similarly	similarly	ADV
ejpam-6528	191	2	,	,	PUNCT
ejpam-6528	191	3	in	in	ADP
ejpam-6528	191	4	topological	topological	ADJ
ejpam-6528	191	5	data	datum	NOUN
ejpam-6528	191	6	analysis	analysis	NOUN
ejpam-6528	191	7	,	,	PUNCT
ejpam-6528	191	8	persistent	persistent	ADJ
ejpam-6528	191	9	homology	homology	NOUN
ejpam-6528	191	10	could	could	AUX
ejpam-6528	191	11	be	be	AUX
ejpam-6528	191	12	extended	extend	VERB
ejpam-6528	191	13	using	use	VERB
ejpam-6528	191	14	mr	mr	NOUN
ejpam-6528	191	15	-	-	PUNCT
ejpam-6528	191	16	metrics	metric	NOUN
ejpam-6528	191	17	to	to	PART
ejpam-6528	191	18	capture	capture	VERB
ejpam-6528	191	19	ternary	ternary	ADJ
ejpam-6528	191	20	relationships	relationship	NOUN
ejpam-6528	191	21	in	in	ADP
ejpam-6528	191	22	simplicial	simplicial	ADJ
ejpam-6528	191	23	complexes	complex	NOUN
ejpam-6528	191	24	.	.	PUNCT
ejpam-6528	192	1	references	reference	NOUN
ejpam-6528	192	2	[	[	X
ejpam-6528	192	3	1	1	NUM
ejpam-6528	192	4	]	]	PUNCT
ejpam-6528	192	5	a.	a.	NOUN
ejpam-6528	192	6	malkawi	malkawi	PROPN
ejpam-6528	192	7	,	,	PUNCT
ejpam-6528	192	8	a.	a.	NOUN
ejpam-6528	192	9	talafhah	talafhah	PROPN
ejpam-6528	192	10	,	,	PUNCT
ejpam-6528	192	11	and	and	CCONJ
ejpam-6528	192	12	w.	w.	PROPN
ejpam-6528	192	13	shatanawi	shatanawi	PROPN
ejpam-6528	192	14	.	.	PUNCT
ejpam-6528	193	1	coincidence	coincidence	NOUN
ejpam-6528	193	2	and	and	CCONJ
ejpam-6528	193	3	fixed	fix	VERB
ejpam-6528	193	4	point	point	NOUN
ejpam-6528	193	5	results	result	NOUN
ejpam-6528	193	6	for	for	ADP
ejpam-6528	193	7	(	(	PUNCT
ejpam-6528	193	8	ψ	ψ	X
ejpam-6528	193	9	,	,	PUNCT
ejpam-6528	193	10	l)-mweak	l)-mweak	NOUN
ejpam-6528	193	11	contraction	contraction	NOUN
ejpam-6528	193	12	mapping	mapping	NOUN
ejpam-6528	193	13	on	on	ADP
ejpam-6528	193	14	mb	mb	ADJ
ejpam-6528	193	15	-	-	ADJ
ejpam-6528	193	16	metric	metric	ADJ
ejpam-6528	193	17	spaces	space	NOUN
ejpam-6528	193	18	.	.	PUNCT
ejpam-6528	194	1	italian	italian	ADJ
ejpam-6528	194	2	journal	journal	NOUN
ejpam-6528	194	3	of	of	ADP
ejpam-6528	194	4	pure	pure	ADJ
ejpam-6528	194	5	and	and	CCONJ
ejpam-6528	194	6	applied	applied	ADJ
ejpam-6528	194	7	mathematics	mathematic	NOUN
ejpam-6528	194	8	,	,	PUNCT
ejpam-6528	194	9	(	(	PUNCT
ejpam-6528	194	10	47):751–768	47):751–768	NOUN
ejpam-6528	194	11	,	,	PUNCT
ejpam-6528	194	12	2022	2022	NUM
ejpam-6528	194	13	.	.	PUNCT
ejpam-6528	195	1	[	[	X
ejpam-6528	195	2	2	2	X
ejpam-6528	195	3	]	]	PUNCT
ejpam-6528	195	4	t.	t.	NOUN
ejpam-6528	195	5	qawasmeh	qawasmeh	NOUN
ejpam-6528	195	6	.	.	PUNCT
ejpam-6528	196	1	(	(	PUNCT
ejpam-6528	196	2	h	h	NOUN
ejpam-6528	196	3	,	,	PUNCT
ejpam-6528	196	4	ωb)-interpolative	ωb)-interpolative	ADJ
ejpam-6528	196	5	contractions	contraction	NOUN
ejpam-6528	196	6	in	in	ADP
ejpam-6528	196	7	ωb	ωb	NOUN
ejpam-6528	196	8	-	-	PUNCT
ejpam-6528	196	9	distance	distance	NOUN
ejpam-6528	196	10	mappings	mapping	NOUN
ejpam-6528	196	11	with	with	ADP
ejpam-6528	196	12	applications	application	NOUN
ejpam-6528	196	13	.	.	PUNCT
ejpam-6528	197	1	european	european	ADJ
ejpam-6528	197	2	journal	journal	PROPN
ejpam-6528	197	3	of	of	ADP
ejpam-6528	197	4	pure	pure	ADJ
ejpam-6528	197	5	and	and	CCONJ
ejpam-6528	197	6	applied	applied	ADJ
ejpam-6528	197	7	mathematics	mathematic	NOUN
ejpam-6528	197	8	,	,	PUNCT
ejpam-6528	197	9	16(3):1717–1730	16(3):1717–1730	NUM
ejpam-6528	197	10	,	,	PUNCT
ejpam-6528	197	11	2023	2023	NUM
ejpam-6528	197	12	.	.	PUNCT
ejpam-6528	198	1	[	[	X
ejpam-6528	198	2	3	3	NUM
ejpam-6528	198	3	]	]	X
ejpam-6528	198	4	a.	a.	NOUN
ejpam-6528	198	5	malkawi	malkawi	PROPN
ejpam-6528	198	6	,	,	PUNCT
ejpam-6528	198	7	a.	a.	NOUN
ejpam-6528	198	8	tallafha	tallafha	NOUN
ejpam-6528	198	9	,	,	PUNCT
ejpam-6528	198	10	and	and	CCONJ
ejpam-6528	198	11	w.	w.	PROPN
ejpam-6528	198	12	shatanawi	shatanawi	PROPN
ejpam-6528	198	13	.	.	PUNCT
ejpam-6528	199	1	coincidence	coincidence	NOUN
ejpam-6528	199	2	and	and	CCONJ
ejpam-6528	199	3	fixed	fix	VERB
ejpam-6528	199	4	point	point	NOUN
ejpam-6528	199	5	results	result	NOUN
ejpam-6528	199	6	for	for	ADP
ejpam-6528	199	7	generalized	generalized	ADJ
ejpam-6528	199	8	weak	weak	ADJ
ejpam-6528	199	9	contraction	contraction	NOUN
ejpam-6528	199	10	mapping	mapping	NOUN
ejpam-6528	199	11	on	on	ADP
ejpam-6528	199	12	b	b	NOUN
ejpam-6528	199	13	-	-	PUNCT
ejpam-6528	199	14	metric	metric	ADJ
ejpam-6528	199	15	spaces	space	NOUN
ejpam-6528	199	16	.	.	PUNCT
ejpam-6528	200	1	nonlinear	nonlinear	ADJ
ejpam-6528	200	2	functional	functional	ADJ
ejpam-6528	200	3	analysis	analysis	NOUN
ejpam-6528	200	4	and	and	CCONJ
ejpam-6528	200	5	applications	application	NOUN
ejpam-6528	200	6	,	,	PUNCT
ejpam-6528	200	7	26(1):177–195	26(1):177–195	NOUN
ejpam-6528	200	8	,	,	PUNCT
ejpam-6528	200	9	2021	2021	NUM
ejpam-6528	200	10	.	.	PUNCT
ejpam-6528	201	1	a.	a.	PROPN
ejpam-6528	201	2	malkawi	malkawi	PROPN
ejpam-6528	201	3	,	,	PUNCT
ejpam-6528	201	4	a.	a.	PROPN
ejpam-6528	201	5	rabaiah	rabaiah	PROPN
ejpam-6528	201	6	/	/	SYM
ejpam-6528	201	7	eur	eur	PROPN
ejpam-6528	201	8	.	.	PUNCT
ejpam-6528	202	1	j.	j.	PROPN
ejpam-6528	202	2	pure	pure	PROPN
ejpam-6528	202	3	appl	appl	PROPN
ejpam-6528	202	4	.	.	PROPN
ejpam-6528	202	5	math	math	PROPN
ejpam-6528	202	6	,	,	PUNCT
ejpam-6528	202	7	18	18	NUM
ejpam-6528	202	8	(	(	PUNCT
ejpam-6528	202	9	3	3	NUM
ejpam-6528	202	10	)	)	PUNCT
ejpam-6528	202	11	(	(	PUNCT
ejpam-6528	202	12	2025	2025	NUM
ejpam-6528	202	13	)	)	PUNCT
ejpam-6528	202	14	,	,	PUNCT
ejpam-6528	202	15	6528	6528	NUM
ejpam-6528	202	16	11	11	NUM
ejpam-6528	202	17	of	of	ADP
ejpam-6528	202	18	12	12	NUM
ejpam-6528	202	19	[	[	SYM
ejpam-6528	202	20	4	4	NUM
ejpam-6528	202	21	]	]	PUNCT
ejpam-6528	202	22	r.	r.	PROPN
ejpam-6528	202	23	al	al	PROPN
ejpam-6528	202	24	-	-	PUNCT
ejpam-6528	202	25	deiakeh	deiakeh	ADJ
ejpam-6528	202	26	,	,	PUNCT
ejpam-6528	202	27	m.	m.	NOUN
ejpam-6528	202	28	alquran	alquran	PROPN
ejpam-6528	202	29	,	,	PUNCT
ejpam-6528	202	30	m.	m.	PROPN
ejpam-6528	202	31	ali	ali	PROPN
ejpam-6528	202	32	,	,	PUNCT
ejpam-6528	202	33	s.	s.	PROPN
ejpam-6528	202	34	qureshi	qureshi	PROPN
ejpam-6528	202	35	,	,	PUNCT
ejpam-6528	202	36	s.	s.	PROPN
ejpam-6528	202	37	momani	momani	PROPN
ejpam-6528	202	38	,	,	PUNCT
ejpam-6528	202	39	and	and	CCONJ
ejpam-6528	202	40	a.	a.	NOUN
ejpam-6528	202	41	a.	a.	PROPN
ejpam-6528	202	42	r.	r.	PROPN
ejpam-6528	202	43	malkawi	malkawi	PROPN
ejpam-6528	202	44	.	.	PROPN
ejpam-6528	203	1	lie	lie	PROPN
ejpam-6528	203	2	symmetry	symmetry	NOUN
ejpam-6528	203	3	,	,	PUNCT
ejpam-6528	203	4	convergence	convergence	NOUN
ejpam-6528	203	5	analysis	analysis	NOUN
ejpam-6528	203	6	,	,	PUNCT
ejpam-6528	203	7	explicit	explicit	ADJ
ejpam-6528	203	8	solutions	solution	NOUN
ejpam-6528	203	9	,	,	PUNCT
ejpam-6528	203	10	and	and	CCONJ
ejpam-6528	203	11	conservation	conservation	NOUN
ejpam-6528	203	12	laws	law	NOUN
ejpam-6528	203	13	for	for	ADP
ejpam-6528	203	14	the	the	DET
ejpam-6528	203	15	timefractional	timefractional	ADJ
ejpam-6528	203	16	modified	modify	VERB
ejpam-6528	203	17	benjamin	benjamin	PROPN
ejpam-6528	203	18	-	-	PUNCT
ejpam-6528	203	19	bona	bona	ADJ
ejpam-6528	203	20	-	-	PUNCT
ejpam-6528	203	21	mahony	mahony	NOUN
ejpam-6528	203	22	equation	equation	NOUN
ejpam-6528	203	23	.	.	PUNCT
ejpam-6528	204	1	journal	journal	PROPN
ejpam-6528	204	2	of	of	ADP
ejpam-6528	204	3	applied	apply	VERB
ejpam-6528	204	4	mathematics	mathematic	NOUN
ejpam-6528	204	5	and	and	CCONJ
ejpam-6528	204	6	computational	computational	ADJ
ejpam-6528	204	7	mechanics	mechanic	NOUN
ejpam-6528	204	8	,	,	PUNCT
ejpam-6528	204	9	23(1):19–31	23(1):19–31	NUM
ejpam-6528	204	10	,	,	PUNCT
ejpam-6528	204	11	2024	2024	NUM
ejpam-6528	204	12	.	.	PUNCT
ejpam-6528	205	1	[	[	X
ejpam-6528	205	2	5	5	X
ejpam-6528	205	3	]	]	PUNCT
ejpam-6528	205	4	t.	t.	NOUN
ejpam-6528	205	5	qawasmeh	qawasmeh	NOUN
ejpam-6528	205	6	.	.	PUNCT
ejpam-6528	206	1	h	h	NOUN
ejpam-6528	206	2	-	-	PUNCT
ejpam-6528	206	3	simulation	simulation	NOUN
ejpam-6528	206	4	functions	function	NOUN
ejpam-6528	206	5	and	and	CCONJ
ejpam-6528	206	6	ωb	ωb	NOUN
ejpam-6528	206	7	-	-	PUNCT
ejpam-6528	206	8	distance	distance	NOUN
ejpam-6528	206	9	mappings	mapping	NOUN
ejpam-6528	206	10	in	in	ADP
ejpam-6528	206	11	the	the	DET
ejpam-6528	206	12	setting	setting	NOUN
ejpam-6528	206	13	of	of	ADP
ejpam-6528	206	14	gb	gb	ADV
ejpam-6528	206	15	-	-	PUNCT
ejpam-6528	206	16	metric	metric	ADJ
ejpam-6528	206	17	spaces	space	NOUN
ejpam-6528	206	18	and	and	CCONJ
ejpam-6528	206	19	application	application	NOUN
ejpam-6528	206	20	.	.	PUNCT
ejpam-6528	207	1	nonlinear	nonlinear	ADJ
ejpam-6528	207	2	functional	functional	ADJ
ejpam-6528	207	3	analysis	analysis	NOUN
ejpam-6528	207	4	and	and	CCONJ
ejpam-6528	207	5	applications	application	NOUN
ejpam-6528	207	6	,	,	PUNCT
ejpam-6528	207	7	28(2):557–570	28(2):557–570	NOUN
ejpam-6528	207	8	,	,	PUNCT
ejpam-6528	207	9	2023	2023	NUM
ejpam-6528	207	10	.	.	PUNCT
ejpam-6528	208	1	[	[	X
ejpam-6528	208	2	6	6	NUM
ejpam-6528	208	3	]	]	PUNCT
ejpam-6528	208	4	a.	a.	NOUN
ejpam-6528	208	5	bataihah	bataihah	PROPN
ejpam-6528	208	6	and	and	CCONJ
ejpam-6528	208	7	t.	t.	NOUN
ejpam-6528	208	8	qawasmeh	qawasmeh	NOUN
ejpam-6528	208	9	.	.	PUNCT
ejpam-6528	209	1	a	a	DET
ejpam-6528	209	2	new	new	ADJ
ejpam-6528	209	3	type	type	NOUN
ejpam-6528	209	4	of	of	ADP
ejpam-6528	209	5	distance	distance	NOUN
ejpam-6528	209	6	spaces	space	NOUN
ejpam-6528	209	7	and	and	CCONJ
ejpam-6528	209	8	fixed	fix	VERB
ejpam-6528	209	9	point	point	NOUN
ejpam-6528	209	10	results	result	NOUN
ejpam-6528	209	11	.	.	PUNCT
ejpam-6528	210	1	journal	journal	NOUN
ejpam-6528	210	2	of	of	ADP
ejpam-6528	210	3	mathematical	mathematical	ADJ
ejpam-6528	210	4	analysis	analysis	NOUN
ejpam-6528	210	5	,	,	PUNCT
ejpam-6528	210	6	15(4):81–90	15(4):81–90	NUM
ejpam-6528	210	7	,	,	PUNCT
ejpam-6528	210	8	2024	2024	NUM
ejpam-6528	210	9	.	.	PUNCT
ejpam-6528	211	1	[	[	X
ejpam-6528	211	2	7	7	X
ejpam-6528	211	3	]	]	PUNCT
ejpam-6528	211	4	t.	t.	NOUN
ejpam-6528	211	5	qawasmeh	qawasmeh	NOUN
ejpam-6528	211	6	,	,	PUNCT
ejpam-6528	211	7	w.	w.	PROPN
ejpam-6528	211	8	shatanawi	shatanawi	PROPN
ejpam-6528	211	9	,	,	PUNCT
ejpam-6528	211	10	a.	a.	NOUN
ejpam-6528	211	11	bataihah	bataihah	PROPN
ejpam-6528	211	12	,	,	PUNCT
ejpam-6528	211	13	and	and	CCONJ
ejpam-6528	211	14	a.	a.	NOUN
ejpam-6528	211	15	tallafha	tallafha	NOUN
ejpam-6528	211	16	.	.	PUNCT
ejpam-6528	212	1	fixed	fix	VERB
ejpam-6528	212	2	point	point	NOUN
ejpam-6528	212	3	results	result	NOUN
ejpam-6528	212	4	and	and	CCONJ
ejpam-6528	212	5	(	(	PUNCT
ejpam-6528	212	6	α	α	NOUN
ejpam-6528	212	7	,	,	PUNCT
ejpam-6528	212	8	β)-triangular	β)-triangular	ADJ
ejpam-6528	212	9	admissibility	admissibility	NOUN
ejpam-6528	212	10	in	in	ADP
ejpam-6528	212	11	the	the	DET
ejpam-6528	212	12	frame	frame	NOUN
ejpam-6528	212	13	of	of	ADP
ejpam-6528	212	14	complete	complete	ADJ
ejpam-6528	212	15	extended	extended	ADJ
ejpam-6528	212	16	b	b	NOUN
ejpam-6528	212	17	-	-	PUNCT
ejpam-6528	212	18	metric	metric	ADJ
ejpam-6528	212	19	spaces	space	NOUN
ejpam-6528	212	20	and	and	CCONJ
ejpam-6528	212	21	application	application	NOUN
ejpam-6528	212	22	.	.	PUNCT
ejpam-6528	213	1	u.p.b	u.p.b	PROPN
ejpam-6528	213	2	.	.	PUNCT
ejpam-6528	214	1	scientific	scientific	ADJ
ejpam-6528	214	2	bulletin	bulletin	NOUN
ejpam-6528	214	3	,	,	PUNCT
ejpam-6528	214	4	series	series	PROPN
ejpam-6528	214	5	a	a	PROPN
ejpam-6528	214	6	,	,	PUNCT
ejpam-6528	214	7	83(1):113–124	83(1):113–124	PROPN
ejpam-6528	214	8	,	,	PUNCT
ejpam-6528	214	9	2021	2021	NUM
ejpam-6528	214	10	.	.	PUNCT
ejpam-6528	215	1	[	[	X
ejpam-6528	215	2	8	8	NUM
ejpam-6528	215	3	]	]	PUNCT
ejpam-6528	215	4	a.	a.	NOUN
ejpam-6528	215	5	bataihah	bataihah	PROPN
ejpam-6528	215	6	,	,	PUNCT
ejpam-6528	215	7	w.	w.	PROPN
ejpam-6528	215	8	shatanawi	shatanawi	PROPN
ejpam-6528	215	9	,	,	PUNCT
ejpam-6528	215	10	and	and	CCONJ
ejpam-6528	215	11	a.	a.	NOUN
ejpam-6528	215	12	tallafha	tallafha	NOUN
ejpam-6528	215	13	.	.	PUNCT
ejpam-6528	216	1	fixed	fix	VERB
ejpam-6528	216	2	point	point	NOUN
ejpam-6528	216	3	results	result	NOUN
ejpam-6528	216	4	with	with	ADP
ejpam-6528	216	5	simulation	simulation	NOUN
ejpam-6528	216	6	functions	function	NOUN
ejpam-6528	216	7	.	.	PUNCT
ejpam-6528	217	1	nonlinear	nonlinear	ADJ
ejpam-6528	217	2	functional	functional	ADJ
ejpam-6528	217	3	analysis	analysis	NOUN
ejpam-6528	217	4	and	and	CCONJ
ejpam-6528	217	5	applications	application	NOUN
ejpam-6528	217	6	,	,	PUNCT
ejpam-6528	217	7	25(1):13–23	25(1):13–23	NUM
ejpam-6528	217	8	,	,	PUNCT
ejpam-6528	217	9	2020	2020	NUM
ejpam-6528	217	10	.	.	PUNCT
ejpam-6528	218	1	[	[	X
ejpam-6528	218	2	9	9	NUM
ejpam-6528	218	3	]	]	PUNCT
ejpam-6528	218	4	k.	k.	PROPN
ejpam-6528	218	5	abodayeh	abodayeh	PROPN
ejpam-6528	218	6	,	,	PUNCT
ejpam-6528	218	7	a.	a.	PROPN
ejpam-6528	218	8	bataihah	bataihah	PROPN
ejpam-6528	218	9	,	,	PUNCT
ejpam-6528	218	10	w.	w.	PROPN
ejpam-6528	218	11	shatanawi	shatanawi	PROPN
ejpam-6528	218	12	,	,	PUNCT
ejpam-6528	218	13	and	and	CCONJ
ejpam-6528	218	14	a.	a.	PROPN
ejpam-6528	218	15	h.	h.	PROPN
ejpam-6528	218	16	ansari	ansari	PROPN
ejpam-6528	218	17	.	.	PUNCT
ejpam-6528	219	1	some	some	DET
ejpam-6528	219	2	fixed	fix	VERB
ejpam-6528	219	3	point	point	NOUN
ejpam-6528	219	4	and	and	CCONJ
ejpam-6528	219	5	common	common	ADJ
ejpam-6528	219	6	fixed	fix	VERB
ejpam-6528	219	7	point	point	NOUN
ejpam-6528	219	8	results	result	NOUN
ejpam-6528	219	9	through	through	ADP
ejpam-6528	219	10	ω	ω	NOUN
ejpam-6528	219	11	-	-	PUNCT
ejpam-6528	219	12	distance	distance	NOUN
ejpam-6528	219	13	under	under	ADP
ejpam-6528	219	14	nonlinear	nonlinear	ADJ
ejpam-6528	219	15	contractions	contraction	NOUN
ejpam-6528	219	16	.	.	PUNCT
ejpam-6528	220	1	gazi	gazi	PROPN
ejpam-6528	220	2	university	university	PROPN
ejpam-6528	220	3	journal	journal	PROPN
ejpam-6528	220	4	of	of	ADP
ejpam-6528	220	5	science	science	NOUN
ejpam-6528	220	6	,	,	PUNCT
ejpam-6528	220	7	30(1):293–302	30(1):293–302	NOUN
ejpam-6528	220	8	,	,	PUNCT
ejpam-6528	220	9	2017	2017	NUM
ejpam-6528	220	10	.	.	PUNCT
ejpam-6528	221	1	[	[	X
ejpam-6528	221	2	10	10	NUM
ejpam-6528	221	3	]	]	X
ejpam-6528	221	4	a.	a.	NOUN
ejpam-6528	221	5	bataihah	bataihah	PROPN
ejpam-6528	221	6	,	,	PUNCT
ejpam-6528	221	7	a.	a.	NOUN
ejpam-6528	221	8	tallafha	tallafha	NOUN
ejpam-6528	221	9	,	,	PUNCT
ejpam-6528	221	10	and	and	CCONJ
ejpam-6528	221	11	w.	w.	PROPN
ejpam-6528	221	12	shatanawi	shatanawi	PROPN
ejpam-6528	221	13	.	.	PUNCT
ejpam-6528	222	1	fixed	fix	VERB
ejpam-6528	222	2	point	point	NOUN
ejpam-6528	222	3	results	result	NOUN
ejpam-6528	222	4	with	with	ADP
ejpam-6528	222	5	ω	ω	NOUN
ejpam-6528	222	6	-	-	PUNCT
ejpam-6528	222	7	distance	distance	NOUN
ejpam-6528	222	8	by	by	ADP
ejpam-6528	222	9	utilizing	utilize	VERB
ejpam-6528	222	10	simulation	simulation	NOUN
ejpam-6528	222	11	functions	function	NOUN
ejpam-6528	222	12	.	.	PUNCT
ejpam-6528	223	1	italian	italian	ADJ
ejpam-6528	223	2	journal	journal	NOUN
ejpam-6528	223	3	of	of	ADP
ejpam-6528	223	4	pure	pure	ADJ
ejpam-6528	223	5	and	and	CCONJ
ejpam-6528	223	6	applied	applied	ADJ
ejpam-6528	223	7	mathematics	mathematic	NOUN
ejpam-6528	223	8	,	,	PUNCT
ejpam-6528	223	9	(	(	PUNCT
ejpam-6528	223	10	43):185–196	43):185–196	NOUN
ejpam-6528	223	11	,	,	PUNCT
ejpam-6528	223	12	2017	2017	NUM
ejpam-6528	223	13	.	.	PUNCT
ejpam-6528	224	1	[	[	X
ejpam-6528	224	2	11	11	NUM
ejpam-6528	224	3	]	]	PUNCT
ejpam-6528	224	4	a.	a.	NOUN
ejpam-6528	224	5	rabaiah	rabaiah	PROPN
ejpam-6528	224	6	,	,	PUNCT
ejpam-6528	224	7	a.	a.	NOUN
ejpam-6528	224	8	tallafha	tallafha	NOUN
ejpam-6528	224	9	,	,	PUNCT
ejpam-6528	224	10	and	and	CCONJ
ejpam-6528	224	11	w.	w.	PROPN
ejpam-6528	224	12	shatanawi	shatanawi	PROPN
ejpam-6528	224	13	.	.	PUNCT
ejpam-6528	225	1	common	common	ADJ
ejpam-6528	225	2	fixed	fix	VERB
ejpam-6528	225	3	point	point	NOUN
ejpam-6528	225	4	results	result	NOUN
ejpam-6528	225	5	for	for	ADP
ejpam-6528	225	6	mappings	mapping	NOUN
ejpam-6528	225	7	under	under	ADP
ejpam-6528	225	8	nonlinear	nonlinear	ADJ
ejpam-6528	225	9	contraction	contraction	NOUN
ejpam-6528	225	10	of	of	ADP
ejpam-6528	225	11	cyclic	cyclic	ADJ
ejpam-6528	225	12	form	form	NOUN
ejpam-6528	225	13	in	in	ADP
ejpam-6528	225	14	b	b	NOUN
ejpam-6528	225	15	-	-	ADJ
ejpam-6528	225	16	metric	metric	ADJ
ejpam-6528	225	17	spaces	space	NOUN
ejpam-6528	225	18	.	.	PUNCT
ejpam-6528	226	1	advances	advance	NOUN
ejpam-6528	226	2	in	in	ADP
ejpam-6528	226	3	mathematics	mathematic	NOUN
ejpam-6528	226	4	:	:	PUNCT
ejpam-6528	226	5	scientific	scientific	ADJ
ejpam-6528	226	6	journal	journal	NOUN
ejpam-6528	226	7	,	,	PUNCT
ejpam-6528	226	8	26(2):289–301	26(2):289–301	PROPN
ejpam-6528	226	9	,	,	PUNCT
ejpam-6528	226	10	2021	2021	NUM
ejpam-6528	226	11	.	.	PUNCT
ejpam-6528	227	1	[	[	X
ejpam-6528	227	2	12	12	NUM
ejpam-6528	227	3	]	]	X
ejpam-6528	227	4	w.	w.	PROPN
ejpam-6528	227	5	shatanawi	shatanawi	PROPN
ejpam-6528	227	6	,	,	PUNCT
ejpam-6528	227	7	t.	t.	NOUN
ejpam-6528	227	8	qawasmeh	qawasmeh	NOUN
ejpam-6528	227	9	,	,	PUNCT
ejpam-6528	227	10	a.	a.	NOUN
ejpam-6528	227	11	bataihah	bataihah	PROPN
ejpam-6528	227	12	,	,	PUNCT
ejpam-6528	227	13	and	and	CCONJ
ejpam-6528	227	14	a.	a.	NOUN
ejpam-6528	227	15	tallafha	tallafha	NOUN
ejpam-6528	227	16	.	.	PUNCT
ejpam-6528	228	1	new	new	ADJ
ejpam-6528	228	2	contractions	contraction	NOUN
ejpam-6528	228	3	and	and	CCONJ
ejpam-6528	228	4	some	some	DET
ejpam-6528	228	5	fixed	fix	VERB
ejpam-6528	228	6	point	point	NOUN
ejpam-6528	228	7	results	result	NOUN
ejpam-6528	228	8	with	with	ADP
ejpam-6528	228	9	application	application	NOUN
ejpam-6528	228	10	based	base	VERB
ejpam-6528	228	11	on	on	ADP
ejpam-6528	228	12	extended	extended	ADJ
ejpam-6528	228	13	quasi	quasi	ADJ
ejpam-6528	228	14	b	b	NOUN
ejpam-6528	228	15	-	-	ADJ
ejpam-6528	228	16	metric	metric	ADJ
ejpam-6528	228	17	spaces	space	NOUN
ejpam-6528	228	18	.	.	PUNCT
ejpam-6528	229	1	u.p.b	u.p.b	ADJ
ejpam-6528	229	2	.	.	PUNCT
ejpam-6528	230	1	scientific	scientific	ADJ
ejpam-6528	230	2	bulletin	bulletin	NOUN
ejpam-6528	230	3	,	,	PUNCT
ejpam-6528	230	4	series	series	PROPN
ejpam-6528	230	5	a	a	PROPN
ejpam-6528	230	6	,	,	PUNCT
ejpam-6528	230	7	83(2):1223–7027	83(2):1223–7027	NUM
ejpam-6528	230	8	,	,	PUNCT
ejpam-6528	230	9	2021	2021	NUM
ejpam-6528	230	10	.	.	PUNCT
ejpam-6528	231	1	[	[	X
ejpam-6528	231	2	13	13	NUM
ejpam-6528	231	3	]	]	PUNCT
ejpam-6528	231	4	k.	k.	PROPN
ejpam-6528	231	5	abodayeh	abodayeh	PROPN
ejpam-6528	231	6	,	,	PUNCT
ejpam-6528	231	7	a.	a.	PROPN
ejpam-6528	231	8	bataihah	bataihah	PROPN
ejpam-6528	231	9	,	,	PUNCT
ejpam-6528	231	10	and	and	CCONJ
ejpam-6528	231	11	w.	w.	PROPN
ejpam-6528	231	12	shatanawi	shatanawi	PROPN
ejpam-6528	231	13	.	.	PUNCT
ejpam-6528	232	1	generalized	generalize	VERB
ejpam-6528	232	2	ω	ω	NUM
ejpam-6528	232	3	-	-	PUNCT
ejpam-6528	232	4	distance	distance	NOUN
ejpam-6528	232	5	mappings	mapping	NOUN
ejpam-6528	232	6	and	and	CCONJ
ejpam-6528	232	7	some	some	DET
ejpam-6528	232	8	fixed	fix	VERB
ejpam-6528	232	9	point	point	NOUN
ejpam-6528	232	10	theorems	theorem	NOUN
ejpam-6528	232	11	.	.	PUNCT
ejpam-6528	233	1	u.p.b	u.p.b	PROPN
ejpam-6528	233	2	.	.	PUNCT
ejpam-6528	234	1	scientific	scientific	ADJ
ejpam-6528	234	2	bulletin	bulletin	NOUN
ejpam-6528	234	3	,	,	PUNCT
ejpam-6528	234	4	series	series	PROPN
ejpam-6528	234	5	a	a	PROPN
ejpam-6528	234	6	,	,	PUNCT
ejpam-6528	234	7	79:223–232	79:223–232	PROPN
ejpam-6528	234	8	,	,	PUNCT
ejpam-6528	234	9	2017	2017	NUM
ejpam-6528	234	10	.	.	PUNCT
ejpam-6528	235	1	[	[	X
ejpam-6528	235	2	14	14	NUM
ejpam-6528	235	3	]	]	PUNCT
ejpam-6528	235	4	t.	t.	NOUN
ejpam-6528	235	5	qawasmeh	qawasmeh	NOUN
ejpam-6528	235	6	,	,	PUNCT
ejpam-6528	235	7	w.	w.	PROPN
ejpam-6528	235	8	shatanawi	shatanawi	PROPN
ejpam-6528	235	9	,	,	PUNCT
ejpam-6528	235	10	and	and	CCONJ
ejpam-6528	235	11	a.	a.	NOUN
ejpam-6528	235	12	bataihah	bataihah	PROPN
ejpam-6528	235	13	.	.	PUNCT
ejpam-6528	236	1	common	common	ADJ
ejpam-6528	236	2	fixed	fix	VERB
ejpam-6528	236	3	point	point	NOUN
ejpam-6528	236	4	results	result	NOUN
ejpam-6528	236	5	for	for	ADP
ejpam-6528	236	6	rational	rational	ADJ
ejpam-6528	236	7	(	(	PUNCT
ejpam-6528	236	8	α	α	NOUN
ejpam-6528	236	9	,	,	PUNCT
ejpam-6528	236	10	β)ϕ-mω	β)ϕ-mω	NOUN
ejpam-6528	236	11	contractions	contraction	NOUN
ejpam-6528	236	12	in	in	ADP
ejpam-6528	236	13	complete	complete	ADJ
ejpam-6528	236	14	quasi	quasi	ADJ
ejpam-6528	236	15	metric	metric	ADJ
ejpam-6528	236	16	spaces	space	NOUN
ejpam-6528	236	17	.	.	PUNCT
ejpam-6528	237	1	mathematics	mathematic	NOUN
ejpam-6528	237	2	,	,	PUNCT
ejpam-6528	237	3	7(5):392	7(5):392	NUM
ejpam-6528	237	4	,	,	PUNCT
ejpam-6528	237	5	2017	2017	NUM
ejpam-6528	237	6	.	.	PUNCT
ejpam-6528	238	1	[	[	X
ejpam-6528	238	2	15	15	NUM
ejpam-6528	238	3	]	]	X
ejpam-6528	238	4	a.	a.	NOUN
ejpam-6528	238	5	a.	a.	PROPN
ejpam-6528	238	6	r.	r.	PROPN
ejpam-6528	238	7	m.	m.	PROPN
ejpam-6528	238	8	malkawi	malkawi	PROPN
ejpam-6528	238	9	.	.	PROPN
ejpam-6528	239	1	existence	existence	NOUN
ejpam-6528	239	2	and	and	CCONJ
ejpam-6528	239	3	uniqueness	uniqueness	NOUN
ejpam-6528	239	4	of	of	ADP
ejpam-6528	239	5	fixed	fix	VERB
ejpam-6528	239	6	points	point	NOUN
ejpam-6528	239	7	in	in	ADP
ejpam-6528	239	8	mr	mr	PROPN
ejpam-6528	239	9	-	-	PUNCT
ejpam-6528	239	10	metric	metric	ADJ
ejpam-6528	239	11	spaces	space	NOUN
ejpam-6528	239	12	and	and	CCONJ
ejpam-6528	239	13	their	their	PRON
ejpam-6528	239	14	applications	application	NOUN
ejpam-6528	239	15	.	.	PUNCT
ejpam-6528	240	1	european	european	ADJ
ejpam-6528	240	2	journal	journal	PROPN
ejpam-6528	240	3	of	of	ADP
ejpam-6528	240	4	pure	pure	ADJ
ejpam-6528	240	5	and	and	CCONJ
ejpam-6528	240	6	applied	applied	ADJ
ejpam-6528	240	7	mathematics	mathematic	NOUN
ejpam-6528	240	8	,	,	PUNCT
ejpam-6528	240	9	18(2):6077	18(2):6077	NUM
ejpam-6528	240	10	,	,	PUNCT
ejpam-6528	240	11	2025	2025	NUM
ejpam-6528	240	12	.	.	PUNCT
ejpam-6528	241	1	[	[	X
ejpam-6528	241	2	16	16	NUM
ejpam-6528	241	3	]	]	PUNCT
ejpam-6528	241	4	a.	a.	NOUN
ejpam-6528	241	5	a.	a.	PROPN
ejpam-6528	241	6	r.	r.	PROPN
ejpam-6528	241	7	m.	m.	PROPN
ejpam-6528	241	8	malkawi	malkawi	PROPN
ejpam-6528	241	9	.	.	PROPN
ejpam-6528	242	1	convergence	convergence	NOUN
ejpam-6528	242	2	and	and	CCONJ
ejpam-6528	242	3	fixed	fix	VERB
ejpam-6528	242	4	points	point	NOUN
ejpam-6528	242	5	of	of	ADP
ejpam-6528	242	6	self	self	NOUN
ejpam-6528	242	7	-	-	PUNCT
ejpam-6528	242	8	mappings	mapping	NOUN
ejpam-6528	242	9	in	in	ADP
ejpam-6528	242	10	mr	mr	PROPN
ejpam-6528	242	11	-	-	PUNCT
ejpam-6528	242	12	metric	metric	ADJ
ejpam-6528	242	13	spaces	space	NOUN
ejpam-6528	242	14	:	:	PUNCT
ejpam-6528	242	15	theory	theory	NOUN
ejpam-6528	242	16	and	and	CCONJ
ejpam-6528	242	17	applications	application	NOUN
ejpam-6528	242	18	.	.	PUNCT
ejpam-6528	243	1	european	european	ADJ
ejpam-6528	243	2	journal	journal	PROPN
ejpam-6528	243	3	of	of	ADP
ejpam-6528	243	4	pure	pure	ADJ
ejpam-6528	243	5	and	and	CCONJ
ejpam-6528	243	6	applied	applied	ADJ
ejpam-6528	243	7	mathematics	mathematic	NOUN
ejpam-6528	243	8	,	,	PUNCT
ejpam-6528	243	9	18(2):5952	18(2):5952	NUM
ejpam-6528	243	10	,	,	PUNCT
ejpam-6528	243	11	2025	2025	NUM
ejpam-6528	243	12	.	.	PUNCT
ejpam-6528	244	1	[	[	X
ejpam-6528	244	2	17	17	NUM
ejpam-6528	244	3	]	]	PUNCT
ejpam-6528	244	4	a.	a.	NOUN
ejpam-6528	244	5	a.	a.	PROPN
ejpam-6528	244	6	r.	r.	PROPN
ejpam-6528	244	7	m.	m.	PROPN
ejpam-6528	244	8	malkawi	malkawi	PROPN
ejpam-6528	244	9	.	.	PUNCT
ejpam-6528	244	10	fixed	fix	VERB
ejpam-6528	244	11	point	point	NOUN
ejpam-6528	244	12	theorem	theorem	VERB
ejpam-6528	244	13	in	in	ADP
ejpam-6528	244	14	mr	mr	PROPN
ejpam-6528	244	15	-	-	PUNCT
ejpam-6528	244	16	metric	metric	ADJ
ejpam-6528	244	17	spaces	space	NOUN
ejpam-6528	244	18	via	via	ADP
ejpam-6528	244	19	integral	integral	ADJ
ejpam-6528	244	20	type	type	NOUN
ejpam-6528	244	21	contraction	contraction	NOUN
ejpam-6528	244	22	.	.	PUNCT
ejpam-6528	245	1	wseas	wseas	VERB
ejpam-6528	245	2	transactions	transaction	NOUN
ejpam-6528	245	3	on	on	ADP
ejpam-6528	245	4	mathematics	mathematic	NOUN
ejpam-6528	245	5	,	,	PUNCT
ejpam-6528	245	6	24:295–299	24:295–299	PROPN
ejpam-6528	245	7	,	,	PUNCT
ejpam-6528	245	8	2025	2025	NUM
ejpam-6528	245	9	.	.	PUNCT
ejpam-6528	246	1	[	[	X
ejpam-6528	246	2	18	18	NUM
ejpam-6528	246	3	]	]	PUNCT
ejpam-6528	246	4	a.	a.	NOUN
ejpam-6528	246	5	a.	a.	PROPN
ejpam-6528	246	6	r.	r.	PROPN
ejpam-6528	246	7	m.	m.	PROPN
ejpam-6528	246	8	malkawi	malkawi	PROPN
ejpam-6528	246	9	,	,	PUNCT
ejpam-6528	246	10	d.	d.	PROPN
ejpam-6528	246	11	mahmoud	mahmoud	PROPN
ejpam-6528	246	12	,	,	PUNCT
ejpam-6528	246	13	a.	a.	PROPN
ejpam-6528	246	14	m.	m.	PROPN
ejpam-6528	246	15	rabaiah	rabaiah	PROPN
ejpam-6528	246	16	,	,	PUNCT
ejpam-6528	246	17	r.	r.	PROPN
ejpam-6528	246	18	al	al	PROPN
ejpam-6528	246	19	-	-	PUNCT
ejpam-6528	246	20	deiakeh	deiakeh	PROPN
ejpam-6528	246	21	,	,	PUNCT
ejpam-6528	246	22	andw	andw	NOUN
ejpam-6528	246	23	.	.	PUNCT
ejpam-6528	246	24	shatanawi	shatanawi	PROPN
ejpam-6528	246	25	.	.	PUNCT
ejpam-6528	247	1	on	on	ADP
ejpam-6528	247	2	fixed	fix	VERB
ejpam-6528	247	3	point	point	NOUN
ejpam-6528	247	4	theorems	theorem	NOUN
ejpam-6528	247	5	in	in	ADP
ejpam-6528	247	6	mr	mr	PROPN
ejpam-6528	247	7	-	-	PUNCT
ejpam-6528	247	8	metric	metric	ADJ
ejpam-6528	247	9	spaces	space	NOUN
ejpam-6528	247	10	.	.	PUNCT
ejpam-6528	248	1	nonlinear	nonlinear	ADJ
ejpam-6528	248	2	functional	functional	ADJ
ejpam-6528	248	3	analysis	analysis	NOUN
ejpam-6528	248	4	and	and	CCONJ
ejpam-6528	248	5	applications	application	NOUN
ejpam-6528	248	6	,	,	PUNCT
ejpam-6528	248	7	29(4):1125–1136	29(4):1125–1136	NUM
ejpam-6528	248	8	,	,	PUNCT
ejpam-6528	248	9	2024	2024	NUM
ejpam-6528	248	10	.	.	PUNCT
ejpam-6528	249	1	[	[	X
ejpam-6528	249	2	19	19	NUM
ejpam-6528	249	3	]	]	X
ejpam-6528	249	4	g.	g.	PROPN
ejpam-6528	249	5	gharib	gharib	PROPN
ejpam-6528	249	6	,	,	PUNCT
ejpam-6528	249	7	a.	a.	PROPN
ejpam-6528	249	8	malkawi	malkawi	PROPN
ejpam-6528	249	9	,	,	PUNCT
ejpam-6528	249	10	a.	a.	PROPN
ejpam-6528	249	11	rabaiah	rabaiah	PROPN
ejpam-6528	249	12	,	,	PUNCT
ejpam-6528	249	13	w.	w.	PROPN
ejpam-6528	249	14	shatanawi	shatanawi	PROPN
ejpam-6528	249	15	,	,	PUNCT
ejpam-6528	249	16	and	and	CCONJ
ejpam-6528	249	17	m.	m.	NOUN
ejpam-6528	249	18	alsauodi	alsauodi	PROPN
ejpam-6528	249	19	.	.	PUNCT
ejpam-6528	250	1	a	a	DET
ejpam-6528	250	2	common	common	ADJ
ejpam-6528	250	3	a.	a.	NOUN
ejpam-6528	250	4	malkawi	malkawi	PROPN
ejpam-6528	250	5	,	,	PUNCT
ejpam-6528	250	6	a.	a.	PROPN
ejpam-6528	250	7	rabaiah	rabaiah	PROPN
ejpam-6528	250	8	/	/	SYM
ejpam-6528	250	9	eur	eur	PROPN
ejpam-6528	250	10	.	.	PUNCT
ejpam-6528	251	1	j.	j.	PROPN
ejpam-6528	251	2	pure	pure	PROPN
ejpam-6528	251	3	appl	appl	PROPN
ejpam-6528	251	4	.	.	PROPN
ejpam-6528	251	5	math	math	PROPN
ejpam-6528	251	6	,	,	PUNCT
ejpam-6528	251	7	18	18	NUM
ejpam-6528	251	8	(	(	PUNCT
ejpam-6528	251	9	3	3	NUM
ejpam-6528	251	10	)	)	PUNCT
ejpam-6528	251	11	(	(	PUNCT
ejpam-6528	251	12	2025	2025	NUM
ejpam-6528	251	13	)	)	PUNCT
ejpam-6528	251	14	,	,	PUNCT
ejpam-6528	251	15	6528	6528	NUM
ejpam-6528	251	16	12	12	NUM
ejpam-6528	251	17	of	of	ADP
ejpam-6528	251	18	12	12	NUM
ejpam-6528	251	19	fixed	fix	VERB
ejpam-6528	251	20	point	point	NOUN
ejpam-6528	251	21	theorem	theorem	VERB
ejpam-6528	251	22	in	in	ADP
ejpam-6528	251	23	m*-metric	m*-metric	ADV
ejpam-6528	252	1	space	space	NOUN
ejpam-6528	252	2	and	and	CCONJ
ejpam-6528	252	3	an	an	DET
ejpam-6528	252	4	application	application	NOUN
ejpam-6528	252	5	.	.	PUNCT
ejpam-6528	253	1	nonlinear	nonlinear	ADJ
ejpam-6528	253	2	functional	functional	ADJ
ejpam-6528	253	3	analysis	analysis	NOUN
ejpam-6528	253	4	and	and	CCONJ
ejpam-6528	253	5	applications	application	NOUN
ejpam-6528	253	6	,	,	PUNCT
ejpam-6528	253	7	27(2):289–308	27(2):289–308	NUM
ejpam-6528	253	8	,	,	PUNCT
ejpam-6528	253	9	2022	2022	NUM
ejpam-6528	253	10	.	.	PUNCT
ejpam-6528	254	1	[	[	X
ejpam-6528	254	2	20	20	NUM
ejpam-6528	254	3	]	]	PUNCT
ejpam-6528	254	4	s.	s.	PROPN
ejpam-6528	254	5	al	al	PROPN
ejpam-6528	254	6	-	-	PUNCT
ejpam-6528	254	7	sharif	sharif	PROPN
ejpam-6528	254	8	and	and	CCONJ
ejpam-6528	254	9	a.	a.	NOUN
ejpam-6528	254	10	malkawi	malkawi	PROPN
ejpam-6528	254	11	.	.	PUNCT
ejpam-6528	255	1	modification	modification	NOUN
ejpam-6528	255	2	of	of	ADP
ejpam-6528	255	3	conformable	conformable	ADJ
ejpam-6528	255	4	fractional	fractional	ADJ
ejpam-6528	255	5	derivative	derivative	NOUN
ejpam-6528	255	6	with	with	ADP
ejpam-6528	255	7	classical	classical	ADJ
ejpam-6528	255	8	properties	property	NOUN
ejpam-6528	255	9	.	.	PUNCT
ejpam-6528	256	1	italian	italian	ADJ
ejpam-6528	256	2	journal	journal	NOUN
ejpam-6528	256	3	of	of	ADP
ejpam-6528	256	4	pure	pure	ADJ
ejpam-6528	256	5	and	and	CCONJ
ejpam-6528	256	6	applied	applied	ADJ
ejpam-6528	256	7	mathematics	mathematic	NOUN
ejpam-6528	256	8	,	,	PUNCT
ejpam-6528	256	9	44:30–39	44:30–39	PROPN
ejpam-6528	256	10	,	,	PUNCT
ejpam-6528	256	11	2020	2020	NUM
ejpam-6528	256	12	.	.	PUNCT
ejpam-6528	257	1	[	[	X
ejpam-6528	257	2	21	21	NUM
ejpam-6528	257	3	]	]	X
ejpam-6528	257	4	g.	g.	PROPN
ejpam-6528	257	5	m.	m.	PROPN
ejpam-6528	257	6	gharib	gharib	PROPN
ejpam-6528	257	7	,	,	PUNCT
ejpam-6528	257	8	m.	m.	PROPN
ejpam-6528	257	9	s.	s.	PROPN
ejpam-6528	257	10	alsauodi	alsauodi	PROPN
ejpam-6528	257	11	,	,	PUNCT
ejpam-6528	257	12	a.	a.	NOUN
ejpam-6528	257	13	guiatni	guiatni	PROPN
ejpam-6528	257	14	,	,	PUNCT
ejpam-6528	257	15	m.	m.	NOUN
ejpam-6528	257	16	a.	a.	PROPN
ejpam-6528	257	17	al	al	PROPN
ejpam-6528	257	18	-	-	PUNCT
ejpam-6528	257	19	omari	omari	PROPN
ejpam-6528	257	20	,	,	PUNCT
ejpam-6528	257	21	and	and	CCONJ
ejpam-6528	257	22	a.	a.	PROPN
ejpam-6528	257	23	a.	a.	PROPN
ejpam-6528	257	24	r.	r.	PROPN
ejpam-6528	257	25	m.	m.	PROPN
ejpam-6528	257	26	malkawi	malkawi	PROPN
ejpam-6528	257	27	.	.	PUNCT
ejpam-6528	258	1	using	use	VERB
ejpam-6528	258	2	atomic	atomic	ADJ
ejpam-6528	258	3	solution	solution	NOUN
ejpam-6528	258	4	method	method	NOUN
ejpam-6528	258	5	to	to	PART
ejpam-6528	258	6	solve	solve	VERB
ejpam-6528	258	7	the	the	DET
ejpam-6528	258	8	fractional	fractional	ADJ
ejpam-6528	258	9	equations	equation	NOUN
ejpam-6528	258	10	.	.	PUNCT
ejpam-6528	259	1	springer	springer	NOUN
ejpam-6528	259	2	proceedings	proceeding	NOUN
ejpam-6528	259	3	in	in	ADP
ejpam-6528	259	4	mathematics	mathematic	NOUN
ejpam-6528	259	5	and	and	CCONJ
ejpam-6528	259	6	statistics	statistic	NOUN
ejpam-6528	259	7	,	,	PUNCT
ejpam-6528	259	8	418:123–129	418:123–129	NUM
ejpam-6528	259	9	,	,	PUNCT
ejpam-6528	259	10	2023	2023	NUM
ejpam-6528	259	11	.	.	PUNCT
ejpam-6528	260	1	[	[	X
ejpam-6528	260	2	22	22	NUM
ejpam-6528	260	3	]	]	X
ejpam-6528	260	4	a.	a.	NOUN
ejpam-6528	260	5	malkawi	malkawi	PROPN
ejpam-6528	260	6	,	,	PUNCT
ejpam-6528	260	7	a.	a.	PROPN
ejpam-6528	260	8	rabaiah	rabaiah	PROPN
ejpam-6528	260	9	,	,	PUNCT
ejpam-6528	260	10	w.	w.	PROPN
ejpam-6528	260	11	shatanawi	shatanawi	PROPN
ejpam-6528	260	12	,	,	PUNCT
ejpam-6528	260	13	and	and	CCONJ
ejpam-6528	260	14	a.	a.	NOUN
ejpam-6528	260	15	talafhah	talafhah	PROPN
ejpam-6528	260	16	.	.	PUNCT
ejpam-6528	261	1	mr	mr	PROPN
ejpam-6528	261	2	-	-	PUNCT
ejpam-6528	261	3	metric	metric	ADJ
ejpam-6528	261	4	spaces	space	NOUN
ejpam-6528	261	5	and	and	CCONJ
ejpam-6528	261	6	an	an	DET
ejpam-6528	261	7	application	application	NOUN
ejpam-6528	261	8	.	.	PUNCT
ejpam-6528	262	1	preprint	preprint	NOUN
ejpam-6528	262	2	,	,	PUNCT
ejpam-6528	262	3	2021	2021	NUM
ejpam-6528	262	4	.	.	PUNCT
ejpam-6528	263	1	[	[	X
ejpam-6528	263	2	23	23	NUM
ejpam-6528	263	3	]	]	X
ejpam-6528	263	4	w.	w.	PROPN
ejpam-6528	263	5	rudin	rudin	PROPN
ejpam-6528	263	6	.	.	PUNCT
ejpam-6528	264	1	real	real	ADJ
ejpam-6528	264	2	and	and	CCONJ
ejpam-6528	264	3	complex	complex	ADJ
ejpam-6528	264	4	analysis	analysis	NOUN
ejpam-6528	264	5	.	.	PUNCT
ejpam-6528	265	1	mcgraw	mcgraw	PROPN
ejpam-6528	265	2	-	-	PUNCT
ejpam-6528	265	3	hill	hill	PROPN
ejpam-6528	265	4	,	,	PUNCT
ejpam-6528	265	5	3rd	3rd	ADJ
ejpam-6528	265	6	edition	edition	NOUN
ejpam-6528	265	7	,	,	PUNCT
ejpam-6528	265	8	1987	1987	NUM
ejpam-6528	265	9	.	.	PUNCT
ejpam-6528	266	1	[	[	X
ejpam-6528	266	2	24	24	NUM
ejpam-6528	266	3	]	]	PUNCT
ejpam-6528	266	4	p.	p.	PROPN
ejpam-6528	266	5	r.	r.	PROPN
ejpam-6528	266	6	halmos	halmos	PROPN
ejpam-6528	266	7	.	.	PUNCT
ejpam-6528	267	1	measure	measure	NOUN
ejpam-6528	267	2	theory	theory	NOUN
ejpam-6528	267	3	.	.	PUNCT
ejpam-6528	268	1	springer	springer	NOUN
ejpam-6528	268	2	,	,	PUNCT
ejpam-6528	268	3	1974	1974	NUM
ejpam-6528	268	4	.	.	PUNCT
ejpam-6528	269	1	[	[	X
ejpam-6528	269	2	25	25	NUM
ejpam-6528	269	3	]	]	X
ejpam-6528	269	4	h.	h.	PROPN
ejpam-6528	269	5	l.	l.	PROPN
ejpam-6528	269	6	royden	royden	PROPN
ejpam-6528	269	7	and	and	CCONJ
ejpam-6528	269	8	p.	p.	PROPN
ejpam-6528	269	9	m.	m.	PROPN
ejpam-6528	269	10	fitzpatrick	fitzpatrick	PROPN
ejpam-6528	269	11	.	.	PUNCT
ejpam-6528	270	1	real	real	ADJ
ejpam-6528	270	2	analysis	analysis	NOUN
ejpam-6528	270	3	.	.	PUNCT
ejpam-6528	271	1	pearson	pearson	PROPN
ejpam-6528	271	2	,	,	PUNCT
ejpam-6528	271	3	4th	4th	ADJ
ejpam-6528	271	4	edition	edition	NOUN
ejpam-6528	271	5	,	,	PUNCT
ejpam-6528	271	6	2010	2010	NUM
ejpam-6528	271	7	.	.	PUNCT