id	sid	tid	token	lemma	pos
ejpam-7124	1	1	european	european	PROPN
ejpam-7124	1	2	journal	journal	PROPN
ejpam-7124	1	3	of	of	ADP
ejpam-7124	1	4	pure	pure	ADJ
ejpam-7124	1	5	and	and	CCONJ
ejpam-7124	1	6	applied	applied	ADJ
ejpam-7124	1	7	mathematics	mathematic	NOUN
ejpam-7124	1	8	2025	2025	NUM
ejpam-7124	1	9	,	,	PUNCT
ejpam-7124	1	10	vol	vol	NOUN
ejpam-7124	1	11	.	.	PROPN
ejpam-7124	1	12	18	18	NUM
ejpam-7124	1	13	,	,	PUNCT
ejpam-7124	1	14	issue	issue	NOUN
ejpam-7124	1	15	4	4	NUM
ejpam-7124	1	16	,	,	PUNCT
ejpam-7124	1	17	article	article	NOUN
ejpam-7124	1	18	number	number	NOUN
ejpam-7124	1	19	7124	7124	NUM
ejpam-7124	1	20	issn	issn	PROPN
ejpam-7124	1	21	1307	1307	NUM
ejpam-7124	1	22	-	-	SYM
ejpam-7124	1	23	5543	5543	NUM
ejpam-7124	1	24	–	–	PUNCT
ejpam-7124	1	25	ejpam.com	ejpam.com	X
ejpam-7124	1	26	published	publish	VERB
ejpam-7124	1	27	by	by	ADP
ejpam-7124	1	28	new	new	PROPN
ejpam-7124	1	29	york	york	PROPN
ejpam-7124	1	30	business	business	PROPN
ejpam-7124	1	31	global	global	ADJ
ejpam-7124	1	32	fixed	fix	VERB
ejpam-7124	1	33	point	point	NOUN
ejpam-7124	1	34	theorems	theorem	NOUN
ejpam-7124	1	35	for	for	ADP
ejpam-7124	1	36	(	(	PUNCT
ejpam-7124	1	37	β	β	X
ejpam-7124	1	38	,	,	PUNCT
ejpam-7124	1	39	φ)-expansive	φ)-expansive	PUNCT
ejpam-7124	1	40	mappings	mapping	NOUN
ejpam-7124	1	41	in	in	ADP
ejpam-7124	1	42	controlled	control	VERB
ejpam-7124	1	43	metric	metric	ADJ
ejpam-7124	1	44	space	space	NOUN
ejpam-7124	1	45	manoj	manoj	PROPN
ejpam-7124	1	46	kumar1	kumar1	PROPN
ejpam-7124	1	47	,	,	PUNCT
ejpam-7124	1	48	neha	neha	NOUN
ejpam-7124	1	49	bhardwaj1	bhardwaj1	NOUN
ejpam-7124	1	50	,	,	PUNCT
ejpam-7124	1	51	gunaseelan	gunaseelan	PROPN
ejpam-7124	1	52	mani2	mani2	PROPN
ejpam-7124	1	53	,	,	PUNCT
ejpam-7124	1	54	rajagopalan	rajagopalan	VERB
ejpam-7124	1	55	ramaswamy3,∗	ramaswamy3,∗	NOUN
ejpam-7124	1	56	,	,	PUNCT
ejpam-7124	1	57	khizar	khizar	NOUN
ejpam-7124	1	58	hayat	hayat	PROPN
ejpam-7124	1	59	khan3	khan3	PROPN
ejpam-7124	1	60	,	,	PUNCT
ejpam-7124	1	61	ola	ola	PROPN
ejpam-7124	1	62	ashour	ashour	NOUN
ejpam-7124	1	63	abdelnaby3,4	abdelnaby3,4	PROPN
ejpam-7124	1	64	1	1	NUM
ejpam-7124	1	65	department	department	NOUN
ejpam-7124	1	66	of	of	ADP
ejpam-7124	1	67	mathematics	mathematics	PROPN
ejpam-7124	1	68	,	,	PUNCT
ejpam-7124	1	69	maharishi	maharishi	PROPN
ejpam-7124	1	70	markandeshwar	markandeshwar	PROPN
ejpam-7124	1	71	(	(	PUNCT
ejpam-7124	1	72	deemed	deem	VERB
ejpam-7124	1	73	to	to	PART
ejpam-7124	1	74	be	be	AUX
ejpam-7124	1	75	university	university	NOUN
ejpam-7124	1	76	)	)	PUNCT
ejpam-7124	1	77	,	,	PUNCT
ejpam-7124	1	78	mullana	mullana	PROPN
ejpam-7124	1	79	,	,	PUNCT
ejpam-7124	1	80	ambala	ambala	PROPN
ejpam-7124	1	81	133207	133207	NUM
ejpam-7124	1	82	,	,	PUNCT
ejpam-7124	1	83	haryana	haryana	PROPN
ejpam-7124	1	84	,	,	PUNCT
ejpam-7124	1	85	india	india	PROPN
ejpam-7124	1	86	2	2	NUM
ejpam-7124	1	87	department	department	NOUN
ejpam-7124	1	88	of	of	ADP
ejpam-7124	1	89	mathematics	mathematic	NOUN
ejpam-7124	1	90	,	,	PUNCT
ejpam-7124	1	91	vel	vel	ADJ
ejpam-7124	1	92	tech	tech	NOUN
ejpam-7124	1	93	rengarajan	rengarajan	PROPN
ejpam-7124	1	94	dr	dr	PROPN
ejpam-7124	1	95	sagunthala	sagunthala	PROPN
ejpam-7124	1	96	r	r	PROPN
ejpam-7124	1	97	&	&	CCONJ
ejpam-7124	1	98	d	d	PROPN
ejpam-7124	1	99	institute	institute	PROPN
ejpam-7124	1	100	of	of	ADP
ejpam-7124	1	101	sciences	sciences	PROPN
ejpam-7124	1	102	,	,	PUNCT
ejpam-7124	1	103	chennai	chennai	NOUN
ejpam-7124	1	104	600062	600062	NUM
ejpam-7124	1	105	,	,	PUNCT
ejpam-7124	1	106	tamilnadu	tamilnadu	PROPN
ejpam-7124	1	107	,	,	PUNCT
ejpam-7124	1	108	india	india	PROPN
ejpam-7124	1	109	3	3	PROPN
ejpam-7124	1	110	department	department	PROPN
ejpam-7124	1	111	of	of	ADP
ejpam-7124	1	112	mathematics	mathematics	PROPN
ejpam-7124	1	113	,	,	PUNCT
ejpam-7124	1	114	college	college	NOUN
ejpam-7124	1	115	of	of	ADP
ejpam-7124	1	116	science	science	NOUN
ejpam-7124	1	117	and	and	CCONJ
ejpam-7124	1	118	humanities	humanity	NOUN
ejpam-7124	1	119	,	,	PUNCT
ejpam-7124	1	120	prince	prince	PROPN
ejpam-7124	1	121	sattam	sattam	PROPN
ejpam-7124	1	122	bin	bin	PROPN
ejpam-7124	1	123	abdulaziz	abdulaziz	PROPN
ejpam-7124	1	124	university	university	PROPN
ejpam-7124	1	125	,	,	PUNCT
ejpam-7124	1	126	alkharj	alkharj	VERB
ejpam-7124	1	127	11942	11942	NUM
ejpam-7124	1	128	,	,	PUNCT
ejpam-7124	1	129	saudi	saudi	PROPN
ejpam-7124	1	130	arabia	arabia	PROPN
ejpam-7124	1	131	4	4	NUM
ejpam-7124	1	132	department	department	NOUN
ejpam-7124	1	133	of	of	ADP
ejpam-7124	1	134	mathematics	mathematic	NOUN
ejpam-7124	1	135	,	,	PUNCT
ejpam-7124	1	136	faculty	faculty	NOUN
ejpam-7124	1	137	of	of	ADP
ejpam-7124	1	138	science	science	NOUN
ejpam-7124	1	139	,	,	PUNCT
ejpam-7124	1	140	university	university	PROPN
ejpam-7124	1	141	of	of	ADP
ejpam-7124	1	142	cairo	cairo	PROPN
ejpam-7124	1	143	,	,	PUNCT
ejpam-7124	1	144	egypt	egypt	PROPN
ejpam-7124	1	145	abstract	abstract	PROPN
ejpam-7124	1	146	.	.	PUNCT
ejpam-7124	2	1	in	in	ADP
ejpam-7124	2	2	the	the	DET
ejpam-7124	2	3	present	present	ADJ
ejpam-7124	2	4	manuscript	manuscript	NOUN
ejpam-7124	2	5	,	,	PUNCT
ejpam-7124	2	6	we	we	PRON
ejpam-7124	2	7	have	have	AUX
ejpam-7124	2	8	introduced	introduce	VERB
ejpam-7124	2	9	a	a	DET
ejpam-7124	2	10	new	new	ADJ
ejpam-7124	2	11	notion	notion	NOUN
ejpam-7124	2	12	of	of	ADP
ejpam-7124	2	13	(	(	PUNCT
ejpam-7124	2	14	β	β	X
ejpam-7124	2	15	,	,	PUNCT
ejpam-7124	2	16	φ)-expansive	φ)-expansive	PUNCT
ejpam-7124	2	17	mappings	mapping	NOUN
ejpam-7124	2	18	in	in	ADP
ejpam-7124	2	19	controlled	control	VERB
ejpam-7124	2	20	metric	metric	ADJ
ejpam-7124	2	21	space	space	NOUN
ejpam-7124	2	22	.	.	PUNCT
ejpam-7124	3	1	in	in	ADP
ejpam-7124	3	2	addition	addition	NOUN
ejpam-7124	3	3	to	to	ADP
ejpam-7124	3	4	this	this	PRON
ejpam-7124	3	5	,	,	PUNCT
ejpam-7124	3	6	some	some	DET
ejpam-7124	3	7	fixed	fix	VERB
ejpam-7124	3	8	point	point	NOUN
ejpam-7124	3	9	results	result	NOUN
ejpam-7124	3	10	are	be	AUX
ejpam-7124	3	11	also	also	ADV
ejpam-7124	3	12	proved	prove	VERB
ejpam-7124	3	13	with	with	ADP
ejpam-7124	3	14	the	the	DET
ejpam-7124	3	15	help	help	NOUN
ejpam-7124	3	16	of	of	ADP
ejpam-7124	3	17	this	this	DET
ejpam-7124	3	18	notion	notion	NOUN
ejpam-7124	3	19	.	.	PUNCT
ejpam-7124	4	1	some	some	DET
ejpam-7124	4	2	results	result	NOUN
ejpam-7124	4	3	from	from	ADP
ejpam-7124	4	4	the	the	DET
ejpam-7124	4	5	literature	literature	NOUN
ejpam-7124	4	6	are	be	AUX
ejpam-7124	4	7	also	also	ADV
ejpam-7124	4	8	deduced	deduce	VERB
ejpam-7124	4	9	from	from	ADP
ejpam-7124	4	10	our	our	PRON
ejpam-7124	4	11	main	main	ADJ
ejpam-7124	4	12	results	result	NOUN
ejpam-7124	4	13	.	.	PUNCT
ejpam-7124	5	1	an	an	DET
ejpam-7124	5	2	example	example	NOUN
ejpam-7124	5	3	is	be	AUX
ejpam-7124	5	4	also	also	ADV
ejpam-7124	5	5	provided	provide	VERB
ejpam-7124	5	6	to	to	PART
ejpam-7124	5	7	prove	prove	VERB
ejpam-7124	5	8	the	the	DET
ejpam-7124	5	9	validity	validity	NOUN
ejpam-7124	5	10	of	of	ADP
ejpam-7124	5	11	our	our	PRON
ejpam-7124	5	12	result	result	NOUN
ejpam-7124	5	13	.	.	PUNCT
ejpam-7124	6	1	as	as	ADP
ejpam-7124	6	2	an	an	DET
ejpam-7124	6	3	application	application	NOUN
ejpam-7124	6	4	an	an	DET
ejpam-7124	6	5	integral	integral	ADJ
ejpam-7124	6	6	equation	equation	NOUN
ejpam-7124	6	7	is	be	AUX
ejpam-7124	6	8	also	also	ADV
ejpam-7124	6	9	solved	solve	VERB
ejpam-7124	6	10	with	with	ADP
ejpam-7124	6	11	the	the	DET
ejpam-7124	6	12	help	help	NOUN
ejpam-7124	6	13	of	of	ADP
ejpam-7124	6	14	our	our	PRON
ejpam-7124	6	15	main	main	ADJ
ejpam-7124	6	16	result	result	NOUN
ejpam-7124	6	17	.	.	PUNCT
ejpam-7124	7	1	2020	2020	NUM
ejpam-7124	7	2	mathematics	mathematic	NOUN
ejpam-7124	7	3	subject	subject	NOUN
ejpam-7124	7	4	classifications	classification	NOUN
ejpam-7124	7	5	:	:	PUNCT
ejpam-7124	7	6	47h10	47h10	NUM
ejpam-7124	7	7	,	,	PUNCT
ejpam-7124	7	8	54h25	54h25	NUM
ejpam-7124	7	9	key	key	ADJ
ejpam-7124	7	10	words	word	NOUN
ejpam-7124	7	11	and	and	CCONJ
ejpam-7124	7	12	phrases	phrase	NOUN
ejpam-7124	7	13	:	:	PUNCT
ejpam-7124	7	14	controlled	control	VERB
ejpam-7124	7	15	metric	metric	ADJ
ejpam-7124	7	16	space	space	NOUN
ejpam-7124	7	17	,	,	PUNCT
ejpam-7124	7	18	fixed	fix	VERB
ejpam-7124	7	19	point	point	NOUN
ejpam-7124	7	20	,	,	PUNCT
ejpam-7124	7	21	(	(	PUNCT
ejpam-7124	7	22	β	β	X
ejpam-7124	7	23	,	,	PUNCT
ejpam-7124	7	24	φ)-expansive	φ)-expansive	PUNCT
ejpam-7124	7	25	mappings	mapping	NOUN
ejpam-7124	7	26	1	1	NUM
ejpam-7124	7	27	.	.	PUNCT
ejpam-7124	7	28	introduction	introduction	NOUN
ejpam-7124	7	29	in	in	ADP
ejpam-7124	7	30	1922	1922	NUM
ejpam-7124	7	31	,	,	PUNCT
ejpam-7124	7	32	banach	banach	NOUN
ejpam-7124	7	33	[	[	X
ejpam-7124	7	34	1	1	X
ejpam-7124	7	35	]	]	PUNCT
ejpam-7124	7	36	proved	prove	VERB
ejpam-7124	7	37	an	an	DET
ejpam-7124	7	38	interesting	interesting	ADJ
ejpam-7124	7	39	result	result	NOUN
ejpam-7124	7	40	in	in	ADP
ejpam-7124	7	41	fixed	fix	VERB
ejpam-7124	7	42	point	point	NOUN
ejpam-7124	7	43	theory	theory	NOUN
ejpam-7124	7	44	known	know	VERB
ejpam-7124	7	45	as	as	ADP
ejpam-7124	7	46	banach	banach	NOUN
ejpam-7124	7	47	contraction	contraction	NOUN
ejpam-7124	7	48	principle	principle	NOUN
ejpam-7124	7	49	.	.	PUNCT
ejpam-7124	8	1	this	this	DET
ejpam-7124	8	2	principle	principle	NOUN
ejpam-7124	8	3	proved	prove	VERB
ejpam-7124	8	4	like	like	ADP
ejpam-7124	8	5	a	a	DET
ejpam-7124	8	6	boom	boom	NOUN
ejpam-7124	8	7	in	in	ADP
ejpam-7124	8	8	the	the	DET
ejpam-7124	8	9	area	area	NOUN
ejpam-7124	8	10	of	of	ADP
ejpam-7124	8	11	non	non	ADJ
ejpam-7124	8	12	-	-	ADJ
ejpam-7124	8	13	linear	linear	ADJ
ejpam-7124	8	14	analysis	analysis	NOUN
ejpam-7124	8	15	and	and	CCONJ
ejpam-7124	8	16	its	its	PRON
ejpam-7124	8	17	applications	application	NOUN
ejpam-7124	8	18	.	.	PUNCT
ejpam-7124	9	1	in	in	ADP
ejpam-7124	9	2	1989	1989	NUM
ejpam-7124	9	3	,	,	PUNCT
ejpam-7124	9	4	bakhtin	bakhtin	NOUN
ejpam-7124	9	5	[	[	X
ejpam-7124	9	6	2	2	NUM
ejpam-7124	9	7	]	]	PUNCT
ejpam-7124	9	8	presented	present	VERB
ejpam-7124	9	9	b	b	X
ejpam-7124	9	10	-	-	PUNCT
ejpam-7124	9	11	metric	metric	ADJ
ejpam-7124	9	12	space	space	NOUN
ejpam-7124	9	13	,	,	PUNCT
ejpam-7124	9	14	an	an	DET
ejpam-7124	9	15	extension	extension	NOUN
ejpam-7124	9	16	of	of	ADP
ejpam-7124	9	17	metric	metric	ADJ
ejpam-7124	9	18	space	space	NOUN
ejpam-7124	9	19	and	and	CCONJ
ejpam-7124	9	20	numerous	numerous	ADJ
ejpam-7124	9	21	desirable	desirable	ADJ
ejpam-7124	9	22	fixed	fix	VERB
ejpam-7124	9	23	point	point	NOUN
ejpam-7124	9	24	solutions	solution	NOUN
ejpam-7124	9	25	for	for	ADP
ejpam-7124	9	26	expansive	expansive	ADJ
ejpam-7124	9	27	mappings	mapping	NOUN
ejpam-7124	9	28	were	be	AUX
ejpam-7124	9	29	examined	examine	VERB
ejpam-7124	9	30	in	in	ADP
ejpam-7124	9	31	b	b	NOUN
ejpam-7124	9	32	-	-	PUNCT
ejpam-7124	9	33	metric	metric	ADJ
ejpam-7124	9	34	space	space	NOUN
ejpam-7124	9	35	.	.	PUNCT
ejpam-7124	10	1	also	also	ADV
ejpam-7124	10	2	in	in	ADP
ejpam-7124	10	3	1993	1993	NUM
ejpam-7124	10	4	,	,	PUNCT
ejpam-7124	10	5	czerwik	czerwik	PROPN
ejpam-7124	11	1	[	[	X
ejpam-7124	11	2	3	3	NUM
ejpam-7124	11	3	]	]	PUNCT
ejpam-7124	11	4	expanded	expand	VERB
ejpam-7124	11	5	on	on	ADP
ejpam-7124	11	6	the	the	DET
ejpam-7124	11	7	findings	finding	NOUN
ejpam-7124	11	8	of	of	ADP
ejpam-7124	11	9	b	b	NOUN
ejpam-7124	11	10	-	-	PUNCT
ejpam-7124	11	11	metric	metric	ADJ
ejpam-7124	11	12	space	space	NOUN
ejpam-7124	11	13	as	as	ADV
ejpam-7124	11	14	well	well	ADV
ejpam-7124	11	15	.	.	PUNCT
ejpam-7124	12	1	the	the	DET
ejpam-7124	12	2	idea	idea	NOUN
ejpam-7124	12	3	of	of	ADP
ejpam-7124	12	4	expansive	expansive	ADJ
ejpam-7124	12	5	mapping	mapping	NOUN
ejpam-7124	12	6	of	of	ADP
ejpam-7124	12	7	wang	wang	PROPN
ejpam-7124	12	8	et	et	PROPN
ejpam-7124	12	9	al	al	PROPN
ejpam-7124	12	10	.	.	PUNCT
ejpam-7124	13	1	[	[	X
ejpam-7124	13	2	4	4	X
ejpam-7124	13	3	]	]	PUNCT
ejpam-7124	13	4	as	as	SCONJ
ejpam-7124	13	5	follows	follow	VERB
ejpam-7124	13	6	“	"	PUNCT
ejpam-7124	13	7	let	let	VERB
ejpam-7124	13	8	(	(	PUNCT
ejpam-7124	13	9	x	x	NOUN
ejpam-7124	13	10	,	,	PUNCT
ejpam-7124	13	11	d	d	NOUN
ejpam-7124	13	12	)	)	PUNCT
ejpam-7124	13	13	be	be	AUX
ejpam-7124	13	14	a	a	DET
ejpam-7124	13	15	metric	metric	ADJ
ejpam-7124	13	16	space	space	NOUN
ejpam-7124	13	17	.	.	PUNCT
ejpam-7124	14	1	a	a	DET
ejpam-7124	14	2	mapping	mapping	NOUN
ejpam-7124	14	3	t	t	NOUN
ejpam-7124	14	4	:	:	PUNCT
ejpam-7124	14	5	ω	ω	PROPN
ejpam-7124	14	6	→	→	SYM
ejpam-7124	14	7	ω	ω	PROPN
ejpam-7124	14	8	on	on	ADP
ejpam-7124	14	9	(	(	PUNCT
ejpam-7124	14	10	ω	ω	PROPN
ejpam-7124	14	11	,	,	PUNCT
ejpam-7124	14	12	d̄	d̄	PROPN
ejpam-7124	14	13	)	)	PUNCT
ejpam-7124	14	14	such	such	ADJ
ejpam-7124	14	15	that	that	SCONJ
ejpam-7124	14	16	∀	∀	NOUN
ejpam-7124	14	17	u	u	NOUN
ejpam-7124	14	18	,	,	PUNCT
ejpam-7124	14	19	e	e	PROPN
ejpam-7124	14	20	∈	∈	PROPN
ejpam-7124	14	21	ω	ω	NOUN
ejpam-7124	14	22	:	:	PUNCT
ejpam-7124	14	23	d̄(tu	d̄(tu	PROPN
ejpam-7124	14	24	,	,	PUNCT
ejpam-7124	14	25	te	te	PROPN
ejpam-7124	14	26	)	)	PUNCT
ejpam-7124	14	27	≥	≥	NOUN
ejpam-7124	14	28	d(u	d(u	PROPN
ejpam-7124	14	29	,	,	PUNCT
ejpam-7124	14	30	e	e	NOUN
ejpam-7124	14	31	)	)	PUNCT
ejpam-7124	14	32	”	"	PUNCT
ejpam-7124	14	33	.	.	PUNCT
ejpam-7124	15	1	after	after	ADP
ejpam-7124	15	2	∗corresponding	∗corresponde	VERB
ejpam-7124	15	3	author	author	NOUN
ejpam-7124	15	4	.	.	PUNCT
ejpam-7124	16	1	doi	doi	NOUN
ejpam-7124	16	2	:	:	PUNCT
ejpam-7124	16	3	https://doi.org/10.29020/nybg.ejpam.v18i4.7124	https://doi.org/10.29020/nybg.ejpam.v18i4.7124	VERB
ejpam-7124	16	4	email	email	NOUN
ejpam-7124	16	5	addresses	address	NOUN
ejpam-7124	16	6	:	:	PUNCT
ejpam-7124	17	1	manojantil18@gmail.com	manojantil18@gmail.com	X
ejpam-7124	17	2	(	(	PUNCT
ejpam-7124	17	3	m.	m.	NOUN
ejpam-7124	17	4	kumar	kumar	PROPN
ejpam-7124	17	5	)	)	PUNCT
ejpam-7124	17	6	,	,	PUNCT
ejpam-7124	17	7	bhardwajneha15@gmail.com	bhardwajneha15@gmail.com	PROPN
ejpam-7124	17	8	(	(	PUNCT
ejpam-7124	17	9	n.	n.	PROPN
ejpam-7124	17	10	bhardwaj	bhardwaj	PROPN
ejpam-7124	17	11	)	)	PUNCT
ejpam-7124	17	12	,	,	PUNCT
ejpam-7124	17	13	mathsguna@yahoo.com	mathsguna@yahoo.com	X
ejpam-7124	17	14	(	(	PUNCT
ejpam-7124	17	15	g.	g.	PROPN
ejpam-7124	17	16	mani	mani	PROPN
ejpam-7124	17	17	)	)	PUNCT
ejpam-7124	17	18	,	,	PUNCT
ejpam-7124	17	19	r.gopalan@psau.edu.sa	r.gopalan@psau.edu.sa	PROPN
ejpam-7124	17	20	(	(	PUNCT
ejpam-7124	17	21	r.	r.	PROPN
ejpam-7124	17	22	ramaswamy	ramaswamy	PROPN
ejpam-7124	17	23	)	)	PUNCT
ejpam-7124	17	24	,	,	PUNCT
ejpam-7124	17	25	k.khan@psau.edu.sa	k.khan@psau.edu.sa	PROPN
ejpam-7124	17	26	(	(	PUNCT
ejpam-7124	17	27	k.	k.	PROPN
ejpam-7124	17	28	h.	h.	PROPN
ejpam-7124	17	29	khan	khan	PROPN
ejpam-7124	17	30	)	)	PUNCT
ejpam-7124	17	31	,	,	PUNCT
ejpam-7124	17	32	o.abdelnaby@psau.edu.sa	o.abdelnaby@psau.edu.sa	PROPN
ejpam-7124	17	33	(	(	PUNCT
ejpam-7124	17	34	o.	o.	PROPN
ejpam-7124	17	35	a.	a.	PROPN
ejpam-7124	17	36	abdelnaby	abdelnaby	PROPN
ejpam-7124	17	37	)	)	PUNCT
ejpam-7124	17	38	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-7124	17	39	1	1	NUM
ejpam-7124	17	40	copyright	copyright	NOUN
ejpam-7124	17	41	:	:	PUNCT
ejpam-7124	17	42	©	©	PROPN
ejpam-7124	17	43	2025	2025	NUM
ejpam-7124	17	44	the	the	DET
ejpam-7124	17	45	author(s	author(s	NOUN
ejpam-7124	17	46	)	)	PUNCT
ejpam-7124	17	47	.	.	PUNCT
ejpam-7124	18	1	(	(	PUNCT
ejpam-7124	18	2	cc	cc	NOUN
ejpam-7124	18	3	by	by	ADP
ejpam-7124	18	4	-	-	PUNCT
ejpam-7124	18	5	nc	nc	PROPN
ejpam-7124	18	6	4.0	4.0	NUM
ejpam-7124	18	7	)	)	PUNCT
ejpam-7124	18	8	m.	m.	NOUN
ejpam-7124	18	9	kumar	kumar	PROPN
ejpam-7124	18	10	et	et	PROPN
ejpam-7124	18	11	al	al	PROPN
ejpam-7124	18	12	.	.	PUNCT
ejpam-7124	18	13	/	/	SYM
ejpam-7124	18	14	eur	eur	PROPN
ejpam-7124	18	15	.	.	PUNCT
ejpam-7124	19	1	j.	j.	PROPN
ejpam-7124	19	2	pure	pure	PROPN
ejpam-7124	19	3	appl	appl	PROPN
ejpam-7124	19	4	.	.	PROPN
ejpam-7124	19	5	math	math	PROPN
ejpam-7124	19	6	,	,	PUNCT
ejpam-7124	19	7	18	18	NUM
ejpam-7124	19	8	(	(	PUNCT
ejpam-7124	19	9	4	4	NUM
ejpam-7124	19	10	)	)	PUNCT
ejpam-7124	19	11	(	(	PUNCT
ejpam-7124	19	12	2025	2025	NUM
ejpam-7124	19	13	)	)	PUNCT
ejpam-7124	19	14	,	,	PUNCT
ejpam-7124	19	15	7124	7124	NUM
ejpam-7124	19	16	2	2	NUM
ejpam-7124	19	17	of	of	ADP
ejpam-7124	19	18	14	14	NUM
ejpam-7124	19	19	getting	get	VERB
ejpam-7124	19	20	the	the	DET
ejpam-7124	19	21	motivation	motivation	NOUN
ejpam-7124	19	22	of	of	ADP
ejpam-7124	19	23	this	this	PRON
ejpam-7124	19	24	,	,	PUNCT
ejpam-7124	19	25	shahi	shahi	PROPN
ejpam-7124	19	26	et	et	PROPN
ejpam-7124	19	27	al	al	PROPN
ejpam-7124	19	28	.	.	PUNCT
ejpam-7124	20	1	[	[	X
ejpam-7124	20	2	5	5	NUM
ejpam-7124	20	3	]	]	PUNCT
ejpam-7124	20	4	proved	prove	VERB
ejpam-7124	20	5	fixed	fix	VERB
ejpam-7124	20	6	point	point	NOUN
ejpam-7124	20	7	theorems	theorem	NOUN
ejpam-7124	20	8	and	and	CCONJ
ejpam-7124	20	9	gave	give	VERB
ejpam-7124	20	10	some	some	DET
ejpam-7124	20	11	applications	application	NOUN
ejpam-7124	20	12	for	for	ADP
ejpam-7124	20	13	(	(	PUNCT
ejpam-7124	20	14	ξ	ξ	PROPN
ejpam-7124	20	15	,	,	PUNCT
ejpam-7124	20	16	α)-expansive	α)-expansive	ADJ
ejpam-7124	20	17	mappings	mapping	NOUN
ejpam-7124	20	18	in	in	ADP
ejpam-7124	20	19	complete	complete	ADJ
ejpam-7124	20	20	metric	metric	ADJ
ejpam-7124	20	21	spaces	space	NOUN
ejpam-7124	20	22	.	.	PUNCT
ejpam-7124	21	1	recently	recently	ADV
ejpam-7124	21	2	,	,	PUNCT
ejpam-7124	21	3	many	many	ADJ
ejpam-7124	21	4	research	research	NOUN
ejpam-7124	21	5	studies	study	NOUN
ejpam-7124	21	6	were	be	AUX
ejpam-7124	21	7	conducted	conduct	VERB
ejpam-7124	21	8	on	on	ADP
ejpam-7124	21	9	b	b	X
ejpam-7124	21	10	-	-	PUNCT
ejpam-7124	21	11	metric	metric	ADJ
ejpam-7124	21	12	space	space	NOUN
ejpam-7124	21	13	under	under	ADP
ejpam-7124	21	14	different	different	ADJ
ejpam-7124	21	15	expansive	expansive	ADJ
ejpam-7124	21	16	conditions	condition	NOUN
ejpam-7124	21	17	.	.	PUNCT
ejpam-7124	22	1	after	after	ADP
ejpam-7124	22	2	that	that	PRON
ejpam-7124	22	3	,	,	PUNCT
ejpam-7124	22	4	many	many	ADJ
ejpam-7124	22	5	authors	author	NOUN
ejpam-7124	22	6	used	use	VERB
ejpam-7124	22	7	different	different	ADJ
ejpam-7124	22	8	types	type	NOUN
ejpam-7124	22	9	of	of	ADP
ejpam-7124	22	10	contractive	contractive	ADJ
ejpam-7124	22	11	conditions	condition	NOUN
ejpam-7124	22	12	including	include	VERB
ejpam-7124	22	13	(	(	PUNCT
ejpam-7124	22	14	α	α	NOUN
ejpam-7124	22	15	,	,	PUNCT
ejpam-7124	22	16	ψ)expansive	ψ)expansive	ADJ
ejpam-7124	22	17	mappings	mapping	NOUN
ejpam-7124	22	18	in	in	ADP
ejpam-7124	22	19	different	different	ADJ
ejpam-7124	22	20	metric	metric	ADJ
ejpam-7124	22	21	and	and	CCONJ
ejpam-7124	22	22	metric	metric	ADJ
ejpam-7124	22	23	like	like	ADP
ejpam-7124	22	24	spaces	space	NOUN
ejpam-7124	22	25	(	(	PUNCT
ejpam-7124	22	26	see	see	VERB
ejpam-7124	22	27	[	[	X
ejpam-7124	22	28	6–14	6–14	PROPN
ejpam-7124	22	29	]	]	PUNCT
ejpam-7124	22	30	)	)	PUNCT
ejpam-7124	22	31	.	.	PUNCT
ejpam-7124	23	1	in	in	ADP
ejpam-7124	23	2	2017	2017	NUM
ejpam-7124	23	3	,	,	PUNCT
ejpam-7124	23	4	kamran	kamran	PROPN
ejpam-7124	23	5	et	et	PROPN
ejpam-7124	23	6	al	al	PROPN
ejpam-7124	23	7	.	.	PUNCT
ejpam-7124	24	1	[	[	X
ejpam-7124	24	2	8	8	NUM
ejpam-7124	24	3	]	]	PUNCT
ejpam-7124	24	4	introduced	introduce	VERB
ejpam-7124	24	5	an	an	DET
ejpam-7124	24	6	extended	extended	ADJ
ejpam-7124	24	7	generalization	generalization	NOUN
ejpam-7124	24	8	of	of	ADP
ejpam-7124	24	9	the	the	DET
ejpam-7124	24	10	b	b	NOUN
ejpam-7124	24	11	-	-	PUNCT
ejpam-7124	24	12	metric	metric	ADJ
ejpam-7124	24	13	space	space	NOUN
ejpam-7124	24	14	and	and	CCONJ
ejpam-7124	24	15	many	many	ADJ
ejpam-7124	24	16	results	result	NOUN
ejpam-7124	24	17	in	in	ADP
ejpam-7124	24	18	the	the	DET
ejpam-7124	24	19	literature	literature	NOUN
ejpam-7124	24	20	were	be	AUX
ejpam-7124	24	21	generalized	generalize	VERB
ejpam-7124	24	22	by	by	ADP
ejpam-7124	24	23	their	their	PRON
ejpam-7124	24	24	study	study	NOUN
ejpam-7124	24	25	.	.	PUNCT
ejpam-7124	25	1	mlaiki	mlaiki	PROPN
ejpam-7124	25	2	et	et	PROPN
ejpam-7124	25	3	al	al	PROPN
ejpam-7124	25	4	.	.	PUNCT
ejpam-7124	26	1	[	[	X
ejpam-7124	26	2	15	15	NUM
ejpam-7124	26	3	]	]	PUNCT
ejpam-7124	26	4	produced	produce	VERB
ejpam-7124	26	5	a	a	DET
ejpam-7124	26	6	controlled	control	VERB
ejpam-7124	26	7	metric	metric	ADJ
ejpam-7124	26	8	type	type	NOUN
ejpam-7124	26	9	space	space	NOUN
ejpam-7124	26	10	,	,	PUNCT
ejpam-7124	26	11	which	which	PRON
ejpam-7124	26	12	is	be	AUX
ejpam-7124	26	13	an	an	DET
ejpam-7124	26	14	extension	extension	NOUN
ejpam-7124	26	15	of	of	ADP
ejpam-7124	26	16	the	the	DET
ejpam-7124	26	17	extended	extended	ADJ
ejpam-7124	26	18	b	b	X
ejpam-7124	26	19	-	-	PUNCT
ejpam-7124	26	20	metric	metric	ADJ
ejpam-7124	26	21	space	space	NOUN
ejpam-7124	26	22	.	.	PUNCT
ejpam-7124	27	1	2	2	X
ejpam-7124	27	2	.	.	X
ejpam-7124	27	3	preliminaries	preliminary	NOUN
ejpam-7124	27	4	definition	definition	NOUN
ejpam-7124	27	5	1	1	NUM
ejpam-7124	27	6	(	(	PUNCT
ejpam-7124	27	7	[	[	X
ejpam-7124	27	8	8	8	NUM
ejpam-7124	27	9	]	]	PUNCT
ejpam-7124	27	10	)	)	PUNCT
ejpam-7124	27	11	.	.	PUNCT
ejpam-7124	28	1	let	let	VERB
ejpam-7124	28	2	ω	ω	NUM
ejpam-7124	28	3	be	be	AUX
ejpam-7124	28	4	a	a	DET
ejpam-7124	28	5	non	non	ADJ
ejpam-7124	28	6	-	-	ADJ
ejpam-7124	28	7	empty	empty	ADJ
ejpam-7124	28	8	set	set	NOUN
ejpam-7124	28	9	and	and	CCONJ
ejpam-7124	28	10	define	define	VERB
ejpam-7124	28	11	the	the	DET
ejpam-7124	28	12	mappings	mapping	NOUN
ejpam-7124	28	13	µ	µ	X
ejpam-7124	28	14	:	:	PUNCT
ejpam-7124	28	15	ω×ω	ω×ω	NUM
ejpam-7124	28	16	→	→	SYM
ejpam-7124	28	17	[	[	X
ejpam-7124	28	18	1,∞	1,∞	NUM
ejpam-7124	28	19	)	)	PUNCT
ejpam-7124	28	20	and	and	CCONJ
ejpam-7124	28	21	d̄	d̄	NOUN
ejpam-7124	28	22	:	:	PUNCT
ejpam-7124	28	23	ω×	ω×	NUM
ejpam-7124	28	24	ω	ω	NUM
ejpam-7124	28	25	→	→	SYM
ejpam-7124	28	26	[	[	X
ejpam-7124	28	27	0,∞	0,∞	NUM
ejpam-7124	28	28	)	)	PUNCT
ejpam-7124	28	29	such	such	ADJ
ejpam-7124	28	30	that	that	PRON
ejpam-7124	28	31	for	for	ADP
ejpam-7124	28	32	all	all	DET
ejpam-7124	28	33	u	u	NOUN
ejpam-7124	28	34	,	,	PUNCT
ejpam-7124	28	35	e	e	NOUN
ejpam-7124	28	36	,	,	PUNCT
ejpam-7124	28	37	f	f	PROPN
ejpam-7124	28	38	∈	∈	PROPN
ejpam-7124	28	39	ω	ω	PROPN
ejpam-7124	29	1	[	[	X
ejpam-7124	29	2	label=(i),itemsep=-.16em	label=(i),itemsep=-.16em	NOUN
ejpam-7124	29	3	,	,	PUNCT
ejpam-7124	29	4	topsep=2pt	topsep=2pt	PROPN
ejpam-7124	29	5	]	]	PUNCT
ejpam-7124	29	6	(	(	PUNCT
ejpam-7124	29	7	i	i	NOUN
ejpam-7124	29	8	)	)	PUNCT
ejpam-7124	29	9	d̄(u	d̄(u	NOUN
ejpam-7124	29	10	,	,	PUNCT
ejpam-7124	29	11	e	e	NOUN
ejpam-7124	29	12	)	)	PUNCT
ejpam-7124	29	13	=	=	SYM
ejpam-7124	29	14	0	0	NUM
ejpam-7124	29	15	⇐	⇐	ADJ
ejpam-7124	29	16	⇒	⇒	NOUN
ejpam-7124	29	17	u	u	NOUN
ejpam-7124	29	18	=	=	SYM
ejpam-7124	29	19	e	e	PROPN
ejpam-7124	29	20	,	,	PUNCT
ejpam-7124	29	21	(	(	PUNCT
ejpam-7124	29	22	ii	ii	NOUN
ejpam-7124	29	23	)	)	PUNCT
ejpam-7124	29	24	d̄(u	d̄(u	NOUN
ejpam-7124	29	25	,	,	PUNCT
ejpam-7124	29	26	e	e	NOUN
ejpam-7124	29	27	)	)	PUNCT
ejpam-7124	29	28	=	=	SYM
ejpam-7124	29	29	d̄(e	d̄(e	NOUN
ejpam-7124	29	30	,	,	PUNCT
ejpam-7124	29	31	u	u	NOUN
ejpam-7124	29	32	)	)	PUNCT
ejpam-7124	29	33	,	,	PUNCT
ejpam-7124	29	34	(	(	PUNCT
ejpam-7124	29	35	iii	iii	NOUN
ejpam-7124	29	36	)	)	PUNCT
ejpam-7124	29	37	d̄(u	d̄(u	NOUN
ejpam-7124	29	38	,	,	PUNCT
ejpam-7124	29	39	e	e	NOUN
ejpam-7124	29	40	)	)	PUNCT
ejpam-7124	29	41	≤	≤	NUM
ejpam-7124	29	42	µ(u	µ(u	NOUN
ejpam-7124	29	43	,	,	PUNCT
ejpam-7124	29	44	e)[d̄(u	e)[d̄(u	PROPN
ejpam-7124	29	45	,	,	PUNCT
ejpam-7124	29	46	f	f	X
ejpam-7124	29	47	)	)	PUNCT
ejpam-7124	29	48	+	+	CCONJ
ejpam-7124	29	49	d̄(f	d̄(f	ADJ
ejpam-7124	29	50	,	,	PUNCT
ejpam-7124	29	51	e	e	NOUN
ejpam-7124	29	52	)	)	PUNCT
ejpam-7124	29	53	]	]	PUNCT
ejpam-7124	29	54	.	.	PUNCT
ejpam-7124	30	1	then	then	ADV
ejpam-7124	30	2	the	the	DET
ejpam-7124	30	3	pair	pair	NOUN
ejpam-7124	30	4	(	(	PUNCT
ejpam-7124	30	5	ω	ω	NOUN
ejpam-7124	30	6	,	,	PUNCT
ejpam-7124	30	7	d̄	d̄	PROPN
ejpam-7124	30	8	)	)	PUNCT
ejpam-7124	30	9	is	be	AUX
ejpam-7124	30	10	known	know	VERB
ejpam-7124	30	11	as	as	ADP
ejpam-7124	30	12	an	an	DET
ejpam-7124	30	13	extended	extended	ADJ
ejpam-7124	30	14	b	b	NOUN
ejpam-7124	30	15	-	-	PUNCT
ejpam-7124	30	16	metric	metric	ADJ
ejpam-7124	30	17	space	space	NOUN
ejpam-7124	30	18	.	.	PUNCT
ejpam-7124	31	1	further	far	ADV
ejpam-7124	31	2	,	,	PUNCT
ejpam-7124	31	3	mlaiki	mlaiki	PROPN
ejpam-7124	31	4	et	et	PROPN
ejpam-7124	31	5	al	al	PROPN
ejpam-7124	31	6	.	.	PUNCT
ejpam-7124	32	1	[	[	X
ejpam-7124	32	2	15	15	NUM
ejpam-7124	32	3	]	]	PUNCT
ejpam-7124	32	4	introduced	introduce	VERB
ejpam-7124	32	5	the	the	DET
ejpam-7124	32	6	notion	notion	NOUN
ejpam-7124	32	7	of	of	ADP
ejpam-7124	32	8	controlled	control	VERB
ejpam-7124	32	9	metric	metric	ADJ
ejpam-7124	32	10	type	type	NOUN
ejpam-7124	32	11	space	space	NOUN
ejpam-7124	32	12	which	which	PRON
ejpam-7124	32	13	is	be	AUX
ejpam-7124	32	14	defined	define	VERB
ejpam-7124	32	15	as	as	SCONJ
ejpam-7124	32	16	follows	follow	VERB
ejpam-7124	32	17	:	:	PUNCT
ejpam-7124	32	18	definition	definition	NOUN
ejpam-7124	32	19	2	2	NUM
ejpam-7124	32	20	(	(	PUNCT
ejpam-7124	32	21	[	[	X
ejpam-7124	32	22	15	15	NUM
ejpam-7124	32	23	]	]	NUM
ejpam-7124	32	24	)	)	PUNCT
ejpam-7124	32	25	.	.	PUNCT
ejpam-7124	33	1	on	on	ADP
ejpam-7124	33	2	a	a	DET
ejpam-7124	33	3	non	non	ADJ
ejpam-7124	33	4	-	-	ADJ
ejpam-7124	33	5	empty	empty	ADJ
ejpam-7124	33	6	set	set	ADJ
ejpam-7124	33	7	ω	ω	PROPN
ejpam-7124	33	8	,	,	PUNCT
ejpam-7124	33	9	define	define	VERB
ejpam-7124	33	10	the	the	DET
ejpam-7124	33	11	mappings	mapping	NOUN
ejpam-7124	33	12	µ	µ	X
ejpam-7124	33	13	:	:	PUNCT
ejpam-7124	33	14	ω×ω	ω×ω	NUM
ejpam-7124	33	15	→	→	SYM
ejpam-7124	33	16	[	[	X
ejpam-7124	33	17	1,∞	1,∞	NUM
ejpam-7124	33	18	)	)	PUNCT
ejpam-7124	33	19	and	and	CCONJ
ejpam-7124	33	20	d̄	d̄	NOUN
ejpam-7124	33	21	:	:	PUNCT
ejpam-7124	33	22	ω×	ω×	NUM
ejpam-7124	33	23	ω	ω	NUM
ejpam-7124	33	24	→	→	SYM
ejpam-7124	33	25	[	[	X
ejpam-7124	33	26	0,∞	0,∞	NUM
ejpam-7124	33	27	)	)	PUNCT
ejpam-7124	33	28	such	such	ADJ
ejpam-7124	33	29	that	that	PRON
ejpam-7124	33	30	for	for	ADP
ejpam-7124	33	31	all	all	DET
ejpam-7124	33	32	u	u	NOUN
ejpam-7124	33	33	,	,	PUNCT
ejpam-7124	33	34	e	e	NOUN
ejpam-7124	33	35	,	,	PUNCT
ejpam-7124	33	36	f	f	PROPN
ejpam-7124	33	37	∈	∈	PROPN
ejpam-7124	33	38	ω	ω	PROPN
ejpam-7124	33	39	the	the	DET
ejpam-7124	33	40	following	follow	VERB
ejpam-7124	33	41	condition	condition	NOUN
ejpam-7124	33	42	holds	hold	VERB
ejpam-7124	33	43	[	[	X
ejpam-7124	33	44	label=(s2),itemsep=-.16em	label=(s2),itemsep=-.16em	NOUN
ejpam-7124	33	45	,	,	PUNCT
ejpam-7124	33	46	topsep=2pt	topsep=2pt	PROPN
ejpam-7124	33	47	]	]	PUNCT
ejpam-7124	33	48	(	(	PUNCT
ejpam-7124	33	49	i	i	NOUN
ejpam-7124	33	50	)	)	PUNCT
ejpam-7124	33	51	d̄(u	d̄(u	NOUN
ejpam-7124	33	52	,	,	PUNCT
ejpam-7124	33	53	e	e	NOUN
ejpam-7124	33	54	)	)	PUNCT
ejpam-7124	33	55	=	=	SYM
ejpam-7124	33	56	0	0	NUM
ejpam-7124	33	57	⇐	⇐	ADJ
ejpam-7124	33	58	⇒	⇒	NOUN
ejpam-7124	33	59	u	u	NOUN
ejpam-7124	33	60	=	=	SYM
ejpam-7124	33	61	e	e	PROPN
ejpam-7124	33	62	,	,	PUNCT
ejpam-7124	33	63	(	(	PUNCT
ejpam-7124	33	64	ii	ii	NOUN
ejpam-7124	33	65	)	)	PUNCT
ejpam-7124	33	66	d̄(u	d̄(u	NOUN
ejpam-7124	33	67	,	,	PUNCT
ejpam-7124	33	68	e	e	NOUN
ejpam-7124	33	69	)	)	PUNCT
ejpam-7124	33	70	=	=	SYM
ejpam-7124	33	71	d̄(e	d̄(e	NOUN
ejpam-7124	33	72	,	,	PUNCT
ejpam-7124	33	73	u	u	NOUN
ejpam-7124	33	74	)	)	PUNCT
ejpam-7124	33	75	,	,	PUNCT
ejpam-7124	33	76	(	(	PUNCT
ejpam-7124	33	77	iii	iii	NOUN
ejpam-7124	33	78	)	)	PUNCT
ejpam-7124	33	79	d̄(u	d̄(u	NOUN
ejpam-7124	33	80	,	,	PUNCT
ejpam-7124	33	81	e	e	NOUN
ejpam-7124	33	82	)	)	PUNCT
ejpam-7124	33	83	≤	≤	NUM
ejpam-7124	33	84	µ(u	µ(u	NOUN
ejpam-7124	33	85	,	,	PUNCT
ejpam-7124	33	86	f)d̄(u	f)d̄(u	NUM
ejpam-7124	33	87	,	,	PUNCT
ejpam-7124	33	88	f	f	X
ejpam-7124	33	89	)	)	PUNCT
ejpam-7124	34	1	+	+	CCONJ
ejpam-7124	34	2	µ(f	µ(f	PROPN
ejpam-7124	34	3	,	,	PUNCT
ejpam-7124	34	4	e)d̄(f	e)d̄(f	PROPN
ejpam-7124	34	5	,	,	PUNCT
ejpam-7124	34	6	e	e	NOUN
ejpam-7124	34	7	)	)	PUNCT
ejpam-7124	34	8	.	.	PUNCT
ejpam-7124	35	1	then	then	ADV
ejpam-7124	35	2	the	the	DET
ejpam-7124	35	3	pair	pair	NOUN
ejpam-7124	35	4	(	(	PUNCT
ejpam-7124	35	5	ω	ω	NOUN
ejpam-7124	35	6	,	,	PUNCT
ejpam-7124	35	7	d̄	d̄	PROPN
ejpam-7124	35	8	)	)	PUNCT
ejpam-7124	35	9	is	be	AUX
ejpam-7124	35	10	called	call	VERB
ejpam-7124	35	11	a	a	DET
ejpam-7124	35	12	controlled	control	VERB
ejpam-7124	35	13	metric	metric	ADJ
ejpam-7124	35	14	type	type	NOUN
ejpam-7124	35	15	space	space	NOUN
ejpam-7124	35	16	.	.	PUNCT
ejpam-7124	36	1	to	to	PART
ejpam-7124	36	2	prove	prove	VERB
ejpam-7124	36	3	the	the	DET
ejpam-7124	36	4	above	above	ADJ
ejpam-7124	36	5	definition	definition	NOUN
ejpam-7124	36	6	,	,	PUNCT
ejpam-7124	36	7	we	we	PRON
ejpam-7124	36	8	provide	provide	VERB
ejpam-7124	36	9	the	the	DET
ejpam-7124	36	10	following	follow	VERB
ejpam-7124	36	11	examples	example	NOUN
ejpam-7124	36	12	.	.	PUNCT
ejpam-7124	37	1	example	example	NOUN
ejpam-7124	37	2	1	1	NUM
ejpam-7124	37	3	(	(	PUNCT
ejpam-7124	37	4	[	[	X
ejpam-7124	37	5	15	15	NUM
ejpam-7124	37	6	]	]	NUM
ejpam-7124	37	7	)	)	PUNCT
ejpam-7124	37	8	.	.	PUNCT
ejpam-7124	38	1	choose	choose	VERB
ejpam-7124	38	2	ω	ω	X
ejpam-7124	38	3	=	=	SYM
ejpam-7124	38	4	{	{	PUNCT
ejpam-7124	38	5	1	1	NUM
ejpam-7124	38	6	,	,	PUNCT
ejpam-7124	38	7	2	2	NUM
ejpam-7124	38	8	,	,	PUNCT
ejpam-7124	38	9	3	3	NUM
ejpam-7124	38	10	,	,	PUNCT
ejpam-7124	38	11	.	.	PUNCT
ejpam-7124	38	12	.	.	PUNCT
ejpam-7124	38	13	.	.	PUNCT
ejpam-7124	39	1	}	}	PUNCT
ejpam-7124	39	2	.	.	PUNCT
ejpam-7124	40	1	take	take	VERB
ejpam-7124	40	2	d̄	d̄	NOUN
ejpam-7124	40	3	:	:	PUNCT
ejpam-7124	40	4	ω×	ω×	PROPN
ejpam-7124	40	5	ω	ω	NUM
ejpam-7124	40	6	→	→	SYM
ejpam-7124	40	7	[	[	X
ejpam-7124	40	8	0,∞	0,∞	NUM
ejpam-7124	40	9	)	)	PUNCT
ejpam-7124	40	10	such	such	ADJ
ejpam-7124	40	11	that	that	DET
ejpam-7124	40	12	d̄(u	d̄(u	NOUN
ejpam-7124	40	13	,	,	PUNCT
ejpam-7124	40	14	e	e	NOUN
ejpam-7124	40	15	)	)	PUNCT
ejpam-7124	40	16	=	=	SYM
ejpam-7124	40	17			PROPN
ejpam-7124	40	18	0	0	NUM
ejpam-7124	40	19	⇔	⇔	X
ejpam-7124	40	20	u	u	NOUN
ejpam-7124	40	21	=	=	SYM
ejpam-7124	40	22	e	e	PROPN
ejpam-7124	40	23	1	1	NUM
ejpam-7124	40	24	u	u	NOUN
ejpam-7124	40	25	,	,	PUNCT
ejpam-7124	40	26	if	if	SCONJ
ejpam-7124	40	27	u	u	NOUN
ejpam-7124	40	28	=	=	NOUN
ejpam-7124	40	29	2j	2j	NUM
ejpam-7124	40	30	and	and	CCONJ
ejpam-7124	40	31	e	e	NOUN
ejpam-7124	40	32	=	=	NOUN
ejpam-7124	40	33	2j	2j	PROPN
ejpam-7124	41	1	+	+	CCONJ
ejpam-7124	41	2	1	1	NUM
ejpam-7124	41	3	,	,	PUNCT
ejpam-7124	41	4	1	1	NUM
ejpam-7124	41	5	e	e	NOUN
ejpam-7124	41	6	,	,	PUNCT
ejpam-7124	41	7	if	if	SCONJ
ejpam-7124	41	8	u	u	NOUN
ejpam-7124	41	9	=	=	NOUN
ejpam-7124	41	10	2j	2j	NUM
ejpam-7124	41	11	+	+	CCONJ
ejpam-7124	41	12	1	1	NUM
ejpam-7124	41	13	and	and	CCONJ
ejpam-7124	41	14	e	e	NOUN
ejpam-7124	41	15	=	=	NOUN
ejpam-7124	41	16	2j	2j	NUM
ejpam-7124	41	17	,	,	PUNCT
ejpam-7124	41	18	1	1	NUM
ejpam-7124	41	19	,	,	PUNCT
ejpam-7124	41	20	otherwise	otherwise	ADV
ejpam-7124	41	21	.	.	PUNCT
ejpam-7124	42	1	(	(	PUNCT
ejpam-7124	42	2	1	1	X
ejpam-7124	42	3	)	)	PUNCT
ejpam-7124	42	4	m.	m.	NOUN
ejpam-7124	42	5	kumar	kumar	PROPN
ejpam-7124	42	6	et	et	PROPN
ejpam-7124	42	7	al	al	PROPN
ejpam-7124	42	8	.	.	PUNCT
ejpam-7124	42	9	/	/	SYM
ejpam-7124	42	10	eur	eur	PROPN
ejpam-7124	42	11	.	.	PUNCT
ejpam-7124	43	1	j.	j.	PROPN
ejpam-7124	43	2	pure	pure	PROPN
ejpam-7124	43	3	appl	appl	PROPN
ejpam-7124	43	4	.	.	PROPN
ejpam-7124	43	5	math	math	PROPN
ejpam-7124	43	6	,	,	PUNCT
ejpam-7124	43	7	18	18	NUM
ejpam-7124	43	8	(	(	PUNCT
ejpam-7124	43	9	4	4	NUM
ejpam-7124	43	10	)	)	PUNCT
ejpam-7124	43	11	(	(	PUNCT
ejpam-7124	43	12	2025	2025	NUM
ejpam-7124	43	13	)	)	PUNCT
ejpam-7124	43	14	,	,	PUNCT
ejpam-7124	43	15	7124	7124	NUM
ejpam-7124	43	16	3	3	NUM
ejpam-7124	43	17	of	of	ADP
ejpam-7124	43	18	14	14	NUM
ejpam-7124	43	19	consider	consider	VERB
ejpam-7124	43	20	µ	µ	NOUN
ejpam-7124	43	21	:	:	PUNCT
ejpam-7124	43	22	ω×	ω×	PROPN
ejpam-7124	43	23	ω	ω	PROPN
ejpam-7124	43	24	→	→	SYM
ejpam-7124	43	25	[	[	X
ejpam-7124	43	26	1,∞	1,∞	NUM
ejpam-7124	43	27	)	)	PUNCT
ejpam-7124	43	28	as	as	ADP
ejpam-7124	43	29	µ(u	µ(u	NOUN
ejpam-7124	43	30	,	,	PUNCT
ejpam-7124	43	31	e	e	NOUN
ejpam-7124	43	32	)	)	PUNCT
ejpam-7124	43	33	=	=	SYM
ejpam-7124	44	1			PRON
ejpam-7124	44	2	u	u	NOUN
ejpam-7124	44	3	,	,	PUNCT
ejpam-7124	44	4	if	if	SCONJ
ejpam-7124	44	5	u	u	NOUN
ejpam-7124	44	6	=	=	NOUN
ejpam-7124	44	7	2j	2j	NUM
ejpam-7124	44	8	and	and	CCONJ
ejpam-7124	44	9	e	e	NOUN
ejpam-7124	44	10	=	=	NOUN
ejpam-7124	44	11	2j	2j	PROPN
ejpam-7124	45	1	+	+	CCONJ
ejpam-7124	45	2	1	1	NUM
ejpam-7124	45	3	,	,	PUNCT
ejpam-7124	45	4	e	e	NOUN
ejpam-7124	45	5	,	,	PUNCT
ejpam-7124	45	6	if	if	SCONJ
ejpam-7124	45	7	u	u	NOUN
ejpam-7124	45	8	=	=	X
ejpam-7124	45	9	2j	2j	NUM
ejpam-7124	45	10	+	+	CCONJ
ejpam-7124	45	11	1	1	NUM
ejpam-7124	45	12	and	and	CCONJ
ejpam-7124	45	13	e	e	NOUN
ejpam-7124	45	14	=	=	NOUN
ejpam-7124	45	15	2j	2j	NUM
ejpam-7124	45	16	,	,	PUNCT
ejpam-7124	45	17	1	1	NUM
ejpam-7124	45	18	,	,	PUNCT
ejpam-7124	45	19	otherwise	otherwise	ADV
ejpam-7124	45	20	.	.	PUNCT
ejpam-7124	46	1	(	(	PUNCT
ejpam-7124	46	2	2	2	X
ejpam-7124	46	3	)	)	PUNCT
ejpam-7124	46	4	it	it	PRON
ejpam-7124	46	5	is	be	AUX
ejpam-7124	46	6	clear	clear	ADJ
ejpam-7124	46	7	that	that	SCONJ
ejpam-7124	46	8	the	the	DET
ejpam-7124	46	9	conditions	condition	NOUN
ejpam-7124	46	10	(	(	PUNCT
ejpam-7124	46	11	s1	s1	NOUN
ejpam-7124	46	12	)	)	PUNCT
ejpam-7124	46	13	and	and	CCONJ
ejpam-7124	46	14	(	(	PUNCT
ejpam-7124	46	15	s2	s2	PROPN
ejpam-7124	46	16	)	)	PUNCT
ejpam-7124	46	17	are	be	AUX
ejpam-7124	46	18	satisfied	satisfied	ADJ
ejpam-7124	46	19	.	.	PUNCT
ejpam-7124	47	1	now	now	ADV
ejpam-7124	47	2	we	we	PRON
ejpam-7124	47	3	investigate	investigate	VERB
ejpam-7124	47	4	(	(	PUNCT
ejpam-7124	47	5	s3	s3	PROPN
ejpam-7124	47	6	)	)	PUNCT
ejpam-7124	47	7	.	.	PUNCT
ejpam-7124	48	1	[	[	X
ejpam-7124	48	2	label	label	NOUN
ejpam-7124	48	3	=	=	NOUN
ejpam-7124	48	4	case	case	NOUN
ejpam-7124	48	5	i:,itemsep=-.16em	i:,itemsep=-.16em	NOUN
ejpam-7124	48	6	,	,	PUNCT
ejpam-7124	48	7	topsep=2pt	topsep=2pt	PROPN
ejpam-7124	48	8	,	,	PUNCT
ejpam-7124	48	9	itemindent=-1.75em	itemindent=-1.75em	NOUN
ejpam-7124	48	10	,	,	PUNCT
ejpam-7124	48	11	align	align	NOUN
ejpam-7124	48	12	=	=	NOUN
ejpam-7124	48	13	left	leave	VERB
ejpam-7124	48	14	]	]	PUNCT
ejpam-7124	48	15	(	(	PUNCT
ejpam-7124	48	16	i	i	NOUN
ejpam-7124	48	17	)	)	PUNCT
ejpam-7124	48	18	if	if	SCONJ
ejpam-7124	48	19	f	f	PROPN
ejpam-7124	48	20	=	=	SYM
ejpam-7124	48	21	u	u	PROPN
ejpam-7124	48	22	or	or	CCONJ
ejpam-7124	48	23	f	f	NOUN
ejpam-7124	48	24	=	=	SYM
ejpam-7124	48	25	e	e	PROPN
ejpam-7124	48	26	,	,	PUNCT
ejpam-7124	48	27	(	(	PUNCT
ejpam-7124	48	28	s3	s3	NOUN
ejpam-7124	48	29	)	)	PUNCT
ejpam-7124	48	30	is	be	AUX
ejpam-7124	48	31	satisfied	satisfied	ADJ
ejpam-7124	48	32	.	.	PUNCT
ejpam-7124	49	1	(	(	PUNCT
ejpam-7124	49	2	ii	ii	NOUN
ejpam-7124	49	3	)	)	PUNCT
ejpam-7124	49	4	if	if	SCONJ
ejpam-7124	49	5	f	f	PROPN
ejpam-7124	49	6	̸=	̸=	PROPN
ejpam-7124	49	7	u	u	PROPN
ejpam-7124	49	8	and	and	CCONJ
ejpam-7124	49	9	f	f	PROPN
ejpam-7124	49	10	̸=	̸=	PROPN
ejpam-7124	49	11	e.	e.	PROPN
ejpam-7124	49	12	(	(	PUNCT
ejpam-7124	49	13	s3	s3	PROPN
ejpam-7124	49	14	)	)	PUNCT
ejpam-7124	49	15	holds	hold	VERB
ejpam-7124	49	16	when	when	SCONJ
ejpam-7124	49	17	u	u	PROPN
ejpam-7124	49	18	=	=	PROPN
ejpam-7124	49	19	e.	e.	PROPN
ejpam-7124	49	20	now	now	ADV
ejpam-7124	49	21	we	we	PRON
ejpam-7124	49	22	may	may	AUX
ejpam-7124	49	23	assume	assume	VERB
ejpam-7124	49	24	that	that	SCONJ
ejpam-7124	49	25	u	u	PROPN
ejpam-7124	49	26	̸=	̸=	PROPN
ejpam-7124	49	27	e.	e.	PROPN
ejpam-7124	49	28	then	then	ADV
ejpam-7124	49	29	,	,	PUNCT
ejpam-7124	49	30	we	we	PRON
ejpam-7124	49	31	have	have	VERB
ejpam-7124	49	32	u	u	NOUN
ejpam-7124	49	33	̸=	̸=	PROPN
ejpam-7124	49	34	e	e	PROPN
ejpam-7124	49	35	̸=	̸=	PROPN
ejpam-7124	49	36	f	f	PROPN
ejpam-7124	49	37	.	.	PUNCT
ejpam-7124	50	1	it	it	PRON
ejpam-7124	50	2	is	be	AUX
ejpam-7124	50	3	clear	clear	ADJ
ejpam-7124	50	4	that	that	SCONJ
ejpam-7124	50	5	(	(	PUNCT
ejpam-7124	50	6	s3	s3	PROPN
ejpam-7124	50	7	)	)	PUNCT
ejpam-7124	50	8	holds	hold	VERB
ejpam-7124	50	9	in	in	ADP
ejpam-7124	50	10	each	each	PRON
ejpam-7124	50	11	of	of	ADP
ejpam-7124	50	12	the	the	DET
ejpam-7124	50	13	following	follow	VERB
ejpam-7124	50	14	subcases	subcase	NOUN
ejpam-7124	50	15	:	:	PUNCT
ejpam-7124	51	1	[	[	X
ejpam-7124	51	2	label=(1),itemsep=-.16em	label=(1),itemsep=-.16em	X
ejpam-7124	51	3	,	,	PUNCT
ejpam-7124	51	4	topsep=2pt	topsep=2pt	PROPN
ejpam-7124	51	5	]	]	PUNCT
ejpam-7124	51	6	(	(	PUNCT
ejpam-7124	51	7	i	i	NOUN
ejpam-7124	51	8	)	)	PUNCT
ejpam-7124	51	9	u	u	PROPN
ejpam-7124	51	10	,	,	PUNCT
ejpam-7124	51	11	f	f	PROPN
ejpam-7124	51	12	are	be	AUX
ejpam-7124	51	13	even	even	ADV
ejpam-7124	51	14	and	and	CCONJ
ejpam-7124	51	15	e	e	X
ejpam-7124	51	16	=	=	NOUN
ejpam-7124	51	17	2j	2j	X
ejpam-7124	52	1	+	+	CCONJ
ejpam-7124	52	2	1	1	X
ejpam-7124	52	3	.	.	X
ejpam-7124	52	4	(	(	PUNCT
ejpam-7124	52	5	ii	ii	NOUN
ejpam-7124	52	6	)	)	PUNCT
ejpam-7124	52	7	u	u	NOUN
ejpam-7124	52	8	=	=	NOUN
ejpam-7124	52	9	2j	2j	NUM
ejpam-7124	52	10	are	be	AUX
ejpam-7124	52	11	and	and	CCONJ
ejpam-7124	52	12	e	e	NOUN
ejpam-7124	52	13	,	,	PUNCT
ejpam-7124	52	14	f	f	PROPN
ejpam-7124	52	15	are	be	AUX
ejpam-7124	52	16	odd	odd	ADJ
ejpam-7124	52	17	.	.	PUNCT
ejpam-7124	53	1	(	(	PUNCT
ejpam-7124	53	2	iii	iii	X
ejpam-7124	53	3	)	)	PUNCT
ejpam-7124	53	4	u	u	NOUN
ejpam-7124	53	5	,	,	PUNCT
ejpam-7124	53	6	f	f	PROPN
ejpam-7124	53	7	are	be	AUX
ejpam-7124	53	8	odd	odd	ADJ
ejpam-7124	53	9	and	and	CCONJ
ejpam-7124	53	10	e	e	X
ejpam-7124	53	11	=	=	NOUN
ejpam-7124	53	12	2j	2j	X
ejpam-7124	53	13	.	.	PUNCT
ejpam-7124	54	1	(	(	PUNCT
ejpam-7124	54	2	iv	iv	X
ejpam-7124	54	3	)	)	PUNCT
ejpam-7124	54	4	u	u	NOUN
ejpam-7124	54	5	,	,	PUNCT
ejpam-7124	54	6	e	e	PROPN
ejpam-7124	54	7	,	,	PUNCT
ejpam-7124	54	8	f	f	PROPN
ejpam-7124	54	9	are	be	AUX
ejpam-7124	54	10	even	even	ADV
ejpam-7124	54	11	.	.	PUNCT
ejpam-7124	55	1	(	(	PUNCT
ejpam-7124	55	2	v	v	NOUN
ejpam-7124	55	3	)	)	PUNCT
ejpam-7124	55	4	u	u	NOUN
ejpam-7124	55	5	,	,	PUNCT
ejpam-7124	55	6	e	e	NOUN
ejpam-7124	55	7	are	be	AUX
ejpam-7124	55	8	even	even	ADV
ejpam-7124	55	9	and	and	CCONJ
ejpam-7124	55	10	f	f	X
ejpam-7124	55	11	=	=	SYM
ejpam-7124	55	12	2j	2j	PROPN
ejpam-7124	56	1	+	+	CCONJ
ejpam-7124	56	2	1	1	X
ejpam-7124	56	3	.	.	PUNCT
ejpam-7124	56	4	(	(	PUNCT
ejpam-7124	56	5	vi	vi	NOUN
ejpam-7124	56	6	)	)	PUNCT
ejpam-7124	56	7	u	u	NOUN
ejpam-7124	56	8	,	,	PUNCT
ejpam-7124	56	9	e	e	PROPN
ejpam-7124	56	10	are	be	AUX
ejpam-7124	56	11	odd	odd	ADJ
ejpam-7124	56	12	and	and	CCONJ
ejpam-7124	56	13	f	f	NOUN
ejpam-7124	56	14	=	=	PUNCT
ejpam-7124	56	15	2j	2j	X
ejpam-7124	56	16	.	.	PUNCT
ejpam-7124	57	1	(	(	PUNCT
ejpam-7124	57	2	vii	vii	PROPN
ejpam-7124	57	3	)	)	PUNCT
ejpam-7124	57	4	u	u	PROPN
ejpam-7124	57	5	,	,	PUNCT
ejpam-7124	57	6	f	f	PROPN
ejpam-7124	57	7	are	be	AUX
ejpam-7124	57	8	odd	odd	ADJ
ejpam-7124	57	9	.	.	PUNCT
ejpam-7124	58	1	thus	thus	ADV
ejpam-7124	58	2	d̄	d̄	PROPN
ejpam-7124	58	3	is	be	AUX
ejpam-7124	58	4	a	a	DET
ejpam-7124	58	5	controlled	control	VERB
ejpam-7124	58	6	metric	metric	ADJ
ejpam-7124	58	7	type	type	NOUN
ejpam-7124	58	8	space	space	NOUN
ejpam-7124	58	9	.	.	PUNCT
ejpam-7124	59	1	moreover	moreover	ADV
ejpam-7124	59	2	for	for	ADP
ejpam-7124	59	3	j	j	PROPN
ejpam-7124	59	4	=	=	SYM
ejpam-7124	59	5	2	2	NUM
ejpam-7124	59	6	,	,	PUNCT
ejpam-7124	59	7	3	3	NUM
ejpam-7124	59	8	,	,	PUNCT
ejpam-7124	59	9	.	.	PUNCT
ejpam-7124	59	10	.	.	PUNCT
ejpam-7124	60	1	.	.	PUNCT
ejpam-7124	61	1	we	we	PRON
ejpam-7124	61	2	have	have	VERB
ejpam-7124	61	3	d̄(2j	d̄(2j	PRON
ejpam-7124	61	4	+	+	CCONJ
ejpam-7124	61	5	1	1	NUM
ejpam-7124	61	6	,	,	PUNCT
ejpam-7124	61	7	4j	4j	NOUN
ejpam-7124	61	8	+	+	CCONJ
ejpam-7124	61	9	1	1	X
ejpam-7124	61	10	)	)	PUNCT
ejpam-7124	61	11	=	=	SYM
ejpam-7124	61	12	1	1	NUM
ejpam-7124	61	13	>	>	SYM
ejpam-7124	61	14	1	1	NUM
ejpam-7124	61	15	j	j	PROPN
ejpam-7124	61	16	µ(2j	µ(2j	ADP
ejpam-7124	61	17	+	+	NUM
ejpam-7124	61	18	1	1	NUM
ejpam-7124	61	19	,	,	PUNCT
ejpam-7124	61	20	4j	4j	NOUN
ejpam-7124	61	21	+	+	CCONJ
ejpam-7124	61	22	1)[d̄(2j	1)[d̄(2j	NUM
ejpam-7124	61	23	+	+	SYM
ejpam-7124	61	24	1	1	NUM
ejpam-7124	61	25	,	,	PUNCT
ejpam-7124	61	26	2j	2j	NUM
ejpam-7124	61	27	)	)	PUNCT
ejpam-7124	62	1	+	+	CCONJ
ejpam-7124	62	2	d̄(2j	d̄(2j	NOUN
ejpam-7124	62	3	,	,	PUNCT
ejpam-7124	62	4	4j	4j	NOUN
ejpam-7124	62	5	+	+	CCONJ
ejpam-7124	62	6	1	1	NUM
ejpam-7124	62	7	)	)	PUNCT
ejpam-7124	62	8	]	]	PUNCT
ejpam-7124	62	9	.	.	PUNCT
ejpam-7124	63	1	(	(	PUNCT
ejpam-7124	63	2	3	3	X
ejpam-7124	63	3	)	)	PUNCT
ejpam-7124	63	4	therefore	therefore	ADV
ejpam-7124	63	5	d̄	d̄	PROPN
ejpam-7124	63	6	is	be	AUX
ejpam-7124	63	7	not	not	PART
ejpam-7124	63	8	an	an	DET
ejpam-7124	63	9	extended	extended	ADJ
ejpam-7124	63	10	b	b	X
ejpam-7124	63	11	-	-	PUNCT
ejpam-7124	63	12	metric	metric	ADJ
ejpam-7124	63	13	space	space	NOUN
ejpam-7124	63	14	.	.	PUNCT
ejpam-7124	64	1	example	example	NOUN
ejpam-7124	64	2	2	2	NUM
ejpam-7124	64	3	(	(	PUNCT
ejpam-7124	64	4	[	[	X
ejpam-7124	64	5	15	15	NUM
ejpam-7124	64	6	]	]	NUM
ejpam-7124	64	7	)	)	PUNCT
ejpam-7124	64	8	.	.	PUNCT
ejpam-7124	65	1	take	take	VERB
ejpam-7124	65	2	ω	ω	NOUN
ejpam-7124	65	3	=	=	SYM
ejpam-7124	65	4	{	{	PUNCT
ejpam-7124	65	5	0	0	NUM
ejpam-7124	65	6	,	,	PUNCT
ejpam-7124	65	7	1	1	NUM
ejpam-7124	65	8	,	,	PUNCT
ejpam-7124	65	9	2	2	NUM
ejpam-7124	65	10	}	}	PUNCT
ejpam-7124	65	11	.	.	PUNCT
ejpam-7124	66	1	consider	consider	VERB
ejpam-7124	66	2	the	the	DET
ejpam-7124	66	3	function	function	NOUN
ejpam-7124	66	4	d̄	d̄	NOUN
ejpam-7124	66	5	given	give	VERB
ejpam-7124	66	6	as	as	ADP
ejpam-7124	66	7	d̄(0	d̄(0	NOUN
ejpam-7124	66	8	,	,	PUNCT
ejpam-7124	66	9	0	0	NUM
ejpam-7124	66	10	)	)	PUNCT
ejpam-7124	66	11	=	=	SYM
ejpam-7124	67	1	d̄(1	d̄(1	NOUN
ejpam-7124	67	2	,	,	PUNCT
ejpam-7124	67	3	1	1	X
ejpam-7124	67	4	)	)	PUNCT
ejpam-7124	67	5	=	=	SYM
ejpam-7124	67	6	d̄(2	d̄(2	ADJ
ejpam-7124	67	7	,	,	PUNCT
ejpam-7124	67	8	2	2	NUM
ejpam-7124	67	9	)	)	PUNCT
ejpam-7124	67	10	=	=	SYM
ejpam-7124	67	11	0	0	NUM
ejpam-7124	67	12	,	,	PUNCT
ejpam-7124	67	13	d̄(0	d̄(0	NUM
ejpam-7124	67	14	,	,	PUNCT
ejpam-7124	67	15	1	1	NUM
ejpam-7124	67	16	)	)	PUNCT
ejpam-7124	67	17	=	=	SYM
ejpam-7124	67	18	d̄(1	d̄(1	NOUN
ejpam-7124	67	19	,	,	PUNCT
ejpam-7124	67	20	0	0	NUM
ejpam-7124	67	21	)	)	PUNCT
ejpam-7124	67	22	,	,	PUNCT
ejpam-7124	67	23	d̄(0	d̄(0	NUM
ejpam-7124	67	24	,	,	PUNCT
ejpam-7124	67	25	2	2	NUM
ejpam-7124	67	26	)	)	PUNCT
ejpam-7124	67	27	=	=	SYM
ejpam-7124	67	28	d̄(2	d̄(2	ADJ
ejpam-7124	67	29	,	,	PUNCT
ejpam-7124	67	30	0	0	NUM
ejpam-7124	67	31	)	)	PUNCT
ejpam-7124	67	32	=	=	SYM
ejpam-7124	67	33	1	1	NUM
ejpam-7124	67	34	2	2	NUM
ejpam-7124	67	35	,	,	PUNCT
ejpam-7124	67	36	d̄(1	d̄(1	NUM
ejpam-7124	67	37	,	,	PUNCT
ejpam-7124	67	38	2	2	X
ejpam-7124	67	39	)	)	PUNCT
ejpam-7124	67	40	=	=	SYM
ejpam-7124	67	41	d̄(2	d̄(2	ADJ
ejpam-7124	67	42	,	,	PUNCT
ejpam-7124	67	43	1	1	X
ejpam-7124	67	44	)	)	PUNCT
ejpam-7124	67	45	=	=	SYM
ejpam-7124	67	46	2	2	NUM
ejpam-7124	67	47	5	5	NUM
ejpam-7124	67	48	.	.	PUNCT
ejpam-7124	68	1	(	(	PUNCT
ejpam-7124	68	2	4	4	X
ejpam-7124	68	3	)	)	PUNCT
ejpam-7124	68	4	m.	m.	NOUN
ejpam-7124	68	5	kumar	kumar	PROPN
ejpam-7124	68	6	et	et	PROPN
ejpam-7124	68	7	al	al	PROPN
ejpam-7124	68	8	.	.	PUNCT
ejpam-7124	68	9	/	/	SYM
ejpam-7124	68	10	eur	eur	PROPN
ejpam-7124	68	11	.	.	PUNCT
ejpam-7124	69	1	j.	j.	PROPN
ejpam-7124	69	2	pure	pure	PROPN
ejpam-7124	69	3	appl	appl	PROPN
ejpam-7124	69	4	.	.	PROPN
ejpam-7124	69	5	math	math	PROPN
ejpam-7124	69	6	,	,	PUNCT
ejpam-7124	69	7	18	18	NUM
ejpam-7124	69	8	(	(	PUNCT
ejpam-7124	69	9	4	4	NUM
ejpam-7124	69	10	)	)	PUNCT
ejpam-7124	69	11	(	(	PUNCT
ejpam-7124	69	12	2025	2025	NUM
ejpam-7124	69	13	)	)	PUNCT
ejpam-7124	69	14	,	,	PUNCT
ejpam-7124	69	15	7124	7124	NUM
ejpam-7124	69	16	4	4	NUM
ejpam-7124	69	17	of	of	ADP
ejpam-7124	69	18	14	14	NUM
ejpam-7124	69	19	define	define	VERB
ejpam-7124	69	20	a	a	DET
ejpam-7124	69	21	symmetric	symmetric	ADJ
ejpam-7124	69	22	function	function	NOUN
ejpam-7124	69	23	µ	µ	NOUN
ejpam-7124	69	24	:	:	PUNCT
ejpam-7124	69	25	ω×	ω×	PROPN
ejpam-7124	69	26	ω	ω	PROPN
ejpam-7124	69	27	→	→	SYM
ejpam-7124	69	28	[	[	X
ejpam-7124	69	29	1,∞	1,∞	NUM
ejpam-7124	69	30	)	)	PUNCT
ejpam-7124	69	31	such	such	ADJ
ejpam-7124	69	32	that	that	SCONJ
ejpam-7124	69	33	µ(0	µ(0	NOUN
ejpam-7124	69	34	,	,	PUNCT
ejpam-7124	69	35	0	0	NUM
ejpam-7124	69	36	)	)	PUNCT
ejpam-7124	69	37	=	=	SYM
ejpam-7124	69	38	µ(1	µ(1	PROPN
ejpam-7124	69	39	,	,	PUNCT
ejpam-7124	69	40	1	1	NUM
ejpam-7124	69	41	)	)	PUNCT
ejpam-7124	69	42	=	=	SYM
ejpam-7124	69	43	µ(2	µ(2	PROPN
ejpam-7124	69	44	,	,	PUNCT
ejpam-7124	69	45	2	2	NUM
ejpam-7124	69	46	)	)	PUNCT
ejpam-7124	69	47	=	=	PUNCT
ejpam-7124	69	48	µ(0	µ(0	NOUN
ejpam-7124	69	49	,	,	PUNCT
ejpam-7124	69	50	2	2	NUM
ejpam-7124	69	51	)	)	PUNCT
ejpam-7124	69	52	=	=	SYM
ejpam-7124	69	53	1	1	NUM
ejpam-7124	69	54	,	,	PUNCT
ejpam-7124	69	55	µ(1	µ(1	PROPN
ejpam-7124	69	56	,	,	PUNCT
ejpam-7124	69	57	2	2	NUM
ejpam-7124	69	58	)	)	PUNCT
ejpam-7124	69	59	=	=	SYM
ejpam-7124	69	60	5	5	NUM
ejpam-7124	69	61	4	4	NUM
ejpam-7124	69	62	,	,	PUNCT
ejpam-7124	69	63	µ(0	µ(0	NOUN
ejpam-7124	69	64	,	,	PUNCT
ejpam-7124	69	65	1	1	NUM
ejpam-7124	69	66	)	)	PUNCT
ejpam-7124	69	67	=	=	SYM
ejpam-7124	69	68	11	11	NUM
ejpam-7124	69	69	10	10	NUM
ejpam-7124	69	70	.	.	PUNCT
ejpam-7124	70	1	(	(	PUNCT
ejpam-7124	70	2	5	5	X
ejpam-7124	70	3	)	)	PUNCT
ejpam-7124	70	4	one	one	NOUN
ejpam-7124	70	5	can	can	AUX
ejpam-7124	70	6	easily	easily	ADV
ejpam-7124	70	7	verify	verify	VERB
ejpam-7124	70	8	that	that	DET
ejpam-7124	70	9	d̄	d̄	NOUN
ejpam-7124	70	10	is	be	AUX
ejpam-7124	70	11	a	a	DET
ejpam-7124	70	12	controlled	control	VERB
ejpam-7124	70	13	metric	metric	ADJ
ejpam-7124	70	14	type	type	NOUN
ejpam-7124	70	15	space	space	NOUN
ejpam-7124	70	16	.	.	PUNCT
ejpam-7124	71	1	since	since	SCONJ
ejpam-7124	71	2	d̄(0	d̄(0	NUM
ejpam-7124	71	3	,	,	PUNCT
ejpam-7124	71	4	1	1	NUM
ejpam-7124	71	5	)	)	PUNCT
ejpam-7124	71	6	=	=	SYM
ejpam-7124	71	7	1	1	NUM
ejpam-7124	71	8	>	>	SYM
ejpam-7124	71	9	99	99	NUM
ejpam-7124	71	10	100	100	NUM
ejpam-7124	71	11	µ(0	µ(0	NOUN
ejpam-7124	71	12	,	,	PUNCT
ejpam-7124	71	13	1)[d̄(0	1)[d̄(0	NUM
ejpam-7124	71	14	,	,	PUNCT
ejpam-7124	71	15	2	2	NUM
ejpam-7124	71	16	)	)	PUNCT
ejpam-7124	71	17	+	+	CCONJ
ejpam-7124	71	18	d̄(2	d̄(2	ADJ
ejpam-7124	71	19	,	,	PUNCT
ejpam-7124	71	20	1	1	NUM
ejpam-7124	71	21	)	)	PUNCT
ejpam-7124	71	22	]	]	PUNCT
ejpam-7124	71	23	.	.	PUNCT
ejpam-7124	72	1	(	(	PUNCT
ejpam-7124	72	2	6	6	NUM
ejpam-7124	72	3	)	)	PUNCT
ejpam-7124	72	4	d̄	d̄	NOUN
ejpam-7124	72	5	is	be	AUX
ejpam-7124	72	6	not	not	PART
ejpam-7124	72	7	an	an	DET
ejpam-7124	72	8	extended	extended	ADJ
ejpam-7124	72	9	b	b	NOUN
ejpam-7124	72	10	-	-	NOUN
ejpam-7124	72	11	metric	metric	ADJ
ejpam-7124	72	12	.	.	PUNCT
ejpam-7124	73	1	the	the	DET
ejpam-7124	73	2	concept	concept	NOUN
ejpam-7124	73	3	of	of	ADP
ejpam-7124	73	4	cauchy	cauchy	NOUN
ejpam-7124	73	5	and	and	CCONJ
ejpam-7124	73	6	convergent	convergent	ADJ
ejpam-7124	73	7	sequences	sequence	NOUN
ejpam-7124	73	8	in	in	ADP
ejpam-7124	73	9	controlled	control	VERB
ejpam-7124	73	10	metric	metric	ADJ
ejpam-7124	73	11	type	type	NOUN
ejpam-7124	73	12	spaces	space	NOUN
ejpam-7124	73	13	are	be	AUX
ejpam-7124	73	14	defined	define	VERB
ejpam-7124	73	15	as	as	SCONJ
ejpam-7124	73	16	follows	follow	VERB
ejpam-7124	73	17	:	:	PUNCT
ejpam-7124	73	18	definition	definition	NOUN
ejpam-7124	73	19	3	3	NUM
ejpam-7124	73	20	(	(	PUNCT
ejpam-7124	73	21	[	[	X
ejpam-7124	73	22	15	15	NUM
ejpam-7124	73	23	]	]	NUM
ejpam-7124	73	24	)	)	PUNCT
ejpam-7124	73	25	.	.	PUNCT
ejpam-7124	74	1	let	let	AUX
ejpam-7124	74	2	(	(	PUNCT
ejpam-7124	74	3	ω	ω	NOUN
ejpam-7124	74	4	,	,	PUNCT
ejpam-7124	74	5	d̄	d̄	PROPN
ejpam-7124	74	6	)	)	PUNCT
ejpam-7124	74	7	be	be	VERB
ejpam-7124	74	8	a	a	DET
ejpam-7124	74	9	controlled	control	VERB
ejpam-7124	74	10	metric	metric	ADJ
ejpam-7124	74	11	type	type	NOUN
ejpam-7124	74	12	space	space	NOUN
ejpam-7124	74	13	and	and	CCONJ
ejpam-7124	74	14	{	{	PUNCT
ejpam-7124	74	15	uj}j≥0	uj}j≥0	NOUN
ejpam-7124	74	16	be	be	AUX
ejpam-7124	74	17	a	a	DET
ejpam-7124	74	18	sequence	sequence	NOUN
ejpam-7124	74	19	in	in	ADP
ejpam-7124	74	20	ω	ω	NUM
ejpam-7124	74	21	.	.	PUNCT
ejpam-7124	75	1	[	[	X
ejpam-7124	75	2	label=(iii),itemsep=-.16em	label=(iii),itemsep=-.16em	X
ejpam-7124	75	3	,	,	PUNCT
ejpam-7124	75	4	topsep=2pt	topsep=2pt	PROPN
ejpam-7124	75	5	]	]	PUNCT
ejpam-7124	75	6	(	(	PUNCT
ejpam-7124	75	7	i	i	NOUN
ejpam-7124	75	8	)	)	PUNCT
ejpam-7124	75	9	the	the	DET
ejpam-7124	75	10	sequence	sequence	NOUN
ejpam-7124	75	11	{	{	PUNCT
ejpam-7124	75	12	uj	uj	PROPN
ejpam-7124	75	13	}	}	PUNCT
ejpam-7124	75	14	converges	converge	NOUN
ejpam-7124	75	15	to	to	ADP
ejpam-7124	75	16	some	some	DET
ejpam-7124	75	17	u	u	NOUN
ejpam-7124	75	18	∈	∈	PROPN
ejpam-7124	75	19	ω	ω	NOUN
ejpam-7124	75	20	if	if	SCONJ
ejpam-7124	75	21	for	for	ADP
ejpam-7124	75	22	all	all	PRON
ejpam-7124	75	23	ε	ε	PROPN
ejpam-7124	75	24	>	>	X
ejpam-7124	75	25	0	0	NUM
ejpam-7124	76	1	∃	∃	PROPN
ejpam-7124	76	2	p	p	PROPN
ejpam-7124	76	3	=	=	X
ejpam-7124	76	4	p	p	X
ejpam-7124	76	5	(	(	PUNCT
ejpam-7124	76	6	ε	ε	PROPN
ejpam-7124	76	7	)	)	PUNCT
ejpam-7124	76	8	∈	∈	PROPN
ejpam-7124	76	9	n	n	PRON
ejpam-7124	76	10	such	such	ADJ
ejpam-7124	76	11	that	that	DET
ejpam-7124	76	12	d̄(uj	d̄(uj	NOUN
ejpam-7124	76	13	,	,	PUNCT
ejpam-7124	76	14	u	u	NOUN
ejpam-7124	76	15	)	)	PUNCT
ejpam-7124	76	16	<	<	X
ejpam-7124	76	17	ε	ε	PROPN
ejpam-7124	76	18	for	for	ADP
ejpam-7124	76	19	all	all	DET
ejpam-7124	76	20	j	j	PROPN
ejpam-7124	76	21	≥	≥	PROPN
ejpam-7124	76	22	p	p	NOUN
ejpam-7124	76	23	,	,	PUNCT
ejpam-7124	76	24	we	we	PRON
ejpam-7124	76	25	write	write	VERB
ejpam-7124	76	26	lim	lim	PROPN
ejpam-7124	76	27	j→0	j→0	X
ejpam-7124	77	1	uj	uj	PROPN
ejpam-7124	77	2	=	=	PUNCT
ejpam-7124	77	3	u.	u.	PROPN
ejpam-7124	77	4	(	(	PUNCT
ejpam-7124	77	5	ii	ii	NOUN
ejpam-7124	77	6	)	)	PUNCT
ejpam-7124	77	7	we	we	PRON
ejpam-7124	77	8	say	say	VERB
ejpam-7124	77	9	that	that	SCONJ
ejpam-7124	77	10	{	{	PUNCT
ejpam-7124	77	11	uj	uj	PROPN
ejpam-7124	77	12	}	}	PUNCT
ejpam-7124	77	13	is	be	AUX
ejpam-7124	77	14	cauchy	cauchy	ADJ
ejpam-7124	77	15	if	if	SCONJ
ejpam-7124	77	16	for	for	ADP
ejpam-7124	77	17	all	all	DET
ejpam-7124	77	18	ε	ε	PROPN
ejpam-7124	77	19	>	>	X
ejpam-7124	77	20	0	0	NUM
ejpam-7124	78	1	∃	∃	PROPN
ejpam-7124	78	2	p	p	PROPN
ejpam-7124	78	3	=	=	X
ejpam-7124	78	4	p	p	X
ejpam-7124	78	5	(	(	PUNCT
ejpam-7124	78	6	ε	ε	PROPN
ejpam-7124	78	7	)	)	PUNCT
ejpam-7124	78	8	∈	∈	PROPN
ejpam-7124	78	9	n	n	CCONJ
ejpam-7124	78	10	such	such	ADJ
ejpam-7124	78	11	that	that	DET
ejpam-7124	78	12	d̄(ui	d̄(ui	NOUN
ejpam-7124	78	13	,	,	PUNCT
ejpam-7124	78	14	uj	uj	PROPN
ejpam-7124	78	15	)	)	PUNCT
ejpam-7124	78	16	<	<	X
ejpam-7124	78	17	ε	ε	PROPN
ejpam-7124	78	18	for	for	ADP
ejpam-7124	78	19	all	all	DET
ejpam-7124	78	20	j	j	PROPN
ejpam-7124	78	21	≥	≥	PROPN
ejpam-7124	78	22	p	p	NOUN
ejpam-7124	78	23	.	.	PUNCT
ejpam-7124	79	1	(	(	PUNCT
ejpam-7124	79	2	iii	iii	X
ejpam-7124	79	3	)	)	PUNCT
ejpam-7124	79	4	if	if	SCONJ
ejpam-7124	79	5	every	every	DET
ejpam-7124	79	6	cauchy	cauchy	ADJ
ejpam-7124	79	7	sequence	sequence	NOUN
ejpam-7124	79	8	is	be	AUX
ejpam-7124	79	9	convergent	convergent	NOUN
ejpam-7124	79	10	then	then	ADV
ejpam-7124	79	11	the	the	DET
ejpam-7124	79	12	space	space	NOUN
ejpam-7124	79	13	(	(	PUNCT
ejpam-7124	79	14	ω	ω	PROPN
ejpam-7124	79	15	,	,	PUNCT
ejpam-7124	79	16	d̄	d̄	PROPN
ejpam-7124	79	17	)	)	PUNCT
ejpam-7124	79	18	is	be	AUX
ejpam-7124	79	19	called	call	VERB
ejpam-7124	79	20	complete	complete	ADJ
ejpam-7124	79	21	.	.	PUNCT
ejpam-7124	80	1	definition	definition	NOUN
ejpam-7124	80	2	4	4	NUM
ejpam-7124	80	3	(	(	PUNCT
ejpam-7124	80	4	[	[	X
ejpam-7124	80	5	15	15	NUM
ejpam-7124	80	6	]	]	NUM
ejpam-7124	80	7	)	)	PUNCT
ejpam-7124	80	8	.	.	PUNCT
ejpam-7124	81	1	let	let	AUX
ejpam-7124	81	2	(	(	PUNCT
ejpam-7124	81	3	ω	ω	NOUN
ejpam-7124	81	4	,	,	PUNCT
ejpam-7124	81	5	d̄	d̄	PROPN
ejpam-7124	81	6	)	)	PUNCT
ejpam-7124	81	7	be	be	VERB
ejpam-7124	81	8	a	a	DET
ejpam-7124	81	9	controlled	control	VERB
ejpam-7124	81	10	metric	metric	ADJ
ejpam-7124	81	11	type	type	NOUN
ejpam-7124	81	12	space	space	NOUN
ejpam-7124	81	13	.	.	PUNCT
ejpam-7124	82	1	let	let	VERB
ejpam-7124	82	2	u	u	PRON
ejpam-7124	82	3	∈	∈	PROPN
ejpam-7124	82	4	ω	ω	PROPN
ejpam-7124	82	5	and	and	CCONJ
ejpam-7124	82	6	ε	ε	PROPN
ejpam-7124	82	7	>	>	X
ejpam-7124	82	8	0	0	PROPN
ejpam-7124	82	9	,	,	PUNCT
ejpam-7124	82	10	c(u	c(u	PROPN
ejpam-7124	82	11	,	,	PUNCT
ejpam-7124	82	12	ε	ε	PROPN
ejpam-7124	82	13	)	)	PUNCT
ejpam-7124	82	14	is	be	AUX
ejpam-7124	82	15	defined	define	VERB
ejpam-7124	82	16	as	as	ADP
ejpam-7124	82	17	[	[	X
ejpam-7124	82	18	label=(iv),itemsep=-.16em	label=(iv),itemsep=-.16em	X
ejpam-7124	82	19	,	,	PUNCT
ejpam-7124	82	20	topsep=2pt	topsep=2pt	PROPN
ejpam-7124	82	21	]	]	PUNCT
ejpam-7124	82	22	(	(	PUNCT
ejpam-7124	82	23	i	i	NOUN
ejpam-7124	82	24	)	)	PUNCT
ejpam-7124	82	25	the	the	DET
ejpam-7124	82	26	open	open	ADJ
ejpam-7124	82	27	ball	ball	NOUN
ejpam-7124	82	28	c(u	c(u	PROPN
ejpam-7124	82	29	,	,	PUNCT
ejpam-7124	82	30	ε	ε	PROPN
ejpam-7124	82	31	)	)	PUNCT
ejpam-7124	82	32	=	=	SYM
ejpam-7124	82	33	{	{	PUNCT
ejpam-7124	82	34	e	e	PROPN
ejpam-7124	82	35	∈	∈	PROPN
ejpam-7124	82	36	ω	ω	PROPN
ejpam-7124	82	37	,	,	PUNCT
ejpam-7124	82	38	d̄(u	d̄(u	NOUN
ejpam-7124	82	39	,	,	PUNCT
ejpam-7124	82	40	e	e	NOUN
ejpam-7124	82	41	)	)	PUNCT
ejpam-7124	82	42	<	<	X
ejpam-7124	82	43	ε	ε	PROPN
ejpam-7124	82	44	}	}	PUNCT
ejpam-7124	82	45	.	.	PUNCT
ejpam-7124	83	1	(	(	PUNCT
ejpam-7124	83	2	7	7	NUM
ejpam-7124	83	3	)	)	PUNCT
ejpam-7124	83	4	(	(	PUNCT
ejpam-7124	83	5	ii	ii	NOUN
ejpam-7124	83	6	)	)	PUNCT
ejpam-7124	83	7	a	a	DET
ejpam-7124	83	8	self	self	NOUN
ejpam-7124	83	9	mapping	mapping	NOUN
ejpam-7124	83	10	t	t	NOUN
ejpam-7124	83	11	on	on	ADP
ejpam-7124	83	12	ω	ω	PROPN
ejpam-7124	83	13	is	be	AUX
ejpam-7124	83	14	said	say	VERB
ejpam-7124	83	15	to	to	PART
ejpam-7124	83	16	be	be	AUX
ejpam-7124	83	17	continuous	continuous	ADJ
ejpam-7124	83	18	at	at	ADP
ejpam-7124	83	19	u	u	PROPN
ejpam-7124	83	20	∈	∈	PROPN
ejpam-7124	83	21	ω	ω	NOUN
ejpam-7124	83	22	if	if	SCONJ
ejpam-7124	83	23	∀	∀	X
ejpam-7124	83	24	ε	ε	X
ejpam-7124	83	25	>	>	X
ejpam-7124	83	26	0	0	NUM
ejpam-7124	84	1	∃	∃	PROPN
ejpam-7124	84	2	δ	δ	PROPN
ejpam-7124	84	3	>	>	X
ejpam-7124	84	4	0	0	NUM
ejpam-7124	84	5	such	such	ADJ
ejpam-7124	84	6	that	that	SCONJ
ejpam-7124	84	7	t	t	PROPN
ejpam-7124	84	8	(	(	PUNCT
ejpam-7124	84	9	c(u	c(u	PROPN
ejpam-7124	84	10	,	,	PUNCT
ejpam-7124	84	11	δ	δ	PROPN
ejpam-7124	84	12	)	)	PUNCT
ejpam-7124	84	13	)	)	PUNCT
ejpam-7124	85	1	⊆	⊆	X
ejpam-7124	85	2	c(tu	c(tu	PROPN
ejpam-7124	85	3	,	,	PUNCT
ejpam-7124	85	4	ε	ε	PROPN
ejpam-7124	85	5	)	)	PUNCT
ejpam-7124	85	6	.	.	PUNCT
ejpam-7124	86	1	remark	remark	PROPN
ejpam-7124	86	2	1	1	NUM
ejpam-7124	86	3	.	.	PUNCT
ejpam-7124	87	1	if	if	SCONJ
ejpam-7124	87	2	for	for	ADP
ejpam-7124	87	3	all	all	DET
ejpam-7124	87	4	u	u	NOUN
ejpam-7124	87	5	,	,	PUNCT
ejpam-7124	87	6	e	e	NOUN
ejpam-7124	87	7	in	in	ADP
ejpam-7124	87	8	ω	ω	PROPN
ejpam-7124	87	9	,	,	PUNCT
ejpam-7124	87	10	µ(u	µ(u	NOUN
ejpam-7124	87	11	,	,	PUNCT
ejpam-7124	87	12	e	e	NOUN
ejpam-7124	87	13	)	)	PUNCT
ejpam-7124	87	14	=	=	SYM
ejpam-7124	87	15	t	t	PROPN
ejpam-7124	87	16	≥	≥	NUM
ejpam-7124	87	17	1	1	NUM
ejpam-7124	87	18	,	,	PUNCT
ejpam-7124	87	19	then	then	ADV
ejpam-7124	87	20	it	it	PRON
ejpam-7124	87	21	is	be	AUX
ejpam-7124	87	22	a	a	DET
ejpam-7124	87	23	b	b	NOUN
ejpam-7124	87	24	-	-	PUNCT
ejpam-7124	87	25	metric	metric	ADJ
ejpam-7124	87	26	space	space	NOUN
ejpam-7124	87	27	.	.	PUNCT
ejpam-7124	88	1	therefore	therefore	ADV
ejpam-7124	88	2	,	,	PUNCT
ejpam-7124	88	3	we	we	PRON
ejpam-7124	88	4	conclude	conclude	VERB
ejpam-7124	88	5	that	that	SCONJ
ejpam-7124	88	6	every	every	DET
ejpam-7124	88	7	b	b	X
ejpam-7124	88	8	-	-	PUNCT
ejpam-7124	88	9	metric	metric	ADJ
ejpam-7124	88	10	space	space	NOUN
ejpam-7124	88	11	is	be	AUX
ejpam-7124	88	12	controlled	control	VERB
ejpam-7124	88	13	metric	metric	ADJ
ejpam-7124	88	14	space	space	NOUN
ejpam-7124	88	15	.	.	PUNCT
ejpam-7124	89	1	however	however	ADV
ejpam-7124	89	2	the	the	DET
ejpam-7124	89	3	converse	converse	NOUN
ejpam-7124	89	4	is	be	AUX
ejpam-7124	89	5	not	not	PART
ejpam-7124	89	6	always	always	ADV
ejpam-7124	89	7	true	true	ADJ
ejpam-7124	89	8	.	.	PUNCT
ejpam-7124	90	1	clearly	clearly	ADV
ejpam-7124	90	2	,	,	PUNCT
ejpam-7124	90	3	if	if	SCONJ
ejpam-7124	90	4	a	a	DET
ejpam-7124	90	5	mapping	mapping	NOUN
ejpam-7124	90	6	t	t	NOUN
ejpam-7124	90	7	is	be	AUX
ejpam-7124	90	8	continuous	continuous	ADJ
ejpam-7124	90	9	at	at	ADP
ejpam-7124	90	10	u	u	NOUN
ejpam-7124	90	11	in	in	ADP
ejpam-7124	90	12	the	the	DET
ejpam-7124	90	13	controlled	control	VERB
ejpam-7124	90	14	metric	metric	ADJ
ejpam-7124	90	15	type	type	NOUN
ejpam-7124	90	16	space	space	NOUN
ejpam-7124	90	17	then	then	ADV
ejpam-7124	90	18	uj	uj	PROPN
ejpam-7124	90	19	→	→	SYM
ejpam-7124	90	20	u⇒	u⇒	PROPN
ejpam-7124	90	21	tuj	tuj	PROPN
ejpam-7124	90	22	→	→	SYM
ejpam-7124	90	23	u	u	PROPN
ejpam-7124	90	24	as	as	ADP
ejpam-7124	90	25	j→	j→	PROPN
ejpam-7124	90	26	∞	∞	PROPN
ejpam-7124	90	27	m.	m.	NOUN
ejpam-7124	90	28	kumar	kumar	PROPN
ejpam-7124	90	29	et	et	PROPN
ejpam-7124	90	30	al	al	PROPN
ejpam-7124	90	31	.	.	PUNCT
ejpam-7124	90	32	/	/	SYM
ejpam-7124	90	33	eur	eur	PROPN
ejpam-7124	90	34	.	.	PUNCT
ejpam-7124	91	1	j.	j.	PROPN
ejpam-7124	91	2	pure	pure	PROPN
ejpam-7124	91	3	appl	appl	PROPN
ejpam-7124	91	4	.	.	PROPN
ejpam-7124	91	5	math	math	PROPN
ejpam-7124	91	6	,	,	PUNCT
ejpam-7124	91	7	18	18	NUM
ejpam-7124	91	8	(	(	PUNCT
ejpam-7124	91	9	4	4	NUM
ejpam-7124	91	10	)	)	PUNCT
ejpam-7124	91	11	(	(	PUNCT
ejpam-7124	91	12	2025	2025	NUM
ejpam-7124	91	13	)	)	PUNCT
ejpam-7124	91	14	,	,	PUNCT
ejpam-7124	91	15	7124	7124	NUM
ejpam-7124	91	16	5	5	NUM
ejpam-7124	91	17	of	of	ADP
ejpam-7124	91	18	14	14	NUM
ejpam-7124	91	19	definition	definition	NOUN
ejpam-7124	91	20	5	5	NUM
ejpam-7124	91	21	(	(	PUNCT
ejpam-7124	91	22	[	[	X
ejpam-7124	91	23	16	16	NUM
ejpam-7124	91	24	]	]	PUNCT
ejpam-7124	91	25	)	)	PUNCT
ejpam-7124	91	26	.	.	PUNCT
ejpam-7124	92	1	let	let	VERB
ejpam-7124	92	2	φ	φ	PROPN
ejpam-7124	92	3	denote	denote	VERB
ejpam-7124	92	4	the	the	DET
ejpam-7124	92	5	set	set	NOUN
ejpam-7124	92	6	of	of	ADP
ejpam-7124	92	7	all	all	DET
ejpam-7124	92	8	functions	function	NOUN
ejpam-7124	92	9	,	,	PUNCT
ejpam-7124	92	10	φ	φ	X
ejpam-7124	92	11	:	:	PUNCT
ejpam-7124	93	1	[	[	X
ejpam-7124	93	2	0,∞	0,∞	NUM
ejpam-7124	93	3	)	)	PUNCT
ejpam-7124	93	4	→	→	PUNCT
ejpam-7124	94	1	[	[	X
ejpam-7124	94	2	0,∞	0,∞	NUM
ejpam-7124	94	3	)	)	PUNCT
ejpam-7124	94	4	such	such	ADJ
ejpam-7124	94	5	that	that	SCONJ
ejpam-7124	94	6	[	[	X
ejpam-7124	94	7	label=(v),itemsep=-.16em	label=(v),itemsep=-.16em	NOUN
ejpam-7124	94	8	,	,	PUNCT
ejpam-7124	94	9	topsep=2pt	topsep=2pt	PROPN
ejpam-7124	94	10	]	]	PUNCT
ejpam-7124	94	11	(	(	PUNCT
ejpam-7124	94	12	i	i	NOUN
ejpam-7124	94	13	)	)	PUNCT
ejpam-7124	94	14	φ	φ	PROPN
ejpam-7124	94	15	is	be	AUX
ejpam-7124	94	16	non	non	ADJ
ejpam-7124	94	17	-	-	ADJ
ejpam-7124	94	18	decreasing	decrease	VERB
ejpam-7124	94	19	.	.	PUNCT
ejpam-7124	95	1	(	(	PUNCT
ejpam-7124	95	2	ii	ii	NOUN
ejpam-7124	95	3	)	)	PUNCT
ejpam-7124	95	4	for	for	ADP
ejpam-7124	95	5	all	all	PRON
ejpam-7124	95	6	q	q	PROPN
ejpam-7124	95	7	>	>	X
ejpam-7124	95	8	0	0	NUM
ejpam-7124	95	9	where	where	SCONJ
ejpam-7124	95	10	φj	φj	PROPN
ejpam-7124	95	11	is	be	AUX
ejpam-7124	95	12	the	the	DET
ejpam-7124	95	13	jth	jth	PROPN
ejpam-7124	95	14	iterate	iterate	NOUN
ejpam-7124	95	15	of	of	ADP
ejpam-7124	95	16	φ	φ	PROPN
ejpam-7124	95	17	.	.	PUNCT
ejpam-7124	96	1	now	now	ADV
ejpam-7124	96	2	,	,	PUNCT
ejpam-7124	96	3	we	we	PRON
ejpam-7124	96	4	recall	recall	VERB
ejpam-7124	96	5	the	the	DET
ejpam-7124	96	6	following	follow	VERB
ejpam-7124	96	7	lemma	lemma	PROPN
ejpam-7124	96	8	.	.	PUNCT
ejpam-7124	97	1	lemma	lemma	PROPN
ejpam-7124	97	2	1	1	NUM
ejpam-7124	97	3	(	(	PUNCT
ejpam-7124	97	4	[	[	X
ejpam-7124	97	5	5	5	NUM
ejpam-7124	97	6	]	]	PUNCT
ejpam-7124	97	7	)	)	PUNCT
ejpam-7124	97	8	.	.	PUNCT
ejpam-7124	98	1	if	if	SCONJ
ejpam-7124	98	2	φ	φ	PROPN
ejpam-7124	98	3	∈	∈	PROPN
ejpam-7124	98	4	φ	φ	X
ejpam-7124	98	5	then	then	ADV
ejpam-7124	98	6	φ(q	φ(q	NUM
ejpam-7124	98	7	)	)	PUNCT
ejpam-7124	98	8	<	<	X
ejpam-7124	98	9	q	q	X
ejpam-7124	98	10	for	for	ADP
ejpam-7124	98	11	all	all	DET
ejpam-7124	98	12	q	q	PROPN
ejpam-7124	98	13	∈	∈	PROPN
ejpam-7124	98	14	(	(	PUNCT
ejpam-7124	98	15	0,∞	0,∞	NOUN
ejpam-7124	98	16	)	)	PUNCT
ejpam-7124	98	17	.	.	PUNCT
ejpam-7124	99	1	next	next	ADV
ejpam-7124	99	2	,	,	PUNCT
ejpam-7124	99	3	mlaiki	mlaiki	PROPN
ejpam-7124	99	4	et	et	PROPN
ejpam-7124	99	5	al	al	PROPN
ejpam-7124	99	6	.	.	PUNCT
ejpam-7124	100	1	[	[	X
ejpam-7124	100	2	?	?	X
ejpam-7124	100	3	]	]	PUNCT
ejpam-7124	100	4	introduced	introduce	VERB
ejpam-7124	100	5	the	the	DET
ejpam-7124	100	6	following	following	ADJ
ejpam-7124	100	7	notion	notion	NOUN
ejpam-7124	100	8	.	.	PUNCT
ejpam-7124	101	1	definition	definition	NOUN
ejpam-7124	101	2	6	6	NUM
ejpam-7124	101	3	(	(	PUNCT
ejpam-7124	101	4	[	[	X
ejpam-7124	101	5	15	15	NUM
ejpam-7124	101	6	]	]	NUM
ejpam-7124	101	7	)	)	PUNCT
ejpam-7124	101	8	.	.	PUNCT
ejpam-7124	102	1	let	let	VERB
ejpam-7124	102	2	ω	ω	NUM
ejpam-7124	102	3	be	be	AUX
ejpam-7124	102	4	a	a	DET
ejpam-7124	102	5	non	non	ADJ
ejpam-7124	102	6	-	-	ADJ
ejpam-7124	102	7	empty	empty	ADJ
ejpam-7124	102	8	set	set	NOUN
ejpam-7124	102	9	and	and	CCONJ
ejpam-7124	102	10	µ	µ	NOUN
ejpam-7124	102	11	:	:	PUNCT
ejpam-7124	102	12	ω	ω	NUM
ejpam-7124	102	13	×	×	PROPN
ejpam-7124	102	14	ω	ω	X
ejpam-7124	102	15	→	→	SYM
ejpam-7124	102	16	[	[	X
ejpam-7124	102	17	1,∞	1,∞	NUM
ejpam-7124	102	18	)	)	PUNCT
ejpam-7124	102	19	be	be	AUX
ejpam-7124	102	20	a	a	DET
ejpam-7124	102	21	mapping	mapping	NOUN
ejpam-7124	102	22	.	.	PUNCT
ejpam-7124	103	1	a	a	DET
ejpam-7124	103	2	function	function	NOUN
ejpam-7124	103	3	φ	φ	NOUN
ejpam-7124	103	4	:	:	PUNCT
ejpam-7124	104	1	[	[	X
ejpam-7124	104	2	0,∞	0,∞	NUM
ejpam-7124	104	3	)	)	PUNCT
ejpam-7124	104	4	→	→	PUNCT
ejpam-7124	105	1	[	[	X
ejpam-7124	105	2	0,∞	0,∞	NOUN
ejpam-7124	105	3	)	)	PUNCT
ejpam-7124	105	4	is	be	AUX
ejpam-7124	105	5	said	say	VERB
ejpam-7124	105	6	to	to	PART
ejpam-7124	105	7	be	be	AUX
ejpam-7124	105	8	controlled	control	VERB
ejpam-7124	105	9	comparison	comparison	NOUN
ejpam-7124	105	10	function	function	NOUN
ejpam-7124	105	11	if	if	SCONJ
ejpam-7124	105	12	φ	φ	PROPN
ejpam-7124	105	13	satisfies	satisfy	VERB
ejpam-7124	105	14	the	the	DET
ejpam-7124	105	15	following	follow	VERB
ejpam-7124	105	16	conditions	condition	NOUN
ejpam-7124	105	17	[	[	X
ejpam-7124	105	18	label=(vi),itemsep=-.16em	label=(vi),itemsep=-.16em	NOUN
ejpam-7124	105	19	,	,	PUNCT
ejpam-7124	105	20	topsep=2pt	topsep=2pt	PROPN
ejpam-7124	105	21	]	]	PUNCT
ejpam-7124	105	22	(	(	PUNCT
ejpam-7124	105	23	i	i	NOUN
ejpam-7124	105	24	)	)	PUNCT
ejpam-7124	105	25	φ	φ	PROPN
ejpam-7124	105	26	is	be	AUX
ejpam-7124	105	27	non	non	ADJ
ejpam-7124	105	28	-	-	ADJ
ejpam-7124	105	29	decreasing	decrease	VERB
ejpam-7124	105	30	.	.	PUNCT
ejpam-7124	106	1	(	(	PUNCT
ejpam-7124	106	2	ii	ii	NOUN
ejpam-7124	106	3	)	)	PUNCT
ejpam-7124	106	4	∞∑	∞∑	NUM
ejpam-7124	106	5	j=1	j=1	NOUN
ejpam-7124	106	6	φj(q	φj(q	ADV
ejpam-7124	106	7	)	)	PUNCT
ejpam-7124	107	1	j∏	j∏	PROPN
ejpam-7124	107	2	m=1	m=1	PROPN
ejpam-7124	107	3	µ(um	µ(um	PROPN
ejpam-7124	107	4	,	,	PUNCT
ejpam-7124	107	5	ui)µ(uj	ui)µ(uj	PROPN
ejpam-7124	107	6	,	,	PUNCT
ejpam-7124	107	7	uj+1	uj+1	NUM
ejpam-7124	107	8	)	)	PUNCT
ejpam-7124	107	9	<	<	X
ejpam-7124	107	10	∞	∞	PROPN
ejpam-7124	107	11	and	and	CCONJ
ejpam-7124	107	12	lim	lim	PROPN
ejpam-7124	107	13	j→0	j→0	X
ejpam-7124	107	14	φj(q)µ(uj	φj(q)µ(uj	PROPN
ejpam-7124	107	15	,	,	PUNCT
ejpam-7124	107	16	uj+1	uj+1	NUM
ejpam-7124	107	17	)	)	PUNCT
ejpam-7124	107	18	<	<	X
ejpam-7124	107	19	∞	∞	PROPN
ejpam-7124	107	20	for	for	ADP
ejpam-7124	107	21	any	any	DET
ejpam-7124	107	22	sequence	sequence	NOUN
ejpam-7124	107	23	{	{	PUNCT
ejpam-7124	107	24	uj	uj	PROPN
ejpam-7124	107	25	}	}	PUNCT
ejpam-7124	107	26	in	in	ADP
ejpam-7124	107	27	ω	ω	PROPN
ejpam-7124	107	28	for	for	ADP
ejpam-7124	107	29	all	all	DET
ejpam-7124	107	30	ε	ε	PROPN
ejpam-7124	107	31	>	>	X
ejpam-7124	107	32	0	0	PUNCT
ejpam-7124	107	33	and	and	CCONJ
ejpam-7124	107	34	non	non	ADJ
ejpam-7124	107	35	-	-	ADJ
ejpam-7124	107	36	negative	negative	ADJ
ejpam-7124	107	37	integer	integer	NOUN
ejpam-7124	107	38	i	i	PRON
ejpam-7124	107	39	,	,	PUNCT
ejpam-7124	107	40	φi	φi	ADV
ejpam-7124	107	41	is	be	AUX
ejpam-7124	107	42	the	the	DET
ejpam-7124	107	43	ith	ith	PROPN
ejpam-7124	107	44	iterate	iterate	NOUN
ejpam-7124	107	45	of	of	ADP
ejpam-7124	107	46	φ	φ	PROPN
ejpam-7124	107	47	.	.	PUNCT
ejpam-7124	108	1	the	the	DET
ejpam-7124	108	2	set	set	NOUN
ejpam-7124	108	3	of	of	ADP
ejpam-7124	108	4	all	all	DET
ejpam-7124	108	5	controlled	control	VERB
ejpam-7124	108	6	comparison	comparison	NOUN
ejpam-7124	108	7	functions	function	NOUN
ejpam-7124	108	8	is	be	AUX
ejpam-7124	108	9	denoted	denote	VERB
ejpam-7124	108	10	by	by	ADP
ejpam-7124	108	11	φ	φ	NUM
ejpam-7124	108	12	which	which	PRON
ejpam-7124	108	13	is	be	AUX
ejpam-7124	108	14	an	an	DET
ejpam-7124	108	15	extension	extension	NOUN
ejpam-7124	108	16	of	of	ADP
ejpam-7124	108	17	b	b	NOUN
ejpam-7124	108	18	-	-	PUNCT
ejpam-7124	108	19	comparison	comparison	NOUN
ejpam-7124	108	20	function	function	NOUN
ejpam-7124	108	21	of	of	ADP
ejpam-7124	108	22	berinde	berinde	NOUN
ejpam-7124	108	23	.	.	PUNCT
ejpam-7124	109	1	note	note	VERB
ejpam-7124	109	2	that	that	SCONJ
ejpam-7124	109	3	if	if	SCONJ
ejpam-7124	109	4	φ	φ	PROPN
ejpam-7124	109	5	∈	∈	PROPN
ejpam-7124	109	6	φ	φ	NOUN
ejpam-7124	109	7	then	then	ADV
ejpam-7124	109	8	we	we	PRON
ejpam-7124	109	9	have	have	VERB
ejpam-7124	109	10	∑∞	∑∞	NOUN
ejpam-7124	109	11	j=1	j=1	PROPN
ejpam-7124	109	12	φ	φ	PROPN
ejpam-7124	109	13	j	j	X
ejpam-7124	109	14	<	<	X
ejpam-7124	109	15	∞	∞	PROPN
ejpam-7124	109	16	,	,	PUNCT
ejpam-7124	109	17	since	since	SCONJ
ejpam-7124	109	18	∞∑	∞∑	NUM
ejpam-7124	109	19	j=1	j=1	NOUN
ejpam-7124	109	20	φj	φj	ADP
ejpam-7124	109	21	j∏	j∏	PROPN
ejpam-7124	109	22	m=1	m=1	PROPN
ejpam-7124	109	23	µ(um	µ(um	PROPN
ejpam-7124	109	24	,	,	PUNCT
ejpam-7124	109	25	ui	ui	PROPN
ejpam-7124	109	26	)	)	PUNCT
ejpam-7124	109	27	≥	≥	NOUN
ejpam-7124	109	28	φj(q	φj(q	NOUN
ejpam-7124	109	29	)	)	PUNCT
ejpam-7124	109	30	∀	∀	X
ejpam-7124	109	31	q	q	X
ejpam-7124	109	32	≥	≥	NOUN
ejpam-7124	109	33	0	0	NUM
ejpam-7124	109	34	.	.	PUNCT
ejpam-7124	110	1	hence	hence	ADV
ejpam-7124	110	2	by	by	ADP
ejpam-7124	110	3	lemma	lemma	PROPN
ejpam-7124	110	4	1	1	NUM
ejpam-7124	110	5	,	,	PUNCT
ejpam-7124	110	6	we	we	PRON
ejpam-7124	110	7	have	have	VERB
ejpam-7124	110	8	φ(q	φ(q	ADV
ejpam-7124	110	9	)	)	PUNCT
ejpam-7124	110	10	<	<	X
ejpam-7124	110	11	q.	q.	PROPN
ejpam-7124	110	12	to	to	PART
ejpam-7124	110	13	show	show	VERB
ejpam-7124	110	14	that	that	SCONJ
ejpam-7124	110	15	the	the	DET
ejpam-7124	110	16	family	family	NOUN
ejpam-7124	110	17	φ	φ	PROPN
ejpam-7124	110	18	is	be	AUX
ejpam-7124	110	19	a	a	DET
ejpam-7124	110	20	non	non	ADJ
ejpam-7124	110	21	-	-	ADJ
ejpam-7124	110	22	empty	empty	ADJ
ejpam-7124	110	23	set	set	NOUN
ejpam-7124	110	24	.	.	PUNCT
ejpam-7124	111	1	we	we	PRON
ejpam-7124	111	2	present	present	VERB
ejpam-7124	111	3	the	the	DET
ejpam-7124	111	4	following	follow	VERB
ejpam-7124	111	5	example	example	NOUN
ejpam-7124	111	6	.	.	PUNCT
ejpam-7124	112	1	example	example	NOUN
ejpam-7124	112	2	3	3	NUM
ejpam-7124	112	3	(	(	PUNCT
ejpam-7124	112	4	[	[	X
ejpam-7124	112	5	15	15	NUM
ejpam-7124	112	6	]	]	NUM
ejpam-7124	112	7	)	)	PUNCT
ejpam-7124	112	8	.	.	PUNCT
ejpam-7124	113	1	consider	consider	VERB
ejpam-7124	113	2	the	the	DET
ejpam-7124	113	3	controlled	controlled	ADJ
ejpam-7124	113	4	b	b	X
ejpam-7124	113	5	-	-	PUNCT
ejpam-7124	113	6	metric	metric	ADJ
ejpam-7124	113	7	space	space	NOUN
ejpam-7124	113	8	(	(	PUNCT
ejpam-7124	113	9	ω	ω	PROPN
ejpam-7124	113	10	,	,	PUNCT
ejpam-7124	113	11	d̄	d̄	PROPN
ejpam-7124	113	12	)	)	PUNCT
ejpam-7124	113	13	which	which	PRON
ejpam-7124	113	14	was	be	AUX
ejpam-7124	113	15	defined	define	VERB
ejpam-7124	113	16	in	in	ADP
ejpam-7124	113	17	example	example	NOUN
ejpam-7124	113	18	2	2	NUM
ejpam-7124	113	19	.	.	PUNCT
ejpam-7124	113	20	define	define	VERB
ejpam-7124	113	21	the	the	DET
ejpam-7124	113	22	mapping	mapping	NOUN
ejpam-7124	113	23	[	[	X
ejpam-7124	113	24	17	17	NUM
ejpam-7124	113	25	]	]	PUNCT
ejpam-7124	113	26	φ	φ	NOUN
ejpam-7124	113	27	:	:	PUNCT
ejpam-7124	114	1	[	[	X
ejpam-7124	114	2	0,∞	0,∞	NUM
ejpam-7124	114	3	)	)	PUNCT
ejpam-7124	114	4	→	→	PUNCT
ejpam-7124	115	1	[	[	X
ejpam-7124	115	2	0,∞	0,∞	NOUN
ejpam-7124	115	3	)	)	PUNCT
ejpam-7124	115	4	by	by	ADP
ejpam-7124	115	5	φ(q	φ(q	NOUN
ejpam-7124	115	6	)	)	PUNCT
ejpam-7124	115	7	=	=	PUNCT
ejpam-7124	116	1	(	(	PUNCT
ejpam-7124	116	2	rq	rq	INTJ
ejpam-7124	116	3	2	2	NUM
ejpam-7124	116	4	)	)	PUNCT
ejpam-7124	116	5	where	where	SCONJ
ejpam-7124	116	6	r	r	NOUN
ejpam-7124	116	7	<	<	X
ejpam-7124	116	8	1	1	NUM
ejpam-7124	116	9	.	.	PUNCT
ejpam-7124	116	10	note	note	VERB
ejpam-7124	116	11	that	that	SCONJ
ejpam-7124	116	12	µ(u	µ(u	NOUN
ejpam-7124	116	13	,	,	PUNCT
ejpam-7124	116	14	e	e	NOUN
ejpam-7124	116	15	)	)	PUNCT
ejpam-7124	116	16	≤	≤	NUM
ejpam-7124	116	17	2	2	NUM
ejpam-7124	116	18	.	.	PUNCT
ejpam-7124	117	1	then	then	ADV
ejpam-7124	117	2	,	,	PUNCT
ejpam-7124	117	3	we	we	PRON
ejpam-7124	117	4	have	have	VERB
ejpam-7124	117	5	φj(q	φj(q	NOUN
ejpam-7124	117	6	)	)	PUNCT
ejpam-7124	118	1	j∏	j∏	PROPN
ejpam-7124	118	2	m=1	m=1	PROPN
ejpam-7124	118	3	µ(um	µ(um	PROPN
ejpam-7124	118	4	,	,	PUNCT
ejpam-7124	118	5	ui)µ(uj	ui)µ(uj	PROPN
ejpam-7124	118	6	,	,	PUNCT
ejpam-7124	118	7	uj+1	uj+1	X
ejpam-7124	118	8	)	)	PUNCT
ejpam-7124	118	9	≤	≤	NOUN
ejpam-7124	118	10	(	(	PUNCT
ejpam-7124	118	11	rjq	rjq	NOUN
ejpam-7124	118	12	2j	2j	NOUN
ejpam-7124	118	13	)	)	PUNCT
ejpam-7124	118	14	·	·	PUNCT
ejpam-7124	119	1	2j+1	2j+1	NUM
ejpam-7124	119	2	=	=	SYM
ejpam-7124	119	3	2rjq	2rjq	NUM
ejpam-7124	119	4	.	.	PUNCT
ejpam-7124	120	1	therefore	therefore	ADV
ejpam-7124	120	2	,	,	PUNCT
ejpam-7124	120	3	∞∑	∞∑	ADJ
ejpam-7124	120	4	j=1	j=1	NOUN
ejpam-7124	120	5	φj(q	φj(q	ADV
ejpam-7124	120	6	)	)	PUNCT
ejpam-7124	121	1	j∏	j∏	PROPN
ejpam-7124	121	2	m=1	m=1	PROPN
ejpam-7124	121	3	µ(um	µ(um	PROPN
ejpam-7124	121	4	,	,	PUNCT
ejpam-7124	121	5	ui)µ(uj	ui)µ(uj	PROPN
ejpam-7124	121	6	,	,	PUNCT
ejpam-7124	121	7	uj+1	uj+1	X
ejpam-7124	121	8	)	)	PUNCT
ejpam-7124	121	9	≤	≤	NOUN
ejpam-7124	122	1	∞∑	∞∑	NUM
ejpam-7124	122	2	j=1	j=1	NOUN
ejpam-7124	122	3	2rjq	2rjq	NUM
ejpam-7124	122	4	<	<	X
ejpam-7124	122	5	∞	∞	NUM
ejpam-7124	122	6	.	.	PUNCT
ejpam-7124	123	1	similarly	similarly	ADV
ejpam-7124	123	2	,	,	PUNCT
ejpam-7124	123	3	it	it	PRON
ejpam-7124	123	4	is	be	AUX
ejpam-7124	123	5	not	not	PART
ejpam-7124	123	6	difficult	difficult	ADJ
ejpam-7124	123	7	to	to	PART
ejpam-7124	123	8	see	see	VERB
ejpam-7124	123	9	that	that	DET
ejpam-7124	123	10	lim	lim	PROPN
ejpam-7124	123	11	j→0	j→0	X
ejpam-7124	123	12	φj(q)µ(uj	φj(q)µ(uj	PROPN
ejpam-7124	123	13	,	,	PUNCT
ejpam-7124	123	14	uj+1	uj+1	NUM
ejpam-7124	123	15	)	)	PUNCT
ejpam-7124	124	1	<	<	X
ejpam-7124	124	2	∞.	∞.	PROPN
ejpam-7124	124	3	m.	m.	NOUN
ejpam-7124	124	4	kumar	kumar	PROPN
ejpam-7124	124	5	et	et	PROPN
ejpam-7124	124	6	al	al	PROPN
ejpam-7124	124	7	.	.	PUNCT
ejpam-7124	124	8	/	/	SYM
ejpam-7124	124	9	eur	eur	PROPN
ejpam-7124	124	10	.	.	PUNCT
ejpam-7124	125	1	j.	j.	PROPN
ejpam-7124	125	2	pure	pure	PROPN
ejpam-7124	125	3	appl	appl	PROPN
ejpam-7124	125	4	.	.	PROPN
ejpam-7124	125	5	math	math	PROPN
ejpam-7124	125	6	,	,	PUNCT
ejpam-7124	125	7	18	18	NUM
ejpam-7124	125	8	(	(	PUNCT
ejpam-7124	125	9	4	4	NUM
ejpam-7124	125	10	)	)	PUNCT
ejpam-7124	125	11	(	(	PUNCT
ejpam-7124	125	12	2025	2025	NUM
ejpam-7124	125	13	)	)	PUNCT
ejpam-7124	125	14	,	,	PUNCT
ejpam-7124	125	15	7124	7124	NUM
ejpam-7124	125	16	6	6	NUM
ejpam-7124	125	17	of	of	ADP
ejpam-7124	125	18	14	14	NUM
ejpam-7124	125	19	3	3	NUM
ejpam-7124	125	20	.	.	PUNCT
ejpam-7124	125	21	main	main	ADJ
ejpam-7124	125	22	result	result	NOUN
ejpam-7124	125	23	in	in	ADP
ejpam-7124	125	24	this	this	DET
ejpam-7124	125	25	section	section	NOUN
ejpam-7124	125	26	,	,	PUNCT
ejpam-7124	125	27	we	we	PRON
ejpam-7124	125	28	shall	shall	AUX
ejpam-7124	125	29	introduce	introduce	VERB
ejpam-7124	125	30	a	a	DET
ejpam-7124	125	31	new	new	ADJ
ejpam-7124	125	32	notion	notion	NOUN
ejpam-7124	125	33	of	of	ADP
ejpam-7124	125	34	(	(	PUNCT
ejpam-7124	125	35	β	β	X
ejpam-7124	125	36	,	,	PUNCT
ejpam-7124	125	37	φ)-expansive	φ)-expansive	PUNCT
ejpam-7124	125	38	mapping	mapping	NOUN
ejpam-7124	125	39	in	in	ADP
ejpam-7124	125	40	controlled	control	VERB
ejpam-7124	125	41	metric	metric	ADJ
ejpam-7124	125	42	space	space	NOUN
ejpam-7124	125	43	and	and	CCONJ
ejpam-7124	125	44	proved	prove	VERB
ejpam-7124	125	45	some	some	DET
ejpam-7124	125	46	fixed	fix	VERB
ejpam-7124	125	47	point	point	NOUN
ejpam-7124	125	48	result	result	NOUN
ejpam-7124	125	49	by	by	ADP
ejpam-7124	125	50	making	make	VERB
ejpam-7124	125	51	use	use	NOUN
ejpam-7124	125	52	of	of	ADP
ejpam-7124	125	53	this	this	DET
ejpam-7124	125	54	notion	notion	NOUN
ejpam-7124	125	55	.	.	PUNCT
ejpam-7124	126	1	definition	definition	NOUN
ejpam-7124	126	2	7	7	NUM
ejpam-7124	126	3	(	(	PUNCT
ejpam-7124	126	4	[	[	X
ejpam-7124	126	5	16	16	NUM
ejpam-7124	126	6	]	]	PUNCT
ejpam-7124	126	7	)	)	PUNCT
ejpam-7124	126	8	.	.	PUNCT
ejpam-7124	127	1	let	let	VERB
ejpam-7124	127	2	φ	φ	PROPN
ejpam-7124	127	3	denote	denote	VERB
ejpam-7124	127	4	all	all	DET
ejpam-7124	127	5	functions	function	NOUN
ejpam-7124	127	6	,	,	PUNCT
ejpam-7124	127	7	φ	φ	X
ejpam-7124	127	8	:	:	PUNCT
ejpam-7124	128	1	[	[	X
ejpam-7124	128	2	0,∞	0,∞	NUM
ejpam-7124	128	3	)	)	PUNCT
ejpam-7124	128	4	→	→	PUNCT
ejpam-7124	129	1	[	[	X
ejpam-7124	129	2	0,∞	0,∞	NOUN
ejpam-7124	129	3	)	)	PUNCT
ejpam-7124	129	4	which	which	PRON
ejpam-7124	129	5	satisfy	satisfy	VERB
ejpam-7124	129	6	the	the	DET
ejpam-7124	129	7	following	follow	VERB
ejpam-7124	129	8	properties	property	NOUN
ejpam-7124	129	9	[	[	X
ejpam-7124	129	10	label=(vii),itemsep=-.16em	label=(vii),itemsep=-.16em	INTJ
ejpam-7124	129	11	,	,	PUNCT
ejpam-7124	129	12	topsep=2pt	topsep=2pt	PROPN
ejpam-7124	129	13	]	]	PUNCT
ejpam-7124	129	14	(	(	PUNCT
ejpam-7124	129	15	i	i	NOUN
ejpam-7124	129	16	)	)	PUNCT
ejpam-7124	129	17	φ	φ	PROPN
ejpam-7124	129	18	is	be	AUX
ejpam-7124	129	19	non	non	ADJ
ejpam-7124	129	20	-	-	ADJ
ejpam-7124	129	21	decreasing	decrease	VERB
ejpam-7124	129	22	.	.	PUNCT
ejpam-7124	130	1	(	(	PUNCT
ejpam-7124	130	2	ii	ii	NOUN
ejpam-7124	130	3	)	)	PUNCT
ejpam-7124	130	4	∞∑	∞∑	NUM
ejpam-7124	130	5	j=1	j=1	NOUN
ejpam-7124	130	6	φj(a	φj(a	NOUN
ejpam-7124	130	7	)	)	PUNCT
ejpam-7124	131	1	<	<	X
ejpam-7124	131	2	∞	∞	NUM
ejpam-7124	131	3	for	for	ADP
ejpam-7124	131	4	each	each	DET
ejpam-7124	131	5	a	a	PRON
ejpam-7124	131	6	>	>	X
ejpam-7124	131	7	0	0	NUM
ejpam-7124	131	8	where	where	SCONJ
ejpam-7124	131	9	φj	φj	PROPN
ejpam-7124	131	10	is	be	AUX
ejpam-7124	131	11	the	the	DET
ejpam-7124	131	12	jth	jth	PROPN
ejpam-7124	131	13	iterate	iterate	NOUN
ejpam-7124	131	14	of	of	ADP
ejpam-7124	131	15	φ	φ	PROPN
ejpam-7124	131	16	.	.	PUNCT
ejpam-7124	132	1	lemma	lemma	PROPN
ejpam-7124	132	2	2	2	NUM
ejpam-7124	132	3	(	(	PUNCT
ejpam-7124	132	4	[	[	X
ejpam-7124	132	5	16	16	NUM
ejpam-7124	132	6	]	]	PUNCT
ejpam-7124	132	7	)	)	PUNCT
ejpam-7124	132	8	.	.	PUNCT
ejpam-7124	133	1	if	if	SCONJ
ejpam-7124	133	2	φ	φ	PROPN
ejpam-7124	133	3	:	:	PUNCT
ejpam-7124	134	1	[	[	X
ejpam-7124	134	2	0,∞	0,∞	NUM
ejpam-7124	134	3	)	)	PUNCT
ejpam-7124	134	4	→	→	PUNCT
ejpam-7124	135	1	[	[	X
ejpam-7124	135	2	0,∞	0,∞	NUM
ejpam-7124	135	3	)	)	PUNCT
ejpam-7124	135	4	is	be	AUX
ejpam-7124	135	5	a	a	DET
ejpam-7124	135	6	non	non	ADJ
ejpam-7124	135	7	-	-	ADJ
ejpam-7124	135	8	decreasing	decrease	VERB
ejpam-7124	135	9	function	function	NOUN
ejpam-7124	135	10	then	then	ADV
ejpam-7124	135	11	for	for	ADP
ejpam-7124	135	12	each	each	DET
ejpam-7124	135	13	a	a	PRON
ejpam-7124	135	14	>	>	X
ejpam-7124	135	15	0	0	PROPN
ejpam-7124	135	16	,	,	PUNCT
ejpam-7124	135	17	lim	lim	PROPN
ejpam-7124	135	18	j→0	j→0	X
ejpam-7124	135	19	φj(a	φj(a	NOUN
ejpam-7124	135	20	)	)	PUNCT
ejpam-7124	136	1	=	=	SYM
ejpam-7124	136	2	0	0	NUM
ejpam-7124	136	3	⇒	⇒	NOUN
ejpam-7124	136	4	φ(a	φ(a	PROPN
ejpam-7124	136	5	)	)	PUNCT
ejpam-7124	136	6	<	<	X
ejpam-7124	136	7	a	a	DET
ejpam-7124	136	8	definition	definition	NOUN
ejpam-7124	136	9	8	8	NUM
ejpam-7124	136	10	(	(	PUNCT
ejpam-7124	136	11	[	[	X
ejpam-7124	136	12	5	5	NUM
ejpam-7124	136	13	]	]	PUNCT
ejpam-7124	136	14	)	)	PUNCT
ejpam-7124	136	15	.	.	PUNCT
ejpam-7124	137	1	let	let	AUX
ejpam-7124	137	2	(	(	PUNCT
ejpam-7124	137	3	ω	ω	NOUN
ejpam-7124	137	4	,	,	PUNCT
ejpam-7124	137	5	d̄	d̄	PROPN
ejpam-7124	137	6	)	)	PUNCT
ejpam-7124	137	7	be	be	VERB
ejpam-7124	137	8	a	a	DET
ejpam-7124	137	9	metric	metric	ADJ
ejpam-7124	137	10	space	space	NOUN
ejpam-7124	137	11	and	and	CCONJ
ejpam-7124	137	12	t	t	PROPN
ejpam-7124	137	13	:	:	PUNCT
ejpam-7124	137	14	ω	ω	PROPN
ejpam-7124	137	15	→	→	SYM
ejpam-7124	137	16	ω	ω	X
ejpam-7124	137	17	be	be	AUX
ejpam-7124	137	18	a	a	DET
ejpam-7124	137	19	given	give	VERB
ejpam-7124	137	20	mapping	mapping	NOUN
ejpam-7124	137	21	.	.	PUNCT
ejpam-7124	138	1	we	we	PRON
ejpam-7124	138	2	say	say	VERB
ejpam-7124	138	3	that	that	SCONJ
ejpam-7124	138	4	t	t	PROPN
ejpam-7124	138	5	is	be	AUX
ejpam-7124	138	6	an	an	DET
ejpam-7124	138	7	(	(	PUNCT
ejpam-7124	138	8	β	β	NOUN
ejpam-7124	138	9	,	,	PUNCT
ejpam-7124	138	10	φ)-expansive	φ)-expansive	PUNCT
ejpam-7124	138	11	mapping	mapping	NOUN
ejpam-7124	138	12	if	if	SCONJ
ejpam-7124	138	13	∃	∃	PROPN
ejpam-7124	138	14	two	two	NUM
ejpam-7124	138	15	functions	function	NOUN
ejpam-7124	138	16	φ	φ	PROPN
ejpam-7124	138	17	∈	∈	PROPN
ejpam-7124	138	18	φ	φ	PROPN
ejpam-7124	138	19	and	and	CCONJ
ejpam-7124	138	20	β	β	X
ejpam-7124	138	21	:	:	PUNCT
ejpam-7124	138	22	ω×ω	ω×ω	PUNCT
ejpam-7124	138	23	→	→	SYM
ejpam-7124	139	1	[	[	X
ejpam-7124	139	2	0,∞	0,∞	NUM
ejpam-7124	139	3	)	)	PUNCT
ejpam-7124	139	4	such	such	ADJ
ejpam-7124	139	5	that	that	SCONJ
ejpam-7124	139	6	φ(d̄(tu	φ(d̄(tu	NOUN
ejpam-7124	139	7	,	,	PUNCT
ejpam-7124	139	8	te	te	PROPN
ejpam-7124	139	9	)	)	PUNCT
ejpam-7124	139	10	)	)	PUNCT
ejpam-7124	139	11	≥	≥	NOUN
ejpam-7124	139	12	β(u	β(u	PROPN
ejpam-7124	139	13	,	,	PUNCT
ejpam-7124	139	14	e)d̄(u	e)d̄(u	PROPN
ejpam-7124	139	15	,	,	PUNCT
ejpam-7124	139	16	e	e	NOUN
ejpam-7124	139	17	)	)	PUNCT
ejpam-7124	139	18	,	,	PUNCT
ejpam-7124	139	19	∀	∀	X
ejpam-7124	139	20	u	u	NOUN
ejpam-7124	139	21	,	,	PUNCT
ejpam-7124	139	22	e	e	PROPN
ejpam-7124	139	23	∈	∈	PROPN
ejpam-7124	139	24	ω	ω	X
ejpam-7124	139	25	(	(	PUNCT
ejpam-7124	139	26	8)	8)	NUM
ejpam-7124	139	27	definition	definition	NOUN
ejpam-7124	139	28	9	9	NUM
ejpam-7124	139	29	(	(	PUNCT
ejpam-7124	139	30	[	[	X
ejpam-7124	139	31	5	5	NUM
ejpam-7124	139	32	]	]	PUNCT
ejpam-7124	139	33	)	)	PUNCT
ejpam-7124	139	34	.	.	PUNCT
ejpam-7124	140	1	let	let	VERB
ejpam-7124	140	2	t	t	NOUN
ejpam-7124	140	3	:	:	PUNCT
ejpam-7124	140	4	ω	ω	PROPN
ejpam-7124	140	5	→	→	SYM
ejpam-7124	140	6	ω	ω	PROPN
ejpam-7124	140	7	and	and	CCONJ
ejpam-7124	140	8	β	β	X
ejpam-7124	140	9	:	:	PUNCT
ejpam-7124	140	10	ω	ω	NUM
ejpam-7124	140	11	×	×	PROPN
ejpam-7124	140	12	ω	ω	X
ejpam-7124	140	13	→	→	PUNCT
ejpam-7124	141	1	[	[	X
ejpam-7124	141	2	0,∞	0,∞	NOUN
ejpam-7124	141	3	)	)	PUNCT
ejpam-7124	141	4	.	.	PUNCT
ejpam-7124	142	1	we	we	PRON
ejpam-7124	142	2	say	say	VERB
ejpam-7124	142	3	t	t	PROPN
ejpam-7124	142	4	is	be	AUX
ejpam-7124	142	5	said	say	VERB
ejpam-7124	142	6	to	to	PART
ejpam-7124	142	7	be	be	AUX
ejpam-7124	142	8	β	β	X
ejpam-7124	142	9	admissible	admissible	ADJ
ejpam-7124	142	10	if	if	SCONJ
ejpam-7124	142	11	∀	∀	X
ejpam-7124	142	12	u	u	NOUN
ejpam-7124	142	13	,	,	PUNCT
ejpam-7124	142	14	e	e	PROPN
ejpam-7124	142	15	∈	∈	PROPN
ejpam-7124	142	16	ω	ω	PROPN
ejpam-7124	142	17	with	with	ADP
ejpam-7124	142	18	β(u	β(u	PROPN
ejpam-7124	142	19	,	,	PUNCT
ejpam-7124	142	20	e	e	NOUN
ejpam-7124	142	21	)	)	PUNCT
ejpam-7124	142	22	≥	≥	NOUN
ejpam-7124	142	23	1	1	NUM
ejpam-7124	142	24	then	then	ADV
ejpam-7124	142	25	β(tu	β(tu	NUM
ejpam-7124	142	26	,	,	PUNCT
ejpam-7124	142	27	te	te	PROPN
ejpam-7124	142	28	)	)	PUNCT
ejpam-7124	142	29	≥	≥	NOUN
ejpam-7124	142	30	1	1	NUM
ejpam-7124	142	31	.	.	PUNCT
ejpam-7124	143	1	now	now	ADV
ejpam-7124	143	2	we	we	PRON
ejpam-7124	143	3	prove	prove	VERB
ejpam-7124	143	4	our	our	PRON
ejpam-7124	143	5	first	first	ADJ
ejpam-7124	143	6	result	result	NOUN
ejpam-7124	143	7	.	.	PUNCT
ejpam-7124	144	1	theorem	theorem	NOUN
ejpam-7124	144	2	1	1	X
ejpam-7124	144	3	.	.	PUNCT
ejpam-7124	145	1	let	let	AUX
ejpam-7124	145	2	(	(	PUNCT
ejpam-7124	145	3	ω	ω	NOUN
ejpam-7124	145	4	,	,	PUNCT
ejpam-7124	145	5	d̄	d̄	PROPN
ejpam-7124	145	6	)	)	PUNCT
ejpam-7124	145	7	be	be	VERB
ejpam-7124	145	8	a	a	DET
ejpam-7124	145	9	complete	complete	ADJ
ejpam-7124	145	10	controlled	control	VERB
ejpam-7124	145	11	metric	metric	ADJ
ejpam-7124	145	12	type	type	NOUN
ejpam-7124	145	13	space	space	NOUN
ejpam-7124	145	14	and	and	CCONJ
ejpam-7124	145	15	t	t	PROPN
ejpam-7124	145	16	:	:	PUNCT
ejpam-7124	145	17	ω	ω	PROPN
ejpam-7124	145	18	→	→	SYM
ejpam-7124	145	19	ω	ω	X
ejpam-7124	145	20	be	be	AUX
ejpam-7124	145	21	a	a	DET
ejpam-7124	145	22	bijective	bijective	ADJ
ejpam-7124	145	23	and	and	CCONJ
ejpam-7124	145	24	(	(	PUNCT
ejpam-7124	145	25	β	β	X
ejpam-7124	145	26	,	,	PUNCT
ejpam-7124	145	27	φ)-expansive	φ)-expansive	PUNCT
ejpam-7124	145	28	mapping	mapping	NOUN
ejpam-7124	145	29	satisfying	satisfy	VERB
ejpam-7124	145	30	the	the	DET
ejpam-7124	145	31	following	follow	VERB
ejpam-7124	145	32	conditions	condition	NOUN
ejpam-7124	145	33	:	:	PUNCT
ejpam-7124	146	1	[	[	X
ejpam-7124	146	2	label=(i),itemsep=-.16em	label=(i),itemsep=-.16em	NOUN
ejpam-7124	146	3	,	,	PUNCT
ejpam-7124	146	4	topsep=2pt	topsep=2pt	PROPN
ejpam-7124	146	5	]	]	PUNCT
ejpam-7124	146	6	(	(	PUNCT
ejpam-7124	146	7	i	i	NOUN
ejpam-7124	146	8	)	)	PUNCT
ejpam-7124	146	9	t−1	t−1	PROPN
ejpam-7124	146	10	is	be	AUX
ejpam-7124	146	11	β	β	NOUN
ejpam-7124	146	12	admissible	admissible	ADJ
ejpam-7124	146	13	,	,	PUNCT
ejpam-7124	146	14	(	(	PUNCT
ejpam-7124	146	15	ii	ii	NOUN
ejpam-7124	146	16	)	)	PUNCT
ejpam-7124	146	17	∃	∃	PROPN
ejpam-7124	146	18	u0	u0	PROPN
ejpam-7124	146	19	∈	∈	PROPN
ejpam-7124	146	20	ω	ω	NOUN
ejpam-7124	146	21	such	such	ADJ
ejpam-7124	146	22	that	that	SCONJ
ejpam-7124	146	23	β(u0	β(u0	NOUN
ejpam-7124	146	24	,	,	PUNCT
ejpam-7124	146	25	t	t	PROPN
ejpam-7124	146	26	−1u0	−1u0	NUM
ejpam-7124	146	27	)	)	PUNCT
ejpam-7124	146	28	≥	≥	NOUN
ejpam-7124	146	29	1	1	NUM
ejpam-7124	146	30	,	,	PUNCT
ejpam-7124	146	31	(	(	PUNCT
ejpam-7124	146	32	iii	iii	X
ejpam-7124	146	33	)	)	PUNCT
ejpam-7124	146	34	t	t	PROPN
ejpam-7124	146	35	is	be	AUX
ejpam-7124	146	36	continuous	continuous	ADJ
ejpam-7124	146	37	.	.	PUNCT
ejpam-7124	147	1	then	then	ADV
ejpam-7124	147	2	t	t	PROPN
ejpam-7124	147	3	has	have	VERB
ejpam-7124	147	4	a	a	DET
ejpam-7124	147	5	fixed	fix	VERB
ejpam-7124	147	6	point	point	NOUN
ejpam-7124	147	7	,	,	PUNCT
ejpam-7124	147	8	i.e.	i.e.	X
ejpam-7124	147	9	∃	∃	PROPN
ejpam-7124	147	10	u	u	PROPN
ejpam-7124	147	11	∈	∈	PROPN
ejpam-7124	147	12	ω	ω	NUM
ejpam-7124	147	13	such	such	ADJ
ejpam-7124	147	14	that	that	SCONJ
ejpam-7124	147	15	tu	tu	PROPN
ejpam-7124	147	16	=	=	PUNCT
ejpam-7124	147	17	u.	u.	PROPN
ejpam-7124	147	18	moreover	moreover	ADV
ejpam-7124	147	19	,	,	PUNCT
ejpam-7124	147	20	if	if	SCONJ
ejpam-7124	147	21	for	for	ADP
ejpam-7124	147	22	any	any	DET
ejpam-7124	147	23	two	two	NUM
ejpam-7124	147	24	fixed	fix	VERB
ejpam-7124	147	25	points	point	NOUN
ejpam-7124	147	26	of	of	ADP
ejpam-7124	147	27	t	t	PROPN
ejpam-7124	147	28	in	in	ADP
ejpam-7124	147	29	ω	ω	PROPN
ejpam-7124	147	30	say	say	VERB
ejpam-7124	147	31	a	a	DET
ejpam-7124	147	32	,	,	PUNCT
ejpam-7124	147	33	b	b	NOUN
ejpam-7124	147	34	,	,	PUNCT
ejpam-7124	147	35	we	we	PRON
ejpam-7124	147	36	have	have	VERB
ejpam-7124	147	37	β(a	β(a	PROPN
ejpam-7124	147	38	,	,	PUNCT
ejpam-7124	147	39	b	b	NOUN
ejpam-7124	147	40	)	)	PUNCT
ejpam-7124	147	41	≥	≥	NOUN
ejpam-7124	148	1	1	1	NUM
ejpam-7124	148	2	then	then	ADV
ejpam-7124	148	3	t	t	PROPN
ejpam-7124	148	4	has	have	VERB
ejpam-7124	148	5	a	a	DET
ejpam-7124	148	6	unique	unique	ADJ
ejpam-7124	148	7	fixed	fix	VERB
ejpam-7124	148	8	point	point	NOUN
ejpam-7124	148	9	in	in	ADP
ejpam-7124	148	10	ω	ω	PROPN
ejpam-7124	148	11	.	.	PUNCT
ejpam-7124	149	1	m.	m.	PROPN
ejpam-7124	149	2	kumar	kumar	PROPN
ejpam-7124	149	3	et	et	PROPN
ejpam-7124	149	4	al	al	PROPN
ejpam-7124	149	5	.	.	PUNCT
ejpam-7124	149	6	/	/	SYM
ejpam-7124	149	7	eur	eur	PROPN
ejpam-7124	149	8	.	.	PUNCT
ejpam-7124	150	1	j.	j.	PROPN
ejpam-7124	150	2	pure	pure	PROPN
ejpam-7124	150	3	appl	appl	PROPN
ejpam-7124	150	4	.	.	PROPN
ejpam-7124	150	5	math	math	PROPN
ejpam-7124	150	6	,	,	PUNCT
ejpam-7124	150	7	18	18	NUM
ejpam-7124	150	8	(	(	PUNCT
ejpam-7124	150	9	4	4	NUM
ejpam-7124	150	10	)	)	PUNCT
ejpam-7124	150	11	(	(	PUNCT
ejpam-7124	150	12	2025	2025	NUM
ejpam-7124	150	13	)	)	PUNCT
ejpam-7124	150	14	,	,	PUNCT
ejpam-7124	150	15	7124	7124	NUM
ejpam-7124	150	16	7	7	NUM
ejpam-7124	150	17	of	of	ADP
ejpam-7124	150	18	14	14	NUM
ejpam-7124	150	19	proof	proof	NOUN
ejpam-7124	150	20	.	.	PUNCT
ejpam-7124	151	1	let	let	VERB
ejpam-7124	151	2	u0	u0	ADJ
ejpam-7124	151	3	be	be	AUX
ejpam-7124	151	4	the	the	DET
ejpam-7124	151	5	point	point	NOUN
ejpam-7124	151	6	in	in	ADP
ejpam-7124	151	7	ω	ω	NUM
ejpam-7124	151	8	satisfying	satisfy	VERB
ejpam-7124	151	9	condition	condition	NOUN
ejpam-7124	151	10	(	(	PUNCT
ejpam-7124	151	11	ii	ii	NOUN
ejpam-7124	151	12	)	)	PUNCT
ejpam-7124	151	13	in	in	ADP
ejpam-7124	151	14	our	our	PRON
ejpam-7124	151	15	theorem	theorem	NOUN
ejpam-7124	151	16	.	.	PROPN
ejpam-7124	151	17	define	define	VERB
ejpam-7124	151	18	a	a	DET
ejpam-7124	151	19	sequence	sequence	NOUN
ejpam-7124	151	20	{	{	PUNCT
ejpam-7124	151	21	uj	uj	PROPN
ejpam-7124	151	22	}	}	PUNCT
ejpam-7124	151	23	in	in	ADP
ejpam-7124	151	24	ω	ω	NUM
ejpam-7124	151	25	by	by	ADP
ejpam-7124	151	26	tuj	tuj	PROPN
ejpam-7124	151	27	=	=	PROPN
ejpam-7124	151	28	uj−1	uj−1	PROPN
ejpam-7124	151	29	∀	∀	NOUN
ejpam-7124	151	30	j	j	PROPN
ejpam-7124	151	31	∈	∈	PROPN
ejpam-7124	151	32	n	n	ADV
ejpam-7124	151	33	.	.	PUNCT
ejpam-7124	152	1	first	first	ADV
ejpam-7124	152	2	of	of	ADP
ejpam-7124	152	3	all	all	PRON
ejpam-7124	152	4	,	,	PUNCT
ejpam-7124	152	5	note	note	VERB
ejpam-7124	152	6	if	if	SCONJ
ejpam-7124	152	7	∃	∃	PROPN
ejpam-7124	152	8	j	j	PROPN
ejpam-7124	152	9	such	such	ADJ
ejpam-7124	152	10	that	that	DET
ejpam-7124	152	11	uj	uj	PROPN
ejpam-7124	152	12	=	=	SYM
ejpam-7124	152	13	uj+1	uj+1	PROPN
ejpam-7124	152	14	,	,	PUNCT
ejpam-7124	152	15	then	then	ADV
ejpam-7124	152	16	we	we	PRON
ejpam-7124	152	17	are	be	AUX
ejpam-7124	152	18	done	do	VERB
ejpam-7124	152	19	and	and	CCONJ
ejpam-7124	152	20	uj	uj	PROPN
ejpam-7124	152	21	is	be	AUX
ejpam-7124	152	22	the	the	DET
ejpam-7124	152	23	fixed	fixed	ADJ
ejpam-7124	152	24	point	point	NOUN
ejpam-7124	152	25	of	of	ADP
ejpam-7124	152	26	t	t	PROPN
ejpam-7124	152	27	.	.	PUNCT
ejpam-7124	153	1	so	so	ADV
ejpam-7124	153	2	we	we	PRON
ejpam-7124	153	3	may	may	AUX
ejpam-7124	153	4	assume	assume	VERB
ejpam-7124	153	5	that	that	SCONJ
ejpam-7124	153	6	uj	uj	PROPN
ejpam-7124	153	7	̸=	̸=	PROPN
ejpam-7124	153	8	uj+1	uj+1	NUM
ejpam-7124	153	9	∀	∀	NOUN
ejpam-7124	153	10	j	j	X
ejpam-7124	153	11	≥	≥	PROPN
ejpam-7124	153	12	0	0	NUM
ejpam-7124	153	13	.	.	PUNCT
ejpam-7124	154	1	also	also	ADV
ejpam-7124	154	2	from	from	ADP
ejpam-7124	154	3	the	the	DET
ejpam-7124	154	4	hypothesis	hypothesis	NOUN
ejpam-7124	154	5	of	of	ADP
ejpam-7124	154	6	our	our	PRON
ejpam-7124	154	7	theorem	theorem	NOUN
ejpam-7124	154	8	,	,	PUNCT
ejpam-7124	154	9	we	we	PRON
ejpam-7124	154	10	know	know	VERB
ejpam-7124	154	11	that	that	SCONJ
ejpam-7124	154	12	β(u0	β(u0	NOUN
ejpam-7124	154	13	,	,	PUNCT
ejpam-7124	154	14	t	t	PROPN
ejpam-7124	154	15	−1u0	−1u0	NUM
ejpam-7124	154	16	)	)	PUNCT
ejpam-7124	155	1	=	=	SYM
ejpam-7124	155	2	β(u0	β(u0	NOUN
ejpam-7124	155	3	,	,	PUNCT
ejpam-7124	155	4	u1	u1	PROPN
ejpam-7124	155	5	)	)	PUNCT
ejpam-7124	155	6	≥	≥	NOUN
ejpam-7124	155	7	1	1	NUM
ejpam-7124	155	8	,	,	PUNCT
ejpam-7124	155	9	using	use	VERB
ejpam-7124	155	10	the	the	DET
ejpam-7124	155	11	result	result	NOUN
ejpam-7124	155	12	t−1	t−1	PROPN
ejpam-7124	155	13	is	be	AUX
ejpam-7124	155	14	β	β	NOUN
ejpam-7124	155	15	admissible	admissible	ADJ
ejpam-7124	155	16	.	.	PUNCT
ejpam-7124	156	1	we	we	PRON
ejpam-7124	156	2	can	can	AUX
ejpam-7124	156	3	easily	easily	ADV
ejpam-7124	156	4	deduce	deduce	VERB
ejpam-7124	156	5	that	that	SCONJ
ejpam-7124	156	6	for	for	ADP
ejpam-7124	156	7	all	all	DET
ejpam-7124	156	8	j	j	PROPN
ejpam-7124	156	9	≥	≥	X
ejpam-7124	156	10	0	0	NUM
ejpam-7124	156	11	β(uj	β(uj	X
ejpam-7124	156	12	,	,	PUNCT
ejpam-7124	156	13	uj+1	uj+1	NUM
ejpam-7124	156	14	)	)	PUNCT
ejpam-7124	156	15	≥	≥	NOUN
ejpam-7124	156	16	1	1	NUM
ejpam-7124	156	17	.	.	PUNCT
ejpam-7124	157	1	now	now	ADV
ejpam-7124	157	2	using	use	VERB
ejpam-7124	157	3	the	the	DET
ejpam-7124	157	4	fact	fact	NOUN
ejpam-7124	157	5	that	that	SCONJ
ejpam-7124	157	6	t	t	PROPN
ejpam-7124	157	7	is	be	AUX
ejpam-7124	157	8	(	(	PUNCT
ejpam-7124	157	9	β	β	X
ejpam-7124	157	10	,	,	PUNCT
ejpam-7124	157	11	φ)-expansive	φ)-expansive	PUNCT
ejpam-7124	157	12	mapping	mapping	NOUN
ejpam-7124	157	13	.	.	PUNCT
ejpam-7124	158	1	we	we	PRON
ejpam-7124	158	2	deduce	deduce	VERB
ejpam-7124	158	3	that	that	DET
ejpam-7124	158	4	d̄(uj	d̄(uj	NOUN
ejpam-7124	158	5	,	,	PUNCT
ejpam-7124	158	6	uj+1	uj+1	X
ejpam-7124	158	7	)	)	PUNCT
ejpam-7124	158	8	≤	≤	NOUN
ejpam-7124	158	9	β(uj	β(uj	X
ejpam-7124	158	10	,	,	PUNCT
ejpam-7124	158	11	uj+1)d̄(uj	uj+1)d̄(uj	PROPN
ejpam-7124	158	12	,	,	PUNCT
ejpam-7124	158	13	uj+1	uj+1	X
ejpam-7124	158	14	)	)	PUNCT
ejpam-7124	158	15	≤	≤	NOUN
ejpam-7124	158	16	φ(d̄(tuj	φ(d̄(tuj	X
ejpam-7124	158	17	,	,	PUNCT
ejpam-7124	158	18	tuj+1	tuj+1	NUM
ejpam-7124	158	19	)	)	PUNCT
ejpam-7124	158	20	)	)	PUNCT
ejpam-7124	159	1	=	=	SYM
ejpam-7124	159	2	φ(d̄(uj−1	φ(d̄(uj−1	PROPN
ejpam-7124	159	3	,	,	PUNCT
ejpam-7124	159	4	tuj	tuj	PROPN
ejpam-7124	159	5	)	)	PUNCT
ejpam-7124	159	6	)	)	PUNCT
ejpam-7124	159	7	.	.	PUNCT
ejpam-7124	160	1	by	by	ADP
ejpam-7124	160	2	repeating	repeat	VERB
ejpam-7124	160	3	the	the	DET
ejpam-7124	160	4	same	same	ADJ
ejpam-7124	160	5	process	process	NOUN
ejpam-7124	160	6	,	,	PUNCT
ejpam-7124	160	7	we	we	PRON
ejpam-7124	160	8	get	get	VERB
ejpam-7124	160	9	d̄(uj	d̄(uj	NOUN
ejpam-7124	160	10	,	,	PUNCT
ejpam-7124	160	11	uj+1	uj+1	X
ejpam-7124	160	12	)	)	PUNCT
ejpam-7124	160	13	≤	≤	NUM
ejpam-7124	160	14	φjt(d̄(u0	φjt(d̄(u0	NOUN
ejpam-7124	160	15	,	,	PUNCT
ejpam-7124	160	16	u1	u1	NOUN
ejpam-7124	160	17	)	)	PUNCT
ejpam-7124	160	18	)	)	PUNCT
ejpam-7124	160	19	,	,	PUNCT
ejpam-7124	160	20	∀	∀	PUNCT
ejpam-7124	160	21	j	j	PROPN
ejpam-7124	160	22	∈	∈	PROPN
ejpam-7124	160	23	n	n	CCONJ
ejpam-7124	160	24	(	(	PUNCT
ejpam-7124	160	25	9	9	NUM
ejpam-7124	160	26	)	)	PUNCT
ejpam-7124	160	27	hence	hence	ADV
ejpam-7124	160	28	for	for	ADP
ejpam-7124	160	29	all	all	DET
ejpam-7124	160	30	i	i	PRON
ejpam-7124	160	31	,	,	PUNCT
ejpam-7124	160	32	j	j	PROPN
ejpam-7124	160	33	∈	∈	PROPN
ejpam-7124	160	34	n	n	ADV
ejpam-7124	160	35	with	with	ADP
ejpam-7124	160	36	i	i	PRON
ejpam-7124	160	37	>	>	X
ejpam-7124	160	38	j	j	PROPN
ejpam-7124	160	39	,	,	PUNCT
ejpam-7124	160	40	we	we	PRON
ejpam-7124	160	41	have	have	VERB
ejpam-7124	160	42	d̄(uj	d̄(uj	NOUN
ejpam-7124	160	43	,	,	PUNCT
ejpam-7124	160	44	ui	ui	PROPN
ejpam-7124	160	45	)	)	PUNCT
ejpam-7124	160	46	≤	≤	NOUN
ejpam-7124	161	1	µ(uj	µ(uj	PROPN
ejpam-7124	161	2	,	,	PUNCT
ejpam-7124	161	3	uj+1)d̄(uj	uj+1)d̄(uj	PROPN
ejpam-7124	161	4	,	,	PUNCT
ejpam-7124	161	5	uj+1	uj+1	X
ejpam-7124	161	6	)	)	PUNCT
ejpam-7124	161	7	+	+	CCONJ
ejpam-7124	161	8	µ(uj+1	µ(uj+1	PROPN
ejpam-7124	161	9	,	,	PUNCT
ejpam-7124	161	10	ui)d̄(uj+1	ui)d̄(uj+1	PROPN
ejpam-7124	161	11	,	,	PUNCT
ejpam-7124	161	12	ui	ui	NOUN
ejpam-7124	161	13	)	)	PUNCT
ejpam-7124	161	14	≤	≤	NOUN
ejpam-7124	161	15	µ(uj	µ(uj	PROPN
ejpam-7124	161	16	,	,	PUNCT
ejpam-7124	161	17	uj+1)d̄(uj	uj+1)d̄(uj	PROPN
ejpam-7124	161	18	,	,	PUNCT
ejpam-7124	161	19	uj+1	uj+1	X
ejpam-7124	161	20	)	)	PUNCT
ejpam-7124	161	21	+	+	CCONJ
ejpam-7124	161	22	µ(uj+1	µ(uj+1	PROPN
ejpam-7124	161	23	,	,	PUNCT
ejpam-7124	161	24	ui)µ(uj+1	ui)µ(uj+1	PROPN
ejpam-7124	161	25	,	,	PUNCT
ejpam-7124	161	26	uj+2)d̄(uj+1	uj+2)d̄(uj+1	NOUN
ejpam-7124	161	27	,	,	PUNCT
ejpam-7124	161	28	uj+2	uj+2	ADJ
ejpam-7124	161	29	)	)	PUNCT
ejpam-7124	161	30	+	+	CCONJ
ejpam-7124	161	31	µ(uj+1	µ(uj+1	ADJ
ejpam-7124	161	32	,	,	PUNCT
ejpam-7124	161	33	ui)µ(uj+2	ui)µ(uj+2	ADJ
ejpam-7124	161	34	,	,	PUNCT
ejpam-7124	161	35	ui)d̄(uj+2	ui)d̄(uj+2	PROPN
ejpam-7124	161	36	,	,	PUNCT
ejpam-7124	161	37	ui	ui	NOUN
ejpam-7124	161	38	)	)	PUNCT
ejpam-7124	161	39	≤	≤	NOUN
ejpam-7124	161	40	µ(uj	µ(uj	PROPN
ejpam-7124	161	41	,	,	PUNCT
ejpam-7124	161	42	uj+1)d̄(uj	uj+1)d̄(uj	PROPN
ejpam-7124	161	43	,	,	PUNCT
ejpam-7124	161	44	uj+1	uj+1	X
ejpam-7124	161	45	)	)	PUNCT
ejpam-7124	161	46	+	+	CCONJ
ejpam-7124	161	47	µ(uj+1	µ(uj+1	PROPN
ejpam-7124	161	48	,	,	PUNCT
ejpam-7124	161	49	ui)µ(uj+1	ui)µ(uj+1	PROPN
ejpam-7124	161	50	,	,	PUNCT
ejpam-7124	161	51	uj+2)d̄(uj+1	uj+2)d̄(uj+1	NOUN
ejpam-7124	161	52	,	,	PUNCT
ejpam-7124	161	53	uj+2	uj+2	ADJ
ejpam-7124	161	54	)	)	PUNCT
ejpam-7124	161	55	+	+	CCONJ
ejpam-7124	161	56	µ(uj+1	µ(uj+1	ADJ
ejpam-7124	161	57	,	,	PUNCT
ejpam-7124	161	58	ui)µ(uj+2	ui)µ(uj+2	ADJ
ejpam-7124	161	59	,	,	PUNCT
ejpam-7124	161	60	ui)µ(uj+2	ui)µ(uj+2	ADJ
ejpam-7124	161	61	,	,	PUNCT
ejpam-7124	161	62	uj+3)d̄(uj+2	uj+3)d̄(uj+2	NOUN
ejpam-7124	161	63	,	,	PUNCT
ejpam-7124	161	64	uj+3	uj+3	NOUN
ejpam-7124	161	65	)	)	PUNCT
ejpam-7124	161	66	+	+	CCONJ
ejpam-7124	161	67	µ(uj+1	µ(uj+1	ADJ
ejpam-7124	161	68	,	,	PUNCT
ejpam-7124	161	69	ui)µ(uj+2	ui)µ(uj+2	ADJ
ejpam-7124	161	70	,	,	PUNCT
ejpam-7124	161	71	ui)µ(uj+3	ui)µ(uj+3	PROPN
ejpam-7124	161	72	,	,	PUNCT
ejpam-7124	161	73	ui)d̄(uj+3	ui)d̄(uj+3	SYM
ejpam-7124	161	74	,	,	PUNCT
ejpam-7124	161	75	ui	ui	NOUN
ejpam-7124	161	76	)	)	PUNCT
ejpam-7124	161	77	≤	≤	PUNCT
ejpam-7124	161	78	µ(uj	µ(uj	PROPN
ejpam-7124	161	79	,	,	PUNCT
ejpam-7124	161	80	uj+1)φ	uj+1)φ	PRON
ejpam-7124	161	81	j	j	PROPN
ejpam-7124	161	82	d̄(u0	d̄(u0	ADV
ejpam-7124	161	83	,	,	PUNCT
ejpam-7124	161	84	u1	u1	PROPN
ejpam-7124	161	85	)	)	PUNCT
ejpam-7124	161	86	+	+	CCONJ
ejpam-7124	161	87	µ(uj+1	µ(uj+1	ADJ
ejpam-7124	161	88	,	,	PUNCT
ejpam-7124	161	89	uj+2)µ(uj+1	uj+2)µ(uj+1	NOUN
ejpam-7124	161	90	,	,	PUNCT
ejpam-7124	161	91	ui)φ	ui)φ	PROPN
ejpam-7124	161	92	j+1d̄(u0	j+1d̄(u0	PROPN
ejpam-7124	161	93	,	,	PUNCT
ejpam-7124	161	94	u1	u1	NOUN
ejpam-7124	161	95	)	)	PUNCT
ejpam-7124	161	96	+	+	CCONJ
ejpam-7124	161	97	µ(uj+1	µ(uj+1	ADJ
ejpam-7124	161	98	,	,	PUNCT
ejpam-7124	161	99	ui)µ(uj+2	ui)µ(uj+2	ADJ
ejpam-7124	161	100	,	,	PUNCT
ejpam-7124	161	101	ui)µ(uj+2	ui)µ(uj+2	ADJ
ejpam-7124	161	102	,	,	PUNCT
ejpam-7124	161	103	uj+3)φ	uj+3)φ	PROPN
ejpam-7124	161	104	j+2d̄(u0	j+2d̄(u0	PROPN
ejpam-7124	161	105	,	,	PUNCT
ejpam-7124	161	106	u1	u1	PROPN
ejpam-7124	161	107	)	)	PUNCT
ejpam-7124	161	108	+	+	CCONJ
ejpam-7124	161	109	.	.	PUNCT
ejpam-7124	161	110	.	.	PUNCT
ejpam-7124	161	111	.	.	PUNCT
ejpam-7124	162	1	+	+	CCONJ
ejpam-7124	162	2	µ(uj+1	µ(uj+1	X
ejpam-7124	162	3	,	,	PUNCT
ejpam-7124	162	4	ui)µ(uj+2	ui)µ(uj+2	ADJ
ejpam-7124	162	5	,	,	PUNCT
ejpam-7124	162	6	ui)µ(uj+3	ui)µ(uj+3	PROPN
ejpam-7124	162	7	,	,	PUNCT
ejpam-7124	162	8	ui	ui	PROPN
ejpam-7124	162	9	)	)	PUNCT
ejpam-7124	162	10	.	.	PUNCT
ejpam-7124	162	11	.	.	PUNCT
ejpam-7124	162	12	.	.	PUNCT
ejpam-7124	163	1	µ(ui−2	µ(ui−2	PROPN
ejpam-7124	163	2	,	,	PUNCT
ejpam-7124	163	3	ui−1)φ	ui−1)φ	PRON
ejpam-7124	163	4	i−1d̄(u0	i−1d̄(u0	ADJ
ejpam-7124	163	5	,	,	PUNCT
ejpam-7124	163	6	u1	u1	NOUN
ejpam-7124	163	7	)	)	PUNCT
ejpam-7124	164	1	=	=	PUNCT
ejpam-7124	165	1	µ(uj	µ(uj	PROPN
ejpam-7124	165	2	,	,	PUNCT
ejpam-7124	165	3	uj+1)φ	uj+1)φ	PRON
ejpam-7124	165	4	j	j	PROPN
ejpam-7124	165	5	d̄(u0	d̄(u0	ADV
ejpam-7124	165	6	,	,	PUNCT
ejpam-7124	165	7	u1	u1	PROPN
ejpam-7124	165	8	)	)	PUNCT
ejpam-7124	165	9	+	+	CCONJ
ejpam-7124	165	10	i−2∑	i−2∑	PROPN
ejpam-7124	165	11	n	n	CCONJ
ejpam-7124	165	12	=	=	NOUN
ejpam-7124	165	13	j+1	j+1	X
ejpam-7124	165	14	φnd̄(u0	φnd̄(u0	ADJ
ejpam-7124	165	15	,	,	PUNCT
ejpam-7124	165	16	u1	u1	NOUN
ejpam-7124	165	17	)	)	PUNCT
ejpam-7124	165	18	n∏	n∏	PROPN
ejpam-7124	165	19	m	m	NOUN
ejpam-7124	165	20	=	=	NOUN
ejpam-7124	165	21	j+1	j+1	X
ejpam-7124	165	22	µ(um	µ(um	PROPN
ejpam-7124	165	23	,	,	PUNCT
ejpam-7124	165	24	ui)µ(un	ui)µ(un	PROPN
ejpam-7124	165	25	,	,	PUNCT
ejpam-7124	165	26	un+1	un+1	NOUN
ejpam-7124	165	27	)	)	PUNCT
ejpam-7124	165	28	=	=	SYM
ejpam-7124	166	1	µ(uj	µ(uj	PROPN
ejpam-7124	166	2	,	,	PUNCT
ejpam-7124	166	3	uj+1)φ	uj+1)φ	PRON
ejpam-7124	166	4	j	j	PROPN
ejpam-7124	166	5	d̄(u0	d̄(u0	ADV
ejpam-7124	166	6	,	,	PUNCT
ejpam-7124	166	7	u1	u1	PROPN
ejpam-7124	166	8	)	)	PUNCT
ejpam-7124	167	1	+	+	CCONJ
ejpam-7124	167	2	i−2∑	i−2∑	PROPN
ejpam-7124	167	3	n=1	n=1	PROPN
ejpam-7124	167	4	φnd̄(u0	φnd̄(u0	ADJ
ejpam-7124	167	5	,	,	PUNCT
ejpam-7124	167	6	u1	u1	NOUN
ejpam-7124	167	7	)	)	PUNCT
ejpam-7124	167	8	n∏	n∏	PROPN
ejpam-7124	168	1	m=1	m=1	PROPN
ejpam-7124	168	2	µ(um	µ(um	PROPN
ejpam-7124	168	3	,	,	PUNCT
ejpam-7124	168	4	ui)µ(un	ui)µ(un	PROPN
ejpam-7124	168	5	,	,	PUNCT
ejpam-7124	168	6	un+1	un+1	NOUN
ejpam-7124	168	7	)	)	PUNCT
ejpam-7124	168	8	−	−	PROPN
ejpam-7124	169	1	j∑	j∑	PROPN
ejpam-7124	169	2	n=1	n=1	PUNCT
ejpam-7124	169	3	φnd̄(u0	φnd̄(u0	ADJ
ejpam-7124	169	4	,	,	PUNCT
ejpam-7124	169	5	u1	u1	NOUN
ejpam-7124	169	6	)	)	PUNCT
ejpam-7124	169	7	n∏	n∏	PROPN
ejpam-7124	169	8	m=1	m=1	PROPN
ejpam-7124	169	9	µ(um	µ(um	PROPN
ejpam-7124	169	10	,	,	PUNCT
ejpam-7124	169	11	ui)µ(un	ui)µ(un	PROPN
ejpam-7124	169	12	,	,	PUNCT
ejpam-7124	169	13	un+1	un+1	NOUN
ejpam-7124	169	14	)	)	PUNCT
ejpam-7124	169	15	=	=	SYM
ejpam-7124	170	1	si−2	si−2	ADJ
ejpam-7124	170	2	−	−	NOUN
ejpam-7124	170	3	sj	sj	INTJ
ejpam-7124	170	4	(	(	PUNCT
ejpam-7124	170	5	10	10	NUM
ejpam-7124	170	6	)	)	PUNCT
ejpam-7124	170	7	where	where	SCONJ
ejpam-7124	170	8	sj	sj	NOUN
ejpam-7124	170	9	=	=	NOUN
ejpam-7124	170	10	j∑	j∑	ADJ
ejpam-7124	170	11	n=1	n=1	PROPN
ejpam-7124	170	12	φnd̄(u0	φnd̄(u0	ADJ
ejpam-7124	170	13	,	,	PUNCT
ejpam-7124	170	14	u1	u1	NOUN
ejpam-7124	170	15	)	)	PUNCT
ejpam-7124	170	16	n∏	n∏	PROPN
ejpam-7124	170	17	m=1	m=1	PROPN
ejpam-7124	170	18	µ(um	µ(um	PROPN
ejpam-7124	170	19	,	,	PUNCT
ejpam-7124	170	20	ui)µ(un	ui)µ(un	PROPN
ejpam-7124	170	21	,	,	PUNCT
ejpam-7124	170	22	un+1	un+1	NOUN
ejpam-7124	170	23	)	)	PUNCT
ejpam-7124	170	24	.	.	PUNCT
ejpam-7124	171	1	(	(	PUNCT
ejpam-7124	171	2	11	11	NUM
ejpam-7124	171	3	)	)	PUNCT
ejpam-7124	171	4	m.	m.	NOUN
ejpam-7124	171	5	kumar	kumar	PROPN
ejpam-7124	171	6	et	et	PROPN
ejpam-7124	171	7	al	al	PROPN
ejpam-7124	171	8	.	.	PUNCT
ejpam-7124	171	9	/	/	SYM
ejpam-7124	171	10	eur	eur	PROPN
ejpam-7124	171	11	.	.	PUNCT
ejpam-7124	172	1	j.	j.	PROPN
ejpam-7124	172	2	pure	pure	PROPN
ejpam-7124	172	3	appl	appl	PROPN
ejpam-7124	172	4	.	.	PROPN
ejpam-7124	172	5	math	math	PROPN
ejpam-7124	172	6	,	,	PUNCT
ejpam-7124	172	7	18	18	NUM
ejpam-7124	172	8	(	(	PUNCT
ejpam-7124	172	9	4	4	NUM
ejpam-7124	172	10	)	)	PUNCT
ejpam-7124	172	11	(	(	PUNCT
ejpam-7124	172	12	2025	2025	NUM
ejpam-7124	172	13	)	)	PUNCT
ejpam-7124	172	14	,	,	PUNCT
ejpam-7124	172	15	7124	7124	NUM
ejpam-7124	172	16	8	8	NUM
ejpam-7124	172	17	of	of	ADP
ejpam-7124	172	18	14	14	NUM
ejpam-7124	172	19	since	since	SCONJ
ejpam-7124	172	20	φ	φ	PROPN
ejpam-7124	172	21	∈	∈	PROPN
ejpam-7124	172	22	φ	φ	NOUN
ejpam-7124	172	23	,	,	PUNCT
ejpam-7124	172	24	we	we	PRON
ejpam-7124	172	25	deduce	deduce	VERB
ejpam-7124	172	26	that	that	SCONJ
ejpam-7124	172	27	lim	lim	PROPN
ejpam-7124	172	28	i	i	PRON
ejpam-7124	172	29	,	,	PUNCT
ejpam-7124	172	30	j→∞	j→∞	NOUN
ejpam-7124	173	1	[	[	X
ejpam-7124	173	2	si−2	si−2	PROPN
ejpam-7124	173	3	,	,	PUNCT
ejpam-7124	173	4	sj	sj	INTJ
ejpam-7124	173	5	]	]	PUNCT
ejpam-7124	173	6	=	=	SYM
ejpam-7124	173	7	0	0	PUNCT
ejpam-7124	173	8	and	and	CCONJ
ejpam-7124	173	9	lim	lim	PROPN
ejpam-7124	174	1	j→∞	j→∞	NOUN
ejpam-7124	175	1	[	[	X
ejpam-7124	175	2	µj	µj	X
ejpam-7124	175	3	,	,	PUNCT
ejpam-7124	175	4	µj+1]φ	µj+1]φ	PROPN
ejpam-7124	175	5	j	j	PROPN
ejpam-7124	175	6	d̄(u0	d̄(u0	PROPN
ejpam-7124	175	7	,	,	PUNCT
ejpam-7124	175	8	u1	u1	PROPN
ejpam-7124	175	9	)	)	PUNCT
ejpam-7124	175	10	<	<	X
ejpam-7124	175	11	∞	∞	NUM
ejpam-7124	175	12	.	.	PUNCT
ejpam-7124	176	1	(	(	PUNCT
ejpam-7124	176	2	12	12	NUM
ejpam-7124	176	3	)	)	PUNCT
ejpam-7124	176	4	thus	thus	ADV
ejpam-7124	176	5	the	the	DET
ejpam-7124	176	6	sequence	sequence	NOUN
ejpam-7124	176	7	{	{	PUNCT
ejpam-7124	176	8	uj}j≥0	uj}j≥0	NOUN
ejpam-7124	176	9	is	be	AUX
ejpam-7124	176	10	a	a	DET
ejpam-7124	176	11	cauchy	cauchy	ADJ
ejpam-7124	176	12	sequence	sequence	NOUN
ejpam-7124	176	13	in	in	ADP
ejpam-7124	176	14	ω	ω	PROPN
ejpam-7124	176	15	and	and	CCONJ
ejpam-7124	176	16	(	(	PUNCT
ejpam-7124	176	17	ω	ω	PROPN
ejpam-7124	176	18	,	,	PUNCT
ejpam-7124	176	19	d̄	d̄	NOUN
ejpam-7124	176	20	)	)	PUNCT
ejpam-7124	176	21	being	be	AUX
ejpam-7124	176	22	the	the	DET
ejpam-7124	176	23	complete	complete	ADJ
ejpam-7124	176	24	controlled	control	VERB
ejpam-7124	176	25	metric	metric	ADJ
ejpam-7124	176	26	type	type	NOUN
ejpam-7124	176	27	space	space	NOUN
ejpam-7124	176	28	.	.	PUNCT
ejpam-7124	177	1	the	the	DET
ejpam-7124	177	2	{	{	PUNCT
ejpam-7124	177	3	uj	uj	PROPN
ejpam-7124	177	4	}	}	PUNCT
ejpam-7124	177	5	converges	converge	NOUN
ejpam-7124	177	6	to	to	ADP
ejpam-7124	177	7	some	some	DET
ejpam-7124	177	8	u	u	PROPN
ejpam-7124	177	9	∈	∈	PROPN
ejpam-7124	177	10	ω	ω	PROPN
ejpam-7124	177	11	.	.	PUNCT
ejpam-7124	177	12	also	also	ADV
ejpam-7124	177	13	note	note	VERB
ejpam-7124	177	14	that	that	SCONJ
ejpam-7124	177	15	d̄(u	d̄(u	NOUN
ejpam-7124	177	16	,	,	PUNCT
ejpam-7124	177	17	t−1	t−1	PROPN
ejpam-7124	177	18	)	)	PUNCT
ejpam-7124	177	19	≤	≤	NOUN
ejpam-7124	177	20	µ(u	µ(u	NOUN
ejpam-7124	177	21	,	,	PUNCT
ejpam-7124	177	22	uj+1)d̄(u	uj+1)d̄(u	NUM
ejpam-7124	177	23	,	,	PUNCT
ejpam-7124	177	24	uj+1	uj+1	NOUN
ejpam-7124	177	25	)	)	PUNCT
ejpam-7124	178	1	+	+	CCONJ
ejpam-7124	178	2	µ(uj+1,t	µ(uj+1,t	NOUN
ejpam-7124	178	3	−1u)d̄(uj+1	−1u)d̄(uj+1	PROPN
ejpam-7124	178	4	,	,	PUNCT
ejpam-7124	178	5	t	t	PROPN
ejpam-7124	178	6	−1u	−1u	PROPN
ejpam-7124	178	7	)	)	PUNCT
ejpam-7124	178	8	=	=	SYM
ejpam-7124	178	9	µ(u	µ(u	NOUN
ejpam-7124	178	10	,	,	PUNCT
ejpam-7124	178	11	u)d̄(u	u)d̄(u	PROPN
ejpam-7124	178	12	,	,	PUNCT
ejpam-7124	178	13	u	u	NOUN
ejpam-7124	178	14	)	)	PUNCT
ejpam-7124	178	15	+	+	CCONJ
ejpam-7124	178	16	µ(u	µ(u	NOUN
ejpam-7124	178	17	,	,	PUNCT
ejpam-7124	178	18	t−1u)d̄(u	t−1u)d̄(u	PROPN
ejpam-7124	178	19	,	,	PUNCT
ejpam-7124	178	20	t−1u	t−1u	NOUN
ejpam-7124	178	21	)	)	PUNCT
ejpam-7124	178	22	.	.	PUNCT
ejpam-7124	179	1	(	(	PUNCT
ejpam-7124	179	2	13	13	X
ejpam-7124	179	3	)	)	PUNCT
ejpam-7124	179	4	proceeding	proceed	VERB
ejpam-7124	179	5	the	the	DET
ejpam-7124	179	6	limit	limit	NOUN
ejpam-7124	179	7	as	as	ADP
ejpam-7124	179	8	j	j	PROPN
ejpam-7124	179	9	→	→	SYM
ejpam-7124	179	10	∞	∞	PROPN
ejpam-7124	179	11	,	,	PUNCT
ejpam-7124	179	12	we	we	PRON
ejpam-7124	179	13	get	get	VERB
ejpam-7124	179	14	d̄(u	d̄(u	NOUN
ejpam-7124	179	15	,	,	PUNCT
ejpam-7124	179	16	uj+1	uj+1	NUM
ejpam-7124	179	17	)	)	PUNCT
ejpam-7124	179	18	=	=	SYM
ejpam-7124	180	1	0	0	X
ejpam-7124	180	2	.	.	PUNCT
ejpam-7124	181	1	and	and	CCONJ
ejpam-7124	181	2	being	be	AUX
ejpam-7124	181	3	t	t	PROPN
ejpam-7124	181	4	is	be	AUX
ejpam-7124	181	5	continuous	continuous	ADJ
ejpam-7124	181	6	,	,	PUNCT
ejpam-7124	181	7	we	we	PRON
ejpam-7124	181	8	conclude	conclude	VERB
ejpam-7124	181	9	that	that	SCONJ
ejpam-7124	181	10	d̄(u	d̄(u	NOUN
ejpam-7124	181	11	,	,	PUNCT
ejpam-7124	181	12	t−1u	t−1u	NOUN
ejpam-7124	181	13	)	)	PUNCT
ejpam-7124	181	14	=	=	SYM
ejpam-7124	181	15	0	0	NUM
ejpam-7124	181	16	,	,	PUNCT
ejpam-7124	181	17	i.e.	i.e.	X
ejpam-7124	181	18	,	,	PUNCT
ejpam-7124	182	1	t−1u	t−1u	PRON
ejpam-7124	182	2	=	=	SYM
ejpam-7124	182	3	u	u	PROPN
ejpam-7124	182	4	⇒	⇒	X
ejpam-7124	182	5	t	t	PROPN
ejpam-7124	182	6	(	(	PUNCT
ejpam-7124	182	7	t−1u	t−1u	NOUN
ejpam-7124	182	8	)	)	PUNCT
ejpam-7124	182	9	=	=	SYM
ejpam-7124	182	10	tu	tu	PROPN
ejpam-7124	182	11	⇒	⇒	NOUN
ejpam-7124	182	12	(	(	PUNCT
ejpam-7124	182	13	tt−1)u	tt−1)u	X
ejpam-7124	182	14	=	=	PUNCT
ejpam-7124	182	15	tu	tu	PROPN
ejpam-7124	182	16	⇒	⇒	NOUN
ejpam-7124	182	17	u	u	PROPN
ejpam-7124	182	18	=	=	PROPN
ejpam-7124	182	19	tu	tu	PROPN
ejpam-7124	182	20	.	.	PUNCT
ejpam-7124	183	1	thus	thus	ADV
ejpam-7124	183	2	u	u	NOUN
ejpam-7124	183	3	is	be	AUX
ejpam-7124	183	4	a	a	DET
ejpam-7124	183	5	fixed	fix	VERB
ejpam-7124	183	6	point	point	NOUN
ejpam-7124	183	7	t	t	PROPN
ejpam-7124	183	8	.	.	PUNCT
ejpam-7124	184	1	now	now	ADV
ejpam-7124	184	2	suppose	suppose	VERB
ejpam-7124	184	3	that	that	SCONJ
ejpam-7124	184	4	t	t	PROPN
ejpam-7124	184	5	has	have	VERB
ejpam-7124	184	6	two	two	NUM
ejpam-7124	184	7	fixed	fix	VERB
ejpam-7124	184	8	points	point	NOUN
ejpam-7124	184	9	a	a	PRON
ejpam-7124	184	10	,	,	PUNCT
ejpam-7124	184	11	b	b	X
ejpam-7124	184	12	such	such	ADJ
ejpam-7124	184	13	that	that	SCONJ
ejpam-7124	184	14	β(a	β(a	PROPN
ejpam-7124	184	15	,	,	PUNCT
ejpam-7124	184	16	b	b	NOUN
ejpam-7124	184	17	)	)	PUNCT
ejpam-7124	184	18	≥	≥	NOUN
ejpam-7124	184	19	1	1	NUM
ejpam-7124	184	20	.	.	PUNCT
ejpam-7124	185	1	now	now	ADV
ejpam-7124	185	2	using	use	VERB
ejpam-7124	185	3	the	the	DET
ejpam-7124	185	4	fact	fact	NOUN
ejpam-7124	185	5	that	that	SCONJ
ejpam-7124	185	6	t	t	PROPN
ejpam-7124	185	7	is	be	AUX
ejpam-7124	185	8	an	an	DET
ejpam-7124	185	9	(	(	PUNCT
ejpam-7124	185	10	β	β	NOUN
ejpam-7124	185	11	,	,	PUNCT
ejpam-7124	185	12	φ)-expansive	φ)-expansive	PUNCT
ejpam-7124	185	13	mapping	mapping	NOUN
ejpam-7124	185	14	and	and	CCONJ
ejpam-7124	185	15	β	β	NOUN
ejpam-7124	185	16	-	-	ADJ
ejpam-7124	185	17	admisible	admisible	ADJ
ejpam-7124	185	18	,	,	PUNCT
ejpam-7124	185	19	we	we	PRON
ejpam-7124	185	20	obtain	obtain	VERB
ejpam-7124	185	21	,	,	PUNCT
ejpam-7124	185	22	d̄(a	d̄(a	NOUN
ejpam-7124	185	23	,	,	PUNCT
ejpam-7124	185	24	b	b	NOUN
ejpam-7124	185	25	)	)	PUNCT
ejpam-7124	185	26	=	=	PUNCT
ejpam-7124	185	27	d̄(t−1a	d̄(t−1a	NOUN
ejpam-7124	185	28	,	,	PUNCT
ejpam-7124	185	29	t−1b	t−1b	NOUN
ejpam-7124	185	30	)	)	PUNCT
ejpam-7124	185	31	≤	≤	ADJ
ejpam-7124	185	32	β(a	β(a	PROPN
ejpam-7124	185	33	,	,	PUNCT
ejpam-7124	185	34	b)d̄(t−1a	b)d̄(t−1a	PROPN
ejpam-7124	185	35	,	,	PUNCT
ejpam-7124	185	36	t−1b	t−1b	NOUN
ejpam-7124	185	37	)	)	PUNCT
ejpam-7124	185	38	≤	≤	NOUN
ejpam-7124	185	39	φd̄(a	φd̄(a	NUM
ejpam-7124	185	40	,	,	PUNCT
ejpam-7124	185	41	b	b	NOUN
ejpam-7124	185	42	)	)	PUNCT
ejpam-7124	185	43	...	...	PUNCT
ejpam-7124	186	1	≤	≤	NUM
ejpam-7124	186	2	φj	φj	ADP
ejpam-7124	186	3	d̄(a	d̄(a	PROPN
ejpam-7124	186	4	,	,	PUNCT
ejpam-7124	186	5	b	b	NOUN
ejpam-7124	186	6	)	)	PUNCT
ejpam-7124	186	7	.	.	PUNCT
ejpam-7124	187	1	(	(	PUNCT
ejpam-7124	187	2	14	14	NUM
ejpam-7124	187	3	)	)	PUNCT
ejpam-7124	187	4	since	since	SCONJ
ejpam-7124	187	5	φ	φ	PROPN
ejpam-7124	187	6	∈	∈	PROPN
ejpam-7124	187	7	φ	φ	PROPN
ejpam-7124	187	8	,	,	PUNCT
ejpam-7124	187	9	taking	take	VERB
ejpam-7124	187	10	the	the	DET
ejpam-7124	187	11	limit	limit	NOUN
ejpam-7124	187	12	in	in	ADP
ejpam-7124	187	13	the	the	DET
ejpam-7124	187	14	above	above	ADJ
ejpam-7124	187	15	inequality	inequality	NOUN
ejpam-7124	187	16	,	,	PUNCT
ejpam-7124	187	17	we	we	PRON
ejpam-7124	187	18	deduce	deduce	VERB
ejpam-7124	187	19	that	that	SCONJ
ejpam-7124	187	20	d̄(a	d̄(a	NOUN
ejpam-7124	187	21	,	,	PUNCT
ejpam-7124	187	22	b	b	NOUN
ejpam-7124	187	23	)	)	PUNCT
ejpam-7124	187	24	=	=	SYM
ejpam-7124	187	25	0	0	NUM
ejpam-7124	187	26	which	which	PRON
ejpam-7124	187	27	implies	imply	VERB
ejpam-7124	187	28	that	that	SCONJ
ejpam-7124	187	29	a	a	DET
ejpam-7124	187	30	=	=	X
ejpam-7124	187	31	b.	b.	PROPN
ejpam-7124	187	32	thus	thus	ADV
ejpam-7124	187	33	t	t	PROPN
ejpam-7124	187	34	has	have	VERB
ejpam-7124	187	35	a	a	DET
ejpam-7124	187	36	unique	unique	ADJ
ejpam-7124	187	37	fixed	fix	VERB
ejpam-7124	187	38	point	point	NOUN
ejpam-7124	187	39	.	.	PUNCT
ejpam-7124	188	1	in	in	ADP
ejpam-7124	188	2	the	the	DET
ejpam-7124	188	3	next	next	ADJ
ejpam-7124	188	4	theorem	theorem	NOUN
ejpam-7124	188	5	,	,	PUNCT
ejpam-7124	188	6	we	we	PRON
ejpam-7124	188	7	shall	shall	AUX
ejpam-7124	188	8	replace	replace	VERB
ejpam-7124	188	9	the	the	DET
ejpam-7124	188	10	continuity	continuity	NOUN
ejpam-7124	188	11	of	of	ADP
ejpam-7124	188	12	t	t	NOUN
ejpam-7124	188	13	by	by	ADP
ejpam-7124	188	14	the	the	DET
ejpam-7124	188	15	following	follow	VERB
ejpam-7124	188	16	weaker	weak	ADJ
ejpam-7124	188	17	condition	condition	NOUN
ejpam-7124	188	18	.	.	PUNCT
ejpam-7124	189	1	if	if	SCONJ
ejpam-7124	189	2	{	{	PUNCT
ejpam-7124	189	3	uj}∞j=1	uj}∞j=1	X
ejpam-7124	189	4	is	be	AUX
ejpam-7124	189	5	a	a	DET
ejpam-7124	189	6	sequence	sequence	NOUN
ejpam-7124	189	7	in	in	ADP
ejpam-7124	189	8	z	z	NOUN
ejpam-7124	189	9	such	such	ADJ
ejpam-7124	189	10	that	that	SCONJ
ejpam-7124	189	11	β(uj	β(uj	NUM
ejpam-7124	189	12	,	,	PUNCT
ejpam-7124	189	13	uj+1	uj+1	NUM
ejpam-7124	189	14	)	)	PUNCT
ejpam-7124	189	15	≥	≥	NOUN
ejpam-7124	189	16	1	1	NUM
ejpam-7124	189	17	for	for	ADP
ejpam-7124	189	18	all	all	DET
ejpam-7124	189	19	j	j	PROPN
ejpam-7124	189	20	and	and	CCONJ
ejpam-7124	189	21	uj	uj	PROPN
ejpam-7124	189	22	→	→	SYM
ejpam-7124	189	23	u	u	PROPN
ejpam-7124	189	24	as	as	ADP
ejpam-7124	189	25	j	j	PROPN
ejpam-7124	189	26	→	→	SYM
ejpam-7124	189	27	∞	∞	PROPN
ejpam-7124	189	28	then	then	ADV
ejpam-7124	189	29	β(uj	β(uj	PRON
ejpam-7124	189	30	,	,	PUNCT
ejpam-7124	189	31	u	u	NOUN
ejpam-7124	189	32	)	)	PUNCT
ejpam-7124	189	33	≥	≥	NOUN
ejpam-7124	189	34	1	1	NUM
ejpam-7124	189	35	∀	∀	PUNCT
ejpam-7124	189	36	j.	j.	PROPN
ejpam-7124	190	1	now	now	ADV
ejpam-7124	190	2	we	we	PRON
ejpam-7124	190	3	present	present	VERB
ejpam-7124	190	4	an	an	DET
ejpam-7124	190	5	example	example	NOUN
ejpam-7124	190	6	to	to	PART
ejpam-7124	190	7	prove	prove	VERB
ejpam-7124	190	8	the	the	DET
ejpam-7124	190	9	validity	validity	NOUN
ejpam-7124	190	10	of	of	ADP
ejpam-7124	190	11	our	our	PRON
ejpam-7124	190	12	result	result	NOUN
ejpam-7124	190	13	.	.	PUNCT
ejpam-7124	191	1	m.	m.	NOUN
ejpam-7124	191	2	kumar	kumar	PROPN
ejpam-7124	191	3	et	et	PROPN
ejpam-7124	191	4	al	al	PROPN
ejpam-7124	191	5	.	.	PUNCT
ejpam-7124	191	6	/	/	SYM
ejpam-7124	191	7	eur	eur	PROPN
ejpam-7124	191	8	.	.	PUNCT
ejpam-7124	192	1	j.	j.	PROPN
ejpam-7124	192	2	pure	pure	PROPN
ejpam-7124	192	3	appl	appl	PROPN
ejpam-7124	192	4	.	.	PROPN
ejpam-7124	192	5	math	math	PROPN
ejpam-7124	192	6	,	,	PUNCT
ejpam-7124	192	7	18	18	NUM
ejpam-7124	192	8	(	(	PUNCT
ejpam-7124	192	9	4	4	NUM
ejpam-7124	192	10	)	)	PUNCT
ejpam-7124	192	11	(	(	PUNCT
ejpam-7124	192	12	2025	2025	NUM
ejpam-7124	192	13	)	)	PUNCT
ejpam-7124	192	14	,	,	PUNCT
ejpam-7124	192	15	7124	7124	NUM
ejpam-7124	192	16	9	9	NUM
ejpam-7124	192	17	of	of	ADP
ejpam-7124	192	18	14	14	NUM
ejpam-7124	192	19	example	example	NOUN
ejpam-7124	192	20	4	4	NUM
ejpam-7124	192	21	.	.	PUNCT
ejpam-7124	193	1	let	let	VERB
ejpam-7124	193	2	(	(	PUNCT
ejpam-7124	193	3	ω	ω	NOUN
ejpam-7124	193	4	,	,	PUNCT
ejpam-7124	193	5	d̄	d̄	PROPN
ejpam-7124	193	6	)	)	PUNCT
ejpam-7124	193	7	be	be	AUX
ejpam-7124	193	8	the	the	DET
ejpam-7124	193	9	controlled	control	VERB
ejpam-7124	193	10	metric	metric	ADJ
ejpam-7124	193	11	type	type	NOUN
ejpam-7124	193	12	space	space	NOUN
ejpam-7124	193	13	on	on	ADP
ejpam-7124	193	14	ω	ω	PROPN
ejpam-7124	193	15	=	=	SYM
ejpam-7124	193	16	{	{	PUNCT
ejpam-7124	193	17	0	0	NUM
ejpam-7124	193	18	,	,	PUNCT
ejpam-7124	193	19	1	1	NUM
ejpam-7124	193	20	,	,	PUNCT
ejpam-7124	193	21	2	2	NUM
ejpam-7124	193	22	}	}	PUNCT
ejpam-7124	193	23	.	.	PUNCT
ejpam-7124	194	1	define	define	VERB
ejpam-7124	194	2	the	the	DET
ejpam-7124	194	3	function	function	NOUN
ejpam-7124	194	4	β	β	NOUN
ejpam-7124	194	5	:	:	PUNCT
ejpam-7124	194	6	ω×	ω×	PROPN
ejpam-7124	194	7	ω	ω	PROPN
ejpam-7124	194	8	→	→	SYM
ejpam-7124	194	9	(	(	PUNCT
ejpam-7124	194	10	−∞,∞	−∞,∞	NOUN
ejpam-7124	194	11	)	)	PUNCT
ejpam-7124	194	12	such	such	ADJ
ejpam-7124	194	13	that	that	SCONJ
ejpam-7124	194	14	β(u	β(u	PROPN
ejpam-7124	194	15	,	,	PUNCT
ejpam-7124	194	16	e	e	NOUN
ejpam-7124	194	17	)	)	PUNCT
ejpam-7124	194	18	=	=	NOUN
ejpam-7124	194	19	{	{	PUNCT
ejpam-7124	194	20	1	1	NUM
ejpam-7124	194	21	,	,	PUNCT
ejpam-7124	194	22	if	if	SCONJ
ejpam-7124	194	23	(	(	PUNCT
ejpam-7124	194	24	u	u	NOUN
ejpam-7124	194	25	,	,	PUNCT
ejpam-7124	194	26	e	e	NOUN
ejpam-7124	194	27	)	)	PUNCT
ejpam-7124	194	28	=	=	SYM
ejpam-7124	194	29	(	(	PUNCT
ejpam-7124	194	30	1	1	NUM
ejpam-7124	194	31	,	,	PUNCT
ejpam-7124	194	32	1	1	NUM
ejpam-7124	194	33	)	)	PUNCT
ejpam-7124	194	34	,	,	PUNCT
ejpam-7124	194	35	1	1	NUM
ejpam-7124	194	36	20	20	NUM
ejpam-7124	194	37	,	,	PUNCT
ejpam-7124	194	38	if	if	SCONJ
ejpam-7124	194	39	(	(	PUNCT
ejpam-7124	194	40	u	u	NOUN
ejpam-7124	194	41	,	,	PUNCT
ejpam-7124	194	42	e	e	NOUN
ejpam-7124	194	43	)	)	PUNCT
ejpam-7124	194	44	̸=	̸=	PROPN
ejpam-7124	194	45	(	(	PUNCT
ejpam-7124	194	46	1	1	NUM
ejpam-7124	194	47	,	,	PUNCT
ejpam-7124	194	48	1	1	NUM
ejpam-7124	194	49	)	)	PUNCT
ejpam-7124	194	50	.	.	PUNCT
ejpam-7124	195	1	(	(	PUNCT
ejpam-7124	195	2	15	15	X
ejpam-7124	195	3	)	)	PUNCT
ejpam-7124	195	4	define	define	VERB
ejpam-7124	195	5	the	the	DET
ejpam-7124	195	6	self	self	NOUN
ejpam-7124	195	7	mapping	mapping	NOUN
ejpam-7124	195	8	t	t	NOUN
ejpam-7124	195	9	on	on	ADP
ejpam-7124	195	10	ω	ω	PROPN
ejpam-7124	195	11	and	and	CCONJ
ejpam-7124	195	12	ω	ω	NUM
ejpam-7124	195	13	=	=	SYM
ejpam-7124	195	14	{	{	PUNCT
ejpam-7124	195	15	0	0	NUM
ejpam-7124	195	16	,	,	PUNCT
ejpam-7124	195	17	1	1	NUM
ejpam-7124	195	18	,	,	PUNCT
ejpam-7124	195	19	2	2	NUM
ejpam-7124	195	20	}	}	PUNCT
ejpam-7124	195	21	by	by	ADP
ejpam-7124	195	22	t	t	PROPN
ejpam-7124	195	23	(	(	PUNCT
ejpam-7124	195	24	0	0	NUM
ejpam-7124	195	25	)	)	PUNCT
ejpam-7124	195	26	=	=	SYM
ejpam-7124	195	27	2	2	NUM
ejpam-7124	195	28	,	,	PUNCT
ejpam-7124	195	29	t	t	PROPN
ejpam-7124	195	30	(	(	PUNCT
ejpam-7124	195	31	1)1	1)1	NUM
ejpam-7124	195	32	,	,	PUNCT
ejpam-7124	195	33	t	t	PROPN
ejpam-7124	195	34	(	(	PUNCT
ejpam-7124	195	35	2	2	NUM
ejpam-7124	195	36	)	)	PUNCT
ejpam-7124	195	37	=	=	SYM
ejpam-7124	195	38	0	0	NUM
ejpam-7124	195	39	and	and	CCONJ
ejpam-7124	195	40	the	the	DET
ejpam-7124	195	41	function	function	NOUN
ejpam-7124	195	42	φ(q	φ(q	NUM
ejpam-7124	195	43	)	)	PUNCT
ejpam-7124	195	44	=	=	SYM
ejpam-7124	195	45	1	1	NUM
ejpam-7124	195	46	8q	8q	NOUN
ejpam-7124	195	47	and	and	CCONJ
ejpam-7124	195	48	d̄	d̄	NOUN
ejpam-7124	195	49	:	:	PUNCT
ejpam-7124	195	50	ω×	ω×	NUM
ejpam-7124	195	51	ω	ω	NUM
ejpam-7124	195	52	→	→	SYM
ejpam-7124	195	53	[	[	X
ejpam-7124	195	54	0,∞	0,∞	NOUN
ejpam-7124	195	55	)	)	PUNCT
ejpam-7124	195	56	defined	define	VERB
ejpam-7124	195	57	by	by	ADP
ejpam-7124	195	58	d̄(0	d̄(0	NUM
ejpam-7124	195	59	,	,	PUNCT
ejpam-7124	195	60	0	0	NUM
ejpam-7124	195	61	)	)	PUNCT
ejpam-7124	195	62	=	=	SYM
ejpam-7124	195	63	d̄(1	d̄(1	NOUN
ejpam-7124	195	64	,	,	PUNCT
ejpam-7124	195	65	1	1	X
ejpam-7124	195	66	)	)	PUNCT
ejpam-7124	195	67	=	=	SYM
ejpam-7124	195	68	d̄(2	d̄(2	ADJ
ejpam-7124	195	69	,	,	PUNCT
ejpam-7124	195	70	2	2	NUM
ejpam-7124	195	71	)	)	PUNCT
ejpam-7124	195	72	=	=	SYM
ejpam-7124	195	73	0	0	NUM
ejpam-7124	195	74	,	,	PUNCT
ejpam-7124	195	75	d̄(0	d̄(0	NUM
ejpam-7124	195	76	,	,	PUNCT
ejpam-7124	195	77	1	1	NUM
ejpam-7124	195	78	)	)	PUNCT
ejpam-7124	195	79	=	=	SYM
ejpam-7124	195	80	d̄(1	d̄(1	NOUN
ejpam-7124	195	81	,	,	PUNCT
ejpam-7124	195	82	0	0	NUM
ejpam-7124	195	83	)	)	PUNCT
ejpam-7124	195	84	,	,	PUNCT
ejpam-7124	195	85	d̄(0	d̄(0	NUM
ejpam-7124	195	86	,	,	PUNCT
ejpam-7124	195	87	2	2	NUM
ejpam-7124	195	88	)	)	PUNCT
ejpam-7124	195	89	=	=	SYM
ejpam-7124	195	90	d̄(2	d̄(2	ADJ
ejpam-7124	195	91	,	,	PUNCT
ejpam-7124	195	92	0	0	NUM
ejpam-7124	195	93	)	)	PUNCT
ejpam-7124	195	94	=	=	SYM
ejpam-7124	195	95	1	1	NUM
ejpam-7124	195	96	2	2	NUM
ejpam-7124	195	97	,	,	PUNCT
ejpam-7124	195	98	d̄(1	d̄(1	NUM
ejpam-7124	195	99	,	,	PUNCT
ejpam-7124	195	100	2	2	X
ejpam-7124	195	101	)	)	PUNCT
ejpam-7124	195	102	=	=	SYM
ejpam-7124	195	103	d̄(2	d̄(2	ADJ
ejpam-7124	195	104	,	,	PUNCT
ejpam-7124	195	105	1	1	X
ejpam-7124	195	106	)	)	PUNCT
ejpam-7124	195	107	=	=	SYM
ejpam-7124	195	108	2	2	NUM
ejpam-7124	195	109	5	5	NUM
ejpam-7124	195	110	.	.	PUNCT
ejpam-7124	196	1	we	we	PRON
ejpam-7124	196	2	want	want	VERB
ejpam-7124	196	3	to	to	PART
ejpam-7124	196	4	verify	verify	VERB
ejpam-7124	196	5	that	that	SCONJ
ejpam-7124	196	6	t	t	PROPN
ejpam-7124	196	7	satisfies	satisfy	VERB
ejpam-7124	196	8	the	the	DET
ejpam-7124	196	9	conditions	condition	NOUN
ejpam-7124	196	10	of	of	ADP
ejpam-7124	196	11	theorem	theorem	NOUN
ejpam-7124	196	12	1	1	NUM
ejpam-7124	196	13	.	.	PUNCT
ejpam-7124	197	1	it	it	PRON
ejpam-7124	197	2	is	be	AUX
ejpam-7124	197	3	clear	clear	ADJ
ejpam-7124	197	4	that	that	SCONJ
ejpam-7124	197	5	t	t	PROPN
ejpam-7124	197	6	is	be	AUX
ejpam-7124	197	7	continuous	continuous	ADJ
ejpam-7124	197	8	for	for	ADP
ejpam-7124	197	9	u0	u0	ADJ
ejpam-7124	197	10	=	=	ADJ
ejpam-7124	197	11	1	1	X
ejpam-7124	197	12	.	.	X
ejpam-7124	198	1	we	we	PRON
ejpam-7124	198	2	have	have	VERB
ejpam-7124	198	3	β(1	β(1	PROPN
ejpam-7124	198	4	,	,	PUNCT
ejpam-7124	198	5	t−1(1	t−1(1	NUM
ejpam-7124	198	6	)	)	PUNCT
ejpam-7124	198	7	)	)	PUNCT
ejpam-7124	199	1	=	=	SYM
ejpam-7124	199	2	β(1	β(1	PROPN
ejpam-7124	199	3	,	,	PUNCT
ejpam-7124	199	4	1	1	NUM
ejpam-7124	199	5	)	)	PUNCT
ejpam-7124	199	6	=	=	SYM
ejpam-7124	199	7	1	1	NUM
ejpam-7124	199	8	≥	≥	NOUN
ejpam-7124	199	9	1	1	NUM
ejpam-7124	199	10	.	.	PUNCT
ejpam-7124	200	1	so	so	ADV
ejpam-7124	200	2	t	t	PROPN
ejpam-7124	200	3	is	be	AUX
ejpam-7124	200	4	β	β	NOUN
ejpam-7124	200	5	-	-	ADJ
ejpam-7124	200	6	admissible	admissible	ADJ
ejpam-7124	200	7	.	.	PUNCT
ejpam-7124	201	1	now	now	ADV
ejpam-7124	201	2	we	we	PRON
ejpam-7124	201	3	verify	verify	VERB
ejpam-7124	201	4	that	that	SCONJ
ejpam-7124	201	5	t	t	PROPN
ejpam-7124	201	6	is	be	AUX
ejpam-7124	201	7	(	(	PUNCT
ejpam-7124	201	8	β	β	X
ejpam-7124	201	9	,	,	PUNCT
ejpam-7124	201	10	φ)-expansive	φ)-expansive	PUNCT
ejpam-7124	201	11	mapping	mapping	NOUN
ejpam-7124	201	12	.	.	PUNCT
ejpam-7124	202	1	note	note	VERB
ejpam-7124	202	2	that	that	SCONJ
ejpam-7124	202	3	β(u	β(u	PROPN
ejpam-7124	202	4	,	,	PUNCT
ejpam-7124	202	5	e	e	NOUN
ejpam-7124	202	6	)	)	PUNCT
ejpam-7124	202	7	=	=	SYM
ejpam-7124	203	1	β(e	β(e	PROPN
ejpam-7124	203	2	,	,	PUNCT
ejpam-7124	203	3	u	u	NOUN
ejpam-7124	203	4	)	)	PUNCT
ejpam-7124	204	1	[	[	X
ejpam-7124	204	2	label=(iv),itemsep=-.16em	label=(iv),itemsep=-.16em	X
ejpam-7124	204	3	,	,	PUNCT
ejpam-7124	204	4	topsep=2pt	topsep=2pt	PROPN
ejpam-7124	204	5	]	]	PUNCT
ejpam-7124	204	6	(	(	PUNCT
ejpam-7124	204	7	i	i	NOUN
ejpam-7124	204	8	)	)	PUNCT
ejpam-7124	204	9	β(u	β(u	PROPN
ejpam-7124	204	10	,	,	PUNCT
ejpam-7124	204	11	u)d̄(tu	u)d̄(tu	PROPN
ejpam-7124	204	12	,	,	PUNCT
ejpam-7124	204	13	tu	tu	PROPN
ejpam-7124	204	14	)	)	PUNCT
ejpam-7124	204	15	=	=	SYM
ejpam-7124	204	16	0	0	X
ejpam-7124	204	17	≤	≤	NOUN
ejpam-7124	204	18	φ(d̄(u	φ(d̄(u	ADJ
ejpam-7124	204	19	,	,	PUNCT
ejpam-7124	204	20	u	u	NOUN
ejpam-7124	204	21	)	)	PUNCT
ejpam-7124	204	22	)	)	PUNCT
ejpam-7124	205	1	=	=	PUNCT
ejpam-7124	205	2	φ(0	φ(0	ADJ
ejpam-7124	205	3	)	)	PUNCT
ejpam-7124	205	4	=	=	SYM
ejpam-7124	205	5	0	0	NUM
ejpam-7124	205	6	∀	∀	NOUN
ejpam-7124	205	7	u	u	NOUN
ejpam-7124	205	8	∈	∈	PROPN
ejpam-7124	205	9	ω	ω	PROPN
ejpam-7124	205	10	,	,	PUNCT
ejpam-7124	205	11	(	(	PUNCT
ejpam-7124	205	12	ii	ii	NOUN
ejpam-7124	205	13	)	)	PUNCT
ejpam-7124	205	14	β(1	β(1	PROPN
ejpam-7124	205	15	,	,	PUNCT
ejpam-7124	205	16	0)d̄(t1	0)d̄(t1	NOUN
ejpam-7124	205	17	,	,	PUNCT
ejpam-7124	205	18	t0	t0	PROPN
ejpam-7124	205	19	)	)	PUNCT
ejpam-7124	205	20	=	=	SYM
ejpam-7124	205	21	1	1	NUM
ejpam-7124	205	22	20(d̄(1	20(d̄(1	NUM
ejpam-7124	205	23	,	,	PUNCT
ejpam-7124	205	24	2	2	NUM
ejpam-7124	205	25	)	)	PUNCT
ejpam-7124	205	26	)	)	PUNCT
ejpam-7124	205	27	=	=	SYM
ejpam-7124	205	28	1	1	NUM
ejpam-7124	205	29	50	50	NUM
ejpam-7124	205	30	≤	≤	NUM
ejpam-7124	205	31	φd̄(1	φd̄(1	NOUN
ejpam-7124	205	32	,	,	PUNCT
ejpam-7124	205	33	2	2	NUM
ejpam-7124	205	34	)	)	PUNCT
ejpam-7124	205	35	=	=	SYM
ejpam-7124	205	36	φ	φ	PROPN
ejpam-7124	205	37	(	(	PUNCT
ejpam-7124	205	38	2	2	NUM
ejpam-7124	205	39	5	5	NUM
ejpam-7124	205	40	)	)	PUNCT
ejpam-7124	205	41	=	=	SYM
ejpam-7124	205	42	1	1	NUM
ejpam-7124	205	43	20	20	NUM
ejpam-7124	205	44	.	.	PUNCT
ejpam-7124	206	1	(	(	PUNCT
ejpam-7124	206	2	iii	iii	NOUN
ejpam-7124	206	3	)	)	PUNCT
ejpam-7124	206	4	β(1	β(1	PROPN
ejpam-7124	206	5	,	,	PUNCT
ejpam-7124	206	6	2)d̄(t1	2)d̄(t1	NOUN
ejpam-7124	206	7	,	,	PUNCT
ejpam-7124	206	8	t2	t2	NOUN
ejpam-7124	206	9	)	)	PUNCT
ejpam-7124	206	10	=	=	SYM
ejpam-7124	206	11	1	1	NUM
ejpam-7124	206	12	20(d̄(1	20(d̄(1	NUM
ejpam-7124	206	13	,	,	PUNCT
ejpam-7124	206	14	0	0	NUM
ejpam-7124	206	15	)	)	PUNCT
ejpam-7124	206	16	)	)	PUNCT
ejpam-7124	207	1	=	=	SYM
ejpam-7124	207	2	1	1	NUM
ejpam-7124	207	3	20	20	NUM
ejpam-7124	207	4	≤	≤	NUM
ejpam-7124	207	5	φd̄(1	φd̄(1	NOUN
ejpam-7124	207	6	,	,	PUNCT
ejpam-7124	207	7	0	0	NUM
ejpam-7124	207	8	)	)	PUNCT
ejpam-7124	207	9	=	=	SYM
ejpam-7124	207	10	φ(1	φ(1	PROPN
ejpam-7124	207	11	)	)	PUNCT
ejpam-7124	207	12	=	=	SYM
ejpam-7124	207	13	1	1	NUM
ejpam-7124	207	14	8	8	NUM
ejpam-7124	207	15	.	.	PUNCT
ejpam-7124	208	1	(	(	PUNCT
ejpam-7124	208	2	iv	iv	X
ejpam-7124	208	3	)	)	PUNCT
ejpam-7124	208	4	β(2	β(2	PROPN
ejpam-7124	208	5	,	,	PUNCT
ejpam-7124	208	6	0)d̄(t2	0)d̄(t2	NOUN
ejpam-7124	208	7	,	,	PUNCT
ejpam-7124	208	8	t0	t0	NOUN
ejpam-7124	208	9	)	)	PUNCT
ejpam-7124	208	10	=	=	SYM
ejpam-7124	208	11	1	1	NUM
ejpam-7124	208	12	20(d̄(0	20(d̄(0	NOUN
ejpam-7124	208	13	,	,	PUNCT
ejpam-7124	208	14	2	2	NUM
ejpam-7124	208	15	)	)	PUNCT
ejpam-7124	208	16	)	)	PUNCT
ejpam-7124	209	1	=	=	SYM
ejpam-7124	209	2	1	1	NUM
ejpam-7124	209	3	40	40	NUM
ejpam-7124	209	4	≤	≤	NOUN
ejpam-7124	209	5	φd̄(0	φd̄(0	PROPN
ejpam-7124	209	6	,	,	PUNCT
ejpam-7124	209	7	2	2	NUM
ejpam-7124	209	8	)	)	PUNCT
ejpam-7124	209	9	=	=	SYM
ejpam-7124	209	10	φ	φ	PROPN
ejpam-7124	209	11	(	(	PUNCT
ejpam-7124	209	12	1	1	NUM
ejpam-7124	209	13	2	2	NUM
ejpam-7124	209	14	)	)	PUNCT
ejpam-7124	209	15	=	=	SYM
ejpam-7124	209	16	1	1	NUM
ejpam-7124	209	17	16	16	NUM
ejpam-7124	209	18	.	.	PUNCT
ejpam-7124	210	1	therefore	therefore	ADV
ejpam-7124	210	2	,	,	PUNCT
ejpam-7124	210	3	t	t	PROPN
ejpam-7124	210	4	satisfies	satisfy	VERB
ejpam-7124	210	5	the	the	DET
ejpam-7124	210	6	conditions	condition	NOUN
ejpam-7124	210	7	in	in	ADP
ejpam-7124	210	8	theorem	theorem	NOUN
ejpam-7124	210	9	1	1	NUM
ejpam-7124	210	10	and	and	CCONJ
ejpam-7124	210	11	hence	hence	ADV
ejpam-7124	210	12	it	it	PRON
ejpam-7124	210	13	has	have	VERB
ejpam-7124	210	14	a	a	DET
ejpam-7124	210	15	unique	unique	ADJ
ejpam-7124	210	16	fixed	fix	VERB
ejpam-7124	210	17	point	point	NOUN
ejpam-7124	210	18	u	u	NOUN
ejpam-7124	210	19	=	=	NOUN
ejpam-7124	210	20	1	1	NUM
ejpam-7124	210	21	.	.	PUNCT
ejpam-7124	210	22	theorem	theorem	NOUN
ejpam-7124	210	23	2	2	NUM
ejpam-7124	210	24	.	.	PUNCT
ejpam-7124	211	1	let	let	AUX
ejpam-7124	211	2	(	(	PUNCT
ejpam-7124	211	3	ω	ω	NOUN
ejpam-7124	211	4	,	,	PUNCT
ejpam-7124	211	5	d̄	d̄	PROPN
ejpam-7124	211	6	)	)	PUNCT
ejpam-7124	211	7	be	be	VERB
ejpam-7124	211	8	a	a	DET
ejpam-7124	211	9	complete	complete	ADJ
ejpam-7124	211	10	controlled	control	VERB
ejpam-7124	211	11	metric	metric	ADJ
ejpam-7124	211	12	type	type	NOUN
ejpam-7124	211	13	space	space	NOUN
ejpam-7124	211	14	and	and	CCONJ
ejpam-7124	211	15	be	be	AUX
ejpam-7124	211	16	a	a	DET
ejpam-7124	211	17	t	t	NOUN
ejpam-7124	211	18	:	:	PUNCT
ejpam-7124	211	19	ω	ω	PROPN
ejpam-7124	211	20	→	→	SYM
ejpam-7124	211	21	ω	ω	X
ejpam-7124	211	22	be	be	AUX
ejpam-7124	211	23	a	a	DET
ejpam-7124	211	24	bijective	bijective	ADJ
ejpam-7124	211	25	and	and	CCONJ
ejpam-7124	211	26	(	(	PUNCT
ejpam-7124	211	27	β	β	X
ejpam-7124	211	28	,	,	PUNCT
ejpam-7124	211	29	φ)-expansive	φ)-expansive	PUNCT
ejpam-7124	211	30	mapping	mapping	NOUN
ejpam-7124	211	31	for	for	ADP
ejpam-7124	211	32	some	some	DET
ejpam-7124	211	33	φ	φ	PROPN
ejpam-7124	211	34	∈	∈	PROPN
ejpam-7124	211	35	φ	φ	PROPN
ejpam-7124	211	36	.	.	PUNCT
ejpam-7124	211	37	suppose	suppose	VERB
ejpam-7124	211	38	that	that	SCONJ
ejpam-7124	211	39	the	the	DET
ejpam-7124	211	40	following	follow	VERB
ejpam-7124	211	41	condition	condition	NOUN
ejpam-7124	211	42	holds	hold	VERB
ejpam-7124	211	43	:	:	PUNCT
ejpam-7124	212	1	[	[	X
ejpam-7124	212	2	label=(ii),itemsep=-.16em	label=(ii),itemsep=-.16em	X
ejpam-7124	212	3	,	,	PUNCT
ejpam-7124	212	4	topsep=2pt	topsep=2pt	PROPN
ejpam-7124	212	5	]	]	PUNCT
ejpam-7124	212	6	(	(	PUNCT
ejpam-7124	212	7	i	i	NOUN
ejpam-7124	212	8	)	)	PUNCT
ejpam-7124	212	9	t−1	t−1	PROPN
ejpam-7124	212	10	is	be	AUX
ejpam-7124	212	11	β	β	NOUN
ejpam-7124	212	12	-	-	ADJ
ejpam-7124	212	13	admissible	admissible	ADJ
ejpam-7124	212	14	,	,	PUNCT
ejpam-7124	212	15	(	(	PUNCT
ejpam-7124	212	16	ii	ii	NOUN
ejpam-7124	212	17	)	)	PUNCT
ejpam-7124	212	18	∃	∃	PROPN
ejpam-7124	212	19	u0	u0	PROPN
ejpam-7124	212	20	∈	∈	PROPN
ejpam-7124	212	21	ω	ω	NOUN
ejpam-7124	212	22	such	such	ADJ
ejpam-7124	212	23	that	that	SCONJ
ejpam-7124	212	24	β(u0	β(u0	NOUN
ejpam-7124	212	25	,	,	PUNCT
ejpam-7124	212	26	t	t	PROPN
ejpam-7124	212	27	−1u0	−1u0	NUM
ejpam-7124	212	28	)	)	PUNCT
ejpam-7124	212	29	≥	≥	NOUN
ejpam-7124	212	30	1	1	NUM
ejpam-7124	212	31	,	,	PUNCT
ejpam-7124	212	32	(	(	PUNCT
ejpam-7124	212	33	iii	iii	NOUN
ejpam-7124	212	34	)	)	PUNCT
ejpam-7124	212	35	if	if	SCONJ
ejpam-7124	212	36	{	{	PUNCT
ejpam-7124	212	37	uj}∞j=1	uj}∞j=1	X
ejpam-7124	212	38	is	be	AUX
ejpam-7124	212	39	a	a	DET
ejpam-7124	212	40	sequence	sequence	NOUN
ejpam-7124	212	41	in	in	ADP
ejpam-7124	212	42	ω	ω	NUM
ejpam-7124	212	43	such	such	ADJ
ejpam-7124	212	44	that	that	SCONJ
ejpam-7124	212	45	β(uj	β(uj	NUM
ejpam-7124	212	46	,	,	PUNCT
ejpam-7124	212	47	uj+1	uj+1	NUM
ejpam-7124	212	48	)	)	PUNCT
ejpam-7124	212	49	≥	≥	NOUN
ejpam-7124	212	50	1	1	NUM
ejpam-7124	212	51	for	for	ADP
ejpam-7124	212	52	all	all	DET
ejpam-7124	212	53	j	j	PROPN
ejpam-7124	212	54	and	and	CCONJ
ejpam-7124	212	55	{	{	PUNCT
ejpam-7124	212	56	uj	uj	PROPN
ejpam-7124	212	57	}	}	PUNCT
ejpam-7124	212	58	→	→	SYM
ejpam-7124	212	59	u	u	NOUN
ejpam-7124	212	60	as	as	ADP
ejpam-7124	212	61	j	j	PROPN
ejpam-7124	212	62	→	→	SYM
ejpam-7124	212	63	∞	∞	PROPN
ejpam-7124	212	64	then	then	ADV
ejpam-7124	212	65	β(t−1uj	β(t−1uj	PROPN
ejpam-7124	212	66	,	,	PUNCT
ejpam-7124	212	67	t	t	PROPN
ejpam-7124	212	68	−1u	−1u	PROPN
ejpam-7124	212	69	)	)	PUNCT
ejpam-7124	212	70	≥	≥	NOUN
ejpam-7124	212	71	1	1	NUM
ejpam-7124	212	72	∀	∀	NOUN
ejpam-7124	213	1	j.	j.	PROPN
ejpam-7124	213	2	m.	m.	PROPN
ejpam-7124	213	3	kumar	kumar	PROPN
ejpam-7124	213	4	et	et	PROPN
ejpam-7124	213	5	al	al	PROPN
ejpam-7124	213	6	.	.	PUNCT
ejpam-7124	213	7	/	/	SYM
ejpam-7124	213	8	eur	eur	PROPN
ejpam-7124	213	9	.	.	PUNCT
ejpam-7124	214	1	j.	j.	PROPN
ejpam-7124	214	2	pure	pure	PROPN
ejpam-7124	214	3	appl	appl	PROPN
ejpam-7124	214	4	.	.	PROPN
ejpam-7124	214	5	math	math	PROPN
ejpam-7124	214	6	,	,	PUNCT
ejpam-7124	214	7	18	18	NUM
ejpam-7124	214	8	(	(	PUNCT
ejpam-7124	214	9	4	4	NUM
ejpam-7124	214	10	)	)	PUNCT
ejpam-7124	214	11	(	(	PUNCT
ejpam-7124	214	12	2025	2025	NUM
ejpam-7124	214	13	)	)	PUNCT
ejpam-7124	214	14	,	,	PUNCT
ejpam-7124	214	15	7124	7124	NUM
ejpam-7124	214	16	10	10	NUM
ejpam-7124	214	17	of	of	ADP
ejpam-7124	214	18	14	14	NUM
ejpam-7124	214	19	then	then	ADV
ejpam-7124	214	20	t	t	PROPN
ejpam-7124	214	21	has	have	VERB
ejpam-7124	214	22	a	a	DET
ejpam-7124	214	23	fixed	fix	VERB
ejpam-7124	214	24	point	point	NOUN
ejpam-7124	214	25	.	.	PUNCT
ejpam-7124	215	1	proof	proof	NOUN
ejpam-7124	215	2	.	.	PUNCT
ejpam-7124	216	1	in	in	ADP
ejpam-7124	216	2	proving	prove	VERB
ejpam-7124	216	3	the	the	DET
ejpam-7124	216	4	result	result	NOUN
ejpam-7124	216	5	,	,	PUNCT
ejpam-7124	216	6	we	we	PRON
ejpam-7124	216	7	follow	follow	VERB
ejpam-7124	216	8	the	the	DET
ejpam-7124	216	9	same	same	ADJ
ejpam-7124	216	10	steps	step	NOUN
ejpam-7124	216	11	as	as	ADP
ejpam-7124	216	12	in	in	ADP
ejpam-7124	216	13	the	the	DET
ejpam-7124	216	14	proof	proof	NOUN
ejpam-7124	216	15	of	of	ADP
ejpam-7124	216	16	theorem	theorem	NOUN
ejpam-7124	216	17	1	1	NUM
ejpam-7124	216	18	to	to	PART
ejpam-7124	216	19	construct	construct	VERB
ejpam-7124	216	20	a	a	DET
ejpam-7124	216	21	sequence	sequence	NOUN
ejpam-7124	216	22	{	{	PUNCT
ejpam-7124	216	23	uj}∞j=1	uj}∞j=1	NOUN
ejpam-7124	216	24	that	that	PRON
ejpam-7124	216	25	converges	converge	VERB
ejpam-7124	216	26	to	to	ADP
ejpam-7124	216	27	a	a	DET
ejpam-7124	216	28	point	point	NOUN
ejpam-7124	216	29	u	u	NOUN
ejpam-7124	216	30	,	,	PUNCT
ejpam-7124	216	31	u	u	PROPN
ejpam-7124	216	32	∈	∈	PROPN
ejpam-7124	216	33	ω	ω	PROPN
ejpam-7124	216	34	.	.	PUNCT
ejpam-7124	217	1	the	the	DET
ejpam-7124	217	2	constructed	construct	VERB
ejpam-7124	217	3	sequence	sequence	NOUN
ejpam-7124	217	4	has	have	VERB
ejpam-7124	217	5	the	the	DET
ejpam-7124	217	6	property	property	NOUN
ejpam-7124	217	7	β(uj	β(uj	X
ejpam-7124	217	8	,	,	PUNCT
ejpam-7124	217	9	uj+1	uj+1	NUM
ejpam-7124	217	10	)	)	PUNCT
ejpam-7124	217	11	≥	≥	NOUN
ejpam-7124	217	12	1	1	NUM
ejpam-7124	217	13	,	,	PUNCT
ejpam-7124	217	14	for	for	ADP
ejpam-7124	217	15	all	all	DET
ejpam-7124	217	16	natural	natural	ADJ
ejpam-7124	217	17	numbers	number	NOUN
ejpam-7124	217	18	j.	j.	PROPN
ejpam-7124	217	19	the	the	DET
ejpam-7124	217	20	last	last	ADJ
ejpam-7124	217	21	assumption	assumption	NOUN
ejpam-7124	217	22	of	of	ADP
ejpam-7124	217	23	the	the	DET
ejpam-7124	217	24	result	result	NOUN
ejpam-7124	217	25	implies	imply	VERB
ejpam-7124	217	26	that	that	SCONJ
ejpam-7124	217	27	β(t−1uj	β(t−1uj	PROPN
ejpam-7124	217	28	,	,	PUNCT
ejpam-7124	217	29	t	t	PROPN
ejpam-7124	217	30	−1u	−1u	PROPN
ejpam-7124	217	31	)	)	PUNCT
ejpam-7124	217	32	≥	≥	NOUN
ejpam-7124	217	33	1	1	NUM
ejpam-7124	217	34	.	.	PUNCT
ejpam-7124	218	1	now	now	ADV
ejpam-7124	218	2	,	,	PUNCT
ejpam-7124	218	3	we	we	PRON
ejpam-7124	218	4	will	will	AUX
ejpam-7124	218	5	prove	prove	VERB
ejpam-7124	218	6	that	that	SCONJ
ejpam-7124	218	7	u	u	NOUN
ejpam-7124	218	8	is	be	AUX
ejpam-7124	218	9	a	a	DET
ejpam-7124	218	10	fixed	fix	VERB
ejpam-7124	218	11	point	point	NOUN
ejpam-7124	218	12	of	of	ADP
ejpam-7124	218	13	t	t	PROPN
ejpam-7124	218	14	.	.	PUNCT
ejpam-7124	219	1	the	the	DET
ejpam-7124	219	2	triangle	triangle	NOUN
ejpam-7124	219	3	inequality	inequality	NOUN
ejpam-7124	219	4	implies	imply	VERB
ejpam-7124	219	5	that	that	SCONJ
ejpam-7124	219	6	d̄(t−1u	d̄(t−1u	PROPN
ejpam-7124	219	7	,	,	PUNCT
ejpam-7124	219	8	u	u	NOUN
ejpam-7124	219	9	)	)	PUNCT
ejpam-7124	219	10	≤	≤	NOUN
ejpam-7124	219	11	µ(t−1u	µ(t−1u	PROPN
ejpam-7124	219	12	,	,	PUNCT
ejpam-7124	219	13	uj)d̄(t	uj)d̄(t	ADJ
ejpam-7124	219	14	−1u	−1u	PROPN
ejpam-7124	219	15	,	,	PUNCT
ejpam-7124	219	16	uj	uj	NOUN
ejpam-7124	219	17	)	)	PUNCT
ejpam-7124	220	1	+	+	CCONJ
ejpam-7124	220	2	µ(uj	µ(uj	PROPN
ejpam-7124	220	3	,	,	PUNCT
ejpam-7124	220	4	u)d̄(uj	u)d̄(uj	PROPN
ejpam-7124	220	5	,	,	PUNCT
ejpam-7124	220	6	u	u	NOUN
ejpam-7124	220	7	)	)	PUNCT
ejpam-7124	220	8	=	=	SYM
ejpam-7124	220	9	µ(t−1u	µ(t−1u	PROPN
ejpam-7124	220	10	,	,	PUNCT
ejpam-7124	220	11	uj)d̄(t	uj)d̄(t	ADJ
ejpam-7124	220	12	−1u	−1u	PROPN
ejpam-7124	220	13	,	,	PUNCT
ejpam-7124	220	14	uj	uj	NOUN
ejpam-7124	220	15	)	)	PUNCT
ejpam-7124	220	16	+	+	CCONJ
ejpam-7124	220	17	µ(u	µ(u	NOUN
ejpam-7124	220	18	,	,	PUNCT
ejpam-7124	220	19	u)d̄(u	u)d̄(u	PROPN
ejpam-7124	220	20	,	,	PUNCT
ejpam-7124	220	21	u	u	NOUN
ejpam-7124	220	22	)	)	PUNCT
ejpam-7124	220	23	=	=	SYM
ejpam-7124	220	24	µ(t−1u	µ(t−1u	PROPN
ejpam-7124	220	25	,	,	PUNCT
ejpam-7124	220	26	uj)d̄(t	uj)d̄(t	ADJ
ejpam-7124	220	27	−1u	−1u	PROPN
ejpam-7124	220	28	,	,	PUNCT
ejpam-7124	220	29	uj	uj	NOUN
ejpam-7124	220	30	)	)	PUNCT
ejpam-7124	220	31	+	+	NOUN
ejpam-7124	220	32	0	0	X
ejpam-7124	220	33	.	.	PUNCT
ejpam-7124	221	1	thus	thus	ADV
ejpam-7124	221	2	,	,	PUNCT
ejpam-7124	221	3	we	we	PRON
ejpam-7124	221	4	get	get	VERB
ejpam-7124	221	5	d̄(t−1u	d̄(t−1u	PRON
ejpam-7124	221	6	,	,	PUNCT
ejpam-7124	221	7	u	u	NOUN
ejpam-7124	221	8	)	)	PUNCT
ejpam-7124	221	9	≤	≤	NOUN
ejpam-7124	221	10	µ(t−1u	µ(t−1u	PROPN
ejpam-7124	221	11	,	,	PUNCT
ejpam-7124	221	12	uj)d̄(t	uj)d̄(t	ADJ
ejpam-7124	221	13	−1u	−1u	PROPN
ejpam-7124	221	14	,	,	PUNCT
ejpam-7124	221	15	uj	uj	PROPN
ejpam-7124	221	16	)	)	PUNCT
ejpam-7124	221	17	≤	≤	NOUN
ejpam-7124	221	18	µ(t−1u	µ(t−1u	PROPN
ejpam-7124	221	19	,	,	PUNCT
ejpam-7124	221	20	uj)d̄(t	uj)d̄(t	ADJ
ejpam-7124	221	21	−1u	−1u	PROPN
ejpam-7124	221	22	,	,	PUNCT
ejpam-7124	221	23	uj)β(t	uj)β(t	X
ejpam-7124	221	24	−1u	−1u	PROPN
ejpam-7124	221	25	,	,	PUNCT
ejpam-7124	221	26	uj	uj	PROPN
ejpam-7124	221	27	)	)	PUNCT
ejpam-7124	221	28	≤	≤	NOUN
ejpam-7124	221	29	µ(t−1u	µ(t−1u	PROPN
ejpam-7124	221	30	,	,	PUNCT
ejpam-7124	221	31	uj)φ(d̄(u	uj)φ(d̄(u	ADJ
ejpam-7124	221	32	,	,	PUNCT
ejpam-7124	221	33	tuj	tuj	PROPN
ejpam-7124	221	34	)	)	PUNCT
ejpam-7124	221	35	)	)	PUNCT
ejpam-7124	221	36	≤	≤	PROPN
ejpam-7124	221	37	µ(t−1u	µ(t−1u	PROPN
ejpam-7124	221	38	,	,	PUNCT
ejpam-7124	221	39	u)φ(d̄(u	u)φ(d̄(u	PROPN
ejpam-7124	221	40	,	,	PUNCT
ejpam-7124	221	41	tu	tu	PROPN
ejpam-7124	221	42	)	)	PUNCT
ejpam-7124	221	43	)	)	PUNCT
ejpam-7124	221	44	.	.	PUNCT
ejpam-7124	222	1	(	(	PUNCT
ejpam-7124	222	2	16	16	X
ejpam-7124	222	3	)	)	PUNCT
ejpam-7124	222	4	continuity	continuity	NOUN
ejpam-7124	222	5	of	of	ADP
ejpam-7124	222	6	φ	φ	PROPN
ejpam-7124	222	7	at	at	ADP
ejpam-7124	222	8	u	u	NOUN
ejpam-7124	222	9	=	=	SYM
ejpam-7124	222	10	0	0	NUM
ejpam-7124	222	11	implies	imply	VERB
ejpam-7124	222	12	that	that	SCONJ
ejpam-7124	222	13	d̄(t−1u	d̄(t−1u	PROPN
ejpam-7124	222	14	,	,	PUNCT
ejpam-7124	222	15	u	u	NOUN
ejpam-7124	222	16	)	)	PUNCT
ejpam-7124	222	17	=	=	SYM
ejpam-7124	222	18	0	0	PUNCT
ejpam-7124	223	1	as	as	SCONJ
ejpam-7124	223	2	j	j	PROPN
ejpam-7124	223	3	→	→	SYM
ejpam-7124	223	4	+	+	ADJ
ejpam-7124	223	5	∞	∞	PROPN
ejpam-7124	223	6	i.e.	i.e.	X
ejpam-7124	223	7	t−1u	t−1u	X
ejpam-7124	223	8	=	=	X
ejpam-7124	223	9	u.	u.	NOUN
ejpam-7124	223	10	consider	consider	VERB
ejpam-7124	223	11	tu	tu	PROPN
ejpam-7124	223	12	=	=	PROPN
ejpam-7124	223	13	t	t	PROPN
ejpam-7124	223	14	(	(	PUNCT
ejpam-7124	223	15	t−1u	t−1u	NOUN
ejpam-7124	223	16	)	)	PUNCT
ejpam-7124	223	17	=	=	NOUN
ejpam-7124	223	18	(	(	PUNCT
ejpam-7124	223	19	tt−1)u	tt−1)u	PUNCT
ejpam-7124	223	20	=	=	VERB
ejpam-7124	223	21	u.	u.	NOUN
ejpam-7124	223	22	hence	hence	ADV
ejpam-7124	223	23	the	the	DET
ejpam-7124	223	24	result	result	NOUN
ejpam-7124	223	25	is	be	AUX
ejpam-7124	223	26	proved	prove	VERB
ejpam-7124	223	27	.	.	PUNCT
ejpam-7124	224	1	example	example	NOUN
ejpam-7124	225	1	5	5	NUM
ejpam-7124	225	2	.	.	PUNCT
ejpam-7124	225	3	let	let	VERB
ejpam-7124	225	4	ω	ω	NOUN
ejpam-7124	225	5	=	=	PUNCT
ejpam-7124	226	1	[	[	X
ejpam-7124	226	2	0,∞	0,∞	NOUN
ejpam-7124	226	3	)	)	PUNCT
ejpam-7124	226	4	with	with	ADP
ejpam-7124	226	5	controlled	control	VERB
ejpam-7124	226	6	metric	metric	ADJ
ejpam-7124	226	7	space	space	NOUN
ejpam-7124	226	8	d̄	d̄	NOUN
ejpam-7124	226	9	:	:	PUNCT
ejpam-7124	227	1	ω	ω	NUM
ejpam-7124	227	2	×	×	PROPN
ejpam-7124	227	3	ω	ω	X
ejpam-7124	227	4	→	→	PUNCT
ejpam-7124	227	5	[	[	X
ejpam-7124	227	6	0,∞	0,∞	NOUN
ejpam-7124	227	7	)	)	PUNCT
ejpam-7124	227	8	defined	define	VERB
ejpam-7124	227	9	by	by	ADP
ejpam-7124	227	10	d̄	d̄	NOUN
ejpam-7124	227	11	=	=	PUNCT
ejpam-7124	227	12	|u	|u	ADJ
ejpam-7124	227	13	−	−	PROPN
ejpam-7124	227	14	e|	e|	PROPN
ejpam-7124	227	15	∀	∀	X
ejpam-7124	227	16	u	u	NOUN
ejpam-7124	227	17	,	,	PUNCT
ejpam-7124	227	18	e	e	PROPN
ejpam-7124	227	19	∈	∈	PROPN
ejpam-7124	227	20	ω	ω	X
ejpam-7124	227	21	along	along	ADP
ejpam-7124	227	22	with	with	ADP
ejpam-7124	227	23	the	the	DET
ejpam-7124	227	24	mapping	mapping	NOUN
ejpam-7124	227	25	µ	µ	X
ejpam-7124	227	26	:	:	PUNCT
ejpam-7124	227	27	ω	ω	NUM
ejpam-7124	227	28	×	×	PROPN
ejpam-7124	227	29	ω	ω	X
ejpam-7124	227	30	→	→	SYM
ejpam-7124	227	31	[	[	X
ejpam-7124	227	32	1,∞	1,∞	NUM
ejpam-7124	227	33	)	)	PUNCT
ejpam-7124	227	34	and	and	CCONJ
ejpam-7124	227	35	define	define	VERB
ejpam-7124	227	36	t	t	PROPN
ejpam-7124	227	37	:	:	PUNCT
ejpam-7124	227	38	ω	ω	PROPN
ejpam-7124	227	39	→	→	SYM
ejpam-7124	227	40	ω	ω	PROPN
ejpam-7124	227	41	is	be	AUX
ejpam-7124	227	42	(	(	PUNCT
ejpam-7124	227	43	β	β	X
ejpam-7124	227	44	,	,	PUNCT
ejpam-7124	227	45	φ)-expansive	φ)-expansive	PUNCT
ejpam-7124	227	46	mapping	mapping	NOUN
ejpam-7124	227	47	with	with	ADP
ejpam-7124	227	48	φ	φ	PROPN
ejpam-7124	227	49	∈	∈	PROPN
ejpam-7124	227	50	φ	φ	PROPN
ejpam-7124	227	51	by	by	ADP
ejpam-7124	227	52	t	t	PROPN
ejpam-7124	227	53	(	(	PUNCT
ejpam-7124	227	54	u	u	NOUN
ejpam-7124	227	55	)	)	PUNCT
ejpam-7124	227	56	=	=	SYM
ejpam-7124	227	57	{	{	PUNCT
ejpam-7124	227	58	u3	u3	PROPN
ejpam-7124	227	59	,	,	PUNCT
ejpam-7124	227	60	u	u	NOUN
ejpam-7124	227	61	≥	≥	NUM
ejpam-7124	227	62	1	1	NUM
ejpam-7124	227	63	2	2	NUM
ejpam-7124	227	64	,	,	PUNCT
ejpam-7124	227	65	u	u	NOUN
ejpam-7124	227	66	2	2	NUM
ejpam-7124	227	67	,	,	PUNCT
ejpam-7124	227	68	0	0	NUM
ejpam-7124	227	69	≤	≤	NUM
ejpam-7124	227	70	u	u	NOUN
ejpam-7124	227	71	<	<	X
ejpam-7124	227	72	1	1	NUM
ejpam-7124	227	73	2	2	NUM
ejpam-7124	227	74	and	and	CCONJ
ejpam-7124	227	75	β(u	β(u	PROPN
ejpam-7124	227	76	,	,	PUNCT
ejpam-7124	227	77	e	e	NOUN
ejpam-7124	227	78	)	)	PUNCT
ejpam-7124	227	79	=	=	SYM
ejpam-7124	227	80	{	{	PUNCT
ejpam-7124	227	81	0	0	NUM
ejpam-7124	227	82	,	,	PUNCT
ejpam-7124	227	83	u	u	NOUN
ejpam-7124	227	84	,	,	PUNCT
ejpam-7124	227	85	e	e	X
ejpam-7124	227	86	≥	≥	X
ejpam-7124	227	87	[	[	PUNCT
ejpam-7124	227	88	0	0	NUM
ejpam-7124	227	89	,	,	PUNCT
ejpam-7124	227	90	12	12	NUM
ejpam-7124	227	91	)	)	PUNCT
ejpam-7124	227	92	,	,	PUNCT
ejpam-7124	227	93	1	1	NUM
ejpam-7124	227	94	,	,	PUNCT
ejpam-7124	227	95	otherwise	otherwise	ADV
ejpam-7124	227	96	.	.	PUNCT
ejpam-7124	228	1	clearly	clearly	ADV
ejpam-7124	228	2	t	t	PROPN
ejpam-7124	228	3	is	be	AUX
ejpam-7124	228	4	not	not	PART
ejpam-7124	228	5	continuous	continuous	ADJ
ejpam-7124	228	6	at	at	ADP
ejpam-7124	228	7	1	1	NUM
ejpam-7124	228	8	2	2	NUM
ejpam-7124	228	9	and	and	CCONJ
ejpam-7124	228	10	t	t	PROPN
ejpam-7124	228	11	is	be	AUX
ejpam-7124	228	12	(	(	PUNCT
ejpam-7124	228	13	β	β	X
ejpam-7124	228	14	,	,	PUNCT
ejpam-7124	228	15	φ)-expansive	φ)-expansive	PUNCT
ejpam-7124	228	16	mapping	mapping	NOUN
ejpam-7124	228	17	with	with	ADP
ejpam-7124	228	18	φ(r	φ(r	ADJ
ejpam-7124	228	19	)	)	PUNCT
ejpam-7124	229	1	=	=	SYM
ejpam-7124	229	2	r	r	NOUN
ejpam-7124	229	3	2	2	NUM
ejpam-7124	229	4	∀	∀	NOUN
ejpam-7124	229	5	r	r	NOUN
ejpam-7124	229	6	≥	≥	NOUN
ejpam-7124	229	7	0	0	NUM
ejpam-7124	229	8	.	.	PUNCT
ejpam-7124	230	1	moreover	moreover	ADV
ejpam-7124	230	2	for	for	ADP
ejpam-7124	230	3	all	all	DET
ejpam-7124	230	4	u	u	NOUN
ejpam-7124	230	5	,	,	PUNCT
ejpam-7124	230	6	e	e	PROPN
ejpam-7124	230	7	∈	∈	PROPN
ejpam-7124	230	8	ω	ω	PROPN
ejpam-7124	230	9	,	,	PUNCT
ejpam-7124	230	10	we	we	PRON
ejpam-7124	230	11	have	have	VERB
ejpam-7124	230	12	1	1	NUM
ejpam-7124	230	13	2	2	NUM
ejpam-7124	230	14	(	(	PUNCT
ejpam-7124	230	15	d̄(tu	d̄(tu	NOUN
ejpam-7124	230	16	,	,	PUNCT
ejpam-7124	230	17	te	te	PROPN
ejpam-7124	230	18	)	)	PUNCT
ejpam-7124	230	19	)	)	PUNCT
ejpam-7124	230	20	≥	≥	NOUN
ejpam-7124	230	21	β(u	β(u	PROPN
ejpam-7124	230	22	,	,	PUNCT
ejpam-7124	230	23	e)d̄(u	e)d̄(u	PROPN
ejpam-7124	230	24	,	,	PUNCT
ejpam-7124	230	25	e	e	NOUN
ejpam-7124	230	26	)	)	PUNCT
ejpam-7124	230	27	.	.	PUNCT
ejpam-7124	231	1	also	also	ADV
ejpam-7124	231	2	there	there	PRON
ejpam-7124	231	3	exist	exist	VERB
ejpam-7124	231	4	u0	u0	ADJ
ejpam-7124	231	5	∈	∈	PROPN
ejpam-7124	231	6	ω	ω	NOUN
ejpam-7124	231	7	such	such	ADJ
ejpam-7124	231	8	that	that	SCONJ
ejpam-7124	231	9	β(u0	β(u0	NOUN
ejpam-7124	231	10	,	,	PUNCT
ejpam-7124	231	11	t	t	PROPN
ejpam-7124	231	12	−1u0	−1u0	NUM
ejpam-7124	231	13	)	)	PUNCT
ejpam-7124	231	14	≥	≥	NOUN
ejpam-7124	231	15	1	1	NUM
ejpam-7124	231	16	.	.	X
ejpam-7124	231	17	infact	infact	VERB
ejpam-7124	231	18	for	for	ADP
ejpam-7124	231	19	u0	u0	ADJ
ejpam-7124	231	20	=	=	ADJ
ejpam-7124	231	21	1	1	NUM
ejpam-7124	231	22	,	,	PUNCT
ejpam-7124	231	23	we	we	PRON
ejpam-7124	231	24	have	have	VERB
ejpam-7124	231	25	β(1	β(1	PROPN
ejpam-7124	231	26	,	,	PUNCT
ejpam-7124	231	27	t−11	t−11	PROPN
ejpam-7124	231	28	)	)	PUNCT
ejpam-7124	232	1	=	=	SYM
ejpam-7124	232	2	1	1	X
ejpam-7124	232	3	.	.	PUNCT
ejpam-7124	232	4	now	now	ADV
ejpam-7124	232	5	,	,	PUNCT
ejpam-7124	232	6	let	let	VERB
ejpam-7124	232	7	u	u	PRON
ejpam-7124	232	8	,	,	PUNCT
ejpam-7124	232	9	e	e	PROPN
ejpam-7124	232	10	∈	∈	PROPN
ejpam-7124	232	11	ω	ω	NUM
ejpam-7124	232	12	such	such	ADJ
ejpam-7124	232	13	that	that	SCONJ
ejpam-7124	232	14	β(u	β(u	PROPN
ejpam-7124	232	15	,	,	PUNCT
ejpam-7124	232	16	e	e	NOUN
ejpam-7124	232	17	)	)	PUNCT
ejpam-7124	232	18	≥	≥	NOUN
ejpam-7124	232	19	1	1	NUM
ejpam-7124	232	20	.	.	PUNCT
ejpam-7124	233	1	this	this	PRON
ejpam-7124	233	2	implies	imply	VERB
ejpam-7124	233	3	that	that	SCONJ
ejpam-7124	233	4	u	u	PRON
ejpam-7124	233	5	≥	≥	NOUN
ejpam-7124	233	6	1	1	NUM
ejpam-7124	233	7	,	,	PUNCT
ejpam-7124	233	8	e	e	X
ejpam-7124	233	9	≥	≥	NUM
ejpam-7124	233	10	1	1	NUM
ejpam-7124	233	11	.	.	PUNCT
ejpam-7124	233	12	m.	m.	PROPN
ejpam-7124	233	13	kumar	kumar	PROPN
ejpam-7124	233	14	et	et	PROPN
ejpam-7124	233	15	al	al	PROPN
ejpam-7124	233	16	.	.	PUNCT
ejpam-7124	233	17	/	/	SYM
ejpam-7124	233	18	eur	eur	PROPN
ejpam-7124	233	19	.	.	PUNCT
ejpam-7124	234	1	j.	j.	PROPN
ejpam-7124	234	2	pure	pure	PROPN
ejpam-7124	234	3	appl	appl	PROPN
ejpam-7124	234	4	.	.	PROPN
ejpam-7124	234	5	math	math	PROPN
ejpam-7124	234	6	,	,	PUNCT
ejpam-7124	234	7	18	18	NUM
ejpam-7124	234	8	(	(	PUNCT
ejpam-7124	234	9	4	4	NUM
ejpam-7124	234	10	)	)	PUNCT
ejpam-7124	234	11	(	(	PUNCT
ejpam-7124	234	12	2025	2025	NUM
ejpam-7124	234	13	)	)	PUNCT
ejpam-7124	234	14	,	,	PUNCT
ejpam-7124	234	15	7124	7124	NUM
ejpam-7124	234	16	11	11	NUM
ejpam-7124	234	17	of	of	ADP
ejpam-7124	234	18	14	14	NUM
ejpam-7124	234	19	and	and	CCONJ
ejpam-7124	234	20	by	by	ADP
ejpam-7124	234	21	definition	definition	NOUN
ejpam-7124	234	22	of	of	ADP
ejpam-7124	234	23	t−1	t−1	PROPN
ejpam-7124	234	24	and	and	CCONJ
ejpam-7124	234	25	β	β	NOUN
ejpam-7124	234	26	,	,	PUNCT
ejpam-7124	234	27	we	we	PRON
ejpam-7124	234	28	have	have	VERB
ejpam-7124	234	29	t−1u	t−1u	NOUN
ejpam-7124	234	30	=	=	SYM
ejpam-7124	234	31	u	u	NOUN
ejpam-7124	234	32	1	1	NUM
ejpam-7124	234	33	3	3	NUM
ejpam-7124	234	34	≥	≥	NOUN
ejpam-7124	234	35	1	1	NUM
ejpam-7124	234	36	,	,	PUNCT
ejpam-7124	234	37	t−1e	t−1e	NOUN
ejpam-7124	235	1	=	=	PUNCT
ejpam-7124	236	1	e	e	NOUN
ejpam-7124	236	2	1	1	NUM
ejpam-7124	236	3	3	3	NUM
ejpam-7124	236	4	≥	≥	NOUN
ejpam-7124	236	5	1	1	NUM
ejpam-7124	236	6	and	and	CCONJ
ejpam-7124	236	7	β(t−1u	β(t−1u	PROPN
ejpam-7124	236	8	,	,	PUNCT
ejpam-7124	236	9	t−1	t−1	PROPN
ejpam-7124	236	10	)	)	PUNCT
ejpam-7124	236	11	=	=	SYM
ejpam-7124	237	1	1	1	NUM
ejpam-7124	237	2	i.e.	i.e.	X
ejpam-7124	237	3	t−1	t−1	PROPN
ejpam-7124	237	4	is	be	AUX
ejpam-7124	237	5	β	β	NOUN
ejpam-7124	237	6	-	-	ADJ
ejpam-7124	237	7	admissible	admissible	ADJ
ejpam-7124	237	8	.	.	PUNCT
ejpam-7124	238	1	finally	finally	ADV
ejpam-7124	238	2	,	,	PUNCT
ejpam-7124	238	3	let	let	VERB
ejpam-7124	238	4	{	{	PUNCT
ejpam-7124	238	5	uj	uj	PROPN
ejpam-7124	238	6	}	}	PUNCT
ejpam-7124	238	7	be	be	AUX
ejpam-7124	238	8	a	a	DET
ejpam-7124	238	9	sequence	sequence	NOUN
ejpam-7124	238	10	in	in	ADP
ejpam-7124	238	11	ω	ω	NUM
ejpam-7124	238	12	such	such	ADJ
ejpam-7124	238	13	that	that	SCONJ
ejpam-7124	238	14	β(uj	β(uj	NUM
ejpam-7124	238	15	,	,	PUNCT
ejpam-7124	238	16	uj+1	uj+1	NUM
ejpam-7124	238	17	)	)	PUNCT
ejpam-7124	238	18	≥	≥	NOUN
ejpam-7124	238	19	1	1	NUM
ejpam-7124	238	20	∀	∀	NOUN
ejpam-7124	238	21	j	j	PROPN
ejpam-7124	238	22	and	and	CCONJ
ejpam-7124	238	23	{	{	PUNCT
ejpam-7124	238	24	uj	uj	PROPN
ejpam-7124	238	25	}	}	PUNCT
ejpam-7124	238	26	→	→	SYM
ejpam-7124	238	27	u	u	NOUN
ejpam-7124	238	28	∈	∈	PROPN
ejpam-7124	238	29	ω	ω	NOUN
ejpam-7124	238	30	as	as	ADP
ejpam-7124	238	31	j	j	PROPN
ejpam-7124	238	32	→	→	SYM
ejpam-7124	238	33	∞.	∞.	PROPN
ejpam-7124	238	34	since	since	SCONJ
ejpam-7124	238	35	β(uj	β(uj	NUM
ejpam-7124	238	36	,	,	PUNCT
ejpam-7124	238	37	uj+1	uj+1	NUM
ejpam-7124	238	38	)	)	PUNCT
ejpam-7124	238	39	≥	≥	NOUN
ejpam-7124	238	40	1	1	NUM
ejpam-7124	238	41	∀	∀	NOUN
ejpam-7124	238	42	j	j	NOUN
ejpam-7124	238	43	by	by	ADP
ejpam-7124	238	44	the	the	DET
ejpam-7124	238	45	definition	definition	NOUN
ejpam-7124	238	46	of	of	ADP
ejpam-7124	238	47	β	β	X
ejpam-7124	238	48	,	,	PUNCT
ejpam-7124	238	49	we	we	PRON
ejpam-7124	238	50	have	have	VERB
ejpam-7124	238	51	uj	uj	VERB
ejpam-7124	238	52	≥	≥	NUM
ejpam-7124	238	53	1	1	NUM
ejpam-7124	238	54	∀	∀	NOUN
ejpam-7124	238	55	j	j	NOUN
ejpam-7124	238	56	and	and	CCONJ
ejpam-7124	238	57	u	u	PRON
ejpam-7124	238	58	≥	≥	NOUN
ejpam-7124	238	59	1	1	NUM
ejpam-7124	238	60	then	then	ADV
ejpam-7124	238	61	β(t−1	β(t−1	PROPN
ejpam-7124	238	62	uj	uj	PROPN
ejpam-7124	238	63	,	,	PUNCT
ejpam-7124	238	64	t	t	PROPN
ejpam-7124	238	65	−1u	−1u	PROPN
ejpam-7124	238	66	)	)	PUNCT
ejpam-7124	238	67	=	=	SYM
ejpam-7124	239	1	1	1	X
ejpam-7124	239	2	.	.	PUNCT
ejpam-7124	239	3	therefore	therefore	ADV
ejpam-7124	239	4	,	,	PUNCT
ejpam-7124	239	5	all	all	DET
ejpam-7124	239	6	the	the	DET
ejpam-7124	239	7	conditions	condition	NOUN
ejpam-7124	239	8	of	of	ADP
ejpam-7124	239	9	theorem	theorem	ADJ
ejpam-7124	239	10	2	2	NUM
ejpam-7124	239	11	are	be	AUX
ejpam-7124	239	12	satisfied	satisfied	ADJ
ejpam-7124	239	13	.	.	PUNCT
ejpam-7124	240	1	so	so	ADV
ejpam-7124	240	2	that	that	SCONJ
ejpam-7124	240	3	t	t	PROPN
ejpam-7124	240	4	has	have	VERB
ejpam-7124	240	5	the	the	DET
ejpam-7124	240	6	fixed	fix	VERB
ejpam-7124	240	7	points	point	NOUN
ejpam-7124	240	8	.	.	PUNCT
ejpam-7124	241	1	here	here	ADV
ejpam-7124	241	2	0	0	NUM
ejpam-7124	241	3	,	,	PUNCT
ejpam-7124	241	4	1	1	NUM
ejpam-7124	241	5	are	be	AUX
ejpam-7124	241	6	two	two	NUM
ejpam-7124	241	7	fixed	fix	VERB
ejpam-7124	241	8	points	point	NOUN
ejpam-7124	241	9	of	of	ADP
ejpam-7124	241	10	t	t	PROPN
ejpam-7124	241	11	.	.	PUNCT
ejpam-7124	242	1	to	to	PART
ejpam-7124	242	2	ensure	ensure	VERB
ejpam-7124	242	3	the	the	DET
ejpam-7124	242	4	uniqueness	uniqueness	NOUN
ejpam-7124	242	5	of	of	ADP
ejpam-7124	242	6	the	the	DET
ejpam-7124	242	7	fixed	fix	VERB
ejpam-7124	242	8	point	point	NOUN
ejpam-7124	242	9	in	in	ADP
ejpam-7124	242	10	theorem	theorem	NOUN
ejpam-7124	242	11	2	2	NUM
ejpam-7124	242	12	,	,	PUNCT
ejpam-7124	242	13	we	we	PRON
ejpam-7124	242	14	consider	consider	VERB
ejpam-7124	242	15	the	the	DET
ejpam-7124	242	16	following	follow	VERB
ejpam-7124	242	17	condition	condition	NOUN
ejpam-7124	242	18	:	:	PUNCT
ejpam-7124	242	19	(	(	PUNCT
ejpam-7124	242	20	n	n	CCONJ
ejpam-7124	242	21	):	):	PUNCT
ejpam-7124	242	22	for	for	ADP
ejpam-7124	242	23	all	all	DET
ejpam-7124	242	24	g	g	NOUN
ejpam-7124	242	25	,	,	PUNCT
ejpam-7124	242	26	e	e	PROPN
ejpam-7124	242	27	∈	∈	PROPN
ejpam-7124	242	28	ω	ω	PROPN
ejpam-7124	242	29	,	,	PUNCT
ejpam-7124	242	30	there	there	PRON
ejpam-7124	242	31	exist	exist	VERB
ejpam-7124	242	32	w	w	PROPN
ejpam-7124	242	33	∈	∈	PROPN
ejpam-7124	242	34	ω	ω	NOUN
ejpam-7124	242	35	such	such	ADJ
ejpam-7124	242	36	that	that	SCONJ
ejpam-7124	242	37	β(g	β(g	PROPN
ejpam-7124	242	38	,	,	PUNCT
ejpam-7124	242	39	w	w	PROPN
ejpam-7124	242	40	)	)	PUNCT
ejpam-7124	242	41	≥	≥	NOUN
ejpam-7124	242	42	1	1	NUM
ejpam-7124	242	43	and	and	CCONJ
ejpam-7124	242	44	β(e	β(e	PROPN
ejpam-7124	242	45	,	,	PUNCT
ejpam-7124	242	46	w	w	PROPN
ejpam-7124	242	47	)	)	PUNCT
ejpam-7124	242	48	≥	≥	NOUN
ejpam-7124	242	49	1	1	NUM
ejpam-7124	242	50	.	.	PUNCT
ejpam-7124	242	51	theorem	theorem	NOUN
ejpam-7124	242	52	3	3	NUM
ejpam-7124	242	53	.	.	PUNCT
ejpam-7124	242	54	adding	add	VERB
ejpam-7124	242	55	the	the	DET
ejpam-7124	242	56	hypothesis	hypothesis	NOUN
ejpam-7124	242	57	of	of	ADP
ejpam-7124	242	58	theorem	theorem	NOUN
ejpam-7124	242	59	2	2	NUM
ejpam-7124	242	60	to	to	ADP
ejpam-7124	242	61	the	the	DET
ejpam-7124	242	62	condition	condition	NOUN
ejpam-7124	242	63	(	(	PUNCT
ejpam-7124	242	64	n	n	CCONJ
ejpam-7124	242	65	)	)	PUNCT
ejpam-7124	242	66	,	,	PUNCT
ejpam-7124	242	67	we	we	PRON
ejpam-7124	242	68	yield	yield	VERB
ejpam-7124	242	69	the	the	DET
ejpam-7124	242	70	uniqueness	uniqueness	NOUN
ejpam-7124	242	71	of	of	ADP
ejpam-7124	242	72	the	the	DET
ejpam-7124	242	73	fixed	fix	VERB
ejpam-7124	242	74	point	point	NOUN
ejpam-7124	242	75	of	of	ADP
ejpam-7124	242	76	t	t	PROPN
ejpam-7124	242	77	.	.	PUNCT
ejpam-7124	243	1	proof	proof	NOUN
ejpam-7124	243	2	.	.	PUNCT
ejpam-7124	244	1	in	in	ADP
ejpam-7124	244	2	this	this	DET
ejpam-7124	244	3	theorem	theorem	NOUN
ejpam-7124	244	4	,	,	PUNCT
ejpam-7124	244	5	we	we	PRON
ejpam-7124	244	6	will	will	AUX
ejpam-7124	244	7	the	the	DET
ejpam-7124	244	8	prove	prove	NOUN
ejpam-7124	244	9	of	of	ADP
ejpam-7124	244	10	uniqueness	uniqueness	NOUN
ejpam-7124	244	11	of	of	ADP
ejpam-7124	244	12	fixed	fix	VERB
ejpam-7124	244	13	points	point	NOUN
ejpam-7124	244	14	,	,	PUNCT
ejpam-7124	244	15	i.e.	i.e.	X
ejpam-7124	244	16	t	t	X
ejpam-7124	244	17	(	(	PUNCT
ejpam-7124	244	18	g	g	NOUN
ejpam-7124	244	19	)	)	PUNCT
ejpam-7124	244	20	=	=	SYM
ejpam-7124	244	21	g	g	PROPN
ejpam-7124	244	22	,	,	PUNCT
ejpam-7124	244	23	t	t	PROPN
ejpam-7124	244	24	(	(	PUNCT
ejpam-7124	244	25	e	e	NOUN
ejpam-7124	244	26	)	)	PUNCT
ejpam-7124	244	27	=	=	SYM
ejpam-7124	244	28	e	e	NOUN
ejpam-7124	244	29	,	,	PUNCT
ejpam-7124	244	30	∀	∀	X
ejpam-7124	244	31	g	g	NOUN
ejpam-7124	244	32	,	,	PUNCT
ejpam-7124	244	33	e	e	PROPN
ejpam-7124	244	34	∈	∈	PROPN
ejpam-7124	244	35	ω	ω	PROPN
ejpam-7124	244	36	.	.	PROPN
ejpam-7124	245	1	from	from	ADP
ejpam-7124	245	2	the	the	DET
ejpam-7124	245	3	condition	condition	NOUN
ejpam-7124	245	4	(	(	PUNCT
ejpam-7124	245	5	n	n	CCONJ
ejpam-7124	245	6	)	)	PUNCT
ejpam-7124	245	7	there	there	PRON
ejpam-7124	245	8	exist	exist	VERB
ejpam-7124	245	9	w	w	PROPN
ejpam-7124	245	10	∈	∈	PROPN
ejpam-7124	245	11	ω	ω	NOUN
ejpam-7124	245	12	such	such	ADJ
ejpam-7124	245	13	that	that	SCONJ
ejpam-7124	245	14	β(g	β(g	PROPN
ejpam-7124	245	15	,	,	PUNCT
ejpam-7124	245	16	w	w	PROPN
ejpam-7124	245	17	)	)	PUNCT
ejpam-7124	245	18	≥	≥	NOUN
ejpam-7124	245	19	1	1	NUM
ejpam-7124	245	20	and	and	CCONJ
ejpam-7124	245	21	β(e	β(e	PROPN
ejpam-7124	245	22	,	,	PUNCT
ejpam-7124	245	23	w	w	PROPN
ejpam-7124	245	24	)	)	PUNCT
ejpam-7124	245	25	≥	≥	NOUN
ejpam-7124	245	26	1	1	NUM
ejpam-7124	245	27	.	.	PUNCT
ejpam-7124	245	28	using	use	VERB
ejpam-7124	245	29	the	the	DET
ejpam-7124	245	30	β	β	NOUN
ejpam-7124	245	31	-	-	ADJ
ejpam-7124	245	32	admissible	admissible	ADJ
ejpam-7124	245	33	property	property	NOUN
ejpam-7124	245	34	of	of	ADP
ejpam-7124	245	35	t−1	t−1	PROPN
ejpam-7124	245	36	,	,	PUNCT
ejpam-7124	245	37	we	we	PRON
ejpam-7124	245	38	get	get	VERB
ejpam-7124	245	39	β(g	β(g	PROPN
ejpam-7124	245	40	,	,	PUNCT
ejpam-7124	245	41	t−1w	t−1w	NOUN
ejpam-7124	245	42	)	)	PUNCT
ejpam-7124	245	43	≥	≥	NOUN
ejpam-7124	245	44	1	1	NUM
ejpam-7124	245	45	and	and	CCONJ
ejpam-7124	245	46	β(e	β(e	PROPN
ejpam-7124	245	47	,	,	PUNCT
ejpam-7124	245	48	t−1w	t−1w	NOUN
ejpam-7124	245	49	)	)	PUNCT
ejpam-7124	245	50	≥	≥	NOUN
ejpam-7124	245	51	1	1	NUM
ejpam-7124	245	52	.	.	PUNCT
ejpam-7124	245	53	therefore	therefore	ADV
ejpam-7124	245	54	by	by	ADP
ejpam-7124	245	55	repeatedly	repeatedly	ADV
ejpam-7124	245	56	using	use	VERB
ejpam-7124	245	57	β	β	NOUN
ejpam-7124	245	58	-	-	ADJ
ejpam-7124	245	59	admissible	admissible	ADJ
ejpam-7124	245	60	property	property	NOUN
ejpam-7124	245	61	of	of	ADP
ejpam-7124	245	62	t−1	t−1	PROPN
ejpam-7124	245	63	,	,	PUNCT
ejpam-7124	245	64	we	we	PRON
ejpam-7124	245	65	get	get	VERB
ejpam-7124	245	66	β(g	β(g	PROPN
ejpam-7124	245	67	,	,	PUNCT
ejpam-7124	245	68	t−jw	t−jw	PROPN
ejpam-7124	245	69	)	)	PUNCT
ejpam-7124	245	70	≥	≥	NOUN
ejpam-7124	245	71	1	1	NUM
ejpam-7124	245	72	and	and	CCONJ
ejpam-7124	245	73	β(e	β(e	PROPN
ejpam-7124	245	74	,	,	PUNCT
ejpam-7124	245	75	t−jw	t−jw	PROPN
ejpam-7124	245	76	)	)	PUNCT
ejpam-7124	245	77	≥	≥	NOUN
ejpam-7124	245	78	1	1	NUM
ejpam-7124	245	79	∀	∀	NOUN
ejpam-7124	245	80	j	j	PROPN
ejpam-7124	245	81	∈	∈	PROPN
ejpam-7124	245	82	n.	n.	NOUN
ejpam-7124	245	83	(	(	PUNCT
ejpam-7124	245	84	17	17	NUM
ejpam-7124	245	85	)	)	PUNCT
ejpam-7124	245	86	using	use	VERB
ejpam-7124	245	87	the	the	DET
ejpam-7124	245	88	inequality	inequality	NOUN
ejpam-7124	245	89	of	of	ADP
ejpam-7124	245	90	equations	equation	NOUN
ejpam-7124	245	91	(	(	PUNCT
ejpam-7124	245	92	8)	8)	NUM
ejpam-7124	245	93	and	and	CCONJ
ejpam-7124	245	94	(	(	PUNCT
ejpam-7124	245	95	17	17	NUM
ejpam-7124	245	96	)	)	PUNCT
ejpam-7124	245	97	,	,	PUNCT
ejpam-7124	245	98	we	we	PRON
ejpam-7124	245	99	get	get	VERB
ejpam-7124	245	100	(	(	PUNCT
ejpam-7124	245	101	d̄(g	d̄(g	NOUN
ejpam-7124	245	102	,	,	PUNCT
ejpam-7124	245	103	t−jw	t−jw	PROPN
ejpam-7124	245	104	)	)	PUNCT
ejpam-7124	245	105	)	)	PUNCT
ejpam-7124	246	1	≤	≤	NUM
ejpam-7124	246	2	β(g	β(g	PROPN
ejpam-7124	246	3	,	,	PUNCT
ejpam-7124	246	4	t−jw)d̄(g	t−jw)d̄(g	NOUN
ejpam-7124	246	5	,	,	PUNCT
ejpam-7124	246	6	t−jw	t−jw	PROPN
ejpam-7124	246	7	)	)	PUNCT
ejpam-7124	246	8	≤	≤	NOUN
ejpam-7124	246	9	φ(d̄(tg	φ(d̄(tg	NOUN
ejpam-7124	246	10	,	,	PUNCT
ejpam-7124	246	11	t−j+1w	t−j+1w	ADJ
ejpam-7124	246	12	)	)	PUNCT
ejpam-7124	246	13	)	)	PUNCT
ejpam-7124	246	14	=	=	SYM
ejpam-7124	246	15	φ(d̄(g	φ(d̄(g	NOUN
ejpam-7124	246	16	,	,	PUNCT
ejpam-7124	246	17	t−j+1w	t−j+1w	PROPN
ejpam-7124	246	18	)	)	PUNCT
ejpam-7124	246	19	)	)	PUNCT
ejpam-7124	246	20	≤	≤	NUM
ejpam-7124	246	21	φ(d̄(g	φ(d̄(g	NOUN
ejpam-7124	246	22	,	,	PUNCT
ejpam-7124	246	23	t−j+1w	t−j+1w	PROPN
ejpam-7124	246	24	)	)	PUNCT
ejpam-7124	246	25	)	)	PUNCT
ejpam-7124	246	26	.	.	PUNCT
ejpam-7124	247	1	by	by	ADP
ejpam-7124	247	2	the	the	DET
ejpam-7124	247	3	repetition	repetition	NOUN
ejpam-7124	247	4	of	of	ADP
ejpam-7124	247	5	above	above	ADP
ejpam-7124	247	6	inequality	inequality	NOUN
ejpam-7124	247	7	,	,	PUNCT
ejpam-7124	247	8	we	we	PRON
ejpam-7124	247	9	get	get	AUX
ejpam-7124	247	10	(	(	PUNCT
ejpam-7124	247	11	d̄(g	d̄(g	NOUN
ejpam-7124	247	12	,	,	PUNCT
ejpam-7124	247	13	t−jw	t−jw	PROPN
ejpam-7124	247	14	)	)	PUNCT
ejpam-7124	247	15	)	)	PUNCT
ejpam-7124	248	1	≤	≤	NUM
ejpam-7124	248	2	φ(d̄(g	φ(d̄(g	NOUN
ejpam-7124	248	3	,	,	PUNCT
ejpam-7124	248	4	w	w	NOUN
ejpam-7124	248	5	)	)	PUNCT
ejpam-7124	248	6	)	)	PUNCT
ejpam-7124	248	7	∀	∀	PUNCT
ejpam-7124	249	1	j	j	PROPN
ejpam-7124	249	2	∈	∈	PROPN
ejpam-7124	249	3	n.	n.	NOUN
ejpam-7124	249	4	thus	thus	ADV
ejpam-7124	249	5	we	we	PRON
ejpam-7124	249	6	have	have	VERB
ejpam-7124	249	7	,	,	PUNCT
ejpam-7124	249	8	t−jw	t−jw	PROPN
ejpam-7124	249	9	→	→	SYM
ejpam-7124	249	10	g	g	PROPN
ejpam-7124	249	11	as	as	ADP
ejpam-7124	249	12	j	j	PROPN
ejpam-7124	249	13	→	→	SYM
ejpam-7124	249	14	∞.	∞.	PROPN
ejpam-7124	249	15	m.	m.	NOUN
ejpam-7124	249	16	kumar	kumar	PROPN
ejpam-7124	249	17	et	et	PROPN
ejpam-7124	249	18	al	al	PROPN
ejpam-7124	249	19	.	.	PUNCT
ejpam-7124	249	20	/	/	SYM
ejpam-7124	249	21	eur	eur	PROPN
ejpam-7124	249	22	.	.	PUNCT
ejpam-7124	250	1	j.	j.	PROPN
ejpam-7124	250	2	pure	pure	PROPN
ejpam-7124	250	3	appl	appl	PROPN
ejpam-7124	250	4	.	.	PROPN
ejpam-7124	250	5	math	math	PROPN
ejpam-7124	250	6	,	,	PUNCT
ejpam-7124	250	7	18	18	NUM
ejpam-7124	250	8	(	(	PUNCT
ejpam-7124	250	9	4	4	NUM
ejpam-7124	250	10	)	)	PUNCT
ejpam-7124	250	11	(	(	PUNCT
ejpam-7124	250	12	2025	2025	NUM
ejpam-7124	250	13	)	)	PUNCT
ejpam-7124	250	14	,	,	PUNCT
ejpam-7124	250	15	7124	7124	NUM
ejpam-7124	250	16	12	12	NUM
ejpam-7124	250	17	of	of	ADP
ejpam-7124	250	18	14	14	NUM
ejpam-7124	250	19	similarly	similarly	ADV
ejpam-7124	250	20	we	we	PRON
ejpam-7124	250	21	may	may	AUX
ejpam-7124	250	22	obtain	obtain	VERB
ejpam-7124	250	23	,	,	PUNCT
ejpam-7124	250	24	t−jw	t−jw	PROPN
ejpam-7124	250	25	→	→	SYM
ejpam-7124	250	26	e	e	NOUN
ejpam-7124	250	27	as	as	ADP
ejpam-7124	250	28	j	j	PROPN
ejpam-7124	250	29	→	→	SYM
ejpam-7124	250	30	∞.	∞.	PROPN
ejpam-7124	250	31	from	from	ADP
ejpam-7124	250	32	the	the	DET
ejpam-7124	250	33	uniqueness	uniqueness	NOUN
ejpam-7124	250	34	of	of	ADP
ejpam-7124	250	35	the	the	DET
ejpam-7124	250	36	limit	limit	NOUN
ejpam-7124	250	37	of	of	ADP
ejpam-7124	250	38	t−jw	t−jw	PROPN
ejpam-7124	250	39	,	,	PUNCT
ejpam-7124	250	40	we	we	PRON
ejpam-7124	250	41	get	get	VERB
ejpam-7124	250	42	g	g	NOUN
ejpam-7124	250	43	=	=	PUNCT
ejpam-7124	250	44	e.	e.	PROPN
ejpam-7124	250	45	hence	hence	ADV
ejpam-7124	250	46	the	the	DET
ejpam-7124	250	47	proof	proof	NOUN
ejpam-7124	250	48	.	.	PUNCT
ejpam-7124	251	1	4	4	X
ejpam-7124	251	2	.	.	X
ejpam-7124	251	3	consequences	consequence	NOUN
ejpam-7124	251	4	now	now	ADV
ejpam-7124	251	5	we	we	PRON
ejpam-7124	251	6	present	present	VERB
ejpam-7124	251	7	the	the	DET
ejpam-7124	251	8	following	following	NOUN
ejpam-7124	251	9	as	as	ADP
ejpam-7124	251	10	an	an	DET
ejpam-7124	251	11	immediate	immediate	ADJ
ejpam-7124	251	12	consequence	consequence	NOUN
ejpam-7124	251	13	of	of	ADP
ejpam-7124	251	14	theorem	theorem	NOUN
ejpam-7124	251	15	1	1	NUM
ejpam-7124	251	16	.	.	PUNCT
ejpam-7124	251	17	corollary	corollary	ADJ
ejpam-7124	251	18	1	1	NUM
ejpam-7124	251	19	.	.	PUNCT
ejpam-7124	252	1	let	let	AUX
ejpam-7124	252	2	(	(	PUNCT
ejpam-7124	252	3	ω	ω	NOUN
ejpam-7124	252	4	,	,	PUNCT
ejpam-7124	252	5	d̄	d̄	PROPN
ejpam-7124	252	6	)	)	PUNCT
ejpam-7124	252	7	be	be	VERB
ejpam-7124	252	8	a	a	DET
ejpam-7124	252	9	complete	complete	ADJ
ejpam-7124	252	10	controlled	control	VERB
ejpam-7124	252	11	metric	metric	ADJ
ejpam-7124	252	12	type	type	NOUN
ejpam-7124	252	13	spaces	space	NOUN
ejpam-7124	252	14	and	and	CCONJ
ejpam-7124	252	15	t	t	PROPN
ejpam-7124	252	16	:	:	PUNCT
ejpam-7124	252	17	ω	ω	PROPN
ejpam-7124	252	18	→	→	SYM
ejpam-7124	252	19	ω	ω	X
ejpam-7124	252	20	be	be	AUX
ejpam-7124	252	21	a	a	DET
ejpam-7124	252	22	bijective	bijective	NOUN
ejpam-7124	252	23	and	and	CCONJ
ejpam-7124	252	24	β	β	X
ejpam-7124	252	25	:	:	PUNCT
ejpam-7124	253	1	ω×	ω×	PROPN
ejpam-7124	253	2	ω	ω	PROPN
ejpam-7124	253	3	→	→	SYM
ejpam-7124	253	4	[	[	X
ejpam-7124	253	5	0,∞	0,∞	NOUN
ejpam-7124	253	6	)	)	PUNCT
ejpam-7124	253	7	mappings	mapping	NOUN
ejpam-7124	253	8	satisfying	satisfy	VERB
ejpam-7124	253	9	the	the	DET
ejpam-7124	253	10	following	follow	VERB
ejpam-7124	253	11	conditions	condition	NOUN
ejpam-7124	253	12	:	:	PUNCT
ejpam-7124	254	1	[	[	X
ejpam-7124	254	2	label=(i),itemsep=-.16em	label=(i),itemsep=-.16em	NOUN
ejpam-7124	254	3	,	,	PUNCT
ejpam-7124	254	4	topsep=2pt	topsep=2pt	PROPN
ejpam-7124	254	5	]	]	PUNCT
ejpam-7124	254	6	(	(	PUNCT
ejpam-7124	254	7	i	i	NOUN
ejpam-7124	254	8	)	)	PUNCT
ejpam-7124	254	9	t	t	PROPN
ejpam-7124	254	10	is	be	AUX
ejpam-7124	254	11	continuous	continuous	ADJ
ejpam-7124	254	12	.	.	PUNCT
ejpam-7124	255	1	(	(	PUNCT
ejpam-7124	255	2	ii	ii	NOUN
ejpam-7124	255	3	)	)	PUNCT
ejpam-7124	255	4	∃	∃	PROPN
ejpam-7124	255	5	φ	φ	PROPN
ejpam-7124	255	6	∈	∈	PROPN
ejpam-7124	255	7	φ	φ	PROPN
ejpam-7124	255	8	such	such	ADJ
ejpam-7124	255	9	that	that	SCONJ
ejpam-7124	255	10	φ(d̄(tu	φ(d̄(tu	NOUN
ejpam-7124	255	11	,	,	PUNCT
ejpam-7124	255	12	te	te	PROPN
ejpam-7124	255	13	)	)	PUNCT
ejpam-7124	255	14	)	)	PUNCT
ejpam-7124	255	15	≥	≥	NOUN
ejpam-7124	255	16	β(u	β(u	PROPN
ejpam-7124	255	17	,	,	PUNCT
ejpam-7124	255	18	e)d̄(u	e)d̄(u	PROPN
ejpam-7124	255	19	,	,	PUNCT
ejpam-7124	255	20	e	e	NOUN
ejpam-7124	255	21	)	)	PUNCT
ejpam-7124	255	22	∀	∀	X
ejpam-7124	255	23	u	u	NOUN
ejpam-7124	255	24	,	,	PUNCT
ejpam-7124	255	25	e	e	PROPN
ejpam-7124	255	26	∈	∈	PROPN
ejpam-7124	255	27	ω	ω	PROPN
ejpam-7124	255	28	.	.	PUNCT
ejpam-7124	256	1	then	then	ADV
ejpam-7124	256	2	t	t	PROPN
ejpam-7124	256	3	has	have	VERB
ejpam-7124	256	4	a	a	DET
ejpam-7124	256	5	unique	unique	ADJ
ejpam-7124	256	6	fixed	fix	VERB
ejpam-7124	256	7	point	point	NOUN
ejpam-7124	256	8	.	.	PUNCT
ejpam-7124	257	1	proof	proof	NOUN
ejpam-7124	257	2	.	.	PUNCT
ejpam-7124	258	1	define	define	VERB
ejpam-7124	258	2	the	the	DET
ejpam-7124	258	3	function	function	NOUN
ejpam-7124	258	4	β	β	NOUN
ejpam-7124	258	5	:	:	PUNCT
ejpam-7124	258	6	ω×	ω×	PROPN
ejpam-7124	258	7	ω	ω	NUM
ejpam-7124	258	8	→	→	SYM
ejpam-7124	258	9	[	[	X
ejpam-7124	258	10	0,∞	0,∞	NOUN
ejpam-7124	258	11	)	)	PUNCT
ejpam-7124	258	12	by	by	ADP
ejpam-7124	258	13	β(u	β(u	PROPN
ejpam-7124	258	14	,	,	PUNCT
ejpam-7124	258	15	e	e	NOUN
ejpam-7124	258	16	)	)	PUNCT
ejpam-7124	258	17	=	=	SYM
ejpam-7124	258	18	1	1	X
ejpam-7124	258	19	.	.	X
ejpam-7124	258	20	note	note	VERB
ejpam-7124	258	21	that	that	SCONJ
ejpam-7124	258	22	t−1	t−1	PROPN
ejpam-7124	258	23	is	be	AUX
ejpam-7124	258	24	β	β	NOUN
ejpam-7124	258	25	-	-	ADJ
ejpam-7124	258	26	admissible	admissible	ADJ
ejpam-7124	258	27	.	.	PUNCT
ejpam-7124	259	1	moreover	moreover	ADV
ejpam-7124	259	2	,	,	PUNCT
ejpam-7124	259	3	t	t	PROPN
ejpam-7124	259	4	satisfies	satisfy	VERB
ejpam-7124	259	5	all	all	DET
ejpam-7124	259	6	the	the	DET
ejpam-7124	259	7	conditions	condition	NOUN
ejpam-7124	259	8	of	of	ADP
ejpam-7124	259	9	theorem	theorem	NOUN
ejpam-7124	259	10	1	1	NUM
ejpam-7124	259	11	,	,	PUNCT
ejpam-7124	259	12	so	so	SCONJ
ejpam-7124	259	13	t	t	PROPN
ejpam-7124	259	14	has	have	VERB
ejpam-7124	259	15	a	a	DET
ejpam-7124	259	16	unique	unique	ADJ
ejpam-7124	259	17	fixed	fix	VERB
ejpam-7124	259	18	point	point	NOUN
ejpam-7124	259	19	.	.	PUNCT
ejpam-7124	260	1	5	5	X
ejpam-7124	260	2	.	.	X
ejpam-7124	260	3	application	application	NOUN
ejpam-7124	260	4	consider	consider	VERB
ejpam-7124	260	5	the	the	DET
ejpam-7124	260	6	non	non	ADJ
ejpam-7124	260	7	-	-	ADJ
ejpam-7124	260	8	linear	linear	ADJ
ejpam-7124	260	9	volterra	volterra	NOUN
ejpam-7124	260	10	integral	integral	ADJ
ejpam-7124	260	11	equation	equation	NOUN
ejpam-7124	260	12	u(q	u(q	ADV
ejpam-7124	260	13	)	)	PUNCT
ejpam-7124	261	1	=	=	PUNCT
ejpam-7124	261	2	∫	∫	PROPN
ejpam-7124	262	1	q	q	NOUN
ejpam-7124	262	2	0	0	NUM
ejpam-7124	262	3	u(l	u(l	ADJ
ejpam-7124	262	4	)	)	PUNCT
ejpam-7124	262	5	1	1	NUM
ejpam-7124	263	1	+	+	CCONJ
ejpam-7124	263	2	|u(l)|	|u(l)|	PROPN
ejpam-7124	263	3	dl	dl	PROPN
ejpam-7124	263	4	,	,	PUNCT
ejpam-7124	263	5	q	q	PROPN
ejpam-7124	263	6	∈	∈	PROPN
ejpam-7124	264	1	[	[	X
ejpam-7124	264	2	0	0	NUM
ejpam-7124	264	3	,	,	PUNCT
ejpam-7124	264	4	2	2	NUM
ejpam-7124	264	5	]	]	PUNCT
ejpam-7124	264	6	.	.	PUNCT
ejpam-7124	265	1	let	let	VERB
ejpam-7124	265	2	ω	ω	NOUN
ejpam-7124	265	3	=	=	SYM
ejpam-7124	265	4	c([0	c([0	PROPN
ejpam-7124	265	5	,	,	PUNCT
ejpam-7124	265	6	2],r	2],r	NUM
ejpam-7124	265	7	)	)	PUNCT
ejpam-7124	265	8	space	space	NOUN
ejpam-7124	265	9	of	of	ADP
ejpam-7124	265	10	continuous	continuous	ADJ
ejpam-7124	265	11	real	real	ADV
ejpam-7124	265	12	-	-	PUNCT
ejpam-7124	265	13	valued	value	VERB
ejpam-7124	265	14	functions	function	NOUN
ejpam-7124	265	15	on	on	ADP
ejpam-7124	265	16	[	[	X
ejpam-7124	265	17	0	0	NUM
ejpam-7124	265	18	,	,	PUNCT
ejpam-7124	265	19	2	2	NUM
ejpam-7124	265	20	]	]	PUNCT
ejpam-7124	265	21	and	and	CCONJ
ejpam-7124	265	22	d̄(u	d̄(u	NOUN
ejpam-7124	265	23	,	,	PUNCT
ejpam-7124	265	24	e	e	NOUN
ejpam-7124	265	25	)	)	PUNCT
ejpam-7124	265	26	=	=	SYM
ejpam-7124	265	27	sup	sup	NOUN
ejpam-7124	265	28	|u(q	|u(q	PROPN
ejpam-7124	265	29	)	)	PUNCT
ejpam-7124	265	30	,	,	PUNCT
ejpam-7124	265	31	e(q)|	e(q)|	PROPN
ejpam-7124	265	32	,	,	PUNCT
ejpam-7124	265	33	∀	∀	VERB
ejpam-7124	265	34	q	q	NOUN
ejpam-7124	265	35	∈	∈	PROPN
ejpam-7124	266	1	[	[	X
ejpam-7124	266	2	0	0	NUM
ejpam-7124	266	3	,	,	PUNCT
ejpam-7124	266	4	2	2	NUM
ejpam-7124	266	5	]	]	PUNCT
ejpam-7124	266	6	.	.	PUNCT
ejpam-7124	267	1	define	define	VERB
ejpam-7124	267	2	t	t	PROPN
ejpam-7124	267	3	:	:	PUNCT
ejpam-7124	267	4	ω	ω	PROPN
ejpam-7124	267	5	→	→	SYM
ejpam-7124	267	6	ω	ω	X
ejpam-7124	267	7	by	by	ADP
ejpam-7124	267	8	(	(	PUNCT
ejpam-7124	267	9	tu)q	tu)q	NOUN
ejpam-7124	267	10	=	=	SYM
ejpam-7124	267	11	∫	∫	PROPN
ejpam-7124	267	12	q	q	NOUN
ejpam-7124	267	13	0	0	NUM
ejpam-7124	267	14	u(l	u(l	ADJ
ejpam-7124	267	15	)	)	PUNCT
ejpam-7124	267	16	1	1	NUM
ejpam-7124	268	1	+	+	CCONJ
ejpam-7124	268	2	|u(l)|	|u(l)|	PROPN
ejpam-7124	268	3	dl	dl	PROPN
ejpam-7124	268	4	,	,	PUNCT
ejpam-7124	268	5	q	q	PROPN
ejpam-7124	268	6	∈	∈	PROPN
ejpam-7124	269	1	[	[	X
ejpam-7124	269	2	0	0	NUM
ejpam-7124	269	3	,	,	PUNCT
ejpam-7124	269	4	2	2	NUM
ejpam-7124	269	5	]	]	PUNCT
ejpam-7124	269	6	.	.	PUNCT
ejpam-7124	270	1	define	define	VERB
ejpam-7124	270	2	β	β	NOUN
ejpam-7124	270	3	:	:	PUNCT
ejpam-7124	270	4	ω×	ω×	PROPN
ejpam-7124	270	5	ω	ω	NUM
ejpam-7124	270	6	→	→	SYM
ejpam-7124	270	7	[	[	X
ejpam-7124	270	8	0,∞	0,∞	NOUN
ejpam-7124	270	9	)	)	PUNCT
ejpam-7124	270	10	by	by	ADP
ejpam-7124	270	11	β(u	β(u	PROPN
ejpam-7124	270	12	,	,	PUNCT
ejpam-7124	270	13	e	e	NOUN
ejpam-7124	270	14	)	)	PUNCT
ejpam-7124	270	15	=	=	SYM
ejpam-7124	270	16	1	1	NUM
ejpam-7124	270	17	4	4	NUM
ejpam-7124	270	18	and	and	CCONJ
ejpam-7124	270	19	φ	φ	NUM
ejpam-7124	270	20	:	:	PUNCT
ejpam-7124	271	1	[	[	X
ejpam-7124	271	2	0,∞	0,∞	NUM
ejpam-7124	271	3	)	)	PUNCT
ejpam-7124	271	4	→	→	PUNCT
ejpam-7124	272	1	[	[	X
ejpam-7124	272	2	0,∞	0,∞	NOUN
ejpam-7124	272	3	)	)	PUNCT
ejpam-7124	272	4	by	by	ADP
ejpam-7124	272	5	φ(q	φ(q	NOUN
ejpam-7124	272	6	)	)	PUNCT
ejpam-7124	272	7	=	=	PUNCT
ejpam-7124	273	1	q	q	PROPN
ejpam-7124	273	2	4	4	X
ejpam-7124	273	3	.	.	PUNCT
ejpam-7124	274	1	define	define	VERB
ejpam-7124	274	2	a	a	DET
ejpam-7124	274	3	symmetric	symmetric	ADJ
ejpam-7124	274	4	function	function	NOUN
ejpam-7124	274	5	µ	µ	NOUN
ejpam-7124	274	6	:	:	PUNCT
ejpam-7124	274	7	ω×	ω×	PROPN
ejpam-7124	274	8	ω	ω	PROPN
ejpam-7124	274	9	→	→	SYM
ejpam-7124	274	10	[	[	X
ejpam-7124	274	11	1,∞	1,∞	NUM
ejpam-7124	274	12	)	)	PUNCT
ejpam-7124	274	13	by	by	ADP
ejpam-7124	274	14	µ(u	µ(u	NOUN
ejpam-7124	274	15	,	,	PUNCT
ejpam-7124	274	16	e	e	NOUN
ejpam-7124	274	17	)	)	PUNCT
ejpam-7124	274	18	=	=	SYM
ejpam-7124	274	19	2	2	NUM
ejpam-7124	274	20	∀	∀	NOUN
ejpam-7124	274	21	u	u	NOUN
ejpam-7124	274	22	,	,	PUNCT
ejpam-7124	274	23	e	e	PROPN
ejpam-7124	274	24	∈	∈	PROPN
ejpam-7124	275	1	[	[	X
ejpam-7124	275	2	0	0	NUM
ejpam-7124	275	3	,	,	PUNCT
ejpam-7124	275	4	2	2	NUM
ejpam-7124	275	5	]	]	PUNCT
ejpam-7124	275	6	.	.	PUNCT
ejpam-7124	276	1	clearly	clearly	ADV
ejpam-7124	276	2	φ(q	φ(q	NUM
ejpam-7124	276	3	)	)	PUNCT
ejpam-7124	276	4	is	be	AUX
ejpam-7124	276	5	an	an	DET
ejpam-7124	276	6	increasing	increase	VERB
ejpam-7124	276	7	and	and	CCONJ
ejpam-7124	276	8	continuous	continuous	ADJ
ejpam-7124	276	9	function	function	NOUN
ejpam-7124	276	10	of	of	ADP
ejpam-7124	276	11	q	q	NOUN
ejpam-7124	276	12	and	and	CCONJ
ejpam-7124	276	13	φ(0	φ(0	ADJ
ejpam-7124	276	14	)	)	PUNCT
ejpam-7124	276	15	=	=	SYM
ejpam-7124	276	16	0	0	NUM
ejpam-7124	276	17	and	and	CCONJ
ejpam-7124	276	18	φ(q	φ(q	NUM
ejpam-7124	276	19	)	)	PUNCT
ejpam-7124	276	20	<	<	X
ejpam-7124	276	21	q	q	X
ejpam-7124	276	22	∀	∀	X
ejpam-7124	276	23	q	q	X
ejpam-7124	276	24	>	>	X
ejpam-7124	276	25	0	0	X
ejpam-7124	276	26	.	.	PUNCT
ejpam-7124	276	27	m.	m.	PROPN
ejpam-7124	276	28	kumar	kumar	PROPN
ejpam-7124	276	29	et	et	PROPN
ejpam-7124	276	30	al	al	PROPN
ejpam-7124	276	31	.	.	PUNCT
ejpam-7124	276	32	/	/	SYM
ejpam-7124	276	33	eur	eur	PROPN
ejpam-7124	276	34	.	.	PUNCT
ejpam-7124	277	1	j.	j.	PROPN
ejpam-7124	277	2	pure	pure	PROPN
ejpam-7124	277	3	appl	appl	PROPN
ejpam-7124	277	4	.	.	PROPN
ejpam-7124	277	5	math	math	PROPN
ejpam-7124	277	6	,	,	PUNCT
ejpam-7124	277	7	18	18	NUM
ejpam-7124	277	8	(	(	PUNCT
ejpam-7124	277	9	4	4	NUM
ejpam-7124	277	10	)	)	PUNCT
ejpam-7124	277	11	(	(	PUNCT
ejpam-7124	277	12	2025	2025	NUM
ejpam-7124	277	13	)	)	PUNCT
ejpam-7124	277	14	,	,	PUNCT
ejpam-7124	277	15	7124	7124	NUM
ejpam-7124	277	16	13	13	NUM
ejpam-7124	277	17	of	of	ADP
ejpam-7124	277	18	14	14	NUM
ejpam-7124	277	19	now	now	ADV
ejpam-7124	277	20	d̄(u	d̄(u	NOUN
ejpam-7124	277	21	,	,	PUNCT
ejpam-7124	277	22	e	e	NOUN
ejpam-7124	277	23	)	)	PUNCT
ejpam-7124	277	24	=	=	SYM
ejpam-7124	277	25	sup	sup	NUM
ejpam-7124	277	26	|u(q)−	|u(q)−	NUM
ejpam-7124	277	27	e(q)|	e(q)|	X
ejpam-7124	277	28	∀	∀	PUNCT
ejpam-7124	278	1	q	q	X
ejpam-7124	278	2	∈	∈	PROPN
ejpam-7124	279	1	[	[	X
ejpam-7124	279	2	0	0	NUM
ejpam-7124	279	3	,	,	PUNCT
ejpam-7124	279	4	2	2	NUM
ejpam-7124	279	5	]	]	PUNCT
ejpam-7124	279	6	=	=	PUNCT
ejpam-7124	279	7	sup	sup	NOUN
ejpam-7124	279	8	∣∣∣∣(∫	∣∣∣∣(∫	NOUN
ejpam-7124	279	9	q	q	NOUN
ejpam-7124	279	10	0	0	NUM
ejpam-7124	279	11	u(l	u(l	ADJ
ejpam-7124	279	12	)	)	PUNCT
ejpam-7124	279	13	1	1	NUM
ejpam-7124	280	1	+	+	NUM
ejpam-7124	280	2	|u(l)|	|u(l)|	NOUN
ejpam-7124	280	3	dl	dl	PROPN
ejpam-7124	280	4	∫	∫	PROPN
ejpam-7124	280	5	q	q	PROPN
ejpam-7124	280	6	0	0	PUNCT
ejpam-7124	280	7	e(l	e(l	PROPN
ejpam-7124	280	8	)	)	PUNCT
ejpam-7124	280	9	1	1	NUM
ejpam-7124	281	1	+	+	NUM
ejpam-7124	281	2	|e(l)|	|e(l)|	PROPN
ejpam-7124	281	3	dl	dl	PROPN
ejpam-7124	281	4	)	)	PUNCT
ejpam-7124	281	5	∣∣∣∣	∣∣∣∣	PROPN
ejpam-7124	281	6	=	=	SYM
ejpam-7124	281	7	sup	sup	NOUN
ejpam-7124	281	8	|(tu)(q)(te)(q)|	|(tu)(q)(te)(q)|	SYM
ejpam-7124	281	9	≤	≤	PROPN
ejpam-7124	281	10	|(tu)(q)(te)(q)|	|(tu)(q)(te)(q)|	PUNCT
ejpam-7124	281	11	=	=	SYM
ejpam-7124	281	12	d̄(tu	d̄(tu	NOUN
ejpam-7124	281	13	,	,	PUNCT
ejpam-7124	281	14	te	te	PROPN
ejpam-7124	281	15	)	)	PUNCT
ejpam-7124	281	16	⇒	⇒	NOUN
ejpam-7124	281	17	d̄(u	d̄(u	PROPN
ejpam-7124	281	18	,	,	PUNCT
ejpam-7124	281	19	e	e	NOUN
ejpam-7124	281	20	)	)	PUNCT
ejpam-7124	281	21	≤	≤	NOUN
ejpam-7124	281	22	d̄(tu	d̄(tu	NOUN
ejpam-7124	281	23	,	,	PUNCT
ejpam-7124	281	24	te	te	PROPN
ejpam-7124	281	25	)	)	PUNCT
ejpam-7124	281	26	.	.	PUNCT
ejpam-7124	282	1	now	now	ADV
ejpam-7124	282	2	φ(d̄(tu	φ(d̄(tu	NUM
ejpam-7124	282	3	,	,	PUNCT
ejpam-7124	282	4	te	te	ADJ
ejpam-7124	282	5	)	)	PUNCT
ejpam-7124	282	6	)	)	PUNCT
ejpam-7124	283	1	=	=	SYM
ejpam-7124	283	2	d̄(tu	d̄(tu	PROPN
ejpam-7124	283	3	,	,	PUNCT
ejpam-7124	283	4	te	te	PROPN
ejpam-7124	283	5	)	)	PUNCT
ejpam-7124	283	6	4	4	NUM
ejpam-7124	283	7	≥	≥	NOUN
ejpam-7124	283	8	d̄(u	d̄(u	NOUN
ejpam-7124	283	9	,	,	PUNCT
ejpam-7124	283	10	e	e	NOUN
ejpam-7124	283	11	)	)	PUNCT
ejpam-7124	283	12	4	4	NUM
ejpam-7124	283	13	=	=	SYM
ejpam-7124	283	14	β(u	β(u	PROPN
ejpam-7124	283	15	,	,	PUNCT
ejpam-7124	283	16	e	e	NOUN
ejpam-7124	283	17	)	)	PUNCT
ejpam-7124	283	18	d̄(u	d̄(u	NOUN
ejpam-7124	283	19	,	,	PUNCT
ejpam-7124	283	20	e	e	NOUN
ejpam-7124	283	21	)	)	PUNCT
ejpam-7124	283	22	⇒	⇒	PROPN
ejpam-7124	283	23	φ(d̄(tu	φ(d̄(tu	NUM
ejpam-7124	283	24	,	,	PUNCT
ejpam-7124	283	25	te	te	PROPN
ejpam-7124	283	26	)	)	PUNCT
ejpam-7124	283	27	)	)	PUNCT
ejpam-7124	283	28	≥	≥	NOUN
ejpam-7124	283	29	β(u	β(u	PROPN
ejpam-7124	283	30	,	,	PUNCT
ejpam-7124	283	31	e)d̄(u	e)d̄(u	PROPN
ejpam-7124	283	32	,	,	PUNCT
ejpam-7124	283	33	e	e	NOUN
ejpam-7124	283	34	)	)	PUNCT
ejpam-7124	283	35	.	.	PUNCT
ejpam-7124	284	1	thus	thus	ADV
ejpam-7124	284	2	t	t	X
ejpam-7124	284	3	:	:	PUNCT
ejpam-7124	284	4	ω	ω	PROPN
ejpam-7124	284	5	→	→	SYM
ejpam-7124	284	6	ω	ω	PROPN
ejpam-7124	284	7	is	be	AUX
ejpam-7124	284	8	(	(	PUNCT
ejpam-7124	284	9	β	β	X
ejpam-7124	284	10	,	,	PUNCT
ejpam-7124	284	11	φ)-expansive	φ)-expansive	PUNCT
ejpam-7124	284	12	mapping	mapping	NOUN
ejpam-7124	284	13	.	.	PUNCT
ejpam-7124	285	1	thus	thus	ADV
ejpam-7124	285	2	t	t	PROPN
ejpam-7124	285	3	has	have	AUX
ejpam-7124	285	4	atleast	atleast	VERB
ejpam-7124	285	5	one	one	NUM
ejpam-7124	285	6	fixed	fix	VERB
ejpam-7124	285	7	point	point	NOUN
ejpam-7124	285	8	.	.	PUNCT
ejpam-7124	286	1	clearly	clearly	ADV
ejpam-7124	286	2	t	t	X
ejpam-7124	286	3	(	(	PUNCT
ejpam-7124	286	4	0	0	NUM
ejpam-7124	286	5	)	)	PUNCT
ejpam-7124	286	6	=	=	SYM
ejpam-7124	286	7	0	0	PUNCT
ejpam-7124	287	1	and	and	CCONJ
ejpam-7124	287	2	hence	hence	ADV
ejpam-7124	287	3	0	0	NUM
ejpam-7124	287	4	is	be	AUX
ejpam-7124	287	5	the	the	DET
ejpam-7124	287	6	unique	unique	ADJ
ejpam-7124	287	7	fixed	fix	VERB
ejpam-7124	287	8	point	point	NOUN
ejpam-7124	287	9	of	of	ADP
ejpam-7124	287	10	t	t	PROPN
ejpam-7124	287	11	.	.	PUNCT
ejpam-7124	288	1	6	6	X
ejpam-7124	288	2	.	.	X
ejpam-7124	288	3	conclusion	conclusion	NOUN
ejpam-7124	288	4	fixed	fix	VERB
ejpam-7124	288	5	point	point	NOUN
ejpam-7124	288	6	results	result	NOUN
ejpam-7124	288	7	in	in	ADP
ejpam-7124	288	8	the	the	DET
ejpam-7124	288	9	setting	setting	NOUN
ejpam-7124	288	10	of	of	ADP
ejpam-7124	288	11	controlled	control	VERB
ejpam-7124	288	12	metric	metric	ADJ
ejpam-7124	288	13	space	space	NOUN
ejpam-7124	288	14	using	use	VERB
ejpam-7124	288	15	(	(	PUNCT
ejpam-7124	288	16	β	β	X
ejpam-7124	288	17	,	,	PUNCT
ejpam-7124	288	18	φ)-expansive	φ)-expansive	PUNCT
ejpam-7124	288	19	mappings	mapping	NOUN
ejpam-7124	288	20	have	have	AUX
ejpam-7124	288	21	been	be	AUX
ejpam-7124	288	22	established	establish	VERB
ejpam-7124	288	23	in	in	ADP
ejpam-7124	288	24	this	this	DET
ejpam-7124	288	25	manuscript	manuscript	NOUN
ejpam-7124	288	26	.	.	PUNCT
ejpam-7124	289	1	the	the	DET
ejpam-7124	289	2	derived	derive	VERB
ejpam-7124	289	3	results	result	NOUN
ejpam-7124	289	4	have	have	AUX
ejpam-7124	289	5	been	be	AUX
ejpam-7124	289	6	supplemented	supplement	VERB
ejpam-7124	289	7	with	with	ADP
ejpam-7124	289	8	suitable	suitable	ADJ
ejpam-7124	289	9	non	non	ADJ
ejpam-7124	289	10	trivial	trivial	ADJ
ejpam-7124	289	11	example	example	NOUN
ejpam-7124	289	12	and	and	CCONJ
ejpam-7124	289	13	the	the	DET
ejpam-7124	289	14	result	result	NOUN
ejpam-7124	289	15	is	be	AUX
ejpam-7124	289	16	applied	apply	VERB
ejpam-7124	289	17	to	to	PART
ejpam-7124	289	18	find	find	VERB
ejpam-7124	289	19	analytical	analytical	ADJ
ejpam-7124	289	20	solution	solution	NOUN
ejpam-7124	289	21	of	of	ADP
ejpam-7124	289	22	integral	integral	ADJ
ejpam-7124	289	23	equation	equation	NOUN
ejpam-7124	289	24	.	.	PUNCT
ejpam-7124	290	1	it	it	PRON
ejpam-7124	290	2	will	will	AUX
ejpam-7124	290	3	be	be	AUX
ejpam-7124	290	4	open	open	ADJ
ejpam-7124	290	5	problem	problem	NOUN
ejpam-7124	290	6	to	to	PART
ejpam-7124	290	7	extend	extend	VERB
ejpam-7124	290	8	the	the	DET
ejpam-7124	290	9	result	result	NOUN
ejpam-7124	290	10	in	in	ADP
ejpam-7124	290	11	the	the	DET
ejpam-7124	290	12	generalised	generalise	VERB
ejpam-7124	290	13	forms	form	NOUN
ejpam-7124	290	14	of	of	ADP
ejpam-7124	290	15	controlled	control	VERB
ejpam-7124	290	16	metric	metric	ADJ
ejpam-7124	290	17	and	and	CCONJ
ejpam-7124	290	18	metric	metric	ADJ
ejpam-7124	290	19	like	like	ADP
ejpam-7124	290	20	spaces	space	NOUN
ejpam-7124	290	21	and	and	CCONJ
ejpam-7124	290	22	find	find	VERB
ejpam-7124	290	23	application	application	NOUN
ejpam-7124	290	24	to	to	ADP
ejpam-7124	290	25	integro	integro	ADJ
ejpam-7124	290	26	-	-	PUNCT
ejpam-7124	290	27	differential	differential	NOUN
ejpam-7124	290	28	equations	equation	NOUN
ejpam-7124	290	29	.	.	PUNCT
ejpam-7124	291	1	acknowledgements	acknowledgement	NOUN
ejpam-7124	291	2	”	"	PUNCT
ejpam-7124	291	3	the	the	DET
ejpam-7124	291	4	authors	author	NOUN
ejpam-7124	291	5	extend	extend	VERB
ejpam-7124	291	6	their	their	PRON
ejpam-7124	291	7	appreciation	appreciation	NOUN
ejpam-7124	291	8	to	to	ADP
ejpam-7124	291	9	prince	prince	PROPN
ejpam-7124	291	10	sattam	sattam	PROPN
ejpam-7124	291	11	bin	bin	PROPN
ejpam-7124	291	12	abdulaziz	abdulaziz	PROPN
ejpam-7124	291	13	university	university	PROPN
ejpam-7124	291	14	for	for	ADP
ejpam-7124	291	15	funding	fund	VERB
ejpam-7124	291	16	this	this	DET
ejpam-7124	291	17	research	research	NOUN
ejpam-7124	291	18	work	work	NOUN
ejpam-7124	291	19	through	through	ADP
ejpam-7124	291	20	the	the	DET
ejpam-7124	291	21	project	project	NOUN
ejpam-7124	291	22	number	number	NOUN
ejpam-7124	291	23	(	(	PUNCT
ejpam-7124	291	24	psau/2025/01/33096	psau/2025/01/33096	NOUN
ejpam-7124	291	25	)	)	PUNCT
ejpam-7124	291	26	”	"	PUNCT
ejpam-7124	291	27	.	.	PUNCT
ejpam-7124	292	1	author	author	NOUN
ejpam-7124	292	2	contributions	contribution	NOUN
ejpam-7124	292	3	conceptualisation	conceptualisation	NOUN
ejpam-7124	292	4	m	m	PROPN
ejpam-7124	292	5	kumar	kumar	PROPN
ejpam-7124	292	6	,	,	PUNCT
ejpam-7124	292	7	g	g	PROPN
ejpam-7124	292	8	mani	mani	NOUN
ejpam-7124	292	9	,	,	PUNCT
ejpam-7124	292	10	ramaswamy	ramaswamy	ADJ
ejpam-7124	292	11	r	r	NOUN
ejpam-7124	292	12	,	,	PUNCT
ejpam-7124	292	13	methodology	methodology	NOUN
ejpam-7124	292	14	m	m	PROPN
ejpam-7124	292	15	kumar	kumar	PROPN
ejpam-7124	292	16	,	,	PUNCT
ejpam-7124	292	17	g	g	PROPN
ejpam-7124	292	18	mani	mani	PROPN
ejpam-7124	292	19	sofaware	sofaware	PROPN
ejpam-7124	292	20	neha	neha	PROPN
ejpam-7124	292	21	b	b	PROPN
ejpam-7124	292	22	,	,	PUNCT
ejpam-7124	292	23	ola	ola	PROPN
ejpam-7124	292	24	a	a	DET
ejpam-7124	292	25	a	a	PROPN
ejpam-7124	292	26	,	,	PUNCT
ejpam-7124	292	27	supervision	supervision	NOUN
ejpam-7124	292	28	g	g	PROPN
ejpam-7124	292	29	mani	mani	NOUN
ejpam-7124	292	30	,	,	PUNCT
ejpam-7124	292	31	ramaswamy	ramaswamy	ADJ
ejpam-7124	292	32	r	r	NOUN
ejpam-7124	292	33	,	,	PUNCT
ejpam-7124	292	34	writing	write	VERB
ejpam-7124	292	35	orignial	orignial	ADJ
ejpam-7124	292	36	draft	draft	NOUN
ejpam-7124	292	37	m	m	NOUN
ejpam-7124	292	38	kumar	kumar	PROPN
ejpam-7124	292	39	,	,	PUNCT
ejpam-7124	292	40	g	g	PROPN
ejpam-7124	292	41	mani	mani	PROPN
ejpam-7124	292	42	,	,	PUNCT
ejpam-7124	292	43	khizar	khizar	PROPN
ejpam-7124	292	44	hk	hk	PROPN
ejpam-7124	292	45	,	,	PUNCT
ejpam-7124	292	46	editing	edit	VERB
ejpam-7124	292	47	m	m	PROPN
ejpam-7124	292	48	kumar	kumar	PROPN
ejpam-7124	292	49	,	,	PUNCT
ejpam-7124	292	50	neha	neha	NOUN
ejpam-7124	292	51	b	b	PROPN
ejpam-7124	292	52	,	,	PUNCT
ejpam-7124	292	53	g	g	PROPN
ejpam-7124	292	54	mani	mani	PROPN
ejpam-7124	292	55	,	,	PUNCT
ejpam-7124	292	56	rajagopalan	rajagopalan	VERB
ejpam-7124	292	57	r	r	NOUN
ejpam-7124	292	58	,	,	PUNCT
ejpam-7124	292	59	ola	ola	PROPN
ejpam-7124	292	60	aa	aa	PROPN
ejpam-7124	292	61	,	,	PUNCT
ejpam-7124	292	62	khizar	khizar	PROPN
ejpam-7124	292	63	hk	hk	PROPN
ejpam-7124	292	64	m.	m.	PROPN
ejpam-7124	292	65	kumar	kumar	PROPN
ejpam-7124	292	66	et	et	PROPN
ejpam-7124	292	67	al	al	PROPN
ejpam-7124	292	68	.	.	PUNCT
ejpam-7124	292	69	/	/	SYM
ejpam-7124	292	70	eur	eur	PROPN
ejpam-7124	292	71	.	.	PUNCT
ejpam-7124	293	1	j.	j.	PROPN
ejpam-7124	293	2	pure	pure	PROPN
ejpam-7124	293	3	appl	appl	PROPN
ejpam-7124	293	4	.	.	PROPN
ejpam-7124	293	5	math	math	PROPN
ejpam-7124	293	6	,	,	PUNCT
ejpam-7124	293	7	18	18	NUM
ejpam-7124	293	8	(	(	PUNCT
ejpam-7124	293	9	4	4	NUM
ejpam-7124	293	10	)	)	PUNCT
ejpam-7124	293	11	(	(	PUNCT
ejpam-7124	293	12	2025	2025	NUM
ejpam-7124	293	13	)	)	PUNCT
ejpam-7124	293	14	,	,	PUNCT
ejpam-7124	293	15	7124	7124	NUM
ejpam-7124	293	16	14	14	NUM
ejpam-7124	293	17	of	of	ADP
ejpam-7124	293	18	14	14	NUM
ejpam-7124	293	19	references	reference	NOUN
ejpam-7124	293	20	[	[	X
ejpam-7124	293	21	1	1	NUM
ejpam-7124	293	22	]	]	PUNCT
ejpam-7124	293	23	banach	banach	NOUN
ejpam-7124	293	24	s.	s.	PROPN
ejpam-7124	293	25	sur	sur	PROPN
ejpam-7124	293	26	les	les	PROPN
ejpam-7124	293	27	op´erations	op´eration	NOUN
ejpam-7124	293	28	dans	dans	PROPN
ejpam-7124	293	29	les	le	NOUN
ejpam-7124	293	30	ensembles	ensemble	NOUN
ejpam-7124	293	31	abstraits	abstrait	NOUN
ejpam-7124	293	32	et	et	PROPN
ejpam-7124	293	33	leurs	leurs	PROPN
ejpam-7124	293	34	applications	applications	PROPN
ejpam-7124	293	35	aux	aux	PROPN
ejpam-7124	293	36	equations	equation	NOUN
ejpam-7124	293	37	integrales	integrale	NOUN
ejpam-7124	293	38	.	.	PUNCT
ejpam-7124	294	1	fund	fund	NOUN
ejpam-7124	294	2	math	math	PROPN
ejpam-7124	294	3	,	,	PUNCT
ejpam-7124	294	4	,	,	PUNCT
ejpam-7124	294	5	3:133–181	3:133–181	NUM
ejpam-7124	294	6	,	,	PUNCT
ejpam-7124	294	7	1922	1922	NUM
ejpam-7124	294	8	.	.	PUNCT
ejpam-7124	295	1	[	[	X
ejpam-7124	295	2	2	2	NUM
ejpam-7124	295	3	]	]	PUNCT
ejpam-7124	295	4	bakhtin	bakhtin	NOUN
ejpam-7124	295	5	i	i	PRON
ejpam-7124	295	6	a.	a.	VERB
ejpam-7124	296	1	the	the	DET
ejpam-7124	296	2	contraction	contraction	NOUN
ejpam-7124	296	3	mapping	map	VERB
ejpam-7124	296	4	principle	principle	NOUN
ejpam-7124	296	5	in	in	ADP
ejpam-7124	296	6	almost	almost	ADV
ejpam-7124	296	7	metric	metric	ADJ
ejpam-7124	296	8	spaces	space	NOUN
ejpam-7124	296	9	.	.	PUNCT
ejpam-7124	297	1	funct	funct	ADJ
ejpam-7124	297	2	anal	anal	PROPN
ejpam-7124	297	3	,	,	PUNCT
ejpam-7124	297	4	,	,	PUNCT
ejpam-7124	297	5	30:26–37	30:26–37	PROPN
ejpam-7124	297	6	,	,	PUNCT
ejpam-7124	297	7	1989	1989	NUM
ejpam-7124	297	8	.	.	PUNCT
ejpam-7124	298	1	[	[	X
ejpam-7124	298	2	3	3	X
ejpam-7124	298	3	]	]	X
ejpam-7124	298	4	czerwik	czerwik	PROPN
ejpam-7124	298	5	s.	s.	PROPN
ejpam-7124	298	6	contraction	contraction	PROPN
ejpam-7124	298	7	mappings	mapping	NOUN
ejpam-7124	298	8	in	in	ADP
ejpam-7124	298	9	b	b	NOUN
ejpam-7124	298	10	-	-	ADJ
ejpam-7124	298	11	metric	metric	ADJ
ejpam-7124	298	12	spaces	space	NOUN
ejpam-7124	298	13	,	,	PUNCT
ejpam-7124	298	14	.	.	PUNCT
ejpam-7124	299	1	acta	acta	PROPN
ejpam-7124	299	2	math	math	PROPN
ejpam-7124	299	3	.	.	PUNCT
ejpam-7124	300	1	inform	inform	NOUN
ejpam-7124	300	2	.	.	PUNCT
ejpam-7124	301	1	univ	univ	PROPN
ejpam-7124	301	2	.	.	PUNCT
ejpam-7124	301	3	ostrav	ostrav	PROPN
ejpam-7124	301	4	.	.	PUNCT
ejpam-7124	302	1	,	,	PUNCT
ejpam-7124	302	2	,	,	PUNCT
ejpam-7124	302	3	30:5–11	30:5–11	NUM
ejpam-7124	302	4	,	,	PUNCT
ejpam-7124	302	5	1993	1993	NUM
ejpam-7124	302	6	.	.	PUNCT
ejpam-7124	303	1	[	[	X
ejpam-7124	303	2	4	4	NUM
ejpam-7124	303	3	]	]	X
ejpam-7124	303	4	gao	gao	PROPN
ejpam-7124	303	5	z	z	PROPN
ejpam-7124	303	6	m.	m.	PROPN
ejpam-7124	303	7	wang	wang	PROPN
ejpam-7124	303	8	s	s	PROPN
ejpam-7124	303	9	z.	z.	PROPN
ejpam-7124	303	10	,	,	PUNCT
ejpam-7124	303	11	li	li	PROPN
ejpam-7124	303	12	b	b	PROPN
ejpam-7124	303	13	y.	y.	PROPN
ejpam-7124	303	14	and	and	CCONJ
ejpam-7124	303	15	iseki	iseki	PROPN
ejpam-7124	303	16	k.	k.	PROPN
ejpam-7124	304	1	some	some	DET
ejpam-7124	304	2	fixed	fix	VERB
ejpam-7124	304	3	point	point	NOUN
ejpam-7124	304	4	theorems	theorem	NOUN
ejpam-7124	304	5	on	on	ADP
ejpam-7124	304	6	expansion	expansion	NOUN
ejpam-7124	304	7	mappings	mapping	NOUN
ejpam-7124	304	8	,	,	PUNCT
ejpam-7124	304	9	.	.	PUNCT
ejpam-7124	305	1	math	math	NOUN
ejpam-7124	305	2	.	.	PUNCT
ejpam-7124	306	1	japon	japon	PROPN
ejpam-7124	306	2	,	,	PUNCT
ejpam-7124	306	3	,	,	PUNCT
ejpam-7124	306	4	29:631–636	29:631–636	PROPN
ejpam-7124	306	5	,	,	PUNCT
ejpam-7124	306	6	1984	1984	NUM
ejpam-7124	306	7	.	.	PUNCT
ejpam-7124	307	1	[	[	X
ejpam-7124	307	2	5	5	X
ejpam-7124	307	3	]	]	X
ejpam-7124	307	4	kaur	kaur	PROPN
ejpam-7124	307	5	j.	j.	PROPN
ejpam-7124	307	6	shahi	shahi	PROPN
ejpam-7124	307	7	,	,	PUNCT
ejpam-7124	307	8	p.	p.	NOUN
ejpam-7124	307	9	and	and	CCONJ
ejpam-7124	307	10	s.s	s.s	PROPN
ejpam-7124	307	11	.	.	PROPN
ejpam-7124	307	12	bhatia	bhatia	PROPN
ejpam-7124	307	13	.	.	PUNCT
ejpam-7124	308	1	fixed	fix	VERB
ejpam-7124	308	2	point	point	NOUN
ejpam-7124	308	3	theorems	theorem	VERB
ejpam-7124	308	4	for	for	ADP
ejpam-7124	308	5	(	(	PUNCT
ejpam-7124	308	6	α	α	NOUN
ejpam-7124	308	7	,	,	PUNCT
ejpam-7124	308	8	ψ)-expansive	ψ)-expansive	ADJ
ejpam-7124	308	9	mappings	mapping	NOUN
ejpam-7124	308	10	in	in	ADP
ejpam-7124	308	11	complete	complete	ADJ
ejpam-7124	308	12	metric	metric	ADJ
ejpam-7124	308	13	spaces	space	NOUN
ejpam-7124	308	14	,	,	PUNCT
ejpam-7124	308	15	.	.	PUNCT
ejpam-7124	309	1	fixed	fix	VERB
ejpam-7124	309	2	point	point	NOUN
ejpam-7124	309	3	theory	theory	NOUN
ejpam-7124	309	4	and	and	CCONJ
ejpam-7124	309	5	applications	application	NOUN
ejpam-7124	309	6	,	,	PUNCT
ejpam-7124	309	7	,	,	PUNCT
ejpam-7124	309	8	2012:157	2012:157	NUM
ejpam-7124	309	9	,	,	PUNCT
ejpam-7124	309	10	2012	2012	NUM
ejpam-7124	309	11	.	.	PUNCT
ejpam-7124	310	1	[	[	X
ejpam-7124	310	2	6	6	NUM
ejpam-7124	310	3	]	]	PUNCT
ejpam-7124	310	4	bhaskar	bhaskar	NOUN
ejpam-7124	310	5	t	t	PROPN
ejpam-7124	310	6	g	g	PROPN
ejpam-7124	310	7	and	and	CCONJ
ejpam-7124	310	8	lakshmikantham	lakshmikantham	VERB
ejpam-7124	310	9	v.	v.	CCONJ
ejpam-7124	310	10	fixed	fix	VERB
ejpam-7124	310	11	point	point	NOUN
ejpam-7124	310	12	theory	theory	NOUN
ejpam-7124	310	13	in	in	ADP
ejpam-7124	310	14	partially	partially	ADV
ejpam-7124	310	15	ordered	order	VERB
ejpam-7124	310	16	metric	metric	ADJ
ejpam-7124	310	17	spaces	space	NOUN
ejpam-7124	310	18	and	and	CCONJ
ejpam-7124	310	19	applications	application	NOUN
ejpam-7124	310	20	,	,	PUNCT
ejpam-7124	310	21	.	.	PUNCT
ejpam-7124	311	1	non	non	PROPN
ejpam-7124	311	2	linear	linear	PROPN
ejpam-7124	311	3	anal	anal	PROPN
ejpam-7124	311	4	,	,	PUNCT
ejpam-7124	311	5	,	,	PUNCT
ejpam-7124	311	6	65:1379–1393	65:1379–1393	PROPN
ejpam-7124	311	7	,	,	PUNCT
ejpam-7124	311	8	2006	2006	NUM
ejpam-7124	311	9	.	.	PUNCT
ejpam-7124	312	1	[	[	X
ejpam-7124	312	2	7	7	X
ejpam-7124	312	3	]	]	X
ejpam-7124	312	4	mani	mani	PROPN
ejpam-7124	312	5	g.	g.	PROPN
ejpam-7124	312	6	ege	ege	PROPN
ejpam-7124	312	7	o.	o.	PROPN
ejpam-7124	312	8	gnanaprakasam	gnanaprakasam	PROPN
ejpam-7124	312	9	aj	aj	PROPN
ejpam-7124	312	10	.	.	PROPN
ejpam-7124	312	11	,	,	PUNCT
ejpam-7124	312	12	prakasam	prakasam	PROPN
ejpam-7124	312	13	s	s	PROPN
ejpam-7124	312	14	k.	k.	PROPN
ejpam-7124	312	15	fixed	fix	VERB
ejpam-7124	312	16	point	point	NOUN
ejpam-7124	312	17	results	result	NOUN
ejpam-7124	312	18	via	via	ADP
ejpam-7124	312	19	of	of	ADP
ejpam-7124	312	20	contraction	contraction	NOUN
ejpam-7124	312	21	and	and	CCONJ
ejpam-7124	312	22	applications	application	NOUN
ejpam-7124	312	23	to	to	PART
ejpam-7124	312	24	fredholm	fredholm	VERB
ejpam-7124	312	25	and	and	CCONJ
ejpam-7124	312	26	integro	integro	ADJ
ejpam-7124	312	27	-	-	PUNCT
ejpam-7124	312	28	differential	differential	NOUN
ejpam-7124	312	29	equations	equation	NOUN
ejpam-7124	312	30	,	,	PUNCT
ejpam-7124	312	31	.	.	PUNCT
ejpam-7124	313	1	filomat	filomat	PROPN
ejpam-7124	313	2	,	,	PUNCT
ejpam-7124	313	3	,	,	PUNCT
ejpam-7124	313	4	38(29	38(29	NUM
ejpam-7124	313	5	)	)	PUNCT
ejpam-7124	313	6	,	,	PUNCT
ejpam-7124	313	7	2024	2024	NUM
ejpam-7124	313	8	.	.	PUNCT
ejpam-7124	314	1	[	[	X
ejpam-7124	314	2	8	8	NUM
ejpam-7124	314	3	]	]	SYM
ejpam-7124	314	4	samreen	samreen	NUM
ejpam-7124	314	5	m.	m.	NOUN
ejpam-7124	314	6	kamran	kamran	PROPN
ejpam-7124	314	7	t.	t.	PROPN
ejpam-7124	314	8	and	and	CCONJ
ejpam-7124	314	9	ul	ul	INTJ
ejpam-7124	314	10	ain	ain	PROPN
ejpam-7124	314	11	q.	q.	PROPN
ejpam-7124	314	12	a	a	DET
ejpam-7124	314	13	generalization	generalization	NOUN
ejpam-7124	314	14	of	of	ADP
ejpam-7124	314	15	b	b	NOUN
ejpam-7124	314	16	-	-	PUNCT
ejpam-7124	314	17	metric	metric	ADJ
ejpam-7124	314	18	space	space	NOUN
ejpam-7124	314	19	and	and	CCONJ
ejpam-7124	314	20	some	some	DET
ejpam-7124	314	21	fixed	fix	VERB
ejpam-7124	314	22	point	point	NOUN
ejpam-7124	314	23	theorems	theorem	NOUN
ejpam-7124	314	24	,	,	PUNCT
ejpam-7124	314	25	.	.	PUNCT
ejpam-7124	315	1	mathematics	mathematic	NOUN
ejpam-7124	315	2	,	,	PUNCT
ejpam-7124	315	3	,	,	PUNCT
ejpam-7124	315	4	5(2):19	5(2):19	NUM
ejpam-7124	315	5	,	,	PUNCT
ejpam-7124	315	6	2017	2017	NUM
ejpam-7124	315	7	.	.	PUNCT
ejpam-7124	316	1	[	[	X
ejpam-7124	316	2	9	9	NUM
ejpam-7124	316	3	]	]	X
ejpam-7124	316	4	petrusel	petrusel	NOUN
ejpam-7124	316	5	a.	a.	NOUN
ejpam-7124	316	6	karapinar	karapinar	PROPN
ejpam-7124	316	7	e.	e.	PROPN
ejpam-7124	316	8	and	and	CCONJ
ejpam-7124	316	9	petrusel	petrusel	PROPN
ejpam-7124	316	10	g.	g.	NOUN
ejpam-7124	316	11	on	on	ADP
ejpam-7124	316	12	admissible	admissible	ADJ
ejpam-7124	316	13	hybrid	hybrid	ADJ
ejpam-7124	316	14	geraghty	geraghty	VERB
ejpam-7124	316	15	contractions	contraction	NOUN
ejpam-7124	316	16	,	,	PUNCT
ejpam-7124	316	17	.	.	PUNCT
ejpam-7124	317	1	carpathian	carpathian	ADJ
ejpam-7124	317	2	journal	journal	PROPN
ejpam-7124	317	3	of	of	ADP
ejpam-7124	317	4	mathematics	mathematic	NOUN
ejpam-7124	317	5	,	,	PUNCT
ejpam-7124	317	6	,	,	PUNCT
ejpam-7124	317	7	36(3):433–442	36(3):433–442	NUM
ejpam-7124	317	8	,	,	PUNCT
ejpam-7124	317	9	2020	2020	NUM
ejpam-7124	317	10	.	.	PUNCT
ejpam-7124	318	1	[	[	X
ejpam-7124	318	2	10	10	NUM
ejpam-7124	318	3	]	]	X
ejpam-7124	318	4	jleli	jleli	PROPN
ejpam-7124	318	5	m	m	PROPN
ejpam-7124	318	6	and	and	CCONJ
ejpam-7124	318	7	samet	samet	PROPN
ejpam-7124	318	8	b.	b.	PROPN
ejpam-7124	319	1	a	a	DET
ejpam-7124	319	2	new	new	ADJ
ejpam-7124	319	3	generalization	generalization	NOUN
ejpam-7124	319	4	of	of	ADP
ejpam-7124	319	5	the	the	DET
ejpam-7124	319	6	banach	banach	NOUN
ejpam-7124	319	7	contraction	contraction	NOUN
ejpam-7124	319	8	principle	principle	NOUN
ejpam-7124	319	9	,	,	PUNCT
ejpam-7124	319	10	.	.	PUNCT
ejpam-7124	320	1	journal	journal	PROPN
ejpam-7124	320	2	of	of	ADP
ejpam-7124	320	3	inequalities	inequality	NOUN
ejpam-7124	320	4	and	and	CCONJ
ejpam-7124	320	5	applications	application	NOUN
ejpam-7124	320	6	,	,	PUNCT
ejpam-7124	320	7	,	,	PUNCT
ejpam-7124	320	8	2014:38	2014:38	NUM
ejpam-7124	320	9	,	,	PUNCT
ejpam-7124	320	10	2014	2014	NUM
ejpam-7124	320	11	.	.	PUNCT
ejpam-7124	321	1	[	[	X
ejpam-7124	321	2	11	11	NUM
ejpam-7124	321	3	]	]	PUNCT
ejpam-7124	321	4	rohen	rohen	PROPN
ejpam-7124	321	5	y.	y.	PROPN
ejpam-7124	321	6	souayah	souayah	PROPN
ejpam-7124	321	7	n.	n.	PROPN
ejpam-7124	321	8	mlaiki	mlaiki	PROPN
ejpam-7124	321	9	n.	n.	PROPN
ejpam-7124	321	10	,	,	PUNCT
ejpam-7124	321	11	mukheimer	mukheimer	PROPN
ejpam-7124	321	12	a.	a.	NOUN
ejpam-7124	321	13	and	and	CCONJ
ejpam-7124	321	14	abdeljawad	abdeljawad	NOUN
ejpam-7124	321	15	t.	t.	PROPN
ejpam-7124	321	16	fixed	fix	VERB
ejpam-7124	321	17	point	point	NOUN
ejpam-7124	321	18	theorems	theorem	NOUN
ejpam-7124	321	19	for	for	ADP
ejpam-7124	321	20	α	α	NOUN
ejpam-7124	321	21	-	-	PUNCT
ejpam-7124	321	22	ψ	ψ	NOUN
ejpam-7124	321	23	-	-	ADJ
ejpam-7124	321	24	contractive	contractive	ADJ
ejpam-7124	321	25	mapping	mapping	NOUN
ejpam-7124	321	26	in	in	ADP
ejpam-7124	321	27	sb	sb	NOUN
ejpam-7124	321	28	-	-	ADJ
ejpam-7124	321	29	metric	metric	ADJ
ejpam-7124	321	30	spaces	space	NOUN
ejpam-7124	321	31	,	,	PUNCT
ejpam-7124	321	32	.	.	PUNCT
ejpam-7124	322	1	journal	journal	PROPN
ejpam-7124	322	2	of	of	ADP
ejpam-7124	322	3	mathematical	mathematical	ADJ
ejpam-7124	322	4	analysis	analysis	NOUN
ejpam-7124	322	5	,	,	PUNCT
ejpam-7124	322	6	,	,	PUNCT
ejpam-7124	322	7	8(5):40–46	8(5):40–46	NUM
ejpam-7124	322	8	,	,	PUNCT
ejpam-7124	322	9	2017	2017	NUM
ejpam-7124	322	10	.	.	PUNCT
ejpam-7124	323	1	[	[	X
ejpam-7124	323	2	12	12	NUM
ejpam-7124	323	3	]	]	PUNCT
ejpam-7124	323	4	souayah	souayah	NOUN
ejpam-7124	323	5	n.	n.	NOUN
ejpam-7124	323	6	and	and	CCONJ
ejpam-7124	323	7	mlaiki	mlaiki	PROPN
ejpam-7124	323	8	a.	a.	NOUN
ejpam-7124	324	1	some	some	DET
ejpam-7124	324	2	fixed	fix	VERB
ejpam-7124	324	3	point	point	NOUN
ejpam-7124	324	4	theorem	theorem	VERB
ejpam-7124	324	5	in	in	ADP
ejpam-7124	324	6	sb	sb	PROPN
ejpam-7124	324	7	metric	metric	ADJ
ejpam-7124	324	8	spaces	space	NOUN
ejpam-7124	324	9	,	,	PUNCT
ejpam-7124	324	10	.	.	PUNCT
ejpam-7124	325	1	j.	j.	PROPN
ejpam-7124	325	2	math	math	PROPN
ejpam-7124	325	3	.	.	PUNCT
ejpam-7124	326	1	computer	computer	PROPN
ejpam-7124	326	2	sci	sci	PROPN
ejpam-7124	326	3	,	,	PUNCT
ejpam-7124	326	4	,	,	PUNCT
ejpam-7124	326	5	16:131–139	16:131–139	NUM
ejpam-7124	326	6	,	,	PUNCT
ejpam-7124	326	7	2016	2016	NUM
ejpam-7124	326	8	.	.	PUNCT
ejpam-7124	327	1	[	[	X
ejpam-7124	327	2	13	13	NUM
ejpam-7124	327	3	]	]	PUNCT
ejpam-7124	327	4	abodayeh	abodayeh	PROPN
ejpam-7124	327	5	k.	k.	PROPN
ejpam-7124	327	6	shatanawi	shatanawi	PROPN
ejpam-7124	327	7	w.	w.	PROPN
ejpam-7124	327	8	and	and	CCONJ
ejpam-7124	327	9	mukheimer	mukheimer	PROPN
ejpam-7124	327	10	a.	a.	PROPN
ejpam-7124	327	11	some	some	DET
ejpam-7124	327	12	fixed	fix	VERB
ejpam-7124	327	13	point	point	NOUN
ejpam-7124	327	14	theorems	theorem	NOUN
ejpam-7124	327	15	in	in	ADP
ejpam-7124	327	16	extended	extended	ADJ
ejpam-7124	327	17	b	b	X
ejpam-7124	327	18	-	-	ADJ
ejpam-7124	327	19	metric	metric	ADJ
ejpam-7124	327	20	spaces	space	NOUN
ejpam-7124	327	21	,	,	PUNCT
ejpam-7124	327	22	.	.	PUNCT
ejpam-7124	328	1	upb	upb	ADJ
ejpam-7124	328	2	scientific	scientific	ADJ
ejpam-7124	328	3	bulletin	bulletin	NOUN
ejpam-7124	328	4	.	.	PUNCT
ejpam-7124	329	1	,	,	PUNCT
ejpam-7124	329	2	,	,	PUNCT
ejpam-7124	329	3	80	80	NUM
ejpam-7124	329	4	,	,	PUNCT
ejpam-7124	329	5	2018	2018	NUM
ejpam-7124	329	6	.	.	PUNCT
ejpam-7124	330	1	[	[	X
ejpam-7124	330	2	14	14	NUM
ejpam-7124	330	3	]	]	X
ejpam-7124	330	4	gao	gao	PROPN
ejpam-7124	330	5	z	z	PROPN
ejpam-7124	330	6	m.	m.	PROPN
ejpam-7124	330	7	wang	wang	PROPN
ejpam-7124	330	8	s	s	PROPN
ejpam-7124	330	9	z.	z.	PROPN
ejpam-7124	330	10	,	,	PUNCT
ejpam-7124	330	11	li	li	PROPN
ejpam-7124	330	12	b	b	PROPN
ejpam-7124	330	13	y.	y.	PROPN
ejpam-7124	330	14	and	and	CCONJ
ejpam-7124	330	15	iseki	iseki	PROPN
ejpam-7124	330	16	k.	k.	PROPN
ejpam-7124	330	17	on	on	ADP
ejpam-7124	330	18	elliptic	elliptic	ADJ
ejpam-7124	330	19	valued	value	VERB
ejpam-7124	330	20	b	b	NOUN
ejpam-7124	330	21	-	-	PUNCT
ejpam-7124	330	22	metric	metric	ADJ
ejpam-7124	330	23	spaces	space	NOUN
ejpam-7124	330	24	and	and	CCONJ
ejpam-7124	330	25	some	some	DET
ejpam-7124	330	26	new	new	ADJ
ejpam-7124	330	27	fixed	fix	VERB
ejpam-7124	330	28	point	point	NOUN
ejpam-7124	330	29	results	result	NOUN
ejpam-7124	330	30	with	with	ADP
ejpam-7124	330	31	an	an	DET
ejpam-7124	330	32	application	application	NOUN
ejpam-7124	330	33	,	,	PUNCT
ejpam-7124	330	34	.	.	PUNCT
ejpam-7124	331	1	aims	aim	VERB
ejpam-7124	331	2	mathematics	mathematic	NOUN
ejpam-7124	331	3	,	,	PUNCT
ejpam-7124	331	4	,	,	PUNCT
ejpam-7124	331	5	9(7	9(7	NUM
ejpam-7124	331	6	)	)	PUNCT
ejpam-7124	331	7	,	,	PUNCT
ejpam-7124	331	8	2024	2024	NUM
ejpam-7124	331	9	.	.	PUNCT
ejpam-7124	332	1	[	[	X
ejpam-7124	332	2	15	15	NUM
ejpam-7124	332	3	]	]	X
ejpam-7124	332	4	souayah	souayah	NOUN
ejpam-7124	332	5	n.	n.	PROPN
ejpam-7124	332	6	mlaiki	mlaiki	PROPN
ejpam-7124	332	7	n.	n.	PROPN
ejpam-7124	332	8	,	,	PUNCT
ejpam-7124	332	9	aydi	aydi	VERB
ejpam-7124	332	10	h.	h.	NOUN
ejpam-7124	332	11	and	and	CCONJ
ejpam-7124	332	12	abdeljawad	abdeljawad	PROPN
ejpam-7124	332	13	t.	t.	PROPN
ejpam-7124	332	14	controlled	control	VERB
ejpam-7124	332	15	metric	metric	ADJ
ejpam-7124	332	16	type	type	NOUN
ejpam-7124	332	17	spaces	space	NOUN
ejpam-7124	332	18	and	and	CCONJ
ejpam-7124	332	19	the	the	DET
ejpam-7124	332	20	related	related	ADJ
ejpam-7124	332	21	contraction	contraction	NOUN
ejpam-7124	332	22	principle	principle	NOUN
ejpam-7124	332	23	,	,	PUNCT
ejpam-7124	332	24	.	.	PUNCT
ejpam-7124	333	1	mathematics	mathematic	NOUN
ejpam-7124	333	2	,	,	PUNCT
ejpam-7124	333	3	,	,	PUNCT
ejpam-7124	333	4	6(10):194	6(10):194	NUM
ejpam-7124	333	5	,	,	PUNCT
ejpam-7124	333	6	2018	2018	NUM
ejpam-7124	333	7	.	.	PUNCT
ejpam-7124	334	1	[	[	X
ejpam-7124	334	2	16	16	NUM
ejpam-7124	334	3	]	]	X
ejpam-7124	334	4	vetro	vetro	PROPN
ejpam-7124	334	5	c.	c.	PROPN
ejpam-7124	334	6	samet	samet	PROPN
ejpam-7124	334	7	b.	b.	PROPN
ejpam-7124	334	8	and	and	CCONJ
ejpam-7124	334	9	vetro	vetro	PROPN
ejpam-7124	334	10	p.	p.	NOUN
ejpam-7124	334	11	fixed	fix	VERB
ejpam-7124	334	12	point	point	NOUN
ejpam-7124	334	13	theorems	theorem	NOUN
ejpam-7124	334	14	for	for	ADP
ejpam-7124	334	15	α	α	NOUN
ejpam-7124	334	16	-	-	PUNCT
ejpam-7124	334	17	ψ	ψ	NOUN
ejpam-7124	334	18	-	-	ADJ
ejpam-7124	334	19	contractive	contractive	ADJ
ejpam-7124	334	20	type	type	NOUN
ejpam-7124	334	21	mappings	mapping	NOUN
ejpam-7124	334	22	,	,	PUNCT
ejpam-7124	334	23	.	.	PUNCT
ejpam-7124	335	1	nonlinear	nonlinear	ADJ
ejpam-7124	335	2	analysis	analysis	NOUN
ejpam-7124	335	3	:	:	PUNCT
ejpam-7124	335	4	theory	theory	NOUN
ejpam-7124	335	5	,	,	PUNCT
ejpam-7124	335	6	methods	method	NOUN
ejpam-7124	335	7	applications	application	NOUN
ejpam-7124	335	8	,	,	PUNCT
ejpam-7124	335	9	75(4):2154–2165	75(4):2154–2165	NOUN
ejpam-7124	335	10	,	,	PUNCT
ejpam-7124	335	11	2012	2012	NUM
ejpam-7124	335	12	.	.	PUNCT
ejpam-7124	336	1	[	[	X
ejpam-7124	336	2	17	17	NUM
ejpam-7124	336	3	]	]	PUNCT
ejpam-7124	336	4	abodayeh	abodayeh	PROPN
ejpam-7124	336	5	k	k	PROPN
ejpam-7124	336	6	mlaiki	mlaiki	PROPN
ejpam-7124	336	7	n	n	PROPN
ejpam-7124	336	8	abuloha	abuloha	PROPN
ejpam-7124	336	9	m	m	PROPN
ejpam-7124	336	10	,	,	PUNCT
ejpam-7124	336	11	rizk	rizk	PROPN
ejpam-7124	336	12	d	d	PROPN
ejpam-7124	336	13	and	and	CCONJ
ejpam-7124	336	14	abdeljawad	abdeljawad	NOUN
ejpam-7124	336	15	.	.	PUNCT
ejpam-7124	337	1	new	new	ADJ
ejpam-7124	337	2	results	result	NOUN
ejpam-7124	337	3	in	in	ADP
ejpam-7124	337	4	controlled	control	VERB
ejpam-7124	337	5	metric	metric	ADJ
ejpam-7124	337	6	type	type	NOUN
ejpam-7124	337	7	spaces	space	NOUN
ejpam-7124	337	8	.	.	PUNCT
ejpam-7124	338	1	journal	journal	NOUN
ejpam-7124	338	2	of	of	ADP
ejpam-7124	338	3	mathematics	mathematic	NOUN
ejpam-7124	338	4	,	,	PUNCT
ejpam-7124	338	5	1:5575512	1:5575512	NUM
ejpam-7124	338	6	,	,	PUNCT
ejpam-7124	338	7	2021	2021	NUM
ejpam-7124	338	8	.	.	PUNCT