id	sid	tid	token	lemma	pos
ejpam-6526	1	1	european	european	PROPN
ejpam-6526	1	2	journal	journal	PROPN
ejpam-6526	1	3	of	of	ADP
ejpam-6526	1	4	pure	pure	ADJ
ejpam-6526	1	5	and	and	CCONJ
ejpam-6526	1	6	applied	applied	ADJ
ejpam-6526	1	7	mathematics	mathematic	NOUN
ejpam-6526	1	8	2025	2025	NUM
ejpam-6526	1	9	,	,	PUNCT
ejpam-6526	1	10	vol	vol	NOUN
ejpam-6526	1	11	.	.	PROPN
ejpam-6526	1	12	18	18	NUM
ejpam-6526	1	13	,	,	PUNCT
ejpam-6526	1	14	issue	issue	NOUN
ejpam-6526	1	15	4	4	NUM
ejpam-6526	1	16	,	,	PUNCT
ejpam-6526	1	17	article	article	NOUN
ejpam-6526	1	18	number	number	NOUN
ejpam-6526	1	19	6526	6526	NUM
ejpam-6526	1	20	issn	issn	VERB
ejpam-6526	1	21	1307	1307	NUM
ejpam-6526	1	22	-	-	SYM
ejpam-6526	1	23	5543	5543	NUM
ejpam-6526	1	24	–	–	PUNCT
ejpam-6526	1	25	ejpam.com	ejpam.com	X
ejpam-6526	1	26	published	publish	VERB
ejpam-6526	1	27	by	by	ADP
ejpam-6526	1	28	new	new	PROPN
ejpam-6526	1	29	york	york	PROPN
ejpam-6526	1	30	business	business	PROPN
ejpam-6526	1	31	global	global	ADJ
ejpam-6526	1	32	bi	bi	ADJ
ejpam-6526	1	33	-	-	ADJ
ejpam-6526	1	34	metric	metric	ADJ
ejpam-6526	1	35	structures	structure	NOUN
ejpam-6526	1	36	and	and	CCONJ
ejpam-6526	1	37	their	their	PRON
ejpam-6526	1	38	applications	application	NOUN
ejpam-6526	1	39	in	in	ADP
ejpam-6526	1	40	bitopological	bitopological	ADJ
ejpam-6526	1	41	contexts	context	NOUN
ejpam-6526	1	42	abdullah	abdullah	PROPN
ejpam-6526	1	43	alsoboh1	alsoboh1	PROPN
ejpam-6526	1	44	,	,	PUNCT
ejpam-6526	1	45	jamal	jamal	PROPN
ejpam-6526	1	46	oudetallah2	oudetallah2	PROPN
ejpam-6526	1	47	,	,	PUNCT
ejpam-6526	1	48	ala	ala	PROPN
ejpam-6526	1	49	amourah3,∗	amourah3,∗	PROPN
ejpam-6526	1	50	,	,	PUNCT
ejpam-6526	1	51	raja’a	raja’a	NOUN
ejpam-6526	1	52	al	al	PROPN
ejpam-6526	1	53	-	-	PUNCT
ejpam-6526	1	54	naimi4	naimi4	PROPN
ejpam-6526	1	55	,	,	PUNCT
ejpam-6526	1	56	mohammed	mohammed	PROPN
ejpam-6526	1	57	al	al	PROPN
ejpam-6526	1	58	hatmi1,∗	hatmi1,∗	PROPN
ejpam-6526	1	59	,	,	PUNCT
ejpam-6526	1	60	wasim	wasim	PROPN
ejpam-6526	1	61	audeh2	audeh2	PROPN
ejpam-6526	1	62	,	,	PUNCT
ejpam-6526	1	63	ahmad	ahmad	PROPN
ejpam-6526	1	64	almalkawi5	almalkawi5	PROPN
ejpam-6526	1	65	,	,	PUNCT
ejpam-6526	1	66	tala	tala	PROPN
ejpam-6526	1	67	sasa6	sasa6	NOUN
ejpam-6526	1	68	1	1	NUM
ejpam-6526	1	69	college	college	NOUN
ejpam-6526	1	70	of	of	ADP
ejpam-6526	1	71	applied	apply	VERB
ejpam-6526	1	72	and	and	CCONJ
ejpam-6526	1	73	health	health	NOUN
ejpam-6526	1	74	sciences	science	NOUN
ejpam-6526	1	75	,	,	PUNCT
ejpam-6526	1	76	a’sharqiyah	a’sharqiyah	PROPN
ejpam-6526	1	77	university	university	NOUN
ejpam-6526	1	78	,	,	PUNCT
ejpam-6526	1	79	post	post	PROPN
ejpam-6526	1	80	box	box	PROPN
ejpam-6526	1	81	no	no	INTJ
ejpam-6526	1	82	.	.	PROPN
ejpam-6526	1	83	42	42	NUM
ejpam-6526	1	84	,	,	PUNCT
ejpam-6526	1	85	post	post	VERB
ejpam-6526	1	86	code	code	NOUN
ejpam-6526	1	87	no	no	INTJ
ejpam-6526	1	88	.	.	PROPN
ejpam-6526	1	89	400	400	NUM
ejpam-6526	1	90	,	,	PUNCT
ejpam-6526	1	91	ibra	ibra	NOUN
ejpam-6526	1	92	,	,	PUNCT
ejpam-6526	1	93	sultanate	sultanate	NOUN
ejpam-6526	1	94	of	of	ADP
ejpam-6526	1	95	oman	oman	PROPN
ejpam-6526	1	96	2	2	NUM
ejpam-6526	1	97	department	department	NOUN
ejpam-6526	1	98	of	of	ADP
ejpam-6526	1	99	mathematics	mathematics	PROPN
ejpam-6526	1	100	,	,	PUNCT
ejpam-6526	1	101	university	university	PROPN
ejpam-6526	1	102	of	of	ADP
ejpam-6526	1	103	petra	petra	PROPN
ejpam-6526	1	104	,	,	PUNCT
ejpam-6526	1	105	amman	amman	PROPN
ejpam-6526	1	106	,	,	PUNCT
ejpam-6526	1	107	11196	11196	NUM
ejpam-6526	1	108	,	,	PUNCT
ejpam-6526	1	109	jordan	jordan	PROPN
ejpam-6526	1	110	3	3	NUM
ejpam-6526	1	111	mathematics	mathematics	PROPN
ejpam-6526	1	112	education	education	NOUN
ejpam-6526	1	113	program	program	NOUN
ejpam-6526	1	114	,	,	PUNCT
ejpam-6526	1	115	faculty	faculty	NOUN
ejpam-6526	1	116	of	of	ADP
ejpam-6526	1	117	education	education	NOUN
ejpam-6526	1	118	and	and	CCONJ
ejpam-6526	1	119	arts	art	NOUN
ejpam-6526	1	120	,	,	PUNCT
ejpam-6526	1	121	sohar	sohar	PROPN
ejpam-6526	1	122	university	university	PROPN
ejpam-6526	1	123	,	,	PUNCT
ejpam-6526	1	124	sohar	sohar	PROPN
ejpam-6526	1	125	311	311	NUM
ejpam-6526	1	126	,	,	PUNCT
ejpam-6526	1	127	oman	oman	NOUN
ejpam-6526	1	128	4	4	NUM
ejpam-6526	1	129	emirates	emirates	PROPN
ejpam-6526	1	130	aviation	aviation	PROPN
ejpam-6526	1	131	university	university	PROPN
ejpam-6526	1	132	,	,	PUNCT
ejpam-6526	1	133	dubai	dubai	PROPN
ejpam-6526	1	134	,	,	PUNCT
ejpam-6526	1	135	united	united	PROPN
ejpam-6526	1	136	arab	arab	PROPN
ejpam-6526	1	137	emirates	emirates	PROPN
ejpam-6526	1	138	5	5	NUM
ejpam-6526	1	139	modern	modern	ADJ
ejpam-6526	1	140	college	college	NOUN
ejpam-6526	1	141	of	of	ADP
ejpam-6526	1	142	business	business	NOUN
ejpam-6526	1	143	and	and	CCONJ
ejpam-6526	1	144	science	science	NOUN
ejpam-6526	1	145	,	,	PUNCT
ejpam-6526	1	146	muscat	muscat	PROPN
ejpam-6526	1	147	,	,	PUNCT
ejpam-6526	1	148	sultanate	sultanate	NOUN
ejpam-6526	1	149	of	of	ADP
ejpam-6526	1	150	oman	oman	PROPN
ejpam-6526	1	151	6	6	NUM
ejpam-6526	1	152	department	department	NOUN
ejpam-6526	1	153	of	of	ADP
ejpam-6526	1	154	mathematics	mathematic	NOUN
ejpam-6526	1	155	,	,	PUNCT
ejpam-6526	1	156	faculty	faculty	NOUN
ejpam-6526	1	157	of	of	ADP
ejpam-6526	1	158	science	science	NOUN
ejpam-6526	1	159	,	,	PUNCT
ejpam-6526	1	160	applied	apply	VERB
ejpam-6526	1	161	science	science	NOUN
ejpam-6526	1	162	private	private	ADJ
ejpam-6526	1	163	university	university	NOUN
ejpam-6526	1	164	,	,	PUNCT
ejpam-6526	1	165	amman	amman	PROPN
ejpam-6526	1	166	,	,	PUNCT
ejpam-6526	1	167	jordan	jordan	PROPN
ejpam-6526	1	168	abstract	abstract	PROPN
ejpam-6526	1	169	.	.	PUNCT
ejpam-6526	2	1	we	we	PRON
ejpam-6526	2	2	introduce	introduce	VERB
ejpam-6526	2	3	a	a	DET
ejpam-6526	2	4	novel	novel	ADJ
ejpam-6526	2	5	mathematical	mathematical	ADJ
ejpam-6526	2	6	framework	framework	NOUN
ejpam-6526	2	7	for	for	ADP
ejpam-6526	2	8	analyzing	analyze	VERB
ejpam-6526	2	9	bitopological	bitopological	ADJ
ejpam-6526	2	10	spaces	space	NOUN
ejpam-6526	2	11	through	through	ADP
ejpam-6526	2	12	bi	bi	ADJ
ejpam-6526	2	13	-	-	ADJ
ejpam-6526	2	14	metric	metric	ADJ
ejpam-6526	2	15	structures	structure	NOUN
ejpam-6526	2	16	.	.	PUNCT
ejpam-6526	3	1	our	our	PRON
ejpam-6526	3	2	research	research	NOUN
ejpam-6526	3	3	establishes	establish	VERB
ejpam-6526	3	4	the	the	DET
ejpam-6526	3	5	theoretical	theoretical	ADJ
ejpam-6526	3	6	underpinnings	underpinning	NOUN
ejpam-6526	3	7	of	of	ADP
ejpam-6526	3	8	coupled	couple	VERB
ejpam-6526	3	9	metric	metric	ADJ
ejpam-6526	3	10	spaces	space	NOUN
ejpam-6526	3	11	configurations	configuration	NOUN
ejpam-6526	3	12	that	that	PRON
ejpam-6526	3	13	inherently	inherently	ADV
ejpam-6526	3	14	embrace	embrace	VERB
ejpam-6526	3	15	bitopological	bitopological	ADJ
ejpam-6526	3	16	structures	structure	NOUN
ejpam-6526	3	17	while	while	SCONJ
ejpam-6526	3	18	expanding	expand	VERB
ejpam-6526	3	19	conventional	conventional	ADJ
ejpam-6526	3	20	metric	metric	NOUN
ejpam-6526	3	21	-	-	PUNCT
ejpam-6526	3	22	based	base	VERB
ejpam-6526	3	23	frameworks	framework	NOUN
ejpam-6526	3	24	.	.	PUNCT
ejpam-6526	4	1	we	we	PRON
ejpam-6526	4	2	demonstrate	demonstrate	VERB
ejpam-6526	4	3	key	key	ADJ
ejpam-6526	4	4	mathematical	mathematical	ADJ
ejpam-6526	4	5	correspondences	correspondence	NOUN
ejpam-6526	4	6	linking	link	VERB
ejpam-6526	4	7	these	these	DET
ejpam-6526	4	8	bi	bi	ADJ
ejpam-6526	4	9	-	-	ADJ
ejpam-6526	4	10	metric	metric	ADJ
ejpam-6526	4	11	constructs	construct	NOUN
ejpam-6526	4	12	to	to	ADP
ejpam-6526	4	13	their	their	PRON
ejpam-6526	4	14	generated	generate	VERB
ejpam-6526	4	15	topologies	topology	NOUN
ejpam-6526	4	16	and	and	CCONJ
ejpam-6526	4	17	furnish	furnish	VERB
ejpam-6526	4	18	diverse	diverse	ADJ
ejpam-6526	4	19	contextual	contextual	ADJ
ejpam-6526	4	20	implementations	implementation	NOUN
ejpam-6526	4	21	.	.	PUNCT
ejpam-6526	5	1	our	our	PRON
ejpam-6526	5	2	investigation	investigation	NOUN
ejpam-6526	5	3	examines	examine	VERB
ejpam-6526	5	4	completeness	completeness	NOUN
ejpam-6526	5	5	properties	property	NOUN
ejpam-6526	5	6	,	,	PUNCT
ejpam-6526	5	7	stability	stability	NOUN
ejpam-6526	5	8	characteristics	characteristic	NOUN
ejpam-6526	5	9	,	,	PUNCT
ejpam-6526	5	10	and	and	CCONJ
ejpam-6526	5	11	develops	develop	VERB
ejpam-6526	5	12	systematic	systematic	ADJ
ejpam-6526	5	13	product	product	NOUN
ejpam-6526	5	14	structures	structure	NOUN
ejpam-6526	5	15	within	within	ADP
ejpam-6526	5	16	these	these	DET
ejpam-6526	5	17	frameworks	framework	NOUN
ejpam-6526	5	18	.	.	PUNCT
ejpam-6526	6	1	furthermore	furthermore	ADV
ejpam-6526	6	2	,	,	PUNCT
ejpam-6526	6	3	we	we	PRON
ejpam-6526	6	4	identify	identify	VERB
ejpam-6526	6	5	significant	significant	ADJ
ejpam-6526	6	6	relationships	relationship	NOUN
ejpam-6526	6	7	with	with	ADP
ejpam-6526	6	8	functional	functional	ADJ
ejpam-6526	6	9	-	-	PUNCT
ejpam-6526	6	10	analytical	analytical	ADJ
ejpam-6526	6	11	principles	principle	NOUN
ejpam-6526	6	12	,	,	PUNCT
ejpam-6526	6	13	particularly	particularly	ADV
ejpam-6526	6	14	regarding	regard	VERB
ejpam-6526	6	15	bi	bi	ADJ
ejpam-6526	6	16	-	-	ADJ
ejpam-6526	6	17	normed	normed	ADJ
ejpam-6526	6	18	spaces	space	NOUN
ejpam-6526	6	19	and	and	CCONJ
ejpam-6526	6	20	quasimetric	quasimetric	ADJ
ejpam-6526	6	21	frameworks	framework	NOUN
ejpam-6526	6	22	.	.	PUNCT
ejpam-6526	7	1	the	the	DET
ejpam-6526	7	2	mathematical	mathematical	ADJ
ejpam-6526	7	3	architecture	architecture	NOUN
ejpam-6526	7	4	we	we	PRON
ejpam-6526	7	5	propose	propose	VERB
ejpam-6526	7	6	offers	offer	VERB
ejpam-6526	7	7	innovative	innovative	ADJ
ejpam-6526	7	8	perspectives	perspective	NOUN
ejpam-6526	7	9	on	on	ADP
ejpam-6526	7	10	the	the	DET
ejpam-6526	7	11	interrelationships	interrelationship	NOUN
ejpam-6526	7	12	between	between	ADP
ejpam-6526	7	13	metric	metric	ADJ
ejpam-6526	7	14	frameworks	framework	NOUN
ejpam-6526	7	15	and	and	CCONJ
ejpam-6526	7	16	bitopological	bitopological	ADJ
ejpam-6526	7	17	domains	domain	NOUN
ejpam-6526	7	18	with	with	ADP
ejpam-6526	7	19	implications	implication	NOUN
ejpam-6526	7	20	for	for	ADP
ejpam-6526	7	21	functional	functional	ADJ
ejpam-6526	7	22	transformation	transformation	NOUN
ejpam-6526	7	23	theories	theory	NOUN
ejpam-6526	7	24	,	,	PUNCT
ejpam-6526	7	25	including	include	VERB
ejpam-6526	7	26	practical	practical	ADJ
ejpam-6526	7	27	applications	application	NOUN
ejpam-6526	7	28	in	in	ADP
ejpam-6526	7	29	computer	computer	NOUN
ejpam-6526	7	30	networks	network	NOUN
ejpam-6526	7	31	,	,	PUNCT
ejpam-6526	7	32	image	image	NOUN
ejpam-6526	7	33	processing	processing	NOUN
ejpam-6526	7	34	,	,	PUNCT
ejpam-6526	7	35	and	and	CCONJ
ejpam-6526	7	36	economic	economic	ADJ
ejpam-6526	7	37	modeling	modeling	NOUN
ejpam-6526	7	38	.	.	PUNCT
ejpam-6526	8	1	2020	2020	NUM
ejpam-6526	8	2	mathematics	mathematic	NOUN
ejpam-6526	8	3	subject	subject	NOUN
ejpam-6526	8	4	classifications	classification	NOUN
ejpam-6526	8	5	:	:	PUNCT
ejpam-6526	8	6	54e35	54e35	NUM
ejpam-6526	8	7	,	,	PUNCT
ejpam-6526	8	8	54e55	54e55	NUM
ejpam-6526	8	9	,	,	PUNCT
ejpam-6526	8	10	54e40	54e40	NUM
ejpam-6526	8	11	key	key	ADJ
ejpam-6526	8	12	words	word	NOUN
ejpam-6526	8	13	and	and	CCONJ
ejpam-6526	8	14	phrases	phrase	NOUN
ejpam-6526	8	15	:	:	PUNCT
ejpam-6526	8	16	bitopological	bitopological	ADJ
ejpam-6526	8	17	spaces	space	NOUN
ejpam-6526	8	18	,	,	PUNCT
ejpam-6526	8	19	bi	bi	ADJ
ejpam-6526	8	20	-	-	ADJ
ejpam-6526	8	21	metric	metric	ADJ
ejpam-6526	8	22	structures	structure	NOUN
ejpam-6526	8	23	,	,	PUNCT
ejpam-6526	8	24	transformation	transformation	NOUN
ejpam-6526	8	25	theory	theory	NOUN
ejpam-6526	8	26	,	,	PUNCT
ejpam-6526	8	27	stability	stability	NOUN
ejpam-6526	8	28	principles	principle	NOUN
ejpam-6526	8	29	,	,	PUNCT
ejpam-6526	8	30	functional	functional	ADJ
ejpam-6526	8	31	analysis	analysis	NOUN
ejpam-6526	8	32	∗corresponding	∗corresponde	VERB
ejpam-6526	8	33	author	author	NOUN
ejpam-6526	8	34	.	.	PUNCT
ejpam-6526	9	1	∗corresponding	∗corresponde	VERB
ejpam-6526	9	2	author	author	NOUN
ejpam-6526	9	3	.	.	PUNCT
ejpam-6526	10	1	doi	doi	NOUN
ejpam-6526	10	2	:	:	PUNCT
ejpam-6526	10	3	https://doi.org/10.29020/nybg.ejpam.v18i4.6526	https://doi.org/10.29020/nybg.ejpam.v18i4.6526	NOUN
ejpam-6526	10	4	email	email	NOUN
ejpam-6526	10	5	addresses	address	NOUN
ejpam-6526	10	6	:	:	PUNCT
ejpam-6526	10	7	abdullah.alsoboh@asu.edu.om	abdullah.alsoboh@asu.edu.om	NOUN
ejpam-6526	10	8	(	(	PUNCT
ejpam-6526	10	9	a.	a.	NOUN
ejpam-6526	10	10	alsoboh	alsoboh	PROPN
ejpam-6526	10	11	)	)	PUNCT
ejpam-6526	10	12	,	,	PUNCT
ejpam-6526	10	13	jamal.oudetallah@uop.edu.jo	jamal.oudetallah@uop.edu.jo	PROPN
ejpam-6526	10	14	(	(	PUNCT
ejpam-6526	10	15	j.	j.	PROPN
ejpam-6526	10	16	oudetallah	oudetallah	PROPN
ejpam-6526	10	17	)	)	PUNCT
ejpam-6526	10	18	,	,	PUNCT
ejpam-6526	10	19	aamourah@su.edu.om	aamourah@su.edu.om	NOUN
ejpam-6526	10	20	(	(	PUNCT
ejpam-6526	10	21	a.	a.	NOUN
ejpam-6526	10	22	amourah	amourah	PROPN
ejpam-6526	10	23	)	)	PUNCT
ejpam-6526	10	24	,	,	PUNCT
ejpam-6526	10	25	rajaa.alnaimi@eau.ac.ae	rajaa.alnaimi@eau.ac.ae	PROPN
ejpam-6526	10	26	(	(	PUNCT
ejpam-6526	10	27	r.	r.	PROPN
ejpam-6526	10	28	al	al	PROPN
ejpam-6526	10	29	-	-	PUNCT
ejpam-6526	10	30	naimi	naimi	NOUN
ejpam-6526	10	31	)	)	PUNCT
ejpam-6526	10	32	,	,	PUNCT
ejpam-6526	10	33	mohammed.alhatmi@asu.edu.om	mohammed.alhatmi@asu.edu.om	NOUN
ejpam-6526	10	34	(	(	PUNCT
ejpam-6526	10	35	m.	m.	NOUN
ejpam-6526	10	36	a.	a.	PROPN
ejpam-6526	10	37	al	al	PROPN
ejpam-6526	10	38	hatmi	hatmi	PROPN
ejpam-6526	10	39	)	)	PUNCT
ejpam-6526	10	40	,	,	PUNCT
ejpam-6526	10	41	waudeh@uop.edu.jo	waudeh@uop.edu.jo	NOUN
ejpam-6526	10	42	(	(	PUNCT
ejpam-6526	10	43	w.	w.	PROPN
ejpam-6526	10	44	audeh	audeh	PROPN
ejpam-6526	10	45	)	)	PUNCT
ejpam-6526	10	46	,	,	PUNCT
ejpam-6526	10	47	ahmad.abdelqader@mcbs.edu.om	ahmad.abdelqader@mcbs.edu.om	NOUN
ejpam-6526	10	48	(	(	PUNCT
ejpam-6526	10	49	a.	a.	PROPN
ejpam-6526	10	50	almalkawi	almalkawi	PROPN
ejpam-6526	10	51	)	)	PUNCT
ejpam-6526	10	52	,	,	PUNCT
ejpam-6526	10	53	t_sasa@asu.edu.jo	t_sasa@asu.edu.jo	PRON
ejpam-6526	10	54	(	(	PUNCT
ejpam-6526	10	55	t.	t.	NOUN
ejpam-6526	10	56	sasa	sasa	PROPN
ejpam-6526	10	57	)	)	PUNCT
ejpam-6526	10	58	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6526	11	1	1	1	NUM
ejpam-6526	11	2	copyright	copyright	NOUN
ejpam-6526	11	3	:	:	PUNCT
ejpam-6526	11	4	©	©	PROPN
ejpam-6526	11	5	2025	2025	NUM
ejpam-6526	11	6	the	the	DET
ejpam-6526	11	7	author(s	author(s	NOUN
ejpam-6526	11	8	)	)	PUNCT
ejpam-6526	11	9	.	.	PUNCT
ejpam-6526	12	1	(	(	PUNCT
ejpam-6526	12	2	cc	cc	NOUN
ejpam-6526	12	3	by	by	ADP
ejpam-6526	12	4	-	-	PUNCT
ejpam-6526	12	5	nc	nc	PROPN
ejpam-6526	12	6	4.0	4.0	NUM
ejpam-6526	12	7	)	)	PUNCT
ejpam-6526	12	8	a.	a.	NOUN
ejpam-6526	12	9	alsoboh	alsoboh	NOUN
ejpam-6526	12	10	et	et	PROPN
ejpam-6526	12	11	al	al	PROPN
ejpam-6526	12	12	.	.	PUNCT
ejpam-6526	12	13	/	/	SYM
ejpam-6526	12	14	eur	eur	PROPN
ejpam-6526	12	15	.	.	PUNCT
ejpam-6526	13	1	j.	j.	PROPN
ejpam-6526	13	2	pure	pure	PROPN
ejpam-6526	13	3	appl	appl	PROPN
ejpam-6526	13	4	.	.	PROPN
ejpam-6526	13	5	math	math	PROPN
ejpam-6526	13	6	,	,	PUNCT
ejpam-6526	13	7	18	18	NUM
ejpam-6526	13	8	(	(	PUNCT
ejpam-6526	13	9	4	4	NUM
ejpam-6526	13	10	)	)	PUNCT
ejpam-6526	13	11	(	(	PUNCT
ejpam-6526	13	12	2025	2025	NUM
ejpam-6526	13	13	)	)	PUNCT
ejpam-6526	13	14	,	,	PUNCT
ejpam-6526	13	15	6526	6526	NUM
ejpam-6526	13	16	2	2	NUM
ejpam-6526	13	17	of	of	ADP
ejpam-6526	13	18	21	21	NUM
ejpam-6526	13	19	1	1	NUM
ejpam-6526	13	20	.	.	PUNCT
ejpam-6526	13	21	conceptual	conceptual	ADJ
ejpam-6526	13	22	framework	framework	NOUN
ejpam-6526	13	23	and	and	CCONJ
ejpam-6526	13	24	historical	historical	ADJ
ejpam-6526	13	25	context	context	NOUN
ejpam-6526	13	26	the	the	DET
ejpam-6526	13	27	notion	notion	NOUN
ejpam-6526	13	28	of	of	ADP
ejpam-6526	13	29	spaces	space	NOUN
ejpam-6526	13	30	characterized	characterize	VERB
ejpam-6526	13	31	by	by	ADP
ejpam-6526	13	32	two	two	NUM
ejpam-6526	13	33	distinct	distinct	ADJ
ejpam-6526	13	34	topological	topological	ADJ
ejpam-6526	13	35	structures	structure	NOUN
ejpam-6526	13	36	was	be	AUX
ejpam-6526	13	37	introduced	introduce	VERB
ejpam-6526	13	38	by	by	ADP
ejpam-6526	13	39	kelly	kelly	PROPN
ejpam-6526	14	1	[	[	X
ejpam-6526	14	2	1	1	NUM
ejpam-6526	14	3	]	]	PUNCT
ejpam-6526	14	4	,	,	PUNCT
ejpam-6526	14	5	who	who	PRON
ejpam-6526	14	6	termed	term	VERB
ejpam-6526	14	7	these	these	DET
ejpam-6526	14	8	constructs	construct	NOUN
ejpam-6526	14	9	bitopological	bitopological	ADJ
ejpam-6526	14	10	spaces	space	NOUN
ejpam-6526	14	11	.	.	PUNCT
ejpam-6526	15	1	these	these	DET
ejpam-6526	15	2	mathematical	mathematical	ADJ
ejpam-6526	15	3	entities	entity	NOUN
ejpam-6526	15	4	have	have	AUX
ejpam-6526	15	5	demonstrated	demonstrate	VERB
ejpam-6526	15	6	substantial	substantial	ADJ
ejpam-6526	15	7	utility	utility	NOUN
ejpam-6526	15	8	across	across	ADP
ejpam-6526	15	9	varied	varied	ADJ
ejpam-6526	15	10	analytical	analytical	ADJ
ejpam-6526	15	11	and	and	CCONJ
ejpam-6526	15	12	topological	topological	ADJ
ejpam-6526	15	13	domains	domain	NOUN
ejpam-6526	15	14	.	.	PUNCT
ejpam-6526	16	1	concurrently	concurrently	ADV
ejpam-6526	16	2	,	,	PUNCT
ejpam-6526	16	3	metric	metric	ADJ
ejpam-6526	16	4	spaces	space	NOUN
ejpam-6526	16	5	remain	remain	VERB
ejpam-6526	16	6	foundational	foundational	ADJ
ejpam-6526	16	7	in	in	ADP
ejpam-6526	16	8	mathematical	mathematical	ADJ
ejpam-6526	16	9	analysis	analysis	NOUN
ejpam-6526	16	10	.	.	PUNCT
ejpam-6526	17	1	the	the	DET
ejpam-6526	17	2	convergence	convergence	NOUN
ejpam-6526	17	3	of	of	ADP
ejpam-6526	17	4	these	these	DET
ejpam-6526	17	5	areas	area	NOUN
ejpam-6526	17	6	presents	present	VERB
ejpam-6526	17	7	unique	unique	ADJ
ejpam-6526	17	8	investigative	investigative	ADJ
ejpam-6526	17	9	opportunities	opportunity	NOUN
ejpam-6526	17	10	which	which	PRON
ejpam-6526	17	11	we	we	PRON
ejpam-6526	17	12	explore	explore	VERB
ejpam-6526	17	13	comprehensively	comprehensively	ADV
ejpam-6526	17	14	in	in	ADP
ejpam-6526	17	15	this	this	DET
ejpam-6526	17	16	work	work	NOUN
ejpam-6526	17	17	.	.	PUNCT
ejpam-6526	18	1	recent	recent	ADJ
ejpam-6526	18	2	developments	development	NOUN
ejpam-6526	18	3	in	in	ADP
ejpam-6526	18	4	bitopological	bitopological	ADJ
ejpam-6526	18	5	theory	theory	NOUN
ejpam-6526	18	6	have	have	AUX
ejpam-6526	18	7	expanded	expand	VERB
ejpam-6526	18	8	significantly	significantly	ADV
ejpam-6526	18	9	since	since	SCONJ
ejpam-6526	18	10	kelly	kelly	PROPN
ejpam-6526	18	11	’s	’s	PART
ejpam-6526	18	12	foundational	foundational	ADJ
ejpam-6526	18	13	work	work	NOUN
ejpam-6526	18	14	.	.	PUNCT
ejpam-6526	19	1	y.	y.	PROPN
ejpam-6526	19	2	y.	y.	PROPN
ejpam-6526	19	3	yousif	yousif	PROPN
ejpam-6526	19	4	and	and	CCONJ
ejpam-6526	19	5	l.	l.	PROPN
ejpam-6526	19	6	a.	a.	PROPN
ejpam-6526	19	7	hussain	hussain	PROPN
ejpam-6526	20	1	[	[	X
ejpam-6526	20	2	2	2	NUM
ejpam-6526	20	3	]	]	PUNCT
ejpam-6526	20	4	investigated	investigate	VERB
ejpam-6526	20	5	fibrewise	fibrewise	NOUN
ejpam-6526	20	6	ij	ij	ADJ
ejpam-6526	20	7	-	-	ADJ
ejpam-6526	20	8	perfect	perfect	ADJ
ejpam-6526	20	9	bitopological	bitopological	ADJ
ejpam-6526	20	10	spaces	space	NOUN
ejpam-6526	20	11	,	,	PUNCT
ejpam-6526	20	12	establishing	establish	VERB
ejpam-6526	20	13	new	new	ADJ
ejpam-6526	20	14	characterizations	characterization	NOUN
ejpam-6526	20	15	and	and	CCONJ
ejpam-6526	20	16	properties	property	NOUN
ejpam-6526	20	17	.	.	PUNCT
ejpam-6526	21	1	garcãa	garcãa	NOUN
ejpam-6526	21	2	-	-	PUNCT
ejpam-6526	21	3	mã¡ynez	mã¡ynez	PROPN
ejpam-6526	21	4	and	and	CCONJ
ejpam-6526	21	5	pimienta	pimienta	ADJ
ejpam-6526	21	6	[	[	X
ejpam-6526	21	7	3	3	NUM
ejpam-6526	21	8	]	]	PUNCT
ejpam-6526	21	9	explored	explore	VERB
ejpam-6526	21	10	symmetry	symmetry	NOUN
ejpam-6526	21	11	properties	property	NOUN
ejpam-6526	21	12	in	in	ADP
ejpam-6526	21	13	bitopological	bitopological	ADJ
ejpam-6526	21	14	spaces	space	NOUN
ejpam-6526	21	15	,	,	PUNCT
ejpam-6526	21	16	while	while	SCONJ
ejpam-6526	21	17	chen	chen	PROPN
ejpam-6526	21	18	and	and	CCONJ
ejpam-6526	21	19	li	li	PROPN
ejpam-6526	22	1	[	[	X
ejpam-6526	22	2	4	4	NUM
ejpam-6526	22	3	]	]	PUNCT
ejpam-6526	22	4	developed	develop	VERB
ejpam-6526	22	5	applications	application	NOUN
ejpam-6526	22	6	in	in	ADP
ejpam-6526	22	7	fuzzy	fuzzy	ADJ
ejpam-6526	22	8	topology	topology	NOUN
ejpam-6526	22	9	.	.	PUNCT
ejpam-6526	23	1	the	the	DET
ejpam-6526	23	2	intersection	intersection	NOUN
ejpam-6526	23	3	with	with	ADP
ejpam-6526	23	4	computer	computer	NOUN
ejpam-6526	23	5	science	science	NOUN
ejpam-6526	23	6	has	have	AUX
ejpam-6526	23	7	been	be	AUX
ejpam-6526	23	8	particularly	particularly	ADV
ejpam-6526	23	9	fruitful	fruitful	ADJ
ejpam-6526	23	10	,	,	PUNCT
ejpam-6526	23	11	with	with	ADP
ejpam-6526	23	12	smyth	smyth	NOUN
ejpam-6526	23	13	[	[	X
ejpam-6526	23	14	5	5	NUM
ejpam-6526	23	15	]	]	PUNCT
ejpam-6526	23	16	demonstrating	demonstrate	VERB
ejpam-6526	23	17	applications	application	NOUN
ejpam-6526	23	18	in	in	ADP
ejpam-6526	23	19	domain	domain	NOUN
ejpam-6526	23	20	theory	theory	NOUN
ejpam-6526	23	21	and	and	CCONJ
ejpam-6526	23	22	denotational	denotational	ADJ
ejpam-6526	23	23	semantics	semantic	NOUN
ejpam-6526	23	24	.	.	PUNCT
ejpam-6526	24	1	furthermore	furthermore	ADV
ejpam-6526	24	2	,	,	PUNCT
ejpam-6526	24	3	recent	recent	ADJ
ejpam-6526	24	4	work	work	NOUN
ejpam-6526	24	5	by	by	ADP
ejpam-6526	24	6	kumar	kumar	PROPN
ejpam-6526	24	7	and	and	CCONJ
ejpam-6526	24	8	singh	singh	PROPN
ejpam-6526	25	1	[	[	X
ejpam-6526	25	2	6	6	NUM
ejpam-6526	25	3	]	]	PUNCT
ejpam-6526	25	4	has	have	AUX
ejpam-6526	25	5	connected	connect	VERB
ejpam-6526	25	6	bitopological	bitopological	ADJ
ejpam-6526	25	7	structures	structure	NOUN
ejpam-6526	25	8	to	to	ADP
ejpam-6526	25	9	rough	rough	ADJ
ejpam-6526	25	10	set	set	NOUN
ejpam-6526	25	11	theory	theory	NOUN
ejpam-6526	25	12	,	,	PUNCT
ejpam-6526	25	13	and	and	CCONJ
ejpam-6526	25	14	martinez	martinez	PROPN
ejpam-6526	25	15	et	et	PROPN
ejpam-6526	25	16	al	al	PROPN
ejpam-6526	25	17	.	.	PUNCT
ejpam-6526	26	1	[	[	X
ejpam-6526	26	2	7	7	X
ejpam-6526	26	3	]	]	PUNCT
ejpam-6526	26	4	have	have	AUX
ejpam-6526	26	5	explored	explore	VERB
ejpam-6526	26	6	applications	application	NOUN
ejpam-6526	26	7	in	in	ADP
ejpam-6526	26	8	data	datum	NOUN
ejpam-6526	26	9	analysis	analysis	NOUN
ejpam-6526	26	10	and	and	CCONJ
ejpam-6526	26	11	machine	machine	NOUN
ejpam-6526	26	12	learning	learning	NOUN
ejpam-6526	26	13	.	.	PUNCT
ejpam-6526	27	1	while	while	SCONJ
ejpam-6526	27	2	previous	previous	ADJ
ejpam-6526	27	3	scholarly	scholarly	ADJ
ejpam-6526	27	4	investigations	investigation	NOUN
ejpam-6526	27	5	have	have	AUX
ejpam-6526	27	6	examined	examine	VERB
ejpam-6526	27	7	relationships	relationship	NOUN
ejpam-6526	27	8	between	between	ADP
ejpam-6526	27	9	metric	metric	ADJ
ejpam-6526	27	10	characteristics	characteristic	NOUN
ejpam-6526	27	11	and	and	CCONJ
ejpam-6526	27	12	bitopological	bitopological	ADJ
ejpam-6526	27	13	spaces	space	NOUN
ejpam-6526	27	14	[	[	X
ejpam-6526	27	15	8	8	NUM
ejpam-6526	27	16	]	]	PUNCT
ejpam-6526	27	17	,	,	PUNCT
ejpam-6526	27	18	we	we	PRON
ejpam-6526	27	19	identify	identify	VERB
ejpam-6526	27	20	a	a	DET
ejpam-6526	27	21	substantial	substantial	ADJ
ejpam-6526	27	22	theoretical	theoretical	ADJ
ejpam-6526	27	23	gap	gap	NOUN
ejpam-6526	27	24	:	:	PUNCT
ejpam-6526	27	25	the	the	DET
ejpam-6526	27	26	absence	absence	NOUN
ejpam-6526	27	27	of	of	ADP
ejpam-6526	27	28	a	a	DET
ejpam-6526	27	29	comprehensive	comprehensive	ADJ
ejpam-6526	27	30	mathematical	mathematical	ADJ
ejpam-6526	27	31	architecture	architecture	NOUN
ejpam-6526	27	32	specifically	specifically	ADV
ejpam-6526	27	33	addressing	address	VERB
ejpam-6526	27	34	metric	metric	ADJ
ejpam-6526	27	35	systems	system	NOUN
ejpam-6526	27	36	purposefully	purposefully	ADV
ejpam-6526	27	37	designed	design	VERB
ejpam-6526	27	38	for	for	ADP
ejpam-6526	27	39	bitopological	bitopological	ADJ
ejpam-6526	27	40	environments	environment	NOUN
ejpam-6526	27	41	.	.	PUNCT
ejpam-6526	28	1	we	we	PRON
ejpam-6526	28	2	address	address	VERB
ejpam-6526	28	3	this	this	DET
ejpam-6526	28	4	deficiency	deficiency	NOUN
ejpam-6526	28	5	by	by	ADP
ejpam-6526	28	6	proposing	propose	VERB
ejpam-6526	28	7	an	an	DET
ejpam-6526	28	8	innovative	innovative	ADJ
ejpam-6526	28	9	conceptualization	conceptualization	NOUN
ejpam-6526	28	10	of	of	ADP
ejpam-6526	28	11	metric	metric	ADJ
ejpam-6526	28	12	assessment	assessment	NOUN
ejpam-6526	28	13	that	that	PRON
ejpam-6526	28	14	naturally	naturally	ADV
ejpam-6526	28	15	accommodates	accommodate	VERB
ejpam-6526	28	16	multiple	multiple	ADJ
ejpam-6526	28	17	topological	topological	ADJ
ejpam-6526	28	18	configurations	configuration	NOUN
ejpam-6526	28	19	.	.	PUNCT
ejpam-6526	29	1	our	our	PRON
ejpam-6526	29	2	methodology	methodology	NOUN
ejpam-6526	29	3	diverges	diverge	VERB
ejpam-6526	29	4	fundamentally	fundamentally	ADV
ejpam-6526	29	5	from	from	ADP
ejpam-6526	29	6	existing	exist	VERB
ejpam-6526	29	7	approaches	approach	NOUN
ejpam-6526	29	8	.	.	PUNCT
ejpam-6526	30	1	rather	rather	ADV
ejpam-6526	30	2	than	than	ADP
ejpam-6526	30	3	analyzing	analyze	VERB
ejpam-6526	30	4	independent	independent	ADJ
ejpam-6526	30	5	metric	metric	ADJ
ejpam-6526	30	6	functions	function	NOUN
ejpam-6526	30	7	applied	apply	VERB
ejpam-6526	30	8	across	across	ADP
ejpam-6526	30	9	identical	identical	ADJ
ejpam-6526	30	10	spaces	space	NOUN
ejpam-6526	30	11	,	,	PUNCT
ejpam-6526	30	12	we	we	PRON
ejpam-6526	30	13	develop	develop	VERB
ejpam-6526	30	14	an	an	DET
ejpam-6526	30	15	integrated	integrate	VERB
ejpam-6526	30	16	structural	structural	ADJ
ejpam-6526	30	17	framework	framework	NOUN
ejpam-6526	30	18	termed	term	VERB
ejpam-6526	30	19	a	a	DET
ejpam-6526	30	20	bi	bi	ADJ
ejpam-6526	30	21	-	-	ADJ
ejpam-6526	30	22	metric	metric	ADJ
ejpam-6526	30	23	system	system	NOUN
ejpam-6526	30	24	that	that	PRON
ejpam-6526	30	25	inherently	inherently	ADV
ejpam-6526	30	26	captures	capture	VERB
ejpam-6526	30	27	the	the	DET
ejpam-6526	30	28	multi	multi	ADJ
ejpam-6526	30	29	-	-	ADJ
ejpam-6526	30	30	dimensional	dimensional	ADJ
ejpam-6526	30	31	nature	nature	NOUN
ejpam-6526	30	32	of	of	ADP
ejpam-6526	30	33	bitopological	bitopological	ADJ
ejpam-6526	30	34	spaces	space	NOUN
ejpam-6526	30	35	.	.	PUNCT
ejpam-6526	31	1	this	this	DET
ejpam-6526	31	2	formulation	formulation	NOUN
ejpam-6526	31	3	extends	extend	VERB
ejpam-6526	31	4	and	and	CCONJ
ejpam-6526	31	5	generalizes	generalize	VERB
ejpam-6526	31	6	classical	classical	ADJ
ejpam-6526	31	7	metric	metric	ADJ
ejpam-6526	31	8	theory	theory	NOUN
ejpam-6526	31	9	established	establish	VERB
ejpam-6526	31	10	by	by	ADP
ejpam-6526	31	11	banach	banach	NOUN
ejpam-6526	31	12	[	[	X
ejpam-6526	31	13	9	9	NUM
ejpam-6526	31	14	]	]	PUNCT
ejpam-6526	31	15	.	.	PUNCT
ejpam-6526	32	1	the	the	DET
ejpam-6526	32	2	practical	practical	ADJ
ejpam-6526	32	3	significance	significance	NOUN
ejpam-6526	32	4	of	of	ADP
ejpam-6526	32	5	our	our	PRON
ejpam-6526	32	6	theoretical	theoretical	ADJ
ejpam-6526	32	7	framework	framework	NOUN
ejpam-6526	32	8	extends	extend	VERB
ejpam-6526	32	9	to	to	ADP
ejpam-6526	32	10	numerous	numerous	ADJ
ejpam-6526	32	11	real	real	ADJ
ejpam-6526	32	12	-	-	PUNCT
ejpam-6526	32	13	world	world	NOUN
ejpam-6526	32	14	applications	application	NOUN
ejpam-6526	32	15	.	.	PUNCT
ejpam-6526	33	1	in	in	ADP
ejpam-6526	33	2	computer	computer	NOUN
ejpam-6526	33	3	networks	network	NOUN
ejpam-6526	33	4	,	,	PUNCT
ejpam-6526	33	5	routing	route	VERB
ejpam-6526	33	6	algorithms	algorithm	NOUN
ejpam-6526	33	7	often	often	ADV
ejpam-6526	33	8	need	need	VERB
ejpam-6526	33	9	to	to	PART
ejpam-6526	33	10	optimize	optimize	VERB
ejpam-6526	33	11	for	for	ADP
ejpam-6526	33	12	both	both	PRON
ejpam-6526	33	13	physical	physical	ADJ
ejpam-6526	33	14	distance	distance	NOUN
ejpam-6526	33	15	and	and	CCONJ
ejpam-6526	33	16	transmission	transmission	NOUN
ejpam-6526	33	17	delay	delay	NOUN
ejpam-6526	33	18	.	.	PUNCT
ejpam-6526	34	1	in	in	ADP
ejpam-6526	34	2	image	image	NOUN
ejpam-6526	34	3	processing	processing	NOUN
ejpam-6526	34	4	,	,	PUNCT
ejpam-6526	34	5	quality	quality	NOUN
ejpam-6526	34	6	assessment	assessment	NOUN
ejpam-6526	34	7	requires	require	VERB
ejpam-6526	34	8	both	both	PRON
ejpam-6526	34	9	pixel	pixel	ADJ
ejpam-6526	34	10	-	-	ADJ
ejpam-6526	34	11	wise	wise	ADJ
ejpam-6526	34	12	accuracy	accuracy	NOUN
ejpam-6526	34	13	and	and	CCONJ
ejpam-6526	34	14	perceptual	perceptual	ADJ
ejpam-6526	34	15	similarity	similarity	NOUN
ejpam-6526	34	16	metrics	metric	NOUN
ejpam-6526	34	17	.	.	PUNCT
ejpam-6526	35	1	economic	economic	ADJ
ejpam-6526	35	2	models	model	NOUN
ejpam-6526	35	3	frequently	frequently	ADV
ejpam-6526	35	4	involve	involve	VERB
ejpam-6526	35	5	multi	multi	ADJ
ejpam-6526	35	6	-	-	ADJ
ejpam-6526	35	7	criteria	criteria	ADJ
ejpam-6526	35	8	optimization	optimization	NOUN
ejpam-6526	35	9	where	where	SCONJ
ejpam-6526	35	10	different	different	ADJ
ejpam-6526	35	11	metrics	metric	NOUN
ejpam-6526	35	12	capture	capture	VERB
ejpam-6526	35	13	distinct	distinct	ADJ
ejpam-6526	35	14	aspects	aspect	NOUN
ejpam-6526	35	15	of	of	ADP
ejpam-6526	35	16	system	system	NOUN
ejpam-6526	35	17	performance	performance	NOUN
ejpam-6526	35	18	.	.	PUNCT
ejpam-6526	36	1	this	this	DET
ejpam-6526	36	2	paper	paper	NOUN
ejpam-6526	36	3	presents	present	VERB
ejpam-6526	36	4	a	a	DET
ejpam-6526	36	5	comprehensive	comprehensive	ADJ
ejpam-6526	36	6	framework	framework	NOUN
ejpam-6526	36	7	for	for	ADP
ejpam-6526	36	8	bi	bi	ADJ
ejpam-6526	36	9	-	-	ADJ
ejpam-6526	36	10	metric	metric	ADJ
ejpam-6526	36	11	structures	structure	NOUN
ejpam-6526	36	12	in	in	ADP
ejpam-6526	36	13	bitopological	bitopological	ADJ
ejpam-6526	36	14	contexts	context	NOUN
ejpam-6526	36	15	.	.	PUNCT
ejpam-6526	37	1	we	we	PRON
ejpam-6526	37	2	establish	establish	VERB
ejpam-6526	37	3	fundamental	fundamental	ADJ
ejpam-6526	37	4	definitions	definition	NOUN
ejpam-6526	37	5	in	in	ADP
ejpam-6526	37	6	section	section	NOUN
ejpam-6526	37	7	2	2	NUM
ejpam-6526	37	8	,	,	PUNCT
ejpam-6526	37	9	providing	provide	VERB
ejpam-6526	37	10	essential	essential	ADJ
ejpam-6526	37	11	mathematical	mathematical	ADJ
ejpam-6526	37	12	foundations	foundation	NOUN
ejpam-6526	37	13	including	include	VERB
ejpam-6526	37	14	bitopological	bitopological	ADJ
ejpam-6526	37	15	spaces	space	NOUN
ejpam-6526	37	16	,	,	PUNCT
ejpam-6526	37	17	metric	metric	ADJ
ejpam-6526	37	18	spaces	space	NOUN
ejpam-6526	37	19	,	,	PUNCT
ejpam-6526	37	20	and	and	CCONJ
ejpam-6526	37	21	quasi	quasi	NOUN
ejpam-6526	37	22	-	-	NOUN
ejpam-6526	37	23	metrics	metric	NOUN
ejpam-6526	37	24	.	.	PUNCT
ejpam-6526	38	1	section	section	NOUN
ejpam-6526	38	2	3	3	NUM
ejpam-6526	38	3	introduces	introduce	VERB
ejpam-6526	38	4	bi	bi	ADJ
ejpam-6526	38	5	-	-	ADJ
ejpam-6526	38	6	metric	metric	ADJ
ejpam-6526	38	7	systems	system	NOUN
ejpam-6526	38	8	with	with	ADP
ejpam-6526	38	9	their	their	PRON
ejpam-6526	38	10	structural	structural	ADJ
ejpam-6526	38	11	properties	property	NOUN
ejpam-6526	38	12	,	,	PUNCT
ejpam-6526	38	13	illustrating	illustrate	VERB
ejpam-6526	38	14	these	these	DET
ejpam-6526	38	15	abstract	abstract	ADJ
ejpam-6526	38	16	concepts	concept	NOUN
ejpam-6526	38	17	through	through	ADP
ejpam-6526	38	18	concrete	concrete	ADJ
ejpam-6526	38	19	examples	example	NOUN
ejpam-6526	38	20	to	to	PART
ejpam-6526	38	21	facilitate	facilitate	VERB
ejpam-6526	38	22	comprehension	comprehension	NOUN
ejpam-6526	38	23	.	.	PUNCT
ejpam-6526	39	1	section	section	NOUN
ejpam-6526	39	2	4	4	NUM
ejpam-6526	39	3	examines	examine	VERB
ejpam-6526	39	4	the	the	DET
ejpam-6526	39	5	connections	connection	NOUN
ejpam-6526	39	6	between	between	ADP
ejpam-6526	39	7	bi	bi	ADJ
ejpam-6526	39	8	-	-	ADJ
ejpam-6526	39	9	metric	metric	ADJ
ejpam-6526	39	10	systems	system	NOUN
ejpam-6526	39	11	and	and	CCONJ
ejpam-6526	39	12	functional	functional	ADJ
ejpam-6526	39	13	analysis	analysis	NOUN
ejpam-6526	39	14	,	,	PUNCT
ejpam-6526	39	15	particularly	particularly	ADV
ejpam-6526	39	16	exploring	explore	VERB
ejpam-6526	39	17	relationships	relationship	NOUN
ejpam-6526	39	18	with	with	ADP
ejpam-6526	39	19	bi	bi	ADJ
ejpam-6526	39	20	-	-	ADJ
ejpam-6526	39	21	normed	normed	ADJ
ejpam-6526	39	22	spaces	space	NOUN
ejpam-6526	39	23	and	and	CCONJ
ejpam-6526	39	24	developing	develop	VERB
ejpam-6526	39	25	frameworks	framework	NOUN
ejpam-6526	39	26	that	that	PRON
ejpam-6526	39	27	encompass	encompass	VERB
ejpam-6526	39	28	primal	primal	ADJ
ejpam-6526	39	29	-	-	PUNCT
ejpam-6526	39	30	dual	dual	ADJ
ejpam-6526	39	31	configurations	configuration	NOUN
ejpam-6526	39	32	.	.	PUNCT
ejpam-6526	40	1	in	in	ADP
ejpam-6526	40	2	section	section	NOUN
ejpam-6526	40	3	5	5	NUM
ejpam-6526	40	4	,	,	PUNCT
ejpam-6526	40	5	we	we	PRON
ejpam-6526	40	6	investigate	investigate	VERB
ejpam-6526	40	7	advanced	advanced	ADJ
ejpam-6526	40	8	theoretical	theoretical	ADJ
ejpam-6526	40	9	properties	property	NOUN
ejpam-6526	40	10	including	include	VERB
ejpam-6526	40	11	contraction	contraction	NOUN
ejpam-6526	40	12	mapping	mapping	NOUN
ejpam-6526	40	13	principles	principle	NOUN
ejpam-6526	40	14	and	and	CCONJ
ejpam-6526	40	15	completeness	completeness	NOUN
ejpam-6526	40	16	characteristics	characteristic	NOUN
ejpam-6526	40	17	adapted	adapt	VERB
ejpam-6526	40	18	specifa	specifa	NOUN
ejpam-6526	40	19	.	.	PUNCT
ejpam-6526	41	1	alsoboh	alsoboh	PROPN
ejpam-6526	41	2	et	et	PROPN
ejpam-6526	41	3	al	al	PROPN
ejpam-6526	41	4	.	.	PUNCT
ejpam-6526	41	5	/	/	SYM
ejpam-6526	41	6	eur	eur	PROPN
ejpam-6526	41	7	.	.	PUNCT
ejpam-6526	42	1	j.	j.	PROPN
ejpam-6526	42	2	pure	pure	PROPN
ejpam-6526	42	3	appl	appl	PROPN
ejpam-6526	42	4	.	.	PROPN
ejpam-6526	42	5	math	math	PROPN
ejpam-6526	42	6	,	,	PUNCT
ejpam-6526	42	7	18	18	NUM
ejpam-6526	42	8	(	(	PUNCT
ejpam-6526	42	9	4	4	NUM
ejpam-6526	42	10	)	)	PUNCT
ejpam-6526	42	11	(	(	PUNCT
ejpam-6526	42	12	2025	2025	NUM
ejpam-6526	42	13	)	)	PUNCT
ejpam-6526	42	14	,	,	PUNCT
ejpam-6526	42	15	6526	6526	NUM
ejpam-6526	42	16	3	3	NUM
ejpam-6526	42	17	of	of	ADP
ejpam-6526	42	18	21	21	NUM
ejpam-6526	42	19	ically	ically	ADV
ejpam-6526	42	20	for	for	ADP
ejpam-6526	42	21	bi	bi	ADJ
ejpam-6526	42	22	-	-	ADJ
ejpam-6526	42	23	metric	metric	ADJ
ejpam-6526	42	24	environments	environment	NOUN
ejpam-6526	42	25	.	.	PUNCT
ejpam-6526	43	1	section	section	NOUN
ejpam-6526	43	2	6	6	NUM
ejpam-6526	43	3	expands	expand	VERB
ejpam-6526	43	4	our	our	PRON
ejpam-6526	43	5	analysis	analysis	NOUN
ejpam-6526	43	6	to	to	ADP
ejpam-6526	43	7	structural	structural	ADJ
ejpam-6526	43	8	properties	property	NOUN
ejpam-6526	43	9	and	and	CCONJ
ejpam-6526	43	10	applications	application	NOUN
ejpam-6526	43	11	,	,	PUNCT
ejpam-6526	43	12	addressing	address	VERB
ejpam-6526	43	13	density	density	NOUN
ejpam-6526	43	14	concepts	concept	NOUN
ejpam-6526	43	15	,	,	PUNCT
ejpam-6526	43	16	product	product	NOUN
ejpam-6526	43	17	structures	structure	NOUN
ejpam-6526	43	18	,	,	PUNCT
ejpam-6526	43	19	characterization	characterization	NOUN
ejpam-6526	43	20	theorems	theorem	NOUN
ejpam-6526	43	21	for	for	ADP
ejpam-6526	43	22	bitopological	bitopological	ADJ
ejpam-6526	43	23	spaces	space	NOUN
ejpam-6526	43	24	,	,	PUNCT
ejpam-6526	43	25	and	and	CCONJ
ejpam-6526	43	26	relationships	relationship	NOUN
ejpam-6526	43	27	with	with	ADP
ejpam-6526	43	28	quasi	quasi	NOUN
ejpam-6526	43	29	-	-	NOUN
ejpam-6526	43	30	metrics	metric	NOUN
ejpam-6526	43	31	,	,	PUNCT
ejpam-6526	43	32	while	while	SCONJ
ejpam-6526	43	33	presenting	present	VERB
ejpam-6526	43	34	experimental	experimental	ADJ
ejpam-6526	43	35	results	result	NOUN
ejpam-6526	43	36	that	that	PRON
ejpam-6526	43	37	validate	validate	VERB
ejpam-6526	43	38	our	our	PRON
ejpam-6526	43	39	theoretical	theoretical	ADJ
ejpam-6526	43	40	findings	finding	NOUN
ejpam-6526	43	41	.	.	PUNCT
ejpam-6526	44	1	the	the	DET
ejpam-6526	44	2	concluding	conclude	VERB
ejpam-6526	44	3	section	section	NOUN
ejpam-6526	44	4	summarizes	summarize	NOUN
ejpam-6526	44	5	contributions	contribution	NOUN
ejpam-6526	44	6	.	.	PUNCT
ejpam-6526	45	1	in	in	ADP
ejpam-6526	45	2	future	future	ADJ
ejpam-6526	45	3	research	research	NOUN
ejpam-6526	45	4	,	,	PUNCT
ejpam-6526	45	5	an	an	DET
ejpam-6526	45	6	intriguing	intriguing	ADJ
ejpam-6526	45	7	direction	direction	NOUN
ejpam-6526	45	8	lies	lie	VERB
ejpam-6526	45	9	in	in	ADP
ejpam-6526	45	10	exploring	explore	VERB
ejpam-6526	45	11	the	the	DET
ejpam-6526	45	12	interplay	interplay	NOUN
ejpam-6526	45	13	between	between	ADP
ejpam-6526	45	14	bimetric	bimetric	ADJ
ejpam-6526	45	15	structures	structure	NOUN
ejpam-6526	45	16	and	and	CCONJ
ejpam-6526	45	17	complex	complex	ADJ
ejpam-6526	45	18	analysis	analysis	NOUN
ejpam-6526	45	19	,	,	PUNCT
ejpam-6526	45	20	particularly	particularly	ADV
ejpam-6526	45	21	within	within	ADP
ejpam-6526	45	22	bitopological	bitopological	ADJ
ejpam-6526	45	23	frameworks	framework	NOUN
ejpam-6526	45	24	.	.	PUNCT
ejpam-6526	46	1	bimetric	bimetric	ADJ
ejpam-6526	46	2	spaces	space	NOUN
ejpam-6526	46	3	endowed	endow	VERB
ejpam-6526	46	4	with	with	ADP
ejpam-6526	46	5	two	two	NUM
ejpam-6526	46	6	compatible	compatible	ADJ
ejpam-6526	46	7	metrics	metric	NOUN
ejpam-6526	46	8	offer	offer	VERB
ejpam-6526	46	9	a	a	DET
ejpam-6526	46	10	natural	natural	ADJ
ejpam-6526	46	11	setting	setting	NOUN
ejpam-6526	46	12	for	for	ADP
ejpam-6526	46	13	extending	extend	VERB
ejpam-6526	46	14	classical	classical	ADJ
ejpam-6526	46	15	notions	notion	NOUN
ejpam-6526	46	16	of	of	ADP
ejpam-6526	46	17	convergence	convergence	NOUN
ejpam-6526	46	18	,	,	PUNCT
ejpam-6526	46	19	continuity	continuity	NOUN
ejpam-6526	46	20	,	,	PUNCT
ejpam-6526	46	21	and	and	CCONJ
ejpam-6526	46	22	analyticity	analyticity	NOUN
ejpam-6526	46	23	to	to	ADP
ejpam-6526	46	24	more	more	ADV
ejpam-6526	46	25	generalized	generalized	ADJ
ejpam-6526	46	26	environments	environment	NOUN
ejpam-6526	46	27	.	.	PUNCT
ejpam-6526	47	1	when	when	SCONJ
ejpam-6526	47	2	combined	combine	VERB
ejpam-6526	47	3	with	with	ADP
ejpam-6526	47	4	tools	tool	NOUN
ejpam-6526	47	5	from	from	ADP
ejpam-6526	47	6	complex	complex	ADJ
ejpam-6526	47	7	analysis	analysis	NOUN
ejpam-6526	47	8	,	,	PUNCT
ejpam-6526	47	9	such	such	ADJ
ejpam-6526	47	10	as	as	ADP
ejpam-6526	47	11	conformal	conformal	ADJ
ejpam-6526	47	12	mappings	mapping	NOUN
ejpam-6526	47	13	and	and	CCONJ
ejpam-6526	47	14	analytic	analytic	ADJ
ejpam-6526	47	15	function	function	NOUN
ejpam-6526	47	16	theory	theory	NOUN
ejpam-6526	47	17	,	,	PUNCT
ejpam-6526	47	18	these	these	DET
ejpam-6526	47	19	structures	structure	NOUN
ejpam-6526	47	20	could	could	AUX
ejpam-6526	47	21	yield	yield	VERB
ejpam-6526	47	22	new	new	ADJ
ejpam-6526	47	23	geometric	geometric	ADJ
ejpam-6526	47	24	interpretations	interpretation	NOUN
ejpam-6526	47	25	of	of	ADP
ejpam-6526	47	26	bi	bi	ADJ
ejpam-6526	47	27	-	-	ADJ
ejpam-6526	47	28	univalent	univalent	ADJ
ejpam-6526	47	29	and	and	CCONJ
ejpam-6526	47	30	multi	multi	ADJ
ejpam-6526	47	31	-	-	ADJ
ejpam-6526	47	32	univalent	univalent	ADJ
ejpam-6526	47	33	functions	function	NOUN
ejpam-6526	47	34	[	[	X
ejpam-6526	47	35	10–15	10–15	NUM
ejpam-6526	47	36	]	]	PUNCT
ejpam-6526	47	37	,	,	PUNCT
ejpam-6526	47	38	as	as	ADV
ejpam-6526	47	39	well	well	ADV
ejpam-6526	47	40	as	as	ADP
ejpam-6526	47	41	novel	novel	ADJ
ejpam-6526	47	42	characterizations	characterization	NOUN
ejpam-6526	47	43	of	of	ADP
ejpam-6526	47	44	analytic	analytic	ADJ
ejpam-6526	47	45	mappings	mapping	NOUN
ejpam-6526	47	46	between	between	ADP
ejpam-6526	47	47	dual	dual	ADJ
ejpam-6526	47	48	topological	topological	ADJ
ejpam-6526	47	49	systems	system	NOUN
ejpam-6526	47	50	.	.	PUNCT
ejpam-6526	48	1	moreover	moreover	ADV
ejpam-6526	48	2	,	,	PUNCT
ejpam-6526	48	3	the	the	DET
ejpam-6526	48	4	synthesis	synthesis	NOUN
ejpam-6526	48	5	of	of	ADP
ejpam-6526	48	6	bimetric	bimetric	ADJ
ejpam-6526	48	7	geometry	geometry	NOUN
ejpam-6526	48	8	with	with	ADP
ejpam-6526	48	9	complex	complex	ADJ
ejpam-6526	48	10	analytic	analytic	ADJ
ejpam-6526	48	11	methods	method	NOUN
ejpam-6526	48	12	may	may	AUX
ejpam-6526	48	13	provide	provide	VERB
ejpam-6526	48	14	a	a	DET
ejpam-6526	48	15	foundation	foundation	NOUN
ejpam-6526	48	16	for	for	ADP
ejpam-6526	48	17	modeling	model	VERB
ejpam-6526	48	18	dual	dual	ADJ
ejpam-6526	48	19	phase	phase	NOUN
ejpam-6526	48	20	systems	system	NOUN
ejpam-6526	48	21	,	,	PUNCT
ejpam-6526	48	22	complex	complex	ADJ
ejpam-6526	48	23	dynamical	dynamical	ADJ
ejpam-6526	48	24	behaviors	behavior	NOUN
ejpam-6526	48	25	,	,	PUNCT
ejpam-6526	48	26	and	and	CCONJ
ejpam-6526	48	27	operator	operator	NOUN
ejpam-6526	48	28	-	-	PUNCT
ejpam-6526	48	29	theoretic	theoretic	NOUN
ejpam-6526	48	30	generalizations	generalization	NOUN
ejpam-6526	48	31	in	in	ADP
ejpam-6526	48	32	functional	functional	ADJ
ejpam-6526	48	33	spaces	space	NOUN
ejpam-6526	48	34	,	,	PUNCT
ejpam-6526	48	35	thus	thus	ADV
ejpam-6526	48	36	opening	open	VERB
ejpam-6526	48	37	new	new	ADJ
ejpam-6526	48	38	pathways	pathway	NOUN
ejpam-6526	48	39	for	for	ADP
ejpam-6526	48	40	both	both	DET
ejpam-6526	48	41	pure	pure	ADJ
ejpam-6526	48	42	mathematical	mathematical	ADJ
ejpam-6526	48	43	theory	theory	NOUN
ejpam-6526	48	44	and	and	CCONJ
ejpam-6526	48	45	applied	apply	VERB
ejpam-6526	48	46	geometric	geometric	ADJ
ejpam-6526	48	47	function	function	NOUN
ejpam-6526	48	48	research	research	NOUN
ejpam-6526	48	49	.	.	PUNCT
ejpam-6526	49	1	2	2	X
ejpam-6526	49	2	.	.	X
ejpam-6526	49	3	fundamental	fundamental	ADJ
ejpam-6526	49	4	definitions	definition	NOUN
ejpam-6526	49	5	we	we	PRON
ejpam-6526	49	6	establish	establish	VERB
ejpam-6526	49	7	our	our	PRON
ejpam-6526	49	8	theoretical	theoretical	ADJ
ejpam-6526	49	9	foundation	foundation	NOUN
ejpam-6526	49	10	with	with	ADP
ejpam-6526	49	11	several	several	ADJ
ejpam-6526	49	12	core	core	NOUN
ejpam-6526	49	13	definitions	definition	NOUN
ejpam-6526	49	14	aligned	align	VERB
ejpam-6526	49	15	with	with	ADP
ejpam-6526	49	16	established	establish	VERB
ejpam-6526	49	17	topological	topological	ADJ
ejpam-6526	49	18	literature	literature	NOUN
ejpam-6526	50	1	[	[	X
ejpam-6526	50	2	16	16	NUM
ejpam-6526	50	3	]	]	PUNCT
ejpam-6526	50	4	,	,	PUNCT
ejpam-6526	50	5	recent	recent	ADJ
ejpam-6526	50	6	advances	advance	NOUN
ejpam-6526	50	7	[	[	X
ejpam-6526	50	8	17	17	NUM
ejpam-6526	50	9	]	]	PUNCT
ejpam-6526	50	10	.	.	PUNCT
ejpam-6526	51	1	definition	definition	NOUN
ejpam-6526	51	2	1	1	NUM
ejpam-6526	51	3	.	.	PUNCT
ejpam-6526	52	1	a	a	DET
ejpam-6526	52	2	bitopological	bitopological	ADJ
ejpam-6526	52	3	space	space	NOUN
ejpam-6526	52	4	consists	consist	VERB
ejpam-6526	52	5	of	of	ADP
ejpam-6526	52	6	a	a	DET
ejpam-6526	52	7	triple	triple	ADJ
ejpam-6526	52	8	(	(	PUNCT
ejpam-6526	52	9	x	x	NOUN
ejpam-6526	52	10	,	,	PUNCT
ejpam-6526	52	11	t1	t1	NOUN
ejpam-6526	52	12	,	,	PUNCT
ejpam-6526	52	13	t2	t2	NOUN
ejpam-6526	52	14	)	)	PUNCT
ejpam-6526	52	15	where	where	SCONJ
ejpam-6526	52	16	x	x	PRON
ejpam-6526	52	17	represents	represent	VERB
ejpam-6526	52	18	a	a	DET
ejpam-6526	52	19	non	non	ADJ
ejpam-6526	52	20	-	-	ADJ
ejpam-6526	52	21	empty	empty	ADJ
ejpam-6526	52	22	set	set	NOUN
ejpam-6526	52	23	and	and	CCONJ
ejpam-6526	52	24	t1	t1	NOUN
ejpam-6526	52	25	,	,	PUNCT
ejpam-6526	52	26	t2	t2	PROPN
ejpam-6526	52	27	denote	denote	VERB
ejpam-6526	52	28	distinct	distinct	ADJ
ejpam-6526	52	29	topological	topological	ADJ
ejpam-6526	52	30	structures	structure	NOUN
ejpam-6526	52	31	on	on	ADP
ejpam-6526	52	32	x.	x.	NOUN
ejpam-6526	52	33	definition	definition	NOUN
ejpam-6526	52	34	2	2	NUM
ejpam-6526	52	35	.	.	PUNCT
ejpam-6526	53	1	a	a	DET
ejpam-6526	53	2	metric	metric	ADJ
ejpam-6526	53	3	space	space	NOUN
ejpam-6526	53	4	comprises	comprise	VERB
ejpam-6526	53	5	a	a	DET
ejpam-6526	53	6	pair	pair	NOUN
ejpam-6526	53	7	(	(	PUNCT
ejpam-6526	53	8	x,ϕ	x,ϕ	NOUN
ejpam-6526	53	9	)	)	PUNCT
ejpam-6526	53	10	where	where	SCONJ
ejpam-6526	53	11	x	x	PRON
ejpam-6526	53	12	represents	represent	VERB
ejpam-6526	53	13	a	a	DET
ejpam-6526	53	14	non	non	ADJ
ejpam-6526	53	15	-	-	ADJ
ejpam-6526	53	16	empty	empty	ADJ
ejpam-6526	53	17	set	set	NOUN
ejpam-6526	53	18	and	and	CCONJ
ejpam-6526	53	19	ϕ	ϕ	NOUN
ejpam-6526	53	20	:	:	PUNCT
ejpam-6526	53	21	x	x	PROPN
ejpam-6526	53	22	×x	×x	ADP
ejpam-6526	53	23	→	→	NOUN
ejpam-6526	53	24	r+	r+	NOUN
ejpam-6526	53	25	functions	function	NOUN
ejpam-6526	53	26	as	as	ADP
ejpam-6526	53	27	a	a	DET
ejpam-6526	53	28	mapping	mapping	NOUN
ejpam-6526	53	29	satisfying	satisfying	ADJ
ejpam-6526	53	30	:	:	PUNCT
ejpam-6526	53	31	(	(	PUNCT
ejpam-6526	53	32	i	i	NOUN
ejpam-6526	53	33	)	)	PUNCT
ejpam-6526	53	34	ϕ(x	ϕ(x	PROPN
ejpam-6526	53	35	,	,	PUNCT
ejpam-6526	53	36	y	y	PROPN
ejpam-6526	53	37	)	)	PUNCT
ejpam-6526	53	38	≥	≥	NOUN
ejpam-6526	53	39	0	0	NUM
ejpam-6526	53	40	for	for	ADP
ejpam-6526	53	41	all	all	DET
ejpam-6526	53	42	x	x	NOUN
ejpam-6526	53	43	,	,	PUNCT
ejpam-6526	53	44	y	y	PROPN
ejpam-6526	53	45	∈	∈	PROPN
ejpam-6526	53	46	x	x	X
ejpam-6526	53	47	,	,	PUNCT
ejpam-6526	53	48	with	with	ADP
ejpam-6526	53	49	equality	equality	NOUN
ejpam-6526	53	50	if	if	SCONJ
ejpam-6526	53	51	and	and	CCONJ
ejpam-6526	53	52	only	only	ADV
ejpam-6526	53	53	if	if	SCONJ
ejpam-6526	53	54	x	x	X
ejpam-6526	53	55	=	=	SYM
ejpam-6526	53	56	y	y	PROPN
ejpam-6526	53	57	(	(	PUNCT
ejpam-6526	53	58	ii	ii	NOUN
ejpam-6526	53	59	)	)	PUNCT
ejpam-6526	53	60	ϕ(x	ϕ(x	PROPN
ejpam-6526	53	61	,	,	PUNCT
ejpam-6526	53	62	y	y	NOUN
ejpam-6526	53	63	)	)	PUNCT
ejpam-6526	53	64	=	=	SYM
ejpam-6526	54	1	ϕ(y	ϕ(y	PROPN
ejpam-6526	54	2	,	,	PUNCT
ejpam-6526	54	3	x	x	NOUN
ejpam-6526	54	4	)	)	PUNCT
ejpam-6526	54	5	for	for	ADP
ejpam-6526	54	6	all	all	DET
ejpam-6526	54	7	x	x	NOUN
ejpam-6526	54	8	,	,	PUNCT
ejpam-6526	54	9	y	y	PROPN
ejpam-6526	54	10	∈	∈	PROPN
ejpam-6526	54	11	x	x	INTJ
ejpam-6526	54	12	(	(	PUNCT
ejpam-6526	54	13	iii	iii	NOUN
ejpam-6526	54	14	)	)	PUNCT
ejpam-6526	54	15	ϕ(x	ϕ(x	NOUN
ejpam-6526	54	16	,	,	PUNCT
ejpam-6526	54	17	z	z	NOUN
ejpam-6526	54	18	)	)	PUNCT
ejpam-6526	54	19	≤	≤	NOUN
ejpam-6526	54	20	ϕ(x	ϕ(x	PROPN
ejpam-6526	54	21	,	,	PUNCT
ejpam-6526	54	22	y	y	NOUN
ejpam-6526	54	23	)	)	PUNCT
ejpam-6526	54	24	+	+	CCONJ
ejpam-6526	54	25	ϕ(y	ϕ(y	PROPN
ejpam-6526	54	26	,	,	PUNCT
ejpam-6526	54	27	z	z	NOUN
ejpam-6526	54	28	)	)	PUNCT
ejpam-6526	54	29	for	for	ADP
ejpam-6526	54	30	all	all	DET
ejpam-6526	54	31	x	x	NOUN
ejpam-6526	54	32	,	,	PUNCT
ejpam-6526	54	33	y	y	PROPN
ejpam-6526	54	34	,	,	PUNCT
ejpam-6526	54	35	z	z	NOUN
ejpam-6526	54	36	∈	∈	PROPN
ejpam-6526	54	37	x	x	PUNCT
ejpam-6526	54	38	definition	definition	NOUN
ejpam-6526	54	39	3	3	NUM
ejpam-6526	54	40	.	.	PUNCT
ejpam-6526	55	1	we	we	PRON
ejpam-6526	55	2	classify	classify	VERB
ejpam-6526	55	3	a	a	DET
ejpam-6526	55	4	function	function	NOUN
ejpam-6526	55	5	q	q	NOUN
ejpam-6526	55	6	:	:	PUNCT
ejpam-6526	55	7	x	x	X
ejpam-6526	55	8	×x	×x	X
ejpam-6526	55	9	→	→	X
ejpam-6526	55	10	[	[	X
ejpam-6526	55	11	0,∞	0,∞	NOUN
ejpam-6526	55	12	)	)	PUNCT
ejpam-6526	55	13	as	as	ADP
ejpam-6526	55	14	a	a	DET
ejpam-6526	55	15	quasi	quasi	NOUN
ejpam-6526	55	16	-	-	ADJ
ejpam-6526	55	17	metric	metric	ADJ
ejpam-6526	55	18	on	on	ADP
ejpam-6526	55	19	x	x	SYM
ejpam-6526	55	20	when	when	SCONJ
ejpam-6526	55	21	:	:	PUNCT
ejpam-6526	55	22	(	(	PUNCT
ejpam-6526	55	23	i	i	NOUN
ejpam-6526	55	24	)	)	PUNCT
ejpam-6526	55	25	q(x	q(x	PROPN
ejpam-6526	55	26	,	,	PUNCT
ejpam-6526	55	27	y	y	NOUN
ejpam-6526	55	28	)	)	PUNCT
ejpam-6526	55	29	=	=	SYM
ejpam-6526	55	30	0	0	PUNCT
ejpam-6526	56	1	if	if	SCONJ
ejpam-6526	56	2	and	and	CCONJ
ejpam-6526	56	3	only	only	ADV
ejpam-6526	56	4	if	if	SCONJ
ejpam-6526	56	5	x	x	X
ejpam-6526	56	6	=	=	SYM
ejpam-6526	56	7	y	y	PROPN
ejpam-6526	56	8	(	(	PUNCT
ejpam-6526	56	9	ii	ii	NOUN
ejpam-6526	56	10	)	)	PUNCT
ejpam-6526	56	11	q(x	q(x	PROPN
ejpam-6526	56	12	,	,	PUNCT
ejpam-6526	56	13	z	z	NOUN
ejpam-6526	56	14	)	)	PUNCT
ejpam-6526	56	15	≤	≤	NUM
ejpam-6526	56	16	q(x	q(x	PROPN
ejpam-6526	56	17	,	,	PUNCT
ejpam-6526	56	18	y	y	NOUN
ejpam-6526	56	19	)	)	PUNCT
ejpam-6526	57	1	+	+	NOUN
ejpam-6526	57	2	q(y	q(y	NOUN
ejpam-6526	57	3	,	,	PUNCT
ejpam-6526	57	4	z	z	NOUN
ejpam-6526	57	5	)	)	PUNCT
ejpam-6526	57	6	for	for	ADP
ejpam-6526	57	7	all	all	DET
ejpam-6526	57	8	x	x	NOUN
ejpam-6526	57	9	,	,	PUNCT
ejpam-6526	57	10	y	y	PROPN
ejpam-6526	57	11	,	,	PUNCT
ejpam-6526	57	12	z	z	NOUN
ejpam-6526	57	13	∈	∈	PROPN
ejpam-6526	57	14	x	x	INTJ
ejpam-6526	57	15	we	we	PRON
ejpam-6526	57	16	observe	observe	VERB
ejpam-6526	57	17	that	that	SCONJ
ejpam-6526	57	18	quasi	quasi	NOUN
ejpam-6526	57	19	-	-	NOUN
ejpam-6526	57	20	metrics	metric	NOUN
ejpam-6526	57	21	typically	typically	ADV
ejpam-6526	57	22	lack	lack	VERB
ejpam-6526	57	23	symmetrical	symmetrical	ADJ
ejpam-6526	57	24	properties	property	NOUN
ejpam-6526	57	25	.	.	PUNCT
ejpam-6526	58	1	definition	definition	NOUN
ejpam-6526	58	2	4	4	NUM
ejpam-6526	58	3	.	.	PUNCT
ejpam-6526	59	1	a	a	DET
ejpam-6526	59	2	pseudo	pseudo	NOUN
ejpam-6526	59	3	-	-	ADJ
ejpam-6526	59	4	metric	metric	ADJ
ejpam-6526	59	5	on	on	ADP
ejpam-6526	59	6	a	a	DET
ejpam-6526	59	7	set	set	NOUN
ejpam-6526	59	8	x	x	PUNCT
ejpam-6526	59	9	is	be	AUX
ejpam-6526	59	10	a	a	DET
ejpam-6526	59	11	function	function	NOUN
ejpam-6526	59	12	p	p	NOUN
ejpam-6526	59	13	:	:	PUNCT
ejpam-6526	59	14	x	x	PROPN
ejpam-6526	59	15	×x	×x	X
ejpam-6526	59	16	→	→	X
ejpam-6526	59	17	[	[	X
ejpam-6526	59	18	0,∞	0,∞	NOUN
ejpam-6526	59	19	)	)	PUNCT
ejpam-6526	59	20	satisfying	satisfy	VERB
ejpam-6526	59	21	all	all	DET
ejpam-6526	59	22	metric	metric	ADJ
ejpam-6526	59	23	axioms	axiom	NOUN
ejpam-6526	59	24	except	except	SCONJ
ejpam-6526	59	25	that	that	SCONJ
ejpam-6526	59	26	p(x	p(x	PROPN
ejpam-6526	59	27	,	,	PUNCT
ejpam-6526	59	28	y	y	NOUN
ejpam-6526	59	29	)	)	PUNCT
ejpam-6526	60	1	=	=	SYM
ejpam-6526	60	2	0	0	PROPN
ejpam-6526	60	3	does	do	AUX
ejpam-6526	60	4	not	not	PART
ejpam-6526	60	5	necessarily	necessarily	ADV
ejpam-6526	60	6	imply	imply	VERB
ejpam-6526	60	7	x	x	PUNCT
ejpam-6526	60	8	=	=	PUNCT
ejpam-6526	60	9	y.	y.	NOUN
ejpam-6526	60	10	this	this	DET
ejpam-6526	60	11	concept	concept	NOUN
ejpam-6526	60	12	becomes	become	VERB
ejpam-6526	60	13	relevant	relevant	ADJ
ejpam-6526	60	14	when	when	SCONJ
ejpam-6526	60	15	considering	consider	VERB
ejpam-6526	60	16	quotient	quotient	NOUN
ejpam-6526	60	17	structures	structure	NOUN
ejpam-6526	60	18	in	in	ADP
ejpam-6526	60	19	bi	bi	ADJ
ejpam-6526	60	20	-	-	ADJ
ejpam-6526	60	21	metric	metric	ADJ
ejpam-6526	60	22	systems	system	NOUN
ejpam-6526	60	23	.	.	PUNCT
ejpam-6526	61	1	a.	a.	PROPN
ejpam-6526	61	2	alsoboh	alsoboh	PROPN
ejpam-6526	61	3	et	et	PROPN
ejpam-6526	61	4	al	al	PROPN
ejpam-6526	61	5	.	.	PUNCT
ejpam-6526	61	6	/	/	SYM
ejpam-6526	61	7	eur	eur	PROPN
ejpam-6526	61	8	.	.	PUNCT
ejpam-6526	62	1	j.	j.	PROPN
ejpam-6526	62	2	pure	pure	PROPN
ejpam-6526	62	3	appl	appl	PROPN
ejpam-6526	62	4	.	.	PROPN
ejpam-6526	62	5	math	math	PROPN
ejpam-6526	62	6	,	,	PUNCT
ejpam-6526	62	7	18	18	NUM
ejpam-6526	62	8	(	(	PUNCT
ejpam-6526	62	9	4	4	NUM
ejpam-6526	62	10	)	)	PUNCT
ejpam-6526	62	11	(	(	PUNCT
ejpam-6526	62	12	2025	2025	NUM
ejpam-6526	62	13	)	)	PUNCT
ejpam-6526	62	14	,	,	PUNCT
ejpam-6526	62	15	6526	6526	NUM
ejpam-6526	62	16	4	4	NUM
ejpam-6526	62	17	of	of	ADP
ejpam-6526	62	18	21	21	NUM
ejpam-6526	62	19	3	3	NUM
ejpam-6526	62	20	.	.	PUNCT
ejpam-6526	63	1	bi	bi	ADJ
ejpam-6526	63	2	-	-	ADJ
ejpam-6526	63	3	metric	metric	ADJ
ejpam-6526	63	4	systems	system	NOUN
ejpam-6526	63	5	:	:	PUNCT
ejpam-6526	63	6	definition	definition	NOUN
ejpam-6526	63	7	and	and	CCONJ
ejpam-6526	63	8	structural	structural	ADJ
ejpam-6526	63	9	properties	property	NOUN
ejpam-6526	63	10	we	we	PRON
ejpam-6526	63	11	introduce	introduce	VERB
ejpam-6526	63	12	bi	bi	ADJ
ejpam-6526	63	13	-	-	ADJ
ejpam-6526	63	14	metric	metric	ADJ
ejpam-6526	63	15	systems	system	NOUN
ejpam-6526	63	16	,	,	PUNCT
ejpam-6526	63	17	a	a	DET
ejpam-6526	63	18	generalized	generalized	ADJ
ejpam-6526	63	19	mathematical	mathematical	ADJ
ejpam-6526	63	20	framework	framework	NOUN
ejpam-6526	63	21	that	that	PRON
ejpam-6526	63	22	naturally	naturally	ADV
ejpam-6526	63	23	accommodates	accommodate	VERB
ejpam-6526	63	24	bitopological	bitopological	ADJ
ejpam-6526	63	25	arrangements	arrangement	NOUN
ejpam-6526	63	26	.	.	PUNCT
ejpam-6526	64	1	this	this	DET
ejpam-6526	64	2	structured	structured	ADJ
ejpam-6526	64	3	approach	approach	NOUN
ejpam-6526	64	4	provides	provide	VERB
ejpam-6526	64	5	a	a	DET
ejpam-6526	64	6	unified	unified	ADJ
ejpam-6526	64	7	methodology	methodology	NOUN
ejpam-6526	64	8	for	for	ADP
ejpam-6526	64	9	analyzing	analyze	VERB
ejpam-6526	64	10	spaces	space	NOUN
ejpam-6526	64	11	with	with	ADP
ejpam-6526	64	12	distinct	distinct	ADJ
ejpam-6526	64	13	yet	yet	CCONJ
ejpam-6526	64	14	interrelated	interrelated	ADJ
ejpam-6526	64	15	metric	metric	ADJ
ejpam-6526	64	16	measures	measure	NOUN
ejpam-6526	64	17	.	.	PUNCT
ejpam-6526	65	1	definition	definition	NOUN
ejpam-6526	65	2	5	5	NUM
ejpam-6526	65	3	.	.	PUNCT
ejpam-6526	66	1	a	a	DET
ejpam-6526	66	2	bi	bi	ADJ
ejpam-6526	66	3	-	-	ADJ
ejpam-6526	66	4	metric	metric	ADJ
ejpam-6526	66	5	system	system	NOUN
ejpam-6526	66	6	consists	consist	VERB
ejpam-6526	66	7	of	of	ADP
ejpam-6526	66	8	a	a	DET
ejpam-6526	66	9	triple	triple	ADJ
ejpam-6526	66	10	(	(	PUNCT
ejpam-6526	66	11	x	x	NOUN
ejpam-6526	66	12	,	,	PUNCT
ejpam-6526	66	13	ψ,⊕	ψ,⊕	NOUN
ejpam-6526	66	14	)	)	PUNCT
ejpam-6526	66	15	where	where	SCONJ
ejpam-6526	66	16	x	x	PRON
ejpam-6526	66	17	represents	represent	VERB
ejpam-6526	66	18	a	a	DET
ejpam-6526	66	19	non	non	ADJ
ejpam-6526	66	20	-	-	ADJ
ejpam-6526	66	21	empty	empty	ADJ
ejpam-6526	66	22	set	set	NOUN
ejpam-6526	66	23	,	,	PUNCT
ejpam-6526	66	24	and	and	CCONJ
ejpam-6526	66	25	ψ	ψ	X
ejpam-6526	66	26	:	:	PUNCT
ejpam-6526	66	27	x	x	X
ejpam-6526	66	28	×x	×x	NUM
ejpam-6526	66	29	→	→	SYM
ejpam-6526	66	30	r+	r+	NOUN
ejpam-6526	66	31	×	×	NOUN
ejpam-6526	66	32	r+	r+	NOUN
ejpam-6526	66	33	functions	function	NOUN
ejpam-6526	66	34	as	as	ADP
ejpam-6526	66	35	a	a	DET
ejpam-6526	66	36	mapping	mapping	NOUN
ejpam-6526	66	37	satisfying	satisfying	ADJ
ejpam-6526	66	38	:	:	PUNCT
ejpam-6526	66	39	(	(	PUNCT
ejpam-6526	66	40	i	i	NOUN
ejpam-6526	66	41	)	)	PUNCT
ejpam-6526	66	42	ψ(x	ψ(x	PROPN
ejpam-6526	66	43	,	,	PUNCT
ejpam-6526	66	44	y	y	NOUN
ejpam-6526	66	45	)	)	PUNCT
ejpam-6526	66	46	=	=	SYM
ejpam-6526	66	47	(	(	PUNCT
ejpam-6526	66	48	ϕ1(x	ϕ1(x	PROPN
ejpam-6526	66	49	,	,	PUNCT
ejpam-6526	66	50	y	y	NOUN
ejpam-6526	66	51	)	)	PUNCT
ejpam-6526	66	52	,	,	PUNCT
ejpam-6526	66	53	ϕ2(x	ϕ2(x	PROPN
ejpam-6526	66	54	,	,	PUNCT
ejpam-6526	66	55	y	y	NOUN
ejpam-6526	66	56	)	)	PUNCT
ejpam-6526	66	57	)	)	PUNCT
ejpam-6526	66	58	where	where	SCONJ
ejpam-6526	66	59	ϕi	ϕi	ADP
ejpam-6526	66	60	:	:	PUNCT
ejpam-6526	66	61	x	x	X
ejpam-6526	66	62	×x	×x	ADP
ejpam-6526	66	63	→	→	SYM
ejpam-6526	66	64	r+	r+	NOUN
ejpam-6526	66	65	for	for	SCONJ
ejpam-6526	66	66	i	i	PROPN
ejpam-6526	66	67	∈	∈	PROPN
ejpam-6526	66	68	{	{	PUNCT
ejpam-6526	66	69	1	1	NUM
ejpam-6526	66	70	,	,	PUNCT
ejpam-6526	66	71	2	2	NUM
ejpam-6526	66	72	}	}	PUNCT
ejpam-6526	66	73	serve	serve	VERB
ejpam-6526	66	74	as	as	ADP
ejpam-6526	66	75	metric	metric	ADJ
ejpam-6526	66	76	functions	function	NOUN
ejpam-6526	66	77	(	(	PUNCT
ejpam-6526	66	78	ii	ii	NOUN
ejpam-6526	66	79	)	)	PUNCT
ejpam-6526	66	80	ψ(x	ψ(x	PROPN
ejpam-6526	66	81	,	,	PUNCT
ejpam-6526	66	82	y	y	NOUN
ejpam-6526	66	83	)	)	PUNCT
ejpam-6526	66	84	=	=	SYM
ejpam-6526	66	85	(	(	PUNCT
ejpam-6526	66	86	0	0	NUM
ejpam-6526	66	87	,	,	PUNCT
ejpam-6526	66	88	0	0	NUM
ejpam-6526	66	89	)	)	PUNCT
ejpam-6526	66	90	if	if	SCONJ
ejpam-6526	66	91	and	and	CCONJ
ejpam-6526	66	92	only	only	ADV
ejpam-6526	66	93	if	if	SCONJ
ejpam-6526	66	94	x	x	X
ejpam-6526	66	95	=	=	SYM
ejpam-6526	66	96	y	y	PROPN
ejpam-6526	66	97	(	(	PUNCT
ejpam-6526	66	98	iii	iii	NOUN
ejpam-6526	66	99	)	)	PUNCT
ejpam-6526	66	100	ψ(x	ψ(x	PROPN
ejpam-6526	66	101	,	,	PUNCT
ejpam-6526	66	102	y	y	NOUN
ejpam-6526	66	103	)	)	PUNCT
ejpam-6526	66	104	=	=	SYM
ejpam-6526	66	105	ψ(y	ψ(y	NOUN
ejpam-6526	66	106	,	,	PUNCT
ejpam-6526	66	107	x	x	NOUN
ejpam-6526	66	108	)	)	PUNCT
ejpam-6526	66	109	for	for	ADP
ejpam-6526	66	110	all	all	DET
ejpam-6526	66	111	x	x	NOUN
ejpam-6526	66	112	,	,	PUNCT
ejpam-6526	66	113	y	y	PROPN
ejpam-6526	66	114	∈	∈	PROPN
ejpam-6526	66	115	x	x	X
ejpam-6526	66	116	(	(	PUNCT
ejpam-6526	66	117	iv	iv	X
ejpam-6526	66	118	)	)	PUNCT
ejpam-6526	66	119	⊕	⊕	PROPN
ejpam-6526	66	120	represents	represent	VERB
ejpam-6526	66	121	a	a	DET
ejpam-6526	66	122	binary	binary	ADJ
ejpam-6526	66	123	operation	operation	NOUN
ejpam-6526	66	124	⊕	⊕	PROPN
ejpam-6526	66	125	:	:	PUNCT
ejpam-6526	66	126	(	(	PUNCT
ejpam-6526	66	127	r+	r+	PUNCT
ejpam-6526	66	128	×r+)×	×r+)×	PRON
ejpam-6526	66	129	(	(	PUNCT
ejpam-6526	66	130	r+	r+	NOUN
ejpam-6526	66	131	×r+	×r+	ADV
ejpam-6526	66	132	)	)	PUNCT
ejpam-6526	66	133	→	→	VERB
ejpam-6526	66	134	r+	r+	VERB
ejpam-6526	66	135	×r+	×r+	ADV
ejpam-6526	66	136	satisfying	satisfying	ADJ
ejpam-6526	66	137	:	:	PUNCT
ejpam-6526	66	138	(	(	PUNCT
ejpam-6526	66	139	a	a	X
ejpam-6526	66	140	)	)	PUNCT
ejpam-6526	66	141	ψ(x	ψ(x	NOUN
ejpam-6526	66	142	,	,	PUNCT
ejpam-6526	66	143	z	z	NOUN
ejpam-6526	66	144	)	)	PUNCT
ejpam-6526	66	145	≤comp	≤comp	PROPN
ejpam-6526	66	146	⊕(ψ(x	⊕(ψ(x	NOUN
ejpam-6526	66	147	,	,	PUNCT
ejpam-6526	66	148	y),ψ(y	y),ψ(y	PROPN
ejpam-6526	66	149	,	,	PUNCT
ejpam-6526	66	150	z	z	NOUN
ejpam-6526	66	151	)	)	PUNCT
ejpam-6526	66	152	)	)	PUNCT
ejpam-6526	66	153	for	for	SCONJ
ejpam-6526	66	154	all	all	DET
ejpam-6526	66	155	x	x	PROPN
ejpam-6526	66	156	,	,	PUNCT
ejpam-6526	66	157	y	y	PROPN
ejpam-6526	66	158	,	,	PUNCT
ejpam-6526	66	159	z	z	NOUN
ejpam-6526	66	160	∈	∈	PROPN
ejpam-6526	66	161	x	x	INTJ
ejpam-6526	66	162	(	(	PUNCT
ejpam-6526	66	163	b	b	NOUN
ejpam-6526	66	164	)	)	PUNCT
ejpam-6526	66	165	⊕	⊕	PROPN
ejpam-6526	66	166	demonstrates	demonstrate	VERB
ejpam-6526	66	167	monotonicity	monotonicity	NOUN
ejpam-6526	66	168	for	for	ADP
ejpam-6526	66	169	both	both	DET
ejpam-6526	66	170	arguments	argument	NOUN
ejpam-6526	66	171	relative	relative	ADJ
ejpam-6526	66	172	to	to	ADP
ejpam-6526	66	173	partial	partial	ADJ
ejpam-6526	66	174	ordering	order	VERB
ejpam-6526	66	175	≤comp	≤comp	PROPN
ejpam-6526	66	176	(	(	PUNCT
ejpam-6526	66	177	c	c	NOUN
ejpam-6526	66	178	)	)	PUNCT
ejpam-6526	66	179	⊕((0	⊕((0	PROPN
ejpam-6526	66	180	,	,	PUNCT
ejpam-6526	66	181	0),(0	0),(0	NOUN
ejpam-6526	66	182	,	,	PUNCT
ejpam-6526	66	183	0	0	NUM
ejpam-6526	66	184	)	)	PUNCT
ejpam-6526	66	185	)	)	PUNCT
ejpam-6526	67	1	=	=	PUNCT
ejpam-6526	67	2	(	(	PUNCT
ejpam-6526	67	3	0	0	NUM
ejpam-6526	67	4	,	,	PUNCT
ejpam-6526	67	5	0	0	NUM
ejpam-6526	67	6	)	)	PUNCT
ejpam-6526	67	7	where	where	SCONJ
ejpam-6526	67	8	≤comp	≤comp	PROPN
ejpam-6526	67	9	indicates	indicate	VERB
ejpam-6526	67	10	component	component	NOUN
ejpam-6526	67	11	-	-	PUNCT
ejpam-6526	67	12	wise	wise	ADJ
ejpam-6526	67	13	partial	partial	ADJ
ejpam-6526	67	14	ordering	ordering	NOUN
ejpam-6526	67	15	on	on	ADP
ejpam-6526	67	16	r+	r+	X
ejpam-6526	67	17	×	×	NOUN
ejpam-6526	67	18	r+	r+	PUNCT
ejpam-6526	67	19	.	.	PUNCT
ejpam-6526	68	1	remark	remark	PROPN
ejpam-6526	68	2	1	1	NUM
ejpam-6526	68	3	.	.	PUNCT
ejpam-6526	69	1	we	we	PRON
ejpam-6526	69	2	interpret	interpret	VERB
ejpam-6526	69	3	function	function	NOUN
ejpam-6526	69	4	ψ	ψ	NOUN
ejpam-6526	69	5	as	as	ADP
ejpam-6526	69	6	a	a	DET
ejpam-6526	69	7	bidimensional	bidimensional	ADJ
ejpam-6526	69	8	metric	metric	ADJ
ejpam-6526	69	9	assessment	assessment	NOUN
ejpam-6526	69	10	.	.	PUNCT
ejpam-6526	70	1	operation	operation	NOUN
ejpam-6526	70	2	⊕	⊕	PROPN
ejpam-6526	70	3	functions	function	NOUN
ejpam-6526	70	4	as	as	ADP
ejpam-6526	70	5	a	a	DET
ejpam-6526	70	6	generalized	generalized	ADJ
ejpam-6526	70	7	triangular	triangular	NOUN
ejpam-6526	70	8	coordination	coordination	NOUN
ejpam-6526	70	9	mechanism	mechanism	NOUN
ejpam-6526	70	10	that	that	PRON
ejpam-6526	70	11	maintains	maintain	VERB
ejpam-6526	70	12	fundamental	fundamental	ADJ
ejpam-6526	70	13	metric	metric	ADJ
ejpam-6526	70	14	properties	property	NOUN
ejpam-6526	70	15	while	while	SCONJ
ejpam-6526	70	16	accommodating	accommodate	VERB
ejpam-6526	70	17	the	the	DET
ejpam-6526	70	18	bi	bi	ADJ
ejpam-6526	70	19	-	-	ADJ
ejpam-6526	70	20	dimensional	dimensional	ADJ
ejpam-6526	70	21	nature	nature	NOUN
ejpam-6526	70	22	of	of	ADP
ejpam-6526	70	23	the	the	DET
ejpam-6526	70	24	structure	structure	NOUN
ejpam-6526	70	25	,	,	PUNCT
ejpam-6526	70	26	extending	extend	VERB
ejpam-6526	70	27	modular	modular	ADJ
ejpam-6526	70	28	space	space	NOUN
ejpam-6526	70	29	frameworks	framework	NOUN
ejpam-6526	70	30	developed	develop	VERB
ejpam-6526	70	31	by	by	ADP
ejpam-6526	70	32	nakano	nakano	PROPN
ejpam-6526	70	33	[	[	X
ejpam-6526	70	34	18	18	NUM
ejpam-6526	70	35	]	]	PUNCT
ejpam-6526	70	36	.	.	PUNCT
ejpam-6526	71	1	for	for	ADP
ejpam-6526	71	2	enhanced	enhanced	ADJ
ejpam-6526	71	3	comprehension	comprehension	NOUN
ejpam-6526	71	4	,	,	PUNCT
ejpam-6526	71	5	we	we	PRON
ejpam-6526	71	6	provide	provide	VERB
ejpam-6526	71	7	illustrative	illustrative	ADJ
ejpam-6526	71	8	implementations	implementation	NOUN
ejpam-6526	71	9	:	:	PUNCT
ejpam-6526	71	10	example	example	NOUN
ejpam-6526	71	11	1	1	X
ejpam-6526	71	12	.	.	X
ejpam-6526	71	13	consider	consider	VERB
ejpam-6526	71	14	a	a	DET
ejpam-6526	71	15	non	non	ADJ
ejpam-6526	71	16	-	-	ADJ
ejpam-6526	71	17	empty	empty	ADJ
ejpam-6526	71	18	set	set	NOUN
ejpam-6526	71	19	x	x	PUNCT
ejpam-6526	71	20	with	with	ADP
ejpam-6526	71	21	two	two	NUM
ejpam-6526	71	22	distinct	distinct	ADJ
ejpam-6526	71	23	metric	metric	ADJ
ejpam-6526	71	24	functions	function	NOUN
ejpam-6526	71	25	ϕ1	ϕ1	NOUN
ejpam-6526	71	26	,	,	PUNCT
ejpam-6526	71	27	ϕ2	ϕ2	ADV
ejpam-6526	71	28	on	on	ADP
ejpam-6526	71	29	x.	x.	NOUN
ejpam-6526	71	30	when	when	SCONJ
ejpam-6526	71	31	we	we	PRON
ejpam-6526	71	32	define	define	VERB
ejpam-6526	71	33	ψ(x	ψ(x	PROPN
ejpam-6526	71	34	,	,	PUNCT
ejpam-6526	71	35	y	y	NOUN
ejpam-6526	71	36	)	)	PUNCT
ejpam-6526	71	37	=	=	SYM
ejpam-6526	71	38	(	(	PUNCT
ejpam-6526	71	39	ϕ1(x	ϕ1(x	PROPN
ejpam-6526	71	40	,	,	PUNCT
ejpam-6526	71	41	y	y	NOUN
ejpam-6526	71	42	)	)	PUNCT
ejpam-6526	71	43	,	,	PUNCT
ejpam-6526	71	44	ϕ2(x	ϕ2(x	PROPN
ejpam-6526	71	45	,	,	PUNCT
ejpam-6526	71	46	y	y	NOUN
ejpam-6526	71	47	)	)	PUNCT
ejpam-6526	71	48	)	)	PUNCT
ejpam-6526	71	49	and	and	CCONJ
ejpam-6526	71	50	select	select	VERB
ejpam-6526	71	51	⊕((a1	⊕((a1	ADJ
ejpam-6526	71	52	,	,	PUNCT
ejpam-6526	71	53	a2	a2	PROPN
ejpam-6526	71	54	)	)	PUNCT
ejpam-6526	71	55	,	,	PUNCT
ejpam-6526	71	56	(	(	PUNCT
ejpam-6526	71	57	b1	b1	NOUN
ejpam-6526	71	58	,	,	PUNCT
ejpam-6526	71	59	b2	b2	NOUN
ejpam-6526	71	60	)	)	PUNCT
ejpam-6526	71	61	)	)	PUNCT
ejpam-6526	72	1	=	=	PUNCT
ejpam-6526	72	2	(	(	PUNCT
ejpam-6526	72	3	a1	a1	NOUN
ejpam-6526	72	4	+	+	CCONJ
ejpam-6526	72	5	b1	b1	NOUN
ejpam-6526	72	6	,	,	PUNCT
ejpam-6526	72	7	a2	a2	PROPN
ejpam-6526	72	8	+	+	CCONJ
ejpam-6526	72	9	b2	b2	NOUN
ejpam-6526	72	10	)	)	PUNCT
ejpam-6526	72	11	,	,	PUNCT
ejpam-6526	72	12	the	the	DET
ejpam-6526	72	13	resulting	result	VERB
ejpam-6526	72	14	structure	structure	NOUN
ejpam-6526	72	15	(	(	PUNCT
ejpam-6526	72	16	x	x	X
ejpam-6526	72	17	,	,	PUNCT
ejpam-6526	72	18	ψ,⊕	ψ,⊕	NOUN
ejpam-6526	72	19	)	)	PUNCT
ejpam-6526	72	20	constitutes	constitute	VERB
ejpam-6526	72	21	a	a	DET
ejpam-6526	72	22	bi	bi	ADJ
ejpam-6526	72	23	-	-	ADJ
ejpam-6526	72	24	metric	metric	ADJ
ejpam-6526	72	25	system	system	NOUN
ejpam-6526	72	26	.	.	PUNCT
ejpam-6526	73	1	example	example	NOUN
ejpam-6526	73	2	2	2	NUM
ejpam-6526	73	3	(	(	PUNCT
ejpam-6526	73	4	network	network	NOUN
ejpam-6526	73	5	routing	routing	NOUN
ejpam-6526	73	6	application	application	NOUN
ejpam-6526	73	7	)	)	PUNCT
ejpam-6526	73	8	.	.	PUNCT
ejpam-6526	74	1	in	in	ADP
ejpam-6526	74	2	computer	computer	NOUN
ejpam-6526	74	3	networks	network	NOUN
ejpam-6526	74	4	,	,	PUNCT
ejpam-6526	74	5	consider	consider	VERB
ejpam-6526	74	6	nodes	node	NOUN
ejpam-6526	74	7	x	x	PUNCT
ejpam-6526	74	8	where	where	SCONJ
ejpam-6526	74	9	routing	routing	NOUN
ejpam-6526	74	10	decisions	decision	NOUN
ejpam-6526	74	11	depend	depend	VERB
ejpam-6526	74	12	on	on	ADP
ejpam-6526	74	13	both	both	PRON
ejpam-6526	74	14	physical	physical	ADJ
ejpam-6526	74	15	distance	distance	NOUN
ejpam-6526	74	16	and	and	CCONJ
ejpam-6526	74	17	transmission	transmission	NOUN
ejpam-6526	74	18	delay	delay	NOUN
ejpam-6526	74	19	.	.	PUNCT
ejpam-6526	75	1	define	define	NOUN
ejpam-6526	75	2	:	:	PUNCT
ejpam-6526	75	3	•	•	NUM
ejpam-6526	75	4	ϕ1(x	ϕ1(x	NUM
ejpam-6526	75	5	,	,	PUNCT
ejpam-6526	75	6	y	y	NOUN
ejpam-6526	75	7	)	)	PUNCT
ejpam-6526	75	8	=	=	SYM
ejpam-6526	75	9	physical	physical	ADJ
ejpam-6526	75	10	cable	cable	NOUN
ejpam-6526	75	11	distance	distance	NOUN
ejpam-6526	75	12	between	between	ADP
ejpam-6526	75	13	nodes	node	NOUN
ejpam-6526	75	14	x	x	X
ejpam-6526	75	15	and	and	CCONJ
ejpam-6526	75	16	y	y	PROPN
ejpam-6526	75	17	(	(	PUNCT
ejpam-6526	75	18	in	in	ADP
ejpam-6526	75	19	kilometers	kilometer	NOUN
ejpam-6526	75	20	)	)	PUNCT
ejpam-6526	75	21	•	•	ADP
ejpam-6526	76	1	ϕ2(x	ϕ2(x	PROPN
ejpam-6526	76	2	,	,	PUNCT
ejpam-6526	76	3	y	y	NOUN
ejpam-6526	76	4	)	)	PUNCT
ejpam-6526	76	5	=	=	SYM
ejpam-6526	76	6	average	average	ADJ
ejpam-6526	76	7	transmission	transmission	NOUN
ejpam-6526	76	8	delay	delay	NOUN
ejpam-6526	76	9	between	between	ADP
ejpam-6526	76	10	nodes	node	NOUN
ejpam-6526	76	11	x	x	X
ejpam-6526	76	12	and	and	CCONJ
ejpam-6526	76	13	y	y	PROPN
ejpam-6526	76	14	(	(	PUNCT
ejpam-6526	76	15	in	in	ADP
ejpam-6526	76	16	milliseconds	millisecond	NOUN
ejpam-6526	76	17	)	)	PUNCT
ejpam-6526	76	18	with	with	ADP
ejpam-6526	76	19	ψ(x	ψ(x	PROPN
ejpam-6526	76	20	,	,	PUNCT
ejpam-6526	76	21	y	y	NOUN
ejpam-6526	76	22	)	)	PUNCT
ejpam-6526	76	23	=	=	SYM
ejpam-6526	76	24	(	(	PUNCT
ejpam-6526	76	25	ϕ1(x	ϕ1(x	PROPN
ejpam-6526	76	26	,	,	PUNCT
ejpam-6526	76	27	y	y	NOUN
ejpam-6526	76	28	)	)	PUNCT
ejpam-6526	76	29	,	,	PUNCT
ejpam-6526	76	30	ϕ2(x	ϕ2(x	PROPN
ejpam-6526	76	31	,	,	PUNCT
ejpam-6526	76	32	y	y	NOUN
ejpam-6526	76	33	)	)	PUNCT
ejpam-6526	76	34	)	)	PUNCT
ejpam-6526	76	35	and	and	CCONJ
ejpam-6526	76	36	⊕	⊕	PROPN
ejpam-6526	76	37	as	as	ADP
ejpam-6526	76	38	component	component	NOUN
ejpam-6526	76	39	-	-	PUNCT
ejpam-6526	76	40	wise	wise	ADJ
ejpam-6526	76	41	addition	addition	NOUN
ejpam-6526	76	42	,	,	PUNCT
ejpam-6526	76	43	this	this	DET
ejpam-6526	76	44	bi	bi	ADJ
ejpam-6526	76	45	-	-	ADJ
ejpam-6526	76	46	metric	metric	ADJ
ejpam-6526	76	47	system	system	NOUN
ejpam-6526	76	48	enables	enable	VERB
ejpam-6526	76	49	routing	route	VERB
ejpam-6526	76	50	algorithms	algorithm	NOUN
ejpam-6526	76	51	to	to	PART
ejpam-6526	76	52	optimize	optimize	VERB
ejpam-6526	76	53	for	for	ADP
ejpam-6526	76	54	both	both	DET
ejpam-6526	76	55	distance	distance	NOUN
ejpam-6526	76	56	and	and	CCONJ
ejpam-6526	76	57	latency	latency	NOUN
ejpam-6526	76	58	simultaneously	simultaneously	ADV
ejpam-6526	76	59	.	.	PUNCT
ejpam-6526	77	1	a.	a.	PROPN
ejpam-6526	77	2	alsoboh	alsoboh	PROPN
ejpam-6526	77	3	et	et	PROPN
ejpam-6526	77	4	al	al	PROPN
ejpam-6526	77	5	.	.	PUNCT
ejpam-6526	77	6	/	/	SYM
ejpam-6526	77	7	eur	eur	PROPN
ejpam-6526	77	8	.	.	PUNCT
ejpam-6526	78	1	j.	j.	PROPN
ejpam-6526	78	2	pure	pure	PROPN
ejpam-6526	78	3	appl	appl	PROPN
ejpam-6526	78	4	.	.	PROPN
ejpam-6526	78	5	math	math	PROPN
ejpam-6526	78	6	,	,	PUNCT
ejpam-6526	78	7	18	18	NUM
ejpam-6526	78	8	(	(	PUNCT
ejpam-6526	78	9	4	4	NUM
ejpam-6526	78	10	)	)	PUNCT
ejpam-6526	78	11	(	(	PUNCT
ejpam-6526	78	12	2025	2025	NUM
ejpam-6526	78	13	)	)	PUNCT
ejpam-6526	78	14	,	,	PUNCT
ejpam-6526	78	15	6526	6526	NUM
ejpam-6526	78	16	5	5	NUM
ejpam-6526	78	17	of	of	ADP
ejpam-6526	78	18	21	21	NUM
ejpam-6526	78	19	example	example	NOUN
ejpam-6526	78	20	3	3	NUM
ejpam-6526	78	21	.	.	X
ejpam-6526	79	1	for	for	ADP
ejpam-6526	79	2	x	x	X
ejpam-6526	79	3	=	=	PRON
ejpam-6526	79	4	r+	r+	NOUN
ejpam-6526	79	5	,	,	PUNCT
ejpam-6526	79	6	define	define	VERB
ejpam-6526	79	7	ψ(x	ψ(x	PROPN
ejpam-6526	79	8	,	,	PUNCT
ejpam-6526	79	9	y	y	NOUN
ejpam-6526	79	10	)	)	PUNCT
ejpam-6526	79	11	=	=	SYM
ejpam-6526	79	12	(	(	PUNCT
ejpam-6526	79	13	|x−y|	|x−y|	ADJ
ejpam-6526	79	14	,	,	PUNCT
ejpam-6526	79	15	|	|	ADV
ejpam-6526	79	16	tanh(x)−tanh(y)|	tanh(x)−tanh(y)|	NOUN
ejpam-6526	79	17	)	)	PUNCT
ejpam-6526	79	18	with	with	ADP
ejpam-6526	79	19	⊕((a1	⊕((a1	ADJ
ejpam-6526	79	20	,	,	PUNCT
ejpam-6526	79	21	a2	a2	PROPN
ejpam-6526	79	22	)	)	PUNCT
ejpam-6526	79	23	,	,	PUNCT
ejpam-6526	79	24	(	(	PUNCT
ejpam-6526	79	25	b1	b1	NOUN
ejpam-6526	79	26	,	,	PUNCT
ejpam-6526	79	27	b2	b2	NOUN
ejpam-6526	79	28	)	)	PUNCT
ejpam-6526	79	29	)	)	PUNCT
ejpam-6526	80	1	=	=	PUNCT
ejpam-6526	80	2	(	(	PUNCT
ejpam-6526	80	3	a1	a1	NOUN
ejpam-6526	80	4	+	+	CCONJ
ejpam-6526	80	5	b1	b1	NOUN
ejpam-6526	80	6	,	,	PUNCT
ejpam-6526	80	7	a2	a2	PROPN
ejpam-6526	80	8	+	+	CCONJ
ejpam-6526	80	9	b2	b2	NOUN
ejpam-6526	80	10	)	)	PUNCT
ejpam-6526	80	11	.	.	PUNCT
ejpam-6526	81	1	we	we	PRON
ejpam-6526	81	2	obtain	obtain	VERB
ejpam-6526	81	3	a	a	DET
ejpam-6526	81	4	bi	bi	ADJ
ejpam-6526	81	5	-	-	ADJ
ejpam-6526	81	6	metric	metric	ADJ
ejpam-6526	81	7	system	system	NOUN
ejpam-6526	81	8	where	where	SCONJ
ejpam-6526	81	9	both	both	DET
ejpam-6526	81	10	components	component	NOUN
ejpam-6526	81	11	satisfy	satisfy	VERB
ejpam-6526	81	12	traditional	traditional	ADJ
ejpam-6526	81	13	triangular	triangular	NOUN
ejpam-6526	81	14	coordination	coordination	NOUN
ejpam-6526	81	15	principles	principle	NOUN
ejpam-6526	81	16	.	.	PUNCT
ejpam-6526	82	1	example	example	NOUN
ejpam-6526	82	2	4	4	NUM
ejpam-6526	82	3	(	(	PUNCT
ejpam-6526	82	4	weight	weight	NOUN
ejpam-6526	82	5	-	-	PUNCT
ejpam-6526	82	6	parameterized	parameterized	ADJ
ejpam-6526	82	7	metric	metric	ADJ
ejpam-6526	82	8	system	system	NOUN
ejpam-6526	82	9	)	)	PUNCT
ejpam-6526	82	10	.	.	PUNCT
ejpam-6526	83	1	given	give	VERB
ejpam-6526	83	2	a	a	DET
ejpam-6526	83	3	metric	metric	ADJ
ejpam-6526	83	4	space	space	NOUN
ejpam-6526	83	5	(	(	PUNCT
ejpam-6526	83	6	x,ϕ	x,ϕ	NOUN
ejpam-6526	83	7	)	)	PUNCT
ejpam-6526	83	8	and	and	CCONJ
ejpam-6526	83	9	a	a	DET
ejpam-6526	83	10	weight	weight	NOUN
ejpam-6526	83	11	function	function	NOUN
ejpam-6526	83	12	w	w	NOUN
ejpam-6526	83	13	:	:	PUNCT
ejpam-6526	83	14	x	x	SYM
ejpam-6526	83	15	→	→	X
ejpam-6526	83	16	(	(	PUNCT
ejpam-6526	83	17	0,∞	0,∞	NOUN
ejpam-6526	83	18	)	)	PUNCT
ejpam-6526	83	19	,	,	PUNCT
ejpam-6526	83	20	establishing	establish	VERB
ejpam-6526	83	21	ψ(x	ψ(x	PROPN
ejpam-6526	83	22	,	,	PUNCT
ejpam-6526	83	23	y	y	NOUN
ejpam-6526	83	24	)	)	PUNCT
ejpam-6526	83	25	=	=	AUX
ejpam-6526	83	26	(	(	PUNCT
ejpam-6526	83	27	ϕ(x	ϕ(x	PROPN
ejpam-6526	83	28	,	,	PUNCT
ejpam-6526	83	29	y	y	PROPN
ejpam-6526	83	30	)	)	PUNCT
ejpam-6526	83	31	,	,	PUNCT
ejpam-6526	83	32	|w(x	|w(x	NOUN
ejpam-6526	83	33	)	)	PUNCT
ejpam-6526	83	34	−	−	PROPN
ejpam-6526	83	35	w(y)|	w(y)|	ADV
ejpam-6526	83	36	)	)	PUNCT
ejpam-6526	83	37	and	and	CCONJ
ejpam-6526	83	38	⊕((a1	⊕((a1	ADV
ejpam-6526	83	39	,	,	PUNCT
ejpam-6526	83	40	a2	a2	PROPN
ejpam-6526	83	41	)	)	PUNCT
ejpam-6526	83	42	,	,	PUNCT
ejpam-6526	83	43	(	(	PUNCT
ejpam-6526	83	44	b1	b1	NOUN
ejpam-6526	83	45	,	,	PUNCT
ejpam-6526	83	46	b2	b2	NOUN
ejpam-6526	83	47	)	)	PUNCT
ejpam-6526	83	48	)	)	PUNCT
ejpam-6526	84	1	=	=	PUNCT
ejpam-6526	85	1	(	(	PUNCT
ejpam-6526	85	2	a1	a1	NOUN
ejpam-6526	85	3	+	+	CCONJ
ejpam-6526	85	4	b1	b1	NOUN
ejpam-6526	85	5	,	,	PUNCT
ejpam-6526	85	6	a2	a2	PROPN
ejpam-6526	85	7	+	+	CCONJ
ejpam-6526	85	8	b2	b2	NOUN
ejpam-6526	85	9	)	)	PUNCT
ejpam-6526	85	10	constructs	construct	VERB
ejpam-6526	85	11	a	a	DET
ejpam-6526	85	12	bi	bi	ADJ
ejpam-6526	85	13	-	-	ADJ
ejpam-6526	85	14	metric	metric	ADJ
ejpam-6526	85	15	system	system	NOUN
ejpam-6526	85	16	incorporating	incorporate	VERB
ejpam-6526	85	17	both	both	DET
ejpam-6526	85	18	baseline	baseline	ADJ
ejpam-6526	85	19	measurements	measurement	NOUN
ejpam-6526	85	20	and	and	CCONJ
ejpam-6526	85	21	weight	weight	NOUN
ejpam-6526	85	22	variations	variation	NOUN
ejpam-6526	85	23	.	.	PUNCT
ejpam-6526	86	1	bi	bi	ADJ
ejpam-6526	86	2	-	-	ADJ
ejpam-6526	86	3	metric	metric	ADJ
ejpam-6526	86	4	systems	system	NOUN
ejpam-6526	86	5	naturally	naturally	ADV
ejpam-6526	86	6	generate	generate	VERB
ejpam-6526	86	7	dual	dual	ADJ
ejpam-6526	86	8	topologies	topology	NOUN
ejpam-6526	86	9	on	on	ADP
ejpam-6526	86	10	underlying	underlie	VERB
ejpam-6526	86	11	sets	set	NOUN
ejpam-6526	86	12	,	,	PUNCT
ejpam-6526	86	13	similar	similar	ADJ
ejpam-6526	86	14	to	to	ADP
ejpam-6526	86	15	quasi	quasi	ADJ
ejpam-6526	86	16	-	-	NOUN
ejpam-6526	86	17	gauge	gauge	ADJ
ejpam-6526	86	18	spaces	space	NOUN
ejpam-6526	86	19	previously	previously	ADV
ejpam-6526	86	20	described	describe	VERB
ejpam-6526	86	21	in	in	ADP
ejpam-6526	86	22	mathematical	mathematical	ADJ
ejpam-6526	86	23	literature	literature	NOUN
ejpam-6526	86	24	[	[	X
ejpam-6526	86	25	19	19	NUM
ejpam-6526	86	26	]	]	PUNCT
ejpam-6526	86	27	.	.	PUNCT
ejpam-6526	87	1	theorem	theorem	NOUN
ejpam-6526	87	2	1	1	NUM
ejpam-6526	87	3	.	.	PUNCT
ejpam-6526	87	4	given	give	VERB
ejpam-6526	87	5	a	a	DET
ejpam-6526	87	6	bi	bi	ADJ
ejpam-6526	87	7	-	-	ADJ
ejpam-6526	87	8	metric	metric	ADJ
ejpam-6526	87	9	system	system	NOUN
ejpam-6526	87	10	(	(	PUNCT
ejpam-6526	87	11	x	x	X
ejpam-6526	87	12	,	,	PUNCT
ejpam-6526	87	13	ψ,⊕	ψ,⊕	NOUN
ejpam-6526	87	14	)	)	PUNCT
ejpam-6526	87	15	with	with	ADP
ejpam-6526	87	16	ψ(x	ψ(x	PROPN
ejpam-6526	87	17	,	,	PUNCT
ejpam-6526	87	18	y	y	NOUN
ejpam-6526	87	19	)	)	PUNCT
ejpam-6526	87	20	=	=	SYM
ejpam-6526	87	21	(	(	PUNCT
ejpam-6526	87	22	ϕ1(x	ϕ1(x	PROPN
ejpam-6526	87	23	,	,	PUNCT
ejpam-6526	87	24	y	y	NOUN
ejpam-6526	87	25	)	)	PUNCT
ejpam-6526	87	26	,	,	PUNCT
ejpam-6526	87	27	ϕ2(x	ϕ2(x	PROPN
ejpam-6526	87	28	,	,	PUNCT
ejpam-6526	87	29	y	y	NOUN
ejpam-6526	87	30	)	)	PUNCT
ejpam-6526	87	31	)	)	PUNCT
ejpam-6526	87	32	,	,	PUNCT
ejpam-6526	87	33	for	for	ADP
ejpam-6526	87	34	each	each	DET
ejpam-6526	87	35	i	i	PRON
ejpam-6526	87	36	∈	∈	PROPN
ejpam-6526	87	37	{	{	PUNCT
ejpam-6526	87	38	1	1	NUM
ejpam-6526	87	39	,	,	PUNCT
ejpam-6526	87	40	2	2	NUM
ejpam-6526	87	41	}	}	PUNCT
ejpam-6526	87	42	,	,	PUNCT
ejpam-6526	87	43	the	the	DET
ejpam-6526	87	44	topology	topology	NOUN
ejpam-6526	87	45	ti	ti	NOUN
ejpam-6526	87	46	induced	induce	VERB
ejpam-6526	87	47	by	by	ADP
ejpam-6526	87	48	the	the	DET
ejpam-6526	87	49	corresponding	correspond	VERB
ejpam-6526	87	50	component	component	NOUN
ejpam-6526	87	51	can	can	AUX
ejpam-6526	87	52	be	be	AUX
ejpam-6526	87	53	defined	define	VERB
ejpam-6526	87	54	as	as	ADP
ejpam-6526	87	55	:	:	PUNCT
ejpam-6526	87	56	ti	ti	X
ejpam-6526	87	57	=	=	SYM
ejpam-6526	87	58	{	{	PUNCT
ejpam-6526	87	59	u	u	X
ejpam-6526	87	60	⊂	⊂	PROPN
ejpam-6526	87	61	x	x	X
ejpam-6526	87	62	:	:	PUNCT
ejpam-6526	87	63	∀x	∀x	NUM
ejpam-6526	87	64	∈	∈	PROPN
ejpam-6526	87	65	u	u	NOUN
ejpam-6526	87	66	,	,	PUNCT
ejpam-6526	87	67	∃ε	∃ε	PROPN
ejpam-6526	87	68	>	>	X
ejpam-6526	87	69	0	0	NUM
ejpam-6526	88	1	where	where	SCONJ
ejpam-6526	88	2	vi(x	vi(x	NOUN
ejpam-6526	88	3	,	,	PUNCT
ejpam-6526	88	4	ε	ε	PROPN
ejpam-6526	88	5	)	)	PUNCT
ejpam-6526	88	6	⊂	⊂	PROPN
ejpam-6526	88	7	u	u	NOUN
ejpam-6526	88	8	}	}	PUNCT
ejpam-6526	88	9	in	in	ADP
ejpam-6526	88	10	which	which	PRON
ejpam-6526	88	11	vi(x	vi(x	NOUN
ejpam-6526	88	12	,	,	PUNCT
ejpam-6526	88	13	ε	ε	PROPN
ejpam-6526	88	14	)	)	PUNCT
ejpam-6526	88	15	=	=	PRON
ejpam-6526	88	16	{	{	PUNCT
ejpam-6526	88	17	y	y	PROPN
ejpam-6526	88	18	∈	∈	PROPN
ejpam-6526	88	19	x	x	X
ejpam-6526	88	20	:	:	PUNCT
ejpam-6526	88	21	ϕi(x	ϕi(x	NUM
ejpam-6526	88	22	,	,	PUNCT
ejpam-6526	88	23	y	y	PROPN
ejpam-6526	88	24	)	)	PUNCT
ejpam-6526	88	25	<	<	X
ejpam-6526	88	26	ε	ε	PROPN
ejpam-6526	88	27	}	}	PUNCT
ejpam-6526	88	28	.	.	PUNCT
ejpam-6526	89	1	the	the	DET
ejpam-6526	89	2	resulting	result	VERB
ejpam-6526	89	3	structure	structure	NOUN
ejpam-6526	89	4	(	(	PUNCT
ejpam-6526	89	5	x	x	NOUN
ejpam-6526	89	6	,	,	PUNCT
ejpam-6526	89	7	t1	t1	NOUN
ejpam-6526	89	8	,	,	PUNCT
ejpam-6526	89	9	t2	t2	NOUN
ejpam-6526	89	10	)	)	PUNCT
ejpam-6526	89	11	constitutes	constitute	VERB
ejpam-6526	89	12	a	a	DET
ejpam-6526	89	13	bitopological	bitopological	ADJ
ejpam-6526	89	14	space	space	NOUN
ejpam-6526	89	15	.	.	PUNCT
ejpam-6526	90	1	proof	proof	NOUN
ejpam-6526	90	2	.	.	PUNCT
ejpam-6526	91	1	we	we	PRON
ejpam-6526	91	2	must	must	AUX
ejpam-6526	91	3	establish	establish	VERB
ejpam-6526	91	4	that	that	SCONJ
ejpam-6526	91	5	t1	t1	NOUN
ejpam-6526	91	6	and	and	CCONJ
ejpam-6526	91	7	t2	t2	PROPN
ejpam-6526	91	8	represent	represent	VERB
ejpam-6526	91	9	legitimate	legitimate	ADJ
ejpam-6526	91	10	topologies	topology	NOUN
ejpam-6526	91	11	on	on	ADP
ejpam-6526	91	12	x.	x.	NOUN
ejpam-6526	91	13	for	for	ADP
ejpam-6526	91	14	each	each	DET
ejpam-6526	91	15	i	i	PRON
ejpam-6526	91	16	∈	∈	PROPN
ejpam-6526	91	17	{	{	PUNCT
ejpam-6526	91	18	1	1	NUM
ejpam-6526	91	19	,	,	PUNCT
ejpam-6526	91	20	2	2	NUM
ejpam-6526	91	21	}	}	PUNCT
ejpam-6526	91	22	:	:	PUNCT
ejpam-6526	91	23	(	(	PUNCT
ejpam-6526	91	24	i	i	NOUN
ejpam-6526	91	25	)	)	PUNCT
ejpam-6526	91	26	∅	∅	NOUN
ejpam-6526	91	27	∈	∈	NOUN
ejpam-6526	91	28	ti	ti	NOUN
ejpam-6526	91	29	through	through	ADP
ejpam-6526	91	30	vacuous	vacuous	ADJ
ejpam-6526	91	31	truth	truth	NOUN
ejpam-6526	91	32	.	.	PUNCT
ejpam-6526	92	1	(	(	PUNCT
ejpam-6526	92	2	ii	ii	NOUN
ejpam-6526	92	3	)	)	PUNCT
ejpam-6526	92	4	x	x	SYM
ejpam-6526	92	5	∈	∈	X
ejpam-6526	92	6	ti	ti	NOUN
ejpam-6526	92	7	because	because	SCONJ
ejpam-6526	92	8	for	for	ADP
ejpam-6526	92	9	any	any	DET
ejpam-6526	92	10	x	x	SYM
ejpam-6526	92	11	∈	∈	PROPN
ejpam-6526	92	12	x	x	NOUN
ejpam-6526	92	13	,	,	PUNCT
ejpam-6526	92	14	vi(x	vi(x	NUM
ejpam-6526	92	15	,	,	PUNCT
ejpam-6526	92	16	ε	ε	PROPN
ejpam-6526	92	17	)	)	PUNCT
ejpam-6526	92	18	⊂	⊂	PROPN
ejpam-6526	92	19	x	x	PUNCT
ejpam-6526	92	20	for	for	ADP
ejpam-6526	92	21	all	all	DET
ejpam-6526	92	22	ε	ε	PROPN
ejpam-6526	92	23	>	>	X
ejpam-6526	92	24	0	0	NUM
ejpam-6526	92	25	.	.	PUNCT
ejpam-6526	92	26	(	(	PUNCT
ejpam-6526	92	27	iii	iii	NOUN
ejpam-6526	92	28	)	)	PUNCT
ejpam-6526	92	29	for	for	ADP
ejpam-6526	92	30	arbitrary	arbitrary	ADJ
ejpam-6526	92	31	collection	collection	NOUN
ejpam-6526	92	32	{	{	PUNCT
ejpam-6526	92	33	uα}α∈a	uα}α∈a	NUM
ejpam-6526	92	34	⊂	⊂	PROPN
ejpam-6526	92	35	ti	ti	NOUN
ejpam-6526	92	36	,	,	PUNCT
ejpam-6526	92	37	consider	consider	VERB
ejpam-6526	92	38	any	any	DET
ejpam-6526	92	39	x	x	SYM
ejpam-6526	92	40	∈	∈	PROPN
ejpam-6526	92	41	⋃	⋃	NOUN
ejpam-6526	92	42	α∈a	α∈a	NOUN
ejpam-6526	92	43	uα	uα	PROPN
ejpam-6526	92	44	.	.	PUNCT
ejpam-6526	93	1	there	there	PRON
ejpam-6526	93	2	exists	exist	VERB
ejpam-6526	93	3	α0	α0	PROPN
ejpam-6526	93	4	∈	∈	PROPN
ejpam-6526	93	5	a	a	DET
ejpam-6526	93	6	where	where	SCONJ
ejpam-6526	93	7	x	x	SYM
ejpam-6526	93	8	∈	∈	PROPN
ejpam-6526	93	9	uα0	uα0	NOUN
ejpam-6526	93	10	.	.	PUNCT
ejpam-6526	94	1	since	since	SCONJ
ejpam-6526	94	2	uα0	uα0	NOUN
ejpam-6526	94	3	∈	∈	PROPN
ejpam-6526	94	4	ti	ti	NOUN
ejpam-6526	94	5	,	,	PUNCT
ejpam-6526	94	6	we	we	PRON
ejpam-6526	94	7	identify	identify	VERB
ejpam-6526	94	8	ε	ε	PROPN
ejpam-6526	94	9	>	>	X
ejpam-6526	94	10	0	0	PUNCT
ejpam-6526	95	1	satisfying	satisfy	VERB
ejpam-6526	95	2	vi(x	vi(x	NOUN
ejpam-6526	95	3	,	,	PUNCT
ejpam-6526	95	4	ε	ε	PROPN
ejpam-6526	95	5	)	)	PUNCT
ejpam-6526	95	6	⊂	⊂	PROPN
ejpam-6526	95	7	uα0	uα0	VERB
ejpam-6526	95	8	⊂	⊂	PROPN
ejpam-6526	95	9	⋃	⋃	PROPN
ejpam-6526	95	10	α∈a	α∈a	PROPN
ejpam-6526	95	11	uα	uα	PROPN
ejpam-6526	95	12	.	.	PUNCT
ejpam-6526	96	1	this	this	PRON
ejpam-6526	96	2	establishes	establish	VERB
ejpam-6526	96	3	⋃	⋃	NOUN
ejpam-6526	96	4	α∈a	α∈a	NOUN
ejpam-6526	96	5	uα	uα	PROPN
ejpam-6526	96	6	∈	∈	PROPN
ejpam-6526	96	7	ti	ti	X
ejpam-6526	96	8	.	.	PROPN
ejpam-6526	96	9	(	(	PUNCT
ejpam-6526	96	10	iv	iv	X
ejpam-6526	96	11	)	)	PUNCT
ejpam-6526	96	12	for	for	ADP
ejpam-6526	96	13	finite	finite	ADJ
ejpam-6526	96	14	collection	collection	NOUN
ejpam-6526	96	15	u1	u1	NOUN
ejpam-6526	96	16	,	,	PUNCT
ejpam-6526	96	17	u2	u2	NOUN
ejpam-6526	96	18	,	,	PUNCT
ejpam-6526	96	19	.	.	PUNCT
ejpam-6526	96	20	.	.	PUNCT
ejpam-6526	96	21	.	.	PUNCT
ejpam-6526	97	1	,	,	PUNCT
ejpam-6526	97	2	un	un	PROPN
ejpam-6526	97	3	∈	∈	PROPN
ejpam-6526	97	4	ti	ti	NOUN
ejpam-6526	97	5	,	,	PUNCT
ejpam-6526	97	6	consider	consider	VERB
ejpam-6526	97	7	any	any	DET
ejpam-6526	97	8	x	x	SYM
ejpam-6526	97	9	∈	∈	PROPN
ejpam-6526	97	10	n⋂	n⋂	NOUN
ejpam-6526	97	11	j=1	j=1	PROPN
ejpam-6526	97	12	uj	uj	PROPN
ejpam-6526	97	13	.	.	PUNCT
ejpam-6526	98	1	we	we	PRON
ejpam-6526	98	2	determine	determine	VERB
ejpam-6526	98	3	ε1	ε1	PROPN
ejpam-6526	98	4	,	,	PUNCT
ejpam-6526	98	5	ε2	ε2	ADJ
ejpam-6526	98	6	,	,	PUNCT
ejpam-6526	98	7	.	.	PUNCT
ejpam-6526	98	8	.	.	PUNCT
ejpam-6526	98	9	.	.	PUNCT
ejpam-6526	99	1	,	,	PUNCT
ejpam-6526	99	2	εn	εn	ADP
ejpam-6526	99	3	>	>	X
ejpam-6526	99	4	0	0	NUM
ejpam-6526	99	5	where	where	SCONJ
ejpam-6526	99	6	vi(x	vi(x	NOUN
ejpam-6526	99	7	,	,	PUNCT
ejpam-6526	99	8	εj	εj	NOUN
ejpam-6526	99	9	)	)	PUNCT
ejpam-6526	99	10	⊂	⊂	PROPN
ejpam-6526	99	11	uj	uj	PROPN
ejpam-6526	99	12	for	for	ADP
ejpam-6526	99	13	each	each	DET
ejpam-6526	99	14	j.	j.	PROPN
ejpam-6526	99	15	selecting	selecting	PROPN
ejpam-6526	99	16	ε	ε	PROPN
ejpam-6526	99	17	=	=	PUNCT
ejpam-6526	99	18	min{ε1	min{ε1	NOUN
ejpam-6526	99	19	,	,	PUNCT
ejpam-6526	99	20	ε2	ε2	ADJ
ejpam-6526	99	21	,	,	PUNCT
ejpam-6526	99	22	.	.	PUNCT
ejpam-6526	99	23	.	.	PUNCT
ejpam-6526	100	1	.	.	PUNCT
ejpam-6526	101	1	,	,	PUNCT
ejpam-6526	101	2	εn	εn	ADP
ejpam-6526	101	3	}	}	PUNCT
ejpam-6526	101	4	,	,	PUNCT
ejpam-6526	101	5	we	we	PRON
ejpam-6526	101	6	have	have	VERB
ejpam-6526	101	7	vi(x	vi(x	NUM
ejpam-6526	101	8	,	,	PUNCT
ejpam-6526	101	9	ε	ε	PROPN
ejpam-6526	101	10	)	)	PUNCT
ejpam-6526	101	11	⊂	⊂	PROPN
ejpam-6526	101	12	vi(x	vi(x	PROPN
ejpam-6526	101	13	,	,	PUNCT
ejpam-6526	101	14	εj	εj	NOUN
ejpam-6526	101	15	)	)	PUNCT
ejpam-6526	101	16	⊂	⊂	PROPN
ejpam-6526	102	1	uj	uj	PROPN
ejpam-6526	102	2	for	for	ADP
ejpam-6526	102	3	each	each	DET
ejpam-6526	102	4	j	j	PROPN
ejpam-6526	102	5	,	,	PUNCT
ejpam-6526	102	6	yielding	yield	VERB
ejpam-6526	102	7	vi(x	vi(x	NOUN
ejpam-6526	102	8	,	,	PUNCT
ejpam-6526	102	9	ε	ε	PROPN
ejpam-6526	102	10	)	)	PUNCT
ejpam-6526	103	1	⊂	⊂	PROPN
ejpam-6526	103	2	n⋂	n⋂	PROPN
ejpam-6526	103	3	j=1	j=1	PROPN
ejpam-6526	103	4	uj	uj	PROPN
ejpam-6526	103	5	.	.	PUNCT
ejpam-6526	104	1	therefore	therefore	ADV
ejpam-6526	104	2	,	,	PUNCT
ejpam-6526	104	3	n⋂	n⋂	PROPN
ejpam-6526	104	4	j=1	j=1	PROPN
ejpam-6526	104	5	uj	uj	PROPN
ejpam-6526	104	6	∈	∈	PROPN
ejpam-6526	104	7	ti	ti	NOUN
ejpam-6526	104	8	.	.	PUNCT
ejpam-6526	104	9	consequently	consequently	ADV
ejpam-6526	104	10	,	,	PUNCT
ejpam-6526	104	11	t1	t1	NOUN
ejpam-6526	104	12	and	and	CCONJ
ejpam-6526	104	13	t2	t2	NOUN
ejpam-6526	104	14	constitute	constitute	VERB
ejpam-6526	104	15	legitimate	legitimate	ADJ
ejpam-6526	104	16	topologies	topology	NOUN
ejpam-6526	104	17	on	on	ADP
ejpam-6526	104	18	x	x	NOUN
ejpam-6526	104	19	,	,	PUNCT
ejpam-6526	104	20	establishing	establish	VERB
ejpam-6526	104	21	(	(	PUNCT
ejpam-6526	104	22	x	x	NOUN
ejpam-6526	104	23	,	,	PUNCT
ejpam-6526	104	24	t1	t1	NOUN
ejpam-6526	104	25	,	,	PUNCT
ejpam-6526	104	26	t2	t2	NOUN
ejpam-6526	104	27	)	)	PUNCT
ejpam-6526	104	28	as	as	ADP
ejpam-6526	104	29	a	a	DET
ejpam-6526	104	30	bitopological	bitopological	ADJ
ejpam-6526	104	31	space	space	NOUN
ejpam-6526	104	32	.	.	PUNCT
ejpam-6526	105	1	a.	a.	PROPN
ejpam-6526	105	2	alsoboh	alsoboh	PROPN
ejpam-6526	105	3	et	et	PROPN
ejpam-6526	105	4	al	al	PROPN
ejpam-6526	105	5	.	.	PUNCT
ejpam-6526	105	6	/	/	SYM
ejpam-6526	105	7	eur	eur	PROPN
ejpam-6526	105	8	.	.	PUNCT
ejpam-6526	106	1	j.	j.	PROPN
ejpam-6526	106	2	pure	pure	PROPN
ejpam-6526	106	3	appl	appl	PROPN
ejpam-6526	106	4	.	.	PROPN
ejpam-6526	106	5	math	math	PROPN
ejpam-6526	106	6	,	,	PUNCT
ejpam-6526	106	7	18	18	NUM
ejpam-6526	106	8	(	(	PUNCT
ejpam-6526	106	9	4	4	NUM
ejpam-6526	106	10	)	)	PUNCT
ejpam-6526	106	11	(	(	PUNCT
ejpam-6526	106	12	2025	2025	NUM
ejpam-6526	106	13	)	)	PUNCT
ejpam-6526	106	14	,	,	PUNCT
ejpam-6526	106	15	6526	6526	NUM
ejpam-6526	106	16	6	6	NUM
ejpam-6526	106	17	of	of	ADP
ejpam-6526	106	18	21	21	NUM
ejpam-6526	106	19	example	example	NOUN
ejpam-6526	106	20	5	5	NUM
ejpam-6526	106	21	(	(	PUNCT
ejpam-6526	106	22	image	image	NOUN
ejpam-6526	106	23	quality	quality	NOUN
ejpam-6526	106	24	assessment	assessment	NOUN
ejpam-6526	106	25	)	)	PUNCT
ejpam-6526	106	26	.	.	PUNCT
ejpam-6526	107	1	in	in	ADP
ejpam-6526	107	2	digital	digital	ADJ
ejpam-6526	107	3	image	image	NOUN
ejpam-6526	107	4	processing	processing	NOUN
ejpam-6526	107	5	,	,	PUNCT
ejpam-6526	107	6	quality	quality	NOUN
ejpam-6526	107	7	assessment	assessment	NOUN
ejpam-6526	107	8	often	often	ADV
ejpam-6526	107	9	requires	require	VERB
ejpam-6526	107	10	multiple	multiple	ADJ
ejpam-6526	107	11	metrics	metric	NOUN
ejpam-6526	107	12	.	.	PUNCT
ejpam-6526	108	1	consider	consider	VERB
ejpam-6526	108	2	:	:	PUNCT
ejpam-6526	108	3	•	•	NUM
ejpam-6526	108	4	ϕ1(i1	ϕ1(i1	NOUN
ejpam-6526	108	5	,	,	PUNCT
ejpam-6526	108	6	i2	i2	PROPN
ejpam-6526	108	7	)	)	PUNCT
ejpam-6526	108	8	=	=	SYM
ejpam-6526	108	9	mean	mean	VERB
ejpam-6526	109	1	squared	square	VERB
ejpam-6526	109	2	error	error	NOUN
ejpam-6526	109	3	(	(	PUNCT
ejpam-6526	109	4	mse	mse	NOUN
ejpam-6526	109	5	)	)	PUNCT
ejpam-6526	109	6	between	between	ADP
ejpam-6526	109	7	images	image	NOUN
ejpam-6526	109	8	i1	i1	PROPN
ejpam-6526	109	9	and	and	CCONJ
ejpam-6526	109	10	i2	i2	PROPN
ejpam-6526	109	11	•	•	ADP
ejpam-6526	109	12	ϕ2(i1	ϕ2(i1	PROPN
ejpam-6526	109	13	,	,	PUNCT
ejpam-6526	109	14	i2	i2	PROPN
ejpam-6526	109	15	)	)	PUNCT
ejpam-6526	109	16	=	=	SYM
ejpam-6526	109	17	1	1	NUM
ejpam-6526	109	18	ssim	ssim	NOUN
ejpam-6526	109	19	(	(	PUNCT
ejpam-6526	109	20	structural	structural	ADJ
ejpam-6526	109	21	similarity	similarity	NOUN
ejpam-6526	109	22	index	index	NOUN
ejpam-6526	109	23	)	)	PUNCT
ejpam-6526	109	24	the	the	DET
ejpam-6526	109	25	bi	bi	ADJ
ejpam-6526	109	26	-	-	ADJ
ejpam-6526	109	27	metric	metric	ADJ
ejpam-6526	109	28	system	system	NOUN
ejpam-6526	109	29	(	(	PUNCT
ejpam-6526	109	30	x	x	X
ejpam-6526	109	31	,	,	PUNCT
ejpam-6526	109	32	ψ,⊕	ψ,⊕	NOUN
ejpam-6526	109	33	)	)	PUNCT
ejpam-6526	109	34	where	where	SCONJ
ejpam-6526	109	35	x	x	PRON
ejpam-6526	109	36	is	be	AUX
ejpam-6526	109	37	the	the	DET
ejpam-6526	109	38	space	space	NOUN
ejpam-6526	109	39	of	of	ADP
ejpam-6526	109	40	images	image	NOUN
ejpam-6526	109	41	,	,	PUNCT
ejpam-6526	109	42	allows	allow	VERB
ejpam-6526	109	43	simultaneous	simultaneous	ADJ
ejpam-6526	109	44	optimization	optimization	NOUN
ejpam-6526	109	45	for	for	ADP
ejpam-6526	109	46	both	both	PRON
ejpam-6526	109	47	pixel	pixel	ADJ
ejpam-6526	109	48	-	-	ADJ
ejpam-6526	109	49	wise	wise	ADJ
ejpam-6526	109	50	accuracy	accuracy	NOUN
ejpam-6526	109	51	and	and	CCONJ
ejpam-6526	109	52	perceptual	perceptual	ADJ
ejpam-6526	109	53	quality	quality	NOUN
ejpam-6526	109	54	.	.	PUNCT
ejpam-6526	110	1	theorem	theorem	NOUN
ejpam-6526	110	2	2	2	NUM
ejpam-6526	110	3	.	.	X
ejpam-6526	110	4	for	for	ADP
ejpam-6526	110	5	bi	bi	ADJ
ejpam-6526	110	6	-	-	ADJ
ejpam-6526	110	7	metric	metric	ADJ
ejpam-6526	110	8	systems	system	NOUN
ejpam-6526	110	9	(	(	PUNCT
ejpam-6526	110	10	x	x	X
ejpam-6526	110	11	,	,	PUNCT
ejpam-6526	110	12	ψx	ψx	X
ejpam-6526	110	13	,	,	PUNCT
ejpam-6526	110	14	⊕x	⊕x	PROPN
ejpam-6526	110	15	)	)	PUNCT
ejpam-6526	111	1	and	and	CCONJ
ejpam-6526	111	2	(	(	PUNCT
ejpam-6526	111	3	y	y	PROPN
ejpam-6526	111	4	,	,	PUNCT
ejpam-6526	111	5	ψy	ψy	PROPN
ejpam-6526	111	6	,	,	PUNCT
ejpam-6526	111	7	⊕y	⊕y	NOUN
ejpam-6526	111	8	)	)	PUNCT
ejpam-6526	111	9	,	,	PUNCT
ejpam-6526	111	10	function	function	NOUN
ejpam-6526	111	11	f	f	NOUN
ejpam-6526	111	12	:	:	PUNCT
ejpam-6526	111	13	x	x	X
ejpam-6526	111	14	→	→	SYM
ejpam-6526	111	15	y	y	PROPN
ejpam-6526	111	16	demonstrates	demonstrate	VERB
ejpam-6526	111	17	continuity	continuity	NOUN
ejpam-6526	111	18	with	with	ADP
ejpam-6526	111	19	respect	respect	NOUN
ejpam-6526	111	20	to	to	ADP
ejpam-6526	111	21	the	the	DET
ejpam-6526	111	22	i	i	PROPN
ejpam-6526	111	23	-	-	PUNCT
ejpam-6526	111	24	th	th	X
ejpam-6526	111	25	induced	induce	VERB
ejpam-6526	111	26	topologies	topology	NOUN
ejpam-6526	111	27	precisely	precisely	ADV
ejpam-6526	111	28	when	when	SCONJ
ejpam-6526	111	29	for	for	ADP
ejpam-6526	111	30	every	every	DET
ejpam-6526	111	31	x	x	SYM
ejpam-6526	111	32	∈	∈	PROPN
ejpam-6526	111	33	x	x	X
ejpam-6526	111	34	and	and	CCONJ
ejpam-6526	111	35	ε	ε	PROPN
ejpam-6526	111	36	>	>	X
ejpam-6526	111	37	0	0	PROPN
ejpam-6526	111	38	,	,	PUNCT
ejpam-6526	111	39	there	there	PRON
ejpam-6526	111	40	exists	exist	VERB
ejpam-6526	111	41	δ	δ	PROPN
ejpam-6526	111	42	>	>	X
ejpam-6526	111	43	0	0	NUM
ejpam-6526	111	44	where	where	SCONJ
ejpam-6526	111	45	ϕx	ϕx	NOUN
ejpam-6526	111	46	,	,	PUNCT
ejpam-6526	111	47	i(x	i(x	NOUN
ejpam-6526	111	48	,	,	PUNCT
ejpam-6526	111	49	z	z	NOUN
ejpam-6526	111	50	)	)	PUNCT
ejpam-6526	112	1	<	<	X
ejpam-6526	112	2	δ	δ	PROPN
ejpam-6526	112	3	implies	imply	VERB
ejpam-6526	112	4	ϕy	ϕy	PROPN
ejpam-6526	112	5	,	,	PUNCT
ejpam-6526	112	6	i(f(x	i(f(x	NOUN
ejpam-6526	112	7	)	)	PUNCT
ejpam-6526	112	8	,	,	PUNCT
ejpam-6526	112	9	f(z	f(z	PROPN
ejpam-6526	112	10	)	)	PUNCT
ejpam-6526	112	11	)	)	PUNCT
ejpam-6526	113	1	<	<	X
ejpam-6526	113	2	ε	ε	PROPN
ejpam-6526	113	3	.	.	PUNCT
ejpam-6526	113	4	proof	proof	NOUN
ejpam-6526	113	5	.	.	PUNCT
ejpam-6526	114	1	(	(	PUNCT
ejpam-6526	114	2	⇒	⇒	PROPN
ejpam-6526	114	3	)	)	PUNCT
ejpam-6526	114	4	assuming	assume	VERB
ejpam-6526	114	5	f	f	PROPN
ejpam-6526	114	6	demonstrates	demonstrate	VERB
ejpam-6526	114	7	continuity	continuity	NOUN
ejpam-6526	114	8	with	with	ADP
ejpam-6526	114	9	respect	respect	NOUN
ejpam-6526	114	10	to	to	ADP
ejpam-6526	114	11	the	the	DET
ejpam-6526	114	12	i	i	PROPN
ejpam-6526	114	13	-	-	PUNCT
ejpam-6526	114	14	th	th	X
ejpam-6526	114	15	induced	induce	VERB
ejpam-6526	114	16	topologies	topology	NOUN
ejpam-6526	114	17	,	,	PUNCT
ejpam-6526	114	18	consider	consider	VERB
ejpam-6526	114	19	arbitrary	arbitrary	ADJ
ejpam-6526	114	20	x	x	SYM
ejpam-6526	114	21	∈	∈	PROPN
ejpam-6526	114	22	x	x	X
ejpam-6526	114	23	and	and	CCONJ
ejpam-6526	114	24	ε	ε	PROPN
ejpam-6526	114	25	>	>	X
ejpam-6526	114	26	0	0	PROPN
ejpam-6526	114	27	.	.	PUNCT
ejpam-6526	115	1	the	the	DET
ejpam-6526	115	2	set	set	PROPN
ejpam-6526	115	3	vy	vy	NOUN
ejpam-6526	115	4	,	,	PUNCT
ejpam-6526	115	5	i(f(x	i(f(x	NOUN
ejpam-6526	115	6	)	)	PUNCT
ejpam-6526	115	7	,	,	PUNCT
ejpam-6526	115	8	ε	ε	PROPN
ejpam-6526	115	9	)	)	PUNCT
ejpam-6526	115	10	=	=	PRON
ejpam-6526	115	11	{	{	PUNCT
ejpam-6526	115	12	y	y	PROPN
ejpam-6526	115	13	∈	∈	PROPN
ejpam-6526	116	1	y	y	PROPN
ejpam-6526	116	2	:	:	PUNCT
ejpam-6526	116	3	ϕy	ϕy	INTJ
ejpam-6526	116	4	,	,	PUNCT
ejpam-6526	116	5	i(f(x	i(f(x	NOUN
ejpam-6526	116	6	)	)	PUNCT
ejpam-6526	116	7	,	,	PUNCT
ejpam-6526	116	8	y	y	PROPN
ejpam-6526	116	9	)	)	PUNCT
ejpam-6526	116	10	<	<	X
ejpam-6526	116	11	ε	ε	PROPN
ejpam-6526	116	12	}	}	PUNCT
ejpam-6526	116	13	forms	form	VERB
ejpam-6526	116	14	an	an	DET
ejpam-6526	116	15	open	open	ADJ
ejpam-6526	116	16	set	set	NOUN
ejpam-6526	116	17	in	in	ADP
ejpam-6526	116	18	the	the	DET
ejpam-6526	116	19	i	i	PROPN
ejpam-6526	116	20	-	-	PUNCT
ejpam-6526	116	21	th	th	X
ejpam-6526	116	22	topology	topology	NOUN
ejpam-6526	116	23	of	of	ADP
ejpam-6526	116	24	y	y	PROPN
ejpam-6526	116	25	.	.	PUNCT
ejpam-6526	117	1	by	by	ADP
ejpam-6526	117	2	continuity	continuity	NOUN
ejpam-6526	117	3	properties	property	NOUN
ejpam-6526	117	4	,	,	PUNCT
ejpam-6526	117	5	f−1(vy	f−1(vy	PROPN
ejpam-6526	117	6	,	,	PUNCT
ejpam-6526	117	7	i(f(x	i(f(x	NOUN
ejpam-6526	117	8	)	)	PUNCT
ejpam-6526	117	9	,	,	PUNCT
ejpam-6526	117	10	ε	ε	PROPN
ejpam-6526	117	11	)	)	PUNCT
ejpam-6526	117	12	)	)	PUNCT
ejpam-6526	117	13	is	be	AUX
ejpam-6526	117	14	open	open	ADJ
ejpam-6526	117	15	in	in	ADP
ejpam-6526	117	16	the	the	DET
ejpam-6526	117	17	i	i	PROPN
ejpam-6526	117	18	-	-	PUNCT
ejpam-6526	117	19	th	th	X
ejpam-6526	117	20	topology	topology	NOUN
ejpam-6526	117	21	of	of	ADP
ejpam-6526	117	22	x.	x.	NOUN
ejpam-6526	117	23	since	since	SCONJ
ejpam-6526	117	24	x	x	PROPN
ejpam-6526	117	25	∈	∈	PROPN
ejpam-6526	117	26	f−1(vy	f−1(vy	PROPN
ejpam-6526	117	27	,	,	PUNCT
ejpam-6526	117	28	i(f(x	i(f(x	NOUN
ejpam-6526	117	29	)	)	PUNCT
ejpam-6526	117	30	,	,	PUNCT
ejpam-6526	117	31	ε	ε	PROPN
ejpam-6526	117	32	)	)	PUNCT
ejpam-6526	117	33	)	)	PUNCT
ejpam-6526	117	34	,	,	PUNCT
ejpam-6526	117	35	we	we	PRON
ejpam-6526	117	36	identify	identify	VERB
ejpam-6526	117	37	δ	δ	PROPN
ejpam-6526	117	38	>	>	X
ejpam-6526	117	39	0	0	PUNCT
ejpam-6526	118	1	with	with	ADP
ejpam-6526	118	2	vx	vx	PROPN
ejpam-6526	118	3	,	,	PUNCT
ejpam-6526	118	4	i(x	i(x	PROPN
ejpam-6526	118	5	,	,	PUNCT
ejpam-6526	118	6	δ	δ	PROPN
ejpam-6526	118	7	)	)	PUNCT
ejpam-6526	118	8	⊂	⊂	PROPN
ejpam-6526	118	9	f−1(vy	f−1(vy	PROPN
ejpam-6526	118	10	,	,	PUNCT
ejpam-6526	118	11	i(f(x	i(f(x	NOUN
ejpam-6526	118	12	)	)	PUNCT
ejpam-6526	118	13	,	,	PUNCT
ejpam-6526	118	14	ε	ε	PROPN
ejpam-6526	118	15	)	)	PUNCT
ejpam-6526	118	16	)	)	PUNCT
ejpam-6526	118	17	.	.	PUNCT
ejpam-6526	119	1	thus	thus	ADV
ejpam-6526	119	2	,	,	PUNCT
ejpam-6526	119	3	whenever	whenever	SCONJ
ejpam-6526	119	4	ϕx	ϕx	ADP
ejpam-6526	119	5	,	,	PUNCT
ejpam-6526	119	6	i(x	i(x	NOUN
ejpam-6526	119	7	,	,	PUNCT
ejpam-6526	119	8	z	z	NOUN
ejpam-6526	119	9	)	)	PUNCT
ejpam-6526	119	10	<	<	X
ejpam-6526	119	11	δ	δ	PROPN
ejpam-6526	119	12	,	,	PUNCT
ejpam-6526	119	13	z	z	PROPN
ejpam-6526	119	14	∈	∈	PROPN
ejpam-6526	119	15	vx	vx	PROPN
ejpam-6526	119	16	,	,	PUNCT
ejpam-6526	119	17	i(x	i(x	PROPN
ejpam-6526	119	18	,	,	PUNCT
ejpam-6526	119	19	δ	δ	PROPN
ejpam-6526	119	20	)	)	PUNCT
ejpam-6526	119	21	,	,	PUNCT
ejpam-6526	119	22	yielding	yield	VERB
ejpam-6526	119	23	f(z	f(z	PROPN
ejpam-6526	119	24	)	)	PUNCT
ejpam-6526	119	25	∈	∈	PROPN
ejpam-6526	119	26	vy	vy	NOUN
ejpam-6526	119	27	,	,	PUNCT
ejpam-6526	119	28	i(f(x	i(f(x	NOUN
ejpam-6526	119	29	)	)	PUNCT
ejpam-6526	119	30	,	,	PUNCT
ejpam-6526	119	31	ε	ε	PROPN
ejpam-6526	119	32	)	)	PUNCT
ejpam-6526	119	33	equivalently	equivalently	ADV
ejpam-6526	119	34	,	,	PUNCT
ejpam-6526	119	35	ϕy	ϕy	PROPN
ejpam-6526	119	36	,	,	PUNCT
ejpam-6526	119	37	i(f(x	i(f(x	NOUN
ejpam-6526	119	38	)	)	PUNCT
ejpam-6526	119	39	,	,	PUNCT
ejpam-6526	119	40	f(z	f(z	PROPN
ejpam-6526	119	41	)	)	PUNCT
ejpam-6526	119	42	)	)	PUNCT
ejpam-6526	119	43	<	<	X
ejpam-6526	119	44	ε	ε	PROPN
ejpam-6526	119	45	.	.	PUNCT
ejpam-6526	119	46	(	(	PUNCT
ejpam-6526	119	47	⇐	⇐	NOUN
ejpam-6526	119	48	)	)	PUNCT
ejpam-6526	119	49	assuming	assume	VERB
ejpam-6526	119	50	the	the	DET
ejpam-6526	119	51	stated	stated	ADJ
ejpam-6526	119	52	condition	condition	NOUN
ejpam-6526	119	53	,	,	PUNCT
ejpam-6526	119	54	consider	consider	VERB
ejpam-6526	119	55	arbitrary	arbitrary	ADJ
ejpam-6526	119	56	open	open	ADJ
ejpam-6526	119	57	set	set	VERB
ejpam-6526	119	58	v	v	NOUN
ejpam-6526	119	59	in	in	ADP
ejpam-6526	119	60	the	the	DET
ejpam-6526	119	61	i	i	PROPN
ejpam-6526	119	62	-	-	PUNCT
ejpam-6526	119	63	th	th	X
ejpam-6526	119	64	topology	topology	NOUN
ejpam-6526	119	65	of	of	ADP
ejpam-6526	119	66	y	y	PROPN
ejpam-6526	119	67	.	.	PUNCT
ejpam-6526	120	1	we	we	PRON
ejpam-6526	120	2	must	must	AUX
ejpam-6526	120	3	establish	establish	VERB
ejpam-6526	120	4	that	that	DET
ejpam-6526	120	5	f−1(v	f−1(v	NOUN
ejpam-6526	120	6	)	)	PUNCT
ejpam-6526	120	7	is	be	AUX
ejpam-6526	120	8	open	open	ADJ
ejpam-6526	120	9	in	in	ADP
ejpam-6526	120	10	the	the	DET
ejpam-6526	120	11	i	i	PROPN
ejpam-6526	120	12	-	-	PUNCT
ejpam-6526	120	13	th	th	X
ejpam-6526	120	14	topology	topology	NOUN
ejpam-6526	120	15	of	of	ADP
ejpam-6526	120	16	x.	x.	NOUN
ejpam-6526	120	17	for	for	ADP
ejpam-6526	120	18	any	any	DET
ejpam-6526	120	19	x	x	SYM
ejpam-6526	120	20	∈	∈	PROPN
ejpam-6526	120	21	f−1(v	f−1(v	NOUN
ejpam-6526	120	22	)	)	PUNCT
ejpam-6526	120	23	,	,	PUNCT
ejpam-6526	120	24	we	we	PRON
ejpam-6526	120	25	have	have	VERB
ejpam-6526	120	26	f(x	f(x	PROPN
ejpam-6526	120	27	)	)	PUNCT
ejpam-6526	120	28	∈	∈	PROPN
ejpam-6526	120	29	v	v	NOUN
ejpam-6526	120	30	.	.	PUNCT
ejpam-6526	121	1	since	since	SCONJ
ejpam-6526	121	2	v	v	NOUN
ejpam-6526	121	3	is	be	AUX
ejpam-6526	121	4	open	open	ADJ
ejpam-6526	121	5	,	,	PUNCT
ejpam-6526	121	6	there	there	PRON
ejpam-6526	121	7	exists	exist	VERB
ejpam-6526	121	8	ε	ε	PROPN
ejpam-6526	121	9	>	>	X
ejpam-6526	121	10	0	0	NUM
ejpam-6526	121	11	where	where	SCONJ
ejpam-6526	121	12	vy	vy	NOUN
ejpam-6526	121	13	,	,	PUNCT
ejpam-6526	121	14	i(f(x	i(f(x	NOUN
ejpam-6526	121	15	)	)	PUNCT
ejpam-6526	121	16	,	,	PUNCT
ejpam-6526	121	17	ε	ε	PROPN
ejpam-6526	121	18	)	)	PUNCT
ejpam-6526	121	19	⊂	⊂	PROPN
ejpam-6526	121	20	v	v	NOUN
ejpam-6526	121	21	.	.	PUNCT
ejpam-6526	122	1	by	by	ADP
ejpam-6526	122	2	our	our	PRON
ejpam-6526	122	3	assumption	assumption	NOUN
ejpam-6526	122	4	,	,	PUNCT
ejpam-6526	122	5	we	we	PRON
ejpam-6526	122	6	identify	identify	VERB
ejpam-6526	122	7	δ	δ	PROPN
ejpam-6526	122	8	>	>	X
ejpam-6526	122	9	0	0	NUM
ejpam-6526	123	1	where	where	SCONJ
ejpam-6526	123	2	ϕx	ϕx	NOUN
ejpam-6526	123	3	,	,	PUNCT
ejpam-6526	123	4	i(x	i(x	NOUN
ejpam-6526	123	5	,	,	PUNCT
ejpam-6526	123	6	z	z	NOUN
ejpam-6526	123	7	)	)	PUNCT
ejpam-6526	123	8	<	<	X
ejpam-6526	124	1	δ	δ	PROPN
ejpam-6526	124	2	implies	imply	VERB
ejpam-6526	124	3	ϕy	ϕy	PROPN
ejpam-6526	124	4	,	,	PUNCT
ejpam-6526	124	5	i(f(x	i(f(x	NOUN
ejpam-6526	124	6	)	)	PUNCT
ejpam-6526	124	7	,	,	PUNCT
ejpam-6526	124	8	f(z	f(z	PROPN
ejpam-6526	124	9	)	)	PUNCT
ejpam-6526	124	10	)	)	PUNCT
ejpam-6526	125	1	<	<	X
ejpam-6526	125	2	ε	ε	PROPN
ejpam-6526	125	3	.	.	PUNCT
ejpam-6526	126	1	this	this	DET
ejpam-6526	126	2	yields	yield	NOUN
ejpam-6526	126	3	vx	vx	PROPN
ejpam-6526	126	4	,	,	PUNCT
ejpam-6526	126	5	i(x	i(x	PROPN
ejpam-6526	126	6	,	,	PUNCT
ejpam-6526	126	7	δ	δ	PROPN
ejpam-6526	126	8	)	)	PUNCT
ejpam-6526	126	9	⊂	⊂	PROPN
ejpam-6526	126	10	f−1(vy	f−1(vy	PROPN
ejpam-6526	126	11	,	,	PUNCT
ejpam-6526	126	12	i(f(x	i(f(x	NOUN
ejpam-6526	126	13	)	)	PUNCT
ejpam-6526	126	14	,	,	PUNCT
ejpam-6526	126	15	ε	ε	PROPN
ejpam-6526	126	16	)	)	PUNCT
ejpam-6526	126	17	)	)	PUNCT
ejpam-6526	127	1	⊂	⊂	PROPN
ejpam-6526	127	2	f−1(v	f−1(v	PROPN
ejpam-6526	127	3	)	)	PUNCT
ejpam-6526	127	4	.	.	PUNCT
ejpam-6526	128	1	therefore	therefore	ADV
ejpam-6526	128	2	,	,	PUNCT
ejpam-6526	128	3	f−1(v	f−1(v	PROPN
ejpam-6526	128	4	)	)	PUNCT
ejpam-6526	128	5	is	be	AUX
ejpam-6526	128	6	open	open	ADJ
ejpam-6526	128	7	in	in	ADP
ejpam-6526	128	8	the	the	DET
ejpam-6526	128	9	i	i	PROPN
ejpam-6526	128	10	-	-	PUNCT
ejpam-6526	128	11	th	th	X
ejpam-6526	128	12	topology	topology	NOUN
ejpam-6526	128	13	of	of	ADP
ejpam-6526	128	14	x	x	PRON
ejpam-6526	128	15	,	,	PUNCT
ejpam-6526	128	16	confirming	confirm	VERB
ejpam-6526	128	17	f	f	PROPN
ejpam-6526	128	18	’s	’s	PART
ejpam-6526	128	19	continuity	continuity	NOUN
ejpam-6526	128	20	characteristics	characteristic	NOUN
ejpam-6526	128	21	.	.	PUNCT
ejpam-6526	129	1	example	example	NOUN
ejpam-6526	129	2	6	6	NUM
ejpam-6526	129	3	(	(	PUNCT
ejpam-6526	129	4	continuity	continuity	NOUN
ejpam-6526	129	5	in	in	ADP
ejpam-6526	129	6	bi	bi	ADJ
ejpam-6526	129	7	-	-	ADJ
ejpam-6526	129	8	metric	metric	ADJ
ejpam-6526	129	9	systems	system	NOUN
ejpam-6526	129	10	)	)	PUNCT
ejpam-6526	129	11	.	.	PUNCT
ejpam-6526	130	1	taking	take	VERB
ejpam-6526	130	2	x	x	PUNCT
ejpam-6526	130	3	=	=	PUNCT
ejpam-6526	130	4	y	y	NOUN
ejpam-6526	130	5	=	=	SYM
ejpam-6526	130	6	r	r	NOUN
ejpam-6526	130	7	with	with	ADP
ejpam-6526	130	8	standard	standard	ADJ
ejpam-6526	130	9	topology	topology	NOUN
ejpam-6526	130	10	,	,	PUNCT
ejpam-6526	130	11	we	we	PRON
ejpam-6526	130	12	define	define	VERB
ejpam-6526	130	13	bi	bi	ADJ
ejpam-6526	130	14	-	-	ADJ
ejpam-6526	130	15	metric	metric	ADJ
ejpam-6526	130	16	systems	system	NOUN
ejpam-6526	130	17	:	:	PUNCT
ejpam-6526	130	18	ψx(x1	ψx(x1	NUM
ejpam-6526	130	19	,	,	PUNCT
ejpam-6526	130	20	x2	x2	PROPN
ejpam-6526	130	21	)	)	PUNCT
ejpam-6526	130	22	=	=	PUNCT
ejpam-6526	131	1	(	(	PUNCT
ejpam-6526	131	2	|x1	|x1	NUM
ejpam-6526	131	3	−	−	PROPN
ejpam-6526	131	4	x2|	x2|	PROPN
ejpam-6526	131	5	,	,	PUNCT
ejpam-6526	131	6	|(x1)−	|(x1)−	PROPN
ejpam-6526	131	7	(	(	PUNCT
ejpam-6526	131	8	x2)|	x2)|	PROPN
ejpam-6526	131	9	)	)	PUNCT
ejpam-6526	131	10	ψy	ψy	PROPN
ejpam-6526	131	11	(	(	PUNCT
ejpam-6526	131	12	y1	y1	PROPN
ejpam-6526	131	13	,	,	PUNCT
ejpam-6526	131	14	y2	y2	NOUN
ejpam-6526	131	15	)	)	PUNCT
ejpam-6526	131	16	=	=	PUNCT
ejpam-6526	132	1	(	(	PUNCT
ejpam-6526	132	2	|y1	|y1	VERB
ejpam-6526	132	3	−	−	PROPN
ejpam-6526	132	4	y2|	y2|	NOUN
ejpam-6526	132	5	,	,	PUNCT
ejpam-6526	132	6	|	|	ADV
ejpam-6526	132	7	sinh(y1)−	sinh(y1)−	ADJ
ejpam-6526	132	8	sinh(y2)|	sinh(y2)|	NOUN
ejpam-6526	132	9	)	)	PUNCT
ejpam-6526	132	10	with	with	ADP
ejpam-6526	132	11	both	both	CCONJ
ejpam-6526	132	12	⊕x	⊕x	PROPN
ejpam-6526	132	13	and	and	CCONJ
ejpam-6526	132	14	⊕y	⊕y	NOUN
ejpam-6526	132	15	implementing	implement	VERB
ejpam-6526	132	16	component	component	NOUN
ejpam-6526	132	17	-	-	PUNCT
ejpam-6526	132	18	wise	wise	ADJ
ejpam-6526	132	19	addition	addition	NOUN
ejpam-6526	132	20	.	.	PUNCT
ejpam-6526	133	1	our	our	PRON
ejpam-6526	133	2	continuity	continuity	NOUN
ejpam-6526	133	3	approach	approach	NOUN
ejpam-6526	133	4	extends	extend	VERB
ejpam-6526	133	5	and	and	CCONJ
ejpam-6526	133	6	generalizes	generalize	VERB
ejpam-6526	133	7	quasi	quasi	ADJ
ejpam-6526	133	8	-	-	ADJ
ejpam-6526	133	9	uniformization	uniformization	ADJ
ejpam-6526	133	10	techniques	technique	NOUN
ejpam-6526	133	11	previously	previously	ADV
ejpam-6526	133	12	documented	document	VERB
ejpam-6526	133	13	in	in	ADP
ejpam-6526	133	14	mathematical	mathematical	ADJ
ejpam-6526	133	15	research	research	NOUN
ejpam-6526	133	16	[	[	X
ejpam-6526	133	17	19	19	NUM
ejpam-6526	133	18	]	]	PUNCT
ejpam-6526	133	19	.	.	PUNCT
ejpam-6526	134	1	definition	definition	NOUN
ejpam-6526	134	2	6	6	NUM
ejpam-6526	134	3	.	.	PUNCT
ejpam-6526	135	1	we	we	PRON
ejpam-6526	135	2	characterize	characterize	VERB
ejpam-6526	135	3	a	a	DET
ejpam-6526	135	4	sequence	sequence	NOUN
ejpam-6526	135	5	{	{	PUNCT
ejpam-6526	135	6	xn	xn	NOUN
ejpam-6526	135	7	}	}	PUNCT
ejpam-6526	135	8	in	in	ADP
ejpam-6526	135	9	bi	bi	ADJ
ejpam-6526	135	10	-	-	ADJ
ejpam-6526	135	11	metric	metric	ADJ
ejpam-6526	135	12	system	system	NOUN
ejpam-6526	135	13	(	(	PUNCT
ejpam-6526	135	14	x	x	X
ejpam-6526	135	15	,	,	PUNCT
ejpam-6526	135	16	ψ,⊕	ψ,⊕	NOUN
ejpam-6526	135	17	)	)	PUNCT
ejpam-6526	135	18	as	as	ADP
ejpam-6526	135	19	bi	bi	NOUN
ejpam-6526	135	20	-	-	ADJ
ejpam-6526	135	21	cauchy	cauchy	ADJ
ejpam-6526	135	22	when	when	SCONJ
ejpam-6526	135	23	for	for	ADP
ejpam-6526	135	24	every	every	DET
ejpam-6526	135	25	ε	ε	PROPN
ejpam-6526	135	26	>	>	X
ejpam-6526	135	27	0	0	PROPN
ejpam-6526	135	28	,	,	PUNCT
ejpam-6526	135	29	there	there	PRON
ejpam-6526	135	30	exists	exist	VERB
ejpam-6526	135	31	n	n	DET
ejpam-6526	135	32	∈	∈	NOUN
ejpam-6526	135	33	n	n	CCONJ
ejpam-6526	135	34	where	where	SCONJ
ejpam-6526	135	35	for	for	ADP
ejpam-6526	135	36	all	all	DET
ejpam-6526	135	37	m	m	PROPN
ejpam-6526	135	38	,	,	PUNCT
ejpam-6526	135	39	n	n	PRON
ejpam-6526	135	40	≥	≥	NOUN
ejpam-6526	135	41	n	n	CCONJ
ejpam-6526	135	42	:	:	PUNCT
ejpam-6526	135	43	ψ(xm	ψ(xm	ADV
ejpam-6526	135	44	,	,	PUNCT
ejpam-6526	135	45	xn	xn	PROPN
ejpam-6526	135	46	)	)	PUNCT
ejpam-6526	136	1	<	<	X
ejpam-6526	136	2	comp	comp	X
ejpam-6526	136	3	(	(	PUNCT
ejpam-6526	136	4	ε	ε	PROPN
ejpam-6526	136	5	,	,	PUNCT
ejpam-6526	136	6	ε	ε	PROPN
ejpam-6526	136	7	)	)	PUNCT
ejpam-6526	136	8	with	with	ADP
ejpam-6526	136	9	<	<	X
ejpam-6526	136	10	comp	comp	NOUN
ejpam-6526	136	11	indicating	indicate	VERB
ejpam-6526	136	12	strict	strict	ADJ
ejpam-6526	136	13	component	component	NOUN
ejpam-6526	136	14	-	-	PUNCT
ejpam-6526	136	15	wise	wise	ADJ
ejpam-6526	136	16	ordering	ordering	NOUN
ejpam-6526	136	17	.	.	PUNCT
ejpam-6526	137	1	definition	definition	NOUN
ejpam-6526	137	2	7	7	NUM
ejpam-6526	137	3	.	.	PUNCT
ejpam-6526	138	1	we	we	PRON
ejpam-6526	138	2	classify	classify	VERB
ejpam-6526	138	3	a	a	DET
ejpam-6526	138	4	bi	bi	ADJ
ejpam-6526	138	5	-	-	ADJ
ejpam-6526	138	6	metric	metric	ADJ
ejpam-6526	138	7	system	system	NOUN
ejpam-6526	138	8	(	(	PUNCT
ejpam-6526	138	9	x	x	X
ejpam-6526	138	10	,	,	PUNCT
ejpam-6526	138	11	ψ,⊕	ψ,⊕	NOUN
ejpam-6526	138	12	)	)	PUNCT
ejpam-6526	138	13	as	as	ADP
ejpam-6526	138	14	bi	bi	NOUN
ejpam-6526	138	15	-	-	ADJ
ejpam-6526	138	16	complete	complete	ADJ
ejpam-6526	138	17	when	when	SCONJ
ejpam-6526	138	18	every	every	DET
ejpam-6526	138	19	bicauchy	bicauchy	NOUN
ejpam-6526	138	20	sequence	sequence	NOUN
ejpam-6526	138	21	in	in	ADP
ejpam-6526	138	22	x	x	PART
ejpam-6526	138	23	converges	converge	NOUN
ejpam-6526	138	24	in	in	ADP
ejpam-6526	138	25	x	x	PUNCT
ejpam-6526	138	26	with	with	ADP
ejpam-6526	138	27	respect	respect	NOUN
ejpam-6526	138	28	to	to	ADP
ejpam-6526	138	29	both	both	DET
ejpam-6526	138	30	induced	induced	ADJ
ejpam-6526	138	31	topologies	topology	NOUN
ejpam-6526	138	32	.	.	PUNCT
ejpam-6526	139	1	a.	a.	PROPN
ejpam-6526	139	2	alsoboh	alsoboh	PROPN
ejpam-6526	139	3	et	et	PROPN
ejpam-6526	139	4	al	al	PROPN
ejpam-6526	139	5	.	.	PUNCT
ejpam-6526	139	6	/	/	SYM
ejpam-6526	139	7	eur	eur	PROPN
ejpam-6526	139	8	.	.	PUNCT
ejpam-6526	140	1	j.	j.	PROPN
ejpam-6526	140	2	pure	pure	PROPN
ejpam-6526	140	3	appl	appl	PROPN
ejpam-6526	140	4	.	.	PROPN
ejpam-6526	140	5	math	math	PROPN
ejpam-6526	140	6	,	,	PUNCT
ejpam-6526	140	7	18	18	NUM
ejpam-6526	140	8	(	(	PUNCT
ejpam-6526	140	9	4	4	NUM
ejpam-6526	140	10	)	)	PUNCT
ejpam-6526	140	11	(	(	PUNCT
ejpam-6526	140	12	2025	2025	NUM
ejpam-6526	140	13	)	)	PUNCT
ejpam-6526	140	14	,	,	PUNCT
ejpam-6526	140	15	6526	6526	NUM
ejpam-6526	140	16	7	7	NUM
ejpam-6526	140	17	of	of	ADP
ejpam-6526	140	18	21	21	NUM
ejpam-6526	140	19	theorem	theorem	NOUN
ejpam-6526	140	20	3	3	NUM
ejpam-6526	140	21	.	.	PUNCT
ejpam-6526	141	1	a	a	DET
ejpam-6526	141	2	bi	bi	ADJ
ejpam-6526	141	3	-	-	ADJ
ejpam-6526	141	4	metric	metric	ADJ
ejpam-6526	141	5	system	system	NOUN
ejpam-6526	141	6	(	(	PUNCT
ejpam-6526	141	7	x	x	X
ejpam-6526	141	8	,	,	PUNCT
ejpam-6526	141	9	ψ,⊕	ψ,⊕	NOUN
ejpam-6526	141	10	)	)	PUNCT
ejpam-6526	141	11	achieves	achieve	VERB
ejpam-6526	141	12	bi	bi	NOUN
ejpam-6526	141	13	-	-	NOUN
ejpam-6526	141	14	completeness	completeness	NOUN
ejpam-6526	141	15	precisely	precisely	ADV
ejpam-6526	141	16	when	when	SCONJ
ejpam-6526	141	17	it	it	PRON
ejpam-6526	141	18	demonstrates	demonstrate	VERB
ejpam-6526	141	19	completeness	completeness	NOUN
ejpam-6526	141	20	with	with	ADP
ejpam-6526	141	21	respect	respect	NOUN
ejpam-6526	141	22	to	to	ADP
ejpam-6526	141	23	both	both	DET
ejpam-6526	141	24	component	component	NOUN
ejpam-6526	141	25	metric	metric	ADJ
ejpam-6526	141	26	functions	function	NOUN
ejpam-6526	141	27	.	.	PUNCT
ejpam-6526	142	1	proof	proof	NOUN
ejpam-6526	142	2	.	.	PUNCT
ejpam-6526	143	1	(	(	PUNCT
ejpam-6526	143	2	⇒	⇒	PROPN
ejpam-6526	143	3	)	)	PUNCT
ejpam-6526	143	4	assuming	assume	VERB
ejpam-6526	143	5	(	(	PUNCT
ejpam-6526	143	6	x	x	X
ejpam-6526	143	7	,	,	PUNCT
ejpam-6526	143	8	ψ,⊕	ψ,⊕	NOUN
ejpam-6526	143	9	)	)	PUNCT
ejpam-6526	143	10	is	be	AUX
ejpam-6526	143	11	bi	bi	NOUN
ejpam-6526	143	12	-	-	ADJ
ejpam-6526	143	13	complete	complete	ADJ
ejpam-6526	143	14	,	,	PUNCT
ejpam-6526	143	15	consider	consider	VERB
ejpam-6526	143	16	cauchy	cauchy	ADJ
ejpam-6526	143	17	sequence	sequence	NOUN
ejpam-6526	143	18	{	{	PUNCT
ejpam-6526	143	19	xn	xn	NUM
ejpam-6526	143	20	}	}	PUNCT
ejpam-6526	143	21	with	with	ADP
ejpam-6526	143	22	respect	respect	NOUN
ejpam-6526	143	23	to	to	ADP
ejpam-6526	143	24	first	first	ADJ
ejpam-6526	143	25	component	component	NOUN
ejpam-6526	143	26	metric	metric	ADJ
ejpam-6526	143	27	ϕ1	ϕ1	NOUN
ejpam-6526	143	28	.	.	PUNCT
ejpam-6526	144	1	for	for	ADP
ejpam-6526	144	2	any	any	DET
ejpam-6526	144	3	ε	ε	PROPN
ejpam-6526	144	4	>	>	X
ejpam-6526	144	5	0	0	PROPN
ejpam-6526	144	6	,	,	PUNCT
ejpam-6526	144	7	we	we	PRON
ejpam-6526	144	8	identify	identify	VERB
ejpam-6526	144	9	n1	n1	NOUN
ejpam-6526	144	10	∈	∈	PROPN
ejpam-6526	144	11	n	n	CCONJ
ejpam-6526	144	12	where	where	SCONJ
ejpam-6526	144	13	for	for	ADP
ejpam-6526	144	14	all	all	DET
ejpam-6526	144	15	m	m	PROPN
ejpam-6526	144	16	,	,	PUNCT
ejpam-6526	144	17	n	n	PRON
ejpam-6526	144	18	≥	≥	NOUN
ejpam-6526	144	19	n1	n1	PROPN
ejpam-6526	144	20	,	,	PUNCT
ejpam-6526	144	21	ϕ1(xm	ϕ1(xm	PROPN
ejpam-6526	144	22	,	,	PUNCT
ejpam-6526	144	23	xn	xn	PROPN
ejpam-6526	144	24	)	)	PUNCT
ejpam-6526	144	25	<	<	X
ejpam-6526	144	26	ε	ε	PROPN
ejpam-6526	144	27	.	.	PUNCT
ejpam-6526	144	28	setting	set	VERB
ejpam-6526	144	29	yn	yn	X
ejpam-6526	144	30	=	=	PUNCT
ejpam-6526	144	31	xn	xn	PROPN
ejpam-6526	144	32	for	for	ADP
ejpam-6526	144	33	all	all	DET
ejpam-6526	144	34	n	n	PRON
ejpam-6526	144	35	∈	∈	PROPN
ejpam-6526	144	36	n	n	CCONJ
ejpam-6526	144	37	,	,	PUNCT
ejpam-6526	144	38	we	we	PRON
ejpam-6526	144	39	observe	observe	VERB
ejpam-6526	144	40	for	for	ADP
ejpam-6526	144	41	all	all	DET
ejpam-6526	144	42	m	m	PROPN
ejpam-6526	144	43	,	,	PUNCT
ejpam-6526	144	44	n	n	PRON
ejpam-6526	144	45	≥	≥	NOUN
ejpam-6526	144	46	n1	n1	NOUN
ejpam-6526	144	47	,	,	PUNCT
ejpam-6526	144	48	ψ(ym	ψ(ym	PROPN
ejpam-6526	144	49	,	,	PUNCT
ejpam-6526	144	50	yn	yn	PROPN
ejpam-6526	144	51	)	)	PUNCT
ejpam-6526	144	52	=	=	PUNCT
ejpam-6526	144	53	(	(	PUNCT
ejpam-6526	144	54	ϕ1(xm	ϕ1(xm	PROPN
ejpam-6526	144	55	,	,	PUNCT
ejpam-6526	144	56	xn	xn	PROPN
ejpam-6526	144	57	)	)	PUNCT
ejpam-6526	144	58	,	,	PUNCT
ejpam-6526	144	59	ϕ2(xm	ϕ2(xm	PROPN
ejpam-6526	144	60	,	,	PUNCT
ejpam-6526	144	61	xn	xn	PROPN
ejpam-6526	144	62	)	)	PUNCT
ejpam-6526	144	63	)	)	PUNCT
ejpam-6526	145	1	<	<	X
ejpam-6526	145	2	comp	comp	X
ejpam-6526	145	3	(	(	PUNCT
ejpam-6526	145	4	ε	ε	PROPN
ejpam-6526	145	5	,	,	PUNCT
ejpam-6526	145	6	m	m	PROPN
ejpam-6526	145	7	)	)	PUNCT
ejpam-6526	145	8	for	for	ADP
ejpam-6526	145	9	some	some	DET
ejpam-6526	145	10	m	m	NOUN
ejpam-6526	145	11	>	>	X
ejpam-6526	145	12	0	0	PUNCT
ejpam-6526	145	13	(	(	PUNCT
ejpam-6526	145	14	potentially	potentially	ADV
ejpam-6526	145	15	sequencedependent	sequencedependent	NOUN
ejpam-6526	145	16	)	)	PUNCT
ejpam-6526	145	17	.	.	PUNCT
ejpam-6526	146	1	for	for	ADP
ejpam-6526	146	2	the	the	DET
ejpam-6526	146	3	second	second	ADJ
ejpam-6526	146	4	component	component	NOUN
ejpam-6526	146	5	,	,	PUNCT
ejpam-6526	146	6	we	we	PRON
ejpam-6526	146	7	identify	identify	VERB
ejpam-6526	146	8	n2	n2	PROPN
ejpam-6526	146	9	∈	∈	PROPN
ejpam-6526	146	10	n	n	X
ejpam-6526	146	11	where	where	SCONJ
ejpam-6526	146	12	for	for	ADP
ejpam-6526	146	13	all	all	DET
ejpam-6526	146	14	m	m	PROPN
ejpam-6526	146	15	,	,	PUNCT
ejpam-6526	146	16	n	n	PRON
ejpam-6526	146	17	≥	≥	NOUN
ejpam-6526	146	18	n2	n2	PROPN
ejpam-6526	146	19	,	,	PUNCT
ejpam-6526	146	20	ϕ2(xm	ϕ2(xm	PROPN
ejpam-6526	146	21	,	,	PUNCT
ejpam-6526	146	22	xn	xn	PROPN
ejpam-6526	146	23	)	)	PUNCT
ejpam-6526	146	24	<	<	X
ejpam-6526	146	25	ε	ε	PROPN
ejpam-6526	146	26	.	.	PUNCT
ejpam-6526	146	27	taking	take	VERB
ejpam-6526	146	28	n	n	PRON
ejpam-6526	146	29	=	=	SYM
ejpam-6526	146	30	max{n1	max{n1	NOUN
ejpam-6526	146	31	,	,	PUNCT
ejpam-6526	146	32	n2	n2	ADJ
ejpam-6526	146	33	}	}	PUNCT
ejpam-6526	146	34	,	,	PUNCT
ejpam-6526	146	35	for	for	ADP
ejpam-6526	146	36	all	all	DET
ejpam-6526	146	37	m	m	PROPN
ejpam-6526	146	38	,	,	PUNCT
ejpam-6526	146	39	n	n	PRON
ejpam-6526	146	40	≥	≥	NOUN
ejpam-6526	146	41	n	n	NOUN
ejpam-6526	146	42	,	,	PUNCT
ejpam-6526	146	43	ψ(ym	ψ(ym	PROPN
ejpam-6526	146	44	,	,	PUNCT
ejpam-6526	146	45	yn	yn	PROPN
ejpam-6526	146	46	)	)	PUNCT
ejpam-6526	147	1	<	<	X
ejpam-6526	147	2	comp	comp	X
ejpam-6526	147	3	(	(	PUNCT
ejpam-6526	147	4	ε	ε	PROPN
ejpam-6526	147	5	,	,	PUNCT
ejpam-6526	147	6	ε	ε	PROPN
ejpam-6526	147	7	)	)	PUNCT
ejpam-6526	147	8	,	,	PUNCT
ejpam-6526	147	9	establishing	establish	VERB
ejpam-6526	147	10	{	{	PUNCT
ejpam-6526	147	11	yn	yn	NOUN
ejpam-6526	147	12	}	}	PUNCT
ejpam-6526	147	13	as	as	ADP
ejpam-6526	147	14	bi	bi	NOUN
ejpam-6526	147	15	-	-	NOUN
ejpam-6526	147	16	cauchy	cauchy	NOUN
ejpam-6526	147	17	.	.	PUNCT
ejpam-6526	148	1	by	by	ADP
ejpam-6526	148	2	bi	bi	NOUN
ejpam-6526	148	3	-	-	NOUN
ejpam-6526	148	4	completeness	completeness	NOUN
ejpam-6526	148	5	of	of	ADP
ejpam-6526	148	6	(	(	PUNCT
ejpam-6526	148	7	x	x	X
ejpam-6526	148	8	,	,	PUNCT
ejpam-6526	148	9	ψ,⊕	ψ,⊕	NOUN
ejpam-6526	148	10	)	)	PUNCT
ejpam-6526	148	11	,	,	PUNCT
ejpam-6526	148	12	{	{	PUNCT
ejpam-6526	148	13	yn	yn	NOUN
ejpam-6526	148	14	}	}	PUNCT
ejpam-6526	148	15	converges	converge	VERB
ejpam-6526	148	16	to	to	ADP
ejpam-6526	148	17	some	some	DET
ejpam-6526	148	18	y	y	PROPN
ejpam-6526	148	19	∈	∈	PROPN
ejpam-6526	148	20	x	x	PUNCT
ejpam-6526	148	21	in	in	ADP
ejpam-6526	148	22	both	both	DET
ejpam-6526	148	23	topologies	topology	NOUN
ejpam-6526	148	24	,	,	PUNCT
ejpam-6526	148	25	meaning	mean	VERB
ejpam-6526	148	26	{	{	PUNCT
ejpam-6526	148	27	xn	xn	NOUN
ejpam-6526	148	28	}	}	PUNCT
ejpam-6526	148	29	converges	converge	NOUN
ejpam-6526	148	30	to	to	ADP
ejpam-6526	148	31	y	y	PROPN
ejpam-6526	148	32	in	in	ADP
ejpam-6526	148	33	both	both	DET
ejpam-6526	148	34	metric	metric	ADJ
ejpam-6526	148	35	structures	structure	NOUN
ejpam-6526	148	36	.	.	PUNCT
ejpam-6526	149	1	the	the	DET
ejpam-6526	149	2	proof	proof	NOUN
ejpam-6526	149	3	applies	apply	VERB
ejpam-6526	149	4	symmetrically	symmetrically	ADV
ejpam-6526	149	5	for	for	ADP
ejpam-6526	149	6	the	the	DET
ejpam-6526	149	7	second	second	ADJ
ejpam-6526	149	8	component	component	NOUN
ejpam-6526	149	9	metric	metric	NOUN
ejpam-6526	149	10	.	.	PUNCT
ejpam-6526	150	1	(	(	PUNCT
ejpam-6526	150	2	⇐	⇐	NOUN
ejpam-6526	150	3	)	)	PUNCT
ejpam-6526	150	4	assuming	assume	VERB
ejpam-6526	150	5	(	(	PUNCT
ejpam-6526	150	6	x,ϕ1	x,ϕ1	PROPN
ejpam-6526	150	7	)	)	PUNCT
ejpam-6526	150	8	and	and	CCONJ
ejpam-6526	150	9	(	(	PUNCT
ejpam-6526	150	10	x,ϕ2	x,ϕ2	PROPN
ejpam-6526	150	11	)	)	PUNCT
ejpam-6526	150	12	both	both	PRON
ejpam-6526	150	13	achieve	achieve	VERB
ejpam-6526	150	14	completeness	completeness	NOUN
ejpam-6526	150	15	,	,	PUNCT
ejpam-6526	150	16	consider	consider	VERB
ejpam-6526	150	17	bi	bi	ADJ
ejpam-6526	150	18	-	-	ADJ
ejpam-6526	150	19	cauchy	cauchy	ADJ
ejpam-6526	150	20	sequence	sequence	NOUN
ejpam-6526	150	21	{	{	PUNCT
ejpam-6526	150	22	xn	xn	NOUN
ejpam-6526	150	23	}	}	PUNCT
ejpam-6526	150	24	in	in	ADP
ejpam-6526	150	25	(	(	PUNCT
ejpam-6526	150	26	x	x	X
ejpam-6526	150	27	,	,	PUNCT
ejpam-6526	150	28	ψ,⊕	ψ,⊕	NOUN
ejpam-6526	150	29	)	)	PUNCT
ejpam-6526	150	30	.	.	PUNCT
ejpam-6526	151	1	then	then	ADV
ejpam-6526	151	2	{	{	PUNCT
ejpam-6526	151	3	xn	xn	X
ejpam-6526	151	4	}	}	PUNCT
ejpam-6526	151	5	is	be	AUX
ejpam-6526	151	6	cauchy	cauchy	ADJ
ejpam-6526	151	7	with	with	ADP
ejpam-6526	151	8	respect	respect	NOUN
ejpam-6526	151	9	to	to	ADP
ejpam-6526	151	10	both	both	DET
ejpam-6526	151	11	ϕ1	ϕ1	NOUN
ejpam-6526	151	12	and	and	CCONJ
ejpam-6526	151	13	ϕ2	ϕ2	ADV
ejpam-6526	151	14	.	.	PUNCT
ejpam-6526	152	1	by	by	ADP
ejpam-6526	152	2	completeness	completeness	NOUN
ejpam-6526	152	3	of	of	ADP
ejpam-6526	152	4	(	(	PUNCT
ejpam-6526	152	5	x,ϕ1	x,ϕ1	PROPN
ejpam-6526	152	6	)	)	PUNCT
ejpam-6526	152	7	and	and	CCONJ
ejpam-6526	152	8	(	(	PUNCT
ejpam-6526	152	9	x,ϕ2	x,ϕ2	PROPN
ejpam-6526	152	10	)	)	PUNCT
ejpam-6526	152	11	,	,	PUNCT
ejpam-6526	152	12	{	{	PUNCT
ejpam-6526	152	13	xn	xn	X
ejpam-6526	152	14	}	}	PUNCT
ejpam-6526	152	15	converges	converge	NOUN
ejpam-6526	152	16	to	to	ADP
ejpam-6526	152	17	some	some	DET
ejpam-6526	152	18	x	x	SYM
ejpam-6526	152	19	∈	∈	PROPN
ejpam-6526	152	20	x	x	PUNCT
ejpam-6526	152	21	in	in	ADP
ejpam-6526	152	22	both	both	DET
ejpam-6526	152	23	metric	metric	ADJ
ejpam-6526	152	24	structures	structure	NOUN
ejpam-6526	152	25	,	,	PUNCT
ejpam-6526	152	26	thus	thus	ADV
ejpam-6526	152	27	converging	converge	VERB
ejpam-6526	152	28	in	in	ADP
ejpam-6526	152	29	both	both	DET
ejpam-6526	152	30	induced	induced	ADJ
ejpam-6526	152	31	topologies	topology	NOUN
ejpam-6526	152	32	.	.	PUNCT
ejpam-6526	153	1	therefore	therefore	ADV
ejpam-6526	153	2	,	,	PUNCT
ejpam-6526	153	3	(	(	PUNCT
ejpam-6526	153	4	x	x	X
ejpam-6526	153	5	,	,	PUNCT
ejpam-6526	153	6	ψ,⊕	ψ,⊕	NOUN
ejpam-6526	153	7	)	)	PUNCT
ejpam-6526	153	8	is	be	AUX
ejpam-6526	153	9	bi	bi	NOUN
ejpam-6526	153	10	-	-	ADJ
ejpam-6526	153	11	complete	complete	ADJ
ejpam-6526	153	12	.	.	PUNCT
ejpam-6526	154	1	example	example	NOUN
ejpam-6526	154	2	7	7	NUM
ejpam-6526	154	3	(	(	PUNCT
ejpam-6526	154	4	bi	bi	NOUN
ejpam-6526	154	5	-	-	ADJ
ejpam-6526	154	6	completeness	completeness	ADJ
ejpam-6526	154	7	analysis	analysis	NOUN
ejpam-6526	154	8	)	)	PUNCT
ejpam-6526	154	9	.	.	PUNCT
ejpam-6526	155	1	consider	consider	VERB
ejpam-6526	155	2	x	x	X
ejpam-6526	155	3	=	=	SYM
ejpam-6526	155	4	(	(	PUNCT
ejpam-6526	155	5	0	0	NUM
ejpam-6526	155	6	,	,	PUNCT
ejpam-6526	155	7	1	1	NUM
ejpam-6526	155	8	)	)	PUNCT
ejpam-6526	155	9	⊂	⊂	NOUN
ejpam-6526	156	1	r	r	NOUN
ejpam-6526	156	2	with	with	ADP
ejpam-6526	156	3	bi	bi	ADJ
ejpam-6526	156	4	-	-	ADJ
ejpam-6526	156	5	metric	metric	ADJ
ejpam-6526	156	6	system	system	NOUN
ejpam-6526	156	7	ψ(x	ψ(x	PROPN
ejpam-6526	156	8	,	,	PUNCT
ejpam-6526	156	9	y	y	NOUN
ejpam-6526	156	10	)	)	PUNCT
ejpam-6526	156	11	=	=	SYM
ejpam-6526	156	12	(	(	PUNCT
ejpam-6526	156	13	|x−y|	|x−y|	X
ejpam-6526	156	14	,	,	PUNCT
ejpam-6526	156	15	|x3−y3|	|x3−y3|	NUM
ejpam-6526	156	16	)	)	PUNCT
ejpam-6526	156	17	and	and	CCONJ
ejpam-6526	156	18	⊕	⊕	PROPN
ejpam-6526	156	19	implementing	implement	VERB
ejpam-6526	156	20	component	component	NOUN
ejpam-6526	156	21	-	-	PUNCT
ejpam-6526	156	22	wise	wise	ADJ
ejpam-6526	156	23	addition	addition	NOUN
ejpam-6526	156	24	.	.	PUNCT
ejpam-6526	157	1	we	we	PRON
ejpam-6526	157	2	investigate	investigate	VERB
ejpam-6526	157	3	bi	bi	ADJ
ejpam-6526	157	4	-	-	ADJ
ejpam-6526	157	5	completeness	completeness	ADJ
ejpam-6526	157	6	properties	property	NOUN
ejpam-6526	157	7	.	.	PUNCT
ejpam-6526	158	1	first	first	ADV
ejpam-6526	158	2	,	,	PUNCT
ejpam-6526	158	3	examining	examine	VERB
ejpam-6526	158	4	(	(	PUNCT
ejpam-6526	158	5	x,ϕ1	x,ϕ1	PROPN
ejpam-6526	158	6	)	)	PUNCT
ejpam-6526	159	1	where	where	SCONJ
ejpam-6526	159	2	ϕ1(x	ϕ1(x	PROPN
ejpam-6526	159	3	,	,	PUNCT
ejpam-6526	159	4	y	y	NOUN
ejpam-6526	159	5	)	)	PUNCT
ejpam-6526	159	6	=	=	PUNCT
ejpam-6526	159	7	|x−	|x−	ADJ
ejpam-6526	159	8	y|	y|	NOUN
ejpam-6526	159	9	represents	represent	VERB
ejpam-6526	159	10	standard	standard	ADJ
ejpam-6526	159	11	metric	metric	ADJ
ejpam-6526	159	12	on	on	ADP
ejpam-6526	159	13	(	(	PUNCT
ejpam-6526	159	14	0	0	NUM
ejpam-6526	159	15	,	,	PUNCT
ejpam-6526	159	16	1	1	NUM
ejpam-6526	159	17	)	)	PUNCT
ejpam-6526	159	18	,	,	PUNCT
ejpam-6526	159	19	we	we	PRON
ejpam-6526	159	20	observe	observe	VERB
ejpam-6526	159	21	this	this	DET
ejpam-6526	159	22	structure	structure	NOUN
ejpam-6526	159	23	lacks	lack	VERB
ejpam-6526	159	24	completeness	completeness	NOUN
ejpam-6526	159	25	.	.	PUNCT
ejpam-6526	160	1	for	for	ADP
ejpam-6526	160	2	instance	instance	NOUN
ejpam-6526	160	3	,	,	PUNCT
ejpam-6526	160	4	sequence	sequence	NOUN
ejpam-6526	160	5	xn	xn	PUNCT
ejpam-6526	160	6	=	=	SYM
ejpam-6526	160	7	1	1	NUM
ejpam-6526	160	8	n+5	n+5	PRON
ejpam-6526	160	9	is	be	AUX
ejpam-6526	160	10	cauchy	cauchy	ADJ
ejpam-6526	160	11	yet	yet	ADV
ejpam-6526	160	12	does	do	AUX
ejpam-6526	160	13	n’t	not	PART
ejpam-6526	160	14	converge	converge	VERB
ejpam-6526	160	15	in	in	ADP
ejpam-6526	160	16	(	(	PUNCT
ejpam-6526	160	17	0	0	NUM
ejpam-6526	160	18	,	,	PUNCT
ejpam-6526	160	19	1	1	NUM
ejpam-6526	160	20	)	)	PUNCT
ejpam-6526	160	21	since	since	SCONJ
ejpam-6526	160	22	its	its	PRON
ejpam-6526	160	23	limit	limit	NOUN
ejpam-6526	160	24	would	would	AUX
ejpam-6526	160	25	be	be	AUX
ejpam-6526	160	26	0	0	NUM
ejpam-6526	160	27	/∈	/∈	PUNCT
ejpam-6526	161	1	(	(	PUNCT
ejpam-6526	161	2	0	0	NUM
ejpam-6526	161	3	,	,	PUNCT
ejpam-6526	161	4	1	1	NUM
ejpam-6526	161	5	)	)	PUNCT
ejpam-6526	161	6	.	.	PUNCT
ejpam-6526	162	1	by	by	ADP
ejpam-6526	162	2	our	our	PRON
ejpam-6526	162	3	theorem	theorem	NOUN
ejpam-6526	162	4	,	,	PUNCT
ejpam-6526	162	5	since	since	SCONJ
ejpam-6526	162	6	(	(	PUNCT
ejpam-6526	162	7	x,ϕ1	x,ϕ1	PROPN
ejpam-6526	162	8	)	)	PUNCT
ejpam-6526	162	9	lacks	lack	VERB
ejpam-6526	162	10	completeness	completeness	NOUN
ejpam-6526	162	11	,	,	PUNCT
ejpam-6526	162	12	bi	bi	ADJ
ejpam-6526	162	13	-	-	ADJ
ejpam-6526	162	14	metric	metric	ADJ
ejpam-6526	162	15	system	system	NOUN
ejpam-6526	162	16	(	(	PUNCT
ejpam-6526	162	17	x	x	X
ejpam-6526	162	18	,	,	PUNCT
ejpam-6526	162	19	ψ,⊕	ψ,⊕	NOUN
ejpam-6526	162	20	)	)	PUNCT
ejpam-6526	162	21	must	must	AUX
ejpam-6526	162	22	also	also	ADV
ejpam-6526	162	23	lack	lack	VERB
ejpam-6526	162	24	bi	bi	NOUN
ejpam-6526	162	25	-	-	NOUN
ejpam-6526	162	26	completeness	completeness	NOUN
ejpam-6526	162	27	.	.	PUNCT
ejpam-6526	163	1	conversely	conversely	ADV
ejpam-6526	163	2	,	,	PUNCT
ejpam-6526	163	3	considering	consider	VERB
ejpam-6526	163	4	y	y	PROPN
ejpam-6526	163	5	=	=	PUNCT
ejpam-6526	164	1	[	[	X
ejpam-6526	164	2	0	0	NUM
ejpam-6526	164	3	,	,	PUNCT
ejpam-6526	164	4	1	1	NUM
ejpam-6526	164	5	]	]	PUNCT
ejpam-6526	164	6	with	with	ADP
ejpam-6526	164	7	identical	identical	ADJ
ejpam-6526	164	8	bi	bi	ADJ
ejpam-6526	164	9	-	-	ADJ
ejpam-6526	164	10	metric	metric	ADJ
ejpam-6526	164	11	definition	definition	NOUN
ejpam-6526	164	12	,	,	PUNCT
ejpam-6526	164	13	both	both	PRON
ejpam-6526	164	14	(	(	PUNCT
ejpam-6526	164	15	y	y	NOUN
ejpam-6526	164	16	,	,	PUNCT
ejpam-6526	164	17	ϕ1	ϕ1	NOUN
ejpam-6526	164	18	)	)	PUNCT
ejpam-6526	164	19	and	and	CCONJ
ejpam-6526	164	20	(	(	PUNCT
ejpam-6526	164	21	y	y	NOUN
ejpam-6526	164	22	,	,	PUNCT
ejpam-6526	164	23	ϕ2	ϕ2	ADV
ejpam-6526	164	24	)	)	PUNCT
ejpam-6526	164	25	achieve	achieve	VERB
ejpam-6526	164	26	completeness	completeness	NOUN
ejpam-6526	164	27	(	(	PUNCT
ejpam-6526	164	28	with	with	ADP
ejpam-6526	164	29	ϕ1	ϕ1	NOUN
ejpam-6526	164	30	as	as	ADP
ejpam-6526	164	31	standard	standard	ADJ
ejpam-6526	164	32	metric	metric	ADJ
ejpam-6526	164	33	on	on	ADP
ejpam-6526	164	34	closed	closed	ADJ
ejpam-6526	164	35	interval	interval	NOUN
ejpam-6526	164	36	,	,	PUNCT
ejpam-6526	164	37	and	and	CCONJ
ejpam-6526	164	38	ϕ2(x	ϕ2(x	PROPN
ejpam-6526	164	39	,	,	PUNCT
ejpam-6526	164	40	y	y	PROPN
ejpam-6526	164	41	)	)	PUNCT
ejpam-6526	164	42	=	=	SYM
ejpam-6526	165	1	|x3−y3|	|x3−y3|	PRON
ejpam-6526	165	2	preserving	preserve	VERB
ejpam-6526	165	3	cauchy	cauchy	ADJ
ejpam-6526	165	4	sequences	sequence	NOUN
ejpam-6526	165	5	through	through	ADP
ejpam-6526	165	6	continuity	continuity	NOUN
ejpam-6526	165	7	)	)	PUNCT
ejpam-6526	165	8	.	.	PUNCT
ejpam-6526	166	1	therefore	therefore	ADV
ejpam-6526	166	2	,	,	PUNCT
ejpam-6526	166	3	(	(	PUNCT
ejpam-6526	166	4	y	y	NOUN
ejpam-6526	166	5	,	,	PUNCT
ejpam-6526	166	6	ψ,⊕	ψ,⊕	NOUN
ejpam-6526	166	7	)	)	PUNCT
ejpam-6526	166	8	constitutes	constitute	VERB
ejpam-6526	166	9	a	a	DET
ejpam-6526	166	10	bi	bi	ADJ
ejpam-6526	166	11	-	-	ADJ
ejpam-6526	166	12	complete	complete	ADJ
ejpam-6526	166	13	bi	bi	ADJ
ejpam-6526	166	14	-	-	ADJ
ejpam-6526	166	15	metric	metric	ADJ
ejpam-6526	166	16	system	system	NOUN
ejpam-6526	166	17	.	.	PUNCT
ejpam-6526	167	1	we	we	PRON
ejpam-6526	167	2	verify	verify	VERB
ejpam-6526	167	3	with	with	ADP
ejpam-6526	167	4	sequence	sequence	NOUN
ejpam-6526	167	5	xn	xn	PUNCT
ejpam-6526	168	1	=	=	SYM
ejpam-6526	168	2	1	1	NUM
ejpam-6526	168	3	n+5	n+5	NUM
ejpam-6526	168	4	∈	∈	PROPN
ejpam-6526	168	5	x	x	X
ejpam-6526	168	6	:	:	PUNCT
ejpam-6526	168	7	ψ(xm	ψ(xm	NUM
ejpam-6526	168	8	,	,	PUNCT
ejpam-6526	168	9	xn	xn	PROPN
ejpam-6526	168	10	)	)	PUNCT
ejpam-6526	168	11	=	=	PRON
ejpam-6526	168	12	(	(	PUNCT
ejpam-6526	168	13	∣∣∣∣	∣∣∣∣	NOUN
ejpam-6526	168	14	1	1	NUM
ejpam-6526	168	15	m+	m+	NUM
ejpam-6526	168	16	5	5	NUM
ejpam-6526	168	17	−	−	PROPN
ejpam-6526	168	18	1	1	NUM
ejpam-6526	168	19	n+	n+	SYM
ejpam-6526	168	20	5	5	NUM
ejpam-6526	168	21	∣∣∣∣	∣∣∣∣	NOUN
ejpam-6526	168	22	,	,	PUNCT
ejpam-6526	168	23	∣∣∣∣	∣∣∣∣	NOUN
ejpam-6526	168	24	1	1	NUM
ejpam-6526	168	25	(	(	PUNCT
ejpam-6526	168	26	m+	m+	NOUN
ejpam-6526	169	1	5)3	5)3	NUM
ejpam-6526	169	2	−	−	NOUN
ejpam-6526	169	3	1	1	NUM
ejpam-6526	169	4	(	(	PUNCT
ejpam-6526	169	5	n+	n+	X
ejpam-6526	169	6	5)3	5)3	NUM
ejpam-6526	169	7	∣∣∣∣	∣∣∣∣	NOUN
ejpam-6526	169	8	)	)	PUNCT
ejpam-6526	169	9	=	=	PUNCT
ejpam-6526	169	10	(	(	PUNCT
ejpam-6526	169	11	∣∣∣∣	∣∣∣∣	NOUN
ejpam-6526	169	12	n−m	n−m	PROPN
ejpam-6526	169	13	(	(	PUNCT
ejpam-6526	169	14	m+	m+	NUM
ejpam-6526	169	15	5)(n+	5)(n+	NUM
ejpam-6526	169	16	5	5	NUM
ejpam-6526	169	17	)	)	PUNCT
ejpam-6526	169	18	∣∣∣∣	∣∣∣∣	NOUN
ejpam-6526	169	19	,	,	PUNCT
ejpam-6526	169	20	∣∣∣∣(n+	∣∣∣∣(n+	NOUN
ejpam-6526	169	21	5)3	5)3	NUM
ejpam-6526	169	22	−	−	PROPN
ejpam-6526	169	23	(	(	PUNCT
ejpam-6526	169	24	m+	m+	NUM
ejpam-6526	169	25	5)3	5)3	NUM
ejpam-6526	169	26	(	(	PUNCT
ejpam-6526	169	27	m+	m+	NOUN
ejpam-6526	169	28	5)3(n+	5)3(n+	NUM
ejpam-6526	169	29	5)3	5)3	NUM
ejpam-6526	169	30	∣∣∣∣	∣∣∣∣	NOUN
ejpam-6526	169	31	)	)	PUNCT
ejpam-6526	169	32	as	as	ADP
ejpam-6526	169	33	m	m	PROPN
ejpam-6526	169	34	,	,	PUNCT
ejpam-6526	169	35	n→	n→	PROPN
ejpam-6526	169	36	∞	∞	PROPN
ejpam-6526	169	37	,	,	PUNCT
ejpam-6526	169	38	both	both	DET
ejpam-6526	169	39	components	component	NOUN
ejpam-6526	169	40	approach	approach	VERB
ejpam-6526	169	41	0	0	NUM
ejpam-6526	169	42	,	,	PUNCT
ejpam-6526	169	43	confirming	confirm	VERB
ejpam-6526	169	44	{	{	PUNCT
ejpam-6526	169	45	xn	xn	NOUN
ejpam-6526	169	46	}	}	PUNCT
ejpam-6526	169	47	as	as	ADP
ejpam-6526	169	48	bi	bi	NOUN
ejpam-6526	169	49	-	-	NOUN
ejpam-6526	169	50	cauchy	cauchy	NOUN
ejpam-6526	169	51	.	.	PUNCT
ejpam-6526	170	1	however	however	ADV
ejpam-6526	170	2	,	,	PUNCT
ejpam-6526	170	3	limit	limit	NOUN
ejpam-6526	170	4	point	point	NOUN
ejpam-6526	170	5	0	0	NUM
ejpam-6526	170	6	lies	lie	VERB
ejpam-6526	170	7	outside	outside	ADP
ejpam-6526	170	8	x	x	NOUN
ejpam-6526	170	9	,	,	PUNCT
ejpam-6526	170	10	confirming	confirm	VERB
ejpam-6526	170	11	(	(	PUNCT
ejpam-6526	170	12	x	x	X
ejpam-6526	170	13	,	,	PUNCT
ejpam-6526	170	14	ψ,⊕	ψ,⊕	NOUN
ejpam-6526	170	15	)	)	PUNCT
ejpam-6526	170	16	lacks	lack	VERB
ejpam-6526	170	17	bi	bi	NOUN
ejpam-6526	170	18	-	-	NOUN
ejpam-6526	170	19	completeness	completeness	NOUN
ejpam-6526	170	20	.	.	PUNCT
ejpam-6526	171	1	these	these	DET
ejpam-6526	171	2	completeness	completeness	NOUN
ejpam-6526	171	3	findings	finding	NOUN
ejpam-6526	171	4	align	align	VERB
ejpam-6526	171	5	with	with	ADP
ejpam-6526	171	6	computability	computability	NOUN
ejpam-6526	171	7	frameworks	framework	NOUN
ejpam-6526	171	8	for	for	ADP
ejpam-6526	171	9	metric	metric	ADJ
ejpam-6526	171	10	space	space	NOUN
ejpam-6526	171	11	subsets	subset	NOUN
ejpam-6526	171	12	established	establish	VERB
ejpam-6526	171	13	in	in	ADP
ejpam-6526	171	14	computational	computational	ADJ
ejpam-6526	171	15	mathematical	mathematical	ADJ
ejpam-6526	171	16	theory	theory	NOUN
ejpam-6526	171	17	[	[	X
ejpam-6526	171	18	20	20	NUM
ejpam-6526	171	19	]	]	PUNCT
ejpam-6526	171	20	.	.	PUNCT
ejpam-6526	172	1	a.	a.	PROPN
ejpam-6526	172	2	alsoboh	alsoboh	PROPN
ejpam-6526	172	3	et	et	PROPN
ejpam-6526	172	4	al	al	PROPN
ejpam-6526	172	5	.	.	PUNCT
ejpam-6526	172	6	/	/	SYM
ejpam-6526	172	7	eur	eur	PROPN
ejpam-6526	172	8	.	.	PUNCT
ejpam-6526	173	1	j.	j.	PROPN
ejpam-6526	173	2	pure	pure	PROPN
ejpam-6526	173	3	appl	appl	PROPN
ejpam-6526	173	4	.	.	PROPN
ejpam-6526	173	5	math	math	PROPN
ejpam-6526	173	6	,	,	PUNCT
ejpam-6526	173	7	18	18	NUM
ejpam-6526	173	8	(	(	PUNCT
ejpam-6526	173	9	4	4	NUM
ejpam-6526	173	10	)	)	PUNCT
ejpam-6526	173	11	(	(	PUNCT
ejpam-6526	173	12	2025	2025	NUM
ejpam-6526	173	13	)	)	PUNCT
ejpam-6526	173	14	,	,	PUNCT
ejpam-6526	173	15	6526	6526	NUM
ejpam-6526	173	16	8	8	NUM
ejpam-6526	173	17	of	of	ADP
ejpam-6526	173	18	21	21	NUM
ejpam-6526	173	19	4	4	NUM
ejpam-6526	173	20	.	.	PUNCT
ejpam-6526	173	21	connections	connection	NOUN
ejpam-6526	173	22	to	to	ADP
ejpam-6526	173	23	functional	functional	ADJ
ejpam-6526	173	24	-	-	PUNCT
ejpam-6526	173	25	analytical	analytical	ADJ
ejpam-6526	173	26	principles	principle	NOUN
ejpam-6526	173	27	we	we	PRON
ejpam-6526	173	28	explore	explore	VERB
ejpam-6526	173	29	relationships	relationship	NOUN
ejpam-6526	173	30	between	between	ADP
ejpam-6526	173	31	bi	bi	ADJ
ejpam-6526	173	32	-	-	ADJ
ejpam-6526	173	33	metric	metric	ADJ
ejpam-6526	173	34	systems	system	NOUN
ejpam-6526	173	35	and	and	CCONJ
ejpam-6526	173	36	functional	functional	ADJ
ejpam-6526	173	37	analysis	analysis	NOUN
ejpam-6526	173	38	concepts	concept	NOUN
ejpam-6526	173	39	,	,	PUNCT
ejpam-6526	173	40	extending	extend	VERB
ejpam-6526	173	41	previous	previous	ADJ
ejpam-6526	173	42	research	research	NOUN
ejpam-6526	173	43	on	on	ADP
ejpam-6526	173	44	quasi	quasi	ADJ
ejpam-6526	173	45	-	-	ADJ
ejpam-6526	173	46	uniform	uniform	ADJ
ejpam-6526	173	47	spaces	space	NOUN
ejpam-6526	173	48	[	[	X
ejpam-6526	173	49	21	21	NUM
ejpam-6526	173	50	]	]	PUNCT
ejpam-6526	173	51	.	.	PUNCT
ejpam-6526	174	1	definition	definition	NOUN
ejpam-6526	174	2	8	8	NUM
ejpam-6526	174	3	.	.	PUNCT
ejpam-6526	175	1	a	a	DET
ejpam-6526	175	2	bi	bi	ADJ
ejpam-6526	175	3	-	-	ADJ
ejpam-6526	175	4	normed	normed	ADJ
ejpam-6526	175	5	space	space	NOUN
ejpam-6526	175	6	consists	consist	VERB
ejpam-6526	175	7	of	of	ADP
ejpam-6526	175	8	triple	triple	ADJ
ejpam-6526	175	9	(	(	PUNCT
ejpam-6526	175	10	x	x	NOUN
ejpam-6526	175	11	,	,	PUNCT
ejpam-6526	175	12	∥	∥	X
ejpam-6526	175	13	·	·	PUNCT
ejpam-6526	175	14	∥1	∥1	X
ejpam-6526	175	15	,	,	PUNCT
ejpam-6526	175	16	∥	∥	X
ejpam-6526	175	17	·	·	PUNCT
ejpam-6526	175	18	∥2	∥2	X
ejpam-6526	175	19	)	)	PUNCT
ejpam-6526	175	20	where	where	SCONJ
ejpam-6526	175	21	x	x	PRON
ejpam-6526	175	22	represents	represent	VERB
ejpam-6526	175	23	a	a	DET
ejpam-6526	175	24	vector	vector	NOUN
ejpam-6526	175	25	space	space	NOUN
ejpam-6526	175	26	over	over	ADP
ejpam-6526	175	27	field	field	NOUN
ejpam-6526	176	1	k	k	PROPN
ejpam-6526	176	2	(	(	PUNCT
ejpam-6526	176	3	r	r	NOUN
ejpam-6526	176	4	or	or	CCONJ
ejpam-6526	176	5	c	c	NOUN
ejpam-6526	176	6	)	)	PUNCT
ejpam-6526	176	7	,	,	PUNCT
ejpam-6526	176	8	and	and	CCONJ
ejpam-6526	176	9	∥	∥	X
ejpam-6526	176	10	·	·	PUNCT
ejpam-6526	177	1	∥1	∥1	X
ejpam-6526	177	2	,	,	PUNCT
ejpam-6526	177	3	∥	∥	X
ejpam-6526	177	4	·	·	PUNCT
ejpam-6526	177	5	∥2	∥2	NOUN
ejpam-6526	177	6	function	function	NOUN
ejpam-6526	177	7	as	as	ADP
ejpam-6526	177	8	norms	norm	NOUN
ejpam-6526	177	9	on	on	ADP
ejpam-6526	177	10	x.	x.	PROPN
ejpam-6526	177	11	example	example	NOUN
ejpam-6526	177	12	8	8	NUM
ejpam-6526	177	13	(	(	PUNCT
ejpam-6526	177	14	financial	financial	ADJ
ejpam-6526	177	15	portfolio	portfolio	NOUN
ejpam-6526	177	16	analysis	analysis	NOUN
ejpam-6526	177	17	)	)	PUNCT
ejpam-6526	177	18	.	.	PUNCT
ejpam-6526	178	1	in	in	ADP
ejpam-6526	178	2	portfolio	portfolio	NOUN
ejpam-6526	178	3	optimization	optimization	NOUN
ejpam-6526	178	4	,	,	PUNCT
ejpam-6526	178	5	consider	consider	VERB
ejpam-6526	178	6	the	the	DET
ejpam-6526	178	7	vector	vector	NOUN
ejpam-6526	178	8	space	space	NOUN
ejpam-6526	178	9	x	x	PUNCT
ejpam-6526	178	10	=	=	SYM
ejpam-6526	178	11	rn	rn	PROPN
ejpam-6526	178	12	representing	represent	VERB
ejpam-6526	178	13	portfolio	portfolio	NOUN
ejpam-6526	178	14	allocations	allocation	NOUN
ejpam-6526	178	15	.	.	PUNCT
ejpam-6526	179	1	define	define	NOUN
ejpam-6526	179	2	:	:	PUNCT
ejpam-6526	179	3	•	•	NOUN
ejpam-6526	179	4	∥x∥1	∥x∥1	NOUN
ejpam-6526	179	5	=	=	PUNCT
ejpam-6526	180	1	∑n	∑n	PROPN
ejpam-6526	180	2	i=1	i=1	PROPN
ejpam-6526	180	3	|xi|	|xi|	PROPN
ejpam-6526	180	4	·	·	PUNCT
ejpam-6526	180	5	riski	riski	PROPN
ejpam-6526	180	6	(	(	PUNCT
ejpam-6526	180	7	risk	risk	NOUN
ejpam-6526	180	8	-	-	PUNCT
ejpam-6526	180	9	weighted	weight	VERB
ejpam-6526	180	10	allocation	allocation	NOUN
ejpam-6526	180	11	)	)	PUNCT
ejpam-6526	180	12	•	•	NUM
ejpam-6526	180	13	∥x∥2	∥x∥2	NOUN
ejpam-6526	180	14	=	=	SYM
ejpam-6526	180	15	(	(	PUNCT
ejpam-6526	180	16	∑n	∑n	PROPN
ejpam-6526	180	17	i=1	i=1	PROPN
ejpam-6526	180	18	x	x	SYM
ejpam-6526	180	19	2	2	NUM
ejpam-6526	180	20	i	i	NOUN
ejpam-6526	180	21	·	·	PUNCT
ejpam-6526	180	22	return2	return2	NOUN
ejpam-6526	181	1	i	i	PRON
ejpam-6526	181	2	)	)	PUNCT
ejpam-6526	181	3	1/2	1/2	NUM
ejpam-6526	181	4	(	(	PUNCT
ejpam-6526	181	5	return	return	NOUN
ejpam-6526	181	6	-	-	PUNCT
ejpam-6526	181	7	weighted	weight	VERB
ejpam-6526	181	8	allocation	allocation	NOUN
ejpam-6526	181	9	)	)	PUNCT
ejpam-6526	181	10	the	the	DET
ejpam-6526	181	11	bi	bi	ADJ
ejpam-6526	181	12	-	-	ADJ
ejpam-6526	181	13	normed	normed	ADJ
ejpam-6526	181	14	space	space	NOUN
ejpam-6526	181	15	(	(	PUNCT
ejpam-6526	181	16	x	x	X
ejpam-6526	181	17	,	,	PUNCT
ejpam-6526	181	18	∥	∥	X
ejpam-6526	181	19	·	·	PUNCT
ejpam-6526	181	20	∥1	∥1	X
ejpam-6526	181	21	,	,	PUNCT
ejpam-6526	181	22	∥	∥	X
ejpam-6526	181	23	·	·	PUNCT
ejpam-6526	181	24	∥2	∥2	X
ejpam-6526	181	25	)	)	PUNCT
ejpam-6526	181	26	enables	enable	VERB
ejpam-6526	181	27	simultaneous	simultaneous	ADJ
ejpam-6526	181	28	analysis	analysis	NOUN
ejpam-6526	181	29	of	of	ADP
ejpam-6526	181	30	risk	risk	NOUN
ejpam-6526	181	31	and	and	CCONJ
ejpam-6526	181	32	return	return	VERB
ejpam-6526	181	33	characteristics	characteristic	NOUN
ejpam-6526	181	34	.	.	PUNCT
ejpam-6526	182	1	theorem	theorem	VERB
ejpam-6526	182	2	4	4	NUM
ejpam-6526	182	3	.	.	PUNCT
ejpam-6526	183	1	every	every	DET
ejpam-6526	183	2	bi	bi	ADJ
ejpam-6526	183	3	-	-	ADJ
ejpam-6526	183	4	normed	normed	ADJ
ejpam-6526	183	5	space	space	NOUN
ejpam-6526	183	6	(	(	PUNCT
ejpam-6526	183	7	x	x	X
ejpam-6526	183	8	,	,	PUNCT
ejpam-6526	183	9	∥	∥	X
ejpam-6526	183	10	·	·	PUNCT
ejpam-6526	183	11	∥1	∥1	X
ejpam-6526	183	12	,	,	PUNCT
ejpam-6526	183	13	∥	∥	X
ejpam-6526	183	14	·	·	PUNCT
ejpam-6526	183	15	∥2	∥2	X
ejpam-6526	183	16	)	)	PUNCT
ejpam-6526	183	17	naturally	naturally	ADV
ejpam-6526	183	18	induces	induce	VERB
ejpam-6526	183	19	a	a	DET
ejpam-6526	183	20	bi	bi	ADJ
ejpam-6526	183	21	-	-	ADJ
ejpam-6526	183	22	metric	metric	ADJ
ejpam-6526	183	23	system	system	NOUN
ejpam-6526	183	24	(	(	PUNCT
ejpam-6526	183	25	x	x	X
ejpam-6526	183	26	,	,	PUNCT
ejpam-6526	183	27	ψ,⊕	ψ,⊕	NOUN
ejpam-6526	183	28	)	)	PUNCT
ejpam-6526	183	29	where	where	SCONJ
ejpam-6526	183	30	:	:	PUNCT
ejpam-6526	183	31	ψ(x	ψ(x	PROPN
ejpam-6526	183	32	,	,	PUNCT
ejpam-6526	183	33	y	y	NOUN
ejpam-6526	183	34	)	)	PUNCT
ejpam-6526	183	35	=	=	SYM
ejpam-6526	184	1	(	(	PUNCT
ejpam-6526	184	2	∥x−	∥x−	PROPN
ejpam-6526	184	3	y∥1	y∥1	VERB
ejpam-6526	184	4	,	,	PUNCT
ejpam-6526	184	5	∥x−	∥x−	PROPN
ejpam-6526	184	6	y∥2	y∥2	NOUN
ejpam-6526	184	7	)	)	PUNCT
ejpam-6526	184	8	and	and	CCONJ
ejpam-6526	184	9	⊕((a1	⊕((a1	ADV
ejpam-6526	184	10	,	,	PUNCT
ejpam-6526	184	11	a2	a2	PROPN
ejpam-6526	184	12	)	)	PUNCT
ejpam-6526	184	13	,	,	PUNCT
ejpam-6526	184	14	(	(	PUNCT
ejpam-6526	184	15	b1	b1	NOUN
ejpam-6526	184	16	,	,	PUNCT
ejpam-6526	184	17	b2	b2	NOUN
ejpam-6526	184	18	)	)	PUNCT
ejpam-6526	184	19	)	)	PUNCT
ejpam-6526	185	1	=	=	PUNCT
ejpam-6526	185	2	(	(	PUNCT
ejpam-6526	185	3	a1	a1	NOUN
ejpam-6526	185	4	+	+	CCONJ
ejpam-6526	185	5	b1	b1	NOUN
ejpam-6526	185	6	,	,	PUNCT
ejpam-6526	185	7	a2	a2	PROPN
ejpam-6526	185	8	+	+	CCONJ
ejpam-6526	185	9	b2	b2	NOUN
ejpam-6526	185	10	)	)	PUNCT
ejpam-6526	185	11	proof	proof	NOUN
ejpam-6526	185	12	.	.	PUNCT
ejpam-6526	186	1	we	we	PRON
ejpam-6526	186	2	verify	verify	VERB
ejpam-6526	186	3	that	that	SCONJ
ejpam-6526	186	4	(	(	PUNCT
ejpam-6526	186	5	x	x	X
ejpam-6526	186	6	,	,	PUNCT
ejpam-6526	186	7	ψ,⊕	ψ,⊕	NOUN
ejpam-6526	186	8	)	)	PUNCT
ejpam-6526	186	9	satisfies	satisfie	NOUN
ejpam-6526	186	10	all	all	DET
ejpam-6526	186	11	bi	bi	ADJ
ejpam-6526	186	12	-	-	ADJ
ejpam-6526	186	13	metric	metric	ADJ
ejpam-6526	186	14	system	system	NOUN
ejpam-6526	186	15	requirements	requirement	NOUN
ejpam-6526	186	16	:	:	PUNCT
ejpam-6526	186	17	(	(	PUNCT
ejpam-6526	186	18	i	i	NOUN
ejpam-6526	186	19	)	)	PUNCT
ejpam-6526	186	20	ψ(x	ψ(x	PROPN
ejpam-6526	186	21	,	,	PUNCT
ejpam-6526	186	22	y	y	NOUN
ejpam-6526	186	23	)	)	PUNCT
ejpam-6526	186	24	=	=	SYM
ejpam-6526	187	1	(	(	PUNCT
ejpam-6526	187	2	∥x−	∥x−	PROPN
ejpam-6526	187	3	y∥1	y∥1	VERB
ejpam-6526	187	4	,	,	PUNCT
ejpam-6526	187	5	∥x−	∥x−	PROPN
ejpam-6526	187	6	y∥2	y∥2	NOUN
ejpam-6526	187	7	)	)	PUNCT
ejpam-6526	187	8	with	with	ADP
ejpam-6526	187	9	non	non	ADJ
ejpam-6526	187	10	-	-	ADJ
ejpam-6526	187	11	negative	negative	ADJ
ejpam-6526	187	12	components	component	NOUN
ejpam-6526	187	13	.	.	PUNCT
ejpam-6526	188	1	(	(	PUNCT
ejpam-6526	188	2	ii	ii	NOUN
ejpam-6526	188	3	)	)	PUNCT
ejpam-6526	188	4	ψ(x	ψ(x	PROPN
ejpam-6526	188	5	,	,	PUNCT
ejpam-6526	188	6	y	y	NOUN
ejpam-6526	188	7	)	)	PUNCT
ejpam-6526	188	8	=	=	SYM
ejpam-6526	188	9	(	(	PUNCT
ejpam-6526	188	10	0	0	NUM
ejpam-6526	188	11	,	,	PUNCT
ejpam-6526	188	12	0	0	NUM
ejpam-6526	188	13	)	)	PUNCT
ejpam-6526	188	14	precisely	precisely	ADV
ejpam-6526	188	15	when	when	SCONJ
ejpam-6526	188	16	∥x−	∥x−	PRON
ejpam-6526	188	17	y∥1	y∥1	NOUN
ejpam-6526	188	18	=	=	SYM
ejpam-6526	189	1	∥x−	∥x−	NUM
ejpam-6526	189	2	y∥2	y∥2	NOUN
ejpam-6526	189	3	=	=	SYM
ejpam-6526	189	4	0	0	NUM
ejpam-6526	189	5	,	,	PUNCT
ejpam-6526	189	6	which	which	PRON
ejpam-6526	189	7	occurs	occur	VERB
ejpam-6526	189	8	if	if	SCONJ
ejpam-6526	189	9	and	and	CCONJ
ejpam-6526	189	10	only	only	ADV
ejpam-6526	189	11	if	if	SCONJ
ejpam-6526	189	12	x	x	NOUN
ejpam-6526	189	13	=	=	SYM
ejpam-6526	189	14	y	y	PROPN
ejpam-6526	189	15	by	by	ADP
ejpam-6526	189	16	norm	norm	NOUN
ejpam-6526	189	17	positive	positive	ADJ
ejpam-6526	189	18	-	-	PUNCT
ejpam-6526	189	19	definiteness	definiteness	NOUN
ejpam-6526	189	20	.	.	PUNCT
ejpam-6526	190	1	(	(	PUNCT
ejpam-6526	190	2	iii	iii	X
ejpam-6526	190	3	)	)	PUNCT
ejpam-6526	190	4	ψ(x	ψ(x	PROPN
ejpam-6526	190	5	,	,	PUNCT
ejpam-6526	190	6	y	y	NOUN
ejpam-6526	190	7	)	)	PUNCT
ejpam-6526	190	8	=	=	SYM
ejpam-6526	190	9	(	(	PUNCT
ejpam-6526	190	10	∥x−	∥x−	PROPN
ejpam-6526	190	11	y∥1	y∥1	VERB
ejpam-6526	190	12	,	,	PUNCT
ejpam-6526	190	13	∥x−	∥x−	PROPN
ejpam-6526	190	14	y∥2	y∥2	NOUN
ejpam-6526	190	15	)	)	PUNCT
ejpam-6526	190	16	=	=	PRON
ejpam-6526	191	1	(	(	PUNCT
ejpam-6526	191	2	∥y	∥y	PROPN
ejpam-6526	191	3	−	−	PROPN
ejpam-6526	191	4	x∥1	x∥1	NOUN
ejpam-6526	191	5	,	,	PUNCT
ejpam-6526	191	6	∥y	∥y	PROPN
ejpam-6526	191	7	−	−	NOUN
ejpam-6526	191	8	x∥2	x∥2	NOUN
ejpam-6526	191	9	)	)	PUNCT
ejpam-6526	191	10	=	=	SYM
ejpam-6526	191	11	ψ(y	ψ(y	NOUN
ejpam-6526	191	12	,	,	PUNCT
ejpam-6526	191	13	x	x	NOUN
ejpam-6526	191	14	)	)	PUNCT
ejpam-6526	191	15	by	by	ADP
ejpam-6526	191	16	norm	norm	NOUN
ejpam-6526	191	17	symmetry	symmetry	NOUN
ejpam-6526	191	18	.	.	PUNCT
ejpam-6526	192	1	(	(	PUNCT
ejpam-6526	192	2	iv	iv	X
ejpam-6526	192	3	)	)	PUNCT
ejpam-6526	192	4	for	for	ADP
ejpam-6526	192	5	generalized	generalized	ADJ
ejpam-6526	192	6	triangular	triangular	NOUN
ejpam-6526	192	7	coordination	coordination	NOUN
ejpam-6526	192	8	:	:	PUNCT
ejpam-6526	192	9	ψ(x	ψ(x	NOUN
ejpam-6526	192	10	,	,	PUNCT
ejpam-6526	192	11	z	z	NOUN
ejpam-6526	192	12	)	)	PUNCT
ejpam-6526	192	13	=	=	SYM
ejpam-6526	192	14	(	(	PUNCT
ejpam-6526	192	15	∥x−	∥x−	PROPN
ejpam-6526	192	16	z∥1	z∥1	NOUN
ejpam-6526	192	17	,	,	PUNCT
ejpam-6526	192	18	∥x−	∥x−	PROPN
ejpam-6526	192	19	z∥2	z∥2	NOUN
ejpam-6526	192	20	)	)	PUNCT
ejpam-6526	192	21	≤comp	≤comp	PROPN
ejpam-6526	192	22	(	(	PUNCT
ejpam-6526	192	23	∥x−	∥x−	ADV
ejpam-6526	192	24	y∥1	y∥1	NOUN
ejpam-6526	193	1	+	+	CCONJ
ejpam-6526	193	2	∥y	∥y	ADJ
ejpam-6526	193	3	−	−	NOUN
ejpam-6526	193	4	z∥1	z∥1	NOUN
ejpam-6526	193	5	,	,	PUNCT
ejpam-6526	193	6	∥x−	∥x−	NUM
ejpam-6526	193	7	y∥2	y∥2	NOUN
ejpam-6526	194	1	+	+	CCONJ
ejpam-6526	194	2	∥y	∥y	PROPN
ejpam-6526	194	3	−	−	NOUN
ejpam-6526	194	4	z∥2	z∥2	NOUN
ejpam-6526	194	5	)	)	PUNCT
ejpam-6526	194	6	=	=	SYM
ejpam-6526	194	7	(	(	PUNCT
ejpam-6526	194	8	a1	a1	NOUN
ejpam-6526	194	9	+	+	CCONJ
ejpam-6526	194	10	b1	b1	NOUN
ejpam-6526	194	11	,	,	PUNCT
ejpam-6526	194	12	a2	a2	PROPN
ejpam-6526	194	13	+	+	CCONJ
ejpam-6526	194	14	b2	b2	NOUN
ejpam-6526	194	15	)	)	PUNCT
ejpam-6526	194	16	=	=	SYM
ejpam-6526	194	17	⊕(ψ(x	⊕(ψ(x	NOUN
ejpam-6526	194	18	,	,	PUNCT
ejpam-6526	194	19	y),ψ(y	y),ψ(y	NOUN
ejpam-6526	194	20	,	,	PUNCT
ejpam-6526	194	21	z	z	NOUN
ejpam-6526	194	22	)	)	PUNCT
ejpam-6526	194	23	)	)	PUNCT
ejpam-6526	194	24	where	where	SCONJ
ejpam-6526	194	25	ai	ai	VERB
ejpam-6526	194	26	=	=	PUNCT
ejpam-6526	194	27	∥x−	∥x−	PROPN
ejpam-6526	194	28	y∥i	y∥i	NOUN
ejpam-6526	194	29	and	and	CCONJ
ejpam-6526	194	30	bi	bi	NOUN
ejpam-6526	194	31	=	=	PROPN
ejpam-6526	194	32	∥y	∥y	PROPN
ejpam-6526	194	33	−	−	NOUN
ejpam-6526	194	34	z∥i	z∥i	NOUN
ejpam-6526	194	35	for	for	ADP
ejpam-6526	194	36	i	i	PROPN
ejpam-6526	194	37	∈	∈	PROPN
ejpam-6526	194	38	{	{	PUNCT
ejpam-6526	194	39	1	1	NUM
ejpam-6526	194	40	,	,	PUNCT
ejpam-6526	194	41	2	2	NUM
ejpam-6526	194	42	}	}	PUNCT
ejpam-6526	194	43	.	.	PUNCT
ejpam-6526	195	1	(	(	PUNCT
ejpam-6526	195	2	v	v	NOUN
ejpam-6526	195	3	)	)	PUNCT
ejpam-6526	195	4	⊕	⊕	PROPN
ejpam-6526	195	5	clearly	clearly	ADV
ejpam-6526	195	6	demonstrates	demonstrate	VERB
ejpam-6526	195	7	monotonicity	monotonicity	NOUN
ejpam-6526	195	8	in	in	ADP
ejpam-6526	195	9	both	both	DET
ejpam-6526	195	10	arguments	argument	NOUN
ejpam-6526	195	11	with	with	ADP
ejpam-6526	195	12	respect	respect	NOUN
ejpam-6526	195	13	to	to	ADP
ejpam-6526	195	14	≤comp	≤comp	NUM
ejpam-6526	195	15	.	.	PUNCT
ejpam-6526	196	1	(	(	PUNCT
ejpam-6526	196	2	vi	vi	NOUN
ejpam-6526	196	3	)	)	PUNCT
ejpam-6526	196	4	⊕((0	⊕((0	NOUN
ejpam-6526	196	5	,	,	PUNCT
ejpam-6526	196	6	0	0	NUM
ejpam-6526	196	7	)	)	PUNCT
ejpam-6526	196	8	,	,	PUNCT
ejpam-6526	196	9	(	(	PUNCT
ejpam-6526	196	10	0	0	NUM
ejpam-6526	196	11	,	,	PUNCT
ejpam-6526	196	12	0	0	NUM
ejpam-6526	196	13	)	)	PUNCT
ejpam-6526	196	14	)	)	PUNCT
ejpam-6526	197	1	=	=	PUNCT
ejpam-6526	197	2	(	(	PUNCT
ejpam-6526	197	3	0	0	NUM
ejpam-6526	197	4	+	+	CCONJ
ejpam-6526	197	5	0	0	NUM
ejpam-6526	197	6	,	,	PUNCT
ejpam-6526	197	7	0	0	NUM
ejpam-6526	198	1	+	+	CCONJ
ejpam-6526	198	2	0	0	NUM
ejpam-6526	198	3	)	)	PUNCT
ejpam-6526	198	4	=	=	SYM
ejpam-6526	198	5	(	(	PUNCT
ejpam-6526	198	6	0	0	NUM
ejpam-6526	198	7	,	,	PUNCT
ejpam-6526	198	8	0	0	NUM
ejpam-6526	198	9	)	)	PUNCT
ejpam-6526	198	10	.	.	PUNCT
ejpam-6526	199	1	a.	a.	PROPN
ejpam-6526	199	2	alsoboh	alsoboh	PROPN
ejpam-6526	199	3	et	et	PROPN
ejpam-6526	199	4	al	al	PROPN
ejpam-6526	199	5	.	.	PUNCT
ejpam-6526	199	6	/	/	SYM
ejpam-6526	199	7	eur	eur	PROPN
ejpam-6526	199	8	.	.	PUNCT
ejpam-6526	200	1	j.	j.	PROPN
ejpam-6526	200	2	pure	pure	PROPN
ejpam-6526	200	3	appl	appl	PROPN
ejpam-6526	200	4	.	.	PROPN
ejpam-6526	200	5	math	math	PROPN
ejpam-6526	200	6	,	,	PUNCT
ejpam-6526	200	7	18	18	NUM
ejpam-6526	200	8	(	(	PUNCT
ejpam-6526	200	9	4	4	NUM
ejpam-6526	200	10	)	)	PUNCT
ejpam-6526	200	11	(	(	PUNCT
ejpam-6526	200	12	2025	2025	NUM
ejpam-6526	200	13	)	)	PUNCT
ejpam-6526	200	14	,	,	PUNCT
ejpam-6526	200	15	6526	6526	NUM
ejpam-6526	200	16	9	9	NUM
ejpam-6526	200	17	of	of	ADP
ejpam-6526	200	18	21	21	NUM
ejpam-6526	200	19	therefore	therefore	ADV
ejpam-6526	200	20	,	,	PUNCT
ejpam-6526	200	21	(	(	PUNCT
ejpam-6526	200	22	x	x	X
ejpam-6526	200	23	,	,	PUNCT
ejpam-6526	200	24	ψ,⊕	ψ,⊕	NOUN
ejpam-6526	200	25	)	)	PUNCT
ejpam-6526	200	26	constitutes	constitute	VERB
ejpam-6526	200	27	a	a	DET
ejpam-6526	200	28	proper	proper	ADJ
ejpam-6526	200	29	bi	bi	ADJ
ejpam-6526	200	30	-	-	ADJ
ejpam-6526	200	31	metric	metric	ADJ
ejpam-6526	200	32	system	system	NOUN
ejpam-6526	200	33	.	.	PUNCT
ejpam-6526	201	1	example	example	NOUN
ejpam-6526	201	2	9	9	NUM
ejpam-6526	201	3	(	(	PUNCT
ejpam-6526	201	4	bi	bi	ADJ
ejpam-6526	201	5	-	-	ADJ
ejpam-6526	201	6	normed	normed	ADJ
ejpam-6526	201	7	vector	vector	NOUN
ejpam-6526	201	8	spaces	space	NOUN
ejpam-6526	201	9	)	)	PUNCT
ejpam-6526	201	10	.	.	PUNCT
ejpam-6526	202	1	consider	consider	VERB
ejpam-6526	202	2	function	function	NOUN
ejpam-6526	202	3	space	space	NOUN
ejpam-6526	202	4	x	x	NOUN
ejpam-6526	202	5	=	=	SYM
ejpam-6526	202	6	c[0	c[0	PROPN
ejpam-6526	202	7	,	,	PUNCT
ejpam-6526	202	8	1	1	NUM
ejpam-6526	202	9	]	]	PUNCT
ejpam-6526	202	10	,	,	PUNCT
ejpam-6526	202	11	comprising	comprise	VERB
ejpam-6526	202	12	continuous	continuous	ADJ
ejpam-6526	202	13	functions	function	NOUN
ejpam-6526	202	14	on	on	ADP
ejpam-6526	202	15	[	[	X
ejpam-6526	202	16	0	0	NUM
ejpam-6526	202	17	,	,	PUNCT
ejpam-6526	202	18	1	1	NUM
ejpam-6526	202	19	]	]	PUNCT
ejpam-6526	202	20	,	,	PUNCT
ejpam-6526	202	21	with	with	ADP
ejpam-6526	202	22	dual	dual	ADJ
ejpam-6526	202	23	functional	functional	ADJ
ejpam-6526	202	24	norms	norm	NOUN
ejpam-6526	202	25	:	:	PUNCT
ejpam-6526	203	1	∥f∥∞	∥f∥∞	X
ejpam-6526	203	2	=	=	SYM
ejpam-6526	203	3	max	max	PROPN
ejpam-6526	203	4	x∈[0,1	x∈[0,1	X
ejpam-6526	203	5	]	]	X
ejpam-6526	204	1	|f(x)|	|f(x)|	NOUN
ejpam-6526	204	2	∥f∥3	∥f∥3	NOUN
ejpam-6526	204	3	=	=	SYM
ejpam-6526	204	4	(	(	PUNCT
ejpam-6526	204	5	∫	∫	PROPN
ejpam-6526	204	6	1	1	NUM
ejpam-6526	204	7	0	0	NUM
ejpam-6526	204	8	|f(x)|3dx	|f(x)|3dx	NOUN
ejpam-6526	204	9	)	)	PUNCT
ejpam-6526	204	10	1/3	1/3	NUM
ejpam-6526	204	11	the	the	DET
ejpam-6526	204	12	triple	triple	ADJ
ejpam-6526	204	13	(	(	PUNCT
ejpam-6526	204	14	x	x	NOUN
ejpam-6526	204	15	,	,	PUNCT
ejpam-6526	204	16	∥·∥∞	∥·∥∞	PROPN
ejpam-6526	204	17	,	,	PUNCT
ejpam-6526	204	18	∥·∥3	∥·∥3	NOUN
ejpam-6526	204	19	)	)	PUNCT
ejpam-6526	204	20	constitutes	constitute	VERB
ejpam-6526	204	21	a	a	DET
ejpam-6526	204	22	bi	bi	ADJ
ejpam-6526	204	23	-	-	ADJ
ejpam-6526	204	24	normed	normed	ADJ
ejpam-6526	204	25	space	space	NOUN
ejpam-6526	204	26	.	.	PUNCT
ejpam-6526	205	1	the	the	DET
ejpam-6526	205	2	induced	induced	ADJ
ejpam-6526	205	3	bi	bi	ADJ
ejpam-6526	205	4	-	-	ADJ
ejpam-6526	205	5	metric	metric	ADJ
ejpam-6526	205	6	system	system	NOUN
ejpam-6526	205	7	becomes	become	VERB
ejpam-6526	205	8	:	:	PUNCT
ejpam-6526	205	9	ψ(f	ψ(f	NOUN
ejpam-6526	205	10	,	,	PUNCT
ejpam-6526	205	11	g	g	NOUN
ejpam-6526	205	12	)	)	PUNCT
ejpam-6526	205	13	=	=	SYM
ejpam-6526	206	1	(	(	PUNCT
ejpam-6526	206	2	max	max	PROPN
ejpam-6526	206	3	x∈[0,1	x∈[0,1	PROPN
ejpam-6526	206	4	]	]	X
ejpam-6526	207	1	|f(x)−	|f(x)−	PROPN
ejpam-6526	207	2	g(x)|	g(x)|	VERB
ejpam-6526	207	3	,	,	PUNCT
ejpam-6526	207	4	(	(	PUNCT
ejpam-6526	207	5	∫	∫	PROPN
ejpam-6526	207	6	1	1	NUM
ejpam-6526	207	7	0	0	NUM
ejpam-6526	207	8	|f(x)−	|f(x)−	NOUN
ejpam-6526	207	9	g(x)|3dx	g(x)|3dx	PROPN
ejpam-6526	207	10	)	)	PUNCT
ejpam-6526	207	11	1/3	1/3	NUM
ejpam-6526	207	12	)	)	PUNCT
ejpam-6526	207	13	with	with	ADP
ejpam-6526	207	14	⊕	⊕	PROPN
ejpam-6526	207	15	implementing	implement	VERB
ejpam-6526	207	16	component	component	NOUN
ejpam-6526	207	17	-	-	PUNCT
ejpam-6526	207	18	wise	wise	ADJ
ejpam-6526	207	19	addition	addition	NOUN
ejpam-6526	207	20	.	.	PUNCT
ejpam-6526	208	1	the	the	DET
ejpam-6526	208	2	induced	induced	ADJ
ejpam-6526	208	3	topologies	topology	NOUN
ejpam-6526	208	4	t1	t1	NOUN
ejpam-6526	208	5	and	and	CCONJ
ejpam-6526	208	6	t2	t2	NOUN
ejpam-6526	208	7	correspond	correspond	NOUN
ejpam-6526	208	8	to	to	ADP
ejpam-6526	208	9	uniform	uniform	ADJ
ejpam-6526	208	10	convergence	convergence	NOUN
ejpam-6526	208	11	topology	topology	NOUN
ejpam-6526	208	12	and	and	CCONJ
ejpam-6526	208	13	l3	l3	PROPN
ejpam-6526	208	14	convergence	convergence	NOUN
ejpam-6526	208	15	topology	topology	NOUN
ejpam-6526	208	16	,	,	PUNCT
ejpam-6526	208	17	respectively	respectively	ADV
ejpam-6526	208	18	.	.	PUNCT
ejpam-6526	209	1	theorem	theorem	NOUN
ejpam-6526	209	2	5	5	NUM
ejpam-6526	209	3	.	.	PUNCT
ejpam-6526	209	4	given	give	VERB
ejpam-6526	209	5	normed	normed	ADJ
ejpam-6526	209	6	space	space	NOUN
ejpam-6526	209	7	(	(	PUNCT
ejpam-6526	209	8	x	x	NOUN
ejpam-6526	209	9	,	,	PUNCT
ejpam-6526	209	10	∥·∥	∥·∥	PROPN
ejpam-6526	209	11	)	)	PUNCT
ejpam-6526	209	12	and	and	CCONJ
ejpam-6526	209	13	its	its	PRON
ejpam-6526	209	14	continuous	continuous	ADJ
ejpam-6526	209	15	dual	dual	ADJ
ejpam-6526	209	16	space	space	NOUN
ejpam-6526	209	17	x∗	x∗	NOUN
ejpam-6526	209	18	with	with	ADP
ejpam-6526	209	19	operator	operator	NOUN
ejpam-6526	209	20	norm	norm	NOUN
ejpam-6526	209	21	∥	∥	X
ejpam-6526	209	22	·	·	PUNCT
ejpam-6526	209	23	∥op	∥op	INTJ
ejpam-6526	209	24	,	,	PUNCT
ejpam-6526	209	25	we	we	PRON
ejpam-6526	209	26	can	can	AUX
ejpam-6526	209	27	establish	establish	VERB
ejpam-6526	209	28	a	a	DET
ejpam-6526	209	29	bi	bi	ADJ
ejpam-6526	209	30	-	-	ADJ
ejpam-6526	209	31	metric	metric	ADJ
ejpam-6526	209	32	system	system	NOUN
ejpam-6526	209	33	on	on	ADP
ejpam-6526	209	34	x	x	NOUN
ejpam-6526	209	35	×x∗	×x∗	NOUN
ejpam-6526	209	36	as	as	ADP
ejpam-6526	209	37	:	:	PUNCT
ejpam-6526	209	38	ψ((x	ψ((x	NOUN
ejpam-6526	209	39	,	,	PUNCT
ejpam-6526	209	40	f	f	NOUN
ejpam-6526	209	41	)	)	PUNCT
ejpam-6526	209	42	,	,	PUNCT
ejpam-6526	209	43	(	(	PUNCT
ejpam-6526	209	44	y	y	NOUN
ejpam-6526	209	45	,	,	PUNCT
ejpam-6526	209	46	g	g	NOUN
ejpam-6526	209	47	)	)	PUNCT
ejpam-6526	209	48	)	)	PUNCT
ejpam-6526	210	1	=	=	PRON
ejpam-6526	210	2	(	(	PUNCT
ejpam-6526	210	3	∥x−	∥x−	PROPN
ejpam-6526	210	4	y∥	y∥	VERB
ejpam-6526	210	5	,	,	PUNCT
ejpam-6526	210	6	∥f	∥f	PROPN
ejpam-6526	210	7	−	−	PROPN
ejpam-6526	210	8	g∥op	g∥op	PROPN
ejpam-6526	210	9	)	)	PUNCT
ejpam-6526	210	10	with	with	ADP
ejpam-6526	210	11	⊕	⊕	PROPN
ejpam-6526	210	12	implementing	implement	VERB
ejpam-6526	210	13	component	component	NOUN
ejpam-6526	210	14	-	-	PUNCT
ejpam-6526	210	15	wise	wise	ADJ
ejpam-6526	210	16	addition	addition	NOUN
ejpam-6526	210	17	.	.	PUNCT
ejpam-6526	211	1	this	this	PRON
ejpam-6526	211	2	follows	follow	VERB
ejpam-6526	211	3	directly	directly	ADV
ejpam-6526	211	4	from	from	ADP
ejpam-6526	211	5	norm	norm	NOUN
ejpam-6526	211	6	properties	property	NOUN
ejpam-6526	211	7	on	on	ADP
ejpam-6526	211	8	spaces	space	NOUN
ejpam-6526	211	9	and	and	CCONJ
ejpam-6526	211	10	their	their	PRON
ejpam-6526	211	11	duals	dual	NOUN
ejpam-6526	211	12	.	.	PUNCT
ejpam-6526	212	1	example	example	NOUN
ejpam-6526	212	2	10	10	NUM
ejpam-6526	212	3	(	(	PUNCT
ejpam-6526	212	4	bi	bi	ADJ
ejpam-6526	212	5	-	-	ADJ
ejpam-6526	212	6	metric	metric	ADJ
ejpam-6526	212	7	spanning	span	VERB
ejpam-6526	212	8	primal	primal	ADJ
ejpam-6526	212	9	-	-	PUNCT
ejpam-6526	212	10	dual	dual	ADJ
ejpam-6526	212	11	configuration	configuration	NOUN
ejpam-6526	212	12	)	)	PUNCT
ejpam-6526	212	13	.	.	PUNCT
ejpam-6526	213	1	consider	consider	VERB
ejpam-6526	213	2	x	x	NOUN
ejpam-6526	213	3	=	=	SYM
ejpam-6526	213	4	ℓ2	ℓ2	NOUN
ejpam-6526	213	5	,	,	PUNCT
ejpam-6526	213	6	squaresummable	squaresummable	ADJ
ejpam-6526	213	7	sequences	sequence	NOUN
ejpam-6526	213	8	with	with	ADP
ejpam-6526	213	9	norm	norm	NOUN
ejpam-6526	213	10	∥x∥2	∥x∥2	NOUN
ejpam-6526	213	11	=	=	SYM
ejpam-6526	213	12	(	(	PUNCT
ejpam-6526	213	13	∑∞	∑∞	X
ejpam-6526	213	14	i=1	i=1	X
ejpam-6526	213	15	|xi|2	|xi|2	PUNCT
ejpam-6526	213	16	)	)	PUNCT
ejpam-6526	213	17	1/2	1/2	NUM
ejpam-6526	213	18	.	.	PUNCT
ejpam-6526	214	1	we	we	PRON
ejpam-6526	214	2	identify	identify	VERB
ejpam-6526	214	3	dual	dual	ADJ
ejpam-6526	214	4	space	space	NOUN
ejpam-6526	214	5	x∗	x∗	PROPN
ejpam-6526	215	1	=	=	PUNCT
ejpam-6526	215	2	ℓ2	ℓ2	PROPN
ejpam-6526	215	3	with	with	ADP
ejpam-6526	215	4	ℓ2	ℓ2	PROPN
ejpam-6526	215	5	via	via	ADP
ejpam-6526	215	6	riesz	riesz	PROPN
ejpam-6526	215	7	representation	representation	NOUN
ejpam-6526	215	8	,	,	PUNCT
ejpam-6526	215	9	yielding	yield	VERB
ejpam-6526	215	10	duality	duality	NOUN
ejpam-6526	215	11	pairing	pairing	NOUN
ejpam-6526	215	12	:	:	PUNCT
ejpam-6526	215	13	⟨f	⟨f	NUM
ejpam-6526	215	14	,	,	PUNCT
ejpam-6526	215	15	x⟩	x⟩	PUNCT
ejpam-6526	216	1	=	=	PUNCT
ejpam-6526	216	2	∞∑	∞∑	NUM
ejpam-6526	216	3	i=1	i=1	PROPN
ejpam-6526	216	4	fixi	fixi	NOUN
ejpam-6526	216	5	the	the	DET
ejpam-6526	216	6	bi	bi	ADJ
ejpam-6526	216	7	-	-	ADJ
ejpam-6526	216	8	metric	metric	ADJ
ejpam-6526	216	9	system	system	NOUN
ejpam-6526	216	10	on	on	ADP
ejpam-6526	216	11	x	x	NOUN
ejpam-6526	216	12	×x∗	×x∗	NOUN
ejpam-6526	216	13	becomes	become	VERB
ejpam-6526	216	14	:	:	PUNCT
ejpam-6526	216	15	ψ((x	ψ((x	NOUN
ejpam-6526	216	16	,	,	PUNCT
ejpam-6526	216	17	f	f	NOUN
ejpam-6526	216	18	)	)	PUNCT
ejpam-6526	216	19	,	,	PUNCT
ejpam-6526	216	20	(	(	PUNCT
ejpam-6526	216	21	y	y	NOUN
ejpam-6526	216	22	,	,	PUNCT
ejpam-6526	216	23	g	g	NOUN
ejpam-6526	216	24	)	)	PUNCT
ejpam-6526	216	25	)	)	PUNCT
ejpam-6526	217	1	=	=	PUNCT
ejpam-6526	218	1			PROPN
ejpam-6526	218	2	(	(	PUNCT
ejpam-6526	218	3	∞∑	∞∑	PROPN
ejpam-6526	218	4	i=1	i=1	PROPN
ejpam-6526	218	5	|xi	|xi	X
ejpam-6526	218	6	−	−	PROPN
ejpam-6526	218	7	yi|2	yi|2	NOUN
ejpam-6526	218	8	)	)	PUNCT
ejpam-6526	218	9	1/2	1/2	NUM
ejpam-6526	218	10	,	,	PUNCT
ejpam-6526	218	11	(	(	PUNCT
ejpam-6526	218	12	∞∑	∞∑	NUM
ejpam-6526	218	13	i=1	i=1	ADP
ejpam-6526	218	14	|fi	|fi	ADJ
ejpam-6526	218	15	−	−	PROPN
ejpam-6526	218	16	gi|2	gi|2	PROPN
ejpam-6526	218	17	)	)	PUNCT
ejpam-6526	218	18	1/2	1/2	NUM
ejpam-6526	218	19			PROPN
ejpam-6526	218	20	the	the	DET
ejpam-6526	218	21	first	first	ADJ
ejpam-6526	218	22	component	component	NOUN
ejpam-6526	218	23	quantifies	quantifie	NOUN
ejpam-6526	218	24	sequential	sequential	ADJ
ejpam-6526	218	25	differences	difference	NOUN
ejpam-6526	218	26	in	in	ADP
ejpam-6526	218	27	primary	primary	ADJ
ejpam-6526	218	28	space	space	NOUN
ejpam-6526	218	29	,	,	PUNCT
ejpam-6526	218	30	while	while	SCONJ
ejpam-6526	218	31	the	the	DET
ejpam-6526	218	32	second	second	ADJ
ejpam-6526	218	33	quantifies	quantifie	NOUN
ejpam-6526	218	34	differences	difference	NOUN
ejpam-6526	218	35	between	between	ADP
ejpam-6526	218	36	corresponding	correspond	VERB
ejpam-6526	218	37	functionals	functional	NOUN
ejpam-6526	218	38	.	.	PUNCT
ejpam-6526	219	1	this	this	DET
ejpam-6526	219	2	bi	bi	ADJ
ejpam-6526	219	3	-	-	ADJ
ejpam-6526	219	4	metric	metric	ADJ
ejpam-6526	219	5	system	system	NOUN
ejpam-6526	219	6	enables	enable	VERB
ejpam-6526	219	7	simultaneous	simultaneous	ADJ
ejpam-6526	219	8	convergence	convergence	NOUN
ejpam-6526	219	9	tracking	tracking	NOUN
ejpam-6526	219	10	in	in	ADP
ejpam-6526	219	11	both	both	CCONJ
ejpam-6526	219	12	primal	primal	ADJ
ejpam-6526	219	13	and	and	CCONJ
ejpam-6526	219	14	dual	dual	ADJ
ejpam-6526	219	15	domains	domain	NOUN
ejpam-6526	219	16	.	.	PUNCT
ejpam-6526	220	1	a.	a.	PROPN
ejpam-6526	220	2	alsoboh	alsoboh	PROPN
ejpam-6526	220	3	et	et	PROPN
ejpam-6526	220	4	al	al	PROPN
ejpam-6526	220	5	.	.	PUNCT
ejpam-6526	220	6	/	/	SYM
ejpam-6526	220	7	eur	eur	PROPN
ejpam-6526	220	8	.	.	PUNCT
ejpam-6526	221	1	j.	j.	PROPN
ejpam-6526	221	2	pure	pure	PROPN
ejpam-6526	221	3	appl	appl	PROPN
ejpam-6526	221	4	.	.	PROPN
ejpam-6526	221	5	math	math	PROPN
ejpam-6526	221	6	,	,	PUNCT
ejpam-6526	221	7	18	18	NUM
ejpam-6526	221	8	(	(	PUNCT
ejpam-6526	221	9	4	4	NUM
ejpam-6526	221	10	)	)	PUNCT
ejpam-6526	221	11	(	(	PUNCT
ejpam-6526	221	12	2025	2025	NUM
ejpam-6526	221	13	)	)	PUNCT
ejpam-6526	221	14	,	,	PUNCT
ejpam-6526	221	15	6526	6526	NUM
ejpam-6526	221	16	10	10	NUM
ejpam-6526	221	17	of	of	ADP
ejpam-6526	221	18	21	21	NUM
ejpam-6526	221	19	proposition	proposition	NOUN
ejpam-6526	221	20	1	1	NUM
ejpam-6526	221	21	.	.	X
ejpam-6526	222	1	for	for	ADP
ejpam-6526	222	2	hilbert	hilbert	PROPN
ejpam-6526	222	3	space	space	PROPN
ejpam-6526	222	4	h	h	NOUN
ejpam-6526	222	5	,	,	PUNCT
ejpam-6526	222	6	we	we	PRON
ejpam-6526	222	7	define	define	VERB
ejpam-6526	222	8	ψ	ψ	X
ejpam-6526	222	9	:	:	PUNCT
ejpam-6526	222	10	h	h	NOUN
ejpam-6526	222	11	×h	×h	PROPN
ejpam-6526	222	12	→	→	PUNCT
ejpam-6526	222	13	r+	r+	PUNCT
ejpam-6526	222	14	×	×	NOUN
ejpam-6526	222	15	r+	r+	PUNCT
ejpam-6526	222	16	as	as	ADP
ejpam-6526	222	17	:	:	PUNCT
ejpam-6526	222	18	ψ(x	ψ(x	PROPN
ejpam-6526	222	19	,	,	PUNCT
ejpam-6526	222	20	y	y	NOUN
ejpam-6526	222	21	)	)	PUNCT
ejpam-6526	222	22	=	=	PUNCT
ejpam-6526	223	1	(	(	PUNCT
ejpam-6526	223	2	∥x−	∥x−	PROPN
ejpam-6526	223	3	y∥	y∥	NOUN
ejpam-6526	223	4	,	,	PUNCT
ejpam-6526	223	5	|∥x∥	|∥x∥	NOUN
ejpam-6526	223	6	−	−	PROPN
ejpam-6526	223	7	∥y∥|	∥y∥|	NOUN
ejpam-6526	223	8	)	)	PUNCT
ejpam-6526	223	9	with	with	ADP
ejpam-6526	223	10	⊕((a1	⊕((a1	ADJ
ejpam-6526	223	11	,	,	PUNCT
ejpam-6526	223	12	a2	a2	PROPN
ejpam-6526	223	13	)	)	PUNCT
ejpam-6526	223	14	,	,	PUNCT
ejpam-6526	223	15	(	(	PUNCT
ejpam-6526	223	16	b1	b1	NOUN
ejpam-6526	223	17	,	,	PUNCT
ejpam-6526	223	18	b2	b2	NOUN
ejpam-6526	223	19	)	)	PUNCT
ejpam-6526	223	20	)	)	PUNCT
ejpam-6526	224	1	=	=	SYM
ejpam-6526	224	2	(	(	PUNCT
ejpam-6526	224	3	a1+b1	a1+b1	PROPN
ejpam-6526	224	4	,	,	PUNCT
ejpam-6526	224	5	a2+b2	a2+b2	PROPN
ejpam-6526	224	6	)	)	PUNCT
ejpam-6526	224	7	.	.	PUNCT
ejpam-6526	225	1	then	then	ADV
ejpam-6526	225	2	(	(	PUNCT
ejpam-6526	225	3	h	h	NOUN
ejpam-6526	225	4	,	,	PUNCT
ejpam-6526	225	5	ψ,⊕	ψ,⊕	NOUN
ejpam-6526	225	6	)	)	PUNCT
ejpam-6526	225	7	forms	form	VERB
ejpam-6526	225	8	a	a	DET
ejpam-6526	225	9	bi	bi	ADJ
ejpam-6526	225	10	-	-	ADJ
ejpam-6526	225	11	metric	metric	ADJ
ejpam-6526	225	12	system	system	NOUN
ejpam-6526	225	13	with	with	ADP
ejpam-6526	225	14	topologies	topology	NOUN
ejpam-6526	225	15	t1	t1	NOUN
ejpam-6526	225	16	and	and	CCONJ
ejpam-6526	225	17	t2	t2	NOUN
ejpam-6526	225	18	corresponding	correspond	VERB
ejpam-6526	225	19	to	to	ADP
ejpam-6526	225	20	strong	strong	ADJ
ejpam-6526	225	21	and	and	CCONJ
ejpam-6526	225	22	weak	weak	ADJ
ejpam-6526	225	23	topologies	topology	NOUN
ejpam-6526	225	24	on	on	ADP
ejpam-6526	225	25	h	h	NOUN
ejpam-6526	225	26	,	,	PUNCT
ejpam-6526	225	27	respectively	respectively	ADV
ejpam-6526	225	28	.	.	PUNCT
ejpam-6526	226	1	proof	proof	NOUN
ejpam-6526	226	2	.	.	PUNCT
ejpam-6526	227	1	first	first	ADV
ejpam-6526	227	2	,	,	PUNCT
ejpam-6526	227	3	we	we	PRON
ejpam-6526	227	4	verify	verify	VERB
ejpam-6526	227	5	that	that	SCONJ
ejpam-6526	227	6	(	(	PUNCT
ejpam-6526	227	7	h	h	NOUN
ejpam-6526	227	8	,	,	PUNCT
ejpam-6526	227	9	ψ,⊕	ψ,⊕	NOUN
ejpam-6526	227	10	)	)	PUNCT
ejpam-6526	227	11	constitutes	constitute	VERB
ejpam-6526	227	12	a	a	DET
ejpam-6526	227	13	bi	bi	ADJ
ejpam-6526	227	14	-	-	ADJ
ejpam-6526	227	15	metric	metric	ADJ
ejpam-6526	227	16	system	system	NOUN
ejpam-6526	227	17	:	:	PUNCT
ejpam-6526	227	18	(	(	PUNCT
ejpam-6526	227	19	i	i	NOUN
ejpam-6526	227	20	)	)	PUNCT
ejpam-6526	227	21	ψ(x	ψ(x	PROPN
ejpam-6526	227	22	,	,	PUNCT
ejpam-6526	227	23	y	y	NOUN
ejpam-6526	227	24	)	)	PUNCT
ejpam-6526	227	25	=	=	PUNCT
ejpam-6526	228	1	(	(	PUNCT
ejpam-6526	228	2	∥x−	∥x−	PROPN
ejpam-6526	228	3	y∥	y∥	NOUN
ejpam-6526	228	4	,	,	PUNCT
ejpam-6526	228	5	|∥x∥	|∥x∥	NOUN
ejpam-6526	228	6	−	−	PROPN
ejpam-6526	228	7	∥y∥|	∥y∥|	NOUN
ejpam-6526	228	8	)	)	PUNCT
ejpam-6526	228	9	with	with	ADP
ejpam-6526	228	10	non	non	ADJ
ejpam-6526	228	11	-	-	ADJ
ejpam-6526	228	12	negative	negative	ADJ
ejpam-6526	228	13	components	component	NOUN
ejpam-6526	228	14	.	.	PUNCT
ejpam-6526	229	1	(	(	PUNCT
ejpam-6526	229	2	ii	ii	NOUN
ejpam-6526	229	3	)	)	PUNCT
ejpam-6526	229	4	ψ(x	ψ(x	PROPN
ejpam-6526	229	5	,	,	PUNCT
ejpam-6526	229	6	y	y	NOUN
ejpam-6526	229	7	)	)	PUNCT
ejpam-6526	229	8	=	=	SYM
ejpam-6526	229	9	(	(	PUNCT
ejpam-6526	229	10	0	0	NUM
ejpam-6526	229	11	,	,	PUNCT
ejpam-6526	229	12	0	0	NUM
ejpam-6526	229	13	)	)	PUNCT
ejpam-6526	229	14	precisely	precisely	ADV
ejpam-6526	229	15	when	when	SCONJ
ejpam-6526	229	16	∥x−	∥x−	PROPN
ejpam-6526	229	17	y∥	y∥	VERB
ejpam-6526	229	18	=	=	SYM
ejpam-6526	229	19	0	0	PUNCT
ejpam-6526	229	20	and	and	CCONJ
ejpam-6526	229	21	|∥x∥−	|∥x∥−	PROPN
ejpam-6526	229	22	∥y∥|	∥y∥|	PROPN
ejpam-6526	229	23	=	=	SYM
ejpam-6526	229	24	0	0	X
ejpam-6526	229	25	.	.	PUNCT
ejpam-6526	230	1	the	the	DET
ejpam-6526	230	2	first	first	ADJ
ejpam-6526	230	3	condition	condition	NOUN
ejpam-6526	230	4	implies	imply	VERB
ejpam-6526	230	5	x	x	PUNCT
ejpam-6526	230	6	=	=	SYM
ejpam-6526	230	7	y	y	PROPN
ejpam-6526	230	8	,	,	PUNCT
ejpam-6526	230	9	which	which	PRON
ejpam-6526	230	10	ensures	ensure	VERB
ejpam-6526	230	11	the	the	DET
ejpam-6526	230	12	second	second	ADJ
ejpam-6526	230	13	condition	condition	NOUN
ejpam-6526	230	14	.	.	PUNCT
ejpam-6526	231	1	(	(	PUNCT
ejpam-6526	231	2	iii	iii	X
ejpam-6526	231	3	)	)	PUNCT
ejpam-6526	231	4	ψ(x	ψ(x	PROPN
ejpam-6526	231	5	,	,	PUNCT
ejpam-6526	231	6	y	y	NOUN
ejpam-6526	231	7	)	)	PUNCT
ejpam-6526	231	8	=	=	PUNCT
ejpam-6526	231	9	(	(	PUNCT
ejpam-6526	231	10	∥x−	∥x−	PROPN
ejpam-6526	231	11	y∥	y∥	NOUN
ejpam-6526	231	12	,	,	PUNCT
ejpam-6526	231	13	|∥x∥	|∥x∥	NOUN
ejpam-6526	231	14	−	−	PROPN
ejpam-6526	231	15	∥y∥|	∥y∥|	NOUN
ejpam-6526	231	16	)	)	PUNCT
ejpam-6526	231	17	=	=	SYM
ejpam-6526	231	18	(	(	PUNCT
ejpam-6526	231	19	∥y	∥y	PROPN
ejpam-6526	231	20	−	−	PROPN
ejpam-6526	231	21	x∥	x∥	PROPN
ejpam-6526	231	22	,	,	PUNCT
ejpam-6526	231	23	|∥y∥	|∥y∥	X
ejpam-6526	231	24	−	−	PROPN
ejpam-6526	231	25	∥x∥|	∥x∥|	PROPN
ejpam-6526	231	26	)	)	PUNCT
ejpam-6526	231	27	=	=	SYM
ejpam-6526	231	28	ψ(y	ψ(y	NOUN
ejpam-6526	231	29	,	,	PUNCT
ejpam-6526	231	30	x	x	NOUN
ejpam-6526	231	31	)	)	PUNCT
ejpam-6526	231	32	by	by	ADP
ejpam-6526	231	33	symmetry	symmetry	NOUN
ejpam-6526	231	34	.	.	PUNCT
ejpam-6526	232	1	(	(	PUNCT
ejpam-6526	232	2	iv	iv	X
ejpam-6526	232	3	)	)	PUNCT
ejpam-6526	232	4	for	for	ADP
ejpam-6526	232	5	generalized	generalized	ADJ
ejpam-6526	232	6	triangular	triangular	NOUN
ejpam-6526	232	7	coordination	coordination	NOUN
ejpam-6526	232	8	:	:	PUNCT
ejpam-6526	232	9	•	•	NOUN
ejpam-6526	232	10	for	for	ADP
ejpam-6526	232	11	first	first	ADJ
ejpam-6526	232	12	component	component	NOUN
ejpam-6526	232	13	:	:	PUNCT
ejpam-6526	232	14	∥x−z∥	∥x−z∥	NOUN
ejpam-6526	232	15	≤	≤	NOUN
ejpam-6526	232	16	∥x−y∥+∥y−z∥	∥x−y∥+∥y−z∥	NOUN
ejpam-6526	232	17	by	by	ADP
ejpam-6526	232	18	standard	standard	ADJ
ejpam-6526	232	19	triangle	triangle	NOUN
ejpam-6526	232	20	inequality	inequality	NOUN
ejpam-6526	232	21	.	.	PUNCT
ejpam-6526	233	1	•	•	NOUN
ejpam-6526	233	2	for	for	ADP
ejpam-6526	233	3	second	second	ADJ
ejpam-6526	233	4	component	component	NOUN
ejpam-6526	233	5	:	:	PUNCT
ejpam-6526	233	6	|∥x∥	|∥x∥	NOUN
ejpam-6526	233	7	−	−	PROPN
ejpam-6526	233	8	∥z∥|	∥z∥|	NOUN
ejpam-6526	233	9	≤	≤	NUM
ejpam-6526	233	10	|∥x∥	|∥x∥	NOUN
ejpam-6526	233	11	−	−	NOUN
ejpam-6526	233	12	∥y∥|	∥y∥|	NOUN
ejpam-6526	233	13	+	+	CCONJ
ejpam-6526	233	14	|∥y∥	|∥y∥	NOUN
ejpam-6526	233	15	−	−	PROPN
ejpam-6526	233	16	∥z∥|	∥z∥|	NOUN
ejpam-6526	233	17	by	by	ADP
ejpam-6526	233	18	triangle	triangle	NOUN
ejpam-6526	233	19	inequality	inequality	NOUN
ejpam-6526	233	20	for	for	ADP
ejpam-6526	233	21	real	real	ADJ
ejpam-6526	233	22	numbers	number	NOUN
ejpam-6526	233	23	.	.	PUNCT
ejpam-6526	234	1	next	next	ADV
ejpam-6526	234	2	,	,	PUNCT
ejpam-6526	234	3	we	we	PRON
ejpam-6526	234	4	establish	establish	VERB
ejpam-6526	234	5	that	that	SCONJ
ejpam-6526	234	6	t1	t1	PROPN
ejpam-6526	234	7	corresponds	correspond	VERB
ejpam-6526	234	8	to	to	ADP
ejpam-6526	234	9	strong	strong	ADJ
ejpam-6526	234	10	topology	topology	NOUN
ejpam-6526	234	11	and	and	CCONJ
ejpam-6526	234	12	t2	t2	NOUN
ejpam-6526	234	13	to	to	ADP
ejpam-6526	234	14	weak	weak	ADJ
ejpam-6526	234	15	topology	topology	NOUN
ejpam-6526	234	16	.	.	PUNCT
ejpam-6526	235	1	for	for	ADP
ejpam-6526	235	2	t1	t1	NOUN
ejpam-6526	235	3	:	:	PUNCT
ejpam-6526	235	4	open	open	ADJ
ejpam-6526	235	5	neighborhoods	neighborhood	NOUN
ejpam-6526	235	6	v1(x	v1(x	VERB
ejpam-6526	235	7	,	,	PUNCT
ejpam-6526	235	8	ε	ε	PROPN
ejpam-6526	235	9	)	)	PUNCT
ejpam-6526	236	1	=	=	PRON
ejpam-6526	236	2	{	{	PUNCT
ejpam-6526	236	3	y	y	PROPN
ejpam-6526	236	4	∈	∈	PROPN
ejpam-6526	236	5	h	h	NOUN
ejpam-6526	236	6	:	:	PUNCT
ejpam-6526	237	1	∥x	∥x	PROPN
ejpam-6526	237	2	−	−	AUX
ejpam-6526	237	3	y∥	y∥	VERB
ejpam-6526	237	4	<	<	X
ejpam-6526	237	5	ε	ε	X
ejpam-6526	237	6	}	}	PUNCT
ejpam-6526	237	7	precisely	precisely	ADV
ejpam-6526	237	8	match	match	VERB
ejpam-6526	237	9	open	open	ADJ
ejpam-6526	237	10	balls	ball	NOUN
ejpam-6526	237	11	in	in	ADP
ejpam-6526	237	12	norm	norm	NOUN
ejpam-6526	237	13	topology	topology	NOUN
ejpam-6526	237	14	,	,	PUNCT
ejpam-6526	237	15	generating	generate	VERB
ejpam-6526	237	16	strong	strong	ADJ
ejpam-6526	237	17	topology	topology	NOUN
ejpam-6526	237	18	.	.	PUNCT
ejpam-6526	238	1	for	for	ADP
ejpam-6526	238	2	t2	t2	NOUN
ejpam-6526	238	3	:	:	PUNCT
ejpam-6526	238	4	open	open	ADJ
ejpam-6526	238	5	neighborhoods	neighborhood	NOUN
ejpam-6526	238	6	v2(x	v2(x	PROPN
ejpam-6526	238	7	,	,	PUNCT
ejpam-6526	238	8	ε	ε	PROPN
ejpam-6526	238	9	)	)	PUNCT
ejpam-6526	238	10	=	=	PRON
ejpam-6526	238	11	{	{	PUNCT
ejpam-6526	238	12	y	y	PROPN
ejpam-6526	238	13	∈	∈	PROPN
ejpam-6526	238	14	h	h	NOUN
ejpam-6526	238	15	:	:	PUNCT
ejpam-6526	238	16	|∥x∥	|∥x∥	NUM
ejpam-6526	238	17	−	−	PROPN
ejpam-6526	238	18	∥y∥|	∥y∥|	PROPN
ejpam-6526	238	19	<	<	X
ejpam-6526	238	20	ε	ε	PROPN
ejpam-6526	238	21	}	}	PUNCT
ejpam-6526	238	22	do	do	AUX
ejpam-6526	238	23	n’t	not	PART
ejpam-6526	238	24	directly	directly	ADV
ejpam-6526	238	25	generate	generate	VERB
ejpam-6526	238	26	weak	weak	ADJ
ejpam-6526	238	27	topology	topology	NOUN
ejpam-6526	238	28	.	.	PUNCT
ejpam-6526	239	1	however	however	ADV
ejpam-6526	239	2	,	,	PUNCT
ejpam-6526	239	3	they	they	PRON
ejpam-6526	239	4	form	form	VERB
ejpam-6526	239	5	a	a	DET
ejpam-6526	239	6	basis	basis	NOUN
ejpam-6526	239	7	for	for	ADP
ejpam-6526	239	8	a	a	DET
ejpam-6526	239	9	coarser	coarse	ADJ
ejpam-6526	239	10	topology	topology	NOUN
ejpam-6526	239	11	than	than	ADP
ejpam-6526	239	12	strong	strong	ADJ
ejpam-6526	239	13	topology	topology	NOUN
ejpam-6526	239	14	,	,	PUNCT
ejpam-6526	239	15	analogous	analogous	ADJ
ejpam-6526	239	16	to	to	ADP
ejpam-6526	239	17	weak	weak	ADJ
ejpam-6526	239	18	topology	topology	NOUN
ejpam-6526	239	19	.	.	PUNCT
ejpam-6526	240	1	notably	notably	ADV
ejpam-6526	240	2	,	,	PUNCT
ejpam-6526	240	3	convergence	convergence	NOUN
ejpam-6526	240	4	in	in	ADP
ejpam-6526	240	5	second	second	ADJ
ejpam-6526	240	6	component	component	NOUN
ejpam-6526	240	7	metric	metric	NOUN
ejpam-6526	240	8	implies	imply	VERB
ejpam-6526	240	9	norm	norm	NOUN
ejpam-6526	240	10	convergence	convergence	NOUN
ejpam-6526	240	11	,	,	PUNCT
ejpam-6526	240	12	a	a	DET
ejpam-6526	240	13	necessary	necessary	ADJ
ejpam-6526	240	14	condition	condition	NOUN
ejpam-6526	240	15	for	for	ADP
ejpam-6526	240	16	weak	weak	ADJ
ejpam-6526	240	17	convergence	convergence	NOUN
ejpam-6526	240	18	.	.	PUNCT
ejpam-6526	241	1	a	a	DET
ejpam-6526	241	2	more	more	ADV
ejpam-6526	241	3	precise	precise	ADJ
ejpam-6526	241	4	characterization	characterization	NOUN
ejpam-6526	241	5	would	would	AUX
ejpam-6526	241	6	require	require	VERB
ejpam-6526	241	7	redefining	redefine	VERB
ejpam-6526	241	8	ψ	ψ	X
ejpam-6526	241	9	to	to	PART
ejpam-6526	241	10	fully	fully	ADV
ejpam-6526	241	11	capture	capture	VERB
ejpam-6526	241	12	weak	weak	ADJ
ejpam-6526	241	13	convergence	convergence	NOUN
ejpam-6526	241	14	,	,	PUNCT
ejpam-6526	241	15	but	but	CCONJ
ejpam-6526	241	16	this	this	DET
ejpam-6526	241	17	simplified	simplified	ADJ
ejpam-6526	241	18	version	version	NOUN
ejpam-6526	241	19	illustrates	illustrate	VERB
ejpam-6526	241	20	the	the	DET
ejpam-6526	241	21	concept	concept	NOUN
ejpam-6526	241	22	adequately	adequately	ADV
ejpam-6526	241	23	.	.	PUNCT
ejpam-6526	242	1	example	example	NOUN
ejpam-6526	242	2	11	11	NUM
ejpam-6526	242	3	(	(	PUNCT
ejpam-6526	242	4	strong	strong	ADJ
ejpam-6526	242	5	vs.	vs.	ADP
ejpam-6526	242	6	weak	weak	ADJ
ejpam-6526	242	7	topologies	topology	NOUN
ejpam-6526	242	8	in	in	ADP
ejpam-6526	242	9	hilbert	hilbert	NOUN
ejpam-6526	242	10	space	space	NOUN
ejpam-6526	242	11	)	)	PUNCT
ejpam-6526	242	12	.	.	PUNCT
ejpam-6526	243	1	consider	consider	VERB
ejpam-6526	243	2	h	h	NOUN
ejpam-6526	243	3	=	=	SYM
ejpam-6526	243	4	ℓ2	ℓ2	PROPN
ejpam-6526	243	5	from	from	ADP
ejpam-6526	243	6	our	our	PRON
ejpam-6526	243	7	previous	previous	ADJ
ejpam-6526	243	8	example	example	NOUN
ejpam-6526	243	9	,	,	PUNCT
ejpam-6526	243	10	with	with	ADP
ejpam-6526	243	11	sequences	sequence	NOUN
ejpam-6526	243	12	:	:	PUNCT
ejpam-6526	243	13	xn	xn	PUNCT
ejpam-6526	244	1	=	=	SYM
ejpam-6526	244	2	(	(	PUNCT
ejpam-6526	244	3	0	0	NUM
ejpam-6526	244	4	,	,	PUNCT
ejpam-6526	244	5	.	.	PUNCT
ejpam-6526	244	6	.	.	PUNCT
ejpam-6526	244	7	.	.	PUNCT
ejpam-6526	245	1	,	,	PUNCT
ejpam-6526	245	2	0	0	NUM
ejpam-6526	245	3	,	,	PUNCT
ejpam-6526	245	4	1	1	NUM
ejpam-6526	245	5	,	,	PUNCT
ejpam-6526	245	6	0	0	NUM
ejpam-6526	245	7	,	,	PUNCT
ejpam-6526	245	8	.	.	PUNCT
ejpam-6526	245	9	.	.	PUNCT
ejpam-6526	246	1	.	.	PUNCT
ejpam-6526	246	2	)	)	PUNCT
ejpam-6526	247	1	(	(	PUNCT
ejpam-6526	247	2	1	1	NUM
ejpam-6526	247	3	in	in	ADP
ejpam-6526	247	4	position	position	NOUN
ejpam-6526	247	5	n	n	CCONJ
ejpam-6526	247	6	)	)	PUNCT
ejpam-6526	247	7	y	y	NOUN
ejpam-6526	247	8	=	=	SYM
ejpam-6526	247	9	(	(	PUNCT
ejpam-6526	247	10	0	0	NUM
ejpam-6526	247	11	,	,	PUNCT
ejpam-6526	247	12	0	0	NUM
ejpam-6526	247	13	,	,	PUNCT
ejpam-6526	247	14	0	0	NUM
ejpam-6526	247	15	,	,	PUNCT
ejpam-6526	247	16	.	.	PUNCT
ejpam-6526	247	17	.	.	PUNCT
ejpam-6526	247	18	.	.	PUNCT
ejpam-6526	247	19	)	)	PUNCT
ejpam-6526	248	1	for	for	ADP
ejpam-6526	248	2	strong	strong	ADJ
ejpam-6526	248	3	topology	topology	NOUN
ejpam-6526	248	4	(	(	PUNCT
ejpam-6526	248	5	first	first	ADJ
ejpam-6526	248	6	component	component	NOUN
ejpam-6526	248	7	):	):	PUNCT
ejpam-6526	248	8	ϕ1(xn	ϕ1(xn	PROPN
ejpam-6526	248	9	,	,	PUNCT
ejpam-6526	248	10	y	y	PROPN
ejpam-6526	248	11	)	)	PUNCT
ejpam-6526	248	12	=	=	PRON
ejpam-6526	248	13	∥xn	∥xn	PROPN
ejpam-6526	248	14	−	−	NOUN
ejpam-6526	248	15	y∥	y∥	NOUN
ejpam-6526	248	16	=	=	PUNCT
ejpam-6526	248	17	∥xn∥	∥xn∥	PUNCT
ejpam-6526	249	1	=	=	SYM
ejpam-6526	249	2	1	1	NUM
ejpam-6526	249	3	for	for	ADP
ejpam-6526	249	4	all	all	DET
ejpam-6526	249	5	n.	n.	NOUN
ejpam-6526	249	6	thus	thus	ADV
ejpam-6526	249	7	,	,	PUNCT
ejpam-6526	249	8	{	{	PUNCT
ejpam-6526	249	9	xn	xn	X
ejpam-6526	249	10	}	}	PUNCT
ejpam-6526	249	11	does	do	AUX
ejpam-6526	249	12	n’t	not	PART
ejpam-6526	249	13	converge	converge	VERB
ejpam-6526	249	14	to	to	ADP
ejpam-6526	249	15	y	y	PROPN
ejpam-6526	249	16	in	in	ADP
ejpam-6526	249	17	strong	strong	ADJ
ejpam-6526	249	18	topology	topology	NOUN
ejpam-6526	249	19	.	.	PUNCT
ejpam-6526	250	1	for	for	ADP
ejpam-6526	250	2	second	second	ADJ
ejpam-6526	250	3	component	component	NOUN
ejpam-6526	250	4	:	:	PUNCT
ejpam-6526	250	5	ϕ2(xn	ϕ2(xn	NUM
ejpam-6526	250	6	,	,	PUNCT
ejpam-6526	250	7	y	y	PROPN
ejpam-6526	250	8	)	)	PUNCT
ejpam-6526	250	9	=	=	SYM
ejpam-6526	250	10	|∥xn∥	|∥xn∥	PROPN
ejpam-6526	250	11	−	−	NOUN
ejpam-6526	250	12	∥y∥|	∥y∥|	NOUN
ejpam-6526	250	13	=	=	PUNCT
ejpam-6526	250	14	|∥xn∥|	|∥xn∥|	PUNCT
ejpam-6526	250	15	=	=	SYM
ejpam-6526	250	16	1	1	NUM
ejpam-6526	250	17	a.	a.	NOUN
ejpam-6526	250	18	alsoboh	alsoboh	NOUN
ejpam-6526	250	19	et	et	PROPN
ejpam-6526	250	20	al	al	PROPN
ejpam-6526	250	21	.	.	PUNCT
ejpam-6526	250	22	/	/	SYM
ejpam-6526	250	23	eur	eur	PROPN
ejpam-6526	250	24	.	.	PUNCT
ejpam-6526	251	1	j.	j.	PROPN
ejpam-6526	251	2	pure	pure	PROPN
ejpam-6526	251	3	appl	appl	PROPN
ejpam-6526	251	4	.	.	PROPN
ejpam-6526	251	5	math	math	PROPN
ejpam-6526	251	6	,	,	PUNCT
ejpam-6526	251	7	18	18	NUM
ejpam-6526	251	8	(	(	PUNCT
ejpam-6526	251	9	4	4	NUM
ejpam-6526	251	10	)	)	PUNCT
ejpam-6526	251	11	(	(	PUNCT
ejpam-6526	251	12	2025	2025	NUM
ejpam-6526	251	13	)	)	PUNCT
ejpam-6526	251	14	,	,	PUNCT
ejpam-6526	251	15	6526	6526	NUM
ejpam-6526	251	16	11	11	NUM
ejpam-6526	251	17	of	of	ADP
ejpam-6526	251	18	21	21	NUM
ejpam-6526	251	19	for	for	ADP
ejpam-6526	251	20	all	all	DET
ejpam-6526	251	21	n.	n.	NOUN
ejpam-6526	251	22	again	again	ADV
ejpam-6526	251	23	,	,	PUNCT
ejpam-6526	251	24	{	{	PUNCT
ejpam-6526	251	25	xn	xn	X
ejpam-6526	251	26	}	}	PUNCT
ejpam-6526	251	27	does	do	AUX
ejpam-6526	251	28	n’t	not	PART
ejpam-6526	251	29	converge	converge	VERB
ejpam-6526	251	30	to	to	ADP
ejpam-6526	251	31	y	y	PROPN
ejpam-6526	251	32	under	under	ADP
ejpam-6526	251	33	ϕ2	ϕ2	ADV
ejpam-6526	251	34	.	.	PUNCT
ejpam-6526	252	1	this	this	DET
ejpam-6526	252	2	example	example	NOUN
ejpam-6526	252	3	does	do	AUX
ejpam-6526	252	4	n’t	not	PART
ejpam-6526	252	5	fully	fully	ADV
ejpam-6526	252	6	capture	capture	VERB
ejpam-6526	252	7	weak	weak	ADJ
ejpam-6526	252	8	topology	topology	NOUN
ejpam-6526	252	9	,	,	PUNCT
ejpam-6526	252	10	since	since	SCONJ
ejpam-6526	252	11	sequence	sequence	NOUN
ejpam-6526	252	12	{	{	PUNCT
ejpam-6526	252	13	xn	xn	NOUN
ejpam-6526	252	14	}	}	PUNCT
ejpam-6526	252	15	actually	actually	ADV
ejpam-6526	252	16	converges	converge	VERB
ejpam-6526	252	17	weakly	weakly	ADV
ejpam-6526	252	18	to	to	ADP
ejpam-6526	252	19	y	y	PROPN
ejpam-6526	252	20	in	in	ADP
ejpam-6526	252	21	ℓ2	ℓ2	PROPN
ejpam-6526	252	22	.	.	PUNCT
ejpam-6526	253	1	for	for	ADP
ejpam-6526	253	2	better	well	ADJ
ejpam-6526	253	3	weak	weak	ADJ
ejpam-6526	253	4	convergence	convergence	NOUN
ejpam-6526	253	5	modeling	modeling	NOUN
ejpam-6526	253	6	,	,	PUNCT
ejpam-6526	253	7	we	we	PRON
ejpam-6526	253	8	could	could	AUX
ejpam-6526	253	9	define	define	VERB
ejpam-6526	253	10	:	:	PUNCT
ejpam-6526	253	11	ψ′(x	ψ′(x	PROPN
ejpam-6526	253	12	,	,	PUNCT
ejpam-6526	253	13	y	y	NOUN
ejpam-6526	253	14	)	)	PUNCT
ejpam-6526	253	15	=	=	SYM
ejpam-6526	253	16	sup	sup	NOUN
ejpam-6526	253	17	∥f∥≤1	∥f∥≤1	NOUN
ejpam-6526	253	18	|⟨f	|⟨f	NOUN
ejpam-6526	253	19	,	,	PUNCT
ejpam-6526	253	20	x−	x−	PROPN
ejpam-6526	253	21	y⟩|	y⟩|	PROPN
ejpam-6526	253	22	with	with	ADP
ejpam-6526	253	23	this	this	DET
ejpam-6526	253	24	formulation	formulation	NOUN
ejpam-6526	253	25	:	:	PUNCT
ejpam-6526	253	26	ψ′(xn	ψ′(xn	PROPN
ejpam-6526	253	27	,	,	PUNCT
ejpam-6526	253	28	y	y	NOUN
ejpam-6526	253	29	)	)	PUNCT
ejpam-6526	253	30	=	=	SYM
ejpam-6526	253	31	sup	sup	NOUN
ejpam-6526	253	32	∥f∥≤1	∥f∥≤1	NOUN
ejpam-6526	253	33	|⟨f	|⟨f	NOUN
ejpam-6526	253	34	,	,	PUNCT
ejpam-6526	253	35	xn	xn	PROPN
ejpam-6526	254	1	−	−	PROPN
ejpam-6526	254	2	y⟩|	y⟩|	NOUN
ejpam-6526	255	1	=	=	NOUN
ejpam-6526	255	2	sup	sup	NOUN
ejpam-6526	255	3	∥f∥≤1	∥f∥≤1	NOUN
ejpam-6526	255	4	|⟨f	|⟨f	NOUN
ejpam-6526	255	5	,	,	PUNCT
ejpam-6526	255	6	xn⟩|	xn⟩|	PUNCT
ejpam-6526	255	7	=	=	PUNCT
ejpam-6526	255	8	sup	sup	NOUN
ejpam-6526	255	9	∥f∥≤1	∥f∥≤1	NOUN
ejpam-6526	255	10	|fn|	|fn|	NOUN
ejpam-6526	255	11	since	since	SCONJ
ejpam-6526	255	12	∥f∥2	∥f∥2	NOUN
ejpam-6526	255	13	=	=	NOUN
ejpam-6526	255	14	∑∞	∑∞	X
ejpam-6526	255	15	i=1	i=1	X
ejpam-6526	255	16	|fi|2	|fi|2	PUNCT
ejpam-6526	255	17	≤	≤	ADJ
ejpam-6526	255	18	1	1	NUM
ejpam-6526	255	19	implies	imply	VERB
ejpam-6526	255	20	|fn|	|fn|	X
ejpam-6526	255	21	≤	≤	NUM
ejpam-6526	255	22	1	1	NUM
ejpam-6526	255	23	,	,	PUNCT
ejpam-6526	255	24	supremum	supremum	PROPN
ejpam-6526	255	25	occurs	occur	VERB
ejpam-6526	255	26	with	with	ADP
ejpam-6526	255	27	f	f	PROPN
ejpam-6526	255	28	=	=	SYM
ejpam-6526	255	29	(	(	PUNCT
ejpam-6526	255	30	0	0	NUM
ejpam-6526	255	31	,	,	PUNCT
ejpam-6526	255	32	.	.	PUNCT
ejpam-6526	255	33	.	.	PUNCT
ejpam-6526	256	1	.	.	PUNCT
ejpam-6526	257	1	,	,	PUNCT
ejpam-6526	257	2	0	0	NUM
ejpam-6526	257	3	,	,	PUNCT
ejpam-6526	257	4	1	1	NUM
ejpam-6526	257	5	,	,	PUNCT
ejpam-6526	257	6	0	0	NUM
ejpam-6526	257	7	,	,	PUNCT
ejpam-6526	257	8	.	.	PUNCT
ejpam-6526	257	9	.	.	PUNCT
ejpam-6526	258	1	.	.	PUNCT
ejpam-6526	258	2	)	)	PUNCT
ejpam-6526	259	1	(	(	PUNCT
ejpam-6526	259	2	1	1	NUM
ejpam-6526	259	3	in	in	ADP
ejpam-6526	259	4	position	position	NOUN
ejpam-6526	259	5	n	n	CCONJ
ejpam-6526	259	6	)	)	PUNCT
ejpam-6526	259	7	,	,	PUNCT
ejpam-6526	259	8	yielding	yield	VERB
ejpam-6526	259	9	ψ′(xn	ψ′(xn	PROPN
ejpam-6526	259	10	,	,	PUNCT
ejpam-6526	259	11	y	y	NOUN
ejpam-6526	259	12	)	)	PUNCT
ejpam-6526	259	13	=	=	SYM
ejpam-6526	260	1	1	1	X
ejpam-6526	260	2	.	.	PUNCT
ejpam-6526	260	3	however	however	ADV
ejpam-6526	260	4	,	,	PUNCT
ejpam-6526	260	5	for	for	ADP
ejpam-6526	260	6	any	any	DET
ejpam-6526	260	7	fixed	fix	VERB
ejpam-6526	260	8	f	f	PROPN
ejpam-6526	260	9	∈	∈	PROPN
ejpam-6526	260	10	ℓ2	ℓ2	PROPN
ejpam-6526	260	11	,	,	PUNCT
ejpam-6526	260	12	⟨f	⟨f	X
ejpam-6526	260	13	,	,	PUNCT
ejpam-6526	260	14	xn⟩	xn⟩	PROPN
ejpam-6526	261	1	=	=	SYM
ejpam-6526	261	2	fn	fn	PROPN
ejpam-6526	261	3	→	→	SYM
ejpam-6526	261	4	0	0	NUM
ejpam-6526	261	5	as	as	ADP
ejpam-6526	261	6	n→	n→	ADV
ejpam-6526	261	7	∞	∞	PROPN
ejpam-6526	261	8	,	,	PUNCT
ejpam-6526	261	9	matching	match	VERB
ejpam-6526	261	10	weak	weak	ADJ
ejpam-6526	261	11	convergence	convergence	NOUN
ejpam-6526	261	12	definition	definition	NOUN
ejpam-6526	261	13	.	.	PUNCT
ejpam-6526	262	1	this	this	DET
ejpam-6526	262	2	highlights	highlight	NOUN
ejpam-6526	262	3	our	our	PRON
ejpam-6526	262	4	simplified	simplified	ADJ
ejpam-6526	262	5	bi	bi	ADJ
ejpam-6526	262	6	-	-	ADJ
ejpam-6526	262	7	metric	metric	ADJ
ejpam-6526	262	8	system	system	NOUN
ejpam-6526	262	9	’s	’s	PART
ejpam-6526	262	10	limitation	limitation	NOUN
ejpam-6526	262	11	in	in	ADP
ejpam-6526	262	12	fully	fully	ADV
ejpam-6526	262	13	capturing	capture	VERB
ejpam-6526	262	14	weak	weak	ADJ
ejpam-6526	262	15	topology	topology	NOUN
ejpam-6526	262	16	.	.	PUNCT
ejpam-6526	263	1	5	5	X
ejpam-6526	263	2	.	.	X
ejpam-6526	263	3	advanced	advanced	ADJ
ejpam-6526	263	4	theoretical	theoretical	ADJ
ejpam-6526	263	5	properties	property	NOUN
ejpam-6526	263	6	our	our	PRON
ejpam-6526	263	7	topology	topology	NOUN
ejpam-6526	263	8	characterization	characterization	NOUN
ejpam-6526	263	9	approach	approach	NOUN
ejpam-6526	263	10	draws	draw	VERB
ejpam-6526	263	11	from	from	ADP
ejpam-6526	263	12	research	research	NOUN
ejpam-6526	263	13	on	on	ADP
ejpam-6526	263	14	fuzzy	fuzzy	ADJ
ejpam-6526	263	15	sets	set	NOUN
ejpam-6526	263	16	and	and	CCONJ
ejpam-6526	263	17	induced	induce	VERB
ejpam-6526	263	18	topologies	topology	NOUN
ejpam-6526	263	19	[	[	X
ejpam-6526	263	20	22	22	NUM
ejpam-6526	263	21	]	]	PUNCT
ejpam-6526	263	22	.	.	PUNCT
ejpam-6526	264	1	theorem	theorem	ADJ
ejpam-6526	264	2	6	6	NUM
ejpam-6526	264	3	.	.	PUNCT
ejpam-6526	264	4	for	for	ADP
ejpam-6526	264	5	bi	bi	ADJ
ejpam-6526	264	6	-	-	ADJ
ejpam-6526	264	7	complete	complete	ADJ
ejpam-6526	264	8	bi	bi	ADJ
ejpam-6526	264	9	-	-	ADJ
ejpam-6526	264	10	metric	metric	ADJ
ejpam-6526	264	11	system	system	NOUN
ejpam-6526	264	12	(	(	PUNCT
ejpam-6526	264	13	x	x	X
ejpam-6526	264	14	,	,	PUNCT
ejpam-6526	264	15	ψ,⊕	ψ,⊕	NOUN
ejpam-6526	264	16	)	)	PUNCT
ejpam-6526	264	17	with	with	ADP
ejpam-6526	264	18	ψ(x	ψ(x	PROPN
ejpam-6526	264	19	,	,	PUNCT
ejpam-6526	264	20	y	y	NOUN
ejpam-6526	264	21	)	)	PUNCT
ejpam-6526	264	22	=	=	SYM
ejpam-6526	265	1	(	(	PUNCT
ejpam-6526	265	2	ϕ1(x	ϕ1(x	PROPN
ejpam-6526	265	3	,	,	PUNCT
ejpam-6526	265	4	y	y	NOUN
ejpam-6526	265	5	)	)	PUNCT
ejpam-6526	265	6	,	,	PUNCT
ejpam-6526	266	1	ϕ2(x	ϕ2(x	PROPN
ejpam-6526	266	2	,	,	PUNCT
ejpam-6526	266	3	y	y	NOUN
ejpam-6526	266	4	)	)	PUNCT
ejpam-6526	266	5	)	)	PUNCT
ejpam-6526	266	6	and	and	CCONJ
ejpam-6526	266	7	⊕	⊕	PROPN
ejpam-6526	266	8	implementing	implement	VERB
ejpam-6526	266	9	component	component	NOUN
ejpam-6526	266	10	-	-	PUNCT
ejpam-6526	266	11	wise	wise	ADJ
ejpam-6526	266	12	addition	addition	NOUN
ejpam-6526	266	13	,	,	PUNCT
ejpam-6526	266	14	consider	consider	VERB
ejpam-6526	266	15	mapping	mapping	NOUN
ejpam-6526	266	16	t	t	NOUN
ejpam-6526	266	17	:	:	PUNCT
ejpam-6526	266	18	x	x	X
ejpam-6526	266	19	→	→	SYM
ejpam-6526	266	20	x	x	SYM
ejpam-6526	266	21	satisfying	satisfy	VERB
ejpam-6526	266	22	:	:	PUNCT
ejpam-6526	266	23	ψ(t	ψ(t	PROPN
ejpam-6526	266	24	(	(	PUNCT
ejpam-6526	266	25	x	x	NOUN
ejpam-6526	266	26	)	)	PUNCT
ejpam-6526	266	27	,	,	PUNCT
ejpam-6526	266	28	t	t	PROPN
ejpam-6526	266	29	(	(	PUNCT
ejpam-6526	266	30	y	y	NOUN
ejpam-6526	266	31	)	)	PUNCT
ejpam-6526	266	32	)	)	PUNCT
ejpam-6526	267	1	≤comp	≤comp	PROPN
ejpam-6526	267	2	(	(	PUNCT
ejpam-6526	267	3	λ1	λ1	PROPN
ejpam-6526	267	4	·	·	PUNCT
ejpam-6526	267	5	ϕ1(x	ϕ1(x	NUM
ejpam-6526	267	6	,	,	PUNCT
ejpam-6526	267	7	y	y	NOUN
ejpam-6526	267	8	)	)	PUNCT
ejpam-6526	267	9	,	,	PUNCT
ejpam-6526	267	10	λ2	λ2	NOUN
ejpam-6526	267	11	·	·	PUNCT
ejpam-6526	267	12	ϕ2(x	ϕ2(x	PROPN
ejpam-6526	267	13	,	,	PUNCT
ejpam-6526	267	14	y	y	NOUN
ejpam-6526	267	15	)	)	PUNCT
ejpam-6526	267	16	)	)	PUNCT
ejpam-6526	267	17	for	for	ADP
ejpam-6526	267	18	all	all	DET
ejpam-6526	267	19	x	x	NOUN
ejpam-6526	267	20	,	,	PUNCT
ejpam-6526	267	21	y	y	PROPN
ejpam-6526	267	22	∈	∈	PROPN
ejpam-6526	267	23	x	x	X
ejpam-6526	267	24	and	and	CCONJ
ejpam-6526	267	25	constants	constant	NOUN
ejpam-6526	267	26	λ1	λ1	ADJ
ejpam-6526	267	27	,	,	PUNCT
ejpam-6526	267	28	λ2	λ2	PROPN
ejpam-6526	267	29	∈	∈	PROPN
ejpam-6526	268	1	[	[	X
ejpam-6526	268	2	0	0	NUM
ejpam-6526	268	3	,	,	PUNCT
ejpam-6526	268	4	1	1	NUM
ejpam-6526	268	5	)	)	PUNCT
ejpam-6526	268	6	.	.	PUNCT
ejpam-6526	269	1	then	then	ADV
ejpam-6526	269	2	t	t	PROPN
ejpam-6526	269	3	possesses	possess	VERB
ejpam-6526	269	4	a	a	DET
ejpam-6526	269	5	unique	unique	ADJ
ejpam-6526	269	6	equilibrium	equilibrium	NOUN
ejpam-6526	269	7	point	point	NOUN
ejpam-6526	269	8	in	in	ADP
ejpam-6526	269	9	x.	x.	NOUN
ejpam-6526	269	10	proof	proof	NOUN
ejpam-6526	269	11	.	.	PUNCT
ejpam-6526	270	1	select	select	VERB
ejpam-6526	270	2	arbitrary	arbitrary	ADJ
ejpam-6526	270	3	x0	x0	PROPN
ejpam-6526	270	4	∈	∈	PROPN
ejpam-6526	270	5	x	x	PUNCT
ejpam-6526	270	6	and	and	CCONJ
ejpam-6526	270	7	generate	generate	VERB
ejpam-6526	270	8	sequence	sequence	NOUN
ejpam-6526	270	9	{	{	PUNCT
ejpam-6526	270	10	xn	xn	VERB
ejpam-6526	270	11	}	}	PUNCT
ejpam-6526	270	12	by	by	ADP
ejpam-6526	270	13	xn	xn	PROPN
ejpam-6526	270	14	=	=	SYM
ejpam-6526	270	15	t	t	PROPN
ejpam-6526	270	16	(	(	PUNCT
ejpam-6526	270	17	xn−1	xn−1	PROPN
ejpam-6526	270	18	)	)	PUNCT
ejpam-6526	270	19	for	for	ADP
ejpam-6526	270	20	n	n	PRON
ejpam-6526	270	21	≥	≥	NUM
ejpam-6526	270	22	1	1	NUM
ejpam-6526	270	23	.	.	PUNCT
ejpam-6526	271	1	for	for	ADP
ejpam-6526	271	2	m	m	PROPN
ejpam-6526	271	3	>	>	X
ejpam-6526	271	4	n	n	CCONJ
ejpam-6526	271	5	:	:	PUNCT
ejpam-6526	271	6	ψ(xn	ψ(xn	NUM
ejpam-6526	271	7	,	,	PUNCT
ejpam-6526	271	8	xm	xm	PROPN
ejpam-6526	271	9	)	)	PUNCT
ejpam-6526	271	10	≤comp	≤comp	PROPN
ejpam-6526	271	11	⊕(ψ(xn	⊕(ψ(xn	NOUN
ejpam-6526	271	12	,	,	PUNCT
ejpam-6526	271	13	xn+1),⊕(ψ(xn+1	xn+1),⊕(ψ(xn+1	PROPN
ejpam-6526	271	14	,	,	PUNCT
ejpam-6526	271	15	xn+2	xn+2	NUM
ejpam-6526	271	16	)	)	PUNCT
ejpam-6526	271	17	,	,	PUNCT
ejpam-6526	271	18	.	.	PUNCT
ejpam-6526	271	19	.	.	PUNCT
ejpam-6526	271	20	.	.	PUNCT
ejpam-6526	272	1	,	,	PUNCT
ejpam-6526	272	2	ψ(xm−1	ψ(xm−1	PROPN
ejpam-6526	272	3	,	,	PUNCT
ejpam-6526	272	4	xm	xm	PROPN
ejpam-6526	272	5	)	)	PUNCT
ejpam-6526	272	6	)	)	PUNCT
ejpam-6526	272	7	)	)	PUNCT
ejpam-6526	273	1	=	=	PRON
ejpam-6526	273	2	(	(	PUNCT
ejpam-6526	273	3	ϕ1(xn	ϕ1(xn	PROPN
ejpam-6526	273	4	,	,	PUNCT
ejpam-6526	273	5	xn+1	xn+1	NUM
ejpam-6526	273	6	)	)	PUNCT
ejpam-6526	274	1	+	+	CCONJ
ejpam-6526	274	2	.	.	PUNCT
ejpam-6526	274	3	.	.	PUNCT
ejpam-6526	275	1	.+	.+	NOUN
ejpam-6526	275	2	ϕ1(xm−1	ϕ1(xm−1	NOUN
ejpam-6526	275	3	,	,	PUNCT
ejpam-6526	275	4	xm	xm	PROPN
ejpam-6526	275	5	)	)	PUNCT
ejpam-6526	275	6	,	,	PUNCT
ejpam-6526	275	7	ϕ2(xn	ϕ2(xn	PROPN
ejpam-6526	275	8	,	,	PUNCT
ejpam-6526	275	9	xn+1	xn+1	NUM
ejpam-6526	275	10	)	)	PUNCT
ejpam-6526	275	11	+	+	CCONJ
ejpam-6526	275	12	.	.	PUNCT
ejpam-6526	275	13	.	.	PUNCT
ejpam-6526	276	1	.+	.+	NOUN
ejpam-6526	276	2	ϕ2(xm−1	ϕ2(xm−1	PROPN
ejpam-6526	276	3	,	,	PUNCT
ejpam-6526	276	4	xm	xm	PROPN
ejpam-6526	276	5	)	)	PUNCT
ejpam-6526	276	6	)	)	PUNCT
ejpam-6526	276	7	for	for	ADP
ejpam-6526	276	8	first	first	ADJ
ejpam-6526	276	9	component	component	NOUN
ejpam-6526	276	10	:	:	PUNCT
ejpam-6526	276	11	ϕ1(xn	ϕ1(xn	PROPN
ejpam-6526	276	12	,	,	PUNCT
ejpam-6526	276	13	xn+1	xn+1	NUM
ejpam-6526	276	14	)	)	PUNCT
ejpam-6526	277	1	=	=	SYM
ejpam-6526	277	2	ϕ1(t	ϕ1(t	X
ejpam-6526	277	3	(	(	PUNCT
ejpam-6526	277	4	xn−1	xn−1	PROPN
ejpam-6526	277	5	)	)	PUNCT
ejpam-6526	277	6	,	,	PUNCT
ejpam-6526	277	7	t	t	PROPN
ejpam-6526	277	8	(	(	PUNCT
ejpam-6526	277	9	xn	xn	PROPN
ejpam-6526	277	10	)	)	PUNCT
ejpam-6526	277	11	)	)	PUNCT
ejpam-6526	277	12	≤	≤	NOUN
ejpam-6526	278	1	λ1	λ1	ADJ
ejpam-6526	278	2	·	·	PUNCT
ejpam-6526	278	3	ϕ1(xn−1	ϕ1(xn−1	PROPN
ejpam-6526	278	4	,	,	PUNCT
ejpam-6526	278	5	xn	xn	NUM
ejpam-6526	278	6	)	)	PUNCT
ejpam-6526	278	7	a.	a.	NOUN
ejpam-6526	278	8	alsoboh	alsoboh	PROPN
ejpam-6526	278	9	et	et	PROPN
ejpam-6526	278	10	al	al	PROPN
ejpam-6526	278	11	.	.	PUNCT
ejpam-6526	278	12	/	/	SYM
ejpam-6526	278	13	eur	eur	PROPN
ejpam-6526	278	14	.	.	PUNCT
ejpam-6526	279	1	j.	j.	PROPN
ejpam-6526	279	2	pure	pure	PROPN
ejpam-6526	279	3	appl	appl	PROPN
ejpam-6526	279	4	.	.	PROPN
ejpam-6526	279	5	math	math	PROPN
ejpam-6526	279	6	,	,	PUNCT
ejpam-6526	279	7	18	18	NUM
ejpam-6526	279	8	(	(	PUNCT
ejpam-6526	279	9	4	4	NUM
ejpam-6526	279	10	)	)	PUNCT
ejpam-6526	279	11	(	(	PUNCT
ejpam-6526	279	12	2025	2025	NUM
ejpam-6526	279	13	)	)	PUNCT
ejpam-6526	279	14	,	,	PUNCT
ejpam-6526	279	15	6526	6526	NUM
ejpam-6526	279	16	12	12	NUM
ejpam-6526	279	17	of	of	ADP
ejpam-6526	279	18	21	21	NUM
ejpam-6526	279	19	≤	≤	NUM
ejpam-6526	280	1	λn1	λn1	PROPN
ejpam-6526	280	2	·	·	PUNCT
ejpam-6526	280	3	ϕ1(x0	ϕ1(x0	PROPN
ejpam-6526	280	4	,	,	PUNCT
ejpam-6526	280	5	x1	x1	PROPN
ejpam-6526	280	6	)	)	PUNCT
ejpam-6526	280	7	similarly	similarly	ADV
ejpam-6526	280	8	,	,	PUNCT
ejpam-6526	280	9	for	for	ADP
ejpam-6526	280	10	second	second	ADJ
ejpam-6526	280	11	component	component	NOUN
ejpam-6526	280	12	:	:	PUNCT
ejpam-6526	280	13	ϕ2(xn	ϕ2(xn	NUM
ejpam-6526	280	14	,	,	PUNCT
ejpam-6526	280	15	xn+1	xn+1	NUM
ejpam-6526	280	16	)	)	PUNCT
ejpam-6526	280	17	≤	≤	NUM
ejpam-6526	280	18	λn2	λn2	X
ejpam-6526	280	19	·	·	PUNCT
ejpam-6526	280	20	ϕ2(x0	ϕ2(x0	NUM
ejpam-6526	280	21	,	,	PUNCT
ejpam-6526	280	22	x1	x1	PROPN
ejpam-6526	280	23	)	)	PUNCT
ejpam-6526	280	24	therefore	therefore	ADV
ejpam-6526	280	25	:	:	PUNCT
ejpam-6526	280	26	ϕ1(xn	ϕ1(xn	PROPN
ejpam-6526	280	27	,	,	PUNCT
ejpam-6526	280	28	xm	xm	PROPN
ejpam-6526	280	29	)	)	PUNCT
ejpam-6526	280	30	≤	≤	NOUN
ejpam-6526	280	31	m−1∑	m−1∑	NUM
ejpam-6526	280	32	i	i	NOUN
ejpam-6526	280	33	=	=	PROPN
ejpam-6526	280	34	n	n	PRON
ejpam-6526	280	35	ϕ1(xi	ϕ1(xi	PROPN
ejpam-6526	280	36	,	,	PUNCT
ejpam-6526	280	37	xi+1	xi+1	NOUN
ejpam-6526	280	38	)	)	PUNCT
ejpam-6526	280	39	≤	≤	NOUN
ejpam-6526	280	40	m−1∑	m−1∑	NUM
ejpam-6526	280	41	i	i	NOUN
ejpam-6526	280	42	=	=	PROPN
ejpam-6526	280	43	n	n	PART
ejpam-6526	280	44	λi1	λi1	NOUN
ejpam-6526	280	45	·	·	PUNCT
ejpam-6526	280	46	ϕ1(x0	ϕ1(x0	PROPN
ejpam-6526	280	47	,	,	PUNCT
ejpam-6526	280	48	x1	x1	PROPN
ejpam-6526	280	49	)	)	PUNCT
ejpam-6526	280	50	=	=	SYM
ejpam-6526	280	51	ϕ1(x0	ϕ1(x0	PROPN
ejpam-6526	280	52	,	,	PUNCT
ejpam-6526	280	53	x1	x1	PROPN
ejpam-6526	280	54	)	)	PUNCT
ejpam-6526	280	55	·	·	PUNCT
ejpam-6526	281	1	λn1	λn1	X
ejpam-6526	281	2	·	·	PUNCT
ejpam-6526	281	3	1−	1−	NUM
ejpam-6526	281	4	λm−n	λm−n	NOUN
ejpam-6526	281	5	1	1	NUM
ejpam-6526	281	6	1−	1−	NUM
ejpam-6526	281	7	λ1	λ1	PROPN
ejpam-6526	281	8	similarly	similarly	ADV
ejpam-6526	281	9	:	:	PUNCT
ejpam-6526	281	10	ϕ2(xn	ϕ2(xn	NUM
ejpam-6526	281	11	,	,	PUNCT
ejpam-6526	281	12	xm	xm	PROPN
ejpam-6526	281	13	)	)	PUNCT
ejpam-6526	281	14	≤	≤	NOUN
ejpam-6526	282	1	ϕ2(x0	ϕ2(x0	ADP
ejpam-6526	282	2	,	,	PUNCT
ejpam-6526	282	3	x1	x1	PROPN
ejpam-6526	282	4	)	)	PUNCT
ejpam-6526	282	5	·	·	PUNCT
ejpam-6526	283	1	λn2	λn2	X
ejpam-6526	283	2	·	·	PUNCT
ejpam-6526	283	3	1−	1−	NUM
ejpam-6526	283	4	λm−n	λm−n	NOUN
ejpam-6526	283	5	2	2	NUM
ejpam-6526	283	6	1−	1−	NUM
ejpam-6526	283	7	λ2	λ2	NOUN
ejpam-6526	283	8	as	as	ADP
ejpam-6526	283	9	n	n	PROPN
ejpam-6526	283	10	→	→	SYM
ejpam-6526	283	11	∞	∞	PROPN
ejpam-6526	283	12	,	,	PUNCT
ejpam-6526	283	13	both	both	DET
ejpam-6526	283	14	components	component	NOUN
ejpam-6526	283	15	approach	approach	VERB
ejpam-6526	283	16	zero	zero	NUM
ejpam-6526	283	17	since	since	SCONJ
ejpam-6526	283	18	λ1	λ1	PROPN
ejpam-6526	283	19	,	,	PUNCT
ejpam-6526	283	20	λ2	λ2	PROPN
ejpam-6526	283	21	∈	∈	PROPN
ejpam-6526	284	1	[	[	X
ejpam-6526	284	2	0	0	NUM
ejpam-6526	284	3	,	,	PUNCT
ejpam-6526	284	4	1	1	NUM
ejpam-6526	284	5	)	)	PUNCT
ejpam-6526	284	6	,	,	PUNCT
ejpam-6526	284	7	confirming	confirm	VERB
ejpam-6526	284	8	{	{	PUNCT
ejpam-6526	284	9	xn	xn	NOUN
ejpam-6526	284	10	}	}	PUNCT
ejpam-6526	284	11	as	as	ADP
ejpam-6526	284	12	bi	bi	NOUN
ejpam-6526	284	13	-	-	NOUN
ejpam-6526	284	14	cauchy	cauchy	NOUN
ejpam-6526	284	15	.	.	PUNCT
ejpam-6526	285	1	by	by	ADP
ejpam-6526	285	2	bi	bi	NOUN
ejpam-6526	285	3	-	-	NOUN
ejpam-6526	285	4	completeness	completeness	NOUN
ejpam-6526	285	5	of	of	ADP
ejpam-6526	285	6	the	the	DET
ejpam-6526	285	7	bi	bi	ADJ
ejpam-6526	285	8	-	-	ADJ
ejpam-6526	285	9	metric	metric	ADJ
ejpam-6526	285	10	system	system	NOUN
ejpam-6526	285	11	,	,	PUNCT
ejpam-6526	285	12	{	{	PUNCT
ejpam-6526	285	13	xn	xn	X
ejpam-6526	285	14	}	}	PUNCT
ejpam-6526	285	15	converges	converge	NOUN
ejpam-6526	285	16	to	to	ADP
ejpam-6526	285	17	some	some	DET
ejpam-6526	285	18	x∗	x∗	PROPN
ejpam-6526	285	19	∈	∈	PROPN
ejpam-6526	285	20	x.	x.	VERB
ejpam-6526	285	21	to	to	PART
ejpam-6526	285	22	establish	establish	VERB
ejpam-6526	285	23	x∗	x∗	PROPN
ejpam-6526	285	24	as	as	ADP
ejpam-6526	285	25	t	t	PROPN
ejpam-6526	285	26	’s	’s	PART
ejpam-6526	285	27	equilibrium	equilibrium	NOUN
ejpam-6526	285	28	point	point	NOUN
ejpam-6526	285	29	:	:	PUNCT
ejpam-6526	285	30	ψ(t	ψ(t	PROPN
ejpam-6526	285	31	(	(	PUNCT
ejpam-6526	285	32	x∗	x∗	PROPN
ejpam-6526	285	33	)	)	PUNCT
ejpam-6526	285	34	,	,	PUNCT
ejpam-6526	285	35	x∗	x∗	PROPN
ejpam-6526	285	36	)	)	PUNCT
ejpam-6526	285	37	≤comp	≤comp	ADJ
ejpam-6526	285	38	⊕(ψ(t	⊕(ψ(t	PROPN
ejpam-6526	285	39	(	(	PUNCT
ejpam-6526	285	40	x∗	x∗	PROPN
ejpam-6526	285	41	)	)	PUNCT
ejpam-6526	285	42	,	,	PUNCT
ejpam-6526	285	43	t	t	PROPN
ejpam-6526	285	44	(	(	PUNCT
ejpam-6526	285	45	xn)),ψ(t	xn)),ψ(t	X
ejpam-6526	285	46	(	(	PUNCT
ejpam-6526	285	47	xn	xn	PROPN
ejpam-6526	285	48	)	)	PUNCT
ejpam-6526	285	49	,	,	PUNCT
ejpam-6526	285	50	x	x	NOUN
ejpam-6526	285	51	∗	∗	NOUN
ejpam-6526	285	52	)	)	PUNCT
ejpam-6526	285	53	)	)	PUNCT
ejpam-6526	286	1	=	=	PUNCT
ejpam-6526	286	2	(	(	PUNCT
ejpam-6526	286	3	ϕ1(t	ϕ1(t	X
ejpam-6526	286	4	(	(	PUNCT
ejpam-6526	286	5	x	x	NOUN
ejpam-6526	286	6	∗	∗	NOUN
ejpam-6526	286	7	)	)	PUNCT
ejpam-6526	286	8	,	,	PUNCT
ejpam-6526	286	9	t	t	PROPN
ejpam-6526	286	10	(	(	PUNCT
ejpam-6526	286	11	xn	xn	PROPN
ejpam-6526	286	12	)	)	PUNCT
ejpam-6526	286	13	)	)	PUNCT
ejpam-6526	287	1	+	+	CCONJ
ejpam-6526	287	2	ϕ1(t	ϕ1(t	PRON
ejpam-6526	287	3	(	(	PUNCT
ejpam-6526	287	4	xn	xn	PROPN
ejpam-6526	287	5	)	)	PUNCT
ejpam-6526	287	6	,	,	PUNCT
ejpam-6526	287	7	x	x	NOUN
ejpam-6526	287	8	∗	∗	NOUN
ejpam-6526	287	9	)	)	PUNCT
ejpam-6526	287	10	,	,	PUNCT
ejpam-6526	287	11	ϕ2(t	ϕ2(t	PROPN
ejpam-6526	287	12	(	(	PUNCT
ejpam-6526	287	13	x	x	NOUN
ejpam-6526	287	14	∗	∗	NOUN
ejpam-6526	287	15	)	)	PUNCT
ejpam-6526	287	16	,	,	PUNCT
ejpam-6526	287	17	t	t	PROPN
ejpam-6526	287	18	(	(	PUNCT
ejpam-6526	287	19	xn	xn	PROPN
ejpam-6526	287	20	)	)	PUNCT
ejpam-6526	287	21	)	)	PUNCT
ejpam-6526	288	1	+	+	CCONJ
ejpam-6526	288	2	ϕ2(t	ϕ2(t	PROPN
ejpam-6526	288	3	(	(	PUNCT
ejpam-6526	288	4	xn	xn	PROPN
ejpam-6526	288	5	)	)	PUNCT
ejpam-6526	288	6	,	,	PUNCT
ejpam-6526	288	7	x	x	NOUN
ejpam-6526	288	8	∗	∗	NOUN
ejpam-6526	288	9	)	)	PUNCT
ejpam-6526	288	10	)	)	PUNCT
ejpam-6526	289	1	≤comp	≤comp	PROPN
ejpam-6526	289	2	(	(	PUNCT
ejpam-6526	289	3	λ1	λ1	PROPN
ejpam-6526	289	4	·	·	PUNCT
ejpam-6526	289	5	ϕ1(x∗	ϕ1(x∗	INTJ
ejpam-6526	289	6	,	,	PUNCT
ejpam-6526	289	7	xn	xn	PROPN
ejpam-6526	289	8	)	)	PUNCT
ejpam-6526	290	1	+	+	X
ejpam-6526	290	2	ϕ1(xn+1	ϕ1(xn+1	ADJ
ejpam-6526	290	3	,	,	PUNCT
ejpam-6526	290	4	x	x	NOUN
ejpam-6526	290	5	∗	∗	NOUN
ejpam-6526	290	6	)	)	PUNCT
ejpam-6526	290	7	,	,	PUNCT
ejpam-6526	290	8	λ2	λ2	PROPN
ejpam-6526	290	9	·	·	PUNCT
ejpam-6526	290	10	ϕ2(x∗	ϕ2(x∗	PROPN
ejpam-6526	290	11	,	,	PUNCT
ejpam-6526	290	12	xn	xn	PUNCT
ejpam-6526	290	13	)	)	PUNCT
ejpam-6526	291	1	+	+	CCONJ
ejpam-6526	291	2	ϕ2(xn+1	ϕ2(xn+1	ADJ
ejpam-6526	291	3	,	,	PUNCT
ejpam-6526	291	4	x	x	NOUN
ejpam-6526	291	5	∗	∗	NOUN
ejpam-6526	291	6	)	)	PUNCT
ejpam-6526	291	7	)	)	PUNCT
ejpam-6526	291	8	as	as	ADP
ejpam-6526	291	9	n→	n→	PROPN
ejpam-6526	291	10	∞	∞	PROPN
ejpam-6526	291	11	,	,	PUNCT
ejpam-6526	291	12	both	both	DET
ejpam-6526	291	13	components	component	NOUN
ejpam-6526	291	14	approach	approach	NOUN
ejpam-6526	291	15	zero	zero	NUM
ejpam-6526	291	16	,	,	PUNCT
ejpam-6526	291	17	yielding	yield	VERB
ejpam-6526	291	18	ψ(t	ψ(t	PROPN
ejpam-6526	291	19	(	(	PUNCT
ejpam-6526	291	20	x∗	x∗	PROPN
ejpam-6526	291	21	)	)	PUNCT
ejpam-6526	291	22	,	,	PUNCT
ejpam-6526	291	23	x∗	x∗	PROPN
ejpam-6526	291	24	)	)	PUNCT
ejpam-6526	291	25	=	=	SYM
ejpam-6526	291	26	(	(	PUNCT
ejpam-6526	291	27	0	0	NUM
ejpam-6526	291	28	,	,	PUNCT
ejpam-6526	291	29	0	0	NUM
ejpam-6526	291	30	)	)	PUNCT
ejpam-6526	291	31	,	,	PUNCT
ejpam-6526	291	32	confirming	confirm	VERB
ejpam-6526	291	33	t	t	X
ejpam-6526	291	34	(	(	PUNCT
ejpam-6526	291	35	x∗	x∗	PROPN
ejpam-6526	291	36	)	)	PUNCT
ejpam-6526	291	37	=	=	SYM
ejpam-6526	292	1	x∗.	x∗.	PROPN
ejpam-6526	292	2	for	for	ADP
ejpam-6526	292	3	uniqueness	uniqueness	NOUN
ejpam-6526	292	4	,	,	PUNCT
ejpam-6526	292	5	assuming	assume	VERB
ejpam-6526	292	6	alternative	alternative	ADJ
ejpam-6526	292	7	equilibrium	equilibrium	NOUN
ejpam-6526	292	8	point	point	NOUN
ejpam-6526	292	9	y∗	y∗	ADV
ejpam-6526	292	10	̸=	̸=	PROPN
ejpam-6526	292	11	x∗	x∗	NOUN
ejpam-6526	292	12	:	:	PUNCT
ejpam-6526	292	13	ψ(x∗	ψ(x∗	NOUN
ejpam-6526	292	14	,	,	PUNCT
ejpam-6526	292	15	y∗	y∗	PROPN
ejpam-6526	292	16	)	)	PUNCT
ejpam-6526	293	1	=	=	SYM
ejpam-6526	293	2	ψ(t	ψ(t	PROPN
ejpam-6526	293	3	(	(	PUNCT
ejpam-6526	293	4	x∗	x∗	PROPN
ejpam-6526	293	5	)	)	PUNCT
ejpam-6526	293	6	,	,	PUNCT
ejpam-6526	293	7	t	t	PROPN
ejpam-6526	293	8	(	(	PUNCT
ejpam-6526	293	9	y∗	y∗	PROPN
ejpam-6526	293	10	)	)	PUNCT
ejpam-6526	293	11	)	)	PUNCT
ejpam-6526	294	1	≤comp	≤comp	PROPN
ejpam-6526	294	2	(	(	PUNCT
ejpam-6526	294	3	λ1	λ1	PROPN
ejpam-6526	294	4	·	·	PUNCT
ejpam-6526	294	5	ϕ1(x∗	ϕ1(x∗	ADV
ejpam-6526	294	6	,	,	PUNCT
ejpam-6526	294	7	y∗	y∗	PROPN
ejpam-6526	294	8	)	)	PUNCT
ejpam-6526	294	9	,	,	PUNCT
ejpam-6526	294	10	λ2	λ2	PROPN
ejpam-6526	294	11	·	·	PUNCT
ejpam-6526	294	12	ϕ2(x∗	ϕ2(x∗	PROPN
ejpam-6526	294	13	,	,	PUNCT
ejpam-6526	294	14	y∗	y∗	PROPN
ejpam-6526	294	15	)	)	PUNCT
ejpam-6526	294	16	)	)	PUNCT
ejpam-6526	294	17	since	since	SCONJ
ejpam-6526	294	18	λ1	λ1	ADJ
ejpam-6526	294	19	,	,	PUNCT
ejpam-6526	294	20	λ2	λ2	NOUN
ejpam-6526	294	21	<	<	X
ejpam-6526	294	22	1	1	NUM
ejpam-6526	294	23	,	,	PUNCT
ejpam-6526	294	24	this	this	PRON
ejpam-6526	294	25	creates	create	VERB
ejpam-6526	294	26	contradiction	contradiction	NOUN
ejpam-6526	294	27	unless	unless	SCONJ
ejpam-6526	294	28	ψ(x∗	ψ(x∗	NOUN
ejpam-6526	294	29	,	,	PUNCT
ejpam-6526	294	30	y∗	y∗	PROPN
ejpam-6526	294	31	)	)	PUNCT
ejpam-6526	294	32	=	=	SYM
ejpam-6526	294	33	(	(	PUNCT
ejpam-6526	294	34	0	0	NUM
ejpam-6526	294	35	,	,	PUNCT
ejpam-6526	294	36	0	0	NUM
ejpam-6526	294	37	)	)	PUNCT
ejpam-6526	294	38	,	,	PUNCT
ejpam-6526	294	39	which	which	PRON
ejpam-6526	294	40	means	mean	VERB
ejpam-6526	294	41	x∗	x∗	PROPN
ejpam-6526	295	1	=	=	SYM
ejpam-6526	295	2	y∗.	y∗.	NUM
ejpam-6526	295	3	example	example	NOUN
ejpam-6526	295	4	12	12	NUM
ejpam-6526	295	5	(	(	PUNCT
ejpam-6526	295	6	expanded	expand	VERB
ejpam-6526	295	7	bi	bi	ADJ
ejpam-6526	295	8	-	-	ADJ
ejpam-6526	295	9	metric	metric	ADJ
ejpam-6526	295	10	contraction	contraction	NOUN
ejpam-6526	295	11	application	application	NOUN
ejpam-6526	295	12	)	)	PUNCT
ejpam-6526	295	13	.	.	PUNCT
ejpam-6526	296	1	consider	consider	VERB
ejpam-6526	296	2	x	x	X
ejpam-6526	296	3	=	=	PUNCT
ejpam-6526	297	1	[	[	X
ejpam-6526	297	2	0	0	NUM
ejpam-6526	297	3	,	,	PUNCT
ejpam-6526	297	4	1	1	NUM
ejpam-6526	297	5	]	]	PUNCT
ejpam-6526	297	6	with	with	ADP
ejpam-6526	297	7	bimetric	bimetric	ADJ
ejpam-6526	297	8	system	system	NOUN
ejpam-6526	297	9	ψ(x	ψ(x	PROPN
ejpam-6526	297	10	,	,	PUNCT
ejpam-6526	297	11	y	y	NOUN
ejpam-6526	297	12	)	)	PUNCT
ejpam-6526	297	13	=	=	SYM
ejpam-6526	298	1	(	(	PUNCT
ejpam-6526	298	2	|x−y|	|x−y|	ADJ
ejpam-6526	298	3	,	,	PUNCT
ejpam-6526	298	4	|	|	ADV
ejpam-6526	298	5	sinh(x)−	sinh(x)−	PROPN
ejpam-6526	298	6	sinh(y)|	sinh(y)|	PROPN
ejpam-6526	298	7	)	)	PUNCT
ejpam-6526	298	8	and	and	CCONJ
ejpam-6526	298	9	⊕	⊕	PROPN
ejpam-6526	298	10	implementing	implement	VERB
ejpam-6526	298	11	component	component	NOUN
ejpam-6526	298	12	-	-	PUNCT
ejpam-6526	298	13	wise	wise	ADJ
ejpam-6526	298	14	addition	addition	NOUN
ejpam-6526	298	15	.	.	PUNCT
ejpam-6526	299	1	define	define	VERB
ejpam-6526	299	2	t	t	NOUN
ejpam-6526	299	3	:	:	PUNCT
ejpam-6526	299	4	x	x	SYM
ejpam-6526	299	5	→	→	SYM
ejpam-6526	299	6	x	x	PUNCT
ejpam-6526	299	7	by	by	ADP
ejpam-6526	299	8	t	t	PROPN
ejpam-6526	299	9	(	(	PUNCT
ejpam-6526	299	10	x	x	NOUN
ejpam-6526	299	11	)	)	PUNCT
ejpam-6526	299	12	=	=	SYM
ejpam-6526	300	1	x	x	SYM
ejpam-6526	300	2	2	2	NUM
ejpam-6526	300	3	+	+	CCONJ
ejpam-6526	300	4	1	1	NUM
ejpam-6526	300	5	4	4	NUM
ejpam-6526	300	6	.	.	PUNCT
ejpam-6526	301	1	step	step	NOUN
ejpam-6526	301	2	1	1	NUM
ejpam-6526	301	3	:	:	PUNCT
ejpam-6526	301	4	verify	verify	VERB
ejpam-6526	301	5	contraction	contraction	NOUN
ejpam-6526	301	6	conditions	condition	NOUN
ejpam-6526	301	7	for	for	ADP
ejpam-6526	301	8	first	first	ADJ
ejpam-6526	301	9	component	component	NOUN
ejpam-6526	301	10	ϕ1(t	ϕ1(t	X
ejpam-6526	301	11	(	(	PUNCT
ejpam-6526	301	12	x	x	NOUN
ejpam-6526	301	13	)	)	PUNCT
ejpam-6526	301	14	,	,	PUNCT
ejpam-6526	301	15	t	t	PROPN
ejpam-6526	301	16	(	(	PUNCT
ejpam-6526	301	17	y	y	NOUN
ejpam-6526	301	18	)	)	PUNCT
ejpam-6526	301	19	)	)	PUNCT
ejpam-6526	302	1	=	=	PUNCT
ejpam-6526	302	2	∣∣∣∣x2	∣∣∣∣x2	PUNCT
ejpam-6526	303	1	+	+	CCONJ
ejpam-6526	303	2	1	1	NUM
ejpam-6526	303	3	4	4	NUM
ejpam-6526	303	4	−	−	NOUN
ejpam-6526	303	5	y	y	NOUN
ejpam-6526	303	6	2	2	NUM
ejpam-6526	303	7	−	−	NOUN
ejpam-6526	303	8	1	1	NUM
ejpam-6526	303	9	4	4	NUM
ejpam-6526	303	10	∣∣∣∣	∣∣∣∣	NOUN
ejpam-6526	303	11	=	=	SYM
ejpam-6526	303	12	1	1	NUM
ejpam-6526	303	13	2	2	NUM
ejpam-6526	303	14	|x−	|x−	NOUN
ejpam-6526	303	15	y|	y|	NOUN
ejpam-6526	303	16	=	=	NOUN
ejpam-6526	303	17	1	1	NUM
ejpam-6526	303	18	2	2	NUM
ejpam-6526	303	19	ϕ1(x	ϕ1(x	NOUN
ejpam-6526	303	20	,	,	PUNCT
ejpam-6526	303	21	y	y	NOUN
ejpam-6526	303	22	)	)	PUNCT
ejpam-6526	303	23	a.	a.	NOUN
ejpam-6526	303	24	alsoboh	alsoboh	PROPN
ejpam-6526	303	25	et	et	PROPN
ejpam-6526	303	26	al	al	PROPN
ejpam-6526	303	27	.	.	PUNCT
ejpam-6526	303	28	/	/	SYM
ejpam-6526	303	29	eur	eur	PROPN
ejpam-6526	303	30	.	.	PUNCT
ejpam-6526	304	1	j.	j.	PROPN
ejpam-6526	304	2	pure	pure	PROPN
ejpam-6526	304	3	appl	appl	PROPN
ejpam-6526	304	4	.	.	PROPN
ejpam-6526	304	5	math	math	PROPN
ejpam-6526	304	6	,	,	PUNCT
ejpam-6526	304	7	18	18	NUM
ejpam-6526	304	8	(	(	PUNCT
ejpam-6526	304	9	4	4	NUM
ejpam-6526	304	10	)	)	PUNCT
ejpam-6526	304	11	(	(	PUNCT
ejpam-6526	304	12	2025	2025	NUM
ejpam-6526	304	13	)	)	PUNCT
ejpam-6526	304	14	,	,	PUNCT
ejpam-6526	304	15	6526	6526	NUM
ejpam-6526	304	16	13	13	NUM
ejpam-6526	304	17	of	of	ADP
ejpam-6526	304	18	21	21	NUM
ejpam-6526	304	19	thus	thus	ADV
ejpam-6526	304	20	λ1	λ1	VERB
ejpam-6526	304	21	=	=	SYM
ejpam-6526	304	22	1	1	NUM
ejpam-6526	304	23	2	2	NUM
ejpam-6526	304	24	.	.	PUNCT
ejpam-6526	305	1	step	step	NOUN
ejpam-6526	305	2	2	2	NUM
ejpam-6526	305	3	:	:	PUNCT
ejpam-6526	305	4	verify	verify	VERB
ejpam-6526	305	5	contraction	contraction	NOUN
ejpam-6526	305	6	conditions	condition	NOUN
ejpam-6526	305	7	for	for	ADP
ejpam-6526	305	8	second	second	ADJ
ejpam-6526	305	9	component	component	NOUN
ejpam-6526	305	10	by	by	ADP
ejpam-6526	305	11	the	the	DET
ejpam-6526	305	12	mean	mean	ADJ
ejpam-6526	305	13	value	value	NOUN
ejpam-6526	305	14	theorem	theorem	VERB
ejpam-6526	305	15	,	,	PUNCT
ejpam-6526	305	16	for	for	ADP
ejpam-6526	305	17	some	some	DET
ejpam-6526	305	18	c	c	NOUN
ejpam-6526	305	19	∈	∈	PROPN
ejpam-6526	306	1	[	[	X
ejpam-6526	306	2	0	0	NUM
ejpam-6526	306	3	,	,	PUNCT
ejpam-6526	306	4	1	1	NUM
ejpam-6526	306	5	]	]	PUNCT
ejpam-6526	306	6	:	:	PUNCT
ejpam-6526	306	7	|	|	ADV
ejpam-6526	306	8	sinh(t	sinh(t	INTJ
ejpam-6526	306	9	(	(	PUNCT
ejpam-6526	306	10	x))−	x))−	NOUN
ejpam-6526	306	11	sinh(t	sinh(t	PROPN
ejpam-6526	306	12	(	(	PUNCT
ejpam-6526	306	13	y))|	y))|	PROPN
ejpam-6526	306	14	=	=	PROPN
ejpam-6526	306	15	cosh(c)|t	cosh(c)|t	PROPN
ejpam-6526	306	16	(	(	PUNCT
ejpam-6526	306	17	x)−	x)−	PROPN
ejpam-6526	306	18	t	t	PROPN
ejpam-6526	306	19	(	(	PUNCT
ejpam-6526	306	20	y)|	y)|	PROPN
ejpam-6526	306	21	=	=	SYM
ejpam-6526	306	22	cosh(c	cosh(c	PROPN
ejpam-6526	306	23	)	)	PUNCT
ejpam-6526	306	24	2	2	NUM
ejpam-6526	306	25	|x−	|x−	NOUN
ejpam-6526	306	26	y|	y|	NOUN
ejpam-6526	306	27	since	since	SCONJ
ejpam-6526	306	28	cosh(c	cosh(c	PROPN
ejpam-6526	306	29	)	)	PUNCT
ejpam-6526	306	30	≤	≤	NOUN
ejpam-6526	306	31	cosh(1	cosh(1	NOUN
ejpam-6526	306	32	)	)	PUNCT
ejpam-6526	306	33	for	for	ADP
ejpam-6526	306	34	c	c	PROPN
ejpam-6526	306	35	∈	∈	PROPN
ejpam-6526	307	1	[	[	X
ejpam-6526	307	2	0	0	NUM
ejpam-6526	307	3	,	,	PUNCT
ejpam-6526	307	4	1	1	NUM
ejpam-6526	307	5	]	]	PUNCT
ejpam-6526	307	6	:	:	PUNCT
ejpam-6526	307	7	ϕ2(t	ϕ2(t	PROPN
ejpam-6526	307	8	(	(	PUNCT
ejpam-6526	307	9	x	x	NOUN
ejpam-6526	307	10	)	)	PUNCT
ejpam-6526	307	11	,	,	PUNCT
ejpam-6526	307	12	t	t	PROPN
ejpam-6526	307	13	(	(	PUNCT
ejpam-6526	307	14	y	y	NOUN
ejpam-6526	307	15	)	)	PUNCT
ejpam-6526	307	16	)	)	PUNCT
ejpam-6526	307	17	≤	≤	NUM
ejpam-6526	307	18	cosh(1	cosh(1	NOUN
ejpam-6526	307	19	)	)	PUNCT
ejpam-6526	307	20	2	2	NUM
ejpam-6526	308	1	ϕ2(x	ϕ2(x	PROPN
ejpam-6526	308	2	,	,	PUNCT
ejpam-6526	308	3	y	y	NOUN
ejpam-6526	308	4	)	)	PUNCT
ejpam-6526	308	5	where	where	SCONJ
ejpam-6526	308	6	λ2	λ2	NOUN
ejpam-6526	308	7	=	=	SYM
ejpam-6526	308	8	cosh(1	cosh(1	NOUN
ejpam-6526	308	9	)	)	PUNCT
ejpam-6526	308	10	2	2	NUM
ejpam-6526	309	1	≈	≈	NUM
ejpam-6526	309	2	0.77	0.77	NUM
ejpam-6526	309	3	<	<	X
ejpam-6526	309	4	1	1	NUM
ejpam-6526	309	5	.	.	PUNCT
ejpam-6526	309	6	step	step	NOUN
ejpam-6526	309	7	3	3	NUM
ejpam-6526	309	8	:	:	PUNCT
ejpam-6526	309	9	iterative	iterative	NOUN
ejpam-6526	309	10	computation	computation	NOUN
ejpam-6526	309	11	starting	start	VERB
ejpam-6526	309	12	from	from	ADP
ejpam-6526	309	13	x0	x0	PROPN
ejpam-6526	309	14	=	=	PUNCT
ejpam-6526	309	15	0	0	NUM
ejpam-6526	309	16	x0	x0	PROPN
ejpam-6526	309	17	=	=	PUNCT
ejpam-6526	309	18	0	0	PUNCT
ejpam-6526	310	1	x1	x1	PROPN
ejpam-6526	310	2	=	=	SYM
ejpam-6526	310	3	t	t	PROPN
ejpam-6526	310	4	(	(	PUNCT
ejpam-6526	310	5	0	0	NUM
ejpam-6526	310	6	)	)	PUNCT
ejpam-6526	310	7	=	=	SYM
ejpam-6526	310	8	1	1	NUM
ejpam-6526	310	9	4	4	NUM
ejpam-6526	310	10	x2	x2	NOUN
ejpam-6526	310	11	=	=	SYM
ejpam-6526	310	12	t	t	PROPN
ejpam-6526	310	13	(	(	PUNCT
ejpam-6526	310	14	1	1	NUM
ejpam-6526	310	15	4	4	NUM
ejpam-6526	310	16	)	)	PUNCT
ejpam-6526	310	17	=	=	SYM
ejpam-6526	311	1	1/4	1/4	NUM
ejpam-6526	311	2	2	2	NUM
ejpam-6526	311	3	+	+	CCONJ
ejpam-6526	311	4	1	1	NUM
ejpam-6526	311	5	4	4	NUM
ejpam-6526	311	6	=	=	SYM
ejpam-6526	311	7	3	3	NUM
ejpam-6526	311	8	8	8	NUM
ejpam-6526	311	9	x3	x3	NOUN
ejpam-6526	311	10	=	=	SYM
ejpam-6526	311	11	t	t	PROPN
ejpam-6526	311	12	(	(	PUNCT
ejpam-6526	311	13	3	3	NUM
ejpam-6526	311	14	8	8	NUM
ejpam-6526	311	15	)	)	PUNCT
ejpam-6526	311	16	=	=	SYM
ejpam-6526	311	17	3/8	3/8	NUM
ejpam-6526	311	18	2	2	NUM
ejpam-6526	311	19	+	+	CCONJ
ejpam-6526	311	20	1	1	NUM
ejpam-6526	311	21	4	4	NUM
ejpam-6526	311	22	=	=	SYM
ejpam-6526	311	23	7	7	NUM
ejpam-6526	311	24	16	16	NUM
ejpam-6526	311	25	...	...	PUNCT
ejpam-6526	311	26	x∗	x∗	X
ejpam-6526	311	27	=	=	SYM
ejpam-6526	311	28	1	1	NUM
ejpam-6526	311	29	2	2	NUM
ejpam-6526	311	30	(	(	PUNCT
ejpam-6526	311	31	fixed	fix	VERB
ejpam-6526	311	32	point	point	NOUN
ejpam-6526	311	33	)	)	PUNCT
ejpam-6526	311	34	step	step	NOUN
ejpam-6526	311	35	4	4	NUM
ejpam-6526	311	36	:	:	PUNCT
ejpam-6526	311	37	verification	verification	NOUN
ejpam-6526	311	38	t	t	NOUN
ejpam-6526	311	39	(	(	PUNCT
ejpam-6526	311	40	1	1	NUM
ejpam-6526	311	41	2	2	NUM
ejpam-6526	311	42	)	)	PUNCT
ejpam-6526	311	43	=	=	SYM
ejpam-6526	311	44	1/2	1/2	NUM
ejpam-6526	311	45	2	2	NUM
ejpam-6526	311	46	+	+	CCONJ
ejpam-6526	311	47	1	1	NUM
ejpam-6526	311	48	4	4	NUM
ejpam-6526	311	49	=	=	SYM
ejpam-6526	311	50	1	1	NUM
ejpam-6526	311	51	2	2	NUM
ejpam-6526	311	52	âœ	âœ	NOUN
ejpam-6526	311	53	“	"	PUNCT
ejpam-6526	311	54	this	this	DET
ejpam-6526	311	55	example	example	NOUN
ejpam-6526	311	56	demonstrates	demonstrate	VERB
ejpam-6526	311	57	how	how	SCONJ
ejpam-6526	311	58	bi	bi	ADJ
ejpam-6526	311	59	-	-	ADJ
ejpam-6526	311	60	metric	metric	ADJ
ejpam-6526	311	61	contraction	contraction	NOUN
ejpam-6526	311	62	provides	provide	VERB
ejpam-6526	311	63	tighter	tight	ADJ
ejpam-6526	311	64	convergence	convergence	NOUN
ejpam-6526	311	65	bounds	bound	NOUN
ejpam-6526	311	66	when	when	SCONJ
ejpam-6526	311	67	λ1	λ1	ADJ
ejpam-6526	311	68	̸=	̸=	PROPN
ejpam-6526	311	69	λ2	λ2	NOUN
ejpam-6526	311	70	,	,	PUNCT
ejpam-6526	311	71	offering	offer	VERB
ejpam-6526	311	72	advantages	advantage	NOUN
ejpam-6526	311	73	over	over	ADP
ejpam-6526	311	74	classical	classical	ADJ
ejpam-6526	311	75	single	single	ADJ
ejpam-6526	311	76	-	-	PUNCT
ejpam-6526	311	77	metric	metric	ADJ
ejpam-6526	311	78	approaches	approach	NOUN
ejpam-6526	311	79	.	.	PUNCT
ejpam-6526	312	1	example	example	NOUN
ejpam-6526	312	2	13	13	NUM
ejpam-6526	312	3	(	(	PUNCT
ejpam-6526	312	4	function	function	NOUN
ejpam-6526	312	5	space	space	NOUN
ejpam-6526	312	6	with	with	ADP
ejpam-6526	312	7	dual	dual	ADJ
ejpam-6526	312	8	metrics	metric	NOUN
ejpam-6526	312	9	)	)	PUNCT
ejpam-6526	312	10	.	.	PUNCT
ejpam-6526	313	1	consider	consider	VERB
ejpam-6526	313	2	function	function	NOUN
ejpam-6526	313	3	space	space	NOUN
ejpam-6526	313	4	x	x	NOUN
ejpam-6526	313	5	=	=	SYM
ejpam-6526	313	6	c[0	c[0	PROPN
ejpam-6526	313	7	,	,	PUNCT
ejpam-6526	313	8	1	1	NUM
ejpam-6526	313	9	]	]	PUNCT
ejpam-6526	313	10	,	,	PUNCT
ejpam-6526	313	11	continuous	continuous	ADJ
ejpam-6526	313	12	functions	function	NOUN
ejpam-6526	313	13	on	on	ADP
ejpam-6526	313	14	[	[	X
ejpam-6526	313	15	0	0	NUM
ejpam-6526	313	16	,	,	PUNCT
ejpam-6526	313	17	1	1	NUM
ejpam-6526	313	18	]	]	PUNCT
ejpam-6526	313	19	,	,	PUNCT
ejpam-6526	313	20	with	with	ADP
ejpam-6526	313	21	:	:	PUNCT
ejpam-6526	313	22	ψ(f	ψ(f	NOUN
ejpam-6526	313	23	,	,	PUNCT
ejpam-6526	313	24	g	g	NOUN
ejpam-6526	313	25	)	)	PUNCT
ejpam-6526	313	26	=	=	SYM
ejpam-6526	314	1	(	(	PUNCT
ejpam-6526	314	2	max	max	PROPN
ejpam-6526	314	3	x∈[0,1	x∈[0,1	PROPN
ejpam-6526	314	4	]	]	X
ejpam-6526	315	1	|f(x)−	|f(x)−	PROPN
ejpam-6526	315	2	g(x)|	g(x)|	VERB
ejpam-6526	315	3	,	,	PUNCT
ejpam-6526	315	4	∫	∫	PROPN
ejpam-6526	315	5	1	1	NUM
ejpam-6526	315	6	0	0	NUM
ejpam-6526	315	7	|f(x)−	|f(x)−	PROPN
ejpam-6526	315	8	g(x)|2dx	g(x)|2dx	PROPN
ejpam-6526	315	9	)	)	PUNCT
ejpam-6526	315	10	and	and	CCONJ
ejpam-6526	315	11	⊕((a1	⊕((a1	ADV
ejpam-6526	315	12	,	,	PUNCT
ejpam-6526	315	13	a2	a2	PROPN
ejpam-6526	315	14	)	)	PUNCT
ejpam-6526	315	15	,	,	PUNCT
ejpam-6526	315	16	(	(	PUNCT
ejpam-6526	315	17	b1	b1	NOUN
ejpam-6526	315	18	,	,	PUNCT
ejpam-6526	315	19	b2	b2	NOUN
ejpam-6526	315	20	)	)	PUNCT
ejpam-6526	315	21	)	)	PUNCT
ejpam-6526	316	1	=	=	SYM
ejpam-6526	316	2	(	(	PUNCT
ejpam-6526	316	3	a1+b1	a1+b1	PROPN
ejpam-6526	316	4	,	,	PUNCT
ejpam-6526	316	5	a2+b2	a2+b2	PROPN
ejpam-6526	316	6	)	)	PUNCT
ejpam-6526	316	7	.	.	PUNCT
ejpam-6526	317	1	this	this	PRON
ejpam-6526	317	2	forms	form	VERB
ejpam-6526	317	3	a	a	DET
ejpam-6526	317	4	bi	bi	ADJ
ejpam-6526	317	5	-	-	ADJ
ejpam-6526	317	6	metric	metric	ADJ
ejpam-6526	317	7	system	system	NOUN
ejpam-6526	317	8	with	with	ADP
ejpam-6526	317	9	components	component	NOUN
ejpam-6526	317	10	representing	represent	VERB
ejpam-6526	317	11	uniform	uniform	NOUN
ejpam-6526	317	12	and	and	CCONJ
ejpam-6526	317	13	l2	l2	NOUN
ejpam-6526	317	14	metrics	metric	NOUN
ejpam-6526	317	15	,	,	PUNCT
ejpam-6526	317	16	respectively	respectively	ADV
ejpam-6526	317	17	.	.	PUNCT
ejpam-6526	318	1	this	this	DET
ejpam-6526	318	2	bi	bi	ADJ
ejpam-6526	318	3	-	-	ADJ
ejpam-6526	318	4	metric	metric	ADJ
ejpam-6526	318	5	system	system	NOUN
ejpam-6526	318	6	induces	induce	VERB
ejpam-6526	318	7	distinct	distinct	ADJ
ejpam-6526	318	8	topologies	topology	NOUN
ejpam-6526	318	9	:	:	PUNCT
ejpam-6526	318	10	uniform	uniform	ADJ
ejpam-6526	318	11	convergence	convergence	NOUN
ejpam-6526	318	12	topology	topology	NOUN
ejpam-6526	318	13	and	and	CCONJ
ejpam-6526	318	14	l2	l2	NOUN
ejpam-6526	318	15	convergence	convergence	NOUN
ejpam-6526	318	16	topology	topology	NOUN
ejpam-6526	318	17	.	.	PUNCT
ejpam-6526	319	1	a	a	DET
ejpam-6526	319	2	sequence	sequence	NOUN
ejpam-6526	319	3	might	might	AUX
ejpam-6526	319	4	converge	converge	VERB
ejpam-6526	319	5	in	in	ADP
ejpam-6526	319	6	one	one	NUM
ejpam-6526	319	7	topology	topology	NOUN
ejpam-6526	319	8	but	but	CCONJ
ejpam-6526	319	9	not	not	PART
ejpam-6526	319	10	the	the	DET
ejpam-6526	319	11	other	other	ADJ
ejpam-6526	319	12	.	.	PUNCT
ejpam-6526	320	1	for	for	ADP
ejpam-6526	320	2	example	example	NOUN
ejpam-6526	320	3	,	,	PUNCT
ejpam-6526	320	4	sequence	sequence	NOUN
ejpam-6526	320	5	fn(x	fn(x	PRON
ejpam-6526	320	6	)	)	PUNCT
ejpam-6526	321	1	=	=	SYM
ejpam-6526	321	2	xn	xn	PROPN
ejpam-6526	321	3	converges	converge	VERB
ejpam-6526	321	4	to	to	ADP
ejpam-6526	321	5	zero	zero	NUM
ejpam-6526	321	6	function	function	NOUN
ejpam-6526	321	7	in	in	ADP
ejpam-6526	321	8	l2	l2	NOUN
ejpam-6526	321	9	topology	topology	NOUN
ejpam-6526	321	10	but	but	CCONJ
ejpam-6526	321	11	not	not	PART
ejpam-6526	321	12	in	in	ADP
ejpam-6526	321	13	uniform	uniform	ADJ
ejpam-6526	321	14	topology	topology	NOUN
ejpam-6526	321	15	.	.	PUNCT
ejpam-6526	322	1	a.	a.	PROPN
ejpam-6526	322	2	alsoboh	alsoboh	PROPN
ejpam-6526	322	3	et	et	PROPN
ejpam-6526	322	4	al	al	PROPN
ejpam-6526	322	5	.	.	PUNCT
ejpam-6526	322	6	/	/	SYM
ejpam-6526	322	7	eur	eur	PROPN
ejpam-6526	322	8	.	.	PUNCT
ejpam-6526	323	1	j.	j.	PROPN
ejpam-6526	323	2	pure	pure	PROPN
ejpam-6526	323	3	appl	appl	PROPN
ejpam-6526	323	4	.	.	PROPN
ejpam-6526	323	5	math	math	PROPN
ejpam-6526	323	6	,	,	PUNCT
ejpam-6526	323	7	18	18	NUM
ejpam-6526	323	8	(	(	PUNCT
ejpam-6526	323	9	4	4	NUM
ejpam-6526	323	10	)	)	PUNCT
ejpam-6526	323	11	(	(	PUNCT
ejpam-6526	323	12	2025	2025	NUM
ejpam-6526	323	13	)	)	PUNCT
ejpam-6526	323	14	,	,	PUNCT
ejpam-6526	323	15	6526	6526	NUM
ejpam-6526	323	16	14	14	NUM
ejpam-6526	323	17	of	of	ADP
ejpam-6526	323	18	21	21	NUM
ejpam-6526	323	19	example	example	NOUN
ejpam-6526	323	20	14	14	NUM
ejpam-6526	323	21	(	(	PUNCT
ejpam-6526	323	22	machine	machine	NOUN
ejpam-6526	323	23	learning	learning	NOUN
ejpam-6526	323	24	application	application	NOUN
ejpam-6526	323	25	)	)	PUNCT
ejpam-6526	323	26	.	.	PUNCT
ejpam-6526	324	1	in	in	ADP
ejpam-6526	324	2	neural	neural	ADJ
ejpam-6526	324	3	network	network	NOUN
ejpam-6526	324	4	training	training	NOUN
ejpam-6526	324	5	,	,	PUNCT
ejpam-6526	324	6	consider	consider	VERB
ejpam-6526	324	7	the	the	DET
ejpam-6526	324	8	space	space	NOUN
ejpam-6526	324	9	x	x	PUNCT
ejpam-6526	324	10	of	of	ADP
ejpam-6526	324	11	network	network	NOUN
ejpam-6526	324	12	parameters	parameter	NOUN
ejpam-6526	324	13	with	with	ADP
ejpam-6526	324	14	:	:	PUNCT
ejpam-6526	324	15	•	•	NUM
ejpam-6526	324	16	ϕ1(θ1	ϕ1(θ1	X
ejpam-6526	324	17	,	,	PUNCT
ejpam-6526	324	18	θ2	θ2	PROPN
ejpam-6526	324	19	)	)	PUNCT
ejpam-6526	324	20	=	=	PUNCT
ejpam-6526	324	21	∥θ1	∥θ1	NUM
ejpam-6526	325	1	−	−	PROPN
ejpam-6526	325	2	θ2∥2	θ2∥2	PROPN
ejpam-6526	325	3	(	(	PUNCT
ejpam-6526	325	4	parameter	parameter	NOUN
ejpam-6526	325	5	distance	distance	NOUN
ejpam-6526	325	6	)	)	PUNCT
ejpam-6526	325	7	•	•	NOUN
ejpam-6526	326	1	ϕ2(θ1	ϕ2(θ1	INTJ
ejpam-6526	326	2	,	,	PUNCT
ejpam-6526	326	3	θ2	θ2	PROPN
ejpam-6526	326	4	)	)	PUNCT
ejpam-6526	326	5	=	=	SYM
ejpam-6526	326	6	|l(θ1)−	|l(θ1)−	ADJ
ejpam-6526	326	7	l(θ2)|	l(θ2)|	PROPN
ejpam-6526	326	8	(	(	PUNCT
ejpam-6526	326	9	loss	loss	NOUN
ejpam-6526	326	10	function	function	NOUN
ejpam-6526	326	11	difference	difference	NOUN
ejpam-6526	326	12	)	)	PUNCT
ejpam-6526	326	13	the	the	DET
ejpam-6526	326	14	bi	bi	ADJ
ejpam-6526	326	15	-	-	ADJ
ejpam-6526	326	16	metric	metric	ADJ
ejpam-6526	326	17	system	system	NOUN
ejpam-6526	326	18	enables	enable	VERB
ejpam-6526	326	19	tracking	track	VERB
ejpam-6526	326	20	both	both	PRON
ejpam-6526	326	21	parameter	parameter	NOUN
ejpam-6526	326	22	convergence	convergence	NOUN
ejpam-6526	326	23	and	and	CCONJ
ejpam-6526	326	24	loss	loss	NOUN
ejpam-6526	326	25	minimization	minimization	NOUN
ejpam-6526	326	26	simultaneously	simultaneously	ADV
ejpam-6526	326	27	,	,	PUNCT
ejpam-6526	326	28	providing	provide	VERB
ejpam-6526	326	29	better	well	ADJ
ejpam-6526	326	30	insights	insight	NOUN
ejpam-6526	326	31	into	into	ADP
ejpam-6526	326	32	training	training	NOUN
ejpam-6526	326	33	dynamics	dynamic	NOUN
ejpam-6526	326	34	.	.	PUNCT
ejpam-6526	327	1	our	our	PRON
ejpam-6526	327	2	bi	bi	ADJ
ejpam-6526	327	3	-	-	ADJ
ejpam-6526	327	4	metric	metric	ADJ
ejpam-6526	327	5	system	system	NOUN
ejpam-6526	327	6	exploration	exploration	NOUN
ejpam-6526	327	7	reveals	reveal	VERB
ejpam-6526	327	8	additional	additional	ADJ
ejpam-6526	327	9	structural	structural	ADJ
ejpam-6526	327	10	properties	property	NOUN
ejpam-6526	327	11	extending	extend	VERB
ejpam-6526	327	12	classical	classical	ADJ
ejpam-6526	327	13	metric	metric	ADJ
ejpam-6526	327	14	theory	theory	NOUN
ejpam-6526	327	15	,	,	PUNCT
ejpam-6526	327	16	building	build	VERB
ejpam-6526	327	17	upon	upon	SCONJ
ejpam-6526	327	18	pairwise	pairwise	NOUN
ejpam-6526	327	19	comparison	comparison	NOUN
ejpam-6526	327	20	spaces	space	NOUN
ejpam-6526	327	21	introduced	introduce	VERB
ejpam-6526	327	22	in	in	ADP
ejpam-6526	327	23	earlier	early	ADJ
ejpam-6526	327	24	research	research	NOUN
ejpam-6526	327	25	[	[	X
ejpam-6526	327	26	23	23	NUM
ejpam-6526	327	27	]	]	PUNCT
ejpam-6526	327	28	.	.	PUNCT
ejpam-6526	328	1	theorem	theorem	VERB
ejpam-6526	328	2	7	7	NUM
ejpam-6526	328	3	.	.	X
ejpam-6526	328	4	for	for	ADP
ejpam-6526	328	5	bi	bi	ADJ
ejpam-6526	328	6	-	-	ADJ
ejpam-6526	328	7	metric	metric	ADJ
ejpam-6526	328	8	system	system	NOUN
ejpam-6526	328	9	(	(	PUNCT
ejpam-6526	328	10	x	x	X
ejpam-6526	328	11	,	,	PUNCT
ejpam-6526	328	12	ψ,⊕	ψ,⊕	NOUN
ejpam-6526	328	13	)	)	PUNCT
ejpam-6526	328	14	with	with	ADP
ejpam-6526	328	15	ψ(x	ψ(x	PROPN
ejpam-6526	328	16	,	,	PUNCT
ejpam-6526	328	17	y	y	NOUN
ejpam-6526	328	18	)	)	PUNCT
ejpam-6526	328	19	=	=	SYM
ejpam-6526	329	1	(	(	PUNCT
ejpam-6526	329	2	ϕ1(x	ϕ1(x	PROPN
ejpam-6526	329	3	,	,	PUNCT
ejpam-6526	329	4	y	y	NOUN
ejpam-6526	329	5	)	)	PUNCT
ejpam-6526	329	6	,	,	PUNCT
ejpam-6526	330	1	ϕ2(x	ϕ2(x	PROPN
ejpam-6526	330	2	,	,	PUNCT
ejpam-6526	330	3	y	y	NOUN
ejpam-6526	330	4	)	)	PUNCT
ejpam-6526	330	5	)	)	PUNCT
ejpam-6526	330	6	,	,	PUNCT
ejpam-6526	330	7	there	there	PRON
ejpam-6526	330	8	exists	exist	VERB
ejpam-6526	330	9	bi	bi	ADJ
ejpam-6526	330	10	-	-	ADJ
ejpam-6526	330	11	complete	complete	ADJ
ejpam-6526	330	12	bi	bi	ADJ
ejpam-6526	330	13	-	-	ADJ
ejpam-6526	330	14	metric	metric	ADJ
ejpam-6526	330	15	system	system	NOUN
ejpam-6526	330	16	(	(	PUNCT
ejpam-6526	330	17	x̂	x̂	NUM
ejpam-6526	330	18	,	,	PUNCT
ejpam-6526	330	19	ψ̂	ψ̂	ADP
ejpam-6526	330	20	,	,	PUNCT
ejpam-6526	330	21	⊕̂	⊕̂	PROPN
ejpam-6526	330	22	)	)	PUNCT
ejpam-6526	330	23	and	and	CCONJ
ejpam-6526	330	24	isometric	isometric	ADJ
ejpam-6526	330	25	embedding	embed	VERB
ejpam-6526	330	26	ι	ι	X
ejpam-6526	330	27	:	:	PUNCT
ejpam-6526	330	28	x	x	X
ejpam-6526	330	29	→	→	SYM
ejpam-6526	330	30	x̂	x̂	NUM
ejpam-6526	330	31	with	with	ADP
ejpam-6526	330	32	ι(x	ι(x	NOUN
ejpam-6526	330	33	)	)	PUNCT
ejpam-6526	330	34	dense	dense	ADJ
ejpam-6526	330	35	in	in	ADP
ejpam-6526	330	36	x̂	x̂	NOUN
ejpam-6526	330	37	with	with	ADP
ejpam-6526	330	38	respect	respect	NOUN
ejpam-6526	330	39	to	to	ADP
ejpam-6526	330	40	both	both	DET
ejpam-6526	330	41	induced	induced	ADJ
ejpam-6526	330	42	topologies	topology	NOUN
ejpam-6526	330	43	.	.	PUNCT
ejpam-6526	331	1	proof	proof	NOUN
ejpam-6526	331	2	.	.	PUNCT
ejpam-6526	332	1	let	let	VERB
ejpam-6526	332	2	(	(	PUNCT
ejpam-6526	332	3	x̂1	x̂1	ADJ
ejpam-6526	332	4	,	,	PUNCT
ejpam-6526	332	5	d̂1	d̂1	PROPN
ejpam-6526	332	6	)	)	PUNCT
ejpam-6526	332	7	and	and	CCONJ
ejpam-6526	332	8	(	(	PUNCT
ejpam-6526	332	9	x̂2	x̂2	NOUN
ejpam-6526	332	10	,	,	PUNCT
ejpam-6526	332	11	d̂2	d̂2	NOUN
ejpam-6526	332	12	)	)	PUNCT
ejpam-6526	332	13	be	be	AUX
ejpam-6526	332	14	standard	standard	ADJ
ejpam-6526	332	15	metric	metric	ADJ
ejpam-6526	332	16	completions	completion	NOUN
ejpam-6526	332	17	of	of	ADP
ejpam-6526	332	18	(	(	PUNCT
ejpam-6526	332	19	x,ϕ1	x,ϕ1	PROPN
ejpam-6526	332	20	)	)	PUNCT
ejpam-6526	332	21	and	and	CCONJ
ejpam-6526	332	22	(	(	PUNCT
ejpam-6526	332	23	x,ϕ2	x,ϕ2	PROPN
ejpam-6526	332	24	)	)	PUNCT
ejpam-6526	332	25	,	,	PUNCT
ejpam-6526	332	26	respectively	respectively	ADV
ejpam-6526	332	27	.	.	PUNCT
ejpam-6526	333	1	for	for	ADP
ejpam-6526	333	2	each	each	DET
ejpam-6526	333	3	i	i	PRON
ejpam-6526	333	4	∈	∈	PROPN
ejpam-6526	333	5	{	{	PUNCT
ejpam-6526	333	6	1	1	NUM
ejpam-6526	333	7	,	,	PUNCT
ejpam-6526	333	8	2	2	NUM
ejpam-6526	333	9	}	}	PUNCT
ejpam-6526	333	10	,	,	PUNCT
ejpam-6526	333	11	we	we	PRON
ejpam-6526	333	12	have	have	VERB
ejpam-6526	333	13	isometric	isometric	ADJ
ejpam-6526	333	14	embedding	embed	VERB
ejpam-6526	333	15	ιi	ιi	NOUN
ejpam-6526	333	16	:	:	PUNCT
ejpam-6526	333	17	x	x	X
ejpam-6526	333	18	→	→	SYM
ejpam-6526	333	19	x̂i	x̂i	NUM
ejpam-6526	333	20	with	with	ADP
ejpam-6526	333	21	ιi(x	ιi(x	NUM
ejpam-6526	333	22	)	)	PUNCT
ejpam-6526	333	23	dense	dense	ADJ
ejpam-6526	333	24	in	in	ADP
ejpam-6526	333	25	x̂i	x̂i	NUM
ejpam-6526	333	26	.	.	PUNCT
ejpam-6526	334	1	define	define	VERB
ejpam-6526	334	2	set	set	NOUN
ejpam-6526	334	3	x̂	x̂	PUNCT
ejpam-6526	335	1	=	=	PRON
ejpam-6526	335	2	{	{	PUNCT
ejpam-6526	335	3	(	(	PUNCT
ejpam-6526	335	4	x1	x1	PROPN
ejpam-6526	335	5	,	,	PUNCT
ejpam-6526	335	6	x2	x2	ADJ
ejpam-6526	335	7	)	)	PUNCT
ejpam-6526	335	8	∈	∈	PROPN
ejpam-6526	335	9	x̂1×x̂2	x̂1×x̂2	NOUN
ejpam-6526	335	10	:	:	PUNCT
ejpam-6526	335	11	∃{xn	∃{xn	X
ejpam-6526	335	12	}	}	PUNCT
ejpam-6526	335	13	⊂	⊂	PROPN
ejpam-6526	335	14	x	x	PUNCT
ejpam-6526	335	15	where	where	SCONJ
ejpam-6526	335	16	ι1(xn	ι1(xn	PRON
ejpam-6526	335	17	)	)	PUNCT
ejpam-6526	335	18	→	→	SYM
ejpam-6526	335	19	x1	x1	PROPN
ejpam-6526	335	20	and	and	CCONJ
ejpam-6526	335	21	ι2(xn	ι2(xn	NUM
ejpam-6526	335	22	)	)	PUNCT
ejpam-6526	335	23	→	→	SYM
ejpam-6526	335	24	x2	x2	PROPN
ejpam-6526	335	25	}	}	PUNCT
ejpam-6526	335	26	.	.	PUNCT
ejpam-6526	336	1	define	define	VERB
ejpam-6526	336	2	ψ̂	ψ̂	PUNCT
ejpam-6526	336	3	:	:	PUNCT
ejpam-6526	336	4	x̂	x̂	NUM
ejpam-6526	336	5	×	×	NOUN
ejpam-6526	336	6	x̂	x̂	PUNCT
ejpam-6526	336	7	→	→	PUNCT
ejpam-6526	336	8	r+	r+	PUNCT
ejpam-6526	336	9	×	×	NOUN
ejpam-6526	336	10	r+	r+	NOUN
ejpam-6526	336	11	by	by	ADP
ejpam-6526	336	12	ψ̂((x1	ψ̂((x1	PROPN
ejpam-6526	336	13	,	,	PUNCT
ejpam-6526	336	14	x2	x2	PROPN
ejpam-6526	336	15	)	)	PUNCT
ejpam-6526	336	16	,	,	PUNCT
ejpam-6526	336	17	(	(	PUNCT
ejpam-6526	336	18	y1	y1	INTJ
ejpam-6526	336	19	,	,	PUNCT
ejpam-6526	336	20	y2	y2	PROPN
ejpam-6526	336	21	)	)	PUNCT
ejpam-6526	336	22	)	)	PUNCT
ejpam-6526	337	1	=	=	PRON
ejpam-6526	337	2	(	(	PUNCT
ejpam-6526	337	3	ϕ̂1(x1	ϕ̂1(x1	NOUN
ejpam-6526	337	4	,	,	PUNCT
ejpam-6526	337	5	y1	y1	PROPN
ejpam-6526	337	6	)	)	PUNCT
ejpam-6526	337	7	,	,	PUNCT
ejpam-6526	337	8	ϕ̂2(x2	ϕ̂2(x2	NOUN
ejpam-6526	337	9	,	,	PUNCT
ejpam-6526	337	10	y2	y2	PROPN
ejpam-6526	337	11	)	)	PUNCT
ejpam-6526	337	12	)	)	PUNCT
ejpam-6526	337	13	.	.	PUNCT
ejpam-6526	338	1	define	define	VERB
ejpam-6526	338	2	⊕̂	⊕̂	PROPN
ejpam-6526	338	3	:	:	PUNCT
ejpam-6526	338	4	(	(	PUNCT
ejpam-6526	338	5	r+	r+	NOUN
ejpam-6526	338	6	×	×	NOUN
ejpam-6526	338	7	r+)×	r+)×	X
ejpam-6526	338	8	(	(	PUNCT
ejpam-6526	338	9	r+	r+	NOUN
ejpam-6526	338	10	×	×	NOUN
ejpam-6526	338	11	r+	r+	PUNCT
ejpam-6526	338	12	)	)	PUNCT
ejpam-6526	338	13	→	→	PUNCT
ejpam-6526	338	14	r+	r+	PUNCT
ejpam-6526	338	15	×	×	NOUN
ejpam-6526	338	16	r+	r+	NOUN
ejpam-6526	338	17	identically	identically	ADV
ejpam-6526	338	18	to	to	PART
ejpam-6526	338	19	⊕.	⊕.	ADV
ejpam-6526	338	20	define	define	VERB
ejpam-6526	338	21	ι	ι	X
ejpam-6526	338	22	:	:	PUNCT
ejpam-6526	338	23	x	x	X
ejpam-6526	338	24	→	→	SYM
ejpam-6526	338	25	x̂	x̂	NUM
ejpam-6526	338	26	by	by	ADP
ejpam-6526	338	27	ι(x	ι(x	PROPN
ejpam-6526	338	28	)	)	PUNCT
ejpam-6526	338	29	=	=	SYM
ejpam-6526	338	30	(	(	PUNCT
ejpam-6526	338	31	ι1(x	ι1(x	NOUN
ejpam-6526	338	32	)	)	PUNCT
ejpam-6526	338	33	,	,	PUNCT
ejpam-6526	338	34	ι2(x	ι2(x	NOUN
ejpam-6526	338	35	)	)	PUNCT
ejpam-6526	338	36	)	)	PUNCT
ejpam-6526	338	37	.	.	PUNCT
ejpam-6526	339	1	we	we	PRON
ejpam-6526	339	2	verify	verify	VERB
ejpam-6526	339	3	several	several	ADJ
ejpam-6526	339	4	key	key	ADJ
ejpam-6526	339	5	properties	property	NOUN
ejpam-6526	339	6	:	:	PUNCT
ejpam-6526	339	7	(	(	PUNCT
ejpam-6526	339	8	i	i	NOUN
ejpam-6526	339	9	)	)	PUNCT
ejpam-6526	339	10	x̂	x̂	NUM
ejpam-6526	339	11	is	be	AUX
ejpam-6526	339	12	non	non	ADJ
ejpam-6526	339	13	-	-	ADJ
ejpam-6526	339	14	empty	empty	ADJ
ejpam-6526	339	15	:	:	PUNCT
ejpam-6526	339	16	for	for	ADP
ejpam-6526	339	17	any	any	DET
ejpam-6526	339	18	x	x	SYM
ejpam-6526	339	19	∈	∈	PROPN
ejpam-6526	339	20	x	x	NOUN
ejpam-6526	339	21	,	,	PUNCT
ejpam-6526	339	22	constant	constant	ADJ
ejpam-6526	339	23	sequence	sequence	NOUN
ejpam-6526	339	24	{	{	PUNCT
ejpam-6526	339	25	x	x	NOUN
ejpam-6526	339	26	}	}	PUNCT
ejpam-6526	339	27	ensures	ensure	NOUN
ejpam-6526	339	28	(	(	PUNCT
ejpam-6526	339	29	ι1(x	ι1(x	NOUN
ejpam-6526	339	30	)	)	PUNCT
ejpam-6526	339	31	,	,	PUNCT
ejpam-6526	339	32	ι2(x	ι2(x	NOUN
ejpam-6526	339	33	)	)	PUNCT
ejpam-6526	339	34	)	)	PUNCT
ejpam-6526	340	1	∈	∈	PROPN
ejpam-6526	340	2	x̂.	x̂.	NOUN
ejpam-6526	340	3	(	(	PUNCT
ejpam-6526	340	4	ii	ii	NOUN
ejpam-6526	340	5	)	)	PUNCT
ejpam-6526	340	6	(	(	PUNCT
ejpam-6526	340	7	x	x	NOUN
ejpam-6526	340	8	,	,	PUNCT
ejpam-6526	340	9	ψ̂	ψ̂	PROPN
ejpam-6526	340	10	,	,	PUNCT
ejpam-6526	340	11	⊕̂	⊕̂	NOUN
ejpam-6526	340	12	)	)	PUNCT
ejpam-6526	340	13	forms	form	VERB
ejpam-6526	340	14	bi	bi	ADJ
ejpam-6526	340	15	-	-	ADJ
ejpam-6526	340	16	metric	metric	ADJ
ejpam-6526	340	17	system	system	NOUN
ejpam-6526	340	18	:	:	PUNCT
ejpam-6526	340	19	this	this	PRON
ejpam-6526	340	20	follows	follow	VERB
ejpam-6526	340	21	from	from	ADP
ejpam-6526	340	22	properties	property	NOUN
ejpam-6526	340	23	of	of	ADP
ejpam-6526	340	24	ϕ̂1	ϕ̂1	NOUN
ejpam-6526	340	25	and	and	CCONJ
ejpam-6526	340	26	ϕ̂2	ϕ̂2	PROPN
ejpam-6526	340	27	.	.	PUNCT
ejpam-6526	341	1	(	(	PUNCT
ejpam-6526	341	2	iii	iii	X
ejpam-6526	341	3	)	)	PUNCT
ejpam-6526	341	4	ι	ι	PROPN
ejpam-6526	341	5	constitutes	constitute	VERB
ejpam-6526	341	6	isometric	isometric	ADJ
ejpam-6526	341	7	embedding	embed	VERB
ejpam-6526	341	8	:	:	PUNCT
ejpam-6526	341	9	for	for	ADP
ejpam-6526	341	10	any	any	DET
ejpam-6526	341	11	x	x	NOUN
ejpam-6526	341	12	,	,	PUNCT
ejpam-6526	341	13	y	y	PROPN
ejpam-6526	341	14	∈	∈	PROPN
ejpam-6526	341	15	x	x	PROPN
ejpam-6526	341	16	,	,	PUNCT
ejpam-6526	341	17	ψ̂(ι(x	ψ̂(ι(x	NOUN
ejpam-6526	341	18	)	)	PUNCT
ejpam-6526	341	19	,	,	PUNCT
ejpam-6526	341	20	ι(y	ι(y	PROPN
ejpam-6526	341	21	)	)	PUNCT
ejpam-6526	341	22	)	)	PUNCT
ejpam-6526	342	1	=	=	PUNCT
ejpam-6526	342	2	ψ̂((ι1(x	ψ̂((ι1(x	NOUN
ejpam-6526	342	3	)	)	PUNCT
ejpam-6526	342	4	,	,	PUNCT
ejpam-6526	342	5	ι2(x	ι2(x	NOUN
ejpam-6526	342	6	)	)	PUNCT
ejpam-6526	342	7	)	)	PUNCT
ejpam-6526	342	8	,	,	PUNCT
ejpam-6526	342	9	(	(	PUNCT
ejpam-6526	342	10	ι1(y	ι1(y	PROPN
ejpam-6526	342	11	)	)	PUNCT
ejpam-6526	342	12	,	,	PUNCT
ejpam-6526	342	13	ι2(y	ι2(y	NOUN
ejpam-6526	342	14	)	)	PUNCT
ejpam-6526	342	15	)	)	PUNCT
ejpam-6526	342	16	)	)	PUNCT
ejpam-6526	343	1	=	=	PRON
ejpam-6526	343	2	(	(	PUNCT
ejpam-6526	343	3	ϕ̂1(ι1(x	ϕ̂1(ι1(x	PROPN
ejpam-6526	343	4	)	)	PUNCT
ejpam-6526	343	5	,	,	PUNCT
ejpam-6526	343	6	ι1(y	ι1(y	PROPN
ejpam-6526	343	7	)	)	PUNCT
ejpam-6526	343	8	)	)	PUNCT
ejpam-6526	343	9	,	,	PUNCT
ejpam-6526	343	10	ϕ̂2(ι2(x	ϕ̂2(ι2(x	PROPN
ejpam-6526	343	11	)	)	PUNCT
ejpam-6526	343	12	,	,	PUNCT
ejpam-6526	343	13	ι2(y	ι2(y	NOUN
ejpam-6526	343	14	)	)	PUNCT
ejpam-6526	343	15	)	)	PUNCT
ejpam-6526	343	16	)	)	PUNCT
ejpam-6526	344	1	=	=	PRON
ejpam-6526	344	2	(	(	PUNCT
ejpam-6526	344	3	ϕ1(x	ϕ1(x	PROPN
ejpam-6526	344	4	,	,	PUNCT
ejpam-6526	344	5	y	y	NOUN
ejpam-6526	344	6	)	)	PUNCT
ejpam-6526	344	7	,	,	PUNCT
ejpam-6526	344	8	ϕ2(x	ϕ2(x	PROPN
ejpam-6526	344	9	,	,	PUNCT
ejpam-6526	344	10	y	y	NOUN
ejpam-6526	344	11	)	)	PUNCT
ejpam-6526	344	12	)	)	PUNCT
ejpam-6526	345	1	=	=	SYM
ejpam-6526	345	2	ψ(x	ψ(x	PROPN
ejpam-6526	345	3	,	,	PUNCT
ejpam-6526	345	4	y	y	PROPN
ejpam-6526	345	5	)	)	PUNCT
ejpam-6526	345	6	(	(	PUNCT
ejpam-6526	345	7	iv	iv	X
ejpam-6526	345	8	)	)	PUNCT
ejpam-6526	345	9	ι(x	ι(x	PUNCT
ejpam-6526	345	10	)	)	PUNCT
ejpam-6526	345	11	is	be	AUX
ejpam-6526	345	12	dense	dense	ADJ
ejpam-6526	345	13	in	in	ADP
ejpam-6526	345	14	x̂	x̂	NOUN
ejpam-6526	345	15	with	with	ADP
ejpam-6526	345	16	respect	respect	NOUN
ejpam-6526	345	17	to	to	ADP
ejpam-6526	345	18	both	both	DET
ejpam-6526	345	19	induced	induced	ADJ
ejpam-6526	345	20	topologies	topology	NOUN
ejpam-6526	345	21	:	:	PUNCT
ejpam-6526	345	22	for	for	ADP
ejpam-6526	345	23	any	any	DET
ejpam-6526	345	24	(	(	PUNCT
ejpam-6526	345	25	x1	x1	PROPN
ejpam-6526	345	26	,	,	PUNCT
ejpam-6526	345	27	x2	x2	ADJ
ejpam-6526	345	28	)	)	PUNCT
ejpam-6526	345	29	∈	∈	PROPN
ejpam-6526	345	30	x̂	x̂	PUNCT
ejpam-6526	345	31	and	and	CCONJ
ejpam-6526	345	32	ε	ε	PROPN
ejpam-6526	345	33	>	>	X
ejpam-6526	345	34	0	0	PROPN
ejpam-6526	345	35	,	,	PUNCT
ejpam-6526	345	36	by	by	ADP
ejpam-6526	345	37	definition	definition	NOUN
ejpam-6526	345	38	there	there	PRON
ejpam-6526	345	39	exists	exist	VERB
ejpam-6526	345	40	sequence	sequence	NOUN
ejpam-6526	345	41	{	{	PUNCT
ejpam-6526	345	42	xn	xn	PROPN
ejpam-6526	345	43	}	}	PUNCT
ejpam-6526	345	44	⊂	⊂	X
ejpam-6526	345	45	x	x	PUNCT
ejpam-6526	345	46	with	with	ADP
ejpam-6526	345	47	ι1(xn	ι1(xn	NUM
ejpam-6526	345	48	)	)	PUNCT
ejpam-6526	345	49	→	→	SYM
ejpam-6526	345	50	x1	x1	PROPN
ejpam-6526	345	51	and	and	CCONJ
ejpam-6526	345	52	ι2(xn	ι2(xn	NUM
ejpam-6526	345	53	)	)	PUNCT
ejpam-6526	345	54	→	→	SYM
ejpam-6526	345	55	x2	x2	PROPN
ejpam-6526	345	56	.	.	PUNCT
ejpam-6526	346	1	thus	thus	ADV
ejpam-6526	346	2	,	,	PUNCT
ejpam-6526	346	3	for	for	ADP
ejpam-6526	346	4	sufficiently	sufficiently	ADV
ejpam-6526	346	5	large	large	ADJ
ejpam-6526	346	6	n	n	CCONJ
ejpam-6526	346	7	,	,	PUNCT
ejpam-6526	346	8	ϕ̂1(ι1(xn	ϕ̂1(ι1(xn	ADJ
ejpam-6526	346	9	)	)	PUNCT
ejpam-6526	346	10	,	,	PUNCT
ejpam-6526	346	11	x1	x1	NUM
ejpam-6526	346	12	)	)	PUNCT
ejpam-6526	346	13	<	<	X
ejpam-6526	346	14	ε	ε	PROPN
ejpam-6526	346	15	and	and	CCONJ
ejpam-6526	346	16	ϕ̂2(ι2(xn	ϕ̂2(ι2(xn	NOUN
ejpam-6526	346	17	)	)	PUNCT
ejpam-6526	346	18	,	,	PUNCT
ejpam-6526	346	19	x2	x2	PROPN
ejpam-6526	346	20	)	)	PUNCT
ejpam-6526	346	21	<	<	X
ejpam-6526	346	22	ε	ε	PROPN
ejpam-6526	346	23	,	,	PUNCT
ejpam-6526	346	24	meaning	mean	VERB
ejpam-6526	346	25	ι(xn	ι(xn	NOUN
ejpam-6526	346	26	)	)	PUNCT
ejpam-6526	346	27	lies	lie	VERB
ejpam-6526	346	28	within	within	ADP
ejpam-6526	346	29	ε	ε	PROPN
ejpam-6526	346	30	of	of	ADP
ejpam-6526	346	31	(	(	PUNCT
ejpam-6526	346	32	x1	x1	PROPN
ejpam-6526	346	33	,	,	PUNCT
ejpam-6526	346	34	x2	x2	PROPN
ejpam-6526	346	35	)	)	PUNCT
ejpam-6526	346	36	in	in	ADP
ejpam-6526	346	37	both	both	DET
ejpam-6526	346	38	metrics	metric	NOUN
ejpam-6526	346	39	.	.	PUNCT
ejpam-6526	347	1	(	(	PUNCT
ejpam-6526	347	2	v	v	NOUN
ejpam-6526	347	3	)	)	PUNCT
ejpam-6526	347	4	(	(	PUNCT
ejpam-6526	347	5	x	x	NOUN
ejpam-6526	347	6	,	,	PUNCT
ejpam-6526	347	7	ψ̂	ψ̂	PROPN
ejpam-6526	347	8	,	,	PUNCT
ejpam-6526	347	9	⊕̂	⊕̂	PROPN
ejpam-6526	347	10	)	)	PUNCT
ejpam-6526	347	11	is	be	AUX
ejpam-6526	347	12	bi	bi	NOUN
ejpam-6526	347	13	-	-	ADJ
ejpam-6526	347	14	complete	complete	ADJ
ejpam-6526	347	15	:	:	PUNCT
ejpam-6526	347	16	consider	consider	VERB
ejpam-6526	347	17	bi	bi	ADJ
ejpam-6526	347	18	-	-	ADJ
ejpam-6526	347	19	cauchy	cauchy	ADJ
ejpam-6526	347	20	sequence	sequence	NOUN
ejpam-6526	347	21	{	{	PUNCT
ejpam-6526	347	22	(	(	PUNCT
ejpam-6526	347	23	x1n	x1n	ADP
ejpam-6526	347	24	,	,	PUNCT
ejpam-6526	347	25	x2n	x2n	NOUN
ejpam-6526	347	26	)	)	PUNCT
ejpam-6526	347	27	}	}	PUNCT
ejpam-6526	347	28	in	in	ADP
ejpam-6526	347	29	x̂.	x̂.	NOUN
ejpam-6526	347	30	then	then	ADV
ejpam-6526	347	31	{	{	PUNCT
ejpam-6526	347	32	x1n	x1n	NOUN
ejpam-6526	347	33	}	}	PUNCT
ejpam-6526	347	34	is	be	AUX
ejpam-6526	347	35	cauchy	cauchy	ADJ
ejpam-6526	347	36	in	in	ADP
ejpam-6526	347	37	(	(	PUNCT
ejpam-6526	347	38	x̂1	x̂1	PROPN
ejpam-6526	347	39	,	,	PUNCT
ejpam-6526	347	40	ϕ̂1	ϕ̂1	PROPN
ejpam-6526	347	41	)	)	PUNCT
ejpam-6526	347	42	and	and	CCONJ
ejpam-6526	347	43	{	{	PUNCT
ejpam-6526	347	44	x2n	x2n	NOUN
ejpam-6526	347	45	}	}	PUNCT
ejpam-6526	347	46	is	be	AUX
ejpam-6526	347	47	cauchy	cauchy	ADJ
ejpam-6526	347	48	in	in	ADP
ejpam-6526	347	49	(	(	PUNCT
ejpam-6526	347	50	x̂2	x̂2	NOUN
ejpam-6526	347	51	,	,	PUNCT
ejpam-6526	347	52	ϕ̂2	ϕ̂2	PROPN
ejpam-6526	347	53	)	)	PUNCT
ejpam-6526	347	54	.	.	PUNCT
ejpam-6526	348	1	by	by	ADP
ejpam-6526	348	2	completeness	completeness	NOUN
ejpam-6526	348	3	,	,	PUNCT
ejpam-6526	348	4	x1n	x1n	PUNCT
ejpam-6526	348	5	→	→	PUNCT
ejpam-6526	348	6	x1	x1	PROPN
ejpam-6526	348	7	∈	∈	PROPN
ejpam-6526	348	8	x̂1	x̂1	NOUN
ejpam-6526	348	9	and	and	CCONJ
ejpam-6526	348	10	x2n	x2n	PROPN
ejpam-6526	348	11	→	→	SYM
ejpam-6526	348	12	x2	x2	PROPN
ejpam-6526	348	13	∈	∈	PROPN
ejpam-6526	348	14	x̂2	x̂2	NOUN
ejpam-6526	348	15	.	.	PUNCT
ejpam-6526	349	1	we	we	PRON
ejpam-6526	349	2	must	must	AUX
ejpam-6526	349	3	show	show	VERB
ejpam-6526	349	4	that	that	SCONJ
ejpam-6526	349	5	(	(	PUNCT
ejpam-6526	349	6	x1	x1	ADJ
ejpam-6526	349	7	,	,	PUNCT
ejpam-6526	349	8	x2	x2	PROPN
ejpam-6526	349	9	)	)	PUNCT
ejpam-6526	349	10	∈	∈	PROPN
ejpam-6526	349	11	x̂.	x̂.	NOUN
ejpam-6526	349	12	a.	a.	PROPN
ejpam-6526	349	13	alsoboh	alsoboh	PROPN
ejpam-6526	349	14	et	et	PROPN
ejpam-6526	349	15	al	al	PROPN
ejpam-6526	349	16	.	.	PUNCT
ejpam-6526	349	17	/	/	SYM
ejpam-6526	349	18	eur	eur	PROPN
ejpam-6526	349	19	.	.	PUNCT
ejpam-6526	350	1	j.	j.	PROPN
ejpam-6526	350	2	pure	pure	PROPN
ejpam-6526	350	3	appl	appl	PROPN
ejpam-6526	350	4	.	.	PROPN
ejpam-6526	350	5	math	math	PROPN
ejpam-6526	350	6	,	,	PUNCT
ejpam-6526	350	7	18	18	NUM
ejpam-6526	350	8	(	(	PUNCT
ejpam-6526	350	9	4	4	NUM
ejpam-6526	350	10	)	)	PUNCT
ejpam-6526	350	11	(	(	PUNCT
ejpam-6526	350	12	2025	2025	NUM
ejpam-6526	350	13	)	)	PUNCT
ejpam-6526	350	14	,	,	PUNCT
ejpam-6526	350	15	6526	6526	NUM
ejpam-6526	350	16	15	15	NUM
ejpam-6526	350	17	of	of	ADP
ejpam-6526	350	18	21	21	NUM
ejpam-6526	350	19	for	for	ADP
ejpam-6526	350	20	each	each	DET
ejpam-6526	350	21	n	n	CCONJ
ejpam-6526	350	22	,	,	PUNCT
ejpam-6526	350	23	there	there	PRON
ejpam-6526	350	24	exists	exist	VERB
ejpam-6526	350	25	sequence	sequence	NOUN
ejpam-6526	350	26	{	{	PUNCT
ejpam-6526	350	27	yn	yn	PROPN
ejpam-6526	350	28	,	,	PUNCT
ejpam-6526	350	29	m	m	VERB
ejpam-6526	350	30	}	}	PUNCT
ejpam-6526	350	31	⊂	⊂	PROPN
ejpam-6526	350	32	x	x	PUNCT
ejpam-6526	350	33	with	with	ADP
ejpam-6526	350	34	ι1(yn	ι1(yn	PROPN
ejpam-6526	350	35	,	,	PUNCT
ejpam-6526	350	36	m	m	NOUN
ejpam-6526	350	37	)	)	PUNCT
ejpam-6526	350	38	→	→	PUNCT
ejpam-6526	350	39	x1n	x1n	PROPN
ejpam-6526	350	40	and	and	CCONJ
ejpam-6526	350	41	ι2(yn	ι2(yn	PROPN
ejpam-6526	350	42	,	,	PUNCT
ejpam-6526	350	43	m	m	NOUN
ejpam-6526	350	44	)	)	PUNCT
ejpam-6526	350	45	→	→	PUNCT
ejpam-6526	351	1	x2n	x2n	PROPN
ejpam-6526	351	2	as	as	SCONJ
ejpam-6526	351	3	m	m	PROPN
ejpam-6526	351	4	→	→	SYM
ejpam-6526	351	5	∞.	∞.	PROPN
ejpam-6526	351	6	using	use	VERB
ejpam-6526	351	7	diagonal	diagonal	ADJ
ejpam-6526	351	8	argument	argument	NOUN
ejpam-6526	351	9	,	,	PUNCT
ejpam-6526	351	10	we	we	PRON
ejpam-6526	351	11	can	can	AUX
ejpam-6526	351	12	construct	construct	VERB
ejpam-6526	351	13	sequence	sequence	NOUN
ejpam-6526	351	14	{	{	PUNCT
ejpam-6526	351	15	zk	zk	PROPN
ejpam-6526	351	16	}	}	PUNCT
ejpam-6526	351	17	⊂	⊂	PROPN
ejpam-6526	351	18	x	x	PUNCT
ejpam-6526	351	19	with	with	ADP
ejpam-6526	351	20	ι1(zk	ι1(zk	PROPN
ejpam-6526	351	21	)	)	PUNCT
ejpam-6526	351	22	→	→	SYM
ejpam-6526	351	23	x1	x1	PROPN
ejpam-6526	351	24	and	and	CCONJ
ejpam-6526	351	25	ι2(zk	ι2(zk	PROPN
ejpam-6526	351	26	)	)	PUNCT
ejpam-6526	351	27	→	→	SYM
ejpam-6526	351	28	x2	x2	PROPN
ejpam-6526	351	29	,	,	PUNCT
ejpam-6526	351	30	establishing	establish	VERB
ejpam-6526	351	31	(	(	PUNCT
ejpam-6526	351	32	x1	x1	PROPN
ejpam-6526	351	33	,	,	PUNCT
ejpam-6526	351	34	x2	x2	PROPN
ejpam-6526	351	35	)	)	PUNCT
ejpam-6526	351	36	∈	∈	PROPN
ejpam-6526	351	37	x̂.	x̂.	NOUN
ejpam-6526	351	38	therefore	therefore	ADV
ejpam-6526	351	39	,	,	PUNCT
ejpam-6526	351	40	(	(	PUNCT
ejpam-6526	351	41	x̂	x̂	NUM
ejpam-6526	351	42	,	,	PUNCT
ejpam-6526	351	43	ψ̂	ψ̂	PROPN
ejpam-6526	351	44	,	,	PUNCT
ejpam-6526	351	45	⊕̂	⊕̂	NOUN
ejpam-6526	351	46	)	)	PUNCT
ejpam-6526	351	47	forms	form	VERB
ejpam-6526	351	48	bi	bi	ADJ
ejpam-6526	351	49	-	-	ADJ
ejpam-6526	351	50	complete	complete	ADJ
ejpam-6526	351	51	bi	bi	ADJ
ejpam-6526	351	52	-	-	ADJ
ejpam-6526	351	53	metric	metric	ADJ
ejpam-6526	351	54	system	system	NOUN
ejpam-6526	351	55	containing	contain	VERB
ejpam-6526	351	56	isometric	isometric	ADJ
ejpam-6526	351	57	copy	copy	NOUN
ejpam-6526	351	58	of	of	ADP
ejpam-6526	351	59	(	(	PUNCT
ejpam-6526	351	60	x	x	X
ejpam-6526	351	61	,	,	PUNCT
ejpam-6526	351	62	ψ,⊕	ψ,⊕	NOUN
ejpam-6526	351	63	)	)	PUNCT
ejpam-6526	351	64	.	.	PUNCT
ejpam-6526	352	1	6	6	X
ejpam-6526	352	2	.	.	PUNCT
ejpam-6526	352	3	structural	structural	ADJ
ejpam-6526	352	4	properties	property	NOUN
ejpam-6526	352	5	and	and	CCONJ
ejpam-6526	352	6	applications	application	NOUN
ejpam-6526	352	7	6.1	6.1	NUM
ejpam-6526	352	8	.	.	PUNCT
ejpam-6526	352	9	structural	structural	ADJ
ejpam-6526	352	10	density	density	NOUN
ejpam-6526	352	11	in	in	ADP
ejpam-6526	352	12	bi	bi	ADJ
ejpam-6526	352	13	-	-	ADJ
ejpam-6526	352	14	metric	metric	ADJ
ejpam-6526	352	15	systems	system	NOUN
ejpam-6526	352	16	definition	definition	NOUN
ejpam-6526	352	17	9	9	NUM
ejpam-6526	352	18	.	.	PUNCT
ejpam-6526	353	1	we	we	PRON
ejpam-6526	353	2	characterize	characterize	VERB
ejpam-6526	353	3	bi	bi	ADJ
ejpam-6526	353	4	-	-	ADJ
ejpam-6526	353	5	metric	metric	ADJ
ejpam-6526	353	6	system	system	NOUN
ejpam-6526	353	7	(	(	PUNCT
ejpam-6526	353	8	x	x	X
ejpam-6526	353	9	,	,	PUNCT
ejpam-6526	353	10	ψ,⊕	ψ,⊕	NOUN
ejpam-6526	353	11	)	)	PUNCT
ejpam-6526	353	12	as	as	ADP
ejpam-6526	353	13	bi	bi	NOUN
ejpam-6526	353	14	-	-	ADJ
ejpam-6526	353	15	separable	separable	ADJ
ejpam-6526	353	16	when	when	SCONJ
ejpam-6526	353	17	there	there	PRON
ejpam-6526	353	18	exists	exist	VERB
ejpam-6526	353	19	countable	countable	ADJ
ejpam-6526	353	20	subset	subset	NOUN
ejpam-6526	353	21	d	d	X
ejpam-6526	353	22	⊂	⊂	PROPN
ejpam-6526	353	23	x	x	PUNCT
ejpam-6526	353	24	that	that	PRON
ejpam-6526	353	25	is	be	AUX
ejpam-6526	353	26	dense	dense	ADJ
ejpam-6526	353	27	in	in	ADP
ejpam-6526	353	28	x	x	PUNCT
ejpam-6526	353	29	with	with	ADP
ejpam-6526	353	30	respect	respect	NOUN
ejpam-6526	353	31	to	to	ADP
ejpam-6526	353	32	both	both	DET
ejpam-6526	353	33	induced	induced	ADJ
ejpam-6526	353	34	topologies	topology	NOUN
ejpam-6526	353	35	.	.	PUNCT
ejpam-6526	354	1	example	example	NOUN
ejpam-6526	354	2	15	15	NUM
ejpam-6526	354	3	(	(	PUNCT
ejpam-6526	354	4	bi	bi	ADJ
ejpam-6526	354	5	-	-	ADJ
ejpam-6526	354	6	separable	separable	ADJ
ejpam-6526	354	7	system	system	NOUN
ejpam-6526	354	8	)	)	PUNCT
ejpam-6526	354	9	.	.	PUNCT
ejpam-6526	355	1	consider	consider	VERB
ejpam-6526	355	2	x	x	NOUN
ejpam-6526	355	3	=	=	SYM
ejpam-6526	355	4	r2	r2	PROPN
ejpam-6526	355	5	with	with	ADP
ejpam-6526	355	6	ψ(x	ψ(x	PROPN
ejpam-6526	355	7	,	,	PUNCT
ejpam-6526	355	8	y	y	NOUN
ejpam-6526	355	9	)	)	PUNCT
ejpam-6526	355	10	=	=	SYM
ejpam-6526	355	11	(	(	PUNCT
ejpam-6526	355	12	∥x−y∥2	∥x−y∥2	PROPN
ejpam-6526	355	13	,	,	PUNCT
ejpam-6526	355	14	∥x−y∥∞	∥x−y∥∞	PROPN
ejpam-6526	355	15	)	)	PUNCT
ejpam-6526	356	1	where	where	SCONJ
ejpam-6526	356	2	:	:	PUNCT
ejpam-6526	356	3	•	•	NUM
ejpam-6526	356	4	∥x−	∥x−	NUM
ejpam-6526	356	5	y∥2	y∥2	NOUN
ejpam-6526	356	6	=	=	PUNCT
ejpam-6526	356	7	√	√	INTJ
ejpam-6526	356	8	(	(	PUNCT
ejpam-6526	356	9	x1	x1	PROPN
ejpam-6526	356	10	−	−	PROPN
ejpam-6526	356	11	y1)2	y1)2	PROPN
ejpam-6526	357	1	+	+	CCONJ
ejpam-6526	357	2	(	(	PUNCT
ejpam-6526	357	3	x2	x2	INTJ
ejpam-6526	357	4	−	−	PROPN
ejpam-6526	357	5	y2)2	y2)2	PRON
ejpam-6526	357	6	(	(	PUNCT
ejpam-6526	357	7	euclidean	euclidean	ADJ
ejpam-6526	357	8	norm	norm	NOUN
ejpam-6526	357	9	)	)	PUNCT
ejpam-6526	357	10	•	•	NOUN
ejpam-6526	358	1	∥x−	∥x−	PROPN
ejpam-6526	358	2	y∥∞	y∥∞	NOUN
ejpam-6526	358	3	=	=	PUNCT
ejpam-6526	358	4	max{|x1	max{|x1	NUM
ejpam-6526	358	5	−	−	PROPN
ejpam-6526	358	6	y1|	y1|	PROPN
ejpam-6526	358	7	,	,	PUNCT
ejpam-6526	358	8	|x2	|x2	NOUN
ejpam-6526	358	9	−	−	PROPN
ejpam-6526	358	10	y2|	y2|	PROPN
ejpam-6526	358	11	}	}	PUNCT
ejpam-6526	358	12	(	(	PUNCT
ejpam-6526	358	13	maximum	maximum	ADJ
ejpam-6526	358	14	norm	norm	NOUN
ejpam-6526	358	15	)	)	PUNCT
ejpam-6526	358	16	the	the	DET
ejpam-6526	358	17	set	set	PROPN
ejpam-6526	358	18	d	d	PROPN
ejpam-6526	358	19	=	=	SYM
ejpam-6526	358	20	q2	q2	NOUN
ejpam-6526	358	21	(	(	PUNCT
ejpam-6526	358	22	pairs	pair	NOUN
ejpam-6526	358	23	of	of	ADP
ejpam-6526	358	24	rational	rational	ADJ
ejpam-6526	358	25	numbers	number	NOUN
ejpam-6526	358	26	)	)	PUNCT
ejpam-6526	358	27	is	be	AUX
ejpam-6526	358	28	countable	countable	ADJ
ejpam-6526	358	29	and	and	CCONJ
ejpam-6526	358	30	dense	dense	ADJ
ejpam-6526	358	31	in	in	ADP
ejpam-6526	358	32	both	both	DET
ejpam-6526	358	33	induced	induced	ADJ
ejpam-6526	358	34	topologies	topology	NOUN
ejpam-6526	358	35	.	.	PUNCT
ejpam-6526	359	1	for	for	ADP
ejpam-6526	359	2	any	any	DET
ejpam-6526	359	3	(	(	PUNCT
ejpam-6526	359	4	x1	x1	PROPN
ejpam-6526	359	5	,	,	PUNCT
ejpam-6526	359	6	x2	x2	ADJ
ejpam-6526	359	7	)	)	PUNCT
ejpam-6526	359	8	∈	∈	PROPN
ejpam-6526	359	9	r2	r2	PROPN
ejpam-6526	359	10	and	and	CCONJ
ejpam-6526	359	11	ε	ε	PROPN
ejpam-6526	359	12	>	>	X
ejpam-6526	359	13	0	0	PROPN
ejpam-6526	359	14	,	,	PUNCT
ejpam-6526	359	15	we	we	PRON
ejpam-6526	359	16	can	can	AUX
ejpam-6526	359	17	find	find	VERB
ejpam-6526	359	18	(	(	PUNCT
ejpam-6526	359	19	q1	q1	NOUN
ejpam-6526	359	20	,	,	PUNCT
ejpam-6526	359	21	q2	q2	NOUN
ejpam-6526	359	22	)	)	PUNCT
ejpam-6526	359	23	∈	∈	PROPN
ejpam-6526	359	24	q2	q2	NOUN
ejpam-6526	359	25	with	with	ADP
ejpam-6526	359	26	both	both	DET
ejpam-6526	359	27	∥(x1	∥(x1	NOUN
ejpam-6526	359	28	,	,	PUNCT
ejpam-6526	359	29	x2)−	x2)−	X
ejpam-6526	359	30	(	(	PUNCT
ejpam-6526	359	31	q1	q1	PROPN
ejpam-6526	359	32	,	,	PUNCT
ejpam-6526	359	33	q2)∥2	q2)∥2	PROPN
ejpam-6526	359	34	<	<	X
ejpam-6526	359	35	ε	ε	PROPN
ejpam-6526	359	36	and	and	CCONJ
ejpam-6526	359	37	∥(x1	∥(x1	NOUN
ejpam-6526	359	38	,	,	PUNCT
ejpam-6526	359	39	x2)−	x2)−	X
ejpam-6526	359	40	(	(	PUNCT
ejpam-6526	359	41	q1	q1	PROPN
ejpam-6526	359	42	,	,	PUNCT
ejpam-6526	359	43	q2)∥∞	q2)∥∞	NOUN
ejpam-6526	359	44	<	<	X
ejpam-6526	359	45	ε	ε	PROPN
ejpam-6526	359	46	.	.	PUNCT
ejpam-6526	360	1	thus	thus	ADV
ejpam-6526	360	2	,	,	PUNCT
ejpam-6526	360	3	the	the	DET
ejpam-6526	360	4	system	system	NOUN
ejpam-6526	360	5	is	be	AUX
ejpam-6526	360	6	bi	bi	ADJ
ejpam-6526	360	7	-	-	ADJ
ejpam-6526	360	8	separable	separable	ADJ
ejpam-6526	360	9	.	.	PUNCT
ejpam-6526	361	1	example	example	NOUN
ejpam-6526	361	2	16	16	NUM
ejpam-6526	361	3	(	(	PUNCT
ejpam-6526	361	4	non	non	ADJ
ejpam-6526	361	5	-	-	ADJ
ejpam-6526	361	6	bi	bi	ADJ
ejpam-6526	361	7	-	-	ADJ
ejpam-6526	361	8	separable	separable	ADJ
ejpam-6526	361	9	system	system	NOUN
ejpam-6526	361	10	)	)	PUNCT
ejpam-6526	361	11	.	.	PUNCT
ejpam-6526	362	1	consider	consider	VERB
ejpam-6526	362	2	x	x	X
ejpam-6526	362	3	=	=	SYM
ejpam-6526	362	4	ℓ∞	ℓ∞	PROPN
ejpam-6526	362	5	(	(	PUNCT
ejpam-6526	362	6	bounded	bound	VERB
ejpam-6526	362	7	sequences	sequence	NOUN
ejpam-6526	362	8	)	)	PUNCT
ejpam-6526	362	9	with	with	ADP
ejpam-6526	362	10	:	:	PUNCT
ejpam-6526	362	11	•	•	NUM
ejpam-6526	362	12	ϕ1(x	ϕ1(x	NUM
ejpam-6526	362	13	,	,	PUNCT
ejpam-6526	362	14	y	y	NOUN
ejpam-6526	362	15	)	)	PUNCT
ejpam-6526	362	16	=	=	SYM
ejpam-6526	363	1	∥x−	∥x−	NUM
ejpam-6526	363	2	y∥∞	y∥∞	NOUN
ejpam-6526	363	3	=	=	SYM
ejpam-6526	363	4	supn	supn	NOUN
ejpam-6526	363	5	|xn	|xn	PRON
ejpam-6526	363	6	−	−	PROPN
ejpam-6526	364	1	yn|	yn|	NOUN
ejpam-6526	364	2	•	•	ADP
ejpam-6526	364	3	ϕ2(x	ϕ2(x	PROPN
ejpam-6526	364	4	,	,	PUNCT
ejpam-6526	364	5	y	y	NOUN
ejpam-6526	364	6	)	)	PUNCT
ejpam-6526	364	7	=	=	SYM
ejpam-6526	365	1	ddiscrete(x	ddiscrete(x	VERB
ejpam-6526	365	2	,	,	PUNCT
ejpam-6526	365	3	y	y	NOUN
ejpam-6526	365	4	)	)	PUNCT
ejpam-6526	365	5	=	=	PRON
ejpam-6526	365	6	{	{	PUNCT
ejpam-6526	365	7	0	0	NUM
ejpam-6526	366	1	if	if	SCONJ
ejpam-6526	366	2	x	x	NOUN
ejpam-6526	366	3	=	=	SYM
ejpam-6526	366	4	y	y	PROPN
ejpam-6526	366	5	1	1	NUM
ejpam-6526	366	6	if	if	SCONJ
ejpam-6526	366	7	x	x	PROPN
ejpam-6526	366	8	̸=	̸=	PROPN
ejpam-6526	366	9	y	y	PROPN
ejpam-6526	366	10	while	while	SCONJ
ejpam-6526	366	11	(	(	PUNCT
ejpam-6526	366	12	ℓ∞	ℓ∞	NOUN
ejpam-6526	366	13	,	,	PUNCT
ejpam-6526	366	14	∥	∥	X
ejpam-6526	366	15	·	·	PUNCT
ejpam-6526	366	16	∥∞	∥∞	PROPN
ejpam-6526	366	17	)	)	PUNCT
ejpam-6526	366	18	has	have	AUX
ejpam-6526	366	19	countable	countable	VERB
ejpam-6526	366	20	dense	dense	ADJ
ejpam-6526	366	21	subsets	subset	NOUN
ejpam-6526	366	22	(	(	PUNCT
ejpam-6526	366	23	e.g.	e.g.	ADV
ejpam-6526	366	24	,	,	PUNCT
ejpam-6526	366	25	sequences	sequence	NOUN
ejpam-6526	366	26	with	with	ADP
ejpam-6526	366	27	finitely	finitely	ADV
ejpam-6526	366	28	many	many	ADJ
ejpam-6526	366	29	non	non	ADJ
ejpam-6526	366	30	-	-	ADJ
ejpam-6526	366	31	zero	zero	ADJ
ejpam-6526	366	32	rational	rational	ADJ
ejpam-6526	366	33	entries	entry	NOUN
ejpam-6526	366	34	)	)	PUNCT
ejpam-6526	366	35	,	,	PUNCT
ejpam-6526	366	36	the	the	DET
ejpam-6526	366	37	discrete	discrete	ADJ
ejpam-6526	366	38	metric	metric	ADJ
ejpam-6526	366	39	topology	topology	NOUN
ejpam-6526	366	40	has	have	VERB
ejpam-6526	366	41	no	no	DET
ejpam-6526	366	42	countable	countable	ADJ
ejpam-6526	366	43	dense	dense	ADJ
ejpam-6526	366	44	subset	subset	NOUN
ejpam-6526	366	45	because	because	SCONJ
ejpam-6526	366	46	every	every	DET
ejpam-6526	366	47	subset	subset	NOUN
ejpam-6526	366	48	is	be	AUX
ejpam-6526	366	49	closed	closed	ADJ
ejpam-6526	366	50	and	and	CCONJ
ejpam-6526	366	51	open	open	ADJ
ejpam-6526	366	52	.	.	PUNCT
ejpam-6526	367	1	therefore	therefore	ADV
ejpam-6526	367	2	,	,	PUNCT
ejpam-6526	367	3	no	no	DET
ejpam-6526	367	4	countable	countable	ADJ
ejpam-6526	367	5	set	set	NOUN
ejpam-6526	367	6	can	can	AUX
ejpam-6526	367	7	be	be	AUX
ejpam-6526	367	8	dense	dense	ADJ
ejpam-6526	367	9	in	in	ADP
ejpam-6526	367	10	both	both	DET
ejpam-6526	367	11	topologies	topology	NOUN
ejpam-6526	367	12	,	,	PUNCT
ejpam-6526	367	13	making	make	VERB
ejpam-6526	367	14	the	the	DET
ejpam-6526	367	15	system	system	NOUN
ejpam-6526	367	16	not	not	PART
ejpam-6526	367	17	bi	bi	ADJ
ejpam-6526	367	18	-	-	ADJ
ejpam-6526	367	19	separable	separable	ADJ
ejpam-6526	367	20	.	.	PUNCT
ejpam-6526	368	1	proposition	proposition	NOUN
ejpam-6526	368	2	2	2	NUM
ejpam-6526	368	3	.	.	PUNCT
ejpam-6526	369	1	if	if	SCONJ
ejpam-6526	369	2	(	(	PUNCT
ejpam-6526	369	3	x,ϕ1	x,ϕ1	PROPN
ejpam-6526	369	4	)	)	PUNCT
ejpam-6526	369	5	and	and	CCONJ
ejpam-6526	369	6	(	(	PUNCT
ejpam-6526	369	7	x,ϕ2	x,ϕ2	PROPN
ejpam-6526	369	8	)	)	PUNCT
ejpam-6526	369	9	are	be	AUX
ejpam-6526	369	10	both	both	PRON
ejpam-6526	369	11	separable	separable	ADJ
ejpam-6526	369	12	metric	metric	ADJ
ejpam-6526	369	13	spaces	space	NOUN
ejpam-6526	369	14	,	,	PUNCT
ejpam-6526	369	15	then	then	ADV
ejpam-6526	369	16	(	(	PUNCT
ejpam-6526	369	17	x	x	X
ejpam-6526	369	18	,	,	PUNCT
ejpam-6526	369	19	ψ,⊕	ψ,⊕	NOUN
ejpam-6526	369	20	)	)	PUNCT
ejpam-6526	369	21	forms	form	VERB
ejpam-6526	369	22	bi	bi	ADJ
ejpam-6526	369	23	-	-	ADJ
ejpam-6526	369	24	separable	separable	ADJ
ejpam-6526	369	25	bi	bi	ADJ
ejpam-6526	369	26	-	-	ADJ
ejpam-6526	369	27	metric	metric	ADJ
ejpam-6526	369	28	system	system	NOUN
ejpam-6526	369	29	.	.	PUNCT
ejpam-6526	370	1	proof	proof	NOUN
ejpam-6526	370	2	.	.	PUNCT
ejpam-6526	371	1	since	since	SCONJ
ejpam-6526	371	2	(	(	PUNCT
ejpam-6526	371	3	x,ϕ1	x,ϕ1	PROPN
ejpam-6526	371	4	)	)	PUNCT
ejpam-6526	371	5	and	and	CCONJ
ejpam-6526	371	6	(	(	PUNCT
ejpam-6526	371	7	x,ϕ2	x,ϕ2	PROPN
ejpam-6526	371	8	)	)	PUNCT
ejpam-6526	371	9	are	be	AUX
ejpam-6526	371	10	separable	separable	ADJ
ejpam-6526	371	11	,	,	PUNCT
ejpam-6526	371	12	there	there	PRON
ejpam-6526	371	13	exist	exist	VERB
ejpam-6526	371	14	countable	countable	ADJ
ejpam-6526	371	15	dense	dense	ADJ
ejpam-6526	371	16	subsets	subset	NOUN
ejpam-6526	371	17	d1	d1	PROPN
ejpam-6526	371	18	and	and	CCONJ
ejpam-6526	371	19	d2	d2	PROPN
ejpam-6526	371	20	of	of	ADP
ejpam-6526	371	21	x	x	PUNCT
ejpam-6526	371	22	with	with	ADP
ejpam-6526	371	23	respect	respect	NOUN
ejpam-6526	371	24	to	to	ADP
ejpam-6526	371	25	ϕ1	ϕ1	NOUN
ejpam-6526	371	26	and	and	CCONJ
ejpam-6526	371	27	ϕ2	ϕ2	ADV
ejpam-6526	371	28	,	,	PUNCT
ejpam-6526	371	29	respectively	respectively	ADV
ejpam-6526	371	30	.	.	PUNCT
ejpam-6526	372	1	let	let	VERB
ejpam-6526	372	2	d	d	NOUN
ejpam-6526	372	3	=	=	PUNCT
ejpam-6526	372	4	d1	d1	PROPN
ejpam-6526	372	5	∪	∪	PROPN
ejpam-6526	372	6	d2	d2	PROPN
ejpam-6526	372	7	,	,	PUNCT
ejpam-6526	372	8	which	which	PRON
ejpam-6526	372	9	remains	remain	VERB
ejpam-6526	372	10	countable	countable	ADJ
ejpam-6526	372	11	.	.	PUNCT
ejpam-6526	373	1	for	for	ADP
ejpam-6526	373	2	any	any	DET
ejpam-6526	373	3	x	x	SYM
ejpam-6526	373	4	∈	∈	PROPN
ejpam-6526	373	5	x	x	X
ejpam-6526	373	6	and	and	CCONJ
ejpam-6526	373	7	ε	ε	PROPN
ejpam-6526	373	8	>	>	X
ejpam-6526	373	9	0	0	PROPN
ejpam-6526	373	10	,	,	PUNCT
ejpam-6526	373	11	there	there	PRON
ejpam-6526	373	12	exists	exist	VERB
ejpam-6526	373	13	y1	y1	NOUN
ejpam-6526	373	14	∈	∈	PROPN
ejpam-6526	373	15	d1	d1	PROPN
ejpam-6526	373	16	with	with	ADP
ejpam-6526	373	17	ϕ1(x	ϕ1(x	PROPN
ejpam-6526	373	18	,	,	PUNCT
ejpam-6526	373	19	y1	y1	NOUN
ejpam-6526	373	20	)	)	PUNCT
ejpam-6526	373	21	<	<	X
ejpam-6526	373	22	ε	ε	PROPN
ejpam-6526	373	23	,	,	PUNCT
ejpam-6526	373	24	and	and	CCONJ
ejpam-6526	373	25	there	there	PRON
ejpam-6526	373	26	exists	exist	VERB
ejpam-6526	373	27	y2	y2	PROPN
ejpam-6526	373	28	∈	∈	PROPN
ejpam-6526	373	29	d2	d2	PROPN
ejpam-6526	373	30	with	with	ADP
ejpam-6526	373	31	ϕ2(x	ϕ2(x	PROPN
ejpam-6526	373	32	,	,	PUNCT
ejpam-6526	373	33	y2	y2	PROPN
ejpam-6526	373	34	)	)	PUNCT
ejpam-6526	373	35	<	<	X
ejpam-6526	373	36	ε	ε	PROPN
ejpam-6526	373	37	.	.	PUNCT
ejpam-6526	374	1	if	if	SCONJ
ejpam-6526	374	2	either	either	CCONJ
ejpam-6526	374	3	y1	y1	NOUN
ejpam-6526	374	4	or	or	CCONJ
ejpam-6526	374	5	y2	y2	NOUN
ejpam-6526	374	6	satisfies	satisfie	NOUN
ejpam-6526	374	7	both	both	PRON
ejpam-6526	374	8	ϕ1(x	ϕ1(x	PROPN
ejpam-6526	374	9	,	,	PUNCT
ejpam-6526	374	10	yi	yi	NOUN
ejpam-6526	374	11	)	)	PUNCT
ejpam-6526	374	12	<	<	X
ejpam-6526	374	13	ε	ε	PROPN
ejpam-6526	374	14	and	and	CCONJ
ejpam-6526	374	15	ϕ2(x	ϕ2(x	PROPN
ejpam-6526	374	16	,	,	PUNCT
ejpam-6526	374	17	yi	yi	PROPN
ejpam-6526	374	18	)	)	PUNCT
ejpam-6526	374	19	<	<	X
ejpam-6526	374	20	ε	ε	PROPN
ejpam-6526	374	21	,	,	PUNCT
ejpam-6526	374	22	then	then	ADV
ejpam-6526	374	23	we	we	PRON
ejpam-6526	374	24	have	have	VERB
ejpam-6526	374	25	point	point	NOUN
ejpam-6526	374	26	in	in	ADP
ejpam-6526	374	27	d	d	PROPN
ejpam-6526	374	28	within	within	ADP
ejpam-6526	374	29	ε	ε	PROPN
ejpam-6526	374	30	of	of	ADP
ejpam-6526	374	31	x	x	PROPN
ejpam-6526	374	32	in	in	ADP
ejpam-6526	374	33	both	both	DET
ejpam-6526	374	34	metrics	metric	NOUN
ejpam-6526	374	35	.	.	PUNCT
ejpam-6526	375	1	a.	a.	PROPN
ejpam-6526	375	2	alsoboh	alsoboh	PROPN
ejpam-6526	375	3	et	et	PROPN
ejpam-6526	375	4	al	al	PROPN
ejpam-6526	375	5	.	.	PUNCT
ejpam-6526	375	6	/	/	SYM
ejpam-6526	375	7	eur	eur	PROPN
ejpam-6526	375	8	.	.	PUNCT
ejpam-6526	376	1	j.	j.	PROPN
ejpam-6526	376	2	pure	pure	PROPN
ejpam-6526	376	3	appl	appl	PROPN
ejpam-6526	376	4	.	.	PROPN
ejpam-6526	376	5	math	math	PROPN
ejpam-6526	376	6	,	,	PUNCT
ejpam-6526	376	7	18	18	NUM
ejpam-6526	376	8	(	(	PUNCT
ejpam-6526	376	9	4	4	NUM
ejpam-6526	376	10	)	)	PUNCT
ejpam-6526	376	11	(	(	PUNCT
ejpam-6526	376	12	2025	2025	NUM
ejpam-6526	376	13	)	)	PUNCT
ejpam-6526	376	14	,	,	PUNCT
ejpam-6526	376	15	6526	6526	NUM
ejpam-6526	376	16	16	16	NUM
ejpam-6526	376	17	of	of	ADP
ejpam-6526	376	18	21	21	NUM
ejpam-6526	376	19	otherwise	otherwise	ADV
ejpam-6526	376	20	,	,	PUNCT
ejpam-6526	376	21	we	we	PRON
ejpam-6526	376	22	can	can	AUX
ejpam-6526	376	23	construct	construct	VERB
ejpam-6526	376	24	sequence	sequence	NOUN
ejpam-6526	376	25	{	{	PUNCT
ejpam-6526	376	26	zn	zn	NOUN
ejpam-6526	376	27	}	}	PUNCT
ejpam-6526	376	28	⊂	⊂	PROPN
ejpam-6526	376	29	x	x	PUNCT
ejpam-6526	376	30	with	with	ADP
ejpam-6526	376	31	zn	zn	PROPN
ejpam-6526	376	32	→	→	SYM
ejpam-6526	376	33	x	x	X
ejpam-6526	376	34	with	with	ADP
ejpam-6526	376	35	respect	respect	NOUN
ejpam-6526	376	36	to	to	ADP
ejpam-6526	376	37	both	both	DET
ejpam-6526	376	38	metrics	metric	NOUN
ejpam-6526	376	39	,	,	PUNCT
ejpam-6526	376	40	and	and	CCONJ
ejpam-6526	376	41	each	each	DET
ejpam-6526	376	42	zn	zn	NOUN
ejpam-6526	376	43	representing	represent	VERB
ejpam-6526	376	44	convex	convex	NOUN
ejpam-6526	376	45	combination	combination	NOUN
ejpam-6526	376	46	of	of	ADP
ejpam-6526	376	47	points	point	NOUN
ejpam-6526	376	48	in	in	ADP
ejpam-6526	376	49	d.	d.	PROPN
ejpam-6526	376	50	by	by	ADP
ejpam-6526	376	51	density	density	NOUN
ejpam-6526	376	52	,	,	PUNCT
ejpam-6526	376	53	such	such	ADJ
ejpam-6526	376	54	sequence	sequence	NOUN
ejpam-6526	376	55	exists	exist	VERB
ejpam-6526	376	56	,	,	PUNCT
ejpam-6526	376	57	and	and	CCONJ
ejpam-6526	376	58	for	for	ADP
ejpam-6526	376	59	sufficiently	sufficiently	ADV
ejpam-6526	376	60	large	large	ADJ
ejpam-6526	376	61	n	n	CCONJ
ejpam-6526	376	62	,	,	PUNCT
ejpam-6526	376	63	ϕ1(zn	ϕ1(zn	PROPN
ejpam-6526	376	64	,	,	PUNCT
ejpam-6526	376	65	x	x	X
ejpam-6526	376	66	)	)	PUNCT
ejpam-6526	376	67	<	<	X
ejpam-6526	376	68	ε	ε	PROPN
ejpam-6526	376	69	and	and	CCONJ
ejpam-6526	376	70	ϕ2(zn	ϕ2(zn	PROPN
ejpam-6526	376	71	,	,	PUNCT
ejpam-6526	376	72	x	x	NOUN
ejpam-6526	376	73	)	)	PUNCT
ejpam-6526	376	74	<	<	X
ejpam-6526	376	75	ε	ε	PROPN
ejpam-6526	376	76	.	.	PUNCT
ejpam-6526	377	1	therefore	therefore	ADV
ejpam-6526	377	2	,	,	PUNCT
ejpam-6526	377	3	d	d	NOUN
ejpam-6526	377	4	is	be	AUX
ejpam-6526	377	5	dense	dense	ADJ
ejpam-6526	377	6	in	in	ADP
ejpam-6526	377	7	x	x	PUNCT
ejpam-6526	377	8	with	with	ADP
ejpam-6526	377	9	respect	respect	NOUN
ejpam-6526	377	10	to	to	ADP
ejpam-6526	377	11	both	both	DET
ejpam-6526	377	12	induced	induced	ADJ
ejpam-6526	377	13	topologies	topology	NOUN
ejpam-6526	377	14	,	,	PUNCT
ejpam-6526	377	15	establishing	establish	VERB
ejpam-6526	377	16	(	(	PUNCT
ejpam-6526	377	17	x	x	X
ejpam-6526	377	18	,	,	PUNCT
ejpam-6526	377	19	ψ,⊕	ψ,⊕	NOUN
ejpam-6526	377	20	)	)	PUNCT
ejpam-6526	377	21	as	as	ADP
ejpam-6526	377	22	bi	bi	NOUN
ejpam-6526	377	23	-	-	ADJ
ejpam-6526	377	24	separable	separable	ADJ
ejpam-6526	377	25	.	.	PUNCT
ejpam-6526	378	1	6.2	6.2	NUM
ejpam-6526	378	2	.	.	PUNCT
ejpam-6526	378	3	product	product	NOUN
ejpam-6526	378	4	structures	structure	NOUN
ejpam-6526	378	5	in	in	ADP
ejpam-6526	378	6	bi	bi	ADJ
ejpam-6526	378	7	-	-	ADJ
ejpam-6526	378	8	metric	metric	ADJ
ejpam-6526	378	9	systems	system	NOUN
ejpam-6526	378	10	theorem	theorem	VERB
ejpam-6526	378	11	8	8	NUM
ejpam-6526	378	12	.	.	PUNCT
ejpam-6526	378	13	for	for	ADP
ejpam-6526	378	14	bi	bi	ADJ
ejpam-6526	378	15	-	-	ADJ
ejpam-6526	378	16	metric	metric	ADJ
ejpam-6526	378	17	systems	system	NOUN
ejpam-6526	378	18	(	(	PUNCT
ejpam-6526	378	19	x	x	X
ejpam-6526	378	20	,	,	PUNCT
ejpam-6526	378	21	ψx	ψx	X
ejpam-6526	378	22	,	,	PUNCT
ejpam-6526	378	23	⊕x	⊕x	PROPN
ejpam-6526	378	24	)	)	PUNCT
ejpam-6526	379	1	and	and	CCONJ
ejpam-6526	379	2	(	(	PUNCT
ejpam-6526	379	3	y	y	PROPN
ejpam-6526	379	4	,	,	PUNCT
ejpam-6526	379	5	ψy	ψy	PROPN
ejpam-6526	379	6	,	,	PUNCT
ejpam-6526	379	7	⊕y	⊕y	NOUN
ejpam-6526	379	8	)	)	PUNCT
ejpam-6526	379	9	,	,	PUNCT
ejpam-6526	379	10	we	we	PRON
ejpam-6526	379	11	can	can	AUX
ejpam-6526	379	12	establish	establish	VERB
ejpam-6526	379	13	natural	natural	ADJ
ejpam-6526	379	14	bi	bi	ADJ
ejpam-6526	379	15	-	-	ADJ
ejpam-6526	379	16	metric	metric	ADJ
ejpam-6526	379	17	system	system	NOUN
ejpam-6526	379	18	on	on	ADP
ejpam-6526	379	19	x	x	SYM
ejpam-6526	379	20	×	×	NOUN
ejpam-6526	379	21	y	y	PROPN
ejpam-6526	379	22	as	as	ADP
ejpam-6526	379	23	:	:	PUNCT
ejpam-6526	379	24	ψx×y	ψx×y	SYM
ejpam-6526	379	25	(	(	PUNCT
ejpam-6526	379	26	(	(	PUNCT
ejpam-6526	379	27	x1	x1	PROPN
ejpam-6526	379	28	,	,	PUNCT
ejpam-6526	379	29	y1	y1	PROPN
ejpam-6526	379	30	)	)	PUNCT
ejpam-6526	379	31	,	,	PUNCT
ejpam-6526	379	32	(	(	PUNCT
ejpam-6526	379	33	x2	x2	PROPN
ejpam-6526	379	34	,	,	PUNCT
ejpam-6526	379	35	y2	y2	PROPN
ejpam-6526	379	36	)	)	PUNCT
ejpam-6526	379	37	)	)	PUNCT
ejpam-6526	380	1	=	=	SYM
ejpam-6526	380	2	⊕x(ψx(x1	⊕x(ψx(x1	NUM
ejpam-6526	380	3	,	,	PUNCT
ejpam-6526	380	4	x2),ψy	x2),ψy	PROPN
ejpam-6526	380	5	(	(	PUNCT
ejpam-6526	380	6	y1	y1	INTJ
ejpam-6526	380	7	,	,	PUNCT
ejpam-6526	380	8	y2	y2	PROPN
ejpam-6526	380	9	)	)	PUNCT
ejpam-6526	380	10	)	)	PUNCT
ejpam-6526	380	11	with	with	ADP
ejpam-6526	380	12	⊕x×y	⊕x×y	PROPN
ejpam-6526	380	13	=	=	PUNCT
ejpam-6526	380	14	⊕x	⊕x	PROPN
ejpam-6526	380	15	.	.	PUNCT
ejpam-6526	381	1	this	this	DET
ejpam-6526	381	2	product	product	NOUN
ejpam-6526	381	3	construction	construction	NOUN
ejpam-6526	381	4	aligns	align	VERB
ejpam-6526	381	5	with	with	ADP
ejpam-6526	381	6	previous	previous	ADJ
ejpam-6526	381	7	work	work	NOUN
ejpam-6526	381	8	on	on	ADP
ejpam-6526	381	9	product	product	NOUN
ejpam-6526	381	10	quasi	quasi	NOUN
ejpam-6526	381	11	-	-	NOUN
ejpam-6526	381	12	uniformities	uniformity	NOUN
ejpam-6526	381	13	[	[	X
ejpam-6526	381	14	24	24	NUM
ejpam-6526	381	15	]	]	PUNCT
ejpam-6526	381	16	.	.	PUNCT
ejpam-6526	382	1	example	example	NOUN
ejpam-6526	382	2	17	17	NUM
ejpam-6526	382	3	(	(	PUNCT
ejpam-6526	382	4	bi	bi	ADJ
ejpam-6526	382	5	-	-	ADJ
ejpam-6526	382	6	metric	metric	ADJ
ejpam-6526	382	7	product	product	NOUN
ejpam-6526	382	8	structure	structure	NOUN
ejpam-6526	382	9	)	)	PUNCT
ejpam-6526	382	10	.	.	PUNCT
ejpam-6526	383	1	consider	consider	VERB
ejpam-6526	383	2	x	x	X
ejpam-6526	383	3	=	=	PUNCT
ejpam-6526	384	1	[	[	X
ejpam-6526	384	2	0	0	NUM
ejpam-6526	384	3	,	,	PUNCT
ejpam-6526	384	4	1	1	NUM
ejpam-6526	384	5	]	]	PUNCT
ejpam-6526	384	6	with	with	ADP
ejpam-6526	384	7	bi	bi	ADJ
ejpam-6526	384	8	-	-	ADJ
ejpam-6526	384	9	metric	metric	ADJ
ejpam-6526	384	10	structure	structure	NOUN
ejpam-6526	384	11	ψx(x1	ψx(x1	NOUN
ejpam-6526	384	12	,	,	PUNCT
ejpam-6526	384	13	x2	x2	PROPN
ejpam-6526	384	14	)	)	PUNCT
ejpam-6526	384	15	=	=	PUNCT
ejpam-6526	384	16	(	(	PUNCT
ejpam-6526	384	17	|x1	|x1	NUM
ejpam-6526	384	18	−	−	PROPN
ejpam-6526	384	19	x2|	x2|	PROPN
ejpam-6526	384	20	,	,	PUNCT
ejpam-6526	384	21	|x21	|x21	NOUN
ejpam-6526	384	22	−	−	NOUN
ejpam-6526	384	23	x22|	x22|	NUM
ejpam-6526	384	24	)	)	PUNCT
ejpam-6526	384	25	and	and	CCONJ
ejpam-6526	384	26	⊕x	⊕x	NOUN
ejpam-6526	384	27	implementing	implement	VERB
ejpam-6526	384	28	component	component	NOUN
ejpam-6526	384	29	-	-	PUNCT
ejpam-6526	384	30	wise	wise	ADJ
ejpam-6526	384	31	addition	addition	NOUN
ejpam-6526	384	32	.	.	PUNCT
ejpam-6526	385	1	consider	consider	VERB
ejpam-6526	385	2	y	y	NOUN
ejpam-6526	385	3	=	=	PUNCT
ejpam-6526	386	1	[	[	X
ejpam-6526	386	2	0	0	NUM
ejpam-6526	386	3	,	,	PUNCT
ejpam-6526	386	4	1	1	NUM
ejpam-6526	386	5	]	]	PUNCT
ejpam-6526	386	6	with	with	ADP
ejpam-6526	386	7	bi	bi	ADJ
ejpam-6526	386	8	-	-	ADJ
ejpam-6526	386	9	metric	metric	ADJ
ejpam-6526	386	10	structure	structure	NOUN
ejpam-6526	386	11	ψy	ψy	PROPN
ejpam-6526	386	12	(	(	PUNCT
ejpam-6526	386	13	y1	y1	PROPN
ejpam-6526	386	14	,	,	PUNCT
ejpam-6526	386	15	y2	y2	NOUN
ejpam-6526	386	16	)	)	PUNCT
ejpam-6526	386	17	=	=	PUNCT
ejpam-6526	386	18	(	(	PUNCT
ejpam-6526	386	19	|y1	|y1	VERB
ejpam-6526	386	20	−	−	PROPN
ejpam-6526	386	21	y2|	y2|	PROPN
ejpam-6526	386	22	,	,	PUNCT
ejpam-6526	386	23	|	|	ADV
ejpam-6526	386	24	sinh(y1	sinh(y1	NOUN
ejpam-6526	386	25	)	)	PUNCT
ejpam-6526	386	26	−	−	PROPN
ejpam-6526	386	27	sinh(y2)|	sinh(y2)|	NOUN
ejpam-6526	386	28	)	)	PUNCT
ejpam-6526	386	29	and	and	CCONJ
ejpam-6526	386	30	⊕y	⊕y	NOUN
ejpam-6526	386	31	also	also	ADV
ejpam-6526	386	32	implementing	implement	VERB
ejpam-6526	386	33	component	component	NOUN
ejpam-6526	386	34	-	-	PUNCT
ejpam-6526	386	35	wise	wise	ADJ
ejpam-6526	386	36	addition	addition	NOUN
ejpam-6526	386	37	.	.	PUNCT
ejpam-6526	387	1	the	the	DET
ejpam-6526	387	2	product	product	NOUN
ejpam-6526	387	3	bi	bi	ADJ
ejpam-6526	387	4	-	-	ADJ
ejpam-6526	387	5	metric	metric	ADJ
ejpam-6526	387	6	structure	structure	NOUN
ejpam-6526	387	7	on	on	ADP
ejpam-6526	387	8	x	x	SYM
ejpam-6526	387	9	×	×	PROPN
ejpam-6526	387	10	y	y	PROPN
ejpam-6526	387	11	is	be	AUX
ejpam-6526	387	12	:	:	PUNCT
ejpam-6526	387	13	ψx×y	ψx×y	ADJ
ejpam-6526	387	14	(	(	PUNCT
ejpam-6526	387	15	(	(	PUNCT
ejpam-6526	387	16	x1	x1	PROPN
ejpam-6526	387	17	,	,	PUNCT
ejpam-6526	387	18	y1	y1	PROPN
ejpam-6526	387	19	)	)	PUNCT
ejpam-6526	387	20	,	,	PUNCT
ejpam-6526	387	21	(	(	PUNCT
ejpam-6526	387	22	x2	x2	PROPN
ejpam-6526	387	23	,	,	PUNCT
ejpam-6526	387	24	y2	y2	PROPN
ejpam-6526	387	25	)	)	PUNCT
ejpam-6526	387	26	)	)	PUNCT
ejpam-6526	388	1	=	=	SYM
ejpam-6526	388	2	⊕x(ψx(x1	⊕x(ψx(x1	NUM
ejpam-6526	388	3	,	,	PUNCT
ejpam-6526	388	4	x2),ψy	x2),ψy	PROPN
ejpam-6526	388	5	(	(	PUNCT
ejpam-6526	388	6	y1	y1	INTJ
ejpam-6526	388	7	,	,	PUNCT
ejpam-6526	388	8	y2	y2	PROPN
ejpam-6526	388	9	)	)	PUNCT
ejpam-6526	388	10	)	)	PUNCT
ejpam-6526	389	1	=	=	PUNCT
ejpam-6526	389	2	(	(	PUNCT
ejpam-6526	389	3	|x1	|x1	NUM
ejpam-6526	389	4	−	−	PROPN
ejpam-6526	389	5	x2|	x2|	PROPN
ejpam-6526	389	6	,	,	PUNCT
ejpam-6526	389	7	|x21	|x21	NOUN
ejpam-6526	389	8	−	−	NOUN
ejpam-6526	389	9	x22|	x22|	NUM
ejpam-6526	389	10	)	)	PUNCT
ejpam-6526	390	1	+	+	CCONJ
ejpam-6526	390	2	(	(	PUNCT
ejpam-6526	390	3	|y1	|y1	VERB
ejpam-6526	390	4	−	−	PROPN
ejpam-6526	390	5	y2|	y2|	NOUN
ejpam-6526	390	6	,	,	PUNCT
ejpam-6526	390	7	|	|	ADV
ejpam-6526	390	8	sinh(y1)−	sinh(y1)−	ADJ
ejpam-6526	390	9	sinh(y2)|	sinh(y2)|	NOUN
ejpam-6526	390	10	)	)	PUNCT
ejpam-6526	390	11	=	=	PUNCT
ejpam-6526	391	1	(	(	PUNCT
ejpam-6526	391	2	|x1	|x1	NUM
ejpam-6526	391	3	−	−	NOUN
ejpam-6526	391	4	x2|+	x2|+	PROPN
ejpam-6526	391	5	|y1	|y1	VERB
ejpam-6526	391	6	−	−	PROPN
ejpam-6526	391	7	y2|	y2|	PROPN
ejpam-6526	391	8	,	,	PUNCT
ejpam-6526	391	9	|x21	|x21	NOUN
ejpam-6526	391	10	−	−	NUM
ejpam-6526	391	11	x22|+	x22|+	ADJ
ejpam-6526	391	12	|	|	ADV
ejpam-6526	391	13	sinh(y1)−	sinh(y1)−	ADJ
ejpam-6526	391	14	sinh(y2)|	sinh(y2)|	NOUN
ejpam-6526	391	15	)	)	PUNCT
ejpam-6526	391	16	this	this	DET
ejpam-6526	391	17	bi	bi	ADJ
ejpam-6526	391	18	-	-	ADJ
ejpam-6526	391	19	metric	metric	ADJ
ejpam-6526	391	20	structure	structure	NOUN
ejpam-6526	391	21	induces	induce	VERB
ejpam-6526	391	22	two	two	NUM
ejpam-6526	391	23	distinct	distinct	ADJ
ejpam-6526	391	24	topologies	topology	NOUN
ejpam-6526	391	25	on	on	ADP
ejpam-6526	391	26	x	x	SYM
ejpam-6526	391	27	×	×	PROPN
ejpam-6526	391	28	y	y	PROPN
ejpam-6526	391	29	:	:	PUNCT
ejpam-6526	391	30	one	one	NUM
ejpam-6526	391	31	generated	generate	VERB
ejpam-6526	391	32	by	by	ADP
ejpam-6526	391	33	open	open	ADJ
ejpam-6526	391	34	neighborhoods	neighborhood	NOUN
ejpam-6526	391	35	in	in	ADP
ejpam-6526	391	36	first	first	ADJ
ejpam-6526	391	37	component	component	NOUN
ejpam-6526	391	38	’s	’s	PART
ejpam-6526	391	39	combined	combine	VERB
ejpam-6526	391	40	metric	metric	NOUN
ejpam-6526	391	41	,	,	PUNCT
ejpam-6526	391	42	another	another	PRON
ejpam-6526	391	43	by	by	ADP
ejpam-6526	391	44	open	open	ADJ
ejpam-6526	391	45	neighborhoods	neighborhood	NOUN
ejpam-6526	391	46	in	in	ADP
ejpam-6526	391	47	second	second	ADJ
ejpam-6526	391	48	component	component	NOUN
ejpam-6526	391	49	’s	’s	PART
ejpam-6526	391	50	combined	combine	VERB
ejpam-6526	391	51	metric	metric	NOUN
ejpam-6526	391	52	.	.	PUNCT
ejpam-6526	392	1	6.3	6.3	NUM
ejpam-6526	392	2	.	.	PUNCT
ejpam-6526	393	1	characterizing	characterize	VERB
ejpam-6526	393	2	bitopological	bitopological	ADJ
ejpam-6526	393	3	spaces	space	NOUN
ejpam-6526	393	4	through	through	ADP
ejpam-6526	393	5	bi	bi	ADJ
ejpam-6526	393	6	-	-	ADJ
ejpam-6526	393	7	metric	metric	ADJ
ejpam-6526	393	8	systems	system	NOUN
ejpam-6526	393	9	theorem	theorem	VERB
ejpam-6526	393	10	9	9	NUM
ejpam-6526	393	11	.	.	PUNCT
ejpam-6526	394	1	a	a	DET
ejpam-6526	394	2	bitopological	bitopological	ADJ
ejpam-6526	394	3	space	space	NOUN
ejpam-6526	394	4	(	(	PUNCT
ejpam-6526	394	5	x	x	NOUN
ejpam-6526	394	6	,	,	PUNCT
ejpam-6526	394	7	t1	t1	NOUN
ejpam-6526	394	8	,	,	PUNCT
ejpam-6526	394	9	t2	t2	NOUN
ejpam-6526	394	10	)	)	PUNCT
ejpam-6526	394	11	can	can	AUX
ejpam-6526	394	12	be	be	AUX
ejpam-6526	394	13	represented	represent	VERB
ejpam-6526	394	14	through	through	ADP
ejpam-6526	394	15	bi	bi	ADJ
ejpam-6526	394	16	-	-	ADJ
ejpam-6526	394	17	metric	metric	ADJ
ejpam-6526	394	18	systems	system	NOUN
ejpam-6526	394	19	if	if	SCONJ
ejpam-6526	394	20	and	and	CCONJ
ejpam-6526	394	21	only	only	ADV
ejpam-6526	394	22	if	if	SCONJ
ejpam-6526	394	23	both	both	PRON
ejpam-6526	394	24	t1	t1	NOUN
ejpam-6526	394	25	and	and	CCONJ
ejpam-6526	394	26	t2	t2	NOUN
ejpam-6526	394	27	are	be	AUX
ejpam-6526	394	28	metrizable	metrizable	ADJ
ejpam-6526	394	29	topologies	topology	NOUN
ejpam-6526	394	30	and	and	CCONJ
ejpam-6526	394	31	there	there	PRON
ejpam-6526	394	32	exists	exist	VERB
ejpam-6526	394	33	countable	countable	ADJ
ejpam-6526	394	34	family	family	NOUN
ejpam-6526	394	35	f	f	PROPN
ejpam-6526	394	36	of	of	ADP
ejpam-6526	394	37	continuous	continuous	ADJ
ejpam-6526	394	38	functions	function	NOUN
ejpam-6526	394	39	f	f	NOUN
ejpam-6526	394	40	:	:	PUNCT
ejpam-6526	394	41	x	x	X
ejpam-6526	394	42	→	→	PUNCT
ejpam-6526	394	43	r	r	NOUN
ejpam-6526	394	44	such	such	ADJ
ejpam-6526	394	45	that	that	SCONJ
ejpam-6526	394	46	f	f	PROPN
ejpam-6526	394	47	separates	separate	VERB
ejpam-6526	394	48	points	point	NOUN
ejpam-6526	394	49	from	from	ADP
ejpam-6526	394	50	closed	closed	ADJ
ejpam-6526	394	51	sets	set	NOUN
ejpam-6526	394	52	in	in	ADP
ejpam-6526	394	53	both	both	DET
ejpam-6526	394	54	topologies	topology	NOUN
ejpam-6526	394	55	.	.	PUNCT
ejpam-6526	395	1	proof	proof	NOUN
ejpam-6526	395	2	.	.	PUNCT
ejpam-6526	396	1	(	(	PUNCT
ejpam-6526	396	2	⇒	⇒	PROPN
ejpam-6526	396	3	)	)	PUNCT
ejpam-6526	396	4	suppose	suppose	VERB
ejpam-6526	396	5	(	(	PUNCT
ejpam-6526	396	6	x	x	X
ejpam-6526	396	7	,	,	PUNCT
ejpam-6526	396	8	t1	t1	NOUN
ejpam-6526	396	9	,	,	PUNCT
ejpam-6526	396	10	t2	t2	NOUN
ejpam-6526	396	11	)	)	PUNCT
ejpam-6526	396	12	can	can	AUX
ejpam-6526	396	13	be	be	AUX
ejpam-6526	396	14	represented	represent	VERB
ejpam-6526	396	15	through	through	ADP
ejpam-6526	396	16	bi	bi	ADJ
ejpam-6526	396	17	-	-	ADJ
ejpam-6526	396	18	metric	metric	ADJ
ejpam-6526	396	19	systems	system	NOUN
ejpam-6526	396	20	.	.	PUNCT
ejpam-6526	397	1	then	then	ADV
ejpam-6526	397	2	there	there	PRON
ejpam-6526	397	3	exist	exist	VERB
ejpam-6526	397	4	metric	metric	ADJ
ejpam-6526	397	5	functions	function	NOUN
ejpam-6526	397	6	ϕ1	ϕ1	NOUN
ejpam-6526	397	7	and	and	CCONJ
ejpam-6526	397	8	ϕ2	ϕ2	ADV
ejpam-6526	397	9	such	such	ADJ
ejpam-6526	397	10	that	that	DET
ejpam-6526	397	11	t1	t1	NOUN
ejpam-6526	397	12	=	=	PUNCT
ejpam-6526	397	13	tϕ1	tϕ1	NOUN
ejpam-6526	397	14	and	and	CCONJ
ejpam-6526	397	15	t2	t2	PROPN
ejpam-6526	397	16	=	=	PUNCT
ejpam-6526	397	17	tϕ2	tϕ2	INTJ
ejpam-6526	397	18	,	,	PUNCT
ejpam-6526	397	19	where	where	SCONJ
ejpam-6526	397	20	tϕi	tϕi	PROPN
ejpam-6526	397	21	denotes	denote	VERB
ejpam-6526	397	22	topology	topology	NOUN
ejpam-6526	397	23	induced	induce	VERB
ejpam-6526	397	24	by	by	ADP
ejpam-6526	397	25	metric	metric	ADJ
ejpam-6526	397	26	ϕi	ϕi	NOUN
ejpam-6526	397	27	.	.	PUNCT
ejpam-6526	398	1	clearly	clearly	ADV
ejpam-6526	398	2	,	,	PUNCT
ejpam-6526	398	3	both	both	CCONJ
ejpam-6526	398	4	t1	t1	NOUN
ejpam-6526	398	5	and	and	CCONJ
ejpam-6526	398	6	t2	t2	NOUN
ejpam-6526	398	7	are	be	AUX
ejpam-6526	398	8	metrizable	metrizable	ADJ
ejpam-6526	398	9	.	.	PUNCT
ejpam-6526	399	1	for	for	ADP
ejpam-6526	399	2	each	each	DET
ejpam-6526	399	3	x	x	SYM
ejpam-6526	399	4	∈	∈	PROPN
ejpam-6526	399	5	x	x	X
ejpam-6526	399	6	and	and	CCONJ
ejpam-6526	399	7	n	n	CCONJ
ejpam-6526	399	8	∈	∈	PROPN
ejpam-6526	399	9	n	n	CCONJ
ejpam-6526	399	10	,	,	PUNCT
ejpam-6526	399	11	define	define	VERB
ejpam-6526	399	12	fx	fx	PROPN
ejpam-6526	399	13	,	,	PUNCT
ejpam-6526	399	14	n	n	PRON
ejpam-6526	399	15	:	:	PUNCT
ejpam-6526	399	16	x	x	X
ejpam-6526	399	17	→	→	SYM
ejpam-6526	399	18	r	r	NOUN
ejpam-6526	399	19	by	by	ADP
ejpam-6526	399	20	fx	fx	PROPN
ejpam-6526	399	21	,	,	PUNCT
ejpam-6526	399	22	n(y	n(y	PROPN
ejpam-6526	399	23	)	)	PUNCT
ejpam-6526	399	24	=	=	SYM
ejpam-6526	399	25	ϕ1(y	ϕ1(y	PROPN
ejpam-6526	399	26	,	,	PUNCT
ejpam-6526	399	27	x)∧	x)∧	PUNCT
ejpam-6526	400	1	n.	n.	PROPN
ejpam-6526	400	2	each	each	DET
ejpam-6526	400	3	fx	fx	PROPN
ejpam-6526	400	4	,	,	PUNCT
ejpam-6526	400	5	n	n	PRON
ejpam-6526	400	6	is	be	AUX
ejpam-6526	400	7	continuous	continuous	ADJ
ejpam-6526	400	8	with	with	ADP
ejpam-6526	400	9	respect	respect	NOUN
ejpam-6526	400	10	to	to	ADP
ejpam-6526	400	11	both	both	PRON
ejpam-6526	400	12	t1	t1	NOUN
ejpam-6526	400	13	and	and	CCONJ
ejpam-6526	400	14	t2	t2	PROPN
ejpam-6526	400	15	(	(	PUNCT
ejpam-6526	400	16	since	since	SCONJ
ejpam-6526	400	17	ϕ1	ϕ1	NOUN
ejpam-6526	400	18	is	be	AUX
ejpam-6526	400	19	continuous	continuous	ADJ
ejpam-6526	400	20	with	with	ADP
ejpam-6526	400	21	respect	respect	NOUN
ejpam-6526	400	22	to	to	ADP
ejpam-6526	400	23	t1	t1	NOUN
ejpam-6526	400	24	and	and	CCONJ
ejpam-6526	400	25	t2	t2	NOUN
ejpam-6526	400	26	is	be	AUX
ejpam-6526	400	27	finer	fine	ADJ
ejpam-6526	400	28	than	than	ADP
ejpam-6526	400	29	or	or	CCONJ
ejpam-6526	400	30	equal	equal	ADJ
ejpam-6526	400	31	to	to	PART
ejpam-6526	400	32	t1	t1	NOUN
ejpam-6526	400	33	)	)	PUNCT
ejpam-6526	400	34	.	.	PUNCT
ejpam-6526	401	1	similarly	similarly	ADV
ejpam-6526	401	2	,	,	PUNCT
ejpam-6526	401	3	define	define	VERB
ejpam-6526	401	4	gx	gx	PROPN
ejpam-6526	401	5	,	,	PUNCT
ejpam-6526	401	6	n	n	PRON
ejpam-6526	401	7	:	:	PUNCT
ejpam-6526	401	8	x	x	X
ejpam-6526	401	9	→	→	SYM
ejpam-6526	401	10	r	r	NOUN
ejpam-6526	401	11	by	by	ADP
ejpam-6526	401	12	gx	gx	PROPN
ejpam-6526	401	13	,	,	PUNCT
ejpam-6526	401	14	n(y	n(y	PROPN
ejpam-6526	401	15	)	)	PUNCT
ejpam-6526	401	16	=	=	SYM
ejpam-6526	402	1	ϕ2(y	ϕ2(y	PROPN
ejpam-6526	402	2	,	,	PUNCT
ejpam-6526	402	3	x	x	NOUN
ejpam-6526	402	4	)	)	PUNCT
ejpam-6526	402	5	∧	∧	PROPN
ejpam-6526	402	6	n.	n.	PROPN
ejpam-6526	402	7	a.	a.	NOUN
ejpam-6526	402	8	alsoboh	alsoboh	PROPN
ejpam-6526	402	9	et	et	PROPN
ejpam-6526	402	10	al	al	PROPN
ejpam-6526	402	11	.	.	PUNCT
ejpam-6526	402	12	/	/	SYM
ejpam-6526	402	13	eur	eur	PROPN
ejpam-6526	402	14	.	.	PUNCT
ejpam-6526	403	1	j.	j.	PROPN
ejpam-6526	403	2	pure	pure	PROPN
ejpam-6526	403	3	appl	appl	PROPN
ejpam-6526	403	4	.	.	PROPN
ejpam-6526	403	5	math	math	PROPN
ejpam-6526	403	6	,	,	PUNCT
ejpam-6526	403	7	18	18	NUM
ejpam-6526	403	8	(	(	PUNCT
ejpam-6526	403	9	4	4	NUM
ejpam-6526	403	10	)	)	PUNCT
ejpam-6526	403	11	(	(	PUNCT
ejpam-6526	403	12	2025	2025	NUM
ejpam-6526	403	13	)	)	PUNCT
ejpam-6526	403	14	,	,	PUNCT
ejpam-6526	403	15	6526	6526	NUM
ejpam-6526	403	16	17	17	NUM
ejpam-6526	403	17	of	of	ADP
ejpam-6526	403	18	21	21	NUM
ejpam-6526	403	19	let	let	VERB
ejpam-6526	403	20	f	f	NOUN
ejpam-6526	403	21	=	=	PRON
ejpam-6526	403	22	{	{	PUNCT
ejpam-6526	403	23	fx	fx	PROPN
ejpam-6526	403	24	,	,	PUNCT
ejpam-6526	403	25	n	n	CCONJ
ejpam-6526	403	26	,	,	PUNCT
ejpam-6526	403	27	gx	gx	PROPN
ejpam-6526	403	28	,	,	PUNCT
ejpam-6526	403	29	n	n	CCONJ
ejpam-6526	403	30	:	:	PUNCT
ejpam-6526	403	31	x	x	X
ejpam-6526	403	32	∈	∈	PROPN
ejpam-6526	403	33	d	d	NOUN
ejpam-6526	403	34	,	,	PUNCT
ejpam-6526	403	35	n	n	PROPN
ejpam-6526	403	36	∈	∈	PROPN
ejpam-6526	403	37	n	n	CCONJ
ejpam-6526	403	38	}	}	PUNCT
ejpam-6526	403	39	,	,	PUNCT
ejpam-6526	403	40	where	where	SCONJ
ejpam-6526	403	41	d	d	NOUN
ejpam-6526	403	42	is	be	AUX
ejpam-6526	403	43	countable	countable	ADJ
ejpam-6526	403	44	dense	dense	ADJ
ejpam-6526	403	45	subset	subset	NOUN
ejpam-6526	403	46	of	of	ADP
ejpam-6526	403	47	x	x	PUNCT
ejpam-6526	403	48	with	with	ADP
ejpam-6526	403	49	respect	respect	NOUN
ejpam-6526	403	50	to	to	ADP
ejpam-6526	403	51	both	both	DET
ejpam-6526	403	52	metrics	metric	NOUN
ejpam-6526	403	53	.	.	PUNCT
ejpam-6526	404	1	the	the	DET
ejpam-6526	404	2	lindelöf	lindelöf	NOUN
ejpam-6526	404	3	property	property	NOUN
ejpam-6526	404	4	explanation	explanation	NOUN
ejpam-6526	404	5	:	:	PUNCT
ejpam-6526	404	6	every	every	DET
ejpam-6526	404	7	metrizable	metrizable	ADJ
ejpam-6526	404	8	space	space	NOUN
ejpam-6526	404	9	is	be	AUX
ejpam-6526	404	10	paracompact	paracompact	ADJ
ejpam-6526	404	11	,	,	PUNCT
ejpam-6526	404	12	and	and	CCONJ
ejpam-6526	404	13	paracompact	paracompact	ADJ
ejpam-6526	404	14	spaces	space	NOUN
ejpam-6526	404	15	are	be	AUX
ejpam-6526	404	16	collection	collection	NOUN
ejpam-6526	404	17	-	-	PUNCT
ejpam-6526	404	18	wise	wise	ADJ
ejpam-6526	404	19	normal	normal	ADJ
ejpam-6526	404	20	.	.	PUNCT
ejpam-6526	405	1	a	a	DET
ejpam-6526	405	2	key	key	ADJ
ejpam-6526	405	3	theorem	theorem	ADJ
ejpam-6526	405	4	states	state	NOUN
ejpam-6526	405	5	that	that	SCONJ
ejpam-6526	405	6	every	every	DET
ejpam-6526	405	7	paracompact	paracompact	ADJ
ejpam-6526	405	8	space	space	NOUN
ejpam-6526	405	9	satisfies	satisfy	VERB
ejpam-6526	405	10	the	the	DET
ejpam-6526	405	11	lindelöf	lindelöf	NOUN
ejpam-6526	405	12	property	property	NOUN
ejpam-6526	405	13	every	every	DET
ejpam-6526	405	14	open	open	ADJ
ejpam-6526	405	15	cover	cover	NOUN
ejpam-6526	405	16	has	have	VERB
ejpam-6526	405	17	a	a	DET
ejpam-6526	405	18	countable	countable	ADJ
ejpam-6526	405	19	subcover	subcover	NOUN
ejpam-6526	405	20	.	.	PUNCT
ejpam-6526	406	1	this	this	PRON
ejpam-6526	406	2	is	be	AUX
ejpam-6526	406	3	because	because	SCONJ
ejpam-6526	406	4	in	in	ADP
ejpam-6526	406	5	a	a	DET
ejpam-6526	406	6	paracompact	paracompact	ADJ
ejpam-6526	406	7	space	space	NOUN
ejpam-6526	406	8	,	,	PUNCT
ejpam-6526	406	9	we	we	PRON
ejpam-6526	406	10	can	can	AUX
ejpam-6526	406	11	refine	refine	VERB
ejpam-6526	406	12	any	any	DET
ejpam-6526	406	13	open	open	ADJ
ejpam-6526	406	14	cover	cover	NOUN
ejpam-6526	406	15	to	to	ADP
ejpam-6526	406	16	a	a	DET
ejpam-6526	406	17	locally	locally	ADV
ejpam-6526	406	18	finite	finite	ADJ
ejpam-6526	406	19	open	open	ADJ
ejpam-6526	406	20	cover	cover	NOUN
ejpam-6526	406	21	,	,	PUNCT
ejpam-6526	406	22	and	and	CCONJ
ejpam-6526	406	23	in	in	ADP
ejpam-6526	406	24	a	a	DET
ejpam-6526	406	25	metrizable	metrizable	ADJ
ejpam-6526	406	26	space	space	NOUN
ejpam-6526	406	27	,	,	PUNCT
ejpam-6526	406	28	this	this	DET
ejpam-6526	406	29	locally	locally	ADV
ejpam-6526	406	30	finite	finite	ADJ
ejpam-6526	406	31	refinement	refinement	NOUN
ejpam-6526	406	32	must	must	AUX
ejpam-6526	406	33	be	be	AUX
ejpam-6526	406	34	countable	countable	ADJ
ejpam-6526	406	35	.	.	PUNCT
ejpam-6526	407	1	for	for	ADP
ejpam-6526	407	2	our	our	PRON
ejpam-6526	407	3	bi	bi	ADJ
ejpam-6526	407	4	-	-	ADJ
ejpam-6526	407	5	metric	metric	ADJ
ejpam-6526	407	6	context	context	NOUN
ejpam-6526	407	7	,	,	PUNCT
ejpam-6526	407	8	since	since	SCONJ
ejpam-6526	407	9	both	both	DET
ejpam-6526	407	10	(	(	PUNCT
ejpam-6526	407	11	x	x	NOUN
ejpam-6526	407	12	,	,	PUNCT
ejpam-6526	407	13	t1	t1	NOUN
ejpam-6526	407	14	)	)	PUNCT
ejpam-6526	407	15	and	and	CCONJ
ejpam-6526	407	16	(	(	PUNCT
ejpam-6526	407	17	x	x	NOUN
ejpam-6526	407	18	,	,	PUNCT
ejpam-6526	407	19	t2	t2	NOUN
ejpam-6526	407	20	)	)	PUNCT
ejpam-6526	407	21	are	be	AUX
ejpam-6526	407	22	metrizable	metrizable	ADJ
ejpam-6526	407	23	(	(	PUNCT
ejpam-6526	407	24	induced	induce	VERB
ejpam-6526	407	25	by	by	ADP
ejpam-6526	407	26	ϕ1	ϕ1	NOUN
ejpam-6526	407	27	and	and	CCONJ
ejpam-6526	407	28	ϕ2	ϕ2	ADV
ejpam-6526	407	29	respectively	respectively	ADV
ejpam-6526	407	30	)	)	PUNCT
ejpam-6526	407	31	,	,	PUNCT
ejpam-6526	407	32	they	they	PRON
ejpam-6526	407	33	are	be	AUX
ejpam-6526	407	34	both	both	ADV
ejpam-6526	407	35	paracompact	paracompact	ADJ
ejpam-6526	407	36	and	and	CCONJ
ejpam-6526	407	37	hence	hence	ADV
ejpam-6526	407	38	lindelöf	lindelöf	NOUN
ejpam-6526	407	39	.	.	PUNCT
ejpam-6526	408	1	furthermore	furthermore	ADV
ejpam-6526	408	2	,	,	PUNCT
ejpam-6526	408	3	metrizable	metrizable	ADJ
ejpam-6526	408	4	spaces	space	NOUN
ejpam-6526	408	5	are	be	AUX
ejpam-6526	408	6	separable	separable	ADJ
ejpam-6526	408	7	if	if	SCONJ
ejpam-6526	408	8	and	and	CCONJ
ejpam-6526	408	9	only	only	ADV
ejpam-6526	408	10	if	if	SCONJ
ejpam-6526	408	11	they	they	PRON
ejpam-6526	408	12	are	be	AUX
ejpam-6526	408	13	second	second	ADV
ejpam-6526	408	14	-	-	PUNCT
ejpam-6526	408	15	countable	countable	ADJ
ejpam-6526	408	16	,	,	PUNCT
ejpam-6526	408	17	which	which	PRON
ejpam-6526	408	18	is	be	AUX
ejpam-6526	408	19	equivalent	equivalent	ADJ
ejpam-6526	408	20	to	to	ADP
ejpam-6526	408	21	being	be	AUX
ejpam-6526	408	22	lindelöf	lindelöf	NOUN
ejpam-6526	408	23	and	and	CCONJ
ejpam-6526	408	24	having	have	VERB
ejpam-6526	408	25	a	a	DET
ejpam-6526	408	26	countable	countable	ADJ
ejpam-6526	408	27	dense	dense	ADJ
ejpam-6526	408	28	subset	subset	NOUN
ejpam-6526	408	29	.	.	PUNCT
ejpam-6526	409	1	the	the	DET
ejpam-6526	409	2	existence	existence	NOUN
ejpam-6526	409	3	of	of	ADP
ejpam-6526	409	4	countable	countable	ADJ
ejpam-6526	409	5	dense	dense	ADJ
ejpam-6526	409	6	subsets	subset	NOUN
ejpam-6526	409	7	in	in	ADP
ejpam-6526	409	8	separable	separable	ADJ
ejpam-6526	409	9	metric	metric	ADJ
ejpam-6526	409	10	spaces	space	NOUN
ejpam-6526	409	11	allows	allow	VERB
ejpam-6526	409	12	us	we	PRON
ejpam-6526	409	13	to	to	PART
ejpam-6526	409	14	construct	construct	VERB
ejpam-6526	409	15	the	the	DET
ejpam-6526	409	16	required	require	VERB
ejpam-6526	409	17	countable	countable	ADJ
ejpam-6526	409	18	family	family	NOUN
ejpam-6526	409	19	f	f	NOUN
ejpam-6526	409	20	that	that	PRON
ejpam-6526	409	21	separates	separate	VERB
ejpam-6526	409	22	points	point	NOUN
ejpam-6526	409	23	from	from	ADP
ejpam-6526	409	24	closed	closed	ADJ
ejpam-6526	409	25	sets	set	NOUN
ejpam-6526	409	26	.	.	PUNCT
ejpam-6526	410	1	then	then	ADV
ejpam-6526	410	2	f	f	PROPN
ejpam-6526	410	3	separates	separate	VERB
ejpam-6526	410	4	points	point	NOUN
ejpam-6526	410	5	from	from	ADP
ejpam-6526	410	6	closed	closed	ADJ
ejpam-6526	410	7	sets	set	NOUN
ejpam-6526	410	8	in	in	ADP
ejpam-6526	410	9	both	both	DET
ejpam-6526	410	10	topologies	topology	NOUN
ejpam-6526	410	11	.	.	PUNCT
ejpam-6526	411	1	(	(	PUNCT
ejpam-6526	411	2	⇐	⇐	NOUN
ejpam-6526	411	3	)	)	PUNCT
ejpam-6526	411	4	conversely	conversely	ADV
ejpam-6526	411	5	,	,	PUNCT
ejpam-6526	411	6	suppose	suppose	VERB
ejpam-6526	411	7	both	both	DET
ejpam-6526	411	8	t1	t1	NOUN
ejpam-6526	411	9	and	and	CCONJ
ejpam-6526	411	10	t2	t2	NOUN
ejpam-6526	411	11	are	be	AUX
ejpam-6526	411	12	metrizable	metrizable	ADJ
ejpam-6526	411	13	and	and	CCONJ
ejpam-6526	411	14	there	there	PRON
ejpam-6526	411	15	exists	exist	VERB
ejpam-6526	411	16	countable	countable	ADJ
ejpam-6526	411	17	family	family	NOUN
ejpam-6526	411	18	f	f	NOUN
ejpam-6526	412	1	=	=	PRON
ejpam-6526	412	2	{	{	PUNCT
ejpam-6526	412	3	fn	fn	NOUN
ejpam-6526	412	4	:	:	PUNCT
ejpam-6526	412	5	n	n	CCONJ
ejpam-6526	412	6	∈	∈	PROPN
ejpam-6526	412	7	n	n	CCONJ
ejpam-6526	412	8	}	}	PUNCT
ejpam-6526	412	9	of	of	ADP
ejpam-6526	412	10	continuous	continuous	ADJ
ejpam-6526	412	11	functions	function	NOUN
ejpam-6526	412	12	fn	fn	NOUN
ejpam-6526	412	13	:	:	PUNCT
ejpam-6526	412	14	x	x	X
ejpam-6526	412	15	→	→	PUNCT
ejpam-6526	412	16	r	r	NOUN
ejpam-6526	412	17	such	such	ADJ
ejpam-6526	412	18	that	that	SCONJ
ejpam-6526	412	19	f	f	PROPN
ejpam-6526	412	20	separates	separate	VERB
ejpam-6526	412	21	points	point	NOUN
ejpam-6526	412	22	from	from	ADP
ejpam-6526	412	23	closed	closed	ADJ
ejpam-6526	412	24	sets	set	NOUN
ejpam-6526	412	25	in	in	ADP
ejpam-6526	412	26	both	both	DET
ejpam-6526	412	27	topologies	topology	NOUN
ejpam-6526	412	28	.	.	PUNCT
ejpam-6526	413	1	let	let	VERB
ejpam-6526	413	2	ϕ1	ϕ1	NOUN
ejpam-6526	413	3	be	be	AUX
ejpam-6526	413	4	metric	metric	ADJ
ejpam-6526	413	5	function	function	NOUN
ejpam-6526	413	6	that	that	PRON
ejpam-6526	413	7	generates	generate	VERB
ejpam-6526	413	8	t1	t1	NOUN
ejpam-6526	413	9	.	.	PUNCT
ejpam-6526	414	1	define	define	VERB
ejpam-6526	414	2	new	new	ADJ
ejpam-6526	414	3	metric	metric	ADJ
ejpam-6526	414	4	ϕ′1	ϕ′1	NOUN
ejpam-6526	414	5	by	by	ADP
ejpam-6526	414	6	:	:	PUNCT
ejpam-6526	414	7	ϕ′1(x	ϕ′1(x	ADJ
ejpam-6526	414	8	,	,	PUNCT
ejpam-6526	414	9	y	y	NOUN
ejpam-6526	414	10	)	)	PUNCT
ejpam-6526	414	11	=	=	SYM
ejpam-6526	414	12	ϕ1(x	ϕ1(x	PROPN
ejpam-6526	414	13	,	,	PUNCT
ejpam-6526	414	14	y	y	NOUN
ejpam-6526	414	15	)	)	PUNCT
ejpam-6526	415	1	+	+	CCONJ
ejpam-6526	415	2	∞∑	∞∑	NUM
ejpam-6526	415	3	n=1	n=1	ADP
ejpam-6526	415	4	1	1	NUM
ejpam-6526	415	5	2n	2n	NUM
ejpam-6526	415	6	|fn(x)−	|fn(x)−	PROPN
ejpam-6526	415	7	fn(y)|	fn(y)|	AUX
ejpam-6526	415	8	1	1	NUM
ejpam-6526	415	9	+	+	CCONJ
ejpam-6526	415	10	|fn(x)−	|fn(x)−	PROPN
ejpam-6526	415	11	fn(y)|	fn(y)|	PUNCT
ejpam-6526	415	12	similarly	similarly	ADV
ejpam-6526	415	13	,	,	PUNCT
ejpam-6526	415	14	let	let	VERB
ejpam-6526	415	15	ϕ2	ϕ2	ADV
ejpam-6526	415	16	be	be	AUX
ejpam-6526	415	17	metric	metric	ADJ
ejpam-6526	415	18	function	function	NOUN
ejpam-6526	415	19	that	that	PRON
ejpam-6526	415	20	generates	generate	VERB
ejpam-6526	415	21	t2	t2	NOUN
ejpam-6526	415	22	and	and	CCONJ
ejpam-6526	415	23	define	define	VERB
ejpam-6526	415	24	:	:	PUNCT
ejpam-6526	415	25	ϕ′2(x	ϕ′2(x	PROPN
ejpam-6526	415	26	,	,	PUNCT
ejpam-6526	415	27	y	y	PROPN
ejpam-6526	415	28	)	)	PUNCT
ejpam-6526	415	29	=	=	SYM
ejpam-6526	416	1	ϕ2(x	ϕ2(x	PROPN
ejpam-6526	416	2	,	,	PUNCT
ejpam-6526	416	3	y	y	PROPN
ejpam-6526	416	4	)	)	PUNCT
ejpam-6526	417	1	+	+	CCONJ
ejpam-6526	417	2	∞∑	∞∑	NUM
ejpam-6526	417	3	n=1	n=1	ADP
ejpam-6526	417	4	1	1	NUM
ejpam-6526	417	5	2n	2n	NUM
ejpam-6526	417	6	|fn(x)−	|fn(x)−	PROPN
ejpam-6526	417	7	fn(y)|	fn(y)|	AUX
ejpam-6526	417	8	1	1	NUM
ejpam-6526	417	9	+	+	CCONJ
ejpam-6526	417	10	|fn(x)−	|fn(x)−	PROPN
ejpam-6526	417	11	fn(y)|	fn(y)|	AUX
ejpam-6526	417	12	we	we	PRON
ejpam-6526	417	13	can	can	AUX
ejpam-6526	417	14	verify	verify	VERB
ejpam-6526	417	15	that	that	PRON
ejpam-6526	417	16	ϕ′1	ϕ′1	NOUN
ejpam-6526	417	17	generates	generate	VERB
ejpam-6526	417	18	t1	t1	NOUN
ejpam-6526	417	19	and	and	CCONJ
ejpam-6526	417	20	ϕ′2	ϕ′2	NOUN
ejpam-6526	417	21	generates	generate	VERB
ejpam-6526	417	22	t2	t2	NOUN
ejpam-6526	417	23	,	,	PUNCT
ejpam-6526	417	24	thus	thus	ADV
ejpam-6526	417	25	(	(	PUNCT
ejpam-6526	417	26	x	x	X
ejpam-6526	417	27	,	,	PUNCT
ejpam-6526	417	28	t1	t1	NOUN
ejpam-6526	417	29	,	,	PUNCT
ejpam-6526	417	30	t2	t2	NOUN
ejpam-6526	417	31	)	)	PUNCT
ejpam-6526	417	32	can	can	AUX
ejpam-6526	417	33	be	be	AUX
ejpam-6526	417	34	represented	represent	VERB
ejpam-6526	417	35	through	through	ADP
ejpam-6526	417	36	bi	bi	ADJ
ejpam-6526	417	37	-	-	ADJ
ejpam-6526	417	38	metric	metric	ADJ
ejpam-6526	417	39	systems	system	NOUN
ejpam-6526	417	40	.	.	PUNCT
ejpam-6526	418	1	6.4	6.4	NUM
ejpam-6526	418	2	.	.	PUNCT
ejpam-6526	418	3	applications	application	NOUN
ejpam-6526	418	4	to	to	ADP
ejpam-6526	418	5	quasi	quasi	ADJ
ejpam-6526	418	6	-	-	ADJ
ejpam-6526	418	7	metric	metric	ADJ
ejpam-6526	418	8	structures	structure	NOUN
ejpam-6526	418	9	a	a	DET
ejpam-6526	418	10	natural	natural	ADJ
ejpam-6526	418	11	application	application	NOUN
ejpam-6526	418	12	of	of	ADP
ejpam-6526	418	13	bi	bi	ADJ
ejpam-6526	418	14	-	-	ADJ
ejpam-6526	418	15	metric	metric	ADJ
ejpam-6526	418	16	systems	system	NOUN
ejpam-6526	418	17	is	be	AUX
ejpam-6526	418	18	in	in	ADP
ejpam-6526	418	19	studying	study	VERB
ejpam-6526	418	20	quasi	quasi	ADJ
ejpam-6526	418	21	-	-	ADJ
ejpam-6526	418	22	metric	metric	ADJ
ejpam-6526	418	23	structures	structure	NOUN
ejpam-6526	418	24	.	.	PUNCT
ejpam-6526	419	1	theorem	theorem	ADJ
ejpam-6526	419	2	10	10	NUM
ejpam-6526	419	3	.	.	PUNCT
ejpam-6526	420	1	every	every	DET
ejpam-6526	420	2	quasi	quasi	ADJ
ejpam-6526	420	3	-	-	ADJ
ejpam-6526	420	4	metric	metric	ADJ
ejpam-6526	420	5	structure	structure	NOUN
ejpam-6526	420	6	(	(	PUNCT
ejpam-6526	420	7	x	x	X
ejpam-6526	420	8	,	,	PUNCT
ejpam-6526	420	9	q	q	NOUN
ejpam-6526	420	10	)	)	PUNCT
ejpam-6526	420	11	induces	induce	VERB
ejpam-6526	420	12	bi	bi	ADJ
ejpam-6526	420	13	-	-	ADJ
ejpam-6526	420	14	metric	metric	ADJ
ejpam-6526	420	15	system	system	NOUN
ejpam-6526	420	16	(	(	PUNCT
ejpam-6526	420	17	x,ϕ1	x,ϕ1	PROPN
ejpam-6526	420	18	,	,	PUNCT
ejpam-6526	420	19	ϕ2	ϕ2	ADV
ejpam-6526	420	20	)	)	PUNCT
ejpam-6526	420	21	where	where	SCONJ
ejpam-6526	420	22	:	:	PUNCT
ejpam-6526	420	23	ϕ1(x	ϕ1(x	NUM
ejpam-6526	420	24	,	,	PUNCT
ejpam-6526	420	25	y	y	NOUN
ejpam-6526	420	26	)	)	PUNCT
ejpam-6526	420	27	=	=	SYM
ejpam-6526	420	28	q(x	q(x	PROPN
ejpam-6526	420	29	,	,	PUNCT
ejpam-6526	420	30	y	y	NOUN
ejpam-6526	420	31	)	)	PUNCT
ejpam-6526	421	1	+	+	NOUN
ejpam-6526	421	2	q(y	q(y	NOUN
ejpam-6526	421	3	,	,	PUNCT
ejpam-6526	421	4	x	x	X
ejpam-6526	421	5	)	)	PUNCT
ejpam-6526	421	6	ϕ2(x	ϕ2(x	PROPN
ejpam-6526	421	7	,	,	PUNCT
ejpam-6526	421	8	y	y	NOUN
ejpam-6526	421	9	)	)	PUNCT
ejpam-6526	421	10	=	=	PUNCT
ejpam-6526	421	11	max{q(x	max{q(x	PROPN
ejpam-6526	421	12	,	,	PUNCT
ejpam-6526	421	13	y	y	NOUN
ejpam-6526	421	14	)	)	PUNCT
ejpam-6526	421	15	,	,	PUNCT
ejpam-6526	421	16	q(y	q(y	X
ejpam-6526	421	17	,	,	PUNCT
ejpam-6526	421	18	x	x	NOUN
ejpam-6526	421	19	)	)	PUNCT
ejpam-6526	421	20	}	}	PUNCT
ejpam-6526	421	21	both	both	CCONJ
ejpam-6526	421	22	ϕ1	ϕ1	NOUN
ejpam-6526	421	23	and	and	CCONJ
ejpam-6526	421	24	ϕ2	ϕ2	ADV
ejpam-6526	421	25	represent	represent	VERB
ejpam-6526	421	26	legitimate	legitimate	ADJ
ejpam-6526	421	27	metric	metric	ADJ
ejpam-6526	421	28	functions	function	NOUN
ejpam-6526	421	29	that	that	PRON
ejpam-6526	421	30	generally	generally	ADV
ejpam-6526	421	31	generate	generate	VERB
ejpam-6526	421	32	different	different	ADJ
ejpam-6526	421	33	topologies	topology	NOUN
ejpam-6526	421	34	,	,	PUNCT
ejpam-6526	421	35	thus	thus	ADV
ejpam-6526	421	36	forming	form	VERB
ejpam-6526	421	37	bi	bi	ADJ
ejpam-6526	421	38	-	-	ADJ
ejpam-6526	421	39	metric	metric	ADJ
ejpam-6526	421	40	system	system	NOUN
ejpam-6526	421	41	.	.	PUNCT
ejpam-6526	422	1	this	this	DET
ejpam-6526	422	2	product	product	NOUN
ejpam-6526	422	3	construction	construction	NOUN
ejpam-6526	422	4	aligns	align	VERB
ejpam-6526	422	5	with	with	ADP
ejpam-6526	422	6	previous	previous	ADJ
ejpam-6526	422	7	work	work	NOUN
ejpam-6526	422	8	on	on	ADP
ejpam-6526	422	9	product	product	NOUN
ejpam-6526	422	10	quasi	quasi	NOUN
ejpam-6526	422	11	-	-	NOUN
ejpam-6526	422	12	uniformities	uniformity	NOUN
ejpam-6526	422	13	[	[	X
ejpam-6526	422	14	24	24	NUM
ejpam-6526	422	15	]	]	PUNCT
ejpam-6526	422	16	.	.	PUNCT
ejpam-6526	423	1	a.	a.	PROPN
ejpam-6526	423	2	alsoboh	alsoboh	PROPN
ejpam-6526	423	3	et	et	PROPN
ejpam-6526	423	4	al	al	PROPN
ejpam-6526	423	5	.	.	PUNCT
ejpam-6526	423	6	/	/	SYM
ejpam-6526	423	7	eur	eur	PROPN
ejpam-6526	423	8	.	.	PUNCT
ejpam-6526	424	1	j.	j.	PROPN
ejpam-6526	424	2	pure	pure	PROPN
ejpam-6526	424	3	appl	appl	PROPN
ejpam-6526	424	4	.	.	PROPN
ejpam-6526	424	5	math	math	PROPN
ejpam-6526	424	6	,	,	PUNCT
ejpam-6526	424	7	18	18	NUM
ejpam-6526	424	8	(	(	PUNCT
ejpam-6526	424	9	4	4	NUM
ejpam-6526	424	10	)	)	PUNCT
ejpam-6526	424	11	(	(	PUNCT
ejpam-6526	424	12	2025	2025	NUM
ejpam-6526	424	13	)	)	PUNCT
ejpam-6526	424	14	,	,	PUNCT
ejpam-6526	424	15	6526	6526	NUM
ejpam-6526	424	16	18	18	NUM
ejpam-6526	424	17	of	of	ADP
ejpam-6526	424	18	21	21	NUM
ejpam-6526	424	19	6.5	6.5	NUM
ejpam-6526	424	20	.	.	PUNCT
ejpam-6526	425	1	applications	application	NOUN
ejpam-6526	425	2	in	in	ADP
ejpam-6526	425	3	equilibrium	equilibrium	NOUN
ejpam-6526	425	4	theory	theory	NOUN
ejpam-6526	425	5	and	and	CCONJ
ejpam-6526	425	6	differential	differential	ADJ
ejpam-6526	425	7	equations	equation	NOUN
ejpam-6526	425	8	the	the	DET
ejpam-6526	425	9	bi	bi	ADJ
ejpam-6526	425	10	-	-	ADJ
ejpam-6526	425	11	metric	metric	ADJ
ejpam-6526	425	12	framework	framework	NOUN
ejpam-6526	425	13	provides	provide	VERB
ejpam-6526	425	14	powerful	powerful	ADJ
ejpam-6526	425	15	tools	tool	NOUN
ejpam-6526	425	16	for	for	ADP
ejpam-6526	425	17	analyzing	analyze	VERB
ejpam-6526	425	18	equilibrium	equilibrium	NOUN
ejpam-6526	425	19	problems	problem	NOUN
ejpam-6526	425	20	and	and	CCONJ
ejpam-6526	425	21	differential	differential	ADJ
ejpam-6526	425	22	equations	equation	NOUN
ejpam-6526	425	23	with	with	ADP
ejpam-6526	425	24	mixed	mixed	ADJ
ejpam-6526	425	25	conditions	condition	NOUN
ejpam-6526	425	26	.	.	PUNCT
ejpam-6526	426	1	example	example	NOUN
ejpam-6526	426	2	18	18	NUM
ejpam-6526	426	3	(	(	PUNCT
ejpam-6526	426	4	nash	nash	PROPN
ejpam-6526	426	5	equilibrium	equilibrium	NOUN
ejpam-6526	426	6	with	with	ADP
ejpam-6526	426	7	dual	dual	ADJ
ejpam-6526	426	8	performance	performance	NOUN
ejpam-6526	426	9	metrics	metric	NOUN
ejpam-6526	426	10	)	)	PUNCT
ejpam-6526	426	11	.	.	PUNCT
ejpam-6526	427	1	consider	consider	VERB
ejpam-6526	427	2	a	a	DET
ejpam-6526	427	3	game	game	NOUN
ejpam-6526	427	4	where	where	SCONJ
ejpam-6526	427	5	player	player	NOUN
ejpam-6526	427	6	strategies	strategy	NOUN
ejpam-6526	427	7	s	s	VERB
ejpam-6526	427	8	∈	∈	X
ejpam-6526	427	9	s	s	X
ejpam-6526	427	10	⊂	⊂	PROPN
ejpam-6526	427	11	rn	rn	PROPN
ejpam-6526	427	12	are	be	AUX
ejpam-6526	427	13	evaluated	evaluate	VERB
ejpam-6526	427	14	by	by	ADP
ejpam-6526	427	15	:	:	PUNCT
ejpam-6526	427	16	•	•	NUM
ejpam-6526	427	17	ϕ1(s1	ϕ1(s1	PROPN
ejpam-6526	427	18	,	,	PUNCT
ejpam-6526	427	19	s2	s2	PROPN
ejpam-6526	427	20	)	)	PUNCT
ejpam-6526	427	21	=	=	NOUN
ejpam-6526	427	22	price	price	NOUN
ejpam-6526	427	23	deviation	deviation	NOUN
ejpam-6526	427	24	between	between	ADP
ejpam-6526	427	25	strategies	strategy	NOUN
ejpam-6526	427	26	•	•	NOUN
ejpam-6526	427	27	ϕ2(s1	ϕ2(s1	ADJ
ejpam-6526	427	28	,	,	PUNCT
ejpam-6526	427	29	s2	s2	PROPN
ejpam-6526	427	30	)	)	PUNCT
ejpam-6526	427	31	=	=	NOUN
ejpam-6526	427	32	market	market	NOUN
ejpam-6526	427	33	share	share	NOUN
ejpam-6526	427	34	difference	difference	NOUN
ejpam-6526	427	35	the	the	DET
ejpam-6526	427	36	bi	bi	ADJ
ejpam-6526	427	37	-	-	ADJ
ejpam-6526	427	38	metric	metric	ADJ
ejpam-6526	427	39	contraction	contraction	NOUN
ejpam-6526	427	40	theorem	theorem	VERB
ejpam-6526	427	41	guarantees	guarantee	NOUN
ejpam-6526	427	42	unique	unique	ADJ
ejpam-6526	427	43	nash	nash	ADJ
ejpam-6526	427	44	equilibrium	equilibrium	NOUN
ejpam-6526	427	45	when	when	SCONJ
ejpam-6526	427	46	best	good	ADJ
ejpam-6526	427	47	-	-	PUNCT
ejpam-6526	427	48	response	response	NOUN
ejpam-6526	427	49	mappings	mapping	NOUN
ejpam-6526	427	50	contract	contract	NOUN
ejpam-6526	427	51	in	in	ADP
ejpam-6526	427	52	both	both	DET
ejpam-6526	427	53	metrics	metric	NOUN
ejpam-6526	427	54	with	with	ADP
ejpam-6526	427	55	λ1	λ1	ADJ
ejpam-6526	427	56	,	,	PUNCT
ejpam-6526	427	57	λ2	λ2	NOUN
ejpam-6526	427	58	<	<	X
ejpam-6526	427	59	1	1	NUM
ejpam-6526	427	60	.	.	PUNCT
ejpam-6526	427	61	example	example	NOUN
ejpam-6526	428	1	19	19	NUM
ejpam-6526	428	2	(	(	PUNCT
ejpam-6526	428	3	heat	heat	NOUN
ejpam-6526	428	4	equation	equation	NOUN
ejpam-6526	428	5	with	with	ADP
ejpam-6526	428	6	mixed	mixed	ADJ
ejpam-6526	428	7	boundary	boundary	ADJ
ejpam-6526	428	8	conditions	condition	NOUN
ejpam-6526	428	9	)	)	PUNCT
ejpam-6526	428	10	.	.	PUNCT
ejpam-6526	428	11	consider	consider	VERB
ejpam-6526	428	12	the	the	DET
ejpam-6526	428	13	heat	heat	NOUN
ejpam-6526	428	14	equation	equation	NOUN
ejpam-6526	428	15	on	on	ADP
ejpam-6526	428	16	domain	domain	NOUN
ejpam-6526	428	17	ω	ω	PROPN
ejpam-6526	428	18	with	with	ADP
ejpam-6526	428	19	:	:	PUNCT
ejpam-6526	428	20	•	•	NUM
ejpam-6526	428	21	boundary	boundary	ADJ
ejpam-6526	428	22	temperature	temperature	NOUN
ejpam-6526	428	23	measured	measure	VERB
ejpam-6526	428	24	in	in	ADP
ejpam-6526	428	25	l∞	l∞	NOUN
ejpam-6526	428	26	norm	norm	NOUN
ejpam-6526	428	27	:	:	PUNCT
ejpam-6526	428	28	ϕ1(u	ϕ1(u	PROPN
ejpam-6526	428	29	,	,	PUNCT
ejpam-6526	428	30	v	v	NOUN
ejpam-6526	428	31	)	)	PUNCT
ejpam-6526	428	32	=	=	SYM
ejpam-6526	428	33	∥u−	∥u−	NUM
ejpam-6526	428	34	v∥l∞(∂ω	v∥l∞(∂ω	NOUN
ejpam-6526	428	35	)	)	PUNCT
ejpam-6526	428	36	•	•	ADP
ejpam-6526	428	37	internal	internal	ADJ
ejpam-6526	428	38	energy	energy	NOUN
ejpam-6526	428	39	measured	measure	VERB
ejpam-6526	428	40	in	in	ADP
ejpam-6526	428	41	l2	l2	NOUN
ejpam-6526	428	42	norm	norm	NOUN
ejpam-6526	428	43	:	:	PUNCT
ejpam-6526	428	44	ϕ2(u	ϕ2(u	NUM
ejpam-6526	428	45	,	,	PUNCT
ejpam-6526	428	46	v	v	NOUN
ejpam-6526	428	47	)	)	PUNCT
ejpam-6526	428	48	=	=	SYM
ejpam-6526	428	49	∥u−	∥u−	NUM
ejpam-6526	428	50	v∥l2(ω	v∥l2(ω	NOUN
ejpam-6526	428	51	)	)	PUNCT
ejpam-6526	428	52	the	the	DET
ejpam-6526	428	53	bi	bi	ADJ
ejpam-6526	428	54	-	-	ADJ
ejpam-6526	428	55	metric	metric	ADJ
ejpam-6526	428	56	system	system	NOUN
ejpam-6526	428	57	(	(	PUNCT
ejpam-6526	428	58	c(ω̄),ψ,⊕	c(ω̄),ψ,⊕	NOUN
ejpam-6526	428	59	)	)	PUNCT
ejpam-6526	428	60	enables	enable	VERB
ejpam-6526	428	61	analysis	analysis	NOUN
ejpam-6526	428	62	of	of	ADP
ejpam-6526	428	63	convergence	convergence	NOUN
ejpam-6526	428	64	in	in	ADP
ejpam-6526	428	65	both	both	PRON
ejpam-6526	428	66	boundary	boundary	ADJ
ejpam-6526	428	67	behavior	behavior	NOUN
ejpam-6526	428	68	and	and	CCONJ
ejpam-6526	428	69	energy	energy	NOUN
ejpam-6526	428	70	dissipation	dissipation	NOUN
ejpam-6526	428	71	simultaneously	simultaneously	ADV
ejpam-6526	428	72	,	,	PUNCT
ejpam-6526	428	73	providing	provide	VERB
ejpam-6526	428	74	sharper	sharp	ADJ
ejpam-6526	428	75	error	error	NOUN
ejpam-6526	428	76	estimates	estimate	NOUN
ejpam-6526	428	77	for	for	ADP
ejpam-6526	428	78	numerical	numerical	ADJ
ejpam-6526	428	79	methods	method	NOUN
ejpam-6526	428	80	.	.	PUNCT
ejpam-6526	429	1	specifically	specifically	ADV
ejpam-6526	429	2	,	,	PUNCT
ejpam-6526	429	3	for	for	ADP
ejpam-6526	429	4	the	the	DET
ejpam-6526	429	5	discrete	discrete	ADJ
ejpam-6526	429	6	heat	heat	NOUN
ejpam-6526	429	7	equation	equation	NOUN
ejpam-6526	429	8	un+1	un+1	NOUN
ejpam-6526	429	9	=	=	PUNCT
ejpam-6526	429	10	aun	aun	PROPN
ejpam-6526	429	11	+	+	CCONJ
ejpam-6526	429	12	f	f	PROPN
ejpam-6526	429	13	,	,	PUNCT
ejpam-6526	429	14	if	if	SCONJ
ejpam-6526	429	15	the	the	DET
ejpam-6526	429	16	iteration	iteration	NOUN
ejpam-6526	429	17	operator	operator	NOUN
ejpam-6526	429	18	satisfies	satisfie	NOUN
ejpam-6526	429	19	:	:	PUNCT
ejpam-6526	429	20	ψ(au1	ψ(au1	NOUN
ejpam-6526	429	21	+	+	PROPN
ejpam-6526	429	22	f	f	PROPN
ejpam-6526	429	23	,	,	PUNCT
ejpam-6526	429	24	au2	au2	NOUN
ejpam-6526	429	25	+	+	CCONJ
ejpam-6526	429	26	f	f	X
ejpam-6526	429	27	)	)	PUNCT
ejpam-6526	429	28	≤comp	≤comp	NOUN
ejpam-6526	429	29	(	(	PUNCT
ejpam-6526	429	30	λ1ϕ1(u1	λ1ϕ1(u1	X
ejpam-6526	429	31	,	,	PUNCT
ejpam-6526	429	32	u2	u2	PROPN
ejpam-6526	429	33	)	)	PUNCT
ejpam-6526	429	34	,	,	PUNCT
ejpam-6526	429	35	λ2ϕ2(u1	λ2ϕ2(u1	NOUN
ejpam-6526	429	36	,	,	PUNCT
ejpam-6526	429	37	u2	u2	PROPN
ejpam-6526	429	38	)	)	PUNCT
ejpam-6526	429	39	)	)	PUNCT
ejpam-6526	430	1	then	then	ADV
ejpam-6526	430	2	we	we	PRON
ejpam-6526	430	3	obtain	obtain	VERB
ejpam-6526	430	4	convergence	convergence	NOUN
ejpam-6526	430	5	rates	rate	NOUN
ejpam-6526	430	6	:	:	PUNCT
ejpam-6526	430	7	•	•	NUM
ejpam-6526	430	8	boundary	boundary	ADJ
ejpam-6526	430	9	error	error	NOUN
ejpam-6526	430	10	:	:	PUNCT
ejpam-6526	430	11	∥un	∥un	PROPN
ejpam-6526	430	12	−	−	NOUN
ejpam-6526	430	13	u∗∥l∞(∂ω	u∗∥l∞(∂ω	VERB
ejpam-6526	430	14	)	)	PUNCT
ejpam-6526	430	15	≤	≤	NOUN
ejpam-6526	430	16	λn1∥u0	λn1∥u0	NOUN
ejpam-6526	430	17	−	−	NOUN
ejpam-6526	430	18	u∗∥l∞(∂ω	u∗∥l∞(∂ω	VERB
ejpam-6526	430	19	)	)	PUNCT
ejpam-6526	430	20	•	•	NOUN
ejpam-6526	430	21	energy	energy	NOUN
ejpam-6526	430	22	error	error	NOUN
ejpam-6526	430	23	:	:	PUNCT
ejpam-6526	430	24	∥un	∥un	PROPN
ejpam-6526	430	25	−	−	PROPN
ejpam-6526	430	26	u∗∥l2(ω	u∗∥l2(ω	ADV
ejpam-6526	430	27	)	)	PUNCT
ejpam-6526	430	28	≤	≤	NOUN
ejpam-6526	430	29	λn2∥u0	λn2∥u0	X
ejpam-6526	430	30	−	−	X
ejpam-6526	430	31	u∗∥l2(ω	u∗∥l2(ω	ADJ
ejpam-6526	430	32	)	)	PUNCT
ejpam-6526	430	33	6.6	6.6	NUM
ejpam-6526	430	34	.	.	PUNCT
ejpam-6526	431	1	experimental	experimental	ADJ
ejpam-6526	431	2	results	result	NOUN
ejpam-6526	431	3	to	to	PART
ejpam-6526	431	4	illustrate	illustrate	VERB
ejpam-6526	431	5	our	our	PRON
ejpam-6526	431	6	theoretical	theoretical	ADJ
ejpam-6526	431	7	findings	finding	NOUN
ejpam-6526	431	8	,	,	PUNCT
ejpam-6526	431	9	we	we	PRON
ejpam-6526	431	10	present	present	VERB
ejpam-6526	431	11	numerical	numerical	ADJ
ejpam-6526	431	12	experiments	experiment	NOUN
ejpam-6526	431	13	on	on	ADP
ejpam-6526	431	14	specific	specific	ADJ
ejpam-6526	431	15	bi	bi	ADJ
ejpam-6526	431	16	-	-	ADJ
ejpam-6526	431	17	metric	metric	ADJ
ejpam-6526	431	18	systems	system	NOUN
ejpam-6526	431	19	.	.	PUNCT
ejpam-6526	432	1	table	table	NOUN
ejpam-6526	432	2	1	1	NUM
ejpam-6526	432	3	:	:	PUNCT
ejpam-6526	432	4	convergence	convergence	NOUN
ejpam-6526	432	5	characteristics	characteristic	NOUN
ejpam-6526	432	6	for	for	ADP
ejpam-6526	432	7	equilibrium	equilibrium	NOUN
ejpam-6526	432	8	point	point	NOUN
ejpam-6526	432	9	iterations	iteration	NOUN
ejpam-6526	432	10	in	in	ADP
ejpam-6526	432	11	various	various	ADJ
ejpam-6526	432	12	bi	bi	ADJ
ejpam-6526	432	13	-	-	ADJ
ejpam-6526	432	14	metric	metric	ADJ
ejpam-6526	432	15	systems	system	NOUN
ejpam-6526	432	16	system	system	NOUN
ejpam-6526	432	17	contraction	contraction	NOUN
ejpam-6526	432	18	parameters	parameter	VERB
ejpam-6526	432	19	iteration	iteration	NOUN
ejpam-6526	432	20	count	count	NOUN
ejpam-6526	432	21	precision	precision	NOUN
ejpam-6526	432	22	level	level	NOUN
ejpam-6526	432	23	(	(	PUNCT
ejpam-6526	432	24	r2	r2	NOUN
ejpam-6526	432	25	,	,	PUNCT
ejpam-6526	432	26	ϕ1	ϕ1	NOUN
ejpam-6526	432	27	,	,	PUNCT
ejpam-6526	432	28	ϕ2	ϕ2	ADV
ejpam-6526	432	29	)	)	PUNCT
ejpam-6526	432	30	λ1	λ1	PROPN
ejpam-6526	432	31	=	=	SYM
ejpam-6526	432	32	0.3	0.3	NUM
ejpam-6526	432	33	,	,	PUNCT
ejpam-6526	432	34	λ2	λ2	NOUN
ejpam-6526	432	35	=	=	NOUN
ejpam-6526	432	36	0.4	0.4	NUM
ejpam-6526	432	37	12	12	NUM
ejpam-6526	432	38	1.2×	1.2×	NUM
ejpam-6526	432	39	10−6	10−6	NUM
ejpam-6526	432	40	(	(	PUNCT
ejpam-6526	432	41	r3	r3	PROPN
ejpam-6526	432	42	,	,	PUNCT
ejpam-6526	432	43	ϕ1	ϕ1	NOUN
ejpam-6526	432	44	,	,	PUNCT
ejpam-6526	432	45	ϕ2	ϕ2	ADV
ejpam-6526	432	46	)	)	PUNCT
ejpam-6526	432	47	λ1	λ1	PROPN
ejpam-6526	432	48	=	=	SYM
ejpam-6526	432	49	0.2	0.2	NUM
ejpam-6526	432	50	,	,	PUNCT
ejpam-6526	432	51	λ2	λ2	NOUN
ejpam-6526	432	52	=	=	NOUN
ejpam-6526	432	53	0.3	0.3	NUM
ejpam-6526	432	54	9	9	NUM
ejpam-6526	432	55	5.7×	5.7×	NUM
ejpam-6526	432	56	10−7	10−7	NUM
ejpam-6526	432	57	(	(	PUNCT
ejpam-6526	432	58	ℓ2	ℓ2	NOUN
ejpam-6526	432	59	,	,	PUNCT
ejpam-6526	432	60	ϕ1	ϕ1	NOUN
ejpam-6526	432	61	,	,	PUNCT
ejpam-6526	432	62	ϕ2	ϕ2	ADV
ejpam-6526	432	63	)	)	PUNCT
ejpam-6526	432	64	λ1	λ1	PROPN
ejpam-6526	432	65	=	=	SYM
ejpam-6526	432	66	0.4	0.4	NUM
ejpam-6526	432	67	,	,	PUNCT
ejpam-6526	432	68	λ2	λ2	NOUN
ejpam-6526	432	69	=	=	NOUN
ejpam-6526	432	70	0.3	0.3	NUM
ejpam-6526	432	71	15	15	NUM
ejpam-6526	432	72	3.9×	3.9×	NUM
ejpam-6526	432	73	10−6	10−6	NUM
ejpam-6526	432	74	(	(	PUNCT
ejpam-6526	432	75	c[0	c[0	PROPN
ejpam-6526	432	76	,	,	PUNCT
ejpam-6526	432	77	1	1	NUM
ejpam-6526	432	78	]	]	PUNCT
ejpam-6526	432	79	,	,	PUNCT
ejpam-6526	432	80	ϕ1	ϕ1	NOUN
ejpam-6526	432	81	,	,	PUNCT
ejpam-6526	432	82	ϕ2	ϕ2	ADV
ejpam-6526	432	83	)	)	PUNCT
ejpam-6526	432	84	λ1	λ1	PROPN
ejpam-6526	432	85	=	=	SYM
ejpam-6526	432	86	0.5	0.5	NUM
ejpam-6526	432	87	,	,	PUNCT
ejpam-6526	432	88	λ2	λ2	NOUN
ejpam-6526	432	89	=	=	NOUN
ejpam-6526	432	90	0.2	0.2	NUM
ejpam-6526	432	91	18	18	NUM
ejpam-6526	432	92	8.3×	8.3×	NUM
ejpam-6526	432	93	10−6	10−6	NUM
ejpam-6526	432	94	we	we	PRON
ejpam-6526	432	95	observe	observe	VERB
ejpam-6526	432	96	that	that	SCONJ
ejpam-6526	432	97	convergence	convergence	NOUN
ejpam-6526	432	98	characteristics	characteristic	NOUN
ejpam-6526	432	99	depend	depend	VERB
ejpam-6526	432	100	on	on	ADP
ejpam-6526	432	101	combined	combined	ADJ
ejpam-6526	432	102	contraction	contraction	NOUN
ejpam-6526	432	103	parameters	parameter	NOUN
ejpam-6526	433	1	λ1	λ1	ADJ
ejpam-6526	433	2	+	+	CCONJ
ejpam-6526	433	3	λ2	λ2	NOUN
ejpam-6526	433	4	,	,	PUNCT
ejpam-6526	433	5	with	with	ADP
ejpam-6526	433	6	lower	low	ADJ
ejpam-6526	433	7	parameter	parameter	NOUN
ejpam-6526	433	8	values	value	NOUN
ejpam-6526	433	9	yielding	yield	VERB
ejpam-6526	433	10	faster	fast	ADJ
ejpam-6526	433	11	convergence	convergence	NOUN
ejpam-6526	433	12	,	,	PUNCT
ejpam-6526	433	13	aligning	align	VERB
ejpam-6526	433	14	with	with	ADP
ejpam-6526	433	15	our	our	PRON
ejpam-6526	433	16	theoretical	theoretical	ADJ
ejpam-6526	433	17	analysis	analysis	NOUN
ejpam-6526	433	18	in	in	ADP
ejpam-6526	433	19	the	the	DET
ejpam-6526	433	20	contraction	contraction	NOUN
ejpam-6526	433	21	theorem	theorem	VERB
ejpam-6526	433	22	.	.	PUNCT
ejpam-6526	433	23	a.	a.	PROPN
ejpam-6526	433	24	alsoboh	alsoboh	PROPN
ejpam-6526	433	25	et	et	PROPN
ejpam-6526	433	26	al	al	PROPN
ejpam-6526	433	27	.	.	PUNCT
ejpam-6526	433	28	/	/	SYM
ejpam-6526	433	29	eur	eur	PROPN
ejpam-6526	433	30	.	.	PUNCT
ejpam-6526	434	1	j.	j.	PROPN
ejpam-6526	434	2	pure	pure	PROPN
ejpam-6526	434	3	appl	appl	PROPN
ejpam-6526	434	4	.	.	PROPN
ejpam-6526	434	5	math	math	PROPN
ejpam-6526	434	6	,	,	PUNCT
ejpam-6526	434	7	18	18	NUM
ejpam-6526	434	8	(	(	PUNCT
ejpam-6526	434	9	4	4	NUM
ejpam-6526	434	10	)	)	PUNCT
ejpam-6526	434	11	(	(	PUNCT
ejpam-6526	434	12	2025	2025	NUM
ejpam-6526	434	13	)	)	PUNCT
ejpam-6526	434	14	,	,	PUNCT
ejpam-6526	434	15	6526	6526	NUM
ejpam-6526	434	16	19	19	NUM
ejpam-6526	434	17	of	of	ADP
ejpam-6526	434	18	21	21	NUM
ejpam-6526	434	19	7	7	NUM
ejpam-6526	434	20	.	.	PUNCT
ejpam-6526	435	1	conclusions	conclusion	NOUN
ejpam-6526	435	2	and	and	CCONJ
ejpam-6526	435	3	future	future	ADJ
ejpam-6526	435	4	research	research	NOUN
ejpam-6526	435	5	directions	direction	NOUN
ejpam-6526	435	6	we	we	PRON
ejpam-6526	435	7	have	have	AUX
ejpam-6526	435	8	established	establish	VERB
ejpam-6526	435	9	and	and	CCONJ
ejpam-6526	435	10	developed	develop	VERB
ejpam-6526	435	11	bi	bi	ADJ
ejpam-6526	435	12	-	-	ADJ
ejpam-6526	435	13	metric	metric	ADJ
ejpam-6526	435	14	system	system	NOUN
ejpam-6526	435	15	theory	theory	NOUN
ejpam-6526	435	16	as	as	ADP
ejpam-6526	435	17	natural	natural	ADJ
ejpam-6526	435	18	extension	extension	NOUN
ejpam-6526	435	19	of	of	ADP
ejpam-6526	435	20	classical	classical	ADJ
ejpam-6526	435	21	metric	metric	ADJ
ejpam-6526	435	22	spaces	space	NOUN
ejpam-6526	435	23	within	within	ADP
ejpam-6526	435	24	bitopological	bitopological	ADJ
ejpam-6526	435	25	environments	environment	NOUN
ejpam-6526	435	26	.	.	PUNCT
ejpam-6526	436	1	our	our	PRON
ejpam-6526	436	2	research	research	NOUN
ejpam-6526	436	3	has	have	AUX
ejpam-6526	436	4	validated	validate	VERB
ejpam-6526	436	5	fundamental	fundamental	ADJ
ejpam-6526	436	6	properties	property	NOUN
ejpam-6526	436	7	,	,	PUNCT
ejpam-6526	436	8	explored	explore	VERB
ejpam-6526	436	9	functional	functional	ADJ
ejpam-6526	436	10	-	-	PUNCT
ejpam-6526	436	11	analytical	analytical	ADJ
ejpam-6526	436	12	connections	connection	NOUN
ejpam-6526	436	13	,	,	PUNCT
ejpam-6526	436	14	and	and	CCONJ
ejpam-6526	436	15	provided	provide	VERB
ejpam-6526	436	16	applications	application	NOUN
ejpam-6526	436	17	in	in	ADP
ejpam-6526	436	18	equilibrium	equilibrium	NOUN
ejpam-6526	436	19	theory	theory	NOUN
ejpam-6526	436	20	and	and	CCONJ
ejpam-6526	436	21	differential	differential	ADJ
ejpam-6526	436	22	equations	equation	NOUN
ejpam-6526	436	23	.	.	PUNCT
ejpam-6526	437	1	the	the	DET
ejpam-6526	437	2	bi	bi	ADJ
ejpam-6526	437	3	-	-	ADJ
ejpam-6526	437	4	metric	metric	ADJ
ejpam-6526	437	5	approach	approach	NOUN
ejpam-6526	437	6	offers	offer	VERB
ejpam-6526	437	7	several	several	ADJ
ejpam-6526	437	8	significant	significant	ADJ
ejpam-6526	437	9	advantages	advantage	NOUN
ejpam-6526	437	10	:	:	PUNCT
ejpam-6526	437	11	(	(	PUNCT
ejpam-6526	437	12	i	i	NOUN
ejpam-6526	437	13	)	)	PUNCT
ejpam-6526	437	14	it	it	PRON
ejpam-6526	437	15	provides	provide	VERB
ejpam-6526	437	16	an	an	DET
ejpam-6526	437	17	integrated	integrate	VERB
ejpam-6526	437	18	framework	framework	NOUN
ejpam-6526	437	19	for	for	ADP
ejpam-6526	437	20	analyzing	analyze	VERB
ejpam-6526	437	21	spaces	space	NOUN
ejpam-6526	437	22	with	with	ADP
ejpam-6526	437	23	bitopological	bitopological	ADJ
ejpam-6526	437	24	structures	structure	NOUN
ejpam-6526	437	25	.	.	PUNCT
ejpam-6526	438	1	(	(	PUNCT
ejpam-6526	438	2	ii	ii	X
ejpam-6526	438	3	)	)	PUNCT
ejpam-6526	438	4	it	it	PRON
ejpam-6526	438	5	extends	extend	VERB
ejpam-6526	438	6	classical	classical	ADJ
ejpam-6526	438	7	metric	metric	ADJ
ejpam-6526	438	8	theory	theory	NOUN
ejpam-6526	438	9	in	in	ADP
ejpam-6526	438	10	natural	natural	ADJ
ejpam-6526	438	11	and	and	CCONJ
ejpam-6526	438	12	intuitive	intuitive	ADJ
ejpam-6526	438	13	manner	manner	NOUN
ejpam-6526	438	14	.	.	PUNCT
ejpam-6526	439	1	(	(	PUNCT
ejpam-6526	439	2	iii	iii	X
ejpam-6526	439	3	)	)	PUNCT
ejpam-6526	439	4	it	it	PRON
ejpam-6526	439	5	offers	offer	VERB
ejpam-6526	439	6	innovative	innovative	ADJ
ejpam-6526	439	7	perspectives	perspective	NOUN
ejpam-6526	439	8	into	into	ADP
ejpam-6526	439	9	relationships	relationship	NOUN
ejpam-6526	439	10	between	between	ADP
ejpam-6526	439	11	different	different	ADJ
ejpam-6526	439	12	convergence	convergence	NOUN
ejpam-6526	439	13	types	type	NOUN
ejpam-6526	439	14	in	in	ADP
ejpam-6526	439	15	function	function	NOUN
ejpam-6526	439	16	spaces	space	NOUN
ejpam-6526	439	17	.	.	PUNCT
ejpam-6526	440	1	(	(	PUNCT
ejpam-6526	440	2	iv	iv	X
ejpam-6526	440	3	)	)	PUNCT
ejpam-6526	440	4	it	it	PRON
ejpam-6526	440	5	enables	enable	VERB
ejpam-6526	440	6	enhanced	enhanced	ADJ
ejpam-6526	440	7	equilibrium	equilibrium	NOUN
ejpam-6526	440	8	theorems	theorem	NOUN
ejpam-6526	440	9	accommodating	accommodate	VERB
ejpam-6526	440	10	mixed	mixed	ADJ
ejpam-6526	440	11	contractivity	contractivity	NOUN
ejpam-6526	440	12	conditions	condition	NOUN
ejpam-6526	440	13	.	.	PUNCT
ejpam-6526	441	1	future	future	ADJ
ejpam-6526	441	2	research	research	NOUN
ejpam-6526	441	3	directions	direction	NOUN
ejpam-6526	441	4	include	include	VERB
ejpam-6526	441	5	:	:	PUNCT
ejpam-6526	441	6	•	•	NUM
ejpam-6526	441	7	further	further	ADJ
ejpam-6526	441	8	investigation	investigation	NOUN
ejpam-6526	441	9	of	of	ADP
ejpam-6526	441	10	relationships	relationship	NOUN
ejpam-6526	441	11	between	between	ADP
ejpam-6526	441	12	bi	bi	ADJ
ejpam-6526	441	13	-	-	ADJ
ejpam-6526	441	14	metric	metric	ADJ
ejpam-6526	441	15	systems	system	NOUN
ejpam-6526	441	16	and	and	CCONJ
ejpam-6526	441	17	other	other	ADJ
ejpam-6526	441	18	mathematical	mathematical	ADJ
ejpam-6526	441	19	frameworks	framework	NOUN
ejpam-6526	441	20	such	such	ADJ
ejpam-6526	441	21	as	as	ADP
ejpam-6526	441	22	fuzzy	fuzzy	ADJ
ejpam-6526	441	23	topological	topological	ADJ
ejpam-6526	441	24	spaces	space	NOUN
ejpam-6526	441	25	and	and	CCONJ
ejpam-6526	441	26	rough	rough	ADJ
ejpam-6526	441	27	sets	set	NOUN
ejpam-6526	441	28	.	.	PUNCT
ejpam-6526	442	1	•	•	NUM
ejpam-6526	442	2	development	development	NOUN
ejpam-6526	442	3	of	of	ADP
ejpam-6526	442	4	computational	computational	ADJ
ejpam-6526	442	5	methodologies	methodology	NOUN
ejpam-6526	442	6	based	base	VERB
ejpam-6526	442	7	on	on	ADP
ejpam-6526	442	8	bi	bi	ADJ
ejpam-6526	442	9	-	-	ADJ
ejpam-6526	442	10	metric	metric	ADJ
ejpam-6526	442	11	equilibrium	equilibrium	NOUN
ejpam-6526	442	12	theorems	theorem	NOUN
ejpam-6526	442	13	.	.	PUNCT
ejpam-6526	443	1	•	•	NUM
ejpam-6526	443	2	application	application	NOUN
ejpam-6526	443	3	of	of	ADP
ejpam-6526	443	4	bi	bi	ADJ
ejpam-6526	443	5	-	-	ADJ
ejpam-6526	443	6	metric	metric	ADJ
ejpam-6526	443	7	systems	system	NOUN
ejpam-6526	443	8	to	to	PART
ejpam-6526	443	9	differential	differential	VERB
ejpam-6526	443	10	equations	equation	NOUN
ejpam-6526	443	11	with	with	ADP
ejpam-6526	443	12	heterogeneous	heterogeneous	ADJ
ejpam-6526	443	13	boundary	boundary	ADJ
ejpam-6526	443	14	conditions	condition	NOUN
ejpam-6526	443	15	.	.	PUNCT
ejpam-6526	444	1	•	•	NUM
ejpam-6526	444	2	extension	extension	NOUN
ejpam-6526	444	3	of	of	ADP
ejpam-6526	444	4	theoretical	theoretical	ADJ
ejpam-6526	444	5	framework	framework	NOUN
ejpam-6526	444	6	to	to	ADP
ejpam-6526	444	7	multi	multi	ADJ
ejpam-6526	444	8	-	-	ADJ
ejpam-6526	444	9	metric	metric	ADJ
ejpam-6526	444	10	structures	structure	NOUN
ejpam-6526	444	11	with	with	ADP
ejpam-6526	444	12	more	more	ADJ
ejpam-6526	444	13	than	than	ADP
ejpam-6526	444	14	two	two	NUM
ejpam-6526	444	15	components	component	NOUN
ejpam-6526	444	16	.	.	PUNCT
ejpam-6526	445	1	•	•	NUM
ejpam-6526	445	2	exploration	exploration	NOUN
ejpam-6526	445	3	of	of	ADP
ejpam-6526	445	4	applications	application	NOUN
ejpam-6526	445	5	in	in	ADP
ejpam-6526	445	6	quantum	quantum	ADJ
ejpam-6526	445	7	information	information	NOUN
ejpam-6526	445	8	theory	theory	NOUN
ejpam-6526	445	9	,	,	PUNCT
ejpam-6526	445	10	machine	machine	NOUN
ejpam-6526	445	11	learning	learning	NOUN
ejpam-6526	445	12	algorithms	algorithm	NOUN
ejpam-6526	445	13	,	,	PUNCT
ejpam-6526	445	14	and	and	CCONJ
ejpam-6526	445	15	data	datum	NOUN
ejpam-6526	445	16	science	science	NOUN
ejpam-6526	445	17	.	.	PUNCT
ejpam-6526	446	1	the	the	DET
ejpam-6526	446	2	mathematical	mathematical	ADJ
ejpam-6526	446	3	architecture	architecture	NOUN
ejpam-6526	446	4	described	describe	VERB
ejpam-6526	446	5	in	in	ADP
ejpam-6526	446	6	this	this	DET
ejpam-6526	446	7	paper	paper	NOUN
ejpam-6526	446	8	enables	enable	VERB
ejpam-6526	446	9	innovative	innovative	ADJ
ejpam-6526	446	10	perspectives	perspective	NOUN
ejpam-6526	446	11	on	on	ADP
ejpam-6526	446	12	interrelationships	interrelationship	NOUN
ejpam-6526	446	13	between	between	ADP
ejpam-6526	446	14	metric	metric	ADJ
ejpam-6526	446	15	structures	structure	NOUN
ejpam-6526	446	16	and	and	CCONJ
ejpam-6526	446	17	bitopological	bitopological	ADJ
ejpam-6526	446	18	domains	domain	NOUN
ejpam-6526	446	19	,	,	PUNCT
ejpam-6526	446	20	with	with	ADP
ejpam-6526	446	21	significant	significant	ADJ
ejpam-6526	446	22	applications	application	NOUN
ejpam-6526	446	23	to	to	ADP
ejpam-6526	446	24	functional	functional	ADJ
ejpam-6526	446	25	transformation	transformation	NOUN
ejpam-6526	446	26	theory	theory	NOUN
ejpam-6526	446	27	and	and	CCONJ
ejpam-6526	446	28	other	other	ADJ
ejpam-6526	446	29	mathematical	mathematical	ADJ
ejpam-6526	446	30	analysis	analysis	NOUN
ejpam-6526	446	31	domains	domain	NOUN
ejpam-6526	446	32	.	.	PUNCT
ejpam-6526	447	1	references	reference	NOUN
ejpam-6526	447	2	[	[	X
ejpam-6526	447	3	1	1	X
ejpam-6526	447	4	]	]	PUNCT
ejpam-6526	447	5	j.	j.	PROPN
ejpam-6526	447	6	c.	c.	PROPN
ejpam-6526	447	7	kelly	kelly	PROPN
ejpam-6526	447	8	.	.	PUNCT
ejpam-6526	448	1	bitopological	bitopological	ADJ
ejpam-6526	448	2	spaces	space	NOUN
ejpam-6526	448	3	.	.	PUNCT
ejpam-6526	449	1	proceedings	proceeding	NOUN
ejpam-6526	449	2	of	of	ADP
ejpam-6526	449	3	the	the	DET
ejpam-6526	449	4	london	london	PROPN
ejpam-6526	449	5	mathematical	mathematical	ADJ
ejpam-6526	449	6	society	society	NOUN
ejpam-6526	449	7	,	,	PUNCT
ejpam-6526	449	8	3(1):71–89	3(1):71–89	NUM
ejpam-6526	449	9	,	,	PUNCT
ejpam-6526	449	10	1963	1963	NUM
ejpam-6526	449	11	.	.	PUNCT
ejpam-6526	450	1	[	[	X
ejpam-6526	450	2	2	2	X
ejpam-6526	450	3	]	]	PUNCT
ejpam-6526	450	4	y.	y.	PROPN
ejpam-6526	450	5	y.	y.	PROPN
ejpam-6526	450	6	yousif	yousif	PROPN
ejpam-6526	450	7	and	and	CCONJ
ejpam-6526	450	8	l.	l.	PROPN
ejpam-6526	450	9	a.	a.	PROPN
ejpam-6526	450	10	hussain	hussain	PROPN
ejpam-6526	450	11	.	.	PUNCT
ejpam-6526	451	1	fibrewise	fibrewise	PROPN
ejpam-6526	451	2	ij	ij	ADJ
ejpam-6526	451	3	-	-	ADJ
ejpam-6526	451	4	perfect	perfect	ADJ
ejpam-6526	451	5	bitopological	bitopological	ADJ
ejpam-6526	451	6	spaces	space	NOUN
ejpam-6526	451	7	.	.	PUNCT
ejpam-6526	452	1	journal	journal	PROPN
ejpam-6526	452	2	of	of	ADP
ejpam-6526	452	3	physics	physics	PROPN
ejpam-6526	452	4	:	:	PUNCT
ejpam-6526	452	5	conference	conference	NOUN
ejpam-6526	452	6	series	series	NOUN
ejpam-6526	452	7	,	,	PUNCT
ejpam-6526	452	8	1003:1–12	1003:1–12	PROPN
ejpam-6526	452	9	,	,	PUNCT
ejpam-6526	452	10	2018	2018	NUM
ejpam-6526	452	11	.	.	PUNCT
ejpam-6526	453	1	a.	a.	PROPN
ejpam-6526	453	2	alsoboh	alsoboh	PROPN
ejpam-6526	453	3	et	et	PROPN
ejpam-6526	453	4	al	al	PROPN
ejpam-6526	453	5	.	.	PUNCT
ejpam-6526	453	6	/	/	SYM
ejpam-6526	453	7	eur	eur	PROPN
ejpam-6526	453	8	.	.	PUNCT
ejpam-6526	454	1	j.	j.	PROPN
ejpam-6526	454	2	pure	pure	PROPN
ejpam-6526	454	3	appl	appl	PROPN
ejpam-6526	454	4	.	.	PROPN
ejpam-6526	454	5	math	math	PROPN
ejpam-6526	454	6	,	,	PUNCT
ejpam-6526	454	7	18	18	NUM
ejpam-6526	454	8	(	(	PUNCT
ejpam-6526	454	9	4	4	NUM
ejpam-6526	454	10	)	)	PUNCT
ejpam-6526	454	11	(	(	PUNCT
ejpam-6526	454	12	2025	2025	NUM
ejpam-6526	454	13	)	)	PUNCT
ejpam-6526	454	14	,	,	PUNCT
ejpam-6526	454	15	6526	6526	NUM
ejpam-6526	454	16	20	20	NUM
ejpam-6526	454	17	of	of	ADP
ejpam-6526	454	18	21	21	NUM
ejpam-6526	454	19	[	[	X
ejpam-6526	454	20	3	3	NUM
ejpam-6526	454	21	]	]	PUNCT
ejpam-6526	454	22	a.	a.	NOUN
ejpam-6526	454	23	garcía	garcía	NOUN
ejpam-6526	454	24	-	-	PUNCT
ejpam-6526	454	25	máynez	máynez	NOUN
ejpam-6526	454	26	and	and	CCONJ
ejpam-6526	454	27	a.	a.	PROPN
ejpam-6526	454	28	pimienta	pimienta	PROPN
ejpam-6526	454	29	.	.	PUNCT
ejpam-6526	454	30	symmetry	symmetry	NOUN
ejpam-6526	454	31	in	in	ADP
ejpam-6526	454	32	bitopological	bitopological	ADJ
ejpam-6526	454	33	spaces	space	NOUN
ejpam-6526	454	34	.	.	PUNCT
ejpam-6526	455	1	topology	topology	NOUN
ejpam-6526	455	2	and	and	CCONJ
ejpam-6526	455	3	its	its	PRON
ejpam-6526	455	4	applications	application	NOUN
ejpam-6526	455	5	,	,	PUNCT
ejpam-6526	455	6	258:112–125	258:112–125	NUM
ejpam-6526	455	7	,	,	PUNCT
ejpam-6526	455	8	2019	2019	NUM
ejpam-6526	455	9	.	.	PUNCT
ejpam-6526	456	1	[	[	X
ejpam-6526	456	2	4	4	X
ejpam-6526	456	3	]	]	PUNCT
ejpam-6526	456	4	j.	j.	PROPN
ejpam-6526	456	5	chen	chen	PROPN
ejpam-6526	456	6	and	and	CCONJ
ejpam-6526	456	7	s.	s.	PROPN
ejpam-6526	456	8	li	li	PROPN
ejpam-6526	456	9	.	.	PROPN
ejpam-6526	456	10	fuzzy	fuzzy	ADJ
ejpam-6526	456	11	bitopological	bitopological	ADJ
ejpam-6526	456	12	spaces	space	NOUN
ejpam-6526	456	13	and	and	CCONJ
ejpam-6526	456	14	separation	separation	NOUN
ejpam-6526	456	15	axioms	axiom	NOUN
ejpam-6526	456	16	.	.	PUNCT
ejpam-6526	457	1	fuzzy	fuzzy	ADJ
ejpam-6526	457	2	sets	set	NOUN
ejpam-6526	457	3	and	and	CCONJ
ejpam-6526	457	4	systems	system	NOUN
ejpam-6526	457	5	,	,	PUNCT
ejpam-6526	457	6	389:96–113	389:96–113	NUM
ejpam-6526	457	7	,	,	PUNCT
ejpam-6526	457	8	2020	2020	NUM
ejpam-6526	457	9	.	.	PUNCT
ejpam-6526	458	1	[	[	X
ejpam-6526	458	2	5	5	NUM
ejpam-6526	458	3	]	]	PUNCT
ejpam-6526	458	4	m.	m.	PROPN
ejpam-6526	458	5	b.	b.	PROPN
ejpam-6526	458	6	smyth	smyth	PROPN
ejpam-6526	458	7	.	.	PUNCT
ejpam-6526	459	1	bitopology	bitopology	NOUN
ejpam-6526	459	2	and	and	CCONJ
ejpam-6526	459	3	domain	domain	NOUN
ejpam-6526	459	4	theory	theory	NOUN
ejpam-6526	459	5	.	.	PUNCT
ejpam-6526	460	1	electronic	electronic	ADJ
ejpam-6526	460	2	notes	note	NOUN
ejpam-6526	460	3	in	in	ADP
ejpam-6526	460	4	theoretical	theoretical	ADJ
ejpam-6526	460	5	computer	computer	NOUN
ejpam-6526	460	6	science	science	NOUN
ejpam-6526	460	7	,	,	PUNCT
ejpam-6526	460	8	345:221–235	345:221–235	NUM
ejpam-6526	460	9	,	,	PUNCT
ejpam-6526	460	10	2018	2018	NUM
ejpam-6526	460	11	.	.	PUNCT
ejpam-6526	461	1	[	[	X
ejpam-6526	461	2	6	6	NUM
ejpam-6526	461	3	]	]	PUNCT
ejpam-6526	461	4	s.	s.	PROPN
ejpam-6526	461	5	kumar	kumar	PROPN
ejpam-6526	461	6	and	and	CCONJ
ejpam-6526	461	7	p.	p.	PROPN
ejpam-6526	461	8	singh	singh	PROPN
ejpam-6526	461	9	.	.	PUNCT
ejpam-6526	462	1	rough	rough	ADJ
ejpam-6526	462	2	sets	set	NOUN
ejpam-6526	462	3	in	in	ADP
ejpam-6526	462	4	bitopological	bitopological	ADJ
ejpam-6526	462	5	spaces	space	NOUN
ejpam-6526	462	6	.	.	PUNCT
ejpam-6526	463	1	information	information	NOUN
ejpam-6526	463	2	sciences	sciences	PROPN
ejpam-6526	463	3	,	,	PUNCT
ejpam-6526	463	4	556:289–307	556:289–307	NUM
ejpam-6526	463	5	,	,	PUNCT
ejpam-6526	463	6	2021	2021	NUM
ejpam-6526	463	7	.	.	PUNCT
ejpam-6526	464	1	[	[	X
ejpam-6526	464	2	7	7	X
ejpam-6526	464	3	]	]	PUNCT
ejpam-6526	464	4	j.	j.	PROPN
ejpam-6526	464	5	martinez	martinez	PROPN
ejpam-6526	464	6	,	,	PUNCT
ejpam-6526	464	7	c.	c.	PROPN
ejpam-6526	464	8	rodriguez	rodriguez	PROPN
ejpam-6526	464	9	,	,	PUNCT
ejpam-6526	464	10	and	and	CCONJ
ejpam-6526	464	11	a.	a.	NOUN
ejpam-6526	464	12	lopez	lopez	PROPN
ejpam-6526	464	13	.	.	PUNCT
ejpam-6526	465	1	machine	machine	NOUN
ejpam-6526	465	2	learning	learn	VERB
ejpam-6526	465	3	applications	application	NOUN
ejpam-6526	465	4	of	of	ADP
ejpam-6526	465	5	bitopological	bitopological	ADJ
ejpam-6526	465	6	data	datum	NOUN
ejpam-6526	465	7	structures	structure	NOUN
ejpam-6526	465	8	.	.	PUNCT
ejpam-6526	466	1	journal	journal	NOUN
ejpam-6526	466	2	of	of	ADP
ejpam-6526	466	3	machine	machine	NOUN
ejpam-6526	466	4	learning	learn	VERB
ejpam-6526	466	5	research	research	NOUN
ejpam-6526	466	6	,	,	PUNCT
ejpam-6526	466	7	23(142):1–28	23(142):1–28	PROPN
ejpam-6526	466	8	,	,	PUNCT
ejpam-6526	466	9	2022	2022	NUM
ejpam-6526	466	10	.	.	PUNCT
ejpam-6526	467	1	[	[	X
ejpam-6526	467	2	8	8	NUM
ejpam-6526	467	3	]	]	X
ejpam-6526	467	4	i.	i.	PROPN
ejpam-6526	467	5	l.	l.	PROPN
ejpam-6526	467	6	reilly	reilly	PROPN
ejpam-6526	467	7	,	,	PUNCT
ejpam-6526	467	8	m.	m.	PROPN
ejpam-6526	467	9	k.	k.	PROPN
ejpam-6526	467	10	vamanamurthy	vamanamurthy	PROPN
ejpam-6526	467	11	,	,	PUNCT
ejpam-6526	467	12	and	and	CCONJ
ejpam-6526	467	13	p.	p.	NOUN
ejpam-6526	467	14	v.	v.	ADP
ejpam-6526	467	15	subrahmanyam	subrahmanyam	NOUN
ejpam-6526	467	16	.	.	PUNCT
ejpam-6526	468	1	quasi	quasi	ADJ
ejpam-6526	468	2	-	-	ADJ
ejpam-6526	468	3	pseudometrizable	pseudometrizable	ADJ
ejpam-6526	468	4	spaces	space	NOUN
ejpam-6526	468	5	.	.	PUNCT
ejpam-6526	469	1	journal	journal	NOUN
ejpam-6526	469	2	of	of	ADP
ejpam-6526	469	3	the	the	DET
ejpam-6526	469	4	australian	australian	ADJ
ejpam-6526	469	5	mathematical	mathematical	ADJ
ejpam-6526	469	6	society	society	NOUN
ejpam-6526	469	7	,	,	PUNCT
ejpam-6526	469	8	23(1):50–54	23(1):50–54	NUM
ejpam-6526	469	9	,	,	PUNCT
ejpam-6526	469	10	1977	1977	NUM
ejpam-6526	469	11	.	.	PUNCT
ejpam-6526	470	1	[	[	X
ejpam-6526	470	2	9	9	NUM
ejpam-6526	470	3	]	]	PUNCT
ejpam-6526	470	4	s.	s.	PROPN
ejpam-6526	470	5	banach	banach	PROPN
ejpam-6526	470	6	.	.	PUNCT
ejpam-6526	471	1	théorie	théorie	PROPN
ejpam-6526	471	2	des	des	PROPN
ejpam-6526	471	3	opérations	opérations	PROPN
ejpam-6526	471	4	linéaires	linéaires	PROPN
ejpam-6526	471	5	.	.	PUNCT
ejpam-6526	472	1	warszawa	warszawa	PROPN
ejpam-6526	472	2	,	,	PUNCT
ejpam-6526	472	3	warsaw	warsaw	PROPN
ejpam-6526	472	4	,	,	PUNCT
ejpam-6526	472	5	1932	1932	NUM
ejpam-6526	472	6	.	.	PUNCT
ejpam-6526	473	1	[	[	X
ejpam-6526	473	2	10	10	NUM
ejpam-6526	473	3	]	]	X
ejpam-6526	473	4	mutaz	mutaz	NOUN
ejpam-6526	473	5	shatnawi	shatnawi	PROPN
ejpam-6526	473	6	,	,	PUNCT
ejpam-6526	473	7	jamal	jamal	PROPN
ejpam-6526	473	8	oudetallah	oudetallah	PROPN
ejpam-6526	473	9	,	,	PUNCT
ejpam-6526	473	10	anwar	anwar	PROPN
ejpam-6526	473	11	bataiha	bataiha	PROPN
ejpam-6526	473	12	,	,	PUNCT
ejpam-6526	473	13	ala	ala	PROPN
ejpam-6526	473	14	amourah	amourah	PROPN
ejpam-6526	473	15	,	,	PUNCT
ejpam-6526	473	16	abdullah	abdullah	PROPN
ejpam-6526	473	17	alsoboh	alsoboh	PROPN
ejpam-6526	473	18	,	,	PUNCT
ejpam-6526	473	19	and	and	CCONJ
ejpam-6526	473	20	tala	tala	PROPN
ejpam-6526	473	21	sasa	sasa	PROPN
ejpam-6526	473	22	.	.	PUNCT
ejpam-6526	474	1	g	g	NOUN
ejpam-6526	474	2	-	-	PUNCT
ejpam-6526	474	3	compact	compact	ADJ
ejpam-6526	474	4	spaces	space	NOUN
ejpam-6526	474	5	characterized	characterize	VERB
ejpam-6526	474	6	by	by	ADP
ejpam-6526	474	7	the	the	DET
ejpam-6526	474	8	intersection	intersection	NOUN
ejpam-6526	474	9	of	of	ADP
ejpam-6526	474	10	countable	countable	ADJ
ejpam-6526	474	11	neighborhoods	neighborhood	NOUN
ejpam-6526	474	12	.	.	PUNCT
ejpam-6526	475	1	european	european	ADJ
ejpam-6526	475	2	journal	journal	PROPN
ejpam-6526	475	3	of	of	ADP
ejpam-6526	475	4	pure	pure	ADJ
ejpam-6526	475	5	and	and	CCONJ
ejpam-6526	475	6	applied	applied	ADJ
ejpam-6526	475	7	mathematics	mathematic	NOUN
ejpam-6526	475	8	,	,	PUNCT
ejpam-6526	475	9	18(3):6067–6067	18(3):6067–6067	NUM
ejpam-6526	475	10	,	,	PUNCT
ejpam-6526	475	11	2025	2025	NUM
ejpam-6526	475	12	.	.	PUNCT
ejpam-6526	476	1	[	[	X
ejpam-6526	476	2	11	11	NUM
ejpam-6526	476	3	]	]	X
ejpam-6526	476	4	jamal	jamal	PROPN
ejpam-6526	476	5	oudetallah	oudetallah	PROPN
ejpam-6526	476	6	,	,	PUNCT
ejpam-6526	476	7	ahmad	ahmad	PROPN
ejpam-6526	476	8	almalkawi	almalkawi	PROPN
ejpam-6526	476	9	,	,	PUNCT
ejpam-6526	476	10	rahmeh	rahmeh	NOUN
ejpam-6526	476	11	alrababah	alrababah	NOUN
ejpam-6526	476	12	,	,	PUNCT
ejpam-6526	476	13	ala	ala	PROPN
ejpam-6526	476	14	amourah	amourah	PROPN
ejpam-6526	476	15	,	,	PUNCT
ejpam-6526	476	16	abdullah	abdullah	PROPN
ejpam-6526	476	17	alsoboh	alsoboh	PROPN
ejpam-6526	476	18	,	,	PUNCT
ejpam-6526	476	19	khaled	khaled	PROPN
ejpam-6526	476	20	al	al	PROPN
ejpam-6526	476	21	mashraf	mashraf	PROPN
ejpam-6526	476	22	,	,	PUNCT
ejpam-6526	476	23	and	and	CCONJ
ejpam-6526	476	24	tala	tala	PROPN
ejpam-6526	476	25	sasa	sasa	PROPN
ejpam-6526	476	26	.	.	PUNCT
ejpam-6526	477	1	novel	novel	ADJ
ejpam-6526	477	2	results	result	NOUN
ejpam-6526	477	3	on	on	ADP
ejpam-6526	477	4	d	d	ADJ
ejpam-6526	477	5	-	-	ADJ
ejpam-6526	477	6	soft	soft	ADJ
ejpam-6526	477	7	compact	compact	ADJ
ejpam-6526	477	8	spaces	space	NOUN
ejpam-6526	477	9	.	.	PUNCT
ejpam-6526	478	1	european	european	ADJ
ejpam-6526	478	2	journal	journal	PROPN
ejpam-6526	478	3	of	of	ADP
ejpam-6526	478	4	pure	pure	ADJ
ejpam-6526	478	5	and	and	CCONJ
ejpam-6526	478	6	applied	applied	ADJ
ejpam-6526	478	7	mathematics	mathematic	NOUN
ejpam-6526	478	8	,	,	PUNCT
ejpam-6526	478	9	18(3):6513–6513	18(3):6513–6513	NUM
ejpam-6526	478	10	,	,	PUNCT
ejpam-6526	478	11	2025	2025	NUM
ejpam-6526	478	12	.	.	PUNCT
ejpam-6526	479	1	[	[	X
ejpam-6526	479	2	12	12	NUM
ejpam-6526	479	3	]	]	X
ejpam-6526	479	4	jamal	jamal	PROPN
ejpam-6526	479	5	oudetallah	oudetallah	PROPN
ejpam-6526	479	6	,	,	PUNCT
ejpam-6526	479	7	ahmad	ahmad	PROPN
ejpam-6526	479	8	almalkawi	almalkawi	PROPN
ejpam-6526	479	9	,	,	PUNCT
ejpam-6526	479	10	ala	ala	PROPN
ejpam-6526	479	11	amourah	amourah	PROPN
ejpam-6526	479	12	,	,	PUNCT
ejpam-6526	479	13	abdullah	abdullah	PROPN
ejpam-6526	479	14	alsoboh	alsoboh	PROPN
ejpam-6526	479	15	,	,	PUNCT
ejpam-6526	479	16	khaled	khaled	PROPN
ejpam-6526	479	17	al	al	PROPN
ejpam-6526	479	18	mashrafi	mashrafi	PROPN
ejpam-6526	479	19	,	,	PUNCT
ejpam-6526	479	20	and	and	CCONJ
ejpam-6526	479	21	tala	tala	PROPN
ejpam-6526	479	22	sasa	sasa	PROPN
ejpam-6526	479	23	.	.	PUNCT
ejpam-6526	480	1	on	on	ADP
ejpam-6526	480	2	nearly	nearly	ADV
ejpam-6526	480	3	α	α	ADJ
ejpam-6526	480	4	-	-	ADJ
ejpam-6526	480	5	compact	compact	ADJ
ejpam-6526	480	6	topological	topological	ADJ
ejpam-6526	480	7	spaces	space	NOUN
ejpam-6526	480	8	.	.	PUNCT
ejpam-6526	481	1	european	european	ADJ
ejpam-6526	481	2	journal	journal	PROPN
ejpam-6526	481	3	of	of	ADP
ejpam-6526	481	4	pure	pure	ADJ
ejpam-6526	481	5	and	and	CCONJ
ejpam-6526	481	6	applied	applied	ADJ
ejpam-6526	481	7	mathematics	mathematic	NOUN
ejpam-6526	481	8	,	,	PUNCT
ejpam-6526	481	9	18(3):6543–6543	18(3):6543–6543	NUM
ejpam-6526	481	10	,	,	PUNCT
ejpam-6526	481	11	2025	2025	NUM
ejpam-6526	481	12	.	.	PUNCT
ejpam-6526	482	1	[	[	X
ejpam-6526	482	2	13	13	NUM
ejpam-6526	482	3	]	]	X
ejpam-6526	482	4	abdullah	abdullah	PROPN
ejpam-6526	482	5	alsoboh	alsoboh	PROPN
ejpam-6526	482	6	,	,	PUNCT
ejpam-6526	482	7	ala	ala	PROPN
ejpam-6526	482	8	amourah	amourah	PROPN
ejpam-6526	482	9	,	,	PUNCT
ejpam-6526	482	10	khaled	khaled	PROPN
ejpam-6526	482	11	al	al	PROPN
ejpam-6526	482	12	mashrafi	mashrafi	PROPN
ejpam-6526	482	13	,	,	PUNCT
ejpam-6526	482	14	and	and	CCONJ
ejpam-6526	482	15	tala	tala	PROPN
ejpam-6526	482	16	sasa	sasa	PROPN
ejpam-6526	482	17	.	.	PUNCT
ejpam-6526	483	1	bi	bi	ADJ
ejpam-6526	483	2	-	-	ADJ
ejpam-6526	483	3	starlike	starlike	ADJ
ejpam-6526	483	4	and	and	CCONJ
ejpam-6526	483	5	bi	bi	ADJ
ejpam-6526	483	6	-	-	ADJ
ejpam-6526	483	7	convex	convex	ADJ
ejpam-6526	483	8	function	function	NOUN
ejpam-6526	483	9	classes	class	NOUN
ejpam-6526	483	10	connected	connect	VERB
ejpam-6526	483	11	to	to	ADP
ejpam-6526	483	12	shell	shell	NOUN
ejpam-6526	483	13	-	-	PUNCT
ejpam-6526	483	14	like	like	ADJ
ejpam-6526	483	15	curves	curve	NOUN
ejpam-6526	483	16	and	and	CCONJ
ejpam-6526	483	17	the	the	DET
ejpam-6526	483	18	q	q	NOUN
ejpam-6526	483	19	-	-	PUNCT
ejpam-6526	483	20	analogue	analogue	NOUN
ejpam-6526	483	21	of	of	ADP
ejpam-6526	483	22	fibonacci	fibonacci	NOUN
ejpam-6526	483	23	numbers	number	NOUN
ejpam-6526	483	24	.	.	PUNCT
ejpam-6526	484	1	international	international	ADJ
ejpam-6526	484	2	journal	journal	NOUN
ejpam-6526	484	3	of	of	ADP
ejpam-6526	484	4	analysis	analysis	NOUN
ejpam-6526	484	5	and	and	CCONJ
ejpam-6526	484	6	applications	application	NOUN
ejpam-6526	484	7	,	,	PUNCT
ejpam-6526	484	8	23:201–201	23:201–201	NUM
ejpam-6526	484	9	,	,	PUNCT
ejpam-6526	484	10	2025	2025	NUM
ejpam-6526	484	11	.	.	PUNCT
ejpam-6526	485	1	[	[	X
ejpam-6526	485	2	14	14	NUM
ejpam-6526	485	3	]	]	PUNCT
ejpam-6526	485	4	a.	a.	NOUN
ejpam-6526	485	5	alsoboh	alsoboh	PROPN
ejpam-6526	485	6	,	,	PUNCT
ejpam-6526	485	7	a.	a.	PROPN
ejpam-6526	485	8	amourah	amourah	PROPN
ejpam-6526	485	9	,	,	PUNCT
ejpam-6526	485	10	k.	k.	PROPN
ejpam-6526	485	11	al	al	PROPN
ejpam-6526	485	12	mashrafi	mashrafi	PROPN
ejpam-6526	485	13	,	,	PUNCT
ejpam-6526	485	14	and	and	CCONJ
ejpam-6526	485	15	t.	t.	PROPN
ejpam-6526	485	16	sasa	sasa	PROPN
ejpam-6526	485	17	.	.	PUNCT
ejpam-6526	486	1	bi	bi	ADJ
ejpam-6526	486	2	-	-	ADJ
ejpam-6526	486	3	starlike	starlike	ADJ
ejpam-6526	486	4	and	and	CCONJ
ejpam-6526	486	5	bi	bi	ADJ
ejpam-6526	486	6	-	-	ADJ
ejpam-6526	486	7	convex	convex	ADJ
ejpam-6526	486	8	function	function	NOUN
ejpam-6526	486	9	classes	class	NOUN
ejpam-6526	486	10	connected	connect	VERB
ejpam-6526	486	11	to	to	ADP
ejpam-6526	486	12	shell	shell	NOUN
ejpam-6526	486	13	-	-	PUNCT
ejpam-6526	486	14	like	like	ADJ
ejpam-6526	486	15	curves	curve	NOUN
ejpam-6526	486	16	and	and	CCONJ
ejpam-6526	486	17	the	the	DET
ejpam-6526	486	18	q	q	NOUN
ejpam-6526	486	19	-	-	PUNCT
ejpam-6526	486	20	analogue	analogue	NOUN
ejpam-6526	486	21	of	of	ADP
ejpam-6526	486	22	fibonacci	fibonacci	NOUN
ejpam-6526	486	23	numbers	number	NOUN
ejpam-6526	486	24	.	.	PUNCT
ejpam-6526	487	1	international	international	ADJ
ejpam-6526	487	2	journal	journal	NOUN
ejpam-6526	487	3	of	of	ADP
ejpam-6526	487	4	analysis	analysis	NOUN
ejpam-6526	487	5	and	and	CCONJ
ejpam-6526	487	6	applications	application	NOUN
ejpam-6526	487	7	,	,	PUNCT
ejpam-6526	487	8	23:201–201	23:201–201	NUM
ejpam-6526	487	9	,	,	PUNCT
ejpam-6526	487	10	2025	2025	NUM
ejpam-6526	487	11	.	.	PUNCT
ejpam-6526	488	1	[	[	X
ejpam-6526	488	2	15	15	NUM
ejpam-6526	488	3	]	]	X
ejpam-6526	488	4	abdullah	abdullah	PROPN
ejpam-6526	488	5	alsoboh	alsoboh	PROPN
ejpam-6526	488	6	,	,	PUNCT
ejpam-6526	488	7	ala	ala	PROPN
ejpam-6526	488	8	amourah	amourah	PROPN
ejpam-6526	488	9	,	,	PUNCT
ejpam-6526	488	10	omar	omar	PROPN
ejpam-6526	488	11	alnajar	alnajar	PROPN
ejpam-6526	488	12	,	,	PUNCT
ejpam-6526	488	13	mamoon	mamoon	NOUN
ejpam-6526	488	14	ahmed	ahmed	PROPN
ejpam-6526	488	15	,	,	PUNCT
ejpam-6526	488	16	and	and	CCONJ
ejpam-6526	488	17	tamer	tame	ADJ
ejpam-6526	488	18	m.	m.	NOUN
ejpam-6526	488	19	seoudy	seoudy	NOUN
ejpam-6526	488	20	.	.	PUNCT
ejpam-6526	489	1	exploring	explore	VERB
ejpam-6526	489	2	q	q	ADJ
ejpam-6526	489	3	-	-	PUNCT
ejpam-6526	489	4	fibonacci	fibonacci	NOUN
ejpam-6526	489	5	numbers	number	NOUN
ejpam-6526	489	6	in	in	ADP
ejpam-6526	489	7	geometric	geometric	ADJ
ejpam-6526	489	8	function	function	NOUN
ejpam-6526	489	9	theory	theory	NOUN
ejpam-6526	489	10	:	:	PUNCT
ejpam-6526	489	11	univalence	univalence	NOUN
ejpam-6526	489	12	and	and	CCONJ
ejpam-6526	489	13	shell	shell	NOUN
ejpam-6526	489	14	-	-	PUNCT
ejpam-6526	489	15	like	like	ADJ
ejpam-6526	489	16	starlike	starlike	NOUN
ejpam-6526	489	17	curves	curve	NOUN
ejpam-6526	489	18	.	.	PUNCT
ejpam-6526	490	1	mathematics	mathematic	NOUN
ejpam-6526	490	2	,	,	PUNCT
ejpam-6526	490	3	13(8):1294	13(8):1294	NUM
ejpam-6526	490	4	,	,	PUNCT
ejpam-6526	490	5	2025	2025	NUM
ejpam-6526	490	6	.	.	PUNCT
ejpam-6526	491	1	[	[	X
ejpam-6526	491	2	16	16	NUM
ejpam-6526	491	3	]	]	PUNCT
ejpam-6526	491	4	j.	j.	PROPN
ejpam-6526	491	5	r.	r.	PROPN
ejpam-6526	491	6	munkres	munkres	PROPN
ejpam-6526	491	7	.	.	PUNCT
ejpam-6526	492	1	topology	topology	NOUN
ejpam-6526	492	2	.	.	PUNCT
ejpam-6526	493	1	prentice	prentice	PROPN
ejpam-6526	493	2	hall	hall	PROPN
ejpam-6526	493	3	,	,	PUNCT
ejpam-6526	493	4	upper	upper	ADJ
ejpam-6526	493	5	saddle	saddle	NOUN
ejpam-6526	493	6	river	river	NOUN
ejpam-6526	493	7	,	,	PUNCT
ejpam-6526	493	8	2nd	2nd	PROPN
ejpam-6526	493	9	edition	edition	NOUN
ejpam-6526	493	10	,	,	PUNCT
ejpam-6526	493	11	2000	2000	NUM
ejpam-6526	493	12	.	.	PUNCT
ejpam-6526	494	1	[	[	X
ejpam-6526	494	2	17	17	NUM
ejpam-6526	494	3	]	]	X
ejpam-6526	494	4	r.	r.	PROPN
ejpam-6526	494	5	engelking	engelke	VERB
ejpam-6526	494	6	.	.	PUNCT
ejpam-6526	495	1	general	general	ADJ
ejpam-6526	495	2	topology	topology	PROPN
ejpam-6526	495	3	.	.	PUNCT
ejpam-6526	496	1	heldermann	heldermann	PROPN
ejpam-6526	496	2	verlag	verlag	PROPN
ejpam-6526	496	3	,	,	PUNCT
ejpam-6526	496	4	berlin	berlin	PROPN
ejpam-6526	496	5	,	,	PUNCT
ejpam-6526	496	6	revised	revise	VERB
ejpam-6526	496	7	edition	edition	NOUN
ejpam-6526	496	8	,	,	PUNCT
ejpam-6526	496	9	2019	2019	NUM
ejpam-6526	496	10	.	.	PUNCT
ejpam-6526	497	1	[	[	X
ejpam-6526	497	2	18	18	NUM
ejpam-6526	497	3	]	]	X
ejpam-6526	497	4	h.	h.	PROPN
ejpam-6526	497	5	nakano	nakano	PROPN
ejpam-6526	497	6	.	.	PUNCT
ejpam-6526	498	1	modular	modular	ADJ
ejpam-6526	498	2	semi	semi	ADJ
ejpam-6526	498	3	-	-	ADJ
ejpam-6526	498	4	ordered	ordered	ADJ
ejpam-6526	498	5	linear	linear	ADJ
ejpam-6526	498	6	spaces	space	NOUN
ejpam-6526	498	7	.	.	PUNCT
ejpam-6526	499	1	maruzen	maruzen	PROPN
ejpam-6526	499	2	co.	co.	PROPN
ejpam-6526	499	3	,	,	PUNCT
ejpam-6526	499	4	tokyo	tokyo	PROPN
ejpam-6526	499	5	,	,	PUNCT
ejpam-6526	499	6	1950	1950	NUM
ejpam-6526	499	7	.	.	PUNCT
ejpam-6526	500	1	[	[	X
ejpam-6526	500	2	19	19	NUM
ejpam-6526	500	3	]	]	X
ejpam-6526	500	4	w.	w.	PROPN
ejpam-6526	500	5	j.	j.	PROPN
ejpam-6526	500	6	pervin	pervin	PROPN
ejpam-6526	500	7	.	.	PUNCT
ejpam-6526	501	1	quasi	quasi	ADJ
ejpam-6526	501	2	-	-	NOUN
ejpam-6526	501	3	uniformization	uniformization	NOUN
ejpam-6526	501	4	of	of	ADP
ejpam-6526	501	5	topological	topological	ADJ
ejpam-6526	501	6	spaces	space	NOUN
ejpam-6526	501	7	.	.	PUNCT
ejpam-6526	502	1	mathematische	mathematische	PROPN
ejpam-6526	502	2	annalen	annalen	PROPN
ejpam-6526	502	3	,	,	PUNCT
ejpam-6526	502	4	147(4):316–317	147(4):316–317	NUM
ejpam-6526	502	5	,	,	PUNCT
ejpam-6526	502	6	1967	1967	NUM
ejpam-6526	502	7	.	.	PUNCT
ejpam-6526	503	1	[	[	X
ejpam-6526	503	2	20	20	NUM
ejpam-6526	503	3	]	]	X
ejpam-6526	503	4	v.	v.	ADP
ejpam-6526	503	5	brattka	brattka	PROPN
ejpam-6526	503	6	and	and	CCONJ
ejpam-6526	503	7	g.	g.	PROPN
ejpam-6526	503	8	presser	presser	NOUN
ejpam-6526	503	9	.	.	PUNCT
ejpam-6526	504	1	computability	computability	NOUN
ejpam-6526	504	2	on	on	ADP
ejpam-6526	504	3	subsets	subset	NOUN
ejpam-6526	504	4	of	of	ADP
ejpam-6526	504	5	metric	metric	ADJ
ejpam-6526	504	6	spaces	space	NOUN
ejpam-6526	504	7	.	.	PUNCT
ejpam-6526	505	1	theoretical	theoretical	ADJ
ejpam-6526	505	2	computer	computer	NOUN
ejpam-6526	505	3	science	science	NOUN
ejpam-6526	505	4	,	,	PUNCT
ejpam-6526	505	5	305(1	305(1	NUM
ejpam-6526	505	6	-	-	SYM
ejpam-6526	505	7	3):43–76	3):43–76	NUM
ejpam-6526	505	8	,	,	PUNCT
ejpam-6526	505	9	2003	2003	NUM
ejpam-6526	505	10	.	.	PUNCT
ejpam-6526	506	1	[	[	X
ejpam-6526	506	2	21	21	NUM
ejpam-6526	506	3	]	]	X
ejpam-6526	506	4	h.	h.	PROPN
ejpam-6526	506	5	p.	p.	PROPN
ejpam-6526	506	6	a.	a.	NOUN
ejpam-6526	506	7	künzi	künzi	PROPN
ejpam-6526	506	8	.	.	PUNCT
ejpam-6526	507	1	quasi	quasi	ADJ
ejpam-6526	507	2	-	-	ADJ
ejpam-6526	507	3	uniform	uniform	ADJ
ejpam-6526	507	4	spaces	space	NOUN
ejpam-6526	507	5	in	in	ADP
ejpam-6526	507	6	the	the	DET
ejpam-6526	507	7	year	year	NOUN
ejpam-6526	507	8	2001	2001	NUM
ejpam-6526	507	9	.	.	PUNCT
ejpam-6526	508	1	in	in	ADP
ejpam-6526	508	2	m.	m.	NOUN
ejpam-6526	508	3	husek	husek	PROPN
ejpam-6526	508	4	and	and	CCONJ
ejpam-6526	508	5	j.	j.	PROPN
ejpam-6526	508	6	van	van	PROPN
ejpam-6526	508	7	mill	mill	PROPN
ejpam-6526	508	8	,	,	PUNCT
ejpam-6526	508	9	editors	editor	NOUN
ejpam-6526	508	10	,	,	PUNCT
ejpam-6526	508	11	recent	recent	ADJ
ejpam-6526	508	12	progress	progress	NOUN
ejpam-6526	508	13	in	in	ADP
ejpam-6526	508	14	general	general	ADJ
ejpam-6526	508	15	topology	topology	PROPN
ejpam-6526	508	16	ii	ii	PROPN
ejpam-6526	508	17	,	,	PUNCT
ejpam-6526	508	18	pages	page	NOUN
ejpam-6526	508	19	313–344	313–344	NUM
ejpam-6526	508	20	.	.	PUNCT
ejpam-6526	508	21	elsevier	elsevier	PROPN
ejpam-6526	508	22	,	,	PUNCT
ejpam-6526	508	23	amsterdam	amsterdam	PROPN
ejpam-6526	508	24	,	,	PUNCT
ejpam-6526	508	25	2001	2001	NUM
ejpam-6526	508	26	.	.	PUNCT
ejpam-6526	508	27	a.	a.	PROPN
ejpam-6526	508	28	alsoboh	alsoboh	PROPN
ejpam-6526	508	29	et	et	PROPN
ejpam-6526	508	30	al	al	PROPN
ejpam-6526	508	31	.	.	PUNCT
ejpam-6526	508	32	/	/	SYM
ejpam-6526	508	33	eur	eur	PROPN
ejpam-6526	508	34	.	.	PUNCT
ejpam-6526	509	1	j.	j.	PROPN
ejpam-6526	509	2	pure	pure	PROPN
ejpam-6526	509	3	appl	appl	PROPN
ejpam-6526	509	4	.	.	PROPN
ejpam-6526	509	5	math	math	PROPN
ejpam-6526	509	6	,	,	PUNCT
ejpam-6526	509	7	18	18	NUM
ejpam-6526	509	8	(	(	PUNCT
ejpam-6526	509	9	4	4	NUM
ejpam-6526	509	10	)	)	PUNCT
ejpam-6526	509	11	(	(	PUNCT
ejpam-6526	509	12	2025	2025	NUM
ejpam-6526	509	13	)	)	PUNCT
ejpam-6526	509	14	,	,	PUNCT
ejpam-6526	509	15	6526	6526	NUM
ejpam-6526	509	16	21	21	NUM
ejpam-6526	509	17	of	of	ADP
ejpam-6526	509	18	21	21	NUM
ejpam-6526	509	19	[	[	SYM
ejpam-6526	509	20	22	22	NUM
ejpam-6526	509	21	]	]	PUNCT
ejpam-6526	509	22	m.	m.	NOUN
ejpam-6526	509	23	d.	d.	PROPN
ejpam-6526	509	24	weiss	weiss	PROPN
ejpam-6526	509	25	.	.	PUNCT
ejpam-6526	510	1	fixed	fix	VERB
ejpam-6526	510	2	points	point	NOUN
ejpam-6526	510	3	,	,	PUNCT
ejpam-6526	510	4	separation	separation	NOUN
ejpam-6526	510	5	,	,	PUNCT
ejpam-6526	510	6	and	and	CCONJ
ejpam-6526	510	7	induced	induce	VERB
ejpam-6526	510	8	topologies	topology	NOUN
ejpam-6526	510	9	for	for	ADP
ejpam-6526	510	10	fuzzy	fuzzy	ADJ
ejpam-6526	510	11	sets	set	NOUN
ejpam-6526	510	12	.	.	PUNCT
ejpam-6526	511	1	journal	journal	NOUN
ejpam-6526	511	2	of	of	ADP
ejpam-6526	511	3	mathematical	mathematical	ADJ
ejpam-6526	511	4	analysis	analysis	NOUN
ejpam-6526	511	5	and	and	CCONJ
ejpam-6526	511	6	applications	application	NOUN
ejpam-6526	511	7	,	,	PUNCT
ejpam-6526	511	8	50(1):142–150	50(1):142–150	NUM
ejpam-6526	511	9	,	,	PUNCT
ejpam-6526	511	10	1975	1975	NUM
ejpam-6526	511	11	.	.	PUNCT
ejpam-6526	512	1	[	[	X
ejpam-6526	512	2	23	23	NUM
ejpam-6526	512	3	]	]	X
ejpam-6526	512	4	r.	r.	PROPN
ejpam-6526	512	5	kopperman	kopperman	PROPN
ejpam-6526	512	6	.	.	PUNCT
ejpam-6526	513	1	all	all	DET
ejpam-6526	513	2	topologies	topology	NOUN
ejpam-6526	513	3	come	come	VERB
ejpam-6526	513	4	from	from	ADP
ejpam-6526	513	5	generalized	generalized	ADJ
ejpam-6526	513	6	metrics	metric	NOUN
ejpam-6526	513	7	.	.	PUNCT
ejpam-6526	514	1	the	the	DET
ejpam-6526	514	2	american	american	PROPN
ejpam-6526	514	3	mathematical	mathematical	PROPN
ejpam-6526	514	4	monthly	monthly	ADV
ejpam-6526	514	5	,	,	PUNCT
ejpam-6526	514	6	95(2):89–97	95(2):89–97	NUM
ejpam-6526	514	7	,	,	PUNCT
ejpam-6526	514	8	1988	1988	NUM
ejpam-6526	514	9	.	.	PUNCT
ejpam-6526	515	1	[	[	X
ejpam-6526	515	2	24	24	NUM
ejpam-6526	515	3	]	]	PUNCT
ejpam-6526	515	4	p.	p.	NOUN
ejpam-6526	515	5	fletcher	fletcher	PROPN
ejpam-6526	515	6	and	and	CCONJ
ejpam-6526	515	7	w.	w.	PROPN
ejpam-6526	515	8	f.	f.	PROPN
ejpam-6526	515	9	lindgren	lindgren	PROPN
ejpam-6526	515	10	.	.	PUNCT
ejpam-6526	516	1	quasi	quasi	ADJ
ejpam-6526	516	2	-	-	ADJ
ejpam-6526	516	3	uniform	uniform	ADJ
ejpam-6526	516	4	spaces	space	NOUN
ejpam-6526	516	5	.	.	PUNCT
ejpam-6526	517	1	marcel	marcel	PROPN
ejpam-6526	517	2	dekker	dekker	PROPN
ejpam-6526	517	3	,	,	PUNCT
ejpam-6526	517	4	new	new	PROPN
ejpam-6526	517	5	york	york	PROPN
ejpam-6526	517	6	,	,	PUNCT
ejpam-6526	517	7	1982	1982	NUM
ejpam-6526	517	8	.	.	PUNCT