id	sid	tid	token	lemma	pos
ejpam-6802	1	1	european	european	PROPN
ejpam-6802	1	2	journal	journal	PROPN
ejpam-6802	1	3	of	of	ADP
ejpam-6802	1	4	pure	pure	ADJ
ejpam-6802	1	5	and	and	CCONJ
ejpam-6802	1	6	applied	applied	ADJ
ejpam-6802	1	7	mathematics	mathematic	NOUN
ejpam-6802	1	8	2025	2025	NUM
ejpam-6802	1	9	,	,	PUNCT
ejpam-6802	1	10	vol	vol	NOUN
ejpam-6802	1	11	.	.	PROPN
ejpam-6802	1	12	18	18	NUM
ejpam-6802	1	13	,	,	PUNCT
ejpam-6802	1	14	issue	issue	NOUN
ejpam-6802	1	15	4	4	NUM
ejpam-6802	1	16	,	,	PUNCT
ejpam-6802	1	17	article	article	NOUN
ejpam-6802	1	18	number	number	NOUN
ejpam-6802	1	19	6802	6802	NUM
ejpam-6802	1	20	issn	issn	PROPN
ejpam-6802	1	21	1307	1307	NUM
ejpam-6802	1	22	-	-	SYM
ejpam-6802	1	23	5543	5543	NUM
ejpam-6802	1	24	–	–	PUNCT
ejpam-6802	1	25	ejpam.com	ejpam.com	X
ejpam-6802	1	26	published	publish	VERB
ejpam-6802	1	27	by	by	ADP
ejpam-6802	1	28	new	new	PROPN
ejpam-6802	1	29	york	york	PROPN
ejpam-6802	1	30	business	business	PROPN
ejpam-6802	1	31	global	global	ADJ
ejpam-6802	1	32	fixed	fix	VERB
ejpam-6802	1	33	point	point	NOUN
ejpam-6802	1	34	theorems	theorem	NOUN
ejpam-6802	1	35	on	on	ADP
ejpam-6802	1	36	multiplicative	multiplicative	ADJ
ejpam-6802	1	37	cone	cone	NOUN
ejpam-6802	1	38	bipolar	bipolar	ADJ
ejpam-6802	1	39	metric	metric	ADJ
ejpam-6802	1	40	space	space	NOUN
ejpam-6802	1	41	and	and	CCONJ
ejpam-6802	1	42	their	their	PRON
ejpam-6802	1	43	applications	application	NOUN
ejpam-6802	1	44	rajagopalan	rajagopalan	VERB
ejpam-6802	1	45	ramaswamy	ramaswamy	ADJ
ejpam-6802	1	46	department	department	NOUN
ejpam-6802	1	47	of	of	ADP
ejpam-6802	1	48	mathematics	mathematics	PROPN
ejpam-6802	1	49	,	,	PUNCT
ejpam-6802	1	50	college	college	NOUN
ejpam-6802	1	51	of	of	ADP
ejpam-6802	1	52	science	science	NOUN
ejpam-6802	1	53	and	and	CCONJ
ejpam-6802	1	54	humanities	humanity	NOUN
ejpam-6802	1	55	in	in	ADP
ejpam-6802	1	56	alkharj	alkharj	NOUN
ejpam-6802	1	57	,	,	PUNCT
ejpam-6802	1	58	prince	prince	PROPN
ejpam-6802	1	59	sattam	sattam	PROPN
ejpam-6802	1	60	bin	bin	PROPN
ejpam-6802	1	61	abdulaziz	abdulaziz	PROPN
ejpam-6802	1	62	university	university	PROPN
ejpam-6802	1	63	,	,	PUNCT
ejpam-6802	1	64	alkharj	alkharj	VERB
ejpam-6802	1	65	11942	11942	NUM
ejpam-6802	1	66	,	,	PUNCT
ejpam-6802	1	67	saudi	saudi	PROPN
ejpam-6802	1	68	arabia	arabia	PROPN
ejpam-6802	1	69	abstract	abstract	NOUN
ejpam-6802	1	70	.	.	PUNCT
ejpam-6802	2	1	in	in	ADP
ejpam-6802	2	2	this	this	DET
ejpam-6802	2	3	article	article	NOUN
ejpam-6802	2	4	,	,	PUNCT
ejpam-6802	2	5	the	the	DET
ejpam-6802	2	6	notion	notion	NOUN
ejpam-6802	2	7	of	of	ADP
ejpam-6802	2	8	a	a	DET
ejpam-6802	2	9	multiplicative	multiplicative	ADJ
ejpam-6802	2	10	cone	cone	NOUN
ejpam-6802	2	11	bipolar	bipolar	ADJ
ejpam-6802	2	12	metric	metric	ADJ
ejpam-6802	2	13	space	space	NOUN
ejpam-6802	2	14	is	be	AUX
ejpam-6802	2	15	introduced	introduce	VERB
ejpam-6802	2	16	.	.	PUNCT
ejpam-6802	3	1	fixed	fix	VERB
ejpam-6802	3	2	point	point	NOUN
ejpam-6802	3	3	results	result	NOUN
ejpam-6802	3	4	in	in	ADP
ejpam-6802	3	5	the	the	DET
ejpam-6802	3	6	setting	setting	NOUN
ejpam-6802	3	7	of	of	ADP
ejpam-6802	3	8	these	these	DET
ejpam-6802	3	9	spaces	space	NOUN
ejpam-6802	3	10	have	have	AUX
ejpam-6802	3	11	been	be	AUX
ejpam-6802	3	12	established	establish	VERB
ejpam-6802	3	13	and	and	CCONJ
ejpam-6802	3	14	supported	support	VERB
ejpam-6802	3	15	with	with	ADP
ejpam-6802	3	16	suitable	suitable	ADJ
ejpam-6802	3	17	non	non	ADJ
ejpam-6802	3	18	-	-	ADJ
ejpam-6802	3	19	trivial	trivial	ADJ
ejpam-6802	3	20	examples	example	NOUN
ejpam-6802	3	21	.	.	PUNCT
ejpam-6802	4	1	our	our	PRON
ejpam-6802	4	2	results	result	NOUN
ejpam-6802	4	3	generalize	generalize	VERB
ejpam-6802	4	4	and	and	CCONJ
ejpam-6802	4	5	extend	extend	VERB
ejpam-6802	4	6	proven	prove	VERB
ejpam-6802	4	7	results	result	NOUN
ejpam-6802	4	8	from	from	ADP
ejpam-6802	4	9	previous	previous	ADJ
ejpam-6802	4	10	studies	study	NOUN
ejpam-6802	4	11	.	.	PUNCT
ejpam-6802	5	1	the	the	DET
ejpam-6802	5	2	main	main	ADJ
ejpam-6802	5	3	results	result	NOUN
ejpam-6802	5	4	are	be	AUX
ejpam-6802	5	5	applied	apply	VERB
ejpam-6802	5	6	to	to	PART
ejpam-6802	5	7	obtain	obtain	VERB
ejpam-6802	5	8	analytical	analytical	ADJ
ejpam-6802	5	9	solutions	solution	NOUN
ejpam-6802	5	10	to	to	ADP
ejpam-6802	5	11	integral	integral	ADJ
ejpam-6802	5	12	equations	equation	NOUN
ejpam-6802	5	13	and	and	CCONJ
ejpam-6802	5	14	fractional	fractional	ADJ
ejpam-6802	5	15	differential	differential	ADJ
ejpam-6802	5	16	equations	equation	NOUN
ejpam-6802	5	17	.	.	PUNCT
ejpam-6802	6	1	these	these	DET
ejpam-6802	6	2	findings	finding	NOUN
ejpam-6802	6	3	contribute	contribute	VERB
ejpam-6802	6	4	to	to	ADP
ejpam-6802	6	5	the	the	DET
ejpam-6802	6	6	growing	grow	VERB
ejpam-6802	6	7	body	body	NOUN
ejpam-6802	6	8	of	of	ADP
ejpam-6802	6	9	literature	literature	NOUN
ejpam-6802	6	10	on	on	ADP
ejpam-6802	6	11	fixed	fix	VERB
ejpam-6802	6	12	point	point	NOUN
ejpam-6802	6	13	theory	theory	NOUN
ejpam-6802	6	14	and	and	CCONJ
ejpam-6802	6	15	its	its	PRON
ejpam-6802	6	16	applications	application	NOUN
ejpam-6802	6	17	in	in	ADP
ejpam-6802	6	18	analysis	analysis	NOUN
ejpam-6802	6	19	.	.	PUNCT
ejpam-6802	7	1	2020	2020	NUM
ejpam-6802	7	2	mathematics	mathematic	NOUN
ejpam-6802	7	3	subject	subject	NOUN
ejpam-6802	7	4	classifications	classification	NOUN
ejpam-6802	7	5	:	:	PUNCT
ejpam-6802	7	6	47h10	47h10	NUM
ejpam-6802	7	7	,	,	PUNCT
ejpam-6802	7	8	54h25	54h25	NUM
ejpam-6802	7	9	key	key	ADJ
ejpam-6802	7	10	words	word	NOUN
ejpam-6802	7	11	and	and	CCONJ
ejpam-6802	7	12	phrases	phrase	NOUN
ejpam-6802	7	13	:	:	PUNCT
ejpam-6802	7	14	multiplicative	multiplicative	ADJ
ejpam-6802	7	15	cone	cone	NOUN
ejpam-6802	7	16	bipolar	bipolar	ADJ
ejpam-6802	7	17	metric	metric	ADJ
ejpam-6802	7	18	space	space	NOUN
ejpam-6802	7	19	,	,	PUNCT
ejpam-6802	7	20	fixed	fix	VERB
ejpam-6802	7	21	point	point	NOUN
ejpam-6802	7	22	,	,	PUNCT
ejpam-6802	7	23	covariant	covariant	PROPN
ejpam-6802	7	24	map	map	NOUN
ejpam-6802	7	25	,	,	PUNCT
ejpam-6802	7	26	contravariant	contravariant	ADJ
ejpam-6802	7	27	map	map	NOUN
ejpam-6802	7	28	1	1	NUM
ejpam-6802	7	29	.	.	PUNCT
ejpam-6802	8	1	introduction	introduction	NOUN
ejpam-6802	8	2	it	it	PRON
ejpam-6802	8	3	would	would	AUX
ejpam-6802	8	4	be	be	AUX
ejpam-6802	8	5	fair	fair	ADJ
ejpam-6802	8	6	to	to	PART
ejpam-6802	8	7	say	say	VERB
ejpam-6802	8	8	that	that	SCONJ
ejpam-6802	8	9	the	the	DET
ejpam-6802	8	10	concept	concept	NOUN
ejpam-6802	8	11	of	of	ADP
ejpam-6802	8	12	metric	metric	ADJ
ejpam-6802	8	13	fixed	fix	VERB
ejpam-6802	8	14	point	point	NOUN
ejpam-6802	8	15	theory	theory	NOUN
ejpam-6802	8	16	started	start	VERB
ejpam-6802	8	17	with	with	ADP
ejpam-6802	8	18	the	the	DET
ejpam-6802	8	19	famous	famous	ADJ
ejpam-6802	8	20	contraction	contraction	NOUN
ejpam-6802	8	21	mapping	mapping	NOUN
ejpam-6802	8	22	theorem	theorem	NOUN
ejpam-6802	8	23	of	of	ADP
ejpam-6802	8	24	s.	s.	PROPN
ejpam-6802	8	25	banach	banach	PROPN
ejpam-6802	9	1	[	[	X
ejpam-6802	9	2	1	1	NUM
ejpam-6802	9	3	]	]	PUNCT
ejpam-6802	9	4	,	,	PUNCT
ejpam-6802	9	5	that	that	PRON
ejpam-6802	9	6	was	be	AUX
ejpam-6802	9	7	generalised	generalise	VERB
ejpam-6802	9	8	by	by	ADP
ejpam-6802	9	9	kannan	kannan	PROPN
ejpam-6802	9	10	[	[	X
ejpam-6802	9	11	2	2	NUM
ejpam-6802	9	12	]	]	PUNCT
ejpam-6802	9	13	,	,	PUNCT
ejpam-6802	9	14	reich	reich	PROPN
ejpam-6802	10	1	[	[	X
ejpam-6802	10	2	3	3	NUM
ejpam-6802	10	3	]	]	PUNCT
ejpam-6802	10	4	,	,	PUNCT
ejpam-6802	10	5	junck[4	junck[4	PROPN
ejpam-6802	10	6	]	]	PUNCT
ejpam-6802	10	7	,	,	PUNCT
ejpam-6802	10	8	to	to	PART
ejpam-6802	10	9	name	name	VERB
ejpam-6802	10	10	a	a	DET
ejpam-6802	10	11	few	few	ADJ
ejpam-6802	10	12	.	.	PUNCT
ejpam-6802	11	1	this	this	DET
ejpam-6802	11	2	theory	theory	NOUN
ejpam-6802	11	3	has	have	AUX
ejpam-6802	11	4	seen	see	VERB
ejpam-6802	11	5	rapid	rapid	ADJ
ejpam-6802	11	6	development	development	NOUN
ejpam-6802	11	7	in	in	ADP
ejpam-6802	11	8	the	the	DET
ejpam-6802	11	9	past	past	ADJ
ejpam-6802	11	10	nineteenth	nineteenth	ADJ
ejpam-6802	11	11	and	and	CCONJ
ejpam-6802	11	12	twentieth	twentieth	ADJ
ejpam-6802	11	13	centuries	century	NOUN
ejpam-6802	11	14	.	.	PUNCT
ejpam-6802	12	1	in	in	ADP
ejpam-6802	12	2	the	the	DET
ejpam-6802	12	3	overlaps	overlap	NOUN
ejpam-6802	12	4	made	make	VERB
ejpam-6802	12	5	in	in	ADP
ejpam-6802	12	6	these	these	DET
ejpam-6802	12	7	centuries	century	NOUN
ejpam-6802	12	8	,	,	PUNCT
ejpam-6802	12	9	while	while	SCONJ
ejpam-6802	12	10	metric	metric	ADJ
ejpam-6802	12	11	spaces	space	NOUN
ejpam-6802	12	12	and	and	CCONJ
ejpam-6802	12	13	normed	normed	ADJ
ejpam-6802	12	14	spaces	space	NOUN
ejpam-6802	12	15	developed	develop	VERB
ejpam-6802	12	16	,	,	PUNCT
ejpam-6802	12	17	the	the	DET
ejpam-6802	12	18	domains	domain	NOUN
ejpam-6802	12	19	were	be	AUX
ejpam-6802	12	20	only	only	ADV
ejpam-6802	12	21	taken	take	VERB
ejpam-6802	12	22	as	as	ADP
ejpam-6802	12	23	value	value	NOUN
ejpam-6802	12	24	regions	region	NOUN
ejpam-6802	12	25	with	with	ADP
ejpam-6802	12	26	single	single	ADJ
ejpam-6802	12	27	variables	variable	NOUN
ejpam-6802	12	28	and	and	CCONJ
ejpam-6802	12	29	real	real	ADJ
ejpam-6802	12	30	positive	positive	ADJ
ejpam-6802	12	31	numbers	number	NOUN
ejpam-6802	12	32	.	.	PUNCT
ejpam-6802	13	1	in	in	ADP
ejpam-6802	13	2	other	other	ADJ
ejpam-6802	13	3	words	word	NOUN
ejpam-6802	13	4	,	,	PUNCT
ejpam-6802	13	5	new	new	ADJ
ejpam-6802	13	6	metric	metric	ADJ
ejpam-6802	13	7	spaces	space	NOUN
ejpam-6802	13	8	are	be	AUX
ejpam-6802	13	9	produced	produce	VERB
ejpam-6802	13	10	by	by	ADP
ejpam-6802	13	11	taking	take	VERB
ejpam-6802	13	12	the	the	DET
ejpam-6802	13	13	domains	domain	NOUN
ejpam-6802	13	14	x	x	SYM
ejpam-6802	13	15	,	,	PUNCT
ejpam-6802	13	16	x2	x2	PROPN
ejpam-6802	13	17	,	,	PUNCT
ejpam-6802	13	18	and	and	CCONJ
ejpam-6802	13	19	x3	x3	ADJ
ejpam-6802	13	20	.	.	PUNCT
ejpam-6802	14	1	however	however	ADV
ejpam-6802	14	2	,	,	PUNCT
ejpam-6802	14	3	bipolar	bipolar	ADJ
ejpam-6802	14	4	metric	metric	ADJ
ejpam-6802	14	5	space	space	NOUN
ejpam-6802	14	6	is	be	AUX
ejpam-6802	14	7	defined	define	VERB
ejpam-6802	14	8	as	as	ADP
ejpam-6802	14	9	a	a	DET
ejpam-6802	14	10	new	new	ADJ
ejpam-6802	14	11	space	space	NOUN
ejpam-6802	14	12	by	by	ADP
ejpam-6802	14	13	going	go	VERB
ejpam-6802	14	14	beyond	beyond	ADP
ejpam-6802	14	15	the	the	DET
ejpam-6802	14	16	conventional	conventional	ADJ
ejpam-6802	14	17	definition	definition	NOUN
ejpam-6802	14	18	of	of	ADP
ejpam-6802	14	19	metric	metric	ADJ
ejpam-6802	14	20	spaces	space	NOUN
ejpam-6802	14	21	that	that	PRON
ejpam-6802	14	22	have	have	AUX
ejpam-6802	14	23	been	be	AUX
ejpam-6802	14	24	defined	define	VERB
ejpam-6802	14	25	for	for	ADP
ejpam-6802	14	26	years	year	NOUN
ejpam-6802	14	27	.	.	PUNCT
ejpam-6802	15	1	at	at	ADP
ejpam-6802	15	2	the	the	DET
ejpam-6802	15	3	same	same	ADJ
ejpam-6802	15	4	time	time	NOUN
ejpam-6802	15	5	,	,	PUNCT
ejpam-6802	15	6	this	this	DET
ejpam-6802	15	7	theory	theory	NOUN
ejpam-6802	15	8	has	have	AUX
ejpam-6802	15	9	been	be	AUX
ejpam-6802	15	10	applied	apply	VERB
ejpam-6802	15	11	to	to	ADP
ejpam-6802	15	12	real	real	ADJ
ejpam-6802	15	13	life	life	NOUN
ejpam-6802	15	14	and	and	CCONJ
ejpam-6802	15	15	various	various	ADJ
ejpam-6802	15	16	fields	field	NOUN
ejpam-6802	15	17	of	of	ADP
ejpam-6802	15	18	science	science	NOUN
ejpam-6802	15	19	,	,	PUNCT
ejpam-6802	15	20	namely	namely	ADV
ejpam-6802	15	21	engineering	engineering	NOUN
ejpam-6802	15	22	,	,	PUNCT
ejpam-6802	15	23	economics	economic	NOUN
ejpam-6802	15	24	,	,	PUNCT
ejpam-6802	15	25	medical	medical	ADJ
ejpam-6802	15	26	sciences	science	NOUN
ejpam-6802	15	27	,	,	PUNCT
ejpam-6802	15	28	and	and	CCONJ
ejpam-6802	15	29	computer	computer	NOUN
ejpam-6802	15	30	,	,	PUNCT
ejpam-6802	15	31	etc	etc	X
ejpam-6802	15	32	.	.	X
ejpam-6802	15	33	metric	metric	ADJ
ejpam-6802	15	34	fixed	fix	VERB
ejpam-6802	15	35	point	point	NOUN
ejpam-6802	15	36	theory	theory	NOUN
ejpam-6802	15	37	has	have	VERB
ejpam-6802	15	38	vast	vast	ADJ
ejpam-6802	15	39	applications	application	NOUN
ejpam-6802	15	40	.	.	PUNCT
ejpam-6802	16	1	the	the	DET
ejpam-6802	16	2	attraction	attraction	NOUN
ejpam-6802	16	3	of	of	ADP
ejpam-6802	16	4	productive	productive	ADJ
ejpam-6802	16	5	research	research	NOUN
ejpam-6802	16	6	activity	activity	NOUN
ejpam-6802	16	7	in	in	ADP
ejpam-6802	16	8	the	the	DET
ejpam-6802	16	9	fixed	fix	VERB
ejpam-6802	16	10	point	point	NOUN
ejpam-6802	16	11	theory	theory	NOUN
ejpam-6802	16	12	has	have	AUX
ejpam-6802	16	13	taken	take	VERB
ejpam-6802	16	14	shape	shape	NOUN
ejpam-6802	16	15	in	in	ADP
ejpam-6802	16	16	the	the	DET
ejpam-6802	16	17	form	form	NOUN
ejpam-6802	16	18	of	of	ADP
ejpam-6802	16	19	the	the	DET
ejpam-6802	16	20	search	search	NOUN
ejpam-6802	16	21	for	for	ADP
ejpam-6802	16	22	fixed	fix	VERB
ejpam-6802	16	23	points	point	NOUN
ejpam-6802	16	24	of	of	ADP
ejpam-6802	16	25	generalized	generalized	ADJ
ejpam-6802	16	26	contraction	contraction	NOUN
ejpam-6802	16	27	mappings	mapping	NOUN
ejpam-6802	16	28	.	.	PUNCT
ejpam-6802	17	1	not	not	PART
ejpam-6802	17	2	to	to	PART
ejpam-6802	17	3	mention	mention	VERB
ejpam-6802	17	4	,	,	PUNCT
ejpam-6802	17	5	a	a	DET
ejpam-6802	17	6	lot	lot	NOUN
ejpam-6802	17	7	of	of	ADP
ejpam-6802	17	8	investigators	investigator	NOUN
ejpam-6802	17	9	have	have	AUX
ejpam-6802	17	10	released	release	VERB
ejpam-6802	17	11	many	many	ADJ
ejpam-6802	17	12	publications	publication	NOUN
ejpam-6802	17	13	on	on	ADP
ejpam-6802	17	14	fixed	fix	VERB
ejpam-6802	17	15	point	point	NOUN
ejpam-6802	17	16	theory	theory	NOUN
ejpam-6802	17	17	in	in	ADP
ejpam-6802	17	18	various	various	ADJ
ejpam-6802	17	19	ways	way	NOUN
ejpam-6802	17	20	.	.	PUNCT
ejpam-6802	18	1	and	and	CCONJ
ejpam-6802	18	2	the	the	DET
ejpam-6802	18	3	existence	existence	NOUN
ejpam-6802	18	4	of	of	ADP
ejpam-6802	18	5	fixed	fix	VERB
ejpam-6802	18	6	points	point	NOUN
ejpam-6802	18	7	doi	doi	NOUN
ejpam-6802	18	8	:	:	PUNCT
ejpam-6802	18	9	https://doi.org/10.29020/nybg.ejpam.v18i4.6802	https://doi.org/10.29020/nybg.ejpam.v18i4.6802	NOUN
ejpam-6802	18	10	email	email	NOUN
ejpam-6802	18	11	address	address	NOUN
ejpam-6802	18	12	:	:	PUNCT
ejpam-6802	18	13	r.gopalan@psau.edu.sa	r.gopalan@psau.edu.sa	PROPN
ejpam-6802	18	14	(	(	PUNCT
ejpam-6802	18	15	r.	r.	PROPN
ejpam-6802	18	16	ramaswamy	ramaswamy	PROPN
ejpam-6802	18	17	)	)	PUNCT
ejpam-6802	18	18	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6802	18	19	1	1	NUM
ejpam-6802	18	20	copyright	copyright	NOUN
ejpam-6802	18	21	:	:	PUNCT
ejpam-6802	18	22	©	©	PROPN
ejpam-6802	18	23	2025	2025	NUM
ejpam-6802	18	24	the	the	DET
ejpam-6802	18	25	author(s	author(s	NOUN
ejpam-6802	18	26	)	)	PUNCT
ejpam-6802	18	27	.	.	PUNCT
ejpam-6802	19	1	(	(	PUNCT
ejpam-6802	19	2	cc	cc	NOUN
ejpam-6802	19	3	by	by	ADP
ejpam-6802	19	4	-	-	PUNCT
ejpam-6802	19	5	nc	nc	PROPN
ejpam-6802	19	6	4.0	4.0	NUM
ejpam-6802	19	7	)	)	PUNCT
ejpam-6802	19	8	r.	r.	PROPN
ejpam-6802	19	9	ramaswamy	ramaswamy	PROPN
ejpam-6802	19	10	/	/	SYM
ejpam-6802	19	11	eur	eur	PROPN
ejpam-6802	19	12	.	.	PUNCT
ejpam-6802	20	1	j.	j.	PROPN
ejpam-6802	20	2	pure	pure	PROPN
ejpam-6802	20	3	appl	appl	PROPN
ejpam-6802	20	4	.	.	PROPN
ejpam-6802	20	5	math	math	PROPN
ejpam-6802	20	6	,	,	PUNCT
ejpam-6802	20	7	18	18	NUM
ejpam-6802	20	8	(	(	PUNCT
ejpam-6802	20	9	4	4	NUM
ejpam-6802	20	10	)	)	PUNCT
ejpam-6802	20	11	(	(	PUNCT
ejpam-6802	20	12	2025	2025	NUM
ejpam-6802	20	13	)	)	PUNCT
ejpam-6802	20	14	,	,	PUNCT
ejpam-6802	20	15	6802	6802	NUM
ejpam-6802	20	16	2	2	NUM
ejpam-6802	20	17	of	of	ADP
ejpam-6802	20	18	20	20	NUM
ejpam-6802	20	19	of	of	ADP
ejpam-6802	20	20	contraction	contraction	NOUN
ejpam-6802	20	21	functions	function	NOUN
ejpam-6802	20	22	has	have	AUX
ejpam-6802	20	23	been	be	AUX
ejpam-6802	20	24	recently	recently	ADV
ejpam-6802	20	25	one	one	NUM
ejpam-6802	20	26	of	of	ADP
ejpam-6802	20	27	the	the	DET
ejpam-6802	20	28	popular	popular	ADJ
ejpam-6802	20	29	topics	topic	NOUN
ejpam-6802	20	30	in	in	ADP
ejpam-6802	20	31	the	the	DET
ejpam-6802	20	32	discipline	discipline	NOUN
ejpam-6802	20	33	of	of	ADP
ejpam-6802	20	34	fixed	fix	VERB
ejpam-6802	20	35	point	point	NOUN
ejpam-6802	20	36	theory	theory	NOUN
ejpam-6802	20	37	.	.	PUNCT
ejpam-6802	21	1	one	one	NUM
ejpam-6802	21	2	of	of	ADP
ejpam-6802	21	3	the	the	DET
ejpam-6802	21	4	generalisation	generalisation	NOUN
ejpam-6802	21	5	of	of	ADP
ejpam-6802	21	6	the	the	DET
ejpam-6802	21	7	metric	metric	ADJ
ejpam-6802	21	8	spaces	space	NOUN
ejpam-6802	21	9	was	be	AUX
ejpam-6802	21	10	the	the	DET
ejpam-6802	21	11	cone	cone	NOUN
ejpam-6802	21	12	metric	metric	ADJ
ejpam-6802	21	13	space	space	NOUN
ejpam-6802	21	14	,	,	PUNCT
ejpam-6802	21	15	introduced	introduce	VERB
ejpam-6802	21	16	in	in	ADP
ejpam-6802	21	17	2007	2007	NUM
ejpam-6802	21	18	by	by	ADP
ejpam-6802	21	19	huang	huang	PROPN
ejpam-6802	21	20	and	and	CCONJ
ejpam-6802	21	21	zhang	zhang	PROPN
ejpam-6802	22	1	[	[	X
ejpam-6802	22	2	5	5	NUM
ejpam-6802	22	3	]	]	PUNCT
ejpam-6802	22	4	who	who	PRON
ejpam-6802	22	5	presented	present	VERB
ejpam-6802	22	6	fixed	fix	VERB
ejpam-6802	22	7	point	point	NOUN
ejpam-6802	22	8	theorems	theorem	NOUN
ejpam-6802	22	9	of	of	ADP
ejpam-6802	22	10	contractive	contractive	ADJ
ejpam-6802	22	11	mappings	mapping	NOUN
ejpam-6802	22	12	in	in	ADP
ejpam-6802	22	13	the	the	DET
ejpam-6802	22	14	setting	setting	NOUN
ejpam-6802	22	15	of	of	ADP
ejpam-6802	22	16	cone	cone	NOUN
ejpam-6802	22	17	metric	metric	ADJ
ejpam-6802	22	18	space	space	NOUN
ejpam-6802	22	19	.	.	PUNCT
ejpam-6802	23	1	subsequently	subsequently	ADV
ejpam-6802	23	2	,	,	PUNCT
ejpam-6802	23	3	fixed	fix	VERB
ejpam-6802	23	4	point	point	NOUN
ejpam-6802	23	5	results	result	NOUN
ejpam-6802	23	6	were	be	AUX
ejpam-6802	23	7	reported	report	VERB
ejpam-6802	23	8	by	by	ADP
ejpam-6802	23	9	generalising	generalise	VERB
ejpam-6802	23	10	the	the	DET
ejpam-6802	23	11	cone	cone	NOUN
ejpam-6802	23	12	metric	metric	ADJ
ejpam-6802	23	13	space	space	NOUN
ejpam-6802	23	14	such	such	ADJ
ejpam-6802	23	15	as	as	ADP
ejpam-6802	23	16	dislocated	dislocate	VERB
ejpam-6802	23	17	cone	cone	NOUN
ejpam-6802	23	18	metric	metric	ADJ
ejpam-6802	23	19	space	space	NOUN
ejpam-6802	23	20	,	,	PUNCT
ejpam-6802	23	21	cone	cone	NOUN
ejpam-6802	23	22	b	b	X
ejpam-6802	23	23	-	-	PUNCT
ejpam-6802	23	24	metric	metric	ADJ
ejpam-6802	23	25	space	space	NOUN
ejpam-6802	23	26	,	,	PUNCT
ejpam-6802	23	27	rectangular	rectangular	ADJ
ejpam-6802	23	28	cone	cone	NOUN
ejpam-6802	23	29	metric	metric	ADJ
ejpam-6802	23	30	space	space	NOUN
ejpam-6802	23	31	,	,	PUNCT
ejpam-6802	23	32	rectangular	rectangular	ADJ
ejpam-6802	23	33	cone	cone	NOUN
ejpam-6802	23	34	b	b	X
ejpam-6802	23	35	metric	metric	ADJ
ejpam-6802	23	36	space	space	NOUN
ejpam-6802	23	37	etc	etc	X
ejpam-6802	23	38	were	be	AUX
ejpam-6802	23	39	reported	report	VERB
ejpam-6802	23	40	in	in	ADP
ejpam-6802	23	41	literature	literature	NOUN
ejpam-6802	23	42	,	,	PUNCT
ejpam-6802	23	43	for	for	ADP
ejpam-6802	23	44	example	example	NOUN
ejpam-6802	23	45	,	,	PUNCT
ejpam-6802	24	1	[	[	X
ejpam-6802	24	2	6–9	6–9	NOUN
ejpam-6802	24	3	]	]	PUNCT
ejpam-6802	24	4	.	.	PUNCT
ejpam-6802	25	1	in	in	ADP
ejpam-6802	25	2	2016	2016	NUM
ejpam-6802	25	3	,	,	PUNCT
ejpam-6802	25	4	the	the	DET
ejpam-6802	25	5	concept	concept	NOUN
ejpam-6802	25	6	of	of	ADP
ejpam-6802	25	7	bipolar	bipolar	ADJ
ejpam-6802	25	8	metric	metric	ADJ
ejpam-6802	25	9	space	space	NOUN
ejpam-6802	25	10	has	have	AUX
ejpam-6802	25	11	been	be	AUX
ejpam-6802	25	12	established	establish	VERB
ejpam-6802	25	13	by	by	ADP
ejpam-6802	25	14	mutlu	mutlu	PROPN
ejpam-6802	25	15	&	&	CCONJ
ejpam-6802	25	16	u.	u.	PROPN
ejpam-6802	25	17	gürdal,[10	gürdal,[10	PROPN
ejpam-6802	25	18	]	]	PUNCT
ejpam-6802	25	19	and	and	CCONJ
ejpam-6802	25	20	the	the	DET
ejpam-6802	25	21	authors	author	NOUN
ejpam-6802	25	22	explored	explore	VERB
ejpam-6802	25	23	several	several	ADJ
ejpam-6802	25	24	basic	basic	ADJ
ejpam-6802	25	25	fixed	fix	VERB
ejpam-6802	25	26	point	point	NOUN
ejpam-6802	25	27	and	and	CCONJ
ejpam-6802	25	28	coupled	couple	VERB
ejpam-6802	25	29	fixed	fix	VERB
ejpam-6802	25	30	point	point	NOUN
ejpam-6802	25	31	theorems	theorem	NOUN
ejpam-6802	25	32	for	for	ADP
ejpam-6802	25	33	co	co	NOUN
ejpam-6802	25	34	-	-	NOUN
ejpam-6802	25	35	variant	variant	ADJ
ejpam-6802	25	36	and	and	CCONJ
ejpam-6802	25	37	contra	contra	ADJ
ejpam-6802	25	38	-	-	ADJ
ejpam-6802	25	39	variant	variant	ADJ
ejpam-6802	25	40	mappings	mapping	NOUN
ejpam-6802	25	41	subject	subject	ADJ
ejpam-6802	25	42	to	to	ADP
ejpam-6802	25	43	contractive	contractive	ADJ
ejpam-6802	25	44	conditions	condition	NOUN
ejpam-6802	25	45	;	;	PUNCT
ejpam-6802	25	46	see	see	VERB
ejpam-6802	25	47	[	[	X
ejpam-6802	25	48	11	11	NUM
ejpam-6802	25	49	]	]	PUNCT
ejpam-6802	25	50	.	.	PUNCT
ejpam-6802	26	1	many	many	ADJ
ejpam-6802	26	2	important	important	ADJ
ejpam-6802	26	3	contributions	contribution	NOUN
ejpam-6802	26	4	by	by	ADP
ejpam-6802	26	5	many	many	ADJ
ejpam-6802	26	6	authors	author	NOUN
ejpam-6802	26	7	have	have	AUX
ejpam-6802	26	8	been	be	AUX
ejpam-6802	26	9	made	make	VERB
ejpam-6802	26	10	in	in	ADP
ejpam-6802	26	11	bipolar	bipolar	ADJ
ejpam-6802	26	12	metric	metric	ADJ
ejpam-6802	26	13	spaces	space	NOUN
ejpam-6802	26	14	see	see	VERB
ejpam-6802	26	15	,	,	PUNCT
ejpam-6802	26	16	[	[	X
ejpam-6802	26	17	12–19	12–19	NOUN
ejpam-6802	26	18	]	]	PUNCT
ejpam-6802	26	19	.	.	PUNCT
ejpam-6802	27	1	it	it	PRON
ejpam-6802	27	2	was	be	AUX
ejpam-6802	27	3	in	in	ADP
ejpam-6802	27	4	2021	2021	NUM
ejpam-6802	27	5	that	that	PRON
ejpam-6802	27	6	gaba	gaba	PROPN
ejpam-6802	27	7	aphane	aphane	ADJ
ejpam-6802	27	8	and	and	CCONJ
ejpam-6802	27	9	aydi	aydi	VERB
ejpam-6802	27	10	,	,	PUNCT
ejpam-6802	27	11	[	[	X
ejpam-6802	27	12	20	20	NUM
ejpam-6802	27	13	]	]	PUNCT
ejpam-6802	27	14	showed	show	VERB
ejpam-6802	27	15	fixed	fix	VERB
ejpam-6802	27	16	point	point	NOUN
ejpam-6802	27	17	theorems	theorem	NOUN
ejpam-6802	27	18	on	on	ADP
ejpam-6802	27	19	bipolar	bipolar	ADJ
ejpam-6802	27	20	metric	metric	ADJ
ejpam-6802	27	21	space	space	NOUN
ejpam-6802	27	22	.	.	PUNCT
ejpam-6802	28	1	ampadu	ampadu	ADJ
ejpam-6802	28	2	in	in	ADP
ejpam-6802	28	3	2007[21	2007[21	NUM
ejpam-6802	28	4	]	]	PUNCT
ejpam-6802	29	1	introduced	introduce	VERB
ejpam-6802	29	2	to	to	ADP
ejpam-6802	29	3	the	the	DET
ejpam-6802	29	4	concept	concept	NOUN
ejpam-6802	29	5	of	of	ADP
ejpam-6802	29	6	multiplicative	multiplicative	ADJ
ejpam-6802	29	7	cone	cone	NOUN
ejpam-6802	29	8	metric	metric	ADJ
ejpam-6802	29	9	space	space	NOUN
ejpam-6802	29	10	&	&	CCONJ
ejpam-6802	29	11	proved	prove	VERB
ejpam-6802	29	12	coupled	couple	VERB
ejpam-6802	29	13	version	version	NOUN
ejpam-6802	29	14	of	of	ADP
ejpam-6802	29	15	the	the	DET
ejpam-6802	29	16	higher	high	ADJ
ejpam-6802	29	17	order	order	NOUN
ejpam-6802	29	18	banach	banach	NOUN
ejpam-6802	29	19	contracton	contracton	PROPN
ejpam-6802	29	20	principle	principle	NOUN
ejpam-6802	29	21	.	.	PUNCT
ejpam-6802	30	1	in	in	ADP
ejpam-6802	30	2	this	this	DET
ejpam-6802	30	3	article	article	NOUN
ejpam-6802	30	4	,	,	PUNCT
ejpam-6802	30	5	the	the	DET
ejpam-6802	30	6	definition	definition	NOUN
ejpam-6802	30	7	to	to	ADP
ejpam-6802	30	8	the	the	DET
ejpam-6802	30	9	concept	concept	NOUN
ejpam-6802	30	10	of	of	ADP
ejpam-6802	30	11	multiplicative	multiplicative	ADJ
ejpam-6802	30	12	cone	cone	NOUN
ejpam-6802	30	13	bipolar	bipolar	ADJ
ejpam-6802	30	14	metric	metric	ADJ
ejpam-6802	30	15	space	space	NOUN
ejpam-6802	30	16	is	be	AUX
ejpam-6802	30	17	being	be	AUX
ejpam-6802	30	18	introduced	introduce	VERB
ejpam-6802	30	19	&	&	CCONJ
ejpam-6802	30	20	fixed	fix	VERB
ejpam-6802	30	21	point	point	NOUN
ejpam-6802	30	22	theorems	theorem	NOUN
ejpam-6802	30	23	are	be	AUX
ejpam-6802	30	24	established	establish	VERB
ejpam-6802	30	25	.	.	PUNCT
ejpam-6802	31	1	the	the	DET
ejpam-6802	31	2	derived	derive	VERB
ejpam-6802	31	3	results	result	NOUN
ejpam-6802	31	4	extend	extend	VERB
ejpam-6802	31	5	/	/	SYM
ejpam-6802	31	6	generalise	generalise	VERB
ejpam-6802	31	7	proven	prove	VERB
ejpam-6802	31	8	results	result	NOUN
ejpam-6802	31	9	of	of	ADP
ejpam-6802	31	10	the	the	DET
ejpam-6802	31	11	past	past	NOUN
ejpam-6802	31	12	and	and	CCONJ
ejpam-6802	31	13	are	be	AUX
ejpam-6802	31	14	supplemented	supplement	VERB
ejpam-6802	31	15	with	with	ADP
ejpam-6802	31	16	non	non	ADJ
ejpam-6802	31	17	-	-	ADJ
ejpam-6802	31	18	trivial	trivial	ADJ
ejpam-6802	31	19	examples	example	NOUN
ejpam-6802	31	20	.	.	PUNCT
ejpam-6802	32	1	the	the	DET
ejpam-6802	32	2	results	result	NOUN
ejpam-6802	32	3	are	be	AUX
ejpam-6802	32	4	applied	apply	VERB
ejpam-6802	32	5	to	to	PART
ejpam-6802	32	6	find	find	VERB
ejpam-6802	32	7	solutions	solution	NOUN
ejpam-6802	32	8	of	of	ADP
ejpam-6802	32	9	integral	integral	ADJ
ejpam-6802	32	10	equation	equation	NOUN
ejpam-6802	32	11	and	and	CCONJ
ejpam-6802	32	12	fractional	fractional	ADJ
ejpam-6802	32	13	differential	differential	ADJ
ejpam-6802	32	14	equations	equation	NOUN
ejpam-6802	32	15	.	.	PUNCT
ejpam-6802	33	1	the	the	DET
ejpam-6802	33	2	rest	rest	NOUN
ejpam-6802	33	3	of	of	ADP
ejpam-6802	33	4	the	the	DET
ejpam-6802	33	5	paper	paper	NOUN
ejpam-6802	33	6	is	be	AUX
ejpam-6802	33	7	organised	organise	VERB
ejpam-6802	33	8	as	as	SCONJ
ejpam-6802	33	9	follows	follow	VERB
ejpam-6802	33	10	:	:	PUNCT
ejpam-6802	33	11	in	in	ADP
ejpam-6802	33	12	section	section	NOUN
ejpam-6802	33	13	2	2	NUM
ejpam-6802	33	14	,	,	PUNCT
ejpam-6802	33	15	some	some	DET
ejpam-6802	33	16	basic	basic	ADJ
ejpam-6802	33	17	definitions	definition	NOUN
ejpam-6802	33	18	are	be	AUX
ejpam-6802	33	19	given	give	VERB
ejpam-6802	33	20	.	.	PUNCT
ejpam-6802	34	1	in	in	ADP
ejpam-6802	34	2	section	section	NOUN
ejpam-6802	34	3	-3	-3	PUNCT
ejpam-6802	34	4	fixed	fix	VERB
ejpam-6802	34	5	point	point	NOUN
ejpam-6802	34	6	results	result	NOUN
ejpam-6802	34	7	in	in	ADP
ejpam-6802	34	8	the	the	DET
ejpam-6802	34	9	setting	setting	NOUN
ejpam-6802	34	10	of	of	ADP
ejpam-6802	34	11	mcbms	mcbms	NOUN
ejpam-6802	34	12	are	be	AUX
ejpam-6802	34	13	established	establish	VERB
ejpam-6802	34	14	and	and	CCONJ
ejpam-6802	34	15	the	the	DET
ejpam-6802	34	16	derived	derive	VERB
ejpam-6802	34	17	results	result	NOUN
ejpam-6802	34	18	are	be	AUX
ejpam-6802	34	19	supplemented	supplement	VERB
ejpam-6802	34	20	using	use	VERB
ejpam-6802	34	21	non	non	ADJ
ejpam-6802	34	22	-	-	ADJ
ejpam-6802	34	23	trivial	trivial	ADJ
ejpam-6802	34	24	examples	example	NOUN
ejpam-6802	34	25	.	.	PUNCT
ejpam-6802	35	1	fractional	fractional	ADJ
ejpam-6802	35	2	calculus	calculus	NOUN
ejpam-6802	35	3	plays	play	VERB
ejpam-6802	35	4	an	an	DET
ejpam-6802	35	5	important	important	ADJ
ejpam-6802	35	6	role	role	NOUN
ejpam-6802	35	7	in	in	ADP
ejpam-6802	35	8	designing	designing	NOUN
ejpam-6802	35	9	and	and	CCONJ
ejpam-6802	35	10	analysis	analysis	NOUN
ejpam-6802	35	11	of	of	ADP
ejpam-6802	35	12	various	various	ADJ
ejpam-6802	35	13	mathematical	mathematical	ADJ
ejpam-6802	35	14	models	model	NOUN
ejpam-6802	35	15	that	that	PRON
ejpam-6802	35	16	help	help	VERB
ejpam-6802	35	17	in	in	ADP
ejpam-6802	35	18	achieving	achieve	VERB
ejpam-6802	35	19	sustainable	sustainable	ADJ
ejpam-6802	35	20	development	development	NOUN
ejpam-6802	35	21	goals	goal	NOUN
ejpam-6802	35	22	of	of	ADP
ejpam-6802	35	23	the	the	DET
ejpam-6802	35	24	united	united	PROPN
ejpam-6802	35	25	nations	nations	PROPN
ejpam-6802	35	26	and	and	CCONJ
ejpam-6802	35	27	fixed	fix	VERB
ejpam-6802	35	28	point	point	NOUN
ejpam-6802	35	29	theory	theory	NOUN
ejpam-6802	35	30	plays	play	VERB
ejpam-6802	35	31	a	a	DET
ejpam-6802	35	32	vital	vital	ADJ
ejpam-6802	35	33	role	role	NOUN
ejpam-6802	35	34	in	in	ADP
ejpam-6802	35	35	analysing	analyse	VERB
ejpam-6802	35	36	the	the	DET
ejpam-6802	35	37	existence	existence	NOUN
ejpam-6802	35	38	of	of	ADP
ejpam-6802	35	39	unique	unique	ADJ
ejpam-6802	35	40	solutions	solution	NOUN
ejpam-6802	35	41	to	to	ADP
ejpam-6802	35	42	those	those	DET
ejpam-6802	35	43	systems	system	NOUN
ejpam-6802	35	44	,	,	PUNCT
ejpam-6802	35	45	see	see	VERB
ejpam-6802	35	46	[	[	X
ejpam-6802	35	47	22–26	22–26	NOUN
ejpam-6802	35	48	]	]	PUNCT
ejpam-6802	35	49	.	.	PUNCT
ejpam-6802	36	1	accordingly	accordingly	ADV
ejpam-6802	36	2	,	,	PUNCT
ejpam-6802	36	3	in	in	ADP
ejpam-6802	36	4	section-4	section-4	NUM
ejpam-6802	36	5	,	,	PUNCT
ejpam-6802	36	6	the	the	DET
ejpam-6802	36	7	fixed	fix	VERB
ejpam-6802	36	8	point	point	NOUN
ejpam-6802	36	9	results	result	NOUN
ejpam-6802	36	10	are	be	AUX
ejpam-6802	36	11	applied	apply	VERB
ejpam-6802	36	12	to	to	PART
ejpam-6802	36	13	find	find	VERB
ejpam-6802	36	14	analytical	analytical	ADJ
ejpam-6802	36	15	solution	solution	NOUN
ejpam-6802	36	16	to	to	ADP
ejpam-6802	36	17	integral	integral	ADJ
ejpam-6802	36	18	and	and	CCONJ
ejpam-6802	36	19	fractional	fractional	ADJ
ejpam-6802	36	20	differential	differential	ADJ
ejpam-6802	36	21	equations	equation	NOUN
ejpam-6802	36	22	.	.	PUNCT
ejpam-6802	37	1	finally	finally	ADV
ejpam-6802	37	2	,	,	PUNCT
ejpam-6802	37	3	the	the	DET
ejpam-6802	37	4	manuscript	manuscript	NOUN
ejpam-6802	37	5	is	be	AUX
ejpam-6802	37	6	concluded	conclude	VERB
ejpam-6802	37	7	by	by	ADP
ejpam-6802	37	8	placing	place	VERB
ejpam-6802	37	9	some	some	DET
ejpam-6802	37	10	open	open	ADJ
ejpam-6802	37	11	problems	problem	NOUN
ejpam-6802	37	12	for	for	ADP
ejpam-6802	37	13	future	future	ADJ
ejpam-6802	37	14	research	research	NOUN
ejpam-6802	37	15	in	in	ADP
ejpam-6802	37	16	section-5	section-5	PROPN
ejpam-6802	37	17	.	.	PROPN
ejpam-6802	37	18	2	2	NUM
ejpam-6802	37	19	.	.	NOUN
ejpam-6802	37	20	preliminaries	preliminary	NOUN
ejpam-6802	37	21	the	the	DET
ejpam-6802	37	22	following	follow	VERB
ejpam-6802	37	23	definitions	definition	NOUN
ejpam-6802	37	24	and	and	CCONJ
ejpam-6802	37	25	monograph	monograph	NOUN
ejpam-6802	37	26	are	be	AUX
ejpam-6802	37	27	required	require	VERB
ejpam-6802	37	28	in	in	ADP
ejpam-6802	37	29	the	the	DET
ejpam-6802	37	30	sequel	sequel	NOUN
ejpam-6802	37	31	.	.	PUNCT
ejpam-6802	38	1	let	let	VERB
ejpam-6802	38	2	a	a	PRON
ejpam-6802	38	3	always	always	ADV
ejpam-6802	38	4	be	be	AUX
ejpam-6802	38	5	a	a	DET
ejpam-6802	38	6	real	real	ADJ
ejpam-6802	38	7	banach	banach	NOUN
ejpam-6802	38	8	space	space	NOUN
ejpam-6802	38	9	&	&	CCONJ
ejpam-6802	38	10	z	z	PROPN
ejpam-6802	38	11	⊆	⊆	NUM
ejpam-6802	38	12	a.	a.	NOUN
ejpam-6802	38	13	z	z	NOUN
ejpam-6802	38	14	is	be	AUX
ejpam-6802	38	15	known	know	VERB
ejpam-6802	38	16	as	as	ADP
ejpam-6802	38	17	a	a	DET
ejpam-6802	38	18	cone	cone	NOUN
ejpam-6802	38	19	iff	iff	NOUN
ejpam-6802	38	20	(	(	PUNCT
ejpam-6802	38	21	i	i	NOUN
ejpam-6802	38	22	)	)	PUNCT
ejpam-6802	38	23	z	z	PROPN
ejpam-6802	38	24	is	be	AUX
ejpam-6802	38	25	closed	closed	ADJ
ejpam-6802	38	26	,	,	PUNCT
ejpam-6802	38	27	non	non	ADJ
ejpam-6802	38	28	-	-	ADJ
ejpam-6802	38	29	void	void	ADJ
ejpam-6802	38	30	,	,	PUNCT
ejpam-6802	38	31	&	&	CCONJ
ejpam-6802	38	32	z	z	PROPN
ejpam-6802	38	33	̸=	̸=	PROPN
ejpam-6802	38	34	{	{	PUNCT
ejpam-6802	38	35	0	0	NUM
ejpam-6802	38	36	}	}	PUNCT
ejpam-6802	38	37	;	;	PUNCT
ejpam-6802	38	38	(	(	PUNCT
ejpam-6802	38	39	ii	ii	NOUN
ejpam-6802	38	40	)	)	PUNCT
ejpam-6802	38	41	a	a	PROPN
ejpam-6802	38	42	,	,	PUNCT
ejpam-6802	38	43	c	c	PROPN
ejpam-6802	38	44	∈	∈	PROPN
ejpam-6802	38	45	r	r	NOUN
ejpam-6802	38	46	,	,	PUNCT
ejpam-6802	38	47	a	a	PRON
ejpam-6802	38	48	,	,	PUNCT
ejpam-6802	38	49	c	c	X
ejpam-6802	38	50	≥	≥	NOUN
ejpam-6802	38	51	0	0	NUM
ejpam-6802	38	52	,	,	PUNCT
ejpam-6802	38	53	q	q	NOUN
ejpam-6802	38	54	,	,	PUNCT
ejpam-6802	38	55	ϑ	ϑ	X
ejpam-6802	38	56	∈	∈	PROPN
ejpam-6802	38	57	z	z	NOUN
ejpam-6802	38	58	=	=	NOUN
ejpam-6802	38	59	⇒	⇒	NOUN
ejpam-6802	38	60	aq+	aq+	ADJ
ejpam-6802	38	61	cϑ	cϑ	ADJ
ejpam-6802	38	62	∈	∈	PROPN
ejpam-6802	38	63	z	z	NOUN
ejpam-6802	38	64	;	;	PUNCT
ejpam-6802	38	65	(	(	PUNCT
ejpam-6802	38	66	iii	iii	X
ejpam-6802	38	67	)	)	PUNCT
ejpam-6802	39	1	q	q	NOUN
ejpam-6802	39	2	∈	∈	PROPN
ejpam-6802	39	3	z	z	NOUN
ejpam-6802	39	4	and	and	CCONJ
ejpam-6802	39	5	−q	−q	ADJ
ejpam-6802	39	6	∈	∈	NOUN
ejpam-6802	39	7	z	z	NOUN
ejpam-6802	39	8	=	=	NOUN
ejpam-6802	39	9	⇒	⇒	X
ejpam-6802	39	10	q	q	NOUN
ejpam-6802	40	1	=	=	PUNCT
ejpam-6802	40	2	0	0	NUM
ejpam-6802	40	3	.	.	PUNCT
ejpam-6802	41	1	given	give	VERB
ejpam-6802	41	2	a	a	DET
ejpam-6802	41	3	cone	cone	NOUN
ejpam-6802	41	4	z	z	X
ejpam-6802	41	5	⊂	⊂	PROPN
ejpam-6802	41	6	a	a	X
ejpam-6802	41	7	,	,	PUNCT
ejpam-6802	41	8	we	we	PRON
ejpam-6802	41	9	define	define	VERB
ejpam-6802	41	10	a	a	DET
ejpam-6802	41	11	partial	partial	ADJ
ejpam-6802	41	12	ordering	ordering	NOUN
ejpam-6802	41	13	≤	≤	NOUN
ejpam-6802	41	14	with	with	ADP
ejpam-6802	41	15	respect	respect	NOUN
ejpam-6802	41	16	to	to	ADP
ejpam-6802	41	17	z	z	NOUN
ejpam-6802	41	18	as	as	ADP
ejpam-6802	41	19	q	q	PROPN
ejpam-6802	41	20	≤	≤	PROPN
ejpam-6802	41	21	ϑ	ϑ	X
ejpam-6802	41	22	iff	iff	PROPN
ejpam-6802	41	23	ϑ−	ϑ−	PROPN
ejpam-6802	41	24	q	q	PROPN
ejpam-6802	41	25	∈	∈	PROPN
ejpam-6802	41	26	z.	z.	NOUN
ejpam-6802	42	1	we	we	PRON
ejpam-6802	42	2	shall	shall	AUX
ejpam-6802	42	3	write	write	VERB
ejpam-6802	42	4	q	q	PROPN
ejpam-6802	42	5	<	<	X
ejpam-6802	42	6	ϑ	ϑ	X
ejpam-6802	42	7	to	to	PART
ejpam-6802	42	8	indicate	indicate	VERB
ejpam-6802	42	9	that	that	DET
ejpam-6802	42	10	q	q	PROPN
ejpam-6802	42	11	≤	≤	X
ejpam-6802	42	12	ϑ	ϑ	X
ejpam-6802	42	13	and	and	CCONJ
ejpam-6802	42	14	q	q	PROPN
ejpam-6802	42	15	̸=	̸=	PROPN
ejpam-6802	42	16	ϑ	ϑ	NOUN
ejpam-6802	42	17	,	,	PUNCT
ejpam-6802	42	18	while	while	SCONJ
ejpam-6802	42	19	q	q	NOUN
ejpam-6802	42	20	≪	≪	PUNCT
ejpam-6802	42	21	ϑ	ϑ	X
ejpam-6802	42	22	will	will	AUX
ejpam-6802	42	23	stand	stand	VERB
ejpam-6802	42	24	for	for	ADP
ejpam-6802	42	25	ϑ−	ϑ−	PROPN
ejpam-6802	42	26	q	q	PROPN
ejpam-6802	42	27	∈	∈	PROPN
ejpam-6802	42	28	intz	intz	NOUN
ejpam-6802	42	29	,	,	PUNCT
ejpam-6802	42	30	intz	intz	PROPN
ejpam-6802	42	31	is	be	AUX
ejpam-6802	42	32	the	the	DET
ejpam-6802	42	33	interior	interior	NOUN
ejpam-6802	42	34	of	of	ADP
ejpam-6802	42	35	z.	z.	PROPN
ejpam-6802	43	1	the	the	DET
ejpam-6802	43	2	cone	cone	PROPN
ejpam-6802	43	3	z	z	PROPN
ejpam-6802	43	4	is	be	AUX
ejpam-6802	43	5	referred	refer	VERB
ejpam-6802	43	6	to	to	ADP
ejpam-6802	43	7	normal	normal	ADJ
ejpam-6802	43	8	if	if	SCONJ
ejpam-6802	43	9	there	there	PRON
ejpam-6802	43	10	is	be	VERB
ejpam-6802	43	11	a	a	DET
ejpam-6802	43	12	number	number	NOUN
ejpam-6802	43	13	w	w	ADP
ejpam-6802	43	14	>	>	X
ejpam-6802	43	15	0	0	PUNCT
ejpam-6802	44	1	s.t	s.t	PROPN
ejpam-6802	44	2	for	for	ADP
ejpam-6802	44	3	all	all	DET
ejpam-6802	44	4	q	q	PROPN
ejpam-6802	44	5	,	,	PUNCT
ejpam-6802	44	6	ϑ	ϑ	X
ejpam-6802	44	7	∈	∈	PROPN
ejpam-6802	44	8	a	a	PRON
ejpam-6802	44	9	,	,	PUNCT
ejpam-6802	44	10	0	0	NUM
ejpam-6802	44	11	≤	≤	NUM
ejpam-6802	44	12	q	q	PROPN
ejpam-6802	44	13	≤	≤	NUM
ejpam-6802	44	14	ϑ	ϑ	PRON
ejpam-6802	44	15	⇒	⇒	NOUN
ejpam-6802	44	16	||q||	||q||	PROPN
ejpam-6802	44	17	≤	≤	NUM
ejpam-6802	44	18	||ϑ||	||ϑ||	NOUN
ejpam-6802	44	19	.	.	PUNCT
ejpam-6802	45	1	r.	r.	PROPN
ejpam-6802	45	2	ramaswamy	ramaswamy	PROPN
ejpam-6802	45	3	/	/	SYM
ejpam-6802	45	4	eur	eur	PROPN
ejpam-6802	45	5	.	.	PUNCT
ejpam-6802	46	1	j.	j.	PROPN
ejpam-6802	46	2	pure	pure	PROPN
ejpam-6802	46	3	appl	appl	PROPN
ejpam-6802	46	4	.	.	PROPN
ejpam-6802	46	5	math	math	PROPN
ejpam-6802	46	6	,	,	PUNCT
ejpam-6802	46	7	18	18	NUM
ejpam-6802	46	8	(	(	PUNCT
ejpam-6802	46	9	4	4	NUM
ejpam-6802	46	10	)	)	PUNCT
ejpam-6802	46	11	(	(	PUNCT
ejpam-6802	46	12	2025	2025	NUM
ejpam-6802	46	13	)	)	PUNCT
ejpam-6802	46	14	,	,	PUNCT
ejpam-6802	46	15	6802	6802	NUM
ejpam-6802	46	16	3	3	NUM
ejpam-6802	46	17	of	of	ADP
ejpam-6802	46	18	20	20	NUM
ejpam-6802	46	19	the	the	DET
ejpam-6802	46	20	smallest	small	ADJ
ejpam-6802	46	21	non	non	ADJ
ejpam-6802	46	22	negative	negative	ADJ
ejpam-6802	46	23	number	number	NOUN
ejpam-6802	46	24	that	that	PRON
ejpam-6802	46	25	satisfies	satisfy	VERB
ejpam-6802	46	26	the	the	DET
ejpam-6802	46	27	above	above	ADJ
ejpam-6802	46	28	is	be	AUX
ejpam-6802	46	29	known	know	VERB
ejpam-6802	46	30	as	as	ADP
ejpam-6802	46	31	normal	normal	ADJ
ejpam-6802	46	32	constant	constant	ADJ
ejpam-6802	46	33	z.	z.	NOUN
ejpam-6802	46	34	the	the	DET
ejpam-6802	46	35	cone	cone	PROPN
ejpam-6802	46	36	z	z	PROPN
ejpam-6802	46	37	is	be	AUX
ejpam-6802	46	38	regular	regular	ADJ
ejpam-6802	46	39	,	,	PUNCT
ejpam-6802	46	40	if	if	SCONJ
ejpam-6802	46	41	every	every	DET
ejpam-6802	46	42	monotonically	monotonically	ADV
ejpam-6802	46	43	increasing	increase	VERB
ejpam-6802	46	44	and	and	CCONJ
ejpam-6802	46	45	upper	upper	ADJ
ejpam-6802	46	46	bounded	bounded	ADJ
ejpam-6802	46	47	sequence	sequence	NOUN
ejpam-6802	46	48	is	be	AUX
ejpam-6802	46	49	convergent	convergent	ADJ
ejpam-6802	46	50	.	.	PUNCT
ejpam-6802	47	1	to	to	PART
ejpam-6802	47	2	be	be	AUX
ejpam-6802	47	3	specific	specific	ADJ
ejpam-6802	47	4	,	,	PUNCT
ejpam-6802	47	5	if	if	SCONJ
ejpam-6802	47	6	{	{	PUNCT
ejpam-6802	47	7	qℓ	qℓ	AUX
ejpam-6802	47	8	}	}	PUNCT
ejpam-6802	47	9	is	be	AUX
ejpam-6802	47	10	sequence	sequence	NOUN
ejpam-6802	47	11	for	for	ADP
ejpam-6802	47	12	which	which	PRON
ejpam-6802	47	13	q1	q1	VERB
ejpam-6802	47	14	≤	≤	NUM
ejpam-6802	47	15	q2	q2	NOUN
ejpam-6802	47	16	≤	≤	PUNCT
ejpam-6802	47	17	·	·	PUNCT
ejpam-6802	48	1	·	·	PUNCT
ejpam-6802	48	2	·	·	PUNCT
ejpam-6802	48	3	≤	≤	NUM
ejpam-6802	48	4	qℓ	qℓ	ADP
ejpam-6802	48	5	≤	≤	NOUN
ejpam-6802	48	6	·	·	PUNCT
ejpam-6802	48	7	·	·	PUNCT
ejpam-6802	48	8	·	·	PUNCT
ejpam-6802	49	1	≤	≤	NUM
ejpam-6802	49	2	ϑ	ϑ	X
ejpam-6802	49	3	for	for	ADP
ejpam-6802	49	4	some	some	DET
ejpam-6802	49	5	ϑ	ϑ	NOUN
ejpam-6802	49	6	∈	∈	NOUN
ejpam-6802	49	7	a	a	PRON
ejpam-6802	49	8	,	,	PUNCT
ejpam-6802	49	9	there	there	PRON
ejpam-6802	49	10	is	be	VERB
ejpam-6802	49	11	q	q	PROPN
ejpam-6802	49	12	∈	∈	PROPN
ejpam-6802	49	13	a	a	DET
ejpam-6802	49	14	such	such	ADJ
ejpam-6802	49	15	that	that	PRON
ejpam-6802	49	16	||qℓ	||qℓ	VERB
ejpam-6802	49	17	−	−	PROPN
ejpam-6802	49	18	q||	q||	PROPN
ejpam-6802	49	19	→	→	SYM
ejpam-6802	49	20	0(ℓ	0(ℓ	NUM
ejpam-6802	49	21	→	→	SYM
ejpam-6802	49	22	∞	∞	NUM
ejpam-6802	49	23	)	)	PUNCT
ejpam-6802	49	24	.	.	PUNCT
ejpam-6802	50	1	in	in	ADP
ejpam-6802	50	2	other	other	ADJ
ejpam-6802	50	3	words	word	NOUN
ejpam-6802	50	4	the	the	DET
ejpam-6802	50	5	cone	cone	NOUN
ejpam-6802	50	6	z	z	PROPN
ejpam-6802	50	7	is	be	AUX
ejpam-6802	50	8	regular	regular	ADJ
ejpam-6802	50	9	iff	iff	NOUN
ejpam-6802	50	10	every	every	DET
ejpam-6802	50	11	sequence	sequence	NOUN
ejpam-6802	50	12	which	which	PRON
ejpam-6802	50	13	is	be	AUX
ejpam-6802	50	14	non	non	ADJ
ejpam-6802	50	15	-	-	ADJ
ejpam-6802	50	16	increasing	increase	VERB
ejpam-6802	50	17	and	and	CCONJ
ejpam-6802	50	18	bounded	bound	VERB
ejpam-6802	50	19	from	from	ADP
ejpam-6802	50	20	below	below	ADV
ejpam-6802	50	21	is	be	AUX
ejpam-6802	50	22	convergent	convergent	NOUN
ejpam-6802	50	23	.	.	PUNCT
ejpam-6802	51	1	it	it	PRON
ejpam-6802	51	2	is	be	AUX
ejpam-6802	51	3	a	a	DET
ejpam-6802	51	4	common	common	ADJ
ejpam-6802	51	5	knowledge	knowledge	NOUN
ejpam-6802	51	6	that	that	SCONJ
ejpam-6802	51	7	a	a	DET
ejpam-6802	51	8	regular	regular	ADJ
ejpam-6802	51	9	cone	cone	NOUN
ejpam-6802	51	10	is	be	AUX
ejpam-6802	51	11	a	a	DET
ejpam-6802	51	12	normal	normal	ADJ
ejpam-6802	51	13	one	one	NOUN
ejpam-6802	51	14	.	.	PUNCT
ejpam-6802	52	1	in	in	ADP
ejpam-6802	52	2	what	what	PRON
ejpam-6802	52	3	follows	follow	VERB
ejpam-6802	52	4	we	we	PRON
ejpam-6802	52	5	always	always	ADV
ejpam-6802	52	6	have	have	VERB
ejpam-6802	52	7	in	in	ADP
ejpam-6802	52	8	mind	mind	NOUN
ejpam-6802	52	9	that	that	SCONJ
ejpam-6802	52	10	a	a	PRON
ejpam-6802	52	11	is	be	AUX
ejpam-6802	52	12	a	a	DET
ejpam-6802	52	13	banach	banach	NOUN
ejpam-6802	52	14	space	space	NOUN
ejpam-6802	52	15	,	,	PUNCT
ejpam-6802	52	16	z	z	PROPN
ejpam-6802	52	17	is	be	AUX
ejpam-6802	52	18	a	a	DET
ejpam-6802	52	19	cone	cone	NOUN
ejpam-6802	52	20	in	in	ADP
ejpam-6802	52	21	a	a	PRON
ejpam-6802	52	22	with	with	ADP
ejpam-6802	52	23	intz	intz	NOUN
ejpam-6802	52	24	̸=	̸=	PROPN
ejpam-6802	52	25	∅	∅	NOUN
ejpam-6802	52	26	and	and	CCONJ
ejpam-6802	52	27	≤	≤	NOUN
ejpam-6802	52	28	is	be	AUX
ejpam-6802	52	29	partial	partial	ADJ
ejpam-6802	52	30	ordering	ordering	NOUN
ejpam-6802	52	31	with	with	ADP
ejpam-6802	52	32	respect	respect	NOUN
ejpam-6802	52	33	to	to	ADP
ejpam-6802	52	34	z.	z.	PROPN
ejpam-6802	52	35	definition	definition	NOUN
ejpam-6802	52	36	2.1	2.1	NUM
ejpam-6802	52	37	.	.	PUNCT
ejpam-6802	53	1	consider	consider	VERB
ejpam-6802	53	2	b	b	NOUN
ejpam-6802	53	3	and	and	CCONJ
ejpam-6802	53	4	f	f	PROPN
ejpam-6802	53	5	be	be	AUX
ejpam-6802	53	6	non	non	ADJ
ejpam-6802	53	7	-	-	ADJ
ejpam-6802	53	8	void	void	ADJ
ejpam-6802	53	9	sets	set	NOUN
ejpam-6802	53	10	and	and	CCONJ
ejpam-6802	53	11	α	α	NOUN
ejpam-6802	53	12	:	:	PUNCT
ejpam-6802	53	13	b	b	X
ejpam-6802	53	14	×	×	NOUN
ejpam-6802	53	15	f	f	X
ejpam-6802	53	16	→	→	PUNCT
ejpam-6802	53	17	a	a	DET
ejpam-6802	53	18	be	be	AUX
ejpam-6802	53	19	a	a	DET
ejpam-6802	53	20	function	function	NOUN
ejpam-6802	53	21	such	such	ADJ
ejpam-6802	53	22	that	that	SCONJ
ejpam-6802	53	23	(	(	PUNCT
ejpam-6802	53	24	i	i	NOUN
ejpam-6802	53	25	)	)	PUNCT
ejpam-6802	53	26	if	if	SCONJ
ejpam-6802	53	27	α(q	α(q	PROPN
ejpam-6802	53	28	,	,	PUNCT
ejpam-6802	53	29	ϑ	ϑ	X
ejpam-6802	53	30	)	)	PUNCT
ejpam-6802	53	31	=	=	SYM
ejpam-6802	53	32	1	1	NUM
ejpam-6802	53	33	then	then	ADV
ejpam-6802	53	34	q	q	X
ejpam-6802	53	35	=	=	SYM
ejpam-6802	53	36	ϑ	ϑ	NOUN
ejpam-6802	53	37	,	,	PUNCT
ejpam-6802	53	38	for	for	ADP
ejpam-6802	53	39	all	all	PRON
ejpam-6802	53	40	(	(	PUNCT
ejpam-6802	53	41	q	q	ADJ
ejpam-6802	53	42	,	,	PUNCT
ejpam-6802	53	43	ϑ	ϑ	NOUN
ejpam-6802	53	44	)	)	PUNCT
ejpam-6802	53	45	∈	∈	PROPN
ejpam-6802	53	46	b×f	b×f	PROPN
ejpam-6802	53	47	.	.	PUNCT
ejpam-6802	54	1	,	,	PUNCT
ejpam-6802	54	2	where	where	SCONJ
ejpam-6802	54	3	1	1	NUM
ejpam-6802	54	4	represents	represent	VERB
ejpam-6802	54	5	the	the	DET
ejpam-6802	54	6	unit	unit	NOUN
ejpam-6802	54	7	element	element	NOUN
ejpam-6802	54	8	.	.	PUNCT
ejpam-6802	55	1	(	(	PUNCT
ejpam-6802	55	2	ii	ii	NOUN
ejpam-6802	55	3	)	)	PUNCT
ejpam-6802	55	4	if	if	SCONJ
ejpam-6802	55	5	q	q	PROPN
ejpam-6802	55	6	=	=	SYM
ejpam-6802	55	7	ϑ	ϑ	NOUN
ejpam-6802	55	8	,	,	PUNCT
ejpam-6802	55	9	then	then	ADV
ejpam-6802	55	10	α(q	α(q	PROPN
ejpam-6802	55	11	,	,	PUNCT
ejpam-6802	55	12	ϑ	ϑ	X
ejpam-6802	55	13	)	)	PUNCT
ejpam-6802	55	14	=	=	SYM
ejpam-6802	55	15	1	1	NUM
ejpam-6802	55	16	,	,	PUNCT
ejpam-6802	55	17	for	for	ADP
ejpam-6802	55	18	all	all	DET
ejpam-6802	55	19	(	(	PUNCT
ejpam-6802	55	20	q	q	ADJ
ejpam-6802	55	21	,	,	PUNCT
ejpam-6802	55	22	ϑ	ϑ	NOUN
ejpam-6802	55	23	)	)	PUNCT
ejpam-6802	55	24	∈	∈	PROPN
ejpam-6802	55	25	b	b	PROPN
ejpam-6802	55	26	×	×	PROPN
ejpam-6802	55	27	f	f	PROPN
ejpam-6802	55	28	(	(	PUNCT
ejpam-6802	55	29	iii	iii	NOUN
ejpam-6802	55	30	)	)	PUNCT
ejpam-6802	55	31	α(q	α(q	PROPN
ejpam-6802	55	32	,	,	PUNCT
ejpam-6802	55	33	ϑ	ϑ	X
ejpam-6802	55	34	)	)	PUNCT
ejpam-6802	55	35	=	=	SYM
ejpam-6802	55	36	α(ϑ	α(ϑ	VERB
ejpam-6802	55	37	,	,	PUNCT
ejpam-6802	55	38	q	q	NOUN
ejpam-6802	55	39	)	)	PUNCT
ejpam-6802	55	40	,	,	PUNCT
ejpam-6802	55	41	for	for	ADP
ejpam-6802	55	42	all	all	DET
ejpam-6802	55	43	q	q	PROPN
ejpam-6802	55	44	,	,	PUNCT
ejpam-6802	55	45	ϑ	ϑ	PROPN
ejpam-6802	55	46	∈	∈	PROPN
ejpam-6802	55	47	b	b	PROPN
ejpam-6802	55	48	∩	∩	PROPN
ejpam-6802	55	49	f	f	PROPN
ejpam-6802	55	50	(	(	PUNCT
ejpam-6802	55	51	iv	iv	X
ejpam-6802	55	52	)	)	PUNCT
ejpam-6802	55	53	α(q	α(q	PROPN
ejpam-6802	55	54	,	,	PUNCT
ejpam-6802	55	55	ϑ	ϑ	NOUN
ejpam-6802	55	56	)	)	PUNCT
ejpam-6802	55	57	≤	≤	NOUN
ejpam-6802	55	58	α(q	α(q	PROPN
ejpam-6802	55	59	,	,	PUNCT
ejpam-6802	55	60	ω)α(β	ω)α(β	NOUN
ejpam-6802	55	61	,	,	PUNCT
ejpam-6802	55	62	ω)α(β	ω)α(β	NOUN
ejpam-6802	55	63	,	,	PUNCT
ejpam-6802	55	64	ϑ	ϑ	NOUN
ejpam-6802	55	65	)	)	PUNCT
ejpam-6802	55	66	,	,	PUNCT
ejpam-6802	55	67	for	for	ADP
ejpam-6802	55	68	all	all	DET
ejpam-6802	55	69	q	q	PROPN
ejpam-6802	55	70	,	,	PUNCT
ejpam-6802	55	71	β	β	X
ejpam-6802	55	72	∈	∈	PROPN
ejpam-6802	55	73	b	b	PROPN
ejpam-6802	55	74	and	and	CCONJ
ejpam-6802	55	75	ω	ω	NUM
ejpam-6802	55	76	,	,	PUNCT
ejpam-6802	55	77	ϑ	ϑ	X
ejpam-6802	55	78	∈	∈	PROPN
ejpam-6802	55	79	f	f	X
ejpam-6802	55	80	.	.	PUNCT
ejpam-6802	56	1	then	then	ADV
ejpam-6802	56	2	the	the	DET
ejpam-6802	56	3	triplet	triplet	NOUN
ejpam-6802	56	4	(	(	PUNCT
ejpam-6802	56	5	b	b	NOUN
ejpam-6802	56	6	,	,	PUNCT
ejpam-6802	56	7	f	f	PROPN
ejpam-6802	56	8	,	,	PUNCT
ejpam-6802	56	9	α	α	PROPN
ejpam-6802	56	10	)	)	PUNCT
ejpam-6802	56	11	is	be	AUX
ejpam-6802	56	12	called	call	VERB
ejpam-6802	56	13	a	a	DET
ejpam-6802	56	14	multiplicative	multiplicative	ADJ
ejpam-6802	56	15	cone	cone	NOUN
ejpam-6802	56	16	bipolar	bipolar	ADJ
ejpam-6802	56	17	metric	metric	ADJ
ejpam-6802	56	18	space(mcbms	space(mcbms	PROPN
ejpam-6802	56	19	)	)	PUNCT
ejpam-6802	56	20	.	.	PUNCT
ejpam-6802	57	1	example	example	NOUN
ejpam-6802	58	1	1	1	X
ejpam-6802	58	2	.	.	X
ejpam-6802	58	3	consider	consider	VERB
ejpam-6802	58	4	a	a	DET
ejpam-6802	58	5	=	=	NOUN
ejpam-6802	58	6	r2	r2	NOUN
ejpam-6802	58	7	,	,	PUNCT
ejpam-6802	58	8	z	z	NOUN
ejpam-6802	58	9	=	=	SYM
ejpam-6802	58	10	{	{	PUNCT
ejpam-6802	58	11	(	(	PUNCT
ejpam-6802	58	12	q	q	NOUN
ejpam-6802	58	13	,	,	PUNCT
ejpam-6802	58	14	ϑ	ϑ	NOUN
ejpam-6802	58	15	)	)	PUNCT
ejpam-6802	58	16	∈	∈	PROPN
ejpam-6802	58	17	a|q	a|q	PROPN
ejpam-6802	58	18	,	,	PUNCT
ejpam-6802	58	19	ϑ	ϑ	X
ejpam-6802	58	20	≥	≥	NOUN
ejpam-6802	58	21	0	0	NUM
ejpam-6802	58	22	}	}	PUNCT
ejpam-6802	58	23	⊂	⊂	ADJ
ejpam-6802	58	24	r2	r2	PROPN
ejpam-6802	58	25	,	,	PUNCT
ejpam-6802	58	26	b	b	X
ejpam-6802	59	1	=	=	SYM
ejpam-6802	60	1	[	[	X
ejpam-6802	60	2	0	0	NUM
ejpam-6802	60	3	,	,	PUNCT
ejpam-6802	60	4	1	1	NUM
ejpam-6802	60	5	]	]	PUNCT
ejpam-6802	60	6	,	,	PUNCT
ejpam-6802	60	7	f	f	PROPN
ejpam-6802	60	8	=	=	PUNCT
ejpam-6802	61	1	[	[	X
ejpam-6802	61	2	1	1	NUM
ejpam-6802	61	3	,	,	PUNCT
ejpam-6802	61	4	2	2	NUM
ejpam-6802	61	5	]	]	PUNCT
ejpam-6802	61	6	and	and	CCONJ
ejpam-6802	61	7	α	α	NOUN
ejpam-6802	61	8	:	:	PUNCT
ejpam-6802	61	9	b	b	X
ejpam-6802	61	10	×	×	NOUN
ejpam-6802	61	11	f	f	X
ejpam-6802	61	12	→	→	PUNCT
ejpam-6802	61	13	a	a	DET
ejpam-6802	61	14	such	such	ADJ
ejpam-6802	61	15	that	that	DET
ejpam-6802	61	16	α(q	α(q	PROPN
ejpam-6802	61	17	,	,	PUNCT
ejpam-6802	61	18	ϑ	ϑ	X
ejpam-6802	61	19	)	)	PUNCT
ejpam-6802	61	20	=	=	SYM
ejpam-6802	61	21	e(|q−ϑ|,β|q−ϑ|	e(|q−ϑ|,β|q−ϑ|	NOUN
ejpam-6802	61	22	)	)	PUNCT
ejpam-6802	61	23	,	,	PUNCT
ejpam-6802	61	24	where	where	SCONJ
ejpam-6802	61	25	β	β	X
ejpam-6802	61	26	≥	≥	X
ejpam-6802	61	27	0	0	NUM
ejpam-6802	61	28	is	be	AUX
ejpam-6802	61	29	a	a	DET
ejpam-6802	61	30	constant	constant	ADJ
ejpam-6802	61	31	.	.	PUNCT
ejpam-6802	62	1	then	then	ADV
ejpam-6802	62	2	(	(	PUNCT
ejpam-6802	62	3	b	b	X
ejpam-6802	62	4	,	,	PUNCT
ejpam-6802	62	5	f	f	PROPN
ejpam-6802	62	6	,	,	PUNCT
ejpam-6802	62	7	α	α	PROPN
ejpam-6802	62	8	)	)	PUNCT
ejpam-6802	62	9	is	be	AUX
ejpam-6802	62	10	a	a	DET
ejpam-6802	62	11	mcbms	mcbms	NOUN
ejpam-6802	62	12	.	.	PUNCT
ejpam-6802	63	1	definition	definition	NOUN
ejpam-6802	63	2	2.2	2.2	NUM
ejpam-6802	63	3	.	.	PUNCT
ejpam-6802	64	1	(	(	PUNCT
ejpam-6802	64	2	i	i	NOUN
ejpam-6802	64	3	)	)	PUNCT
ejpam-6802	64	4	let	let	VERB
ejpam-6802	64	5	us	we	PRON
ejpam-6802	64	6	consider	consider	VERB
ejpam-6802	64	7	a	a	DET
ejpam-6802	64	8	mcbms	mcbms	NOUN
ejpam-6802	64	9	(	(	PUNCT
ejpam-6802	64	10	b	b	NOUN
ejpam-6802	64	11	,	,	PUNCT
ejpam-6802	64	12	f	f	PROPN
ejpam-6802	64	13	,	,	PUNCT
ejpam-6802	64	14	α	α	PROPN
ejpam-6802	64	15	)	)	PUNCT
ejpam-6802	64	16	.	.	PUNCT
ejpam-6802	65	1	then	then	ADV
ejpam-6802	65	2	the	the	DET
ejpam-6802	65	3	sets	set	NOUN
ejpam-6802	65	4	points	point	NOUN
ejpam-6802	65	5	are	be	AUX
ejpam-6802	65	6	b	b	NOUN
ejpam-6802	65	7	,	,	PUNCT
ejpam-6802	65	8	f	f	PROPN
ejpam-6802	65	9	and	and	CCONJ
ejpam-6802	65	10	b∩f	b∩f	NOUN
ejpam-6802	65	11	are	be	AUX
ejpam-6802	65	12	named	name	VERB
ejpam-6802	65	13	as	as	ADP
ejpam-6802	65	14	left	leave	VERB
ejpam-6802	65	15	,	,	PUNCT
ejpam-6802	65	16	right	right	ADJ
ejpam-6802	65	17	and	and	CCONJ
ejpam-6802	65	18	central	central	ADJ
ejpam-6802	65	19	points	point	NOUN
ejpam-6802	65	20	,	,	PUNCT
ejpam-6802	65	21	respectively	respectively	ADV
ejpam-6802	65	22	,	,	PUNCT
ejpam-6802	65	23	and	and	CCONJ
ejpam-6802	65	24	any	any	DET
ejpam-6802	65	25	sequence	sequence	NOUN
ejpam-6802	65	26	,	,	PUNCT
ejpam-6802	65	27	that	that	PRON
ejpam-6802	65	28	is	be	AUX
ejpam-6802	65	29	consisted	consist	VERB
ejpam-6802	65	30	of	of	ADP
ejpam-6802	65	31	only	only	ADV
ejpam-6802	65	32	left	leave	VERB
ejpam-6802	65	33	(	(	PUNCT
ejpam-6802	65	34	or	or	CCONJ
ejpam-6802	65	35	right	right	ADJ
ejpam-6802	65	36	,	,	PUNCT
ejpam-6802	65	37	or	or	CCONJ
ejpam-6802	65	38	central	central	ADJ
ejpam-6802	65	39	)	)	PUNCT
ejpam-6802	65	40	points	point	NOUN
ejpam-6802	65	41	is	be	AUX
ejpam-6802	65	42	called	call	VERB
ejpam-6802	65	43	a	a	DET
ejpam-6802	65	44	left	left	NOUN
ejpam-6802	65	45	(	(	PUNCT
ejpam-6802	65	46	or	or	CCONJ
ejpam-6802	65	47	right	right	ADJ
ejpam-6802	65	48	,	,	PUNCT
ejpam-6802	65	49	or	or	CCONJ
ejpam-6802	65	50	central	central	ADJ
ejpam-6802	65	51	)	)	PUNCT
ejpam-6802	65	52	sequence	sequence	NOUN
ejpam-6802	65	53	on	on	ADP
ejpam-6802	65	54	(	(	PUNCT
ejpam-6802	65	55	b	b	NOUN
ejpam-6802	65	56	,	,	PUNCT
ejpam-6802	65	57	f	f	PROPN
ejpam-6802	65	58	,	,	PUNCT
ejpam-6802	65	59	α	α	PROPN
ejpam-6802	65	60	)	)	PUNCT
ejpam-6802	65	61	.	.	PUNCT
ejpam-6802	66	1	(	(	PUNCT
ejpam-6802	66	2	ii	ii	NOUN
ejpam-6802	66	3	)	)	PUNCT
ejpam-6802	66	4	let	let	VERB
ejpam-6802	66	5	(	(	PUNCT
ejpam-6802	66	6	b1,f1	b1,f1	PROPN
ejpam-6802	66	7	,	,	PUNCT
ejpam-6802	66	8	α1	α1	PROPN
ejpam-6802	66	9	)	)	PUNCT
ejpam-6802	66	10	&	&	CCONJ
ejpam-6802	66	11	(	(	PUNCT
ejpam-6802	66	12	b2,f2	b2,f2	PROPN
ejpam-6802	66	13	,	,	PUNCT
ejpam-6802	66	14	α2	α2	PROPN
ejpam-6802	66	15	)	)	PUNCT
ejpam-6802	66	16	be	be	AUX
ejpam-6802	66	17	mcbms	mcbms	PROPN
ejpam-6802	66	18	&	&	CCONJ
ejpam-6802	66	19	ω	ω	PROPN
ejpam-6802	66	20	:	:	PUNCT
ejpam-6802	66	21	b1	b1	NOUN
ejpam-6802	66	22	∪	∪	NOUN
ejpam-6802	66	23	f1	f1	PROPN
ejpam-6802	66	24	→	→	SYM
ejpam-6802	66	25	b2	b2	NOUN
ejpam-6802	66	26	∪	∪	NOUN
ejpam-6802	66	27	f2	f2	ADV
ejpam-6802	66	28	be	be	VERB
ejpam-6802	66	29	a	a	DET
ejpam-6802	66	30	function	function	NOUN
ejpam-6802	66	31	.	.	PUNCT
ejpam-6802	67	1	if	if	SCONJ
ejpam-6802	67	2	ω(b1	ω(b1	NOUN
ejpam-6802	67	3	)	)	PUNCT
ejpam-6802	67	4	⊆	⊆	NUM
ejpam-6802	67	5	b2	b2	NOUN
ejpam-6802	67	6	and	and	CCONJ
ejpam-6802	67	7	ω(f1	ω(f1	ADJ
ejpam-6802	67	8	)	)	PUNCT
ejpam-6802	68	1	⊆	⊆	NUM
ejpam-6802	68	2	f2	f2	PROPN
ejpam-6802	68	3	,	,	PUNCT
ejpam-6802	68	4	then	then	ADV
ejpam-6802	68	5	ω	ω	PROPN
ejpam-6802	68	6	is	be	AUX
ejpam-6802	68	7	called	call	VERB
ejpam-6802	68	8	a	a	DET
ejpam-6802	68	9	covariant	covariant	ADJ
ejpam-6802	68	10	map	map	NOUN
ejpam-6802	68	11	,	,	PUNCT
ejpam-6802	68	12	or	or	CCONJ
ejpam-6802	68	13	a	a	DET
ejpam-6802	68	14	map	map	NOUN
ejpam-6802	68	15	from	from	ADP
ejpam-6802	68	16	(	(	PUNCT
ejpam-6802	68	17	b1,f1	b1,f1	PROPN
ejpam-6802	68	18	,	,	PUNCT
ejpam-6802	68	19	α1	α1	PROPN
ejpam-6802	68	20	)	)	PUNCT
ejpam-6802	68	21	to	to	ADP
ejpam-6802	68	22	(	(	PUNCT
ejpam-6802	68	23	b2,f2	b2,f2	PROPN
ejpam-6802	68	24	,	,	PUNCT
ejpam-6802	68	25	α2	α2	PROPN
ejpam-6802	68	26	)	)	PUNCT
ejpam-6802	68	27	and	and	CCONJ
ejpam-6802	68	28	this	this	PRON
ejpam-6802	68	29	is	be	AUX
ejpam-6802	68	30	written	write	VERB
ejpam-6802	68	31	as	as	ADP
ejpam-6802	68	32	ω	ω	PROPN
ejpam-6802	68	33	:	:	PUNCT
ejpam-6802	68	34	(	(	PUNCT
ejpam-6802	68	35	b1,f1	b1,f1	PROPN
ejpam-6802	68	36	,	,	PUNCT
ejpam-6802	68	37	α1	α1	PROPN
ejpam-6802	68	38	)	)	PUNCT
ejpam-6802	68	39	⇒	⇒	NOUN
ejpam-6802	68	40	(	(	PUNCT
ejpam-6802	68	41	b2,f2	b2,f2	PROPN
ejpam-6802	68	42	,	,	PUNCT
ejpam-6802	68	43	α2	α2	PROPN
ejpam-6802	68	44	)	)	PUNCT
ejpam-6802	68	45	.	.	PUNCT
ejpam-6802	69	1	if	if	SCONJ
ejpam-6802	69	2	ω	ω	PROPN
ejpam-6802	69	3	:	:	PUNCT
ejpam-6802	69	4	(	(	PUNCT
ejpam-6802	69	5	b1,f1	b1,f1	PROPN
ejpam-6802	69	6	,	,	PUNCT
ejpam-6802	69	7	α1	α1	PROPN
ejpam-6802	69	8	)	)	PUNCT
ejpam-6802	69	9	⇒	⇒	NOUN
ejpam-6802	69	10	(	(	PUNCT
ejpam-6802	69	11	f2,b2	f2,b2	PROPN
ejpam-6802	69	12	,	,	PUNCT
ejpam-6802	69	13	α2	α2	PROPN
ejpam-6802	69	14	)	)	PUNCT
ejpam-6802	69	15	is	be	AUX
ejpam-6802	69	16	a	a	DET
ejpam-6802	69	17	map	map	NOUN
ejpam-6802	69	18	,	,	PUNCT
ejpam-6802	69	19	then	then	ADV
ejpam-6802	69	20	ω	ω	PROPN
ejpam-6802	69	21	is	be	AUX
ejpam-6802	69	22	called	call	VERB
ejpam-6802	69	23	a	a	DET
ejpam-6802	69	24	contravariant	contravariant	ADJ
ejpam-6802	69	25	map	map	NOUN
ejpam-6802	69	26	from	from	ADP
ejpam-6802	69	27	(	(	PUNCT
ejpam-6802	69	28	b1,f1	b1,f1	PROPN
ejpam-6802	69	29	,	,	PUNCT
ejpam-6802	69	30	α1	α1	PROPN
ejpam-6802	69	31	)	)	PUNCT
ejpam-6802	69	32	to	to	ADP
ejpam-6802	69	33	(	(	PUNCT
ejpam-6802	69	34	b2,f2	b2,f2	PROPN
ejpam-6802	69	35	,	,	PUNCT
ejpam-6802	69	36	α2	α2	PROPN
ejpam-6802	69	37	)	)	PUNCT
ejpam-6802	69	38	and	and	CCONJ
ejpam-6802	69	39	this	this	PRON
ejpam-6802	69	40	is	be	AUX
ejpam-6802	69	41	denoted	denote	VERB
ejpam-6802	69	42	as	as	ADP
ejpam-6802	69	43	ω	ω	NUM
ejpam-6802	69	44	:	:	PUNCT
ejpam-6802	69	45	(	(	PUNCT
ejpam-6802	69	46	b1,f1	b1,f1	PROPN
ejpam-6802	69	47	,	,	PUNCT
ejpam-6802	69	48	α1	α1	PROPN
ejpam-6802	69	49	)	)	PUNCT
ejpam-6802	69	50	⇄	⇄	PROPN
ejpam-6802	69	51	(	(	PUNCT
ejpam-6802	69	52	b2,f2	b2,f2	PROPN
ejpam-6802	69	53	,	,	PUNCT
ejpam-6802	69	54	α2	α2	PROPN
ejpam-6802	69	55	)	)	PUNCT
ejpam-6802	69	56	.	.	PUNCT
ejpam-6802	70	1	definition	definition	NOUN
ejpam-6802	70	2	2.3	2.3	NUM
ejpam-6802	70	3	.	.	PUNCT
ejpam-6802	71	1	let	let	VERB
ejpam-6802	71	2	(	(	PUNCT
ejpam-6802	71	3	b	b	X
ejpam-6802	71	4	,	,	PUNCT
ejpam-6802	71	5	f	f	PROPN
ejpam-6802	71	6	,	,	PUNCT
ejpam-6802	71	7	α	α	PROPN
ejpam-6802	71	8	)	)	PUNCT
ejpam-6802	71	9	be	be	VERB
ejpam-6802	71	10	a	a	DET
ejpam-6802	71	11	mcbms	mcbms	NOUN
ejpam-6802	71	12	.	.	PUNCT
ejpam-6802	72	1	a	a	DET
ejpam-6802	72	2	left	left	ADJ
ejpam-6802	72	3	sequence	sequence	NOUN
ejpam-6802	72	4	{	{	PUNCT
ejpam-6802	72	5	qℓ	qℓ	PROPN
ejpam-6802	72	6	}	}	PUNCT
ejpam-6802	72	7	converges	converge	VERB
ejpam-6802	72	8	to	to	ADP
ejpam-6802	72	9	a	a	DET
ejpam-6802	72	10	right	right	ADJ
ejpam-6802	72	11	point	point	NOUN
ejpam-6802	72	12	ϑ	ϑ	X
ejpam-6802	72	13	iff	iff	NOUN
ejpam-6802	72	14	for	for	ADP
ejpam-6802	72	15	every	every	DET
ejpam-6802	72	16	z	z	PROPN
ejpam-6802	72	17	∈	∈	PROPN
ejpam-6802	72	18	a	a	PRON
ejpam-6802	72	19	with	with	ADP
ejpam-6802	72	20	0	0	NUM
ejpam-6802	72	21	≪	≪	PUNCT
ejpam-6802	72	22	z	z	PROPN
ejpam-6802	72	23	∃	∃	PROPN
ejpam-6802	72	24	an	an	DET
ejpam-6802	72	25	ℓ0	ℓ0	PROPN
ejpam-6802	72	26	∈	∈	PROPN
ejpam-6802	72	27	n	n	PROPN
ejpam-6802	72	28	s.t	s.t	PROPN
ejpam-6802	72	29	α(qℓ	α(qℓ	PROPN
ejpam-6802	72	30	,	,	PUNCT
ejpam-6802	72	31	ϑ	ϑ	NOUN
ejpam-6802	72	32	)	)	PUNCT
ejpam-6802	72	33	≪	≪	PUNCT
ejpam-6802	72	34	z	z	NOUN
ejpam-6802	72	35	for	for	ADP
ejpam-6802	72	36	all	all	DET
ejpam-6802	72	37	ℓ	ℓ	PROPN
ejpam-6802	72	38	≥	≥	NUM
ejpam-6802	72	39	ℓ0	ℓ0	ADV
ejpam-6802	72	40	.	.	PUNCT
ejpam-6802	73	1	similarly	similarly	ADV
ejpam-6802	73	2	,	,	PUNCT
ejpam-6802	73	3	a	a	DET
ejpam-6802	73	4	right	right	ADJ
ejpam-6802	73	5	seqence	seqence	NOUN
ejpam-6802	73	6	{	{	PUNCT
ejpam-6802	73	7	ϑℓ	ϑℓ	NOUN
ejpam-6802	73	8	}	}	PUNCT
ejpam-6802	73	9	→	→	SYM
ejpam-6802	73	10	q	q	X
ejpam-6802	73	11	iff	iff	NOUN
ejpam-6802	73	12	,	,	PUNCT
ejpam-6802	73	13	for	for	ADP
ejpam-6802	73	14	every	every	DET
ejpam-6802	73	15	z	z	NOUN
ejpam-6802	73	16	∈	∈	PROPN
ejpam-6802	73	17	a	a	PRON
ejpam-6802	73	18	with	with	ADP
ejpam-6802	73	19	0	0	NUM
ejpam-6802	73	20	≪	≪	PUNCT
ejpam-6802	73	21	z	z	PROPN
ejpam-6802	73	22	∃	∃	PROPN
ejpam-6802	73	23	an	an	DET
ejpam-6802	73	24	ℓ0	ℓ0	PROPN
ejpam-6802	73	25	∈	∈	PROPN
ejpam-6802	73	26	n	n	PRON
ejpam-6802	73	27	s.t	s.t	PROPN
ejpam-6802	73	28	,	,	PUNCT
ejpam-6802	73	29	whenever	whenever	SCONJ
ejpam-6802	73	30	ℓ	ℓ	PROPN
ejpam-6802	73	31	≥	≥	X
ejpam-6802	73	32	ℓ0	ℓ0	ADV
ejpam-6802	73	33	,	,	PUNCT
ejpam-6802	73	34	α(q	α(q	PROPN
ejpam-6802	73	35	,	,	PUNCT
ejpam-6802	73	36	ϑℓ	ϑℓ	PROPN
ejpam-6802	73	37	)	)	PUNCT
ejpam-6802	73	38	≪	≪	PUNCT
ejpam-6802	73	39	z.	z.	PROPN
ejpam-6802	73	40	definition	definition	NOUN
ejpam-6802	73	41	2.4	2.4	NUM
ejpam-6802	73	42	.	.	PUNCT
ejpam-6802	74	1	let	let	AUX
ejpam-6802	74	2	(	(	PUNCT
ejpam-6802	74	3	b1,f1	b1,f1	PROPN
ejpam-6802	74	4	,	,	PUNCT
ejpam-6802	74	5	α1	α1	PROPN
ejpam-6802	74	6	)	)	PUNCT
ejpam-6802	74	7	and	and	CCONJ
ejpam-6802	74	8	(	(	PUNCT
ejpam-6802	74	9	b2,f2	b2,f2	PROPN
ejpam-6802	74	10	,	,	PUNCT
ejpam-6802	74	11	α2	α2	PROPN
ejpam-6802	74	12	)	)	PUNCT
ejpam-6802	74	13	be	be	AUX
ejpam-6802	74	14	a	a	DET
ejpam-6802	74	15	mcbms	mcbms	NOUN
ejpam-6802	74	16	.	.	PUNCT
ejpam-6802	75	1	r.	r.	PROPN
ejpam-6802	75	2	ramaswamy	ramaswamy	PROPN
ejpam-6802	75	3	/	/	SYM
ejpam-6802	75	4	eur	eur	PROPN
ejpam-6802	75	5	.	.	PUNCT
ejpam-6802	76	1	j.	j.	PROPN
ejpam-6802	76	2	pure	pure	PROPN
ejpam-6802	76	3	appl	appl	PROPN
ejpam-6802	76	4	.	.	PROPN
ejpam-6802	76	5	math	math	PROPN
ejpam-6802	76	6	,	,	PUNCT
ejpam-6802	76	7	18	18	NUM
ejpam-6802	76	8	(	(	PUNCT
ejpam-6802	76	9	4	4	NUM
ejpam-6802	76	10	)	)	PUNCT
ejpam-6802	76	11	(	(	PUNCT
ejpam-6802	76	12	2025	2025	NUM
ejpam-6802	76	13	)	)	PUNCT
ejpam-6802	76	14	,	,	PUNCT
ejpam-6802	76	15	6802	6802	NUM
ejpam-6802	76	16	4	4	NUM
ejpam-6802	76	17	of	of	ADP
ejpam-6802	76	18	20	20	NUM
ejpam-6802	76	19	(	(	PUNCT
ejpam-6802	76	20	i	i	NOUN
ejpam-6802	76	21	)	)	PUNCT
ejpam-6802	76	22	a	a	DET
ejpam-6802	76	23	map	map	NOUN
ejpam-6802	76	24	ω	ω	NOUN
ejpam-6802	76	25	:	:	PUNCT
ejpam-6802	76	26	(	(	PUNCT
ejpam-6802	76	27	b1,f1	b1,f1	PROPN
ejpam-6802	76	28	,	,	PUNCT
ejpam-6802	76	29	α1	α1	PROPN
ejpam-6802	76	30	)	)	PUNCT
ejpam-6802	76	31	⇒	⇒	NOUN
ejpam-6802	76	32	(	(	PUNCT
ejpam-6802	76	33	b2,f2	b2,f2	PROPN
ejpam-6802	76	34	,	,	PUNCT
ejpam-6802	76	35	α2	α2	PROPN
ejpam-6802	76	36	)	)	PUNCT
ejpam-6802	76	37	is	be	AUX
ejpam-6802	76	38	known	know	VERB
ejpam-6802	76	39	to	to	PART
ejpam-6802	76	40	be	be	AUX
ejpam-6802	76	41	continuous	continuous	ADJ
ejpam-6802	76	42	at	at	ADP
ejpam-6802	76	43	a	a	DET
ejpam-6802	76	44	point	point	NOUN
ejpam-6802	77	1	q0	q0	PROPN
ejpam-6802	77	2	∈	∈	PROPN
ejpam-6802	77	3	b	b	NOUN
ejpam-6802	77	4	,	,	PUNCT
ejpam-6802	77	5	if	if	SCONJ
ejpam-6802	77	6	for	for	ADP
ejpam-6802	77	7	every	every	DET
ejpam-6802	77	8	z	z	PROPN
ejpam-6802	77	9	,	,	PUNCT
ejpam-6802	77	10	δ	δ	PROPN
ejpam-6802	77	11	∈	∈	PROPN
ejpam-6802	77	12	a	a	DET
ejpam-6802	77	13	with	with	ADP
ejpam-6802	77	14	0	0	NUM
ejpam-6802	77	15	≪	≪	PROPN
ejpam-6802	77	16	z	z	PROPN
ejpam-6802	77	17	,	,	PUNCT
ejpam-6802	77	18	∃	∃	PROPN
ejpam-6802	77	19	a	a	PRON
ejpam-6802	77	20	0	0	NUM
ejpam-6802	77	21	≪	≪	PUNCT
ejpam-6802	77	22	δ	δ	PROPN
ejpam-6802	77	23	s.t	s.t	PROPN
ejpam-6802	77	24	whenever	whenever	SCONJ
ejpam-6802	77	25	ϑ	ϑ	X
ejpam-6802	77	26	∈	∈	PROPN
ejpam-6802	77	27	f1	f1	NOUN
ejpam-6802	77	28	&	&	CCONJ
ejpam-6802	77	29	α1(q0	α1(q0	NUM
ejpam-6802	77	30	,	,	PUNCT
ejpam-6802	77	31	ϑ	ϑ	NOUN
ejpam-6802	77	32	)	)	PUNCT
ejpam-6802	77	33	≪	≪	PUNCT
ejpam-6802	77	34	δ	δ	PROPN
ejpam-6802	77	35	,	,	PUNCT
ejpam-6802	77	36	α2(ω(q0	α2(ω(q0	NUM
ejpam-6802	77	37	)	)	PUNCT
ejpam-6802	77	38	,	,	PUNCT
ejpam-6802	77	39	ω(ϑ	ω(ϑ	NUM
ejpam-6802	77	40	)	)	PUNCT
ejpam-6802	77	41	)	)	PUNCT
ejpam-6802	78	1	≪	≪	PUNCT
ejpam-6802	78	2	z.	z.	PROPN
ejpam-6802	79	1	it	it	PRON
ejpam-6802	79	2	is	be	AUX
ejpam-6802	79	3	continuous	continuous	ADJ
ejpam-6802	79	4	at	at	ADP
ejpam-6802	79	5	a	a	DET
ejpam-6802	79	6	point	point	NOUN
ejpam-6802	79	7	ϑ0	ϑ0	NOUN
ejpam-6802	79	8	∈	∈	PROPN
ejpam-6802	79	9	f1	f1	NOUN
ejpam-6802	79	10	if	if	SCONJ
ejpam-6802	79	11	for	for	ADP
ejpam-6802	79	12	every	every	DET
ejpam-6802	79	13	z	z	PROPN
ejpam-6802	79	14	,	,	PUNCT
ejpam-6802	79	15	δ	δ	PROPN
ejpam-6802	79	16	∈	∈	PROPN
ejpam-6802	79	17	a	a	DET
ejpam-6802	79	18	with	with	ADP
ejpam-6802	79	19	0	0	NUM
ejpam-6802	79	20	≪	≪	PROPN
ejpam-6802	79	21	z	z	PROPN
ejpam-6802	79	22	,	,	PUNCT
ejpam-6802	79	23	∃	∃	PROPN
ejpam-6802	79	24	a	a	PRON
ejpam-6802	79	25	0	0	NUM
ejpam-6802	79	26	≪	≪	PUNCT
ejpam-6802	79	27	δ	δ	PROPN
ejpam-6802	79	28	s.t	s.t	VERB
ejpam-6802	79	29	whenever	whenever	SCONJ
ejpam-6802	79	30	q	q	PROPN
ejpam-6802	79	31	∈	∈	PROPN
ejpam-6802	79	32	b1	b1	NOUN
ejpam-6802	79	33	and	and	CCONJ
ejpam-6802	79	34	α1(q	α1(q	NOUN
ejpam-6802	79	35	,	,	PUNCT
ejpam-6802	79	36	ϑ0	ϑ0	NOUN
ejpam-6802	79	37	)	)	PUNCT
ejpam-6802	79	38	≪	≪	PUNCT
ejpam-6802	79	39	δ	δ	PROPN
ejpam-6802	79	40	,	,	PUNCT
ejpam-6802	79	41	α2(ω(q),ω(ϑ0	α2(ω(q),ω(ϑ0	PROPN
ejpam-6802	79	42	)	)	PUNCT
ejpam-6802	79	43	)	)	PUNCT
ejpam-6802	79	44	≪	≪	PUNCT
ejpam-6802	79	45	z.	z.	PROPN
ejpam-6802	79	46	in	in	ADP
ejpam-6802	79	47	the	the	DET
ejpam-6802	79	48	case	case	NOUN
ejpam-6802	79	49	where	where	SCONJ
ejpam-6802	79	50	f	f	PROPN
ejpam-6802	79	51	is	be	AUX
ejpam-6802	79	52	continuous	continuous	ADJ
ejpam-6802	79	53	at	at	ADP
ejpam-6802	79	54	any	any	DET
ejpam-6802	79	55	point	point	NOUN
ejpam-6802	79	56	q	q	X
ejpam-6802	79	57	∈	∈	PROPN
ejpam-6802	79	58	b1	b1	NOUN
ejpam-6802	79	59	&	&	CCONJ
ejpam-6802	79	60	ϑ	ϑ	X
ejpam-6802	79	61	∈	∈	PROPN
ejpam-6802	79	62	f1	f1	NOUN
ejpam-6802	79	63	,	,	PUNCT
ejpam-6802	79	64	then	then	ADV
ejpam-6802	79	65	it	it	PRON
ejpam-6802	79	66	is	be	AUX
ejpam-6802	79	67	called	call	VERB
ejpam-6802	79	68	continuous	continuous	ADJ
ejpam-6802	79	69	.	.	PUNCT
ejpam-6802	80	1	(	(	PUNCT
ejpam-6802	80	2	ii	ii	NOUN
ejpam-6802	80	3	)	)	PUNCT
ejpam-6802	80	4	a	a	DET
ejpam-6802	80	5	contravariant	contravariant	ADJ
ejpam-6802	80	6	function	function	NOUN
ejpam-6802	80	7	ω	ω	NOUN
ejpam-6802	80	8	:	:	PUNCT
ejpam-6802	80	9	(	(	PUNCT
ejpam-6802	80	10	b1,f1	b1,f1	PROPN
ejpam-6802	80	11	,	,	PUNCT
ejpam-6802	80	12	α1	α1	PROPN
ejpam-6802	80	13	)	)	PUNCT
ejpam-6802	80	14	⇆	⇆	PROPN
ejpam-6802	80	15	(	(	PUNCT
ejpam-6802	80	16	b2,f2	b2,f2	PROPN
ejpam-6802	80	17	,	,	PUNCT
ejpam-6802	80	18	α2	α2	PROPN
ejpam-6802	80	19	)	)	PUNCT
ejpam-6802	80	20	is	be	AUX
ejpam-6802	80	21	continuous	continuous	ADJ
ejpam-6802	80	22	iff	iff	PROPN
ejpam-6802	80	23	it	it	PRON
ejpam-6802	80	24	is	be	AUX
ejpam-6802	80	25	continuous	continuous	ADJ
ejpam-6802	80	26	as	as	ADP
ejpam-6802	80	27	a	a	DET
ejpam-6802	80	28	covariant	covariant	ADJ
ejpam-6802	80	29	function	function	NOUN
ejpam-6802	80	30	ω	ω	NOUN
ejpam-6802	80	31	:	:	PUNCT
ejpam-6802	80	32	(	(	PUNCT
ejpam-6802	80	33	b1,f1	b1,f1	PROPN
ejpam-6802	80	34	,	,	PUNCT
ejpam-6802	80	35	α1	α1	PROPN
ejpam-6802	80	36	)	)	PUNCT
ejpam-6802	80	37	⇒	⇒	NOUN
ejpam-6802	80	38	(	(	PUNCT
ejpam-6802	80	39	b2,f2	b2,f2	PROPN
ejpam-6802	80	40	,	,	PUNCT
ejpam-6802	80	41	ᾱ2	ᾱ2	ADV
ejpam-6802	80	42	)	)	PUNCT
ejpam-6802	80	43	.	.	PUNCT
ejpam-6802	81	1	this	this	DET
ejpam-6802	81	2	definition	definition	NOUN
ejpam-6802	81	3	assumes	assume	VERB
ejpam-6802	81	4	that	that	SCONJ
ejpam-6802	81	5	a	a	DET
ejpam-6802	81	6	covariant	covariant	NOUN
ejpam-6802	81	7	or	or	CCONJ
ejpam-6802	81	8	a	a	DET
ejpam-6802	81	9	contravariant	contravariant	ADJ
ejpam-6802	81	10	map	map	NOUN
ejpam-6802	81	11	ω	ω	PROPN
ejpam-6802	81	12	from	from	ADP
ejpam-6802	81	13	(	(	PUNCT
ejpam-6802	81	14	b1,f1	b1,f1	PROPN
ejpam-6802	81	15	,	,	PUNCT
ejpam-6802	81	16	α1	α1	PROPN
ejpam-6802	81	17	)	)	PUNCT
ejpam-6802	81	18	to	to	ADP
ejpam-6802	81	19	(	(	PUNCT
ejpam-6802	81	20	b2,f2	b2,f2	PROPN
ejpam-6802	81	21	,	,	PUNCT
ejpam-6802	81	22	α2	α2	PROPN
ejpam-6802	81	23	)	)	PUNCT
ejpam-6802	81	24	is	be	AUX
ejpam-6802	81	25	continuous	continuous	ADJ
ejpam-6802	81	26	,	,	PUNCT
ejpam-6802	81	27	if	if	SCONJ
ejpam-6802	81	28	and	and	CCONJ
ejpam-6802	81	29	only	only	ADV
ejpam-6802	81	30	if	if	SCONJ
ejpam-6802	81	31	{	{	PUNCT
ejpam-6802	81	32	xℓ	xℓ	NOUN
ejpam-6802	81	33	}	}	PUNCT
ejpam-6802	81	34	→	→	SYM
ejpam-6802	81	35	z	z	NOUN
ejpam-6802	81	36	on	on	ADP
ejpam-6802	81	37	(	(	PUNCT
ejpam-6802	81	38	b1,f1	b1,f1	PROPN
ejpam-6802	81	39	,	,	PUNCT
ejpam-6802	81	40	α1	α1	PROPN
ejpam-6802	81	41	)	)	PUNCT
ejpam-6802	81	42	implies	imply	VERB
ejpam-6802	81	43	{	{	PUNCT
ejpam-6802	81	44	ω(xℓ	ω(xℓ	NOUN
ejpam-6802	81	45	)	)	PUNCT
ejpam-6802	81	46	}	}	PUNCT
ejpam-6802	81	47	→	→	SYM
ejpam-6802	81	48	ω(z	ω(z	PROPN
ejpam-6802	81	49	)	)	PUNCT
ejpam-6802	81	50	on	on	ADP
ejpam-6802	81	51	(	(	PUNCT
ejpam-6802	81	52	b2,f2	b2,f2	PROPN
ejpam-6802	81	53	,	,	PUNCT
ejpam-6802	81	54	α2	α2	PROPN
ejpam-6802	81	55	)	)	PUNCT
ejpam-6802	81	56	.	.	PUNCT
ejpam-6802	82	1	definition	definition	NOUN
ejpam-6802	82	2	2.5	2.5	NUM
ejpam-6802	82	3	.	.	PUNCT
ejpam-6802	83	1	let	let	VERB
ejpam-6802	83	2	(	(	PUNCT
ejpam-6802	83	3	b1,f1	b1,f1	PROPN
ejpam-6802	83	4	,	,	PUNCT
ejpam-6802	83	5	α1	α1	PROPN
ejpam-6802	83	6	)	)	PUNCT
ejpam-6802	83	7	and	and	CCONJ
ejpam-6802	83	8	(	(	PUNCT
ejpam-6802	83	9	b2,f2	b2,f2	PROPN
ejpam-6802	83	10	,	,	PUNCT
ejpam-6802	83	11	α2	α2	PROPN
ejpam-6802	83	12	)	)	PUNCT
ejpam-6802	83	13	be	be	AUX
ejpam-6802	83	14	mcbms	mcbms	NOUN
ejpam-6802	83	15	.	.	PUNCT
ejpam-6802	84	1	a	a	DET
ejpam-6802	84	2	covariant	covariant	ADJ
ejpam-6802	84	3	map	map	NOUN
ejpam-6802	84	4	ω	ω	NOUN
ejpam-6802	84	5	:	:	PUNCT
ejpam-6802	84	6	(	(	PUNCT
ejpam-6802	84	7	b1,f1	b1,f1	PROPN
ejpam-6802	84	8	,	,	PUNCT
ejpam-6802	84	9	α1)⇒(b2,f2	α1)⇒(b2,f2	ADV
ejpam-6802	84	10	,	,	PUNCT
ejpam-6802	84	11	α2	α2	ADJ
ejpam-6802	84	12	)	)	PUNCT
ejpam-6802	84	13	such	such	ADJ
ejpam-6802	84	14	that	that	SCONJ
ejpam-6802	84	15	α(ω(q),ω(ϑ	α(ω(q),ω(ϑ	PROPN
ejpam-6802	84	16	)	)	PUNCT
ejpam-6802	84	17	)	)	PUNCT
ejpam-6802	84	18	≤	≤	NOUN
ejpam-6802	84	19	(	(	PUNCT
ejpam-6802	84	20	α(q	α(q	ADJ
ejpam-6802	84	21	,	,	PUNCT
ejpam-6802	84	22	ϑ))λ	ϑ))λ	VERB
ejpam-6802	84	23	for	for	ADP
ejpam-6802	84	24	all	all	DET
ejpam-6802	84	25	q	q	PROPN
ejpam-6802	84	26	∈	∈	PROPN
ejpam-6802	84	27	b1	b1	NOUN
ejpam-6802	84	28	,	,	PUNCT
ejpam-6802	84	29	ϑ	ϑ	X
ejpam-6802	84	30	∈	∈	PROPN
ejpam-6802	84	31	f1	f1	NOUN
ejpam-6802	84	32	or	or	CCONJ
ejpam-6802	84	33	a	a	DET
ejpam-6802	84	34	contravariant	contravariant	ADJ
ejpam-6802	84	35	map	map	NOUN
ejpam-6802	84	36	ω	ω	INTJ
ejpam-6802	84	37	:	:	PUNCT
ejpam-6802	84	38	(	(	PUNCT
ejpam-6802	84	39	b1,f1	b1,f1	PROPN
ejpam-6802	84	40	,	,	PUNCT
ejpam-6802	84	41	α1	α1	PROPN
ejpam-6802	84	42	)	)	PUNCT
ejpam-6802	85	1	⇆	⇆	PROPN
ejpam-6802	85	2	(	(	PUNCT
ejpam-6802	85	3	b2,f2	b2,f2	PROPN
ejpam-6802	85	4	,	,	PUNCT
ejpam-6802	85	5	α2	α2	PROPN
ejpam-6802	85	6	)	)	PUNCT
ejpam-6802	85	7	such	such	ADJ
ejpam-6802	85	8	that	that	SCONJ
ejpam-6802	85	9	α(ω(q),ω(ϑ	α(ω(q),ω(ϑ	PROPN
ejpam-6802	85	10	)	)	PUNCT
ejpam-6802	85	11	)	)	PUNCT
ejpam-6802	85	12	≤	≤	NOUN
ejpam-6802	85	13	(	(	PUNCT
ejpam-6802	85	14	α(q	α(q	ADJ
ejpam-6802	85	15	,	,	PUNCT
ejpam-6802	85	16	ϑ))λ	ϑ))λ	VERB
ejpam-6802	85	17	for	for	ADP
ejpam-6802	85	18	all	all	DET
ejpam-6802	85	19	q	q	PROPN
ejpam-6802	85	20	∈	∈	PROPN
ejpam-6802	85	21	b1	b1	NOUN
ejpam-6802	85	22	,	,	PUNCT
ejpam-6802	85	23	ϑ	ϑ	PROPN
ejpam-6802	85	24	∈	∈	PROPN
ejpam-6802	85	25	f1	f1	NOUN
ejpam-6802	85	26	is	be	AUX
ejpam-6802	85	27	known	know	VERB
ejpam-6802	85	28	as	as	ADP
ejpam-6802	85	29	lipschitz	lipschitz	NOUN
ejpam-6802	85	30	continuous	continuous	ADJ
ejpam-6802	85	31	.	.	PUNCT
ejpam-6802	86	1	if	if	SCONJ
ejpam-6802	86	2	λ=1	λ=1	NOUN
ejpam-6802	86	3	,	,	PUNCT
ejpam-6802	86	4	then	then	ADV
ejpam-6802	86	5	the	the	DET
ejpam-6802	86	6	covariant	covariant	ADJ
ejpam-6802	86	7	or	or	CCONJ
ejpam-6802	86	8	contravariant	contravariant	ADJ
ejpam-6802	86	9	function	function	NOUN
ejpam-6802	86	10	is	be	AUX
ejpam-6802	86	11	known	know	VERB
ejpam-6802	86	12	as	as	ADP
ejpam-6802	86	13	non	non	ADJ
ejpam-6802	86	14	-	-	ADJ
ejpam-6802	86	15	expansive	expansive	ADJ
ejpam-6802	86	16	,	,	PUNCT
ejpam-6802	86	17	&	&	CCONJ
ejpam-6802	86	18	it	it	PRON
ejpam-6802	86	19	is	be	AUX
ejpam-6802	86	20	termed	term	VERB
ejpam-6802	86	21	as	as	ADP
ejpam-6802	86	22	contraction	contraction	NOUN
ejpam-6802	86	23	if	if	SCONJ
ejpam-6802	86	24	it	it	PRON
ejpam-6802	86	25	is	be	AUX
ejpam-6802	86	26	confirmed	confirm	VERB
ejpam-6802	86	27	for	for	ADP
ejpam-6802	86	28	λ	λ	PROPN
ejpam-6802	86	29	∈	∈	PROPN
ejpam-6802	86	30	(	(	PUNCT
ejpam-6802	86	31	0	0	NUM
ejpam-6802	86	32	,	,	PUNCT
ejpam-6802	86	33	1	1	NUM
ejpam-6802	86	34	)	)	PUNCT
ejpam-6802	86	35	.	.	PUNCT
ejpam-6802	87	1	definition	definition	NOUN
ejpam-6802	87	2	2.6	2.6	NUM
ejpam-6802	87	3	.	.	PUNCT
ejpam-6802	88	1	consider	consider	VERB
ejpam-6802	88	2	(	(	PUNCT
ejpam-6802	88	3	b	b	NOUN
ejpam-6802	88	4	,	,	PUNCT
ejpam-6802	88	5	f	f	PROPN
ejpam-6802	88	6	,	,	PUNCT
ejpam-6802	88	7	α	α	PROPN
ejpam-6802	88	8	)	)	PUNCT
ejpam-6802	88	9	be	be	VERB
ejpam-6802	88	10	a	a	DET
ejpam-6802	88	11	mcbms	mcbms	NOUN
ejpam-6802	88	12	.	.	PUNCT
ejpam-6802	89	1	(	(	PUNCT
ejpam-6802	89	2	i	i	NOUN
ejpam-6802	89	3	)	)	PUNCT
ejpam-6802	89	4	a	a	DET
ejpam-6802	89	5	sequence	sequence	NOUN
ejpam-6802	89	6	(	(	PUNCT
ejpam-6802	89	7	{	{	PUNCT
ejpam-6802	89	8	qn	qn	NOUN
ejpam-6802	89	9	}	}	PUNCT
ejpam-6802	89	10	,	,	PUNCT
ejpam-6802	89	11	{	{	PUNCT
ejpam-6802	89	12	ϑn	ϑn	NOUN
ejpam-6802	89	13	}	}	PUNCT
ejpam-6802	89	14	)	)	PUNCT
ejpam-6802	89	15	on	on	ADP
ejpam-6802	89	16	the	the	DET
ejpam-6802	89	17	set	set	NOUN
ejpam-6802	89	18	b	b	PROPN
ejpam-6802	89	19	×	×	PROPN
ejpam-6802	89	20	f	f	PROPN
ejpam-6802	89	21	is	be	AUX
ejpam-6802	89	22	known	know	VERB
ejpam-6802	89	23	as	as	ADP
ejpam-6802	89	24	bisequence	bisequence	NOUN
ejpam-6802	89	25	on	on	ADP
ejpam-6802	89	26	(	(	PUNCT
ejpam-6802	89	27	b	b	NOUN
ejpam-6802	89	28	,	,	PUNCT
ejpam-6802	89	29	f	f	PROPN
ejpam-6802	89	30	,	,	PUNCT
ejpam-6802	89	31	α	α	PROPN
ejpam-6802	89	32	)	)	PUNCT
ejpam-6802	89	33	.	.	PUNCT
ejpam-6802	90	1	(	(	PUNCT
ejpam-6802	90	2	ii	ii	NOUN
ejpam-6802	90	3	)	)	PUNCT
ejpam-6802	90	4	if	if	SCONJ
ejpam-6802	90	5	both	both	PRON
ejpam-6802	90	6	{	{	PUNCT
ejpam-6802	90	7	qn	qn	NOUN
ejpam-6802	90	8	}	}	PUNCT
ejpam-6802	90	9	&	&	CCONJ
ejpam-6802	90	10	{	{	PUNCT
ejpam-6802	90	11	ϑn	ϑn	NOUN
ejpam-6802	90	12	}	}	PUNCT
ejpam-6802	90	13	converge	converge	NOUN
ejpam-6802	90	14	,	,	PUNCT
ejpam-6802	90	15	then	then	ADV
ejpam-6802	90	16	the	the	DET
ejpam-6802	90	17	bisequence	bisequence	NOUN
ejpam-6802	90	18	(	(	PUNCT
ejpam-6802	90	19	qn	qn	INTJ
ejpam-6802	90	20	,	,	PUNCT
ejpam-6802	90	21	ϑn	ϑn	NOUN
ejpam-6802	90	22	)	)	PUNCT
ejpam-6802	90	23	is	be	AUX
ejpam-6802	90	24	known	know	VERB
ejpam-6802	90	25	as	as	ADP
ejpam-6802	90	26	convergent	convergent	NOUN
ejpam-6802	90	27	.	.	PUNCT
ejpam-6802	91	1	if	if	SCONJ
ejpam-6802	91	2	{	{	PUNCT
ejpam-6802	91	3	qn	qn	NOUN
ejpam-6802	91	4	}	}	PUNCT
ejpam-6802	91	5	and	and	CCONJ
ejpam-6802	91	6	{	{	PUNCT
ejpam-6802	91	7	ϑn	ϑn	NOUN
ejpam-6802	91	8	}	}	PUNCT
ejpam-6802	91	9	both	both	PRON
ejpam-6802	91	10	converge	converge	VERB
ejpam-6802	91	11	to	to	ADP
ejpam-6802	91	12	a	a	DET
ejpam-6802	91	13	same	same	ADJ
ejpam-6802	91	14	point	point	NOUN
ejpam-6802	91	15	u	u	PROPN
ejpam-6802	91	16	∈	∈	PROPN
ejpam-6802	91	17	b	b	PROPN
ejpam-6802	91	18	∩	∩	PROPN
ejpam-6802	91	19	f	f	PROPN
ejpam-6802	91	20	,	,	PUNCT
ejpam-6802	91	21	then	then	ADV
ejpam-6802	91	22	this	this	DET
ejpam-6802	91	23	bisequence	bisequence	NOUN
ejpam-6802	91	24	is	be	AUX
ejpam-6802	91	25	known	know	VERB
ejpam-6802	91	26	as	as	ADP
ejpam-6802	91	27	biconvergent	biconvergent	NOUN
ejpam-6802	91	28	.	.	PUNCT
ejpam-6802	92	1	(	(	PUNCT
ejpam-6802	92	2	iii	iii	X
ejpam-6802	92	3	)	)	PUNCT
ejpam-6802	92	4	a	a	DET
ejpam-6802	92	5	bisequence	bisequence	NOUN
ejpam-6802	92	6	(	(	PUNCT
ejpam-6802	92	7	{	{	PUNCT
ejpam-6802	92	8	qn	qn	NOUN
ejpam-6802	92	9	}	}	PUNCT
ejpam-6802	92	10	,	,	PUNCT
ejpam-6802	92	11	{	{	PUNCT
ejpam-6802	92	12	ϑn	ϑn	NOUN
ejpam-6802	92	13	}	}	PUNCT
ejpam-6802	92	14	)	)	PUNCT
ejpam-6802	92	15	on	on	ADP
ejpam-6802	92	16	(	(	PUNCT
ejpam-6802	92	17	b	b	X
ejpam-6802	92	18	,	,	PUNCT
ejpam-6802	92	19	f	f	PROPN
ejpam-6802	92	20	,	,	PUNCT
ejpam-6802	92	21	α	α	PROPN
ejpam-6802	92	22	)	)	PUNCT
ejpam-6802	92	23	is	be	AUX
ejpam-6802	92	24	known	know	VERB
ejpam-6802	92	25	as	as	ADP
ejpam-6802	92	26	a	a	DET
ejpam-6802	92	27	cauchy	cauchy	ADJ
ejpam-6802	92	28	bisequence	bisequence	NOUN
ejpam-6802	92	29	,	,	PUNCT
ejpam-6802	92	30	if	if	SCONJ
ejpam-6802	92	31	for	for	ADP
ejpam-6802	92	32	each	each	PRON
ejpam-6802	92	33	ϵ	ϵ	X
ejpam-6802	92	34	>	>	X
ejpam-6802	92	35	0	0	PROPN
ejpam-6802	92	36	,	,	PUNCT
ejpam-6802	92	37	∃	∃	PROPN
ejpam-6802	92	38	a	a	DET
ejpam-6802	92	39	number	number	NOUN
ejpam-6802	92	40	ℓ0	ℓ0	PROPN
ejpam-6802	92	41	∈	∈	PROPN
ejpam-6802	92	42	n	n	AUX
ejpam-6802	92	43	,	,	PUNCT
ejpam-6802	92	44	s.t	s.t	PROPN
ejpam-6802	92	45	for	for	ADP
ejpam-6802	92	46	all	all	DET
ejpam-6802	92	47	positive	positive	ADJ
ejpam-6802	92	48	integers	integer	NOUN
ejpam-6802	92	49	ℓ	ℓ	NOUN
ejpam-6802	92	50	,	,	PUNCT
ejpam-6802	92	51	r	r	NOUN
ejpam-6802	92	52	≥	≥	NOUN
ejpam-6802	92	53	ℓ0	ℓ0	ADV
ejpam-6802	92	54	,	,	PUNCT
ejpam-6802	92	55	α(qℓ	α(qℓ	PRON
ejpam-6802	92	56	,	,	PUNCT
ejpam-6802	92	57	ϑr	ϑr	PROPN
ejpam-6802	92	58	)	)	PUNCT
ejpam-6802	92	59	<	<	X
ejpam-6802	92	60	ϵ.	ϵ.	NOUN
ejpam-6802	92	61	definition	definition	NOUN
ejpam-6802	92	62	2.7	2.7	NUM
ejpam-6802	92	63	.	.	PUNCT
ejpam-6802	93	1	a	a	DET
ejpam-6802	93	2	mcbms	mcbms	NOUN
ejpam-6802	93	3	is	be	AUX
ejpam-6802	93	4	complete	complete	ADJ
ejpam-6802	93	5	,	,	PUNCT
ejpam-6802	93	6	if	if	SCONJ
ejpam-6802	93	7	every	every	DET
ejpam-6802	93	8	cauchy	cauchy	ADJ
ejpam-6802	93	9	bisequence	bisequence	NOUN
ejpam-6802	93	10	is	be	AUX
ejpam-6802	93	11	convergent	convergent	ADJ
ejpam-6802	93	12	.	.	PUNCT
ejpam-6802	94	1	in	in	ADP
ejpam-6802	94	2	the	the	DET
ejpam-6802	94	3	next	next	ADJ
ejpam-6802	94	4	section	section	NOUN
ejpam-6802	94	5	fixed	fix	VERB
ejpam-6802	94	6	point	point	NOUN
ejpam-6802	94	7	results	result	NOUN
ejpam-6802	94	8	in	in	ADP
ejpam-6802	94	9	the	the	DET
ejpam-6802	94	10	setting	setting	NOUN
ejpam-6802	94	11	of	of	ADP
ejpam-6802	94	12	mcbms	mcbms	NOUN
ejpam-6802	94	13	are	be	AUX
ejpam-6802	94	14	presented	present	VERB
ejpam-6802	94	15	.	.	PUNCT
ejpam-6802	95	1	3	3	X
ejpam-6802	95	2	.	.	X
ejpam-6802	95	3	main	main	ADJ
ejpam-6802	95	4	results	result	NOUN
ejpam-6802	95	5	now	now	ADV
ejpam-6802	95	6	we	we	PRON
ejpam-6802	95	7	present	present	VERB
ejpam-6802	95	8	our	our	PRON
ejpam-6802	95	9	first	first	ADJ
ejpam-6802	95	10	result	result	NOUN
ejpam-6802	95	11	.	.	PUNCT
ejpam-6802	96	1	theorem	theorem	VERB
ejpam-6802	96	2	3.1	3.1	NUM
ejpam-6802	96	3	.	.	PUNCT
ejpam-6802	97	1	let	let	VERB
ejpam-6802	97	2	(	(	PUNCT
ejpam-6802	97	3	b	b	X
ejpam-6802	97	4	,	,	PUNCT
ejpam-6802	97	5	f	f	PROPN
ejpam-6802	97	6	,	,	PUNCT
ejpam-6802	97	7	α	α	PROPN
ejpam-6802	97	8	)	)	PUNCT
ejpam-6802	97	9	be	be	VERB
ejpam-6802	97	10	a	a	DET
ejpam-6802	97	11	complete	complete	ADJ
ejpam-6802	97	12	mcbms	mcbms	NOUN
ejpam-6802	97	13	,	,	PUNCT
ejpam-6802	97	14	z	z	PROPN
ejpam-6802	97	15	be	be	AUX
ejpam-6802	97	16	a	a	DET
ejpam-6802	97	17	cone	cone	NOUN
ejpam-6802	97	18	with	with	ADP
ejpam-6802	97	19	constant	constant	ADJ
ejpam-6802	97	20	w	w	PROPN
ejpam-6802	97	21	&	&	CCONJ
ejpam-6802	97	22	.	.	PUNCT
ejpam-6802	98	1	given	give	VERB
ejpam-6802	98	2	a	a	DET
ejpam-6802	98	3	contraction	contraction	NOUN
ejpam-6802	98	4	function	function	NOUN
ejpam-6802	98	5	ω	ω	NOUN
ejpam-6802	98	6	:	:	PUNCT
ejpam-6802	98	7	(	(	PUNCT
ejpam-6802	98	8	b	b	X
ejpam-6802	98	9	,	,	PUNCT
ejpam-6802	98	10	f	f	PROPN
ejpam-6802	98	11	,	,	PUNCT
ejpam-6802	98	12	α	α	X
ejpam-6802	98	13	)	)	PUNCT
ejpam-6802	98	14	⇒	⇒	NOUN
ejpam-6802	98	15	(	(	PUNCT
ejpam-6802	98	16	b	b	X
ejpam-6802	98	17	,	,	PUNCT
ejpam-6802	98	18	f	f	PROPN
ejpam-6802	98	19	,	,	PUNCT
ejpam-6802	98	20	α	α	PROPN
ejpam-6802	98	21	)	)	PUNCT
ejpam-6802	98	22	,	,	PUNCT
ejpam-6802	98	23	the	the	DET
ejpam-6802	98	24	mapping	mapping	NOUN
ejpam-6802	98	25	ω	ω	PROPN
ejpam-6802	98	26	:	:	PUNCT
ejpam-6802	98	27	b	b	X
ejpam-6802	98	28	∪	∪	X
ejpam-6802	98	29	f	f	PROPN
ejpam-6802	98	30	→	→	SYM
ejpam-6802	98	31	b	b	X
ejpam-6802	98	32	∪	∪	X
ejpam-6802	98	33	f	f	PROPN
ejpam-6802	98	34	possesses	possess	VERB
ejpam-6802	98	35	a	a	DET
ejpam-6802	98	36	ufp(unique	ufp(unique	ADJ
ejpam-6802	98	37	fixed	fix	VERB
ejpam-6802	98	38	point	point	NOUN
ejpam-6802	98	39	)	)	PUNCT
ejpam-6802	98	40	.	.	PUNCT
ejpam-6802	99	1	r.	r.	PROPN
ejpam-6802	99	2	ramaswamy	ramaswamy	PROPN
ejpam-6802	99	3	/	/	SYM
ejpam-6802	99	4	eur	eur	PROPN
ejpam-6802	99	5	.	.	PUNCT
ejpam-6802	100	1	j.	j.	PROPN
ejpam-6802	100	2	pure	pure	PROPN
ejpam-6802	100	3	appl	appl	PROPN
ejpam-6802	100	4	.	.	PROPN
ejpam-6802	100	5	math	math	PROPN
ejpam-6802	100	6	,	,	PUNCT
ejpam-6802	100	7	18	18	NUM
ejpam-6802	100	8	(	(	PUNCT
ejpam-6802	100	9	4	4	NUM
ejpam-6802	100	10	)	)	PUNCT
ejpam-6802	100	11	(	(	PUNCT
ejpam-6802	100	12	2025	2025	NUM
ejpam-6802	100	13	)	)	PUNCT
ejpam-6802	100	14	,	,	PUNCT
ejpam-6802	100	15	6802	6802	NUM
ejpam-6802	100	16	5	5	NUM
ejpam-6802	100	17	of	of	ADP
ejpam-6802	100	18	20	20	NUM
ejpam-6802	100	19	proof	proof	NOUN
ejpam-6802	100	20	.	.	PUNCT
ejpam-6802	101	1	let	let	VERB
ejpam-6802	101	2	q0∈	q0∈	PROPN
ejpam-6802	101	3	b	b	PROPN
ejpam-6802	101	4	&	&	CCONJ
ejpam-6802	101	5	ϑ0∈f	ϑ0∈f	NOUN
ejpam-6802	101	6	.	.	PUNCT
ejpam-6802	102	1	for	for	ADP
ejpam-6802	102	2	all	all	DET
ejpam-6802	102	3	ℓ	ℓ	PROPN
ejpam-6802	102	4	∈	∈	PROPN
ejpam-6802	102	5	n	n	CCONJ
ejpam-6802	102	6	,	,	PUNCT
ejpam-6802	102	7	define	define	VERB
ejpam-6802	102	8	ω(qℓ	ω(qℓ	NUM
ejpam-6802	102	9	)	)	PUNCT
ejpam-6802	103	1	=	=	SYM
ejpam-6802	103	2	qℓ+1	qℓ+1	X
ejpam-6802	103	3	.	.	PUNCT
ejpam-6802	104	1	then	then	ADV
ejpam-6802	104	2	(	(	PUNCT
ejpam-6802	104	3	{	{	PUNCT
ejpam-6802	104	4	qℓ	qℓ	NOUN
ejpam-6802	104	5	}	}	PUNCT
ejpam-6802	104	6	,	,	PUNCT
ejpam-6802	104	7	{	{	PUNCT
ejpam-6802	104	8	ϑℓ	ϑℓ	NOUN
ejpam-6802	104	9	}	}	PUNCT
ejpam-6802	104	10	)	)	PUNCT
ejpam-6802	104	11	be	be	AUX
ejpam-6802	104	12	a	a	DET
ejpam-6802	104	13	bisequence	bisequence	NOUN
ejpam-6802	104	14	on	on	ADP
ejpam-6802	104	15	(	(	PUNCT
ejpam-6802	104	16	b	b	NOUN
ejpam-6802	104	17	,	,	PUNCT
ejpam-6802	104	18	f	f	PROPN
ejpam-6802	104	19	,	,	PUNCT
ejpam-6802	104	20	α	α	PROPN
ejpam-6802	104	21	)	)	PUNCT
ejpam-6802	104	22	.	.	PUNCT
ejpam-6802	105	1	consider	consider	VERB
ejpam-6802	105	2	m	m	PRON
ejpam-6802	105	3	:	:	PUNCT
ejpam-6802	105	4	=	=	SYM
ejpam-6802	105	5	α(q0	α(q0	X
ejpam-6802	105	6	,	,	PUNCT
ejpam-6802	105	7	ϑ0)α(q0	ϑ0)α(q0	NOUN
ejpam-6802	105	8	,	,	PUNCT
ejpam-6802	105	9	ϑ1	ϑ1	PROPN
ejpam-6802	105	10	)	)	PUNCT
ejpam-6802	105	11	.	.	PUNCT
ejpam-6802	106	1	then	then	ADV
ejpam-6802	106	2	,	,	PUNCT
ejpam-6802	106	3	for	for	ADP
ejpam-6802	106	4	all	all	DET
ejpam-6802	106	5	positive	positive	ADJ
ejpam-6802	106	6	integer	integer	NOUN
ejpam-6802	106	7	ℓ	ℓ	PROPN
ejpam-6802	106	8	and	and	CCONJ
ejpam-6802	106	9	p	p	X
ejpam-6802	106	10	,	,	PUNCT
ejpam-6802	106	11	α(qℓ	α(qℓ	PROPN
ejpam-6802	106	12	,	,	PUNCT
ejpam-6802	106	13	ϑℓ	ϑℓ	PROPN
ejpam-6802	106	14	)	)	PUNCT
ejpam-6802	106	15	=	=	SYM
ejpam-6802	106	16	α(ω(qℓ−1),ω(ϑℓ−1	α(ω(qℓ−1),ω(ϑℓ−1	NOUN
ejpam-6802	106	17	)	)	PUNCT
ejpam-6802	106	18	)	)	PUNCT
ejpam-6802	106	19	≤	≤	NOUN
ejpam-6802	106	20	(	(	PUNCT
ejpam-6802	106	21	α(qℓ−1	α(qℓ−1	NOUN
ejpam-6802	106	22	,	,	PUNCT
ejpam-6802	106	23	ϑℓ−1	ϑℓ−1	PROPN
ejpam-6802	106	24	)	)	PUNCT
ejpam-6802	106	25	)	)	PUNCT
ejpam-6802	107	1	λ	λ	NOUN
ejpam-6802	107	2	...	...	PUNCT
ejpam-6802	107	3	≤	≤	X
ejpam-6802	107	4	(	(	PUNCT
ejpam-6802	107	5	α(q0	α(q0	NOUN
ejpam-6802	107	6	,	,	PUNCT
ejpam-6802	107	7	ϑ0	ϑ0	NOUN
ejpam-6802	107	8	)	)	PUNCT
ejpam-6802	107	9	)	)	PUNCT
ejpam-6802	108	1	λℓ	λℓ	ADP
ejpam-6802	108	2	and	and	CCONJ
ejpam-6802	108	3	also	also	ADV
ejpam-6802	108	4	,	,	PUNCT
ejpam-6802	108	5	α(qℓ	α(qℓ	PROPN
ejpam-6802	108	6	,	,	PUNCT
ejpam-6802	108	7	ϑℓ+1	ϑℓ+1	NUM
ejpam-6802	108	8	)	)	PUNCT
ejpam-6802	108	9	=	=	SYM
ejpam-6802	108	10	α(ω(qℓ−1),ω(ϑℓ	α(ω(qℓ−1),ω(ϑℓ	NUM
ejpam-6802	108	11	)	)	PUNCT
ejpam-6802	108	12	)	)	PUNCT
ejpam-6802	108	13	≤	≤	NOUN
ejpam-6802	108	14	(	(	PUNCT
ejpam-6802	108	15	α(qℓ−1	α(qℓ−1	PROPN
ejpam-6802	108	16	,	,	PUNCT
ejpam-6802	108	17	ϑℓ	ϑℓ	NOUN
ejpam-6802	108	18	)	)	PUNCT
ejpam-6802	108	19	)	)	PUNCT
ejpam-6802	109	1	λ	λ	NOUN
ejpam-6802	109	2	...	...	PUNCT
ejpam-6802	109	3	≤	≤	X
ejpam-6802	109	4	(	(	PUNCT
ejpam-6802	109	5	α(q0	α(q0	NOUN
ejpam-6802	109	6	,	,	PUNCT
ejpam-6802	109	7	ϑ1	ϑ1	PROPN
ejpam-6802	109	8	)	)	PUNCT
ejpam-6802	109	9	)	)	PUNCT
ejpam-6802	110	1	λℓ	λℓ	INTJ
ejpam-6802	110	2	.	.	PUNCT
ejpam-6802	110	3	α(qℓ+p	α(qℓ+p	PROPN
ejpam-6802	110	4	,	,	PUNCT
ejpam-6802	110	5	ϑℓ	ϑℓ	NOUN
ejpam-6802	110	6	)	)	PUNCT
ejpam-6802	110	7	≤	≤	NOUN
ejpam-6802	110	8	α(qℓ+p	α(qℓ+p	PROPN
ejpam-6802	110	9	,	,	PUNCT
ejpam-6802	110	10	ϑℓ+1)α(qℓ	ϑℓ+1)α(qℓ	PROPN
ejpam-6802	110	11	,	,	PUNCT
ejpam-6802	110	12	ϑℓ+1)α(qℓ	ϑℓ+1)α(qℓ	PROPN
ejpam-6802	110	13	,	,	PUNCT
ejpam-6802	110	14	ϑℓ	ϑℓ	PROPN
ejpam-6802	110	15	)	)	PUNCT
ejpam-6802	110	16	≤	≤	NOUN
ejpam-6802	110	17	α(qℓ+p	α(qℓ+p	NOUN
ejpam-6802	110	18	,	,	PUNCT
ejpam-6802	110	19	ϑℓ+1)mλℓ	ϑℓ+1)mλℓ	ADJ
ejpam-6802	110	20	≤	≤	PROPN
ejpam-6802	110	21	α(qℓ+p	α(qℓ+p	NOUN
ejpam-6802	110	22	,	,	PUNCT
ejpam-6802	110	23	ϑℓ+2)α(qℓ+1	ϑℓ+2)α(qℓ+1	NOUN
ejpam-6802	110	24	,	,	PUNCT
ejpam-6802	110	25	ϑℓ+2)α(qℓ+1	ϑℓ+2)α(qℓ+1	NOUN
ejpam-6802	110	26	,	,	PUNCT
ejpam-6802	110	27	ϑℓ+1)mλℓ	ϑℓ+1)mλℓ	ADJ
ejpam-6802	110	28	≤	≤	NUM
ejpam-6802	110	29	α(qℓ+p	α(qℓ+p	NOUN
ejpam-6802	110	30	,	,	PUNCT
ejpam-6802	110	31	ϑℓ+2)m(λℓ+1+λℓ	ϑℓ+2)m(λℓ+1+λℓ	NOUN
ejpam-6802	110	32	)	)	PUNCT
ejpam-6802	110	33	...	...	PUNCT
ejpam-6802	110	34	≤	≤	NUM
ejpam-6802	110	35	α(qℓ+p	α(qℓ+p	NOUN
ejpam-6802	110	36	,	,	PUNCT
ejpam-6802	110	37	ϑℓ+p)m(λℓ+p−1+···+λℓ+1+λℓ	ϑℓ+p)m(λℓ+p−1+···+λℓ+1+λℓ	NOUN
ejpam-6802	110	38	)	)	PUNCT
ejpam-6802	110	39	≤	≤	NUM
ejpam-6802	110	40	m(λℓ+p+···+λℓ+1+λℓ	m(λℓ+p+···+λℓ+1+λℓ	NOUN
ejpam-6802	110	41	)	)	PUNCT
ejpam-6802	110	42	≤	≤	NUM
ejpam-6802	110	43	m	m	VERB
ejpam-6802	110	44	λℓ	λℓ	ADP
ejpam-6802	110	45	1−λ	1−λ	NUM
ejpam-6802	110	46	and	and	CCONJ
ejpam-6802	110	47	similarly	similarly	ADV
ejpam-6802	110	48	α(qℓ	α(qℓ	NUM
ejpam-6802	110	49	,	,	PUNCT
ejpam-6802	110	50	ϑℓ+p	ϑℓ+p	PROPN
ejpam-6802	110	51	)	)	PUNCT
ejpam-6802	110	52	≤	≤	NUM
ejpam-6802	110	53	m	m	VERB
ejpam-6802	110	54	λℓ	λℓ	ADP
ejpam-6802	110	55	1−λ	1−λ	NUM
ejpam-6802	110	56	.	.	PUNCT
ejpam-6802	111	1	now	now	ADV
ejpam-6802	111	2	,	,	PUNCT
ejpam-6802	111	3	α(qℓ	α(qℓ	PROPN
ejpam-6802	111	4	,	,	PUNCT
ejpam-6802	111	5	ϑr	ϑr	PROPN
ejpam-6802	111	6	)	)	PUNCT
ejpam-6802	111	7	≤	≤	NUM
ejpam-6802	111	8	α(qℓ	α(qℓ	PROPN
ejpam-6802	111	9	,	,	PUNCT
ejpam-6802	111	10	ϑℓ0)α(qℓ0	ϑℓ0)α(qℓ0	NOUN
ejpam-6802	111	11	,	,	PUNCT
ejpam-6802	111	12	ϑℓ0)α(qℓ0	ϑℓ0)α(qℓ0	NOUN
ejpam-6802	111	13	,	,	PUNCT
ejpam-6802	111	14	ϑr	ϑr	PROPN
ejpam-6802	111	15	)	)	PUNCT
ejpam-6802	111	16	≤	≤	NUM
ejpam-6802	112	1	3	3	NUM
ejpam-6802	112	2	m	m	NOUN
ejpam-6802	112	3	λℓ	λℓ	ADP
ejpam-6802	112	4	1−λ	1−λ	NUM
ejpam-6802	112	5	.	.	PUNCT
ejpam-6802	113	1	therefore	therefore	ADV
ejpam-6802	113	2	,	,	PUNCT
ejpam-6802	113	3	α(qℓ	α(qℓ	PROPN
ejpam-6802	113	4	,	,	PUNCT
ejpam-6802	113	5	ϑr	ϑr	PROPN
ejpam-6802	113	6	)	)	PUNCT
ejpam-6802	113	7	→	→	SYM
ejpam-6802	113	8	0(ℓ	0(ℓ	NUM
ejpam-6802	113	9	,	,	PUNCT
ejpam-6802	113	10	r	r	NOUN
ejpam-6802	113	11	→	→	SYM
ejpam-6802	113	12	+	+	PROPN
ejpam-6802	113	13	+	+	NOUN
ejpam-6802	113	14	∞	∞	NUM
ejpam-6802	113	15	)	)	PUNCT
ejpam-6802	113	16	.	.	PUNCT
ejpam-6802	114	1	hence	hence	ADV
ejpam-6802	114	2	(	(	PUNCT
ejpam-6802	114	3	{	{	PUNCT
ejpam-6802	114	4	qℓ	qℓ	NOUN
ejpam-6802	114	5	}	}	PUNCT
ejpam-6802	114	6	,	,	PUNCT
ejpam-6802	114	7	{	{	PUNCT
ejpam-6802	114	8	ϑℓ	ϑℓ	NOUN
ejpam-6802	114	9	}	}	PUNCT
ejpam-6802	114	10	)	)	PUNCT
ejpam-6802	114	11	)	)	PUNCT
ejpam-6802	114	12	is	be	AUX
ejpam-6802	114	13	a	a	DET
ejpam-6802	114	14	cauchy	cauchy	ADJ
ejpam-6802	114	15	bisequence	bisequence	NOUN
ejpam-6802	114	16	.	.	PUNCT
ejpam-6802	115	1	since	since	SCONJ
ejpam-6802	115	2	(	(	PUNCT
ejpam-6802	115	3	b	b	X
ejpam-6802	115	4	,	,	PUNCT
ejpam-6802	115	5	f	f	PROPN
ejpam-6802	115	6	,	,	PUNCT
ejpam-6802	115	7	α	α	PROPN
ejpam-6802	115	8	)	)	PUNCT
ejpam-6802	115	9	is	be	AUX
ejpam-6802	115	10	complete	complete	ADJ
ejpam-6802	115	11	,	,	PUNCT
ejpam-6802	115	12	(	(	PUNCT
ejpam-6802	115	13	{	{	PUNCT
ejpam-6802	115	14	qℓ	qℓ	NOUN
ejpam-6802	115	15	}	}	PUNCT
ejpam-6802	115	16	,	,	PUNCT
ejpam-6802	115	17	{	{	PUNCT
ejpam-6802	115	18	ϑℓ	ϑℓ	NOUN
ejpam-6802	115	19	}	}	PUNCT
ejpam-6802	115	20	)	)	PUNCT
ejpam-6802	115	21	converges	converge	NOUN
ejpam-6802	115	22	,	,	PUNCT
ejpam-6802	115	23	&	&	CCONJ
ejpam-6802	115	24	biconverges	biconverge	NOUN
ejpam-6802	115	25	to	to	ADP
ejpam-6802	115	26	a	a	DET
ejpam-6802	115	27	point	point	NOUN
ejpam-6802	115	28	x	x	X
ejpam-6802	115	29	∈	∈	PROPN
ejpam-6802	115	30	b	b	PROPN
ejpam-6802	115	31	∩	∩	PROPN
ejpam-6802	115	32	f	f	PROPN
ejpam-6802	115	33	,	,	PUNCT
ejpam-6802	115	34	{	{	PUNCT
ejpam-6802	115	35	ω(ϑℓ	ω(ϑℓ	NOUN
ejpam-6802	115	36	)	)	PUNCT
ejpam-6802	115	37	}	}	PUNCT
ejpam-6802	115	38	=	=	SYM
ejpam-6802	115	39	{	{	PUNCT
ejpam-6802	115	40	ϑℓ+1	ϑℓ+1	X
ejpam-6802	115	41	}	}	PUNCT
ejpam-6802	115	42	→	→	SYM
ejpam-6802	115	43	x	x	SYM
ejpam-6802	115	44	∈	∈	PROPN
ejpam-6802	115	45	b	b	NOUN
ejpam-6802	115	46	∩	∩	PROPN
ejpam-6802	115	47	f	f	X
ejpam-6802	115	48	.	.	PUNCT
ejpam-6802	116	1	since	since	SCONJ
ejpam-6802	116	2	ω	ω	PROPN
ejpam-6802	116	3	is	be	AUX
ejpam-6802	116	4	continuous	continuous	ADJ
ejpam-6802	116	5	,	,	PUNCT
ejpam-6802	116	6	it	it	PRON
ejpam-6802	116	7	follows	follow	VERB
ejpam-6802	116	8	that	that	SCONJ
ejpam-6802	116	9	ω(ϑℓ	ω(ϑℓ	NOUN
ejpam-6802	116	10	)	)	PUNCT
ejpam-6802	116	11	→	→	SYM
ejpam-6802	116	12	ω(x	ω(x	NOUN
ejpam-6802	116	13	)	)	PUNCT
ejpam-6802	116	14	,	,	PUNCT
ejpam-6802	116	15	then	then	ADV
ejpam-6802	116	16	ω(x	ω(x	NOUN
ejpam-6802	116	17	)	)	PUNCT
ejpam-6802	116	18	=	=	PUNCT
ejpam-6802	117	1	x.	x.	NOUN
ejpam-6802	117	2	hence	hence	ADV
ejpam-6802	117	3	x	x	PROPN
ejpam-6802	117	4	is	be	AUX
ejpam-6802	117	5	a	a	DET
ejpam-6802	117	6	fp	fp	PROPN
ejpam-6802	117	7	of	of	ADP
ejpam-6802	117	8	ω	ω	PROPN
ejpam-6802	117	9	.	.	PUNCT
ejpam-6802	118	1	if	if	SCONJ
ejpam-6802	118	2	z	z	NOUN
ejpam-6802	118	3	is	be	AUX
ejpam-6802	118	4	any	any	DET
ejpam-6802	118	5	fp	fp	PROPN
ejpam-6802	118	6	of	of	ADP
ejpam-6802	118	7	ω	ω	PROPN
ejpam-6802	118	8	,	,	PUNCT
ejpam-6802	118	9	then	then	ADV
ejpam-6802	118	10	ω(z	ω(z	PUNCT
ejpam-6802	118	11	)	)	PUNCT
ejpam-6802	118	12	=	=	SYM
ejpam-6802	118	13	z	z	NOUN
ejpam-6802	118	14	⇒	⇒	NOUN
ejpam-6802	118	15	z	z	PROPN
ejpam-6802	118	16	∈	∈	PROPN
ejpam-6802	118	17	b	b	PROPN
ejpam-6802	118	18	∩	∩	PROPN
ejpam-6802	118	19	f	f	PROPN
ejpam-6802	118	20	&	&	CCONJ
ejpam-6802	118	21	α(x	α(x	PROPN
ejpam-6802	118	22	,	,	PUNCT
ejpam-6802	118	23	z	z	NOUN
ejpam-6802	118	24	)	)	PUNCT
ejpam-6802	118	25	=	=	SYM
ejpam-6802	118	26	α(ω(x),ω(z	α(ω(x),ω(z	PROPN
ejpam-6802	118	27	)	)	PUNCT
ejpam-6802	118	28	)	)	PUNCT
ejpam-6802	118	29	≤	≤	NOUN
ejpam-6802	118	30	(	(	PUNCT
ejpam-6802	118	31	α(x	α(x	PROPN
ejpam-6802	118	32	,	,	PUNCT
ejpam-6802	118	33	z))λ	z))λ	PROPN
ejpam-6802	118	34	r.	r.	PROPN
ejpam-6802	118	35	ramaswamy	ramaswamy	PROPN
ejpam-6802	118	36	/	/	SYM
ejpam-6802	118	37	eur	eur	PROPN
ejpam-6802	118	38	.	.	PUNCT
ejpam-6802	119	1	j.	j.	PROPN
ejpam-6802	119	2	pure	pure	PROPN
ejpam-6802	119	3	appl	appl	PROPN
ejpam-6802	119	4	.	.	PROPN
ejpam-6802	119	5	math	math	PROPN
ejpam-6802	119	6	,	,	PUNCT
ejpam-6802	119	7	18	18	NUM
ejpam-6802	119	8	(	(	PUNCT
ejpam-6802	119	9	4	4	NUM
ejpam-6802	119	10	)	)	PUNCT
ejpam-6802	119	11	(	(	PUNCT
ejpam-6802	119	12	2025	2025	NUM
ejpam-6802	119	13	)	)	PUNCT
ejpam-6802	119	14	,	,	PUNCT
ejpam-6802	119	15	6802	6802	NUM
ejpam-6802	119	16	6	6	NUM
ejpam-6802	119	17	of	of	ADP
ejpam-6802	119	18	20	20	NUM
ejpam-6802	119	19	where	where	SCONJ
ejpam-6802	119	20	0	0	NUM
ejpam-6802	119	21	<	<	X
ejpam-6802	119	22	λ	λ	X
ejpam-6802	119	23	<	<	X
ejpam-6802	119	24	1	1	NUM
ejpam-6802	119	25	,	,	PUNCT
ejpam-6802	119	26	which	which	PRON
ejpam-6802	119	27	implies	imply	VERB
ejpam-6802	119	28	α(x	α(x	PROPN
ejpam-6802	119	29	,	,	PUNCT
ejpam-6802	119	30	z	z	NOUN
ejpam-6802	119	31	)	)	PUNCT
ejpam-6802	119	32	=	=	SYM
ejpam-6802	119	33	0	0	NUM
ejpam-6802	119	34	,	,	PUNCT
ejpam-6802	119	35	and	and	CCONJ
ejpam-6802	119	36	so	so	ADV
ejpam-6802	119	37	x	x	X
ejpam-6802	119	38	=	=	PUNCT
ejpam-6802	119	39	z.	z.	PROPN
ejpam-6802	119	40	example	example	NOUN
ejpam-6802	120	1	2	2	X
ejpam-6802	120	2	.	.	PUNCT
ejpam-6802	120	3	let	let	VERB
ejpam-6802	120	4	a	a	DET
ejpam-6802	120	5	=	=	PUNCT
ejpam-6802	120	6	mℓ×ℓ(r	mℓ×ℓ(r	NOUN
ejpam-6802	120	7	)	)	PUNCT
ejpam-6802	120	8	be	be	VERB
ejpam-6802	120	9	a	a	DET
ejpam-6802	120	10	set	set	NOUN
ejpam-6802	120	11	all	all	DET
ejpam-6802	120	12	real	real	ADJ
ejpam-6802	120	13	enteries	enterie	NOUN
ejpam-6802	120	14	and	and	CCONJ
ejpam-6802	120	15	z	z	NOUN
ejpam-6802	120	16	=	=	SYM
ejpam-6802	120	17	mℓ×ℓ(r	mℓ×ℓ(r	NOUN
ejpam-6802	120	18	)	)	PUNCT
ejpam-6802	120	19	be	be	VERB
ejpam-6802	120	20	a	a	DET
ejpam-6802	120	21	set	set	NOUN
ejpam-6802	120	22	all	all	DET
ejpam-6802	120	23	non	non	PRON
ejpam-6802	120	24	negative	negative	ADJ
ejpam-6802	120	25	real	real	ADJ
ejpam-6802	120	26	enteries	enterie	NOUN
ejpam-6802	120	27	.	.	PUNCT
ejpam-6802	121	1	let	let	VERB
ejpam-6802	121	2	b	b	NOUN
ejpam-6802	121	3	=	=	PRON
ejpam-6802	121	4	{	{	PUNCT
ejpam-6802	121	5	jℓ(r	jℓ(r	NOUN
ejpam-6802	121	6	)	)	PUNCT
ejpam-6802	121	7	:	:	PUNCT
ejpam-6802	122	1	jℓ(r	jℓ(r	X
ejpam-6802	122	2	)	)	PUNCT
ejpam-6802	123	1	is	be	AUX
ejpam-6802	123	2	an	an	DET
ejpam-6802	123	3	upper	upper	ADJ
ejpam-6802	123	4	triangular	triangular	NOUN
ejpam-6802	123	5	matrices	matrix	NOUN
ejpam-6802	123	6	over	over	ADP
ejpam-6802	123	7	r	r	NOUN
ejpam-6802	123	8	}	}	PUNCT
ejpam-6802	123	9	,	,	PUNCT
ejpam-6802	123	10	f	f	PROPN
ejpam-6802	123	11	=	=	PRON
ejpam-6802	123	12	{	{	PUNCT
ejpam-6802	123	13	lℓ(r	lℓ(r	NOUN
ejpam-6802	123	14	)	)	PUNCT
ejpam-6802	123	15	:	:	PUNCT
ejpam-6802	124	1	lℓ(r	lℓ(r	X
ejpam-6802	124	2	)	)	PUNCT
ejpam-6802	124	3	is	be	AUX
ejpam-6802	124	4	an	an	DET
ejpam-6802	124	5	upper	upper	ADJ
ejpam-6802	124	6	triangular	triangular	NOUN
ejpam-6802	124	7	matrices	matrix	NOUN
ejpam-6802	124	8	over	over	ADP
ejpam-6802	124	9	r	r	NOUN
ejpam-6802	124	10	}	}	PUNCT
ejpam-6802	124	11	and	and	CCONJ
ejpam-6802	124	12	the	the	DET
ejpam-6802	124	13	function	function	NOUN
ejpam-6802	124	14	α	α	NOUN
ejpam-6802	124	15	:	:	PUNCT
ejpam-6802	124	16	b	b	X
ejpam-6802	124	17	×	×	NOUN
ejpam-6802	124	18	f	f	X
ejpam-6802	124	19	→	→	PUNCT
ejpam-6802	124	20	a	a	PRON
ejpam-6802	124	21	is	be	AUX
ejpam-6802	124	22	defined	define	VERB
ejpam-6802	124	23	as	as	ADP
ejpam-6802	124	24	α(u	α(u	NOUN
ejpam-6802	124	25	,	,	PUNCT
ejpam-6802	124	26	v	v	NOUN
ejpam-6802	124	27	)	)	PUNCT
ejpam-6802	125	1	=	=	PUNCT
ejpam-6802	125	2	e	e	X
ejpam-6802	125	3	∑ℓ	∑ℓ	PROPN
ejpam-6802	125	4	i	i	PROPN
ejpam-6802	125	5	,	,	PUNCT
ejpam-6802	125	6	j=1	j=1	ADJ
ejpam-6802	125	7	|ςij−mij|	|ςij−mij|	NOUN
ejpam-6802	125	8	for	for	ADP
ejpam-6802	125	9	all	all	DET
ejpam-6802	125	10	u	u	NOUN
ejpam-6802	125	11	=	=	PUNCT
ejpam-6802	125	12	(	(	PUNCT
ejpam-6802	125	13	ςij)ℓ×ℓ	ςij)ℓ×ℓ	NOUN
ejpam-6802	125	14	∈	∈	PROPN
ejpam-6802	125	15	b	b	PROPN
ejpam-6802	125	16	and	and	CCONJ
ejpam-6802	125	17	v	v	NOUN
ejpam-6802	125	18	=	=	PUNCT
ejpam-6802	125	19	(	(	PUNCT
ejpam-6802	125	20	mij)ℓ×ℓ	mij)ℓ×ℓ	PROPN
ejpam-6802	125	21	∈	∈	PROPN
ejpam-6802	125	22	f	f	PROPN
ejpam-6802	125	23	.	.	PUNCT
ejpam-6802	126	1	then	then	ADV
ejpam-6802	126	2	(	(	PUNCT
ejpam-6802	126	3	b	b	X
ejpam-6802	126	4	,	,	PUNCT
ejpam-6802	126	5	f	f	PROPN
ejpam-6802	126	6	,	,	PUNCT
ejpam-6802	126	7	α	α	PROPN
ejpam-6802	126	8	)	)	PUNCT
ejpam-6802	126	9	is	be	AUX
ejpam-6802	126	10	a	a	DET
ejpam-6802	126	11	complete	complete	ADJ
ejpam-6802	126	12	mcbms	mcbms	NOUN
ejpam-6802	126	13	.	.	PUNCT
ejpam-6802	127	1	also	also	ADV
ejpam-6802	127	2	define	define	VERB
ejpam-6802	127	3	t	t	NOUN
ejpam-6802	127	4	:	:	PUNCT
ejpam-6802	127	5	(	(	PUNCT
ejpam-6802	127	6	b	b	X
ejpam-6802	127	7	,	,	PUNCT
ejpam-6802	127	8	f	f	PROPN
ejpam-6802	127	9	,	,	PUNCT
ejpam-6802	127	10	α	α	X
ejpam-6802	127	11	)	)	PUNCT
ejpam-6802	127	12	⇒	⇒	NOUN
ejpam-6802	127	13	(	(	PUNCT
ejpam-6802	127	14	b	b	X
ejpam-6802	127	15	,	,	PUNCT
ejpam-6802	127	16	f	f	PROPN
ejpam-6802	127	17	,	,	PUNCT
ejpam-6802	127	18	α	α	PROPN
ejpam-6802	127	19	)	)	PUNCT
ejpam-6802	127	20	as	as	ADP
ejpam-6802	127	21	t	t	PROPN
ejpam-6802	127	22	(	(	PUNCT
ejpam-6802	127	23	u	u	NOUN
ejpam-6802	127	24	)	)	PUNCT
ejpam-6802	127	25	=	=	SYM
ejpam-6802	127	26	(	(	PUNCT
ejpam-6802	127	27	ςij	ςij	NOUN
ejpam-6802	127	28	4	4	NUM
ejpam-6802	127	29	)	)	PUNCT
ejpam-6802	127	30	ℓ×ℓ	ℓ×ℓ	PROPN
ejpam-6802	127	31	for	for	ADP
ejpam-6802	127	32	all	all	DET
ejpam-6802	127	33	u	u	NOUN
ejpam-6802	127	34	=	=	PUNCT
ejpam-6802	127	35	(	(	PUNCT
ejpam-6802	127	36	ςij)ℓ×ℓ	ςij)ℓ×ℓ	NOUN
ejpam-6802	127	37	∈	∈	PROPN
ejpam-6802	127	38	jℓ(r	jℓ(r	NOUN
ejpam-6802	127	39	)	)	PUNCT
ejpam-6802	127	40	∪	∪	ADP
ejpam-6802	127	41	lℓ(r	lℓ(r	NOUN
ejpam-6802	127	42	)	)	PUNCT
ejpam-6802	127	43	.	.	PUNCT
ejpam-6802	128	1	now	now	ADV
ejpam-6802	128	2	,	,	PUNCT
ejpam-6802	128	3	α(t	α(t	PROPN
ejpam-6802	128	4	(	(	PUNCT
ejpam-6802	128	5	u	u	NOUN
ejpam-6802	128	6	)	)	PUNCT
ejpam-6802	128	7	,	,	PUNCT
ejpam-6802	128	8	t	t	PROPN
ejpam-6802	128	9	(	(	PUNCT
ejpam-6802	128	10	v	v	NOUN
ejpam-6802	128	11	)	)	PUNCT
ejpam-6802	128	12	)	)	PUNCT
ejpam-6802	129	1	=	=	PUNCT
ejpam-6802	129	2	e	e	NOUN
ejpam-6802	129	3	1	1	NUM
ejpam-6802	129	4	4	4	NUM
ejpam-6802	129	5	∑ℓ	∑ℓ	ADJ
ejpam-6802	129	6	i	i	PROPN
ejpam-6802	129	7	,	,	PUNCT
ejpam-6802	129	8	j=1	j=1	PROPN
ejpam-6802	129	9	|ςij−mij|	|ςij−mij|	NOUN
ejpam-6802	129	10	≤	≤	NUM
ejpam-6802	129	11	e	e	NOUN
ejpam-6802	129	12	1	1	NUM
ejpam-6802	129	13	2	2	NUM
ejpam-6802	129	14	∑ℓ	∑ℓ	PROPN
ejpam-6802	129	15	i	i	PROPN
ejpam-6802	129	16	,	,	PUNCT
ejpam-6802	129	17	j=1	j=1	PROPN
ejpam-6802	129	18	|ςij−mij|	|ςij−mij|	NOUN
ejpam-6802	129	19	=	=	SYM
ejpam-6802	129	20	(	(	PUNCT
ejpam-6802	129	21	e	e	X
ejpam-6802	129	22	∑ℓ	∑ℓ	PROPN
ejpam-6802	129	23	i	i	PROPN
ejpam-6802	129	24	,	,	PUNCT
ejpam-6802	129	25	j=1	j=1	ADJ
ejpam-6802	129	26	|ςij−mij|	|ςij−mij|	NOUN
ejpam-6802	129	27	)	)	PUNCT
ejpam-6802	129	28	1	1	NUM
ejpam-6802	129	29	2	2	NUM
ejpam-6802	129	30	=	=	SYM
ejpam-6802	129	31	(	(	PUNCT
ejpam-6802	129	32	α(u	α(u	NOUN
ejpam-6802	129	33	,	,	PUNCT
ejpam-6802	129	34	v))λ	v))λ	VERB
ejpam-6802	129	35	for	for	ADP
ejpam-6802	129	36	all	all	DET
ejpam-6802	129	37	u	u	NOUN
ejpam-6802	129	38	=	=	PUNCT
ejpam-6802	129	39	(	(	PUNCT
ejpam-6802	129	40	ςij)ℓ×ℓ	ςij)ℓ×ℓ	NOUN
ejpam-6802	129	41	∈	∈	PROPN
ejpam-6802	129	42	b	b	PROPN
ejpam-6802	129	43	and	and	CCONJ
ejpam-6802	129	44	v	v	NOUN
ejpam-6802	129	45	=	=	PUNCT
ejpam-6802	129	46	(	(	PUNCT
ejpam-6802	129	47	mij)ℓ×ℓ	mij)ℓ×ℓ	PROPN
ejpam-6802	129	48	∈	∈	PROPN
ejpam-6802	129	49	f	f	PROPN
ejpam-6802	129	50	.	.	PUNCT
ejpam-6802	130	1	all	all	DET
ejpam-6802	130	2	the	the	DET
ejpam-6802	130	3	criteria	criterion	NOUN
ejpam-6802	130	4	of	of	ADP
ejpam-6802	130	5	theorem	theorem	ADJ
ejpam-6802	130	6	3.1	3.1	NUM
ejpam-6802	130	7	are	be	AUX
ejpam-6802	130	8	satisfied	satisfied	ADJ
ejpam-6802	130	9	with	with	ADP
ejpam-6802	130	10	λ	λ	X
ejpam-6802	130	11	=	=	SYM
ejpam-6802	130	12	1	1	NUM
ejpam-6802	130	13	2	2	NUM
ejpam-6802	130	14	&	&	CCONJ
ejpam-6802	130	15	t	t	PROPN
ejpam-6802	130	16	possesses	possess	VERB
ejpam-6802	130	17	a	a	DET
ejpam-6802	130	18	ufp	ufp	NOUN
ejpam-6802	130	19	(	(	PUNCT
ejpam-6802	130	20	0ℓ×ℓ	0ℓ×ℓ	NOUN
ejpam-6802	130	21	,	,	PUNCT
ejpam-6802	130	22	0ℓ×ℓ	0ℓ×ℓ	NOUN
ejpam-6802	130	23	)	)	PUNCT
ejpam-6802	130	24	∈	∈	PROPN
ejpam-6802	130	25	jℓ(r	jℓ(r	NOUN
ejpam-6802	130	26	)	)	PUNCT
ejpam-6802	130	27	∪	∪	ADP
ejpam-6802	130	28	lℓ(r	lℓ(r	NOUN
ejpam-6802	130	29	)	)	PUNCT
ejpam-6802	130	30	where	where	SCONJ
ejpam-6802	130	31	0ℓ×ℓ	0ℓ×ℓ	NOUN
ejpam-6802	130	32	is	be	AUX
ejpam-6802	130	33	the	the	DET
ejpam-6802	130	34	null	null	ADJ
ejpam-6802	130	35	matrix	matrix	NOUN
ejpam-6802	130	36	.	.	PUNCT
ejpam-6802	131	1	example	example	NOUN
ejpam-6802	132	1	3	3	X
ejpam-6802	132	2	.	.	PUNCT
ejpam-6802	132	3	let	let	VERB
ejpam-6802	132	4	a	a	DET
ejpam-6802	132	5	=	=	SYM
ejpam-6802	132	6	r	r	NOUN
ejpam-6802	132	7	,	,	PUNCT
ejpam-6802	132	8	z	z	NOUN
ejpam-6802	132	9	=	=	PRON
ejpam-6802	132	10	{	{	PUNCT
ejpam-6802	132	11	q	q	NOUN
ejpam-6802	132	12	∈	∈	PROPN
ejpam-6802	132	13	a|q	a|q	PROPN
ejpam-6802	132	14	≥	≥	NOUN
ejpam-6802	132	15	0	0	NUM
ejpam-6802	132	16	}	}	PUNCT
ejpam-6802	132	17	.	.	PUNCT
ejpam-6802	133	1	take	take	VERB
ejpam-6802	133	2	b	b	NOUN
ejpam-6802	133	3	=	=	PUNCT
ejpam-6802	134	1	[	[	X
ejpam-6802	134	2	0	0	NUM
ejpam-6802	134	3	,	,	PUNCT
ejpam-6802	134	4	1	1	NUM
ejpam-6802	134	5	]	]	PUNCT
ejpam-6802	134	6	&	&	CCONJ
ejpam-6802	134	7	f	f	PROPN
ejpam-6802	134	8	=	=	NOUN
ejpam-6802	134	9	{	{	PUNCT
ejpam-6802	134	10	0	0	NUM
ejpam-6802	134	11	}	}	PUNCT
ejpam-6802	134	12	∪	∪	NOUN
ejpam-6802	134	13	n	n	PRON
ejpam-6802	134	14	−	−	PROPN
ejpam-6802	134	15	{	{	PUNCT
ejpam-6802	134	16	1	1	NUM
ejpam-6802	134	17	}	}	PUNCT
ejpam-6802	134	18	be	be	AUX
ejpam-6802	134	19	equipped	equip	VERB
ejpam-6802	134	20	with	with	ADP
ejpam-6802	134	21	α(q	α(q	PROPN
ejpam-6802	134	22	,	,	PUNCT
ejpam-6802	134	23	ϑ	ϑ	NOUN
ejpam-6802	134	24	)	)	PUNCT
ejpam-6802	134	25	=	=	SYM
ejpam-6802	134	26	e|q−ϑ|	e|q−ϑ|	PROPN
ejpam-6802	134	27	for	for	ADP
ejpam-6802	134	28	all	all	DET
ejpam-6802	134	29	q	q	PROPN
ejpam-6802	134	30	∈	∈	PROPN
ejpam-6802	134	31	b	b	PROPN
ejpam-6802	134	32	,	,	PUNCT
ejpam-6802	134	33	ϑ	ϑ	X
ejpam-6802	134	34	∈	∈	PROPN
ejpam-6802	134	35	f	f	X
ejpam-6802	134	36	.	.	PUNCT
ejpam-6802	135	1	then	then	ADV
ejpam-6802	135	2	,	,	PUNCT
ejpam-6802	135	3	(	(	PUNCT
ejpam-6802	135	4	b	b	X
ejpam-6802	135	5	,	,	PUNCT
ejpam-6802	135	6	f	f	PROPN
ejpam-6802	135	7	,	,	PUNCT
ejpam-6802	135	8	α	α	PROPN
ejpam-6802	135	9	)	)	PUNCT
ejpam-6802	135	10	is	be	AUX
ejpam-6802	135	11	a	a	DET
ejpam-6802	135	12	complete	complete	ADJ
ejpam-6802	135	13	mcbms	mcbms	NOUN
ejpam-6802	135	14	.	.	PUNCT
ejpam-6802	136	1	also	also	ADV
ejpam-6802	136	2	define	define	VERB
ejpam-6802	136	3	ω	ω	NOUN
ejpam-6802	136	4	:	:	PUNCT
ejpam-6802	136	5	b	b	X
ejpam-6802	136	6	∪	∪	ADP
ejpam-6802	136	7	f	f	PROPN
ejpam-6802	136	8	⇒	⇒	PROPN
ejpam-6802	136	9	b	b	PROPN
ejpam-6802	136	10	∪	∪	VERB
ejpam-6802	136	11	f	f	NOUN
ejpam-6802	136	12	as	as	ADP
ejpam-6802	136	13	ω(q	ω(q	NOUN
ejpam-6802	136	14	)	)	PUNCT
ejpam-6802	137	1	=	=	PRON
ejpam-6802	137	2	{	{	PUNCT
ejpam-6802	137	3	q	q	NOUN
ejpam-6802	137	4	5	5	NUM
ejpam-6802	137	5	,	,	PUNCT
ejpam-6802	137	6	if	if	SCONJ
ejpam-6802	137	7	q	q	X
ejpam-6802	137	8	∈	∈	PROPN
ejpam-6802	137	9	(	(	PUNCT
ejpam-6802	137	10	0	0	NUM
ejpam-6802	137	11	,	,	PUNCT
ejpam-6802	137	12	1	1	NUM
ejpam-6802	137	13	]	]	PUNCT
ejpam-6802	137	14	,	,	PUNCT
ejpam-6802	137	15	0	0	NUM
ejpam-6802	137	16	,	,	PUNCT
ejpam-6802	137	17	if	if	SCONJ
ejpam-6802	137	18	q	q	X
ejpam-6802	137	19	∈	∈	PROPN
ejpam-6802	137	20	{	{	PUNCT
ejpam-6802	137	21	0	0	NUM
ejpam-6802	137	22	}	}	PUNCT
ejpam-6802	137	23	∪	∪	ADP
ejpam-6802	137	24	n−	n−	PROPN
ejpam-6802	137	25	{	{	PUNCT
ejpam-6802	137	26	1	1	NUM
ejpam-6802	137	27	}	}	PUNCT
ejpam-6802	137	28	,	,	PUNCT
ejpam-6802	137	29	for	for	ADP
ejpam-6802	137	30	all	all	DET
ejpam-6802	137	31	q	q	PROPN
ejpam-6802	137	32	∈	∈	PROPN
ejpam-6802	137	33	b	b	X
ejpam-6802	137	34	∪	∪	PROPN
ejpam-6802	137	35	f	f	PROPN
ejpam-6802	137	36	.	.	PUNCT
ejpam-6802	138	1	let	let	VERB
ejpam-6802	138	2	q	q	PROPN
ejpam-6802	138	3	∈	∈	PROPN
ejpam-6802	138	4	b	b	PROPN
ejpam-6802	138	5	and	and	CCONJ
ejpam-6802	138	6	ϑ	ϑ	PROPN
ejpam-6802	138	7	∈	∈	PROPN
ejpam-6802	138	8	f	f	X
ejpam-6802	138	9	,	,	PUNCT
ejpam-6802	138	10	then	then	ADV
ejpam-6802	138	11	α(ωq	α(ωq	NUM
ejpam-6802	138	12	,	,	PUNCT
ejpam-6802	138	13	ωϑ	ωϑ	NOUN
ejpam-6802	138	14	)	)	PUNCT
ejpam-6802	138	15	=	=	SYM
ejpam-6802	139	1	e|	e|	PROPN
ejpam-6802	139	2	q	q	PROPN
ejpam-6802	139	3	5	5	NUM
ejpam-6802	139	4	−0|	−0|	NOUN
ejpam-6802	139	5	≤	≤	NUM
ejpam-6802	139	6	e	e	NOUN
ejpam-6802	139	7	1	1	NUM
ejpam-6802	139	8	2	2	NUM
ejpam-6802	139	9	|q−ϑ|	|q−ϑ|	PUNCT
ejpam-6802	139	10	=	=	NOUN
ejpam-6802	139	11	(	(	PUNCT
ejpam-6802	139	12	e|q−ϑ|	e|q−ϑ|	PROPN
ejpam-6802	139	13	)	)	PUNCT
ejpam-6802	139	14	1	1	NUM
ejpam-6802	139	15	2	2	NUM
ejpam-6802	139	16	.	.	PUNCT
ejpam-6802	140	1	therefore	therefore	ADV
ejpam-6802	140	2	,	,	PUNCT
ejpam-6802	140	3	conditions	condition	NOUN
ejpam-6802	140	4	of	of	ADP
ejpam-6802	140	5	theorem	theorem	ADJ
ejpam-6802	140	6	3.1	3.1	NUM
ejpam-6802	140	7	are	be	AUX
ejpam-6802	140	8	satisfied	satisfied	ADJ
ejpam-6802	140	9	&	&	CCONJ
ejpam-6802	140	10	ω	ω	PROPN
ejpam-6802	140	11	possesses	possess	VERB
ejpam-6802	140	12	a	a	DET
ejpam-6802	140	13	ufp	ufp	NOUN
ejpam-6802	140	14	q	q	NOUN
ejpam-6802	141	1	=	=	NOUN
ejpam-6802	141	2	0	0	X
ejpam-6802	141	3	.	.	PUNCT
ejpam-6802	142	1	below	below	ADP
ejpam-6802	142	2	we	we	PRON
ejpam-6802	142	3	prove	prove	VERB
ejpam-6802	142	4	a	a	DET
ejpam-6802	142	5	similar	similar	ADJ
ejpam-6802	142	6	result	result	NOUN
ejpam-6802	142	7	for	for	ADP
ejpam-6802	142	8	contravariant	contravariant	ADJ
ejpam-6802	142	9	maps	map	NOUN
ejpam-6802	142	10	.	.	PUNCT
ejpam-6802	143	1	r.	r.	PROPN
ejpam-6802	143	2	ramaswamy	ramaswamy	PROPN
ejpam-6802	143	3	/	/	SYM
ejpam-6802	143	4	eur	eur	PROPN
ejpam-6802	143	5	.	.	PUNCT
ejpam-6802	144	1	j.	j.	PROPN
ejpam-6802	144	2	pure	pure	PROPN
ejpam-6802	144	3	appl	appl	PROPN
ejpam-6802	144	4	.	.	PROPN
ejpam-6802	144	5	math	math	PROPN
ejpam-6802	144	6	,	,	PUNCT
ejpam-6802	144	7	18	18	NUM
ejpam-6802	144	8	(	(	PUNCT
ejpam-6802	144	9	4	4	NUM
ejpam-6802	144	10	)	)	PUNCT
ejpam-6802	144	11	(	(	PUNCT
ejpam-6802	144	12	2025	2025	NUM
ejpam-6802	144	13	)	)	PUNCT
ejpam-6802	144	14	,	,	PUNCT
ejpam-6802	144	15	6802	6802	NUM
ejpam-6802	144	16	7	7	NUM
ejpam-6802	144	17	of	of	ADP
ejpam-6802	144	18	20	20	NUM
ejpam-6802	144	19	theorem	theorem	ADJ
ejpam-6802	144	20	3.2	3.2	NUM
ejpam-6802	144	21	.	.	PUNCT
ejpam-6802	145	1	let	let	VERB
ejpam-6802	145	2	(	(	PUNCT
ejpam-6802	145	3	b	b	X
ejpam-6802	145	4	,	,	PUNCT
ejpam-6802	145	5	f	f	PROPN
ejpam-6802	145	6	,	,	PUNCT
ejpam-6802	145	7	α	α	PROPN
ejpam-6802	145	8	)	)	PUNCT
ejpam-6802	145	9	be	be	VERB
ejpam-6802	145	10	a	a	DET
ejpam-6802	145	11	complete	complete	ADJ
ejpam-6802	145	12	mcbms	mcbms	NOUN
ejpam-6802	145	13	,	,	PUNCT
ejpam-6802	145	14	z	z	PROPN
ejpam-6802	145	15	be	be	AUX
ejpam-6802	145	16	a	a	DET
ejpam-6802	145	17	cone	cone	NOUN
ejpam-6802	145	18	with	with	ADP
ejpam-6802	145	19	constant	constant	ADJ
ejpam-6802	145	20	w	w	NOUN
ejpam-6802	145	21	&	&	CCONJ
ejpam-6802	145	22	given	give	VERB
ejpam-6802	145	23	a	a	DET
ejpam-6802	145	24	contravariant	contravariant	ADJ
ejpam-6802	145	25	contraction	contraction	NOUN
ejpam-6802	146	1	function	function	NOUN
ejpam-6802	146	2	ω	ω	NOUN
ejpam-6802	146	3	:	:	PUNCT
ejpam-6802	146	4	(	(	PUNCT
ejpam-6802	146	5	b	b	X
ejpam-6802	146	6	,	,	PUNCT
ejpam-6802	146	7	f	f	PROPN
ejpam-6802	146	8	,	,	PUNCT
ejpam-6802	146	9	α	α	NOUN
ejpam-6802	146	10	)	)	PUNCT
ejpam-6802	146	11	⇆	⇆	PROPN
ejpam-6802	146	12	(	(	PUNCT
ejpam-6802	146	13	b	b	NOUN
ejpam-6802	146	14	,	,	PUNCT
ejpam-6802	146	15	f	f	PROPN
ejpam-6802	146	16	,	,	PUNCT
ejpam-6802	146	17	α	α	PROPN
ejpam-6802	146	18	)	)	PUNCT
ejpam-6802	146	19	.	.	PUNCT
ejpam-6802	147	1	then	then	ADV
ejpam-6802	147	2	the	the	DET
ejpam-6802	147	3	function	function	PROPN
ejpam-6802	147	4	ω	ω	NOUN
ejpam-6802	147	5	:	:	PUNCT
ejpam-6802	147	6	b	b	X
ejpam-6802	147	7	∪	∪	X
ejpam-6802	147	8	f	f	PROPN
ejpam-6802	147	9	→	→	SYM
ejpam-6802	147	10	b	b	X
ejpam-6802	147	11	∪	∪	X
ejpam-6802	147	12	f	f	PROPN
ejpam-6802	147	13	has	have	VERB
ejpam-6802	147	14	a	a	DET
ejpam-6802	147	15	ufp	ufp	NOUN
ejpam-6802	147	16	.	.	PUNCT
ejpam-6802	148	1	proof	proof	NOUN
ejpam-6802	148	2	.	.	PUNCT
ejpam-6802	149	1	let	let	VERB
ejpam-6802	149	2	q0	q0	PROPN
ejpam-6802	149	3	∈	∈	PROPN
ejpam-6802	149	4	b.	b.	PROPN
ejpam-6802	149	5	for	for	ADP
ejpam-6802	149	6	all	all	DET
ejpam-6802	149	7	ℓ	ℓ	PROPN
ejpam-6802	149	8	∈	∈	PROPN
ejpam-6802	149	9	n	n	CCONJ
ejpam-6802	149	10	,	,	PUNCT
ejpam-6802	149	11	define	define	VERB
ejpam-6802	149	12	ω(qℓ	ω(qℓ	NUM
ejpam-6802	149	13	)	)	PUNCT
ejpam-6802	149	14	=	=	SYM
ejpam-6802	149	15	ϑℓ	ϑℓ	NOUN
ejpam-6802	149	16	and	and	CCONJ
ejpam-6802	149	17	ω(ϑℓ	ω(ϑℓ	ADV
ejpam-6802	149	18	)	)	PUNCT
ejpam-6802	149	19	=	=	SYM
ejpam-6802	150	1	qℓ+1	qℓ+1	X
ejpam-6802	150	2	.	.	PUNCT
ejpam-6802	151	1	then	then	ADV
ejpam-6802	151	2	(	(	PUNCT
ejpam-6802	151	3	{	{	PUNCT
ejpam-6802	151	4	qℓ	qℓ	NOUN
ejpam-6802	151	5	}	}	PUNCT
ejpam-6802	151	6	,	,	PUNCT
ejpam-6802	151	7	{	{	PUNCT
ejpam-6802	151	8	ϑℓ	ϑℓ	NOUN
ejpam-6802	151	9	}	}	PUNCT
ejpam-6802	151	10	)	)	PUNCT
ejpam-6802	151	11	is	be	AUX
ejpam-6802	151	12	a	a	DET
ejpam-6802	151	13	bisequence	bisequence	NOUN
ejpam-6802	151	14	on	on	ADP
ejpam-6802	151	15	(	(	PUNCT
ejpam-6802	151	16	b	b	NOUN
ejpam-6802	151	17	,	,	PUNCT
ejpam-6802	151	18	f	f	PROPN
ejpam-6802	151	19	,	,	PUNCT
ejpam-6802	151	20	α	α	PROPN
ejpam-6802	151	21	)	)	PUNCT
ejpam-6802	151	22	.	.	PUNCT
ejpam-6802	152	1	then	then	ADV
ejpam-6802	152	2	for	for	ADP
ejpam-6802	152	3	all	all	DET
ejpam-6802	152	4	ℓ	ℓ	NOUN
ejpam-6802	152	5	,	,	PUNCT
ejpam-6802	152	6	p	p	PROPN
ejpam-6802	152	7	∈	∈	PROPN
ejpam-6802	152	8	z+	z+	NUM
ejpam-6802	152	9	,	,	PUNCT
ejpam-6802	152	10	α(qℓ	α(qℓ	PROPN
ejpam-6802	152	11	,	,	PUNCT
ejpam-6802	152	12	ϑℓ	ϑℓ	PROPN
ejpam-6802	152	13	)	)	PUNCT
ejpam-6802	152	14	=	=	PUNCT
ejpam-6802	152	15	α(ω(ϑℓ−1),ω(qℓ	α(ω(ϑℓ−1),ω(qℓ	NUM
ejpam-6802	152	16	)	)	PUNCT
ejpam-6802	152	17	)	)	PUNCT
ejpam-6802	152	18	≤	≤	PROPN
ejpam-6802	152	19	α(qℓ	α(qℓ	PROPN
ejpam-6802	152	20	,	,	PUNCT
ejpam-6802	152	21	ϑℓ−1	ϑℓ−1	PROPN
ejpam-6802	152	22	)	)	PUNCT
ejpam-6802	152	23	=	=	SYM
ejpam-6802	152	24	(	(	PUNCT
ejpam-6802	152	25	α(ω(ϑℓ−1),ω(qℓ−1	α(ω(ϑℓ−1),ω(qℓ−1	PROPN
ejpam-6802	152	26	)	)	PUNCT
ejpam-6802	152	27	)	)	PUNCT
ejpam-6802	152	28	)	)	PUNCT
ejpam-6802	153	1	λ	λ	NOUN
ejpam-6802	153	2	≤	≤	NUM
ejpam-6802	153	3	(	(	PUNCT
ejpam-6802	153	4	α(qℓ−1	α(qℓ−1	PROPN
ejpam-6802	153	5	,	,	PUNCT
ejpam-6802	153	6	ϑℓ−1	ϑℓ−1	PROPN
ejpam-6802	153	7	)	)	PUNCT
ejpam-6802	153	8	)	)	PUNCT
ejpam-6802	154	1	λ2	λ2	NOUN
ejpam-6802	154	2	...	...	PUNCT
ejpam-6802	154	3	≤	≤	NUM
ejpam-6802	154	4	(	(	PUNCT
ejpam-6802	154	5	α(q0	α(q0	NOUN
ejpam-6802	154	6	,	,	PUNCT
ejpam-6802	154	7	ϑ0	ϑ0	NOUN
ejpam-6802	154	8	)	)	PUNCT
ejpam-6802	154	9	)	)	PUNCT
ejpam-6802	155	1	λ2ℓ	λ2ℓ	PROPN
ejpam-6802	155	2	α(qℓ+1	α(qℓ+1	X
ejpam-6802	155	3	,	,	PUNCT
ejpam-6802	155	4	ϑℓ	ϑℓ	NOUN
ejpam-6802	155	5	)	)	PUNCT
ejpam-6802	155	6	=	=	SYM
ejpam-6802	155	7	α(ω(ϑℓ),ω(qℓ	α(ω(ϑℓ),ω(qℓ	NUM
ejpam-6802	155	8	)	)	PUNCT
ejpam-6802	155	9	)	)	PUNCT
ejpam-6802	156	1	≤	≤	NOUN
ejpam-6802	156	2	(	(	PUNCT
ejpam-6802	156	3	α(qℓ	α(qℓ	PROPN
ejpam-6802	156	4	,	,	PUNCT
ejpam-6802	156	5	ϑℓ	ϑℓ	PROPN
ejpam-6802	156	6	)	)	PUNCT
ejpam-6802	156	7	)	)	PUNCT
ejpam-6802	157	1	λ	λ	NOUN
ejpam-6802	157	2	...	...	PUNCT
ejpam-6802	157	3	≤	≤	X
ejpam-6802	157	4	(	(	PUNCT
ejpam-6802	157	5	α(q0	α(q0	NOUN
ejpam-6802	157	6	,	,	PUNCT
ejpam-6802	157	7	ϑ0	ϑ0	NOUN
ejpam-6802	157	8	)	)	PUNCT
ejpam-6802	157	9	)	)	PUNCT
ejpam-6802	158	1	λ2ℓ+1	λ2ℓ+1	PROPN
ejpam-6802	158	2	.	.	PUNCT
ejpam-6802	159	1	α(qℓ+p	α(qℓ+p	PROPN
ejpam-6802	159	2	,	,	PUNCT
ejpam-6802	159	3	ϑℓ	ϑℓ	NOUN
ejpam-6802	159	4	)	)	PUNCT
ejpam-6802	159	5	≤	≤	NUM
ejpam-6802	159	6	α(qℓp	α(qℓp	NOUN
ejpam-6802	159	7	,	,	PUNCT
ejpam-6802	159	8	ϑℓ+1)α(qℓ+1	ϑℓ+1)α(qℓ+1	ADJ
ejpam-6802	159	9	,	,	PUNCT
ejpam-6802	159	10	ϑℓ+1)α(qℓ+1	ϑℓ+1)α(qℓ+1	ADJ
ejpam-6802	159	11	,	,	PUNCT
ejpam-6802	159	12	ϑℓ	ϑℓ	NOUN
ejpam-6802	159	13	)	)	PUNCT
ejpam-6802	159	14	≤	≤	NOUN
ejpam-6802	159	15	α(qℓ+p	α(qℓ+p	PROPN
ejpam-6802	159	16	,	,	PUNCT
ejpam-6802	159	17	ϑℓ+1)(α(q0	ϑℓ+1)(α(q0	NOUN
ejpam-6802	159	18	,	,	PUNCT
ejpam-6802	159	19	ϑ0	ϑ0	NOUN
ejpam-6802	159	20	)	)	PUNCT
ejpam-6802	159	21	)	)	PUNCT
ejpam-6802	159	22	(	(	PUNCT
ejpam-6802	159	23	λ2ℓ+2+λ2ℓ+1	λ2ℓ+2+λ2ℓ+1	X
ejpam-6802	159	24	)	)	PUNCT
ejpam-6802	159	25	≤	≤	NOUN
ejpam-6802	159	26	α(qℓ+p	α(qℓ+p	NOUN
ejpam-6802	159	27	,	,	PUNCT
ejpam-6802	159	28	ϑℓ+2)α(qℓ+2	ϑℓ+2)α(qℓ+2	NOUN
ejpam-6802	159	29	,	,	PUNCT
ejpam-6802	159	30	ϑℓ+2)α(qℓ+2	ϑℓ+2)α(qℓ+2	X
ejpam-6802	159	31	,	,	PUNCT
ejpam-6802	159	32	ϑℓ+1	ϑℓ+1	NUM
ejpam-6802	159	33	)	)	PUNCT
ejpam-6802	159	34	(	(	PUNCT
ejpam-6802	159	35	α(q0	α(q0	NOUN
ejpam-6802	159	36	,	,	PUNCT
ejpam-6802	159	37	ϑ0	ϑ0	NOUN
ejpam-6802	159	38	)	)	PUNCT
ejpam-6802	159	39	)	)	PUNCT
ejpam-6802	159	40	(	(	PUNCT
ejpam-6802	159	41	λ2ℓ+2+λ2ℓ+1	λ2ℓ+2+λ2ℓ+1	X
ejpam-6802	159	42	)	)	PUNCT
ejpam-6802	159	43	≤	≤	NOUN
ejpam-6802	159	44	α(qℓ+p	α(qℓ+p	NOUN
ejpam-6802	159	45	,	,	PUNCT
ejpam-6802	159	46	ϑℓ+2)(α(q0	ϑℓ+2)(α(q0	NOUN
ejpam-6802	159	47	,	,	PUNCT
ejpam-6802	159	48	ϑ0	ϑ0	NOUN
ejpam-6802	159	49	)	)	PUNCT
ejpam-6802	159	50	)	)	PUNCT
ejpam-6802	159	51	(	(	PUNCT
ejpam-6802	159	52	λ2ℓ+4+λ2ℓ+3+λ2ℓ+2+λ2ℓ+1	λ2ℓ+4+λ2ℓ+3+λ2ℓ+2+λ2ℓ+1	PROPN
ejpam-6802	159	53	)	)	PUNCT
ejpam-6802	159	54	...	...	PUNCT
ejpam-6802	160	1	≤	≤	NUM
ejpam-6802	160	2	α(qℓ+p	α(qℓ+p	PROPN
ejpam-6802	160	3	,	,	PUNCT
ejpam-6802	160	4	ϑℓ+p−1)(α(q0	ϑℓ+p−1)(α(q0	NOUN
ejpam-6802	160	5	,	,	PUNCT
ejpam-6802	160	6	ϑ0	ϑ0	NOUN
ejpam-6802	160	7	)	)	PUNCT
ejpam-6802	160	8	)	)	PUNCT
ejpam-6802	160	9	(	(	PUNCT
ejpam-6802	160	10	λ2ℓ+2p−2+	λ2ℓ+2p−2+	X
ejpam-6802	160	11	....	....	PUNCT
ejpam-6802	160	12	+λ2ℓ+1	+λ2ℓ+1	NUM
ejpam-6802	160	13	)	)	PUNCT
ejpam-6802	160	14	≤	≤	NOUN
ejpam-6802	160	15	(	(	PUNCT
ejpam-6802	160	16	α(q0	α(q0	NOUN
ejpam-6802	160	17	,	,	PUNCT
ejpam-6802	160	18	ϑ0	ϑ0	NOUN
ejpam-6802	160	19	)	)	PUNCT
ejpam-6802	160	20	)	)	PUNCT
ejpam-6802	160	21	(	(	PUNCT
ejpam-6802	160	22	λ2ℓ+2p−1+λ2ℓ+2p−2+λ2ℓ+2p−3+	λ2ℓ+2p−1+λ2ℓ+2p−2+λ2ℓ+2p−3+	PROPN
ejpam-6802	160	23	....	....	X
ejpam-6802	160	24	+λ2ℓ+1	+λ2ℓ+1	ADJ
ejpam-6802	160	25	)	)	PUNCT
ejpam-6802	160	26	≤	≤	NOUN
ejpam-6802	160	27	(	(	PUNCT
ejpam-6802	160	28	α(q0	α(q0	NOUN
ejpam-6802	160	29	,	,	PUNCT
ejpam-6802	160	30	ϑ0	ϑ0	NOUN
ejpam-6802	160	31	)	)	PUNCT
ejpam-6802	160	32	)	)	PUNCT
ejpam-6802	161	1	λ2ℓ+1	λ2ℓ+1	PROPN
ejpam-6802	161	2	1−λ	1−λ	NUM
ejpam-6802	161	3	.	.	PUNCT
ejpam-6802	162	1	α(qℓ	α(qℓ	PROPN
ejpam-6802	162	2	,	,	PUNCT
ejpam-6802	162	3	ϑℓ+p	ϑℓ+p	PROPN
ejpam-6802	162	4	)	)	PUNCT
ejpam-6802	162	5	≤	≤	NUM
ejpam-6802	162	6	α(qℓ	α(qℓ	PROPN
ejpam-6802	162	7	,	,	PUNCT
ejpam-6802	162	8	ϑℓ)α(qℓ+1	ϑℓ)α(qℓ+1	ADV
ejpam-6802	162	9	,	,	PUNCT
ejpam-6802	162	10	ϑℓ)α(qℓ+1	ϑℓ)α(qℓ+1	ADV
ejpam-6802	162	11	,	,	PUNCT
ejpam-6802	162	12	ϑℓ+p	ϑℓ+p	PROPN
ejpam-6802	162	13	)	)	PUNCT
ejpam-6802	162	14	≤	≤	NOUN
ejpam-6802	162	15	(	(	PUNCT
ejpam-6802	162	16	α(q0	α(q0	NOUN
ejpam-6802	162	17	,	,	PUNCT
ejpam-6802	162	18	ϑ0	ϑ0	NOUN
ejpam-6802	162	19	)	)	PUNCT
ejpam-6802	162	20	)	)	PUNCT
ejpam-6802	162	21	(	(	PUNCT
ejpam-6802	162	22	λ2ℓ+λ2ℓ+1)α(qℓ+1	λ2ℓ+λ2ℓ+1)α(qℓ+1	X
ejpam-6802	162	23	,	,	PUNCT
ejpam-6802	162	24	ϑℓ+p	ϑℓ+p	NUM
ejpam-6802	162	25	)	)	PUNCT
ejpam-6802	162	26	≤	≤	NOUN
ejpam-6802	162	27	(	(	PUNCT
ejpam-6802	162	28	α(q0	α(q0	NOUN
ejpam-6802	162	29	,	,	PUNCT
ejpam-6802	162	30	ϑ0	ϑ0	NOUN
ejpam-6802	162	31	)	)	PUNCT
ejpam-6802	162	32	)	)	PUNCT
ejpam-6802	162	33	(	(	PUNCT
ejpam-6802	162	34	λ2ℓ+λ2ℓ+1)α(qℓ+1	λ2ℓ+λ2ℓ+1)α(qℓ+1	X
ejpam-6802	162	35	,	,	PUNCT
ejpam-6802	162	36	ϑℓ+1)α(qℓ+2	ϑℓ+1)α(qℓ+2	NOUN
ejpam-6802	162	37	,	,	PUNCT
ejpam-6802	162	38	ϑℓ+1	ϑℓ+1	NUM
ejpam-6802	162	39	)	)	PUNCT
ejpam-6802	162	40	α(qℓ+2	α(qℓ+2	PROPN
ejpam-6802	162	41	,	,	PUNCT
ejpam-6802	162	42	ϑℓ+p	ϑℓ+p	NUM
ejpam-6802	162	43	)	)	PUNCT
ejpam-6802	162	44	≤	≤	NOUN
ejpam-6802	162	45	(	(	PUNCT
ejpam-6802	162	46	α(q0	α(q0	NOUN
ejpam-6802	162	47	,	,	PUNCT
ejpam-6802	162	48	ϑ0	ϑ0	NOUN
ejpam-6802	162	49	)	)	PUNCT
ejpam-6802	162	50	)	)	PUNCT
ejpam-6802	162	51	(	(	PUNCT
ejpam-6802	162	52	λ2ℓ+λ2ℓ+1+λ2ℓ+2+λ2ℓ+3)α(qℓ+2	λ2ℓ+λ2ℓ+1+λ2ℓ+2+λ2ℓ+3)α(qℓ+2	NOUN
ejpam-6802	162	53	,	,	PUNCT
ejpam-6802	162	54	ϑℓ+p	ϑℓ+p	PROPN
ejpam-6802	162	55	)	)	PUNCT
ejpam-6802	162	56	r.	r.	PROPN
ejpam-6802	162	57	ramaswamy	ramaswamy	PROPN
ejpam-6802	162	58	/	/	SYM
ejpam-6802	162	59	eur	eur	PROPN
ejpam-6802	162	60	.	.	PUNCT
ejpam-6802	163	1	j.	j.	PROPN
ejpam-6802	163	2	pure	pure	PROPN
ejpam-6802	163	3	appl	appl	PROPN
ejpam-6802	163	4	.	.	PROPN
ejpam-6802	163	5	math	math	PROPN
ejpam-6802	163	6	,	,	PUNCT
ejpam-6802	163	7	18	18	NUM
ejpam-6802	163	8	(	(	PUNCT
ejpam-6802	163	9	4	4	NUM
ejpam-6802	163	10	)	)	PUNCT
ejpam-6802	163	11	(	(	PUNCT
ejpam-6802	163	12	2025	2025	NUM
ejpam-6802	163	13	)	)	PUNCT
ejpam-6802	163	14	,	,	PUNCT
ejpam-6802	163	15	6802	6802	NUM
ejpam-6802	163	16	8	8	NUM
ejpam-6802	163	17	of	of	ADP
ejpam-6802	163	18	20	20	NUM
ejpam-6802	163	19	...	...	PUNCT
ejpam-6802	163	20	≤	≤	NUM
ejpam-6802	163	21	(	(	PUNCT
ejpam-6802	163	22	α(q0ϑ0	α(q0ϑ0	NOUN
ejpam-6802	163	23	)	)	PUNCT
ejpam-6802	163	24	)	)	PUNCT
ejpam-6802	163	25	(	(	PUNCT
ejpam-6802	163	26	λ2ℓ+λ2ℓ+1+···+λ2ℓ+2p−1)α(qℓ+p	λ2ℓ+λ2ℓ+1+···+λ2ℓ+2p−1)α(qℓ+p	PROPN
ejpam-6802	163	27	,	,	PUNCT
ejpam-6802	163	28	ϑℓ+p	ϑℓ+p	PROPN
ejpam-6802	163	29	)	)	PUNCT
ejpam-6802	163	30	≤	≤	NOUN
ejpam-6802	163	31	α(q0	α(q0	NUM
ejpam-6802	163	32	,	,	PUNCT
ejpam-6802	163	33	ϑ0	ϑ0	PROPN
ejpam-6802	163	34	)	)	PUNCT
ejpam-6802	163	35	(	(	PUNCT
ejpam-6802	163	36	λ2ℓ+λ2ℓ+1+···+λ2ℓ+2p−1+λ2ℓ+2p	λ2ℓ+λ2ℓ+1+···+λ2ℓ+2p−1+λ2ℓ+2p	NOUN
ejpam-6802	163	37	)	)	PUNCT
ejpam-6802	163	38	≤	≤	NOUN
ejpam-6802	163	39	(	(	PUNCT
ejpam-6802	163	40	α(q0	α(q0	NOUN
ejpam-6802	163	41	,	,	PUNCT
ejpam-6802	163	42	ϑ0	ϑ0	NOUN
ejpam-6802	163	43	)	)	PUNCT
ejpam-6802	163	44	)	)	PUNCT
ejpam-6802	164	1	λ2ℓ	λ2ℓ	PROPN
ejpam-6802	164	2	1−λ	1−λ	NUM
ejpam-6802	164	3	.	.	PUNCT
ejpam-6802	165	1	now	now	ADV
ejpam-6802	165	2	,	,	PUNCT
ejpam-6802	165	3	α(qℓ	α(qℓ	PROPN
ejpam-6802	165	4	,	,	PUNCT
ejpam-6802	165	5	ϑr	ϑr	PROPN
ejpam-6802	165	6	)	)	PUNCT
ejpam-6802	165	7	≤	≤	NUM
ejpam-6802	165	8	α(qℓ	α(qℓ	PROPN
ejpam-6802	165	9	,	,	PUNCT
ejpam-6802	165	10	ϑℓ0)α(qℓ0	ϑℓ0)α(qℓ0	NOUN
ejpam-6802	165	11	,	,	PUNCT
ejpam-6802	165	12	ϑℓ0)α(qℓ0	ϑℓ0)α(qℓ0	NOUN
ejpam-6802	165	13	,	,	PUNCT
ejpam-6802	165	14	ϑr	ϑr	PROPN
ejpam-6802	165	15	)	)	PUNCT
ejpam-6802	165	16	≤	≤	NOUN
ejpam-6802	165	17	(	(	PUNCT
ejpam-6802	165	18	α(q0	α(q0	NOUN
ejpam-6802	165	19	,	,	PUNCT
ejpam-6802	165	20	ϑ0	ϑ0	NOUN
ejpam-6802	165	21	)	)	PUNCT
ejpam-6802	165	22	)	)	PUNCT
ejpam-6802	165	23	3	3	NUM
ejpam-6802	165	24	λ2ℓ	λ2ℓ	PROPN
ejpam-6802	165	25	1−λ	1−λ	NUM
ejpam-6802	165	26	.	.	PUNCT
ejpam-6802	166	1	therefore	therefore	ADV
ejpam-6802	166	2	,	,	PUNCT
ejpam-6802	166	3	α(qℓ	α(qℓ	PROPN
ejpam-6802	166	4	,	,	PUNCT
ejpam-6802	166	5	ϑr	ϑr	PROPN
ejpam-6802	166	6	)	)	PUNCT
ejpam-6802	166	7	→	→	SYM
ejpam-6802	166	8	0(ℓ	0(ℓ	NUM
ejpam-6802	166	9	,	,	PUNCT
ejpam-6802	166	10	r	r	NOUN
ejpam-6802	166	11	→	→	SYM
ejpam-6802	166	12	+	+	CCONJ
ejpam-6802	166	13	+	+	X
ejpam-6802	166	14	∞	∞	NUM
ejpam-6802	166	15	)	)	PUNCT
ejpam-6802	166	16	.	.	PUNCT
ejpam-6802	167	1	therefore	therefore	ADV
ejpam-6802	167	2	,	,	PUNCT
ejpam-6802	167	3	(	(	PUNCT
ejpam-6802	167	4	{	{	PUNCT
ejpam-6802	167	5	qℓ	qℓ	NOUN
ejpam-6802	167	6	}	}	PUNCT
ejpam-6802	167	7	,	,	PUNCT
ejpam-6802	167	8	{	{	PUNCT
ejpam-6802	167	9	ϑℓ	ϑℓ	NOUN
ejpam-6802	167	10	}	}	PUNCT
ejpam-6802	167	11	)	)	PUNCT
ejpam-6802	167	12	is	be	AUX
ejpam-6802	167	13	a	a	DET
ejpam-6802	167	14	cauchy	cauchy	ADJ
ejpam-6802	167	15	bisequence	bisequence	NOUN
ejpam-6802	167	16	.	.	PUNCT
ejpam-6802	168	1	since	since	SCONJ
ejpam-6802	168	2	(	(	PUNCT
ejpam-6802	168	3	b	b	X
ejpam-6802	168	4	,	,	PUNCT
ejpam-6802	168	5	f	f	PROPN
ejpam-6802	168	6	,	,	PUNCT
ejpam-6802	168	7	α	α	PROPN
ejpam-6802	168	8	)	)	PUNCT
ejpam-6802	168	9	is	be	AUX
ejpam-6802	168	10	complete	complete	ADJ
ejpam-6802	168	11	,	,	PUNCT
ejpam-6802	168	12	(	(	PUNCT
ejpam-6802	168	13	{	{	PUNCT
ejpam-6802	168	14	qℓ	qℓ	NOUN
ejpam-6802	168	15	}	}	PUNCT
ejpam-6802	168	16	,	,	PUNCT
ejpam-6802	168	17	{	{	PUNCT
ejpam-6802	168	18	ϑℓ	ϑℓ	NOUN
ejpam-6802	168	19	}	}	PUNCT
ejpam-6802	168	20	)	)	PUNCT
ejpam-6802	168	21	converges	converge	NOUN
ejpam-6802	168	22	,	,	PUNCT
ejpam-6802	168	23	then	then	ADV
ejpam-6802	168	24	{	{	PUNCT
ejpam-6802	168	25	qℓ	qℓ	NOUN
ejpam-6802	168	26	}	}	PUNCT
ejpam-6802	168	27	→	→	SYM
ejpam-6802	168	28	x	x	X
ejpam-6802	168	29	,	,	PUNCT
ejpam-6802	168	30	{	{	PUNCT
ejpam-6802	168	31	ϑℓ	ϑℓ	X
ejpam-6802	168	32	}	}	PUNCT
ejpam-6802	168	33	→	→	SYM
ejpam-6802	168	34	x	x	X
ejpam-6802	168	35	,	,	PUNCT
ejpam-6802	168	36	where	where	SCONJ
ejpam-6802	168	37	x	x	PUNCT
ejpam-6802	168	38	∈	∈	PROPN
ejpam-6802	168	39	b	b	PROPN
ejpam-6802	168	40	∩	∩	PROPN
ejpam-6802	168	41	f	f	X
ejpam-6802	168	42	.	.	PUNCT
ejpam-6802	169	1	since	since	SCONJ
ejpam-6802	169	2	the	the	DET
ejpam-6802	169	3	contravariant	contravariant	PROPN
ejpam-6802	169	4	mapping	mapping	PROPN
ejpam-6802	169	5	ω	ω	PROPN
ejpam-6802	169	6	is	be	AUX
ejpam-6802	169	7	continuous	continuous	ADJ
ejpam-6802	169	8	{	{	PUNCT
ejpam-6802	169	9	qn	qn	NOUN
ejpam-6802	169	10	}	}	PUNCT
ejpam-6802	169	11	→	→	SYM
ejpam-6802	169	12	x	x	X
ejpam-6802	169	13	,	,	PUNCT
ejpam-6802	169	14	⇒	⇒	PROPN
ejpam-6802	169	15	{	{	PUNCT
ejpam-6802	169	16	ϑℓ	ϑℓ	PROPN
ejpam-6802	169	17	}	}	PUNCT
ejpam-6802	169	18	=	=	PUNCT
ejpam-6802	169	19	{	{	PUNCT
ejpam-6802	169	20	ω(qℓ	ω(qℓ	NUM
ejpam-6802	169	21	)	)	PUNCT
ejpam-6802	169	22	}	}	PUNCT
ejpam-6802	169	23	→	→	SYM
ejpam-6802	169	24	ω(x	ω(x	X
ejpam-6802	169	25	)	)	PUNCT
ejpam-6802	169	26	and	and	CCONJ
ejpam-6802	169	27	combining	combine	VERB
ejpam-6802	169	28	this	this	PRON
ejpam-6802	169	29	with	with	ADP
ejpam-6802	169	30	{	{	PUNCT
ejpam-6802	169	31	ϑℓ	ϑℓ	NOUN
ejpam-6802	169	32	}	}	PUNCT
ejpam-6802	169	33	→	→	SYM
ejpam-6802	169	34	x	x	PUNCT
ejpam-6802	169	35	gives	give	VERB
ejpam-6802	169	36	ω(x	ω(x	NOUN
ejpam-6802	169	37	)	)	PUNCT
ejpam-6802	169	38	=	=	SYM
ejpam-6802	169	39	x.	x.	NOUN
ejpam-6802	169	40	suppose	suppose	VERB
ejpam-6802	169	41	z	z	NOUN
ejpam-6802	169	42	is	be	AUX
ejpam-6802	169	43	a	a	DET
ejpam-6802	169	44	fp	fp	PROPN
ejpam-6802	169	45	of	of	ADP
ejpam-6802	169	46	ω	ω	PROPN
ejpam-6802	169	47	,	,	PUNCT
ejpam-6802	169	48	then	then	ADV
ejpam-6802	169	49	ω(z	ω(z	PUNCT
ejpam-6802	169	50	)	)	PUNCT
ejpam-6802	169	51	=	=	SYM
ejpam-6802	169	52	z	z	NOUN
ejpam-6802	169	53	implies	imply	VERB
ejpam-6802	169	54	z	z	PROPN
ejpam-6802	169	55	∈	∈	PROPN
ejpam-6802	169	56	b	b	PROPN
ejpam-6802	169	57	∩	∩	X
ejpam-6802	169	58	f	f	PROPN
ejpam-6802	169	59	so	so	SCONJ
ejpam-6802	169	60	that	that	SCONJ
ejpam-6802	169	61	α(x	α(x	NOUN
ejpam-6802	169	62	,	,	PUNCT
ejpam-6802	169	63	z	z	NOUN
ejpam-6802	169	64	)	)	PUNCT
ejpam-6802	169	65	=	=	SYM
ejpam-6802	169	66	α(ω(x),ω(z	α(ω(x),ω(z	PROPN
ejpam-6802	169	67	)	)	PUNCT
ejpam-6802	169	68	)	)	PUNCT
ejpam-6802	169	69	≤	≤	NOUN
ejpam-6802	169	70	(	(	PUNCT
ejpam-6802	169	71	α(x	α(x	NOUN
ejpam-6802	169	72	,	,	PUNCT
ejpam-6802	169	73	z))λ	z))λ	PROPN
ejpam-6802	169	74	,	,	PUNCT
ejpam-6802	169	75	which	which	PRON
ejpam-6802	169	76	gives	give	VERB
ejpam-6802	169	77	α(x	α(x	PROPN
ejpam-6802	169	78	,	,	PUNCT
ejpam-6802	169	79	z	z	NOUN
ejpam-6802	169	80	)	)	PUNCT
ejpam-6802	170	1	=	=	SYM
ejpam-6802	170	2	1	1	X
ejpam-6802	170	3	.	.	PUNCT
ejpam-6802	170	4	hence	hence	ADV
ejpam-6802	170	5	x	x	X
ejpam-6802	170	6	=	=	PUNCT
ejpam-6802	170	7	z.	z.	PROPN
ejpam-6802	170	8	example	example	NOUN
ejpam-6802	170	9	4	4	X
ejpam-6802	170	10	.	.	PUNCT
ejpam-6802	171	1	let	let	VERB
ejpam-6802	171	2	a	a	DET
ejpam-6802	171	3	=	=	SYM
ejpam-6802	171	4	r	r	NOUN
ejpam-6802	171	5	,	,	PUNCT
ejpam-6802	171	6	z	z	NOUN
ejpam-6802	171	7	=	=	PRON
ejpam-6802	171	8	{	{	PUNCT
ejpam-6802	171	9	q	q	NOUN
ejpam-6802	171	10	∈	∈	PROPN
ejpam-6802	171	11	a|q	a|q	PROPN
ejpam-6802	171	12	≥	≥	NOUN
ejpam-6802	171	13	0	0	NUM
ejpam-6802	171	14	}	}	PUNCT
ejpam-6802	171	15	.	.	PUNCT
ejpam-6802	172	1	let	let	VERB
ejpam-6802	172	2	b	b	NOUN
ejpam-6802	172	3	=	=	PRON
ejpam-6802	172	4	{	{	PUNCT
ejpam-6802	172	5	0	0	NUM
ejpam-6802	172	6	,	,	PUNCT
ejpam-6802	172	7	1	1	NUM
ejpam-6802	172	8	,	,	PUNCT
ejpam-6802	172	9	2	2	NUM
ejpam-6802	172	10	,	,	PUNCT
ejpam-6802	172	11	7	7	NUM
ejpam-6802	172	12	}	}	PUNCT
ejpam-6802	172	13	and	and	CCONJ
ejpam-6802	172	14	f	f	X
ejpam-6802	172	15	=	=	PUNCT
ejpam-6802	172	16	{	{	PUNCT
ejpam-6802	172	17	0	0	NUM
ejpam-6802	172	18	,	,	PUNCT
ejpam-6802	172	19	14	14	NUM
ejpam-6802	172	20	,	,	PUNCT
ejpam-6802	172	21	1	1	NUM
ejpam-6802	172	22	2	2	NUM
ejpam-6802	172	23	,	,	PUNCT
ejpam-6802	172	24	3	3	NUM
ejpam-6802	172	25	}	}	PUNCT
ejpam-6802	172	26	be	be	AUX
ejpam-6802	172	27	equipped	equip	VERB
ejpam-6802	172	28	with	with	ADP
ejpam-6802	172	29	α(q	α(q	PROPN
ejpam-6802	172	30	,	,	PUNCT
ejpam-6802	172	31	ϑ	ϑ	NOUN
ejpam-6802	172	32	)	)	PUNCT
ejpam-6802	173	1	=	=	SYM
ejpam-6802	173	2	e|q−ϑ|	e|q−ϑ|	PROPN
ejpam-6802	173	3	for	for	ADP
ejpam-6802	173	4	all	all	DET
ejpam-6802	173	5	q	q	PROPN
ejpam-6802	173	6	∈	∈	PROPN
ejpam-6802	173	7	b	b	PROPN
ejpam-6802	173	8	,	,	PUNCT
ejpam-6802	173	9	ϑ	ϑ	X
ejpam-6802	173	10	∈	∈	PROPN
ejpam-6802	173	11	f	f	X
ejpam-6802	173	12	.	.	PUNCT
ejpam-6802	174	1	then	then	ADV
ejpam-6802	174	2	,	,	PUNCT
ejpam-6802	174	3	(	(	PUNCT
ejpam-6802	174	4	b	b	X
ejpam-6802	174	5	,	,	PUNCT
ejpam-6802	174	6	f	f	PROPN
ejpam-6802	174	7	,	,	PUNCT
ejpam-6802	174	8	α	α	PROPN
ejpam-6802	174	9	)	)	PUNCT
ejpam-6802	174	10	is	be	AUX
ejpam-6802	174	11	a	a	DET
ejpam-6802	174	12	complete	complete	ADJ
ejpam-6802	174	13	mcbms	mcbms	NOUN
ejpam-6802	174	14	.	.	PUNCT
ejpam-6802	175	1	also	also	ADV
ejpam-6802	175	2	define	define	VERB
ejpam-6802	175	3	ω	ω	NOUN
ejpam-6802	175	4	:	:	PUNCT
ejpam-6802	175	5	b	b	X
ejpam-6802	175	6	∪	∪	X
ejpam-6802	175	7	f	f	PROPN
ejpam-6802	175	8	⇄	⇄	PROPN
ejpam-6802	175	9	b	b	PROPN
ejpam-6802	175	10	∪	∪	ADJ
ejpam-6802	175	11	f	f	NOUN
ejpam-6802	175	12	as	as	ADP
ejpam-6802	175	13	ω(q	ω(q	NOUN
ejpam-6802	175	14	)	)	PUNCT
ejpam-6802	176	1	=	=	PRON
ejpam-6802	176	2	{	{	PUNCT
ejpam-6802	176	3	1	1	NUM
ejpam-6802	176	4	4	4	NUM
ejpam-6802	176	5	,	,	PUNCT
ejpam-6802	176	6	if	if	SCONJ
ejpam-6802	176	7	q	q	X
ejpam-6802	176	8	∈	∈	PROPN
ejpam-6802	176	9	{	{	PUNCT
ejpam-6802	176	10	2	2	NUM
ejpam-6802	176	11	,	,	PUNCT
ejpam-6802	176	12	7	7	NUM
ejpam-6802	176	13	}	}	PUNCT
ejpam-6802	176	14	,	,	PUNCT
ejpam-6802	176	15	0	0	NUM
ejpam-6802	176	16	,	,	PUNCT
ejpam-6802	176	17	if	if	SCONJ
ejpam-6802	176	18	q	q	X
ejpam-6802	176	19	∈	∈	PROPN
ejpam-6802	176	20	{	{	PUNCT
ejpam-6802	176	21	0	0	NUM
ejpam-6802	176	22	,	,	PUNCT
ejpam-6802	176	23	14	14	NUM
ejpam-6802	176	24	,	,	PUNCT
ejpam-6802	176	25	1	1	NUM
ejpam-6802	176	26	2	2	NUM
ejpam-6802	176	27	,	,	PUNCT
ejpam-6802	176	28	1	1	NUM
ejpam-6802	176	29	,	,	PUNCT
ejpam-6802	176	30	3	3	NUM
ejpam-6802	176	31	}	}	PUNCT
ejpam-6802	176	32	,	,	PUNCT
ejpam-6802	176	33	for	for	ADP
ejpam-6802	176	34	all	all	DET
ejpam-6802	176	35	q	q	PROPN
ejpam-6802	176	36	∈	∈	PROPN
ejpam-6802	176	37	b	b	X
ejpam-6802	176	38	∪	∪	PROPN
ejpam-6802	176	39	f	f	PROPN
ejpam-6802	176	40	.	.	PUNCT
ejpam-6802	177	1	let	let	VERB
ejpam-6802	177	2	q	q	PROPN
ejpam-6802	177	3	∈	∈	PROPN
ejpam-6802	177	4	b	b	PROPN
ejpam-6802	177	5	and	and	CCONJ
ejpam-6802	177	6	ϑ	ϑ	PROPN
ejpam-6802	177	7	∈	∈	PROPN
ejpam-6802	177	8	f	f	NOUN
ejpam-6802	177	9	,	,	PUNCT
ejpam-6802	177	10	then	then	ADV
ejpam-6802	177	11	we	we	PRON
ejpam-6802	177	12	get	get	VERB
ejpam-6802	177	13	α(ωq	α(ωq	NOUN
ejpam-6802	177	14	,	,	PUNCT
ejpam-6802	177	15	ωϑ	ωϑ	NOUN
ejpam-6802	177	16	)	)	PUNCT
ejpam-6802	177	17	≤	≤	NOUN
ejpam-6802	177	18	(	(	PUNCT
ejpam-6802	177	19	α(q	α(q	PROPN
ejpam-6802	177	20	,	,	PUNCT
ejpam-6802	177	21	ϑ	ϑ	NOUN
ejpam-6802	177	22	)	)	PUNCT
ejpam-6802	177	23	)	)	PUNCT
ejpam-6802	177	24	1	1	NUM
ejpam-6802	177	25	2	2	NUM
ejpam-6802	177	26	.	.	PUNCT
ejpam-6802	178	1	therefore	therefore	ADV
ejpam-6802	178	2	,	,	PUNCT
ejpam-6802	178	3	criteria	criterion	NOUN
ejpam-6802	178	4	of	of	ADP
ejpam-6802	178	5	theorem	theorem	ADJ
ejpam-6802	178	6	3.2	3.2	NUM
ejpam-6802	178	7	are	be	AUX
ejpam-6802	178	8	verified	verify	VERB
ejpam-6802	178	9	&	&	CCONJ
ejpam-6802	178	10	ω	ω	PROPN
ejpam-6802	178	11	possesses	possess	VERB
ejpam-6802	178	12	a	a	DET
ejpam-6802	178	13	ufp	ufp	NOUN
ejpam-6802	178	14	q	q	NOUN
ejpam-6802	178	15	=	=	SYM
ejpam-6802	178	16	0	0	X
ejpam-6802	178	17	.	.	PUNCT
ejpam-6802	179	1	finally	finally	ADV
ejpam-6802	179	2	,	,	PUNCT
ejpam-6802	179	3	we	we	PRON
ejpam-6802	179	4	express	express	VERB
ejpam-6802	179	5	a	a	DET
ejpam-6802	179	6	theorem	theorem	NOUN
ejpam-6802	179	7	based	base	VERB
ejpam-6802	179	8	of	of	ADP
ejpam-6802	179	9	kannan	kannan	PROPN
ejpam-6802	179	10	’s	’s	PART
ejpam-6802	179	11	fp	fp	PROPN
ejpam-6802	179	12	result	result	NOUN
ejpam-6802	179	13	[	[	X
ejpam-6802	179	14	2	2	NUM
ejpam-6802	179	15	]	]	PUNCT
ejpam-6802	179	16	.	.	PUNCT
ejpam-6802	180	1	r.	r.	PROPN
ejpam-6802	180	2	ramaswamy	ramaswamy	PROPN
ejpam-6802	180	3	/	/	SYM
ejpam-6802	180	4	eur	eur	PROPN
ejpam-6802	180	5	.	.	PUNCT
ejpam-6802	181	1	j.	j.	PROPN
ejpam-6802	181	2	pure	pure	PROPN
ejpam-6802	181	3	appl	appl	PROPN
ejpam-6802	181	4	.	.	PROPN
ejpam-6802	181	5	math	math	PROPN
ejpam-6802	181	6	,	,	PUNCT
ejpam-6802	181	7	18	18	NUM
ejpam-6802	181	8	(	(	PUNCT
ejpam-6802	181	9	4	4	NUM
ejpam-6802	181	10	)	)	PUNCT
ejpam-6802	181	11	(	(	PUNCT
ejpam-6802	181	12	2025	2025	NUM
ejpam-6802	181	13	)	)	PUNCT
ejpam-6802	181	14	,	,	PUNCT
ejpam-6802	181	15	6802	6802	NUM
ejpam-6802	181	16	9	9	NUM
ejpam-6802	181	17	of	of	ADP
ejpam-6802	181	18	20	20	NUM
ejpam-6802	181	19	theorem	theorem	VERB
ejpam-6802	181	20	3.3	3.3	NUM
ejpam-6802	181	21	.	.	PUNCT
ejpam-6802	182	1	consider	consider	VERB
ejpam-6802	182	2	ω	ω	NOUN
ejpam-6802	182	3	:	:	PUNCT
ejpam-6802	182	4	(	(	PUNCT
ejpam-6802	182	5	b	b	X
ejpam-6802	182	6	,	,	PUNCT
ejpam-6802	182	7	f	f	PROPN
ejpam-6802	182	8	,	,	PUNCT
ejpam-6802	182	9	α	α	NOUN
ejpam-6802	182	10	)	)	PUNCT
ejpam-6802	182	11	⇆	⇆	PROPN
ejpam-6802	182	12	(	(	PUNCT
ejpam-6802	182	13	b	b	NOUN
ejpam-6802	182	14	,	,	PUNCT
ejpam-6802	182	15	f	f	PROPN
ejpam-6802	182	16	,	,	PUNCT
ejpam-6802	182	17	α	α	PROPN
ejpam-6802	182	18	)	)	PUNCT
ejpam-6802	182	19	,	,	PUNCT
ejpam-6802	182	20	where	where	SCONJ
ejpam-6802	182	21	(	(	PUNCT
ejpam-6802	182	22	b	b	NOUN
ejpam-6802	182	23	,	,	PUNCT
ejpam-6802	182	24	f	f	PROPN
ejpam-6802	182	25	,	,	PUNCT
ejpam-6802	182	26	α	α	PROPN
ejpam-6802	182	27	)	)	PUNCT
ejpam-6802	182	28	is	be	AUX
ejpam-6802	182	29	a	a	DET
ejpam-6802	182	30	complete	complete	ADJ
ejpam-6802	182	31	mcbms	mcbms	NOUN
ejpam-6802	182	32	,	,	PUNCT
ejpam-6802	182	33	z	z	PROPN
ejpam-6802	182	34	be	be	AUX
ejpam-6802	182	35	a	a	DET
ejpam-6802	182	36	cone	cone	NOUN
ejpam-6802	182	37	with	with	ADP
ejpam-6802	182	38	constant	constant	ADJ
ejpam-6802	182	39	w	w	NOUN
ejpam-6802	182	40	and	and	CCONJ
ejpam-6802	182	41	let	let	VERB
ejpam-6802	182	42	β	β	X
ejpam-6802	182	43	∈	∈	PROPN
ejpam-6802	182	44	(	(	PUNCT
ejpam-6802	182	45	0	0	NUM
ejpam-6802	182	46	,	,	PUNCT
ejpam-6802	182	47	12	12	NUM
ejpam-6802	182	48	)	)	PUNCT
ejpam-6802	182	49	s.t	s.t	PROPN
ejpam-6802	182	50	the	the	DET
ejpam-6802	182	51	inequality	inequality	NOUN
ejpam-6802	182	52	α(ωϑ,ωq	α(ωϑ,ωq	PROPN
ejpam-6802	182	53	)	)	PUNCT
ejpam-6802	182	54	≤	≤	NOUN
ejpam-6802	182	55	(	(	PUNCT
ejpam-6802	182	56	α(q	α(q	PROPN
ejpam-6802	182	57	,	,	PUNCT
ejpam-6802	182	58	ωq)α(ωϑ	ωq)α(ωϑ	PROPN
ejpam-6802	182	59	,	,	PUNCT
ejpam-6802	182	60	ϑ))β	ϑ))β	NOUN
ejpam-6802	182	61	holds	hold	VERB
ejpam-6802	182	62	for	for	ADP
ejpam-6802	182	63	all	all	DET
ejpam-6802	182	64	q	q	PROPN
ejpam-6802	182	65	∈	∈	PROPN
ejpam-6802	182	66	b	b	PROPN
ejpam-6802	182	67	and	and	CCONJ
ejpam-6802	182	68	ϑ	ϑ	X
ejpam-6802	182	69	∈	∈	PROPN
ejpam-6802	182	70	f	f	X
ejpam-6802	182	71	.	.	PUNCT
ejpam-6802	183	1	then	then	ADV
ejpam-6802	183	2	the	the	DET
ejpam-6802	183	3	function	function	PROPN
ejpam-6802	183	4	ω	ω	NOUN
ejpam-6802	183	5	:	:	PUNCT
ejpam-6802	183	6	b∪f	b∪f	PROPN
ejpam-6802	183	7	→	→	SYM
ejpam-6802	183	8	b	b	X
ejpam-6802	183	9	∪	∪	X
ejpam-6802	183	10	f	f	PROPN
ejpam-6802	183	11	possesses	possess	VERB
ejpam-6802	183	12	a	a	DET
ejpam-6802	183	13	ufp	ufp	NOUN
ejpam-6802	183	14	.	.	PUNCT
ejpam-6802	184	1	proof	proof	NOUN
ejpam-6802	184	2	.	.	PUNCT
ejpam-6802	185	1	consider	consider	VERB
ejpam-6802	185	2	q0	q0	PROPN
ejpam-6802	185	3	∈	∈	PROPN
ejpam-6802	185	4	b	b	PROPN
ejpam-6802	185	5	,	,	PUNCT
ejpam-6802	185	6	for	for	ADP
ejpam-6802	185	7	each	each	DET
ejpam-6802	185	8	positive	positive	ADJ
ejpam-6802	185	9	integer	integer	NOUN
ejpam-6802	185	10	ℓ	ℓ	NOUN
ejpam-6802	185	11	,	,	PUNCT
ejpam-6802	185	12	we	we	PRON
ejpam-6802	185	13	define	define	VERB
ejpam-6802	185	14	ϑℓ	ϑℓ	PROPN
ejpam-6802	185	15	=	=	SYM
ejpam-6802	185	16	ωqℓ	ωqℓ	PROPN
ejpam-6802	185	17	&	&	CCONJ
ejpam-6802	185	18	qℓ+1	qℓ+1	NUM
ejpam-6802	185	19	=	=	NOUN
ejpam-6802	185	20	ωϑℓ.	ωϑℓ.	PRON
ejpam-6802	185	21	then	then	ADV
ejpam-6802	185	22	we	we	PRON
ejpam-6802	185	23	have	have	VERB
ejpam-6802	185	24	α(qℓ	α(qℓ	PROPN
ejpam-6802	185	25	,	,	PUNCT
ejpam-6802	185	26	ϑℓ	ϑℓ	PROPN
ejpam-6802	185	27	)	)	PUNCT
ejpam-6802	185	28	=	=	SYM
ejpam-6802	185	29	α(ωϑℓ−1,ωqℓ	α(ωϑℓ−1,ωqℓ	NOUN
ejpam-6802	185	30	)	)	PUNCT
ejpam-6802	185	31	≤	≤	NOUN
ejpam-6802	185	32	(	(	PUNCT
ejpam-6802	185	33	α(qℓ,ωqℓ)α(ωϑℓ−1	α(qℓ,ωqℓ)α(ωϑℓ−1	NOUN
ejpam-6802	185	34	,	,	PUNCT
ejpam-6802	185	35	ϑℓ−1	ϑℓ−1	PROPN
ejpam-6802	185	36	)	)	PUNCT
ejpam-6802	185	37	)	)	PUNCT
ejpam-6802	186	1	β	β	X
ejpam-6802	186	2	=	=	PUNCT
ejpam-6802	186	3	(	(	PUNCT
ejpam-6802	186	4	α(qℓ	α(qℓ	PROPN
ejpam-6802	186	5	,	,	PUNCT
ejpam-6802	186	6	ϑℓ)α(qℓ	ϑℓ)α(qℓ	NOUN
ejpam-6802	186	7	,	,	PUNCT
ejpam-6802	186	8	ϑℓ−1	ϑℓ−1	PROPN
ejpam-6802	186	9	)	)	PUNCT
ejpam-6802	186	10	)	)	PUNCT
ejpam-6802	187	1	β	β	NOUN
ejpam-6802	187	2	for	for	ADP
ejpam-6802	187	3	all	all	DET
ejpam-6802	187	4	integers	integer	NOUN
ejpam-6802	187	5	ℓ	ℓ	NOUN
ejpam-6802	187	6	≥	≥	NUM
ejpam-6802	187	7	1	1	NUM
ejpam-6802	187	8	.	.	PUNCT
ejpam-6802	188	1	then	then	ADV
ejpam-6802	188	2	,	,	PUNCT
ejpam-6802	188	3	α(qℓ	α(qℓ	PROPN
ejpam-6802	188	4	,	,	PUNCT
ejpam-6802	188	5	ϑℓ	ϑℓ	NOUN
ejpam-6802	188	6	)	)	PUNCT
ejpam-6802	188	7	≤	≤	NOUN
ejpam-6802	188	8	(	(	PUNCT
ejpam-6802	188	9	α(qℓ	α(qℓ	PROPN
ejpam-6802	188	10	,	,	PUNCT
ejpam-6802	188	11	ϑℓ−1	ϑℓ−1	PROPN
ejpam-6802	188	12	)	)	PUNCT
ejpam-6802	188	13	)	)	PUNCT
ejpam-6802	188	14	β	β	PROPN
ejpam-6802	188	15	1−β	1−β	NUM
ejpam-6802	188	16	,	,	PUNCT
ejpam-6802	188	17	and	and	CCONJ
ejpam-6802	188	18	α(qℓ	α(qℓ	PROPN
ejpam-6802	188	19	,	,	PUNCT
ejpam-6802	188	20	ϑℓ−1	ϑℓ−1	PROPN
ejpam-6802	188	21	)	)	PUNCT
ejpam-6802	188	22	=	=	SYM
ejpam-6802	188	23	α(ωϑℓ−1,ωqℓ−1	α(ωϑℓ−1,ωqℓ−1	PROPN
ejpam-6802	188	24	)	)	PUNCT
ejpam-6802	188	25	≤	≤	NOUN
ejpam-6802	188	26	(	(	PUNCT
ejpam-6802	188	27	α(qℓ−1,ωqℓ−1)α(ωϑℓ−1	α(qℓ−1,ωqℓ−1)α(ωϑℓ−1	PROPN
ejpam-6802	188	28	,	,	PUNCT
ejpam-6802	188	29	ϑℓ−1	ϑℓ−1	PROPN
ejpam-6802	188	30	)	)	PUNCT
ejpam-6802	188	31	)	)	PUNCT
ejpam-6802	188	32	β	β	X
ejpam-6802	188	33	=	=	SYM
ejpam-6802	188	34	(	(	PUNCT
ejpam-6802	188	35	α(qℓ−1	α(qℓ−1	PROPN
ejpam-6802	188	36	,	,	PUNCT
ejpam-6802	188	37	ϑℓ−1)α(qℓ	ϑℓ−1)α(qℓ	ADJ
ejpam-6802	188	38	,	,	PUNCT
ejpam-6802	188	39	ϑℓ−1	ϑℓ−1	PROPN
ejpam-6802	188	40	)	)	PUNCT
ejpam-6802	188	41	)	)	PUNCT
ejpam-6802	189	1	β	β	NOUN
ejpam-6802	189	2	,	,	PUNCT
ejpam-6802	189	3	so	so	SCONJ
ejpam-6802	189	4	α(qℓ	α(qℓ	PROPN
ejpam-6802	189	5	,	,	PUNCT
ejpam-6802	189	6	ϑℓ−1	ϑℓ−1	PROPN
ejpam-6802	189	7	)	)	PUNCT
ejpam-6802	189	8	≤	≤	NOUN
ejpam-6802	189	9	(	(	PUNCT
ejpam-6802	189	10	α(qℓ−1	α(qℓ−1	NOUN
ejpam-6802	189	11	,	,	PUNCT
ejpam-6802	189	12	ϑℓ−1	ϑℓ−1	PROPN
ejpam-6802	189	13	)	)	PUNCT
ejpam-6802	189	14	)	)	PUNCT
ejpam-6802	189	15	β	β	PROPN
ejpam-6802	189	16	1−β	1−β	NUM
ejpam-6802	189	17	.	.	PUNCT
ejpam-6802	190	1	take	take	VERB
ejpam-6802	190	2	ȷ	ȷ	NOUN
ejpam-6802	190	3	:	:	PUNCT
ejpam-6802	190	4	=	=	PUNCT
ejpam-6802	190	5	β	β	X
ejpam-6802	190	6	1−β	1−β	NUM
ejpam-6802	190	7	,	,	PUNCT
ejpam-6802	190	8	then	then	ADV
ejpam-6802	190	9	we	we	PRON
ejpam-6802	190	10	have	have	VERB
ejpam-6802	190	11	ȷ	ȷ	PRON
ejpam-6802	190	12	∈	∈	NOUN
ejpam-6802	190	13	(	(	PUNCT
ejpam-6802	190	14	0	0	NUM
ejpam-6802	190	15	,	,	PUNCT
ejpam-6802	190	16	1	1	NUM
ejpam-6802	190	17	)	)	PUNCT
ejpam-6802	190	18	since	since	SCONJ
ejpam-6802	190	19	β	β	X
ejpam-6802	190	20	∈	∈	PROPN
ejpam-6802	190	21	(	(	PUNCT
ejpam-6802	190	22	0	0	NUM
ejpam-6802	190	23	,	,	PUNCT
ejpam-6802	190	24	12	12	NUM
ejpam-6802	190	25	)	)	PUNCT
ejpam-6802	190	26	.	.	PUNCT
ejpam-6802	191	1	α(qℓ	α(qℓ	PROPN
ejpam-6802	191	2	,	,	PUNCT
ejpam-6802	191	3	ϑℓ	ϑℓ	PROPN
ejpam-6802	191	4	)	)	PUNCT
ejpam-6802	191	5	≤	≤	NOUN
ejpam-6802	191	6	(	(	PUNCT
ejpam-6802	191	7	α(q0	α(q0	NOUN
ejpam-6802	191	8	,	,	PUNCT
ejpam-6802	191	9	ϑ0	ϑ0	NOUN
ejpam-6802	191	10	)	)	PUNCT
ejpam-6802	191	11	)	)	PUNCT
ejpam-6802	192	1	ȷ2ℓ	ȷ2ℓ	NOUN
ejpam-6802	192	2	,	,	PUNCT
ejpam-6802	192	3	α(qℓ	α(qℓ	PROPN
ejpam-6802	192	4	,	,	PUNCT
ejpam-6802	192	5	ϑℓ−1	ϑℓ−1	PROPN
ejpam-6802	192	6	)	)	PUNCT
ejpam-6802	192	7	≤	≤	NOUN
ejpam-6802	192	8	(	(	PUNCT
ejpam-6802	192	9	α(q0	α(q0	NOUN
ejpam-6802	192	10	,	,	PUNCT
ejpam-6802	192	11	ϑ0	ϑ0	NOUN
ejpam-6802	192	12	)	)	PUNCT
ejpam-6802	192	13	)	)	PUNCT
ejpam-6802	193	1	ȷ2ℓ−1	ȷ2ℓ−1	PROPN
ejpam-6802	193	2	.	.	PUNCT
ejpam-6802	194	1	for	for	ADP
ejpam-6802	194	2	each	each	DET
ejpam-6802	194	3	r	r	NOUN
ejpam-6802	194	4	>	>	PUNCT
ejpam-6802	194	5	ℓ	ℓ	PROPN
ejpam-6802	194	6	,	,	PUNCT
ejpam-6802	194	7	α(qℓ	α(qℓ	PROPN
ejpam-6802	194	8	,	,	PUNCT
ejpam-6802	194	9	ϑr	ϑr	PROPN
ejpam-6802	194	10	)	)	PUNCT
ejpam-6802	194	11	≤	≤	NUM
ejpam-6802	194	12	α(qℓ	α(qℓ	PROPN
ejpam-6802	194	13	,	,	PUNCT
ejpam-6802	194	14	ϑℓ)α(qℓ+1	ϑℓ)α(qℓ+1	ADV
ejpam-6802	194	15	,	,	PUNCT
ejpam-6802	194	16	ϑℓ)α(qℓ+1	ϑℓ)α(qℓ+1	ADV
ejpam-6802	194	17	,	,	PUNCT
ejpam-6802	194	18	ϑr	ϑr	ADJ
ejpam-6802	194	19	)	)	PUNCT
ejpam-6802	194	20	≤	≤	NOUN
ejpam-6802	194	21	(	(	PUNCT
ejpam-6802	194	22	α(q0	α(q0	NUM
ejpam-6802	194	23	,	,	PUNCT
ejpam-6802	194	24	ϑ0)α(qℓ+1	ϑ0)α(qℓ+1	X
ejpam-6802	194	25	,	,	PUNCT
ejpam-6802	194	26	ϑr	ϑr	NOUN
ejpam-6802	194	27	)	)	PUNCT
ejpam-6802	194	28	)	)	PUNCT
ejpam-6802	194	29	(	(	PUNCT
ejpam-6802	195	1	ȷ2ℓ+ȷ2ℓ+1	ȷ2ℓ+ȷ2ℓ+1	NUM
ejpam-6802	195	2	)	)	PUNCT
ejpam-6802	195	3	...	...	PUNCT
ejpam-6802	196	1	≤	≤	NUM
ejpam-6802	196	2	(	(	PUNCT
ejpam-6802	196	3	α(q0	α(q0	NOUN
ejpam-6802	196	4	,	,	PUNCT
ejpam-6802	196	5	ϑ0	ϑ0	NOUN
ejpam-6802	196	6	)	)	PUNCT
ejpam-6802	196	7	)	)	PUNCT
ejpam-6802	196	8	(	(	PUNCT
ejpam-6802	196	9	ȷ2ℓ+ȷ2ℓ+1+···+ȷ2r−1+ȷ2r	ȷ2ℓ+ȷ2ℓ+1+···+ȷ2r−1+ȷ2r	ADJ
ejpam-6802	196	10	)	)	PUNCT
ejpam-6802	196	11	=	=	SYM
ejpam-6802	196	12	(	(	PUNCT
ejpam-6802	196	13	α(q0	α(q0	NOUN
ejpam-6802	196	14	,	,	PUNCT
ejpam-6802	196	15	ϑ0	ϑ0	NOUN
ejpam-6802	196	16	)	)	PUNCT
ejpam-6802	196	17	)	)	PUNCT
ejpam-6802	196	18	(	(	PUNCT
ejpam-6802	196	19	ȷ2ℓ+···+ȷ2r	ȷ2ℓ+···+ȷ2r	NOUN
ejpam-6802	196	20	)	)	PUNCT
ejpam-6802	196	21	≤	≤	NOUN
ejpam-6802	196	22	(	(	PUNCT
ejpam-6802	196	23	α(q0	α(q0	NOUN
ejpam-6802	196	24	,	,	PUNCT
ejpam-6802	196	25	ϑ0	ϑ0	NOUN
ejpam-6802	196	26	)	)	PUNCT
ejpam-6802	196	27	)	)	PUNCT
ejpam-6802	197	1	ȷ2ℓ	ȷ2ℓ	ADP
ejpam-6802	197	2	1−ȷ	1−ȷ	NUM
ejpam-6802	197	3	.	.	PUNCT
ejpam-6802	198	1	r.	r.	PROPN
ejpam-6802	198	2	ramaswamy	ramaswamy	PROPN
ejpam-6802	198	3	/	/	SYM
ejpam-6802	198	4	eur	eur	PROPN
ejpam-6802	198	5	.	.	PUNCT
ejpam-6802	199	1	j.	j.	PROPN
ejpam-6802	199	2	pure	pure	PROPN
ejpam-6802	199	3	appl	appl	PROPN
ejpam-6802	199	4	.	.	PROPN
ejpam-6802	199	5	math	math	PROPN
ejpam-6802	199	6	,	,	PUNCT
ejpam-6802	199	7	18	18	NUM
ejpam-6802	199	8	(	(	PUNCT
ejpam-6802	199	9	4	4	NUM
ejpam-6802	199	10	)	)	PUNCT
ejpam-6802	199	11	(	(	PUNCT
ejpam-6802	199	12	2025	2025	NUM
ejpam-6802	199	13	)	)	PUNCT
ejpam-6802	199	14	,	,	PUNCT
ejpam-6802	199	15	6802	6802	NUM
ejpam-6802	199	16	10	10	NUM
ejpam-6802	199	17	of	of	ADP
ejpam-6802	199	18	20	20	NUM
ejpam-6802	199	19	as	as	ADP
ejpam-6802	199	20	ℓ	ℓ	NOUN
ejpam-6802	199	21	,	,	PUNCT
ejpam-6802	199	22	r	r	NOUN
ejpam-6802	199	23	→	→	SYM
ejpam-6802	199	24	+	+	PROPN
ejpam-6802	199	25	+	+	NOUN
ejpam-6802	199	26	∞	∞	ADJ
ejpam-6802	199	27	,	,	PUNCT
ejpam-6802	199	28	we	we	PRON
ejpam-6802	199	29	get	get	VERB
ejpam-6802	199	30	α(qℓ	α(qℓ	NUM
ejpam-6802	199	31	,	,	PUNCT
ejpam-6802	199	32	ϑr	ϑr	ADJ
ejpam-6802	199	33	)	)	PUNCT
ejpam-6802	199	34	→	→	SYM
ejpam-6802	199	35	1	1	X
ejpam-6802	199	36	.	.	PUNCT
ejpam-6802	199	37	consequently	consequently	ADV
ejpam-6802	199	38	,	,	PUNCT
ejpam-6802	199	39	for	for	ADP
ejpam-6802	199	40	each	each	DET
ejpam-6802	199	41	r	r	NOUN
ejpam-6802	199	42	<	<	X
ejpam-6802	199	43	ℓ	ℓ	PROPN
ejpam-6802	199	44	α(qℓ	α(qℓ	PROPN
ejpam-6802	199	45	,	,	PUNCT
ejpam-6802	199	46	ϑr	ϑr	PROPN
ejpam-6802	199	47	)	)	PUNCT
ejpam-6802	199	48	≤	≤	NOUN
ejpam-6802	199	49	α(qr+1	α(qr+1	PROPN
ejpam-6802	199	50	,	,	PUNCT
ejpam-6802	199	51	ϑr)α(qr+1	ϑr)α(qr+1	PROPN
ejpam-6802	199	52	,	,	PUNCT
ejpam-6802	199	53	ϑr+1)α(qℓ	ϑr+1)α(qℓ	PROPN
ejpam-6802	199	54	,	,	PUNCT
ejpam-6802	199	55	ϑr+1	ϑr+1	NOUN
ejpam-6802	199	56	)	)	PUNCT
ejpam-6802	199	57	≤	≤	NOUN
ejpam-6802	199	58	(	(	PUNCT
ejpam-6802	199	59	α(q0	α(q0	NOUN
ejpam-6802	199	60	,	,	PUNCT
ejpam-6802	199	61	ϑ0	ϑ0	NOUN
ejpam-6802	199	62	)	)	PUNCT
ejpam-6802	199	63	)	)	PUNCT
ejpam-6802	199	64	(	(	PUNCT
ejpam-6802	199	65	ȷ2r+1+ȷ2r+2)α(qℓ	ȷ2r+1+ȷ2r+2)α(qℓ	INTJ
ejpam-6802	199	66	,	,	PUNCT
ejpam-6802	199	67	ϑr+1	ϑr+1	NOUN
ejpam-6802	199	68	)	)	PUNCT
ejpam-6802	199	69	...	...	PUNCT
ejpam-6802	200	1	≤	≤	NUM
ejpam-6802	200	2	(	(	PUNCT
ejpam-6802	200	3	α(q0	α(q0	NOUN
ejpam-6802	200	4	,	,	PUNCT
ejpam-6802	200	5	ϑ0	ϑ0	NOUN
ejpam-6802	200	6	)	)	PUNCT
ejpam-6802	200	7	)	)	PUNCT
ejpam-6802	200	8	(	(	PUNCT
ejpam-6802	200	9	ȷ2r+1+ȷ2r+2+···+ȷ2ℓ)α(qℓ	ȷ2r+1+ȷ2r+2+···+ȷ2ℓ)α(qℓ	ADV
ejpam-6802	200	10	,	,	PUNCT
ejpam-6802	200	11	ϑℓ	ϑℓ	NOUN
ejpam-6802	200	12	)	)	PUNCT
ejpam-6802	200	13	≤	≤	NOUN
ejpam-6802	200	14	(	(	PUNCT
ejpam-6802	200	15	α(q0	α(q0	NOUN
ejpam-6802	200	16	,	,	PUNCT
ejpam-6802	200	17	ϑ0	ϑ0	NOUN
ejpam-6802	200	18	)	)	PUNCT
ejpam-6802	200	19	)	)	PUNCT
ejpam-6802	200	20	(	(	PUNCT
ejpam-6802	200	21	ȷ2r+1+ȷ2r+2+···+ȷ2ℓ+1	ȷ2r+1+ȷ2r+2+···+ȷ2ℓ+1	PROPN
ejpam-6802	200	22	)	)	PUNCT
ejpam-6802	200	23	=	=	SYM
ejpam-6802	200	24	(	(	PUNCT
ejpam-6802	200	25	α(q0	α(q0	NOUN
ejpam-6802	200	26	,	,	PUNCT
ejpam-6802	200	27	ϑ0	ϑ0	NOUN
ejpam-6802	200	28	)	)	PUNCT
ejpam-6802	200	29	)	)	PUNCT
ejpam-6802	200	30	(	(	PUNCT
ejpam-6802	200	31	ȷ2r+1+···+ȷ2ℓ+1	ȷ2r+1+···+ȷ2ℓ+1	NOUN
ejpam-6802	200	32	)	)	PUNCT
ejpam-6802	200	33	≤	≤	NOUN
ejpam-6802	200	34	(	(	PUNCT
ejpam-6802	200	35	α(q0	α(q0	NOUN
ejpam-6802	200	36	,	,	PUNCT
ejpam-6802	200	37	ϑ0	ϑ0	NOUN
ejpam-6802	200	38	)	)	PUNCT
ejpam-6802	200	39	)	)	PUNCT
ejpam-6802	201	1	ȷ2r+1	ȷ2r+1	INTJ
ejpam-6802	201	2	1−ȷ	1−ȷ	NUM
ejpam-6802	201	3	.	.	PUNCT
ejpam-6802	202	1	as	as	ADP
ejpam-6802	202	2	ℓ	ℓ	X
ejpam-6802	202	3	,	,	PUNCT
ejpam-6802	202	4	r	r	NOUN
ejpam-6802	202	5	→	→	SYM
ejpam-6802	202	6	+	+	PROPN
ejpam-6802	202	7	+	+	NOUN
ejpam-6802	202	8	∞	∞	ADJ
ejpam-6802	202	9	,	,	PUNCT
ejpam-6802	202	10	we	we	PRON
ejpam-6802	202	11	get	get	VERB
ejpam-6802	202	12	α(qℓ	α(qℓ	NUM
ejpam-6802	202	13	,	,	PUNCT
ejpam-6802	202	14	ϑr	ϑr	ADJ
ejpam-6802	202	15	)	)	PUNCT
ejpam-6802	202	16	→	→	SYM
ejpam-6802	202	17	1	1	X
ejpam-6802	202	18	.	.	X
ejpam-6802	203	1	therefore	therefore	ADV
ejpam-6802	203	2	,	,	PUNCT
ejpam-6802	203	3	(	(	PUNCT
ejpam-6802	203	4	{	{	PUNCT
ejpam-6802	203	5	qℓ	qℓ	NOUN
ejpam-6802	203	6	}	}	PUNCT
ejpam-6802	203	7	,	,	PUNCT
ejpam-6802	203	8	{	{	PUNCT
ejpam-6802	203	9	ϑm	ϑm	NOUN
ejpam-6802	203	10	}	}	PUNCT
ejpam-6802	203	11	)	)	PUNCT
ejpam-6802	203	12	is	be	AUX
ejpam-6802	203	13	a	a	DET
ejpam-6802	203	14	cauchy	cauchy	ADJ
ejpam-6802	203	15	bisequence	bisequence	NOUN
ejpam-6802	203	16	.	.	PUNCT
ejpam-6802	204	1	since	since	SCONJ
ejpam-6802	204	2	(	(	PUNCT
ejpam-6802	204	3	b	b	X
ejpam-6802	204	4	,	,	PUNCT
ejpam-6802	204	5	f	f	PROPN
ejpam-6802	204	6	,	,	PUNCT
ejpam-6802	204	7	α	α	PROPN
ejpam-6802	204	8	)	)	PUNCT
ejpam-6802	204	9	is	be	AUX
ejpam-6802	204	10	complete	complete	ADJ
ejpam-6802	204	11	,	,	PUNCT
ejpam-6802	204	12	{	{	PUNCT
ejpam-6802	204	13	qℓ	qℓ	NOUN
ejpam-6802	204	14	}	}	PUNCT
ejpam-6802	204	15	→	→	SYM
ejpam-6802	204	16	x	x	X
ejpam-6802	204	17	,	,	PUNCT
ejpam-6802	204	18	{	{	PUNCT
ejpam-6802	204	19	ϑr	ϑr	VERB
ejpam-6802	204	20	}	}	PUNCT
ejpam-6802	204	21	→	→	SYM
ejpam-6802	204	22	x	x	X
ejpam-6802	204	23	,	,	PUNCT
ejpam-6802	204	24	where	where	SCONJ
ejpam-6802	204	25	x	x	PUNCT
ejpam-6802	204	26	∈	∈	PROPN
ejpam-6802	204	27	b	b	X
ejpam-6802	204	28	∪	∪	PROPN
ejpam-6802	204	29	f	f	PROPN
ejpam-6802	204	30	.	.	PUNCT
ejpam-6802	205	1	since	since	SCONJ
ejpam-6802	205	2	{	{	PUNCT
ejpam-6802	205	3	ωqℓ	ωqℓ	NOUN
ejpam-6802	205	4	}	}	PUNCT
ejpam-6802	205	5	=	=	SYM
ejpam-6802	205	6	{	{	PUNCT
ejpam-6802	205	7	ϑℓ	ϑℓ	NOUN
ejpam-6802	205	8	}	}	PUNCT
ejpam-6802	205	9	→	→	PUNCT
ejpam-6802	205	10	x.	x.	NOUN
ejpam-6802	205	11	on	on	ADP
ejpam-6802	205	12	the	the	DET
ejpam-6802	205	13	other	other	ADJ
ejpam-6802	205	14	hand	hand	NOUN
ejpam-6802	205	15	,	,	PUNCT
ejpam-6802	205	16	α(ωx	α(ωx	NUM
ejpam-6802	205	17	,	,	PUNCT
ejpam-6802	205	18	ωqℓ	ωqℓ	NOUN
ejpam-6802	205	19	)	)	PUNCT
ejpam-6802	205	20	≤	≤	NOUN
ejpam-6802	205	21	(	(	PUNCT
ejpam-6802	205	22	α(qℓ,ωqℓ)α(ωx	α(qℓ,ωqℓ)α(ωx	NOUN
ejpam-6802	205	23	,	,	PUNCT
ejpam-6802	205	24	x	x	NOUN
ejpam-6802	205	25	)	)	PUNCT
ejpam-6802	205	26	)	)	PUNCT
ejpam-6802	205	27	β	β	X
ejpam-6802	205	28	=	=	PUNCT
ejpam-6802	205	29	(	(	PUNCT
ejpam-6802	205	30	α(qℓ	α(qℓ	PROPN
ejpam-6802	205	31	,	,	PUNCT
ejpam-6802	205	32	ϑℓ)α(ωx	ϑℓ)α(ωx	PROPN
ejpam-6802	205	33	,	,	PUNCT
ejpam-6802	205	34	x	x	NOUN
ejpam-6802	205	35	)	)	PUNCT
ejpam-6802	205	36	)	)	PUNCT
ejpam-6802	206	1	β	β	X
ejpam-6802	206	2	.	.	PUNCT
ejpam-6802	207	1	as	as	ADP
ejpam-6802	207	2	ℓ	ℓ	PROPN
ejpam-6802	207	3	→	→	PUNCT
ejpam-6802	207	4	+	+	PROPN
ejpam-6802	207	5	+	+	PROPN
ejpam-6802	207	6	∞	∞	PROPN
ejpam-6802	207	7	,	,	PUNCT
ejpam-6802	207	8	α(ωx	α(ωx	NUM
ejpam-6802	207	9	,	,	PUNCT
ejpam-6802	207	10	x	x	NOUN
ejpam-6802	207	11	)	)	PUNCT
ejpam-6802	207	12	≤	≤	NOUN
ejpam-6802	207	13	(	(	PUNCT
ejpam-6802	207	14	α(ωx	α(ωx	NUM
ejpam-6802	207	15	,	,	PUNCT
ejpam-6802	207	16	x))β	x))β	ADV
ejpam-6802	207	17	therefore	therefore	ADV
ejpam-6802	207	18	,	,	PUNCT
ejpam-6802	207	19	α(ωx	α(ωx	NUM
ejpam-6802	207	20	,	,	PUNCT
ejpam-6802	207	21	x	x	X
ejpam-6802	207	22	)	)	PUNCT
ejpam-6802	207	23	=	=	SYM
ejpam-6802	207	24	1	1	X
ejpam-6802	207	25	.	.	X
ejpam-6802	207	26	hence	hence	ADV
ejpam-6802	207	27	ωx	ωx	PROPN
ejpam-6802	208	1	=	=	PUNCT
ejpam-6802	208	2	x.	x.	NOUN
ejpam-6802	208	3	if	if	SCONJ
ejpam-6802	208	4	z	z	NOUN
ejpam-6802	208	5	is	be	AUX
ejpam-6802	208	6	any	any	DET
ejpam-6802	208	7	fp	fp	PROPN
ejpam-6802	208	8	of	of	ADP
ejpam-6802	208	9	ω	ω	PROPN
ejpam-6802	208	10	,	,	PUNCT
ejpam-6802	208	11	then	then	ADV
ejpam-6802	208	12	ωz	ωz	ADP
ejpam-6802	208	13	=	=	SYM
ejpam-6802	208	14	z	z	NOUN
ejpam-6802	208	15	⇒	⇒	NOUN
ejpam-6802	208	16	z	z	PROPN
ejpam-6802	208	17	∈	∈	PROPN
ejpam-6802	208	18	b	b	PROPN
ejpam-6802	208	19	∩	∩	PROPN
ejpam-6802	208	20	f	f	PROPN
ejpam-6802	208	21	.	.	PUNCT
ejpam-6802	209	1	then	then	ADV
ejpam-6802	209	2	α(x	α(x	PROPN
ejpam-6802	209	3	,	,	PUNCT
ejpam-6802	209	4	z	z	NOUN
ejpam-6802	209	5	)	)	PUNCT
ejpam-6802	209	6	=	=	SYM
ejpam-6802	209	7	α(ωx	α(ωx	NUM
ejpam-6802	209	8	,	,	PUNCT
ejpam-6802	209	9	ωz	ωz	NOUN
ejpam-6802	209	10	)	)	PUNCT
ejpam-6802	209	11	≤	≤	NOUN
ejpam-6802	209	12	(	(	PUNCT
ejpam-6802	209	13	α(x	α(x	NOUN
ejpam-6802	209	14	,	,	PUNCT
ejpam-6802	209	15	ωx)α(ωz	ωx)α(ωz	NUM
ejpam-6802	209	16	,	,	PUNCT
ejpam-6802	209	17	z))β	z))β	NOUN
ejpam-6802	209	18	=	=	SYM
ejpam-6802	209	19	(	(	PUNCT
ejpam-6802	209	20	α(x	α(x	PROPN
ejpam-6802	209	21	,	,	PUNCT
ejpam-6802	209	22	x)α(v	x)α(v	PROPN
ejpam-6802	209	23	,	,	PUNCT
ejpam-6802	209	24	z))β	z))β	NOUN
ejpam-6802	209	25	=	=	SYM
ejpam-6802	209	26	1	1	X
ejpam-6802	209	27	.	.	PUNCT
ejpam-6802	210	1	consequently	consequently	ADV
ejpam-6802	210	2	x	x	PUNCT
ejpam-6802	210	3	=	=	SYM
ejpam-6802	210	4	z.	z.	PROPN
ejpam-6802	210	5	theorem	theorem	VERB
ejpam-6802	210	6	3.4	3.4	NUM
ejpam-6802	210	7	.	.	PUNCT
ejpam-6802	211	1	let	let	VERB
ejpam-6802	211	2	(	(	PUNCT
ejpam-6802	211	3	b	b	X
ejpam-6802	211	4	,	,	PUNCT
ejpam-6802	211	5	f	f	PROPN
ejpam-6802	211	6	,	,	PUNCT
ejpam-6802	211	7	α	α	PROPN
ejpam-6802	211	8	)	)	PUNCT
ejpam-6802	211	9	be	be	VERB
ejpam-6802	211	10	a	a	DET
ejpam-6802	211	11	complete	complete	ADJ
ejpam-6802	211	12	mcbms	mcbms	NOUN
ejpam-6802	211	13	,	,	PUNCT
ejpam-6802	211	14	z	z	PROPN
ejpam-6802	211	15	be	be	AUX
ejpam-6802	211	16	a	a	DET
ejpam-6802	211	17	cone	cone	NOUN
ejpam-6802	211	18	with	with	ADP
ejpam-6802	211	19	constant	constant	PROPN
ejpam-6802	211	20	w	w	PROPN
ejpam-6802	211	21	&	&	CCONJ
ejpam-6802	211	22	ω	ω	PROPN
ejpam-6802	211	23	,	,	PUNCT
ejpam-6802	211	24	s	s	PART
ejpam-6802	211	25	:	:	PUNCT
ejpam-6802	211	26	(	(	PUNCT
ejpam-6802	211	27	b	b	X
ejpam-6802	211	28	,	,	PUNCT
ejpam-6802	211	29	f	f	PROPN
ejpam-6802	211	30	,	,	PUNCT
ejpam-6802	211	31	α	α	NOUN
ejpam-6802	211	32	)	)	PUNCT
ejpam-6802	211	33	⇄	⇄	PROPN
ejpam-6802	211	34	(	(	PUNCT
ejpam-6802	211	35	b	b	NOUN
ejpam-6802	211	36	,	,	PUNCT
ejpam-6802	211	37	f	f	PROPN
ejpam-6802	211	38	,	,	PUNCT
ejpam-6802	211	39	α	α	PROPN
ejpam-6802	211	40	)	)	PUNCT
ejpam-6802	211	41	be	be	VERB
ejpam-6802	211	42	a	a	DET
ejpam-6802	211	43	contravariant	contravariant	ADJ
ejpam-6802	211	44	mapping	mapping	NOUN
ejpam-6802	211	45	satisfying	satisfy	VERB
ejpam-6802	211	46	α(sϑ,ωq	α(sϑ,ωq	PROPN
ejpam-6802	211	47	)	)	PUNCT
ejpam-6802	211	48	≤	≤	NOUN
ejpam-6802	211	49	(	(	PUNCT
ejpam-6802	211	50	α(q	α(q	PROPN
ejpam-6802	211	51	,	,	PUNCT
ejpam-6802	211	52	ωq)α(sϑ	ωq)α(sϑ	NUM
ejpam-6802	211	53	,	,	PUNCT
ejpam-6802	211	54	ϑ	ϑ	NOUN
ejpam-6802	211	55	)	)	PUNCT
ejpam-6802	211	56	α(q	α(q	PROPN
ejpam-6802	211	57	,	,	PUNCT
ejpam-6802	211	58	ϑ	ϑ	NOUN
ejpam-6802	211	59	)	)	PUNCT
ejpam-6802	211	60	)	)	PUNCT
ejpam-6802	212	1	β	β	X
ejpam-6802	212	2	(	(	PUNCT
ejpam-6802	212	3	α(q	α(q	PROPN
ejpam-6802	212	4	,	,	PUNCT
ejpam-6802	212	5	ϑ))γ(α(q	ϑ))γ(α(q	ADJ
ejpam-6802	212	6	,	,	PUNCT
ejpam-6802	212	7	ωq)α(sϑ	ωq)α(sϑ	PROPN
ejpam-6802	212	8	,	,	PUNCT
ejpam-6802	212	9	ϑ))ı	ϑ))ı	NOUN
ejpam-6802	212	10	,	,	PUNCT
ejpam-6802	212	11	(	(	PUNCT
ejpam-6802	212	12	1	1	X
ejpam-6802	212	13	)	)	PUNCT
ejpam-6802	212	14	r.	r.	PROPN
ejpam-6802	212	15	ramaswamy	ramaswamy	PROPN
ejpam-6802	212	16	/	/	SYM
ejpam-6802	212	17	eur	eur	PROPN
ejpam-6802	212	18	.	.	PUNCT
ejpam-6802	213	1	j.	j.	PROPN
ejpam-6802	213	2	pure	pure	PROPN
ejpam-6802	213	3	appl	appl	PROPN
ejpam-6802	213	4	.	.	PROPN
ejpam-6802	213	5	math	math	PROPN
ejpam-6802	213	6	,	,	PUNCT
ejpam-6802	213	7	18	18	NUM
ejpam-6802	213	8	(	(	PUNCT
ejpam-6802	213	9	4	4	NUM
ejpam-6802	213	10	)	)	PUNCT
ejpam-6802	213	11	(	(	PUNCT
ejpam-6802	213	12	2025	2025	NUM
ejpam-6802	213	13	)	)	PUNCT
ejpam-6802	213	14	,	,	PUNCT
ejpam-6802	213	15	6802	6802	NUM
ejpam-6802	213	16	11	11	NUM
ejpam-6802	213	17	of	of	ADP
ejpam-6802	213	18	20	20	NUM
ejpam-6802	213	19	for	for	ADP
ejpam-6802	213	20	all	all	PRON
ejpam-6802	213	21	(	(	PUNCT
ejpam-6802	213	22	q	q	ADJ
ejpam-6802	213	23	,	,	PUNCT
ejpam-6802	213	24	ϑ	ϑ	NOUN
ejpam-6802	213	25	)	)	PUNCT
ejpam-6802	214	1	∈	∈	PROPN
ejpam-6802	214	2	b	b	PROPN
ejpam-6802	214	3	×	×	PROPN
ejpam-6802	214	4	f	f	X
ejpam-6802	214	5	,	,	PUNCT
ejpam-6802	214	6	with	with	ADP
ejpam-6802	214	7	q	q	PROPN
ejpam-6802	214	8	̸=	̸=	PROPN
ejpam-6802	214	9	ϑ	ϑ	NOUN
ejpam-6802	214	10	,	,	PUNCT
ejpam-6802	214	11	β	β	X
ejpam-6802	214	12	,	,	PUNCT
ejpam-6802	214	13	γ	γ	PROPN
ejpam-6802	214	14	,	,	PUNCT
ejpam-6802	214	15	ı	ı	PRON
ejpam-6802	214	16	≥	≥	NOUN
ejpam-6802	214	17	0	0	NUM
ejpam-6802	214	18	,	,	PUNCT
ejpam-6802	214	19	γ	γ	X
ejpam-6802	214	20	<	<	X
ejpam-6802	214	21	β	β	X
ejpam-6802	214	22	&	&	CCONJ
ejpam-6802	214	23	0	0	NUM
ejpam-6802	214	24	≤	≤	NUM
ejpam-6802	214	25	β	β	X
ejpam-6802	214	26	+	+	CCONJ
ejpam-6802	214	27	γ	γ	X
ejpam-6802	214	28	+	+	NOUN
ejpam-6802	214	29	2ı	2ı	NOUN
ejpam-6802	214	30	<	<	X
ejpam-6802	214	31	1	1	NUM
ejpam-6802	214	32	.	.	PUNCT
ejpam-6802	215	1	then	then	ADV
ejpam-6802	215	2	ω	ω	NUM
ejpam-6802	215	3	,	,	PUNCT
ejpam-6802	215	4	s	s	PART
ejpam-6802	215	5	:	:	PUNCT
ejpam-6802	215	6	b	b	X
ejpam-6802	215	7	∪	∪	X
ejpam-6802	215	8	f	f	PROPN
ejpam-6802	215	9	→	→	SYM
ejpam-6802	215	10	b	b	X
ejpam-6802	215	11	∪	∪	X
ejpam-6802	215	12	f	f	X
ejpam-6802	215	13	have	have	VERB
ejpam-6802	215	14	a	a	DET
ejpam-6802	215	15	unique	unique	ADJ
ejpam-6802	215	16	cfp	cfp	NOUN
ejpam-6802	215	17	.	.	PUNCT
ejpam-6802	216	1	proof	proof	NOUN
ejpam-6802	216	2	.	.	PUNCT
ejpam-6802	217	1	consider	consider	VERB
ejpam-6802	217	2	q0	q0	PROPN
ejpam-6802	217	3	∈	∈	PROPN
ejpam-6802	217	4	b	b	PROPN
ejpam-6802	217	5	&	&	CCONJ
ejpam-6802	217	6	ϑ0	ϑ0	PROPN
ejpam-6802	217	7	∈	∈	PROPN
ejpam-6802	218	1	f	f	X
ejpam-6802	218	2	then	then	ADV
ejpam-6802	218	3	for	for	ADP
ejpam-6802	218	4	each	each	DET
ejpam-6802	218	5	ℓ	ℓ	PROPN
ejpam-6802	218	6	∈	∈	PROPN
ejpam-6802	218	7	n	n	PART
ejpam-6802	218	8	∪	∪	X
ejpam-6802	218	9	{	{	PUNCT
ejpam-6802	218	10	0	0	NUM
ejpam-6802	218	11	}	}	PUNCT
ejpam-6802	218	12	,	,	PUNCT
ejpam-6802	218	13	we	we	PRON
ejpam-6802	218	14	define	define	VERB
ejpam-6802	218	15	sq2ℓ	sq2ℓ	NOUN
ejpam-6802	218	16	=	=	SYM
ejpam-6802	218	17	ϑ2ℓ,ωq2ℓ+1	ϑ2ℓ,ωq2ℓ+1	NOUN
ejpam-6802	218	18	=	=	PUNCT
ejpam-6802	218	19	ϑ2ℓ+1,sϑ2ℓ	ϑ2ℓ+1,sϑ2ℓ	NOUN
ejpam-6802	218	20	=	=	SYM
ejpam-6802	218	21	q2ℓ+1,ωϑ2ℓ+1	q2ℓ+1,ωϑ2ℓ+1	ADJ
ejpam-6802	218	22	=	=	SYM
ejpam-6802	218	23	q2ℓ+2	q2ℓ+2	PROPN
ejpam-6802	218	24	.	.	PUNCT
ejpam-6802	218	25	now	now	ADV
ejpam-6802	218	26	by	by	ADP
ejpam-6802	218	27	(	(	PUNCT
ejpam-6802	218	28	1	1	NUM
ejpam-6802	218	29	)	)	PUNCT
ejpam-6802	218	30	,	,	PUNCT
ejpam-6802	218	31	we	we	PRON
ejpam-6802	218	32	get	get	VERB
ejpam-6802	218	33	α(q2ℓ+1	α(q2ℓ+1	NUM
ejpam-6802	218	34	,	,	PUNCT
ejpam-6802	218	35	ϑ2ℓ+1	ϑ2ℓ+1	NOUN
ejpam-6802	218	36	)	)	PUNCT
ejpam-6802	218	37	=	=	NOUN
ejpam-6802	218	38	α(sϑ2ℓ,ωq2ℓ+1	α(sϑ2ℓ,ωq2ℓ+1	NUM
ejpam-6802	218	39	)	)	PUNCT
ejpam-6802	218	40	≤	≤	NOUN
ejpam-6802	218	41	(	(	PUNCT
ejpam-6802	218	42	α(q2ℓ+1,ωq2ℓ+1)α(sϑ2ℓ+1	α(q2ℓ+1,ωq2ℓ+1)α(sϑ2ℓ+1	X
ejpam-6802	218	43	,	,	PUNCT
ejpam-6802	218	44	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	218	45	)	)	PUNCT
ejpam-6802	218	46	α(q2ℓ+1	α(q2ℓ+1	NUM
ejpam-6802	218	47	,	,	PUNCT
ejpam-6802	218	48	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	218	49	)	)	PUNCT
ejpam-6802	218	50	)	)	PUNCT
ejpam-6802	219	1	β	β	X
ejpam-6802	219	2	(	(	PUNCT
ejpam-6802	219	3	α(q2ℓ+1	α(q2ℓ+1	NUM
ejpam-6802	219	4	,	,	PUNCT
ejpam-6802	219	5	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	219	6	)	)	PUNCT
ejpam-6802	219	7	)	)	PUNCT
ejpam-6802	219	8	γ	γ	PROPN
ejpam-6802	219	9	(	(	PUNCT
ejpam-6802	219	10	α(q2ℓ+1,ωq2ℓ+1	α(q2ℓ+1,ωq2ℓ+1	NOUN
ejpam-6802	219	11	)	)	PUNCT
ejpam-6802	219	12	+	+	NUM
ejpam-6802	219	13	α(sϑ2ℓ	α(sϑ2ℓ	NOUN
ejpam-6802	219	14	,	,	PUNCT
ejpam-6802	219	15	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	219	16	)	)	PUNCT
ejpam-6802	219	17	)	)	PUNCT
ejpam-6802	219	18	ı	ı	NOUN
ejpam-6802	220	1	=	=	X
ejpam-6802	220	2	(	(	PUNCT
ejpam-6802	220	3	α(q2ℓ+1	α(q2ℓ+1	NUM
ejpam-6802	220	4	,	,	PUNCT
ejpam-6802	220	5	ϑ2ℓ+1)α(q2ℓ+1	ϑ2ℓ+1)α(q2ℓ+1	ADJ
ejpam-6802	220	6	,	,	PUNCT
ejpam-6802	220	7	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	220	8	)	)	PUNCT
ejpam-6802	220	9	α(q2ℓ+1	α(q2ℓ+1	NUM
ejpam-6802	220	10	,	,	PUNCT
ejpam-6802	220	11	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	220	12	)	)	PUNCT
ejpam-6802	220	13	)	)	PUNCT
ejpam-6802	220	14	β	β	X
ejpam-6802	220	15	(	(	PUNCT
ejpam-6802	220	16	α(q2ℓ+1	α(q2ℓ+1	NUM
ejpam-6802	220	17	,	,	PUNCT
ejpam-6802	220	18	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	220	19	)	)	PUNCT
ejpam-6802	220	20	)	)	PUNCT
ejpam-6802	220	21	γ	γ	X
ejpam-6802	220	22	(	(	PUNCT
ejpam-6802	220	23	α(q2ℓ+1	α(q2ℓ+1	NUM
ejpam-6802	220	24	,	,	PUNCT
ejpam-6802	220	25	ϑ2ℓ+1)α(q2ℓ+1	ϑ2ℓ+1)α(q2ℓ+1	ADJ
ejpam-6802	220	26	,	,	PUNCT
ejpam-6802	220	27	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	220	28	)	)	PUNCT
ejpam-6802	220	29	)	)	PUNCT
ejpam-6802	220	30	ı	ı	PROPN
ejpam-6802	220	31	=(	=(	NOUN
ejpam-6802	220	32	α(q2ℓ+1	α(q2ℓ+1	NUM
ejpam-6802	220	33	,	,	PUNCT
ejpam-6802	220	34	ϑ2ℓ+1	ϑ2ℓ+1	NOUN
ejpam-6802	220	35	)	)	PUNCT
ejpam-6802	220	36	)	)	PUNCT
ejpam-6802	220	37	β(α(q2ℓ+1	β(α(q2ℓ+1	NOUN
ejpam-6802	220	38	,	,	PUNCT
ejpam-6802	220	39	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	220	40	)	)	PUNCT
ejpam-6802	220	41	)	)	PUNCT
ejpam-6802	220	42	γ	γ	X
ejpam-6802	220	43	(	(	PUNCT
ejpam-6802	220	44	α(q2ℓ+1	α(q2ℓ+1	NUM
ejpam-6802	220	45	,	,	PUNCT
ejpam-6802	220	46	ϑ2ℓ+1)α(q2ℓ+1	ϑ2ℓ+1)α(q2ℓ+1	ADJ
ejpam-6802	220	47	,	,	PUNCT
ejpam-6802	220	48	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	220	49	)	)	PUNCT
ejpam-6802	220	50	)	)	PUNCT
ejpam-6802	220	51	ı	ı	PROPN
ejpam-6802	220	52	,	,	PUNCT
ejpam-6802	220	53	which	which	PRON
ejpam-6802	220	54	implies	imply	VERB
ejpam-6802	220	55	that	that	SCONJ
ejpam-6802	220	56	α(q2ℓ+1	α(q2ℓ+1	NUM
ejpam-6802	220	57	,	,	PUNCT
ejpam-6802	220	58	ϑ2ℓ+1	ϑ2ℓ+1	NOUN
ejpam-6802	220	59	)	)	PUNCT
ejpam-6802	220	60	≤	≤	NOUN
ejpam-6802	220	61	(	(	PUNCT
ejpam-6802	220	62	α(q2ℓ+1	α(q2ℓ+1	NUM
ejpam-6802	220	63	,	,	PUNCT
ejpam-6802	220	64	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	220	65	)	)	PUNCT
ejpam-6802	220	66	)	)	PUNCT
ejpam-6802	220	67	γ+ı	γ+ı	NUM
ejpam-6802	221	1	1−β−ı	1−β−ı	INTJ
ejpam-6802	221	2	.	.	PUNCT
ejpam-6802	222	1	(	(	PUNCT
ejpam-6802	222	2	2	2	X
ejpam-6802	222	3	)	)	PUNCT
ejpam-6802	222	4	also	also	ADV
ejpam-6802	222	5	,	,	PUNCT
ejpam-6802	222	6	we	we	PRON
ejpam-6802	222	7	have	have	VERB
ejpam-6802	222	8	α(q2ℓ+1	α(q2ℓ+1	NUM
ejpam-6802	222	9	,	,	PUNCT
ejpam-6802	222	10	ϑ2ℓ	ϑ2ℓ	ADJ
ejpam-6802	222	11	)	)	PUNCT
ejpam-6802	222	12	=	=	SYM
ejpam-6802	222	13	α(sϑ2ℓ,ωq2ℓ	α(sϑ2ℓ,ωq2ℓ	NOUN
ejpam-6802	222	14	)	)	PUNCT
ejpam-6802	222	15	≤	≤	NOUN
ejpam-6802	222	16	(	(	PUNCT
ejpam-6802	222	17	α(q2ℓ,ωq2ℓ)α(sϑ2ℓ	α(q2ℓ,ωq2ℓ)α(sϑ2ℓ	NOUN
ejpam-6802	222	18	,	,	PUNCT
ejpam-6802	222	19	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	222	20	)	)	PUNCT
ejpam-6802	222	21	α(q2ℓ	α(q2ℓ	NOUN
ejpam-6802	222	22	,	,	PUNCT
ejpam-6802	222	23	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	222	24	)	)	PUNCT
ejpam-6802	222	25	)	)	PUNCT
ejpam-6802	223	1	β	β	X
ejpam-6802	223	2	(	(	PUNCT
ejpam-6802	223	3	α(q2ℓ	α(q2ℓ	NOUN
ejpam-6802	223	4	,	,	PUNCT
ejpam-6802	223	5	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	223	6	)	)	PUNCT
ejpam-6802	223	7	)	)	PUNCT
ejpam-6802	223	8	γ	γ	X
ejpam-6802	223	9	(	(	PUNCT
ejpam-6802	223	10	α(q2ℓ,ωq2ℓ	α(q2ℓ,ωq2ℓ	NUM
ejpam-6802	223	11	)	)	PUNCT
ejpam-6802	223	12	+	+	NUM
ejpam-6802	223	13	α(sϑ2ℓ	α(sϑ2ℓ	NOUN
ejpam-6802	223	14	,	,	PUNCT
ejpam-6802	223	15	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	223	16	)	)	PUNCT
ejpam-6802	223	17	)	)	PUNCT
ejpam-6802	223	18	ı	ı	NOUN
ejpam-6802	223	19	=	=	PUNCT
ejpam-6802	223	20	(	(	PUNCT
ejpam-6802	223	21	α(q2ℓ	α(q2ℓ	NOUN
ejpam-6802	223	22	,	,	PUNCT
ejpam-6802	223	23	ϑ2ℓ)α(q2ℓ+1	ϑ2ℓ)α(q2ℓ+1	X
ejpam-6802	223	24	,	,	PUNCT
ejpam-6802	223	25	ϑ2ℓ	ϑ2ℓ	ADJ
ejpam-6802	223	26	)	)	PUNCT
ejpam-6802	223	27	α(q2ℓ	α(q2ℓ	NOUN
ejpam-6802	223	28	,	,	PUNCT
ejpam-6802	223	29	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	223	30	)	)	PUNCT
ejpam-6802	223	31	)	)	PUNCT
ejpam-6802	223	32	β	β	X
ejpam-6802	223	33	(	(	PUNCT
ejpam-6802	223	34	α(q2ℓ	α(q2ℓ	NOUN
ejpam-6802	223	35	,	,	PUNCT
ejpam-6802	223	36	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	223	37	)	)	PUNCT
ejpam-6802	223	38	)	)	PUNCT
ejpam-6802	223	39	γ	γ	X
ejpam-6802	223	40	(	(	PUNCT
ejpam-6802	223	41	α(q2ℓ	α(q2ℓ	NOUN
ejpam-6802	223	42	,	,	PUNCT
ejpam-6802	223	43	ϑ2ℓ)α(q2ℓ+1	ϑ2ℓ)α(q2ℓ+1	X
ejpam-6802	223	44	,	,	PUNCT
ejpam-6802	223	45	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	223	46	)	)	PUNCT
ejpam-6802	223	47	)	)	PUNCT
ejpam-6802	224	1	ı	ı	PROPN
ejpam-6802	224	2	=(	=(	NOUN
ejpam-6802	224	3	α(q2ℓ+1	α(q2ℓ+1	NUM
ejpam-6802	224	4	,	,	PUNCT
ejpam-6802	224	5	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	224	6	)	)	PUNCT
ejpam-6802	224	7	)	)	PUNCT
ejpam-6802	224	8	β(α(q2ℓ	β(α(q2ℓ	NOUN
ejpam-6802	224	9	,	,	PUNCT
ejpam-6802	224	10	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	224	11	)	)	PUNCT
ejpam-6802	224	12	)	)	PUNCT
ejpam-6802	224	13	γ(α(q2ℓ	γ(α(q2ℓ	PROPN
ejpam-6802	224	14	,	,	PUNCT
ejpam-6802	224	15	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	224	16	)	)	PUNCT
ejpam-6802	224	17	)	)	PUNCT
ejpam-6802	224	18	ı	ı	PROPN
ejpam-6802	224	19	(	(	PUNCT
ejpam-6802	224	20	α(q2ℓ+1	α(q2ℓ+1	NUM
ejpam-6802	224	21	,	,	PUNCT
ejpam-6802	224	22	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	224	23	)	)	PUNCT
ejpam-6802	224	24	)	)	PUNCT
ejpam-6802	224	25	ı	ı	PROPN
ejpam-6802	224	26	,	,	PUNCT
ejpam-6802	224	27	which	which	PRON
ejpam-6802	224	28	implies	imply	VERB
ejpam-6802	224	29	that	that	SCONJ
ejpam-6802	224	30	α(q2ℓ+1	α(q2ℓ+1	NUM
ejpam-6802	224	31	,	,	PUNCT
ejpam-6802	224	32	ϑ2ℓ	ϑ2ℓ	ADJ
ejpam-6802	224	33	)	)	PUNCT
ejpam-6802	224	34	≤	≤	NOUN
ejpam-6802	224	35	(	(	PUNCT
ejpam-6802	224	36	α(q2ℓ	α(q2ℓ	NOUN
ejpam-6802	224	37	,	,	PUNCT
ejpam-6802	224	38	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	224	39	)	)	PUNCT
ejpam-6802	224	40	)	)	PUNCT
ejpam-6802	224	41	γ+ı	γ+ı	NUM
ejpam-6802	225	1	1−β−ı	1−β−ı	INTJ
ejpam-6802	225	2	.	.	PUNCT
ejpam-6802	226	1	(	(	PUNCT
ejpam-6802	226	2	3	3	X
ejpam-6802	226	3	)	)	PUNCT
ejpam-6802	226	4	since	since	SCONJ
ejpam-6802	226	5	β	β	X
ejpam-6802	226	6	+	+	CCONJ
ejpam-6802	226	7	γ	γ	X
ejpam-6802	226	8	+2ı	+2ı	NUM
ejpam-6802	226	9	∈	∈	PROPN
ejpam-6802	227	1	[	[	X
ejpam-6802	227	2	0	0	NUM
ejpam-6802	227	3	,	,	PUNCT
ejpam-6802	227	4	1	1	NUM
ejpam-6802	227	5	)	)	PUNCT
ejpam-6802	227	6	and	and	CCONJ
ejpam-6802	227	7	γ+ı	γ+ı	NUM
ejpam-6802	228	1	1−β−ı	1−β−ı	INTJ
ejpam-6802	228	2	=	=	PUNCT
ejpam-6802	228	3	℘	℘	PROPN
ejpam-6802	228	4	(	(	PUNCT
ejpam-6802	228	5	say	say	PROPN
ejpam-6802	228	6	)	)	PUNCT
ejpam-6802	228	7	,	,	PUNCT
ejpam-6802	228	8	℘	℘	X
ejpam-6802	228	9	∈	∈	PROPN
ejpam-6802	229	1	[	[	X
ejpam-6802	229	2	0	0	NUM
ejpam-6802	229	3	,	,	PUNCT
ejpam-6802	229	4	1	1	NUM
ejpam-6802	229	5	)	)	PUNCT
ejpam-6802	229	6	.	.	PUNCT
ejpam-6802	230	1	hence	hence	ADV
ejpam-6802	230	2	,	,	PUNCT
ejpam-6802	230	3	from	from	ADP
ejpam-6802	230	4	(	(	PUNCT
ejpam-6802	230	5	2	2	NUM
ejpam-6802	230	6	)	)	PUNCT
ejpam-6802	230	7	and	and	CCONJ
ejpam-6802	230	8	(	(	PUNCT
ejpam-6802	230	9	3	3	NUM
ejpam-6802	230	10	)	)	PUNCT
ejpam-6802	230	11	,	,	PUNCT
ejpam-6802	230	12	we	we	PRON
ejpam-6802	230	13	get	get	VERB
ejpam-6802	230	14	α(q2ℓ+1	α(q2ℓ+1	NUM
ejpam-6802	230	15	,	,	PUNCT
ejpam-6802	230	16	ϑ2ℓ+1	ϑ2ℓ+1	NOUN
ejpam-6802	230	17	)	)	PUNCT
ejpam-6802	230	18	≤	≤	NOUN
ejpam-6802	230	19	(	(	PUNCT
ejpam-6802	230	20	α(q0	α(q0	NOUN
ejpam-6802	230	21	,	,	PUNCT
ejpam-6802	230	22	ϑ0	ϑ0	NOUN
ejpam-6802	230	23	)	)	PUNCT
ejpam-6802	230	24	)	)	PUNCT
ejpam-6802	230	25	℘4ℓ+2	℘4ℓ+2	PROPN
ejpam-6802	230	26	and	and	CCONJ
ejpam-6802	230	27	α(q2ℓ+1	α(q2ℓ+1	NUM
ejpam-6802	230	28	,	,	PUNCT
ejpam-6802	230	29	ϑ2ℓ	ϑ2ℓ	ADJ
ejpam-6802	230	30	)	)	PUNCT
ejpam-6802	230	31	≤	≤	NOUN
ejpam-6802	230	32	(	(	PUNCT
ejpam-6802	230	33	α(q0	α(q0	NOUN
ejpam-6802	230	34	,	,	PUNCT
ejpam-6802	230	35	ϑ0	ϑ0	NOUN
ejpam-6802	230	36	)	)	PUNCT
ejpam-6802	230	37	)	)	PUNCT
ejpam-6802	230	38	℘4ℓ+1	℘4ℓ+1	X
ejpam-6802	230	39	.	.	PUNCT
ejpam-6802	231	1	(	(	PUNCT
ejpam-6802	231	2	4	4	X
ejpam-6802	231	3	)	)	PUNCT
ejpam-6802	231	4	r.	r.	PROPN
ejpam-6802	231	5	ramaswamy	ramaswamy	PROPN
ejpam-6802	231	6	/	/	SYM
ejpam-6802	231	7	eur	eur	PROPN
ejpam-6802	231	8	.	.	PUNCT
ejpam-6802	232	1	j.	j.	PROPN
ejpam-6802	232	2	pure	pure	PROPN
ejpam-6802	232	3	appl	appl	PROPN
ejpam-6802	232	4	.	.	PROPN
ejpam-6802	232	5	math	math	PROPN
ejpam-6802	232	6	,	,	PUNCT
ejpam-6802	232	7	18	18	NUM
ejpam-6802	232	8	(	(	PUNCT
ejpam-6802	232	9	4	4	NUM
ejpam-6802	232	10	)	)	PUNCT
ejpam-6802	232	11	(	(	PUNCT
ejpam-6802	232	12	2025	2025	NUM
ejpam-6802	232	13	)	)	PUNCT
ejpam-6802	232	14	,	,	PUNCT
ejpam-6802	232	15	6802	6802	NUM
ejpam-6802	232	16	12	12	NUM
ejpam-6802	232	17	of	of	ADP
ejpam-6802	232	18	20	20	NUM
ejpam-6802	232	19	now	now	ADV
ejpam-6802	232	20	,	,	PUNCT
ejpam-6802	232	21	for	for	ADP
ejpam-6802	232	22	any	any	DET
ejpam-6802	232	23	ℓ	ℓ	PROPN
ejpam-6802	232	24	∈	∈	PROPN
ejpam-6802	232	25	n	n	CCONJ
ejpam-6802	232	26	,	,	PUNCT
ejpam-6802	232	27	α(qℓ+1	α(qℓ+1	NUM
ejpam-6802	232	28	,	,	PUNCT
ejpam-6802	232	29	ϑℓ+1	ϑℓ+1	NUM
ejpam-6802	232	30	)	)	PUNCT
ejpam-6802	232	31	≤	≤	NOUN
ejpam-6802	232	32	(	(	PUNCT
ejpam-6802	232	33	α(q0	α(q0	NOUN
ejpam-6802	232	34	,	,	PUNCT
ejpam-6802	232	35	ϑ0	ϑ0	NOUN
ejpam-6802	232	36	)	)	PUNCT
ejpam-6802	232	37	)	)	PUNCT
ejpam-6802	233	1	℘2ℓ+2	℘2ℓ+2	NOUN
ejpam-6802	233	2	,	,	PUNCT
ejpam-6802	233	3	α(qℓ+1	α(qℓ+1	NUM
ejpam-6802	233	4	,	,	PUNCT
ejpam-6802	233	5	ϑℓ	ϑℓ	NOUN
ejpam-6802	233	6	)	)	PUNCT
ejpam-6802	233	7	≤	≤	NOUN
ejpam-6802	233	8	(	(	PUNCT
ejpam-6802	233	9	α(q0	α(q0	NOUN
ejpam-6802	233	10	,	,	PUNCT
ejpam-6802	233	11	ϑ0	ϑ0	NOUN
ejpam-6802	233	12	)	)	PUNCT
ejpam-6802	233	13	)	)	PUNCT
ejpam-6802	233	14	℘2ℓ+1	℘2ℓ+1	PUNCT
ejpam-6802	233	15	and	and	CCONJ
ejpam-6802	233	16	α(qℓ	α(qℓ	PROPN
ejpam-6802	233	17	,	,	PUNCT
ejpam-6802	233	18	ϑℓ	ϑℓ	NOUN
ejpam-6802	233	19	)	)	PUNCT
ejpam-6802	233	20	≤	≤	NOUN
ejpam-6802	233	21	(	(	PUNCT
ejpam-6802	233	22	α(q0	α(q0	NOUN
ejpam-6802	233	23	,	,	PUNCT
ejpam-6802	233	24	ϑ0	ϑ0	NOUN
ejpam-6802	233	25	)	)	PUNCT
ejpam-6802	233	26	)	)	PUNCT
ejpam-6802	233	27	℘2ℓ	℘2ℓ	VERB
ejpam-6802	233	28	.	.	PUNCT
ejpam-6802	234	1	for	for	ADP
ejpam-6802	234	2	all	all	DET
ejpam-6802	234	3	r	r	NOUN
ejpam-6802	234	4	,	,	PUNCT
ejpam-6802	234	5	ℓ	ℓ	PROPN
ejpam-6802	234	6	∈	∈	PROPN
ejpam-6802	234	7	n	n	CCONJ
ejpam-6802	234	8	,	,	PUNCT
ejpam-6802	234	9	case	case	NOUN
ejpam-6802	234	10	1	1	NUM
ejpam-6802	234	11	.	.	PUNCT
ejpam-6802	235	1	if	if	SCONJ
ejpam-6802	235	2	r	r	NOUN
ejpam-6802	235	3	>	>	X
ejpam-6802	235	4	ℓ	ℓ	PROPN
ejpam-6802	235	5	,	,	PUNCT
ejpam-6802	235	6	α(qℓ	α(qℓ	PROPN
ejpam-6802	235	7	,	,	PUNCT
ejpam-6802	235	8	ϑr	ϑr	PROPN
ejpam-6802	235	9	)	)	PUNCT
ejpam-6802	235	10	≤α(qℓ	≤α(qℓ	NOUN
ejpam-6802	235	11	,	,	PUNCT
ejpam-6802	235	12	ϑℓ)α(qℓ+1	ϑℓ)α(qℓ+1	ADV
ejpam-6802	235	13	,	,	PUNCT
ejpam-6802	235	14	ϑℓ)α(qℓ+1	ϑℓ)α(qℓ+1	ADV
ejpam-6802	235	15	,	,	PUNCT
ejpam-6802	235	16	ϑr	ϑr	ADJ
ejpam-6802	235	17	)	)	PUNCT
ejpam-6802	235	18	≤(α(q0	≤(α(q0	ADJ
ejpam-6802	235	19	,	,	PUNCT
ejpam-6802	235	20	ϑ0	ϑ0	NOUN
ejpam-6802	235	21	)	)	PUNCT
ejpam-6802	235	22	)	)	PUNCT
ejpam-6802	236	1	℘2ℓ	℘2ℓ	PROPN
ejpam-6802	236	2	(	(	PUNCT
ejpam-6802	236	3	α(q0	α(q0	NOUN
ejpam-6802	236	4	,	,	PUNCT
ejpam-6802	236	5	ϑ0	ϑ0	NOUN
ejpam-6802	236	6	)	)	PUNCT
ejpam-6802	236	7	)	)	PUNCT
ejpam-6802	236	8	℘2ℓ+1	℘2ℓ+1	ADP
ejpam-6802	236	9	α(qℓ+1	α(qℓ+1	ADJ
ejpam-6802	236	10	,	,	PUNCT
ejpam-6802	236	11	ϑr	ϑr	ADJ
ejpam-6802	236	12	)	)	PUNCT
ejpam-6802	236	13	≤(α(q0	≤(α(q0	ADJ
ejpam-6802	236	14	,	,	PUNCT
ejpam-6802	236	15	ϑ0	ϑ0	NOUN
ejpam-6802	236	16	)	)	PUNCT
ejpam-6802	236	17	)	)	PUNCT
ejpam-6802	236	18	(	(	PUNCT
ejpam-6802	236	19	℘2ℓ+℘2ℓ+1)α(qℓ+1	℘2ℓ+℘2ℓ+1)α(qℓ+1	X
ejpam-6802	236	20	,	,	PUNCT
ejpam-6802	236	21	ϑℓ+1	ϑℓ+1	NUM
ejpam-6802	236	22	)	)	PUNCT
ejpam-6802	236	23	α(qℓ+2	α(qℓ+2	PROPN
ejpam-6802	236	24	,	,	PUNCT
ejpam-6802	236	25	ϑℓ+1)α(qℓ+2	ϑℓ+1)α(qℓ+2	NOUN
ejpam-6802	236	26	,	,	PUNCT
ejpam-6802	236	27	ϑr	ϑr	NOUN
ejpam-6802	236	28	)	)	PUNCT
ejpam-6802	236	29	≤(α(q0	≤(α(q0	ADJ
ejpam-6802	236	30	,	,	PUNCT
ejpam-6802	236	31	ϑ0	ϑ0	NOUN
ejpam-6802	236	32	)	)	PUNCT
ejpam-6802	236	33	)	)	PUNCT
ejpam-6802	236	34	(	(	PUNCT
ejpam-6802	236	35	℘2ℓ+℘2ℓ+1)(α(q0	℘2ℓ+℘2ℓ+1)(α(q0	X
ejpam-6802	236	36	,	,	PUNCT
ejpam-6802	236	37	ϑ0	ϑ0	NOUN
ejpam-6802	236	38	)	)	PUNCT
ejpam-6802	236	39	)	)	PUNCT
ejpam-6802	237	1	℘2ℓ+2	℘2ℓ+2	PROPN
ejpam-6802	237	2	(	(	PUNCT
ejpam-6802	237	3	α(q0	α(q0	NOUN
ejpam-6802	237	4	,	,	PUNCT
ejpam-6802	237	5	ϑ0	ϑ0	NOUN
ejpam-6802	237	6	)	)	PUNCT
ejpam-6802	237	7	)	)	PUNCT
ejpam-6802	237	8	℘2ℓ+3	℘2ℓ+3	PROPN
ejpam-6802	237	9	α(qℓ+2	α(qℓ+2	PROPN
ejpam-6802	237	10	,	,	PUNCT
ejpam-6802	237	11	ϑr	ϑr	PROPN
ejpam-6802	237	12	)	)	PUNCT
ejpam-6802	237	13	...	...	PUNCT
ejpam-6802	238	1	≤α(q0	≤α(q0	ADP
ejpam-6802	238	2	,	,	PUNCT
ejpam-6802	238	3	ϑ0	ϑ0	NOUN
ejpam-6802	238	4	)	)	PUNCT
ejpam-6802	238	5	℘2ℓ(1+℘+℘2+℘3	℘2ℓ(1+℘+℘2+℘3	NOUN
ejpam-6802	238	6	+	+	PROPN
ejpam-6802	238	7	...	...	PUNCT
ejpam-6802	238	8	)	)	PUNCT
ejpam-6802	238	9	=(	=(	PROPN
ejpam-6802	238	10	α(q0	α(q0	NOUN
ejpam-6802	238	11	,	,	PUNCT
ejpam-6802	238	12	ϑ0	ϑ0	NOUN
ejpam-6802	238	13	)	)	PUNCT
ejpam-6802	238	14	)	)	PUNCT
ejpam-6802	239	1	℘2ℓ	℘2ℓ	PROPN
ejpam-6802	239	2	(	(	PUNCT
ejpam-6802	239	3	1	1	NUM
ejpam-6802	239	4	1−℘	1−℘	PROPN
ejpam-6802	239	5	)	)	PUNCT
ejpam-6802	239	6	.	.	PUNCT
ejpam-6802	240	1	since	since	SCONJ
ejpam-6802	240	2	℘	℘	PROPN
ejpam-6802	240	3	<	<	X
ejpam-6802	240	4	1	1	NUM
ejpam-6802	240	5	,	,	PUNCT
ejpam-6802	240	6	limℓ,r→++∞	limℓ,r→++∞	NOUN
ejpam-6802	240	7	α(qℓ	α(qℓ	PROPN
ejpam-6802	240	8	,	,	PUNCT
ejpam-6802	240	9	ϑr	ϑr	ADJ
ejpam-6802	240	10	)	)	PUNCT
ejpam-6802	240	11	=	=	SYM
ejpam-6802	240	12	1	1	X
ejpam-6802	240	13	.	.	PUNCT
ejpam-6802	240	14	case	case	NOUN
ejpam-6802	240	15	2	2	NUM
ejpam-6802	240	16	.	.	PUNCT
ejpam-6802	241	1	if	if	SCONJ
ejpam-6802	241	2	r	r	NOUN
ejpam-6802	241	3	<	<	X
ejpam-6802	241	4	ℓ	ℓ	NUM
ejpam-6802	241	5	,	,	PUNCT
ejpam-6802	241	6	we	we	PRON
ejpam-6802	241	7	have	have	VERB
ejpam-6802	241	8	α(qℓ	α(qℓ	NUM
ejpam-6802	241	9	,	,	PUNCT
ejpam-6802	241	10	ϑr	ϑr	ADJ
ejpam-6802	241	11	)	)	PUNCT
ejpam-6802	241	12	≤α(qr+1	≤α(qr+1	PROPN
ejpam-6802	241	13	,	,	PUNCT
ejpam-6802	241	14	ϑr)α(qr+1	ϑr)α(qr+1	PROPN
ejpam-6802	241	15	,	,	PUNCT
ejpam-6802	241	16	ϑr+1)α(qℓ	ϑr+1)α(qℓ	PROPN
ejpam-6802	241	17	,	,	PUNCT
ejpam-6802	241	18	ϑr+1	ϑr+1	NOUN
ejpam-6802	241	19	)	)	PUNCT
ejpam-6802	241	20	≤(α(q0	≤(α(q0	NOUN
ejpam-6802	241	21	,	,	PUNCT
ejpam-6802	241	22	ϑ0	ϑ0	NOUN
ejpam-6802	241	23	)	)	PUNCT
ejpam-6802	241	24	)	)	PUNCT
ejpam-6802	242	1	℘2r+1	℘2r+1	PROPN
ejpam-6802	242	2	(	(	PUNCT
ejpam-6802	242	3	α(q0	α(q0	NOUN
ejpam-6802	242	4	,	,	PUNCT
ejpam-6802	242	5	ϑ0	ϑ0	NOUN
ejpam-6802	242	6	)	)	PUNCT
ejpam-6802	242	7	)	)	PUNCT
ejpam-6802	242	8	℘2r+2	℘2r+2	PUNCT
ejpam-6802	242	9	α(qℓ	α(qℓ	PROPN
ejpam-6802	242	10	,	,	PUNCT
ejpam-6802	242	11	ϑr+1	ϑr+1	NOUN
ejpam-6802	242	12	)	)	PUNCT
ejpam-6802	242	13	≤(α(q0	≤(α(q0	NOUN
ejpam-6802	242	14	,	,	PUNCT
ejpam-6802	242	15	ϑ0	ϑ0	NOUN
ejpam-6802	242	16	)	)	PUNCT
ejpam-6802	242	17	)	)	PUNCT
ejpam-6802	242	18	(	(	PUNCT
ejpam-6802	242	19	℘2r+1+℘2r+2)α(qr+2	℘2r+1+℘2r+2)α(qr+2	NOUN
ejpam-6802	242	20	,	,	PUNCT
ejpam-6802	242	21	ϑr+1	ϑr+1	NOUN
ejpam-6802	242	22	)	)	PUNCT
ejpam-6802	242	23	α(qr+2	α(qr+2	PROPN
ejpam-6802	242	24	,	,	PUNCT
ejpam-6802	242	25	ϑr+2)α(qℓ	ϑr+2)α(qℓ	NOUN
ejpam-6802	242	26	,	,	PUNCT
ejpam-6802	242	27	ϑr+2	ϑr+2	NOUN
ejpam-6802	242	28	)	)	PUNCT
ejpam-6802	242	29	...	...	PUNCT
ejpam-6802	243	1	≤(α(q0	≤(α(q0	CCONJ
ejpam-6802	243	2	,	,	PUNCT
ejpam-6802	243	3	ϑ0	ϑ0	NOUN
ejpam-6802	243	4	)	)	PUNCT
ejpam-6802	243	5	)	)	PUNCT
ejpam-6802	243	6	(	(	PUNCT
ejpam-6802	243	7	℘2r+1+℘2r+2+℘2r+3+℘2r+4	℘2r+1+℘2r+2+℘2r+3+℘2r+4	NOUN
ejpam-6802	243	8	+	+	NOUN
ejpam-6802	243	9	...	...	PUNCT
ejpam-6802	243	10	)	)	PUNCT
ejpam-6802	243	11	=(	=(	PROPN
ejpam-6802	243	12	α(q0	α(q0	NOUN
ejpam-6802	243	13	,	,	PUNCT
ejpam-6802	243	14	ϑ0	ϑ0	NOUN
ejpam-6802	243	15	)	)	PUNCT
ejpam-6802	243	16	)	)	PUNCT
ejpam-6802	244	1	℘2r+1	℘2r+1	PROPN
ejpam-6802	244	2	(	(	PUNCT
ejpam-6802	244	3	1	1	NUM
ejpam-6802	244	4	1−℘	1−℘	PROPN
ejpam-6802	244	5	)	)	PUNCT
ejpam-6802	244	6	.	.	PUNCT
ejpam-6802	245	1	again	again	ADV
ejpam-6802	245	2	,	,	PUNCT
ejpam-6802	245	3	since	since	SCONJ
ejpam-6802	245	4	℘	℘	PROPN
ejpam-6802	245	5	<	<	X
ejpam-6802	245	6	1	1	NUM
ejpam-6802	245	7	,	,	PUNCT
ejpam-6802	245	8	limℓ,r→++∞	limℓ,r→++∞	NOUN
ejpam-6802	245	9	α(qℓ	α(qℓ	PROPN
ejpam-6802	245	10	,	,	PUNCT
ejpam-6802	245	11	ϑr	ϑr	ADJ
ejpam-6802	245	12	)	)	PUNCT
ejpam-6802	245	13	=	=	SYM
ejpam-6802	245	14	1	1	X
ejpam-6802	245	15	.	.	PUNCT
ejpam-6802	246	1	therefore	therefore	ADV
ejpam-6802	246	2	,	,	PUNCT
ejpam-6802	246	3	(	(	PUNCT
ejpam-6802	246	4	{	{	PUNCT
ejpam-6802	246	5	qℓ	qℓ	NOUN
ejpam-6802	246	6	}	}	PUNCT
ejpam-6802	246	7	,	,	PUNCT
ejpam-6802	246	8	{	{	PUNCT
ejpam-6802	246	9	ϑm	ϑm	NOUN
ejpam-6802	246	10	}	}	PUNCT
ejpam-6802	246	11	)	)	PUNCT
ejpam-6802	246	12	is	be	AUX
ejpam-6802	246	13	a	a	DET
ejpam-6802	246	14	cauchy	cauchy	ADJ
ejpam-6802	246	15	bisequence	bisequence	NOUN
ejpam-6802	246	16	.	.	PUNCT
ejpam-6802	247	1	since	since	SCONJ
ejpam-6802	247	2	(	(	PUNCT
ejpam-6802	247	3	b	b	X
ejpam-6802	247	4	,	,	PUNCT
ejpam-6802	247	5	f	f	PROPN
ejpam-6802	247	6	,	,	PUNCT
ejpam-6802	247	7	α	α	PROPN
ejpam-6802	247	8	)	)	PUNCT
ejpam-6802	247	9	is	be	AUX
ejpam-6802	247	10	complete	complete	ADJ
ejpam-6802	247	11	,	,	PUNCT
ejpam-6802	247	12	{	{	PUNCT
ejpam-6802	247	13	qℓ	qℓ	NOUN
ejpam-6802	247	14	}	}	PUNCT
ejpam-6802	247	15	→	→	SYM
ejpam-6802	247	16	q∗	q∗	NOUN
ejpam-6802	247	17	,	,	PUNCT
ejpam-6802	247	18	{	{	PUNCT
ejpam-6802	247	19	ϑr	ϑr	VERB
ejpam-6802	247	20	}	}	PUNCT
ejpam-6802	247	21	→	→	SYM
ejpam-6802	247	22	q∗	q∗	NOUN
ejpam-6802	247	23	,	,	PUNCT
ejpam-6802	247	24	where	where	SCONJ
ejpam-6802	247	25	q∗	q∗	PROPN
ejpam-6802	247	26	∈	∈	PROPN
ejpam-6802	247	27	b	b	X
ejpam-6802	247	28	∪	∪	PROPN
ejpam-6802	247	29	f	f	PROPN
ejpam-6802	247	30	.	.	PUNCT
ejpam-6802	248	1	also	also	ADV
ejpam-6802	248	2	,	,	PUNCT
ejpam-6802	248	3	{	{	PUNCT
ejpam-6802	248	4	s(q2ℓ	s(q2ℓ	NOUN
ejpam-6802	248	5	)	)	PUNCT
ejpam-6802	248	6	}	}	PUNCT
ejpam-6802	248	7	=	=	SYM
ejpam-6802	248	8	{	{	PUNCT
ejpam-6802	248	9	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	248	10	}	}	PUNCT
ejpam-6802	248	11	→	→	SYM
ejpam-6802	248	12	q∗	q∗	NOUN
ejpam-6802	248	13	∈	∈	PROPN
ejpam-6802	248	14	b	b	NOUN
ejpam-6802	248	15	∩	∩	PROPN
ejpam-6802	248	16	f	f	PROPN
ejpam-6802	248	17	⇒	⇒	NOUN
ejpam-6802	248	18	s(q2ℓ	s(q2ℓ	PROPN
ejpam-6802	248	19	)	)	PUNCT
ejpam-6802	248	20	has	have	AUX
ejpam-6802	248	21	a	a	DET
ejpam-6802	248	22	unique	unique	ADJ
ejpam-6802	248	23	limit	limit	NOUN
ejpam-6802	248	24	q∗	q∗	NOUN
ejpam-6802	248	25	,	,	PUNCT
ejpam-6802	248	26	and	and	CCONJ
ejpam-6802	248	27	{	{	PUNCT
ejpam-6802	248	28	qℓ	qℓ	NOUN
ejpam-6802	248	29	}	}	PUNCT
ejpam-6802	248	30	→	→	SYM
ejpam-6802	248	31	q∗	q∗	NOUN
ejpam-6802	248	32	⇒	⇒	NOUN
ejpam-6802	248	33	{	{	PUNCT
ejpam-6802	248	34	q2ℓ	q2ℓ	NOUN
ejpam-6802	248	35	}	}	PUNCT
ejpam-6802	248	36	→	→	SYM
ejpam-6802	248	37	q∗.	q∗.	NOUN
ejpam-6802	248	38	since	since	SCONJ
ejpam-6802	248	39	s	s	PROPN
ejpam-6802	248	40	is	be	AUX
ejpam-6802	248	41	a	a	DET
ejpam-6802	248	42	continuous	continuous	ADJ
ejpam-6802	248	43	,	,	PUNCT
ejpam-6802	248	44	s(q2ℓ	s(q2ℓ	NOUN
ejpam-6802	248	45	)	)	PUNCT
ejpam-6802	248	46	→	→	SYM
ejpam-6802	248	47	sq∗.	sq∗.	NOUN
ejpam-6802	248	48	(	(	PUNCT
ejpam-6802	248	49	i.e	i.e	NOUN
ejpam-6802	248	50	)	)	PUNCT
ejpam-6802	248	51	sq∗	sq∗	NOUN
ejpam-6802	248	52	=	=	PUNCT
ejpam-6802	248	53	q∗.	q∗.	NOUN
ejpam-6802	248	54	correspondingly	correspondingly	ADV
ejpam-6802	248	55	,	,	PUNCT
ejpam-6802	248	56	{	{	PUNCT
ejpam-6802	248	57	ω(ϑ2ℓ+1	ω(ϑ2ℓ+1	NUM
ejpam-6802	248	58	)	)	PUNCT
ejpam-6802	248	59	}	}	PUNCT
ejpam-6802	248	60	=	=	SYM
ejpam-6802	248	61	{	{	PUNCT
ejpam-6802	248	62	q2ℓ+2	q2ℓ+2	NOUN
ejpam-6802	248	63	}	}	PUNCT
ejpam-6802	248	64	→	→	SYM
ejpam-6802	248	65	q∗	q∗	NOUN
ejpam-6802	248	66	∈	∈	PROPN
ejpam-6802	248	67	b	b	NOUN
ejpam-6802	248	68	∩	∩	PROPN
ejpam-6802	248	69	f	f	PROPN
ejpam-6802	248	70	⇒	⇒	PROPN
ejpam-6802	248	71	ω(ϑ2ℓ+1	ω(ϑ2ℓ+1	NUM
ejpam-6802	248	72	)	)	PUNCT
ejpam-6802	248	73	has	have	VERB
ejpam-6802	248	74	a	a	DET
ejpam-6802	248	75	unique	unique	ADJ
ejpam-6802	248	76	limit	limit	NOUN
ejpam-6802	248	77	q∗	q∗	NOUN
ejpam-6802	248	78	,	,	PUNCT
ejpam-6802	248	79	and	and	CCONJ
ejpam-6802	248	80	{	{	PUNCT
ejpam-6802	248	81	ϑℓ	ϑℓ	NOUN
ejpam-6802	248	82	}	}	PUNCT
ejpam-6802	248	83	→	→	SYM
ejpam-6802	248	84	q∗	q∗	NOUN
ejpam-6802	248	85	⇒	⇒	NOUN
ejpam-6802	248	86	{	{	PUNCT
ejpam-6802	248	87	ϑ2ℓ+1	ϑ2ℓ+1	NOUN
ejpam-6802	248	88	}	}	PUNCT
ejpam-6802	248	89	→	→	SYM
ejpam-6802	248	90	q∗.	q∗.	NOUN
ejpam-6802	248	91	now	now	ADV
ejpam-6802	248	92	,	,	PUNCT
ejpam-6802	248	93	the	the	DET
ejpam-6802	248	94	stability	stability	NOUN
ejpam-6802	248	95	of	of	ADP
ejpam-6802	248	96	ω	ω	PROPN
ejpam-6802	248	97	⇒	⇒	PROPN
ejpam-6802	248	98	{	{	PUNCT
ejpam-6802	248	99	ω(ϑ2ℓ+1	ω(ϑ2ℓ+1	NUM
ejpam-6802	248	100	)	)	PUNCT
ejpam-6802	248	101	}	}	PUNCT
ejpam-6802	248	102	→	→	PUNCT
ejpam-6802	248	103	ωq∗.	ωq∗.	PROPN
ejpam-6802	248	104	therefore	therefore	ADV
ejpam-6802	248	105	,	,	PUNCT
ejpam-6802	248	106	r.	r.	PROPN
ejpam-6802	248	107	ramaswamy	ramaswamy	PROPN
ejpam-6802	248	108	/	/	SYM
ejpam-6802	248	109	eur	eur	PROPN
ejpam-6802	248	110	.	.	PUNCT
ejpam-6802	249	1	j.	j.	PROPN
ejpam-6802	249	2	pure	pure	PROPN
ejpam-6802	249	3	appl	appl	PROPN
ejpam-6802	249	4	.	.	PROPN
ejpam-6802	249	5	math	math	PROPN
ejpam-6802	249	6	,	,	PUNCT
ejpam-6802	249	7	18	18	NUM
ejpam-6802	249	8	(	(	PUNCT
ejpam-6802	249	9	4	4	NUM
ejpam-6802	249	10	)	)	PUNCT
ejpam-6802	249	11	(	(	PUNCT
ejpam-6802	249	12	2025	2025	NUM
ejpam-6802	249	13	)	)	PUNCT
ejpam-6802	249	14	,	,	PUNCT
ejpam-6802	249	15	6802	6802	NUM
ejpam-6802	249	16	13	13	NUM
ejpam-6802	249	17	of	of	ADP
ejpam-6802	249	18	20	20	NUM
ejpam-6802	249	19	ωq∗	ωq∗	NOUN
ejpam-6802	249	20	=	=	NOUN
ejpam-6802	249	21	q∗.	q∗.	NOUN
ejpam-6802	249	22	let	let	VERB
ejpam-6802	249	23	ϑ∗	ϑ∗	PROPN
ejpam-6802	249	24	∈	∈	PROPN
ejpam-6802	249	25	b	b	NOUN
ejpam-6802	249	26	∩	∩	PROPN
ejpam-6802	249	27	f	f	X
ejpam-6802	249	28	such	such	ADJ
ejpam-6802	249	29	that	that	SCONJ
ejpam-6802	249	30	sϑ∗	sϑ∗	NOUN
ejpam-6802	249	31	=	=	PUNCT
ejpam-6802	249	32	ωϑ∗	ωϑ∗	NOUN
ejpam-6802	249	33	=	=	SYM
ejpam-6802	249	34	ϑ∗	ϑ∗	PROPN
ejpam-6802	249	35	∈	∈	PROPN
ejpam-6802	249	36	b	b	PROPN
ejpam-6802	249	37	∩	∩	PROPN
ejpam-6802	249	38	f	f	PROPN
ejpam-6802	249	39	.	.	PUNCT
ejpam-6802	250	1	then	then	ADV
ejpam-6802	250	2	,	,	PUNCT
ejpam-6802	250	3	we	we	PRON
ejpam-6802	250	4	get	get	VERB
ejpam-6802	250	5	α(ϑ∗	α(ϑ∗	NOUN
ejpam-6802	250	6	,	,	PUNCT
ejpam-6802	250	7	q∗	q∗	NOUN
ejpam-6802	250	8	)	)	PUNCT
ejpam-6802	250	9	=	=	SYM
ejpam-6802	250	10	α(sϑ∗,ωq∗	α(sϑ∗,ωq∗	NOUN
ejpam-6802	250	11	)	)	PUNCT
ejpam-6802	250	12	≤	≤	NOUN
ejpam-6802	250	13	(	(	PUNCT
ejpam-6802	250	14	α(q∗,ωq∗)α(sϑ∗	α(q∗,ωq∗)α(sϑ∗	NOUN
ejpam-6802	250	15	,	,	PUNCT
ejpam-6802	250	16	ϑ∗	ϑ∗	PROPN
ejpam-6802	250	17	)	)	PUNCT
ejpam-6802	250	18	α(q∗	α(q∗	NOUN
ejpam-6802	250	19	,	,	PUNCT
ejpam-6802	250	20	ϑ∗	ϑ∗	PROPN
ejpam-6802	250	21	)	)	PUNCT
ejpam-6802	250	22	)	)	PUNCT
ejpam-6802	251	1	β	β	X
ejpam-6802	251	2	(	(	PUNCT
ejpam-6802	251	3	α(q∗	α(q∗	ADP
ejpam-6802	251	4	,	,	PUNCT
ejpam-6802	251	5	ϑ∗))γ(α(q∗,ωq∗)α(sϑ∗	ϑ∗))γ(α(q∗,ωq∗)α(sϑ∗	ADJ
ejpam-6802	251	6	,	,	PUNCT
ejpam-6802	251	7	ϑ∗))ı	ϑ∗))ı	NOUN
ejpam-6802	251	8	=	=	PUNCT
ejpam-6802	251	9	(	(	PUNCT
ejpam-6802	251	10	α(q∗	α(q∗	ADP
ejpam-6802	251	11	,	,	PUNCT
ejpam-6802	251	12	q∗)α(ϑ∗	q∗)α(ϑ∗	PROPN
ejpam-6802	251	13	,	,	PUNCT
ejpam-6802	251	14	ϑ∗	ϑ∗	PROPN
ejpam-6802	251	15	)	)	PUNCT
ejpam-6802	251	16	α(q∗	α(q∗	NOUN
ejpam-6802	251	17	,	,	PUNCT
ejpam-6802	251	18	ϑ∗	ϑ∗	PROPN
ejpam-6802	251	19	)	)	PUNCT
ejpam-6802	251	20	)	)	PUNCT
ejpam-6802	252	1	β	β	X
ejpam-6802	252	2	(	(	PUNCT
ejpam-6802	252	3	α(q∗	α(q∗	ADP
ejpam-6802	252	4	,	,	PUNCT
ejpam-6802	252	5	ϑ∗))γ(α(q∗	ϑ∗))γ(α(q∗	NOUN
ejpam-6802	252	6	,	,	PUNCT
ejpam-6802	252	7	q∗)α(ϑ∗	q∗)α(ϑ∗	NOUN
ejpam-6802	252	8	,	,	PUNCT
ejpam-6802	252	9	ϑ∗))ı	ϑ∗))ı	NOUN
ejpam-6802	252	10	=	=	PUNCT
ejpam-6802	252	11	(	(	PUNCT
ejpam-6802	252	12	α(q∗	α(q∗	ADP
ejpam-6802	252	13	,	,	PUNCT
ejpam-6802	252	14	ϑ∗))γ−β	ϑ∗))γ−β	NOUN
ejpam-6802	252	15	.	.	PUNCT
ejpam-6802	253	1	hence	hence	ADV
ejpam-6802	253	2	,	,	PUNCT
ejpam-6802	253	3	q∗	q∗	PROPN
ejpam-6802	253	4	=	=	SYM
ejpam-6802	253	5	ϑ∗.	ϑ∗.	PRON
ejpam-6802	253	6	theorem	theorem	VERB
ejpam-6802	253	7	3.5	3.5	NUM
ejpam-6802	253	8	.	.	PUNCT
ejpam-6802	254	1	let	let	VERB
ejpam-6802	254	2	(	(	PUNCT
ejpam-6802	254	3	b	b	X
ejpam-6802	254	4	,	,	PUNCT
ejpam-6802	254	5	f	f	PROPN
ejpam-6802	254	6	,	,	PUNCT
ejpam-6802	254	7	α	α	PROPN
ejpam-6802	254	8	)	)	PUNCT
ejpam-6802	254	9	be	be	VERB
ejpam-6802	254	10	a	a	DET
ejpam-6802	254	11	complete	complete	ADJ
ejpam-6802	254	12	mcbms	mcbms	NOUN
ejpam-6802	254	13	,	,	PUNCT
ejpam-6802	254	14	z	z	PROPN
ejpam-6802	254	15	be	be	AUX
ejpam-6802	254	16	a	a	DET
ejpam-6802	254	17	cone	cone	NOUN
ejpam-6802	254	18	with	with	ADP
ejpam-6802	254	19	constant	constant	ADJ
ejpam-6802	254	20	w	w	NOUN
ejpam-6802	254	21	and	and	CCONJ
ejpam-6802	254	22	ω	ω	NUM
ejpam-6802	254	23	,	,	PUNCT
ejpam-6802	254	24	s	s	PART
ejpam-6802	254	25	:	:	PUNCT
ejpam-6802	254	26	(	(	PUNCT
ejpam-6802	254	27	b	b	X
ejpam-6802	254	28	,	,	PUNCT
ejpam-6802	254	29	f	f	PROPN
ejpam-6802	254	30	,	,	PUNCT
ejpam-6802	254	31	α	α	NOUN
ejpam-6802	254	32	)	)	PUNCT
ejpam-6802	254	33	⇄	⇄	PROPN
ejpam-6802	254	34	(	(	PUNCT
ejpam-6802	254	35	b	b	NOUN
ejpam-6802	254	36	,	,	PUNCT
ejpam-6802	254	37	f	f	PROPN
ejpam-6802	254	38	,	,	PUNCT
ejpam-6802	254	39	α	α	PROPN
ejpam-6802	254	40	)	)	PUNCT
ejpam-6802	254	41	be	be	VERB
ejpam-6802	254	42	a	a	DET
ejpam-6802	254	43	contravariant	contravariant	ADJ
ejpam-6802	254	44	function	function	NOUN
ejpam-6802	254	45	satisfying	satisfy	VERB
ejpam-6802	254	46	α(sϑ,ωq	α(sϑ,ωq	PROPN
ejpam-6802	254	47	)	)	PUNCT
ejpam-6802	254	48	≤	≤	NOUN
ejpam-6802	254	49	(	(	PUNCT
ejpam-6802	254	50	α(q	α(q	PROPN
ejpam-6802	254	51	,	,	PUNCT
ejpam-6802	254	52	ωq)α(q	ωq)α(q	NUM
ejpam-6802	254	53	,	,	PUNCT
ejpam-6802	254	54	sϑ)α(sϑ	sϑ)α(sϑ	NOUN
ejpam-6802	254	55	,	,	PUNCT
ejpam-6802	254	56	ϑ)α(ϑ,ωq	ϑ)α(ϑ,ωq	NOUN
ejpam-6802	254	57	)	)	PUNCT
ejpam-6802	254	58	α(q	α(q	NUM
ejpam-6802	254	59	,	,	PUNCT
ejpam-6802	254	60	sϑ)α(ϑ,ωq	sϑ)α(ϑ,ωq	NOUN
ejpam-6802	254	61	)	)	PUNCT
ejpam-6802	254	62	)	)	PUNCT
ejpam-6802	255	1	β	β	X
ejpam-6802	255	2	,	,	PUNCT
ejpam-6802	255	3	(	(	PUNCT
ejpam-6802	255	4	5	5	NUM
ejpam-6802	255	5	)	)	PUNCT
ejpam-6802	255	6	for	for	ADP
ejpam-6802	255	7	all	all	PRON
ejpam-6802	255	8	(	(	PUNCT
ejpam-6802	255	9	q	q	ADJ
ejpam-6802	255	10	,	,	PUNCT
ejpam-6802	255	11	ϑ	ϑ	NOUN
ejpam-6802	255	12	)	)	PUNCT
ejpam-6802	255	13	∈	∈	PROPN
ejpam-6802	255	14	b	b	PROPN
ejpam-6802	255	15	×	×	PROPN
ejpam-6802	255	16	f	f	X
ejpam-6802	255	17	,	,	PUNCT
ejpam-6802	255	18	with	with	ADP
ejpam-6802	255	19	q	q	PROPN
ejpam-6802	255	20	̸=	̸=	PROPN
ejpam-6802	255	21	ϑ	ϑ	X
ejpam-6802	255	22	and	and	CCONJ
ejpam-6802	255	23	β	β	X
ejpam-6802	255	24	∈	∈	PROPN
ejpam-6802	255	25	(	(	PUNCT
ejpam-6802	255	26	0	0	NUM
ejpam-6802	255	27	,	,	PUNCT
ejpam-6802	255	28	12	12	NUM
ejpam-6802	255	29	)	)	PUNCT
ejpam-6802	255	30	.	.	PUNCT
ejpam-6802	256	1	then	then	ADV
ejpam-6802	256	2	ω	ω	NUM
ejpam-6802	256	3	,	,	PUNCT
ejpam-6802	256	4	s	s	PART
ejpam-6802	256	5	:	:	PUNCT
ejpam-6802	256	6	b	b	X
ejpam-6802	256	7	∪	∪	X
ejpam-6802	256	8	f	f	PROPN
ejpam-6802	256	9	→	→	SYM
ejpam-6802	256	10	b	b	X
ejpam-6802	256	11	∪	∪	X
ejpam-6802	256	12	f	f	X
ejpam-6802	256	13	have	have	VERB
ejpam-6802	256	14	a	a	DET
ejpam-6802	256	15	unique	unique	ADJ
ejpam-6802	256	16	cfp	cfp	NOUN
ejpam-6802	256	17	.	.	PUNCT
ejpam-6802	257	1	proof	proof	NOUN
ejpam-6802	257	2	.	.	PUNCT
ejpam-6802	258	1	let	let	VERB
ejpam-6802	258	2	q0	q0	PROPN
ejpam-6802	258	3	∈	∈	PROPN
ejpam-6802	258	4	b	b	PROPN
ejpam-6802	258	5	&	&	CCONJ
ejpam-6802	258	6	ϑ0	ϑ0	PROPN
ejpam-6802	258	7	∈	∈	PROPN
ejpam-6802	258	8	f	f	X
ejpam-6802	258	9	then	then	ADV
ejpam-6802	258	10	for	for	SCONJ
ejpam-6802	258	11	each	each	DET
ejpam-6802	258	12	ℓ	ℓ	PROPN
ejpam-6802	258	13	∈	∈	PROPN
ejpam-6802	258	14	n	n	PART
ejpam-6802	258	15	∪	∪	X
ejpam-6802	258	16	{	{	PUNCT
ejpam-6802	258	17	0	0	NUM
ejpam-6802	258	18	}	}	PUNCT
ejpam-6802	258	19	,	,	PUNCT
ejpam-6802	258	20	define	define	VERB
ejpam-6802	258	21	sq2ℓ	sq2ℓ	NOUN
ejpam-6802	258	22	=	=	SYM
ejpam-6802	258	23	ϑ2ℓ,ωq2ℓ+1	ϑ2ℓ,ωq2ℓ+1	NOUN
ejpam-6802	258	24	=	=	PUNCT
ejpam-6802	258	25	ϑ2ℓ+1,sϑ2ℓ	ϑ2ℓ+1,sϑ2ℓ	NOUN
ejpam-6802	258	26	=	=	SYM
ejpam-6802	258	27	q2ℓ+1,ωϑ2ℓ+1	q2ℓ+1,ωϑ2ℓ+1	ADJ
ejpam-6802	258	28	=	=	SYM
ejpam-6802	258	29	q2ℓ+2	q2ℓ+2	PROPN
ejpam-6802	258	30	.	.	PUNCT
ejpam-6802	258	31	now	now	ADV
ejpam-6802	258	32	by	by	ADP
ejpam-6802	258	33	(	(	PUNCT
ejpam-6802	258	34	5	5	NUM
ejpam-6802	258	35	)	)	PUNCT
ejpam-6802	258	36	,	,	PUNCT
ejpam-6802	258	37	we	we	PRON
ejpam-6802	258	38	get	get	VERB
ejpam-6802	258	39	α(q2ℓ+1	α(q2ℓ+1	NUM
ejpam-6802	258	40	,	,	PUNCT
ejpam-6802	258	41	ϑ2ℓ+1	ϑ2ℓ+1	NOUN
ejpam-6802	258	42	)	)	PUNCT
ejpam-6802	258	43	=	=	NOUN
ejpam-6802	258	44	α(sϑ2ℓ,ωq2ℓ+1	α(sϑ2ℓ,ωq2ℓ+1	NUM
ejpam-6802	258	45	)	)	PUNCT
ejpam-6802	258	46	≤	≤	NOUN
ejpam-6802	258	47	(	(	PUNCT
ejpam-6802	258	48	α(q2ℓ+1,ωq2ℓ+1)α(q2ℓ+1,sϑ2ℓ)α(sϑ2ℓ	α(q2ℓ+1,ωq2ℓ+1)α(q2ℓ+1,sϑ2ℓ)α(sϑ2ℓ	NOUN
ejpam-6802	258	49	,	,	PUNCT
ejpam-6802	258	50	ϑ2ℓ)α(ϑ2ℓ,ωq2ℓ+1	ϑ2ℓ)α(ϑ2ℓ,ωq2ℓ+1	NOUN
ejpam-6802	258	51	)	)	PUNCT
ejpam-6802	258	52	α(q2ℓ+1,sϑ2ℓ)α(ϑ2ℓ,ωq2ℓ+1	α(q2ℓ+1,sϑ2ℓ)α(ϑ2ℓ,ωq2ℓ+1	NUM
ejpam-6802	258	53	)	)	PUNCT
ejpam-6802	258	54	)	)	PUNCT
ejpam-6802	259	1	β	β	X
ejpam-6802	259	2	=	=	PUNCT
ejpam-6802	259	3	(	(	PUNCT
ejpam-6802	259	4	α(q2ℓ+1	α(q2ℓ+1	NUM
ejpam-6802	259	5	,	,	PUNCT
ejpam-6802	259	6	ϑ2ℓ+1)α(q2ℓ+1	ϑ2ℓ+1)α(q2ℓ+1	ADJ
ejpam-6802	259	7	,	,	PUNCT
ejpam-6802	259	8	q2ℓ+1)α(q2ℓ+1	q2ℓ+1)α(q2ℓ+1	NOUN
ejpam-6802	259	9	,	,	PUNCT
ejpam-6802	259	10	ϑ2ℓ)α(ϑ2ℓ	ϑ2ℓ)α(ϑ2ℓ	NOUN
ejpam-6802	259	11	,	,	PUNCT
ejpam-6802	259	12	ϑ2ℓ+1	ϑ2ℓ+1	NOUN
ejpam-6802	259	13	)	)	PUNCT
ejpam-6802	259	14	α(q2ℓ+1	α(q2ℓ+1	NUM
ejpam-6802	259	15	,	,	PUNCT
ejpam-6802	259	16	q2ℓ+1)α(ϑ2ℓ	q2ℓ+1)α(ϑ2ℓ	NOUN
ejpam-6802	259	17	,	,	PUNCT
ejpam-6802	259	18	ϑ2ℓ+1	ϑ2ℓ+1	NOUN
ejpam-6802	259	19	)	)	PUNCT
ejpam-6802	259	20	)	)	PUNCT
ejpam-6802	260	1	β	β	PROPN
ejpam-6802	260	2	=(	=(	PROPN
ejpam-6802	260	3	α(q2ℓ+1	α(q2ℓ+1	NUM
ejpam-6802	260	4	,	,	PUNCT
ejpam-6802	260	5	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	260	6	)	)	PUNCT
ejpam-6802	260	7	)	)	PUNCT
ejpam-6802	260	8	β	β	PROPN
ejpam-6802	260	9	1−β	1−β	NUM
ejpam-6802	260	10	,	,	PUNCT
ejpam-6802	260	11	which	which	PRON
ejpam-6802	260	12	implies	imply	VERB
ejpam-6802	260	13	that	that	SCONJ
ejpam-6802	260	14	α(q2ℓ+1	α(q2ℓ+1	NUM
ejpam-6802	260	15	,	,	PUNCT
ejpam-6802	260	16	ϑ2ℓ+1	ϑ2ℓ+1	NOUN
ejpam-6802	260	17	)	)	PUNCT
ejpam-6802	260	18	≤	≤	NOUN
ejpam-6802	260	19	(	(	PUNCT
ejpam-6802	260	20	α(q2ℓ+1	α(q2ℓ+1	NUM
ejpam-6802	260	21	,	,	PUNCT
ejpam-6802	260	22	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	260	23	)	)	PUNCT
ejpam-6802	260	24	)	)	PUNCT
ejpam-6802	260	25	β	β	PROPN
ejpam-6802	260	26	1−β	1−β	NUM
ejpam-6802	260	27	.	.	PUNCT
ejpam-6802	261	1	(	(	PUNCT
ejpam-6802	261	2	6	6	NUM
ejpam-6802	261	3	)	)	PUNCT
ejpam-6802	261	4	also	also	ADV
ejpam-6802	261	5	,	,	PUNCT
ejpam-6802	261	6	we	we	PRON
ejpam-6802	261	7	have	have	VERB
ejpam-6802	261	8	α(q2ℓ+1	α(q2ℓ+1	NUM
ejpam-6802	261	9	,	,	PUNCT
ejpam-6802	261	10	ϑ2ℓ	ϑ2ℓ	ADJ
ejpam-6802	261	11	)	)	PUNCT
ejpam-6802	261	12	=	=	SYM
ejpam-6802	261	13	α(sϑ2ℓ,ωq2ℓ	α(sϑ2ℓ,ωq2ℓ	NOUN
ejpam-6802	261	14	)	)	PUNCT
ejpam-6802	261	15	≤	≤	NOUN
ejpam-6802	261	16	(	(	PUNCT
ejpam-6802	261	17	α(q2ℓ,ωq2ℓ)α(q2ℓ,sϑ2ℓ)α(sϑ2ℓ	α(q2ℓ,ωq2ℓ)α(q2ℓ,sϑ2ℓ)α(sϑ2ℓ	NOUN
ejpam-6802	261	18	,	,	PUNCT
ejpam-6802	261	19	ϑ2ℓ)α(ϑ2ℓ,ωq2ℓ	ϑ2ℓ)α(ϑ2ℓ,ωq2ℓ	NUM
ejpam-6802	261	20	)	)	PUNCT
ejpam-6802	261	21	α(q2ℓ,sϑ2ℓ)α(ϑ2ℓ,sq2ℓ	α(q2ℓ,sϑ2ℓ)α(ϑ2ℓ,sq2ℓ	PROPN
ejpam-6802	261	22	)	)	PUNCT
ejpam-6802	261	23	)	)	PUNCT
ejpam-6802	262	1	β	β	PROPN
ejpam-6802	262	2	r.	r.	PROPN
ejpam-6802	262	3	ramaswamy	ramaswamy	PROPN
ejpam-6802	262	4	/	/	SYM
ejpam-6802	262	5	eur	eur	PROPN
ejpam-6802	262	6	.	.	PUNCT
ejpam-6802	263	1	j.	j.	PROPN
ejpam-6802	263	2	pure	pure	PROPN
ejpam-6802	263	3	appl	appl	PROPN
ejpam-6802	263	4	.	.	PROPN
ejpam-6802	263	5	math	math	PROPN
ejpam-6802	263	6	,	,	PUNCT
ejpam-6802	263	7	18	18	NUM
ejpam-6802	263	8	(	(	PUNCT
ejpam-6802	263	9	4	4	NUM
ejpam-6802	263	10	)	)	PUNCT
ejpam-6802	263	11	(	(	PUNCT
ejpam-6802	263	12	2025	2025	NUM
ejpam-6802	263	13	)	)	PUNCT
ejpam-6802	263	14	,	,	PUNCT
ejpam-6802	263	15	6802	6802	NUM
ejpam-6802	263	16	14	14	NUM
ejpam-6802	263	17	of	of	ADP
ejpam-6802	263	18	20	20	NUM
ejpam-6802	263	19	=	=	SYM
ejpam-6802	263	20	(	(	PUNCT
ejpam-6802	263	21	α(q2ℓ	α(q2ℓ	NOUN
ejpam-6802	263	22	,	,	PUNCT
ejpam-6802	263	23	ϑ2ℓ)α(q2ℓ	ϑ2ℓ)α(q2ℓ	NOUN
ejpam-6802	263	24	,	,	PUNCT
ejpam-6802	263	25	q2ℓ+1)α(q2ℓ+1	q2ℓ+1)α(q2ℓ+1	NOUN
ejpam-6802	263	26	,	,	PUNCT
ejpam-6802	263	27	ϑ2ℓ)α(ϑ2ℓ	ϑ2ℓ)α(ϑ2ℓ	NOUN
ejpam-6802	263	28	,	,	PUNCT
ejpam-6802	263	29	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	263	30	)	)	PUNCT
ejpam-6802	263	31	α(q2ℓ	α(q2ℓ	NOUN
ejpam-6802	263	32	,	,	PUNCT
ejpam-6802	263	33	q2ℓ+1)α(ϑ2ℓ	q2ℓ+1)α(ϑ2ℓ	NOUN
ejpam-6802	263	34	,	,	PUNCT
ejpam-6802	263	35	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	263	36	)	)	PUNCT
ejpam-6802	263	37	)	)	PUNCT
ejpam-6802	264	1	β	β	PROPN
ejpam-6802	264	2	=(	=(	X
ejpam-6802	264	3	α(q2ℓ	α(q2ℓ	NOUN
ejpam-6802	264	4	,	,	PUNCT
ejpam-6802	264	5	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	264	6	)	)	PUNCT
ejpam-6802	264	7	)	)	PUNCT
ejpam-6802	265	1	β	β	PROPN
ejpam-6802	265	2	1−β	1−β	NUM
ejpam-6802	265	3	,	,	PUNCT
ejpam-6802	265	4	which	which	PRON
ejpam-6802	265	5	implies	imply	VERB
ejpam-6802	265	6	that	that	SCONJ
ejpam-6802	265	7	α(q2ℓ+1	α(q2ℓ+1	NUM
ejpam-6802	265	8	,	,	PUNCT
ejpam-6802	265	9	ϑ2ℓ	ϑ2ℓ	ADJ
ejpam-6802	265	10	)	)	PUNCT
ejpam-6802	265	11	≤	≤	NOUN
ejpam-6802	265	12	(	(	PUNCT
ejpam-6802	265	13	α(q2ℓ	α(q2ℓ	NOUN
ejpam-6802	265	14	,	,	PUNCT
ejpam-6802	265	15	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	265	16	)	)	PUNCT
ejpam-6802	265	17	)	)	PUNCT
ejpam-6802	265	18	β	β	PROPN
ejpam-6802	265	19	1−β	1−β	NUM
ejpam-6802	265	20	.	.	PUNCT
ejpam-6802	266	1	(	(	PUNCT
ejpam-6802	266	2	7	7	X
ejpam-6802	266	3	)	)	PUNCT
ejpam-6802	266	4	if	if	SCONJ
ejpam-6802	266	5	we	we	PRON
ejpam-6802	266	6	say	say	VERB
ejpam-6802	266	7	ȷ	ȷ	X
ejpam-6802	266	8	:	:	PUNCT
ejpam-6802	266	9	=	=	PUNCT
ejpam-6802	266	10	β	β	X
ejpam-6802	266	11	1−β	1−β	NUM
ejpam-6802	266	12	,	,	PUNCT
ejpam-6802	266	13	then	then	ADV
ejpam-6802	266	14	we	we	PRON
ejpam-6802	266	15	have	have	VERB
ejpam-6802	266	16	ȷ	ȷ	PRON
ejpam-6802	266	17	∈	∈	NOUN
ejpam-6802	266	18	(	(	PUNCT
ejpam-6802	266	19	0	0	NUM
ejpam-6802	266	20	,	,	PUNCT
ejpam-6802	266	21	1	1	NUM
ejpam-6802	266	22	)	)	PUNCT
ejpam-6802	266	23	since	since	SCONJ
ejpam-6802	266	24	β	β	X
ejpam-6802	266	25	∈	∈	PROPN
ejpam-6802	266	26	(	(	PUNCT
ejpam-6802	266	27	0	0	NUM
ejpam-6802	266	28	,	,	PUNCT
ejpam-6802	266	29	12	12	NUM
ejpam-6802	266	30	)	)	PUNCT
ejpam-6802	266	31	.	.	PUNCT
ejpam-6802	267	1	hence	hence	ADV
ejpam-6802	267	2	,	,	PUNCT
ejpam-6802	267	3	from	from	ADP
ejpam-6802	267	4	the	the	DET
ejpam-6802	267	5	previous	previous	ADJ
ejpam-6802	267	6	two	two	NUM
ejpam-6802	267	7	inequalities	inequality	NOUN
ejpam-6802	267	8	(	(	PUNCT
ejpam-6802	267	9	6	6	NUM
ejpam-6802	267	10	)	)	PUNCT
ejpam-6802	267	11	and	and	CCONJ
ejpam-6802	267	12	(	(	PUNCT
ejpam-6802	267	13	7	7	NUM
ejpam-6802	267	14	)	)	PUNCT
ejpam-6802	267	15	,	,	PUNCT
ejpam-6802	267	16	we	we	PRON
ejpam-6802	267	17	get	get	VERB
ejpam-6802	267	18	α(q2ℓ+1	α(q2ℓ+1	NUM
ejpam-6802	267	19	,	,	PUNCT
ejpam-6802	267	20	ϑ2ℓ+1	ϑ2ℓ+1	NOUN
ejpam-6802	267	21	)	)	PUNCT
ejpam-6802	267	22	≤	≤	NOUN
ejpam-6802	267	23	(	(	PUNCT
ejpam-6802	267	24	α(q0	α(q0	NOUN
ejpam-6802	267	25	,	,	PUNCT
ejpam-6802	267	26	ϑ0	ϑ0	NOUN
ejpam-6802	267	27	)	)	PUNCT
ejpam-6802	267	28	)	)	PUNCT
ejpam-6802	268	1	ȷ4ℓ+2	ȷ4ℓ+2	NOUN
ejpam-6802	268	2	and	and	CCONJ
ejpam-6802	268	3	α(q2ℓ+1	α(q2ℓ+1	NUM
ejpam-6802	268	4	,	,	PUNCT
ejpam-6802	268	5	ϑ2ℓ	ϑ2ℓ	ADJ
ejpam-6802	268	6	)	)	PUNCT
ejpam-6802	268	7	≤	≤	NOUN
ejpam-6802	268	8	(	(	PUNCT
ejpam-6802	268	9	α(q0	α(q0	NOUN
ejpam-6802	268	10	,	,	PUNCT
ejpam-6802	268	11	ϑ0	ϑ0	NOUN
ejpam-6802	268	12	)	)	PUNCT
ejpam-6802	268	13	)	)	PUNCT
ejpam-6802	269	1	ȷ4ℓ+1	ȷ4ℓ+1	NOUN
ejpam-6802	269	2	.	.	PUNCT
ejpam-6802	270	1	(	(	PUNCT
ejpam-6802	270	2	8)	8)	NUM
ejpam-6802	270	3	now	now	ADV
ejpam-6802	270	4	,	,	PUNCT
ejpam-6802	270	5	we	we	PRON
ejpam-6802	270	6	can	can	AUX
ejpam-6802	270	7	get	get	VERB
ejpam-6802	270	8	that	that	PRON
ejpam-6802	270	9	for	for	ADP
ejpam-6802	270	10	any	any	DET
ejpam-6802	270	11	ℓ	ℓ	PROPN
ejpam-6802	270	12	∈	∈	PROPN
ejpam-6802	270	13	n	n	CCONJ
ejpam-6802	270	14	,	,	PUNCT
ejpam-6802	270	15	α(qℓ+1	α(qℓ+1	NUM
ejpam-6802	270	16	,	,	PUNCT
ejpam-6802	270	17	ϑℓ+1	ϑℓ+1	NUM
ejpam-6802	270	18	)	)	PUNCT
ejpam-6802	270	19	≤	≤	NOUN
ejpam-6802	270	20	(	(	PUNCT
ejpam-6802	270	21	α(q0	α(q0	NOUN
ejpam-6802	270	22	,	,	PUNCT
ejpam-6802	270	23	ϑ0	ϑ0	NOUN
ejpam-6802	270	24	)	)	PUNCT
ejpam-6802	270	25	)	)	PUNCT
ejpam-6802	271	1	ȷ2ℓ+2	ȷ2ℓ+2	VERB
ejpam-6802	271	2	,	,	PUNCT
ejpam-6802	271	3	α(qℓ+1	α(qℓ+1	NUM
ejpam-6802	271	4	,	,	PUNCT
ejpam-6802	271	5	ϑℓ	ϑℓ	NOUN
ejpam-6802	271	6	)	)	PUNCT
ejpam-6802	271	7	≤	≤	NOUN
ejpam-6802	271	8	(	(	PUNCT
ejpam-6802	271	9	α(q0	α(q0	NOUN
ejpam-6802	271	10	,	,	PUNCT
ejpam-6802	271	11	ϑ0	ϑ0	NOUN
ejpam-6802	271	12	)	)	PUNCT
ejpam-6802	271	13	)	)	PUNCT
ejpam-6802	271	14	ȷ2ℓ+1	ȷ2ℓ+1	NOUN
ejpam-6802	271	15	and	and	CCONJ
ejpam-6802	271	16	α(qℓ	α(qℓ	PROPN
ejpam-6802	271	17	,	,	PUNCT
ejpam-6802	271	18	ϑℓ	ϑℓ	NOUN
ejpam-6802	271	19	)	)	PUNCT
ejpam-6802	271	20	≤	≤	NOUN
ejpam-6802	271	21	(	(	PUNCT
ejpam-6802	271	22	α(q0	α(q0	NOUN
ejpam-6802	271	23	,	,	PUNCT
ejpam-6802	271	24	ϑ0	ϑ0	NOUN
ejpam-6802	271	25	)	)	PUNCT
ejpam-6802	271	26	)	)	PUNCT
ejpam-6802	272	1	ȷ2ℓ	ȷ2ℓ	NOUN
ejpam-6802	272	2	.	.	PUNCT
ejpam-6802	273	1	for	for	ADP
ejpam-6802	273	2	all	all	DET
ejpam-6802	273	3	r	r	NOUN
ejpam-6802	273	4	,	,	PUNCT
ejpam-6802	273	5	ℓ	ℓ	PROPN
ejpam-6802	273	6	∈	∈	PROPN
ejpam-6802	273	7	n	n	CCONJ
ejpam-6802	273	8	,	,	PUNCT
ejpam-6802	273	9	case	case	NOUN
ejpam-6802	273	10	1	1	NUM
ejpam-6802	273	11	.	.	PUNCT
ejpam-6802	274	1	if	if	SCONJ
ejpam-6802	274	2	r	r	NOUN
ejpam-6802	274	3	>	>	X
ejpam-6802	274	4	ℓ	ℓ	PROPN
ejpam-6802	274	5	,	,	PUNCT
ejpam-6802	274	6	α(qℓ	α(qℓ	PROPN
ejpam-6802	274	7	,	,	PUNCT
ejpam-6802	274	8	ϑr	ϑr	PROPN
ejpam-6802	274	9	)	)	PUNCT
ejpam-6802	274	10	≤α(qℓ	≤α(qℓ	NOUN
ejpam-6802	274	11	,	,	PUNCT
ejpam-6802	274	12	ϑℓ)α(qℓ+1	ϑℓ)α(qℓ+1	ADV
ejpam-6802	274	13	,	,	PUNCT
ejpam-6802	274	14	ϑℓ)α(qℓ+1	ϑℓ)α(qℓ+1	ADV
ejpam-6802	274	15	,	,	PUNCT
ejpam-6802	274	16	ϑr	ϑr	ADJ
ejpam-6802	274	17	)	)	PUNCT
ejpam-6802	274	18	≤(α(q0	≤(α(q0	ADJ
ejpam-6802	274	19	,	,	PUNCT
ejpam-6802	274	20	ϑ0	ϑ0	NOUN
ejpam-6802	274	21	)	)	PUNCT
ejpam-6802	274	22	)	)	PUNCT
ejpam-6802	275	1	ȷ2ℓ(α(q0	ȷ2ℓ(α(q0	ADP
ejpam-6802	275	2	,	,	PUNCT
ejpam-6802	275	3	ϑ0	ϑ0	NOUN
ejpam-6802	275	4	)	)	PUNCT
ejpam-6802	275	5	)	)	PUNCT
ejpam-6802	275	6	ȷ2ℓ+1	ȷ2ℓ+1	NOUN
ejpam-6802	275	7	α(qℓ+1	α(qℓ+1	X
ejpam-6802	275	8	,	,	PUNCT
ejpam-6802	275	9	ϑr	ϑr	ADJ
ejpam-6802	275	10	)	)	PUNCT
ejpam-6802	275	11	≤(α(q0	≤(α(q0	ADJ
ejpam-6802	275	12	,	,	PUNCT
ejpam-6802	275	13	ϑ0	ϑ0	NOUN
ejpam-6802	275	14	)	)	PUNCT
ejpam-6802	275	15	)	)	PUNCT
ejpam-6802	275	16	(	(	PUNCT
ejpam-6802	275	17	ȷ2ℓ+ȷ2ℓ+1)α(qℓ+1	ȷ2ℓ+ȷ2ℓ+1)α(qℓ+1	X
ejpam-6802	275	18	,	,	PUNCT
ejpam-6802	275	19	ϑℓ+1	ϑℓ+1	NUM
ejpam-6802	275	20	)	)	PUNCT
ejpam-6802	275	21	α(qℓ+2	α(qℓ+2	PROPN
ejpam-6802	275	22	,	,	PUNCT
ejpam-6802	275	23	ϑℓ+1)α(qℓ+2	ϑℓ+1)α(qℓ+2	NOUN
ejpam-6802	275	24	,	,	PUNCT
ejpam-6802	275	25	ϑr	ϑr	NOUN
ejpam-6802	275	26	)	)	PUNCT
ejpam-6802	275	27	≤(α(q0	≤(α(q0	ADJ
ejpam-6802	275	28	,	,	PUNCT
ejpam-6802	275	29	ϑ0	ϑ0	NOUN
ejpam-6802	275	30	)	)	PUNCT
ejpam-6802	275	31	)	)	PUNCT
ejpam-6802	275	32	(	(	PUNCT
ejpam-6802	275	33	ȷ2ℓ+ȷ2ℓ+1)(α(q0	ȷ2ℓ+ȷ2ℓ+1)(α(q0	PROPN
ejpam-6802	275	34	,	,	PUNCT
ejpam-6802	275	35	ϑ0	ϑ0	NOUN
ejpam-6802	275	36	)	)	PUNCT
ejpam-6802	275	37	)	)	PUNCT
ejpam-6802	275	38	ȷ2ℓ+2	ȷ2ℓ+2	PROPN
ejpam-6802	275	39	(	(	PUNCT
ejpam-6802	275	40	α(q0	α(q0	NOUN
ejpam-6802	275	41	,	,	PUNCT
ejpam-6802	275	42	ϑ0	ϑ0	NOUN
ejpam-6802	275	43	)	)	PUNCT
ejpam-6802	275	44	)	)	PUNCT
ejpam-6802	276	1	ȷ2ℓ+3	ȷ2ℓ+3	PUNCT
ejpam-6802	276	2	α(qℓ+2	α(qℓ+2	PROPN
ejpam-6802	276	3	,	,	PUNCT
ejpam-6802	276	4	ϑr	ϑr	PROPN
ejpam-6802	276	5	)	)	PUNCT
ejpam-6802	276	6	...	...	PUNCT
ejpam-6802	277	1	≤(α(q0	≤(α(q0	CCONJ
ejpam-6802	277	2	,	,	PUNCT
ejpam-6802	277	3	ϑ0	ϑ0	NOUN
ejpam-6802	277	4	)	)	PUNCT
ejpam-6802	277	5	)	)	PUNCT
ejpam-6802	277	6	(	(	PUNCT
ejpam-6802	277	7	ȷ2ℓ+ȷ2ℓ+1+ȷ2ℓ+2+ȷ2ℓ+3	ȷ2ℓ+ȷ2ℓ+1+ȷ2ℓ+2+ȷ2ℓ+3	NOUN
ejpam-6802	277	8	+	+	NOUN
ejpam-6802	277	9	·	·	SYM
ejpam-6802	277	10	·	·	SYM
ejpam-6802	277	11	·	·	PUNCT
ejpam-6802	277	12	)	)	PUNCT
ejpam-6802	277	13	=(	=(	PROPN
ejpam-6802	277	14	α(q0	α(q0	NOUN
ejpam-6802	277	15	,	,	PUNCT
ejpam-6802	277	16	ϑ0	ϑ0	NOUN
ejpam-6802	277	17	)	)	PUNCT
ejpam-6802	277	18	)	)	PUNCT
ejpam-6802	277	19	ȷ2ℓ	ȷ2ℓ	NOUN
ejpam-6802	277	20	(	(	PUNCT
ejpam-6802	277	21	1	1	NUM
ejpam-6802	277	22	1−ȷ	1−ȷ	NUM
ejpam-6802	277	23	)	)	PUNCT
ejpam-6802	277	24	.	.	PUNCT
ejpam-6802	278	1	since	since	SCONJ
ejpam-6802	278	2	ȷ	ȷ	X
ejpam-6802	278	3	<	<	X
ejpam-6802	278	4	1	1	NUM
ejpam-6802	278	5	,	,	PUNCT
ejpam-6802	278	6	limℓ,r→++∞	limℓ,r→++∞	NOUN
ejpam-6802	278	7	α(qℓ	α(qℓ	PROPN
ejpam-6802	278	8	,	,	PUNCT
ejpam-6802	278	9	ϑr	ϑr	ADJ
ejpam-6802	278	10	)	)	PUNCT
ejpam-6802	278	11	=	=	SYM
ejpam-6802	279	1	0	0	X
ejpam-6802	279	2	.	.	PUNCT
ejpam-6802	279	3	case	case	NOUN
ejpam-6802	279	4	2	2	NUM
ejpam-6802	279	5	.	.	PUNCT
ejpam-6802	280	1	if	if	SCONJ
ejpam-6802	280	2	r	r	NOUN
ejpam-6802	280	3	<	<	X
ejpam-6802	280	4	ℓ	ℓ	PROPN
ejpam-6802	280	5	,	,	PUNCT
ejpam-6802	280	6	α(qℓ	α(qℓ	PROPN
ejpam-6802	280	7	,	,	PUNCT
ejpam-6802	280	8	ϑr	ϑr	ADJ
ejpam-6802	280	9	)	)	PUNCT
ejpam-6802	280	10	≤α(qr+1	≤α(qr+1	PROPN
ejpam-6802	280	11	,	,	PUNCT
ejpam-6802	280	12	ϑr)α(qr+1	ϑr)α(qr+1	PROPN
ejpam-6802	280	13	,	,	PUNCT
ejpam-6802	280	14	ϑr+1)α(qℓ	ϑr+1)α(qℓ	PROPN
ejpam-6802	280	15	,	,	PUNCT
ejpam-6802	280	16	ϑr+1	ϑr+1	NOUN
ejpam-6802	280	17	)	)	PUNCT
ejpam-6802	280	18	≤(α(q0	≤(α(q0	NOUN
ejpam-6802	280	19	,	,	PUNCT
ejpam-6802	280	20	ϑ0	ϑ0	NOUN
ejpam-6802	280	21	)	)	PUNCT
ejpam-6802	280	22	)	)	PUNCT
ejpam-6802	281	1	ȷ2r+1	ȷ2r+1	INTJ
ejpam-6802	281	2	(	(	PUNCT
ejpam-6802	281	3	α(q0	α(q0	NOUN
ejpam-6802	281	4	,	,	PUNCT
ejpam-6802	281	5	ϑ0	ϑ0	NOUN
ejpam-6802	281	6	)	)	PUNCT
ejpam-6802	281	7	)	)	PUNCT
ejpam-6802	282	1	ȷ2r+2	ȷ2r+2	ADP
ejpam-6802	282	2	α(qℓ	α(qℓ	PROPN
ejpam-6802	282	3	,	,	PUNCT
ejpam-6802	282	4	ϑr+1	ϑr+1	NOUN
ejpam-6802	282	5	)	)	PUNCT
ejpam-6802	282	6	≤(α(q0	≤(α(q0	NOUN
ejpam-6802	282	7	,	,	PUNCT
ejpam-6802	282	8	ϑ0	ϑ0	NOUN
ejpam-6802	282	9	)	)	PUNCT
ejpam-6802	282	10	)	)	PUNCT
ejpam-6802	282	11	(	(	PUNCT
ejpam-6802	282	12	ȷ2r+1+ȷ2r+2)α(qr+2	ȷ2r+1+ȷ2r+2)α(qr+2	INTJ
ejpam-6802	282	13	,	,	PUNCT
ejpam-6802	282	14	ϑr+1	ϑr+1	NOUN
ejpam-6802	282	15	)	)	PUNCT
ejpam-6802	282	16	r.	r.	PROPN
ejpam-6802	282	17	ramaswamy	ramaswamy	PROPN
ejpam-6802	282	18	/	/	SYM
ejpam-6802	282	19	eur	eur	PROPN
ejpam-6802	282	20	.	.	PUNCT
ejpam-6802	283	1	j.	j.	PROPN
ejpam-6802	283	2	pure	pure	PROPN
ejpam-6802	283	3	appl	appl	PROPN
ejpam-6802	283	4	.	.	PROPN
ejpam-6802	283	5	math	math	PROPN
ejpam-6802	283	6	,	,	PUNCT
ejpam-6802	283	7	18	18	NUM
ejpam-6802	283	8	(	(	PUNCT
ejpam-6802	283	9	4	4	NUM
ejpam-6802	283	10	)	)	PUNCT
ejpam-6802	283	11	(	(	PUNCT
ejpam-6802	283	12	2025	2025	NUM
ejpam-6802	283	13	)	)	PUNCT
ejpam-6802	283	14	,	,	PUNCT
ejpam-6802	283	15	6802	6802	NUM
ejpam-6802	283	16	15	15	NUM
ejpam-6802	283	17	of	of	ADP
ejpam-6802	283	18	20	20	NUM
ejpam-6802	283	19	α(qr+2	α(qr+2	NOUN
ejpam-6802	283	20	,	,	PUNCT
ejpam-6802	283	21	ϑr+2)α(qℓ	ϑr+2)α(qℓ	NOUN
ejpam-6802	283	22	,	,	PUNCT
ejpam-6802	283	23	ϑr+2	ϑr+2	NOUN
ejpam-6802	283	24	)	)	PUNCT
ejpam-6802	283	25	...	...	PUNCT
ejpam-6802	284	1	≤(α(q0	≤(α(q0	CCONJ
ejpam-6802	284	2	,	,	PUNCT
ejpam-6802	284	3	ϑ0	ϑ0	NOUN
ejpam-6802	284	4	)	)	PUNCT
ejpam-6802	284	5	)	)	PUNCT
ejpam-6802	284	6	(	(	PUNCT
ejpam-6802	285	1	ȷ2r+1+ȷ2r+2+ȷ2r+3+ȷ2r+4	ȷ2r+1+ȷ2r+2+ȷ2r+3+ȷ2r+4	NOUN
ejpam-6802	285	2	+	+	NOUN
ejpam-6802	285	3	...	...	PUNCT
ejpam-6802	285	4	)	)	PUNCT
ejpam-6802	285	5	=(	=(	PROPN
ejpam-6802	285	6	α(q0	α(q0	NOUN
ejpam-6802	285	7	,	,	PUNCT
ejpam-6802	285	8	ϑ0	ϑ0	NOUN
ejpam-6802	285	9	)	)	PUNCT
ejpam-6802	285	10	)	)	PUNCT
ejpam-6802	286	1	ȷ2r+1	ȷ2r+1	PROPN
ejpam-6802	286	2	(	(	PUNCT
ejpam-6802	286	3	1	1	NUM
ejpam-6802	286	4	1−ȷ	1−ȷ	NUM
ejpam-6802	286	5	)	)	PUNCT
ejpam-6802	286	6	.	.	PUNCT
ejpam-6802	287	1	again	again	ADV
ejpam-6802	287	2	,	,	PUNCT
ejpam-6802	287	3	since	since	SCONJ
ejpam-6802	287	4	ȷ	ȷ	X
ejpam-6802	287	5	<	<	X
ejpam-6802	287	6	1	1	NUM
ejpam-6802	287	7	,	,	PUNCT
ejpam-6802	287	8	limℓ,r→++∞	limℓ,r→++∞	NOUN
ejpam-6802	287	9	α(qℓ	α(qℓ	PROPN
ejpam-6802	287	10	,	,	PUNCT
ejpam-6802	287	11	ϑr	ϑr	ADJ
ejpam-6802	287	12	)	)	PUNCT
ejpam-6802	287	13	=	=	SYM
ejpam-6802	288	1	0	0	X
ejpam-6802	288	2	.	.	PUNCT
ejpam-6802	289	1	therefore	therefore	ADV
ejpam-6802	289	2	,	,	PUNCT
ejpam-6802	289	3	(	(	PUNCT
ejpam-6802	289	4	{	{	PUNCT
ejpam-6802	289	5	qℓ	qℓ	NOUN
ejpam-6802	289	6	}	}	PUNCT
ejpam-6802	289	7	,	,	PUNCT
ejpam-6802	289	8	{	{	PUNCT
ejpam-6802	289	9	ϑm	ϑm	NOUN
ejpam-6802	289	10	}	}	PUNCT
ejpam-6802	289	11	)	)	PUNCT
ejpam-6802	289	12	is	be	AUX
ejpam-6802	289	13	a	a	DET
ejpam-6802	289	14	cauchy	cauchy	ADJ
ejpam-6802	289	15	bisequence	bisequence	NOUN
ejpam-6802	289	16	.	.	PUNCT
ejpam-6802	290	1	since	since	SCONJ
ejpam-6802	290	2	(	(	PUNCT
ejpam-6802	290	3	b	b	X
ejpam-6802	290	4	,	,	PUNCT
ejpam-6802	290	5	f	f	PROPN
ejpam-6802	290	6	,	,	PUNCT
ejpam-6802	290	7	α	α	PROPN
ejpam-6802	290	8	)	)	PUNCT
ejpam-6802	290	9	is	be	AUX
ejpam-6802	290	10	complete	complete	ADJ
ejpam-6802	290	11	,	,	PUNCT
ejpam-6802	290	12	{	{	PUNCT
ejpam-6802	290	13	qℓ	qℓ	NOUN
ejpam-6802	290	14	}	}	PUNCT
ejpam-6802	290	15	→	→	SYM
ejpam-6802	290	16	q∗	q∗	NOUN
ejpam-6802	290	17	,	,	PUNCT
ejpam-6802	290	18	{	{	PUNCT
ejpam-6802	290	19	ϑr	ϑr	VERB
ejpam-6802	290	20	}	}	PUNCT
ejpam-6802	290	21	→	→	SYM
ejpam-6802	290	22	q∗	q∗	NOUN
ejpam-6802	290	23	,	,	PUNCT
ejpam-6802	290	24	where	where	SCONJ
ejpam-6802	290	25	q∗	q∗	PROPN
ejpam-6802	290	26	∈	∈	PROPN
ejpam-6802	290	27	b	b	X
ejpam-6802	290	28	∪	∪	PROPN
ejpam-6802	290	29	f	f	PROPN
ejpam-6802	290	30	.	.	PUNCT
ejpam-6802	291	1	also	also	ADV
ejpam-6802	291	2	,	,	PUNCT
ejpam-6802	291	3	{	{	PUNCT
ejpam-6802	291	4	s(q2ℓ	s(q2ℓ	NOUN
ejpam-6802	291	5	)	)	PUNCT
ejpam-6802	291	6	}	}	PUNCT
ejpam-6802	291	7	=	=	SYM
ejpam-6802	291	8	{	{	PUNCT
ejpam-6802	291	9	ϑ2ℓ	ϑ2ℓ	NOUN
ejpam-6802	291	10	}	}	PUNCT
ejpam-6802	291	11	→	→	SYM
ejpam-6802	291	12	q∗	q∗	NOUN
ejpam-6802	291	13	∈	∈	PROPN
ejpam-6802	291	14	b	b	NOUN
ejpam-6802	291	15	∩	∩	PROPN
ejpam-6802	291	16	f	f	PROPN
ejpam-6802	291	17	⇒	⇒	NOUN
ejpam-6802	291	18	s(q2ℓ	s(q2ℓ	PROPN
ejpam-6802	291	19	)	)	PUNCT
ejpam-6802	291	20	has	have	AUX
ejpam-6802	291	21	a	a	DET
ejpam-6802	291	22	unique	unique	ADJ
ejpam-6802	291	23	limit	limit	NOUN
ejpam-6802	291	24	q∗	q∗	NOUN
ejpam-6802	291	25	,	,	PUNCT
ejpam-6802	291	26	and	and	CCONJ
ejpam-6802	291	27	{	{	PUNCT
ejpam-6802	291	28	qℓ	qℓ	NOUN
ejpam-6802	291	29	}	}	PUNCT
ejpam-6802	291	30	→	→	SYM
ejpam-6802	291	31	q∗	q∗	NOUN
ejpam-6802	291	32	⇒	⇒	NOUN
ejpam-6802	291	33	{	{	PUNCT
ejpam-6802	291	34	q2ℓ	q2ℓ	NOUN
ejpam-6802	291	35	}	}	PUNCT
ejpam-6802	291	36	→	→	SYM
ejpam-6802	291	37	q∗.	q∗.	NOUN
ejpam-6802	291	38	since	since	SCONJ
ejpam-6802	291	39	s	s	PROPN
ejpam-6802	291	40	is	be	AUX
ejpam-6802	291	41	a	a	DET
ejpam-6802	291	42	continuous	continuous	ADJ
ejpam-6802	291	43	,	,	PUNCT
ejpam-6802	291	44	{	{	PUNCT
ejpam-6802	291	45	s(q2ℓ	s(q2ℓ	NOUN
ejpam-6802	291	46	)	)	PUNCT
ejpam-6802	291	47	}	}	PUNCT
ejpam-6802	291	48	→	→	SYM
ejpam-6802	291	49	sq∗.	sq∗.	NOUN
ejpam-6802	291	50	(	(	PUNCT
ejpam-6802	291	51	i.e	i.e	NOUN
ejpam-6802	291	52	)	)	PUNCT
ejpam-6802	291	53	sq∗	sq∗	NOUN
ejpam-6802	291	54	=	=	PUNCT
ejpam-6802	291	55	q∗.	q∗.	NOUN
ejpam-6802	291	56	similarly	similarly	ADV
ejpam-6802	291	57	,	,	PUNCT
ejpam-6802	291	58	{	{	PUNCT
ejpam-6802	291	59	ω(ϑ2ℓ+1	ω(ϑ2ℓ+1	NUM
ejpam-6802	291	60	)	)	PUNCT
ejpam-6802	291	61	}	}	PUNCT
ejpam-6802	291	62	=	=	SYM
ejpam-6802	291	63	{	{	PUNCT
ejpam-6802	291	64	q2ℓ+2	q2ℓ+2	NOUN
ejpam-6802	291	65	}	}	PUNCT
ejpam-6802	291	66	→	→	SYM
ejpam-6802	291	67	q∗	q∗	NOUN
ejpam-6802	291	68	∈	∈	PROPN
ejpam-6802	291	69	b	b	NOUN
ejpam-6802	291	70	∩	∩	PROPN
ejpam-6802	291	71	f	f	PROPN
ejpam-6802	291	72	⇒	⇒	PROPN
ejpam-6802	291	73	ω(ϑ2ℓ+1	ω(ϑ2ℓ+1	NUM
ejpam-6802	291	74	)	)	PUNCT
ejpam-6802	291	75	has	have	VERB
ejpam-6802	291	76	a	a	DET
ejpam-6802	291	77	unique	unique	ADJ
ejpam-6802	291	78	limit	limit	NOUN
ejpam-6802	291	79	q∗	q∗	NOUN
ejpam-6802	291	80	,	,	PUNCT
ejpam-6802	291	81	and	and	CCONJ
ejpam-6802	291	82	{	{	PUNCT
ejpam-6802	291	83	ϑℓ	ϑℓ	NOUN
ejpam-6802	291	84	}	}	PUNCT
ejpam-6802	291	85	→	→	SYM
ejpam-6802	291	86	q∗	q∗	NOUN
ejpam-6802	291	87	⇒	⇒	NOUN
ejpam-6802	291	88	{	{	PUNCT
ejpam-6802	291	89	ϑ2ℓ+1	ϑ2ℓ+1	NOUN
ejpam-6802	291	90	}	}	PUNCT
ejpam-6802	291	91	→	→	SYM
ejpam-6802	291	92	q∗.	q∗.	NOUN
ejpam-6802	291	93	now	now	ADV
ejpam-6802	291	94	,	,	PUNCT
ejpam-6802	291	95	ω	ω	PROPN
ejpam-6802	291	96	⇒	⇒	X
ejpam-6802	291	97	{	{	PUNCT
ejpam-6802	291	98	ω(ϑ2ℓ+1	ω(ϑ2ℓ+1	NUM
ejpam-6802	291	99	)	)	PUNCT
ejpam-6802	291	100	}	}	PUNCT
ejpam-6802	291	101	→	→	PUNCT
ejpam-6802	291	102	ωq∗.	ωq∗.	PROPN
ejpam-6802	291	103	therefore	therefore	ADV
ejpam-6802	291	104	,	,	PUNCT
ejpam-6802	291	105	ωq∗	ωq∗	NOUN
ejpam-6802	291	106	=	=	SYM
ejpam-6802	291	107	q∗.	q∗.	SCONJ
ejpam-6802	291	108	hence	hence	ADV
ejpam-6802	291	109	,	,	PUNCT
ejpam-6802	291	110	ω	ω	PROPN
ejpam-6802	291	111	&	&	CCONJ
ejpam-6802	291	112	s	s	PART
ejpam-6802	291	113	have	have	VERB
ejpam-6802	291	114	a	a	DET
ejpam-6802	291	115	cfp	cfp	NOUN
ejpam-6802	291	116	.	.	PUNCT
ejpam-6802	292	1	let	let	VERB
ejpam-6802	292	2	ϑ∗	ϑ∗	PROPN
ejpam-6802	292	3	∈	∈	PROPN
ejpam-6802	292	4	b∩f	b∩f	NOUN
ejpam-6802	292	5	s.t	s.t	PROPN
ejpam-6802	292	6	sϑ∗	sϑ∗	NOUN
ejpam-6802	292	7	=	=	PUNCT
ejpam-6802	292	8	ωϑ∗	ωϑ∗	NOUN
ejpam-6802	292	9	=	=	SYM
ejpam-6802	292	10	ϑ∗	ϑ∗	PROPN
ejpam-6802	292	11	∈	∈	PROPN
ejpam-6802	292	12	b∩f	b∩f	NOUN
ejpam-6802	292	13	.	.	PUNCT
ejpam-6802	293	1	then	then	ADV
ejpam-6802	293	2	,	,	PUNCT
ejpam-6802	293	3	we	we	PRON
ejpam-6802	293	4	get	get	VERB
ejpam-6802	293	5	α(ϑ∗	α(ϑ∗	NOUN
ejpam-6802	293	6	,	,	PUNCT
ejpam-6802	293	7	ϖ∗	ϖ∗	NOUN
ejpam-6802	293	8	)	)	PUNCT
ejpam-6802	294	1	=	=	SYM
ejpam-6802	294	2	α(sϑ∗,ωϖ∗	α(sϑ∗,ωϖ∗	NOUN
ejpam-6802	294	3	)	)	PUNCT
ejpam-6802	294	4	≤	≤	NOUN
ejpam-6802	294	5	(	(	PUNCT
ejpam-6802	294	6	α(ϖ∗,ωϖ∗)α(ϖ∗,sϑ∗)α(sϑ∗	α(ϖ∗,ωϖ∗)α(ϖ∗,sϑ∗)α(sϑ∗	NUM
ejpam-6802	294	7	,	,	PUNCT
ejpam-6802	294	8	ϑ∗)α(ϑ∗,ωϖ∗	ϑ∗)α(ϑ∗,ωϖ∗	NUM
ejpam-6802	294	9	)	)	PUNCT
ejpam-6802	294	10	α(ϖ∗,sϑ∗)α(ϑ∗,ωϖ∗	α(ϖ∗,sϑ∗)α(ϑ∗,ωϖ∗	NUM
ejpam-6802	294	11	)	)	PUNCT
ejpam-6802	294	12	)	)	PUNCT
ejpam-6802	295	1	ȷ	ȷ	NOUN
ejpam-6802	295	2	=	=	PRON
ejpam-6802	295	3	(	(	PUNCT
ejpam-6802	295	4	α(ϖ∗	α(ϖ∗	NOUN
ejpam-6802	295	5	,	,	PUNCT
ejpam-6802	295	6	ϖ∗)α(ϖ∗	ϖ∗)α(ϖ∗	PROPN
ejpam-6802	295	7	,	,	PUNCT
ejpam-6802	295	8	ϑ∗)α(ϑ∗	ϑ∗)α(ϑ∗	PROPN
ejpam-6802	295	9	,	,	PUNCT
ejpam-6802	295	10	ϑ∗)α(ϑ∗	ϑ∗)α(ϑ∗	PROPN
ejpam-6802	295	11	,	,	PUNCT
ejpam-6802	295	12	ϖ∗	ϖ∗	NOUN
ejpam-6802	295	13	)	)	PUNCT
ejpam-6802	295	14	α(ϖ∗	α(ϖ∗	NOUN
ejpam-6802	295	15	,	,	PUNCT
ejpam-6802	295	16	ϑ∗)α(ϑ∗	ϑ∗)α(ϑ∗	PROPN
ejpam-6802	295	17	,	,	PUNCT
ejpam-6802	295	18	ϖ∗	ϖ∗	NOUN
ejpam-6802	295	19	)	)	PUNCT
ejpam-6802	295	20	)	)	PUNCT
ejpam-6802	295	21	ȷ	ȷ	X
ejpam-6802	296	1	=	=	NOUN
ejpam-6802	296	2	1	1	NUM
ejpam-6802	296	3	.	.	PUNCT
ejpam-6802	297	1	therefore	therefore	ADV
ejpam-6802	297	2	,	,	PUNCT
ejpam-6802	297	3	q∗	q∗	PROPN
ejpam-6802	297	4	=	=	PUNCT
ejpam-6802	297	5	ϑ∗.	ϑ∗.	PROPN
ejpam-6802	297	6	4	4	NUM
ejpam-6802	297	7	.	.	X
ejpam-6802	297	8	application	application	NOUN
ejpam-6802	297	9	in	in	ADP
ejpam-6802	297	10	this	this	DET
ejpam-6802	297	11	section	section	NOUN
ejpam-6802	297	12	,	,	PUNCT
ejpam-6802	297	13	the	the	DET
ejpam-6802	297	14	presence	presence	NOUN
ejpam-6802	297	15	&	&	CCONJ
ejpam-6802	297	16	the	the	DET
ejpam-6802	297	17	uniqueness	uniqueness	NOUN
ejpam-6802	297	18	of	of	ADP
ejpam-6802	297	19	the	the	DET
ejpam-6802	297	20	solution	solution	NOUN
ejpam-6802	297	21	to	to	ADP
ejpam-6802	297	22	an	an	DET
ejpam-6802	297	23	integral	integral	ADJ
ejpam-6802	297	24	equations	equation	NOUN
ejpam-6802	297	25	is	be	AUX
ejpam-6802	297	26	revealed	reveal	VERB
ejpam-6802	297	27	as	as	ADP
ejpam-6802	297	28	an	an	DET
ejpam-6802	297	29	application	application	NOUN
ejpam-6802	297	30	of	of	ADP
ejpam-6802	297	31	theorem	theorem	ADJ
ejpam-6802	297	32	3.1	3.1	NUM
ejpam-6802	297	33	.	.	PUNCT
ejpam-6802	297	34	theorem	theorem	VERB
ejpam-6802	297	35	4.1	4.1	NUM
ejpam-6802	297	36	.	.	PUNCT
ejpam-6802	298	1	let	let	VERB
ejpam-6802	298	2	us	we	PRON
ejpam-6802	298	3	consider	consider	VERB
ejpam-6802	298	4	the	the	DET
ejpam-6802	298	5	integral	integral	ADJ
ejpam-6802	298	6	equation	equation	NOUN
ejpam-6802	298	7	q(φ	q(φ	VERB
ejpam-6802	298	8	)	)	PUNCT
ejpam-6802	298	9	=	=	SYM
ejpam-6802	299	1	b(φ	b(φ	ADJ
ejpam-6802	299	2	)	)	PUNCT
ejpam-6802	300	1	+	+	CCONJ
ejpam-6802	300	2	∫	∫	PROPN
ejpam-6802	300	3	e1e2	e1e2	NOUN
ejpam-6802	300	4	g(φ	g(φ	PROPN
ejpam-6802	300	5	,	,	PUNCT
ejpam-6802	300	6	s	s	PROPN
ejpam-6802	300	7	,	,	PUNCT
ejpam-6802	300	8	q(s))ds	q(s))ds	PROPN
ejpam-6802	300	9	,	,	PUNCT
ejpam-6802	300	10	φ	φ	PROPN
ejpam-6802	300	11	∈	∈	PROPN
ejpam-6802	300	12	e1	e1	PROPN
ejpam-6802	300	13	∪	∪	PROPN
ejpam-6802	300	14	e2	e2	PROPN
ejpam-6802	300	15	,	,	PUNCT
ejpam-6802	300	16	(	(	PUNCT
ejpam-6802	300	17	9	9	X
ejpam-6802	300	18	)	)	PUNCT
ejpam-6802	300	19	where	where	SCONJ
ejpam-6802	300	20	e1	e1	PROPN
ejpam-6802	300	21	∪	∪	NOUN
ejpam-6802	300	22	e2	e2	PROPN
ejpam-6802	300	23	is	be	AUX
ejpam-6802	300	24	a	a	DET
ejpam-6802	300	25	lebesgue	lebesgue	ADJ
ejpam-6802	300	26	measurable	measurable	ADJ
ejpam-6802	300	27	set	set	NOUN
ejpam-6802	300	28	.	.	PUNCT
ejpam-6802	301	1	assume	assume	VERB
ejpam-6802	301	2	(	(	PUNCT
ejpam-6802	301	3	i	i	NOUN
ejpam-6802	301	4	)	)	PUNCT
ejpam-6802	301	5	g	g	NOUN
ejpam-6802	301	6	:	:	PUNCT
ejpam-6802	301	7	(	(	PUNCT
ejpam-6802	301	8	e2	e2	NOUN
ejpam-6802	301	9	1	1	NUM
ejpam-6802	301	10	∪	∪	PROPN
ejpam-6802	301	11	e2	e2	PROPN
ejpam-6802	301	12	2	2	NUM
ejpam-6802	301	13	)	)	PUNCT
ejpam-6802	301	14	×	×	NOUN
ejpam-6802	302	1	[	[	X
ejpam-6802	302	2	0,+∞	0,+∞	NUM
ejpam-6802	302	3	)	)	PUNCT
ejpam-6802	302	4	→	→	PUNCT
ejpam-6802	303	1	[	[	X
ejpam-6802	303	2	0,+∞	0,+∞	NUM
ejpam-6802	303	3	)	)	PUNCT
ejpam-6802	303	4	&	&	CCONJ
ejpam-6802	303	5	b	b	PROPN
ejpam-6802	303	6	∈	∈	PROPN
ejpam-6802	303	7	l+∞(e1	l+∞(e1	PROPN
ejpam-6802	303	8	)	)	PUNCT
ejpam-6802	303	9	∪	∪	ADP
ejpam-6802	303	10	l+∞(e2	l+∞(e2	PROPN
ejpam-6802	303	11	)	)	PUNCT
ejpam-6802	303	12	,	,	PUNCT
ejpam-6802	303	13	(	(	PUNCT
ejpam-6802	303	14	ii	ii	NOUN
ejpam-6802	303	15	)	)	PUNCT
ejpam-6802	303	16	∃	∃	PROPN
ejpam-6802	303	17	continuous	continuous	ADJ
ejpam-6802	303	18	function	function	NOUN
ejpam-6802	303	19	θ	θ	NOUN
ejpam-6802	303	20	:	:	PUNCT
ejpam-6802	303	21	e2	e2	NOUN
ejpam-6802	303	22	1	1	NUM
ejpam-6802	303	23	∪	∪	PROPN
ejpam-6802	303	24	e2	e2	X
ejpam-6802	303	25	2	2	NUM
ejpam-6802	303	26	→	→	SYM
ejpam-6802	303	27	[	[	X
ejpam-6802	303	28	0,+∞	0,+∞	NUM
ejpam-6802	303	29	)	)	PUNCT
ejpam-6802	303	30	&	&	CCONJ
ejpam-6802	303	31	λ	λ	PROPN
ejpam-6802	303	32	∈	∈	PROPN
ejpam-6802	303	33	(	(	PUNCT
ejpam-6802	303	34	0	0	NUM
ejpam-6802	303	35	,	,	PUNCT
ejpam-6802	303	36	1	1	X
ejpam-6802	303	37	)	)	PUNCT
ejpam-6802	303	38	s.t	s.t	PROPN
ejpam-6802	303	39	|g(φ	|g(φ	PROPN
ejpam-6802	303	40	,	,	PUNCT
ejpam-6802	303	41	s	s	PART
ejpam-6802	303	42	,	,	PUNCT
ejpam-6802	303	43	q(s))−	q(s))−	VERB
ejpam-6802	303	44	g(φ	g(φ	PROPN
ejpam-6802	303	45	,	,	PUNCT
ejpam-6802	303	46	s	s	AUX
ejpam-6802	303	47	,	,	PUNCT
ejpam-6802	303	48	ϑ(s)|	ϑ(s)|	VERB
ejpam-6802	303	49	≤	≤	NUM
ejpam-6802	303	50	λθ(φ	λθ(φ	NOUN
ejpam-6802	303	51	,	,	PUNCT
ejpam-6802	303	52	s)(|q(s)−	s)(|q(s)−	ADP
ejpam-6802	303	53	ϑ(s)|	ϑ(s)|	VERB
ejpam-6802	303	54	,	,	PUNCT
ejpam-6802	303	55	for	for	ADP
ejpam-6802	303	56	φ	φ	NUM
ejpam-6802	303	57	,	,	PUNCT
ejpam-6802	303	58	s	s	PROPN
ejpam-6802	303	59	∈	∈	PROPN
ejpam-6802	303	60	e2	e2	PROPN
ejpam-6802	303	61	1	1	NUM
ejpam-6802	303	62	∪	∪	PROPN
ejpam-6802	303	63	e2	e2	PROPN
ejpam-6802	303	64	2	2	NUM
ejpam-6802	303	65	,	,	PUNCT
ejpam-6802	303	66	(	(	PUNCT
ejpam-6802	303	67	iii	iii	X
ejpam-6802	303	68	)	)	PUNCT
ejpam-6802	303	69	supφ∈e1∪e2	supφ∈e1∪e2	PROPN
ejpam-6802	303	70	∫	∫	PROPN
ejpam-6802	303	71	e1∪e2	e1∪e2	PROPN
ejpam-6802	303	72	θ(φ	θ(φ	PROPN
ejpam-6802	303	73	,	,	PUNCT
ejpam-6802	303	74	s)ds	s)ds	PROPN
ejpam-6802	303	75	≤	≤	PROPN
ejpam-6802	303	76	1	1	NUM
ejpam-6802	303	77	.	.	PUNCT
ejpam-6802	304	1	r.	r.	PROPN
ejpam-6802	304	2	ramaswamy	ramaswamy	PROPN
ejpam-6802	304	3	/	/	SYM
ejpam-6802	304	4	eur	eur	PROPN
ejpam-6802	304	5	.	.	PUNCT
ejpam-6802	305	1	j.	j.	PROPN
ejpam-6802	305	2	pure	pure	PROPN
ejpam-6802	305	3	appl	appl	PROPN
ejpam-6802	305	4	.	.	PROPN
ejpam-6802	305	5	math	math	PROPN
ejpam-6802	305	6	,	,	PUNCT
ejpam-6802	305	7	18	18	NUM
ejpam-6802	305	8	(	(	PUNCT
ejpam-6802	305	9	4	4	NUM
ejpam-6802	305	10	)	)	PUNCT
ejpam-6802	305	11	(	(	PUNCT
ejpam-6802	305	12	2025	2025	NUM
ejpam-6802	305	13	)	)	PUNCT
ejpam-6802	305	14	,	,	PUNCT
ejpam-6802	305	15	6802	6802	NUM
ejpam-6802	305	16	16	16	NUM
ejpam-6802	305	17	of	of	ADP
ejpam-6802	305	18	20	20	NUM
ejpam-6802	305	19	then	then	ADV
ejpam-6802	305	20	the	the	DET
ejpam-6802	305	21	integral	integral	ADJ
ejpam-6802	305	22	equation	equation	NOUN
ejpam-6802	305	23	9	9	NUM
ejpam-6802	305	24	has	have	VERB
ejpam-6802	305	25	a	a	DET
ejpam-6802	305	26	unique	unique	ADJ
ejpam-6802	305	27	solution	solution	NOUN
ejpam-6802	305	28	in	in	ADP
ejpam-6802	305	29	l+∞(e1	l+∞(e1	NOUN
ejpam-6802	305	30	)	)	PUNCT
ejpam-6802	305	31	∪	∪	ADP
ejpam-6802	305	32	l+∞(e2	l+∞(e2	PROPN
ejpam-6802	305	33	)	)	PUNCT
ejpam-6802	305	34	.	.	PUNCT
ejpam-6802	306	1	proof	proof	NOUN
ejpam-6802	306	2	.	.	PUNCT
ejpam-6802	307	1	let	let	VERB
ejpam-6802	307	2	a	a	DET
ejpam-6802	307	3	=	=	SYM
ejpam-6802	307	4	r	r	NOUN
ejpam-6802	307	5	,	,	PUNCT
ejpam-6802	307	6	z	z	NOUN
ejpam-6802	307	7	=	=	SYM
ejpam-6802	307	8	{	{	PUNCT
ejpam-6802	307	9	(	(	PUNCT
ejpam-6802	307	10	q	q	NOUN
ejpam-6802	307	11	,	,	PUNCT
ejpam-6802	307	12	ϑ	ϑ	NOUN
ejpam-6802	307	13	)	)	PUNCT
ejpam-6802	307	14	∈	∈	PROPN
ejpam-6802	307	15	a|q	a|q	PROPN
ejpam-6802	307	16	,	,	PUNCT
ejpam-6802	307	17	ϑ	ϑ	X
ejpam-6802	307	18	≥	≥	NOUN
ejpam-6802	307	19	0	0	NUM
ejpam-6802	307	20	}	}	PUNCT
ejpam-6802	307	21	.	.	PUNCT
ejpam-6802	308	1	let	let	VERB
ejpam-6802	308	2	b	b	NOUN
ejpam-6802	308	3	=	=	SYM
ejpam-6802	308	4	l+∞(e1	l+∞(e1	PROPN
ejpam-6802	308	5	)	)	PUNCT
ejpam-6802	308	6	and	and	CCONJ
ejpam-6802	308	7	f	f	PROPN
ejpam-6802	308	8	=	=	SYM
ejpam-6802	308	9	l+∞(e2	l+∞(e2	PROPN
ejpam-6802	308	10	)	)	PUNCT
ejpam-6802	308	11	be	be	VERB
ejpam-6802	308	12	two	two	NUM
ejpam-6802	308	13	normed	normed	ADJ
ejpam-6802	308	14	linear	linear	ADJ
ejpam-6802	308	15	spaces	space	NOUN
ejpam-6802	308	16	,	,	PUNCT
ejpam-6802	308	17	where	where	SCONJ
ejpam-6802	308	18	e1	e1	PROPN
ejpam-6802	308	19	,	,	PUNCT
ejpam-6802	308	20	e2	e2	PROPN
ejpam-6802	308	21	are	be	AUX
ejpam-6802	308	22	lebesgue	lebesgue	ADJ
ejpam-6802	308	23	measurable	measurable	ADJ
ejpam-6802	308	24	sets	set	NOUN
ejpam-6802	308	25	&	&	CCONJ
ejpam-6802	308	26	m(e1	m(e1	VERB
ejpam-6802	308	27	∪e2	∪e2	NOUN
ejpam-6802	308	28	)	)	PUNCT
ejpam-6802	308	29	<	<	X
ejpam-6802	309	1	+	+	PUNCT
ejpam-6802	309	2	∞.	∞.	PROPN
ejpam-6802	309	3	consider	consider	VERB
ejpam-6802	309	4	α	α	NOUN
ejpam-6802	309	5	:	:	PUNCT
ejpam-6802	309	6	b	b	X
ejpam-6802	309	7	×f	×f	PROPN
ejpam-6802	309	8	→	→	PUNCT
ejpam-6802	309	9	a	a	PRON
ejpam-6802	309	10	to	to	PART
ejpam-6802	309	11	be	be	AUX
ejpam-6802	309	12	defined	define	VERB
ejpam-6802	309	13	by	by	ADP
ejpam-6802	309	14	α(q	α(q	PROPN
ejpam-6802	309	15	,	,	PUNCT
ejpam-6802	309	16	ϑ	ϑ	X
ejpam-6802	309	17	)	)	PUNCT
ejpam-6802	309	18	=	=	SYM
ejpam-6802	309	19	esupφ∈e1∪e2	esupφ∈e1∪e2	NOUN
ejpam-6802	309	20	|q−ϑ|	|q−ϑ|	PUNCT
ejpam-6802	309	21	for	for	ADP
ejpam-6802	309	22	all	all	PRON
ejpam-6802	309	23	(	(	PUNCT
ejpam-6802	309	24	q	q	ADJ
ejpam-6802	309	25	,	,	PUNCT
ejpam-6802	309	26	ϑ	ϑ	NOUN
ejpam-6802	309	27	)	)	PUNCT
ejpam-6802	309	28	∈	∈	PROPN
ejpam-6802	309	29	b	b	PROPN
ejpam-6802	309	30	×f	×f	PROPN
ejpam-6802	309	31	.	.	PUNCT
ejpam-6802	310	1	then	then	ADV
ejpam-6802	310	2	(	(	PUNCT
ejpam-6802	310	3	b	b	X
ejpam-6802	310	4	,	,	PUNCT
ejpam-6802	310	5	f	f	PROPN
ejpam-6802	310	6	,	,	PUNCT
ejpam-6802	310	7	α	α	PROPN
ejpam-6802	310	8	)	)	PUNCT
ejpam-6802	310	9	is	be	AUX
ejpam-6802	310	10	a	a	DET
ejpam-6802	310	11	complete	complete	ADJ
ejpam-6802	310	12	mcbms	mcbms	NOUN
ejpam-6802	310	13	.	.	PUNCT
ejpam-6802	311	1	also	also	ADV
ejpam-6802	311	2	define	define	VERB
ejpam-6802	311	3	the	the	DET
ejpam-6802	311	4	covariant	covariant	ADJ
ejpam-6802	311	5	function	function	NOUN
ejpam-6802	311	6	ω	ω	NOUN
ejpam-6802	311	7	:	:	PUNCT
ejpam-6802	311	8	l+∞(e1	l+∞(e1	NOUN
ejpam-6802	311	9	)	)	PUNCT
ejpam-6802	311	10	∪	∪	ADP
ejpam-6802	311	11	l+∞(e2	l+∞(e2	PROPN
ejpam-6802	311	12	)	)	PUNCT
ejpam-6802	311	13	→	→	SYM
ejpam-6802	311	14	l+∞(e1	l+∞(e1	NOUN
ejpam-6802	311	15	)	)	PUNCT
ejpam-6802	311	16	∪	∪	PROPN
ejpam-6802	311	17	l+∞(e2	l+∞(e2	PROPN
ejpam-6802	311	18	)	)	PUNCT
ejpam-6802	311	19	by	by	ADP
ejpam-6802	311	20	ω(q(φ	ω(q(φ	NOUN
ejpam-6802	311	21	)	)	PUNCT
ejpam-6802	311	22	)	)	PUNCT
ejpam-6802	312	1	=	=	SYM
ejpam-6802	312	2	b(φ	b(φ	ADJ
ejpam-6802	312	3	)	)	PUNCT
ejpam-6802	313	1	+	+	CCONJ
ejpam-6802	313	2	∫	∫	PROPN
ejpam-6802	313	3	e1∪e2	e1∪e2	PROPN
ejpam-6802	313	4	g(φ	g(φ	PROPN
ejpam-6802	313	5	,	,	PUNCT
ejpam-6802	313	6	s	s	PROPN
ejpam-6802	313	7	,	,	PUNCT
ejpam-6802	313	8	q(s))ds	q(s))ds	PROPN
ejpam-6802	313	9	,	,	PUNCT
ejpam-6802	313	10	φ	φ	PROPN
ejpam-6802	313	11	∈	∈	PROPN
ejpam-6802	313	12	e1	e1	PROPN
ejpam-6802	313	13	∪	∪	PROPN
ejpam-6802	313	14	e2	e2	PROPN
ejpam-6802	313	15	.	.	PUNCT
ejpam-6802	314	1	now	now	ADV
ejpam-6802	314	2	,	,	PUNCT
ejpam-6802	314	3	α(ωq(φ),ωϑ(φ	α(ωq(φ),ωϑ(φ	NOUN
ejpam-6802	314	4	)	)	PUNCT
ejpam-6802	314	5	)	)	PUNCT
ejpam-6802	315	1	=	=	PRON
ejpam-6802	315	2	esupφ∈e1∪e2	esupφ∈e1∪e2	PRON
ejpam-6802	315	3	|ωq(φ)−ωϑ(φ)|	|ωq(φ)−ωϑ(φ)|	NOUN
ejpam-6802	315	4	=	=	SYM
ejpam-6802	315	5	e	e	NOUN
ejpam-6802	315	6	supφ∈e1∪e2	supφ∈e1∪e2	X
ejpam-6802	315	7	∣∣∣∣b(φ)+∫	∣∣∣∣b(φ)+∫	PROPN
ejpam-6802	315	8	e1∪e2	e1∪e2	PROPN
ejpam-6802	315	9	g(φ	g(φ	PROPN
ejpam-6802	315	10	,	,	PUNCT
ejpam-6802	315	11	s	s	PART
ejpam-6802	315	12	,	,	PUNCT
ejpam-6802	315	13	q(s))ds−	q(s))ds−	NOUN
ejpam-6802	315	14	(	(	PUNCT
ejpam-6802	315	15	b(φ)+	b(φ)+	PROPN
ejpam-6802	315	16	∫	∫	PROPN
ejpam-6802	315	17	e1∪e2	e1∪e2	PROPN
ejpam-6802	315	18	g(φ	g(φ	PROPN
ejpam-6802	315	19	,	,	PUNCT
ejpam-6802	315	20	s	s	NOUN
ejpam-6802	315	21	,	,	PUNCT
ejpam-6802	315	22	q(s))ds	q(s))ds	PROPN
ejpam-6802	315	23	)	)	PUNCT
ejpam-6802	315	24	∣∣∣∣	∣∣∣∣	NOUN
ejpam-6802	315	25	≤	≤	NUM
ejpam-6802	315	26	e	e	NOUN
ejpam-6802	315	27	supφ∈e1∪e2	supφ∈e1∪e2	NOUN
ejpam-6802	315	28	∫	∫	PROPN
ejpam-6802	315	29	e1∪e2	e1∪e2	PROPN
ejpam-6802	315	30	|g(φ	|g(φ	PROPN
ejpam-6802	315	31	,	,	PUNCT
ejpam-6802	315	32	s	s	NOUN
ejpam-6802	315	33	,	,	PUNCT
ejpam-6802	315	34	q(s))−g(φ	q(s))−g(φ	NUM
ejpam-6802	315	35	,	,	PUNCT
ejpam-6802	315	36	s,ϑ(s))|ds	s,ϑ(s))|ds	VERB
ejpam-6802	315	37	≤	≤	NUM
ejpam-6802	316	1	e	e	NOUN
ejpam-6802	316	2	supφ∈e1∪e2	supφ∈e1∪e2	PROPN
ejpam-6802	316	3	∫	∫	PROPN
ejpam-6802	316	4	e1∪e2	e1∪e2	PROPN
ejpam-6802	316	5	λθ(φ	λθ(φ	PROPN
ejpam-6802	316	6	,	,	PUNCT
ejpam-6802	316	7	s)(|q(s)−ϑ(s)|)ds	s)(|q(s)−ϑ(s)|)d	NOUN
ejpam-6802	316	8	≤	≤	NUM
ejpam-6802	316	9	e	e	ADP
ejpam-6802	316	10	λ(supφ∈e1∪e2	λ(supφ∈e1∪e2	NOUN
ejpam-6802	316	11	|q(s)−ϑ(s)|	|q(s)−ϑ(s)|	NOUN
ejpam-6802	316	12	supφ∈e1∪e2	supφ∈e1∪e2	PROPN
ejpam-6802	316	13	∫	∫	PROPN
ejpam-6802	316	14	e1∪e2	e1∪e2	PROPN
ejpam-6802	316	15	θ(φ	θ(φ	PROPN
ejpam-6802	316	16	,	,	PUNCT
ejpam-6802	316	17	s)ds	s)ds	PROPN
ejpam-6802	316	18	)	)	PUNCT
ejpam-6802	316	19	=	=	PUNCT
ejpam-6802	316	20	(	(	PUNCT
ejpam-6802	316	21	α(q	α(q	PROPN
ejpam-6802	316	22	,	,	PUNCT
ejpam-6802	316	23	ϑ))λ	ϑ))λ	VERB
ejpam-6802	316	24	therefore	therefore	ADV
ejpam-6802	316	25	,	,	PUNCT
ejpam-6802	316	26	all	all	DET
ejpam-6802	316	27	the	the	DET
ejpam-6802	316	28	hypothesis	hypothesis	NOUN
ejpam-6802	316	29	of	of	ADP
ejpam-6802	316	30	theorem	theorem	ADJ
ejpam-6802	316	31	3.1	3.1	NUM
ejpam-6802	316	32	are	be	AUX
ejpam-6802	316	33	satisfied	satisfied	ADJ
ejpam-6802	316	34	&	&	CCONJ
ejpam-6802	316	35	as	as	ADP
ejpam-6802	316	36	a	a	DET
ejpam-6802	316	37	result	result	NOUN
ejpam-6802	316	38	,	,	PUNCT
ejpam-6802	316	39	the	the	DET
ejpam-6802	316	40	integral	integral	ADJ
ejpam-6802	316	41	equation	equation	NOUN
ejpam-6802	316	42	possesses	possess	VERB
ejpam-6802	316	43	a	a	DET
ejpam-6802	316	44	unique	unique	ADJ
ejpam-6802	316	45	solution	solution	NOUN
ejpam-6802	316	46	.	.	PUNCT
ejpam-6802	317	1	4.1	4.1	NUM
ejpam-6802	317	2	.	.	PUNCT
ejpam-6802	317	3	application	application	NOUN
ejpam-6802	317	4	to	to	ADP
ejpam-6802	317	5	fractional	fractional	ADJ
ejpam-6802	317	6	differential	differential	ADJ
ejpam-6802	317	7	equations	equation	NOUN
ejpam-6802	317	8	we	we	PRON
ejpam-6802	317	9	recall	recall	VERB
ejpam-6802	317	10	many	many	ADJ
ejpam-6802	317	11	important	important	ADJ
ejpam-6802	317	12	definitions	definition	NOUN
ejpam-6802	317	13	from	from	ADP
ejpam-6802	317	14	fractional	fractional	ADJ
ejpam-6802	317	15	calculus	calculus	NOUN
ejpam-6802	317	16	theory	theory	NOUN
ejpam-6802	317	17	[	[	X
ejpam-6802	317	18	?	?	PUNCT
ejpam-6802	318	1	]	]	X
ejpam-6802	318	2	-	-	PUNCT
ejpam-6802	318	3	[	[	X
ejpam-6802	318	4	?	?	PUNCT
ejpam-6802	319	1	]	]	PUNCT
ejpam-6802	319	2	.	.	PUNCT
ejpam-6802	320	1	the	the	DET
ejpam-6802	320	2	reiman	reiman	NOUN
ejpam-6802	320	3	-	-	PUNCT
ejpam-6802	320	4	liouville	liouville	NOUN
ejpam-6802	320	5	fraction	fraction	NOUN
ejpam-6802	320	6	derivative	derivative	NOUN
ejpam-6802	320	7	of	of	ADP
ejpam-6802	320	8	a	a	DET
ejpam-6802	320	9	function	function	NOUN
ejpam-6802	320	10	s	s	X
ejpam-6802	320	11	∈	∈	PROPN
ejpam-6802	320	12	c[0	c[0	PROPN
ejpam-6802	320	13	,	,	PUNCT
ejpam-6802	320	14	1	1	NUM
ejpam-6802	320	15	]	]	PUNCT
ejpam-6802	320	16	,	,	PUNCT
ejpam-6802	320	17	of	of	ADP
ejpam-6802	320	18	order	order	NOUN
ejpam-6802	320	19	δ	δ	X
ejpam-6802	320	20	>	>	X
ejpam-6802	320	21	0	0	PUNCT
ejpam-6802	320	22	is	be	AUX
ejpam-6802	320	23	as	as	SCONJ
ejpam-6802	320	24	follows	follow	VERB
ejpam-6802	320	25	:	:	PUNCT
ejpam-6802	320	26	1	1	NUM
ejpam-6802	320	27	γ(ℓ−	γ(ℓ−	ADP
ejpam-6802	320	28	δ	δ	PROPN
ejpam-6802	320	29	)	)	PUNCT
ejpam-6802	320	30	dℓ	dℓ	PROPN
ejpam-6802	320	31	dηℓ	dηℓ	NOUN
ejpam-6802	320	32	∫	∫	PROPN
ejpam-6802	320	33	η	η	PROPN
ejpam-6802	320	34	0	0	PROPN
ejpam-6802	320	35	s(e)de	s(e)de	PART
ejpam-6802	320	36	(	(	PUNCT
ejpam-6802	320	37	η	η	PROPN
ejpam-6802	320	38	−	−	PROPN
ejpam-6802	320	39	e)δ−ℓ+1	e)δ−ℓ+1	PROPN
ejpam-6802	320	40	=	=	SYM
ejpam-6802	320	41	dδs(η	dδs(η	PROPN
ejpam-6802	320	42	)	)	PUNCT
ejpam-6802	320	43	,	,	PUNCT
ejpam-6802	320	44	taking	take	VERB
ejpam-6802	320	45	the	the	DET
ejpam-6802	320	46	right	right	ADJ
ejpam-6802	320	47	hand	hand	NOUN
ejpam-6802	320	48	side	side	NOUN
ejpam-6802	320	49	as	as	SCONJ
ejpam-6802	320	50	defined	define	VERB
ejpam-6802	320	51	point	point	NOUN
ejpam-6802	320	52	-	-	PUNCT
ejpam-6802	320	53	wise	wise	ADJ
ejpam-6802	320	54	on	on	ADP
ejpam-6802	320	55	[	[	X
ejpam-6802	320	56	0	0	NUM
ejpam-6802	320	57	,	,	PUNCT
ejpam-6802	320	58	1	1	NUM
ejpam-6802	320	59	]	]	PUNCT
ejpam-6802	320	60	where	where	SCONJ
ejpam-6802	320	61	,	,	PUNCT
ejpam-6802	320	62	[	[	X
ejpam-6802	320	63	δ	δ	X
ejpam-6802	320	64	]	]	X
ejpam-6802	320	65	denotes	denote	VERB
ejpam-6802	320	66	the	the	DET
ejpam-6802	320	67	integer	integer	NOUN
ejpam-6802	320	68	part	part	NOUN
ejpam-6802	320	69	of	of	ADP
ejpam-6802	320	70	number	number	NOUN
ejpam-6802	320	71	δ	δ	PROPN
ejpam-6802	320	72	,	,	PUNCT
ejpam-6802	320	73	γ	γ	PROPN
ejpam-6802	320	74	is	be	AUX
ejpam-6802	320	75	euler	euler	NOUN
ejpam-6802	320	76	gamma	gamma	PROPN
ejpam-6802	320	77	function	function	PROPN
ejpam-6802	320	78	.	.	PUNCT
ejpam-6802	321	1	let	let	VERB
ejpam-6802	321	2	us	we	PRON
ejpam-6802	321	3	consider	consider	VERB
ejpam-6802	321	4	the	the	DET
ejpam-6802	321	5	fractional	fractional	ADJ
ejpam-6802	321	6	differential	differential	ADJ
ejpam-6802	321	7	equation	equation	NOUN
ejpam-6802	321	8	:	:	PUNCT
ejpam-6802	322	1	edqs(η	edqs(η	NOUN
ejpam-6802	322	2	)	)	PUNCT
ejpam-6802	323	1	+	+	NUM
ejpam-6802	323	2	f(η	f(η	PROPN
ejpam-6802	323	3	,	,	PUNCT
ejpam-6802	323	4	s(η	s(η	PROPN
ejpam-6802	323	5	)	)	PUNCT
ejpam-6802	323	6	)	)	PUNCT
ejpam-6802	324	1	=	=	SYM
ejpam-6802	324	2	0	0	NUM
ejpam-6802	324	3	,	,	PUNCT
ejpam-6802	324	4	1	1	NUM
ejpam-6802	324	5	≤	≤	NUM
ejpam-6802	324	6	η	η	PROPN
ejpam-6802	324	7	≤	≤	NOUN
ejpam-6802	324	8	0	0	NUM
ejpam-6802	324	9	,	,	PUNCT
ejpam-6802	324	10	2	2	NUM
ejpam-6802	324	11	≤	≤	NOUN
ejpam-6802	324	12	q	q	NOUN
ejpam-6802	324	13	>	>	X
ejpam-6802	324	14	1	1	NUM
ejpam-6802	324	15	;	;	PUNCT
ejpam-6802	324	16	s(0	s(0	PROPN
ejpam-6802	324	17	)	)	PUNCT
ejpam-6802	324	18	=	=	SYM
ejpam-6802	325	1	s(1	s(1	PROPN
ejpam-6802	325	2	)	)	PUNCT
ejpam-6802	325	3	=	=	SYM
ejpam-6802	325	4	0	0	NUM
ejpam-6802	325	5	,	,	PUNCT
ejpam-6802	325	6	(	(	PUNCT
ejpam-6802	325	7	10	10	NUM
ejpam-6802	325	8	)	)	PUNCT
ejpam-6802	325	9	where	where	SCONJ
ejpam-6802	325	10	f	f	PROPN
ejpam-6802	325	11	is	be	AUX
ejpam-6802	325	12	a	a	DET
ejpam-6802	325	13	continuous	continuous	ADJ
ejpam-6802	325	14	mapping	mapping	NOUN
ejpam-6802	325	15	from	from	ADP
ejpam-6802	325	16	[	[	X
ejpam-6802	325	17	0	0	NUM
ejpam-6802	325	18	,	,	PUNCT
ejpam-6802	325	19	1	1	NUM
ejpam-6802	325	20	]	]	SYM
ejpam-6802	325	21	×	×	NOUN
ejpam-6802	325	22	r	r	NOUN
ejpam-6802	325	23	to	to	ADP
ejpam-6802	325	24	r	r	NOUN
ejpam-6802	325	25	and	and	CCONJ
ejpam-6802	325	26	edq	edq	NOUN
ejpam-6802	325	27	represents	represent	VERB
ejpam-6802	325	28	the	the	DET
ejpam-6802	325	29	caputo	caputo	PROPN
ejpam-6802	325	30	fractional	fractional	PROPN
ejpam-6802	325	31	derivative	derivative	NOUN
ejpam-6802	325	32	of	of	ADP
ejpam-6802	325	33	order	order	NOUN
ejpam-6802	325	34	q	q	PROPN
ejpam-6802	326	1	&	&	CCONJ
ejpam-6802	326	2	it	it	PRON
ejpam-6802	326	3	is	be	AUX
ejpam-6802	326	4	defined	define	VERB
ejpam-6802	326	5	as	as	ADP
ejpam-6802	326	6	edq	edq	NOUN
ejpam-6802	326	7	=	=	SYM
ejpam-6802	326	8	1	1	NUM
ejpam-6802	326	9	γ(ℓ−	γ(ℓ−	ADP
ejpam-6802	326	10	q	q	NOUN
ejpam-6802	326	11	)	)	PUNCT
ejpam-6802	326	12	∫	∫	PROPN
ejpam-6802	326	13	ζ	ζ	NOUN
ejpam-6802	326	14	0	0	NUM
ejpam-6802	326	15	sℓ(e)de	sℓ(e)de	NOUN
ejpam-6802	326	16	(	(	PUNCT
ejpam-6802	326	17	η	η	PROPN
ejpam-6802	326	18	−	−	PROPN
ejpam-6802	326	19	e)q−ℓ+1	e)q−ℓ+1	PROPN
ejpam-6802	326	20	r.	r.	PROPN
ejpam-6802	326	21	ramaswamy	ramaswamy	PROPN
ejpam-6802	326	22	/	/	SYM
ejpam-6802	326	23	eur	eur	PROPN
ejpam-6802	326	24	.	.	PUNCT
ejpam-6802	327	1	j.	j.	PROPN
ejpam-6802	327	2	pure	pure	PROPN
ejpam-6802	327	3	appl	appl	PROPN
ejpam-6802	327	4	.	.	PROPN
ejpam-6802	327	5	math	math	PROPN
ejpam-6802	327	6	,	,	PUNCT
ejpam-6802	327	7	18	18	NUM
ejpam-6802	327	8	(	(	PUNCT
ejpam-6802	327	9	4	4	NUM
ejpam-6802	327	10	)	)	PUNCT
ejpam-6802	327	11	(	(	PUNCT
ejpam-6802	327	12	2025	2025	NUM
ejpam-6802	327	13	)	)	PUNCT
ejpam-6802	327	14	,	,	PUNCT
ejpam-6802	327	15	6802	6802	NUM
ejpam-6802	327	16	17	17	NUM
ejpam-6802	327	17	of	of	ADP
ejpam-6802	327	18	20	20	NUM
ejpam-6802	327	19	let	let	VERB
ejpam-6802	327	20	a	a	DET
ejpam-6802	327	21	=	=	X
ejpam-6802	327	22	(	(	PUNCT
ejpam-6802	327	23	c[0	c[0	PROPN
ejpam-6802	327	24	,	,	PUNCT
ejpam-6802	327	25	1	1	NUM
ejpam-6802	327	26	]	]	PUNCT
ejpam-6802	327	27	,	,	PUNCT
ejpam-6802	327	28	[	[	X
ejpam-6802	327	29	0,+∞	0,+∞	NUM
ejpam-6802	327	30	)	)	PUNCT
ejpam-6802	327	31	)	)	PUNCT
ejpam-6802	327	32	,	,	PUNCT
ejpam-6802	327	33	z	z	NOUN
ejpam-6802	327	34	=	=	PRON
ejpam-6802	327	35	{	{	PUNCT
ejpam-6802	327	36	g	g	PROPN
ejpam-6802	327	37	∈	∈	PROPN
ejpam-6802	327	38	a|g(η	a|g(η	PROPN
ejpam-6802	327	39	)	)	PUNCT
ejpam-6802	327	40	on	on	ADP
ejpam-6802	327	41	[	[	X
ejpam-6802	327	42	0	0	NUM
ejpam-6802	327	43	,	,	PUNCT
ejpam-6802	327	44	1	1	NUM
ejpam-6802	327	45	]	]	PUNCT
ejpam-6802	327	46	}	}	PUNCT
ejpam-6802	327	47	.	.	PUNCT
ejpam-6802	328	1	suppose	suppose	VERB
ejpam-6802	328	2	b	b	X
ejpam-6802	328	3	=	=	SYM
ejpam-6802	328	4	(	(	PUNCT
ejpam-6802	328	5	c[0	c[0	PROPN
ejpam-6802	328	6	,	,	PUNCT
ejpam-6802	328	7	1	1	NUM
ejpam-6802	328	8	]	]	PUNCT
ejpam-6802	328	9	,	,	PUNCT
ejpam-6802	328	10	(	(	PUNCT
ejpam-6802	328	11	−+∞	−+∞	ADV
ejpam-6802	328	12	,	,	PUNCT
ejpam-6802	328	13	0	0	NUM
ejpam-6802	328	14	]	]	PUNCT
ejpam-6802	328	15	)	)	PUNCT
ejpam-6802	328	16	is	be	AUX
ejpam-6802	328	17	all	all	DET
ejpam-6802	328	18	continuous	continuous	ADJ
ejpam-6802	328	19	functions	function	NOUN
ejpam-6802	328	20	defined	define	VERB
ejpam-6802	328	21	on	on	ADP
ejpam-6802	328	22	[	[	X
ejpam-6802	328	23	0	0	NUM
ejpam-6802	328	24	,	,	PUNCT
ejpam-6802	328	25	1	1	NUM
ejpam-6802	328	26	]	]	PUNCT
ejpam-6802	328	27	that	that	PRON
ejpam-6802	328	28	have	have	VERB
ejpam-6802	328	29	their	their	PRON
ejpam-6802	328	30	values	value	NOUN
ejpam-6802	328	31	in	in	ADP
ejpam-6802	328	32	(	(	PUNCT
ejpam-6802	328	33	−+	−+	PROPN
ejpam-6802	328	34	,	,	PUNCT
ejpam-6802	328	35	0	0	NUM
ejpam-6802	328	36	]	]	PUNCT
ejpam-6802	328	37	,	,	PUNCT
ejpam-6802	328	38	where	where	SCONJ
ejpam-6802	328	39	as	as	ADP
ejpam-6802	328	40	f	f	PROPN
ejpam-6802	328	41	=	=	SYM
ejpam-6802	328	42	(	(	PUNCT
ejpam-6802	328	43	c[0	c[0	PROPN
ejpam-6802	328	44	,	,	PUNCT
ejpam-6802	328	45	1	1	NUM
ejpam-6802	328	46	]	]	PUNCT
ejpam-6802	328	47	,	,	PUNCT
ejpam-6802	328	48	[	[	X
ejpam-6802	328	49	0,+∞	0,+∞	NUM
ejpam-6802	328	50	)	)	PUNCT
ejpam-6802	328	51	)	)	PUNCT
ejpam-6802	328	52	is	be	AUX
ejpam-6802	328	53	all	all	DET
ejpam-6802	328	54	continuous	continuous	ADJ
ejpam-6802	328	55	functions	function	NOUN
ejpam-6802	328	56	defined	define	VERB
ejpam-6802	328	57	on	on	ADP
ejpam-6802	328	58	[	[	X
ejpam-6802	328	59	0,+∞	0,+∞	NUM
ejpam-6802	328	60	)	)	PUNCT
ejpam-6802	328	61	)	)	PUNCT
ejpam-6802	328	62	.	.	PUNCT
ejpam-6802	329	1	consider	consider	VERB
ejpam-6802	329	2	α	α	NOUN
ejpam-6802	329	3	:	:	PUNCT
ejpam-6802	329	4	b×f	b×f	PROPN
ejpam-6802	329	5	→	→	PUNCT
ejpam-6802	329	6	a	a	PRON
ejpam-6802	329	7	to	to	PART
ejpam-6802	329	8	be	be	AUX
ejpam-6802	329	9	defined	define	VERB
ejpam-6802	329	10	as	as	ADP
ejpam-6802	329	11	α(s	α(s	PROPN
ejpam-6802	329	12	,	,	PUNCT
ejpam-6802	329	13	s	s	PART
ejpam-6802	329	14	′	′	NOUN
ejpam-6802	329	15	)	)	PUNCT
ejpam-6802	330	1	=	=	PUNCT
ejpam-6802	330	2	esupη∈[0,1	esupη∈[0,1	PROPN
ejpam-6802	330	3	]	]	PUNCT
ejpam-6802	330	4	|s(η)−s	|s(η)−s	PUNCT
ejpam-6802	330	5	′	′	NUM
ejpam-6802	330	6	(	(	PUNCT
ejpam-6802	330	7	η)|	η)|	NOUN
ejpam-6802	330	8	for	for	ADP
ejpam-6802	330	9	all	all	PRON
ejpam-6802	330	10	(	(	PUNCT
ejpam-6802	330	11	s	s	X
ejpam-6802	330	12	,	,	PUNCT
ejpam-6802	330	13	s	s	PART
ejpam-6802	330	14	′	′	NOUN
ejpam-6802	330	15	)	)	PUNCT
ejpam-6802	331	1	∈	∈	PROPN
ejpam-6802	332	1	b	b	PROPN
ejpam-6802	332	2	×	×	PROPN
ejpam-6802	332	3	f	f	PROPN
ejpam-6802	332	4	.	.	PUNCT
ejpam-6802	333	1	then	then	ADV
ejpam-6802	333	2	(	(	PUNCT
ejpam-6802	333	3	b	b	X
ejpam-6802	333	4	,	,	PUNCT
ejpam-6802	333	5	f	f	PROPN
ejpam-6802	333	6	,	,	PUNCT
ejpam-6802	333	7	α	α	PROPN
ejpam-6802	333	8	)	)	PUNCT
ejpam-6802	333	9	is	be	AUX
ejpam-6802	333	10	a	a	DET
ejpam-6802	333	11	complete	complete	ADJ
ejpam-6802	333	12	mcbms	mcbms	NOUN
ejpam-6802	333	13	.	.	PUNCT
ejpam-6802	334	1	theorem	theorem	VERB
ejpam-6802	334	2	4.2	4.2	NUM
ejpam-6802	334	3	.	.	PUNCT
ejpam-6802	335	1	consider	consider	VERB
ejpam-6802	335	2	the	the	DET
ejpam-6802	335	3	nonlinear	nonlinear	ADJ
ejpam-6802	335	4	fractional	fractional	ADJ
ejpam-6802	335	5	differential	differential	ADJ
ejpam-6802	335	6	equation	equation	NOUN
ejpam-6802	335	7	(	(	PUNCT
ejpam-6802	335	8	10	10	NUM
ejpam-6802	335	9	)	)	PUNCT
ejpam-6802	335	10	.	.	PUNCT
ejpam-6802	336	1	suppose	suppose	VERB
ejpam-6802	336	2	that	that	SCONJ
ejpam-6802	336	3	the	the	DET
ejpam-6802	336	4	following	follow	VERB
ejpam-6802	336	5	conditions	condition	NOUN
ejpam-6802	336	6	are	be	AUX
ejpam-6802	336	7	satisfies	satisfie	NOUN
ejpam-6802	336	8	:	:	PUNCT
ejpam-6802	336	9	[	[	X
ejpam-6802	336	10	label=(ii	label=(ii	NOUN
ejpam-6802	336	11	)	)	PUNCT
ejpam-6802	336	12	]	]	PUNCT
ejpam-6802	336	13	(	(	PUNCT
ejpam-6802	336	14	i	i	NOUN
ejpam-6802	336	15	)	)	PUNCT
ejpam-6802	336	16	∃	∃	PROPN
ejpam-6802	336	17	η	η	PROPN
ejpam-6802	336	18	∈	∈	PROPN
ejpam-6802	337	1	[	[	X
ejpam-6802	337	2	0	0	NUM
ejpam-6802	337	3	,	,	PUNCT
ejpam-6802	337	4	1	1	NUM
ejpam-6802	337	5	]	]	PUNCT
ejpam-6802	337	6	,	,	PUNCT
ejpam-6802	337	7	λ	λ	PROPN
ejpam-6802	337	8	∈	∈	PROPN
ejpam-6802	337	9	(	(	PUNCT
ejpam-6802	337	10	0	0	NUM
ejpam-6802	337	11	,	,	PUNCT
ejpam-6802	337	12	1	1	NUM
ejpam-6802	337	13	)	)	PUNCT
ejpam-6802	337	14	&	&	CCONJ
ejpam-6802	337	15	(	(	PUNCT
ejpam-6802	337	16	s	s	X
ejpam-6802	337	17	,	,	PUNCT
ejpam-6802	337	18	s	s	PART
ejpam-6802	337	19	′	′	NOUN
ejpam-6802	337	20	)	)	PUNCT
ejpam-6802	337	21	∈	∈	PROPN
ejpam-6802	338	1	b	b	X
ejpam-6802	338	2	×	×	NOUN
ejpam-6802	338	3	f	f	PROPN
ejpam-6802	338	4	such	such	ADJ
ejpam-6802	338	5	that	that	DET
ejpam-6802	338	6	|f(η	|f(η	PROPN
ejpam-6802	338	7	,	,	PUNCT
ejpam-6802	338	8	s)−	s)−	PROPN
ejpam-6802	338	9	f(η	f(η	PROPN
ejpam-6802	338	10	,	,	PUNCT
ejpam-6802	338	11	s	s	PART
ejpam-6802	338	12	′	′	NOUN
ejpam-6802	338	13	)	)	PUNCT
ejpam-6802	339	1	|	|	ADV
ejpam-6802	339	2	≤	≤	NUM
ejpam-6802	339	3	λ|s(η)−	λ|s(η)−	NOUN
ejpam-6802	339	4	s	s	VERB
ejpam-6802	339	5	′	′	NOUN
ejpam-6802	339	6	(	(	PUNCT
ejpam-6802	339	7	η)|	η)|	NOUN
ejpam-6802	339	8	;	;	PUNCT
ejpam-6802	339	9	(	(	PUNCT
ejpam-6802	339	10	ii	ii	NOUN
ejpam-6802	339	11	)	)	PUNCT
ejpam-6802	339	12	sup	sup	NOUN
ejpam-6802	339	13	η∈[0,1	η∈[0,1	PROPN
ejpam-6802	339	14	]	]	PUNCT
ejpam-6802	339	15	∫	∫	PROPN
ejpam-6802	339	16	1	1	NUM
ejpam-6802	339	17	0	0	X
ejpam-6802	339	18	|g(η	|g(η	PROPN
ejpam-6802	339	19	,	,	PUNCT
ejpam-6802	339	20	e)|dm	e)|dm	PROPN
ejpam-6802	339	21	≤	≤	ADV
ejpam-6802	339	22	1	1	NUM
ejpam-6802	339	23	.	.	PUNCT
ejpam-6802	340	1	then	then	ADV
ejpam-6802	340	2	the	the	DET
ejpam-6802	340	3	fractional	fractional	ADJ
ejpam-6802	340	4	differential	differential	ADJ
ejpam-6802	340	5	equation	equation	NOUN
ejpam-6802	340	6	(	(	PUNCT
ejpam-6802	340	7	10	10	NUM
ejpam-6802	340	8	)	)	PUNCT
ejpam-6802	340	9	possesses	possess	VERB
ejpam-6802	340	10	a	a	DET
ejpam-6802	340	11	unique	unique	ADJ
ejpam-6802	340	12	solution	solution	NOUN
ejpam-6802	340	13	in	in	ADP
ejpam-6802	340	14	b	b	NOUN
ejpam-6802	340	15	∪	∪	PROPN
ejpam-6802	340	16	f	f	PROPN
ejpam-6802	340	17	.	.	PUNCT
ejpam-6802	341	1	proof	proof	NOUN
ejpam-6802	341	2	.	.	PUNCT
ejpam-6802	342	1	the	the	DET
ejpam-6802	342	2	given	give	VERB
ejpam-6802	342	3	fractional	fractional	ADJ
ejpam-6802	342	4	differential	differential	ADJ
ejpam-6802	342	5	equation	equation	NOUN
ejpam-6802	342	6	(	(	PUNCT
ejpam-6802	342	7	10	10	NUM
ejpam-6802	342	8	)	)	PUNCT
ejpam-6802	342	9	is	be	AUX
ejpam-6802	342	10	equivalent	equivalent	ADJ
ejpam-6802	342	11	to	to	ADP
ejpam-6802	342	12	the	the	DET
ejpam-6802	342	13	succeeding	succeed	VERB
ejpam-6802	342	14	integral	integral	ADJ
ejpam-6802	342	15	equation	equation	NOUN
ejpam-6802	342	16	s(η	s(η	PROPN
ejpam-6802	342	17	)	)	PUNCT
ejpam-6802	343	1	=	=	SYM
ejpam-6802	343	2	∫	∫	PROPN
ejpam-6802	343	3	1	1	NUM
ejpam-6802	343	4	0	0	NUM
ejpam-6802	343	5	g(η	g(η	VERB
ejpam-6802	343	6	,	,	PUNCT
ejpam-6802	343	7	e)f(m	e)f(m	NOUN
ejpam-6802	343	8	,	,	PUNCT
ejpam-6802	343	9	s(e))de	s(e))de	NOUN
ejpam-6802	343	10	,	,	PUNCT
ejpam-6802	343	11	where	where	SCONJ
ejpam-6802	343	12	g(η	g(η	PROPN
ejpam-6802	343	13	,	,	PUNCT
ejpam-6802	343	14	e	e	NOUN
ejpam-6802	343	15	)	)	PUNCT
ejpam-6802	343	16	=	=	SYM
ejpam-6802	343	17	{	{	PUNCT
ejpam-6802	344	1	[	[	X
ejpam-6802	344	2	η(1−e)]q−1−(η−e)q−1	η(1−e)]q−1−(η−e)q−1	NOUN
ejpam-6802	344	3	γ(q	γ(q	NOUN
ejpam-6802	344	4	)	)	PUNCT
ejpam-6802	344	5	,	,	PUNCT
ejpam-6802	344	6	0	0	NUM
ejpam-6802	344	7	≤	≤	NUM
ejpam-6802	344	8	e	e	X
ejpam-6802	344	9	≤	≤	NUM
ejpam-6802	344	10	η	η	PROPN
ejpam-6802	344	11	≤	≤	PROPN
ejpam-6802	344	12	1	1	NUM
ejpam-6802	344	13	,	,	PUNCT
ejpam-6802	344	14	[	[	X
ejpam-6802	344	15	η(1−e)]q−1	η(1−e)]q−1	NOUN
ejpam-6802	344	16	γ(q	γ(q	NOUN
ejpam-6802	344	17	)	)	PUNCT
ejpam-6802	344	18	,	,	PUNCT
ejpam-6802	345	1	0	0	NUM
ejpam-6802	345	2	≤	≤	NUM
ejpam-6802	345	3	η	η	PROPN
ejpam-6802	345	4	≤	≤	X
ejpam-6802	345	5	e	e	X
ejpam-6802	345	6	≤	≤	NOUN
ejpam-6802	345	7	1	1	NUM
ejpam-6802	345	8	.	.	PUNCT
ejpam-6802	345	9	define	define	VERB
ejpam-6802	345	10	the	the	DET
ejpam-6802	345	11	covariant	covariant	PROPN
ejpam-6802	345	12	function	function	NOUN
ejpam-6802	345	13	ω	ω	PROPN
ejpam-6802	345	14	:	:	PUNCT
ejpam-6802	345	15	b	b	X
ejpam-6802	345	16	∪	∪	X
ejpam-6802	345	17	f	f	PROPN
ejpam-6802	345	18	→	→	SYM
ejpam-6802	345	19	b	b	X
ejpam-6802	345	20	∪	∪	X
ejpam-6802	345	21	f	f	PROPN
ejpam-6802	345	22	defined	define	VERB
ejpam-6802	345	23	by	by	ADP
ejpam-6802	345	24	ωs(η	ωs(η	NOUN
ejpam-6802	345	25	)	)	PUNCT
ejpam-6802	345	26	=	=	SYM
ejpam-6802	345	27	∫	∫	PROPN
ejpam-6802	345	28	1	1	NUM
ejpam-6802	345	29	0	0	NUM
ejpam-6802	345	30	g(η	g(η	VERB
ejpam-6802	345	31	,	,	PUNCT
ejpam-6802	345	32	e)f(m	e)f(m	NOUN
ejpam-6802	345	33	,	,	PUNCT
ejpam-6802	345	34	s(e))de	s(e))de	NOUN
ejpam-6802	345	35	.	.	PUNCT
ejpam-6802	346	1	now	now	ADV
ejpam-6802	346	2	|ωs(η)−	|ωs(η)−	NOUN
ejpam-6802	346	3	ωs	ωs	ADP
ejpam-6802	346	4	′	′	NUM
ejpam-6802	346	5	(	(	PUNCT
ejpam-6802	346	6	η)|	η)|	NOUN
ejpam-6802	346	7	=	=	PUNCT
ejpam-6802	346	8	∣∣∣∣	∣∣∣∣	NOUN
ejpam-6802	346	9	∫	∫	PROPN
ejpam-6802	346	10	1	1	NUM
ejpam-6802	346	11	0	0	NUM
ejpam-6802	346	12	g(η	g(η	VERB
ejpam-6802	346	13	,	,	PUNCT
ejpam-6802	346	14	e)f(m	e)f(m	NOUN
ejpam-6802	346	15	,	,	PUNCT
ejpam-6802	346	16	s(e))de−	s(e))de−	PROPN
ejpam-6802	346	17	∫	∫	PROPN
ejpam-6802	346	18	1	1	NUM
ejpam-6802	346	19	o	o	NOUN
ejpam-6802	346	20	g(η	g(η	PROPN
ejpam-6802	346	21	,	,	PUNCT
ejpam-6802	346	22	e)f(m	e)f(m	NOUN
ejpam-6802	346	23	,	,	PUNCT
ejpam-6802	346	24	s	s	PART
ejpam-6802	346	25	′	′	NOUN
ejpam-6802	346	26	(	(	PUNCT
ejpam-6802	346	27	e))de	e))de	PROPN
ejpam-6802	346	28	∣∣∣∣	∣∣∣∣	PROPN
ejpam-6802	346	29	≤	≤	NUM
ejpam-6802	346	30	∫	∫	NOUN
ejpam-6802	346	31	1	1	NUM
ejpam-6802	346	32	0	0	X
ejpam-6802	346	33	|g(η	|g(η	PROPN
ejpam-6802	346	34	,	,	PUNCT
ejpam-6802	346	35	e)|de	e)|de	PROPN
ejpam-6802	346	36	·	·	PUNCT
ejpam-6802	346	37	∫	∫	PROPN
ejpam-6802	346	38	1	1	NUM
ejpam-6802	346	39	0	0	X
ejpam-6802	347	1	∣∣∣∣f(m	∣∣∣∣f(m	NOUN
ejpam-6802	347	2	,	,	PUNCT
ejpam-6802	347	3	s(e))−	s(e))−	ADP
ejpam-6802	347	4	f(m	f(m	PROPN
ejpam-6802	347	5	,	,	PUNCT
ejpam-6802	347	6	s	s	PART
ejpam-6802	347	7	′	′	NOUN
ejpam-6802	347	8	(	(	PUNCT
ejpam-6802	347	9	e	e	NOUN
ejpam-6802	347	10	)	)	PUNCT
ejpam-6802	347	11	)	)	PUNCT
ejpam-6802	348	1	∣∣∣∣de	∣∣∣∣de	VERB
ejpam-6802	348	2	r.	r.	PROPN
ejpam-6802	348	3	ramaswamy	ramaswamy	PROPN
ejpam-6802	348	4	/	/	SYM
ejpam-6802	348	5	eur	eur	PROPN
ejpam-6802	348	6	.	.	PUNCT
ejpam-6802	349	1	j.	j.	PROPN
ejpam-6802	349	2	pure	pure	PROPN
ejpam-6802	349	3	appl	appl	PROPN
ejpam-6802	349	4	.	.	PROPN
ejpam-6802	349	5	math	math	PROPN
ejpam-6802	349	6	,	,	PUNCT
ejpam-6802	349	7	18	18	NUM
ejpam-6802	349	8	(	(	PUNCT
ejpam-6802	349	9	4	4	NUM
ejpam-6802	349	10	)	)	PUNCT
ejpam-6802	349	11	(	(	PUNCT
ejpam-6802	349	12	2025	2025	NUM
ejpam-6802	349	13	)	)	PUNCT
ejpam-6802	349	14	,	,	PUNCT
ejpam-6802	349	15	6802	6802	NUM
ejpam-6802	349	16	18	18	NUM
ejpam-6802	349	17	of	of	ADP
ejpam-6802	349	18	20	20	NUM
ejpam-6802	349	19	≤	≤	NUM
ejpam-6802	349	20	λ	λ	X
ejpam-6802	349	21	∣∣s(η)−	∣∣s(η)−	PROPN
ejpam-6802	349	22	s	s	PROPN
ejpam-6802	349	23	′	′	NUM
ejpam-6802	349	24	(	(	PUNCT
ejpam-6802	349	25	η	η	PROPN
ejpam-6802	349	26	)	)	PUNCT
ejpam-6802	349	27	∣∣.	∣∣.	PROPN
ejpam-6802	349	28	therefore	therefore	ADV
ejpam-6802	349	29	,	,	PUNCT
ejpam-6802	349	30	α(ωs	α(ωs	NUM
ejpam-6802	349	31	,	,	PUNCT
ejpam-6802	349	32	ωs	ωs	X
ejpam-6802	349	33	′	′	NUM
ejpam-6802	349	34	)	)	PUNCT
ejpam-6802	350	1	=	=	PUNCT
ejpam-6802	351	1	esupη∈[0,1	esupη∈[0,1	ADP
ejpam-6802	351	2	]	]	X
ejpam-6802	351	3	|ωs(η)−ωs	|ωs(η)−ωs	PROPN
ejpam-6802	351	4	′	′	NUM
ejpam-6802	351	5	(	(	PUNCT
ejpam-6802	351	6	η)|	η)|	NOUN
ejpam-6802	351	7	≤	≤	PROPN
ejpam-6802	351	8	eλ	eλ	ADP
ejpam-6802	352	1	supη∈[0,1	supη∈[0,1	ADJ
ejpam-6802	352	2	]	]	PUNCT
ejpam-6802	352	3	|s(η)−s	|s(η)−s	PUNCT
ejpam-6802	352	4	′	′	NUM
ejpam-6802	352	5	(	(	PUNCT
ejpam-6802	352	6	η)|	η)|	NOUN
ejpam-6802	352	7	=	=	SYM
ejpam-6802	352	8	(	(	PUNCT
ejpam-6802	352	9	α(s	α(s	PROPN
ejpam-6802	352	10	,	,	PUNCT
ejpam-6802	352	11	s	s	PART
ejpam-6802	352	12	′	′	NOUN
ejpam-6802	352	13	)	)	PUNCT
ejpam-6802	352	14	)	)	PUNCT
ejpam-6802	353	1	λ	λ	X
ejpam-6802	353	2	.	.	PUNCT
ejpam-6802	353	3	therefore	therefore	ADV
ejpam-6802	353	4	,	,	PUNCT
ejpam-6802	353	5	all	all	DET
ejpam-6802	353	6	the	the	DET
ejpam-6802	353	7	criteria	criterion	NOUN
ejpam-6802	353	8	of	of	ADP
ejpam-6802	353	9	theorem	theorem	ADJ
ejpam-6802	353	10	3.1	3.1	NUM
ejpam-6802	353	11	are	be	AUX
ejpam-6802	353	12	satisfied	satisfied	ADJ
ejpam-6802	353	13	&	&	CCONJ
ejpam-6802	353	14	as	as	ADP
ejpam-6802	353	15	a	a	DET
ejpam-6802	353	16	result	result	NOUN
ejpam-6802	353	17	,	,	PUNCT
ejpam-6802	353	18	the	the	DET
ejpam-6802	353	19	fractional	fractional	ADJ
ejpam-6802	353	20	differential	differential	ADJ
ejpam-6802	353	21	equation	equation	NOUN
ejpam-6802	353	22	(	(	PUNCT
ejpam-6802	353	23	10	10	NUM
ejpam-6802	353	24	)	)	PUNCT
ejpam-6802	353	25	possesses	possess	VERB
ejpam-6802	353	26	a	a	DET
ejpam-6802	353	27	unique	unique	ADJ
ejpam-6802	353	28	solution	solution	NOUN
ejpam-6802	353	29	.	.	PUNCT
ejpam-6802	354	1	5	5	X
ejpam-6802	354	2	.	.	X
ejpam-6802	354	3	conclusion	conclusion	NOUN
ejpam-6802	354	4	in	in	ADP
ejpam-6802	354	5	this	this	DET
ejpam-6802	354	6	article	article	NOUN
ejpam-6802	354	7	,	,	PUNCT
ejpam-6802	354	8	the	the	DET
ejpam-6802	354	9	concept	concept	NOUN
ejpam-6802	354	10	of	of	ADP
ejpam-6802	354	11	mcbms	mcbms	NOUN
ejpam-6802	354	12	is	be	AUX
ejpam-6802	354	13	introduced	introduce	VERB
ejpam-6802	354	14	and	and	CCONJ
ejpam-6802	354	15	analogues	analogue	NOUN
ejpam-6802	354	16	of	of	ADP
ejpam-6802	354	17	fixed	fix	VERB
ejpam-6802	354	18	point	point	NOUN
ejpam-6802	354	19	theorems	theorem	NOUN
ejpam-6802	354	20	of	of	ADP
ejpam-6802	354	21	banach	banach	NOUN
ejpam-6802	354	22	and	and	CCONJ
ejpam-6802	354	23	kannan	kannan	PROPN
ejpam-6802	354	24	are	be	AUX
ejpam-6802	354	25	proved	prove	VERB
ejpam-6802	354	26	.	.	PUNCT
ejpam-6802	355	1	the	the	DET
ejpam-6802	355	2	derived	derive	VERB
ejpam-6802	355	3	results	result	NOUN
ejpam-6802	355	4	have	have	AUX
ejpam-6802	355	5	been	be	AUX
ejpam-6802	355	6	supplemented	supplement	VERB
ejpam-6802	355	7	with	with	ADP
ejpam-6802	355	8	non	non	ADJ
ejpam-6802	355	9	trivial	trivial	ADJ
ejpam-6802	355	10	example	example	NOUN
ejpam-6802	355	11	.	.	PUNCT
ejpam-6802	356	1	the	the	DET
ejpam-6802	356	2	practical	practical	ADJ
ejpam-6802	356	3	applicability	applicability	NOUN
ejpam-6802	356	4	for	for	ADP
ejpam-6802	356	5	finding	find	VERB
ejpam-6802	356	6	solution	solution	NOUN
ejpam-6802	356	7	to	to	ADP
ejpam-6802	356	8	integral	integral	ADJ
ejpam-6802	356	9	and	and	CCONJ
ejpam-6802	356	10	fractional	fractional	ADJ
ejpam-6802	356	11	differential	differential	ADJ
ejpam-6802	356	12	equations	equation	NOUN
ejpam-6802	356	13	is	be	AUX
ejpam-6802	356	14	also	also	ADV
ejpam-6802	356	15	presented	present	VERB
ejpam-6802	356	16	.	.	PUNCT
ejpam-6802	357	1	it	it	PRON
ejpam-6802	357	2	will	will	AUX
ejpam-6802	357	3	be	be	AUX
ejpam-6802	357	4	an	an	DET
ejpam-6802	357	5	open	open	ADJ
ejpam-6802	357	6	problem	problem	NOUN
ejpam-6802	357	7	to	to	PART
ejpam-6802	357	8	extend	extend	VERB
ejpam-6802	357	9	the	the	DET
ejpam-6802	357	10	fixed	fix	VERB
ejpam-6802	357	11	point	point	NOUN
ejpam-6802	357	12	results	result	NOUN
ejpam-6802	357	13	using	use	VERB
ejpam-6802	357	14	other	other	ADJ
ejpam-6802	357	15	contractive	contractive	ADJ
ejpam-6802	357	16	conditions	condition	NOUN
ejpam-6802	357	17	such	such	ADJ
ejpam-6802	357	18	as	as	ADP
ejpam-6802	357	19	ciric	ciric	ADJ
ejpam-6802	357	20	,	,	PUNCT
ejpam-6802	357	21	reich	reich	PROPN
ejpam-6802	357	22	,	,	PUNCT
ejpam-6802	357	23	junck	junck	ADJ
ejpam-6802	357	24	etc	etc	X
ejpam-6802	357	25	.	.	X
ejpam-6802	358	1	it	it	PRON
ejpam-6802	358	2	is	be	AUX
ejpam-6802	358	3	also	also	ADV
ejpam-6802	358	4	an	an	DET
ejpam-6802	358	5	open	open	ADJ
ejpam-6802	358	6	problem	problem	NOUN
ejpam-6802	358	7	to	to	PART
ejpam-6802	358	8	examine	examine	VERB
ejpam-6802	358	9	whether	whether	SCONJ
ejpam-6802	358	10	the	the	DET
ejpam-6802	358	11	space	space	NOUN
ejpam-6802	358	12	can	can	AUX
ejpam-6802	358	13	be	be	AUX
ejpam-6802	358	14	generalised	generalise	VERB
ejpam-6802	358	15	in	in	ADP
ejpam-6802	358	16	the	the	DET
ejpam-6802	358	17	form	form	NOUN
ejpam-6802	358	18	of	of	ADP
ejpam-6802	358	19	multiplicative	multiplicative	ADJ
ejpam-6802	358	20	cone	cone	NOUN
ejpam-6802	358	21	bipoloar	bipoloar	NOUN
ejpam-6802	358	22	b	b	X
ejpam-6802	358	23	-	-	PUNCT
ejpam-6802	358	24	metric	metric	ADJ
ejpam-6802	358	25	,	,	PUNCT
ejpam-6802	358	26	dislocated	dislocated	ADJ
ejpam-6802	358	27	mcbs	mcb	NOUN
ejpam-6802	358	28	,	,	PUNCT
ejpam-6802	358	29	retangular	retangular	ADJ
ejpam-6802	358	30	mcbs	mcb	NOUN
ejpam-6802	358	31	etc	etc	X
ejpam-6802	358	32	.	.	X
ejpam-6802	358	33	funding	fund	VERB
ejpam-6802	358	34	the	the	DET
ejpam-6802	358	35	author	author	NOUN
ejpam-6802	358	36	extends	extend	VERB
ejpam-6802	358	37	his	his	PRON
ejpam-6802	358	38	appreciation	appreciation	NOUN
ejpam-6802	358	39	to	to	ADP
ejpam-6802	358	40	prince	prince	PROPN
ejpam-6802	358	41	sattam	sattam	PROPN
ejpam-6802	358	42	bin	bin	PROPN
ejpam-6802	358	43	abdulaziz	abdulaziz	PROPN
ejpam-6802	358	44	university	university	PROPN
ejpam-6802	358	45	for	for	ADP
ejpam-6802	358	46	funding	fund	VERB
ejpam-6802	358	47	this	this	DET
ejpam-6802	358	48	research	research	NOUN
ejpam-6802	358	49	work	work	NOUN
ejpam-6802	358	50	through	through	ADP
ejpam-6802	358	51	the	the	DET
ejpam-6802	358	52	project	project	NOUN
ejpam-6802	358	53	number	number	NOUN
ejpam-6802	358	54	psau/2025/01/33096	psau/2025/01/33096	NOUN
ejpam-6802	358	55	.	.	PUNCT
ejpam-6802	359	1	acknowledgements	acknowledgement	VERB
ejpam-6802	359	2	the	the	DET
ejpam-6802	359	3	author	author	NOUN
ejpam-6802	359	4	extend	extend	VERB
ejpam-6802	359	5	his	his	PRON
ejpam-6802	359	6	appreciation	appreciation	NOUN
ejpam-6802	359	7	to	to	ADP
ejpam-6802	359	8	the	the	DET
ejpam-6802	359	9	deanship	deanship	NOUN
ejpam-6802	359	10	of	of	ADP
ejpam-6802	359	11	scientific	scientific	ADJ
ejpam-6802	359	12	research	research	NOUN
ejpam-6802	359	13	,	,	PUNCT
ejpam-6802	359	14	psau	psau	NOUN
ejpam-6802	359	15	,	,	PUNCT
ejpam-6802	359	16	alkharj	alkharj	VERB
ejpam-6802	359	17	for	for	ADP
ejpam-6802	359	18	supporting	support	VERB
ejpam-6802	359	19	the	the	DET
ejpam-6802	359	20	study	study	NOUN
ejpam-6802	359	21	.	.	PUNCT
ejpam-6802	360	1	conflict	conflict	NOUN
ejpam-6802	360	2	of	of	ADP
ejpam-6802	360	3	interests	interest	NOUN
ejpam-6802	360	4	the	the	DET
ejpam-6802	360	5	author	author	NOUN
ejpam-6802	360	6	declare	declare	VERB
ejpam-6802	360	7	no	no	DET
ejpam-6802	360	8	conflicts	conflict	NOUN
ejpam-6802	360	9	of	of	ADP
ejpam-6802	360	10	interest	interest	NOUN
ejpam-6802	360	11	.	.	PUNCT
ejpam-6802	361	1	references	reference	NOUN
ejpam-6802	361	2	[	[	X
ejpam-6802	361	3	1	1	X
ejpam-6802	361	4	]	]	PUNCT
ejpam-6802	361	5	s.	s.	PROPN
ejpam-6802	361	6	banach	banach	PROPN
ejpam-6802	361	7	.	.	PUNCT
ejpam-6802	362	1	sur	sur	PROPN
ejpam-6802	362	2	les	les	X
ejpam-6802	362	3	opérations	opération	NOUN
ejpam-6802	362	4	dans	dan	NOUN
ejpam-6802	362	5	les	les	X
ejpam-6802	362	6	ensembles	ensemble	NOUN
ejpam-6802	362	7	abstraits	abstrait	NOUN
ejpam-6802	362	8	et	et	PROPN
ejpam-6802	362	9	leur	leur	X
ejpam-6802	362	10	application	application	PROPN
ejpam-6802	362	11	aux	aux	PROPN
ejpam-6802	362	12	équations	équations	PROPN
ejpam-6802	362	13	intégrales	intégrale	NOUN
ejpam-6802	362	14	.	.	PUNCT
ejpam-6802	363	1	fundamenta	fundamenta	PROPN
ejpam-6802	363	2	mathematicae	mathematicae	PROPN
ejpam-6802	363	3	,	,	PUNCT
ejpam-6802	363	4	3(1):133–181	3(1):133–181	NUM
ejpam-6802	363	5	,	,	PUNCT
ejpam-6802	363	6	1922	1922	NUM
ejpam-6802	363	7	.	.	PUNCT
ejpam-6802	364	1	[	[	X
ejpam-6802	364	2	2	2	NUM
ejpam-6802	364	3	]	]	PUNCT
ejpam-6802	364	4	r.	r.	PROPN
ejpam-6802	364	5	kannan	kannan	PROPN
ejpam-6802	364	6	.	.	PUNCT
ejpam-6802	365	1	some	some	DET
ejpam-6802	365	2	results	result	NOUN
ejpam-6802	365	3	on	on	ADP
ejpam-6802	365	4	fixed	fix	VERB
ejpam-6802	365	5	points	point	NOUN
ejpam-6802	365	6	.	.	PUNCT
ejpam-6802	366	1	bulletin	bulletin	NOUN
ejpam-6802	366	2	of	of	ADP
ejpam-6802	366	3	the	the	DET
ejpam-6802	366	4	calcutta	calcutta	NOUN
ejpam-6802	366	5	mathematical	mathematical	ADJ
ejpam-6802	366	6	society	society	NOUN
ejpam-6802	366	7	,	,	PUNCT
ejpam-6802	366	8	60:71–76	60:71–76	NUM
ejpam-6802	366	9	,	,	PUNCT
ejpam-6802	366	10	1968	1968	NUM
ejpam-6802	366	11	.	.	PUNCT
ejpam-6802	367	1	[	[	X
ejpam-6802	367	2	3	3	X
ejpam-6802	367	3	]	]	PUNCT
ejpam-6802	367	4	s.	s.	PROPN
ejpam-6802	367	5	reich	reich	PROPN
ejpam-6802	367	6	.	.	PUNCT
ejpam-6802	368	1	kannan	kannan	PROPN
ejpam-6802	368	2	’s	’s	PART
ejpam-6802	368	3	fixed	fix	VERB
ejpam-6802	368	4	point	point	NOUN
ejpam-6802	368	5	theorem	theorem	VERB
ejpam-6802	368	6	.	.	PUNCT
ejpam-6802	369	1	bollettino	bollettino	PROPN
ejpam-6802	369	2	dell’unione	dell’unione	PROPN
ejpam-6802	369	3	matematica	matematica	PROPN
ejpam-6802	369	4	italiana	italiana	PROPN
ejpam-6802	369	5	,	,	PUNCT
ejpam-6802	369	6	4(4):1–11	4(4):1–11	NUM
ejpam-6802	369	7	,	,	PUNCT
ejpam-6802	369	8	1971	1971	NUM
ejpam-6802	369	9	.	.	PUNCT
ejpam-6802	370	1	[	[	X
ejpam-6802	370	2	4	4	X
ejpam-6802	370	3	]	]	X
ejpam-6802	370	4	g.	g.	PROPN
ejpam-6802	370	5	jungck	jungck	PROPN
ejpam-6802	370	6	.	.	PUNCT
ejpam-6802	371	1	compatible	compatible	ADJ
ejpam-6802	371	2	mappings	mapping	NOUN
ejpam-6802	371	3	and	and	CCONJ
ejpam-6802	371	4	common	common	ADJ
ejpam-6802	371	5	fixed	fix	VERB
ejpam-6802	371	6	point	point	NOUN
ejpam-6802	371	7	.	.	PUNCT
ejpam-6802	372	1	international	international	ADJ
ejpam-6802	372	2	journal	journal	PROPN
ejpam-6802	372	3	of	of	ADP
ejpam-6802	372	4	mathematics	mathematics	PROPN
ejpam-6802	372	5	and	and	CCONJ
ejpam-6802	372	6	mathematical	mathematical	ADJ
ejpam-6802	372	7	sciences	sciences	PROPN
ejpam-6802	372	8	,	,	PUNCT
ejpam-6802	372	9	9(4):771–779	9(4):771–779	NUM
ejpam-6802	372	10	,	,	PUNCT
ejpam-6802	372	11	1986	1986	NUM
ejpam-6802	372	12	.	.	PUNCT
ejpam-6802	373	1	r.	r.	PROPN
ejpam-6802	373	2	ramaswamy	ramaswamy	PROPN
ejpam-6802	373	3	/	/	SYM
ejpam-6802	373	4	eur	eur	PROPN
ejpam-6802	373	5	.	.	PUNCT
ejpam-6802	374	1	j.	j.	PROPN
ejpam-6802	374	2	pure	pure	PROPN
ejpam-6802	374	3	appl	appl	PROPN
ejpam-6802	374	4	.	.	PROPN
ejpam-6802	374	5	math	math	PROPN
ejpam-6802	374	6	,	,	PUNCT
ejpam-6802	374	7	18	18	NUM
ejpam-6802	374	8	(	(	PUNCT
ejpam-6802	374	9	4	4	NUM
ejpam-6802	374	10	)	)	PUNCT
ejpam-6802	374	11	(	(	PUNCT
ejpam-6802	374	12	2025	2025	NUM
ejpam-6802	374	13	)	)	PUNCT
ejpam-6802	374	14	,	,	PUNCT
ejpam-6802	374	15	6802	6802	NUM
ejpam-6802	374	16	19	19	NUM
ejpam-6802	374	17	of	of	ADP
ejpam-6802	374	18	20	20	NUM
ejpam-6802	374	19	[	[	SYM
ejpam-6802	374	20	5	5	NUM
ejpam-6802	374	21	]	]	PUNCT
ejpam-6802	374	22	l.-g	l.-g	PROPN
ejpam-6802	374	23	.	.	PUNCT
ejpam-6802	375	1	huang	huang	PROPN
ejpam-6802	375	2	and	and	CCONJ
ejpam-6802	375	3	x.	x.	PROPN
ejpam-6802	375	4	zhang	zhang	PROPN
ejpam-6802	375	5	.	.	PUNCT
ejpam-6802	376	1	cone	cone	PROPN
ejpam-6802	376	2	metric	metric	ADJ
ejpam-6802	376	3	spaces	space	NOUN
ejpam-6802	376	4	and	and	CCONJ
ejpam-6802	376	5	fixed	fix	VERB
ejpam-6802	376	6	point	point	NOUN
ejpam-6802	376	7	theorems	theorem	NOUN
ejpam-6802	376	8	of	of	ADP
ejpam-6802	376	9	contractive	contractive	ADJ
ejpam-6802	376	10	mappings	mapping	NOUN
ejpam-6802	376	11	.	.	PUNCT
ejpam-6802	377	1	journal	journal	PROPN
ejpam-6802	377	2	of	of	ADP
ejpam-6802	377	3	mathematical	mathematical	ADJ
ejpam-6802	377	4	analysis	analysis	NOUN
ejpam-6802	377	5	and	and	CCONJ
ejpam-6802	377	6	applications	application	NOUN
ejpam-6802	377	7	,	,	PUNCT
ejpam-6802	377	8	332(2):1468–1476	332(2):1468–1476	PROPN
ejpam-6802	377	9	,	,	PUNCT
ejpam-6802	377	10	2007	2007	NUM
ejpam-6802	377	11	.	.	PUNCT
ejpam-6802	378	1	[	[	X
ejpam-6802	378	2	6	6	NUM
ejpam-6802	378	3	]	]	PUNCT
ejpam-6802	378	4	r.	r.	PROPN
ejpam-6802	378	5	george	george	PROPN
ejpam-6802	378	6	,	,	PUNCT
ejpam-6802	378	7	r.	r.	PROPN
ejpam-6802	378	8	rajagopalan	rajagopalan	PROPN
ejpam-6802	378	9	,	,	PUNCT
ejpam-6802	378	10	h.	h.	PROPN
ejpam-6802	378	11	a.	a.	PROPN
ejpam-6802	378	12	nabway	nabway	PROPN
ejpam-6802	378	13	,	,	PUNCT
ejpam-6802	378	14	and	and	CCONJ
ejpam-6802	378	15	s.	s.	PROPN
ejpam-6802	378	16	radenovic	radenovic	PROPN
ejpam-6802	378	17	.	.	PUNCT
ejpam-6802	379	1	dislocated	dislocate	VERB
ejpam-6802	379	2	cone	cone	NOUN
ejpam-6802	379	3	metric	metric	ADJ
ejpam-6802	379	4	space	space	NOUN
ejpam-6802	379	5	over	over	ADP
ejpam-6802	379	6	banach	banach	NOUN
ejpam-6802	379	7	algebra	algebra	NOUN
ejpam-6802	379	8	and	and	CCONJ
ejpam-6802	379	9	α	α	NOUN
ejpam-6802	379	10	-	-	PUNCT
ejpam-6802	379	11	quasi	quasi	ADJ
ejpam-6802	379	12	contraction	contraction	NOUN
ejpam-6802	379	13	mappings	mapping	NOUN
ejpam-6802	379	14	of	of	ADP
ejpam-6802	379	15	perov	perov	PROPN
ejpam-6802	379	16	type	type	NOUN
ejpam-6802	379	17	.	.	PUNCT
ejpam-6802	380	1	fixed	fix	VERB
ejpam-6802	380	2	point	point	NOUN
ejpam-6802	380	3	theory	theory	NOUN
ejpam-6802	380	4	and	and	CCONJ
ejpam-6802	380	5	applications	application	NOUN
ejpam-6802	380	6	,	,	PUNCT
ejpam-6802	380	7	2017:24	2017:24	NUM
ejpam-6802	380	8	,	,	PUNCT
ejpam-6802	380	9	2017	2017	NUM
ejpam-6802	380	10	.	.	PUNCT
ejpam-6802	381	1	[	[	X
ejpam-6802	381	2	7	7	X
ejpam-6802	381	3	]	]	X
ejpam-6802	381	4	r.	r.	PROPN
ejpam-6802	381	5	george	george	PROPN
ejpam-6802	381	6	,	,	PUNCT
ejpam-6802	381	7	h.	h.	PROPN
ejpam-6802	381	8	a.	a.	PROPN
ejpam-6802	381	9	nabway	nabway	PROPN
ejpam-6802	381	10	,	,	PUNCT
ejpam-6802	381	11	k.	k.	PROPN
ejpam-6802	382	1	p.	p.	PROPN
ejpam-6802	382	2	reshma	reshma	PROPN
ejpam-6802	382	3	,	,	PUNCT
ejpam-6802	382	4	and	and	CCONJ
ejpam-6802	382	5	r.	r.	PROPN
ejpam-6802	382	6	rajagopalan	rajagopalan	PROPN
ejpam-6802	382	7	.	.	PUNCT
ejpam-6802	383	1	generalized	generalize	VERB
ejpam-6802	383	2	cone	cone	NOUN
ejpam-6802	383	3	b	b	X
ejpam-6802	383	4	-	-	PUNCT
ejpam-6802	383	5	metric	metric	ADJ
ejpam-6802	383	6	spaces	space	NOUN
ejpam-6802	383	7	and	and	CCONJ
ejpam-6802	383	8	contraction	contraction	NOUN
ejpam-6802	383	9	principles	principle	NOUN
ejpam-6802	383	10	.	.	PUNCT
ejpam-6802	384	1	matematički	matematički	PROPN
ejpam-6802	384	2	vesnik	vesnik	X
ejpam-6802	384	3	,	,	PUNCT
ejpam-6802	384	4	67(4):246–257	67(4):246–257	NOUN
ejpam-6802	384	5	,	,	PUNCT
ejpam-6802	384	6	2015	2015	NUM
ejpam-6802	384	7	.	.	PUNCT
ejpam-6802	385	1	[	[	X
ejpam-6802	385	2	8	8	NUM
ejpam-6802	385	3	]	]	X
ejpam-6802	385	4	r.	r.	PROPN
ejpam-6802	385	5	george	george	PROPN
ejpam-6802	385	6	,	,	PUNCT
ejpam-6802	385	7	k.	k.	PROPN
ejpam-6802	386	1	p.	p.	PROPN
ejpam-6802	386	2	reshma	reshma	PROPN
ejpam-6802	386	3	,	,	PUNCT
ejpam-6802	386	4	and	and	CCONJ
ejpam-6802	386	5	r.	r.	PROPN
ejpam-6802	386	6	rajagopalan	rajagopalan	PROPN
ejpam-6802	386	7	.	.	PUNCT
ejpam-6802	387	1	a	a	DET
ejpam-6802	387	2	generalised	generalise	VERB
ejpam-6802	387	3	fixed	fix	VERB
ejpam-6802	387	4	point	point	NOUN
ejpam-6802	387	5	theorem	theorem	NOUN
ejpam-6802	387	6	of	of	ADP
ejpam-6802	387	7	presic	presic	ADJ
ejpam-6802	387	8	type	type	NOUN
ejpam-6802	387	9	in	in	ADP
ejpam-6802	387	10	cone	cone	NOUN
ejpam-6802	387	11	metric	metric	ADJ
ejpam-6802	387	12	spaces	space	NOUN
ejpam-6802	387	13	and	and	CCONJ
ejpam-6802	387	14	application	application	NOUN
ejpam-6802	387	15	to	to	ADP
ejpam-6802	387	16	markov	markov	NOUN
ejpam-6802	387	17	process	process	NOUN
ejpam-6802	387	18	.	.	PUNCT
ejpam-6802	388	1	fixed	fix	VERB
ejpam-6802	388	2	point	point	NOUN
ejpam-6802	388	3	theory	theory	NOUN
ejpam-6802	388	4	and	and	CCONJ
ejpam-6802	388	5	applications	application	NOUN
ejpam-6802	388	6	,	,	PUNCT
ejpam-6802	388	7	2011:85	2011:85	NUM
ejpam-6802	388	8	,	,	PUNCT
ejpam-6802	388	9	2011	2011	NUM
ejpam-6802	388	10	.	.	PUNCT
ejpam-6802	389	1	[	[	X
ejpam-6802	389	2	9	9	NUM
ejpam-6802	389	3	]	]	X
ejpam-6802	389	4	r.	r.	PROPN
ejpam-6802	389	5	george	george	PROPN
ejpam-6802	389	6	,	,	PUNCT
ejpam-6802	389	7	h.	h.	PROPN
ejpam-6802	389	8	a.	a.	PROPN
ejpam-6802	389	9	nabway	nabway	PROPN
ejpam-6802	389	10	,	,	PUNCT
ejpam-6802	389	11	r.	r.	PROPN
ejpam-6802	389	12	rajagopalan	rajagopalan	PROPN
ejpam-6802	389	13	,	,	PUNCT
ejpam-6802	389	14	s.	s.	PROPN
ejpam-6802	389	15	radenovic	radenovic	PROPN
ejpam-6802	389	16	,	,	PUNCT
ejpam-6802	389	17	and	and	CCONJ
ejpam-6802	389	18	k.	k.	PROPN
ejpam-6802	389	19	p.	p.	PROPN
ejpam-6802	389	20	reshma	reshma	PROPN
ejpam-6802	389	21	.	.	PUNCT
ejpam-6802	390	1	rectangular	rectangular	ADJ
ejpam-6802	390	2	cone	cone	NOUN
ejpam-6802	390	3	b	b	X
ejpam-6802	390	4	-	-	ADJ
ejpam-6802	390	5	metric	metric	ADJ
ejpam-6802	390	6	spaces	space	NOUN
ejpam-6802	390	7	over	over	ADP
ejpam-6802	390	8	banach	banach	NOUN
ejpam-6802	390	9	algebra	algebra	NOUN
ejpam-6802	390	10	and	and	CCONJ
ejpam-6802	390	11	contraction	contraction	NOUN
ejpam-6802	390	12	principle	principle	NOUN
ejpam-6802	390	13	.	.	PUNCT
ejpam-6802	391	1	fixed	fix	VERB
ejpam-6802	391	2	point	point	NOUN
ejpam-6802	391	3	theory	theory	NOUN
ejpam-6802	391	4	and	and	CCONJ
ejpam-6802	391	5	applications	application	NOUN
ejpam-6802	391	6	,	,	PUNCT
ejpam-6802	391	7	2017:14	2017:14	NUM
ejpam-6802	391	8	,	,	PUNCT
ejpam-6802	391	9	2017	2017	NUM
ejpam-6802	391	10	.	.	PUNCT
ejpam-6802	392	1	[	[	X
ejpam-6802	392	2	10	10	NUM
ejpam-6802	392	3	]	]	PUNCT
ejpam-6802	392	4	a.	a.	NOUN
ejpam-6802	392	5	mutlu	mutlu	PROPN
ejpam-6802	392	6	and	and	CCONJ
ejpam-6802	392	7	u.	u.	PROPN
ejpam-6802	392	8	gürdal	gürdal	PROPN
ejpam-6802	392	9	.	.	PUNCT
ejpam-6802	393	1	bipolar	bipolar	ADJ
ejpam-6802	393	2	metric	metric	ADJ
ejpam-6802	393	3	spaces	space	NOUN
ejpam-6802	393	4	and	and	CCONJ
ejpam-6802	393	5	some	some	DET
ejpam-6802	393	6	fixed	fix	VERB
ejpam-6802	393	7	point	point	NOUN
ejpam-6802	393	8	theorems	theorem	NOUN
ejpam-6802	393	9	.	.	PUNCT
ejpam-6802	393	10	journal	journal	PROPN
ejpam-6802	393	11	of	of	ADP
ejpam-6802	393	12	nonlinear	nonlinear	PROPN
ejpam-6802	393	13	sciences	sciences	PROPN
ejpam-6802	393	14	and	and	CCONJ
ejpam-6802	393	15	applications	application	NOUN
ejpam-6802	393	16	,	,	PUNCT
ejpam-6802	393	17	9:5362–5373	9:5362–5373	NUM
ejpam-6802	393	18	,	,	PUNCT
ejpam-6802	393	19	2016	2016	NUM
ejpam-6802	393	20	.	.	PUNCT
ejpam-6802	394	1	[	[	X
ejpam-6802	394	2	11	11	NUM
ejpam-6802	394	3	]	]	PUNCT
ejpam-6802	394	4	a.	a.	NOUN
ejpam-6802	394	5	mutlu	mutlu	PROPN
ejpam-6802	394	6	,	,	PUNCT
ejpam-6802	394	7	k.	k.	PROPN
ejpam-6802	394	8	özkan	özkan	PROPN
ejpam-6802	394	9	,	,	PUNCT
ejpam-6802	394	10	and	and	CCONJ
ejpam-6802	394	11	u.	u.	PROPN
ejpam-6802	394	12	gürdal	gürdal	PROPN
ejpam-6802	394	13	.	.	PUNCT
ejpam-6802	394	14	coupled	couple	VERB
ejpam-6802	394	15	fixed	fix	VERB
ejpam-6802	394	16	point	point	NOUN
ejpam-6802	394	17	theorems	theorem	NOUN
ejpam-6802	394	18	on	on	ADP
ejpam-6802	394	19	bipolar	bipolar	ADJ
ejpam-6802	394	20	metric	metric	ADJ
ejpam-6802	394	21	spaces	space	NOUN
ejpam-6802	394	22	.	.	PUNCT
ejpam-6802	395	1	european	european	ADJ
ejpam-6802	395	2	journal	journal	PROPN
ejpam-6802	395	3	of	of	ADP
ejpam-6802	395	4	pure	pure	ADJ
ejpam-6802	395	5	and	and	CCONJ
ejpam-6802	395	6	applied	applied	ADJ
ejpam-6802	395	7	mathematics	mathematic	NOUN
ejpam-6802	395	8	,	,	PUNCT
ejpam-6802	395	9	10(4):655–667	10(4):655–667	PROPN
ejpam-6802	395	10	,	,	PUNCT
ejpam-6802	395	11	2017	2017	NUM
ejpam-6802	395	12	.	.	PUNCT
ejpam-6802	396	1	[	[	X
ejpam-6802	396	2	12	12	NUM
ejpam-6802	396	3	]	]	PUNCT
ejpam-6802	396	4	a.	a.	NOUN
ejpam-6802	396	5	mutlu	mutlu	PROPN
ejpam-6802	396	6	,	,	PUNCT
ejpam-6802	396	7	k.	k.	PROPN
ejpam-6802	396	8	özkan	özkan	PROPN
ejpam-6802	396	9	,	,	PUNCT
ejpam-6802	396	10	and	and	CCONJ
ejpam-6802	396	11	u.	u.	PROPN
ejpam-6802	396	12	gürdal	gürdal	PROPN
ejpam-6802	396	13	.	.	PUNCT
ejpam-6802	397	1	locally	locally	ADV
ejpam-6802	397	2	and	and	CCONJ
ejpam-6802	397	3	weakly	weakly	ADJ
ejpam-6802	397	4	contractive	contractive	ADJ
ejpam-6802	397	5	principle	principle	NOUN
ejpam-6802	397	6	in	in	ADP
ejpam-6802	397	7	bipolar	bipolar	ADJ
ejpam-6802	397	8	metric	metric	ADJ
ejpam-6802	397	9	spaces	space	NOUN
ejpam-6802	397	10	.	.	PUNCT
ejpam-6802	398	1	twms	twms	PROPN
ejpam-6802	398	2	journal	journal	PROPN
ejpam-6802	398	3	of	of	ADP
ejpam-6802	398	4	applied	apply	VERB
ejpam-6802	398	5	and	and	CCONJ
ejpam-6802	398	6	engineering	engineering	NOUN
ejpam-6802	398	7	mathematics	mathematic	NOUN
ejpam-6802	398	8	,	,	PUNCT
ejpam-6802	398	9	10(2):379–388	10(2):379–388	NUM
ejpam-6802	398	10	,	,	PUNCT
ejpam-6802	398	11	2020	2020	NUM
ejpam-6802	398	12	.	.	PUNCT
ejpam-6802	399	1	[	[	X
ejpam-6802	399	2	13	13	NUM
ejpam-6802	399	3	]	]	X
ejpam-6802	399	4	b.	b.	PROPN
ejpam-6802	399	5	s.	s.	PROPN
ejpam-6802	399	6	rao	rao	PROPN
ejpam-6802	399	7	,	,	PUNCT
ejpam-6802	399	8	g.	g.	PROPN
ejpam-6802	399	9	n.	n.	PROPN
ejpam-6802	399	10	v.	v.	PROPN
ejpam-6802	399	11	kishore	kishore	PROPN
ejpam-6802	399	12	,	,	PUNCT
ejpam-6802	399	13	and	and	CCONJ
ejpam-6802	399	14	g.	g.	PROPN
ejpam-6802	399	15	k.	k.	PROPN
ejpam-6802	399	16	kumar	kumar	PROPN
ejpam-6802	399	17	.	.	PUNCT
ejpam-6802	400	1	geraghty	geraghty	PROPN
ejpam-6802	400	2	type	type	NOUN
ejpam-6802	400	3	contraction	contraction	NOUN
ejpam-6802	400	4	and	and	CCONJ
ejpam-6802	400	5	common	common	ADJ
ejpam-6802	400	6	coupled	couple	VERB
ejpam-6802	400	7	fixed	fix	VERB
ejpam-6802	400	8	point	point	NOUN
ejpam-6802	400	9	theorems	theorem	NOUN
ejpam-6802	400	10	in	in	ADP
ejpam-6802	400	11	bipolar	bipolar	ADJ
ejpam-6802	400	12	metric	metric	ADJ
ejpam-6802	400	13	spaces	space	NOUN
ejpam-6802	400	14	with	with	ADP
ejpam-6802	400	15	applications	application	NOUN
ejpam-6802	400	16	to	to	PART
ejpam-6802	400	17	homotopy	homotopy	VERB
ejpam-6802	400	18	.	.	PUNCT
ejpam-6802	401	1	international	international	ADJ
ejpam-6802	401	2	journal	journal	PROPN
ejpam-6802	401	3	of	of	ADP
ejpam-6802	401	4	mathematics	mathematics	NOUN
ejpam-6802	401	5	trends	trend	NOUN
ejpam-6802	401	6	and	and	CCONJ
ejpam-6802	401	7	technology	technology	NOUN
ejpam-6802	401	8	,	,	PUNCT
ejpam-6802	401	9	63	63	NUM
ejpam-6802	401	10	,	,	PUNCT
ejpam-6802	401	11	2018	2018	NUM
ejpam-6802	401	12	.	.	PUNCT
ejpam-6802	402	1	[	[	X
ejpam-6802	402	2	14	14	NUM
ejpam-6802	402	3	]	]	X
ejpam-6802	402	4	g.	g.	PROPN
ejpam-6802	402	5	mani	mani	PROPN
ejpam-6802	402	6	,	,	PUNCT
ejpam-6802	402	7	s.	s.	PROPN
ejpam-6802	402	8	s.	s.	PROPN
ejpam-6802	402	9	ramulu	ramulu	PROPN
ejpam-6802	402	10	,	,	PUNCT
ejpam-6802	402	11	s.	s.	PROPN
ejpam-6802	402	12	aljohani	aljohani	PROPN
ejpam-6802	402	13	,	,	PUNCT
ejpam-6802	402	14	z.	z.	PROPN
ejpam-6802	402	15	d.	d.	PROPN
ejpam-6802	402	16	mitrovic	mitrovic	PROPN
ejpam-6802	402	17	,	,	PUNCT
ejpam-6802	402	18	and	and	CCONJ
ejpam-6802	402	19	n.	n.	PROPN
ejpam-6802	402	20	mlaiki	mlaiki	PROPN
ejpam-6802	402	21	.	.	PUNCT
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ejpam-6802	402	23	on	on	ADP
ejpam-6802	402	24	fixed	fix	VERB
ejpam-6802	402	25	points	point	NOUN
ejpam-6802	402	26	and	and	CCONJ
ejpam-6802	402	27	common	common	ADJ
ejpam-6802	402	28	fixed	fix	VERB
ejpam-6802	402	29	points	point	NOUN
ejpam-6802	402	30	on	on	ADP
ejpam-6802	402	31	bipolar	bipolar	ADJ
ejpam-6802	402	32	b	b	NOUN
ejpam-6802	402	33	-	-	PUNCT
ejpam-6802	402	34	metric	metric	ADJ
ejpam-6802	402	35	space	space	NOUN
ejpam-6802	402	36	with	with	ADP
ejpam-6802	402	37	applications	application	NOUN
ejpam-6802	402	38	.	.	PUNCT
ejpam-6802	403	1	journal	journal	NOUN
ejpam-6802	403	2	of	of	ADP
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ejpam-6802	403	4	and	and	CCONJ
ejpam-6802	403	5	computer	computer	NOUN
ejpam-6802	403	6	science	science	NOUN
ejpam-6802	403	7	,	,	PUNCT
ejpam-6802	403	8	37(3):274–286	37(3):274–286	PROPN
ejpam-6802	403	9	,	,	PUNCT
ejpam-6802	403	10	2025	2025	NUM
ejpam-6802	403	11	.	.	PUNCT
ejpam-6802	404	1	[	[	X
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ejpam-6802	404	3	]	]	X
ejpam-6802	404	4	g.	g.	PROPN
ejpam-6802	404	5	n.	n.	PROPN
ejpam-6802	404	6	v.	v.	PROPN
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ejpam-6802	404	8	,	,	PUNCT
ejpam-6802	404	9	k.	k.	PROPN
ejpam-6802	405	1	p.	p.	PROPN
ejpam-6802	405	2	r.	r.	PROPN
ejpam-6802	405	3	rao	rao	PROPN
ejpam-6802	405	4	,	,	PUNCT
ejpam-6802	405	5	b.	b.	PROPN
ejpam-6802	405	6	s.	s.	PROPN
ejpam-6802	405	7	rao	rao	PROPN
ejpam-6802	405	8	,	,	PUNCT
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ejpam-6802	405	10	a.	a.	PROPN
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ejpam-6802	405	12	.	.	PUNCT
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ejpam-6802	406	2	mappings	mapping	NOUN
ejpam-6802	406	3	and	and	CCONJ
ejpam-6802	406	4	coupled	couple	VERB
ejpam-6802	406	5	fixed	fix	VERB
ejpam-6802	406	6	point	point	NOUN
ejpam-6802	406	7	results	result	NOUN
ejpam-6802	406	8	in	in	ADP
ejpam-6802	406	9	bipolar	bipolar	ADJ
ejpam-6802	406	10	metric	metric	ADJ
ejpam-6802	406	11	spaces	space	NOUN
ejpam-6802	406	12	.	.	PUNCT
ejpam-6802	407	1	international	international	ADJ
ejpam-6802	407	2	journal	journal	PROPN
ejpam-6802	407	3	of	of	ADP
ejpam-6802	407	4	nonlinear	nonlinear	ADJ
ejpam-6802	407	5	analysis	analysis	NOUN
ejpam-6802	407	6	and	and	CCONJ
ejpam-6802	407	7	applications	application	NOUN
ejpam-6802	407	8	,	,	PUNCT
ejpam-6802	407	9	12(1):1–15	12(1):1–15	NUM
ejpam-6802	407	10	,	,	PUNCT
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ejpam-6802	407	12	.	.	PUNCT
ejpam-6802	408	1	[	[	X
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ejpam-6802	408	3	]	]	X
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ejpam-6802	408	8	,	,	PUNCT
ejpam-6802	408	9	r.	r.	PROPN
ejpam-6802	408	10	p.	p.	PROPN
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ejpam-6802	408	13	b.	b.	PROPN
ejpam-6802	408	14	s.	s.	PROPN
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ejpam-6802	408	20	n.	n.	PROPN
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ejpam-6802	409	2	type	type	NOUN
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ejpam-6802	409	4	contraction	contraction	NOUN
ejpam-6802	409	5	and	and	CCONJ
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ejpam-6802	409	8	point	point	NOUN
ejpam-6802	409	9	theorems	theorem	NOUN
ejpam-6802	409	10	in	in	ADP
ejpam-6802	409	11	bipolar	bipolar	ADJ
ejpam-6802	409	12	metric	metric	ADJ
ejpam-6802	409	13	spaces	space	NOUN
ejpam-6802	409	14	with	with	ADP
ejpam-6802	409	15	applications	application	NOUN
ejpam-6802	409	16	.	.	PUNCT
ejpam-6802	410	1	fixed	fix	VERB
ejpam-6802	410	2	point	point	NOUN
ejpam-6802	410	3	theory	theory	NOUN
ejpam-6802	410	4	and	and	CCONJ
ejpam-6802	410	5	applications	application	NOUN
ejpam-6802	410	6	,	,	PUNCT
ejpam-6802	410	7	2018(21	2018(21	NUM
ejpam-6802	410	8	)	)	PUNCT
ejpam-6802	410	9	,	,	PUNCT
ejpam-6802	410	10	2018	2018	NUM
ejpam-6802	410	11	.	.	PUNCT
ejpam-6802	411	1	[	[	X
ejpam-6802	411	2	17	17	NUM
ejpam-6802	411	3	]	]	X
ejpam-6802	411	4	g.	g.	PROPN
ejpam-6802	411	5	n.	n.	PROPN
ejpam-6802	411	6	v.	v.	PROPN
ejpam-6802	411	7	kishore	kishore	PROPN
ejpam-6802	411	8	,	,	PUNCT
ejpam-6802	411	9	d.	d.	PROPN
ejpam-6802	411	10	r.	r.	PROPN
ejpam-6802	411	11	prasad	prasad	PROPN
ejpam-6802	411	12	,	,	PUNCT
ejpam-6802	411	13	b.	b.	PROPN
ejpam-6802	411	14	s.	s.	PROPN
ejpam-6802	411	15	rao	rao	PROPN
ejpam-6802	411	16	,	,	PUNCT
ejpam-6802	411	17	and	and	CCONJ
ejpam-6802	411	18	v.	v.	PROPN
ejpam-6802	411	19	s.	s.	PROPN
ejpam-6802	411	20	baghavan	baghavan	PROPN
ejpam-6802	411	21	.	.	PUNCT
ejpam-6802	412	1	some	some	DET
ejpam-6802	412	2	applications	application	NOUN
ejpam-6802	412	3	via	via	ADP
ejpam-6802	412	4	common	common	ADJ
ejpam-6802	412	5	coupled	couple	VERB
ejpam-6802	412	6	fixed	fix	VERB
ejpam-6802	412	7	point	point	NOUN
ejpam-6802	412	8	theorems	theorem	NOUN
ejpam-6802	412	9	in	in	ADP
ejpam-6802	412	10	bipolar	bipolar	ADJ
ejpam-6802	412	11	metric	metric	ADJ
ejpam-6802	412	12	spaces	space	NOUN
ejpam-6802	412	13	.	.	PUNCT
ejpam-6802	413	1	journal	journal	NOUN
ejpam-6802	413	2	of	of	ADP
ejpam-6802	413	3	critical	critical	ADJ
ejpam-6802	413	4	reviews	review	NOUN
ejpam-6802	413	5	,	,	PUNCT
ejpam-6802	413	6	7(2):601–607	7(2):601–607	NUM
ejpam-6802	413	7	,	,	PUNCT
ejpam-6802	413	8	2019	2019	NUM
ejpam-6802	413	9	.	.	PUNCT
ejpam-6802	414	1	[	[	X
ejpam-6802	414	2	18	18	NUM
ejpam-6802	414	3	]	]	X
ejpam-6802	414	4	g.	g.	PROPN
ejpam-6802	414	5	n.	n.	PROPN
ejpam-6802	414	6	v.	v.	PROPN
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ejpam-6802	414	8	,	,	PUNCT
ejpam-6802	414	9	k.	k.	PROPN
ejpam-6802	415	1	p.	p.	PROPN
ejpam-6802	415	2	r.	r.	PROPN
ejpam-6802	415	3	rao	rao	PROPN
ejpam-6802	415	4	,	,	PUNCT
ejpam-6802	415	5	h.	h.	PROPN
ejpam-6802	415	6	isik	isik	PROPN
ejpam-6802	415	7	,	,	PUNCT
ejpam-6802	415	8	b.	b.	PROPN
ejpam-6802	415	9	s.	s.	PROPN
ejpam-6802	415	10	rao	rao	PROPN
ejpam-6802	415	11	,	,	PUNCT
ejpam-6802	415	12	and	and	CCONJ
ejpam-6802	415	13	a.	a.	PROPN
ejpam-6802	415	14	sombabu	sombabu	PROPN
ejpam-6802	415	15	.	.	PUNCT
ejpam-6802	416	1	covariant	covariant	ADJ
ejpam-6802	416	2	mappings	mapping	NOUN
ejpam-6802	416	3	and	and	CCONJ
ejpam-6802	416	4	coupled	couple	VERB
ejpam-6802	416	5	fixed	fix	VERB
ejpam-6802	416	6	point	point	NOUN
ejpam-6802	416	7	results	result	NOUN
ejpam-6802	416	8	in	in	ADP
ejpam-6802	416	9	bipolar	bipolar	ADJ
ejpam-6802	416	10	metric	metric	ADJ
ejpam-6802	416	11	spaces	space	NOUN
ejpam-6802	416	12	.	.	PUNCT
ejpam-6802	417	1	international	international	ADJ
ejpam-6802	417	2	journal	journal	PROPN
ejpam-6802	417	3	of	of	ADP
ejpam-6802	417	4	nonlinear	nonlinear	ADJ
ejpam-6802	417	5	analysis	analysis	NOUN
ejpam-6802	417	6	and	and	CCONJ
ejpam-6802	417	7	applications	application	NOUN
ejpam-6802	417	8	,	,	PUNCT
ejpam-6802	417	9	12(1):1–15	12(1):1–15	NUM
ejpam-6802	417	10	,	,	PUNCT
ejpam-6802	417	11	2021	2021	NUM
ejpam-6802	417	12	.	.	PUNCT
ejpam-6802	418	1	[	[	X
ejpam-6802	418	2	19	19	NUM
ejpam-6802	418	3	]	]	X
ejpam-6802	418	4	u.	u.	NOUN
ejpam-6802	418	5	gürdal	gürdal	PROPN
ejpam-6802	418	6	,	,	PUNCT
ejpam-6802	418	7	a.	a.	NOUN
ejpam-6802	418	8	mutlu	mutlu	PROPN
ejpam-6802	418	9	,	,	PUNCT
ejpam-6802	418	10	and	and	CCONJ
ejpam-6802	418	11	a.	a.	NOUN
ejpam-6802	418	12	özkan	özkan	PROPN
ejpam-6802	418	13	.	.	PUNCT
ejpam-6802	418	14	fixed	fix	VERB
ejpam-6802	418	15	point	point	NOUN
ejpam-6802	418	16	results	result	NOUN
ejpam-6802	418	17	for	for	SCONJ
ejpam-6802	418	18	β−ϕ	β−ϕ	NOUN
ejpam-6802	418	19	contractive	contractive	ADJ
ejpam-6802	418	20	mappings	mapping	NOUN
ejpam-6802	418	21	in	in	ADP
ejpam-6802	418	22	bipolar	bipolar	ADJ
ejpam-6802	418	23	metric	metric	ADJ
ejpam-6802	418	24	spaces	space	NOUN
ejpam-6802	418	25	.	.	PUNCT
ejpam-6802	419	1	journal	journal	NOUN
ejpam-6802	419	2	of	of	ADP
ejpam-6802	419	3	inequalities	inequality	NOUN
ejpam-6802	419	4	and	and	CCONJ
ejpam-6802	419	5	special	special	ADJ
ejpam-6802	419	6	functions	function	NOUN
ejpam-6802	419	7	,	,	PUNCT
ejpam-6802	419	8	11(1	11(1	NUM
ejpam-6802	419	9	)	)	PUNCT
ejpam-6802	419	10	,	,	PUNCT
ejpam-6802	419	11	2020	2020	NUM
ejpam-6802	419	12	.	.	PUNCT
ejpam-6802	420	1	[	[	X
ejpam-6802	420	2	20	20	NUM
ejpam-6802	420	3	]	]	X
ejpam-6802	420	4	y.	y.	PROPN
ejpam-6802	420	5	u.	u.	PROPN
ejpam-6802	420	6	gaba	gaba	PROPN
ejpam-6802	420	7	,	,	PUNCT
ejpam-6802	420	8	m.	m.	NOUN
ejpam-6802	420	9	aphane	aphane	PROPN
ejpam-6802	420	10	,	,	PUNCT
ejpam-6802	420	11	and	and	CCONJ
ejpam-6802	420	12	h.	h.	PROPN
ejpam-6802	420	13	aydi	aydi	VERB
ejpam-6802	420	14	.	.	PUNCT
ejpam-6802	421	1	contractions	contraction	NOUN
ejpam-6802	421	2	in	in	ADP
ejpam-6802	421	3	bipolar	bipolar	ADJ
ejpam-6802	421	4	metric	metric	ADJ
ejpam-6802	421	5	spaces	space	NOUN
ejpam-6802	421	6	.	.	PUNCT
ejpam-6802	422	1	journal	journal	NOUN
ejpam-6802	422	2	of	of	ADP
ejpam-6802	422	3	mathematics	mathematic	NOUN
ejpam-6802	422	4	,	,	PUNCT
ejpam-6802	422	5	2021(5562651	2021(5562651	NUM
ejpam-6802	422	6	)	)	PUNCT
ejpam-6802	422	7	,	,	PUNCT
ejpam-6802	422	8	2021	2021	NUM
ejpam-6802	422	9	.	.	PUNCT
ejpam-6802	423	1	r.	r.	PROPN
ejpam-6802	423	2	ramaswamy	ramaswamy	PROPN
ejpam-6802	423	3	/	/	SYM
ejpam-6802	423	4	eur	eur	PROPN
ejpam-6802	423	5	.	.	PUNCT
ejpam-6802	424	1	j.	j.	PROPN
ejpam-6802	424	2	pure	pure	PROPN
ejpam-6802	424	3	appl	appl	PROPN
ejpam-6802	424	4	.	.	PROPN
ejpam-6802	424	5	math	math	PROPN
ejpam-6802	424	6	,	,	PUNCT
ejpam-6802	424	7	18	18	NUM
ejpam-6802	424	8	(	(	PUNCT
ejpam-6802	424	9	4	4	NUM
ejpam-6802	424	10	)	)	PUNCT
ejpam-6802	424	11	(	(	PUNCT
ejpam-6802	424	12	2025	2025	NUM
ejpam-6802	424	13	)	)	PUNCT
ejpam-6802	424	14	,	,	PUNCT
ejpam-6802	424	15	6802	6802	NUM
ejpam-6802	424	16	20	20	NUM
ejpam-6802	424	17	of	of	ADP
ejpam-6802	424	18	20	20	NUM
ejpam-6802	425	1	[	[	SYM
ejpam-6802	425	2	21	21	NUM
ejpam-6802	425	3	]	]	X
ejpam-6802	425	4	c.	c.	PROPN
ejpam-6802	425	5	b.	b.	PROPN
ejpam-6802	425	6	ampadu	ampadu	PROPN
ejpam-6802	425	7	.	.	PUNCT
ejpam-6802	426	1	a	a	DET
ejpam-6802	426	2	coupled	couple	VERB
ejpam-6802	426	3	version	version	NOUN
ejpam-6802	426	4	of	of	ADP
ejpam-6802	426	5	the	the	DET
ejpam-6802	426	6	higher	high	ADJ
ejpam-6802	426	7	-	-	PUNCT
ejpam-6802	426	8	order	order	NOUN
ejpam-6802	426	9	banach	banach	NOUN
ejpam-6802	426	10	contraction	contraction	NOUN
ejpam-6802	426	11	principle	principle	NOUN
ejpam-6802	426	12	in	in	ADP
ejpam-6802	426	13	multiplicative	multiplicative	ADJ
ejpam-6802	426	14	cone	cone	NOUN
ejpam-6802	426	15	metric	metric	ADJ
ejpam-6802	426	16	space	space	NOUN
ejpam-6802	426	17	.	.	PUNCT
ejpam-6802	427	1	jp	jp	PROPN
ejpam-6802	427	2	journal	journal	PROPN
ejpam-6802	427	3	of	of	ADP
ejpam-6802	427	4	applied	apply	VERB
ejpam-6802	427	5	mathematics	mathematic	NOUN
ejpam-6802	427	6	,	,	PUNCT
ejpam-6802	427	7	15:63–88	15:63–88	NUM
ejpam-6802	427	8	,	,	PUNCT
ejpam-6802	427	9	2017	2017	NUM
ejpam-6802	427	10	.	.	PUNCT
ejpam-6802	428	1	[	[	X
ejpam-6802	428	2	22	22	NUM
ejpam-6802	428	3	]	]	PUNCT
ejpam-6802	428	4	a.	a.	NOUN
ejpam-6802	428	5	j.	j.	PROPN
ejpam-6802	428	6	gnanaprakasam	gnanaprakasam	PROPN
ejpam-6802	428	7	,	,	PUNCT
ejpam-6802	428	8	b.	b.	PROPN
ejpam-6802	428	9	ramalingam	ramalingam	PROPN
ejpam-6802	428	10	,	,	PUNCT
ejpam-6802	428	11	g.	g.	PROPN
ejpam-6802	428	12	mani	mani	PROPN
ejpam-6802	428	13	,	,	PUNCT
ejpam-6802	428	14	o.	o.	PROPN
ejpam-6802	428	15	ege	ege	PROPN
ejpam-6802	428	16	,	,	PUNCT
ejpam-6802	428	17	and	and	CCONJ
ejpam-6802	428	18	r.	r.	PROPN
ejpam-6802	428	19	george	george	PROPN
ejpam-6802	428	20	.	.	PUNCT
ejpam-6802	429	1	a	a	DET
ejpam-6802	429	2	numerical	numerical	ADJ
ejpam-6802	429	3	scheme	scheme	NOUN
ejpam-6802	429	4	and	and	CCONJ
ejpam-6802	429	5	application	application	NOUN
ejpam-6802	429	6	to	to	ADP
ejpam-6802	429	7	the	the	DET
ejpam-6802	429	8	fractional	fractional	ADJ
ejpam-6802	429	9	integro	integro	ADJ
ejpam-6802	429	10	-	-	PUNCT
ejpam-6802	429	11	differential	differential	NOUN
ejpam-6802	429	12	equation	equation	NOUN
ejpam-6802	429	13	using	use	VERB
ejpam-6802	429	14	fixed	fix	VERB
ejpam-6802	429	15	-	-	PUNCT
ejpam-6802	429	16	point	point	NOUN
ejpam-6802	429	17	techniques	technique	NOUN
ejpam-6802	429	18	.	.	PUNCT
ejpam-6802	430	1	fractal	fractal	ADJ
ejpam-6802	430	2	and	and	CCONJ
ejpam-6802	430	3	fractional	fractional	ADJ
ejpam-6802	430	4	,	,	PUNCT
ejpam-6802	430	5	8(34	8(34	NUM
ejpam-6802	430	6	)	)	PUNCT
ejpam-6802	430	7	,	,	PUNCT
ejpam-6802	430	8	2024	2024	NUM
ejpam-6802	430	9	.	.	PUNCT
ejpam-6802	431	1	[	[	X
ejpam-6802	431	2	23	23	NUM
ejpam-6802	431	3	]	]	X
ejpam-6802	431	4	g.	g.	PROPN
ejpam-6802	431	5	janardhanan	janardhanan	PROPN
ejpam-6802	431	6	,	,	PUNCT
ejpam-6802	431	7	g.	g.	PROPN
ejpam-6802	431	8	mani	mani	PROPN
ejpam-6802	431	9	,	,	PUNCT
ejpam-6802	431	10	d.	d.	PROPN
ejpam-6802	431	11	santina	santina	PROPN
ejpam-6802	431	12	,	,	PUNCT
ejpam-6802	431	13	and	and	CCONJ
ejpam-6802	431	14	n.	n.	PROPN
ejpam-6802	431	15	mlaiki	mlaiki	PROPN
ejpam-6802	431	16	.	.	PUNCT
ejpam-6802	432	1	existence	existence	NOUN
ejpam-6802	432	2	and	and	CCONJ
ejpam-6802	432	3	uniqueness	uniqueness	ADJ
ejpam-6802	432	4	theorems	theorem	NOUN
ejpam-6802	432	5	for	for	ADP
ejpam-6802	432	6	nonlinear	nonlinear	ADJ
ejpam-6802	432	7	coupled	couple	VERB
ejpam-6802	432	8	boundary	boundary	ADJ
ejpam-6802	432	9	value	value	NOUN
ejpam-6802	432	10	problem	problem	NOUN
ejpam-6802	432	11	of	of	ADP
ejpam-6802	432	12	the	the	DET
ejpam-6802	432	13	abc	abc	PROPN
ejpam-6802	432	14	fractional	fractional	PROPN
ejpam-6802	432	15	differential	differential	NOUN
ejpam-6802	432	16	equation	equation	NOUN
ejpam-6802	432	17	.	.	PUNCT
ejpam-6802	433	1	journal	journal	NOUN
ejpam-6802	433	2	of	of	ADP
ejpam-6802	433	3	mathematics	mathematics	PROPN
ejpam-6802	433	4	and	and	CCONJ
ejpam-6802	433	5	computer	computer	NOUN
ejpam-6802	433	6	science	science	NOUN
ejpam-6802	433	7	,	,	PUNCT
ejpam-6802	433	8	37(3):297–318	37(3):297–318	ADV
ejpam-6802	433	9	,	,	PUNCT
ejpam-6802	433	10	2025	2025	NUM
ejpam-6802	433	11	.	.	PUNCT
ejpam-6802	434	1	[	[	X
ejpam-6802	434	2	24	24	NUM
ejpam-6802	434	3	]	]	X
ejpam-6802	434	4	g.	g.	PROPN
ejpam-6802	434	5	janardhanan	janardhanan	PROPN
ejpam-6802	434	6	,	,	PUNCT
ejpam-6802	434	7	g.	g.	PROPN
ejpam-6802	434	8	mani	mani	PROPN
ejpam-6802	434	9	,	,	PUNCT
ejpam-6802	434	10	z.	z.	PROPN
ejpam-6802	434	11	d.	d.	PROPN
ejpam-6802	434	12	mitrovic	mitrovic	PROPN
ejpam-6802	434	13	,	,	PUNCT
ejpam-6802	434	14	a.	a.	NOUN
ejpam-6802	434	15	aloqaily	aloqaily	ADV
ejpam-6802	434	16	,	,	PUNCT
ejpam-6802	434	17	and	and	CCONJ
ejpam-6802	434	18	n.	n.	PROPN
ejpam-6802	434	19	mlaiki	mlaiki	PROPN
ejpam-6802	434	20	.	.	PUNCT
ejpam-6802	435	1	best	good	ADJ
ejpam-6802	435	2	proximity	proximity	NOUN
ejpam-6802	435	3	point	point	NOUN
ejpam-6802	435	4	results	result	NOUN
ejpam-6802	435	5	on	on	ADP
ejpam-6802	435	6	r	r	NOUN
ejpam-6802	435	7	-	-	ADJ
ejpam-6802	435	8	metric	metric	ADJ
ejpam-6802	435	9	spaces	space	NOUN
ejpam-6802	435	10	with	with	ADP
ejpam-6802	435	11	applications	application	NOUN
ejpam-6802	435	12	to	to	ADP
ejpam-6802	435	13	fractional	fractional	ADJ
ejpam-6802	435	14	differential	differential	ADJ
ejpam-6802	435	15	equation	equation	NOUN
ejpam-6802	435	16	and	and	CCONJ
ejpam-6802	435	17	production	production	NOUN
ejpam-6802	435	18	-	-	PUNCT
ejpam-6802	435	19	consumption	consumption	NOUN
ejpam-6802	435	20	equilibrium	equilibrium	NOUN
ejpam-6802	435	21	.	.	PUNCT
ejpam-6802	436	1	journal	journal	NOUN
ejpam-6802	436	2	of	of	ADP
ejpam-6802	436	3	mathematics	mathematic	NOUN
ejpam-6802	436	4	and	and	CCONJ
ejpam-6802	436	5	computer	computer	NOUN
ejpam-6802	436	6	science	science	NOUN
ejpam-6802	436	7	,	,	PUNCT
ejpam-6802	436	8	38(1):45–55	38(1):45–55	NUM
ejpam-6802	436	9	,	,	PUNCT
ejpam-6802	436	10	2025	2025	NUM
ejpam-6802	436	11	.	.	PUNCT
ejpam-6802	437	1	[	[	X
ejpam-6802	437	2	25	25	NUM
ejpam-6802	437	3	]	]	PUNCT
ejpam-6802	437	4	m.	m.	NOUN
ejpam-6802	437	5	dhanraj	dhanraj	ADJ
ejpam-6802	437	6	,	,	PUNCT
ejpam-6802	437	7	a.	a.	PROPN
ejpam-6802	437	8	j.	j.	PROPN
ejpam-6802	437	9	gnanaprakasam	gnanaprakasam	PROPN
ejpam-6802	437	10	,	,	PUNCT
ejpam-6802	437	11	g.	g.	PROPN
ejpam-6802	437	12	mani	mani	PROPN
ejpam-6802	437	13	,	,	PUNCT
ejpam-6802	437	14	o.	o.	PROPN
ejpam-6802	437	15	ege	ege	PROPN
ejpam-6802	437	16	,	,	PUNCT
ejpam-6802	437	17	and	and	CCONJ
ejpam-6802	437	18	m.	m.	PROPN
ejpam-6802	437	19	de	de	PROPN
ejpam-6802	437	20	la	la	PROPN
ejpam-6802	437	21	sen	sen	PROPN
ejpam-6802	437	22	.	.	PROPN
ejpam-6802	437	23	solution	solution	NOUN
ejpam-6802	437	24	to	to	ADP
ejpam-6802	437	25	integral	integral	ADJ
ejpam-6802	437	26	equation	equation	NOUN
ejpam-6802	437	27	in	in	ADP
ejpam-6802	437	28	an	an	DET
ejpam-6802	437	29	o	o	NOUN
ejpam-6802	437	30	-	-	NOUN
ejpam-6802	437	31	complete	complete	ADJ
ejpam-6802	437	32	branciari	branciari	ADJ
ejpam-6802	437	33	b	b	X
ejpam-6802	437	34	-	-	PUNCT
ejpam-6802	437	35	metric	metric	ADJ
ejpam-6802	437	36	spaces	space	NOUN
ejpam-6802	437	37	.	.	PUNCT
ejpam-6802	438	1	axioms	axiom	NOUN
ejpam-6802	438	2	,	,	PUNCT
ejpam-6802	438	3	11(728	11(728	NOUN
ejpam-6802	438	4	)	)	PUNCT
ejpam-6802	438	5	,	,	PUNCT
ejpam-6802	438	6	2022	2022	NUM
ejpam-6802	438	7	.	.	PUNCT
ejpam-6802	439	1	[	[	X
ejpam-6802	439	2	26	26	NUM
ejpam-6802	439	3	]	]	X
ejpam-6802	439	4	r.	r.	PROPN
ejpam-6802	439	5	rajagopalan	rajagopalan	PROPN
ejpam-6802	439	6	,	,	PUNCT
ejpam-6802	439	7	m.	m.	NOUN
ejpam-6802	439	8	gunaseelan	gunaseelan	PROPN
ejpam-6802	439	9	,	,	PUNCT
ejpam-6802	439	10	d.	d.	PROPN
ejpam-6802	439	11	kumar	kumar	PROPN
ejpam-6802	439	12	,	,	PUNCT
ejpam-6802	439	13	and	and	CCONJ
ejpam-6802	439	14	o.	o.	PROPN
ejpam-6802	439	15	ege	ege	PROPN
ejpam-6802	439	16	.	.	PUNCT
ejpam-6802	440	1	mathematical	mathematical	ADJ
ejpam-6802	440	2	model	model	NOUN
ejpam-6802	440	3	of	of	ADP
ejpam-6802	440	4	the	the	DET
ejpam-6802	440	5	monkeypox	monkeypox	PROPN
ejpam-6802	440	6	virus	virus	NOUN
ejpam-6802	440	7	disease	disease	NOUN
ejpam-6802	440	8	via	via	ADP
ejpam-6802	440	9	abc	abc	PROPN
ejpam-6802	440	10	fractional	fractional	PROPN
ejpam-6802	440	11	order	order	NOUN
ejpam-6802	440	12	derivative	derivative	NOUN
ejpam-6802	440	13	.	.	PUNCT
ejpam-6802	441	1	computer	computer	NOUN
ejpam-6802	441	2	modeling	modeling	NOUN
ejpam-6802	441	3	in	in	ADP
ejpam-6802	441	4	engineering	engineering	NOUN
ejpam-6802	441	5	&	&	CCONJ
ejpam-6802	441	6	sciences	sciences	PROPN
ejpam-6802	441	7	,	,	PUNCT
ejpam-6802	441	8	143(2):1843–1894	143(2):1843–1894	NUM
ejpam-6802	441	9	,	,	PUNCT
ejpam-6802	441	10	2025	2025	NUM
ejpam-6802	441	11	.	.	PUNCT