id	sid	tid	token	lemma	pos
ejpam-6106	1	1	european	european	PROPN
ejpam-6106	1	2	journal	journal	PROPN
ejpam-6106	1	3	of	of	ADP
ejpam-6106	1	4	pure	pure	ADJ
ejpam-6106	1	5	and	and	CCONJ
ejpam-6106	1	6	applied	applied	ADJ
ejpam-6106	1	7	mathematics	mathematic	NOUN
ejpam-6106	1	8	2025	2025	NUM
ejpam-6106	1	9	,	,	PUNCT
ejpam-6106	1	10	vol	vol	NOUN
ejpam-6106	1	11	.	.	PROPN
ejpam-6106	1	12	18	18	NUM
ejpam-6106	1	13	,	,	PUNCT
ejpam-6106	1	14	issue	issue	NOUN
ejpam-6106	1	15	4	4	NUM
ejpam-6106	1	16	,	,	PUNCT
ejpam-6106	1	17	article	article	NOUN
ejpam-6106	1	18	number	number	NOUN
ejpam-6106	1	19	6106	6106	NUM
ejpam-6106	1	20	issn	issn	VERB
ejpam-6106	1	21	1307	1307	NUM
ejpam-6106	1	22	-	-	SYM
ejpam-6106	1	23	5543	5543	NUM
ejpam-6106	1	24	–	–	PUNCT
ejpam-6106	1	25	ejpam.com	ejpam.com	X
ejpam-6106	1	26	published	publish	VERB
ejpam-6106	1	27	by	by	ADP
ejpam-6106	1	28	new	new	PROPN
ejpam-6106	1	29	york	york	PROPN
ejpam-6106	1	30	business	business	PROPN
ejpam-6106	1	31	global	global	ADJ
ejpam-6106	1	32	fixed	fix	VERB
ejpam-6106	1	33	points	point	NOUN
ejpam-6106	1	34	of	of	ADP
ejpam-6106	1	35	mappings	mapping	NOUN
ejpam-6106	1	36	contracting	contract	VERB
ejpam-6106	1	37	perimeters	perimeter	NOUN
ejpam-6106	1	38	of	of	ADP
ejpam-6106	1	39	polygons	polygon	NOUN
ejpam-6106	1	40	:	:	PUNCT
ejpam-6106	1	41	a	a	DET
ejpam-6106	1	42	geometric	geometric	ADJ
ejpam-6106	1	43	generalization	generalization	NOUN
ejpam-6106	1	44	of	of	ADP
ejpam-6106	1	45	banach	banach	NOUN
ejpam-6106	1	46	’s	’s	PART
ejpam-6106	1	47	principle	principle	NOUN
ejpam-6106	1	48	muhammad	muhammad	PROPN
ejpam-6106	1	49	nazam1,∗	nazam1,∗	PROPN
ejpam-6106	1	50	,	,	PUNCT
ejpam-6106	1	51	umme	umme	ADJ
ejpam-6106	1	52	habiba1	habiba1	NOUN
ejpam-6106	1	53	,	,	PUNCT
ejpam-6106	1	54	manuel	manuel	PROPN
ejpam-6106	1	55	de	de	X
ejpam-6106	1	56	la	la	PROPN
ejpam-6106	1	57	sen2,∗	sen2,∗	PROPN
ejpam-6106	1	58	1	1	NUM
ejpam-6106	1	59	department	department	NOUN
ejpam-6106	1	60	of	of	ADP
ejpam-6106	1	61	mathematics	mathematic	NOUN
ejpam-6106	1	62	,	,	PUNCT
ejpam-6106	1	63	allama	allama	PROPN
ejpam-6106	1	64	iqbal	iqbal	PROPN
ejpam-6106	1	65	open	open	PROPN
ejpam-6106	1	66	university	university	PROPN
ejpam-6106	1	67	,	,	PUNCT
ejpam-6106	1	68	h-8	h-8	NOUN
ejpam-6106	1	69	,	,	PUNCT
ejpam-6106	1	70	islamabad	islamabad	PROPN
ejpam-6106	1	71	,	,	PUNCT
ejpam-6106	1	72	pakistan	pakistan	PROPN
ejpam-6106	1	73	2	2	NUM
ejpam-6106	1	74	department	department	NOUN
ejpam-6106	1	75	of	of	ADP
ejpam-6106	1	76	electricity	electricity	NOUN
ejpam-6106	1	77	and	and	CCONJ
ejpam-6106	1	78	electronics	electronic	NOUN
ejpam-6106	1	79	,	,	PUNCT
ejpam-6106	1	80	faculty	faculty	NOUN
ejpam-6106	1	81	of	of	ADP
ejpam-6106	1	82	science	science	NOUN
ejpam-6106	1	83	and	and	CCONJ
ejpam-6106	1	84	technology	technology	NOUN
ejpam-6106	1	85	,	,	PUNCT
ejpam-6106	1	86	university	university	NOUN
ejpam-6106	1	87	of	of	ADP
ejpam-6106	1	88	the	the	DET
ejpam-6106	1	89	basque	basque	ADJ
ejpam-6106	1	90	country	country	NOUN
ejpam-6106	1	91	,	,	PUNCT
ejpam-6106	1	92	campus	campus	NOUN
ejpam-6106	1	93	of	of	ADP
ejpam-6106	1	94	leioa	leioa	ADJ
ejpam-6106	1	95	,	,	PUNCT
ejpam-6106	1	96	leioa	leioa	PROPN
ejpam-6106	1	97	(	(	PUNCT
ejpam-6106	1	98	bizkaia	bizkaia	PROPN
ejpam-6106	1	99	)	)	PUNCT
ejpam-6106	1	100	,	,	PUNCT
ejpam-6106	1	101	48940	48940	NUM
ejpam-6106	1	102	,	,	PUNCT
ejpam-6106	1	103	spain	spain	PROPN
ejpam-6106	1	104	abstract	abstract	NOUN
ejpam-6106	1	105	.	.	PUNCT
ejpam-6106	2	1	in	in	ADP
ejpam-6106	2	2	this	this	DET
ejpam-6106	2	3	paper	paper	NOUN
ejpam-6106	2	4	,	,	PUNCT
ejpam-6106	2	5	we	we	PRON
ejpam-6106	2	6	investigate	investigate	VERB
ejpam-6106	2	7	fixed	fix	VERB
ejpam-6106	2	8	points	point	NOUN
ejpam-6106	2	9	of	of	ADP
ejpam-6106	2	10	mappings	mapping	NOUN
ejpam-6106	2	11	that	that	PRON
ejpam-6106	2	12	contract	contract	VERB
ejpam-6106	2	13	the	the	DET
ejpam-6106	2	14	perimeters	perimeter	NOUN
ejpam-6106	2	15	of	of	ADP
ejpam-6106	2	16	polygons	polygon	NOUN
ejpam-6106	2	17	.	.	PUNCT
ejpam-6106	3	1	our	our	PRON
ejpam-6106	3	2	study	study	NOUN
ejpam-6106	3	3	is	be	AUX
ejpam-6106	3	4	motivated	motivate	VERB
ejpam-6106	3	5	by	by	ADP
ejpam-6106	3	6	the	the	DET
ejpam-6106	3	7	idea	idea	NOUN
ejpam-6106	3	8	that	that	SCONJ
ejpam-6106	3	9	the	the	DET
ejpam-6106	3	10	perimeter	perimeter	NOUN
ejpam-6106	3	11	,	,	PUNCT
ejpam-6106	3	12	as	as	ADP
ejpam-6106	3	13	a	a	DET
ejpam-6106	3	14	global	global	ADJ
ejpam-6106	3	15	geometric	geometric	ADJ
ejpam-6106	3	16	measure	measure	NOUN
ejpam-6106	3	17	,	,	PUNCT
ejpam-6106	3	18	provides	provide	VERB
ejpam-6106	3	19	a	a	DET
ejpam-6106	3	20	more	more	ADV
ejpam-6106	3	21	natural	natural	ADJ
ejpam-6106	3	22	and	and	CCONJ
ejpam-6106	3	23	flexible	flexible	ADJ
ejpam-6106	3	24	framework	framework	NOUN
ejpam-6106	3	25	than	than	ADP
ejpam-6106	3	26	the	the	DET
ejpam-6106	3	27	individual	individual	ADJ
ejpam-6106	3	28	edge	edge	NOUN
ejpam-6106	3	29	lengths	length	NOUN
ejpam-6106	3	30	when	when	SCONJ
ejpam-6106	3	31	analyzing	analyze	VERB
ejpam-6106	3	32	contraction	contraction	NOUN
ejpam-6106	3	33	properties	property	NOUN
ejpam-6106	3	34	in	in	ADP
ejpam-6106	3	35	metric	metric	ADJ
ejpam-6106	3	36	spaces	space	NOUN
ejpam-6106	3	37	.	.	PUNCT
ejpam-6106	4	1	we	we	PRON
ejpam-6106	4	2	extend	extend	VERB
ejpam-6106	4	3	petrov	petrov	PROPN
ejpam-6106	4	4	’s	’s	PART
ejpam-6106	4	5	fixed	fix	VERB
ejpam-6106	4	6	point	point	NOUN
ejpam-6106	4	7	theorem	theorem	VERB
ejpam-6106	4	8	from	from	ADP
ejpam-6106	4	9	triangles	triangle	NOUN
ejpam-6106	4	10	to	to	ADP
ejpam-6106	4	11	polygons	polygon	NOUN
ejpam-6106	4	12	with	with	ADP
ejpam-6106	4	13	an	an	DET
ejpam-6106	4	14	arbitrary	arbitrary	ADJ
ejpam-6106	4	15	number	number	NOUN
ejpam-6106	4	16	of	of	ADP
ejpam-6106	4	17	vertices	vertex	NOUN
ejpam-6106	4	18	and	and	CCONJ
ejpam-6106	4	19	establish	establish	VERB
ejpam-6106	4	20	conditions	condition	NOUN
ejpam-6106	4	21	under	under	ADP
ejpam-6106	4	22	which	which	PRON
ejpam-6106	4	23	such	such	ADJ
ejpam-6106	4	24	mappings	mapping	NOUN
ejpam-6106	4	25	admit	admit	VERB
ejpam-6106	4	26	unique	unique	ADJ
ejpam-6106	4	27	fixed	fix	VERB
ejpam-6106	4	28	points	point	NOUN
ejpam-6106	4	29	.	.	PUNCT
ejpam-6106	5	1	the	the	DET
ejpam-6106	5	2	methodology	methodology	NOUN
ejpam-6106	5	3	relies	rely	VERB
ejpam-6106	5	4	on	on	ADP
ejpam-6106	5	5	generalizations	generalization	NOUN
ejpam-6106	5	6	of	of	ADP
ejpam-6106	5	7	contraction	contraction	NOUN
ejpam-6106	5	8	mappings	mapping	NOUN
ejpam-6106	5	9	and	and	CCONJ
ejpam-6106	5	10	properties	property	NOUN
ejpam-6106	5	11	of	of	ADP
ejpam-6106	5	12	metric	metric	ADJ
ejpam-6106	5	13	spaces	space	NOUN
ejpam-6106	5	14	.	.	PUNCT
ejpam-6106	6	1	our	our	PRON
ejpam-6106	6	2	main	main	ADJ
ejpam-6106	6	3	contributions	contribution	NOUN
ejpam-6106	6	4	include	include	VERB
ejpam-6106	6	5	a	a	DET
ejpam-6106	6	6	new	new	ADJ
ejpam-6106	6	7	perimeter	perimeter	NOUN
ejpam-6106	6	8	-	-	PUNCT
ejpam-6106	6	9	based	base	VERB
ejpam-6106	6	10	contraction	contraction	NOUN
ejpam-6106	6	11	principle	principle	NOUN
ejpam-6106	6	12	,	,	PUNCT
ejpam-6106	6	13	a	a	DET
ejpam-6106	6	14	demonstration	demonstration	NOUN
ejpam-6106	6	15	of	of	ADP
ejpam-6106	6	16	its	its	PRON
ejpam-6106	6	17	application	application	NOUN
ejpam-6106	6	18	in	in	ADP
ejpam-6106	6	19	proving	prove	VERB
ejpam-6106	6	20	banach	banach	NOUN
ejpam-6106	6	21	’s	’s	PART
ejpam-6106	6	22	contraction	contraction	NOUN
ejpam-6106	6	23	theorem	theorem	VERB
ejpam-6106	6	24	,	,	PUNCT
ejpam-6106	6	25	and	and	CCONJ
ejpam-6106	6	26	examples	example	NOUN
ejpam-6106	6	27	that	that	PRON
ejpam-6106	6	28	validate	validate	VERB
ejpam-6106	6	29	the	the	DET
ejpam-6106	6	30	theoretical	theoretical	ADJ
ejpam-6106	6	31	results	result	NOUN
ejpam-6106	6	32	.	.	PUNCT
ejpam-6106	7	1	this	this	DET
ejpam-6106	7	2	generalization	generalization	NOUN
ejpam-6106	7	3	enriches	enrich	VERB
ejpam-6106	7	4	the	the	DET
ejpam-6106	7	5	existing	exist	VERB
ejpam-6106	7	6	literature	literature	NOUN
ejpam-6106	7	7	on	on	ADP
ejpam-6106	7	8	fixed	fix	VERB
ejpam-6106	7	9	point	point	NOUN
ejpam-6106	7	10	theory	theory	NOUN
ejpam-6106	7	11	and	and	CCONJ
ejpam-6106	7	12	opens	open	VERB
ejpam-6106	7	13	avenues	avenue	NOUN
ejpam-6106	7	14	for	for	ADP
ejpam-6106	7	15	further	further	ADJ
ejpam-6106	7	16	applications	application	NOUN
ejpam-6106	7	17	in	in	ADP
ejpam-6106	7	18	geometric	geometric	ADJ
ejpam-6106	7	19	analysis	analysis	NOUN
ejpam-6106	7	20	and	and	CCONJ
ejpam-6106	7	21	related	related	ADJ
ejpam-6106	7	22	areas	area	NOUN
ejpam-6106	7	23	.	.	PUNCT
ejpam-6106	8	1	2020	2020	NUM
ejpam-6106	8	2	mathematics	mathematic	NOUN
ejpam-6106	8	3	subject	subject	NOUN
ejpam-6106	8	4	classifications	classification	NOUN
ejpam-6106	8	5	:	:	PUNCT
ejpam-6106	8	6	47h09	47h09	NUM
ejpam-6106	8	7	,	,	PUNCT
ejpam-6106	8	8	47h10	47h10	NUM
ejpam-6106	8	9	,	,	PUNCT
ejpam-6106	8	10	30l15	30l15	NUM
ejpam-6106	8	11	key	key	ADJ
ejpam-6106	8	12	words	word	NOUN
ejpam-6106	8	13	and	and	CCONJ
ejpam-6106	8	14	phrases	phrase	NOUN
ejpam-6106	8	15	:	:	PUNCT
ejpam-6106	8	16	fixed	fix	VERB
ejpam-6106	8	17	point	point	NOUN
ejpam-6106	8	18	,	,	PUNCT
ejpam-6106	8	19	perimeter	perimeter	NOUN
ejpam-6106	8	20	of	of	ADP
ejpam-6106	8	21	polygons	polygon	NOUN
ejpam-6106	8	22	,	,	PUNCT
ejpam-6106	8	23	contractions	contraction	NOUN
ejpam-6106	8	24	,	,	PUNCT
ejpam-6106	8	25	complete	complete	ADJ
ejpam-6106	8	26	metric	metric	ADJ
ejpam-6106	8	27	space	space	NOUN
ejpam-6106	8	28	1	1	NUM
ejpam-6106	8	29	.	.	PUNCT
ejpam-6106	8	30	introduction	introduction	NOUN
ejpam-6106	8	31	the	the	DET
ejpam-6106	8	32	choice	choice	NOUN
ejpam-6106	8	33	of	of	ADP
ejpam-6106	8	34	considering	consider	VERB
ejpam-6106	8	35	the	the	DET
ejpam-6106	8	36	perimeter	perimeter	NOUN
ejpam-6106	8	37	instead	instead	ADV
ejpam-6106	8	38	of	of	ADP
ejpam-6106	8	39	the	the	DET
ejpam-6106	8	40	side	side	NOUN
ejpam-6106	8	41	lengths	length	NOUN
ejpam-6106	8	42	individually	individually	ADV
ejpam-6106	8	43	is	be	AUX
ejpam-6106	8	44	motivated	motivate	VERB
ejpam-6106	8	45	by	by	ADP
ejpam-6106	8	46	both	both	CCONJ
ejpam-6106	8	47	mathematical	mathematical	ADJ
ejpam-6106	8	48	and	and	CCONJ
ejpam-6106	8	49	geometric	geometric	ADJ
ejpam-6106	8	50	reasons	reason	NOUN
ejpam-6106	8	51	.	.	PUNCT
ejpam-6106	9	1	the	the	DET
ejpam-6106	9	2	perimeter	perimeter	NOUN
ejpam-6106	9	3	provides	provide	VERB
ejpam-6106	9	4	a	a	DET
ejpam-6106	9	5	single	single	ADJ
ejpam-6106	9	6	aggregate	aggregate	ADJ
ejpam-6106	9	7	measure	measure	NOUN
ejpam-6106	9	8	that	that	PRON
ejpam-6106	9	9	captures	capture	VERB
ejpam-6106	9	10	the	the	DET
ejpam-6106	9	11	overall	overall	ADJ
ejpam-6106	9	12	contraction	contraction	NOUN
ejpam-6106	9	13	of	of	ADP
ejpam-6106	9	14	a	a	DET
ejpam-6106	9	15	polygon	polygon	NOUN
ejpam-6106	9	16	,	,	PUNCT
ejpam-6106	9	17	avoiding	avoid	VERB
ejpam-6106	9	18	the	the	DET
ejpam-6106	9	19	need	need	NOUN
ejpam-6106	9	20	to	to	PART
ejpam-6106	9	21	track	track	VERB
ejpam-6106	9	22	each	each	DET
ejpam-6106	9	23	side	side	NOUN
ejpam-6106	9	24	separately	separately	ADV
ejpam-6106	9	25	.	.	PUNCT
ejpam-6106	10	1	this	this	DET
ejpam-6106	10	2	global	global	ADJ
ejpam-6106	10	3	perspective	perspective	NOUN
ejpam-6106	10	4	not	not	PART
ejpam-6106	10	5	only	only	ADV
ejpam-6106	10	6	simplifies	simplify	VERB
ejpam-6106	10	7	the	the	DET
ejpam-6106	10	8	analysis	analysis	NOUN
ejpam-6106	10	9	but	but	CCONJ
ejpam-6106	10	10	also	also	ADV
ejpam-6106	10	11	aligns	align	VERB
ejpam-6106	10	12	naturally	naturally	ADV
ejpam-6106	10	13	with	with	ADP
ejpam-6106	10	14	contraction	contraction	NOUN
ejpam-6106	10	15	principles	principle	NOUN
ejpam-6106	10	16	,	,	PUNCT
ejpam-6106	10	17	which	which	PRON
ejpam-6106	10	18	are	be	AUX
ejpam-6106	10	19	often	often	ADV
ejpam-6106	10	20	defined	define	VERB
ejpam-6106	10	21	in	in	ADP
ejpam-6106	10	22	terms	term	NOUN
ejpam-6106	10	23	of	of	ADP
ejpam-6106	10	24	distances	distance	NOUN
ejpam-6106	10	25	between	between	ADP
ejpam-6106	10	26	objects	object	NOUN
ejpam-6106	10	27	rather	rather	ADV
ejpam-6106	10	28	than	than	ADP
ejpam-6106	10	29	individual	individual	ADJ
ejpam-6106	10	30	components	component	NOUN
ejpam-6106	10	31	.	.	PUNCT
ejpam-6106	11	1	furthermore	furthermore	ADV
ejpam-6106	11	2	,	,	PUNCT
ejpam-6106	11	3	by	by	ADP
ejpam-6106	11	4	contracting	contract	VERB
ejpam-6106	11	5	perimeters	perimeter	NOUN
ejpam-6106	11	6	,	,	PUNCT
ejpam-6106	11	7	we	we	PRON
ejpam-6106	11	8	establish	establish	VERB
ejpam-6106	11	9	a	a	DET
ejpam-6106	11	10	unified	unified	ADJ
ejpam-6106	11	11	framework	framework	NOUN
ejpam-6106	11	12	that	that	PRON
ejpam-6106	11	13	generalizes	generalize	VERB
ejpam-6106	11	14	earlier	early	ADJ
ejpam-6106	11	15	results	result	NOUN
ejpam-6106	11	16	(	(	PUNCT
ejpam-6106	11	17	such	such	ADJ
ejpam-6106	11	18	as	as	ADP
ejpam-6106	11	19	∗corresponding	∗corresponde	VERB
ejpam-6106	11	20	author	author	NOUN
ejpam-6106	11	21	.	.	PUNCT
ejpam-6106	12	1	∗corresponding	∗corresponde	VERB
ejpam-6106	12	2	author	author	NOUN
ejpam-6106	12	3	.	.	PUNCT
ejpam-6106	13	1	doi	doi	NOUN
ejpam-6106	13	2	:	:	PUNCT
ejpam-6106	13	3	https://doi.org/10.29020/nybg.ejpam.v18i4.6106	https://doi.org/10.29020/nybg.ejpam.v18i4.6106	NUM
ejpam-6106	13	4	email	email	NOUN
ejpam-6106	13	5	addresses	address	NOUN
ejpam-6106	13	6	:	:	PUNCT
ejpam-6106	14	1	muhammad.nazam@aiou.edu.pk	muhammad.nazam@aiou.edu.pk	PROPN
ejpam-6106	14	2	(	(	PUNCT
ejpam-6106	14	3	m.	m.	NOUN
ejpam-6106	14	4	nazam	nazam	PROPN
ejpam-6106	14	5	)	)	PUNCT
ejpam-6106	14	6	,	,	PUNCT
ejpam-6106	14	7	umme6427@gmail.com	umme6427@gmail.com	X
ejpam-6106	14	8	(	(	PUNCT
ejpam-6106	14	9	u.	u.	PROPN
ejpam-6106	14	10	habiba	habiba	PROPN
ejpam-6106	14	11	)	)	PUNCT
ejpam-6106	14	12	,	,	PUNCT
ejpam-6106	14	13	manuel.delasen@ehu.eus	manuel.delasen@ehu.eus	PROPN
ejpam-6106	14	14	(	(	PUNCT
ejpam-6106	14	15	m.	m.	NOUN
ejpam-6106	14	16	de	de	X
ejpam-6106	14	17	la	la	X
ejpam-6106	14	18	sen	sen	PROPN
ejpam-6106	14	19	)	)	PUNCT
ejpam-6106	14	20	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-6106	14	21	1	1	NUM
ejpam-6106	14	22	copyright	copyright	NOUN
ejpam-6106	14	23	:	:	PUNCT
ejpam-6106	15	1	©	©	PROPN
ejpam-6106	15	2	2025	2025	NUM
ejpam-6106	15	3	the	the	DET
ejpam-6106	15	4	author(s	author(s	NOUN
ejpam-6106	15	5	)	)	PUNCT
ejpam-6106	15	6	.	.	PUNCT
ejpam-6106	16	1	(	(	PUNCT
ejpam-6106	16	2	cc	cc	NOUN
ejpam-6106	16	3	by	by	ADP
ejpam-6106	16	4	-	-	PUNCT
ejpam-6106	16	5	nc	nc	PROPN
ejpam-6106	16	6	4.0	4.0	NUM
ejpam-6106	16	7	)	)	PUNCT
ejpam-6106	16	8	m.	m.	NOUN
ejpam-6106	16	9	nazam	nazam	PROPN
ejpam-6106	16	10	,	,	PUNCT
ejpam-6106	16	11	u.	u.	PROPN
ejpam-6106	16	12	habiba	habiba	PROPN
ejpam-6106	16	13	,	,	PUNCT
ejpam-6106	16	14	m.	m.	PROPN
ejpam-6106	16	15	de	de	X
ejpam-6106	16	16	la	la	PROPN
ejpam-6106	16	17	sen	sen	PROPN
ejpam-6106	16	18	/	/	SYM
ejpam-6106	16	19	eur	eur	PROPN
ejpam-6106	16	20	.	.	PUNCT
ejpam-6106	17	1	j.	j.	PROPN
ejpam-6106	17	2	pure	pure	PROPN
ejpam-6106	17	3	appl	appl	PROPN
ejpam-6106	17	4	.	.	PROPN
ejpam-6106	17	5	math	math	PROPN
ejpam-6106	17	6	,	,	PUNCT
ejpam-6106	17	7	18	18	NUM
ejpam-6106	17	8	(	(	PUNCT
ejpam-6106	17	9	4	4	NUM
ejpam-6106	17	10	)	)	PUNCT
ejpam-6106	17	11	(	(	PUNCT
ejpam-6106	17	12	2025	2025	NUM
ejpam-6106	17	13	)	)	PUNCT
ejpam-6106	17	14	,	,	PUNCT
ejpam-6106	17	15	6106	6106	NUM
ejpam-6106	17	16	2	2	NUM
ejpam-6106	17	17	of	of	ADP
ejpam-6106	17	18	10	10	NUM
ejpam-6106	17	19	petrov	petrov	PROPN
ejpam-6106	17	20	’s	’s	PART
ejpam-6106	17	21	theorem	theorem	NOUN
ejpam-6106	17	22	for	for	ADP
ejpam-6106	17	23	triangles	triangle	NOUN
ejpam-6106	17	24	)	)	PUNCT
ejpam-6106	17	25	while	while	SCONJ
ejpam-6106	17	26	offering	offer	VERB
ejpam-6106	17	27	broader	broad	ADJ
ejpam-6106	17	28	applicability	applicability	NOUN
ejpam-6106	17	29	to	to	ADP
ejpam-6106	17	30	polygons	polygon	NOUN
ejpam-6106	17	31	with	with	ADP
ejpam-6106	17	32	arbitrary	arbitrary	ADJ
ejpam-6106	17	33	numbers	number	NOUN
ejpam-6106	17	34	of	of	ADP
ejpam-6106	17	35	vertices	vertex	NOUN
ejpam-6106	17	36	.	.	PUNCT
ejpam-6106	18	1	this	this	DET
ejpam-6106	18	2	approach	approach	NOUN
ejpam-6106	18	3	highlights	highlight	VERB
ejpam-6106	18	4	the	the	DET
ejpam-6106	18	5	novelty	novelty	NOUN
ejpam-6106	18	6	and	and	CCONJ
ejpam-6106	18	7	significance	significance	NOUN
ejpam-6106	18	8	of	of	ADP
ejpam-6106	18	9	our	our	PRON
ejpam-6106	18	10	contribution	contribution	NOUN
ejpam-6106	18	11	in	in	ADP
ejpam-6106	18	12	comparison	comparison	NOUN
ejpam-6106	18	13	to	to	ADP
ejpam-6106	18	14	other	other	ADJ
ejpam-6106	18	15	extensions	extension	NOUN
ejpam-6106	18	16	of	of	ADP
ejpam-6106	18	17	banach	banach	NOUN
ejpam-6106	18	18	’s	’s	PART
ejpam-6106	18	19	fixed	fix	VERB
ejpam-6106	18	20	point	point	NOUN
ejpam-6106	18	21	theorem	theorem	VERB
ejpam-6106	18	22	.	.	PUNCT
ejpam-6106	19	1	we	we	PRON
ejpam-6106	19	2	consider	consider	VERB
ejpam-6106	19	3	a	a	DET
ejpam-6106	19	4	mapping	mapping	NOUN
ejpam-6106	19	5	contracting	contracting	NOUN
ejpam-6106	19	6	perimeters	perimeter	NOUN
ejpam-6106	19	7	of	of	ADP
ejpam-6106	19	8	polygons	polygon	NOUN
ejpam-6106	19	9	and	and	CCONJ
ejpam-6106	19	10	demonstrate	demonstrate	VERB
ejpam-6106	19	11	the	the	DET
ejpam-6106	19	12	continuity	continuity	NOUN
ejpam-6106	19	13	of	of	ADP
ejpam-6106	19	14	such	such	ADJ
ejpam-6106	19	15	mapping	mapping	NOUN
ejpam-6106	19	16	.	.	PUNCT
ejpam-6106	20	1	we	we	PRON
ejpam-6106	20	2	prove	prove	VERB
ejpam-6106	20	3	a	a	DET
ejpam-6106	20	4	fixed	fix	VERB
ejpam-6106	20	5	point	point	NOUN
ejpam-6106	20	6	theorem	theorem	NOUN
ejpam-6106	20	7	for	for	ADP
ejpam-6106	20	8	this	this	DET
ejpam-6106	20	9	mapping	mapping	NOUN
ejpam-6106	20	10	.	.	PUNCT
ejpam-6106	21	1	through	through	ADP
ejpam-6106	21	2	an	an	DET
ejpam-6106	21	3	example	example	NOUN
ejpam-6106	21	4	we	we	PRON
ejpam-6106	21	5	illustrate	illustrate	VERB
ejpam-6106	21	6	that	that	DET
ejpam-6106	21	7	maximum	maximum	ADJ
ejpam-6106	21	8	number	number	NOUN
ejpam-6106	21	9	of	of	ADP
ejpam-6106	21	10	the	the	DET
ejpam-6106	21	11	fixed	fix	VERB
ejpam-6106	21	12	points	point	NOUN
ejpam-6106	21	13	can	can	AUX
ejpam-6106	21	14	be	be	AUX
ejpam-6106	21	15	n−	n−	NOUN
ejpam-6106	21	16	1	1	NUM
ejpam-6106	21	17	.	.	PUNCT
ejpam-6106	22	1	we	we	PRON
ejpam-6106	22	2	deduce	deduce	VERB
ejpam-6106	22	3	the	the	DET
ejpam-6106	22	4	well	well	ADV
ejpam-6106	22	5	-	-	PUNCT
ejpam-6106	22	6	known	know	VERB
ejpam-6106	22	7	banach	banach	NOUN
ejpam-6106	22	8	fixed	fix	VERB
ejpam-6106	22	9	point	point	NOUN
ejpam-6106	22	10	theorem	theorem	VERB
ejpam-6106	22	11	.	.	PUNCT
ejpam-6106	23	1	our	our	PRON
ejpam-6106	23	2	work	work	NOUN
ejpam-6106	23	3	is	be	AUX
ejpam-6106	23	4	generalization	generalization	NOUN
ejpam-6106	23	5	of	of	ADP
ejpam-6106	23	6	petrov	petrov	PROPN
ejpam-6106	23	7	fixed	fix	VERB
ejpam-6106	23	8	point	point	NOUN
ejpam-6106	23	9	theorem	theorem	VERB
ejpam-6106	23	10	[	[	X
ejpam-6106	23	11	1	1	NUM
ejpam-6106	23	12	]	]	PUNCT
ejpam-6106	23	13	.	.	PUNCT
ejpam-6106	24	1	by	by	ADP
ejpam-6106	24	2	defining	define	VERB
ejpam-6106	24	3	the	the	DET
ejpam-6106	24	4	concept	concept	NOUN
ejpam-6106	24	5	of	of	ADP
ejpam-6106	24	6	mappings	mapping	NOUN
ejpam-6106	24	7	contracting	contract	VERB
ejpam-6106	24	8	perimeters	perimeter	NOUN
ejpam-6106	24	9	of	of	ADP
ejpam-6106	24	10	triangles	triangle	NOUN
ejpam-6106	24	11	and	and	CCONJ
ejpam-6106	24	12	proving	prove	VERB
ejpam-6106	24	13	the	the	DET
ejpam-6106	24	14	fixed	fix	VERB
ejpam-6106	24	15	point	point	NOUN
ejpam-6106	24	16	theorem	theorem	VERB
ejpam-6106	24	17	for	for	ADP
ejpam-6106	24	18	such	such	ADJ
ejpam-6106	24	19	mappings	mapping	NOUN
ejpam-6106	24	20	,	,	PUNCT
ejpam-6106	24	21	petrov[1	petrov[1	NOUN
ejpam-6106	24	22	]	]	PUNCT
ejpam-6106	24	23	(	(	PUNCT
ejpam-6106	24	24	2023	2023	NUM
ejpam-6106	24	25	)	)	PUNCT
ejpam-6106	24	26	provided	provide	VERB
ejpam-6106	24	27	a	a	DET
ejpam-6106	24	28	new	new	ADJ
ejpam-6106	24	29	generalization	generalization	NOUN
ejpam-6106	24	30	of	of	ADP
ejpam-6106	24	31	the	the	DET
ejpam-6106	24	32	banach	banach	ADV
ejpam-6106	24	33	fixed	fix	VERB
ejpam-6106	24	34	point	point	NOUN
ejpam-6106	24	35	theorem	theorem	NOUN
ejpam-6106	24	36	(	(	PUNCT
ejpam-6106	24	37	bfpt	bfpt	PROPN
ejpam-6106	24	38	)	)	PUNCT
ejpam-6106	24	39	.	.	PUNCT
ejpam-6106	25	1	the	the	DET
ejpam-6106	25	2	key	key	ADJ
ejpam-6106	25	3	distinction	distinction	NOUN
ejpam-6106	25	4	is	be	AUX
ejpam-6106	25	5	that	that	SCONJ
ejpam-6106	25	6	,	,	PUNCT
ejpam-6106	25	7	though	though	SCONJ
ejpam-6106	25	8	the	the	DET
ejpam-6106	25	9	central	central	ADJ
ejpam-6106	25	10	hypothesis	hypothesis	NOUN
ejpam-6106	25	11	of	of	ADP
ejpam-6106	25	12	his	his	PRON
ejpam-6106	25	13	research	research	NOUN
ejpam-6106	25	14	is	be	AUX
ejpam-6106	25	15	predicated	predicate	VERB
ejpam-6106	25	16	on	on	ADP
ejpam-6106	25	17	the	the	DET
ejpam-6106	25	18	concepts	concept	NOUN
ejpam-6106	25	19	of	of	ADP
ejpam-6106	25	20	banach	banach	NOUN
ejpam-6106	25	21	’s	’s	PART
ejpam-6106	25	22	classical	classical	ADJ
ejpam-6106	25	23	theorem	theorem	NOUN
ejpam-6106	25	24	,	,	PUNCT
ejpam-6106	25	25	the	the	DET
ejpam-6106	25	26	definition	definition	NOUN
ejpam-6106	25	27	of	of	ADP
ejpam-6106	25	28	this	this	DET
ejpam-6106	25	29	mapping	mapping	NOUN
ejpam-6106	25	30	is	be	AUX
ejpam-6106	25	31	based	base	VERB
ejpam-6106	25	32	on	on	ADP
ejpam-6106	25	33	the	the	DET
ejpam-6106	25	34	mapping	mapping	NOUN
ejpam-6106	25	35	of	of	ADP
ejpam-6106	25	36	three	three	NUM
ejpam-6106	25	37	points	point	NOUN
ejpam-6106	25	38	of	of	ADP
ejpam-6106	25	39	space	space	NOUN
ejpam-6106	25	40	rather	rather	ADV
ejpam-6106	25	41	than	than	ADP
ejpam-6106	25	42	two	two	NUM
ejpam-6106	25	43	.	.	PUNCT
ejpam-6106	26	1	further	far	ADV
ejpam-6106	26	2	a	a	DET
ejpam-6106	26	3	condition	condition	NOUN
ejpam-6106	26	4	is	be	AUX
ejpam-6106	26	5	imposed	impose	VERB
ejpam-6106	26	6	on	on	ADP
ejpam-6106	26	7	mapping	mapping	NOUN
ejpam-6106	26	8	l	l	NOUN
ejpam-6106	26	9	,	,	PUNCT
ejpam-6106	26	10	l(l(y	l(l(y	NOUN
ejpam-6106	26	11	)	)	PUNCT
ejpam-6106	26	12	)	)	PUNCT
ejpam-6106	27	1	̸=	̸=	PROPN
ejpam-6106	27	2	y	y	PROPN
ejpam-6106	27	3	for	for	ADP
ejpam-6106	27	4	all	all	DET
ejpam-6106	27	5	y	y	PROPN
ejpam-6106	27	6	∈	∈	PROPN
ejpam-6106	27	7	y	y	PROPN
ejpam-6106	27	8	such	such	ADJ
ejpam-6106	27	9	that	that	PRON
ejpam-6106	27	10	ly	ly	ADP
ejpam-6106	27	11	̸=	̸=	PROPN
ejpam-6106	27	12	y.	y.	NOUN
ejpam-6106	27	13	one	one	NUM
ejpam-6106	27	14	significant	significant	ADJ
ejpam-6106	27	15	subclass	subclass	NOUN
ejpam-6106	27	16	of	of	ADP
ejpam-6106	27	17	these	these	DET
ejpam-6106	27	18	mappings	mapping	NOUN
ejpam-6106	27	19	is	be	AUX
ejpam-6106	27	20	the	the	DET
ejpam-6106	27	21	ordinary	ordinary	ADJ
ejpam-6106	27	22	contraction	contraction	NOUN
ejpam-6106	27	23	mapping	mapping	NOUN
ejpam-6106	27	24	,	,	PUNCT
ejpam-6106	27	25	from	from	ADP
ejpam-6106	27	26	which	which	PRON
ejpam-6106	27	27	we	we	PRON
ejpam-6106	27	28	are	be	AUX
ejpam-6106	27	29	able	able	ADJ
ejpam-6106	27	30	to	to	PART
ejpam-6106	27	31	derive	derive	VERB
ejpam-6106	27	32	the	the	DET
ejpam-6106	27	33	classical	classical	ADJ
ejpam-6106	27	34	banach	banach	NOUN
ejpam-6106	27	35	theorem	theorem	VERB
ejpam-6106	27	36	as	as	ADP
ejpam-6106	27	37	a	a	DET
ejpam-6106	27	38	straightforward	straightforward	ADJ
ejpam-6106	27	39	corollary	corollary	NOUN
ejpam-6106	27	40	right	right	ADV
ejpam-6106	27	41	away	away	ADV
ejpam-6106	27	42	.	.	PUNCT
ejpam-6106	28	1	banach	banach	PROPN
ejpam-6106	28	2	developed	develop	VERB
ejpam-6106	28	3	the	the	DET
ejpam-6106	28	4	contraction	contraction	NOUN
ejpam-6106	28	5	principle	principle	NOUN
ejpam-6106	28	6	in	in	ADP
ejpam-6106	28	7	1922	1922	NUM
ejpam-6106	28	8	[	[	X
ejpam-6106	28	9	2	2	NUM
ejpam-6106	28	10	]	]	PUNCT
ejpam-6106	28	11	.	.	PUNCT
ejpam-6106	29	1	though	though	SCONJ
ejpam-6106	29	2	the	the	DET
ejpam-6106	29	3	concept	concept	NOUN
ejpam-6106	29	4	of	of	ADP
ejpam-6106	29	5	successive	successive	ADJ
ejpam-6106	29	6	approximations	approximation	NOUN
ejpam-6106	29	7	had	have	AUX
ejpam-6106	29	8	previously	previously	ADV
ejpam-6106	29	9	been	be	AUX
ejpam-6106	29	10	mentioned	mention	VERB
ejpam-6106	29	11	in	in	ADP
ejpam-6106	29	12	the	the	DET
ejpam-6106	29	13	writings	writing	NOUN
ejpam-6106	29	14	of	of	ADP
ejpam-6106	29	15	chebysher	chebysher	NOUN
ejpam-6106	29	16	,	,	PUNCT
ejpam-6106	29	17	picard	picard	NOUN
ejpam-6106	29	18	,	,	PUNCT
ejpam-6106	29	19	and	and	CCONJ
ejpam-6106	29	20	others	other	NOUN
ejpam-6106	29	21	,	,	PUNCT
ejpam-6106	29	22	it	it	PRON
ejpam-6106	29	23	was	be	AUX
ejpam-6106	29	24	applied	apply	VERB
ejpam-6106	29	25	in	in	ADP
ejpam-6106	29	26	approximation	approximation	NOUN
ejpam-6106	29	27	theory	theory	NOUN
ejpam-6106	29	28	,	,	PUNCT
ejpam-6106	29	29	solutions	solution	NOUN
ejpam-6106	29	30	to	to	PART
ejpam-6106	29	31	differential	differential	VERB
ejpam-6106	29	32	and	and	CCONJ
ejpam-6106	29	33	integral	integral	ADJ
ejpam-6106	29	34	equations	equation	NOUN
ejpam-6106	29	35	,	,	PUNCT
ejpam-6106	29	36	etc	etc	X
ejpam-6106	29	37	.	.	X
ejpam-6106	30	1	the	the	DET
ejpam-6106	30	2	first	first	ADJ
ejpam-6106	30	3	person	person	NOUN
ejpam-6106	30	4	to	to	PART
ejpam-6106	30	5	correctly	correctly	ADV
ejpam-6106	30	6	formulate	formulate	VERB
ejpam-6106	30	7	this	this	DET
ejpam-6106	30	8	outcome	outcome	NOUN
ejpam-6106	30	9	was	be	AUX
ejpam-6106	30	10	banach	banach	ADV
ejpam-6106	30	11	.	.	PUNCT
ejpam-6106	31	1	over	over	ADP
ejpam-6106	31	2	time	time	NOUN
ejpam-6106	31	3	,	,	PUNCT
ejpam-6106	31	4	there	there	PRON
ejpam-6106	31	5	have	have	AUX
ejpam-6106	31	6	been	be	AUX
ejpam-6106	31	7	numerous	numerous	ADJ
ejpam-6106	31	8	ways	way	NOUN
ejpam-6106	31	9	to	to	PART
ejpam-6106	31	10	generalize	generalize	VERB
ejpam-6106	31	11	the	the	DET
ejpam-6106	31	12	banach	banach	NOUN
ejpam-6106	31	13	contraction	contraction	NOUN
ejpam-6106	31	14	principle	principle	NOUN
ejpam-6106	31	15	.	.	PUNCT
ejpam-6106	32	1	additionally	additionally	ADV
ejpam-6106	32	2	,	,	PUNCT
ejpam-6106	32	3	metric	metric	ADJ
ejpam-6106	32	4	extensions	extension	NOUN
ejpam-6106	32	5	are	be	AUX
ejpam-6106	32	6	typically	typically	ADV
ejpam-6106	32	7	verified	verify	VERB
ejpam-6106	32	8	against	against	ADP
ejpam-6106	32	9	three	three	NUM
ejpam-6106	32	10	classical	classical	ADJ
ejpam-6106	32	11	fixed	fix	VERB
ejpam-6106	32	12	point	point	NOUN
ejpam-6106	32	13	theorems[3	theorems[3	NOUN
ejpam-6106	32	14	]	]	PUNCT
ejpam-6106	32	15	.	.	PUNCT
ejpam-6106	33	1	these	these	PRON
ejpam-6106	33	2	are	be	AUX
ejpam-6106	33	3	the	the	DET
ejpam-6106	33	4	extension	extension	NOUN
ejpam-6106	33	5	of	of	ADP
ejpam-6106	33	6	banach	banach	NOUN
ejpam-6106	33	7	’s	’s	PUNCT
ejpam-6106	33	8	theorem	theorem	VERB
ejpam-6106	33	9	to	to	PART
ejpam-6106	33	10	nonexpansive	nonexpansive	VERB
ejpam-6106	33	11	mappings[4	mappings[4	PROPN
ejpam-6106	33	12	]	]	PUNCT
ejpam-6106	33	13	,	,	PUNCT
ejpam-6106	33	14	caristi	caristi	PROPN
ejpam-6106	33	15	’s	’s	PART
ejpam-6106	33	16	theorem[5	theorem[5	NOUN
ejpam-6106	33	17	]	]	PUNCT
ejpam-6106	33	18	,	,	PUNCT
ejpam-6106	33	19	and	and	CCONJ
ejpam-6106	33	20	nadler	nadler	PROPN
ejpam-6106	33	21	’s	’s	PART
ejpam-6106	33	22	well	well	ADV
ejpam-6106	33	23	-	-	PUNCT
ejpam-6106	33	24	known	know	VERB
ejpam-6106	33	25	set	set	NOUN
ejpam-6106	33	26	-	-	PUNCT
ejpam-6106	33	27	valued	value	VERB
ejpam-6106	33	28	extension	extension	NOUN
ejpam-6106	33	29	of	of	ADP
ejpam-6106	33	30	banach	banach	NOUN
ejpam-6106	33	31	’s	’s	NOUN
ejpam-6106	33	32	theorem[6	theorem[6	PRON
ejpam-6106	33	33	]	]	PUNCT
ejpam-6106	33	34	.	.	PUNCT
ejpam-6106	34	1	in	in	ADP
ejpam-6106	34	2	present	present	ADJ
ejpam-6106	34	3	paper	paper	NOUN
ejpam-6106	34	4	,	,	PUNCT
ejpam-6106	34	5	we	we	PRON
ejpam-6106	34	6	extend	extend	VERB
ejpam-6106	34	7	petrov	petrov	PROPN
ejpam-6106	34	8	’s	’s	PART
ejpam-6106	34	9	idea	idea	NOUN
ejpam-6106	34	10	and	and	CCONJ
ejpam-6106	34	11	prove	prove	VERB
ejpam-6106	34	12	banach	banach	NOUN
ejpam-6106	34	13	’s	’s	ADV
ejpam-6106	34	14	theorem	theorem	NOUN
ejpam-6106	34	15	as	as	ADP
ejpam-6106	34	16	a	a	DET
ejpam-6106	34	17	sub	sub	NOUN
ejpam-6106	34	18	-	-	NOUN
ejpam-6106	34	19	result	result	NOUN
ejpam-6106	34	20	.	.	PUNCT
ejpam-6106	35	1	the	the	DET
ejpam-6106	35	2	fundamental	fundamental	ADJ
ejpam-6106	35	3	distinction	distinction	NOUN
ejpam-6106	35	4	is	be	AUX
ejpam-6106	35	5	that	that	SCONJ
ejpam-6106	35	6	mentioned	mention	VERB
ejpam-6106	35	7	contraction	contraction	NOUN
ejpam-6106	35	8	mapping	mapping	NOUN
ejpam-6106	35	9	is	be	AUX
ejpam-6106	35	10	predicated	predicate	VERB
ejpam-6106	35	11	on	on	ADP
ejpam-6106	35	12	n	n	NUM
ejpam-6106	35	13	points	point	NOUN
ejpam-6106	35	14	in	in	ADP
ejpam-6106	35	15	the	the	DET
ejpam-6106	35	16	space	space	NOUN
ejpam-6106	35	17	.	.	PUNCT
ejpam-6106	36	1	a	a	DET
ejpam-6106	36	2	mapping	mapping	NOUN
ejpam-6106	36	3	contracting	contract	VERB
ejpam-6106	36	4	perimeter	perimeter	NOUN
ejpam-6106	36	5	of	of	ADP
ejpam-6106	36	6	polygon	polygon	PROPN
ejpam-6106	36	7	that	that	PRON
ejpam-6106	36	8	is	be	AUX
ejpam-6106	36	9	not	not	PART
ejpam-6106	36	10	a	a	DET
ejpam-6106	36	11	contraction	contraction	NOUN
ejpam-6106	36	12	mapping	mapping	NOUN
ejpam-6106	36	13	is	be	AUX
ejpam-6106	36	14	defined	define	VERB
ejpam-6106	36	15	with	with	ADP
ejpam-6106	36	16	|y	|y	NOUN
ejpam-6106	36	17	|	|	NOUN
ejpam-6106	36	18	=	=	SYM
ejpam-6106	36	19	n0	n0	PROPN
ejpam-6106	36	20	,	,	PUNCT
ejpam-6106	36	21	where	where	SCONJ
ejpam-6106	36	22	n0	n0	PROPN
ejpam-6106	36	23	is	be	AUX
ejpam-6106	36	24	the	the	DET
ejpam-6106	36	25	cardinality	cardinality	NOUN
ejpam-6106	36	26	of	of	ADP
ejpam-6106	36	27	set	set	PROPN
ejpam-6106	36	28	y	y	PROPN
ejpam-6106	36	29	.	.	PUNCT
ejpam-6106	37	1	recent	recent	ADJ
ejpam-6106	37	2	developments	development	NOUN
ejpam-6106	37	3	in	in	ADP
ejpam-6106	37	4	fixed	fix	VERB
ejpam-6106	37	5	point	point	NOUN
ejpam-6106	37	6	theory	theory	NOUN
ejpam-6106	37	7	and	and	CCONJ
ejpam-6106	37	8	ulam	ulam	ADJ
ejpam-6106	37	9	-	-	PUNCT
ejpam-6106	37	10	type	type	NOUN
ejpam-6106	37	11	stabilities	stability	NOUN
ejpam-6106	37	12	include	include	VERB
ejpam-6106	37	13	results	result	NOUN
ejpam-6106	37	14	on	on	ADP
ejpam-6106	37	15	dislocated	dislocated	ADJ
ejpam-6106	37	16	quasi	quasi	NOUN
ejpam-6106	37	17	-	-	ADJ
ejpam-6106	37	18	metric	metric	ADJ
ejpam-6106	37	19	and	and	CCONJ
ejpam-6106	37	20	b	b	NOUN
ejpam-6106	37	21	-	-	PUNCT
ejpam-6106	37	22	metric	metric	ADJ
ejpam-6106	37	23	settings	setting	NOUN
ejpam-6106	37	24	[	[	X
ejpam-6106	37	25	7	7	NUM
ejpam-6106	37	26	,	,	PUNCT
ejpam-6106	37	27	8	8	NUM
ejpam-6106	37	28	]	]	PUNCT
ejpam-6106	37	29	,	,	PUNCT
ejpam-6106	37	30	and	and	CCONJ
ejpam-6106	37	31	on	on	ADP
ejpam-6106	37	32	ulam	ulam	ADJ
ejpam-6106	37	33	-	-	PUNCT
ejpam-6106	37	34	type	type	NOUN
ejpam-6106	37	35	stabilities	stability	NOUN
ejpam-6106	37	36	for	for	ADP
ejpam-6106	37	37	integral	integral	ADJ
ejpam-6106	37	38	and	and	CCONJ
ejpam-6106	37	39	integro	integro	ADJ
ejpam-6106	37	40	-	-	PUNCT
ejpam-6106	37	41	differential	differential	NOUN
ejpam-6106	37	42	equations	equation	NOUN
ejpam-6106	37	43	with	with	ADP
ejpam-6106	37	44	delays	delay	NOUN
ejpam-6106	37	45	[	[	X
ejpam-6106	37	46	9–13	9–13	NOUN
ejpam-6106	37	47	]	]	PUNCT
ejpam-6106	37	48	.	.	PUNCT
ejpam-6106	38	1	recent	recent	ADJ
ejpam-6106	38	2	works	work	NOUN
ejpam-6106	38	3	have	have	AUX
ejpam-6106	38	4	also	also	ADV
ejpam-6106	38	5	emphasized	emphasize	VERB
ejpam-6106	38	6	the	the	DET
ejpam-6106	38	7	importance	importance	NOUN
ejpam-6106	38	8	of	of	ADP
ejpam-6106	38	9	stability	stability	NOUN
ejpam-6106	38	10	and	and	CCONJ
ejpam-6106	38	11	qualitative	qualitative	ADJ
ejpam-6106	38	12	analyses	analysis	NOUN
ejpam-6106	38	13	in	in	ADP
ejpam-6106	38	14	related	related	ADJ
ejpam-6106	38	15	contexts	context	NOUN
ejpam-6106	38	16	.	.	PUNCT
ejpam-6106	39	1	for	for	ADP
ejpam-6106	39	2	instance	instance	NOUN
ejpam-6106	39	3	,	,	PUNCT
ejpam-6106	39	4	several	several	ADJ
ejpam-6106	39	5	studies	study	NOUN
ejpam-6106	39	6	on	on	ADP
ejpam-6106	39	7	ulam	ulam	PROPN
ejpam-6106	39	8	stability	stability	NOUN
ejpam-6106	39	9	have	have	AUX
ejpam-6106	39	10	investigated	investigate	VERB
ejpam-6106	39	11	nonlinear	nonlinear	ADJ
ejpam-6106	39	12	integral	integral	ADJ
ejpam-6106	39	13	equations	equation	NOUN
ejpam-6106	39	14	and	and	CCONJ
ejpam-6106	39	15	differential	differential	ADJ
ejpam-6106	39	16	equations	equation	NOUN
ejpam-6106	39	17	with	with	ADP
ejpam-6106	39	18	delays	delay	NOUN
ejpam-6106	39	19	[	[	X
ejpam-6106	39	20	9–13	9–13	NOUN
ejpam-6106	39	21	]	]	PUNCT
ejpam-6106	39	22	.	.	PUNCT
ejpam-6106	40	1	these	these	DET
ejpam-6106	40	2	contributions	contribution	NOUN
ejpam-6106	40	3	highlight	highlight	VERB
ejpam-6106	40	4	how	how	SCONJ
ejpam-6106	40	5	stability	stability	NOUN
ejpam-6106	40	6	conditions	condition	NOUN
ejpam-6106	40	7	can	can	AUX
ejpam-6106	40	8	broaden	broaden	VERB
ejpam-6106	40	9	the	the	DET
ejpam-6106	40	10	applicability	applicability	NOUN
ejpam-6106	40	11	of	of	ADP
ejpam-6106	40	12	fixed	fix	VERB
ejpam-6106	40	13	point	point	NOUN
ejpam-6106	40	14	techniques	technique	NOUN
ejpam-6106	40	15	.	.	PUNCT
ejpam-6106	41	1	motivated	motivate	VERB
ejpam-6106	41	2	by	by	ADP
ejpam-6106	41	3	these	these	DET
ejpam-6106	41	4	directions	direction	NOUN
ejpam-6106	41	5	,	,	PUNCT
ejpam-6106	41	6	our	our	PRON
ejpam-6106	41	7	paper	paper	NOUN
ejpam-6106	41	8	proposes	propose	VERB
ejpam-6106	41	9	a	a	DET
ejpam-6106	41	10	geometric	geometric	ADJ
ejpam-6106	41	11	generalization	generalization	NOUN
ejpam-6106	41	12	where	where	SCONJ
ejpam-6106	41	13	contraction	contraction	NOUN
ejpam-6106	41	14	acts	act	VERB
ejpam-6106	41	15	on	on	ADP
ejpam-6106	41	16	the	the	DET
ejpam-6106	41	17	perimeters	perimeter	NOUN
ejpam-6106	41	18	of	of	ADP
ejpam-6106	41	19	polygons	polygon	NOUN
ejpam-6106	41	20	rather	rather	ADV
ejpam-6106	41	21	than	than	ADP
ejpam-6106	41	22	point	point	NOUN
ejpam-6106	41	23	-	-	PUNCT
ejpam-6106	41	24	to	to	ADP
ejpam-6106	41	25	-	-	PUNCT
ejpam-6106	41	26	point	point	NOUN
ejpam-6106	41	27	distances	distance	NOUN
ejpam-6106	41	28	.	.	PUNCT
ejpam-6106	42	1	2	2	X
ejpam-6106	42	2	.	.	X
ejpam-6106	42	3	comparison	comparison	NOUN
ejpam-6106	42	4	with	with	ADP
ejpam-6106	42	5	other	other	ADJ
ejpam-6106	42	6	generalizations	generalization	NOUN
ejpam-6106	42	7	the	the	DET
ejpam-6106	42	8	banach	banach	NOUN
ejpam-6106	42	9	contraction	contraction	NOUN
ejpam-6106	42	10	principle	principle	NOUN
ejpam-6106	42	11	has	have	AUX
ejpam-6106	42	12	been	be	AUX
ejpam-6106	42	13	generalized	generalize	VERB
ejpam-6106	42	14	in	in	ADP
ejpam-6106	42	15	many	many	ADJ
ejpam-6106	42	16	directions	direction	NOUN
ejpam-6106	42	17	,	,	PUNCT
ejpam-6106	42	18	including	include	VERB
ejpam-6106	42	19	in	in	ADP
ejpam-6106	42	20	b	b	NOUN
ejpam-6106	42	21	-	-	ADJ
ejpam-6106	42	22	metric	metric	ADJ
ejpam-6106	42	23	spaces	space	NOUN
ejpam-6106	42	24	,	,	PUNCT
ejpam-6106	42	25	dislocated	dislocate	VERB
ejpam-6106	42	26	quasi	quasi	ADJ
ejpam-6106	42	27	-	-	ADJ
ejpam-6106	42	28	metric	metric	ADJ
ejpam-6106	42	29	spaces	space	NOUN
ejpam-6106	42	30	,	,	PUNCT
ejpam-6106	42	31	ordered	order	VERB
ejpam-6106	42	32	metric	metric	ADJ
ejpam-6106	42	33	spaces	space	NOUN
ejpam-6106	42	34	,	,	PUNCT
ejpam-6106	42	35	and	and	CCONJ
ejpam-6106	42	36	multivalued	multivalue	VERB
ejpam-6106	42	37	mapping	mapping	NOUN
ejpam-6106	42	38	settings	setting	NOUN
ejpam-6106	42	39	.	.	PUNCT
ejpam-6106	43	1	for	for	ADP
ejpam-6106	43	2	example	example	NOUN
ejpam-6106	43	3	,	,	PUNCT
ejpam-6106	43	4	shoaib	shoaib	PROPN
ejpam-6106	43	5	and	and	CCONJ
ejpam-6106	43	6	mehmood	mehmood	PROPN
ejpam-6106	44	1	[	[	X
ejpam-6106	44	2	7	7	NUM
ejpam-6106	44	3	]	]	PUNCT
ejpam-6106	44	4	established	establish	VERB
ejpam-6106	44	5	fixed	fix	VERB
ejpam-6106	44	6	point	point	NOUN
ejpam-6106	44	7	results	result	NOUN
ejpam-6106	44	8	for	for	ADP
ejpam-6106	44	9	mappings	mapping	NOUN
ejpam-6106	44	10	satisfying	satisfy	VERB
ejpam-6106	44	11	(	(	PUNCT
ejpam-6106	44	12	δ	δ	NOUN
ejpam-6106	44	13	,	,	PUNCT
ejpam-6106	44	14	φ)-domination	φ)-domination	NOUN
ejpam-6106	44	15	and	and	CCONJ
ejpam-6106	44	16	continuity	continuity	NOUN
ejpam-6106	44	17	in	in	ADP
ejpam-6106	44	18	dislocated	dislocated	ADJ
ejpam-6106	44	19	quasi	quasi	ADJ
ejpam-6106	44	20	-	-	ADJ
ejpam-6106	44	21	metric	metric	ADJ
ejpam-6106	44	22	spaces	space	NOUN
ejpam-6106	44	23	,	,	PUNCT
ejpam-6106	44	24	m.	m.	NOUN
ejpam-6106	44	25	nazam	nazam	PROPN
ejpam-6106	44	26	,	,	PUNCT
ejpam-6106	44	27	u.	u.	PROPN
ejpam-6106	44	28	habiba	habiba	PROPN
ejpam-6106	44	29	,	,	PUNCT
ejpam-6106	44	30	m.	m.	PROPN
ejpam-6106	44	31	de	de	X
ejpam-6106	44	32	la	la	PROPN
ejpam-6106	44	33	sen	sen	PROPN
ejpam-6106	44	34	/	/	SYM
ejpam-6106	44	35	eur	eur	PROPN
ejpam-6106	44	36	.	.	PUNCT
ejpam-6106	45	1	j.	j.	PROPN
ejpam-6106	45	2	pure	pure	PROPN
ejpam-6106	45	3	appl	appl	PROPN
ejpam-6106	45	4	.	.	PROPN
ejpam-6106	45	5	math	math	PROPN
ejpam-6106	45	6	,	,	PUNCT
ejpam-6106	45	7	18	18	NUM
ejpam-6106	45	8	(	(	PUNCT
ejpam-6106	45	9	4	4	NUM
ejpam-6106	45	10	)	)	PUNCT
ejpam-6106	45	11	(	(	PUNCT
ejpam-6106	45	12	2025	2025	NUM
ejpam-6106	45	13	)	)	PUNCT
ejpam-6106	45	14	,	,	PUNCT
ejpam-6106	45	15	6106	6106	NUM
ejpam-6106	45	16	3	3	NUM
ejpam-6106	45	17	of	of	ADP
ejpam-6106	45	18	10	10	NUM
ejpam-6106	45	19	while	while	SCONJ
ejpam-6106	45	20	shoaib	shoaib	PROPN
ejpam-6106	45	21	and	and	CCONJ
ejpam-6106	45	22	mir	mir	NOUN
ejpam-6106	45	23	[	[	X
ejpam-6106	45	24	8	8	NUM
ejpam-6106	45	25	]	]	PUNCT
ejpam-6106	45	26	studied	study	VERB
ejpam-6106	45	27	interpolative	interpolative	ADJ
ejpam-6106	45	28	multivalued	multivalue	VERB
ejpam-6106	45	29	α∗-dominated	α∗-dominate	VERB
ejpam-6106	45	30	contractive	contractive	ADJ
ejpam-6106	45	31	mappings	mapping	NOUN
ejpam-6106	45	32	in	in	ADP
ejpam-6106	45	33	dislocated	dislocated	ADJ
ejpam-6106	45	34	b	b	NOUN
ejpam-6106	45	35	-	-	ADJ
ejpam-6106	45	36	metric	metric	ADJ
ejpam-6106	45	37	spaces	space	NOUN
ejpam-6106	45	38	.	.	PUNCT
ejpam-6106	46	1	these	these	DET
ejpam-6106	46	2	works	work	NOUN
ejpam-6106	46	3	extend	extend	VERB
ejpam-6106	46	4	banach	banach	NOUN
ejpam-6106	46	5	’s	’s	ADV
ejpam-6106	46	6	theorem	theorem	VERB
ejpam-6106	46	7	by	by	ADP
ejpam-6106	46	8	relaxing	relax	VERB
ejpam-6106	46	9	the	the	DET
ejpam-6106	46	10	underlying	underlying	ADJ
ejpam-6106	46	11	distance	distance	NOUN
ejpam-6106	46	12	structure	structure	NOUN
ejpam-6106	46	13	or	or	CCONJ
ejpam-6106	46	14	by	by	ADP
ejpam-6106	46	15	considering	consider	VERB
ejpam-6106	46	16	generalized	generalized	ADJ
ejpam-6106	46	17	contraction	contraction	NOUN
ejpam-6106	46	18	conditions	condition	NOUN
ejpam-6106	46	19	.	.	PUNCT
ejpam-6106	47	1	our	our	PRON
ejpam-6106	47	2	approach	approach	NOUN
ejpam-6106	47	3	differs	differ	VERB
ejpam-6106	47	4	in	in	ADP
ejpam-6106	47	5	a	a	DET
ejpam-6106	47	6	fundamental	fundamental	ADJ
ejpam-6106	47	7	aspect	aspect	NOUN
ejpam-6106	47	8	:	:	PUNCT
ejpam-6106	47	9	instead	instead	ADV
ejpam-6106	47	10	of	of	ADP
ejpam-6106	47	11	directly	directly	ADV
ejpam-6106	47	12	contracting	contract	VERB
ejpam-6106	47	13	distances	distance	NOUN
ejpam-6106	47	14	between	between	ADP
ejpam-6106	47	15	two	two	NUM
ejpam-6106	47	16	points	point	NOUN
ejpam-6106	47	17	,	,	PUNCT
ejpam-6106	47	18	we	we	PRON
ejpam-6106	47	19	require	require	VERB
ejpam-6106	47	20	contraction	contraction	NOUN
ejpam-6106	47	21	of	of	ADP
ejpam-6106	47	22	the	the	DET
ejpam-6106	47	23	perimeter	perimeter	NOUN
ejpam-6106	47	24	of	of	ADP
ejpam-6106	47	25	polygons	polygon	NOUN
ejpam-6106	47	26	with	with	ADP
ejpam-6106	47	27	n	n	ADP
ejpam-6106	47	28	vertices	vertex	NOUN
ejpam-6106	47	29	.	.	PUNCT
ejpam-6106	48	1	this	this	DET
ejpam-6106	48	2	perimeter	perimeter	NOUN
ejpam-6106	48	3	-	-	PUNCT
ejpam-6106	48	4	based	base	VERB
ejpam-6106	48	5	contraction	contraction	NOUN
ejpam-6106	48	6	condition	condition	NOUN
ejpam-6106	48	7	naturally	naturally	ADV
ejpam-6106	48	8	generalizes	generalize	VERB
ejpam-6106	48	9	petrov	petrov	PROPN
ejpam-6106	48	10	’s	’s	PART
ejpam-6106	48	11	theorem	theorem	NOUN
ejpam-6106	48	12	(	(	PUNCT
ejpam-6106	48	13	which	which	PRON
ejpam-6106	48	14	focused	focus	VERB
ejpam-6106	48	15	on	on	ADP
ejpam-6106	48	16	triangles	triangle	NOUN
ejpam-6106	48	17	,	,	PUNCT
ejpam-6106	48	18	n	n	NOUN
ejpam-6106	48	19	=	=	SYM
ejpam-6106	48	20	3	3	NUM
ejpam-6106	48	21	)	)	PUNCT
ejpam-6106	48	22	and	and	CCONJ
ejpam-6106	48	23	provides	provide	VERB
ejpam-6106	48	24	a	a	DET
ejpam-6106	48	25	geometric	geometric	ADJ
ejpam-6106	48	26	perspective	perspective	NOUN
ejpam-6106	48	27	distinct	distinct	ADJ
ejpam-6106	48	28	from	from	ADP
ejpam-6106	48	29	the	the	DET
ejpam-6106	48	30	analytic	analytic	ADJ
ejpam-6106	48	31	contractive	contractive	ADJ
ejpam-6106	48	32	inequalities	inequality	NOUN
ejpam-6106	48	33	found	find	VERB
ejpam-6106	48	34	in	in	ADP
ejpam-6106	48	35	earlier	early	ADJ
ejpam-6106	48	36	literature	literature	NOUN
ejpam-6106	48	37	.	.	PUNCT
ejpam-6106	49	1	in	in	ADP
ejpam-6106	49	2	particular	particular	ADJ
ejpam-6106	49	3	,	,	PUNCT
ejpam-6106	49	4	our	our	PRON
ejpam-6106	49	5	framework	framework	NOUN
ejpam-6106	49	6	allows	allow	VERB
ejpam-6106	49	7	us	we	PRON
ejpam-6106	49	8	to	to	PART
ejpam-6106	49	9	re	re	VERB
ejpam-6106	49	10	-	-	VERB
ejpam-6106	49	11	derive	derive	ADJ
ejpam-6106	49	12	banach	banach	NOUN
ejpam-6106	49	13	’s	’s	PART
ejpam-6106	49	14	contraction	contraction	NOUN
ejpam-6106	49	15	theorem	theorem	VERB
ejpam-6106	49	16	as	as	ADP
ejpam-6106	49	17	a	a	DET
ejpam-6106	49	18	corollary	corollary	ADJ
ejpam-6106	49	19	while	while	NOUN
ejpam-6106	49	20	also	also	ADV
ejpam-6106	49	21	offering	offer	VERB
ejpam-6106	49	22	new	new	ADJ
ejpam-6106	49	23	geometric	geometric	ADJ
ejpam-6106	49	24	insights	insight	NOUN
ejpam-6106	49	25	that	that	PRON
ejpam-6106	49	26	are	be	AUX
ejpam-6106	49	27	not	not	PART
ejpam-6106	49	28	captured	capture	VERB
ejpam-6106	49	29	by	by	ADP
ejpam-6106	49	30	distance	distance	NOUN
ejpam-6106	49	31	-	-	PUNCT
ejpam-6106	49	32	based	base	VERB
ejpam-6106	49	33	or	or	CCONJ
ejpam-6106	49	34	order	order	NOUN
ejpam-6106	49	35	-	-	PUNCT
ejpam-6106	49	36	based	base	VERB
ejpam-6106	49	37	generalizations	generalization	NOUN
ejpam-6106	49	38	.	.	PUNCT
ejpam-6106	50	1	therefore	therefore	ADV
ejpam-6106	50	2	,	,	PUNCT
ejpam-6106	50	3	while	while	SCONJ
ejpam-6106	50	4	earlier	early	ADJ
ejpam-6106	50	5	results	result	NOUN
ejpam-6106	50	6	broadened	broaden	VERB
ejpam-6106	50	7	the	the	DET
ejpam-6106	50	8	applicability	applicability	NOUN
ejpam-6106	50	9	of	of	ADP
ejpam-6106	50	10	banach	banach	NOUN
ejpam-6106	50	11	’s	’s	PART
ejpam-6106	50	12	principle	principle	NOUN
ejpam-6106	50	13	by	by	ADP
ejpam-6106	50	14	altering	alter	VERB
ejpam-6106	50	15	the	the	DET
ejpam-6106	50	16	ambient	ambient	ADJ
ejpam-6106	50	17	space	space	NOUN
ejpam-6106	50	18	or	or	CCONJ
ejpam-6106	50	19	the	the	DET
ejpam-6106	50	20	contractive	contractive	ADJ
ejpam-6106	50	21	condition	condition	NOUN
ejpam-6106	50	22	,	,	PUNCT
ejpam-6106	50	23	the	the	DET
ejpam-6106	50	24	present	present	ADJ
ejpam-6106	50	25	work	work	NOUN
ejpam-6106	50	26	contributes	contribute	VERB
ejpam-6106	50	27	a	a	DET
ejpam-6106	50	28	complementary	complementary	ADJ
ejpam-6106	50	29	direction	direction	NOUN
ejpam-6106	50	30	by	by	ADP
ejpam-6106	50	31	showing	show	VERB
ejpam-6106	50	32	that	that	DET
ejpam-6106	50	33	geometric	geometric	ADJ
ejpam-6106	50	34	contraction	contraction	NOUN
ejpam-6106	50	35	of	of	ADP
ejpam-6106	50	36	polygons	polygon	NOUN
ejpam-6106	50	37	leads	lead	VERB
ejpam-6106	50	38	to	to	ADP
ejpam-6106	50	39	robust	robust	ADJ
ejpam-6106	50	40	fixed	fix	VERB
ejpam-6106	50	41	point	point	NOUN
ejpam-6106	50	42	results	result	NOUN
ejpam-6106	50	43	.	.	PUNCT
ejpam-6106	51	1	this	this	DET
ejpam-6106	51	2	highlights	highlight	VERB
ejpam-6106	51	3	the	the	DET
ejpam-6106	51	4	uniqueness	uniqueness	NOUN
ejpam-6106	51	5	of	of	ADP
ejpam-6106	51	6	our	our	PRON
ejpam-6106	51	7	contribution	contribution	NOUN
ejpam-6106	51	8	relative	relative	ADJ
ejpam-6106	51	9	to	to	ADP
ejpam-6106	51	10	other	other	ADJ
ejpam-6106	51	11	existing	exist	VERB
ejpam-6106	51	12	generalizations	generalization	NOUN
ejpam-6106	51	13	.	.	PUNCT
ejpam-6106	52	1	3	3	X
ejpam-6106	52	2	.	.	NOUN
ejpam-6106	52	3	mappings	mapping	NOUN
ejpam-6106	52	4	contracting	contract	VERB
ejpam-6106	52	5	perimeters	perimeter	NOUN
ejpam-6106	52	6	of	of	ADP
ejpam-6106	52	7	polygons	polygon	NOUN
ejpam-6106	52	8	in	in	ADP
ejpam-6106	52	9	this	this	DET
ejpam-6106	52	10	section	section	NOUN
ejpam-6106	52	11	,	,	PUNCT
ejpam-6106	52	12	we	we	PRON
ejpam-6106	52	13	recall	recall	VERB
ejpam-6106	52	14	some	some	DET
ejpam-6106	52	15	fundamental	fundamental	ADJ
ejpam-6106	52	16	notions	notion	NOUN
ejpam-6106	52	17	in	in	ADP
ejpam-6106	52	18	metric	metric	ADJ
ejpam-6106	52	19	fixed	fix	VERB
ejpam-6106	52	20	point	point	NOUN
ejpam-6106	52	21	theory	theory	NOUN
ejpam-6106	52	22	and	and	CCONJ
ejpam-6106	52	23	give	give	VERB
ejpam-6106	52	24	some	some	DET
ejpam-6106	52	25	properties	property	NOUN
ejpam-6106	52	26	of	of	ADP
ejpam-6106	52	27	mappings	mapping	NOUN
ejpam-6106	52	28	contracting	contract	VERB
ejpam-6106	52	29	perimeters	perimeter	NOUN
ejpam-6106	52	30	of	of	ADP
ejpam-6106	52	31	polygons	polygon	NOUN
ejpam-6106	52	32	.	.	PUNCT
ejpam-6106	53	1	definition	definition	NOUN
ejpam-6106	53	2	1	1	NUM
ejpam-6106	53	3	.	.	PUNCT
ejpam-6106	54	1	let	let	VERB
ejpam-6106	54	2	z	z	NOUN
ejpam-6106	54	3	be	be	AUX
ejpam-6106	54	4	any	any	DET
ejpam-6106	54	5	non	non	ADJ
ejpam-6106	54	6	-	-	ADJ
ejpam-6106	54	7	empty	empty	ADJ
ejpam-6106	54	8	set	set	NOUN
ejpam-6106	54	9	.	.	PUNCT
ejpam-6106	55	1	a	a	DET
ejpam-6106	55	2	function	function	NOUN
ejpam-6106	55	3	s	s	VERB
ejpam-6106	55	4	:	:	PUNCT
ejpam-6106	55	5	z	z	NOUN
ejpam-6106	55	6	×	×	NOUN
ejpam-6106	55	7	z	z	PROPN
ejpam-6106	56	1	→	→	SYM
ejpam-6106	56	2	r	r	NOUN
ejpam-6106	56	3	is	be	AUX
ejpam-6106	56	4	a	a	DET
ejpam-6106	56	5	metric	metric	NOUN
ejpam-6106	56	6	on	on	ADP
ejpam-6106	56	7	z	z	NOUN
ejpam-6106	56	8	if	if	SCONJ
ejpam-6106	56	9	it	it	PRON
ejpam-6106	56	10	satisfies	satisfy	VERB
ejpam-6106	56	11	the	the	DET
ejpam-6106	56	12	following	follow	VERB
ejpam-6106	56	13	axioms	axiom	NOUN
ejpam-6106	56	14	:	:	PUNCT
ejpam-6106	56	15	(	(	PUNCT
ejpam-6106	56	16	1	1	X
ejpam-6106	56	17	)	)	PUNCT
ejpam-6106	56	18	s(l1	s(l1	NOUN
ejpam-6106	56	19	,	,	PUNCT
ejpam-6106	56	20	l2	l2	NOUN
ejpam-6106	56	21	)	)	PUNCT
ejpam-6106	56	22	≥	≥	NOUN
ejpam-6106	56	23	0	0	NUM
ejpam-6106	56	24	;	;	PUNCT
ejpam-6106	56	25	(	(	PUNCT
ejpam-6106	56	26	2	2	X
ejpam-6106	56	27	)	)	PUNCT
ejpam-6106	56	28	s(l1	s(l1	NOUN
ejpam-6106	56	29	,	,	PUNCT
ejpam-6106	56	30	l2	l2	NOUN
ejpam-6106	56	31	)	)	PUNCT
ejpam-6106	57	1	=	=	SYM
ejpam-6106	57	2	0	0	NUM
ejpam-6106	57	3	⇐	⇐	ADJ
ejpam-6106	57	4	⇒	⇒	PROPN
ejpam-6106	57	5	l1	l1	PROPN
ejpam-6106	57	6	=	=	PUNCT
ejpam-6106	57	7	l2	l2	PROPN
ejpam-6106	57	8	;	;	PUNCT
ejpam-6106	57	9	(	(	PUNCT
ejpam-6106	57	10	3	3	X
ejpam-6106	57	11	)	)	PUNCT
ejpam-6106	57	12	s(l1	s(l1	NOUN
ejpam-6106	57	13	,	,	PUNCT
ejpam-6106	57	14	l2	l2	NOUN
ejpam-6106	57	15	)	)	PUNCT
ejpam-6106	57	16	=	=	SYM
ejpam-6106	57	17	s(l2	s(l2	PROPN
ejpam-6106	57	18	,	,	PUNCT
ejpam-6106	57	19	l1	l1	PROPN
ejpam-6106	57	20	)	)	PUNCT
ejpam-6106	57	21	for	for	ADP
ejpam-6106	57	22	all	all	DET
ejpam-6106	57	23	l1	l1	PROPN
ejpam-6106	57	24	,	,	PUNCT
ejpam-6106	57	25	l2	l2	NOUN
ejpam-6106	57	26	∈	∈	PROPN
ejpam-6106	57	27	z	z	NOUN
ejpam-6106	57	28	;	;	PUNCT
ejpam-6106	57	29	(	(	PUNCT
ejpam-6106	57	30	4	4	NUM
ejpam-6106	57	31	)	)	PUNCT
ejpam-6106	57	32	s(l1	s(l1	NOUN
ejpam-6106	57	33	,	,	PUNCT
ejpam-6106	57	34	l2	l2	NOUN
ejpam-6106	57	35	)	)	PUNCT
ejpam-6106	57	36	≤	≤	NUM
ejpam-6106	57	37	s(l1	s(l1	ADP
ejpam-6106	57	38	,	,	PUNCT
ejpam-6106	57	39	l	l	NOUN
ejpam-6106	57	40	)	)	PUNCT
ejpam-6106	57	41	+	+	SYM
ejpam-6106	57	42	s(l	s(l	X
ejpam-6106	57	43	,	,	PUNCT
ejpam-6106	57	44	l2	l2	NOUN
ejpam-6106	57	45	)	)	PUNCT
ejpam-6106	57	46	for	for	ADP
ejpam-6106	57	47	all	all	DET
ejpam-6106	57	48	l1	l1	PROPN
ejpam-6106	57	49	,	,	PUNCT
ejpam-6106	57	50	l2	l2	NOUN
ejpam-6106	57	51	,	,	PUNCT
ejpam-6106	57	52	l	l	PROPN
ejpam-6106	57	53	∈	∈	PROPN
ejpam-6106	57	54	z	z	NOUN
ejpam-6106	57	55	,	,	PUNCT
ejpam-6106	57	56	the	the	DET
ejpam-6106	57	57	pair	pair	NOUN
ejpam-6106	57	58	(	(	PUNCT
ejpam-6106	57	59	z	z	NOUN
ejpam-6106	57	60	,	,	PUNCT
ejpam-6106	57	61	s	s	PART
ejpam-6106	57	62	)	)	PUNCT
ejpam-6106	57	63	is	be	AUX
ejpam-6106	57	64	known	know	VERB
ejpam-6106	57	65	as	as	ADP
ejpam-6106	57	66	metric	metric	ADJ
ejpam-6106	57	67	space	space	NOUN
ejpam-6106	57	68	.	.	PUNCT
ejpam-6106	58	1	definition	definition	NOUN
ejpam-6106	58	2	2	2	NUM
ejpam-6106	58	3	.	.	PUNCT
ejpam-6106	59	1	let	let	VERB
ejpam-6106	59	2	|z|	|z|	NOUN
ejpam-6106	59	3	≥	≥	PRON
ejpam-6106	59	4	n	n	NOUN
ejpam-6106	60	1	and	and	CCONJ
ejpam-6106	60	2	(	(	PUNCT
ejpam-6106	60	3	z	z	NOUN
ejpam-6106	60	4	,	,	PUNCT
ejpam-6106	60	5	s	s	PART
ejpam-6106	60	6	)	)	PUNCT
ejpam-6106	60	7	be	be	AUX
ejpam-6106	60	8	a	a	DET
ejpam-6106	60	9	metric	metric	ADJ
ejpam-6106	60	10	space	space	NOUN
ejpam-6106	60	11	.	.	PUNCT
ejpam-6106	61	1	we	we	PRON
ejpam-6106	61	2	refer	refer	VERB
ejpam-6106	61	3	to	to	ADP
ejpam-6106	61	4	l	l	NOUN
ejpam-6106	61	5	:	:	PUNCT
ejpam-6106	61	6	z	z	X
ejpam-6106	61	7	→	→	SYM
ejpam-6106	61	8	z	z	NOUN
ejpam-6106	61	9	as	as	ADP
ejpam-6106	61	10	a	a	DET
ejpam-6106	61	11	mapping	mapping	NOUN
ejpam-6106	61	12	contracting	contracting	NOUN
ejpam-6106	61	13	perimeters	perimeter	NOUN
ejpam-6106	61	14	of	of	ADP
ejpam-6106	61	15	polygons	polygon	NOUN
ejpam-6106	61	16	on	on	ADP
ejpam-6106	61	17	z	z	PROPN
ejpam-6106	61	18	,	,	PUNCT
ejpam-6106	61	19	if	if	SCONJ
ejpam-6106	61	20	there	there	PRON
ejpam-6106	61	21	exists	exist	VERB
ejpam-6106	61	22	β	β	X
ejpam-6106	61	23	∈	∈	PROPN
ejpam-6106	62	1	[	[	X
ejpam-6106	62	2	0	0	NUM
ejpam-6106	62	3	,	,	PUNCT
ejpam-6106	62	4	1	1	NUM
ejpam-6106	62	5	)	)	PUNCT
ejpam-6106	62	6	such	such	ADJ
ejpam-6106	62	7	that	that	SCONJ
ejpam-6106	62	8	the	the	DET
ejpam-6106	62	9	inequality	inequality	NOUN
ejpam-6106	62	10	s	s	PART
ejpam-6106	62	11	(	(	PUNCT
ejpam-6106	62	12	ll1	ll1	ADV
ejpam-6106	62	13	,	,	PUNCT
ejpam-6106	62	14	ll2	ll2	NOUN
ejpam-6106	62	15	)	)	PUNCT
ejpam-6106	63	1	+	+	SYM
ejpam-6106	63	2	s	s	X
ejpam-6106	63	3	(	(	PUNCT
ejpam-6106	63	4	ll2	ll2	NOUN
ejpam-6106	63	5	,	,	PUNCT
ejpam-6106	63	6	ll3	ll3	NOUN
ejpam-6106	63	7	)	)	PUNCT
ejpam-6106	63	8	+	+	CCONJ
ejpam-6106	63	9	·	·	PUNCT
ejpam-6106	63	10	·	·	PUNCT
ejpam-6106	63	11	·	·	PUNCT
ejpam-6106	63	12	+	+	SYM
ejpam-6106	63	13	s	s	X
ejpam-6106	63	14	(	(	PUNCT
ejpam-6106	63	15	lln−1	lln−1	PROPN
ejpam-6106	63	16	,	,	PUNCT
ejpam-6106	63	17	lln	lln	PROPN
ejpam-6106	63	18	)	)	PUNCT
ejpam-6106	63	19	≤	≤	NUM
ejpam-6106	63	20	β	β	X
ejpam-6106	63	21	(	(	PUNCT
ejpam-6106	63	22	s	s	X
ejpam-6106	63	23	(	(	PUNCT
ejpam-6106	63	24	l1	l1	PROPN
ejpam-6106	63	25	,	,	PUNCT
ejpam-6106	63	26	l2	l2	NOUN
ejpam-6106	63	27	)	)	PUNCT
ejpam-6106	64	1	+	+	SYM
ejpam-6106	64	2	s	s	X
ejpam-6106	64	3	(	(	PUNCT
ejpam-6106	64	4	l2	l2	NOUN
ejpam-6106	64	5	,	,	PUNCT
ejpam-6106	64	6	l3	l3	PROPN
ejpam-6106	64	7	)	)	PUNCT
ejpam-6106	64	8	+	+	CCONJ
ejpam-6106	64	9	·	·	PUNCT
ejpam-6106	64	10	·	·	PUNCT
ejpam-6106	64	11	·	·	PUNCT
ejpam-6106	64	12	+	+	SYM
ejpam-6106	64	13	s	s	X
ejpam-6106	64	14	(	(	PUNCT
ejpam-6106	64	15	ln−1	ln−1	PROPN
ejpam-6106	64	16	,	,	PUNCT
ejpam-6106	64	17	ln	ln	ADJ
ejpam-6106	64	18	)	)	PUNCT
ejpam-6106	64	19	)	)	PUNCT
ejpam-6106	64	20	(	(	PUNCT
ejpam-6106	64	21	1	1	X
ejpam-6106	64	22	)	)	PUNCT
ejpam-6106	64	23	holds	hold	VERB
ejpam-6106	64	24	for	for	ADP
ejpam-6106	64	25	every	every	DET
ejpam-6106	64	26	n	n	NUM
ejpam-6106	64	27	points	point	NOUN
ejpam-6106	64	28	l1	l1	PROPN
ejpam-6106	64	29	,	,	PUNCT
ejpam-6106	64	30	l2	l2	NOUN
ejpam-6106	64	31	,	,	PUNCT
ejpam-6106	64	32	l3	l3	PROPN
ejpam-6106	64	33	,	,	PUNCT
ejpam-6106	64	34	·	·	PUNCT
ejpam-6106	64	35	·	·	PUNCT
ejpam-6106	64	36	·	·	PUNCT
ejpam-6106	64	37	,	,	PUNCT
ejpam-6106	64	38	ln	ln	PROPN
ejpam-6106	64	39	∈	∈	PROPN
ejpam-6106	64	40	z.	z.	PROPN
ejpam-6106	64	41	proposition	proposition	NOUN
ejpam-6106	64	42	3	3	NUM
ejpam-6106	64	43	.	.	PUNCT
ejpam-6106	64	44	mappings	mapping	NOUN
ejpam-6106	64	45	contracting	contract	VERB
ejpam-6106	64	46	perimeters	perimeter	NOUN
ejpam-6106	64	47	of	of	ADP
ejpam-6106	64	48	polygons	polygon	NOUN
ejpam-6106	64	49	are	be	AUX
ejpam-6106	64	50	continuous	continuous	ADJ
ejpam-6106	64	51	.	.	PUNCT
ejpam-6106	65	1	proof	proof	NOUN
ejpam-6106	65	2	.	.	PUNCT
ejpam-6106	66	1	given	give	VERB
ejpam-6106	66	2	a	a	DET
ejpam-6106	66	3	metric	metric	ADJ
ejpam-6106	66	4	space	space	NOUN
ejpam-6106	66	5	(	(	PUNCT
ejpam-6106	66	6	z	z	NOUN
ejpam-6106	66	7	,	,	PUNCT
ejpam-6106	66	8	s	s	NOUN
ejpam-6106	66	9	)	)	PUNCT
ejpam-6106	66	10	,	,	PUNCT
ejpam-6106	66	11	and	and	CCONJ
ejpam-6106	66	12	|z|	|z|	NOUN
ejpam-6106	66	13	≥	≥	NOUN
ejpam-6106	66	14	n	n	CCONJ
ejpam-6106	66	15	,	,	PUNCT
ejpam-6106	66	16	and	and	CCONJ
ejpam-6106	66	17	l	l	NOUN
ejpam-6106	66	18	:	:	PUNCT
ejpam-6106	66	19	z	z	X
ejpam-6106	66	20	→	→	SYM
ejpam-6106	66	21	z	z	X
ejpam-6106	66	22	a	a	DET
ejpam-6106	66	23	mapping	mapping	NOUN
ejpam-6106	66	24	contracting	contracting	NOUN
ejpam-6106	66	25	perimeters	perimeter	NOUN
ejpam-6106	66	26	of	of	ADP
ejpam-6106	66	27	polygons	polygon	NOUN
ejpam-6106	66	28	on	on	ADP
ejpam-6106	66	29	z	z	NOUN
ejpam-6106	66	30	and	and	CCONJ
ejpam-6106	66	31	consider	consider	VERB
ejpam-6106	66	32	l0	l0	PROPN
ejpam-6106	66	33	as	as	ADP
ejpam-6106	66	34	an	an	DET
ejpam-6106	66	35	isolated	isolated	ADJ
ejpam-6106	66	36	point	point	NOUN
ejpam-6106	66	37	in	in	ADP
ejpam-6106	66	38	z.	z.	PROPN
ejpam-6106	66	39	the	the	DET
ejpam-6106	66	40	mapping	mapping	NOUN
ejpam-6106	66	41	l	l	NOUN
ejpam-6106	66	42	is	be	AUX
ejpam-6106	66	43	m.	m.	NOUN
ejpam-6106	66	44	nazam	nazam	PROPN
ejpam-6106	66	45	,	,	PUNCT
ejpam-6106	66	46	u.	u.	PROPN
ejpam-6106	66	47	habiba	habiba	PROPN
ejpam-6106	66	48	,	,	PUNCT
ejpam-6106	66	49	m.	m.	PROPN
ejpam-6106	66	50	de	de	X
ejpam-6106	66	51	la	la	PROPN
ejpam-6106	66	52	sen	sen	PROPN
ejpam-6106	66	53	/	/	SYM
ejpam-6106	66	54	eur	eur	PROPN
ejpam-6106	66	55	.	.	PUNCT
ejpam-6106	67	1	j.	j.	PROPN
ejpam-6106	67	2	pure	pure	PROPN
ejpam-6106	67	3	appl	appl	PROPN
ejpam-6106	67	4	.	.	PROPN
ejpam-6106	67	5	math	math	PROPN
ejpam-6106	67	6	,	,	PUNCT
ejpam-6106	67	7	18	18	NUM
ejpam-6106	67	8	(	(	PUNCT
ejpam-6106	67	9	4	4	NUM
ejpam-6106	67	10	)	)	PUNCT
ejpam-6106	67	11	(	(	PUNCT
ejpam-6106	67	12	2025	2025	NUM
ejpam-6106	67	13	)	)	PUNCT
ejpam-6106	67	14	,	,	PUNCT
ejpam-6106	67	15	6106	6106	NUM
ejpam-6106	67	16	4	4	NUM
ejpam-6106	67	17	of	of	ADP
ejpam-6106	67	18	10	10	NUM
ejpam-6106	67	19	thus	thus	ADV
ejpam-6106	67	20	obviously	obviously	ADV
ejpam-6106	67	21	continuous	continuous	ADJ
ejpam-6106	67	22	at	at	ADP
ejpam-6106	67	23	l0	l0	PROPN
ejpam-6106	67	24	.	.	PUNCT
ejpam-6106	68	1	it	it	PRON
ejpam-6106	68	2	can	can	AUX
ejpam-6106	68	3	be	be	AUX
ejpam-6106	68	4	shown	show	VERB
ejpam-6106	68	5	that	that	SCONJ
ejpam-6106	68	6	for	for	ADP
ejpam-6106	68	7	each	each	DET
ejpam-6106	68	8	ϵ	ϵ	PRON
ejpam-6106	68	9	≥	≥	NOUN
ejpam-6106	68	10	0	0	NUM
ejpam-6106	68	11	,	,	PUNCT
ejpam-6106	68	12	there	there	PRON
ejpam-6106	68	13	exists	exist	VERB
ejpam-6106	68	14	a	a	DET
ejpam-6106	68	15	δ	δ	PROPN
ejpam-6106	68	16	≥	≥	X
ejpam-6106	68	17	0	0	NUM
ejpam-6106	68	18	such	such	ADJ
ejpam-6106	68	19	that	that	DET
ejpam-6106	68	20	s	s	X
ejpam-6106	68	21	(	(	PUNCT
ejpam-6106	68	22	ll0	ll0	ADJ
ejpam-6106	68	23	,	,	PUNCT
ejpam-6106	68	24	ll	ll	NOUN
ejpam-6106	68	25	)	)	PUNCT
ejpam-6106	68	26	≤	≤	NOUN
ejpam-6106	69	1	ϵ	ϵ	ADP
ejpam-6106	69	2	,	,	PUNCT
ejpam-6106	69	3	whenever	whenever	SCONJ
ejpam-6106	69	4	s	s	X
ejpam-6106	69	5	(	(	PUNCT
ejpam-6106	69	6	l0	l0	PROPN
ejpam-6106	69	7	,	,	PUNCT
ejpam-6106	69	8	l	l	NOUN
ejpam-6106	69	9	)	)	PUNCT
ejpam-6106	69	10	≤	≤	NUM
ejpam-6106	69	11	δ	δ	PROPN
ejpam-6106	69	12	.	.	PUNCT
ejpam-6106	69	13	given	give	VERB
ejpam-6106	69	14	l0	l0	PROPN
ejpam-6106	69	15	as	as	ADP
ejpam-6106	69	16	an	an	DET
ejpam-6106	69	17	accumulation	accumulation	NOUN
ejpam-6106	69	18	point	point	NOUN
ejpam-6106	69	19	.	.	PUNCT
ejpam-6106	70	1	therefore	therefore	ADV
ejpam-6106	70	2	,	,	PUNCT
ejpam-6106	70	3	for	for	ADP
ejpam-6106	70	4	every	every	DET
ejpam-6106	70	5	δ	δ	PROPN
ejpam-6106	70	6	≥	≥	NUM
ejpam-6106	70	7	0	0	NUM
ejpam-6106	70	8	,	,	PUNCT
ejpam-6106	70	9	∃	∃	PROPN
ejpam-6106	70	10	l	l	PROPN
ejpam-6106	70	11	∈	∈	PROPN
ejpam-6106	70	12	z	z	NOUN
ejpam-6106	70	13	such	such	ADJ
ejpam-6106	70	14	that	that	SCONJ
ejpam-6106	70	15	s(li	s(li	NOUN
ejpam-6106	70	16	,	,	PUNCT
ejpam-6106	70	17	l	l	NOUN
ejpam-6106	70	18	)	)	PUNCT
ejpam-6106	70	19	≤	≤	NUM
ejpam-6106	70	20	δ	δ	PROPN
ejpam-6106	70	21	for	for	ADP
ejpam-6106	70	22	all	all	PRON
ejpam-6106	70	23	i	i	PRON
ejpam-6106	70	24	≥	≥	VERB
ejpam-6106	70	25	0	0	NUM
ejpam-6106	71	1	s(ll0	s(ll0	ADJ
ejpam-6106	71	2	,	,	PUNCT
ejpam-6106	71	3	ll	ll	NOUN
ejpam-6106	71	4	)	)	PUNCT
ejpam-6106	71	5	≤	≤	NOUN
ejpam-6106	71	6	s(ll0	s(ll0	NOUN
ejpam-6106	71	7	,	,	PUNCT
ejpam-6106	71	8	ll	ll	NOUN
ejpam-6106	71	9	)	)	PUNCT
ejpam-6106	71	10	+	+	CCONJ
ejpam-6106	71	11	s(ll	s(ll	NOUN
ejpam-6106	71	12	,	,	PUNCT
ejpam-6106	71	13	ll1	ll1	ADV
ejpam-6106	71	14	)	)	PUNCT
ejpam-6106	71	15	+	+	CCONJ
ejpam-6106	71	16	s(ll1	s(ll1	ADJ
ejpam-6106	71	17	,	,	PUNCT
ejpam-6106	71	18	ll2	ll2	NOUN
ejpam-6106	71	19	)	)	PUNCT
ejpam-6106	71	20	+	+	CCONJ
ejpam-6106	71	21	·	·	PUNCT
ejpam-6106	71	22	·	·	PUNCT
ejpam-6106	71	23	·	·	PUNCT
ejpam-6106	71	24	+	+	CCONJ
ejpam-6106	71	25	s(lln−1	s(lln−1	ADJ
ejpam-6106	71	26	,	,	PUNCT
ejpam-6106	71	27	ll0	ll0	NOUN
ejpam-6106	71	28	)	)	PUNCT
ejpam-6106	71	29	≤	≤	NOUN
ejpam-6106	71	30	β(s(lo	β(s(lo	NOUN
ejpam-6106	71	31	,	,	PUNCT
ejpam-6106	71	32	l	l	NOUN
ejpam-6106	71	33	)	)	PUNCT
ejpam-6106	72	1	+	+	SYM
ejpam-6106	72	2	s(l	s(l	X
ejpam-6106	72	3	,	,	PUNCT
ejpam-6106	72	4	l1	l1	PROPN
ejpam-6106	72	5	)	)	PUNCT
ejpam-6106	73	1	+	+	CCONJ
ejpam-6106	73	2	s(l1	s(l1	PROPN
ejpam-6106	73	3	,	,	PUNCT
ejpam-6106	73	4	l2	l2	NOUN
ejpam-6106	73	5	)	)	PUNCT
ejpam-6106	74	1	+	+	CCONJ
ejpam-6106	75	1	·	·	PUNCT
ejpam-6106	75	2	·	·	PUNCT
ejpam-6106	75	3	·	·	PUNCT
ejpam-6106	75	4	+	+	X
ejpam-6106	75	5	s(ln−1	s(ln−1	PROPN
ejpam-6106	75	6	,	,	PUNCT
ejpam-6106	75	7	l0	l0	PROPN
ejpam-6106	75	8	)	)	PUNCT
ejpam-6106	75	9	)	)	PUNCT
ejpam-6106	75	10	≤	≤	PUNCT
ejpam-6106	76	1	β(s(l0	β(s(l0	PROPN
ejpam-6106	76	2	,	,	PUNCT
ejpam-6106	76	3	l	l	NOUN
ejpam-6106	76	4	)	)	PUNCT
ejpam-6106	76	5	+	+	SYM
ejpam-6106	76	6	s(l	s(l	X
ejpam-6106	76	7	,	,	PUNCT
ejpam-6106	76	8	l0	l0	PROPN
ejpam-6106	76	9	)	)	PUNCT
ejpam-6106	76	10	+	+	CCONJ
ejpam-6106	76	11	s(l0	s(l0	NOUN
ejpam-6106	76	12	,	,	PUNCT
ejpam-6106	76	13	l1	l1	PROPN
ejpam-6106	76	14	)	)	PUNCT
ejpam-6106	76	15	+	+	CCONJ
ejpam-6106	76	16	s(l1	s(l1	PROPN
ejpam-6106	76	17	,	,	PUNCT
ejpam-6106	76	18	l0	l0	PROPN
ejpam-6106	76	19	)	)	PUNCT
ejpam-6106	76	20	+	+	CCONJ
ejpam-6106	76	21	s(l0	s(l0	ADJ
ejpam-6106	76	22	,	,	PUNCT
ejpam-6106	76	23	l2	l2	NOUN
ejpam-6106	76	24	)	)	PUNCT
ejpam-6106	76	25	+	+	CCONJ
ejpam-6106	76	26	·	·	PUNCT
ejpam-6106	76	27	·	·	PUNCT
ejpam-6106	76	28	·	·	PUNCT
ejpam-6106	76	29	+	+	NUM
ejpam-6106	76	30	s(ln−2	s(ln−2	PROPN
ejpam-6106	76	31	,	,	PUNCT
ejpam-6106	76	32	l0	l0	PROPN
ejpam-6106	76	33	)	)	PUNCT
ejpam-6106	76	34	)	)	PUNCT
ejpam-6106	76	35	≤	≤	NOUN
ejpam-6106	76	36	2β(n−	2β(n−	NUM
ejpam-6106	76	37	1)δ	1)δ	NUM
ejpam-6106	76	38	.	.	PUNCT
ejpam-6106	77	1	set	set	VERB
ejpam-6106	77	2	δ	δ	X
ejpam-6106	77	3	=	=	PUNCT
ejpam-6106	77	4	ϵ/2(n−	ϵ/2(n−	PROPN
ejpam-6106	77	5	1)β	1)β	NUM
ejpam-6106	77	6	,	,	PUNCT
ejpam-6106	77	7	the	the	DET
ejpam-6106	77	8	desired	desire	VERB
ejpam-6106	77	9	inequality	inequality	NOUN
ejpam-6106	77	10	is	be	AUX
ejpam-6106	77	11	obtained	obtain	VERB
ejpam-6106	77	12	.	.	PUNCT
ejpam-6106	78	1	theorem	theorem	ADJ
ejpam-6106	78	2	4	4	NUM
ejpam-6106	78	3	.	.	PUNCT
ejpam-6106	78	4	given	give	VERB
ejpam-6106	78	5	a	a	DET
ejpam-6106	78	6	complete	complete	ADJ
ejpam-6106	78	7	metric	metric	ADJ
ejpam-6106	78	8	space	space	NOUN
ejpam-6106	78	9	(	(	PUNCT
ejpam-6106	78	10	z	z	NOUN
ejpam-6106	78	11	,	,	PUNCT
ejpam-6106	78	12	s	s	PART
ejpam-6106	78	13	)	)	PUNCT
ejpam-6106	78	14	,	,	PUNCT
ejpam-6106	78	15	|z|	|z|	VERB
ejpam-6106	78	16	≥	≥	NOUN
ejpam-6106	78	17	n	n	CCONJ
ejpam-6106	78	18	,	,	PUNCT
ejpam-6106	78	19	if	if	SCONJ
ejpam-6106	78	20	the	the	DET
ejpam-6106	78	21	mapping	mapping	NOUN
ejpam-6106	78	22	l	l	NOUN
ejpam-6106	78	23	:	:	PUNCT
ejpam-6106	78	24	z	z	X
ejpam-6106	78	25	→	→	SYM
ejpam-6106	78	26	z	z	AUX
ejpam-6106	78	27	satisfy	satisfy	VERB
ejpam-6106	78	28	the	the	DET
ejpam-6106	78	29	following	following	ADJ
ejpam-6106	78	30	conditions	condition	NOUN
ejpam-6106	78	31	.	.	PUNCT
ejpam-6106	79	1	(	(	PUNCT
ejpam-6106	79	2	i	i	NOUN
ejpam-6106	79	3	)	)	PUNCT
ejpam-6106	79	4	l(l(l	l(l(l	PROPN
ejpam-6106	79	5	)	)	PUNCT
ejpam-6106	79	6	)	)	PUNCT
ejpam-6106	80	1	̸=	̸=	PROPN
ejpam-6106	80	2	l	l	NOUN
ejpam-6106	80	3	∀	∀	X
ejpam-6106	80	4	l	l	NOUN
ejpam-6106	80	5	∈	∈	PROPN
ejpam-6106	80	6	z	z	NOUN
ejpam-6106	80	7	such	such	ADJ
ejpam-6106	80	8	that	that	SCONJ
ejpam-6106	80	9	l(l	l(l	NOUN
ejpam-6106	80	10	)	)	PUNCT
ejpam-6106	80	11	̸=	̸=	PROPN
ejpam-6106	80	12	l.	l.	PROPN
ejpam-6106	80	13	(	(	PUNCT
ejpam-6106	80	14	ii	ii	PROPN
ejpam-6106	80	15	)	)	PUNCT
ejpam-6106	80	16	l	l	NOUN
ejpam-6106	80	17	is	be	AUX
ejpam-6106	80	18	a	a	DET
ejpam-6106	80	19	mapping	mapping	NOUN
ejpam-6106	80	20	contracting	contracting	NOUN
ejpam-6106	80	21	perimeters	perimeter	NOUN
ejpam-6106	80	22	of	of	ADP
ejpam-6106	80	23	polygons	polygon	NOUN
ejpam-6106	80	24	.	.	PUNCT
ejpam-6106	81	1	then	then	ADV
ejpam-6106	81	2	it	it	PRON
ejpam-6106	81	3	admits	admit	VERB
ejpam-6106	81	4	a	a	DET
ejpam-6106	81	5	fixed	fixed	ADJ
ejpam-6106	81	6	point	point	NOUN
ejpam-6106	81	7	,	,	PUNCT
ejpam-6106	81	8	moreover	moreover	ADV
ejpam-6106	81	9	,	,	PUNCT
ejpam-6106	81	10	maximum	maximum	ADJ
ejpam-6106	81	11	number	number	NOUN
ejpam-6106	81	12	of	of	ADP
ejpam-6106	81	13	fixed	fix	VERB
ejpam-6106	81	14	points	point	NOUN
ejpam-6106	81	15	is	be	AUX
ejpam-6106	81	16	(	(	PUNCT
ejpam-6106	81	17	n−	n−	NOUN
ejpam-6106	81	18	1	1	NUM
ejpam-6106	81	19	)	)	PUNCT
ejpam-6106	81	20	.	.	PUNCT
ejpam-6106	82	1	proof	proof	NOUN
ejpam-6106	82	2	.	.	PUNCT
ejpam-6106	83	1	let	let	VERB
ejpam-6106	83	2	l0	l0	PROPN
ejpam-6106	83	3	∈	∈	PROPN
ejpam-6106	83	4	z	z	PROPN
ejpam-6106	83	5	,	,	PUNCT
ejpam-6106	83	6	ll0	ll0	PROPN
ejpam-6106	83	7	=	=	SYM
ejpam-6106	83	8	l1	l1	PROPN
ejpam-6106	83	9	,	,	PUNCT
ejpam-6106	83	10	ll1	ll1	NOUN
ejpam-6106	83	11	=	=	SYM
ejpam-6106	83	12	l2	l2	NOUN
ejpam-6106	83	13	,	,	PUNCT
ejpam-6106	83	14	·	·	PUNCT
ejpam-6106	83	15	·	·	PUNCT
ejpam-6106	83	16	·	·	PUNCT
ejpam-6106	83	17	,	,	PUNCT
ejpam-6106	83	18	lln	lln	PROPN
ejpam-6106	83	19	=	=	SYM
ejpam-6106	83	20	ln+1	ln+1	PROPN
ejpam-6106	83	21	.	.	PUNCT
ejpam-6106	84	1	let	let	VERB
ejpam-6106	84	2	li	li	PROPN
ejpam-6106	84	3	̸=	̸=	PROPN
ejpam-6106	84	4	l(li	l(li	VERB
ejpam-6106	84	5	)	)	PUNCT
ejpam-6106	84	6	for	for	ADP
ejpam-6106	84	7	each	each	DET
ejpam-6106	84	8	i	i	NOUN
ejpam-6106	84	9	=	=	NOUN
ejpam-6106	84	10	0	0	NUM
ejpam-6106	84	11	,	,	PUNCT
ejpam-6106	84	12	1	1	NUM
ejpam-6106	84	13	,	,	PUNCT
ejpam-6106	84	14	2	2	NUM
ejpam-6106	84	15	,	,	PUNCT
ejpam-6106	84	16	....	....	PUNCT
ejpam-6106	85	1	so	so	SCONJ
ejpam-6106	85	2	that	that	SCONJ
ejpam-6106	85	3	,	,	PUNCT
ejpam-6106	85	4	we	we	PRON
ejpam-6106	85	5	have	have	VERB
ejpam-6106	85	6	li	li	PROPN
ejpam-6106	85	7	̸=	̸=	PROPN
ejpam-6106	85	8	li+1	li+1	NOUN
ejpam-6106	85	9	=	=	SYM
ejpam-6106	85	10	lli	lli	PROPN
ejpam-6106	85	11	.	.	PUNCT
ejpam-6106	86	1	as	as	ADP
ejpam-6106	86	2	li+2	li+2	NUM
ejpam-6106	86	3	=	=	SYM
ejpam-6106	86	4	l(lli	l(lli	ADJ
ejpam-6106	86	5	)	)	PUNCT
ejpam-6106	86	6	̸=	̸=	PROPN
ejpam-6106	86	7	li	li	PROPN
ejpam-6106	86	8	,	,	PUNCT
ejpam-6106	86	9	by	by	ADP
ejpam-6106	86	10	condition	condition	NOUN
ejpam-6106	86	11	(	(	PUNCT
ejpam-6106	86	12	i	i	NOUN
ejpam-6106	86	13	)	)	PUNCT
ejpam-6106	86	14	and	and	CCONJ
ejpam-6106	86	15	assuming	assume	VERB
ejpam-6106	86	16	that	that	SCONJ
ejpam-6106	86	17	li+1	li+1	PROPN
ejpam-6106	86	18	is	be	AUX
ejpam-6106	86	19	not	not	PART
ejpam-6106	86	20	fixed	fix	VERB
ejpam-6106	86	21	,	,	PUNCT
ejpam-6106	86	22	we	we	PRON
ejpam-6106	86	23	obtain	obtain	VERB
ejpam-6106	86	24	li+1	li+1	NOUN
ejpam-6106	86	25	̸=	̸=	PROPN
ejpam-6106	86	26	li+2	li+2	NUM
ejpam-6106	86	27	=	=	SYM
ejpam-6106	86	28	lli+1	lli+1	PROPN
ejpam-6106	86	29	.	.	PUNCT
ejpam-6106	87	1	continuing	continue	VERB
ejpam-6106	87	2	the	the	DET
ejpam-6106	87	3	process	process	NOUN
ejpam-6106	87	4	,	,	PUNCT
ejpam-6106	87	5	we	we	PRON
ejpam-6106	87	6	have	have	VERB
ejpam-6106	87	7	li	li	PROPN
ejpam-6106	87	8	,	,	PUNCT
ejpam-6106	87	9	li+1	li+1	PROPN
ejpam-6106	87	10	,	,	PUNCT
ejpam-6106	87	11	·	·	PUNCT
ejpam-6106	87	12	·	·	PUNCT
ejpam-6106	87	13	·	·	PUNCT
ejpam-6106	87	14	,	,	PUNCT
ejpam-6106	87	15	li+n	li+n	PROPN
ejpam-6106	87	16	pairwise	pairwise	NOUN
ejpam-6106	87	17	distinct	distinct	NOUN
ejpam-6106	87	18	.	.	PUNCT
ejpam-6106	88	1	moreover	moreover	ADV
ejpam-6106	88	2	,	,	PUNCT
ejpam-6106	88	3	set	set	VERB
ejpam-6106	88	4	p0	p0	NOUN
ejpam-6106	88	5	=	=	SYM
ejpam-6106	88	6	s(l0	s(l0	NOUN
ejpam-6106	88	7	,	,	PUNCT
ejpam-6106	88	8	l1	l1	PROPN
ejpam-6106	88	9	)	)	PUNCT
ejpam-6106	89	1	+	+	CCONJ
ejpam-6106	89	2	s(l1	s(l1	PROPN
ejpam-6106	89	3	,	,	PUNCT
ejpam-6106	89	4	l2	l2	NOUN
ejpam-6106	89	5	)	)	PUNCT
ejpam-6106	90	1	+	+	CCONJ
ejpam-6106	90	2	s(l2	s(l2	PROPN
ejpam-6106	90	3	,	,	PUNCT
ejpam-6106	90	4	l3	l3	PROPN
ejpam-6106	90	5	)	)	PUNCT
ejpam-6106	91	1	+	+	CCONJ
ejpam-6106	91	2	·	·	PUNCT
ejpam-6106	91	3	·	·	PUNCT
ejpam-6106	91	4	·	·	PUNCT
ejpam-6106	91	5	+	+	X
ejpam-6106	91	6	s(ln−1	s(ln−1	PROPN
ejpam-6106	91	7	,	,	PUNCT
ejpam-6106	91	8	l0	l0	PROPN
ejpam-6106	91	9	)	)	PUNCT
ejpam-6106	91	10	p1	p1	NOUN
ejpam-6106	91	11	=	=	SYM
ejpam-6106	91	12	s(l1	s(l1	PROPN
ejpam-6106	91	13	,	,	PUNCT
ejpam-6106	91	14	l2	l2	NOUN
ejpam-6106	91	15	)	)	PUNCT
ejpam-6106	92	1	+	+	CCONJ
ejpam-6106	92	2	s(l2	s(l2	PROPN
ejpam-6106	92	3	,	,	PUNCT
ejpam-6106	92	4	l3	l3	PROPN
ejpam-6106	92	5	)	)	PUNCT
ejpam-6106	92	6	+	+	CCONJ
ejpam-6106	92	7	s(l3	s(l3	PROPN
ejpam-6106	92	8	,	,	PUNCT
ejpam-6106	92	9	l4	l4	PROPN
ejpam-6106	92	10	)	)	PUNCT
ejpam-6106	92	11	+	+	CCONJ
ejpam-6106	92	12	·	·	PUNCT
ejpam-6106	92	13	·	·	PUNCT
ejpam-6106	92	14	·	·	PUNCT
ejpam-6106	92	15	+	+	NUM
ejpam-6106	92	16	s(ln	s(ln	NOUN
ejpam-6106	92	17	,	,	PUNCT
ejpam-6106	92	18	l1	l1	PROPN
ejpam-6106	92	19	)	)	PUNCT
ejpam-6106	92	20	·	·	PUNCT
ejpam-6106	92	21	·	·	PUNCT
ejpam-6106	92	22	·	·	PUNCT
ejpam-6106	92	23	pk	pk	NOUN
ejpam-6106	92	24	=	=	SYM
ejpam-6106	92	25	s(lk	s(lk	PROPN
ejpam-6106	92	26	,	,	PUNCT
ejpam-6106	92	27	lk+1	lk+1	NUM
ejpam-6106	92	28	)	)	PUNCT
ejpam-6106	92	29	+	+	NUM
ejpam-6106	92	30	s(lk+1	s(lk+1	NOUN
ejpam-6106	92	31	,	,	PUNCT
ejpam-6106	92	32	lk+2	lk+2	NOUN
ejpam-6106	92	33	)	)	PUNCT
ejpam-6106	92	34	+	+	CCONJ
ejpam-6106	92	35	·	·	PUNCT
ejpam-6106	92	36	·	·	PUNCT
ejpam-6106	92	37	·	·	PUNCT
ejpam-6106	92	38	+	+	NUM
ejpam-6106	92	39	s(ln+(k−1	s(ln+(k−1	NOUN
ejpam-6106	92	40	)	)	PUNCT
ejpam-6106	92	41	,	,	PUNCT
ejpam-6106	92	42	lk	lk	PROPN
ejpam-6106	92	43	)	)	PUNCT
ejpam-6106	92	44	,	,	PUNCT
ejpam-6106	92	45	by	by	ADP
ejpam-6106	92	46	(	(	PUNCT
ejpam-6106	92	47	1	1	X
ejpam-6106	92	48	)	)	PUNCT
ejpam-6106	92	49	we	we	PRON
ejpam-6106	92	50	have	have	AUX
ejpam-6106	92	51	,	,	PUNCT
ejpam-6106	92	52	p1	p1	VERB
ejpam-6106	92	53	≤	≤	NUM
ejpam-6106	92	54	βp0	βp0	PROPN
ejpam-6106	92	55	,	,	PUNCT
ejpam-6106	92	56	p2	p2	VERB
ejpam-6106	92	57	≤	≤	NUM
ejpam-6106	92	58	βp1	βp1	NOUN
ejpam-6106	92	59	,	,	PUNCT
ejpam-6106	92	60	·	·	PUNCT
ejpam-6106	92	61	·	·	PUNCT
ejpam-6106	92	62	·	·	PUNCT
ejpam-6106	93	1	pk	pk	NOUN
ejpam-6106	93	2	≤	≤	X
ejpam-6106	93	3	βpk−1	βpk−1	PROPN
ejpam-6106	93	4	and	and	CCONJ
ejpam-6106	93	5	p0	p0	NOUN
ejpam-6106	93	6	>	>	X
ejpam-6106	93	7	p1	p1	PROPN
ejpam-6106	93	8	>	>	X
ejpam-6106	93	9	p2	p2	X
ejpam-6106	93	10	·	·	PUNCT
ejpam-6106	93	11	·	·	PUNCT
ejpam-6106	93	12	·	·	PUNCT
ejpam-6106	93	13	>	>	PUNCT
ejpam-6106	93	14	pk	pk	X
ejpam-6106	93	15	>	>	X
ejpam-6106	93	16	·	·	PUNCT
ejpam-6106	93	17	·	·	PUNCT
ejpam-6106	93	18	·	·	PUNCT
ejpam-6106	93	19	.	.	PUNCT
ejpam-6106	94	1	(	(	PUNCT
ejpam-6106	94	2	2	2	X
ejpam-6106	94	3	)	)	PUNCT
ejpam-6106	94	4	assume	assume	VERB
ejpam-6106	94	5	that	that	SCONJ
ejpam-6106	94	6	j	j	PROPN
ejpam-6106	94	7	≥	≥	PRON
ejpam-6106	94	8	n	n	X
ejpam-6106	94	9	is	be	AUX
ejpam-6106	94	10	such	such	ADJ
ejpam-6106	94	11	that	that	SCONJ
ejpam-6106	94	12	lj	lj	PROPN
ejpam-6106	94	13	=	=	SYM
ejpam-6106	94	14	li	li	PROPN
ejpam-6106	94	15	for	for	ADP
ejpam-6106	94	16	0	0	NUM
ejpam-6106	94	17	≤	≤	NUM
ejpam-6106	95	1	i	i	PRON
ejpam-6106	95	2	≤	≤	NUM
ejpam-6106	96	1	j	j	PROPN
ejpam-6106	96	2	−	−	PROPN
ejpam-6106	96	3	(	(	PUNCT
ejpam-6106	96	4	n	n	CCONJ
ejpam-6106	96	5	−	−	PROPN
ejpam-6106	96	6	1	1	NUM
ejpam-6106	96	7	)	)	PUNCT
ejpam-6106	96	8	then	then	ADV
ejpam-6106	96	9	lj+1	lj+1	ADJ
ejpam-6106	96	10	=	=	SYM
ejpam-6106	96	11	li+1	li+1	NOUN
ejpam-6106	96	12	,	,	PUNCT
ejpam-6106	96	13	lj+2	lj+2	ADJ
ejpam-6106	96	14	=	=	SYM
ejpam-6106	96	15	li+2	li+2	NOUN
ejpam-6106	96	16	,	,	PUNCT
ejpam-6106	96	17	lj+3	lj+3	PROPN
ejpam-6106	96	18	=	=	SYM
ejpam-6106	96	19	li+3	li+3	X
ejpam-6106	96	20	,	,	PUNCT
ejpam-6106	96	21	·	·	PUNCT
ejpam-6106	96	22	·	·	PUNCT
ejpam-6106	96	23	·	·	PUNCT
ejpam-6106	96	24	.	.	PUNCT
ejpam-6106	97	1	thus	thus	ADV
ejpam-6106	97	2	,	,	PUNCT
ejpam-6106	97	3	pi	pi	NOUN
ejpam-6106	97	4	=	=	SYM
ejpam-6106	97	5	pj	pj	PROPN
ejpam-6106	97	6	,	,	PUNCT
ejpam-6106	97	7	which	which	PRON
ejpam-6106	97	8	defies	defy	VERB
ejpam-6106	97	9	(	(	PUNCT
ejpam-6106	97	10	2	2	NUM
ejpam-6106	97	11	)	)	PUNCT
ejpam-6106	97	12	.	.	PUNCT
ejpam-6106	97	13	let	let	VERB
ejpam-6106	97	14	’s	’s	PRON
ejpam-6106	97	15	verify	verify	VERB
ejpam-6106	97	16	that	that	SCONJ
ejpam-6106	97	17	the	the	DET
ejpam-6106	97	18	{	{	PUNCT
ejpam-6106	97	19	li	li	NOUN
ejpam-6106	97	20	}	}	PUNCT
ejpam-6106	97	21	is	be	AUX
ejpam-6106	97	22	a	a	DET
ejpam-6106	97	23	cauchy	cauchy	ADJ
ejpam-6106	97	24	sequence	sequence	NOUN
ejpam-6106	97	25	.	.	PUNCT
ejpam-6106	98	1	clearly	clearly	ADV
ejpam-6106	98	2	,	,	PUNCT
ejpam-6106	98	3	s(l1	s(l1	INTJ
ejpam-6106	98	4	,	,	PUNCT
ejpam-6106	98	5	l2	l2	NOUN
ejpam-6106	98	6	)	)	PUNCT
ejpam-6106	98	7	≤	≤	NOUN
ejpam-6106	98	8	p0	p0	NOUN
ejpam-6106	98	9	,	,	PUNCT
ejpam-6106	98	10	s(l2	s(l2	PROPN
ejpam-6106	98	11	,	,	PUNCT
ejpam-6106	98	12	l3	l3	PROPN
ejpam-6106	98	13	)	)	PUNCT
ejpam-6106	98	14	≤	≤	PROPN
ejpam-6106	98	15	p1	p1	NOUN
ejpam-6106	98	16	≤	≤	PROPN
ejpam-6106	98	17	βp0	βp0	PROPN
ejpam-6106	98	18	,	,	PUNCT
ejpam-6106	98	19	m.	m.	NOUN
ejpam-6106	98	20	nazam	nazam	PROPN
ejpam-6106	98	21	,	,	PUNCT
ejpam-6106	98	22	u.	u.	PROPN
ejpam-6106	98	23	habiba	habiba	PROPN
ejpam-6106	98	24	,	,	PUNCT
ejpam-6106	98	25	m.	m.	PROPN
ejpam-6106	98	26	de	de	X
ejpam-6106	98	27	la	la	PROPN
ejpam-6106	98	28	sen	sen	PROPN
ejpam-6106	98	29	/	/	SYM
ejpam-6106	98	30	eur	eur	PROPN
ejpam-6106	98	31	.	.	PUNCT
ejpam-6106	99	1	j.	j.	PROPN
ejpam-6106	99	2	pure	pure	PROPN
ejpam-6106	99	3	appl	appl	PROPN
ejpam-6106	99	4	.	.	PROPN
ejpam-6106	99	5	math	math	PROPN
ejpam-6106	99	6	,	,	PUNCT
ejpam-6106	99	7	18	18	NUM
ejpam-6106	99	8	(	(	PUNCT
ejpam-6106	99	9	4	4	NUM
ejpam-6106	99	10	)	)	PUNCT
ejpam-6106	99	11	(	(	PUNCT
ejpam-6106	99	12	2025	2025	NUM
ejpam-6106	99	13	)	)	PUNCT
ejpam-6106	99	14	,	,	PUNCT
ejpam-6106	99	15	6106	6106	NUM
ejpam-6106	99	16	5	5	NUM
ejpam-6106	99	17	of	of	ADP
ejpam-6106	99	18	10	10	NUM
ejpam-6106	99	19	s(l3	s(l3	NOUN
ejpam-6106	99	20	,	,	PUNCT
ejpam-6106	99	21	l4	l4	PROPN
ejpam-6106	99	22	)	)	PUNCT
ejpam-6106	99	23	≤	≤	NUM
ejpam-6106	99	24	p2	p2	X
ejpam-6106	99	25	≤	≤	NUM
ejpam-6106	99	26	β2p0	β2p0	PUNCT
ejpam-6106	99	27	,	,	PUNCT
ejpam-6106	99	28	·	·	PUNCT
ejpam-6106	99	29	·	·	PUNCT
ejpam-6106	99	30	·	·	PUNCT
ejpam-6106	100	1	s(lk	s(lk	NUM
ejpam-6106	100	2	,	,	PUNCT
ejpam-6106	100	3	lk+1	lk+1	NUM
ejpam-6106	100	4	)	)	PUNCT
ejpam-6106	100	5	≤	≤	PUNCT
ejpam-6106	100	6	pk−1	pk−1	PROPN
ejpam-6106	100	7	≤	≤	NUM
ejpam-6106	100	8	βk−1p0	βk−1p0	AUX
ejpam-6106	100	9	,	,	PUNCT
ejpam-6106	100	10	by	by	ADP
ejpam-6106	100	11	triangular	triangular	NOUN
ejpam-6106	100	12	inequality	inequality	NOUN
ejpam-6106	100	13	,	,	PUNCT
ejpam-6106	100	14	s(lk	s(lk	NOUN
ejpam-6106	100	15	,	,	PUNCT
ejpam-6106	100	16	lk+p	lk+p	PROPN
ejpam-6106	100	17	)	)	PUNCT
ejpam-6106	100	18	≤	≤	PROPN
ejpam-6106	100	19	s(lk	s(lk	ADJ
ejpam-6106	100	20	,	,	PUNCT
ejpam-6106	100	21	lk+1	lk+1	NUM
ejpam-6106	100	22	)	)	PUNCT
ejpam-6106	100	23	+	+	NUM
ejpam-6106	100	24	s(lk+1	s(lk+1	NOUN
ejpam-6106	100	25	,	,	PUNCT
ejpam-6106	100	26	lk+2	lk+2	NOUN
ejpam-6106	100	27	)	)	PUNCT
ejpam-6106	100	28	+	+	CCONJ
ejpam-6106	100	29	·	·	PUNCT
ejpam-6106	100	30	·	·	PUNCT
ejpam-6106	100	31	·	·	PUNCT
ejpam-6106	100	32	+	+	PUNCT
ejpam-6106	100	33	s(lk+p−1	s(lk+p−1	X
ejpam-6106	100	34	,	,	PUNCT
ejpam-6106	100	35	lk+p	lk+p	PROPN
ejpam-6106	100	36	)	)	PUNCT
ejpam-6106	100	37	,	,	PUNCT
ejpam-6106	100	38	≤	≤	NUM
ejpam-6106	100	39	βk−1p0	βk−1p0	VERB
ejpam-6106	101	1	+	+	CCONJ
ejpam-6106	101	2	βkp0	βkp0	PROPN
ejpam-6106	101	3	+	+	NUM
ejpam-6106	101	4	·	·	PUNCT
ejpam-6106	101	5	·	·	PUNCT
ejpam-6106	101	6	·	·	PUNCT
ejpam-6106	102	1	+	+	CCONJ
ejpam-6106	102	2	βk+p−2p0	βk+p−2p0	ADJ
ejpam-6106	102	3	,	,	PUNCT
ejpam-6106	102	4	=	=	PUNCT
ejpam-6106	102	5	βk−1(1	βk−1(1	PUNCT
ejpam-6106	102	6	+	+	X
ejpam-6106	102	7	β	β	X
ejpam-6106	102	8	+	+	X
ejpam-6106	102	9	·	·	PUNCT
ejpam-6106	102	10	·	·	PUNCT
ejpam-6106	102	11	·	·	PUNCT
ejpam-6106	102	12	+	+	NUM
ejpam-6106	102	13	βp−1)p0	βp−1)p0	PROPN
ejpam-6106	102	14	,	,	PUNCT
ejpam-6106	102	15	=	=	PUNCT
ejpam-6106	102	16	βk−1	βk−1	PROPN
ejpam-6106	102	17	(	(	PUNCT
ejpam-6106	102	18	1−	1−	NUM
ejpam-6106	102	19	βp	βp	NUM
ejpam-6106	102	20	1−	1−	NUM
ejpam-6106	102	21	β	β	X
ejpam-6106	102	22	)	)	PUNCT
ejpam-6106	102	23	p0	p0	NOUN
ejpam-6106	102	24	.	.	PUNCT
ejpam-6106	103	1	since	since	SCONJ
ejpam-6106	103	2	,	,	PUNCT
ejpam-6106	103	3	0	0	NUM
ejpam-6106	103	4	≤	≤	NUM
ejpam-6106	103	5	β	β	X
ejpam-6106	103	6	<	<	X
ejpam-6106	103	7	1	1	NUM
ejpam-6106	103	8	,	,	PUNCT
ejpam-6106	103	9	we	we	PRON
ejpam-6106	103	10	have	have	VERB
ejpam-6106	103	11	s(lk	s(lk	NUM
ejpam-6106	103	12	,	,	PUNCT
ejpam-6106	103	13	lk+p	lk+p	PROPN
ejpam-6106	103	14	)	)	PUNCT
ejpam-6106	103	15	<	<	X
ejpam-6106	104	1	βk−1	βk−1	PROPN
ejpam-6106	104	2	1	1	NUM
ejpam-6106	104	3	1−βp0	1−βp0	NUM
ejpam-6106	104	4	.	.	PUNCT
ejpam-6106	105	1	hence	hence	ADV
ejpam-6106	105	2	,	,	PUNCT
ejpam-6106	105	3	s(lk	s(lk	NOUN
ejpam-6106	105	4	,	,	PUNCT
ejpam-6106	105	5	lk+p	lk+p	NOUN
ejpam-6106	105	6	)	)	PUNCT
ejpam-6106	105	7	→	→	SYM
ejpam-6106	105	8	0	0	PUNCT
ejpam-6106	105	9	as	as	ADP
ejpam-6106	105	10	k	k	PROPN
ejpam-6106	105	11	→	→	SYM
ejpam-6106	105	12	∞	∞	PROPN
ejpam-6106	105	13	where	where	SCONJ
ejpam-6106	105	14	p	p	NOUN
ejpam-6106	105	15	is	be	AUX
ejpam-6106	105	16	positive	positive	ADJ
ejpam-6106	105	17	.	.	PUNCT
ejpam-6106	106	1	the	the	DET
ejpam-6106	106	2	sequence	sequence	NOUN
ejpam-6106	106	3	lk	lk	NOUN
ejpam-6106	106	4	is	be	AUX
ejpam-6106	106	5	therefore	therefore	ADV
ejpam-6106	106	6	cauchy	cauchy	PROPN
ejpam-6106	106	7	.	.	PUNCT
ejpam-6106	107	1	according	accord	VERB
ejpam-6106	107	2	to	to	ADP
ejpam-6106	107	3	(	(	PUNCT
ejpam-6106	107	4	z	z	NOUN
ejpam-6106	107	5	,	,	PUNCT
ejpam-6106	107	6	s	s	NOUN
ejpam-6106	107	7	)	)	PUNCT
ejpam-6106	107	8	completeness	completeness	NOUN
ejpam-6106	107	9	,	,	PUNCT
ejpam-6106	107	10	this	this	DET
ejpam-6106	107	11	sequence	sequence	NOUN
ejpam-6106	107	12	’s	’s	PART
ejpam-6106	107	13	limit	limit	NOUN
ejpam-6106	107	14	is	be	AUX
ejpam-6106	107	15	l∗	l∗	PROPN
ejpam-6106	107	16	∈	∈	PROPN
ejpam-6106	107	17	z.	z.	PROPN
ejpam-6106	107	18	let	let	VERB
ejpam-6106	107	19	us	we	PRON
ejpam-6106	107	20	establish	establish	VERB
ejpam-6106	107	21	ll∗	ll∗	NOUN
ejpam-6106	107	22	=	=	PUNCT
ejpam-6106	107	23	l∗.	l∗.	NOUN
ejpam-6106	107	24	using	use	VERB
ejpam-6106	107	25	inequality	inequality	NOUN
ejpam-6106	107	26	(	(	PUNCT
ejpam-6106	107	27	1	1	NUM
ejpam-6106	107	28	)	)	PUNCT
ejpam-6106	107	29	and	and	CCONJ
ejpam-6106	107	30	triangular	triangular	NOUN
ejpam-6106	107	31	inequality	inequality	NOUN
ejpam-6106	107	32	,	,	PUNCT
ejpam-6106	107	33	we	we	PRON
ejpam-6106	107	34	have	have	VERB
ejpam-6106	107	35	s(l∗	s(l∗	NOUN
ejpam-6106	107	36	,	,	PUNCT
ejpam-6106	107	37	ll∗	ll∗	NOUN
ejpam-6106	107	38	)	)	PUNCT
ejpam-6106	107	39	≤	≤	NUM
ejpam-6106	107	40	s(l∗	s(l∗	NOUN
ejpam-6106	107	41	,	,	PUNCT
ejpam-6106	107	42	lk	lk	PROPN
ejpam-6106	107	43	)	)	PUNCT
ejpam-6106	107	44	+	+	SYM
ejpam-6106	107	45	s(lk	s(lk	NUM
ejpam-6106	107	46	,	,	PUNCT
ejpam-6106	107	47	l	l	NOUN
ejpam-6106	107	48	∗	∗	NOUN
ejpam-6106	107	49	)	)	PUNCT
ejpam-6106	107	50	=	=	SYM
ejpam-6106	107	51	s(l∗	s(l∗	NOUN
ejpam-6106	107	52	,	,	PUNCT
ejpam-6106	107	53	lk	lk	PROPN
ejpam-6106	107	54	)	)	PUNCT
ejpam-6106	107	55	+	+	NUM
ejpam-6106	107	56	s(llk−1	s(llk−1	NOUN
ejpam-6106	107	57	,	,	PUNCT
ejpam-6106	107	58	l	l	NOUN
ejpam-6106	107	59	∗	∗	NOUN
ejpam-6106	107	60	)	)	PUNCT
ejpam-6106	107	61	≤	≤	NUM
ejpam-6106	107	62	s(l∗	s(l∗	NOUN
ejpam-6106	107	63	,	,	PUNCT
ejpam-6106	107	64	lk	lk	PROPN
ejpam-6106	107	65	)	)	PUNCT
ejpam-6106	107	66	+	+	NUM
ejpam-6106	107	67	s(llk−1	s(llk−1	NOUN
ejpam-6106	107	68	,	,	PUNCT
ejpam-6106	107	69	ll	ll	NOUN
ejpam-6106	107	70	∗	∗	NOUN
ejpam-6106	107	71	)	)	PUNCT
ejpam-6106	108	1	+	+	SYM
ejpam-6106	108	2	s(llk−1	s(llk−1	ADJ
ejpam-6106	108	3	,	,	PUNCT
ejpam-6106	108	4	llk	llk	NOUN
ejpam-6106	108	5	)	)	PUNCT
ejpam-6106	109	1	+	+	CCONJ
ejpam-6106	109	2	·	·	PUNCT
ejpam-6106	109	3	·	·	PUNCT
ejpam-6106	109	4	·	·	PUNCT
ejpam-6106	109	5	+	+	CCONJ
ejpam-6106	109	6	s(llk+n−3	s(llk+n−3	ADJ
ejpam-6106	109	7	,	,	PUNCT
ejpam-6106	109	8	ll	ll	NOUN
ejpam-6106	109	9	∗	∗	NUM
ejpam-6106	109	10	)	)	PUNCT
ejpam-6106	109	11	≤	≤	NUM
ejpam-6106	109	12	s(l∗	s(l∗	NOUN
ejpam-6106	109	13	,	,	PUNCT
ejpam-6106	109	14	lk	lk	PROPN
ejpam-6106	109	15	)	)	PUNCT
ejpam-6106	109	16	+	+	CCONJ
ejpam-6106	109	17	β(s(lk−1	β(s(lk−1	ADJ
ejpam-6106	109	18	,	,	PUNCT
ejpam-6106	109	19	l	l	NOUN
ejpam-6106	109	20	∗	∗	NOUN
ejpam-6106	109	21	)	)	PUNCT
ejpam-6106	110	1	+	+	NUM
ejpam-6106	110	2	s(lk−1	s(lk−1	PROPN
ejpam-6106	110	3	,	,	PUNCT
ejpam-6106	110	4	lk	lk	NOUN
ejpam-6106	110	5	)	)	PUNCT
ejpam-6106	110	6	+	+	CCONJ
ejpam-6106	110	7	·	·	PUNCT
ejpam-6106	110	8	·	·	PUNCT
ejpam-6106	110	9	·	·	PUNCT
ejpam-6106	111	1	+	+	CCONJ
ejpam-6106	111	2	s(lk+n−3	s(lk+n−3	PROPN
ejpam-6106	111	3	,	,	PUNCT
ejpam-6106	111	4	l	l	NOUN
ejpam-6106	111	5	∗	∗	NOUN
ejpam-6106	111	6	)	)	PUNCT
ejpam-6106	111	7	)	)	PUNCT
ejpam-6106	111	8	.	.	PUNCT
ejpam-6106	112	1	we	we	PRON
ejpam-6106	112	2	obtain	obtain	VERB
ejpam-6106	112	3	s(l∗	s(l∗	NOUN
ejpam-6106	112	4	,	,	PUNCT
ejpam-6106	112	5	ll∗	ll∗	NOUN
ejpam-6106	112	6	)	)	PUNCT
ejpam-6106	113	1	=	=	SYM
ejpam-6106	113	2	0	0	NUM
ejpam-6106	113	3	,	,	PUNCT
ejpam-6106	113	4	as	as	SCONJ
ejpam-6106	113	5	every	every	DET
ejpam-6106	113	6	term	term	NOUN
ejpam-6106	113	7	in	in	ADP
ejpam-6106	113	8	the	the	DET
ejpam-6106	113	9	preceding	precede	VERB
ejpam-6106	113	10	sum	sum	NOUN
ejpam-6106	113	11	tends	tend	VERB
ejpam-6106	113	12	to	to	ADP
ejpam-6106	113	13	zero	zero	NUM
ejpam-6106	113	14	as	as	SCONJ
ejpam-6106	113	15	k	k	PROPN
ejpam-6106	113	16	→	→	SYM
ejpam-6106	113	17	∞.	∞.	PROPN
ejpam-6106	113	18	let	let	VERB
ejpam-6106	113	19	us	we	PRON
ejpam-6106	113	20	assume	assume	VERB
ejpam-6106	113	21	that	that	SCONJ
ejpam-6106	113	22	there	there	PRON
ejpam-6106	113	23	are	be	VERB
ejpam-6106	113	24	a	a	DET
ejpam-6106	113	25	minimum	minimum	NOUN
ejpam-6106	113	26	of	of	ADP
ejpam-6106	113	27	n	n	PRON
ejpam-6106	113	28	distinct	distinct	ADJ
ejpam-6106	113	29	pairwise	pairwise	NOUN
ejpam-6106	113	30	fixed	fix	VERB
ejpam-6106	113	31	points	point	NOUN
ejpam-6106	113	32	,	,	PUNCT
ejpam-6106	113	33	denoted	denote	VERB
ejpam-6106	113	34	as	as	ADP
ejpam-6106	113	35	l1	l1	PROPN
ejpam-6106	113	36	,	,	PUNCT
ejpam-6106	113	37	l2	l2	NOUN
ejpam-6106	113	38	,	,	PUNCT
ejpam-6106	113	39	·	·	PUNCT
ejpam-6106	113	40	·	·	PUNCT
ejpam-6106	113	41	·	·	PUNCT
ejpam-6106	113	42	,	,	PUNCT
ejpam-6106	113	43	ln	ln	X
ejpam-6106	113	44	.	.	PUNCT
ejpam-6106	113	45	thus	thus	ADV
ejpam-6106	113	46	,	,	PUNCT
ejpam-6106	113	47	ll1	ll1	PROPN
ejpam-6106	113	48	=	=	SYM
ejpam-6106	113	49	l1	l1	PROPN
ejpam-6106	113	50	,	,	PUNCT
ejpam-6106	113	51	ll2	ll2	NOUN
ejpam-6106	113	52	=	=	SYM
ejpam-6106	113	53	l2	l2	NOUN
ejpam-6106	113	54	,	,	PUNCT
ejpam-6106	113	55	·	·	PUNCT
ejpam-6106	113	56	·	·	PUNCT
ejpam-6106	113	57	·	·	PUNCT
ejpam-6106	113	58	,	,	PUNCT
ejpam-6106	113	59	lln	lln	PROPN
ejpam-6106	113	60	=	=	SYM
ejpam-6106	113	61	ln	ln	PROPN
ejpam-6106	113	62	,	,	PUNCT
ejpam-6106	113	63	which	which	PRON
ejpam-6106	113	64	is	be	AUX
ejpam-6106	113	65	in	in	ADP
ejpam-6106	113	66	opposition	opposition	NOUN
ejpam-6106	113	67	to	to	ADP
ejpam-6106	113	68	(	(	PUNCT
ejpam-6106	113	69	1	1	NUM
ejpam-6106	113	70	)	)	PUNCT
ejpam-6106	113	71	.	.	PUNCT
ejpam-6106	114	1	the	the	DET
ejpam-6106	114	2	main	main	ADJ
ejpam-6106	114	3	result	result	NOUN
ejpam-6106	114	4	shows	show	VERB
ejpam-6106	114	5	that	that	SCONJ
ejpam-6106	114	6	if	if	SCONJ
ejpam-6106	114	7	a	a	DET
ejpam-6106	114	8	mapping	mapping	NOUN
ejpam-6106	114	9	always	always	ADV
ejpam-6106	114	10	reduces	reduce	VERB
ejpam-6106	114	11	the	the	DET
ejpam-6106	114	12	perimeter	perimeter	NOUN
ejpam-6106	114	13	of	of	ADP
ejpam-6106	114	14	any	any	DET
ejpam-6106	114	15	polygon	polygon	NOUN
ejpam-6106	114	16	,	,	PUNCT
ejpam-6106	114	17	then	then	ADV
ejpam-6106	114	18	applying	apply	VERB
ejpam-6106	114	19	the	the	DET
ejpam-6106	114	20	mapping	mapping	NOUN
ejpam-6106	114	21	repeatedly	repeatedly	ADV
ejpam-6106	114	22	forces	force	VERB
ejpam-6106	114	23	the	the	DET
ejpam-6106	114	24	polygon	polygon	NOUN
ejpam-6106	114	25	to	to	PART
ejpam-6106	114	26	shrink	shrink	VERB
ejpam-6106	114	27	down	down	ADP
ejpam-6106	114	28	to	to	ADP
ejpam-6106	114	29	a	a	DET
ejpam-6106	114	30	single	single	ADJ
ejpam-6106	114	31	point	point	NOUN
ejpam-6106	114	32	.	.	PUNCT
ejpam-6106	115	1	that	that	DET
ejpam-6106	115	2	point	point	NOUN
ejpam-6106	115	3	is	be	AUX
ejpam-6106	115	4	the	the	DET
ejpam-6106	115	5	fixed	fixed	ADJ
ejpam-6106	115	6	point	point	NOUN
ejpam-6106	115	7	of	of	ADP
ejpam-6106	115	8	the	the	DET
ejpam-6106	115	9	mapping	mapping	NOUN
ejpam-6106	115	10	.	.	PUNCT
ejpam-6106	116	1	this	this	PRON
ejpam-6106	116	2	provides	provide	VERB
ejpam-6106	116	3	a	a	DET
ejpam-6106	116	4	geometric	geometric	ADJ
ejpam-6106	116	5	way	way	NOUN
ejpam-6106	116	6	of	of	ADP
ejpam-6106	116	7	understanding	understanding	NOUN
ejpam-6106	116	8	banach	banach	NOUN
ejpam-6106	116	9	’s	’s	PART
ejpam-6106	116	10	contraction	contraction	NOUN
ejpam-6106	116	11	principle	principle	NOUN
ejpam-6106	116	12	and	and	CCONJ
ejpam-6106	116	13	extends	extend	VERB
ejpam-6106	116	14	petrov	petrov	PROPN
ejpam-6106	116	15	’s	’s	PART
ejpam-6106	116	16	triangle	triangle	NOUN
ejpam-6106	116	17	-	-	PUNCT
ejpam-6106	116	18	based	base	VERB
ejpam-6106	116	19	theorem	theorem	NOUN
ejpam-6106	116	20	to	to	PART
ejpam-6106	116	21	polygons	polygon	NOUN
ejpam-6106	116	22	with	with	ADP
ejpam-6106	116	23	any	any	DET
ejpam-6106	116	24	number	number	NOUN
ejpam-6106	116	25	of	of	ADP
ejpam-6106	116	26	vertices	vertex	NOUN
ejpam-6106	116	27	.	.	PUNCT
ejpam-6106	117	1	remark	remark	NOUN
ejpam-6106	117	2	5	5	NUM
ejpam-6106	117	3	.	.	PUNCT
ejpam-6106	118	1	let	let	VERB
ejpam-6106	118	2	us	we	PRON
ejpam-6106	118	3	assume	assume	VERB
ejpam-6106	118	4	the	the	DET
ejpam-6106	118	5	following	following	NOUN
ejpam-6106	118	6	:	:	PUNCT
ejpam-6106	118	7	the	the	DET
ejpam-6106	118	8	mapping	mapping	NOUN
ejpam-6106	118	9	l	l	NOUN
ejpam-6106	118	10	has	have	VERB
ejpam-6106	118	11	a	a	DET
ejpam-6106	118	12	fixed	fix	VERB
ejpam-6106	118	13	point	point	NOUN
ejpam-6106	118	14	point	point	NOUN
ejpam-6106	118	15	l∗	l∗	PROPN
ejpam-6106	118	16	,	,	PUNCT
ejpam-6106	118	17	a	a	DET
ejpam-6106	118	18	limit	limit	NOUN
ejpam-6106	118	19	of	of	ADP
ejpam-6106	118	20	a	a	DET
ejpam-6106	118	21	sequence	sequence	NOUN
ejpam-6106	118	22	{	{	PUNCT
ejpam-6106	118	23	ln	ln	ADJ
ejpam-6106	118	24	}	}	PUNCT
ejpam-6106	118	25	,	,	PUNCT
ejpam-6106	118	26	consider	consider	VERB
ejpam-6106	118	27	l1	l1	PROPN
ejpam-6106	118	28	=	=	PUNCT
ejpam-6106	118	29	ll0,l2	ll0,l2	NOUN
ejpam-6106	118	30	=	=	SYM
ejpam-6106	118	31	ll1	ll1	ADV
ejpam-6106	118	32	...	...	PUNCT
ejpam-6106	118	33	such	such	ADJ
ejpam-6106	118	34	that	that	SCONJ
ejpam-6106	118	35	ln	ln	ADJ
ejpam-6106	118	36	̸=	̸=	PROPN
ejpam-6106	118	37	l∗	l∗	NOUN
ejpam-6106	118	38	for	for	ADP
ejpam-6106	118	39	all	all	PRON
ejpam-6106	118	40	n	n	PRON
ejpam-6106	118	41	∈	∈	PROPN
ejpam-6106	118	42	n	n	NOUN
ejpam-6106	118	43	.	.	PUNCT
ejpam-6106	119	1	then	then	ADV
ejpam-6106	119	2	there	there	PRON
ejpam-6106	119	3	is	be	VERB
ejpam-6106	119	4	only	only	ADV
ejpam-6106	119	5	one	one	NUM
ejpam-6106	119	6	fixed	fix	VERB
ejpam-6106	119	7	point	point	NOUN
ejpam-6106	119	8	,	,	PUNCT
ejpam-6106	119	9	l∗.	l∗.	NOUN
ejpam-6106	119	10	suppose	suppose	VERB
ejpam-6106	119	11	l	l	NOUN
ejpam-6106	119	12	has	have	VERB
ejpam-6106	119	13	an	an	DET
ejpam-6106	119	14	additional	additional	ADJ
ejpam-6106	119	15	fixed	fix	VERB
ejpam-6106	119	16	point	point	NOUN
ejpam-6106	119	17	l2∗	l2∗	NOUN
ejpam-6106	119	18	̸=	̸=	PROPN
ejpam-6106	119	19	l∗.	l∗.	NOUN
ejpam-6106	119	20	it	it	PRON
ejpam-6106	119	21	is	be	AUX
ejpam-6106	119	22	evident	evident	ADJ
ejpam-6106	119	23	that	that	SCONJ
ejpam-6106	119	24	ln	ln	ADJ
ejpam-6106	119	25	̸=	̸=	PROPN
ejpam-6106	119	26	l2∗	l2∗	NOUN
ejpam-6106	119	27	∀	∀	X
ejpam-6106	120	1	n	n	PRON
ejpam-6106	120	2	≥	≥	NOUN
ejpam-6106	120	3	1	1	NUM
ejpam-6106	120	4	.	.	PUNCT
ejpam-6106	121	1	likewise	likewise	ADV
ejpam-6106	121	2	,	,	PUNCT
ejpam-6106	121	3	assume	assume	VERB
ejpam-6106	121	4	that	that	SCONJ
ejpam-6106	121	5	l3∗	l3∗	PROPN
ejpam-6106	121	6	,	,	PUNCT
ejpam-6106	121	7	l4∗	l4∗	NUM
ejpam-6106	121	8	,	,	PUNCT
ejpam-6106	121	9	·	·	PUNCT
ejpam-6106	121	10	·	·	PUNCT
ejpam-6106	121	11	·	·	PUNCT
ejpam-6106	121	12	,	,	PUNCT
ejpam-6106	121	13	ln∗	ln∗	NOUN
ejpam-6106	121	14	are	be	AUX
ejpam-6106	121	15	fixed	fix	VERB
ejpam-6106	121	16	points	point	NOUN
ejpam-6106	121	17	.	.	PUNCT
ejpam-6106	122	1	then	then	ADV
ejpam-6106	122	2	ln	ln	ADJ
ejpam-6106	122	3	̸=	̸=	PROPN
ejpam-6106	122	4	l3∗	l3∗	PROPN
ejpam-6106	122	5	̸=	̸=	PROPN
ejpam-6106	122	6	l4∗	l4∗	NUM
ejpam-6106	122	7	,	,	PUNCT
ejpam-6106	122	8	·	·	PUNCT
ejpam-6106	122	9	·	·	PUNCT
ejpam-6106	122	10	·	·	PUNCT
ejpam-6106	122	11	,	,	PUNCT
ejpam-6106	122	12	̸=	̸=	PROPN
ejpam-6106	122	13	ln∗	ln∗	NOUN
ejpam-6106	122	14	∀	∀	X
ejpam-6106	122	15	n	n	PRON
ejpam-6106	122	16	≥	≥	NOUN
ejpam-6106	122	17	1	1	NUM
ejpam-6106	122	18	.	.	PUNCT
ejpam-6106	123	1	thus	thus	ADV
ejpam-6106	123	2	,	,	PUNCT
ejpam-6106	123	3	∀	∀	X
ejpam-6106	123	4	n	n	PRON
ejpam-6106	123	5	≥	≥	NOUN
ejpam-6106	123	6	1	1	NUM
ejpam-6106	123	7	,	,	PUNCT
ejpam-6106	123	8	we	we	PRON
ejpam-6106	123	9	have	have	VERB
ejpam-6106	123	10	that	that	SCONJ
ejpam-6106	123	11	the	the	DET
ejpam-6106	123	12	points	point	NOUN
ejpam-6106	123	13	ln	ln	ADJ
ejpam-6106	123	14	,	,	PUNCT
ejpam-6106	123	15	l	l	NOUN
ejpam-6106	123	16	∗	∗	NOUN
ejpam-6106	123	17	,	,	PUNCT
ejpam-6106	123	18	l2∗	l2∗	NOUN
ejpam-6106	123	19	,	,	PUNCT
ejpam-6106	123	20	·	·	PUNCT
ejpam-6106	123	21	·	·	PUNCT
ejpam-6106	123	22	·	·	PUNCT
ejpam-6106	123	23	,	,	PUNCT
ejpam-6106	123	24	l(n−1)∗	l(n−1)∗	X
ejpam-6106	123	25	are	be	AUX
ejpam-6106	123	26	pairwise	pairwise	NOUN
ejpam-6106	123	27	distinct	distinct	ADJ
ejpam-6106	123	28	.	.	PUNCT
ejpam-6106	124	1	consider	consider	VERB
ejpam-6106	124	2	the	the	DET
ejpam-6106	124	3	ratio	ratio	NOUN
ejpam-6106	124	4	rn	rn	PROPN
ejpam-6106	124	5	=	=	SYM
ejpam-6106	124	6	s(lln	s(lln	PROPN
ejpam-6106	124	7	,	,	PUNCT
ejpam-6106	124	8	ll	ll	NOUN
ejpam-6106	124	9	∗	∗	NOUN
ejpam-6106	124	10	)	)	PUNCT
ejpam-6106	125	1	+	+	CCONJ
ejpam-6106	125	2	s(ll∗	s(ll∗	NOUN
ejpam-6106	125	3	,	,	PUNCT
ejpam-6106	125	4	ll2∗	ll2∗	PROPN
ejpam-6106	125	5	)	)	PUNCT
ejpam-6106	125	6	+	+	CCONJ
ejpam-6106	125	7	·	·	PUNCT
ejpam-6106	125	8	·	·	PUNCT
ejpam-6106	125	9	·	·	PUNCT
ejpam-6106	125	10	+	+	NUM
ejpam-6106	125	11	s(ll(n−1)∗	s(ll(n−1)∗	ADJ
ejpam-6106	125	12	,	,	PUNCT
ejpam-6106	125	13	lln	lln	NOUN
ejpam-6106	125	14	)	)	PUNCT
ejpam-6106	125	15	s(ln	s(ln	NOUN
ejpam-6106	125	16	,	,	PUNCT
ejpam-6106	125	17	l∗	l∗	PROPN
ejpam-6106	125	18	)	)	PUNCT
ejpam-6106	126	1	+	+	PUNCT
ejpam-6106	126	2	s(l∗	s(l∗	X
ejpam-6106	126	3	,	,	PUNCT
ejpam-6106	126	4	l2∗	l2∗	NOUN
ejpam-6106	126	5	)	)	PUNCT
ejpam-6106	126	6	+	+	CCONJ
ejpam-6106	126	7	·	·	PUNCT
ejpam-6106	126	8	·	·	PUNCT
ejpam-6106	126	9	·	·	PUNCT
ejpam-6106	126	10	+	+	NUM
ejpam-6106	126	11	s(l(n−1)∗	s(l(n−1)∗	NOUN
ejpam-6106	126	12	,	,	PUNCT
ejpam-6106	126	13	ln	ln	ADJ
ejpam-6106	126	14	)	)	PUNCT
ejpam-6106	126	15	=	=	SYM
ejpam-6106	126	16	s(ln+1	s(ln+1	PROPN
ejpam-6106	126	17	,	,	PUNCT
ejpam-6106	126	18	l	l	NOUN
ejpam-6106	126	19	∗	∗	NOUN
ejpam-6106	126	20	)	)	PUNCT
ejpam-6106	126	21	+	+	NUM
ejpam-6106	126	22	s(l∗	s(l∗	ADJ
ejpam-6106	126	23	,	,	PUNCT
ejpam-6106	126	24	l2∗	l2∗	NOUN
ejpam-6106	126	25	)	)	PUNCT
ejpam-6106	126	26	+	+	CCONJ
ejpam-6106	126	27	·	·	PUNCT
ejpam-6106	126	28	·	·	PUNCT
ejpam-6106	126	29	·	·	PUNCT
ejpam-6106	126	30	+	+	NUM
ejpam-6106	126	31	s(l(n−1)∗	s(l(n−1)∗	NOUN
ejpam-6106	126	32	,	,	PUNCT
ejpam-6106	126	33	ln+1	ln+1	ADJ
ejpam-6106	126	34	)	)	PUNCT
ejpam-6106	126	35	s(ln	s(ln	NOUN
ejpam-6106	126	36	,	,	PUNCT
ejpam-6106	126	37	l∗	l∗	PROPN
ejpam-6106	126	38	)	)	PUNCT
ejpam-6106	126	39	+	+	PUNCT
ejpam-6106	126	40	s(l∗	s(l∗	X
ejpam-6106	126	41	,	,	PUNCT
ejpam-6106	126	42	l2∗	l2∗	NOUN
ejpam-6106	126	43	)	)	PUNCT
ejpam-6106	126	44	+	+	CCONJ
ejpam-6106	126	45	·	·	PUNCT
ejpam-6106	126	46	·	·	PUNCT
ejpam-6106	126	47	·	·	PUNCT
ejpam-6106	126	48	+	+	NUM
ejpam-6106	126	49	s(l(n−1)∗	s(l(n−1)∗	NOUN
ejpam-6106	126	50	,	,	PUNCT
ejpam-6106	126	51	ln	ln	ADJ
ejpam-6106	126	52	)	)	PUNCT
ejpam-6106	126	53	.	.	PUNCT
ejpam-6106	127	1	we	we	PRON
ejpam-6106	127	2	obtain	obtain	VERB
ejpam-6106	127	3	rn	rn	PROPN
ejpam-6106	127	4	→	→	SYM
ejpam-6106	127	5	1	1	NUM
ejpam-6106	127	6	as	as	ADP
ejpam-6106	127	7	n	n	NUM
ejpam-6106	127	8	→	→	SYM
ejpam-6106	127	9	∞	∞	PROPN
ejpam-6106	127	10	,	,	PUNCT
ejpam-6106	127	11	which	which	PRON
ejpam-6106	127	12	does	do	AUX
ejpam-6106	127	13	not	not	PART
ejpam-6106	127	14	satisfy	satisfy	VERB
ejpam-6106	127	15	1	1	NUM
ejpam-6106	127	16	.	.	PUNCT
ejpam-6106	128	1	m.	m.	NOUN
ejpam-6106	128	2	nazam	nazam	PROPN
ejpam-6106	128	3	,	,	PUNCT
ejpam-6106	128	4	u.	u.	PROPN
ejpam-6106	128	5	habiba	habiba	PROPN
ejpam-6106	128	6	,	,	PUNCT
ejpam-6106	128	7	m.	m.	PROPN
ejpam-6106	128	8	de	de	X
ejpam-6106	128	9	la	la	PROPN
ejpam-6106	128	10	sen	sen	PROPN
ejpam-6106	128	11	/	/	SYM
ejpam-6106	128	12	eur	eur	PROPN
ejpam-6106	128	13	.	.	PUNCT
ejpam-6106	129	1	j.	j.	PROPN
ejpam-6106	129	2	pure	pure	PROPN
ejpam-6106	129	3	appl	appl	PROPN
ejpam-6106	129	4	.	.	PROPN
ejpam-6106	129	5	math	math	PROPN
ejpam-6106	129	6	,	,	PUNCT
ejpam-6106	129	7	18	18	NUM
ejpam-6106	129	8	(	(	PUNCT
ejpam-6106	129	9	4	4	NUM
ejpam-6106	129	10	)	)	PUNCT
ejpam-6106	129	11	(	(	PUNCT
ejpam-6106	129	12	2025	2025	NUM
ejpam-6106	129	13	)	)	PUNCT
ejpam-6106	129	14	,	,	PUNCT
ejpam-6106	129	15	6106	6106	NUM
ejpam-6106	129	16	6	6	NUM
ejpam-6106	129	17	of	of	ADP
ejpam-6106	129	18	10	10	NUM
ejpam-6106	129	19	example	example	NOUN
ejpam-6106	129	20	6	6	NUM
ejpam-6106	129	21	.	.	PUNCT
ejpam-6106	130	1	let	let	VERB
ejpam-6106	130	2	’s	’s	PRON
ejpam-6106	130	3	take	take	VERB
ejpam-6106	130	4	an	an	DET
ejpam-6106	130	5	example	example	NOUN
ejpam-6106	130	6	of	of	ADP
ejpam-6106	130	7	mapping	mapping	NOUN
ejpam-6106	130	8	l	l	NOUN
ejpam-6106	130	9	having	have	VERB
ejpam-6106	130	10	precisely	precisely	ADV
ejpam-6106	130	11	n−	n−	NOUN
ejpam-6106	130	12	1	1	NUM
ejpam-6106	130	13	fixed	fix	VERB
ejpam-6106	130	14	points	point	NOUN
ejpam-6106	130	15	.	.	PUNCT
ejpam-6106	131	1	let	let	VERB
ejpam-6106	131	2	z	z	NOUN
ejpam-6106	131	3	=	=	PRON
ejpam-6106	131	4	{	{	PUNCT
ejpam-6106	131	5	l1	l1	PROPN
ejpam-6106	131	6	,	,	PUNCT
ejpam-6106	131	7	l2	l2	NOUN
ejpam-6106	131	8	,	,	PUNCT
ejpam-6106	131	9	·	·	PUNCT
ejpam-6106	131	10	·	·	PUNCT
ejpam-6106	131	11	·	·	PUNCT
ejpam-6106	131	12	,	,	PUNCT
ejpam-6106	131	13	ln	ln	ADJ
ejpam-6106	131	14	}	}	PUNCT
ejpam-6106	131	15	,	,	PUNCT
ejpam-6106	131	16	s(l1	s(l1	INTJ
ejpam-6106	131	17	,	,	PUNCT
ejpam-6106	131	18	l2	l2	NOUN
ejpam-6106	131	19	)	)	PUNCT
ejpam-6106	131	20	=	=	SYM
ejpam-6106	131	21	s(l2	s(l2	PROPN
ejpam-6106	131	22	,	,	PUNCT
ejpam-6106	131	23	l3	l3	X
ejpam-6106	131	24	)	)	PUNCT
ejpam-6106	131	25	=	=	SYM
ejpam-6106	131	26	s(l3	s(l3	PROPN
ejpam-6106	131	27	,	,	PUNCT
ejpam-6106	131	28	l4	l4	PROPN
ejpam-6106	131	29	)	)	PUNCT
ejpam-6106	131	30	=	=	SYM
ejpam-6106	131	31	·	·	PUNCT
ejpam-6106	131	32	·	·	PUNCT
ejpam-6106	131	33	·	·	PUNCT
ejpam-6106	132	1	=	=	SYM
ejpam-6106	132	2	s(ln	s(ln	NOUN
ejpam-6106	132	3	,	,	PUNCT
ejpam-6106	132	4	l1	l1	PROPN
ejpam-6106	132	5	)	)	PUNCT
ejpam-6106	132	6	=	=	SYM
ejpam-6106	132	7	1	1	NUM
ejpam-6106	132	8	and	and	CCONJ
ejpam-6106	132	9	let	let	VERB
ejpam-6106	132	10	l	l	NOUN
ejpam-6106	132	11	:	:	PUNCT
ejpam-6106	132	12	z	z	X
ejpam-6106	132	13	→	→	SYM
ejpam-6106	132	14	z	z	NOUN
ejpam-6106	132	15	be	be	AUX
ejpam-6106	132	16	such	such	ADJ
ejpam-6106	132	17	that	that	DET
ejpam-6106	132	18	l(l1	l(l1	NOUN
ejpam-6106	132	19	)	)	PUNCT
ejpam-6106	133	1	=	=	SYM
ejpam-6106	133	2	l1	l1	PROPN
ejpam-6106	133	3	,	,	PUNCT
ejpam-6106	133	4	l(l2	l(l2	NOUN
ejpam-6106	133	5	)	)	PUNCT
ejpam-6106	133	6	=	=	SYM
ejpam-6106	133	7	l2	l2	NOUN
ejpam-6106	133	8	,	,	PUNCT
ejpam-6106	133	9	·	·	PUNCT
ejpam-6106	133	10	·	·	PUNCT
ejpam-6106	133	11	·	·	PUNCT
ejpam-6106	133	12	,	,	PUNCT
ejpam-6106	133	13	l(ln−1	l(ln−1	PROPN
ejpam-6106	133	14	)	)	PUNCT
ejpam-6106	133	15	=	=	SYM
ejpam-6106	133	16	ln−1	ln−1	PROPN
ejpam-6106	133	17	,	,	PUNCT
ejpam-6106	133	18	lln	lln	PROPN
ejpam-6106	133	19	=	=	PROPN
ejpam-6106	133	20	l1	l1	PROPN
ejpam-6106	133	21	.	.	PUNCT
ejpam-6106	134	1	it	it	PRON
ejpam-6106	134	2	is	be	AUX
ejpam-6106	134	3	evident	evident	ADJ
ejpam-6106	134	4	that	that	SCONJ
ejpam-6106	134	5	both	both	PRON
ejpam-6106	134	6	of	of	ADP
ejpam-6106	134	7	the	the	DET
ejpam-6106	134	8	requirements	requirement	NOUN
ejpam-6106	134	9	of	of	ADP
ejpam-6106	134	10	theorem	theorem	NOUN
ejpam-6106	134	11	(	(	PUNCT
ejpam-6106	134	12	4	4	NUM
ejpam-6106	134	13	)	)	PUNCT
ejpam-6106	134	14	are	be	AUX
ejpam-6106	134	15	met	meet	VERB
ejpam-6106	134	16	.	.	PUNCT
ejpam-6106	134	17	example	example	NOUN
ejpam-6106	135	1	7	7	NUM
ejpam-6106	135	2	.	.	PUNCT
ejpam-6106	135	3	we	we	PRON
ejpam-6106	135	4	will	will	AUX
ejpam-6106	135	5	demonstrate	demonstrate	VERB
ejpam-6106	135	6	that	that	PRON
ejpam-6106	135	7	assumption	assumption	NOUN
ejpam-6106	135	8	(	(	PUNCT
ejpam-6106	135	9	i	i	NOUN
ejpam-6106	135	10	)	)	PUNCT
ejpam-6106	135	11	of	of	ADP
ejpam-6106	135	12	theorem	theorem	NOUN
ejpam-6106	135	13	(	(	PUNCT
ejpam-6106	135	14	4	4	NUM
ejpam-6106	135	15	)	)	PUNCT
ejpam-6106	135	16	is	be	AUX
ejpam-6106	135	17	necessary	necessary	ADJ
ejpam-6106	135	18	.	.	PUNCT
ejpam-6106	136	1	let	let	VERB
ejpam-6106	136	2	y	y	PROPN
ejpam-6106	136	3	=	=	PRON
ejpam-6106	136	4	{	{	PUNCT
ejpam-6106	136	5	l1	l1	PROPN
ejpam-6106	136	6	,	,	PUNCT
ejpam-6106	136	7	l2	l2	NOUN
ejpam-6106	136	8	,	,	PUNCT
ejpam-6106	136	9	l3	l3	PROPN
ejpam-6106	136	10	,	,	PUNCT
ejpam-6106	136	11	·	·	PUNCT
ejpam-6106	136	12	·	·	PUNCT
ejpam-6106	136	13	·	·	PUNCT
ejpam-6106	136	14	,	,	PUNCT
ejpam-6106	136	15	ln	ln	ADJ
ejpam-6106	136	16	}	}	PUNCT
ejpam-6106	136	17	,	,	PUNCT
ejpam-6106	136	18	s(l1	s(l1	INTJ
ejpam-6106	136	19	,	,	PUNCT
ejpam-6106	136	20	l2	l2	NOUN
ejpam-6106	136	21	)	)	PUNCT
ejpam-6106	136	22	=	=	SYM
ejpam-6106	136	23	s(l2	s(l2	PROPN
ejpam-6106	136	24	,	,	PUNCT
ejpam-6106	136	25	l3	l3	X
ejpam-6106	136	26	)	)	PUNCT
ejpam-6106	136	27	=	=	SYM
ejpam-6106	136	28	·	·	PUNCT
ejpam-6106	136	29	·	·	PUNCT
ejpam-6106	136	30	·	·	PUNCT
ejpam-6106	137	1	=	=	SYM
ejpam-6106	137	2	s(ln	s(ln	NOUN
ejpam-6106	137	3	,	,	PUNCT
ejpam-6106	137	4	l1	l1	PROPN
ejpam-6106	137	5	)	)	PUNCT
ejpam-6106	137	6	and	and	CCONJ
ejpam-6106	137	7	let	let	VERB
ejpam-6106	137	8	l	l	NOUN
ejpam-6106	137	9	:	:	PUNCT
ejpam-6106	137	10	z	z	X
ejpam-6106	137	11	→	→	SYM
ejpam-6106	137	12	z	z	NOUN
ejpam-6106	137	13	be	be	AUX
ejpam-6106	137	14	such	such	ADJ
ejpam-6106	137	15	that	that	DET
ejpam-6106	137	16	ll1	ll1	NOUN
ejpam-6106	137	17	=	=	SYM
ejpam-6106	137	18	l2	l2	NOUN
ejpam-6106	137	19	,	,	PUNCT
ejpam-6106	137	20	ll2	ll2	NOUN
ejpam-6106	137	21	=	=	SYM
ejpam-6106	137	22	l1	l1	PROPN
ejpam-6106	137	23	,	,	PUNCT
ejpam-6106	137	24	ll3	ll3	NOUN
ejpam-6106	137	25	=	=	SYM
ejpam-6106	137	26	l4	l4	PROPN
ejpam-6106	137	27	,	,	PUNCT
ejpam-6106	137	28	ll4	ll4	PROPN
ejpam-6106	137	29	=	=	SYM
ejpam-6106	137	30	l3	l3	PROPN
ejpam-6106	137	31	,	,	PUNCT
ejpam-6106	137	32	·	·	PUNCT
ejpam-6106	137	33	·	·	PUNCT
ejpam-6106	137	34	·	·	PUNCT
ejpam-6106	137	35	,	,	PUNCT
ejpam-6106	137	36	lln	lln	PROPN
ejpam-6106	137	37	=	=	PROPN
ejpam-6106	137	38	l1	l1	PROPN
ejpam-6106	137	39	.	.	PUNCT
ejpam-6106	138	1	it	it	PRON
ejpam-6106	138	2	is	be	AUX
ejpam-6106	138	3	evident	evident	ADJ
ejpam-6106	138	4	that	that	SCONJ
ejpam-6106	138	5	assumption	assumption	NOUN
ejpam-6106	138	6	(	(	PUNCT
ejpam-6106	138	7	ii	ii	NOUN
ejpam-6106	138	8	)	)	PUNCT
ejpam-6106	138	9	of	of	ADP
ejpam-6106	138	10	theorem	theorem	NOUN
ejpam-6106	138	11	(	(	PUNCT
ejpam-6106	138	12	4	4	NUM
ejpam-6106	138	13	)	)	PUNCT
ejpam-6106	138	14	is	be	AUX
ejpam-6106	138	15	satisfied	satisfied	ADJ
ejpam-6106	138	16	,	,	PUNCT
ejpam-6106	138	17	but	but	CCONJ
ejpam-6106	138	18	the	the	DET
ejpam-6106	138	19	mapping	mapping	NOUN
ejpam-6106	138	20	l	l	NOUN
ejpam-6106	138	21	lacks	lack	VERB
ejpam-6106	138	22	a	a	DET
ejpam-6106	138	23	fixed	fix	VERB
ejpam-6106	138	24	point	point	NOUN
ejpam-6106	138	25	.	.	PUNCT
ejpam-6106	139	1	given	give	VERB
ejpam-6106	139	2	a	a	DET
ejpam-6106	139	3	metric	metric	ADJ
ejpam-6106	139	4	space	space	NOUN
ejpam-6106	139	5	(	(	PUNCT
ejpam-6106	139	6	z	z	NOUN
ejpam-6106	139	7	,	,	PUNCT
ejpam-6106	139	8	s	s	PROPN
ejpam-6106	139	9	)	)	PUNCT
ejpam-6106	139	10	,	,	PUNCT
ejpam-6106	139	11	a	a	DET
ejpam-6106	139	12	mapping	mapping	NOUN
ejpam-6106	139	13	l	l	NOUN
ejpam-6106	139	14	:	:	PUNCT
ejpam-6106	139	15	z	z	X
ejpam-6106	139	16	→	→	SYM
ejpam-6106	139	17	z	z	NOUN
ejpam-6106	139	18	is	be	AUX
ejpam-6106	139	19	known	know	VERB
ejpam-6106	139	20	as	as	ADP
ejpam-6106	139	21	a	a	DET
ejpam-6106	139	22	contraction	contraction	NOUN
ejpam-6106	139	23	mapping	mapping	NOUN
ejpam-6106	139	24	on	on	ADP
ejpam-6106	139	25	z	z	NOUN
ejpam-6106	139	26	if	if	SCONJ
ejpam-6106	140	1	and	and	CCONJ
ejpam-6106	140	2	only	only	ADV
ejpam-6106	140	3	if	if	SCONJ
ejpam-6106	140	4	there	there	PRON
ejpam-6106	140	5	is	be	VERB
ejpam-6106	140	6	a	a	DET
ejpam-6106	140	7	value	value	NOUN
ejpam-6106	140	8	for	for	ADP
ejpam-6106	140	9	β	β	X
ejpam-6106	140	10	∈	∈	PROPN
ejpam-6106	141	1	[	[	X
ejpam-6106	141	2	0	0	NUM
ejpam-6106	141	3	,	,	PUNCT
ejpam-6106	141	4	1	1	NUM
ejpam-6106	141	5	)	)	PUNCT
ejpam-6106	141	6	such	such	ADJ
ejpam-6106	141	7	that	that	SCONJ
ejpam-6106	141	8	s(lx	s(lx	NOUN
ejpam-6106	141	9	,	,	PUNCT
ejpam-6106	141	10	ll	ll	NOUN
ejpam-6106	141	11	)	)	PUNCT
ejpam-6106	141	12	≤	≤	NOUN
ejpam-6106	141	13	βs(x	βs(x	PUNCT
ejpam-6106	141	14	,	,	PUNCT
ejpam-6106	141	15	l	l	NOUN
ejpam-6106	141	16	)	)	PUNCT
ejpam-6106	141	17	.	.	PUNCT
ejpam-6106	142	1	(	(	PUNCT
ejpam-6106	142	2	3	3	X
ejpam-6106	142	3	)	)	PUNCT
ejpam-6106	142	4	where	where	SCONJ
ejpam-6106	142	5	x	x	X
ejpam-6106	142	6	,	,	PUNCT
ejpam-6106	142	7	l	l	PROPN
ejpam-6106	142	8	∈	∈	PROPN
ejpam-6106	142	9	z.	z.	PROPN
ejpam-6106	142	10	corollary	corollary	NOUN
ejpam-6106	142	11	1	1	PROPN
ejpam-6106	142	12	.	.	PUNCT
ejpam-6106	142	13	(	(	PUNCT
ejpam-6106	142	14	banach	banach	ADV
ejpam-6106	142	15	fixed	fix	VERB
ejpam-6106	142	16	point	point	NOUN
ejpam-6106	142	17	theorem	theorem	VERB
ejpam-6106	142	18	)	)	PUNCT
ejpam-6106	142	19	let	let	VERB
ejpam-6106	142	20	(	(	PUNCT
ejpam-6106	142	21	z	z	NOUN
ejpam-6106	142	22	,	,	PUNCT
ejpam-6106	142	23	s	s	PART
ejpam-6106	142	24	)	)	PUNCT
ejpam-6106	142	25	be	be	AUX
ejpam-6106	142	26	a	a	DET
ejpam-6106	142	27	complete	complete	ADJ
ejpam-6106	142	28	metric	metric	ADJ
ejpam-6106	142	29	space	space	NOUN
ejpam-6106	142	30	,	,	PUNCT
ejpam-6106	142	31	and	and	CCONJ
ejpam-6106	142	32	let	let	VERB
ejpam-6106	142	33	l	l	NOUN
ejpam-6106	143	1	:	:	PUNCT
ejpam-6106	143	2	z	z	X
ejpam-6106	143	3	→	→	SYM
ejpam-6106	143	4	z	z	AUX
ejpam-6106	143	5	be	be	AUX
ejpam-6106	143	6	a	a	DET
ejpam-6106	143	7	contraction	contraction	NOUN
ejpam-6106	143	8	mapping	mapping	NOUN
ejpam-6106	143	9	,	,	PUNCT
ejpam-6106	143	10	then	then	ADV
ejpam-6106	143	11	it	it	PRON
ejpam-6106	143	12	admits	admit	VERB
ejpam-6106	143	13	a	a	DET
ejpam-6106	143	14	fixed	fixed	ADJ
ejpam-6106	143	15	point	point	NOUN
ejpam-6106	143	16	.	.	PUNCT
ejpam-6106	144	1	proof	proof	NOUN
ejpam-6106	144	2	.	.	PUNCT
ejpam-6106	145	1	let	let	VERB
ejpam-6106	145	2	|z|	|z|	NOUN
ejpam-6106	145	3	≥	≥	PRON
ejpam-6106	145	4	n.	n.	NOUN
ejpam-6106	145	5	assume	assume	VERB
ejpam-6106	145	6	that	that	SCONJ
ejpam-6106	145	7	∃	∃	PROPN
ejpam-6106	145	8	l	l	PROPN
ejpam-6106	145	9	∈	∈	PROPN
ejpam-6106	145	10	z	z	X
ejpam-6106	145	11	which	which	PRON
ejpam-6106	145	12	satisfy	satisfy	VERB
ejpam-6106	145	13	l(l(l	l(l(l	PROPN
ejpam-6106	145	14	)	)	PUNCT
ejpam-6106	145	15	)	)	PUNCT
ejpam-6106	146	1	=	=	PUNCT
ejpam-6106	146	2	l.	l.	NOUN
ejpam-6106	146	3	as	as	ADP
ejpam-6106	146	4	a	a	DET
ejpam-6106	146	5	result	result	NOUN
ejpam-6106	146	6	,	,	PUNCT
ejpam-6106	146	7	s(x	s(x	NOUN
ejpam-6106	146	8	,	,	PUNCT
ejpam-6106	146	9	lx	lx	NOUN
ejpam-6106	146	10	)	)	PUNCT
ejpam-6106	146	11	=	=	SYM
ejpam-6106	146	12	s(lx	s(lx	NOUN
ejpam-6106	146	13	,	,	PUNCT
ejpam-6106	146	14	x	x	NOUN
ejpam-6106	146	15	)	)	PUNCT
ejpam-6106	146	16	=	=	SYM
ejpam-6106	146	17	s(lx	s(lx	NOUN
ejpam-6106	146	18	,	,	PUNCT
ejpam-6106	146	19	l(lx	l(lx	NOUN
ejpam-6106	146	20	)	)	PUNCT
ejpam-6106	146	21	)	)	PUNCT
ejpam-6106	146	22	,	,	PUNCT
ejpam-6106	146	23	which	which	PRON
ejpam-6106	146	24	contradicts	contradict	VERB
ejpam-6106	146	25	to	to	ADP
ejpam-6106	146	26	(	(	PUNCT
ejpam-6106	146	27	3	3	NUM
ejpam-6106	146	28	)	)	PUNCT
ejpam-6106	146	29	.	.	PUNCT
ejpam-6106	147	1	condition	condition	NOUN
ejpam-6106	147	2	(	(	PUNCT
ejpam-6106	147	3	i	i	NOUN
ejpam-6106	147	4	)	)	PUNCT
ejpam-6106	147	5	of	of	ADP
ejpam-6106	147	6	theorem(4	theorem(4	PROPN
ejpam-6106	147	7	)	)	PUNCT
ejpam-6106	147	8	is	be	AUX
ejpam-6106	147	9	thus	thus	ADV
ejpam-6106	147	10	met	meet	VERB
ejpam-6106	147	11	.	.	PUNCT
ejpam-6106	148	1	assume	assume	VERB
ejpam-6106	148	2	that	that	SCONJ
ejpam-6106	148	3	l1	l1	PROPN
ejpam-6106	148	4	,	,	PUNCT
ejpam-6106	148	5	l2	l2	NOUN
ejpam-6106	148	6	,	,	PUNCT
ejpam-6106	148	7	l3	l3	PROPN
ejpam-6106	148	8	·	·	PUNCT
ejpam-6106	148	9	·	·	PUNCT
ejpam-6106	148	10	·	·	PUNCT
ejpam-6106	148	11	ln	ln	PROPN
ejpam-6106	148	12	∈	∈	PROPN
ejpam-6106	149	1	z	z	NOUN
ejpam-6106	149	2	are	be	AUX
ejpam-6106	149	3	distinctive	distinctive	ADJ
ejpam-6106	149	4	pairs	pair	NOUN
ejpam-6106	149	5	.	.	PUNCT
ejpam-6106	150	1	by	by	ADP
ejpam-6106	150	2	using	use	VERB
ejpam-6106	150	3	the	the	DET
ejpam-6106	150	4	contraction	contraction	NOUN
ejpam-6106	150	5	definition	definition	NOUN
ejpam-6106	150	6	,	,	PUNCT
ejpam-6106	150	7	we	we	PRON
ejpam-6106	150	8	have	have	AUX
ejpam-6106	150	9	s(ll1	s(ll1	VERB
ejpam-6106	150	10	,	,	PUNCT
ejpam-6106	150	11	ll2	ll2	NOUN
ejpam-6106	150	12	)	)	PUNCT
ejpam-6106	150	13	≤	≤	PROPN
ejpam-6106	150	14	αs(l1	αs(l1	PROPN
ejpam-6106	150	15	,	,	PUNCT
ejpam-6106	150	16	l2	l2	NOUN
ejpam-6106	150	17	)	)	PUNCT
ejpam-6106	150	18	,	,	PUNCT
ejpam-6106	150	19	s(ll2	s(ll2	PROPN
ejpam-6106	150	20	,	,	PUNCT
ejpam-6106	150	21	ll3	ll3	NOUN
ejpam-6106	150	22	)	)	PUNCT
ejpam-6106	150	23	≤	≤	NOUN
ejpam-6106	150	24	αs(l2	αs(l2	NUM
ejpam-6106	150	25	,	,	PUNCT
ejpam-6106	150	26	l3	l3	PROPN
ejpam-6106	150	27	)	)	PUNCT
ejpam-6106	150	28	,	,	PUNCT
ejpam-6106	150	29	·	·	PUNCT
ejpam-6106	150	30	·	·	PUNCT
ejpam-6106	150	31	·	·	PUNCT
ejpam-6106	150	32	,	,	PUNCT
ejpam-6106	150	33	s(lln−1	s(lln−1	X
ejpam-6106	150	34	,	,	PUNCT
ejpam-6106	150	35	lln	lln	PROPN
ejpam-6106	150	36	)	)	PUNCT
ejpam-6106	150	37	)	)	PUNCT
ejpam-6106	150	38	≤	≤	NUM
ejpam-6106	151	1	αs(ln−1	αs(ln−1	NUM
ejpam-6106	151	2	,	,	PUNCT
ejpam-6106	151	3	ln	ln	ADJ
ejpam-6106	151	4	)	)	PUNCT
ejpam-6106	151	5	and	and	CCONJ
ejpam-6106	151	6	which	which	PRON
ejpam-6106	151	7	implies	imply	VERB
ejpam-6106	151	8	condition	condition	NOUN
ejpam-6106	151	9	(	(	PUNCT
ejpam-6106	151	10	ii	ii	NOUN
ejpam-6106	151	11	)	)	PUNCT
ejpam-6106	151	12	of	of	ADP
ejpam-6106	151	13	theorem(4	theorem(4	NOUN
ejpam-6106	151	14	)	)	PUNCT
ejpam-6106	151	15	.	.	PUNCT
ejpam-6106	152	1	this	this	PRON
ejpam-6106	152	2	concludes	conclude	VERB
ejpam-6106	152	3	the	the	DET
ejpam-6106	152	4	evidence	evidence	NOUN
ejpam-6106	152	5	for	for	ADP
ejpam-6106	152	6	the	the	DET
ejpam-6106	152	7	existence	existence	NOUN
ejpam-6106	152	8	of	of	ADP
ejpam-6106	152	9	a	a	DET
ejpam-6106	152	10	fixed	fix	VERB
ejpam-6106	152	11	point	point	NOUN
ejpam-6106	152	12	.	.	PUNCT
ejpam-6106	153	1	assume	assume	VERB
ejpam-6106	153	2	there	there	PRON
ejpam-6106	153	3	are	be	VERB
ejpam-6106	153	4	x	x	NOUN
ejpam-6106	153	5	,	,	PUNCT
ejpam-6106	153	6	y	y	PROPN
ejpam-6106	153	7	∈	∈	PROPN
ejpam-6106	153	8	z	z	NOUN
ejpam-6106	153	9	such	such	ADJ
ejpam-6106	153	10	that	that	PRON
ejpam-6106	154	1	lx	lx	NOUN
ejpam-6106	154	2	=	=	PUNCT
ejpam-6106	154	3	x	x	X
ejpam-6106	154	4	and	and	CCONJ
ejpam-6106	154	5	ly	ly	X
ejpam-6106	154	6	=	=	SYM
ejpam-6106	154	7	y.	y.	PROPN
ejpam-6106	154	8	hence	hence	ADV
ejpam-6106	154	9	,	,	PUNCT
ejpam-6106	154	10	s(lx	s(lx	NOUN
ejpam-6106	154	11	,	,	PUNCT
ejpam-6106	154	12	ly	ly	NOUN
ejpam-6106	154	13	)	)	PUNCT
ejpam-6106	154	14	=	=	SYM
ejpam-6106	154	15	s(x	s(x	PROPN
ejpam-6106	154	16	,	,	PUNCT
ejpam-6106	154	17	y	y	PROPN
ejpam-6106	154	18	)	)	PUNCT
ejpam-6106	154	19	which	which	PRON
ejpam-6106	154	20	contradicts	contradict	VERB
ejpam-6106	154	21	to	to	ADP
ejpam-6106	154	22	(	(	PUNCT
ejpam-6106	154	23	3	3	NUM
ejpam-6106	154	24	)	)	PUNCT
ejpam-6106	154	25	.	.	PUNCT
ejpam-6106	155	1	example	example	NOUN
ejpam-6106	156	1	8	8	NUM
ejpam-6106	156	2	.	.	PUNCT
ejpam-6106	157	1	let	let	VERB
ejpam-6106	157	2	us	we	PRON
ejpam-6106	157	3	define	define	VERB
ejpam-6106	157	4	a	a	DET
ejpam-6106	157	5	mapping	mapping	NOUN
ejpam-6106	157	6	l	l	NOUN
ejpam-6106	157	7	:	:	PUNCT
ejpam-6106	157	8	z	z	X
ejpam-6106	157	9	→	→	SYM
ejpam-6106	157	10	z	z	NOUN
ejpam-6106	157	11	contracting	contracting	NOUN
ejpam-6106	157	12	perimeters	perimeter	NOUN
ejpam-6106	157	13	of	of	ADP
ejpam-6106	157	14	polygons	polygon	NOUN
ejpam-6106	157	15	for	for	ADP
ejpam-6106	157	16	a	a	DET
ejpam-6106	157	17	metric	metric	ADJ
ejpam-6106	157	18	space	space	NOUN
ejpam-6106	157	19	z	z	NOUN
ejpam-6106	157	20	,	,	PUNCT
ejpam-6106	157	21	that	that	PRON
ejpam-6106	157	22	is	be	AUX
ejpam-6106	157	23	not	not	PART
ejpam-6106	157	24	a	a	DET
ejpam-6106	157	25	contraction	contraction	NOUN
ejpam-6106	157	26	mapping	mapping	NOUN
ejpam-6106	157	27	with	with	ADP
ejpam-6106	157	28	|z|	|z|	NOUN
ejpam-6106	157	29	=	=	SYM
ejpam-6106	157	30	n0	n0	PROPN
ejpam-6106	157	31	.	.	PUNCT
ejpam-6106	158	1	let	let	VERB
ejpam-6106	158	2	z	z	NOUN
ejpam-6106	158	3	=	=	PRON
ejpam-6106	158	4	{	{	PUNCT
ejpam-6106	158	5	l∗	l∗	PROPN
ejpam-6106	158	6	,	,	PUNCT
ejpam-6106	158	7	l1	l1	PROPN
ejpam-6106	158	8	,	,	PUNCT
ejpam-6106	158	9	l2	l2	NOUN
ejpam-6106	158	10	...	...	PUNCT
ejpam-6106	158	11	}	}	PUNCT
ejpam-6106	158	12	and	and	CCONJ
ejpam-6106	158	13	let	let	VERB
ejpam-6106	158	14	b	b	PROPN
ejpam-6106	158	15	∈	∈	PROPN
ejpam-6106	158	16	r+	r+	X
ejpam-6106	158	17	.	.	PUNCT
ejpam-6106	159	1	consider	consider	VERB
ejpam-6106	159	2	a	a	DET
ejpam-6106	159	3	metric	metric	NOUN
ejpam-6106	159	4	s	s	NOUN
ejpam-6106	159	5	on	on	ADP
ejpam-6106	159	6	z	z	NOUN
ejpam-6106	159	7	such	such	ADJ
ejpam-6106	159	8	that	that	SCONJ
ejpam-6106	159	9	s(x	s(x	PROPN
ejpam-6106	159	10	,	,	PUNCT
ejpam-6106	159	11	l	l	NOUN
ejpam-6106	159	12	)	)	PUNCT
ejpam-6106	159	13	=	=	SYM
ejpam-6106	159	14			NUM
ejpam-6106	159	15	b	b	NOUN
ejpam-6106	159	16	2⌊i/(n−1)⌋	2⌊i/(n−1)⌋	NUM
ejpam-6106	160	1	if	if	SCONJ
ejpam-6106	160	2	x	x	X
ejpam-6106	160	3	=	=	SYM
ejpam-6106	160	4	li	li	PROPN
ejpam-6106	160	5	,	,	PUNCT
ejpam-6106	160	6	l	l	PROPN
ejpam-6106	160	7	=	=	SYM
ejpam-6106	160	8	li+1	li+1	PROPN
ejpam-6106	160	9	,	,	PUNCT
ejpam-6106	160	10	i	i	PRON
ejpam-6106	160	11	∈	∈	VERB
ejpam-6106	160	12	n	n	PRON
ejpam-6106	160	13	;	;	PUNCT
ejpam-6106	160	14	s(li	s(li	NUM
ejpam-6106	160	15	,	,	PUNCT
ejpam-6106	160	16	li+1	li+1	NOUN
ejpam-6106	160	17	)	)	PUNCT
ejpam-6106	160	18	+	+	CCONJ
ejpam-6106	160	19	·	·	PUNCT
ejpam-6106	160	20	·	·	PUNCT
ejpam-6106	160	21	·	·	PUNCT
ejpam-6106	161	1	+	+	NUM
ejpam-6106	161	2	s(lj−1	s(lj−1	NOUN
ejpam-6106	161	3	,	,	PUNCT
ejpam-6106	161	4	lj	lj	PROPN
ejpam-6106	161	5	)	)	PUNCT
ejpam-6106	161	6	if	if	SCONJ
ejpam-6106	161	7	x	x	X
ejpam-6106	161	8	=	=	SYM
ejpam-6106	161	9	li	li	PROPN
ejpam-6106	161	10	,	,	PUNCT
ejpam-6106	161	11	l	l	PROPN
ejpam-6106	162	1	=	=	PUNCT
ejpam-6106	162	2	lj	lj	INTJ
ejpam-6106	162	3	,	,	PUNCT
ejpam-6106	162	4	i+	i+	NUM
ejpam-6106	162	5	1	1	NUM
ejpam-6106	162	6	<	<	X
ejpam-6106	162	7	j	j	PROPN
ejpam-6106	162	8	;	;	PUNCT
ejpam-6106	162	9	2(n−	2(n−	NUM
ejpam-6106	162	10	1)b−	1)b−	NUM
ejpam-6106	162	11	s(l0	s(l0	NOUN
ejpam-6106	162	12	,	,	PUNCT
ejpam-6106	162	13	li	li	PROPN
ejpam-6106	162	14	)	)	PUNCT
ejpam-6106	163	1	if	if	SCONJ
ejpam-6106	163	2	x	x	X
ejpam-6106	163	3	=	=	SYM
ejpam-6106	163	4	li	li	X
ejpam-6106	163	5	,	,	PUNCT
ejpam-6106	163	6	l	l	NOUN
ejpam-6106	163	7	=	=	PUNCT
ejpam-6106	163	8	l∗	l∗	PROPN
ejpam-6106	163	9	0	0	PUNCT
ejpam-6106	164	1	if	if	SCONJ
ejpam-6106	164	2	x	x	X
ejpam-6106	164	3	=	=	SYM
ejpam-6106	164	4	l	l	NOUN
ejpam-6106	164	5	,	,	PUNCT
ejpam-6106	164	6	where	where	SCONJ
ejpam-6106	164	7	⌊.⌋	⌊.⌋	PROPN
ejpam-6106	164	8	is	be	AUX
ejpam-6106	164	9	a	a	DET
ejpam-6106	164	10	floor	floor	NOUN
ejpam-6106	164	11	function	function	NOUN
ejpam-6106	164	12	.	.	PUNCT
ejpam-6106	165	1	define	define	VERB
ejpam-6106	165	2	a	a	DET
ejpam-6106	165	3	mapping	mapping	NOUN
ejpam-6106	165	4	l	l	NOUN
ejpam-6106	165	5	:	:	PUNCT
ejpam-6106	165	6	z	z	X
ejpam-6106	165	7	→	→	SYM
ejpam-6106	165	8	z	z	PROPN
ejpam-6106	165	9	as	as	ADP
ejpam-6106	165	10	lli	lli	PROPN
ejpam-6106	165	11	=	=	SYM
ejpam-6106	165	12	li+1	li+1	PROPN
ejpam-6106	165	13	for	for	ADP
ejpam-6106	165	14	all	all	DET
ejpam-6106	165	15	i	i	PRON
ejpam-6106	165	16	=	=	NOUN
ejpam-6106	165	17	0	0	NUM
ejpam-6106	165	18	,	,	PUNCT
ejpam-6106	165	19	1	1	NUM
ejpam-6106	165	20	,	,	PUNCT
ejpam-6106	165	21	...	...	PUNCT
ejpam-6106	165	22	and	and	CCONJ
ejpam-6106	165	23	ll∗	ll∗	NOUN
ejpam-6106	165	24	=	=	PUNCT
ejpam-6106	165	25	l∗.	l∗.	NOUN
ejpam-6106	165	26	since	since	SCONJ
ejpam-6106	165	27	s(l2n	s(l2n	NOUN
ejpam-6106	165	28	,	,	PUNCT
ejpam-6106	165	29	l2n+1	l2n+1	PROPN
ejpam-6106	165	30	)	)	PUNCT
ejpam-6106	165	31	=	=	SYM
ejpam-6106	165	32	s(ll2n	s(ll2n	NOUN
ejpam-6106	165	33	,	,	PUNCT
ejpam-6106	165	34	ll2n+1	ll2n+1	PROPN
ejpam-6106	165	35	)	)	PUNCT
ejpam-6106	165	36	,	,	PUNCT
ejpam-6106	165	37	n	n	NOUN
ejpam-6106	165	38	=	=	SYM
ejpam-6106	165	39	0	0	NUM
ejpam-6106	165	40	,	,	PUNCT
ejpam-6106	165	41	1	1	NUM
ejpam-6106	165	42	,	,	PUNCT
ejpam-6106	165	43	....	....	PUNCT
ejpam-6106	166	1	we	we	PRON
ejpam-6106	166	2	note	note	VERB
ejpam-6106	166	3	that	that	SCONJ
ejpam-6106	166	4	l	l	NOUN
ejpam-6106	166	5	does	do	AUX
ejpam-6106	166	6	not	not	PART
ejpam-6106	166	7	satisfy	satisfy	VERB
ejpam-6106	166	8	(	(	PUNCT
ejpam-6106	166	9	1	1	NUM
ejpam-6106	166	10	)	)	PUNCT
ejpam-6106	166	11	.	.	PUNCT
ejpam-6106	167	1	let	let	VERB
ejpam-6106	167	2	us	we	PRON
ejpam-6106	167	3	demonstrate	demonstrate	VERB
ejpam-6106	167	4	that	that	SCONJ
ejpam-6106	167	5	inequality(1	inequality(1	PROPN
ejpam-6106	167	6	)	)	PUNCT
ejpam-6106	167	7	is	be	AUX
ejpam-6106	167	8	true	true	ADJ
ejpam-6106	167	9	for	for	SCONJ
ejpam-6106	167	10	each	each	PRON
ejpam-6106	167	11	of	of	ADP
ejpam-6106	167	12	the	the	DET
ejpam-6106	167	13	n	n	CCONJ
ejpam-6106	167	14	distinct	distinct	ADJ
ejpam-6106	167	15	pairwise	pairwise	NOUN
ejpam-6106	167	16	points	point	NOUN
ejpam-6106	167	17	in	in	ADP
ejpam-6106	167	18	the	the	DET
ejpam-6106	167	19	space	space	NOUN
ejpam-6106	167	20	z.	z.	PROPN
ejpam-6106	167	21	consider	consider	VERB
ejpam-6106	167	22	the	the	DET
ejpam-6106	167	23	points	point	NOUN
ejpam-6106	167	24	l1	l1	PROPN
ejpam-6106	167	25	,	,	PUNCT
ejpam-6106	167	26	l2	l2	NOUN
ejpam-6106	167	27	,	,	PUNCT
ejpam-6106	167	28	·	·	PUNCT
ejpam-6106	167	29	·	·	PUNCT
ejpam-6106	167	30	·	·	PUNCT
ejpam-6106	167	31	,	,	PUNCT
ejpam-6106	167	32	ln−1	ln−1	PROPN
ejpam-6106	167	33	,	,	PUNCT
ejpam-6106	167	34	l	l	PROPN
ejpam-6106	167	35	∗	∗	NOUN
ejpam-6106	167	36	∈	∈	PROPN
ejpam-6106	167	37	z.	z.	NOUN
ejpam-6106	167	38	using	use	VERB
ejpam-6106	167	39	structure	structure	NOUN
ejpam-6106	167	40	of	of	ADP
ejpam-6106	167	41	metric	metric	NOUN
ejpam-6106	167	42	s	s	X
ejpam-6106	167	43	we	we	PRON
ejpam-6106	167	44	have	have	VERB
ejpam-6106	167	45	s(li	s(li	NUM
ejpam-6106	167	46	,	,	PUNCT
ejpam-6106	167	47	li+1	li+1	NOUN
ejpam-6106	167	48	)	)	PUNCT
ejpam-6106	168	1	+	+	X
ejpam-6106	168	2	s(li+1	s(li+1	X
ejpam-6106	168	3	,	,	PUNCT
ejpam-6106	168	4	li+2	li+2	X
ejpam-6106	168	5	)	)	PUNCT
ejpam-6106	169	1	+	+	CCONJ
ejpam-6106	169	2	·	·	PUNCT
ejpam-6106	169	3	·	·	PUNCT
ejpam-6106	169	4	·	·	PUNCT
ejpam-6106	169	5	+	+	NUM
ejpam-6106	169	6	s(l∗	s(l∗	ADJ
ejpam-6106	169	7	,	,	PUNCT
ejpam-6106	169	8	li+(n−1	li+(n−1	ADJ
ejpam-6106	169	9	)	)	PUNCT
ejpam-6106	169	10	)	)	PUNCT
ejpam-6106	170	1	+	+	CCONJ
ejpam-6106	170	2	s(li+(n−1	s(li+(n−1	ADJ
ejpam-6106	170	3	)	)	PUNCT
ejpam-6106	170	4	,	,	PUNCT
ejpam-6106	170	5	li+n	li+n	PROPN
ejpam-6106	170	6	)	)	PUNCT
ejpam-6106	170	7	m.	m.	NOUN
ejpam-6106	170	8	nazam	nazam	PROPN
ejpam-6106	170	9	,	,	PUNCT
ejpam-6106	170	10	u.	u.	PROPN
ejpam-6106	170	11	habiba	habiba	PROPN
ejpam-6106	170	12	,	,	PUNCT
ejpam-6106	170	13	m.	m.	PROPN
ejpam-6106	170	14	de	de	X
ejpam-6106	170	15	la	la	PROPN
ejpam-6106	170	16	sen	sen	PROPN
ejpam-6106	170	17	/	/	SYM
ejpam-6106	170	18	eur	eur	PROPN
ejpam-6106	170	19	.	.	PUNCT
ejpam-6106	171	1	j.	j.	PROPN
ejpam-6106	171	2	pure	pure	PROPN
ejpam-6106	171	3	appl	appl	PROPN
ejpam-6106	171	4	.	.	PROPN
ejpam-6106	171	5	math	math	PROPN
ejpam-6106	171	6	,	,	PUNCT
ejpam-6106	171	7	18	18	NUM
ejpam-6106	171	8	(	(	PUNCT
ejpam-6106	171	9	4	4	NUM
ejpam-6106	171	10	)	)	PUNCT
ejpam-6106	171	11	(	(	PUNCT
ejpam-6106	171	12	2025	2025	NUM
ejpam-6106	171	13	)	)	PUNCT
ejpam-6106	171	14	,	,	PUNCT
ejpam-6106	171	15	6106	6106	NUM
ejpam-6106	171	16	7	7	NUM
ejpam-6106	171	17	of	of	ADP
ejpam-6106	171	18	10	10	NUM
ejpam-6106	171	19	=	=	SYM
ejpam-6106	171	20	(	(	PUNCT
ejpam-6106	171	21	n−	n−	NOUN
ejpam-6106	171	22	1)s(li	1)s(li	NUM
ejpam-6106	171	23	,	,	PUNCT
ejpam-6106	171	24	l	l	NOUN
ejpam-6106	171	25	∗	∗	NOUN
ejpam-6106	171	26	)	)	PUNCT
ejpam-6106	171	27	=	=	SYM
ejpam-6106	172	1	(	(	PUNCT
ejpam-6106	172	2	n−	n−	NOUN
ejpam-6106	172	3	1)(2(n−	1)(2(n−	NUM
ejpam-6106	172	4	1)a−	1)a−	NUM
ejpam-6106	172	5	s(li	s(li	NOUN
ejpam-6106	172	6	,	,	PUNCT
ejpam-6106	172	7	l	l	NOUN
ejpam-6106	172	8	∗	∗	NOUN
ejpam-6106	172	9	)	)	PUNCT
ejpam-6106	172	10	)	)	PUNCT
ejpam-6106	172	11	,	,	PUNCT
ejpam-6106	172	12	and	and	CCONJ
ejpam-6106	172	13	s(lli	s(lli	ADV
ejpam-6106	172	14	,	,	PUNCT
ejpam-6106	172	15	lli+1)+s(lli+1	lli+1)+s(lli+1	PROPN
ejpam-6106	172	16	,	,	PUNCT
ejpam-6106	172	17	lli+2)+	lli+2)+	PROPN
ejpam-6106	172	18	·	·	PUNCT
ejpam-6106	172	19	·	·	PUNCT
ejpam-6106	172	20	·	·	PUNCT
ejpam-6106	173	1	+	+	NOUN
ejpam-6106	173	2	s(ll∗	s(ll∗	NOUN
ejpam-6106	173	3	,	,	PUNCT
ejpam-6106	173	4	lli+(n−1))+s(lli+(n−1	lli+(n−1))+s(lli+(n−1	NOUN
ejpam-6106	173	5	)	)	PUNCT
ejpam-6106	173	6	,	,	PUNCT
ejpam-6106	173	7	lli+n	lli+n	PROPN
ejpam-6106	173	8	)	)	PUNCT
ejpam-6106	173	9	=	=	SYM
ejpam-6106	173	10	(	(	PUNCT
ejpam-6106	173	11	n−1)s(lli+1	n−1)s(lli+1	PROPN
ejpam-6106	173	12	,	,	PUNCT
ejpam-6106	173	13	ll	ll	NOUN
ejpam-6106	173	14	∗	∗	NOUN
ejpam-6106	173	15	)	)	PUNCT
ejpam-6106	174	1	=	=	SYM
ejpam-6106	174	2	(	(	PUNCT
ejpam-6106	174	3	n−	n−	NOUN
ejpam-6106	174	4	1)(2(n−	1)(2(n−	VERB
ejpam-6106	174	5	1)b−	1)b−	NUM
ejpam-6106	174	6	s(li+1	s(li+1	PROPN
ejpam-6106	174	7	,	,	PUNCT
ejpam-6106	174	8	l	l	NOUN
ejpam-6106	174	9	∗	∗	NOUN
ejpam-6106	174	10	)	)	PUNCT
ejpam-6106	174	11	)	)	PUNCT
ejpam-6106	174	12	.	.	PUNCT
ejpam-6106	175	1	moreover	moreover	ADV
ejpam-6106	175	2	,	,	PUNCT
ejpam-6106	175	3	s(l0	s(l0	NOUN
ejpam-6106	175	4	,	,	PUNCT
ejpam-6106	175	5	li	li	NOUN
ejpam-6106	175	6	)	)	PUNCT
ejpam-6106	175	7	=	=	PUNCT
ejpam-6106	175	8			NUM
ejpam-6106	175	9	2(n−	2(n−	NUM
ejpam-6106	175	10	1)b(1−	1)b(1−	NUM
ejpam-6106	175	11	1	1	NUM
ejpam-6106	175	12	2n	2n	NUM
ejpam-6106	175	13	)	)	PUNCT
ejpam-6106	176	1	if	if	SCONJ
ejpam-6106	176	2	i	i	PRON
ejpam-6106	176	3	=	=	VERB
ejpam-6106	176	4	n(n−	n(n−	ADJ
ejpam-6106	176	5	1	1	NUM
ejpam-6106	176	6	)	)	PUNCT
ejpam-6106	176	7	2(n−	2(n−	NUM
ejpam-6106	176	8	1)b(1−	1)b(1−	NUM
ejpam-6106	176	9	1	1	NUM
ejpam-6106	176	10	2n	2n	NUM
ejpam-6106	176	11	)	)	PUNCT
ejpam-6106	176	12	−	−	PROPN
ejpam-6106	177	1	b	b	X
ejpam-6106	177	2	2n−1	2n−1	NUM
ejpam-6106	177	3	if	if	SCONJ
ejpam-6106	177	4	i	i	PRON
ejpam-6106	177	5	=	=	VERB
ejpam-6106	177	6	n(n−	n(n−	PROPN
ejpam-6106	177	7	1)−	1)−	PROPN
ejpam-6106	177	8	1	1	NUM
ejpam-6106	177	9	,	,	PUNCT
ejpam-6106	177	10	n=1,2	n=1,2	ADJ
ejpam-6106	177	11	,	,	PUNCT
ejpam-6106	177	12	...	...	PUNCT
ejpam-6106	177	13	,	,	PUNCT
ejpam-6106	177	14	observe	observe	VERB
ejpam-6106	177	15	also	also	ADV
ejpam-6106	177	16	that	that	DET
ejpam-6106	177	17	s(t0	s(t0	NOUN
ejpam-6106	177	18	,	,	PUNCT
ejpam-6106	177	19	ti+1	ti+1	NOUN
ejpam-6106	177	20	)	)	PUNCT
ejpam-6106	177	21	=	=	SYM
ejpam-6106	177	22	s(t0	s(t0	NOUN
ejpam-6106	177	23	,	,	PUNCT
ejpam-6106	177	24	ti	ti	NOUN
ejpam-6106	177	25	)	)	PUNCT
ejpam-6106	177	26	+	+	SYM
ejpam-6106	177	27	b	b	X
ejpam-6106	177	28	2⌊i	2⌊i	NUM
ejpam-6106	177	29	/	/	SYM
ejpam-6106	177	30	n−1⌋	n−1⌋	NOUN
ejpam-6106	177	31	.	.	PUNCT
ejpam-6106	178	1	consider	consider	VERB
ejpam-6106	178	2	the	the	DET
ejpam-6106	178	3	ratio	ratio	NOUN
ejpam-6106	178	4	s(lli	s(lli	ADV
ejpam-6106	178	5	,	,	PUNCT
ejpam-6106	178	6	lli+1	lli+1	PROPN
ejpam-6106	178	7	)	)	PUNCT
ejpam-6106	178	8	+	+	CCONJ
ejpam-6106	178	9	s(lli+1	s(lli+1	NUM
ejpam-6106	178	10	,	,	PUNCT
ejpam-6106	178	11	lli+2	lli+2	PROPN
ejpam-6106	178	12	)	)	PUNCT
ejpam-6106	178	13	+	+	CCONJ
ejpam-6106	178	14	·	·	PUNCT
ejpam-6106	178	15	·	·	PUNCT
ejpam-6106	179	1	·	·	PUNCT
ejpam-6106	179	2	+	+	NUM
ejpam-6106	179	3	s(ll∗	s(ll∗	NOUN
ejpam-6106	179	4	,	,	PUNCT
ejpam-6106	179	5	lli+n−1	lli+n−1	PROPN
ejpam-6106	179	6	)	)	PUNCT
ejpam-6106	179	7	+	+	X
ejpam-6106	179	8	s(lli+n−1	s(lli+n−1	ADJ
ejpam-6106	179	9	,	,	PUNCT
ejpam-6106	179	10	lli+n	lli+n	PROPN
ejpam-6106	179	11	)	)	PUNCT
ejpam-6106	179	12	s(li	s(li	NOUN
ejpam-6106	179	13	,	,	PUNCT
ejpam-6106	179	14	li+1	li+1	NOUN
ejpam-6106	179	15	)	)	PUNCT
ejpam-6106	179	16	+	+	X
ejpam-6106	179	17	s(li+1	s(li+1	X
ejpam-6106	179	18	,	,	PUNCT
ejpam-6106	179	19	li+2	li+2	X
ejpam-6106	179	20	)	)	PUNCT
ejpam-6106	180	1	+	+	CCONJ
ejpam-6106	180	2	·	·	PUNCT
ejpam-6106	180	3	·	·	PUNCT
ejpam-6106	180	4	·	·	PUNCT
ejpam-6106	180	5	+	+	NUM
ejpam-6106	180	6	s(l∗	s(l∗	ADJ
ejpam-6106	180	7	,	,	PUNCT
ejpam-6106	180	8	li+n−1	li+n−1	ADJ
ejpam-6106	180	9	)	)	PUNCT
ejpam-6106	181	1	+	+	X
ejpam-6106	181	2	s(li+n−1	s(li+n−1	PROPN
ejpam-6106	181	3	,	,	PUNCT
ejpam-6106	181	4	li+n	li+n	PROPN
ejpam-6106	181	5	=	=	PUNCT
ejpam-6106	181	6	(	(	PUNCT
ejpam-6106	181	7	n−	n−	NOUN
ejpam-6106	181	8	1)(2(n−	1)(2(n−	VERB
ejpam-6106	181	9	1)b−	1)b−	NUM
ejpam-6106	181	10	s(l0	s(l0	NOUN
ejpam-6106	181	11	,	,	PUNCT
ejpam-6106	181	12	li+1	li+1	NOUN
ejpam-6106	181	13	)	)	PUNCT
ejpam-6106	181	14	)	)	PUNCT
ejpam-6106	182	1	(	(	PUNCT
ejpam-6106	182	2	n−	n−	NOUN
ejpam-6106	182	3	1)(2(n−	1)(2(n−	VERB
ejpam-6106	182	4	1)b−	1)b−	NUM
ejpam-6106	182	5	s(l0	s(l0	NOUN
ejpam-6106	182	6	,	,	PUNCT
ejpam-6106	182	7	li	li	NOUN
ejpam-6106	182	8	)	)	PUNCT
ejpam-6106	182	9	)	)	PUNCT
ejpam-6106	183	1	=	=	PUNCT
ejpam-6106	184	1	2(n−	2(n−	NUM
ejpam-6106	184	2	1)b−	1)b−	NUM
ejpam-6106	184	3	s(l0	s(l0	NOUN
ejpam-6106	184	4	,	,	PUNCT
ejpam-6106	184	5	li+1	li+1	NOUN
ejpam-6106	184	6	)	)	PUNCT
ejpam-6106	184	7	2(n−	2(n−	NUM
ejpam-6106	184	8	1)b−	1)b−	NUM
ejpam-6106	184	9	s(l0	s(l0	NOUN
ejpam-6106	184	10	,	,	PUNCT
ejpam-6106	184	11	li	li	PROPN
ejpam-6106	184	12	)	)	PUNCT
ejpam-6106	184	13	=	=	PUNCT
ejpam-6106	185	1	2(n−	2(n−	NUM
ejpam-6106	185	2	1)b−	1)b−	NUM
ejpam-6106	185	3	s(l0	s(l0	NOUN
ejpam-6106	185	4	,	,	PUNCT
ejpam-6106	185	5	li)−	li)−	NOUN
ejpam-6106	185	6	b/2⌊i	b/2⌊i	NOUN
ejpam-6106	185	7	/	/	SYM
ejpam-6106	185	8	n−1⌋	n−1⌋	PROPN
ejpam-6106	185	9	2(n−	2(n−	NUM
ejpam-6106	185	10	1)b−	1)b−	NUM
ejpam-6106	185	11	s(l0	s(l0	NOUN
ejpam-6106	185	12	,	,	PUNCT
ejpam-6106	185	13	li	li	NOUN
ejpam-6106	185	14	)	)	PUNCT
ejpam-6106	185	15	=	=	SYM
ejpam-6106	185	16			NUM
ejpam-6106	185	17	2(n−1)b−2(n−1)b(1−1/2n)−b/2⌊i	2(n−1)b−2(n−1)b(1−1/2n)−b/2⌊i	NUM
ejpam-6106	185	18	/	/	SYM
ejpam-6106	185	19	n−1⌋	n−1⌋	NOUN
ejpam-6106	185	20	2(n−1)b−2b(n−1)(1−1/2n	2(n−1)b−2b(n−1)(1−1/2n	NUM
ejpam-6106	185	21	)	)	PUNCT
ejpam-6106	186	1	if	if	SCONJ
ejpam-6106	186	2	i	i	PRON
ejpam-6106	186	3	=	=	VERB
ejpam-6106	186	4	n(n−	n(n−	ADJ
ejpam-6106	186	5	1	1	NUM
ejpam-6106	186	6	)	)	PUNCT
ejpam-6106	186	7	;	;	PUNCT
ejpam-6106	186	8	2(n−1)b−2(n−1)b(1−1/2n)+a/2n−1−b/2⌊i	2(n−1)b−2(n−1)b(1−1/2n)+a/2n−1−b/2⌊i	NUM
ejpam-6106	186	9	/	/	SYM
ejpam-6106	187	1	n−1⌋	n−1⌋	VERB
ejpam-6106	187	2	2(n−1)b−2(n−1)b(1−1/2n)−b/2n−1	2(n−1)b−2(n−1)b(1−1/2n)−b/2n−1	NUM
ejpam-6106	187	3	if	if	SCONJ
ejpam-6106	187	4	i	i	PRON
ejpam-6106	187	5	=	=	VERB
ejpam-6106	187	6	n(n−	n(n−	PROPN
ejpam-6106	187	7	1)−	1)−	NUM
ejpam-6106	187	8	1	1	NUM
ejpam-6106	187	9	=	=	PUNCT
ejpam-6106	187	10			PROPN
ejpam-6106	187	11	2n−3	2n−3	NUM
ejpam-6106	187	12	2n−2	2n−2	NUM
ejpam-6106	187	13	if	if	SCONJ
ejpam-6106	187	14	i	i	PRON
ejpam-6106	187	15	=	=	VERB
ejpam-6106	187	16	n(n−	n(n−	ADJ
ejpam-6106	187	17	1	1	NUM
ejpam-6106	187	18	)	)	PUNCT
ejpam-6106	187	19	n−1	n−1	PROPN
ejpam-6106	187	20	n−2	n−2	PROPN
ejpam-6106	187	21	if	if	SCONJ
ejpam-6106	187	22	i	i	PRON
ejpam-6106	187	23	=	=	VERB
ejpam-6106	187	24	n(n−	n(n−	PROPN
ejpam-6106	187	25	1)−	1)−	PROPN
ejpam-6106	187	26	1	1	NUM
ejpam-6106	187	27	.	.	PUNCT
ejpam-6106	188	1	let	let	VERB
ejpam-6106	188	2	li+1	li+1	NOUN
ejpam-6106	188	3	,	,	PUNCT
ejpam-6106	188	4	li+2	li+2	PROPN
ejpam-6106	188	5	,	,	PUNCT
ejpam-6106	188	6	·	·	PUNCT
ejpam-6106	188	7	·	·	PUNCT
ejpam-6106	189	1	·	·	PUNCT
ejpam-6106	189	2	,	,	PUNCT
ejpam-6106	189	3	li+n	li+n	PROPN
ejpam-6106	189	4	∈	∈	PROPN
ejpam-6106	189	5	z.	z.	NOUN
ejpam-6106	189	6	we	we	PRON
ejpam-6106	189	7	see	see	VERB
ejpam-6106	189	8	that	that	PRON
ejpam-6106	189	9	s(li+1	s(li+1	ADV
ejpam-6106	189	10	,	,	PUNCT
ejpam-6106	189	11	li+2	li+2	PROPN
ejpam-6106	189	12	)	)	PUNCT
ejpam-6106	189	13	+	+	CCONJ
ejpam-6106	189	14	s(li+2	s(li+2	PROPN
ejpam-6106	189	15	,	,	PUNCT
ejpam-6106	189	16	li+3	li+3	X
ejpam-6106	189	17	)	)	PUNCT
ejpam-6106	189	18	+	+	CCONJ
ejpam-6106	189	19	·	·	PUNCT
ejpam-6106	189	20	·	·	PUNCT
ejpam-6106	189	21	·	·	PUNCT
ejpam-6106	189	22	+	+	CCONJ
ejpam-6106	189	23	s(li+n−1	s(li+n−1	PROPN
ejpam-6106	189	24	,	,	PUNCT
ejpam-6106	189	25	li+n	li+n	PROPN
ejpam-6106	189	26	)	)	PUNCT
ejpam-6106	189	27	−	−	PROPN
ejpam-6106	190	1	(	(	PUNCT
ejpam-6106	190	2	s(lli+1	s(lli+1	PROPN
ejpam-6106	190	3	,	,	PUNCT
ejpam-6106	190	4	lli+2	lli+2	PROPN
ejpam-6106	190	5	)	)	PUNCT
ejpam-6106	191	1	+	+	CCONJ
ejpam-6106	191	2	s(lli+2	s(lli+2	ADJ
ejpam-6106	191	3	,	,	PUNCT
ejpam-6106	191	4	lli+3	lli+3	PROPN
ejpam-6106	191	5	)	)	PUNCT
ejpam-6106	191	6	+	+	NUM
ejpam-6106	191	7	·	·	PUNCT
ejpam-6106	191	8	·	·	PUNCT
ejpam-6106	191	9	·	·	PUNCT
ejpam-6106	191	10	+	+	NUM
ejpam-6106	191	11	s(lli+n−1	s(lli+n−1	ADJ
ejpam-6106	191	12	,	,	PUNCT
ejpam-6106	191	13	lli+n	lli+n	PROPN
ejpam-6106	191	14	)	)	PUNCT
ejpam-6106	191	15	)	)	PUNCT
ejpam-6106	192	1	=	=	PUNCT
ejpam-6106	192	2	(	(	PUNCT
ejpam-6106	192	3	n−	n−	NOUN
ejpam-6106	192	4	1)(b/2⌊i/(n−1)⌋	1)(b/2⌊i/(n−1)⌋	NOUN
ejpam-6106	192	5	−	−	ADV
ejpam-6106	193	1	b/2⌊m/(n−1)⌋	b/2⌊m/(n−1)⌋	NOUN
ejpam-6106	193	2	)	)	PUNCT
ejpam-6106	193	3	.	.	PUNCT
ejpam-6106	194	1	consider	consider	VERB
ejpam-6106	194	2	the	the	DET
ejpam-6106	194	3	ratio	ratio	NOUN
ejpam-6106	194	4	ri	ri	PROPN
ejpam-6106	194	5	,	,	PUNCT
ejpam-6106	194	6	k	k	PROPN
ejpam-6106	194	7	=	=	SYM
ejpam-6106	194	8	s(lli+1	s(lli+1	PROPN
ejpam-6106	194	9	,	,	PUNCT
ejpam-6106	194	10	lli+2	lli+2	PROPN
ejpam-6106	194	11	)	)	PUNCT
ejpam-6106	194	12	+	+	CCONJ
ejpam-6106	194	13	s(lli+2	s(lli+2	ADJ
ejpam-6106	194	14	,	,	PUNCT
ejpam-6106	194	15	lli+3	lli+3	PROPN
ejpam-6106	194	16	)	)	PUNCT
ejpam-6106	194	17	+	+	NUM
ejpam-6106	194	18	·	·	PUNCT
ejpam-6106	194	19	·	·	PUNCT
ejpam-6106	194	20	·	·	PUNCT
ejpam-6106	195	1	+	+	NUM
ejpam-6106	195	2	s(lli+n−1	s(lli+n−1	ADJ
ejpam-6106	195	3	,	,	PUNCT
ejpam-6106	195	4	lli+n	lli+n	PROPN
ejpam-6106	195	5	)	)	PUNCT
ejpam-6106	195	6	s(li+1	s(li+1	NUM
ejpam-6106	195	7	,	,	PUNCT
ejpam-6106	195	8	li+2	li+2	PROPN
ejpam-6106	195	9	)	)	PUNCT
ejpam-6106	195	10	+	+	CCONJ
ejpam-6106	195	11	s(li+2	s(li+2	PROPN
ejpam-6106	195	12	,	,	PUNCT
ejpam-6106	195	13	li+3	li+3	X
ejpam-6106	195	14	)	)	PUNCT
ejpam-6106	195	15	+	+	CCONJ
ejpam-6106	195	16	·	·	PUNCT
ejpam-6106	195	17	·	·	PUNCT
ejpam-6106	195	18	·	·	PUNCT
ejpam-6106	196	1	+	+	CCONJ
ejpam-6106	196	2	s(li+n−1	s(li+n−1	PROPN
ejpam-6106	196	3	,	,	PUNCT
ejpam-6106	196	4	li+n	li+n	PROPN
ejpam-6106	196	5	)	)	PUNCT
ejpam-6106	196	6	=	=	SYM
ejpam-6106	196	7	s(li+1	s(li+1	ADJ
ejpam-6106	196	8	,	,	PUNCT
ejpam-6106	196	9	li+2	li+2	PROPN
ejpam-6106	196	10	)	)	PUNCT
ejpam-6106	196	11	+	+	CCONJ
ejpam-6106	196	12	s(li+2	s(li+2	PROPN
ejpam-6106	196	13	,	,	PUNCT
ejpam-6106	196	14	li+3	li+3	X
ejpam-6106	196	15	)	)	PUNCT
ejpam-6106	196	16	+	+	CCONJ
ejpam-6106	196	17	·	·	PUNCT
ejpam-6106	196	18	·	·	PUNCT
ejpam-6106	196	19	·	·	PUNCT
ejpam-6106	196	20	+	+	PUNCT
ejpam-6106	196	21	s(li+n−1	s(li+n−1	ADJ
ejpam-6106	196	22	,	,	PUNCT
ejpam-6106	196	23	li+n)−	li+n)−	NOUN
ejpam-6106	196	24	(	(	PUNCT
ejpam-6106	196	25	n−	n−	PROPN
ejpam-6106	196	26	1)(b/2⌊i/(n−1)⌋	1)(b/2⌊i/(n−1)⌋	NOUN
ejpam-6106	196	27	−	−	ADV
ejpam-6106	197	1	b/2⌊m/(n−1)⌋	b/2⌊m/(n−1)⌋	NOUN
ejpam-6106	197	2	)	)	PUNCT
ejpam-6106	197	3	s(li+1	s(li+1	ADV
ejpam-6106	197	4	,	,	PUNCT
ejpam-6106	197	5	li+2	li+2	VERB
ejpam-6106	198	1	+	+	CCONJ
ejpam-6106	198	2	s(li+2	s(li+2	PROPN
ejpam-6106	198	3	,	,	PUNCT
ejpam-6106	198	4	li+3	li+3	X
ejpam-6106	198	5	)	)	PUNCT
ejpam-6106	199	1	+	+	CCONJ
ejpam-6106	199	2	·	·	PUNCT
ejpam-6106	199	3	·	·	PUNCT
ejpam-6106	199	4	·	·	PUNCT
ejpam-6106	199	5	+	+	CCONJ
ejpam-6106	199	6	s(li+n−1	s(li+n−1	PROPN
ejpam-6106	199	7	,	,	PUNCT
ejpam-6106	199	8	li+n	li+n	PROPN
ejpam-6106	199	9	)	)	PUNCT
ejpam-6106	199	10	m.	m.	NOUN
ejpam-6106	199	11	nazam	nazam	PROPN
ejpam-6106	199	12	,	,	PUNCT
ejpam-6106	199	13	u.	u.	PROPN
ejpam-6106	199	14	habiba	habiba	PROPN
ejpam-6106	199	15	,	,	PUNCT
ejpam-6106	199	16	m.	m.	PROPN
ejpam-6106	199	17	de	de	X
ejpam-6106	199	18	la	la	PROPN
ejpam-6106	199	19	sen	sen	PROPN
ejpam-6106	199	20	/	/	SYM
ejpam-6106	199	21	eur	eur	PROPN
ejpam-6106	199	22	.	.	PUNCT
ejpam-6106	200	1	j.	j.	PROPN
ejpam-6106	200	2	pure	pure	PROPN
ejpam-6106	200	3	appl	appl	PROPN
ejpam-6106	200	4	.	.	PROPN
ejpam-6106	200	5	math	math	PROPN
ejpam-6106	200	6	,	,	PUNCT
ejpam-6106	200	7	18	18	NUM
ejpam-6106	200	8	(	(	PUNCT
ejpam-6106	200	9	4	4	NUM
ejpam-6106	200	10	)	)	PUNCT
ejpam-6106	200	11	(	(	PUNCT
ejpam-6106	200	12	2025	2025	NUM
ejpam-6106	200	13	)	)	PUNCT
ejpam-6106	200	14	,	,	PUNCT
ejpam-6106	200	15	6106	6106	NUM
ejpam-6106	200	16	8	8	NUM
ejpam-6106	200	17	of	of	ADP
ejpam-6106	200	18	10	10	NUM
ejpam-6106	200	19	=	=	SYM
ejpam-6106	200	20	1−	1−	NUM
ejpam-6106	200	21	(	(	PUNCT
ejpam-6106	200	22	n−	n−	NOUN
ejpam-6106	200	23	1)(b/2⌊i/(n−1)⌋	1)(b/2⌊i/(n−1)⌋	NOUN
ejpam-6106	200	24	−	−	NOUN
ejpam-6106	200	25	b/2⌊m/(n−1)⌋	b/2⌊m/(n−1)⌋	NOUN
ejpam-6106	200	26	s(li+1	s(li+1	ADP
ejpam-6106	200	27	,	,	PUNCT
ejpam-6106	200	28	li+2	li+2	PROPN
ejpam-6106	200	29	)	)	PUNCT
ejpam-6106	200	30	+	+	CCONJ
ejpam-6106	200	31	s(li+2	s(li+2	PROPN
ejpam-6106	200	32	,	,	PUNCT
ejpam-6106	200	33	li+3	li+3	X
ejpam-6106	200	34	)	)	PUNCT
ejpam-6106	200	35	+	+	CCONJ
ejpam-6106	200	36	·	·	PUNCT
ejpam-6106	200	37	·	·	PUNCT
ejpam-6106	200	38	·	·	PUNCT
ejpam-6106	201	1	+	+	CCONJ
ejpam-6106	201	2	s(li+n−1	s(li+n−1	PROPN
ejpam-6106	201	3	,	,	PUNCT
ejpam-6106	201	4	li+n	li+n	PROPN
ejpam-6106	201	5	)	)	PUNCT
ejpam-6106	201	6	.	.	PUNCT
ejpam-6106	202	1	note	note	VERB
ejpam-6106	202	2	that	that	SCONJ
ejpam-6106	202	3	i+	i+	NUM
ejpam-6106	202	4	n−	n−	NOUN
ejpam-6106	202	5	2	2	NUM
ejpam-6106	202	6	<	<	X
ejpam-6106	202	7	k	k	NOUN
ejpam-6106	202	8	,	,	PUNCT
ejpam-6106	202	9	hence	hence	ADV
ejpam-6106	202	10	,	,	PUNCT
ejpam-6106	202	11	b/2⌊k/(n−1)⌋	b/2⌊k/(n−1)⌋	ADJ
ejpam-6106	202	12	≤	≤	ADJ
ejpam-6106	202	13	b/2.b/2⌊i/(n−1)⌋.	b/2.b/2⌊i/(n−1)⌋.	NOUN
ejpam-6106	202	14	(	(	PUNCT
ejpam-6106	202	15	4	4	X
ejpam-6106	202	16	)	)	PUNCT
ejpam-6106	202	17	it	it	PRON
ejpam-6106	202	18	can	can	AUX
ejpam-6106	202	19	be	be	AUX
ejpam-6106	202	20	demonstrated	demonstrate	VERB
ejpam-6106	202	21	that	that	SCONJ
ejpam-6106	202	22	by	by	ADP
ejpam-6106	202	23	using	use	VERB
ejpam-6106	202	24	the	the	DET
ejpam-6106	202	25	structure	structure	NOUN
ejpam-6106	202	26	of	of	ADP
ejpam-6106	202	27	the	the	DET
ejpam-6106	202	28	space	space	NOUN
ejpam-6106	202	29	(	(	PUNCT
ejpam-6106	202	30	l	l	NOUN
ejpam-6106	202	31	,	,	PUNCT
ejpam-6106	202	32	s	s	PART
ejpam-6106	202	33	)	)	PUNCT
ejpam-6106	202	34	,	,	PUNCT
ejpam-6106	202	35	s(li+k	s(li+k	PROPN
ejpam-6106	202	36	,	,	PUNCT
ejpam-6106	202	37	l	l	NOUN
ejpam-6106	202	38	∗	∗	NOUN
ejpam-6106	202	39	)	)	PUNCT
ejpam-6106	202	40	≤	≤	NOUN
ejpam-6106	203	1	2(n−	2(n−	NUM
ejpam-6106	203	2	1)s(li+1	1)s(li+1	NUM
ejpam-6106	203	3	,	,	PUNCT
ejpam-6106	203	4	li+2	li+2	PROPN
ejpam-6106	203	5	)	)	PUNCT
ejpam-6106	203	6	.	.	PUNCT
ejpam-6106	204	1	clearly	clearly	ADV
ejpam-6106	204	2	s(li+k	s(li+k	VERB
ejpam-6106	204	3	,	,	PUNCT
ejpam-6106	204	4	li+m	li+m	PROPN
ejpam-6106	204	5	)	)	PUNCT
ejpam-6106	204	6	≤	≤	NOUN
ejpam-6106	204	7	s(li+k	s(li+k	PROPN
ejpam-6106	204	8	,	,	PUNCT
ejpam-6106	204	9	l	l	NOUN
ejpam-6106	204	10	∗	∗	NOUN
ejpam-6106	204	11	)	)	PUNCT
ejpam-6106	204	12	.	.	PUNCT
ejpam-6106	205	1	hence	hence	ADV
ejpam-6106	205	2	s(li+m	s(li+m	PROPN
ejpam-6106	205	3	,	,	PUNCT
ejpam-6106	205	4	li+k	li+k	NOUN
ejpam-6106	205	5	)	)	PUNCT
ejpam-6106	205	6	≤	≤	NOUN
ejpam-6106	205	7	2(n−	2(n−	NUM
ejpam-6106	205	8	1)s(li+m	1)s(li+m	NOUN
ejpam-6106	205	9	,	,	PUNCT
ejpam-6106	205	10	li+m+1	li+m+1	NUM
ejpam-6106	205	11	)	)	PUNCT
ejpam-6106	205	12	.	.	PUNCT
ejpam-6106	206	1	it	it	PRON
ejpam-6106	206	2	can	can	AUX
ejpam-6106	206	3	be	be	AUX
ejpam-6106	206	4	concluded	conclude	VERB
ejpam-6106	206	5	by	by	ADP
ejpam-6106	206	6	using	use	VERB
ejpam-6106	206	7	the	the	DET
ejpam-6106	206	8	last	last	ADJ
ejpam-6106	206	9	inequality	inequality	NOUN
ejpam-6106	206	10	s(li+1	s(li+1	ADP
ejpam-6106	206	11	,	,	PUNCT
ejpam-6106	206	12	li+2	li+2	PROPN
ejpam-6106	206	13	)	)	PUNCT
ejpam-6106	206	14	+	+	CCONJ
ejpam-6106	206	15	s(li+2	s(li+2	PROPN
ejpam-6106	206	16	,	,	PUNCT
ejpam-6106	206	17	li+3	li+3	X
ejpam-6106	206	18	)	)	PUNCT
ejpam-6106	206	19	+	+	CCONJ
ejpam-6106	206	20	·	·	PUNCT
ejpam-6106	206	21	·	·	PUNCT
ejpam-6106	206	22	·	·	PUNCT
ejpam-6106	207	1	+	+	CCONJ
ejpam-6106	207	2	s(li+n−1	s(li+n−1	PROPN
ejpam-6106	207	3	,	,	PUNCT
ejpam-6106	207	4	li+n	li+n	PROPN
ejpam-6106	207	5	)	)	PUNCT
ejpam-6106	207	6	=	=	PUNCT
ejpam-6106	208	1	(	(	PUNCT
ejpam-6106	208	2	n−	n−	NOUN
ejpam-6106	208	3	1)s(li+m	1)s(li+m	NOUN
ejpam-6106	208	4	,	,	PUNCT
ejpam-6106	208	5	li+k	li+k	NOUN
ejpam-6106	208	6	)	)	PUNCT
ejpam-6106	208	7	≤	≤	NOUN
ejpam-6106	208	8	2(n−	2(n−	NUM
ejpam-6106	208	9	1)2s(li+1	1)2s(li+1	NUM
ejpam-6106	208	10	,	,	PUNCT
ejpam-6106	208	11	li+2	li+2	NUM
ejpam-6106	208	12	)	)	PUNCT
ejpam-6106	208	13	=	=	NOUN
ejpam-6106	209	1	2(n−	2(n−	NUM
ejpam-6106	209	2	1)2b	1)2b	PROPN
ejpam-6106	209	3	2⌊i/(n−1)⌋	2⌊i/(n−1)⌋	NOUN
ejpam-6106	209	4	.	.	PUNCT
ejpam-6106	210	1	by	by	ADP
ejpam-6106	210	2	applying	apply	VERB
ejpam-6106	210	3	the	the	DET
ejpam-6106	210	4	inequality	inequality	NOUN
ejpam-6106	210	5	(	(	PUNCT
ejpam-6106	210	6	4	4	NUM
ejpam-6106	210	7	)	)	PUNCT
ejpam-6106	210	8	,	,	PUNCT
ejpam-6106	210	9	we	we	PRON
ejpam-6106	210	10	get	get	VERB
ejpam-6106	210	11	ri	ri	PROPN
ejpam-6106	210	12	,	,	PUNCT
ejpam-6106	210	13	k	k	PROPN
ejpam-6106	210	14	≤	≤	PROPN
ejpam-6106	210	15	1−	1−	NUM
ejpam-6106	211	1	(	(	PUNCT
ejpam-6106	211	2	n−	n−	NOUN
ejpam-6106	211	3	1	1	NUM
ejpam-6106	211	4	)	)	PUNCT
ejpam-6106	211	5	(	(	PUNCT
ejpam-6106	211	6	b/2⌊i/(n−1)⌋	b/2⌊i/(n−1)⌋	NOUN
ejpam-6106	211	7	−	−	PROPN
ejpam-6106	211	8	b/2.2⌊i/(n−1)⌋	b/2.2⌊i/(n−1)⌋	X
ejpam-6106	211	9	2(n−	2(n−	NUM
ejpam-6106	211	10	1)2b/2⌊i/(n−1)⌋	1)2b/2⌊i/(n−1)⌋	NUM
ejpam-6106	211	11	)	)	PUNCT
ejpam-6106	212	1	=	=	PUNCT
ejpam-6106	213	1	4n−	4n−	NUM
ejpam-6106	213	2	5	5	NUM
ejpam-6106	213	3	4n−	4n−	NUM
ejpam-6106	213	4	4	4	NUM
ejpam-6106	213	5	.	.	PUNCT
ejpam-6106	214	1	hence	hence	ADV
ejpam-6106	214	2	the	the	DET
ejpam-6106	214	3	inequality	inequality	NOUN
ejpam-6106	214	4	(	(	PUNCT
ejpam-6106	214	5	1	1	X
ejpam-6106	214	6	)	)	PUNCT
ejpam-6106	214	7	holds	hold	VERB
ejpam-6106	214	8	true	true	ADJ
ejpam-6106	214	9	.	.	PUNCT
ejpam-6106	215	1	the	the	DET
ejpam-6106	215	2	following	follow	VERB
ejpam-6106	215	3	diagrams	diagram	NOUN
ejpam-6106	215	4	effectively	effectively	ADV
ejpam-6106	215	5	demonstrate	demonstrate	VERB
ejpam-6106	215	6	:	:	PUNCT
ejpam-6106	216	1	1	1	X
ejpam-6106	216	2	.	.	X
ejpam-6106	216	3	how	how	SCONJ
ejpam-6106	216	4	a	a	DET
ejpam-6106	216	5	sequence	sequence	NOUN
ejpam-6106	216	6	of	of	ADP
ejpam-6106	216	7	polygons	polygon	NOUN
ejpam-6106	216	8	shrinks	shrink	VERB
ejpam-6106	216	9	under	under	ADP
ejpam-6106	216	10	a	a	DET
ejpam-6106	216	11	contracting	contracting	NOUN
ejpam-6106	216	12	transformation	transformation	NOUN
ejpam-6106	216	13	.	.	PUNCT
ejpam-6106	217	1	2	2	X
ejpam-6106	217	2	.	.	X
ejpam-6106	217	3	the	the	DET
ejpam-6106	217	4	idea	idea	NOUN
ejpam-6106	217	5	that	that	SCONJ
ejpam-6106	217	6	a	a	DET
ejpam-6106	217	7	sequence	sequence	NOUN
ejpam-6106	217	8	of	of	ADP
ejpam-6106	217	9	transformations	transformation	NOUN
ejpam-6106	217	10	converges	converge	NOUN
ejpam-6106	217	11	to	to	ADP
ejpam-6106	217	12	a	a	DET
ejpam-6106	217	13	unique	unique	ADJ
ejpam-6106	217	14	fixed	fix	VERB
ejpam-6106	217	15	point	point	NOUN
ejpam-6106	217	16	.	.	PUNCT
ejpam-6106	218	1	3	3	X
ejpam-6106	218	2	.	.	X
ejpam-6106	218	3	the	the	DET
ejpam-6106	218	4	concept	concept	NOUN
ejpam-6106	218	5	of	of	ADP
ejpam-6106	218	6	the	the	DET
ejpam-6106	218	7	fixed	fix	VERB
ejpam-6106	218	8	point	point	NOUN
ejpam-6106	218	9	theorem	theorem	VERB
ejpam-6106	218	10	in	in	ADP
ejpam-6106	218	11	action	action	NOUN
ejpam-6106	218	12	.	.	PUNCT
ejpam-6106	219	1	polygon	polygon	PROPN
ejpam-6106	219	2	p0	p0	PROPN
ejpam-6106	219	3	polygon	polygon	PROPN
ejpam-6106	219	4	p1	p1	NOUN
ejpam-6106	219	5	fixed	fix	VERB
ejpam-6106	219	6	point	point	NOUN
ejpam-6106	219	7	p	p	NOUN
ejpam-6106	219	8	∗	∗	NOUN
ejpam-6106	219	9	m.	m.	NOUN
ejpam-6106	219	10	nazam	nazam	PROPN
ejpam-6106	219	11	,	,	PUNCT
ejpam-6106	219	12	u.	u.	PROPN
ejpam-6106	219	13	habiba	habiba	PROPN
ejpam-6106	219	14	,	,	PUNCT
ejpam-6106	219	15	m.	m.	PROPN
ejpam-6106	219	16	de	de	X
ejpam-6106	219	17	la	la	PROPN
ejpam-6106	219	18	sen	sen	PROPN
ejpam-6106	219	19	/	/	SYM
ejpam-6106	219	20	eur	eur	PROPN
ejpam-6106	219	21	.	.	PUNCT
ejpam-6106	220	1	j.	j.	PROPN
ejpam-6106	220	2	pure	pure	PROPN
ejpam-6106	220	3	appl	appl	PROPN
ejpam-6106	220	4	.	.	PROPN
ejpam-6106	220	5	math	math	PROPN
ejpam-6106	220	6	,	,	PUNCT
ejpam-6106	220	7	18	18	NUM
ejpam-6106	220	8	(	(	PUNCT
ejpam-6106	220	9	4	4	NUM
ejpam-6106	220	10	)	)	PUNCT
ejpam-6106	220	11	(	(	PUNCT
ejpam-6106	220	12	2025	2025	NUM
ejpam-6106	220	13	)	)	PUNCT
ejpam-6106	220	14	,	,	PUNCT
ejpam-6106	220	15	6106	6106	NUM
ejpam-6106	220	16	9	9	NUM
ejpam-6106	220	17	of	of	ADP
ejpam-6106	220	18	10	10	NUM
ejpam-6106	220	19	hexagon	hexagon	NOUN
ejpam-6106	220	20	h0	h0	NOUN
ejpam-6106	220	21	hexagon	hexagon	PROPN
ejpam-6106	220	22	h1	h1	PROPN
ejpam-6106	220	23	fixed	fix	VERB
ejpam-6106	220	24	point	point	NOUN
ejpam-6106	220	25	h∗	h∗	PROPN
ejpam-6106	220	26	heptagon	heptagon	PROPN
ejpam-6106	220	27	g0	g0	PROPN
ejpam-6106	220	28	heptagon	heptagon	PROPN
ejpam-6106	220	29	g1	g1	PROPN
ejpam-6106	220	30	fixed	fix	VERB
ejpam-6106	220	31	point	point	NOUN
ejpam-6106	220	32	g∗	g∗	VERB
ejpam-6106	220	33	4	4	NUM
ejpam-6106	220	34	.	.	PUNCT
ejpam-6106	220	35	conclusion	conclusion	NOUN
ejpam-6106	220	36	in	in	ADP
ejpam-6106	220	37	this	this	DET
ejpam-6106	220	38	paper	paper	NOUN
ejpam-6106	220	39	,	,	PUNCT
ejpam-6106	220	40	we	we	PRON
ejpam-6106	220	41	studied	study	VERB
ejpam-6106	220	42	fixed	fix	VERB
ejpam-6106	220	43	point	point	NOUN
ejpam-6106	220	44	results	result	NOUN
ejpam-6106	220	45	for	for	ADP
ejpam-6106	220	46	mappings	mapping	NOUN
ejpam-6106	220	47	that	that	PRON
ejpam-6106	220	48	contract	contract	VERB
ejpam-6106	220	49	the	the	DET
ejpam-6106	220	50	perimeters	perimeter	NOUN
ejpam-6106	220	51	of	of	ADP
ejpam-6106	220	52	polygons	polygon	NOUN
ejpam-6106	220	53	.	.	PUNCT
ejpam-6106	221	1	by	by	ADP
ejpam-6106	221	2	extending	extend	VERB
ejpam-6106	221	3	petrov	petrov	PROPN
ejpam-6106	221	4	’s	’s	PART
ejpam-6106	221	5	theorem	theorem	NOUN
ejpam-6106	221	6	from	from	ADP
ejpam-6106	221	7	the	the	DET
ejpam-6106	221	8	case	case	NOUN
ejpam-6106	221	9	of	of	ADP
ejpam-6106	221	10	triangles	triangle	NOUN
ejpam-6106	221	11	(	(	PUNCT
ejpam-6106	221	12	n	n	NOUN
ejpam-6106	221	13	=	=	SYM
ejpam-6106	221	14	3	3	NUM
ejpam-6106	221	15	)	)	PUNCT
ejpam-6106	221	16	to	to	ADP
ejpam-6106	221	17	general	general	ADJ
ejpam-6106	221	18	polygons	polygon	NOUN
ejpam-6106	221	19	with	with	ADP
ejpam-6106	221	20	n	n	ADP
ejpam-6106	221	21	vertices	vertex	NOUN
ejpam-6106	221	22	,	,	PUNCT
ejpam-6106	221	23	we	we	PRON
ejpam-6106	221	24	developed	develop	VERB
ejpam-6106	221	25	a	a	DET
ejpam-6106	221	26	new	new	ADJ
ejpam-6106	221	27	geometric	geometric	ADJ
ejpam-6106	221	28	perspective	perspective	NOUN
ejpam-6106	221	29	on	on	ADP
ejpam-6106	221	30	contraction	contraction	NOUN
ejpam-6106	221	31	mappings	mapping	NOUN
ejpam-6106	221	32	.	.	PUNCT
ejpam-6106	222	1	our	our	PRON
ejpam-6106	222	2	results	result	NOUN
ejpam-6106	222	3	show	show	VERB
ejpam-6106	222	4	that	that	SCONJ
ejpam-6106	222	5	requiring	require	VERB
ejpam-6106	222	6	the	the	DET
ejpam-6106	222	7	contraction	contraction	NOUN
ejpam-6106	222	8	of	of	ADP
ejpam-6106	222	9	polygonal	polygonal	ADJ
ejpam-6106	222	10	perimeters	perimeter	NOUN
ejpam-6106	222	11	is	be	AUX
ejpam-6106	222	12	sufficient	sufficient	ADJ
ejpam-6106	222	13	to	to	PART
ejpam-6106	222	14	ensure	ensure	VERB
ejpam-6106	222	15	the	the	DET
ejpam-6106	222	16	existence	existence	NOUN
ejpam-6106	222	17	of	of	ADP
ejpam-6106	222	18	fixed	fix	VERB
ejpam-6106	222	19	points	point	NOUN
ejpam-6106	222	20	,	,	PUNCT
ejpam-6106	222	21	thereby	thereby	ADV
ejpam-6106	222	22	providing	provide	VERB
ejpam-6106	222	23	a	a	DET
ejpam-6106	222	24	natural	natural	ADJ
ejpam-6106	222	25	generalization	generalization	NOUN
ejpam-6106	222	26	of	of	ADP
ejpam-6106	222	27	both	both	DET
ejpam-6106	222	28	petrov	petrov	PROPN
ejpam-6106	222	29	’s	’s	PART
ejpam-6106	222	30	theorem	theorem	NOUN
ejpam-6106	222	31	and	and	CCONJ
ejpam-6106	222	32	the	the	DET
ejpam-6106	222	33	banach	banach	NOUN
ejpam-6106	222	34	contraction	contraction	NOUN
ejpam-6106	222	35	principle	principle	NOUN
ejpam-6106	222	36	.	.	PUNCT
ejpam-6106	223	1	we	we	PRON
ejpam-6106	223	2	also	also	ADV
ejpam-6106	223	3	demonstrated	demonstrate	VERB
ejpam-6106	223	4	that	that	DET
ejpam-6106	223	5	banach	banach	NOUN
ejpam-6106	223	6	’s	’s	PART
ejpam-6106	223	7	contraction	contraction	NOUN
ejpam-6106	223	8	theorem	theorem	NOUN
ejpam-6106	223	9	can	can	AUX
ejpam-6106	223	10	be	be	AUX
ejpam-6106	223	11	re	re	VERB
ejpam-6106	223	12	-	-	VERB
ejpam-6106	223	13	derived	derive	VERB
ejpam-6106	223	14	as	as	ADP
ejpam-6106	223	15	a	a	DET
ejpam-6106	223	16	corollary	corollary	NOUN
ejpam-6106	223	17	of	of	ADP
ejpam-6106	223	18	our	our	PRON
ejpam-6106	223	19	framework	framework	NOUN
ejpam-6106	223	20	,	,	PUNCT
ejpam-6106	223	21	which	which	PRON
ejpam-6106	223	22	highlights	highlight	VERB
ejpam-6106	223	23	the	the	DET
ejpam-6106	223	24	strength	strength	NOUN
ejpam-6106	223	25	and	and	CCONJ
ejpam-6106	223	26	versatility	versatility	NOUN
ejpam-6106	223	27	of	of	ADP
ejpam-6106	223	28	this	this	DET
ejpam-6106	223	29	perimeter	perimeter	NOUN
ejpam-6106	223	30	-	-	PUNCT
ejpam-6106	223	31	based	base	VERB
ejpam-6106	223	32	approach	approach	NOUN
ejpam-6106	223	33	.	.	PUNCT
ejpam-6106	224	1	the	the	DET
ejpam-6106	224	2	comparison	comparison	NOUN
ejpam-6106	224	3	with	with	ADP
ejpam-6106	224	4	other	other	ADJ
ejpam-6106	224	5	generalizations	generalization	NOUN
ejpam-6106	224	6	of	of	ADP
ejpam-6106	224	7	banach	banach	NOUN
ejpam-6106	224	8	’s	’s	PART
ejpam-6106	224	9	principle	principle	NOUN
ejpam-6106	224	10	shows	show	VERB
ejpam-6106	224	11	that	that	SCONJ
ejpam-6106	224	12	,	,	PUNCT
ejpam-6106	224	13	unlike	unlike	ADP
ejpam-6106	224	14	distanceor	distanceor	PROPN
ejpam-6106	224	15	order	order	NOUN
ejpam-6106	224	16	-	-	PUNCT
ejpam-6106	224	17	based	base	VERB
ejpam-6106	224	18	extensions	extension	NOUN
ejpam-6106	224	19	,	,	PUNCT
ejpam-6106	224	20	our	our	PRON
ejpam-6106	224	21	method	method	NOUN
ejpam-6106	224	22	provides	provide	VERB
ejpam-6106	224	23	geometric	geometric	ADJ
ejpam-6106	224	24	insights	insight	NOUN
ejpam-6106	224	25	not	not	PART
ejpam-6106	224	26	captured	capture	VERB
ejpam-6106	224	27	in	in	ADP
ejpam-6106	224	28	previous	previous	ADJ
ejpam-6106	224	29	studies	study	NOUN
ejpam-6106	224	30	.	.	PUNCT
ejpam-6106	225	1	in	in	ADP
ejpam-6106	225	2	addition	addition	NOUN
ejpam-6106	225	3	to	to	ADP
ejpam-6106	225	4	the	the	DET
ejpam-6106	225	5	theoretical	theoretical	ADJ
ejpam-6106	225	6	contributions	contribution	NOUN
ejpam-6106	225	7	,	,	PUNCT
ejpam-6106	225	8	several	several	ADJ
ejpam-6106	225	9	illustrative	illustrative	ADJ
ejpam-6106	225	10	examples	example	NOUN
ejpam-6106	225	11	were	be	AUX
ejpam-6106	225	12	given	give	VERB
ejpam-6106	225	13	to	to	PART
ejpam-6106	225	14	confirm	confirm	VERB
ejpam-6106	225	15	the	the	DET
ejpam-6106	225	16	applicability	applicability	NOUN
ejpam-6106	225	17	of	of	ADP
ejpam-6106	225	18	the	the	DET
ejpam-6106	225	19	main	main	ADJ
ejpam-6106	225	20	results	result	NOUN
ejpam-6106	225	21	.	.	PUNCT
ejpam-6106	226	1	the	the	DET
ejpam-6106	226	2	presented	present	VERB
ejpam-6106	226	3	framework	framework	NOUN
ejpam-6106	226	4	opens	open	VERB
ejpam-6106	226	5	avenues	avenue	NOUN
ejpam-6106	226	6	for	for	ADP
ejpam-6106	226	7	further	further	ADJ
ejpam-6106	226	8	work	work	NOUN
ejpam-6106	226	9	on	on	ADP
ejpam-6106	226	10	fixed	fix	VERB
ejpam-6106	226	11	point	point	NOUN
ejpam-6106	226	12	theory	theory	NOUN
ejpam-6106	226	13	in	in	ADP
ejpam-6106	226	14	geometric	geometric	ADJ
ejpam-6106	226	15	and	and	CCONJ
ejpam-6106	226	16	applied	applied	ADJ
ejpam-6106	226	17	settings	setting	NOUN
ejpam-6106	226	18	,	,	PUNCT
ejpam-6106	226	19	such	such	ADJ
ejpam-6106	226	20	as	as	ADP
ejpam-6106	226	21	in	in	ADP
ejpam-6106	226	22	dynamical	dynamical	ADJ
ejpam-6106	226	23	systems	system	NOUN
ejpam-6106	226	24	,	,	PUNCT
ejpam-6106	226	25	optimization	optimization	NOUN
ejpam-6106	226	26	,	,	PUNCT
ejpam-6106	226	27	and	and	CCONJ
ejpam-6106	226	28	problems	problem	NOUN
ejpam-6106	226	29	involving	involve	VERB
ejpam-6106	226	30	polygonal	polygonal	ADJ
ejpam-6106	226	31	or	or	CCONJ
ejpam-6106	226	32	polyhedral	polyhedral	ADJ
ejpam-6106	226	33	structures	structure	NOUN
ejpam-6106	226	34	.	.	PUNCT
ejpam-6106	227	1	in	in	ADP
ejpam-6106	227	2	summary	summary	NOUN
ejpam-6106	227	3	,	,	PUNCT
ejpam-6106	227	4	the	the	DET
ejpam-6106	227	5	present	present	ADJ
ejpam-6106	227	6	study	study	NOUN
ejpam-6106	227	7	contributes	contribute	VERB
ejpam-6106	227	8	a	a	DET
ejpam-6106	227	9	complementary	complementary	ADJ
ejpam-6106	227	10	and	and	CCONJ
ejpam-6106	227	11	geometrically	geometrically	ADV
ejpam-6106	227	12	motivated	motivated	ADJ
ejpam-6106	227	13	extension	extension	NOUN
ejpam-6106	227	14	of	of	ADP
ejpam-6106	227	15	fixed	fix	VERB
ejpam-6106	227	16	point	point	NOUN
ejpam-6106	227	17	theory	theory	NOUN
ejpam-6106	227	18	.	.	PUNCT
ejpam-6106	228	1	future	future	ADJ
ejpam-6106	228	2	work	work	NOUN
ejpam-6106	228	3	may	may	AUX
ejpam-6106	228	4	explore	explore	VERB
ejpam-6106	228	5	practical	practical	ADJ
ejpam-6106	228	6	applications	application	NOUN
ejpam-6106	228	7	of	of	ADP
ejpam-6106	228	8	perimeter	perimeter	NOUN
ejpam-6106	228	9	-	-	PUNCT
ejpam-6106	228	10	contraction	contraction	NOUN
ejpam-6106	228	11	mappings	mapping	NOUN
ejpam-6106	228	12	,	,	PUNCT
ejpam-6106	228	13	connections	connection	NOUN
ejpam-6106	228	14	with	with	ADP
ejpam-6106	228	15	ulam	ulam	PROPN
ejpam-6106	228	16	stability	stability	PROPN
ejpam-6106	228	17	problems	problem	NOUN
ejpam-6106	228	18	,	,	PUNCT
ejpam-6106	228	19	and	and	CCONJ
ejpam-6106	228	20	the	the	DET
ejpam-6106	228	21	extension	extension	NOUN
ejpam-6106	228	22	of	of	ADP
ejpam-6106	228	23	these	these	DET
ejpam-6106	228	24	ideas	idea	NOUN
ejpam-6106	228	25	to	to	ADP
ejpam-6106	228	26	higher	higher	ADV
ejpam-6106	228	27	-	-	PUNCT
ejpam-6106	228	28	dimensional	dimensional	ADJ
ejpam-6106	228	29	polyhedra	polyhedra	NOUN
ejpam-6106	228	30	.	.	PUNCT
ejpam-6106	229	1	acknowledgements	acknowledgement	NOUN
ejpam-6106	229	2	the	the	DET
ejpam-6106	229	3	authors	author	NOUN
ejpam-6106	229	4	extend	extend	VERB
ejpam-6106	229	5	their	their	PRON
ejpam-6106	229	6	sincere	sincere	ADJ
ejpam-6106	229	7	gratitude	gratitude	NOUN
ejpam-6106	229	8	to	to	ADP
ejpam-6106	229	9	the	the	DET
ejpam-6106	229	10	honorable	honorable	ADJ
ejpam-6106	229	11	reviewers	reviewer	NOUN
ejpam-6106	229	12	of	of	ADP
ejpam-6106	229	13	the	the	DET
ejpam-6106	229	14	european	european	PROPN
ejpam-6106	229	15	journal	journal	PROPN
ejpam-6106	229	16	of	of	ADP
ejpam-6106	229	17	pure	pure	ADJ
ejpam-6106	229	18	and	and	CCONJ
ejpam-6106	229	19	applied	applied	ADJ
ejpam-6106	229	20	mathematics	mathematic	NOUN
ejpam-6106	229	21	for	for	ADP
ejpam-6106	229	22	their	their	PRON
ejpam-6106	229	23	valuable	valuable	ADJ
ejpam-6106	229	24	insights	insight	NOUN
ejpam-6106	229	25	and	and	CCONJ
ejpam-6106	229	26	constructive	constructive	ADJ
ejpam-6106	229	27	feedback	feedback	NOUN
ejpam-6106	229	28	,	,	PUNCT
ejpam-6106	229	29	which	which	PRON
ejpam-6106	229	30	greatly	greatly	ADV
ejpam-6106	229	31	enhanced	enhance	VERB
ejpam-6106	229	32	the	the	DET
ejpam-6106	229	33	quality	quality	NOUN
ejpam-6106	229	34	of	of	ADP
ejpam-6106	229	35	our	our	PRON
ejpam-6106	229	36	submission	submission	NOUN
ejpam-6106	229	37	.	.	PUNCT
ejpam-6106	230	1	m.	m.	NOUN
ejpam-6106	230	2	nazam	nazam	PROPN
ejpam-6106	230	3	,	,	PUNCT
ejpam-6106	230	4	u.	u.	PROPN
ejpam-6106	230	5	habiba	habiba	PROPN
ejpam-6106	230	6	,	,	PUNCT
ejpam-6106	230	7	m.	m.	PROPN
ejpam-6106	230	8	de	de	X
ejpam-6106	230	9	la	la	PROPN
ejpam-6106	230	10	sen	sen	PROPN
ejpam-6106	230	11	/	/	SYM
ejpam-6106	230	12	eur	eur	PROPN
ejpam-6106	230	13	.	.	PUNCT
ejpam-6106	231	1	j.	j.	PROPN
ejpam-6106	231	2	pure	pure	PROPN
ejpam-6106	231	3	appl	appl	PROPN
ejpam-6106	231	4	.	.	PROPN
ejpam-6106	231	5	math	math	PROPN
ejpam-6106	231	6	,	,	PUNCT
ejpam-6106	231	7	18	18	NUM
ejpam-6106	231	8	(	(	PUNCT
ejpam-6106	231	9	4	4	NUM
ejpam-6106	231	10	)	)	PUNCT
ejpam-6106	231	11	(	(	PUNCT
ejpam-6106	231	12	2025	2025	NUM
ejpam-6106	231	13	)	)	PUNCT
ejpam-6106	231	14	,	,	PUNCT
ejpam-6106	231	15	6106	6106	NUM
ejpam-6106	231	16	10	10	NUM
ejpam-6106	231	17	of	of	ADP
ejpam-6106	231	18	10	10	NUM
ejpam-6106	231	19	5	5	NUM
ejpam-6106	231	20	.	.	PUNCT
ejpam-6106	232	1	authors	author	NOUN
ejpam-6106	232	2	’	’	PART
ejpam-6106	232	3	contributions	contribution	NOUN
ejpam-6106	232	4	m.n	m.n	PROPN
ejpam-6106	232	5	.	.	PROPN
ejpam-6106	232	6	tabled	table	VERB
ejpam-6106	232	7	the	the	DET
ejpam-6106	232	8	main	main	ADJ
ejpam-6106	232	9	idea	idea	NOUN
ejpam-6106	232	10	of	of	ADP
ejpam-6106	232	11	this	this	DET
ejpam-6106	232	12	paper	paper	NOUN
ejpam-6106	232	13	;	;	PUNCT
ejpam-6106	232	14	u.h	u.h	PROPN
ejpam-6106	232	15	.	.	PROPN
ejpam-6106	232	16	wrote	write	VERB
ejpam-6106	232	17	the	the	DET
ejpam-6106	232	18	first	first	ADJ
ejpam-6106	232	19	draft	draft	NOUN
ejpam-6106	232	20	of	of	ADP
ejpam-6106	232	21	this	this	DET
ejpam-6106	232	22	paper	paper	NOUN
ejpam-6106	232	23	;	;	PUNCT
ejpam-6106	233	1	m.n	m.n	PROPN
ejpam-6106	233	2	.	.	PROPN
ejpam-6106	233	3	and	and	CCONJ
ejpam-6106	233	4	m.	m.	PROPN
ejpam-6106	233	5	d.	d.	PROPN
ejpam-6106	233	6	s.	s.	PROPN
ejpam-6106	233	7	reviewed	review	VERB
ejpam-6106	233	8	and	and	CCONJ
ejpam-6106	233	9	prepared	prepare	VERB
ejpam-6106	233	10	the	the	DET
ejpam-6106	233	11	second	second	ADJ
ejpam-6106	233	12	draft	draft	NOUN
ejpam-6106	233	13	;	;	PUNCT
ejpam-6106	233	14	m.	m.	PROPN
ejpam-6106	233	15	d.	d.	PROPN
ejpam-6106	233	16	s.	s.	PROPN
ejpam-6106	233	17	supervised	supervise	VERB
ejpam-6106	233	18	the	the	DET
ejpam-6106	233	19	project	project	NOUN
ejpam-6106	233	20	.	.	PUNCT
ejpam-6106	234	1	all	all	DET
ejpam-6106	234	2	authors	author	NOUN
ejpam-6106	234	3	have	have	AUX
ejpam-6106	234	4	read	read	VERB
ejpam-6106	234	5	and	and	CCONJ
ejpam-6106	234	6	agreed	agree	VERB
ejpam-6106	234	7	to	to	ADP
ejpam-6106	234	8	the	the	DET
ejpam-6106	234	9	published	publish	VERB
ejpam-6106	234	10	version	version	NOUN
ejpam-6106	234	11	of	of	ADP
ejpam-6106	234	12	the	the	DET
ejpam-6106	234	13	manuscript	manuscript	NOUN
ejpam-6106	234	14	.	.	PUNCT
ejpam-6106	235	1	references	reference	NOUN
ejpam-6106	235	2	[	[	X
ejpam-6106	235	3	1	1	NUM
ejpam-6106	235	4	]	]	PUNCT
ejpam-6106	235	5	evgeniy	evgeniy	ADJ
ejpam-6106	235	6	petrov	petrov	PROPN
ejpam-6106	235	7	.	.	PUNCT
ejpam-6106	236	1	fixed	fix	VERB
ejpam-6106	236	2	point	point	NOUN
ejpam-6106	236	3	theorem	theorem	NOUN
ejpam-6106	236	4	for	for	ADP
ejpam-6106	236	5	mappings	mapping	NOUN
ejpam-6106	236	6	contracting	contract	VERB
ejpam-6106	236	7	perimeters	perimeter	NOUN
ejpam-6106	236	8	of	of	ADP
ejpam-6106	236	9	triangles	triangle	NOUN
ejpam-6106	236	10	.	.	PUNCT
ejpam-6106	237	1	journal	journal	NOUN
ejpam-6106	237	2	of	of	ADP
ejpam-6106	237	3	fixed	fix	VERB
ejpam-6106	237	4	point	point	NOUN
ejpam-6106	237	5	theory	theory	NOUN
ejpam-6106	237	6	and	and	CCONJ
ejpam-6106	237	7	applications	application	NOUN
ejpam-6106	237	8	,	,	PUNCT
ejpam-6106	237	9	25(74	25(74	NUM
ejpam-6106	237	10	)	)	PUNCT
ejpam-6106	237	11	,	,	PUNCT
ejpam-6106	237	12	2023	2023	NUM
ejpam-6106	237	13	.	.	PUNCT
ejpam-6106	238	1	[	[	X
ejpam-6106	238	2	2	2	X
ejpam-6106	238	3	]	]	PUNCT
ejpam-6106	238	4	s.	s.	PROPN
ejpam-6106	238	5	banach	banach	PROPN
ejpam-6106	238	6	.	.	PUNCT
ejpam-6106	239	1	sur	sur	PROPN
ejpam-6106	239	2	les	les	PROPN
ejpam-6106	239	3	options	option	NOUN
ejpam-6106	239	4	dans	dan	NOUN
ejpam-6106	239	5	les	les	X
ejpam-6106	239	6	ensembles	ensemble	NOUN
ejpam-6106	239	7	abstraits	abstrait	NOUN
ejpam-6106	239	8	et	et	PROPN
ejpam-6106	239	9	leur	leur	X
ejpam-6106	239	10	application	application	PROPN
ejpam-6106	239	11	aux	aux	PROPN
ejpam-6106	239	12	ations	ations	PROPN
ejpam-6106	239	13	intales	intales	PROPN
ejpam-6106	239	14	.	.	PUNCT
ejpam-6106	240	1	fundamenta	fundamenta	PROPN
ejpam-6106	240	2	mathematicae	mathematicae	PROPN
ejpam-6106	240	3	,	,	PUNCT
ejpam-6106	240	4	3:133–181	3:133–181	NUM
ejpam-6106	240	5	,	,	PUNCT
ejpam-6106	240	6	1922	1922	NUM
ejpam-6106	240	7	.	.	PUNCT
ejpam-6106	241	1	[	[	X
ejpam-6106	241	2	3	3	X
ejpam-6106	241	3	]	]	PUNCT
ejpam-6106	241	4	w.	w.	PROPN
ejpam-6106	241	5	a.	a.	PROPN
ejpam-6106	241	6	kirk	kirk	PROPN
ejpam-6106	241	7	and	and	CCONJ
ejpam-6106	241	8	n.	n.	PROPN
ejpam-6106	241	9	shahzad	shahzad	PROPN
ejpam-6106	241	10	.	.	PUNCT
ejpam-6106	242	1	fixed	fix	VERB
ejpam-6106	242	2	point	point	NOUN
ejpam-6106	242	3	theory	theory	NOUN
ejpam-6106	242	4	in	in	ADP
ejpam-6106	242	5	distance	distance	NOUN
ejpam-6106	242	6	spaces	space	NOUN
ejpam-6106	242	7	.	.	PUNCT
ejpam-6106	243	1	springer	springer	NOUN
ejpam-6106	243	2	,	,	PUNCT
ejpam-6106	243	3	2014	2014	NUM
ejpam-6106	243	4	.	.	PUNCT
ejpam-6106	244	1	[	[	X
ejpam-6106	244	2	4	4	X
ejpam-6106	244	3	]	]	PUNCT
ejpam-6106	244	4	w.	w.	PROPN
ejpam-6106	244	5	a.	a.	PROPN
ejpam-6106	244	6	kirk	kirk	PROPN
ejpam-6106	244	7	.	.	PUNCT
ejpam-6106	245	1	a	a	DET
ejpam-6106	245	2	fixed	fix	VERB
ejpam-6106	245	3	point	point	NOUN
ejpam-6106	245	4	theorem	theorem	NOUN
ejpam-6106	245	5	for	for	ADP
ejpam-6106	245	6	mappings	mapping	NOUN
ejpam-6106	245	7	which	which	PRON
ejpam-6106	245	8	do	do	AUX
ejpam-6106	245	9	not	not	PART
ejpam-6106	245	10	increase	increase	VERB
ejpam-6106	245	11	distances	distance	NOUN
ejpam-6106	245	12	.	.	PUNCT
ejpam-6106	246	1	american	american	PROPN
ejpam-6106	246	2	mathematical	mathematical	PROPN
ejpam-6106	246	3	monthly	monthly	ADV
ejpam-6106	246	4	,	,	PUNCT
ejpam-6106	246	5	72:1004–1006	72:1004–1006	NOUN
ejpam-6106	246	6	,	,	PUNCT
ejpam-6106	246	7	1965	1965	NUM
ejpam-6106	246	8	.	.	PUNCT
ejpam-6106	247	1	[	[	X
ejpam-6106	247	2	5	5	X
ejpam-6106	247	3	]	]	X
ejpam-6106	247	4	james	james	PROPN
ejpam-6106	247	5	caristi	caristi	PROPN
ejpam-6106	247	6	.	.	PUNCT
ejpam-6106	248	1	fixed	fix	VERB
ejpam-6106	248	2	point	point	NOUN
ejpam-6106	248	3	theorems	theorem	NOUN
ejpam-6106	248	4	for	for	ADP
ejpam-6106	248	5	mappings	mapping	NOUN
ejpam-6106	248	6	satisfying	satisfy	VERB
ejpam-6106	248	7	inwardness	inwardness	NOUN
ejpam-6106	248	8	conditions	condition	NOUN
ejpam-6106	248	9	.	.	PUNCT
ejpam-6106	249	1	transactions	transaction	NOUN
ejpam-6106	249	2	of	of	ADP
ejpam-6106	249	3	the	the	DET
ejpam-6106	249	4	american	american	PROPN
ejpam-6106	249	5	mathematical	mathematical	PROPN
ejpam-6106	249	6	society	society	NOUN
ejpam-6106	249	7	,	,	PUNCT
ejpam-6106	249	8	215:241–251	215:241–251	NUM
ejpam-6106	249	9	,	,	PUNCT
ejpam-6106	249	10	1976	1976	NUM
ejpam-6106	249	11	.	.	PUNCT
ejpam-6106	250	1	[	[	X
ejpam-6106	250	2	6	6	NUM
ejpam-6106	250	3	]	]	PUNCT
ejpam-6106	250	4	s.	s.	PROPN
ejpam-6106	250	5	b.	b.	PROPN
ejpam-6106	250	6	nadler	nadler	PROPN
ejpam-6106	250	7	jr	jr	PROPN
ejpam-6106	250	8	.	.	PUNCT
ejpam-6106	250	9	multi	multi	ADJ
ejpam-6106	250	10	-	-	ADJ
ejpam-6106	250	11	valued	value	VERB
ejpam-6106	250	12	contraction	contraction	NOUN
ejpam-6106	250	13	mappings	mapping	NOUN
ejpam-6106	250	14	.	.	PUNCT
ejpam-6106	251	1	pacific	pacific	PROPN
ejpam-6106	251	2	journal	journal	PROPN
ejpam-6106	251	3	of	of	ADP
ejpam-6106	251	4	mathematics	mathematic	NOUN
ejpam-6106	251	5	,	,	PUNCT
ejpam-6106	251	6	30:475–488	30:475–488	NUM
ejpam-6106	251	7	,	,	PUNCT
ejpam-6106	251	8	1969	1969	NUM
ejpam-6106	251	9	.	.	PUNCT
ejpam-6106	252	1	[	[	X
ejpam-6106	252	2	7	7	NUM
ejpam-6106	252	3	]	]	PUNCT
ejpam-6106	252	4	a.	a.	NOUN
ejpam-6106	252	5	shoaib	shoaib	PROPN
ejpam-6106	252	6	and	and	CCONJ
ejpam-6106	252	7	m.	m.	PROPN
ejpam-6106	252	8	mehmood	mehmood	PROPN
ejpam-6106	252	9	.	.	PUNCT
ejpam-6106	253	1	fixed	fix	VERB
ejpam-6106	253	2	point	point	NOUN
ejpam-6106	253	3	results	result	NOUN
ejpam-6106	253	4	on	on	ADP
ejpam-6106	253	5	locally	locally	ADV
ejpam-6106	253	6	(	(	PUNCT
ejpam-6106	253	7	δ	δ	PROPN
ejpam-6106	253	8	,	,	PUNCT
ejpam-6106	253	9	φ)-dominated	φ)-dominate	VERB
ejpam-6106	253	10	and	and	CCONJ
ejpam-6106	253	11	(	(	PUNCT
ejpam-6106	253	12	δ	δ	PROPN
ejpam-6106	253	13	,	,	PUNCT
ejpam-6106	253	14	φ)-continuous	φ)-continuous	ADJ
ejpam-6106	253	15	mappings	mapping	NOUN
ejpam-6106	253	16	in	in	ADP
ejpam-6106	253	17	dislocated	dislocated	ADJ
ejpam-6106	253	18	quasi	quasi	ADJ
ejpam-6106	253	19	metric	metric	ADJ
ejpam-6106	253	20	spaces	space	NOUN
ejpam-6106	253	21	.	.	PUNCT
ejpam-6106	254	1	journal	journal	PROPN
ejpam-6106	254	2	of	of	ADP
ejpam-6106	254	3	inequalities	inequality	NOUN
ejpam-6106	254	4	and	and	CCONJ
ejpam-6106	254	5	applications	application	NOUN
ejpam-6106	254	6	,	,	PUNCT
ejpam-6106	254	7	2024(157	2024(157	NUM
ejpam-6106	254	8	)	)	PUNCT
ejpam-6106	254	9	,	,	PUNCT
ejpam-6106	254	10	2024	2024	NUM
ejpam-6106	254	11	.	.	PUNCT
ejpam-6106	255	1	[	[	X
ejpam-6106	255	2	8	8	NUM
ejpam-6106	255	3	]	]	PUNCT
ejpam-6106	255	4	a.	a.	NOUN
ejpam-6106	255	5	shoaib	shoaib	PROPN
ejpam-6106	255	6	and	and	CCONJ
ejpam-6106	255	7	u.	u.	PROPN
ejpam-6106	255	8	mir	mir	PROPN
ejpam-6106	255	9	.	.	PROPN
ejpam-6106	255	10	interpolative	interpolative	ADJ
ejpam-6106	255	11	multivalued	multivalue	VERB
ejpam-6106	255	12	α∗-dominated	α∗-dominate	VERB
ejpam-6106	255	13	contractive	contractive	ADJ
ejpam-6106	255	14	functions	function	NOUN
ejpam-6106	255	15	in	in	ADP
ejpam-6106	255	16	dislocated	dislocated	ADJ
ejpam-6106	255	17	b	b	X
ejpam-6106	255	18	-	-	ADJ
ejpam-6106	255	19	metric	metric	ADJ
ejpam-6106	255	20	spaces	space	NOUN
ejpam-6106	255	21	and	and	CCONJ
ejpam-6106	255	22	some	some	DET
ejpam-6106	255	23	fixed	fix	VERB
ejpam-6106	255	24	point	point	NOUN
ejpam-6106	255	25	results	result	NOUN
ejpam-6106	255	26	.	.	PUNCT
ejpam-6106	256	1	mathematical	mathematical	ADJ
ejpam-6106	256	2	sciences	science	NOUN
ejpam-6106	256	3	,	,	PUNCT
ejpam-6106	256	4	18(1):9–16	18(1):9–16	NUM
ejpam-6106	256	5	,	,	PUNCT
ejpam-6106	256	6	2024	2024	NUM
ejpam-6106	256	7	.	.	PUNCT
ejpam-6106	257	1	[	[	X
ejpam-6106	257	2	9	9	NUM
ejpam-6106	257	3	]	]	X
ejpam-6106	257	4	c.	c.	NOUN
ejpam-6106	257	5	tunç	tunç	PROPN
ejpam-6106	257	6	,	,	PUNCT
ejpam-6106	257	7	o.	o.	PROPN
ejpam-6106	257	8	tunç	tunç	PROPN
ejpam-6106	257	9	,	,	PUNCT
ejpam-6106	257	10	g.	g.	PROPN
ejpam-6106	257	11	petruşel	petruşel	NUM
ejpam-6106	257	12	,	,	PUNCT
ejpam-6106	257	13	and	and	CCONJ
ejpam-6106	257	14	j.-c	j.-c	PROPN
ejpam-6106	257	15	.	.	PUNCT
ejpam-6106	258	1	yao	yao	PROPN
ejpam-6106	258	2	.	.	PUNCT
ejpam-6106	259	1	on	on	ADP
ejpam-6106	259	2	the	the	DET
ejpam-6106	259	3	ulam	ulam	PROPN
ejpam-6106	259	4	stabilities	stability	NOUN
ejpam-6106	259	5	of	of	ADP
ejpam-6106	259	6	nonlinear	nonlinear	ADJ
ejpam-6106	259	7	integral	integral	ADJ
ejpam-6106	259	8	equations	equation	NOUN
ejpam-6106	259	9	and	and	CCONJ
ejpam-6106	259	10	integro	integro	ADJ
ejpam-6106	259	11	-	-	PUNCT
ejpam-6106	259	12	differential	differential	NOUN
ejpam-6106	259	13	equations	equation	NOUN
ejpam-6106	259	14	.	.	PUNCT
ejpam-6106	260	1	mathematical	mathematical	ADJ
ejpam-6106	260	2	methods	method	NOUN
ejpam-6106	260	3	in	in	ADP
ejpam-6106	260	4	the	the	DET
ejpam-6106	260	5	applied	apply	VERB
ejpam-6106	260	6	sciences	science	NOUN
ejpam-6106	260	7	,	,	PUNCT
ejpam-6106	260	8	47:4014–4028	47:4014–4028	NUM
ejpam-6106	260	9	,	,	PUNCT
ejpam-6106	260	10	2024	2024	NUM
ejpam-6106	260	11	.	.	PUNCT
ejpam-6106	261	1	[	[	X
ejpam-6106	261	2	10	10	NUM
ejpam-6106	261	3	]	]	X
ejpam-6106	261	4	o.	o.	NOUN
ejpam-6106	261	5	tunç	tunç	NOUN
ejpam-6106	261	6	and	and	CCONJ
ejpam-6106	261	7	c.	c.	NOUN
ejpam-6106	261	8	tunç.	tunç.	NOUN
ejpam-6106	261	9	on	on	ADP
ejpam-6106	261	10	ulam	ulam	PROPN
ejpam-6106	261	11	stabilities	stability	NOUN
ejpam-6106	261	12	of	of	ADP
ejpam-6106	261	13	delay	delay	PROPN
ejpam-6106	261	14	hammerstein	hammerstein	PROPN
ejpam-6106	261	15	integral	integral	ADJ
ejpam-6106	261	16	equation	equation	NOUN
ejpam-6106	261	17	.	.	PUNCT
ejpam-6106	262	1	symmetry	symmetry	NOUN
ejpam-6106	262	2	,	,	PUNCT
ejpam-6106	262	3	15(9):1736	15(9):1736	NUM
ejpam-6106	262	4	,	,	PUNCT
ejpam-6106	262	5	2023	2023	NUM
ejpam-6106	262	6	.	.	PUNCT
ejpam-6106	263	1	[	[	X
ejpam-6106	263	2	11	11	NUM
ejpam-6106	263	3	]	]	X
ejpam-6106	263	4	o.	o.	NOUN
ejpam-6106	263	5	tunç.	tunç.	NOUN
ejpam-6106	263	6	on	on	ADP
ejpam-6106	263	7	the	the	DET
ejpam-6106	263	8	behaviors	behavior	NOUN
ejpam-6106	263	9	of	of	ADP
ejpam-6106	263	10	solutions	solution	NOUN
ejpam-6106	263	11	of	of	ADP
ejpam-6106	263	12	systems	system	NOUN
ejpam-6106	263	13	of	of	ADP
ejpam-6106	263	14	non	non	ADJ
ejpam-6106	263	15	-	-	ADJ
ejpam-6106	263	16	linear	linear	ADJ
ejpam-6106	263	17	differential	differential	ADJ
ejpam-6106	263	18	equations	equation	NOUN
ejpam-6106	263	19	with	with	ADP
ejpam-6106	263	20	multiple	multiple	ADJ
ejpam-6106	263	21	constant	constant	ADJ
ejpam-6106	263	22	delays	delay	NOUN
ejpam-6106	263	23	.	.	PUNCT
ejpam-6106	264	1	revista	revista	PROPN
ejpam-6106	264	2	de	de	X
ejpam-6106	264	3	la	la	PROPN
ejpam-6106	264	4	real	real	PROPN
ejpam-6106	264	5	academia	academia	PROPN
ejpam-6106	264	6	de	de	PROPN
ejpam-6106	264	7	ciencias	ciencias	PROPN
ejpam-6106	264	8	exactas	exactas	PROPN
ejpam-6106	264	9	,	,	PUNCT
ejpam-6106	264	10	fcas	fcas	PROPN
ejpam-6106	264	11	y	y	PROPN
ejpam-6106	264	12	naturales	naturales	PROPN
ejpam-6106	264	13	.	.	PUNCT
ejpam-6106	265	1	serie	serie	PROPN
ejpam-6106	265	2	a.	a.	NOUN
ejpam-6106	265	3	matemcas	matemcas	PROPN
ejpam-6106	265	4	(	(	PUNCT
ejpam-6106	265	5	racsam	racsam	ADJ
ejpam-6106	265	6	)	)	PUNCT
ejpam-6106	265	7	,	,	PUNCT
ejpam-6106	265	8	115(4):paper	115(4):paper	NUM
ejpam-6106	265	9	no	no	NOUN
ejpam-6106	265	10	.	.	NOUN
ejpam-6106	265	11	164	164	NUM
ejpam-6106	265	12	,	,	PUNCT
ejpam-6106	265	13	22	22	NUM
ejpam-6106	265	14	pp	pp	NOUN
ejpam-6106	265	15	.	.	PUNCT
ejpam-6106	265	16	,	,	PUNCT
ejpam-6106	265	17	2021	2021	NUM
ejpam-6106	265	18	.	.	PUNCT
ejpam-6106	266	1	[	[	X
ejpam-6106	266	2	12	12	NUM
ejpam-6106	266	3	]	]	X
ejpam-6106	266	4	c.	c.	NOUN
ejpam-6106	266	5	tunç	tunç	PROPN
ejpam-6106	266	6	,	,	PUNCT
ejpam-6106	266	7	c.-f	c.-f	NOUN
ejpam-6106	266	8	.	.	PUNCT
ejpam-6106	267	1	wang	wang	PROPN
ejpam-6106	267	2	,	,	PUNCT
ejpam-6106	267	3	and	and	CCONJ
ejpam-6106	267	4	j.-c	j.-c	PROPN
ejpam-6106	267	5	.	.	PUNCT
ejpam-6106	268	1	yao	yao	PROPN
ejpam-6106	268	2	.	.	PUNCT
ejpam-6106	269	1	qualitative	qualitative	ADJ
ejpam-6106	269	2	analyses	analysis	NOUN
ejpam-6106	269	3	of	of	ADP
ejpam-6106	269	4	differential	differential	ADJ
ejpam-6106	269	5	systems	system	NOUN
ejpam-6106	269	6	with	with	ADP
ejpam-6106	269	7	time	time	NOUN
ejpam-6106	269	8	-	-	PUNCT
ejpam-6106	269	9	varying	vary	VERB
ejpam-6106	269	10	delays	delay	NOUN
ejpam-6106	269	11	via	via	ADP
ejpam-6106	269	12	lyapunov	lyapunov	PROPN
ejpam-6106	269	13	–	–	PUNCT
ejpam-6106	269	14	krasovskĭı	krasovskĭı	PROPN
ejpam-6106	269	15	approach	approach	NOUN
ejpam-6106	269	16	.	.	PUNCT
ejpam-6106	270	1	mathematics	mathematic	NOUN
ejpam-6106	270	2	,	,	PUNCT
ejpam-6106	270	3	9(11):1355	9(11):1355	NUM
ejpam-6106	270	4	,	,	PUNCT
ejpam-6106	270	5	2021	2021	NUM
ejpam-6106	270	6	.	.	PUNCT
ejpam-6106	271	1	[	[	X
ejpam-6106	271	2	13	13	NUM
ejpam-6106	271	3	]	]	X
ejpam-6106	271	4	c.	c.	NOUN
ejpam-6106	271	5	tunç	tunç	NOUN
ejpam-6106	271	6	and	and	CCONJ
ejpam-6106	271	7	o.	o.	NOUN
ejpam-6106	271	8	tunç.	tunç.	NOUN
ejpam-6106	271	9	new	new	ADJ
ejpam-6106	271	10	results	result	NOUN
ejpam-6106	271	11	on	on	ADP
ejpam-6106	271	12	the	the	DET
ejpam-6106	271	13	qualitative	qualitative	ADJ
ejpam-6106	271	14	analysis	analysis	NOUN
ejpam-6106	271	15	of	of	ADP
ejpam-6106	271	16	integro	integro	ADJ
ejpam-6106	271	17	-	-	PUNCT
ejpam-6106	271	18	differential	differential	NOUN
ejpam-6106	271	19	equations	equation	NOUN
ejpam-6106	271	20	with	with	ADP
ejpam-6106	271	21	constant	constant	ADJ
ejpam-6106	271	22	time	time	NOUN
ejpam-6106	271	23	-	-	PUNCT
ejpam-6106	271	24	delay	delay	NOUN
ejpam-6106	271	25	.	.	PUNCT
ejpam-6106	272	1	journal	journal	PROPN
ejpam-6106	272	2	of	of	ADP
ejpam-6106	272	3	nonlinear	nonlinear	ADJ
ejpam-6106	272	4	and	and	CCONJ
ejpam-6106	272	5	convex	convex	ADJ
ejpam-6106	272	6	analysis	analysis	NOUN
ejpam-6106	272	7	,	,	PUNCT
ejpam-6106	272	8	23(3):435–448	23(3):435–448	NUM
ejpam-6106	272	9	,	,	PUNCT
ejpam-6106	272	10	2022	2022	NUM
ejpam-6106	272	11	.	.	PUNCT