id	sid	tid	token	lemma	pos
ejpam-6175	1	1	european	european	PROPN
ejpam-6175	1	2	journal	journal	PROPN
ejpam-6175	1	3	of	of	ADP
ejpam-6175	1	4	pure	pure	ADJ
ejpam-6175	1	5	and	and	CCONJ
ejpam-6175	1	6	applied	applied	ADJ
ejpam-6175	1	7	mathematics	mathematic	NOUN
ejpam-6175	1	8	2025	2025	NUM
ejpam-6175	1	9	,	,	PUNCT
ejpam-6175	1	10	vol	vol	NOUN
ejpam-6175	1	11	.	.	PROPN
ejpam-6175	1	12	18	18	NUM
ejpam-6175	1	13	,	,	PUNCT
ejpam-6175	1	14	issue	issue	NOUN
ejpam-6175	1	15	3	3	NUM
ejpam-6175	1	16	,	,	PUNCT
ejpam-6175	1	17	article	article	NOUN
ejpam-6175	1	18	number	number	NOUN
ejpam-6175	1	19	6175	6175	NUM
ejpam-6175	1	20	issn	issn	PROPN
ejpam-6175	1	21	1307	1307	NUM
ejpam-6175	1	22	-	-	SYM
ejpam-6175	1	23	5543	5543	NUM
ejpam-6175	1	24	–	–	PUNCT
ejpam-6175	1	25	ejpam.com	ejpam.com	X
ejpam-6175	1	26	published	publish	VERB
ejpam-6175	1	27	by	by	ADP
ejpam-6175	1	28	new	new	PROPN
ejpam-6175	1	29	york	york	PROPN
ejpam-6175	1	30	business	business	PROPN
ejpam-6175	1	31	global	global	ADJ
ejpam-6175	1	32	enhancing	enhance	VERB
ejpam-6175	1	33	shape	shape	NOUN
ejpam-6175	1	34	analysis	analysis	NOUN
ejpam-6175	1	35	with	with	ADP
ejpam-6175	1	36	persistent	persistent	ADJ
ejpam-6175	1	37	homology	homology	NOUN
ejpam-6175	1	38	for	for	ADP
ejpam-6175	1	39	edge	edge	NOUN
ejpam-6175	1	40	detection	detection	NOUN
ejpam-6175	1	41	and	and	CCONJ
ejpam-6175	1	42	topological	topological	ADJ
ejpam-6175	1	43	feature	feature	NOUN
ejpam-6175	1	44	extraction	extraction	NOUN
ejpam-6175	1	45	shivam	shivam	PROPN
ejpam-6175	1	46	kumar	kumar	PROPN
ejpam-6175	1	47	jha	jha	PROPN
ejpam-6175	1	48	sanjay1	sanjay1	PROPN
ejpam-6175	1	49	,	,	PUNCT
ejpam-6175	1	50	mohana	mohana	PROPN
ejpam-6175	1	51	natarajan1∗	natarajan1∗	PROPN
ejpam-6175	1	52	1	1	NUM
ejpam-6175	1	53	department	department	NOUN
ejpam-6175	1	54	of	of	ADP
ejpam-6175	1	55	mathematics	mathematic	NOUN
ejpam-6175	1	56	,	,	PUNCT
ejpam-6175	1	57	school	school	NOUN
ejpam-6175	1	58	of	of	ADP
ejpam-6175	1	59	advanced	advanced	ADJ
ejpam-6175	1	60	sciences	science	NOUN
ejpam-6175	1	61	,	,	PUNCT
ejpam-6175	1	62	vellore	vellore	PROPN
ejpam-6175	1	63	institute	institute	PROPN
ejpam-6175	1	64	of	of	ADP
ejpam-6175	1	65	technology	technology	PROPN
ejpam-6175	1	66	,	,	PUNCT
ejpam-6175	1	67	chennai	chennai	VERB
ejpam-6175	1	68	600	600	NUM
ejpam-6175	1	69	127	127	NUM
ejpam-6175	1	70	,	,	PUNCT
ejpam-6175	1	71	india	india	PROPN
ejpam-6175	1	72	abstract	abstract	NOUN
ejpam-6175	1	73	.	.	PUNCT
ejpam-6175	2	1	this	this	DET
ejpam-6175	2	2	article	article	NOUN
ejpam-6175	2	3	examines	examine	VERB
ejpam-6175	2	4	the	the	DET
ejpam-6175	2	5	application	application	NOUN
ejpam-6175	2	6	of	of	ADP
ejpam-6175	2	7	persistent	persistent	ADJ
ejpam-6175	2	8	homology	homology	NOUN
ejpam-6175	2	9	in	in	ADP
ejpam-6175	2	10	analyzing	analyze	VERB
ejpam-6175	2	11	the	the	DET
ejpam-6175	2	12	structure	structure	NOUN
ejpam-6175	2	13	of	of	ADP
ejpam-6175	2	14	various	various	ADJ
ejpam-6175	2	15	shapes	shape	NOUN
ejpam-6175	2	16	,	,	PUNCT
ejpam-6175	2	17	including	include	VERB
ejpam-6175	2	18	both	both	CCONJ
ejpam-6175	2	19	standard	standard	ADJ
ejpam-6175	2	20	and	and	CCONJ
ejpam-6175	2	21	random	random	ADJ
ejpam-6175	2	22	forms	form	NOUN
ejpam-6175	2	23	.	.	PUNCT
ejpam-6175	3	1	it	it	PRON
ejpam-6175	3	2	focuses	focus	VERB
ejpam-6175	3	3	on	on	ADP
ejpam-6175	3	4	the	the	DET
ejpam-6175	3	5	computation	computation	NOUN
ejpam-6175	3	6	of	of	ADP
ejpam-6175	3	7	genus	genus	NOUN
ejpam-6175	3	8	,	,	PUNCT
ejpam-6175	3	9	betti	betti	NOUN
ejpam-6175	3	10	numbers	number	NOUN
ejpam-6175	3	11	,	,	PUNCT
ejpam-6175	3	12	and	and	CCONJ
ejpam-6175	3	13	persistent	persistent	ADJ
ejpam-6175	3	14	homology	homology	NOUN
ejpam-6175	3	15	,	,	PUNCT
ejpam-6175	3	16	demonstrating	demonstrate	VERB
ejpam-6175	3	17	how	how	SCONJ
ejpam-6175	3	18	this	this	DET
ejpam-6175	3	19	method	method	NOUN
ejpam-6175	3	20	simplifies	simplify	VERB
ejpam-6175	3	21	the	the	DET
ejpam-6175	3	22	analysis	analysis	NOUN
ejpam-6175	3	23	of	of	ADP
ejpam-6175	3	24	complex	complex	ADJ
ejpam-6175	3	25	structures	structure	NOUN
ejpam-6175	3	26	.	.	PUNCT
ejpam-6175	4	1	the	the	DET
ejpam-6175	4	2	approach	approach	NOUN
ejpam-6175	4	3	proves	prove	VERB
ejpam-6175	4	4	valuable	valuable	ADJ
ejpam-6175	4	5	for	for	ADP
ejpam-6175	4	6	tasks	task	NOUN
ejpam-6175	4	7	such	such	ADJ
ejpam-6175	4	8	as	as	ADP
ejpam-6175	4	9	edge	edge	NOUN
ejpam-6175	4	10	detection	detection	NOUN
ejpam-6175	4	11	,	,	PUNCT
ejpam-6175	4	12	thinning	thinning	NOUN
ejpam-6175	4	13	,	,	PUNCT
ejpam-6175	4	14	restructuring	restructuring	NOUN
ejpam-6175	4	15	,	,	PUNCT
ejpam-6175	4	16	and	and	CCONJ
ejpam-6175	4	17	identifying	identify	VERB
ejpam-6175	4	18	genus	genus	ADJ
ejpam-6175	4	19	deformations	deformation	NOUN
ejpam-6175	4	20	.	.	PUNCT
ejpam-6175	5	1	furthermore	furthermore	ADV
ejpam-6175	5	2	,	,	PUNCT
ejpam-6175	5	3	the	the	DET
ejpam-6175	5	4	study	study	NOUN
ejpam-6175	5	5	outlines	outline	VERB
ejpam-6175	5	6	future	future	ADJ
ejpam-6175	5	7	directions	direction	NOUN
ejpam-6175	5	8	,	,	PUNCT
ejpam-6175	5	9	including	include	VERB
ejpam-6175	5	10	the	the	DET
ejpam-6175	5	11	development	development	NOUN
ejpam-6175	5	12	of	of	ADP
ejpam-6175	5	13	a	a	DET
ejpam-6175	5	14	model	model	NOUN
ejpam-6175	5	15	leveraging	leverage	VERB
ejpam-6175	5	16	persistent	persistent	ADJ
ejpam-6175	5	17	homology	homology	NOUN
ejpam-6175	5	18	for	for	ADP
ejpam-6175	5	19	tumor	tumor	NOUN
ejpam-6175	5	20	cell	cell	NOUN
ejpam-6175	5	21	detection	detection	NOUN
ejpam-6175	5	22	and	and	CCONJ
ejpam-6175	5	23	structural	structural	ADJ
ejpam-6175	5	24	genus	genus	NOUN
ejpam-6175	5	25	estimation	estimation	NOUN
ejpam-6175	5	26	.	.	PUNCT
ejpam-6175	6	1	these	these	DET
ejpam-6175	6	2	findings	finding	NOUN
ejpam-6175	6	3	hold	hold	VERB
ejpam-6175	6	4	significant	significant	ADJ
ejpam-6175	6	5	potential	potential	NOUN
ejpam-6175	6	6	for	for	ADP
ejpam-6175	6	7	advancing	advance	VERB
ejpam-6175	6	8	image	image	NOUN
ejpam-6175	6	9	processing	processing	NOUN
ejpam-6175	6	10	and	and	CCONJ
ejpam-6175	6	11	biomedical	biomedical	ADJ
ejpam-6175	6	12	analysis	analysis	NOUN
ejpam-6175	6	13	.	.	PUNCT
ejpam-6175	7	1	2020	2020	NUM
ejpam-6175	7	2	mathematics	mathematic	NOUN
ejpam-6175	7	3	subject	subject	NOUN
ejpam-6175	7	4	classifications	classification	NOUN
ejpam-6175	7	5	:	:	PUNCT
ejpam-6175	7	6	55n31	55n31	NUM
ejpam-6175	7	7	,	,	PUNCT
ejpam-6175	7	8	62r40	62r40	NUM
ejpam-6175	7	9	,	,	PUNCT
ejpam-6175	7	10	68u10	68u10	NUM
ejpam-6175	7	11	key	key	ADJ
ejpam-6175	7	12	words	word	NOUN
ejpam-6175	7	13	and	and	CCONJ
ejpam-6175	7	14	phrases	phrase	NOUN
ejpam-6175	7	15	:	:	PUNCT
ejpam-6175	7	16	digital	digital	ADJ
ejpam-6175	7	17	homology	homology	NOUN
ejpam-6175	7	18	,	,	PUNCT
ejpam-6175	7	19	digital	digital	ADJ
ejpam-6175	7	20	image	image	NOUN
ejpam-6175	7	21	,	,	PUNCT
ejpam-6175	7	22	edge	edge	NOUN
ejpam-6175	7	23	detection	detection	NOUN
ejpam-6175	7	24	,	,	PUNCT
ejpam-6175	7	25	genus	genus	NOUN
ejpam-6175	7	26	,	,	PUNCT
ejpam-6175	7	27	persistent	persistent	ADJ
ejpam-6175	7	28	homology	homology	NOUN
ejpam-6175	7	29	,	,	PUNCT
ejpam-6175	7	30	simplices	simplice	NOUN
ejpam-6175	7	31	1	1	NUM
ejpam-6175	7	32	.	.	PUNCT
ejpam-6175	8	1	introduction	introduction	NOUN
ejpam-6175	8	2	in	in	ADP
ejpam-6175	8	3	digital	digital	ADJ
ejpam-6175	8	4	topology	topology	NOUN
ejpam-6175	8	5	,	,	PUNCT
ejpam-6175	8	6	objects	object	NOUN
ejpam-6175	8	7	are	be	AUX
ejpam-6175	8	8	represented	represent	VERB
ejpam-6175	8	9	as	as	ADV
ejpam-6175	8	10	discrete	discrete	ADJ
ejpam-6175	8	11	or	or	CCONJ
ejpam-6175	8	12	lattice	lattice	NOUN
ejpam-6175	8	13	sets	set	NOUN
ejpam-6175	8	14	of	of	ADP
ejpam-6175	8	15	points	point	NOUN
ejpam-6175	8	16	as	as	ADP
ejpam-6175	8	17	pixels	pixel	NOUN
ejpam-6175	8	18	(	(	PUNCT
ejpam-6175	8	19	2	2	NUM
ejpam-6175	8	20	-	-	PUNCT
ejpam-6175	8	21	dimensional	dimensional	ADJ
ejpam-6175	8	22	array	array	NOUN
ejpam-6175	8	23	)	)	PUNCT
ejpam-6175	8	24	or	or	CCONJ
ejpam-6175	8	25	voxels	voxel	NOUN
ejpam-6175	8	26	(	(	PUNCT
ejpam-6175	8	27	3	3	NUM
ejpam-6175	8	28	-	-	PUNCT
ejpam-6175	8	29	dimensional	dimensional	ADJ
ejpam-6175	8	30	array	array	NOUN
ejpam-6175	8	31	)	)	PUNCT
ejpam-6175	8	32	in	in	ADP
ejpam-6175	8	33	an	an	DET
ejpam-6175	8	34	euclidean	euclidean	ADJ
ejpam-6175	8	35	n	n	CCONJ
ejpam-6175	8	36	-	-	PUNCT
ejpam-6175	8	37	dimensional	dimensional	ADJ
ejpam-6175	8	38	space	space	NOUN
ejpam-6175	8	39	.	.	PUNCT
ejpam-6175	9	1	many	many	ADJ
ejpam-6175	9	2	more	more	ADV
ejpam-6175	9	3	beautiful	beautiful	ADJ
ejpam-6175	9	4	papers	paper	NOUN
ejpam-6175	9	5	are	be	AUX
ejpam-6175	9	6	available	available	ADJ
ejpam-6175	9	7	,	,	PUNCT
ejpam-6175	9	8	which	which	PRON
ejpam-6175	9	9	make	make	VERB
ejpam-6175	9	10	us	we	PRON
ejpam-6175	9	11	understand	understand	VERB
ejpam-6175	9	12	and	and	CCONJ
ejpam-6175	9	13	motivate	motivate	VERB
ejpam-6175	9	14	us	we	PRON
ejpam-6175	9	15	to	to	PART
ejpam-6175	9	16	proceed	proceed	VERB
ejpam-6175	9	17	with	with	ADP
ejpam-6175	9	18	digital	digital	ADJ
ejpam-6175	9	19	topology	topology	NOUN
ejpam-6175	9	20	.	.	PUNCT
ejpam-6175	10	1	there	there	PRON
ejpam-6175	10	2	has	have	AUX
ejpam-6175	10	3	already	already	ADV
ejpam-6175	10	4	been	be	AUX
ejpam-6175	10	5	a	a	DET
ejpam-6175	10	6	lot	lot	NOUN
ejpam-6175	10	7	of	of	ADP
ejpam-6175	10	8	work	work	NOUN
ejpam-6175	10	9	done	do	VERB
ejpam-6175	10	10	in	in	ADP
ejpam-6175	10	11	this	this	DET
ejpam-6175	10	12	area	area	NOUN
ejpam-6175	10	13	,	,	PUNCT
ejpam-6175	10	14	and	and	CCONJ
ejpam-6175	10	15	we	we	PRON
ejpam-6175	10	16	are	be	AUX
ejpam-6175	10	17	still	still	ADV
ejpam-6175	10	18	counting	count	VERB
ejpam-6175	10	19	on	on	ADP
ejpam-6175	10	20	it	it	PRON
ejpam-6175	10	21	.	.	PUNCT
ejpam-6175	11	1	all	all	DET
ejpam-6175	11	2	thanks	thank	NOUN
ejpam-6175	11	3	to	to	ADP
ejpam-6175	11	4	its	its	PRON
ejpam-6175	11	5	wide	wide	ADJ
ejpam-6175	11	6	variety	variety	NOUN
ejpam-6175	11	7	of	of	ADP
ejpam-6175	11	8	practical	practical	ADJ
ejpam-6175	11	9	applications	application	NOUN
ejpam-6175	11	10	such	such	ADJ
ejpam-6175	11	11	as	as	ADP
ejpam-6175	11	12	image	image	NOUN
ejpam-6175	11	13	analysis	analysis	NOUN
ejpam-6175	11	14	,	,	PUNCT
ejpam-6175	11	15	digital	digital	ADJ
ejpam-6175	11	16	geometry	geometry	NOUN
ejpam-6175	11	17	,	,	PUNCT
ejpam-6175	11	18	computer	computer	NOUN
ejpam-6175	11	19	vision	vision	NOUN
ejpam-6175	11	20	,	,	PUNCT
ejpam-6175	11	21	shape	shape	NOUN
ejpam-6175	11	22	recognition	recognition	NOUN
ejpam-6175	11	23	,	,	PUNCT
ejpam-6175	11	24	etc	etc	X
ejpam-6175	11	25	.	.	X
ejpam-6175	11	26	,	,	PUNCT
ejpam-6175	11	27	which	which	PRON
ejpam-6175	11	28	offer	offer	VERB
ejpam-6175	11	29	tools	tool	NOUN
ejpam-6175	11	30	and	and	CCONJ
ejpam-6175	11	31	techniques	technique	NOUN
ejpam-6175	11	32	for	for	ADP
ejpam-6175	11	33	analysing	analyse	VERB
ejpam-6175	11	34	the	the	DET
ejpam-6175	11	35	structure	structure	NOUN
ejpam-6175	11	36	and	and	CCONJ
ejpam-6175	11	37	features	feature	NOUN
ejpam-6175	11	38	of	of	ADP
ejpam-6175	11	39	digital	digital	ADJ
ejpam-6175	11	40	objects	object	NOUN
ejpam-6175	11	41	,	,	PUNCT
ejpam-6175	11	42	as	as	ADV
ejpam-6175	11	43	well	well	ADV
ejpam-6175	11	44	as	as	ADP
ejpam-6175	11	45	the	the	DET
ejpam-6175	11	46	ability	ability	NOUN
ejpam-6175	11	47	to	to	PART
ejpam-6175	11	48	create	create	VERB
ejpam-6175	11	49	algorithms	algorithm	NOUN
ejpam-6175	11	50	for	for	ADP
ejpam-6175	11	51	tasks	task	NOUN
ejpam-6175	11	52	such	such	ADJ
ejpam-6175	11	53	as	as	ADP
ejpam-6175	11	54	image	image	NOUN
ejpam-6175	11	55	thickening	thickening	NOUN
ejpam-6175	11	56	and	and	CCONJ
ejpam-6175	11	57	thinning	thinning	NOUN
ejpam-6175	11	58	,	,	PUNCT
ejpam-6175	11	59	contour	contour	NOUN
ejpam-6175	11	60	filling	filling	NOUN
ejpam-6175	11	61	,	,	PUNCT
ejpam-6175	11	62	object	object	VERB
ejpam-6175	11	63	tracking	tracking	NOUN
ejpam-6175	11	64	and	and	CCONJ
ejpam-6175	11	65	counting	counting	NOUN
ejpam-6175	11	66	,	,	PUNCT
ejpam-6175	11	67	image	image	NOUN
ejpam-6175	11	68	segmentation	segmentation	NOUN
ejpam-6175	11	69	,	,	PUNCT
ejpam-6175	11	70	boundary	boundary	ADJ
ejpam-6175	11	71	extraction	extraction	NOUN
ejpam-6175	11	72	,	,	PUNCT
ejpam-6175	11	73	and	and	CCONJ
ejpam-6175	11	74	shape	shape	NOUN
ejpam-6175	11	75	matching	matching	NOUN
ejpam-6175	11	76	.	.	PUNCT
ejpam-6175	12	1	generally	generally	ADV
ejpam-6175	12	2	topology	topology	NOUN
ejpam-6175	12	3	deals	deal	NOUN
ejpam-6175	12	4	with	with	ADP
ejpam-6175	12	5	continuous	continuous	ADJ
ejpam-6175	12	6	function[1	function[1	PROPN
ejpam-6175	12	7	]	]	PUNCT
ejpam-6175	12	8	,	,	PUNCT
ejpam-6175	12	9	and	and	CCONJ
ejpam-6175	12	10	in	in	ADP
ejpam-6175	12	11	order	order	NOUN
ejpam-6175	12	12	to	to	PART
ejpam-6175	12	13	make	make	VERB
ejpam-6175	12	14	use	use	NOUN
ejpam-6175	12	15	of	of	ADP
ejpam-6175	12	16	the	the	DET
ejpam-6175	12	17	digital	digital	ADJ
ejpam-6175	12	18	algebraic	algebraic	ADJ
ejpam-6175	12	19	topology	topology	NOUN
ejpam-6175	12	20	where	where	SCONJ
ejpam-6175	12	21	the	the	DET
ejpam-6175	12	22	invariants	invariant	NOUN
ejpam-6175	12	23	are	be	AUX
ejpam-6175	12	24	digitilized	digitilize	VERB
ejpam-6175	12	25	or	or	CCONJ
ejpam-6175	12	26	discreted	discrete	VERB
ejpam-6175	12	27	,	,	PUNCT
ejpam-6175	12	28	we	we	PRON
ejpam-6175	12	29	will	will	AUX
ejpam-6175	12	30	be	be	AUX
ejpam-6175	12	31	using	use	VERB
ejpam-6175	12	32	∗corresponding	∗corresponde	VERB
ejpam-6175	12	33	author	author	NOUN
ejpam-6175	12	34	.	.	PUNCT
ejpam-6175	13	1	doi	doi	NOUN
ejpam-6175	13	2	:	:	PUNCT
ejpam-6175	13	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6175	https://doi.org/10.29020/nybg.ejpam.v18i3.6175	NOUN
ejpam-6175	13	4	email	email	NOUN
ejpam-6175	13	5	addresses	address	NOUN
ejpam-6175	13	6	:	:	PUNCT
ejpam-6175	13	7	shivamkumar.jhas2020@vitstudent.ac.in	shivamkumar.jhas2020@vitstudent.ac.in	INTJ
ejpam-6175	13	8	(	(	PUNCT
ejpam-6175	13	9	s.	s.	PROPN
ejpam-6175	13	10	kumar	kumar	PROPN
ejpam-6175	13	11	jha	jha	PROPN
ejpam-6175	13	12	s.	s.	PROPN
ejpam-6175	13	13	)	)	PUNCT
ejpam-6175	13	14	,	,	PUNCT
ejpam-6175	13	15	mohana.n@vit.ac.in	mohana.n@vit.ac.in	PROPN
ejpam-6175	13	16	(	(	PUNCT
ejpam-6175	13	17	m.	m.	PROPN
ejpam-6175	13	18	natarajan	natarajan	PROPN
ejpam-6175	13	19	)	)	PUNCT
ejpam-6175	13	20	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6175	14	1	1	1	NUM
ejpam-6175	14	2	copyright	copyright	NOUN
ejpam-6175	14	3	:	:	PUNCT
ejpam-6175	14	4	©	©	PROPN
ejpam-6175	14	5	2025	2025	NUM
ejpam-6175	14	6	the	the	DET
ejpam-6175	14	7	author(s	author(s	NOUN
ejpam-6175	14	8	)	)	PUNCT
ejpam-6175	14	9	.	.	PUNCT
ejpam-6175	15	1	(	(	PUNCT
ejpam-6175	15	2	cc	cc	NOUN
ejpam-6175	15	3	by	by	ADP
ejpam-6175	15	4	-	-	PUNCT
ejpam-6175	15	5	nc	nc	PROPN
ejpam-6175	15	6	4.0	4.0	NUM
ejpam-6175	15	7	)	)	PUNCT
ejpam-6175	15	8	s.k	s.k	PROPN
ejpam-6175	15	9	.	.	PUNCT
ejpam-6175	15	10	jha	jha	PROPN
ejpam-6175	15	11	sanjay	sanjay	PROPN
ejpam-6175	15	12	,	,	PUNCT
ejpam-6175	15	13	m	m	PROPN
ejpam-6175	15	14	,	,	PUNCT
ejpam-6175	15	15	natarajan	natarajan	PROPN
ejpam-6175	15	16	/	/	SYM
ejpam-6175	15	17	eur	eur	PROPN
ejpam-6175	15	18	.	.	PUNCT
ejpam-6175	16	1	j.	j.	PROPN
ejpam-6175	16	2	pure	pure	PROPN
ejpam-6175	16	3	appl	appl	PROPN
ejpam-6175	16	4	.	.	PROPN
ejpam-6175	16	5	math	math	PROPN
ejpam-6175	16	6	,	,	PUNCT
ejpam-6175	16	7	18	18	NUM
ejpam-6175	16	8	(	(	PUNCT
ejpam-6175	16	9	3	3	NUM
ejpam-6175	16	10	)	)	PUNCT
ejpam-6175	16	11	(	(	PUNCT
ejpam-6175	16	12	2025	2025	NUM
ejpam-6175	16	13	)	)	PUNCT
ejpam-6175	16	14	,	,	PUNCT
ejpam-6175	16	15	6175	6175	NUM
ejpam-6175	16	16	2	2	NUM
ejpam-6175	16	17	of	of	ADP
ejpam-6175	16	18	18	18	NUM
ejpam-6175	16	19	z1	z1	ADJ
ejpam-6175	16	20	z2	z2	PROPN
ejpam-6175	16	21	z3	z3	PROPN
ejpam-6175	16	22	·	·	PUNCT
ejpam-6175	16	23	·	·	PUNCT
ejpam-6175	17	1	·	·	PUNCT
ejpam-6175	17	2	κ1	κ1	NOUN
ejpam-6175	17	3	2	2	NUM
ejpam-6175	17	4	4	4	NUM
ejpam-6175	17	5	6	6	NUM
ejpam-6175	17	6	·	·	PUNCT
ejpam-6175	17	7	·	·	PUNCT
ejpam-6175	17	8	·	·	PUNCT
ejpam-6175	17	9	κ2	κ2	NOUN
ejpam-6175	17	10	−	−	NOUN
ejpam-6175	17	11	8	8	NUM
ejpam-6175	17	12	18	18	NUM
ejpam-6175	17	13	·	·	PUNCT
ejpam-6175	17	14	·	·	PUNCT
ejpam-6175	17	15	·	·	PUNCT
ejpam-6175	17	16	κ3	κ3	VERB
ejpam-6175	17	17	−	−	NOUN
ejpam-6175	17	18	−	−	PROPN
ejpam-6175	17	19	26	26	NUM
ejpam-6175	17	20	·	·	PUNCT
ejpam-6175	17	21	·	·	PUNCT
ejpam-6175	17	22	·	·	PUNCT
ejpam-6175	17	23	table	table	NOUN
ejpam-6175	17	24	1	1	NUM
ejpam-6175	17	25	:	:	PUNCT
ejpam-6175	17	26	adjacency	adjacency	PROPN
ejpam-6175	17	27	relations	relation	NOUN
ejpam-6175	17	28	for	for	ADP
ejpam-6175	17	29	different	different	ADJ
ejpam-6175	17	30	dimensions	dimension	NOUN
ejpam-6175	17	31	(	(	PUNCT
ejpam-6175	17	32	z1	z1	PROPN
ejpam-6175	17	33	,	,	PUNCT
ejpam-6175	17	34	z2	z2	PROPN
ejpam-6175	17	35	,	,	PUNCT
ejpam-6175	17	36	z3	z3	PROPN
ejpam-6175	17	37	)	)	PUNCT
ejpam-6175	17	38	.	.	PUNCT
ejpam-6175	18	1	each	each	DET
ejpam-6175	18	2	entry	entry	NOUN
ejpam-6175	18	3	κi	κi	NOUN
ejpam-6175	18	4	denotes	denote	VERB
ejpam-6175	18	5	the	the	DET
ejpam-6175	18	6	number	number	NOUN
ejpam-6175	18	7	of	of	ADP
ejpam-6175	18	8	adjacencies	adjacency	NOUN
ejpam-6175	18	9	in	in	ADP
ejpam-6175	18	10	the	the	DET
ejpam-6175	18	11	respective	respective	ADJ
ejpam-6175	18	12	dimension	dimension	NOUN
ejpam-6175	18	13	.	.	PUNCT
ejpam-6175	19	1	zero	zero	NUM
ejpam-6175	19	2	entries	entry	NOUN
ejpam-6175	19	3	denotes	denote	VERB
ejpam-6175	19	4	the	the	DET
ejpam-6175	19	5	digital	digital	ADJ
ejpam-6175	19	6	continuity	continuity	NOUN
ejpam-6175	19	7	which	which	PRON
ejpam-6175	19	8	is	be	AUX
ejpam-6175	19	9	nothing	nothing	PRON
ejpam-6175	19	10	but	but	CCONJ
ejpam-6175	19	11	nearer	near	ADJ
ejpam-6175	19	12	points	point	NOUN
ejpam-6175	19	13	to	to	ADP
ejpam-6175	19	14	nearer	near	ADJ
ejpam-6175	19	15	points	point	NOUN
ejpam-6175	19	16	or	or	CCONJ
ejpam-6175	19	17	neighbouring	neighbouring	NOUN
ejpam-6175	19	18	points	point	NOUN
ejpam-6175	19	19	which	which	PRON
ejpam-6175	19	20	are	be	AUX
ejpam-6175	19	21	next	next	ADJ
ejpam-6175	19	22	to	to	ADP
ejpam-6175	19	23	each	each	DET
ejpam-6175	19	24	other	other	ADJ
ejpam-6175	19	25	,	,	PUNCT
ejpam-6175	19	26	and	and	CCONJ
ejpam-6175	19	27	similarly	similarly	ADV
ejpam-6175	19	28	all	all	DET
ejpam-6175	19	29	other	other	ADJ
ejpam-6175	19	30	notions	notion	NOUN
ejpam-6175	19	31	including	include	VERB
ejpam-6175	19	32	digital	digital	ADJ
ejpam-6175	19	33	homology	homology	NOUN
ejpam-6175	19	34	.	.	PUNCT
ejpam-6175	20	1	further	far	ADV
ejpam-6175	20	2	we	we	PRON
ejpam-6175	20	3	will	will	AUX
ejpam-6175	20	4	see	see	VERB
ejpam-6175	20	5	the	the	DET
ejpam-6175	20	6	digital	digital	ADJ
ejpam-6175	20	7	method	method	NOUN
ejpam-6175	20	8	to	to	PART
ejpam-6175	20	9	find	find	VERB
ejpam-6175	20	10	the	the	DET
ejpam-6175	20	11	genus	genus	NOUN
ejpam-6175	20	12	of	of	ADP
ejpam-6175	20	13	a	a	DET
ejpam-6175	20	14	structure	structure	NOUN
ejpam-6175	20	15	and	and	CCONJ
ejpam-6175	20	16	have	have	VERB
ejpam-6175	20	17	some	some	DET
ejpam-6175	20	18	beautiful	beautiful	ADJ
ejpam-6175	20	19	result	result	NOUN
ejpam-6175	20	20	,	,	PUNCT
ejpam-6175	20	21	which	which	PRON
ejpam-6175	20	22	have	have	VERB
ejpam-6175	20	23	some	some	DET
ejpam-6175	20	24	justification	justification	NOUN
ejpam-6175	20	25	in	in	ADP
ejpam-6175	20	26	the	the	DET
ejpam-6175	20	27	contribution	contribution	NOUN
ejpam-6175	20	28	of	of	ADP
ejpam-6175	20	29	digital	digital	ADJ
ejpam-6175	20	30	topology	topology	NOUN
ejpam-6175	20	31	.	.	PUNCT
ejpam-6175	21	1	1.1	1.1	NUM
ejpam-6175	21	2	.	.	PUNCT
ejpam-6175	21	3	basic	basic	ADJ
ejpam-6175	21	4	definitions	definition	NOUN
ejpam-6175	21	5	on	on	ADP
ejpam-6175	21	6	digital	digital	ADJ
ejpam-6175	21	7	algebraic	algebraic	ADJ
ejpam-6175	21	8	topology	topology	NOUN
ejpam-6175	21	9	let	let	VERB
ejpam-6175	21	10	z	z	NOUN
ejpam-6175	21	11	be	be	AUX
ejpam-6175	21	12	the	the	DET
ejpam-6175	21	13	set	set	NOUN
ejpam-6175	21	14	of	of	ADP
ejpam-6175	21	15	all	all	DET
ejpam-6175	21	16	integers	integer	NOUN
ejpam-6175	21	17	.	.	PUNCT
ejpam-6175	22	1	in	in	ADP
ejpam-6175	22	2	euclidean	euclidean	ADJ
ejpam-6175	22	3	n	n	CCONJ
ejpam-6175	22	4	-	-	PUNCT
ejpam-6175	22	5	dimensional	dimensional	ADJ
ejpam-6175	22	6	space	space	NOUN
ejpam-6175	22	7	,	,	PUNCT
ejpam-6175	22	8	zn	zn	PROPN
ejpam-6175	22	9	is	be	AUX
ejpam-6175	22	10	the	the	DET
ejpam-6175	22	11	set	set	NOUN
ejpam-6175	22	12	of	of	ADP
ejpam-6175	22	13	lattice	lattice	NOUN
ejpam-6175	22	14	points	point	NOUN
ejpam-6175	22	15	.	.	PUNCT
ejpam-6175	23	1	a	a	DET
ejpam-6175	23	2	finite	finite	PROPN
ejpam-6175	23	3	subset	subset	NOUN
ejpam-6175	23	4	of	of	ADP
ejpam-6175	23	5	zn	zn	PROPN
ejpam-6175	23	6	with	with	ADP
ejpam-6175	23	7	an	an	DET
ejpam-6175	23	8	adjacent	adjacent	ADJ
ejpam-6175	23	9	relation	relation	NOUN
ejpam-6175	23	10	is	be	AUX
ejpam-6175	23	11	called	call	VERB
ejpam-6175	23	12	a	a	DET
ejpam-6175	23	13	(	(	PUNCT
ejpam-6175	23	14	binary)digital	binary)digital	ADJ
ejpam-6175	23	15	image	image	NOUN
ejpam-6175	23	16	.	.	PUNCT
ejpam-6175	24	1	definition	definition	NOUN
ejpam-6175	24	2	1	1	NUM
ejpam-6175	24	3	.	.	PUNCT
ejpam-6175	25	1	[	[	X
ejpam-6175	25	2	2	2	X
ejpam-6175	25	3	]	]	PUNCT
ejpam-6175	25	4	for	for	ADP
ejpam-6175	25	5	a	a	DET
ejpam-6175	25	6	<	<	X
ejpam-6175	25	7	b	b	PROPN
ejpam-6175	25	8	,	,	PUNCT
ejpam-6175	25	9	the	the	DET
ejpam-6175	25	10	set	set	NOUN
ejpam-6175	25	11	[	[	X
ejpam-6175	25	12	a	a	X
ejpam-6175	25	13	,	,	PUNCT
ejpam-6175	25	14	b]z	b]z	NOUN
ejpam-6175	25	15	=	=	SYM
ejpam-6175	25	16	{	{	PUNCT
ejpam-6175	25	17	z	z	NOUN
ejpam-6175	25	18	∈	∈	PROPN
ejpam-6175	25	19	z|a	z|a	X
ejpam-6175	26	1	≤	≤	NUM
ejpam-6175	26	2	z	z	NOUN
ejpam-6175	26	3	≤	≤	NUM
ejpam-6175	27	1	b	b	X
ejpam-6175	27	2	}	}	PUNCT
ejpam-6175	27	3	is	be	AUX
ejpam-6175	27	4	called	call	VERB
ejpam-6175	27	5	a	a	DET
ejpam-6175	27	6	digital	digital	ADJ
ejpam-6175	27	7	interval	interval	NOUN
ejpam-6175	27	8	where	where	SCONJ
ejpam-6175	27	9	a	a	DET
ejpam-6175	27	10	,	,	PUNCT
ejpam-6175	27	11	b	b	PROPN
ejpam-6175	27	12	∈	∈	PROPN
ejpam-6175	27	13	z.	z.	PROPN
ejpam-6175	27	14	digital	digital	PROPN
ejpam-6175	27	15	image	image	NOUN
ejpam-6175	27	16	analysis	analysis	NOUN
ejpam-6175	27	17	utilizes	utilize	VERB
ejpam-6175	27	18	various	various	ADJ
ejpam-6175	27	19	types	type	NOUN
ejpam-6175	27	20	of	of	ADP
ejpam-6175	27	21	adjacency	adjacency	NOUN
ejpam-6175	27	22	relations	relation	NOUN
ejpam-6175	27	23	[	[	X
ejpam-6175	27	24	3	3	NUM
ejpam-6175	27	25	]	]	PUNCT
ejpam-6175	27	26	.	.	PUNCT
ejpam-6175	28	1	in	in	ADP
ejpam-6175	28	2	z1	z1	PROPN
ejpam-6175	28	3	,	,	PUNCT
ejpam-6175	28	4	two	two	NUM
ejpam-6175	28	5	points	point	NOUN
ejpam-6175	28	6	p	p	NOUN
ejpam-6175	28	7	and	and	CCONJ
ejpam-6175	28	8	q	q	NOUN
ejpam-6175	28	9	are	be	AUX
ejpam-6175	28	10	considered	consider	VERB
ejpam-6175	28	11	1	1	NUM
ejpam-6175	28	12	-	-	NOUN
ejpam-6175	28	13	adjacent	adjacent	ADJ
ejpam-6175	28	14	if	if	SCONJ
ejpam-6175	28	15	they	they	PRON
ejpam-6175	28	16	differ	differ	VERB
ejpam-6175	28	17	by	by	ADP
ejpam-6175	28	18	exactly	exactly	ADV
ejpam-6175	28	19	one	one	NUM
ejpam-6175	28	20	coordinate	coordinate	NOUN
ejpam-6175	28	21	.	.	PUNCT
ejpam-6175	29	1	in	in	ADP
ejpam-6175	29	2	z2	z2	PROPN
ejpam-6175	29	3	,	,	PUNCT
ejpam-6175	29	4	points	point	VERB
ejpam-6175	29	5	p	p	NOUN
ejpam-6175	29	6	and	and	CCONJ
ejpam-6175	29	7	q	q	NOUN
ejpam-6175	29	8	are	be	AUX
ejpam-6175	29	9	4	4	NUM
ejpam-6175	29	10	-	-	NOUN
ejpam-6175	29	11	adjacent	adjacent	ADJ
ejpam-6175	29	12	if	if	SCONJ
ejpam-6175	29	13	they	they	PRON
ejpam-6175	29	14	are	be	AUX
ejpam-6175	29	15	distinct	distinct	ADJ
ejpam-6175	29	16	and	and	CCONJ
ejpam-6175	29	17	differ	differ	VERB
ejpam-6175	29	18	in	in	ADP
ejpam-6175	29	19	exactly	exactly	ADV
ejpam-6175	29	20	one	one	NUM
ejpam-6175	29	21	coordinate	coordinate	NOUN
ejpam-6175	29	22	,	,	PUNCT
ejpam-6175	29	23	and	and	CCONJ
ejpam-6175	29	24	they	they	PRON
ejpam-6175	29	25	are	be	AUX
ejpam-6175	29	26	8	8	NUM
ejpam-6175	29	27	-	-	PUNCT
ejpam-6175	29	28	adjacent	adjacent	ADJ
ejpam-6175	29	29	if	if	SCONJ
ejpam-6175	29	30	they	they	PRON
ejpam-6175	29	31	differ	differ	VERB
ejpam-6175	29	32	by	by	ADP
ejpam-6175	29	33	at	at	ADV
ejpam-6175	29	34	most	most	ADV
ejpam-6175	29	35	one	one	NUM
ejpam-6175	29	36	in	in	ADP
ejpam-6175	29	37	each	each	DET
ejpam-6175	29	38	coordinate	coordinate	NOUN
ejpam-6175	29	39	.	.	PUNCT
ejpam-6175	30	1	in	in	ADP
ejpam-6175	30	2	z3	z3	PROPN
ejpam-6175	30	3	,	,	PUNCT
ejpam-6175	30	4	points	point	VERB
ejpam-6175	30	5	p	p	NOUN
ejpam-6175	30	6	and	and	CCONJ
ejpam-6175	30	7	q	q	NOUN
ejpam-6175	30	8	are	be	AUX
ejpam-6175	30	9	deemed	deem	VERB
ejpam-6175	30	10	26	26	NUM
ejpam-6175	30	11	-	-	PUNCT
ejpam-6175	30	12	adjacent	adjacent	ADJ
ejpam-6175	30	13	if	if	SCONJ
ejpam-6175	30	14	they	they	PRON
ejpam-6175	30	15	are	be	AUX
ejpam-6175	30	16	distinct	distinct	ADJ
ejpam-6175	30	17	and	and	CCONJ
ejpam-6175	30	18	differ	differ	VERB
ejpam-6175	30	19	by	by	ADP
ejpam-6175	30	20	no	no	DET
ejpam-6175	30	21	more	more	ADJ
ejpam-6175	30	22	than	than	ADP
ejpam-6175	30	23	one	one	NUM
ejpam-6175	30	24	coordinate	coordinate	NOUN
ejpam-6175	30	25	;	;	PUNCT
ejpam-6175	30	26	they	they	PRON
ejpam-6175	30	27	are	be	AUX
ejpam-6175	30	28	18	18	NUM
ejpam-6175	30	29	-	-	PUNCT
ejpam-6175	30	30	adjacent	adjacent	ADJ
ejpam-6175	30	31	if	if	SCONJ
ejpam-6175	30	32	they	they	PRON
ejpam-6175	30	33	are	be	AUX
ejpam-6175	30	34	26	26	NUM
ejpam-6175	30	35	-	-	PUNCT
ejpam-6175	30	36	adjacent	adjacent	ADJ
ejpam-6175	30	37	and	and	CCONJ
ejpam-6175	30	38	differ	differ	VERB
ejpam-6175	30	39	by	by	ADP
ejpam-6175	30	40	at	at	ADP
ejpam-6175	30	41	most	most	ADV
ejpam-6175	30	42	two	two	NUM
ejpam-6175	30	43	coordinates	coordinate	NOUN
ejpam-6175	30	44	;	;	PUNCT
ejpam-6175	30	45	and	and	CCONJ
ejpam-6175	30	46	they	they	PRON
ejpam-6175	30	47	are	be	AUX
ejpam-6175	30	48	6	6	NUM
ejpam-6175	30	49	-	-	PUNCT
ejpam-6175	30	50	adjacent	adjacent	ADJ
ejpam-6175	30	51	if	if	SCONJ
ejpam-6175	30	52	they	they	PRON
ejpam-6175	30	53	are	be	AUX
ejpam-6175	30	54	18	18	NUM
ejpam-6175	30	55	-	-	PUNCT
ejpam-6175	30	56	adjacent	adjacent	ADJ
ejpam-6175	30	57	and	and	CCONJ
ejpam-6175	30	58	differ	differ	VERB
ejpam-6175	30	59	by	by	ADP
ejpam-6175	30	60	exactly	exactly	ADV
ejpam-6175	30	61	one	one	NUM
ejpam-6175	30	62	coordinate	coordinate	NOUN
ejpam-6175	30	63	.	.	PUNCT
ejpam-6175	31	1	the	the	DET
ejpam-6175	31	2	table[1	table[1	NOUN
ejpam-6175	31	3	]	]	PUNCT
ejpam-6175	31	4	illustrates	illustrate	VERB
ejpam-6175	31	5	the	the	DET
ejpam-6175	31	6	different	different	ADJ
ejpam-6175	31	7	adjacency	adjacency	NOUN
ejpam-6175	31	8	relations	relation	NOUN
ejpam-6175	31	9	for	for	ADP
ejpam-6175	31	10	their	their	PRON
ejpam-6175	31	11	respective	respective	ADJ
ejpam-6175	31	12	dimensions	dimension	NOUN
ejpam-6175	31	13	.	.	PUNCT
ejpam-6175	32	1	a	a	DET
ejpam-6175	32	2	point	point	NOUN
ejpam-6175	32	3	that	that	PRON
ejpam-6175	32	4	is	be	AUX
ejpam-6175	32	5	κ	κ	NOUN
ejpam-6175	32	6	-	-	ADJ
ejpam-6175	32	7	adjacent	adjacent	ADJ
ejpam-6175	32	8	to	to	ADP
ejpam-6175	32	9	a	a	DET
ejpam-6175	32	10	lattice	lattice	NOUN
ejpam-6175	32	11	point	point	NOUN
ejpam-6175	32	12	p	p	NOUN
ejpam-6175	32	13	is	be	AUX
ejpam-6175	32	14	referred	refer	VERB
ejpam-6175	32	15	to	to	ADP
ejpam-6175	32	16	as	as	ADP
ejpam-6175	32	17	its	its	PRON
ejpam-6175	32	18	κ	κ	NOUN
ejpam-6175	32	19	-	-	NOUN
ejpam-6175	32	20	neighbor	neighbor	NOUN
ejpam-6175	32	21	for	for	ADP
ejpam-6175	32	22	κ	κ	PROPN
ejpam-6175	32	23	∈	∈	PROPN
ejpam-6175	32	24	{	{	PUNCT
ejpam-6175	32	25	2	2	NUM
ejpam-6175	32	26	,	,	PUNCT
ejpam-6175	32	27	4	4	NUM
ejpam-6175	32	28	,	,	PUNCT
ejpam-6175	32	29	8	8	NUM
ejpam-6175	32	30	,	,	PUNCT
ejpam-6175	32	31	6	6	NUM
ejpam-6175	32	32	,	,	PUNCT
ejpam-6175	32	33	18	18	NUM
ejpam-6175	32	34	,	,	PUNCT
ejpam-6175	32	35	26	26	NUM
ejpam-6175	32	36	}	}	PUNCT
ejpam-6175	32	37	.	.	PUNCT
ejpam-6175	33	1	these	these	DET
ejpam-6175	33	2	adjacencies	adjacency	NOUN
ejpam-6175	33	3	are	be	AUX
ejpam-6175	33	4	generalized	generalize	VERB
ejpam-6175	33	5	as	as	SCONJ
ejpam-6175	33	6	follows	follow	VERB
ejpam-6175	33	7	:	:	PUNCT
ejpam-6175	33	8	consider	consider	VERB
ejpam-6175	33	9	positive	positive	ADJ
ejpam-6175	33	10	numbers	number	NOUN
ejpam-6175	33	11	l	l	NOUN
ejpam-6175	33	12	and	and	CCONJ
ejpam-6175	33	13	n	n	CCONJ
ejpam-6175	33	14	such	such	ADJ
ejpam-6175	33	15	that	that	SCONJ
ejpam-6175	33	16	1	1	NUM
ejpam-6175	33	17	≤	≤	NUM
ejpam-6175	33	18	l	l	NOUN
ejpam-6175	33	19	≤	≤	PUNCT
ejpam-6175	33	20	n.	n.	NOUN
ejpam-6175	33	21	if	if	SCONJ
ejpam-6175	33	22	there	there	PRON
ejpam-6175	33	23	exist	exist	VERB
ejpam-6175	33	24	at	at	ADP
ejpam-6175	33	25	most	most	ADJ
ejpam-6175	33	26	l	l	NOUN
ejpam-6175	33	27	unique	unique	ADJ
ejpam-6175	33	28	coordinates	coordinate	NOUN
ejpam-6175	33	29	j	j	PROPN
ejpam-6175	34	1	for	for	ADP
ejpam-6175	34	2	which	which	PRON
ejpam-6175	34	3	|pj	|pj	ADP
ejpam-6175	34	4	−	−	PROPN
ejpam-6175	34	5	qj	qj	PROPN
ejpam-6175	34	6	|	|	NOUN
ejpam-6175	34	7	=	=	NOUN
ejpam-6175	34	8	1	1	NUM
ejpam-6175	34	9	,	,	PUNCT
ejpam-6175	34	10	and	and	CCONJ
ejpam-6175	34	11	for	for	ADP
ejpam-6175	34	12	all	all	DET
ejpam-6175	34	13	other	other	ADJ
ejpam-6175	34	14	coordinates	coordinate	NOUN
ejpam-6175	34	15	j	j	PROPN
ejpam-6175	34	16	,	,	PUNCT
ejpam-6175	34	17	pj	pj	PROPN
ejpam-6175	34	18	=	=	SYM
ejpam-6175	34	19	qj	qj	PROPN
ejpam-6175	34	20	,	,	PUNCT
ejpam-6175	34	21	subsequently	subsequently	ADV
ejpam-6175	34	22	distinct	distinct	ADJ
ejpam-6175	34	23	points	point	NOUN
ejpam-6175	34	24	p	p	X
ejpam-6175	34	25	,	,	PUNCT
ejpam-6175	34	26	q	q	PROPN
ejpam-6175	34	27	∈	∈	PROPN
ejpam-6175	34	28	zn	zn	X
ejpam-6175	34	29	are	be	AUX
ejpam-6175	34	30	said	say	VERB
ejpam-6175	34	31	to	to	PART
ejpam-6175	34	32	be	be	AUX
ejpam-6175	34	33	kl	kl	NOUN
ejpam-6175	34	34	-	-	NOUN
ejpam-6175	34	35	adjacent	adjacent	ADJ
ejpam-6175	34	36	.	.	PUNCT
ejpam-6175	35	1	in	in	ADP
ejpam-6175	35	2	this	this	DET
ejpam-6175	35	3	sense	sense	NOUN
ejpam-6175	35	4	,	,	PUNCT
ejpam-6175	35	5	the	the	DET
ejpam-6175	35	6	notation	notation	NOUN
ejpam-6175	35	7	kl	kl	PROPN
ejpam-6175	35	8	denotes	denote	VERB
ejpam-6175	35	9	the	the	DET
ejpam-6175	35	10	number	number	NOUN
ejpam-6175	35	11	of	of	ADP
ejpam-6175	35	12	points	point	NOUN
ejpam-6175	35	13	q	q	PROPN
ejpam-6175	35	14	∈	∈	PROPN
ejpam-6175	35	15	zn	zn	X
ejpam-6175	35	16	that	that	PRON
ejpam-6175	35	17	are	be	AUX
ejpam-6175	35	18	close	close	ADJ
ejpam-6175	35	19	to	to	ADP
ejpam-6175	35	20	a	a	DET
ejpam-6175	35	21	given	give	VERB
ejpam-6175	35	22	point	point	NOUN
ejpam-6175	35	23	p	p	X
ejpam-6175	35	24	∈	∈	PROPN
ejpam-6175	36	1	zn	zn	X
ejpam-6175	36	2	.	.	PUNCT
ejpam-6175	37	1	therefore	therefore	ADV
ejpam-6175	37	2	,	,	PUNCT
ejpam-6175	37	3	the	the	DET
ejpam-6175	37	4	numbers	number	NOUN
ejpam-6175	37	5	listed	list	VERB
ejpam-6175	37	6	above	above	ADV
ejpam-6175	37	7	are	be	AUX
ejpam-6175	37	8	as	as	SCONJ
ejpam-6175	37	9	follows	follow	VERB
ejpam-6175	37	10	:	:	PUNCT
ejpam-6175	38	1	adjacency	adjacency	PROPN
ejpam-6175	38	2	relation	relation	NOUN
ejpam-6175	38	3	=	=	PROPN
ejpam-6175	39	1			PROPN
ejpam-6175	39	2	κ1	κ1	NOUN
ejpam-6175	39	3	=	=	SYM
ejpam-6175	39	4	2n	2n	NUM
ejpam-6175	39	5	;	;	PUNCT
ejpam-6175	39	6	n	n	NUM
ejpam-6175	39	7	≥	≥	NOUN
ejpam-6175	39	8	1	1	NUM
ejpam-6175	39	9	κ2	κ2	PROPN
ejpam-6175	39	10	=	=	SYM
ejpam-6175	39	11	n2	n2	PROPN
ejpam-6175	39	12	+	+	CCONJ
ejpam-6175	39	13	5n−	5n−	NUM
ejpam-6175	39	14	6	6	NUM
ejpam-6175	39	15	;	;	PUNCT
ejpam-6175	39	16	n	n	PRON
ejpam-6175	39	17	≥	≥	NOUN
ejpam-6175	39	18	2	2	NUM
ejpam-6175	40	1	κ3	κ3	PROPN
ejpam-6175	40	2	=	=	SYM
ejpam-6175	41	1	13n2	13n2	NUM
ejpam-6175	41	2	−	−	NUM
ejpam-6175	41	3	39n+	39n+	NUM
ejpam-6175	41	4	26	26	NUM
ejpam-6175	41	5	;	;	PUNCT
ejpam-6175	41	6	n	n	PRON
ejpam-6175	41	7	≥	≥	NOUN
ejpam-6175	41	8	3	3	NUM
ejpam-6175	41	9	additionally	additionally	ADV
ejpam-6175	42	1	,	,	PUNCT
ejpam-6175	42	2	a	a	DET
ejpam-6175	42	3	few	few	ADJ
ejpam-6175	42	4	general	general	ADJ
ejpam-6175	42	5	adjacency	adjacency	NOUN
ejpam-6175	42	6	relations	relation	NOUN
ejpam-6175	42	7	are	be	AUX
ejpam-6175	42	8	covered	cover	VERB
ejpam-6175	42	9	in	in	ADP
ejpam-6175	42	10	[	[	X
ejpam-6175	42	11	4	4	NUM
ejpam-6175	42	12	,	,	PUNCT
ejpam-6175	42	13	5	5	NUM
ejpam-6175	42	14	]	]	PUNCT
ejpam-6175	42	15	.	.	PUNCT
ejpam-6175	43	1	in	in	ADP
ejpam-6175	43	2	digital	digital	ADJ
ejpam-6175	43	3	topology	topology	NOUN
ejpam-6175	43	4	,	,	PUNCT
ejpam-6175	43	5	κ	κ	NOUN
ejpam-6175	43	6	-	-	PUNCT
ejpam-6175	43	7	connectivity	connectivity	NOUN
ejpam-6175	43	8	describes	describe	VERB
ejpam-6175	43	9	the	the	DET
ejpam-6175	43	10	relationship	relationship	NOUN
ejpam-6175	43	11	between	between	ADP
ejpam-6175	43	12	points	point	NOUN
ejpam-6175	43	13	in	in	ADP
ejpam-6175	43	14	a	a	DET
ejpam-6175	43	15	digital	digital	ADJ
ejpam-6175	43	16	image	image	NOUN
ejpam-6175	43	17	where	where	SCONJ
ejpam-6175	43	18	each	each	DET
ejpam-6175	43	19	point	point	NOUN
ejpam-6175	43	20	is	be	AUX
ejpam-6175	43	21	connected	connect	VERB
ejpam-6175	43	22	to	to	ADP
ejpam-6175	43	23	its	its	PRON
ejpam-6175	43	24	neighbors	neighbor	NOUN
ejpam-6175	43	25	based	base	VERB
ejpam-6175	43	26	on	on	ADP
ejpam-6175	43	27	a	a	DET
ejpam-6175	43	28	specified	specify	VERB
ejpam-6175	43	29	adjacency	adjacency	NOUN
ejpam-6175	43	30	[	[	X
ejpam-6175	43	31	5	5	NUM
ejpam-6175	43	32	,	,	PUNCT
ejpam-6175	43	33	6	6	NUM
ejpam-6175	43	34	]	]	PUNCT
ejpam-6175	43	35	.	.	PUNCT
ejpam-6175	44	1	a	a	DET
ejpam-6175	44	2	κ	κ	ADV
ejpam-6175	44	3	-	-	PUNCT
ejpam-6175	44	4	connected	connect	VERB
ejpam-6175	44	5	s.k	s.k	PROPN
ejpam-6175	44	6	.	.	PUNCT
ejpam-6175	44	7	jha	jha	PROPN
ejpam-6175	44	8	sanjay	sanjay	PROPN
ejpam-6175	44	9	,	,	PUNCT
ejpam-6175	44	10	m	m	PROPN
ejpam-6175	44	11	,	,	PUNCT
ejpam-6175	44	12	natarajan	natarajan	PROPN
ejpam-6175	44	13	/	/	SYM
ejpam-6175	44	14	eur	eur	PROPN
ejpam-6175	44	15	.	.	PUNCT
ejpam-6175	45	1	j.	j.	PROPN
ejpam-6175	45	2	pure	pure	PROPN
ejpam-6175	45	3	appl	appl	PROPN
ejpam-6175	45	4	.	.	PROPN
ejpam-6175	45	5	math	math	PROPN
ejpam-6175	45	6	,	,	PUNCT
ejpam-6175	45	7	18	18	NUM
ejpam-6175	45	8	(	(	PUNCT
ejpam-6175	45	9	3	3	NUM
ejpam-6175	45	10	)	)	PUNCT
ejpam-6175	45	11	(	(	PUNCT
ejpam-6175	45	12	2025	2025	NUM
ejpam-6175	45	13	)	)	PUNCT
ejpam-6175	45	14	,	,	PUNCT
ejpam-6175	45	15	6175	6175	NUM
ejpam-6175	45	16	3	3	NUM
ejpam-6175	45	17	of	of	ADP
ejpam-6175	45	18	18	18	NUM
ejpam-6175	45	19	image	image	NOUN
ejpam-6175	45	20	ensures	ensure	VERB
ejpam-6175	45	21	that	that	SCONJ
ejpam-6175	45	22	there	there	PRON
ejpam-6175	45	23	is	be	VERB
ejpam-6175	45	24	a	a	DET
ejpam-6175	45	25	path	path	NOUN
ejpam-6175	45	26	between	between	ADP
ejpam-6175	45	27	any	any	DET
ejpam-6175	45	28	two	two	NUM
ejpam-6175	45	29	distinct	distinct	ADJ
ejpam-6175	45	30	points	point	NOUN
ejpam-6175	45	31	,	,	PUNCT
ejpam-6175	45	32	defined	define	VERB
ejpam-6175	45	33	by	by	ADP
ejpam-6175	45	34	a	a	DET
ejpam-6175	45	35	sequence	sequence	NOUN
ejpam-6175	45	36	of	of	ADP
ejpam-6175	45	37	neighboring	neighboring	NOUN
ejpam-6175	45	38	points	point	NOUN
ejpam-6175	45	39	.	.	PUNCT
ejpam-6175	46	1	the	the	DET
ejpam-6175	46	2	concept	concept	NOUN
ejpam-6175	46	3	of	of	ADP
ejpam-6175	46	4	κ	κ	NOUN
ejpam-6175	46	5	-	-	PUNCT
ejpam-6175	46	6	connectivity	connectivity	NOUN
ejpam-6175	46	7	extends	extend	VERB
ejpam-6175	46	8	to	to	ADP
ejpam-6175	46	9	functions	function	NOUN
ejpam-6175	46	10	between	between	ADP
ejpam-6175	46	11	digital	digital	ADJ
ejpam-6175	46	12	images	image	NOUN
ejpam-6175	46	13	,	,	PUNCT
ejpam-6175	46	14	where	where	SCONJ
ejpam-6175	46	15	a	a	DET
ejpam-6175	46	16	function	function	NOUN
ejpam-6175	46	17	is	be	AUX
ejpam-6175	46	18	considered	consider	VERB
ejpam-6175	46	19	continuous	continuous	ADJ
ejpam-6175	46	20	if	if	SCONJ
ejpam-6175	46	21	it	it	PRON
ejpam-6175	46	22	preserves	preserve	VERB
ejpam-6175	46	23	the	the	DET
ejpam-6175	46	24	connectivity	connectivity	NOUN
ejpam-6175	46	25	of	of	ADP
ejpam-6175	46	26	subsets	subset	NOUN
ejpam-6175	46	27	from	from	ADP
ejpam-6175	46	28	one	one	NUM
ejpam-6175	46	29	image	image	NOUN
ejpam-6175	46	30	to	to	ADP
ejpam-6175	46	31	another	another	PRON
ejpam-6175	46	32	[	[	X
ejpam-6175	46	33	6	6	NUM
ejpam-6175	46	34	,	,	PUNCT
ejpam-6175	46	35	7	7	NUM
ejpam-6175	46	36	]	]	PUNCT
ejpam-6175	46	37	.	.	PUNCT
ejpam-6175	47	1	this	this	DET
ejpam-6175	47	2	notion	notion	NOUN
ejpam-6175	47	3	of	of	ADP
ejpam-6175	47	4	continuity	continuity	NOUN
ejpam-6175	47	5	leads	lead	VERB
ejpam-6175	47	6	to	to	ADP
ejpam-6175	47	7	the	the	DET
ejpam-6175	47	8	idea	idea	NOUN
ejpam-6175	47	9	of	of	ADP
ejpam-6175	47	10	κ	κ	NOUN
ejpam-6175	47	11	-	-	PUNCT
ejpam-6175	47	12	isomorphisms	isomorphism	NOUN
ejpam-6175	47	13	,	,	PUNCT
ejpam-6175	47	14	where	where	SCONJ
ejpam-6175	47	15	two	two	NUM
ejpam-6175	47	16	images	image	NOUN
ejpam-6175	47	17	are	be	AUX
ejpam-6175	47	18	considered	consider	VERB
ejpam-6175	47	19	structurally	structurally	ADV
ejpam-6175	47	20	equivalent	equivalent	ADJ
ejpam-6175	47	21	if	if	SCONJ
ejpam-6175	47	22	there	there	PRON
ejpam-6175	47	23	is	be	VERB
ejpam-6175	47	24	a	a	DET
ejpam-6175	47	25	continuous	continuous	ADJ
ejpam-6175	47	26	,	,	PUNCT
ejpam-6175	47	27	bijective	bijective	ADJ
ejpam-6175	47	28	function	function	NOUN
ejpam-6175	47	29	between	between	ADP
ejpam-6175	47	30	them	they	PRON
ejpam-6175	47	31	that	that	PRON
ejpam-6175	47	32	also	also	ADV
ejpam-6175	47	33	has	have	VERB
ejpam-6175	47	34	a	a	DET
ejpam-6175	47	35	continuous	continuous	ADJ
ejpam-6175	47	36	inverse	inverse	NOUN
ejpam-6175	47	37	[	[	X
ejpam-6175	47	38	8	8	NUM
ejpam-6175	47	39	]	]	PUNCT
ejpam-6175	47	40	.	.	PUNCT
ejpam-6175	48	1	additionally	additionally	ADV
ejpam-6175	48	2	,	,	PUNCT
ejpam-6175	48	3	specific	specific	ADJ
ejpam-6175	48	4	structures	structure	NOUN
ejpam-6175	48	5	within	within	ADP
ejpam-6175	48	6	images	image	NOUN
ejpam-6175	48	7	,	,	PUNCT
ejpam-6175	48	8	such	such	ADJ
ejpam-6175	48	9	as	as	ADP
ejpam-6175	48	10	simple	simple	ADJ
ejpam-6175	48	11	closed	closed	ADJ
ejpam-6175	48	12	κ	κ	NOUN
ejpam-6175	48	13	-	-	PUNCT
ejpam-6175	48	14	curves	curve	NOUN
ejpam-6175	48	15	and	and	CCONJ
ejpam-6175	48	16	κ	κ	NOUN
ejpam-6175	48	17	-	-	PUNCT
ejpam-6175	48	18	corners	corner	NOUN
ejpam-6175	48	19	,	,	PUNCT
ejpam-6175	48	20	help	help	NOUN
ejpam-6175	48	21	to	to	PART
ejpam-6175	48	22	further	far	ADV
ejpam-6175	48	23	characterize	characterize	VERB
ejpam-6175	48	24	the	the	DET
ejpam-6175	48	25	topology	topology	NOUN
ejpam-6175	48	26	of	of	ADP
ejpam-6175	48	27	digital	digital	ADJ
ejpam-6175	48	28	spaces	space	NOUN
ejpam-6175	48	29	by	by	ADP
ejpam-6175	48	30	examining	examine	VERB
ejpam-6175	48	31	how	how	SCONJ
ejpam-6175	48	32	points	point	NOUN
ejpam-6175	48	33	are	be	AUX
ejpam-6175	48	34	connected	connect	VERB
ejpam-6175	48	35	and	and	CCONJ
ejpam-6175	48	36	how	how	SCONJ
ejpam-6175	48	37	these	these	DET
ejpam-6175	48	38	connections	connection	NOUN
ejpam-6175	48	39	can	can	AUX
ejpam-6175	48	40	be	be	AUX
ejpam-6175	48	41	simplified	simplify	VERB
ejpam-6175	48	42	without	without	ADP
ejpam-6175	48	43	altering	alter	VERB
ejpam-6175	48	44	the	the	DET
ejpam-6175	48	45	essential	essential	ADJ
ejpam-6175	48	46	structure	structure	NOUN
ejpam-6175	48	47	.	.	PUNCT
ejpam-6175	49	1	2	2	X
ejpam-6175	49	2	.	.	X
ejpam-6175	49	3	simplices	simplice	NOUN
ejpam-6175	49	4	and	and	CCONJ
ejpam-6175	49	5	homology	homology	NOUN
ejpam-6175	49	6	in	in	ADP
ejpam-6175	49	7	digital	digital	ADJ
ejpam-6175	49	8	topology	topology	NOUN
ejpam-6175	49	9	in	in	ADP
ejpam-6175	49	10	the	the	DET
ejpam-6175	49	11	realm	realm	NOUN
ejpam-6175	49	12	of	of	ADP
ejpam-6175	49	13	digital	digital	ADJ
ejpam-6175	49	14	simplicial	simplicial	ADJ
ejpam-6175	49	15	complexes	complex	NOUN
ejpam-6175	49	16	,	,	PUNCT
ejpam-6175	49	17	the	the	DET
ejpam-6175	49	18	concept	concept	NOUN
ejpam-6175	49	19	of	of	ADP
ejpam-6175	49	20	faces	face	NOUN
ejpam-6175	49	21	plays	play	VERB
ejpam-6175	49	22	a	a	DET
ejpam-6175	49	23	crucial	crucial	ADJ
ejpam-6175	49	24	role	role	NOUN
ejpam-6175	49	25	in	in	ADP
ejpam-6175	49	26	understanding	understand	VERB
ejpam-6175	49	27	the	the	DET
ejpam-6175	49	28	structure	structure	NOUN
ejpam-6175	49	29	and	and	CCONJ
ejpam-6175	49	30	properties	property	NOUN
ejpam-6175	49	31	of	of	ADP
ejpam-6175	49	32	these	these	DET
ejpam-6175	49	33	complexes	complex	NOUN
ejpam-6175	49	34	.	.	PUNCT
ejpam-6175	50	1	a	a	DET
ejpam-6175	50	2	face	face	NOUN
ejpam-6175	50	3	is	be	AUX
ejpam-6175	50	4	essentially	essentially	ADV
ejpam-6175	50	5	a	a	DET
ejpam-6175	50	6	subset	subset	NOUN
ejpam-6175	50	7	of	of	ADP
ejpam-6175	50	8	a	a	DET
ejpam-6175	50	9	simplex	simplex	NOUN
ejpam-6175	50	10	that	that	PRON
ejpam-6175	50	11	forms	form	VERB
ejpam-6175	50	12	a	a	DET
ejpam-6175	50	13	lower	lower	ADV
ejpam-6175	50	14	-	-	PUNCT
ejpam-6175	50	15	dimensional	dimensional	ADJ
ejpam-6175	50	16	simplex	simplex	NOUN
ejpam-6175	50	17	.	.	PUNCT
ejpam-6175	51	1	0	0	NUM
ejpam-6175	51	2	-	-	PUNCT
ejpam-6175	51	3	simplex	simplex	NOUN
ejpam-6175	51	4	(	(	PUNCT
ejpam-6175	51	5	point	point	NOUN
ejpam-6175	51	6	)	)	PUNCT
ejpam-6175	51	7	a	a	DET
ejpam-6175	51	8	0	0	NUM
ejpam-6175	51	9	-	-	PUNCT
ejpam-6175	51	10	simplex	simplex	NOUN
ejpam-6175	51	11	,	,	PUNCT
ejpam-6175	51	12	representing	represent	VERB
ejpam-6175	51	13	a	a	DET
ejpam-6175	51	14	single	single	ADJ
ejpam-6175	51	15	point	point	NOUN
ejpam-6175	51	16	in	in	ADP
ejpam-6175	51	17	zn	zn	PROPN
ejpam-6175	51	18	,	,	PUNCT
ejpam-6175	51	19	has	have	VERB
ejpam-6175	51	20	no	no	DET
ejpam-6175	51	21	faces	face	NOUN
ejpam-6175	51	22	other	other	ADJ
ejpam-6175	51	23	than	than	ADP
ejpam-6175	51	24	itself	itself	PRON
ejpam-6175	51	25	.	.	PUNCT
ejpam-6175	52	1	for	for	ADP
ejpam-6175	52	2	example	example	NOUN
ejpam-6175	52	3	,	,	PUNCT
ejpam-6175	52	4	the	the	DET
ejpam-6175	52	5	simplex	simplex	NOUN
ejpam-6175	52	6	{	{	PUNCT
ejpam-6175	52	7	p	p	NOUN
ejpam-6175	52	8	}	}	PUNCT
ejpam-6175	52	9	has	have	VERB
ejpam-6175	52	10	the	the	DET
ejpam-6175	52	11	face	face	NOUN
ejpam-6175	52	12	{	{	PUNCT
ejpam-6175	52	13	p	p	NOUN
ejpam-6175	52	14	}	}	PUNCT
ejpam-6175	52	15	.	.	PUNCT
ejpam-6175	53	1	1	1	NUM
ejpam-6175	53	2	-	-	PUNCT
ejpam-6175	53	3	simplex	simplex	NOUN
ejpam-6175	53	4	(	(	PUNCT
ejpam-6175	53	5	line	line	NOUN
ejpam-6175	53	6	segment	segment	NOUN
ejpam-6175	53	7	)	)	PUNCT
ejpam-6175	53	8	a	a	DET
ejpam-6175	53	9	1	1	NUM
ejpam-6175	53	10	-	-	PUNCT
ejpam-6175	53	11	simplex	simplex	NOUN
ejpam-6175	53	12	,	,	PUNCT
ejpam-6175	53	13	which	which	PRON
ejpam-6175	53	14	corresponds	correspond	VERB
ejpam-6175	53	15	to	to	ADP
ejpam-6175	53	16	a	a	DET
ejpam-6175	53	17	line	line	NOUN
ejpam-6175	53	18	segment	segment	NOUN
ejpam-6175	53	19	,	,	PUNCT
ejpam-6175	53	20	has	have	VERB
ejpam-6175	53	21	two	two	NUM
ejpam-6175	53	22	0	0	NUM
ejpam-6175	53	23	-	-	PUNCT
ejpam-6175	53	24	simplex	simplex	NOUN
ejpam-6175	53	25	faces	face	NOUN
ejpam-6175	53	26	.	.	PUNCT
ejpam-6175	54	1	for	for	ADP
ejpam-6175	54	2	instance	instance	NOUN
ejpam-6175	54	3	,	,	PUNCT
ejpam-6175	54	4	the	the	DET
ejpam-6175	54	5	simplex	simplex	NOUN
ejpam-6175	54	6	{	{	PUNCT
ejpam-6175	54	7	p	p	X
ejpam-6175	54	8	,	,	PUNCT
ejpam-6175	54	9	q	q	NOUN
ejpam-6175	54	10	}	}	PUNCT
ejpam-6175	54	11	,	,	PUNCT
ejpam-6175	54	12	where	where	SCONJ
ejpam-6175	54	13	p	p	NOUN
ejpam-6175	54	14	and	and	CCONJ
ejpam-6175	54	15	q	q	NOUN
ejpam-6175	54	16	are	be	AUX
ejpam-6175	54	17	κ	κ	PRON
ejpam-6175	54	18	-	-	ADJ
ejpam-6175	54	19	adjacent	adjacent	ADJ
ejpam-6175	54	20	pixels	pixel	NOUN
ejpam-6175	54	21	,	,	PUNCT
ejpam-6175	54	22	has	have	AUX
ejpam-6175	54	23	faces	face	NOUN
ejpam-6175	54	24	{	{	PUNCT
ejpam-6175	54	25	p	p	X
ejpam-6175	54	26	}	}	PUNCT
ejpam-6175	54	27	and	and	CCONJ
ejpam-6175	54	28	{	{	PUNCT
ejpam-6175	54	29	q	q	NOUN
ejpam-6175	54	30	}	}	PUNCT
ejpam-6175	54	31	.	.	PUNCT
ejpam-6175	55	1	2	2	NUM
ejpam-6175	55	2	-	-	PUNCT
ejpam-6175	55	3	simplex	simplex	NOUN
ejpam-6175	55	4	(	(	PUNCT
ejpam-6175	55	5	triangle	triangle	NOUN
ejpam-6175	55	6	)	)	PUNCT
ejpam-6175	55	7	a	a	DET
ejpam-6175	55	8	2	2	NUM
ejpam-6175	55	9	-	-	PUNCT
ejpam-6175	55	10	simplex	simplex	NOUN
ejpam-6175	55	11	,	,	PUNCT
ejpam-6175	55	12	which	which	PRON
ejpam-6175	55	13	corresponds	correspond	VERB
ejpam-6175	55	14	to	to	ADP
ejpam-6175	55	15	a	a	DET
ejpam-6175	55	16	triangle	triangle	NOUN
ejpam-6175	55	17	,	,	PUNCT
ejpam-6175	55	18	has	have	VERB
ejpam-6175	55	19	three	three	NUM
ejpam-6175	55	20	1simplex	1simplex	NUM
ejpam-6175	55	21	faces	face	NOUN
ejpam-6175	55	22	and	and	CCONJ
ejpam-6175	55	23	their	their	PRON
ejpam-6175	55	24	associated	associate	VERB
ejpam-6175	55	25	0	0	NUM
ejpam-6175	55	26	-	-	PUNCT
ejpam-6175	55	27	simplex	simplex	NOUN
ejpam-6175	55	28	faces	face	NOUN
ejpam-6175	55	29	.	.	PUNCT
ejpam-6175	56	1	for	for	ADP
ejpam-6175	56	2	instance	instance	NOUN
ejpam-6175	56	3	,	,	PUNCT
ejpam-6175	56	4	the	the	DET
ejpam-6175	56	5	simplex	simplex	NOUN
ejpam-6175	56	6	{	{	PUNCT
ejpam-6175	56	7	p	p	X
ejpam-6175	56	8	,	,	PUNCT
ejpam-6175	56	9	q	q	ADJ
ejpam-6175	56	10	,	,	PUNCT
ejpam-6175	56	11	r	r	NOUN
ejpam-6175	56	12	}	}	PUNCT
ejpam-6175	56	13	,	,	PUNCT
ejpam-6175	56	14	where	where	SCONJ
ejpam-6175	56	15	p	p	X
ejpam-6175	56	16	,	,	PUNCT
ejpam-6175	56	17	q	q	X
ejpam-6175	56	18	,	,	PUNCT
ejpam-6175	56	19	and	and	CCONJ
ejpam-6175	56	20	r	r	NOUN
ejpam-6175	56	21	are	be	AUX
ejpam-6175	56	22	κ	κ	NOUN
ejpam-6175	56	23	-	-	ADJ
ejpam-6175	56	24	adjacent	adjacent	ADJ
ejpam-6175	56	25	pixels	pixel	NOUN
ejpam-6175	56	26	,	,	PUNCT
ejpam-6175	56	27	has	have	VERB
ejpam-6175	56	28	three	three	NUM
ejpam-6175	56	29	1	1	NUM
ejpam-6175	56	30	-	-	PUNCT
ejpam-6175	56	31	simplex	simplex	NOUN
ejpam-6175	56	32	faces	face	NOUN
ejpam-6175	56	33	as	as	ADP
ejpam-6175	56	34	{	{	PUNCT
ejpam-6175	56	35	p	p	X
ejpam-6175	56	36	,	,	PUNCT
ejpam-6175	56	37	q	q	NOUN
ejpam-6175	56	38	}	}	PUNCT
ejpam-6175	56	39	,	,	PUNCT
ejpam-6175	56	40	{	{	PUNCT
ejpam-6175	56	41	q	q	X
ejpam-6175	56	42	,	,	PUNCT
ejpam-6175	56	43	r	r	NOUN
ejpam-6175	56	44	}	}	PUNCT
ejpam-6175	56	45	,	,	PUNCT
ejpam-6175	56	46	{	{	PUNCT
ejpam-6175	56	47	r	r	NOUN
ejpam-6175	56	48	,	,	PUNCT
ejpam-6175	56	49	p	p	NOUN
ejpam-6175	56	50	}	}	PUNCT
ejpam-6175	56	51	and	and	CCONJ
ejpam-6175	56	52	three	three	NUM
ejpam-6175	56	53	0	0	NUM
ejpam-6175	56	54	-	-	PUNCT
ejpam-6175	56	55	simplex	simplex	NOUN
ejpam-6175	56	56	faces	face	VERB
ejpam-6175	56	57	namely	namely	ADV
ejpam-6175	56	58	{	{	PUNCT
ejpam-6175	56	59	p	p	NOUN
ejpam-6175	56	60	}	}	PUNCT
ejpam-6175	56	61	,	,	PUNCT
ejpam-6175	56	62	{	{	PUNCT
ejpam-6175	56	63	q	q	X
ejpam-6175	56	64	}	}	PUNCT
ejpam-6175	56	65	,	,	PUNCT
ejpam-6175	56	66	{	{	PUNCT
ejpam-6175	56	67	r	r	NOUN
ejpam-6175	56	68	}	}	SYM
ejpam-6175	56	69	3	3	NUM
ejpam-6175	56	70	-	-	PUNCT
ejpam-6175	56	71	simplex	simplex	NOUN
ejpam-6175	56	72	(	(	PUNCT
ejpam-6175	56	73	tetrahedron	tetrahedron	NOUN
ejpam-6175	56	74	)	)	PUNCT
ejpam-6175	56	75	a	a	DET
ejpam-6175	56	76	3	3	NUM
ejpam-6175	56	77	-	-	PUNCT
ejpam-6175	56	78	simplex	simplex	NOUN
ejpam-6175	56	79	,	,	PUNCT
ejpam-6175	56	80	corresponding	correspond	VERB
ejpam-6175	56	81	to	to	ADP
ejpam-6175	56	82	a	a	DET
ejpam-6175	56	83	tetrahedron	tetrahedron	NOUN
ejpam-6175	56	84	,	,	PUNCT
ejpam-6175	56	85	has	have	VERB
ejpam-6175	56	86	four	four	NUM
ejpam-6175	56	87	2	2	NUM
ejpam-6175	56	88	-	-	PUNCT
ejpam-6175	56	89	simplex	simplex	NOUN
ejpam-6175	56	90	faces	face	NOUN
ejpam-6175	56	91	,	,	PUNCT
ejpam-6175	56	92	six	six	NUM
ejpam-6175	56	93	1	1	NUM
ejpam-6175	56	94	-	-	PUNCT
ejpam-6175	56	95	simplex	simplex	NOUN
ejpam-6175	56	96	faces	face	NOUN
ejpam-6175	56	97	,	,	PUNCT
ejpam-6175	56	98	and	and	CCONJ
ejpam-6175	56	99	four	four	NUM
ejpam-6175	56	100	0	0	NUM
ejpam-6175	56	101	-	-	PUNCT
ejpam-6175	56	102	simplex	simplex	NOUN
ejpam-6175	56	103	faces	face	NOUN
ejpam-6175	56	104	.	.	PUNCT
ejpam-6175	57	1	for	for	ADP
ejpam-6175	57	2	instance	instance	NOUN
ejpam-6175	57	3	,	,	PUNCT
ejpam-6175	57	4	the	the	DET
ejpam-6175	57	5	simplex	simplex	NOUN
ejpam-6175	57	6	{	{	PUNCT
ejpam-6175	57	7	p	p	X
ejpam-6175	57	8	,	,	PUNCT
ejpam-6175	57	9	q	q	ADJ
ejpam-6175	57	10	,	,	PUNCT
ejpam-6175	57	11	r	r	NOUN
ejpam-6175	57	12	,	,	PUNCT
ejpam-6175	57	13	s	s	PART
ejpam-6175	57	14	}	}	PUNCT
ejpam-6175	57	15	,	,	PUNCT
ejpam-6175	57	16	where	where	SCONJ
ejpam-6175	57	17	p	p	X
ejpam-6175	57	18	,	,	PUNCT
ejpam-6175	57	19	q	q	ADJ
ejpam-6175	57	20	,	,	PUNCT
ejpam-6175	57	21	r	r	NOUN
ejpam-6175	57	22	,	,	PUNCT
ejpam-6175	57	23	and	and	CCONJ
ejpam-6175	57	24	s	s	VERB
ejpam-6175	57	25	are	be	AUX
ejpam-6175	57	26	26	26	NUM
ejpam-6175	57	27	-	-	PUNCT
ejpam-6175	57	28	adjacent	adjacent	ADJ
ejpam-6175	57	29	voxels	voxel	NOUN
ejpam-6175	57	30	,	,	PUNCT
ejpam-6175	57	31	has	have	VERB
ejpam-6175	57	32	four	four	NUM
ejpam-6175	57	33	2	2	NUM
ejpam-6175	57	34	-	-	PUNCT
ejpam-6175	57	35	simplex	simplex	NOUN
ejpam-6175	57	36	faces	face	NOUN
ejpam-6175	57	37	,	,	PUNCT
ejpam-6175	57	38	six	six	NUM
ejpam-6175	57	39	1simplex	1simplex	NUM
ejpam-6175	57	40	faces	face	NOUN
ejpam-6175	57	41	and	and	CCONJ
ejpam-6175	57	42	four	four	NUM
ejpam-6175	57	43	0	0	NUM
ejpam-6175	57	44	-	-	PUNCT
ejpam-6175	57	45	simplex	simplex	NOUN
ejpam-6175	57	46	faces	face	NOUN
ejpam-6175	57	47	.	.	PUNCT
ejpam-6175	58	1	an	an	PRON
ejpam-6175	58	2	m	m	NOUN
ejpam-6175	58	3	-	-	NOUN
ejpam-6175	58	4	simplex	simplex	NOUN
ejpam-6175	58	5	is	be	AUX
ejpam-6175	58	6	defined	define	VERB
ejpam-6175	58	7	as	as	ADP
ejpam-6175	58	8	a	a	DET
ejpam-6175	58	9	simplex	simplex	NOUN
ejpam-6175	58	10	with	with	ADP
ejpam-6175	58	11	a	a	DET
ejpam-6175	58	12	cardinality	cardinality	NOUN
ejpam-6175	58	13	of	of	ADP
ejpam-6175	58	14	m	m	PROPN
ejpam-6175	58	15	+	+	PROPN
ejpam-6175	58	16	1	1	X
ejpam-6175	58	17	.	.	X
ejpam-6175	58	18	for	for	ADP
ejpam-6175	58	19	a	a	DET
ejpam-6175	58	20	digital	digital	ADJ
ejpam-6175	58	21	msimplex	msimplex	NOUN
ejpam-6175	59	1	i	i	PRON
ejpam-6175	59	2	,	,	PUNCT
ejpam-6175	59	3	if	if	SCONJ
ejpam-6175	59	4	i	i	PRON
ejpam-6175	59	5	′	′	VERB
ejpam-6175	59	6	is	be	AUX
ejpam-6175	59	7	a	a	DET
ejpam-6175	59	8	nonempty	nonempty	ADJ
ejpam-6175	59	9	proper	proper	ADJ
ejpam-6175	59	10	subset	subset	NOUN
ejpam-6175	59	11	of	of	ADP
ejpam-6175	59	12	p	p	PROPN
ejpam-6175	59	13	,	,	PUNCT
ejpam-6175	59	14	then	then	ADV
ejpam-6175	59	15	i	i	PRON
ejpam-6175	59	16	′	′	VERB
ejpam-6175	59	17	is	be	AUX
ejpam-6175	59	18	considered	consider	VERB
ejpam-6175	59	19	a	a	DET
ejpam-6175	59	20	face	face	NOUN
ejpam-6175	59	21	of	of	ADP
ejpam-6175	59	22	i.	i.	NOUN
ejpam-6175	59	23	definition	definition	NOUN
ejpam-6175	59	24	2	2	NUM
ejpam-6175	59	25	.	.	PUNCT
ejpam-6175	59	26	suppose	suppose	VERB
ejpam-6175	59	27	(	(	PUNCT
ejpam-6175	59	28	x	x	X
ejpam-6175	59	29	,	,	PUNCT
ejpam-6175	59	30	κ	κ	NOUN
ejpam-6175	59	31	)	)	PUNCT
ejpam-6175	59	32	is	be	AUX
ejpam-6175	59	33	a	a	DET
ejpam-6175	59	34	finite	finite	ADJ
ejpam-6175	59	35	collection	collection	NOUN
ejpam-6175	59	36	of	of	ADP
ejpam-6175	59	37	digital	digital	ADJ
ejpam-6175	59	38	m	m	PROPN
ejpam-6175	59	39	-	-	PUNCT
ejpam-6175	59	40	simplices	simplice	NOUN
ejpam-6175	59	41	,	,	PUNCT
ejpam-6175	59	42	where	where	SCONJ
ejpam-6175	59	43	0	0	NUM
ejpam-6175	59	44	≤	≤	NUM
ejpam-6175	59	45	m	m	VERB
ejpam-6175	59	46	≤	≤	NOUN
ejpam-6175	59	47	d	d	NOUN
ejpam-6175	59	48	for	for	ADP
ejpam-6175	59	49	some	some	DET
ejpam-6175	59	50	non	non	ADJ
ejpam-6175	59	51	-	-	ADJ
ejpam-6175	59	52	negative	negative	ADJ
ejpam-6175	59	53	integer	integer	NOUN
ejpam-6175	59	54	d.	d.	NOUN
ejpam-6175	59	55	if	if	SCONJ
ejpam-6175	59	56	the	the	DET
ejpam-6175	59	57	following	follow	VERB
ejpam-6175	59	58	conditions	condition	NOUN
ejpam-6175	59	59	hold	hold	VERB
ejpam-6175	59	60	,	,	PUNCT
ejpam-6175	59	61	then	then	ADV
ejpam-6175	59	62	(	(	PUNCT
ejpam-6175	59	63	x	x	X
ejpam-6175	59	64	,	,	PUNCT
ejpam-6175	59	65	κ	κ	NOUN
ejpam-6175	59	66	)	)	PUNCT
ejpam-6175	59	67	is	be	AUX
ejpam-6175	59	68	referred	refer	VERB
ejpam-6175	59	69	to	to	ADP
ejpam-6175	59	70	as	as	ADP
ejpam-6175	59	71	a	a	DET
ejpam-6175	59	72	finite	finite	ADJ
ejpam-6175	59	73	digital	digital	PROPN
ejpam-6175	59	74	∆-complex	∆-complex	PROPN
ejpam-6175	59	75	:	:	PUNCT
ejpam-6175	59	76	•	•	SCONJ
ejpam-6175	59	77	if	if	SCONJ
ejpam-6175	59	78	p	p	NOUN
ejpam-6175	59	79	is	be	AUX
ejpam-6175	59	80	a	a	DET
ejpam-6175	59	81	member	member	NOUN
ejpam-6175	59	82	of	of	ADP
ejpam-6175	59	83	x	x	PROPN
ejpam-6175	59	84	,	,	PUNCT
ejpam-6175	59	85	then	then	ADV
ejpam-6175	59	86	all	all	DET
ejpam-6175	59	87	the	the	DET
ejpam-6175	59	88	faces	face	NOUN
ejpam-6175	59	89	of	of	ADP
ejpam-6175	59	90	p	p	NOUN
ejpam-6175	59	91	are	be	AUX
ejpam-6175	59	92	also	also	ADV
ejpam-6175	59	93	contained	contain	VERB
ejpam-6175	59	94	within	within	ADP
ejpam-6175	59	95	x.	x.	NOUN
ejpam-6175	59	96	•	•	NOUN
ejpam-6175	59	97	if	if	SCONJ
ejpam-6175	59	98	p	p	PROPN
ejpam-6175	59	99	and	and	CCONJ
ejpam-6175	59	100	q	q	NOUN
ejpam-6175	59	101	are	be	AUX
ejpam-6175	59	102	members	member	NOUN
ejpam-6175	59	103	of	of	ADP
ejpam-6175	59	104	x	x	NOUN
ejpam-6175	59	105	,	,	PUNCT
ejpam-6175	59	106	then	then	ADV
ejpam-6175	59	107	their	their	PRON
ejpam-6175	59	108	intersection	intersection	NOUN
ejpam-6175	59	109	p	p	NOUN
ejpam-6175	59	110	∩	∩	NOUN
ejpam-6175	59	111	q	q	X
ejpam-6175	59	112	is	be	AUX
ejpam-6175	59	113	either	either	CCONJ
ejpam-6175	59	114	empty	empty	ADJ
ejpam-6175	59	115	or	or	CCONJ
ejpam-6175	59	116	multiple	multiple	ADJ
ejpam-6175	59	117	simplexes	simplexe	NOUN
ejpam-6175	59	118	(	(	PUNCT
ejpam-6175	59	119	need	need	AUX
ejpam-6175	59	120	not	not	PART
ejpam-6175	59	121	to	to	PART
ejpam-6175	59	122	be	be	AUX
ejpam-6175	59	123	a	a	DET
ejpam-6175	59	124	single	single	ADJ
ejpam-6175	59	125	or	or	CCONJ
ejpam-6175	59	126	common	common	ADJ
ejpam-6175	59	127	simplex	simplex	NOUN
ejpam-6175	59	128	)	)	PUNCT
ejpam-6175	59	129	.	.	PUNCT
ejpam-6175	60	1	if	if	SCONJ
ejpam-6175	60	2	it	it	PRON
ejpam-6175	60	3	has	have	VERB
ejpam-6175	60	4	a	a	DET
ejpam-6175	60	5	common	common	ADJ
ejpam-6175	60	6	face	face	NOUN
ejpam-6175	60	7	of	of	ADP
ejpam-6175	60	8	both	both	CCONJ
ejpam-6175	60	9	p	p	NOUN
ejpam-6175	60	10	and	and	CCONJ
ejpam-6175	60	11	q	q	NOUN
ejpam-6175	60	12	then	then	ADV
ejpam-6175	60	13	it	it	PRON
ejpam-6175	60	14	is	be	AUX
ejpam-6175	60	15	called	call	VERB
ejpam-6175	60	16	as	as	ADP
ejpam-6175	60	17	finite	finite	ADJ
ejpam-6175	60	18	digital	digital	ADJ
ejpam-6175	60	19	simplicial	simplicial	ADJ
ejpam-6175	60	20	complex	complex	NOUN
ejpam-6175	60	21	.	.	PUNCT
ejpam-6175	61	1	a	a	DET
ejpam-6175	61	2	digital	digital	ADJ
ejpam-6175	61	3	image	image	NOUN
ejpam-6175	61	4	can	can	AUX
ejpam-6175	61	5	be	be	AUX
ejpam-6175	61	6	considered	consider	VERB
ejpam-6175	61	7	a	a	DET
ejpam-6175	61	8	union	union	NOUN
ejpam-6175	61	9	of	of	ADP
ejpam-6175	61	10	0	0	NOUN
ejpam-6175	61	11	-	-	PUNCT
ejpam-6175	61	12	simplices	simplice	NOUN
ejpam-6175	61	13	,	,	PUNCT
ejpam-6175	61	14	meaning	mean	VERB
ejpam-6175	61	15	that	that	SCONJ
ejpam-6175	61	16	every	every	DET
ejpam-6175	61	17	digital	digital	ADJ
ejpam-6175	61	18	image	image	NOUN
ejpam-6175	61	19	forms	form	VERB
ejpam-6175	61	20	a	a	DET
ejpam-6175	61	21	digital	digital	ADJ
ejpam-6175	61	22	simplicial	simplicial	ADJ
ejpam-6175	61	23	complex	complex	NOUN
ejpam-6175	61	24	.	.	PUNCT
ejpam-6175	62	1	consequently	consequently	ADV
ejpam-6175	62	2	,	,	PUNCT
ejpam-6175	62	3	we	we	PRON
ejpam-6175	62	4	use	use	VERB
ejpam-6175	62	5	the	the	DET
ejpam-6175	62	6	same	same	ADJ
ejpam-6175	62	7	notation	notation	NOUN
ejpam-6175	62	8	to	to	ADP
ejpam-6175	62	9	s.k	s.k	PROPN
ejpam-6175	62	10	.	.	PUNCT
ejpam-6175	62	11	jha	jha	PROPN
ejpam-6175	62	12	sanjay	sanjay	PROPN
ejpam-6175	62	13	,	,	PUNCT
ejpam-6175	62	14	m	m	PROPN
ejpam-6175	62	15	,	,	PUNCT
ejpam-6175	62	16	natarajan	natarajan	PROPN
ejpam-6175	62	17	/	/	SYM
ejpam-6175	62	18	eur	eur	PROPN
ejpam-6175	62	19	.	.	PUNCT
ejpam-6175	63	1	j.	j.	PROPN
ejpam-6175	63	2	pure	pure	PROPN
ejpam-6175	63	3	appl	appl	PROPN
ejpam-6175	63	4	.	.	PROPN
ejpam-6175	63	5	math	math	PROPN
ejpam-6175	63	6	,	,	PUNCT
ejpam-6175	63	7	18	18	NUM
ejpam-6175	63	8	(	(	PUNCT
ejpam-6175	63	9	3	3	NUM
ejpam-6175	63	10	)	)	PUNCT
ejpam-6175	63	11	(	(	PUNCT
ejpam-6175	63	12	2025	2025	NUM
ejpam-6175	63	13	)	)	PUNCT
ejpam-6175	63	14	,	,	PUNCT
ejpam-6175	63	15	6175	6175	NUM
ejpam-6175	63	16	4	4	NUM
ejpam-6175	63	17	of	of	ADP
ejpam-6175	63	18	18	18	NUM
ejpam-6175	63	19	describe	describe	VERB
ejpam-6175	63	20	both	both	CCONJ
ejpam-6175	63	21	a	a	DET
ejpam-6175	63	22	digital	digital	ADJ
ejpam-6175	63	23	image	image	NOUN
ejpam-6175	63	24	and	and	CCONJ
ejpam-6175	63	25	its	its	PRON
ejpam-6175	63	26	associated	associated	ADJ
ejpam-6175	63	27	simplicial	simplicial	ADJ
ejpam-6175	63	28	complex	complex	NOUN
ejpam-6175	63	29	.	.	PUNCT
ejpam-6175	64	1	the	the	DET
ejpam-6175	64	2	dimension	dimension	NOUN
ejpam-6175	64	3	of	of	ADP
ejpam-6175	64	4	a	a	DET
ejpam-6175	64	5	digital	digital	ADJ
ejpam-6175	64	6	simplicial	simplicial	ADJ
ejpam-6175	64	7	complex	complex	NOUN
ejpam-6175	64	8	x	x	PRON
ejpam-6175	64	9	is	be	AUX
ejpam-6175	64	10	defined	define	VERB
ejpam-6175	64	11	as	as	ADP
ejpam-6175	64	12	the	the	DET
ejpam-6175	64	13	largest	large	ADJ
ejpam-6175	64	14	integer	integer	NOUN
ejpam-6175	64	15	m	m	NOUN
ejpam-6175	64	16	for	for	ADP
ejpam-6175	64	17	which	which	PRON
ejpam-6175	64	18	x	x	PRON
ejpam-6175	64	19	contains	contain	VERB
ejpam-6175	64	20	an	an	DET
ejpam-6175	64	21	m	m	NOUN
ejpam-6175	64	22	-	-	NOUN
ejpam-6175	64	23	simplex	simplex	NOUN
ejpam-6175	64	24	.	.	PUNCT
ejpam-6175	65	1	the	the	DET
ejpam-6175	65	2	group	group	NOUN
ejpam-6175	65	3	cκ	cκ	ADP
ejpam-6175	65	4	q	q	PROPN
ejpam-6175	65	5	(	(	PUNCT
ejpam-6175	65	6	x	x	X
ejpam-6175	65	7	)	)	PUNCT
ejpam-6175	65	8	is	be	AUX
ejpam-6175	65	9	a	a	DET
ejpam-6175	65	10	free	free	ADJ
ejpam-6175	65	11	abelian	abelian	ADJ
ejpam-6175	65	12	group	group	NOUN
ejpam-6175	66	1	[	[	X
ejpam-6175	66	2	9	9	NUM
ejpam-6175	66	3	]	]	PUNCT
ejpam-6175	66	4	that	that	PRON
ejpam-6175	66	5	has	have	VERB
ejpam-6175	66	6	a	a	DET
ejpam-6175	66	7	basis	basis	NOUN
ejpam-6175	66	8	composed	compose	VERB
ejpam-6175	66	9	of	of	ADP
ejpam-6175	66	10	all	all	DET
ejpam-6175	66	11	digital	digital	ADJ
ejpam-6175	66	12	(	(	PUNCT
ejpam-6175	66	13	κ	κ	NOUN
ejpam-6175	66	14	,	,	PUNCT
ejpam-6175	66	15	q)-simplices	q)-simplice	VERB
ejpam-6175	66	16	in	in	ADP
ejpam-6175	66	17	x.	x.	NOUN
ejpam-6175	66	18	if	if	SCONJ
ejpam-6175	66	19	(	(	PUNCT
ejpam-6175	66	20	x	x	NOUN
ejpam-6175	66	21	,	,	PUNCT
ejpam-6175	66	22	κ	κ	NOUN
ejpam-6175	66	23	)	)	PUNCT
ejpam-6175	66	24	⊆	⊆	NUM
ejpam-6175	66	25	zn	zn	PROPN
ejpam-6175	66	26	represents	represent	VERB
ejpam-6175	66	27	a	a	DET
ejpam-6175	66	28	digital	digital	ADJ
ejpam-6175	66	29	simplicial	simplicial	ADJ
ejpam-6175	66	30	complex	complex	NOUN
ejpam-6175	66	31	of	of	ADP
ejpam-6175	66	32	dimension	dimension	NOUN
ejpam-6175	66	33	m	m	PROPN
ejpam-6175	66	34	,	,	PUNCT
ejpam-6175	66	35	then	then	ADV
ejpam-6175	66	36	for	for	ADP
ejpam-6175	66	37	any	any	DET
ejpam-6175	66	38	q	q	X
ejpam-6175	66	39	>	>	X
ejpam-6175	66	40	m	m	PROPN
ejpam-6175	66	41	,	,	PUNCT
ejpam-6175	66	42	the	the	DET
ejpam-6175	66	43	group	group	NOUN
ejpam-6175	66	44	cκ	cκ	ADP
ejpam-6175	66	45	q	q	PROPN
ejpam-6175	66	46	(	(	PUNCT
ejpam-6175	66	47	x	x	X
ejpam-6175	66	48	)	)	PUNCT
ejpam-6175	66	49	is	be	AUX
ejpam-6175	66	50	trivial	trivial	ADJ
ejpam-6175	66	51	.	.	PUNCT
ejpam-6175	67	1	every	every	DET
ejpam-6175	67	2	digital	digital	ADJ
ejpam-6175	67	3	m	m	PROPN
ejpam-6175	67	4	-	-	NOUN
ejpam-6175	67	5	simplex	simplex	NOUN
ejpam-6175	67	6	is	be	AUX
ejpam-6175	67	7	naturally	naturally	ADV
ejpam-6175	67	8	oriented	orient	VERB
ejpam-6175	67	9	based	base	VERB
ejpam-6175	67	10	on	on	ADP
ejpam-6175	67	11	its	its	PRON
ejpam-6175	67	12	vertices	vertex	NOUN
ejpam-6175	67	13	.	.	PUNCT
ejpam-6175	68	1	this	this	DET
ejpam-6175	68	2	orientation	orientation	NOUN
ejpam-6175	68	3	is	be	AUX
ejpam-6175	68	4	described	describe	VERB
ejpam-6175	68	5	by	by	ADP
ejpam-6175	68	6	the	the	DET
ejpam-6175	68	7	formula	formula	NOUN
ejpam-6175	68	8	:	:	PUNCT
ejpam-6175	68	9	∂	∂	NUM
ejpam-6175	68	10	(	(	PUNCT
ejpam-6175	68	11	<	<	X
ejpam-6175	68	12	p0	p0	PROPN
ejpam-6175	68	13	,	,	PUNCT
ejpam-6175	68	14	p1	p1	NOUN
ejpam-6175	68	15	,	,	PUNCT
ejpam-6175	68	16	.	.	PUNCT
ejpam-6175	68	17	.	.	PUNCT
ejpam-6175	69	1	.	.	PUNCT
ejpam-6175	70	1	,	,	PUNCT
ejpam-6175	70	2	pq	pq	INTJ
ejpam-6175	70	3	>	>	PUNCT
ejpam-6175	70	4	)	)	PUNCT
ejpam-6175	71	1	=	=	PUNCT
ejpam-6175	71	2	q∑	q∑	PROPN
ejpam-6175	72	1	i=0	i=0	PROPN
ejpam-6175	72	2	(	(	PUNCT
ejpam-6175	72	3	−1)i	−1)i	X
ejpam-6175	72	4	<	<	X
ejpam-6175	72	5	p0	p0	NOUN
ejpam-6175	72	6	,	,	PUNCT
ejpam-6175	72	7	p1	p1	NOUN
ejpam-6175	72	8	,	,	PUNCT
ejpam-6175	72	9	.	.	PUNCT
ejpam-6175	72	10	.	.	PUNCT
ejpam-6175	72	11	.	.	PUNCT
ejpam-6175	73	1	,	,	PUNCT
ejpam-6175	73	2	p̂i	p̂i	PROPN
ejpam-6175	73	3	,	,	PUNCT
ejpam-6175	73	4	.	.	PUNCT
ejpam-6175	73	5	.	.	PUNCT
ejpam-6175	74	1	.	.	PUNCT
ejpam-6175	75	1	,	,	PUNCT
ejpam-6175	75	2	pq	pq	INTJ
ejpam-6175	75	3	>	>	X
ejpam-6175	75	4	where	where	SCONJ
ejpam-6175	75	5	p̂i	p̂i	PROPN
ejpam-6175	75	6	indicates	indicate	VERB
ejpam-6175	75	7	that	that	SCONJ
ejpam-6175	75	8	the	the	DET
ejpam-6175	75	9	point	point	NOUN
ejpam-6175	75	10	pi	pi	NOUN
ejpam-6175	75	11	is	be	AUX
ejpam-6175	75	12	removed	remove	VERB
ejpam-6175	75	13	.	.	PUNCT
ejpam-6175	76	1	this	this	DET
ejpam-6175	76	2	notation	notation	NOUN
ejpam-6175	76	3	allows	allow	VERB
ejpam-6175	76	4	us	we	PRON
ejpam-6175	76	5	to	to	PART
ejpam-6175	76	6	define	define	VERB
ejpam-6175	76	7	the	the	DET
ejpam-6175	76	8	boundary	boundary	ADJ
ejpam-6175	76	9	homomorphism	homomorphism	NOUN
ejpam-6175	77	1	∂q	∂q	NOUN
ejpam-6175	77	2	:	:	PUNCT
ejpam-6175	77	3	c	c	NOUN
ejpam-6175	77	4	κ	κ	X
ejpam-6175	77	5	q	q	X
ejpam-6175	77	6	(	(	PUNCT
ejpam-6175	77	7	x	x	NOUN
ejpam-6175	77	8	)	)	PUNCT
ejpam-6175	77	9	→	→	SYM
ejpam-6175	77	10	cκ	cκ	PROPN
ejpam-6175	77	11	q−1(x	q−1(x	NOUN
ejpam-6175	77	12	)	)	PUNCT
ejpam-6175	77	13	as	as	SCONJ
ejpam-6175	77	14	follows	follow	VERB
ejpam-6175	77	15	:	:	PUNCT
ejpam-6175	77	16	∂q(⟨p0	∂q(⟨p0	NUM
ejpam-6175	77	17	,	,	PUNCT
ejpam-6175	77	18	p1	p1	NOUN
ejpam-6175	77	19	,	,	PUNCT
ejpam-6175	77	20	.	.	PUNCT
ejpam-6175	77	21	.	.	PUNCT
ejpam-6175	77	22	.	.	PUNCT
ejpam-6175	78	1	,	,	PUNCT
ejpam-6175	78	2	pq⟩	pq⟩	NOUN
ejpam-6175	78	3	)	)	PUNCT
ejpam-6175	78	4	=	=	PRON
ejpam-6175	78	5	{	{	PUNCT
ejpam-6175	78	6	∑q	∑q	ADJ
ejpam-6175	78	7	i=0(−1)i⟨p0	i=0(−1)i⟨p0	ADJ
ejpam-6175	78	8	,	,	PUNCT
ejpam-6175	78	9	p1	p1	NOUN
ejpam-6175	78	10	,	,	PUNCT
ejpam-6175	78	11	.	.	PUNCT
ejpam-6175	78	12	.	.	PUNCT
ejpam-6175	78	13	.	.	PUNCT
ejpam-6175	79	1	,	,	PUNCT
ejpam-6175	79	2	p̂i	p̂i	PROPN
ejpam-6175	79	3	,	,	PUNCT
ejpam-6175	79	4	.	.	PUNCT
ejpam-6175	79	5	.	.	PUNCT
ejpam-6175	80	1	.	.	PUNCT
ejpam-6175	81	1	,	,	PUNCT
ejpam-6175	81	2	pq⟩	pq⟩	NOUN
ejpam-6175	81	3	,	,	PUNCT
ejpam-6175	81	4	q	q	PROPN
ejpam-6175	81	5	≤	≤	NUM
ejpam-6175	81	6	m	m	NOUN
ejpam-6175	81	7	0	0	NUM
ejpam-6175	81	8	,	,	PUNCT
ejpam-6175	81	9	q	q	X
ejpam-6175	81	10	>	>	X
ejpam-6175	81	11	m	m	VERB
ejpam-6175	81	12	.	.	PUNCT
ejpam-6175	82	1	for	for	ADP
ejpam-6175	82	2	example	example	NOUN
ejpam-6175	82	3	,	,	PUNCT
ejpam-6175	82	4	consider	consider	VERB
ejpam-6175	82	5	a	a	DET
ejpam-6175	82	6	digital	digital	ADJ
ejpam-6175	82	7	interval	interval	NOUN
ejpam-6175	83	1	[	[	X
ejpam-6175	83	2	a	a	X
ejpam-6175	83	3	,	,	PUNCT
ejpam-6175	83	4	b]z	b]z	NOUN
ejpam-6175	83	5	where	where	SCONJ
ejpam-6175	83	6	a	a	DET
ejpam-6175	83	7	<	<	X
ejpam-6175	83	8	b	b	NOUN
ejpam-6175	83	9	and	and	CCONJ
ejpam-6175	83	10	the	the	DET
ejpam-6175	83	11	direction	direction	NOUN
ejpam-6175	83	12	is	be	AUX
ejpam-6175	83	13	from	from	ADP
ejpam-6175	83	14	a	a	PRON
ejpam-6175	83	15	to	to	ADP
ejpam-6175	83	16	b.	b.	PROPN
ejpam-6175	83	17	there	there	PRON
ejpam-6175	83	18	are	be	VERB
ejpam-6175	83	19	two	two	NUM
ejpam-6175	83	20	0	0	NUM
ejpam-6175	83	21	-	-	PUNCT
ejpam-6175	83	22	simplexes	simplexe	NOUN
ejpam-6175	83	23	,	,	PUNCT
ejpam-6175	83	24	<	<	X
ejpam-6175	83	25	a	a	X
ejpam-6175	83	26	>	>	X
ejpam-6175	83	27	and	and	CCONJ
ejpam-6175	83	28	<	<	X
ejpam-6175	83	29	b	b	X
ejpam-6175	83	30	>	>	PUNCT
ejpam-6175	83	31	,	,	PUNCT
ejpam-6175	83	32	and	and	CCONJ
ejpam-6175	83	33	one	one	NUM
ejpam-6175	83	34	1	1	NUM
ejpam-6175	83	35	-	-	PUNCT
ejpam-6175	83	36	simplex	simplex	NOUN
ejpam-6175	83	37	,	,	PUNCT
ejpam-6175	83	38	<	<	X
ejpam-6175	83	39	a	a	DET
ejpam-6175	83	40	,	,	PUNCT
ejpam-6175	83	41	b	b	X
ejpam-6175	83	42	>	>	PUNCT
ejpam-6175	83	43	.	.	PUNCT
ejpam-6175	84	1	thus	thus	ADV
ejpam-6175	84	2	,	,	PUNCT
ejpam-6175	84	3	the	the	DET
ejpam-6175	84	4	boundary	boundary	ADJ
ejpam-6175	84	5	homomorphism	homomorphism	NOUN
ejpam-6175	84	6	∂1	∂1	ADJ
ejpam-6175	84	7	is	be	AUX
ejpam-6175	84	8	defined	define	VERB
ejpam-6175	84	9	by	by	ADP
ejpam-6175	84	10	:	:	PUNCT
ejpam-6175	84	11	∂1(u	∂1(u	X
ejpam-6175	84	12	<	<	X
ejpam-6175	84	13	a	a	PROPN
ejpam-6175	84	14	,	,	PUNCT
ejpam-6175	84	15	b	b	NOUN
ejpam-6175	84	16	>	>	PUNCT
ejpam-6175	84	17	)	)	PUNCT
ejpam-6175	85	1	=	=	SYM
ejpam-6175	85	2	u∂1	u∂1	PROPN
ejpam-6175	85	3	(	(	PUNCT
ejpam-6175	85	4	<	<	X
ejpam-6175	85	5	a	a	PROPN
ejpam-6175	85	6	,	,	PUNCT
ejpam-6175	85	7	b	b	NOUN
ejpam-6175	85	8	>	>	PUNCT
ejpam-6175	85	9	)	)	PUNCT
ejpam-6175	85	10	=	=	SYM
ejpam-6175	85	11	u	u	NOUN
ejpam-6175	85	12	(	(	PUNCT
ejpam-6175	85	13	<	<	X
ejpam-6175	85	14	b	b	X
ejpam-6175	85	15	>	>	X
ejpam-6175	85	16	−	−	X
ejpam-6175	85	17	<	<	X
ejpam-6175	85	18	a	a	X
ejpam-6175	85	19	>	>	X
ejpam-6175	85	20	)	)	PUNCT
ejpam-6175	85	21	where	where	SCONJ
ejpam-6175	85	22	u	u	NOUN
ejpam-6175	85	23	is	be	AUX
ejpam-6175	85	24	a	a	DET
ejpam-6175	85	25	scalar	scalar	NOUN
ejpam-6175	85	26	.	.	PUNCT
ejpam-6175	86	1	it	it	PRON
ejpam-6175	86	2	holds	hold	VERB
ejpam-6175	86	3	that	that	SCONJ
ejpam-6175	86	4	∂q−1	∂q−1	AUX
ejpam-6175	86	5	◦	◦	NOUN
ejpam-6175	86	6	∂q	∂q	NOUN
ejpam-6175	86	7	=	=	NOUN
ejpam-6175	86	8	0	0	PROPN
ejpam-6175	86	9	for	for	ADP
ejpam-6175	86	10	all	all	PRON
ejpam-6175	86	11	1	1	NUM
ejpam-6175	86	12	≤	≤	NUM
ejpam-6175	86	13	q	q	ADJ
ejpam-6175	86	14	≤	≤	ADJ
ejpam-6175	86	15	m.	m.	NOUN
ejpam-6175	86	16	this	this	DET
ejpam-6175	86	17	composition	composition	NOUN
ejpam-6175	86	18	is	be	AUX
ejpam-6175	86	19	the	the	DET
ejpam-6175	86	20	natural	natural	ADJ
ejpam-6175	86	21	composition	composition	NOUN
ejpam-6175	86	22	of	of	ADP
ejpam-6175	86	23	two	two	NUM
ejpam-6175	86	24	homomorphisms	homomorphism	NOUN
ejpam-6175	86	25	.	.	PUNCT
ejpam-6175	87	1	definition	definition	NOUN
ejpam-6175	87	2	3	3	NUM
ejpam-6175	87	3	.	.	PUNCT
ejpam-6175	88	1	[	[	X
ejpam-6175	88	2	9	9	NUM
ejpam-6175	88	3	]	]	X
ejpam-6175	88	4	let	let	VERB
ejpam-6175	88	5	(	(	PUNCT
ejpam-6175	88	6	x	x	NOUN
ejpam-6175	88	7	,	,	PUNCT
ejpam-6175	88	8	κ	κ	NOUN
ejpam-6175	88	9	)	)	PUNCT
ejpam-6175	88	10	be	be	VERB
ejpam-6175	88	11	a	a	DET
ejpam-6175	88	12	digital	digital	ADJ
ejpam-6175	88	13	simplicial	simplicial	ADJ
ejpam-6175	88	14	complex	complex	NOUN
ejpam-6175	88	15	.	.	PUNCT
ejpam-6175	89	1	•	•	NUM
ejpam-6175	89	2	zκ	zκ	INTJ
ejpam-6175	89	3	q	q	X
ejpam-6175	89	4	(	(	PUNCT
ejpam-6175	89	5	x	x	NOUN
ejpam-6175	89	6	)	)	PUNCT
ejpam-6175	89	7	=	=	SYM
ejpam-6175	89	8	ker	ker	NOUN
ejpam-6175	90	1	∂q	∂q	PROPN
ejpam-6175	90	2	is	be	AUX
ejpam-6175	90	3	the	the	DET
ejpam-6175	90	4	group	group	NOUN
ejpam-6175	90	5	of	of	ADP
ejpam-6175	90	6	digital	digital	ADJ
ejpam-6175	90	7	simplicial	simplicial	ADJ
ejpam-6175	90	8	q	q	NOUN
ejpam-6175	90	9	-	-	PUNCT
ejpam-6175	90	10	cycles	cycle	NOUN
ejpam-6175	90	11	.	.	PUNCT
ejpam-6175	91	1	•	•	NUM
ejpam-6175	91	2	bκ	bκ	VERB
ejpam-6175	91	3	q	q	PROPN
ejpam-6175	91	4	(	(	PUNCT
ejpam-6175	91	5	x	x	NOUN
ejpam-6175	91	6	)	)	PUNCT
ejpam-6175	91	7	=	=	SYM
ejpam-6175	91	8	im(∂q+1	im(∂q+1	PROPN
ejpam-6175	91	9	)	)	PUNCT
ejpam-6175	91	10	is	be	AUX
ejpam-6175	91	11	the	the	DET
ejpam-6175	91	12	group	group	NOUN
ejpam-6175	91	13	of	of	ADP
ejpam-6175	91	14	digital	digital	ADJ
ejpam-6175	91	15	simplicial	simplicial	ADJ
ejpam-6175	91	16	q	q	NOUN
ejpam-6175	91	17	-	-	NOUN
ejpam-6175	91	18	boundaries	boundary	NOUN
ejpam-6175	91	19	.	.	PUNCT
ejpam-6175	92	1	•	•	NUM
ejpam-6175	92	2	hκ	hκ	PROPN
ejpam-6175	92	3	q	q	X
ejpam-6175	92	4	(	(	PUNCT
ejpam-6175	92	5	x	x	NOUN
ejpam-6175	92	6	)	)	PUNCT
ejpam-6175	92	7	=	=	SYM
ejpam-6175	92	8	zκ	zκ	PROPN
ejpam-6175	92	9	q	q	X
ejpam-6175	92	10	(	(	PUNCT
ejpam-6175	92	11	x)/bκ	x)/bκ	PROPN
ejpam-6175	92	12	q	q	X
ejpam-6175	92	13	(	(	PUNCT
ejpam-6175	92	14	x	x	X
ejpam-6175	92	15	)	)	PUNCT
ejpam-6175	92	16	is	be	AUX
ejpam-6175	92	17	known	know	VERB
ejpam-6175	92	18	as	as	ADP
ejpam-6175	92	19	the	the	DET
ejpam-6175	92	20	q	q	NOUN
ejpam-6175	92	21	-	-	PUNCT
ejpam-6175	92	22	th	th	VERB
ejpam-6175	92	23	digital	digital	ADJ
ejpam-6175	92	24	simplicial	simplicial	ADJ
ejpam-6175	92	25	homology	homology	NOUN
ejpam-6175	92	26	group	group	NOUN
ejpam-6175	92	27	.	.	PUNCT
ejpam-6175	93	1	it	it	PRON
ejpam-6175	93	2	has	have	AUX
ejpam-6175	93	3	been	be	AUX
ejpam-6175	93	4	demonstrated	demonstrate	VERB
ejpam-6175	93	5	in	in	ADP
ejpam-6175	93	6	[	[	X
ejpam-6175	93	7	9	9	NUM
ejpam-6175	93	8	]	]	PUNCT
ejpam-6175	93	9	that	that	SCONJ
ejpam-6175	93	10	if	if	SCONJ
ejpam-6175	93	11	f	f	X
ejpam-6175	93	12	:	:	PUNCT
ejpam-6175	93	13	x	x	X
ejpam-6175	93	14	→	→	SYM
ejpam-6175	93	15	y	y	PROPN
ejpam-6175	93	16	is	be	AUX
ejpam-6175	93	17	a	a	DET
ejpam-6175	93	18	digital	digital	ADJ
ejpam-6175	93	19	(	(	PUNCT
ejpam-6175	93	20	κ	κ	NOUN
ejpam-6175	93	21	,	,	PUNCT
ejpam-6175	93	22	λ)-isomorphism	λ)-isomorphism	PUNCT
ejpam-6175	93	23	,	,	PUNCT
ejpam-6175	93	24	then	then	ADV
ejpam-6175	93	25	for	for	ADP
ejpam-6175	93	26	all	all	PRON
ejpam-6175	93	27	q	q	PROPN
ejpam-6175	93	28	≤	≤	NUM
ejpam-6175	93	29	m	m	NOUN
ejpam-6175	93	30	,	,	PUNCT
ejpam-6175	93	31	hκ	hκ	PROPN
ejpam-6175	93	32	q	q	X
ejpam-6175	93	33	(	(	PUNCT
ejpam-6175	93	34	x	x	NOUN
ejpam-6175	93	35	)	)	PUNCT
ejpam-6175	93	36	∼=(κ	∼=(κ	NOUN
ejpam-6175	93	37	,	,	PUNCT
ejpam-6175	93	38	λ	λ	NOUN
ejpam-6175	93	39	)	)	PUNCT
ejpam-6175	93	40	h	h	NOUN
ejpam-6175	94	1	λ	λ	X
ejpam-6175	94	2	q	q	PROPN
ejpam-6175	94	3	(	(	PUNCT
ejpam-6175	94	4	y	y	PROPN
ejpam-6175	94	5	)	)	PUNCT
ejpam-6175	94	6	.	.	PUNCT
ejpam-6175	95	1	theorem	theorem	NOUN
ejpam-6175	95	2	1	1	NUM
ejpam-6175	95	3	.	.	PUNCT
ejpam-6175	96	1	[	[	X
ejpam-6175	96	2	9	9	NUM
ejpam-6175	96	3	]	]	X
ejpam-6175	96	4	if	if	SCONJ
ejpam-6175	96	5	(	(	PUNCT
ejpam-6175	96	6	x	x	NOUN
ejpam-6175	96	7	,	,	PUNCT
ejpam-6175	96	8	κ	κ	NOUN
ejpam-6175	96	9	)	)	PUNCT
ejpam-6175	96	10	⊆	⊆	NUM
ejpam-6175	96	11	zn	zn	PROPN
ejpam-6175	96	12	is	be	AUX
ejpam-6175	96	13	a	a	DET
ejpam-6175	96	14	nonempty	nonempty	ADJ
ejpam-6175	96	15	and	and	CCONJ
ejpam-6175	96	16	κ	κ	NOUN
ejpam-6175	96	17	-	-	PUNCT
ejpam-6175	96	18	connected	connected	ADJ
ejpam-6175	96	19	digital	digital	ADJ
ejpam-6175	96	20	image	image	NOUN
ejpam-6175	96	21	,	,	PUNCT
ejpam-6175	96	22	then	then	ADV
ejpam-6175	96	23	hκ	hκ	PROPN
ejpam-6175	96	24	0	0	PROPN
ejpam-6175	96	25	(	(	PUNCT
ejpam-6175	96	26	x	x	NOUN
ejpam-6175	96	27	)	)	PUNCT
ejpam-6175	96	28	=	=	PUNCT
ejpam-6175	97	1	z.	z.	PROPN
ejpam-6175	97	2	let	let	AUX
ejpam-6175	97	3	(	(	PUNCT
ejpam-6175	97	4	x	x	NOUN
ejpam-6175	97	5	,	,	PUNCT
ejpam-6175	97	6	κ	κ	NOUN
ejpam-6175	97	7	)	)	PUNCT
ejpam-6175	97	8	be	be	VERB
ejpam-6175	97	9	a	a	DET
ejpam-6175	97	10	digital	digital	ADJ
ejpam-6175	97	11	simplicial	simplicial	ADJ
ejpam-6175	97	12	complex	complex	NOUN
ejpam-6175	97	13	with	with	ADP
ejpam-6175	97	14	dimension	dimension	NOUN
ejpam-6175	97	15	n.	n.	PROPN
ejpam-6175	97	16	consider	consider	VERB
ejpam-6175	97	17	the	the	DET
ejpam-6175	97	18	following	follow	VERB
ejpam-6175	97	19	filtration	filtration	NOUN
ejpam-6175	97	20	on	on	ADP
ejpam-6175	97	21	x	x	PUNCT
ejpam-6175	97	22	based	base	VERB
ejpam-6175	97	23	on	on	ADP
ejpam-6175	97	24	κ	κ	NOUN
ejpam-6175	97	25	-	-	NOUN
ejpam-6175	97	26	adjacency	adjacency	NOUN
ejpam-6175	97	27	.	.	PUNCT
ejpam-6175	98	1	∅	∅	NOUN
ejpam-6175	98	2	=	=	PUNCT
ejpam-6175	98	3	k0	k0	PROPN
ejpam-6175	98	4	⊆	⊆	NUM
ejpam-6175	98	5	i1	i1	PROPN
ejpam-6175	98	6	⊆	⊆	NUM
ejpam-6175	98	7	i2	i2	PROPN
ejpam-6175	98	8	⊆	⊆	NUM
ejpam-6175	98	9	·	·	PUNCT
ejpam-6175	98	10	·	·	PUNCT
ejpam-6175	98	11	·	·	PUNCT
ejpam-6175	99	1	⊆	⊆	X
ejpam-6175	99	2	in	in	ADP
ejpam-6175	99	3	=	=	SYM
ejpam-6175	99	4	x	x	X
ejpam-6175	99	5	(	(	PUNCT
ejpam-6175	99	6	1	1	X
ejpam-6175	99	7	)	)	PUNCT
ejpam-6175	99	8	s.k	s.k	PROPN
ejpam-6175	99	9	.	.	PUNCT
ejpam-6175	99	10	jha	jha	PROPN
ejpam-6175	99	11	sanjay	sanjay	PROPN
ejpam-6175	99	12	,	,	PUNCT
ejpam-6175	99	13	m	m	PROPN
ejpam-6175	99	14	,	,	PUNCT
ejpam-6175	99	15	natarajan	natarajan	PROPN
ejpam-6175	99	16	/	/	SYM
ejpam-6175	99	17	eur	eur	PROPN
ejpam-6175	99	18	.	.	PUNCT
ejpam-6175	100	1	j.	j.	PROPN
ejpam-6175	100	2	pure	pure	PROPN
ejpam-6175	100	3	appl	appl	PROPN
ejpam-6175	100	4	.	.	PROPN
ejpam-6175	100	5	math	math	PROPN
ejpam-6175	100	6	,	,	PUNCT
ejpam-6175	100	7	18	18	NUM
ejpam-6175	100	8	(	(	PUNCT
ejpam-6175	100	9	3	3	NUM
ejpam-6175	100	10	)	)	PUNCT
ejpam-6175	100	11	(	(	PUNCT
ejpam-6175	100	12	2025	2025	NUM
ejpam-6175	100	13	)	)	PUNCT
ejpam-6175	100	14	,	,	PUNCT
ejpam-6175	100	15	6175	6175	NUM
ejpam-6175	100	16	5	5	NUM
ejpam-6175	100	17	of	of	ADP
ejpam-6175	100	18	18	18	NUM
ejpam-6175	100	19	this	this	DET
ejpam-6175	100	20	filtration	filtration	NOUN
ejpam-6175	100	21	is	be	AUX
ejpam-6175	100	22	constructed	construct	VERB
ejpam-6175	100	23	by	by	ADP
ejpam-6175	100	24	successively	successively	ADV
ejpam-6175	100	25	removing	remove	VERB
ejpam-6175	100	26	points	point	NOUN
ejpam-6175	100	27	without	without	ADP
ejpam-6175	100	28	affecting	affect	VERB
ejpam-6175	100	29	adjacency	adjacency	NOUN
ejpam-6175	100	30	between	between	ADP
ejpam-6175	100	31	remaining	remain	VERB
ejpam-6175	100	32	points	point	NOUN
ejpam-6175	100	33	.	.	PUNCT
ejpam-6175	101	1	the	the	DET
ejpam-6175	101	2	sequence	sequence	NOUN
ejpam-6175	101	3	of	of	ADP
ejpam-6175	101	4	inclusion	inclusion	NOUN
ejpam-6175	101	5	maps	map	NOUN
ejpam-6175	101	6	in	in	ADP
ejpam-6175	101	7	(	(	PUNCT
ejpam-6175	101	8	1	1	X
ejpam-6175	101	9	)	)	PUNCT
ejpam-6175	101	10	induces	induce	VERB
ejpam-6175	101	11	maps	map	NOUN
ejpam-6175	101	12	on	on	ADP
ejpam-6175	101	13	digital	digital	ADJ
ejpam-6175	101	14	homology	homology	NOUN
ejpam-6175	101	15	for	for	ADP
ejpam-6175	101	16	any	any	DET
ejpam-6175	101	17	dimension	dimension	NOUN
ejpam-6175	101	18	q	q	NOUN
ejpam-6175	101	19	,	,	PUNCT
ejpam-6175	101	20	0	0	PUNCT
ejpam-6175	101	21	→	→	SYM
ejpam-6175	101	22	hκ	hκ	PROPN
ejpam-6175	101	23	q	q	PROPN
ejpam-6175	101	24	(	(	PUNCT
ejpam-6175	101	25	i1	i1	PROPN
ejpam-6175	101	26	)	)	PUNCT
ejpam-6175	101	27	→	→	PUNCT
ejpam-6175	101	28	hκ	hκ	PROPN
ejpam-6175	101	29	q	q	X
ejpam-6175	101	30	(	(	PUNCT
ejpam-6175	101	31	i2	i2	PROPN
ejpam-6175	101	32	)	)	PUNCT
ejpam-6175	101	33	→	→	SYM
ejpam-6175	101	34	·	·	PUNCT
ejpam-6175	101	35	·	·	PUNCT
ejpam-6175	101	36	·	·	PUNCT
ejpam-6175	102	1	→	→	PUNCT
ejpam-6175	102	2	hκ	hκ	PART
ejpam-6175	102	3	q	q	X
ejpam-6175	102	4	(	(	PUNCT
ejpam-6175	102	5	in	in	ADP
ejpam-6175	102	6	)	)	PUNCT
ejpam-6175	102	7	.	.	PUNCT
ejpam-6175	103	1	(	(	PUNCT
ejpam-6175	103	2	2	2	X
ejpam-6175	103	3	)	)	PUNCT
ejpam-6175	103	4	let	let	VERB
ejpam-6175	103	5	f	f	PROPN
ejpam-6175	103	6	i	i	PROPN
ejpam-6175	103	7	,	,	PUNCT
ejpam-6175	103	8	j	j	PROPN
ejpam-6175	103	9	q	q	NOUN
ejpam-6175	103	10	:	:	PUNCT
ejpam-6175	103	11	hκ	hκ	PROPN
ejpam-6175	103	12	q	q	PROPN
ejpam-6175	103	13	(	(	PUNCT
ejpam-6175	103	14	ii	ii	NOUN
ejpam-6175	103	15	)	)	PUNCT
ejpam-6175	103	16	→	→	SYM
ejpam-6175	103	17	hκ	hκ	PROPN
ejpam-6175	103	18	q	q	X
ejpam-6175	103	19	(	(	PUNCT
ejpam-6175	103	20	ij	ij	NOUN
ejpam-6175	103	21	)	)	PUNCT
ejpam-6175	103	22	represent	represent	VERB
ejpam-6175	103	23	the	the	DET
ejpam-6175	103	24	composition	composition	NOUN
ejpam-6175	103	25	of	of	ADP
ejpam-6175	103	26	maps	map	NOUN
ejpam-6175	103	27	in	in	ADP
ejpam-6175	103	28	(	(	PUNCT
ejpam-6175	103	29	2	2	NUM
ejpam-6175	103	30	)	)	PUNCT
ejpam-6175	103	31	.	.	PUNCT
ejpam-6175	104	1	the	the	DET
ejpam-6175	104	2	groups	group	NOUN
ejpam-6175	104	3	zi	zi	PROPN
ejpam-6175	104	4	,	,	PUNCT
ejpam-6175	104	5	κ	κ	PROPN
ejpam-6175	104	6	q	q	NOUN
ejpam-6175	104	7	and	and	CCONJ
ejpam-6175	104	8	bi	bi	NOUN
ejpam-6175	104	9	,	,	PUNCT
ejpam-6175	104	10	κ	κ	PRON
ejpam-6175	104	11	q	q	X
ejpam-6175	104	12	,	,	PUNCT
ejpam-6175	104	13	known	know	VERB
ejpam-6175	104	14	as	as	ADP
ejpam-6175	104	15	the	the	DET
ejpam-6175	104	16	group	group	NOUN
ejpam-6175	104	17	of	of	ADP
ejpam-6175	104	18	q	q	NOUN
ejpam-6175	104	19	-	-	PUNCT
ejpam-6175	104	20	cycles	cycle	NOUN
ejpam-6175	104	21	and	and	CCONJ
ejpam-6175	104	22	q	q	NOUN
ejpam-6175	104	23	-	-	PUNCT
ejpam-6175	104	24	boundaries	boundary	NOUN
ejpam-6175	104	25	of	of	ADP
ejpam-6175	104	26	ii	ii	NOUN
ejpam-6175	104	27	respectively	respectively	ADV
ejpam-6175	104	28	,	,	PUNCT
ejpam-6175	104	29	and	and	CCONJ
ejpam-6175	104	30	the	the	DET
ejpam-6175	104	31	j	j	PROPN
ejpam-6175	104	32	-	-	PUNCT
ejpam-6175	104	33	persistent	persistent	ADJ
ejpam-6175	104	34	qth	qth	NOUN
ejpam-6175	104	35	homology	homology	NOUN
ejpam-6175	104	36	group	group	NOUN
ejpam-6175	104	37	of	of	ADP
ejpam-6175	104	38	ii	ii	PROPN
ejpam-6175	105	1	[	[	X
ejpam-6175	105	2	4	4	NUM
ejpam-6175	105	3	]	]	PUNCT
ejpam-6175	105	4	,	,	PUNCT
ejpam-6175	105	5	denoted	denote	VERB
ejpam-6175	105	6	h	h	NOUN
ejpam-6175	105	7	i	i	PROPN
ejpam-6175	105	8	,	,	PUNCT
ejpam-6175	105	9	j	j	PROPN
ejpam-6175	105	10	,	,	PUNCT
ejpam-6175	105	11	κ	κ	X
ejpam-6175	105	12	q	q	PROPN
ejpam-6175	105	13	(	(	PUNCT
ejpam-6175	105	14	ii	ii	NOUN
ejpam-6175	105	15	)	)	PUNCT
ejpam-6175	105	16	,	,	PUNCT
ejpam-6175	105	17	is	be	AUX
ejpam-6175	105	18	defined	define	VERB
ejpam-6175	105	19	as	as	ADP
ejpam-6175	105	20	:	:	PUNCT
ejpam-6175	105	21	hj	hj	PROPN
ejpam-6175	105	22	,	,	PUNCT
ejpam-6175	105	23	κ	κ	PRON
ejpam-6175	105	24	q	q	PROPN
ejpam-6175	105	25	(	(	PUNCT
ejpam-6175	105	26	ii	ii	NOUN
ejpam-6175	105	27	)	)	PUNCT
ejpam-6175	105	28	=	=	SYM
ejpam-6175	105	29	zj	zj	PROPN
ejpam-6175	105	30	,	,	PUNCT
ejpam-6175	105	31	κ	κ	X
ejpam-6175	105	32	q	q	X
ejpam-6175	105	33	(	(	PUNCT
ejpam-6175	105	34	ii)/b	ii)/b	PROPN
ejpam-6175	105	35	i+j	i+j	NUM
ejpam-6175	105	36	,	,	PUNCT
ejpam-6175	105	37	κ	κ	PRON
ejpam-6175	105	38	q	q	PROPN
ejpam-6175	105	39	(	(	PUNCT
ejpam-6175	105	40	ii	ii	NOUN
ejpam-6175	105	41	)	)	PUNCT
ejpam-6175	105	42	∩	∩	PROPN
ejpam-6175	105	43	zi	zi	PROPN
ejpam-6175	105	44	,	,	PUNCT
ejpam-6175	105	45	κ	κ	PROPN
ejpam-6175	105	46	q	q	PROPN
ejpam-6175	105	47	(	(	PUNCT
ejpam-6175	105	48	ii	ii	NOUN
ejpam-6175	105	49	)	)	PUNCT
ejpam-6175	105	50	.	.	PUNCT
ejpam-6175	106	1	one	one	PRON
ejpam-6175	106	2	can	can	AUX
ejpam-6175	106	3	find	find	VERB
ejpam-6175	106	4	some	some	DET
ejpam-6175	106	5	alternative	alternative	ADJ
ejpam-6175	106	6	definition	definition	NOUN
ejpam-6175	106	7	of	of	ADP
ejpam-6175	106	8	digital	digital	ADJ
ejpam-6175	106	9	persistent	persistent	ADJ
ejpam-6175	106	10	homology	homology	NOUN
ejpam-6175	106	11	along	along	ADP
ejpam-6175	106	12	with	with	ADP
ejpam-6175	106	13	examples	example	NOUN
ejpam-6175	106	14	in	in	ADP
ejpam-6175	106	15	[	[	X
ejpam-6175	106	16	4	4	NUM
ejpam-6175	106	17	]	]	PUNCT
ejpam-6175	106	18	.	.	PUNCT
ejpam-6175	107	1	definition	definition	NOUN
ejpam-6175	107	2	4	4	NUM
ejpam-6175	107	3	.	.	PUNCT
ejpam-6175	108	1	[	[	X
ejpam-6175	108	2	4	4	X
ejpam-6175	108	3	]	]	X
ejpam-6175	108	4	for	for	ADP
ejpam-6175	108	5	0	0	NUM
ejpam-6175	108	6	≤	≤	NUM
ejpam-6175	108	7	i	i	PRON
ejpam-6175	108	8	≤	≤	NUM
ejpam-6175	108	9	j	j	PROPN
ejpam-6175	108	10	≤	≤	NUM
ejpam-6175	108	11	n	n	CCONJ
ejpam-6175	108	12	,	,	PUNCT
ejpam-6175	108	13	the	the	DET
ejpam-6175	108	14	j	j	PROPN
ejpam-6175	108	15	-	-	PUNCT
ejpam-6175	108	16	persistent	persistent	ADJ
ejpam-6175	108	17	q	q	NOUN
ejpam-6175	108	18	-	-	PUNCT
ejpam-6175	108	19	th	th	VERB
ejpam-6175	108	20	homology	homology	NOUN
ejpam-6175	108	21	group	group	NOUN
ejpam-6175	108	22	of	of	ADP
ejpam-6175	108	23	ki	ki	PROPN
ejpam-6175	108	24	is	be	AUX
ejpam-6175	108	25	defined	define	VERB
ejpam-6175	108	26	by	by	ADP
ejpam-6175	108	27	hj	hj	PROPN
ejpam-6175	108	28	,	,	PUNCT
ejpam-6175	108	29	κ	κ	PRON
ejpam-6175	108	30	q	q	PROPN
ejpam-6175	108	31	(	(	PUNCT
ejpam-6175	108	32	ii	ii	NOUN
ejpam-6175	108	33	)	)	PUNCT
ejpam-6175	109	1	=	=	SYM
ejpam-6175	109	2	imf	imf	PROPN
ejpam-6175	109	3	i	i	PRON
ejpam-6175	109	4	,	,	PUNCT
ejpam-6175	109	5	j	j	PROPN
ejpam-6175	109	6	q	q	PROPN
ejpam-6175	109	7	.	.	PUNCT
ejpam-6175	110	1	betti	betti	ADJ
ejpam-6175	110	2	numbers	number	NOUN
ejpam-6175	110	3	classify	classify	VERB
ejpam-6175	110	4	topological	topological	ADJ
ejpam-6175	110	5	spaces	space	NOUN
ejpam-6175	110	6	by	by	ADP
ejpam-6175	110	7	counting	count	VERB
ejpam-6175	110	8	n	n	CCONJ
ejpam-6175	110	9	-	-	PUNCT
ejpam-6175	110	10	dimensional	dimensional	ADJ
ejpam-6175	110	11	holes	hole	NOUN
ejpam-6175	110	12	,	,	PUNCT
ejpam-6175	110	13	reflecting	reflect	VERB
ejpam-6175	110	14	the	the	DET
ejpam-6175	110	15	connectivity	connectivity	NOUN
ejpam-6175	110	16	structure	structure	NOUN
ejpam-6175	110	17	of	of	ADP
ejpam-6175	110	18	simplicial	simplicial	ADJ
ejpam-6175	110	19	complexes	complex	NOUN
ejpam-6175	110	20	.	.	PUNCT
ejpam-6175	111	1	definition	definition	NOUN
ejpam-6175	111	2	5	5	NUM
ejpam-6175	111	3	.	.	PUNCT
ejpam-6175	112	1	[	[	X
ejpam-6175	112	2	4	4	X
ejpam-6175	112	3	]	]	PUNCT
ejpam-6175	112	4	the	the	DET
ejpam-6175	112	5	p	p	NOUN
ejpam-6175	112	6	-	-	PUNCT
ejpam-6175	112	7	persistent	persistent	ADJ
ejpam-6175	112	8	q	q	NOUN
ejpam-6175	112	9	-	-	PUNCT
ejpam-6175	112	10	th	th	VERB
ejpam-6175	112	11	betti	betti	NOUN
ejpam-6175	112	12	number	number	NOUN
ejpam-6175	112	13	,	,	PUNCT
ejpam-6175	112	14	denoted	denote	VERB
ejpam-6175	112	15	by	by	ADP
ejpam-6175	112	16	βi+p	βi+p	PROPN
ejpam-6175	112	17	,	,	PUNCT
ejpam-6175	112	18	κ	κ	PRON
ejpam-6175	112	19	q	q	PROPN
ejpam-6175	112	20	(	(	PUNCT
ejpam-6175	112	21	ii	ii	NOUN
ejpam-6175	112	22	)	)	PUNCT
ejpam-6175	112	23	,	,	PUNCT
ejpam-6175	112	24	is	be	AUX
ejpam-6175	112	25	defined	define	VERB
ejpam-6175	112	26	as	as	ADP
ejpam-6175	112	27	βi+p	βi+p	PROPN
ejpam-6175	112	28	,	,	PUNCT
ejpam-6175	112	29	κ	κ	PRON
ejpam-6175	112	30	q	q	PROPN
ejpam-6175	112	31	(	(	PUNCT
ejpam-6175	112	32	ii	ii	NOUN
ejpam-6175	112	33	)	)	PUNCT
ejpam-6175	112	34	=	=	SYM
ejpam-6175	112	35	rank	rank	NOUN
ejpam-6175	112	36	hp	hp	PROPN
ejpam-6175	112	37	,	,	PUNCT
ejpam-6175	112	38	κ	κ	PRON
ejpam-6175	112	39	q	q	PROPN
ejpam-6175	112	40	(	(	PUNCT
ejpam-6175	112	41	ii	ii	NOUN
ejpam-6175	112	42	)	)	PUNCT
ejpam-6175	112	43	.	.	PUNCT
ejpam-6175	113	1	example	example	NOUN
ejpam-6175	114	1	1	1	NUM
ejpam-6175	114	2	.	.	PUNCT
ejpam-6175	114	3	let	let	VERB
ejpam-6175	114	4	δ	δ	PROPN
ejpam-6175	114	5	=	=	VERB
ejpam-6175	114	6	8	8	NUM
ejpam-6175	114	7	,	,	PUNCT
ejpam-6175	114	8	considering	consider	VERB
ejpam-6175	114	9	a	a	DET
ejpam-6175	114	10	graph	graph	NOUN
ejpam-6175	114	11	-	-	PUNCT
ejpam-6175	114	12	like	like	ADJ
ejpam-6175	114	13	digital	digital	ADJ
ejpam-6175	114	14	image	image	NOUN
ejpam-6175	114	15	with	with	ADP
ejpam-6175	114	16	δ	δ	PROPN
ejpam-6175	114	17	-	-	NOUN
ejpam-6175	114	18	adjacency	adjacency	PROPN
ejpam-6175	114	19	and	and	CCONJ
ejpam-6175	114	20	i	i	PRON
ejpam-6175	114	21	=	=	PUNCT
ejpam-6175	114	22	{	{	PUNCT
ejpam-6175	114	23	a1	a1	NOUN
ejpam-6175	114	24	=	=	SYM
ejpam-6175	114	25	(	(	PUNCT
ejpam-6175	114	26	−1,−1	−1,−1	NOUN
ejpam-6175	114	27	)	)	PUNCT
ejpam-6175	114	28	,	,	PUNCT
ejpam-6175	114	29	a2	a2	PROPN
ejpam-6175	114	30	=	=	PUNCT
ejpam-6175	114	31	(	(	PUNCT
ejpam-6175	114	32	−1	−1	NOUN
ejpam-6175	114	33	,	,	PUNCT
ejpam-6175	114	34	0	0	NUM
ejpam-6175	114	35	)	)	PUNCT
ejpam-6175	114	36	,	,	PUNCT
ejpam-6175	114	37	a3	a3	NOUN
ejpam-6175	114	38	=	=	SYM
ejpam-6175	114	39	(	(	PUNCT
ejpam-6175	114	40	−1	−1	NOUN
ejpam-6175	114	41	,	,	PUNCT
ejpam-6175	114	42	1	1	NUM
ejpam-6175	114	43	)	)	PUNCT
ejpam-6175	114	44	,	,	PUNCT
ejpam-6175	114	45	a4	a4	NOUN
ejpam-6175	114	46	=	=	SYM
ejpam-6175	114	47	(	(	PUNCT
ejpam-6175	114	48	0	0	NUM
ejpam-6175	114	49	,	,	PUNCT
ejpam-6175	114	50	1	1	NUM
ejpam-6175	114	51	)	)	PUNCT
ejpam-6175	114	52	,	,	PUNCT
ejpam-6175	114	53	a5	a5	PROPN
ejpam-6175	114	54	=	=	SYM
ejpam-6175	114	55	(	(	PUNCT
ejpam-6175	114	56	0	0	NUM
ejpam-6175	114	57	,	,	PUNCT
ejpam-6175	114	58	0	0	NUM
ejpam-6175	114	59	)	)	PUNCT
ejpam-6175	114	60	,	,	PUNCT
ejpam-6175	114	61	a6	a6	NOUN
ejpam-6175	114	62	=	=	SYM
ejpam-6175	114	63	(	(	PUNCT
ejpam-6175	114	64	0,−1	0,−1	PROPN
ejpam-6175	114	65	)	)	PUNCT
ejpam-6175	114	66	,	,	PUNCT
ejpam-6175	114	67	a7	a7	PROPN
ejpam-6175	114	68	=	=	SYM
ejpam-6175	114	69	(	(	PUNCT
ejpam-6175	114	70	1,−1	1,−1	NUM
ejpam-6175	114	71	)	)	PUNCT
ejpam-6175	114	72	,	,	PUNCT
ejpam-6175	114	73	a8	a8	PROPN
ejpam-6175	114	74	=	=	SYM
ejpam-6175	114	75	(	(	PUNCT
ejpam-6175	114	76	1	1	NUM
ejpam-6175	114	77	,	,	PUNCT
ejpam-6175	114	78	0	0	NUM
ejpam-6175	114	79	)	)	PUNCT
ejpam-6175	114	80	,	,	PUNCT
ejpam-6175	114	81	a9	a9	NOUN
ejpam-6175	114	82	=	=	SYM
ejpam-6175	114	83	(	(	PUNCT
ejpam-6175	114	84	1	1	NUM
ejpam-6175	114	85	,	,	PUNCT
ejpam-6175	114	86	1	1	NUM
ejpam-6175	114	87	)	)	PUNCT
ejpam-6175	114	88	}	}	PUNCT
ejpam-6175	114	89	.	.	PUNCT
ejpam-6175	115	1	let	let	VERB
ejpam-6175	115	2	us	we	PRON
ejpam-6175	115	3	examine	examine	VERB
ejpam-6175	115	4	the	the	DET
ejpam-6175	115	5	following	follow	VERB
ejpam-6175	115	6	filtration	filtration	NOUN
ejpam-6175	115	7	on	on	ADP
ejpam-6175	115	8	i	i	PRON
ejpam-6175	115	9	:	:	PUNCT
ejpam-6175	115	10	∅	∅	NOUN
ejpam-6175	115	11	=	=	PUNCT
ejpam-6175	115	12	i0	i0	PROPN
ejpam-6175	115	13	⊆	⊆	NUM
ejpam-6175	115	14	i1	i1	PROPN
ejpam-6175	115	15	=	=	PUNCT
ejpam-6175	115	16	{	{	PUNCT
ejpam-6175	115	17	a1	a1	PROPN
ejpam-6175	115	18	}	}	PUNCT
ejpam-6175	115	19	⊆	⊆	NUM
ejpam-6175	115	20	i2	i2	NOUN
ejpam-6175	115	21	=	=	SYM
ejpam-6175	115	22	{	{	PUNCT
ejpam-6175	115	23	a1	a1	PROPN
ejpam-6175	115	24	,	,	PUNCT
ejpam-6175	115	25	a2	a2	PROPN
ejpam-6175	115	26	}	}	PUNCT
ejpam-6175	115	27	⊆	⊆	NUM
ejpam-6175	115	28	i3	i3	NOUN
ejpam-6175	115	29	=	=	SYM
ejpam-6175	115	30	{	{	PUNCT
ejpam-6175	115	31	a1	a1	PROPN
ejpam-6175	115	32	,	,	PUNCT
ejpam-6175	115	33	a2	a2	PROPN
ejpam-6175	115	34	,	,	PUNCT
ejpam-6175	115	35	a3	a3	NOUN
ejpam-6175	115	36	}	}	PUNCT
ejpam-6175	115	37	⊆	⊆	NUM
ejpam-6175	115	38	i4	i4	PROPN
ejpam-6175	115	39	=	=	SYM
ejpam-6175	115	40	{	{	PUNCT
ejpam-6175	115	41	a1	a1	PROPN
ejpam-6175	115	42	,	,	PUNCT
ejpam-6175	115	43	a2	a2	PROPN
ejpam-6175	115	44	,	,	PUNCT
ejpam-6175	115	45	a3	a3	NOUN
ejpam-6175	115	46	,	,	PUNCT
ejpam-6175	115	47	a4	a4	PROPN
ejpam-6175	115	48	}	}	PUNCT
ejpam-6175	115	49	⊆	⊆	NUM
ejpam-6175	115	50	i5	i5	NOUN
ejpam-6175	115	51	=	=	SYM
ejpam-6175	115	52	{	{	PUNCT
ejpam-6175	115	53	a1	a1	PROPN
ejpam-6175	115	54	,	,	PUNCT
ejpam-6175	115	55	a2	a2	PROPN
ejpam-6175	115	56	,	,	PUNCT
ejpam-6175	115	57	a3	a3	NOUN
ejpam-6175	115	58	,	,	PUNCT
ejpam-6175	115	59	a4	a4	PROPN
ejpam-6175	115	60	,	,	PUNCT
ejpam-6175	115	61	a5	a5	PROPN
ejpam-6175	115	62	}	}	PUNCT
ejpam-6175	115	63	⊆	⊆	NUM
ejpam-6175	115	64	·	·	PUNCT
ejpam-6175	115	65	·	·	PUNCT
ejpam-6175	115	66	·	·	PUNCT
ejpam-6175	115	67	⊆	⊆	NUM
ejpam-6175	115	68	i8	i8	NOUN
ejpam-6175	115	69	=	=	SYM
ejpam-6175	115	70	{	{	PUNCT
ejpam-6175	115	71	a1	a1	PROPN
ejpam-6175	115	72	,	,	PUNCT
ejpam-6175	115	73	a2	a2	PROPN
ejpam-6175	115	74	,	,	PUNCT
ejpam-6175	115	75	a3	a3	NOUN
ejpam-6175	115	76	,	,	PUNCT
ejpam-6175	115	77	a4	a4	PROPN
ejpam-6175	115	78	,	,	PUNCT
ejpam-6175	115	79	a5	a5	NOUN
ejpam-6175	115	80	,	,	PUNCT
ejpam-6175	115	81	a6	a6	NOUN
ejpam-6175	115	82	,	,	PUNCT
ejpam-6175	115	83	a7	a7	PROPN
ejpam-6175	115	84	,	,	PUNCT
ejpam-6175	115	85	a8	a8	PROPN
ejpam-6175	115	86	}	}	PUNCT
ejpam-6175	115	87	⊆	⊆	NUM
ejpam-6175	115	88	i9	i9	NOUN
ejpam-6175	115	89	=	=	PUNCT
ejpam-6175	115	90	x.	x.	NOUN
ejpam-6175	115	91	for	for	ADP
ejpam-6175	115	92	all	all	PRON
ejpam-6175	115	93	0	0	NUM
ejpam-6175	115	94	≤	≤	NUM
ejpam-6175	116	1	i	i	PRON
ejpam-6175	116	2	<	<	X
ejpam-6175	116	3	j	j	PROPN
ejpam-6175	116	4	≤	≤	ADV
ejpam-6175	116	5	9	9	NUM
ejpam-6175	116	6	,	,	PUNCT
ejpam-6175	116	7	f	f	PROPN
ejpam-6175	117	1	i	i	PROPN
ejpam-6175	117	2	,	,	PUNCT
ejpam-6175	117	3	j	j	PROPN
ejpam-6175	117	4	q	q	X
ejpam-6175	117	5	:	:	PUNCT
ejpam-6175	117	6	ii	ii	PROPN
ejpam-6175	117	7	→	→	SYM
ejpam-6175	117	8	ij	ij	INTJ
ejpam-6175	117	9	being	be	AUX
ejpam-6175	117	10	the	the	DET
ejpam-6175	117	11	inclusion	inclusion	NOUN
ejpam-6175	117	12	map	map	NOUN
ejpam-6175	117	13	,	,	PUNCT
ejpam-6175	117	14	we	we	PRON
ejpam-6175	117	15	obtain	obtain	VERB
ejpam-6175	117	16	homomorphism	homomorphism	PROPN
ejpam-6175	117	17	f	f	PROPN
ejpam-6175	117	18	i	i	PROPN
ejpam-6175	117	19	,	,	PUNCT
ejpam-6175	117	20	j	j	PROPN
ejpam-6175	117	21	q	q	NOUN
ejpam-6175	117	22	:	:	PUNCT
ejpam-6175	117	23	hκ	hκ	PROPN
ejpam-6175	117	24	q	q	PROPN
ejpam-6175	117	25	(	(	PUNCT
ejpam-6175	117	26	ii	ii	NOUN
ejpam-6175	117	27	)	)	PUNCT
ejpam-6175	117	28	→	→	SYM
ejpam-6175	117	29	hκ	hκ	PROPN
ejpam-6175	117	30	q	q	X
ejpam-6175	117	31	(	(	PUNCT
ejpam-6175	117	32	ij	ij	NOUN
ejpam-6175	117	33	)	)	PUNCT
ejpam-6175	117	34	.	.	PUNCT
ejpam-6175	118	1	the	the	DET
ejpam-6175	118	2	digital	digital	ADJ
ejpam-6175	118	3	homology	homology	NOUN
ejpam-6175	118	4	groups	group	NOUN
ejpam-6175	118	5	of	of	ADP
ejpam-6175	118	6	i	i	PRON
ejpam-6175	118	7	are	be	AUX
ejpam-6175	118	8	given	give	VERB
ejpam-6175	118	9	as	as	SCONJ
ejpam-6175	118	10	follows	follow	VERB
ejpam-6175	118	11	[	[	X
ejpam-6175	118	12	9	9	NUM
ejpam-6175	118	13	]	]	PUNCT
ejpam-6175	118	14	:	:	PUNCT
ejpam-6175	118	15	hδ	hδ	ADP
ejpam-6175	118	16	q	q	PROPN
ejpam-6175	118	17	(	(	PUNCT
ejpam-6175	118	18	ii	ii	NOUN
ejpam-6175	118	19	)	)	PUNCT
ejpam-6175	118	20	=	=	SYM
ejpam-6175	119	1			NOUN
ejpam-6175	119	2	0	0	NUM
ejpam-6175	119	3	,	,	PUNCT
ejpam-6175	119	4	i	i	PRON
ejpam-6175	119	5	=	=	NOUN
ejpam-6175	119	6	0	0	PUNCT
ejpam-6175	120	1	z	z	NOUN
ejpam-6175	120	2	,	,	PUNCT
ejpam-6175	120	3	q	q	X
ejpam-6175	120	4	=	=	SYM
ejpam-6175	120	5	0	0	NUM
ejpam-6175	120	6	,	,	PUNCT
ejpam-6175	120	7	i	i	PRON
ejpam-6175	120	8	≥	≥	VERB
ejpam-6175	120	9	1	1	NUM
ejpam-6175	120	10	0	0	NUM
ejpam-6175	120	11	,	,	PUNCT
ejpam-6175	120	12	q	q	PROPN
ejpam-6175	120	13	̸=	̸=	PROPN
ejpam-6175	120	14	0	0	NUM
ejpam-6175	120	15	,	,	PUNCT
ejpam-6175	120	16	i	i	PRON
ejpam-6175	120	17	≥	≥	VERB
ejpam-6175	120	18	1	1	NUM
ejpam-6175	120	19	.	.	PUNCT
ejpam-6175	121	1	s.k	s.k	PROPN
ejpam-6175	121	2	.	.	PUNCT
ejpam-6175	121	3	jha	jha	PROPN
ejpam-6175	121	4	sanjay	sanjay	PROPN
ejpam-6175	121	5	,	,	PUNCT
ejpam-6175	121	6	m	m	PROPN
ejpam-6175	121	7	,	,	PUNCT
ejpam-6175	121	8	natarajan	natarajan	PROPN
ejpam-6175	121	9	/	/	SYM
ejpam-6175	121	10	eur	eur	PROPN
ejpam-6175	121	11	.	.	PUNCT
ejpam-6175	122	1	j.	j.	PROPN
ejpam-6175	122	2	pure	pure	PROPN
ejpam-6175	122	3	appl	appl	PROPN
ejpam-6175	122	4	.	.	PROPN
ejpam-6175	122	5	math	math	PROPN
ejpam-6175	122	6	,	,	PUNCT
ejpam-6175	122	7	18	18	NUM
ejpam-6175	122	8	(	(	PUNCT
ejpam-6175	122	9	3	3	NUM
ejpam-6175	122	10	)	)	PUNCT
ejpam-6175	122	11	(	(	PUNCT
ejpam-6175	122	12	2025	2025	NUM
ejpam-6175	122	13	)	)	PUNCT
ejpam-6175	122	14	,	,	PUNCT
ejpam-6175	122	15	6175	6175	NUM
ejpam-6175	122	16	6	6	NUM
ejpam-6175	122	17	of	of	ADP
ejpam-6175	122	18	18	18	NUM
ejpam-6175	122	19	therefore	therefore	ADV
ejpam-6175	122	20	,	,	PUNCT
ejpam-6175	122	21	we	we	PRON
ejpam-6175	122	22	have	have	AUX
ejpam-6175	122	23	im(f	im(f	VERB
ejpam-6175	122	24	i	i	PRON
ejpam-6175	122	25	,	,	PUNCT
ejpam-6175	122	26	j	j	PROPN
ejpam-6175	122	27	q	q	X
ejpam-6175	122	28	)	)	PUNCT
ejpam-6175	123	1	=	=	SYM
ejpam-6175	124	1			NOUN
ejpam-6175	124	2	0	0	NUM
ejpam-6175	124	3	,	,	PUNCT
ejpam-6175	124	4	i	i	PRON
ejpam-6175	124	5	=	=	NOUN
ejpam-6175	124	6	0	0	NUM
ejpam-6175	124	7	,	,	PUNCT
ejpam-6175	124	8	1	1	NUM
ejpam-6175	124	9	≤	≤	NUM
ejpam-6175	124	10	j	j	PROPN
ejpam-6175	124	11	≤	≤	ADV
ejpam-6175	124	12	9	9	NUM
ejpam-6175	124	13	z	z	NOUN
ejpam-6175	124	14	,	,	PUNCT
ejpam-6175	124	15	q	q	X
ejpam-6175	124	16	=	=	SYM
ejpam-6175	124	17	0	0	NUM
ejpam-6175	124	18	,	,	PUNCT
ejpam-6175	124	19	1	1	NUM
ejpam-6175	124	20	≤	≤	NUM
ejpam-6175	125	1	i	i	PRON
ejpam-6175	125	2	<	<	X
ejpam-6175	125	3	j	j	PROPN
ejpam-6175	125	4	≥	≥	NUM
ejpam-6175	125	5	9	9	NUM
ejpam-6175	125	6	0	0	NUM
ejpam-6175	125	7	,	,	PUNCT
ejpam-6175	125	8	q	q	PUNCT
ejpam-6175	125	9	̸=	̸=	PROPN
ejpam-6175	125	10	0	0	NUM
ejpam-6175	125	11	,	,	PUNCT
ejpam-6175	125	12	1	1	NUM
ejpam-6175	125	13	≤	≤	NUM
ejpam-6175	126	1	i	i	PRON
ejpam-6175	126	2	<	<	X
ejpam-6175	126	3	j	j	PROPN
ejpam-6175	126	4	≥	≥	NUM
ejpam-6175	126	5	9	9	NUM
ejpam-6175	126	6	.	.	PUNCT
ejpam-6175	127	1	consequently	consequently	ADV
ejpam-6175	127	2	,	,	PUNCT
ejpam-6175	127	3	we	we	PRON
ejpam-6175	127	4	get	get	VERB
ejpam-6175	127	5	hj	hj	PROPN
ejpam-6175	127	6	,	,	PUNCT
ejpam-6175	127	7	δ	δ	PROPN
ejpam-6175	127	8	q	q	PROPN
ejpam-6175	127	9	(	(	PUNCT
ejpam-6175	127	10	ii	ii	NOUN
ejpam-6175	127	11	)	)	PUNCT
ejpam-6175	127	12	=	=	PRON
ejpam-6175	128	1	{	{	PUNCT
ejpam-6175	128	2	z	z	NOUN
ejpam-6175	128	3	,	,	PUNCT
ejpam-6175	128	4	i	i	PRON
ejpam-6175	128	5	̸=	̸=	PROPN
ejpam-6175	128	6	0	0	PUNCT
ejpam-6175	128	7	and	and	CCONJ
ejpam-6175	128	8	q	q	NOUN
ejpam-6175	129	1	=	=	NOUN
ejpam-6175	129	2	0	0	NUM
ejpam-6175	129	3	0	0	NUM
ejpam-6175	129	4	,	,	PUNCT
ejpam-6175	129	5	i	i	PRON
ejpam-6175	129	6	=	=	NOUN
ejpam-6175	129	7	0	0	NUM
ejpam-6175	129	8	or	or	CCONJ
ejpam-6175	129	9	q	q	ADJ
ejpam-6175	129	10	̸=	̸=	PROPN
ejpam-6175	129	11	0	0	NUM
ejpam-6175	129	12	.	.	PUNCT
ejpam-6175	130	1	theorem	theorem	NOUN
ejpam-6175	130	2	2	2	NUM
ejpam-6175	130	3	.	.	PUNCT
ejpam-6175	131	1	if	if	SCONJ
ejpam-6175	131	2	(	(	PUNCT
ejpam-6175	131	3	x	x	NOUN
ejpam-6175	131	4	,	,	PUNCT
ejpam-6175	131	5	κ	κ	NOUN
ejpam-6175	131	6	)	)	PUNCT
ejpam-6175	131	7	be	be	VERB
ejpam-6175	131	8	any	any	DET
ejpam-6175	131	9	κ	κ	NOUN
ejpam-6175	131	10	-	-	PUNCT
ejpam-6175	131	11	connected	connect	VERB
ejpam-6175	131	12	graph	graph	NOUN
ejpam-6175	131	13	-	-	PUNCT
ejpam-6175	131	14	like	like	ADJ
ejpam-6175	131	15	digital	digital	ADJ
ejpam-6175	131	16	image	image	NOUN
ejpam-6175	131	17	with	with	ADP
ejpam-6175	131	18	κ	κ	NOUN
ejpam-6175	131	19	-	-	NOUN
ejpam-6175	131	20	adjacency	adjacency	NOUN
ejpam-6175	131	21	then	then	ADV
ejpam-6175	131	22	hj	hj	PROPN
ejpam-6175	131	23	,	,	PUNCT
ejpam-6175	131	24	κ	κ	X
ejpam-6175	131	25	q	q	X
ejpam-6175	131	26	(	(	PUNCT
ejpam-6175	131	27	xi	xi	NOUN
ejpam-6175	131	28	)	)	PUNCT
ejpam-6175	131	29	=	=	PRON
ejpam-6175	131	30	{	{	PUNCT
ejpam-6175	131	31	z	z	NOUN
ejpam-6175	131	32	,	,	PUNCT
ejpam-6175	131	33	i	i	PRON
ejpam-6175	131	34	̸=	̸=	PROPN
ejpam-6175	131	35	0	0	PUNCT
ejpam-6175	131	36	and	and	CCONJ
ejpam-6175	131	37	q	q	NOUN
ejpam-6175	132	1	=	=	NOUN
ejpam-6175	132	2	0	0	NUM
ejpam-6175	132	3	0	0	NUM
ejpam-6175	132	4	,	,	PUNCT
ejpam-6175	132	5	i	i	PRON
ejpam-6175	132	6	=	=	NOUN
ejpam-6175	132	7	0	0	NUM
ejpam-6175	132	8	or	or	CCONJ
ejpam-6175	132	9	q	q	ADJ
ejpam-6175	132	10	̸=	̸=	PROPN
ejpam-6175	132	11	0	0	NUM
ejpam-6175	132	12	.	.	PUNCT
ejpam-6175	133	1	proof	proof	NOUN
ejpam-6175	133	2	.	.	PUNCT
ejpam-6175	134	1	let	let	AUX
ejpam-6175	134	2	(	(	PUNCT
ejpam-6175	134	3	x	x	NOUN
ejpam-6175	134	4	,	,	PUNCT
ejpam-6175	134	5	κ	κ	NOUN
ejpam-6175	134	6	)	)	PUNCT
ejpam-6175	134	7	be	be	VERB
ejpam-6175	134	8	a	a	DET
ejpam-6175	134	9	digital	digital	ADJ
ejpam-6175	134	10	image	image	NOUN
ejpam-6175	134	11	that	that	PRON
ejpam-6175	134	12	resembles	resemble	VERB
ejpam-6175	134	13	a	a	DET
ejpam-6175	134	14	κ	κ	ADV
ejpam-6175	134	15	-	-	PUNCT
ejpam-6175	134	16	connected	connected	ADJ
ejpam-6175	134	17	graph	graph	NOUN
ejpam-6175	134	18	.	.	PUNCT
ejpam-6175	135	1	in	in	ADP
ejpam-6175	135	2	order	order	NOUN
ejpam-6175	135	3	to	to	PART
ejpam-6175	135	4	develop	develop	VERB
ejpam-6175	135	5	a	a	DET
ejpam-6175	135	6	filtration	filtration	NOUN
ejpam-6175	135	7	of	of	ADP
ejpam-6175	135	8	x	x	PRON
ejpam-6175	135	9	,	,	PUNCT
ejpam-6175	135	10	we	we	PRON
ejpam-6175	135	11	need	need	VERB
ejpam-6175	135	12	to	to	PART
ejpam-6175	135	13	eliminate	eliminate	VERB
ejpam-6175	135	14	a	a	DET
ejpam-6175	135	15	few	few	ADJ
ejpam-6175	135	16	points	point	NOUN
ejpam-6175	135	17	from	from	ADP
ejpam-6175	135	18	this	this	DET
ejpam-6175	135	19	digital	digital	ADJ
ejpam-6175	135	20	image	image	NOUN
ejpam-6175	135	21	.	.	PUNCT
ejpam-6175	136	1	assuming	assume	VERB
ejpam-6175	136	2	that	that	SCONJ
ejpam-6175	136	3	the	the	DET
ejpam-6175	136	4	adjacency	adjacency	NOUN
ejpam-6175	136	5	between	between	ADP
ejpam-6175	136	6	the	the	DET
ejpam-6175	136	7	points	point	NOUN
ejpam-6175	136	8	remains	remain	VERB
ejpam-6175	136	9	unchanged	unchanged	ADJ
ejpam-6175	136	10	,	,	PUNCT
ejpam-6175	136	11	we	we	PRON
ejpam-6175	136	12	may	may	AUX
ejpam-6175	136	13	conclude	conclude	VERB
ejpam-6175	136	14	that	that	SCONJ
ejpam-6175	136	15	each	each	DET
ejpam-6175	136	16	digital	digital	ADJ
ejpam-6175	136	17	image	image	NOUN
ejpam-6175	136	18	in	in	ADP
ejpam-6175	136	19	the	the	DET
ejpam-6175	136	20	x	x	NOUN
ejpam-6175	136	21	filtration	filtration	NOUN
ejpam-6175	136	22	is	be	AUX
ejpam-6175	136	23	κ	κ	PRON
ejpam-6175	136	24	-	-	PUNCT
ejpam-6175	136	25	connected	connect	VERB
ejpam-6175	136	26	.	.	PUNCT
ejpam-6175	137	1	let	let	VERB
ejpam-6175	137	2	us	we	PRON
ejpam-6175	137	3	suppose	suppose	VERB
ejpam-6175	137	4	two	two	NUM
ejpam-6175	137	5	cases	case	NOUN
ejpam-6175	137	6	:	:	PUNCT
ejpam-6175	137	7	case	case	NOUN
ejpam-6175	137	8	1	1	NUM
ejpam-6175	137	9	:	:	PUNCT
ejpam-6175	137	10	if	if	SCONJ
ejpam-6175	137	11	(	(	PUNCT
ejpam-6175	137	12	x	x	NOUN
ejpam-6175	137	13	,	,	PUNCT
ejpam-6175	137	14	κ	κ	NOUN
ejpam-6175	137	15	)	)	PUNCT
ejpam-6175	137	16	is	be	AUX
ejpam-6175	137	17	a	a	DET
ejpam-6175	137	18	singleton	singleton	NOUN
ejpam-6175	137	19	point	point	NOUN
ejpam-6175	137	20	,	,	PUNCT
ejpam-6175	137	21	then	then	ADV
ejpam-6175	137	22	there	there	PRON
ejpam-6175	137	23	is	be	VERB
ejpam-6175	137	24	only	only	ADV
ejpam-6175	137	25	one	one	NUM
ejpam-6175	137	26	filtration	filtration	NOUN
ejpam-6175	137	27	on	on	ADP
ejpam-6175	137	28	∅	∅	NOUN
ejpam-6175	137	29	=	=	PUNCT
ejpam-6175	137	30	x0	x0	PROPN
ejpam-6175	138	1	⊆	⊆	NUM
ejpam-6175	138	2	x1	x1	NOUN
ejpam-6175	138	3	=	=	SYM
ejpam-6175	138	4	x	x	NOUN
ejpam-6175	138	5	,	,	PUNCT
ejpam-6175	138	6	which	which	PRON
ejpam-6175	138	7	leads	lead	VERB
ejpam-6175	138	8	us	we	PRON
ejpam-6175	138	9	to	to	PART
ejpam-6175	138	10	have	have	VERB
ejpam-6175	138	11	the	the	DET
ejpam-6175	138	12	digital	digital	ADJ
ejpam-6175	138	13	homology	homology	NOUN
ejpam-6175	138	14	group	group	NOUN
ejpam-6175	138	15	of	of	ADP
ejpam-6175	138	16	xi	xi	PROPN
ejpam-6175	138	17	as	as	ADP
ejpam-6175	138	18	hq(x0	hq(x0	NOUN
ejpam-6175	138	19	)	)	PUNCT
ejpam-6175	138	20	=	=	SYM
ejpam-6175	138	21	0	0	NUM
ejpam-6175	138	22	and	and	CCONJ
ejpam-6175	138	23	hj	hj	PROPN
ejpam-6175	138	24	,	,	PUNCT
ejpam-6175	138	25	κ	κ	X
ejpam-6175	138	26	q	q	X
ejpam-6175	138	27	(	(	PUNCT
ejpam-6175	138	28	xi	xi	NOUN
ejpam-6175	138	29	)	)	PUNCT
ejpam-6175	138	30	=	=	PRON
ejpam-6175	138	31	{	{	PUNCT
ejpam-6175	138	32	z	z	NOUN
ejpam-6175	138	33	,	,	PUNCT
ejpam-6175	138	34	q	q	X
ejpam-6175	138	35	=	=	SYM
ejpam-6175	138	36	0	0	NUM
ejpam-6175	138	37	0	0	NUM
ejpam-6175	138	38	,	,	PUNCT
ejpam-6175	138	39	q	q	PROPN
ejpam-6175	138	40	̸=	̸=	PROPN
ejpam-6175	138	41	0	0	NUM
ejpam-6175	138	42	.	.	PUNCT
ejpam-6175	139	1	case	case	NOUN
ejpam-6175	139	2	2	2	NUM
ejpam-6175	139	3	:	:	PUNCT
ejpam-6175	139	4	if	if	SCONJ
ejpam-6175	139	5	(	(	PUNCT
ejpam-6175	139	6	x	x	NOUN
ejpam-6175	139	7	,	,	PUNCT
ejpam-6175	139	8	κ	κ	NOUN
ejpam-6175	139	9	)	)	PUNCT
ejpam-6175	139	10	is	be	AUX
ejpam-6175	139	11	a	a	DET
ejpam-6175	139	12	non	non	ADJ
ejpam-6175	139	13	-	-	ADJ
ejpam-6175	139	14	singleton	singleton	ADJ
ejpam-6175	139	15	point	point	NOUN
ejpam-6175	139	16	,	,	PUNCT
ejpam-6175	139	17	then	then	ADV
ejpam-6175	139	18	without	without	ADP
ejpam-6175	139	19	changing	change	VERB
ejpam-6175	139	20	the	the	DET
ejpam-6175	139	21	adjacency	adjacency	NOUN
ejpam-6175	139	22	relation	relation	NOUN
ejpam-6175	139	23	,	,	PUNCT
ejpam-6175	139	24	we	we	PRON
ejpam-6175	139	25	perform	perform	VERB
ejpam-6175	139	26	filtration	filtration	NOUN
ejpam-6175	139	27	by	by	ADP
ejpam-6175	139	28	removing	remove	VERB
ejpam-6175	139	29	points	point	NOUN
ejpam-6175	139	30	.	.	PUNCT
ejpam-6175	140	1	following	follow	VERB
ejpam-6175	140	2	theorem	theorem	NOUN
ejpam-6175	140	3	1	1	NUM
ejpam-6175	140	4	,	,	PUNCT
ejpam-6175	140	5	we	we	PRON
ejpam-6175	140	6	see	see	VERB
ejpam-6175	140	7	that	that	SCONJ
ejpam-6175	140	8	hκ	hκ	PROPN
ejpam-6175	140	9	0	0	X
ejpam-6175	140	10	(	(	PUNCT
ejpam-6175	140	11	x	x	NOUN
ejpam-6175	140	12	)	)	PUNCT
ejpam-6175	140	13	=	=	SYM
ejpam-6175	140	14	z	z	NOUN
ejpam-6175	140	15	for	for	ADP
ejpam-6175	140	16	every	every	DET
ejpam-6175	140	17	κ	κ	ADV
ejpam-6175	140	18	-	-	PUNCT
ejpam-6175	140	19	connected	connected	ADJ
ejpam-6175	140	20	graph	graph	NOUN
ejpam-6175	140	21	like	like	ADP
ejpam-6175	140	22	digital	digital	ADJ
ejpam-6175	140	23	image	image	NOUN
ejpam-6175	140	24	,	,	PUNCT
ejpam-6175	140	25	this	this	PRON
ejpam-6175	140	26	completes	complete	VERB
ejpam-6175	140	27	the	the	DET
ejpam-6175	140	28	proof	proof	NOUN
ejpam-6175	140	29	.	.	PUNCT
ejpam-6175	141	1	theorem	theorem	NOUN
ejpam-6175	141	2	3	3	X
ejpam-6175	141	3	.	.	PUNCT
ejpam-6175	142	1	let	let	VERB
ejpam-6175	142	2	i	i	PRON
ejpam-6175	142	3	:	:	PUNCT
ejpam-6175	142	4	(	(	PUNCT
ejpam-6175	142	5	x	x	NOUN
ejpam-6175	142	6	,	,	PUNCT
ejpam-6175	142	7	κ	κ	NOUN
ejpam-6175	142	8	)	)	PUNCT
ejpam-6175	142	9	→	→	SYM
ejpam-6175	142	10	(	(	PUNCT
ejpam-6175	142	11	x	x	X
ejpam-6175	142	12	,	,	PUNCT
ejpam-6175	142	13	κ	κ	NOUN
ejpam-6175	142	14	)	)	PUNCT
ejpam-6175	142	15	be	be	VERB
ejpam-6175	142	16	an	an	DET
ejpam-6175	142	17	identity	identity	NOUN
ejpam-6175	142	18	map	map	NOUN
ejpam-6175	142	19	for	for	ADP
ejpam-6175	142	20	a	a	DET
ejpam-6175	142	21	graph	graph	NOUN
ejpam-6175	142	22	-	-	PUNCT
ejpam-6175	142	23	like	like	ADJ
ejpam-6175	142	24	digital	digital	ADJ
ejpam-6175	142	25	image	image	NOUN
ejpam-6175	142	26	(	(	PUNCT
ejpam-6175	142	27	x	x	NOUN
ejpam-6175	142	28	,	,	PUNCT
ejpam-6175	142	29	κ	κ	NOUN
ejpam-6175	142	30	)	)	PUNCT
ejpam-6175	142	31	.	.	PUNCT
ejpam-6175	143	1	then	then	ADV
ejpam-6175	143	2	for	for	ADP
ejpam-6175	143	3	each	each	DET
ejpam-6175	143	4	integer	integer	NOUN
ejpam-6175	143	5	q	q	NOUN
ejpam-6175	143	6	,	,	PUNCT
ejpam-6175	143	7	we	we	PRON
ejpam-6175	143	8	have	have	VERB
ejpam-6175	143	9	(	(	PUNCT
ejpam-6175	143	10	iq)∗	iq)∗	PROPN
ejpam-6175	143	11	:	:	PUNCT
ejpam-6175	143	12	h	h	NOUN
ejpam-6175	143	13	κ	κ	X
ejpam-6175	143	14	q	q	X
ejpam-6175	143	15	(	(	PUNCT
ejpam-6175	143	16	xi	xi	NOUN
ejpam-6175	143	17	)	)	PUNCT
ejpam-6175	143	18	→	→	SYM
ejpam-6175	143	19	hκ	hκ	PROPN
ejpam-6175	143	20	q	q	PROPN
ejpam-6175	143	21	(	(	PUNCT
ejpam-6175	143	22	xi	xi	PROPN
ejpam-6175	143	23	)	)	PUNCT
ejpam-6175	143	24	,	,	PUNCT
ejpam-6175	143	25	where	where	SCONJ
ejpam-6175	143	26	xi	xi	PROPN
ejpam-6175	143	27	is	be	AUX
ejpam-6175	143	28	a	a	DET
ejpam-6175	143	29	member	member	NOUN
ejpam-6175	143	30	of	of	ADP
ejpam-6175	143	31	the	the	DET
ejpam-6175	143	32	filtration	filtration	NOUN
ejpam-6175	143	33	on	on	ADP
ejpam-6175	143	34	x.	x.	NOUN
ejpam-6175	143	35	proof	proof	NOUN
ejpam-6175	143	36	.	.	PUNCT
ejpam-6175	144	1	let	let	VERB
ejpam-6175	144	2	x	x	X
ejpam-6175	144	3	′	′	NOUN
ejpam-6175	144	4	is	be	AUX
ejpam-6175	144	5	be	be	AUX
ejpam-6175	144	6	the	the	DET
ejpam-6175	144	7	filtration	filtration	NOUN
ejpam-6175	144	8	on	on	ADP
ejpam-6175	144	9	x	x	NOUN
ejpam-6175	144	10	,	,	PUNCT
ejpam-6175	144	11	following	follow	VERB
ejpam-6175	144	12	theorem	theorem	ADJ
ejpam-6175	144	13	[	[	X
ejpam-6175	144	14	2	2	NUM
ejpam-6175	144	15	]	]	PUNCT
ejpam-6175	144	16	,	,	PUNCT
ejpam-6175	144	17	we	we	PRON
ejpam-6175	144	18	get	get	VERB
ejpam-6175	144	19	the	the	DET
ejpam-6175	144	20	digital	digital	ADJ
ejpam-6175	144	21	homology	homology	NOUN
ejpam-6175	144	22	group	group	NOUN
ejpam-6175	144	23	of	of	ADP
ejpam-6175	144	24	each	each	PRON
ejpam-6175	144	25	x	x	PROPN
ejpam-6175	144	26	′	′	NUM
ejpam-6175	144	27	is	be	AUX
ejpam-6175	144	28	to	to	PART
ejpam-6175	144	29	be	be	AUX
ejpam-6175	144	30	0	0	NUM
ejpam-6175	144	31	or	or	CCONJ
ejpam-6175	144	32	z	z	NOUN
ejpam-6175	144	33	depending	depend	VERB
ejpam-6175	144	34	upon	upon	SCONJ
ejpam-6175	144	35	i	i	PRON
ejpam-6175	144	36	and	and	CCONJ
ejpam-6175	144	37	q.	q.	PROPN
ejpam-6175	144	38	since	since	SCONJ
ejpam-6175	144	39	each	each	DET
ejpam-6175	144	40	mapping	mapping	NOUN
ejpam-6175	144	41	in	in	ADP
ejpam-6175	144	42	the	the	DET
ejpam-6175	144	43	filtration	filtration	NOUN
ejpam-6175	144	44	is	be	AUX
ejpam-6175	144	45	the	the	DET
ejpam-6175	144	46	identity	identity	NOUN
ejpam-6175	144	47	,	,	PUNCT
ejpam-6175	144	48	they	they	PRON
ejpam-6175	144	49	share	share	VERB
ejpam-6175	144	50	the	the	DET
ejpam-6175	144	51	same	same	ADJ
ejpam-6175	144	52	digital	digital	ADJ
ejpam-6175	144	53	homology	homology	NOUN
ejpam-6175	144	54	group	group	NOUN
ejpam-6175	144	55	.	.	PUNCT
ejpam-6175	145	1	thus	thus	ADV
ejpam-6175	145	2	,	,	PUNCT
ejpam-6175	145	3	the	the	DET
ejpam-6175	145	4	induced	induced	ADJ
ejpam-6175	145	5	map	map	NOUN
ejpam-6175	145	6	(	(	PUNCT
ejpam-6175	145	7	iq)∗	iq)∗	PROPN
ejpam-6175	145	8	is	be	AUX
ejpam-6175	145	9	an	an	DET
ejpam-6175	145	10	isomorphism	isomorphism	NOUN
ejpam-6175	145	11	from	from	ADP
ejpam-6175	145	12	hκ	hκ	PROPN
ejpam-6175	145	13	q	q	PROPN
ejpam-6175	145	14	(	(	PUNCT
ejpam-6175	145	15	xi	xi	NOUN
ejpam-6175	145	16	)	)	PUNCT
ejpam-6175	145	17	to	to	ADP
ejpam-6175	145	18	hκ	hκ	PROPN
ejpam-6175	145	19	q	q	PROPN
ejpam-6175	145	20	(	(	PUNCT
ejpam-6175	145	21	xi	xi	NOUN
ejpam-6175	145	22	)	)	PUNCT
ejpam-6175	145	23	.	.	PUNCT
ejpam-6175	146	1	s.k	s.k	PROPN
ejpam-6175	146	2	.	.	PUNCT
ejpam-6175	146	3	jha	jha	PROPN
ejpam-6175	146	4	sanjay	sanjay	PROPN
ejpam-6175	146	5	,	,	PUNCT
ejpam-6175	146	6	m	m	PROPN
ejpam-6175	146	7	,	,	PUNCT
ejpam-6175	146	8	natarajan	natarajan	PROPN
ejpam-6175	146	9	/	/	SYM
ejpam-6175	146	10	eur	eur	PROPN
ejpam-6175	146	11	.	.	PUNCT
ejpam-6175	147	1	j.	j.	PROPN
ejpam-6175	147	2	pure	pure	PROPN
ejpam-6175	147	3	appl	appl	PROPN
ejpam-6175	147	4	.	.	PROPN
ejpam-6175	147	5	math	math	PROPN
ejpam-6175	147	6	,	,	PUNCT
ejpam-6175	147	7	18	18	NUM
ejpam-6175	147	8	(	(	PUNCT
ejpam-6175	147	9	3	3	NUM
ejpam-6175	147	10	)	)	PUNCT
ejpam-6175	147	11	(	(	PUNCT
ejpam-6175	147	12	2025	2025	NUM
ejpam-6175	147	13	)	)	PUNCT
ejpam-6175	147	14	,	,	PUNCT
ejpam-6175	147	15	6175	6175	NUM
ejpam-6175	147	16	7	7	NUM
ejpam-6175	147	17	of	of	ADP
ejpam-6175	147	18	18	18	NUM
ejpam-6175	147	19	figure	figure	NOUN
ejpam-6175	147	20	1	1	NUM
ejpam-6175	147	21	:	:	PUNCT
ejpam-6175	147	22	analyzing	analyze	VERB
ejpam-6175	147	23	a	a	DET
ejpam-6175	147	24	four	four	NUM
ejpam-6175	147	25	-	-	PUNCT
ejpam-6175	147	26	leaf	leaf	NOUN
ejpam-6175	147	27	clover	clover	NOUN
ejpam-6175	147	28	:	:	PUNCT
ejpam-6175	147	29	original	original	ADJ
ejpam-6175	147	30	image	image	NOUN
ejpam-6175	147	31	,	,	PUNCT
ejpam-6175	147	32	edge	edge	NOUN
ejpam-6175	147	33	detection	detection	NOUN
ejpam-6175	147	34	,	,	PUNCT
ejpam-6175	147	35	and	and	CCONJ
ejpam-6175	147	36	graph	graph	NOUN
ejpam-6175	147	37	representation	representation	NOUN
ejpam-6175	147	38	.	.	PUNCT
ejpam-6175	148	1	3	3	X
ejpam-6175	148	2	.	.	X
ejpam-6175	148	3	some	some	DET
ejpam-6175	148	4	digital	digital	ADJ
ejpam-6175	148	5	homology	homology	NOUN
ejpam-6175	148	6	applications	application	NOUN
ejpam-6175	148	7	to	to	PART
ejpam-6175	148	8	image	image	NOUN
ejpam-6175	148	9	processing	processing	NOUN
ejpam-6175	148	10	in	in	ADP
ejpam-6175	148	11	this	this	DET
ejpam-6175	148	12	section	section	NOUN
ejpam-6175	148	13	,	,	PUNCT
ejpam-6175	148	14	we	we	PRON
ejpam-6175	148	15	explore	explore	VERB
ejpam-6175	148	16	diverse	diverse	ADJ
ejpam-6175	148	17	applications	application	NOUN
ejpam-6175	148	18	of	of	ADP
ejpam-6175	148	19	digital	digital	ADJ
ejpam-6175	148	20	persistent	persistent	ADJ
ejpam-6175	148	21	homology	homology	NOUN
ejpam-6175	148	22	groups	group	NOUN
ejpam-6175	148	23	within	within	ADP
ejpam-6175	148	24	the	the	DET
ejpam-6175	148	25	domain	domain	NOUN
ejpam-6175	148	26	of	of	ADP
ejpam-6175	148	27	image	image	NOUN
ejpam-6175	148	28	processing	processing	NOUN
ejpam-6175	148	29	,	,	PUNCT
ejpam-6175	148	30	emphasizing	emphasize	VERB
ejpam-6175	148	31	tasks	task	NOUN
ejpam-6175	148	32	such	such	ADJ
ejpam-6175	148	33	as	as	ADP
ejpam-6175	148	34	calculating	calculate	VERB
ejpam-6175	148	35	the	the	DET
ejpam-6175	148	36	genus	genus	NOUN
ejpam-6175	148	37	and	and	CCONJ
ejpam-6175	148	38	processing	process	VERB
ejpam-6175	148	39	various	various	ADJ
ejpam-6175	148	40	standard	standard	ADJ
ejpam-6175	148	41	shapes	shape	NOUN
ejpam-6175	148	42	.	.	PUNCT
ejpam-6175	149	1	these	these	DET
ejpam-6175	149	2	applications	application	NOUN
ejpam-6175	149	3	demonstrate	demonstrate	VERB
ejpam-6175	149	4	how	how	SCONJ
ejpam-6175	149	5	persistent	persistent	ADJ
ejpam-6175	149	6	homology	homology	NOUN
ejpam-6175	149	7	offers	offer	VERB
ejpam-6175	149	8	a	a	DET
ejpam-6175	149	9	mathematically	mathematically	ADV
ejpam-6175	149	10	grounded	ground	VERB
ejpam-6175	149	11	framework	framework	NOUN
ejpam-6175	149	12	for	for	ADP
ejpam-6175	149	13	analyzing	analyze	VERB
ejpam-6175	149	14	structural	structural	ADJ
ejpam-6175	149	15	features	feature	NOUN
ejpam-6175	149	16	in	in	ADP
ejpam-6175	149	17	digital	digital	ADJ
ejpam-6175	149	18	images	image	NOUN
ejpam-6175	149	19	.	.	PUNCT
ejpam-6175	150	1	for	for	ADP
ejpam-6175	150	2	instance	instance	NOUN
ejpam-6175	150	3	,	,	PUNCT
ejpam-6175	150	4	consider	consider	VERB
ejpam-6175	150	5	the	the	DET
ejpam-6175	150	6	case	case	NOUN
ejpam-6175	150	7	of	of	ADP
ejpam-6175	150	8	a	a	DET
ejpam-6175	150	9	four	four	NUM
ejpam-6175	150	10	-	-	PUNCT
ejpam-6175	150	11	leaf	leaf	NOUN
ejpam-6175	150	12	clover	clover	NOUN
ejpam-6175	150	13	shape	shape	NOUN
ejpam-6175	150	14	,	,	PUNCT
ejpam-6175	150	15	denoted	denote	VERB
ejpam-6175	150	16	as	as	ADP
ejpam-6175	150	17	cn	cn	PROPN
ejpam-6175	150	18	.	.	PUNCT
ejpam-6175	151	1	using	use	VERB
ejpam-6175	151	2	our	our	PRON
ejpam-6175	151	3	proposed	propose	VERB
ejpam-6175	151	4	method	method	NOUN
ejpam-6175	151	5	,	,	PUNCT
ejpam-6175	151	6	cn	cn	PROPN
ejpam-6175	151	7	is	be	AUX
ejpam-6175	151	8	transformed	transform	VERB
ejpam-6175	151	9	into	into	ADP
ejpam-6175	151	10	a	a	DET
ejpam-6175	151	11	digital	digital	ADJ
ejpam-6175	151	12	graph	graph	NOUN
ejpam-6175	151	13	representation	representation	NOUN
ejpam-6175	151	14	while	while	SCONJ
ejpam-6175	151	15	retaining	retain	VERB
ejpam-6175	151	16	its	its	PRON
ejpam-6175	151	17	topological	topological	ADJ
ejpam-6175	151	18	characteristics	characteristic	NOUN
ejpam-6175	151	19	.	.	PUNCT
ejpam-6175	152	1	cmin	cmin	NOUN
ejpam-6175	152	2	=	=	PUNCT
ejpam-6175	152	3	{	{	PUNCT
ejpam-6175	152	4	c0	c0	NOUN
ejpam-6175	152	5	=	=	PUNCT
ejpam-6175	152	6	(	(	PUNCT
ejpam-6175	152	7	−1	−1	NOUN
ejpam-6175	152	8	,	,	PUNCT
ejpam-6175	152	9	0	0	NUM
ejpam-6175	152	10	)	)	PUNCT
ejpam-6175	152	11	,	,	PUNCT
ejpam-6175	152	12	c1	c1	NOUN
ejpam-6175	152	13	=	=	SYM
ejpam-6175	152	14	(	(	PUNCT
ejpam-6175	152	15	0,−1	0,−1	PROPN
ejpam-6175	152	16	)	)	PUNCT
ejpam-6175	152	17	,	,	PUNCT
ejpam-6175	152	18	c2	c2	PROPN
ejpam-6175	152	19	=	=	SYM
ejpam-6175	152	20	(	(	PUNCT
ejpam-6175	152	21	0	0	NUM
ejpam-6175	152	22	,	,	PUNCT
ejpam-6175	152	23	1	1	NUM
ejpam-6175	152	24	)	)	PUNCT
ejpam-6175	152	25	,	,	PUNCT
ejpam-6175	152	26	c3	c3	PROPN
ejpam-6175	152	27	=	=	PUNCT
ejpam-6175	152	28	(	(	PUNCT
ejpam-6175	152	29	1	1	NUM
ejpam-6175	152	30	,	,	PUNCT
ejpam-6175	152	31	0	0	NUM
ejpam-6175	152	32	)	)	PUNCT
ejpam-6175	152	33	}	}	PUNCT
ejpam-6175	152	34	.	.	PUNCT
ejpam-6175	153	1	the	the	DET
ejpam-6175	153	2	filtration	filtration	NOUN
ejpam-6175	153	3	on	on	ADP
ejpam-6175	153	4	cmin	cmin	NOUN
ejpam-6175	153	5	is	be	AUX
ejpam-6175	153	6	given	give	VERB
ejpam-6175	153	7	by	by	ADP
ejpam-6175	153	8	:	:	PUNCT
ejpam-6175	153	9	∅	∅	NOUN
ejpam-6175	153	10	=	=	SYM
ejpam-6175	153	11	∆0	∆0	NOUN
ejpam-6175	154	1	⊆	⊆	NUM
ejpam-6175	154	2	∆1	∆1	PUNCT
ejpam-6175	154	3	=	=	SYM
ejpam-6175	154	4	{	{	PUNCT
ejpam-6175	154	5	c0	c0	NOUN
ejpam-6175	154	6	}	}	PUNCT
ejpam-6175	154	7	⊆	⊆	NUM
ejpam-6175	154	8	∆2	∆2	X
ejpam-6175	154	9	=	=	SYM
ejpam-6175	154	10	{	{	PUNCT
ejpam-6175	154	11	c0	c0	NOUN
ejpam-6175	154	12	,	,	PUNCT
ejpam-6175	154	13	c1	c1	PROPN
ejpam-6175	154	14	}	}	PUNCT
ejpam-6175	154	15	⊆	⊆	NUM
ejpam-6175	154	16	∆3	∆3	PROPN
ejpam-6175	154	17	=	=	SYM
ejpam-6175	154	18	{	{	PUNCT
ejpam-6175	154	19	c0	c0	NOUN
ejpam-6175	154	20	,	,	PUNCT
ejpam-6175	154	21	c1	c1	PROPN
ejpam-6175	154	22	,	,	PUNCT
ejpam-6175	154	23	c2	c2	PROPN
ejpam-6175	154	24	}	}	PUNCT
ejpam-6175	154	25	⊆	⊆	NUM
ejpam-6175	154	26	∆4	∆4	NOUN
ejpam-6175	154	27	=	=	SYM
ejpam-6175	154	28	cmin	cmin	PROPN
ejpam-6175	154	29	.	.	PUNCT
ejpam-6175	155	1	the	the	DET
ejpam-6175	155	2	minimal	minimal	ADJ
ejpam-6175	155	3	simple	simple	ADJ
ejpam-6175	155	4	closed	closed	ADJ
ejpam-6175	155	5	curve	curve	NOUN
ejpam-6175	155	6	,	,	PUNCT
ejpam-6175	155	7	denoted	denote	VERB
ejpam-6175	155	8	cmin	cmin	NOUN
ejpam-6175	155	9	,	,	PUNCT
ejpam-6175	155	10	is	be	AUX
ejpam-6175	155	11	then	then	ADV
ejpam-6175	155	12	computed	compute	VERB
ejpam-6175	155	13	within	within	ADP
ejpam-6175	155	14	the	the	DET
ejpam-6175	155	15	framework	framework	NOUN
ejpam-6175	155	16	of	of	ADP
ejpam-6175	155	17	8	8	NUM
ejpam-6175	155	18	-	-	PUNCT
ejpam-6175	155	19	adjacency	adjacency	NOUN
ejpam-6175	155	20	.	.	PUNCT
ejpam-6175	156	1	this	this	DET
ejpam-6175	156	2	process	process	NOUN
ejpam-6175	156	3	ensures	ensure	VERB
ejpam-6175	156	4	that	that	SCONJ
ejpam-6175	156	5	the	the	DET
ejpam-6175	156	6	structural	structural	ADJ
ejpam-6175	156	7	and	and	CCONJ
ejpam-6175	156	8	topological	topological	ADJ
ejpam-6175	156	9	features	feature	NOUN
ejpam-6175	156	10	,	,	PUNCT
ejpam-6175	156	11	such	such	ADJ
ejpam-6175	156	12	as	as	ADP
ejpam-6175	156	13	loops	loop	NOUN
ejpam-6175	156	14	and	and	CCONJ
ejpam-6175	156	15	connected	connected	ADJ
ejpam-6175	156	16	components	component	NOUN
ejpam-6175	156	17	,	,	PUNCT
ejpam-6175	156	18	are	be	AUX
ejpam-6175	156	19	preserved	preserve	VERB
ejpam-6175	156	20	accurately	accurately	ADV
ejpam-6175	156	21	.	.	PUNCT
ejpam-6175	157	1	figure[1	figure[1	PUNCT
ejpam-6175	157	2	]	]	PUNCT
ejpam-6175	157	3	illustrates	illustrate	VERB
ejpam-6175	157	4	this	this	DET
ejpam-6175	157	5	transformation	transformation	NOUN
ejpam-6175	157	6	,	,	PUNCT
ejpam-6175	157	7	where	where	SCONJ
ejpam-6175	157	8	the	the	DET
ejpam-6175	157	9	original	original	ADJ
ejpam-6175	157	10	shape	shape	NOUN
ejpam-6175	157	11	is	be	AUX
ejpam-6175	157	12	represented	represent	VERB
ejpam-6175	157	13	in	in	ADP
ejpam-6175	157	14	a	a	DET
ejpam-6175	157	15	discrete	discrete	ADJ
ejpam-6175	157	16	digital	digital	ADJ
ejpam-6175	157	17	format	format	NOUN
ejpam-6175	157	18	,	,	PUNCT
ejpam-6175	157	19	and	and	CCONJ
ejpam-6175	157	20	the	the	DET
ejpam-6175	157	21	minimal	minimal	ADJ
ejpam-6175	157	22	curve	curve	NOUN
ejpam-6175	157	23	is	be	AUX
ejpam-6175	157	24	identified	identify	VERB
ejpam-6175	157	25	and	and	CCONJ
ejpam-6175	157	26	visualized	visualize	VERB
ejpam-6175	157	27	.	.	PUNCT
ejpam-6175	158	1	thus	thus	ADV
ejpam-6175	158	2	,	,	PUNCT
ejpam-6175	158	3	h8	h8	PROPN
ejpam-6175	158	4	q	q	PROPN
ejpam-6175	158	5	(	(	PUNCT
ejpam-6175	158	6	∆i	∆i	PROPN
ejpam-6175	158	7	)	)	PUNCT
ejpam-6175	158	8	=	=	SYM
ejpam-6175	159	1			NOUN
ejpam-6175	159	2	0	0	NUM
ejpam-6175	159	3	,	,	PUNCT
ejpam-6175	159	4	for	for	ADP
ejpam-6175	159	5	q	q	PROPN
ejpam-6175	159	6	∈	∈	PROPN
ejpam-6175	159	7	z	z	PROPN
ejpam-6175	159	8	and	and	CCONJ
ejpam-6175	159	9	i	i	PRON
ejpam-6175	159	10	=	=	NOUN
ejpam-6175	159	11	0	0	PUNCT
ejpam-6175	160	1	z	z	NOUN
ejpam-6175	160	2	,	,	PUNCT
ejpam-6175	160	3	for	for	ADP
ejpam-6175	160	4	q	q	NOUN
ejpam-6175	160	5	=	=	SYM
ejpam-6175	160	6	0	0	NUM
ejpam-6175	160	7	and	and	CCONJ
ejpam-6175	160	8	0	0	NUM
ejpam-6175	161	1	<	<	X
ejpam-6175	161	2	i	i	X
ejpam-6175	161	3	≤	≤	ADV
ejpam-6175	161	4	4	4	NUM
ejpam-6175	161	5	z	z	NOUN
ejpam-6175	161	6	,	,	PUNCT
ejpam-6175	161	7	q	q	NOUN
ejpam-6175	161	8	=	=	SYM
ejpam-6175	161	9	1	1	NUM
ejpam-6175	162	1	and	and	CCONJ
ejpam-6175	162	2	i	i	PRON
ejpam-6175	162	3	=	=	NOUN
ejpam-6175	162	4	4	4	X
ejpam-6175	162	5	.	.	PUNCT
ejpam-6175	163	1	using	use	VERB
ejpam-6175	163	2	this	this	DET
ejpam-6175	163	3	result	result	NOUN
ejpam-6175	163	4	,	,	PUNCT
ejpam-6175	163	5	we	we	PRON
ejpam-6175	163	6	calculate	calculate	VERB
ejpam-6175	163	7	the	the	DET
ejpam-6175	163	8	p	p	NOUN
ejpam-6175	163	9	-	-	PUNCT
ejpam-6175	163	10	persistent	persistent	ADJ
ejpam-6175	163	11	q	q	NOUN
ejpam-6175	163	12	-	-	PUNCT
ejpam-6175	163	13	th	th	X
ejpam-6175	163	14	betti	betti	NOUN
ejpam-6175	163	15	number	number	NOUN
ejpam-6175	163	16	:	:	PUNCT
ejpam-6175	163	17	•	•	NOUN
ejpam-6175	163	18	for	for	ADP
ejpam-6175	163	19	q	q	NOUN
ejpam-6175	163	20	=	=	SYM
ejpam-6175	163	21	0	0	PUNCT
ejpam-6175	164	1	and	and	CCONJ
ejpam-6175	164	2	i	i	PRON
ejpam-6175	164	3	̸=	̸=	PROPN
ejpam-6175	164	4	0	0	NUM
ejpam-6175	164	5	,	,	PUNCT
ejpam-6175	164	6	we	we	PRON
ejpam-6175	164	7	get	get	VERB
ejpam-6175	164	8	βi+p	βi+p	PROPN
ejpam-6175	164	9	,	,	PUNCT
ejpam-6175	164	10	κ	κ	X
ejpam-6175	164	11	q	q	X
ejpam-6175	164	12	(	(	PUNCT
ejpam-6175	164	13	∆i	∆i	PROPN
ejpam-6175	164	14	)	)	PUNCT
ejpam-6175	165	1	=	=	PUNCT
ejpam-6175	166	1	rank(h	rank(h	VERB
ejpam-6175	166	2	i	i	PRON
ejpam-6175	166	3	,	,	PUNCT
ejpam-6175	166	4	κ	κ	PROPN
ejpam-6175	166	5	0	0	NUM
ejpam-6175	166	6	(	(	PUNCT
ejpam-6175	166	7	∆i	∆i	PROPN
ejpam-6175	166	8	)	)	PUNCT
ejpam-6175	166	9	)	)	PUNCT
ejpam-6175	167	1	=	=	SYM
ejpam-6175	167	2	rank(z	rank(z	X
ejpam-6175	167	3	)	)	PUNCT
ejpam-6175	167	4	=	=	SYM
ejpam-6175	168	1	1	1	NUM
ejpam-6175	168	2	.	.	NOUN
ejpam-6175	168	3	•	•	NUM
ejpam-6175	168	4	for	for	ADP
ejpam-6175	168	5	q	q	PROPN
ejpam-6175	168	6	̸=	̸=	PROPN
ejpam-6175	168	7	0	0	PUNCT
ejpam-6175	169	1	and	and	CCONJ
ejpam-6175	169	2	i	i	PRON
ejpam-6175	169	3	=	=	NOUN
ejpam-6175	169	4	0	0	NUM
ejpam-6175	169	5	,	,	PUNCT
ejpam-6175	169	6	we	we	PRON
ejpam-6175	169	7	have	have	VERB
ejpam-6175	169	8	βi+p	βi+p	PROPN
ejpam-6175	169	9	,	,	PUNCT
ejpam-6175	169	10	κ	κ	X
ejpam-6175	169	11	q	q	X
ejpam-6175	169	12	(	(	PUNCT
ejpam-6175	169	13	∆i	∆i	PROPN
ejpam-6175	169	14	)	)	PUNCT
ejpam-6175	170	1	=	=	PUNCT
ejpam-6175	171	1	rank(h	rank(h	VERB
ejpam-6175	171	2	i	i	PRON
ejpam-6175	171	3	,	,	PUNCT
ejpam-6175	171	4	κ	κ	PROPN
ejpam-6175	171	5	q	q	X
ejpam-6175	171	6	(	(	PUNCT
ejpam-6175	171	7	∆i	∆i	PROPN
ejpam-6175	171	8	)	)	PUNCT
ejpam-6175	171	9	)	)	PUNCT
ejpam-6175	172	1	=	=	SYM
ejpam-6175	172	2	rank(0	rank(0	NOUN
ejpam-6175	172	3	)	)	PUNCT
ejpam-6175	172	4	=	=	SYM
ejpam-6175	172	5	0	0	X
ejpam-6175	172	6	.	.	PUNCT
ejpam-6175	173	1	let	let	VERB
ejpam-6175	173	2	κ1	κ1	NOUN
ejpam-6175	173	3	and	and	CCONJ
ejpam-6175	173	4	κ2	κ2	PROPN
ejpam-6175	173	5	represent	represent	VERB
ejpam-6175	173	6	the	the	DET
ejpam-6175	173	7	two	two	NUM
ejpam-6175	173	8	adjacency	adjacency	NOUN
ejpam-6175	173	9	relations	relation	NOUN
ejpam-6175	173	10	in	in	ADP
ejpam-6175	173	11	zn	zn	PROPN
ejpam-6175	173	12	,	,	PUNCT
ejpam-6175	173	13	with	with	ADP
ejpam-6175	173	14	x	x	SYM
ejpam-6175	173	15	denoting	denote	VERB
ejpam-6175	173	16	a	a	DET
ejpam-6175	173	17	digital	digital	ADJ
ejpam-6175	173	18	image	image	NOUN
ejpam-6175	173	19	within	within	ADP
ejpam-6175	173	20	zn	zn	PROPN
ejpam-6175	173	21	.	.	PUNCT
ejpam-6175	174	1	when	when	SCONJ
ejpam-6175	174	2	κ1	κ1	PROPN
ejpam-6175	174	3	>	>	X
ejpam-6175	174	4	κ2	κ2	PROPN
ejpam-6175	174	5	,	,	PUNCT
ejpam-6175	174	6	any	any	DET
ejpam-6175	174	7	pair	pair	NOUN
ejpam-6175	174	8	of	of	ADP
ejpam-6175	174	9	κ2	κ2	NOUN
ejpam-6175	174	10	-	-	PUNCT
ejpam-6175	174	11	adjacent	adjacent	ADJ
ejpam-6175	174	12	points	point	NOUN
ejpam-6175	174	13	will	will	AUX
ejpam-6175	174	14	also	also	ADV
ejpam-6175	174	15	be	be	AUX
ejpam-6175	174	16	κ1	κ1	NOUN
ejpam-6175	174	17	-	-	PUNCT
ejpam-6175	174	18	adjacent	adjacent	NOUN
ejpam-6175	174	19	.	.	PUNCT
ejpam-6175	175	1	the	the	DET
ejpam-6175	175	2	impact	impact	NOUN
ejpam-6175	175	3	of	of	ADP
ejpam-6175	175	4	applying	apply	VERB
ejpam-6175	175	5	different	different	ADJ
ejpam-6175	175	6	adjacency	adjacency	NOUN
ejpam-6175	175	7	relations	relation	NOUN
ejpam-6175	175	8	to	to	ADP
ejpam-6175	175	9	the	the	DET
ejpam-6175	175	10	same	same	ADJ
ejpam-6175	175	11	set	set	NOUN
ejpam-6175	175	12	of	of	ADP
ejpam-6175	175	13	points	point	NOUN
ejpam-6175	175	14	is	be	AUX
ejpam-6175	175	15	demonstrated	demonstrate	VERB
ejpam-6175	175	16	in	in	ADP
ejpam-6175	175	17	the	the	DET
ejpam-6175	175	18	subsequent	subsequent	ADJ
ejpam-6175	175	19	example	example	NOUN
ejpam-6175	175	20	.	.	PUNCT
ejpam-6175	176	1	s.k	s.k	PROPN
ejpam-6175	176	2	.	.	PUNCT
ejpam-6175	176	3	jha	jha	PROPN
ejpam-6175	176	4	sanjay	sanjay	PROPN
ejpam-6175	176	5	,	,	PUNCT
ejpam-6175	176	6	m	m	PROPN
ejpam-6175	176	7	,	,	PUNCT
ejpam-6175	176	8	natarajan	natarajan	PROPN
ejpam-6175	176	9	/	/	SYM
ejpam-6175	176	10	eur	eur	PROPN
ejpam-6175	176	11	.	.	PUNCT
ejpam-6175	177	1	j.	j.	PROPN
ejpam-6175	177	2	pure	pure	PROPN
ejpam-6175	177	3	appl	appl	PROPN
ejpam-6175	177	4	.	.	PROPN
ejpam-6175	177	5	math	math	PROPN
ejpam-6175	177	6	,	,	PUNCT
ejpam-6175	177	7	18	18	NUM
ejpam-6175	177	8	(	(	PUNCT
ejpam-6175	177	9	3	3	NUM
ejpam-6175	177	10	)	)	PUNCT
ejpam-6175	177	11	(	(	PUNCT
ejpam-6175	177	12	2025	2025	NUM
ejpam-6175	177	13	)	)	PUNCT
ejpam-6175	177	14	,	,	PUNCT
ejpam-6175	177	15	6175	6175	NUM
ejpam-6175	177	16	8	8	NUM
ejpam-6175	177	17	of	of	ADP
ejpam-6175	177	18	18	18	NUM
ejpam-6175	177	19	example	example	NOUN
ejpam-6175	177	20	2	2	NUM
ejpam-6175	177	21	.	.	PUNCT
ejpam-6175	178	1	let	let	VERB
ejpam-6175	178	2	us	we	PRON
ejpam-6175	178	3	again	again	ADV
ejpam-6175	178	4	consider	consider	VERB
ejpam-6175	178	5	the	the	DET
ejpam-6175	178	6	same	same	ADJ
ejpam-6175	178	7	cmin	cmin	NOUN
ejpam-6175	178	8	but	but	CCONJ
ejpam-6175	178	9	this	this	DET
ejpam-6175	178	10	time	time	NOUN
ejpam-6175	178	11	with	with	ADP
ejpam-6175	178	12	4	4	NUM
ejpam-6175	178	13	-	-	PUNCT
ejpam-6175	178	14	adjacency	adjacency	NOUN
ejpam-6175	178	15	,	,	PUNCT
ejpam-6175	178	16	the	the	DET
ejpam-6175	178	17	digital	digital	ADJ
ejpam-6175	178	18	homology	homology	NOUN
ejpam-6175	178	19	groups	group	NOUN
ejpam-6175	178	20	of	of	ADP
ejpam-6175	178	21	cmin	cmin	PROPN
ejpam-6175	178	22	are	be	AUX
ejpam-6175	178	23	:	:	PUNCT
ejpam-6175	178	24	h4	h4	PROPN
ejpam-6175	178	25	q	q	PROPN
ejpam-6175	178	26	(	(	PUNCT
ejpam-6175	178	27	cmin	cmin	NOUN
ejpam-6175	178	28	)	)	PUNCT
ejpam-6175	178	29	=	=	PRON
ejpam-6175	178	30	{	{	PUNCT
ejpam-6175	178	31	0	0	NUM
ejpam-6175	178	32	,	,	PUNCT
ejpam-6175	178	33	q	q	PROPN
ejpam-6175	178	34	̸=	̸=	PROPN
ejpam-6175	178	35	0⊕i	0⊕i	PUNCT
ejpam-6175	178	36	n=1	n=1	PROPN
ejpam-6175	178	37	z	z	PROPN
ejpam-6175	178	38	,	,	PUNCT
ejpam-6175	178	39	q	q	X
ejpam-6175	179	1	=	=	NOUN
ejpam-6175	179	2	0	0	NUM
ejpam-6175	179	3	.	.	PUNCT
ejpam-6175	180	1	the	the	DET
ejpam-6175	180	2	digital	digital	ADJ
ejpam-6175	180	3	homology	homology	NOUN
ejpam-6175	180	4	groups	group	NOUN
ejpam-6175	180	5	of	of	ADP
ejpam-6175	180	6	∆i	∆i	PROPN
ejpam-6175	180	7	vary	vary	VERB
ejpam-6175	180	8	based	base	VERB
ejpam-6175	180	9	on	on	ADP
ejpam-6175	180	10	the	the	DET
ejpam-6175	180	11	chosen	choose	VERB
ejpam-6175	180	12	adjacency	adjacency	PROPN
ejpam-6175	180	13	relation	relation	PROPN
ejpam-6175	180	14	,	,	PUNCT
ejpam-6175	180	15	which	which	PRON
ejpam-6175	180	16	in	in	ADP
ejpam-6175	180	17	turn	turn	NOUN
ejpam-6175	180	18	affects	affect	VERB
ejpam-6175	180	19	the	the	DET
ejpam-6175	180	20	digital	digital	ADJ
ejpam-6175	180	21	persistent	persistent	ADJ
ejpam-6175	180	22	homology	homology	NOUN
ejpam-6175	180	23	groups	group	NOUN
ejpam-6175	180	24	of	of	ADP
ejpam-6175	180	25	cmin	cmin	PROPN
ejpam-6175	180	26	.	.	PUNCT
ejpam-6175	181	1	for	for	ADP
ejpam-6175	181	2	example	example	NOUN
ejpam-6175	181	3	,	,	PUNCT
ejpam-6175	181	4	the	the	DET
ejpam-6175	181	5	function	function	NOUN
ejpam-6175	181	6	f1,2	f1,2	PROPN
ejpam-6175	181	7	0	0	NUM
ejpam-6175	181	8	:	:	PUNCT
ejpam-6175	181	9	h4	h4	PROPN
ejpam-6175	181	10	0	0	NUM
ejpam-6175	181	11	(	(	PUNCT
ejpam-6175	181	12	∆1	∆1	NOUN
ejpam-6175	181	13	)	)	PUNCT
ejpam-6175	181	14	→	→	SYM
ejpam-6175	181	15	h4	h4	PROPN
ejpam-6175	181	16	0	0	NUM
ejpam-6175	181	17	(	(	PUNCT
ejpam-6175	181	18	∆2	∆2	X
ejpam-6175	181	19	)	)	PUNCT
ejpam-6175	181	20	as	as	ADP
ejpam-6175	181	21	an	an	DET
ejpam-6175	181	22	image	image	NOUN
ejpam-6175	181	23	of	of	ADP
ejpam-6175	181	24	z⊕	z⊕	PROPN
ejpam-6175	181	25	z	z	PROPN
ejpam-6175	181	26	,	,	PUNCT
ejpam-6175	181	27	leading	lead	VERB
ejpam-6175	181	28	to	to	ADP
ejpam-6175	181	29	the	the	DET
ejpam-6175	181	30	conclusion	conclusion	NOUN
ejpam-6175	181	31	that	that	SCONJ
ejpam-6175	181	32	h2,4	h2,4	PROPN
ejpam-6175	181	33	0	0	NUM
ejpam-6175	181	34	(	(	PUNCT
ejpam-6175	181	35	∆2	∆2	NOUN
ejpam-6175	181	36	)	)	PUNCT
ejpam-6175	181	37	equals	equal	VERB
ejpam-6175	181	38	z⊕	z⊕	PROPN
ejpam-6175	181	39	z.	z.	PROPN
ejpam-6175	181	40	however	however	ADV
ejpam-6175	181	41	,	,	PUNCT
ejpam-6175	181	42	as	as	SCONJ
ejpam-6175	181	43	illustrated	illustrate	VERB
ejpam-6175	181	44	above	above	ADV
ejpam-6175	181	45	,	,	PUNCT
ejpam-6175	181	46	determined	determine	VERB
ejpam-6175	181	47	to	to	PART
ejpam-6175	181	48	be	be	AUX
ejpam-6175	181	49	z.	z.	PROPN
ejpam-6175	181	50	in	in	ADP
ejpam-6175	181	51	the	the	DET
ejpam-6175	181	52	following	follow	VERB
ejpam-6175	181	53	examples	example	NOUN
ejpam-6175	181	54	,	,	PUNCT
ejpam-6175	181	55	we	we	PRON
ejpam-6175	181	56	will	will	AUX
ejpam-6175	181	57	examine	examine	VERB
ejpam-6175	181	58	a	a	DET
ejpam-6175	181	59	variety	variety	NOUN
ejpam-6175	181	60	of	of	ADP
ejpam-6175	181	61	special	special	ADJ
ejpam-6175	181	62	structures	structure	NOUN
ejpam-6175	181	63	,	,	PUNCT
ejpam-6175	181	64	including	include	VERB
ejpam-6175	181	65	spheres	sphere	NOUN
ejpam-6175	181	66	and	and	CCONJ
ejpam-6175	181	67	toruses	torus	NOUN
ejpam-6175	181	68	.	.	PUNCT
ejpam-6175	182	1	these	these	DET
ejpam-6175	182	2	examples	example	NOUN
ejpam-6175	182	3	are	be	AUX
ejpam-6175	182	4	carefully	carefully	ADV
ejpam-6175	182	5	chosen	choose	VERB
ejpam-6175	182	6	to	to	PART
ejpam-6175	182	7	clearly	clearly	ADV
ejpam-6175	182	8	demonstrate	demonstrate	VERB
ejpam-6175	182	9	their	their	PRON
ejpam-6175	182	10	unique	unique	ADJ
ejpam-6175	182	11	properties	property	NOUN
ejpam-6175	182	12	and	and	CCONJ
ejpam-6175	182	13	the	the	DET
ejpam-6175	182	14	relationships	relationship	NOUN
ejpam-6175	182	15	between	between	ADP
ejpam-6175	182	16	them	they	PRON
ejpam-6175	182	17	.	.	PUNCT
ejpam-6175	183	1	3.1	3.1	NUM
ejpam-6175	183	2	.	.	PUNCT
ejpam-6175	183	3	case	case	NOUN
ejpam-6175	183	4	of	of	ADP
ejpam-6175	183	5	a	a	DET
ejpam-6175	183	6	digitized	digitize	VERB
ejpam-6175	183	7	sphere	sphere	NOUN
ejpam-6175	183	8	smin	smin	NOUN
ejpam-6175	183	9	to	to	PART
ejpam-6175	183	10	further	far	ADV
ejpam-6175	183	11	elaborate	elaborate	VERB
ejpam-6175	183	12	on	on	ADP
ejpam-6175	183	13	the	the	DET
ejpam-6175	183	14	application	application	NOUN
ejpam-6175	183	15	of	of	ADP
ejpam-6175	183	16	digital	digital	ADJ
ejpam-6175	183	17	persistent	persistent	ADJ
ejpam-6175	183	18	homology	homology	NOUN
ejpam-6175	183	19	in	in	ADP
ejpam-6175	183	20	analyzing	analyze	VERB
ejpam-6175	183	21	topological	topological	ADJ
ejpam-6175	183	22	structures	structure	NOUN
ejpam-6175	183	23	,	,	PUNCT
ejpam-6175	183	24	we	we	PRON
ejpam-6175	183	25	consider	consider	VERB
ejpam-6175	183	26	the	the	DET
ejpam-6175	183	27	case	case	NOUN
ejpam-6175	183	28	of	of	ADP
ejpam-6175	183	29	a	a	DET
ejpam-6175	183	30	digitized	digitize	VERB
ejpam-6175	183	31	sphere	sphere	NOUN
ejpam-6175	183	32	smin	smin	PROPN
ejpam-6175	183	33	,	,	PUNCT
ejpam-6175	183	34	as	as	SCONJ
ejpam-6175	183	35	depicted	depict	VERB
ejpam-6175	183	36	in	in	ADP
ejpam-6175	183	37	figure[2	figure[2	PROPN
ejpam-6175	183	38	]	]	PUNCT
ejpam-6175	183	39	.	.	PUNCT
ejpam-6175	184	1	this	this	DET
ejpam-6175	184	2	sphere	sphere	NOUN
ejpam-6175	184	3	with	with	ADP
ejpam-6175	184	4	vertices	vertex	NOUN
ejpam-6175	184	5	v	v	NOUN
ejpam-6175	184	6	=	=	SYM
ejpam-6175	184	7	{	{	PUNCT
ejpam-6175	184	8	v1	v1	PROPN
ejpam-6175	184	9	,	,	PUNCT
ejpam-6175	184	10	v2	v2	PROPN
ejpam-6175	184	11	,	,	PUNCT
ejpam-6175	184	12	·	·	PUNCT
ejpam-6175	184	13	·	·	PUNCT
ejpam-6175	184	14	·	·	PUNCT
ejpam-6175	184	15	,	,	PUNCT
ejpam-6175	184	16	v6	v6	NOUN
ejpam-6175	184	17	}	}	PUNCT
ejpam-6175	184	18	,	,	PUNCT
ejpam-6175	184	19	edges	edge	NOUN
ejpam-6175	184	20	e	e	NOUN
ejpam-6175	184	21	=	=	SYM
ejpam-6175	184	22	{	{	PUNCT
ejpam-6175	184	23	e1	e1	PROPN
ejpam-6175	184	24	,	,	PUNCT
ejpam-6175	184	25	e2	e2	PROPN
ejpam-6175	184	26	,	,	PUNCT
ejpam-6175	184	27	·	·	PUNCT
ejpam-6175	184	28	·	·	PUNCT
ejpam-6175	184	29	·	·	PUNCT
ejpam-6175	184	30	,	,	PUNCT
ejpam-6175	184	31	e12	e12	NOUN
ejpam-6175	184	32	}	}	PUNCT
ejpam-6175	184	33	and	and	CCONJ
ejpam-6175	184	34	faces	face	VERB
ejpam-6175	184	35	f	f	PROPN
ejpam-6175	184	36	=	=	SYM
ejpam-6175	184	37	{	{	PUNCT
ejpam-6175	184	38	f1	f1	NOUN
ejpam-6175	184	39	,	,	PUNCT
ejpam-6175	184	40	f2	f2	PROPN
ejpam-6175	184	41	,	,	PUNCT
ejpam-6175	184	42	·	·	PUNCT
ejpam-6175	184	43	·	·	PUNCT
ejpam-6175	184	44	·	·	PUNCT
ejpam-6175	184	45	,	,	PUNCT
ejpam-6175	184	46	f8	f8	PROPN
ejpam-6175	184	47	}	}	PUNCT
ejpam-6175	184	48	is	be	AUX
ejpam-6175	184	49	represented	represent	VERB
ejpam-6175	184	50	as	as	ADP
ejpam-6175	184	51	a	a	DET
ejpam-6175	184	52	simplicial	simplicial	ADJ
ejpam-6175	184	53	complex	complex	NOUN
ejpam-6175	184	54	,	,	PUNCT
ejpam-6175	184	55	the	the	DET
ejpam-6175	184	56	genus	genus	NOUN
ejpam-6175	184	57	g	g	NOUN
ejpam-6175	184	58	of	of	ADP
ejpam-6175	184	59	a	a	DET
ejpam-6175	184	60	surface	surface	NOUN
ejpam-6175	184	61	can	can	AUX
ejpam-6175	184	62	be	be	AUX
ejpam-6175	184	63	derived	derive	VERB
ejpam-6175	184	64	from	from	ADP
ejpam-6175	184	65	the	the	DET
ejpam-6175	184	66	euler	euler	PROPN
ejpam-6175	184	67	characteristic	characteristic	PROPN
ejpam-6175	184	68	χ	χ	NOUN
ejpam-6175	184	69	,	,	PUNCT
ejpam-6175	184	70	which	which	PRON
ejpam-6175	184	71	is	be	AUX
ejpam-6175	184	72	given	give	VERB
ejpam-6175	184	73	by	by	ADP
ejpam-6175	184	74	χ	χ	PROPN
ejpam-6175	184	75	=	=	SYM
ejpam-6175	184	76	v	v	ADP
ejpam-6175	184	77	−	−	PROPN
ejpam-6175	184	78	e	e	NOUN
ejpam-6175	185	1	+	+	NUM
ejpam-6175	185	2	f	f	X
ejpam-6175	185	3	.	.	PUNCT
ejpam-6175	186	1	for	for	ADP
ejpam-6175	186	2	a	a	DET
ejpam-6175	186	3	genus	genus	NOUN
ejpam-6175	186	4	g	g	PROPN
ejpam-6175	186	5	surface	surface	NOUN
ejpam-6175	186	6	,	,	PUNCT
ejpam-6175	186	7	the	the	DET
ejpam-6175	186	8	relationship	relationship	NOUN
ejpam-6175	186	9	is	be	AUX
ejpam-6175	186	10	χ	χ	NOUN
ejpam-6175	186	11	=	=	SYM
ejpam-6175	186	12	2−	2−	NUM
ejpam-6175	186	13	2	2	NUM
ejpam-6175	186	14	g.	g.	NOUN
ejpam-6175	186	15	thus	thus	ADV
ejpam-6175	186	16	,	,	PUNCT
ejpam-6175	186	17	the	the	DET
ejpam-6175	186	18	genus	genus	NOUN
ejpam-6175	186	19	g	g	PROPN
ejpam-6175	186	20	can	can	AUX
ejpam-6175	186	21	be	be	AUX
ejpam-6175	186	22	calculated	calculate	VERB
ejpam-6175	186	23	as	as	ADP
ejpam-6175	186	24	g	g	PROPN
ejpam-6175	186	25	=	=	PUNCT
ejpam-6175	187	1	2−χ	2−χ	NUM
ejpam-6175	187	2	2	2	NUM
ejpam-6175	187	3	.	.	PUNCT
ejpam-6175	188	1	here	here	ADV
ejpam-6175	188	2	χ	χ	X
ejpam-6175	188	3	=	=	SYM
ejpam-6175	188	4	6−	6−	NUM
ejpam-6175	188	5	12	12	NUM
ejpam-6175	188	6	+	+	CCONJ
ejpam-6175	188	7	8	8	NUM
ejpam-6175	188	8	,	,	PUNCT
ejpam-6175	188	9	that	that	PRON
ejpam-6175	188	10	is	be	AUX
ejpam-6175	188	11	χ	χ	NOUN
ejpam-6175	188	12	=	=	SYM
ejpam-6175	188	13	2	2	NUM
ejpam-6175	188	14	.	.	NOUN
ejpam-6175	188	15	which	which	PRON
ejpam-6175	188	16	leads	lead	VERB
ejpam-6175	188	17	to	to	ADP
ejpam-6175	188	18	genus	genus	NOUN
ejpam-6175	188	19	of	of	ADP
ejpam-6175	188	20	a	a	DET
ejpam-6175	188	21	torus	torus	NOUN
ejpam-6175	188	22	g	g	PROPN
ejpam-6175	188	23	=	=	NOUN
ejpam-6175	188	24	0	0	PROPN
ejpam-6175	188	25	.	.	PUNCT
ejpam-6175	189	1	in	in	ADP
ejpam-6175	189	2	the	the	DET
ejpam-6175	189	3	filtration	filtration	NOUN
ejpam-6175	189	4	process	process	NOUN
ejpam-6175	189	5	of	of	ADP
ejpam-6175	189	6	a	a	DET
ejpam-6175	189	7	sphere	sphere	NOUN
ejpam-6175	189	8	,	,	PUNCT
ejpam-6175	189	9	we	we	PRON
ejpam-6175	189	10	start	start	VERB
ejpam-6175	189	11	with	with	ADP
ejpam-6175	189	12	an	an	DET
ejpam-6175	189	13	empty	empty	ADJ
ejpam-6175	189	14	set	set	NOUN
ejpam-6175	189	15	∆0	∆0	NOUN
ejpam-6175	189	16	=	=	PUNCT
ejpam-6175	189	17	∅.	∅.	NOUN
ejpam-6175	189	18	as	as	SCONJ
ejpam-6175	189	19	we	we	PRON
ejpam-6175	189	20	proceed	proceed	VERB
ejpam-6175	189	21	to	to	ADP
ejpam-6175	189	22	∆1	∆1	VERB
ejpam-6175	189	23	,	,	PUNCT
ejpam-6175	189	24	we	we	PRON
ejpam-6175	189	25	introduce	introduce	VERB
ejpam-6175	189	26	the	the	DET
ejpam-6175	189	27	first	first	ADJ
ejpam-6175	189	28	vertex	vertex	NOUN
ejpam-6175	189	29	,	,	PUNCT
ejpam-6175	189	30	{	{	PUNCT
ejpam-6175	189	31	v1	v1	NOUN
ejpam-6175	189	32	}	}	PUNCT
ejpam-6175	189	33	.	.	PUNCT
ejpam-6175	190	1	moving	move	VERB
ejpam-6175	190	2	to	to	ADP
ejpam-6175	190	3	∆2	∆2	PRON
ejpam-6175	190	4	,	,	PUNCT
ejpam-6175	190	5	we	we	PRON
ejpam-6175	190	6	add	add	VERB
ejpam-6175	190	7	another	another	DET
ejpam-6175	190	8	vertex	vertex	NOUN
ejpam-6175	190	9	and	and	CCONJ
ejpam-6175	190	10	an	an	DET
ejpam-6175	190	11	edge	edge	NOUN
ejpam-6175	190	12	,	,	PUNCT
ejpam-6175	190	13	forming	form	VERB
ejpam-6175	190	14	{	{	PUNCT
ejpam-6175	190	15	v1	v1	NOUN
ejpam-6175	190	16	,	,	PUNCT
ejpam-6175	190	17	v2	v2	NOUN
ejpam-6175	190	18	}	}	PUNCT
ejpam-6175	190	19	∪	∪	NOUN
ejpam-6175	190	20	{	{	PUNCT
ejpam-6175	190	21	e1	e1	NOUN
ejpam-6175	190	22	}	}	PUNCT
ejpam-6175	190	23	.	.	PUNCT
ejpam-6175	191	1	at	at	ADP
ejpam-6175	191	2	∆3	∆3	NOUN
ejpam-6175	191	3	,	,	PUNCT
ejpam-6175	191	4	the	the	DET
ejpam-6175	191	5	set	set	NOUN
ejpam-6175	191	6	expands	expand	VERB
ejpam-6175	191	7	to	to	PART
ejpam-6175	191	8	include	include	VERB
ejpam-6175	191	9	three	three	NUM
ejpam-6175	191	10	vertices	vertex	NOUN
ejpam-6175	191	11	and	and	CCONJ
ejpam-6175	191	12	three	three	NUM
ejpam-6175	191	13	edges	edge	NOUN
ejpam-6175	191	14	,	,	PUNCT
ejpam-6175	191	15	{	{	PUNCT
ejpam-6175	191	16	v1	v1	NOUN
ejpam-6175	191	17	,	,	PUNCT
ejpam-6175	191	18	v2	v2	PROPN
ejpam-6175	191	19	,	,	PUNCT
ejpam-6175	191	20	v3	v3	PROPN
ejpam-6175	191	21	}	}	PUNCT
ejpam-6175	191	22	∪	∪	NOUN
ejpam-6175	191	23	{	{	PUNCT
ejpam-6175	191	24	e1	e1	PROPN
ejpam-6175	191	25	,	,	PUNCT
ejpam-6175	191	26	e2	e2	PROPN
ejpam-6175	191	27	,	,	PUNCT
ejpam-6175	191	28	e3	e3	NOUN
ejpam-6175	191	29	}	}	PUNCT
ejpam-6175	191	30	.	.	PUNCT
ejpam-6175	192	1	continuing	continue	VERB
ejpam-6175	192	2	this	this	DET
ejpam-6175	192	3	progression	progression	NOUN
ejpam-6175	192	4	,	,	PUNCT
ejpam-6175	192	5	∆4	∆4	NOUN
ejpam-6175	192	6	comprises	comprise	VERB
ejpam-6175	192	7	four	four	NUM
ejpam-6175	192	8	vertices	vertex	NOUN
ejpam-6175	192	9	and	and	CCONJ
ejpam-6175	192	10	five	five	NUM
ejpam-6175	192	11	edges	edge	NOUN
ejpam-6175	192	12	,	,	PUNCT
ejpam-6175	192	13	{	{	PUNCT
ejpam-6175	192	14	v1	v1	NOUN
ejpam-6175	192	15	,	,	PUNCT
ejpam-6175	192	16	.	.	PUNCT
ejpam-6175	192	17	.	.	PUNCT
ejpam-6175	193	1	.	.	PUNCT
ejpam-6175	194	1	,	,	PUNCT
ejpam-6175	194	2	v4	v4	NOUN
ejpam-6175	194	3	}	}	PUNCT
ejpam-6175	194	4	∪	∪	NOUN
ejpam-6175	194	5	{	{	PUNCT
ejpam-6175	194	6	e1	e1	NOUN
ejpam-6175	194	7	,	,	PUNCT
ejpam-6175	194	8	.	.	PUNCT
ejpam-6175	194	9	.	.	PUNCT
ejpam-6175	195	1	.	.	PUNCT
ejpam-6175	196	1	,	,	PUNCT
ejpam-6175	196	2	e5	e5	PROPN
ejpam-6175	196	3	}	}	PUNCT
ejpam-6175	196	4	.	.	PUNCT
ejpam-6175	197	1	in	in	ADP
ejpam-6175	197	2	∆5	∆5	PROPN
ejpam-6175	197	3	,	,	PUNCT
ejpam-6175	197	4	we	we	PRON
ejpam-6175	197	5	incorporate	incorporate	VERB
ejpam-6175	197	6	five	five	NUM
ejpam-6175	197	7	vertices	vertex	NOUN
ejpam-6175	197	8	and	and	CCONJ
ejpam-6175	197	9	eight	eight	NUM
ejpam-6175	197	10	edges	edge	NOUN
ejpam-6175	197	11	,	,	PUNCT
ejpam-6175	197	12	{	{	PUNCT
ejpam-6175	197	13	v1	v1	NOUN
ejpam-6175	197	14	,	,	PUNCT
ejpam-6175	197	15	.	.	PUNCT
ejpam-6175	197	16	.	.	PUNCT
ejpam-6175	197	17	.	.	PUNCT
ejpam-6175	198	1	,	,	PUNCT
ejpam-6175	198	2	v5	v5	NOUN
ejpam-6175	198	3	}	}	PUNCT
ejpam-6175	198	4	∪	∪	NOUN
ejpam-6175	198	5	{	{	PUNCT
ejpam-6175	198	6	e1	e1	NOUN
ejpam-6175	198	7	,	,	PUNCT
ejpam-6175	198	8	.	.	PUNCT
ejpam-6175	198	9	.	.	PUNCT
ejpam-6175	199	1	.	.	PUNCT
ejpam-6175	200	1	,	,	PUNCT
ejpam-6175	200	2	e8	e8	PROPN
ejpam-6175	200	3	}	}	PUNCT
ejpam-6175	200	4	.	.	PUNCT
ejpam-6175	201	1	the	the	DET
ejpam-6175	201	2	complexity	complexity	NOUN
ejpam-6175	201	3	increases	increase	VERB
ejpam-6175	201	4	significantly	significantly	ADV
ejpam-6175	201	5	at	at	ADP
ejpam-6175	201	6	∆6	∆6	NOUN
ejpam-6175	201	7	with	with	ADP
ejpam-6175	201	8	six	six	NUM
ejpam-6175	201	9	vertices	vertex	NOUN
ejpam-6175	201	10	,	,	PUNCT
ejpam-6175	201	11	twelve	twelve	NUM
ejpam-6175	201	12	edges	edge	NOUN
ejpam-6175	201	13	,	,	PUNCT
ejpam-6175	201	14	and	and	CCONJ
ejpam-6175	201	15	the	the	DET
ejpam-6175	201	16	introduction	introduction	NOUN
ejpam-6175	201	17	of	of	ADP
ejpam-6175	201	18	faces	face	NOUN
ejpam-6175	201	19	,	,	PUNCT
ejpam-6175	201	20	{	{	PUNCT
ejpam-6175	201	21	v1	v1	NOUN
ejpam-6175	201	22	,	,	PUNCT
ejpam-6175	201	23	.	.	PUNCT
ejpam-6175	201	24	.	.	PUNCT
ejpam-6175	202	1	.	.	PUNCT
ejpam-6175	203	1	,	,	PUNCT
ejpam-6175	203	2	v6	v6	PROPN
ejpam-6175	203	3	}	}	PUNCT
ejpam-6175	203	4	∪	∪	NOUN
ejpam-6175	203	5	{	{	PUNCT
ejpam-6175	203	6	e1	e1	NOUN
ejpam-6175	203	7	,	,	PUNCT
ejpam-6175	203	8	.	.	PUNCT
ejpam-6175	203	9	.	.	PUNCT
ejpam-6175	203	10	.	.	PUNCT
ejpam-6175	204	1	,	,	PUNCT
ejpam-6175	204	2	e12	e12	NOUN
ejpam-6175	204	3	}	}	PUNCT
ejpam-6175	204	4	∪	∪	NOUN
ejpam-6175	204	5	{	{	PUNCT
ejpam-6175	204	6	f1	f1	NOUN
ejpam-6175	204	7	,	,	PUNCT
ejpam-6175	204	8	.	.	PUNCT
ejpam-6175	204	9	.	.	PUNCT
ejpam-6175	205	1	.	.	PUNCT
ejpam-6175	206	1	,	,	PUNCT
ejpam-6175	206	2	f8	f8	PROPN
ejpam-6175	206	3	}	}	PUNCT
ejpam-6175	206	4	.	.	PUNCT
ejpam-6175	207	1	with	with	ADP
ejpam-6175	207	2	each	each	DET
ejpam-6175	207	3	increment	increment	NOUN
ejpam-6175	207	4	in	in	ADP
ejpam-6175	207	5	i	i	PRON
ejpam-6175	207	6	,	,	PUNCT
ejpam-6175	207	7	additional	additional	ADJ
ejpam-6175	207	8	vertices	vertex	NOUN
ejpam-6175	207	9	,	,	PUNCT
ejpam-6175	207	10	edges	edge	NOUN
ejpam-6175	207	11	,	,	PUNCT
ejpam-6175	207	12	and	and	CCONJ
ejpam-6175	207	13	faces	face	NOUN
ejpam-6175	207	14	are	be	AUX
ejpam-6175	207	15	incorporated	incorporate	VERB
ejpam-6175	207	16	,	,	PUNCT
ejpam-6175	207	17	gradually	gradually	ADV
ejpam-6175	207	18	forming	form	VERB
ejpam-6175	207	19	a	a	DET
ejpam-6175	207	20	complete	complete	ADJ
ejpam-6175	207	21	sphere	sphere	NOUN
ejpam-6175	207	22	structure	structure	NOUN
ejpam-6175	207	23	.	.	PUNCT
ejpam-6175	208	1	the	the	DET
ejpam-6175	208	2	betti	betti	NOUN
ejpam-6175	208	3	numbers	number	NOUN
ejpam-6175	208	4	,	,	PUNCT
ejpam-6175	208	5	which	which	PRON
ejpam-6175	208	6	are	be	AUX
ejpam-6175	208	7	topological	topological	ADJ
ejpam-6175	208	8	invariants	invariant	NOUN
ejpam-6175	208	9	,	,	PUNCT
ejpam-6175	208	10	count	count	VERB
ejpam-6175	208	11	the	the	DET
ejpam-6175	208	12	number	number	NOUN
ejpam-6175	208	13	of	of	ADP
ejpam-6175	208	14	independent	independent	ADJ
ejpam-6175	208	15	cycles	cycle	NOUN
ejpam-6175	208	16	of	of	ADP
ejpam-6175	208	17	different	different	ADJ
ejpam-6175	208	18	dimensions	dimension	NOUN
ejpam-6175	208	19	.	.	PUNCT
ejpam-6175	209	1	for	for	ADP
ejpam-6175	209	2	a	a	DET
ejpam-6175	209	3	sphere	sphere	NOUN
ejpam-6175	209	4	,	,	PUNCT
ejpam-6175	209	5	the	the	DET
ejpam-6175	209	6	betti	betti	NOUN
ejpam-6175	209	7	numbers	number	NOUN
ejpam-6175	209	8	are	be	AUX
ejpam-6175	209	9	:	:	PUNCT
ejpam-6175	209	10	β0	β0	ADJ
ejpam-6175	209	11	(	(	PUNCT
ejpam-6175	209	12	number	number	NOUN
ejpam-6175	209	13	of	of	ADP
ejpam-6175	209	14	connected	connected	ADJ
ejpam-6175	209	15	components	component	NOUN
ejpam-6175	209	16	)	)	PUNCT
ejpam-6175	209	17	is	be	AUX
ejpam-6175	209	18	1	1	NUM
ejpam-6175	209	19	,	,	PUNCT
ejpam-6175	209	20	β1	β1	PROPN
ejpam-6175	209	21	(	(	PUNCT
ejpam-6175	209	22	number	number	NOUN
ejpam-6175	209	23	of	of	ADP
ejpam-6175	209	24	1dimensional	1dimensional	NUM
ejpam-6175	209	25	holes	hole	NOUN
ejpam-6175	209	26	or	or	CCONJ
ejpam-6175	209	27	loops	loop	NOUN
ejpam-6175	209	28	)	)	PUNCT
ejpam-6175	209	29	is	be	AUX
ejpam-6175	209	30	0	0	NUM
ejpam-6175	209	31	,	,	PUNCT
ejpam-6175	209	32	and	and	CCONJ
ejpam-6175	209	33	β2	β2	NOUN
ejpam-6175	209	34	(	(	PUNCT
ejpam-6175	209	35	number	number	NOUN
ejpam-6175	209	36	of	of	ADP
ejpam-6175	209	37	2	2	NUM
ejpam-6175	209	38	-	-	PUNCT
ejpam-6175	209	39	dimensional	dimensional	ADJ
ejpam-6175	209	40	holes	hole	NOUN
ejpam-6175	209	41	or	or	CCONJ
ejpam-6175	209	42	voids	void	NOUN
ejpam-6175	209	43	)	)	PUNCT
ejpam-6175	209	44	is	be	AUX
ejpam-6175	209	45	1	1	NUM
ejpam-6175	209	46	.	.	NOUN
ejpam-6175	209	47	h18	h18	PROPN
ejpam-6175	209	48	q	q	PROPN
ejpam-6175	209	49	(	(	PUNCT
ejpam-6175	209	50	∆i	∆i	PROPN
ejpam-6175	209	51	)	)	PUNCT
ejpam-6175	210	1	=	=	PUNCT
ejpam-6175	211	1			PROPN
ejpam-6175	211	2	z	z	NOUN
ejpam-6175	211	3	,	,	PUNCT
ejpam-6175	211	4	for	for	ADP
ejpam-6175	211	5	q	q	NOUN
ejpam-6175	211	6	=	=	SYM
ejpam-6175	211	7	0	0	NUM
ejpam-6175	211	8	and	and	CCONJ
ejpam-6175	211	9	0	0	NUM
ejpam-6175	211	10	<	<	X
ejpam-6175	212	1	i	i	X
ejpam-6175	212	2	≤	≤	ADV
ejpam-6175	212	3	8	8	NUM
ejpam-6175	212	4	0	0	NUM
ejpam-6175	212	5	,	,	PUNCT
ejpam-6175	212	6	for	for	ADP
ejpam-6175	212	7	q	q	PROPN
ejpam-6175	212	8	∈	∈	PROPN
ejpam-6175	212	9	z	z	PROPN
ejpam-6175	212	10	and	and	CCONJ
ejpam-6175	212	11	i	i	PRON
ejpam-6175	212	12	=	=	NOUN
ejpam-6175	212	13	0	0	PUNCT
ejpam-6175	213	1	z	z	NOUN
ejpam-6175	213	2	,	,	PUNCT
ejpam-6175	213	3	q	q	X
ejpam-6175	213	4	=	=	SYM
ejpam-6175	213	5	2	2	NUM
ejpam-6175	213	6	and	and	CCONJ
ejpam-6175	213	7	i	i	PRON
ejpam-6175	213	8	=	=	NOUN
ejpam-6175	213	9	8	8	X
ejpam-6175	213	10	.	.	PUNCT
ejpam-6175	214	1	s.k	s.k	PROPN
ejpam-6175	214	2	.	.	PUNCT
ejpam-6175	214	3	jha	jha	PROPN
ejpam-6175	214	4	sanjay	sanjay	PROPN
ejpam-6175	214	5	,	,	PUNCT
ejpam-6175	214	6	m	m	PROPN
ejpam-6175	214	7	,	,	PUNCT
ejpam-6175	214	8	natarajan	natarajan	PROPN
ejpam-6175	214	9	/	/	SYM
ejpam-6175	214	10	eur	eur	PROPN
ejpam-6175	214	11	.	.	PUNCT
ejpam-6175	215	1	j.	j.	PROPN
ejpam-6175	215	2	pure	pure	PROPN
ejpam-6175	215	3	appl	appl	PROPN
ejpam-6175	215	4	.	.	PROPN
ejpam-6175	215	5	math	math	PROPN
ejpam-6175	215	6	,	,	PUNCT
ejpam-6175	215	7	18	18	NUM
ejpam-6175	215	8	(	(	PUNCT
ejpam-6175	215	9	3	3	NUM
ejpam-6175	215	10	)	)	PUNCT
ejpam-6175	215	11	(	(	PUNCT
ejpam-6175	215	12	2025	2025	NUM
ejpam-6175	215	13	)	)	PUNCT
ejpam-6175	215	14	,	,	PUNCT
ejpam-6175	215	15	6175	6175	NUM
ejpam-6175	215	16	9	9	NUM
ejpam-6175	215	17	of	of	ADP
ejpam-6175	215	18	18	18	NUM
ejpam-6175	215	19	figure	figure	NOUN
ejpam-6175	215	20	2	2	NUM
ejpam-6175	215	21	:	:	PUNCT
ejpam-6175	215	22	from	from	ADP
ejpam-6175	215	23	smooth	smooth	ADJ
ejpam-6175	215	24	surface	surface	NOUN
ejpam-6175	215	25	to	to	ADP
ejpam-6175	215	26	simplified	simplified	ADJ
ejpam-6175	215	27	geometry	geometry	NOUN
ejpam-6175	215	28	:	:	PUNCT
ejpam-6175	215	29	transformation	transformation	NOUN
ejpam-6175	215	30	stages	stage	NOUN
ejpam-6175	215	31	of	of	ADP
ejpam-6175	215	32	a	a	DET
ejpam-6175	215	33	sphere	sphere	NOUN
ejpam-6175	215	34	using	use	VERB
ejpam-6175	215	35	the	the	DET
ejpam-6175	215	36	homology	homology	NOUN
ejpam-6175	215	37	groups	group	NOUN
ejpam-6175	215	38	h18	h18	VERB
ejpam-6175	215	39	q	q	PROPN
ejpam-6175	215	40	(	(	PUNCT
ejpam-6175	215	41	∆i	∆i	PROPN
ejpam-6175	215	42	)	)	PUNCT
ejpam-6175	215	43	,	,	PUNCT
ejpam-6175	215	44	we	we	PRON
ejpam-6175	215	45	compute	compute	VERB
ejpam-6175	215	46	the	the	DET
ejpam-6175	215	47	persistent	persistent	ADJ
ejpam-6175	215	48	q	q	ADJ
ejpam-6175	215	49	-	-	PUNCT
ejpam-6175	215	50	th	th	VERB
ejpam-6175	215	51	betti	betti	NOUN
ejpam-6175	215	52	numbers	number	NOUN
ejpam-6175	215	53	as	as	SCONJ
ejpam-6175	215	54	follows	follow	VERB
ejpam-6175	215	55	:	:	PUNCT
ejpam-6175	215	56	•	•	NOUN
ejpam-6175	215	57	for	for	ADP
ejpam-6175	215	58	q	q	NOUN
ejpam-6175	215	59	=	=	SYM
ejpam-6175	215	60	0	0	NUM
ejpam-6175	215	61	and	and	CCONJ
ejpam-6175	215	62	0	0	NUM
ejpam-6175	215	63	<	<	X
ejpam-6175	215	64	i	i	X
ejpam-6175	215	65	≤	≤	ADV
ejpam-6175	215	66	8	8	NUM
ejpam-6175	215	67	,	,	PUNCT
ejpam-6175	215	68	we	we	PRON
ejpam-6175	215	69	have	have	VERB
ejpam-6175	215	70	βi+p	βi+p	PROPN
ejpam-6175	215	71	,	,	PUNCT
ejpam-6175	215	72	κ	κ	PROPN
ejpam-6175	215	73	0	0	NUM
ejpam-6175	215	74	(	(	PUNCT
ejpam-6175	215	75	∆i	∆i	PROPN
ejpam-6175	215	76	)	)	PUNCT
ejpam-6175	216	1	=	=	PUNCT
ejpam-6175	217	1	rank(h	rank(h	NOUN
ejpam-6175	217	2	i	i	PRON
ejpam-6175	217	3	0(∆i	0(∆i	PROPN
ejpam-6175	217	4	)	)	PUNCT
ejpam-6175	217	5	)	)	PUNCT
ejpam-6175	218	1	=	=	SYM
ejpam-6175	218	2	rank(z	rank(z	X
ejpam-6175	218	3	)	)	PUNCT
ejpam-6175	218	4	=	=	SYM
ejpam-6175	219	1	1	1	X
ejpam-6175	219	2	.	.	PUNCT
ejpam-6175	219	3	this	this	PRON
ejpam-6175	219	4	indicates	indicate	VERB
ejpam-6175	219	5	the	the	DET
ejpam-6175	219	6	presence	presence	NOUN
ejpam-6175	219	7	of	of	ADP
ejpam-6175	219	8	a	a	DET
ejpam-6175	219	9	single	single	ADJ
ejpam-6175	219	10	connected	connected	ADJ
ejpam-6175	219	11	component	component	NOUN
ejpam-6175	219	12	for	for	ADP
ejpam-6175	219	13	0	0	NUM
ejpam-6175	219	14	<	<	X
ejpam-6175	219	15	i	i	X
ejpam-6175	219	16	≤	≤	ADV
ejpam-6175	219	17	8	8	NUM
ejpam-6175	219	18	.	.	NOUN
ejpam-6175	219	19	•	•	NOUN
ejpam-6175	219	20	for	for	ADP
ejpam-6175	219	21	q	q	NOUN
ejpam-6175	219	22	=	=	SYM
ejpam-6175	219	23	1	1	NUM
ejpam-6175	219	24	,	,	PUNCT
ejpam-6175	219	25	we	we	PRON
ejpam-6175	219	26	have	have	VERB
ejpam-6175	219	27	h18	h18	NOUN
ejpam-6175	219	28	1	1	NUM
ejpam-6175	219	29	(	(	PUNCT
ejpam-6175	219	30	∆i	∆i	PROPN
ejpam-6175	219	31	)	)	PUNCT
ejpam-6175	220	1	=	=	SYM
ejpam-6175	220	2	0	0	NUM
ejpam-6175	220	3	for	for	ADP
ejpam-6175	220	4	all	all	DET
ejpam-6175	220	5	i	i	PROPN
ejpam-6175	220	6	,	,	PUNCT
ejpam-6175	220	7	implying	imply	VERB
ejpam-6175	220	8	that	that	SCONJ
ejpam-6175	220	9	there	there	PRON
ejpam-6175	220	10	are	be	VERB
ejpam-6175	220	11	no	no	DET
ejpam-6175	220	12	1	1	NUM
ejpam-6175	220	13	-	-	PUNCT
ejpam-6175	220	14	dimensional	dimensional	ADJ
ejpam-6175	220	15	holes	hole	NOUN
ejpam-6175	220	16	at	at	ADP
ejpam-6175	220	17	any	any	DET
ejpam-6175	220	18	filtration	filtration	NOUN
ejpam-6175	220	19	step	step	NOUN
ejpam-6175	220	20	.	.	PUNCT
ejpam-6175	221	1	•	•	NOUN
ejpam-6175	221	2	for	for	ADP
ejpam-6175	221	3	q	q	NOUN
ejpam-6175	221	4	=	=	SYM
ejpam-6175	221	5	2	2	NUM
ejpam-6175	221	6	and	and	CCONJ
ejpam-6175	221	7	i	i	PRON
ejpam-6175	221	8	=	=	NOUN
ejpam-6175	221	9	8	8	NUM
ejpam-6175	221	10	,	,	PUNCT
ejpam-6175	221	11	we	we	PRON
ejpam-6175	221	12	have	have	VERB
ejpam-6175	221	13	βi+p	βi+p	PROPN
ejpam-6175	221	14	,	,	PUNCT
ejpam-6175	221	15	κ	κ	PROPN
ejpam-6175	221	16	2	2	NUM
ejpam-6175	221	17	(	(	PUNCT
ejpam-6175	221	18	∆i	∆i	PROPN
ejpam-6175	221	19	)	)	PUNCT
ejpam-6175	222	1	=	=	PUNCT
ejpam-6175	223	1	rank(h	rank(h	VERB
ejpam-6175	223	2	i	i	PRON
ejpam-6175	223	3	2(∆i	2(∆i	NUM
ejpam-6175	223	4	)	)	PUNCT
ejpam-6175	223	5	)	)	PUNCT
ejpam-6175	224	1	=	=	SYM
ejpam-6175	224	2	rank(z	rank(z	X
ejpam-6175	224	3	)	)	PUNCT
ejpam-6175	224	4	=	=	SYM
ejpam-6175	224	5	1	1	NUM
ejpam-6175	224	6	,	,	PUNCT
ejpam-6175	224	7	indicating	indicate	VERB
ejpam-6175	224	8	the	the	DET
ejpam-6175	224	9	presence	presence	NOUN
ejpam-6175	224	10	of	of	ADP
ejpam-6175	224	11	a	a	DET
ejpam-6175	224	12	single	single	ADJ
ejpam-6175	224	13	2	2	NUM
ejpam-6175	224	14	-	-	PUNCT
ejpam-6175	224	15	dimensional	dimensional	ADJ
ejpam-6175	224	16	hole	hole	NOUN
ejpam-6175	224	17	at	at	ADP
ejpam-6175	224	18	i	i	NOUN
ejpam-6175	224	19	=	=	NOUN
ejpam-6175	224	20	8	8	X
ejpam-6175	224	21	.	.	PUNCT
ejpam-6175	225	1	consequently	consequently	ADV
ejpam-6175	225	2	,	,	PUNCT
ejpam-6175	225	3	the	the	DET
ejpam-6175	225	4	persistent	persistent	ADJ
ejpam-6175	225	5	betti	betti	NOUN
ejpam-6175	225	6	numbers	number	NOUN
ejpam-6175	225	7	are	be	AUX
ejpam-6175	225	8	:	:	PUNCT
ejpam-6175	225	9	•	•	ADP
ejpam-6175	225	10	for	for	ADP
ejpam-6175	225	11	q	q	NOUN
ejpam-6175	225	12	=	=	SYM
ejpam-6175	225	13	0	0	NUM
ejpam-6175	225	14	and	and	CCONJ
ejpam-6175	225	15	0	0	NUM
ejpam-6175	225	16	<	<	X
ejpam-6175	225	17	i	i	X
ejpam-6175	225	18	≤	≤	ADJ
ejpam-6175	225	19	8	8	NUM
ejpam-6175	225	20	:	:	PUNCT
ejpam-6175	225	21	βi+p	βi+p	NOUN
ejpam-6175	225	22	,	,	PUNCT
ejpam-6175	225	23	κ	κ	PROPN
ejpam-6175	225	24	0	0	NUM
ejpam-6175	225	25	(	(	PUNCT
ejpam-6175	225	26	∆i	∆i	PROPN
ejpam-6175	225	27	)	)	PUNCT
ejpam-6175	225	28	=	=	PUNCT
ejpam-6175	226	1	1	1	X
ejpam-6175	226	2	.	.	NOUN
ejpam-6175	226	3	•	•	NUM
ejpam-6175	226	4	for	for	ADP
ejpam-6175	226	5	q	q	PROPN
ejpam-6175	226	6	̸=	̸=	PROPN
ejpam-6175	226	7	0	0	PUNCT
ejpam-6175	227	1	and	and	CCONJ
ejpam-6175	227	2	i	i	PRON
ejpam-6175	227	3	=	=	NOUN
ejpam-6175	227	4	0	0	NUM
ejpam-6175	227	5	:	:	PUNCT
ejpam-6175	227	6	βi+p	βi+p	NOUN
ejpam-6175	227	7	,	,	PUNCT
ejpam-6175	227	8	κ	κ	PRON
ejpam-6175	227	9	q	q	X
ejpam-6175	227	10	(	(	PUNCT
ejpam-6175	227	11	∆i	∆i	PROPN
ejpam-6175	227	12	)	)	PUNCT
ejpam-6175	228	1	=	=	PUNCT
ejpam-6175	228	2	0	0	NUM
ejpam-6175	228	3	.	.	NOUN
ejpam-6175	228	4	3.2	3.2	NUM
ejpam-6175	228	5	.	.	PUNCT
ejpam-6175	229	1	case	case	NOUN
ejpam-6175	229	2	of	of	ADP
ejpam-6175	229	3	torus	torus	PROPN
ejpam-6175	229	4	t	t	PROPN
ejpam-6175	229	5	2	2	NUM
ejpam-6175	229	6	let	let	VERB
ejpam-6175	229	7	v	v	NOUN
ejpam-6175	229	8	=	=	SYM
ejpam-6175	229	9	{	{	PUNCT
ejpam-6175	229	10	v1	v1	NOUN
ejpam-6175	229	11	,	,	PUNCT
ejpam-6175	229	12	.	.	PUNCT
ejpam-6175	229	13	.	.	PUNCT
ejpam-6175	230	1	.	.	PUNCT
ejpam-6175	231	1	,	,	PUNCT
ejpam-6175	231	2	v9	v9	PROPN
ejpam-6175	231	3	}	}	PUNCT
ejpam-6175	231	4	,	,	PUNCT
ejpam-6175	231	5	e	e	X
ejpam-6175	231	6	=	=	PRON
ejpam-6175	231	7	{	{	PUNCT
ejpam-6175	231	8	e1	e1	PROPN
ejpam-6175	231	9	,	,	PUNCT
ejpam-6175	231	10	.	.	PUNCT
ejpam-6175	231	11	.	.	PUNCT
ejpam-6175	231	12	.	.	PUNCT
ejpam-6175	232	1	,	,	PUNCT
ejpam-6175	232	2	e18	e18	PROPN
ejpam-6175	232	3	}	}	PUNCT
ejpam-6175	232	4	,	,	PUNCT
ejpam-6175	232	5	and	and	CCONJ
ejpam-6175	232	6	f	f	X
ejpam-6175	232	7	=	=	NOUN
ejpam-6175	232	8	{	{	PUNCT
ejpam-6175	232	9	f1	f1	NOUN
ejpam-6175	232	10	,	,	PUNCT
ejpam-6175	232	11	.	.	PUNCT
ejpam-6175	232	12	.	.	PUNCT
ejpam-6175	233	1	.	.	PUNCT
ejpam-6175	234	1	,	,	PUNCT
ejpam-6175	234	2	f9	f9	PROPN
ejpam-6175	234	3	}	}	PUNCT
ejpam-6175	234	4	denote	denote	VERB
ejpam-6175	234	5	the	the	DET
ejpam-6175	234	6	sets	set	NOUN
ejpam-6175	234	7	of	of	ADP
ejpam-6175	234	8	vertices	vertex	NOUN
ejpam-6175	234	9	,	,	PUNCT
ejpam-6175	234	10	edges	edge	NOUN
ejpam-6175	234	11	,	,	PUNCT
ejpam-6175	234	12	and	and	CCONJ
ejpam-6175	234	13	faces	face	NOUN
ejpam-6175	234	14	,	,	PUNCT
ejpam-6175	234	15	respectively	respectively	ADV
ejpam-6175	234	16	.	.	PUNCT
ejpam-6175	235	1	the	the	DET
ejpam-6175	235	2	genus	genus	ADJ
ejpam-6175	235	3	g	g	NOUN
ejpam-6175	235	4	of	of	ADP
ejpam-6175	235	5	the	the	DET
ejpam-6175	235	6	surface	surface	NOUN
ejpam-6175	235	7	is	be	AUX
ejpam-6175	235	8	derived	derive	VERB
ejpam-6175	235	9	from	from	ADP
ejpam-6175	235	10	the	the	DET
ejpam-6175	235	11	euler	euler	PROPN
ejpam-6175	235	12	characteristic	characteristic	PROPN
ejpam-6175	235	13	χ	χ	PROPN
ejpam-6175	235	14	,	,	PUNCT
ejpam-6175	235	15	given	give	VERB
ejpam-6175	235	16	by	by	ADP
ejpam-6175	235	17	χ	χ	X
ejpam-6175	235	18	=	=	SYM
ejpam-6175	235	19	v	v	ADP
ejpam-6175	235	20	−	−	PROPN
ejpam-6175	235	21	e	e	PROPN
ejpam-6175	235	22	+	+	PROPN
ejpam-6175	235	23	f.	f.	PROPN
ejpam-6175	235	24	for	for	ADP
ejpam-6175	235	25	a	a	DET
ejpam-6175	235	26	genus	genus	NOUN
ejpam-6175	235	27	g	g	PROPN
ejpam-6175	235	28	surface	surface	NOUN
ejpam-6175	235	29	,	,	PUNCT
ejpam-6175	235	30	χ	χ	X
ejpam-6175	235	31	=	=	SYM
ejpam-6175	235	32	2−	2−	NUM
ejpam-6175	235	33	2	2	NUM
ejpam-6175	235	34	g	g	NOUN
ejpam-6175	235	35	,	,	PUNCT
ejpam-6175	235	36	so	so	SCONJ
ejpam-6175	235	37	g	g	PROPN
ejpam-6175	235	38	=	=	PROPN
ejpam-6175	235	39	2−	2−	NUM
ejpam-6175	235	40	χ	χ	DET
ejpam-6175	235	41	2	2	NUM
ejpam-6175	235	42	.	.	PUNCT
ejpam-6175	236	1	here	here	ADV
ejpam-6175	236	2	,	,	PUNCT
ejpam-6175	236	3	χ	χ	X
ejpam-6175	236	4	=	=	SYM
ejpam-6175	236	5	9−	9−	NUM
ejpam-6175	236	6	18	18	NUM
ejpam-6175	236	7	+	+	CCONJ
ejpam-6175	236	8	9	9	NUM
ejpam-6175	236	9	=	=	SYM
ejpam-6175	236	10	0	0	NUM
ejpam-6175	236	11	,	,	PUNCT
ejpam-6175	236	12	giving	give	VERB
ejpam-6175	236	13	g	g	NOUN
ejpam-6175	236	14	=	=	SYM
ejpam-6175	236	15	1	1	NUM
ejpam-6175	236	16	,	,	PUNCT
ejpam-6175	236	17	the	the	DET
ejpam-6175	236	18	genus	genus	NOUN
ejpam-6175	236	19	of	of	ADP
ejpam-6175	236	20	a	a	DET
ejpam-6175	236	21	torus	torus	NOUN
ejpam-6175	236	22	.	.	PUNCT
ejpam-6175	237	1	s.k	s.k	PROPN
ejpam-6175	237	2	.	.	PUNCT
ejpam-6175	237	3	jha	jha	PROPN
ejpam-6175	237	4	sanjay	sanjay	PROPN
ejpam-6175	237	5	,	,	PUNCT
ejpam-6175	237	6	m	m	PROPN
ejpam-6175	237	7	,	,	PUNCT
ejpam-6175	237	8	natarajan	natarajan	PROPN
ejpam-6175	237	9	/	/	SYM
ejpam-6175	237	10	eur	eur	PROPN
ejpam-6175	237	11	.	.	PUNCT
ejpam-6175	238	1	j.	j.	PROPN
ejpam-6175	238	2	pure	pure	PROPN
ejpam-6175	238	3	appl	appl	PROPN
ejpam-6175	238	4	.	.	PROPN
ejpam-6175	238	5	math	math	PROPN
ejpam-6175	238	6	,	,	PUNCT
ejpam-6175	238	7	18	18	NUM
ejpam-6175	238	8	(	(	PUNCT
ejpam-6175	238	9	3	3	NUM
ejpam-6175	238	10	)	)	PUNCT
ejpam-6175	238	11	(	(	PUNCT
ejpam-6175	238	12	2025	2025	NUM
ejpam-6175	238	13	)	)	PUNCT
ejpam-6175	238	14	,	,	PUNCT
ejpam-6175	238	15	6175	6175	NUM
ejpam-6175	238	16	10	10	NUM
ejpam-6175	238	17	of	of	ADP
ejpam-6175	238	18	18	18	NUM
ejpam-6175	238	19	the	the	DET
ejpam-6175	238	20	filtration	filtration	NOUN
ejpam-6175	238	21	of	of	ADP
ejpam-6175	238	22	a	a	DET
ejpam-6175	238	23	torus	torus	NOUN
ejpam-6175	238	24	is	be	AUX
ejpam-6175	238	25	a	a	DET
ejpam-6175	238	26	sequence	sequence	NOUN
ejpam-6175	238	27	of	of	ADP
ejpam-6175	238	28	subcomplexes	subcomplexe	NOUN
ejpam-6175	238	29	,	,	PUNCT
ejpam-6175	238	30	each	each	PRON
ejpam-6175	238	31	successively	successively	ADV
ejpam-6175	238	32	adding	add	VERB
ejpam-6175	238	33	vertices	vertex	NOUN
ejpam-6175	238	34	vi	vi	PROPN
ejpam-6175	238	35	,	,	PUNCT
ejpam-6175	238	36	edges	edge	VERB
ejpam-6175	238	37	ei	ei	NOUN
ejpam-6175	238	38	,	,	PUNCT
ejpam-6175	238	39	and	and	CCONJ
ejpam-6175	238	40	faces	face	VERB
ejpam-6175	238	41	fi	fi	NOUN
ejpam-6175	238	42	,	,	PUNCT
ejpam-6175	238	43	such	such	ADJ
ejpam-6175	238	44	that	that	DET
ejpam-6175	238	45	∅	∅	NOUN
ejpam-6175	238	46	=	=	PUNCT
ejpam-6175	238	47	∆0	∆0	NOUN
ejpam-6175	239	1	⊆	⊆	NUM
ejpam-6175	239	2	∆1	∆1	NUM
ejpam-6175	239	3	⊆	⊆	NUM
ejpam-6175	239	4	∆2	∆2	X
ejpam-6175	239	5	⊆	⊆	NUM
ejpam-6175	239	6	·	·	PUNCT
ejpam-6175	239	7	·	·	PUNCT
ejpam-6175	239	8	·	·	PUNCT
ejpam-6175	239	9	⊆	⊆	X
ejpam-6175	239	10	∆9	∆9	NUM
ejpam-6175	239	11	=	=	SYM
ejpam-6175	239	12	t	t	PROPN
ejpam-6175	239	13	2	2	NUM
ejpam-6175	239	14	.	.	PUNCT
ejpam-6175	239	15	in	in	ADP
ejpam-6175	239	16	the	the	DET
ejpam-6175	239	17	filtration	filtration	NOUN
ejpam-6175	239	18	process	process	NOUN
ejpam-6175	239	19	of	of	ADP
ejpam-6175	239	20	the	the	DET
ejpam-6175	239	21	torus	torus	NOUN
ejpam-6175	239	22	,	,	PUNCT
ejpam-6175	239	23	we	we	PRON
ejpam-6175	239	24	start	start	VERB
ejpam-6175	239	25	with	with	ADP
ejpam-6175	239	26	an	an	DET
ejpam-6175	239	27	empty	empty	ADJ
ejpam-6175	239	28	set	set	NOUN
ejpam-6175	239	29	∆0	∆0	NOUN
ejpam-6175	239	30	=	=	PUNCT
ejpam-6175	239	31	∅.	∅.	NOUN
ejpam-6175	239	32	as	as	SCONJ
ejpam-6175	239	33	we	we	PRON
ejpam-6175	239	34	proceed	proceed	VERB
ejpam-6175	239	35	to	to	ADP
ejpam-6175	239	36	∆1	∆1	VERB
ejpam-6175	239	37	,	,	PUNCT
ejpam-6175	239	38	we	we	PRON
ejpam-6175	239	39	introduce	introduce	VERB
ejpam-6175	239	40	the	the	DET
ejpam-6175	239	41	first	first	ADJ
ejpam-6175	239	42	vertex	vertex	NOUN
ejpam-6175	239	43	,	,	PUNCT
ejpam-6175	239	44	{	{	PUNCT
ejpam-6175	239	45	v1	v1	NOUN
ejpam-6175	239	46	}	}	PUNCT
ejpam-6175	239	47	.	.	PUNCT
ejpam-6175	240	1	moving	move	VERB
ejpam-6175	240	2	to	to	ADP
ejpam-6175	240	3	∆2	∆2	PRON
ejpam-6175	240	4	,	,	PUNCT
ejpam-6175	240	5	we	we	PRON
ejpam-6175	240	6	add	add	VERB
ejpam-6175	240	7	another	another	DET
ejpam-6175	240	8	vertex	vertex	NOUN
ejpam-6175	240	9	and	and	CCONJ
ejpam-6175	240	10	an	an	DET
ejpam-6175	240	11	edge	edge	NOUN
ejpam-6175	240	12	,	,	PUNCT
ejpam-6175	240	13	forming	form	VERB
ejpam-6175	240	14	{	{	PUNCT
ejpam-6175	240	15	v1	v1	NOUN
ejpam-6175	240	16	,	,	PUNCT
ejpam-6175	240	17	v2	v2	NOUN
ejpam-6175	240	18	}	}	PUNCT
ejpam-6175	240	19	∪	∪	NOUN
ejpam-6175	240	20	{	{	PUNCT
ejpam-6175	240	21	e1	e1	NOUN
ejpam-6175	240	22	}	}	PUNCT
ejpam-6175	240	23	.	.	PUNCT
ejpam-6175	241	1	at	at	ADP
ejpam-6175	241	2	∆3	∆3	NOUN
ejpam-6175	241	3	,	,	PUNCT
ejpam-6175	241	4	the	the	DET
ejpam-6175	241	5	set	set	NOUN
ejpam-6175	241	6	expands	expand	VERB
ejpam-6175	241	7	to	to	PART
ejpam-6175	241	8	include	include	VERB
ejpam-6175	241	9	three	three	NUM
ejpam-6175	241	10	vertices	vertex	NOUN
ejpam-6175	241	11	and	and	CCONJ
ejpam-6175	241	12	three	three	NUM
ejpam-6175	241	13	edges	edge	NOUN
ejpam-6175	241	14	,	,	PUNCT
ejpam-6175	241	15	{	{	PUNCT
ejpam-6175	241	16	v1	v1	NOUN
ejpam-6175	241	17	,	,	PUNCT
ejpam-6175	241	18	v2	v2	PROPN
ejpam-6175	241	19	,	,	PUNCT
ejpam-6175	241	20	v3	v3	PROPN
ejpam-6175	241	21	}	}	PUNCT
ejpam-6175	241	22	∪	∪	NOUN
ejpam-6175	241	23	{	{	PUNCT
ejpam-6175	241	24	e1	e1	PROPN
ejpam-6175	241	25	,	,	PUNCT
ejpam-6175	241	26	e2	e2	PROPN
ejpam-6175	241	27	,	,	PUNCT
ejpam-6175	241	28	e3	e3	NOUN
ejpam-6175	241	29	}	}	PUNCT
ejpam-6175	241	30	.	.	PUNCT
ejpam-6175	242	1	continuing	continue	VERB
ejpam-6175	242	2	this	this	DET
ejpam-6175	242	3	progression	progression	NOUN
ejpam-6175	242	4	,	,	PUNCT
ejpam-6175	242	5	∆4	∆4	NOUN
ejpam-6175	242	6	comprises	comprise	VERB
ejpam-6175	242	7	four	four	NUM
ejpam-6175	242	8	vertices	vertex	NOUN
ejpam-6175	242	9	and	and	CCONJ
ejpam-6175	242	10	five	five	NUM
ejpam-6175	242	11	edges	edge	NOUN
ejpam-6175	242	12	,	,	PUNCT
ejpam-6175	242	13	{	{	PUNCT
ejpam-6175	242	14	v1	v1	NOUN
ejpam-6175	242	15	,	,	PUNCT
ejpam-6175	242	16	·	·	PUNCT
ejpam-6175	242	17	·	·	PUNCT
ejpam-6175	242	18	·	·	PUNCT
ejpam-6175	242	19	,	,	PUNCT
ejpam-6175	242	20	v4	v4	PROPN
ejpam-6175	242	21	}	}	PUNCT
ejpam-6175	242	22	∪	∪	NOUN
ejpam-6175	242	23	{	{	PUNCT
ejpam-6175	242	24	e1	e1	NOUN
ejpam-6175	242	25	,	,	PUNCT
ejpam-6175	242	26	·	·	PUNCT
ejpam-6175	242	27	·	·	PUNCT
ejpam-6175	242	28	·	·	PUNCT
ejpam-6175	242	29	,	,	PUNCT
ejpam-6175	242	30	e5	e5	PROPN
ejpam-6175	242	31	}	}	PUNCT
ejpam-6175	242	32	.	.	PUNCT
ejpam-6175	243	1	in	in	ADP
ejpam-6175	243	2	∆5	∆5	PROPN
ejpam-6175	243	3	,	,	PUNCT
ejpam-6175	243	4	we	we	PRON
ejpam-6175	243	5	incorporate	incorporate	VERB
ejpam-6175	243	6	five	five	NUM
ejpam-6175	243	7	vertices	vertex	NOUN
ejpam-6175	243	8	and	and	CCONJ
ejpam-6175	243	9	eight	eight	NUM
ejpam-6175	243	10	edges	edge	NOUN
ejpam-6175	243	11	,	,	PUNCT
ejpam-6175	243	12	{	{	PUNCT
ejpam-6175	243	13	v1	v1	NOUN
ejpam-6175	243	14	,	,	PUNCT
ejpam-6175	243	15	·	·	PUNCT
ejpam-6175	243	16	·	·	PUNCT
ejpam-6175	243	17	·	·	PUNCT
ejpam-6175	243	18	,	,	PUNCT
ejpam-6175	243	19	v5	v5	PROPN
ejpam-6175	243	20	}	}	PUNCT
ejpam-6175	243	21	∪	∪	ADJ
ejpam-6175	243	22	{	{	PUNCT
ejpam-6175	243	23	e1	e1	NOUN
ejpam-6175	243	24	,	,	PUNCT
ejpam-6175	243	25	·	·	PUNCT
ejpam-6175	243	26	·	·	PUNCT
ejpam-6175	243	27	·	·	PUNCT
ejpam-6175	243	28	,	,	PUNCT
ejpam-6175	243	29	e8	e8	PROPN
ejpam-6175	243	30	}	}	PUNCT
ejpam-6175	243	31	.	.	PUNCT
ejpam-6175	244	1	the	the	DET
ejpam-6175	244	2	complexity	complexity	NOUN
ejpam-6175	244	3	increases	increase	VERB
ejpam-6175	244	4	significantly	significantly	ADV
ejpam-6175	244	5	at	at	ADP
ejpam-6175	244	6	∆6	∆6	NOUN
ejpam-6175	244	7	with	with	ADP
ejpam-6175	244	8	six	six	NUM
ejpam-6175	244	9	vertices	vertex	NOUN
ejpam-6175	244	10	,	,	PUNCT
ejpam-6175	244	11	twelve	twelve	NUM
ejpam-6175	244	12	edges	edge	NOUN
ejpam-6175	244	13	,	,	PUNCT
ejpam-6175	244	14	and	and	CCONJ
ejpam-6175	244	15	the	the	DET
ejpam-6175	244	16	introduction	introduction	NOUN
ejpam-6175	244	17	of	of	ADP
ejpam-6175	244	18	faces	face	NOUN
ejpam-6175	244	19	,	,	PUNCT
ejpam-6175	244	20	{	{	PUNCT
ejpam-6175	244	21	v1	v1	NOUN
ejpam-6175	244	22	,	,	PUNCT
ejpam-6175	244	23	·	·	PUNCT
ejpam-6175	244	24	·	·	PUNCT
ejpam-6175	244	25	·	·	PUNCT
ejpam-6175	244	26	,	,	PUNCT
ejpam-6175	244	27	v6	v6	NOUN
ejpam-6175	244	28	}	}	PUNCT
ejpam-6175	244	29	∪	∪	NOUN
ejpam-6175	244	30	{	{	PUNCT
ejpam-6175	244	31	e1	e1	NOUN
ejpam-6175	244	32	,	,	PUNCT
ejpam-6175	244	33	·	·	PUNCT
ejpam-6175	244	34	·	·	PUNCT
ejpam-6175	244	35	·	·	PUNCT
ejpam-6175	244	36	,	,	PUNCT
ejpam-6175	244	37	e12	e12	NOUN
ejpam-6175	244	38	}	}	PUNCT
ejpam-6175	244	39	∪	∪	NOUN
ejpam-6175	244	40	{	{	PUNCT
ejpam-6175	244	41	f1	f1	NOUN
ejpam-6175	244	42	,	,	PUNCT
ejpam-6175	244	43	f2	f2	PROPN
ejpam-6175	244	44	,	,	PUNCT
ejpam-6175	244	45	f3	f3	PROPN
ejpam-6175	244	46	}	}	PUNCT
ejpam-6175	244	47	.	.	PUNCT
ejpam-6175	245	1	this	this	DET
ejpam-6175	245	2	pattern	pattern	NOUN
ejpam-6175	245	3	continues	continue	VERB
ejpam-6175	245	4	as	as	SCONJ
ejpam-6175	245	5	we	we	PRON
ejpam-6175	245	6	add	add	VERB
ejpam-6175	245	7	more	more	ADJ
ejpam-6175	245	8	elements	element	NOUN
ejpam-6175	245	9	;	;	PUNCT
ejpam-6175	245	10	∆7	∆7	PROPN
ejpam-6175	245	11	includes	include	VERB
ejpam-6175	245	12	seven	seven	NUM
ejpam-6175	245	13	vertices	vertex	NOUN
ejpam-6175	245	14	,	,	PUNCT
ejpam-6175	245	15	fourteen	fourteen	NUM
ejpam-6175	245	16	edges	edge	NOUN
ejpam-6175	245	17	,	,	PUNCT
ejpam-6175	245	18	and	and	CCONJ
ejpam-6175	245	19	four	four	NUM
ejpam-6175	245	20	faces	face	NOUN
ejpam-6175	245	21	,	,	PUNCT
ejpam-6175	245	22	{	{	PUNCT
ejpam-6175	245	23	v1	v1	NOUN
ejpam-6175	245	24	,	,	PUNCT
ejpam-6175	245	25	·	·	PUNCT
ejpam-6175	245	26	·	·	PUNCT
ejpam-6175	245	27	·	·	PUNCT
ejpam-6175	245	28	,	,	PUNCT
ejpam-6175	245	29	v7}∪{e1	v7}∪{e1	PROPN
ejpam-6175	245	30	,	,	PUNCT
ejpam-6175	245	31	·	·	PUNCT
ejpam-6175	245	32	·	·	PUNCT
ejpam-6175	245	33	·	·	PUNCT
ejpam-6175	245	34	,	,	PUNCT
ejpam-6175	245	35	e14}∪{f1	e14}∪{f1	PROPN
ejpam-6175	245	36	,	,	PUNCT
ejpam-6175	245	37	·	·	PUNCT
ejpam-6175	245	38	·	·	PUNCT
ejpam-6175	245	39	·	·	PUNCT
ejpam-6175	245	40	,	,	PUNCT
ejpam-6175	245	41	f4	f4	PROPN
ejpam-6175	245	42	}	}	PUNCT
ejpam-6175	245	43	.	.	PUNCT
ejpam-6175	246	1	by	by	ADP
ejpam-6175	246	2	∆8	∆8	NOUN
ejpam-6175	246	3	,	,	PUNCT
ejpam-6175	246	4	we	we	PRON
ejpam-6175	246	5	have	have	VERB
ejpam-6175	246	6	eight	eight	NUM
ejpam-6175	246	7	vertices	vertex	NOUN
ejpam-6175	246	8	,	,	PUNCT
ejpam-6175	246	9	sixteen	sixteen	NUM
ejpam-6175	246	10	edges	edge	NOUN
ejpam-6175	246	11	,	,	PUNCT
ejpam-6175	246	12	and	and	CCONJ
ejpam-6175	246	13	five	five	NUM
ejpam-6175	246	14	faces	face	NOUN
ejpam-6175	246	15	,	,	PUNCT
ejpam-6175	246	16	{	{	PUNCT
ejpam-6175	246	17	v1	v1	NOUN
ejpam-6175	246	18	,	,	PUNCT
ejpam-6175	246	19	·	·	PUNCT
ejpam-6175	246	20	·	·	PUNCT
ejpam-6175	246	21	·	·	PUNCT
ejpam-6175	246	22	,	,	PUNCT
ejpam-6175	246	23	v8	v8	PROPN
ejpam-6175	246	24	}	}	PUNCT
ejpam-6175	246	25	∪	∪	NOUN
ejpam-6175	246	26	{	{	PUNCT
ejpam-6175	246	27	e1	e1	NOUN
ejpam-6175	246	28	,	,	PUNCT
ejpam-6175	246	29	·	·	PUNCT
ejpam-6175	246	30	·	·	PUNCT
ejpam-6175	246	31	·	·	PUNCT
ejpam-6175	246	32	,	,	PUNCT
ejpam-6175	246	33	e16	e16	PROPN
ejpam-6175	246	34	}	}	PUNCT
ejpam-6175	246	35	∪	∪	NOUN
ejpam-6175	246	36	{	{	PUNCT
ejpam-6175	246	37	f1	f1	NOUN
ejpam-6175	246	38	,	,	PUNCT
ejpam-6175	246	39	·	·	PUNCT
ejpam-6175	246	40	·	·	PUNCT
ejpam-6175	246	41	·	·	PUNCT
ejpam-6175	246	42	,	,	PUNCT
ejpam-6175	246	43	f5	f5	PROPN
ejpam-6175	246	44	}	}	PUNCT
ejpam-6175	246	45	.	.	PUNCT
ejpam-6175	247	1	finally	finally	ADV
ejpam-6175	247	2	,	,	PUNCT
ejpam-6175	247	3	∆9	∆9	NUM
ejpam-6175	247	4	completes	complete	VERB
ejpam-6175	247	5	the	the	DET
ejpam-6175	247	6	structure	structure	NOUN
ejpam-6175	247	7	with	with	ADP
ejpam-6175	247	8	nine	nine	NUM
ejpam-6175	247	9	vertices	vertex	NOUN
ejpam-6175	247	10	,	,	PUNCT
ejpam-6175	247	11	eighteen	eighteen	NUM
ejpam-6175	247	12	edges	edge	NOUN
ejpam-6175	247	13	,	,	PUNCT
ejpam-6175	247	14	and	and	CCONJ
ejpam-6175	247	15	nine	nine	NUM
ejpam-6175	247	16	faces	face	NOUN
ejpam-6175	247	17	,	,	PUNCT
ejpam-6175	247	18	{	{	PUNCT
ejpam-6175	247	19	v1	v1	NOUN
ejpam-6175	247	20	,	,	PUNCT
ejpam-6175	247	21	·	·	PUNCT
ejpam-6175	247	22	·	·	PUNCT
ejpam-6175	247	23	·	·	PUNCT
ejpam-6175	247	24	,	,	PUNCT
ejpam-6175	247	25	v9}∪{e1	v9}∪{e1	PROPN
ejpam-6175	247	26	,	,	PUNCT
ejpam-6175	247	27	·	·	PUNCT
ejpam-6175	247	28	·	·	PUNCT
ejpam-6175	247	29	·	·	PUNCT
ejpam-6175	247	30	,	,	PUNCT
ejpam-6175	247	31	e18}∪{f1	e18}∪{f1	PROPN
ejpam-6175	247	32	,	,	PUNCT
ejpam-6175	247	33	·	·	PUNCT
ejpam-6175	247	34	·	·	PUNCT
ejpam-6175	247	35	·	·	PUNCT
ejpam-6175	247	36	,	,	PUNCT
ejpam-6175	247	37	f9	f9	PROPN
ejpam-6175	247	38	}	}	PUNCT
ejpam-6175	247	39	.	.	PUNCT
ejpam-6175	248	1	with	with	ADP
ejpam-6175	248	2	each	each	DET
ejpam-6175	248	3	increment	increment	NOUN
ejpam-6175	248	4	in	in	ADP
ejpam-6175	248	5	i	i	PRON
ejpam-6175	248	6	,	,	PUNCT
ejpam-6175	248	7	additional	additional	ADJ
ejpam-6175	248	8	vertices	vertex	NOUN
ejpam-6175	248	9	and	and	CCONJ
ejpam-6175	248	10	edges	edge	NOUN
ejpam-6175	248	11	are	be	AUX
ejpam-6175	248	12	incorporated	incorporate	VERB
ejpam-6175	248	13	,	,	PUNCT
ejpam-6175	248	14	gradually	gradually	ADV
ejpam-6175	248	15	forming	form	VERB
ejpam-6175	248	16	loops	loop	NOUN
ejpam-6175	248	17	and	and	CCONJ
ejpam-6175	248	18	eventually	eventually	ADV
ejpam-6175	248	19	utilizing	utilize	VERB
ejpam-6175	248	20	faces	face	NOUN
ejpam-6175	248	21	to	to	PART
ejpam-6175	248	22	accurately	accurately	ADV
ejpam-6175	248	23	construct	construct	VERB
ejpam-6175	248	24	the	the	DET
ejpam-6175	248	25	complete	complete	ADJ
ejpam-6175	248	26	structure	structure	NOUN
ejpam-6175	248	27	,	,	PUNCT
ejpam-6175	248	28	the	the	DET
ejpam-6175	248	29	betti	betti	NOUN
ejpam-6175	248	30	numbers	number	NOUN
ejpam-6175	248	31	are	be	AUX
ejpam-6175	248	32	topological	topological	ADJ
ejpam-6175	248	33	invariants	invariant	NOUN
ejpam-6175	248	34	that	that	PRON
ejpam-6175	248	35	count	count	VERB
ejpam-6175	248	36	the	the	DET
ejpam-6175	248	37	number	number	NOUN
ejpam-6175	248	38	of	of	ADP
ejpam-6175	248	39	independent	independent	ADJ
ejpam-6175	248	40	cycles	cycle	NOUN
ejpam-6175	248	41	of	of	ADP
ejpam-6175	248	42	different	different	ADJ
ejpam-6175	248	43	dimensions	dimension	NOUN
ejpam-6175	248	44	.	.	PUNCT
ejpam-6175	249	1	in	in	ADP
ejpam-6175	249	2	this	this	DET
ejpam-6175	249	3	case	case	NOUN
ejpam-6175	249	4	β0(number	β0(number	NOUN
ejpam-6175	249	5	of	of	ADP
ejpam-6175	249	6	connected	connected	ADJ
ejpam-6175	249	7	components	component	NOUN
ejpam-6175	249	8	is	be	AUX
ejpam-6175	249	9	1	1	NUM
ejpam-6175	249	10	)	)	PUNCT
ejpam-6175	249	11	,	,	PUNCT
ejpam-6175	249	12	β1(number	β1(number	NUM
ejpam-6175	249	13	of	of	ADP
ejpam-6175	249	14	1	1	NUM
ejpam-6175	249	15	-	-	PUNCT
ejpam-6175	249	16	dimensional	dimensional	ADJ
ejpam-6175	249	17	holes	hole	NOUN
ejpam-6175	249	18	(	(	PUNCT
ejpam-6175	249	19	loops	loop	NOUN
ejpam-6175	249	20	)	)	PUNCT
ejpam-6175	249	21	is	be	AUX
ejpam-6175	249	22	2	2	NUM
ejpam-6175	249	23	)	)	PUNCT
ejpam-6175	249	24	and	and	CCONJ
ejpam-6175	249	25	β2	β2	PROPN
ejpam-6175	249	26	(	(	PUNCT
ejpam-6175	249	27	number	number	NOUN
ejpam-6175	249	28	of	of	ADP
ejpam-6175	249	29	2	2	NUM
ejpam-6175	249	30	-	-	PUNCT
ejpam-6175	249	31	dimensional	dimensional	ADJ
ejpam-6175	249	32	holes	hole	NOUN
ejpam-6175	249	33	(	(	PUNCT
ejpam-6175	249	34	voids	voids	NOUN
ejpam-6175	249	35	)	)	PUNCT
ejpam-6175	249	36	is	be	AUX
ejpam-6175	249	37	1	1	NUM
ejpam-6175	249	38	)	)	PUNCT
ejpam-6175	249	39	.	.	PUNCT
ejpam-6175	250	1	from	from	ADP
ejpam-6175	250	2	figure[3	figure[3	PROPN
ejpam-6175	250	3	]	]	PUNCT
ejpam-6175	250	4	,	,	PUNCT
ejpam-6175	250	5	let	let	VERB
ejpam-6175	250	6	us	we	PRON
ejpam-6175	250	7	see	see	VERB
ejpam-6175	250	8	the	the	DET
ejpam-6175	250	9	digital	digital	ADJ
ejpam-6175	250	10	homology	homology	NOUN
ejpam-6175	250	11	group	group	NOUN
ejpam-6175	250	12	of	of	ADP
ejpam-6175	250	13	a	a	DET
ejpam-6175	250	14	torus	torus	PROPN
ejpam-6175	250	15	t	t	PROPN
ejpam-6175	250	16	2	2	NUM
ejpam-6175	250	17	:	:	PUNCT
ejpam-6175	250	18	h0(t	h0(t	PROPN
ejpam-6175	250	19	2	2	X
ejpam-6175	250	20	)	)	PUNCT
ejpam-6175	250	21	∼=	∼=	PROPN
ejpam-6175	250	22	z	z	NOUN
ejpam-6175	250	23	,	,	PUNCT
ejpam-6175	250	24	h1(t	h1(t	NOUN
ejpam-6175	250	25	2	2	X
ejpam-6175	250	26	)	)	PUNCT
ejpam-6175	250	27	∼=	∼=	PROPN
ejpam-6175	250	28	z⊕	z⊕	PROPN
ejpam-6175	250	29	z	z	NOUN
ejpam-6175	250	30	,	,	PUNCT
ejpam-6175	250	31	h2(t	h2(t	PRON
ejpam-6175	250	32	2	2	X
ejpam-6175	250	33	)	)	PUNCT
ejpam-6175	250	34	∼=	∼=	PROPN
ejpam-6175	250	35	z	z	NOUN
ejpam-6175	250	36	.	.	PUNCT
ejpam-6175	251	1	using	use	VERB
ejpam-6175	251	2	the	the	DET
ejpam-6175	251	3	homology	homology	NOUN
ejpam-6175	251	4	groups	group	NOUN
ejpam-6175	251	5	h8	h8	PROPN
ejpam-6175	251	6	q	q	PROPN
ejpam-6175	251	7	(	(	PUNCT
ejpam-6175	251	8	∆i	∆i	PROPN
ejpam-6175	251	9	)	)	PUNCT
ejpam-6175	251	10	,	,	PUNCT
ejpam-6175	251	11	we	we	PRON
ejpam-6175	251	12	compute	compute	VERB
ejpam-6175	251	13	the	the	DET
ejpam-6175	251	14	persistent	persistent	ADJ
ejpam-6175	251	15	q	q	ADJ
ejpam-6175	251	16	-	-	PUNCT
ejpam-6175	251	17	th	th	VERB
ejpam-6175	251	18	betti	betti	NOUN
ejpam-6175	251	19	number	number	NOUN
ejpam-6175	251	20	as	as	SCONJ
ejpam-6175	251	21	follows	follow	VERB
ejpam-6175	251	22	:	:	PUNCT
ejpam-6175	251	23	•	•	NOUN
ejpam-6175	251	24	for	for	ADP
ejpam-6175	251	25	q	q	NOUN
ejpam-6175	251	26	=	=	SYM
ejpam-6175	251	27	0	0	NUM
ejpam-6175	251	28	and	and	CCONJ
ejpam-6175	251	29	0	0	NUM
ejpam-6175	251	30	<	<	X
ejpam-6175	252	1	i	i	X
ejpam-6175	252	2	≤	≤	ADJ
ejpam-6175	252	3	6	6	NUM
ejpam-6175	252	4	:	:	PUNCT
ejpam-6175	252	5	βi+p	βi+p	PROPN
ejpam-6175	252	6	,	,	PUNCT
ejpam-6175	252	7	κ	κ	NOUN
ejpam-6175	252	8	0	0	NUM
ejpam-6175	252	9	(	(	PUNCT
ejpam-6175	252	10	∆i	∆i	PROPN
ejpam-6175	252	11	)	)	PUNCT
ejpam-6175	253	1	=	=	PUNCT
ejpam-6175	254	1	rank(h	rank(h	NOUN
ejpam-6175	254	2	i	i	PRON
ejpam-6175	254	3	0(∆i	0(∆i	PROPN
ejpam-6175	254	4	)	)	PUNCT
ejpam-6175	254	5	)	)	PUNCT
ejpam-6175	255	1	=	=	SYM
ejpam-6175	255	2	rank(z	rank(z	X
ejpam-6175	255	3	)	)	PUNCT
ejpam-6175	255	4	=	=	SYM
ejpam-6175	255	5	1	1	NUM
ejpam-6175	256	1	this	this	DET
ejpam-6175	256	2	result	result	NOUN
ejpam-6175	256	3	indicates	indicate	VERB
ejpam-6175	256	4	the	the	DET
ejpam-6175	256	5	presence	presence	NOUN
ejpam-6175	256	6	of	of	ADP
ejpam-6175	256	7	a	a	DET
ejpam-6175	256	8	single	single	ADJ
ejpam-6175	256	9	connected	connected	ADJ
ejpam-6175	256	10	component	component	NOUN
ejpam-6175	256	11	for	for	ADP
ejpam-6175	256	12	0	0	NUM
ejpam-6175	256	13	<	<	X
ejpam-6175	256	14	i	i	X
ejpam-6175	256	15	≤	≤	ADV
ejpam-6175	256	16	6	6	NUM
ejpam-6175	256	17	.	.	NOUN
ejpam-6175	256	18	•	•	NOUN
ejpam-6175	256	19	for	for	ADP
ejpam-6175	256	20	q	q	NOUN
ejpam-6175	256	21	=	=	SYM
ejpam-6175	256	22	1	1	NUM
ejpam-6175	256	23	and	and	CCONJ
ejpam-6175	256	24	i	i	PRON
ejpam-6175	256	25	=	=	NOUN
ejpam-6175	256	26	3	3	NUM
ejpam-6175	256	27	,	,	PUNCT
ejpam-6175	256	28	4	4	NUM
ejpam-6175	256	29	,	,	PUNCT
ejpam-6175	256	30	5	5	NUM
ejpam-6175	256	31	:	:	PUNCT
ejpam-6175	256	32	βi+p	βi+p	PROPN
ejpam-6175	256	33	,	,	PUNCT
ejpam-6175	256	34	κ	κ	PROPN
ejpam-6175	256	35	1	1	NUM
ejpam-6175	256	36	(	(	PUNCT
ejpam-6175	256	37	∆i	∆i	PROPN
ejpam-6175	256	38	)	)	PUNCT
ejpam-6175	257	1	=	=	PUNCT
ejpam-6175	258	1	rank(h	rank(h	VERB
ejpam-6175	258	2	i	i	PRON
ejpam-6175	258	3	1(∆i	1(∆i	NUM
ejpam-6175	258	4	)	)	PUNCT
ejpam-6175	258	5	)	)	PUNCT
ejpam-6175	259	1	=	=	PRON
ejpam-6175	259	2	{	{	PUNCT
ejpam-6175	259	3	1	1	NUM
ejpam-6175	259	4	,	,	PUNCT
ejpam-6175	259	5	for	for	ADP
ejpam-6175	259	6	i	i	PRON
ejpam-6175	259	7	=	=	NOUN
ejpam-6175	259	8	3	3	NUM
ejpam-6175	259	9	2	2	NUM
ejpam-6175	259	10	,	,	PUNCT
ejpam-6175	259	11	for	for	ADP
ejpam-6175	259	12	i	i	PROPN
ejpam-6175	259	13	=	=	NOUN
ejpam-6175	259	14	4	4	NUM
ejpam-6175	259	15	,	,	PUNCT
ejpam-6175	259	16	5	5	NUM
ejpam-6175	259	17	this	this	PRON
ejpam-6175	259	18	reflects	reflect	VERB
ejpam-6175	259	19	the	the	DET
ejpam-6175	259	20	formation	formation	NOUN
ejpam-6175	259	21	of	of	ADP
ejpam-6175	259	22	one	one	NUM
ejpam-6175	259	23	1	1	NUM
ejpam-6175	259	24	-	-	PUNCT
ejpam-6175	259	25	dimensional	dimensional	ADJ
ejpam-6175	259	26	loop	loop	NOUN
ejpam-6175	259	27	at	at	ADP
ejpam-6175	259	28	i	i	NOUN
ejpam-6175	259	29	=	=	PROPN
ejpam-6175	259	30	3	3	NUM
ejpam-6175	259	31	and	and	CCONJ
ejpam-6175	259	32	two	two	NUM
ejpam-6175	259	33	independent	independent	ADJ
ejpam-6175	259	34	1	1	NUM
ejpam-6175	259	35	-	-	PUNCT
ejpam-6175	259	36	dimensional	dimensional	ADJ
ejpam-6175	259	37	loops	loop	NOUN
ejpam-6175	259	38	at	at	ADP
ejpam-6175	259	39	i	i	NOUN
ejpam-6175	259	40	=	=	NOUN
ejpam-6175	259	41	4	4	NUM
ejpam-6175	259	42	and	and	CCONJ
ejpam-6175	259	43	i	i	PRON
ejpam-6175	259	44	=	=	NOUN
ejpam-6175	259	45	5	5	X
ejpam-6175	259	46	.	.	PUNCT
ejpam-6175	259	47	s.k	s.k	PROPN
ejpam-6175	259	48	.	.	PUNCT
ejpam-6175	259	49	jha	jha	PROPN
ejpam-6175	259	50	sanjay	sanjay	PROPN
ejpam-6175	259	51	,	,	PUNCT
ejpam-6175	259	52	m	m	PROPN
ejpam-6175	259	53	,	,	PUNCT
ejpam-6175	259	54	natarajan	natarajan	PROPN
ejpam-6175	259	55	/	/	SYM
ejpam-6175	259	56	eur	eur	PROPN
ejpam-6175	259	57	.	.	PUNCT
ejpam-6175	260	1	j.	j.	PROPN
ejpam-6175	260	2	pure	pure	PROPN
ejpam-6175	260	3	appl	appl	PROPN
ejpam-6175	260	4	.	.	PROPN
ejpam-6175	260	5	math	math	PROPN
ejpam-6175	260	6	,	,	PUNCT
ejpam-6175	260	7	18	18	NUM
ejpam-6175	260	8	(	(	PUNCT
ejpam-6175	260	9	3	3	NUM
ejpam-6175	260	10	)	)	PUNCT
ejpam-6175	260	11	(	(	PUNCT
ejpam-6175	260	12	2025	2025	NUM
ejpam-6175	260	13	)	)	PUNCT
ejpam-6175	260	14	,	,	PUNCT
ejpam-6175	260	15	6175	6175	NUM
ejpam-6175	260	16	11	11	NUM
ejpam-6175	260	17	of	of	ADP
ejpam-6175	260	18	18	18	NUM
ejpam-6175	260	19	figure	figure	NOUN
ejpam-6175	260	20	3	3	NUM
ejpam-6175	260	21	:	:	PUNCT
ejpam-6175	260	22	sequential	sequential	ADJ
ejpam-6175	260	23	stages	stage	NOUN
ejpam-6175	260	24	of	of	ADP
ejpam-6175	260	25	a	a	DET
ejpam-6175	260	26	torus	torus	NOUN
ejpam-6175	260	27	mesh	mesh	NOUN
ejpam-6175	260	28	transformation	transformation	NOUN
ejpam-6175	260	29	:	:	PUNCT
ejpam-6175	260	30	starting	start	VERB
ejpam-6175	260	31	with	with	ADP
ejpam-6175	260	32	a	a	DET
ejpam-6175	260	33	smooth	smooth	ADJ
ejpam-6175	260	34	torus	torus	NOUN
ejpam-6175	260	35	shape	shape	NOUN
ejpam-6175	260	36	,	,	PUNCT
ejpam-6175	260	37	progressing	progress	VERB
ejpam-6175	260	38	through	through	ADP
ejpam-6175	260	39	various	various	ADJ
ejpam-6175	260	40	stages	stage	NOUN
ejpam-6175	260	41	of	of	ADP
ejpam-6175	260	42	mesh	mesh	NOUN
ejpam-6175	260	43	refinement	refinement	NOUN
ejpam-6175	260	44	and	and	CCONJ
ejpam-6175	260	45	simplification	simplification	NOUN
ejpam-6175	260	46	,	,	PUNCT
ejpam-6175	260	47	culminating	culminate	VERB
ejpam-6175	260	48	in	in	ADP
ejpam-6175	260	49	a	a	DET
ejpam-6175	260	50	simplified	simplified	ADJ
ejpam-6175	260	51	geometric	geometric	ADJ
ejpam-6175	260	52	representation	representation	NOUN
ejpam-6175	260	53	.	.	PUNCT
ejpam-6175	261	1	•	•	NOUN
ejpam-6175	261	2	for	for	ADP
ejpam-6175	261	3	q	q	NOUN
ejpam-6175	261	4	=	=	SYM
ejpam-6175	261	5	2	2	NUM
ejpam-6175	261	6	and	and	CCONJ
ejpam-6175	261	7	i	i	PRON
ejpam-6175	261	8	=	=	NOUN
ejpam-6175	262	1	6	6	NUM
ejpam-6175	262	2	:	:	PUNCT
ejpam-6175	262	3	βi+p	βi+p	PROPN
ejpam-6175	262	4	,	,	PUNCT
ejpam-6175	262	5	κ	κ	PROPN
ejpam-6175	262	6	2	2	NUM
ejpam-6175	262	7	(	(	PUNCT
ejpam-6175	262	8	∆i	∆i	PROPN
ejpam-6175	262	9	)	)	PUNCT
ejpam-6175	263	1	=	=	PUNCT
ejpam-6175	264	1	rank(h	rank(h	VERB
ejpam-6175	264	2	i	i	PRON
ejpam-6175	264	3	2(∆i	2(∆i	NUM
ejpam-6175	264	4	)	)	PUNCT
ejpam-6175	264	5	)	)	PUNCT
ejpam-6175	265	1	=	=	SYM
ejpam-6175	265	2	rank(z	rank(z	X
ejpam-6175	265	3	)	)	PUNCT
ejpam-6175	265	4	=	=	SYM
ejpam-6175	265	5	1	1	NUM
ejpam-6175	266	1	this	this	PRON
ejpam-6175	266	2	indicates	indicate	VERB
ejpam-6175	266	3	the	the	DET
ejpam-6175	266	4	presence	presence	NOUN
ejpam-6175	266	5	of	of	ADP
ejpam-6175	266	6	a	a	DET
ejpam-6175	266	7	single	single	ADJ
ejpam-6175	266	8	2	2	NUM
ejpam-6175	266	9	-	-	PUNCT
ejpam-6175	266	10	dimensional	dimensional	ADJ
ejpam-6175	266	11	hole	hole	NOUN
ejpam-6175	266	12	at	at	ADP
ejpam-6175	266	13	i	i	NOUN
ejpam-6175	266	14	=	=	PROPN
ejpam-6175	266	15	6	6	NUM
ejpam-6175	266	16	.	.	PUNCT
ejpam-6175	266	17	earlier	early	ADJ
ejpam-6175	266	18	examples	example	NOUN
ejpam-6175	266	19	in	in	ADP
ejpam-6175	266	20	this	this	DET
ejpam-6175	266	21	section	section	NOUN
ejpam-6175	266	22	have	have	AUX
ejpam-6175	266	23	demonstrated	demonstrate	VERB
ejpam-6175	266	24	the	the	DET
ejpam-6175	266	25	genus	genus	NOUN
ejpam-6175	266	26	,	,	PUNCT
ejpam-6175	266	27	betti	betti	NOUN
ejpam-6175	266	28	numbers	number	NOUN
ejpam-6175	266	29	,	,	PUNCT
ejpam-6175	266	30	and	and	CCONJ
ejpam-6175	266	31	persistent	persistent	ADJ
ejpam-6175	266	32	homology	homology	NOUN
ejpam-6175	266	33	for	for	ADP
ejpam-6175	266	34	various	various	ADJ
ejpam-6175	266	35	standard	standard	ADJ
ejpam-6175	266	36	shapes	shape	NOUN
ejpam-6175	266	37	.	.	PUNCT
ejpam-6175	267	1	now	now	ADV
ejpam-6175	267	2	,	,	PUNCT
ejpam-6175	267	3	we	we	PRON
ejpam-6175	267	4	turn	turn	VERB
ejpam-6175	267	5	our	our	PRON
ejpam-6175	267	6	attention	attention	NOUN
ejpam-6175	267	7	to	to	ADP
ejpam-6175	267	8	random	random	ADJ
ejpam-6175	267	9	shapes	shape	NOUN
ejpam-6175	267	10	as	as	SCONJ
ejpam-6175	267	11	shown	show	VERB
ejpam-6175	267	12	in	in	ADP
ejpam-6175	267	13	the	the	DET
ejpam-6175	267	14	figure[4	figure[4	NOUN
ejpam-6175	267	15	]	]	X
ejpam-6175	267	16	.	.	PUNCT
ejpam-6175	268	1	our	our	PRON
ejpam-6175	268	2	method	method	NOUN
ejpam-6175	268	3	focuses	focus	VERB
ejpam-6175	268	4	on	on	ADP
ejpam-6175	268	5	the	the	DET
ejpam-6175	268	6	nodes	node	NOUN
ejpam-6175	268	7	and	and	CCONJ
ejpam-6175	268	8	vertices	vertex	NOUN
ejpam-6175	268	9	of	of	ADP
ejpam-6175	268	10	any	any	DET
ejpam-6175	268	11	object	object	NOUN
ejpam-6175	268	12	to	to	PART
ejpam-6175	268	13	provide	provide	VERB
ejpam-6175	268	14	its	its	PRON
ejpam-6175	268	15	digital	digital	ADJ
ejpam-6175	268	16	homology	homology	NOUN
ejpam-6175	268	17	,	,	PUNCT
ejpam-6175	268	18	allowing	allow	VERB
ejpam-6175	268	19	us	we	PRON
ejpam-6175	268	20	to	to	PART
ejpam-6175	268	21	gain	gain	VERB
ejpam-6175	268	22	a	a	DET
ejpam-6175	268	23	deeper	deep	ADJ
ejpam-6175	268	24	understanding	understanding	NOUN
ejpam-6175	268	25	of	of	ADP
ejpam-6175	268	26	any	any	DET
ejpam-6175	268	27	given	give	VERB
ejpam-6175	268	28	structure	structure	NOUN
ejpam-6175	268	29	.	.	PUNCT
ejpam-6175	269	1	using	use	VERB
ejpam-6175	269	2	this	this	DET
ejpam-6175	269	3	method	method	NOUN
ejpam-6175	269	4	,	,	PUNCT
ejpam-6175	269	5	it	it	PRON
ejpam-6175	269	6	becomes	become	VERB
ejpam-6175	269	7	significantly	significantly	ADV
ejpam-6175	269	8	easier	easy	ADJ
ejpam-6175	269	9	to	to	PART
ejpam-6175	269	10	comprehend	comprehend	VERB
ejpam-6175	269	11	complex	complex	ADJ
ejpam-6175	269	12	objects	object	NOUN
ejpam-6175	269	13	and	and	CCONJ
ejpam-6175	269	14	shapes	shape	NOUN
ejpam-6175	269	15	.	.	PUNCT
ejpam-6175	270	1	this	this	DET
ejpam-6175	270	2	approach	approach	NOUN
ejpam-6175	270	3	is	be	AUX
ejpam-6175	270	4	particularly	particularly	ADV
ejpam-6175	270	5	beneficial	beneficial	ADJ
ejpam-6175	270	6	for	for	ADP
ejpam-6175	270	7	edge	edge	NOUN
ejpam-6175	270	8	detection	detection	NOUN
ejpam-6175	270	9	,	,	PUNCT
ejpam-6175	270	10	thinning	thinning	NOUN
ejpam-6175	270	11	,	,	PUNCT
ejpam-6175	270	12	and	and	CCONJ
ejpam-6175	270	13	restructuring	restructuring	NOUN
ejpam-6175	270	14	,	,	PUNCT
ejpam-6175	270	15	which	which	PRON
ejpam-6175	270	16	are	be	AUX
ejpam-6175	270	17	critical	critical	ADJ
ejpam-6175	270	18	processes	process	NOUN
ejpam-6175	270	19	for	for	ADP
ejpam-6175	270	20	identifying	identify	VERB
ejpam-6175	270	21	the	the	DET
ejpam-6175	270	22	genus	genus	ADJ
ejpam-6175	270	23	deformation	deformation	NOUN
ejpam-6175	270	24	of	of	ADP
ejpam-6175	270	25	such	such	ADJ
ejpam-6175	270	26	shapes	shape	NOUN
ejpam-6175	270	27	.	.	PUNCT
ejpam-6175	271	1	by	by	ADP
ejpam-6175	271	2	analyzing	analyze	VERB
ejpam-6175	271	3	the	the	DET
ejpam-6175	271	4	digital	digital	ADJ
ejpam-6175	271	5	homology	homology	NOUN
ejpam-6175	271	6	,	,	PUNCT
ejpam-6175	271	7	we	we	PRON
ejpam-6175	271	8	can	can	AUX
ejpam-6175	271	9	accurately	accurately	ADV
ejpam-6175	271	10	detect	detect	VERB
ejpam-6175	271	11	and	and	CCONJ
ejpam-6175	271	12	categorize	categorize	VERB
ejpam-6175	271	13	structural	structural	ADJ
ejpam-6175	271	14	changes	change	NOUN
ejpam-6175	271	15	and	and	CCONJ
ejpam-6175	271	16	deformations	deformation	NOUN
ejpam-6175	271	17	,	,	PUNCT
ejpam-6175	271	18	providing	provide	VERB
ejpam-6175	271	19	valuable	valuable	ADJ
ejpam-6175	271	20	insights	insight	NOUN
ejpam-6175	271	21	into	into	ADP
ejpam-6175	271	22	the	the	DET
ejpam-6175	271	23	underlying	underlie	VERB
ejpam-6175	271	24	geometry	geometry	NOUN
ejpam-6175	271	25	and	and	CCONJ
ejpam-6175	271	26	topology	topology	NOUN
ejpam-6175	271	27	of	of	ADP
ejpam-6175	271	28	the	the	DET
ejpam-6175	271	29	objects	object	NOUN
ejpam-6175	271	30	.	.	PUNCT
ejpam-6175	272	1	figure	figure	VERB
ejpam-6175	272	2	4	4	NUM
ejpam-6175	272	3	:	:	PUNCT
ejpam-6175	272	4	transformation	transformation	NOUN
ejpam-6175	272	5	of	of	ADP
ejpam-6175	272	6	a	a	DET
ejpam-6175	272	7	snowflake	snowflake	NOUN
ejpam-6175	272	8	image	image	NOUN
ejpam-6175	272	9	into	into	ADP
ejpam-6175	272	10	a	a	DET
ejpam-6175	272	11	graph	graph	NOUN
ejpam-6175	272	12	representation	representation	NOUN
ejpam-6175	272	13	:	:	PUNCT
ejpam-6175	272	14	from	from	ADP
ejpam-6175	272	15	the	the	DET
ejpam-6175	272	16	original	original	ADJ
ejpam-6175	272	17	snowflake	snowflake	NOUN
ejpam-6175	272	18	image	image	NOUN
ejpam-6175	272	19	,	,	PUNCT
ejpam-6175	272	20	progressing	progress	VERB
ejpam-6175	272	21	through	through	ADP
ejpam-6175	272	22	edge	edge	NOUN
ejpam-6175	272	23	detection	detection	NOUN
ejpam-6175	272	24	and	and	CCONJ
ejpam-6175	272	25	keypoint	keypoint	NOUN
ejpam-6175	272	26	extraction	extraction	NOUN
ejpam-6175	272	27	,	,	PUNCT
ejpam-6175	272	28	to	to	ADP
ejpam-6175	272	29	the	the	DET
ejpam-6175	272	30	construction	construction	NOUN
ejpam-6175	272	31	and	and	CCONJ
ejpam-6175	272	32	refinement	refinement	NOUN
ejpam-6175	272	33	of	of	ADP
ejpam-6175	272	34	a	a	DET
ejpam-6175	272	35	graph	graph	NOUN
ejpam-6175	272	36	representing	represent	VERB
ejpam-6175	272	37	the	the	DET
ejpam-6175	272	38	snowflake	snowflake	NOUN
ejpam-6175	272	39	’s	’s	PART
ejpam-6175	272	40	structural	structural	ADJ
ejpam-6175	272	41	geometry	geometry	NOUN
ejpam-6175	272	42	.	.	PUNCT
ejpam-6175	273	1	s.k	s.k	PROPN
ejpam-6175	273	2	.	.	PUNCT
ejpam-6175	273	3	jha	jha	PROPN
ejpam-6175	273	4	sanjay	sanjay	PROPN
ejpam-6175	273	5	,	,	PUNCT
ejpam-6175	273	6	m	m	PROPN
ejpam-6175	273	7	,	,	PUNCT
ejpam-6175	273	8	natarajan	natarajan	PROPN
ejpam-6175	273	9	/	/	SYM
ejpam-6175	273	10	eur	eur	PROPN
ejpam-6175	273	11	.	.	PUNCT
ejpam-6175	274	1	j.	j.	PROPN
ejpam-6175	274	2	pure	pure	PROPN
ejpam-6175	274	3	appl	appl	PROPN
ejpam-6175	274	4	.	.	PROPN
ejpam-6175	274	5	math	math	PROPN
ejpam-6175	274	6	,	,	PUNCT
ejpam-6175	274	7	18	18	NUM
ejpam-6175	274	8	(	(	PUNCT
ejpam-6175	274	9	3	3	NUM
ejpam-6175	274	10	)	)	PUNCT
ejpam-6175	274	11	(	(	PUNCT
ejpam-6175	274	12	2025	2025	NUM
ejpam-6175	274	13	)	)	PUNCT
ejpam-6175	274	14	,	,	PUNCT
ejpam-6175	274	15	6175	6175	NUM
ejpam-6175	274	16	12	12	NUM
ejpam-6175	274	17	of	of	ADP
ejpam-6175	274	18	18	18	NUM
ejpam-6175	274	19	algorithm	algorithm	NOUN
ejpam-6175	274	20	1	1	NUM
ejpam-6175	274	21	pseudo	pseudo	NOUN
ejpam-6175	274	22	code	code	NOUN
ejpam-6175	274	23	:	:	PUNCT
ejpam-6175	274	24	1	1	NUM
ejpam-6175	274	25	require	require	NOUN
ejpam-6175	274	26	:	:	PUNCT
ejpam-6175	274	27	import	import	NOUN
ejpam-6175	274	28	required	require	VERB
ejpam-6175	274	29	libraries	library	NOUN
ejpam-6175	274	30	for	for	ADP
ejpam-6175	274	31	image	image	NOUN
ejpam-6175	274	32	processing	processing	NOUN
ejpam-6175	274	33	,	,	PUNCT
ejpam-6175	274	34	clustering	clustering	NOUN
ejpam-6175	274	35	,	,	PUNCT
ejpam-6175	274	36	and	and	CCONJ
ejpam-6175	274	37	persistent	persistent	ADJ
ejpam-6175	274	38	homology	homology	NOUN
ejpam-6175	274	39	visualization	visualization	NOUN
ejpam-6175	274	40	.	.	PUNCT
ejpam-6175	275	1	load	load	VERB
ejpam-6175	275	2	an	an	DET
ejpam-6175	275	3	image	image	NOUN
ejpam-6175	275	4	i(x	i(x	NOUN
ejpam-6175	275	5	,	,	PUNCT
ejpam-6175	275	6	y	y	NOUN
ejpam-6175	275	7	)	)	PUNCT
ejpam-6175	275	8	as	as	ADP
ejpam-6175	275	9	input	input	NOUN
ejpam-6175	275	10	.	.	PUNCT
ejpam-6175	276	1	edge	edge	NOUN
ejpam-6175	276	2	detection	detection	NOUN
ejpam-6175	276	3	:	:	PUNCT
ejpam-6175	276	4	load	load	NOUN
ejpam-6175	276	5	i(x	i(x	PROPN
ejpam-6175	276	6	,	,	PUNCT
ejpam-6175	276	7	y	y	PROPN
ejpam-6175	276	8	)	)	PUNCT
ejpam-6175	276	9	as	as	ADP
ejpam-6175	276	10	a	a	DET
ejpam-6175	276	11	grayscale	grayscale	NOUN
ejpam-6175	276	12	image	image	NOUN
ejpam-6175	276	13	.	.	PUNCT
ejpam-6175	277	1	apply	apply	VERB
ejpam-6175	277	2	canny	canny	ADJ
ejpam-6175	277	3	edge	edge	NOUN
ejpam-6175	277	4	detection	detection	NOUN
ejpam-6175	277	5	:	:	PUNCT
ejpam-6175	277	6	e(x	e(x	NUM
ejpam-6175	277	7	,	,	PUNCT
ejpam-6175	277	8	y	y	NOUN
ejpam-6175	277	9	)	)	PUNCT
ejpam-6175	277	10	=	=	SYM
ejpam-6175	277	11	canny(i	canny(i	PROPN
ejpam-6175	277	12	,	,	PUNCT
ejpam-6175	277	13	t1	t1	NOUN
ejpam-6175	277	14	,	,	PUNCT
ejpam-6175	277	15	t2	t2	NOUN
ejpam-6175	277	16	)	)	PUNCT
ejpam-6175	277	17	,	,	PUNCT
ejpam-6175	277	18	where	where	SCONJ
ejpam-6175	277	19	t1	t1	NOUN
ejpam-6175	277	20	,	,	PUNCT
ejpam-6175	277	21	t2	t2	PROPN
ejpam-6175	277	22	are	be	AUX
ejpam-6175	277	23	thresholds	threshold	NOUN
ejpam-6175	277	24	.	.	PUNCT
ejpam-6175	278	1	return	return	VERB
ejpam-6175	278	2	the	the	DET
ejpam-6175	278	3	edge	edge	NOUN
ejpam-6175	278	4	-	-	PUNCT
ejpam-6175	278	5	detected	detect	VERB
ejpam-6175	278	6	image	image	NOUN
ejpam-6175	278	7	e(x	e(x	NUM
ejpam-6175	278	8	,	,	PUNCT
ejpam-6175	278	9	y	y	NOUN
ejpam-6175	278	10	)	)	PUNCT
ejpam-6175	278	11	.	.	PUNCT
ejpam-6175	279	1	keypoint	keypoint	PROPN
ejpam-6175	279	2	detection	detection	NOUN
ejpam-6175	279	3	and	and	CCONJ
ejpam-6175	279	4	clustering	clustering	NOUN
ejpam-6175	279	5	:	:	PUNCT
ejpam-6175	279	6	detect	detect	VERB
ejpam-6175	279	7	corners	corner	NOUN
ejpam-6175	279	8	using	use	VERB
ejpam-6175	279	9	the	the	DET
ejpam-6175	279	10	shi	shi	PROPN
ejpam-6175	279	11	-	-	PUNCT
ejpam-6175	279	12	tomasi	tomasi	PROPN
ejpam-6175	279	13	method	method	NOUN
ejpam-6175	279	14	:	:	PUNCT
ejpam-6175	279	15	λ	λ	X
ejpam-6175	279	16	=	=	SYM
ejpam-6175	279	17	min(λ1	min(λ1	PROPN
ejpam-6175	279	18	,	,	PUNCT
ejpam-6175	279	19	λ2	λ2	PROPN
ejpam-6175	279	20	)	)	PUNCT
ejpam-6175	279	21	,	,	PUNCT
ejpam-6175	279	22	λ1	λ1	ADJ
ejpam-6175	279	23	,	,	PUNCT
ejpam-6175	279	24	λ2	λ2	NOUN
ejpam-6175	279	25	are	be	AUX
ejpam-6175	279	26	eigenvalues	eigenvalue	NOUN
ejpam-6175	279	27	of	of	ADP
ejpam-6175	279	28	the	the	DET
ejpam-6175	279	29	structure	structure	NOUN
ejpam-6175	279	30	tensor	tensor	NOUN
ejpam-6175	279	31	.	.	PUNCT
ejpam-6175	280	1	extract	extract	VERB
ejpam-6175	280	2	corner	corner	NOUN
ejpam-6175	280	3	coordinates	coordinate	NOUN
ejpam-6175	280	4	{	{	PUNCT
ejpam-6175	280	5	c(xi	c(xi	PROPN
ejpam-6175	280	6	,	,	PUNCT
ejpam-6175	280	7	yi	yi	PROPN
ejpam-6175	280	8	)	)	PUNCT
ejpam-6175	280	9	}	}	PUNCT
ejpam-6175	280	10	.	.	PUNCT
ejpam-6175	281	1	apply	apply	VERB
ejpam-6175	281	2	k	k	NOUN
ejpam-6175	281	3	-	-	PUNCT
ejpam-6175	281	4	means	means	NOUN
ejpam-6175	281	5	clustering	cluster	VERB
ejpam-6175	281	6	to	to	PART
ejpam-6175	281	7	reduce	reduce	VERB
ejpam-6175	281	8	points	point	NOUN
ejpam-6175	281	9	to	to	ADP
ejpam-6175	281	10	k	k	PROPN
ejpam-6175	281	11	clusters	cluster	NOUN
ejpam-6175	281	12	:	:	PUNCT
ejpam-6175	281	13	min	min	PROPN
ejpam-6175	281	14	c1,c2,	c1,c2,	PROPN
ejpam-6175	281	15	...	...	PUNCT
ejpam-6175	281	16	,ck	,ck	PUNCT
ejpam-6175	281	17	k∑	k∑	VERB
ejpam-6175	281	18	i=1	i=1	PROPN
ejpam-6175	281	19	∑	∑	PROPN
ejpam-6175	281	20	x∈ci	x∈ci	PROPN
ejpam-6175	281	21	||x−	||x−	PROPN
ejpam-6175	281	22	µi||2	µi||2	PROPN
ejpam-6175	281	23	,	,	PUNCT
ejpam-6175	281	24	µi	µi	PROPN
ejpam-6175	281	25	=	=	NOUN
ejpam-6175	281	26	cluster	cluster	NOUN
ejpam-6175	281	27	center	center	NOUN
ejpam-6175	281	28	.	.	PUNCT
ejpam-6175	282	1	return	return	VERB
ejpam-6175	282	2	keypoint	keypoint	NOUN
ejpam-6175	282	3	centroids	centroid	NOUN
ejpam-6175	282	4	{	{	PUNCT
ejpam-6175	282	5	kc	kc	PROPN
ejpam-6175	282	6	=	=	SYM
ejpam-6175	282	7	(	(	PUNCT
ejpam-6175	282	8	xc	xc	PROPN
ejpam-6175	282	9	,	,	PUNCT
ejpam-6175	282	10	yc	yc	NOUN
ejpam-6175	282	11	)	)	PUNCT
ejpam-6175	282	12	}	}	PUNCT
ejpam-6175	282	13	.	.	PUNCT
ejpam-6175	283	1	rips	rip	NOUN
ejpam-6175	283	2	complex	complex	ADJ
ejpam-6175	283	3	construction	construction	NOUN
ejpam-6175	283	4	:	:	PUNCT
ejpam-6175	283	5	compute	compute	NOUN
ejpam-6175	283	6	pairwise	pairwise	NOUN
ejpam-6175	283	7	distances	distance	NOUN
ejpam-6175	283	8	between	between	ADP
ejpam-6175	283	9	keypoints	keypoint	NOUN
ejpam-6175	283	10	:	:	PUNCT
ejpam-6175	283	11	dij	dij	PROPN
ejpam-6175	283	12	=	=	SYM
ejpam-6175	283	13	||ki	||ki	NOUN
ejpam-6175	284	1	c	c	NOUN
ejpam-6175	284	2	−kj	−kj	INTJ
ejpam-6175	284	3	c	c	NOUN
ejpam-6175	284	4	||	||	PROPN
ejpam-6175	284	5	,	,	PUNCT
ejpam-6175	284	6	ki	ki	PROPN
ejpam-6175	284	7	c	c	X
ejpam-6175	284	8	,	,	PUNCT
ejpam-6175	284	9	k	k	PROPN
ejpam-6175	284	10	j	j	PROPN
ejpam-6175	284	11	c	c	PROPN
ejpam-6175	284	12	∈	∈	PROPN
ejpam-6175	284	13	r2	r2	PROPN
ejpam-6175	284	14	.	.	PUNCT
ejpam-6175	285	1	for	for	ADP
ejpam-6175	285	2	each	each	DET
ejpam-6175	285	3	radius	radius	NOUN
ejpam-6175	285	4	r	r	NOUN
ejpam-6175	285	5	:	:	PUNCT
ejpam-6175	285	6	plot	plot	NOUN
ejpam-6175	285	7	points	point	NOUN
ejpam-6175	285	8	{	{	PUNCT
ejpam-6175	285	9	kc	kc	NOUN
ejpam-6175	285	10	}	}	PUNCT
ejpam-6175	285	11	as	as	ADP
ejpam-6175	285	12	vertices	vertex	NOUN
ejpam-6175	285	13	.	.	PUNCT
ejpam-6175	286	1	draw	draw	VERB
ejpam-6175	286	2	circles	circle	NOUN
ejpam-6175	286	3	of	of	ADP
ejpam-6175	286	4	radius	radius	NOUN
ejpam-6175	286	5	r	r	NOUN
ejpam-6175	286	6	around	around	ADP
ejpam-6175	286	7	each	each	DET
ejpam-6175	286	8	vertex	vertex	NOUN
ejpam-6175	286	9	:	:	PUNCT
ejpam-6175	287	1	c(kc	c(kc	X
ejpam-6175	287	2	,	,	PUNCT
ejpam-6175	287	3	r	r	NOUN
ejpam-6175	287	4	)	)	PUNCT
ejpam-6175	287	5	=	=	SYM
ejpam-6175	287	6	{	{	PUNCT
ejpam-6175	287	7	y	y	PROPN
ejpam-6175	287	8	∈	∈	PROPN
ejpam-6175	287	9	r2	r2	NOUN
ejpam-6175	287	10	|	|	ADV
ejpam-6175	287	11	||kc	||kc	VERB
ejpam-6175	287	12	−	−	PROPN
ejpam-6175	287	13	y||	y||	NOUN
ejpam-6175	287	14	≤	≤	X
ejpam-6175	287	15	r	r	NOUN
ejpam-6175	287	16	}	}	PUNCT
ejpam-6175	287	17	.	.	PUNCT
ejpam-6175	288	1	connect	connect	VERB
ejpam-6175	288	2	vertices	vertex	NOUN
ejpam-6175	288	3	i	i	PRON
ejpam-6175	288	4	,	,	PUNCT
ejpam-6175	288	5	j	j	PROPN
ejpam-6175	288	6	if	if	SCONJ
ejpam-6175	288	7	dij	dij	PROPN
ejpam-6175	288	8	≤	≤	NUM
ejpam-6175	288	9	2r	2r	NUM
ejpam-6175	288	10	.	.	PUNCT
ejpam-6175	289	1	persistent	persistent	ADJ
ejpam-6175	289	2	homology	homology	NOUN
ejpam-6175	289	3	:	:	PUNCT
ejpam-6175	289	4	use	use	VERB
ejpam-6175	289	5	pairwise	pairwise	NOUN
ejpam-6175	289	6	distances	distance	NOUN
ejpam-6175	289	7	dij	dij	VERB
ejpam-6175	289	8	to	to	PART
ejpam-6175	289	9	compute	compute	VERB
ejpam-6175	289	10	persistent	persistent	ADJ
ejpam-6175	289	11	homology	homology	NOUN
ejpam-6175	289	12	:	:	PUNCT
ejpam-6175	289	13	birth	birth	NOUN
ejpam-6175	289	14	,	,	PUNCT
ejpam-6175	289	15	death	death	NOUN
ejpam-6175	289	16	of	of	ADP
ejpam-6175	289	17	h0	h0	PROPN
ejpam-6175	289	18	(	(	PUNCT
ejpam-6175	289	19	connected	connected	ADJ
ejpam-6175	289	20	components	component	NOUN
ejpam-6175	289	21	)	)	PUNCT
ejpam-6175	289	22	and	and	CCONJ
ejpam-6175	289	23	h1	h1	PROPN
ejpam-6175	289	24	(	(	PUNCT
ejpam-6175	289	25	loops	loop	NOUN
ejpam-6175	289	26	)	)	PUNCT
ejpam-6175	289	27	.	.	PUNCT
ejpam-6175	290	1	plot	plot	VERB
ejpam-6175	290	2	the	the	DET
ejpam-6175	290	3	persistence	persistence	NOUN
ejpam-6175	290	4	diagram	diagram	NOUN
ejpam-6175	290	5	for	for	ADP
ejpam-6175	290	6	h0	h0	NOUN
ejpam-6175	290	7	and	and	CCONJ
ejpam-6175	290	8	h1	h1	PROPN
ejpam-6175	290	9	.	.	PUNCT
ejpam-6175	291	1	main	main	ADJ
ejpam-6175	291	2	execution	execution	NOUN
ejpam-6175	291	3	:	:	PUNCT
ejpam-6175	291	4	load	load	NOUN
ejpam-6175	291	5	image	image	NOUN
ejpam-6175	291	6	i(x	i(x	PROPN
ejpam-6175	291	7	,	,	PUNCT
ejpam-6175	291	8	y	y	NOUN
ejpam-6175	291	9	)	)	PUNCT
ejpam-6175	291	10	and	and	CCONJ
ejpam-6175	291	11	detect	detect	VERB
ejpam-6175	291	12	edges	edge	NOUN
ejpam-6175	291	13	e(x	e(x	NUM
ejpam-6175	291	14	,	,	PUNCT
ejpam-6175	291	15	y	y	NOUN
ejpam-6175	291	16	)	)	PUNCT
ejpam-6175	291	17	.	.	PUNCT
ejpam-6175	292	1	extract	extract	NOUN
ejpam-6175	292	2	and	and	CCONJ
ejpam-6175	292	3	cluster	cluster	NOUN
ejpam-6175	292	4	keypoints	keypoint	NOUN
ejpam-6175	292	5	{	{	PUNCT
ejpam-6175	292	6	kc	kc	PROPN
ejpam-6175	292	7	}	}	PUNCT
ejpam-6175	292	8	.	.	PUNCT
ejpam-6175	293	1	construct	construct	VERB
ejpam-6175	293	2	rips	rip	NOUN
ejpam-6175	293	3	complexes	complex	NOUN
ejpam-6175	293	4	for	for	ADP
ejpam-6175	293	5	radii	radii	NOUN
ejpam-6175	293	6	r	r	NOUN
ejpam-6175	293	7	∈	∈	PROPN
ejpam-6175	293	8	radii	radius	NOUN
ejpam-6175	293	9	.	.	PUNCT
ejpam-6175	294	1	visualize	visualize	VERB
ejpam-6175	294	2	rips	rip	NOUN
ejpam-6175	294	3	complexes	complex	NOUN
ejpam-6175	294	4	and	and	CCONJ
ejpam-6175	294	5	persistence	persistence	NOUN
ejpam-6175	294	6	diagram	diagram	NOUN
ejpam-6175	294	7	.	.	PUNCT
ejpam-6175	295	1	edge	edge	NOUN
ejpam-6175	295	2	map	map	NOUN
ejpam-6175	295	3	e(x	e(x	NUM
ejpam-6175	295	4	,	,	PUNCT
ejpam-6175	295	5	y	y	NOUN
ejpam-6175	295	6	)	)	PUNCT
ejpam-6175	295	7	,	,	PUNCT
ejpam-6175	295	8	rips	rip	VERB
ejpam-6175	295	9	complex	complex	ADJ
ejpam-6175	295	10	visualizations	visualization	NOUN
ejpam-6175	295	11	,	,	PUNCT
ejpam-6175	295	12	and	and	CCONJ
ejpam-6175	295	13	persistence	persistence	NOUN
ejpam-6175	295	14	diagram	diagram	NOUN
ejpam-6175	295	15	.	.	PUNCT
ejpam-6175	296	1	=	=	NOUN
ejpam-6175	296	2	0	0	NUM
ejpam-6175	296	3	s.k	s.k	PROPN
ejpam-6175	296	4	.	.	PUNCT
ejpam-6175	296	5	jha	jha	PROPN
ejpam-6175	296	6	sanjay	sanjay	PROPN
ejpam-6175	296	7	,	,	PUNCT
ejpam-6175	296	8	m	m	PROPN
ejpam-6175	296	9	,	,	PUNCT
ejpam-6175	296	10	natarajan	natarajan	PROPN
ejpam-6175	296	11	/	/	SYM
ejpam-6175	296	12	eur	eur	PROPN
ejpam-6175	296	13	.	.	PUNCT
ejpam-6175	297	1	j.	j.	PROPN
ejpam-6175	297	2	pure	pure	PROPN
ejpam-6175	297	3	appl	appl	PROPN
ejpam-6175	297	4	.	.	PROPN
ejpam-6175	297	5	math	math	PROPN
ejpam-6175	297	6	,	,	PUNCT
ejpam-6175	297	7	18	18	NUM
ejpam-6175	297	8	(	(	PUNCT
ejpam-6175	297	9	3	3	NUM
ejpam-6175	297	10	)	)	PUNCT
ejpam-6175	297	11	(	(	PUNCT
ejpam-6175	297	12	2025	2025	NUM
ejpam-6175	297	13	)	)	PUNCT
ejpam-6175	297	14	,	,	PUNCT
ejpam-6175	297	15	6175	6175	NUM
ejpam-6175	297	16	13	13	NUM
ejpam-6175	297	17	of	of	ADP
ejpam-6175	297	18	18	18	NUM
ejpam-6175	297	19	figure	figure	NOUN
ejpam-6175	297	20	5	5	NUM
ejpam-6175	297	21	:	:	PUNCT
ejpam-6175	297	22	evolution	evolution	NOUN
ejpam-6175	297	23	of	of	ADP
ejpam-6175	297	24	vietoris	vietoris	ADJ
ejpam-6175	297	25	-	-	PUNCT
ejpam-6175	297	26	rips	rip	NOUN
ejpam-6175	297	27	complexes	complex	NOUN
ejpam-6175	297	28	at	at	ADP
ejpam-6175	297	29	varying	vary	VERB
ejpam-6175	297	30	radii	radii	NOUN
ejpam-6175	297	31	r	r	NOUN
ejpam-6175	297	32	,	,	PUNCT
ejpam-6175	297	33	showing	show	VERB
ejpam-6175	297	34	increasing	increase	VERB
ejpam-6175	297	35	connectivity	connectivity	NOUN
ejpam-6175	297	36	3.3	3.3	NUM
ejpam-6175	297	37	.	.	PUNCT
ejpam-6175	298	1	topological	topological	ADJ
ejpam-6175	298	2	data	datum	NOUN
ejpam-6175	298	3	analysis	analysis	NOUN
ejpam-6175	298	4	and	and	CCONJ
ejpam-6175	298	5	rips	rip	NOUN
ejpam-6175	298	6	complexes	complexe	VERB
ejpam-6175	298	7	topological	topological	ADJ
ejpam-6175	298	8	data	datum	NOUN
ejpam-6175	298	9	analysis	analysis	NOUN
ejpam-6175	298	10	(	(	PUNCT
ejpam-6175	298	11	tda	tda	NOUN
ejpam-6175	298	12	)	)	PUNCT
ejpam-6175	298	13	provides	provide	VERB
ejpam-6175	298	14	a	a	DET
ejpam-6175	298	15	robust	robust	ADJ
ejpam-6175	298	16	mathematical	mathematical	ADJ
ejpam-6175	298	17	framework	framework	NOUN
ejpam-6175	298	18	for	for	ADP
ejpam-6175	298	19	extracting	extract	VERB
ejpam-6175	298	20	the	the	DET
ejpam-6175	298	21	shape	shape	NOUN
ejpam-6175	298	22	and	and	CCONJ
ejpam-6175	298	23	structure	structure	NOUN
ejpam-6175	298	24	of	of	ADP
ejpam-6175	298	25	datasets	dataset	NOUN
ejpam-6175	298	26	by	by	ADP
ejpam-6175	298	27	analyzing	analyze	VERB
ejpam-6175	298	28	their	their	PRON
ejpam-6175	298	29	topological	topological	ADJ
ejpam-6175	298	30	features	feature	NOUN
ejpam-6175	298	31	.	.	PUNCT
ejpam-6175	299	1	at	at	ADP
ejpam-6175	299	2	the	the	DET
ejpam-6175	299	3	core	core	NOUN
ejpam-6175	299	4	of	of	ADP
ejpam-6175	299	5	tda	tda	PROPN
ejpam-6175	299	6	lies	lie	VERB
ejpam-6175	299	7	the	the	DET
ejpam-6175	299	8	concept	concept	NOUN
ejpam-6175	299	9	of	of	ADP
ejpam-6175	299	10	the	the	DET
ejpam-6175	299	11	rips	rip	NOUN
ejpam-6175	299	12	complex	complex	NOUN
ejpam-6175	299	13	,	,	PUNCT
ejpam-6175	299	14	which	which	PRON
ejpam-6175	299	15	encodes	encode	VERB
ejpam-6175	299	16	the	the	DET
ejpam-6175	299	17	geometric	geometric	ADJ
ejpam-6175	299	18	proximity	proximity	NOUN
ejpam-6175	299	19	of	of	ADP
ejpam-6175	299	20	a	a	DET
ejpam-6175	299	21	dataset	dataset	NOUN
ejpam-6175	299	22	into	into	ADP
ejpam-6175	299	23	a	a	DET
ejpam-6175	299	24	simplicial	simplicial	ADJ
ejpam-6175	299	25	complex	complex	NOUN
ejpam-6175	299	26	.	.	PUNCT
ejpam-6175	300	1	for	for	ADP
ejpam-6175	300	2	a	a	DET
ejpam-6175	300	3	given	give	VERB
ejpam-6175	300	4	set	set	NOUN
ejpam-6175	300	5	of	of	ADP
ejpam-6175	300	6	points	point	NOUN
ejpam-6175	300	7	p	p	X
ejpam-6175	300	8	=	=	X
ejpam-6175	300	9	{	{	PUNCT
ejpam-6175	300	10	p1	p1	NOUN
ejpam-6175	300	11	,	,	PUNCT
ejpam-6175	300	12	p2	p2	NOUN
ejpam-6175	300	13	,	,	PUNCT
ejpam-6175	300	14	.	.	PUNCT
ejpam-6175	300	15	.	.	PUNCT
ejpam-6175	300	16	.	.	PUNCT
ejpam-6175	301	1	,	,	PUNCT
ejpam-6175	301	2	pn	pn	NOUN
ejpam-6175	301	3	}	}	PUNCT
ejpam-6175	301	4	in	in	ADP
ejpam-6175	301	5	rd	rd	PROPN
ejpam-6175	301	6	,	,	PUNCT
ejpam-6175	301	7	the	the	DET
ejpam-6175	301	8	rips	rip	NOUN
ejpam-6175	301	9	complex	complex	ADJ
ejpam-6175	301	10	r(p	r(p	NOUN
ejpam-6175	301	11	,	,	PUNCT
ejpam-6175	301	12	r	r	NOUN
ejpam-6175	301	13	)	)	PUNCT
ejpam-6175	301	14	at	at	ADP
ejpam-6175	301	15	a	a	DET
ejpam-6175	301	16	radius	radius	NOUN
ejpam-6175	301	17	r	r	NOUN
ejpam-6175	301	18	is	be	AUX
ejpam-6175	301	19	defined	define	VERB
ejpam-6175	301	20	as	as	ADP
ejpam-6175	301	21	:	:	PUNCT
ejpam-6175	301	22	r(p	r(p	NOUN
ejpam-6175	301	23	,	,	PUNCT
ejpam-6175	301	24	r	r	NOUN
ejpam-6175	301	25	)	)	PUNCT
ejpam-6175	301	26	=	=	SYM
ejpam-6175	301	27	{	{	PUNCT
ejpam-6175	301	28	σ	σ	PROPN
ejpam-6175	301	29	⊆	⊆	NUM
ejpam-6175	301	30	p	p	NOUN
ejpam-6175	302	1	|	|	NOUN
ejpam-6175	302	2	∥pi	∥pi	NOUN
ejpam-6175	302	3	−	−	NOUN
ejpam-6175	302	4	pj∥	pj∥	PROPN
ejpam-6175	302	5	≤	≤	NOUN
ejpam-6175	302	6	r	r	NOUN
ejpam-6175	302	7	for	for	ADP
ejpam-6175	302	8	all	all	DET
ejpam-6175	302	9	pi	pi	NOUN
ejpam-6175	302	10	,	,	PUNCT
ejpam-6175	302	11	pj	pj	PROPN
ejpam-6175	302	12	∈	∈	PROPN
ejpam-6175	302	13	σ	σ	PROPN
ejpam-6175	302	14	}	}	PUNCT
ejpam-6175	302	15	.	.	PUNCT
ejpam-6175	303	1	here	here	ADV
ejpam-6175	303	2	,	,	PUNCT
ejpam-6175	303	3	σ	σ	PROPN
ejpam-6175	303	4	represents	represent	VERB
ejpam-6175	303	5	a	a	DET
ejpam-6175	303	6	simplex	simplex	NOUN
ejpam-6175	303	7	,	,	PUNCT
ejpam-6175	303	8	which	which	PRON
ejpam-6175	303	9	could	could	AUX
ejpam-6175	303	10	be	be	AUX
ejpam-6175	303	11	a	a	DET
ejpam-6175	303	12	vertex	vertex	NOUN
ejpam-6175	303	13	,	,	PUNCT
ejpam-6175	303	14	edge	edge	NOUN
ejpam-6175	303	15	,	,	PUNCT
ejpam-6175	303	16	or	or	CCONJ
ejpam-6175	303	17	higher	higher	ADV
ejpam-6175	303	18	-	-	PUNCT
ejpam-6175	303	19	dimensional	dimensional	ADJ
ejpam-6175	303	20	face	face	NOUN
ejpam-6175	303	21	,	,	PUNCT
ejpam-6175	303	22	depending	depend	VERB
ejpam-6175	303	23	on	on	ADP
ejpam-6175	303	24	the	the	DET
ejpam-6175	303	25	proximity	proximity	NOUN
ejpam-6175	303	26	condition	condition	NOUN
ejpam-6175	303	27	.	.	PUNCT
ejpam-6175	304	1	a	a	DET
ejpam-6175	304	2	sequence	sequence	NOUN
ejpam-6175	304	3	of	of	ADP
ejpam-6175	304	4	rips	rip	NOUN
ejpam-6175	304	5	complexes	complex	NOUN
ejpam-6175	304	6	constructed	construct	VERB
ejpam-6175	304	7	for	for	ADP
ejpam-6175	304	8	increasing	increase	VERB
ejpam-6175	304	9	radii	radius	NOUN
ejpam-6175	304	10	forms	form	NOUN
ejpam-6175	304	11	a	a	DET
ejpam-6175	304	12	filtration	filtration	NOUN
ejpam-6175	304	13	:	:	PUNCT
ejpam-6175	304	14	r(p	r(p	NOUN
ejpam-6175	304	15	,	,	PUNCT
ejpam-6175	304	16	r1	r1	PROPN
ejpam-6175	304	17	)	)	PUNCT
ejpam-6175	304	18	⊆	⊆	NUM
ejpam-6175	304	19	r(p	r(p	NOUN
ejpam-6175	304	20	,	,	PUNCT
ejpam-6175	304	21	r2	r2	PROPN
ejpam-6175	304	22	)	)	PUNCT
ejpam-6175	304	23	⊆	⊆	NUM
ejpam-6175	304	24	.	.	PUNCT
ejpam-6175	304	25	.	.	PUNCT
ejpam-6175	304	26	.	.	PUNCT
ejpam-6175	305	1	⊆	⊆	NUM
ejpam-6175	305	2	r(p	r(p	NOUN
ejpam-6175	305	3	,	,	PUNCT
ejpam-6175	305	4	rk	rk	NOUN
ejpam-6175	305	5	)	)	PUNCT
ejpam-6175	305	6	,	,	PUNCT
ejpam-6175	305	7	allowing	allow	VERB
ejpam-6175	305	8	the	the	DET
ejpam-6175	305	9	computation	computation	NOUN
ejpam-6175	305	10	of	of	ADP
ejpam-6175	305	11	persistent	persistent	ADJ
ejpam-6175	305	12	homology	homology	NOUN
ejpam-6175	305	13	.	.	PUNCT
ejpam-6175	306	1	persistent	persistent	ADJ
ejpam-6175	306	2	homology	homology	NOUN
ejpam-6175	306	3	identifies	identify	VERB
ejpam-6175	306	4	and	and	CCONJ
ejpam-6175	306	5	tracks	track	VERB
ejpam-6175	306	6	k	k	ADJ
ejpam-6175	306	7	-	-	ADJ
ejpam-6175	306	8	dimensional	dimensional	ADJ
ejpam-6175	306	9	topological	topological	ADJ
ejpam-6175	306	10	features	feature	NOUN
ejpam-6175	306	11	,	,	PUNCT
ejpam-6175	306	12	such	such	ADJ
ejpam-6175	306	13	as	as	ADP
ejpam-6175	306	14	connected	connected	ADJ
ejpam-6175	306	15	components	component	NOUN
ejpam-6175	306	16	(	(	PUNCT
ejpam-6175	306	17	h0	h0	NOUN
ejpam-6175	306	18	)	)	PUNCT
ejpam-6175	306	19	and	and	CCONJ
ejpam-6175	306	20	loops	loop	NOUN
ejpam-6175	306	21	(	(	PUNCT
ejpam-6175	306	22	h1	h1	PROPN
ejpam-6175	306	23	)	)	PUNCT
ejpam-6175	306	24	,	,	PUNCT
ejpam-6175	306	25	across	across	ADP
ejpam-6175	306	26	different	different	ADJ
ejpam-6175	306	27	radii	radius	NOUN
ejpam-6175	306	28	.	.	PUNCT
ejpam-6175	307	1	these	these	DET
ejpam-6175	307	2	features	feature	NOUN
ejpam-6175	307	3	are	be	AUX
ejpam-6175	307	4	summarized	summarize	VERB
ejpam-6175	307	5	in	in	ADP
ejpam-6175	307	6	a	a	DET
ejpam-6175	307	7	persistence	persistence	NOUN
ejpam-6175	307	8	diagram	diagram	NOUN
ejpam-6175	307	9	,	,	PUNCT
ejpam-6175	307	10	where	where	SCONJ
ejpam-6175	307	11	each	each	DET
ejpam-6175	307	12	point	point	NOUN
ejpam-6175	307	13	(	(	PUNCT
ejpam-6175	307	14	b	b	X
ejpam-6175	307	15	,	,	PUNCT
ejpam-6175	307	16	d	d	NOUN
ejpam-6175	307	17	)	)	PUNCT
ejpam-6175	307	18	represents	represent	VERB
ejpam-6175	307	19	the	the	DET
ejpam-6175	307	20	birth	birth	NOUN
ejpam-6175	307	21	and	and	CCONJ
ejpam-6175	307	22	death	death	NOUN
ejpam-6175	307	23	radii	radius	NOUN
ejpam-6175	307	24	of	of	ADP
ejpam-6175	307	25	a	a	DET
ejpam-6175	307	26	topological	topological	ADJ
ejpam-6175	307	27	feature	feature	NOUN
ejpam-6175	307	28	.	.	PUNCT
ejpam-6175	308	1	figure[5	figure[5	PUNCT
ejpam-6175	308	2	]	]	PUNCT
ejpam-6175	308	3	demonstrates	demonstrate	VERB
ejpam-6175	308	4	this	this	DET
ejpam-6175	308	5	process	process	NOUN
ejpam-6175	308	6	for	for	ADP
ejpam-6175	308	7	a	a	DET
ejpam-6175	308	8	2d	2d	NUM
ejpam-6175	308	9	point	point	NOUN
ejpam-6175	308	10	cloud	cloud	NOUN
ejpam-6175	308	11	,	,	PUNCT
ejpam-6175	308	12	illustrating	illustrate	VERB
ejpam-6175	308	13	how	how	SCONJ
ejpam-6175	308	14	the	the	DET
ejpam-6175	308	15	topology	topology	NOUN
ejpam-6175	308	16	evolves	evolve	VERB
ejpam-6175	308	17	as	as	ADP
ejpam-6175	308	18	the	the	DET
ejpam-6175	308	19	radius	radius	NOUN
ejpam-6175	308	20	r	r	NOUN
ejpam-6175	308	21	increases	increase	NOUN
ejpam-6175	308	22	,	,	PUNCT
ejpam-6175	308	23	eventually	eventually	ADV
ejpam-6175	308	24	leading	lead	VERB
ejpam-6175	308	25	to	to	ADP
ejpam-6175	308	26	full	full	ADJ
ejpam-6175	308	27	connectivity	connectivity	NOUN
ejpam-6175	308	28	.	.	PUNCT
ejpam-6175	309	1	the	the	DET
ejpam-6175	309	2	persistence	persistence	NOUN
ejpam-6175	309	3	diagram	diagram	NOUN
ejpam-6175	309	4	summarizes	summarize	NOUN
ejpam-6175	309	5	this	this	DET
ejpam-6175	309	6	evolution	evolution	NOUN
ejpam-6175	309	7	,	,	PUNCT
ejpam-6175	309	8	providing	provide	VERB
ejpam-6175	309	9	a	a	DET
ejpam-6175	309	10	concise	concise	ADJ
ejpam-6175	309	11	description	description	NOUN
ejpam-6175	309	12	of	of	ADP
ejpam-6175	309	13	the	the	DET
ejpam-6175	309	14	dataset	dataset	NOUN
ejpam-6175	309	15	’s	’s	PART
ejpam-6175	309	16	topological	topological	ADJ
ejpam-6175	309	17	structure	structure	NOUN
ejpam-6175	309	18	(	(	PUNCT
ejpam-6175	309	19	see	see	VERB
ejpam-6175	309	20	references[10–12	references[10–12	NOUN
ejpam-6175	309	21	]	]	PUNCT
ejpam-6175	309	22	)	)	PUNCT
ejpam-6175	309	23	.	.	PUNCT
ejpam-6175	310	1	3.4	3.4	NUM
ejpam-6175	310	2	.	.	PUNCT
ejpam-6175	310	3	algorithmic	algorithmic	ADJ
ejpam-6175	310	4	implementation	implementation	NOUN
ejpam-6175	310	5	and	and	CCONJ
ejpam-6175	310	6	applications	application	NOUN
ejpam-6175	310	7	the	the	DET
ejpam-6175	310	8	construction	construction	NOUN
ejpam-6175	310	9	of	of	ADP
ejpam-6175	310	10	rips	rip	NOUN
ejpam-6175	310	11	complexes	complex	NOUN
ejpam-6175	310	12	and	and	CCONJ
ejpam-6175	310	13	persistence	persistence	NOUN
ejpam-6175	310	14	diagrams	diagram	NOUN
ejpam-6175	310	15	relies	rely	VERB
ejpam-6175	310	16	on	on	ADP
ejpam-6175	310	17	an	an	DET
ejpam-6175	310	18	algorithmic	algorithmic	ADJ
ejpam-6175	310	19	approach	approach	NOUN
ejpam-6175	310	20	that	that	PRON
ejpam-6175	310	21	begins	begin	VERB
ejpam-6175	310	22	with	with	ADP
ejpam-6175	310	23	computing	compute	VERB
ejpam-6175	310	24	pairwise	pairwise	NOUN
ejpam-6175	310	25	distances	distance	NOUN
ejpam-6175	310	26	between	between	ADP
ejpam-6175	310	27	points	point	NOUN
ejpam-6175	310	28	.	.	PUNCT
ejpam-6175	311	1	as	as	ADP
ejpam-6175	311	2	the	the	DET
ejpam-6175	311	3	radius	radius	NOUN
ejpam-6175	311	4	s.k	s.k	PROPN
ejpam-6175	311	5	.	.	PUNCT
ejpam-6175	311	6	jha	jha	PROPN
ejpam-6175	311	7	sanjay	sanjay	PROPN
ejpam-6175	311	8	,	,	PUNCT
ejpam-6175	311	9	m	m	PROPN
ejpam-6175	311	10	,	,	PUNCT
ejpam-6175	311	11	natarajan	natarajan	PROPN
ejpam-6175	311	12	/	/	SYM
ejpam-6175	311	13	eur	eur	PROPN
ejpam-6175	311	14	.	.	PUNCT
ejpam-6175	312	1	j.	j.	PROPN
ejpam-6175	312	2	pure	pure	PROPN
ejpam-6175	312	3	appl	appl	PROPN
ejpam-6175	312	4	.	.	PROPN
ejpam-6175	312	5	math	math	PROPN
ejpam-6175	312	6	,	,	PUNCT
ejpam-6175	312	7	18	18	NUM
ejpam-6175	312	8	(	(	PUNCT
ejpam-6175	312	9	3	3	NUM
ejpam-6175	312	10	)	)	PUNCT
ejpam-6175	312	11	(	(	PUNCT
ejpam-6175	312	12	2025	2025	NUM
ejpam-6175	312	13	)	)	PUNCT
ejpam-6175	312	14	,	,	PUNCT
ejpam-6175	312	15	6175	6175	NUM
ejpam-6175	312	16	14	14	NUM
ejpam-6175	312	17	of	of	ADP
ejpam-6175	312	18	18	18	NUM
ejpam-6175	312	19	figure	figure	NOUN
ejpam-6175	312	20	6	6	NUM
ejpam-6175	312	21	:	:	PUNCT
ejpam-6175	312	22	brain	brain	NOUN
ejpam-6175	312	23	like	like	ADP
ejpam-6175	312	24	complex	complex	ADJ
ejpam-6175	312	25	structure	structure	NOUN
ejpam-6175	312	26	and	and	CCONJ
ejpam-6175	312	27	the	the	DET
ejpam-6175	312	28	corresponding	corresponding	ADJ
ejpam-6175	312	29	persistence	persistence	NOUN
ejpam-6175	312	30	diagram	diagram	NOUN
ejpam-6175	312	31	summarizing	summarize	VERB
ejpam-6175	312	32	topological	topological	ADJ
ejpam-6175	312	33	features	feature	NOUN
ejpam-6175	312	34	(	(	PUNCT
ejpam-6175	312	35	h0	h0	NOUN
ejpam-6175	312	36	and	and	CCONJ
ejpam-6175	312	37	h1	h1	PROPN
ejpam-6175	312	38	)	)	PUNCT
ejpam-6175	312	39	across	across	ADP
ejpam-6175	312	40	scales	scale	NOUN
ejpam-6175	312	41	.	.	PUNCT
ejpam-6175	313	1	increases	increase	NOUN
ejpam-6175	313	2	,	,	PUNCT
ejpam-6175	313	3	the	the	DET
ejpam-6175	313	4	algorithm	algorithm	NOUN
ejpam-6175	313	5	iteratively	iteratively	ADV
ejpam-6175	313	6	constructs	construct	VERB
ejpam-6175	313	7	rips	rip	NOUN
ejpam-6175	313	8	complexes	complex	NOUN
ejpam-6175	313	9	and	and	CCONJ
ejpam-6175	313	10	computes	compute	VERB
ejpam-6175	313	11	the	the	DET
ejpam-6175	313	12	corresponding	corresponding	ADJ
ejpam-6175	313	13	k	k	PROPN
ejpam-6175	313	14	-	-	PUNCT
ejpam-6175	313	15	th	th	VERB
ejpam-6175	313	16	homology	homology	NOUN
ejpam-6175	313	17	groups	group	NOUN
ejpam-6175	313	18	hk(r(p	hk(r(p	NOUN
ejpam-6175	313	19	,	,	PUNCT
ejpam-6175	313	20	r	r	NOUN
ejpam-6175	313	21	)	)	PUNCT
ejpam-6175	313	22	)	)	PUNCT
ejpam-6175	313	23	.	.	PUNCT
ejpam-6175	314	1	these	these	DET
ejpam-6175	314	2	homology	homology	NOUN
ejpam-6175	314	3	groups	group	NOUN
ejpam-6175	314	4	capture	capture	VERB
ejpam-6175	314	5	the	the	DET
ejpam-6175	314	6	features	feature	NOUN
ejpam-6175	314	7	of	of	ADP
ejpam-6175	314	8	different	different	ADJ
ejpam-6175	314	9	dimensions	dimension	NOUN
ejpam-6175	314	10	,	,	PUNCT
ejpam-6175	314	11	which	which	PRON
ejpam-6175	314	12	are	be	AUX
ejpam-6175	314	13	recorded	record	VERB
ejpam-6175	314	14	in	in	ADP
ejpam-6175	314	15	the	the	DET
ejpam-6175	314	16	persistence	persistence	NOUN
ejpam-6175	314	17	diagram	diagram	NOUN
ejpam-6175	314	18	.	.	PUNCT
ejpam-6175	315	1	this	this	DET
ejpam-6175	315	2	algorithmic	algorithmic	ADJ
ejpam-6175	315	3	framework	framework	NOUN
ejpam-6175	315	4	enables	enable	VERB
ejpam-6175	315	5	a	a	DET
ejpam-6175	315	6	systematic	systematic	ADJ
ejpam-6175	315	7	exploration	exploration	NOUN
ejpam-6175	315	8	of	of	ADP
ejpam-6175	315	9	data	datum	NOUN
ejpam-6175	315	10	topology[13	topology[13	X
ejpam-6175	315	11	]	]	X
ejpam-6175	315	12	.	.	PUNCT
ejpam-6175	316	1	this	this	DET
ejpam-6175	316	2	framework	framework	NOUN
ejpam-6175	316	3	is	be	AUX
ejpam-6175	316	4	also	also	ADV
ejpam-6175	316	5	extended	extend	VERB
ejpam-6175	316	6	to	to	ADP
ejpam-6175	316	7	complex	complex	ADJ
ejpam-6175	316	8	datasets	dataset	NOUN
ejpam-6175	316	9	,	,	PUNCT
ejpam-6175	316	10	as	as	SCONJ
ejpam-6175	316	11	shown	show	VERB
ejpam-6175	316	12	in	in	ADP
ejpam-6175	316	13	figure[6	figure[6	NOUN
ejpam-6175	316	14	]	]	PUNCT
ejpam-6175	316	15	.	.	PUNCT
ejpam-6175	317	1	here	here	ADV
ejpam-6175	317	2	,	,	PUNCT
ejpam-6175	317	3	each	each	DET
ejpam-6175	317	4	pixel	pixel	NOUN
ejpam-6175	317	5	of	of	ADP
ejpam-6175	317	6	a	a	DET
ejpam-6175	317	7	brain	brain	NOUN
ejpam-6175	317	8	image	image	NOUN
ejpam-6175	317	9	is	be	AUX
ejpam-6175	317	10	treated	treat	VERB
ejpam-6175	317	11	as	as	ADP
ejpam-6175	317	12	a	a	DET
ejpam-6175	317	13	point	point	NOUN
ejpam-6175	317	14	in	in	ADP
ejpam-6175	317	15	a	a	DET
ejpam-6175	317	16	high	high	ADJ
ejpam-6175	317	17	-	-	PUNCT
ejpam-6175	317	18	dimensional	dimensional	ADJ
ejpam-6175	317	19	space	space	NOUN
ejpam-6175	317	20	.	.	PUNCT
ejpam-6175	318	1	the	the	DET
ejpam-6175	318	2	algorithm	algorithm	NOUN
ejpam-6175	318	3	constructs	construct	VERB
ejpam-6175	318	4	a	a	DET
ejpam-6175	318	5	rips	rip	NOUN
ejpam-6175	318	6	complex	complex	NOUN
ejpam-6175	318	7	that	that	PRON
ejpam-6175	318	8	evolves	evolve	VERB
ejpam-6175	318	9	across	across	ADP
ejpam-6175	318	10	radii	radii	NOUN
ejpam-6175	318	11	,	,	PUNCT
ejpam-6175	318	12	and	and	CCONJ
ejpam-6175	318	13	the	the	DET
ejpam-6175	318	14	persistence	persistence	NOUN
ejpam-6175	318	15	diagram	diagram	NOUN
ejpam-6175	318	16	captures	capture	VERB
ejpam-6175	318	17	intricate	intricate	ADJ
ejpam-6175	318	18	connectivity	connectivity	NOUN
ejpam-6175	318	19	patterns	pattern	NOUN
ejpam-6175	318	20	within	within	ADP
ejpam-6175	318	21	the	the	DET
ejpam-6175	318	22	image	image	NOUN
ejpam-6175	318	23	.	.	PUNCT
ejpam-6175	319	1	the	the	DET
ejpam-6175	319	2	shaded	shade	VERB
ejpam-6175	319	3	circles	circle	NOUN
ejpam-6175	319	4	and	and	CCONJ
ejpam-6175	319	5	edges	edge	NOUN
ejpam-6175	319	6	in	in	ADP
ejpam-6175	319	7	the	the	DET
ejpam-6175	319	8	rips	rip	NOUN
ejpam-6175	319	9	complex	complex	NOUN
ejpam-6175	319	10	visually	visually	ADV
ejpam-6175	319	11	illustrate	illustrate	VERB
ejpam-6175	319	12	the	the	DET
ejpam-6175	319	13	evolution	evolution	NOUN
ejpam-6175	319	14	of	of	ADP
ejpam-6175	319	15	topological	topological	ADJ
ejpam-6175	319	16	features	feature	NOUN
ejpam-6175	319	17	,	,	PUNCT
ejpam-6175	319	18	even	even	ADV
ejpam-6175	319	19	in	in	ADP
ejpam-6175	319	20	noisy	noisy	ADJ
ejpam-6175	319	21	and	and	CCONJ
ejpam-6175	319	22	high	high	ADJ
ejpam-6175	319	23	-	-	PUNCT
ejpam-6175	319	24	dimensional	dimensional	ADJ
ejpam-6175	319	25	data	datum	NOUN
ejpam-6175	319	26	.	.	PUNCT
ejpam-6175	320	1	the	the	DET
ejpam-6175	320	2	computational	computational	ADJ
ejpam-6175	320	3	steps	step	NOUN
ejpam-6175	320	4	for	for	ADP
ejpam-6175	320	5	generating	generate	VERB
ejpam-6175	320	6	figure[6	figure[6	NOUN
ejpam-6175	320	7	]	]	X
ejpam-6175	320	8	are	be	AUX
ejpam-6175	320	9	explicitly	explicitly	ADV
ejpam-6175	320	10	detailed	detail	VERB
ejpam-6175	320	11	in	in	ADP
ejpam-6175	320	12	the	the	DET
ejpam-6175	320	13	provided	provide	VERB
ejpam-6175	320	14	algorithmic	algorithmic	ADJ
ejpam-6175	320	15	model[1	model[1	NOUN
ejpam-6175	320	16	]	]	PUNCT
ejpam-6175	320	17	.	.	PUNCT
ejpam-6175	321	1	this	this	DET
ejpam-6175	321	2	model	model	NOUN
ejpam-6175	321	3	outlines	outline	VERB
ejpam-6175	321	4	the	the	DET
ejpam-6175	321	5	method	method	NOUN
ejpam-6175	321	6	for	for	ADP
ejpam-6175	321	7	preprocessing	preprocesse	VERB
ejpam-6175	321	8	the	the	DET
ejpam-6175	321	9	image	image	NOUN
ejpam-6175	321	10	,	,	PUNCT
ejpam-6175	321	11	constructing	construct	VERB
ejpam-6175	321	12	rips	rip	NOUN
ejpam-6175	321	13	complexes	complex	NOUN
ejpam-6175	321	14	,	,	PUNCT
ejpam-6175	321	15	and	and	CCONJ
ejpam-6175	321	16	calculating	calculate	VERB
ejpam-6175	321	17	persistent	persistent	ADJ
ejpam-6175	321	18	homology	homology	NOUN
ejpam-6175	321	19	,	,	PUNCT
ejpam-6175	321	20	providing	provide	VERB
ejpam-6175	321	21	a	a	DET
ejpam-6175	321	22	step	step	NOUN
ejpam-6175	321	23	-	-	PUNCT
ejpam-6175	321	24	bystep	bystep	NOUN
ejpam-6175	321	25	explanation	explanation	NOUN
ejpam-6175	321	26	of	of	ADP
ejpam-6175	321	27	how	how	SCONJ
ejpam-6175	321	28	tda	tda	NOUN
ejpam-6175	321	29	is	be	AUX
ejpam-6175	321	30	applied	apply	VERB
ejpam-6175	321	31	to	to	ADP
ejpam-6175	321	32	complex	complex	ADJ
ejpam-6175	321	33	image	image	NOUN
ejpam-6175	321	34	data	datum	NOUN
ejpam-6175	321	35	.	.	PUNCT
ejpam-6175	322	1	by	by	ADP
ejpam-6175	322	2	referring	refer	VERB
ejpam-6175	322	3	to	to	ADP
ejpam-6175	322	4	this	this	DET
ejpam-6175	322	5	pseudocode	pseudocode	NOUN
ejpam-6175	322	6	,	,	PUNCT
ejpam-6175	322	7	readers	reader	NOUN
ejpam-6175	322	8	can	can	AUX
ejpam-6175	322	9	replicate	replicate	VERB
ejpam-6175	322	10	the	the	DET
ejpam-6175	322	11	methodology	methodology	NOUN
ejpam-6175	322	12	for	for	ADP
ejpam-6175	322	13	other	other	ADJ
ejpam-6175	322	14	high	high	ADJ
ejpam-6175	322	15	-	-	PUNCT
ejpam-6175	322	16	dimensional	dimensional	ADJ
ejpam-6175	322	17	datasets	dataset	NOUN
ejpam-6175	322	18	.	.	PUNCT
ejpam-6175	323	1	these	these	DET
ejpam-6175	323	2	results	result	NOUN
ejpam-6175	323	3	emphasize	emphasize	VERB
ejpam-6175	323	4	the	the	DET
ejpam-6175	323	5	robustness	robustness	NOUN
ejpam-6175	323	6	of	of	ADP
ejpam-6175	323	7	tda	tda	NOUN
ejpam-6175	323	8	in	in	ADP
ejpam-6175	323	9	capturing	capture	VERB
ejpam-6175	323	10	meaningful	meaningful	ADJ
ejpam-6175	323	11	topological	topological	ADJ
ejpam-6175	323	12	features	feature	NOUN
ejpam-6175	323	13	,	,	PUNCT
ejpam-6175	323	14	making	make	VERB
ejpam-6175	323	15	it	it	PRON
ejpam-6175	323	16	a	a	DET
ejpam-6175	323	17	valuable	valuable	ADJ
ejpam-6175	323	18	tool	tool	NOUN
ejpam-6175	323	19	across	across	ADP
ejpam-6175	323	20	domains	domain	NOUN
ejpam-6175	323	21	such	such	ADJ
ejpam-6175	323	22	as	as	ADP
ejpam-6175	323	23	shape	shape	NOUN
ejpam-6175	323	24	analysis	analysis	NOUN
ejpam-6175	323	25	,	,	PUNCT
ejpam-6175	323	26	image	image	NOUN
ejpam-6175	323	27	processing	processing	NOUN
ejpam-6175	323	28	,	,	PUNCT
ejpam-6175	323	29	and	and	CCONJ
ejpam-6175	323	30	data	datum	NOUN
ejpam-6175	323	31	clustering[14	clustering[14	PROPN
ejpam-6175	323	32	,	,	PUNCT
ejpam-6175	323	33	15	15	NUM
ejpam-6175	323	34	]	]	PUNCT
ejpam-6175	323	35	.	.	PUNCT
ejpam-6175	324	1	the	the	DET
ejpam-6175	324	2	computation	computation	NOUN
ejpam-6175	324	3	of	of	ADP
ejpam-6175	324	4	simplicial	simplicial	ADJ
ejpam-6175	324	5	homology	homology	NOUN
ejpam-6175	324	6	groups	group	NOUN
ejpam-6175	324	7	for	for	ADP
ejpam-6175	324	8	2d	2d	NUM
ejpam-6175	324	9	digital	digital	ADJ
ejpam-6175	324	10	images	image	NOUN
ejpam-6175	324	11	has	have	AUX
ejpam-6175	324	12	been	be	AUX
ejpam-6175	324	13	explored	explore	VERB
ejpam-6175	324	14	in	in	ADP
ejpam-6175	324	15	foundational	foundational	ADJ
ejpam-6175	324	16	work	work	NOUN
ejpam-6175	324	17	by	by	ADP
ejpam-6175	324	18	karaca	karaca	PROPN
ejpam-6175	324	19	and	and	CCONJ
ejpam-6175	324	20	ege	ege	PROPN
ejpam-6175	325	1	[	[	X
ejpam-6175	325	2	16	16	NUM
ejpam-6175	325	3	]	]	PUNCT
ejpam-6175	325	4	,	,	PUNCT
ejpam-6175	325	5	and	and	CCONJ
ejpam-6175	325	6	further	far	ADV
ejpam-6175	325	7	extended	extend	VERB
ejpam-6175	325	8	through	through	ADP
ejpam-6175	325	9	efficient	efficient	ADJ
ejpam-6175	325	10	algorithms	algorithm	NOUN
ejpam-6175	325	11	for	for	ADP
ejpam-6175	325	12	calculating	calculate	VERB
ejpam-6175	325	13	betti	betti	NOUN
ejpam-6175	325	14	numbers	number	NOUN
ejpam-6175	325	15	and	and	CCONJ
ejpam-6175	325	16	euler	euler	NOUN
ejpam-6175	325	17	characteristics	characteristic	NOUN
ejpam-6175	325	18	in	in	ADP
ejpam-6175	325	19	recent	recent	ADJ
ejpam-6175	325	20	studies	study	NOUN
ejpam-6175	325	21	[	[	X
ejpam-6175	325	22	17	17	NUM
ejpam-6175	325	23	]	]	PUNCT
ejpam-6175	325	24	.	.	PUNCT
ejpam-6175	326	1	as	as	ADP
ejpam-6175	326	2	a	a	DET
ejpam-6175	326	3	result	result	NOUN
ejpam-6175	326	4	shown	show	VERB
ejpam-6175	326	5	in	in	ADP
ejpam-6175	326	6	figure[7	figure[7	NOUN
ejpam-6175	326	7	]	]	PUNCT
ejpam-6175	326	8	.	.	PUNCT
ejpam-6175	327	1	this	this	DET
ejpam-6175	327	2	technique	technique	NOUN
ejpam-6175	327	3	focuses	focus	VERB
ejpam-6175	327	4	on	on	ADP
ejpam-6175	327	5	the	the	DET
ejpam-6175	327	6	regions	region	NOUN
ejpam-6175	327	7	identified	identify	VERB
ejpam-6175	327	8	through	through	ADP
ejpam-6175	327	9	mapping	mapping	NOUN
ejpam-6175	327	10	,	,	PUNCT
ejpam-6175	327	11	enabling	enable	VERB
ejpam-6175	327	12	precise	precise	ADJ
ejpam-6175	327	13	marking	marking	NOUN
ejpam-6175	327	14	of	of	ADP
ejpam-6175	327	15	areas	area	NOUN
ejpam-6175	327	16	affected	affect	VERB
ejpam-6175	327	17	by	by	ADP
ejpam-6175	327	18	tumor	tumor	NOUN
ejpam-6175	327	19	cells	cell	NOUN
ejpam-6175	327	20	.	.	PUNCT
ejpam-6175	328	1	by	by	ADP
ejpam-6175	328	2	transitioning	transition	VERB
ejpam-6175	328	3	from	from	ADP
ejpam-6175	328	4	the	the	DET
ejpam-6175	328	5	original	original	ADJ
ejpam-6175	328	6	mri	mri	NOUN
ejpam-6175	328	7	scans	scan	NOUN
ejpam-6175	328	8	to	to	ADP
ejpam-6175	328	9	edge	edge	NOUN
ejpam-6175	328	10	-	-	PUNCT
ejpam-6175	328	11	detected	detect	VERB
ejpam-6175	328	12	images[18	images[18	NOUN
ejpam-6175	328	13	]	]	PUNCT
ejpam-6175	328	14	and	and	CCONJ
ejpam-6175	328	15	finally	finally	ADV
ejpam-6175	328	16	to	to	PART
ejpam-6175	328	17	graph	graph	VERB
ejpam-6175	328	18	representations	representation	NOUN
ejpam-6175	328	19	,	,	PUNCT
ejpam-6175	328	20	this	this	DET
ejpam-6175	328	21	approach	approach	NOUN
ejpam-6175	328	22	provides	provide	VERB
ejpam-6175	328	23	a	a	DET
ejpam-6175	328	24	comprehensive	comprehensive	ADJ
ejpam-6175	328	25	analysis	analysis	NOUN
ejpam-6175	328	26	of	of	ADP
ejpam-6175	328	27	brain	brain	NOUN
ejpam-6175	328	28	structures	structure	NOUN
ejpam-6175	328	29	.	.	PUNCT
ejpam-6175	329	1	figure	figure	NOUN
ejpam-6175	330	1	[	[	X
ejpam-6175	330	2	8	8	NUM
ejpam-6175	330	3	]	]	PUNCT
ejpam-6175	330	4	illustrates	illustrate	VERB
ejpam-6175	330	5	the	the	DET
ejpam-6175	330	6	process	process	NOUN
ejpam-6175	330	7	of	of	ADP
ejpam-6175	330	8	identifying	identify	VERB
ejpam-6175	330	9	and	and	CCONJ
ejpam-6175	330	10	marking	mark	VERB
ejpam-6175	330	11	the	the	DET
ejpam-6175	330	12	affected	affected	ADJ
ejpam-6175	330	13	area	area	NOUN
ejpam-6175	330	14	within	within	ADP
ejpam-6175	330	15	an	an	DET
ejpam-6175	330	16	mri	mri	NOUN
ejpam-6175	330	17	-	-	PUNCT
ejpam-6175	330	18	scanned	scan	VERB
ejpam-6175	330	19	image	image	NOUN
ejpam-6175	330	20	.	.	PUNCT
ejpam-6175	331	1	s.k	s.k	PROPN
ejpam-6175	331	2	.	.	PUNCT
ejpam-6175	331	3	jha	jha	PROPN
ejpam-6175	331	4	sanjay	sanjay	PROPN
ejpam-6175	331	5	,	,	PUNCT
ejpam-6175	331	6	m	m	PROPN
ejpam-6175	331	7	,	,	PUNCT
ejpam-6175	331	8	natarajan	natarajan	PROPN
ejpam-6175	331	9	/	/	SYM
ejpam-6175	331	10	eur	eur	PROPN
ejpam-6175	331	11	.	.	PUNCT
ejpam-6175	332	1	j.	j.	PROPN
ejpam-6175	332	2	pure	pure	PROPN
ejpam-6175	332	3	appl	appl	PROPN
ejpam-6175	332	4	.	.	PROPN
ejpam-6175	332	5	math	math	PROPN
ejpam-6175	332	6	,	,	PUNCT
ejpam-6175	332	7	18	18	NUM
ejpam-6175	332	8	(	(	PUNCT
ejpam-6175	332	9	3	3	NUM
ejpam-6175	332	10	)	)	PUNCT
ejpam-6175	332	11	(	(	PUNCT
ejpam-6175	332	12	2025	2025	NUM
ejpam-6175	332	13	)	)	PUNCT
ejpam-6175	332	14	,	,	PUNCT
ejpam-6175	332	15	6175	6175	NUM
ejpam-6175	332	16	15	15	NUM
ejpam-6175	332	17	of	of	ADP
ejpam-6175	332	18	18	18	NUM
ejpam-6175	332	19	the	the	DET
ejpam-6175	332	20	precise	precise	ADJ
ejpam-6175	332	21	delineation	delineation	NOUN
ejpam-6175	332	22	of	of	ADP
ejpam-6175	332	23	this	this	DET
ejpam-6175	332	24	region	region	NOUN
ejpam-6175	332	25	is	be	AUX
ejpam-6175	332	26	achieved	achieve	VERB
ejpam-6175	332	27	through	through	ADP
ejpam-6175	332	28	advanced	advanced	ADJ
ejpam-6175	332	29	image	image	NOUN
ejpam-6175	332	30	processing	processing	NOUN
ejpam-6175	332	31	techniques	technique	NOUN
ejpam-6175	332	32	,	,	PUNCT
ejpam-6175	332	33	ensuring	ensure	VERB
ejpam-6175	332	34	accurate	accurate	ADJ
ejpam-6175	332	35	correspondence	correspondence	NOUN
ejpam-6175	332	36	between	between	ADP
ejpam-6175	332	37	the	the	DET
ejpam-6175	332	38	affected	affected	ADJ
ejpam-6175	332	39	area	area	NOUN
ejpam-6175	332	40	and	and	CCONJ
ejpam-6175	332	41	its	its	PRON
ejpam-6175	332	42	representation	representation	NOUN
ejpam-6175	332	43	on	on	ADP
ejpam-6175	332	44	a	a	DET
ejpam-6175	332	45	graphical	graphical	ADJ
ejpam-6175	332	46	map	map	NOUN
ejpam-6175	332	47	.	.	PUNCT
ejpam-6175	333	1	this	this	DET
ejpam-6175	333	2	approach	approach	NOUN
ejpam-6175	333	3	enhances	enhance	VERB
ejpam-6175	333	4	the	the	DET
ejpam-6175	333	5	ability	ability	NOUN
ejpam-6175	333	6	to	to	PART
ejpam-6175	333	7	visualize	visualize	VERB
ejpam-6175	333	8	and	and	CCONJ
ejpam-6175	333	9	analyze	analyze	VERB
ejpam-6175	333	10	the	the	DET
ejpam-6175	333	11	extent	extent	NOUN
ejpam-6175	333	12	of	of	ADP
ejpam-6175	333	13	the	the	DET
ejpam-6175	333	14	impact	impact	NOUN
ejpam-6175	333	15	,	,	PUNCT
ejpam-6175	333	16	thereby	thereby	ADV
ejpam-6175	333	17	facilitating	facilitate	VERB
ejpam-6175	333	18	more	more	ADV
ejpam-6175	333	19	targeted	targeted	ADJ
ejpam-6175	333	20	and	and	CCONJ
ejpam-6175	333	21	effective	effective	ADJ
ejpam-6175	333	22	assessments	assessment	NOUN
ejpam-6175	333	23	.	.	PUNCT
ejpam-6175	334	1	similar	similar	ADJ
ejpam-6175	334	2	work	work	NOUN
ejpam-6175	334	3	can	can	AUX
ejpam-6175	334	4	be	be	AUX
ejpam-6175	334	5	seen	see	VERB
ejpam-6175	334	6	in	in	ADP
ejpam-6175	334	7	[	[	X
ejpam-6175	334	8	19	19	NUM
ejpam-6175	334	9	]	]	PUNCT
ejpam-6175	334	10	,	,	PUNCT
ejpam-6175	334	11	where	where	SCONJ
ejpam-6175	334	12	they	they	PRON
ejpam-6175	334	13	used	use	VERB
ejpam-6175	334	14	the	the	DET
ejpam-6175	334	15	linear	linear	PROPN
ejpam-6175	334	16	homotopy	homotopy	NOUN
ejpam-6175	334	17	concept	concept	NOUN
ejpam-6175	334	18	to	to	PART
ejpam-6175	334	19	capture	capture	VERB
ejpam-6175	334	20	fine	fine	ADJ
ejpam-6175	334	21	edges	edge	NOUN
ejpam-6175	334	22	and	and	CCONJ
ejpam-6175	334	23	provide	provide	VERB
ejpam-6175	334	24	enhanced	enhanced	ADJ
ejpam-6175	334	25	edge	edge	NOUN
ejpam-6175	334	26	detection	detection	NOUN
ejpam-6175	334	27	.	.	PUNCT
ejpam-6175	335	1	this	this	DET
ejpam-6175	335	2	method	method	NOUN
ejpam-6175	335	3	enhances	enhance	VERB
ejpam-6175	335	4	the	the	DET
ejpam-6175	335	5	ability	ability	NOUN
ejpam-6175	335	6	to	to	PART
ejpam-6175	335	7	detect	detect	VERB
ejpam-6175	335	8	and	and	CCONJ
ejpam-6175	335	9	understand	understand	VERB
ejpam-6175	335	10	the	the	DET
ejpam-6175	335	11	underlying	underlying	ADJ
ejpam-6175	335	12	connections	connection	NOUN
ejpam-6175	335	13	and	and	CCONJ
ejpam-6175	335	14	abnormalities	abnormality	NOUN
ejpam-6175	335	15	,	,	PUNCT
ejpam-6175	335	16	facilitating	facilitate	VERB
ejpam-6175	335	17	accurate	accurate	ADJ
ejpam-6175	335	18	identification	identification	NOUN
ejpam-6175	335	19	and	and	CCONJ
ejpam-6175	335	20	diagnosis	diagnosis	NOUN
ejpam-6175	335	21	of	of	ADP
ejpam-6175	335	22	affected	affected	ADJ
ejpam-6175	335	23	regions	region	NOUN
ejpam-6175	335	24	.	.	PUNCT
ejpam-6175	336	1	the	the	DET
ejpam-6175	336	2	detailed	detailed	ADJ
ejpam-6175	336	3	graph	graph	NOUN
ejpam-6175	336	4	representations	representation	NOUN
ejpam-6175	336	5	are	be	AUX
ejpam-6175	336	6	particularly	particularly	ADV
ejpam-6175	336	7	useful	useful	ADJ
ejpam-6175	336	8	for	for	ADP
ejpam-6175	336	9	visualizing	visualize	VERB
ejpam-6175	336	10	the	the	DET
ejpam-6175	336	11	connectivity	connectivity	NOUN
ejpam-6175	336	12	within	within	ADP
ejpam-6175	336	13	the	the	DET
ejpam-6175	336	14	brain	brain	NOUN
ejpam-6175	336	15	,	,	PUNCT
ejpam-6175	336	16	aiding	aid	VERB
ejpam-6175	336	17	in	in	ADP
ejpam-6175	336	18	the	the	DET
ejpam-6175	336	19	study	study	NOUN
ejpam-6175	336	20	of	of	ADP
ejpam-6175	336	21	its	its	PRON
ejpam-6175	336	22	structural	structural	ADJ
ejpam-6175	336	23	integrity	integrity	NOUN
ejpam-6175	336	24	and	and	CCONJ
ejpam-6175	336	25	potential	potential	ADJ
ejpam-6175	336	26	pathological	pathological	ADJ
ejpam-6175	336	27	changes	change	NOUN
ejpam-6175	336	28	.	.	PUNCT
ejpam-6175	337	1	here	here	ADV
ejpam-6175	337	2	’s	’	VERB
ejpam-6175	337	3	a	a	DET
ejpam-6175	337	4	pseudocode	pseudocode	NOUN
ejpam-6175	337	5	of	of	ADP
ejpam-6175	337	6	the	the	DET
ejpam-6175	337	7	following	following	ADJ
ejpam-6175	337	8	model[2	model[2	NOUN
ejpam-6175	337	9	]	]	PUNCT
ejpam-6175	337	10	.	.	PUNCT
ejpam-6175	338	1	algorithm	algorithm	NOUN
ejpam-6175	338	2	2	2	NUM
ejpam-6175	338	3	pseudo	pseudo	NOUN
ejpam-6175	338	4	code	code	NOUN
ejpam-6175	338	5	:	:	PUNCT
ejpam-6175	338	6	2	2	NUM
ejpam-6175	338	7	require	require	NOUN
ejpam-6175	338	8	:	:	PUNCT
ejpam-6175	338	9	import	import	NOUN
ejpam-6175	338	10	required	require	VERB
ejpam-6175	338	11	libraries	library	NOUN
ejpam-6175	338	12	for	for	ADP
ejpam-6175	338	13	image	image	NOUN
ejpam-6175	338	14	processing	processing	NOUN
ejpam-6175	338	15	and	and	CCONJ
ejpam-6175	338	16	visualization	visualization	NOUN
ejpam-6175	338	17	define	define	VERB
ejpam-6175	338	18	edgedetection	edgedetection	NOUN
ejpam-6175	338	19	to	to	PART
ejpam-6175	338	20	load	load	VERB
ejpam-6175	338	21	image	image	NOUN
ejpam-6175	338	22	i(x	i(x	PROPN
ejpam-6175	338	23	,	,	PUNCT
ejpam-6175	338	24	y	y	NOUN
ejpam-6175	338	25	)	)	PUNCT
ejpam-6175	338	26	raise	raise	VERB
ejpam-6175	338	27	error	error	NOUN
ejpam-6175	338	28	if	if	SCONJ
ejpam-6175	338	29	i(x	i(x	PROPN
ejpam-6175	338	30	,	,	PUNCT
ejpam-6175	338	31	y	y	NOUN
ejpam-6175	338	32	)	)	PUNCT
ejpam-6175	338	33	fails	fail	VERB
ejpam-6175	338	34	to	to	PART
ejpam-6175	338	35	load	load	VERB
ejpam-6175	338	36	.	.	PUNCT
ejpam-6175	339	1	while	while	SCONJ
ejpam-6175	339	2	apply	apply	VERB
ejpam-6175	339	3	canny	canny	ADJ
ejpam-6175	339	4	edge	edge	NOUN
ejpam-6175	339	5	detection	detection	NOUN
ejpam-6175	339	6	to	to	PART
ejpam-6175	339	7	find	find	VERB
ejpam-6175	339	8	gradient	gradient	ADJ
ejpam-6175	339	9	magnitude	magnitude	NOUN
ejpam-6175	339	10	do	do	AUX
ejpam-6175	339	11	return	return	VERB
ejpam-6175	339	12	edge	edge	NOUN
ejpam-6175	339	13	map	map	NOUN
ejpam-6175	339	14	e(x	e(x	NUM
ejpam-6175	339	15	,	,	PUNCT
ejpam-6175	339	16	y	y	NOUN
ejpam-6175	339	17	)	)	PUNCT
ejpam-6175	339	18	where	where	SCONJ
ejpam-6175	339	19	gradients	gradient	NOUN
ejpam-6175	339	20	are	be	AUX
ejpam-6175	339	21	strong	strong	ADJ
ejpam-6175	339	22	define	define	VERB
ejpam-6175	339	23	find	find	NOUN
ejpam-6175	339	24	-	-	PUNCT
ejpam-6175	339	25	keypoints	keypoint	NOUN
ejpam-6175	339	26	to	to	PART
ejpam-6175	339	27	detect	detect	VERB
ejpam-6175	339	28	corners	corner	NOUN
ejpam-6175	339	29	using	use	VERB
ejpam-6175	339	30	harris	harris	PROPN
ejpam-6175	339	31	-	-	PUNCT
ejpam-6175	339	32	stephens	stephens	PROPN
ejpam-6175	339	33	method	method	NOUN
ejpam-6175	339	34	convert	convert	NOUN
ejpam-6175	339	35	corners	corner	NOUN
ejpam-6175	339	36	c(xi	c(xi	PROPN
ejpam-6175	339	37	,	,	PUNCT
ejpam-6175	339	38	yi	yi	PROPN
ejpam-6175	339	39	to	to	ADP
ejpam-6175	339	40	integer	integer	PROPN
ejpam-6175	339	41	coordinates	coordinate	NOUN
ejpam-6175	339	42	apply	apply	VERB
ejpam-6175	339	43	kmeans	kmean	NOUN
ejpam-6175	339	44	to	to	PART
ejpam-6175	339	45	cluster	cluster	VERB
ejpam-6175	339	46	key	key	ADJ
ejpam-6175	339	47	points	point	NOUN
ejpam-6175	339	48	ki	ki	PROPN
ejpam-6175	339	49	into	into	ADP
ejpam-6175	339	50	k	k	PROPN
ejpam-6175	339	51	clusters	clusters	PROPN
ejpam-6175	339	52	return	return	VERB
ejpam-6175	339	53	centroids	centroid	NOUN
ejpam-6175	339	54	kc	kc	PROPN
ejpam-6175	339	55	=	=	SYM
ejpam-6175	339	56	(	(	PUNCT
ejpam-6175	339	57	xc	xc	PROPN
ejpam-6175	339	58	,	,	PUNCT
ejpam-6175	339	59	yc	yc	NOUN
ejpam-6175	339	60	)	)	PUNCT
ejpam-6175	339	61	of	of	ADP
ejpam-6175	339	62	clusters	cluster	NOUN
ejpam-6175	339	63	.	.	PUNCT
ejpam-6175	340	1	define	define	VERB
ejpam-6175	340	2	create	create	NOUN
ejpam-6175	340	3	-	-	PUNCT
ejpam-6175	340	4	simplicial	simplicial	NOUN
ejpam-6175	340	5	-	-	PUNCT
ejpam-6175	340	6	complex	complex	NOUN
ejpam-6175	340	7	for	for	SCONJ
ejpam-6175	340	8	delaunay	delaunay	PROPN
ejpam-6175	340	9	triangulation	triangulation	NOUN
ejpam-6175	340	10	construct	construct	VERB
ejpam-6175	340	11	simplicial	simplicial	ADJ
ejpam-6175	340	12	complex	complex	NOUN
ejpam-6175	340	13	∆(s	∆(s	NOUN
ejpam-6175	340	14	)	)	PUNCT
ejpam-6175	340	15	from	from	ADP
ejpam-6175	340	16	points	point	NOUN
ejpam-6175	340	17	kc	kc	PROPN
ejpam-6175	340	18	graph	graph	NOUN
ejpam-6175	340	19	g(v	g(v	PROPN
ejpam-6175	340	20	,	,	PUNCT
ejpam-6175	340	21	e	e	NOUN
ejpam-6175	340	22	)	)	PUNCT
ejpam-6175	340	23	is	be	AUX
ejpam-6175	340	24	formed	form	VERB
ejpam-6175	340	25	,	,	PUNCT
ejpam-6175	340	26	where	where	SCONJ
ejpam-6175	340	27	v	v	NOUN
ejpam-6175	340	28	are	be	AUX
ejpam-6175	340	29	vertices	vertex	NOUN
ejpam-6175	340	30	.	.	PUNCT
ejpam-6175	341	1	return	return	NOUN
ejpam-6175	341	2	graph	graph	NOUN
ejpam-6175	341	3	g	g	NOUN
ejpam-6175	341	4	and	and	CCONJ
ejpam-6175	341	5	positions	position	NOUN
ejpam-6175	341	6	p	p	X
ejpam-6175	341	7	(	(	PUNCT
ejpam-6175	341	8	x	x	NOUN
ejpam-6175	341	9	,	,	PUNCT
ejpam-6175	341	10	y	y	NOUN
ejpam-6175	341	11	)	)	PUNCT
ejpam-6175	341	12	define	define	VERB
ejpam-6175	341	13	draw	draw	ADJ
ejpam-6175	341	14	-	-	PUNCT
ejpam-6175	341	15	simplicial	simplicial	ADJ
ejpam-6175	341	16	-	-	PUNCT
ejpam-6175	341	17	complex	complex	NOUN
ejpam-6175	341	18	to	to	PART
ejpam-6175	341	19	plot	plot	VERB
ejpam-6175	341	20	g(v	g(v	PROPN
ejpam-6175	341	21	,	,	PUNCT
ejpam-6175	341	22	e	e	NOUN
ejpam-6175	341	23	)	)	PUNCT
ejpam-6175	341	24	plot	plot	NOUN
ejpam-6175	341	25	nodes	node	NOUN
ejpam-6175	341	26	v	v	NOUN
ejpam-6175	341	27	as	as	ADP
ejpam-6175	341	28	points	point	NOUN
ejpam-6175	341	29	in	in	ADP
ejpam-6175	341	30	2d	2d	NUM
ejpam-6175	341	31	space	space	NOUN
ejpam-6175	341	32	plot	plot	NOUN
ejpam-6175	341	33	edges	edge	NOUN
ejpam-6175	341	34	e	e	NOUN
ejpam-6175	341	35	as	as	ADP
ejpam-6175	341	36	connections	connection	NOUN
ejpam-6175	341	37	between	between	ADP
ejpam-6175	341	38	nodes	node	NOUN
ejpam-6175	341	39	.	.	PUNCT
ejpam-6175	342	1	adjust	adjust	VERB
ejpam-6175	342	2	plot	plot	NOUN
ejpam-6175	342	3	to	to	PART
ejpam-6175	342	4	match	match	VERB
ejpam-6175	342	5	image	image	NOUN
ejpam-6175	342	6	dimensions	dimension	NOUN
ejpam-6175	342	7	(	(	PUNCT
ejpam-6175	342	8	w	w	PROPN
ejpam-6175	342	9	,	,	PUNCT
ejpam-6175	342	10	h	h	NOUN
ejpam-6175	342	11	)	)	PUNCT
ejpam-6175	342	12	display	display	NOUN
ejpam-6175	342	13	visualization	visualization	NOUN
ejpam-6175	342	14	of	of	ADP
ejpam-6175	342	15	g(v	g(v	PROPN
ejpam-6175	342	16	,	,	PUNCT
ejpam-6175	342	17	e	e	NOUN
ejpam-6175	342	18	)	)	PUNCT
ejpam-6175	342	19	define	define	VERB
ejpam-6175	342	20	main	main	ADJ
ejpam-6175	342	21	to	to	PART
ejpam-6175	342	22	execute	execute	VERB
ejpam-6175	342	23	detection	detection	NOUN
ejpam-6175	342	24	,	,	PUNCT
ejpam-6175	342	25	clustering	clustering	NOUN
ejpam-6175	342	26	,	,	PUNCT
ejpam-6175	342	27	and	and	CCONJ
ejpam-6175	342	28	complex	complex	ADJ
ejpam-6175	342	29	creation	creation	NOUN
ejpam-6175	342	30	call	call	NOUN
ejpam-6175	342	31	main	main	ADJ
ejpam-6175	342	32	with	with	ADP
ejpam-6175	342	33	image	image	NOUN
ejpam-6175	342	34	path	path	NOUN
ejpam-6175	342	35	i(x	i(x	PROPN
ejpam-6175	342	36	,	,	PUNCT
ejpam-6175	342	37	y	y	PROPN
ejpam-6175	342	38	)	)	PUNCT
ejpam-6175	342	39	and	and	CCONJ
ejpam-6175	342	40	max	max	PROPN
ejpam-6175	342	41	points	point	NOUN
ejpam-6175	342	42	k	k	PROPN
ejpam-6175	342	43	display	display	PROPN
ejpam-6175	342	44	e(x	e(x	NUM
ejpam-6175	342	45	,	,	PUNCT
ejpam-6175	342	46	y	y	NOUN
ejpam-6175	342	47	)	)	PUNCT
ejpam-6175	342	48	and	and	CCONJ
ejpam-6175	342	49	simplicial	simplicial	ADJ
ejpam-6175	342	50	complex	complex	ADJ
ejpam-6175	342	51	∆(s	∆(s	NOUN
ejpam-6175	342	52	)	)	PUNCT
ejpam-6175	342	53	end	end	VERB
ejpam-6175	342	54	whileanatomical	whileanatomical	ADJ
ejpam-6175	342	55	connections	connection	NOUN
ejpam-6175	342	56	and	and	CCONJ
ejpam-6175	342	57	graphical	graphical	ADJ
ejpam-6175	342	58	representation	representation	NOUN
ejpam-6175	342	59	.	.	PUNCT
ejpam-6175	343	1	=	=	X
ejpam-6175	343	2	0	0	NUM
ejpam-6175	343	3	s.k	s.k	PROPN
ejpam-6175	343	4	.	.	PUNCT
ejpam-6175	343	5	jha	jha	PROPN
ejpam-6175	343	6	sanjay	sanjay	PROPN
ejpam-6175	343	7	,	,	PUNCT
ejpam-6175	343	8	m	m	PROPN
ejpam-6175	343	9	,	,	PUNCT
ejpam-6175	343	10	natarajan	natarajan	PROPN
ejpam-6175	343	11	/	/	SYM
ejpam-6175	343	12	eur	eur	PROPN
ejpam-6175	343	13	.	.	PUNCT
ejpam-6175	344	1	j.	j.	PROPN
ejpam-6175	344	2	pure	pure	PROPN
ejpam-6175	344	3	appl	appl	PROPN
ejpam-6175	344	4	.	.	PROPN
ejpam-6175	344	5	math	math	PROPN
ejpam-6175	344	6	,	,	PUNCT
ejpam-6175	344	7	18	18	NUM
ejpam-6175	344	8	(	(	PUNCT
ejpam-6175	344	9	3	3	NUM
ejpam-6175	344	10	)	)	PUNCT
ejpam-6175	344	11	(	(	PUNCT
ejpam-6175	344	12	2025	2025	NUM
ejpam-6175	344	13	)	)	PUNCT
ejpam-6175	344	14	,	,	PUNCT
ejpam-6175	344	15	6175	6175	NUM
ejpam-6175	344	16	16	16	NUM
ejpam-6175	344	17	of	of	ADP
ejpam-6175	344	18	18	18	NUM
ejpam-6175	344	19	figure	figure	NOUN
ejpam-6175	344	20	7	7	NUM
ejpam-6175	344	21	:	:	PUNCT
ejpam-6175	344	22	analysis	analysis	NOUN
ejpam-6175	344	23	of	of	ADP
ejpam-6175	344	24	mri	mri	NOUN
ejpam-6175	344	25	brain	brain	NOUN
ejpam-6175	344	26	scans	scan	NOUN
ejpam-6175	344	27	.	.	PUNCT
ejpam-6175	345	1	left	leave	VERB
ejpam-6175	345	2	:	:	PUNCT
ejpam-6175	345	3	original	original	ADJ
ejpam-6175	345	4	axial	axial	ADJ
ejpam-6175	345	5	and	and	CCONJ
ejpam-6175	345	6	sagittal	sagittal	ADJ
ejpam-6175	345	7	mri	mri	NOUN
ejpam-6175	345	8	views	view	NOUN
ejpam-6175	345	9	.	.	PUNCT
ejpam-6175	346	1	middle	middle	ADJ
ejpam-6175	346	2	:	:	PUNCT
ejpam-6175	346	3	edge	edge	NOUN
ejpam-6175	346	4	detection	detection	NOUN
ejpam-6175	346	5	highlighting	highlight	VERB
ejpam-6175	346	6	brain	brain	NOUN
ejpam-6175	346	7	structures	structure	NOUN
ejpam-6175	346	8	.	.	PUNCT
ejpam-6175	347	1	right	right	ADV
ejpam-6175	347	2	:	:	PUNCT
ejpam-6175	347	3	graph	graph	VERB
ejpam-6175	347	4	representations	representation	NOUN
ejpam-6175	347	5	showing	show	VERB
ejpam-6175	347	6	anatomical	anatomical	ADJ
ejpam-6175	347	7	connections	connection	NOUN
ejpam-6175	347	8	.	.	PUNCT
ejpam-6175	348	1	figure	figure	VERB
ejpam-6175	348	2	8	8	NUM
ejpam-6175	348	3	:	:	PUNCT
ejpam-6175	348	4	original	original	ADJ
ejpam-6175	348	5	scan	scan	NOUN
ejpam-6175	348	6	showing	show	VERB
ejpam-6175	348	7	the	the	DET
ejpam-6175	348	8	presence	presence	NOUN
ejpam-6175	348	9	of	of	ADP
ejpam-6175	348	10	an	an	DET
ejpam-6175	348	11	abnormality	abnormality	NOUN
ejpam-6175	348	12	.	.	PUNCT
ejpam-6175	349	1	right	right	ADV
ejpam-6175	349	2	:	:	PUNCT
ejpam-6175	349	3	the	the	DET
ejpam-6175	349	4	affected	affected	ADJ
ejpam-6175	349	5	area	area	NOUN
ejpam-6175	349	6	is	be	AUX
ejpam-6175	349	7	precisely	precisely	ADV
ejpam-6175	349	8	marked	mark	VERB
ejpam-6175	349	9	for	for	ADP
ejpam-6175	349	10	further	further	ADJ
ejpam-6175	349	11	analysis	analysis	NOUN
ejpam-6175	349	12	.	.	PUNCT
ejpam-6175	350	1	when	when	SCONJ
ejpam-6175	350	2	the	the	DET
ejpam-6175	350	3	script	script	NOUN
ejpam-6175	350	4	is	be	AUX
ejpam-6175	350	5	executed	execute	VERB
ejpam-6175	350	6	,	,	PUNCT
ejpam-6175	350	7	it	it	PRON
ejpam-6175	350	8	processes	process	VERB
ejpam-6175	350	9	the	the	DET
ejpam-6175	350	10	given	give	VERB
ejpam-6175	350	11	image	image	NOUN
ejpam-6175	350	12	path	path	NOUN
ejpam-6175	350	13	and	and	CCONJ
ejpam-6175	350	14	specified	specify	VERB
ejpam-6175	350	15	number	number	NOUN
ejpam-6175	350	16	of	of	ADP
ejpam-6175	350	17	keypoints	keypoint	NOUN
ejpam-6175	350	18	,	,	PUNCT
ejpam-6175	350	19	seamlessly	seamlessly	ADV
ejpam-6175	350	20	executing	execute	VERB
ejpam-6175	350	21	the	the	DET
ejpam-6175	350	22	entire	entire	ADJ
ejpam-6175	350	23	workflow	workflow	NOUN
ejpam-6175	350	24	.	.	PUNCT
ejpam-6175	351	1	looking	look	VERB
ejpam-6175	351	2	ahead	ahead	ADV
ejpam-6175	351	3	,	,	PUNCT
ejpam-6175	351	4	our	our	PRON
ejpam-6175	351	5	future	future	ADJ
ejpam-6175	351	6	work	work	NOUN
ejpam-6175	351	7	aims	aim	VERB
ejpam-6175	351	8	to	to	PART
ejpam-6175	351	9	develop	develop	VERB
ejpam-6175	351	10	a	a	DET
ejpam-6175	351	11	model	model	NOUN
ejpam-6175	351	12	that	that	PRON
ejpam-6175	351	13	can	can	AUX
ejpam-6175	351	14	train	train	VERB
ejpam-6175	351	15	on	on	ADP
ejpam-6175	351	16	datasets	dataset	NOUN
ejpam-6175	351	17	and	and	CCONJ
ejpam-6175	351	18	utilize	utilize	VERB
ejpam-6175	351	19	persistent	persistent	ADJ
ejpam-6175	351	20	homology	homology	NOUN
ejpam-6175	351	21	to	to	PART
ejpam-6175	351	22	detect	detect	VERB
ejpam-6175	351	23	tumor	tumor	NOUN
ejpam-6175	351	24	cells	cell	NOUN
ejpam-6175	351	25	and	and	CCONJ
ejpam-6175	351	26	determine	determine	VERB
ejpam-6175	351	27	the	the	DET
ejpam-6175	351	28	genus	genus	NOUN
ejpam-6175	351	29	of	of	ADP
ejpam-6175	351	30	different	different	ADJ
ejpam-6175	351	31	structures	structure	NOUN
ejpam-6175	351	32	.	.	PUNCT
ejpam-6175	352	1	4	4	X
ejpam-6175	352	2	.	.	X
ejpam-6175	352	3	conclusion	conclusion	NOUN
ejpam-6175	352	4	in	in	ADP
ejpam-6175	352	5	this	this	DET
ejpam-6175	352	6	article	article	NOUN
ejpam-6175	352	7	,	,	PUNCT
ejpam-6175	352	8	we	we	PRON
ejpam-6175	352	9	demonstrated	demonstrate	VERB
ejpam-6175	352	10	the	the	DET
ejpam-6175	352	11	effectiveness	effectiveness	NOUN
ejpam-6175	352	12	of	of	ADP
ejpam-6175	352	13	persistent	persistent	ADJ
ejpam-6175	352	14	homology	homology	NOUN
ejpam-6175	352	15	in	in	ADP
ejpam-6175	352	16	analyzing	analyze	VERB
ejpam-6175	352	17	both	both	CCONJ
ejpam-6175	352	18	structured	structured	ADJ
ejpam-6175	352	19	and	and	CCONJ
ejpam-6175	352	20	irregular	irregular	ADJ
ejpam-6175	352	21	geometric	geometric	ADJ
ejpam-6175	352	22	forms	form	NOUN
ejpam-6175	352	23	.	.	PUNCT
ejpam-6175	353	1	by	by	ADP
ejpam-6175	353	2	examining	examine	VERB
ejpam-6175	353	3	the	the	DET
ejpam-6175	353	4	evolution	evolution	NOUN
ejpam-6175	353	5	of	of	ADP
ejpam-6175	353	6	homology	homology	NOUN
ejpam-6175	353	7	groups	group	NOUN
ejpam-6175	353	8	across	across	ADP
ejpam-6175	353	9	filtration	filtration	NOUN
ejpam-6175	353	10	stages	stage	NOUN
ejpam-6175	353	11	,	,	PUNCT
ejpam-6175	353	12	our	our	PRON
ejpam-6175	353	13	method	method	NOUN
ejpam-6175	353	14	captures	capture	VERB
ejpam-6175	353	15	topological	topological	ADJ
ejpam-6175	353	16	features	feature	NOUN
ejpam-6175	353	17	such	such	ADJ
ejpam-6175	353	18	as	as	ADP
ejpam-6175	353	19	connected	connected	ADJ
ejpam-6175	353	20	components	component	NOUN
ejpam-6175	353	21	,	,	PUNCT
ejpam-6175	353	22	tunnels	tunnel	NOUN
ejpam-6175	353	23	,	,	PUNCT
ejpam-6175	353	24	and	and	CCONJ
ejpam-6175	353	25	voids	void	NOUN
ejpam-6175	353	26	with	with	ADP
ejpam-6175	353	27	high	high	ADJ
ejpam-6175	353	28	fidelity	fidelity	NOUN
ejpam-6175	353	29	.	.	PUNCT
ejpam-6175	354	1	the	the	DET
ejpam-6175	354	2	approach	approach	NOUN
ejpam-6175	354	3	emphasizes	emphasize	VERB
ejpam-6175	354	4	vertexand	vertexand	ADP
ejpam-6175	354	5	edge	edge	NOUN
ejpam-6175	354	6	-	-	PUNCT
ejpam-6175	354	7	level	level	NOUN
ejpam-6175	354	8	analysis	analysis	NOUN
ejpam-6175	354	9	to	to	PART
ejpam-6175	354	10	compute	compute	VERB
ejpam-6175	354	11	digital	digital	ADJ
ejpam-6175	354	12	homology	homology	NOUN
ejpam-6175	354	13	,	,	PUNCT
ejpam-6175	354	14	making	make	VERB
ejpam-6175	354	15	it	it	PRON
ejpam-6175	354	16	suitable	suitable	ADJ
ejpam-6175	354	17	for	for	ADP
ejpam-6175	354	18	tasks	task	NOUN
ejpam-6175	354	19	such	such	ADJ
ejpam-6175	354	20	as	as	ADP
ejpam-6175	354	21	edge	edge	NOUN
ejpam-6175	354	22	detection	detection	NOUN
ejpam-6175	354	23	,	,	PUNCT
ejpam-6175	354	24	skeletonization	skeletonization	NOUN
ejpam-6175	354	25	,	,	PUNCT
ejpam-6175	354	26	and	and	CCONJ
ejpam-6175	354	27	geometric	geometric	ADJ
ejpam-6175	354	28	restructuring	restructuring	NOUN
ejpam-6175	354	29	—	—	PUNCT
ejpam-6175	354	30	particularly	particularly	ADV
ejpam-6175	354	31	in	in	ADP
ejpam-6175	354	32	contexts	context	NOUN
ejpam-6175	354	33	where	where	SCONJ
ejpam-6175	354	34	genus	genus	ADJ
ejpam-6175	354	35	variations	variation	NOUN
ejpam-6175	354	36	are	be	AUX
ejpam-6175	354	37	of	of	ADP
ejpam-6175	354	38	interest	interest	NOUN
ejpam-6175	354	39	.	.	PUNCT
ejpam-6175	355	1	the	the	DET
ejpam-6175	355	2	presented	present	VERB
ejpam-6175	355	3	examples	example	NOUN
ejpam-6175	355	4	highlight	highlight	VERB
ejpam-6175	355	5	the	the	DET
ejpam-6175	355	6	practical	practical	ADJ
ejpam-6175	355	7	utility	utility	NOUN
ejpam-6175	355	8	of	of	ADP
ejpam-6175	355	9	this	this	DET
ejpam-6175	355	10	method	method	NOUN
ejpam-6175	355	11	in	in	ADP
ejpam-6175	355	12	uncovering	uncover	VERB
ejpam-6175	355	13	the	the	DET
ejpam-6175	355	14	underlying	underlying	ADJ
ejpam-6175	355	15	topology	topology	NOUN
ejpam-6175	355	16	of	of	ADP
ejpam-6175	355	17	complex	complex	ADJ
ejpam-6175	355	18	shapes	shape	NOUN
ejpam-6175	355	19	.	.	PUNCT
ejpam-6175	356	1	building	build	VERB
ejpam-6175	356	2	on	on	ADP
ejpam-6175	356	3	this	this	DET
ejpam-6175	356	4	foundation	foundation	NOUN
ejpam-6175	356	5	,	,	PUNCT
ejpam-6175	356	6	future	future	ADJ
ejpam-6175	356	7	research	research	NOUN
ejpam-6175	356	8	will	will	AUX
ejpam-6175	356	9	focus	focus	VERB
ejpam-6175	356	10	on	on	ADP
ejpam-6175	356	11	developing	develop	VERB
ejpam-6175	356	12	hybrid	hybrid	ADJ
ejpam-6175	356	13	models	model	NOUN
ejpam-6175	356	14	that	that	PRON
ejpam-6175	356	15	incorporate	incorporate	VERB
ejpam-6175	356	16	persistent	persistent	ADJ
ejpam-6175	356	17	homology	homology	NOUN
ejpam-6175	356	18	as	as	ADP
ejpam-6175	356	19	a	a	DET
ejpam-6175	356	20	learning	learning	NOUN
ejpam-6175	356	21	mechanism	mechanism	NOUN
ejpam-6175	356	22	within	within	ADP
ejpam-6175	356	23	training	training	NOUN
ejpam-6175	356	24	datasets	dataset	NOUN
ejpam-6175	356	25	.	.	PUNCT
ejpam-6175	357	1	this	this	PRON
ejpam-6175	357	2	will	will	AUX
ejpam-6175	357	3	be	be	AUX
ejpam-6175	357	4	particularly	particularly	ADV
ejpam-6175	357	5	directed	direct	VERB
ejpam-6175	357	6	toward	toward	ADP
ejpam-6175	357	7	biomedical	biomedical	ADJ
ejpam-6175	357	8	imaging	imaging	NOUN
ejpam-6175	357	9	applications	application	NOUN
ejpam-6175	357	10	,	,	PUNCT
ejpam-6175	357	11	such	such	ADJ
ejpam-6175	357	12	as	as	ADP
ejpam-6175	357	13	the	the	DET
ejpam-6175	357	14	detection	detection	NOUN
ejpam-6175	357	15	of	of	ADP
ejpam-6175	357	16	tumor	tumor	NOUN
ejpam-6175	357	17	cells	cell	NOUN
ejpam-6175	357	18	and	and	CCONJ
ejpam-6175	357	19	the	the	DET
ejpam-6175	357	20	classification	classification	NOUN
ejpam-6175	357	21	of	of	ADP
ejpam-6175	357	22	topological	topological	ADJ
ejpam-6175	357	23	signatures	signature	NOUN
ejpam-6175	357	24	associated	associate	VERB
ejpam-6175	357	25	with	with	ADP
ejpam-6175	357	26	structural	structural	ADJ
ejpam-6175	357	27	abnormalities	abnormality	NOUN
ejpam-6175	357	28	.	.	PUNCT
ejpam-6175	358	1	s.k	s.k	PROPN
ejpam-6175	358	2	.	.	PUNCT
ejpam-6175	358	3	jha	jha	PROPN
ejpam-6175	358	4	sanjay	sanjay	PROPN
ejpam-6175	358	5	,	,	PUNCT
ejpam-6175	358	6	m	m	PROPN
ejpam-6175	358	7	,	,	PUNCT
ejpam-6175	358	8	natarajan	natarajan	PROPN
ejpam-6175	358	9	/	/	SYM
ejpam-6175	358	10	eur	eur	PROPN
ejpam-6175	358	11	.	.	PUNCT
ejpam-6175	359	1	j.	j.	PROPN
ejpam-6175	359	2	pure	pure	PROPN
ejpam-6175	359	3	appl	appl	PROPN
ejpam-6175	359	4	.	.	PROPN
ejpam-6175	359	5	math	math	PROPN
ejpam-6175	359	6	,	,	PUNCT
ejpam-6175	359	7	18	18	NUM
ejpam-6175	359	8	(	(	PUNCT
ejpam-6175	359	9	3	3	NUM
ejpam-6175	359	10	)	)	PUNCT
ejpam-6175	359	11	(	(	PUNCT
ejpam-6175	359	12	2025	2025	NUM
ejpam-6175	359	13	)	)	PUNCT
ejpam-6175	359	14	,	,	PUNCT
ejpam-6175	359	15	6175	6175	NUM
ejpam-6175	359	16	17	17	NUM
ejpam-6175	359	17	of	of	ADP
ejpam-6175	359	18	18	18	NUM
ejpam-6175	359	19	open	open	ADJ
ejpam-6175	359	20	problems	problem	NOUN
ejpam-6175	359	21	that	that	PRON
ejpam-6175	359	22	remain	remain	VERB
ejpam-6175	359	23	include	include	VERB
ejpam-6175	359	24	the	the	DET
ejpam-6175	359	25	optimization	optimization	NOUN
ejpam-6175	359	26	of	of	ADP
ejpam-6175	359	27	persistence	persistence	NOUN
ejpam-6175	359	28	-	-	PUNCT
ejpam-6175	359	29	based	base	VERB
ejpam-6175	359	30	metrics	metric	NOUN
ejpam-6175	359	31	for	for	ADP
ejpam-6175	359	32	noisy	noisy	ADJ
ejpam-6175	359	33	or	or	CCONJ
ejpam-6175	359	34	high	high	ADV
ejpam-6175	359	35	-	-	PUNCT
ejpam-6175	359	36	dimensional	dimensional	ADJ
ejpam-6175	359	37	data	datum	NOUN
ejpam-6175	359	38	,	,	PUNCT
ejpam-6175	359	39	the	the	DET
ejpam-6175	359	40	formalization	formalization	NOUN
ejpam-6175	359	41	of	of	ADP
ejpam-6175	359	42	stability	stability	NOUN
ejpam-6175	359	43	bounds	bound	VERB
ejpam-6175	359	44	under	under	ADP
ejpam-6175	359	45	random	random	ADJ
ejpam-6175	359	46	perturbations	perturbation	NOUN
ejpam-6175	359	47	in	in	ADP
ejpam-6175	359	48	simplicial	simplicial	ADJ
ejpam-6175	359	49	structures	structure	NOUN
ejpam-6175	359	50	,	,	PUNCT
ejpam-6175	359	51	and	and	CCONJ
ejpam-6175	359	52	the	the	DET
ejpam-6175	359	53	extension	extension	NOUN
ejpam-6175	359	54	of	of	ADP
ejpam-6175	359	55	the	the	DET
ejpam-6175	359	56	framework	framework	NOUN
ejpam-6175	359	57	to	to	ADP
ejpam-6175	359	58	non	non	ADJ
ejpam-6175	359	59	-	-	ADJ
ejpam-6175	359	60	euclidean	euclidean	ADJ
ejpam-6175	359	61	settings	setting	NOUN
ejpam-6175	359	62	and	and	CCONJ
ejpam-6175	359	63	manifold	manifold	ADJ
ejpam-6175	359	64	-	-	PUNCT
ejpam-6175	359	65	valued	value	VERB
ejpam-6175	359	66	data	datum	NOUN
ejpam-6175	359	67	.	.	PUNCT
ejpam-6175	360	1	addressing	address	VERB
ejpam-6175	360	2	these	these	DET
ejpam-6175	360	3	challenges	challenge	NOUN
ejpam-6175	360	4	could	could	AUX
ejpam-6175	360	5	significantly	significantly	ADV
ejpam-6175	360	6	broaden	broaden	VERB
ejpam-6175	360	7	the	the	DET
ejpam-6175	360	8	scope	scope	NOUN
ejpam-6175	360	9	and	and	CCONJ
ejpam-6175	360	10	precision	precision	NOUN
ejpam-6175	360	11	of	of	ADP
ejpam-6175	360	12	topological	topological	ADJ
ejpam-6175	360	13	data	datum	NOUN
ejpam-6175	360	14	analysis	analysis	NOUN
ejpam-6175	360	15	in	in	ADP
ejpam-6175	360	16	both	both	CCONJ
ejpam-6175	360	17	theoretical	theoretical	ADJ
ejpam-6175	360	18	and	and	CCONJ
ejpam-6175	360	19	applied	apply	VERB
ejpam-6175	360	20	domains	domain	NOUN
ejpam-6175	360	21	.	.	PUNCT
ejpam-6175	361	1	conflict	conflict	NOUN
ejpam-6175	361	2	of	of	ADP
ejpam-6175	361	3	interest	interest	NOUN
ejpam-6175	361	4	the	the	DET
ejpam-6175	361	5	authors	author	NOUN
ejpam-6175	361	6	declare	declare	VERB
ejpam-6175	361	7	no	no	DET
ejpam-6175	361	8	conflict	conflict	NOUN
ejpam-6175	361	9	of	of	ADP
ejpam-6175	361	10	interest	interest	NOUN
ejpam-6175	361	11	.	.	PUNCT
ejpam-6175	362	1	availability	availability	NOUN
ejpam-6175	362	2	of	of	ADP
ejpam-6175	362	3	data	datum	NOUN
ejpam-6175	362	4	and	and	CCONJ
ejpam-6175	362	5	material	material	NOUN
ejpam-6175	362	6	all	all	DET
ejpam-6175	362	7	the	the	DET
ejpam-6175	362	8	data	datum	NOUN
ejpam-6175	362	9	of	of	ADP
ejpam-6175	362	10	the	the	DET
ejpam-6175	362	11	study	study	NOUN
ejpam-6175	362	12	are	be	AUX
ejpam-6175	362	13	included	include	VERB
ejpam-6175	362	14	.	.	PUNCT
ejpam-6175	363	1	references	reference	NOUN
ejpam-6175	363	2	[	[	X
ejpam-6175	363	3	1	1	NUM
ejpam-6175	363	4	]	]	PUNCT
ejpam-6175	363	5	e.	e.	PROPN
ejpam-6175	363	6	spanier	spanier	PROPN
ejpam-6175	363	7	.	.	PUNCT
ejpam-6175	364	1	algebraic	algebraic	ADJ
ejpam-6175	364	2	topology	topology	PROPN
ejpam-6175	364	3	.	.	PUNCT
ejpam-6175	365	1	mcgraw	mcgraw	PROPN
ejpam-6175	365	2	-	-	PUNCT
ejpam-6175	365	3	hill	hill	PROPN
ejpam-6175	365	4	,	,	PUNCT
ejpam-6175	365	5	1966	1966	NUM
ejpam-6175	365	6	.	.	PUNCT
ejpam-6175	366	1	[	[	X
ejpam-6175	366	2	2	2	NUM
ejpam-6175	366	3	]	]	PUNCT
ejpam-6175	366	4	l.	l.	PROPN
ejpam-6175	366	5	boxer	boxer	PROPN
ejpam-6175	366	6	.	.	PUNCT
ejpam-6175	367	1	digitally	digitally	ADV
ejpam-6175	367	2	continuous	continuous	ADJ
ejpam-6175	367	3	functions	function	NOUN
ejpam-6175	367	4	.	.	PUNCT
ejpam-6175	368	1	pattern	pattern	NOUN
ejpam-6175	368	2	recognit	recognit	VERB
ejpam-6175	368	3	.	.	PUNCT
ejpam-6175	369	1	lett	lett	PROPN
ejpam-6175	369	2	.	.	PROPN
ejpam-6175	369	3	,	,	PUNCT
ejpam-6175	369	4	15:833–839	15:833–839	PROPN
ejpam-6175	369	5	,	,	PUNCT
ejpam-6175	369	6	1994	1994	NUM
ejpam-6175	369	7	.	.	PUNCT
ejpam-6175	370	1	[	[	X
ejpam-6175	370	2	3	3	X
ejpam-6175	370	3	]	]	PUNCT
ejpam-6175	370	4	t.	t.	PROPN
ejpam-6175	370	5	y.	y.	PROPN
ejpam-6175	370	6	kong	kong	PROPN
ejpam-6175	370	7	.	.	PUNCT
ejpam-6175	371	1	a	a	DET
ejpam-6175	371	2	digital	digital	ADJ
ejpam-6175	371	3	fundamental	fundamental	ADJ
ejpam-6175	371	4	group	group	NOUN
ejpam-6175	371	5	.	.	PUNCT
ejpam-6175	372	1	comput	comput	NOUN
ejpam-6175	372	2	.	.	PUNCT
ejpam-6175	373	1	graph	graph	NOUN
ejpam-6175	373	2	.	.	PUNCT
ejpam-6175	373	3	,	,	PUNCT
ejpam-6175	373	4	13:159–166	13:159–166	NUM
ejpam-6175	373	5	,	,	PUNCT
ejpam-6175	373	6	1989	1989	NUM
ejpam-6175	373	7	.	.	PUNCT
ejpam-6175	374	1	[	[	X
ejpam-6175	374	2	4	4	X
ejpam-6175	374	3	]	]	X
ejpam-6175	374	4	o.	o.	PROPN
ejpam-6175	374	5	ege	ege	PROPN
ejpam-6175	374	6	and	and	CCONJ
ejpam-6175	374	7	i.	i.	PROPN
ejpam-6175	374	8	karaca	karaca	PROPN
ejpam-6175	374	9	.	.	PUNCT
ejpam-6175	375	1	persistent	persistent	ADJ
ejpam-6175	375	2	homology	homology	NOUN
ejpam-6175	375	3	of	of	ADP
ejpam-6175	375	4	graph	graph	NOUN
ejpam-6175	375	5	-	-	PUNCT
ejpam-6175	375	6	like	like	ADJ
ejpam-6175	375	7	digital	digital	ADJ
ejpam-6175	375	8	images	image	NOUN
ejpam-6175	375	9	.	.	PUNCT
ejpam-6175	376	1	annali	annali	PROPN
ejpam-6175	376	2	di	di	PROPN
ejpam-6175	376	3	matematica	matematica	PROPN
ejpam-6175	376	4	pura	pura	PROPN
ejpam-6175	376	5	ed	ed	PROPN
ejpam-6175	376	6	applicata	applicata	NOUN
ejpam-6175	376	7	,	,	PUNCT
ejpam-6175	376	8	199:2167–2179	199:2167–2179	NUM
ejpam-6175	376	9	,	,	PUNCT
ejpam-6175	376	10	2020	2020	NUM
ejpam-6175	376	11	.	.	PUNCT
ejpam-6175	377	1	doi	doi	NOUN
ejpam-6175	377	2	:	:	PUNCT
ejpam-6175	377	3	https://doi.org/10	https://doi.org/10	PROPN
ejpam-6175	377	4	.	.	PUNCT
ejpam-6175	378	1	1007	1007	NUM
ejpam-6175	378	2	/	/	SYM
ejpam-6175	378	3	s10231	s10231	NOUN
ejpam-6175	378	4	-	-	PUNCT
ejpam-6175	378	5	020	020	NUM
ejpam-6175	378	6	-	-	PUNCT
ejpam-6175	378	7	00962	00962	NUM
ejpam-6175	378	8	-	-	PUNCT
ejpam-6175	378	9	x.	x.	NOUN
ejpam-6175	379	1	[	[	X
ejpam-6175	379	2	5	5	X
ejpam-6175	379	3	]	]	X
ejpam-6175	379	4	g.	g.	PROPN
ejpam-6175	379	5	t.	t.	PROPN
ejpam-6175	379	6	herman	herman	PROPN
ejpam-6175	379	7	.	.	PUNCT
ejpam-6175	380	1	oriented	orient	VERB
ejpam-6175	380	2	surfaces	surface	NOUN
ejpam-6175	380	3	in	in	ADP
ejpam-6175	380	4	digital	digital	ADJ
ejpam-6175	380	5	spaces	space	NOUN
ejpam-6175	380	6	.	.	PUNCT
ejpam-6175	381	1	cvgip	cvgip	ADJ
ejpam-6175	381	2	:	:	PUNCT
ejpam-6175	381	3	graph	graph	NOUN
ejpam-6175	381	4	.	.	PUNCT
ejpam-6175	382	1	models	model	NOUN
ejpam-6175	382	2	image	image	NOUN
ejpam-6175	382	3	process	process	NOUN
ejpam-6175	382	4	.	.	PUNCT
ejpam-6175	383	1	,	,	PUNCT
ejpam-6175	383	2	55:381–396	55:381–396	NUM
ejpam-6175	383	3	,	,	PUNCT
ejpam-6175	383	4	1993	1993	NUM
ejpam-6175	383	5	.	.	PUNCT
ejpam-6175	384	1	[	[	X
ejpam-6175	384	2	6	6	NUM
ejpam-6175	384	3	]	]	PUNCT
ejpam-6175	384	4	a.	a.	NOUN
ejpam-6175	384	5	rosenfeld	rosenfeld	PROPN
ejpam-6175	384	6	.	.	PUNCT
ejpam-6175	385	1	continuous	continuous	ADJ
ejpam-6175	385	2	functions	function	NOUN
ejpam-6175	385	3	on	on	ADP
ejpam-6175	385	4	digital	digital	ADJ
ejpam-6175	385	5	pictures	picture	NOUN
ejpam-6175	385	6	.	.	PUNCT
ejpam-6175	386	1	pattern	pattern	NOUN
ejpam-6175	386	2	recognit	recognit	VERB
ejpam-6175	386	3	.	.	PUNCT
ejpam-6175	387	1	lett	lett	PROPN
ejpam-6175	387	2	.	.	PROPN
ejpam-6175	387	3	,	,	PUNCT
ejpam-6175	387	4	4:177	4:177	NUM
ejpam-6175	387	5	–	–	PUNCT
ejpam-6175	387	6	184	184	NUM
ejpam-6175	387	7	,	,	PUNCT
ejpam-6175	387	8	1986	1986	NUM
ejpam-6175	387	9	.	.	PUNCT
ejpam-6175	388	1	[	[	X
ejpam-6175	388	2	7	7	X
ejpam-6175	388	3	]	]	X
ejpam-6175	388	4	l.	l.	PROPN
ejpam-6175	388	5	boxer	boxer	PROPN
ejpam-6175	388	6	.	.	PUNCT
ejpam-6175	389	1	a	a	DET
ejpam-6175	389	2	classical	classical	ADJ
ejpam-6175	389	3	construction	construction	NOUN
ejpam-6175	389	4	for	for	ADP
ejpam-6175	389	5	the	the	DET
ejpam-6175	389	6	digital	digital	ADJ
ejpam-6175	389	7	fundamental	fundamental	ADJ
ejpam-6175	389	8	group	group	NOUN
ejpam-6175	389	9	.	.	PUNCT
ejpam-6175	390	1	j.	j.	PROPN
ejpam-6175	390	2	math	math	PROPN
ejpam-6175	390	3	.	.	PUNCT
ejpam-6175	391	1	imaging	imaging	NOUN
ejpam-6175	391	2	vis	vis	X
ejpam-6175	391	3	.	.	PROPN
ejpam-6175	391	4	,	,	PUNCT
ejpam-6175	391	5	10:51–62	10:51–62	NUM
ejpam-6175	391	6	,	,	PUNCT
ejpam-6175	391	7	1999	1999	NUM
ejpam-6175	391	8	.	.	PUNCT
ejpam-6175	392	1	[	[	X
ejpam-6175	392	2	8	8	NUM
ejpam-6175	392	3	]	]	X
ejpam-6175	392	4	l.	l.	PROPN
ejpam-6175	392	5	boxer	boxer	PROPN
ejpam-6175	392	6	.	.	PUNCT
ejpam-6175	393	1	digital	digital	ADJ
ejpam-6175	393	2	products	product	NOUN
ejpam-6175	393	3	,	,	PUNCT
ejpam-6175	393	4	wedges	wedge	NOUN
ejpam-6175	393	5	and	and	CCONJ
ejpam-6175	393	6	covering	covering	NOUN
ejpam-6175	393	7	spaces	space	NOUN
ejpam-6175	393	8	.	.	PUNCT
ejpam-6175	394	1	j.	j.	PROPN
ejpam-6175	394	2	math	math	PROPN
ejpam-6175	394	3	.	.	PUNCT
ejpam-6175	395	1	imaging	imaging	NOUN
ejpam-6175	395	2	vis	vis	X
ejpam-6175	395	3	.	.	PROPN
ejpam-6175	395	4	,	,	PUNCT
ejpam-6175	395	5	25:159–171	25:159–171	PROPN
ejpam-6175	395	6	,	,	PUNCT
ejpam-6175	395	7	2006	2006	NUM
ejpam-6175	395	8	.	.	PUNCT
ejpam-6175	396	1	[	[	X
ejpam-6175	396	2	9	9	NUM
ejpam-6175	396	3	]	]	X
ejpam-6175	396	4	h.	h.	PROPN
ejpam-6175	396	5	arslan	arslan	PROPN
ejpam-6175	396	6	and	and	CCONJ
ejpam-6175	396	7	i.	i.	PROPN
ejpam-6175	396	8	karaca	karaca	PROPN
ejpam-6175	396	9	.	.	PUNCT
ejpam-6175	397	1	homology	homology	NOUN
ejpam-6175	397	2	groups	group	NOUN
ejpam-6175	397	3	of	of	ADP
ejpam-6175	397	4	n	n	CCONJ
ejpam-6175	397	5	-	-	PUNCT
ejpam-6175	397	6	dimensional	dimensional	ADJ
ejpam-6175	397	7	digital	digital	ADJ
ejpam-6175	397	8	images	image	NOUN
ejpam-6175	397	9	.	.	PUNCT
ejpam-6175	398	1	xxi	xxi	PROPN
ejpam-6175	398	2	.	.	PUNCT
ejpam-6175	398	3	turk	turk	PROPN
ejpam-6175	398	4	.	.	PUNCT
ejpam-6175	399	1	natl	natl	PROPN
ejpam-6175	399	2	.	.	PUNCT
ejpam-6175	399	3	math	math	PROPN
ejpam-6175	399	4	.	.	PUNCT
ejpam-6175	400	1	symp	symp	PROPN
ejpam-6175	400	2	.	.	PUNCT
ejpam-6175	401	1	b	b	X
ejpam-6175	401	2	,	,	PUNCT
ejpam-6175	401	3	pages	page	NOUN
ejpam-6175	401	4	1–13	1–13	NOUN
ejpam-6175	401	5	,	,	PUNCT
ejpam-6175	401	6	2008	2008	NUM
ejpam-6175	401	7	.	.	PUNCT
ejpam-6175	402	1	[	[	X
ejpam-6175	402	2	10	10	NUM
ejpam-6175	402	3	]	]	X
ejpam-6175	402	4	a.	a.	NOUN
ejpam-6175	402	5	zomorodian	zomorodian	PROPN
ejpam-6175	402	6	and	and	CCONJ
ejpam-6175	402	7	g.	g.	PROPN
ejpam-6175	402	8	carlsson	carlsson	PROPN
ejpam-6175	402	9	.	.	PUNCT
ejpam-6175	403	1	computing	compute	VERB
ejpam-6175	403	2	persistent	persistent	ADJ
ejpam-6175	403	3	homology	homology	NOUN
ejpam-6175	403	4	.	.	PUNCT
ejpam-6175	404	1	discrete	discrete	ADJ
ejpam-6175	404	2	comput	comput	NOUN
ejpam-6175	404	3	.	.	PUNCT
ejpam-6175	405	1	geom	geom	PROPN
ejpam-6175	405	2	.	.	PROPN
ejpam-6175	405	3	,	,	PUNCT
ejpam-6175	405	4	33:249–274	33:249–274	PROPN
ejpam-6175	405	5	,	,	PUNCT
ejpam-6175	405	6	2005	2005	NUM
ejpam-6175	405	7	.	.	PUNCT
ejpam-6175	406	1	doi	doi	NOUN
ejpam-6175	406	2	:	:	PUNCT
ejpam-6175	406	3	https://doi.org/10.1007/s00454-004-1146-y	https://doi.org/10.1007/s00454-004-1146-y	NOUN
ejpam-6175	406	4	.	.	PUNCT
ejpam-6175	407	1	[	[	X
ejpam-6175	407	2	11	11	NUM
ejpam-6175	407	3	]	]	X
ejpam-6175	407	4	h.	h.	NOUN
ejpam-6175	407	5	edelsbrunner	edelsbrunner	NOUN
ejpam-6175	407	6	and	and	CCONJ
ejpam-6175	407	7	j.	j.	PROPN
ejpam-6175	407	8	harer	harer	PROPN
ejpam-6175	407	9	.	.	PUNCT
ejpam-6175	408	1	computational	computational	ADJ
ejpam-6175	408	2	topology	topology	NOUN
ejpam-6175	408	3	:	:	PUNCT
ejpam-6175	408	4	an	an	DET
ejpam-6175	408	5	introduction	introduction	NOUN
ejpam-6175	408	6	.	.	PUNCT
ejpam-6175	409	1	amer	amer	PROPN
ejpam-6175	409	2	.	.	PUNCT
ejpam-6175	409	3	math	math	PROPN
ejpam-6175	409	4	.	.	PUNCT
ejpam-6175	410	1	soc	soc	PROPN
ejpam-6175	410	2	.	.	PROPN
ejpam-6175	410	3	,	,	PUNCT
ejpam-6175	410	4	2010	2010	NUM
ejpam-6175	410	5	.	.	PUNCT
ejpam-6175	411	1	[	[	X
ejpam-6175	411	2	12	12	NUM
ejpam-6175	411	3	]	]	X
ejpam-6175	411	4	n.	n.	NOUN
ejpam-6175	411	5	otter	otter	NOUN
ejpam-6175	411	6	,	,	PUNCT
ejpam-6175	411	7	m.	m.	NOUN
ejpam-6175	411	8	a.	a.	NOUN
ejpam-6175	411	9	porter	porter	NOUN
ejpam-6175	411	10	,	,	PUNCT
ejpam-6175	411	11	and	and	CCONJ
ejpam-6175	411	12	u.	u.	PROPN
ejpam-6175	411	13	tillmann	tillmann	PROPN
ejpam-6175	411	14	et	et	PROPN
ejpam-6175	411	15	al	al	PROPN
ejpam-6175	411	16	.	.	PUNCT
ejpam-6175	412	1	a	a	DET
ejpam-6175	412	2	roadmap	roadmap	NOUN
ejpam-6175	412	3	for	for	ADP
ejpam-6175	412	4	the	the	DET
ejpam-6175	412	5	computation	computation	NOUN
ejpam-6175	412	6	of	of	ADP
ejpam-6175	412	7	persistent	persistent	ADJ
ejpam-6175	412	8	homology	homology	NOUN
ejpam-6175	412	9	.	.	PUNCT
ejpam-6175	413	1	epj	epj	PROPN
ejpam-6175	413	2	data	data	PROPN
ejpam-6175	413	3	sci	sci	PROPN
ejpam-6175	413	4	.	.	PROPN
ejpam-6175	413	5	,	,	PUNCT
ejpam-6175	413	6	6:17	6:17	NUM
ejpam-6175	413	7	,	,	PUNCT
ejpam-6175	413	8	2017	2017	NUM
ejpam-6175	413	9	.	.	PUNCT
ejpam-6175	414	1	doi	doi	NOUN
ejpam-6175	414	2	:	:	PUNCT
ejpam-6175	414	3	https://doi.org/10.1140/	https://doi.org/10.1140/	PROPN
ejpam-6175	414	4	epjds	epjds	ADJ
ejpam-6175	414	5	/	/	SYM
ejpam-6175	414	6	s13688	s13688	NOUN
ejpam-6175	414	7	-	-	PUNCT
ejpam-6175	414	8	017	017	NUM
ejpam-6175	414	9	-	-	PUNCT
ejpam-6175	414	10	0109	0109	NUM
ejpam-6175	414	11	-	-	PUNCT
ejpam-6175	414	12	5	5	NUM
ejpam-6175	414	13	.	.	PUNCT
ejpam-6175	415	1	[	[	X
ejpam-6175	415	2	13	13	NUM
ejpam-6175	415	3	]	]	X
ejpam-6175	415	4	r.	r.	PROPN
ejpam-6175	415	5	ghrist	ghrist	PROPN
ejpam-6175	415	6	.	.	PUNCT
ejpam-6175	416	1	barcodes	barcode	NOUN
ejpam-6175	416	2	:	:	PUNCT
ejpam-6175	416	3	the	the	DET
ejpam-6175	416	4	persistent	persistent	ADJ
ejpam-6175	416	5	topology	topology	NOUN
ejpam-6175	416	6	of	of	ADP
ejpam-6175	416	7	data	data	PROPN
ejpam-6175	416	8	.	.	PUNCT
ejpam-6175	417	1	bull	bull	NOUN
ejpam-6175	417	2	.	.	PUNCT
ejpam-6175	418	1	amer	amer	PROPN
ejpam-6175	418	2	.	.	PUNCT
ejpam-6175	418	3	math	math	PROPN
ejpam-6175	418	4	.	.	PUNCT
ejpam-6175	419	1	soc	soc	PROPN
ejpam-6175	419	2	.	.	PUNCT
ejpam-6175	419	3	,	,	PUNCT
ejpam-6175	419	4	45:61–75	45:61–75	NUM
ejpam-6175	419	5	,	,	PUNCT
ejpam-6175	419	6	2008	2008	NUM
ejpam-6175	419	7	.	.	PUNCT
ejpam-6175	420	1	[	[	X
ejpam-6175	420	2	14	14	NUM
ejpam-6175	420	3	]	]	X
ejpam-6175	420	4	g.	g.	PROPN
ejpam-6175	420	5	carlsson	carlsson	PROPN
ejpam-6175	420	6	.	.	PUNCT
ejpam-6175	421	1	topology	topology	NOUN
ejpam-6175	421	2	and	and	CCONJ
ejpam-6175	421	3	data	datum	NOUN
ejpam-6175	421	4	.	.	PUNCT
ejpam-6175	422	1	bull	bull	PROPN
ejpam-6175	422	2	.	.	PUNCT
ejpam-6175	423	1	amer	amer	PROPN
ejpam-6175	423	2	.	.	PUNCT
ejpam-6175	423	3	math	math	PROPN
ejpam-6175	423	4	.	.	PUNCT
ejpam-6175	424	1	soc	soc	PROPN
ejpam-6175	424	2	.	.	PUNCT
ejpam-6175	424	3	,	,	PUNCT
ejpam-6175	424	4	46:255–308	46:255–308	PROPN
ejpam-6175	424	5	,	,	PUNCT
ejpam-6175	424	6	2009	2009	NUM
ejpam-6175	424	7	.	.	PUNCT
ejpam-6175	425	1	s.k	s.k	PROPN
ejpam-6175	425	2	.	.	PUNCT
ejpam-6175	425	3	jha	jha	PROPN
ejpam-6175	425	4	sanjay	sanjay	PROPN
ejpam-6175	425	5	,	,	PUNCT
ejpam-6175	425	6	m	m	PROPN
ejpam-6175	425	7	,	,	PUNCT
ejpam-6175	425	8	natarajan	natarajan	PROPN
ejpam-6175	425	9	/	/	SYM
ejpam-6175	425	10	eur	eur	PROPN
ejpam-6175	425	11	.	.	PUNCT
ejpam-6175	426	1	j.	j.	PROPN
ejpam-6175	426	2	pure	pure	PROPN
ejpam-6175	426	3	appl	appl	PROPN
ejpam-6175	426	4	.	.	PROPN
ejpam-6175	426	5	math	math	PROPN
ejpam-6175	426	6	,	,	PUNCT
ejpam-6175	426	7	18	18	NUM
ejpam-6175	426	8	(	(	PUNCT
ejpam-6175	426	9	3	3	NUM
ejpam-6175	426	10	)	)	PUNCT
ejpam-6175	426	11	(	(	PUNCT
ejpam-6175	426	12	2025	2025	NUM
ejpam-6175	426	13	)	)	PUNCT
ejpam-6175	426	14	,	,	PUNCT
ejpam-6175	426	15	6175	6175	NUM
ejpam-6175	426	16	18	18	NUM
ejpam-6175	426	17	of	of	ADP
ejpam-6175	426	18	18	18	NUM
ejpam-6175	426	19	[	[	SYM
ejpam-6175	426	20	15	15	NUM
ejpam-6175	426	21	]	]	X
ejpam-6175	426	22	f.	f.	PROPN
ejpam-6175	426	23	chazal	chazal	PROPN
ejpam-6175	426	24	and	and	CCONJ
ejpam-6175	426	25	b.	b.	PROPN
ejpam-6175	426	26	michel	michel	PROPN
ejpam-6175	426	27	.	.	PUNCT
ejpam-6175	427	1	an	an	DET
ejpam-6175	427	2	introduction	introduction	NOUN
ejpam-6175	427	3	to	to	ADP
ejpam-6175	427	4	topological	topological	ADJ
ejpam-6175	427	5	data	datum	NOUN
ejpam-6175	427	6	analysis	analysis	NOUN
ejpam-6175	427	7	:	:	PUNCT
ejpam-6175	427	8	fundamental	fundamental	ADJ
ejpam-6175	427	9	and	and	CCONJ
ejpam-6175	427	10	practical	practical	ADJ
ejpam-6175	427	11	aspects	aspect	NOUN
ejpam-6175	427	12	for	for	ADP
ejpam-6175	427	13	data	datum	NOUN
ejpam-6175	427	14	scientists	scientist	NOUN
ejpam-6175	427	15	.	.	PUNCT
ejpam-6175	428	1	front	front	ADJ
ejpam-6175	428	2	.	.	PUNCT
ejpam-6175	429	1	artif	artif	INTJ
ejpam-6175	429	2	.	.	PUNCT
ejpam-6175	430	1	intell	intell	PROPN
ejpam-6175	430	2	.	.	PUNCT
ejpam-6175	430	3	,	,	PUNCT
ejpam-6175	430	4	4:39	4:39	NUM
ejpam-6175	430	5	,	,	PUNCT
ejpam-6175	430	6	2021	2021	NUM
ejpam-6175	430	7	.	.	PUNCT
ejpam-6175	431	1	doi	doi	NOUN
ejpam-6175	431	2	:	:	PUNCT
ejpam-6175	431	3	https	https	NOUN
ejpam-6175	431	4	:	:	PUNCT
ejpam-6175	431	5	//doi.org/10.3389	//doi.org/10.3389	X
ejpam-6175	431	6	/	/	SYM
ejpam-6175	431	7	frai.2021.667963	frai.2021.667963	NUM
ejpam-6175	431	8	.	.	PUNCT
ejpam-6175	432	1	[	[	X
ejpam-6175	432	2	16	16	NUM
ejpam-6175	432	3	]	]	X
ejpam-6175	432	4	i.	i.	PROPN
ejpam-6175	432	5	karaca	karaca	PROPN
ejpam-6175	432	6	and	and	CCONJ
ejpam-6175	432	7	o.	o.	PROPN
ejpam-6175	432	8	ege	ege	PROPN
ejpam-6175	432	9	.	.	PUNCT
ejpam-6175	433	1	some	some	DET
ejpam-6175	433	2	results	result	NOUN
ejpam-6175	433	3	on	on	ADP
ejpam-6175	433	4	simplicial	simplicial	ADJ
ejpam-6175	433	5	homology	homology	NOUN
ejpam-6175	433	6	groups	group	NOUN
ejpam-6175	433	7	of	of	ADP
ejpam-6175	433	8	2d	2d	NUM
ejpam-6175	433	9	digital	digital	ADJ
ejpam-6175	433	10	images	image	NOUN
ejpam-6175	433	11	.	.	PUNCT
ejpam-6175	434	1	int	int	NOUN
ejpam-6175	434	2	.	.	PUNCT
ejpam-6175	435	1	j.	j.	PROPN
ejpam-6175	435	2	inform	inform	PROPN
ejpam-6175	435	3	.	.	PUNCT
ejpam-6175	436	1	computer	computer	NOUN
ejpam-6175	436	2	sci	sci	PROPN
ejpam-6175	436	3	.	.	PROPN
ejpam-6175	436	4	,	,	PUNCT
ejpam-6175	436	5	1:198–203	1:198–203	NUM
ejpam-6175	436	6	,	,	PUNCT
ejpam-6175	436	7	2012	2012	NUM
ejpam-6175	436	8	.	.	PUNCT
ejpam-6175	437	1	[	[	X
ejpam-6175	437	2	17	17	NUM
ejpam-6175	437	3	]	]	PUNCT
ejpam-6175	437	4	a.	a.	NOUN
ejpam-6175	437	5	öztel	öztel	PROPN
ejpam-6175	437	6	,	,	PUNCT
ejpam-6175	437	7	b.	b.	PROPN
ejpam-6175	437	8	akgül	akgül	PROPN
ejpam-6175	437	9	,	,	PUNCT
ejpam-6175	437	10	i.	i.	PROPN
ejpam-6175	437	11	karaca	karaca	PROPN
ejpam-6175	437	12	,	,	PUNCT
ejpam-6175	437	13	and	and	CCONJ
ejpam-6175	437	14	others	other	NOUN
ejpam-6175	437	15	.	.	PUNCT
ejpam-6175	438	1	efficient	efficient	ADJ
ejpam-6175	438	2	computation	computation	NOUN
ejpam-6175	438	3	of	of	ADP
ejpam-6175	438	4	homology	homology	NOUN
ejpam-6175	438	5	groups	group	NOUN
ejpam-6175	438	6	,	,	PUNCT
ejpam-6175	438	7	betti	betti	NOUN
ejpam-6175	438	8	numbers	number	NOUN
ejpam-6175	438	9	,	,	PUNCT
ejpam-6175	438	10	and	and	CCONJ
ejpam-6175	438	11	euler	euler	NOUN
ejpam-6175	438	12	characteristics	characteristic	NOUN
ejpam-6175	438	13	for	for	ADP
ejpam-6175	438	14	2d	2d	NUM
ejpam-6175	438	15	digital	digital	ADJ
ejpam-6175	438	16	images	image	NOUN
ejpam-6175	438	17	.	.	PUNCT
ejpam-6175	439	1	applicable	applicable	ADJ
ejpam-6175	439	2	algebra	algebra	NOUN
ejpam-6175	439	3	in	in	ADP
ejpam-6175	439	4	engineering	engineering	NOUN
ejpam-6175	439	5	,	,	PUNCT
ejpam-6175	439	6	communication	communication	NOUN
ejpam-6175	439	7	and	and	CCONJ
ejpam-6175	439	8	computing	computing	NOUN
ejpam-6175	439	9	,	,	PUNCT
ejpam-6175	439	10	2025	2025	NUM
ejpam-6175	439	11	.	.	PUNCT
ejpam-6175	440	1	doi	doi	NOUN
ejpam-6175	440	2	:	:	PUNCT
ejpam-6175	440	3	https://doi.org/10	https://doi.org/10	PROPN
ejpam-6175	440	4	.	.	PUNCT
ejpam-6175	441	1	1007	1007	NUM
ejpam-6175	441	2	/	/	SYM
ejpam-6175	441	3	s00200	s00200	PROPN
ejpam-6175	441	4	-	-	PUNCT
ejpam-6175	441	5	025	025	NUM
ejpam-6175	441	6	-	-	PUNCT
ejpam-6175	441	7	00682	00682	NUM
ejpam-6175	441	8	-	-	PUNCT
ejpam-6175	441	9	w.	w.	NOUN
ejpam-6175	442	1	[	[	X
ejpam-6175	442	2	18	18	NUM
ejpam-6175	442	3	]	]	PUNCT
ejpam-6175	442	4	j.	j.	PROPN
ejpam-6175	442	5	canny	canny	PROPN
ejpam-6175	442	6	.	.	PUNCT
ejpam-6175	443	1	a	a	DET
ejpam-6175	443	2	computational	computational	ADJ
ejpam-6175	443	3	approach	approach	NOUN
ejpam-6175	443	4	to	to	ADP
ejpam-6175	443	5	edge	edge	NOUN
ejpam-6175	443	6	detection	detection	NOUN
ejpam-6175	443	7	.	.	PUNCT
ejpam-6175	444	1	ieee	ieee	PROPN
ejpam-6175	444	2	trans	trans	PROPN
ejpam-6175	444	3	.	.	PUNCT
ejpam-6175	444	4	pattern	pattern	PROPN
ejpam-6175	444	5	anal	anal	PROPN
ejpam-6175	444	6	.	.	PUNCT
ejpam-6175	445	1	mach	mach	PROPN
ejpam-6175	445	2	.	.	PUNCT
ejpam-6175	446	1	intell	intell	PROPN
ejpam-6175	446	2	.	.	PUNCT
ejpam-6175	446	3	,	,	PUNCT
ejpam-6175	446	4	pami-8:679–698	pami-8:679–698	PROPN
ejpam-6175	446	5	,	,	PUNCT
ejpam-6175	446	6	1986	1986	NUM
ejpam-6175	446	7	.	.	PUNCT
ejpam-6175	447	1	[	[	X
ejpam-6175	447	2	19	19	NUM
ejpam-6175	447	3	]	]	X
ejpam-6175	447	4	s.	s.	PROPN
ejpam-6175	447	5	shivam	shivam	PROPN
ejpam-6175	447	6	kumar	kumar	PROPN
ejpam-6175	447	7	jha	jha	PROPN
ejpam-6175	447	8	and	and	CCONJ
ejpam-6175	447	9	n.	n.	PROPN
ejpam-6175	447	10	mohana	mohana	PROPN
ejpam-6175	447	11	.	.	PUNCT
ejpam-6175	448	1	an	an	DET
ejpam-6175	448	2	integrated	integrate	VERB
ejpam-6175	448	3	method	method	NOUN
ejpam-6175	448	4	using	use	VERB
ejpam-6175	448	5	linear	linear	PROPN
ejpam-6175	448	6	homotopy	homotopy	NOUN
ejpam-6175	448	7	parametric	parametric	ADJ
ejpam-6175	448	8	value	value	NOUN
ejpam-6175	448	9	for	for	ADP
ejpam-6175	448	10	enhancing	enhance	VERB
ejpam-6175	448	11	image	image	NOUN
ejpam-6175	448	12	edge	edge	NOUN
ejpam-6175	448	13	detection	detection	NOUN
ejpam-6175	448	14	.	.	PUNCT
ejpam-6175	449	1	ieee	ieee	NOUN
ejpam-6175	449	2	access	access	NOUN
ejpam-6175	449	3	,	,	PUNCT
ejpam-6175	449	4	12:107113	12:107113	NUM
ejpam-6175	449	5	–	–	PUNCT
ejpam-6175	449	6	107118	107118	NUM
ejpam-6175	449	7	,	,	PUNCT
ejpam-6175	449	8	2024	2024	NUM
ejpam-6175	449	9	.	.	PUNCT
ejpam-6175	450	1	doi	doi	NOUN
ejpam-6175	450	2	:	:	PUNCT
ejpam-6175	450	3	https://doi.org/10.1109/access.2024.3436661	https://doi.org/10.1109/access.2024.3436661	PROPN
ejpam-6175	450	4	.	.	PUNCT