id	sid	tid	token	lemma	pos
ejpam-4713	1	1	european	european	PROPN
ejpam-4713	1	2	journal	journal	PROPN
ejpam-4713	1	3	of	of	ADP
ejpam-4713	1	4	pure	pure	ADJ
ejpam-4713	1	5	and	and	CCONJ
ejpam-4713	1	6	applied	apply	VERB
ejpam-4713	1	7	mathematics	mathematic	NOUN
ejpam-4713	1	8	vol	vol	NOUN
ejpam-4713	1	9	.	.	PUNCT
ejpam-4713	2	1	16	16	NUM
ejpam-4713	2	2	,	,	PUNCT
ejpam-4713	2	3	no	no	INTJ
ejpam-4713	2	4	.	.	NOUN
ejpam-4713	2	5	2	2	NUM
ejpam-4713	2	6	,	,	PUNCT
ejpam-4713	2	7	2023	2023	NUM
ejpam-4713	2	8	,	,	PUNCT
ejpam-4713	2	9	893	893	NUM
ejpam-4713	2	10	-	-	SYM
ejpam-4713	2	11	898	898	NUM
ejpam-4713	2	12	issn	issn	PROPN
ejpam-4713	2	13	1307	1307	NUM
ejpam-4713	2	14	-	-	SYM
ejpam-4713	2	15	5543	5543	NUM
ejpam-4713	2	16	–	–	PUNCT
ejpam-4713	2	17	ejpam.com	ejpam.com	X
ejpam-4713	2	18	published	publish	VERB
ejpam-4713	2	19	by	by	ADP
ejpam-4713	2	20	new	new	PROPN
ejpam-4713	2	21	york	york	PROPN
ejpam-4713	2	22	business	business	PROPN
ejpam-4713	2	23	global	global	PROPN
ejpam-4713	2	24	on	on	ADP
ejpam-4713	2	25	the	the	DET
ejpam-4713	2	26	partition	partition	NOUN
ejpam-4713	2	27	of	of	ADP
ejpam-4713	2	28	space	space	NOUN
ejpam-4713	2	29	by	by	ADP
ejpam-4713	2	30	hyperplanes	hyperplane	NOUN
ejpam-4713	2	31	armen	armen	VERB
ejpam-4713	2	32	bagdasaryan	bagdasaryan	ADJ
ejpam-4713	2	33	college	college	NOUN
ejpam-4713	2	34	of	of	ADP
ejpam-4713	2	35	engineering	engineering	NOUN
ejpam-4713	2	36	and	and	CCONJ
ejpam-4713	2	37	technology	technology	NOUN
ejpam-4713	2	38	,	,	PUNCT
ejpam-4713	2	39	american	american	PROPN
ejpam-4713	2	40	university	university	PROPN
ejpam-4713	2	41	of	of	ADP
ejpam-4713	2	42	the	the	DET
ejpam-4713	2	43	middle	middle	PROPN
ejpam-4713	2	44	east	east	PROPN
ejpam-4713	2	45	,	,	PUNCT
ejpam-4713	2	46	egaila	egaila	PROPN
ejpam-4713	2	47	54200	54200	NUM
ejpam-4713	2	48	,	,	PUNCT
ejpam-4713	2	49	kuwait	kuwait	PROPN
ejpam-4713	2	50	abstract	abstract	NOUN
ejpam-4713	2	51	.	.	PUNCT
ejpam-4713	3	1	we	we	PRON
ejpam-4713	3	2	consider	consider	VERB
ejpam-4713	3	3	the	the	DET
ejpam-4713	3	4	problem	problem	NOUN
ejpam-4713	3	5	of	of	ADP
ejpam-4713	3	6	partitioning	partitioning	NOUN
ejpam-4713	3	7	of	of	ADP
ejpam-4713	3	8	space	space	NOUN
ejpam-4713	3	9	by	by	ADP
ejpam-4713	3	10	hyperplanes	hyperplane	NOUN
ejpam-4713	3	11	that	that	PRON
ejpam-4713	3	12	arises	arise	VERB
ejpam-4713	3	13	in	in	ADP
ejpam-4713	3	14	many	many	ADJ
ejpam-4713	3	15	application	application	NOUN
ejpam-4713	3	16	areas	area	NOUN
ejpam-4713	3	17	,	,	PUNCT
ejpam-4713	3	18	where	where	SCONJ
ejpam-4713	3	19	the	the	DET
ejpam-4713	3	20	number	number	NOUN
ejpam-4713	3	21	of	of	ADP
ejpam-4713	3	22	regions	region	NOUN
ejpam-4713	3	23	the	the	DET
ejpam-4713	3	24	space	space	NOUN
ejpam-4713	3	25	is	be	AUX
ejpam-4713	3	26	divided	divide	VERB
ejpam-4713	3	27	into	into	ADP
ejpam-4713	3	28	is	be	AUX
ejpam-4713	3	29	required	require	VERB
ejpam-4713	3	30	to	to	PART
ejpam-4713	3	31	be	be	AUX
ejpam-4713	3	32	determined	determine	VERB
ejpam-4713	3	33	,	,	PUNCT
ejpam-4713	3	34	such	such	ADJ
ejpam-4713	3	35	as	as	ADP
ejpam-4713	3	36	speech	speech	NOUN
ejpam-4713	3	37	/	/	SYM
ejpam-4713	3	38	pattern	pattern	NOUN
ejpam-4713	3	39	recognition	recognition	NOUN
ejpam-4713	3	40	,	,	PUNCT
ejpam-4713	3	41	various	various	ADJ
ejpam-4713	3	42	classification	classification	NOUN
ejpam-4713	3	43	problems	problem	NOUN
ejpam-4713	3	44	,	,	PUNCT
ejpam-4713	3	45	data	datum	NOUN
ejpam-4713	3	46	analysis	analysis	NOUN
ejpam-4713	3	47	.	.	PUNCT
ejpam-4713	4	1	we	we	PRON
ejpam-4713	4	2	obtain	obtain	VERB
ejpam-4713	4	3	some	some	DET
ejpam-4713	4	4	relations	relation	NOUN
ejpam-4713	4	5	for	for	ADP
ejpam-4713	4	6	the	the	DET
ejpam-4713	4	7	number	number	NOUN
ejpam-4713	4	8	of	of	ADP
ejpam-4713	4	9	divisions	division	NOUN
ejpam-4713	4	10	and	and	CCONJ
ejpam-4713	4	11	establish	establish	VERB
ejpam-4713	4	12	a	a	DET
ejpam-4713	4	13	recurrence	recurrence	NOUN
ejpam-4713	4	14	relation	relation	NOUN
ejpam-4713	4	15	for	for	ADP
ejpam-4713	4	16	the	the	DET
ejpam-4713	4	17	maximum	maximum	ADJ
ejpam-4713	4	18	number	number	NOUN
ejpam-4713	4	19	of	of	ADP
ejpam-4713	4	20	regions	region	NOUN
ejpam-4713	4	21	in	in	ADP
ejpam-4713	4	22	d	d	ADJ
ejpam-4713	4	23	-	-	ADJ
ejpam-4713	4	24	dimensional	dimensional	ADJ
ejpam-4713	4	25	euclidean	euclidean	ADJ
ejpam-4713	4	26	space	space	NOUN
ejpam-4713	4	27	cut	cut	VERB
ejpam-4713	4	28	by	by	ADP
ejpam-4713	4	29	n	n	DET
ejpam-4713	4	30	hyperplanes	hyperplane	NOUN
ejpam-4713	4	31	.	.	PUNCT
ejpam-4713	5	1	we	we	PRON
ejpam-4713	5	2	also	also	ADV
ejpam-4713	5	3	re	re	VERB
ejpam-4713	5	4	-	-	VERB
ejpam-4713	5	5	derive	derive	VERB
ejpam-4713	5	6	an	an	DET
ejpam-4713	5	7	explicit	explicit	ADJ
ejpam-4713	5	8	formula	formula	NOUN
ejpam-4713	5	9	for	for	ADP
ejpam-4713	5	10	the	the	DET
ejpam-4713	5	11	number	number	NOUN
ejpam-4713	5	12	of	of	ADP
ejpam-4713	5	13	regions	region	NOUN
ejpam-4713	5	14	into	into	ADP
ejpam-4713	5	15	which	which	PRON
ejpam-4713	5	16	the	the	DET
ejpam-4713	5	17	space	space	NOUN
ejpam-4713	5	18	can	can	AUX
ejpam-4713	5	19	be	be	AUX
ejpam-4713	5	20	partitioned	partition	VERB
ejpam-4713	5	21	by	by	ADP
ejpam-4713	5	22	n	n	DET
ejpam-4713	5	23	hyperplanes	hyperplane	NOUN
ejpam-4713	5	24	.	.	PUNCT
ejpam-4713	6	1	2020	2020	NUM
ejpam-4713	6	2	mathematics	mathematic	NOUN
ejpam-4713	6	3	subject	subject	NOUN
ejpam-4713	6	4	classifications	classification	NOUN
ejpam-4713	6	5	:	:	PUNCT
ejpam-4713	6	6	52c35	52c35	NUM
ejpam-4713	6	7	,	,	PUNCT
ejpam-4713	6	8	51m20	51m20	NUM
ejpam-4713	6	9	,	,	PUNCT
ejpam-4713	6	10	05a19	05a19	NUM
ejpam-4713	6	11	,	,	PUNCT
ejpam-4713	6	12	14n20	14n20	NUM
ejpam-4713	6	13	,	,	PUNCT
ejpam-4713	6	14	32s22	32s22	NUM
ejpam-4713	6	15	key	key	ADJ
ejpam-4713	6	16	words	word	NOUN
ejpam-4713	6	17	and	and	CCONJ
ejpam-4713	6	18	phrases	phrase	NOUN
ejpam-4713	6	19	:	:	PUNCT
ejpam-4713	6	20	hyperplanes	hyperplane	NOUN
ejpam-4713	6	21	in	in	ADP
ejpam-4713	6	22	rd	rd	PROPN
ejpam-4713	6	23	,	,	PUNCT
ejpam-4713	6	24	partition	partition	NOUN
ejpam-4713	6	25	of	of	ADP
ejpam-4713	6	26	rd	rd	PROPN
ejpam-4713	6	27	,	,	PUNCT
ejpam-4713	6	28	hyperplane	hyperplane	NOUN
ejpam-4713	6	29	arrangement	arrangement	NOUN
ejpam-4713	6	30	,	,	PUNCT
ejpam-4713	6	31	number	number	NOUN
ejpam-4713	6	32	of	of	ADP
ejpam-4713	6	33	regions	region	NOUN
ejpam-4713	6	34	,	,	PUNCT
ejpam-4713	6	35	k	k	NOUN
ejpam-4713	6	36	-	-	NOUN
ejpam-4713	6	37	edge	edge	NOUN
ejpam-4713	6	38	of	of	ADP
ejpam-4713	6	39	a	a	DET
ejpam-4713	6	40	family	family	NOUN
ejpam-4713	6	41	of	of	ADP
ejpam-4713	6	42	hyperplanes	hyperplane	NOUN
ejpam-4713	6	43	,	,	PUNCT
ejpam-4713	6	44	binomial	binomial	ADJ
ejpam-4713	6	45	coefficients	coefficient	NOUN
ejpam-4713	6	46	1	1	NUM
ejpam-4713	6	47	.	.	PUNCT
ejpam-4713	7	1	introduction	introduction	NOUN
ejpam-4713	7	2	and	and	CCONJ
ejpam-4713	7	3	preliminaries	preliminary	NOUN
ejpam-4713	7	4	the	the	DET
ejpam-4713	7	5	problem	problem	NOUN
ejpam-4713	7	6	of	of	ADP
ejpam-4713	7	7	partitioning	partition	VERB
ejpam-4713	7	8	of	of	ADP
ejpam-4713	7	9	d	d	NOUN
ejpam-4713	7	10	-	-	NOUN
ejpam-4713	7	11	space	space	NOUN
ejpam-4713	7	12	(	(	PUNCT
ejpam-4713	7	13	or	or	CCONJ
ejpam-4713	7	14	,	,	PUNCT
ejpam-4713	7	15	points	point	VERB
ejpam-4713	7	16	in	in	ADP
ejpam-4713	7	17	d	d	NOUN
ejpam-4713	7	18	-	-	NOUN
ejpam-4713	7	19	space	space	NOUN
ejpam-4713	7	20	)	)	PUNCT
ejpam-4713	7	21	by	by	ADP
ejpam-4713	7	22	a	a	DET
ejpam-4713	7	23	set	set	NOUN
ejpam-4713	7	24	of	of	ADP
ejpam-4713	7	25	hyperplanes	hyperplane	NOUN
ejpam-4713	7	26	(	(	PUNCT
ejpam-4713	7	27	a	a	DET
ejpam-4713	7	28	single	single	ADJ
ejpam-4713	7	29	hyperplane	hyperplane	NOUN
ejpam-4713	7	30	)	)	PUNCT
ejpam-4713	7	31	has	have	AUX
ejpam-4713	7	32	been	be	AUX
ejpam-4713	7	33	the	the	DET
ejpam-4713	7	34	subject	subject	NOUN
ejpam-4713	7	35	of	of	ADP
ejpam-4713	7	36	multiple	multiple	ADJ
ejpam-4713	7	37	works	work	NOUN
ejpam-4713	7	38	.	.	PUNCT
ejpam-4713	8	1	it	it	PRON
ejpam-4713	8	2	is	be	AUX
ejpam-4713	8	3	an	an	DET
ejpam-4713	8	4	interesting	interesting	ADJ
ejpam-4713	8	5	problem	problem	NOUN
ejpam-4713	8	6	in	in	ADP
ejpam-4713	8	7	its	its	PRON
ejpam-4713	8	8	own	own	ADJ
ejpam-4713	8	9	right	right	NOUN
ejpam-4713	8	10	(	(	PUNCT
ejpam-4713	8	11	see	see	VERB
ejpam-4713	8	12	[	[	X
ejpam-4713	8	13	1	1	NUM
ejpam-4713	8	14	,	,	PUNCT
ejpam-4713	8	15	4	4	NUM
ejpam-4713	8	16	,	,	PUNCT
ejpam-4713	8	17	8	8	NUM
ejpam-4713	8	18	,	,	PUNCT
ejpam-4713	8	19	11	11	NUM
ejpam-4713	8	20	]	]	NUM
ejpam-4713	8	21	)	)	PUNCT
ejpam-4713	8	22	,	,	PUNCT
ejpam-4713	8	23	but	but	CCONJ
ejpam-4713	8	24	the	the	DET
ejpam-4713	8	25	problem	problem	NOUN
ejpam-4713	8	26	has	have	AUX
ejpam-4713	8	27	been	be	AUX
ejpam-4713	8	28	considered	consider	VERB
ejpam-4713	8	29	not	not	PART
ejpam-4713	8	30	only	only	ADV
ejpam-4713	8	31	for	for	ADP
ejpam-4713	8	32	its	its	PRON
ejpam-4713	8	33	general	general	ADJ
ejpam-4713	8	34	mathematical	mathematical	ADJ
ejpam-4713	8	35	interest	interest	NOUN
ejpam-4713	8	36	.	.	PUNCT
ejpam-4713	9	1	the	the	DET
ejpam-4713	9	2	question	question	NOUN
ejpam-4713	9	3	of	of	ADP
ejpam-4713	9	4	how	how	SCONJ
ejpam-4713	9	5	many	many	ADJ
ejpam-4713	9	6	such	such	ADJ
ejpam-4713	9	7	partitions	partition	NOUN
ejpam-4713	9	8	are	be	AUX
ejpam-4713	9	9	possible	possible	ADJ
ejpam-4713	9	10	arises	arise	NOUN
ejpam-4713	9	11	in	in	ADP
ejpam-4713	9	12	various	various	ADJ
ejpam-4713	9	13	applications	application	NOUN
ejpam-4713	9	14	,	,	PUNCT
ejpam-4713	9	15	for	for	ADP
ejpam-4713	9	16	example	example	NOUN
ejpam-4713	9	17	,	,	PUNCT
ejpam-4713	9	18	in	in	ADP
ejpam-4713	9	19	the	the	DET
ejpam-4713	9	20	theory	theory	NOUN
ejpam-4713	9	21	of	of	ADP
ejpam-4713	9	22	pattern	pattern	NOUN
ejpam-4713	9	23	classification	classification	NOUN
ejpam-4713	9	24	and	and	CCONJ
ejpam-4713	9	25	machine	machine	NOUN
ejpam-4713	9	26	learning	learn	VERB
ejpam-4713	9	27	[	[	X
ejpam-4713	9	28	3	3	NUM
ejpam-4713	9	29	,	,	PUNCT
ejpam-4713	9	30	5	5	NUM
ejpam-4713	9	31	,	,	PUNCT
ejpam-4713	9	32	10	10	NUM
ejpam-4713	9	33	]	]	PUNCT
ejpam-4713	9	34	which	which	PRON
ejpam-4713	9	35	employs	employ	VERB
ejpam-4713	9	36	partitions	partition	NOUN
ejpam-4713	9	37	using	use	VERB
ejpam-4713	9	38	some	some	DET
ejpam-4713	9	39	number	number	NOUN
ejpam-4713	9	40	k	k	NOUN
ejpam-4713	9	41	of	of	ADP
ejpam-4713	9	42	parallel	parallel	ADJ
ejpam-4713	9	43	hyperplanes	hyperplane	NOUN
ejpam-4713	9	44	;	;	PUNCT
ejpam-4713	9	45	in	in	ADP
ejpam-4713	9	46	cluster	cluster	NOUN
ejpam-4713	9	47	analysis	analysis	NOUN
ejpam-4713	9	48	[	[	X
ejpam-4713	9	49	6	6	NUM
ejpam-4713	9	50	]	]	PUNCT
ejpam-4713	9	51	dealing	deal	VERB
ejpam-4713	9	52	with	with	ADP
ejpam-4713	9	53	the	the	DET
ejpam-4713	9	54	methods	method	NOUN
ejpam-4713	9	55	of	of	ADP
ejpam-4713	9	56	partitioning	partition	VERB
ejpam-4713	9	57	of	of	ADP
ejpam-4713	9	58	a	a	DET
ejpam-4713	9	59	set	set	NOUN
ejpam-4713	9	60	of	of	ADP
ejpam-4713	9	61	objects	object	NOUN
ejpam-4713	9	62	on	on	ADP
ejpam-4713	9	63	the	the	DET
ejpam-4713	9	64	basis	basis	NOUN
ejpam-4713	9	65	of	of	ADP
ejpam-4713	9	66	their	their	PRON
ejpam-4713	9	67	characteristics	characteristic	NOUN
ejpam-4713	9	68	or	or	CCONJ
ejpam-4713	9	69	properties	property	NOUN
ejpam-4713	9	70	into	into	ADP
ejpam-4713	9	71	clusters	cluster	NOUN
ejpam-4713	9	72	or	or	CCONJ
ejpam-4713	9	73	groups	group	NOUN
ejpam-4713	9	74	so	so	SCONJ
ejpam-4713	9	75	that	that	SCONJ
ejpam-4713	9	76	the	the	DET
ejpam-4713	9	77	objects	object	NOUN
ejpam-4713	9	78	of	of	ADP
ejpam-4713	9	79	a	a	DET
ejpam-4713	9	80	cluster	cluster	NOUN
ejpam-4713	9	81	are	be	AUX
ejpam-4713	9	82	closely	closely	ADV
ejpam-4713	9	83	related	relate	VERB
ejpam-4713	9	84	according	accord	VERB
ejpam-4713	9	85	to	to	ADP
ejpam-4713	9	86	certain	certain	ADJ
ejpam-4713	9	87	criteria	criterion	NOUN
ejpam-4713	9	88	;	;	PUNCT
ejpam-4713	9	89	data	datum	NOUN
ejpam-4713	9	90	analysis	analysis	NOUN
ejpam-4713	9	91	and	and	CCONJ
ejpam-4713	9	92	classification	classification	NOUN
ejpam-4713	9	93	in	in	ADP
ejpam-4713	9	94	which	which	PRON
ejpam-4713	9	95	space	space	NOUN
ejpam-4713	9	96	partitioning	partition	VERB
ejpam-4713	9	97	by	by	ADP
ejpam-4713	9	98	hyperplanes	hyperplane	NOUN
ejpam-4713	9	99	may	may	AUX
ejpam-4713	9	100	be	be	AUX
ejpam-4713	9	101	used	use	VERB
ejpam-4713	9	102	together	together	ADV
ejpam-4713	9	103	with	with	ADP
ejpam-4713	9	104	linear	linear	ADJ
ejpam-4713	9	105	regression	regression	NOUN
ejpam-4713	9	106	techniques	technique	NOUN
ejpam-4713	9	107	based	base	VERB
ejpam-4713	9	108	on	on	ADP
ejpam-4713	9	109	least	least	ADJ
ejpam-4713	9	110	absolute	absolute	ADJ
ejpam-4713	9	111	value	value	NOUN
ejpam-4713	9	112	estimates	estimate	NOUN
ejpam-4713	9	113	approach	approach	NOUN
ejpam-4713	9	114	and	and	CCONJ
ejpam-4713	9	115	its	its	PRON
ejpam-4713	9	116	generalizations	generalization	NOUN
ejpam-4713	9	117	;	;	PUNCT
ejpam-4713	9	118	in	in	ADP
ejpam-4713	9	119	the	the	DET
ejpam-4713	9	120	theory	theory	NOUN
ejpam-4713	9	121	of	of	ADP
ejpam-4713	9	122	hybrid	hybrid	NOUN
ejpam-4713	9	123	systems	system	NOUN
ejpam-4713	9	124	and	and	CCONJ
ejpam-4713	9	125	control	control	NOUN
ejpam-4713	9	126	[	[	X
ejpam-4713	9	127	9	9	NUM
ejpam-4713	9	128	]	]	PUNCT
ejpam-4713	9	129	.	.	PUNCT
ejpam-4713	10	1	the	the	DET
ejpam-4713	10	2	problem	problem	NOUN
ejpam-4713	10	3	has	have	AUX
ejpam-4713	10	4	been	be	AUX
ejpam-4713	10	5	generalized	generalize	VERB
ejpam-4713	10	6	and	and	CCONJ
ejpam-4713	10	7	analyzed	analyze	VERB
ejpam-4713	10	8	in	in	ADP
ejpam-4713	10	9	different	different	ADJ
ejpam-4713	10	10	ways	way	NOUN
ejpam-4713	10	11	,	,	PUNCT
ejpam-4713	10	12	e.g.	e.g.	ADV
ejpam-4713	10	13	[	[	X
ejpam-4713	10	14	7	7	NUM
ejpam-4713	10	15	]	]	PUNCT
ejpam-4713	10	16	considers	consider	VERB
ejpam-4713	10	17	the	the	DET
ejpam-4713	10	18	division	division	NOUN
ejpam-4713	10	19	of	of	ADP
ejpam-4713	10	20	d	d	NOUN
ejpam-4713	10	21	-	-	NOUN
ejpam-4713	10	22	space	space	NOUN
ejpam-4713	10	23	by	by	ADP
ejpam-4713	10	24	topological	topological	ADJ
ejpam-4713	10	25	hyperplanes	hyperplane	NOUN
ejpam-4713	10	26	,	,	PUNCT
ejpam-4713	10	27	subspaces	subspace	NOUN
ejpam-4713	10	28	of	of	ADP
ejpam-4713	10	29	rd	rd	NOUN
ejpam-4713	10	30	or	or	CCONJ
ejpam-4713	10	31	homeomorphs	homeomorph	NOUN
ejpam-4713	10	32	of	of	ADP
ejpam-4713	10	33	it	it	PRON
ejpam-4713	10	34	,	,	PUNCT
ejpam-4713	10	35	that	that	PRON
ejpam-4713	10	36	is	be	AUX
ejpam-4713	10	37	topological	topological	ADJ
ejpam-4713	10	38	equivalent	equivalent	ADJ
ejpam-4713	10	39	to	to	ADP
ejpam-4713	10	40	an	an	DET
ejpam-4713	10	41	ordinary	ordinary	ADJ
ejpam-4713	10	42	straight	straight	ADJ
ejpam-4713	10	43	hyperplane	hyperplane	NOUN
ejpam-4713	10	44	,	,	PUNCT
ejpam-4713	10	45	and	and	CCONJ
ejpam-4713	10	46	in	in	ADP
ejpam-4713	10	47	[	[	X
ejpam-4713	10	48	2	2	X
ejpam-4713	10	49	]	]	PUNCT
ejpam-4713	10	50	partitioning	partition	VERB
ejpam-4713	10	51	by	by	ADP
ejpam-4713	10	52	the	the	DET
ejpam-4713	10	53	polynomial	polynomial	ADJ
ejpam-4713	10	54	separating	separate	VERB
ejpam-4713	10	55	surfaces	surface	NOUN
ejpam-4713	10	56	has	have	AUX
ejpam-4713	10	57	been	be	AUX
ejpam-4713	10	58	studied	study	VERB
ejpam-4713	10	59	.	.	PUNCT
ejpam-4713	11	1	in	in	ADP
ejpam-4713	11	2	this	this	DET
ejpam-4713	11	3	note	note	NOUN
ejpam-4713	11	4	we	we	PRON
ejpam-4713	11	5	deal	deal	VERB
ejpam-4713	11	6	with	with	ADP
ejpam-4713	11	7	the	the	DET
ejpam-4713	11	8	partitions	partition	NOUN
ejpam-4713	11	9	of	of	ADP
ejpam-4713	11	10	rd	rd	NOUN
ejpam-4713	11	11	that	that	PRON
ejpam-4713	11	12	may	may	AUX
ejpam-4713	11	13	naturally	naturally	ADV
ejpam-4713	11	14	arise	arise	VERB
ejpam-4713	11	15	in	in	ADP
ejpam-4713	11	16	the	the	DET
ejpam-4713	11	17	problems	problem	NOUN
ejpam-4713	11	18	of	of	ADP
ejpam-4713	11	19	clustering	clustering	NOUN
ejpam-4713	11	20	or	or	CCONJ
ejpam-4713	11	21	classification	classification	NOUN
ejpam-4713	11	22	considered	consider	VERB
ejpam-4713	11	23	in	in	ADP
ejpam-4713	11	24	discrete	discrete	ADJ
ejpam-4713	11	25	form	form	NOUN
ejpam-4713	11	26	.	.	PUNCT
ejpam-4713	12	1	doi	doi	NOUN
ejpam-4713	12	2	:	:	PUNCT
ejpam-4713	12	3	https://doi.org/10.29020/nybg.ejpam.v16i2.4713	https://doi.org/10.29020/nybg.ejpam.v16i2.4713	PROPN
ejpam-4713	12	4	email	email	NOUN
ejpam-4713	12	5	address	address	NOUN
ejpam-4713	12	6	:	:	PUNCT
ejpam-4713	12	7	armen.bagdasaryan@aum.edu.kw	armen.bagdasaryan@aum.edu.kw	X
ejpam-4713	12	8	(	(	PUNCT
ejpam-4713	12	9	a.	a.	NOUN
ejpam-4713	12	10	bagdasaryan	bagdasaryan	ADJ
ejpam-4713	12	11	)	)	PUNCT
ejpam-4713	12	12	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4713	12	13	893	893	NUM
ejpam-4713	12	14	©	©	PROPN
ejpam-4713	12	15	2023	2023	NUM
ejpam-4713	12	16	ejpam	ejpam	NOUN
ejpam-4713	12	17	all	all	DET
ejpam-4713	12	18	rights	right	NOUN
ejpam-4713	12	19	reserved	reserve	VERB
ejpam-4713	12	20	.	.	PUNCT
ejpam-4713	13	1	a.	a.	PROPN
ejpam-4713	13	2	bagdasaryan	bagdasaryan	PROPN
ejpam-4713	13	3	/	/	SYM
ejpam-4713	13	4	eur	eur	PROPN
ejpam-4713	13	5	.	.	PUNCT
ejpam-4713	14	1	j.	j.	PROPN
ejpam-4713	14	2	pure	pure	PROPN
ejpam-4713	14	3	appl	appl	PROPN
ejpam-4713	14	4	.	.	PROPN
ejpam-4713	14	5	math	math	PROPN
ejpam-4713	14	6	,	,	PUNCT
ejpam-4713	14	7	16	16	NUM
ejpam-4713	14	8	(	(	PUNCT
ejpam-4713	14	9	2	2	NUM
ejpam-4713	14	10	)	)	PUNCT
ejpam-4713	14	11	(	(	PUNCT
ejpam-4713	14	12	2023	2023	NUM
ejpam-4713	14	13	)	)	PUNCT
ejpam-4713	14	14	,	,	PUNCT
ejpam-4713	14	15	893	893	NUM
ejpam-4713	14	16	-	-	SYM
ejpam-4713	14	17	898	898	NUM
ejpam-4713	14	18	894	894	NUM
ejpam-4713	14	19	let	let	VERB
ejpam-4713	14	20	us	we	PRON
ejpam-4713	14	21	first	first	ADV
ejpam-4713	14	22	introduce	introduce	VERB
ejpam-4713	14	23	the	the	DET
ejpam-4713	14	24	notations	notation	NOUN
ejpam-4713	14	25	.	.	PUNCT
ejpam-4713	15	1	we	we	PRON
ejpam-4713	15	2	use	use	VERB
ejpam-4713	15	3	the	the	DET
ejpam-4713	15	4	term	term	NOUN
ejpam-4713	15	5	hyperplane	hyperplane	NOUN
ejpam-4713	15	6	in	in	ADP
ejpam-4713	15	7	rd	rd	PROPN
ejpam-4713	15	8	to	to	PART
ejpam-4713	15	9	mean	mean	VERB
ejpam-4713	15	10	an	an	DET
ejpam-4713	15	11	affine	affine	NOUN
ejpam-4713	15	12	subspace	subspace	NOUN
ejpam-4713	15	13	of	of	ADP
ejpam-4713	15	14	codimension	codimension	NOUN
ejpam-4713	15	15	one	one	NUM
ejpam-4713	15	16	in	in	ADP
ejpam-4713	15	17	rd	rd	PROPN
ejpam-4713	15	18	,	,	PUNCT
ejpam-4713	15	19	i.e.	i.e.	X
ejpam-4713	15	20	a	a	PRON
ejpam-4713	15	21	(	(	PUNCT
ejpam-4713	15	22	d−	d−	PROPN
ejpam-4713	15	23	1)-dimensional	1)-dimensional	ADJ
ejpam-4713	15	24	straight	straight	ADJ
ejpam-4713	15	25	plane	plane	NOUN
ejpam-4713	15	26	in	in	ADP
ejpam-4713	15	27	rd	rd	NOUN
ejpam-4713	15	28	which	which	PRON
ejpam-4713	15	29	need	need	AUX
ejpam-4713	15	30	not	not	PART
ejpam-4713	15	31	necessarily	necessarily	ADV
ejpam-4713	15	32	pass	pass	VERB
ejpam-4713	15	33	through	through	ADP
ejpam-4713	15	34	the	the	DET
ejpam-4713	15	35	origin	origin	NOUN
ejpam-4713	15	36	.	.	PUNCT
ejpam-4713	16	1	let	let	VERB
ejpam-4713	16	2	h	h	PRON
ejpam-4713	16	3	be	be	AUX
ejpam-4713	16	4	a	a	DET
ejpam-4713	16	5	family	family	NOUN
ejpam-4713	16	6	of	of	ADP
ejpam-4713	16	7	n	n	DET
ejpam-4713	16	8	distinct	distinct	ADJ
ejpam-4713	16	9	hyperplanes	hyperplane	NOUN
ejpam-4713	16	10	{	{	PUNCT
ejpam-4713	16	11	hi}i=1,	hi}i=1,	ADJ
ejpam-4713	16	12	...	...	PUNCT
ejpam-4713	16	13	,n	,n	PUNCT
ejpam-4713	16	14	in	in	ADP
ejpam-4713	16	15	the	the	DET
ejpam-4713	16	16	d	d	ADJ
ejpam-4713	16	17	-	-	ADJ
ejpam-4713	16	18	dimensional	dimensional	ADJ
ejpam-4713	16	19	space	space	NOUN
ejpam-4713	16	20	rd	rd	NOUN
ejpam-4713	16	21	,	,	PUNCT
ejpam-4713	16	22	where	where	SCONJ
ejpam-4713	16	23	d	d	NOUN
ejpam-4713	16	24	and	and	CCONJ
ejpam-4713	16	25	n	n	NUM
ejpam-4713	16	26	are	be	AUX
ejpam-4713	16	27	arbitrary	arbitrary	ADJ
ejpam-4713	16	28	positive	positive	ADJ
ejpam-4713	16	29	integers	integer	NOUN
ejpam-4713	16	30	.	.	PUNCT
ejpam-4713	17	1	we	we	PRON
ejpam-4713	17	2	denote	denote	VERB
ejpam-4713	17	3	by	by	ADP
ejpam-4713	17	4	gh(d	gh(d	PROPN
ejpam-4713	17	5	,	,	PUNCT
ejpam-4713	17	6	n	n	CCONJ
ejpam-4713	17	7	)	)	PUNCT
ejpam-4713	17	8	=	=	SYM
ejpam-4713	17	9	gh1,h2,	gh1,h2,	NOUN
ejpam-4713	17	10	...	...	PUNCT
ejpam-4713	17	11	,hn(d	,hn(d	ADJ
ejpam-4713	17	12	,	,	PUNCT
ejpam-4713	17	13	n	n	CCONJ
ejpam-4713	17	14	)	)	PUNCT
ejpam-4713	17	15	the	the	DET
ejpam-4713	17	16	number	number	NOUN
ejpam-4713	17	17	of	of	ADP
ejpam-4713	17	18	regions	region	NOUN
ejpam-4713	17	19	into	into	ADP
ejpam-4713	17	20	which	which	PRON
ejpam-4713	17	21	the	the	DET
ejpam-4713	17	22	space	space	NOUN
ejpam-4713	17	23	rd	rd	PROPN
ejpam-4713	17	24	is	be	AUX
ejpam-4713	17	25	partitioned	partition	VERB
ejpam-4713	17	26	by	by	ADP
ejpam-4713	17	27	the	the	DET
ejpam-4713	17	28	hyperplanes	hyperplanes	PROPN
ejpam-4713	17	29	h1	h1	PROPN
ejpam-4713	17	30	,	,	PUNCT
ejpam-4713	17	31	h2	h2	PROPN
ejpam-4713	17	32	,	,	PUNCT
ejpam-4713	17	33	.	.	PUNCT
ejpam-4713	17	34	.	.	PUNCT
ejpam-4713	18	1	.	.	PUNCT
ejpam-4713	19	1	,	,	PUNCT
ejpam-4713	19	2	hn	hn	PROPN
ejpam-4713	19	3	,	,	PUNCT
ejpam-4713	19	4	that	that	PRON
ejpam-4713	19	5	is	be	AUX
ejpam-4713	19	6	the	the	DET
ejpam-4713	19	7	number	number	NOUN
ejpam-4713	19	8	of	of	ADP
ejpam-4713	19	9	connected	connected	ADJ
ejpam-4713	19	10	components	component	NOUN
ejpam-4713	19	11	of	of	ADP
ejpam-4713	19	12	the	the	DET
ejpam-4713	19	13	set	set	PROPN
ejpam-4713	19	14	rd	rd	PROPN
ejpam-4713	19	15	\h	\h	PROPN
ejpam-4713	20	1	=	=	SYM
ejpam-4713	20	2	rd	rd	PROPN
ejpam-4713	20	3	\	\	PROPN
ejpam-4713	20	4	(	(	PUNCT
ejpam-4713	20	5	h1	h1	PROPN
ejpam-4713	20	6	∪	∪	VERB
ejpam-4713	20	7	h2	h2	NOUN
ejpam-4713	20	8	∪	∪	NOUN
ejpam-4713	20	9	.	.	PUNCT
ejpam-4713	20	10	.	.	PUNCT
ejpam-4713	20	11	.	.	PUNCT
ejpam-4713	21	1	∪	∪	ADP
ejpam-4713	21	2	hn	hn	PROPN
ejpam-4713	21	3	)	)	PUNCT
ejpam-4713	21	4	.	.	PUNCT
ejpam-4713	22	1	we	we	PRON
ejpam-4713	22	2	also	also	ADV
ejpam-4713	22	3	set	set	VERB
ejpam-4713	22	4	g(d	g(d	PROPN
ejpam-4713	22	5	,	,	PUNCT
ejpam-4713	22	6	n	n	CCONJ
ejpam-4713	22	7	)	)	PUNCT
ejpam-4713	23	1	=	=	SYM
ejpam-4713	23	2	max	max	PROPN
ejpam-4713	23	3	h	h	PROPN
ejpam-4713	23	4	gh(d	gh(d	PROPN
ejpam-4713	23	5	,	,	PUNCT
ejpam-4713	23	6	n	n	CCONJ
ejpam-4713	23	7	)	)	PUNCT
ejpam-4713	23	8	.	.	PUNCT
ejpam-4713	24	1	any	any	DET
ejpam-4713	24	2	nonvoid	nonvoid	ADJ
ejpam-4713	24	3	intersection	intersection	NOUN
ejpam-4713	24	4	of	of	ADP
ejpam-4713	24	5	hyperplanes	hyperplane	NOUN
ejpam-4713	24	6	of	of	ADP
ejpam-4713	24	7	the	the	DET
ejpam-4713	24	8	family	family	NOUN
ejpam-4713	24	9	h	h	NOUN
ejpam-4713	24	10	is	be	AUX
ejpam-4713	24	11	referred	refer	VERB
ejpam-4713	24	12	to	to	ADP
ejpam-4713	24	13	as	as	ADP
ejpam-4713	24	14	an	an	DET
ejpam-4713	24	15	edge	edge	NOUN
ejpam-4713	24	16	of	of	ADP
ejpam-4713	24	17	the	the	DET
ejpam-4713	24	18	family	family	NOUN
ejpam-4713	24	19	.	.	PUNCT
ejpam-4713	25	1	the	the	DET
ejpam-4713	25	2	edge	edge	NOUN
ejpam-4713	25	3	of	of	ADP
ejpam-4713	25	4	dimension	dimension	NOUN
ejpam-4713	25	5	k	k	PROPN
ejpam-4713	25	6	,	,	PUNCT
ejpam-4713	25	7	which	which	PRON
ejpam-4713	25	8	is	be	AUX
ejpam-4713	25	9	a	a	DET
ejpam-4713	25	10	k	k	ADJ
ejpam-4713	25	11	-	-	ADJ
ejpam-4713	25	12	dimensional	dimensional	ADJ
ejpam-4713	25	13	plane	plane	NOUN
ejpam-4713	25	14	,	,	PUNCT
ejpam-4713	25	15	is	be	AUX
ejpam-4713	25	16	called	call	VERB
ejpam-4713	25	17	a	a	DET
ejpam-4713	25	18	k	k	NOUN
ejpam-4713	25	19	-	-	PUNCT
ejpam-4713	25	20	edge	edge	NOUN
ejpam-4713	25	21	and	and	CCONJ
ejpam-4713	25	22	denoted	denote	VERB
ejpam-4713	25	23	by	by	ADP
ejpam-4713	25	24	ek	ek	PROPN
ejpam-4713	25	25	,	,	PUNCT
ejpam-4713	25	26	k	k	PROPN
ejpam-4713	25	27	=	=	SYM
ejpam-4713	25	28	0	0	NUM
ejpam-4713	25	29	,	,	PUNCT
ejpam-4713	25	30	1	1	NUM
ejpam-4713	25	31	,	,	PUNCT
ejpam-4713	25	32	.	.	PUNCT
ejpam-4713	25	33	.	.	PUNCT
ejpam-4713	26	1	.	.	PUNCT
ejpam-4713	27	1	,	,	PUNCT
ejpam-4713	27	2	m.	m.	NOUN
ejpam-4713	27	3	we	we	PRON
ejpam-4713	27	4	let	let	VERB
ejpam-4713	27	5	ek	ek	PRON
ejpam-4713	27	6	[	[	X
ejpam-4713	27	7	t	t	X
ejpam-4713	27	8	]	]	PUNCT
ejpam-4713	27	9	denote	denote	VERB
ejpam-4713	27	10	a	a	DET
ejpam-4713	27	11	k	k	ADJ
ejpam-4713	27	12	-	-	PUNCT
ejpam-4713	27	13	edge	edge	NOUN
ejpam-4713	27	14	made	make	VERB
ejpam-4713	27	15	of	of	ADP
ejpam-4713	27	16	intersections	intersection	NOUN
ejpam-4713	27	17	of	of	ADP
ejpam-4713	27	18	any	any	DET
ejpam-4713	27	19	j	j	NOUN
ejpam-4713	27	20	=	=	SYM
ejpam-4713	27	21	1	1	NUM
ejpam-4713	27	22	,	,	PUNCT
ejpam-4713	27	23	.	.	PUNCT
ejpam-4713	27	24	.	.	PUNCT
ejpam-4713	28	1	.	.	PUNCT
ejpam-4713	29	1	,	,	PUNCT
ejpam-4713	29	2	t	t	PROPN
ejpam-4713	29	3	hyperplanes	hyperplane	NOUN
ejpam-4713	29	4	,	,	PUNCT
ejpam-4713	29	5	ek	ek	PROPN
ejpam-4713	30	1	[	[	X
ejpam-4713	30	2	1,t	1,t	X
ejpam-4713	30	3	]	]	PUNCT
ejpam-4713	30	4	a	a	DET
ejpam-4713	30	5	k	k	ADJ
ejpam-4713	30	6	-	-	PUNCT
ejpam-4713	30	7	edge	edge	NOUN
ejpam-4713	30	8	made	make	VERB
ejpam-4713	30	9	of	of	ADP
ejpam-4713	30	10	intersections	intersection	NOUN
ejpam-4713	30	11	of	of	ADP
ejpam-4713	30	12	all	all	PRON
ejpam-4713	30	13	of	of	ADP
ejpam-4713	30	14	h1	h1	PROPN
ejpam-4713	30	15	,	,	PUNCT
ejpam-4713	30	16	h2	h2	PROPN
ejpam-4713	30	17	,	,	PUNCT
ejpam-4713	30	18	.	.	PUNCT
ejpam-4713	30	19	.	.	PUNCT
ejpam-4713	31	1	.	.	PUNCT
ejpam-4713	32	1	,	,	PUNCT
ejpam-4713	32	2	ht	ht	PROPN
ejpam-4713	32	3	hyperplanes	hyperplane	NOUN
ejpam-4713	32	4	,	,	PUNCT
ejpam-4713	32	5	and	and	CCONJ
ejpam-4713	32	6	ek	ek	X
ejpam-4713	33	1	[	[	X
ejpam-4713	33	2	1,t;n	1,t;n	X
ejpam-4713	33	3	]	]	X
ejpam-4713	33	4	a	a	DET
ejpam-4713	33	5	k	k	ADJ
ejpam-4713	33	6	-	-	PUNCT
ejpam-4713	33	7	edge	edge	NOUN
ejpam-4713	33	8	made	make	VERB
ejpam-4713	33	9	of	of	ADP
ejpam-4713	33	10	intersections	intersection	NOUN
ejpam-4713	33	11	of	of	ADP
ejpam-4713	33	12	each	each	DET
ejpam-4713	33	13	hi	hi	INTJ
ejpam-4713	33	14	with	with	ADP
ejpam-4713	33	15	hn	hn	PROPN
ejpam-4713	33	16	,	,	PUNCT
ejpam-4713	33	17	i	i	PRON
ejpam-4713	33	18	̸=	̸=	PROPN
ejpam-4713	33	19	n.	n.	NOUN
ejpam-4713	33	20	in	in	ADP
ejpam-4713	33	21	these	these	DET
ejpam-4713	33	22	terms	term	NOUN
ejpam-4713	33	23	the	the	DET
ejpam-4713	33	24	space	space	NOUN
ejpam-4713	33	25	rd	rd	PROPN
ejpam-4713	33	26	can	can	AUX
ejpam-4713	33	27	be	be	AUX
ejpam-4713	33	28	considered	consider	VERB
ejpam-4713	33	29	as	as	ADP
ejpam-4713	33	30	a	a	DET
ejpam-4713	33	31	d	d	ADJ
ejpam-4713	33	32	-	-	ADJ
ejpam-4713	33	33	dimensional	dimensional	ADJ
ejpam-4713	33	34	edge	edge	NOUN
ejpam-4713	33	35	of	of	ADP
ejpam-4713	33	36	the	the	DET
ejpam-4713	33	37	family	family	NOUN
ejpam-4713	33	38	h.	h.	NOUN
ejpam-4713	34	1	each	each	PRON
ejpam-4713	34	2	separating	separate	VERB
ejpam-4713	34	3	surface	surface	NOUN
ejpam-4713	34	4	in	in	ADP
ejpam-4713	34	5	rd	rd	PROPN
ejpam-4713	34	6	,	,	PUNCT
ejpam-4713	34	7	represented	represent	VERB
ejpam-4713	34	8	as	as	ADP
ejpam-4713	34	9	a	a	DET
ejpam-4713	34	10	hyperplane	hyperplane	NOUN
ejpam-4713	34	11	,	,	PUNCT
ejpam-4713	34	12	is	be	AUX
ejpam-4713	34	13	given	give	VERB
ejpam-4713	34	14	by	by	ADP
ejpam-4713	34	15	a	a	DET
ejpam-4713	34	16	linear	linear	ADJ
ejpam-4713	34	17	equation	equation	NOUN
ejpam-4713	34	18	hi	hi	INTJ
ejpam-4713	34	19	=	=	SYM
ejpam-4713	34	20	{	{	PUNCT
ejpam-4713	34	21	x	x	PUNCT
ejpam-4713	34	22	∈	∈	PROPN
ejpam-4713	34	23	rd	rd	NOUN
ejpam-4713	34	24	|	|	NOUN
ejpam-4713	34	25	aix	aix	NOUN
ejpam-4713	34	26	=	=	SYM
ejpam-4713	34	27	bi	bi	NOUN
ejpam-4713	34	28	}	}	PUNCT
ejpam-4713	34	29	,	,	PUNCT
ejpam-4713	34	30	or	or	CCONJ
ejpam-4713	34	31	equivalently	equivalently	ADV
ejpam-4713	34	32	hi(x	hi(x	NOUN
ejpam-4713	34	33	)	)	PUNCT
ejpam-4713	35	1	≡	≡	PROPN
ejpam-4713	35	2	d∑	d∑	PROPN
ejpam-4713	36	1	j=1	j=1	NOUN
ejpam-4713	36	2	aijxj	aijxj	ADP
ejpam-4713	36	3	−	−	PROPN
ejpam-4713	36	4	bi	bi	NOUN
ejpam-4713	36	5	=	=	NOUN
ejpam-4713	36	6	0	0	PROPN
ejpam-4713	36	7	,	,	PUNCT
ejpam-4713	36	8	i	i	PRON
ejpam-4713	36	9	=	=	NOUN
ejpam-4713	36	10	1	1	NUM
ejpam-4713	36	11	,	,	PUNCT
ejpam-4713	36	12	.	.	PUNCT
ejpam-4713	36	13	.	.	PUNCT
ejpam-4713	36	14	.	.	PUNCT
ejpam-4713	37	1	,	,	PUNCT
ejpam-4713	37	2	n.	n.	NOUN
ejpam-4713	37	3	(	(	PUNCT
ejpam-4713	37	4	1	1	NUM
ejpam-4713	37	5	)	)	PUNCT
ejpam-4713	37	6	then	then	ADV
ejpam-4713	37	7	the	the	DET
ejpam-4713	37	8	classification	classification	NOUN
ejpam-4713	37	9	problem	problem	NOUN
ejpam-4713	37	10	in	in	ADP
ejpam-4713	37	11	discrete	discrete	ADJ
ejpam-4713	37	12	form	form	NOUN
ejpam-4713	37	13	can	can	AUX
ejpam-4713	37	14	typically	typically	ADV
ejpam-4713	37	15	be	be	AUX
ejpam-4713	37	16	formulated	formulate	VERB
ejpam-4713	37	17	as	as	SCONJ
ejpam-4713	37	18	follows	follow	VERB
ejpam-4713	37	19	.	.	PUNCT
ejpam-4713	38	1	given	give	VERB
ejpam-4713	38	2	the	the	DET
ejpam-4713	38	3	probabilities	probability	NOUN
ejpam-4713	38	4	pτ	pτ	ADP
ejpam-4713	38	5	of	of	ADP
ejpam-4713	38	6	appearance	appearance	NOUN
ejpam-4713	38	7	of	of	ADP
ejpam-4713	38	8	each	each	PRON
ejpam-4713	38	9	of	of	ADP
ejpam-4713	38	10	n	n	X
ejpam-4713	38	11	objects	objects	AUX
ejpam-4713	38	12	x̃1	x̃1	PROPN
ejpam-4713	39	1	=	=	SYM
ejpam-4713	40	1	(	(	PUNCT
ejpam-4713	41	1	x11	x11	NOUN
ejpam-4713	41	2	,	,	PUNCT
ejpam-4713	41	3	.	.	PUNCT
ejpam-4713	41	4	.	.	PUNCT
ejpam-4713	41	5	.	.	PUNCT
ejpam-4713	41	6	,	,	PUNCT
ejpam-4713	41	7	x1d	x1d	PUNCT
ejpam-4713	41	8	)	)	PUNCT
ejpam-4713	41	9	,	,	PUNCT
ejpam-4713	41	10	.	.	PUNCT
ejpam-4713	41	11	.	.	PUNCT
ejpam-4713	42	1	.	.	PUNCT
ejpam-4713	42	2	,	,	PUNCT
ejpam-4713	42	3	x̃τ	x̃τ	PUNCT
ejpam-4713	43	1	=	=	PUNCT
ejpam-4713	43	2	(	(	PUNCT
ejpam-4713	43	3	xτ1	xτ1	PROPN
ejpam-4713	43	4	,	,	PUNCT
ejpam-4713	43	5	.	.	PUNCT
ejpam-4713	43	6	.	.	PUNCT
ejpam-4713	43	7	.	.	PUNCT
ejpam-4713	44	1	,	,	PUNCT
ejpam-4713	44	2	xτd	xτd	PROPN
ejpam-4713	44	3	)	)	PUNCT
ejpam-4713	44	4	,	,	PUNCT
ejpam-4713	44	5	.	.	PUNCT
ejpam-4713	44	6	.	.	PUNCT
ejpam-4713	45	1	.	.	PUNCT
ejpam-4713	46	1	,	,	PUNCT
ejpam-4713	46	2	x̃n	x̃n	PROPN
ejpam-4713	47	1	=	=	PUNCT
ejpam-4713	47	2	(	(	PUNCT
ejpam-4713	47	3	xn1	xn1	PROPN
ejpam-4713	47	4	,	,	PUNCT
ejpam-4713	47	5	.	.	PUNCT
ejpam-4713	47	6	.	.	PUNCT
ejpam-4713	47	7	.	.	PUNCT
ejpam-4713	48	1	,	,	PUNCT
ejpam-4713	48	2	xnd	xnd	PROPN
ejpam-4713	48	3	)	)	PUNCT
ejpam-4713	48	4	and	and	CCONJ
ejpam-4713	48	5	information	information	NOUN
ejpam-4713	48	6	about	about	ADP
ejpam-4713	48	7	belonging	belong	VERB
ejpam-4713	48	8	of	of	ADP
ejpam-4713	48	9	each	each	DET
ejpam-4713	48	10	object	object	NOUN
ejpam-4713	48	11	to	to	ADP
ejpam-4713	48	12	a	a	DET
ejpam-4713	48	13	certain	certain	ADJ
ejpam-4713	48	14	class	class	NOUN
ejpam-4713	48	15	,	,	PUNCT
ejpam-4713	48	16	one	one	PRON
ejpam-4713	48	17	needs	need	VERB
ejpam-4713	48	18	to	to	PART
ejpam-4713	48	19	distinguish	distinguish	VERB
ejpam-4713	48	20	s	s	PART
ejpam-4713	48	21	classes	class	NOUN
ejpam-4713	48	22	c1	c1	NOUN
ejpam-4713	48	23	,	,	PUNCT
ejpam-4713	48	24	.	.	PUNCT
ejpam-4713	48	25	.	.	PUNCT
ejpam-4713	49	1	.	.	PUNCT
ejpam-4713	50	1	,	,	PUNCT
ejpam-4713	50	2	ck	ck	INTJ
ejpam-4713	50	3	,	,	PUNCT
ejpam-4713	50	4	.	.	PUNCT
ejpam-4713	50	5	.	.	PUNCT
ejpam-4713	51	1	.	.	PUNCT
ejpam-4713	52	1	,	,	PUNCT
ejpam-4713	52	2	cs	cs	PROPN
ejpam-4713	52	3	in	in	ADP
ejpam-4713	52	4	the	the	DET
ejpam-4713	52	5	space	space	NOUN
ejpam-4713	52	6	of	of	ADP
ejpam-4713	52	7	parameters	parameters	PROPN
ejpam-4713	52	8	rd	rd	PROPN
ejpam-4713	52	9	.	.	PUNCT
ejpam-4713	53	1	the	the	DET
ejpam-4713	53	2	rule	rule	NOUN
ejpam-4713	53	3	that	that	PRON
ejpam-4713	53	4	determines	determine	VERB
ejpam-4713	53	5	the	the	DET
ejpam-4713	53	6	assignment	assignment	NOUN
ejpam-4713	53	7	of	of	ADP
ejpam-4713	53	8	the	the	DET
ejpam-4713	53	9	object	object	NOUN
ejpam-4713	53	10	x̃	x̃	PROPN
ejpam-4713	53	11	to	to	ADP
ejpam-4713	53	12	a	a	DET
ejpam-4713	53	13	class	class	NOUN
ejpam-4713	53	14	can	can	AUX
ejpam-4713	53	15	be	be	AUX
ejpam-4713	53	16	defined	define	VERB
ejpam-4713	53	17	as	as	ADP
ejpam-4713	53	18	x̃	x̃	PROPN
ejpam-4713	53	19	∈	∈	PROPN
ejpam-4713	53	20			PROPN
ejpam-4713	53	21	c1	c1	NOUN
ejpam-4713	53	22	,	,	PUNCT
ejpam-4713	53	23	hα(x	hα(x	PUNCT
ejpam-4713	53	24	)	)	PUNCT
ejpam-4713	53	25	≥	≥	NOUN
ejpam-4713	53	26	0	0	NUM
ejpam-4713	53	27	,	,	PUNCT
ejpam-4713	53	28	α	α	PROPN
ejpam-4713	53	29	∈	∈	PROPN
ejpam-4713	53	30	i1	i1	PROPN
ejpam-4713	53	31	;	;	PUNCT
ejpam-4713	53	32	hβ(x	hβ(x	NOUN
ejpam-4713	53	33	)	)	PUNCT
ejpam-4713	53	34	<	<	X
ejpam-4713	53	35	0	0	PROPN
ejpam-4713	53	36	,	,	PUNCT
ejpam-4713	53	37	β	β	PROPN
ejpam-4713	53	38	∈	∈	PROPN
ejpam-4713	53	39	j1	j1	PROPN
ejpam-4713	53	40	c2	c2	PROPN
ejpam-4713	53	41	,	,	PUNCT
ejpam-4713	53	42	hα(x	hα(x	PROPN
ejpam-4713	53	43	)	)	PUNCT
ejpam-4713	53	44	≥	≥	NOUN
ejpam-4713	53	45	0	0	NUM
ejpam-4713	53	46	,	,	PUNCT
ejpam-4713	53	47	α	α	PROPN
ejpam-4713	53	48	∈	∈	PROPN
ejpam-4713	53	49	i2	i2	PROPN
ejpam-4713	53	50	;	;	PUNCT
ejpam-4713	53	51	hβ(x	hβ(x	NOUN
ejpam-4713	53	52	)	)	PUNCT
ejpam-4713	53	53	<	<	X
ejpam-4713	53	54	0	0	PROPN
ejpam-4713	53	55	,	,	PUNCT
ejpam-4713	53	56	β	β	PROPN
ejpam-4713	53	57	∈	∈	PROPN
ejpam-4713	53	58	j2	j2	PROPN
ejpam-4713	53	59	...	...	PUNCT
ejpam-4713	53	60	...	...	PUNCT
ejpam-4713	53	61	...	...	PUNCT
ejpam-4713	53	62	...	...	PUNCT
ejpam-4713	53	63	...	...	PUNCT
ejpam-4713	54	1	cs	cs	X
ejpam-4713	54	2	,	,	PUNCT
ejpam-4713	54	3	hα(x	hα(x	ADJ
ejpam-4713	54	4	)	)	PUNCT
ejpam-4713	54	5	≥	≥	NOUN
ejpam-4713	54	6	0	0	NUM
ejpam-4713	54	7	,	,	PUNCT
ejpam-4713	54	8	α	α	PROPN
ejpam-4713	54	9	∈	∈	PROPN
ejpam-4713	54	10	is	be	AUX
ejpam-4713	54	11	;	;	PUNCT
ejpam-4713	54	12	hβ(x	hβ(x	NOUN
ejpam-4713	54	13	)	)	PUNCT
ejpam-4713	54	14	<	<	X
ejpam-4713	54	15	0	0	PROPN
ejpam-4713	54	16	,	,	PUNCT
ejpam-4713	54	17	β	β	X
ejpam-4713	54	18	∈	∈	PROPN
ejpam-4713	54	19	js	js	NOUN
ejpam-4713	54	20	(	(	PUNCT
ejpam-4713	54	21	2	2	NUM
ejpam-4713	54	22	)	)	PUNCT
ejpam-4713	54	23	where	where	SCONJ
ejpam-4713	54	24	ik	ik	X
ejpam-4713	54	25	,	,	PUNCT
ejpam-4713	54	26	jk	jk	PROPN
ejpam-4713	54	27	,	,	PUNCT
ejpam-4713	54	28	k	k	PROPN
ejpam-4713	54	29	=	=	SYM
ejpam-4713	54	30	1	1	NUM
ejpam-4713	54	31	,	,	PUNCT
ejpam-4713	54	32	.	.	PUNCT
ejpam-4713	54	33	.	.	PUNCT
ejpam-4713	54	34	.	.	PUNCT
ejpam-4713	55	1	,	,	PUNCT
ejpam-4713	55	2	s	s	X
ejpam-4713	55	3	,	,	PUNCT
ejpam-4713	55	4	are	be	AUX
ejpam-4713	55	5	the	the	DET
ejpam-4713	55	6	predefined	predefine	VERB
ejpam-4713	55	7	index	index	NOUN
ejpam-4713	55	8	sets	set	NOUN
ejpam-4713	55	9	.	.	PUNCT
ejpam-4713	56	1	hence	hence	ADV
ejpam-4713	56	2	,	,	PUNCT
ejpam-4713	56	3	the	the	DET
ejpam-4713	56	4	points	point	NOUN
ejpam-4713	56	5	x̃′	x̃′	PROPN
ejpam-4713	56	6	and	and	CCONJ
ejpam-4713	56	7	x̃′′	x̃′′	PROPN
ejpam-4713	56	8	lie	lie	NOUN
ejpam-4713	56	9	in	in	ADP
ejpam-4713	56	10	different	different	ADJ
ejpam-4713	56	11	regions	region	NOUN
ejpam-4713	56	12	if	if	SCONJ
ejpam-4713	56	13	and	and	CCONJ
ejpam-4713	56	14	only	only	ADV
ejpam-4713	56	15	if	if	SCONJ
ejpam-4713	56	16	there	there	PRON
ejpam-4713	56	17	exists	exist	VERB
ejpam-4713	56	18	an	an	DET
ejpam-4713	56	19	index	index	NOUN
ejpam-4713	56	20	i	i	PRON
ejpam-4713	56	21	for	for	ADP
ejpam-4713	56	22	which	which	PRON
ejpam-4713	56	23	hi	hi	INTJ
ejpam-4713	56	24	(	(	PUNCT
ejpam-4713	56	25	x̃	x̃	PROPN
ejpam-4713	56	26	′	′	NUM
ejpam-4713	56	27	)	)	PUNCT
ejpam-4713	56	28	and	and	CCONJ
ejpam-4713	56	29	hi	hi	INTJ
ejpam-4713	56	30	(	(	PUNCT
ejpam-4713	56	31	x̃	x̃	PROPN
ejpam-4713	56	32	′′	′′	PROPN
ejpam-4713	56	33	)	)	PUNCT
ejpam-4713	56	34	have	have	VERB
ejpam-4713	56	35	different	different	ADJ
ejpam-4713	56	36	signs	sign	NOUN
ejpam-4713	56	37	.	.	PUNCT
ejpam-4713	57	1	for	for	ADP
ejpam-4713	57	2	this	this	DET
ejpam-4713	57	3	purpose	purpose	NOUN
ejpam-4713	57	4	the	the	DET
ejpam-4713	57	5	sign	sign	NOUN
ejpam-4713	57	6	vector	vector	NOUN
ejpam-4713	57	7	-	-	PUNCT
ejpam-4713	57	8	function	function	NOUN
ejpam-4713	57	9	,	,	PUNCT
ejpam-4713	57	10	the	the	DET
ejpam-4713	57	11	partition	partition	NOUN
ejpam-4713	57	12	signature	signature	NOUN
ejpam-4713	57	13	,	,	PUNCT
ejpam-4713	57	14	f	f	PROPN
ejpam-4713	57	15	:	:	PUNCT
ejpam-4713	57	16	rd	rd	PROPN
ejpam-4713	57	17	→	→	PUNCT
ejpam-4713	57	18	{	{	PUNCT
ejpam-4713	57	19	−,+}n	−,+}n	ADV
ejpam-4713	57	20	defined	define	VERB
ejpam-4713	57	21	as	as	ADP
ejpam-4713	57	22	fi(x	fi(x	NUM
ejpam-4713	57	23	)	)	PUNCT
ejpam-4713	57	24	∈	∈	PROPN
ejpam-4713	57	25	{	{	PUNCT
ejpam-4713	57	26	−	−	NOUN
ejpam-4713	57	27	,	,	PUNCT
ejpam-4713	57	28	if	if	SCONJ
ejpam-4713	57	29	aix	aix	NOUN
ejpam-4713	57	30	≤	≤	PUNCT
ejpam-4713	57	31	bi	bi	NOUN
ejpam-4713	58	1	+	+	PROPN
ejpam-4713	58	2	,	,	PUNCT
ejpam-4713	58	3	if	if	SCONJ
ejpam-4713	58	4	aix	aix	PROPN
ejpam-4713	58	5	>	>	X
ejpam-4713	58	6	bi	bi	NOUN
ejpam-4713	58	7	,	,	PUNCT
ejpam-4713	58	8	i	i	PRON
ejpam-4713	58	9	=	=	NOUN
ejpam-4713	58	10	1	1	NUM
ejpam-4713	58	11	,	,	PUNCT
ejpam-4713	58	12	.	.	PUNCT
ejpam-4713	58	13	.	.	PUNCT
ejpam-4713	58	14	.	.	PUNCT
ejpam-4713	59	1	,	,	PUNCT
ejpam-4713	59	2	n	n	PRON
ejpam-4713	59	3	can	can	AUX
ejpam-4713	59	4	be	be	AUX
ejpam-4713	59	5	employed	employ	VERB
ejpam-4713	59	6	for	for	ADP
ejpam-4713	59	7	the	the	DET
ejpam-4713	59	8	analysis	analysis	NOUN
ejpam-4713	59	9	of	of	ADP
ejpam-4713	59	10	the	the	DET
ejpam-4713	59	11	classification	classification	NOUN
ejpam-4713	59	12	schemes	scheme	NOUN
ejpam-4713	59	13	with	with	ADP
ejpam-4713	59	14	different	different	ADJ
ejpam-4713	59	15	rules	rule	NOUN
ejpam-4713	59	16	.	.	PUNCT
ejpam-4713	60	1	therefore	therefore	ADV
ejpam-4713	60	2	,	,	PUNCT
ejpam-4713	60	3	the	the	DET
ejpam-4713	60	4	problem	problem	NOUN
ejpam-4713	60	5	of	of	ADP
ejpam-4713	60	6	classification	classification	NOUN
ejpam-4713	60	7	has	have	VERB
ejpam-4713	60	8	two	two	NUM
ejpam-4713	60	9	different	different	ADJ
ejpam-4713	60	10	formulations	formulation	NOUN
ejpam-4713	60	11	.	.	PUNCT
ejpam-4713	61	1	direct	direct	ADJ
ejpam-4713	61	2	problem	problem	NOUN
ejpam-4713	61	3	:	:	PUNCT
ejpam-4713	61	4	find	find	VERB
ejpam-4713	61	5	the	the	DET
ejpam-4713	61	6	minimum	minimum	ADJ
ejpam-4713	61	7	number	number	NOUN
ejpam-4713	61	8	of	of	ADP
ejpam-4713	61	9	hyperplanes	hyperplane	NOUN
ejpam-4713	61	10	(	(	PUNCT
ejpam-4713	61	11	1	1	NUM
ejpam-4713	61	12	)	)	PUNCT
ejpam-4713	61	13	that	that	PRON
ejpam-4713	61	14	divide	divide	VERB
ejpam-4713	61	15	rd	rd	PROPN
ejpam-4713	61	16	into	into	ADP
ejpam-4713	61	17	s	s	PART
ejpam-4713	61	18	regions	region	NOUN
ejpam-4713	61	19	according	accord	VERB
ejpam-4713	61	20	to	to	ADP
ejpam-4713	61	21	the	the	DET
ejpam-4713	61	22	rule	rule	NOUN
ejpam-4713	61	23	(	(	PUNCT
ejpam-4713	61	24	2	2	NUM
ejpam-4713	61	25	)	)	PUNCT
ejpam-4713	61	26	;	;	PUNCT
ejpam-4713	61	27	inverse	inverse	ADJ
ejpam-4713	61	28	problem	problem	NOUN
ejpam-4713	61	29	:	:	PUNCT
ejpam-4713	61	30	determine	determine	VERB
ejpam-4713	61	31	the	the	DET
ejpam-4713	61	32	maximum	maximum	ADJ
ejpam-4713	61	33	number	number	NOUN
ejpam-4713	61	34	of	of	ADP
ejpam-4713	61	35	regions	region	NOUN
ejpam-4713	61	36	into	into	ADP
ejpam-4713	61	37	which	which	PRON
ejpam-4713	61	38	rd	rd	NOUN
ejpam-4713	61	39	can	can	AUX
ejpam-4713	61	40	be	be	AUX
ejpam-4713	61	41	divided	divide	VERB
ejpam-4713	61	42	by	by	ADP
ejpam-4713	61	43	means	mean	NOUN
ejpam-4713	61	44	of	of	ADP
ejpam-4713	61	45	n	n	DET
ejpam-4713	61	46	hyperplanes	hyperplane	NOUN
ejpam-4713	61	47	(	(	PUNCT
ejpam-4713	61	48	1	1	NUM
ejpam-4713	61	49	)	)	PUNCT
ejpam-4713	61	50	according	accord	VERB
ejpam-4713	61	51	to	to	ADP
ejpam-4713	61	52	the	the	DET
ejpam-4713	61	53	rule	rule	NOUN
ejpam-4713	61	54	(	(	PUNCT
ejpam-4713	61	55	2	2	NUM
ejpam-4713	61	56	)	)	PUNCT
ejpam-4713	61	57	.	.	PUNCT
ejpam-4713	62	1	a.	a.	PROPN
ejpam-4713	62	2	bagdasaryan	bagdasaryan	PROPN
ejpam-4713	62	3	/	/	SYM
ejpam-4713	62	4	eur	eur	PROPN
ejpam-4713	62	5	.	.	PUNCT
ejpam-4713	63	1	j.	j.	PROPN
ejpam-4713	63	2	pure	pure	PROPN
ejpam-4713	63	3	appl	appl	PROPN
ejpam-4713	63	4	.	.	PROPN
ejpam-4713	63	5	math	math	PROPN
ejpam-4713	63	6	,	,	PUNCT
ejpam-4713	63	7	16	16	NUM
ejpam-4713	63	8	(	(	PUNCT
ejpam-4713	63	9	2	2	NUM
ejpam-4713	63	10	)	)	PUNCT
ejpam-4713	63	11	(	(	PUNCT
ejpam-4713	63	12	2023	2023	NUM
ejpam-4713	63	13	)	)	PUNCT
ejpam-4713	63	14	,	,	PUNCT
ejpam-4713	63	15	893	893	NUM
ejpam-4713	63	16	-	-	SYM
ejpam-4713	63	17	898	898	NUM
ejpam-4713	63	18	895	895	NUM
ejpam-4713	63	19	2	2	NUM
ejpam-4713	63	20	.	.	PUNCT
ejpam-4713	63	21	main	main	ADJ
ejpam-4713	63	22	results	result	NOUN
ejpam-4713	63	23	the	the	DET
ejpam-4713	63	24	main	main	ADJ
ejpam-4713	63	25	results	result	NOUN
ejpam-4713	63	26	of	of	ADP
ejpam-4713	63	27	this	this	DET
ejpam-4713	63	28	paper	paper	NOUN
ejpam-4713	63	29	are	be	AUX
ejpam-4713	63	30	contained	contain	VERB
ejpam-4713	63	31	in	in	ADP
ejpam-4713	63	32	the	the	DET
ejpam-4713	63	33	propositions	proposition	NOUN
ejpam-4713	63	34	presented	present	VERB
ejpam-4713	63	35	below	below	ADV
ejpam-4713	63	36	.	.	PUNCT
ejpam-4713	64	1	theorem	theorem	NOUN
ejpam-4713	64	2	1	1	NUM
ejpam-4713	64	3	.	.	PUNCT
ejpam-4713	64	4	suppose	suppose	VERB
ejpam-4713	64	5	that	that	SCONJ
ejpam-4713	64	6	h(x	h(x	PROPN
ejpam-4713	64	7	)	)	PUNCT
ejpam-4713	64	8	=	=	PRON
ejpam-4713	64	9	{	{	PUNCT
ejpam-4713	64	10	h1(x	h1(x	NOUN
ejpam-4713	64	11	)	)	PUNCT
ejpam-4713	64	12	,	,	PUNCT
ejpam-4713	64	13	h2(x	h2(x	PROPN
ejpam-4713	64	14	)	)	PUNCT
ejpam-4713	64	15	,	,	PUNCT
ejpam-4713	64	16	.	.	PUNCT
ejpam-4713	64	17	.	.	PUNCT
ejpam-4713	65	1	.	.	PUNCT
ejpam-4713	66	1	,	,	PUNCT
ejpam-4713	66	2	hn(x	hn(x	X
ejpam-4713	66	3	)	)	PUNCT
ejpam-4713	66	4	}	}	PUNCT
ejpam-4713	66	5	is	be	AUX
ejpam-4713	66	6	a	a	DET
ejpam-4713	66	7	collection	collection	NOUN
ejpam-4713	66	8	of	of	ADP
ejpam-4713	66	9	hyperplanes	hyperplane	NOUN
ejpam-4713	66	10	in	in	ADP
ejpam-4713	66	11	rd	rd	PROPN
ejpam-4713	66	12	.	.	PUNCT
ejpam-4713	67	1	let	let	VERB
ejpam-4713	67	2	h′	h′	PROPN
ejpam-4713	67	3	⊂	⊂	PROPN
ejpam-4713	67	4	h	h	PROPN
ejpam-4713	67	5	be	be	AUX
ejpam-4713	67	6	the	the	DET
ejpam-4713	67	7	partition	partition	NOUN
ejpam-4713	67	8	of	of	ADP
ejpam-4713	67	9	rd	rd	NOUN
ejpam-4713	67	10	by	by	ADP
ejpam-4713	67	11	n	n	CCONJ
ejpam-4713	67	12	−	−	PROPN
ejpam-4713	67	13	1	1	NUM
ejpam-4713	67	14	hyperplanes	hyperplane	NOUN
ejpam-4713	67	15	and	and	CCONJ
ejpam-4713	67	16	h′′	h′′	PROPN
ejpam-4713	67	17	⊂	⊂	PROPN
ejpam-4713	67	18	h′	h′	PROPN
ejpam-4713	67	19	be	be	AUX
ejpam-4713	67	20	the	the	DET
ejpam-4713	67	21	partition	partition	NOUN
ejpam-4713	67	22	of	of	ADP
ejpam-4713	67	23	rd−1	rd−1	PROPN
ejpam-4713	67	24	by	by	ADP
ejpam-4713	67	25	ℓ	ℓ	PROPN
ejpam-4713	67	26	hyperplanes	hyperplane	NOUN
ejpam-4713	67	27	,	,	PUNCT
ejpam-4713	67	28	ℓ	ℓ	PROPN
ejpam-4713	67	29	<	<	X
ejpam-4713	67	30	n.	n.	NOUN
ejpam-4713	67	31	then	then	ADV
ejpam-4713	67	32	we	we	PRON
ejpam-4713	67	33	have	have	VERB
ejpam-4713	67	34	gh(d	gh(d	PROPN
ejpam-4713	67	35	,	,	PUNCT
ejpam-4713	67	36	n	n	CCONJ
ejpam-4713	67	37	)	)	PUNCT
ejpam-4713	67	38	=	=	SYM
ejpam-4713	67	39	gh′(d	gh′(d	NOUN
ejpam-4713	67	40	,	,	PUNCT
ejpam-4713	67	41	n−	n−	NOUN
ejpam-4713	67	42	1	1	NUM
ejpam-4713	67	43	)	)	PUNCT
ejpam-4713	68	1	+	+	NOUN
ejpam-4713	68	2	gh′′(d−	gh′′(d−	PROPN
ejpam-4713	68	3	1	1	NUM
ejpam-4713	68	4	,	,	PUNCT
ejpam-4713	68	5	ℓ	ℓ	NOUN
ejpam-4713	68	6	)	)	PUNCT
ejpam-4713	68	7	≤	≤	NOUN
ejpam-4713	68	8	g(d	g(d	PROPN
ejpam-4713	68	9	,	,	PUNCT
ejpam-4713	68	10	n−	n−	NOUN
ejpam-4713	68	11	1	1	NUM
ejpam-4713	68	12	)	)	PUNCT
ejpam-4713	68	13	+	+	NOUN
ejpam-4713	68	14	g(d−	g(d−	PROPN
ejpam-4713	68	15	1	1	NUM
ejpam-4713	68	16	,	,	PUNCT
ejpam-4713	68	17	n−	n−	NOUN
ejpam-4713	68	18	1	1	NUM
ejpam-4713	68	19	)	)	PUNCT
ejpam-4713	68	20	.	.	PUNCT
ejpam-4713	69	1	(	(	PUNCT
ejpam-4713	69	2	3	3	X
ejpam-4713	69	3	)	)	PUNCT
ejpam-4713	69	4	proof	proof	NOUN
ejpam-4713	69	5	.	.	PUNCT
ejpam-4713	70	1	without	without	ADP
ejpam-4713	70	2	loss	loss	NOUN
ejpam-4713	70	3	of	of	ADP
ejpam-4713	70	4	generality	generality	NOUN
ejpam-4713	70	5	,	,	PUNCT
ejpam-4713	70	6	we	we	PRON
ejpam-4713	70	7	can	can	AUX
ejpam-4713	70	8	assume	assume	VERB
ejpam-4713	70	9	that	that	SCONJ
ejpam-4713	70	10	the	the	DET
ejpam-4713	70	11	coordinate	coordinate	NOUN
ejpam-4713	70	12	system	system	NOUN
ejpam-4713	70	13	in	in	ADP
ejpam-4713	70	14	rd	rd	PROPN
ejpam-4713	70	15	is	be	AUX
ejpam-4713	70	16	chosen	choose	VERB
ejpam-4713	70	17	in	in	ADP
ejpam-4713	70	18	such	such	DET
ejpam-4713	70	19	a	a	DET
ejpam-4713	70	20	way	way	NOUN
ejpam-4713	70	21	that	that	PRON
ejpam-4713	70	22	hn(x	hn(x	X
ejpam-4713	70	23	)	)	PUNCT
ejpam-4713	70	24	coincides	coincide	VERB
ejpam-4713	70	25	with	with	ADP
ejpam-4713	70	26	rd−1	rd−1	PROPN
ejpam-4713	70	27	⊂	⊂	PROPN
ejpam-4713	70	28	rd	rd	PROPN
ejpam-4713	70	29	and	and	CCONJ
ejpam-4713	70	30	is	be	AUX
ejpam-4713	70	31	defined	define	VERB
ejpam-4713	70	32	by	by	ADP
ejpam-4713	70	33	the	the	DET
ejpam-4713	70	34	equation	equation	NOUN
ejpam-4713	70	35	xd	xd	ADP
ejpam-4713	71	1	=	=	NOUN
ejpam-4713	71	2	0	0	PROPN
ejpam-4713	71	3	.	.	PUNCT
ejpam-4713	72	1	the	the	DET
ejpam-4713	72	2	hyperplanes	hyperplane	NOUN
ejpam-4713	72	3	h1(x	h1(x	NOUN
ejpam-4713	72	4	)	)	PUNCT
ejpam-4713	72	5	,	,	PUNCT
ejpam-4713	72	6	.	.	PUNCT
ejpam-4713	72	7	.	.	PUNCT
ejpam-4713	72	8	.	.	PUNCT
ejpam-4713	73	1	,	,	PUNCT
ejpam-4713	73	2	hn−1(x	hn−1(x	NOUN
ejpam-4713	73	3	)	)	PUNCT
ejpam-4713	73	4	divide	divide	NOUN
ejpam-4713	73	5	rd	rd	PROPN
ejpam-4713	73	6	into	into	ADP
ejpam-4713	73	7	gh′(d	gh′(d	PROPN
ejpam-4713	73	8	,	,	PUNCT
ejpam-4713	73	9	n	n	CCONJ
ejpam-4713	73	10	−	−	PROPN
ejpam-4713	73	11	1	1	NUM
ejpam-4713	73	12	)	)	PUNCT
ejpam-4713	73	13	≤	≤	NOUN
ejpam-4713	73	14	gh(d	gh(d	PROPN
ejpam-4713	73	15	,	,	PUNCT
ejpam-4713	73	16	n	n	CCONJ
ejpam-4713	73	17	−	−	PROPN
ejpam-4713	73	18	1	1	NUM
ejpam-4713	73	19	)	)	PUNCT
ejpam-4713	73	20	regions	region	NOUN
ejpam-4713	73	21	.	.	PUNCT
ejpam-4713	74	1	the	the	DET
ejpam-4713	74	2	hyperplane	hyperplane	PROPN
ejpam-4713	74	3	hn(x	hn(x	X
ejpam-4713	74	4	)	)	PUNCT
ejpam-4713	74	5	divides	divide	VERB
ejpam-4713	74	6	some	some	PRON
ejpam-4713	74	7	of	of	ADP
ejpam-4713	74	8	these	these	DET
ejpam-4713	74	9	regions	region	NOUN
ejpam-4713	74	10	into	into	ADP
ejpam-4713	74	11	two	two	NUM
ejpam-4713	74	12	other	other	ADJ
ejpam-4713	74	13	regions	region	NOUN
ejpam-4713	74	14	.	.	PUNCT
ejpam-4713	75	1	the	the	DET
ejpam-4713	75	2	boundaries	boundary	NOUN
ejpam-4713	75	3	of	of	ADP
ejpam-4713	75	4	these	these	DET
ejpam-4713	75	5	latter	latter	ADJ
ejpam-4713	75	6	regions	region	NOUN
ejpam-4713	75	7	are	be	AUX
ejpam-4713	75	8	(	(	PUNCT
ejpam-4713	75	9	d	d	NOUN
ejpam-4713	75	10	−	−	PROPN
ejpam-4713	75	11	1)-dimensional	1)-dimensional	NUM
ejpam-4713	75	12	regions	region	NOUN
ejpam-4713	75	13	that	that	PRON
ejpam-4713	75	14	lie	lie	VERB
ejpam-4713	75	15	in	in	ADP
ejpam-4713	75	16	hn(x	hn(x	PROPN
ejpam-4713	75	17	)	)	PUNCT
ejpam-4713	75	18	;	;	PUNCT
ejpam-4713	75	19	or	or	CCONJ
ejpam-4713	75	20	,	,	PUNCT
ejpam-4713	75	21	if	if	SCONJ
ejpam-4713	75	22	hn(x	hn(x	X
ejpam-4713	75	23	)	)	PUNCT
ejpam-4713	75	24	is	be	AUX
ejpam-4713	75	25	parallel	parallel	ADJ
ejpam-4713	75	26	to	to	ADP
ejpam-4713	75	27	the	the	DET
ejpam-4713	75	28	all	all	DET
ejpam-4713	75	29	hi(x	hi(x	NUM
ejpam-4713	75	30	)	)	PUNCT
ejpam-4713	75	31	,	,	PUNCT
ejpam-4713	76	1	i	i	PRON
ejpam-4713	76	2	̸=	̸=	PROPN
ejpam-4713	76	3	n	n	CCONJ
ejpam-4713	76	4	,	,	PUNCT
ejpam-4713	76	5	then	then	ADV
ejpam-4713	76	6	the	the	DET
ejpam-4713	76	7	whole	whole	ADJ
ejpam-4713	76	8	hn(x	hn(x	X
ejpam-4713	76	9	)	)	PUNCT
ejpam-4713	76	10	is	be	AUX
ejpam-4713	76	11	the	the	DET
ejpam-4713	76	12	boundary	boundary	ADJ
ejpam-4713	76	13	hyperplane	hyperplane	NOUN
ejpam-4713	76	14	.	.	PUNCT
ejpam-4713	77	1	hence	hence	ADV
ejpam-4713	77	2	,	,	PUNCT
ejpam-4713	77	3	we	we	PRON
ejpam-4713	77	4	have	have	VERB
ejpam-4713	77	5	that	that	SCONJ
ejpam-4713	77	6	the	the	DET
ejpam-4713	77	7	number	number	NOUN
ejpam-4713	77	8	of	of	ADP
ejpam-4713	77	9	regions	region	NOUN
ejpam-4713	77	10	contributed	contribute	VERB
ejpam-4713	77	11	by	by	ADP
ejpam-4713	77	12	the	the	DET
ejpam-4713	77	13	hyperplane	hyperplane	PROPN
ejpam-4713	77	14	hn(x	hn(x	X
ejpam-4713	77	15	)	)	PUNCT
ejpam-4713	77	16	coincides	coincide	VERB
ejpam-4713	77	17	with	with	ADP
ejpam-4713	77	18	the	the	DET
ejpam-4713	77	19	number	number	NOUN
ejpam-4713	77	20	of	of	ADP
ejpam-4713	77	21	regions	region	NOUN
ejpam-4713	77	22	formed	form	VERB
ejpam-4713	77	23	in	in	ADP
ejpam-4713	77	24	hn(x	hn(x	ADP
ejpam-4713	77	25	)	)	PUNCT
ejpam-4713	77	26	by	by	ADP
ejpam-4713	77	27	some	some	DET
ejpam-4713	77	28	number	number	NOUN
ejpam-4713	77	29	ℓ	ℓ	PROPN
ejpam-4713	77	30	of	of	ADP
ejpam-4713	77	31	k	k	NOUN
ejpam-4713	77	32	-	-	PUNCT
ejpam-4713	77	33	edges	edge	VERB
ejpam-4713	77	34	ek	ek	NOUN
ejpam-4713	78	1	[	[	X
ejpam-4713	78	2	1,n−1;n],j	1,n−1;n],j	X
ejpam-4713	78	3	,	,	PUNCT
ejpam-4713	78	4	j	j	PROPN
ejpam-4713	78	5	=	=	SYM
ejpam-4713	78	6	1	1	NUM
ejpam-4713	78	7	,	,	PUNCT
ejpam-4713	78	8	.	.	PUNCT
ejpam-4713	78	9	.	.	PUNCT
ejpam-4713	78	10	.	.	PUNCT
ejpam-4713	79	1	,	,	PUNCT
ejpam-4713	79	2	ℓ	ℓ	X
ejpam-4713	79	3	,	,	PUNCT
ejpam-4713	79	4	k	k	PROPN
ejpam-4713	79	5	≤	≤	PROPN
ejpam-4713	80	1	d	d	ADP
ejpam-4713	80	2	−	−	PROPN
ejpam-4713	80	3	1	1	NUM
ejpam-4713	80	4	,	,	PUNCT
ejpam-4713	80	5	of	of	ADP
ejpam-4713	80	6	intersection	intersection	NOUN
ejpam-4713	80	7	of	of	ADP
ejpam-4713	80	8	hn(x	hn(x	PRON
ejpam-4713	80	9	)	)	PUNCT
ejpam-4713	80	10	with	with	ADP
ejpam-4713	80	11	each	each	DET
ejpam-4713	80	12	hi(x	hi(x	NOUN
ejpam-4713	80	13	)	)	PUNCT
ejpam-4713	80	14	,	,	PUNCT
ejpam-4713	80	15	i	i	PRON
ejpam-4713	80	16	=	=	NOUN
ejpam-4713	80	17	1	1	NUM
ejpam-4713	80	18	,	,	PUNCT
ejpam-4713	80	19	.	.	PUNCT
ejpam-4713	80	20	.	.	PUNCT
ejpam-4713	81	1	.	.	PUNCT
ejpam-4713	82	1	,	,	PUNCT
ejpam-4713	82	2	n	n	CCONJ
ejpam-4713	82	3	−	−	PROPN
ejpam-4713	82	4	1	1	NUM
ejpam-4713	82	5	,	,	PUNCT
ejpam-4713	82	6	or	or	CCONJ
ejpam-4713	82	7	,	,	PUNCT
ejpam-4713	82	8	equivalently	equivalently	ADV
ejpam-4713	82	9	,	,	PUNCT
ejpam-4713	82	10	generated	generate	VERB
ejpam-4713	82	11	by	by	ADP
ejpam-4713	82	12	the	the	DET
ejpam-4713	82	13	“	"	PUNCT
ejpam-4713	82	14	lines	line	NOUN
ejpam-4713	82	15	”	"	PUNCT
ejpam-4713	82	16	of	of	ADP
ejpam-4713	82	17	intersection	intersection	NOUN
ejpam-4713	82	18	of	of	ADP
ejpam-4713	82	19	hn(x	hn(x	PRON
ejpam-4713	82	20	)	)	PUNCT
ejpam-4713	82	21	with	with	ADP
ejpam-4713	82	22	each	each	DET
ejpam-4713	82	23	hi(x	hi(x	NOUN
ejpam-4713	82	24	)	)	PUNCT
ejpam-4713	82	25	;	;	PUNCT
ejpam-4713	82	26	moreover	moreover	ADV
ejpam-4713	82	27	,	,	PUNCT
ejpam-4713	82	28	hn(x	hn(x	X
ejpam-4713	82	29	)	)	PUNCT
ejpam-4713	82	30	contributes	contribute	VERB
ejpam-4713	82	31	only	only	ADV
ejpam-4713	82	32	one	one	NUM
ejpam-4713	82	33	region	region	NOUN
ejpam-4713	82	34	if	if	SCONJ
ejpam-4713	82	35	it	it	PRON
ejpam-4713	82	36	is	be	AUX
ejpam-4713	82	37	parallel	parallel	ADJ
ejpam-4713	82	38	to	to	ADP
ejpam-4713	82	39	the	the	DET
ejpam-4713	82	40	all	all	DET
ejpam-4713	82	41	hi(x	hi(x	NUM
ejpam-4713	82	42	)	)	PUNCT
ejpam-4713	82	43	,	,	PUNCT
ejpam-4713	82	44	i	i	PRON
ejpam-4713	82	45	=	=	NOUN
ejpam-4713	82	46	1	1	NUM
ejpam-4713	82	47	,	,	PUNCT
ejpam-4713	82	48	.	.	PUNCT
ejpam-4713	82	49	.	.	PUNCT
ejpam-4713	83	1	.	.	PUNCT
ejpam-4713	84	1	,	,	PUNCT
ejpam-4713	85	1	n	n	CCONJ
ejpam-4713	85	2	−	−	PROPN
ejpam-4713	85	3	1	1	NUM
ejpam-4713	85	4	.	.	PUNCT
ejpam-4713	86	1	the	the	DET
ejpam-4713	86	2	number	number	NOUN
ejpam-4713	86	3	of	of	ADP
ejpam-4713	86	4	such	such	ADJ
ejpam-4713	86	5	lines	line	NOUN
ejpam-4713	86	6	is	be	AUX
ejpam-4713	86	7	equal	equal	ADJ
ejpam-4713	86	8	to	to	ADP
ejpam-4713	86	9	ℓ	ℓ	PROPN
ejpam-4713	86	10	≤	≤	NOUN
ejpam-4713	86	11	n	n	CCONJ
ejpam-4713	86	12	−	−	PROPN
ejpam-4713	86	13	1	1	NUM
ejpam-4713	86	14	.	.	PUNCT
ejpam-4713	87	1	therefore	therefore	ADV
ejpam-4713	87	2	,	,	PUNCT
ejpam-4713	87	3	the	the	DET
ejpam-4713	87	4	number	number	NOUN
ejpam-4713	87	5	of	of	ADP
ejpam-4713	87	6	new	new	ADJ
ejpam-4713	87	7	regions	region	NOUN
ejpam-4713	87	8	is	be	AUX
ejpam-4713	87	9	gh′′(d	gh′′(d	NOUN
ejpam-4713	87	10	−	−	PROPN
ejpam-4713	87	11	1	1	NUM
ejpam-4713	87	12	,	,	PUNCT
ejpam-4713	87	13	ℓ	ℓ	NOUN
ejpam-4713	87	14	)	)	PUNCT
ejpam-4713	87	15	≤	≤	PUNCT
ejpam-4713	88	1	g(d	g(d	PROPN
ejpam-4713	88	2	−	−	PROPN
ejpam-4713	88	3	1	1	NUM
ejpam-4713	88	4	,	,	PUNCT
ejpam-4713	88	5	n	n	CCONJ
ejpam-4713	88	6	−	−	PROPN
ejpam-4713	88	7	1	1	NUM
ejpam-4713	88	8	)	)	PUNCT
ejpam-4713	88	9	,	,	PUNCT
ejpam-4713	88	10	from	from	ADP
ejpam-4713	88	11	which	which	PRON
ejpam-4713	88	12	the	the	DET
ejpam-4713	88	13	inequality	inequality	NOUN
ejpam-4713	88	14	(	(	PUNCT
ejpam-4713	88	15	3	3	X
ejpam-4713	88	16	)	)	PUNCT
ejpam-4713	88	17	immediately	immediately	ADV
ejpam-4713	88	18	follows	follow	VERB
ejpam-4713	88	19	.	.	PUNCT
ejpam-4713	89	1	the	the	DET
ejpam-4713	89	2	proof	proof	NOUN
ejpam-4713	89	3	is	be	AUX
ejpam-4713	89	4	completed	complete	VERB
ejpam-4713	89	5	.	.	PUNCT
ejpam-4713	90	1	corollary	corollary	ADJ
ejpam-4713	90	2	1	1	PROPN
ejpam-4713	90	3	.	.	PUNCT
ejpam-4713	90	4	suppose	suppose	VERB
ejpam-4713	90	5	that	that	SCONJ
ejpam-4713	90	6	for	for	ADP
ejpam-4713	90	7	the	the	DET
ejpam-4713	90	8	partition	partition	NOUN
ejpam-4713	90	9	h(x	h(x	PROPN
ejpam-4713	90	10	)	)	PUNCT
ejpam-4713	90	11	it	it	PRON
ejpam-4713	90	12	holds	hold	VERB
ejpam-4713	90	13	that	that	SCONJ
ejpam-4713	90	14	gh(d	gh(d	PROPN
ejpam-4713	90	15	,	,	PUNCT
ejpam-4713	90	16	n	n	CCONJ
ejpam-4713	90	17	)	)	PUNCT
ejpam-4713	90	18	=	=	SYM
ejpam-4713	90	19	g(d	g(d	PROPN
ejpam-4713	90	20	,	,	PUNCT
ejpam-4713	90	21	n	n	CCONJ
ejpam-4713	90	22	)	)	PUNCT
ejpam-4713	90	23	.	.	PUNCT
ejpam-4713	91	1	then	then	ADV
ejpam-4713	91	2	we	we	PRON
ejpam-4713	91	3	have	have	VERB
ejpam-4713	91	4	g(d	g(d	PROPN
ejpam-4713	91	5	,	,	PUNCT
ejpam-4713	91	6	n	n	CCONJ
ejpam-4713	91	7	)	)	PUNCT
ejpam-4713	91	8	≤	≤	NOUN
ejpam-4713	91	9	g(d	g(d	PROPN
ejpam-4713	91	10	,	,	PUNCT
ejpam-4713	91	11	n−	n−	NOUN
ejpam-4713	91	12	1	1	NUM
ejpam-4713	91	13	)	)	PUNCT
ejpam-4713	92	1	+	+	NOUN
ejpam-4713	92	2	g(d−	g(d−	PROPN
ejpam-4713	92	3	1	1	NUM
ejpam-4713	92	4	,	,	PUNCT
ejpam-4713	92	5	n−	n−	NOUN
ejpam-4713	92	6	1	1	NUM
ejpam-4713	92	7	)	)	PUNCT
ejpam-4713	92	8	.	.	PUNCT
ejpam-4713	93	1	(	(	PUNCT
ejpam-4713	93	2	4	4	X
ejpam-4713	93	3	)	)	PUNCT
ejpam-4713	93	4	remark	remark	NOUN
ejpam-4713	93	5	1	1	NUM
ejpam-4713	93	6	.	.	PUNCT
ejpam-4713	94	1	since	since	SCONJ
ejpam-4713	94	2	gh′′(d	gh′′(d	PROPN
ejpam-4713	94	3	−	−	PROPN
ejpam-4713	94	4	1	1	NUM
ejpam-4713	94	5	,	,	PUNCT
ejpam-4713	94	6	0	0	NUM
ejpam-4713	94	7	)	)	PUNCT
ejpam-4713	94	8	=	=	SYM
ejpam-4713	94	9	1	1	NUM
ejpam-4713	94	10	,	,	PUNCT
ejpam-4713	94	11	the	the	DET
ejpam-4713	94	12	relation	relation	NOUN
ejpam-4713	94	13	(	(	PUNCT
ejpam-4713	94	14	3	3	X
ejpam-4713	94	15	)	)	PUNCT
ejpam-4713	94	16	holds	hold	VERB
ejpam-4713	94	17	also	also	ADV
ejpam-4713	94	18	in	in	ADP
ejpam-4713	94	19	“	"	PUNCT
ejpam-4713	94	20	parallel	parallel	ADJ
ejpam-4713	94	21	”	"	PUNCT
ejpam-4713	94	22	case	case	NOUN
ejpam-4713	94	23	,	,	PUNCT
ejpam-4713	94	24	that	that	ADV
ejpam-4713	94	25	is	is	ADV
ejpam-4713	94	26	,	,	PUNCT
ejpam-4713	94	27	in	in	ADP
ejpam-4713	94	28	the	the	DET
ejpam-4713	94	29	case	case	NOUN
ejpam-4713	94	30	when	when	SCONJ
ejpam-4713	94	31	hn(x	hn(x	X
ejpam-4713	94	32	)	)	PUNCT
ejpam-4713	94	33	is	be	AUX
ejpam-4713	94	34	parallel	parallel	ADJ
ejpam-4713	94	35	to	to	ADP
ejpam-4713	94	36	the	the	DET
ejpam-4713	94	37	all	all	DET
ejpam-4713	94	38	hi(x	hi(x	NUM
ejpam-4713	94	39	)	)	PUNCT
ejpam-4713	94	40	,	,	PUNCT
ejpam-4713	94	41	i	i	PRON
ejpam-4713	94	42	̸=	̸=	PROPN
ejpam-4713	94	43	n.	n.	NOUN
ejpam-4713	94	44	in	in	ADP
ejpam-4713	94	45	the	the	DET
ejpam-4713	94	46	next	next	ADJ
ejpam-4713	94	47	theorem	theorem	NOUN
ejpam-4713	94	48	we	we	PRON
ejpam-4713	94	49	obtain	obtain	VERB
ejpam-4713	94	50	the	the	DET
ejpam-4713	94	51	recurrence	recurrence	NOUN
ejpam-4713	94	52	formula	formula	NOUN
ejpam-4713	94	53	for	for	ADP
ejpam-4713	94	54	the	the	DET
ejpam-4713	94	55	number	number	NOUN
ejpam-4713	94	56	of	of	ADP
ejpam-4713	94	57	regions	region	NOUN
ejpam-4713	94	58	.	.	PUNCT
ejpam-4713	95	1	theorem	theorem	NOUN
ejpam-4713	95	2	2	2	NUM
ejpam-4713	95	3	.	.	PUNCT
ejpam-4713	96	1	let	let	VERB
ejpam-4713	96	2	a	a	PRON
ejpam-4713	96	3	=	=	PUNCT
ejpam-4713	97	1	[	[	X
ejpam-4713	97	2	aij	aij	X
ejpam-4713	97	3	]	]	PUNCT
ejpam-4713	97	4	,	,	PUNCT
ejpam-4713	97	5	i	i	PRON
ejpam-4713	97	6	=	=	NOUN
ejpam-4713	97	7	1	1	NUM
ejpam-4713	97	8	,	,	PUNCT
ejpam-4713	97	9	.	.	PUNCT
ejpam-4713	97	10	.	.	PUNCT
ejpam-4713	98	1	.	.	PUNCT
ejpam-4713	99	1	,	,	PUNCT
ejpam-4713	99	2	n	n	CCONJ
ejpam-4713	99	3	,	,	PUNCT
ejpam-4713	99	4	j	j	PROPN
ejpam-4713	99	5	=	=	SYM
ejpam-4713	99	6	1	1	NUM
ejpam-4713	99	7	,	,	PUNCT
ejpam-4713	99	8	.	.	PUNCT
ejpam-4713	99	9	.	.	PUNCT
ejpam-4713	99	10	.	.	PUNCT
ejpam-4713	100	1	,	,	PUNCT
ejpam-4713	100	2	d	d	X
ejpam-4713	100	3	be	be	AUX
ejpam-4713	100	4	the	the	DET
ejpam-4713	100	5	matrix	matrix	NOUN
ejpam-4713	100	6	of	of	ADP
ejpam-4713	100	7	coefficients	coefficient	NOUN
ejpam-4713	100	8	of	of	ADP
ejpam-4713	100	9	the	the	DET
ejpam-4713	100	10	equation	equation	NOUN
ejpam-4713	100	11	(	(	PUNCT
ejpam-4713	100	12	1	1	NUM
ejpam-4713	100	13	)	)	PUNCT
ejpam-4713	100	14	.	.	PUNCT
ejpam-4713	101	1	further	far	ADV
ejpam-4713	101	2	,	,	PUNCT
ejpam-4713	101	3	suppose	suppose	VERB
ejpam-4713	101	4	that	that	SCONJ
ejpam-4713	101	5	the	the	DET
ejpam-4713	101	6	partition	partition	NOUN
ejpam-4713	101	7	h(x	h(x	PROPN
ejpam-4713	101	8	)	)	PUNCT
ejpam-4713	101	9	=	=	PRON
ejpam-4713	101	10	{	{	PUNCT
ejpam-4713	101	11	hi(x	hi(x	PROPN
ejpam-4713	101	12	)	)	PUNCT
ejpam-4713	101	13	,	,	PUNCT
ejpam-4713	101	14	i	i	PRON
ejpam-4713	101	15	=	=	NOUN
ejpam-4713	101	16	1	1	NUM
ejpam-4713	101	17	,	,	PUNCT
ejpam-4713	101	18	.	.	PUNCT
ejpam-4713	101	19	.	.	PUNCT
ejpam-4713	101	20	.	.	PUNCT
ejpam-4713	102	1	,	,	PUNCT
ejpam-4713	102	2	n	n	CCONJ
ejpam-4713	102	3	}	}	PUNCT
ejpam-4713	102	4	of	of	ADP
ejpam-4713	102	5	rd	rd	NOUN
ejpam-4713	102	6	is	be	AUX
ejpam-4713	102	7	such	such	ADJ
ejpam-4713	102	8	that	that	SCONJ
ejpam-4713	102	9	every	every	DET
ejpam-4713	102	10	set	set	NOUN
ejpam-4713	102	11	of	of	ADP
ejpam-4713	102	12	k	k	PROPN
ejpam-4713	102	13	rows	row	NOUN
ejpam-4713	102	14	,	,	PUNCT
ejpam-4713	102	15	k	k	PROPN
ejpam-4713	102	16	≤	≤	PROPN
ejpam-4713	102	17	d	d	X
ejpam-4713	102	18	,	,	PUNCT
ejpam-4713	102	19	of	of	ADP
ejpam-4713	102	20	a	a	PRON
ejpam-4713	102	21	has	have	VERB
ejpam-4713	102	22	the	the	DET
ejpam-4713	102	23	rank	rank	PROPN
ejpam-4713	102	24	k	k	PROPN
ejpam-4713	102	25	,	,	PUNCT
ejpam-4713	102	26	i.e.	i.e.	X
ejpam-4713	102	27	rankk≤d	rankk≤d	NOUN
ejpam-4713	102	28	[	[	X
ejpam-4713	102	29	aij	aij	X
ejpam-4713	102	30	]	]	X
ejpam-4713	103	1	=	=	PUNCT
ejpam-4713	103	2	k.	k.	PROPN
ejpam-4713	103	3	then	then	ADV
ejpam-4713	103	4	we	we	PRON
ejpam-4713	103	5	have	have	VERB
ejpam-4713	103	6	gh(d	gh(d	PROPN
ejpam-4713	103	7	,	,	PUNCT
ejpam-4713	103	8	n	n	CCONJ
ejpam-4713	103	9	)	)	PUNCT
ejpam-4713	103	10	=	=	SYM
ejpam-4713	104	1	g(d	g(d	PROPN
ejpam-4713	104	2	,	,	PUNCT
ejpam-4713	104	3	n−	n−	NOUN
ejpam-4713	104	4	1	1	NUM
ejpam-4713	104	5	)	)	PUNCT
ejpam-4713	104	6	+	+	NOUN
ejpam-4713	104	7	g(d−	g(d−	PROPN
ejpam-4713	104	8	1	1	NUM
ejpam-4713	104	9	,	,	PUNCT
ejpam-4713	104	10	n−	n−	NOUN
ejpam-4713	104	11	1	1	NUM
ejpam-4713	104	12	)	)	PUNCT
ejpam-4713	104	13	,	,	PUNCT
ejpam-4713	104	14	(	(	PUNCT
ejpam-4713	104	15	5	5	X
ejpam-4713	104	16	)	)	PUNCT
ejpam-4713	104	17	that	that	PRON
ejpam-4713	104	18	is	be	AUX
ejpam-4713	104	19	,	,	PUNCT
ejpam-4713	104	20	gh(d	gh(d	PROPN
ejpam-4713	104	21	,	,	PUNCT
ejpam-4713	104	22	n	n	CCONJ
ejpam-4713	104	23	)	)	PUNCT
ejpam-4713	104	24	takes	take	VERB
ejpam-4713	104	25	on	on	ADP
ejpam-4713	104	26	the	the	DET
ejpam-4713	104	27	maximal	maximal	ADJ
ejpam-4713	104	28	value	value	NOUN
ejpam-4713	104	29	.	.	PUNCT
ejpam-4713	105	1	proof	proof	NOUN
ejpam-4713	105	2	.	.	PUNCT
ejpam-4713	106	1	we	we	PRON
ejpam-4713	106	2	prove	prove	VERB
ejpam-4713	106	3	the	the	DET
ejpam-4713	106	4	theorem	theorem	NOUN
ejpam-4713	106	5	by	by	ADP
ejpam-4713	106	6	induction	induction	NOUN
ejpam-4713	106	7	over	over	ADP
ejpam-4713	106	8	n.	n.	PROPN
ejpam-4713	106	9	it	it	PRON
ejpam-4713	106	10	is	be	AUX
ejpam-4713	106	11	straightforward	straightforward	ADJ
ejpam-4713	106	12	to	to	PART
ejpam-4713	106	13	check	check	VERB
ejpam-4713	106	14	that	that	PRON
ejpam-4713	106	15	for	for	ADP
ejpam-4713	106	16	n	n	NOUN
ejpam-4713	106	17	=	=	SYM
ejpam-4713	106	18	1	1	NUM
ejpam-4713	106	19	the	the	DET
ejpam-4713	106	20	statement	statement	NOUN
ejpam-4713	106	21	of	of	ADP
ejpam-4713	106	22	theorem	theorem	NOUN
ejpam-4713	106	23	is	be	AUX
ejpam-4713	106	24	true	true	ADJ
ejpam-4713	106	25	,	,	PUNCT
ejpam-4713	106	26	since	since	SCONJ
ejpam-4713	106	27	g(d	g(d	PROPN
ejpam-4713	106	28	,	,	PUNCT
ejpam-4713	106	29	1	1	NUM
ejpam-4713	106	30	)	)	PUNCT
ejpam-4713	106	31	=	=	SYM
ejpam-4713	106	32	2	2	NUM
ejpam-4713	106	33	and	and	CCONJ
ejpam-4713	106	34	g(d	g(d	PROPN
ejpam-4713	106	35	,	,	PUNCT
ejpam-4713	106	36	0	0	NUM
ejpam-4713	106	37	)	)	PUNCT
ejpam-4713	106	38	=	=	SYM
ejpam-4713	106	39	g(d−1	g(d−1	NOUN
ejpam-4713	106	40	,	,	PUNCT
ejpam-4713	106	41	0	0	NUM
ejpam-4713	106	42	)	)	PUNCT
ejpam-4713	106	43	=	=	SYM
ejpam-4713	106	44	1	1	X
ejpam-4713	106	45	.	.	PUNCT
ejpam-4713	106	46	suppose	suppose	VERB
ejpam-4713	106	47	that	that	SCONJ
ejpam-4713	106	48	the	the	DET
ejpam-4713	106	49	theorem	theorem	NOUN
ejpam-4713	106	50	holds	hold	VERB
ejpam-4713	106	51	true	true	ADJ
ejpam-4713	106	52	for	for	ADP
ejpam-4713	106	53	the	the	DET
ejpam-4713	106	54	partition	partition	NOUN
ejpam-4713	106	55	by	by	ADP
ejpam-4713	106	56	(	(	PUNCT
ejpam-4713	106	57	n−	n−	NOUN
ejpam-4713	106	58	1	1	NUM
ejpam-4713	106	59	)	)	PUNCT
ejpam-4713	106	60	hyperplanes	hyperplane	NOUN
ejpam-4713	106	61	.	.	PUNCT
ejpam-4713	107	1	now	now	ADV
ejpam-4713	107	2	let	let	VERB
ejpam-4713	107	3	h1(x	h1(x	NOUN
ejpam-4713	107	4	)	)	PUNCT
ejpam-4713	107	5	,	,	PUNCT
ejpam-4713	107	6	h2(x	h2(x	PROPN
ejpam-4713	107	7	)	)	PUNCT
ejpam-4713	107	8	,	,	PUNCT
ejpam-4713	107	9	.	.	PUNCT
ejpam-4713	107	10	.	.	PUNCT
ejpam-4713	108	1	.	.	PUNCT
ejpam-4713	109	1	,	,	PUNCT
ejpam-4713	109	2	hn(x	hn(x	PROPN
ejpam-4713	109	3	)	)	PUNCT
ejpam-4713	109	4	,	,	PUNCT
ejpam-4713	109	5	where	where	SCONJ
ejpam-4713	109	6	hn(x	hn(x	ADP
ejpam-4713	109	7	)	)	PUNCT
ejpam-4713	109	8	=	=	SYM
ejpam-4713	109	9	rd−1	rd−1	PROPN
ejpam-4713	109	10	,	,	PUNCT
ejpam-4713	109	11	be	be	VERB
ejpam-4713	109	12	the	the	DET
ejpam-4713	109	13	hyperplanes	hyperplane	NOUN
ejpam-4713	109	14	satisfying	satisfy	VERB
ejpam-4713	109	15	the	the	DET
ejpam-4713	109	16	theorem	theorem	PROPN
ejpam-4713	109	17	’s	’s	PART
ejpam-4713	109	18	a.	a.	NOUN
ejpam-4713	109	19	bagdasaryan	bagdasaryan	PROPN
ejpam-4713	109	20	/	/	SYM
ejpam-4713	109	21	eur	eur	PROPN
ejpam-4713	109	22	.	.	PUNCT
ejpam-4713	110	1	j.	j.	PROPN
ejpam-4713	110	2	pure	pure	PROPN
ejpam-4713	110	3	appl	appl	PROPN
ejpam-4713	110	4	.	.	PROPN
ejpam-4713	110	5	math	math	PROPN
ejpam-4713	110	6	,	,	PUNCT
ejpam-4713	110	7	16	16	NUM
ejpam-4713	110	8	(	(	PUNCT
ejpam-4713	110	9	2	2	NUM
ejpam-4713	110	10	)	)	PUNCT
ejpam-4713	110	11	(	(	PUNCT
ejpam-4713	110	12	2023	2023	NUM
ejpam-4713	110	13	)	)	PUNCT
ejpam-4713	110	14	,	,	PUNCT
ejpam-4713	110	15	893	893	NUM
ejpam-4713	110	16	-	-	SYM
ejpam-4713	110	17	898	898	NUM
ejpam-4713	110	18	896	896	NUM
ejpam-4713	110	19	condition	condition	NOUN
ejpam-4713	110	20	.	.	PUNCT
ejpam-4713	111	1	then	then	ADV
ejpam-4713	111	2	it	it	PRON
ejpam-4713	111	3	is	be	AUX
ejpam-4713	111	4	obvious	obvious	ADJ
ejpam-4713	111	5	to	to	PART
ejpam-4713	111	6	see	see	VERB
ejpam-4713	111	7	that	that	SCONJ
ejpam-4713	111	8	the	the	DET
ejpam-4713	111	9	partition	partition	NOUN
ejpam-4713	111	10	h′(x	h′(x	NOUN
ejpam-4713	111	11	)	)	PUNCT
ejpam-4713	111	12	=	=	SYM
ejpam-4713	111	13	{	{	PUNCT
ejpam-4713	111	14	h1(x	h1(x	NOUN
ejpam-4713	111	15	)	)	PUNCT
ejpam-4713	111	16	,	,	PUNCT
ejpam-4713	111	17	h2(x	h2(x	PROPN
ejpam-4713	111	18	)	)	PUNCT
ejpam-4713	111	19	,	,	PUNCT
ejpam-4713	111	20	.	.	PUNCT
ejpam-4713	111	21	.	.	PUNCT
ejpam-4713	112	1	.	.	PUNCT
ejpam-4713	113	1	,	,	PUNCT
ejpam-4713	113	2	hn−1(x	hn−1(x	NOUN
ejpam-4713	113	3	)	)	PUNCT
ejpam-4713	113	4	}	}	PUNCT
ejpam-4713	113	5	satisfies	satisfy	VERB
ejpam-4713	113	6	the	the	DET
ejpam-4713	113	7	condition	condition	NOUN
ejpam-4713	113	8	of	of	ADP
ejpam-4713	113	9	the	the	DET
ejpam-4713	113	10	theorem	theorem	NOUN
ejpam-4713	113	11	as	as	ADV
ejpam-4713	113	12	well	well	ADV
ejpam-4713	113	13	,	,	PUNCT
ejpam-4713	113	14	so	so	SCONJ
ejpam-4713	113	15	that	that	SCONJ
ejpam-4713	113	16	by	by	ADP
ejpam-4713	113	17	induction	induction	NOUN
ejpam-4713	113	18	we	we	PRON
ejpam-4713	113	19	have	have	VERB
ejpam-4713	113	20	gh′(d	gh′(d	NOUN
ejpam-4713	113	21	,	,	PUNCT
ejpam-4713	113	22	n−1	n−1	PROPN
ejpam-4713	113	23	)	)	PUNCT
ejpam-4713	113	24	=	=	SYM
ejpam-4713	113	25	g(d	g(d	PROPN
ejpam-4713	113	26	,	,	PUNCT
ejpam-4713	113	27	n	n	CCONJ
ejpam-4713	113	28	−	−	PROPN
ejpam-4713	113	29	1	1	NUM
ejpam-4713	113	30	)	)	PUNCT
ejpam-4713	113	31	.	.	PUNCT
ejpam-4713	114	1	the	the	DET
ejpam-4713	114	2	equations	equation	NOUN
ejpam-4713	114	3	of	of	ADP
ejpam-4713	114	4	k	k	NOUN
ejpam-4713	114	5	-	-	NOUN
ejpam-4713	114	6	edges	edge	NOUN
ejpam-4713	114	7	,	,	PUNCT
ejpam-4713	114	8	k	k	PROPN
ejpam-4713	114	9	≤	≤	PROPN
ejpam-4713	115	1	d	d	ADP
ejpam-4713	115	2	−	−	PROPN
ejpam-4713	115	3	1	1	NUM
ejpam-4713	115	4	,	,	PUNCT
ejpam-4713	115	5	h′i(x	h′i(x	NOUN
ejpam-4713	115	6	)	)	PUNCT
ejpam-4713	115	7	as	as	ADP
ejpam-4713	115	8	the	the	DET
ejpam-4713	115	9	“	"	PUNCT
ejpam-4713	115	10	lines	line	NOUN
ejpam-4713	115	11	”	"	PUNCT
ejpam-4713	115	12	of	of	ADP
ejpam-4713	115	13	intersection	intersection	NOUN
ejpam-4713	115	14	of	of	ADP
ejpam-4713	115	15	hi(x	hi(x	NOUN
ejpam-4713	115	16	)	)	PUNCT
ejpam-4713	115	17	with	with	ADP
ejpam-4713	115	18	hn(x	hn(x	NOUN
ejpam-4713	115	19	)	)	PUNCT
ejpam-4713	115	20	,	,	PUNCT
ejpam-4713	115	21	i	i	PRON
ejpam-4713	115	22	̸=	̸=	PROPN
ejpam-4713	115	23	n	n	CCONJ
ejpam-4713	115	24	,	,	PUNCT
ejpam-4713	115	25	has	have	VERB
ejpam-4713	115	26	the	the	DET
ejpam-4713	115	27	following	follow	VERB
ejpam-4713	115	28	form	form	NOUN
ejpam-4713	115	29	h′i(x	h′i(x	NOUN
ejpam-4713	115	30	)	)	PUNCT
ejpam-4713	115	31	≡	≡	PROPN
ejpam-4713	115	32	d−1∑	d−1∑	PROPN
ejpam-4713	116	1	j=1	j=1	PROPN
ejpam-4713	116	2	aijxj	aijxj	ADP
ejpam-4713	116	3	−	−	PROPN
ejpam-4713	116	4	bi	bi	NOUN
ejpam-4713	116	5	=	=	NOUN
ejpam-4713	116	6	0	0	PROPN
ejpam-4713	116	7	,	,	PUNCT
ejpam-4713	116	8	i	i	PRON
ejpam-4713	116	9	=	=	NOUN
ejpam-4713	116	10	1	1	NUM
ejpam-4713	116	11	,	,	PUNCT
ejpam-4713	116	12	.	.	PUNCT
ejpam-4713	116	13	.	.	PUNCT
ejpam-4713	116	14	.	.	PUNCT
ejpam-4713	117	1	,	,	PUNCT
ejpam-4713	117	2	n−	n−	NOUN
ejpam-4713	117	3	1	1	NUM
ejpam-4713	117	4	,	,	PUNCT
ejpam-4713	117	5	(	(	PUNCT
ejpam-4713	117	6	6	6	NUM
ejpam-4713	117	7	)	)	PUNCT
ejpam-4713	117	8	regarding	regard	VERB
ejpam-4713	117	9	h′i(x	h′i(x	NOUN
ejpam-4713	117	10	)	)	PUNCT
ejpam-4713	117	11	as	as	ADP
ejpam-4713	117	12	the	the	DET
ejpam-4713	117	13	hyperplanes	hyperplane	NOUN
ejpam-4713	117	14	in	in	ADP
ejpam-4713	117	15	rd−1	rd−1	PROPN
ejpam-4713	117	16	.	.	PUNCT
ejpam-4713	118	1	now	now	ADV
ejpam-4713	118	2	it	it	PRON
ejpam-4713	118	3	is	be	AUX
ejpam-4713	118	4	sufficient	sufficient	ADJ
ejpam-4713	118	5	to	to	PART
ejpam-4713	118	6	show	show	VERB
ejpam-4713	118	7	that	that	SCONJ
ejpam-4713	118	8	the	the	DET
ejpam-4713	118	9	partition	partition	NOUN
ejpam-4713	118	10	h′′(x	h′′(x	NOUN
ejpam-4713	118	11	)	)	PUNCT
ejpam-4713	118	12	=	=	SYM
ejpam-4713	118	13	{	{	PUNCT
ejpam-4713	118	14	h′1(x	h′1(x	NOUN
ejpam-4713	118	15	)	)	PUNCT
ejpam-4713	118	16	,	,	PUNCT
ejpam-4713	118	17	h′2(x	h′2(x	PROPN
ejpam-4713	118	18	)	)	PUNCT
ejpam-4713	118	19	,	,	PUNCT
ejpam-4713	118	20	.	.	PUNCT
ejpam-4713	118	21	.	.	PUNCT
ejpam-4713	119	1	.	.	PUNCT
ejpam-4713	120	1	,	,	PUNCT
ejpam-4713	120	2	h′n−1(x	h′n−1(x	NUM
ejpam-4713	120	3	)	)	PUNCT
ejpam-4713	120	4	}	}	PUNCT
ejpam-4713	120	5	of	of	ADP
ejpam-4713	120	6	hn(x	hn(x	PRON
ejpam-4713	120	7	)	)	PUNCT
ejpam-4713	120	8	=	=	SYM
ejpam-4713	120	9	rd−1	rd−1	ADJ
ejpam-4713	120	10	satisfies	satisfy	VERB
ejpam-4713	120	11	the	the	DET
ejpam-4713	120	12	condition	condition	NOUN
ejpam-4713	120	13	of	of	ADP
ejpam-4713	120	14	the	the	DET
ejpam-4713	120	15	theorem	theorem	NOUN
ejpam-4713	120	16	.	.	PUNCT
ejpam-4713	121	1	indeed	indeed	ADV
ejpam-4713	121	2	,	,	PUNCT
ejpam-4713	121	3	assume	assume	VERB
ejpam-4713	121	4	that	that	SCONJ
ejpam-4713	121	5	the	the	DET
ejpam-4713	121	6	set	set	NOUN
ejpam-4713	121	7	of	of	ADP
ejpam-4713	121	8	k	k	PROPN
ejpam-4713	121	9	≤	≤	PROPN
ejpam-4713	122	1	d	d	ADP
ejpam-4713	122	2	−	−	PROPN
ejpam-4713	122	3	1	1	NUM
ejpam-4713	122	4	rows	row	NOUN
ejpam-4713	122	5	(	(	PUNCT
ejpam-4713	122	6	aνi,1	aνi,1	ADJ
ejpam-4713	122	7	,	,	PUNCT
ejpam-4713	122	8	aνi,2	aνi,2	ADJ
ejpam-4713	122	9	,	,	PUNCT
ejpam-4713	122	10	.	.	PUNCT
ejpam-4713	122	11	.	.	PUNCT
ejpam-4713	122	12	.	.	PUNCT
ejpam-4713	123	1	,	,	PUNCT
ejpam-4713	123	2	aνi	aνi	PROPN
ejpam-4713	123	3	,	,	PUNCT
ejpam-4713	123	4	d−1	d−1	PROPN
ejpam-4713	123	5	)	)	PUNCT
ejpam-4713	123	6	,	,	PUNCT
ejpam-4713	123	7	i	i	NOUN
ejpam-4713	123	8	=	=	NOUN
ejpam-4713	123	9	1	1	NUM
ejpam-4713	123	10	,	,	PUNCT
ejpam-4713	123	11	2	2	NUM
ejpam-4713	123	12	,	,	PUNCT
ejpam-4713	123	13	.	.	PUNCT
ejpam-4713	123	14	.	.	PUNCT
ejpam-4713	124	1	.	.	PUNCT
ejpam-4713	125	1	,	,	PUNCT
ejpam-4713	125	2	k	k	NOUN
ejpam-4713	125	3	,	,	PUNCT
ejpam-4713	125	4	of	of	ADP
ejpam-4713	125	5	coefficients	coefficient	NOUN
ejpam-4713	125	6	{	{	PUNCT
ejpam-4713	125	7	aij	aij	NOUN
ejpam-4713	125	8	}	}	PUNCT
ejpam-4713	125	9	in	in	ADP
ejpam-4713	125	10	(	(	PUNCT
ejpam-4713	125	11	6	6	NUM
ejpam-4713	125	12	)	)	PUNCT
ejpam-4713	125	13	has	have	VERB
ejpam-4713	125	14	the	the	DET
ejpam-4713	125	15	rank	rank	NOUN
ejpam-4713	125	16	r	r	NOUN
ejpam-4713	125	17	<	<	X
ejpam-4713	126	1	k.	k.	PROPN
ejpam-4713	127	1	then	then	ADV
ejpam-4713	127	2	we	we	PRON
ejpam-4713	127	3	get	get	VERB
ejpam-4713	127	4	that	that	SCONJ
ejpam-4713	127	5	the	the	DET
ejpam-4713	127	6	set	set	NOUN
ejpam-4713	127	7	of	of	ADP
ejpam-4713	127	8	rows	row	NOUN
ejpam-4713	127	9	(	(	PUNCT
ejpam-4713	127	10	aνi,1	aνi,1	ADJ
ejpam-4713	127	11	,	,	PUNCT
ejpam-4713	127	12	aνi,2	aνi,2	ADJ
ejpam-4713	127	13	,	,	PUNCT
ejpam-4713	127	14	.	.	PUNCT
ejpam-4713	127	15	.	.	PUNCT
ejpam-4713	128	1	.	.	PUNCT
ejpam-4713	129	1	,	,	PUNCT
ejpam-4713	129	2	aνi	aνi	PROPN
ejpam-4713	129	3	,	,	PUNCT
ejpam-4713	129	4	d−1	d−1	PROPN
ejpam-4713	129	5	,	,	PUNCT
ejpam-4713	129	6	aνi	aνi	NOUN
ejpam-4713	129	7	,	,	PUNCT
ejpam-4713	129	8	d	d	NOUN
ejpam-4713	129	9	)	)	PUNCT
ejpam-4713	129	10	,	,	PUNCT
ejpam-4713	129	11	i	i	PRON
ejpam-4713	129	12	=	=	NOUN
ejpam-4713	129	13	1	1	NUM
ejpam-4713	129	14	,	,	PUNCT
ejpam-4713	129	15	2	2	NUM
ejpam-4713	129	16	,	,	PUNCT
ejpam-4713	129	17	.	.	PUNCT
ejpam-4713	129	18	.	.	PUNCT
ejpam-4713	130	1	.	.	PUNCT
ejpam-4713	131	1	,	,	PUNCT
ejpam-4713	131	2	k	k	NOUN
ejpam-4713	131	3	,	,	PUNCT
ejpam-4713	131	4	together	together	ADV
ejpam-4713	131	5	with	with	ADP
ejpam-4713	131	6	the	the	DET
ejpam-4713	131	7	row	row	NOUN
ejpam-4713	131	8	(	(	PUNCT
ejpam-4713	131	9	0	0	NUM
ejpam-4713	131	10	,	,	PUNCT
ejpam-4713	131	11	.	.	PUNCT
ejpam-4713	131	12	.	.	PUNCT
ejpam-4713	131	13	.	.	PUNCT
ejpam-4713	132	1	,	,	PUNCT
ejpam-4713	132	2	0	0	NUM
ejpam-4713	132	3	,	,	PUNCT
ejpam-4713	132	4	1	1	NUM
ejpam-4713	132	5	)	)	PUNCT
ejpam-4713	132	6	of	of	ADP
ejpam-4713	132	7	coefficients	coefficient	NOUN
ejpam-4713	132	8	of	of	ADP
ejpam-4713	132	9	the	the	DET
ejpam-4713	132	10	equation	equation	NOUN
ejpam-4713	132	11	hn(x	hn(x	ADP
ejpam-4713	132	12	)	)	PUNCT
ejpam-4713	132	13	=	=	SYM
ejpam-4713	132	14	0	0	PROPN
ejpam-4713	132	15	has	have	VERB
ejpam-4713	132	16	the	the	DET
ejpam-4713	132	17	rank	rank	NOUN
ejpam-4713	132	18	r+	r+	PUNCT
ejpam-4713	132	19	1	1	NUM
ejpam-4713	132	20	≤	≤	NUM
ejpam-4713	132	21	k	k	NOUN
ejpam-4713	132	22	,	,	PUNCT
ejpam-4713	132	23	less	less	ADJ
ejpam-4713	132	24	than	than	ADP
ejpam-4713	132	25	the	the	DET
ejpam-4713	132	26	number	number	NOUN
ejpam-4713	132	27	of	of	ADP
ejpam-4713	132	28	rows	row	NOUN
ejpam-4713	132	29	k+	k+	X
ejpam-4713	132	30	1	1	NUM
ejpam-4713	132	31	,	,	PUNCT
ejpam-4713	132	32	which	which	PRON
ejpam-4713	132	33	contradicts	contradict	VERB
ejpam-4713	132	34	the	the	DET
ejpam-4713	132	35	theorem	theorem	NOUN
ejpam-4713	132	36	’s	’s	PART
ejpam-4713	132	37	condition	condition	NOUN
ejpam-4713	132	38	.	.	PUNCT
ejpam-4713	133	1	therefore	therefore	ADV
ejpam-4713	133	2	,	,	PUNCT
ejpam-4713	133	3	by	by	ADP
ejpam-4713	133	4	the	the	DET
ejpam-4713	133	5	induction	induction	NOUN
ejpam-4713	133	6	we	we	PRON
ejpam-4713	133	7	again	again	ADV
ejpam-4713	133	8	have	have	VERB
ejpam-4713	133	9	that	that	DET
ejpam-4713	133	10	gh′′(d−	gh′′(d−	PROPN
ejpam-4713	133	11	1	1	NUM
ejpam-4713	133	12	,	,	PUNCT
ejpam-4713	133	13	n−	n−	NOUN
ejpam-4713	133	14	1	1	NUM
ejpam-4713	133	15	)	)	PUNCT
ejpam-4713	133	16	=	=	SYM
ejpam-4713	133	17	g(d−	g(d−	PROPN
ejpam-4713	133	18	1	1	NUM
ejpam-4713	133	19	,	,	PUNCT
ejpam-4713	133	20	n−	n−	NOUN
ejpam-4713	133	21	1	1	NUM
ejpam-4713	133	22	)	)	PUNCT
ejpam-4713	133	23	,	,	PUNCT
ejpam-4713	133	24	from	from	ADP
ejpam-4713	133	25	which	which	PRON
ejpam-4713	133	26	the	the	DET
ejpam-4713	133	27	statement	statement	NOUN
ejpam-4713	133	28	of	of	ADP
ejpam-4713	133	29	the	the	DET
ejpam-4713	133	30	theorem	theorem	NOUN
ejpam-4713	133	31	follows	follow	VERB
ejpam-4713	133	32	.	.	PUNCT
ejpam-4713	134	1	the	the	DET
ejpam-4713	134	2	proof	proof	NOUN
ejpam-4713	134	3	is	be	AUX
ejpam-4713	134	4	completed	complete	VERB
ejpam-4713	134	5	.	.	PUNCT
ejpam-4713	135	1	remark	remark	NOUN
ejpam-4713	135	2	2	2	NUM
ejpam-4713	135	3	.	.	PUNCT
ejpam-4713	136	1	the	the	DET
ejpam-4713	136	2	existence	existence	NOUN
ejpam-4713	136	3	of	of	ADP
ejpam-4713	136	4	the	the	DET
ejpam-4713	136	5	partition	partition	NOUN
ejpam-4713	136	6	h	h	NOUN
ejpam-4713	136	7	that	that	PRON
ejpam-4713	136	8	satisfies	satisfy	VERB
ejpam-4713	136	9	the	the	DET
ejpam-4713	136	10	condition	condition	NOUN
ejpam-4713	136	11	of	of	ADP
ejpam-4713	136	12	theorem	theorem	ADJ
ejpam-4713	136	13	2	2	NUM
ejpam-4713	136	14	is	be	AUX
ejpam-4713	136	15	provided	provide	VERB
ejpam-4713	136	16	by	by	ADP
ejpam-4713	136	17	and	and	CCONJ
ejpam-4713	136	18	can	can	AUX
ejpam-4713	136	19	be	be	AUX
ejpam-4713	136	20	deduced	deduce	VERB
ejpam-4713	136	21	from	from	ADP
ejpam-4713	136	22	the	the	DET
ejpam-4713	136	23	fact	fact	NOUN
ejpam-4713	136	24	that	that	SCONJ
ejpam-4713	136	25	rd	rd	PROPN
ejpam-4713	136	26	contains	contain	VERB
ejpam-4713	136	27	infinitely	infinitely	ADV
ejpam-4713	136	28	many	many	ADJ
ejpam-4713	136	29	hyperplanes	hyperplane	NOUN
ejpam-4713	136	30	.	.	PUNCT
ejpam-4713	137	1	the	the	DET
ejpam-4713	137	2	following	following	ADJ
ejpam-4713	137	3	result	result	NOUN
ejpam-4713	137	4	is	be	AUX
ejpam-4713	137	5	well	well	ADV
ejpam-4713	137	6	-	-	PUNCT
ejpam-4713	137	7	known	know	VERB
ejpam-4713	137	8	[	[	X
ejpam-4713	137	9	1	1	NUM
ejpam-4713	137	10	,	,	PUNCT
ejpam-4713	137	11	4	4	NUM
ejpam-4713	137	12	,	,	PUNCT
ejpam-4713	137	13	12	12	NUM
ejpam-4713	137	14	]	]	PUNCT
ejpam-4713	137	15	.	.	PUNCT
ejpam-4713	138	1	here	here	ADV
ejpam-4713	138	2	,	,	PUNCT
ejpam-4713	138	3	we	we	PRON
ejpam-4713	138	4	give	give	VERB
ejpam-4713	138	5	an	an	DET
ejpam-4713	138	6	alternative	alternative	ADJ
ejpam-4713	138	7	proof	proof	NOUN
ejpam-4713	138	8	of	of	ADP
ejpam-4713	138	9	this	this	DET
ejpam-4713	138	10	result	result	NOUN
ejpam-4713	138	11	by	by	ADP
ejpam-4713	138	12	re	re	VERB
ejpam-4713	138	13	-	-	VERB
ejpam-4713	138	14	deriving	derive	VERB
ejpam-4713	138	15	the	the	DET
ejpam-4713	138	16	statement	statement	NOUN
ejpam-4713	138	17	using	use	VERB
ejpam-4713	138	18	the	the	DET
ejpam-4713	138	19	formula	formula	NOUN
ejpam-4713	138	20	(	(	PUNCT
ejpam-4713	138	21	5	5	NUM
ejpam-4713	138	22	)	)	PUNCT
ejpam-4713	138	23	of	of	ADP
ejpam-4713	138	24	theorem	theorem	ADJ
ejpam-4713	138	25	2	2	NUM
ejpam-4713	138	26	.	.	PUNCT
ejpam-4713	138	27	theorem	theorem	NOUN
ejpam-4713	138	28	3	3	X
ejpam-4713	138	29	.	.	PUNCT
ejpam-4713	139	1	it	it	PRON
ejpam-4713	139	2	holds	hold	VERB
ejpam-4713	139	3	that	that	SCONJ
ejpam-4713	139	4	g(d	g(d	PROPN
ejpam-4713	139	5	,	,	PUNCT
ejpam-4713	139	6	n	n	CCONJ
ejpam-4713	139	7	)	)	PUNCT
ejpam-4713	139	8	=	=	SYM
ejpam-4713	139	9	d∑	d∑	PROPN
ejpam-4713	139	10	k=0	k=0	PROPN
ejpam-4713	139	11	(	(	PUNCT
ejpam-4713	139	12	n	n	X
ejpam-4713	139	13	k	k	PROPN
ejpam-4713	139	14	)	)	PUNCT
ejpam-4713	139	15	,	,	PUNCT
ejpam-4713	139	16	(	(	PUNCT
ejpam-4713	139	17	7	7	X
ejpam-4713	139	18	)	)	PUNCT
ejpam-4713	139	19	where	where	SCONJ
ejpam-4713	139	20	(	(	PUNCT
ejpam-4713	139	21	n	n	X
ejpam-4713	139	22	k	k	NOUN
ejpam-4713	139	23	)	)	PUNCT
ejpam-4713	139	24	=	=	SYM
ejpam-4713	139	25	n	n	X
ejpam-4713	139	26	!	!	PUNCT
ejpam-4713	139	27	k!(n−k	k!(n−k	NOUN
ejpam-4713	139	28	)	)	PUNCT
ejpam-4713	139	29	!	!	PUNCT
ejpam-4713	140	1	,	,	PUNCT
ejpam-4713	140	2	k	k	PROPN
ejpam-4713	140	3	≤	≤	PROPN
ejpam-4713	140	4	n	n	CCONJ
ejpam-4713	140	5	and	and	CCONJ
ejpam-4713	140	6	(	(	PUNCT
ejpam-4713	140	7	n	n	X
ejpam-4713	140	8	k	k	NOUN
ejpam-4713	140	9	)	)	PUNCT
ejpam-4713	141	1	=	=	SYM
ejpam-4713	141	2	0	0	PUNCT
ejpam-4713	142	1	if	if	SCONJ
ejpam-4713	142	2	k	k	PROPN
ejpam-4713	142	3	>	>	X
ejpam-4713	142	4	n.	n.	PROPN
ejpam-4713	142	5	proof	proof	NOUN
ejpam-4713	142	6	.	.	PUNCT
ejpam-4713	143	1	consider	consider	VERB
ejpam-4713	143	2	the	the	DET
ejpam-4713	143	3	recurrence	recurrence	NOUN
ejpam-4713	143	4	relation	relation	NOUN
ejpam-4713	143	5	g(d	g(d	PROPN
ejpam-4713	143	6	,	,	PUNCT
ejpam-4713	143	7	n	n	CCONJ
ejpam-4713	143	8	)	)	PUNCT
ejpam-4713	143	9	=	=	SYM
ejpam-4713	144	1	g(d	g(d	PROPN
ejpam-4713	144	2	,	,	PUNCT
ejpam-4713	144	3	n−	n−	NOUN
ejpam-4713	144	4	1	1	NUM
ejpam-4713	144	5	)	)	PUNCT
ejpam-4713	144	6	+	+	NOUN
ejpam-4713	144	7	g(d−	g(d−	PROPN
ejpam-4713	144	8	1	1	NUM
ejpam-4713	144	9	,	,	PUNCT
ejpam-4713	144	10	n−	n−	NOUN
ejpam-4713	144	11	1	1	NUM
ejpam-4713	144	12	)	)	PUNCT
ejpam-4713	144	13	(	(	PUNCT
ejpam-4713	144	14	8)	8)	NUM
ejpam-4713	144	15	for	for	ADP
ejpam-4713	144	16	the	the	DET
ejpam-4713	144	17	maximum	maximum	ADJ
ejpam-4713	144	18	number	number	NOUN
ejpam-4713	144	19	of	of	ADP
ejpam-4713	144	20	regions	region	NOUN
ejpam-4713	144	21	that	that	PRON
ejpam-4713	144	22	can	can	AUX
ejpam-4713	144	23	be	be	AUX
ejpam-4713	144	24	obtained	obtain	VERB
ejpam-4713	144	25	by	by	ADP
ejpam-4713	144	26	partitioning	partitioning	NOUN
ejpam-4713	144	27	of	of	ADP
ejpam-4713	144	28	rd	rd	PROPN
ejpam-4713	144	29	by	by	ADP
ejpam-4713	144	30	n	n	CCONJ
ejpam-4713	144	31	hyperplanes	hyperplane	NOUN
ejpam-4713	144	32	(	(	PUNCT
ejpam-4713	144	33	see	see	VERB
ejpam-4713	144	34	eq	eq	ADP
ejpam-4713	144	35	.	.	PUNCT
ejpam-4713	144	36	(	(	PUNCT
ejpam-4713	144	37	5	5	NUM
ejpam-4713	144	38	)	)	PUNCT
ejpam-4713	144	39	)	)	PUNCT
ejpam-4713	144	40	,	,	PUNCT
ejpam-4713	144	41	together	together	ADV
ejpam-4713	144	42	with	with	ADP
ejpam-4713	144	43	the	the	DET
ejpam-4713	144	44	initial	initial	ADJ
ejpam-4713	144	45	conditions	condition	NOUN
ejpam-4713	144	46	g(d	g(d	PROPN
ejpam-4713	144	47	,	,	PUNCT
ejpam-4713	144	48	0	0	NUM
ejpam-4713	144	49	)	)	PUNCT
ejpam-4713	144	50	=	=	SYM
ejpam-4713	144	51	1	1	NUM
ejpam-4713	144	52	,	,	PUNCT
ejpam-4713	144	53	d	d	NOUN
ejpam-4713	144	54	=	=	SYM
ejpam-4713	144	55	1	1	NUM
ejpam-4713	144	56	,	,	PUNCT
ejpam-4713	144	57	2	2	NUM
ejpam-4713	144	58	,	,	PUNCT
ejpam-4713	144	59	.	.	PUNCT
ejpam-4713	144	60	.	.	PUNCT
ejpam-4713	144	61	.	.	PUNCT
ejpam-4713	144	62	.	.	PUNCT
ejpam-4713	145	1	(	(	PUNCT
ejpam-4713	145	2	9	9	X
ejpam-4713	145	3	)	)	PUNCT
ejpam-4713	145	4	the	the	DET
ejpam-4713	145	5	number	number	NOUN
ejpam-4713	145	6	g(d	g(d	PROPN
ejpam-4713	145	7	,	,	PUNCT
ejpam-4713	145	8	n	n	CCONJ
ejpam-4713	145	9	)	)	PUNCT
ejpam-4713	145	10	is	be	AUX
ejpam-4713	145	11	uniquely	uniquely	ADV
ejpam-4713	145	12	defined	define	VERB
ejpam-4713	145	13	by	by	ADP
ejpam-4713	145	14	(	(	PUNCT
ejpam-4713	145	15	8)	8)	NUM
ejpam-4713	145	16	,	,	PUNCT
ejpam-4713	145	17	(	(	PUNCT
ejpam-4713	145	18	9	9	NUM
ejpam-4713	145	19	)	)	PUNCT
ejpam-4713	145	20	.	.	PUNCT
ejpam-4713	146	1	indeed	indeed	ADV
ejpam-4713	146	2	,	,	PUNCT
ejpam-4713	146	3	the	the	DET
ejpam-4713	146	4	values	value	NOUN
ejpam-4713	146	5	of	of	ADP
ejpam-4713	146	6	g(d	g(d	PROPN
ejpam-4713	146	7	,	,	PUNCT
ejpam-4713	146	8	0	0	NUM
ejpam-4713	146	9	)	)	PUNCT
ejpam-4713	146	10	are	be	AUX
ejpam-4713	146	11	known	know	VERB
ejpam-4713	146	12	for	for	ADP
ejpam-4713	146	13	all	all	DET
ejpam-4713	146	14	d.	d.	PROPN
ejpam-4713	146	15	hence	hence	ADV
ejpam-4713	146	16	,	,	PUNCT
ejpam-4713	146	17	if	if	SCONJ
ejpam-4713	146	18	the	the	DET
ejpam-4713	146	19	value	value	NOUN
ejpam-4713	146	20	of	of	ADP
ejpam-4713	146	21	g(d	g(d	PROPN
ejpam-4713	146	22	,	,	PUNCT
ejpam-4713	146	23	n	n	CCONJ
ejpam-4713	146	24	)	)	PUNCT
ejpam-4713	146	25	is	be	AUX
ejpam-4713	146	26	known	know	VERB
ejpam-4713	146	27	for	for	ADP
ejpam-4713	146	28	all	all	DET
ejpam-4713	146	29	pairs	pair	NOUN
ejpam-4713	146	30	(	(	PUNCT
ejpam-4713	146	31	d	d	NOUN
ejpam-4713	146	32	,	,	PUNCT
ejpam-4713	146	33	n	n	CCONJ
ejpam-4713	146	34	)	)	PUNCT
ejpam-4713	146	35	for	for	ADP
ejpam-4713	146	36	which	which	PRON
ejpam-4713	146	37	n	n	NOUN
ejpam-4713	146	38	=	=	SYM
ejpam-4713	147	1	k	k	PROPN
ejpam-4713	147	2	−	−	PROPN
ejpam-4713	147	3	1	1	NUM
ejpam-4713	147	4	,	,	PUNCT
ejpam-4713	147	5	then	then	ADV
ejpam-4713	147	6	g(d	g(d	PROPN
ejpam-4713	147	7	,	,	PUNCT
ejpam-4713	147	8	k	k	PROPN
ejpam-4713	147	9	)	)	PUNCT
ejpam-4713	147	10	can	can	AUX
ejpam-4713	147	11	be	be	AUX
ejpam-4713	147	12	found	find	VERB
ejpam-4713	147	13	from	from	ADP
ejpam-4713	147	14	(	(	PUNCT
ejpam-4713	147	15	8)	8)	NUM
ejpam-4713	147	16	for	for	ADP
ejpam-4713	147	17	any	any	DET
ejpam-4713	147	18	d.	d.	NOUN
ejpam-4713	147	19	now	now	ADV
ejpam-4713	147	20	it	it	PRON
ejpam-4713	147	21	is	be	AUX
ejpam-4713	147	22	sufficient	sufficient	ADJ
ejpam-4713	147	23	to	to	PART
ejpam-4713	147	24	show	show	VERB
ejpam-4713	147	25	that	that	SCONJ
ejpam-4713	147	26	(	(	PUNCT
ejpam-4713	147	27	7	7	X
ejpam-4713	147	28	)	)	PUNCT
ejpam-4713	147	29	satisfies	satisfie	NOUN
ejpam-4713	147	30	(	(	PUNCT
ejpam-4713	147	31	8)	8)	NUM
ejpam-4713	147	32	and	and	CCONJ
ejpam-4713	147	33	(	(	PUNCT
ejpam-4713	147	34	9	9	NUM
ejpam-4713	147	35	)	)	PUNCT
ejpam-4713	147	36	.	.	PUNCT
ejpam-4713	148	1	substituting	substitute	VERB
ejpam-4713	148	2	(	(	PUNCT
ejpam-4713	148	3	7	7	NUM
ejpam-4713	148	4	)	)	PUNCT
ejpam-4713	148	5	into	into	ADP
ejpam-4713	148	6	(	(	PUNCT
ejpam-4713	148	7	8)	8)	NUM
ejpam-4713	148	8	,	,	PUNCT
ejpam-4713	148	9	we	we	PRON
ejpam-4713	148	10	get	get	VERB
ejpam-4713	148	11	d∑	d∑	PROPN
ejpam-4713	148	12	k=0	k=0	PROPN
ejpam-4713	148	13	(	(	PUNCT
ejpam-4713	148	14	n	n	X
ejpam-4713	148	15	k	k	NOUN
ejpam-4713	148	16	)	)	PUNCT
ejpam-4713	149	1	=	=	SYM
ejpam-4713	149	2	d∑	d∑	PROPN
ejpam-4713	149	3	k=0	k=0	PROPN
ejpam-4713	149	4	(	(	PUNCT
ejpam-4713	149	5	n−	n−	NOUN
ejpam-4713	149	6	1	1	NUM
ejpam-4713	149	7	k	k	NOUN
ejpam-4713	149	8	)	)	PUNCT
ejpam-4713	150	1	+	+	CCONJ
ejpam-4713	151	1	d∑	d∑	INTJ
ejpam-4713	151	2	k=1	k=1	X
ejpam-4713	152	1	(	(	PUNCT
ejpam-4713	152	2	n−	n−	NOUN
ejpam-4713	152	3	1	1	NUM
ejpam-4713	152	4	k	k	NOUN
ejpam-4713	152	5	−	−	NOUN
ejpam-4713	152	6	1	1	NUM
ejpam-4713	152	7	)	)	PUNCT
ejpam-4713	152	8	.	.	PUNCT
ejpam-4713	153	1	(	(	PUNCT
ejpam-4713	153	2	10	10	NUM
ejpam-4713	153	3	)	)	PUNCT
ejpam-4713	153	4	references	reference	NOUN
ejpam-4713	153	5	897	897	NUM
ejpam-4713	153	6	the	the	DET
ejpam-4713	153	7	equality	equality	NOUN
ejpam-4713	153	8	(	(	PUNCT
ejpam-4713	153	9	10	10	NUM
ejpam-4713	153	10	)	)	PUNCT
ejpam-4713	153	11	is	be	AUX
ejpam-4713	153	12	in	in	ADP
ejpam-4713	153	13	fact	fact	NOUN
ejpam-4713	153	14	the	the	DET
ejpam-4713	153	15	identity	identity	NOUN
ejpam-4713	153	16	which	which	PRON
ejpam-4713	153	17	can	can	AUX
ejpam-4713	153	18	be	be	AUX
ejpam-4713	153	19	established	establish	VERB
ejpam-4713	153	20	using	use	VERB
ejpam-4713	153	21	the	the	DET
ejpam-4713	153	22	combinatorial	combinatorial	ADJ
ejpam-4713	153	23	formula	formula	NOUN
ejpam-4713	153	24	(	(	PUNCT
ejpam-4713	153	25	n	n	X
ejpam-4713	153	26	k	k	NOUN
ejpam-4713	153	27	)	)	PUNCT
ejpam-4713	154	1	=	=	PUNCT
ejpam-4713	154	2	(	(	PUNCT
ejpam-4713	154	3	n−	n−	NOUN
ejpam-4713	154	4	1	1	NUM
ejpam-4713	154	5	k	k	NOUN
ejpam-4713	154	6	)	)	PUNCT
ejpam-4713	155	1	+	+	CCONJ
ejpam-4713	155	2	(	(	PUNCT
ejpam-4713	155	3	n−	n−	NOUN
ejpam-4713	155	4	1	1	NUM
ejpam-4713	155	5	k	k	NOUN
ejpam-4713	155	6	−	−	NOUN
ejpam-4713	155	7	1	1	NUM
ejpam-4713	155	8	)	)	PUNCT
ejpam-4713	155	9	.	.	PUNCT
ejpam-4713	156	1	it	it	PRON
ejpam-4713	156	2	is	be	AUX
ejpam-4713	156	3	easy	easy	ADJ
ejpam-4713	156	4	to	to	PART
ejpam-4713	156	5	see	see	VERB
ejpam-4713	156	6	that	that	PRON
ejpam-4713	156	7	(	(	PUNCT
ejpam-4713	156	8	7	7	X
ejpam-4713	156	9	)	)	PUNCT
ejpam-4713	156	10	satisfies	satisfy	VERB
ejpam-4713	156	11	the	the	DET
ejpam-4713	156	12	condition	condition	NOUN
ejpam-4713	156	13	(	(	PUNCT
ejpam-4713	156	14	9	9	NUM
ejpam-4713	156	15	)	)	PUNCT
ejpam-4713	156	16	,	,	PUNCT
ejpam-4713	156	17	which	which	PRON
ejpam-4713	156	18	completes	complete	VERB
ejpam-4713	156	19	the	the	DET
ejpam-4713	156	20	proof	proof	NOUN
ejpam-4713	156	21	.	.	PUNCT
ejpam-4713	157	1	3	3	X
ejpam-4713	157	2	.	.	X
ejpam-4713	157	3	conclusions	conclusion	NOUN
ejpam-4713	157	4	and	and	CCONJ
ejpam-4713	157	5	further	further	ADJ
ejpam-4713	157	6	work	work	NOUN
ejpam-4713	157	7	in	in	ADP
ejpam-4713	157	8	this	this	DET
ejpam-4713	157	9	paper	paper	NOUN
ejpam-4713	157	10	we	we	PRON
ejpam-4713	157	11	have	have	AUX
ejpam-4713	157	12	investigated	investigate	VERB
ejpam-4713	157	13	the	the	DET
ejpam-4713	157	14	problem	problem	NOUN
ejpam-4713	157	15	of	of	ADP
ejpam-4713	157	16	space	space	NOUN
ejpam-4713	157	17	partitioning	partition	VERB
ejpam-4713	157	18	by	by	ADP
ejpam-4713	157	19	hyperplanes	hyperplane	NOUN
ejpam-4713	157	20	.	.	PUNCT
ejpam-4713	158	1	we	we	PRON
ejpam-4713	158	2	obtained	obtain	VERB
ejpam-4713	158	3	some	some	DET
ejpam-4713	158	4	relations	relation	NOUN
ejpam-4713	158	5	regarding	regard	VERB
ejpam-4713	158	6	the	the	DET
ejpam-4713	158	7	number	number	NOUN
ejpam-4713	158	8	of	of	ADP
ejpam-4713	158	9	divisions	division	NOUN
ejpam-4713	158	10	and	and	CCONJ
ejpam-4713	158	11	derived	derive	VERB
ejpam-4713	158	12	the	the	DET
ejpam-4713	158	13	recurrence	recurrence	NOUN
ejpam-4713	158	14	formula	formula	NOUN
ejpam-4713	158	15	for	for	ADP
ejpam-4713	158	16	the	the	DET
ejpam-4713	158	17	maximum	maximum	ADJ
ejpam-4713	158	18	number	number	NOUN
ejpam-4713	158	19	of	of	ADP
ejpam-4713	158	20	regions	region	NOUN
ejpam-4713	158	21	in	in	ADP
ejpam-4713	158	22	d	d	ADJ
ejpam-4713	158	23	-	-	ADJ
ejpam-4713	158	24	dimensional	dimensional	ADJ
ejpam-4713	158	25	euclidean	euclidean	ADJ
ejpam-4713	158	26	space	space	NOUN
ejpam-4713	158	27	cut	cut	VERB
ejpam-4713	158	28	by	by	ADP
ejpam-4713	158	29	n	n	NOUN
ejpam-4713	158	30	hyperplanes	hyperplane	NOUN
ejpam-4713	158	31	in	in	ADP
ejpam-4713	158	32	arbitrary	arbitrary	ADJ
ejpam-4713	158	33	position	position	NOUN
ejpam-4713	158	34	.	.	PUNCT
ejpam-4713	159	1	an	an	DET
ejpam-4713	159	2	explicit	explicit	ADJ
ejpam-4713	159	3	formula	formula	NOUN
ejpam-4713	159	4	for	for	ADP
ejpam-4713	159	5	the	the	DET
ejpam-4713	159	6	number	number	NOUN
ejpam-4713	159	7	of	of	ADP
ejpam-4713	159	8	regions	region	NOUN
ejpam-4713	159	9	into	into	ADP
ejpam-4713	159	10	which	which	PRON
ejpam-4713	159	11	the	the	DET
ejpam-4713	159	12	space	space	NOUN
ejpam-4713	159	13	can	can	AUX
ejpam-4713	159	14	be	be	AUX
ejpam-4713	159	15	partitioned	partition	VERB
ejpam-4713	159	16	by	by	ADP
ejpam-4713	159	17	n	n	DET
ejpam-4713	159	18	hyperplanes	hyperplane	NOUN
ejpam-4713	159	19	can	can	AUX
ejpam-4713	159	20	also	also	ADV
ejpam-4713	159	21	be	be	AUX
ejpam-4713	159	22	found	find	VERB
ejpam-4713	159	23	.	.	PUNCT
ejpam-4713	160	1	using	use	VERB
ejpam-4713	160	2	the	the	DET
ejpam-4713	160	3	recurrence	recurrence	NOUN
ejpam-4713	160	4	formula	formula	NOUN
ejpam-4713	160	5	,	,	PUNCT
ejpam-4713	160	6	we	we	PRON
ejpam-4713	160	7	gave	give	VERB
ejpam-4713	160	8	an	an	DET
ejpam-4713	160	9	alternative	alternative	ADJ
ejpam-4713	160	10	derivation	derivation	NOUN
ejpam-4713	160	11	of	of	ADP
ejpam-4713	160	12	the	the	DET
ejpam-4713	160	13	well	well	ADV
ejpam-4713	160	14	known	know	VERB
ejpam-4713	160	15	explicit	explicit	ADJ
ejpam-4713	160	16	formula	formula	NOUN
ejpam-4713	160	17	for	for	ADP
ejpam-4713	160	18	the	the	DET
ejpam-4713	160	19	number	number	NOUN
ejpam-4713	160	20	of	of	ADP
ejpam-4713	160	21	regions	region	NOUN
ejpam-4713	160	22	that	that	PRON
ejpam-4713	160	23	n	n	ADP
ejpam-4713	160	24	hyperplanes	hyperplane	NOUN
ejpam-4713	160	25	in	in	ADP
ejpam-4713	160	26	general	general	ADJ
ejpam-4713	160	27	position	position	NOUN
ejpam-4713	160	28	divide	divide	VERB
ejpam-4713	160	29	the	the	DET
ejpam-4713	160	30	d	d	ADJ
ejpam-4713	160	31	-	-	ADJ
ejpam-4713	160	32	dimensional	dimensional	ADJ
ejpam-4713	160	33	space	space	NOUN
ejpam-4713	160	34	.	.	PUNCT
ejpam-4713	161	1	in	in	ADP
ejpam-4713	161	2	our	our	PRON
ejpam-4713	161	3	subsequent	subsequent	ADJ
ejpam-4713	161	4	works	work	NOUN
ejpam-4713	161	5	we	we	PRON
ejpam-4713	161	6	plan	plan	VERB
ejpam-4713	161	7	to	to	PART
ejpam-4713	161	8	continue	continue	VERB
ejpam-4713	161	9	studying	study	VERB
ejpam-4713	161	10	the	the	DET
ejpam-4713	161	11	combinatorics	combinatoric	NOUN
ejpam-4713	161	12	of	of	ADP
ejpam-4713	161	13	hyperplane	hyperplane	PROPN
ejpam-4713	161	14	configurations	configuration	NOUN
ejpam-4713	161	15	in	in	ADP
ejpam-4713	161	16	rd	rd	PROPN
ejpam-4713	161	17	.	.	PUNCT
ejpam-4713	162	1	further	further	ADJ
ejpam-4713	162	2	work	work	NOUN
ejpam-4713	162	3	may	may	AUX
ejpam-4713	162	4	address	address	VERB
ejpam-4713	162	5	the	the	DET
ejpam-4713	162	6	investigation	investigation	NOUN
ejpam-4713	162	7	of	of	ADP
ejpam-4713	162	8	the	the	DET
ejpam-4713	162	9	number	number	NOUN
ejpam-4713	162	10	of	of	ADP
ejpam-4713	162	11	k	k	NOUN
ejpam-4713	162	12	-	-	NOUN
ejpam-4713	162	13	edges	edge	NOUN
ejpam-4713	162	14	of	of	ADP
ejpam-4713	162	15	hyperplane	hyperplane	NOUN
ejpam-4713	162	16	configurations	configuration	NOUN
ejpam-4713	162	17	,	,	PUNCT
ejpam-4713	162	18	the	the	DET
ejpam-4713	162	19	study	study	NOUN
ejpam-4713	162	20	of	of	ADP
ejpam-4713	162	21	partitioning	partition	VERB
ejpam-4713	162	22	problem	problem	NOUN
ejpam-4713	162	23	of	of	ADP
ejpam-4713	162	24	rd	rd	NOUN
ejpam-4713	162	25	by	by	ADP
ejpam-4713	162	26	hyperplanes	hyperplane	NOUN
ejpam-4713	162	27	with	with	ADP
ejpam-4713	162	28	the	the	DET
ejpam-4713	162	29	use	use	NOUN
ejpam-4713	162	30	of	of	ADP
ejpam-4713	162	31	möbius	möbius	PROPN
ejpam-4713	162	32	function	function	NOUN
ejpam-4713	162	33	defined	define	VERB
ejpam-4713	162	34	on	on	ADP
ejpam-4713	162	35	finite	finite	ADJ
ejpam-4713	162	36	posets	poset	NOUN
ejpam-4713	162	37	connected	connect	VERB
ejpam-4713	162	38	with	with	ADP
ejpam-4713	162	39	the	the	DET
ejpam-4713	162	40	rank	rank	NOUN
ejpam-4713	162	41	of	of	ADP
ejpam-4713	162	42	homology	homology	PROPN
ejpam-4713	162	43	group	group	PROPN
ejpam-4713	162	44	hd	hd	PROPN
ejpam-4713	162	45	(	(	PUNCT
ejpam-4713	162	46	cn	cn	PROPN
ejpam-4713	162	47	\h	\h	PROPN
ejpam-4713	162	48	)	)	PUNCT
ejpam-4713	162	49	,	,	PUNCT
ejpam-4713	162	50	the	the	DET
ejpam-4713	162	51	study	study	NOUN
ejpam-4713	162	52	of	of	ADP
ejpam-4713	162	53	separating	separate	VERB
ejpam-4713	162	54	subgroups	subgroup	NOUN
ejpam-4713	162	55	of	of	ADP
ejpam-4713	162	56	the	the	DET
ejpam-4713	162	57	homology	homology	NOUN
ejpam-4713	162	58	group	group	PROPN
ejpam-4713	162	59	hd	hd	PROPN
ejpam-4713	162	60	(	(	PUNCT
ejpam-4713	162	61	cn	cn	PROPN
ejpam-4713	162	62	\h	\h	PROPN
ejpam-4713	162	63	)	)	PUNCT
ejpam-4713	162	64	,	,	PUNCT
ejpam-4713	162	65	and	and	CCONJ
ejpam-4713	162	66	finding	find	VERB
ejpam-4713	162	67	the	the	DET
ejpam-4713	162	68	formulas	formula	NOUN
ejpam-4713	162	69	for	for	ADP
ejpam-4713	162	70	möbius	möbius	PROPN
ejpam-4713	162	71	function	function	NOUN
ejpam-4713	162	72	of	of	ADP
ejpam-4713	162	73	posets	poset	NOUN
ejpam-4713	162	74	of	of	ADP
ejpam-4713	162	75	edges	edge	NOUN
ejpam-4713	162	76	of	of	ADP
ejpam-4713	162	77	hyperplane	hyperplane	NOUN
ejpam-4713	162	78	configurations	configuration	NOUN
ejpam-4713	162	79	.	.	PUNCT
ejpam-4713	163	1	these	these	DET
ejpam-4713	163	2	results	result	NOUN
ejpam-4713	163	3	may	may	AUX
ejpam-4713	163	4	find	find	VERB
ejpam-4713	163	5	applications	application	NOUN
ejpam-4713	163	6	in	in	ADP
ejpam-4713	163	7	toric	toric	ADJ
ejpam-4713	163	8	geometry	geometry	NOUN
ejpam-4713	163	9	,	,	PUNCT
ejpam-4713	163	10	singularity	singularity	NOUN
ejpam-4713	163	11	theory	theory	NOUN
ejpam-4713	163	12	and	and	CCONJ
ejpam-4713	163	13	multidimensional	multidimensional	ADJ
ejpam-4713	163	14	residues	residue	NOUN
ejpam-4713	163	15	,	,	PUNCT
ejpam-4713	163	16	and	and	CCONJ
ejpam-4713	163	17	the	the	DET
ejpam-4713	163	18	theory	theory	NOUN
ejpam-4713	163	19	of	of	ADP
ejpam-4713	163	20	hypergeometric	hypergeometric	ADJ
ejpam-4713	163	21	functions	function	NOUN
ejpam-4713	163	22	.	.	PUNCT
ejpam-4713	164	1	yet	yet	CCONJ
ejpam-4713	164	2	another	another	DET
ejpam-4713	164	3	direction	direction	NOUN
ejpam-4713	164	4	of	of	ADP
ejpam-4713	164	5	future	future	ADJ
ejpam-4713	164	6	research	research	NOUN
ejpam-4713	164	7	may	may	AUX
ejpam-4713	164	8	be	be	AUX
ejpam-4713	164	9	related	relate	VERB
ejpam-4713	164	10	to	to	ADP
ejpam-4713	164	11	the	the	DET
ejpam-4713	164	12	application	application	NOUN
ejpam-4713	164	13	of	of	ADP
ejpam-4713	164	14	the	the	DET
ejpam-4713	164	15	obtained	obtain	VERB
ejpam-4713	164	16	results	result	NOUN
ejpam-4713	164	17	in	in	ADP
ejpam-4713	164	18	data	data	NOUN
ejpam-4713	164	19	analysis	analysis	NOUN
ejpam-4713	164	20	and	and	CCONJ
ejpam-4713	164	21	classification	classification	NOUN
ejpam-4713	164	22	,	,	PUNCT
ejpam-4713	164	23	and	and	CCONJ
ejpam-4713	164	24	experimental	experimental	ADJ
ejpam-4713	164	25	data	data	NOUN
ejpam-4713	164	26	processing	processing	NOUN
ejpam-4713	164	27	using	use	VERB
ejpam-4713	164	28	linear	linear	PROPN
ejpam-4713	164	29	regression	regression	NOUN
ejpam-4713	164	30	models	model	NOUN
ejpam-4713	164	31	,	,	PUNCT
ejpam-4713	164	32	and	and	CCONJ
ejpam-4713	164	33	in	in	ADP
ejpam-4713	164	34	the	the	DET
ejpam-4713	164	35	problems	problem	NOUN
ejpam-4713	164	36	of	of	ADP
ejpam-4713	164	37	identifiability	identifiability	NOUN
ejpam-4713	164	38	of	of	ADP
ejpam-4713	164	39	linear	linear	PROPN
ejpam-4713	164	40	dynamical	dynamical	ADJ
ejpam-4713	164	41	systems	system	NOUN
ejpam-4713	164	42	in	in	ADP
ejpam-4713	164	43	state	state	NOUN
ejpam-4713	164	44	space	space	NOUN
ejpam-4713	164	45	.	.	PUNCT
ejpam-4713	165	1	acknowledgements	acknowledgement	VERB
ejpam-4713	165	2	the	the	DET
ejpam-4713	165	3	author	author	NOUN
ejpam-4713	165	4	wishes	wish	VERB
ejpam-4713	165	5	to	to	PART
ejpam-4713	165	6	thank	thank	VERB
ejpam-4713	165	7	the	the	DET
ejpam-4713	165	8	anonymous	anonymous	ADJ
ejpam-4713	165	9	referees	referee	NOUN
ejpam-4713	165	10	for	for	ADP
ejpam-4713	165	11	useful	useful	ADJ
ejpam-4713	165	12	comments	comment	NOUN
ejpam-4713	165	13	that	that	PRON
ejpam-4713	165	14	improved	improve	VERB
ejpam-4713	165	15	the	the	DET
ejpam-4713	165	16	presentation	presentation	NOUN
ejpam-4713	165	17	of	of	ADP
ejpam-4713	165	18	this	this	DET
ejpam-4713	165	19	paper	paper	NOUN
ejpam-4713	165	20	.	.	PUNCT
ejpam-4713	166	1	references	reference	NOUN
ejpam-4713	166	2	[	[	X
ejpam-4713	166	3	1	1	NUM
ejpam-4713	166	4	]	]	PUNCT
ejpam-4713	166	5	g	g	PROPN
ejpam-4713	166	6	l	l	NOUN
ejpam-4713	166	7	alexanderson	alexanderson	NOUN
ejpam-4713	166	8	and	and	CCONJ
ejpam-4713	166	9	g	g	PROPN
ejpam-4713	166	10	e	e	PROPN
ejpam-4713	166	11	wetzel	wetzel	PROPN
ejpam-4713	166	12	.	.	PUNCT
ejpam-4713	167	1	divisions	division	NOUN
ejpam-4713	167	2	of	of	ADP
ejpam-4713	167	3	space	space	NOUN
ejpam-4713	167	4	by	by	ADP
ejpam-4713	167	5	parallels	parallel	NOUN
ejpam-4713	167	6	.	.	PUNCT
ejpam-4713	168	1	trans	trans	PROPN
ejpam-4713	168	2	.	.	PUNCT
ejpam-4713	169	1	amer	amer	PROPN
ejpam-4713	169	2	.	.	PUNCT
ejpam-4713	169	3	math	math	PROPN
ejpam-4713	169	4	.	.	PUNCT
ejpam-4713	170	1	soc	soc	PROPN
ejpam-4713	170	2	.	.	PUNCT
ejpam-4713	170	3	,	,	PUNCT
ejpam-4713	170	4	291:363–377	291:363–377	NUM
ejpam-4713	170	5	,	,	PUNCT
ejpam-4713	170	6	1985	1985	NUM
ejpam-4713	170	7	.	.	PUNCT
ejpam-4713	171	1	[	[	X
ejpam-4713	171	2	2	2	NUM
ejpam-4713	171	3	]	]	X
ejpam-4713	171	4	m	m	PROPN
ejpam-4713	171	5	anthony	anthony	PROPN
ejpam-4713	171	6	.	.	PUNCT
ejpam-4713	172	1	classification	classification	NOUN
ejpam-4713	172	2	by	by	ADP
ejpam-4713	172	3	polynomial	polynomial	ADJ
ejpam-4713	172	4	surfaces	surface	NOUN
ejpam-4713	172	5	.	.	PUNCT
ejpam-4713	173	1	discrete	discrete	ADJ
ejpam-4713	173	2	appl	appl	PROPN
ejpam-4713	173	3	.	.	PUNCT
ejpam-4713	173	4	math	math	PROPN
ejpam-4713	173	5	.	.	PUNCT
ejpam-4713	173	6	,	,	PUNCT
ejpam-4713	173	7	61:91–103	61:91–103	NUM
ejpam-4713	173	8	,	,	PUNCT
ejpam-4713	173	9	1995	1995	NUM
ejpam-4713	173	10	.	.	PUNCT
ejpam-4713	174	1	[	[	X
ejpam-4713	174	2	3	3	NUM
ejpam-4713	174	3	]	]	X
ejpam-4713	174	4	m	m	VERB
ejpam-4713	174	5	anthony	anthony	PROPN
ejpam-4713	174	6	and	and	CCONJ
ejpam-4713	174	7	p	p	PROPN
ejpam-4713	174	8	l	l	PROPN
ejpam-4713	174	9	bartlett	bartlett	PROPN
ejpam-4713	174	10	.	.	PUNCT
ejpam-4713	175	1	neural	neural	ADJ
ejpam-4713	175	2	network	network	NOUN
ejpam-4713	175	3	learning	learning	PROPN
ejpam-4713	175	4	:	:	PUNCT
ejpam-4713	175	5	theoretical	theoretical	ADJ
ejpam-4713	175	6	foundations	foundation	NOUN
ejpam-4713	175	7	.	.	PUNCT
ejpam-4713	176	1	cambridge	cambridge	PROPN
ejpam-4713	176	2	university	university	PROPN
ejpam-4713	176	3	press	press	NOUN
ejpam-4713	176	4	,	,	PUNCT
ejpam-4713	176	5	1999	1999	NUM
ejpam-4713	176	6	.	.	PUNCT
ejpam-4713	177	1	[	[	X
ejpam-4713	177	2	4	4	NUM
ejpam-4713	177	3	]	]	X
ejpam-4713	177	4	r	r	NOUN
ejpam-4713	177	5	c	c	NOUN
ejpam-4713	177	6	buck	buck	NOUN
ejpam-4713	177	7	.	.	PUNCT
ejpam-4713	178	1	partition	partition	NOUN
ejpam-4713	178	2	of	of	ADP
ejpam-4713	178	3	space	space	NOUN
ejpam-4713	178	4	.	.	PUNCT
ejpam-4713	179	1	amer	amer	PROPN
ejpam-4713	179	2	.	.	PUNCT
ejpam-4713	179	3	math	math	PROPN
ejpam-4713	179	4	.	.	PUNCT
ejpam-4713	180	1	monthly	monthly	ADJ
ejpam-4713	180	2	,	,	PUNCT
ejpam-4713	180	3	50:541–544	50:541–544	PROPN
ejpam-4713	180	4	,	,	PUNCT
ejpam-4713	180	5	1943	1943	NUM
ejpam-4713	180	6	.	.	PUNCT
ejpam-4713	181	1	references	reference	NOUN
ejpam-4713	181	2	898	898	NUM
ejpam-4713	182	1	[	[	X
ejpam-4713	182	2	5	5	NUM
ejpam-4713	182	3	]	]	SYM
ejpam-4713	182	4	r	r	NOUN
ejpam-4713	182	5	o	o	NOUN
ejpam-4713	182	6	duda	duda	NOUN
ejpam-4713	182	7	,	,	PUNCT
ejpam-4713	182	8	p	p	PROPN
ejpam-4713	182	9	e	e	PROPN
ejpam-4713	182	10	hart	hart	PROPN
ejpam-4713	182	11	,	,	PUNCT
ejpam-4713	182	12	and	and	CCONJ
ejpam-4713	182	13	d	d	ADP
ejpam-4713	182	14	stork	stork	NOUN
ejpam-4713	182	15	.	.	PUNCT
ejpam-4713	183	1	pattern	pattern	NOUN
ejpam-4713	183	2	classification	classification	NOUN
ejpam-4713	183	3	.	.	PUNCT
ejpam-4713	184	1	wiley	wiley	PROPN
ejpam-4713	184	2	,	,	PUNCT
ejpam-4713	184	3	2000	2000	NUM
ejpam-4713	184	4	.	.	PUNCT
ejpam-4713	185	1	[	[	X
ejpam-4713	185	2	6	6	NUM
ejpam-4713	185	3	]	]	SYM
ejpam-4713	185	4	b	b	X
ejpam-4713	185	5	everitt	everitt	NOUN
ejpam-4713	185	6	,	,	PUNCT
ejpam-4713	185	7	s	s	PART
ejpam-4713	185	8	landau	landau	NOUN
ejpam-4713	185	9	,	,	PUNCT
ejpam-4713	185	10	m	m	NOUN
ejpam-4713	185	11	leese	leese	ADJ
ejpam-4713	185	12	,	,	PUNCT
ejpam-4713	185	13	and	and	CCONJ
ejpam-4713	185	14	d	d	PROPN
ejpam-4713	185	15	stahl	stahl	PROPN
ejpam-4713	185	16	.	.	PUNCT
ejpam-4713	186	1	cluster	cluster	NOUN
ejpam-4713	186	2	analysis	analysis	NOUN
ejpam-4713	186	3	.	.	PUNCT
ejpam-4713	187	1	wiley	wiley	PROPN
ejpam-4713	187	2	,	,	PUNCT
ejpam-4713	187	3	2011	2011	NUM
ejpam-4713	187	4	.	.	PUNCT
ejpam-4713	188	1	[	[	X
ejpam-4713	188	2	7	7	X
ejpam-4713	188	3	]	]	X
ejpam-4713	188	4	d	d	X
ejpam-4713	188	5	forge	forge	NOUN
ejpam-4713	188	6	and	and	CCONJ
ejpam-4713	188	7	t	t	PROPN
ejpam-4713	188	8	zaslavsky	zaslavsky	NOUN
ejpam-4713	188	9	.	.	PUNCT
ejpam-4713	189	1	on	on	ADP
ejpam-4713	189	2	the	the	DET
ejpam-4713	189	3	division	division	NOUN
ejpam-4713	189	4	of	of	ADP
ejpam-4713	189	5	space	space	NOUN
ejpam-4713	189	6	by	by	ADP
ejpam-4713	189	7	topological	topological	ADJ
ejpam-4713	189	8	hyperplanes	hyperplane	NOUN
ejpam-4713	189	9	.	.	PUNCT
ejpam-4713	190	1	european	european	PROPN
ejpam-4713	190	2	j.	j.	PROPN
ejpam-4713	190	3	combin	combin	PROPN
ejpam-4713	190	4	.	.	PROPN
ejpam-4713	190	5	,	,	PUNCT
ejpam-4713	190	6	30:1835–1845	30:1835–1845	NUM
ejpam-4713	190	7	,	,	PUNCT
ejpam-4713	190	8	2009	2009	NUM
ejpam-4713	190	9	.	.	PUNCT
ejpam-4713	191	1	[	[	X
ejpam-4713	191	2	8	8	NUM
ejpam-4713	191	3	]	]	X
ejpam-4713	191	4	ch	ch	NOUN
ejpam-4713	191	5	ho	ho	PROPN
ejpam-4713	191	6	and	and	CCONJ
ejpam-4713	191	7	s	s	PROPN
ejpam-4713	191	8	zimmerman	zimmerman	NOUN
ejpam-4713	191	9	.	.	PUNCT
ejpam-4713	192	1	on	on	ADP
ejpam-4713	192	2	the	the	DET
ejpam-4713	192	3	number	number	NOUN
ejpam-4713	192	4	of	of	ADP
ejpam-4713	192	5	regions	region	NOUN
ejpam-4713	192	6	in	in	ADP
ejpam-4713	192	7	an	an	DET
ejpam-4713	192	8	m	m	ADJ
ejpam-4713	192	9	-	-	ADJ
ejpam-4713	192	10	dimensional	dimensional	ADJ
ejpam-4713	192	11	space	space	NOUN
ejpam-4713	192	12	cut	cut	VERB
ejpam-4713	192	13	by	by	ADP
ejpam-4713	192	14	n	n	DET
ejpam-4713	192	15	hyperplanes	hyperplane	NOUN
ejpam-4713	192	16	.	.	PUNCT
ejpam-4713	193	1	gaz	gaz	PROPN
ejpam-4713	193	2	.	.	PROPN
ejpam-4713	194	1	austr	austr	PROPN
ejpam-4713	194	2	.	.	PUNCT
ejpam-4713	194	3	math	math	PROPN
ejpam-4713	194	4	.	.	PUNCT
ejpam-4713	195	1	soc	soc	PROPN
ejpam-4713	195	2	.	.	PUNCT
ejpam-4713	195	3	,	,	PUNCT
ejpam-4713	195	4	8:260–264	8:260–264	NOUN
ejpam-4713	195	5	,	,	PUNCT
ejpam-4713	195	6	2006	2006	NUM
ejpam-4713	195	7	.	.	PUNCT
ejpam-4713	196	1	[	[	X
ejpam-4713	196	2	9	9	NUM
ejpam-4713	196	3	]	]	PUNCT
ejpam-4713	196	4	t	t	PROPN
ejpam-4713	196	5	geyer	geyer	PROPN
ejpam-4713	196	6	,	,	PUNCT
ejpam-4713	196	7	f	f	PROPN
ejpam-4713	196	8	torrisi	torrisi	ADJ
ejpam-4713	196	9	and	and	CCONJ
ejpam-4713	196	10	m	m	NOUN
ejpam-4713	196	11	morari	morari	ADJ
ejpam-4713	196	12	.	.	PUNCT
ejpam-4713	197	1	optimal	optimal	ADJ
ejpam-4713	197	2	complexity	complexity	NOUN
ejpam-4713	197	3	reduction	reduction	NOUN
ejpam-4713	197	4	of	of	ADP
ejpam-4713	197	5	piecewise	piecewise	NOUN
ejpam-4713	197	6	affine	affine	NOUN
ejpam-4713	197	7	models	model	NOUN
ejpam-4713	197	8	based	base	VERB
ejpam-4713	197	9	on	on	ADP
ejpam-4713	197	10	hyperplane	hyperplane	PROPN
ejpam-4713	197	11	arrangements	arrangement	NOUN
ejpam-4713	197	12	.	.	PUNCT
ejpam-4713	198	1	in	in	ADP
ejpam-4713	198	2	proceedings	proceeding	NOUN
ejpam-4713	198	3	of	of	ADP
ejpam-4713	198	4	the	the	DET
ejpam-4713	198	5	2004	2004	NUM
ejpam-4713	198	6	american	american	PROPN
ejpam-4713	198	7	control	control	PROPN
ejpam-4713	198	8	conference	conference	PROPN
ejpam-4713	198	9	.	.	PUNCT
ejpam-4713	198	10	,	,	PUNCT
ejpam-4713	198	11	pages	page	NOUN
ejpam-4713	198	12	1190–1195	1190–1195	NUM
ejpam-4713	198	13	,	,	PUNCT
ejpam-4713	198	14	boston	boston	PROPN
ejpam-4713	198	15	,	,	PUNCT
ejpam-4713	198	16	ma	ma	PROPN
ejpam-4713	198	17	,	,	PUNCT
ejpam-4713	198	18	2004	2004	NUM
ejpam-4713	198	19	.	.	PUNCT
ejpam-4713	199	1	ieee	ieee	NOUN
ejpam-4713	199	2	.	.	PUNCT
ejpam-4713	200	1	[	[	X
ejpam-4713	200	2	10	10	NUM
ejpam-4713	200	3	]	]	SYM
ejpam-4713	200	4	v	v	NOUN
ejpam-4713	200	5	n	n	PRON
ejpam-4713	200	6	vapnik	vapnik	NOUN
ejpam-4713	200	7	.	.	PUNCT
ejpam-4713	201	1	statistical	statistical	ADJ
ejpam-4713	201	2	learning	learning	NOUN
ejpam-4713	201	3	theory	theory	NOUN
ejpam-4713	201	4	.	.	PUNCT
ejpam-4713	202	1	wiley	wiley	PROPN
ejpam-4713	202	2	,	,	PUNCT
ejpam-4713	202	3	1998	1998	NUM
ejpam-4713	202	4	.	.	PUNCT
ejpam-4713	203	1	[	[	X
ejpam-4713	203	2	11	11	NUM
ejpam-4713	203	3	]	]	X
ejpam-4713	203	4	r.o	r.o	PROPN
ejpam-4713	203	5	.	.	PROPN
ejpam-4713	203	6	winder	winder	NOUN
ejpam-4713	203	7	.	.	PUNCT
ejpam-4713	204	1	partitions	partition	NOUN
ejpam-4713	204	2	of	of	ADP
ejpam-4713	204	3	n	n	PRON
ejpam-4713	204	4	-space	-space	NOUN
ejpam-4713	204	5	by	by	ADP
ejpam-4713	204	6	hyperplanes	hyperplane	NOUN
ejpam-4713	204	7	.	.	PUNCT
ejpam-4713	205	1	siam	siam	PROPN
ejpam-4713	205	2	j.	j.	PROPN
ejpam-4713	205	3	appl	appl	PROPN
ejpam-4713	205	4	.	.	PROPN
ejpam-4713	205	5	math	math	PROPN
ejpam-4713	205	6	.	.	PUNCT
ejpam-4713	205	7	,	,	PUNCT
ejpam-4713	205	8	14:811	14:811	NUM
ejpam-4713	205	9	–	–	PUNCT
ejpam-4713	205	10	818	818	NUM
ejpam-4713	205	11	,	,	PUNCT
ejpam-4713	205	12	1966	1966	NUM
ejpam-4713	205	13	.	.	PUNCT
ejpam-4713	206	1	[	[	X
ejpam-4713	206	2	12	12	NUM
ejpam-4713	206	3	]	]	X
ejpam-4713	206	4	s	s	PROPN
ejpam-4713	206	5	zimmerman	zimmerman	PROPN
ejpam-4713	206	6	.	.	PUNCT
ejpam-4713	207	1	slicing	slice	VERB
ejpam-4713	207	2	space	space	NOUN
ejpam-4713	207	3	.	.	PUNCT
ejpam-4713	208	1	college	college	NOUN
ejpam-4713	208	2	math	math	PROPN
ejpam-4713	208	3	.	.	PUNCT
ejpam-4713	209	1	j.	j.	PROPN
ejpam-4713	209	2	,	,	PUNCT
ejpam-4713	209	3	32:126–128	32:126–128	PROPN
ejpam-4713	209	4	,	,	PUNCT
ejpam-4713	209	5	2001	2001	NUM
ejpam-4713	209	6	.	.	PUNCT