id	sid	tid	token	lemma	pos
ap-924	1	1	ap07_2-3.vp	ap07_2-3.vp	VERB
ap-924	1	2	1	1	NUM
ap-924	1	3	introduction	introduction	NOUN
ap-924	1	4	sequences	sequence	NOUN
ap-924	1	5	of	of	ADP
ap-924	1	6	the	the	DET
ap-924	1	7	form	form	NOUN
ap-924	1	8	�	�	PROPN
ap-924	1	9	�	�	PROPN
ap-924	1	10	�	�	PROPN
ap-924	1	11	�	�	PROPN
ap-924	1	12	j	j	PROPN
ap-924	1	13	j	j	PROPN
ap-924	1	14	�	�	PROPN
ap-924	1	15	�	�	PROPN
ap-924	1	16	�	�	PROPN
ap-924	1	17	for	for	ADP
ap-924	1	18	�	�	PROPN
ap-924	1	19	�	�	PROPN
ap-924	1	20	1	1	NUM
ap-924	1	21	,	,	PUNCT
ap-924	1	22	now	now	ADV
ap-924	1	23	known	know	VERB
ap-924	1	24	as	as	ADP
ap-924	1	25	beatty	beatty	PROPN
ap-924	1	26	sequences	sequences	PROPN
ap-924	1	27	,	,	PUNCT
ap-924	1	28	were	be	AUX
ap-924	1	29	first	first	ADV
ap-924	1	30	studied	study	VERB
ap-924	1	31	in	in	ADP
ap-924	1	32	the	the	DET
ap-924	1	33	context	context	NOUN
ap-924	1	34	of	of	ADP
ap-924	1	35	the	the	DET
ap-924	1	36	famous	famous	ADJ
ap-924	1	37	problem	problem	NOUN
ap-924	1	38	of	of	ADP
ap-924	1	39	covering	cover	VERB
ap-924	1	40	the	the	DET
ap-924	1	41	set	set	NOUN
ap-924	1	42	of	of	ADP
ap-924	1	43	positive	positive	ADJ
ap-924	1	44	integers	integer	NOUN
ap-924	1	45	by	by	ADP
ap-924	1	46	disjoint	disjoint	NOUN
ap-924	1	47	sequences	sequence	NOUN
ap-924	2	1	[	[	X
ap-924	2	2	1	1	NUM
ap-924	2	3	]	]	PUNCT
ap-924	2	4	.	.	PUNCT
ap-924	3	1	further	further	ADJ
ap-924	3	2	results	result	NOUN
ap-924	3	3	in	in	ADP
ap-924	3	4	the	the	DET
ap-924	3	5	direction	direction	NOUN
ap-924	3	6	of	of	ADP
ap-924	3	7	so	so	ADV
ap-924	3	8	-	-	PUNCT
ap-924	3	9	called	call	VERB
ap-924	3	10	disjoint	disjoint	NOUN
ap-924	3	11	covering	covering	NOUN
ap-924	3	12	systems	system	NOUN
ap-924	3	13	are	be	AUX
ap-924	3	14	due	due	ADJ
ap-924	3	15	to	to	ADP
ap-924	3	16	[	[	X
ap-924	3	17	2	2	NUM
ap-924	3	18	]	]	PUNCT
ap-924	3	19	,	,	PUNCT
ap-924	3	20	[	[	X
ap-924	3	21	3	3	NUM
ap-924	3	22	]	]	PUNCT
ap-924	3	23	,	,	PUNCT
ap-924	3	24	[	[	X
ap-924	3	25	4	4	X
ap-924	3	26	]	]	PUNCT
ap-924	3	27	and	and	CCONJ
ap-924	3	28	others	other	NOUN
ap-924	3	29	.	.	PUNCT
ap-924	4	1	other	other	ADJ
ap-924	4	2	aspects	aspect	NOUN
ap-924	4	3	of	of	ADP
ap-924	4	4	beatty	beatty	PROPN
ap-924	4	5	sequences	sequence	NOUN
ap-924	4	6	were	be	AUX
ap-924	4	7	then	then	ADV
ap-924	4	8	studied	study	VERB
ap-924	4	9	,	,	PUNCT
ap-924	4	10	such	such	ADJ
ap-924	4	11	as	as	ADP
ap-924	4	12	their	their	PRON
ap-924	4	13	generation	generation	NOUN
ap-924	4	14	using	use	VERB
ap-924	4	15	graphs	graph	NOUN
ap-924	4	16	[	[	X
ap-924	4	17	5	5	NUM
ap-924	4	18	]	]	PUNCT
ap-924	4	19	,	,	PUNCT
ap-924	4	20	their	their	PRON
ap-924	4	21	relation	relation	NOUN
ap-924	4	22	to	to	ADP
ap-924	4	23	generating	generating	NOUN
ap-924	4	24	functions	function	NOUN
ap-924	4	25	[	[	X
ap-924	4	26	6	6	NUM
ap-924	4	27	]	]	PUNCT
ap-924	4	28	,	,	PUNCT
ap-924	5	1	[	[	X
ap-924	5	2	7	7	NUM
ap-924	5	3	]	]	PUNCT
ap-924	5	4	,	,	PUNCT
ap-924	5	5	their	their	PRON
ap-924	5	6	substitution	substitution	NOUN
ap-924	5	7	invariance	invariance	NOUN
ap-924	5	8	[	[	X
ap-924	5	9	8	8	NUM
ap-924	5	10	]	]	PUNCT
ap-924	5	11	,	,	PUNCT
ap-924	5	12	[	[	X
ap-924	5	13	9	9	NUM
ap-924	5	14	]	]	PUNCT
ap-924	5	15	,	,	PUNCT
ap-924	5	16	etc	etc	X
ap-924	5	17	.	.	X
ap-924	5	18	a	a	DET
ap-924	5	19	good	good	ADJ
ap-924	5	20	source	source	NOUN
ap-924	5	21	of	of	ADP
ap-924	5	22	references	reference	NOUN
ap-924	5	23	on	on	ADP
ap-924	5	24	beatty	beatty	PROPN
ap-924	5	25	sequences	sequence	NOUN
ap-924	5	26	and	and	CCONJ
ap-924	5	27	other	other	ADJ
ap-924	5	28	related	related	ADJ
ap-924	5	29	problems	problem	NOUN
ap-924	5	30	can	can	AUX
ap-924	5	31	be	be	AUX
ap-924	5	32	found	find	VERB
ap-924	5	33	in	in	ADP
ap-924	5	34	[	[	X
ap-924	5	35	10	10	NUM
ap-924	5	36	]	]	PUNCT
ap-924	5	37	,	,	PUNCT
ap-924	5	38	[	[	X
ap-924	5	39	11	11	NUM
ap-924	5	40	]	]	PUNCT
ap-924	5	41	.	.	PUNCT
ap-924	6	1	in	in	ADP
ap-924	6	2	[	[	X
ap-924	6	3	12	12	NUM
ap-924	6	4	]	]	PUNCT
ap-924	6	5	the	the	DET
ap-924	6	6	authors	author	NOUN
ap-924	6	7	study	study	VERB
ap-924	6	8	the	the	DET
ap-924	6	9	self	self	NOUN
ap-924	6	10	-	-	PUNCT
ap-924	6	11	matching	match	VERB
ap-924	6	12	properties	property	NOUN
ap-924	6	13	of	of	ADP
ap-924	6	14	the	the	DET
ap-924	6	15	beatty	beatty	PROPN
ap-924	6	16	sequence	sequence	PROPN
ap-924	6	17	�	�	PROPN
ap-924	6	18	�	�	PROPN
ap-924	6	19	�	�	PROPN
ap-924	6	20	�	�	PROPN
ap-924	6	21	j	j	PROPN
ap-924	6	22	j	j	PROPN
ap-924	6	23	�	�	PROPN
ap-924	6	24	�	�	PROPN
ap-924	6	25	�	�	PROPN
ap-924	6	26	for	for	ADP
ap-924	6	27	�	�	PROPN
ap-924	6	28	�	�	PROPN
ap-924	7	1	1	1	NUM
ap-924	7	2	2	2	NUM
ap-924	7	3	5	5	NUM
ap-924	7	4	1	1	NUM
ap-924	7	5	(	(	PUNCT
ap-924	7	6	)	)	PUNCT
ap-924	7	7	,	,	PUNCT
ap-924	7	8	the	the	DET
ap-924	7	9	golden	golden	ADJ
ap-924	7	10	ratio	ratio	NOUN
ap-924	7	11	.	.	PUNCT
ap-924	8	1	their	their	PRON
ap-924	8	2	study	study	NOUN
ap-924	8	3	is	be	AUX
ap-924	8	4	rather	rather	ADV
ap-924	8	5	technical;	technical;	NOUN
ap-924	8	6	they	they	PRON
ap-924	8	7	have	have	AUX
ap-924	8	8	used	use	VERB
ap-924	8	9	for	for	ADP
ap-924	8	10	their	their	PRON
ap-924	8	11	proof	proof	NOUN
ap-924	8	12	the	the	DET
ap-924	8	13	zeckendorf	zeckendorf	PROPN
ap-924	8	14	representation	representation	NOUN
ap-924	8	15	of	of	ADP
ap-924	8	16	integers	integer	NOUN
ap-924	8	17	as	as	ADP
ap-924	8	18	a	a	DET
ap-924	8	19	sum	sum	NOUN
ap-924	8	20	of	of	ADP
ap-924	8	21	distinct	distinct	ADJ
ap-924	8	22	fibonacci	fibonacci	NOUN
ap-924	8	23	numbers	number	NOUN
ap-924	8	24	.	.	PUNCT
ap-924	9	1	the	the	DET
ap-924	9	2	authors	author	NOUN
ap-924	9	3	also	also	ADV
ap-924	9	4	state	state	VERB
ap-924	9	5	an	an	DET
ap-924	9	6	open	open	ADJ
ap-924	9	7	question	question	NOUN
ap-924	9	8	whether	whether	SCONJ
ap-924	9	9	the	the	DET
ap-924	9	10	results	result	NOUN
ap-924	9	11	obtained	obtain	VERB
ap-924	9	12	can	can	AUX
ap-924	9	13	be	be	AUX
ap-924	9	14	generalized	generalize	VERB
ap-924	9	15	to	to	ADP
ap-924	9	16	other	other	ADJ
ap-924	9	17	irrationals	irrational	NOUN
ap-924	9	18	than	than	ADP
ap-924	9	19	�	�	PROPN
ap-924	9	20	.	.	PUNCT
ap-924	10	1	in	in	ADP
ap-924	10	2	our	our	PRON
ap-924	10	3	paper	paper	NOUN
ap-924	10	4	we	we	PRON
ap-924	10	5	answer	answer	VERB
ap-924	10	6	this	this	DET
ap-924	10	7	question	question	NOUN
ap-924	10	8	in	in	ADP
ap-924	10	9	the	the	DET
ap-924	10	10	affirmative	affirmative	NOUN
ap-924	10	11	.	.	PUNCT
ap-924	11	1	we	we	PRON
ap-924	11	2	show	show	VERB
ap-924	11	3	that	that	SCONJ
ap-924	11	4	beatty	beatty	PROPN
ap-924	11	5	sequences	sequences	PROPN
ap-924	11	6	�	�	PROPN
ap-924	11	7	�	�	PROPN
ap-924	11	8	�	�	PROPN
ap-924	11	9	�	�	PROPN
ap-924	11	10	j	j	PROPN
ap-924	11	11	j	j	PROPN
ap-924	11	12	�	�	PROPN
ap-924	11	13	�	�	PROPN
ap-924	11	14	�	�	PROPN
ap-924	11	15	for	for	ADP
ap-924	11	16	quadratic	quadratic	ADJ
ap-924	11	17	pisot	pisot	ADJ
ap-924	11	18	units	unit	NOUN
ap-924	11	19	�	�	PROPN
ap-924	11	20	have	have	VERB
ap-924	11	21	a	a	DET
ap-924	11	22	similar	similar	ADJ
ap-924	11	23	self	self	NOUN
ap-924	11	24	-	-	PUNCT
ap-924	11	25	matching	match	VERB
ap-924	11	26	property	property	NOUN
ap-924	11	27	,	,	PUNCT
ap-924	11	28	and	and	CCONJ
ap-924	11	29	for	for	ADP
ap-924	11	30	our	our	PRON
ap-924	11	31	proof	proof	NOUN
ap-924	11	32	we	we	PRON
ap-924	11	33	use	use	VERB
ap-924	11	34	a	a	DET
ap-924	11	35	simpler	simple	ADJ
ap-924	11	36	method	method	NOUN
ap-924	11	37	,	,	PUNCT
ap-924	11	38	based	base	VERB
ap-924	11	39	on	on	ADP
ap-924	11	40	the	the	DET
ap-924	11	41	cut	cut	NOUN
ap-924	11	42	-	-	PUNCT
ap-924	11	43	and	and	CCONJ
ap-924	11	44	-	-	PUNCT
ap-924	11	45	project	project	NOUN
ap-924	11	46	scheme	scheme	NOUN
ap-924	11	47	.	.	PUNCT
ap-924	12	1	it	it	PRON
ap-924	12	2	is	be	AUX
ap-924	12	3	interesting	interesting	ADJ
ap-924	12	4	to	to	PART
ap-924	12	5	note	note	VERB
ap-924	12	6	that	that	SCONJ
ap-924	12	7	beatty	beatty	PROPN
ap-924	12	8	sequences	sequences	PROPN
ap-924	12	9	,	,	PUNCT
ap-924	12	10	fibonacci	fibonacci	NOUN
ap-924	12	11	numbers	number	NOUN
ap-924	12	12	and	and	CCONJ
ap-924	12	13	the	the	DET
ap-924	12	14	cut	cut	NOUN
ap-924	12	15	-	-	PUNCT
ap-924	12	16	and	and	CCONJ
ap-924	12	17	-	-	PUNCT
ap-924	12	18	project	project	NOUN
ap-924	12	19	scheme	scheme	NOUN
ap-924	12	20	have	have	AUX
ap-924	12	21	attracted	attract	VERB
ap-924	12	22	the	the	DET
ap-924	12	23	attention	attention	NOUN
ap-924	12	24	of	of	ADP
ap-924	12	25	physicists	physicist	NOUN
ap-924	12	26	in	in	ADP
ap-924	12	27	recent	recent	ADJ
ap-924	12	28	years	year	NOUN
ap-924	12	29	because	because	SCONJ
ap-924	12	30	of	of	ADP
ap-924	12	31	their	their	PRON
ap-924	12	32	applications	application	NOUN
ap-924	12	33	for	for	ADP
ap-924	12	34	mathematical	mathematical	ADJ
ap-924	12	35	description	description	NOUN
ap-924	12	36	of	of	ADP
ap-924	12	37	non	non	ADJ
ap-924	12	38	-	-	ADJ
ap-924	12	39	crystallographic	crystallographic	ADJ
ap-924	12	40	solids	solid	NOUN
ap-924	12	41	with	with	ADP
ap-924	12	42	long	long	ADJ
ap-924	12	43	-	-	PUNCT
ap-924	12	44	range	range	NOUN
ap-924	12	45	order	order	NOUN
ap-924	12	46	,	,	PUNCT
ap-924	12	47	so	so	ADV
ap-924	12	48	-	-	PUNCT
ap-924	12	49	called	call	VERB
ap-924	12	50	quasicrystals	quasicrystal	NOUN
ap-924	12	51	,	,	PUNCT
ap-924	12	52	discovered	discover	VERB
ap-924	12	53	in	in	ADP
ap-924	12	54	1982	1982	NUM
ap-924	12	55	[	[	X
ap-924	12	56	13	13	NUM
ap-924	12	57	]	]	PUNCT
ap-924	12	58	.	.	PUNCT
ap-924	13	1	the	the	DET
ap-924	13	2	first	first	ADJ
ap-924	13	3	observed	observe	VERB
ap-924	13	4	quasicrystals	quasicrystal	NOUN
ap-924	13	5	revealed	reveal	VERB
ap-924	13	6	crystallographically	crystallographically	ADV
ap-924	13	7	forbidden	forbid	VERB
ap-924	13	8	rotational	rotational	ADJ
ap-924	13	9	symmetry	symmetry	NOUN
ap-924	13	10	of	of	ADP
ap-924	13	11	order	order	NOUN
ap-924	13	12	5	5	X
ap-924	13	13	.	.	PUNCT
ap-924	14	1	this	this	PRON
ap-924	14	2	necessitates	necessitate	VERB
ap-924	14	3	,	,	PUNCT
ap-924	14	4	for	for	ADP
ap-924	14	5	an	an	DET
ap-924	14	6	algebraic	algebraic	ADJ
ap-924	14	7	description	description	NOUN
ap-924	14	8	of	of	ADP
ap-924	14	9	the	the	DET
ap-924	14	10	mathematical	mathematical	ADJ
ap-924	14	11	model	model	NOUN
ap-924	14	12	of	of	ADP
ap-924	14	13	such	such	DET
ap-924	14	14	a	a	DET
ap-924	14	15	structure	structure	NOUN
ap-924	14	16	,	,	PUNCT
ap-924	14	17	the	the	DET
ap-924	14	18	use	use	NOUN
ap-924	14	19	of	of	ADP
ap-924	14	20	the	the	DET
ap-924	14	21	quadratic	quadratic	ADJ
ap-924	14	22	field	field	NOUN
ap-924	14	23	�	�	PROPN
ap-924	14	24	(	(	PUNCT
ap-924	14	25	�	�	PROPN
ap-924	14	26	)	)	PUNCT
ap-924	14	27	.	.	PUNCT
ap-924	15	1	such	such	DET
ap-924	15	2	a	a	DET
ap-924	15	3	model	model	NOUN
ap-924	15	4	is	be	AUX
ap-924	15	5	self	self	NOUN
ap-924	15	6	-	-	PUNCT
ap-924	15	7	similar	similar	ADJ
ap-924	15	8	with	with	ADP
ap-924	15	9	the	the	DET
ap-924	15	10	scaling	scale	VERB
ap-924	15	11	factor	factor	NOUN
ap-924	15	12	�	�	PROPN
ap-924	15	13	1	1	NUM
ap-924	15	14	.	.	PUNCT
ap-924	16	1	later	later	ADV
ap-924	16	2	,	,	PUNCT
ap-924	16	3	the	the	DET
ap-924	16	4	existence	existence	NOUN
ap-924	16	5	was	be	AUX
ap-924	16	6	observed	observe	VERB
ap-924	16	7	of	of	ADP
ap-924	16	8	quasicrystals	quasicrystal	NOUN
ap-924	16	9	with	with	ADP
ap-924	16	10	8	8	NUM
ap-924	16	11	and	and	CCONJ
ap-924	16	12	12	12	NUM
ap-924	16	13	-	-	ADJ
ap-924	16	14	fold	fold	ADJ
ap-924	16	15	rotational	rotational	ADJ
ap-924	16	16	symmetries	symmetry	NOUN
ap-924	16	17	,	,	PUNCT
ap-924	16	18	corresponding	correspond	VERB
ap-924	16	19	to	to	ADP
ap-924	16	20	mathematical	mathematical	ADJ
ap-924	16	21	models	model	NOUN
ap-924	16	22	with	with	ADP
ap-924	16	23	selfsimilar	selfsimilar	ADJ
ap-924	16	24	factors	factor	NOUN
ap-924	16	25	�	�	X
ap-924	16	26	�	�	PROPN
ap-924	16	27	1	1	NUM
ap-924	16	28	1	1	NUM
ap-924	16	29	2	2	NUM
ap-924	16	30	and	and	CCONJ
ap-924	16	31	�	�	PROPN
ap-924	16	32	�	�	PROPN
ap-924	16	33	1	1	NUM
ap-924	16	34	2	2	NUM
ap-924	16	35	3	3	NUM
ap-924	16	36	.	.	PUNCT
ap-924	17	1	note	note	VERB
ap-924	17	2	that	that	SCONJ
ap-924	17	3	all	all	DET
ap-924	17	4	�	�	PROPN
ap-924	17	5	,	,	PUNCT
ap-924	17	6	�	�	PROPN
ap-924	17	7	,	,	PUNCT
ap-924	17	8	and	and	CCONJ
ap-924	17	9	�	�	PROPN
ap-924	17	10	are	be	AUX
ap-924	17	11	quadratic	quadratic	ADJ
ap-924	17	12	pisot	pisot	ADJ
ap-924	17	13	units	unit	NOUN
ap-924	17	14	,	,	PUNCT
ap-924	17	15	i.e.	i.e.	X
ap-924	17	16	they	they	PRON
ap-924	17	17	belong	belong	VERB
ap-924	17	18	to	to	ADP
ap-924	17	19	the	the	DET
ap-924	17	20	class	class	NOUN
ap-924	17	21	of	of	ADP
ap-924	17	22	numbers	number	NOUN
ap-924	17	23	for	for	ADP
ap-924	17	24	which	which	PRON
ap-924	17	25	the	the	DET
ap-924	17	26	result	result	NOUN
ap-924	17	27	of	of	ADP
ap-924	17	28	bunder	bunder	NOUN
ap-924	17	29	and	and	CCONJ
ap-924	17	30	tognetti	tognetti	NOUN
ap-924	17	31	is	be	AUX
ap-924	17	32	generalized	generalize	VERB
ap-924	17	33	here	here	ADV
ap-924	17	34	.	.	PUNCT
ap-924	18	1	2	2	NUM
ap-924	18	2	quadratic	quadratic	ADJ
ap-924	18	3	pisot	pisot	ADJ
ap-924	18	4	units	unit	NOUN
ap-924	18	5	and	and	CCONJ
ap-924	18	6	the	the	DET
ap-924	18	7	cut	cut	NOUN
ap-924	18	8	-	-	PUNCT
ap-924	18	9	and	and	CCONJ
ap-924	18	10	-	-	PUNCT
ap-924	18	11	project	project	NOUN
ap-924	18	12	scheme	scheme	NOUN
ap-924	18	13	the	the	DET
ap-924	18	14	self	self	NOUN
ap-924	18	15	-	-	PUNCT
ap-924	18	16	matching	match	VERB
ap-924	18	17	properties	property	NOUN
ap-924	18	18	of	of	ADP
ap-924	18	19	the	the	DET
ap-924	18	20	beatty	beatty	PROPN
ap-924	18	21	sequence	sequence	PROPN
ap-924	18	22	�	�	PROPN
ap-924	18	23	�	�	PROPN
ap-924	18	24	�	�	PROPN
ap-924	18	25	�	�	PROPN
ap-924	18	26	j	j	PROPN
ap-924	18	27	j	j	PROPN
ap-924	18	28	�	�	PROPN
ap-924	18	29	�	�	PROPN
ap-924	18	30	�	�	PROPN
ap-924	18	31	are	be	AUX
ap-924	18	32	best	well	ADV
ap-924	18	33	displayed	display	VERB
ap-924	18	34	on	on	ADP
ap-924	18	35	the	the	DET
ap-924	18	36	graph	graph	NOUN
ap-924	18	37	of	of	ADP
ap-924	18	38	�	�	PROPN
ap-924	18	39	�	�	PROPN
ap-924	18	40	j	j	PROPN
ap-924	18	41	�	�	PROPN
ap-924	18	42	against	against	ADP
ap-924	18	43	�	�	PROPN
ap-924	18	44	�	�	PROPN
ap-924	18	45	j	j	PROPN
ap-924	18	46	�	�	PROPN
ap-924	18	47	�	�	PROPN
ap-924	18	48	�	�	PROPN
ap-924	18	49	1	1	NUM
ap-924	18	50	2	2	NUM
ap-924	18	51	3	3	NUM
ap-924	18	52	,	,	PUNCT
ap-924	18	53	,	,	PUNCT
ap-924	18	54	,	,	PUNCT
ap-924	18	55	�	�	PROPN
ap-924	18	56	.	.	PUNCT
ap-924	19	1	an	an	DET
ap-924	19	2	important	important	ADJ
ap-924	19	3	role	role	NOUN
ap-924	19	4	is	be	AUX
ap-924	19	5	played	play	VERB
ap-924	19	6	by	by	ADP
ap-924	19	7	the	the	DET
ap-924	19	8	fibonacci	fibonacci	NOUN
ap-924	19	9	numbers	number	NOUN
ap-924	19	10	,	,	PUNCT
ap-924	19	11	f	f	PROPN
ap-924	20	1	f	f	PROPN
ap-924	20	2	f	f	PROPN
ap-924	21	1	f	f	INTJ
ap-924	21	2	fk	fk	INTJ
ap-924	21	3	k	k	PROPN
ap-924	21	4	k0	k0	PROPN
ap-924	21	5	1	1	NUM
ap-924	21	6	1	1	NUM
ap-924	21	7	10	10	NUM
ap-924	21	8	1	1	NUM
ap-924	21	9	�	�	PROPN
ap-924	21	10	�	�	PROPN
ap-924	21	11	�	�	PROPN
ap-924	21	12	,	,	PUNCT
ap-924	21	13	,	,	PUNCT
ap-924	21	14	,	,	PUNCT
ap-924	21	15	for	for	ADP
ap-924	21	16	k	k	PROPN
ap-924	21	17	1	1	PROPN
ap-924	21	18	.	.	PUNCT
ap-924	22	1	the	the	DET
ap-924	22	2	result	result	NOUN
ap-924	22	3	of	of	ADP
ap-924	22	4	[	[	X
ap-924	22	5	12	12	NUM
ap-924	22	6	]	]	PUNCT
ap-924	22	7	states	state	VERB
ap-924	22	8	that	that	SCONJ
ap-924	22	9	�	�	PROPN
ap-924	22	10	�	�	PROPN
ap-924	22	11	�	�	PROPN
ap-924	22	12	�	�	PROPN
ap-924	22	13	(	(	PUNCT
ap-924	22	14	)	)	PUNCT
ap-924	22	15	j	j	PROPN
ap-924	22	16	f	f	PROPN
ap-924	22	17	j	j	PROPN
ap-924	22	18	fi	fi	PROPN
ap-924	23	1	i	i	PROPN
ap-924	23	2	�	�	PROPN
ap-924	23	3	�	�	PROPN
ap-924	23	4	�	�	PROPN
ap-924	23	5	1	1	NUM
ap-924	23	6	,	,	PUNCT
ap-924	23	7	(	(	PUNCT
ap-924	23	8	1	1	X
ap-924	23	9	)	)	PUNCT
ap-924	23	10	with	with	ADP
ap-924	23	11	the	the	DET
ap-924	23	12	exception	exception	NOUN
ap-924	23	13	of	of	ADP
ap-924	23	14	isolated	isolated	ADJ
ap-924	23	15	mismatches	mismatch	NOUN
ap-924	23	16	of	of	ADP
ap-924	23	17	frequency	frequency	NOUN
ap-924	23	18	�	�	PROPN
ap-924	23	19	i	i	PRON
ap-924	23	20	,	,	PUNCT
ap-924	23	21	namely	namely	ADV
ap-924	23	22	at	at	ADP
ap-924	23	23	points	point	NOUN
ap-924	23	24	of	of	ADP
ap-924	23	25	the	the	DET
ap-924	23	26	form	form	NOUN
ap-924	23	27	�	�	PROPN
ap-924	23	28	�	�	PROPN
ap-924	23	29	j	j	PROPN
ap-924	23	30	kf	kf	PROPN
ap-924	23	31	k	k	PROPN
ap-924	23	32	fi	fi	PROPN
ap-924	23	33	i	i	PRON
ap-924	23	34	�	�	PROPN
ap-924	23	35	1	1	NUM
ap-924	23	36	�	�	PROPN
ap-924	23	37	,	,	PUNCT
ap-924	23	38	k	k	PROPN
ap-924	23	39	�	�	PROPN
ap-924	23	40	�	�	PROPN
ap-924	23	41	.	.	PUNCT
ap-924	24	1	our	our	PRON
ap-924	24	2	aim	aim	NOUN
ap-924	24	3	is	be	AUX
ap-924	24	4	to	to	PART
ap-924	24	5	show	show	VERB
ap-924	24	6	a	a	DET
ap-924	24	7	very	very	ADV
ap-924	24	8	simple	simple	ADJ
ap-924	24	9	proof	proof	NOUN
ap-924	24	10	of	of	ADP
ap-924	24	11	these	these	DET
ap-924	24	12	results	result	NOUN
ap-924	24	13	that	that	PRON
ap-924	24	14	is	be	AUX
ap-924	24	15	valid	valid	ADJ
ap-924	24	16	for	for	ADP
ap-924	24	17	all	all	DET
ap-924	24	18	quadratic	quadratic	ADJ
ap-924	24	19	units	unit	NOUN
ap-924	24	20	�	�	PROPN
ap-924	24	21	�	�	PROPN
ap-924	24	22	(	(	PUNCT
ap-924	24	23	,	,	PUNCT
ap-924	24	24	)	)	PUNCT
ap-924	24	25	0	0	NUM
ap-924	24	26	1	1	NUM
ap-924	24	27	.	.	PUNCT
ap-924	25	1	every	every	DET
ap-924	25	2	such	such	ADJ
ap-924	25	3	unit	unit	NOUN
ap-924	25	4	is	be	AUX
ap-924	25	5	a	a	DET
ap-924	25	6	solution	solution	NOUN
ap-924	25	7	of	of	ADP
ap-924	25	8	the	the	DET
ap-924	25	9	quadratic	quadratic	ADJ
ap-924	25	10	equation	equation	NOUN
ap-924	25	11	x	x	PUNCT
ap-924	25	12	mx	mx	PROPN
ap-924	25	13	m2	m2	PROPN
ap-924	25	14	1	1	NUM
ap-924	25	15	�	�	PROPN
ap-924	25	16	�	�	PROPN
ap-924	25	17	,	,	PUNCT
ap-924	25	18	�	�	PROPN
ap-924	25	19	,	,	PUNCT
ap-924	25	20	or	or	CCONJ
ap-924	25	21	x	x	PROPN
ap-924	25	22	mx	mx	PROPN
ap-924	25	23	m2	m2	PROPN
ap-924	25	24	1	1	NUM
ap-924	25	25	�	�	PROPN
ap-924	25	26	�	�	PROPN
ap-924	25	27	,	,	PUNCT
ap-924	25	28	�	�	PROPN
ap-924	25	29	,	,	PUNCT
ap-924	25	30	m	m	VERB
ap-924	25	31	3	3	NUM
ap-924	25	32	.	.	PUNCT
ap-924	26	1	the	the	DET
ap-924	26	2	considerations	consideration	NOUN
ap-924	26	3	will	will	AUX
ap-924	26	4	differ	differ	VERB
ap-924	26	5	slightly	slightly	ADV
ap-924	26	6	in	in	ADP
ap-924	26	7	the	the	DET
ap-924	26	8	two	two	NUM
ap-924	26	9	cases	case	NOUN
ap-924	26	10	.	.	PUNCT
ap-924	27	1	a	a	PRON
ap-924	27	2	)	)	PUNCT
ap-924	27	3	let	let	VERB
ap-924	27	4	�	�	PROPN
ap-924	27	5	�	�	PROPN
ap-924	27	6	(	(	PUNCT
ap-924	27	7	,	,	PUNCT
ap-924	27	8	)	)	PUNCT
ap-924	27	9	0	0	NUM
ap-924	28	1	1	1	NUM
ap-924	28	2	satisfy	satisfy	PROPN
ap-924	28	3	�	�	PROPN
ap-924	28	4	�	�	PROPN
ap-924	28	5	2	2	NUM
ap-924	28	6	1	1	NUM
ap-924	28	7	�	�	PROPN
ap-924	28	8	m	m	PROPN
ap-924	28	9	for	for	ADP
ap-924	28	10	m	m	PROPN
ap-924	28	11	�	�	PROPN
ap-924	28	12	�	�	PROPN
ap-924	28	13	.	.	PUNCT
ap-924	29	1	the	the	DET
ap-924	29	2	algebraic	algebraic	ADJ
ap-924	29	3	conjugate	conjugate	ADJ
ap-924	29	4	�	�	PROPN
ap-924	29	5	�	�	PROPN
ap-924	29	6	of	of	ADP
ap-924	29	7	�	�	PROPN
ap-924	29	8	,	,	PUNCT
ap-924	29	9	i.e.	i.e.	X
ap-924	29	10	the	the	DET
ap-924	29	11	other	other	ADJ
ap-924	29	12	root	root	PROPN
ap-924	29	13	�	�	PROPN
ap-924	29	14	�	�	PROPN
ap-924	29	15	of	of	ADP
ap-924	29	16	the	the	DET
ap-924	29	17	equation	equation	NOUN
ap-924	29	18	,	,	PUNCT
ap-924	29	19	satisfies	satisfy	VERB
ap-924	29	20	�	�	PROPN
ap-924	29	21	�	�	PROPN
ap-924	29	22	�	�	PROPN
ap-924	29	23	�	�	PROPN
ap-924	29	24	1	1	NUM
ap-924	29	25	.	.	PUNCT
ap-924	30	1	we	we	PRON
ap-924	30	2	define	define	VERB
ap-924	30	3	the	the	DET
ap-924	30	4	generalized	generalized	ADJ
ap-924	30	5	fibonacci	fibonacci	NOUN
ap-924	30	6	sequence	sequence	NOUN
ap-924	30	7	g	g	PROPN
ap-924	30	8	g	g	PROPN
ap-924	30	9	g	g	PROPN
ap-924	30	10	mg	mg	PROPN
ap-924	30	11	g	g	PROPN
ap-924	30	12	nn	nn	PROPN
ap-924	30	13	n	n	PROPN
ap-924	30	14	n0	n0	X
ap-924	30	15	1	1	NUM
ap-924	30	16	2	2	NUM
ap-924	30	17	10	10	NUM
ap-924	30	18	1	1	NUM
ap-924	30	19	0	0	NUM
ap-924	30	20	�	�	PROPN
ap-924	30	21	�	�	PROPN
ap-924	30	22	�	�	PROPN
ap-924	30	23	,	,	PUNCT
ap-924	30	24	,	,	PUNCT
ap-924	30	25	,	,	PUNCT
ap-924	30	26	(	(	PUNCT
ap-924	30	27	2	2	X
ap-924	30	28	)	)	PUNCT
ap-924	30	29	it	it	PRON
ap-924	30	30	is	be	AUX
ap-924	30	31	easy	easy	ADJ
ap-924	30	32	to	to	PART
ap-924	30	33	show	show	VERB
ap-924	30	34	by	by	ADP
ap-924	30	35	induction	induction	NOUN
ap-924	30	36	that	that	PRON
ap-924	30	37	for	for	ADP
ap-924	30	38	i	i	PROPN
ap-924	30	39	�	�	PROPN
ap-924	30	40	�	�	PROPN
ap-924	30	41	,	,	PUNCT
ap-924	30	42	we	we	PRON
ap-924	30	43	have	have	VERB
ap-924	30	44	(	(	PUNCT
ap-924	30	45	)	)	PUNCT
ap-924	30	46	�	�	PROPN
ap-924	30	47	1	1	NUM
ap-924	30	48	1	1	NUM
ap-924	30	49	1	1	NUM
ap-924	31	1	i	i	PRON
ap-924	31	2	i	i	PRON
ap-924	31	3	i	i	PRON
ap-924	31	4	ig	ig	VERB
ap-924	31	5	g	g	PROPN
ap-924	31	6	�	�	PROPN
ap-924	31	7	�	�	PROPN
ap-924	31	8	and	and	CCONJ
ap-924	31	9	(	(	PUNCT
ap-924	31	10	)	)	PUNCT
ap-924	31	11	�	�	PROPN
ap-924	31	12	�	�	PROPN
ap-924	31	13	�	�	PROPN
ap-924	31	14	1	1	NUM
ap-924	31	15	1	1	NUM
ap-924	31	16	1	1	NUM
ap-924	31	17	i	i	PRON
ap-924	32	1	i	i	PRON
ap-924	32	2	i	i	PRON
ap-924	32	3	ig	ig	VERB
ap-924	32	4	g	g	PROPN
ap-924	32	5	�	�	PROPN
ap-924	32	6	�	�	PROPN
ap-924	32	7	.	.	PUNCT
ap-924	33	1	(	(	PUNCT
ap-924	33	2	3	3	X
ap-924	33	3	)	)	PUNCT
ap-924	33	4	b	b	NOUN
ap-924	33	5	)	)	PUNCT
ap-924	33	6	let	let	VERB
ap-924	33	7	�	�	PROPN
ap-924	33	8	�	�	PROPN
ap-924	33	9	(	(	PUNCT
ap-924	33	10	,	,	PUNCT
ap-924	33	11	)	)	PUNCT
ap-924	33	12	0	0	NUM
ap-924	33	13	1	1	NUM
ap-924	33	14	satisfy	satisfy	PROPN
ap-924	33	15	�	�	PROPN
ap-924	33	16	�	�	PROPN
ap-924	33	17	2	2	NUM
ap-924	33	18	1	1	NUM
ap-924	33	19	�	�	PROPN
ap-924	33	20	m	m	PROPN
ap-924	33	21	for	for	ADP
ap-924	33	22	m	m	PROPN
ap-924	33	23	�	�	PROPN
ap-924	33	24	�	�	PROPN
ap-924	33	25	,	,	PUNCT
ap-924	33	26	m	m	VERB
ap-924	33	27	3	3	NUM
ap-924	33	28	.	.	PUNCT
ap-924	34	1	the	the	DET
ap-924	34	2	algebraic	algebraic	ADJ
ap-924	34	3	conjugate	conjugate	ADJ
ap-924	34	4	�	�	PROPN
ap-924	34	5	�	�	PROPN
ap-924	34	6	of	of	ADP
ap-924	34	7	�	�	PROPN
ap-924	34	8	satisfies	satisfy	VERB
ap-924	34	9	�	�	PROPN
ap-924	34	10	�	�	PROPN
ap-924	34	11	�	�	PROPN
ap-924	34	12	1	1	NUM
ap-924	34	13	.	.	PUNCT
ap-924	35	1	we	we	PRON
ap-924	35	2	define	define	VERB
ap-924	35	3	g	g	PROPN
ap-924	35	4	g	g	PROPN
ap-924	35	5	g	g	PROPN
ap-924	35	6	mg	mg	PROPN
ap-924	35	7	g	g	PROPN
ap-924	35	8	nn	nn	PROPN
ap-924	35	9	n	n	PROPN
ap-924	35	10	n0	n0	X
ap-924	35	11	1	1	NUM
ap-924	35	12	2	2	NUM
ap-924	35	13	10	10	NUM
ap-924	35	14	1	1	NUM
ap-924	35	15	0	0	NUM
ap-924	35	16	�	�	PROPN
ap-924	35	17	�	�	PROPN
ap-924	35	18	�	�	PROPN
ap-924	35	19	,	,	PUNCT
ap-924	35	20	,	,	PUNCT
ap-924	35	21	,	,	PUNCT
ap-924	35	22	(	(	PUNCT
ap-924	35	23	4	4	X
ap-924	35	24	)	)	PUNCT
ap-924	35	25	in	in	ADP
ap-924	35	26	this	this	DET
ap-924	35	27	case	case	NOUN
ap-924	35	28	,	,	PUNCT
ap-924	35	29	we	we	PRON
ap-924	35	30	have	have	VERB
ap-924	35	31	for	for	ADP
ap-924	35	32	i	i	PROPN
ap-924	35	33	�	�	PROPN
ap-924	35	34	�	�	PROPN
ap-924	35	35	©	©	PROPN
ap-924	35	36	czech	czech	PROPN
ap-924	35	37	technical	technical	PROPN
ap-924	35	38	university	university	PROPN
ap-924	35	39	publishing	publishing	NOUN
ap-924	35	40	house	house	NOUN
ap-924	35	41	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	PROPN
ap-924	35	42	21	21	NUM
ap-924	35	43	acta	acta	PROPN
ap-924	35	44	polytechnica	polytechnica	PROPN
ap-924	35	45	vol	vol	NOUN
ap-924	35	46	.	.	PUNCT
ap-924	36	1	47	47	NUM
ap-924	36	2	no	no	NOUN
ap-924	36	3	.	.	PUNCT
ap-924	37	1	2–3/2007	2–3/2007	NUM
ap-924	37	2	self	self	NOUN
ap-924	37	3	-	-	PUNCT
ap-924	37	4	matching	match	VERB
ap-924	37	5	properties	property	NOUN
ap-924	37	6	of	of	ADP
ap-924	37	7	beatty	beatty	PROPN
ap-924	37	8	sequences	sequences	PROPN
ap-924	37	9	z.	z.	PROPN
ap-924	37	10	masáková	masáková	PROPN
ap-924	37	11	,	,	PUNCT
ap-924	37	12	e.	e.	PROPN
ap-924	37	13	pelantová	pelantová	PROPN
ap-924	37	14	we	we	PRON
ap-924	37	15	study	study	VERB
ap-924	37	16	the	the	DET
ap-924	37	17	selfmatching	selfmatche	VERB
ap-924	37	18	properties	property	NOUN
ap-924	37	19	of	of	ADP
ap-924	37	20	beatty	beatty	PROPN
ap-924	37	21	sequences	sequences	PROPN
ap-924	37	22	,	,	PUNCT
ap-924	37	23	in	in	ADP
ap-924	37	24	particular	particular	ADJ
ap-924	37	25	of	of	ADP
ap-924	37	26	the	the	DET
ap-924	37	27	graph	graph	NOUN
ap-924	37	28	of	of	ADP
ap-924	37	29	the	the	DET
ap-924	37	30	function	function	NOUN
ap-924	37	31	�	�	PROPN
ap-924	37	32	�	�	PROPN
ap-924	37	33	j	j	PROPN
ap-924	37	34	�	�	PROPN
ap-924	37	35	against	against	ADP
ap-924	37	36	j	j	PROPN
ap-924	37	37	for	for	ADP
ap-924	37	38	every	every	DET
ap-924	37	39	quadratic	quadratic	ADJ
ap-924	37	40	unit	unit	NOUN
ap-924	37	41	�	�	PROPN
ap-924	37	42	�	�	PROPN
ap-924	37	43	(	(	PUNCT
ap-924	37	44	,	,	PUNCT
ap-924	37	45	)	)	PUNCT
ap-924	37	46	0	0	NUM
ap-924	38	1	1	1	NUM
ap-924	38	2	.	.	PUNCT
ap-924	39	1	we	we	PRON
ap-924	39	2	show	show	VERB
ap-924	39	3	that	that	SCONJ
ap-924	39	4	translation	translation	NOUN
ap-924	39	5	in	in	ADP
ap-924	39	6	the	the	DET
ap-924	39	7	argument	argument	NOUN
ap-924	39	8	by	by	ADP
ap-924	39	9	an	an	DET
ap-924	39	10	element	element	NOUN
ap-924	39	11	gi	gi	NOUN
ap-924	39	12	of	of	ADP
ap-924	39	13	a	a	DET
ap-924	39	14	generalized	generalized	ADJ
ap-924	39	15	fibonacci	fibonacci	NOUN
ap-924	39	16	sequence	sequence	NOUN
ap-924	39	17	almost	almost	ADV
ap-924	39	18	always	always	ADV
ap-924	39	19	causes	cause	VERB
ap-924	39	20	the	the	DET
ap-924	39	21	translation	translation	NOUN
ap-924	39	22	of	of	ADP
ap-924	39	23	the	the	DET
ap-924	39	24	value	value	NOUN
ap-924	39	25	of	of	ADP
ap-924	39	26	the	the	DET
ap-924	39	27	function	function	NOUN
ap-924	39	28	by	by	ADP
ap-924	39	29	gi	gi	NOUN
ap-924	39	30	�	�	PROPN
ap-924	39	31	1	1	NUM
ap-924	39	32	.	.	PUNCT
ap-924	40	1	more	more	ADV
ap-924	40	2	precisely	precisely	ADV
ap-924	40	3	,	,	PUNCT
ap-924	40	4	for	for	ADP
ap-924	40	5	fixed	fix	VERB
ap-924	40	6	i	i	PROPN
ap-924	40	7	�	�	PROPN
ap-924	40	8	�	�	PROPN
ap-924	40	9	,	,	PUNCT
ap-924	40	10	we	we	PRON
ap-924	40	11	have	have	VERB
ap-924	40	12	�	�	PROPN
ap-924	40	13	�	�	PROPN
ap-924	40	14	�	�	PROPN
ap-924	40	15	�	�	PROPN
ap-924	40	16	�	�	PROPN
ap-924	40	17	�	�	PROPN
ap-924	40	18	(	(	PUNCT
ap-924	40	19	)	)	PUNCT
ap-924	41	1	j	j	PROPN
ap-924	41	2	g	g	PROPN
ap-924	41	3	j	j	PROPN
ap-924	42	1	gi	gi	INTJ
ap-924	42	2	i	i	PRON
ap-924	42	3	�	�	PROPN
ap-924	42	4	1	1	NUM
ap-924	42	5	,	,	PUNCT
ap-924	42	6	where	where	SCONJ
ap-924	42	7	j	j	PROPN
ap-924	42	8	ui	ui	PROPN
ap-924	42	9	�	�	PROPN
ap-924	42	10	.	.	PUNCT
ap-924	43	1	we	we	PRON
ap-924	43	2	determine	determine	VERB
ap-924	43	3	the	the	DET
ap-924	43	4	set	set	ADJ
ap-924	43	5	ui	ui	NOUN
ap-924	43	6	of	of	ADP
ap-924	43	7	mismatches	mismatch	NOUN
ap-924	43	8	and	and	CCONJ
ap-924	43	9	show	show	VERB
ap-924	43	10	that	that	SCONJ
ap-924	43	11	it	it	PRON
ap-924	43	12	has	have	VERB
ap-924	43	13	a	a	DET
ap-924	43	14	low	low	ADJ
ap-924	43	15	frequency	frequency	NOUN
ap-924	43	16	,	,	PUNCT
ap-924	43	17	namely	namely	ADV
ap-924	43	18	�	�	NOUN
ap-924	43	19	i.	i.	NOUN
ap-924	43	20	keywords	keyword	NOUN
ap-924	43	21	:	:	PUNCT
ap-924	43	22	beatty	beatty	PROPN
ap-924	43	23	sequences	sequences	PROPN
ap-924	43	24	,	,	PUNCT
ap-924	43	25	fibonacci	fibonacci	NOUN
ap-924	43	26	numbers	number	NOUN
ap-924	43	27	,	,	PUNCT
ap-924	43	28	cut	cut	NOUN
ap-924	43	29	-	-	PUNCT
ap-924	43	30	and	and	CCONJ
ap-924	43	31	-	-	PUNCT
ap-924	43	32	project	project	NOUN
ap-924	43	33	scheme	scheme	NOUN
ap-924	43	34	.	.	PUNCT
ap-924	44	1	�	�	PROPN
ap-924	44	2	�	�	PROPN
ap-924	44	3	i	i	PRON
ap-924	44	4	i	i	PRON
ap-924	44	5	ig	ig	VERB
ap-924	44	6	g	g	PROPN
ap-924	44	7	�	�	PROPN
ap-924	44	8	1	1	NUM
ap-924	44	9	and	and	CCONJ
ap-924	44	10	�	�	PROPN
ap-924	44	11	�	�	PROPN
ap-924	44	12	�	�	PROPN
ap-924	44	13	�	�	PROPN
ap-924	44	14	�	�	PROPN
ap-924	44	15	i	i	PRON
ap-924	45	1	i	i	PRON
ap-924	45	2	ig	ig	VERB
ap-924	45	3	g	g	NOUN
ap-924	45	4	1	1	NUM
ap-924	45	5	(	(	PUNCT
ap-924	45	6	5	5	NUM
ap-924	45	7	)	)	PUNCT
ap-924	45	8	the	the	DET
ap-924	45	9	proof	proof	NOUN
ap-924	45	10	we	we	PRON
ap-924	45	11	give	give	VERB
ap-924	45	12	here	here	ADV
ap-924	45	13	is	be	AUX
ap-924	45	14	based	base	VERB
ap-924	45	15	on	on	ADP
ap-924	45	16	the	the	DET
ap-924	45	17	algebraic	algebraic	ADJ
ap-924	45	18	expression	expression	NOUN
ap-924	45	19	for	for	ADP
ap-924	45	20	one	one	NUM
ap-924	45	21	-	-	PUNCT
ap-924	45	22	dimensional	dimensional	ADJ
ap-924	45	23	cut	cut	NOUN
ap-924	45	24	-	-	PUNCT
ap-924	45	25	and	and	CCONJ
ap-924	45	26	-	-	PUNCT
ap-924	45	27	project	project	NOUN
ap-924	45	28	sets	set	NOUN
ap-924	45	29	[	[	X
ap-924	45	30	14	14	NUM
ap-924	45	31	]	]	PUNCT
ap-924	45	32	.	.	PUNCT
ap-924	46	1	let	let	VERB
ap-924	46	2	v1	v1	NOUN
ap-924	46	3	,	,	PUNCT
ap-924	46	4	v2	v2	PROPN
ap-924	46	5	be	be	VERB
ap-924	46	6	straight	straight	ADJ
ap-924	46	7	lines	line	NOUN
ap-924	46	8	in	in	ADP
ap-924	46	9	�	�	NOUN
ap-924	46	10	2	2	NUM
ap-924	46	11	determined	determine	VERB
ap-924	46	12	by	by	ADP
ap-924	46	13	vectors	vector	NOUN
ap-924	46	14	(	(	PUNCT
ap-924	46	15	,	,	PUNCT
ap-924	46	16	)	)	PUNCT
ap-924	46	17	�	�	PROPN
ap-924	46	18	1	1	NUM
ap-924	46	19	and	and	CCONJ
ap-924	46	20	(	(	PUNCT
ap-924	46	21	,	,	PUNCT
ap-924	46	22	)	)	PUNCT
ap-924	46	23	�	�	PROPN
ap-924	46	24	�	�	PROPN
ap-924	46	25	1	1	NUM
ap-924	46	26	,	,	PUNCT
ap-924	46	27	respectively	respectively	ADV
ap-924	46	28	.	.	PUNCT
ap-924	47	1	the	the	DET
ap-924	47	2	projection	projection	NOUN
ap-924	47	3	of	of	ADP
ap-924	47	4	the	the	DET
ap-924	47	5	square	square	PROPN
ap-924	47	6	lattice	lattice	PROPN
ap-924	47	7	�	�	PROPN
ap-924	47	8	2	2	NUM
ap-924	47	9	on	on	ADP
ap-924	47	10	the	the	DET
ap-924	47	11	line	line	NOUN
ap-924	47	12	v1	v1	NOUN
ap-924	47	13	along	along	ADP
ap-924	47	14	the	the	DET
ap-924	47	15	direction	direction	NOUN
ap-924	47	16	of	of	ADP
ap-924	47	17	v2	v2	PROPN
ap-924	47	18	is	be	AUX
ap-924	47	19	given	give	VERB
ap-924	47	20	by	by	ADP
ap-924	47	21	(	(	PUNCT
ap-924	47	22	,	,	PUNCT
ap-924	47	23	)	)	PUNCT
ap-924	47	24	(	(	PUNCT
ap-924	47	25	)	)	PUNCT
ap-924	47	26	(	(	PUNCT
ap-924	47	27	)	)	PUNCT
ap-924	47	28	a	a	DET
ap-924	47	29	b	b	NOUN
ap-924	47	30	a	a	PRON
ap-924	47	31	b	b	NOUN
ap-924	47	32	x	x	SYM
ap-924	47	33	a	a	DET
ap-924	47	34	b	b	X
ap-924	47	35	x	x	SYM
ap-924	47	36	�	�	PROPN
ap-924	47	37	�	�	PROPN
ap-924	47	38	�	�	PROPN
ap-924	47	39	�	�	PROPN
ap-924	47	40	�	�	PROPN
ap-924	47	41	�	�	PROPN
ap-924	47	42	1	1	NUM
ap-924	47	43	2	2	NUM
ap-924	47	44	,	,	PUNCT
ap-924	47	45	for	for	ADP
ap-924	47	46	(	(	PUNCT
ap-924	47	47	,	,	PUNCT
ap-924	47	48	)	)	PUNCT
ap-924	47	49	a	a	DET
ap-924	47	50	b	b	PROPN
ap-924	47	51	�	�	PROPN
ap-924	47	52	�	�	PROPN
ap-924	47	53	2	2	NUM
ap-924	47	54	,	,	PUNCT
ap-924	47	55	where	where	SCONJ
ap-924	47	56	�	�	PROPN
ap-924	47	57	x1	x1	ADJ
ap-924	47	58	1	1	NUM
ap-924	47	59	1	1	NUM
ap-924	47	60	�	�	PROPN
ap-924	47	61	�	�	PROPN
ap-924	47	62	�	�	PROPN
ap-924	47	63	�	�	PROPN
ap-924	47	64	�	�	PROPN
ap-924	47	65	�	�	PROPN
ap-924	47	66	(	(	PUNCT
ap-924	47	67	and	and	CCONJ
ap-924	47	68	�	�	PROPN
ap-924	47	69	x2	x2	NOUN
ap-924	47	70	1	1	NUM
ap-924	47	71	1	1	NUM
ap-924	47	72	�	�	PROPN
ap-924	47	73	�	�	PROPN
ap-924	47	74	�	�	PROPN
ap-924	47	75	�	�	PROPN
ap-924	47	76	�	�	PROPN
ap-924	47	77	�	�	PROPN
ap-924	47	78	�	�	PROPN
ap-924	47	79	(	(	PUNCT
ap-924	47	80	.	.	PUNCT
ap-924	48	1	for	for	ADP
ap-924	48	2	the	the	DET
ap-924	48	3	description	description	NOUN
ap-924	48	4	of	of	ADP
ap-924	48	5	the	the	DET
ap-924	48	6	projection	projection	NOUN
ap-924	48	7	of	of	ADP
ap-924	48	8	�	�	PROPN
ap-924	48	9	2	2	NUM
ap-924	48	10	on	on	ADP
ap-924	48	11	v1	v1	NOUN
ap-924	48	12	it	it	PRON
ap-924	48	13	suffices	suffice	VERB
ap-924	48	14	to	to	PART
ap-924	48	15	consider	consider	VERB
ap-924	48	16	the	the	DET
ap-924	48	17	set	set	VERB
ap-924	48	18	�	�	PROPN
ap-924	48	19	�	�	PROPN
ap-924	48	20	�	�	PROPN
ap-924	48	21	�	�	PROPN
ap-924	48	22	�	�	PROPN
ap-924	48	23	�	�	PROPN
ap-924	48	24	�	�	PROPN
ap-924	48	25	�	�	PROPN
ap-924	48	26	�	�	PROPN
ap-924	48	27	�	�	PROPN
ap-924	48	28	�	�	PROPN
ap-924	48	29	�	�	PROPN
ap-924	48	30	:	:	PUNCT
ap-924	48	31	,	,	PUNCT
ap-924	48	32	a	a	DET
ap-924	48	33	b	b	NOUN
ap-924	48	34	a	a	DET
ap-924	48	35	b	b	NOUN
ap-924	48	36	the	the	DET
ap-924	48	37	integral	integral	ADJ
ap-924	48	38	basis	basis	NOUN
ap-924	48	39	of	of	ADP
ap-924	48	40	this	this	DET
ap-924	48	41	free	free	ADJ
ap-924	48	42	abelian	abelian	PROPN
ap-924	48	43	group	group	NOUN
ap-924	48	44	is	be	AUX
ap-924	48	45	(	(	PUNCT
ap-924	48	46	,	,	PUNCT
ap-924	48	47	)	)	PUNCT
ap-924	48	48	1	1	NUM
ap-924	48	49	�	�	PROPN
ap-924	48	50	�	�	PROPN
ap-924	48	51	,	,	PUNCT
ap-924	48	52	and	and	CCONJ
ap-924	48	53	thus	thus	ADV
ap-924	48	54	every	every	DET
ap-924	48	55	element	element	NOUN
ap-924	48	56	x	x	PROPN
ap-924	48	57	of	of	ADP
ap-924	48	58	�	�	PROPN
ap-924	48	59	�	�	PROPN
ap-924	48	60	�	�	PROPN
ap-924	48	61	�	�	PROPN
ap-924	48	62	�	�	PROPN
ap-924	48	63	has	have	VERB
ap-924	48	64	a	a	DET
ap-924	48	65	unique	unique	ADJ
ap-924	48	66	expression	expression	NOUN
ap-924	48	67	in	in	ADP
ap-924	48	68	this	this	DET
ap-924	48	69	base	base	NOUN
ap-924	48	70	.	.	PUNCT
ap-924	49	1	we	we	PRON
ap-924	49	2	will	will	AUX
ap-924	49	3	say	say	VERB
ap-924	49	4	that	that	SCONJ
ap-924	49	5	a	a	PRON
ap-924	49	6	is	be	AUX
ap-924	49	7	the	the	DET
ap-924	49	8	rational	rational	ADJ
ap-924	49	9	part	part	NOUN
ap-924	49	10	of	of	ADP
ap-924	49	11	x	x	DET
ap-924	49	12	a	a	DET
ap-924	49	13	b	b	PROPN
ap-924	49	14	�	�	PROPN
ap-924	49	15	�	�	PROPN
ap-924	49	16	�	�	PROPN
ap-924	49	17	and	and	CCONJ
ap-924	49	18	b	b	PROPN
ap-924	49	19	is	be	AUX
ap-924	49	20	its	its	PRON
ap-924	49	21	irrational	irrational	ADJ
ap-924	49	22	part	part	NOUN
ap-924	49	23	.	.	PUNCT
ap-924	50	1	since	since	SCONJ
ap-924	50	2	�	�	PROPN
ap-924	50	3	�	�	PROPN
ap-924	50	4	is	be	AUX
ap-924	50	5	a	a	DET
ap-924	50	6	quadratic	quadratic	ADJ
ap-924	50	7	unit	unit	NOUN
ap-924	50	8	,	,	PUNCT
ap-924	50	9	�	�	PROPN
ap-924	50	10	�	�	PROPN
ap-924	50	11	�	�	PROPN
ap-924	50	12	�	�	PROPN
ap-924	50	13	�	�	PROPN
ap-924	50	14	is	be	AUX
ap-924	50	15	a	a	DET
ap-924	50	16	ring	ring	NOUN
ap-924	50	17	and	and	CCONJ
ap-924	50	18	,	,	PUNCT
ap-924	50	19	moreover	moreover	ADV
ap-924	50	20	,	,	PUNCT
ap-924	50	21	it	it	PRON
ap-924	50	22	satisfies	satisfy	VERB
ap-924	50	23	�	�	PROPN
ap-924	50	24	�	�	PROPN
ap-924	50	25	�	�	PROPN
ap-924	50	26	�	�	PROPN
ap-924	50	27	�	�	PROPN
ap-924	50	28	�	�	PROPN
ap-924	50	29	�	�	PROPN
ap-924	50	30	�	�	PROPN
ap-924	50	31	�	�	PROPN
ap-924	50	32	�	�	PROPN
ap-924	50	33	�	�	PROPN
ap-924	50	34	�	�	PROPN
ap-924	50	35	�	�	PROPN
ap-924	50	36	(	(	PUNCT
ap-924	50	37	6	6	NUM
ap-924	50	38	)	)	PUNCT
ap-924	50	39	a	a	DET
ap-924	50	40	cut	cut	NOUN
ap-924	50	41	-	-	PUNCT
ap-924	50	42	and	and	CCONJ
ap-924	50	43	-	-	PUNCT
ap-924	50	44	project	project	NOUN
ap-924	50	45	set	set	NOUN
ap-924	50	46	is	be	AUX
ap-924	50	47	the	the	DET
ap-924	50	48	set	set	NOUN
ap-924	50	49	of	of	ADP
ap-924	50	50	projections	projection	NOUN
ap-924	50	51	of	of	ADP
ap-924	50	52	points	point	NOUN
ap-924	50	53	of	of	ADP
ap-924	50	54	�	�	NOUN
ap-924	50	55	2	2	NUM
ap-924	50	56	to	to	PART
ap-924	50	57	v1	v1	VERB
ap-924	50	58	,	,	PUNCT
ap-924	50	59	that	that	PRON
ap-924	50	60	are	be	AUX
ap-924	50	61	found	find	VERB
ap-924	50	62	in	in	ADP
ap-924	50	63	a	a	DET
ap-924	50	64	strip	strip	NOUN
ap-924	50	65	of	of	ADP
ap-924	50	66	given	give	VERB
ap-924	50	67	bounded	bounded	ADJ
ap-924	50	68	width	width	NOUN
ap-924	50	69	,	,	PUNCT
ap-924	50	70	parallel	parallel	ADJ
ap-924	50	71	to	to	ADP
ap-924	50	72	the	the	DET
ap-924	50	73	straight	straight	ADJ
ap-924	50	74	line	line	NOUN
ap-924	50	75	v1	v1	NOUN
ap-924	50	76	.	.	PUNCT
ap-924	51	1	formally	formally	ADV
ap-924	51	2	,	,	PUNCT
ap-924	51	3	for	for	ADP
ap-924	51	4	a	a	DET
ap-924	51	5	bounded	bound	VERB
ap-924	51	6	interval	interval	NOUN
ap-924	51	7	�	�	PROPN
ap-924	51	8	we	we	PRON
ap-924	51	9	define	define	VERB
ap-924	51	10	�	�	PROPN
ap-924	51	11	�	�	PROPN
ap-924	51	12	�	�	PROPN
ap-924	51	13	�	�	PROPN
ap-924	51	14	(	(	PUNCT
ap-924	51	15	)	)	PUNCT
ap-924	51	16	,	,	PUNCT
ap-924	51	17	,	,	PUNCT
ap-924	51	18	�	�	PROPN
ap-924	51	19	�	�	PROPN
ap-924	51	20	�	�	PROPN
ap-924	51	21	�	�	PROPN
ap-924	51	22	a	a	DET
ap-924	51	23	b	b	NOUN
ap-924	51	24	a	a	DET
ap-924	51	25	b	b	NOUN
ap-924	51	26	a	a	DET
ap-924	51	27	b	b	PROPN
ap-924	51	28	�	�	PROPN
ap-924	51	29	�	�	PROPN
ap-924	51	30	�	�	PROPN
ap-924	51	31	�	�	PROPN
ap-924	51	32	note	note	VERB
ap-924	51	33	that	that	SCONJ
ap-924	51	34	a	a	DET
ap-924	51	35	b	b	PROPN
ap-924	51	36	�	�	PROPN
ap-924	51	37	�	�	PROPN
ap-924	51	38	corresponds	correspond	VERB
ap-924	51	39	to	to	ADP
ap-924	51	40	the	the	DET
ap-924	51	41	projection	projection	NOUN
ap-924	51	42	of	of	ADP
ap-924	51	43	the	the	DET
ap-924	51	44	point	point	NOUN
ap-924	51	45	(	(	PUNCT
ap-924	51	46	a	a	DET
ap-924	51	47	,	,	PUNCT
ap-924	51	48	b	b	NOUN
ap-924	51	49	)	)	PUNCT
ap-924	51	50	to	to	ADP
ap-924	51	51	the	the	DET
ap-924	51	52	straight	straight	ADJ
ap-924	51	53	line	line	NOUN
ap-924	51	54	v1	v1	NOUN
ap-924	51	55	along	along	ADP
ap-924	51	56	v2	v2	PROPN
ap-924	51	57	,	,	PUNCT
ap-924	51	58	whereas	whereas	SCONJ
ap-924	51	59	a	a	DET
ap-924	51	60	b	b	PROPN
ap-924	51	61	�	�	PROPN
ap-924	51	62	corresponds	correspond	VERB
ap-924	51	63	to	to	ADP
ap-924	51	64	the	the	DET
ap-924	51	65	projection	projection	NOUN
ap-924	51	66	of	of	ADP
ap-924	51	67	the	the	DET
ap-924	51	68	same	same	ADJ
ap-924	51	69	lattice	lattice	NOUN
ap-924	51	70	point	point	NOUN
ap-924	51	71	to	to	PART
ap-924	51	72	v2	v2	VERB
ap-924	51	73	along	along	ADP
ap-924	51	74	v1	v1	PROPN
ap-924	51	75	.	.	PUNCT
ap-924	52	1	among	among	ADP
ap-924	52	2	the	the	DET
ap-924	52	3	simple	simple	ADJ
ap-924	52	4	properties	property	NOUN
ap-924	52	5	of	of	ADP
ap-924	52	6	the	the	DET
ap-924	52	7	cut	cut	NOUN
ap-924	52	8	-	-	PUNCT
ap-924	52	9	and	and	CCONJ
ap-924	52	10	-	-	PUNCT
ap-924	52	11	project	project	NOUN
ap-924	52	12	sets	set	NOUN
ap-924	52	13	that	that	PRON
ap-924	52	14	we	we	PRON
ap-924	52	15	use	use	VERB
ap-924	52	16	here	here	ADV
ap-924	52	17	are	be	AUX
ap-924	52	18	�	�	PROPN
ap-924	52	19	�	�	PROPN
ap-924	52	20	�	�	PROPN
ap-924	52	21	�	�	PROPN
ap-924	52	22	(	(	PUNCT
ap-924	52	23	)	)	PUNCT
ap-924	52	24	(	(	PUNCT
ap-924	52	25	)	)	PUNCT
ap-924	52	26	�	�	PROPN
ap-924	52	27	1	1	NUM
ap-924	52	28	1	1	NUM
ap-924	52	29	,	,	PUNCT
ap-924	52	30	�	�	PROPN
ap-924	52	31	�	�	PROPN
ap-924	52	32	�	�	PROPN
ap-924	52	33	�	�	PROPN
ap-924	52	34	�	�	PROPN
ap-924	52	35	�	�	PROPN
ap-924	52	36	�	�	PROPN
ap-924	52	37	�	�	PROPN
ap-924	52	38	(	(	PUNCT
ap-924	52	39	)	)	PUNCT
ap-924	52	40	(	(	PUNCT
ap-924	52	41	)	)	PUNCT
ap-924	52	42	,	,	PUNCT
ap-924	52	43	where	where	SCONJ
ap-924	52	44	the	the	DET
ap-924	52	45	latter	latter	ADJ
ap-924	52	46	is	be	AUX
ap-924	52	47	a	a	DET
ap-924	52	48	consequence	consequence	NOUN
ap-924	52	49	of	of	ADP
ap-924	52	50	(	(	PUNCT
ap-924	52	51	6	6	NUM
ap-924	52	52	)	)	PUNCT
ap-924	52	53	.	.	PUNCT
ap-924	53	1	if	if	SCONJ
ap-924	53	2	the	the	DET
ap-924	53	3	interval	interval	NOUN
ap-924	53	4	�	�	PROPN
ap-924	53	5	is	be	AUX
ap-924	53	6	of	of	ADP
ap-924	53	7	unit	unit	NOUN
ap-924	53	8	length	length	NOUN
ap-924	53	9	,	,	PUNCT
ap-924	53	10	one	one	PRON
ap-924	53	11	can	can	AUX
ap-924	53	12	derive	derive	VERB
ap-924	53	13	directly	directly	ADV
ap-924	53	14	from	from	ADP
ap-924	53	15	the	the	DET
ap-924	53	16	definition	definition	NOUN
ap-924	53	17	a	a	DET
ap-924	53	18	simpler	simple	ADJ
ap-924	53	19	expression	expression	NOUN
ap-924	53	20	for	for	ADP
ap-924	53	21	�	�	PROPN
ap-924	53	22	�	�	PROPN
ap-924	53	23	(	(	PUNCT
ap-924	53	24	)	)	PUNCT
ap-924	53	25	.	.	PUNCT
ap-924	54	1	in	in	ADP
ap-924	54	2	particular	particular	ADJ
ap-924	54	3	,	,	PUNCT
ap-924	54	4	we	we	PRON
ap-924	54	5	have	have	VERB
ap-924	54	6	�	�	PROPN
ap-924	54	7	�	�	PROPN
ap-924	54	8	�	�	PROPN
ap-924	54	9	�	�	PROPN
ap-924	54	10	�	�	PROPN
ap-924	54	11	�	�	PROPN
ap-924	54	12	�	�	PROPN
ap-924	54	13	[	[	PUNCT
ap-924	54	14	,	,	PUNCT
ap-924	54	15	)	)	PUNCT
ap-924	54	16	[	[	PUNCT
ap-924	54	17	,	,	PUNCT
ap-924	54	18	)	)	PUNCT
ap-924	54	19	0	0	NUM
ap-924	54	20	1	1	NUM
ap-924	54	21	0	0	NUM
ap-924	54	22	1	1	NUM
ap-924	54	23	�	�	PROPN
ap-924	54	24	�	�	PROPN
ap-924	54	25	�	�	PROPN
ap-924	54	26	�	�	PROPN
ap-924	54	27	�	�	PROPN
ap-924	54	28	�	�	PROPN
ap-924	54	29	a	a	DET
ap-924	54	30	b	b	NOUN
ap-924	54	31	a	a	DET
ap-924	54	32	b	b	PROPN
ap-924	54	33	b	b	PROPN
ap-924	54	34	b	b	PROPN
ap-924	54	35	b	b	PROPN
ap-924	54	36	�	�	PROPN
ap-924	54	37	�	�	PROPN
ap-924	54	38	�	�	PROPN
ap-924	54	39	�	�	PROPN
ap-924	54	40	�	�	PROPN
ap-924	54	41	,	,	PUNCT
ap-924	54	42	(	(	PUNCT
ap-924	54	43	7	7	X
ap-924	54	44	)	)	PUNCT
ap-924	54	45	where	where	SCONJ
ap-924	54	46	we	we	PRON
ap-924	54	47	use	use	VERB
ap-924	54	48	that	that	SCONJ
ap-924	54	49	the	the	DET
ap-924	54	50	condition0	condition0	ADJ
ap-924	54	51	1	1	NUM
ap-924	54	52	�	�	PROPN
ap-924	54	53	�	�	PROPN
ap-924	54	54	a	a	DET
ap-924	54	55	b	b	PROPN
ap-924	54	56	�	�	PROPN
ap-924	54	57	is	be	AUX
ap-924	54	58	satisfied	satisfied	ADJ
ap-924	54	59	if	if	SCONJ
ap-924	54	60	and	and	CCONJ
ap-924	54	61	only	only	ADV
ap-924	54	62	if	if	SCONJ
ap-924	54	63	�	�	PROPN
ap-924	54	64	�	�	PROPN
ap-924	54	65	�	�	PROPN
ap-924	54	66	�	�	PROPN
ap-924	54	67	a	a	DET
ap-924	54	68	b	b	PROPN
ap-924	54	69	b	b	PROPN
ap-924	54	70	�	�	PROPN
ap-924	54	71	�	�	PROPN
ap-924	54	72	�	�	PROPN
ap-924	54	73	�	�	PROPN
ap-924	54	74	.	.	PUNCT
ap-924	55	1	let	let	VERB
ap-924	55	2	us	we	PRON
ap-924	55	3	mention	mention	VERB
ap-924	55	4	that	that	SCONJ
ap-924	55	5	the	the	DET
ap-924	55	6	above	above	ADJ
ap-924	55	7	properties	property	NOUN
ap-924	55	8	of	of	ADP
ap-924	55	9	one	one	NUM
ap-924	55	10	-	-	PUNCT
ap-924	55	11	dimensional	dimensional	ADJ
ap-924	55	12	cut	cut	NOUN
ap-924	55	13	-	-	PUNCT
ap-924	55	14	and	and	CCONJ
ap-924	55	15	-	-	PUNCT
ap-924	55	16	project	project	NOUN
ap-924	55	17	sets	set	NOUN
ap-924	55	18	,	,	PUNCT
ap-924	55	19	and	and	CCONJ
ap-924	55	20	many	many	ADJ
ap-924	55	21	others	other	NOUN
ap-924	55	22	,	,	PUNCT
ap-924	55	23	are	be	AUX
ap-924	55	24	explained	explain	VERB
ap-924	55	25	in	in	ADP
ap-924	55	26	the	the	DET
ap-924	55	27	review	review	NOUN
ap-924	55	28	article	article	NOUN
ap-924	55	29	[	[	X
ap-924	55	30	14	14	NUM
ap-924	55	31	]	]	SYM
ap-924	55	32	.	.	PUNCT
ap-924	56	1	3	3	NUM
ap-924	56	2	self	self	NOUN
ap-924	56	3	-	-	PUNCT
ap-924	56	4	matching	match	VERB
ap-924	56	5	property	property	NOUN
ap-924	56	6	of	of	ADP
ap-924	56	7	the	the	DET
ap-924	56	8	graph	graph	NOUN
ap-924	56	9	�	�	PROPN
ap-924	56	10	�	�	PROPN
ap-924	56	11	j	j	PROPN
ap-924	56	12	�	�	PROPN
ap-924	56	13	against	against	ADP
ap-924	56	14	j	j	PROPN
ap-924	56	15	an	an	DET
ap-924	56	16	important	important	ADJ
ap-924	56	17	role	role	NOUN
ap-924	56	18	in	in	ADP
ap-924	56	19	the	the	DET
ap-924	56	20	study	study	NOUN
ap-924	56	21	of	of	ADP
ap-924	56	22	the	the	DET
ap-924	56	23	self	self	NOUN
ap-924	56	24	-	-	PUNCT
ap-924	56	25	matching	match	VERB
ap-924	56	26	properties	property	NOUN
ap-924	56	27	of	of	ADP
ap-924	56	28	the	the	DET
ap-924	56	29	graph	graph	NOUN
ap-924	56	30	�	�	PROPN
ap-924	56	31	�	�	PROPN
ap-924	56	32	j	j	PROPN
ap-924	56	33	�	�	PROPN
ap-924	56	34	against	against	ADP
ap-924	56	35	j	j	PROPN
ap-924	56	36	is	be	AUX
ap-924	56	37	played	play	VERB
ap-924	56	38	by	by	ADP
ap-924	56	39	the	the	DET
ap-924	56	40	generalized	generalize	VERB
ap-924	56	41	fibonacci	fibonacci	NOUN
ap-924	56	42	sequence	sequence	NOUN
ap-924	56	43	(	(	PUNCT
ap-924	56	44	)	)	PUNCT
ap-924	56	45	gi	gi	VERB
ap-924	57	1	i	i	PRON
ap-924	57	2	�	�	PROPN
ap-924	57	3	�	�	PROPN
ap-924	57	4	,	,	PUNCT
ap-924	57	5	defined	define	VERB
ap-924	57	6	by	by	ADP
ap-924	57	7	(	(	PUNCT
ap-924	57	8	2	2	NUM
ap-924	57	9	)	)	PUNCT
ap-924	57	10	and	and	CCONJ
ap-924	57	11	(	(	PUNCT
ap-924	57	12	4	4	NUM
ap-924	57	13	)	)	PUNCT
ap-924	57	14	,	,	PUNCT
ap-924	57	15	respectively	respectively	ADV
ap-924	57	16	.	.	PUNCT
ap-924	58	1	it	it	PRON
ap-924	58	2	turns	turn	VERB
ap-924	58	3	out	out	ADP
ap-924	58	4	that	that	SCONJ
ap-924	58	5	shifting	shift	VERB
ap-924	58	6	the	the	DET
ap-924	58	7	argument	argument	NOUN
ap-924	58	8	j	j	PROPN
ap-924	58	9	of	of	ADP
ap-924	58	10	the	the	DET
ap-924	58	11	function	function	NOUN
ap-924	58	12	�	�	PROPN
ap-924	58	13	�	�	PROPN
ap-924	58	14	j	j	PROPN
ap-924	58	15	�	�	PROPN
ap-924	58	16	by	by	ADP
ap-924	58	17	the	the	DET
ap-924	58	18	integer	integer	NOUN
ap-924	58	19	gi	gi	NOUN
ap-924	58	20	results	result	NOUN
ap-924	58	21	in	in	ADP
ap-924	58	22	shifting	shift	VERB
ap-924	58	23	the	the	DET
ap-924	58	24	value	value	NOUN
ap-924	58	25	by	by	ADP
ap-924	58	26	gi	gi	PROPN
ap-924	58	27	1	1	NUM
ap-924	58	28	,	,	PUNCT
ap-924	58	29	with	with	ADP
ap-924	58	30	the	the	DET
ap-924	58	31	exception	exception	NOUN
ap-924	58	32	of	of	ADP
ap-924	58	33	isolated	isolated	ADJ
ap-924	58	34	mismatches	mismatch	NOUN
ap-924	58	35	with	with	ADP
ap-924	58	36	low	low	ADJ
ap-924	58	37	frequency	frequency	NOUN
ap-924	58	38	.	.	PUNCT
ap-924	59	1	the	the	DET
ap-924	59	2	first	first	ADJ
ap-924	59	3	proposition	proposition	NOUN
ap-924	59	4	is	be	AUX
ap-924	59	5	an	an	DET
ap-924	59	6	easy	easy	ADJ
ap-924	59	7	consequence	consequence	NOUN
ap-924	59	8	of	of	ADP
ap-924	59	9	the	the	DET
ap-924	59	10	expressions	expression	NOUN
ap-924	59	11	of	of	ADP
ap-924	59	12	�	�	PROPN
ap-924	59	13	�	�	PROPN
ap-924	59	14	as	as	ADP
ap-924	59	15	an	an	DET
ap-924	59	16	element	element	NOUN
ap-924	59	17	of	of	ADP
ap-924	59	18	the	the	DET
ap-924	59	19	ring	ring	PROPN
ap-924	59	20	�	�	PROPN
ap-924	59	21	�	�	PROPN
ap-924	59	22	�	�	PROPN
ap-924	59	23	�	�	PROPN
ap-924	59	24	in	in	ADP
ap-924	59	25	the	the	DET
ap-924	59	26	integral	integral	ADJ
ap-924	59	27	basis	basis	NOUN
ap-924	59	28	1	1	NUM
ap-924	59	29	,	,	PUNCT
ap-924	59	30	�	�	PROPN
ap-924	59	31	,	,	PUNCT
ap-924	59	32	given	give	VERB
ap-924	59	33	by	by	ADP
ap-924	59	34	(	(	PUNCT
ap-924	59	35	3	3	NUM
ap-924	59	36	)	)	PUNCT
ap-924	59	37	and	and	CCONJ
ap-924	59	38	(	(	PUNCT
ap-924	59	39	5	5	NUM
ap-924	59	40	)	)	PUNCT
ap-924	59	41	.	.	PUNCT
ap-924	60	1	theorem	theorem	NOUN
ap-924	60	2	1	1	NUM
ap-924	60	3	let	let	VERB
ap-924	60	4	�	�	PROPN
ap-924	60	5	�	�	PROPN
ap-924	60	6	(	(	PUNCT
ap-924	60	7	,	,	PUNCT
ap-924	60	8	)	)	PUNCT
ap-924	60	9	0	0	NUM
ap-924	60	10	1	1	NUM
ap-924	60	11	satisfy	satisfy	PROPN
ap-924	60	12	�	�	PROPN
ap-924	60	13	�	�	PROPN
ap-924	60	14	2	2	NUM
ap-924	60	15	1	1	NUM
ap-924	60	16	�	�	NOUN
ap-924	60	17	m	m	PRON
ap-924	60	18	and	and	CCONJ
ap-924	60	19	let	let	VERB
ap-924	60	20	(	(	PUNCT
ap-924	60	21	)	)	PUNCT
ap-924	60	22	gi	gi	VERB
ap-924	60	23	i	i	PRON
ap-924	60	24	�	�	PROPN
ap-924	60	25	�	�	PROPN
ap-924	60	26	0	0	NUM
ap-924	60	27	be	be	AUX
ap-924	60	28	defined	define	VERB
ap-924	60	29	by	by	ADP
ap-924	60	30	(	(	PUNCT
ap-924	60	31	2	2	NUM
ap-924	60	32	)	)	PUNCT
ap-924	60	33	.	.	PUNCT
ap-924	61	1	let	let	VERB
ap-924	61	2	i	i	PRON
ap-924	61	3	�	�	PROPN
ap-924	61	4	�	�	PROPN
ap-924	61	5	.	.	PUNCT
ap-924	62	1	then	then	ADV
ap-924	62	2	for	for	ADP
ap-924	62	3	j	j	PROPN
ap-924	62	4	�	�	PROPN
ap-924	62	5	�	�	PROPN
ap-924	62	6	we	we	PRON
ap-924	62	7	have	have	VERB
ap-924	62	8	�	�	PROPN
ap-924	62	9	�	�	PROPN
ap-924	62	10	�	�	PROPN
ap-924	62	11	�	�	PROPN
ap-924	62	12	�	�	PROPN
ap-924	62	13	�	�	PROPN
ap-924	62	14	(	(	PUNCT
ap-924	62	15	)	)	PUNCT
ap-924	63	1	(	(	PUNCT
ap-924	63	2	)	)	PUNCT
ap-924	63	3	j	j	PROPN
ap-924	63	4	g	g	PROPN
ap-924	63	5	j	j	PROPN
ap-924	63	6	g	g	PROPN
ap-924	64	1	ji	ji	PROPN
ap-924	65	1	i	i	PRON
ap-924	65	2	i	i	PRON
ap-924	65	3	�	�	PROPN
ap-924	65	4	1	1	NUM
ap-924	65	5	where	where	SCONJ
ap-924	65	6	�	�	PROPN
ap-924	65	7	�	�	PROPN
ap-924	65	8	i	i	PRON
ap-924	65	9	ij	ij	NOUN
ap-924	65	10	(	(	PUNCT
ap-924	65	11	)	)	PUNCT
ap-924	65	12	,	,	PUNCT
ap-924	65	13	(	(	PUNCT
ap-924	65	14	)	)	PUNCT
ap-924	65	15	�	�	PROPN
ap-924	65	16	0	0	NUM
ap-924	65	17	1	1	NUM
ap-924	65	18	1	1	NUM
ap-924	65	19	.	.	PUNCT
ap-924	66	1	the	the	DET
ap-924	66	2	frequency	frequency	NOUN
ap-924	66	3	of	of	ADP
ap-924	66	4	integers	integer	NOUN
ap-924	66	5	j	j	PROPN
ap-924	66	6	for	for	ADP
ap-924	66	7	which	which	PRON
ap-924	66	8	the	the	DET
ap-924	66	9	value	value	NOUN
ap-924	66	10	i	i	PRON
ap-924	66	11	j	j	PROPN
ap-924	66	12	(	(	PUNCT
ap-924	66	13	)	)	PUNCT
ap-924	66	14	is	be	AUX
ap-924	66	15	non	non	ADJ
ap-924	66	16	-	-	ADJ
ap-924	66	17	zero	zero	NUM
ap-924	66	18	is	be	AUX
ap-924	66	19	equal	equal	ADJ
ap-924	66	20	to	to	ADP
ap-924	66	21	�	�	PROPN
ap-924	66	22	�	�	PROPN
ap-924	66	23	�	�	PROPN
ap-924	66	24	�	�	PROPN
ap-924	66	25	i	i	PRON
ap-924	67	1	n	n	VERB
ap-924	68	1	i	i	PRON
ap-924	68	2	ij	ij	VERB
ap-924	68	3	n	n	CCONJ
ap-924	68	4	j	j	PROPN
ap-924	68	5	n	n	CCONJ
ap-924	68	6	j	j	PROPN
ap-924	69	1	n	n	PROPN
ap-924	69	2	:	:	PUNCT
ap-924	69	3	lim	lim	PROPN
ap-924	69	4	#	#	NOUN
ap-924	69	5	,	,	PUNCT
ap-924	69	6	(	(	PUNCT
ap-924	69	7	)	)	PUNCT
ap-924	69	8	�	�	PROPN
ap-924	69	9	�	�	PROPN
ap-924	69	10	�	�	PROPN
ap-924	69	11	�	�	PROPN
ap-924	69	12	�	�	PROPN
ap-924	69	13	�	�	PROPN
ap-924	69	14	�	�	PROPN
ap-924	69	15	�	�	PROPN
ap-924	69	16	�	�	PROPN
ap-924	69	17	0	0	NUM
ap-924	69	18	2	2	NUM
ap-924	69	19	1	1	NUM
ap-924	69	20	.	.	PUNCT
ap-924	70	1	proof	proof	NOUN
ap-924	70	2	.	.	PUNCT
ap-924	71	1	the	the	DET
ap-924	71	2	first	first	ADJ
ap-924	71	3	statement	statement	NOUN
ap-924	71	4	is	be	AUX
ap-924	71	5	trivial	trivial	ADJ
ap-924	71	6	.	.	PUNCT
ap-924	72	1	for	for	ADP
ap-924	72	2	,	,	PUNCT
ap-924	72	3	we	we	PRON
ap-924	72	4	have	have	VERB
ap-924	72	5	�	�	PROPN
ap-924	72	6	�	�	PROPN
ap-924	72	7	�	�	PROPN
ap-924	72	8	�	�	PROPN
ap-924	72	9	�	�	PROPN
ap-924	72	10	�	�	PROPN
ap-924	72	11	�	�	PROPN
ap-924	72	12	�	�	PROPN
ap-924	72	13	�	�	PROPN
ap-924	72	14	�	�	PROPN
ap-924	72	15	�	�	PROPN
ap-924	72	16	�	�	PROPN
ap-924	72	17	�	�	PROPN
ap-924	72	18	�	�	PROPN
ap-924	72	19	�	�	PROPN
ap-924	72	20	�	�	PROPN
ap-924	72	21	�	�	PROPN
ap-924	73	1	i	i	PRON
ap-924	74	1	i	i	PRON
ap-924	75	1	i	i	PRON
ap-924	76	1	i	i	PRON
ap-924	77	1	i	i	PRON
ap-924	78	1	i	i	PRON
ap-924	78	2	j	j	PROPN
ap-924	79	1	j	j	PROPN
ap-924	79	2	g	g	PROPN
ap-924	79	3	j	j	PROPN
ap-924	79	4	g	g	PROPN
ap-924	79	5	j	j	PROPN
ap-924	80	1	j	j	PROPN
ap-924	80	2	g	g	PROPN
ap-924	80	3	g	g	PROPN
ap-924	80	4	j	j	PROPN
ap-924	80	5	j	j	PROPN
ap-924	80	6	(	(	PUNCT
ap-924	80	7	)	)	PUNCT
ap-924	80	8	(	(	PUNCT
ap-924	80	9	)	)	PUNCT
ap-924	80	10	(	(	PUNCT
ap-924	80	11	)	)	PUNCT
ap-924	80	12	�	�	PROPN
ap-924	80	13	�	�	PROPN
ap-924	80	14	�	�	PROPN
ap-924	80	15	1	1	NUM
ap-924	80	16	1	1	NUM
ap-924	80	17	11	11	NUM
ap-924	80	18	�	�	PROPN
ap-924	80	19	�	�	PROPN
ap-924	80	20	�	�	PROPN
ap-924	80	21	�	�	PROPN
ap-924	80	22	�	�	PROPN
ap-924	80	23	i	i	PRON
ap-924	80	24	i	i	PROPN
ap-924	80	25	�	�	PROPN
ap-924	80	26	0	0	NUM
ap-924	80	27	1	1	NUM
ap-924	80	28	1	1	NUM
ap-924	80	29	,	,	PUNCT
ap-924	80	30	(	(	PUNCT
ap-924	80	31	)	)	PUNCT
ap-924	80	32	.	.	PUNCT
ap-924	81	1	(	(	PUNCT
ap-924	81	2	8)	8)	NUM
ap-924	81	3	the	the	DET
ap-924	81	4	frequency	frequency	NOUN
ap-924	81	5	�	�	NOUN
ap-924	81	6	i	i	PRON
ap-924	81	7	is	be	AUX
ap-924	81	8	easily	easily	ADV
ap-924	81	9	determined	determined	ADJ
ap-924	81	10	in	in	ADP
ap-924	81	11	the	the	DET
ap-924	81	12	proof	proof	NOUN
ap-924	81	13	of	of	ADP
ap-924	81	14	theorem	theorem	ADJ
ap-924	81	15	1	1	NUM
ap-924	81	16	.	.	X
ap-924	81	17	�	�	PROPN
ap-924	81	18	in	in	ADP
ap-924	81	19	the	the	DET
ap-924	81	20	following	follow	VERB
ap-924	81	21	theorem	theorem	NOUN
ap-924	81	22	we	we	PRON
ap-924	81	23	determine	determine	VERB
ap-924	81	24	the	the	DET
ap-924	81	25	integers	integer	NOUN
ap-924	81	26	j	j	PROPN
ap-924	81	27	for	for	ADP
ap-924	81	28	which	which	PRON
ap-924	81	29	i	i	PRON
ap-924	81	30	j	j	PROPN
ap-924	81	31	(	(	PUNCT
ap-924	81	32	)	)	PUNCT
ap-924	81	33	is	be	AUX
ap-924	81	34	non	non	ADJ
ap-924	81	35	-	-	ADJ
ap-924	81	36	zero	zero	NUM
ap-924	81	37	.	.	PUNCT
ap-924	82	1	from	from	ADP
ap-924	82	2	this	this	PRON
ap-924	82	3	,	,	PUNCT
ap-924	82	4	we	we	PRON
ap-924	82	5	easily	easily	ADV
ap-924	82	6	derive	derive	VERB
ap-924	82	7	the	the	DET
ap-924	82	8	frequency	frequency	NOUN
ap-924	82	9	of	of	ADP
ap-924	82	10	such	such	ADJ
ap-924	82	11	mismatches	mismatch	NOUN
ap-924	82	12	.	.	PUNCT
ap-924	83	1	theorem	theorem	NOUN
ap-924	83	2	2	2	NUM
ap-924	83	3	with	with	ADP
ap-924	83	4	the	the	DET
ap-924	83	5	notation	notation	NOUN
ap-924	83	6	of	of	ADP
ap-924	83	7	theorem	theorem	NOUN
ap-924	83	8	1	1	NUM
ap-924	83	9	,	,	PUNCT
ap-924	83	10	we	we	PRON
ap-924	83	11	have	have	VERB
ap-924	83	12	i	i	PRON
ap-924	84	1	i	i	PRON
ap-924	84	2	ij	ij	INTJ
ap-924	85	1	if	if	SCONJ
ap-924	85	2	j	j	PROPN
ap-924	85	3	u	u	NOUN
ap-924	85	4	otherwise	otherwise	ADV
ap-924	85	5	,	,	PUNCT
ap-924	85	6	(	(	PUNCT
ap-924	85	7	)	)	PUNCT
ap-924	85	8	,	,	PUNCT
ap-924	85	9	(	(	PUNCT
ap-924	85	10	)	)	PUNCT
ap-924	85	11	�	�	PROPN
ap-924	85	12	�	�	PROPN
ap-924	85	13	�	�	PROPN
ap-924	85	14	�	�	PROPN
ap-924	85	15	�	�	PROPN
ap-924	85	16	0	0	NUM
ap-924	85	17	1	1	NUM
ap-924	85	18	1	1	NUM
ap-924	85	19	where	where	SCONJ
ap-924	85	20	�	�	PROPN
ap-924	85	21	�	�	PROPN
ap-924	85	22	�	�	PROPN
ap-924	85	23	�	�	PROPN
ap-924	85	24	u	u	NOUN
ap-924	85	25	kg	kg	NOUN
ap-924	86	1	k	k	NOUN
ap-924	87	1	g	g	PROPN
ap-924	88	1	k	k	PROPN
ap-924	89	1	k	k	NOUN
ap-924	90	1	gi	gi	INTJ
ap-924	91	1	i	i	INTJ
ap-924	92	1	i	i	INTJ
ap-924	93	1	i	i	PRON
ap-924	94	1	i	i	PROPN
ap-924	94	2	�	�	PROPN
ap-924	94	3	�	�	PROPN
ap-924	94	4	�	�	PROPN
ap-924	94	5	�	�	PROPN
ap-924	94	6	�	�	PROPN
ap-924	94	7	�	�	PROPN
ap-924	94	8	�	�	PROPN
ap-924	94	9	�	�	PROPN
ap-924	94	10	�	�	PROPN
ap-924	94	11	�	�	PROPN
ap-924	94	12	1	1	NUM
ap-924	94	13	1	1	NUM
ap-924	94	14	0	0	NUM
ap-924	94	15	1	1	NUM
ap-924	94	16	2	2	NUM
ap-924	94	17	�	�	PROPN
ap-924	94	18	�	�	PROPN
ap-924	94	19	,	,	PUNCT
ap-924	94	20	(	(	PUNCT
ap-924	94	21	)	)	PUNCT
ap-924	94	22	.	.	PUNCT
ap-924	95	1	before	before	ADP
ap-924	95	2	starting	start	VERB
ap-924	95	3	the	the	DET
ap-924	95	4	proof	proof	NOUN
ap-924	95	5	,	,	PUNCT
ap-924	95	6	let	let	VERB
ap-924	95	7	us	we	PRON
ap-924	95	8	mention	mention	VERB
ap-924	95	9	that	that	PRON
ap-924	95	10	for	for	SCONJ
ap-924	95	11	i	i	PRON
ap-924	95	12	even	even	ADV
ap-924	95	13	,	,	PUNCT
ap-924	95	14	the	the	DET
ap-924	95	15	set	set	NOUN
ap-924	95	16	ui	ui	NOUN
ap-924	95	17	can	can	AUX
ap-924	95	18	be	be	AUX
ap-924	95	19	written	write	VERB
ap-924	95	20	simply	simply	ADV
ap-924	95	21	as	as	SCONJ
ap-924	95	22	�	�	PROPN
ap-924	95	23	�	�	PROPN
ap-924	95	24	�	�	PROPN
ap-924	95	25	�	�	PROPN
ap-924	95	26	u	u	NOUN
ap-924	96	1	kg	kg	NOUN
ap-924	96	2	k	k	NOUN
ap-924	97	1	g	g	PROPN
ap-924	97	2	ki	ki	PROPN
ap-924	98	1	i	i	PRON
ap-924	98	2	i	i	PROPN
ap-924	98	3	�	�	PROPN
ap-924	98	4	�	�	PROPN
ap-924	98	5	1	1	NUM
ap-924	98	6	�	�	PROPN
ap-924	98	7	�	�	PROPN
ap-924	98	8	.	.	PUNCT
ap-924	99	1	for	for	SCONJ
ap-924	99	2	i	i	PRON
ap-924	99	3	odd	odd	ADJ
ap-924	99	4	,	,	PUNCT
ap-924	99	5	the	the	DET
ap-924	99	6	element	element	NOUN
ap-924	99	7	corresponding	correspond	VERB
ap-924	99	8	to	to	ADP
ap-924	99	9	k	k	PROPN
ap-924	99	10	�	�	PROPN
ap-924	99	11	0	0	NUM
ap-924	99	12	is	be	AUX
ap-924	99	13	equal	equal	ADJ
ap-924	99	14	to	to	PART
ap-924	99	15	gi	gi	VERB
ap-924	99	16	instead	instead	ADV
ap-924	99	17	of	of	ADP
ap-924	99	18	0	0	NUM
ap-924	99	19	.	.	PUNCT
ap-924	100	1	the	the	DET
ap-924	100	2	distinction	distinction	NOUN
ap-924	100	3	according	accord	VERB
ap-924	100	4	to	to	ADP
ap-924	100	5	the	the	DET
ap-924	100	6	parity	parity	NOUN
ap-924	100	7	of	of	ADP
ap-924	100	8	i	i	PRON
ap-924	100	9	is	be	AUX
ap-924	100	10	necessary	necessary	ADJ
ap-924	100	11	here	here	ADV
ap-924	100	12	,	,	PUNCT
ap-924	100	13	since	since	SCONJ
ap-924	100	14	unlike	unlike	ADP
ap-924	100	15	the	the	DET
ap-924	100	16	paper	paper	NOUN
ap-924	100	17	[	[	X
ap-924	100	18	12	12	NUM
ap-924	100	19	]	]	PUNCT
ap-924	100	20	,	,	PUNCT
ap-924	100	21	we	we	PRON
ap-924	100	22	determine	determine	VERB
ap-924	100	23	the	the	DET
ap-924	100	24	values	value	NOUN
ap-924	100	25	of	of	ADP
ap-924	100	26	i	i	PRON
ap-924	100	27	j	j	PROPN
ap-924	100	28	(	(	PUNCT
ap-924	100	29	)	)	PUNCT
ap-924	100	30	for	for	ADP
ap-924	100	31	j	j	PROPN
ap-924	100	32	�	�	PROPN
ap-924	100	33	�	�	PROPN
ap-924	100	34	,	,	PUNCT
ap-924	100	35	not	not	PART
ap-924	100	36	only	only	ADV
ap-924	100	37	for	for	ADP
ap-924	100	38	.	.	PUNCT
ap-924	101	1	proof	proof	NOUN
ap-924	101	2	.	.	PUNCT
ap-924	102	1	it	it	PRON
ap-924	102	2	is	be	AUX
ap-924	102	3	convenient	convenient	ADJ
ap-924	102	4	to	to	PART
ap-924	102	5	distinguish	distinguish	VERB
ap-924	102	6	two	two	NUM
ap-924	102	7	cases	case	NOUN
ap-924	102	8	according	accord	VERB
ap-924	102	9	to	to	ADP
ap-924	102	10	the	the	DET
ap-924	102	11	parity	parity	NOUN
ap-924	102	12	of	of	ADP
ap-924	102	13	i.	i.	PROPN
ap-924	102	14	�	�	PROPN
ap-924	102	15	first	first	ADV
ap-924	102	16	let	let	VERB
ap-924	102	17	i	i	PRON
ap-924	102	18	be	be	AUX
ap-924	102	19	even	even	ADV
ap-924	102	20	.	.	PUNCT
ap-924	103	1	it	it	PRON
ap-924	103	2	is	be	AUX
ap-924	103	3	obvious	obvious	ADJ
ap-924	103	4	from	from	ADP
ap-924	103	5	(	(	PUNCT
ap-924	103	6	8)	8)	NUM
ap-924	103	7	,	,	PUNCT
ap-924	103	8	that	that	SCONJ
ap-924	103	9	�	�	PROPN
ap-924	103	10	�	�	PROPN
ap-924	103	11	i	i	PRON
ap-924	103	12	j	j	PROPN
ap-924	103	13	(	(	PUNCT
ap-924	103	14	)	)	PUNCT
ap-924	103	15	,	,	PUNCT
ap-924	103	16	�	�	PROPN
ap-924	103	17	0	0	NUM
ap-924	103	18	1	1	NUM
ap-924	103	19	and	and	CCONJ
ap-924	103	20	�	�	PROPN
ap-924	103	21	�	�	PROPN
ap-924	103	22	�	�	PROPN
ap-924	103	23	�	�	PROPN
ap-924	103	24	�	�	PROPN
ap-924	103	25	i	i	PROPN
ap-924	103	26	ij	ij	VERB
ap-924	103	27	j	j	PROPN
ap-924	103	28	j	j	PROPN
ap-924	103	29	(	(	PUNCT
ap-924	103	30	)	)	PUNCT
ap-924	103	31	[	[	PUNCT
ap-924	103	32	,	,	PUNCT
ap-924	103	33	)	)	PUNCT
ap-924	103	34	.	.	PUNCT
ap-924	104	1	�	�	PROPN
ap-924	104	2	�	�	PROPN
ap-924	104	3	1	1	NUM
ap-924	104	4	0if	0if	NOUN
ap-924	105	1	and	and	CCONJ
ap-924	105	2	only	only	ADV
ap-924	105	3	if	if	SCONJ
ap-924	105	4	(	(	PUNCT
ap-924	105	5	9	9	X
ap-924	105	6	)	)	PUNCT
ap-924	105	7	let	let	VERB
ap-924	105	8	us	we	PRON
ap-924	105	9	denote	denote	VERB
ap-924	105	10	by	by	ADP
ap-924	105	11	m	m	PROPN
ap-924	105	12	the	the	DET
ap-924	105	13	set	set	NOUN
ap-924	105	14	of	of	ADP
ap-924	105	15	all	all	PRON
ap-924	105	16	such	such	ADJ
ap-924	105	17	j	j	PROPN
ap-924	105	18	,	,	PUNCT
ap-924	105	19	�	�	PROPN
ap-924	105	20	�	�	PROPN
ap-924	105	21	�	�	PROPN
ap-924	105	22	�	�	PROPN
ap-924	105	23	�	�	PROPN
ap-924	105	24	�	�	PROPN
ap-924	105	25	m	m	PROPN
ap-924	105	26	j	j	PROPN
ap-924	105	27	j	j	PROPN
ap-924	105	28	j	j	PROPN
ap-924	105	29	j	j	PROPN
ap-924	106	1	k	k	PROPN
ap-924	106	2	j	j	PROPN
ap-924	107	1	k	k	INTJ
ap-924	108	1	i	i	PRON
ap-924	108	2	i	i	PROPN
ap-924	108	3	�	�	PROPN
ap-924	108	4	�	�	PROPN
ap-924	108	5	�	�	PROPN
ap-924	108	6	�	�	PROPN
ap-924	108	7	�	�	PROPN
ap-924	108	8	�	�	PROPN
ap-924	108	9	�	�	PROPN
ap-924	108	10	�	�	PROPN
ap-924	108	11	�	�	PROPN
ap-924	108	12	�	�	PROPN
ap-924	108	13	�	�	PROPN
ap-924	108	14	�	�	PROPN
ap-924	108	15	�	�	PROPN
ap-924	108	16	�	�	PROPN
ap-924	108	17	�	�	PROPN
ap-924	108	18	[	[	PUNCT
ap-924	108	19	,	,	PUNCT
ap-924	108	20	)	)	PUNCT
ap-924	108	21	[	[	PUNCT
ap-924	108	22	,	,	PUNCT
ap-924	108	23	)	)	PUNCT
ap-924	108	24	,	,	PUNCT
ap-924	108	25	0	0	NUM
ap-924	108	26	0	0	NUM
ap-924	108	27	for	for	ADP
ap-924	108	28	some	some	DET
ap-924	108	29	22	22	NUM
ap-924	108	30	©	©	PROPN
ap-924	108	31	czech	czech	PROPN
ap-924	108	32	technical	technical	PROPN
ap-924	108	33	university	university	PROPN
ap-924	108	34	publishing	publishing	NOUN
ap-924	108	35	house	house	NOUN
ap-924	108	36	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	PROPN
ap-924	108	37	acta	acta	PROPN
ap-924	108	38	polytechnica	polytechnica	PROPN
ap-924	108	39	vol	vol	NOUN
ap-924	108	40	.	.	PUNCT
ap-924	109	1	47	47	NUM
ap-924	109	2	no	no	INTJ
ap-924	109	3	.	.	PUNCT
ap-924	110	1	2–3/2007	2–3/2007	NUM
ap-924	110	2	therefore	therefore	ADV
ap-924	110	3	m	m	VERB
ap-924	110	4	is	be	AUX
ap-924	110	5	formed	form	VERB
ap-924	110	6	by	by	ADP
ap-924	110	7	the	the	DET
ap-924	110	8	irrational	irrational	ADJ
ap-924	110	9	parts	part	NOUN
ap-924	110	10	of	of	ADP
ap-924	110	11	the	the	DET
ap-924	110	12	elements	element	NOUN
ap-924	110	13	of	of	ADP
ap-924	110	14	the	the	DET
ap-924	110	15	set	set	VERB
ap-924	110	16	�	�	PROPN
ap-924	110	17	�	�	PROPN
ap-924	110	18	k	k	PROPN
ap-924	110	19	j	j	PROPN
ap-924	111	1	k	k	PROPN
ap-924	111	2	j	j	PROPN
ap-924	112	1	g	g	PROPN
ap-924	112	2	g	g	PROPN
ap-924	112	3	k	k	PROPN
ap-924	113	1	i	i	PRON
ap-924	113	2	i	i	PRON
ap-924	114	1	i	i	PRON
ap-924	115	1	i	i	PRON
ap-924	116	1	i	i	PRON
ap-924	116	2	�	�	PROPN
ap-924	116	3	�	�	PROPN
ap-924	116	4	�	�	PROPN
ap-924	116	5	�	�	PROPN
ap-924	116	6	�	�	PROPN
ap-924	116	7	�	�	PROPN
ap-924	116	8	�	�	PROPN
ap-924	116	9	�	�	PROPN
ap-924	116	10	�	�	PROPN
ap-924	116	11	�	�	PROPN
ap-924	116	12	�	�	PROPN
ap-924	116	13	�	�	PROPN
ap-924	116	14	�	�	PROPN
ap-924	116	15	�	�	PROPN
ap-924	116	16	�	�	PROPN
ap-924	116	17	[	[	PUNCT
ap-924	116	18	,	,	PUNCT
ap-924	116	19	)	)	PUNCT
ap-924	116	20	[	[	PUNCT
ap-924	116	21	,	,	PUNCT
ap-924	116	22	)	)	PUNCT
ap-924	116	23	[	[	PUNCT
ap-924	116	24	,	,	PUNCT
ap-924	116	25	)	)	PUNCT
ap-924	116	26	(	(	PUNCT
ap-924	116	27	)	)	PUNCT
ap-924	116	28	0	0	NUM
ap-924	116	29	0	0	NUM
ap-924	116	30	0	0	NUM
ap-924	116	31	1	1	NUM
ap-924	116	32	1	1	NUM
ap-924	116	33	�	�	PROPN
ap-924	116	34	�	�	PROPN
ap-924	116	35	�	�	PROPN
ap-924	116	36	�	�	PROPN
ap-924	116	37	�	�	PROPN
ap-924	116	38	�	�	PROPN
ap-924	116	39	�	�	PROPN
ap-924	116	40	k	k	PROPN
ap-924	116	41	k	k	PROPN
ap-924	116	42	�	�	PROPN
ap-924	116	43	�	�	PROPN
ap-924	116	44	,	,	PUNCT
ap-924	116	45	where	where	SCONJ
ap-924	116	46	the	the	DET
ap-924	116	47	last	last	ADJ
ap-924	116	48	equality	equality	NOUN
ap-924	116	49	follows	follow	VERB
ap-924	116	50	from	from	ADP
ap-924	116	51	(	(	PUNCT
ap-924	116	52	3	3	NUM
ap-924	116	53	)	)	PUNCT
ap-924	116	54	and	and	CCONJ
ap-924	116	55	(	(	PUNCT
ap-924	116	56	7	7	NUM
ap-924	116	57	)	)	PUNCT
ap-924	116	58	.	.	PUNCT
ap-924	117	1	separating	separate	VERB
ap-924	117	2	the	the	DET
ap-924	117	3	irrational	irrational	ADJ
ap-924	117	4	part	part	NOUN
ap-924	117	5	we	we	PRON
ap-924	117	6	obtain	obtain	VERB
ap-924	117	7	�	�	PROPN
ap-924	117	8	�	�	PROPN
ap-924	117	9	�	�	PROPN
ap-924	117	10	�	�	PROPN
ap-924	117	11	�	�	PROPN
ap-924	117	12	�	�	PROPN
ap-924	117	13	�	�	PROPN
ap-924	117	14	�	�	PROPN
ap-924	117	15	m	m	VERB
ap-924	117	16	kg	kg	INTJ
ap-924	117	17	m	m	VERB
ap-924	117	18	kg	kg	INTJ
ap-924	118	1	k	k	NOUN
ap-924	118	2	g	g	PROPN
ap-924	119	1	k	k	PROPN
ap-924	119	2	g	g	PROPN
ap-924	120	1	k	k	PROPN
ap-924	120	2	kg	kg	X
ap-924	121	1	k	k	PROPN
ap-924	121	2	u	u	INTJ
ap-924	122	1	i	i	PRON
ap-924	122	2	i	i	PRON
ap-924	123	1	i	i	PRON
ap-924	123	2	i	i	PRON
ap-924	124	1	i	i	PRON
ap-924	124	2	i	i	PRON
ap-924	124	3	�	�	PROPN
ap-924	124	4	�	�	PROPN
ap-924	124	5	�	�	PROPN
ap-924	124	6	�	�	PROPN
ap-924	124	7	�	�	PROPN
ap-924	124	8	1	1	NUM
ap-924	124	9	1	1	NUM
ap-924	124	10	�	�	PROPN
ap-924	124	11	�	�	PROPN
ap-924	124	12	�	�	PROPN
ap-924	124	13	�	�	PROPN
ap-924	124	14	,	,	PUNCT
ap-924	124	15	where	where	SCONJ
ap-924	124	16	we	we	PRON
ap-924	124	17	have	have	AUX
ap-924	124	18	used	use	VERB
ap-924	124	19	the	the	DET
ap-924	124	20	equations	equation	NOUN
ap-924	124	21	�	�	PROPN
ap-924	124	22	�	�	PROPN
ap-924	124	23	�	�	PROPN
ap-924	124	24	�	�	PROPN
ap-924	124	25	�	�	PROPN
ap-924	124	26	2	2	NUM
ap-924	124	27	1	1	NUM
ap-924	124	28	m	m	NOUN
ap-924	124	29	and	and	CCONJ
ap-924	124	30	mg	mg	PROPN
ap-924	124	31	g	g	PROPN
ap-924	125	1	gi	gi	INTJ
ap-924	125	2	i	i	PRON
ap-924	126	1	i	i	PRON
ap-924	126	2	�	�	PROPN
ap-924	126	3	1	1	NUM
ap-924	126	4	1	1	NUM
ap-924	126	5	.	.	PUNCT
ap-924	126	6	�	�	PROPN
ap-924	126	7	now	now	ADV
ap-924	126	8	let	let	VERB
ap-924	126	9	i	i	PRON
ap-924	126	10	be	be	AUX
ap-924	126	11	odd	odd	ADJ
ap-924	126	12	.	.	PUNCT
ap-924	127	1	then	then	ADV
ap-924	127	2	from	from	ADP
ap-924	127	3	(	(	PUNCT
ap-924	127	4	8)	8)	NUM
ap-924	127	5	,	,	PUNCT
ap-924	127	6	�	�	PROPN
ap-924	127	7	�	�	PROPN
ap-924	127	8	i	i	PRON
ap-924	127	9	j	j	PROPN
ap-924	127	10	(	(	PUNCT
ap-924	127	11	)	)	PUNCT
ap-924	127	12	,	,	PUNCT
ap-924	127	13	�	�	PROPN
ap-924	127	14	0	0	NUM
ap-924	127	15	1	1	NUM
ap-924	127	16	and	and	CCONJ
ap-924	127	17	�	�	PROPN
ap-924	127	18	�	�	PROPN
ap-924	127	19	�	�	PROPN
ap-924	127	20	�	�	PROPN
ap-924	127	21	�	�	PROPN
ap-924	127	22	i	i	PROPN
ap-924	127	23	ij	ij	VERB
ap-924	127	24	j	j	PROPN
ap-924	127	25	j	j	PROPN
ap-924	127	26	(	(	PUNCT
ap-924	127	27	)	)	PUNCT
ap-924	127	28	[	[	PUNCT
ap-924	127	29	,	,	PUNCT
ap-924	127	30	)	)	PUNCT
ap-924	127	31	.	.	PUNCT
ap-924	128	1	�	�	PROPN
ap-924	128	2	�	�	PROPN
ap-924	128	3	1	1	NUM
ap-924	128	4	1	1	NUM
ap-924	128	5	1if	1if	NOUN
ap-924	128	6	and	and	CCONJ
ap-924	128	7	only	only	ADV
ap-924	128	8	if	if	SCONJ
ap-924	128	9	(	(	PUNCT
ap-924	128	10	10	10	NUM
ap-924	128	11	)	)	PUNCT
ap-924	128	12	let	let	VERB
ap-924	128	13	us	we	PRON
ap-924	128	14	denote	denote	VERB
ap-924	128	15	by	by	ADP
ap-924	128	16	m	m	PROPN
ap-924	128	17	the	the	DET
ap-924	128	18	set	set	NOUN
ap-924	128	19	of	of	ADP
ap-924	128	20	all	all	PRON
ap-924	128	21	such	such	ADJ
ap-924	128	22	j	j	PROPN
ap-924	128	23	,	,	PUNCT
ap-924	128	24	�	�	PROPN
ap-924	128	25	�	�	PROPN
ap-924	128	26	�	�	PROPN
ap-924	128	27	�	�	PROPN
ap-924	128	28	�	�	PROPN
ap-924	128	29	�	�	PROPN
ap-924	128	30	m	m	PROPN
ap-924	128	31	j	j	PROPN
ap-924	129	1	j	j	PROPN
ap-924	129	2	j	j	PROPN
ap-924	129	3	j	j	PROPN
ap-924	130	1	k	k	PROPN
ap-924	130	2	j	j	PROPN
ap-924	131	1	k	k	INTJ
ap-924	132	1	i	i	PRON
ap-924	132	2	i	i	PROPN
ap-924	132	3	�	�	PROPN
ap-924	132	4	�	�	PROPN
ap-924	132	5	�	�	PROPN
ap-924	132	6	�	�	PROPN
ap-924	132	7	�	�	PROPN
ap-924	132	8	�	�	PROPN
ap-924	132	9	�	�	PROPN
ap-924	132	10	�	�	PROPN
ap-924	132	11	�	�	PROPN
ap-924	132	12	�	�	PROPN
ap-924	132	13	�	�	PROPN
ap-924	132	14	�	�	PROPN
ap-924	132	15	�	�	PROPN
ap-924	132	16	�	�	PROPN
ap-924	132	17	�	�	PROPN
ap-924	132	18	1	1	NUM
ap-924	132	19	0	0	NUM
ap-924	132	20	0	0	NUM
ap-924	132	21	[	[	PUNCT
ap-924	132	22	,	,	PUNCT
ap-924	132	23	)	)	PUNCT
ap-924	132	24	[	[	PUNCT
ap-924	132	25	,	,	PUNCT
ap-924	132	26	)	)	PUNCT
ap-924	132	27	,	,	PUNCT
ap-924	132	28	.for	.for	PUNCT
ap-924	133	1	some	some	PRON
ap-924	133	2	therefore	therefore	ADV
ap-924	133	3	m	m	VERB
ap-924	133	4	is	be	AUX
ap-924	133	5	formed	form	VERB
ap-924	133	6	by	by	ADP
ap-924	133	7	the	the	DET
ap-924	133	8	irrational	irrational	ADJ
ap-924	133	9	parts	part	NOUN
ap-924	133	10	of	of	ADP
ap-924	133	11	elements	element	NOUN
ap-924	133	12	of	of	ADP
ap-924	133	13	the	the	DET
ap-924	133	14	set	set	VERB
ap-924	133	15	�	�	PROPN
ap-924	133	16	�	�	PROPN
ap-924	133	17	k	k	PROPN
ap-924	133	18	j	j	PROPN
ap-924	134	1	k	k	PROPN
ap-924	134	2	j	j	PROPN
ap-924	135	1	i	i	PRON
ap-924	135	2	i	i	INTJ
ap-924	136	1	i	i	PRON
ap-924	137	1	i	i	PRON
ap-924	137	2	�	�	PROPN
ap-924	137	3	�	�	PROPN
ap-924	137	4	�	�	PROPN
ap-924	137	5	�	�	PROPN
ap-924	137	6	�	�	PROPN
ap-924	137	7	�	�	PROPN
ap-924	137	8	�	�	PROPN
ap-924	137	9	�	�	PROPN
ap-924	137	10	�	�	PROPN
ap-924	137	11	�	�	PROPN
ap-924	137	12	�	�	PROPN
ap-924	137	13	�	�	PROPN
ap-924	137	14	�	�	PROPN
ap-924	137	15	[	[	PUNCT
ap-924	137	16	,	,	PUNCT
ap-924	137	17	)	)	PUNCT
ap-924	137	18	[	[	PUNCT
ap-924	137	19	,	,	PUNCT
ap-924	137	20	)	)	PUNCT
ap-924	137	21	[	[	PUNCT
ap-924	137	22	,	,	PUNCT
ap-924	137	23	)	)	PUNCT
ap-924	137	24	(	(	PUNCT
ap-924	137	25	[	[	PUNCT
ap-924	137	26	,	,	PUNCT
ap-924	137	27	)	)	PUNCT
ap-924	137	28	0	0	NUM
ap-924	137	29	0	0	NUM
ap-924	137	30	1	1	NUM
ap-924	137	31	0	0	NUM
ap-924	137	32	1	1	NUM
ap-924	137	33	0	0	NUM
ap-924	137	34	1	1	NUM
ap-924	137	35	�	�	PROPN
ap-924	137	36	�	�	PROPN
ap-924	137	37	�	�	PROPN
ap-924	137	38	�	�	PROPN
ap-924	137	39	�	�	PROPN
ap-924	137	40	�	�	PROPN
ap-924	137	41	�	�	PROPN
ap-924	137	42	)	)	PUNCT
ap-924	137	43	(	(	PUNCT
ap-924	137	44	)	)	PUNCT
ap-924	137	45	.	.	PUNCT
ap-924	138	1	�	�	PROPN
ap-924	138	2	�	�	PROPN
ap-924	138	3	�	�	PROPN
ap-924	138	4	�	�	PROPN
ap-924	138	5	�	�	PROPN
ap-924	138	6	�	�	PROPN
ap-924	138	7	�	�	PROPN
ap-924	138	8	g	g	PROPN
ap-924	139	1	g	g	PROPN
ap-924	139	2	k	k	PROPN
ap-924	140	1	k	k	PROPN
ap-924	140	2	ki	ki	PROPN
ap-924	141	1	i	i	PRON
ap-924	141	2	1	1	NUM
ap-924	141	3	1	1	NUM
ap-924	141	4	�	�	NOUN
ap-924	141	5	separating	separate	VERB
ap-924	141	6	the	the	DET
ap-924	141	7	irrational	irrational	ADJ
ap-924	141	8	part	part	NOUN
ap-924	141	9	we	we	PRON
ap-924	141	10	obtain	obtain	VERB
ap-924	141	11	�	�	PROPN
ap-924	141	12	�	�	PROPN
ap-924	141	13	�	�	PROPN
ap-924	141	14	�	�	PROPN
ap-924	141	15	�	�	PROPN
ap-924	141	16	�	�	PROPN
ap-924	141	17	�	�	PROPN
ap-924	141	18	�	�	PROPN
ap-924	141	19	m	m	VERB
ap-924	141	20	kg	kg	INTJ
ap-924	141	21	m	m	VERB
ap-924	141	22	kg	kg	INTJ
ap-924	142	1	k	k	NOUN
ap-924	142	2	g	g	PROPN
ap-924	143	1	g	g	PROPN
ap-924	143	2	k	k	PROPN
ap-924	144	1	kg	kg	INTJ
ap-924	145	1	g	g	PROPN
ap-924	145	2	k	k	PROPN
ap-924	145	3	k	k	PROPN
ap-924	146	1	kg	kg	INTJ
ap-924	147	1	i	i	INTJ
ap-924	148	1	i	i	PRON
ap-924	149	1	i	i	PRON
ap-924	150	1	i	i	PRON
ap-924	151	1	i	i	PRON
ap-924	152	1	i	i	PRON
ap-924	153	1	i	i	PRON
ap-924	153	2	�	�	PROPN
ap-924	153	3	�	�	PROPN
ap-924	153	4	�	�	PROPN
ap-924	153	5	�	�	PROPN
ap-924	153	6	�	�	PROPN
ap-924	153	7	1	1	NUM
ap-924	153	8	1	1	NUM
ap-924	153	9	1	1	NUM
ap-924	153	10	1	1	NUM
ap-924	153	11	�	�	PROPN
ap-924	153	12	�	�	PROPN
ap-924	153	13	�	�	PROPN
ap-924	153	14	�	�	PROPN
ap-924	153	15	(	(	PUNCT
ap-924	153	16	)	)	PUNCT
ap-924	153	17	�	�	PROPN
ap-924	153	18	�	�	PROPN
ap-924	153	19	�	�	PROPN
ap-924	153	20	�	�	PROPN
ap-924	153	21	g	g	PROPN
ap-924	153	22	k	k	PROPN
ap-924	153	23	k	k	PROPN
ap-924	153	24	ui	ui	PROPN
ap-924	154	1	i	i	PROPN
ap-924	154	2	(	(	PUNCT
ap-924	154	3	)	)	PUNCT
ap-924	154	4	,	,	PUNCT
ap-924	154	5	�	�	PROPN
ap-924	154	6	�	�	PROPN
ap-924	154	7	�	�	PROPN
ap-924	154	8	1	1	NUM
ap-924	154	9	�	�	NOUN
ap-924	154	10	where	where	SCONJ
ap-924	154	11	we	we	PRON
ap-924	154	12	have	have	AUX
ap-924	154	13	used	use	VERB
ap-924	154	14	the	the	DET
ap-924	154	15	equation	equation	NOUN
ap-924	154	16	�	�	PROPN
ap-924	154	17	�	�	PROPN
ap-924	154	18	�	�	PROPN
ap-924	154	19	�	�	PROPN
ap-924	154	20	�	�	PROPN
ap-924	154	21	2	2	NUM
ap-924	154	22	1	1	NUM
ap-924	154	23	m	m	NOUN
ap-924	154	24	,	,	PUNCT
ap-924	154	25	mg	mg	PROPN
ap-924	154	26	g	g	NOUN
ap-924	154	27	gi	gi	INTJ
ap-924	155	1	i	i	PRON
ap-924	156	1	i	i	PRON
ap-924	156	2	�	�	PROPN
ap-924	156	3	1	1	NUM
ap-924	156	4	1	1	NUM
ap-924	156	5	and	and	CCONJ
ap-924	156	6	�	�	PROPN
ap-924	156	7	�	�	PROPN
ap-924	156	8	�	�	PROPN
ap-924	156	9	�	�	PROPN
ap-924	156	10	�	�	PROPN
ap-924	156	11	k	k	PROPN
ap-924	156	12	k	k	PROPN
ap-924	156	13	�	�	PROPN
ap-924	156	14	�	�	PROPN
ap-924	156	15	.	.	PUNCT
ap-924	157	1	let	let	VERB
ap-924	157	2	us	we	PRON
ap-924	157	3	recall	recall	VERB
ap-924	157	4	that	that	SCONJ
ap-924	157	5	the	the	DET
ap-924	157	6	weyl	weyl	PROPN
ap-924	157	7	theorem	theorem	NOUN
ap-924	157	8	[	[	X
ap-924	157	9	15	15	NUM
ap-924	157	10	]	]	PUNCT
ap-924	157	11	states	state	VERB
ap-924	157	12	that	that	SCONJ
ap-924	157	13	numbers	number	NOUN
ap-924	157	14	of	of	ADP
ap-924	157	15	the	the	DET
ap-924	157	16	form	form	NOUN
ap-924	157	17	�	�	PROPN
ap-924	157	18	�	�	PROPN
ap-924	157	19	j	j	PROPN
ap-924	157	20	j	j	PROPN
ap-924	157	21	�	�	PROPN
ap-924	157	22	�	�	PROPN
ap-924	157	23	,	,	PUNCT
ap-924	157	24	j	j	PROPN
ap-924	157	25	�	�	PROPN
ap-924	157	26	�	�	PROPN
ap-924	157	27	,	,	PUNCT
ap-924	157	28	are	be	AUX
ap-924	157	29	uniformly	uniformly	ADV
ap-924	157	30	distributed	distribute	VERB
ap-924	157	31	in	in	ADP
ap-924	157	32	(	(	PUNCT
ap-924	157	33	0	0	NUM
ap-924	157	34	,	,	PUNCT
ap-924	157	35	1	1	NUM
ap-924	157	36	)	)	PUNCT
ap-924	157	37	for	for	ADP
ap-924	157	38	every	every	DET
ap-924	157	39	irrational	irrational	ADJ
ap-924	157	40	�	�	PROPN
ap-924	157	41	.	.	PUNCT
ap-924	158	1	therefore	therefore	ADV
ap-924	158	2	the	the	DET
ap-924	158	3	frequency	frequency	NOUN
ap-924	158	4	of	of	ADP
ap-924	158	5	those	those	DET
ap-924	158	6	j	j	PROPN
ap-924	158	7	�	�	PROPN
ap-924	158	8	�	�	PROPN
ap-924	158	9	that	that	PRON
ap-924	158	10	satisfy	satisfy	VERB
ap-924	158	11	�	�	PROPN
ap-924	158	12	�	�	PROPN
ap-924	158	13	j	j	PROPN
ap-924	158	14	j	j	PROPN
ap-924	158	15	i	i	PRON
ap-924	158	16	�	�	PROPN
ap-924	158	17	�	�	PROPN
ap-924	158	18	�	�	PROPN
ap-924	158	19	(	(	PUNCT
ap-924	158	20	,	,	PUNCT
ap-924	158	21	)	)	PUNCT
ap-924	158	22	0	0	NUM
ap-924	158	23	1	1	NUM
ap-924	158	24	is	be	AUX
ap-924	158	25	equal	equal	ADJ
ap-924	158	26	to	to	ADP
ap-924	158	27	the	the	DET
ap-924	158	28	length	length	NOUN
ap-924	158	29	of	of	ADP
ap-924	158	30	the	the	DET
ap-924	158	31	interval	interval	NOUN
ap-924	158	32	i.	i.	NOUN
ap-924	158	33	therefore	therefore	ADV
ap-924	158	34	one	one	PRON
ap-924	158	35	can	can	AUX
ap-924	158	36	derive	derive	VERB
ap-924	158	37	from	from	ADP
ap-924	158	38	(	(	PUNCT
ap-924	158	39	9	9	NUM
ap-924	158	40	)	)	PUNCT
ap-924	158	41	and	and	CCONJ
ap-924	158	42	(	(	PUNCT
ap-924	158	43	10	10	NUM
ap-924	158	44	)	)	PUNCT
ap-924	158	45	that	that	SCONJ
ap-924	158	46	the	the	DET
ap-924	158	47	frequency	frequency	NOUN
ap-924	158	48	of	of	ADP
ap-924	158	49	mismatches	mismatch	NOUN
ap-924	158	50	(	(	PUNCT
ap-924	158	51	non	non	ADJ
ap-924	158	52	-	-	ADJ
ap-924	158	53	zero	zero	NUM
ap-924	158	54	values	value	NOUN
ap-924	159	1	i	i	PRON
ap-924	159	2	j	j	PROPN
ap-924	159	3	(	(	PUNCT
ap-924	159	4	)	)	PUNCT
ap-924	159	5	)	)	PUNCT
ap-924	159	6	is	be	AUX
ap-924	159	7	equal	equal	ADJ
ap-924	159	8	to	to	ADP
ap-924	159	9	�	�	PROPN
ap-924	159	10	i	i	PRON
ap-924	159	11	,	,	PUNCT
ap-924	159	12	as	as	SCONJ
ap-924	159	13	stated	state	VERB
ap-924	159	14	by	by	ADP
ap-924	159	15	theorem	theorem	NOUN
ap-924	159	16	1	1	NUM
ap-924	159	17	.	.	PUNCT
ap-924	159	18	�	�	PROPN
ap-924	160	1	if	if	SCONJ
ap-924	160	2	�	�	PROPN
ap-924	160	3	�	�	PROPN
ap-924	160	4	(	(	PUNCT
ap-924	160	5	,	,	PUNCT
ap-924	160	6	)	)	PUNCT
ap-924	160	7	0	0	NUM
ap-924	160	8	1	1	NUM
ap-924	160	9	is	be	AUX
ap-924	160	10	the	the	DET
ap-924	160	11	quadratic	quadratic	ADJ
ap-924	160	12	unit	unit	NOUN
ap-924	160	13	satisfying	satisfy	VERB
ap-924	160	14	�	�	PROPN
ap-924	160	15	�	�	PROPN
ap-924	160	16	2	2	NUM
ap-924	160	17	1	1	NUM
ap-924	160	18	�	�	PROPN
ap-924	160	19	m	m	PROPN
ap-924	160	20	,	,	PUNCT
ap-924	160	21	then	then	ADV
ap-924	160	22	the	the	DET
ap-924	160	23	considerations	consideration	NOUN
ap-924	160	24	are	be	AUX
ap-924	160	25	even	even	ADV
ap-924	160	26	simpler	simple	ADJ
ap-924	160	27	,	,	PUNCT
ap-924	160	28	because	because	SCONJ
ap-924	160	29	expression	expression	NOUN
ap-924	160	30	(	(	PUNCT
ap-924	160	31	5	5	NUM
ap-924	160	32	)	)	PUNCT
ap-924	160	33	does	do	AUX
ap-924	160	34	not	not	PART
ap-924	160	35	depend	depend	VERB
ap-924	160	36	on	on	ADP
ap-924	160	37	the	the	DET
ap-924	160	38	parity	parity	NOUN
ap-924	160	39	of	of	ADP
ap-924	160	40	i.	i.	NOUN
ap-924	160	41	we	we	PRON
ap-924	160	42	state	state	VERB
ap-924	160	43	the	the	DET
ap-924	160	44	result	result	NOUN
ap-924	160	45	as	as	ADP
ap-924	160	46	the	the	DET
ap-924	160	47	following	follow	VERB
ap-924	160	48	theorem	theorem	NOUN
ap-924	160	49	.	.	PUNCT
ap-924	160	50	theorem	theorem	NOUN
ap-924	160	51	3	3	NUM
ap-924	160	52	let	let	VERB
ap-924	160	53	�	�	PROPN
ap-924	160	54	�	�	PROPN
ap-924	160	55	(	(	PUNCT
ap-924	160	56	,	,	PUNCT
ap-924	160	57	)	)	PUNCT
ap-924	160	58	0	0	NUM
ap-924	161	1	1	1	NUM
ap-924	161	2	satisfy	satisfy	PROPN
ap-924	161	3	�	�	PROPN
ap-924	161	4	�	�	PROPN
ap-924	161	5	2	2	NUM
ap-924	161	6	1	1	NUM
ap-924	161	7	�	�	PROPN
ap-924	161	8	m	m	PRON
ap-924	161	9	and	and	CCONJ
ap-924	161	10	let	let	VERB
ap-924	161	11	(	(	PUNCT
ap-924	161	12	)	)	PUNCT
ap-924	161	13	gi	gi	VERB
ap-924	161	14	i	i	PRON
ap-924	161	15	�	�	PROPN
ap-924	161	16	�	�	PROPN
ap-924	161	17	0	0	NUM
ap-924	161	18	be	be	AUX
ap-924	161	19	defined	define	VERB
ap-924	161	20	by	by	ADP
ap-924	161	21	(	(	PUNCT
ap-924	161	22	4	4	NUM
ap-924	161	23	)	)	PUNCT
ap-924	161	24	.	.	PUNCT
ap-924	162	1	for	for	SCONJ
ap-924	162	2	i	i	PROPN
ap-924	162	3	�	�	PROPN
ap-924	162	4	�	�	PROPN
ap-924	162	5	,	,	PUNCT
ap-924	162	6	put	put	VERB
ap-924	162	7	�	�	PROPN
ap-924	162	8	�	�	PROPN
ap-924	162	9	�	�	PROPN
ap-924	162	10	�	�	PROPN
ap-924	162	11	v	v	NOUN
ap-924	162	12	kg	kg	NOUN
ap-924	163	1	k	k	PROPN
ap-924	163	2	g	g	PROPN
ap-924	163	3	ki	ki	PROPN
ap-924	164	1	i	i	PRON
ap-924	164	2	i	i	PROPN
ap-924	164	3	�	�	PROPN
ap-924	164	4	�	�	PROPN
ap-924	164	5	1	1	NUM
ap-924	164	6	1	1	NUM
ap-924	164	7	(	(	PUNCT
ap-924	164	8	)	)	PUNCT
ap-924	164	9	�	�	PROPN
ap-924	164	10	�	�	PROPN
ap-924	164	11	.	.	PUNCT
ap-924	165	1	then	then	ADV
ap-924	165	2	for	for	ADP
ap-924	165	3	j	j	PROPN
ap-924	165	4	�	�	PROPN
ap-924	165	5	�	�	PROPN
ap-924	165	6	we	we	PRON
ap-924	165	7	have	have	VERB
ap-924	165	8	�	�	PROPN
ap-924	165	9	�	�	PROPN
ap-924	165	10	�	�	PROPN
ap-924	165	11	�	�	PROPN
ap-924	165	12	�	�	PROPN
ap-924	165	13	�	�	PROPN
ap-924	165	14	(	(	PUNCT
ap-924	165	15	)	)	PUNCT
ap-924	166	1	(	(	PUNCT
ap-924	166	2	)	)	PUNCT
ap-924	166	3	j	j	PROPN
ap-924	166	4	g	g	PROPN
ap-924	166	5	j	j	PROPN
ap-924	166	6	g	g	PROPN
ap-924	167	1	ji	ji	PROPN
ap-924	168	1	i	i	PRON
ap-924	168	2	i	i	PRON
ap-924	168	3	�	�	PROPN
ap-924	168	4	1	1	NUM
ap-924	168	5	,	,	PUNCT
ap-924	168	6	where	where	SCONJ
ap-924	168	7	i	i	PRON
ap-924	168	8	ij	ij	VERB
ap-924	168	9	if	if	SCONJ
ap-924	168	10	j	j	PROPN
ap-924	168	11	v	v	X
ap-924	168	12	otherwise	otherwise	ADV
ap-924	168	13	.	.	PUNCT
ap-924	169	1	(	(	PUNCT
ap-924	169	2	)	)	PUNCT
ap-924	169	3	,	,	PUNCT
ap-924	169	4	�	�	PROPN
ap-924	169	5	�	�	PROPN
ap-924	169	6	�	�	PROPN
ap-924	169	7	�	�	PROPN
ap-924	169	8	�	�	PROPN
ap-924	169	9	0	0	NUM
ap-924	169	10	1	1	NUM
ap-924	169	11	the	the	DET
ap-924	169	12	density	density	NOUN
ap-924	169	13	of	of	ADP
ap-924	169	14	the	the	DET
ap-924	169	15	set	set	NOUN
ap-924	169	16	ui	ui	NOUN
ap-924	169	17	of	of	ADP
ap-924	169	18	mismatches	mismatch	NOUN
ap-924	169	19	is	be	AUX
ap-924	169	20	equal	equal	ADJ
ap-924	169	21	to	to	ADP
ap-924	169	22	�	�	PROPN
ap-924	169	23	i.	i.	NOUN
ap-924	169	24	proof	proof	NOUN
ap-924	169	25	.	.	PUNCT
ap-924	170	1	the	the	DET
ap-924	170	2	proof	proof	NOUN
ap-924	170	3	follows	follow	VERB
ap-924	170	4	the	the	DET
ap-924	170	5	same	same	ADJ
ap-924	170	6	lines	line	NOUN
ap-924	170	7	as	as	SCONJ
ap-924	170	8	proofs	proof	NOUN
ap-924	170	9	of	of	ADP
ap-924	170	10	theorems	theorem	NOUN
ap-924	170	11	1	1	NUM
ap-924	170	12	and	and	CCONJ
ap-924	170	13	2	2	NUM
ap-924	170	14	.	.	X
ap-924	170	15	�	�	PROPN
ap-924	170	16	4	4	NUM
ap-924	170	17	conclusions	conclusion	NOUN
ap-924	170	18	one	one	NUM
ap-924	170	19	-	-	PUNCT
ap-924	170	20	dimensional	dimensional	ADJ
ap-924	170	21	cut	cut	NOUN
ap-924	170	22	-	-	PUNCT
ap-924	170	23	and	and	CCONJ
ap-924	170	24	-	-	PUNCT
ap-924	170	25	project	project	NOUN
ap-924	170	26	sets	set	NOUN
ap-924	170	27	can	can	AUX
ap-924	170	28	be	be	AUX
ap-924	170	29	constructed	construct	VERB
ap-924	170	30	from	from	ADP
ap-924	170	31	�	�	PROPN
ap-924	170	32	2	2	NUM
ap-924	170	33	for	for	ADP
ap-924	170	34	every	every	DET
ap-924	170	35	choice	choice	NOUN
ap-924	170	36	of	of	ADP
ap-924	170	37	straight	straight	ADJ
ap-924	170	38	lines	line	NOUN
ap-924	170	39	v1	v1	PROPN
ap-924	170	40	,	,	PUNCT
ap-924	170	41	v2	v2	PROPN
ap-924	170	42	,	,	PUNCT
ap-924	170	43	if	if	SCONJ
ap-924	170	44	they	they	PRON
ap-924	170	45	have	have	VERB
ap-924	170	46	irrational	irrational	ADJ
ap-924	170	47	slopes	slope	NOUN
ap-924	170	48	.	.	PUNCT
ap-924	171	1	however	however	ADV
ap-924	171	2	,	,	PUNCT
ap-924	171	3	in	in	ADP
ap-924	171	4	our	our	PRON
ap-924	171	5	proof	proof	NOUN
ap-924	171	6	of	of	ADP
ap-924	171	7	the	the	DET
ap-924	171	8	self	self	NOUN
ap-924	171	9	-	-	PUNCT
ap-924	171	10	matching	match	VERB
ap-924	171	11	properties	property	NOUN
ap-924	171	12	of	of	ADP
ap-924	171	13	the	the	DET
ap-924	171	14	beatty	beatty	PROPN
ap-924	171	15	sequences	sequence	NOUN
ap-924	171	16	we	we	PRON
ap-924	171	17	strongly	strongly	ADV
ap-924	171	18	use	use	VERB
ap-924	171	19	the	the	DET
ap-924	171	20	algebraic	algebraic	ADJ
ap-924	171	21	ring	ring	NOUN
ap-924	171	22	structure	structure	NOUN
ap-924	171	23	of	of	ADP
ap-924	171	24	the	the	DET
ap-924	171	25	set	set	VERB
ap-924	171	26	�	�	PROPN
ap-924	171	27	�	�	PROPN
ap-924	171	28	�	�	PROPN
ap-924	171	29	�	�	PROPN
ap-924	171	30	�	�	PROPN
ap-924	171	31	,	,	PUNCT
ap-924	171	32	and	and	CCONJ
ap-924	171	33	its	its	PRON
ap-924	171	34	scaling	scale	VERB
ap-924	171	35	invariance	invariance	NOUN
ap-924	171	36	with	with	ADP
ap-924	171	37	the	the	DET
ap-924	171	38	factor	factor	NOUN
ap-924	171	39	�	�	PROPN
ap-924	171	40	�	�	PROPN
ap-924	171	41	,	,	PUNCT
ap-924	171	42	namely	namely	ADV
ap-924	171	43	�	�	PROPN
ap-924	171	44	�	�	PROPN
ap-924	171	45	�	�	PROPN
ap-924	171	46	�	�	PROPN
ap-924	171	47	�	�	PROPN
ap-924	171	48	�	�	PROPN
ap-924	171	49	�	�	PROPN
ap-924	171	50	�	�	PROPN
ap-924	171	51	�	�	PROPN
ap-924	171	52	�	�	PROPN
ap-924	171	53	�	�	PROPN
ap-924	171	54	�	�	PROPN
ap-924	171	55	.	.	PUNCT
ap-924	172	1	for	for	ADP
ap-924	172	2	this	this	PRON
ap-924	172	3	,	,	PUNCT
ap-924	172	4	�	�	PROPN
ap-924	172	5	�	�	PROPN
ap-924	172	6	must	must	AUX
ap-924	172	7	necessarily	necessarily	ADV
ap-924	172	8	be	be	AUX
ap-924	172	9	a	a	DET
ap-924	172	10	quadratic	quadratic	ADJ
ap-924	172	11	unit	unit	NOUN
ap-924	172	12	.	.	PUNCT
ap-924	173	1	however	however	ADV
ap-924	173	2	,	,	PUNCT
ap-924	173	3	it	it	PRON
ap-924	173	4	is	be	AUX
ap-924	173	5	plausible	plausible	ADJ
ap-924	173	6	that	that	SCONJ
ap-924	173	7	,	,	PUNCT
ap-924	173	8	even	even	ADV
ap-924	173	9	for	for	ADP
ap-924	173	10	other	other	ADJ
ap-924	173	11	irrationals	irrational	NOUN
ap-924	173	12	�	�	PROPN
ap-924	173	13	,	,	PUNCT
ap-924	173	14	some	some	DET
ap-924	173	15	self	self	NOUN
ap-924	173	16	-	-	PUNCT
ap-924	173	17	matching	match	VERB
ap-924	173	18	property	property	NOUN
ap-924	173	19	is	be	AUX
ap-924	173	20	displayed	display	VERB
ap-924	173	21	by	by	ADP
ap-924	173	22	the	the	DET
ap-924	173	23	graph	graph	NOUN
ap-924	173	24	�	�	PROPN
ap-924	173	25	�	�	PROPN
ap-924	173	26	j	j	PROPN
ap-924	173	27	�	�	PROPN
ap-924	173	28	against	against	ADP
ap-924	173	29	j.	j.	PROPN
ap-924	173	30	to	to	PART
ap-924	173	31	show	show	VERB
ap-924	173	32	that	that	SCONJ
ap-924	173	33	,	,	PUNCT
ap-924	173	34	other	other	ADJ
ap-924	173	35	methods	method	NOUN
ap-924	173	36	would	would	AUX
ap-924	173	37	be	be	AUX
ap-924	173	38	necessary	necessary	ADJ
ap-924	173	39	.	.	PUNCT
ap-924	174	1	5	5	NUM
ap-924	174	2	acknowledgments	acknowledgment	NOUN
ap-924	174	3	the	the	DET
ap-924	174	4	authors	author	NOUN
ap-924	174	5	acknowledge	acknowledge	VERB
ap-924	174	6	financial	financial	ADJ
ap-924	174	7	support	support	NOUN
ap-924	174	8	from	from	ADP
ap-924	174	9	the	the	DET
ap-924	174	10	czech	czech	PROPN
ap-924	174	11	science	science	PROPN
ap-924	174	12	foundation	foundation	PROPN
ap-924	174	13	ga	ga	PROPN
ap-924	174	14	čr	čr	ADJ
ap-924	174	15	201/05/0169	201/05/0169	NUM
ap-924	174	16	,	,	PUNCT
ap-924	174	17	and	and	CCONJ
ap-924	174	18	from	from	ADP
ap-924	174	19	the	the	DET
ap-924	174	20	grant	grant	NOUN
ap-924	174	21	lc06002	lc06002	NOUN
ap-924	174	22	of	of	ADP
ap-924	174	23	the	the	DET
ap-924	174	24	ministry	ministry	PROPN
ap-924	174	25	of	of	ADP
ap-924	174	26	education	education	PROPN
ap-924	174	27	,	,	PUNCT
ap-924	174	28	youth	youth	NOUN
ap-924	174	29	and	and	CCONJ
ap-924	174	30	sports	sport	NOUN
ap-924	174	31	of	of	ADP
ap-924	174	32	the	the	DET
ap-924	174	33	czech	czech	PROPN
ap-924	174	34	republic	republic	NOUN
ap-924	174	35	.	.	PUNCT
ap-924	175	1	references	reference	NOUN
ap-924	175	2	[	[	X
ap-924	175	3	1	1	NUM
ap-924	175	4	]	]	X
ap-924	175	5	beatty	beatty	PROPN
ap-924	175	6	,	,	PUNCT
ap-924	175	7	s.	s.	PROPN
ap-924	175	8	:	:	PUNCT
ap-924	175	9	amer	amer	PROPN
ap-924	175	10	.	.	PROPN
ap-924	175	11	math	math	PROPN
ap-924	175	12	.	.	PUNCT
ap-924	176	1	monthly	monthly	ADJ
ap-924	176	2	,	,	PUNCT
ap-924	176	3	vol	vol	NOUN
ap-924	176	4	.	.	PROPN
ap-924	177	1	33	33	NUM
ap-924	177	2	(	(	PUNCT
ap-924	177	3	1926	1926	NUM
ap-924	177	4	)	)	PUNCT
ap-924	177	5	,	,	PUNCT
ap-924	177	6	no	no	INTJ
ap-924	177	7	.	.	NOUN
ap-924	177	8	2	2	NUM
ap-924	177	9	,	,	PUNCT
ap-924	177	10	p.	p.	NOUN
ap-924	177	11	103–105	103–105	NUM
ap-924	177	12	.	.	PUNCT
ap-924	178	1	[	[	X
ap-924	178	2	2	2	NUM
ap-924	178	3	]	]	PUNCT
ap-924	178	4	fraenkel	fraenkel	NOUN
ap-924	178	5	,	,	PUNCT
ap-924	178	6	a.	a.	PROPN
ap-924	178	7	s.	s.	PROPN
ap-924	178	8	:	:	PUNCT
ap-924	178	9	the	the	DET
ap-924	178	10	bracket	bracket	NOUN
ap-924	178	11	function	function	NOUN
ap-924	178	12	and	and	CCONJ
ap-924	178	13	complementary	complementary	ADJ
ap-924	178	14	sets	set	NOUN
ap-924	178	15	of	of	ADP
ap-924	178	16	integers	integer	NOUN
ap-924	178	17	.	.	PUNCT
ap-924	179	1	canad	canad	PROPN
ap-924	179	2	.	.	PUNCT
ap-924	180	1	j.	j.	PROPN
ap-924	180	2	math	math	PROPN
ap-924	180	3	.	.	PROPN
ap-924	180	4	,	,	PUNCT
ap-924	180	5	21	21	NUM
ap-924	180	6	,	,	PUNCT
ap-924	180	7	1969	1969	NUM
ap-924	180	8	,	,	PUNCT
ap-924	180	9	6–27	6–27	NOUN
ap-924	181	1	[	[	X
ap-924	181	2	3	3	NUM
ap-924	181	3	]	]	X
ap-924	181	4	graham	graham	PROPN
ap-924	181	5	,	,	PUNCT
ap-924	181	6	r.	r.	PROPN
ap-924	181	7	l.	l.	PROPN
ap-924	181	8	:	:	PUNCT
ap-924	181	9	covering	cover	VERB
ap-924	181	10	the	the	DET
ap-924	181	11	positive	positive	ADJ
ap-924	181	12	integers	integer	NOUN
ap-924	181	13	by	by	ADP
ap-924	181	14	disjoint	disjoint	NOUN
ap-924	181	15	sets	set	NOUN
ap-924	181	16	of	of	ADP
ap-924	181	17	the	the	DET
ap-924	181	18	form	form	NOUN
ap-924	181	19	�	�	PROPN
ap-924	181	20	�	�	PROPN
ap-924	181	21	�	�	PROPN
ap-924	181	22	�	�	PROPN
ap-924	181	23	n	n	CCONJ
ap-924	181	24	n	n	CCONJ
ap-924	181	25	�	�	PROPN
ap-924	181	26	�	�	PROPN
ap-924	181	27	�	�	PROPN
ap-924	181	28	:	:	PUNCT
ap-924	181	29	,	,	PUNCT
ap-924	181	30	,	,	PUNCT
ap-924	181	31	1	1	NUM
ap-924	181	32	2	2	NUM
ap-924	181	33	�	�	PROPN
ap-924	181	34	.	.	PUNCT
ap-924	182	1	j.	j.	PROPN
ap-924	182	2	combinatorial	combinatorial	PROPN
ap-924	182	3	theory	theory	NOUN
ap-924	182	4	ser	ser	NOUN
ap-924	182	5	.	.	PUNCT
ap-924	183	1	a	a	DET
ap-924	183	2	,	,	PUNCT
ap-924	183	3	vol	vol	NOUN
ap-924	183	4	.	.	NOUN
ap-924	183	5	15	15	NUM
ap-924	183	6	(	(	PUNCT
ap-924	183	7	1973	1973	NUM
ap-924	183	8	)	)	PUNCT
ap-924	183	9	,	,	PUNCT
ap-924	184	1	p.	p.	NOUN
ap-924	184	2	354–358	354–358	NUM
ap-924	184	3	.	.	PUNCT
ap-924	185	1	[	[	X
ap-924	185	2	4	4	NUM
ap-924	185	3	]	]	PUNCT
ap-924	185	4	tijdeman	tijdeman	NOUN
ap-924	185	5	,	,	PUNCT
ap-924	185	6	r.	r.	PROPN
ap-924	185	7	:	:	PUNCT
ap-924	185	8	exact	exact	ADJ
ap-924	185	9	covers	cover	NOUN
ap-924	185	10	of	of	ADP
ap-924	185	11	balanced	balanced	ADJ
ap-924	185	12	sequences	sequence	NOUN
ap-924	185	13	and	and	CCONJ
ap-924	185	14	fraenkel	fraenkel	NOUN
ap-924	185	15	’s	’s	PART
ap-924	185	16	conjecture	conjecture	NOUN
ap-924	185	17	.	.	PUNCT
ap-924	186	1	in	in	ADP
ap-924	186	2	algebraic	algebraic	ADJ
ap-924	186	3	number	number	NOUN
ap-924	186	4	theory	theory	NOUN
ap-924	186	5	and	and	CCONJ
ap-924	186	6	diophantine	diophantine	VERB
ap-924	186	7	analysis	analysis	NOUN
ap-924	186	8	(	(	PUNCT
ap-924	186	9	graz	graz	PROPN
ap-924	186	10	,	,	PUNCT
ap-924	186	11	1998	1998	NUM
ap-924	186	12	)	)	PUNCT
ap-924	186	13	,	,	PUNCT
ap-924	186	14	berlin	berlin	PROPN
ap-924	186	15	:	:	PUNCT
ap-924	186	16	de	de	X
ap-924	186	17	gruyter	gruyter	NOUN
ap-924	186	18	2000	2000	NUM
ap-924	186	19	,	,	PUNCT
ap-924	186	20	p.	p.	NOUN
ap-924	186	21	467–483	467–483	NUM
ap-924	186	22	.	.	PUNCT
ap-924	187	1	[	[	X
ap-924	187	2	5	5	NUM
ap-924	187	3	]	]	X
ap-924	187	4	de	de	X
ap-924	187	5	bruijn	bruijn	NOUN
ap-924	187	6	,	,	PUNCT
ap-924	187	7	n.	n.	PROPN
ap-924	187	8	g.	g.	PROPN
ap-924	187	9	:	:	PUNCT
ap-924	187	10	updown	updown	ADJ
ap-924	187	11	generation	generation	NOUN
ap-924	187	12	of	of	ADP
ap-924	187	13	beatty	beatty	PROPN
ap-924	187	14	sequences	sequences	PROPN
ap-924	187	15	.	.	PUNCT
ap-924	187	16	nederl	nederl	PROPN
ap-924	187	17	.	.	PUNCT
ap-924	187	18	akad	akad	PROPN
ap-924	187	19	.	.	PUNCT
ap-924	188	1	wetensch	wetensch	PROPN
ap-924	188	2	.	.	PUNCT
ap-924	189	1	indag	indag	PROPN
ap-924	189	2	.	.	PUNCT
ap-924	190	1	math	math	PROPN
ap-924	190	2	.	.	PUNCT
ap-924	191	1	,	,	PUNCT
ap-924	191	2	vol	vol	NOUN
ap-924	191	3	.	.	PROPN
ap-924	192	1	51	51	NUM
ap-924	192	2	(	(	PUNCT
ap-924	192	3	1989	1989	NUM
ap-924	192	4	)	)	PUNCT
ap-924	192	5	,	,	PUNCT
ap-924	193	1	p.	p.	NOUN
ap-924	193	2	385–407	385–407	NUM
ap-924	193	3	.	.	PUNCT
ap-924	194	1	[	[	X
ap-924	194	2	6	6	NUM
ap-924	194	3	]	]	X
ap-924	194	4	komatsu	komatsu	NOUN
ap-924	194	5	,	,	PUNCT
ap-924	194	6	t.	t.	PROPN
ap-924	194	7	:	:	PUNCT
ap-924	194	8	a	a	DET
ap-924	194	9	certain	certain	ADJ
ap-924	194	10	power	power	NOUN
ap-924	194	11	series	series	NOUN
ap-924	194	12	associated	associate	VERB
ap-924	194	13	with	with	ADP
ap-924	194	14	a	a	DET
ap-924	194	15	beatty	beatty	PROPN
ap-924	194	16	sequence	sequence	NOUN
ap-924	194	17	.	.	PUNCT
ap-924	195	1	acta	acta	PROPN
ap-924	195	2	arith	arith	PROPN
ap-924	195	3	.	.	PUNCT
ap-924	196	1	,	,	PUNCT
ap-924	196	2	vol	vol	NOUN
ap-924	196	3	.	.	PROPN
ap-924	197	1	76	76	NUM
ap-924	197	2	(	(	PUNCT
ap-924	197	3	1996	1996	NUM
ap-924	197	4	)	)	PUNCT
ap-924	197	5	,	,	PUNCT
ap-924	198	1	p.	p.	NOUN
ap-924	198	2	109–129	109–129	NUM
ap-924	198	3	.	.	PUNCT
ap-924	199	1	[	[	X
ap-924	199	2	7	7	NUM
ap-924	199	3	]	]	SYM
ap-924	199	4	o’bryant	o’bryant	NOUN
ap-924	199	5	,	,	PUNCT
ap-924	199	6	k.	k.	PROPN
ap-924	199	7	:	:	PUNCT
ap-924	199	8	a	a	DET
ap-924	199	9	generating	generate	VERB
ap-924	199	10	function	function	NOUN
ap-924	199	11	technique	technique	NOUN
ap-924	199	12	for	for	ADP
ap-924	199	13	beatty	beatty	PROPN
ap-924	199	14	sequences	sequence	NOUN
ap-924	199	15	and	and	CCONJ
ap-924	199	16	other	other	ADJ
ap-924	199	17	step	step	NOUN
ap-924	199	18	sequences	sequence	NOUN
ap-924	199	19	.	.	PUNCT
ap-924	200	1	j.	j.	PROPN
ap-924	200	2	number	number	PROPN
ap-924	200	3	theory	theory	NOUN
ap-924	200	4	,	,	PUNCT
ap-924	200	5	vol	vol	NOUN
ap-924	200	6	.	.	PROPN
ap-924	200	7	94	94	NUM
ap-924	200	8	(	(	PUNCT
ap-924	200	9	2002	2002	NUM
ap-924	200	10	)	)	PUNCT
ap-924	200	11	,	,	PUNCT
ap-924	200	12	p.	p.	NOUN
ap-924	200	13	299–319	299–319	NUM
ap-924	200	14	.	.	PUNCT
ap-924	201	1	[	[	X
ap-924	201	2	8	8	NUM
ap-924	201	3	]	]	X
ap-924	201	4	komatsu	komatsu	NOUN
ap-924	201	5	,	,	PUNCT
ap-924	201	6	t.	t.	PROPN
ap-924	201	7	:	:	PUNCT
ap-924	201	8	substitution	substitution	NOUN
ap-924	201	9	invariant	invariant	ADJ
ap-924	201	10	inhomogeneous	inhomogeneous	ADJ
ap-924	201	11	beatty	beatty	PROPN
ap-924	201	12	sequences	sequences	PROPN
ap-924	201	13	.	.	PUNCT
ap-924	202	1	tokyo	tokyo	PROPN
ap-924	202	2	journal	journal	PROPN
ap-924	202	3	math	math	PROPN
ap-924	202	4	.	.	PUNCT
ap-924	203	1	,	,	PUNCT
ap-924	203	2	vol	vol	NOUN
ap-924	203	3	.	.	PROPN
ap-924	204	1	22	22	NUM
ap-924	204	2	(	(	PUNCT
ap-924	204	3	1999	1999	NUM
ap-924	204	4	)	)	PUNCT
ap-924	204	5	,	,	PUNCT
ap-924	204	6	p.	p.	NOUN
ap-924	204	7	235–243	235–243	NUM
ap-924	204	8	.	.	PUNCT
ap-924	205	1	[	[	X
ap-924	205	2	9	9	NUM
ap-924	205	3	]	]	PUNCT
ap-924	205	4	parvaix	parvaix	NOUN
ap-924	205	5	,	,	PUNCT
ap-924	205	6	b.	b.	NOUN
ap-924	205	7	:	:	PUNCT
ap-924	205	8	substitution	substitution	NOUN
ap-924	205	9	invariant	invariant	ADJ
ap-924	205	10	sturmian	sturmian	NOUN
ap-924	205	11	bisequences	bisequence	NOUN
ap-924	205	12	.	.	PUNCT
ap-924	206	1	thor	thor	PROPN
ap-924	206	2	.	.	PUNCT
ap-924	207	1	nombres	nombre	NOUN
ap-924	207	2	bordeaux	bordeaux	PROPN
ap-924	207	3	,	,	PUNCT
ap-924	207	4	vol	vol	NOUN
ap-924	207	5	.	.	PROPN
ap-924	207	6	11	11	NUM
ap-924	207	7	(	(	PUNCT
ap-924	207	8	1999	1999	NUM
ap-924	207	9	)	)	PUNCT
ap-924	207	10	,	,	PUNCT
ap-924	207	11	p.	p.	NOUN
ap-924	207	12	201–210	201–210	NUM
ap-924	207	13	.	.	PUNCT
ap-924	208	1	[	[	X
ap-924	208	2	10	10	NUM
ap-924	208	3	]	]	X
ap-924	208	4	brown	brown	ADJ
ap-924	208	5	,	,	PUNCT
ap-924	208	6	t.	t.	PROPN
ap-924	208	7	:	:	PUNCT
ap-924	208	8	descriptions	description	NOUN
ap-924	208	9	of	of	ADP
ap-924	208	10	the	the	DET
ap-924	208	11	characteristic	characteristic	ADJ
ap-924	208	12	sequence	sequence	NOUN
ap-924	208	13	of	of	ADP
ap-924	208	14	an	an	DET
ap-924	208	15	irrational	irrational	ADJ
ap-924	208	16	.	.	PUNCT
ap-924	209	1	canad	canad	PROPN
ap-924	209	2	.	.	PUNCT
ap-924	210	1	math	math	NOUN
ap-924	210	2	.	.	PUNCT
ap-924	211	1	bull	bull	PROPN
ap-924	211	2	.	.	PUNCT
ap-924	212	1	,	,	PUNCT
ap-924	212	2	vol	vol	NOUN
ap-924	212	3	.	.	PROPN
ap-924	213	1	36	36	NUM
ap-924	213	2	(	(	PUNCT
ap-924	213	3	1993	1993	NUM
ap-924	213	4	)	)	PUNCT
ap-924	213	5	,	,	PUNCT
ap-924	214	1	p.	p.	NOUN
ap-924	214	2	15–21	15–21	NUM
ap-924	214	3	.	.	PUNCT
ap-924	215	1	©	©	PROPN
ap-924	215	2	czech	czech	PROPN
ap-924	215	3	technical	technical	PROPN
ap-924	215	4	university	university	PROPN
ap-924	215	5	publishing	publishing	NOUN
ap-924	215	6	house	house	NOUN
ap-924	215	7	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	PROPN
ap-924	215	8	23	23	NUM
ap-924	215	9	acta	acta	PROPN
ap-924	215	10	polytechnica	polytechnica	PROPN
ap-924	215	11	vol	vol	NOUN
ap-924	215	12	.	.	PUNCT
ap-924	216	1	47	47	NUM
ap-924	216	2	no	no	INTJ
ap-924	216	3	.	.	PUNCT
ap-924	217	1	2–3/2007	2–3/2007	NUM
ap-924	218	1	[	[	X
ap-924	218	2	11	11	NUM
ap-924	218	3	]	]	X
ap-924	218	4	stolarsky	stolarsky	PROPN
ap-924	218	5	,	,	PUNCT
ap-924	218	6	k.	k.	PROPN
ap-924	218	7	:	:	PUNCT
ap-924	218	8	beatty	beatty	PROPN
ap-924	218	9	sequences	sequences	PROPN
ap-924	218	10	,	,	PUNCT
ap-924	218	11	continued	continued	ADJ
ap-924	218	12	fractions	fraction	NOUN
ap-924	218	13	,	,	PUNCT
ap-924	218	14	and	and	CCONJ
ap-924	218	15	certain	certain	ADJ
ap-924	218	16	shift	shift	NOUN
ap-924	218	17	operators	operator	NOUN
ap-924	218	18	.	.	PUNCT
ap-924	219	1	canad	canad	PROPN
ap-924	219	2	.	.	PUNCT
ap-924	220	1	math	math	NOUN
ap-924	220	2	.	.	PUNCT
ap-924	221	1	bull	bull	PROPN
ap-924	221	2	.	.	PUNCT
ap-924	222	1	,	,	PUNCT
ap-924	222	2	vol	vol	NOUN
ap-924	222	3	.	.	PROPN
ap-924	222	4	19	19	NUM
ap-924	222	5	(	(	PUNCT
ap-924	222	6	1976	1976	NUM
ap-924	222	7	)	)	PUNCT
ap-924	222	8	,	,	PUNCT
ap-924	223	1	p.	p.	NOUN
ap-924	223	2	473–482	473–482	NUM
ap-924	223	3	.	.	PUNCT
ap-924	224	1	[	[	X
ap-924	224	2	12	12	NUM
ap-924	224	3	]	]	PUNCT
ap-924	224	4	bunder	bunder	NOUN
ap-924	224	5	,	,	PUNCT
ap-924	224	6	m.	m.	NOUN
ap-924	224	7	,	,	PUNCT
ap-924	224	8	tognetti	tognetti	PROPN
ap-924	224	9	,	,	PUNCT
ap-924	224	10	k.	k.	PROPN
ap-924	224	11	:	:	PUNCT
ap-924	224	12	on	on	ADP
ap-924	224	13	the	the	DET
ap-924	224	14	self	self	NOUN
ap-924	224	15	matching	matching	NOUN
ap-924	224	16	properties	property	NOUN
ap-924	224	17	of	of	ADP
ap-924	224	18	[	[	X
ap-924	224	19	j	j	PROPN
ap-924	224	20	�	�	PROPN
ap-924	224	21	]	]	PUNCT
ap-924	224	22	.	.	PUNCT
ap-924	225	1	discr	discr	PROPN
ap-924	225	2	.	.	PUNCT
ap-924	226	1	math	math	NOUN
ap-924	226	2	.	.	PUNCT
ap-924	227	1	,	,	PUNCT
ap-924	227	2	vol	vol	NOUN
ap-924	227	3	.	.	PUNCT
ap-924	228	1	241	241	NUM
ap-924	228	2	(	(	PUNCT
ap-924	228	3	2001	2001	NUM
ap-924	228	4	)	)	PUNCT
ap-924	228	5	,	,	PUNCT
ap-924	229	1	p.	p.	NOUN
ap-924	229	2	139–151	139–151	NUM
ap-924	229	3	.	.	PUNCT
ap-924	230	1	[	[	X
ap-924	230	2	13	13	NUM
ap-924	230	3	]	]	PUNCT
ap-924	230	4	shechtman	shechtman	NOUN
ap-924	230	5	,	,	PUNCT
ap-924	230	6	d.	d.	PROPN
ap-924	230	7	,	,	PUNCT
ap-924	230	8	blech	blech	NOUN
ap-924	230	9	,	,	PUNCT
ap-924	230	10	i.	i.	NOUN
ap-924	230	11	,	,	PUNCT
ap-924	230	12	gratias	gratias	PROPN
ap-924	230	13	,	,	PUNCT
ap-924	230	14	d.	d.	PROPN
ap-924	230	15	,	,	PUNCT
ap-924	230	16	cahn	cahn	PROPN
ap-924	230	17	,	,	PUNCT
ap-924	230	18	j.	j.	PROPN
ap-924	230	19	w.	w.	PROPN
ap-924	230	20	:	:	PUNCT
ap-924	230	21	metallic	metallic	ADJ
ap-924	230	22	phase	phase	NOUN
ap-924	230	23	with	with	ADP
ap-924	230	24	long	long	ADJ
ap-924	230	25	-	-	PUNCT
ap-924	230	26	range	range	NOUN
ap-924	230	27	orientational	orientational	ADJ
ap-924	230	28	order	order	NOUN
ap-924	230	29	and	and	CCONJ
ap-924	230	30	no	no	DET
ap-924	230	31	translation	translation	NOUN
ap-924	230	32	symmetry	symmetry	NOUN
ap-924	230	33	.	.	PUNCT
ap-924	231	1	phys	phy	NOUN
ap-924	231	2	.	.	PUNCT
ap-924	232	1	rev	rev	PROPN
ap-924	232	2	.	.	PROPN
ap-924	232	3	lett	lett	PROPN
ap-924	232	4	.	.	PROPN
ap-924	232	5	,	,	PUNCT
ap-924	232	6	vol	vol	NOUN
ap-924	232	7	.	.	PROPN
ap-924	233	1	53	53	NUM
ap-924	233	2	(	(	PUNCT
ap-924	233	3	1984	1984	NUM
ap-924	233	4	)	)	PUNCT
ap-924	233	5	,	,	PUNCT
ap-924	233	6	p.	p.	NOUN
ap-924	233	7	1951–1953	1951–1953	NUM
ap-924	233	8	.	.	PUNCT
ap-924	234	1	[	[	X
ap-924	234	2	14	14	NUM
ap-924	234	3	]	]	X
ap-924	234	4	gazeau	gazeau	PROPN
ap-924	234	5	,	,	PUNCT
ap-924	234	6	j.	j.	PROPN
ap-924	234	7	p.	p.	PROPN
ap-924	234	8	,	,	PUNCT
ap-924	234	9	masáková	masáková	PROPN
ap-924	234	10	,	,	PUNCT
ap-924	234	11	z.	z.	PROPN
ap-924	234	12	,	,	PUNCT
ap-924	234	13	pelantová	pelantová	PROPN
ap-924	234	14	,	,	PUNCT
ap-924	234	15	e.	e.	PROPN
ap-924	234	16	:	:	PUNCT
ap-924	234	17	nested	nested	ADJ
ap-924	234	18	quasicrystalline	quasicrystalline	PROPN
ap-924	234	19	discretization	discretization	NOUN
ap-924	234	20	of	of	ADP
ap-924	234	21	the	the	DET
ap-924	234	22	line	line	NOUN
ap-924	234	23	.	.	PUNCT
ap-924	235	1	in	in	ADP
ap-924	235	2	:	:	PUNCT
ap-924	235	3	physics	physics	NOUN
ap-924	235	4	and	and	CCONJ
ap-924	235	5	number	number	NOUN
ap-924	235	6	theory	theory	NOUN
ap-924	235	7	(	(	PUNCT
ap-924	235	8	editor	editor	NOUN
ap-924	235	9	:	:	PUNCT
ap-924	235	10	l.	l.	PROPN
ap-924	235	11	nyssen	nyssen	PROPN
ap-924	235	12	)	)	PUNCT
ap-924	235	13	,	,	PUNCT
ap-924	235	14	vol	vol	NOUN
ap-924	235	15	.	.	PROPN
ap-924	235	16	10	10	NUM
ap-924	235	17	of	of	ADP
ap-924	235	18	irma	irma	PROPN
ap-924	235	19	lectures	lecture	NOUN
ap-924	235	20	in	in	ADP
ap-924	235	21	mathematics	mathematic	NOUN
ap-924	235	22	and	and	CCONJ
ap-924	235	23	theoretical	theoretical	ADJ
ap-924	235	24	physics	physics	PROPN
ap-924	235	25	,	,	PUNCT
ap-924	235	26	zürich	zürich	PROPN
ap-924	235	27	,	,	PUNCT
ap-924	235	28	ems	ems	PROPN
ap-924	235	29	2006	2006	NUM
ap-924	235	30	,	,	PUNCT
ap-924	235	31	p.	p.	NOUN
ap-924	235	32	79–132	79–132	NUM
ap-924	235	33	.	.	PUNCT
ap-924	236	1	[	[	X
ap-924	236	2	15	15	NUM
ap-924	236	3	]	]	X
ap-924	236	4	weyl	weyl	PROPN
ap-924	236	5	,	,	PUNCT
ap-924	236	6	h.	h.	PROPN
ap-924	236	7	:	:	PUNCT
ap-924	236	8	über	über	PROPN
ap-924	236	9	die	die	PROPN
ap-924	236	10	gleichverteilung	gleichverteilung	PROPN
ap-924	236	11	von	von	PROPN
ap-924	236	12	zahlen	zahlen	PROPN
ap-924	236	13	mod	mod	PROPN
ap-924	236	14	.	.	PROPN
ap-924	236	15	eins	eins	PROPN
ap-924	236	16	.	.	PUNCT
ap-924	237	1	math	math	NOUN
ap-924	237	2	.	.	PUNCT
ap-924	238	1	ann	ann	PROPN
ap-924	238	2	.	.	PROPN
ap-924	238	3	,	,	PUNCT
ap-924	238	4	vol	vol	NOUN
ap-924	238	5	.	.	PROPN
ap-924	239	1	77	77	NUM
ap-924	239	2	(	(	PUNCT
ap-924	239	3	1916	1916	NUM
ap-924	239	4	)	)	PUNCT
ap-924	239	5	,	,	PUNCT
ap-924	240	1	p.	p.	NOUN
ap-924	240	2	313–352	313–352	NUM
ap-924	240	3	.	.	PUNCT
ap-924	241	1	doc	doc	PROPN
ap-924	241	2	.	.	PROPN
ap-924	242	1	ing	ing	PROPN
ap-924	242	2	.	.	PUNCT
ap-924	243	1	zuzana	zuzana	PROPN
ap-924	243	2	masáková	masáková	PROPN
ap-924	243	3	,	,	PUNCT
ap-924	243	4	ph.d	ph.d	PROPN
ap-924	243	5	.	.	PUNCT
ap-924	243	6	phone	phone	NOUN
ap-924	243	7	:	:	PUNCT
ap-924	243	8	+420	+420	PROPN
ap-924	243	9	224	224	NUM
ap-924	243	10	358	358	NUM
ap-924	243	11	544	544	NUM
ap-924	243	12	e	e	NOUN
ap-924	243	13	-	-	NOUN
ap-924	243	14	mail	mail	NOUN
ap-924	243	15	:	:	PUNCT
ap-924	243	16	masakova@km1.fjfi.cvut.cz	masakova@km1.fjfi.cvut.cz	NOUN
ap-924	243	17	,	,	PUNCT
ap-924	243	18	prof	prof	NOUN
ap-924	243	19	.	.	PUNCT
ap-924	244	1	ing	ing	PROPN
ap-924	244	2	.	.	PUNCT
ap-924	245	1	edita	edita	PROPN
ap-924	245	2	pelantová	pelantová	PROPN
ap-924	245	3	,	,	PUNCT
ap-924	245	4	csc	csc	PROPN
ap-924	245	5	.	.	PROPN
ap-924	245	6	phone	phone	NOUN
ap-924	245	7	:	:	PUNCT
ap-924	245	8	+420	+420	PROPN
ap-924	245	9	224	224	NUM
ap-924	245	10	358	358	NUM
ap-924	245	11	544	544	NUM
ap-924	245	12	e	e	NOUN
ap-924	245	13	-	-	NOUN
ap-924	245	14	mail	mail	NOUN
ap-924	245	15	:	:	PUNCT
ap-924	245	16	pelantova@km1.fjfi.cvut.cz	pelantova@km1.fjfi.cvut.cz	PROPN
ap-924	245	17	doppler	doppler	NOUN
ap-924	245	18	institute	institute	NOUN
ap-924	245	19	for	for	ADP
ap-924	245	20	mathematical	mathematical	ADJ
ap-924	245	21	physics	physics	NOUN
ap-924	245	22	and	and	CCONJ
ap-924	245	23	applied	apply	VERB
ap-924	245	24	mathematics	mathematic	NOUN
ap-924	245	25	czech	czech	PROPN
ap-924	245	26	technical	technical	PROPN
ap-924	245	27	university	university	PROPN
ap-924	245	28	in	in	ADP
ap-924	245	29	prague	prague	PROPN
ap-924	245	30	faculty	faculty	NOUN
ap-924	245	31	of	of	ADP
ap-924	245	32	nuclear	nuclear	ADJ
ap-924	245	33	sciences	science	NOUN
ap-924	245	34	and	and	CCONJ
ap-924	245	35	physical	physical	ADJ
ap-924	245	36	engineering	engineering	NOUN
ap-924	245	37	trojanova	trojanova	ADP
ap-924	245	38	13	13	NUM
ap-924	245	39	120	120	NUM
ap-924	245	40	00	00	NUM
ap-924	245	41	praha	praha	PROPN
ap-924	245	42	2	2	NUM
ap-924	245	43	,	,	PUNCT
ap-924	245	44	czech	czech	PROPN
ap-924	245	45	republic	republic	NOUN
ap-924	245	46	24	24	NUM
ap-924	245	47	©	©	PROPN
ap-924	245	48	czech	czech	PROPN
ap-924	245	49	technical	technical	PROPN
ap-924	245	50	university	university	PROPN
ap-924	245	51	publishing	publishing	NOUN
ap-924	245	52	house	house	NOUN
ap-924	245	53	http://ctn.cvut.cz/ap/	http://ctn.cvut.cz/ap/	PROPN
ap-924	245	54	acta	acta	PROPN
ap-924	245	55	polytechnica	polytechnica	PROPN
ap-924	245	56	vol	vol	NOUN
ap-924	245	57	.	.	PUNCT
ap-924	246	1	47	47	NUM
ap-924	246	2	no	no	INTJ
ap-924	246	3	.	.	PUNCT
ap-924	247	1	2–3/2007	2–3/2007	NUM