id	sid	tid	token	lemma	pos
ap-4587	1	1	acta	acta	PROPN
ap-4587	1	2	polytechnica	polytechnica	PROPN
ap-4587	1	3	doi:10.14311	doi:10.14311	PROPN
ap-4587	1	4	/	/	SYM
ap-4587	1	5	ap.2017.57.0430	ap.2017.57.0430	PROPN
ap-4587	1	6	acta	acta	PROPN
ap-4587	1	7	polytechnica	polytechnica	PROPN
ap-4587	1	8	57(6):430–445	57(6):430–445	PROPN
ap-4587	1	9	,	,	PUNCT
ap-4587	1	10	2017	2017	NUM
ap-4587	1	11	©	©	PROPN
ap-4587	1	12	czech	czech	PROPN
ap-4587	1	13	technical	technical	PROPN
ap-4587	1	14	university	university	PROPN
ap-4587	1	15	in	in	ADP
ap-4587	1	16	prague	prague	PROPN
ap-4587	1	17	,	,	PUNCT
ap-4587	1	18	2017	2017	NUM
ap-4587	1	19	available	available	ADJ
ap-4587	1	20	online	online	ADV
ap-4587	1	21	at	at	ADP
ap-4587	1	22	http://ojs.cvut.cz/ojs/index.php/ap	http://ojs.cvut.cz/ojs/index.php/ap	PROPN
ap-4587	1	23	on	on	ADP
ap-4587	1	24	self	self	NOUN
ap-4587	1	25	-	-	PUNCT
ap-4587	1	26	similarities	similarity	NOUN
ap-4587	1	27	of	of	ADP
ap-4587	1	28	cut	cut	NOUN
ap-4587	1	29	-	-	PUNCT
ap-4587	1	30	and	and	CCONJ
ap-4587	1	31	-	-	PUNCT
ap-4587	1	32	project	project	NOUN
ap-4587	1	33	sets	set	VERB
ap-4587	1	34	zuzana	zuzana	PROPN
ap-4587	1	35	masáková∗	masáková∗	PROPN
ap-4587	1	36	,	,	PUNCT
ap-4587	1	37	jan	jan	PROPN
ap-4587	1	38	mazáč	mazáč	PROPN
ap-4587	1	39	department	department	PROPN
ap-4587	1	40	of	of	ADP
ap-4587	1	41	mathematics	mathematics	PROPN
ap-4587	1	42	,	,	PUNCT
ap-4587	1	43	fnspe	fnspe	PROPN
ap-4587	1	44	,	,	PUNCT
ap-4587	1	45	czech	czech	PROPN
ap-4587	1	46	technical	technical	PROPN
ap-4587	1	47	university	university	PROPN
ap-4587	1	48	in	in	ADP
ap-4587	1	49	prague	prague	PROPN
ap-4587	1	50	,	,	PUNCT
ap-4587	1	51	trojanova	trojanova	X
ap-4587	1	52	13	13	NUM
ap-4587	1	53	,	,	PUNCT
ap-4587	1	54	120	120	NUM
ap-4587	1	55	00	00	NUM
ap-4587	1	56	praha	praha	PROPN
ap-4587	1	57	2	2	NUM
ap-4587	1	58	,	,	PUNCT
ap-4587	1	59	czech	czech	PROPN
ap-4587	1	60	republic	republic	NOUN
ap-4587	1	61	∗	∗	NOUN
ap-4587	1	62	corresponding	correspond	VERB
ap-4587	1	63	author	author	NOUN
ap-4587	1	64	:	:	PUNCT
ap-4587	1	65	zuzana.masakova@fjfi.cvut.cz	zuzana.masakova@fjfi.cvut.cz	PROPN
ap-4587	1	66	abstract	abstract	PROPN
ap-4587	1	67	.	.	PUNCT
ap-4587	2	1	among	among	ADP
ap-4587	2	2	the	the	DET
ap-4587	2	3	commonly	commonly	ADV
ap-4587	2	4	used	use	VERB
ap-4587	2	5	mathematical	mathematical	ADJ
ap-4587	2	6	models	model	NOUN
ap-4587	2	7	of	of	ADP
ap-4587	2	8	quasicrystals	quasicrystal	NOUN
ap-4587	2	9	are	be	AUX
ap-4587	2	10	delone	delone	NOUN
ap-4587	2	11	sets	set	NOUN
ap-4587	2	12	constructed	construct	VERB
ap-4587	2	13	using	use	VERB
ap-4587	2	14	a	a	DET
ap-4587	2	15	cut	cut	VERB
ap-4587	2	16	-	-	PUNCT
ap-4587	2	17	and	and	CCONJ
ap-4587	2	18	-	-	PUNCT
ap-4587	2	19	project	project	NOUN
ap-4587	2	20	scheme	scheme	NOUN
ap-4587	2	21	,	,	PUNCT
ap-4587	2	22	the	the	DET
ap-4587	2	23	so	so	ADV
ap-4587	2	24	-	-	PUNCT
ap-4587	2	25	called	call	VERB
ap-4587	2	26	cut	cut	NOUN
ap-4587	2	27	-	-	PUNCT
ap-4587	2	28	and	and	CCONJ
ap-4587	2	29	-	-	PUNCT
ap-4587	2	30	project	project	NOUN
ap-4587	2	31	sets	set	NOUN
ap-4587	2	32	.	.	PUNCT
ap-4587	3	1	a	a	DET
ap-4587	3	2	cut	cut	VERB
ap-4587	3	3	-	-	PUNCT
ap-4587	3	4	and	and	CCONJ
ap-4587	3	5	-	-	PUNCT
ap-4587	3	6	project	project	NOUN
ap-4587	3	7	scheme	scheme	NOUN
ap-4587	3	8	(	(	PUNCT
ap-4587	3	9	l	l	NOUN
ap-4587	3	10	,	,	PUNCT
ap-4587	3	11	π1	π1	NOUN
ap-4587	3	12	,	,	PUNCT
ap-4587	3	13	π2	π2	PROPN
ap-4587	3	14	)	)	PUNCT
ap-4587	3	15	is	be	AUX
ap-4587	3	16	given	give	VERB
ap-4587	3	17	by	by	ADP
ap-4587	3	18	a	a	DET
ap-4587	3	19	lattice	lattice	ADJ
ap-4587	3	20	l	l	NOUN
ap-4587	3	21	in	in	ADP
ap-4587	3	22	rs	rs	NOUN
ap-4587	3	23	and	and	CCONJ
ap-4587	3	24	projections	projection	NOUN
ap-4587	3	25	π1	π1	NOUN
ap-4587	3	26	,	,	PUNCT
ap-4587	3	27	π2	π2	ADJ
ap-4587	3	28	to	to	ADP
ap-4587	3	29	suitable	suitable	ADJ
ap-4587	3	30	subspaces	subspace	NOUN
ap-4587	3	31	v1	v1	NOUN
ap-4587	3	32	,	,	PUNCT
ap-4587	3	33	v2	v2	PROPN
ap-4587	3	34	.	.	PUNCT
ap-4587	4	1	in	in	ADP
ap-4587	4	2	this	this	DET
ap-4587	4	3	paper	paper	NOUN
ap-4587	4	4	we	we	PRON
ap-4587	4	5	derive	derive	VERB
ap-4587	4	6	several	several	ADJ
ap-4587	4	7	statements	statement	NOUN
ap-4587	4	8	describing	describe	VERB
ap-4587	4	9	the	the	DET
ap-4587	4	10	connection	connection	NOUN
ap-4587	4	11	between	between	ADP
ap-4587	4	12	self	self	NOUN
ap-4587	4	13	-	-	PUNCT
ap-4587	4	14	similarity	similarity	NOUN
ap-4587	4	15	transformations	transformation	NOUN
ap-4587	4	16	of	of	ADP
ap-4587	4	17	the	the	DET
ap-4587	4	18	lattice	lattice	PROPN
ap-4587	4	19	l	l	NOUN
ap-4587	4	20	and	and	CCONJ
ap-4587	4	21	transformations	transformation	NOUN
ap-4587	4	22	of	of	ADP
ap-4587	4	23	its	its	PRON
ap-4587	4	24	projections	projection	NOUN
ap-4587	4	25	π1(l	π1(l	NOUN
ap-4587	4	26	)	)	PUNCT
ap-4587	4	27	,	,	PUNCT
ap-4587	4	28	π2(l	π2(l	PROPN
ap-4587	4	29	)	)	PUNCT
ap-4587	4	30	.	.	PUNCT
ap-4587	5	1	for	for	ADP
ap-4587	5	2	a	a	DET
ap-4587	5	3	self	self	NOUN
ap-4587	5	4	-	-	PUNCT
ap-4587	5	5	similarity	similarity	NOUN
ap-4587	5	6	of	of	ADP
ap-4587	5	7	a	a	DET
ap-4587	5	8	set	set	NOUN
ap-4587	5	9	σ	σ	NOUN
ap-4587	5	10	we	we	PRON
ap-4587	5	11	take	take	VERB
ap-4587	5	12	any	any	DET
ap-4587	5	13	linear	linear	ADJ
ap-4587	5	14	mapping	mapping	NOUN
ap-4587	5	15	a	a	DET
ap-4587	5	16	such	such	ADJ
ap-4587	5	17	that	that	PRON
ap-4587	5	18	aς	aς	PROPN
ap-4587	5	19	⊂	⊂	PROPN
ap-4587	5	20	σ	σ	PROPN
ap-4587	5	21	,	,	PUNCT
ap-4587	5	22	which	which	PRON
ap-4587	5	23	generalizes	generalize	VERB
ap-4587	5	24	the	the	DET
ap-4587	5	25	notion	notion	NOUN
ap-4587	5	26	of	of	ADP
ap-4587	5	27	self	self	NOUN
ap-4587	5	28	-	-	PUNCT
ap-4587	5	29	similarity	similarity	NOUN
ap-4587	5	30	usually	usually	ADV
ap-4587	5	31	restricted	restrict	VERB
ap-4587	5	32	to	to	ADP
ap-4587	5	33	scaled	scale	VERB
ap-4587	5	34	rotations	rotation	NOUN
ap-4587	5	35	.	.	PUNCT
ap-4587	6	1	we	we	PRON
ap-4587	6	2	describe	describe	VERB
ap-4587	6	3	a	a	DET
ap-4587	6	4	method	method	NOUN
ap-4587	6	5	of	of	ADP
ap-4587	6	6	construction	construction	NOUN
ap-4587	6	7	of	of	ADP
ap-4587	6	8	cut	cut	NOUN
ap-4587	6	9	-	-	PUNCT
ap-4587	6	10	and	and	CCONJ
ap-4587	6	11	-	-	PUNCT
ap-4587	6	12	project	project	NOUN
ap-4587	6	13	schemes	scheme	NOUN
ap-4587	6	14	with	with	ADP
ap-4587	6	15	required	require	VERB
ap-4587	6	16	self	self	NOUN
ap-4587	6	17	-	-	PUNCT
ap-4587	6	18	similarities	similarity	NOUN
ap-4587	6	19	and	and	CCONJ
ap-4587	6	20	apply	apply	VERB
ap-4587	6	21	it	it	PRON
ap-4587	6	22	to	to	PART
ap-4587	6	23	produce	produce	VERB
ap-4587	6	24	a	a	DET
ap-4587	6	25	cut	cut	VERB
ap-4587	6	26	-	-	PUNCT
ap-4587	6	27	and	and	CCONJ
ap-4587	6	28	-	-	PUNCT
ap-4587	6	29	project	project	NOUN
ap-4587	6	30	scheme	scheme	NOUN
ap-4587	6	31	such	such	ADJ
ap-4587	6	32	that	that	PRON
ap-4587	6	33	π1(l	π1(l	X
ap-4587	6	34	)	)	PUNCT
ap-4587	6	35	⊂	⊂	PROPN
ap-4587	6	36	r2	r2	PROPN
ap-4587	6	37	is	be	AUX
ap-4587	6	38	invariant	invariant	ADJ
ap-4587	6	39	under	under	ADP
ap-4587	6	40	an	an	DET
ap-4587	6	41	isometry	isometry	NOUN
ap-4587	6	42	of	of	ADP
ap-4587	6	43	order	order	NOUN
ap-4587	6	44	5	5	X
ap-4587	6	45	.	.	PUNCT
ap-4587	7	1	we	we	PRON
ap-4587	7	2	describe	describe	VERB
ap-4587	7	3	all	all	DET
ap-4587	7	4	linear	linear	ADJ
ap-4587	7	5	self	self	NOUN
ap-4587	7	6	-	-	PUNCT
ap-4587	7	7	similarities	similarity	NOUN
ap-4587	7	8	of	of	ADP
ap-4587	7	9	this	this	DET
ap-4587	7	10	scheme	scheme	NOUN
ap-4587	7	11	and	and	CCONJ
ap-4587	7	12	show	show	VERB
ap-4587	7	13	that	that	SCONJ
ap-4587	7	14	they	they	PRON
ap-4587	7	15	form	form	VERB
ap-4587	7	16	an	an	DET
ap-4587	7	17	8	8	NUM
ap-4587	7	18	-	-	PUNCT
ap-4587	7	19	dimensional	dimensional	ADJ
ap-4587	7	20	associative	associative	ADJ
ap-4587	7	21	algebra	algebra	NOUN
ap-4587	7	22	over	over	ADP
ap-4587	7	23	the	the	DET
ap-4587	7	24	ring	ring	NOUN
ap-4587	7	25	z.	z.	PROPN
ap-4587	7	26	we	we	PRON
ap-4587	7	27	perform	perform	VERB
ap-4587	7	28	an	an	DET
ap-4587	7	29	example	example	NOUN
ap-4587	7	30	of	of	ADP
ap-4587	7	31	a	a	DET
ap-4587	7	32	cut	cut	NOUN
ap-4587	7	33	-	-	PUNCT
ap-4587	7	34	and	and	CCONJ
ap-4587	7	35	-	-	PUNCT
ap-4587	7	36	project	project	NOUN
ap-4587	7	37	set	set	VERB
ap-4587	7	38	with	with	ADP
ap-4587	7	39	linear	linear	ADJ
ap-4587	7	40	self	self	NOUN
ap-4587	7	41	-	-	PUNCT
ap-4587	7	42	similarity	similarity	NOUN
ap-4587	7	43	which	which	PRON
ap-4587	7	44	is	be	AUX
ap-4587	7	45	not	not	PART
ap-4587	7	46	a	a	DET
ap-4587	7	47	scaled	scale	VERB
ap-4587	7	48	rotation	rotation	NOUN
ap-4587	7	49	.	.	PUNCT
ap-4587	8	1	keywords	keyword	NOUN
ap-4587	8	2	:	:	PUNCT
ap-4587	8	3	self	self	NOUN
ap-4587	8	4	-	-	PUNCT
ap-4587	8	5	similarity	similarity	NOUN
ap-4587	8	6	;	;	PUNCT
ap-4587	8	7	quasicrystal	quasicrystal	ADJ
ap-4587	8	8	;	;	PUNCT
ap-4587	8	9	cut	cut	VERB
ap-4587	8	10	-	-	PUNCT
ap-4587	8	11	and	and	CCONJ
ap-4587	8	12	-	-	PUNCT
ap-4587	8	13	project	project	NOUN
ap-4587	8	14	scheme	scheme	NOUN
ap-4587	8	15	.	.	PUNCT
ap-4587	9	1	1	1	X
ap-4587	9	2	.	.	X
ap-4587	9	3	introduction	introduction	NOUN
ap-4587	9	4	quasicrystals	quasicrystal	NOUN
ap-4587	9	5	,	,	PUNCT
ap-4587	9	6	their	their	PRON
ap-4587	9	7	mathematical	mathematical	ADJ
ap-4587	9	8	models	model	NOUN
ap-4587	9	9	and	and	CCONJ
ap-4587	9	10	their	their	PRON
ap-4587	9	11	physical	physical	ADJ
ap-4587	9	12	properties	property	NOUN
ap-4587	9	13	stand	stand	VERB
ap-4587	9	14	in	in	ADP
ap-4587	9	15	the	the	DET
ap-4587	9	16	front	front	ADJ
ap-4587	9	17	row	row	NOUN
ap-4587	9	18	of	of	ADP
ap-4587	9	19	interest	interest	NOUN
ap-4587	9	20	of	of	ADP
ap-4587	9	21	scientists	scientist	NOUN
ap-4587	9	22	since	since	SCONJ
ap-4587	9	23	1984	1984	NUM
ap-4587	9	24	when	when	SCONJ
ap-4587	9	25	shechtmann	shechtmann	PROPN
ap-4587	9	26	and	and	CCONJ
ap-4587	9	27	his	his	PRON
ap-4587	9	28	collegues	collegue	NOUN
ap-4587	9	29	[	[	X
ap-4587	9	30	18	18	NUM
ap-4587	9	31	]	]	PUNCT
ap-4587	9	32	published	publish	VERB
ap-4587	9	33	his	his	PRON
ap-4587	9	34	1982	1982	NUM
ap-4587	9	35	discovery	discovery	NOUN
ap-4587	9	36	of	of	ADP
ap-4587	9	37	non	non	ADJ
ap-4587	9	38	-	-	ADJ
ap-4587	9	39	crystallographic	crystallographic	ADJ
ap-4587	9	40	materials	material	NOUN
ap-4587	9	41	with	with	ADP
ap-4587	9	42	long	long	ADJ
ap-4587	9	43	-	-	PUNCT
ap-4587	9	44	range	range	NOUN
ap-4587	9	45	order	order	NOUN
ap-4587	9	46	.	.	PUNCT
ap-4587	10	1	advances	advance	NOUN
ap-4587	10	2	in	in	ADP
ap-4587	10	3	the	the	DET
ap-4587	10	4	description	description	NOUN
ap-4587	10	5	of	of	ADP
ap-4587	10	6	these	these	DET
ap-4587	10	7	materials	material	NOUN
ap-4587	10	8	obtained	obtain	VERB
ap-4587	10	9	first	first	ADV
ap-4587	10	10	as	as	SCONJ
ap-4587	10	11	rapidly	rapidly	ADV
ap-4587	10	12	solidified	solidify	VERB
ap-4587	10	13	intermetallic	intermetallic	ADJ
ap-4587	10	14	alloys	alloy	NOUN
ap-4587	10	15	have	have	AUX
ap-4587	10	16	been	be	AUX
ap-4587	10	17	since	since	ADV
ap-4587	10	18	achieved	achieve	VERB
ap-4587	10	19	both	both	PRON
ap-4587	10	20	on	on	ADP
ap-4587	10	21	the	the	DET
ap-4587	10	22	mathematical	mathematical	ADJ
ap-4587	10	23	side	side	NOUN
ap-4587	10	24	and	and	CCONJ
ap-4587	10	25	in	in	ADP
ap-4587	10	26	the	the	DET
ap-4587	10	27	experiments	experiment	NOUN
ap-4587	10	28	.	.	PUNCT
ap-4587	11	1	a	a	DET
ap-4587	11	2	number	number	NOUN
ap-4587	11	3	of	of	ADP
ap-4587	11	4	overview	overview	NOUN
ap-4587	11	5	books	book	NOUN
ap-4587	11	6	and	and	CCONJ
ap-4587	11	7	survey	survey	NOUN
ap-4587	11	8	papers	paper	NOUN
ap-4587	11	9	were	be	AUX
ap-4587	11	10	published	publish	VERB
ap-4587	11	11	,	,	PUNCT
ap-4587	11	12	see	see	VERB
ap-4587	11	13	for	for	ADP
ap-4587	11	14	example	example	NOUN
ap-4587	11	15	[	[	X
ap-4587	11	16	1	1	NUM
ap-4587	11	17	]	]	PUNCT
ap-4587	11	18	.	.	PUNCT
ap-4587	12	1	a	a	DET
ap-4587	12	2	fresh	fresh	ADJ
ap-4587	12	3	impuls	impuls	NOUN
ap-4587	12	4	to	to	ADP
ap-4587	12	5	the	the	DET
ap-4587	12	6	research	research	NOUN
ap-4587	12	7	on	on	ADP
ap-4587	12	8	quasicrystals	quasicrystal	NOUN
ap-4587	12	9	was	be	AUX
ap-4587	12	10	given	give	VERB
ap-4587	12	11	by	by	ADP
ap-4587	12	12	awarding	award	VERB
ap-4587	12	13	2011	2011	NUM
ap-4587	12	14	nobel	nobel	PROPN
ap-4587	12	15	prize	prize	PROPN
ap-4587	12	16	for	for	ADP
ap-4587	12	17	chemistry	chemistry	NOUN
ap-4587	12	18	to	to	ADP
ap-4587	12	19	dan	dan	PROPN
ap-4587	12	20	shechtmann	shechtmann	PROPN
ap-4587	12	21	for	for	ADP
ap-4587	12	22	his	his	PRON
ap-4587	12	23	discovery	discovery	NOUN
ap-4587	12	24	.	.	PUNCT
ap-4587	13	1	while	while	SCONJ
ap-4587	13	2	crystals	crystal	NOUN
ap-4587	13	3	are	be	AUX
ap-4587	13	4	modeled	model	VERB
ap-4587	13	5	by	by	ADP
ap-4587	13	6	periodic	periodic	ADJ
ap-4587	13	7	lattices	lattice	NOUN
ap-4587	13	8	,	,	PUNCT
ap-4587	13	9	as	as	SCONJ
ap-4587	13	10	a	a	DET
ap-4587	13	11	suitable	suitable	ADJ
ap-4587	13	12	mathematical	mathematical	ADJ
ap-4587	13	13	model	model	NOUN
ap-4587	13	14	of	of	ADP
ap-4587	13	15	atomic	atomic	ADJ
ap-4587	13	16	positions	position	NOUN
ap-4587	13	17	in	in	ADP
ap-4587	13	18	quasicrystals	quasicrystal	NOUN
ap-4587	13	19	one	one	PRON
ap-4587	13	20	recognizes	recognize	VERB
ap-4587	13	21	the	the	DET
ap-4587	13	22	cut	cut	VERB
ap-4587	13	23	-	-	PUNCT
ap-4587	13	24	and	and	CCONJ
ap-4587	13	25	-	-	PUNCT
ap-4587	13	26	project	project	NOUN
ap-4587	13	27	method	method	NOUN
ap-4587	13	28	that	that	PRON
ap-4587	13	29	stems	stem	VERB
ap-4587	13	30	in	in	ADP
ap-4587	13	31	projecting	project	VERB
ap-4587	13	32	lattice	lattice	NOUN
ap-4587	13	33	points	point	NOUN
ap-4587	13	34	from	from	ADP
ap-4587	13	35	a	a	DET
ap-4587	13	36	higherdimensional	higherdimensional	ADJ
ap-4587	13	37	space	space	NOUN
ap-4587	13	38	to	to	ADP
ap-4587	13	39	suitable	suitable	ADJ
ap-4587	13	40	subspaces	subspace	NOUN
ap-4587	13	41	(	(	PUNCT
ap-4587	13	42	called	call	VERB
ap-4587	13	43	usually	usually	ADV
ap-4587	13	44	physical	physical	ADJ
ap-4587	13	45	and	and	CCONJ
ap-4587	13	46	inner	inner	ADJ
ap-4587	13	47	spaces	space	NOUN
ap-4587	13	48	)	)	PUNCT
ap-4587	13	49	.	.	PUNCT
ap-4587	14	1	a	a	DET
ap-4587	14	2	cut	cut	VERB
ap-4587	14	3	-	-	PUNCT
ap-4587	14	4	and	and	CCONJ
ap-4587	14	5	-	-	PUNCT
ap-4587	14	6	project	project	NOUN
ap-4587	14	7	scheme	scheme	NOUN
ap-4587	14	8	is	be	AUX
ap-4587	14	9	thus	thus	ADV
ap-4587	14	10	given	give	VERB
ap-4587	14	11	by	by	ADP
ap-4587	14	12	a	a	DET
ap-4587	14	13	lattice	lattice	NOUN
ap-4587	14	14	l	l	NOUN
ap-4587	14	15	and	and	CCONJ
ap-4587	14	16	two	two	NUM
ap-4587	14	17	projections	projection	NOUN
ap-4587	14	18	π1	π1	NOUN
ap-4587	14	19	,	,	PUNCT
ap-4587	14	20	π2	π2	PROPN
ap-4587	14	21	,	,	PUNCT
ap-4587	14	22	see	see	VERB
ap-4587	14	23	details	detail	NOUN
ap-4587	14	24	in	in	ADP
ap-4587	14	25	section	section	NOUN
ap-4587	14	26	2	2	NUM
ap-4587	14	27	.	.	PUNCT
ap-4587	14	28	then	then	ADV
ap-4587	14	29	,	,	PUNCT
ap-4587	14	30	choosing	choose	VERB
ap-4587	14	31	properly	properly	ADV
ap-4587	14	32	a	a	DET
ap-4587	14	33	delone	delone	NOUN
ap-4587	14	34	subset	subset	VERB
ap-4587	14	35	σ(ω	σ(ω	PROPN
ap-4587	14	36	)	)	PUNCT
ap-4587	14	37	of	of	ADP
ap-4587	14	38	π1(l	π1(l	PROPN
ap-4587	14	39	)	)	PUNCT
ap-4587	14	40	one	one	NOUN
ap-4587	14	41	obtains	obtain	VERB
ap-4587	14	42	a	a	DET
ap-4587	14	43	quasiperiodic	quasiperiodic	ADJ
ap-4587	14	44	structure	structure	NOUN
ap-4587	14	45	in	in	ADP
ap-4587	14	46	the	the	DET
ap-4587	14	47	physical	physical	ADJ
ap-4587	14	48	space	space	NOUN
ap-4587	14	49	which	which	PRON
ap-4587	14	50	allows	allow	VERB
ap-4587	14	51	symmetries	symmetry	NOUN
ap-4587	14	52	forbidden	forbid	VERB
ap-4587	14	53	for	for	ADP
ap-4587	14	54	periodic	periodic	ADJ
ap-4587	14	55	sets	set	NOUN
ap-4587	14	56	by	by	ADP
ap-4587	14	57	the	the	DET
ap-4587	14	58	crystallographic	crystallographic	ADJ
ap-4587	14	59	restriction	restriction	NOUN
ap-4587	14	60	theorem	theorem	VERB
ap-4587	14	61	.	.	PUNCT
ap-4587	15	1	the	the	DET
ap-4587	15	2	choice	choice	NOUN
ap-4587	15	3	is	be	AUX
ap-4587	15	4	directed	direct	VERB
ap-4587	15	5	by	by	ADP
ap-4587	15	6	a	a	DET
ap-4587	15	7	suitable	suitable	ADJ
ap-4587	15	8	window	window	NOUN
ap-4587	15	9	ω	ω	PROPN
ap-4587	15	10	in	in	ADP
ap-4587	15	11	the	the	DET
ap-4587	15	12	inner	inner	ADJ
ap-4587	15	13	space	space	NOUN
ap-4587	15	14	.	.	PUNCT
ap-4587	16	1	the	the	DET
ap-4587	16	2	set	set	ADJ
ap-4587	16	3	σ(ω	σ(ω	PROPN
ap-4587	16	4	)	)	PUNCT
ap-4587	16	5	is	be	AUX
ap-4587	16	6	then	then	ADV
ap-4587	16	7	called	call	VERB
ap-4587	16	8	a	a	DET
ap-4587	16	9	cut	cut	VERB
ap-4587	16	10	-	-	PUNCT
ap-4587	16	11	and	and	CCONJ
ap-4587	16	12	-	-	PUNCT
ap-4587	16	13	project	project	NOUN
ap-4587	16	14	set	set	NOUN
ap-4587	16	15	.	.	PUNCT
ap-4587	17	1	the	the	DET
ap-4587	17	2	origins	origin	NOUN
ap-4587	17	3	of	of	ADP
ap-4587	17	4	the	the	DET
ap-4587	17	5	idea	idea	NOUN
ap-4587	17	6	can	can	AUX
ap-4587	17	7	be	be	AUX
ap-4587	17	8	stepped	step	VERB
ap-4587	17	9	back	back	ADV
ap-4587	17	10	to	to	ADP
ap-4587	17	11	times	time	NOUN
ap-4587	17	12	long	long	ADV
ap-4587	17	13	before	before	ADP
ap-4587	17	14	quasicrystal	quasicrystal	ADJ
ap-4587	17	15	discovery	discovery	NOUN
ap-4587	17	16	,	,	PUNCT
ap-4587	17	17	to	to	ADP
ap-4587	17	18	bohr	bohr	PROPN
ap-4587	18	1	[	[	X
ap-4587	18	2	6	6	NUM
ap-4587	18	3	]	]	PUNCT
ap-4587	18	4	who	who	PRON
ap-4587	18	5	developed	develop	VERB
ap-4587	18	6	his	his	PRON
ap-4587	18	7	theory	theory	NOUN
ap-4587	18	8	of	of	ADP
ap-4587	18	9	quasiperiodic	quasiperiodic	ADJ
ap-4587	18	10	functions	function	NOUN
ap-4587	18	11	,	,	PUNCT
ap-4587	18	12	and	and	CCONJ
ap-4587	18	13	then	then	ADV
ap-4587	18	14	to	to	ADP
ap-4587	18	15	meyer	meyer	PROPN
ap-4587	18	16	in	in	ADP
ap-4587	18	17	connection	connection	NOUN
ap-4587	18	18	to	to	PART
ap-4587	18	19	harmonic	harmonic	VERB
ap-4587	18	20	analysis	analysis	NOUN
ap-4587	18	21	[	[	X
ap-4587	18	22	15	15	NUM
ap-4587	18	23	]	]	PUNCT
ap-4587	18	24	.	.	PUNCT
ap-4587	19	1	de	de	X
ap-4587	19	2	bruijn	bruijn	PROPN
ap-4587	19	3	performed	perform	VERB
ap-4587	19	4	[	[	X
ap-4587	19	5	8	8	NUM
ap-4587	19	6	]	]	PUNCT
ap-4587	19	7	this	this	DET
ap-4587	19	8	construction	construction	NOUN
ap-4587	19	9	for	for	ADP
ap-4587	19	10	obtaining	obtain	VERB
ap-4587	19	11	the	the	DET
ap-4587	19	12	vertices	vertex	NOUN
ap-4587	19	13	of	of	ADP
ap-4587	19	14	the	the	DET
ap-4587	19	15	famous	famous	ADJ
ap-4587	19	16	penrose	penrose	NOUN
ap-4587	19	17	tiling	tile	VERB
ap-4587	19	18	[	[	X
ap-4587	19	19	17	17	NUM
ap-4587	19	20	]	]	PUNCT
ap-4587	19	21	.	.	PUNCT
ap-4587	20	1	the	the	DET
ap-4587	20	2	utility	utility	NOUN
ap-4587	20	3	of	of	ADP
ap-4587	20	4	this	this	DET
ap-4587	20	5	method	method	NOUN
ap-4587	20	6	for	for	ADP
ap-4587	20	7	constructing	construct	VERB
ap-4587	20	8	quasicrystal	quasicrystal	ADJ
ap-4587	20	9	models	model	NOUN
ap-4587	20	10	was	be	AUX
ap-4587	20	11	then	then	ADV
ap-4587	20	12	recognized	recognize	VERB
ap-4587	20	13	by	by	ADP
ap-4587	20	14	kramer	kramer	PROPN
ap-4587	20	15	and	and	CCONJ
ap-4587	20	16	neri	neri	PROPN
ap-4587	20	17	[	[	X
ap-4587	20	18	10	10	NUM
ap-4587	20	19	]	]	PUNCT
ap-4587	20	20	.	.	PUNCT
ap-4587	21	1	when	when	SCONJ
ap-4587	21	2	studying	study	VERB
ap-4587	21	3	two	two	NUM
ap-4587	21	4	-	-	PUNCT
ap-4587	21	5	dimensional	dimensional	ADJ
ap-4587	21	6	quasicrystal	quasicrystal	ADJ
ap-4587	21	7	models	model	NOUN
ap-4587	21	8	,	,	PUNCT
ap-4587	21	9	one	one	PRON
ap-4587	21	10	is	be	AUX
ap-4587	21	11	interested	interested	ADJ
ap-4587	21	12	in	in	ADP
ap-4587	21	13	those	those	PRON
ap-4587	21	14	displaying	display	VERB
ap-4587	21	15	5-	5-	NUM
ap-4587	21	16	,	,	PUNCT
ap-4587	21	17	8-	8-	PROPN
ap-4587	21	18	,	,	PUNCT
ap-4587	21	19	10and	10and	NOUN
ap-4587	21	20	12	12	NUM
ap-4587	21	21	-	-	ADJ
ap-4587	21	22	fold	fold	ADJ
ap-4587	21	23	rotational	rotational	ADJ
ap-4587	21	24	symmetry	symmetry	NOUN
ap-4587	21	25	which	which	PRON
ap-4587	21	26	corresponds	correspond	VERB
ap-4587	21	27	to	to	ADP
ap-4587	21	28	experimentally	experimentally	ADV
ap-4587	21	29	observed	observe	VERB
ap-4587	21	30	cases	case	NOUN
ap-4587	21	31	[	[	X
ap-4587	21	32	19	19	NUM
ap-4587	21	33	]	]	PUNCT
ap-4587	21	34	.	.	PUNCT
ap-4587	22	1	the	the	DET
ap-4587	22	2	family	family	NOUN
ap-4587	22	3	of	of	ADP
ap-4587	22	4	symmetries	symmetry	NOUN
ap-4587	22	5	is	be	AUX
ap-4587	22	6	however	however	ADV
ap-4587	22	7	much	much	ADV
ap-4587	22	8	more	more	ADV
ap-4587	22	9	rich	rich	ADJ
ap-4587	22	10	;	;	PUNCT
ap-4587	22	11	besides	besides	SCONJ
ap-4587	22	12	rotations	rotation	NOUN
ap-4587	22	13	/	/	SYM
ap-4587	22	14	reflections	reflection	NOUN
ap-4587	22	15	it	it	PRON
ap-4587	22	16	contains	contain	VERB
ap-4587	22	17	scalings	scaling	NOUN
ap-4587	22	18	by	by	ADP
ap-4587	22	19	irrational	irrational	ADJ
ap-4587	22	20	factors	factor	NOUN
ap-4587	22	21	and	and	CCONJ
ap-4587	22	22	other	other	ADJ
ap-4587	22	23	affine	affine	NOUN
ap-4587	22	24	symmetries	symmetry	NOUN
ap-4587	22	25	.	.	PUNCT
ap-4587	23	1	it	it	PRON
ap-4587	23	2	follows	follow	VERB
ap-4587	23	3	from	from	ADP
ap-4587	23	4	the	the	DET
ap-4587	23	5	results	result	NOUN
ap-4587	23	6	of	of	ADP
ap-4587	23	7	lagarias	lagaria	NOUN
ap-4587	23	8	[	[	X
ap-4587	23	9	11	11	NUM
ap-4587	23	10	]	]	PUNCT
ap-4587	23	11	that	that	SCONJ
ap-4587	23	12	if	if	SCONJ
ap-4587	23	13	σ(ω	σ(ω	PROPN
ap-4587	23	14	)	)	PUNCT
ap-4587	23	15	is	be	AUX
ap-4587	23	16	a	a	DET
ap-4587	23	17	cut	cut	VERB
ap-4587	23	18	-	-	PUNCT
ap-4587	23	19	and	and	CCONJ
ap-4587	23	20	-	-	PUNCT
ap-4587	23	21	project	project	NOUN
ap-4587	23	22	set	set	NOUN
ap-4587	23	23	and	and	CCONJ
ap-4587	23	24	η	η	PROPN
ap-4587	23	25	>	>	X
ap-4587	23	26	1	1	NUM
ap-4587	23	27	is	be	AUX
ap-4587	23	28	such	such	ADJ
ap-4587	23	29	that	that	SCONJ
ap-4587	23	30	ης(ω	ης(ω	NOUN
ap-4587	23	31	)	)	PUNCT
ap-4587	24	1	⊂	⊂	PROPN
ap-4587	24	2	σ(ω	σ(ω	PROPN
ap-4587	24	3	)	)	PUNCT
ap-4587	24	4	,	,	PUNCT
ap-4587	24	5	than	than	SCONJ
ap-4587	24	6	η	η	PROPN
ap-4587	24	7	can	can	AUX
ap-4587	24	8	only	only	ADV
ap-4587	24	9	be	be	AUX
ap-4587	24	10	a	a	DET
ap-4587	24	11	pisot	pisot	NOUN
ap-4587	24	12	or	or	CCONJ
ap-4587	24	13	salem	salem	NOUN
ap-4587	24	14	number	number	NOUN
ap-4587	24	15	.	.	PUNCT
ap-4587	25	1	some	some	DET
ap-4587	25	2	authors	author	NOUN
ap-4587	25	3	[	[	X
ap-4587	25	4	2	2	X
ap-4587	25	5	]	]	PUNCT
ap-4587	25	6	have	have	AUX
ap-4587	25	7	considered	consider	VERB
ap-4587	25	8	self	self	NOUN
ap-4587	25	9	-	-	PUNCT
ap-4587	25	10	similarities	similarity	NOUN
ap-4587	25	11	of	of	ADP
ap-4587	25	12	quasilattices	quasilattice	NOUN
ap-4587	25	13	in	in	ADP
ap-4587	25	14	the	the	DET
ap-4587	25	15	form	form	NOUN
ap-4587	25	16	of	of	ADP
ap-4587	25	17	scaled	scale	VERB
ap-4587	25	18	rotations	rotation	NOUN
ap-4587	25	19	,	,	PUNCT
ap-4587	25	20	i.e.	i.e.	X
ap-4587	25	21	,	,	PUNCT
ap-4587	25	22	mappings	mapping	NOUN
ap-4587	25	23	ηr	ηr	NOUN
ap-4587	25	24	,	,	PUNCT
ap-4587	25	25	where	where	SCONJ
ap-4587	25	26	η	η	PROPN
ap-4587	25	27	>	>	X
ap-4587	25	28	1	1	NUM
ap-4587	25	29	and	and	CCONJ
ap-4587	25	30	r	r	NOUN
ap-4587	25	31	is	be	AUX
ap-4587	25	32	an	an	DET
ap-4587	25	33	orthogonal	orthogonal	ADJ
ap-4587	25	34	map	map	NOUN
ap-4587	25	35	.	.	PUNCT
ap-4587	26	1	according	accord	VERB
ap-4587	26	2	to	to	ADP
ap-4587	26	3	our	our	PRON
ap-4587	26	4	knowledge	knowledge	NOUN
ap-4587	26	5	,	,	PUNCT
ap-4587	26	6	no	no	DET
ap-4587	26	7	systematic	systematic	ADJ
ap-4587	26	8	study	study	NOUN
ap-4587	26	9	of	of	ADP
ap-4587	26	10	general	general	ADJ
ap-4587	26	11	affine	affine	NOUN
ap-4587	26	12	self	self	NOUN
ap-4587	26	13	-	-	PUNCT
ap-4587	26	14	similarities	similarity	NOUN
ap-4587	26	15	,	,	PUNCT
ap-4587	26	16	i.e.	i.e.	X
ap-4587	26	17	,	,	PUNCT
ap-4587	26	18	affine	affine	NOUN
ap-4587	26	19	mappings	mapping	VERB
ap-4587	26	20	a	a	DET
ap-4587	26	21	such	such	ADJ
ap-4587	26	22	that	that	DET
ap-4587	26	23	aς(ω	aς(ω	NUM
ap-4587	26	24	)	)	PUNCT
ap-4587	27	1	⊂	⊂	PROPN
ap-4587	27	2	σ(ω	σ(ω	PROPN
ap-4587	27	3	)	)	PUNCT
ap-4587	27	4	,	,	PUNCT
ap-4587	27	5	is	be	AUX
ap-4587	27	6	found	find	VERB
ap-4587	27	7	in	in	ADP
ap-4587	27	8	the	the	DET
ap-4587	27	9	literature	literature	NOUN
ap-4587	27	10	.	.	PUNCT
ap-4587	28	1	in	in	ADP
ap-4587	28	2	this	this	DET
ap-4587	28	3	paper	paper	NOUN
ap-4587	28	4	we	we	PRON
ap-4587	28	5	focus	focus	VERB
ap-4587	28	6	on	on	ADP
ap-4587	28	7	two	two	NUM
ap-4587	28	8	main	main	ADJ
ap-4587	28	9	problems	problem	NOUN
ap-4587	28	10	about	about	ADP
ap-4587	28	11	general	general	ADJ
ap-4587	28	12	linear	linear	ADJ
ap-4587	28	13	self	self	NOUN
ap-4587	28	14	-	-	PUNCT
ap-4587	28	15	similarities	similarity	NOUN
ap-4587	28	16	.	.	PUNCT
ap-4587	29	1	first	first	ADV
ap-4587	29	2	,	,	PUNCT
ap-4587	29	3	given	give	VERB
ap-4587	29	4	a	a	DET
ap-4587	29	5	linear	linear	ADJ
ap-4587	29	6	map	map	NOUN
ap-4587	29	7	a	a	PRON
ap-4587	29	8	,	,	PUNCT
ap-4587	29	9	we	we	PRON
ap-4587	29	10	are	be	AUX
ap-4587	29	11	interested	interested	ADJ
ap-4587	29	12	in	in	ADP
ap-4587	29	13	what	what	PRON
ap-4587	29	14	are	be	AUX
ap-4587	29	15	the	the	DET
ap-4587	29	16	cut	cut	VERB
ap-4587	29	17	-	-	PUNCT
ap-4587	29	18	and	and	CCONJ
ap-4587	29	19	-	-	PUNCT
ap-4587	29	20	project	project	NOUN
ap-4587	29	21	schemes	scheme	NOUN
ap-4587	29	22	that	that	PRON
ap-4587	29	23	allow	allow	VERB
ap-4587	29	24	a	a	PRON
ap-4587	29	25	as	as	ADP
ap-4587	29	26	a	a	DET
ap-4587	29	27	self	self	NOUN
ap-4587	29	28	-	-	PUNCT
ap-4587	29	29	similarity	similarity	NOUN
ap-4587	29	30	,	,	PUNCT
ap-4587	29	31	i.e.	i.e.	X
ap-4587	29	32	,	,	PUNCT
ap-4587	29	33	such	such	ADJ
ap-4587	29	34	that	that	SCONJ
ap-4587	29	35	aπ1(l	aπ1(l	PROPN
ap-4587	29	36	)	)	PUNCT
ap-4587	29	37	⊂	⊂	PROPN
ap-4587	29	38	π1(l	π1(l	NUM
ap-4587	29	39	)	)	PUNCT
ap-4587	29	40	.	.	PUNCT
ap-4587	30	1	then	then	ADV
ap-4587	30	2	,	,	PUNCT
ap-4587	30	3	we	we	PRON
ap-4587	30	4	may	may	AUX
ap-4587	30	5	want	want	VERB
ap-4587	30	6	to	to	PART
ap-4587	30	7	fix	fix	VERB
ap-4587	30	8	the	the	DET
ap-4587	30	9	cut	cut	VERB
ap-4587	30	10	-	-	PUNCT
ap-4587	30	11	and	and	CCONJ
ap-4587	30	12	-	-	PUNCT
ap-4587	30	13	project	project	NOUN
ap-4587	30	14	scheme	scheme	NOUN
ap-4587	30	15	and	and	CCONJ
ap-4587	30	16	ask	ask	VERB
ap-4587	30	17	about	about	ADP
ap-4587	30	18	all	all	DET
ap-4587	30	19	linear	linear	ADJ
ap-4587	30	20	self	self	NOUN
ap-4587	30	21	-	-	PUNCT
ap-4587	30	22	similarities	similarity	NOUN
ap-4587	30	23	allowed	allow	VERB
ap-4587	30	24	by	by	ADP
ap-4587	30	25	this	this	DET
ap-4587	30	26	specific	specific	ADJ
ap-4587	30	27	scheme	scheme	NOUN
ap-4587	30	28	.	.	PUNCT
ap-4587	31	1	to	to	ADP
ap-4587	31	2	this	this	DET
ap-4587	31	3	aim	aim	NOUN
ap-4587	31	4	we	we	PRON
ap-4587	31	5	present	present	VERB
ap-4587	31	6	a	a	DET
ap-4587	31	7	matrix	matrix	NOUN
ap-4587	31	8	formalism	formalism	NOUN
ap-4587	31	9	for	for	ADP
ap-4587	31	10	the	the	DET
ap-4587	31	11	study	study	NOUN
ap-4587	31	12	of	of	ADP
ap-4587	31	13	cut	cut	NOUN
ap-4587	31	14	-	-	PUNCT
ap-4587	31	15	and	and	CCONJ
ap-4587	31	16	-	-	PUNCT
ap-4587	31	17	project	project	NOUN
ap-4587	31	18	schemes	scheme	NOUN
ap-4587	31	19	and	and	CCONJ
ap-4587	31	20	derive	derive	VERB
ap-4587	31	21	several	several	ADJ
ap-4587	31	22	general	general	ADJ
ap-4587	31	23	statements	statement	NOUN
ap-4587	31	24	(	(	PUNCT
ap-4587	31	25	theorem	theorem	VERB
ap-4587	31	26	3.1	3.1	NUM
ap-4587	31	27	and	and	CCONJ
ap-4587	31	28	proposition	proposition	NOUN
ap-4587	31	29	3.3	3.3	NUM
ap-4587	31	30	)	)	PUNCT
ap-4587	31	31	.	.	PUNCT
ap-4587	32	1	these	these	DET
ap-4587	32	2	statements	statement	NOUN
ap-4587	32	3	we	we	PRON
ap-4587	32	4	then	then	ADV
ap-4587	32	5	apply	apply	VERB
ap-4587	32	6	in	in	ADP
ap-4587	32	7	the	the	DET
ap-4587	32	8	context	context	NOUN
ap-4587	32	9	of	of	ADP
ap-4587	32	10	quasicrystal	quasicrystal	ADJ
ap-4587	32	11	models	model	NOUN
ap-4587	32	12	with	with	ADP
ap-4587	32	13	5	5	NUM
ap-4587	32	14	,	,	PUNCT
ap-4587	32	15	resp	resp	NOUN
ap-4587	32	16	.	.	PUNCT
ap-4587	33	1	10	10	NUM
ap-4587	33	2	-	-	ADJ
ap-4587	33	3	fold	fold	ADJ
ap-4587	33	4	symmetry	symmetry	NOUN
ap-4587	33	5	.	.	PUNCT
ap-4587	34	1	we	we	PRON
ap-4587	34	2	derive	derive	VERB
ap-4587	34	3	what	what	PRON
ap-4587	34	4	is	be	AUX
ap-4587	34	5	the	the	DET
ap-4587	34	6	necessary	necessary	ADJ
ap-4587	34	7	form	form	NOUN
ap-4587	34	8	of	of	ADP
ap-4587	34	9	the	the	DET
ap-4587	34	10	cut	cut	NOUN
ap-4587	34	11	-	-	PUNCT
ap-4587	34	12	and	and	CCONJ
ap-4587	34	13	-	-	PUNCT
ap-4587	34	14	project	project	NOUN
ap-4587	34	15	scheme	scheme	NOUN
ap-4587	34	16	(	(	PUNCT
ap-4587	34	17	l	l	NOUN
ap-4587	34	18	⊂	⊂	PROPN
ap-4587	34	19	r4	r4	PROPN
ap-4587	34	20	,	,	PUNCT
ap-4587	34	21	π1	π1	NOUN
ap-4587	34	22	,	,	PUNCT
ap-4587	34	23	π2	π2	NOUN
ap-4587	34	24	)	)	PUNCT
ap-4587	34	25	if	if	SCONJ
ap-4587	34	26	one	one	PRON
ap-4587	34	27	aims	aim	VERB
ap-4587	34	28	to	to	PART
ap-4587	34	29	obtain	obtain	VERB
ap-4587	34	30	a	a	DET
ap-4587	34	31	quasicrystal	quasicrystal	ADJ
ap-4587	34	32	model	model	NOUN
ap-4587	34	33	with	with	ADP
ap-4587	34	34	10	10	NUM
ap-4587	34	35	-	-	ADJ
ap-4587	34	36	fold	fold	ADJ
ap-4587	34	37	symmetry	symmetry	NOUN
ap-4587	34	38	.	.	PUNCT
ap-4587	35	1	it	it	PRON
ap-4587	35	2	turns	turn	VERB
ap-4587	35	3	out	out	ADP
ap-4587	35	4	that	that	SCONJ
ap-4587	35	5	the	the	DET
ap-4587	35	6	requirement	requirement	NOUN
ap-4587	35	7	of	of	ADP
ap-4587	35	8	the	the	DET
ap-4587	35	9	symmetry	symmetry	NOUN
ap-4587	35	10	alone	alone	ADV
ap-4587	35	11	leads	lead	VERB
ap-4587	35	12	to	to	ADP
ap-4587	35	13	430	430	NUM
ap-4587	35	14	http://dx.doi.org/10.14311/ap.2017.57.0430	http://dx.doi.org/10.14311/ap.2017.57.0430	NOUN
ap-4587	35	15	http://ojs.cvut.cz/ojs/index.php/ap	http://ojs.cvut.cz/ojs/index.php/ap	PROPN
ap-4587	35	16	vol	vol	NOUN
ap-4587	35	17	.	.	PUNCT
ap-4587	36	1	57	57	NUM
ap-4587	37	1	no	no	NOUN
ap-4587	37	2	.	.	PUNCT
ap-4587	38	1	6/2017	6/2017	NUM
ap-4587	38	2	on	on	ADP
ap-4587	38	3	self	self	NOUN
ap-4587	38	4	-	-	PUNCT
ap-4587	38	5	similarities	similarity	NOUN
ap-4587	38	6	of	of	ADP
ap-4587	38	7	cut	cut	NOUN
ap-4587	38	8	-	-	PUNCT
ap-4587	38	9	and	and	CCONJ
ap-4587	38	10	-	-	PUNCT
ap-4587	38	11	project	project	NOUN
ap-4587	38	12	sets	set	VERB
ap-4587	38	13	a	a	DET
ap-4587	38	14	construction	construction	NOUN
ap-4587	38	15	equivalent	equivalent	ADJ
ap-4587	38	16	to	to	ADP
ap-4587	38	17	the	the	DET
ap-4587	38	18	classical	classical	ADJ
ap-4587	38	19	one	one	NUM
ap-4587	38	20	[	[	X
ap-4587	38	21	3	3	X
ap-4587	38	22	]	]	PUNCT
ap-4587	38	23	which	which	PRON
ap-4587	38	24	uses	use	VERB
ap-4587	38	25	preliminary	preliminary	ADJ
ap-4587	38	26	knowledge	knowledge	NOUN
ap-4587	38	27	of	of	ADP
ap-4587	38	28	space	space	NOUN
ap-4587	38	29	group	group	NOUN
ap-4587	38	30	description	description	NOUN
ap-4587	38	31	,	,	PUNCT
ap-4587	38	32	provided	provide	VERB
ap-4587	38	33	by	by	ADP
ap-4587	38	34	fedorov	fedorov	NOUN
ap-4587	38	35	and	and	CCONJ
ap-4587	38	36	schönflies	schönflie	NOUN
ap-4587	38	37	in	in	ADP
ap-4587	38	38	dimension	dimension	NOUN
ap-4587	38	39	≤	≤	NUM
ap-4587	38	40	3	3	NUM
ap-4587	38	41	and	and	CCONJ
ap-4587	38	42	then	then	ADV
ap-4587	38	43	generalized	generalize	VERB
ap-4587	38	44	by	by	ADP
ap-4587	38	45	bieberbach	bieberbach	NOUN
ap-4587	38	46	to	to	ADP
ap-4587	38	47	any	any	DET
ap-4587	38	48	dimension	dimension	NOUN
ap-4587	38	49	[	[	X
ap-4587	38	50	5	5	NUM
ap-4587	38	51	]	]	PUNCT
ap-4587	38	52	.	.	PUNCT
ap-4587	39	1	we	we	PRON
ap-4587	39	2	demonstrate	demonstrate	VERB
ap-4587	39	3	the	the	DET
ap-4587	39	4	comparison	comparison	NOUN
ap-4587	39	5	of	of	ADP
ap-4587	39	6	the	the	DET
ap-4587	39	7	two	two	NUM
ap-4587	39	8	constructions	construction	NOUN
ap-4587	39	9	,	,	PUNCT
ap-4587	39	10	see	see	VERB
ap-4587	39	11	section	section	NOUN
ap-4587	39	12	5	5	NUM
ap-4587	39	13	.	.	PUNCT
ap-4587	40	1	next	next	ADV
ap-4587	40	2	,	,	PUNCT
ap-4587	40	3	given	give	VERB
ap-4587	40	4	the	the	DET
ap-4587	40	5	cut	cut	VERB
ap-4587	40	6	-	-	PUNCT
ap-4587	40	7	and	and	CCONJ
ap-4587	40	8	-	-	PUNCT
ap-4587	40	9	project	project	NOUN
ap-4587	40	10	scheme	scheme	NOUN
ap-4587	40	11	allowing	allow	VERB
ap-4587	40	12	5	5	NUM
ap-4587	40	13	-	-	ADJ
ap-4587	40	14	fold	fold	ADJ
ap-4587	40	15	symmetry	symmetry	NOUN
ap-4587	40	16	,	,	PUNCT
ap-4587	40	17	we	we	PRON
ap-4587	40	18	study	study	VERB
ap-4587	40	19	its	its	PRON
ap-4587	40	20	linear	linear	ADJ
ap-4587	40	21	self	self	NOUN
ap-4587	40	22	-	-	PUNCT
ap-4587	40	23	similarities	similarity	NOUN
ap-4587	40	24	in	in	ADP
ap-4587	40	25	section	section	NOUN
ap-4587	40	26	6	6	NUM
ap-4587	40	27	.	.	PUNCT
ap-4587	41	1	we	we	PRON
ap-4587	41	2	show	show	VERB
ap-4587	41	3	that	that	SCONJ
ap-4587	41	4	these	these	DET
ap-4587	41	5	mappings	mapping	NOUN
ap-4587	41	6	form	form	VERB
ap-4587	41	7	an	an	DET
ap-4587	41	8	8	8	NUM
ap-4587	41	9	-	-	PUNCT
ap-4587	41	10	dimensional	dimensional	ADJ
ap-4587	41	11	z	z	NOUN
ap-4587	41	12	-	-	PUNCT
ap-4587	41	13	algebra	algebra	NOUN
ap-4587	41	14	z	z	NOUN
ap-4587	41	15	and	and	CCONJ
ap-4587	41	16	we	we	PRON
ap-4587	41	17	provide	provide	VERB
ap-4587	41	18	explicit	explicit	ADJ
ap-4587	41	19	description	description	NOUN
ap-4587	41	20	of	of	ADP
ap-4587	41	21	its	its	PRON
ap-4587	41	22	elements	element	NOUN
ap-4587	41	23	.	.	PUNCT
ap-4587	42	1	the	the	DET
ap-4587	42	2	z	z	NOUN
ap-4587	42	3	-	-	PUNCT
ap-4587	42	4	algebra	algebra	PROPN
ap-4587	42	5	z	z	NOUN
ap-4587	42	6	has	have	VERB
ap-4587	42	7	a	a	DET
ap-4587	42	8	4	4	NUM
ap-4587	42	9	-	-	PUNCT
ap-4587	42	10	dimensional	dimensional	ADJ
ap-4587	42	11	commutative	commutative	ADJ
ap-4587	42	12	subalgebra	subalgebra	NOUN
ap-4587	42	13	,	,	PUNCT
ap-4587	42	14	whose	whose	DET
ap-4587	42	15	elements	element	NOUN
ap-4587	42	16	give	give	VERB
ap-4587	42	17	scaled	scale	VERB
ap-4587	42	18	rotations	rotation	NOUN
ap-4587	42	19	.	.	PUNCT
ap-4587	43	1	this	this	DET
ap-4587	43	2	subalgebra	subalgebra	NOUN
ap-4587	43	3	is	be	AUX
ap-4587	43	4	ring	ring	NOUN
ap-4587	43	5	-	-	PUNCT
ap-4587	43	6	isomorphic	isomorphic	ADJ
ap-4587	43	7	to	to	ADP
ap-4587	43	8	the	the	DET
ap-4587	43	9	ring	ring	NOUN
ap-4587	43	10	z[ω	z[ω	PROPN
ap-4587	43	11	]	]	PUNCT
ap-4587	43	12	of	of	ADP
ap-4587	43	13	cyclotomic	cyclotomic	ADJ
ap-4587	43	14	integers	integer	NOUN
ap-4587	43	15	,	,	PUNCT
ap-4587	43	16	where	where	SCONJ
ap-4587	43	17	ω	ω	NOUN
ap-4587	43	18	=	=	SYM
ap-4587	43	19	e	e	NOUN
ap-4587	43	20	2πi	2πi	NOUN
ap-4587	43	21	5	5	NUM
ap-4587	43	22	.	.	PUNCT
ap-4587	44	1	not	not	PART
ap-4587	44	2	all	all	DET
ap-4587	44	3	self	self	NOUN
ap-4587	44	4	-	-	PUNCT
ap-4587	44	5	similarities	similarity	NOUN
ap-4587	44	6	of	of	ADP
ap-4587	44	7	the	the	DET
ap-4587	44	8	set	set	NOUN
ap-4587	44	9	π1(l	π1(l	PRON
ap-4587	44	10	)	)	PUNCT
ap-4587	44	11	are	be	AUX
ap-4587	44	12	self	self	NOUN
ap-4587	44	13	-	-	PUNCT
ap-4587	44	14	similarities	similarity	NOUN
ap-4587	44	15	of	of	ADP
ap-4587	44	16	some	some	DET
ap-4587	44	17	cut	cut	VERB
ap-4587	44	18	-	-	PUNCT
ap-4587	44	19	and	and	CCONJ
ap-4587	44	20	-	-	PUNCT
ap-4587	44	21	project	project	NOUN
ap-4587	44	22	set	set	VERB
ap-4587	44	23	σ(ω	σ(ω	PROPN
ap-4587	44	24	)	)	PUNCT
ap-4587	44	25	.	.	PUNCT
ap-4587	45	1	general	general	ADJ
ap-4587	45	2	statements	statement	NOUN
ap-4587	45	3	providing	provide	VERB
ap-4587	45	4	a	a	DET
ap-4587	45	5	necessary	necessary	ADJ
ap-4587	45	6	and	and	CCONJ
ap-4587	45	7	some	some	DET
ap-4587	45	8	sufficient	sufficient	ADJ
ap-4587	45	9	conditions	condition	NOUN
ap-4587	45	10	for	for	ADP
ap-4587	45	11	existence	existence	NOUN
ap-4587	45	12	of	of	ADP
ap-4587	45	13	a	a	DET
ap-4587	45	14	suitable	suitable	ADJ
ap-4587	45	15	window	window	NOUN
ap-4587	45	16	ω	ω	NOUN
ap-4587	45	17	are	be	AUX
ap-4587	45	18	given	give	VERB
ap-4587	45	19	in	in	ADP
ap-4587	45	20	section	section	NOUN
ap-4587	45	21	4	4	NUM
ap-4587	45	22	.	.	PUNCT
ap-4587	45	23	application	application	NOUN
ap-4587	45	24	of	of	ADP
ap-4587	45	25	this	this	DET
ap-4587	45	26	theory	theory	NOUN
ap-4587	45	27	is	be	AUX
ap-4587	45	28	then	then	ADV
ap-4587	45	29	performed	perform	VERB
ap-4587	45	30	in	in	ADP
ap-4587	45	31	section	section	NOUN
ap-4587	45	32	7	7	NUM
ap-4587	45	33	.	.	PUNCT
ap-4587	46	1	we	we	PRON
ap-4587	46	2	focus	focus	VERB
ap-4587	46	3	on	on	ADP
ap-4587	46	4	the	the	DET
ap-4587	46	5	commutative	commutative	ADJ
ap-4587	46	6	subalgebra	subalgebra	NOUN
ap-4587	46	7	of	of	ADP
ap-4587	46	8	the	the	DET
ap-4587	46	9	algebra	algebra	NOUN
ap-4587	46	10	z	z	NOUN
ap-4587	46	11	and	and	CCONJ
ap-4587	46	12	describe	describe	VERB
ap-4587	46	13	the	the	DET
ap-4587	46	14	scaled	scale	VERB
ap-4587	46	15	rotations	rotation	NOUN
ap-4587	46	16	s	s	VERB
ap-4587	46	17	for	for	ADP
ap-4587	46	18	which	which	PRON
ap-4587	46	19	a	a	DET
ap-4587	46	20	window	window	NOUN
ap-4587	46	21	ω	ω	NOUN
ap-4587	46	22	such	such	ADJ
ap-4587	46	23	that	that	PRON
ap-4587	46	24	s	s	X
ap-4587	46	25	(	(	PUNCT
ap-4587	46	26	σ(ω	σ(ω	PROPN
ap-4587	46	27	)	)	PUNCT
ap-4587	46	28	)	)	PUNCT
ap-4587	47	1	⊂	⊂	PROPN
ap-4587	47	2	σ(ω	σ(ω	PROPN
ap-4587	47	3	)	)	PUNCT
ap-4587	47	4	exists	exist	VERB
ap-4587	47	5	.	.	PUNCT
ap-4587	48	1	we	we	PRON
ap-4587	48	2	also	also	ADV
ap-4587	48	3	provide	provide	VERB
ap-4587	48	4	an	an	DET
ap-4587	48	5	example	example	NOUN
ap-4587	48	6	illustrating	illustrate	VERB
ap-4587	48	7	that	that	SCONJ
ap-4587	48	8	a	a	DET
ap-4587	48	9	cut	cut	VERB
ap-4587	48	10	-	-	PUNCT
ap-4587	48	11	and	and	CCONJ
ap-4587	48	12	-	-	PUNCT
ap-4587	48	13	project	project	NOUN
ap-4587	48	14	set	set	NOUN
ap-4587	48	15	may	may	AUX
ap-4587	48	16	have	have	VERB
ap-4587	48	17	a	a	DET
ap-4587	48	18	linear	linear	ADJ
ap-4587	48	19	self	self	NOUN
ap-4587	48	20	-	-	PUNCT
ap-4587	48	21	similarity	similarity	NOUN
ap-4587	48	22	which	which	PRON
ap-4587	48	23	is	be	AUX
ap-4587	48	24	not	not	PART
ap-4587	48	25	a	a	DET
ap-4587	48	26	scaled	scale	VERB
ap-4587	48	27	rotation	rotation	NOUN
ap-4587	48	28	.	.	PUNCT
ap-4587	49	1	2	2	X
ap-4587	49	2	.	.	X
ap-4587	49	3	preliminaries	preliminary	NOUN
ap-4587	49	4	it	it	PRON
ap-4587	49	5	is	be	AUX
ap-4587	49	6	commonly	commonly	ADV
ap-4587	49	7	understood	understand	VERB
ap-4587	49	8	that	that	SCONJ
ap-4587	49	9	a	a	DET
ap-4587	49	10	mathematical	mathematical	ADJ
ap-4587	49	11	model	model	NOUN
ap-4587	49	12	for	for	ADP
ap-4587	49	13	quasicrystals	quasicrystal	NOUN
ap-4587	49	14	should	should	AUX
ap-4587	49	15	satisfy	satisfy	VERB
ap-4587	49	16	the	the	DET
ap-4587	49	17	so	so	ADV
ap-4587	49	18	-	-	PUNCT
ap-4587	49	19	called	call	VERB
ap-4587	49	20	delone	delone	NOUN
ap-4587	49	21	property	property	NOUN
ap-4587	49	22	.	.	PUNCT
ap-4587	50	1	definition	definition	NOUN
ap-4587	50	2	2.1	2.1	NUM
ap-4587	50	3	.	.	PUNCT
ap-4587	51	1	we	we	PRON
ap-4587	51	2	say	say	VERB
ap-4587	51	3	that	that	SCONJ
ap-4587	51	4	x	x	PROPN
ap-4587	51	5	⊂	⊂	PROPN
ap-4587	51	6	rn	rn	PROPN
ap-4587	51	7	is	be	AUX
ap-4587	51	8	delone	delone	VERB
ap-4587	51	9	,	,	PUNCT
ap-4587	51	10	if	if	SCONJ
ap-4587	51	11	the	the	DET
ap-4587	51	12	following	follow	VERB
ap-4587	51	13	conditions	condition	NOUN
ap-4587	51	14	are	be	AUX
ap-4587	51	15	satisfied	satisfied	ADJ
ap-4587	51	16	:	:	PUNCT
ap-4587	51	17	(	(	PUNCT
ap-4587	51	18	1	1	NUM
ap-4587	51	19	.	.	PUNCT
ap-4587	51	20	)	)	PUNCT
ap-4587	52	1	there	there	PRON
ap-4587	52	2	exists	exist	VERB
ap-4587	52	3	r	r	NOUN
ap-4587	52	4	>	>	X
ap-4587	52	5	0	0	NUM
ap-4587	52	6	such	such	ADJ
ap-4587	52	7	that	that	SCONJ
ap-4587	52	8	every	every	DET
ap-4587	52	9	open	open	ADJ
ap-4587	52	10	ball	ball	NOUN
ap-4587	52	11	of	of	ADP
ap-4587	52	12	radius	radius	NOUN
ap-4587	52	13	r	r	NOUN
ap-4587	52	14	contains	contain	VERB
ap-4587	52	15	at	at	ADP
ap-4587	52	16	most	most	ADV
ap-4587	52	17	one	one	NUM
ap-4587	52	18	point	point	NOUN
ap-4587	52	19	of	of	ADP
ap-4587	52	20	x	x	PROPN
ap-4587	52	21	(	(	PUNCT
ap-4587	52	22	uniform	uniform	ADJ
ap-4587	52	23	discreteness	discreteness	PROPN
ap-4587	52	24	)	)	PUNCT
ap-4587	52	25	.	.	PUNCT
ap-4587	53	1	(	(	PUNCT
ap-4587	53	2	2	2	NUM
ap-4587	53	3	.	.	PUNCT
ap-4587	53	4	)	)	PUNCT
ap-4587	54	1	there	there	PRON
ap-4587	54	2	exists	exist	VERB
ap-4587	54	3	r	r	NOUN
ap-4587	54	4	>	>	X
ap-4587	54	5	0	0	NUM
ap-4587	54	6	such	such	ADJ
ap-4587	54	7	that	that	SCONJ
ap-4587	54	8	every	every	DET
ap-4587	54	9	closed	closed	ADJ
ap-4587	54	10	ball	ball	NOUN
ap-4587	54	11	of	of	ADP
ap-4587	54	12	radius	radius	NOUN
ap-4587	54	13	r	r	NOUN
ap-4587	54	14	contains	contain	VERB
ap-4587	54	15	at	at	ADV
ap-4587	54	16	least	least	ADV
ap-4587	54	17	one	one	NUM
ap-4587	54	18	point	point	NOUN
ap-4587	54	19	of	of	ADP
ap-4587	54	20	x	x	PROPN
ap-4587	54	21	(	(	PUNCT
ap-4587	54	22	relative	relative	ADJ
ap-4587	54	23	density	density	NOUN
ap-4587	54	24	)	)	PUNCT
ap-4587	54	25	.	.	PUNCT
ap-4587	55	1	note	note	VERB
ap-4587	55	2	that	that	SCONJ
ap-4587	55	3	the	the	DET
ap-4587	55	4	supremum	supremum	NOUN
ap-4587	55	5	of	of	ADP
ap-4587	55	6	the	the	DET
ap-4587	55	7	values	value	NOUN
ap-4587	55	8	r	r	NOUN
ap-4587	55	9	from	from	ADP
ap-4587	55	10	uniform	uniform	ADJ
ap-4587	55	11	discreteness	discreteness	NOUN
ap-4587	55	12	bounds	bound	VERB
ap-4587	55	13	the	the	DET
ap-4587	55	14	distances	distance	NOUN
ap-4587	55	15	between	between	ADP
ap-4587	55	16	points	point	NOUN
ap-4587	55	17	in	in	ADP
ap-4587	55	18	the	the	DET
ap-4587	55	19	delone	delone	NOUN
ap-4587	55	20	set	set	NOUN
ap-4587	55	21	x	x	PUNCT
ap-4587	55	22	from	from	ADP
ap-4587	55	23	below	below	ADV
ap-4587	55	24	.	.	PUNCT
ap-4587	56	1	if	if	SCONJ
ap-4587	56	2	the	the	DET
ap-4587	56	3	supremum	supremum	NOUN
ap-4587	56	4	is	be	AUX
ap-4587	56	5	achieved	achieve	VERB
ap-4587	56	6	,	,	PUNCT
ap-4587	56	7	then	then	ADV
ap-4587	56	8	it	it	PRON
ap-4587	56	9	is	be	AUX
ap-4587	56	10	the	the	DET
ap-4587	56	11	minimal	minimal	ADJ
ap-4587	56	12	distance	distance	NOUN
ap-4587	56	13	in	in	ADP
ap-4587	56	14	x.	x.	NOUN
ap-4587	56	15	the	the	DET
ap-4587	56	16	infimum	infimum	NOUN
ap-4587	56	17	of	of	ADP
ap-4587	56	18	the	the	DET
ap-4587	56	19	values	value	NOUN
ap-4587	56	20	r	r	NOUN
ap-4587	56	21	from	from	ADP
ap-4587	56	22	relative	relative	ADJ
ap-4587	56	23	density	density	NOUN
ap-4587	56	24	is	be	AUX
ap-4587	56	25	the	the	DET
ap-4587	56	26	so	so	ADV
ap-4587	56	27	-	-	PUNCT
ap-4587	56	28	called	call	VERB
ap-4587	56	29	covering	covering	NOUN
ap-4587	56	30	radius	radius	NOUN
ap-4587	56	31	of	of	ADP
ap-4587	56	32	the	the	DET
ap-4587	56	33	set	set	NOUN
ap-4587	56	34	x.	x.	NOUN
ap-4587	56	35	the	the	DET
ap-4587	56	36	essential	essential	ADJ
ap-4587	56	37	idea	idea	NOUN
ap-4587	56	38	behind	behind	ADP
ap-4587	56	39	the	the	DET
ap-4587	56	40	cut	cut	VERB
ap-4587	56	41	-	-	PUNCT
ap-4587	56	42	and	and	CCONJ
ap-4587	56	43	-	-	PUNCT
ap-4587	56	44	project	project	NOUN
ap-4587	56	45	scheme	scheme	NOUN
ap-4587	56	46	is	be	AUX
ap-4587	56	47	to	to	PART
ap-4587	56	48	project	project	VERB
ap-4587	56	49	elements	element	NOUN
ap-4587	56	50	of	of	ADP
ap-4587	56	51	a	a	DET
ap-4587	56	52	higher	higher	ADV
ap-4587	56	53	-	-	PUNCT
ap-4587	56	54	dimensional	dimensional	ADJ
ap-4587	56	55	lattice	lattice	NOUN
ap-4587	56	56	to	to	ADP
ap-4587	56	57	suitable	suitable	ADJ
ap-4587	56	58	subspaces	subspace	NOUN
ap-4587	56	59	.	.	PUNCT
ap-4587	57	1	there	there	PRON
ap-4587	57	2	are	be	VERB
ap-4587	57	3	two	two	NUM
ap-4587	57	4	basic	basic	ADJ
ap-4587	57	5	approaches	approach	NOUN
ap-4587	57	6	when	when	SCONJ
ap-4587	57	7	doing	do	VERB
ap-4587	57	8	this	this	PRON
ap-4587	57	9	:	:	PUNCT
ap-4587	57	10	either	either	CCONJ
ap-4587	57	11	one	one	NOUN
ap-4587	57	12	takes	take	VERB
ap-4587	57	13	the	the	DET
ap-4587	57	14	standard	standard	ADJ
ap-4587	57	15	lattice	lattice	PROPN
ap-4587	57	16	zn+m	zn+m	NUM
ap-4587	57	17	and	and	CCONJ
ap-4587	57	18	projects	project	NOUN
ap-4587	57	19	to	to	PART
ap-4587	57	20	general	general	VERB
ap-4587	57	21	irrationally	irrationally	ADV
ap-4587	57	22	oriented	orient	VERB
ap-4587	57	23	subspaces	subspace	NOUN
ap-4587	57	24	v1	v1	NOUN
ap-4587	57	25	,	,	PUNCT
ap-4587	57	26	v2	v2	PROPN
ap-4587	57	27	of	of	ADP
ap-4587	57	28	dimensions	dimension	NOUN
ap-4587	57	29	n	n	CCONJ
ap-4587	57	30	,	,	PUNCT
ap-4587	57	31	m	m	PROPN
ap-4587	57	32	,	,	PUNCT
ap-4587	57	33	respectively	respectively	ADV
ap-4587	57	34	,	,	PUNCT
ap-4587	57	35	whose	whose	DET
ap-4587	57	36	direct	direct	ADJ
ap-4587	57	37	sum	sum	NOUN
ap-4587	57	38	is	be	AUX
ap-4587	57	39	equal	equal	ADJ
ap-4587	57	40	to	to	PART
ap-4587	57	41	rn+m	rn+m	VERB
ap-4587	57	42	.	.	PROPN
ap-4587	58	1	or	or	CCONJ
ap-4587	58	2	one	one	NUM
ap-4587	58	3	chooses	choose	VERB
ap-4587	58	4	a	a	DET
ap-4587	58	5	general	general	ADJ
ap-4587	58	6	lattice	lattice	PROPN
ap-4587	58	7	l	l	NOUN
ap-4587	58	8	and	and	CCONJ
ap-4587	58	9	projects	project	NOUN
ap-4587	58	10	to	to	ADP
ap-4587	58	11	subspaces	subspace	NOUN
ap-4587	58	12	spanned	span	VERB
ap-4587	58	13	by	by	ADP
ap-4587	58	14	vectors	vector	NOUN
ap-4587	58	15	of	of	ADP
ap-4587	58	16	the	the	DET
ap-4587	58	17	standard	standard	ADJ
ap-4587	58	18	basis	basis	NOUN
ap-4587	58	19	e1	e1	NOUN
ap-4587	58	20	,	,	PUNCT
ap-4587	58	21	.	.	PUNCT
ap-4587	58	22	.	.	PUNCT
ap-4587	59	1	.	.	PUNCT
ap-4587	60	1	,	,	PUNCT
ap-4587	60	2	en+m	en+m	X
ap-4587	60	3	.	.	PUNCT
ap-4587	61	1	both	both	DET
ap-4587	61	2	methods	method	NOUN
ap-4587	61	3	are	be	AUX
ap-4587	61	4	equivalent	equivalent	ADJ
ap-4587	61	5	in	in	ADP
ap-4587	61	6	principal	principal	NOUN
ap-4587	61	7	.	.	PUNCT
ap-4587	62	1	for	for	ADP
ap-4587	62	2	formal	formal	ADJ
ap-4587	62	3	reasons	reason	NOUN
ap-4587	62	4	,	,	PUNCT
ap-4587	62	5	it	it	PRON
ap-4587	62	6	is	be	AUX
ap-4587	62	7	suitable	suitable	ADJ
ap-4587	62	8	for	for	SCONJ
ap-4587	62	9	us	we	PRON
ap-4587	62	10	to	to	PART
ap-4587	62	11	choose	choose	VERB
ap-4587	62	12	the	the	DET
ap-4587	62	13	second	second	ADJ
ap-4587	62	14	approach	approach	NOUN
ap-4587	62	15	.	.	PUNCT
ap-4587	63	1	definition	definition	NOUN
ap-4587	63	2	2.2	2.2	NUM
ap-4587	63	3	.	.	PUNCT
ap-4587	64	1	let	let	VERB
ap-4587	64	2	l	l	NOUN
ap-4587	64	3	⊂	⊂	PRON
ap-4587	64	4	rn+m	rn+m	AUX
ap-4587	64	5	be	be	AUX
ap-4587	64	6	a	a	DET
ap-4587	64	7	(	(	PUNCT
ap-4587	64	8	n+m)-dimensional	n+m)-dimensional	PROPN
ap-4587	64	9	lattice	lattice	PROPN
ap-4587	64	10	.	.	PUNCT
ap-4587	65	1	let	let	VERB
ap-4587	65	2	further	far	ADV
ap-4587	65	3	v1	v1	VERB
ap-4587	65	4	=	=	SYM
ap-4587	65	5	spanr{e1	spanr{e1	NOUN
ap-4587	65	6	,	,	PUNCT
ap-4587	65	7	e2	e2	PROPN
ap-4587	65	8	,	,	PUNCT
ap-4587	65	9	.	.	PUNCT
ap-4587	65	10	.	.	PUNCT
ap-4587	66	1	.	.	PUNCT
ap-4587	67	1	,	,	PUNCT
ap-4587	67	2	en	en	ADP
ap-4587	67	3	}	}	PUNCT
ap-4587	67	4	and	and	CCONJ
ap-4587	67	5	v2	v2	PROPN
ap-4587	67	6	=	=	SYM
ap-4587	67	7	spanr{en+1	spanr{en+1	PROPN
ap-4587	67	8	,	,	PUNCT
ap-4587	67	9	en+2	en+2	PROPN
ap-4587	67	10	,	,	PUNCT
ap-4587	67	11	.	.	PUNCT
ap-4587	67	12	.	.	PUNCT
ap-4587	67	13	.	.	PUNCT
ap-4587	68	1	,	,	PUNCT
ap-4587	68	2	en+m	en+m	PROPN
ap-4587	68	3	}	}	PUNCT
ap-4587	68	4	.	.	PUNCT
ap-4587	69	1	let	let	VERB
ap-4587	69	2	π1	π1	NOUN
ap-4587	69	3	:	:	PUNCT
ap-4587	69	4	rn+m	rn+m	PROPN
ap-4587	69	5	→	→	SYM
ap-4587	69	6	rn	rn	PROPN
ap-4587	69	7	and	and	CCONJ
ap-4587	69	8	π2	π2	ADV
ap-4587	69	9	:	:	PUNCT
ap-4587	69	10	rn+m	rn+m	PROPN
ap-4587	69	11	→	→	SYM
ap-4587	69	12	rm	rm	PROPN
ap-4587	69	13	be	be	AUX
ap-4587	69	14	projections	projection	NOUN
ap-4587	69	15	to	to	PART
ap-4587	69	16	v1	v1	VERB
ap-4587	69	17	and	and	CCONJ
ap-4587	69	18	v2	v2	NOUN
ap-4587	69	19	,	,	PUNCT
ap-4587	69	20	respectively	respectively	ADV
ap-4587	69	21	.	.	PUNCT
ap-4587	70	1	the	the	DET
ap-4587	70	2	triple	triple	ADJ
ap-4587	70	3	(	(	PUNCT
ap-4587	70	4	l	l	NOUN
ap-4587	70	5	,	,	PUNCT
ap-4587	70	6	π1	π1	NOUN
ap-4587	70	7	,	,	PUNCT
ap-4587	70	8	π2	π2	PROPN
ap-4587	70	9	)	)	PUNCT
ap-4587	70	10	is	be	AUX
ap-4587	70	11	called	call	VERB
ap-4587	70	12	a	a	DET
ap-4587	70	13	cut	cut	VERB
ap-4587	70	14	-	-	PUNCT
ap-4587	70	15	and	and	CCONJ
ap-4587	70	16	-	-	PUNCT
ap-4587	70	17	project	project	NOUN
ap-4587	70	18	scheme	scheme	NOUN
ap-4587	70	19	.	.	PUNCT
ap-4587	71	1	we	we	PRON
ap-4587	71	2	say	say	VERB
ap-4587	71	3	that	that	SCONJ
ap-4587	71	4	the	the	DET
ap-4587	71	5	cut	cut	VERB
ap-4587	71	6	-	-	PUNCT
ap-4587	71	7	and	and	CCONJ
ap-4587	71	8	-	-	PUNCT
ap-4587	71	9	project	project	NOUN
ap-4587	71	10	scheme	scheme	NOUN
ap-4587	71	11	is	be	AUX
ap-4587	71	12	non	non	ADJ
ap-4587	71	13	-	-	ADJ
ap-4587	71	14	degenerated	degenerated	ADJ
ap-4587	71	15	,	,	PUNCT
ap-4587	71	16	if	if	SCONJ
ap-4587	71	17	π1	π1	ADJ
ap-4587	71	18	∣∣	∣∣	PUNCT
ap-4587	71	19	l	l	NOUN
ap-4587	71	20	is	be	AUX
ap-4587	71	21	injective	injective	ADJ
ap-4587	71	22	.	.	PUNCT
ap-4587	72	1	we	we	PRON
ap-4587	72	2	say	say	VERB
ap-4587	72	3	that	that	SCONJ
ap-4587	72	4	the	the	DET
ap-4587	72	5	cut	cut	VERB
ap-4587	72	6	-	-	PUNCT
ap-4587	72	7	andproject	andproject	NOUN
ap-4587	72	8	scheme	scheme	NOUN
ap-4587	72	9	is	be	AUX
ap-4587	72	10	irreducible	irreducible	ADJ
ap-4587	72	11	,	,	PUNCT
ap-4587	72	12	if	if	SCONJ
ap-4587	72	13	π2(l	π2(l	NOUN
ap-4587	72	14	)	)	PUNCT
ap-4587	72	15	is	be	AUX
ap-4587	72	16	dense	dense	ADJ
ap-4587	72	17	in	in	ADP
ap-4587	72	18	rm	rm	PROPN
ap-4587	72	19	.	.	PUNCT
ap-4587	73	1	it	it	PRON
ap-4587	73	2	can	can	AUX
ap-4587	73	3	be	be	AUX
ap-4587	73	4	easily	easily	ADV
ap-4587	73	5	seen	see	VERB
ap-4587	73	6	that	that	SCONJ
ap-4587	73	7	the	the	DET
ap-4587	73	8	set	set	NOUN
ap-4587	73	9	π1(l	π1(l	PRON
ap-4587	73	10	)	)	PUNCT
ap-4587	73	11	is	be	AUX
ap-4587	73	12	a	a	DET
ap-4587	73	13	z	z	NOUN
ap-4587	73	14	-	-	PUNCT
ap-4587	73	15	module	module	NOUN
ap-4587	73	16	in	in	ADP
ap-4587	73	17	rn	rn	PROPN
ap-4587	73	18	and	and	CCONJ
ap-4587	73	19	in	in	ADP
ap-4587	73	20	case	case	NOUN
ap-4587	73	21	that	that	SCONJ
ap-4587	73	22	the	the	DET
ap-4587	73	23	cut	cut	VERB
ap-4587	73	24	-	-	PUNCT
ap-4587	73	25	and	and	CCONJ
ap-4587	73	26	-	-	PUNCT
ap-4587	73	27	project	project	NOUN
ap-4587	73	28	scheme	scheme	NOUN
ap-4587	73	29	is	be	AUX
ap-4587	73	30	non	non	ADJ
ap-4587	73	31	-	-	ADJ
ap-4587	73	32	degenerated	degenerated	ADJ
ap-4587	73	33	,	,	PUNCT
ap-4587	73	34	it	it	PRON
ap-4587	73	35	is	be	AUX
ap-4587	73	36	not	not	PART
ap-4587	73	37	a	a	DET
ap-4587	73	38	discrete	discrete	ADJ
ap-4587	73	39	set	set	NOUN
ap-4587	73	40	.	.	PUNCT
ap-4587	74	1	a	a	DET
ap-4587	74	2	quasicrystal	quasicrystal	ADJ
ap-4587	74	3	model	model	NOUN
ap-4587	74	4	is	be	AUX
ap-4587	74	5	constructed	construct	VERB
ap-4587	74	6	as	as	ADP
ap-4587	74	7	a	a	DET
ap-4587	74	8	suitable	suitable	ADJ
ap-4587	74	9	subset	subset	NOUN
ap-4587	74	10	of	of	ADP
ap-4587	74	11	the	the	DET
ap-4587	74	12	z	z	NOUN
ap-4587	74	13	-	-	PUNCT
ap-4587	74	14	module	module	NOUN
ap-4587	74	15	π1(l	π1(l	NUM
ap-4587	74	16	)	)	PUNCT
ap-4587	74	17	.	.	PUNCT
ap-4587	75	1	in	in	ADP
ap-4587	75	2	order	order	NOUN
ap-4587	75	3	to	to	PART
ap-4587	75	4	choose	choose	VERB
ap-4587	75	5	a	a	DET
ap-4587	75	6	delone	delone	NOUN
ap-4587	75	7	subset	subset	NOUN
ap-4587	75	8	of	of	ADP
ap-4587	75	9	π1(l	π1(l	PROPN
ap-4587	75	10	)	)	PUNCT
ap-4587	75	11	,	,	PUNCT
ap-4587	75	12	one	one	PRON
ap-4587	75	13	puts	put	VERB
ap-4587	75	14	a	a	DET
ap-4587	75	15	condition	condition	NOUN
ap-4587	75	16	on	on	ADP
ap-4587	75	17	the	the	DET
ap-4587	75	18	second	second	ADJ
ap-4587	75	19	projection	projection	NOUN
ap-4587	75	20	of	of	ADP
ap-4587	75	21	lattice	lattice	PROPN
ap-4587	75	22	points	point	NOUN
ap-4587	75	23	.	.	PUNCT
ap-4587	76	1	definition	definition	NOUN
ap-4587	76	2	2.3	2.3	NUM
ap-4587	76	3	.	.	PUNCT
ap-4587	77	1	let	let	AUX
ap-4587	77	2	(	(	PUNCT
ap-4587	77	3	l	l	NOUN
ap-4587	77	4	,	,	PUNCT
ap-4587	77	5	π1	π1	NOUN
ap-4587	77	6	,	,	PUNCT
ap-4587	77	7	π2	π2	PROPN
ap-4587	77	8	)	)	PUNCT
ap-4587	77	9	be	be	AUX
ap-4587	77	10	a	a	DET
ap-4587	77	11	non	non	ADJ
ap-4587	77	12	-	-	ADJ
ap-4587	77	13	degenerated	degenerated	ADJ
ap-4587	77	14	irreducible	irreducible	ADJ
ap-4587	77	15	cut	cut	NOUN
ap-4587	77	16	-	-	PUNCT
ap-4587	77	17	and	and	CCONJ
ap-4587	77	18	-	-	PUNCT
ap-4587	77	19	project	project	NOUN
ap-4587	77	20	scheme	scheme	NOUN
ap-4587	77	21	.	.	PUNCT
ap-4587	78	1	given	give	VERB
ap-4587	78	2	a	a	DET
ap-4587	78	3	bounded	bound	VERB
ap-4587	78	4	set	set	NOUN
ap-4587	78	5	ω	ω	PROPN
ap-4587	78	6	with	with	ADP
ap-4587	78	7	non	non	ADJ
ap-4587	78	8	-	-	ADJ
ap-4587	78	9	empty	empty	ADJ
ap-4587	78	10	interior	interior	NOUN
ap-4587	78	11	,	,	PUNCT
ap-4587	78	12	we	we	PRON
ap-4587	78	13	define	define	VERB
ap-4587	78	14	the	the	DET
ap-4587	78	15	cut	cut	VERB
ap-4587	78	16	-	-	PUNCT
ap-4587	78	17	and	and	CCONJ
ap-4587	78	18	-	-	PUNCT
ap-4587	78	19	project	project	NOUN
ap-4587	78	20	set	set	VERB
ap-4587	78	21	σ(ω	σ(ω	PROPN
ap-4587	78	22	)	)	PUNCT
ap-4587	78	23	with	with	ADP
ap-4587	78	24	acceptance	acceptance	NOUN
ap-4587	78	25	window	window	NOUN
ap-4587	78	26	ω	ω	PROPN
ap-4587	78	27	by	by	ADP
ap-4587	78	28	σ(ω	σ(ω	PROPN
ap-4587	78	29	)	)	PUNCT
ap-4587	78	30	:	:	PUNCT
ap-4587	79	1	=	=	X
ap-4587	79	2	{	{	PUNCT
ap-4587	79	3	π1(x	π1(x	NOUN
ap-4587	79	4	)	)	PUNCT
ap-4587	79	5	:	:	PUNCT
ap-4587	80	1	x	x	PUNCT
ap-4587	80	2	∈	∈	PROPN
ap-4587	80	3	l	l	NOUN
ap-4587	80	4	,	,	PUNCT
ap-4587	80	5	π2(x	π2(x	X
ap-4587	80	6	)	)	PUNCT
ap-4587	80	7	∈	∈	PROPN
ap-4587	80	8	ω	ω	PROPN
ap-4587	80	9	}	}	PUNCT
ap-4587	80	10	.	.	PUNCT
ap-4587	81	1	(	(	PUNCT
ap-4587	81	2	1	1	X
ap-4587	81	3	)	)	PUNCT
ap-4587	81	4	in	in	ADP
ap-4587	81	5	the	the	DET
ap-4587	81	6	literature	literature	NOUN
ap-4587	81	7	,	,	PUNCT
ap-4587	81	8	one	one	PRON
ap-4587	81	9	sometimes	sometimes	ADV
ap-4587	81	10	puts	put	VERB
ap-4587	81	11	different	different	ADJ
ap-4587	81	12	requirements	requirement	NOUN
ap-4587	81	13	on	on	ADP
ap-4587	81	14	the	the	DET
ap-4587	81	15	acceptance	acceptance	NOUN
ap-4587	81	16	window	window	NOUN
ap-4587	81	17	ω	ω	PROPN
ap-4587	81	18	,	,	PUNCT
ap-4587	81	19	for	for	ADP
ap-4587	81	20	example	example	NOUN
ap-4587	81	21	lagarias	lagaria	NOUN
ap-4587	82	1	[	[	X
ap-4587	82	2	11	11	NUM
ap-4587	82	3	]	]	PUNCT
ap-4587	82	4	asks	ask	VERB
ap-4587	82	5	it	it	PRON
ap-4587	82	6	to	to	PART
ap-4587	82	7	be	be	AUX
ap-4587	82	8	bounded	bound	VERB
ap-4587	82	9	and	and	CCONJ
ap-4587	82	10	open	open	ADJ
ap-4587	82	11	.	.	PUNCT
ap-4587	83	1	on	on	ADP
ap-4587	83	2	the	the	DET
ap-4587	83	3	other	other	ADJ
ap-4587	83	4	hand	hand	NOUN
ap-4587	83	5	,	,	PUNCT
ap-4587	83	6	cotfas	cotfas	NOUN
ap-4587	83	7	[	[	X
ap-4587	83	8	7	7	NUM
ap-4587	83	9	]	]	PUNCT
ap-4587	83	10	requires	require	VERB
ap-4587	83	11	compactness	compactness	NOUN
ap-4587	83	12	and	and	CCONJ
ap-4587	83	13	int(ω	int(ω	NOUN
ap-4587	83	14	)	)	PUNCT
ap-4587	83	15	6=	6=	ADP
ap-4587	83	16	∅.	∅.	PRON
ap-4587	83	17	moody	moody	NOUN
ap-4587	83	18	[	[	X
ap-4587	83	19	16	16	NUM
ap-4587	83	20	]	]	PUNCT
ap-4587	83	21	sets	set	VERB
ap-4587	83	22	that	that	SCONJ
ap-4587	83	23	the	the	DET
ap-4587	83	24	bouded	boude	VERB
ap-4587	83	25	set	set	VERB
ap-4587	83	26	ω	ω	PROPN
ap-4587	83	27	satisfies	satisfie	NOUN
ap-4587	83	28	ω	ω	PROPN
ap-4587	83	29	⊂	⊂	PROPN
ap-4587	83	30	int(ω	int(ω	PROPN
ap-4587	83	31	)	)	PUNCT
ap-4587	83	32	.	.	PUNCT
ap-4587	84	1	some	some	DET
ap-4587	84	2	additional	additional	ADJ
ap-4587	84	3	conditions	condition	NOUN
ap-4587	84	4	,	,	PUNCT
ap-4587	84	5	such	such	ADJ
ap-4587	84	6	as	as	ADP
ap-4587	84	7	empty	empty	ADJ
ap-4587	84	8	intersection	intersection	NOUN
ap-4587	84	9	of	of	ADP
ap-4587	84	10	the	the	DET
ap-4587	84	11	boundary	boundary	NOUN
ap-4587	84	12	with	with	ADP
ap-4587	84	13	π1(l	π1(l	NOUN
ap-4587	84	14	)	)	PUNCT
ap-4587	84	15	,	,	PUNCT
ap-4587	84	16	or	or	CCONJ
ap-4587	84	17	convexity	convexity	NOUN
ap-4587	84	18	,	,	PUNCT
ap-4587	84	19	may	may	AUX
ap-4587	84	20	influence	influence	VERB
ap-4587	84	21	some	some	DET
ap-4587	84	22	specific	specific	ADJ
ap-4587	84	23	properties	property	NOUN
ap-4587	84	24	of	of	ADP
ap-4587	84	25	the	the	DET
ap-4587	84	26	cut	cut	NOUN
ap-4587	84	27	-	-	PUNCT
ap-4587	84	28	and	and	CCONJ
ap-4587	84	29	-	-	PUNCT
ap-4587	84	30	project	project	NOUN
ap-4587	84	31	set	set	NOUN
ap-4587	84	32	,	,	PUNCT
ap-4587	84	33	namely	namely	ADV
ap-4587	84	34	repetitivity	repetitivity	NOUN
ap-4587	85	1	[	[	X
ap-4587	85	2	12	12	NUM
ap-4587	85	3	]	]	PUNCT
ap-4587	85	4	,	,	PUNCT
ap-4587	85	5	or	or	CCONJ
ap-4587	85	6	closedness	closedness	ADJ
ap-4587	85	7	under	under	ADP
ap-4587	85	8	quasiaddition	quasiaddition	NOUN
ap-4587	85	9	[	[	X
ap-4587	85	10	4	4	NUM
ap-4587	85	11	]	]	PUNCT
ap-4587	85	12	.	.	PUNCT
ap-4587	86	1	here	here	ADV
ap-4587	86	2	we	we	PRON
ap-4587	86	3	stick	stick	VERB
ap-4587	86	4	to	to	ADP
ap-4587	86	5	the	the	DET
ap-4587	86	6	two	two	NUM
ap-4587	86	7	basic	basic	ADJ
ap-4587	86	8	requirements	requirement	NOUN
ap-4587	86	9	which	which	PRON
ap-4587	86	10	ensure	ensure	VERB
ap-4587	86	11	the	the	DET
ap-4587	86	12	delone	delone	NOUN
ap-4587	86	13	property	property	NOUN
ap-4587	86	14	of	of	ADP
ap-4587	86	15	σ(ω	σ(ω	PROPN
ap-4587	86	16	)	)	PUNCT
ap-4587	86	17	,	,	PUNCT
ap-4587	86	18	see	see	VERB
ap-4587	86	19	[	[	X
ap-4587	86	20	16	16	NUM
ap-4587	86	21	]	]	PUNCT
ap-4587	86	22	.	.	PUNCT
ap-4587	87	1	in	in	ADP
ap-4587	87	2	the	the	DET
ap-4587	87	3	study	study	NOUN
ap-4587	87	4	of	of	ADP
ap-4587	87	5	quasicrystal	quasicrystal	ADJ
ap-4587	87	6	models	model	NOUN
ap-4587	87	7	with	with	ADP
ap-4587	87	8	the	the	DET
ap-4587	87	9	observed	observe	VERB
ap-4587	87	10	rotational	rotational	ADJ
ap-4587	87	11	symmetries	symmetry	NOUN
ap-4587	87	12	,	,	PUNCT
ap-4587	87	13	one	one	NOUN
ap-4587	87	14	necessarily	necessarily	ADV
ap-4587	87	15	encounters	encounter	VERB
ap-4587	87	16	certain	certain	ADJ
ap-4587	87	17	numbertheoretic	numbertheoretic	ADJ
ap-4587	87	18	notions	notion	NOUN
ap-4587	87	19	.	.	PUNCT
ap-4587	88	1	an	an	DET
ap-4587	88	2	algebraic	algebraic	ADJ
ap-4587	88	3	number	number	NOUN
ap-4587	88	4	α	α	NOUN
ap-4587	88	5	is	be	AUX
ap-4587	88	6	a	a	DET
ap-4587	88	7	root	root	NOUN
ap-4587	88	8	of	of	ADP
ap-4587	88	9	a	a	DET
ap-4587	88	10	polynomial	polynomial	NOUN
ap-4587	88	11	with	with	ADP
ap-4587	88	12	rational	rational	ADJ
ap-4587	88	13	coefficients	coefficient	NOUN
ap-4587	88	14	.	.	PUNCT
ap-4587	89	1	if	if	SCONJ
ap-4587	89	2	this	this	DET
ap-4587	89	3	polynomial	polynomial	NOUN
ap-4587	89	4	is	be	AUX
ap-4587	89	5	monic	monic	ADJ
ap-4587	89	6	and	and	CCONJ
ap-4587	89	7	irreducible	irreducible	ADJ
ap-4587	89	8	over	over	ADP
ap-4587	89	9	the	the	DET
ap-4587	89	10	rationals	rational	NOUN
ap-4587	89	11	,	,	PUNCT
ap-4587	89	12	it	it	PRON
ap-4587	89	13	is	be	AUX
ap-4587	89	14	called	call	VERB
ap-4587	89	15	the	the	DET
ap-4587	89	16	minimal	minimal	ADJ
ap-4587	89	17	polynomial	polynomial	NOUN
ap-4587	89	18	of	of	ADP
ap-4587	89	19	α	α	NOUN
ap-4587	89	20	and	and	CCONJ
ap-4587	89	21	its	its	PRON
ap-4587	89	22	degree	degree	NOUN
ap-4587	89	23	is	be	AUX
ap-4587	89	24	the	the	DET
ap-4587	89	25	degree	degree	NOUN
ap-4587	89	26	of	of	ADP
ap-4587	89	27	α	α	NOUN
ap-4587	89	28	.	.	PUNCT
ap-4587	90	1	algebraic	algebraic	ADJ
ap-4587	90	2	numbers	number	NOUN
ap-4587	90	3	with	with	ADP
ap-4587	90	4	the	the	DET
ap-4587	90	5	same	same	ADJ
ap-4587	90	6	minimal	minimal	ADJ
ap-4587	90	7	polynomial	polynomial	NOUN
ap-4587	90	8	are	be	AUX
ap-4587	90	9	called	call	VERB
ap-4587	90	10	algebraic	algebraic	ADJ
ap-4587	90	11	conjugates	conjugate	NOUN
ap-4587	90	12	.	.	PUNCT
ap-4587	91	1	431	431	NUM
ap-4587	91	2	zuzana	zuzana	PROPN
ap-4587	91	3	masáková	masáková	PROPN
ap-4587	91	4	,	,	PUNCT
ap-4587	91	5	jan	jan	PROPN
ap-4587	91	6	mazáč	mazáč	PROPN
ap-4587	91	7	acta	acta	PROPN
ap-4587	91	8	polytechnica	polytechnica	PROPN
ap-4587	91	9	if	if	SCONJ
ap-4587	91	10	the	the	DET
ap-4587	91	11	minimal	minimal	ADJ
ap-4587	91	12	polynomial	polynomial	NOUN
ap-4587	91	13	of	of	ADP
ap-4587	91	14	α	α	PROPN
ap-4587	91	15	has	have	VERB
ap-4587	91	16	integer	integer	NOUN
ap-4587	91	17	coefficients	coefficient	NOUN
ap-4587	91	18	,	,	PUNCT
ap-4587	91	19	then	then	ADV
ap-4587	91	20	α	α	PROPN
ap-4587	91	21	is	be	AUX
ap-4587	91	22	said	say	VERB
ap-4587	91	23	to	to	PART
ap-4587	91	24	be	be	AUX
ap-4587	91	25	an	an	DET
ap-4587	91	26	algebraic	algebraic	ADJ
ap-4587	91	27	integer	integer	NOUN
ap-4587	91	28	.	.	PUNCT
ap-4587	92	1	a	a	DET
ap-4587	92	2	special	special	ADJ
ap-4587	92	3	class	class	NOUN
ap-4587	92	4	of	of	ADP
ap-4587	92	5	algebraic	algebraic	ADJ
ap-4587	92	6	integers	integer	NOUN
ap-4587	92	7	is	be	AUX
ap-4587	92	8	given	give	VERB
ap-4587	92	9	by	by	ADP
ap-4587	92	10	the	the	DET
ap-4587	92	11	so	so	ADV
ap-4587	92	12	-	-	PUNCT
ap-4587	92	13	called	call	VERB
ap-4587	92	14	pisot	pisot	ADJ
ap-4587	92	15	numbers	number	NOUN
ap-4587	92	16	.	.	PUNCT
ap-4587	93	1	a	a	DET
ap-4587	93	2	pisot	pisot	ADJ
ap-4587	93	3	number	number	NOUN
ap-4587	93	4	is	be	AUX
ap-4587	93	5	an	an	DET
ap-4587	93	6	algebraic	algebraic	ADJ
ap-4587	93	7	integer	integer	NOUN
ap-4587	93	8	β	β	X
ap-4587	93	9	>	>	X
ap-4587	93	10	1	1	NUM
ap-4587	93	11	whose	whose	DET
ap-4587	93	12	algebraic	algebraic	ADJ
ap-4587	93	13	conjugates	conjugate	NOUN
ap-4587	93	14	lie	lie	VERB
ap-4587	93	15	in	in	ADP
ap-4587	93	16	the	the	DET
ap-4587	93	17	interior	interior	NOUN
ap-4587	93	18	of	of	ADP
ap-4587	93	19	the	the	DET
ap-4587	93	20	unit	unit	NOUN
ap-4587	93	21	disk	disk	NOUN
ap-4587	93	22	.	.	PUNCT
ap-4587	94	1	the	the	DET
ap-4587	94	2	most	most	ADV
ap-4587	94	3	prominent	prominent	ADJ
ap-4587	94	4	example	example	NOUN
ap-4587	94	5	of	of	ADP
ap-4587	94	6	a	a	DET
ap-4587	94	7	pisot	pisot	ADJ
ap-4587	94	8	number	number	NOUN
ap-4587	94	9	is	be	AUX
ap-4587	94	10	the	the	DET
ap-4587	94	11	golden	golden	ADJ
ap-4587	94	12	ratio	ratio	NOUN
ap-4587	94	13	τ	τ	PROPN
ap-4587	94	14	=	=	SYM
ap-4587	94	15	1	1	NUM
ap-4587	94	16	2	2	NUM
ap-4587	94	17	(	(	PUNCT
ap-4587	94	18	1	1	NUM
ap-4587	94	19	+	+	CCONJ
ap-4587	94	20	√	√	NUM
ap-4587	94	21	5	5	NUM
ap-4587	94	22	)	)	PUNCT
ap-4587	94	23	,	,	PUNCT
ap-4587	94	24	with	with	ADP
ap-4587	94	25	minimal	minimal	ADJ
ap-4587	94	26	polynomial	polynomial	ADJ
ap-4587	95	1	x2	x2	PROPN
ap-4587	95	2	−	−	PROPN
ap-4587	95	3	x−	x−	PROPN
ap-4587	95	4	1	1	NUM
ap-4587	95	5	.	.	PUNCT
ap-4587	96	1	the	the	DET
ap-4587	96	2	golden	golden	ADJ
ap-4587	96	3	ratio	ratio	NOUN
ap-4587	96	4	is	be	AUX
ap-4587	96	5	strongly	strongly	ADV
ap-4587	96	6	linked	link	VERB
ap-4587	96	7	to	to	ADP
ap-4587	96	8	the	the	DET
ap-4587	96	9	5	5	NUM
ap-4587	96	10	-	-	ADJ
ap-4587	96	11	fold	fold	ADJ
ap-4587	96	12	symmetry	symmetry	NOUN
ap-4587	96	13	,	,	PUNCT
ap-4587	96	14	namely	namely	ADV
ap-4587	96	15	by	by	ADP
ap-4587	96	16	the	the	DET
ap-4587	96	17	equality	equality	NOUN
ap-4587	96	18	2	2	NUM
ap-4587	96	19	cos	cos	SCONJ
ap-4587	96	20	2π	2π	PROPN
ap-4587	96	21	5	5	NUM
ap-4587	96	22	=	=	SYM
ap-4587	96	23	τ−1	τ−1	PROPN
ap-4587	96	24	.	.	PROPN
ap-4587	96	25	pisot	pisot	ADJ
ap-4587	96	26	numbers	number	NOUN
ap-4587	96	27	appear	appear	VERB
ap-4587	96	28	as	as	ADP
ap-4587	96	29	self	self	NOUN
ap-4587	96	30	-	-	PUNCT
ap-4587	96	31	similarity	similarity	NOUN
ap-4587	96	32	factors	factor	NOUN
ap-4587	96	33	of	of	ADP
ap-4587	96	34	cut	cut	VERB
ap-4587	96	35	-	-	PUNCT
ap-4587	96	36	and	and	CCONJ
ap-4587	96	37	-	-	PUNCT
ap-4587	96	38	project	project	NOUN
ap-4587	96	39	sets	set	NOUN
ap-4587	96	40	.	.	PUNCT
ap-4587	97	1	another	another	DET
ap-4587	97	2	class	class	NOUN
ap-4587	97	3	of	of	ADP
ap-4587	97	4	important	important	ADJ
ap-4587	97	5	numbers	number	NOUN
ap-4587	97	6	are	be	AUX
ap-4587	97	7	salem	salem	NOUN
ap-4587	97	8	numbers	number	NOUN
ap-4587	97	9	,	,	PUNCT
ap-4587	97	10	algebraic	algebraic	ADJ
ap-4587	97	11	integers	integer	NOUN
ap-4587	97	12	>	>	X
ap-4587	97	13	1	1	NUM
ap-4587	97	14	with	with	ADP
ap-4587	97	15	conjugates	conjugate	NOUN
ap-4587	97	16	in	in	ADP
ap-4587	97	17	the	the	DET
ap-4587	97	18	unit	unit	NOUN
ap-4587	97	19	disk	disk	NOUN
ap-4587	97	20	with	with	ADP
ap-4587	97	21	at	at	ADV
ap-4587	97	22	least	least	ADV
ap-4587	97	23	one	one	NUM
ap-4587	97	24	being	be	AUX
ap-4587	97	25	on	on	ADP
ap-4587	97	26	the	the	DET
ap-4587	97	27	unit	unit	NOUN
ap-4587	97	28	circle	circle	NOUN
ap-4587	97	29	.	.	PUNCT
ap-4587	98	1	the	the	DET
ap-4587	98	2	notion	notion	NOUN
ap-4587	98	3	of	of	ADP
ap-4587	98	4	pisot	pisot	ADJ
ap-4587	98	5	numbers	number	NOUN
ap-4587	98	6	is	be	AUX
ap-4587	98	7	transferred	transfer	VERB
ap-4587	98	8	to	to	ADP
ap-4587	98	9	the	the	DET
ap-4587	98	10	complex	complex	ADJ
ap-4587	98	11	plane	plane	NOUN
ap-4587	98	12	by	by	ADP
ap-4587	98	13	the	the	DET
ap-4587	98	14	term	term	NOUN
ap-4587	98	15	complex	complex	ADJ
ap-4587	98	16	pisot	pisot	ADJ
ap-4587	98	17	number	number	NOUN
ap-4587	98	18	–	–	PUNCT
ap-4587	98	19	a	a	DET
ap-4587	98	20	complex	complex	ADJ
ap-4587	98	21	algebraic	algebraic	ADJ
ap-4587	98	22	integer	integer	NOUN
ap-4587	98	23	β	β	ADP
ap-4587	98	24	such	such	ADJ
ap-4587	98	25	that	that	SCONJ
ap-4587	98	26	all	all	DET
ap-4587	98	27	algebraic	algebraic	ADJ
ap-4587	98	28	conjugates	conjugate	NOUN
ap-4587	98	29	but	but	CCONJ
ap-4587	98	30	β	β	NOUN
ap-4587	98	31	and	and	CCONJ
ap-4587	98	32	its	its	PRON
ap-4587	98	33	complex	complex	ADJ
ap-4587	98	34	conjugate	conjugate	NOUN
ap-4587	98	35	β	β	X
ap-4587	98	36	belong	belong	VERB
ap-4587	98	37	to	to	ADP
ap-4587	98	38	the	the	DET
ap-4587	98	39	interior	interior	NOUN
ap-4587	98	40	of	of	ADP
ap-4587	98	41	the	the	DET
ap-4587	98	42	unit	unit	NOUN
ap-4587	98	43	disk	disk	NOUN
ap-4587	98	44	.	.	PUNCT
ap-4587	99	1	these	these	PRON
ap-4587	99	2	will	will	AUX
ap-4587	99	3	play	play	VERB
ap-4587	99	4	important	important	ADJ
ap-4587	99	5	role	role	NOUN
ap-4587	99	6	in	in	ADP
ap-4587	99	7	section	section	NOUN
ap-4587	99	8	7	7	NUM
ap-4587	99	9	.	.	SYM
ap-4587	99	10	3	3	NUM
ap-4587	99	11	.	.	NOUN
ap-4587	99	12	matrix	matrix	NOUN
ap-4587	99	13	formalism	formalism	NOUN
ap-4587	99	14	for	for	ADP
ap-4587	99	15	the	the	DET
ap-4587	99	16	cut	cut	NOUN
ap-4587	99	17	-	-	PUNCT
ap-4587	99	18	and	and	CCONJ
ap-4587	99	19	-	-	PUNCT
ap-4587	99	20	project	project	NOUN
ap-4587	99	21	method	method	NOUN
ap-4587	99	22	in	in	ADP
ap-4587	99	23	what	what	PRON
ap-4587	99	24	will	will	AUX
ap-4587	99	25	follow	follow	VERB
ap-4587	99	26	,	,	PUNCT
ap-4587	99	27	we	we	PRON
ap-4587	99	28	use	use	VERB
ap-4587	99	29	a	a	DET
ap-4587	99	30	matrix	matrix	NOUN
ap-4587	99	31	formalism	formalism	NOUN
ap-4587	99	32	for	for	ADP
ap-4587	99	33	describing	describe	VERB
ap-4587	99	34	the	the	DET
ap-4587	99	35	cut	cut	VERB
ap-4587	99	36	-	-	PUNCT
ap-4587	99	37	and	and	CCONJ
ap-4587	99	38	-	-	PUNCT
ap-4587	99	39	project	project	NOUN
ap-4587	99	40	sets	set	NOUN
ap-4587	99	41	.	.	PUNCT
ap-4587	100	1	on	on	ADP
ap-4587	100	2	its	its	PRON
ap-4587	100	3	basis	basis	NOUN
ap-4587	100	4	,	,	PUNCT
ap-4587	100	5	we	we	PRON
ap-4587	100	6	will	will	AUX
ap-4587	100	7	set	set	VERB
ap-4587	100	8	the	the	DET
ap-4587	100	9	conditions	condition	NOUN
ap-4587	100	10	on	on	ADP
ap-4587	100	11	the	the	DET
ap-4587	100	12	vectors	vector	NOUN
ap-4587	100	13	generating	generate	VERB
ap-4587	100	14	the	the	DET
ap-4587	100	15	lattice	lattice	NOUN
ap-4587	100	16	l	l	NOUN
ap-4587	100	17	=	=	PUNCT
ap-4587	100	18	{	{	PUNCT
ap-4587	100	19	∑s	∑s	PROPN
ap-4587	100	20	i=1	i=1	PROPN
ap-4587	100	21	aili	aili	NOUN
ap-4587	100	22	:	:	PUNCT
ap-4587	100	23	ai	ai	VERB
ap-4587	100	24	∈	∈	PROPN
ap-4587	100	25	z	z	NOUN
ap-4587	100	26	}	}	PUNCT
ap-4587	100	27	,	,	PUNCT
ap-4587	100	28	so	so	SCONJ
ap-4587	100	29	that	that	SCONJ
ap-4587	100	30	the	the	DET
ap-4587	100	31	scheme	scheme	NOUN
ap-4587	100	32	is	be	AUX
ap-4587	100	33	self	self	NOUN
ap-4587	100	34	-	-	PUNCT
ap-4587	100	35	similar	similar	ADJ
ap-4587	100	36	.	.	PUNCT
ap-4587	101	1	denote	denote	VERB
ap-4587	101	2	by	by	ADP
ap-4587	101	3	v	v	ADP
ap-4587	101	4	the	the	DET
ap-4587	101	5	s×	s×	NOUN
ap-4587	101	6	s	s	PART
ap-4587	101	7	matrix	matrix	NOUN
ap-4587	101	8	formed	form	VERB
ap-4587	101	9	by	by	ADP
ap-4587	101	10	the	the	DET
ap-4587	101	11	vectors	vector	NOUN
ap-4587	101	12	l1	l1	PROPN
ap-4587	101	13	,	,	PUNCT
ap-4587	101	14	.	.	PUNCT
ap-4587	101	15	.	.	PUNCT
ap-4587	102	1	.	.	PUNCT
ap-4587	103	1	,	,	PUNCT
ap-4587	103	2	ls	ls	X
ap-4587	103	3	written	write	VERB
ap-4587	103	4	in	in	ADP
ap-4587	103	5	columns	column	NOUN
ap-4587	103	6	.	.	PUNCT
ap-4587	104	1	every	every	DET
ap-4587	104	2	lattice	lattice	NOUN
ap-4587	104	3	vector	vector	NOUN
ap-4587	104	4	can	can	AUX
ap-4587	104	5	be	be	AUX
ap-4587	104	6	then	then	ADV
ap-4587	104	7	written	write	VERB
ap-4587	104	8	as	as	ADP
ap-4587	104	9	l	l	NOUN
ap-4587	104	10	=	=	SYM
ap-4587	104	11	v	v	NOUN
ap-4587	104	12	x	x	PUNCT
ap-4587	104	13	with	with	ADP
ap-4587	104	14	x	x	PROPN
ap-4587	104	15	∈	∈	PROPN
ap-4587	104	16	zs	zs	PROPN
ap-4587	104	17	.	.	PUNCT
ap-4587	105	1	the	the	DET
ap-4587	105	2	projections	projection	NOUN
ap-4587	105	3	π1	π1	NOUN
ap-4587	105	4	,	,	PUNCT
ap-4587	105	5	π2	π2	PROPN
ap-4587	105	6	then	then	ADV
ap-4587	105	7	act	act	VERB
ap-4587	105	8	on	on	ADP
ap-4587	105	9	a	a	DET
ap-4587	105	10	lattice	lattice	NOUN
ap-4587	105	11	vector	vector	NOUN
ap-4587	105	12	l	l	NOUN
ap-4587	105	13	∈	∈	PROPN
ap-4587	105	14	l	l	NOUN
ap-4587	105	15	as	as	ADP
ap-4587	105	16	π1(l	π1(l	NOUN
ap-4587	105	17	)	)	PUNCT
ap-4587	105	18	=	=	PUNCT
ap-4587	105	19	(	(	PUNCT
ap-4587	105	20	in	in	ADP
ap-4587	105	21	,	,	PUNCT
ap-4587	105	22	o)l	o)l	NOUN
ap-4587	105	23	,	,	PUNCT
ap-4587	105	24	π2(l	π2(l	NOUN
ap-4587	105	25	)	)	PUNCT
ap-4587	105	26	=	=	SYM
ap-4587	105	27	(	(	PUNCT
ap-4587	105	28	o	o	NOUN
ap-4587	105	29	,	,	PUNCT
ap-4587	105	30	is−n)l	is−n)l	NOUN
ap-4587	105	31	,	,	PUNCT
ap-4587	105	32	where	where	SCONJ
ap-4587	105	33	ik	ik	PROPN
ap-4587	105	34	is	be	AUX
ap-4587	105	35	the	the	DET
ap-4587	105	36	identity	identity	NOUN
ap-4587	105	37	matrix	matrix	NOUN
ap-4587	105	38	of	of	ADP
ap-4587	105	39	order	order	NOUN
ap-4587	105	40	k	k	NOUN
ap-4587	105	41	and	and	CCONJ
ap-4587	105	42	o	o	PROPN
ap-4587	105	43	stands	stand	VERB
ap-4587	105	44	for	for	ADP
ap-4587	105	45	the	the	DET
ap-4587	105	46	zero	zero	NUM
ap-4587	105	47	matrix	matrix	NOUN
ap-4587	105	48	of	of	ADP
ap-4587	105	49	the	the	DET
ap-4587	105	50	size	size	NOUN
ap-4587	105	51	n×	n×	PROPN
ap-4587	105	52	(	(	PUNCT
ap-4587	105	53	s−	s−	PROPN
ap-4587	105	54	n	n	CCONJ
ap-4587	105	55	)	)	PUNCT
ap-4587	105	56	or	or	CCONJ
ap-4587	105	57	(	(	PUNCT
ap-4587	105	58	s−	s−	PROPN
ap-4587	105	59	n)×	n)×	PRON
ap-4587	105	60	n	n	CCONJ
ap-4587	105	61	,	,	PUNCT
ap-4587	105	62	respectively	respectively	ADV
ap-4587	105	63	.	.	PUNCT
ap-4587	106	1	assume	assume	VERB
ap-4587	106	2	having	have	VERB
ap-4587	106	3	a	a	DET
ap-4587	106	4	window	window	NOUN
ap-4587	106	5	ω	ω	X
ap-4587	106	6	⊂	⊂	X
ap-4587	106	7	b(0	b(0	PROPN
ap-4587	106	8	,	,	PUNCT
ap-4587	106	9	r	r	NOUN
ap-4587	106	10	)	)	PUNCT
ap-4587	106	11	⊂	⊂	PROPN
ap-4587	106	12	rs−n	rs−n	PROPN
ap-4587	106	13	.	.	PUNCT
ap-4587	107	1	an	an	PRON
ap-4587	107	2	n	n	ADV
ap-4587	107	3	-	-	PUNCT
ap-4587	107	4	dimensional	dimensional	ADJ
ap-4587	107	5	cut	cut	NOUN
ap-4587	107	6	-	-	PUNCT
ap-4587	107	7	and	and	CCONJ
ap-4587	107	8	-	-	PUNCT
ap-4587	107	9	project	project	NOUN
ap-4587	107	10	set	set	VERB
ap-4587	107	11	with	with	ADP
ap-4587	107	12	window	window	NOUN
ap-4587	107	13	ω	ω	PROPN
ap-4587	107	14	can	can	AUX
ap-4587	107	15	be	be	AUX
ap-4587	107	16	expressed	express	VERB
ap-4587	107	17	as	as	ADP
ap-4587	107	18	σ(ω	σ(ω	PROPN
ap-4587	107	19	)	)	PUNCT
ap-4587	107	20	=	=	PRON
ap-4587	107	21	{	{	PUNCT
ap-4587	107	22	(	(	PUNCT
ap-4587	107	23	in	in	ADP
ap-4587	107	24	,	,	PUNCT
ap-4587	107	25	o)v	o)v	NOUN
ap-4587	107	26	x	x	X
ap-4587	107	27	:	:	PUNCT
ap-4587	107	28	x	x	SYM
ap-4587	107	29	∈	∈	PROPN
ap-4587	107	30	zs	zs	PROPN
ap-4587	107	31	,	,	PUNCT
ap-4587	107	32	(	(	PUNCT
ap-4587	107	33	o	o	NOUN
ap-4587	107	34	,	,	PUNCT
ap-4587	107	35	is−n)v	is−n)v	NOUN
ap-4587	107	36	x	x	X
ap-4587	107	37	∈	∈	PROPN
ap-4587	107	38	ω	ω	NUM
ap-4587	107	39	}	}	PUNCT
ap-4587	107	40	.	.	PUNCT
ap-4587	108	1	denoting	denote	VERB
ap-4587	108	2	(	(	PUNCT
ap-4587	108	3	in	in	ADP
ap-4587	108	4	,	,	PUNCT
ap-4587	108	5	o)v	o)v	NOUN
ap-4587	108	6	x	x	PUNCT
ap-4587	108	7	=	=	PUNCT
ap-4587	108	8	π1(v	π1(v	NOUN
ap-4587	108	9	x	x	X
ap-4587	108	10	)	)	PUNCT
ap-4587	108	11	=	=	SYM
ap-4587	108	12	b	b	PROPN
ap-4587	108	13	∈	∈	PROPN
ap-4587	108	14	rn	rn	PROPN
ap-4587	108	15	,	,	PUNCT
ap-4587	108	16	(	(	PUNCT
ap-4587	108	17	o	o	NOUN
ap-4587	108	18	,	,	PUNCT
ap-4587	108	19	is−n)v	is−n)v	NOUN
ap-4587	108	20	x	x	X
ap-4587	108	21	=	=	SYM
ap-4587	108	22	π2(v	π2(v	PROPN
ap-4587	108	23	x	x	X
ap-4587	108	24	)	)	PUNCT
ap-4587	108	25	=	=	SYM
ap-4587	108	26	b∗	b∗	ADJ
ap-4587	108	27	∈	∈	PROPN
ap-4587	108	28	rs−n	rs−n	NOUN
ap-4587	108	29	,	,	PUNCT
ap-4587	108	30	we	we	PRON
ap-4587	108	31	obtain	obtain	VERB
ap-4587	108	32	the	the	DET
ap-4587	108	33	cut	cut	VERB
ap-4587	108	34	-	-	PUNCT
ap-4587	108	35	and	and	CCONJ
ap-4587	108	36	-	-	PUNCT
ap-4587	108	37	project	project	NOUN
ap-4587	108	38	set	set	VERB
ap-4587	108	39	in	in	ADP
ap-4587	108	40	the	the	DET
ap-4587	108	41	form	form	NOUN
ap-4587	108	42	σ(ω	σ(ω	PROPN
ap-4587	108	43	)	)	PUNCT
ap-4587	109	1	=	=	PRON
ap-4587	109	2	{	{	PUNCT
ap-4587	109	3	b	b	X
ap-4587	109	4	∈	∈	PROPN
ap-4587	109	5	π1(l	π1(l	NUM
ap-4587	109	6	)	)	PUNCT
ap-4587	109	7	:	:	PUNCT
ap-4587	109	8	b∗	b∗	PROPN
ap-4587	109	9	∈	∈	PROPN
ap-4587	109	10	ω	ω	PROPN
ap-4587	109	11	}	}	PUNCT
ap-4587	109	12	,	,	PUNCT
ap-4587	109	13	which	which	PRON
ap-4587	109	14	fully	fully	ADV
ap-4587	109	15	corresponds	correspond	VERB
ap-4587	109	16	to	to	ADP
ap-4587	109	17	the	the	DET
ap-4587	109	18	definition	definition	NOUN
ap-4587	109	19	(	(	PUNCT
ap-4587	109	20	1	1	NUM
ap-4587	109	21	)	)	PUNCT
ap-4587	109	22	.	.	PUNCT
ap-4587	110	1	we	we	PRON
ap-4587	110	2	further	far	ADV
ap-4587	110	3	use	use	VERB
ap-4587	110	4	the	the	DET
ap-4587	110	5	above	above	ADJ
ap-4587	110	6	matrix	matrix	NOUN
ap-4587	110	7	formalism	formalism	NOUN
ap-4587	110	8	for	for	ADP
ap-4587	110	9	deriving	derive	VERB
ap-4587	110	10	several	several	ADJ
ap-4587	110	11	statements	statement	NOUN
ap-4587	110	12	about	about	ADP
ap-4587	110	13	self	self	NOUN
ap-4587	110	14	-	-	PUNCT
ap-4587	110	15	similarities	similarity	NOUN
ap-4587	110	16	of	of	ADP
ap-4587	110	17	the	the	DET
ap-4587	110	18	cut	cut	NOUN
ap-4587	110	19	-	-	PUNCT
ap-4587	110	20	and	and	CCONJ
ap-4587	110	21	-	-	PUNCT
ap-4587	110	22	project	project	NOUN
ap-4587	110	23	scheme	scheme	NOUN
ap-4587	110	24	and	and	CCONJ
ap-4587	110	25	cut	cut	NOUN
ap-4587	110	26	-	-	PUNCT
ap-4587	110	27	and	and	CCONJ
ap-4587	110	28	-	-	PUNCT
ap-4587	110	29	project	project	NOUN
ap-4587	110	30	sets	set	NOUN
ap-4587	110	31	.	.	PUNCT
ap-4587	111	1	theorem	theorem	VERB
ap-4587	111	2	3.1	3.1	NUM
ap-4587	111	3	.	.	PUNCT
ap-4587	112	1	let	let	AUX
ap-4587	112	2	(	(	PUNCT
ap-4587	112	3	l	l	NOUN
ap-4587	112	4	,	,	PUNCT
ap-4587	112	5	π1	π1	NOUN
ap-4587	112	6	,	,	PUNCT
ap-4587	112	7	π2	π2	PROPN
ap-4587	112	8	)	)	PUNCT
ap-4587	112	9	be	be	AUX
ap-4587	112	10	a	a	DET
ap-4587	112	11	non	non	ADJ
ap-4587	112	12	-	-	ADJ
ap-4587	112	13	degenerate	degenerate	ADJ
ap-4587	112	14	irreducible	irreducible	ADJ
ap-4587	112	15	cut	cut	NOUN
ap-4587	112	16	-	-	PUNCT
ap-4587	112	17	and	and	CCONJ
ap-4587	112	18	-	-	PUNCT
ap-4587	112	19	project	project	NOUN
ap-4587	112	20	scheme	scheme	NOUN
ap-4587	112	21	with	with	ADP
ap-4587	112	22	l	l	PROPN
ap-4587	112	23	⊂	⊂	PROPN
ap-4587	112	24	rs×s	rs×s	PROPN
ap-4587	112	25	.	.	PUNCT
ap-4587	113	1	let	let	VERB
ap-4587	113	2	a	a	DET
ap-4587	113	3	∈	∈	ADJ
ap-4587	113	4	rn×n	rn×n	NOUN
ap-4587	113	5	satisfy	satisfy	NOUN
ap-4587	113	6	aπ1(l	aπ1(l	PROPN
ap-4587	113	7	)	)	PUNCT
ap-4587	114	1	⊂	⊂	PROPN
ap-4587	114	2	π1(l	π1(l	NUM
ap-4587	114	3	)	)	PUNCT
ap-4587	114	4	.	.	PUNCT
ap-4587	115	1	then	then	ADV
ap-4587	115	2	there	there	PRON
ap-4587	115	3	exists	exist	VERB
ap-4587	115	4	a	a	DET
ap-4587	115	5	matrix	matrix	NOUN
ap-4587	115	6	c	c	NOUN
ap-4587	115	7	∈	∈	PROPN
ap-4587	115	8	zs×s	zs×s	PROPN
ap-4587	115	9	,	,	PUNCT
ap-4587	115	10	similar	similar	ADJ
ap-4587	115	11	to	to	ADP
ap-4587	115	12	a	a	DET
ap-4587	115	13	matrix	matrix	NOUN
ap-4587	115	14	(	(	PUNCT
ap-4587	115	15	a	a	DET
ap-4587	115	16	o	o	X
ap-4587	115	17	o	o	X
ap-4587	115	18	b	b	PROPN
ap-4587	115	19	)	)	PUNCT
ap-4587	115	20	,	,	PUNCT
ap-4587	115	21	where	where	SCONJ
ap-4587	115	22	b	b	X
ap-4587	115	23	∈	∈	PROPN
ap-4587	115	24	r(s−n)×(s−n	r(s−n)×(s−n	PROPN
ap-4587	115	25	)	)	PUNCT
ap-4587	115	26	.	.	PUNCT
ap-4587	116	1	in	in	ADP
ap-4587	116	2	particular	particular	ADJ
ap-4587	116	3	c	c	PROPN
ap-4587	116	4	=	=	SYM
ap-4587	116	5	v	v	PART
ap-4587	116	6	−1	−1	NOUN
ap-4587	116	7	(	(	PUNCT
ap-4587	116	8	a	a	DET
ap-4587	116	9	o	o	X
ap-4587	116	10	o	o	X
ap-4587	116	11	b	b	PROPN
ap-4587	116	12	)	)	PUNCT
ap-4587	116	13	v	v	NOUN
ap-4587	116	14	,	,	PUNCT
ap-4587	116	15	where	where	SCONJ
ap-4587	116	16	v	v	NOUN
ap-4587	116	17	∈	∈	PROPN
ap-4587	116	18	rs×s	rs×s	NOUN
ap-4587	116	19	is	be	AUX
ap-4587	116	20	the	the	DET
ap-4587	116	21	matrix	matrix	NOUN
ap-4587	116	22	formed	form	VERB
ap-4587	116	23	by	by	ADP
ap-4587	116	24	the	the	DET
ap-4587	116	25	vector	vector	NOUN
ap-4587	116	26	generators	generator	NOUN
ap-4587	116	27	of	of	ADP
ap-4587	116	28	the	the	DET
ap-4587	116	29	lattice	lattice	PROPN
ap-4587	116	30	l	l	PROPN
ap-4587	116	31	written	write	VERB
ap-4587	116	32	in	in	ADP
ap-4587	116	33	columns	column	NOUN
ap-4587	116	34	.	.	PUNCT
ap-4587	117	1	proof	proof	NOUN
ap-4587	117	2	.	.	PUNCT
ap-4587	118	1	let	let	VERB
ap-4587	118	2	l1	l1	PROPN
ap-4587	118	3	,	,	PUNCT
ap-4587	118	4	l2	l2	NOUN
ap-4587	118	5	,	,	PUNCT
ap-4587	118	6	.	.	PUNCT
ap-4587	118	7	.	.	PUNCT
ap-4587	119	1	.	.	PUNCT
ap-4587	120	1	,	,	PUNCT
ap-4587	120	2	ls	ls	PROPN
ap-4587	120	3	be	be	AUX
ap-4587	120	4	the	the	DET
ap-4587	120	5	linearly	linearly	ADV
ap-4587	120	6	independent	independent	ADJ
ap-4587	120	7	vectors	vector	NOUN
ap-4587	120	8	in	in	ADP
ap-4587	120	9	rs	rs	PROPN
ap-4587	120	10	generating	generate	VERB
ap-4587	120	11	the	the	DET
ap-4587	120	12	lattice	lattice	PROPN
ap-4587	120	13	l.	l.	PROPN
ap-4587	120	14	since	since	SCONJ
ap-4587	120	15	a	a	PRON
ap-4587	120	16	is	be	AUX
ap-4587	120	17	a	a	DET
ap-4587	120	18	selfsimilarity	selfsimilarity	NOUN
ap-4587	120	19	of	of	ADP
ap-4587	120	20	the	the	DET
ap-4587	120	21	set	set	NOUN
ap-4587	120	22	π1(l	π1(l	PRON
ap-4587	120	23	)	)	PUNCT
ap-4587	120	24	,	,	PUNCT
ap-4587	120	25	for	for	ADP
ap-4587	120	26	every	every	DET
ap-4587	120	27	l	l	NOUN
ap-4587	120	28	∈	∈	NOUN
ap-4587	120	29	l	l	NOUN
ap-4587	120	30	there	there	PRON
ap-4587	120	31	exists	exist	VERB
ap-4587	120	32	l′	l′	NOUN
ap-4587	120	33	∈	∈	PROPN
ap-4587	120	34	l	l	NOUN
ap-4587	121	1	such	such	ADJ
ap-4587	121	2	that	that	SCONJ
ap-4587	121	3	aπ1(l	aπ1(l	PROPN
ap-4587	121	4	)	)	PUNCT
ap-4587	121	5	=	=	NOUN
ap-4587	121	6	π1(l′	π1(l′	PROPN
ap-4587	121	7	)	)	PUNCT
ap-4587	121	8	.	.	PUNCT
ap-4587	122	1	correspondingly	correspondingly	ADV
ap-4587	122	2	,	,	PUNCT
ap-4587	122	3	to	to	ADP
ap-4587	122	4	every	every	DET
ap-4587	122	5	x	x	SYM
ap-4587	122	6	∈	∈	ADJ
ap-4587	122	7	zs	z	NOUN
ap-4587	122	8	there	there	PRON
ap-4587	122	9	exists	exist	VERB
ap-4587	122	10	x′	x′	PROPN
ap-4587	122	11	∈	∈	PROPN
ap-4587	122	12	zs	zs	PROPN
ap-4587	123	1	such	such	ADJ
ap-4587	123	2	that	that	SCONJ
ap-4587	123	3	aπ1(v	aπ1(v	PROPN
ap-4587	123	4	x	x	X
ap-4587	123	5	)	)	PUNCT
ap-4587	123	6	=	=	SYM
ap-4587	123	7	π1(v	π1(v	NOUN
ap-4587	123	8	x′	x′	NUM
ap-4587	123	9	)	)	PUNCT
ap-4587	123	10	.	.	PUNCT
ap-4587	124	1	the	the	DET
ap-4587	124	2	mapping	mapping	NOUN
ap-4587	124	3	x	x	PUNCT
ap-4587	124	4	7→	7→	NUM
ap-4587	124	5	x′	x′	PROPN
ap-4587	124	6	is	be	AUX
ap-4587	124	7	linear	linear	ADJ
ap-4587	124	8	over	over	ADP
ap-4587	124	9	z	z	PROPN
ap-4587	124	10	,	,	PUNCT
ap-4587	124	11	and	and	CCONJ
ap-4587	124	12	thus	thus	ADV
ap-4587	124	13	there	there	PRON
ap-4587	124	14	exists	exist	VERB
ap-4587	124	15	a	a	DET
ap-4587	124	16	matrix	matrix	NOUN
ap-4587	124	17	c	c	NOUN
ap-4587	124	18	∈	∈	PROPN
ap-4587	124	19	zs×s	zs×	VERB
ap-4587	124	20	such	such	ADJ
ap-4587	124	21	that	that	DET
ap-4587	124	22	cx	cx	PROPN
ap-4587	125	1	=	=	PUNCT
ap-4587	125	2	x′.	x′.	PROPN
ap-4587	125	3	we	we	PRON
ap-4587	125	4	further	far	ADV
ap-4587	125	5	define	define	VERB
ap-4587	125	6	a	a	DET
ap-4587	125	7	linear	linear	ADJ
ap-4587	125	8	map	map	NOUN
ap-4587	125	9	b	b	NOUN
ap-4587	125	10	by	by	ADP
ap-4587	125	11	bπ2(v	bπ2(v	PROPN
ap-4587	125	12	x	x	SYM
ap-4587	125	13	)	)	PUNCT
ap-4587	126	1	=	=	SYM
ap-4587	126	2	π2(v	π2(v	PROPN
ap-4587	126	3	x′	x′	NUM
ap-4587	126	4	)	)	PUNCT
ap-4587	126	5	,	,	PUNCT
ap-4587	126	6	for	for	ADP
ap-4587	126	7	x	x	PROPN
ap-4587	126	8	∈	∈	PROPN
ap-4587	126	9	zs	zs	PROPN
ap-4587	126	10	.	.	PUNCT
ap-4587	127	1	together	together	ADV
ap-4587	127	2	,	,	PUNCT
ap-4587	127	3	rewritten	rewrite	VERB
ap-4587	127	4	in	in	ADP
ap-4587	127	5	the	the	DET
ap-4587	127	6	matrix	matrix	NOUN
ap-4587	127	7	formalism	formalism	NOUN
ap-4587	127	8	,	,	PUNCT
ap-4587	127	9	we	we	PRON
ap-4587	127	10	have	have	VERB
ap-4587	127	11	a(in	a(in	NOUN
ap-4587	127	12	,	,	PUNCT
ap-4587	127	13	o)v	o)v	NOUN
ap-4587	127	14	x	x	PUNCT
ap-4587	128	1	=	=	PUNCT
ap-4587	128	2	(	(	PUNCT
ap-4587	128	3	in	in	ADP
ap-4587	128	4	,	,	PUNCT
ap-4587	128	5	o)v	o)v	NOUN
ap-4587	128	6	cx	cx	NOUN
ap-4587	128	7	,	,	PUNCT
ap-4587	128	8	b(o	b(o	PROPN
ap-4587	128	9	,	,	PUNCT
ap-4587	128	10	is−n)v	is−n)v	NOUN
ap-4587	128	11	x	x	X
ap-4587	128	12	=	=	SYM
ap-4587	128	13	(	(	PUNCT
ap-4587	128	14	o	o	NOUN
ap-4587	128	15	,	,	PUNCT
ap-4587	128	16	is−n)v	is−n)v	NOUN
ap-4587	128	17	cx	cx	NOUN
ap-4587	128	18	,	,	PUNCT
ap-4587	128	19	which	which	PRON
ap-4587	128	20	can	can	AUX
ap-4587	128	21	be	be	AUX
ap-4587	128	22	put	put	VERB
ap-4587	128	23	together	together	ADV
ap-4587	128	24	into	into	ADP
ap-4587	128	25	(	(	PUNCT
ap-4587	128	26	a	a	DET
ap-4587	128	27	o	o	X
ap-4587	128	28	o	o	X
ap-4587	128	29	b	b	PROPN
ap-4587	128	30	)	)	PUNCT
ap-4587	128	31	v	v	NOUN
ap-4587	128	32	x	x	SYM
ap-4587	128	33	=	=	PUNCT
ap-4587	128	34	v	v	NUM
ap-4587	128	35	cx	cx	NOUN
ap-4587	128	36	.	.	PUNCT
ap-4587	129	1	since	since	SCONJ
ap-4587	129	2	this	this	PRON
ap-4587	129	3	holds	hold	VERB
ap-4587	129	4	for	for	ADP
ap-4587	129	5	every	every	DET
ap-4587	129	6	x	x	PROPN
ap-4587	129	7	∈	∈	PROPN
ap-4587	129	8	zs	zs	PROPN
ap-4587	129	9	,	,	PUNCT
ap-4587	129	10	we	we	PRON
ap-4587	129	11	derive	derive	VERB
ap-4587	129	12	that	that	SCONJ
ap-4587	129	13	c	c	NOUN
ap-4587	129	14	=	=	SYM
ap-4587	129	15	v	v	NUM
ap-4587	129	16	−1	−1	NOUN
ap-4587	129	17	(	(	PUNCT
ap-4587	129	18	a	a	DET
ap-4587	129	19	o	o	X
ap-4587	129	20	o	o	X
ap-4587	129	21	b	b	PROPN
ap-4587	129	22	)	)	PUNCT
ap-4587	129	23	v	v	NOUN
ap-4587	129	24	,	,	PUNCT
ap-4587	129	25	(	(	PUNCT
ap-4587	129	26	2	2	X
ap-4587	129	27	)	)	PUNCT
ap-4587	129	28	which	which	PRON
ap-4587	129	29	we	we	PRON
ap-4587	129	30	aimed	aim	VERB
ap-4587	129	31	to	to	PART
ap-4587	129	32	show	show	VERB
ap-4587	129	33	.	.	PUNCT
ap-4587	130	1	432	432	NUM
ap-4587	130	2	vol	vol	NOUN
ap-4587	130	3	.	.	PUNCT
ap-4587	130	4	57	57	NUM
ap-4587	130	5	no	no	NOUN
ap-4587	130	6	.	.	PUNCT
ap-4587	131	1	6/2017	6/2017	NUM
ap-4587	131	2	on	on	ADP
ap-4587	131	3	self	self	NOUN
ap-4587	131	4	-	-	PUNCT
ap-4587	131	5	similarities	similarity	NOUN
ap-4587	131	6	of	of	ADP
ap-4587	131	7	cut	cut	NOUN
ap-4587	131	8	-	-	PUNCT
ap-4587	131	9	and	and	CCONJ
ap-4587	131	10	-	-	PUNCT
ap-4587	131	11	project	project	NOUN
ap-4587	131	12	sets	set	NOUN
ap-4587	131	13	we	we	PRON
ap-4587	131	14	have	have	VERB
ap-4587	131	15	an	an	DET
ap-4587	131	16	obvious	obvious	ADJ
ap-4587	131	17	corollary	corollary	NOUN
ap-4587	131	18	to	to	ADP
ap-4587	131	19	the	the	DET
ap-4587	131	20	above	above	ADJ
ap-4587	131	21	theorem	theorem	ADJ
ap-4587	131	22	.	.	PROPN
ap-4587	131	23	corollary	corollary	ADJ
ap-4587	131	24	3.2	3.2	NUM
ap-4587	131	25	.	.	PUNCT
ap-4587	132	1	let	let	AUX
ap-4587	132	2	(	(	PUNCT
ap-4587	132	3	l	l	NOUN
ap-4587	132	4	,	,	PUNCT
ap-4587	132	5	π1	π1	NOUN
ap-4587	132	6	,	,	PUNCT
ap-4587	132	7	π2	π2	PROPN
ap-4587	132	8	)	)	PUNCT
ap-4587	132	9	be	be	AUX
ap-4587	132	10	a	a	DET
ap-4587	132	11	non	non	ADJ
ap-4587	132	12	-	-	ADJ
ap-4587	132	13	degenerate	degenerate	ADJ
ap-4587	132	14	irreducible	irreducible	ADJ
ap-4587	132	15	cut	cut	NOUN
ap-4587	132	16	-	-	PUNCT
ap-4587	132	17	and	and	CCONJ
ap-4587	132	18	-	-	PUNCT
ap-4587	132	19	project	project	NOUN
ap-4587	132	20	scheme	scheme	NOUN
ap-4587	132	21	with	with	ADP
ap-4587	132	22	l	l	PROPN
ap-4587	132	23	⊂	⊂	PROPN
ap-4587	132	24	rs×s	rs×s	PROPN
ap-4587	132	25	.	.	PUNCT
ap-4587	133	1	let	let	VERB
ap-4587	133	2	a	a	DET
ap-4587	133	3	∈	∈	ADJ
ap-4587	133	4	rn×n	rn×n	NOUN
ap-4587	133	5	satisfy	satisfy	NOUN
ap-4587	133	6	aπ1(l	aπ1(l	PROPN
ap-4587	133	7	)	)	PUNCT
ap-4587	134	1	⊂	⊂	PROPN
ap-4587	134	2	π1(l	π1(l	NUM
ap-4587	134	3	)	)	PUNCT
ap-4587	134	4	.	.	PUNCT
ap-4587	135	1	then	then	ADV
ap-4587	135	2	the	the	DET
ap-4587	135	3	eigenvalues	eigenvalue	NOUN
ap-4587	135	4	of	of	ADP
ap-4587	135	5	the	the	DET
ap-4587	135	6	matrix	matrix	NOUN
ap-4587	135	7	a	a	PRON
ap-4587	135	8	are	be	AUX
ap-4587	135	9	algebraic	algebraic	ADJ
ap-4587	135	10	integers	integer	NOUN
ap-4587	135	11	and	and	CCONJ
ap-4587	135	12	their	their	PRON
ap-4587	135	13	minimal	minimal	ADJ
ap-4587	135	14	polynomial	polynomial	NOUN
ap-4587	135	15	divides	divide	VERB
ap-4587	135	16	the	the	DET
ap-4587	135	17	characteristic	characteristic	ADJ
ap-4587	135	18	polynomial	polynomial	NOUN
ap-4587	135	19	of	of	ADP
ap-4587	135	20	the	the	DET
ap-4587	135	21	matrix	matrix	NOUN
ap-4587	135	22	c	c	NOUN
ap-4587	135	23	over	over	ADP
ap-4587	135	24	q.	q.	PROPN
ap-4587	135	25	the	the	DET
ap-4587	135	26	matrix	matrix	NOUN
ap-4587	135	27	framework	framework	NOUN
ap-4587	135	28	also	also	ADV
ap-4587	135	29	enables	enable	VERB
ap-4587	135	30	us	we	PRON
ap-4587	135	31	to	to	PART
ap-4587	135	32	find	find	VERB
ap-4587	135	33	a	a	DET
ap-4587	135	34	cut	cut	VERB
ap-4587	135	35	-	-	PUNCT
ap-4587	135	36	and	and	CCONJ
ap-4587	135	37	-	-	PUNCT
ap-4587	135	38	project	project	NOUN
ap-4587	135	39	scheme	scheme	NOUN
ap-4587	135	40	displaying	display	VERB
ap-4587	135	41	a	a	DET
ap-4587	135	42	self	self	NOUN
ap-4587	135	43	-	-	PUNCT
ap-4587	135	44	similarity	similarity	NOUN
ap-4587	135	45	defined	define	VERB
ap-4587	135	46	by	by	ADP
ap-4587	135	47	a	a	DET
ap-4587	135	48	given	give	VERB
ap-4587	135	49	integer	integer	NOUN
ap-4587	135	50	matrix	matrix	NOUN
ap-4587	135	51	c.	c.	NOUN
ap-4587	135	52	proposition	proposition	NOUN
ap-4587	135	53	3.3	3.3	NUM
ap-4587	135	54	.	.	PUNCT
ap-4587	136	1	let	let	VERB
ap-4587	136	2	for	for	ADP
ap-4587	136	3	c	c	PROPN
ap-4587	136	4	∈	∈	PROPN
ap-4587	136	5	zs×s	zs×s	PROPN
ap-4587	136	6	there	there	PRON
ap-4587	136	7	exist	exist	VERB
ap-4587	136	8	a	a	DET
ap-4587	136	9	matrix	matrix	NOUN
ap-4587	136	10	v	v	ADP
ap-4587	136	11	∈	∈	PROPN
ap-4587	136	12	rs×s	rs×s	PROPN
ap-4587	136	13	of	of	ADP
ap-4587	136	14	rank	rank	PROPN
ap-4587	137	1	s	s	PROPN
ap-4587	137	2	such	such	ADJ
ap-4587	137	3	that	that	SCONJ
ap-4587	137	4	v	v	PROPN
ap-4587	137	5	cv	cv	PROPN
ap-4587	137	6	−1	−1	NOUN
ap-4587	137	7	is	be	AUX
ap-4587	137	8	block	block	NOUN
ap-4587	137	9	diagonal	diagonal	ADJ
ap-4587	137	10	,	,	PUNCT
ap-4587	137	11	i.e.	i.e.	X
ap-4587	137	12	,	,	PUNCT
ap-4587	137	13	v	v	NOUN
ap-4587	137	14	cv	cv	PROPN
ap-4587	137	15	−1	−1	NOUN
ap-4587	137	16	=	=	PUNCT
ap-4587	137	17	(	(	PUNCT
ap-4587	137	18	a	a	DET
ap-4587	137	19	o	o	X
ap-4587	137	20	o	o	X
ap-4587	137	21	b	b	PROPN
ap-4587	137	22	)	)	PUNCT
ap-4587	137	23	,	,	PUNCT
ap-4587	137	24	where	where	SCONJ
ap-4587	137	25	a	a	DET
ap-4587	137	26	∈	∈	NOUN
ap-4587	137	27	rn×n	rn×n	NOUN
ap-4587	137	28	,	,	PUNCT
ap-4587	137	29	b	b	X
ap-4587	137	30	∈	∈	PROPN
ap-4587	137	31	r(s−n)×(s−n	r(s−n)×(s−n	PROPN
ap-4587	137	32	)	)	PUNCT
ap-4587	137	33	.	.	PUNCT
ap-4587	138	1	denote	denote	NOUN
ap-4587	138	2	l	l	NOUN
ap-4587	139	1	=	=	PRON
ap-4587	139	2	{	{	PUNCT
ap-4587	139	3	∑s	∑s	PROPN
ap-4587	139	4	i=1	i=1	PROPN
ap-4587	139	5	aili	aili	NOUN
ap-4587	139	6	:	:	PUNCT
ap-4587	139	7	ai	ai	VERB
ap-4587	139	8	∈	∈	PROPN
ap-4587	139	9	z	z	NOUN
ap-4587	139	10	}	}	PUNCT
ap-4587	139	11	the	the	DET
ap-4587	139	12	lattice	lattice	NOUN
ap-4587	139	13	generated	generate	VERB
ap-4587	139	14	by	by	ADP
ap-4587	139	15	the	the	DET
ap-4587	139	16	linearly	linearly	ADV
ap-4587	139	17	independent	independent	ADJ
ap-4587	139	18	columns	column	NOUN
ap-4587	139	19	li	li	PROPN
ap-4587	139	20	of	of	ADP
ap-4587	139	21	v	v	NOUN
ap-4587	139	22	and	and	CCONJ
ap-4587	139	23	for	for	ADP
ap-4587	139	24	a	a	DET
ap-4587	139	25	lattice	lattice	NOUN
ap-4587	139	26	vector	vector	NOUN
ap-4587	139	27	l	l	NOUN
ap-4587	139	28	∈	∈	PROPN
ap-4587	139	29	l	l	NOUN
ap-4587	139	30	set	set	VERB
ap-4587	139	31	the	the	DET
ap-4587	139	32	projections	projection	NOUN
ap-4587	139	33	π1	π1	NOUN
ap-4587	139	34	:	:	PUNCT
ap-4587	139	35	rs	rs	PROPN
ap-4587	139	36	→	→	SYM
ap-4587	139	37	rn	rn	PROPN
ap-4587	139	38	,	,	PUNCT
ap-4587	139	39	π2	π2	ADV
ap-4587	139	40	:	:	PUNCT
ap-4587	139	41	rs	rs	NOUN
ap-4587	139	42	→	→	SYM
ap-4587	139	43	rs−n	rs−n	NOUN
ap-4587	139	44	to	to	ADP
ap-4587	139	45	π1(l	π1(l	NUM
ap-4587	139	46	)	)	PUNCT
ap-4587	139	47	=	=	PUNCT
ap-4587	139	48	(	(	PUNCT
ap-4587	139	49	in	in	ADP
ap-4587	139	50	,	,	PUNCT
ap-4587	139	51	o)l	o)l	NOUN
ap-4587	139	52	,	,	PUNCT
ap-4587	139	53	π2(l	π2(l	NOUN
ap-4587	139	54	)	)	PUNCT
ap-4587	139	55	=	=	SYM
ap-4587	139	56	(	(	PUNCT
ap-4587	139	57	o	o	NOUN
ap-4587	139	58	,	,	PUNCT
ap-4587	139	59	is−n)l	is−n)l	NOUN
ap-4587	139	60	.	.	PUNCT
ap-4587	140	1	(	(	PUNCT
ap-4587	140	2	1	1	NUM
ap-4587	140	3	.	.	PUNCT
ap-4587	140	4	)	)	PUNCT
ap-4587	141	1	then	then	ADV
ap-4587	141	2	aπ1(l	aπ1(l	PROPN
ap-4587	141	3	)	)	PUNCT
ap-4587	141	4	⊂	⊂	PROPN
ap-4587	141	5	π1(l	π1(l	NUM
ap-4587	141	6	)	)	PUNCT
ap-4587	141	7	.	.	PUNCT
ap-4587	142	1	(	(	PUNCT
ap-4587	142	2	2	2	NUM
ap-4587	142	3	.	.	PUNCT
ap-4587	142	4	)	)	PUNCT
ap-4587	143	1	if	if	SCONJ
ap-4587	143	2	z	z	PROPN
ap-4587	143	3	∈	∈	PROPN
ap-4587	143	4	zs×s	zs×s	PROPN
ap-4587	143	5	is	be	AUX
ap-4587	143	6	another	another	DET
ap-4587	143	7	matrix	matrix	NOUN
ap-4587	143	8	satisfying	satisfy	VERB
ap-4587	143	9	v	v	ADP
ap-4587	143	10	zv	zv	NOUN
ap-4587	143	11	−1	−1	NOUN
ap-4587	143	12	=	=	PUNCT
ap-4587	143	13	(	(	PUNCT
ap-4587	143	14	s	s	NOUN
ap-4587	143	15	o	o	X
ap-4587	143	16	o	o	X
ap-4587	143	17	t	t	PROPN
ap-4587	143	18	)	)	PUNCT
ap-4587	143	19	,	,	PUNCT
ap-4587	143	20	for	for	ADP
ap-4587	143	21	some	some	DET
ap-4587	143	22	s	s	NOUN
ap-4587	143	23	∈	∈	NOUN
ap-4587	143	24	rn×n	rn×n	NOUN
ap-4587	143	25	,	,	PUNCT
ap-4587	143	26	t	t	PROPN
ap-4587	143	27	∈	∈	PROPN
ap-4587	143	28	r(s−n)×(s−n	r(s−n)×(s−n	PROPN
ap-4587	143	29	)	)	PUNCT
ap-4587	143	30	,	,	PUNCT
ap-4587	143	31	then	then	ADV
ap-4587	143	32	sπ1(l	sπ1(l	PROPN
ap-4587	143	33	)	)	PUNCT
ap-4587	143	34	⊂	⊂	PROPN
ap-4587	143	35	π1(l	π1(l	NUM
ap-4587	143	36	)	)	PUNCT
ap-4587	143	37	.	.	PUNCT
ap-4587	144	1	the	the	DET
ap-4587	144	2	above	above	ADJ
ap-4587	144	3	proposition	proposition	NOUN
ap-4587	144	4	follows	follow	VERB
ap-4587	144	5	from	from	ADP
ap-4587	144	6	theorem	theorem	ADJ
ap-4587	144	7	3.1	3.1	NUM
ap-4587	144	8	.	.	PUNCT
ap-4587	145	1	note	note	VERB
ap-4587	145	2	that	that	SCONJ
ap-4587	145	3	the	the	DET
ap-4587	145	4	proposition	proposition	NOUN
ap-4587	145	5	does	do	AUX
ap-4587	145	6	not	not	PART
ap-4587	145	7	state	state	VERB
ap-4587	145	8	anything	anything	PRON
ap-4587	145	9	about	about	ADP
ap-4587	145	10	non	non	ADJ
ap-4587	145	11	-	-	NOUN
ap-4587	145	12	degeneracy	degeneracy	NOUN
ap-4587	145	13	or	or	CCONJ
ap-4587	145	14	irreducibility	irreducibility	NOUN
ap-4587	145	15	of	of	ADP
ap-4587	145	16	the	the	DET
ap-4587	145	17	cut	cut	NOUN
ap-4587	145	18	-	-	PUNCT
ap-4587	145	19	and	and	CCONJ
ap-4587	145	20	-	-	PUNCT
ap-4587	145	21	project	project	NOUN
ap-4587	145	22	scheme	scheme	NOUN
ap-4587	145	23	(	(	PUNCT
ap-4587	145	24	l	l	NOUN
ap-4587	145	25	,	,	PUNCT
ap-4587	145	26	π1	π1	NOUN
ap-4587	145	27	,	,	PUNCT
ap-4587	145	28	π2	π2	PROPN
ap-4587	145	29	)	)	PUNCT
ap-4587	145	30	obtained	obtain	VERB
ap-4587	145	31	as	as	ADP
ap-4587	145	32	shown	show	VERB
ap-4587	145	33	.	.	PUNCT
ap-4587	146	1	examples	example	NOUN
ap-4587	146	2	of	of	ADP
ap-4587	146	3	degenerate	degenerate	ADJ
ap-4587	146	4	or	or	CCONJ
ap-4587	146	5	reducible	reducible	ADJ
ap-4587	146	6	schemes	scheme	NOUN
ap-4587	146	7	may	may	AUX
ap-4587	146	8	be	be	AUX
ap-4587	146	9	constructed	construct	VERB
ap-4587	146	10	.	.	PUNCT
ap-4587	147	1	when	when	SCONJ
ap-4587	147	2	studying	study	VERB
ap-4587	147	3	the	the	DET
ap-4587	147	4	structure	structure	NOUN
ap-4587	147	5	of	of	ADP
ap-4587	147	6	the	the	DET
ap-4587	147	7	set	set	NOUN
ap-4587	147	8	of	of	ADP
ap-4587	147	9	all	all	DET
ap-4587	147	10	self	self	NOUN
ap-4587	147	11	-	-	PUNCT
ap-4587	147	12	similarities	similarity	NOUN
ap-4587	147	13	of	of	ADP
ap-4587	147	14	a	a	DET
ap-4587	147	15	given	give	VERB
ap-4587	147	16	cut	cut	VERB
ap-4587	147	17	-	-	PUNCT
ap-4587	147	18	and	and	CCONJ
ap-4587	147	19	-	-	PUNCT
ap-4587	147	20	project	project	NOUN
ap-4587	147	21	scheme	scheme	NOUN
ap-4587	147	22	,	,	PUNCT
ap-4587	147	23	one	one	PRON
ap-4587	147	24	easily	easily	ADV
ap-4587	147	25	realizes	realize	VERB
ap-4587	147	26	that	that	SCONJ
ap-4587	147	27	they	they	PRON
ap-4587	147	28	form	form	VERB
ap-4587	147	29	an	an	DET
ap-4587	147	30	associative	associative	ADJ
ap-4587	147	31	algebra	algebra	NOUN
ap-4587	147	32	over	over	ADP
ap-4587	147	33	the	the	DET
ap-4587	147	34	ring	ring	NOUN
ap-4587	147	35	z.	z.	PROPN
ap-4587	148	1	this	this	PRON
ap-4587	148	2	follows	follow	VERB
ap-4587	148	3	from	from	ADP
ap-4587	148	4	the	the	DET
ap-4587	148	5	fact	fact	NOUN
ap-4587	148	6	that	that	SCONJ
ap-4587	148	7	π1(l	π1(l	X
ap-4587	148	8	)	)	PUNCT
ap-4587	148	9	is	be	AUX
ap-4587	148	10	a	a	DET
ap-4587	148	11	z	z	NOUN
ap-4587	148	12	-	-	PUNCT
ap-4587	148	13	module	module	NOUN
ap-4587	148	14	.	.	PUNCT
ap-4587	149	1	proposition	proposition	NOUN
ap-4587	149	2	3.4	3.4	NUM
ap-4587	149	3	.	.	PUNCT
ap-4587	150	1	let	let	VERB
ap-4587	150	2	r	r	NOUN
ap-4587	150	3	⊂	⊂	PROPN
ap-4587	150	4	rn	rn	AUX
ap-4587	150	5	be	be	AUX
ap-4587	150	6	a	a	DET
ap-4587	150	7	z	z	NOUN
ap-4587	150	8	-	-	PUNCT
ap-4587	150	9	module	module	NOUN
ap-4587	150	10	.	.	PUNCT
ap-4587	151	1	denote	denote	VERB
ap-4587	151	2	by	by	ADP
ap-4587	151	3	r	r	NOUN
ap-4587	151	4	the	the	DET
ap-4587	151	5	set	set	NOUN
ap-4587	151	6	of	of	ADP
ap-4587	151	7	all	all	DET
ap-4587	151	8	linear	linear	ADJ
ap-4587	151	9	mappings	mapping	NOUN
ap-4587	151	10	s	s	PART
ap-4587	151	11	on	on	ADP
ap-4587	151	12	rn	rn	PROPN
ap-4587	151	13	such	such	ADJ
ap-4587	151	14	that	that	SCONJ
ap-4587	151	15	sr	sr	PROPN
ap-4587	151	16	⊂	⊂	PROPN
ap-4587	151	17	r.	r.	PROPN
ap-4587	152	1	then	then	ADV
ap-4587	152	2	r	r	NOUN
ap-4587	152	3	is	be	AUX
ap-4587	152	4	an	an	DET
ap-4587	152	5	associative	associative	ADJ
ap-4587	152	6	z	z	NOUN
ap-4587	152	7	-	-	NOUN
ap-4587	152	8	algebra	algebra	NOUN
ap-4587	152	9	.	.	PUNCT
ap-4587	153	1	proof	proof	NOUN
ap-4587	153	2	.	.	PUNCT
ap-4587	154	1	let	let	VERB
ap-4587	154	2	s1	s1	NOUN
ap-4587	154	3	,	,	PUNCT
ap-4587	154	4	s2	s2	NOUN
ap-4587	154	5	∈	∈	PROPN
ap-4587	154	6	r	r	NOUN
ap-4587	154	7	,	,	PUNCT
ap-4587	154	8	i.e.	i.e.	X
ap-4587	154	9	,	,	PUNCT
ap-4587	154	10	s1	s1	NOUN
ap-4587	154	11	,	,	PUNCT
ap-4587	154	12	s2	s2	NOUN
ap-4587	154	13	∈	∈	NOUN
ap-4587	154	14	rn×n	rn×n	NOUN
ap-4587	154	15	such	such	ADJ
ap-4587	154	16	that	that	SCONJ
ap-4587	154	17	sir	sir	PROPN
ap-4587	154	18	⊂	⊂	PROPN
ap-4587	154	19	r.	r.	PROPN
ap-4587	154	20	then	then	ADV
ap-4587	154	21	clearly	clearly	ADV
ap-4587	154	22	(	(	PUNCT
ap-4587	154	23	s1	s1	NOUN
ap-4587	154	24	+	+	CCONJ
ap-4587	154	25	s2)r	s2)r	ADJ
ap-4587	154	26	=	=	SYM
ap-4587	154	27	s1r+	s1r+	PROPN
ap-4587	154	28	s2r	s2r	PROPN
ap-4587	154	29	⊂	⊂	X
ap-4587	154	30	r+r	r+r	PUNCT
ap-4587	155	1	=	=	SYM
ap-4587	155	2	r	r	NOUN
ap-4587	155	3	,	,	PUNCT
ap-4587	155	4	(	(	PUNCT
ap-4587	155	5	s1s2)r	s1s2)r	X
ap-4587	155	6	=	=	SYM
ap-4587	155	7	s1(s2r	s1(s2r	X
ap-4587	155	8	)	)	PUNCT
ap-4587	155	9	⊂	⊂	PROPN
ap-4587	156	1	s1r	s1r	PROPN
ap-4587	157	1	⊂	⊂	PUNCT
ap-4587	157	2	r	r	NOUN
ap-4587	157	3	,	,	PUNCT
ap-4587	157	4	where	where	SCONJ
ap-4587	157	5	we	we	PRON
ap-4587	157	6	have	have	AUX
ap-4587	157	7	used	use	VERB
ap-4587	157	8	that	that	PRON
ap-4587	157	9	for	for	ADP
ap-4587	157	10	a	a	DET
ap-4587	157	11	z	z	NOUN
ap-4587	157	12	-	-	PUNCT
ap-4587	157	13	module	module	NOUN
ap-4587	157	14	r	r	NOUN
ap-4587	157	15	we	we	PRON
ap-4587	157	16	have	have	VERB
ap-4587	157	17	r	r	NOUN
ap-4587	157	18	+	+	NOUN
ap-4587	157	19	r	r	NOUN
ap-4587	157	20	=	=	SYM
ap-4587	157	21	r.	r.	NOUN
ap-4587	157	22	this	this	PRON
ap-4587	157	23	means	mean	VERB
ap-4587	157	24	that	that	SCONJ
ap-4587	157	25	s1	s1	PROPN
ap-4587	157	26	+	+	CCONJ
ap-4587	157	27	s2	s2	PROPN
ap-4587	157	28	,	,	PUNCT
ap-4587	157	29	s1s2	s1s2	PROPN
ap-4587	157	30	∈	∈	PROPN
ap-4587	157	31	r	r	NOUN
ap-4587	157	32	,	,	PUNCT
ap-4587	157	33	and	and	CCONJ
ap-4587	157	34	necessarily	necessarily	ADV
ap-4587	157	35	also	also	ADV
ap-4587	157	36	ks1	ks1	PROPN
ap-4587	157	37	∈	∈	PROPN
ap-4587	157	38	r	r	NOUN
ap-4587	157	39	for	for	ADP
ap-4587	157	40	any	any	DET
ap-4587	157	41	k	k	PROPN
ap-4587	157	42	∈	∈	PROPN
ap-4587	157	43	z.	z.	PROPN
ap-4587	157	44	associativity	associativity	PROPN
ap-4587	157	45	is	be	AUX
ap-4587	157	46	obvious	obvious	ADJ
ap-4587	157	47	.	.	PUNCT
ap-4587	158	1	4	4	X
ap-4587	158	2	.	.	X
ap-4587	158	3	self	self	NOUN
ap-4587	158	4	-	-	PUNCT
ap-4587	158	5	similarities	similarity	NOUN
ap-4587	158	6	of	of	ADP
ap-4587	158	7	a	a	DET
ap-4587	158	8	cut	cut	NOUN
ap-4587	158	9	-	-	PUNCT
ap-4587	158	10	and	and	CCONJ
ap-4587	158	11	-	-	PUNCT
ap-4587	158	12	project	project	NOUN
ap-4587	158	13	set	set	VERB
ap-4587	158	14	the	the	DET
ap-4587	158	15	statements	statement	NOUN
ap-4587	158	16	in	in	ADP
ap-4587	158	17	the	the	DET
ap-4587	158	18	previous	previous	ADJ
ap-4587	158	19	section	section	NOUN
ap-4587	158	20	concerned	concern	VERB
ap-4587	158	21	self	self	NOUN
ap-4587	158	22	-	-	PUNCT
ap-4587	158	23	similarities	similarity	NOUN
ap-4587	158	24	of	of	ADP
ap-4587	158	25	the	the	DET
ap-4587	158	26	z	z	NOUN
ap-4587	158	27	-	-	PUNCT
ap-4587	158	28	module	module	NOUN
ap-4587	158	29	π1(l	π1(l	NUM
ap-4587	158	30	)	)	PUNCT
ap-4587	158	31	.	.	PUNCT
ap-4587	159	1	let	let	VERB
ap-4587	159	2	us	we	PRON
ap-4587	159	3	now	now	ADV
ap-4587	159	4	concentrate	concentrate	VERB
ap-4587	159	5	on	on	ADP
ap-4587	159	6	what	what	PRON
ap-4587	159	7	can	can	AUX
ap-4587	159	8	be	be	AUX
ap-4587	159	9	said	say	VERB
ap-4587	159	10	in	in	ADP
ap-4587	159	11	general	general	ADJ
ap-4587	159	12	about	about	ADP
ap-4587	159	13	the	the	DET
ap-4587	159	14	self	self	NOUN
ap-4587	159	15	-	-	PUNCT
ap-4587	159	16	similarities	similarity	NOUN
ap-4587	159	17	of	of	ADP
ap-4587	159	18	cut	cut	NOUN
ap-4587	159	19	-	-	PUNCT
ap-4587	159	20	and	and	CCONJ
ap-4587	159	21	-	-	PUNCT
ap-4587	159	22	project	project	NOUN
ap-4587	159	23	sets	set	NOUN
ap-4587	159	24	,	,	PUNCT
ap-4587	159	25	given	give	VERB
ap-4587	159	26	a	a	DET
ap-4587	159	27	cut	cut	VERB
ap-4587	159	28	-	-	PUNCT
ap-4587	159	29	and	and	CCONJ
ap-4587	159	30	-	-	PUNCT
ap-4587	159	31	project	project	NOUN
ap-4587	159	32	scheme	scheme	NOUN
ap-4587	159	33	(	(	PUNCT
ap-4587	159	34	l	l	NOUN
ap-4587	159	35	,	,	PUNCT
ap-4587	159	36	π1	π1	NOUN
ap-4587	159	37	,	,	PUNCT
ap-4587	159	38	π2	π2	NOUN
ap-4587	159	39	)	)	PUNCT
ap-4587	159	40	with	with	ADP
ap-4587	159	41	a	a	DET
ap-4587	159	42	self	self	NOUN
ap-4587	159	43	-	-	PUNCT
ap-4587	159	44	similarity	similarity	NOUN
ap-4587	159	45	a.	a.	NOUN
ap-4587	159	46	theorem	theorem	NOUN
ap-4587	159	47	4.1	4.1	NUM
ap-4587	159	48	.	.	PUNCT
ap-4587	160	1	let	let	AUX
ap-4587	160	2	(	(	PUNCT
ap-4587	160	3	l	l	NOUN
ap-4587	160	4	,	,	PUNCT
ap-4587	160	5	π1	π1	NOUN
ap-4587	160	6	,	,	PUNCT
ap-4587	160	7	π2	π2	PROPN
ap-4587	160	8	)	)	PUNCT
ap-4587	160	9	be	be	AUX
ap-4587	160	10	a	a	DET
ap-4587	160	11	non	non	ADJ
ap-4587	160	12	-	-	ADJ
ap-4587	160	13	degenerated	degenerated	ADJ
ap-4587	160	14	irreducible	irreducible	ADJ
ap-4587	160	15	cut	cut	NOUN
ap-4587	160	16	-	-	PUNCT
ap-4587	160	17	and	and	CCONJ
ap-4587	160	18	-	-	PUNCT
ap-4587	160	19	project	project	NOUN
ap-4587	160	20	scheme	scheme	NOUN
ap-4587	160	21	with	with	ADP
ap-4587	160	22	a	a	DET
ap-4587	160	23	self	self	NOUN
ap-4587	160	24	-	-	PUNCT
ap-4587	160	25	similarity	similarity	NOUN
ap-4587	160	26	a.	a.	NOUN
ap-4587	160	27	if	if	SCONJ
ap-4587	160	28	there	there	PRON
ap-4587	160	29	exists	exist	VERB
ap-4587	160	30	a	a	DET
ap-4587	160	31	window	window	NOUN
ap-4587	160	32	ω	ω	PROPN
ap-4587	160	33	⊂	⊂	PROPN
ap-4587	160	34	rm	rm	PROPN
ap-4587	160	35	,	,	PUNCT
ap-4587	160	36	such	such	ADJ
ap-4587	160	37	that	that	SCONJ
ap-4587	160	38	aς(ω	aς(ω	NUM
ap-4587	160	39	)	)	PUNCT
ap-4587	161	1	⊂	⊂	PROPN
ap-4587	161	2	σ(ω	σ(ω	PROPN
ap-4587	161	3	)	)	PUNCT
ap-4587	161	4	,	,	PUNCT
ap-4587	161	5	then	then	ADV
ap-4587	161	6	the	the	DET
ap-4587	161	7	eigenvalues	eigenvalue	NOUN
ap-4587	161	8	of	of	ADP
ap-4587	161	9	the	the	DET
ap-4587	161	10	matrix	matrix	NOUN
ap-4587	161	11	b	b	NOUN
ap-4587	161	12	from	from	ADP
ap-4587	161	13	theorem	theorem	ADJ
ap-4587	161	14	3.1	3.1	NUM
ap-4587	161	15	are	be	AUX
ap-4587	161	16	in	in	ADP
ap-4587	161	17	modulus	modulus	NOUN
ap-4587	161	18	smaller	small	ADJ
ap-4587	161	19	or	or	CCONJ
ap-4587	161	20	equal	equal	ADJ
ap-4587	161	21	to	to	ADP
ap-4587	161	22	1	1	NUM
ap-4587	161	23	.	.	PUNCT
ap-4587	162	1	proof	proof	NOUN
ap-4587	162	2	.	.	PUNCT
ap-4587	163	1	assume	assume	VERB
ap-4587	163	2	that	that	SCONJ
ap-4587	163	3	for	for	ADP
ap-4587	163	4	some	some	DET
ap-4587	163	5	ω	ω	NOUN
ap-4587	163	6	⊂	⊂	PROPN
ap-4587	163	7	rm	rm	NOUN
ap-4587	163	8	we	we	PRON
ap-4587	163	9	have	have	VERB
ap-4587	163	10	aς(ω	aς(ω	NUM
ap-4587	163	11	)	)	PUNCT
ap-4587	164	1	⊂	⊂	PROPN
ap-4587	164	2	σ(ω	σ(ω	PROPN
ap-4587	164	3	)	)	PUNCT
ap-4587	164	4	.	.	PUNCT
ap-4587	165	1	this	this	PRON
ap-4587	165	2	means	mean	VERB
ap-4587	165	3	that	that	SCONJ
ap-4587	165	4	akz	akz	PROPN
ap-4587	165	5	∈	∈	PROPN
ap-4587	165	6	σ(ω	σ(ω	PROPN
ap-4587	165	7	)	)	PUNCT
ap-4587	165	8	for	for	ADP
ap-4587	165	9	any	any	DET
ap-4587	165	10	z	z	PROPN
ap-4587	165	11	∈	∈	PROPN
ap-4587	165	12	σ(ω	σ(ω	PROPN
ap-4587	165	13	)	)	PUNCT
ap-4587	165	14	and	and	CCONJ
ap-4587	165	15	k	k	PROPN
ap-4587	165	16	∈	∈	PROPN
ap-4587	165	17	n.	n.	NOUN
ap-4587	165	18	for	for	ADP
ap-4587	165	19	the	the	DET
ap-4587	165	20	integer	integer	NOUN
ap-4587	165	21	matrix	matrix	NOUN
ap-4587	165	22	c	c	PROPN
ap-4587	165	23	∈	∈	PROPN
ap-4587	165	24	zs×s	zs×s	PROPN
ap-4587	165	25	and	and	CCONJ
ap-4587	165	26	the	the	DET
ap-4587	165	27	real	real	ADJ
ap-4587	165	28	matrix	matrix	NOUN
ap-4587	165	29	b	b	NOUN
ap-4587	165	30	∈	∈	PROPN
ap-4587	165	31	r(s−n)×(s−n	r(s−n)×(s−n	PROPN
ap-4587	165	32	)	)	PUNCT
ap-4587	165	33	from	from	ADP
ap-4587	165	34	theorem	theorem	ADJ
ap-4587	165	35	3.1	3.1	NUM
ap-4587	165	36	we	we	PRON
ap-4587	165	37	have	have	VERB
ap-4587	165	38	,	,	PUNCT
ap-4587	165	39	by	by	ADP
ap-4587	165	40	iterating	iterate	VERB
ap-4587	165	41	(	(	PUNCT
ap-4587	165	42	2	2	NUM
ap-4587	165	43	)	)	PUNCT
ap-4587	165	44	,	,	PUNCT
ap-4587	166	1	that	that	SCONJ
ap-4587	166	2	for	for	ADP
ap-4587	166	3	any	any	DET
ap-4587	166	4	k	k	PROPN
ap-4587	166	5	∈	∈	PROPN
ap-4587	166	6	n	n	CCONJ
ap-4587	166	7	(	(	PUNCT
ap-4587	166	8	ak	ak	PROPN
ap-4587	166	9	o	o	X
ap-4587	166	10	o	o	X
ap-4587	166	11	bk	bk	INTJ
ap-4587	166	12	)	)	PUNCT
ap-4587	166	13	=	=	SYM
ap-4587	166	14	v	v	PRON
ap-4587	166	15	ckv	ckv	NOUN
ap-4587	166	16	−1	−1	NOUN
ap-4587	166	17	.	.	PUNCT
ap-4587	167	1	realize	realize	VERB
ap-4587	167	2	that	that	SCONJ
ap-4587	167	3	if	if	SCONJ
ap-4587	167	4	z	z	PROPN
ap-4587	167	5	∈	∈	PROPN
ap-4587	167	6	σ(ω	σ(ω	PROPN
ap-4587	167	7	)	)	PUNCT
ap-4587	167	8	,	,	PUNCT
ap-4587	167	9	then	then	ADV
ap-4587	167	10	z	z	NOUN
ap-4587	167	11	=	=	SYM
ap-4587	167	12	π1(l	π1(l	X
ap-4587	167	13	)	)	PUNCT
ap-4587	167	14	for	for	ADP
ap-4587	167	15	some	some	DET
ap-4587	167	16	l	l	NOUN
ap-4587	167	17	∈	∈	PROPN
ap-4587	167	18	l	l	NOUN
ap-4587	167	19	and	and	CCONJ
ap-4587	167	20	π2(l	π2(l	PROPN
ap-4587	167	21	)	)	PUNCT
ap-4587	167	22	∈	∈	PROPN
ap-4587	167	23	ω	ω	PROPN
ap-4587	167	24	.	.	PUNCT
ap-4587	168	1	thus	thus	ADV
ap-4587	168	2	for	for	ADP
ap-4587	168	3	any	any	DET
ap-4587	168	4	k	k	PROPN
ap-4587	168	5	there	there	PRON
ap-4587	168	6	exist	exist	VERB
ap-4587	168	7	l′	l′	NUM
ap-4587	168	8	∈	∈	NOUN
ap-4587	168	9	l	l	NOUN
ap-4587	168	10	such	such	ADJ
ap-4587	168	11	that	that	DET
ap-4587	168	12	akz	akz	NOUN
ap-4587	168	13	=	=	SYM
ap-4587	168	14	π1(l′	π1(l′	X
ap-4587	168	15	)	)	PUNCT
ap-4587	168	16	and	and	CCONJ
ap-4587	168	17	π2(l′	π2(l′	NOUN
ap-4587	168	18	)	)	PUNCT
ap-4587	168	19	∈	∈	PROPN
ap-4587	168	20	ω	ω	PROPN
ap-4587	168	21	.	.	PUNCT
ap-4587	169	1	rewriting	rewrite	VERB
ap-4587	169	2	in	in	ADP
ap-4587	169	3	the	the	DET
ap-4587	169	4	matrix	matrix	NOUN
ap-4587	169	5	formalism	formalism	NOUN
ap-4587	169	6	,	,	PUNCT
ap-4587	169	7	akπ1(l	akπ1(l	NUM
ap-4587	169	8	)	)	PUNCT
ap-4587	169	9	=	=	PUNCT
ap-4587	169	10	ak(in	ak(in	ADJ
ap-4587	169	11	,	,	PUNCT
ap-4587	169	12	o)l	o)l	NOUN
ap-4587	169	13	=	=	PUNCT
ap-4587	169	14	(	(	PUNCT
ap-4587	169	15	in	in	ADP
ap-4587	169	16	,	,	PUNCT
ap-4587	169	17	o	o	NOUN
ap-4587	169	18	)	)	PUNCT
ap-4587	169	19	(	(	PUNCT
ap-4587	169	20	ak	ak	INTJ
ap-4587	169	21	o	o	X
ap-4587	169	22	o	o	X
ap-4587	169	23	bk	bk	INTJ
ap-4587	169	24	)	)	PUNCT
ap-4587	169	25	l	l	NOUN
ap-4587	170	1	=	=	PUNCT
ap-4587	170	2	(	(	PUNCT
ap-4587	170	3	in	in	ADP
ap-4587	170	4	,	,	PUNCT
ap-4587	170	5	o)v	o)v	ADJ
ap-4587	170	6	ckv	ckv	NOUN
ap-4587	170	7	−1l	−1l	PROPN
ap-4587	170	8	=	=	PUNCT
ap-4587	170	9	π1(l′	π1(l′	X
ap-4587	170	10	)	)	PUNCT
ap-4587	170	11	.	.	PUNCT
ap-4587	171	1	433	433	NUM
ap-4587	171	2	zuzana	zuzana	PROPN
ap-4587	171	3	masáková	masáková	PROPN
ap-4587	171	4	,	,	PUNCT
ap-4587	171	5	jan	jan	PROPN
ap-4587	171	6	mazáč	mazáč	PROPN
ap-4587	171	7	acta	acta	PROPN
ap-4587	171	8	polytechnica	polytechnica	PROPN
ap-4587	171	9	since	since	SCONJ
ap-4587	171	10	the	the	DET
ap-4587	171	11	scheme	scheme	NOUN
ap-4587	171	12	is	be	AUX
ap-4587	171	13	non	non	ADJ
ap-4587	171	14	-	-	ADJ
ap-4587	171	15	degenerate	degenerate	ADJ
ap-4587	171	16	,	,	PUNCT
ap-4587	171	17	the	the	DET
ap-4587	171	18	projection	projection	NOUN
ap-4587	171	19	π1	π1	NOUN
ap-4587	171	20	is	be	AUX
ap-4587	171	21	injective	injective	ADJ
ap-4587	171	22	,	,	PUNCT
ap-4587	171	23	and	and	CCONJ
ap-4587	171	24	thus	thus	ADV
ap-4587	171	25	we	we	PRON
ap-4587	171	26	can	can	AUX
ap-4587	171	27	derive	derive	VERB
ap-4587	171	28	that	that	DET
ap-4587	171	29	l′	l′	NOUN
ap-4587	171	30	=	=	SYM
ap-4587	171	31	v	v	NUM
ap-4587	171	32	ckv	ckv	NOUN
ap-4587	171	33	−1l	−1l	PROPN
ap-4587	171	34	.	.	PUNCT
ap-4587	172	1	the	the	DET
ap-4587	172	2	condition	condition	NOUN
ap-4587	172	3	π2(l′	π2(l′	AUX
ap-4587	172	4	)	)	PUNCT
ap-4587	172	5	∈	∈	PROPN
ap-4587	172	6	ω	ω	NOUN
ap-4587	172	7	is	be	AUX
ap-4587	172	8	therefore	therefore	ADV
ap-4587	172	9	equivalent	equivalent	ADJ
ap-4587	172	10	to	to	ADP
ap-4587	172	11	π2(l′	π2(l′	NOUN
ap-4587	172	12	)	)	PUNCT
ap-4587	172	13	=	=	SYM
ap-4587	173	1	(	(	PUNCT
ap-4587	173	2	o	o	NOUN
ap-4587	173	3	,	,	PUNCT
ap-4587	173	4	is−n)l′	is−n)l′	PROPN
ap-4587	173	5	=	=	SYM
ap-4587	173	6	(	(	PUNCT
ap-4587	173	7	o	o	NOUN
ap-4587	173	8	,	,	PUNCT
ap-4587	173	9	is−n)v	is−n)v	NOUN
ap-4587	173	10	ckv	ckv	NOUN
ap-4587	173	11	−1l	−1l	PROPN
ap-4587	173	12	=	=	PUNCT
ap-4587	173	13	(	(	PUNCT
ap-4587	173	14	o	o	NOUN
ap-4587	173	15	,	,	PUNCT
ap-4587	173	16	is−n	is−n	PROPN
ap-4587	173	17	)	)	PUNCT
ap-4587	173	18	(	(	PUNCT
ap-4587	173	19	ak	ak	INTJ
ap-4587	173	20	o	o	X
ap-4587	173	21	o	o	X
ap-4587	173	22	bk	bk	INTJ
ap-4587	173	23	)	)	PUNCT
ap-4587	173	24	l	l	NOUN
ap-4587	174	1	=	=	PUNCT
ap-4587	174	2	bk(o	bk(o	PROPN
ap-4587	174	3	,	,	PUNCT
ap-4587	174	4	is−n)l	is−n)l	NOUN
ap-4587	174	5	=	=	PUNCT
ap-4587	174	6	bkπ2(l	bkπ2(l	NOUN
ap-4587	174	7	)	)	PUNCT
ap-4587	174	8	.	.	PUNCT
ap-4587	175	1	(	(	PUNCT
ap-4587	175	2	3	3	X
ap-4587	175	3	)	)	PUNCT
ap-4587	175	4	now	now	ADV
ap-4587	175	5	realize	realize	VERB
ap-4587	175	6	that	that	SCONJ
ap-4587	175	7	by	by	ADP
ap-4587	175	8	irreducibility	irreducibility	NOUN
ap-4587	175	9	the	the	DET
ap-4587	175	10	set	set	NOUN
ap-4587	175	11	{	{	PUNCT
ap-4587	175	12	π2(l	π2(l	NOUN
ap-4587	175	13	)	)	PUNCT
ap-4587	175	14	:	:	PUNCT
ap-4587	176	1	l	l	PROPN
ap-4587	176	2	∈	∈	PROPN
ap-4587	176	3	l	l	NOUN
ap-4587	176	4	,	,	PUNCT
ap-4587	176	5	π1(l	π1(l	X
ap-4587	176	6	)	)	PUNCT
ap-4587	176	7	∈	∈	PROPN
ap-4587	176	8	σ(ω	σ(ω	PROPN
ap-4587	176	9	)	)	PUNCT
ap-4587	176	10	}	}	PUNCT
ap-4587	176	11	is	be	AUX
ap-4587	176	12	dense	dense	ADJ
ap-4587	176	13	in	in	ADP
ap-4587	176	14	the	the	DET
ap-4587	176	15	bounded	bounded	ADJ
ap-4587	176	16	window	window	NOUN
ap-4587	176	17	ω	ω	PROPN
ap-4587	176	18	.	.	PUNCT
ap-4587	177	1	by	by	ADP
ap-4587	177	2	linearity	linearity	NOUN
ap-4587	177	3	of	of	ADP
ap-4587	177	4	b	b	PROPN
ap-4587	177	5	,	,	PUNCT
ap-4587	177	6	we	we	PRON
ap-4587	177	7	must	must	AUX
ap-4587	177	8	have	have	VERB
ap-4587	177	9	for	for	ADP
ap-4587	177	10	the	the	DET
ap-4587	177	11	closure	closure	NOUN
ap-4587	177	12	of	of	ADP
ap-4587	177	13	the	the	DET
ap-4587	177	14	window	window	NOUN
ap-4587	177	15	that	that	PRON
ap-4587	177	16	bkω	bkω	ADP
ap-4587	177	17	⊂	⊂	PROPN
ap-4587	177	18	ω	ω	PROPN
ap-4587	177	19	.	.	PROPN
ap-4587	178	1	suppose	suppose	VERB
ap-4587	178	2	that	that	SCONJ
ap-4587	178	3	b	b	PROPN
ap-4587	178	4	has	have	VERB
ap-4587	178	5	a	a	DET
ap-4587	178	6	real	real	ADJ
ap-4587	178	7	eigenvalue	eigenvalue	ADJ
ap-4587	178	8	λ	λ	PROPN
ap-4587	178	9	of	of	ADP
ap-4587	178	10	modulus	modulus	NOUN
ap-4587	178	11	strictly	strictly	ADV
ap-4587	178	12	exceeding	exceed	VERB
ap-4587	178	13	1	1	NUM
ap-4587	178	14	.	.	PUNCT
ap-4587	179	1	as	as	ADP
ap-4587	179	2	b	b	PROPN
ap-4587	179	3	∈	∈	PROPN
ap-4587	179	4	r(s−n)×(s−n	r(s−n)×(s−n	PROPN
ap-4587	179	5	)	)	PUNCT
ap-4587	179	6	,	,	PUNCT
ap-4587	179	7	we	we	PRON
ap-4587	179	8	have	have	VERB
ap-4587	179	9	a	a	DET
ap-4587	179	10	real	real	ADJ
ap-4587	179	11	eigenvector	eigenvector	NOUN
ap-4587	179	12	w	w	PROPN
ap-4587	179	13	of	of	ADP
ap-4587	179	14	b	b	PROPN
ap-4587	179	15	corresponding	correspond	VERB
ap-4587	179	16	to	to	ADP
ap-4587	179	17	the	the	DET
ap-4587	179	18	eigenvalue	eigenvalue	PROPN
ap-4587	179	19	λ	λ	PROPN
ap-4587	179	20	.	.	PUNCT
ap-4587	180	1	iterating	iterate	VERB
ap-4587	180	2	,	,	PUNCT
ap-4587	180	3	we	we	PRON
ap-4587	180	4	obtain	obtain	VERB
ap-4587	180	5	a	a	DET
ap-4587	180	6	contradiction	contradiction	NOUN
ap-4587	180	7	bkw	bkw	PROPN
ap-4587	180	8	=	=	PUNCT
ap-4587	181	1	λkw	λkw	PROPN
ap-4587	181	2	/∈	/∈	PUNCT
ap-4587	182	1	ω	ω	NOUN
ap-4587	182	2	for	for	ADP
ap-4587	182	3	sufficiently	sufficiently	ADV
ap-4587	182	4	large	large	ADJ
ap-4587	182	5	k	k	PROPN
ap-4587	182	6	∈	∈	PROPN
ap-4587	182	7	n.	n.	NOUN
ap-4587	182	8	if	if	SCONJ
ap-4587	182	9	λ	λ	PROPN
ap-4587	182	10	is	be	AUX
ap-4587	182	11	a	a	DET
ap-4587	182	12	non	non	ADJ
ap-4587	182	13	-	-	ADJ
ap-4587	182	14	real	real	ADJ
ap-4587	182	15	eigenvalue	eigenvalue	NOUN
ap-4587	182	16	of	of	ADP
ap-4587	182	17	b	b	PROPN
ap-4587	182	18	with	with	ADP
ap-4587	182	19	|λ|	|λ|	PROPN
ap-4587	182	20	>	>	X
ap-4587	182	21	1	1	NUM
ap-4587	182	22	with	with	ADP
ap-4587	182	23	a	a	DET
ap-4587	182	24	non	non	ADJ
ap-4587	182	25	-	-	ADJ
ap-4587	182	26	real	real	ADJ
ap-4587	182	27	eigenvector	eigenvector	NOUN
ap-4587	182	28	w	w	PROPN
ap-4587	182	29	,	,	PUNCT
ap-4587	182	30	then	then	ADV
ap-4587	182	31	λ	λ	PROPN
ap-4587	182	32	is	be	AUX
ap-4587	182	33	an	an	DET
ap-4587	182	34	eigenvalue	eigenvalue	NOUN
ap-4587	182	35	of	of	ADP
ap-4587	182	36	b	b	PROPN
ap-4587	182	37	corresponding	correspond	VERB
ap-4587	182	38	to	to	ADP
ap-4587	182	39	the	the	DET
ap-4587	182	40	eigenvector	eigenvector	NOUN
ap-4587	182	41	w.	w.	PROPN
ap-4587	182	42	over	over	ADP
ap-4587	182	43	the	the	DET
ap-4587	182	44	real	real	ADJ
ap-4587	182	45	space	space	NOUN
ap-4587	182	46	of	of	ADP
ap-4587	182	47	dimension	dimension	NOUN
ap-4587	182	48	2	2	NUM
ap-4587	182	49	,	,	PUNCT
ap-4587	182	50	spanned	span	VERB
ap-4587	182	51	by	by	ADP
ap-4587	182	52	the	the	DET
ap-4587	182	53	vectors	vector	NOUN
ap-4587	182	54	w	w	ADP
ap-4587	183	1	+	+	CCONJ
ap-4587	183	2	w	w	PROPN
ap-4587	183	3	,	,	PUNCT
ap-4587	183	4	i(w	i(w	PROPN
ap-4587	183	5	−w	−w	NOUN
ap-4587	183	6	)	)	PUNCT
ap-4587	183	7	,	,	PUNCT
ap-4587	183	8	the	the	DET
ap-4587	183	9	mapping	mapping	NOUN
ap-4587	183	10	b	b	PROPN
ap-4587	183	11	acts	act	NOUN
ap-4587	183	12	as	as	ADP
ap-4587	183	13	multiplication	multiplication	NOUN
ap-4587	183	14	by	by	ADP
ap-4587	183	15	|λ|	|λ|	NOUN
ap-4587	183	16	and	and	CCONJ
ap-4587	183	17	rotation	rotation	NOUN
ap-4587	183	18	by	by	ADP
ap-4587	183	19	the	the	DET
ap-4587	183	20	argument	argument	NOUN
ap-4587	183	21	of	of	ADP
ap-4587	183	22	λ	λ	PROPN
ap-4587	183	23	.	.	PUNCT
ap-4587	184	1	we	we	PRON
ap-4587	184	2	obtain	obtain	VERB
ap-4587	184	3	a	a	DET
ap-4587	184	4	similar	similar	ADJ
ap-4587	184	5	contradiction	contradiction	NOUN
ap-4587	184	6	with	with	ADP
ap-4587	184	7	boundedness	boundedness	NOUN
ap-4587	184	8	of	of	ADP
ap-4587	184	9	the	the	DET
ap-4587	184	10	window	window	NOUN
ap-4587	184	11	ω	ω	NOUN
ap-4587	184	12	as	as	ADP
ap-4587	184	13	before	before	ADV
ap-4587	184	14	.	.	PUNCT
ap-4587	185	1	theorem	theorem	VERB
ap-4587	185	2	4.2	4.2	NUM
ap-4587	185	3	.	.	PUNCT
ap-4587	186	1	let	let	AUX
ap-4587	186	2	(	(	PUNCT
ap-4587	186	3	l	l	NOUN
ap-4587	186	4	,	,	PUNCT
ap-4587	186	5	π1	π1	NOUN
ap-4587	186	6	,	,	PUNCT
ap-4587	186	7	π2	π2	PROPN
ap-4587	186	8	)	)	PUNCT
ap-4587	186	9	be	be	AUX
ap-4587	186	10	a	a	DET
ap-4587	186	11	non	non	ADJ
ap-4587	186	12	-	-	ADJ
ap-4587	186	13	degenerated	degenerated	ADJ
ap-4587	186	14	irreducible	irreducible	ADJ
ap-4587	186	15	cut	cut	NOUN
ap-4587	186	16	-	-	PUNCT
ap-4587	186	17	and	and	CCONJ
ap-4587	186	18	-	-	PUNCT
ap-4587	186	19	project	project	NOUN
ap-4587	186	20	scheme	scheme	NOUN
ap-4587	186	21	with	with	ADP
ap-4587	186	22	a	a	DET
ap-4587	186	23	self	self	NOUN
ap-4587	186	24	-	-	PUNCT
ap-4587	186	25	similarity	similarity	NOUN
ap-4587	186	26	a.	a.	NOUN
ap-4587	186	27	if	if	SCONJ
ap-4587	186	28	the	the	DET
ap-4587	186	29	matrix	matrix	NOUN
ap-4587	186	30	b	b	NOUN
ap-4587	186	31	from	from	ADP
ap-4587	186	32	theorem	theorem	ADJ
ap-4587	186	33	3.1	3.1	NUM
ap-4587	186	34	has	have	AUX
ap-4587	186	35	all	all	PRON
ap-4587	186	36	eigenvalues	eigenvalue	NOUN
ap-4587	186	37	in	in	ADP
ap-4587	186	38	modulus	modulus	NOUN
ap-4587	186	39	strictly	strictly	ADV
ap-4587	186	40	smaller	small	ADJ
ap-4587	186	41	than	than	ADP
ap-4587	186	42	1	1	NUM
ap-4587	186	43	,	,	PUNCT
ap-4587	186	44	then	then	ADV
ap-4587	186	45	there	there	PRON
ap-4587	186	46	exists	exist	VERB
ap-4587	186	47	a	a	DET
ap-4587	186	48	window	window	NOUN
ap-4587	186	49	ω	ω	NOUN
ap-4587	186	50	such	such	ADJ
ap-4587	186	51	that	that	SCONJ
ap-4587	186	52	a	a	PRON
ap-4587	186	53	is	be	AUX
ap-4587	186	54	a	a	DET
ap-4587	186	55	self	self	NOUN
ap-4587	186	56	-	-	PUNCT
ap-4587	186	57	similarity	similarity	NOUN
ap-4587	186	58	of	of	ADP
ap-4587	186	59	the	the	DET
ap-4587	186	60	cut	cut	NOUN
ap-4587	186	61	-	-	PUNCT
ap-4587	186	62	and	and	CCONJ
ap-4587	186	63	-	-	PUNCT
ap-4587	186	64	project	project	NOUN
ap-4587	186	65	set	set	VERB
ap-4587	186	66	σ(ω	σ(ω	PROPN
ap-4587	186	67	)	)	PUNCT
ap-4587	186	68	.	.	PUNCT
ap-4587	187	1	proof	proof	NOUN
ap-4587	187	2	.	.	PUNCT
ap-4587	188	1	since	since	SCONJ
ap-4587	188	2	the	the	DET
ap-4587	188	3	eigenvalues	eigenvalue	NOUN
ap-4587	188	4	of	of	ADP
ap-4587	188	5	the	the	DET
ap-4587	188	6	matrix	matrix	NOUN
ap-4587	188	7	b	b	NOUN
ap-4587	188	8	are	be	AUX
ap-4587	188	9	strictly	strictly	ADV
ap-4587	188	10	smaller	small	ADJ
ap-4587	188	11	than	than	ADP
ap-4587	188	12	1	1	NUM
ap-4587	188	13	,	,	PUNCT
ap-4587	188	14	by	by	ADP
ap-4587	188	15	[	[	PUNCT
ap-4587	188	16	9	9	NUM
ap-4587	188	17	,	,	PUNCT
ap-4587	188	18	corollary	corollary	ADJ
ap-4587	188	19	1.2.3	1.2.3	NUM
ap-4587	188	20	]	]	PUNCT
ap-4587	188	21	there	there	PRON
ap-4587	188	22	exists	exist	VERB
ap-4587	188	23	a	a	DET
ap-4587	188	24	metric	metric	ADJ
ap-4587	188	25	ρ	ρ	NOUN
ap-4587	188	26	in	in	ADP
ap-4587	188	27	rs−n	rs−n	NOUN
ap-4587	188	28	such	such	ADJ
ap-4587	188	29	that	that	SCONJ
ap-4587	188	30	the	the	DET
ap-4587	188	31	mapping	mapping	NOUN
ap-4587	188	32	b	b	NOUN
ap-4587	188	33	is	be	AUX
ap-4587	188	34	in	in	ADP
ap-4587	188	35	that	that	DET
ap-4587	188	36	metric	metric	ADJ
ap-4587	188	37	contracting	contracting	NOUN
ap-4587	188	38	,	,	PUNCT
ap-4587	188	39	i.e.	i.e.	X
ap-4587	188	40	,	,	PUNCT
ap-4587	188	41	there	there	PRON
ap-4587	188	42	exists	exist	VERB
ap-4587	188	43	δ	δ	PROPN
ap-4587	188	44	<	<	X
ap-4587	188	45	1	1	NUM
ap-4587	188	46	such	such	ADJ
ap-4587	188	47	that	that	PRON
ap-4587	188	48	for	for	ADP
ap-4587	188	49	every	every	DET
ap-4587	188	50	x	x	NOUN
ap-4587	188	51	,	,	PUNCT
ap-4587	188	52	y	y	PROPN
ap-4587	188	53	∈	∈	PROPN
ap-4587	188	54	rs−n	rs−n	NOUN
ap-4587	188	55	we	we	PRON
ap-4587	188	56	have	have	VERB
ap-4587	188	57	δρ(x	δρ(x	NOUN
ap-4587	188	58	,	,	PUNCT
ap-4587	188	59	y	y	PROPN
ap-4587	188	60	)	)	PUNCT
ap-4587	188	61	>	>	X
ap-4587	189	1	ρ(bx	ρ(bx	PROPN
ap-4587	189	2	,	,	PUNCT
ap-4587	189	3	by	by	ADP
ap-4587	189	4	)	)	PUNCT
ap-4587	189	5	.	.	PUNCT
ap-4587	190	1	choosing	choose	VERB
ap-4587	190	2	for	for	ADP
ap-4587	190	3	the	the	DET
ap-4587	190	4	window	window	NOUN
ap-4587	190	5	the	the	DET
ap-4587	190	6	set	set	NOUN
ap-4587	190	7	ω	ω	PROPN
ap-4587	190	8	=	=	SYM
ap-4587	190	9	{	{	PUNCT
ap-4587	190	10	x	x	PROPN
ap-4587	190	11	∈	∈	PROPN
ap-4587	190	12	rm	rm	NOUN
ap-4587	190	13	:	:	PUNCT
ap-4587	190	14	ρ(x	ρ(x	PROPN
ap-4587	190	15	,	,	PUNCT
ap-4587	190	16	0	0	NUM
ap-4587	190	17	)	)	PUNCT
ap-4587	190	18	≤	≤	NUM
ap-4587	190	19	1	1	NUM
ap-4587	190	20	}	}	PUNCT
ap-4587	190	21	,	,	PUNCT
ap-4587	190	22	we	we	PRON
ap-4587	190	23	have	have	VERB
ap-4587	190	24	for	for	ADP
ap-4587	190	25	any	any	DET
ap-4587	190	26	l	l	NOUN
ap-4587	190	27	∈	∈	PROPN
ap-4587	190	28	l	l	NOUN
ap-4587	190	29	with	with	ADP
ap-4587	190	30	π2(l	π2(l	NOUN
ap-4587	190	31	)	)	PUNCT
ap-4587	190	32	∈	∈	PROPN
ap-4587	190	33	ω	ω	PROPN
ap-4587	190	34	,	,	PUNCT
ap-4587	190	35	that	that	SCONJ
ap-4587	190	36	bπ2(l	bπ2(l	PROPN
ap-4587	190	37	)	)	PUNCT
ap-4587	190	38	∈	∈	PROPN
ap-4587	190	39	ω	ω	PROPN
ap-4587	190	40	.	.	PUNCT
ap-4587	190	41	therefore	therefore	ADV
ap-4587	190	42	aπ1(l	aπ1(l	PROPN
ap-4587	190	43	)	)	PUNCT
ap-4587	190	44	∈	∈	PROPN
ap-4587	190	45	σ(ω	σ(ω	PROPN
ap-4587	190	46	)	)	PUNCT
ap-4587	190	47	for	for	ADP
ap-4587	190	48	any	any	DET
ap-4587	190	49	l	l	NOUN
ap-4587	190	50	∈	∈	NOUN
ap-4587	190	51	l	l	NOUN
ap-4587	190	52	such	such	ADJ
ap-4587	190	53	that	that	SCONJ
ap-4587	190	54	π1(l	π1(l	X
ap-4587	190	55	)	)	PUNCT
ap-4587	190	56	∈	∈	PROPN
ap-4587	190	57	σ(ω	σ(ω	PROPN
ap-4587	190	58	)	)	PUNCT
ap-4587	190	59	.	.	PUNCT
ap-4587	191	1	thus	thus	ADV
ap-4587	191	2	aς(ω	aς(ω	NUM
ap-4587	191	3	)	)	PUNCT
ap-4587	192	1	⊂	⊂	PROPN
ap-4587	192	2	σ(ω	σ(ω	PROPN
ap-4587	192	3	)	)	PUNCT
ap-4587	192	4	.	.	PUNCT
ap-4587	193	1	when	when	SCONJ
ap-4587	193	2	the	the	DET
ap-4587	193	3	matrix	matrix	NOUN
ap-4587	193	4	b	b	NOUN
ap-4587	193	5	is	be	AUX
ap-4587	193	6	diagonalizable	diagonalizable	ADJ
ap-4587	193	7	we	we	PRON
ap-4587	193	8	can	can	AUX
ap-4587	193	9	weaken	weaken	VERB
ap-4587	193	10	the	the	DET
ap-4587	193	11	assumption	assumption	NOUN
ap-4587	193	12	on	on	ADP
ap-4587	193	13	its	its	PRON
ap-4587	193	14	eigenvalues	eigenvalue	NOUN
ap-4587	193	15	.	.	PUNCT
ap-4587	194	1	theorem	theorem	VERB
ap-4587	194	2	4.3	4.3	NUM
ap-4587	194	3	.	.	PUNCT
ap-4587	195	1	let	let	AUX
ap-4587	195	2	(	(	PUNCT
ap-4587	195	3	l	l	NOUN
ap-4587	195	4	,	,	PUNCT
ap-4587	195	5	π1	π1	NOUN
ap-4587	195	6	,	,	PUNCT
ap-4587	195	7	π2	π2	PROPN
ap-4587	195	8	)	)	PUNCT
ap-4587	195	9	be	be	AUX
ap-4587	195	10	a	a	DET
ap-4587	195	11	non	non	ADJ
ap-4587	195	12	-	-	ADJ
ap-4587	195	13	degenerated	degenerated	ADJ
ap-4587	195	14	irreducible	irreducible	ADJ
ap-4587	195	15	cut	cut	NOUN
ap-4587	195	16	-	-	PUNCT
ap-4587	195	17	and	and	CCONJ
ap-4587	195	18	-	-	PUNCT
ap-4587	195	19	project	project	NOUN
ap-4587	195	20	scheme	scheme	NOUN
ap-4587	195	21	with	with	ADP
ap-4587	195	22	a	a	DET
ap-4587	195	23	self	self	NOUN
ap-4587	195	24	-	-	PUNCT
ap-4587	195	25	similarity	similarity	NOUN
ap-4587	195	26	a.	a.	NOUN
ap-4587	195	27	if	if	SCONJ
ap-4587	195	28	the	the	DET
ap-4587	195	29	matrix	matrix	NOUN
ap-4587	195	30	b	b	NOUN
ap-4587	195	31	from	from	ADP
ap-4587	195	32	theorem	theorem	ADJ
ap-4587	195	33	3.1	3.1	NUM
ap-4587	195	34	is	be	AUX
ap-4587	195	35	diagonalizable	diagonalizable	ADJ
ap-4587	195	36	and	and	CCONJ
ap-4587	195	37	all	all	DET
ap-4587	195	38	its	its	PRON
ap-4587	195	39	eigenvalues	eigenvalue	NOUN
ap-4587	195	40	in	in	ADP
ap-4587	195	41	modulus	modulus	NOUN
ap-4587	195	42	smaller	small	ADJ
ap-4587	195	43	than	than	ADP
ap-4587	195	44	or	or	CCONJ
ap-4587	195	45	equal	equal	ADJ
ap-4587	195	46	to	to	ADP
ap-4587	195	47	1	1	NUM
ap-4587	195	48	,	,	PUNCT
ap-4587	195	49	then	then	ADV
ap-4587	195	50	there	there	PRON
ap-4587	195	51	exists	exist	VERB
ap-4587	195	52	a	a	DET
ap-4587	195	53	window	window	NOUN
ap-4587	195	54	ω	ω	NOUN
ap-4587	195	55	such	such	ADJ
ap-4587	195	56	that	that	SCONJ
ap-4587	195	57	a	a	PRON
ap-4587	195	58	is	be	AUX
ap-4587	195	59	a	a	DET
ap-4587	195	60	self	self	NOUN
ap-4587	195	61	-	-	PUNCT
ap-4587	195	62	similarity	similarity	NOUN
ap-4587	195	63	of	of	ADP
ap-4587	195	64	the	the	DET
ap-4587	195	65	cut	cut	NOUN
ap-4587	195	66	-	-	PUNCT
ap-4587	195	67	and	and	CCONJ
ap-4587	195	68	-	-	PUNCT
ap-4587	195	69	project	project	NOUN
ap-4587	195	70	set	set	VERB
ap-4587	195	71	σ(ω	σ(ω	PROPN
ap-4587	195	72	)	)	PUNCT
ap-4587	195	73	.	.	PUNCT
ap-4587	196	1	proof	proof	NOUN
ap-4587	196	2	.	.	PUNCT
ap-4587	197	1	we	we	PRON
ap-4587	197	2	will	will	AUX
ap-4587	197	3	construct	construct	VERB
ap-4587	197	4	a	a	DET
ap-4587	197	5	positive	positive	ADJ
ap-4587	197	6	semi	semi	ADJ
ap-4587	197	7	-	-	ADJ
ap-4587	197	8	definite	definite	ADJ
ap-4587	197	9	matrix	matrix	NOUN
ap-4587	197	10	which	which	PRON
ap-4587	197	11	induces	induce	VERB
ap-4587	197	12	an	an	DET
ap-4587	197	13	inner	inner	ADJ
ap-4587	197	14	product	product	NOUN
ap-4587	197	15	on	on	ADP
ap-4587	197	16	rs−n	rs−n	PROPN
ap-4587	197	17	(	(	PUNCT
ap-4587	197	18	and	and	CCONJ
ap-4587	197	19	consequently	consequently	ADV
ap-4587	197	20	a	a	DET
ap-4587	197	21	metric	metric	NOUN
ap-4587	197	22	)	)	PUNCT
ap-4587	197	23	in	in	ADP
ap-4587	197	24	which	which	PRON
ap-4587	197	25	the	the	DET
ap-4587	197	26	mapping	mapping	NOUN
ap-4587	197	27	b	b	PROPN
ap-4587	197	28	is	be	AUX
ap-4587	197	29	non	non	ADJ
ap-4587	197	30	-	-	ADJ
ap-4587	197	31	expanding	expand	VERB
ap-4587	197	32	,	,	PUNCT
ap-4587	197	33	i.e.	i.e.	X
ap-4587	197	34	,	,	PUNCT
ap-4587	197	35	does	do	AUX
ap-4587	197	36	not	not	PART
ap-4587	197	37	enlarge	enlarge	VERB
ap-4587	197	38	the	the	DET
ap-4587	197	39	distances	distance	NOUN
ap-4587	197	40	.	.	PUNCT
ap-4587	198	1	first	first	ADV
ap-4587	198	2	we	we	PRON
ap-4587	198	3	define	define	VERB
ap-4587	198	4	an	an	DET
ap-4587	198	5	inner	inner	ADJ
ap-4587	198	6	product	product	NOUN
ap-4587	198	7	in	in	ADP
ap-4587	198	8	which	which	PRON
ap-4587	198	9	the	the	DET
ap-4587	198	10	eigenvectors	eigenvector	NOUN
ap-4587	198	11	of	of	ADP
ap-4587	198	12	b	b	NOUN
ap-4587	198	13	form	form	NOUN
ap-4587	198	14	an	an	DET
ap-4587	198	15	orthonormal	orthonormal	ADJ
ap-4587	198	16	basis	basis	NOUN
ap-4587	198	17	of	of	ADP
ap-4587	198	18	rs−n	rs−n	NOUN
ap-4587	198	19	.	.	PUNCT
ap-4587	199	1	denote	denote	VERB
ap-4587	199	2	by	by	ADP
ap-4587	199	3	η1	η1	NOUN
ap-4587	199	4	,	,	PUNCT
ap-4587	199	5	η2	η2	NOUN
ap-4587	199	6	,	,	PUNCT
ap-4587	199	7	.	.	PUNCT
ap-4587	199	8	.	.	PUNCT
ap-4587	200	1	.	.	PUNCT
ap-4587	201	1	,	,	PUNCT
ap-4587	201	2	ηm	ηm	PROPN
ap-4587	201	3	,	,	PUNCT
ap-4587	201	4	m	m	AUX
ap-4587	201	5	:	:	PUNCT
ap-4587	201	6	=	=	SYM
ap-4587	201	7	s−	s−	PROPN
ap-4587	201	8	n	n	CCONJ
ap-4587	201	9	,	,	PUNCT
ap-4587	201	10	the	the	DET
ap-4587	201	11	eigenvalues	eigenvalue	NOUN
ap-4587	201	12	of	of	ADP
ap-4587	201	13	b	b	NOUN
ap-4587	201	14	and	and	CCONJ
ap-4587	201	15	the	the	DET
ap-4587	201	16	corresponding	corresponding	ADJ
ap-4587	201	17	eigenvectors	eigenvector	NOUN
ap-4587	201	18	by	by	ADP
ap-4587	201	19	w1,w2	w1,w2	PROPN
ap-4587	201	20	,	,	PUNCT
ap-4587	201	21	.	.	PUNCT
ap-4587	201	22	.	.	PUNCT
ap-4587	201	23	.	.	PUNCT
ap-4587	202	1	,	,	PUNCT
ap-4587	202	2	wm	wm	PROPN
ap-4587	202	3	.	.	PUNCT
ap-4587	203	1	the	the	DET
ap-4587	203	2	inner	inner	ADJ
ap-4587	203	3	product	product	NOUN
ap-4587	203	4	is	be	AUX
ap-4587	203	5	defined	define	VERB
ap-4587	203	6	using	use	VERB
ap-4587	203	7	the	the	DET
ap-4587	203	8	hermitian	hermitian	ADJ
ap-4587	203	9	matrix	matrix	NOUN
ap-4587	203	10	h	h	NOUN
ap-4587	203	11	=	=	PUNCT
ap-4587	203	12	g∗g	g∗g	PROPN
ap-4587	203	13	where	where	SCONJ
ap-4587	203	14	g∗	g∗	PROPN
ap-4587	203	15	stands	stand	VERB
ap-4587	203	16	for	for	ADP
ap-4587	203	17	conjugate	conjugate	ADJ
ap-4587	203	18	transpose	transpose	NOUN
ap-4587	203	19	.	.	PUNCT
ap-4587	204	1	for	for	ADP
ap-4587	204	2	g	g	PROPN
ap-4587	204	3	we	we	PRON
ap-4587	204	4	take	take	VERB
ap-4587	204	5	the	the	DET
ap-4587	204	6	matrix	matrix	NOUN
ap-4587	204	7	transferring	transfer	VERB
ap-4587	204	8	the	the	DET
ap-4587	204	9	eigenvectors	eigenvector	NOUN
ap-4587	204	10	w1,w2	w1,w2	PROPN
ap-4587	204	11	,	,	PUNCT
ap-4587	204	12	.	.	PUNCT
ap-4587	204	13	.	.	PUNCT
ap-4587	204	14	.	.	PUNCT
ap-4587	205	1	,	,	PUNCT
ap-4587	205	2	wm	wm	PROPN
ap-4587	205	3	into	into	ADP
ap-4587	205	4	the	the	DET
ap-4587	205	5	standard	standard	ADJ
ap-4587	205	6	basis	basis	NOUN
ap-4587	205	7	e1	e1	PROPN
ap-4587	205	8	,	,	PUNCT
ap-4587	205	9	e2	e2	PROPN
ap-4587	205	10	,	,	PUNCT
ap-4587	205	11	.	.	PUNCT
ap-4587	205	12	.	.	PUNCT
ap-4587	206	1	.	.	PUNCT
ap-4587	207	1	,	,	PUNCT
ap-4587	207	2	em	em	PRON
ap-4587	207	3	.	.	PUNCT
ap-4587	208	1	then	then	ADV
ap-4587	208	2	the	the	DET
ap-4587	208	3	inner	inner	ADJ
ap-4587	208	4	product	product	NOUN
ap-4587	208	5	for	for	ADP
ap-4587	208	6	x	x	X
ap-4587	208	7	,	,	PUNCT
ap-4587	208	8	y	y	PROPN
ap-4587	208	9	∈	∈	PROPN
ap-4587	208	10	rm	rm	NOUN
ap-4587	208	11	is	be	AUX
ap-4587	208	12	defined	define	VERB
ap-4587	208	13	〈	〈	PROPN
ap-4587	208	14	x	x	X
ap-4587	208	15	,	,	PUNCT
ap-4587	208	16	y〉h	y〉h	PROPN
ap-4587	208	17	:	:	PUNCT
ap-4587	209	1	=	=	SYM
ap-4587	209	2	x∗g∗gy	x∗g∗gy	PROPN
ap-4587	209	3	.	.	PUNCT
ap-4587	210	1	consider	consider	VERB
ap-4587	210	2	a	a	DET
ap-4587	210	3	general	general	ADJ
ap-4587	210	4	vector	vector	NOUN
ap-4587	210	5	x	x	PUNCT
ap-4587	210	6	=	=	PUNCT
ap-4587	210	7	m∑	m∑	NOUN
ap-4587	211	1	i=1	i=1	PROPN
ap-4587	211	2	αiwi	αiwi	NOUN
ap-4587	211	3	.	.	PUNCT
ap-4587	212	1	then	then	ADV
ap-4587	212	2	〈	〈	PROPN
ap-4587	212	3	x	x	PRON
ap-4587	212	4	,	,	PUNCT
ap-4587	212	5	x〉h	x〉h	PROPN
ap-4587	212	6	=	=	PUNCT
ap-4587	212	7	〈	〈	PROPN
ap-4587	212	8	m∑	m∑	NOUN
ap-4587	212	9	i=1	i=1	PROPN
ap-4587	212	10	αiwi	αiwi	NOUN
ap-4587	212	11	,	,	PUNCT
ap-4587	212	12	m∑	m∑	CCONJ
ap-4587	212	13	j=1	j=1	PROPN
ap-4587	212	14	αjwj	αjwj	PROPN
ap-4587	212	15	〉	〉	NOUN
ap-4587	212	16	h	h	NOUN
ap-4587	212	17	=	=	PUNCT
ap-4587	213	1	m∑	m∑	INTJ
ap-4587	213	2	i	i	PRON
ap-4587	213	3	,	,	PUNCT
ap-4587	213	4	j=1	j=1	PROPN
ap-4587	213	5	αiαj	αiαj	VERB
ap-4587	213	6	〈	〈	PROPN
ap-4587	213	7	wi	wi	PROPN
ap-4587	213	8	,	,	PUNCT
ap-4587	213	9	wj〉h	wj〉h	PROPN
ap-4587	213	10	=	=	PUNCT
ap-4587	213	11	m∑	m∑	INTJ
ap-4587	213	12	i=1	i=1	PROPN
ap-4587	213	13	|αi|2	|αi|2	PROPN
ap-4587	213	14	,	,	PUNCT
ap-4587	213	15	〈	〈	PROPN
ap-4587	213	16	bx	bx	X
ap-4587	213	17	,	,	PUNCT
ap-4587	213	18	bx〉h	bx〉h	PROPN
ap-4587	213	19	=	=	SYM
ap-4587	213	20	〈	〈	PROPN
ap-4587	213	21	m∑	m∑	NOUN
ap-4587	213	22	i=1	i=1	PROPN
ap-4587	213	23	αibwi	αibwi	VERB
ap-4587	213	24	,	,	PUNCT
ap-4587	213	25	m∑	m∑	ADP
ap-4587	214	1	j=1	j=1	PROPN
ap-4587	214	2	αjbwj	αjbwj	ADJ
ap-4587	214	3	〉	〉	NOUN
ap-4587	214	4	h	h	NOUN
ap-4587	214	5	=	=	PUNCT
ap-4587	215	1	m∑	m∑	INTJ
ap-4587	216	1	i	i	PRON
ap-4587	216	2	,	,	PUNCT
ap-4587	216	3	j=1	j=1	PROPN
ap-4587	216	4	αiηiαjηj	αiηiαjηj	PROPN
ap-4587	216	5	〈	〈	PROPN
ap-4587	216	6	wi	wi	PROPN
ap-4587	216	7	,	,	PUNCT
ap-4587	216	8	wj〉h	wj〉h	PROPN
ap-4587	216	9	=	=	PUNCT
ap-4587	216	10	m∑	m∑	INTJ
ap-4587	216	11	i=1	i=1	PROPN
ap-4587	216	12	|ηi|2|αi|2	|ηi|2|αi|2	PROPN
ap-4587	216	13	.	.	PUNCT
ap-4587	217	1	as	as	ADP
ap-4587	217	2	for	for	ADP
ap-4587	217	3	all	all	PRON
ap-4587	217	4	eigenvalues	eigenvalue	VERB
ap-4587	217	5	|η|	|η|	NOUN
ap-4587	217	6	≤	≤	NOUN
ap-4587	217	7	1	1	NUM
ap-4587	217	8	,	,	PUNCT
ap-4587	217	9	we	we	PRON
ap-4587	217	10	thus	thus	ADV
ap-4587	217	11	have	have	VERB
ap-4587	217	12	〈	〈	PROPN
ap-4587	217	13	x	x	X
ap-4587	217	14	,	,	PUNCT
ap-4587	217	15	x〉h	x〉h	PROPN
ap-4587	217	16	≥	≥	NUM
ap-4587	218	1	〈	〈	PROPN
ap-4587	218	2	bx	bx	PROPN
ap-4587	218	3	,	,	PUNCT
ap-4587	218	4	bx〉h	bx〉h	VERB
ap-4587	218	5	for	for	ADP
ap-4587	218	6	any	any	DET
ap-4587	218	7	x	x	SYM
ap-4587	218	8	∈	∈	PROPN
ap-4587	218	9	rm	rm	NOUN
ap-4587	218	10	,	,	PUNCT
ap-4587	218	11	and	and	CCONJ
ap-4587	218	12	therefore	therefore	ADV
ap-4587	218	13	the	the	DET
ap-4587	218	14	mapping	mapping	NOUN
ap-4587	218	15	b	b	PROPN
ap-4587	218	16	is	be	AUX
ap-4587	218	17	not	not	PART
ap-4587	218	18	expanding	expand	VERB
ap-4587	218	19	.	.	PUNCT
ap-4587	219	1	setting	set	VERB
ap-4587	219	2	for	for	ADP
ap-4587	219	3	the	the	DET
ap-4587	219	4	acceptance	acceptance	NOUN
ap-4587	219	5	window	window	NOUN
ap-4587	219	6	ω	ω	PROPN
ap-4587	219	7	a	a	DET
ap-4587	219	8	ball	ball	NOUN
ap-4587	219	9	in	in	ADP
ap-4587	219	10	the	the	DET
ap-4587	219	11	metric	metric	NOUN
ap-4587	219	12	induced	induce	VERB
ap-4587	219	13	by	by	ADP
ap-4587	219	14	this	this	DET
ap-4587	219	15	inner	inner	ADJ
ap-4587	219	16	product	product	NOUN
ap-4587	219	17	,	,	PUNCT
ap-4587	219	18	i.e.	i.e.	X
ap-4587	219	19	,	,	PUNCT
ap-4587	219	20	ω	ω	X
ap-4587	219	21	=	=	SYM
ap-4587	219	22	{	{	PUNCT
ap-4587	219	23	x	x	PUNCT
ap-4587	219	24	∈	∈	PROPN
ap-4587	219	25	rm	rm	NOUN
ap-4587	219	26	:	:	PUNCT
ap-4587	219	27	〈	〈	PROPN
ap-4587	219	28	x	x	PRON
ap-4587	219	29	,	,	PUNCT
ap-4587	219	30	x〉h	x〉h	PROPN
ap-4587	219	31	≤	≤	NUM
ap-4587	219	32	1	1	NUM
ap-4587	219	33	}	}	PUNCT
ap-4587	219	34	,	,	PUNCT
ap-4587	219	35	it	it	PRON
ap-4587	219	36	can	can	AUX
ap-4587	219	37	be	be	AUX
ap-4587	219	38	again	again	ADV
ap-4587	219	39	easily	easily	ADV
ap-4587	219	40	derived	derive	VERB
ap-4587	219	41	that	that	SCONJ
ap-4587	219	42	the	the	DET
ap-4587	219	43	cut	cut	VERB
ap-4587	219	44	-	-	PUNCT
ap-4587	219	45	and	and	CCONJ
ap-4587	219	46	-	-	PUNCT
ap-4587	219	47	project	project	NOUN
ap-4587	219	48	σ(ω	σ(ω	PROPN
ap-4587	219	49	)	)	PUNCT
ap-4587	219	50	has	have	VERB
ap-4587	219	51	self	self	NOUN
ap-4587	219	52	-	-	PUNCT
ap-4587	219	53	similarity	similarity	NOUN
ap-4587	219	54	a.	a.	NOUN
ap-4587	219	55	434	434	NUM
ap-4587	219	56	vol	vol	NOUN
ap-4587	219	57	.	.	PUNCT
ap-4587	220	1	57	57	NUM
ap-4587	220	2	no	no	NOUN
ap-4587	220	3	.	.	PUNCT
ap-4587	221	1	6/2017	6/2017	NUM
ap-4587	221	2	on	on	ADP
ap-4587	221	3	self	self	NOUN
ap-4587	221	4	-	-	PUNCT
ap-4587	221	5	similarities	similarity	NOUN
ap-4587	221	6	of	of	ADP
ap-4587	221	7	cut	cut	NOUN
ap-4587	221	8	-	-	PUNCT
ap-4587	221	9	and	and	CCONJ
ap-4587	221	10	-	-	PUNCT
ap-4587	221	11	project	project	NOUN
ap-4587	221	12	sets	set	NOUN
ap-4587	221	13	5	5	NUM
ap-4587	221	14	.	.	PUNCT
ap-4587	222	1	construction	construction	NOUN
ap-4587	222	2	of	of	ADP
ap-4587	222	3	a	a	DET
ap-4587	222	4	cut	cut	NOUN
ap-4587	222	5	-	-	PUNCT
ap-4587	222	6	and	and	CCONJ
ap-4587	222	7	-	-	PUNCT
ap-4587	222	8	project	project	NOUN
ap-4587	222	9	scheme	scheme	NOUN
ap-4587	222	10	with	with	ADP
ap-4587	222	11	5	5	NUM
ap-4587	222	12	-	-	ADJ
ap-4587	222	13	fold	fold	ADJ
ap-4587	222	14	symmetry	symmetry	NOUN
ap-4587	222	15	in	in	ADP
ap-4587	222	16	the	the	DET
ap-4587	222	17	following	following	NOUN
ap-4587	222	18	,	,	PUNCT
ap-4587	222	19	we	we	PRON
ap-4587	222	20	shall	shall	AUX
ap-4587	222	21	apply	apply	VERB
ap-4587	222	22	proposition	proposition	NOUN
ap-4587	222	23	3.3	3.3	NUM
ap-4587	222	24	in	in	ADP
ap-4587	222	25	order	order	NOUN
ap-4587	222	26	to	to	PART
ap-4587	222	27	construct	construct	VERB
ap-4587	222	28	a	a	DET
ap-4587	222	29	cut	cut	VERB
ap-4587	222	30	-	-	PUNCT
ap-4587	222	31	and	and	CCONJ
ap-4587	222	32	-	-	PUNCT
ap-4587	222	33	project	project	NOUN
ap-4587	222	34	scheme	scheme	NOUN
ap-4587	222	35	allowing	allow	VERB
ap-4587	222	36	5	5	NUM
ap-4587	222	37	-	-	ADJ
ap-4587	222	38	fold	fold	ADJ
ap-4587	222	39	symmetry	symmetry	NOUN
ap-4587	222	40	,	,	PUNCT
ap-4587	222	41	and	and	CCONJ
ap-4587	222	42	subsequently	subsequently	ADV
ap-4587	222	43	,	,	PUNCT
ap-4587	222	44	to	to	PART
ap-4587	222	45	describe	describe	VERB
ap-4587	222	46	all	all	DET
ap-4587	222	47	its	its	PRON
ap-4587	222	48	self	self	NOUN
ap-4587	222	49	-	-	PUNCT
ap-4587	222	50	similarities	similarity	NOUN
ap-4587	222	51	.	.	PUNCT
ap-4587	223	1	the	the	DET
ap-4587	223	2	desired	desire	VERB
ap-4587	223	3	cut	cut	VERB
ap-4587	223	4	-	-	PUNCT
ap-4587	223	5	and	and	CCONJ
ap-4587	223	6	-	-	PUNCT
ap-4587	223	7	project	project	NOUN
ap-4587	223	8	scheme	scheme	NOUN
ap-4587	223	9	must	must	AUX
ap-4587	223	10	admit	admit	VERB
ap-4587	223	11	a	a	DET
ap-4587	223	12	cut	cut	VERB
ap-4587	223	13	-	-	PUNCT
ap-4587	223	14	and	and	CCONJ
ap-4587	223	15	-	-	PUNCT
ap-4587	223	16	project	project	NOUN
ap-4587	223	17	set	set	NOUN
ap-4587	223	18	closed	close	VERB
ap-4587	223	19	under	under	ADP
ap-4587	223	20	an	an	DET
ap-4587	223	21	isometry	isometry	NOUN
ap-4587	223	22	of	of	ADP
ap-4587	223	23	order	order	NOUN
ap-4587	223	24	5	5	NUM
ap-4587	223	25	,	,	PUNCT
ap-4587	223	26	i.e.	i.e.	X
ap-4587	223	27	,	,	PUNCT
ap-4587	223	28	a	a	DET
ap-4587	223	29	mapping	mapping	NOUN
ap-4587	223	30	satisfying	satisfy	VERB
ap-4587	223	31	a5	a5	PROPN
ap-4587	223	32	=	=	PUNCT
ap-4587	223	33	i.	i.	PROPN
ap-4587	223	34	the	the	DET
ap-4587	223	35	minimal	minimal	ADJ
ap-4587	223	36	polynomial	polynomial	NOUN
ap-4587	223	37	of	of	ADP
ap-4587	223	38	the	the	DET
ap-4587	223	39	matrix	matrix	NOUN
ap-4587	223	40	a	a	DET
ap-4587	223	41	over	over	ADP
ap-4587	223	42	z	z	NOUN
ap-4587	223	43	(	(	PUNCT
ap-4587	223	44	monic	monic	ADJ
ap-4587	223	45	polynomial	polynomial	ADJ
ap-4587	223	46	µa	µa	PROPN
ap-4587	223	47	∈	∈	PROPN
ap-4587	223	48	z[x	z[x	NOUN
ap-4587	223	49	]	]	PUNCT
ap-4587	223	50	of	of	ADP
ap-4587	223	51	lowest	low	ADJ
ap-4587	223	52	degree	degree	NOUN
ap-4587	223	53	satisfying	satisfy	VERB
ap-4587	223	54	µ(a	µ(a	PROPN
ap-4587	223	55	)	)	PUNCT
ap-4587	224	1	=	=	SYM
ap-4587	224	2	o	o	X
ap-4587	224	3	)	)	PUNCT
ap-4587	224	4	must	must	AUX
ap-4587	224	5	divide	divide	VERB
ap-4587	224	6	the	the	DET
ap-4587	224	7	polynomial	polynomial	ADJ
ap-4587	224	8	x5	x5	NOUN
ap-4587	224	9	−	−	PROPN
ap-4587	224	10	1	1	NUM
ap-4587	224	11	,	,	PUNCT
ap-4587	224	12	which	which	PRON
ap-4587	224	13	over	over	ADP
ap-4587	224	14	z	z	NOUN
ap-4587	224	15	factors	factor	NOUN
ap-4587	224	16	as	as	ADP
ap-4587	224	17	x5	x5	NOUN
ap-4587	224	18	−	−	PROPN
ap-4587	224	19	1	1	NUM
ap-4587	224	20	=	=	SYM
ap-4587	224	21	(	(	PUNCT
ap-4587	224	22	x	x	SYM
ap-4587	224	23	−	−	NOUN
ap-4587	224	24	1)(x4	1)(x4	ADJ
ap-4587	224	25	+	+	ADJ
ap-4587	224	26	x3	x3	ADJ
ap-4587	224	27	+	+	ADJ
ap-4587	224	28	x2	x2	ADJ
ap-4587	225	1	+	+	NOUN
ap-4587	225	2	x	x	X
ap-4587	225	3	+	+	ADJ
ap-4587	225	4	1	1	NUM
ap-4587	225	5	)	)	PUNCT
ap-4587	225	6	.	.	PUNCT
ap-4587	226	1	the	the	DET
ap-4587	226	2	smallest	small	ADJ
ap-4587	226	3	non	non	ADJ
ap-4587	226	4	-	-	ADJ
ap-4587	226	5	trivial	trivial	ADJ
ap-4587	226	6	example	example	NOUN
ap-4587	226	7	is	be	AUX
ap-4587	226	8	thus	thus	ADV
ap-4587	226	9	the	the	DET
ap-4587	226	10	cyclotomic	cyclotomic	ADJ
ap-4587	226	11	polynomial	polynomial	ADJ
ap-4587	226	12	µa(x	µa(x	NOUN
ap-4587	226	13	)	)	PUNCT
ap-4587	226	14	=	=	PUNCT
ap-4587	227	1	φ5(x	φ5(x	PROPN
ap-4587	227	2	)	)	PUNCT
ap-4587	227	3	:	:	PUNCT
ap-4587	228	1	=	=	SYM
ap-4587	228	2	x4	x4	PROPN
ap-4587	229	1	+	+	NOUN
ap-4587	229	2	x3	x3	ADJ
ap-4587	229	3	+	+	ADJ
ap-4587	229	4	x2	x2	ADJ
ap-4587	229	5	+	+	NOUN
ap-4587	229	6	x	x	PROPN
ap-4587	229	7	+	+	NOUN
ap-4587	229	8	1	1	NUM
ap-4587	229	9	.	.	PUNCT
ap-4587	229	10	the	the	DET
ap-4587	229	11	minimal	minimal	ADJ
ap-4587	229	12	polynomial	polynomial	NOUN
ap-4587	229	13	of	of	ADP
ap-4587	229	14	the	the	DET
ap-4587	229	15	integer	integer	NOUN
ap-4587	229	16	matrix	matrix	NOUN
ap-4587	229	17	c	c	NOUN
ap-4587	229	18	obtained	obtain	VERB
ap-4587	229	19	in	in	ADP
ap-4587	229	20	theorem	theorem	ADJ
ap-4587	229	21	3.1	3.1	NUM
ap-4587	229	22	should	should	AUX
ap-4587	229	23	be	be	AUX
ap-4587	229	24	divisible	divisible	ADJ
ap-4587	229	25	by	by	ADP
ap-4587	229	26	µa	µa	PROPN
ap-4587	229	27	.	.	PUNCT
ap-4587	230	1	thus	thus	ADV
ap-4587	230	2	,	,	PUNCT
ap-4587	230	3	as	as	ADP
ap-4587	230	4	the	the	DET
ap-4587	230	5	simplest	simple	ADJ
ap-4587	230	6	example	example	NOUN
ap-4587	230	7	,	,	PUNCT
ap-4587	230	8	we	we	PRON
ap-4587	230	9	consider	consider	VERB
ap-4587	230	10	for	for	ADP
ap-4587	230	11	c	c	NOUN
ap-4587	230	12	the	the	DET
ap-4587	230	13	companion	companion	NOUN
ap-4587	230	14	matrix	matrix	NOUN
ap-4587	230	15	of	of	ADP
ap-4587	230	16	the	the	DET
ap-4587	230	17	polynomial	polynomial	ADJ
ap-4587	230	18	φ5(x	φ5(x	NOUN
ap-4587	230	19	)	)	PUNCT
ap-4587	230	20	,	,	PUNCT
ap-4587	230	21	namely	namely	ADV
ap-4587	230	22	c	c	X
ap-4587	230	23	=	=	SYM
ap-4587	230	24			ADJ
ap-4587	230	25	0	0	NUM
ap-4587	231	1	1	1	NUM
ap-4587	231	2	0	0	NUM
ap-4587	231	3	0	0	NUM
ap-4587	231	4	0	0	NUM
ap-4587	231	5	0	0	NUM
ap-4587	231	6	1	1	NUM
ap-4587	231	7	0	0	NUM
ap-4587	231	8	0	0	NUM
ap-4587	231	9	0	0	NUM
ap-4587	231	10	0	0	NUM
ap-4587	231	11	1	1	NUM
ap-4587	231	12	−1	−1	NOUN
ap-4587	231	13	−1	−1	NOUN
ap-4587	231	14	−1	−1	NOUN
ap-4587	231	15	−1	−1	NOUN
ap-4587	231	16			NOUN
ap-4587	231	17	.	.	PUNCT
ap-4587	232	1	(	(	PUNCT
ap-4587	232	2	4	4	X
ap-4587	232	3	)	)	PUNCT
ap-4587	232	4	let	let	VERB
ap-4587	232	5	ω	ω	NOUN
ap-4587	232	6	=	=	SYM
ap-4587	232	7	e2πi/5	e2πi/5	ADJ
ap-4587	232	8	.	.	PUNCT
ap-4587	233	1	then	then	ADV
ap-4587	233	2	the	the	DET
ap-4587	233	3	eigenvalues	eigenvalue	NOUN
ap-4587	233	4	of	of	ADP
ap-4587	233	5	c	c	PROPN
ap-4587	233	6	are	be	AUX
ap-4587	233	7	ω	ω	NOUN
ap-4587	233	8	,	,	PUNCT
ap-4587	233	9	ω2	ω2	ADJ
ap-4587	233	10	,	,	PUNCT
ap-4587	233	11	ω3	ω3	NOUN
ap-4587	233	12	=	=	SYM
ap-4587	233	13	ω2	ω2	ADJ
ap-4587	233	14	and	and	CCONJ
ap-4587	233	15	ω4	ω4	NUM
ap-4587	233	16	=	=	SYM
ap-4587	233	17	ω	ω	PROPN
ap-4587	233	18	,	,	PUNCT
ap-4587	233	19	the	the	DET
ap-4587	233	20	four	four	NUM
ap-4587	233	21	roots	root	NOUN
ap-4587	233	22	of	of	ADP
ap-4587	233	23	the	the	DET
ap-4587	233	24	polynomial	polynomial	ADJ
ap-4587	233	25	φ5(x	φ5(x	NOUN
ap-4587	233	26	)	)	PUNCT
ap-4587	233	27	=	=	SYM
ap-4587	233	28	x4	x4	PROPN
ap-4587	234	1	+	+	CCONJ
ap-4587	234	2	x3	x3	ADJ
ap-4587	234	3	+	+	CCONJ
ap-4587	234	4	x2	x2	PROPN
ap-4587	235	1	+	+	CCONJ
ap-4587	235	2	x	x	SYM
ap-4587	235	3	+	+	NUM
ap-4587	235	4	1	1	NUM
ap-4587	235	5	which	which	PRON
ap-4587	235	6	is	be	AUX
ap-4587	235	7	irreducible	irreducible	ADJ
ap-4587	235	8	over	over	ADP
ap-4587	235	9	the	the	DET
ap-4587	235	10	rationals	rational	NOUN
ap-4587	235	11	.	.	PUNCT
ap-4587	236	1	note	note	VERB
ap-4587	236	2	that	that	SCONJ
ap-4587	236	3	these	these	PRON
ap-4587	236	4	are	be	AUX
ap-4587	236	5	precisely	precisely	ADV
ap-4587	236	6	the	the	DET
ap-4587	236	7	primitive	primitive	ADJ
ap-4587	236	8	5th	5th	ADJ
ap-4587	236	9	roots	root	NOUN
ap-4587	236	10	of	of	ADP
ap-4587	236	11	unity	unity	NOUN
ap-4587	236	12	and	and	CCONJ
ap-4587	236	13	they	they	PRON
ap-4587	236	14	generate	generate	VERB
ap-4587	236	15	the	the	DET
ap-4587	236	16	cyclotomic	cyclotomic	ADJ
ap-4587	236	17	field	field	NOUN
ap-4587	236	18	q(ω	q(ω	NOUN
ap-4587	236	19	)	)	PUNCT
ap-4587	236	20	.	.	PUNCT
ap-4587	237	1	since	since	SCONJ
ap-4587	237	2	the	the	DET
ap-4587	237	3	minimal	minimal	ADJ
ap-4587	237	4	polynomial	polynomial	NOUN
ap-4587	237	5	of	of	ADP
ap-4587	237	6	ω	ω	PROPN
ap-4587	237	7	is	be	AUX
ap-4587	237	8	φ5	φ5	ADJ
ap-4587	237	9	and	and	CCONJ
ap-4587	237	10	is	be	AUX
ap-4587	237	11	of	of	ADP
ap-4587	237	12	degree	degree	NOUN
ap-4587	237	13	4	4	NUM
ap-4587	237	14	,	,	PUNCT
ap-4587	237	15	the	the	DET
ap-4587	237	16	cyclotomic	cyclotomic	ADJ
ap-4587	237	17	field	field	NOUN
ap-4587	237	18	is	be	AUX
ap-4587	237	19	expressed	express	VERB
ap-4587	237	20	as	as	ADP
ap-4587	237	21	q(ω	q(ω	NOUN
ap-4587	237	22	)	)	PUNCT
ap-4587	237	23	=	=	PRON
ap-4587	237	24	{	{	PUNCT
ap-4587	237	25	a+	a+	PUNCT
ap-4587	237	26	bω	bω	NOUN
ap-4587	237	27	+	+	CCONJ
ap-4587	237	28	cω2	cω2	NOUN
ap-4587	237	29	+	+	CCONJ
ap-4587	237	30	dω3	dω3	VERB
ap-4587	237	31	:	:	PUNCT
ap-4587	237	32	a	a	DET
ap-4587	237	33	,	,	PUNCT
ap-4587	237	34	b	b	NOUN
ap-4587	237	35	,	,	PUNCT
ap-4587	237	36	c	c	NOUN
ap-4587	237	37	,	,	PUNCT
ap-4587	237	38	d	d	PROPN
ap-4587	237	39	∈	∈	PROPN
ap-4587	237	40	q	q	X
ap-4587	237	41	}	}	PUNCT
ap-4587	237	42	.	.	PUNCT
ap-4587	238	1	the	the	DET
ap-4587	238	2	field	field	NOUN
ap-4587	238	3	q(ω	q(ω	NOUN
ap-4587	238	4	)	)	PUNCT
ap-4587	238	5	has	have	VERB
ap-4587	238	6	four	four	NUM
ap-4587	238	7	automorphisms	automorphism	NOUN
ap-4587	238	8	,	,	PUNCT
ap-4587	238	9	induced	induce	VERB
ap-4587	238	10	by	by	ADP
ap-4587	238	11	σj(ω	σj(ω	NOUN
ap-4587	238	12	)	)	PUNCT
ap-4587	238	13	=	=	SYM
ap-4587	238	14	ωj	ωj	PROPN
ap-4587	238	15	for	for	ADP
ap-4587	238	16	j	j	PROPN
ap-4587	238	17	=	=	SYM
ap-4587	238	18	1	1	NUM
ap-4587	238	19	,	,	PUNCT
ap-4587	238	20	2	2	NUM
ap-4587	238	21	,	,	PUNCT
ap-4587	238	22	3	3	NUM
ap-4587	238	23	,	,	PUNCT
ap-4587	238	24	4	4	NUM
ap-4587	238	25	.	.	X
ap-4587	238	26	recall	recall	VERB
ap-4587	238	27	that	that	SCONJ
ap-4587	238	28	the	the	DET
ap-4587	238	29	isomorphisms	isomorphism	NOUN
ap-4587	238	30	are	be	AUX
ap-4587	238	31	identical	identical	ADJ
ap-4587	238	32	over	over	ADP
ap-4587	238	33	the	the	DET
ap-4587	238	34	rationals	rational	NOUN
ap-4587	238	35	.	.	PUNCT
ap-4587	239	1	we	we	PRON
ap-4587	239	2	can	can	AUX
ap-4587	239	3	diagonalize	diagonalize	VERB
ap-4587	239	4	the	the	DET
ap-4587	239	5	matrix	matrix	NOUN
ap-4587	239	6	c	c	NOUN
ap-4587	239	7	using	use	VERB
ap-4587	239	8	a	a	DET
ap-4587	239	9	matrix	matrix	NOUN
ap-4587	239	10	y	y	NOUN
ap-4587	239	11	composed	compose	VERB
ap-4587	239	12	of	of	ADP
ap-4587	239	13	the	the	DET
ap-4587	239	14	corresponding	corresponding	ADJ
ap-4587	239	15	eigenvectors	eigenvector	NOUN
ap-4587	239	16	written	write	VERB
ap-4587	239	17	in	in	ADP
ap-4587	239	18	columns	column	NOUN
ap-4587	239	19	,	,	PUNCT
ap-4587	239	20	and	and	CCONJ
ap-4587	239	21	its	its	PRON
ap-4587	239	22	inverse	inverse	NOUN
ap-4587	239	23	.	.	PUNCT
ap-4587	240	1	denote	denote	NOUN
ap-4587	240	2	y1	y1	NOUN
ap-4587	240	3	=	=	SYM
ap-4587	240	4	1	1	NUM
ap-4587	240	5	5	5	NUM
ap-4587	240	6			ADJ
ap-4587	240	7	1	1	NUM
ap-4587	240	8	ω	ω	NUM
ap-4587	240	9	ω2	ω2	ADJ
ap-4587	240	10	ω3	ω3	ADJ
ap-4587	240	11			NOUN
ap-4587	240	12	,	,	PUNCT
ap-4587	240	13	y2	y2	NOUN
ap-4587	240	14	=	=	NOUN
ap-4587	240	15	1	1	NUM
ap-4587	240	16	5	5	NUM
ap-4587	240	17			NUM
ap-4587	240	18	1	1	NUM
ap-4587	240	19	ω4	ω4	NUM
ap-4587	240	20	ω3	ω3	PROPN
ap-4587	240	21	ω2	ω2	PROPN
ap-4587	240	22			NOUN
ap-4587	240	23	,	,	PUNCT
ap-4587	240	24	y3	y3	NOUN
ap-4587	240	25	=	=	SYM
ap-4587	240	26	1	1	NUM
ap-4587	240	27	5	5	NUM
ap-4587	240	28			ADJ
ap-4587	240	29	1	1	NUM
ap-4587	240	30	ω2	ω2	NUM
ap-4587	240	31	ω4	ω4	NUM
ap-4587	240	32	ω	ω	NUM
ap-4587	240	33			NOUN
ap-4587	240	34	,	,	PUNCT
ap-4587	240	35	y4	y4	PROPN
ap-4587	240	36	=	=	SYM
ap-4587	240	37	1	1	NUM
ap-4587	240	38	5	5	NUM
ap-4587	240	39			ADJ
ap-4587	240	40	1	1	NUM
ap-4587	240	41	ω3	ω3	PROPN
ap-4587	240	42	ω	ω	NUM
ap-4587	240	43	ω4	ω4	NUM
ap-4587	240	44			NOUN
ap-4587	240	45	,	,	PUNCT
ap-4587	240	46	(	(	PUNCT
ap-4587	240	47	5	5	NUM
ap-4587	240	48	)	)	PUNCT
ap-4587	240	49	where	where	SCONJ
ap-4587	240	50	the	the	DET
ap-4587	240	51	scaling	scale	VERB
ap-4587	240	52	factor	factor	NOUN
ap-4587	240	53	1	1	NUM
ap-4587	240	54	5	5	NUM
ap-4587	240	55	is	be	AUX
ap-4587	240	56	chosen	choose	VERB
ap-4587	240	57	just	just	ADV
ap-4587	240	58	for	for	ADP
ap-4587	240	59	convenience	convenience	NOUN
ap-4587	240	60	.	.	PUNCT
ap-4587	241	1	then	then	ADV
ap-4587	241	2	y	y	PROPN
ap-4587	241	3	and	and	CCONJ
ap-4587	241	4	its	its	PRON
ap-4587	241	5	inverse	inverse	NOUN
ap-4587	241	6	y	y	PROPN
ap-4587	241	7	−1	−1	NOUN
ap-4587	241	8	are	be	AUX
ap-4587	241	9	given	give	VERB
ap-4587	241	10	by	by	ADP
ap-4587	241	11	y	y	PROPN
ap-4587	241	12	=	=	SYM
ap-4587	241	13	1	1	NUM
ap-4587	241	14	5	5	NUM
ap-4587	241	15			NOUN
ap-4587	241	16	1	1	NUM
ap-4587	241	17	1	1	NUM
ap-4587	241	18	1	1	NUM
ap-4587	241	19	1	1	NUM
ap-4587	241	20	ω	ω	NUM
ap-4587	241	21	ω4	ω4	NUM
ap-4587	241	22	ω2	ω2	PROPN
ap-4587	241	23	ω3	ω3	PROPN
ap-4587	241	24	ω2	ω2	PROPN
ap-4587	241	25	ω3	ω3	PROPN
ap-4587	241	26	ω4	ω4	PROPN
ap-4587	241	27	ω	ω	PROPN
ap-4587	241	28	ω3	ω3	PROPN
ap-4587	241	29	ω2	ω2	PROPN
ap-4587	241	30	ω	ω	PROPN
ap-4587	241	31	ω4	ω4	NUM
ap-4587	241	32			NOUN
ap-4587	241	33	,	,	PUNCT
ap-4587	241	34	y	y	PROPN
ap-4587	241	35	−1	−1	NOUN
ap-4587	241	36	=	=	PUNCT
ap-4587	241	37			PROPN
ap-4587	241	38	1−	1−	NUM
ap-4587	241	39	ω	ω	NUM
ap-4587	241	40	ω4	ω4	NUM
ap-4587	242	1	−	−	PROPN
ap-4587	242	2	ω	ω	PROPN
ap-4587	242	3	ω3	ω3	PROPN
ap-4587	242	4	−	−	PROPN
ap-4587	242	5	ω	ω	PROPN
ap-4587	242	6	ω2	ω2	ADP
ap-4587	242	7	−	−	PROPN
ap-4587	242	8	ω	ω	PROPN
ap-4587	242	9	1−	1−	NUM
ap-4587	242	10	ω4	ω4	NUM
ap-4587	242	11	ω	ω	NUM
ap-4587	242	12	−	−	PROPN
ap-4587	242	13	ω4	ω4	PROPN
ap-4587	242	14	ω2	ω2	NOUN
ap-4587	242	15	−	−	PROPN
ap-4587	242	16	ω4	ω4	PROPN
ap-4587	242	17	ω3	ω3	PROPN
ap-4587	242	18	−	−	PROPN
ap-4587	242	19	ω4	ω4	PROPN
ap-4587	242	20	1−	1−	NUM
ap-4587	242	21	ω2	ω2	ADJ
ap-4587	242	22	ω3	ω3	PROPN
ap-4587	242	23	−	−	PROPN
ap-4587	242	24	ω2	ω2	PROPN
ap-4587	242	25	ω	ω	PROPN
ap-4587	242	26	−	−	PROPN
ap-4587	242	27	ω2	ω2	ADJ
ap-4587	242	28	ω4	ω4	PROPN
ap-4587	242	29	−	−	PROPN
ap-4587	242	30	ω2	ω2	PROPN
ap-4587	242	31	1−	1−	NUM
ap-4587	242	32	ω3	ω3	NOUN
ap-4587	242	33	ω2	ω2	ADP
ap-4587	242	34	−	−	PROPN
ap-4587	242	35	ω3	ω3	PROPN
ap-4587	242	36	ω4	ω4	NOUN
ap-4587	242	37	−	−	PROPN
ap-4587	242	38	ω3	ω3	PROPN
ap-4587	242	39	ω	ω	PROPN
ap-4587	243	1	−	−	PROPN
ap-4587	243	2	ω3	ω3	ADJ
ap-4587	243	3			NOUN
ap-4587	243	4	.	.	PUNCT
ap-4587	244	1	(	(	PUNCT
ap-4587	244	2	6	6	X
ap-4587	244	3	)	)	PUNCT
ap-4587	244	4	note	note	NOUN
ap-4587	244	5	that	that	SCONJ
ap-4587	244	6	we	we	PRON
ap-4587	244	7	grouped	group	VERB
ap-4587	244	8	the	the	DET
ap-4587	244	9	columns	column	NOUN
ap-4587	244	10	into	into	ADP
ap-4587	244	11	pairs	pair	NOUN
ap-4587	244	12	that	that	PRON
ap-4587	244	13	are	be	AUX
ap-4587	244	14	complex	complex	ADJ
ap-4587	244	15	conjugates	conjugate	NOUN
ap-4587	244	16	.	.	PUNCT
ap-4587	245	1	then	then	ADV
ap-4587	245	2	y	y	PROPN
ap-4587	245	3	−1cy	−1cy	NOUN
ap-4587	245	4	=	=	SYM
ap-4587	245	5	diag(ω	diag(ω	PROPN
ap-4587	245	6	,	,	PUNCT
ap-4587	245	7	ω4	ω4	NUM
ap-4587	245	8	,	,	PUNCT
ap-4587	245	9	ω2	ω2	ADJ
ap-4587	245	10	,	,	PUNCT
ap-4587	245	11	ω3	ω3	NOUN
ap-4587	245	12	)	)	PUNCT
ap-4587	245	13	is	be	AUX
ap-4587	245	14	a	a	DET
ap-4587	245	15	diagonal	diagonal	ADJ
ap-4587	245	16	matrix	matrix	NOUN
ap-4587	245	17	over	over	ADP
ap-4587	245	18	c.	c.	NOUN
ap-4587	245	19	in	in	ADP
ap-4587	245	20	order	order	NOUN
ap-4587	245	21	to	to	PART
ap-4587	245	22	obtain	obtain	VERB
ap-4587	245	23	a	a	DET
ap-4587	245	24	block	block	NOUN
ap-4587	245	25	diagonal	diagonal	ADJ
ap-4587	245	26	matrix	matrix	NOUN
ap-4587	245	27	over	over	ADP
ap-4587	245	28	r	r	NOUN
ap-4587	245	29	we	we	PRON
ap-4587	245	30	use	use	VERB
ap-4587	245	31	matrices	matrix	NOUN
ap-4587	245	32	p	p	X
ap-4587	245	33	=	=	PUNCT
ap-4587	245	34			ADJ
ap-4587	245	35	1	1	NUM
ap-4587	245	36	−i	−i	NOUN
ap-4587	245	37	0	0	NUM
ap-4587	245	38	0	0	NUM
ap-4587	245	39	1	1	NUM
ap-4587	245	40	i	i	NOUN
ap-4587	245	41	0	0	NUM
ap-4587	245	42	0	0	NUM
ap-4587	245	43	0	0	NUM
ap-4587	245	44	0	0	NUM
ap-4587	245	45	1	1	NUM
ap-4587	245	46	−i	−i	NOUN
ap-4587	245	47	0	0	NUM
ap-4587	245	48	0	0	NUM
ap-4587	245	49	1	1	NUM
ap-4587	245	50	i	i	PRON
ap-4587	245	51			NOUN
ap-4587	245	52	,	,	PUNCT
ap-4587	245	53	p−1	p−1	PROPN
ap-4587	245	54	=	=	NOUN
ap-4587	245	55	1	1	NUM
ap-4587	245	56	2	2	NUM
ap-4587	245	57			NOUN
ap-4587	245	58	1	1	NUM
ap-4587	245	59	1	1	NUM
ap-4587	245	60	0	0	NUM
ap-4587	245	61	0	0	NUM
ap-4587	246	1	i	i	PRON
ap-4587	246	2	−i	−i	ADJ
ap-4587	246	3	0	0	NUM
ap-4587	246	4	0	0	NUM
ap-4587	246	5	0	0	NUM
ap-4587	246	6	0	0	NUM
ap-4587	246	7	1	1	NUM
ap-4587	246	8	1	1	NUM
ap-4587	246	9	0	0	NUM
ap-4587	246	10	0	0	NUM
ap-4587	247	1	i	i	PRON
ap-4587	247	2	−i	−i	PROPN
ap-4587	247	3			NOUN
ap-4587	247	4	.	.	PUNCT
ap-4587	248	1	(	(	PUNCT
ap-4587	248	2	7	7	X
ap-4587	248	3	)	)	PUNCT
ap-4587	248	4	we	we	PRON
ap-4587	248	5	thus	thus	ADV
ap-4587	248	6	have	have	VERB
ap-4587	248	7	p−1y	p−1y	ADJ
ap-4587	248	8	−1cy	−1cy	NOUN
ap-4587	248	9	p	p	NOUN
ap-4587	248	10	=	=	PUNCT
ap-4587	248	11			NOUN
ap-4587	248	12	cos	cos	ADP
ap-4587	248	13	2π	2π	PROPN
ap-4587	248	14	5	5	NUM
ap-4587	248	15	sin	sin	NOUN
ap-4587	248	16	2π	2π	NOUN
ap-4587	248	17	5	5	NUM
ap-4587	248	18	0	0	NUM
ap-4587	248	19	0	0	NUM
ap-4587	249	1	−	−	NOUN
ap-4587	250	1	sin	sin	NOUN
ap-4587	250	2	2π	2π	PROPN
ap-4587	250	3	5	5	NUM
ap-4587	250	4	cos	cos	ADP
ap-4587	250	5	2π	2π	PROPN
ap-4587	250	6	5	5	NUM
ap-4587	250	7	0	0	NUM
ap-4587	250	8	0	0	NUM
ap-4587	250	9	0	0	NUM
ap-4587	250	10	0	0	PUNCT
ap-4587	251	1	cos	cos	ADP
ap-4587	251	2	4π	4π	NUM
ap-4587	251	3	5	5	NUM
ap-4587	251	4	sin	sin	NOUN
ap-4587	251	5	4π	4π	NUM
ap-4587	251	6	5	5	NUM
ap-4587	251	7	0	0	NUM
ap-4587	251	8	0	0	NUM
ap-4587	251	9	−	−	NOUN
ap-4587	252	1	sin	sin	NOUN
ap-4587	252	2	4π	4π	NUM
ap-4587	252	3	5	5	NUM
ap-4587	252	4	cos	cos	ADP
ap-4587	252	5	4π	4π	NUM
ap-4587	252	6	5	5	NUM
ap-4587	252	7			NOUN
ap-4587	252	8	.	.	PUNCT
ap-4587	253	1	(	(	PUNCT
ap-4587	253	2	8)	8)	NUM
ap-4587	253	3	435	435	NUM
ap-4587	253	4	zuzana	zuzana	PROPN
ap-4587	253	5	masáková	masáková	PROPN
ap-4587	253	6	,	,	PUNCT
ap-4587	253	7	jan	jan	PROPN
ap-4587	253	8	mazáč	mazáč	PROPN
ap-4587	253	9	acta	acta	PROPN
ap-4587	253	10	polytechnica	polytechnica	PROPN
ap-4587	253	11	therefore	therefore	ADV
ap-4587	253	12	,	,	PUNCT
ap-4587	253	13	by	by	ADP
ap-4587	253	14	proposition	proposition	NOUN
ap-4587	253	15	3.3	3.3	NUM
ap-4587	253	16	,	,	PUNCT
ap-4587	253	17	as	as	ADP
ap-4587	253	18	the	the	DET
ap-4587	253	19	matrix	matrix	NOUN
ap-4587	253	20	composed	compose	VERB
ap-4587	253	21	of	of	ADP
ap-4587	253	22	vectors	vector	NOUN
ap-4587	253	23	generating	generate	VERB
ap-4587	253	24	the	the	DET
ap-4587	253	25	lattice	lattice	NOUN
ap-4587	253	26	l	l	NOUN
ap-4587	253	27	we	we	PRON
ap-4587	253	28	can	can	AUX
ap-4587	253	29	take	take	VERB
ap-4587	253	30	v	v	NOUN
ap-4587	253	31	=	=	NOUN
ap-4587	253	32	p−1y	p−1y	NOUN
ap-4587	253	33	−1	−1	NOUN
ap-4587	253	34	=	=	PUNCT
ap-4587	253	35			NOUN
ap-4587	253	36	1−	1−	NUM
ap-4587	253	37	cos	cos	PROPN
ap-4587	253	38	2π	2π	PROPN
ap-4587	253	39	5	5	NUM
ap-4587	253	40	0	0	PUNCT
ap-4587	253	41	cos	cos	PROPN
ap-4587	253	42	4π	4π	NUM
ap-4587	253	43	5	5	NUM
ap-4587	253	44	−	−	NOUN
ap-4587	253	45	cos	cos	ADP
ap-4587	253	46	2π	2π	PROPN
ap-4587	253	47	5	5	NUM
ap-4587	253	48	cos	co	NOUN
ap-4587	253	49	4π	4π	NUM
ap-4587	253	50	5	5	NUM
ap-4587	253	51	−	−	NOUN
ap-4587	253	52	cos	cos	ADP
ap-4587	253	53	2π	2π	PROPN
ap-4587	253	54	5	5	NUM
ap-4587	253	55	sin	sin	NOUN
ap-4587	253	56	2π	2π	NOUN
ap-4587	253	57	5	5	NUM
ap-4587	253	58	2	2	NUM
ap-4587	253	59	sin	sin	NOUN
ap-4587	253	60	2π	2π	NOUN
ap-4587	253	61	5	5	NUM
ap-4587	253	62	sin	sin	NOUN
ap-4587	253	63	4π	4π	NUM
ap-4587	253	64	5	5	NUM
ap-4587	253	65	+	+	CCONJ
ap-4587	253	66	sin	sin	NOUN
ap-4587	253	67	2π	2π	PROPN
ap-4587	253	68	5	5	NUM
ap-4587	253	69	sin	sin	NOUN
ap-4587	253	70	2π	2π	NOUN
ap-4587	253	71	5	5	NUM
ap-4587	253	72	−	−	NOUN
ap-4587	253	73	sin	sin	NOUN
ap-4587	253	74	4π	4π	NUM
ap-4587	253	75	5	5	NUM
ap-4587	253	76	1−	1−	NUM
ap-4587	253	77	cos	cos	X
ap-4587	253	78	4π	4π	NUM
ap-4587	253	79	5	5	NUM
ap-4587	253	80	0	0	PUNCT
ap-4587	253	81	cos	cos	ADP
ap-4587	253	82	2π	2π	PROPN
ap-4587	253	83	5	5	NUM
ap-4587	253	84	−	−	NOUN
ap-4587	253	85	cos	cos	PROPN
ap-4587	253	86	4π	4π	NUM
ap-4587	253	87	5	5	NUM
ap-4587	253	88	cos	cos	ADP
ap-4587	253	89	2π	2π	PROPN
ap-4587	253	90	5	5	NUM
ap-4587	253	91	−	−	NOUN
ap-4587	253	92	cos	cos	PROPN
ap-4587	253	93	4π	4π	NUM
ap-4587	253	94	5	5	NUM
ap-4587	253	95	sin	sin	NOUN
ap-4587	253	96	4π	4π	NUM
ap-4587	253	97	5	5	NUM
ap-4587	253	98	2	2	NUM
ap-4587	253	99	sin	sin	NOUN
ap-4587	253	100	4π	4π	NUM
ap-4587	253	101	5	5	NUM
ap-4587	253	102	sin	sin	NOUN
ap-4587	253	103	4π	4π	NUM
ap-4587	253	104	5	5	NUM
ap-4587	253	105	−	−	NOUN
ap-4587	253	106	sin	sin	NOUN
ap-4587	253	107	2π	2π	PROPN
ap-4587	253	108	5	5	NUM
ap-4587	253	109	sin	sin	NOUN
ap-4587	253	110	4π	4π	NUM
ap-4587	253	111	5	5	NUM
ap-4587	253	112	+	+	CCONJ
ap-4587	253	113	sin	sin	NOUN
ap-4587	253	114	2π	2π	NOUN
ap-4587	253	115	5	5	NUM
ap-4587	253	116			NOUN
ap-4587	253	117	=	=	SYM
ap-4587	253	118	1	1	NUM
ap-4587	253	119	2	2	NUM
ap-4587	253	120			NOUN
ap-4587	253	121	2−	2−	NUM
ap-4587	253	122	ω	ω	NOUN
ap-4587	253	123	−	−	PROPN
ap-4587	253	124	ω4	ω4	NOUN
ap-4587	253	125	0	0	NUM
ap-4587	253	126	ω3	ω3	PROPN
ap-4587	253	127	+	+	CCONJ
ap-4587	253	128	ω2	ω2	ADJ
ap-4587	253	129	−	−	PROPN
ap-4587	253	130	(	(	PUNCT
ap-4587	253	131	ω	ω	PROPN
ap-4587	253	132	+	+	NUM
ap-4587	253	133	ω4	ω4	NUM
ap-4587	253	134	)	)	PUNCT
ap-4587	253	135	ω3	ω3	NOUN
ap-4587	253	136	+	+	CCONJ
ap-4587	253	137	ω2	ω2	ADJ
ap-4587	253	138	−	−	PROPN
ap-4587	253	139	(	(	PUNCT
ap-4587	253	140	ω	ω	PROPN
ap-4587	253	141	+	+	NUM
ap-4587	253	142	ω4	ω4	NUM
ap-4587	253	143	)	)	PUNCT
ap-4587	253	144	i(ω4	i(ω4	NOUN
ap-4587	253	145	−	−	PROPN
ap-4587	253	146	ω	ω	NUM
ap-4587	253	147	)	)	PUNCT
ap-4587	253	148	2i(ω4	2i(ω4	NUM
ap-4587	254	1	−	−	PROPN
ap-4587	254	2	ω	ω	NUM
ap-4587	254	3	)	)	PUNCT
ap-4587	254	4	i(ω3	i(ω3	ADP
ap-4587	254	5	−	−	PROPN
ap-4587	254	6	ω2	ω2	PROPN
ap-4587	254	7	+	+	CCONJ
ap-4587	254	8	ω4	ω4	NUM
ap-4587	254	9	−	−	PROPN
ap-4587	254	10	ω	ω	NUM
ap-4587	254	11	)	)	PUNCT
ap-4587	254	12	i(ω2	i(ω2	PUNCT
ap-4587	255	1	−	−	PROPN
ap-4587	255	2	ω3	ω3	NOUN
ap-4587	255	3	+	+	CCONJ
ap-4587	255	4	ω4	ω4	NUM
ap-4587	255	5	−	−	PROPN
ap-4587	255	6	ω	ω	NOUN
ap-4587	255	7	)	)	PUNCT
ap-4587	255	8	2−	2−	NUM
ap-4587	255	9	ω2	ω2	NOUN
ap-4587	255	10	−	−	PROPN
ap-4587	255	11	ω3	ω3	NOUN
ap-4587	256	1	0	0	PROPN
ap-4587	256	2	ω	ω	NUM
ap-4587	256	3	+	+	CCONJ
ap-4587	256	4	ω4	ω4	NUM
ap-4587	256	5	−	−	PROPN
ap-4587	256	6	(	(	PUNCT
ap-4587	256	7	ω2	ω2	ADJ
ap-4587	256	8	+	+	CCONJ
ap-4587	256	9	ω3	ω3	PROPN
ap-4587	256	10	)	)	PUNCT
ap-4587	256	11	ω	ω	PROPN
ap-4587	257	1	+	+	CCONJ
ap-4587	257	2	ω4	ω4	NUM
ap-4587	257	3	−	−	PROPN
ap-4587	257	4	(	(	PUNCT
ap-4587	257	5	ω2	ω2	ADJ
ap-4587	257	6	+	+	CCONJ
ap-4587	257	7	ω3	ω3	NOUN
ap-4587	257	8	)	)	PUNCT
ap-4587	257	9	i(ω3	i(ω3	ADP
ap-4587	257	10	−	−	PROPN
ap-4587	257	11	ω2	ω2	ADJ
ap-4587	257	12	)	)	PUNCT
ap-4587	257	13	2i(ω3	2i(ω3	NUM
ap-4587	257	14	−	−	PROPN
ap-4587	257	15	ω2	ω2	ADJ
ap-4587	257	16	)	)	PUNCT
ap-4587	258	1	i(ω	i(ω	ADP
ap-4587	258	2	−	−	PROPN
ap-4587	258	3	ω4	ω4	PROPN
ap-4587	258	4	+	+	CCONJ
ap-4587	258	5	ω3	ω3	PROPN
ap-4587	258	6	−	−	PROPN
ap-4587	258	7	ω2	ω2	ADJ
ap-4587	258	8	)	)	PUNCT
ap-4587	258	9	i(ω4	i(ω4	NOUN
ap-4587	258	10	−	−	PROPN
ap-4587	258	11	ω	ω	PROPN
ap-4587	258	12	+	+	CCONJ
ap-4587	258	13	ω3	ω3	PROPN
ap-4587	258	14	−	−	PROPN
ap-4587	258	15	ω2	ω2	ADJ
ap-4587	258	16	)	)	PUNCT
ap-4587	258	17			PROPN
ap-4587	258	18	.	.	PUNCT
ap-4587	259	1	(	(	PUNCT
ap-4587	259	2	9	9	X
ap-4587	259	3	)	)	PUNCT
ap-4587	259	4	the	the	DET
ap-4587	259	5	lattice	lattice	PROPN
ap-4587	259	6	l	l	PROPN
ap-4587	259	7	is	be	AUX
ap-4587	259	8	then	then	ADV
ap-4587	259	9	of	of	ADP
ap-4587	259	10	the	the	DET
ap-4587	259	11	form	form	NOUN
ap-4587	259	12	l	l	NOUN
ap-4587	259	13	=	=	PUNCT
ap-4587	260	1	z	z	X
ap-4587	260	2			NOUN
ap-4587	260	3	1−	1−	NUM
ap-4587	260	4	cos	cos	PROPN
ap-4587	260	5	2π	2π	PROPN
ap-4587	260	6	5	5	NUM
ap-4587	260	7	sin	sin	NOUN
ap-4587	260	8	2π	2π	PROPN
ap-4587	260	9	5	5	NUM
ap-4587	260	10	1−	1−	NUM
ap-4587	260	11	cos	cos	X
ap-4587	260	12	4π	4π	NUM
ap-4587	260	13	5	5	NUM
ap-4587	260	14	sin	sin	NOUN
ap-4587	260	15	4π	4π	NUM
ap-4587	260	16	5	5	NUM
ap-4587	260	17			NOUN
ap-4587	260	18	︸	︸	X
ap-4587	260	19	︷︷	︷︷	PROPN
ap-4587	260	20	︸	︸	X
ap-4587	260	21	l1	l1	PROPN
ap-4587	261	1	+	+	CCONJ
ap-4587	261	2	z	z	NOUN
ap-4587	261	3			NOUN
ap-4587	261	4	0	0	NUM
ap-4587	261	5	2	2	NUM
ap-4587	261	6	sin	sin	NOUN
ap-4587	261	7	2π	2π	NOUN
ap-4587	261	8	5	5	NUM
ap-4587	261	9	0	0	NUM
ap-4587	261	10	2	2	NUM
ap-4587	261	11	sin	sin	NOUN
ap-4587	261	12	4π	4π	NUM
ap-4587	261	13	5	5	NUM
ap-4587	261	14			NOUN
ap-4587	261	15	︸	︸	X
ap-4587	261	16	︷︷	︷︷	VERB
ap-4587	261	17	︸	︸	X
ap-4587	261	18	l2	l2	NOUN
ap-4587	261	19	+	+	CCONJ
ap-4587	261	20	z	z	NOUN
ap-4587	261	21			NOUN
ap-4587	262	1	cos	cos	ADP
ap-4587	262	2	4π	4π	NUM
ap-4587	262	3	5	5	NUM
ap-4587	262	4	−	−	NOUN
ap-4587	262	5	cos	cos	ADP
ap-4587	262	6	2π	2π	PROPN
ap-4587	262	7	5	5	NUM
ap-4587	262	8	sin	sin	NOUN
ap-4587	262	9	4π	4π	NUM
ap-4587	262	10	5	5	NUM
ap-4587	262	11	+	+	CCONJ
ap-4587	262	12	sin	sin	NOUN
ap-4587	262	13	2π	2π	PROPN
ap-4587	262	14	5	5	NUM
ap-4587	262	15	cos	cos	ADP
ap-4587	262	16	2π	2π	PROPN
ap-4587	262	17	5	5	NUM
ap-4587	262	18	−	−	NOUN
ap-4587	262	19	cos	cos	PROPN
ap-4587	262	20	4π	4π	NUM
ap-4587	262	21	5	5	NUM
ap-4587	262	22	sin	sin	NOUN
ap-4587	262	23	4π	4π	NUM
ap-4587	262	24	5	5	NUM
ap-4587	262	25	−	−	NOUN
ap-4587	262	26	sin	sin	NOUN
ap-4587	262	27	2π	2π	NOUN
ap-4587	262	28	5	5	NUM
ap-4587	262	29			NOUN
ap-4587	262	30	︸	︸	X
ap-4587	262	31	︷︷	︷︷	NOUN
ap-4587	262	32	︸	︸	ADP
ap-4587	262	33	l3	l3	PROPN
ap-4587	262	34	+	+	PROPN
ap-4587	262	35	z	z	NOUN
ap-4587	262	36			NOUN
ap-4587	263	1	cos	cos	ADP
ap-4587	263	2	4π	4π	NUM
ap-4587	263	3	5	5	NUM
ap-4587	263	4	−	−	NOUN
ap-4587	263	5	cos	cos	ADP
ap-4587	263	6	2π	2π	PROPN
ap-4587	263	7	5	5	NUM
ap-4587	263	8	sin	sin	NOUN
ap-4587	263	9	2π	2π	NOUN
ap-4587	263	10	5	5	NUM
ap-4587	263	11	−	−	NOUN
ap-4587	263	12	sin	sin	NOUN
ap-4587	263	13	4π	4π	NUM
ap-4587	263	14	5	5	NUM
ap-4587	263	15	cos	cos	ADP
ap-4587	263	16	2π	2π	PROPN
ap-4587	263	17	5	5	NUM
ap-4587	263	18	−	−	NOUN
ap-4587	263	19	cos	cos	PROPN
ap-4587	263	20	4π	4π	NUM
ap-4587	263	21	5	5	NUM
ap-4587	263	22	sin	sin	NOUN
ap-4587	263	23	4π	4π	NUM
ap-4587	263	24	5	5	NUM
ap-4587	263	25	+	+	CCONJ
ap-4587	263	26	sin	sin	NOUN
ap-4587	263	27	2π	2π	NOUN
ap-4587	263	28	5	5	NUM
ap-4587	263	29			NOUN
ap-4587	263	30	︸	︸	X
ap-4587	263	31	︷︷	︷︷	PROPN
ap-4587	263	32	︸	︸	ADP
ap-4587	263	33	l4	l4	PROPN
ap-4587	263	34	.	.	PUNCT
ap-4587	264	1	(	(	PUNCT
ap-4587	264	2	10	10	NUM
ap-4587	264	3	)	)	PUNCT
ap-4587	264	4	the	the	DET
ap-4587	264	5	relation	relation	NOUN
ap-4587	264	6	(	(	PUNCT
ap-4587	264	7	8)	8)	NUM
ap-4587	264	8	is	be	AUX
ap-4587	264	9	thus	thus	ADV
ap-4587	264	10	expression	expression	NOUN
ap-4587	264	11	of	of	ADP
ap-4587	264	12	the	the	DET
ap-4587	264	13	matrix	matrix	NOUN
ap-4587	264	14	c	c	NOUN
ap-4587	264	15	in	in	ADP
ap-4587	264	16	the	the	DET
ap-4587	264	17	block	block	NOUN
ap-4587	264	18	diagonal	diagonal	ADJ
ap-4587	264	19	form	form	NOUN
ap-4587	264	20	v	v	ADP
ap-4587	264	21	cv	cv	PROPN
ap-4587	264	22	−1	−1	NOUN
ap-4587	264	23	=	=	PUNCT
ap-4587	264	24	(	(	PUNCT
ap-4587	264	25	a	a	DET
ap-4587	264	26	o	o	X
ap-4587	264	27	o	o	X
ap-4587	264	28	b	b	PROPN
ap-4587	264	29	)	)	PUNCT
ap-4587	264	30	,	,	PUNCT
ap-4587	264	31	where	where	SCONJ
ap-4587	264	32	the	the	DET
ap-4587	264	33	matrices	matrix	NOUN
ap-4587	264	34	a	a	PRON
ap-4587	264	35	=	=	X
ap-4587	264	36	(	(	PUNCT
ap-4587	264	37	cos	cos	ADP
ap-4587	264	38	2π	2π	PROPN
ap-4587	264	39	5	5	NUM
ap-4587	264	40	sin	sin	NOUN
ap-4587	264	41	2π	2π	NOUN
ap-4587	264	42	5	5	NUM
ap-4587	264	43	−	−	NOUN
ap-4587	264	44	sin	sin	NOUN
ap-4587	264	45	2π	2π	PROPN
ap-4587	264	46	5	5	NUM
ap-4587	264	47	cos	cos	ADP
ap-4587	264	48	2π	2π	PROPN
ap-4587	264	49	5	5	NUM
ap-4587	264	50	)	)	PUNCT
ap-4587	264	51	,	,	PUNCT
ap-4587	264	52	b	b	X
ap-4587	265	1	=	=	PRON
ap-4587	265	2	(	(	PUNCT
ap-4587	265	3	cos	cos	INTJ
ap-4587	265	4	4π	4π	NUM
ap-4587	265	5	5	5	NUM
ap-4587	265	6	sin	sin	NOUN
ap-4587	265	7	4π	4π	NUM
ap-4587	265	8	5	5	NUM
ap-4587	265	9	−	−	NOUN
ap-4587	265	10	sin	sin	NOUN
ap-4587	265	11	4π	4π	NUM
ap-4587	265	12	5	5	NUM
ap-4587	265	13	cos	cos	ADP
ap-4587	265	14	4π	4π	NUM
ap-4587	265	15	5	5	NUM
ap-4587	265	16	)	)	PUNCT
ap-4587	265	17	(	(	PUNCT
ap-4587	265	18	11	11	NUM
ap-4587	265	19	)	)	PUNCT
ap-4587	265	20	correspond	correspond	VERB
ap-4587	265	21	to	to	ADP
ap-4587	265	22	rotations	rotation	NOUN
ap-4587	265	23	by	by	ADP
ap-4587	265	24	angle	angle	NOUN
ap-4587	265	25	−	−	PROPN
ap-4587	265	26	2π	2π	NOUN
ap-4587	265	27	5	5	NUM
ap-4587	265	28	and	and	CCONJ
ap-4587	265	29	−	−	NOUN
ap-4587	265	30	4π	4π	NUM
ap-4587	265	31	5	5	NUM
ap-4587	265	32	,	,	PUNCT
ap-4587	265	33	respectively	respectively	ADV
ap-4587	265	34	.	.	PUNCT
ap-4587	266	1	notation	notation	NOUN
ap-4587	266	2	5.1	5.1	NUM
ap-4587	266	3	.	.	PUNCT
ap-4587	267	1	for	for	ADP
ap-4587	267	2	further	further	ADJ
ap-4587	267	3	reference	reference	NOUN
ap-4587	267	4	we	we	PRON
ap-4587	267	5	denote	denote	VERB
ap-4587	267	6	by	by	ADP
ap-4587	267	7	λ	λ	PROPN
ap-4587	267	8	the	the	DET
ap-4587	267	9	cut	cut	NOUN
ap-4587	267	10	-	-	PUNCT
ap-4587	267	11	and	and	CCONJ
ap-4587	267	12	-	-	PUNCT
ap-4587	267	13	project	project	NOUN
ap-4587	267	14	scheme	scheme	NOUN
ap-4587	267	15	λ	λ	NOUN
ap-4587	267	16	:	:	PUNCT
ap-4587	267	17	=	=	SYM
ap-4587	267	18	(	(	PUNCT
ap-4587	267	19	l	l	NOUN
ap-4587	267	20	,	,	PUNCT
ap-4587	267	21	π1	π1	NOUN
ap-4587	267	22	,	,	PUNCT
ap-4587	267	23	π2	π2	NOUN
ap-4587	267	24	)	)	PUNCT
ap-4587	267	25	,	,	PUNCT
ap-4587	267	26	r2	r2	PROPN
ap-4587	267	27	π1←−	π1←−	PROPN
ap-4587	268	1	l	l	NOUN
ap-4587	268	2	⊂	⊂	PROPN
ap-4587	268	3	r4	r4	PROPN
ap-4587	268	4	π2−→	π2−→	ADP
ap-4587	268	5	r2	r2	PROPN
ap-4587	268	6	composed	compose	VERB
ap-4587	268	7	of	of	ADP
ap-4587	268	8	the	the	DET
ap-4587	268	9	lattice	lattice	PROPN
ap-4587	268	10	l	l	PROPN
ap-4587	268	11	defined	define	VERB
ap-4587	268	12	in	in	ADP
ap-4587	268	13	(	(	PUNCT
ap-4587	268	14	10	10	NUM
ap-4587	268	15	)	)	PUNCT
ap-4587	268	16	,	,	PUNCT
ap-4587	268	17	with	with	ADP
ap-4587	268	18	the	the	DET
ap-4587	268	19	projections	projection	NOUN
ap-4587	268	20	π1	π1	NOUN
ap-4587	268	21	,	,	PUNCT
ap-4587	268	22	π2	π2	PROPN
ap-4587	268	23	:	:	PUNCT
ap-4587	268	24	r4	r4	NOUN
ap-4587	268	25	→	→	SYM
ap-4587	268	26	r2	r2	PROPN
ap-4587	268	27	given	give	VERB
ap-4587	268	28	as	as	ADP
ap-4587	268	29	before	before	ADV
ap-4587	268	30	in	in	ADP
ap-4587	268	31	our	our	PRON
ap-4587	268	32	formalism	formalism	NOUN
ap-4587	268	33	,	,	PUNCT
ap-4587	268	34	i.e.	i.e.	X
ap-4587	268	35	,	,	PUNCT
ap-4587	268	36	π1(l	π1(l	X
ap-4587	268	37	)	)	PUNCT
ap-4587	268	38	=	=	SYM
ap-4587	268	39	(	(	PUNCT
ap-4587	268	40	i2	i2	PROPN
ap-4587	268	41	,	,	PUNCT
ap-4587	268	42	o)l	o)l	NOUN
ap-4587	268	43	,	,	PUNCT
ap-4587	268	44	π2(l	π2(l	NOUN
ap-4587	268	45	)	)	PUNCT
ap-4587	268	46	=	=	SYM
ap-4587	268	47	(	(	PUNCT
ap-4587	268	48	o	o	NOUN
ap-4587	268	49	,	,	PUNCT
ap-4587	268	50	i2)l	i2)l	ADJ
ap-4587	268	51	.	.	PUNCT
ap-4587	269	1	in	in	ADP
ap-4587	269	2	order	order	NOUN
ap-4587	269	3	to	to	PART
ap-4587	269	4	understand	understand	VERB
ap-4587	269	5	completely	completely	ADV
ap-4587	269	6	the	the	DET
ap-4587	269	7	structure	structure	NOUN
ap-4587	269	8	of	of	ADP
ap-4587	269	9	cut	cut	NOUN
ap-4587	269	10	-	-	PUNCT
ap-4587	269	11	and	and	CCONJ
ap-4587	269	12	-	-	PUNCT
ap-4587	269	13	project	project	NOUN
ap-4587	269	14	sets	set	NOUN
ap-4587	269	15	defined	define	VERB
ap-4587	269	16	by	by	ADP
ap-4587	269	17	the	the	DET
ap-4587	269	18	cut	cut	NOUN
ap-4587	269	19	-	-	PUNCT
ap-4587	269	20	and	and	CCONJ
ap-4587	269	21	-	-	PUNCT
ap-4587	269	22	project	project	NOUN
ap-4587	269	23	scheme	scheme	NOUN
ap-4587	269	24	λ	λ	PROPN
ap-4587	269	25	,	,	PUNCT
ap-4587	269	26	let	let	VERB
ap-4587	269	27	us	we	PRON
ap-4587	269	28	apply	apply	VERB
ap-4587	269	29	the	the	DET
ap-4587	269	30	projections	projection	NOUN
ap-4587	269	31	to	to	ADP
ap-4587	269	32	the	the	DET
ap-4587	269	33	generating	generate	VERB
ap-4587	269	34	vectors	vector	NOUN
ap-4587	269	35	l1	l1	PROPN
ap-4587	269	36	,	,	PUNCT
ap-4587	269	37	.	.	PUNCT
ap-4587	269	38	.	.	PUNCT
ap-4587	270	1	.	.	PUNCT
ap-4587	271	1	,	,	PUNCT
ap-4587	271	2	l4	l4	PROPN
ap-4587	271	3	.	.	PUNCT
ap-4587	272	1	we	we	PRON
ap-4587	272	2	obtain	obtain	VERB
ap-4587	272	3	for	for	ADP
ap-4587	272	4	the	the	DET
ap-4587	272	5	π1	π1	ADJ
ap-4587	272	6	projection	projection	NOUN
ap-4587	272	7	(	(	PUNCT
ap-4587	272	8	1−	1−	NUM
ap-4587	272	9	cos	cos	ADP
ap-4587	272	10	2π	2π	PROPN
ap-4587	272	11	5	5	NUM
ap-4587	272	12	sin	sin	NOUN
ap-4587	272	13	2π	2π	NOUN
ap-4587	272	14	5	5	NUM
ap-4587	272	15	)	)	PUNCT
ap-4587	272	16	︸	︸	X
ap-4587	273	1	︷︷	︷︷	NOUN
ap-4587	273	2	︸	︸	ADP
ap-4587	273	3	l	l	PROPN
ap-4587	273	4	‖	‖	PROPN
ap-4587	273	5	1	1	NUM
ap-4587	273	6	,	,	PUNCT
ap-4587	273	7	(	(	PUNCT
ap-4587	273	8	0	0	NUM
ap-4587	273	9	2	2	NUM
ap-4587	273	10	sin	sin	NOUN
ap-4587	273	11	2π	2π	NOUN
ap-4587	273	12	5	5	NUM
ap-4587	273	13	)	)	PUNCT
ap-4587	273	14	︸	︸	X
ap-4587	274	1	︷︷	︷︷	NOUN
ap-4587	274	2	︸	︸	ADP
ap-4587	274	3	l	l	PROPN
ap-4587	274	4	‖	‖	PROPN
ap-4587	274	5	2	2	NUM
ap-4587	274	6	,	,	PUNCT
ap-4587	274	7	(	(	PUNCT
ap-4587	274	8	cos	cos	ADP
ap-4587	274	9	4π	4π	NUM
ap-4587	274	10	5	5	NUM
ap-4587	274	11	−	−	NOUN
ap-4587	274	12	cos	cos	ADP
ap-4587	274	13	2π	2π	PROPN
ap-4587	274	14	5	5	NUM
ap-4587	274	15	sin	sin	NOUN
ap-4587	274	16	4π	4π	NUM
ap-4587	274	17	5	5	NUM
ap-4587	274	18	+	+	CCONJ
ap-4587	274	19	sin	sin	NOUN
ap-4587	274	20	2π	2π	NOUN
ap-4587	274	21	5	5	NUM
ap-4587	274	22	)	)	PUNCT
ap-4587	274	23	︸	︸	X
ap-4587	275	1	︷︷	︷︷	NOUN
ap-4587	275	2	︸	︸	ADP
ap-4587	275	3	l	l	PROPN
ap-4587	275	4	‖	‖	PROPN
ap-4587	275	5	3	3	NUM
ap-4587	275	6	,	,	PUNCT
ap-4587	275	7	(	(	PUNCT
ap-4587	275	8	cos	cos	ADP
ap-4587	275	9	4π	4π	NUM
ap-4587	275	10	5	5	NUM
ap-4587	275	11	−	−	NOUN
ap-4587	275	12	cos	cos	ADP
ap-4587	275	13	2π	2π	PROPN
ap-4587	275	14	5	5	NUM
ap-4587	275	15	sin	sin	NOUN
ap-4587	275	16	2π	2π	NOUN
ap-4587	275	17	5	5	NUM
ap-4587	275	18	−	−	NOUN
ap-4587	275	19	sin	sin	NOUN
ap-4587	275	20	4π	4π	NUM
ap-4587	275	21	5	5	NUM
ap-4587	275	22	)	)	PUNCT
ap-4587	275	23	︸	︸	X
ap-4587	276	1	︷︷	︷︷	NOUN
ap-4587	276	2	︸	︸	ADP
ap-4587	276	3	l	l	PROPN
ap-4587	276	4	‖	‖	PROPN
ap-4587	276	5	4	4	NUM
ap-4587	276	6	(	(	PUNCT
ap-4587	276	7	12	12	NUM
ap-4587	276	8	)	)	PUNCT
ap-4587	276	9	and	and	CCONJ
ap-4587	276	10	for	for	ADP
ap-4587	276	11	the	the	DET
ap-4587	276	12	π2	π2	ADJ
ap-4587	276	13	projection	projection	NOUN
ap-4587	276	14	(	(	PUNCT
ap-4587	276	15	1−	1−	NUM
ap-4587	276	16	cos	cos	X
ap-4587	276	17	4π	4π	NUM
ap-4587	276	18	5	5	NUM
ap-4587	276	19	sin	sin	NOUN
ap-4587	276	20	4π	4π	NUM
ap-4587	276	21	5	5	NUM
ap-4587	276	22	)	)	PUNCT
ap-4587	276	23	︸	︸	NOUN
ap-4587	277	1	︷︷	︷︷	NOUN
ap-4587	277	2	︸	︸	X
ap-4587	277	3	l⊥	l⊥	PROPN
ap-4587	277	4	1	1	NUM
ap-4587	277	5	,	,	PUNCT
ap-4587	277	6	(	(	PUNCT
ap-4587	277	7	0	0	NUM
ap-4587	277	8	2	2	NUM
ap-4587	277	9	sin	sin	NOUN
ap-4587	277	10	4π	4π	NUM
ap-4587	277	11	5	5	NUM
ap-4587	277	12	)	)	PUNCT
ap-4587	277	13	︸	︸	NOUN
ap-4587	278	1	︷︷	︷︷	NOUN
ap-4587	278	2	︸	︸	X
ap-4587	278	3	l⊥	l⊥	NOUN
ap-4587	278	4	2	2	NUM
ap-4587	278	5	,	,	PUNCT
ap-4587	278	6	(	(	PUNCT
ap-4587	278	7	cos	cos	ADP
ap-4587	278	8	2π	2π	PROPN
ap-4587	278	9	5	5	NUM
ap-4587	278	10	−	−	NOUN
ap-4587	278	11	cos	cos	PROPN
ap-4587	278	12	4π	4π	NUM
ap-4587	278	13	5	5	NUM
ap-4587	278	14	sin	sin	NOUN
ap-4587	278	15	4π	4π	NUM
ap-4587	278	16	5	5	NUM
ap-4587	278	17	−	−	NOUN
ap-4587	278	18	sin	sin	NOUN
ap-4587	278	19	2π	2π	NOUN
ap-4587	278	20	5	5	NUM
ap-4587	278	21	)	)	PUNCT
ap-4587	278	22	︸	︸	NOUN
ap-4587	279	1	︷︷	︷︷	NOUN
ap-4587	279	2	︸	︸	X
ap-4587	279	3	l⊥	l⊥	NOUN
ap-4587	279	4	3	3	NUM
ap-4587	279	5	,	,	PUNCT
ap-4587	279	6	(	(	PUNCT
ap-4587	279	7	cos	cos	ADP
ap-4587	279	8	2π	2π	PROPN
ap-4587	279	9	5	5	NUM
ap-4587	279	10	−	−	NOUN
ap-4587	279	11	cos	cos	PROPN
ap-4587	279	12	4π	4π	NUM
ap-4587	279	13	5	5	NUM
ap-4587	279	14	sin	sin	NOUN
ap-4587	279	15	4π	4π	NUM
ap-4587	279	16	5	5	NUM
ap-4587	279	17	+	+	CCONJ
ap-4587	279	18	sin	sin	NOUN
ap-4587	279	19	2π	2π	NOUN
ap-4587	279	20	5	5	NUM
ap-4587	279	21	)	)	PUNCT
ap-4587	279	22	︸	︸	NOUN
ap-4587	280	1	︷︷	︷︷	NOUN
ap-4587	280	2	︸	︸	X
ap-4587	280	3	l⊥	l⊥	PROPN
ap-4587	280	4	4	4	NUM
ap-4587	280	5	.	.	PUNCT
ap-4587	281	1	(	(	PUNCT
ap-4587	281	2	13	13	NUM
ap-4587	281	3	)	)	PUNCT
ap-4587	281	4	let	let	VERB
ap-4587	281	5	us	we	PRON
ap-4587	281	6	consider	consider	VERB
ap-4587	281	7	the	the	DET
ap-4587	281	8	projection	projection	NOUN
ap-4587	281	9	π1	π1	NOUN
ap-4587	281	10	.	.	PUNCT
ap-4587	282	1	rewritten	rewrite	VERB
ap-4587	282	2	in	in	ADP
ap-4587	282	3	another	another	DET
ap-4587	282	4	form	form	NOUN
ap-4587	282	5	,	,	PUNCT
ap-4587	282	6	we	we	PRON
ap-4587	282	7	have	have	AUX
ap-4587	282	8	for	for	ADP
ap-4587	282	9	the	the	DET
ap-4587	282	10	projected	project	VERB
ap-4587	282	11	lattice	lattice	PROPN
ap-4587	282	12	vectors	vector	NOUN
ap-4587	282	13	l	l	PROPN
ap-4587	282	14	‖	‖	PROPN
ap-4587	282	15	1	1	NUM
ap-4587	282	16	=	=	SYM
ap-4587	282	17	2	2	NUM
ap-4587	282	18	sin	sin	NOUN
ap-4587	282	19	π5	π5	NOUN
ap-4587	282	20	(	(	PUNCT
ap-4587	282	21	sin	sin	NOUN
ap-4587	282	22	π	π	PROPN
ap-4587	282	23	5	5	NUM
ap-4587	282	24	cos	cos	PROPN
ap-4587	282	25	π5	π5	PROPN
ap-4587	282	26	)	)	PUNCT
ap-4587	282	27	,	,	PUNCT
ap-4587	282	28	l	l	PROPN
ap-4587	282	29	‖	‖	PROPN
ap-4587	282	30	2	2	NUM
ap-4587	282	31	=	=	SYM
ap-4587	282	32	2	2	NUM
ap-4587	282	33	sin	sin	NOUN
ap-4587	282	34	2π	2π	NOUN
ap-4587	282	35	5	5	NUM
ap-4587	282	36	(	(	PUNCT
ap-4587	282	37	0	0	NUM
ap-4587	282	38	1	1	NUM
ap-4587	282	39	)	)	PUNCT
ap-4587	282	40	,	,	PUNCT
ap-4587	282	41	l	l	PROPN
ap-4587	282	42	‖	‖	PROPN
ap-4587	282	43	3	3	NUM
ap-4587	282	44	=	=	SYM
ap-4587	282	45	2	2	NUM
ap-4587	282	46	sin	sin	NOUN
ap-4587	282	47	3π	3π	NOUN
ap-4587	282	48	5	5	NUM
ap-4587	282	49	(	(	PUNCT
ap-4587	282	50	−	−	PROPN
ap-4587	282	51	sin	sin	NOUN
ap-4587	282	52	π	π	PROPN
ap-4587	282	53	5	5	NUM
ap-4587	282	54	cos	cos	PROPN
ap-4587	282	55	π5	π5	PROPN
ap-4587	282	56	)	)	PUNCT
ap-4587	282	57	,	,	PUNCT
ap-4587	282	58	l	l	PROPN
ap-4587	282	59	‖	‖	PROPN
ap-4587	282	60	4	4	NUM
ap-4587	282	61	=	=	SYM
ap-4587	282	62	2	2	NUM
ap-4587	282	63	sin	sin	NOUN
ap-4587	282	64	π5	π5	NOUN
ap-4587	282	65	(	(	PUNCT
ap-4587	282	66	−	−	NOUN
ap-4587	282	67	sin	sin	NOUN
ap-4587	282	68	3π	3π	NOUN
ap-4587	282	69	5	5	NUM
ap-4587	282	70	−	−	NOUN
ap-4587	282	71	cos	cos	ADP
ap-4587	282	72	3π	3π	NOUN
ap-4587	282	73	5	5	NUM
ap-4587	282	74	)	)	PUNCT
ap-4587	282	75	.	.	PUNCT
ap-4587	283	1	in	in	ADP
ap-4587	283	2	this	this	DET
ap-4587	283	3	form	form	NOUN
ap-4587	283	4	,	,	PUNCT
ap-4587	283	5	it	it	PRON
ap-4587	283	6	is	be	AUX
ap-4587	283	7	easily	easily	ADV
ap-4587	283	8	seen	see	VERB
ap-4587	283	9	how	how	SCONJ
ap-4587	283	10	one	one	PRON
ap-4587	283	11	can	can	AUX
ap-4587	283	12	draw	draw	VERB
ap-4587	283	13	the	the	DET
ap-4587	283	14	vectors	vector	NOUN
ap-4587	283	15	π1(lj	π1(lj	NOUN
ap-4587	283	16	)	)	PUNCT
ap-4587	283	17	into	into	ADP
ap-4587	283	18	the	the	DET
ap-4587	283	19	plane	plane	NOUN
ap-4587	283	20	.	.	PUNCT
ap-4587	284	1	436	436	NUM
ap-4587	284	2	vol	vol	NOUN
ap-4587	284	3	.	.	PUNCT
ap-4587	285	1	57	57	NUM
ap-4587	285	2	no	no	NOUN
ap-4587	285	3	.	.	PUNCT
ap-4587	286	1	6/2017	6/2017	NUM
ap-4587	286	2	on	on	ADP
ap-4587	286	3	self	self	NOUN
ap-4587	286	4	-	-	PUNCT
ap-4587	286	5	similarities	similarity	NOUN
ap-4587	286	6	of	of	ADP
ap-4587	286	7	cut	cut	NOUN
ap-4587	286	8	-	-	PUNCT
ap-4587	286	9	and	and	CCONJ
ap-4587	286	10	-	-	PUNCT
ap-4587	286	11	project	project	NOUN
ap-4587	286	12	sets	set	NOUN
ap-4587	286	13	0	0	NUM
ap-4587	286	14	l	l	NOUN
ap-4587	286	15	‖	‖	PROPN
ap-4587	286	16	1	1	NUM
ap-4587	286	17	l	l	NOUN
ap-4587	286	18	‖	‖	NUM
ap-4587	286	19	2	2	NUM
ap-4587	286	20	l	l	NOUN
ap-4587	286	21	‖	‖	PROPN
ap-4587	286	22	3	3	NUM
ap-4587	286	23	l	l	NOUN
ap-4587	286	24	‖	‖	NUM
ap-4587	286	25	4	4	NUM
ap-4587	286	26	π	π	SYM
ap-4587	286	27	5	5	NUM
ap-4587	286	28	π	π	SYM
ap-4587	286	29	5	5	NUM
ap-4587	286	30	π	π	SYM
ap-4587	286	31	5	5	NUM
ap-4587	286	32	as	as	SCONJ
ap-4587	286	33	the	the	DET
ap-4587	286	34	vectors	vector	NOUN
ap-4587	286	35	l	l	PROPN
ap-4587	286	36	‖	‖	PROPN
ap-4587	286	37	i	i	PRON
ap-4587	286	38	together	together	ADV
ap-4587	286	39	with	with	ADP
ap-4587	286	40	the	the	DET
ap-4587	286	41	origin	origin	NOUN
ap-4587	286	42	form	form	VERB
ap-4587	286	43	the	the	DET
ap-4587	286	44	vertices	vertex	NOUN
ap-4587	286	45	of	of	ADP
ap-4587	286	46	a	a	DET
ap-4587	286	47	regular	regular	ADJ
ap-4587	286	48	pentagon	pentagon	NOUN
ap-4587	286	49	,	,	PUNCT
ap-4587	286	50	we	we	PRON
ap-4587	286	51	can	can	AUX
ap-4587	286	52	rewrite	rewrite	VERB
ap-4587	286	53	l	l	NOUN
ap-4587	286	54	‖	‖	PROPN
ap-4587	286	55	2	2	NUM
ap-4587	286	56	=	=	SYM
ap-4587	286	57	l	l	NOUN
ap-4587	286	58	‖	‖	PROPN
ap-4587	286	59	4	4	NUM
ap-4587	286	60	+	+	NUM
ap-4587	286	61	τ	τ	X
ap-4587	286	62	l	l	NOUN
ap-4587	286	63	‖	‖	PROPN
ap-4587	286	64	1	1	NUM
ap-4587	286	65	,	,	PUNCT
ap-4587	286	66	l	l	PROPN
ap-4587	286	67	‖	‖	PROPN
ap-4587	286	68	3	3	NUM
ap-4587	286	69	=	=	SYM
ap-4587	286	70	l	l	NOUN
ap-4587	286	71	‖	‖	PROPN
ap-4587	286	72	1	1	NUM
ap-4587	286	73	+	+	CCONJ
ap-4587	286	74	τ	τ	PROPN
ap-4587	286	75	l	l	NOUN
ap-4587	286	76	‖	‖	PROPN
ap-4587	286	77	4	4	NUM
ap-4587	286	78	,	,	PUNCT
ap-4587	286	79	where	where	SCONJ
ap-4587	286	80	τ	τ	PROPN
ap-4587	286	81	is	be	AUX
ap-4587	286	82	the	the	DET
ap-4587	286	83	golden	golden	ADJ
ap-4587	286	84	ratio	ratio	NOUN
ap-4587	286	85	.	.	PUNCT
ap-4587	287	1	these	these	DET
ap-4587	287	2	relations	relation	NOUN
ap-4587	287	3	can	can	AUX
ap-4587	287	4	be	be	AUX
ap-4587	287	5	verified	verify	VERB
ap-4587	287	6	with	with	ADP
ap-4587	287	7	the	the	DET
ap-4587	287	8	use	use	NOUN
ap-4587	287	9	of	of	ADP
ap-4587	287	10	2	2	NUM
ap-4587	287	11	cos	cos	ADP
ap-4587	287	12	2π	2π	PROPN
ap-4587	287	13	5	5	NUM
ap-4587	287	14	=	=	SYM
ap-4587	287	15	τ−1	τ−1	PROPN
ap-4587	287	16	,	,	PUNCT
ap-4587	287	17	2	2	NUM
ap-4587	287	18	cos	cos	X
ap-4587	287	19	4π	4π	NUM
ap-4587	287	20	5	5	NUM
ap-4587	287	21	=	=	NOUN
ap-4587	287	22	−τ	−τ	NOUN
ap-4587	287	23	.	.	PUNCT
ap-4587	288	1	for	for	ADP
ap-4587	288	2	any	any	DET
ap-4587	288	3	integer	integer	NOUN
ap-4587	288	4	a	a	DET
ap-4587	288	5	,	,	PUNCT
ap-4587	288	6	b	b	NOUN
ap-4587	288	7	,	,	PUNCT
ap-4587	288	8	c	c	NOUN
ap-4587	288	9	,	,	PUNCT
ap-4587	288	10	d	d	PROPN
ap-4587	288	11	∈	∈	PROPN
ap-4587	288	12	z	z	NOUN
ap-4587	288	13	we	we	PRON
ap-4587	288	14	have	have	VERB
ap-4587	288	15	al	al	PROPN
ap-4587	288	16	‖	‖	PROPN
ap-4587	288	17	1	1	PROPN
ap-4587	289	1	+	+	CCONJ
ap-4587	289	2	bl	bl	PROPN
ap-4587	289	3	‖	‖	PROPN
ap-4587	289	4	2	2	NUM
ap-4587	289	5	+	+	CCONJ
ap-4587	289	6	cl	cl	NOUN
ap-4587	289	7	‖	‖	ADJ
ap-4587	289	8	3	3	NUM
ap-4587	289	9	+	+	CCONJ
ap-4587	289	10	dl	dl	PROPN
ap-4587	289	11	‖	‖	PROPN
ap-4587	289	12	4	4	NUM
ap-4587	289	13	=	=	SYM
ap-4587	289	14	(	(	PUNCT
ap-4587	289	15	a+	a+	X
ap-4587	289	16	c+	c+	X
ap-4587	289	17	bτ)l‖1	bτ)l‖1	X
ap-4587	289	18	+	+	CCONJ
ap-4587	289	19	(	(	PUNCT
ap-4587	289	20	b+	b+	X
ap-4587	289	21	d+	d+	NOUN
ap-4587	289	22	cτ)l‖4	cτ)l‖4	PROPN
ap-4587	289	23	,	,	PUNCT
ap-4587	289	24	and	and	CCONJ
ap-4587	289	25	thus	thus	ADV
ap-4587	289	26	π1(l	π1(l	X
ap-4587	289	27	)	)	PUNCT
ap-4587	289	28	=	=	NOUN
ap-4587	289	29	{	{	PUNCT
ap-4587	289	30	al	al	PROPN
ap-4587	289	31	‖	‖	PROPN
ap-4587	289	32	1	1	PROPN
ap-4587	289	33	+	+	CCONJ
ap-4587	289	34	bl	bl	PROPN
ap-4587	289	35	‖	‖	PROPN
ap-4587	289	36	2	2	NUM
ap-4587	289	37	+	+	CCONJ
ap-4587	289	38	cl	cl	NOUN
ap-4587	289	39	‖	‖	ADJ
ap-4587	289	40	3	3	NUM
ap-4587	289	41	+	+	CCONJ
ap-4587	289	42	dl	dl	PROPN
ap-4587	289	43	‖	‖	PROPN
ap-4587	289	44	4	4	NUM
ap-4587	289	45	:	:	PUNCT
ap-4587	289	46	a	a	DET
ap-4587	289	47	,	,	PUNCT
ap-4587	289	48	b	b	NOUN
ap-4587	289	49	,	,	PUNCT
ap-4587	289	50	c	c	NOUN
ap-4587	289	51	,	,	PUNCT
ap-4587	289	52	d	d	PROPN
ap-4587	289	53	∈	∈	PROPN
ap-4587	289	54	z	z	NOUN
ap-4587	289	55	}	}	PUNCT
ap-4587	289	56	=	=	SYM
ap-4587	289	57	z[τ	z[τ	NOUN
ap-4587	289	58	]	]	PUNCT
ap-4587	289	59	l‖1	l‖1	X
ap-4587	289	60	+	+	CCONJ
ap-4587	289	61	z[τ	z[τ	NOUN
ap-4587	289	62	]	]	PUNCT
ap-4587	289	63	l‖4	l‖4	PROPN
ap-4587	289	64	.	.	PUNCT
ap-4587	290	1	(	(	PUNCT
ap-4587	290	2	14	14	NUM
ap-4587	290	3	)	)	PUNCT
ap-4587	290	4	similarly	similarly	ADV
ap-4587	290	5	,	,	PUNCT
ap-4587	290	6	we	we	PRON
ap-4587	290	7	have	have	VERB
ap-4587	290	8	for	for	ADP
ap-4587	290	9	the	the	DET
ap-4587	290	10	second	second	ADJ
ap-4587	290	11	projection	projection	NOUN
ap-4587	290	12	0	0	NUM
ap-4587	290	13	l⊥3	l⊥3	PUNCT
ap-4587	290	14	l⊥1	l⊥1	ADP
ap-4587	290	15	l⊥4	l⊥4	PROPN
ap-4587	290	16	l⊥2	l⊥2	X
ap-4587	290	17	π/5	π/5	NUM
ap-4587	290	18	π	π	NOUN
ap-4587	290	19	5	5	NUM
ap-4587	290	20	π	π	SYM
ap-4587	290	21	5	5	NUM
ap-4587	290	22	and	and	CCONJ
ap-4587	290	23	we	we	PRON
ap-4587	290	24	can	can	AUX
ap-4587	290	25	derive	derive	VERB
ap-4587	290	26	that	that	SCONJ
ap-4587	290	27	π2(l	π2(l	NOUN
ap-4587	290	28	)	)	PUNCT
ap-4587	290	29	=	=	SYM
ap-4587	290	30	l⊥1	l⊥1	PUNCT
ap-4587	290	31	z[τ	z[τ	PROPN
ap-4587	290	32	]	]	PUNCT
ap-4587	291	1	+	+	CCONJ
ap-4587	291	2	l⊥4	l⊥4	NUM
ap-4587	291	3	z[τ	z[τ	PROPN
ap-4587	291	4	]	]	PUNCT
ap-4587	291	5	.	.	PUNCT
ap-4587	292	1	(	(	PUNCT
ap-4587	292	2	15	15	X
ap-4587	292	3	)	)	PUNCT
ap-4587	292	4	we	we	PRON
ap-4587	292	5	will	will	AUX
ap-4587	292	6	now	now	ADV
ap-4587	292	7	show	show	VERB
ap-4587	292	8	that	that	SCONJ
ap-4587	292	9	the	the	DET
ap-4587	292	10	constructed	construct	VERB
ap-4587	292	11	cut	cut	NOUN
ap-4587	292	12	-	-	PUNCT
ap-4587	292	13	and	and	CCONJ
ap-4587	292	14	-	-	PUNCT
ap-4587	292	15	project	project	NOUN
ap-4587	292	16	scheme	scheme	NOUN
ap-4587	292	17	is	be	AUX
ap-4587	292	18	non	non	ADJ
ap-4587	292	19	-	-	ADJ
ap-4587	292	20	degenerate	degenerate	ADJ
ap-4587	292	21	and	and	CCONJ
ap-4587	292	22	irreducible	irreducible	ADJ
ap-4587	292	23	,	,	PUNCT
ap-4587	292	24	which	which	PRON
ap-4587	292	25	is	be	AUX
ap-4587	292	26	obligatory	obligatory	ADJ
ap-4587	292	27	,	,	PUNCT
ap-4587	292	28	in	in	ADP
ap-4587	292	29	order	order	NOUN
ap-4587	292	30	that	that	SCONJ
ap-4587	292	31	it	it	PRON
ap-4587	292	32	allows	allow	VERB
ap-4587	292	33	constructing	construct	VERB
ap-4587	292	34	cut	cut	NOUN
ap-4587	292	35	-	-	PUNCT
ap-4587	292	36	and	and	CCONJ
ap-4587	292	37	-	-	PUNCT
ap-4587	292	38	project	project	NOUN
ap-4587	292	39	sets	set	NOUN
ap-4587	292	40	.	.	PUNCT
ap-4587	293	1	first	first	ADV
ap-4587	293	2	we	we	PRON
ap-4587	293	3	show	show	VERB
ap-4587	293	4	non	non	ADJ
ap-4587	293	5	-	-	NOUN
ap-4587	293	6	degeneracy	degeneracy	NOUN
ap-4587	293	7	,	,	PUNCT
ap-4587	293	8	i.e.	i.e.	X
ap-4587	293	9	,	,	PUNCT
ap-4587	293	10	that	that	SCONJ
ap-4587	293	11	the	the	DET
ap-4587	293	12	projection	projection	NOUN
ap-4587	293	13	π1	π1	NOUN
ap-4587	293	14	restricted	restrict	VERB
ap-4587	293	15	to	to	ADP
ap-4587	293	16	l	l	NOUN
ap-4587	293	17	is	be	AUX
ap-4587	293	18	a	a	DET
ap-4587	293	19	one	one	NUM
ap-4587	293	20	-	-	PUNCT
ap-4587	293	21	to	to	ADP
ap-4587	293	22	-	-	PUNCT
ap-4587	293	23	one	one	NUM
ap-4587	293	24	mapping	mapping	NOUN
ap-4587	293	25	.	.	PUNCT
ap-4587	294	1	lemma	lemma	PROPN
ap-4587	294	2	5.2	5.2	NUM
ap-4587	294	3	.	.	PUNCT
ap-4587	295	1	let	let	VERB
ap-4587	295	2	l	l	NOUN
ap-4587	295	3	be	be	AUX
ap-4587	295	4	given	give	VERB
ap-4587	295	5	in	in	ADP
ap-4587	295	6	(	(	PUNCT
ap-4587	295	7	10	10	NUM
ap-4587	295	8	)	)	PUNCT
ap-4587	295	9	and	and	CCONJ
ap-4587	295	10	π1	π1	NOUN
ap-4587	295	11	:	:	PUNCT
ap-4587	295	12	r4	r4	NOUN
ap-4587	295	13	→	→	SYM
ap-4587	295	14	r2	r2	PROPN
ap-4587	295	15	by	by	ADP
ap-4587	295	16	π1(l	π1(l	PRON
ap-4587	295	17	)	)	PUNCT
ap-4587	295	18	=	=	SYM
ap-4587	295	19	(	(	PUNCT
ap-4587	295	20	i2	i2	PROPN
ap-4587	295	21	,	,	PUNCT
ap-4587	295	22	o)l	o)l	NOUN
ap-4587	295	23	.	.	PUNCT
ap-4587	296	1	then	then	ADV
ap-4587	296	2	π1	π1	PROPN
ap-4587	296	3	restricted	restrict	VERB
ap-4587	296	4	to	to	ADP
ap-4587	296	5	l	l	NOUN
ap-4587	296	6	is	be	AUX
ap-4587	296	7	injective	injective	ADJ
ap-4587	296	8	.	.	PUNCT
ap-4587	297	1	proof	proof	NOUN
ap-4587	297	2	.	.	PUNCT
ap-4587	298	1	in	in	ADP
ap-4587	298	2	order	order	NOUN
ap-4587	298	3	to	to	PART
ap-4587	298	4	verify	verify	VERB
ap-4587	298	5	injectivity	injectivity	NOUN
ap-4587	298	6	of	of	ADP
ap-4587	298	7	π1	π1	ADJ
ap-4587	298	8	∣∣	∣∣	NUM
ap-4587	298	9	l	l	NOUN
ap-4587	298	10	it	it	PRON
ap-4587	298	11	suffices	suffice	VERB
ap-4587	298	12	to	to	PART
ap-4587	298	13	show	show	VERB
ap-4587	298	14	that	that	SCONJ
ap-4587	298	15	the	the	DET
ap-4587	298	16	preimage	preimage	NOUN
ap-4587	298	17	of	of	ADP
ap-4587	298	18	the	the	DET
ap-4587	298	19	zero	zero	NUM
ap-4587	298	20	vector	vector	NOUN
ap-4587	298	21	is	be	AUX
ap-4587	298	22	0	0	NUM
ap-4587	298	23	.	.	PUNCT
ap-4587	299	1	this	this	PRON
ap-4587	299	2	amounts	amount	VERB
ap-4587	299	3	to	to	ADP
ap-4587	299	4	showing	show	VERB
ap-4587	299	5	that	that	SCONJ
ap-4587	299	6	the	the	DET
ap-4587	299	7	vectors	vector	NOUN
ap-4587	299	8	π1(li	π1(li	PROPN
ap-4587	299	9	)	)	PUNCT
ap-4587	299	10	=	=	SYM
ap-4587	299	11	l	l	NOUN
ap-4587	299	12	‖	‖	PROPN
ap-4587	299	13	i	i	PRON
ap-4587	299	14	,	,	PUNCT
ap-4587	299	15	i	i	PRON
ap-4587	299	16	=	=	NOUN
ap-4587	299	17	1	1	X
ap-4587	299	18	.	.	PUNCT
ap-4587	299	19	.	.	PUNCT
ap-4587	299	20	.	.	PUNCT
ap-4587	300	1	4	4	NUM
ap-4587	300	2	,	,	PUNCT
ap-4587	300	3	are	be	AUX
ap-4587	300	4	linearly	linearly	ADV
ap-4587	300	5	independent	independent	ADJ
ap-4587	300	6	over	over	ADP
ap-4587	300	7	q.	q.	PROPN
ap-4587	300	8	recalling	recall	VERB
ap-4587	300	9	(	(	PUNCT
ap-4587	300	10	12	12	NUM
ap-4587	300	11	)	)	PUNCT
ap-4587	300	12	,	,	PUNCT
ap-4587	300	13	this	this	PRON
ap-4587	300	14	can	can	AUX
ap-4587	300	15	be	be	AUX
ap-4587	300	16	verified	verify	VERB
ap-4587	300	17	with	with	ADP
ap-4587	300	18	the	the	DET
ap-4587	300	19	use	use	NOUN
ap-4587	300	20	of	of	ADP
ap-4587	300	21	the	the	DET
ap-4587	300	22	equality	equality	NOUN
ap-4587	300	23	cos	cos	ADP
ap-4587	300	24	2π	2π	PROPN
ap-4587	300	25	5	5	NUM
ap-4587	301	1	+	+	CCONJ
ap-4587	301	2	cos	cos	X
ap-4587	301	3	4π	4π	NUM
ap-4587	301	4	5	5	NUM
ap-4587	301	5	=	=	SYM
ap-4587	301	6	−1	−1	NOUN
ap-4587	301	7	2	2	NUM
ap-4587	301	8	,	,	PUNCT
ap-4587	301	9	(	(	PUNCT
ap-4587	301	10	16	16	NUM
ap-4587	301	11	)	)	PUNCT
ap-4587	301	12	which	which	PRON
ap-4587	301	13	follows	follow	VERB
ap-4587	301	14	from	from	ADP
ap-4587	301	15	the	the	DET
ap-4587	301	16	obvious	obvious	ADJ
ap-4587	301	17	relation	relation	NOUN
ap-4587	301	18	0	0	NUM
ap-4587	301	19	=	=	SYM
ap-4587	301	20	ω4	ω4	PROPN
ap-4587	301	21	+	+	CCONJ
ap-4587	301	22	ω	ω	PROPN
ap-4587	301	23	+	+	CCONJ
ap-4587	301	24	ω3	ω3	NOUN
ap-4587	301	25	+	+	CCONJ
ap-4587	302	1	ω2	ω2	ADJ
ap-4587	302	2	+	+	CCONJ
ap-4587	302	3	1	1	NUM
ap-4587	302	4	=	=	SYM
ap-4587	302	5	2	2	NUM
ap-4587	302	6	cos	cos	ADP
ap-4587	302	7	2π	2π	PROPN
ap-4587	302	8	5	5	NUM
ap-4587	302	9	+	+	SYM
ap-4587	302	10	2	2	NUM
ap-4587	302	11	cos	cos	X
ap-4587	302	12	4π	4π	NUM
ap-4587	302	13	5	5	NUM
ap-4587	302	14	+	+	CCONJ
ap-4587	302	15	1	1	NUM
ap-4587	302	16	.	.	NUM
ap-4587	302	17	437	437	NUM
ap-4587	302	18	zuzana	zuzana	PROPN
ap-4587	302	19	masáková	masáková	PROPN
ap-4587	302	20	,	,	PUNCT
ap-4587	302	21	jan	jan	PROPN
ap-4587	302	22	mazáč	mazáč	PROPN
ap-4587	302	23	acta	acta	PROPN
ap-4587	302	24	polytechnica	polytechnica	PROPN
ap-4587	302	25	it	it	PRON
ap-4587	302	26	remains	remain	VERB
ap-4587	302	27	to	to	PART
ap-4587	302	28	show	show	VERB
ap-4587	302	29	that	that	SCONJ
ap-4587	302	30	the	the	DET
ap-4587	302	31	second	second	ADJ
ap-4587	302	32	projection	projection	NOUN
ap-4587	302	33	of	of	ADP
ap-4587	302	34	the	the	DET
ap-4587	302	35	lattice	lattice	NOUN
ap-4587	302	36	π2(l	π2(l	NOUN
ap-4587	302	37	)	)	PUNCT
ap-4587	302	38	is	be	AUX
ap-4587	302	39	dense	dense	ADJ
ap-4587	302	40	in	in	ADP
ap-4587	302	41	r2	r2	PROPN
ap-4587	302	42	.	.	PUNCT
ap-4587	303	1	lemma	lemma	PROPN
ap-4587	303	2	5.3	5.3	NUM
ap-4587	303	3	.	.	PUNCT
ap-4587	304	1	let	let	VERB
ap-4587	304	2	l	l	NOUN
ap-4587	304	3	be	be	AUX
ap-4587	304	4	given	give	VERB
ap-4587	304	5	in	in	ADP
ap-4587	304	6	(	(	PUNCT
ap-4587	304	7	10	10	NUM
ap-4587	304	8	)	)	PUNCT
ap-4587	304	9	and	and	CCONJ
ap-4587	304	10	π2	π2	ADV
ap-4587	304	11	:	:	PUNCT
ap-4587	304	12	r4	r4	NOUN
ap-4587	304	13	→	→	SYM
ap-4587	304	14	r2	r2	PROPN
ap-4587	304	15	by	by	ADP
ap-4587	304	16	π2(l	π2(l	NOUN
ap-4587	304	17	)	)	PUNCT
ap-4587	304	18	=	=	SYM
ap-4587	304	19	(	(	PUNCT
ap-4587	304	20	o	o	NOUN
ap-4587	304	21	,	,	PUNCT
ap-4587	304	22	i2)l	i2)l	PROPN
ap-4587	304	23	.	.	PUNCT
ap-4587	305	1	then	then	ADV
ap-4587	305	2	π2(l	π2(l	NOUN
ap-4587	305	3	)	)	PUNCT
ap-4587	305	4	is	be	AUX
ap-4587	305	5	dense	dense	ADJ
ap-4587	305	6	in	in	ADP
ap-4587	305	7	r2	r2	PROPN
ap-4587	305	8	.	.	PUNCT
ap-4587	306	1	proof	proof	NOUN
ap-4587	306	2	.	.	PUNCT
ap-4587	307	1	recall	recall	NOUN
ap-4587	307	2	(	(	PUNCT
ap-4587	307	3	15	15	NUM
ap-4587	307	4	)	)	PUNCT
ap-4587	307	5	,	,	PUNCT
ap-4587	307	6	where	where	SCONJ
ap-4587	307	7	vectors	vector	NOUN
ap-4587	307	8	l⊥1	l⊥1	PUNCT
ap-4587	307	9	,	,	PUNCT
ap-4587	307	10	l	l	PROPN
ap-4587	307	11	⊥	⊥	NOUN
ap-4587	307	12	4	4	NUM
ap-4587	307	13	are	be	AUX
ap-4587	307	14	linearly	linearly	ADV
ap-4587	307	15	independent	independent	ADJ
ap-4587	307	16	and	and	CCONJ
ap-4587	307	17	that	that	SCONJ
ap-4587	307	18	,	,	PUNCT
ap-4587	307	19	due	due	ADP
ap-4587	307	20	to	to	ADP
ap-4587	307	21	irrationality	irrationality	NOUN
ap-4587	307	22	of	of	ADP
ap-4587	307	23	the	the	DET
ap-4587	307	24	golden	golden	ADJ
ap-4587	307	25	ratio	ratio	PROPN
ap-4587	307	26	τ	τ	PROPN
ap-4587	307	27	,	,	PUNCT
ap-4587	307	28	the	the	DET
ap-4587	307	29	set	set	NOUN
ap-4587	307	30	z[τ	z[τ	NOUN
ap-4587	307	31	]	]	PUNCT
ap-4587	308	1	=	=	PUNCT
ap-4587	308	2	z	z	NOUN
ap-4587	309	1	+	+	NOUN
ap-4587	309	2	zτ	zτ	X
ap-4587	309	3	is	be	AUX
ap-4587	309	4	dense	dense	ADJ
ap-4587	309	5	in	in	ADP
ap-4587	309	6	r.	r.	PROPN
ap-4587	309	7	thus	thus	ADV
ap-4587	309	8	π2(l	π2(l	NOUN
ap-4587	309	9	)	)	PUNCT
ap-4587	309	10	is	be	AUX
ap-4587	309	11	a	a	DET
ap-4587	309	12	cartesian	cartesian	ADJ
ap-4587	309	13	product	product	NOUN
ap-4587	309	14	of	of	ADP
ap-4587	309	15	two	two	NUM
ap-4587	309	16	sets	set	NOUN
ap-4587	309	17	,	,	PUNCT
ap-4587	309	18	each	each	PRON
ap-4587	309	19	of	of	ADP
ap-4587	309	20	them	they	PRON
ap-4587	309	21	dense	dense	ADJ
ap-4587	309	22	in	in	ADP
ap-4587	309	23	the	the	DET
ap-4587	309	24	subspace	subspace	NOUN
ap-4587	309	25	it	it	PRON
ap-4587	309	26	generates	generate	VERB
ap-4587	309	27	.	.	PUNCT
ap-4587	310	1	whence	whence	NOUN
ap-4587	310	2	,	,	PUNCT
ap-4587	310	3	π2(l	π2(l	NOUN
ap-4587	310	4	)	)	PUNCT
ap-4587	310	5	is	be	AUX
ap-4587	310	6	dense	dense	ADJ
ap-4587	310	7	in	in	ADP
ap-4587	310	8	r2	r2	PROPN
ap-4587	310	9	.	.	PUNCT
ap-4587	311	1	lemmas	lemmas	PROPN
ap-4587	311	2	5.2	5.2	NUM
ap-4587	311	3	and	and	CCONJ
ap-4587	311	4	5.3	5.3	NUM
ap-4587	311	5	can	can	AUX
ap-4587	311	6	be	be	AUX
ap-4587	311	7	summarized	summarize	VERB
ap-4587	311	8	as	as	SCONJ
ap-4587	311	9	follows	follow	VERB
ap-4587	311	10	.	.	PUNCT
ap-4587	312	1	corollary	corollary	ADJ
ap-4587	312	2	5.4	5.4	NUM
ap-4587	312	3	.	.	PUNCT
ap-4587	313	1	the	the	DET
ap-4587	313	2	cut	cut	VERB
ap-4587	313	3	-	-	PUNCT
ap-4587	313	4	and	and	CCONJ
ap-4587	313	5	-	-	PUNCT
ap-4587	313	6	project	project	NOUN
ap-4587	313	7	scheme	scheme	NOUN
ap-4587	313	8	λ	λ	NOUN
ap-4587	313	9	=	=	SYM
ap-4587	313	10	(	(	PUNCT
ap-4587	313	11	l	l	NOUN
ap-4587	313	12	,	,	PUNCT
ap-4587	313	13	π1	π1	NOUN
ap-4587	313	14	,	,	PUNCT
ap-4587	313	15	π2	π2	NUM
ap-4587	313	16	)	)	PUNCT
ap-4587	313	17	defined	define	VERB
ap-4587	313	18	above	above	ADV
ap-4587	313	19	is	be	AUX
ap-4587	313	20	non	non	ADJ
ap-4587	313	21	-	-	ADJ
ap-4587	313	22	degenerate	degenerate	ADJ
ap-4587	313	23	and	and	CCONJ
ap-4587	313	24	irreducible	irreducible	ADJ
ap-4587	313	25	.	.	PUNCT
ap-4587	314	1	if	if	SCONJ
ap-4587	314	2	ω	ω	PROPN
ap-4587	314	3	⊂	⊂	PROPN
ap-4587	314	4	r2	r2	PROPN
ap-4587	314	5	is	be	AUX
ap-4587	314	6	bounded	bound	VERB
ap-4587	314	7	and	and	CCONJ
ap-4587	314	8	such	such	ADJ
ap-4587	314	9	that	that	DET
ap-4587	314	10	int(ω	int(ω	NOUN
ap-4587	314	11	)	)	PUNCT
ap-4587	314	12	6=	6=	NOUN
ap-4587	314	13	∅	∅	NOUN
ap-4587	314	14	,	,	PUNCT
ap-4587	314	15	then	then	ADV
ap-4587	314	16	the	the	DET
ap-4587	314	17	cut	cut	VERB
ap-4587	314	18	-	-	PUNCT
ap-4587	314	19	and	and	CCONJ
ap-4587	314	20	-	-	PUNCT
ap-4587	314	21	project	project	NOUN
ap-4587	314	22	set	set	VERB
ap-4587	314	23	σ(ω	σ(ω	PROPN
ap-4587	314	24	)	)	PUNCT
ap-4587	314	25	can	can	AUX
ap-4587	314	26	be	be	AUX
ap-4587	314	27	written	write	VERB
ap-4587	314	28	as	as	ADP
ap-4587	314	29	σ(ω	σ(ω	PROPN
ap-4587	314	30	)	)	PUNCT
ap-4587	315	1	=	=	PRON
ap-4587	315	2	{	{	PUNCT
ap-4587	315	3	(	(	PUNCT
ap-4587	315	4	a+	a+	X
ap-4587	315	5	bτ)l‖1	bτ)l‖1	X
ap-4587	315	6	+	+	CCONJ
ap-4587	315	7	(	(	PUNCT
ap-4587	315	8	c+	c+	X
ap-4587	315	9	dτ)l‖4	dτ)l‖4	X
ap-4587	315	10	:	:	PUNCT
ap-4587	315	11	a	a	DET
ap-4587	315	12	,	,	PUNCT
ap-4587	315	13	b	b	NOUN
ap-4587	315	14	,	,	PUNCT
ap-4587	315	15	c	c	NOUN
ap-4587	315	16	,	,	PUNCT
ap-4587	316	1	d	d	PROPN
ap-4587	316	2	∈	∈	PROPN
ap-4587	316	3	z	z	PROPN
ap-4587	316	4	,	,	PUNCT
ap-4587	316	5	(	(	PUNCT
ap-4587	316	6	a+	a+	PUNCT
ap-4587	316	7	bτ	bτ	PROPN
ap-4587	316	8	′)l⊥1	′)l⊥1	VERB
ap-4587	316	9	+	+	CCONJ
ap-4587	316	10	(	(	PUNCT
ap-4587	316	11	c+	c+	X
ap-4587	316	12	dτ	dτ	X
ap-4587	316	13	′)l⊥4	′)l⊥4	PROPN
ap-4587	316	14	∈	∈	PROPN
ap-4587	316	15	ω	ω	PROPN
ap-4587	316	16	}	}	PUNCT
ap-4587	316	17	.	.	PUNCT
ap-4587	317	1	one	one	PRON
ap-4587	317	2	can	can	AUX
ap-4587	317	3	review	review	VERB
ap-4587	317	4	the	the	DET
ap-4587	317	5	results	result	NOUN
ap-4587	317	6	of	of	ADP
ap-4587	317	7	barache	barache	NOUN
ap-4587	317	8	et	et	PROPN
ap-4587	317	9	al	al	PROPN
ap-4587	317	10	.	.	PUNCT
ap-4587	318	1	[	[	X
ap-4587	318	2	3	3	X
ap-4587	318	3	]	]	PUNCT
ap-4587	318	4	to	to	PART
ap-4587	318	5	see	see	VERB
ap-4587	318	6	that	that	SCONJ
ap-4587	318	7	our	our	PRON
ap-4587	318	8	method	method	NOUN
ap-4587	318	9	leads	lead	VERB
ap-4587	318	10	to	to	ADP
ap-4587	318	11	the	the	DET
ap-4587	318	12	same	same	ADJ
ap-4587	318	13	model	model	NOUN
ap-4587	318	14	as	as	ADP
ap-4587	318	15	the	the	DET
ap-4587	318	16	classical	classical	ADJ
ap-4587	318	17	method	method	NOUN
ap-4587	318	18	of	of	ADP
ap-4587	318	19	defining	define	VERB
ap-4587	318	20	decagonal	decagonal	ADJ
ap-4587	318	21	cut	cut	NOUN
ap-4587	318	22	-	-	PUNCT
ap-4587	318	23	and	and	CCONJ
ap-4587	318	24	-	-	PUNCT
ap-4587	318	25	project	project	NOUN
ap-4587	318	26	set	set	NOUN
ap-4587	318	27	based	base	VERB
ap-4587	318	28	on	on	ADP
ap-4587	318	29	coxeter	coxet	ADJ
ap-4587	318	30	groups	group	NOUN
ap-4587	318	31	,	,	PUNCT
ap-4587	318	32	where	where	SCONJ
ap-4587	318	33	one	one	NUM
ap-4587	318	34	projects	project	VERB
ap-4587	318	35	the	the	DET
ap-4587	318	36	crystallographic	crystallographic	ADJ
ap-4587	318	37	root	root	NOUN
ap-4587	318	38	system	system	NOUN
ap-4587	318	39	a4	a4	NOUN
ap-4587	318	40	to	to	ADP
ap-4587	318	41	the	the	DET
ap-4587	318	42	non	non	ADJ
ap-4587	318	43	-	-	ADJ
ap-4587	318	44	crystallographic	crystallographic	ADJ
ap-4587	318	45	system	system	NOUN
ap-4587	318	46	h2	h2	NOUN
ap-4587	318	47	.	.	PUNCT
ap-4587	319	1	6	6	NUM
ap-4587	319	2	.	.	X
ap-4587	319	3	self	self	NOUN
ap-4587	319	4	-	-	PUNCT
ap-4587	319	5	similarities	similarity	NOUN
ap-4587	319	6	of	of	ADP
ap-4587	319	7	the	the	DET
ap-4587	319	8	constructed	construct	VERB
ap-4587	319	9	scheme	scheme	NOUN
ap-4587	319	10	let	let	VERB
ap-4587	319	11	us	we	PRON
ap-4587	319	12	now	now	ADV
ap-4587	319	13	study	study	VERB
ap-4587	319	14	in	in	ADP
ap-4587	319	15	according	accord	VERB
ap-4587	319	16	to	to	ADP
ap-4587	319	17	item	item	NOUN
ap-4587	319	18	(	(	PUNCT
ap-4587	319	19	ii	ii	NOUN
ap-4587	319	20	)	)	PUNCT
ap-4587	319	21	of	of	ADP
ap-4587	319	22	proposition	proposition	NOUN
ap-4587	319	23	3.3	3.3	NUM
ap-4587	319	24	what	what	PRON
ap-4587	319	25	other	other	ADJ
ap-4587	319	26	self	self	NOUN
ap-4587	319	27	-	-	PUNCT
ap-4587	319	28	similarities	similarity	NOUN
ap-4587	319	29	are	be	AUX
ap-4587	319	30	present	present	ADJ
ap-4587	319	31	in	in	ADP
ap-4587	319	32	the	the	DET
ap-4587	319	33	cut	cut	NOUN
ap-4587	319	34	-	-	PUNCT
ap-4587	319	35	and	and	CCONJ
ap-4587	319	36	-	-	PUNCT
ap-4587	319	37	project	project	NOUN
ap-4587	319	38	scheme	scheme	NOUN
ap-4587	319	39	λ	λ	NOUN
ap-4587	319	40	constructed	construct	VERB
ap-4587	319	41	in	in	ADP
ap-4587	319	42	section	section	NOUN
ap-4587	319	43	5	5	NUM
ap-4587	319	44	.	.	PUNCT
ap-4587	320	1	we	we	PRON
ap-4587	320	2	thus	thus	ADV
ap-4587	320	3	need	need	VERB
ap-4587	320	4	to	to	PART
ap-4587	320	5	find	find	VERB
ap-4587	320	6	all	all	DET
ap-4587	320	7	integer	integer	NOUN
ap-4587	320	8	matrices	matrix	NOUN
ap-4587	320	9	z	z	NOUN
ap-4587	320	10	which	which	PRON
ap-4587	320	11	by	by	ADP
ap-4587	320	12	similarity	similarity	NOUN
ap-4587	320	13	transformation	transformation	NOUN
ap-4587	320	14	v	v	ADP
ap-4587	320	15	zv	zv	NOUN
ap-4587	320	16	−1	−1	NOUN
ap-4587	320	17	(	(	PUNCT
ap-4587	320	18	with	with	ADP
ap-4587	320	19	the	the	DET
ap-4587	320	20	matrix	matrix	NOUN
ap-4587	320	21	v	v	NOUN
ap-4587	320	22	defined	define	VERB
ap-4587	320	23	in	in	ADP
ap-4587	320	24	(	(	PUNCT
ap-4587	320	25	9	9	NUM
ap-4587	320	26	)	)	PUNCT
ap-4587	320	27	)	)	PUNCT
ap-4587	320	28	becomes	become	VERB
ap-4587	320	29	a	a	DET
ap-4587	320	30	block	block	NOUN
ap-4587	320	31	diagonal	diagonal	ADJ
ap-4587	320	32	matrix	matrix	NOUN
ap-4587	320	33	with	with	ADP
ap-4587	320	34	blocks	block	NOUN
ap-4587	320	35	of	of	ADP
ap-4587	320	36	the	the	DET
ap-4587	320	37	size	size	NOUN
ap-4587	320	38	2	2	NUM
ap-4587	320	39	.	.	PUNCT
ap-4587	321	1	this	this	PRON
ap-4587	321	2	means	mean	VERB
ap-4587	321	3	that	that	SCONJ
ap-4587	321	4	z	z	PROPN
ap-4587	321	5	has	have	VERB
ap-4587	321	6	the	the	DET
ap-4587	321	7	same	same	ADJ
ap-4587	321	8	two	two	NUM
ap-4587	321	9	eigenspaces	eigenspace	NOUN
ap-4587	321	10	of	of	ADP
ap-4587	321	11	dimension	dimension	NOUN
ap-4587	321	12	2	2	NUM
ap-4587	321	13	.	.	PUNCT
ap-4587	322	1	let	let	VERB
ap-4587	322	2	us	we	PRON
ap-4587	322	3	first	first	ADV
ap-4587	322	4	consider	consider	VERB
ap-4587	322	5	those	those	DET
ap-4587	322	6	integer	integer	NOUN
ap-4587	322	7	matrices	matrix	NOUN
ap-4587	322	8	z	z	NOUN
ap-4587	322	9	which	which	PRON
ap-4587	322	10	have	have	VERB
ap-4587	322	11	the	the	DET
ap-4587	322	12	same	same	ADJ
ap-4587	322	13	eigenvectors	eigenvector	NOUN
ap-4587	322	14	yi	yi	NUM
ap-4587	322	15	,	,	PUNCT
ap-4587	322	16	i	i	PRON
ap-4587	322	17	=	=	NOUN
ap-4587	322	18	1	1	NUM
ap-4587	322	19	,	,	PUNCT
ap-4587	322	20	.	.	PUNCT
ap-4587	322	21	.	.	PUNCT
ap-4587	323	1	.	.	PUNCT
ap-4587	324	1	,	,	PUNCT
ap-4587	324	2	4	4	NUM
ap-4587	324	3	,	,	PUNCT
ap-4587	324	4	as	as	ADP
ap-4587	324	5	c	c	PROPN
ap-4587	324	6	defined	define	VERB
ap-4587	324	7	in	in	ADP
ap-4587	324	8	(	(	PUNCT
ap-4587	324	9	4	4	NUM
ap-4587	324	10	)	)	PUNCT
ap-4587	324	11	.	.	PUNCT
ap-4587	325	1	rewriting	rewrite	VERB
ap-4587	325	2	this	this	DET
ap-4587	325	3	requirement	requirement	NOUN
ap-4587	325	4	into	into	ADP
ap-4587	325	5	matrix	matrix	NOUN
ap-4587	325	6	equation	equation	NOUN
ap-4587	325	7	for	for	ADP
ap-4587	325	8	a	a	DET
ap-4587	325	9	general	general	ADJ
ap-4587	325	10	integer	integer	NOUN
ap-4587	325	11	matrix	matrix	NOUN
ap-4587	325	12	z	z	NOUN
ap-4587	325	13	,	,	PUNCT
ap-4587	325	14	we	we	PRON
ap-4587	325	15	obtain	obtain	VERB
ap-4587	325	16	zy1	zy1	NOUN
ap-4587	325	17	=	=	SYM
ap-4587	325	18			PROPN
ap-4587	325	19	a	a	DET
ap-4587	325	20	b	b	NOUN
ap-4587	325	21	c	c	NOUN
ap-4587	325	22	d	d	X
ap-4587	325	23	e	e	X
ap-4587	325	24	f	f	PROPN
ap-4587	325	25	g	g	PROPN
ap-4587	325	26	h	h	PROPN
ap-4587	326	1	j	j	PROPN
ap-4587	327	1	k	k	PROPN
ap-4587	328	1	l	l	PROPN
ap-4587	329	1	m	m	VERB
ap-4587	329	2	n	n	ADV
ap-4587	329	3	o	o	NOUN
ap-4587	329	4	p	p	X
ap-4587	329	5	q	q	NOUN
ap-4587	329	6			NOUN
ap-4587	329	7			ADJ
ap-4587	329	8	1	1	NUM
ap-4587	329	9	ω	ω	NUM
ap-4587	329	10	ω2	ω2	ADJ
ap-4587	329	11	ω3	ω3	ADJ
ap-4587	329	12			NOUN
ap-4587	330	1	=	=	SYM
ap-4587	330	2	ρ	ρ	PROPN
ap-4587	330	3			ADJ
ap-4587	330	4	1	1	NUM
ap-4587	330	5	ω	ω	NUM
ap-4587	330	6	ω2	ω2	ADJ
ap-4587	330	7	ω3	ω3	ADJ
ap-4587	330	8			NOUN
ap-4587	330	9	,	,	PUNCT
ap-4587	330	10	for	for	ADP
ap-4587	330	11	some	some	DET
ap-4587	330	12	ρ	ρ	PROPN
ap-4587	330	13	∈	∈	PROPN
ap-4587	330	14	r.	r.	NOUN
ap-4587	330	15	from	from	ADP
ap-4587	330	16	the	the	DET
ap-4587	330	17	first	first	ADJ
ap-4587	330	18	row	row	NOUN
ap-4587	330	19	,	,	PUNCT
ap-4587	330	20	we	we	PRON
ap-4587	330	21	obtain	obtain	VERB
ap-4587	330	22	ρ	ρ	NOUN
ap-4587	330	23	=	=	SYM
ap-4587	330	24	a+	a+	PUNCT
ap-4587	330	25	bω	bω	NOUN
ap-4587	330	26	+	+	CCONJ
ap-4587	330	27	cω2	cω2	NOUN
ap-4587	330	28	+	+	CCONJ
ap-4587	330	29	dω3	dω3	X
ap-4587	330	30	.	.	PUNCT
ap-4587	330	31	using	use	VERB
ap-4587	330	32	this	this	PRON
ap-4587	330	33	,	,	PUNCT
ap-4587	330	34	and	and	CCONJ
ap-4587	330	35	the	the	DET
ap-4587	330	36	expression	expression	NOUN
ap-4587	330	37	for	for	ADP
ap-4587	330	38	ω4	ω4	NUM
ap-4587	330	39	in	in	ADP
ap-4587	330	40	terms	term	NOUN
ap-4587	330	41	of	of	ADP
ap-4587	330	42	lower	low	ADJ
ap-4587	330	43	powers	power	NOUN
ap-4587	330	44	of	of	ADP
ap-4587	330	45	ω	ω	NUM
ap-4587	330	46	,	,	PUNCT
ap-4587	330	47	namely	namely	ADV
ap-4587	330	48	ω4	ω4	NUM
ap-4587	330	49	=	=	PUNCT
ap-4587	330	50	−1−	−1−	PROPN
ap-4587	330	51	ω	ω	NUM
ap-4587	330	52	−	−	PROPN
ap-4587	331	1	ω2	ω2	ADJ
ap-4587	331	2	−	−	PROPN
ap-4587	331	3	ω3	ω3	PROPN
ap-4587	331	4	,	,	PUNCT
ap-4587	331	5	we	we	PRON
ap-4587	331	6	get	get	VERB
ap-4587	331	7	from	from	ADP
ap-4587	331	8	the	the	DET
ap-4587	331	9	remaining	remain	VERB
ap-4587	331	10	rows	row	NOUN
ap-4587	331	11	the	the	DET
ap-4587	331	12	following	follow	VERB
ap-4587	331	13	relations	relation	NOUN
ap-4587	331	14	between	between	ADP
ap-4587	331	15	the	the	DET
ap-4587	331	16	integer	integer	NOUN
ap-4587	331	17	coefficients	coefficient	VERB
ap-4587	331	18	a	a	PRON
ap-4587	331	19	,	,	PUNCT
ap-4587	331	20	.	.	PUNCT
ap-4587	331	21	.	.	PUNCT
ap-4587	331	22	.	.	PUNCT
ap-4587	332	1	,	,	PUNCT
ap-4587	332	2	q	q	X
ap-4587	332	3	,	,	PUNCT
ap-4587	332	4	e	e	X
ap-4587	332	5	=	=	SYM
ap-4587	332	6	−d	−d	PROPN
ap-4587	332	7	,	,	PUNCT
ap-4587	332	8	j	j	X
ap-4587	332	9	=	=	SYM
ap-4587	332	10	d−	d−	PROPN
ap-4587	332	11	c	c	NOUN
ap-4587	332	12	,	,	PUNCT
ap-4587	332	13	n	n	NOUN
ap-4587	332	14	=	=	SYM
ap-4587	332	15	c−	c−	PROPN
ap-4587	332	16	b	b	PROPN
ap-4587	332	17	,	,	PUNCT
ap-4587	332	18	f	f	NOUN
ap-4587	332	19	=	=	SYM
ap-4587	332	20	a−	a−	PROPN
ap-4587	332	21	d	d	PROPN
ap-4587	332	22	,	,	PUNCT
ap-4587	332	23	k	k	NOUN
ap-4587	332	24	=	=	PUNCT
ap-4587	332	25	−c	−c	NOUN
ap-4587	332	26	,	,	PUNCT
ap-4587	332	27	o	o	NOUN
ap-4587	332	28	=	=	PUNCT
ap-4587	332	29	d−	d−	PROPN
ap-4587	332	30	b	b	PROPN
ap-4587	332	31	,	,	PUNCT
ap-4587	332	32	g	g	PROPN
ap-4587	332	33	=	=	SYM
ap-4587	332	34	b−	b−	PROPN
ap-4587	332	35	d	d	PROPN
ap-4587	332	36	,	,	PUNCT
ap-4587	332	37	l	l	NOUN
ap-4587	332	38	=	=	SYM
ap-4587	332	39	a−	a−	PROPN
ap-4587	332	40	c	c	NOUN
ap-4587	332	41	,	,	PUNCT
ap-4587	332	42	p	p	NOUN
ap-4587	332	43	=	=	X
ap-4587	332	44	−b	−b	ADJ
ap-4587	332	45	,	,	PUNCT
ap-4587	332	46	h	h	NOUN
ap-4587	332	47	=	=	SYM
ap-4587	332	48	c−	c−	PROPN
ap-4587	332	49	d	d	PROPN
ap-4587	332	50	,	,	PUNCT
ap-4587	332	51	m	m	VERB
ap-4587	332	52	=	=	ADJ
ap-4587	332	53	b−	b−	PROPN
ap-4587	332	54	c	c	NOUN
ap-4587	332	55	,	,	PUNCT
ap-4587	332	56	q	q	NOUN
ap-4587	332	57	=	=	SYM
ap-4587	332	58	a−	a−	PROPN
ap-4587	332	59	b.	b.	PROPN
ap-4587	333	1	one	one	PRON
ap-4587	333	2	can	can	AUX
ap-4587	333	3	check	check	VERB
ap-4587	333	4	that	that	PRON
ap-4587	333	5	,	,	PUNCT
ap-4587	333	6	not	not	PART
ap-4587	333	7	surprisingly	surprisingly	ADV
ap-4587	333	8	,	,	PUNCT
ap-4587	333	9	a	a	DET
ap-4587	333	10	matrix	matrix	NOUN
ap-4587	333	11	z	z	AUX
ap-4587	333	12	satisfying	satisfy	VERB
ap-4587	333	13	such	such	ADJ
ap-4587	333	14	conditions	condition	NOUN
ap-4587	333	15	,	,	PUNCT
ap-4587	333	16	i.e.	i.e.	X
ap-4587	333	17	,	,	PUNCT
ap-4587	333	18	z	z	X
ap-4587	333	19	=	=	SYM
ap-4587	333	20			PROPN
ap-4587	333	21	a	a	DET
ap-4587	333	22	b	b	X
ap-4587	333	23	c	c	NOUN
ap-4587	333	24	d	d	X
ap-4587	333	25	−d	−d	VERB
ap-4587	333	26	a−	a−	PROPN
ap-4587	333	27	d	d	PROPN
ap-4587	333	28	b−	b−	PROPN
ap-4587	333	29	d	d	X
ap-4587	333	30	c−	c−	PROPN
ap-4587	334	1	d	d	NUM
ap-4587	334	2	d−	d−	PROPN
ap-4587	334	3	c	c	PROPN
ap-4587	334	4	−c	−c	NOUN
ap-4587	334	5	a−	a−	PROPN
ap-4587	334	6	c	c	NOUN
ap-4587	334	7	b−	b−	NOUN
ap-4587	334	8	c	c	PROPN
ap-4587	334	9	c−	c−	PROPN
ap-4587	334	10	b	b	NUM
ap-4587	334	11	d−	d−	PROPN
ap-4587	334	12	b	b	PROPN
ap-4587	334	13	−b	−b	ADJ
ap-4587	334	14	a−	a−	PROPN
ap-4587	334	15	b	b	PROPN
ap-4587	334	16			NOUN
ap-4587	335	1	=	=	X
ap-4587	335	2	:	:	PUNCT
ap-4587	335	3	za	za	PROPN
ap-4587	335	4	,	,	PUNCT
ap-4587	335	5	b	b	PROPN
ap-4587	335	6	,	,	PUNCT
ap-4587	335	7	c	c	X
ap-4587	335	8	,	,	PUNCT
ap-4587	335	9	d	d	X
ap-4587	335	10	(	(	PUNCT
ap-4587	335	11	17	17	NUM
ap-4587	335	12	)	)	PUNCT
ap-4587	335	13	is	be	AUX
ap-4587	335	14	nothing	nothing	PRON
ap-4587	335	15	else	else	ADV
ap-4587	335	16	then	then	ADV
ap-4587	335	17	an	an	DET
ap-4587	335	18	integer	integer	NOUN
ap-4587	335	19	combination	combination	NOUN
ap-4587	335	20	z	z	NOUN
ap-4587	336	1	=	=	PUNCT
ap-4587	336	2	ai	ai	PROPN
ap-4587	336	3	+	+	PROPN
ap-4587	337	1	bc	bc	PROPN
ap-4587	337	2	+	+	CCONJ
ap-4587	337	3	cc2	cc2	PROPN
ap-4587	337	4	+	+	X
ap-4587	337	5	dc3	dc3	NOUN
ap-4587	337	6	of	of	ADP
ap-4587	337	7	the	the	DET
ap-4587	337	8	powers	power	NOUN
ap-4587	337	9	of	of	ADP
ap-4587	337	10	the	the	DET
ap-4587	337	11	matrix	matrix	NOUN
ap-4587	337	12	c	c	NOUN
ap-4587	337	13	given	give	VERB
ap-4587	337	14	in	in	ADP
ap-4587	337	15	(	(	PUNCT
ap-4587	337	16	4	4	NUM
ap-4587	337	17	)	)	PUNCT
ap-4587	337	18	,	,	PUNCT
ap-4587	337	19	namely	namely	ADV
ap-4587	337	20	i	i	PRON
ap-4587	337	21	=	=	SYM
ap-4587	337	22	c0	c0	NOUN
ap-4587	337	23	=	=	PUNCT
ap-4587	337	24			PROPN
ap-4587	337	25	1	1	NUM
ap-4587	337	26	0	0	NUM
ap-4587	337	27	0	0	NUM
ap-4587	337	28	0	0	NUM
ap-4587	337	29	0	0	NUM
ap-4587	337	30	1	1	NUM
ap-4587	337	31	0	0	NUM
ap-4587	337	32	0	0	NUM
ap-4587	337	33	0	0	NUM
ap-4587	337	34	0	0	NUM
ap-4587	337	35	1	1	NUM
ap-4587	337	36	0	0	NUM
ap-4587	337	37	0	0	NUM
ap-4587	337	38	0	0	NUM
ap-4587	337	39	0	0	NUM
ap-4587	337	40	1	1	NUM
ap-4587	337	41			NOUN
ap-4587	337	42	,	,	PUNCT
ap-4587	337	43	c	c	X
ap-4587	337	44	=	=	SYM
ap-4587	337	45			ADJ
ap-4587	337	46	0	0	NUM
ap-4587	338	1	1	1	NUM
ap-4587	338	2	0	0	NUM
ap-4587	338	3	0	0	NUM
ap-4587	338	4	0	0	NUM
ap-4587	338	5	0	0	NUM
ap-4587	338	6	1	1	NUM
ap-4587	338	7	0	0	NUM
ap-4587	338	8	0	0	NUM
ap-4587	338	9	0	0	NUM
ap-4587	338	10	0	0	NUM
ap-4587	338	11	1	1	NUM
ap-4587	338	12	−1	−1	NOUN
ap-4587	338	13	−1	−1	NOUN
ap-4587	338	14	−1	−1	NOUN
ap-4587	338	15	−1	−1	NOUN
ap-4587	338	16			NOUN
ap-4587	338	17	,	,	PUNCT
ap-4587	338	18	c2	c2	PROPN
ap-4587	338	19	=	=	SYM
ap-4587	338	20			ADJ
ap-4587	338	21	0	0	NUM
ap-4587	338	22	0	0	NUM
ap-4587	338	23	1	1	NUM
ap-4587	338	24	0	0	NUM
ap-4587	338	25	0	0	NUM
ap-4587	338	26	0	0	NUM
ap-4587	338	27	0	0	NUM
ap-4587	338	28	1	1	NUM
ap-4587	338	29	−1	−1	NOUN
ap-4587	338	30	−1	−1	NOUN
ap-4587	338	31	−1	−1	NOUN
ap-4587	338	32	−1	−1	NOUN
ap-4587	338	33	1	1	NUM
ap-4587	338	34	0	0	NUM
ap-4587	338	35	0	0	NUM
ap-4587	338	36	0	0	NUM
ap-4587	338	37			NOUN
ap-4587	338	38	,	,	PUNCT
ap-4587	338	39	c3	c3	X
ap-4587	338	40	=	=	PUNCT
ap-4587	338	41			PROPN
ap-4587	338	42	0	0	NUM
ap-4587	338	43	0	0	SYM
ap-4587	338	44	0	0	NUM
ap-4587	338	45	1	1	NUM
ap-4587	338	46	−1	−1	NOUN
ap-4587	338	47	−1	−1	NOUN
ap-4587	338	48	−1	−1	NOUN
ap-4587	338	49	−1	−1	NOUN
ap-4587	338	50	1	1	NUM
ap-4587	338	51	0	0	NUM
ap-4587	338	52	0	0	NUM
ap-4587	338	53	0	0	NUM
ap-4587	338	54	0	0	NUM
ap-4587	338	55	1	1	NUM
ap-4587	338	56	0	0	NUM
ap-4587	338	57	0	0	NUM
ap-4587	338	58			NOUN
ap-4587	338	59	,	,	PUNCT
ap-4587	338	60	note	note	VERB
ap-4587	338	61	that	that	SCONJ
ap-4587	338	62	we	we	PRON
ap-4587	338	63	do	do	AUX
ap-4587	338	64	not	not	PART
ap-4587	338	65	use	use	VERB
ap-4587	338	66	higher	high	ADJ
ap-4587	338	67	than	than	ADP
ap-4587	338	68	third	third	ADJ
ap-4587	338	69	powers	power	NOUN
ap-4587	338	70	of	of	ADP
ap-4587	338	71	the	the	DET
ap-4587	338	72	matrix	matrix	NOUN
ap-4587	339	1	c	c	NOUN
ap-4587	339	2	,	,	PUNCT
ap-4587	339	3	as	as	ADP
ap-4587	339	4	by	by	ADP
ap-4587	339	5	hamilton	hamilton	PROPN
ap-4587	339	6	-	-	PUNCT
ap-4587	339	7	cayley	cayley	NOUN
ap-4587	339	8	theorem	theorem	ADJ
ap-4587	339	9	,	,	PUNCT
ap-4587	339	10	c4	c4	NOUN
ap-4587	339	11	=	=	SYM
ap-4587	339	12	−c3	−c3	PROPN
ap-4587	339	13	1	1	NUM
ap-4587	339	14	−	−	PROPN
ap-4587	339	15	c2	c2	PROPN
ap-4587	339	16	1	1	NUM
ap-4587	339	17	−	−	PROPN
ap-4587	339	18	c1	c1	PROPN
ap-4587	339	19	−	−	PROPN
ap-4587	339	20	i.	i.	PROPN
ap-4587	339	21	note	note	PROPN
ap-4587	339	22	also	also	ADV
ap-4587	339	23	,	,	PUNCT
ap-4587	339	24	that	that	SCONJ
ap-4587	339	25	we	we	PRON
ap-4587	339	26	do	do	AUX
ap-4587	339	27	not	not	PART
ap-4587	339	28	need	need	VERB
ap-4587	339	29	to	to	PART
ap-4587	339	30	use	use	VERB
ap-4587	339	31	zyi	zyi	NOUN
ap-4587	340	1	=	=	SYM
ap-4587	340	2	ρiyi	ρiyi	NOUN
ap-4587	340	3	,	,	PUNCT
ap-4587	340	4	for	for	ADP
ap-4587	340	5	i	i	PROPN
ap-4587	340	6	=	=	SYM
ap-4587	340	7	2	2	NUM
ap-4587	340	8	,	,	PUNCT
ap-4587	340	9	3	3	NUM
ap-4587	340	10	,	,	PUNCT
ap-4587	340	11	4	4	NUM
ap-4587	340	12	,	,	PUNCT
ap-4587	340	13	since	since	SCONJ
ap-4587	340	14	the	the	DET
ap-4587	340	15	result	result	NOUN
ap-4587	340	16	is	be	AUX
ap-4587	340	17	obtainable	obtainable	ADJ
ap-4587	340	18	applying	apply	VERB
ap-4587	340	19	the	the	DET
ap-4587	340	20	galois	galois	PROPN
ap-4587	340	21	automorphisms	automorphism	NOUN
ap-4587	340	22	of	of	ADP
ap-4587	340	23	the	the	DET
ap-4587	340	24	field	field	NOUN
ap-4587	340	25	q(ω	q(ω	NOUN
ap-4587	340	26	)	)	PUNCT
ap-4587	340	27	,	,	PUNCT
ap-4587	340	28	and	and	CCONJ
ap-4587	340	29	it	it	PRON
ap-4587	340	30	would	would	AUX
ap-4587	340	31	not	not	PART
ap-4587	340	32	provide	provide	VERB
ap-4587	340	33	any	any	DET
ap-4587	340	34	new	new	ADJ
ap-4587	340	35	information	information	NOUN
ap-4587	340	36	.	.	PUNCT
ap-4587	341	1	it	it	PRON
ap-4587	341	2	follows	follow	VERB
ap-4587	341	3	that	that	SCONJ
ap-4587	341	4	the	the	DET
ap-4587	341	5	matrix	matrix	NOUN
ap-4587	341	6	z	z	NOUN
ap-4587	341	7	of	of	ADP
ap-4587	341	8	(	(	PUNCT
ap-4587	341	9	17	17	NUM
ap-4587	341	10	)	)	PUNCT
ap-4587	341	11	can	can	AUX
ap-4587	341	12	be	be	AUX
ap-4587	341	13	diagonalized	diagonalize	VERB
ap-4587	341	14	using	use	VERB
ap-4587	341	15	the	the	DET
ap-4587	341	16	matrix	matrix	NOUN
ap-4587	341	17	y	y	PROPN
ap-4587	341	18	(	(	PUNCT
ap-4587	341	19	of	of	ADP
ap-4587	341	20	(	(	PUNCT
ap-4587	341	21	6	6	NUM
ap-4587	341	22	)	)	PUNCT
ap-4587	341	23	)	)	PUNCT
ap-4587	341	24	composed	compose	VERB
ap-4587	341	25	of	of	ADP
ap-4587	341	26	eigenvectors	eigenvector	NOUN
ap-4587	341	27	yi	yi	PROPN
ap-4587	341	28	,	,	PUNCT
ap-4587	341	29	and	and	CCONJ
ap-4587	341	30	on	on	ADP
ap-4587	341	31	the	the	DET
ap-4587	341	32	diagonal	diagonal	ADJ
ap-4587	341	33	we	we	PRON
ap-4587	341	34	find	find	VERB
ap-4587	341	35	the	the	DET
ap-4587	341	36	numbers	number	NOUN
ap-4587	341	37	σj(a+	σj(a+	PROPN
ap-4587	341	38	bω	bω	NOUN
ap-4587	341	39	+	+	CCONJ
ap-4587	341	40	cω2	cω2	NOUN
ap-4587	341	41	+	+	CCONJ
ap-4587	341	42	dω4	dω4	ADJ
ap-4587	341	43	)	)	PUNCT
ap-4587	341	44	.	.	PUNCT
ap-4587	342	1	these	these	PRON
ap-4587	342	2	are	be	AUX
ap-4587	342	3	thus	thus	ADV
ap-4587	342	4	the	the	DET
ap-4587	342	5	eigenvalues	eigenvalue	NOUN
ap-4587	342	6	of	of	ADP
ap-4587	342	7	z.	z.	PROPN
ap-4587	342	8	438	438	NUM
ap-4587	342	9	vol	vol	NOUN
ap-4587	342	10	.	.	PUNCT
ap-4587	343	1	57	57	NUM
ap-4587	343	2	no	no	NOUN
ap-4587	343	3	.	.	PUNCT
ap-4587	344	1	6/2017	6/2017	NUM
ap-4587	344	2	on	on	ADP
ap-4587	344	3	self	self	NOUN
ap-4587	344	4	-	-	PUNCT
ap-4587	344	5	similarities	similarity	NOUN
ap-4587	344	6	of	of	ADP
ap-4587	344	7	cut	cut	NOUN
ap-4587	344	8	-	-	PUNCT
ap-4587	344	9	and	and	CCONJ
ap-4587	344	10	-	-	PUNCT
ap-4587	344	11	project	project	NOUN
ap-4587	344	12	sets	set	NOUN
ap-4587	344	13	corollary	corollary	ADJ
ap-4587	344	14	6.1	6.1	NUM
ap-4587	344	15	.	.	PUNCT
ap-4587	345	1	the	the	DET
ap-4587	345	2	set	set	NOUN
ap-4587	345	3	zcom	zcom	NOUN
ap-4587	345	4	:	:	PUNCT
ap-4587	345	5	=	=	SYM
ap-4587	345	6	{	{	PUNCT
ap-4587	345	7	za	za	PROPN
ap-4587	345	8	,	,	PUNCT
ap-4587	345	9	b	b	PROPN
ap-4587	345	10	,	,	PUNCT
ap-4587	345	11	c	c	NOUN
ap-4587	345	12	,	,	PUNCT
ap-4587	345	13	d	d	NOUN
ap-4587	345	14	:	:	PUNCT
ap-4587	345	15	a	a	DET
ap-4587	345	16	,	,	PUNCT
ap-4587	345	17	b	b	NOUN
ap-4587	345	18	,	,	PUNCT
ap-4587	345	19	c	c	NOUN
ap-4587	345	20	,	,	PUNCT
ap-4587	345	21	d	d	PROPN
ap-4587	345	22	∈	∈	PROPN
ap-4587	345	23	z	z	NOUN
ap-4587	345	24	}	}	PUNCT
ap-4587	345	25	with	with	ADP
ap-4587	345	26	standard	standard	ADJ
ap-4587	345	27	matrix	matrix	NOUN
ap-4587	345	28	addition	addition	NOUN
ap-4587	345	29	and	and	CCONJ
ap-4587	345	30	multiplication	multiplication	NOUN
ap-4587	345	31	is	be	AUX
ap-4587	345	32	a	a	DET
ap-4587	345	33	commutative	commutative	ADJ
ap-4587	345	34	ring	ring	NOUN
ap-4587	345	35	isomorphic	isomorphic	ADJ
ap-4587	345	36	to	to	ADP
ap-4587	345	37	the	the	DET
ap-4587	345	38	ring	ring	NOUN
ap-4587	345	39	z[ω	z[ω	PROPN
ap-4587	345	40	]	]	PUNCT
ap-4587	345	41	of	of	ADP
ap-4587	345	42	cyclotomic	cyclotomic	ADJ
ap-4587	345	43	integers	integer	NOUN
ap-4587	345	44	.	.	PUNCT
ap-4587	346	1	in	in	ADP
ap-4587	346	2	order	order	NOUN
ap-4587	346	3	to	to	PART
ap-4587	346	4	transform	transform	VERB
ap-4587	346	5	the	the	DET
ap-4587	346	6	matrix	matrix	NOUN
ap-4587	346	7	z	z	NOUN
ap-4587	346	8	into	into	ADP
ap-4587	346	9	the	the	DET
ap-4587	346	10	real	real	ADJ
ap-4587	346	11	block	block	NOUN
ap-4587	346	12	diagonal	diagonal	ADJ
ap-4587	346	13	form	form	NOUN
ap-4587	346	14	,	,	PUNCT
ap-4587	346	15	we	we	PRON
ap-4587	346	16	use	use	VERB
ap-4587	346	17	the	the	DET
ap-4587	346	18	similarity	similarity	NOUN
ap-4587	346	19	transformation	transformation	NOUN
ap-4587	346	20	by	by	ADP
ap-4587	346	21	the	the	DET
ap-4587	346	22	matrix	matrix	NOUN
ap-4587	346	23	p	p	NOUN
ap-4587	346	24	of	of	ADP
ap-4587	346	25	(	(	PUNCT
ap-4587	346	26	7	7	NUM
ap-4587	346	27	)	)	PUNCT
ap-4587	346	28	.	.	PUNCT
ap-4587	347	1	this	this	DET
ap-4587	347	2	yields	yield	NOUN
ap-4587	347	3	p−1y	p−1y	VERB
ap-4587	347	4	−1zyp	−1zyp	ADV
ap-4587	347	5	=	=	PUNCT
ap-4587	347	6	(	(	PUNCT
ap-4587	347	7	s	s	NOUN
ap-4587	347	8	o	o	X
ap-4587	347	9	o	o	X
ap-4587	347	10	t	t	PROPN
ap-4587	347	11	)	)	PUNCT
ap-4587	347	12	,	,	PUNCT
ap-4587	347	13	where	where	SCONJ
ap-4587	347	14	s	s	VERB
ap-4587	347	15	=	=	PUNCT
ap-4587	347	16	(	(	PUNCT
ap-4587	347	17	a+	a+	X
ap-4587	347	18	b	b	X
ap-4587	347	19	cos	cos	ADP
ap-4587	347	20	2π	2π	PROPN
ap-4587	347	21	5	5	NUM
ap-4587	347	22	+	+	CCONJ
ap-4587	347	23	(	(	PUNCT
ap-4587	347	24	c+	c+	NOUN
ap-4587	347	25	d	d	NOUN
ap-4587	347	26	)	)	PUNCT
ap-4587	348	1	cos	cos	PROPN
ap-4587	348	2	4π	4π	NUM
ap-4587	348	3	5	5	NUM
ap-4587	348	4	b	b	NOUN
ap-4587	348	5	sin	sin	NOUN
ap-4587	348	6	2π	2π	NOUN
ap-4587	348	7	5	5	NUM
ap-4587	348	8	+	+	CCONJ
ap-4587	348	9	(	(	PUNCT
ap-4587	348	10	c−	c−	ADJ
ap-4587	348	11	d	d	NOUN
ap-4587	348	12	)	)	PUNCT
ap-4587	348	13	sin	sin	NOUN
ap-4587	348	14	4π	4π	NUM
ap-4587	348	15	5	5	NUM
ap-4587	348	16	−b	−b	ADJ
ap-4587	348	17	sin	sin	VERB
ap-4587	348	18	2π	2π	NOUN
ap-4587	348	19	5	5	NUM
ap-4587	348	20	+	+	CCONJ
ap-4587	348	21	(	(	PUNCT
ap-4587	348	22	d−	d−	PROPN
ap-4587	348	23	c	c	NOUN
ap-4587	348	24	)	)	PUNCT
ap-4587	348	25	sin	sin	NOUN
ap-4587	348	26	4π	4π	NUM
ap-4587	348	27	5	5	NUM
ap-4587	348	28	a+	a+	PRON
ap-4587	348	29	b	b	X
ap-4587	348	30	cos	cos	ADP
ap-4587	348	31	2π	2π	PROPN
ap-4587	348	32	5	5	NUM
ap-4587	348	33	+	+	CCONJ
ap-4587	348	34	(	(	PUNCT
ap-4587	348	35	c+	c+	NOUN
ap-4587	348	36	d	d	NOUN
ap-4587	348	37	)	)	PUNCT
ap-4587	348	38	cos	cos	PROPN
ap-4587	348	39	4π	4π	NUM
ap-4587	348	40	5	5	NUM
ap-4587	348	41	)	)	PUNCT
ap-4587	348	42	,	,	PUNCT
ap-4587	348	43	t	t	NOUN
ap-4587	348	44	=	=	SYM
ap-4587	348	45	(	(	PUNCT
ap-4587	348	46	a+	a+	X
ap-4587	348	47	b	b	X
ap-4587	348	48	cos	cos	PROPN
ap-4587	349	1	4π	4π	NUM
ap-4587	349	2	5	5	NUM
ap-4587	349	3	+	+	CCONJ
ap-4587	349	4	(	(	PUNCT
ap-4587	349	5	c+	c+	NOUN
ap-4587	349	6	d	d	NOUN
ap-4587	349	7	)	)	PUNCT
ap-4587	349	8	cos	cos	ADP
ap-4587	349	9	2π	2π	PROPN
ap-4587	349	10	5	5	NUM
ap-4587	349	11	b	b	NOUN
ap-4587	349	12	sin	sin	NOUN
ap-4587	349	13	4π	4π	NUM
ap-4587	349	14	5	5	NUM
ap-4587	349	15	+	+	CCONJ
ap-4587	349	16	(	(	PUNCT
ap-4587	349	17	c−	c−	ADJ
ap-4587	349	18	d	d	NOUN
ap-4587	349	19	)	)	PUNCT
ap-4587	349	20	sin	sin	NOUN
ap-4587	349	21	2π	2π	NOUN
ap-4587	349	22	5	5	NUM
ap-4587	349	23	−b	−b	ADJ
ap-4587	349	24	sin	sin	NOUN
ap-4587	349	25	4π	4π	NUM
ap-4587	349	26	5	5	NUM
ap-4587	349	27	+	+	CCONJ
ap-4587	349	28	(	(	PUNCT
ap-4587	349	29	d−	d−	PROPN
ap-4587	349	30	c	c	NOUN
ap-4587	349	31	)	)	PUNCT
ap-4587	349	32	sin	sin	NOUN
ap-4587	349	33	2π	2π	PROPN
ap-4587	349	34	5	5	NUM
ap-4587	349	35	a+	a+	SYM
ap-4587	349	36	b	b	PROPN
ap-4587	349	37	cos	cos	PROPN
ap-4587	349	38	4π	4π	NUM
ap-4587	349	39	5	5	NUM
ap-4587	349	40	+	+	CCONJ
ap-4587	349	41	(	(	PUNCT
ap-4587	349	42	c+	c+	NOUN
ap-4587	349	43	d	d	NOUN
ap-4587	349	44	)	)	PUNCT
ap-4587	349	45	cos	cos	ADP
ap-4587	349	46	2π	2π	PROPN
ap-4587	349	47	5	5	NUM
ap-4587	349	48	)	)	PUNCT
ap-4587	349	49	.	.	PUNCT
ap-4587	350	1	the	the	DET
ap-4587	350	2	matrices	matrix	NOUN
ap-4587	350	3	s	s	PART
ap-4587	350	4	,	,	PUNCT
ap-4587	350	5	t	t	PROPN
ap-4587	350	6	are	be	AUX
ap-4587	350	7	in	in	ADP
ap-4587	350	8	the	the	DET
ap-4587	350	9	form	form	NOUN
ap-4587	350	10	λr	λr	ADP
ap-4587	350	11	,	,	PUNCT
ap-4587	350	12	where	where	SCONJ
ap-4587	350	13	λ	λ	X
ap-4587	350	14	>	>	X
ap-4587	350	15	1	1	NUM
ap-4587	350	16	and	and	CCONJ
ap-4587	350	17	r	r	NOUN
ap-4587	350	18	is	be	AUX
ap-4587	350	19	an	an	DET
ap-4587	350	20	orthogonal	orthogonal	ADJ
ap-4587	350	21	matrix	matrix	NOUN
ap-4587	350	22	.	.	PUNCT
ap-4587	351	1	indeed	indeed	ADV
ap-4587	351	2	,	,	PUNCT
ap-4587	351	3	denote	denote	VERB
ap-4587	351	4	η	η	PROPN
ap-4587	351	5	the	the	DET
ap-4587	351	6	cyclotomic	cyclotomic	ADJ
ap-4587	351	7	integer	integer	NOUN
ap-4587	351	8	η	η	PROPN
ap-4587	351	9	=	=	PROPN
ap-4587	351	10	a	a	PROPN
ap-4587	351	11	+	+	NUM
ap-4587	351	12	bω	bω	NOUN
ap-4587	351	13	+	+	CCONJ
ap-4587	351	14	cω2	cω2	NOUN
ap-4587	351	15	+	+	CCONJ
ap-4587	351	16	dω4	dω4	X
ap-4587	351	17	and	and	CCONJ
ap-4587	351	18	find	find	VERB
ap-4587	351	19	its	its	PRON
ap-4587	351	20	goniometric	goniometric	NOUN
ap-4587	351	21	form	form	NOUN
ap-4587	351	22	η	η	NOUN
ap-4587	351	23	=	=	PRON
ap-4587	351	24	|η|(cosϕ	|η|(cosϕ	ADP
ap-4587	351	25	+	+	CCONJ
ap-4587	351	26	i	i	PRON
ap-4587	351	27	sinϕ	sinϕ	VERB
ap-4587	351	28	)	)	PUNCT
ap-4587	351	29	.	.	PUNCT
ap-4587	352	1	then	then	ADV
ap-4587	352	2	by	by	ADP
ap-4587	352	3	construction	construction	NOUN
ap-4587	352	4	,	,	PUNCT
ap-4587	352	5	s	s	PART
ap-4587	352	6	satisfies	satisfie	NOUN
ap-4587	352	7	s	s	PART
ap-4587	352	8	=	=	PUNCT
ap-4587	352	9	|η|	|η|	NOUN
ap-4587	352	10	(	(	PUNCT
ap-4587	352	11	cosϕ	cosϕ	ADJ
ap-4587	352	12	−	−	PROPN
ap-4587	352	13	sinϕ	sinϕ	NOUN
ap-4587	352	14	sinϕ	sinϕ	NOUN
ap-4587	352	15	cosϕ	cosϕ	NOUN
ap-4587	352	16	)	)	PUNCT
ap-4587	352	17	.	.	PUNCT
ap-4587	353	1	(	(	PUNCT
ap-4587	353	2	18	18	NUM
ap-4587	353	3	)	)	PUNCT
ap-4587	353	4	similarly	similarly	ADV
ap-4587	353	5	,	,	PUNCT
ap-4587	353	6	t	t	PROPN
ap-4587	353	7	=	=	PUNCT
ap-4587	353	8	|ν|	|ν|	ADV
ap-4587	353	9	(	(	PUNCT
ap-4587	353	10	cosψ	cosψ	NOUN
ap-4587	353	11	−	−	PROPN
ap-4587	353	12	sinψ	sinψ	PROPN
ap-4587	353	13	sinψ	sinψ	PROPN
ap-4587	353	14	cosψ	cosψ	PROPN
ap-4587	353	15	)	)	PUNCT
ap-4587	353	16	,	,	PUNCT
ap-4587	353	17	(	(	PUNCT
ap-4587	353	18	19	19	NUM
ap-4587	353	19	)	)	PUNCT
ap-4587	353	20	where	where	SCONJ
ap-4587	353	21	ν	ν	X
ap-4587	353	22	=	=	SYM
ap-4587	353	23	σ3(η	σ3(η	X
ap-4587	353	24	)	)	PUNCT
ap-4587	353	25	=	=	NOUN
ap-4587	354	1	|ν|(cosψ	|ν|(cosψ	VERB
ap-4587	355	1	+	+	CCONJ
ap-4587	355	2	i	i	PROPN
ap-4587	355	3	sinψ	sinψ	NOUN
ap-4587	355	4	)	)	PUNCT
ap-4587	355	5	.	.	PUNCT
ap-4587	356	1	we	we	PRON
ap-4587	356	2	thus	thus	ADV
ap-4587	356	3	see	see	VERB
ap-4587	356	4	that	that	SCONJ
ap-4587	356	5	our	our	PRON
ap-4587	356	6	original	original	ADJ
ap-4587	356	7	assumption	assumption	NOUN
ap-4587	356	8	on	on	ADP
ap-4587	356	9	the	the	DET
ap-4587	356	10	integer	integer	NOUN
ap-4587	356	11	matrix	matrix	NOUN
ap-4587	356	12	z	z	VERB
ap-4587	356	13	having	have	VERB
ap-4587	356	14	the	the	DET
ap-4587	356	15	same	same	ADJ
ap-4587	356	16	eigenvectors	eigenvector	NOUN
ap-4587	356	17	as	as	SCONJ
ap-4587	356	18	c	c	PROPN
ap-4587	356	19	leads	lead	VERB
ap-4587	356	20	to	to	ADP
ap-4587	356	21	self	self	NOUN
ap-4587	356	22	-	-	PUNCT
ap-4587	356	23	similarities	similarity	NOUN
ap-4587	356	24	s	s	NOUN
ap-4587	356	25	of	of	ADP
ap-4587	356	26	the	the	DET
ap-4587	356	27	cut	cut	NOUN
ap-4587	356	28	-	-	PUNCT
ap-4587	356	29	and	and	CCONJ
ap-4587	356	30	-	-	PUNCT
ap-4587	356	31	project	project	NOUN
ap-4587	356	32	scheme	scheme	NOUN
ap-4587	356	33	in	in	ADP
ap-4587	356	34	the	the	DET
ap-4587	356	35	form	form	NOUN
ap-4587	356	36	of	of	ADP
ap-4587	356	37	scaled	scale	VERB
ap-4587	356	38	rotations	rotation	NOUN
ap-4587	356	39	.	.	PUNCT
ap-4587	357	1	as	as	SCONJ
ap-4587	357	2	we	we	PRON
ap-4587	357	3	will	will	AUX
ap-4587	357	4	see	see	VERB
ap-4587	357	5	,	,	PUNCT
ap-4587	357	6	these	these	PRON
ap-4587	357	7	are	be	AUX
ap-4587	357	8	not	not	PART
ap-4587	357	9	the	the	DET
ap-4587	357	10	only	only	ADJ
ap-4587	357	11	self	self	NOUN
ap-4587	357	12	-	-	PUNCT
ap-4587	357	13	similarities	similarity	NOUN
ap-4587	357	14	of	of	ADP
ap-4587	357	15	the	the	DET
ap-4587	357	16	constructed	construct	VERB
ap-4587	357	17	cut	cut	NOUN
ap-4587	357	18	-	-	PUNCT
ap-4587	357	19	and	and	CCONJ
ap-4587	357	20	-	-	PUNCT
ap-4587	357	21	project	project	NOUN
ap-4587	357	22	scheme	scheme	NOUN
ap-4587	357	23	.	.	PUNCT
ap-4587	358	1	in	in	ADP
ap-4587	358	2	order	order	NOUN
ap-4587	358	3	to	to	PART
ap-4587	358	4	find	find	VERB
ap-4587	358	5	all	all	PRON
ap-4587	358	6	of	of	ADP
ap-4587	358	7	the	the	DET
ap-4587	358	8	self	self	NOUN
ap-4587	358	9	-	-	PUNCT
ap-4587	358	10	similarities	similarity	NOUN
ap-4587	358	11	,	,	PUNCT
ap-4587	358	12	we	we	PRON
ap-4587	358	13	relax	relax	VERB
ap-4587	358	14	the	the	DET
ap-4587	358	15	condition	condition	NOUN
ap-4587	358	16	on	on	ADP
ap-4587	358	17	eigenvectors	eigenvector	NOUN
ap-4587	358	18	of	of	ADP
ap-4587	358	19	z	z	NOUN
ap-4587	358	20	and	and	CCONJ
ap-4587	358	21	require	require	VERB
ap-4587	358	22	only	only	ADV
ap-4587	358	23	that	that	SCONJ
ap-4587	358	24	c	c	PROPN
ap-4587	358	25	and	and	CCONJ
ap-4587	358	26	z	z	PROPN
ap-4587	358	27	have	have	VERB
ap-4587	358	28	the	the	DET
ap-4587	358	29	same	same	ADJ
ap-4587	358	30	invariant	invariant	ADJ
ap-4587	358	31	subspaces	subspace	NOUN
ap-4587	358	32	of	of	ADP
ap-4587	358	33	dimension	dimension	NOUN
ap-4587	358	34	2	2	NUM
ap-4587	358	35	.	.	PUNCT
ap-4587	358	36	with	with	ADP
ap-4587	358	37	this	this	PRON
ap-4587	358	38	,	,	PUNCT
ap-4587	358	39	we	we	PRON
ap-4587	358	40	obtain	obtain	VERB
ap-4587	358	41	the	the	DET
ap-4587	358	42	following	follow	VERB
ap-4587	358	43	proposition	proposition	NOUN
ap-4587	358	44	.	.	PUNCT
ap-4587	359	1	in	in	ADP
ap-4587	359	2	order	order	NOUN
ap-4587	359	3	to	to	PART
ap-4587	359	4	formulate	formulate	VERB
ap-4587	359	5	the	the	DET
ap-4587	359	6	statement	statement	NOUN
ap-4587	359	7	,	,	PUNCT
ap-4587	359	8	define	define	VERB
ap-4587	359	9	z	z	NOUN
ap-4587	359	10	:	:	PUNCT
ap-4587	359	11	=	=	SYM
ap-4587	359	12	{	{	PUNCT
ap-4587	359	13	z	z	NOUN
ap-4587	359	14	∈	∈	PROPN
ap-4587	359	15	z4×4	z4×4	NOUN
ap-4587	359	16	:	:	PUNCT
ap-4587	360	1	∃s	∃s	PROPN
ap-4587	360	2	,	,	PUNCT
ap-4587	360	3	t	t	PROPN
ap-4587	360	4	∈	∈	PROPN
ap-4587	360	5	r2×2	r2×2	PROPN
ap-4587	360	6	,	,	PUNCT
ap-4587	360	7	v	v	NOUN
ap-4587	360	8	zv	zv	NUM
ap-4587	360	9	−1	−1	NOUN
ap-4587	360	10	=	=	PUNCT
ap-4587	360	11	(	(	PUNCT
ap-4587	360	12	s	s	NOUN
ap-4587	360	13	o	o	X
ap-4587	360	14	o	o	X
ap-4587	360	15	t	t	NOUN
ap-4587	360	16	)	)	PUNCT
ap-4587	360	17	}	}	PUNCT
ap-4587	360	18	,	,	PUNCT
ap-4587	360	19	(	(	PUNCT
ap-4587	360	20	20	20	NUM
ap-4587	360	21	)	)	PUNCT
ap-4587	360	22	where	where	SCONJ
ap-4587	360	23	v	v	NOUN
ap-4587	360	24	is	be	AUX
ap-4587	360	25	the	the	DET
ap-4587	360	26	matrix	matrix	NOUN
ap-4587	360	27	defining	define	VERB
ap-4587	360	28	the	the	DET
ap-4587	360	29	lattice	lattice	ADJ
ap-4587	360	30	l	l	NOUN
ap-4587	360	31	of	of	ADP
ap-4587	360	32	the	the	DET
ap-4587	360	33	cut	cut	NOUN
ap-4587	360	34	-	-	PUNCT
ap-4587	360	35	and	and	CCONJ
ap-4587	360	36	-	-	PUNCT
ap-4587	360	37	project	project	NOUN
ap-4587	360	38	scheme	scheme	NOUN
ap-4587	360	39	λ	λ	PROPN
ap-4587	360	40	.	.	PUNCT
ap-4587	360	41	proposition	proposition	NOUN
ap-4587	360	42	6.2	6.2	NUM
ap-4587	360	43	.	.	PUNCT
ap-4587	361	1	for	for	ADP
ap-4587	361	2	integer	integer	NOUN
ap-4587	361	3	a	a	DET
ap-4587	361	4	,	,	PUNCT
ap-4587	361	5	b	b	NOUN
ap-4587	361	6	,	,	PUNCT
ap-4587	361	7	.	.	PUNCT
ap-4587	361	8	.	.	PUNCT
ap-4587	361	9	.	.	PUNCT
ap-4587	362	1	,	,	PUNCT
ap-4587	362	2	h	h	PROPN
ap-4587	362	3	∈	∈	PROPN
ap-4587	362	4	z	z	AUX
ap-4587	362	5	denote	denote	VERB
ap-4587	362	6	za	za	PROPN
ap-4587	362	7	,	,	PUNCT
ap-4587	362	8	b,	b,	PRON
ap-4587	362	9	...	...	PUNCT
ap-4587	362	10	,h	,h	PUNCT
ap-4587	362	11	:	:	PUNCT
ap-4587	362	12	=	=	SYM
ap-4587	362	13			VERB
ap-4587	362	14	a	a	DET
ap-4587	362	15	b	b	NOUN
ap-4587	362	16	c	c	NOUN
ap-4587	362	17	d	d	X
ap-4587	362	18	e	e	X
ap-4587	362	19	f	f	PROPN
ap-4587	362	20	g	g	PROPN
ap-4587	362	21	h	h	PROPN
ap-4587	362	22	−a−e+f−h	−a−e+f−h	PROPN
ap-4587	362	23	−b+g−h	−b+g−h	PROPN
ap-4587	362	24	−c−e+f	−c−e+f	PRON
ap-4587	362	25	−d−e+g−h	−d−e+g−h	VERB
ap-4587	362	26	a−b+d+e−f+h	a−b+d+e−f+h	PRON
ap-4587	362	27	−c+d−g+h	−c+d−g+h	PROPN
ap-4587	363	1	a−b+e−f	a−b+e−f	VERB
ap-4587	363	2	a−c+d+e−g+h	a−c+d+e−g+h	NUM
ap-4587	363	3			NOUN
ap-4587	363	4	.	.	PUNCT
ap-4587	364	1	then	then	ADV
ap-4587	364	2	z	z	X
ap-4587	364	3	=	=	SYM
ap-4587	364	4	{	{	PUNCT
ap-4587	364	5	za	za	PROPN
ap-4587	364	6	,	,	PUNCT
ap-4587	364	7	b,	b,	PRON
ap-4587	364	8	...	...	PUNCT
ap-4587	364	9	,h	,h	PUNCT
ap-4587	364	10	:	:	PUNCT
ap-4587	364	11	a	a	DET
ap-4587	364	12	,	,	PUNCT
ap-4587	364	13	b	b	NOUN
ap-4587	364	14	,	,	PUNCT
ap-4587	364	15	.	.	PUNCT
ap-4587	364	16	.	.	PUNCT
ap-4587	364	17	.	.	PUNCT
ap-4587	365	1	,	,	PUNCT
ap-4587	365	2	h	h	NOUN
ap-4587	365	3	∈	∈	PROPN
ap-4587	365	4	z	z	X
ap-4587	365	5	}	}	PUNCT
ap-4587	365	6	.	.	PUNCT
ap-4587	366	1	proof	proof	NOUN
ap-4587	366	2	.	.	PUNCT
ap-4587	367	1	recalling	recall	VERB
ap-4587	367	2	the	the	DET
ap-4587	367	3	definition	definition	NOUN
ap-4587	367	4	of	of	ADP
ap-4587	367	5	v	v	NOUN
ap-4587	367	6	in	in	ADP
ap-4587	367	7	(	(	PUNCT
ap-4587	367	8	9	9	NUM
ap-4587	367	9	)	)	PUNCT
ap-4587	367	10	,	,	PUNCT
ap-4587	367	11	we	we	PRON
ap-4587	367	12	rewrite	rewrite	VERB
ap-4587	367	13	the	the	DET
ap-4587	367	14	requirement	requirement	NOUN
ap-4587	367	15	(	(	PUNCT
ap-4587	367	16	s	s	NOUN
ap-4587	367	17	o	o	X
ap-4587	367	18	o	o	X
ap-4587	367	19	t	t	NOUN
ap-4587	367	20	)	)	PUNCT
ap-4587	368	1	=	=	PUNCT
ap-4587	368	2	v	v	X
ap-4587	368	3	zv	zv	NUM
ap-4587	368	4	−1	−1	NOUN
ap-4587	368	5	,	,	PUNCT
ap-4587	368	6	for	for	ADP
ap-4587	368	7	some	some	DET
ap-4587	368	8	matrices	matrix	NOUN
ap-4587	368	9	s	s	PROPN
ap-4587	368	10	,	,	PUNCT
ap-4587	368	11	t	t	PROPN
ap-4587	368	12	∈	∈	PROPN
ap-4587	368	13	r2×2	r2×2	PROPN
ap-4587	368	14	by	by	ADP
ap-4587	368	15	p	p	PROPN
ap-4587	368	16	(	(	PUNCT
ap-4587	368	17	s	s	NOUN
ap-4587	368	18	o	o	X
ap-4587	368	19	o	o	X
ap-4587	368	20	t	t	NOUN
ap-4587	368	21	)	)	PUNCT
ap-4587	369	1	p−1	p−1	PROPN
ap-4587	369	2	=	=	PUNCT
ap-4587	369	3	y	y	PROPN
ap-4587	369	4	−1zy	−1zy	NUM
ap-4587	369	5	⇐	⇐	ADJ
ap-4587	369	6	⇒	⇒	PROPN
ap-4587	369	7	y	y	X
ap-4587	369	8	p	p	X
ap-4587	369	9	(	(	PUNCT
ap-4587	369	10	s	s	NOUN
ap-4587	369	11	o	o	X
ap-4587	369	12	o	o	X
ap-4587	369	13	t	t	NOUN
ap-4587	369	14	)	)	PUNCT
ap-4587	370	1	p−1	p−1	PROPN
ap-4587	370	2	=	=	PUNCT
ap-4587	370	3	zy	zy	PROPN
ap-4587	370	4	.	.	PUNCT
ap-4587	370	5	(	(	PUNCT
ap-4587	370	6	21	21	NUM
ap-4587	370	7	)	)	PUNCT
ap-4587	370	8	since	since	SCONJ
ap-4587	370	9	also	also	ADV
ap-4587	370	10	p	p	PROPN
ap-4587	370	11	and	and	CCONJ
ap-4587	370	12	p−1	p−1	PROPN
ap-4587	370	13	are	be	AUX
ap-4587	370	14	block	block	NOUN
ap-4587	370	15	diagonal	diagonal	ADJ
ap-4587	370	16	,	,	PUNCT
ap-4587	370	17	the	the	DET
ap-4587	370	18	latter	latter	NOUN
ap-4587	370	19	represents	represent	VERB
ap-4587	370	20	the	the	DET
ap-4587	370	21	requirement	requirement	NOUN
ap-4587	370	22	that	that	SCONJ
ap-4587	370	23	the	the	DET
ap-4587	370	24	matrix	matrix	NOUN
ap-4587	370	25	c	c	NOUN
ap-4587	370	26	′	′	PROPN
ap-4587	370	27	has	have	AUX
ap-4587	370	28	two	two	NUM
ap-4587	370	29	invariant	invariant	ADJ
ap-4587	370	30	subspaces	subspace	NOUN
ap-4587	370	31	,	,	PUNCT
ap-4587	370	32	namely	namely	ADV
ap-4587	370	33	spanr{y1,y2	spanr{y1,y2	NOUN
ap-4587	370	34	}	}	PUNCT
ap-4587	370	35	and	and	CCONJ
ap-4587	370	36	spanr{y3,y4	spanr{y3,y4	NOUN
ap-4587	370	37	}	}	PUNCT
ap-4587	370	38	.	.	PUNCT
ap-4587	371	1	stated	state	VERB
ap-4587	371	2	otherwise	otherwise	ADV
ap-4587	371	3	,	,	PUNCT
ap-4587	371	4	we	we	PRON
ap-4587	371	5	require	require	VERB
ap-4587	371	6	existence	existence	NOUN
ap-4587	371	7	of	of	ADP
ap-4587	371	8	complex	complex	ADJ
ap-4587	371	9	coefficients	coefficient	NOUN
ap-4587	371	10	µ	µ	NUM
ap-4587	371	11	,	,	PUNCT
ap-4587	371	12	µ′	µ′	NUM
ap-4587	371	13	,	,	PUNCT
ap-4587	371	14	ν	ν	NOUN
ap-4587	371	15	,	,	PUNCT
ap-4587	371	16	ν′	ν′	NOUN
ap-4587	371	17	,	,	PUNCT
ap-4587	371	18	ζ	ζ	NOUN
ap-4587	371	19	,	,	PUNCT
ap-4587	371	20	ζ	ζ	PROPN
ap-4587	371	21	′	′	NUM
ap-4587	371	22	,	,	PUNCT
ap-4587	371	23	η	η	PROPN
ap-4587	371	24	,	,	PUNCT
ap-4587	371	25	η′	η′	PRON
ap-4587	372	1	such	such	ADJ
ap-4587	372	2	that	that	DET
ap-4587	372	3	zy1	zy1	NOUN
ap-4587	372	4	=	=	PUNCT
ap-4587	372	5	µy1	µy1	PROPN
ap-4587	372	6	+	+	CCONJ
ap-4587	372	7	νy2	νy2	NOUN
ap-4587	372	8	,	,	PUNCT
ap-4587	372	9	zy2	zy2	X
ap-4587	372	10	=	=	PUNCT
ap-4587	372	11	µ′y1	µ′y1	NOUN
ap-4587	372	12	+	+	CCONJ
ap-4587	372	13	ν′y2	ν′y2	PROPN
ap-4587	372	14	,	,	PUNCT
ap-4587	372	15	zy3	zy3	X
ap-4587	372	16	=	=	PUNCT
ap-4587	372	17	ζy3	ζy3	PROPN
ap-4587	372	18	+	+	CCONJ
ap-4587	372	19	ηy4	ηy4	ADJ
ap-4587	372	20	,	,	PUNCT
ap-4587	372	21	zy4	zy4	NOUN
ap-4587	372	22	=	=	SYM
ap-4587	372	23	ζ	ζ	NOUN
ap-4587	372	24	′y3	′y3	NOUN
ap-4587	373	1	+	+	CCONJ
ap-4587	373	2	η′y4	η′y4	NOUN
ap-4587	373	3	.	.	PUNCT
ap-4587	374	1	(	(	PUNCT
ap-4587	374	2	22	22	NUM
ap-4587	374	3	)	)	PUNCT
ap-4587	374	4	439	439	NUM
ap-4587	374	5	zuzana	zuzana	PROPN
ap-4587	374	6	masáková	masáková	PROPN
ap-4587	374	7	,	,	PUNCT
ap-4587	374	8	jan	jan	PROPN
ap-4587	374	9	mazáč	mazáč	PROPN
ap-4587	374	10	acta	acta	PROPN
ap-4587	374	11	polytechnica	polytechnica	PROPN
ap-4587	374	12	since	since	SCONJ
ap-4587	374	13	y1	y1	NOUN
ap-4587	374	14	=	=	PUNCT
ap-4587	374	15	y2	y2	NOUN
ap-4587	374	16	and	and	CCONJ
ap-4587	374	17	y3	y3	NOUN
ap-4587	374	18	=	=	PUNCT
ap-4587	374	19	y4	y4	PROPN
ap-4587	374	20	,	,	PUNCT
ap-4587	374	21	it	it	PRON
ap-4587	374	22	follows	follow	VERB
ap-4587	374	23	that	that	SCONJ
ap-4587	374	24	µ′	µ′	PUNCT
ap-4587	374	25	=	=	SYM
ap-4587	374	26	ν	ν	NOUN
ap-4587	374	27	,	,	PUNCT
ap-4587	374	28	ζ	ζ	NOUN
ap-4587	374	29	′	′	NUM
ap-4587	374	30	=	=	SYM
ap-4587	374	31	η	η	PROPN
ap-4587	374	32	,	,	PUNCT
ap-4587	374	33	ν	ν	X
ap-4587	374	34	=	=	SYM
ap-4587	374	35	µ	µ	NUM
ap-4587	374	36	,	,	PUNCT
ap-4587	374	37	η′	η′	PROPN
ap-4587	374	38	=	=	SYM
ap-4587	374	39	ζ	ζ	X
ap-4587	374	40	.	.	PUNCT
ap-4587	374	41	consider	consider	VERB
ap-4587	374	42	a	a	DET
ap-4587	374	43	general	general	ADJ
ap-4587	374	44	integer	integer	NOUN
ap-4587	374	45	matrix	matrix	NOUN
ap-4587	374	46	z	z	PROPN
ap-4587	374	47	∈	∈	PROPN
ap-4587	374	48	z4×4	z4×4	NOUN
ap-4587	374	49	:	:	PUNCT
ap-4587	374	50	z	z	X
ap-4587	374	51	=	=	SYM
ap-4587	374	52			PROPN
ap-4587	374	53	a	a	DET
ap-4587	374	54	b	b	NOUN
ap-4587	374	55	c	c	NOUN
ap-4587	374	56	d	d	X
ap-4587	374	57	e	e	X
ap-4587	374	58	f	f	PROPN
ap-4587	374	59	g	g	PROPN
ap-4587	374	60	h	h	PROPN
ap-4587	375	1	j	j	PROPN
ap-4587	376	1	k	k	PROPN
ap-4587	377	1	l	l	PROPN
ap-4587	378	1	m	m	VERB
ap-4587	378	2	n	n	ADV
ap-4587	378	3	o	o	NOUN
ap-4587	378	4	p	p	X
ap-4587	378	5	q	q	NOUN
ap-4587	378	6			NOUN
ap-4587	378	7	.	.	PUNCT
ap-4587	379	1	conditions	condition	NOUN
ap-4587	379	2	(	(	PUNCT
ap-4587	379	3	22	22	NUM
ap-4587	379	4	)	)	PUNCT
ap-4587	379	5	are	be	AUX
ap-4587	379	6	conveniently	conveniently	ADV
ap-4587	379	7	rewritten	rewrite	VERB
ap-4587	379	8	as	as	ADP
ap-4587	379	9	(	(	PUNCT
ap-4587	379	10	a	a	DET
ap-4587	379	11	b	b	NOUN
ap-4587	379	12	c	c	NOUN
ap-4587	379	13	d	d	X
ap-4587	379	14	e	e	X
ap-4587	379	15	f	f	PROPN
ap-4587	379	16	g	g	PROPN
ap-4587	379	17	h	h	PROPN
ap-4587	379	18	)	)	PUNCT
ap-4587	379	19	︸	︸	NOUN
ap-4587	380	1	︷︷	︷︷	NOUN
ap-4587	380	2	︸	︸	PUNCT
ap-4587	380	3	zu	zu	PROPN
ap-4587	380	4			PROPN
ap-4587	380	5	1	1	NUM
ap-4587	380	6	ω	ω	NUM
ap-4587	380	7	ω2	ω2	ADJ
ap-4587	380	8	ω3	ω3	ADJ
ap-4587	380	9			NOUN
ap-4587	380	10	=	=	PUNCT
ap-4587	380	11	(	(	PUNCT
ap-4587	380	12	1	1	NUM
ap-4587	380	13	1	1	NUM
ap-4587	380	14	ω	ω	NUM
ap-4587	380	15	ω4	ω4	NUM
ap-4587	380	16	)	)	PUNCT
ap-4587	380	17	︸	︸	X
ap-4587	381	1	︷︷	︷︷	PROPN
ap-4587	381	2	︸	︸	X
ap-4587	381	3	y	y	PROPN
ap-4587	381	4	(	(	PUNCT
ap-4587	381	5	1	1	X
ap-4587	381	6	)	)	PUNCT
ap-4587	381	7	u	u	NOUN
ap-4587	381	8	(	(	PUNCT
ap-4587	381	9	µ	µ	X
ap-4587	381	10	ν	ν	NOUN
ap-4587	381	11	)	)	PUNCT
ap-4587	381	12	,	,	PUNCT
ap-4587	381	13	(	(	PUNCT
ap-4587	381	14	j	j	PROPN
ap-4587	381	15	k	k	PROPN
ap-4587	381	16	l	l	PROPN
ap-4587	381	17	m	m	VERB
ap-4587	381	18	n	n	ADV
ap-4587	381	19	o	o	NOUN
ap-4587	381	20	p	p	X
ap-4587	381	21	q	q	X
ap-4587	381	22	)	)	PUNCT
ap-4587	381	23	︸	︸	NOUN
ap-4587	381	24	︷︷	︷︷	PROPN
ap-4587	381	25	︸	︸	X
ap-4587	381	26	zd	zd	PROPN
ap-4587	381	27			PROPN
ap-4587	381	28	1	1	NUM
ap-4587	381	29	ω	ω	NUM
ap-4587	381	30	ω2	ω2	ADJ
ap-4587	381	31	ω3	ω3	ADJ
ap-4587	381	32			NOUN
ap-4587	381	33	=	=	PUNCT
ap-4587	381	34	(	(	PUNCT
ap-4587	381	35	ω2	ω2	ADJ
ap-4587	381	36	ω3	ω3	PROPN
ap-4587	381	37	ω3	ω3	PROPN
ap-4587	381	38	ω2	ω2	ADJ
ap-4587	381	39	)	)	PUNCT
ap-4587	381	40	︸	︸	X
ap-4587	382	1	︷︷	︷︷	PROPN
ap-4587	382	2	︸	︸	X
ap-4587	382	3	y	y	PROPN
ap-4587	382	4	(	(	PUNCT
ap-4587	382	5	1	1	NUM
ap-4587	382	6	)	)	PUNCT
ap-4587	382	7	d	d	NOUN
ap-4587	382	8	(	(	PUNCT
ap-4587	382	9	µ	µ	X
ap-4587	382	10	ν	ν	NOUN
ap-4587	382	11	)	)	PUNCT
ap-4587	382	12	.	.	PUNCT
ap-4587	383	1	(	(	PUNCT
ap-4587	383	2	23	23	NUM
ap-4587	383	3	)	)	PUNCT
ap-4587	383	4	excluding	exclude	VERB
ap-4587	383	5	parameters	parameter	NOUN
ap-4587	383	6	µ	µ	NUM
ap-4587	383	7	,	,	PUNCT
ap-4587	383	8	ν	ν	PROPN
ap-4587	383	9	,	,	PUNCT
ap-4587	383	10	we	we	PRON
ap-4587	383	11	obtain	obtain	VERB
ap-4587	383	12	relation	relation	NOUN
ap-4587	383	13	between	between	ADP
ap-4587	383	14	coefficients	coefficient	NOUN
ap-4587	383	15	of	of	ADP
ap-4587	383	16	matrices	matrix	NOUN
ap-4587	383	17	fh	fh	PROPN
ap-4587	383	18	and	and	CCONJ
ap-4587	383	19	fd	fd	PROPN
ap-4587	383	20	,	,	PUNCT
ap-4587	383	21	y	y	PROPN
ap-4587	383	22	(	(	PUNCT
ap-4587	383	23	1	1	NUM
ap-4587	383	24	)	)	PUNCT
ap-4587	383	25	d	d	PROPN
ap-4587	383	26	y	y	PROPN
ap-4587	383	27	(	(	PUNCT
ap-4587	383	28	1	1	X
ap-4587	383	29	)	)	PUNCT
ap-4587	383	30	u	u	NOUN
ap-4587	383	31	−1	−1	NOUN
ap-4587	383	32	zuy1	zuy1	NOUN
ap-4587	383	33	=	=	SYM
ap-4587	383	34	zdy1	zdy1	PROPN
ap-4587	383	35	,	,	PUNCT
ap-4587	383	36	1	1	NUM
ap-4587	383	37	ω4	ω4	NUM
ap-4587	383	38	−	−	PROPN
ap-4587	383	39	ω	ω	PROPN
ap-4587	383	40	(	(	PUNCT
ap-4587	383	41	ω2	ω2	PROPN
ap-4587	383	42	ω3	ω3	PROPN
ap-4587	383	43	ω3	ω3	PROPN
ap-4587	383	44	ω2	ω2	PROPN
ap-4587	383	45	)	)	PUNCT
ap-4587	383	46	(	(	PUNCT
ap-4587	383	47	ω4	ω4	NUM
ap-4587	383	48	−1	−1	NOUN
ap-4587	383	49	−ω	−ω	ADJ
ap-4587	383	50	1	1	NUM
ap-4587	383	51	)	)	PUNCT
ap-4587	383	52	(	(	PUNCT
ap-4587	383	53	a	a	DET
ap-4587	383	54	b	b	X
ap-4587	383	55	c	c	NOUN
ap-4587	383	56	d	d	X
ap-4587	383	57	e	e	X
ap-4587	383	58	f	f	PROPN
ap-4587	383	59	g	g	PROPN
ap-4587	383	60	h	h	PROPN
ap-4587	383	61	)	)	PUNCT
ap-4587	383	62			PROPN
ap-4587	383	63	1	1	NUM
ap-4587	383	64	ω	ω	NUM
ap-4587	383	65	ω2	ω2	ADJ
ap-4587	383	66	ω3	ω3	ADJ
ap-4587	383	67			NOUN
ap-4587	384	1	=	=	PUNCT
ap-4587	384	2	(	(	PUNCT
ap-4587	384	3	j	j	PROPN
ap-4587	384	4	k	k	PROPN
ap-4587	384	5	l	l	PROPN
ap-4587	384	6	m	m	VERB
ap-4587	384	7	n	n	ADV
ap-4587	384	8	o	o	NOUN
ap-4587	384	9	p	p	X
ap-4587	384	10	q	q	NOUN
ap-4587	384	11	)	)	PUNCT
ap-4587	384	12			PROPN
ap-4587	384	13	1	1	NUM
ap-4587	384	14	ω	ω	NUM
ap-4587	384	15	ω2	ω2	ADJ
ap-4587	384	16	ω3	ω3	ADJ
ap-4587	384	17			NOUN
ap-4587	384	18	,	,	PUNCT
ap-4587	384	19	(	(	PUNCT
ap-4587	384	20	−1	−1	NOUN
ap-4587	384	21	ω	ω	NOUN
ap-4587	384	22	+	+	CCONJ
ap-4587	384	23	ω4	ω4	NUM
ap-4587	384	24	−ω	−ω	ADJ
ap-4587	384	25	−	−	PROPN
ap-4587	384	26	ω4	ω4	NUM
ap-4587	384	27	−ω	−ω	ADJ
ap-4587	384	28	−	−	PROPN
ap-4587	384	29	ω4	ω4	NUM
ap-4587	384	30	)	)	PUNCT
ap-4587	384	31	(	(	PUNCT
ap-4587	384	32	a+	a+	X
ap-4587	384	33	bω	bω	NOUN
ap-4587	384	34	+	+	CCONJ
ap-4587	384	35	cω2	cω2	NOUN
ap-4587	384	36	+	+	CCONJ
ap-4587	384	37	dω3	dω3	VERB
ap-4587	384	38	e+	e+	ADJ
ap-4587	384	39	fω	fω	X
ap-4587	384	40	+	+	CCONJ
ap-4587	384	41	gω2	gω2	PROPN
ap-4587	384	42	+	+	CCONJ
ap-4587	384	43	hω3	hω3	NOUN
ap-4587	384	44	)	)	PUNCT
ap-4587	384	45	=	=	PRON
ap-4587	385	1	(	(	PUNCT
ap-4587	385	2	j	j	PROPN
ap-4587	385	3	+	+	CCONJ
ap-4587	385	4	kω	kω	X
ap-4587	385	5	+	+	CCONJ
ap-4587	385	6	lω2	lω2	PROPN
ap-4587	386	1	+	+	PROPN
ap-4587	386	2	mω3	mω3	NOUN
ap-4587	386	3	n+	n+	NOUN
ap-4587	386	4	oω	oω	NOUN
ap-4587	386	5	+	+	CCONJ
ap-4587	386	6	pω2	pω2	NOUN
ap-4587	386	7	+	+	CCONJ
ap-4587	386	8	qω3	qω3	NOUN
ap-4587	386	9	)	)	PUNCT
ap-4587	386	10	,	,	PUNCT
ap-4587	386	11			DET
ap-4587	386	12	−a−	−a−	NOUN
ap-4587	386	13	e+	e+	PUNCT
ap-4587	386	14	f	f	PROPN
ap-4587	386	15	−	−	PROPN
ap-4587	386	16	h+	h+	PUNCT
ap-4587	386	17	ω(−b+	ω(−b+	VERB
ap-4587	386	18	g	g	PROPN
ap-4587	386	19	−	−	PROPN
ap-4587	386	20	h	h	PROPN
ap-4587	386	21	)	)	PUNCT
ap-4587	386	22	−	−	PROPN
ap-4587	387	1	a−	a−	PROPN
ap-4587	387	2	e+	e+	VERB
ap-4587	387	3	f	f	PROPN
ap-4587	387	4	−	−	PROPN
ap-4587	387	5	h+	h+	PUNCT
ap-4587	387	6	ω(−b+	ω(−b+	VERB
ap-4587	387	7	g	g	PROPN
ap-4587	387	8	−	−	PROPN
ap-4587	387	9	h	h	NOUN
ap-4587	387	10	)	)	PUNCT
ap-4587	387	11	a−	a−	PROPN
ap-4587	387	12	b+	b+	X
ap-4587	387	13	d+	d+	NOUN
ap-4587	387	14	e−	e−	PROPN
ap-4587	387	15	f	f	PROPN
ap-4587	387	16	+	+	CCONJ
ap-4587	387	17	h+	h+	X
ap-4587	387	18	ω(−c+	ω(−c+	NOUN
ap-4587	387	19	d−	d−	PROPN
ap-4587	387	20	g	g	PROPN
ap-4587	387	21	+	+	CCONJ
ap-4587	387	22	h	h	NOUN
ap-4587	387	23	)	)	PUNCT
ap-4587	387	24	a−	a−	PROPN
ap-4587	387	25	b+	b+	X
ap-4587	387	26	d+	d+	NOUN
ap-4587	387	27	e−	e−	PROPN
ap-4587	387	28	f	f	PROPN
ap-4587	387	29	+	+	CCONJ
ap-4587	387	30	h+	h+	X
ap-4587	387	31	ω(−c+	ω(−c+	NOUN
ap-4587	387	32	d−	d−	PROPN
ap-4587	387	33	g	g	PROPN
ap-4587	387	34	+	+	PROPN
ap-4587	387	35	h	h	NOUN
ap-4587	387	36	)	)	PUNCT
ap-4587	387	37			NOUN
ap-4587	387	38	=	=	SYM
ap-4587	388	1	(	(	PUNCT
ap-4587	388	2	j	j	PROPN
ap-4587	388	3	+	+	CCONJ
ap-4587	388	4	kω	kω	X
ap-4587	388	5	+	+	CCONJ
ap-4587	388	6	lω2	lω2	PROPN
ap-4587	389	1	+	+	PROPN
ap-4587	389	2	mω3	mω3	NOUN
ap-4587	389	3	n+	n+	NOUN
ap-4587	389	4	oω	oω	NOUN
ap-4587	389	5	+	+	CCONJ
ap-4587	389	6	pω2	pω2	NOUN
ap-4587	389	7	+	+	CCONJ
ap-4587	389	8	qω3	qω3	NOUN
ap-4587	389	9	)	)	PUNCT
ap-4587	389	10	.	.	PUNCT
ap-4587	390	1	since	since	SCONJ
ap-4587	390	2	the	the	DET
ap-4587	390	3	entries	entry	NOUN
ap-4587	390	4	are	be	AUX
ap-4587	390	5	–	–	PUNCT
ap-4587	390	6	as	as	ADP
ap-4587	390	7	elements	element	NOUN
ap-4587	390	8	of	of	ADP
ap-4587	390	9	the	the	DET
ap-4587	390	10	cyclotomic	cyclotomic	ADJ
ap-4587	390	11	field	field	NOUN
ap-4587	390	12	q(ω	q(ω	NOUN
ap-4587	390	13	)	)	PUNCT
ap-4587	390	14	–	–	PUNCT
ap-4587	390	15	uniquely	uniquely	ADV
ap-4587	390	16	written	write	VERB
ap-4587	390	17	as	as	ADP
ap-4587	390	18	a	a	DET
ap-4587	390	19	rational	rational	ADJ
ap-4587	390	20	combination	combination	NOUN
ap-4587	390	21	of	of	ADP
ap-4587	390	22	1	1	NUM
ap-4587	390	23	,	,	PUNCT
ap-4587	390	24	ω	ω	PROPN
ap-4587	390	25	,	,	PUNCT
ap-4587	390	26	ω2	ω2	ADJ
ap-4587	390	27	,	,	PUNCT
ap-4587	390	28	ω3	ω3	PROPN
ap-4587	390	29	,	,	PUNCT
ap-4587	390	30	we	we	PRON
ap-4587	390	31	find	find	VERB
ap-4587	390	32	expression	expression	NOUN
ap-4587	390	33	for	for	ADP
ap-4587	390	34	j	j	PROPN
ap-4587	390	35	,	,	PUNCT
ap-4587	390	36	.	.	PUNCT
ap-4587	390	37	.	.	PUNCT
ap-4587	391	1	.	.	PUNCT
ap-4587	392	1	,	,	PUNCT
ap-4587	392	2	q	q	X
ap-4587	392	3	in	in	ADP
ap-4587	392	4	terms	term	NOUN
ap-4587	392	5	of	of	ADP
ap-4587	392	6	a	a	PRON
ap-4587	392	7	,	,	PUNCT
ap-4587	392	8	.	.	PUNCT
ap-4587	392	9	.	.	PUNCT
ap-4587	393	1	.	.	PUNCT
ap-4587	394	1	,	,	PUNCT
ap-4587	394	2	h.	h.	PROPN
ap-4587	394	3	the	the	DET
ap-4587	394	4	matrix	matrix	NOUN
ap-4587	394	5	z	z	PUNCT
ap-4587	394	6	thus	thus	ADV
ap-4587	394	7	has	have	VERB
ap-4587	394	8	only	only	ADV
ap-4587	394	9	eight	eight	NUM
ap-4587	394	10	independent	independent	ADJ
ap-4587	394	11	integer	integer	NOUN
ap-4587	394	12	parameters	parameter	NOUN
ap-4587	394	13	,	,	PUNCT
ap-4587	394	14	z	z	X
ap-4587	394	15	=	=	SYM
ap-4587	394	16			PROPN
ap-4587	394	17	a	a	DET
ap-4587	394	18	b	b	NOUN
ap-4587	394	19	c	c	NOUN
ap-4587	394	20	d	d	X
ap-4587	394	21	e	e	X
ap-4587	394	22	f	f	PROPN
ap-4587	394	23	g	g	PROPN
ap-4587	394	24	h	h	PROPN
ap-4587	394	25	−a−e+f−h	−a−e+f−h	PROPN
ap-4587	394	26	−b+g−h	−b+g−h	PROPN
ap-4587	394	27	−c−e+f	−c−e+f	PRON
ap-4587	394	28	−d−e+g−h	−d−e+g−h	VERB
ap-4587	394	29	a−b+d+e−f+h	a−b+d+e−f+h	PRON
ap-4587	394	30	−c+d−g+h	−c+d−g+h	PROPN
ap-4587	395	1	a−b+e−f	a−b+e−f	VERB
ap-4587	395	2	a−c+d+e−g+h	a−c+d+e−g+h	NUM
ap-4587	395	3			NOUN
ap-4587	395	4	.	.	PUNCT
ap-4587	396	1	(	(	PUNCT
ap-4587	396	2	24	24	NUM
ap-4587	396	3	)	)	PUNCT
ap-4587	396	4	as	as	ADP
ap-4587	396	5	µ	µ	NUM
ap-4587	396	6	,	,	PUNCT
ap-4587	396	7	ν	ν	PROPN
ap-4587	396	8	∈	∈	PROPN
ap-4587	396	9	q(ω	q(ω	PROPN
ap-4587	396	10	)	)	PUNCT
ap-4587	396	11	,	,	PUNCT
ap-4587	396	12	relations	relation	NOUN
ap-4587	396	13	(	(	PUNCT
ap-4587	396	14	22	22	NUM
ap-4587	396	15	)	)	PUNCT
ap-4587	396	16	are	be	AUX
ap-4587	396	17	obtained	obtain	VERB
ap-4587	396	18	from	from	ADP
ap-4587	396	19	the	the	DET
ap-4587	396	20	first	first	ADJ
ap-4587	396	21	of	of	ADP
ap-4587	396	22	them	they	PRON
ap-4587	396	23	by	by	ADP
ap-4587	396	24	application	application	NOUN
ap-4587	396	25	of	of	ADP
ap-4587	396	26	the	the	DET
ap-4587	396	27	field	field	NOUN
ap-4587	396	28	automorphisms	automorphism	NOUN
ap-4587	396	29	.	.	PUNCT
ap-4587	397	1	therefore	therefore	ADV
ap-4587	397	2	µ′	µ′	PUNCT
ap-4587	397	3	=	=	PUNCT
ap-4587	397	4	σ4(µ	σ4(µ	NOUN
ap-4587	397	5	)	)	PUNCT
ap-4587	397	6	,	,	PUNCT
ap-4587	397	7	ν′	ν′	NOUN
ap-4587	397	8	=	=	SYM
ap-4587	397	9	σ4(ν	σ4(ν	NUM
ap-4587	397	10	)	)	PUNCT
ap-4587	397	11	,	,	PUNCT
ap-4587	397	12	ζ	ζ	NOUN
ap-4587	397	13	=	=	SYM
ap-4587	397	14	σ2(µ	σ2(µ	PROPN
ap-4587	397	15	)	)	PUNCT
ap-4587	397	16	,	,	PUNCT
ap-4587	397	17	η	η	PROPN
ap-4587	397	18	=	=	PROPN
ap-4587	397	19	σ2(ν	σ2(ν	PROPN
ap-4587	397	20	)	)	PUNCT
ap-4587	397	21	,	,	PUNCT
ap-4587	397	22	ζ	ζ	NOUN
ap-4587	397	23	′	′	NOUN
ap-4587	397	24	=	=	SYM
ap-4587	397	25	σ3(µ	σ3(µ	PROPN
ap-4587	397	26	)	)	PUNCT
ap-4587	397	27	,	,	PUNCT
ap-4587	397	28	η′	η′	PROPN
ap-4587	397	29	=	=	SYM
ap-4587	397	30	σ3(ν	σ3(ν	PROPN
ap-4587	397	31	)	)	PUNCT
ap-4587	397	32	.	.	PUNCT
ap-4587	398	1	remark	remark	VERB
ap-4587	398	2	6.3	6.3	NUM
ap-4587	398	3	.	.	PUNCT
ap-4587	399	1	note	note	VERB
ap-4587	399	2	that	that	SCONJ
ap-4587	399	3	setting	set	VERB
ap-4587	399	4	e	e	X
ap-4587	399	5	=	=	SYM
ap-4587	399	6	−d	−d	PROPN
ap-4587	399	7	,	,	PUNCT
ap-4587	399	8	f	f	PROPN
ap-4587	399	9	=	=	SYM
ap-4587	399	10	a−	a−	PROPN
ap-4587	399	11	d	d	PROPN
ap-4587	399	12	,	,	PUNCT
ap-4587	399	13	g	g	PROPN
ap-4587	399	14	=	=	SYM
ap-4587	399	15	b−	b−	PROPN
ap-4587	399	16	d	d	PROPN
ap-4587	399	17	,	,	PUNCT
ap-4587	399	18	h	h	NOUN
ap-4587	400	1	=	=	SYM
ap-4587	400	2	c−	c−	PROPN
ap-4587	400	3	d	d	X
ap-4587	400	4	,	,	PUNCT
ap-4587	400	5	we	we	PRON
ap-4587	400	6	obtain	obtain	VERB
ap-4587	400	7	the	the	DET
ap-4587	400	8	matrix	matrix	NOUN
ap-4587	400	9	(	(	PUNCT
ap-4587	400	10	17	17	NUM
ap-4587	400	11	)	)	PUNCT
ap-4587	400	12	.	.	PUNCT
ap-4587	401	1	therefore	therefore	ADV
ap-4587	401	2	the	the	DET
ap-4587	401	3	set	set	NOUN
ap-4587	401	4	zcom	zcom	PROPN
ap-4587	401	5	is	be	AUX
ap-4587	401	6	a	a	DET
ap-4587	401	7	commutative	commutative	ADJ
ap-4587	401	8	z	z	NOUN
ap-4587	401	9	-	-	PUNCT
ap-4587	401	10	subalgebra	subalgebra	NOUN
ap-4587	401	11	of	of	ADP
ap-4587	401	12	the	the	DET
ap-4587	401	13	z	z	PROPN
ap-4587	401	14	-	-	PUNCT
ap-4587	401	15	algebra	algebra	NOUN
ap-4587	401	16	z.	z.	NOUN
ap-4587	401	17	in	in	ADP
ap-4587	401	18	order	order	NOUN
ap-4587	401	19	to	to	PART
ap-4587	401	20	describe	describe	VERB
ap-4587	401	21	self	self	NOUN
ap-4587	401	22	-	-	PUNCT
ap-4587	401	23	similarities	similarity	NOUN
ap-4587	401	24	od	od	ADV
ap-4587	401	25	the	the	DET
ap-4587	401	26	module	module	NOUN
ap-4587	401	27	π1(l	π1(l	PRON
ap-4587	401	28	)	)	PUNCT
ap-4587	401	29	corresponding	correspond	VERB
ap-4587	401	30	to	to	ADP
ap-4587	401	31	the	the	DET
ap-4587	401	32	matrices	matrix	NOUN
ap-4587	401	33	za,	za,	NOUN
ap-4587	401	34	...	...	PUNCT
ap-4587	401	35	,h	,h	PUNCT
ap-4587	401	36	let	let	VERB
ap-4587	401	37	us	we	PRON
ap-4587	401	38	determine	determine	VERB
ap-4587	401	39	the	the	DET
ap-4587	401	40	values	value	NOUN
ap-4587	401	41	of	of	ADP
ap-4587	401	42	µ	µ	NOUN
ap-4587	401	43	,	,	PUNCT
ap-4587	401	44	ν	ν	NOUN
ap-4587	401	45	from	from	ADP
ap-4587	401	46	the	the	DET
ap-4587	401	47	relations	relation	NOUN
ap-4587	401	48	(	(	PUNCT
ap-4587	401	49	23	23	NUM
ap-4587	401	50	)	)	PUNCT
ap-4587	401	51	,	,	PUNCT
ap-4587	401	52	(	(	PUNCT
ap-4587	401	53	µ	µ	X
ap-4587	401	54	ν	ν	NOUN
ap-4587	401	55	)	)	PUNCT
ap-4587	401	56	=	=	SYM
ap-4587	401	57	1	1	NUM
ap-4587	401	58	ω4	ω4	NUM
ap-4587	401	59	−	−	PROPN
ap-4587	401	60	ω	ω	PROPN
ap-4587	401	61	(	(	PUNCT
ap-4587	401	62	ω4	ω4	NUM
ap-4587	401	63	−1	−1	NOUN
ap-4587	401	64	−ω	−ω	ADJ
ap-4587	401	65	1	1	NUM
ap-4587	401	66	)	)	PUNCT
ap-4587	401	67	(	(	PUNCT
ap-4587	401	68	a	a	DET
ap-4587	401	69	b	b	X
ap-4587	401	70	c	c	NOUN
ap-4587	401	71	d	d	X
ap-4587	401	72	e	e	X
ap-4587	401	73	f	f	PROPN
ap-4587	401	74	g	g	PROPN
ap-4587	401	75	h	h	PROPN
ap-4587	401	76	)	)	PUNCT
ap-4587	401	77			PROPN
ap-4587	401	78	1	1	NUM
ap-4587	401	79	ω	ω	NUM
ap-4587	401	80	ω2	ω2	ADJ
ap-4587	401	81	ω3	ω3	ADJ
ap-4587	401	82			NOUN
ap-4587	401	83	=	=	SYM
ap-4587	401	84	1	1	NUM
ap-4587	401	85	5(2	5(2	NUM
ap-4587	401	86	+	+	CCONJ
ap-4587	401	87	4ω	4ω	NOUN
ap-4587	401	88	+	+	CCONJ
ap-4587	402	1	ω2	ω2	ADJ
ap-4587	402	2	+	+	CCONJ
ap-4587	402	3	3ω3	3ω3	NUM
ap-4587	402	4	)	)	PUNCT
ap-4587	402	5	(	(	PUNCT
ap-4587	402	6	b−	b−	PROPN
ap-4587	402	7	e+	e+	VERB
ap-4587	402	8	ω(c−	ω(c−	X
ap-4587	402	9	f	f	NOUN
ap-4587	402	10	)	)	PUNCT
ap-4587	402	11	+	+	NUM
ap-4587	402	12	ω2(d−	ω2(d−	X
ap-4587	402	13	g)−	g)−	ADJ
ap-4587	402	14	hω3	hω3	PROPN
ap-4587	402	15	+	+	CCONJ
ap-4587	402	16	aω4	aω4	CCONJ
ap-4587	402	17	e+	e+	VERB
ap-4587	402	18	ω(f	ω(f	ADJ
ap-4587	402	19	−	−	DET
ap-4587	402	20	a	a	NOUN
ap-4587	402	21	)	)	PUNCT
ap-4587	402	22	+	+	CCONJ
ap-4587	402	23	ω2(g	ω2(g	NUM
ap-4587	402	24	−	−	PROPN
ap-4587	402	25	b	b	NOUN
ap-4587	402	26	)	)	PUNCT
ap-4587	402	27	+	+	CCONJ
ap-4587	402	28	ω3(h−	ω3(h−	PRON
ap-4587	402	29	c)−	c)−	PROPN
ap-4587	402	30	dω4	dω4	X
ap-4587	402	31	)	)	PUNCT
ap-4587	402	32	,	,	PUNCT
ap-4587	402	33	440	440	NUM
ap-4587	402	34	vol	vol	NOUN
ap-4587	402	35	.	.	PUNCT
ap-4587	403	1	57	57	NUM
ap-4587	403	2	no	no	NOUN
ap-4587	403	3	.	.	PUNCT
ap-4587	404	1	6/2017	6/2017	NUM
ap-4587	404	2	on	on	ADP
ap-4587	404	3	self	self	NOUN
ap-4587	404	4	-	-	PUNCT
ap-4587	404	5	similarities	similarity	NOUN
ap-4587	404	6	of	of	ADP
ap-4587	404	7	cut	cut	NOUN
ap-4587	404	8	-	-	PUNCT
ap-4587	404	9	and	and	CCONJ
ap-4587	404	10	-	-	PUNCT
ap-4587	404	11	project	project	NOUN
ap-4587	404	12	sets	set	VERB
ap-4587	404	13	µ	µ	NOUN
ap-4587	404	14	=	=	SYM
ap-4587	404	15	1	1	NUM
ap-4587	404	16	5	5	NUM
ap-4587	404	17	(	(	PUNCT
ap-4587	404	18	2a+	2a+	NUM
ap-4587	404	19	2b−	2b−	NUM
ap-4587	404	20	3c+	3c+	NUM
ap-4587	405	1	2d−	2d−	PROPN
ap-4587	405	2	2e+	2e+	NUM
ap-4587	405	3	3f	3f	PROPN
ap-4587	405	4	−	−	PROPN
ap-4587	405	5	2	2	NUM
ap-4587	405	6	g	g	NOUN
ap-4587	405	7	+	+	NOUN
ap-4587	405	8	3h+	3h+	NUM
ap-4587	405	9	ω(−a+	ω(−a+	VERB
ap-4587	405	10	4b−	4b−	NUM
ap-4587	405	11	c−	c−	NOUN
ap-4587	405	12	d−	d−	PROPN
ap-4587	405	13	4e+	4e+	NUM
ap-4587	405	14	f	f	NOUN
ap-4587	406	1	+	+	CCONJ
ap-4587	406	2	g	g	PROPN
ap-4587	406	3	+	+	NOUN
ap-4587	406	4	h	h	NOUN
ap-4587	406	5	)	)	PUNCT
ap-4587	407	1	+	+	CCONJ
ap-4587	407	2	ω2(a+	ω2(a+	PROPN
ap-4587	407	3	b+	b+	AUX
ap-4587	407	4	c+	c+	VERB
ap-4587	407	5	d−	d−	PROPN
ap-4587	407	6	e−	e−	PROPN
ap-4587	407	7	f	f	PROPN
ap-4587	408	1	−	−	PROPN
ap-4587	408	2	g	g	PROPN
ap-4587	408	3	+	+	CCONJ
ap-4587	408	4	4h	4h	NUM
ap-4587	408	5	)	)	PUNCT
ap-4587	409	1	+	+	CCONJ
ap-4587	409	2	ω3(−2a+	ω3(−2a+	ADP
ap-4587	409	3	3b−	3b−	NUM
ap-4587	409	4	2c+	2c+	NUM
ap-4587	409	5	3d−	3d−	PROPN
ap-4587	409	6	3e+	3e+	NUM
ap-4587	409	7	2f	2f	NUM
ap-4587	409	8	−	−	NOUN
ap-4587	409	9	3	3	NUM
ap-4587	409	10	g	g	NOUN
ap-4587	409	11	+	+	NOUN
ap-4587	409	12	2h	2h	NUM
ap-4587	409	13	)	)	PUNCT
ap-4587	409	14	)	)	PUNCT
ap-4587	409	15	,	,	PUNCT
ap-4587	409	16	ν	ν	X
ap-4587	409	17	=	=	SYM
ap-4587	409	18	1	1	NUM
ap-4587	409	19	5	5	NUM
ap-4587	409	20	(	(	PUNCT
ap-4587	409	21	3a−	3a−	PROPN
ap-4587	409	22	2b+	2b+	NUM
ap-4587	409	23	3c−	3c−	NUM
ap-4587	409	24	2d+	2d+	NUM
ap-4587	409	25	2e−	2e−	PROPN
ap-4587	409	26	3f	3f	NOUN
ap-4587	409	27	+	+	CCONJ
ap-4587	409	28	2	2	NUM
ap-4587	409	29	g	g	NOUN
ap-4587	409	30	−	−	PROPN
ap-4587	409	31	3h+	3h+	NUM
ap-4587	409	32	ω(a+	ω(a+	NOUN
ap-4587	409	33	b+	b+	X
ap-4587	409	34	c+	c+	VERB
ap-4587	409	35	d+	d+	PUNCT
ap-4587	410	1	4e−	4e−	NUM
ap-4587	410	2	f	f	NOUN
ap-4587	410	3	−	−	PROPN
ap-4587	410	4	g	g	PROPN
ap-4587	410	5	−	−	PROPN
ap-4587	410	6	h	h	NOUN
ap-4587	410	7	)	)	PUNCT
ap-4587	411	1	+	+	CCONJ
ap-4587	411	2	ω2(−a−	ω2(−a−	X
ap-4587	411	3	b+	b+	X
ap-4587	411	4	4c−	4c−	NUM
ap-4587	411	5	d+	d+	PUNCT
ap-4587	411	6	e+	e+	PUNCT
ap-4587	411	7	f	f	PROPN
ap-4587	412	1	+	+	CCONJ
ap-4587	412	2	g	g	PROPN
ap-4587	412	3	−	−	PROPN
ap-4587	412	4	4h	4h	NUM
ap-4587	412	5	)	)	PUNCT
ap-4587	413	1	+	+	CCONJ
ap-4587	413	2	ω3(2a−	ω3(2a−	VERB
ap-4587	413	3	3b+	3b+	NUM
ap-4587	413	4	2c+	2c+	NUM
ap-4587	413	5	2d+	2d+	NUM
ap-4587	413	6	3e−	3e−	NUM
ap-4587	413	7	2f	2f	NUM
ap-4587	413	8	+	+	CCONJ
ap-4587	413	9	3	3	NUM
ap-4587	413	10	g	g	NOUN
ap-4587	413	11	−	−	NUM
ap-4587	413	12	2h	2h	NUM
ap-4587	413	13	)	)	PUNCT
ap-4587	413	14	)	)	PUNCT
ap-4587	413	15	.	.	PUNCT
ap-4587	414	1	in	in	ADP
ap-4587	414	2	the	the	DET
ap-4587	414	3	matrix	matrix	NOUN
ap-4587	414	4	formalism	formalism	NOUN
ap-4587	414	5	,	,	PUNCT
ap-4587	414	6	(	(	PUNCT
ap-4587	414	7	22	22	NUM
ap-4587	414	8	)	)	PUNCT
ap-4587	414	9	rewrites	rewrite	NOUN
ap-4587	414	10	as	as	ADP
ap-4587	414	11	zy	zy	PROPN
ap-4587	414	12	=	=	SYM
ap-4587	414	13	y	y	PROPN
ap-4587	414	14			PROPN
ap-4587	414	15	µ	µ	X
ap-4587	414	16	ν	ν	X
ap-4587	414	17	0	0	NUM
ap-4587	414	18	0	0	NUM
ap-4587	414	19	ν	ν	X
ap-4587	414	20	µ	µ	X
ap-4587	414	21	0	0	NUM
ap-4587	414	22	0	0	NUM
ap-4587	414	23	0	0	NUM
ap-4587	414	24	0	0	NUM
ap-4587	414	25	ζ	ζ	PROPN
ap-4587	414	26	η	η	PROPN
ap-4587	414	27	0	0	NUM
ap-4587	414	28	0	0	NUM
ap-4587	414	29	η	η	PROPN
ap-4587	414	30	ζ	ζ	NOUN
ap-4587	414	31			NOUN
ap-4587	414	32	.	.	PUNCT
ap-4587	415	1	comparing	compare	VERB
ap-4587	415	2	to	to	ADP
ap-4587	415	3	(	(	PUNCT
ap-4587	415	4	21	21	NUM
ap-4587	415	5	)	)	PUNCT
ap-4587	415	6	,	,	PUNCT
ap-4587	415	7	we	we	PRON
ap-4587	415	8	obtain	obtain	VERB
ap-4587	415	9	the	the	DET
ap-4587	415	10	expression	expression	NOUN
ap-4587	415	11	for	for	ADP
ap-4587	415	12	s	s	PROPN
ap-4587	415	13	,	,	PUNCT
ap-4587	415	14	t	t	PROPN
ap-4587	415	15	,	,	PUNCT
ap-4587	415	16	(	(	PUNCT
ap-4587	415	17	s	s	VERB
ap-4587	415	18	o	o	X
ap-4587	415	19	o	o	X
ap-4587	415	20	t	t	NOUN
ap-4587	415	21	)	)	PUNCT
ap-4587	416	1	=	=	SYM
ap-4587	416	2	p−1	p−1	PROPN
ap-4587	416	3			PROPN
ap-4587	416	4	µ	µ	PRON
ap-4587	416	5	ν	ν	X
ap-4587	416	6	0	0	NUM
ap-4587	416	7	0	0	NUM
ap-4587	416	8	ν	ν	X
ap-4587	416	9	µ	µ	X
ap-4587	416	10	0	0	NUM
ap-4587	416	11	0	0	NUM
ap-4587	416	12	0	0	NUM
ap-4587	416	13	0	0	NUM
ap-4587	416	14	ζ	ζ	PROPN
ap-4587	416	15	η	η	PROPN
ap-4587	416	16	0	0	NUM
ap-4587	416	17	0	0	NUM
ap-4587	416	18	η	η	PROPN
ap-4587	416	19	ζ	ζ	X
ap-4587	416	20	p	p	X
ap-4587	416	21	=	=	SYM
ap-4587	416	22			PROPN
ap-4587	416	23	re(µ+	re(µ+	PROPN
ap-4587	416	24	ν	ν	PROPN
ap-4587	416	25	)	)	PUNCT
ap-4587	416	26	im(µ+	im(µ+	PROPN
ap-4587	416	27	ν	ν	PROPN
ap-4587	416	28	)	)	PUNCT
ap-4587	416	29	0	0	NUM
ap-4587	416	30	0	0	NUM
ap-4587	416	31	im(ν	im(ν	PUNCT
ap-4587	416	32	−	−	PROPN
ap-4587	416	33	µ	µ	X
ap-4587	416	34	)	)	PUNCT
ap-4587	416	35	re(µ−	re(µ−	PROPN
ap-4587	416	36	ν	ν	NOUN
ap-4587	416	37	)	)	PUNCT
ap-4587	416	38	0	0	NUM
ap-4587	416	39	0	0	NUM
ap-4587	416	40	0	0	NUM
ap-4587	416	41	0	0	NUM
ap-4587	416	42	re(ζ	re(ζ	NOUN
ap-4587	416	43	+	+	CCONJ
ap-4587	416	44	η	η	NOUN
ap-4587	416	45	)	)	PUNCT
ap-4587	416	46	im(ζ	im(ζ	PROPN
ap-4587	416	47	+	+	CCONJ
ap-4587	416	48	η	η	NOUN
ap-4587	416	49	)	)	PUNCT
ap-4587	416	50	0	0	NUM
ap-4587	416	51	0	0	NUM
ap-4587	416	52	im(η	im(η	NOUN
ap-4587	416	53	−	−	PROPN
ap-4587	416	54	ζ	ζ	NOUN
ap-4587	416	55	)	)	PUNCT
ap-4587	416	56	re(ζ	re(ζ	NOUN
ap-4587	416	57	−	−	PROPN
ap-4587	416	58	η	η	PROPN
ap-4587	416	59	)	)	PUNCT
ap-4587	416	60			NOUN
ap-4587	416	61	,	,	PUNCT
ap-4587	416	62	(	(	PUNCT
ap-4587	416	63	25	25	NUM
ap-4587	416	64	)	)	PUNCT
ap-4587	416	65	where	where	SCONJ
ap-4587	416	66	the	the	DET
ap-4587	416	67	coefficients	coefficient	NOUN
ap-4587	416	68	are	be	AUX
ap-4587	416	69	of	of	ADP
ap-4587	416	70	the	the	DET
ap-4587	416	71	form	form	NOUN
ap-4587	416	72	re(µ+	re(µ+	PROPN
ap-4587	416	73	ν	ν	PROPN
ap-4587	416	74	)	)	PUNCT
ap-4587	416	75	=	=	SYM
ap-4587	416	76	a+	a+	PUNCT
ap-4587	416	77	b	b	X
ap-4587	416	78	cos	cos	ADP
ap-4587	416	79	2π	2π	PROPN
ap-4587	416	80	5	5	NUM
ap-4587	416	81	+	+	CCONJ
ap-4587	416	82	(	(	PUNCT
ap-4587	416	83	c+	c+	NOUN
ap-4587	416	84	d	d	NOUN
ap-4587	416	85	)	)	PUNCT
ap-4587	417	1	cos	cos	PROPN
ap-4587	417	2	4π	4π	NUM
ap-4587	417	3	5	5	NUM
ap-4587	417	4	,	,	PUNCT
ap-4587	417	5	im(µ+	im(µ+	PROPN
ap-4587	417	6	ν	ν	PROPN
ap-4587	417	7	)	)	PUNCT
ap-4587	417	8	=	=	SYM
ap-4587	417	9	b	b	NOUN
ap-4587	417	10	sin	sin	NOUN
ap-4587	417	11	2π	2π	NOUN
ap-4587	417	12	5	5	NUM
ap-4587	417	13	+	+	CCONJ
ap-4587	417	14	(	(	PUNCT
ap-4587	417	15	c−	c−	ADJ
ap-4587	417	16	d	d	NOUN
ap-4587	417	17	)	)	PUNCT
ap-4587	417	18	sin	sin	NOUN
ap-4587	417	19	4π	4π	NUM
ap-4587	417	20	5	5	NUM
ap-4587	417	21	,	,	PUNCT
ap-4587	417	22	re(µ−	re(µ−	PROPN
ap-4587	417	23	ν	ν	PROPN
ap-4587	417	24	)	)	PUNCT
ap-4587	417	25	=	=	SYM
ap-4587	417	26	−c+	−c+	NOUN
ap-4587	417	27	d+	d+	PUNCT
ap-4587	417	28	f	f	X
ap-4587	417	29	+	+	NUM
ap-4587	417	30	h+	h+	X
ap-4587	417	31	(	(	PUNCT
ap-4587	417	32	2	2	NUM
ap-4587	417	33	g	g	NOUN
ap-4587	417	34	−	−	PROPN
ap-4587	417	35	b	b	NOUN
ap-4587	417	36	)	)	PUNCT
ap-4587	417	37	cos	cos	ADP
ap-4587	417	38	2π	2π	PROPN
ap-4587	417	39	5	5	NUM
ap-4587	417	40	+	+	CCONJ
ap-4587	417	41	(	(	PUNCT
ap-4587	417	42	−c+	−c+	NOUN
ap-4587	417	43	d+	d+	NOUN
ap-4587	417	44	2h	2h	NUM
ap-4587	417	45	)	)	PUNCT
ap-4587	417	46	cos	cos	PROPN
ap-4587	417	47	4π	4π	NUM
ap-4587	417	48	5	5	NUM
ap-4587	417	49	,	,	PUNCT
ap-4587	417	50	im(ν	im(ν	PUNCT
ap-4587	417	51	−	−	PROPN
ap-4587	417	52	µ	µ	X
ap-4587	417	53	)	)	PUNCT
ap-4587	417	54	=	=	SYM
ap-4587	417	55	1	1	NUM
ap-4587	417	56	5	5	NUM
ap-4587	417	57	(	(	PUNCT
ap-4587	417	58	(	(	PUNCT
ap-4587	417	59	2a−	2a−	NUM
ap-4587	417	60	3b+	3b+	NUM
ap-4587	417	61	2c+	2c+	NUM
ap-4587	417	62	2d+	2d+	NUM
ap-4587	417	63	8e−	8e−	NUM
ap-4587	417	64	2f	2f	NOUN
ap-4587	417	65	−	−	NUM
ap-4587	417	66	2	2	NUM
ap-4587	417	67	g	g	NOUN
ap-4587	417	68	−	−	PROPN
ap-4587	417	69	2h	2h	NUM
ap-4587	417	70	)	)	PUNCT
ap-4587	417	71	sin	sin	NOUN
ap-4587	417	72	2π	2π	NOUN
ap-4587	417	73	5	5	NUM
ap-4587	417	74	+	+	CCONJ
ap-4587	417	75	(	(	PUNCT
ap-4587	417	76	−6a+	−6a+	ADP
ap-4587	417	77	4b−	4b−	NUM
ap-4587	417	78	c−	c−	NOUN
ap-4587	417	79	d−	d−	PROPN
ap-4587	417	80	4e+	4e+	NUM
ap-4587	417	81	6f	6f	NUM
ap-4587	417	82	−	−	NUM
ap-4587	417	83	4	4	NUM
ap-4587	417	84	g	g	NOUN
ap-4587	417	85	−	−	PROPN
ap-4587	417	86	4h	4h	NUM
ap-4587	417	87	)	)	PUNCT
ap-4587	417	88	sin	sin	NOUN
ap-4587	417	89	4π	4π	NUM
ap-4587	417	90	5	5	NUM
ap-4587	417	91	)	)	PUNCT
ap-4587	417	92	,	,	PUNCT
ap-4587	417	93	re(ζ	re(ζ	X
ap-4587	417	94	+	+	CCONJ
ap-4587	417	95	η	η	X
ap-4587	417	96	)	)	PUNCT
ap-4587	417	97	=	=	PRON
ap-4587	417	98	a+	a+	PUNCT
ap-4587	417	99	(	(	PUNCT
ap-4587	417	100	c+	c+	NOUN
ap-4587	417	101	d	d	X
ap-4587	417	102	)	)	PUNCT
ap-4587	417	103	cos	cos	ADP
ap-4587	417	104	2π	2π	PROPN
ap-4587	417	105	5	5	NUM
ap-4587	418	1	+	+	SYM
ap-4587	418	2	b	b	X
ap-4587	418	3	cos	cos	PROPN
ap-4587	418	4	4π	4π	NUM
ap-4587	418	5	5	5	NUM
ap-4587	418	6	,	,	PUNCT
ap-4587	418	7	im(ζ	im(ζ	PROPN
ap-4587	418	8	+	+	CCONJ
ap-4587	418	9	η	η	NOUN
ap-4587	418	10	)	)	PUNCT
ap-4587	418	11	=	=	PUNCT
ap-4587	418	12	(	(	PUNCT
ap-4587	418	13	d−	d−	PROPN
ap-4587	418	14	c	c	NOUN
ap-4587	418	15	)	)	PUNCT
ap-4587	418	16	sin	sin	NOUN
ap-4587	418	17	2π	2π	NOUN
ap-4587	418	18	5	5	NUM
ap-4587	418	19	+	+	CCONJ
ap-4587	418	20	b	b	NOUN
ap-4587	418	21	sin	sin	NOUN
ap-4587	418	22	4π	4π	NUM
ap-4587	418	23	5	5	NUM
ap-4587	418	24	,	,	PUNCT
ap-4587	418	25	re(ζ	re(ζ	NOUN
ap-4587	418	26	−	−	PROPN
ap-4587	418	27	η	η	PROPN
ap-4587	418	28	)	)	PUNCT
ap-4587	418	29	=	=	SYM
ap-4587	418	30	−c+	−c+	NOUN
ap-4587	418	31	d+	d+	PUNCT
ap-4587	419	1	f	f	X
ap-4587	419	2	+	+	NUM
ap-4587	419	3	h+	h+	X
ap-4587	419	4	(	(	PUNCT
ap-4587	419	5	−c+	−c+	NOUN
ap-4587	419	6	d+	d+	NOUN
ap-4587	419	7	2h	2h	NUM
ap-4587	419	8	)	)	PUNCT
ap-4587	419	9	cos	cos	ADP
ap-4587	419	10	2π	2π	PROPN
ap-4587	419	11	5	5	NUM
ap-4587	419	12	+	+	CCONJ
ap-4587	419	13	(	(	PUNCT
ap-4587	419	14	2	2	NUM
ap-4587	419	15	g	g	NOUN
ap-4587	419	16	−	−	PROPN
ap-4587	419	17	b	b	NOUN
ap-4587	419	18	)	)	PUNCT
ap-4587	419	19	cos	cos	PROPN
ap-4587	419	20	4π	4π	NUM
ap-4587	419	21	5	5	NUM
ap-4587	419	22	,	,	PUNCT
ap-4587	419	23	im(η	im(η	VERB
ap-4587	419	24	−	−	PROPN
ap-4587	419	25	ζ	ζ	NOUN
ap-4587	419	26	)	)	PUNCT
ap-4587	419	27	=	=	SYM
ap-4587	419	28	1	1	NUM
ap-4587	419	29	5	5	NUM
ap-4587	419	30	(	(	PUNCT
ap-4587	419	31	(	(	PUNCT
ap-4587	419	32	6a−	6a−	NOUN
ap-4587	419	33	4b+	4b+	NUM
ap-4587	419	34	c+	c+	NOUN
ap-4587	419	35	d+	d+	PUNCT
ap-4587	419	36	4e−	4e−	NUM
ap-4587	419	37	6f	6f	NUM
ap-4587	419	38	+	+	CCONJ
ap-4587	419	39	4	4	NUM
ap-4587	419	40	g	g	NOUN
ap-4587	419	41	+	+	NOUN
ap-4587	419	42	4h	4h	NUM
ap-4587	419	43	)	)	PUNCT
ap-4587	419	44	sin	sin	NOUN
ap-4587	419	45	2π	2π	NOUN
ap-4587	419	46	5	5	NUM
ap-4587	419	47	+	+	CCONJ
ap-4587	419	48	(	(	PUNCT
ap-4587	419	49	2a−	2a−	NUM
ap-4587	419	50	3b+	3b+	NUM
ap-4587	419	51	2c+	2c+	NUM
ap-4587	419	52	2d+	2d+	NUM
ap-4587	419	53	8e−	8e−	NUM
ap-4587	419	54	2f	2f	NOUN
ap-4587	419	55	−	−	NUM
ap-4587	419	56	2	2	NUM
ap-4587	419	57	g	g	NOUN
ap-4587	419	58	−	−	PROPN
ap-4587	419	59	2h	2h	NUM
ap-4587	419	60	)	)	PUNCT
ap-4587	419	61	sin	sin	NOUN
ap-4587	419	62	4π	4π	NUM
ap-4587	419	63	5	5	NUM
ap-4587	419	64	)	)	PUNCT
ap-4587	419	65	.	.	PUNCT
ap-4587	420	1	remark	remark	VERB
ap-4587	420	2	6.4	6.4	NUM
ap-4587	420	3	.	.	PUNCT
ap-4587	421	1	the	the	DET
ap-4587	421	2	matrices	matrix	NOUN
ap-4587	421	3	s	s	PART
ap-4587	421	4	of	of	ADP
ap-4587	421	5	(	(	PUNCT
ap-4587	421	6	25	25	NUM
ap-4587	421	7	)	)	PUNCT
ap-4587	421	8	of	of	ADP
ap-4587	421	9	all	all	DET
ap-4587	421	10	linear	linear	ADJ
ap-4587	421	11	self	self	NOUN
ap-4587	421	12	-	-	PUNCT
ap-4587	421	13	similarities	similarity	NOUN
ap-4587	421	14	of	of	ADP
ap-4587	421	15	the	the	DET
ap-4587	421	16	cut	cut	NOUN
ap-4587	421	17	-	-	PUNCT
ap-4587	421	18	and	and	CCONJ
ap-4587	421	19	-	-	PUNCT
ap-4587	421	20	project	project	NOUN
ap-4587	421	21	scheme	scheme	NOUN
ap-4587	421	22	λ	λ	NOUN
ap-4587	421	23	=	=	SYM
ap-4587	421	24	(	(	PUNCT
ap-4587	421	25	l	l	NOUN
ap-4587	421	26	,	,	PUNCT
ap-4587	421	27	π1	π1	NOUN
ap-4587	421	28	,	,	PUNCT
ap-4587	421	29	π2	π2	NOUN
ap-4587	421	30	)	)	PUNCT
ap-4587	421	31	of	of	ADP
ap-4587	421	32	notation	notation	NOUN
ap-4587	421	33	5.1	5.1	NUM
ap-4587	421	34	form	form	NOUN
ap-4587	421	35	an	an	DET
ap-4587	421	36	8	8	NUM
ap-4587	421	37	-	-	PUNCT
ap-4587	421	38	dimensional	dimensional	ADJ
ap-4587	421	39	associative	associative	ADJ
ap-4587	421	40	z	z	NOUN
ap-4587	421	41	-	-	NOUN
ap-4587	421	42	algebra	algebra	NOUN
ap-4587	421	43	.	.	PUNCT
ap-4587	422	1	7	7	X
ap-4587	422	2	.	.	X
ap-4587	422	3	self	self	NOUN
ap-4587	422	4	-	-	PUNCT
ap-4587	422	5	similarities	similarity	NOUN
ap-4587	422	6	of	of	ADP
ap-4587	422	7	cut	cut	NOUN
ap-4587	422	8	-	-	PUNCT
ap-4587	422	9	and	and	CCONJ
ap-4587	422	10	-	-	PUNCT
ap-4587	422	11	project	project	NOUN
ap-4587	422	12	sets	set	NOUN
ap-4587	422	13	with	with	ADP
ap-4587	422	14	5	5	ADJ
ap-4587	422	15	-	-	ADJ
ap-4587	422	16	fold	fold	ADJ
ap-4587	422	17	symmetry	symmetry	NOUN
ap-4587	422	18	the	the	DET
ap-4587	422	19	requirement	requirement	NOUN
ap-4587	422	20	of	of	ADP
ap-4587	422	21	preserving	preserve	VERB
ap-4587	422	22	the	the	DET
ap-4587	422	23	invariant	invariant	ADJ
ap-4587	422	24	subspaces	subspace	NOUN
ap-4587	422	25	alone	alone	ADV
ap-4587	422	26	is	be	AUX
ap-4587	422	27	not	not	PART
ap-4587	422	28	sufficient	sufficient	ADJ
ap-4587	422	29	for	for	ADP
ap-4587	422	30	providing	provide	VERB
ap-4587	422	31	a	a	DET
ap-4587	422	32	complete	complete	ADJ
ap-4587	422	33	description	description	NOUN
ap-4587	422	34	of	of	ADP
ap-4587	422	35	linear	linear	ADJ
ap-4587	422	36	mappings	mapping	NOUN
ap-4587	422	37	that	that	PRON
ap-4587	422	38	are	be	AUX
ap-4587	422	39	self	self	NOUN
ap-4587	422	40	-	-	PUNCT
ap-4587	422	41	similarities	similarity	NOUN
ap-4587	422	42	of	of	ADP
ap-4587	422	43	some	some	DET
ap-4587	422	44	cut	cut	VERB
ap-4587	422	45	-	-	PUNCT
ap-4587	422	46	and	and	CCONJ
ap-4587	422	47	-	-	PUNCT
ap-4587	422	48	project	project	NOUN
ap-4587	422	49	set	set	NOUN
ap-4587	422	50	.	.	PUNCT
ap-4587	423	1	in	in	ADP
ap-4587	423	2	order	order	NOUN
ap-4587	423	3	that	that	SCONJ
ap-4587	423	4	a	a	DET
ap-4587	423	5	matrix	matrix	NOUN
ap-4587	423	6	z	z	NOUN
ap-4587	423	7	gives	give	VERB
ap-4587	423	8	rise	rise	NOUN
ap-4587	423	9	to	to	ADP
ap-4587	423	10	a	a	DET
ap-4587	423	11	self	self	NOUN
ap-4587	423	12	-	-	PUNCT
ap-4587	423	13	similarity	similarity	NOUN
ap-4587	423	14	of	of	ADP
ap-4587	423	15	σ(ω	σ(ω	PROPN
ap-4587	423	16	)	)	PUNCT
ap-4587	423	17	for	for	ADP
ap-4587	423	18	some	some	DET
ap-4587	423	19	window	window	NOUN
ap-4587	423	20	ω	ω	NOUN
ap-4587	423	21	,	,	PUNCT
ap-4587	423	22	it	it	PRON
ap-4587	423	23	is	be	AUX
ap-4587	423	24	necessary	necessary	ADJ
ap-4587	423	25	to	to	PART
ap-4587	423	26	set	set	VERB
ap-4587	423	27	conditions	condition	NOUN
ap-4587	423	28	on	on	ADP
ap-4587	423	29	the	the	DET
ap-4587	423	30	eigenvalues	eigenvalue	NOUN
ap-4587	423	31	of	of	ADP
ap-4587	423	32	the	the	DET
ap-4587	423	33	matrix	matrix	NOUN
ap-4587	423	34	b	b	NOUN
ap-4587	423	35	,	,	PUNCT
ap-4587	423	36	as	as	SCONJ
ap-4587	423	37	specified	specify	VERB
ap-4587	423	38	in	in	ADP
ap-4587	423	39	proposition	proposition	NOUN
ap-4587	423	40	4.2	4.2	NUM
ap-4587	423	41	.	.	PUNCT
ap-4587	424	1	we	we	PRON
ap-4587	424	2	shall	shall	AUX
ap-4587	424	3	do	do	VERB
ap-4587	424	4	that	that	PRON
ap-4587	424	5	for	for	ADP
ap-4587	424	6	the	the	DET
ap-4587	424	7	self	self	NOUN
ap-4587	424	8	-	-	PUNCT
ap-4587	424	9	similarities	similarity	NOUN
ap-4587	424	10	given	give	VERB
ap-4587	424	11	by	by	ADP
ap-4587	424	12	matrices	matrix	NOUN
ap-4587	424	13	of	of	ADP
ap-4587	424	14	the	the	DET
ap-4587	424	15	z	z	NOUN
ap-4587	424	16	-	-	PUNCT
ap-4587	424	17	algebra	algebra	PROPN
ap-4587	424	18	zcom	zcom	NOUN
ap-4587	424	19	defined	define	VERB
ap-4587	424	20	in	in	ADP
ap-4587	424	21	corollary	corollary	ADJ
ap-4587	424	22	6.1	6.1	NUM
ap-4587	424	23	.	.	PUNCT
ap-4587	425	1	proposition	proposition	NOUN
ap-4587	425	2	7.1	7.1	NUM
ap-4587	425	3	.	.	PUNCT
ap-4587	426	1	let	let	VERB
ap-4587	426	2	z	z	PROPN
ap-4587	426	3	∈	∈	PROPN
ap-4587	426	4	zcom	zcom	NOUN
ap-4587	426	5	and	and	CCONJ
ap-4587	426	6	let	let	VERB
ap-4587	426	7	s	s	PRON
ap-4587	426	8	correspond	correspond	VERB
ap-4587	426	9	to	to	ADP
ap-4587	426	10	z	z	PROPN
ap-4587	426	11	by	by	ADP
ap-4587	426	12	v	v	NUM
ap-4587	426	13	zv	zv	NOUN
ap-4587	426	14	−1	−1	NOUN
ap-4587	426	15	=	=	PUNCT
ap-4587	427	1	(	(	PUNCT
ap-4587	427	2	s	s	NOUN
ap-4587	427	3	o	o	X
ap-4587	427	4	o	o	X
ap-4587	427	5	t	t	NOUN
ap-4587	427	6	)	)	PUNCT
ap-4587	427	7	.	.	PUNCT
ap-4587	428	1	if	if	SCONJ
ap-4587	428	2	s	s	PROPN
ap-4587	428	3	is	be	AUX
ap-4587	428	4	a	a	DET
ap-4587	428	5	self	self	NOUN
ap-4587	428	6	-	-	PUNCT
ap-4587	428	7	similarity	similarity	NOUN
ap-4587	428	8	of	of	ADP
ap-4587	428	9	a	a	DET
ap-4587	428	10	cut	cut	NOUN
ap-4587	428	11	-	-	PUNCT
ap-4587	428	12	and	and	CCONJ
ap-4587	428	13	-	-	PUNCT
ap-4587	428	14	project	project	NOUN
ap-4587	428	15	set	set	VERB
ap-4587	428	16	σ(ω	σ(ω	PROPN
ap-4587	428	17	)	)	PUNCT
ap-4587	428	18	,	,	PUNCT
ap-4587	428	19	then	then	ADV
ap-4587	428	20	there	there	PRON
ap-4587	428	21	exists	exist	VERB
ap-4587	428	22	an	an	DET
ap-4587	428	23	algebraic	algebraic	ADJ
ap-4587	428	24	integer	integer	NOUN
ap-4587	428	25	η	η	PROPN
ap-4587	428	26	=	=	PROPN
ap-4587	428	27	a+bω+cω2+dω3	a+bω+cω2+dω3	PART
ap-4587	428	28	=	=	PUNCT
ap-4587	428	29	|η|(cosϕ+i	|η|(cosϕ+i	NOUN
ap-4587	428	30	sinϕ	sinϕ	NOUN
ap-4587	428	31	)	)	PUNCT
ap-4587	428	32	∈	∈	PROPN
ap-4587	428	33	z[ω	z[ω	PROPN
ap-4587	428	34	]	]	PUNCT
ap-4587	428	35	such	such	ADJ
ap-4587	428	36	that	that	PRON
ap-4587	428	37	s	s	NOUN
ap-4587	428	38	=	=	SYM
ap-4587	428	39	|η|r	|η|r	NOUN
ap-4587	428	40	,	,	PUNCT
ap-4587	428	41	where	where	SCONJ
ap-4587	428	42	r	r	NOUN
ap-4587	428	43	is	be	AUX
ap-4587	428	44	a	a	DET
ap-4587	428	45	rotation	rotation	NOUN
ap-4587	428	46	by	by	ADP
ap-4587	428	47	the	the	DET
ap-4587	428	48	angle	angle	NOUN
ap-4587	428	49	ϕ.	ϕ.	PROPN
ap-4587	428	50	moreover	moreover	ADV
ap-4587	428	51	,	,	PUNCT
ap-4587	428	52	η	η	PROPN
ap-4587	428	53	is	be	AUX
ap-4587	428	54	a	a	DET
ap-4587	428	55	pisot	pisot	ADJ
ap-4587	428	56	number	number	NOUN
ap-4587	428	57	,	,	PUNCT
ap-4587	428	58	complex	complex	ADJ
ap-4587	428	59	pisot	pisot	ADJ
ap-4587	428	60	number	number	NOUN
ap-4587	428	61	or	or	CCONJ
ap-4587	428	62	a	a	DET
ap-4587	428	63	tenth	tenth	ADJ
ap-4587	428	64	root	root	NOUN
ap-4587	428	65	of	of	ADP
ap-4587	428	66	unity	unity	NOUN
ap-4587	428	67	.	.	PUNCT
ap-4587	429	1	441	441	NUM
ap-4587	429	2	zuzana	zuzana	PROPN
ap-4587	429	3	masáková	masáková	PROPN
ap-4587	429	4	,	,	PUNCT
ap-4587	429	5	jan	jan	PROPN
ap-4587	429	6	mazáč	mazáč	PROPN
ap-4587	429	7	acta	acta	PROPN
ap-4587	429	8	polytechnica	polytechnica	PROPN
ap-4587	429	9	proof	proof	NOUN
ap-4587	429	10	.	.	PUNCT
ap-4587	430	1	recall	recall	VERB
ap-4587	430	2	that	that	SCONJ
ap-4587	430	3	a	a	DET
ap-4587	430	4	matrix	matrix	NOUN
ap-4587	430	5	za	za	PROPN
ap-4587	430	6	,	,	PUNCT
ap-4587	430	7	b	b	PROPN
ap-4587	430	8	,	,	PUNCT
ap-4587	430	9	c	c	X
ap-4587	430	10	,	,	PUNCT
ap-4587	430	11	d	d	PROPN
ap-4587	430	12	has	have	VERB
ap-4587	430	13	as	as	ADP
ap-4587	430	14	its	its	PRON
ap-4587	430	15	eigenvalues	eigenvalue	NOUN
ap-4587	430	16	the	the	DET
ap-4587	430	17	numbers	number	NOUN
ap-4587	430	18	σj(η	σj(η	NOUN
ap-4587	430	19	)	)	PUNCT
ap-4587	430	20	,	,	PUNCT
ap-4587	430	21	j	j	PROPN
ap-4587	430	22	=	=	SYM
ap-4587	430	23	1	1	NUM
ap-4587	430	24	,	,	PUNCT
ap-4587	430	25	2	2	NUM
ap-4587	430	26	,	,	PUNCT
ap-4587	430	27	3	3	NUM
ap-4587	430	28	,	,	PUNCT
ap-4587	430	29	4	4	NUM
ap-4587	430	30	,	,	PUNCT
ap-4587	430	31	where	where	SCONJ
ap-4587	430	32	η	η	X
ap-4587	430	33	=	=	PRON
ap-4587	430	34	a+	a+	PUNCT
ap-4587	430	35	bω	bω	NOUN
ap-4587	430	36	+	+	CCONJ
ap-4587	430	37	cω2	cω2	NOUN
ap-4587	430	38	+	+	NOUN
ap-4587	430	39	dω3	dω3	NOUN
ap-4587	430	40	.	.	PUNCT
ap-4587	431	1	the	the	DET
ap-4587	431	2	product	product	NOUN
ap-4587	431	3	∏4	∏4	NOUN
ap-4587	431	4	j=1	j=1	NOUN
ap-4587	431	5	σj(η	σj(η	X
ap-4587	431	6	)	)	PUNCT
ap-4587	431	7	is	be	AUX
ap-4587	431	8	equal	equal	ADJ
ap-4587	431	9	to	to	ADP
ap-4587	431	10	the	the	DET
ap-4587	431	11	determinant	determinant	NOUN
ap-4587	431	12	of	of	ADP
ap-4587	431	13	the	the	DET
ap-4587	431	14	integer	integer	NOUN
ap-4587	431	15	matrix	matrix	NOUN
ap-4587	431	16	za	za	PROPN
ap-4587	431	17	,	,	PUNCT
ap-4587	431	18	b	b	PROPN
ap-4587	431	19	,	,	PUNCT
ap-4587	431	20	c	c	PROPN
ap-4587	431	21	,	,	PUNCT
ap-4587	431	22	d.	d.	PROPN
ap-4587	431	23	the	the	DET
ap-4587	431	24	corresponding	corresponding	ADJ
ap-4587	431	25	matrices	matrix	NOUN
ap-4587	431	26	s	s	PROPN
ap-4587	431	27	,	,	PUNCT
ap-4587	431	28	t	t	PROPN
ap-4587	431	29	from	from	ADP
ap-4587	431	30	(	(	PUNCT
ap-4587	431	31	18	18	NUM
ap-4587	431	32	)	)	PUNCT
ap-4587	431	33	,	,	PUNCT
ap-4587	431	34	(	(	PUNCT
ap-4587	431	35	19	19	NUM
ap-4587	431	36	)	)	PUNCT
ap-4587	431	37	,	,	PUNCT
ap-4587	431	38	have	have	VERB
ap-4587	431	39	the	the	DET
ap-4587	431	40	same	same	ADJ
ap-4587	431	41	eigenvalues	eigenvalue	NOUN
ap-4587	431	42	,	,	PUNCT
ap-4587	431	43	namely	namely	ADV
ap-4587	431	44	σ1(η	σ1(η	PROPN
ap-4587	431	45	)	)	PUNCT
ap-4587	431	46	,	,	PUNCT
ap-4587	431	47	σ4(η	σ4(η	NUM
ap-4587	431	48	)	)	PUNCT
ap-4587	431	49	for	for	ADP
ap-4587	431	50	the	the	DET
ap-4587	431	51	matrix	matrix	NOUN
ap-4587	431	52	s	s	NOUN
ap-4587	431	53	and	and	CCONJ
ap-4587	431	54	σ2(η	σ2(η	NOUN
ap-4587	431	55	)	)	PUNCT
ap-4587	431	56	,	,	PUNCT
ap-4587	431	57	σ3(η	σ3(η	X
ap-4587	431	58	)	)	PUNCT
ap-4587	431	59	for	for	ADP
ap-4587	431	60	the	the	DET
ap-4587	431	61	matrix	matrix	NOUN
ap-4587	431	62	t	t	PROPN
ap-4587	431	63	.	.	PUNCT
ap-4587	432	1	recall	recall	VERB
ap-4587	432	2	that	that	SCONJ
ap-4587	432	3	σ1(η	σ1(η	PROPN
ap-4587	432	4	)	)	PUNCT
ap-4587	432	5	=	=	SYM
ap-4587	432	6	σ4(η	σ4(η	PROPN
ap-4587	432	7	)	)	PUNCT
ap-4587	432	8	and	and	CCONJ
ap-4587	432	9	σ2(η	σ2(η	NOUN
ap-4587	432	10	)	)	PUNCT
ap-4587	432	11	=	=	SYM
ap-4587	432	12	σ3(η	σ3(η	NOUN
ap-4587	432	13	)	)	PUNCT
ap-4587	432	14	.	.	PUNCT
ap-4587	433	1	if	if	SCONJ
ap-4587	433	2	s	s	NOUN
ap-4587	433	3	is	be	AUX
ap-4587	433	4	a	a	DET
ap-4587	433	5	self	self	NOUN
ap-4587	433	6	-	-	PUNCT
ap-4587	433	7	similarity	similarity	NOUN
ap-4587	433	8	of	of	ADP
ap-4587	433	9	a	a	DET
ap-4587	433	10	cut	cut	NOUN
ap-4587	433	11	-	-	PUNCT
ap-4587	433	12	and	and	CCONJ
ap-4587	433	13	-	-	PUNCT
ap-4587	433	14	project	project	NOUN
ap-4587	433	15	set	set	VERB
ap-4587	433	16	σ(ω	σ(ω	PROPN
ap-4587	433	17	)	)	PUNCT
ap-4587	433	18	,	,	PUNCT
ap-4587	433	19	then	then	ADV
ap-4587	433	20	by	by	ADP
ap-4587	433	21	proposition	proposition	NOUN
ap-4587	433	22	4.2	4.2	NUM
ap-4587	433	23	,	,	PUNCT
ap-4587	433	24	the	the	DET
ap-4587	433	25	eigenvalues	eigenvalue	NOUN
ap-4587	433	26	of	of	ADP
ap-4587	433	27	t	t	PROPN
ap-4587	433	28	must	must	AUX
ap-4587	433	29	be	be	AUX
ap-4587	433	30	in	in	ADP
ap-4587	433	31	modulus	modulus	NOUN
ap-4587	433	32	smaller	small	ADJ
ap-4587	433	33	or	or	CCONJ
ap-4587	433	34	equal	equal	ADJ
ap-4587	433	35	to	to	ADP
ap-4587	433	36	1	1	NUM
ap-4587	433	37	.	.	PUNCT
ap-4587	434	1	as	as	ADP
ap-4587	434	2	∏4	∏4	NOUN
ap-4587	434	3	j=1	j=1	NOUN
ap-4587	434	4	σj(η	σj(η	X
ap-4587	434	5	)	)	PUNCT
ap-4587	434	6	∈	∈	PROPN
ap-4587	434	7	z	z	NOUN
ap-4587	434	8	we	we	PRON
ap-4587	434	9	have	have	VERB
ap-4587	434	10	|η|2	|η|2	NOUN
ap-4587	434	11	=	=	SYM
ap-4587	434	12	σ1(η)σ4(η	σ1(η)σ4(η	NUM
ap-4587	434	13	)	)	PUNCT
ap-4587	434	14	≥	≥	NOUN
ap-4587	434	15	1	1	X
ap-4587	434	16	.	.	PUNCT
ap-4587	435	1	assume	assume	VERB
ap-4587	435	2	first	first	ADV
ap-4587	435	3	that	that	SCONJ
ap-4587	435	4	|σ2(η)|	|σ2(η)|	PUNCT
ap-4587	435	5	=	=	SYM
ap-4587	435	6	1	1	X
ap-4587	435	7	.	.	PUNCT
ap-4587	435	8	then	then	ADV
ap-4587	435	9	necessarily	necessarily	ADV
ap-4587	435	10	|σ2(η)σ3(η)|	|σ2(η)σ3(η)|	ADJ
ap-4587	435	11	=	=	SYM
ap-4587	435	12	1	1	NUM
ap-4587	435	13	,	,	PUNCT
ap-4587	435	14	i.e.	i.e.	X
ap-4587	435	15	,	,	PUNCT
ap-4587	435	16	σ2(η	σ2(η	ADJ
ap-4587	435	17	)	)	PUNCT
ap-4587	435	18	=	=	SYM
ap-4587	435	19	σ3(η)−1	σ3(η)−1	NOUN
ap-4587	435	20	.	.	PUNCT
ap-4587	436	1	the	the	DET
ap-4587	436	2	characteristic	characteristic	ADJ
ap-4587	436	3	polynomial	polynomial	NOUN
ap-4587	436	4	of	of	ADP
ap-4587	436	5	the	the	DET
ap-4587	436	6	matrix	matrix	NOUN
ap-4587	436	7	za	za	PROPN
ap-4587	436	8	,	,	PUNCT
ap-4587	436	9	b	b	PROPN
ap-4587	436	10	,	,	PUNCT
ap-4587	436	11	c	c	X
ap-4587	436	12	,	,	PUNCT
ap-4587	436	13	d	d	PROPN
ap-4587	436	14	is	be	AUX
ap-4587	436	15	therefore	therefore	ADV
ap-4587	436	16	reciprocal	reciprocal	ADJ
ap-4587	436	17	and	and	CCONJ
ap-4587	436	18	its	its	PRON
ap-4587	436	19	roots	root	NOUN
ap-4587	436	20	are	be	AUX
ap-4587	436	21	algebraic	algebraic	ADJ
ap-4587	436	22	units	unit	NOUN
ap-4587	436	23	lying	lie	VERB
ap-4587	436	24	on	on	ADP
ap-4587	436	25	the	the	DET
ap-4587	436	26	unit	unit	NOUN
ap-4587	436	27	circle	circle	NOUN
ap-4587	436	28	.	.	PUNCT
ap-4587	437	1	by	by	ADP
ap-4587	437	2	the	the	DET
ap-4587	437	3	well	well	ADV
ap-4587	437	4	-	-	PUNCT
ap-4587	437	5	known	know	VERB
ap-4587	437	6	result	result	NOUN
ap-4587	437	7	of	of	ADP
ap-4587	437	8	kronecker	kronecker	NOUN
ap-4587	437	9	,	,	PUNCT
ap-4587	437	10	these	these	PRON
ap-4587	437	11	must	must	AUX
ap-4587	437	12	be	be	AUX
ap-4587	437	13	roots	root	NOUN
ap-4587	437	14	of	of	ADP
ap-4587	437	15	unity	unity	NOUN
ap-4587	437	16	,	,	PUNCT
ap-4587	437	17	but	but	CCONJ
ap-4587	437	18	the	the	DET
ap-4587	437	19	only	only	ADJ
ap-4587	437	20	roots	root	NOUN
ap-4587	437	21	of	of	ADP
ap-4587	437	22	unity	unity	NOUN
ap-4587	437	23	lying	lie	VERB
ap-4587	437	24	in	in	ADP
ap-4587	437	25	the	the	DET
ap-4587	437	26	field	field	NOUN
ap-4587	437	27	q(ω	q(ω	NOUN
ap-4587	437	28	)	)	PUNCT
ap-4587	437	29	are	be	AUX
ap-4587	437	30	tenth	tenth	ADJ
ap-4587	437	31	roots	root	NOUN
ap-4587	437	32	of	of	ADP
ap-4587	437	33	unity	unity	NOUN
ap-4587	437	34	.	.	PUNCT
ap-4587	438	1	secondly	secondly	ADV
ap-4587	438	2	,	,	PUNCT
ap-4587	438	3	let	let	VERB
ap-4587	438	4	|σ2(η)|	|σ2(η)|	VERB
ap-4587	438	5	<	<	X
ap-4587	438	6	1	1	X
ap-4587	438	7	.	.	PUNCT
ap-4587	439	1	then	then	ADV
ap-4587	439	2	|σ1(η)|	|σ1(η)|	VERB
ap-4587	439	3	>	>	X
ap-4587	439	4	1	1	X
ap-4587	439	5	.	.	PUNCT
ap-4587	439	6	in	in	ADP
ap-4587	439	7	this	this	DET
ap-4587	439	8	case	case	NOUN
ap-4587	439	9	either	either	CCONJ
ap-4587	439	10	the	the	DET
ap-4587	439	11	characteristic	characteristic	ADJ
ap-4587	439	12	polynomial	polynomial	NOUN
ap-4587	439	13	of	of	ADP
ap-4587	439	14	the	the	DET
ap-4587	439	15	matrix	matrix	NOUN
ap-4587	439	16	za	za	PROPN
ap-4587	439	17	,	,	PUNCT
ap-4587	439	18	b	b	PROPN
ap-4587	439	19	,	,	PUNCT
ap-4587	439	20	c	c	X
ap-4587	439	21	,	,	PUNCT
ap-4587	439	22	d	d	NOUN
ap-4587	439	23	is	be	AUX
ap-4587	439	24	irreducible	irreducible	ADJ
ap-4587	439	25	over	over	ADP
ap-4587	439	26	q	q	PROPN
ap-4587	440	1	and	and	CCONJ
ap-4587	440	2	then	then	ADV
ap-4587	440	3	η	η	PROPN
ap-4587	440	4	is	be	AUX
ap-4587	440	5	a	a	DET
ap-4587	440	6	complex	complex	ADJ
ap-4587	440	7	pisot	pisot	ADJ
ap-4587	440	8	number	number	NOUN
ap-4587	440	9	of	of	ADP
ap-4587	440	10	degree	degree	NOUN
ap-4587	440	11	4	4	NUM
ap-4587	440	12	.	.	PUNCT
ap-4587	441	1	on	on	ADP
ap-4587	441	2	the	the	DET
ap-4587	441	3	other	other	ADJ
ap-4587	441	4	hand	hand	NOUN
ap-4587	441	5	,	,	PUNCT
ap-4587	441	6	if	if	SCONJ
ap-4587	441	7	it	it	PRON
ap-4587	441	8	is	be	AUX
ap-4587	441	9	reducible	reducible	ADJ
ap-4587	441	10	,	,	PUNCT
ap-4587	441	11	than	than	SCONJ
ap-4587	441	12	this	this	PRON
ap-4587	441	13	is	be	AUX
ap-4587	441	14	possible	possible	ADJ
ap-4587	441	15	only	only	ADV
ap-4587	441	16	if	if	SCONJ
ap-4587	441	17	η	η	PROPN
ap-4587	441	18	∈	∈	PROPN
ap-4587	441	19	r	r	NOUN
ap-4587	441	20	is	be	AUX
ap-4587	441	21	of	of	ADP
ap-4587	441	22	degree	degree	NOUN
ap-4587	441	23	2	2	NUM
ap-4587	441	24	.	.	PUNCT
ap-4587	442	1	in	in	ADP
ap-4587	442	2	this	this	DET
ap-4587	442	3	case	case	NOUN
ap-4587	442	4	η	η	X
ap-4587	442	5	=	=	SYM
ap-4587	442	6	σ1(η	σ1(η	PROPN
ap-4587	442	7	)	)	PUNCT
ap-4587	442	8	=	=	SYM
ap-4587	442	9	σ4(η	σ4(η	PROPN
ap-4587	442	10	)	)	PUNCT
ap-4587	442	11	and	and	CCONJ
ap-4587	442	12	σ3(η	σ3(η	NUM
ap-4587	442	13	)	)	PUNCT
ap-4587	442	14	=	=	SYM
ap-4587	442	15	σ4(η	σ4(η	X
ap-4587	442	16	)	)	PUNCT
ap-4587	442	17	is	be	AUX
ap-4587	442	18	its	its	PRON
ap-4587	442	19	algebraic	algebraic	ADJ
ap-4587	442	20	conjugate	conjugate	NOUN
ap-4587	442	21	.	.	PUNCT
ap-4587	443	1	it	it	PRON
ap-4587	443	2	follows	follow	VERB
ap-4587	443	3	that	that	SCONJ
ap-4587	443	4	η	η	PROPN
ap-4587	443	5	is	be	AUX
ap-4587	443	6	a	a	DET
ap-4587	443	7	quadratic	quadratic	ADJ
ap-4587	443	8	pisot	pisot	ADJ
ap-4587	443	9	number	number	NOUN
ap-4587	443	10	.	.	PUNCT
ap-4587	444	1	the	the	DET
ap-4587	444	2	above	above	ADJ
ap-4587	444	3	proposition	proposition	NOUN
ap-4587	444	4	states	state	VERB
ap-4587	444	5	that	that	SCONJ
ap-4587	444	6	non	non	ADJ
ap-4587	444	7	-	-	ADJ
ap-4587	444	8	trivial	trivial	ADJ
ap-4587	444	9	scaled	scale	VERB
ap-4587	444	10	rotations	rotation	NOUN
ap-4587	444	11	correspond	correspond	VERB
ap-4587	444	12	to	to	ADP
ap-4587	444	13	complex	complex	ADJ
ap-4587	444	14	pisot	pisot	ADJ
ap-4587	444	15	numbers	number	NOUN
ap-4587	444	16	in	in	ADP
ap-4587	444	17	the	the	DET
ap-4587	444	18	cyclotomic	cyclotomic	ADJ
ap-4587	444	19	integers	integer	NOUN
ap-4587	444	20	ring	re	VERB
ap-4587	444	21	z[ω	z[ω	PROPN
ap-4587	444	22	]	]	PUNCT
ap-4587	444	23	.	.	PUNCT
ap-4587	445	1	it	it	PRON
ap-4587	445	2	is	be	AUX
ap-4587	445	3	clear	clear	ADJ
ap-4587	445	4	that	that	SCONJ
ap-4587	445	5	any	any	DET
ap-4587	445	6	complex	complex	ADJ
ap-4587	445	7	pisot	pisot	ADJ
ap-4587	445	8	number	number	NOUN
ap-4587	445	9	in	in	ADP
ap-4587	445	10	z[ω	z[ω	NOUN
ap-4587	445	11	]	]	PUNCT
ap-4587	445	12	gives	give	VERB
ap-4587	445	13	a	a	DET
ap-4587	445	14	scaled	scale	VERB
ap-4587	445	15	rotation	rotation	NOUN
ap-4587	445	16	of	of	ADP
ap-4587	445	17	some	some	DET
ap-4587	445	18	cut	cut	VERB
ap-4587	445	19	-	-	PUNCT
ap-4587	445	20	and	and	CCONJ
ap-4587	445	21	-	-	PUNCT
ap-4587	445	22	project	project	NOUN
ap-4587	445	23	set	set	NOUN
ap-4587	445	24	.	.	PUNCT
ap-4587	446	1	proposition	proposition	NOUN
ap-4587	446	2	7.2	7.2	NUM
ap-4587	446	3	.	.	PUNCT
ap-4587	447	1	let	let	VERB
ap-4587	447	2	η	η	PROPN
ap-4587	447	3	∈	∈	PROPN
ap-4587	447	4	z[ω	z[ω	PROPN
ap-4587	447	5	]	]	PUNCT
ap-4587	447	6	be	be	AUX
ap-4587	447	7	a	a	DET
ap-4587	447	8	complex	complex	ADJ
ap-4587	447	9	pisot	pisot	ADJ
ap-4587	447	10	number	number	NOUN
ap-4587	447	11	,	,	PUNCT
ap-4587	447	12	η	η	NOUN
ap-4587	447	13	=	=	PRON
ap-4587	447	14	|η|(cosϕ	|η|(cosϕ	ADP
ap-4587	447	15	+	+	CCONJ
ap-4587	447	16	i	i	PRON
ap-4587	447	17	sinϕ	sinϕ	VERB
ap-4587	447	18	)	)	PUNCT
ap-4587	447	19	,	,	PUNCT
ap-4587	447	20	and	and	CCONJ
ap-4587	447	21	denote	denote	VERB
ap-4587	447	22	by	by	ADP
ap-4587	447	23	s	s	PRON
ap-4587	447	24	the	the	DET
ap-4587	447	25	mapping	mapping	NOUN
ap-4587	447	26	s	s	PART
ap-4587	447	27	=	=	SYM
ap-4587	447	28	|η|r	|η|r	NOUN
ap-4587	447	29	,	,	PUNCT
ap-4587	447	30	where	where	SCONJ
ap-4587	447	31	r	r	NOUN
ap-4587	447	32	is	be	AUX
ap-4587	447	33	a	a	DET
ap-4587	447	34	rotation	rotation	NOUN
ap-4587	447	35	by	by	ADP
ap-4587	447	36	the	the	DET
ap-4587	447	37	angle	angle	NOUN
ap-4587	447	38	ϕ.	ϕ.	NOUN
ap-4587	447	39	then	then	ADV
ap-4587	447	40	there	there	PRON
ap-4587	447	41	exists	exist	VERB
ap-4587	447	42	a	a	DET
ap-4587	447	43	window	window	NOUN
ap-4587	447	44	ω	ω	NOUN
ap-4587	447	45	⊂	⊂	PROPN
ap-4587	447	46	r2	r2	PROPN
ap-4587	447	47	such	such	ADJ
ap-4587	447	48	that	that	SCONJ
ap-4587	447	49	the	the	DET
ap-4587	447	50	mapping	mapping	NOUN
ap-4587	447	51	s	s	VERB
ap-4587	447	52	is	be	AUX
ap-4587	447	53	a	a	DET
ap-4587	447	54	self	self	NOUN
ap-4587	447	55	-	-	PUNCT
ap-4587	447	56	similarity	similarity	NOUN
ap-4587	447	57	of	of	ADP
ap-4587	447	58	the	the	DET
ap-4587	447	59	cut	cut	NOUN
ap-4587	447	60	-	-	PUNCT
ap-4587	447	61	and	and	CCONJ
ap-4587	447	62	-	-	PUNCT
ap-4587	447	63	project	project	NOUN
ap-4587	447	64	set	set	VERB
ap-4587	447	65	σ(ω	σ(ω	PROPN
ap-4587	447	66	)	)	PUNCT
ap-4587	447	67	.	.	PUNCT
ap-4587	448	1	however	however	ADV
ap-4587	448	2	,	,	PUNCT
ap-4587	448	3	scaled	scale	VERB
ap-4587	448	4	rotations	rotation	NOUN
ap-4587	448	5	are	be	AUX
ap-4587	448	6	not	not	PART
ap-4587	448	7	the	the	DET
ap-4587	448	8	only	only	ADJ
ap-4587	448	9	linear	linear	ADJ
ap-4587	448	10	self	self	NOUN
ap-4587	448	11	-	-	PUNCT
ap-4587	448	12	similarities	similarity	NOUN
ap-4587	448	13	possible	possible	ADJ
ap-4587	448	14	.	.	PUNCT
ap-4587	449	1	the	the	DET
ap-4587	449	2	following	follow	VERB
ap-4587	449	3	example	example	NOUN
ap-4587	449	4	shows	show	VERB
ap-4587	449	5	a	a	DET
ap-4587	449	6	non	non	ADJ
ap-4587	449	7	-	-	ADJ
ap-4587	449	8	trivial	trivial	ADJ
ap-4587	449	9	linear	linear	NOUN
ap-4587	449	10	map	map	NOUN
ap-4587	449	11	s	s	VERB
ap-4587	449	12	for	for	ADP
ap-4587	449	13	which	which	PRON
ap-4587	449	14	we	we	PRON
ap-4587	449	15	construct	construct	VERB
ap-4587	449	16	a	a	DET
ap-4587	449	17	cut	cut	VERB
ap-4587	449	18	-	-	PUNCT
ap-4587	449	19	and	and	CCONJ
ap-4587	449	20	-	-	PUNCT
ap-4587	449	21	project	project	NOUN
ap-4587	449	22	set	set	VERB
ap-4587	449	23	σ(ω	σ(ω	PROPN
ap-4587	449	24	)	)	PUNCT
ap-4587	449	25	with	with	ADP
ap-4587	449	26	self	self	NOUN
ap-4587	449	27	-	-	PUNCT
ap-4587	449	28	similarity	similarity	NOUN
ap-4587	449	29	s.	s.	PROPN
ap-4587	449	30	consider	consider	VERB
ap-4587	449	31	the	the	DET
ap-4587	449	32	matrix	matrix	NOUN
ap-4587	449	33	za,	za,	NOUN
ap-4587	449	34	...	...	PUNCT
ap-4587	449	35	,h	,h	PUNCT
ap-4587	450	1	∈	∈	PROPN
ap-4587	450	2	z	z	NOUN
ap-4587	450	3	where	where	SCONJ
ap-4587	450	4	we	we	PRON
ap-4587	450	5	choose	choose	VERB
ap-4587	450	6	a	a	DET
ap-4587	450	7	=	=	SYM
ap-4587	450	8	e	e	NOUN
ap-4587	450	9	=	=	SYM
ap-4587	450	10	g	g	PROPN
ap-4587	450	11	=	=	NOUN
ap-4587	450	12	h	h	NOUN
ap-4587	450	13	=	=	SYM
ap-4587	450	14	0	0	NUM
ap-4587	450	15	,	,	PUNCT
ap-4587	450	16	b	b	X
ap-4587	450	17	=	=	SYM
ap-4587	450	18	−1	−1	NOUN
ap-4587	450	19	,	,	PUNCT
ap-4587	450	20	and	and	CCONJ
ap-4587	450	21	c	c	X
ap-4587	450	22	=	=	SYM
ap-4587	451	1	d	d	PROPN
ap-4587	451	2	=	=	SYM
ap-4587	451	3	f	f	PROPN
ap-4587	451	4	=	=	SYM
ap-4587	451	5	1	1	NUM
ap-4587	451	6	,	,	PUNCT
ap-4587	451	7	i.e.	i.e.	X
ap-4587	451	8	,	,	PUNCT
ap-4587	451	9	z	z	NOUN
ap-4587	451	10	=	=	SYM
ap-4587	451	11			ADJ
ap-4587	451	12	0	0	NUM
ap-4587	451	13	−1	−1	NOUN
ap-4587	451	14	1	1	NUM
ap-4587	451	15	1	1	NUM
ap-4587	451	16	0	0	NUM
ap-4587	451	17	1	1	NUM
ap-4587	451	18	0	0	NUM
ap-4587	451	19	0	0	NUM
ap-4587	451	20	1	1	NUM
ap-4587	451	21	1	1	NUM
ap-4587	451	22	0	0	NUM
ap-4587	451	23	−1	−1	NOUN
ap-4587	451	24	1	1	NUM
ap-4587	451	25	0	0	NUM
ap-4587	451	26	0	0	NUM
ap-4587	451	27	0	0	NUM
ap-4587	451	28			NOUN
ap-4587	451	29	.	.	PUNCT
ap-4587	452	1	the	the	DET
ap-4587	452	2	corresponding	correspond	VERB
ap-4587	452	3	mappings	mapping	NOUN
ap-4587	452	4	s	s	PART
ap-4587	452	5	,	,	PUNCT
ap-4587	452	6	t	t	PROPN
ap-4587	452	7	related	relate	VERB
ap-4587	452	8	to	to	ADP
ap-4587	452	9	z	z	NOUN
ap-4587	452	10	by	by	ADP
ap-4587	452	11	v	v	NUM
ap-4587	452	12	zv	zv	NOUN
ap-4587	452	13	−1	−1	NOUN
ap-4587	452	14	=	=	PUNCT
ap-4587	453	1	(	(	PUNCT
ap-4587	453	2	s	s	AUX
ap-4587	453	3	o	o	X
ap-4587	453	4	o	o	X
ap-4587	453	5	t	t	PROPN
ap-4587	453	6	)	)	PUNCT
ap-4587	453	7	are	be	AUX
ap-4587	453	8	of	of	ADP
ap-4587	453	9	the	the	DET
ap-4587	453	10	form	form	NOUN
ap-4587	453	11	s	s	PART
ap-4587	453	12	=	=	PUNCT
ap-4587	453	13	(	(	PUNCT
ap-4587	453	14	−	−	PROPN
ap-4587	453	15	cos	cos	ADP
ap-4587	453	16	2π	2π	PROPN
ap-4587	453	17	5	5	NUM
ap-4587	453	18	+	+	SYM
ap-4587	453	19	2	2	NUM
ap-4587	453	20	cos	cos	X
ap-4587	453	21	4π	4π	NUM
ap-4587	453	22	5	5	NUM
ap-4587	453	23	−	−	NOUN
ap-4587	453	24	sin	sin	NOUN
ap-4587	453	25	2π	2π	PROPN
ap-4587	453	26	5	5	NUM
ap-4587	453	27	sin	sin	NOUN
ap-4587	453	28	2π	2π	PROPN
ap-4587	453	29	5	5	NUM
ap-4587	453	30	cos	cos	ADP
ap-4587	453	31	2π	2π	PROPN
ap-4587	453	32	5	5	NUM
ap-4587	453	33	+	+	CCONJ
ap-4587	453	34	1	1	NUM
ap-4587	453	35	)	)	PUNCT
ap-4587	453	36	=	=	SYM
ap-4587	454	1	(	(	PUNCT
ap-4587	454	2	−τ	−τ	INTJ
ap-4587	454	3	−	−	PROPN
ap-4587	454	4	1	1	NUM
ap-4587	454	5	2τ	2τ	NUM
ap-4587	454	6	−	−	NOUN
ap-4587	454	7	1	1	NUM
ap-4587	454	8	2	2	NUM
ap-4587	454	9	√	√	NOUN
ap-4587	454	10	τ2	τ2	NOUN
ap-4587	455	1	+	+	CCONJ
ap-4587	455	2	1	1	NUM
ap-4587	455	3	1	1	NUM
ap-4587	455	4	2	2	NUM
ap-4587	455	5	√	√	NOUN
ap-4587	455	6	τ2	τ2	NOUN
ap-4587	455	7	+	+	CCONJ
ap-4587	455	8	1	1	NUM
ap-4587	455	9	1	1	NUM
ap-4587	455	10	2τ	2τ	NUM
ap-4587	455	11	+	+	CCONJ
ap-4587	455	12	1	1	NUM
ap-4587	455	13	)	)	PUNCT
ap-4587	455	14	,	,	PUNCT
ap-4587	455	15	t	t	NOUN
ap-4587	455	16	=	=	SYM
ap-4587	455	17	(	(	PUNCT
ap-4587	455	18	2	2	NUM
ap-4587	455	19	cos	cos	ADP
ap-4587	455	20	2π	2π	PROPN
ap-4587	455	21	5	5	NUM
ap-4587	455	22	−	−	NOUN
ap-4587	455	23	cos	cos	PROPN
ap-4587	455	24	4π	4π	NUM
ap-4587	455	25	5	5	NUM
ap-4587	455	26	−	−	NOUN
ap-4587	455	27	sin	sin	NOUN
ap-4587	455	28	4π	4π	NUM
ap-4587	455	29	5	5	NUM
ap-4587	455	30	sin	sin	NOUN
ap-4587	455	31	4π	4π	NUM
ap-4587	455	32	5	5	NUM
ap-4587	455	33	cos	cos	ADP
ap-4587	455	34	4π	4π	NUM
ap-4587	455	35	5	5	NUM
ap-4587	455	36	+	+	CCONJ
ap-4587	455	37	1	1	NUM
ap-4587	455	38	)	)	PUNCT
ap-4587	455	39	=	=	PUNCT
ap-4587	455	40	(	(	PUNCT
ap-4587	455	41	1	1	NUM
ap-4587	455	42	τ	τ	PROPN
ap-4587	455	43	+	+	NUM
ap-4587	455	44	τ	τ	X
ap-4587	455	45	2	2	NUM
ap-4587	455	46	−	−	PROPN
ap-4587	455	47	1	1	NUM
ap-4587	455	48	2τ	2τ	NUM
ap-4587	455	49	√	√	ADP
ap-4587	455	50	τ2	τ2	VERB
ap-4587	455	51	+	+	CCONJ
ap-4587	455	52	1	1	NUM
ap-4587	455	53	1	1	NUM
ap-4587	455	54	2τ	2τ	NUM
ap-4587	455	55	√	√	ADP
ap-4587	455	56	τ2	τ2	VERB
ap-4587	455	57	+	+	CCONJ
ap-4587	455	58	1	1	NUM
ap-4587	455	59	1−	1−	NUM
ap-4587	455	60	τ	τ	X
ap-4587	455	61	2	2	NUM
ap-4587	455	62	)	)	PUNCT
ap-4587	455	63	.	.	PUNCT
ap-4587	456	1	let	let	VERB
ap-4587	456	2	us	we	PRON
ap-4587	456	3	study	study	VERB
ap-4587	456	4	the	the	DET
ap-4587	456	5	action	action	NOUN
ap-4587	456	6	of	of	ADP
ap-4587	456	7	the	the	DET
ap-4587	456	8	linear	linear	PROPN
ap-4587	456	9	map	map	NOUN
ap-4587	456	10	s.	s.	PROPN
ap-4587	456	11	its	its	PRON
ap-4587	456	12	eigenvalues	eigenvalue	NOUN
ap-4587	456	13	are	be	AUX
ap-4587	456	14	λ1	λ1	ADJ
ap-4587	456	15	=	=	SYM
ap-4587	456	16	−τ	−τ	NOUN
ap-4587	456	17	,	,	PUNCT
ap-4587	456	18	λ2	λ2	NOUN
ap-4587	456	19	=	=	SYM
ap-4587	456	20	1	1	NUM
ap-4587	456	21	,	,	PUNCT
ap-4587	456	22	corresponding	correspond	VERB
ap-4587	456	23	to	to	ADP
ap-4587	456	24	the	the	DET
ap-4587	456	25	eigenvectors	eigenvector	NOUN
ap-4587	456	26	w1	w1	NOUN
ap-4587	456	27	=	=	SYM
ap-4587	457	1	(	(	PUNCT
ap-4587	457	2	−	−	PUNCT
ap-4587	457	3	sin	sin	VERB
ap-4587	457	4	2π	2π	PROPN
ap-4587	457	5	5	5	NUM
ap-4587	457	6	cos	cos	ADP
ap-4587	457	7	2π	2π	PROPN
ap-4587	457	8	5	5	NUM
ap-4587	457	9	)	)	PUNCT
ap-4587	457	10	,	,	PUNCT
ap-4587	457	11	w2	w2	NOUN
ap-4587	457	12	=	=	SYM
ap-4587	457	13	(	(	PUNCT
ap-4587	457	14	cos	cos	ADP
ap-4587	457	15	2π	2π	PROPN
ap-4587	457	16	5	5	NUM
ap-4587	457	17	−	−	NOUN
ap-4587	457	18	sin	sin	NOUN
ap-4587	457	19	2π	2π	NOUN
ap-4587	457	20	5	5	NUM
ap-4587	457	21	)	)	PUNCT
ap-4587	457	22	.	.	PUNCT
ap-4587	458	1	the	the	DET
ap-4587	458	2	mapping	mapping	NOUN
ap-4587	458	3	s	s	VERB
ap-4587	458	4	thus	thus	ADV
ap-4587	458	5	acts	act	VERB
ap-4587	458	6	in	in	ADP
ap-4587	458	7	the	the	DET
ap-4587	458	8	direction	direction	NOUN
ap-4587	458	9	of	of	ADP
ap-4587	458	10	w1	w1	NOUN
ap-4587	458	11	as	as	ADP
ap-4587	458	12	a	a	DET
ap-4587	458	13	scaling	scaling	NOUN
ap-4587	458	14	by	by	ADP
ap-4587	458	15	the	the	DET
ap-4587	458	16	factor	factor	NOUN
ap-4587	458	17	−τ	−τ	NOUN
ap-4587	458	18	,	,	PUNCT
ap-4587	458	19	and	and	CCONJ
ap-4587	458	20	in	in	ADP
ap-4587	458	21	the	the	DET
ap-4587	458	22	direction	direction	NOUN
ap-4587	458	23	of	of	ADP
ap-4587	458	24	w2	w2	NOUN
ap-4587	458	25	as	as	ADP
ap-4587	458	26	the	the	DET
ap-4587	458	27	identity	identity	NOUN
ap-4587	458	28	.	.	PUNCT
ap-4587	459	1	figure	figure	NOUN
ap-4587	459	2	1	1	NUM
ap-4587	459	3	shows	show	VERB
ap-4587	459	4	the	the	DET
ap-4587	459	5	action	action	NOUN
ap-4587	459	6	of	of	ADP
ap-4587	459	7	s	s	PRON
ap-4587	459	8	on	on	ADP
ap-4587	459	9	the	the	DET
ap-4587	459	10	regular	regular	ADJ
ap-4587	459	11	decagon	decagon	NOUN
ap-4587	459	12	v0	v0	NOUN
ap-4587	459	13	,	,	PUNCT
ap-4587	459	14	.	.	PUNCT
ap-4587	459	15	.	.	PUNCT
ap-4587	460	1	.v9	.v9	PROPN
ap-4587	460	2	centered	center	VERB
ap-4587	460	3	in	in	ADP
ap-4587	460	4	the	the	DET
ap-4587	460	5	origin	origin	NOUN
ap-4587	460	6	.	.	PUNCT
ap-4587	461	1	let	let	VERB
ap-4587	461	2	us	we	PRON
ap-4587	461	3	study	study	VERB
ap-4587	461	4	the	the	DET
ap-4587	461	5	action	action	NOUN
ap-4587	461	6	of	of	ADP
ap-4587	461	7	the	the	DET
ap-4587	461	8	mapping	mapping	NOUN
ap-4587	461	9	t	t	NOUN
ap-4587	461	10	.	.	PUNCT
ap-4587	462	1	its	its	PRON
ap-4587	462	2	eigenvalues	eigenvalue	NOUN
ap-4587	462	3	are	be	AUX
ap-4587	462	4	η1	η1	NOUN
ap-4587	462	5	=	=	SYM
ap-4587	462	6	1	1	NUM
ap-4587	462	7	τ	τ	PROPN
ap-4587	462	8	,	,	PUNCT
ap-4587	462	9	η2	η2	PROPN
ap-4587	462	10	=	=	SYM
ap-4587	462	11	1	1	NUM
ap-4587	462	12	,	,	PUNCT
ap-4587	462	13	442	442	NUM
ap-4587	462	14	vol	vol	NOUN
ap-4587	462	15	.	.	PUNCT
ap-4587	463	1	57	57	NUM
ap-4587	463	2	no	no	NOUN
ap-4587	463	3	.	.	PUNCT
ap-4587	464	1	6/2017	6/2017	NUM
ap-4587	464	2	on	on	ADP
ap-4587	464	3	self	self	NOUN
ap-4587	464	4	-	-	PUNCT
ap-4587	464	5	similarities	similarity	NOUN
ap-4587	464	6	of	of	ADP
ap-4587	464	7	cut	cut	NOUN
ap-4587	464	8	-	-	PUNCT
ap-4587	464	9	and	and	CCONJ
ap-4587	464	10	-	-	PUNCT
ap-4587	464	11	project	project	NOUN
ap-4587	464	12	sets	set	NOUN
ap-4587	464	13	sv0	sv0	ADJ
ap-4587	464	14	sv1	sv1	PROPN
ap-4587	464	15	sv2	sv2	PROPN
ap-4587	464	16	sv3	sv3	PROPN
ap-4587	464	17	sv4	sv4	PROPN
ap-4587	464	18	sv5	sv5	PROPN
ap-4587	464	19	sv6sv7	sv6sv7	PROPN
ap-4587	464	20	sv8	sv8	PROPN
ap-4587	464	21	sv9	sv9	NOUN
ap-4587	464	22	v0	v0	NOUN
ap-4587	464	23	v1	v1	NOUN
ap-4587	464	24	v2v3	v2v3	PROPN
ap-4587	464	25	v4	v4	PROPN
ap-4587	464	26	v5	v5	PROPN
ap-4587	464	27	v6	v6	NOUN
ap-4587	464	28	v7	v7	VERB
ap-4587	464	29	v8	v8	PROPN
ap-4587	464	30	v9	v9	PROPN
ap-4587	464	31	w1	w1	NOUN
ap-4587	464	32	w2	w2	NOUN
ap-4587	464	33	figure	figure	NOUN
ap-4587	464	34	1	1	NUM
ap-4587	464	35	.	.	PUNCT
ap-4587	465	1	the	the	DET
ap-4587	465	2	action	action	NOUN
ap-4587	465	3	of	of	ADP
ap-4587	465	4	the	the	DET
ap-4587	465	5	mapping	mapping	NOUN
ap-4587	465	6	s	s	NOUN
ap-4587	465	7	on	on	ADP
ap-4587	465	8	the	the	DET
ap-4587	465	9	regular	regular	ADJ
ap-4587	465	10	decagon	decagon	NOUN
ap-4587	465	11	.	.	PUNCT
ap-4587	466	1	its	its	PRON
ap-4587	466	2	vertices	vertex	NOUN
ap-4587	466	3	,	,	PUNCT
ap-4587	466	4	namely	namely	ADV
ap-4587	466	5	the	the	DET
ap-4587	466	6	points	point	NOUN
ap-4587	466	7	vi	vi	PROPN
ap-4587	466	8	,	,	PUNCT
ap-4587	466	9	i	i	PRON
ap-4587	466	10	=	=	NOUN
ap-4587	466	11	0	0	NUM
ap-4587	466	12	,	,	PUNCT
ap-4587	466	13	.	.	PUNCT
ap-4587	466	14	.	.	PUNCT
ap-4587	466	15	.	.	PUNCT
ap-4587	467	1	,	,	PUNCT
ap-4587	467	2	9	9	NUM
ap-4587	467	3	,	,	PUNCT
ap-4587	467	4	are	be	AUX
ap-4587	467	5	marked	mark	VERB
ap-4587	467	6	with	with	ADP
ap-4587	467	7	black	black	ADJ
ap-4587	467	8	dots	dot	NOUN
ap-4587	467	9	.	.	PUNCT
ap-4587	468	1	the	the	DET
ap-4587	468	2	points	point	NOUN
ap-4587	468	3	svi	svi	PROPN
ap-4587	468	4	are	be	AUX
ap-4587	468	5	marked	mark	VERB
ap-4587	468	6	by	by	ADP
ap-4587	468	7	larger	large	ADJ
ap-4587	468	8	grey	grey	ADJ
ap-4587	468	9	dots	dot	NOUN
ap-4587	468	10	.	.	PUNCT
ap-4587	469	1	with	with	ADP
ap-4587	469	2	the	the	DET
ap-4587	469	3	corresponding	corresponding	ADJ
ap-4587	469	4	eigenvectors	eigenvector	NOUN
ap-4587	469	5	z1	z1	X
ap-4587	469	6	=	=	SYM
ap-4587	469	7	(	(	PUNCT
ap-4587	469	8	−	−	PUNCT
ap-4587	469	9	sin	sin	NOUN
ap-4587	469	10	4π	4π	NUM
ap-4587	469	11	5	5	NUM
ap-4587	469	12	cos	cos	ADP
ap-4587	469	13	4π	4π	NUM
ap-4587	469	14	5	5	NUM
ap-4587	469	15	)	)	PUNCT
ap-4587	469	16	,	,	PUNCT
ap-4587	469	17	z2	z2	PROPN
ap-4587	469	18	=	=	SYM
ap-4587	469	19	(	(	PUNCT
ap-4587	469	20	cos	cos	ADP
ap-4587	469	21	4π	4π	NUM
ap-4587	469	22	5	5	NUM
ap-4587	469	23	−	−	NOUN
ap-4587	469	24	sin	sin	NOUN
ap-4587	469	25	4π	4π	NUM
ap-4587	469	26	5	5	NUM
ap-4587	469	27	)	)	PUNCT
ap-4587	469	28	.	.	PUNCT
ap-4587	470	1	since	since	SCONJ
ap-4587	470	2	the	the	DET
ap-4587	470	3	eigenvalues	eigenvalue	NOUN
ap-4587	470	4	of	of	ADP
ap-4587	470	5	the	the	DET
ap-4587	470	6	matrix	matrix	NOUN
ap-4587	470	7	t	t	NOUN
ap-4587	470	8	are	be	AUX
ap-4587	470	9	in	in	ADP
ap-4587	470	10	modulus	modulus	ADJ
ap-4587	470	11	≤	≤	ADJ
ap-4587	470	12	1	1	NUM
ap-4587	470	13	,	,	PUNCT
ap-4587	470	14	by	by	ADP
ap-4587	470	15	proposition	proposition	NOUN
ap-4587	470	16	4.2	4.2	NUM
ap-4587	470	17	,	,	PUNCT
ap-4587	470	18	there	there	PRON
ap-4587	470	19	exists	exist	VERB
ap-4587	470	20	a	a	DET
ap-4587	470	21	window	window	NOUN
ap-4587	470	22	ω	ω	NOUN
ap-4587	470	23	such	such	ADJ
ap-4587	470	24	that	that	SCONJ
ap-4587	470	25	the	the	DET
ap-4587	470	26	cut	cut	VERB
ap-4587	470	27	-	-	PUNCT
ap-4587	470	28	and	and	CCONJ
ap-4587	470	29	-	-	PUNCT
ap-4587	470	30	project	project	NOUN
ap-4587	470	31	set	set	VERB
ap-4587	470	32	σ(ω	σ(ω	PROPN
ap-4587	470	33	)	)	PUNCT
ap-4587	470	34	has	have	VERB
ap-4587	470	35	self	self	NOUN
ap-4587	470	36	-	-	PUNCT
ap-4587	470	37	similarity	similarity	NOUN
ap-4587	470	38	s.	s.	PROPN
ap-4587	470	39	the	the	DET
ap-4587	470	40	window	window	NOUN
ap-4587	470	41	ω	ω	PROPN
ap-4587	470	42	must	must	AUX
ap-4587	470	43	be	be	AUX
ap-4587	470	44	chosen	choose	VERB
ap-4587	470	45	such	such	ADJ
ap-4587	470	46	that	that	SCONJ
ap-4587	470	47	it	it	PRON
ap-4587	470	48	is	be	AUX
ap-4587	470	49	invariant	invariant	ADJ
ap-4587	470	50	under	under	ADP
ap-4587	470	51	the	the	DET
ap-4587	470	52	action	action	NOUN
ap-4587	470	53	of	of	ADP
ap-4587	470	54	t	t	PROPN
ap-4587	470	55	.	.	PUNCT
ap-4587	471	1	obviously	obviously	ADV
ap-4587	471	2	,	,	PUNCT
ap-4587	471	3	we	we	PRON
ap-4587	471	4	could	could	AUX
ap-4587	471	5	choose	choose	VERB
ap-4587	471	6	for	for	ADP
ap-4587	471	7	example	example	NOUN
ap-4587	471	8	a	a	DET
ap-4587	471	9	parallelogram	parallelogram	NOUN
ap-4587	471	10	ω	ω	PROPN
ap-4587	471	11	=	=	PUNCT
ap-4587	471	12	{	{	PUNCT
ap-4587	471	13	α1z1	α1z1	PROPN
ap-4587	471	14	+	+	CCONJ
ap-4587	471	15	α2z2	α2z2	NUM
ap-4587	471	16	:	:	PUNCT
ap-4587	471	17	αi	αi	PROPN
ap-4587	471	18	∈	∈	PROPN
ap-4587	472	1	[	[	X
ap-4587	472	2	−1	−1	NOUN
ap-4587	472	3	,	,	PUNCT
ap-4587	472	4	1	1	NUM
ap-4587	472	5	]	]	PUNCT
ap-4587	472	6	}	}	PUNCT
ap-4587	472	7	.	.	PUNCT
ap-4587	473	1	for	for	ADP
ap-4587	473	2	illustration	illustration	NOUN
ap-4587	473	3	,	,	PUNCT
ap-4587	473	4	let	let	VERB
ap-4587	473	5	us	we	PRON
ap-4587	473	6	find	find	VERB
ap-4587	473	7	a	a	DET
ap-4587	473	8	window	window	NOUN
ap-4587	473	9	according	accord	VERB
ap-4587	473	10	to	to	ADP
ap-4587	473	11	the	the	DET
ap-4587	473	12	construction	construction	NOUN
ap-4587	473	13	presented	present	VERB
ap-4587	473	14	in	in	ADP
ap-4587	473	15	the	the	DET
ap-4587	473	16	proof	proof	NOUN
ap-4587	473	17	of	of	ADP
ap-4587	473	18	proposition	proposition	NOUN
ap-4587	473	19	4.3	4.3	NUM
ap-4587	473	20	,	,	PUNCT
ap-4587	473	21	namely	namely	ADV
ap-4587	473	22	with	with	ADP
ap-4587	473	23	the	the	DET
ap-4587	473	24	use	use	NOUN
ap-4587	473	25	of	of	ADP
ap-4587	473	26	a	a	DET
ap-4587	473	27	special	special	ADJ
ap-4587	473	28	inner	inner	ADJ
ap-4587	473	29	product	product	NOUN
ap-4587	473	30	.	.	PUNCT
ap-4587	474	1	it	it	PRON
ap-4587	474	2	can	can	AUX
ap-4587	474	3	be	be	AUX
ap-4587	474	4	easily	easily	ADV
ap-4587	474	5	shown	show	VERB
ap-4587	474	6	that	that	SCONJ
ap-4587	474	7	a	a	DET
ap-4587	474	8	matrix	matrix	NOUN
ap-4587	474	9	g	g	NOUN
ap-4587	474	10	transferring	transfer	VERB
ap-4587	474	11	the	the	DET
ap-4587	474	12	eigenvectors	eigenvector	NOUN
ap-4587	474	13	zi	zi	X
ap-4587	474	14	of	of	ADP
ap-4587	474	15	t	t	PROPN
ap-4587	474	16	to	to	ADP
ap-4587	474	17	the	the	DET
ap-4587	474	18	vectors	vector	NOUN
ap-4587	474	19	of	of	ADP
ap-4587	474	20	the	the	DET
ap-4587	474	21	standard	standard	ADJ
ap-4587	474	22	basis	basis	NOUN
ap-4587	474	23	is	be	AUX
ap-4587	474	24	of	of	ADP
ap-4587	474	25	the	the	DET
ap-4587	474	26	form	form	NOUN
ap-4587	474	27	g	g	NOUN
ap-4587	474	28	=	=	SYM
ap-4587	474	29	1	1	NUM
ap-4587	474	30	cos	cos	ADP
ap-4587	474	31	2π	2π	PROPN
ap-4587	474	32	5	5	NUM
ap-4587	474	33	(	(	PUNCT
ap-4587	474	34	sin	sin	NOUN
ap-4587	474	35	4π	4π	NUM
ap-4587	474	36	5	5	NUM
ap-4587	475	1	cos	cos	ADP
ap-4587	475	2	4π	4π	NUM
ap-4587	475	3	5	5	NUM
ap-4587	475	4	cos	cos	ADP
ap-4587	475	5	4π	4π	NUM
ap-4587	475	6	5	5	NUM
ap-4587	475	7	sin	sin	NOUN
ap-4587	475	8	4π	4π	NUM
ap-4587	475	9	5	5	NUM
ap-4587	475	10	)	)	PUNCT
ap-4587	475	11	.	.	PUNCT
ap-4587	476	1	it	it	PRON
ap-4587	476	2	is	be	AUX
ap-4587	476	3	a	a	DET
ap-4587	476	4	symmetric	symmetric	ADJ
ap-4587	476	5	real	real	ADJ
ap-4587	476	6	matrix	matrix	NOUN
ap-4587	476	7	,	,	PUNCT
ap-4587	476	8	and	and	CCONJ
ap-4587	476	9	thus	thus	ADV
ap-4587	476	10	g	g	PROPN
ap-4587	476	11	=	=	PUNCT
ap-4587	476	12	g∗.	g∗.	NOUN
ap-4587	476	13	then	then	ADV
ap-4587	476	14	the	the	DET
ap-4587	476	15	matrix	matrix	NOUN
ap-4587	476	16	h	h	NOUN
ap-4587	477	1	=	=	NOUN
ap-4587	477	2	g∗g	g∗g	PROPN
ap-4587	477	3	=	=	PUNCT
ap-4587	477	4	g2	g2	PROPN
ap-4587	477	5	determining	determine	VERB
ap-4587	477	6	the	the	DET
ap-4587	477	7	inner	inner	ADJ
ap-4587	477	8	product	product	NOUN
ap-4587	477	9	is	be	AUX
ap-4587	477	10	given	give	VERB
ap-4587	477	11	by	by	ADP
ap-4587	477	12	h	h	NOUN
ap-4587	477	13	=	=	SYM
ap-4587	477	14	1	1	NUM
ap-4587	477	15	cos2	cos2	NOUN
ap-4587	477	16	2π	2π	PROPN
ap-4587	477	17	5	5	NUM
ap-4587	477	18	(	(	PUNCT
ap-4587	477	19	1	1	NUM
ap-4587	477	20	−	−	NOUN
ap-4587	477	21	sin	sin	NOUN
ap-4587	477	22	2π	2π	NOUN
ap-4587	477	23	5	5	NUM
ap-4587	477	24	−	−	NOUN
ap-4587	477	25	sin	sin	NOUN
ap-4587	477	26	2π	2π	NOUN
ap-4587	477	27	5	5	NUM
ap-4587	477	28	1	1	NUM
ap-4587	477	29	)	)	PUNCT
ap-4587	477	30	.	.	PUNCT
ap-4587	478	1	for	for	ADP
ap-4587	478	2	the	the	DET
ap-4587	478	3	window	window	NOUN
ap-4587	478	4	ω	ω	NOUN
ap-4587	478	5	we	we	PRON
ap-4587	478	6	can	can	AUX
ap-4587	478	7	choose	choose	VERB
ap-4587	478	8	a	a	DET
ap-4587	478	9	ball	ball	NOUN
ap-4587	478	10	in	in	ADP
ap-4587	478	11	the	the	DET
ap-4587	478	12	metric	metric	NOUN
ap-4587	478	13	induced	induce	VERB
ap-4587	478	14	by	by	ADP
ap-4587	478	15	the	the	DET
ap-4587	478	16	new	new	ADJ
ap-4587	478	17	inner	inner	ADJ
ap-4587	478	18	product	product	NOUN
ap-4587	478	19	,	,	PUNCT
ap-4587	478	20	namely	namely	ADV
ap-4587	478	21	ω	ω	NUM
ap-4587	478	22	=	=	SYM
ap-4587	478	23	{	{	PUNCT
ap-4587	478	24	x	x	PUNCT
ap-4587	478	25	∈	∈	PROPN
ap-4587	478	26	r2	r2	NOUN
ap-4587	478	27	:	:	PUNCT
ap-4587	479	1	〈	〈	PROPN
ap-4587	479	2	x	x	PRON
ap-4587	479	3	,	,	PUNCT
ap-4587	479	4	x〉h	x〉h	PROPN
ap-4587	479	5	≤	≤	PROPN
ap-4587	479	6	const	const	NOUN
ap-4587	479	7	.	.	PUNCT
ap-4587	479	8	}	}	PUNCT
ap-4587	479	9	.	.	PUNCT
ap-4587	480	1	in	in	ADP
ap-4587	480	2	particular	particular	ADJ
ap-4587	480	3	,	,	PUNCT
ap-4587	480	4	we	we	PRON
ap-4587	480	5	can	can	AUX
ap-4587	480	6	have	have	VERB
ap-4587	480	7	ω	ω	NUM
ap-4587	480	8	=	=	SYM
ap-4587	480	9	{	{	PUNCT
ap-4587	480	10	(	(	PUNCT
ap-4587	480	11	x	x	SYM
ap-4587	480	12	y	y	PROPN
ap-4587	480	13	)	)	PUNCT
ap-4587	480	14	∈	∈	PROPN
ap-4587	480	15	r2	r2	NOUN
ap-4587	480	16	:	:	PUNCT
ap-4587	481	1	x2	x2	PROPN
ap-4587	481	2	−	−	PROPN
ap-4587	481	3	2xy	2xy	ADJ
ap-4587	481	4	sin	sin	NOUN
ap-4587	481	5	2π	2π	NOUN
ap-4587	481	6	5	5	NUM
ap-4587	481	7	+	+	CCONJ
ap-4587	481	8	y2	y2	NOUN
ap-4587	481	9	≤	≤	ADJ
ap-4587	481	10	1	1	NUM
ap-4587	481	11	}	}	PUNCT
ap-4587	481	12	.	.	PUNCT
ap-4587	482	1	such	such	DET
ap-4587	482	2	a	a	DET
ap-4587	482	3	window	window	NOUN
ap-4587	482	4	ω	ω	NOUN
ap-4587	482	5	satisfies	satisfie	NOUN
ap-4587	482	6	tω	tω	PROPN
ap-4587	482	7	⊂	⊂	PROPN
ap-4587	482	8	ω	ω	PROPN
ap-4587	482	9	and	and	CCONJ
ap-4587	482	10	thus	thus	ADV
ap-4587	482	11	sς(ω	sς(ω	NUM
ap-4587	482	12	)	)	PUNCT
ap-4587	482	13	⊂	⊂	PROPN
ap-4587	482	14	σ(ω	σ(ω	PROPN
ap-4587	482	15	)	)	PUNCT
ap-4587	482	16	.	.	PUNCT
ap-4587	483	1	figure	figure	NOUN
ap-4587	483	2	2	2	NUM
ap-4587	483	3	shows	show	VERB
ap-4587	483	4	the	the	DET
ap-4587	483	5	window	window	NOUN
ap-4587	483	6	ω	ω	PROPN
ap-4587	483	7	and	and	CCONJ
ap-4587	483	8	its	its	PRON
ap-4587	483	9	transformation	transformation	NOUN
ap-4587	483	10	by	by	ADP
ap-4587	483	11	t	t	PROPN
ap-4587	483	12	.	.	PUNCT
ap-4587	484	1	8	8	X
ap-4587	484	2	.	.	PUNCT
ap-4587	484	3	comments	comment	NOUN
ap-4587	484	4	in	in	ADP
ap-4587	484	5	this	this	DET
ap-4587	484	6	article	article	NOUN
ap-4587	484	7	we	we	PRON
ap-4587	484	8	have	have	AUX
ap-4587	484	9	studied	study	VERB
ap-4587	484	10	affine	affine	NOUN
ap-4587	484	11	self	self	NOUN
ap-4587	484	12	-	-	PUNCT
ap-4587	484	13	similarities	similarity	NOUN
ap-4587	484	14	of	of	ADP
ap-4587	484	15	quasicrystal	quasicrystal	ADJ
ap-4587	484	16	models	model	NOUN
ap-4587	484	17	obtained	obtain	VERB
ap-4587	484	18	by	by	ADP
ap-4587	484	19	the	the	DET
ap-4587	484	20	cut	cut	NOUN
ap-4587	484	21	-	-	PUNCT
ap-4587	484	22	and	and	CCONJ
ap-4587	484	23	-	-	PUNCT
ap-4587	484	24	project	project	NOUN
ap-4587	484	25	method	method	NOUN
ap-4587	484	26	.	.	PUNCT
ap-4587	485	1	it	it	PRON
ap-4587	485	2	is	be	AUX
ap-4587	485	3	a	a	DET
ap-4587	485	4	first	first	ADJ
ap-4587	485	5	step	step	NOUN
ap-4587	485	6	towards	towards	ADP
ap-4587	485	7	solving	solve	VERB
ap-4587	485	8	the	the	DET
ap-4587	485	9	general	general	ADJ
ap-4587	485	10	question	question	NOUN
ap-4587	485	11	:	:	PUNCT
ap-4587	485	12	given	give	VERB
ap-4587	485	13	a	a	DET
ap-4587	485	14	linear	linear	ADJ
ap-4587	485	15	map	map	NOUN
ap-4587	485	16	a	a	DET
ap-4587	485	17	:	:	PUNCT
ap-4587	485	18	rn	rn	PROPN
ap-4587	485	19	→	→	SYM
ap-4587	485	20	rn	rn	PROPN
ap-4587	485	21	under	under	ADP
ap-4587	485	22	which	which	PRON
ap-4587	485	23	443	443	NUM
ap-4587	485	24	zuzana	zuzana	PROPN
ap-4587	485	25	masáková	masáková	PROPN
ap-4587	485	26	,	,	PUNCT
ap-4587	485	27	jan	jan	PROPN
ap-4587	485	28	mazáč	mazáč	PROPN
ap-4587	485	29	acta	acta	PROPN
ap-4587	485	30	polytechnica	polytechnica	PROPN
ap-4587	485	31	-3	-3	PUNCT
ap-4587	486	1	-2	-2	INTJ
ap-4587	487	1	-1	-1	INTJ
ap-4587	487	2	1	1	NUM
ap-4587	487	3	2	2	NUM
ap-4587	487	4	3	3	NUM
ap-4587	487	5	-3	-3	NOUN
ap-4587	487	6	-2	-2	INTJ
ap-4587	487	7	-1	-1	NOUN
ap-4587	487	8	1	1	NUM
ap-4587	487	9	2	2	NUM
ap-4587	487	10	3	3	NUM
ap-4587	487	11	figure	figure	NOUN
ap-4587	487	12	2	2	NUM
ap-4587	487	13	.	.	NOUN
ap-4587	487	14	acceptance	acceptance	NOUN
ap-4587	487	15	window	window	NOUN
ap-4587	487	16	ω	ω	PROPN
ap-4587	487	17	is	be	AUX
ap-4587	487	18	marked	mark	VERB
ap-4587	487	19	by	by	ADP
ap-4587	487	20	full	full	ADJ
ap-4587	487	21	line	line	NOUN
ap-4587	487	22	and	and	CCONJ
ap-4587	487	23	the	the	DET
ap-4587	487	24	action	action	NOUN
ap-4587	487	25	of	of	ADP
ap-4587	487	26	the	the	DET
ap-4587	487	27	mapping	mapping	NOUN
ap-4587	487	28	t	t	NOUN
ap-4587	487	29	to	to	ADP
ap-4587	487	30	ω	ω	PROPN
ap-4587	487	31	is	be	AUX
ap-4587	487	32	marked	mark	VERB
ap-4587	487	33	by	by	ADP
ap-4587	487	34	dashed	dash	VERB
ap-4587	487	35	line	line	NOUN
ap-4587	487	36	.	.	PUNCT
ap-4587	488	1	conditions	condition	NOUN
ap-4587	488	2	there	there	PRON
ap-4587	488	3	exist	exist	VERB
ap-4587	488	4	a	a	DET
ap-4587	488	5	cut	cut	NOUN
ap-4587	488	6	-	-	PUNCT
ap-4587	488	7	and	and	CCONJ
ap-4587	488	8	-	-	PUNCT
ap-4587	488	9	project	project	NOUN
ap-4587	488	10	scheme	scheme	NOUN
ap-4587	488	11	that	that	PRON
ap-4587	488	12	admits	admit	VERB
ap-4587	488	13	a	a	DET
ap-4587	488	14	cut	cut	VERB
ap-4587	488	15	-	-	PUNCT
ap-4587	488	16	and	and	CCONJ
ap-4587	488	17	-	-	PUNCT
ap-4587	488	18	project	project	NOUN
ap-4587	488	19	set	set	VERB
ap-4587	488	20	σ(ω	σ(ω	PROPN
ap-4587	488	21	)	)	PUNCT
ap-4587	488	22	such	such	ADJ
ap-4587	488	23	that	that	PRON
ap-4587	488	24	aς(ω	aς(ω	NUM
ap-4587	488	25	)	)	PUNCT
ap-4587	489	1	⊂	⊂	PROPN
ap-4587	489	2	σ(ω	σ(ω	PROPN
ap-4587	489	3	)	)	PUNCT
ap-4587	489	4	?	?	PUNCT
ap-4587	490	1	how	how	SCONJ
ap-4587	490	2	to	to	PART
ap-4587	490	3	find	find	VERB
ap-4587	490	4	such	such	DET
ap-4587	490	5	a	a	DET
ap-4587	490	6	cut	cut	NOUN
ap-4587	490	7	-	-	PUNCT
ap-4587	490	8	and	and	CCONJ
ap-4587	490	9	-	-	PUNCT
ap-4587	490	10	project	project	NOUN
ap-4587	490	11	scheme	scheme	NOUN
ap-4587	490	12	?	?	PUNCT
ap-4587	491	1	which	which	PRON
ap-4587	491	2	other	other	ADJ
ap-4587	491	3	linear	linear	NOUN
ap-4587	491	4	self	self	NOUN
ap-4587	491	5	-	-	PUNCT
ap-4587	491	6	similarities	similarity	NOUN
ap-4587	491	7	such	such	DET
ap-4587	491	8	a	a	DET
ap-4587	491	9	cut	cut	NOUN
ap-4587	491	10	-	-	PUNCT
ap-4587	491	11	and	and	CCONJ
ap-4587	491	12	-	-	PUNCT
ap-4587	491	13	project	project	NOUN
ap-4587	491	14	set	set	NOUN
ap-4587	491	15	has	have	VERB
ap-4587	491	16	?	?	PUNCT
ap-4587	492	1	many	many	ADJ
ap-4587	492	2	problems	problem	NOUN
ap-4587	492	3	that	that	PRON
ap-4587	492	4	concern	concern	VERB
ap-4587	492	5	these	these	DET
ap-4587	492	6	questions	question	NOUN
ap-4587	492	7	remain	remain	VERB
ap-4587	492	8	,	,	PUNCT
ap-4587	492	9	however	however	ADV
ap-4587	492	10	,	,	PUNCT
ap-4587	492	11	unsolved	unsolved	ADJ
ap-4587	492	12	.	.	PUNCT
ap-4587	493	1	for	for	ADP
ap-4587	493	2	example	example	NOUN
ap-4587	493	3	,	,	PUNCT
ap-4587	493	4	what	what	PRON
ap-4587	493	5	are	be	AUX
ap-4587	493	6	the	the	DET
ap-4587	493	7	conditions	condition	NOUN
ap-4587	493	8	on	on	ADP
ap-4587	493	9	the	the	DET
ap-4587	493	10	linear	linear	ADJ
ap-4587	493	11	map	map	NOUN
ap-4587	493	12	a	a	DET
ap-4587	493	13	,	,	PUNCT
ap-4587	493	14	or	or	CCONJ
ap-4587	493	15	the	the	DET
ap-4587	493	16	corresponding	corresponding	ADJ
ap-4587	493	17	integer	integer	NOUN
ap-4587	493	18	matrix	matrix	NOUN
ap-4587	493	19	c	c	NOUN
ap-4587	493	20	,	,	PUNCT
ap-4587	493	21	so	so	SCONJ
ap-4587	493	22	that	that	SCONJ
ap-4587	493	23	the	the	DET
ap-4587	493	24	constructed	construct	VERB
ap-4587	493	25	cut	cut	NOUN
ap-4587	493	26	-	-	PUNCT
ap-4587	493	27	and	and	CCONJ
ap-4587	493	28	-	-	PUNCT
ap-4587	493	29	project	project	NOUN
ap-4587	493	30	scheme	scheme	NOUN
ap-4587	493	31	with	with	ADP
ap-4587	493	32	self	self	NOUN
ap-4587	493	33	-	-	PUNCT
ap-4587	493	34	similarity	similarity	NOUN
ap-4587	493	35	a	a	PRON
ap-4587	493	36	is	be	AUX
ap-4587	493	37	non	non	ADJ
ap-4587	493	38	-	-	ADJ
ap-4587	493	39	degenerate	degenerate	ADJ
ap-4587	493	40	and	and	CCONJ
ap-4587	493	41	irreducible	irreducible	ADJ
ap-4587	493	42	?	?	PUNCT
ap-4587	494	1	a	a	DET
ap-4587	494	2	second	second	ADJ
ap-4587	494	3	important	important	ADJ
ap-4587	494	4	question	question	NOUN
ap-4587	494	5	is	be	AUX
ap-4587	494	6	about	about	ADP
ap-4587	494	7	the	the	DET
ap-4587	494	8	linear	linear	PROPN
ap-4587	494	9	maps	map	NOUN
ap-4587	494	10	a	a	PRON
ap-4587	494	11	that	that	PRON
ap-4587	494	12	admit	admit	VERB
ap-4587	494	13	a	a	DET
ap-4587	494	14	window	window	NOUN
ap-4587	494	15	ω	ω	NOUN
ap-4587	494	16	,	,	PUNCT
ap-4587	494	17	so	so	SCONJ
ap-4587	494	18	that	that	SCONJ
ap-4587	494	19	a	a	DET
ap-4587	494	20	cut	cut	VERB
ap-4587	494	21	-	-	PUNCT
ap-4587	494	22	and	and	CCONJ
ap-4587	494	23	-	-	PUNCT
ap-4587	494	24	project	project	NOUN
ap-4587	494	25	set	set	VERB
ap-4587	494	26	σ(ω	σ(ω	PROPN
ap-4587	494	27	)	)	PUNCT
ap-4587	494	28	satisfies	satisfie	NOUN
ap-4587	494	29	aς(ω	aς(ω	NUM
ap-4587	494	30	)	)	PUNCT
ap-4587	494	31	⊂	⊂	PROPN
ap-4587	495	1	σ(ω	σ(ω	PROPN
ap-4587	495	2	)	)	PUNCT
ap-4587	495	3	.	.	PUNCT
ap-4587	496	1	the	the	DET
ap-4587	496	2	answer	answer	NOUN
ap-4587	496	3	to	to	ADP
ap-4587	496	4	such	such	DET
ap-4587	496	5	a	a	DET
ap-4587	496	6	question	question	NOUN
ap-4587	496	7	could	could	AUX
ap-4587	496	8	be	be	AUX
ap-4587	496	9	viewed	view	VERB
ap-4587	496	10	as	as	ADP
ap-4587	496	11	a	a	DET
ap-4587	496	12	generalization	generalization	NOUN
ap-4587	496	13	of	of	ADP
ap-4587	496	14	lagarias	lagarias	PROPN
ap-4587	496	15	’	'	PUNCT
ap-4587	496	16	result	result	NOUN
ap-4587	496	17	of	of	ADP
ap-4587	496	18	[	[	X
ap-4587	496	19	11	11	NUM
ap-4587	496	20	]	]	PUNCT
ap-4587	496	21	.	.	PUNCT
ap-4587	497	1	one	one	PRON
ap-4587	497	2	can	can	AUX
ap-4587	497	3	also	also	ADV
ap-4587	497	4	ask	ask	VERB
ap-4587	497	5	a	a	DET
ap-4587	497	6	further	further	ADJ
ap-4587	497	7	question	question	NOUN
ap-4587	497	8	,	,	PUNCT
ap-4587	497	9	namely	namely	ADV
ap-4587	497	10	:	:	PUNCT
ap-4587	497	11	given	give	VERB
ap-4587	497	12	a	a	DET
ap-4587	497	13	cut	cut	VERB
ap-4587	497	14	-	-	PUNCT
ap-4587	497	15	and	and	CCONJ
ap-4587	497	16	-	-	PUNCT
ap-4587	497	17	project	project	NOUN
ap-4587	497	18	set	set	VERB
ap-4587	497	19	σ(ω	σ(ω	PROPN
ap-4587	497	20	)	)	PUNCT
ap-4587	497	21	,	,	PUNCT
ap-4587	497	22	what	what	PRON
ap-4587	497	23	are	be	AUX
ap-4587	497	24	its	its	PRON
ap-4587	497	25	possible	possible	ADJ
ap-4587	497	26	self	self	NOUN
ap-4587	497	27	-	-	PUNCT
ap-4587	497	28	similarities	similarity	NOUN
ap-4587	497	29	?	?	PUNCT
ap-4587	498	1	however	however	ADV
ap-4587	498	2	,	,	PUNCT
ap-4587	498	3	answer	answer	VERB
ap-4587	498	4	to	to	ADP
ap-4587	498	5	such	such	DET
ap-4587	498	6	a	a	DET
ap-4587	498	7	question	question	NOUN
ap-4587	498	8	heavily	heavily	ADV
ap-4587	498	9	depends	depend	VERB
ap-4587	498	10	on	on	ADP
ap-4587	498	11	the	the	DET
ap-4587	498	12	form	form	NOUN
ap-4587	498	13	of	of	ADP
ap-4587	498	14	the	the	DET
ap-4587	498	15	acceptance	acceptance	NOUN
ap-4587	498	16	window	window	NOUN
ap-4587	498	17	ω	ω	PROPN
ap-4587	498	18	.	.	PUNCT
ap-4587	499	1	some	some	DET
ap-4587	499	2	research	research	NOUN
ap-4587	499	3	in	in	ADP
ap-4587	499	4	this	this	DET
ap-4587	499	5	direction	direction	NOUN
ap-4587	499	6	has	have	AUX
ap-4587	499	7	been	be	AUX
ap-4587	499	8	done	do	VERB
ap-4587	499	9	for	for	ADP
ap-4587	499	10	pentagonal	pentagonal	ADJ
ap-4587	499	11	quasicrystals	quasicrystal	NOUN
ap-4587	499	12	in	in	ADP
ap-4587	499	13	[	[	X
ap-4587	499	14	4	4	NUM
ap-4587	499	15	]	]	PUNCT
ap-4587	499	16	and	and	CCONJ
ap-4587	499	17	[	[	X
ap-4587	499	18	13	13	NUM
ap-4587	499	19	,	,	PUNCT
ap-4587	499	20	14	14	NUM
ap-4587	499	21	]	]	PUNCT
ap-4587	499	22	,	,	PUNCT
ap-4587	499	23	all	all	PRON
ap-4587	499	24	of	of	ADP
ap-4587	499	25	these	these	PRON
ap-4587	499	26	however	however	ADV
ap-4587	499	27	only	only	ADV
ap-4587	499	28	for	for	ADP
ap-4587	499	29	scaling	scale	VERB
ap-4587	499	30	symmetries	symmetry	NOUN
ap-4587	499	31	.	.	PUNCT
ap-4587	500	1	9	9	X
ap-4587	500	2	.	.	X
ap-4587	500	3	acknowledgements	acknowledgement	NOUN
ap-4587	500	4	this	this	DET
ap-4587	500	5	work	work	NOUN
ap-4587	500	6	was	be	AUX
ap-4587	500	7	supported	support	VERB
ap-4587	500	8	by	by	ADP
ap-4587	500	9	the	the	DET
ap-4587	500	10	czech	czech	PROPN
ap-4587	500	11	science	science	PROPN
ap-4587	500	12	foundation	foundation	PROPN
ap-4587	500	13	,	,	PUNCT
ap-4587	500	14	grant	grant	VERB
ap-4587	500	15	no	no	NOUN
ap-4587	500	16	.	.	NOUN
ap-4587	501	1	13	13	NUM
ap-4587	501	2	-	-	PUNCT
ap-4587	501	3	03538s	03538s	NUM
ap-4587	501	4	.	.	PUNCT
ap-4587	502	1	we	we	PRON
ap-4587	502	2	also	also	ADV
ap-4587	502	3	acknowledge	acknowledge	VERB
ap-4587	502	4	financial	financial	ADJ
ap-4587	502	5	support	support	NOUN
ap-4587	502	6	of	of	ADP
ap-4587	502	7	the	the	DET
ap-4587	502	8	grant	grant	NOUN
ap-4587	502	9	agency	agency	NOUN
ap-4587	502	10	of	of	ADP
ap-4587	502	11	the	the	DET
ap-4587	502	12	czech	czech	PROPN
ap-4587	502	13	technical	technical	PROPN
ap-4587	502	14	university	university	PROPN
ap-4587	502	15	in	in	ADP
ap-4587	502	16	prague	prague	PROPN
ap-4587	502	17	,	,	PUNCT
ap-4587	502	18	grant	grant	VERB
ap-4587	502	19	no	no	INTJ
ap-4587	502	20	.	.	PUNCT
ap-4587	503	1	sgs17/193	sgs17/193	PRON
ap-4587	503	2	/	/	SYM
ap-4587	503	3	ohk4/3t/14	ohk4/3t/14	NOUN
ap-4587	503	4	.	.	PUNCT
ap-4587	504	1	references	reference	NOUN
ap-4587	504	2	[	[	X
ap-4587	504	3	1	1	X
ap-4587	504	4	]	]	PUNCT
ap-4587	504	5	michael	michael	PROPN
ap-4587	504	6	baake	baake	PROPN
ap-4587	504	7	and	and	CCONJ
ap-4587	504	8	uwe	uwe	PROPN
ap-4587	504	9	grimm	grimm	PROPN
ap-4587	504	10	,	,	PUNCT
ap-4587	504	11	aperiodic	aperiodic	ADJ
ap-4587	504	12	order	order	NOUN
ap-4587	504	13	.	.	PUNCT
ap-4587	505	1	vol	vol	NOUN
ap-4587	505	2	.	.	PROPN
ap-4587	506	1	1	1	NUM
ap-4587	506	2	,	,	PUNCT
ap-4587	506	3	encyclopedia	encyclopedia	NOUN
ap-4587	506	4	of	of	ADP
ap-4587	506	5	mathematics	mathematic	NOUN
ap-4587	506	6	and	and	CCONJ
ap-4587	506	7	its	its	PRON
ap-4587	506	8	applications	application	NOUN
ap-4587	506	9	,	,	PUNCT
ap-4587	506	10	vol	vol	NOUN
ap-4587	506	11	.	.	PROPN
ap-4587	506	12	149	149	NUM
ap-4587	506	13	,	,	PUNCT
ap-4587	506	14	cambridge	cambridge	PROPN
ap-4587	506	15	university	university	PROPN
ap-4587	506	16	press	press	PROPN
ap-4587	506	17	,	,	PUNCT
ap-4587	506	18	cambridge	cambridge	PROPN
ap-4587	506	19	,	,	PUNCT
ap-4587	506	20	2013	2013	NUM
ap-4587	506	21	,	,	PUNCT
ap-4587	506	22	a	a	DET
ap-4587	506	23	mathematical	mathematical	ADJ
ap-4587	506	24	invitation	invitation	NOUN
ap-4587	506	25	,	,	PUNCT
ap-4587	506	26	with	with	ADP
ap-4587	506	27	a	a	DET
ap-4587	506	28	foreword	foreword	NOUN
ap-4587	506	29	by	by	ADP
ap-4587	506	30	roger	roger	PROPN
ap-4587	506	31	penrose	penrose	PROPN
ap-4587	506	32	.	.	PUNCT
ap-4587	507	1	mr	mr	PROPN
ap-4587	507	2	3136260	3136260	NUM
ap-4587	507	3	[	[	X
ap-4587	507	4	2	2	NUM
ap-4587	507	5	]	]	PUNCT
ap-4587	507	6	michael	michael	PROPN
ap-4587	507	7	baake	baake	PROPN
ap-4587	507	8	and	and	CCONJ
ap-4587	507	9	robert	robert	PROPN
ap-4587	507	10	v.	v.	ADP
ap-4587	507	11	moody	moody	PROPN
ap-4587	507	12	,	,	PUNCT
ap-4587	507	13	self	self	NOUN
ap-4587	507	14	-	-	PUNCT
ap-4587	507	15	similarities	similarity	NOUN
ap-4587	507	16	and	and	CCONJ
ap-4587	507	17	invariant	invariant	ADJ
ap-4587	507	18	densities	density	NOUN
ap-4587	507	19	for	for	ADP
ap-4587	507	20	model	model	NOUN
ap-4587	507	21	sets	set	NOUN
ap-4587	507	22	,	,	PUNCT
ap-4587	507	23	algebraic	algebraic	ADJ
ap-4587	507	24	methods	method	NOUN
ap-4587	507	25	in	in	ADP
ap-4587	507	26	physics	physics	PROPN
ap-4587	507	27	(	(	PUNCT
ap-4587	507	28	montréal	montréal	PROPN
ap-4587	507	29	,	,	PUNCT
ap-4587	507	30	qc	qc	PROPN
ap-4587	507	31	,	,	PUNCT
ap-4587	507	32	1997	1997	NUM
ap-4587	507	33	)	)	PUNCT
ap-4587	507	34	,	,	PUNCT
ap-4587	507	35	crm	crm	PROPN
ap-4587	507	36	ser	ser	PROPN
ap-4587	507	37	.	.	PROPN
ap-4587	507	38	math	math	NOUN
ap-4587	507	39	.	.	PUNCT
ap-4587	508	1	phys	phy	NOUN
ap-4587	508	2	.	.	PUNCT
ap-4587	508	3	,	,	PUNCT
ap-4587	508	4	springer	springer	NOUN
ap-4587	508	5	,	,	PUNCT
ap-4587	508	6	new	new	PROPN
ap-4587	508	7	york	york	PROPN
ap-4587	508	8	,	,	PUNCT
ap-4587	508	9	2001	2001	NUM
ap-4587	508	10	,	,	PUNCT
ap-4587	508	11	pp	pp	ADJ
ap-4587	508	12	.	.	PUNCT
ap-4587	509	1	1–15	1–15	NUM
ap-4587	509	2	.	.	PUNCT
ap-4587	510	1	mr	mr	PROPN
ap-4587	510	2	1847245	1847245	NUM
ap-4587	510	3	[	[	X
ap-4587	510	4	3	3	NUM
ap-4587	510	5	]	]	X
ap-4587	510	6	damien	damien	PROPN
ap-4587	510	7	barache	barache	PROPN
ap-4587	510	8	,	,	PUNCT
ap-4587	510	9	bernard	bernard	PROPN
ap-4587	510	10	champagne	champagne	PROPN
ap-4587	510	11	,	,	PUNCT
ap-4587	510	12	and	and	CCONJ
ap-4587	510	13	jean	jean	PROPN
ap-4587	510	14	-	-	PUNCT
ap-4587	510	15	pierre	pierre	PROPN
ap-4587	510	16	gazeau	gazeau	NOUN
ap-4587	510	17	,	,	PUNCT
ap-4587	510	18	pisot	pisot	ADJ
ap-4587	510	19	-	-	PUNCT
ap-4587	510	20	cyclotomic	cyclotomic	ADJ
ap-4587	510	21	quasilattices	quasilattice	NOUN
ap-4587	510	22	and	and	CCONJ
ap-4587	510	23	their	their	PRON
ap-4587	510	24	symmetry	symmetry	NOUN
ap-4587	510	25	semigroups	semigroup	NOUN
ap-4587	510	26	,	,	PUNCT
ap-4587	510	27	quasicrystals	quasicrystal	NOUN
ap-4587	510	28	and	and	CCONJ
ap-4587	510	29	discrete	discrete	ADJ
ap-4587	510	30	geometry	geometry	NOUN
ap-4587	510	31	(	(	PUNCT
ap-4587	510	32	toronto	toronto	PROPN
ap-4587	510	33	,	,	PUNCT
ap-4587	510	34	on	on	ADP
ap-4587	510	35	,	,	PUNCT
ap-4587	510	36	1995	1995	NUM
ap-4587	510	37	)	)	PUNCT
ap-4587	510	38	,	,	PUNCT
ap-4587	510	39	fields	field	NOUN
ap-4587	510	40	inst	inst	PROPN
ap-4587	510	41	.	.	PUNCT
ap-4587	511	1	monogr	monogr	PROPN
ap-4587	511	2	.	.	PUNCT
ap-4587	512	1	,	,	PUNCT
ap-4587	512	2	vol	vol	NOUN
ap-4587	512	3	.	.	PROPN
ap-4587	513	1	10	10	NUM
ap-4587	513	2	,	,	PUNCT
ap-4587	513	3	amer	amer	PROPN
ap-4587	513	4	.	.	PROPN
ap-4587	513	5	math	math	PROPN
ap-4587	513	6	.	.	PUNCT
ap-4587	514	1	soc	soc	PROPN
ap-4587	514	2	.	.	PUNCT
ap-4587	514	3	,	,	PUNCT
ap-4587	514	4	providence	providence	NOUN
ap-4587	514	5	,	,	PUNCT
ap-4587	514	6	ri	ri	PROPN
ap-4587	514	7	,	,	PUNCT
ap-4587	514	8	1998	1998	NUM
ap-4587	514	9	,	,	PUNCT
ap-4587	514	10	pp	pp	ADV
ap-4587	514	11	.	.	PUNCT
ap-4587	515	1	15–66	15–66	X
ap-4587	515	2	.	.	PUNCT
ap-4587	516	1	mr	mr	PROPN
ap-4587	516	2	1636775	1636775	NUM
ap-4587	516	3	[	[	X
ap-4587	516	4	4	4	NUM
ap-4587	516	5	]	]	X
ap-4587	516	6	stephen	stephen	PROPN
ap-4587	516	7	berman	berman	PROPN
ap-4587	516	8	and	and	CCONJ
ap-4587	516	9	robert	robert	PROPN
ap-4587	516	10	v.	v.	PROPN
ap-4587	516	11	moody	moody	PROPN
ap-4587	516	12	,	,	PUNCT
ap-4587	516	13	the	the	DET
ap-4587	516	14	algebraic	algebraic	ADJ
ap-4587	516	15	theory	theory	NOUN
ap-4587	516	16	of	of	ADP
ap-4587	516	17	quasicrystals	quasicrystal	NOUN
ap-4587	516	18	with	with	ADP
ap-4587	516	19	five	five	NUM
ap-4587	516	20	-	-	ADJ
ap-4587	516	21	fold	fold	ADJ
ap-4587	516	22	symmetries	symmetry	NOUN
ap-4587	516	23	,	,	PUNCT
ap-4587	516	24	j.	j.	PROPN
ap-4587	516	25	phys	phys	PROPN
ap-4587	516	26	.	.	PUNCT
ap-4587	517	1	a	a	DET
ap-4587	517	2	27	27	NUM
ap-4587	517	3	(	(	PUNCT
ap-4587	517	4	1994	1994	NUM
ap-4587	517	5	)	)	PUNCT
ap-4587	517	6	,	,	PUNCT
ap-4587	517	7	no	no	INTJ
ap-4587	517	8	.	.	NOUN
ap-4587	517	9	1	1	NUM
ap-4587	517	10	,	,	PUNCT
ap-4587	517	11	115–129	115–129	NUM
ap-4587	517	12	.	.	PUNCT
ap-4587	518	1	mr	mr	PROPN
ap-4587	518	2	1288000	1288000	NUM
ap-4587	518	3	[	[	X
ap-4587	518	4	5	5	NUM
ap-4587	518	5	]	]	PUNCT
ap-4587	518	6	ludwig	ludwig	PROPN
ap-4587	518	7	bieberbach	bieberbach	PROPN
ap-4587	518	8	,	,	PUNCT
ap-4587	518	9	über	über	PROPN
ap-4587	518	10	die	die	NOUN
ap-4587	518	11	bewegungsgruppen	bewegungsgruppen	PROPN
ap-4587	518	12	der	der	NOUN
ap-4587	518	13	euklidischen	euklidischen	AUX
ap-4587	518	14	räume	räume	PROPN
ap-4587	518	15	,	,	PUNCT
ap-4587	518	16	math	math	NOUN
ap-4587	518	17	.	.	PUNCT
ap-4587	519	1	ann	ann	PROPN
ap-4587	519	2	.	.	PROPN
ap-4587	520	1	70	70	NUM
ap-4587	520	2	(	(	PUNCT
ap-4587	520	3	1911	1911	NUM
ap-4587	520	4	)	)	PUNCT
ap-4587	520	5	,	,	PUNCT
ap-4587	521	1	no	no	INTJ
ap-4587	521	2	.	.	NOUN
ap-4587	521	3	3	3	NUM
ap-4587	521	4	,	,	PUNCT
ap-4587	521	5	297–336	297–336	NUM
ap-4587	521	6	.	.	PUNCT
ap-4587	522	1	mr	mr	PROPN
ap-4587	522	2	1511623	1511623	NUM
ap-4587	522	3	[	[	X
ap-4587	522	4	6	6	NUM
ap-4587	522	5	]	]	X
ap-4587	522	6	harald	harald	PROPN
ap-4587	522	7	bohr	bohr	PROPN
ap-4587	522	8	,	,	PUNCT
ap-4587	522	9	zur	zur	PROPN
ap-4587	522	10	theorie	theorie	PROPN
ap-4587	522	11	der	der	PROPN
ap-4587	522	12	fastperiodischen	fastperiodischen	PROPN
ap-4587	522	13	funktionen	funktionen	PROPN
ap-4587	522	14	,	,	PUNCT
ap-4587	522	15	acta	acta	PROPN
ap-4587	522	16	math	math	PROPN
ap-4587	522	17	.	.	PUNCT
ap-4587	523	1	45	45	NUM
ap-4587	523	2	(	(	PUNCT
ap-4587	523	3	1924	1924	NUM
ap-4587	523	4	)	)	PUNCT
ap-4587	523	5	,	,	PUNCT
ap-4587	523	6	29–127	29–127	NUM
ap-4587	523	7	.	.	PUNCT
ap-4587	524	1	[	[	X
ap-4587	524	2	7	7	NUM
ap-4587	524	3	]	]	X
ap-4587	524	4	nicolae	nicolae	NOUN
ap-4587	524	5	cotfas	cotfas	NOUN
ap-4587	524	6	,	,	PUNCT
ap-4587	524	7	on	on	ADP
ap-4587	524	8	the	the	DET
ap-4587	524	9	self	self	NOUN
ap-4587	524	10	-	-	PUNCT
ap-4587	524	11	similarities	similarity	NOUN
ap-4587	524	12	of	of	ADP
ap-4587	524	13	a	a	DET
ap-4587	524	14	model	model	NOUN
ap-4587	524	15	set	set	NOUN
ap-4587	524	16	,	,	PUNCT
ap-4587	524	17	j.	j.	PROPN
ap-4587	524	18	phys	phys	PROPN
ap-4587	524	19	.	.	PUNCT
ap-4587	525	1	a	a	DET
ap-4587	525	2	32	32	NUM
ap-4587	525	3	(	(	PUNCT
ap-4587	525	4	1999	1999	NUM
ap-4587	525	5	)	)	PUNCT
ap-4587	525	6	,	,	PUNCT
ap-4587	525	7	no	no	INTJ
ap-4587	525	8	.	.	NOUN
ap-4587	525	9	15	15	NUM
ap-4587	525	10	,	,	PUNCT
ap-4587	525	11	l165	l165	PROPN
ap-4587	525	12	–	–	PUNCT
ap-4587	525	13	l168	l168	PROPN
ap-4587	525	14	.	.	PUNCT
ap-4587	526	1	mr	mr	PROPN
ap-4587	526	2	1685699	1685699	NUM
ap-4587	526	3	[	[	X
ap-4587	526	4	8	8	NUM
ap-4587	526	5	]	]	X
ap-4587	526	6	nicolaas	nicolaas	ADJ
ap-4587	526	7	g.	g.	PROPN
ap-4587	526	8	de	de	PROPN
ap-4587	526	9	bruijn	bruijn	PROPN
ap-4587	526	10	,	,	PUNCT
ap-4587	526	11	algebraic	algebraic	ADJ
ap-4587	526	12	theory	theory	NOUN
ap-4587	526	13	of	of	ADP
ap-4587	526	14	penrose	penrose	PROPN
ap-4587	526	15	’s	’s	PART
ap-4587	526	16	nonperiodic	nonperiodic	ADJ
ap-4587	526	17	tilings	tiling	NOUN
ap-4587	526	18	of	of	ADP
ap-4587	526	19	the	the	DET
ap-4587	526	20	plane	plane	NOUN
ap-4587	526	21	.	.	PUNCT
ap-4587	527	1	i	i	PRON
ap-4587	527	2	,	,	PUNCT
ap-4587	527	3	ii	ii	PROPN
ap-4587	527	4	,	,	PUNCT
ap-4587	527	5	nederl	nederl	PROPN
ap-4587	527	6	.	.	PUNCT
ap-4587	527	7	akad	akad	PROPN
ap-4587	527	8	.	.	PUNCT
ap-4587	528	1	wetensch	wetensch	PROPN
ap-4587	528	2	.	.	PUNCT
ap-4587	529	1	indag	indag	PROPN
ap-4587	529	2	.	.	PUNCT
ap-4587	530	1	math	math	NOUN
ap-4587	530	2	.	.	PUNCT
ap-4587	531	1	43	43	NUM
ap-4587	531	2	(	(	PUNCT
ap-4587	531	3	1981	1981	NUM
ap-4587	531	4	)	)	PUNCT
ap-4587	531	5	,	,	PUNCT
ap-4587	531	6	no	no	INTJ
ap-4587	531	7	.	.	NOUN
ap-4587	531	8	1	1	NUM
ap-4587	531	9	,	,	PUNCT
ap-4587	531	10	39–52	39–52	NUM
ap-4587	531	11	,	,	PUNCT
ap-4587	531	12	53–66	53–66	NUM
ap-4587	531	13	.	.	PUNCT
ap-4587	532	1	mr	mr	PROPN
ap-4587	532	2	609465	609465	NUM
ap-4587	532	3	444	444	NUM
ap-4587	532	4	vol	vol	NOUN
ap-4587	532	5	.	.	PUNCT
ap-4587	532	6	57	57	NUM
ap-4587	532	7	no	no	NOUN
ap-4587	532	8	.	.	PUNCT
ap-4587	533	1	6/2017	6/2017	NUM
ap-4587	533	2	on	on	ADP
ap-4587	533	3	self	self	NOUN
ap-4587	533	4	-	-	PUNCT
ap-4587	533	5	similarities	similarity	NOUN
ap-4587	533	6	of	of	ADP
ap-4587	533	7	cut	cut	NOUN
ap-4587	533	8	-	-	PUNCT
ap-4587	533	9	and	and	CCONJ
ap-4587	533	10	-	-	PUNCT
ap-4587	533	11	project	project	NOUN
ap-4587	533	12	sets	set	NOUN
ap-4587	533	13	[	[	X
ap-4587	533	14	9	9	NUM
ap-4587	533	15	]	]	PUNCT
ap-4587	533	16	anatole	anatole	NOUN
ap-4587	533	17	katok	katok	PROPN
ap-4587	533	18	and	and	CCONJ
ap-4587	533	19	boris	boris	PROPN
ap-4587	533	20	hasselblatt	hasselblatt	PROPN
ap-4587	533	21	,	,	PUNCT
ap-4587	533	22	introduction	introduction	NOUN
ap-4587	533	23	to	to	ADP
ap-4587	533	24	the	the	DET
ap-4587	533	25	modern	modern	ADJ
ap-4587	533	26	theory	theory	NOUN
ap-4587	533	27	of	of	ADP
ap-4587	533	28	dynamical	dynamical	ADJ
ap-4587	533	29	systems	system	NOUN
ap-4587	533	30	,	,	PUNCT
ap-4587	533	31	encyclopedia	encyclopedia	NOUN
ap-4587	533	32	of	of	ADP
ap-4587	533	33	mathematics	mathematic	NOUN
ap-4587	533	34	and	and	CCONJ
ap-4587	533	35	its	its	PRON
ap-4587	533	36	applications	application	NOUN
ap-4587	533	37	,	,	PUNCT
ap-4587	533	38	vol	vol	NOUN
ap-4587	533	39	.	.	PROPN
ap-4587	534	1	54	54	NUM
ap-4587	534	2	,	,	PUNCT
ap-4587	535	1	cambridge	cambridge	PROPN
ap-4587	535	2	university	university	PROPN
ap-4587	535	3	press	press	PROPN
ap-4587	535	4	,	,	PUNCT
ap-4587	535	5	cambridge	cambridge	PROPN
ap-4587	535	6	,	,	PUNCT
ap-4587	535	7	1995	1995	NUM
ap-4587	535	8	,	,	PUNCT
ap-4587	535	9	with	with	ADP
ap-4587	535	10	a	a	DET
ap-4587	535	11	supplementary	supplementary	ADJ
ap-4587	535	12	chapter	chapter	NOUN
ap-4587	535	13	by	by	ADP
ap-4587	535	14	katok	katok	NOUN
ap-4587	535	15	and	and	CCONJ
ap-4587	535	16	leonardo	leonardo	PROPN
ap-4587	535	17	mendoza	mendoza	PROPN
ap-4587	535	18	.	.	PUNCT
ap-4587	536	1	mr	mr	PROPN
ap-4587	536	2	1326374	1326374	NUM
ap-4587	536	3	[	[	X
ap-4587	536	4	10	10	NUM
ap-4587	536	5	]	]	X
ap-4587	536	6	peter	peter	PROPN
ap-4587	536	7	kramer	kramer	PROPN
ap-4587	536	8	and	and	CCONJ
ap-4587	536	9	r.	r.	PROPN
ap-4587	536	10	neri	neri	PROPN
ap-4587	536	11	,	,	PUNCT
ap-4587	536	12	on	on	ADP
ap-4587	536	13	periodic	periodic	ADJ
ap-4587	536	14	and	and	CCONJ
ap-4587	536	15	nonperiodic	nonperiodic	ADJ
ap-4587	536	16	space	space	NOUN
ap-4587	536	17	fillings	filling	NOUN
ap-4587	536	18	of	of	ADP
ap-4587	536	19	em	em	PRON
ap-4587	536	20	obtained	obtain	VERB
ap-4587	536	21	by	by	ADP
ap-4587	536	22	projection	projection	NOUN
ap-4587	536	23	,	,	PUNCT
ap-4587	536	24	acta	acta	PROPN
ap-4587	536	25	cryst	cryst	PROPN
ap-4587	536	26	.	.	PUNCT
ap-4587	537	1	sect	sect	NOUN
ap-4587	537	2	.	.	PUNCT
ap-4587	538	1	a	a	DET
ap-4587	538	2	40	40	NUM
ap-4587	538	3	(	(	PUNCT
ap-4587	538	4	1984	1984	NUM
ap-4587	538	5	)	)	PUNCT
ap-4587	538	6	,	,	PUNCT
ap-4587	538	7	no	no	INTJ
ap-4587	538	8	.	.	NOUN
ap-4587	538	9	5	5	NUM
ap-4587	538	10	,	,	PUNCT
ap-4587	538	11	580–587	580–587	NUM
ap-4587	538	12	.	.	PUNCT
ap-4587	539	1	mr	mr	PROPN
ap-4587	539	2	768042	768042	NUM
ap-4587	539	3	[	[	X
ap-4587	539	4	11	11	NUM
ap-4587	539	5	]	]	X
ap-4587	539	6	jeffrey	jeffrey	PROPN
ap-4587	539	7	c.	c.	PROPN
ap-4587	539	8	lagarias	lagarias	PROPN
ap-4587	539	9	,	,	PUNCT
ap-4587	539	10	geometric	geometric	ADJ
ap-4587	539	11	models	model	NOUN
ap-4587	539	12	for	for	ADP
ap-4587	539	13	quasicrystals	quasicrystal	NOUN
ap-4587	539	14	i.	i.	PROPN
ap-4587	539	15	delone	delone	PROPN
ap-4587	539	16	sets	set	NOUN
ap-4587	539	17	of	of	ADP
ap-4587	539	18	finite	finite	ADJ
ap-4587	539	19	type	type	NOUN
ap-4587	539	20	,	,	PUNCT
ap-4587	539	21	discrete	discrete	ADJ
ap-4587	539	22	comput	comput	NOUN
ap-4587	539	23	.	.	PUNCT
ap-4587	540	1	geom	geom	PROPN
ap-4587	540	2	.	.	PUNCT
ap-4587	541	1	21	21	NUM
ap-4587	541	2	(	(	PUNCT
ap-4587	541	3	1999	1999	NUM
ap-4587	541	4	)	)	PUNCT
ap-4587	541	5	,	,	PUNCT
ap-4587	541	6	no	no	INTJ
ap-4587	541	7	.	.	NOUN
ap-4587	541	8	2	2	NUM
ap-4587	541	9	,	,	PUNCT
ap-4587	541	10	161–191	161–191	NUM
ap-4587	541	11	.	.	PUNCT
ap-4587	542	1	mr	mr	PROPN
ap-4587	542	2	1668082	1668082	NUM
ap-4587	542	3	[	[	X
ap-4587	542	4	12	12	NUM
ap-4587	542	5	]	]	X
ap-4587	542	6	jeffrey	jeffrey	PROPN
ap-4587	542	7	c.	c.	PROPN
ap-4587	542	8	lagarias	lagarias	PROPN
ap-4587	542	9	and	and	CCONJ
ap-4587	542	10	peter	peter	PROPN
ap-4587	542	11	a.	a.	PROPN
ap-4587	542	12	b.	b.	PROPN
ap-4587	542	13	pleasants	pleasants	PROPN
ap-4587	542	14	,	,	PUNCT
ap-4587	542	15	repetitive	repetitive	ADJ
ap-4587	542	16	delone	delone	NOUN
ap-4587	542	17	sets	set	NOUN
ap-4587	542	18	and	and	CCONJ
ap-4587	542	19	quasicrystals	quasicrystal	NOUN
ap-4587	542	20	,	,	PUNCT
ap-4587	542	21	ergodic	ergodic	ADJ
ap-4587	542	22	theory	theory	NOUN
ap-4587	542	23	dynam	dynam	NOUN
ap-4587	542	24	.	.	PUNCT
ap-4587	543	1	systems	system	NOUN
ap-4587	543	2	23	23	NUM
ap-4587	543	3	(	(	PUNCT
ap-4587	543	4	2003	2003	NUM
ap-4587	543	5	)	)	PUNCT
ap-4587	543	6	,	,	PUNCT
ap-4587	543	7	no	no	INTJ
ap-4587	543	8	.	.	NOUN
ap-4587	543	9	3	3	NUM
ap-4587	543	10	,	,	PUNCT
ap-4587	543	11	831–867	831–867	NUM
ap-4587	543	12	.	.	PUNCT
ap-4587	544	1	mr	mr	PROPN
ap-4587	544	2	1992666	1992666	NUM
ap-4587	544	3	[	[	X
ap-4587	544	4	13	13	NUM
ap-4587	544	5	]	]	PUNCT
ap-4587	544	6	zuzana	zuzana	PROPN
ap-4587	544	7	masáková	masáková	PROPN
ap-4587	544	8	,	,	PUNCT
ap-4587	544	9	jiří	jiří	NOUN
ap-4587	544	10	patera	patera	NOUN
ap-4587	544	11	,	,	PUNCT
ap-4587	544	12	and	and	CCONJ
ap-4587	544	13	edita	edita	PROPN
ap-4587	544	14	pelantová	pelantová	NOUN
ap-4587	544	15	,	,	PUNCT
ap-4587	544	16	inflation	inflation	NOUN
ap-4587	544	17	centres	centre	NOUN
ap-4587	544	18	of	of	ADP
ap-4587	544	19	the	the	DET
ap-4587	544	20	cut	cut	NOUN
ap-4587	544	21	and	and	CCONJ
ap-4587	544	22	project	project	NOUN
ap-4587	544	23	quasicrystals	quasicrystal	NOUN
ap-4587	544	24	,	,	PUNCT
ap-4587	544	25	j.	j.	PROPN
ap-4587	544	26	phys	phys	PROPN
ap-4587	544	27	.	.	PUNCT
ap-4587	545	1	a	a	DET
ap-4587	545	2	31	31	NUM
ap-4587	545	3	(	(	PUNCT
ap-4587	545	4	1998	1998	NUM
ap-4587	545	5	)	)	PUNCT
ap-4587	545	6	,	,	PUNCT
ap-4587	545	7	no	no	INTJ
ap-4587	545	8	.	.	NOUN
ap-4587	545	9	5	5	NUM
ap-4587	545	10	,	,	PUNCT
ap-4587	545	11	1443–1453	1443–1453	NUM
ap-4587	545	12	.	.	PUNCT
ap-4587	546	1	mr	mr	PROPN
ap-4587	546	2	1628499	1628499	NUM
ap-4587	546	3	[	[	X
ap-4587	546	4	14	14	NUM
ap-4587	546	5	]	]	PUNCT
ap-4587	546	6	,	,	PUNCT
ap-4587	546	7	self	self	NOUN
ap-4587	546	8	-	-	PUNCT
ap-4587	546	9	similar	similar	ADJ
ap-4587	546	10	delone	delone	NOUN
ap-4587	546	11	sets	set	NOUN
ap-4587	546	12	and	and	CCONJ
ap-4587	546	13	quasicrystals	quasicrystal	NOUN
ap-4587	546	14	,	,	PUNCT
ap-4587	546	15	j.	j.	PROPN
ap-4587	546	16	phys	phys	PROPN
ap-4587	546	17	.	.	PUNCT
ap-4587	547	1	a	a	DET
ap-4587	547	2	31	31	NUM
ap-4587	547	3	(	(	PUNCT
ap-4587	547	4	1998	1998	NUM
ap-4587	547	5	)	)	PUNCT
ap-4587	547	6	,	,	PUNCT
ap-4587	547	7	no	no	INTJ
ap-4587	547	8	.	.	NOUN
ap-4587	547	9	21	21	NUM
ap-4587	547	10	,	,	PUNCT
ap-4587	547	11	4927–4946	4927–4946	NUM
ap-4587	547	12	.	.	PUNCT
ap-4587	548	1	mr	mr	PROPN
ap-4587	548	2	1630499	1630499	NUM
ap-4587	548	3	[	[	X
ap-4587	548	4	15	15	NUM
ap-4587	548	5	]	]	X
ap-4587	548	6	yves	yve	NOUN
ap-4587	548	7	meyer	meyer	PROPN
ap-4587	548	8	,	,	PUNCT
ap-4587	548	9	algebraic	algebraic	ADJ
ap-4587	548	10	numbers	number	NOUN
ap-4587	548	11	and	and	CCONJ
ap-4587	548	12	harmonic	harmonic	ADJ
ap-4587	548	13	analysis	analysis	NOUN
ap-4587	548	14	,	,	PUNCT
ap-4587	548	15	north	north	NOUN
ap-4587	548	16	-	-	PUNCT
ap-4587	548	17	holland	holland	PROPN
ap-4587	548	18	publishing	publishing	PROPN
ap-4587	548	19	co.	co.	PROPN
ap-4587	548	20	,	,	PUNCT
ap-4587	548	21	amsterdam	amsterdam	PROPN
ap-4587	548	22	-	-	PUNCT
ap-4587	548	23	london	london	PROPN
ap-4587	548	24	;	;	PUNCT
ap-4587	548	25	american	american	PROPN
ap-4587	548	26	elsevier	elsevier	PROPN
ap-4587	548	27	publishing	publishing	PROPN
ap-4587	548	28	co.	co.	PROPN
ap-4587	548	29	,	,	PUNCT
ap-4587	548	30	inc	inc	PROPN
ap-4587	548	31	.	.	PROPN
ap-4587	548	32	,	,	PUNCT
ap-4587	548	33	new	new	PROPN
ap-4587	548	34	york	york	PROPN
ap-4587	548	35	,	,	PUNCT
ap-4587	548	36	1972	1972	NUM
ap-4587	548	37	,	,	PUNCT
ap-4587	548	38	north	north	NOUN
ap-4587	548	39	-	-	PUNCT
ap-4587	548	40	holland	holland	PROPN
ap-4587	548	41	mathematical	mathematical	PROPN
ap-4587	548	42	library	library	NOUN
ap-4587	548	43	,	,	PUNCT
ap-4587	548	44	vol	vol	NOUN
ap-4587	548	45	.	.	PROPN
ap-4587	548	46	2	2	NUM
ap-4587	548	47	.	.	X
ap-4587	549	1	mr	mr	PROPN
ap-4587	549	2	0485769	0485769	NUM
ap-4587	549	3	[	[	X
ap-4587	549	4	16	16	NUM
ap-4587	549	5	]	]	X
ap-4587	549	6	robert	robert	PROPN
ap-4587	549	7	v.	v.	PROPN
ap-4587	549	8	moody	moody	PROPN
ap-4587	549	9	,	,	PUNCT
ap-4587	549	10	model	model	NOUN
ap-4587	549	11	sets	set	VERB
ap-4587	549	12	:	:	PUNCT
ap-4587	549	13	a	a	DET
ap-4587	549	14	survey	survey	NOUN
ap-4587	549	15	,	,	PUNCT
ap-4587	549	16	from	from	ADP
ap-4587	549	17	quasiperiodic	quasiperiodic	ADJ
ap-4587	549	18	to	to	ADP
ap-4587	549	19	more	more	ADJ
ap-4587	549	20	complex	complex	ADJ
ap-4587	549	21	systems	system	NOUN
ap-4587	549	22	(	(	PUNCT
ap-4587	549	23	les	les	X
ap-4587	549	24	houches	houche	NOUN
ap-4587	549	25	,	,	PUNCT
ap-4587	549	26	1998	1998	NUM
ap-4587	549	27	)	)	PUNCT
ap-4587	549	28	,	,	PUNCT
ap-4587	549	29	springer	springer	NOUN
ap-4587	549	30	,	,	PUNCT
ap-4587	549	31	berlin	berlin	PROPN
ap-4587	549	32	,	,	PUNCT
ap-4587	549	33	2000	2000	NUM
ap-4587	549	34	,	,	PUNCT
ap-4587	549	35	pp	pp	ADJ
ap-4587	549	36	.	.	PUNCT
ap-4587	550	1	145–166	145–166	NUM
ap-4587	550	2	.	.	PUNCT
ap-4587	551	1	[	[	X
ap-4587	551	2	17	17	NUM
ap-4587	551	3	]	]	X
ap-4587	551	4	robert	robert	PROPN
ap-4587	551	5	penrose	penrose	PROPN
ap-4587	551	6	,	,	PUNCT
ap-4587	551	7	pentaplexity	pentaplexity	NOUN
ap-4587	551	8	:	:	PUNCT
ap-4587	551	9	a	a	DET
ap-4587	551	10	class	class	NOUN
ap-4587	551	11	of	of	ADP
ap-4587	551	12	nonperiodic	nonperiodic	ADJ
ap-4587	551	13	tilings	tiling	NOUN
ap-4587	551	14	of	of	ADP
ap-4587	551	15	the	the	DET
ap-4587	551	16	plane	plane	NOUN
ap-4587	551	17	,	,	PUNCT
ap-4587	551	18	math	math	NOUN
ap-4587	551	19	.	.	PUNCT
ap-4587	552	1	intelligencer	intelligencer	NOUN
ap-4587	552	2	2	2	NUM
ap-4587	552	3	(	(	PUNCT
ap-4587	552	4	1979/80	1979/80	NUM
ap-4587	552	5	)	)	PUNCT
ap-4587	552	6	,	,	PUNCT
ap-4587	552	7	no	no	INTJ
ap-4587	552	8	.	.	NOUN
ap-4587	552	9	1	1	NUM
ap-4587	552	10	,	,	PUNCT
ap-4587	552	11	32–37	32–37	NOUN
ap-4587	552	12	.	.	PUNCT
ap-4587	553	1	mr	mr	PROPN
ap-4587	553	2	558670	558670	NUM
ap-4587	553	3	[	[	X
ap-4587	553	4	18	18	NUM
ap-4587	553	5	]	]	PUNCT
ap-4587	553	6	dan	dan	PROPN
ap-4587	553	7	shechtman	shechtman	PROPN
ap-4587	553	8	,	,	PUNCT
ap-4587	553	9	ilan	ilan	PROPN
ap-4587	553	10	a.	a.	PROPN
ap-4587	553	11	blech	blech	PROPN
ap-4587	553	12	,	,	PUNCT
ap-4587	553	13	denis	denis	PROPN
ap-4587	553	14	gratias	gratia	NOUN
ap-4587	553	15	,	,	PUNCT
ap-4587	553	16	and	and	CCONJ
ap-4587	553	17	john	john	PROPN
ap-4587	553	18	w.	w.	PROPN
ap-4587	553	19	cahn	cahn	PROPN
ap-4587	553	20	,	,	PUNCT
ap-4587	553	21	metallic	metallic	ADJ
ap-4587	553	22	phase	phase	NOUN
ap-4587	553	23	with	with	ADP
ap-4587	553	24	long	long	ADJ
ap-4587	553	25	-	-	PUNCT
ap-4587	553	26	range	range	NOUN
ap-4587	553	27	orientational	orientational	ADJ
ap-4587	553	28	order	order	NOUN
ap-4587	553	29	and	and	CCONJ
ap-4587	553	30	no	no	DET
ap-4587	553	31	translational	translational	ADJ
ap-4587	553	32	symmetry	symmetry	NOUN
ap-4587	553	33	,	,	PUNCT
ap-4587	553	34	phys	phy	NOUN
ap-4587	553	35	.	.	PUNCT
ap-4587	554	1	rev	rev	PROPN
ap-4587	554	2	.	.	PROPN
ap-4587	554	3	lett	lett	PROPN
ap-4587	554	4	.	.	PUNCT
ap-4587	555	1	53	53	NUM
ap-4587	555	2	(	(	PUNCT
ap-4587	555	3	1984	1984	NUM
ap-4587	555	4	)	)	PUNCT
ap-4587	555	5	,	,	PUNCT
ap-4587	556	1	no	no	INTJ
ap-4587	556	2	.	.	NOUN
ap-4587	556	3	20	20	NUM
ap-4587	556	4	,	,	PUNCT
ap-4587	556	5	1951–1954	1951–1954	NUM
ap-4587	556	6	.	.	PUNCT
ap-4587	557	1	[	[	X
ap-4587	557	2	19	19	NUM
ap-4587	557	3	]	]	X
ap-4587	557	4	walter	walter	PROPN
ap-4587	557	5	steurer	steurer	PROPN
ap-4587	557	6	,	,	PUNCT
ap-4587	557	7	twenty	twenty	NUM
ap-4587	557	8	years	year	NOUN
ap-4587	557	9	of	of	ADP
ap-4587	557	10	structure	structure	NOUN
ap-4587	557	11	research	research	NOUN
ap-4587	557	12	on	on	ADP
ap-4587	557	13	quasicrystals	quasicrystal	NOUN
ap-4587	557	14	.	.	PUNCT
ap-4587	558	1	i.	i.	PROPN
ap-4587	558	2	pentagonal	pentagonal	PROPN
ap-4587	558	3	,	,	PUNCT
ap-4587	558	4	octagonal	octagonal	ADJ
ap-4587	558	5	,	,	PUNCT
ap-4587	558	6	decagonal	decagonal	ADJ
ap-4587	558	7	and	and	CCONJ
ap-4587	558	8	dodecagonal	dodecagonal	ADJ
ap-4587	558	9	quasicrystals	quasicrystal	NOUN
ap-4587	558	10	,	,	PUNCT
ap-4587	558	11	z.	z.	PROPN
ap-4587	558	12	krist	krist	PROPN
ap-4587	558	13	.	.	PUNCT
ap-4587	559	1	219	219	NUM
ap-4587	559	2	(	(	PUNCT
ap-4587	559	3	2004	2004	NUM
ap-4587	559	4	)	)	PUNCT
ap-4587	559	5	,	,	PUNCT
ap-4587	560	1	no	no	INTJ
ap-4587	560	2	.	.	NOUN
ap-4587	560	3	7	7	NUM
ap-4587	560	4	,	,	PUNCT
ap-4587	560	5	391–446	391–446	NUM
ap-4587	560	6	.	.	PUNCT
ap-4587	561	1	mr	mr	PROPN
ap-4587	561	2	2082031	2082031	NUM
ap-4587	561	3	445	445	NUM
ap-4587	561	4	acta	acta	PROPN
ap-4587	561	5	polytechnica	polytechnica	PROPN
ap-4587	561	6	57(6):430–445	57(6):430–445	PROPN
ap-4587	561	7	,	,	PUNCT
ap-4587	561	8	2017	2017	NUM
ap-4587	561	9	1	1	NUM
ap-4587	561	10	introduction	introduction	NOUN
ap-4587	561	11	2	2	NUM
ap-4587	561	12	preliminaries	preliminary	NOUN
ap-4587	561	13	3	3	NUM
ap-4587	561	14	matrix	matrix	NOUN
ap-4587	561	15	formalism	formalism	NOUN
ap-4587	561	16	for	for	ADP
ap-4587	561	17	the	the	DET
ap-4587	561	18	cut	cut	NOUN
ap-4587	561	19	-	-	PUNCT
ap-4587	561	20	and	and	CCONJ
ap-4587	561	21	-	-	PUNCT
ap-4587	561	22	project	project	NOUN
ap-4587	561	23	method	method	NOUN
ap-4587	561	24	4	4	NUM
ap-4587	561	25	self	self	NOUN
ap-4587	561	26	-	-	PUNCT
ap-4587	561	27	similarities	similarity	NOUN
ap-4587	561	28	of	of	ADP
ap-4587	561	29	a	a	DET
ap-4587	561	30	cut	cut	NOUN
ap-4587	561	31	-	-	PUNCT
ap-4587	561	32	and	and	CCONJ
ap-4587	561	33	-	-	PUNCT
ap-4587	561	34	project	project	NOUN
ap-4587	561	35	set	set	VERB
ap-4587	561	36	5	5	NUM
ap-4587	561	37	construction	construction	NOUN
ap-4587	561	38	of	of	ADP
ap-4587	561	39	a	a	DET
ap-4587	561	40	cut	cut	NOUN
ap-4587	561	41	-	-	PUNCT
ap-4587	561	42	and	and	CCONJ
ap-4587	561	43	-	-	PUNCT
ap-4587	561	44	project	project	NOUN
ap-4587	561	45	scheme	scheme	NOUN
ap-4587	561	46	with	with	ADP
ap-4587	561	47	5	5	ADJ
ap-4587	561	48	-	-	ADJ
ap-4587	561	49	fold	fold	ADJ
ap-4587	561	50	symmetry	symmetry	NOUN
ap-4587	561	51	6	6	NUM
ap-4587	561	52	self	self	NOUN
ap-4587	561	53	-	-	PUNCT
ap-4587	561	54	similarities	similarity	NOUN
ap-4587	561	55	of	of	ADP
ap-4587	561	56	the	the	DET
ap-4587	561	57	constructed	construct	VERB
ap-4587	561	58	scheme	scheme	NOUN
ap-4587	561	59	7	7	NUM
ap-4587	561	60	self	self	NOUN
ap-4587	561	61	-	-	PUNCT
ap-4587	561	62	similarities	similarity	NOUN
ap-4587	561	63	of	of	ADP
ap-4587	561	64	cut	cut	NOUN
ap-4587	561	65	-	-	PUNCT
ap-4587	561	66	and	and	CCONJ
ap-4587	561	67	-	-	PUNCT
ap-4587	561	68	project	project	NOUN
ap-4587	561	69	sets	set	NOUN
ap-4587	561	70	with	with	ADP
ap-4587	561	71	5	5	ADJ
ap-4587	561	72	-	-	ADJ
ap-4587	561	73	fold	fold	ADJ
ap-4587	561	74	symmetry	symmetry	NOUN
ap-4587	561	75	8	8	NUM
ap-4587	561	76	comments	comment	NOUN
ap-4587	561	77	9	9	NUM
ap-4587	561	78	acknowledgements	acknowledgement	NOUN
ap-4587	561	79	references	reference	NOUN