id	sid	tid	token	lemma	pos
ap-1860	1	1	acta	acta	PROPN
ap-1860	1	2	polytechnica	polytechnica	PROPN
ap-1860	1	3	doi:10.14311	doi:10.14311	PROPN
ap-1860	1	4	/	/	SYM
ap-1860	1	5	ap.2013.53.0444	ap.2013.53.0444	PROPN
ap-1860	1	6	acta	acta	PROPN
ap-1860	1	7	polytechnica	polytechnica	PROPN
ap-1860	1	8	53(5):444–449	53(5):444–449	PROPN
ap-1860	1	9	,	,	PUNCT
ap-1860	1	10	2013	2013	NUM
ap-1860	1	11	©	©	PROPN
ap-1860	1	12	czech	czech	PROPN
ap-1860	1	13	technical	technical	PROPN
ap-1860	1	14	university	university	PROPN
ap-1860	1	15	in	in	ADP
ap-1860	1	16	prague	prague	PROPN
ap-1860	1	17	,	,	PUNCT
ap-1860	1	18	2013	2013	NUM
ap-1860	1	19	available	available	ADJ
ap-1860	1	20	online	online	ADV
ap-1860	1	21	at	at	ADP
ap-1860	1	22	http://ojs.cvut.cz/ojs/index.php/ap	http://ojs.cvut.cz/ojs/index.php/ap	PROPN
ap-1860	1	23	itineraries	itinerary	NOUN
ap-1860	1	24	induced	induce	VERB
ap-1860	1	25	by	by	ADP
ap-1860	1	26	exchange	exchange	NOUN
ap-1860	1	27	of	of	ADP
ap-1860	1	28	two	two	NUM
ap-1860	1	29	intervals	interval	NOUN
ap-1860	1	30	zuzana	zuzana	PROPN
ap-1860	1	31	masáková	masáková	PROPN
ap-1860	1	32	,	,	PUNCT
ap-1860	1	33	edita	edita	PROPN
ap-1860	1	34	pelantová∗	pelantová∗	PROPN
ap-1860	1	35	department	department	PROPN
ap-1860	1	36	of	of	ADP
ap-1860	1	37	mathematics	mathematics	PROPN
ap-1860	1	38	fnspe	fnspe	PROPN
ap-1860	1	39	,	,	PUNCT
ap-1860	1	40	czech	czech	PROPN
ap-1860	1	41	technical	technical	PROPN
ap-1860	1	42	university	university	PROPN
ap-1860	1	43	in	in	ADP
ap-1860	1	44	prague	prague	PROPN
ap-1860	1	45	,	,	PUNCT
ap-1860	1	46	trojanova	trojanova	X
ap-1860	1	47	13	13	NUM
ap-1860	1	48	,	,	PUNCT
ap-1860	1	49	120	120	NUM
ap-1860	1	50	00	00	NUM
ap-1860	1	51	praha	praha	PROPN
ap-1860	1	52	2	2	NUM
ap-1860	1	53	,	,	PUNCT
ap-1860	1	54	czech	czech	PROPN
ap-1860	1	55	republic	republic	NOUN
ap-1860	1	56	∗	∗	NOUN
ap-1860	1	57	corresponding	correspond	VERB
ap-1860	1	58	author	author	NOUN
ap-1860	1	59	:	:	PUNCT
ap-1860	1	60	edita.pelantova@fjfi.cvut.cz	edita.pelantova@fjfi.cvut.cz	NOUN
ap-1860	1	61	abstract	abstract	NOUN
ap-1860	1	62	.	.	PUNCT
ap-1860	2	1	we	we	PRON
ap-1860	2	2	focus	focus	VERB
ap-1860	2	3	on	on	ADP
ap-1860	2	4	the	the	DET
ap-1860	2	5	exchange	exchange	NOUN
ap-1860	2	6	t	t	PROPN
ap-1860	2	7	of	of	ADP
ap-1860	2	8	two	two	NUM
ap-1860	2	9	intervals	interval	NOUN
ap-1860	2	10	with	with	ADP
ap-1860	2	11	an	an	DET
ap-1860	2	12	irrational	irrational	ADJ
ap-1860	2	13	slope	slope	NOUN
ap-1860	2	14	α	α	NOUN
ap-1860	2	15	.	.	PUNCT
ap-1860	3	1	for	for	ADP
ap-1860	3	2	a	a	DET
ap-1860	3	3	general	general	ADJ
ap-1860	3	4	subinterval	subinterval	NOUN
ap-1860	3	5	i	i	PRON
ap-1860	3	6	of	of	ADP
ap-1860	3	7	the	the	DET
ap-1860	3	8	domain	domain	NOUN
ap-1860	3	9	of	of	ADP
ap-1860	3	10	t	t	PROPN
ap-1860	3	11	,	,	PUNCT
ap-1860	3	12	the	the	DET
ap-1860	3	13	first	first	ADJ
ap-1860	3	14	return	return	NOUN
ap-1860	3	15	time	time	NOUN
ap-1860	3	16	to	to	ADP
ap-1860	3	17	i	i	PRON
ap-1860	3	18	takes	take	VERB
ap-1860	3	19	three	three	NUM
ap-1860	3	20	values	value	NOUN
ap-1860	3	21	.	.	PUNCT
ap-1860	4	1	we	we	PRON
ap-1860	4	2	describe	describe	VERB
ap-1860	4	3	the	the	DET
ap-1860	4	4	structure	structure	NOUN
ap-1860	4	5	of	of	ADP
ap-1860	4	6	the	the	DET
ap-1860	4	7	set	set	NOUN
ap-1860	4	8	of	of	ADP
ap-1860	4	9	return	return	NOUN
ap-1860	4	10	itineraries	itinerary	NOUN
ap-1860	4	11	to	to	PART
ap-1860	4	12	i.	i.	VERB
ap-1860	4	13	in	in	ADP
ap-1860	4	14	particular	particular	ADJ
ap-1860	4	15	,	,	PUNCT
ap-1860	4	16	we	we	PRON
ap-1860	4	17	show	show	VERB
ap-1860	4	18	that	that	SCONJ
ap-1860	4	19	it	it	PRON
ap-1860	4	20	is	be	AUX
ap-1860	4	21	equal	equal	ADJ
ap-1860	4	22	to	to	ADP
ap-1860	4	23	{	{	PUNCT
ap-1860	4	24	r1	r1	PROPN
ap-1860	4	25	,	,	PUNCT
ap-1860	4	26	r2	r2	PROPN
ap-1860	4	27	,	,	PUNCT
ap-1860	4	28	r1r2	r1r2	NOUN
ap-1860	4	29	,	,	PUNCT
ap-1860	4	30	q	q	NOUN
ap-1860	4	31	}	}	PUNCT
ap-1860	4	32	where	where	SCONJ
ap-1860	4	33	q	q	NOUN
ap-1860	4	34	is	be	AUX
ap-1860	4	35	amicable	amicable	ADJ
ap-1860	4	36	with	with	ADP
ap-1860	4	37	r1	r1	PROPN
ap-1860	4	38	,	,	PUNCT
ap-1860	4	39	r2	r2	PROPN
ap-1860	4	40	or	or	CCONJ
ap-1860	4	41	r1r2	r1r2	NOUN
ap-1860	4	42	.	.	PUNCT
ap-1860	5	1	keywords	keyword	NOUN
ap-1860	5	2	:	:	PUNCT
ap-1860	5	3	interval	interval	NOUN
ap-1860	5	4	exchange	exchange	NOUN
ap-1860	5	5	,	,	PUNCT
ap-1860	5	6	first	first	ADJ
ap-1860	5	7	return	return	NOUN
ap-1860	5	8	map	map	NOUN
ap-1860	5	9	,	,	PUNCT
ap-1860	5	10	return	return	VERB
ap-1860	5	11	time	time	NOUN
ap-1860	5	12	.	.	PUNCT
ap-1860	6	1	submitted	submit	VERB
ap-1860	6	2	:	:	PUNCT
ap-1860	6	3	27	27	NUM
ap-1860	6	4	march	march	NOUN
ap-1860	6	5	2013	2013	NUM
ap-1860	6	6	.	.	PUNCT
ap-1860	7	1	accepted	accept	VERB
ap-1860	7	2	:	:	PUNCT
ap-1860	7	3	22	22	NUM
ap-1860	7	4	april	april	PROPN
ap-1860	7	5	2013	2013	NUM
ap-1860	7	6	.	.	PUNCT
ap-1860	8	1	1	1	X
ap-1860	8	2	.	.	X
ap-1860	8	3	introduction	introduction	NOUN
ap-1860	8	4	we	we	PRON
ap-1860	8	5	study	study	VERB
ap-1860	8	6	the	the	DET
ap-1860	8	7	symbolic	symbolic	ADJ
ap-1860	8	8	dynamical	dynamical	ADJ
ap-1860	8	9	system	system	NOUN
ap-1860	8	10	given	give	VERB
ap-1860	8	11	by	by	ADP
ap-1860	8	12	the	the	DET
ap-1860	8	13	transformation	transformation	NOUN
ap-1860	8	14	t	t	PROPN
ap-1860	8	15	of	of	ADP
ap-1860	8	16	the	the	DET
ap-1860	8	17	unit	unit	NOUN
ap-1860	8	18	interval	interval	NOUN
ap-1860	8	19	,	,	PUNCT
ap-1860	8	20	t	t	X
ap-1860	8	21	:	:	PUNCT
ap-1860	9	1	[	[	X
ap-1860	9	2	0	0	NUM
ap-1860	9	3	,	,	PUNCT
ap-1860	9	4	1)→	1)→	NUM
ap-1860	10	1	[	[	X
ap-1860	10	2	0	0	NUM
ap-1860	10	3	,	,	PUNCT
ap-1860	10	4	1	1	NUM
ap-1860	10	5	)	)	PUNCT
ap-1860	10	6	,	,	PUNCT
ap-1860	10	7	t	t	PROPN
ap-1860	10	8	(	(	PUNCT
ap-1860	10	9	x	x	X
ap-1860	10	10	)	)	PUNCT
ap-1860	11	1	=	=	PRON
ap-1860	11	2	{	{	PUNCT
ap-1860	11	3	x−	x−	PROPN
ap-1860	11	4	α	α	PROPN
ap-1860	11	5	for	for	ADP
ap-1860	11	6	x	x	PROPN
ap-1860	11	7	∈	∈	PROPN
ap-1860	11	8	[	[	X
ap-1860	11	9	α	α	NOUN
ap-1860	11	10	,	,	PUNCT
ap-1860	11	11	1	1	NUM
ap-1860	11	12	)	)	PUNCT
ap-1860	11	13	,	,	PUNCT
ap-1860	11	14	x+	x+	NUM
ap-1860	11	15	1−	1−	NUM
ap-1860	11	16	α	α	NOUN
ap-1860	11	17	for	for	ADP
ap-1860	11	18	x	x	PROPN
ap-1860	11	19	∈	∈	PROPN
ap-1860	12	1	[	[	X
ap-1860	12	2	0	0	NUM
ap-1860	12	3	,	,	PUNCT
ap-1860	12	4	α	α	NOUN
ap-1860	12	5	)	)	PUNCT
ap-1860	12	6	,	,	PUNCT
ap-1860	12	7	(	(	PUNCT
ap-1860	12	8	1	1	X
ap-1860	12	9	)	)	PUNCT
ap-1860	12	10	where	where	SCONJ
ap-1860	12	11	α	α	NOUN
ap-1860	12	12	is	be	AUX
ap-1860	12	13	a	a	DET
ap-1860	12	14	fixed	fix	VERB
ap-1860	12	15	number	number	NOUN
ap-1860	12	16	in	in	ADP
ap-1860	12	17	[	[	X
ap-1860	12	18	0	0	NUM
ap-1860	12	19	,	,	PUNCT
ap-1860	12	20	1	1	NUM
ap-1860	12	21	)	)	PUNCT
ap-1860	12	22	.	.	PUNCT
ap-1860	13	1	transformation	transformation	NOUN
ap-1860	13	2	t	t	PROPN
ap-1860	13	3	has	have	VERB
ap-1860	13	4	only	only	ADV
ap-1860	13	5	one	one	NUM
ap-1860	13	6	discontinuity	discontinuity	NOUN
ap-1860	13	7	point	point	NOUN
ap-1860	13	8	,	,	PUNCT
ap-1860	13	9	such	such	DET
ap-1860	13	10	a	a	DET
ap-1860	13	11	dynamical	dynamical	ADJ
ap-1860	13	12	system	system	NOUN
ap-1860	13	13	is	be	AUX
ap-1860	13	14	the	the	DET
ap-1860	13	15	simplest	simple	ADJ
ap-1860	13	16	dynamical	dynamical	ADJ
ap-1860	13	17	system	system	NOUN
ap-1860	13	18	with	with	ADP
ap-1860	13	19	discontinuous	discontinuous	ADJ
ap-1860	13	20	transformation	transformation	NOUN
ap-1860	13	21	.	.	PUNCT
ap-1860	14	1	dynamical	dynamical	ADJ
ap-1860	14	2	systems	system	NOUN
ap-1860	14	3	defined	define	VERB
ap-1860	14	4	by	by	ADP
ap-1860	14	5	continuous	continuous	ADJ
ap-1860	14	6	transformations	transformation	NOUN
ap-1860	14	7	f	f	NOUN
ap-1860	14	8	:	:	PUNCT
ap-1860	14	9	j	j	PROPN
ap-1860	14	10	→	→	SYM
ap-1860	14	11	j	j	PROPN
ap-1860	14	12	have	have	VERB
ap-1860	14	13	a	a	DET
ap-1860	14	14	number	number	NOUN
ap-1860	14	15	of	of	ADP
ap-1860	14	16	nice	nice	ADJ
ap-1860	14	17	properties	property	NOUN
ap-1860	14	18	,	,	PUNCT
ap-1860	14	19	for	for	ADP
ap-1860	14	20	example	example	NOUN
ap-1860	14	21	,	,	PUNCT
ap-1860	14	22	there	there	PRON
ap-1860	14	23	exists	exist	VERB
ap-1860	14	24	a	a	DET
ap-1860	14	25	fixed	fixed	ADJ
ap-1860	14	26	point	point	NOUN
ap-1860	14	27	ρ	ρ	PROPN
ap-1860	14	28	∈	∈	PROPN
ap-1860	14	29	j	j	PROPN
ap-1860	14	30	,	,	PUNCT
ap-1860	14	31	f	f	PROPN
ap-1860	14	32	(	(	PUNCT
ap-1860	14	33	ρ	ρ	NOUN
ap-1860	14	34	)	)	PUNCT
ap-1860	14	35	=	=	SYM
ap-1860	14	36	ρ	ρ	PROPN
ap-1860	14	37	.	.	PUNCT
ap-1860	15	1	the	the	DET
ap-1860	15	2	famous	famous	ADJ
ap-1860	15	3	theorem	theorem	NOUN
ap-1860	15	4	of	of	ADP
ap-1860	15	5	sharkovskii	sharkovskii	PROPN
ap-1860	16	1	[	[	X
ap-1860	16	2	1	1	NUM
ap-1860	16	3	]	]	PUNCT
ap-1860	16	4	describes	describe	VERB
ap-1860	16	5	the	the	DET
ap-1860	16	6	structure	structure	NOUN
ap-1860	16	7	of	of	ADP
ap-1860	16	8	periodic	periodic	ADJ
ap-1860	16	9	points	point	NOUN
ap-1860	16	10	,	,	PUNCT
ap-1860	16	11	i.e.	i.e.	X
ap-1860	16	12	,	,	PUNCT
ap-1860	16	13	fixed	fix	VERB
ap-1860	16	14	points	point	NOUN
ap-1860	16	15	of	of	ADP
ap-1860	16	16	f	f	PROPN
ap-1860	16	17	k	k	PROPN
ap-1860	16	18	for	for	ADP
ap-1860	16	19	some	some	DET
ap-1860	16	20	k	k	PROPN
ap-1860	16	21	∈	∈	PROPN
ap-1860	16	22	n.	n.	NOUN
ap-1860	16	23	if	if	SCONJ
ap-1860	16	24	one	one	PRON
ap-1860	16	25	chooses	choose	VERB
ap-1860	16	26	the	the	DET
ap-1860	16	27	parameter	parameter	NOUN
ap-1860	16	28	α	α	PROPN
ap-1860	16	29	in	in	ADP
ap-1860	16	30	(	(	PUNCT
ap-1860	16	31	1	1	NUM
ap-1860	16	32	)	)	PUNCT
ap-1860	16	33	irrational	irrational	ADJ
ap-1860	16	34	,	,	PUNCT
ap-1860	16	35	the	the	DET
ap-1860	16	36	map	map	NOUN
ap-1860	16	37	t	t	PROPN
ap-1860	16	38	has	have	VERB
ap-1860	16	39	no	no	DET
ap-1860	16	40	periodic	periodic	ADJ
ap-1860	16	41	point	point	NOUN
ap-1860	16	42	,	,	PUNCT
ap-1860	16	43	in	in	ADP
ap-1860	16	44	other	other	ADJ
ap-1860	16	45	words	word	NOUN
ap-1860	16	46	,	,	PUNCT
ap-1860	16	47	the	the	DET
ap-1860	16	48	orbit	orbit	NOUN
ap-1860	16	49	{	{	PUNCT
ap-1860	16	50	ρ	ρ	PROPN
ap-1860	16	51	,	,	PUNCT
ap-1860	16	52	t	t	PROPN
ap-1860	16	53	(	(	PUNCT
ap-1860	16	54	ρ	ρ	PROPN
ap-1860	16	55	)	)	PUNCT
ap-1860	16	56	,	,	PUNCT
ap-1860	16	57	t	t	PROPN
ap-1860	16	58	2(ρ	2(ρ	NUM
ap-1860	16	59	)	)	PUNCT
ap-1860	16	60	,	,	PUNCT
ap-1860	16	61	.	.	PUNCT
ap-1860	16	62	.	.	PUNCT
ap-1860	16	63	.	.	PUNCT
ap-1860	17	1	}	}	PUNCT
ap-1860	17	2	is	be	AUX
ap-1860	17	3	infinite	infinite	ADJ
ap-1860	17	4	for	for	ADP
ap-1860	17	5	every	every	DET
ap-1860	17	6	ρ	ρ	PROPN
ap-1860	17	7	∈	∈	PROPN
ap-1860	18	1	[	[	X
ap-1860	18	2	0	0	NUM
ap-1860	18	3	,	,	PUNCT
ap-1860	18	4	1	1	NUM
ap-1860	18	5	)	)	PUNCT
ap-1860	18	6	.	.	PUNCT
ap-1860	19	1	nevertheless	nevertheless	ADV
ap-1860	19	2	,	,	PUNCT
ap-1860	19	3	t	t	PROPN
ap-1860	19	4	has	have	VERB
ap-1860	19	5	a	a	DET
ap-1860	19	6	weaker	weak	ADJ
ap-1860	19	7	property	property	NOUN
ap-1860	19	8	,	,	PUNCT
ap-1860	19	9	namely	namely	ADV
ap-1860	19	10	that	that	SCONJ
ap-1860	19	11	although	although	SCONJ
ap-1860	19	12	t	t	PROPN
ap-1860	19	13	k(ρ	k(ρ	NOUN
ap-1860	19	14	)	)	PUNCT
ap-1860	19	15	6=	6=	ADP
ap-1860	19	16	ρ	ρ	NOUN
ap-1860	19	17	for	for	ADP
ap-1860	19	18	any	any	DET
ap-1860	19	19	k	k	PROPN
ap-1860	19	20	∈	∈	PROPN
ap-1860	19	21	n	n	CCONJ
ap-1860	19	22	,	,	PUNCT
ap-1860	19	23	one	one	PRON
ap-1860	19	24	can	can	AUX
ap-1860	19	25	get	get	VERB
ap-1860	19	26	arbitrarily	arbitrarily	ADV
ap-1860	19	27	close	close	ADJ
ap-1860	19	28	to	to	ADP
ap-1860	19	29	a	a	DET
ap-1860	19	30	point	point	NOUN
ap-1860	19	31	ρ	ρ	NOUN
ap-1860	19	32	with	with	ADP
ap-1860	19	33	some	some	PRON
ap-1860	19	34	of	of	ADP
ap-1860	19	35	its	its	PRON
ap-1860	19	36	iterations	iteration	NOUN
ap-1860	19	37	.	.	PUNCT
ap-1860	20	1	more	more	ADV
ap-1860	20	2	precisely	precisely	ADV
ap-1860	20	3	,	,	PUNCT
ap-1860	20	4	∀ε	∀ε	VERB
ap-1860	20	5	>	>	X
ap-1860	20	6	0	0	PUNCT
ap-1860	21	1	∃n	∃n	PROPN
ap-1860	21	2	∈	∈	PROPN
ap-1860	21	3	n	n	CCONJ
ap-1860	21	4	,	,	PUNCT
ap-1860	21	5	n	n	X
ap-1860	21	6	≥	≥	NOUN
ap-1860	21	7	1	1	NUM
ap-1860	21	8	:	:	PUNCT
ap-1860	21	9	∣∣tn(ρ)−	∣∣tn(ρ)−	PROPN
ap-1860	21	10	ρ	ρ	X
ap-1860	21	11	∣∣	∣∣	X
ap-1860	21	12	<	<	X
ap-1860	21	13	ε	ε	PROPN
ap-1860	21	14	.	.	PUNCT
ap-1860	21	15	(	(	PUNCT
ap-1860	21	16	2	2	NUM
ap-1860	21	17	)	)	PUNCT
ap-1860	21	18	moreover	moreover	ADV
ap-1860	21	19	,	,	PUNCT
ap-1860	21	20	property	property	NOUN
ap-1860	21	21	(	(	PUNCT
ap-1860	21	22	2	2	NUM
ap-1860	21	23	)	)	PUNCT
ap-1860	21	24	holds	hold	VERB
ap-1860	21	25	for	for	ADP
ap-1860	21	26	every	every	DET
ap-1860	21	27	ρ	ρ	PROPN
ap-1860	21	28	∈	∈	PROPN
ap-1860	22	1	[	[	X
ap-1860	22	2	0	0	NUM
ap-1860	22	3	,	,	PUNCT
ap-1860	22	4	1	1	NUM
ap-1860	22	5	)	)	PUNCT
ap-1860	22	6	.	.	PUNCT
ap-1860	23	1	it	it	PRON
ap-1860	23	2	is	be	AUX
ap-1860	23	3	well	well	ADV
ap-1860	23	4	known	know	VERB
ap-1860	23	5	that	that	SCONJ
ap-1860	23	6	every	every	DET
ap-1860	23	7	point	point	NOUN
ap-1860	23	8	ρ	ρ	X
ap-1860	23	9	∈	∈	PROPN
ap-1860	23	10	[	[	X
ap-1860	23	11	0	0	NUM
ap-1860	23	12	,	,	PUNCT
ap-1860	23	13	1	1	NUM
ap-1860	23	14	)	)	PUNCT
ap-1860	23	15	can	can	AUX
ap-1860	23	16	be	be	AUX
ap-1860	23	17	uniquely	uniquely	ADV
ap-1860	23	18	represented	represent	VERB
ap-1860	23	19	using	use	VERB
ap-1860	23	20	the	the	DET
ap-1860	23	21	infinite	infinite	ADJ
ap-1860	23	22	string	string	NOUN
ap-1860	23	23	of	of	ADP
ap-1860	23	24	0	0	NUM
ap-1860	23	25	and	and	CCONJ
ap-1860	23	26	1	1	NUM
ap-1860	23	27	,	,	PUNCT
ap-1860	23	28	which	which	PRON
ap-1860	23	29	constitutes	constitute	VERB
ap-1860	23	30	the	the	DET
ap-1860	23	31	binary	binary	ADJ
ap-1860	23	32	expansion	expansion	NOUN
ap-1860	23	33	of	of	ADP
ap-1860	23	34	the	the	DET
ap-1860	23	35	number	number	NOUN
ap-1860	23	36	ρ	ρ	PROPN
ap-1860	23	37	.	.	PUNCT
ap-1860	24	1	the	the	DET
ap-1860	24	2	mapping	mapping	NOUN
ap-1860	24	3	t	t	PROPN
ap-1860	24	4	of	of	ADP
ap-1860	24	5	(	(	PUNCT
ap-1860	24	6	1	1	X
ap-1860	24	7	)	)	PUNCT
ap-1860	24	8	allows	allow	VERB
ap-1860	24	9	another	another	DET
ap-1860	24	10	type	type	NOUN
ap-1860	24	11	of	of	ADP
ap-1860	24	12	representation	representation	NOUN
ap-1860	24	13	of	of	ADP
ap-1860	24	14	ρ	ρ	PROPN
ap-1860	24	15	,	,	PUNCT
ap-1860	24	16	namely	namely	ADV
ap-1860	24	17	by	by	ADP
ap-1860	24	18	the	the	DET
ap-1860	24	19	coding	coding	NOUN
ap-1860	24	20	of	of	ADP
ap-1860	24	21	the	the	DET
ap-1860	24	22	orbit	orbit	NOUN
ap-1860	24	23	of	of	ADP
ap-1860	24	24	ρ	ρ	PROPN
ap-1860	24	25	under	under	ADP
ap-1860	24	26	t	t	PROPN
ap-1860	24	27	.	.	PUNCT
ap-1860	25	1	denote	denote	VERB
ap-1860	25	2	j0	j0	PROPN
ap-1860	25	3	=	=	PUNCT
ap-1860	26	1	[	[	X
ap-1860	26	2	0	0	NUM
ap-1860	26	3	,	,	PUNCT
ap-1860	26	4	α	α	NOUN
ap-1860	26	5	)	)	PUNCT
ap-1860	26	6	,	,	PUNCT
ap-1860	26	7	j1	j1	PROPN
ap-1860	26	8	=	=	PUNCT
ap-1860	27	1	[	[	X
ap-1860	27	2	α	α	X
ap-1860	27	3	,	,	PUNCT
ap-1860	27	4	1	1	NUM
ap-1860	27	5	)	)	PUNCT
ap-1860	27	6	and	and	CCONJ
ap-1860	27	7	set	set	VERB
ap-1860	27	8	un	un	PROPN
ap-1860	27	9	=	=	X
ap-1860	27	10	{	{	PUNCT
ap-1860	27	11	0	0	NUM
ap-1860	27	12	if	if	SCONJ
ap-1860	27	13	tn(ρ	tn(ρ	NUM
ap-1860	27	14	)	)	PUNCT
ap-1860	27	15	∈	∈	PROPN
ap-1860	27	16	j0	j0	PROPN
ap-1860	27	17	,	,	PUNCT
ap-1860	27	18	1	1	NUM
ap-1860	27	19	if	if	SCONJ
ap-1860	27	20	tn(ρ	tn(ρ	NUM
ap-1860	27	21	)	)	PUNCT
ap-1860	27	22	∈	∈	PROPN
ap-1860	27	23	j1	j1	PROPN
ap-1860	27	24	.	.	PUNCT
ap-1860	28	1	knowledge	knowledge	NOUN
ap-1860	28	2	of	of	ADP
ap-1860	28	3	the	the	DET
ap-1860	28	4	infinite	infinite	ADJ
ap-1860	28	5	word	word	NOUN
ap-1860	28	6	uρ	uρ	INTJ
ap-1860	28	7	:	:	PUNCT
ap-1860	28	8	=	=	SYM
ap-1860	28	9	(	(	PUNCT
ap-1860	28	10	un)∞n=0	un)∞n=0	NOUN
ap-1860	28	11	allows	allow	VERB
ap-1860	28	12	one	one	NUM
ap-1860	28	13	to	to	PART
ap-1860	28	14	determine	determine	VERB
ap-1860	28	15	the	the	DET
ap-1860	28	16	number	number	NOUN
ap-1860	28	17	ρ	ρ	NOUN
ap-1860	28	18	,	,	PUNCT
ap-1860	28	19	i.e.	i.e.	X
ap-1860	28	20	,	,	PUNCT
ap-1860	28	21	the	the	DET
ap-1860	28	22	mapping	mapping	NOUN
ap-1860	28	23	ρ	ρ	NOUN
ap-1860	28	24	7→	7→	NUM
ap-1860	29	1	uρ	uρ	NOUN
ap-1860	29	2	is	be	AUX
ap-1860	29	3	one	one	NUM
ap-1860	29	4	-	-	PUNCT
ap-1860	29	5	to	to	ADP
ap-1860	29	6	-	-	PUNCT
ap-1860	29	7	one	one	NUM
ap-1860	29	8	.	.	PUNCT
ap-1860	30	1	the	the	DET
ap-1860	30	2	above	above	ADV
ap-1860	30	3	defined	define	VERB
ap-1860	30	4	infinite	infinite	ADJ
ap-1860	30	5	words	word	NOUN
ap-1860	30	6	uρ	uρ	AUX
ap-1860	30	7	appear	appear	VERB
ap-1860	30	8	naturally	naturally	ADV
ap-1860	30	9	in	in	ADP
ap-1860	30	10	diverse	diverse	ADJ
ap-1860	30	11	mathematical	mathematical	ADJ
ap-1860	30	12	problems	problem	NOUN
ap-1860	30	13	;	;	PUNCT
ap-1860	30	14	they	they	PRON
ap-1860	30	15	were	be	AUX
ap-1860	30	16	discovered	discover	VERB
ap-1860	30	17	and	and	CCONJ
ap-1860	30	18	re	re	VERB
ap-1860	30	19	-	-	VERB
ap-1860	30	20	discovered	discover	VERB
ap-1860	30	21	several	several	ADJ
ap-1860	30	22	times	time	NOUN
ap-1860	30	23	and	and	CCONJ
ap-1860	30	24	given	give	VERB
ap-1860	30	25	different	different	ADJ
ap-1860	30	26	names	name	NOUN
ap-1860	30	27	.	.	PUNCT
ap-1860	31	1	we	we	PRON
ap-1860	31	2	will	will	AUX
ap-1860	31	3	call	call	VERB
ap-1860	31	4	the	the	DET
ap-1860	31	5	infinite	infinite	ADJ
ap-1860	31	6	word	word	NOUN
ap-1860	31	7	uρ	uρ	ADP
ap-1860	31	8	a	a	DET
ap-1860	31	9	sturmian	sturmian	NOUN
ap-1860	31	10	word	word	NOUN
ap-1860	31	11	with	with	ADP
ap-1860	31	12	slope	slope	NOUN
ap-1860	31	13	α	α	NOUN
ap-1860	31	14	and	and	CCONJ
ap-1860	31	15	intercept	intercept	PROPN
ap-1860	31	16	ρ	ρ	PROPN
ap-1860	31	17	.	.	PUNCT
ap-1860	32	1	let	let	VERB
ap-1860	32	2	us	we	PRON
ap-1860	32	3	point	point	VERB
ap-1860	32	4	out	out	ADP
ap-1860	32	5	one	one	NUM
ap-1860	32	6	important	important	ADJ
ap-1860	32	7	difference	difference	NOUN
ap-1860	32	8	between	between	ADP
ap-1860	32	9	binary	binary	ADJ
ap-1860	32	10	expansion	expansion	NOUN
ap-1860	32	11	of	of	ADP
ap-1860	32	12	numbers	number	NOUN
ap-1860	32	13	and	and	CCONJ
ap-1860	32	14	their	their	PRON
ap-1860	32	15	representation	representation	NOUN
ap-1860	32	16	by	by	ADP
ap-1860	32	17	sturmian	sturmian	ADJ
ap-1860	32	18	words	word	NOUN
ap-1860	32	19	with	with	ADP
ap-1860	32	20	a	a	DET
ap-1860	32	21	fixed	fix	VERB
ap-1860	32	22	slope	slope	NOUN
ap-1860	32	23	α	α	NOUN
ap-1860	32	24	.	.	PUNCT
ap-1860	33	1	every	every	DET
ap-1860	33	2	string	string	NOUN
ap-1860	33	3	of	of	ADP
ap-1860	33	4	length	length	NOUN
ap-1860	33	5	n	n	PROPN
ap-1860	33	6	of	of	ADP
ap-1860	33	7	letters	letter	NOUN
ap-1860	33	8	0	0	PUNCT
ap-1860	33	9	and	and	CCONJ
ap-1860	33	10	1	1	NUM
ap-1860	33	11	appears	appear	VERB
ap-1860	33	12	in	in	ADP
ap-1860	33	13	the	the	DET
ap-1860	33	14	binary	binary	ADJ
ap-1860	33	15	expansion	expansion	NOUN
ap-1860	33	16	of	of	ADP
ap-1860	33	17	some	some	DET
ap-1860	33	18	real	real	ADJ
ap-1860	33	19	number	number	NOUN
ap-1860	33	20	ρ	ρ	PROPN
ap-1860	33	21	∈	∈	PROPN
ap-1860	34	1	[	[	X
ap-1860	34	2	0	0	NUM
ap-1860	34	3	,	,	PUNCT
ap-1860	34	4	1	1	NUM
ap-1860	34	5	)	)	PUNCT
ap-1860	34	6	.	.	PUNCT
ap-1860	35	1	the	the	DET
ap-1860	35	2	number	number	NOUN
ap-1860	35	3	of	of	ADP
ap-1860	35	4	such	such	ADJ
ap-1860	35	5	strings	string	NOUN
ap-1860	35	6	is	be	AUX
ap-1860	35	7	obviously	obviously	ADV
ap-1860	35	8	2n	2n	NUM
ap-1860	35	9	.	.	PUNCT
ap-1860	36	1	by	by	ADP
ap-1860	36	2	contrast	contrast	NOUN
ap-1860	36	3	,	,	PUNCT
ap-1860	36	4	the	the	DET
ap-1860	36	5	list	list	NOUN
ap-1860	36	6	of	of	ADP
ap-1860	36	7	all	all	DET
ap-1860	36	8	strings	string	NOUN
ap-1860	36	9	of	of	ADP
ap-1860	36	10	length	length	NOUN
ap-1860	36	11	n	n	ADP
ap-1860	36	12	appearing	appear	VERB
ap-1860	36	13	in	in	ADP
ap-1860	36	14	the	the	DET
ap-1860	36	15	representation	representation	NOUN
ap-1860	36	16	uρ	uρ	ADP
ap-1860	36	17	of	of	ADP
ap-1860	36	18	all	all	DET
ap-1860	36	19	ρ	ρ	NOUN
ap-1860	36	20	∈	∈	PROPN
ap-1860	37	1	[	[	X
ap-1860	37	2	0	0	NUM
ap-1860	37	3	,	,	PUNCT
ap-1860	37	4	1	1	NUM
ap-1860	37	5	)	)	PUNCT
ap-1860	37	6	has	have	VERB
ap-1860	37	7	exactly	exactly	ADV
ap-1860	37	8	n+	n+	ADP
ap-1860	38	1	1	1	NUM
ap-1860	38	2	elements	element	NOUN
ap-1860	38	3	.	.	PUNCT
ap-1860	39	1	nevertheless	nevertheless	ADV
ap-1860	39	2	,	,	PUNCT
ap-1860	39	3	one	one	PRON
ap-1860	39	4	can	can	AUX
ap-1860	39	5	still	still	ADV
ap-1860	39	6	represent	represent	VERB
ap-1860	39	7	a	a	DET
ap-1860	39	8	continuum	continuum	NOUN
ap-1860	39	9	of	of	ADP
ap-1860	39	10	real	real	ADJ
ap-1860	39	11	numbers	number	NOUN
ap-1860	39	12	ρ	ρ	NOUN
ap-1860	39	13	.	.	PUNCT
ap-1860	40	1	on	on	ADP
ap-1860	40	2	the	the	DET
ap-1860	40	3	other	other	ADJ
ap-1860	40	4	hand	hand	NOUN
ap-1860	40	5	,	,	PUNCT
ap-1860	40	6	any	any	DET
ap-1860	40	7	type	type	NOUN
ap-1860	40	8	of	of	ADP
ap-1860	40	9	representation	representation	NOUN
ap-1860	40	10	using	use	VERB
ap-1860	40	11	at	at	ADP
ap-1860	40	12	most	most	ADJ
ap-1860	40	13	n	n	PRON
ap-1860	40	14	strings	string	NOUN
ap-1860	40	15	of	of	ADP
ap-1860	40	16	0	0	NUM
ap-1860	40	17	and	and	CCONJ
ap-1860	40	18	1	1	NUM
ap-1860	40	19	of	of	ADP
ap-1860	40	20	length	length	NOUN
ap-1860	40	21	n	n	PROPN
ap-1860	40	22	would	would	AUX
ap-1860	40	23	allow	allow	VERB
ap-1860	40	24	representation	representation	NOUN
ap-1860	40	25	of	of	ADP
ap-1860	40	26	only	only	ADV
ap-1860	40	27	countably	countably	ADV
ap-1860	40	28	many	many	ADJ
ap-1860	40	29	numbers	number	NOUN
ap-1860	40	30	.	.	PUNCT
ap-1860	41	1	in	in	ADP
ap-1860	41	2	that	that	DET
ap-1860	41	3	sense	sense	NOUN
ap-1860	41	4	,	,	PUNCT
ap-1860	41	5	sturmian	sturmian	NOUN
ap-1860	41	6	words	word	NOUN
ap-1860	41	7	represent	represent	VERB
ap-1860	41	8	real	real	ADJ
ap-1860	41	9	numbers	number	NOUN
ap-1860	41	10	in	in	ADP
ap-1860	41	11	the	the	DET
ap-1860	41	12	most	most	ADV
ap-1860	41	13	economical	economical	ADJ
ap-1860	41	14	way	way	NOUN
ap-1860	41	15	.	.	PUNCT
ap-1860	42	1	sturmian	sturmian	NOUN
ap-1860	42	2	words	word	NOUN
ap-1860	42	3	have	have	VERB
ap-1860	42	4	many	many	ADJ
ap-1860	42	5	other	other	ADJ
ap-1860	42	6	remarkable	remarkable	ADJ
ap-1860	42	7	properties	property	NOUN
ap-1860	42	8	,	,	PUNCT
ap-1860	42	9	for	for	ADP
ap-1860	42	10	a	a	DET
ap-1860	42	11	review	review	NOUN
ap-1860	42	12	,	,	PUNCT
ap-1860	42	13	see	see	VERB
ap-1860	42	14	[	[	X
ap-1860	42	15	2	2	NUM
ap-1860	42	16	]	]	PUNCT
ap-1860	42	17	.	.	PUNCT
ap-1860	43	1	generalizations	generalization	NOUN
ap-1860	43	2	of	of	ADP
ap-1860	43	3	sturmian	sturmian	ADJ
ap-1860	43	4	words	word	NOUN
ap-1860	43	5	are	be	AUX
ap-1860	43	6	treated	treat	VERB
ap-1860	43	7	in	in	ADP
ap-1860	43	8	[	[	X
ap-1860	43	9	3	3	NUM
ap-1860	43	10	]	]	PUNCT
ap-1860	43	11	.	.	PUNCT
ap-1860	44	1	the	the	DET
ap-1860	44	2	property	property	NOUN
ap-1860	44	3	(	(	PUNCT
ap-1860	44	4	2	2	X
ap-1860	44	5	)	)	PUNCT
ap-1860	44	6	expresses	express	VERB
ap-1860	44	7	the	the	DET
ap-1860	44	8	fact	fact	NOUN
ap-1860	44	9	that	that	SCONJ
ap-1860	44	10	iterations	iteration	NOUN
ap-1860	44	11	tn(ρ	tn(ρ	PUNCT
ap-1860	44	12	)	)	PUNCT
ap-1860	44	13	return	return	VERB
ap-1860	44	14	arbitrarily	arbitrarily	ADV
ap-1860	44	15	close	close	ADJ
ap-1860	44	16	to	to	ADP
ap-1860	44	17	ρ	ρ	NUM
ap-1860	44	18	.	.	PUNCT
ap-1860	45	1	this	this	PRON
ap-1860	45	2	allows	allow	VERB
ap-1860	45	3	one	one	NUM
ap-1860	45	4	to	to	PART
ap-1860	45	5	define	define	VERB
ap-1860	45	6	,	,	PUNCT
ap-1860	45	7	for	for	ADP
ap-1860	45	8	a	a	DET
ap-1860	45	9	subinterval	subinterval	NOUN
ap-1860	46	1	i	i	PRON
ap-1860	46	2	⊂	⊂	PROPN
ap-1860	47	1	[	[	X
ap-1860	47	2	0	0	NUM
ap-1860	47	3	,	,	PUNCT
ap-1860	47	4	1	1	NUM
ap-1860	47	5	)	)	PUNCT
ap-1860	47	6	of	of	ADP
ap-1860	47	7	positive	positive	ADJ
ap-1860	47	8	length	length	NOUN
ap-1860	47	9	,	,	PUNCT
ap-1860	47	10	the	the	DET
ap-1860	47	11	so	so	ADV
ap-1860	47	12	-	-	PUNCT
ap-1860	47	13	called	call	VERB
ap-1860	47	14	return	return	NOUN
ap-1860	47	15	time	time	NOUN
ap-1860	47	16	r	r	NOUN
ap-1860	47	17	:	:	PUNCT
ap-1860	47	18	i	i	PROPN
ap-1860	47	19	→	→	SYM
ap-1860	47	20	n	n	CCONJ
ap-1860	47	21	by	by	ADP
ap-1860	47	22	r(ρ	r(ρ	NOUN
ap-1860	47	23	)	)	PUNCT
ap-1860	47	24	:	:	PUNCT
ap-1860	47	25	=	=	SYM
ap-1860	47	26	min	min	NOUN
ap-1860	47	27	{	{	PUNCT
ap-1860	47	28	n	n	CCONJ
ap-1860	47	29	∈	∈	PROPN
ap-1860	47	30	n	n	CCONJ
ap-1860	47	31	,	,	PUNCT
ap-1860	47	32	n	n	PRON
ap-1860	47	33	≥	≥	NOUN
ap-1860	47	34	1	1	NUM
ap-1860	47	35	,	,	PUNCT
ap-1860	47	36	:	:	PUNCT
ap-1860	47	37	tn(ρ	tn(ρ	X
ap-1860	47	38	)	)	PUNCT
ap-1860	47	39	∈	∈	PROPN
ap-1860	48	1	i	i	PRON
ap-1860	48	2	}	}	PUNCT
ap-1860	48	3	.	.	PUNCT
ap-1860	49	1	the	the	DET
ap-1860	49	2	return	return	NOUN
ap-1860	49	3	time	time	NOUN
ap-1860	49	4	represents	represent	VERB
ap-1860	49	5	the	the	DET
ap-1860	49	6	number	number	NOUN
ap-1860	49	7	of	of	ADP
ap-1860	49	8	iterations	iteration	NOUN
ap-1860	49	9	needed	need	VERB
ap-1860	49	10	for	for	ADP
ap-1860	49	11	a	a	DET
ap-1860	49	12	point	point	NOUN
ap-1860	49	13	ρ	ρ	NOUN
ap-1860	49	14	to	to	PART
ap-1860	49	15	come	come	VERB
ap-1860	49	16	back	back	ADV
ap-1860	49	17	to	to	ADP
ap-1860	49	18	the	the	DET
ap-1860	49	19	interval	interval	NOUN
ap-1860	49	20	where	where	SCONJ
ap-1860	49	21	it	it	PRON
ap-1860	49	22	comes	come	VERB
ap-1860	49	23	from	from	ADP
ap-1860	49	24	.	.	PUNCT
ap-1860	50	1	the	the	DET
ap-1860	50	2	movement	movement	NOUN
ap-1860	50	3	of	of	ADP
ap-1860	50	4	point	point	NOUN
ap-1860	50	5	ρ	ρ	PROPN
ap-1860	50	6	on	on	ADP
ap-1860	50	7	its	its	PRON
ap-1860	50	8	path	path	NOUN
ap-1860	50	9	from	from	ADP
ap-1860	50	10	i	i	PRON
ap-1860	50	11	back	back	ADV
ap-1860	50	12	to	to	ADP
ap-1860	50	13	i	i	PRON
ap-1860	50	14	is	be	AUX
ap-1860	50	15	recorded	record	VERB
ap-1860	50	16	by	by	ADP
ap-1860	50	17	the	the	DET
ap-1860	50	18	so	so	ADV
ap-1860	50	19	-	-	PUNCT
ap-1860	50	20	called	call	VERB
ap-1860	50	21	i	i	NOUN
ap-1860	50	22	-	-	PUNCT
ap-1860	50	23	itinerary	itinerary	NOUN
ap-1860	50	24	of	of	ADP
ap-1860	50	25	ρ	ρ	PROPN
ap-1860	50	26	,	,	PUNCT
ap-1860	50	27	which	which	PRON
ap-1860	50	28	we	we	PRON
ap-1860	50	29	denote	denote	VERB
ap-1860	50	30	by	by	ADP
ap-1860	50	31	r(ρ	r(ρ	PROPN
ap-1860	50	32	)	)	PUNCT
ap-1860	50	33	.	.	PUNCT
ap-1860	51	1	it	it	PRON
ap-1860	51	2	is	be	AUX
ap-1860	51	3	defined	define	VERB
ap-1860	51	4	as	as	ADP
ap-1860	51	5	the	the	DET
ap-1860	51	6	finite	finite	ADJ
ap-1860	51	7	word	word	NOUN
ap-1860	51	8	w0w1	w0w1	X
ap-1860	51	9	·	·	PUNCT
ap-1860	51	10	·	·	PUNCT
ap-1860	52	1	·	·	PUNCT
ap-1860	52	2	wn−1	wn−1	ADJ
ap-1860	52	3	in	in	ADP
ap-1860	52	4	the	the	DET
ap-1860	52	5	alphabet	alphabet	NOUN
ap-1860	52	6	a	a	PROPN
ap-1860	52	7	=	=	X
ap-1860	52	8	{	{	PUNCT
ap-1860	52	9	0	0	NUM
ap-1860	52	10	,	,	PUNCT
ap-1860	52	11	1	1	NUM
ap-1860	52	12	}	}	PUNCT
ap-1860	52	13	of	of	ADP
ap-1860	52	14	length	length	NOUN
ap-1860	52	15	n	n	PROPN
ap-1860	52	16	=	=	SYM
ap-1860	52	17	r(ρ	r(ρ	NOUN
ap-1860	52	18	)	)	PUNCT
ap-1860	52	19	such	such	ADJ
ap-1860	52	20	that	that	SCONJ
ap-1860	52	21	wi	wi	PROPN
ap-1860	52	22	=	=	SYM
ap-1860	52	23	a	a	PROPN
ap-1860	52	24	,	,	PUNCT
ap-1860	52	25	if	if	SCONJ
ap-1860	52	26	t	t	PROPN
ap-1860	52	27	i(ρ	i(ρ	PROPN
ap-1860	52	28	)	)	PUNCT
ap-1860	52	29	∈	∈	PROPN
ap-1860	52	30	ja	ja	PROPN
ap-1860	52	31	,	,	PUNCT
ap-1860	52	32	a	a	PRON
ap-1860	52	33	∈	∈	PROPN
ap-1860	52	34	a.	a.	NOUN
ap-1860	52	35	equivalently	equivalently	ADV
ap-1860	52	36	,	,	PUNCT
ap-1860	52	37	the	the	DET
ap-1860	52	38	i	i	PROPN
ap-1860	52	39	-	-	PUNCT
ap-1860	52	40	itinerary	itinerary	NOUN
ap-1860	52	41	r(ρ	r(ρ	NOUN
ap-1860	52	42	)	)	PUNCT
ap-1860	52	43	of	of	ADP
ap-1860	52	44	ρ	ρ	PROPN
ap-1860	52	45	is	be	AUX
ap-1860	52	46	a	a	DET
ap-1860	52	47	prefix	prefix	NOUN
ap-1860	52	48	of	of	ADP
ap-1860	52	49	the	the	DET
ap-1860	52	50	infinite	infinite	ADJ
ap-1860	52	51	word	word	NOUN
ap-1860	52	52	uρ	uρ	ADP
ap-1860	52	53	of	of	ADP
ap-1860	52	54	length	length	NOUN
ap-1860	52	55	r(ρ	r(ρ	PROPN
ap-1860	52	56	)	)	PUNCT
ap-1860	52	57	.	.	PUNCT
ap-1860	53	1	in	in	ADP
ap-1860	53	2	our	our	PRON
ap-1860	53	3	considerations	consideration	NOUN
ap-1860	53	4	,	,	PUNCT
ap-1860	53	5	the	the	DET
ap-1860	53	6	interval	interval	NOUN
ap-1860	53	7	i	i	PRON
ap-1860	53	8	is	be	AUX
ap-1860	53	9	fixed	fix	VERB
ap-1860	53	10	.	.	PUNCT
ap-1860	54	1	thus	thus	ADV
ap-1860	54	2	,	,	PUNCT
ap-1860	54	3	for	for	ADP
ap-1860	54	4	simplicity	simplicity	NOUN
ap-1860	54	5	444	444	NUM
ap-1860	54	6	http://dx.doi.org/10.14311/ap.2013.53.0444	http://dx.doi.org/10.14311/ap.2013.53.0444	PRON
ap-1860	54	7	http://ojs.cvut.cz/ojs/index.php/ap	http://ojs.cvut.cz/ojs/index.php/ap	PROPN
ap-1860	54	8	vol	vol	NOUN
ap-1860	54	9	.	.	PUNCT
ap-1860	55	1	53	53	NUM
ap-1860	55	2	no	no	NOUN
ap-1860	55	3	.	.	PUNCT
ap-1860	56	1	5/2013	5/2013	NUM
ap-1860	56	2	itineraries	itinerary	NOUN
ap-1860	56	3	induced	induce	VERB
ap-1860	56	4	by	by	ADP
ap-1860	56	5	exchange	exchange	NOUN
ap-1860	56	6	of	of	ADP
ap-1860	56	7	two	two	NUM
ap-1860	56	8	intervals	interval	NOUN
ap-1860	56	9	of	of	ADP
ap-1860	56	10	notation	notation	NOUN
ap-1860	56	11	,	,	PUNCT
ap-1860	56	12	we	we	PRON
ap-1860	56	13	avoid	avoid	VERB
ap-1860	56	14	marking	mark	VERB
ap-1860	56	15	the	the	DET
ap-1860	56	16	dependence	dependence	NOUN
ap-1860	56	17	on	on	ADP
ap-1860	56	18	i	i	PRON
ap-1860	56	19	of	of	ADP
ap-1860	56	20	the	the	DET
ap-1860	56	21	first	first	ADJ
ap-1860	56	22	return	return	NOUN
ap-1860	56	23	time	time	NOUN
ap-1860	56	24	and	and	CCONJ
ap-1860	56	25	return	return	VERB
ap-1860	56	26	itinerary	itinerary	NOUN
ap-1860	56	27	,	,	PUNCT
ap-1860	56	28	i.e.	i.e.	X
ap-1860	56	29	,	,	PUNCT
ap-1860	56	30	we	we	PRON
ap-1860	56	31	write	write	VERB
ap-1860	56	32	r(x	r(x	PROPN
ap-1860	56	33	)	)	PUNCT
ap-1860	56	34	,	,	PUNCT
ap-1860	56	35	r(x	r(x	PROPN
ap-1860	56	36	)	)	PUNCT
ap-1860	56	37	instead	instead	ADV
ap-1860	56	38	of	of	ADP
ap-1860	56	39	ri(x	ri(x	NUM
ap-1860	56	40	)	)	PUNCT
ap-1860	56	41	,	,	PUNCT
ap-1860	56	42	ri(x	ri(x	NUM
ap-1860	56	43	)	)	PUNCT
ap-1860	56	44	,	,	PUNCT
ap-1860	56	45	respectively	respectively	ADV
ap-1860	56	46	.	.	PUNCT
ap-1860	57	1	the	the	DET
ap-1860	57	2	position	position	NOUN
ap-1860	57	3	of	of	ADP
ap-1860	57	4	the	the	DET
ap-1860	57	5	point	point	NOUN
ap-1860	57	6	ρ	ρ	X
ap-1860	57	7	∈	∈	PROPN
ap-1860	58	1	i	i	PRON
ap-1860	58	2	after	after	ADP
ap-1860	58	3	its	its	PRON
ap-1860	58	4	return	return	NOUN
ap-1860	58	5	the	the	DET
ap-1860	58	6	interval	interval	NOUN
ap-1860	58	7	i	i	PRON
ap-1860	58	8	defines	define	VERB
ap-1860	58	9	a	a	DET
ap-1860	58	10	new	new	ADJ
ap-1860	58	11	transformation	transformation	NOUN
ap-1860	58	12	ti	ti	NOUN
ap-1860	58	13	:	:	PUNCT
ap-1860	58	14	i	i	PROPN
ap-1860	58	15	→	→	PUNCT
ap-1860	58	16	i	i	PRON
ap-1860	58	17	by	by	ADP
ap-1860	58	18	ti(ρ	ti(ρ	NOUN
ap-1860	58	19	)	)	PUNCT
ap-1860	58	20	=	=	SYM
ap-1860	58	21	t	t	NOUN
ap-1860	58	22	r(ρ)(ρ	r(ρ)(ρ	NUM
ap-1860	58	23	)	)	PUNCT
ap-1860	58	24	,	,	PUNCT
ap-1860	58	25	(	(	PUNCT
ap-1860	58	26	3	3	X
ap-1860	58	27	)	)	PUNCT
ap-1860	58	28	which	which	PRON
ap-1860	58	29	is	be	AUX
ap-1860	58	30	usually	usually	ADV
ap-1860	58	31	called	call	VERB
ap-1860	58	32	the	the	DET
ap-1860	58	33	first	first	ADJ
ap-1860	58	34	return	return	NOUN
ap-1860	58	35	map	map	NOUN
ap-1860	58	36	or	or	CCONJ
ap-1860	58	37	induced	induced	ADJ
ap-1860	58	38	map	map	NOUN
ap-1860	58	39	.	.	PUNCT
ap-1860	59	1	the	the	DET
ap-1860	59	2	i	i	PROPN
ap-1860	59	3	-	-	PUNCT
ap-1860	59	4	itineraries	itinerary	NOUN
ap-1860	59	5	for	for	ADP
ap-1860	59	6	a	a	DET
ap-1860	59	7	special	special	ADJ
ap-1860	59	8	type	type	NOUN
ap-1860	59	9	of	of	ADP
ap-1860	59	10	interval	interval	NOUN
ap-1860	59	11	i	i	PRON
ap-1860	59	12	were	be	AUX
ap-1860	59	13	studied	study	VERB
ap-1860	59	14	in	in	ADP
ap-1860	59	15	diverse	diverse	ADJ
ap-1860	59	16	contexts	contexts	NOUN
ap-1860	59	17	:	:	PUNCT
ap-1860	59	18	•	•	ADP
ap-1860	59	19	if	if	SCONJ
ap-1860	59	20	the	the	DET
ap-1860	59	21	boundary	boundary	ADJ
ap-1860	59	22	points	point	NOUN
ap-1860	59	23	of	of	ADP
ap-1860	59	24	the	the	DET
ap-1860	59	25	interval	interval	NOUN
ap-1860	59	26	i	i	PRON
ap-1860	59	27	are	be	AUX
ap-1860	59	28	neighbouring	neighbouring	ADJ
ap-1860	59	29	elements	element	NOUN
ap-1860	59	30	of	of	ADP
ap-1860	59	31	the	the	DET
ap-1860	59	32	set	set	NOUN
ap-1860	59	33	{	{	PUNCT
ap-1860	59	34	α	α	NOUN
ap-1860	59	35	,	,	PUNCT
ap-1860	59	36	t−1(α	t−1(α	PROPN
ap-1860	59	37	)	)	PUNCT
ap-1860	59	38	,	,	PUNCT
ap-1860	59	39	.	.	PUNCT
ap-1860	59	40	.	.	PUNCT
ap-1860	60	1	.	.	PUNCT
ap-1860	61	1	,	,	PUNCT
ap-1860	61	2	t−n(α	t−n(α	PROPN
ap-1860	61	3	)	)	PUNCT
ap-1860	61	4	}	}	PUNCT
ap-1860	61	5	for	for	ADP
ap-1860	61	6	some	some	DET
ap-1860	61	7	n	n	PRON
ap-1860	61	8	∈	∈	PROPN
ap-1860	61	9	n	n	CCONJ
ap-1860	61	10	,	,	PUNCT
ap-1860	61	11	then	then	ADV
ap-1860	61	12	the	the	DET
ap-1860	61	13	set	set	NOUN
ap-1860	61	14	of	of	ADP
ap-1860	61	15	iitineraries	iitinerarie	NOUN
ap-1860	61	16	r(ρ	r(ρ	PROPN
ap-1860	61	17	)	)	PUNCT
ap-1860	61	18	for	for	ADP
ap-1860	61	19	ρ	ρ	PROPN
ap-1860	61	20	∈	∈	PROPN
ap-1860	61	21	i	i	PRON
ap-1860	61	22	consists	consist	VERB
ap-1860	61	23	of	of	ADP
ap-1860	61	24	only	only	ADV
ap-1860	61	25	two	two	NUM
ap-1860	61	26	words	word	NOUN
ap-1860	61	27	.	.	PUNCT
ap-1860	62	1	this	this	PRON
ap-1860	62	2	reformulates	reformulate	VERB
ap-1860	62	3	the	the	DET
ap-1860	62	4	result	result	NOUN
ap-1860	62	5	of	of	ADP
ap-1860	62	6	vuillon	vuillon	NOUN
ap-1860	63	1	[	[	X
ap-1860	63	2	4	4	X
ap-1860	63	3	]	]	PUNCT
ap-1860	63	4	about	about	ADP
ap-1860	63	5	the	the	DET
ap-1860	63	6	existence	existence	NOUN
ap-1860	63	7	of	of	ADP
ap-1860	63	8	exactly	exactly	ADV
ap-1860	63	9	two	two	NUM
ap-1860	63	10	return	return	NOUN
ap-1860	63	11	words	word	NOUN
ap-1860	63	12	to	to	ADP
ap-1860	63	13	a	a	DET
ap-1860	63	14	fixed	fix	VERB
ap-1860	63	15	factor	factor	NOUN
ap-1860	63	16	of	of	ADP
ap-1860	63	17	a	a	DET
ap-1860	63	18	sturmian	sturmian	ADJ
ap-1860	63	19	word	word	NOUN
ap-1860	63	20	.	.	PUNCT
ap-1860	64	1	•	•	INTJ
ap-1860	64	2	if	if	SCONJ
ap-1860	64	3	the	the	DET
ap-1860	64	4	sturmian	sturmian	ADJ
ap-1860	64	5	word	word	NOUN
ap-1860	64	6	uρ	uρ	INTJ
ap-1860	64	7	is	be	AUX
ap-1860	64	8	invariant	invariant	ADJ
ap-1860	64	9	under	under	ADP
ap-1860	64	10	a	a	DET
ap-1860	64	11	substitution	substitution	NOUN
ap-1860	64	12	0	0	NUM
ap-1860	64	13	7→	7→	NUM
ap-1860	64	14	ϕ(0	ϕ(0	PROPN
ap-1860	64	15	)	)	PUNCT
ap-1860	64	16	,	,	PUNCT
ap-1860	65	1	1	1	NUM
ap-1860	65	2	7→	7→	NUM
ap-1860	65	3	ϕ(1	ϕ(1	PROPN
ap-1860	65	4	)	)	PUNCT
ap-1860	65	5	,	,	PUNCT
ap-1860	65	6	then	then	ADV
ap-1860	65	7	there	there	PRON
ap-1860	65	8	exists	exist	VERB
ap-1860	65	9	an	an	DET
ap-1860	65	10	interval	interval	NOUN
ap-1860	66	1	i	i	PRON
ap-1860	66	2	⊂	⊂	PROPN
ap-1860	67	1	[	[	X
ap-1860	67	2	0	0	NUM
ap-1860	67	3	,	,	PUNCT
ap-1860	67	4	1	1	NUM
ap-1860	67	5	)	)	PUNCT
ap-1860	68	1	,	,	PUNCT
ap-1860	68	2	ρ	ρ	PROPN
ap-1860	68	3	∈	∈	PROPN
ap-1860	69	1	i	i	PRON
ap-1860	69	2	,	,	PUNCT
ap-1860	69	3	such	such	ADJ
ap-1860	69	4	that	that	SCONJ
ap-1860	69	5	the	the	DET
ap-1860	69	6	induced	induced	ADJ
ap-1860	69	7	map	map	NOUN
ap-1860	69	8	ti	ti	NOUN
ap-1860	69	9	is	be	AUX
ap-1860	69	10	homothetic	homothetic	ADJ
ap-1860	69	11	to	to	ADP
ap-1860	69	12	t	t	PROPN
ap-1860	69	13	,	,	PUNCT
ap-1860	69	14	and	and	CCONJ
ap-1860	69	15	the	the	DET
ap-1860	69	16	finite	finite	ADJ
ap-1860	69	17	words	word	NOUN
ap-1860	69	18	ϕ(0	ϕ(0	PROPN
ap-1860	69	19	)	)	PUNCT
ap-1860	69	20	,	,	PUNCT
ap-1860	69	21	ϕ(1	ϕ(1	PROPN
ap-1860	69	22	)	)	PUNCT
ap-1860	69	23	are	be	AUX
ap-1860	69	24	the	the	DET
ap-1860	69	25	i	i	NOUN
ap-1860	69	26	-	-	PUNCT
ap-1860	69	27	itineraries	itinerary	NOUN
ap-1860	69	28	.	.	PUNCT
ap-1860	70	1	invariance	invariance	NOUN
ap-1860	70	2	of	of	ADP
ap-1860	70	3	sturmian	sturmian	ADJ
ap-1860	70	4	words	word	NOUN
ap-1860	70	5	under	under	ADP
ap-1860	70	6	substitutions	substitution	NOUN
ap-1860	70	7	was	be	AUX
ap-1860	70	8	studied	study	VERB
ap-1860	70	9	by	by	ADP
ap-1860	70	10	yasutomi	yasutomi	NOUN
ap-1860	71	1	[	[	X
ap-1860	71	2	5	5	NUM
ap-1860	71	3	]	]	PUNCT
ap-1860	71	4	.	.	PUNCT
ap-1860	72	1	•	•	NOUN
ap-1860	72	2	an	an	DET
ap-1860	72	3	abelian	abelian	ADJ
ap-1860	72	4	return	return	NOUN
ap-1860	72	5	word	word	NOUN
ap-1860	72	6	to	to	ADP
ap-1860	72	7	a	a	DET
ap-1860	72	8	factor	factor	NOUN
ap-1860	72	9	of	of	ADP
ap-1860	72	10	a	a	DET
ap-1860	72	11	sturmian	sturmian	NOUN
ap-1860	72	12	word	word	NOUN
ap-1860	72	13	is	be	AUX
ap-1860	72	14	an	an	DET
ap-1860	72	15	i	i	NOUN
ap-1860	72	16	-	-	PUNCT
ap-1860	72	17	itinerary	itinerary	NOUN
ap-1860	72	18	for	for	ADP
ap-1860	72	19	i	i	PRON
ap-1860	72	20	=	=	PUNCT
ap-1860	73	1	[	[	X
ap-1860	73	2	0	0	NUM
ap-1860	73	3	,	,	PUNCT
ap-1860	73	4	β	β	NOUN
ap-1860	73	5	)	)	PUNCT
ap-1860	73	6	or	or	CCONJ
ap-1860	73	7	i	i	PRON
ap-1860	73	8	=	=	PUNCT
ap-1860	74	1	[	[	X
ap-1860	74	2	β	β	X
ap-1860	74	3	,	,	PUNCT
ap-1860	74	4	1	1	NUM
ap-1860	74	5	)	)	PUNCT
ap-1860	74	6	for	for	ADP
ap-1860	74	7	some	some	DET
ap-1860	74	8	β	β	X
ap-1860	74	9	∈	∈	PROPN
ap-1860	75	1	[	[	X
ap-1860	75	2	0	0	NUM
ap-1860	75	3	,	,	PUNCT
ap-1860	75	4	1	1	NUM
ap-1860	75	5	)	)	PUNCT
ap-1860	75	6	,	,	PUNCT
ap-1860	75	7	see	see	VERB
ap-1860	75	8	[	[	X
ap-1860	75	9	6	6	NUM
ap-1860	75	10	]	]	PUNCT
ap-1860	75	11	.	.	PUNCT
ap-1860	76	1	as	as	SCONJ
ap-1860	76	2	follows	follow	VERB
ap-1860	76	3	from	from	ADP
ap-1860	76	4	the	the	DET
ap-1860	76	5	result	result	NOUN
ap-1860	76	6	of	of	ADP
ap-1860	76	7	[	[	X
ap-1860	76	8	7	7	NUM
ap-1860	76	9	]	]	PUNCT
ap-1860	76	10	,	,	PUNCT
ap-1860	76	11	the	the	DET
ap-1860	76	12	intervals	interval	NOUN
ap-1860	76	13	of	of	ADP
ap-1860	76	14	the	the	DET
ap-1860	76	15	mentioned	mention	VERB
ap-1860	76	16	form	form	NOUN
ap-1860	76	17	have	have	VERB
ap-1860	76	18	at	at	ADP
ap-1860	76	19	most	most	ADV
ap-1860	76	20	three	three	NUM
ap-1860	76	21	itineraries	itinerary	NOUN
ap-1860	76	22	r1	r1	NOUN
ap-1860	76	23	,	,	PUNCT
ap-1860	76	24	r2	r2	PROPN
ap-1860	76	25	,	,	PUNCT
ap-1860	76	26	r3	r3	PROPN
ap-1860	76	27	and	and	CCONJ
ap-1860	76	28	for	for	ADP
ap-1860	76	29	their	their	PRON
ap-1860	76	30	length	length	NOUN
ap-1860	76	31	one	one	NUM
ap-1860	76	32	has	have	VERB
ap-1860	76	33	|r3|	|r3|	NOUN
ap-1860	76	34	=	=	SYM
ap-1860	76	35	|r1|+	|r1|+	NOUN
ap-1860	76	36	|r2|	|r2|	NOUN
ap-1860	76	37	.	.	PUNCT
ap-1860	77	1	in	in	ADP
ap-1860	77	2	[	[	X
ap-1860	77	3	8	8	NUM
ap-1860	77	4	]	]	PUNCT
ap-1860	77	5	,	,	PUNCT
ap-1860	77	6	we	we	PRON
ap-1860	77	7	have	have	AUX
ap-1860	77	8	shown	show	VERB
ap-1860	77	9	that	that	SCONJ
ap-1860	77	10	a	a	DET
ap-1860	77	11	stronger	strong	ADJ
ap-1860	77	12	statement	statement	NOUN
ap-1860	77	13	holds	hold	VERB
ap-1860	77	14	,	,	PUNCT
ap-1860	77	15	namely	namely	ADV
ap-1860	77	16	that	that	SCONJ
ap-1860	77	17	the	the	DET
ap-1860	77	18	word	word	NOUN
ap-1860	77	19	r3	r3	PROPN
ap-1860	77	20	is	be	AUX
ap-1860	77	21	a	a	DET
ap-1860	77	22	concatenation	concatenation	NOUN
ap-1860	77	23	of	of	ADP
ap-1860	77	24	words	word	NOUN
ap-1860	77	25	r1	r1	PROPN
ap-1860	77	26	and	and	CCONJ
ap-1860	77	27	r2	r2	PROPN
ap-1860	77	28	.	.	PUNCT
ap-1860	78	1	the	the	DET
ap-1860	78	2	aim	aim	NOUN
ap-1860	78	3	of	of	ADP
ap-1860	78	4	this	this	DET
ap-1860	78	5	paper	paper	NOUN
ap-1860	78	6	is	be	AUX
ap-1860	78	7	to	to	PART
ap-1860	78	8	describe	describe	VERB
ap-1860	78	9	the	the	DET
ap-1860	78	10	structure	structure	NOUN
ap-1860	78	11	of	of	ADP
ap-1860	78	12	the	the	DET
ap-1860	78	13	set	set	NOUN
ap-1860	78	14	of	of	ADP
ap-1860	78	15	i	i	PROPN
ap-1860	78	16	-	-	PUNCT
ap-1860	78	17	itineraries	itinerary	NOUN
ap-1860	78	18	for	for	ADP
ap-1860	78	19	a	a	DET
ap-1860	78	20	general	general	ADJ
ap-1860	78	21	position	position	NOUN
ap-1860	78	22	and	and	CCONJ
ap-1860	78	23	length	length	NOUN
ap-1860	78	24	of	of	ADP
ap-1860	78	25	the	the	DET
ap-1860	78	26	subinterval	subinterval	NOUN
ap-1860	79	1	i	i	PRON
ap-1860	79	2	⊂	⊂	PROPN
ap-1860	80	1	[	[	X
ap-1860	80	2	0	0	NUM
ap-1860	80	3	,	,	PUNCT
ap-1860	80	4	1	1	NUM
ap-1860	80	5	)	)	PUNCT
ap-1860	80	6	.	.	PUNCT
ap-1860	81	1	the	the	DET
ap-1860	81	2	set	set	NOUN
ap-1860	81	3	of	of	ADP
ap-1860	81	4	all	all	DET
ap-1860	81	5	i	i	NOUN
ap-1860	81	6	-	-	PUNCT
ap-1860	81	7	itineraries	itinerary	NOUN
ap-1860	81	8	r(x	r(x	PROPN
ap-1860	81	9	)	)	PUNCT
ap-1860	81	10	for	for	ADP
ap-1860	81	11	x	x	X
ap-1860	81	12	∈	∈	PROPN
ap-1860	81	13	i	i	PRON
ap-1860	81	14	is	be	AUX
ap-1860	81	15	denoted	denote	VERB
ap-1860	81	16	by	by	ADP
ap-1860	81	17	iti	iti	PROPN
ap-1860	81	18	.	.	PUNCT
ap-1860	82	1	for	for	ADP
ap-1860	82	2	the	the	DET
ap-1860	82	3	description	description	NOUN
ap-1860	82	4	,	,	PUNCT
ap-1860	82	5	we	we	PRON
ap-1860	82	6	will	will	AUX
ap-1860	82	7	use	use	VERB
ap-1860	82	8	the	the	DET
ap-1860	82	9	notion	notion	NOUN
ap-1860	82	10	of	of	ADP
ap-1860	82	11	word	word	NOUN
ap-1860	82	12	amicability	amicability	NOUN
ap-1860	82	13	.	.	PUNCT
ap-1860	83	1	we	we	PRON
ap-1860	83	2	say	say	VERB
ap-1860	83	3	that	that	SCONJ
ap-1860	83	4	two	two	NUM
ap-1860	83	5	finite	finite	ADJ
ap-1860	83	6	words	word	NOUN
ap-1860	83	7	w	w	PROPN
ap-1860	83	8	and	and	CCONJ
ap-1860	83	9	v	v	NOUN
ap-1860	83	10	over	over	ADP
ap-1860	83	11	the	the	DET
ap-1860	83	12	alphabet	alphabet	NOUN
ap-1860	83	13	{	{	PUNCT
ap-1860	83	14	0	0	NUM
ap-1860	83	15	,	,	PUNCT
ap-1860	83	16	1	1	NUM
ap-1860	83	17	}	}	PUNCT
ap-1860	83	18	are	be	AUX
ap-1860	83	19	amicable	amicable	ADJ
ap-1860	83	20	,	,	PUNCT
ap-1860	83	21	if	if	SCONJ
ap-1860	83	22	there	there	PRON
ap-1860	83	23	exist	exist	VERB
ap-1860	83	24	words	word	NOUN
ap-1860	83	25	p	p	NOUN
ap-1860	83	26	,	,	PUNCT
ap-1860	83	27	q	q	PROPN
ap-1860	83	28	∈	∈	PROPN
ap-1860	83	29	{	{	PUNCT
ap-1860	83	30	0	0	NUM
ap-1860	83	31	,	,	PUNCT
ap-1860	83	32	1}∗	1}∗	VERB
ap-1860	83	33	such	such	ADJ
ap-1860	83	34	that	that	DET
ap-1860	83	35	w	w	NOUN
ap-1860	83	36	=	=	PUNCT
ap-1860	83	37	p01q	p01q	PROPN
ap-1860	83	38	and	and	CCONJ
ap-1860	83	39	v	v	NOUN
ap-1860	83	40	=	=	SYM
ap-1860	83	41	p10q	p10q	NOUN
ap-1860	83	42	or	or	CCONJ
ap-1860	83	43	w	w	NOUN
ap-1860	83	44	=	=	PUNCT
ap-1860	83	45	p10q	p10q	NOUN
ap-1860	83	46	and	and	CCONJ
ap-1860	83	47	v	v	NOUN
ap-1860	83	48	=	=	NOUN
ap-1860	83	49	p01q	p01q	NOUN
ap-1860	83	50	.	.	PUNCT
ap-1860	84	1	in	in	ADP
ap-1860	84	2	other	other	ADJ
ap-1860	84	3	words	word	NOUN
ap-1860	84	4	,	,	PUNCT
ap-1860	84	5	v	v	NOUN
ap-1860	84	6	is	be	AUX
ap-1860	84	7	obtained	obtain	VERB
ap-1860	84	8	from	from	ADP
ap-1860	84	9	w	w	NOUN
ap-1860	84	10	by	by	ADP
ap-1860	84	11	interchanging	interchange	VERB
ap-1860	84	12	the	the	DET
ap-1860	84	13	order	order	NOUN
ap-1860	84	14	of	of	ADP
ap-1860	84	15	letters	letter	NOUN
ap-1860	84	16	0	0	PUNCT
ap-1860	84	17	and	and	CCONJ
ap-1860	84	18	1	1	NUM
ap-1860	84	19	at	at	ADP
ap-1860	84	20	two	two	NUM
ap-1860	84	21	neighbouring	neighbouring	ADJ
ap-1860	84	22	positions	position	NOUN
ap-1860	84	23	i−	i−	ADP
ap-1860	84	24	1	1	NUM
ap-1860	84	25	,	,	PUNCT
ap-1860	84	26	i.	i.	NOUN
ap-1860	84	27	it	it	PRON
ap-1860	84	28	follows	follow	VERB
ap-1860	84	29	from	from	ADP
ap-1860	84	30	[	[	X
ap-1860	84	31	9	9	NUM
ap-1860	84	32	]	]	PUNCT
ap-1860	84	33	that	that	SCONJ
ap-1860	84	34	for	for	ADP
ap-1860	84	35	every	every	DET
ap-1860	84	36	interval	interval	NOUN
ap-1860	84	37	i	i	PRON
ap-1860	84	38	there	there	PRON
ap-1860	84	39	exist	exist	VERB
ap-1860	84	40	at	at	ADP
ap-1860	84	41	most	most	ADJ
ap-1860	84	42	four	four	NUM
ap-1860	84	43	i	i	NOUN
ap-1860	84	44	-	-	PUNCT
ap-1860	84	45	itineraries	itinerary	NOUN
ap-1860	84	46	,	,	PUNCT
ap-1860	84	47	i.e.	i.e.	X
ap-1860	84	48	,	,	PUNCT
ap-1860	84	49	#	#	SYM
ap-1860	84	50	iti	iti	NOUN
ap-1860	84	51	≤	≤	NUM
ap-1860	84	52	4	4	NUM
ap-1860	84	53	.	.	PUNCT
ap-1860	85	1	we	we	PRON
ap-1860	85	2	will	will	AUX
ap-1860	85	3	show	show	VERB
ap-1860	85	4	the	the	DET
ap-1860	85	5	following	follow	VERB
ap-1860	85	6	theorem	theorem	VERB
ap-1860	85	7	.	.	PUNCT
ap-1860	85	8	theorem	theorem	VERB
ap-1860	85	9	1.1	1.1	NUM
ap-1860	85	10	.	.	PUNCT
ap-1860	86	1	let	let	VERB
ap-1860	86	2	t	t	PROPN
ap-1860	86	3	be	be	AUX
ap-1860	86	4	the	the	DET
ap-1860	86	5	transformation	transformation	NOUN
ap-1860	86	6	(	(	PUNCT
ap-1860	86	7	1	1	NUM
ap-1860	86	8	)	)	PUNCT
ap-1860	86	9	for	for	ADP
ap-1860	86	10	some	some	DET
ap-1860	86	11	irrational	irrational	ADJ
ap-1860	86	12	α	α	NOUN
ap-1860	86	13	∈	∈	PROPN
ap-1860	86	14	(	(	PUNCT
ap-1860	86	15	0	0	NUM
ap-1860	86	16	,	,	PUNCT
ap-1860	86	17	1	1	NUM
ap-1860	86	18	)	)	PUNCT
ap-1860	86	19	and	and	CCONJ
ap-1860	86	20	let	let	VERB
ap-1860	86	21	i	i	PRON
ap-1860	86	22	⊂	⊂	PROPN
ap-1860	87	1	[	[	X
ap-1860	87	2	0	0	NUM
ap-1860	87	3	,	,	PUNCT
ap-1860	87	4	1	1	NUM
ap-1860	87	5	)	)	PUNCT
ap-1860	87	6	be	be	AUX
ap-1860	87	7	an	an	DET
ap-1860	87	8	interval	interval	NOUN
ap-1860	87	9	.	.	PUNCT
ap-1860	88	1	then	then	ADV
ap-1860	88	2	there	there	PRON
ap-1860	88	3	exist	exist	VERB
ap-1860	88	4	words	word	NOUN
ap-1860	88	5	r1	r1	PROPN
ap-1860	88	6	,	,	PUNCT
ap-1860	88	7	r2	r2	PROPN
ap-1860	88	8	∈	∈	PROPN
ap-1860	88	9	{	{	PUNCT
ap-1860	88	10	0	0	NUM
ap-1860	88	11	,	,	PUNCT
ap-1860	88	12	1}∗	1}∗	VERB
ap-1860	88	13	such	such	ADJ
ap-1860	88	14	that	that	PRON
ap-1860	88	15	for	for	ADP
ap-1860	88	16	the	the	DET
ap-1860	88	17	set	set	ADJ
ap-1860	88	18	iti	iti	PROPN
ap-1860	88	19	of	of	ADP
ap-1860	88	20	all	all	PRON
ap-1860	88	21	i	i	NOUN
ap-1860	88	22	-	-	PUNCT
ap-1860	88	23	itineraries	itinerary	NOUN
ap-1860	88	24	one	one	NOUN
ap-1860	88	25	has	have	VERB
ap-1860	88	26	iti	iti	PROPN
ap-1860	88	27	⊂	⊂	PROPN
ap-1860	88	28	{	{	PUNCT
ap-1860	88	29	r1	r1	PROPN
ap-1860	88	30	,	,	PUNCT
ap-1860	88	31	r2	r2	PROPN
ap-1860	88	32	,	,	PUNCT
ap-1860	88	33	r1r2	r1r2	NOUN
ap-1860	88	34	,	,	PUNCT
ap-1860	88	35	q	q	NOUN
ap-1860	88	36	}	}	PUNCT
ap-1860	88	37	,	,	PUNCT
ap-1860	88	38	where	where	SCONJ
ap-1860	88	39	q	q	NOUN
ap-1860	88	40	is	be	AUX
ap-1860	88	41	amicable	amicable	ADJ
ap-1860	88	42	with	with	ADP
ap-1860	88	43	r1	r1	PROPN
ap-1860	88	44	,	,	PUNCT
ap-1860	88	45	r2	r2	PROPN
ap-1860	88	46	or	or	CCONJ
ap-1860	88	47	r1r2	r1r2	NOUN
ap-1860	88	48	.	.	PROPN
ap-1860	88	49	from	from	ADP
ap-1860	88	50	the	the	DET
ap-1860	88	51	proof	proof	NOUN
ap-1860	88	52	of	of	ADP
ap-1860	88	53	theorem	theorem	ADJ
ap-1860	88	54	1.1	1.1	NUM
ap-1860	88	55	(	(	PUNCT
ap-1860	88	56	at	at	ADP
ap-1860	88	57	the	the	DET
ap-1860	88	58	end	end	NOUN
ap-1860	88	59	of	of	ADP
ap-1860	88	60	section	section	NOUN
ap-1860	88	61	2	2	NUM
ap-1860	88	62	)	)	PUNCT
ap-1860	88	63	one	one	PRON
ap-1860	88	64	can	can	AUX
ap-1860	88	65	see	see	VERB
ap-1860	88	66	that	that	SCONJ
ap-1860	88	67	in	in	ADP
ap-1860	88	68	the	the	DET
ap-1860	88	69	generic	generic	ADJ
ap-1860	88	70	case	case	NOUN
ap-1860	88	71	,	,	PUNCT
ap-1860	88	72	iti	iti	PROPN
ap-1860	88	73	=	=	PUNCT
ap-1860	88	74	{	{	PUNCT
ap-1860	88	75	r1	r1	PROPN
ap-1860	88	76	,	,	PUNCT
ap-1860	88	77	r2	r2	PROPN
ap-1860	88	78	,	,	PUNCT
ap-1860	88	79	r1r2	r1r2	NOUN
ap-1860	88	80	,	,	PUNCT
ap-1860	88	81	q	q	NOUN
ap-1860	88	82	}	}	PUNCT
ap-1860	88	83	.	.	PUNCT
ap-1860	89	1	in	in	ADP
ap-1860	89	2	section	section	NOUN
ap-1860	89	3	3	3	NUM
ap-1860	89	4	we	we	PRON
ap-1860	89	5	discuss	discuss	VERB
ap-1860	89	6	the	the	DET
ap-1860	89	7	possibilities	possibility	NOUN
ap-1860	89	8	for	for	ADP
ap-1860	89	9	q	q	NOUN
ap-1860	89	10	if	if	SCONJ
ap-1860	89	11	#	#	SYM
ap-1860	89	12	iti	iti	NOUN
ap-1860	89	13	=	=	SYM
ap-1860	89	14	4	4	NUM
ap-1860	89	15	and	and	CCONJ
ap-1860	89	16	determine	determine	VERB
ap-1860	89	17	the	the	DET
ap-1860	89	18	cases	case	NOUN
ap-1860	89	19	for	for	ADP
ap-1860	89	20	which	which	PRON
ap-1860	89	21	the	the	DET
ap-1860	89	22	set	set	NOUN
ap-1860	89	23	iti	iti	PROPN
ap-1860	89	24	has	have	VERB
ap-1860	89	25	less	less	ADJ
ap-1860	89	26	than	than	ADP
ap-1860	89	27	4	4	NUM
ap-1860	89	28	elements	element	NOUN
ap-1860	89	29	.	.	PUNCT
ap-1860	90	1	2	2	X
ap-1860	90	2	.	.	X
ap-1860	90	3	interval	interval	NOUN
ap-1860	90	4	exchange	exchange	NOUN
ap-1860	90	5	transformations	transformation	NOUN
ap-1860	90	6	first	first	ADV
ap-1860	90	7	,	,	PUNCT
ap-1860	90	8	let	let	VERB
ap-1860	90	9	us	we	PRON
ap-1860	90	10	recall	recall	VERB
ap-1860	90	11	the	the	DET
ap-1860	90	12	definition	definition	NOUN
ap-1860	90	13	and	and	CCONJ
ap-1860	90	14	certain	certain	ADJ
ap-1860	90	15	properties	property	NOUN
ap-1860	90	16	of	of	ADP
ap-1860	90	17	k	k	ADJ
ap-1860	90	18	-	-	PUNCT
ap-1860	90	19	interval	interval	NOUN
ap-1860	90	20	exchange	exchange	NOUN
ap-1860	90	21	maps	map	NOUN
ap-1860	90	22	,	,	PUNCT
ap-1860	90	23	which	which	PRON
ap-1860	90	24	we	we	PRON
ap-1860	90	25	use	use	VERB
ap-1860	90	26	for	for	ADP
ap-1860	90	27	k	k	PROPN
ap-1860	90	28	=	=	SYM
ap-1860	90	29	2	2	NUM
ap-1860	90	30	and	and	CCONJ
ap-1860	90	31	3	3	NUM
ap-1860	90	32	.	.	X
ap-1860	90	33	definition	definition	NOUN
ap-1860	90	34	2.1	2.1	NUM
ap-1860	90	35	.	.	PUNCT
ap-1860	91	1	let	let	VERB
ap-1860	91	2	j0∪j1∪	j0∪j1∪	PRON
ap-1860	91	3	·	·	PUNCT
ap-1860	91	4	·	·	PUNCT
ap-1860	91	5	·	·	PUNCT
ap-1860	91	6	∪jk−1	∪jk−1	X
ap-1860	91	7	be	be	AUX
ap-1860	91	8	a	a	DET
ap-1860	91	9	partition	partition	NOUN
ap-1860	91	10	of	of	ADP
ap-1860	91	11	the	the	DET
ap-1860	91	12	interval	interval	NOUN
ap-1860	91	13	j	j	PROPN
ap-1860	91	14	,	,	PUNCT
ap-1860	91	15	where	where	SCONJ
ap-1860	91	16	ji	ji	PROPN
ap-1860	91	17	are	be	AUX
ap-1860	91	18	intervals	interval	NOUN
ap-1860	91	19	closed	close	VERB
ap-1860	91	20	from	from	ADP
ap-1860	91	21	the	the	DET
ap-1860	91	22	left	left	NOUN
ap-1860	91	23	and	and	CCONJ
ap-1860	91	24	open	open	ADJ
ap-1860	91	25	from	from	ADP
ap-1860	91	26	the	the	DET
ap-1860	91	27	right	right	NOUN
ap-1860	91	28	for	for	ADP
ap-1860	91	29	every	every	DET
ap-1860	91	30	i	i	NOUN
ap-1860	91	31	=	=	NOUN
ap-1860	91	32	0	0	NUM
ap-1860	91	33	,	,	PUNCT
ap-1860	91	34	.	.	PUNCT
ap-1860	91	35	.	.	PUNCT
ap-1860	92	1	.	.	PUNCT
ap-1860	93	1	,	,	PUNCT
ap-1860	93	2	k−	k−	PROPN
ap-1860	93	3	1	1	NUM
ap-1860	93	4	.	.	PUNCT
ap-1860	94	1	the	the	DET
ap-1860	94	2	transformation	transformation	NOUN
ap-1860	94	3	t	t	PROPN
ap-1860	94	4	:	:	PUNCT
ap-1860	94	5	j	j	PROPN
ap-1860	94	6	→	→	SYM
ap-1860	94	7	j	j	PROPN
ap-1860	94	8	is	be	AUX
ap-1860	94	9	called	call	VERB
ap-1860	94	10	a	a	DET
ap-1860	94	11	k	k	ADJ
ap-1860	94	12	-	-	PUNCT
ap-1860	94	13	interval	interval	NOUN
ap-1860	94	14	exchange	exchange	NOUN
ap-1860	94	15	if	if	SCONJ
ap-1860	94	16	there	there	PRON
ap-1860	94	17	exist	exist	VERB
ap-1860	94	18	constants	constant	NOUN
ap-1860	94	19	c0	c0	NOUN
ap-1860	94	20	,	,	PUNCT
ap-1860	94	21	c1	c1	PROPN
ap-1860	94	22	,	,	PUNCT
ap-1860	94	23	.	.	PUNCT
ap-1860	94	24	.	.	PUNCT
ap-1860	95	1	.	.	PUNCT
ap-1860	96	1	,	,	PUNCT
ap-1860	96	2	ck−1	ck−1	NOUN
ap-1860	96	3	∈	∈	PROPN
ap-1860	96	4	r	r	NOUN
ap-1860	96	5	such	such	ADJ
ap-1860	96	6	that	that	DET
ap-1860	96	7	t	t	PROPN
ap-1860	96	8	(	(	PUNCT
ap-1860	96	9	x	x	X
ap-1860	96	10	)	)	PUNCT
ap-1860	96	11	=	=	SYM
ap-1860	97	1	x+	x+	PROPN
ap-1860	97	2	cj	cj	INTJ
ap-1860	97	3	,	,	PUNCT
ap-1860	98	1	x	x	PUNCT
ap-1860	98	2	∈	∈	PROPN
ap-1860	98	3	jj	jj	PROPN
ap-1860	98	4	,	,	PUNCT
ap-1860	98	5	and	and	CCONJ
ap-1860	98	6	t	t	PROPN
ap-1860	98	7	is	be	AUX
ap-1860	98	8	a	a	DET
ap-1860	98	9	bijection	bijection	NOUN
ap-1860	98	10	on	on	ADP
ap-1860	98	11	j	j	PROPN
ap-1860	98	12	.	.	PUNCT
ap-1860	99	1	since	since	SCONJ
ap-1860	99	2	t	t	PROPN
ap-1860	99	3	is	be	AUX
ap-1860	99	4	a	a	DET
ap-1860	99	5	bijection	bijection	NOUN
ap-1860	99	6	,	,	PUNCT
ap-1860	99	7	intervals	interval	NOUN
ap-1860	99	8	t	t	PROPN
ap-1860	99	9	(	(	PUNCT
ap-1860	99	10	ji	ji	PROPN
ap-1860	99	11	)	)	PUNCT
ap-1860	99	12	for	for	ADP
ap-1860	99	13	j	j	PROPN
ap-1860	99	14	=	=	SYM
ap-1860	99	15	0	0	PROPN
ap-1860	99	16	,	,	PUNCT
ap-1860	99	17	1	1	NUM
ap-1860	99	18	,	,	PUNCT
ap-1860	99	19	.	.	PUNCT
ap-1860	99	20	.	.	PUNCT
ap-1860	100	1	.	.	PUNCT
ap-1860	101	1	,	,	PUNCT
ap-1860	101	2	k	k	PROPN
ap-1860	101	3	−	−	NOUN
ap-1860	101	4	1	1	NUM
ap-1860	101	5	form	form	VERB
ap-1860	101	6	a	a	DET
ap-1860	101	7	partition	partition	NOUN
ap-1860	101	8	of	of	ADP
ap-1860	101	9	j	j	PROPN
ap-1860	101	10	.	.	PUNCT
ap-1860	102	1	the	the	DET
ap-1860	102	2	order	order	NOUN
ap-1860	102	3	of	of	ADP
ap-1860	102	4	indices	index	NOUN
ap-1860	102	5	j	j	X
ap-1860	102	6	which	which	PRON
ap-1860	102	7	determines	determine	VERB
ap-1860	102	8	the	the	DET
ap-1860	102	9	ordering	ordering	NOUN
ap-1860	102	10	of	of	ADP
ap-1860	102	11	intervals	interval	NOUN
ap-1860	102	12	t	t	PROPN
ap-1860	102	13	(	(	PUNCT
ap-1860	102	14	ji	ji	PROPN
ap-1860	102	15	)	)	PUNCT
ap-1860	102	16	in	in	ADP
ap-1860	102	17	j	j	PROPN
ap-1860	102	18	is	be	AUX
ap-1860	102	19	usually	usually	ADV
ap-1860	102	20	expressed	express	VERB
ap-1860	102	21	by	by	ADP
ap-1860	102	22	a	a	DET
ap-1860	102	23	permutation	permutation	NOUN
ap-1860	102	24	π	π	NOUN
ap-1860	102	25	.	.	PUNCT
ap-1860	103	1	a	a	DET
ap-1860	103	2	trivial	trivial	ADJ
ap-1860	103	3	example	example	NOUN
ap-1860	103	4	of	of	ADP
ap-1860	103	5	a	a	DET
ap-1860	103	6	k	k	ADJ
ap-1860	103	7	-	-	PUNCT
ap-1860	103	8	interval	interval	NOUN
ap-1860	103	9	exchange	exchange	NOUN
ap-1860	103	10	is	be	AUX
ap-1860	103	11	the	the	DET
ap-1860	103	12	choice	choice	NOUN
ap-1860	103	13	cj	cj	NOUN
ap-1860	103	14	=	=	SYM
ap-1860	103	15	0	0	NUM
ap-1860	103	16	for	for	ADP
ap-1860	103	17	j	j	PROPN
ap-1860	103	18	=	=	SYM
ap-1860	103	19	0	0	PROPN
ap-1860	103	20	,	,	PUNCT
ap-1860	103	21	.	.	PUNCT
ap-1860	103	22	.	.	PUNCT
ap-1860	104	1	.	.	PUNCT
ap-1860	105	1	,	,	PUNCT
ap-1860	105	2	k−1	k−1	PROPN
ap-1860	105	3	.	.	PUNCT
ap-1860	106	1	then	then	ADV
ap-1860	106	2	t	t	PROPN
ap-1860	106	3	is	be	AUX
ap-1860	106	4	the	the	DET
ap-1860	106	5	identity	identity	NOUN
ap-1860	106	6	map	map	NOUN
ap-1860	106	7	and	and	CCONJ
ap-1860	106	8	π	π	PROPN
ap-1860	106	9	is	be	AUX
ap-1860	106	10	the	the	DET
ap-1860	106	11	identity	identity	NOUN
ap-1860	106	12	permutation	permutation	NOUN
ap-1860	106	13	.	.	PUNCT
ap-1860	107	1	the	the	DET
ap-1860	107	2	transformation	transformation	NOUN
ap-1860	107	3	t	t	PROPN
ap-1860	107	4	of	of	ADP
ap-1860	107	5	(	(	PUNCT
ap-1860	107	6	1	1	X
ap-1860	107	7	)	)	PUNCT
ap-1860	107	8	is	be	AUX
ap-1860	107	9	a	a	DET
ap-1860	107	10	2	2	NUM
ap-1860	107	11	-	-	PUNCT
ap-1860	107	12	interval	interval	NOUN
ap-1860	107	13	exchange	exchange	NOUN
ap-1860	107	14	with	with	ADP
ap-1860	107	15	permutation	permutation	NOUN
ap-1860	107	16	(	(	PUNCT
ap-1860	107	17	21	21	NUM
ap-1860	107	18	)	)	PUNCT
ap-1860	107	19	.	.	PUNCT
ap-1860	108	1	example	example	NOUN
ap-1860	108	2	2.2	2.2	NUM
ap-1860	108	3	.	.	PUNCT
ap-1860	109	1	consider	consider	VERB
ap-1860	109	2	a	a	DET
ap-1860	109	3	,	,	PUNCT
ap-1860	109	4	b	b	X
ap-1860	109	5	∈	∈	PROPN
ap-1860	109	6	(	(	PUNCT
ap-1860	109	7	0	0	NUM
ap-1860	109	8	,	,	PUNCT
ap-1860	109	9	1	1	NUM
ap-1860	109	10	)	)	PUNCT
ap-1860	109	11	,	,	PUNCT
ap-1860	109	12	a	a	DET
ap-1860	109	13	<	<	X
ap-1860	109	14	b.	b.	PROPN
ap-1860	109	15	put	put	VERB
ap-1860	109	16	i0	i0	PROPN
ap-1860	109	17	=	=	PUNCT
ap-1860	110	1	[	[	X
ap-1860	110	2	0	0	NUM
ap-1860	110	3	,	,	PUNCT
ap-1860	110	4	a	a	PRON
ap-1860	110	5	)	)	PUNCT
ap-1860	110	6	,	,	PUNCT
ap-1860	110	7	i1	i1	PROPN
ap-1860	110	8	=	=	PUNCT
ap-1860	111	1	[	[	X
ap-1860	111	2	a	a	DET
ap-1860	111	3	,	,	PUNCT
ap-1860	111	4	b	b	NOUN
ap-1860	111	5	)	)	PUNCT
ap-1860	111	6	,	,	PUNCT
ap-1860	111	7	i2	i2	PROPN
ap-1860	111	8	=	=	PUNCT
ap-1860	112	1	[	[	X
ap-1860	112	2	b	b	X
ap-1860	112	3	,	,	PUNCT
ap-1860	112	4	1	1	NUM
ap-1860	112	5	)	)	PUNCT
ap-1860	112	6	.	.	PUNCT
ap-1860	113	1	then	then	ADV
ap-1860	113	2	the	the	DET
ap-1860	113	3	transformation	transformation	NOUN
ap-1860	113	4	t	t	NOUN
ap-1860	113	5	:	:	PUNCT
ap-1860	114	1	[	[	X
ap-1860	114	2	0	0	NUM
ap-1860	114	3	,	,	PUNCT
ap-1860	114	4	1)→	1)→	NUM
ap-1860	115	1	[	[	X
ap-1860	115	2	0	0	NUM
ap-1860	115	3	,	,	PUNCT
ap-1860	115	4	1	1	NUM
ap-1860	115	5	)	)	PUNCT
ap-1860	115	6	given	give	VERB
ap-1860	115	7	by	by	ADP
ap-1860	115	8	t	t	PROPN
ap-1860	115	9	(	(	PUNCT
ap-1860	115	10	x	x	NOUN
ap-1860	115	11	)	)	PUNCT
ap-1860	115	12	=	=	SYM
ap-1860	115	13			NOUN
ap-1860	115	14	x+	x+	NUM
ap-1860	115	15	1−	1−	NUM
ap-1860	115	16	a	a	DET
ap-1860	115	17	if	if	NOUN
ap-1860	115	18	x	x	X
ap-1860	115	19	∈	∈	PROPN
ap-1860	116	1	[	[	X
ap-1860	116	2	0	0	NUM
ap-1860	116	3	,	,	PUNCT
ap-1860	116	4	a	a	NOUN
ap-1860	116	5	)	)	PUNCT
ap-1860	116	6	,	,	PUNCT
ap-1860	116	7	x+	x+	X
ap-1860	116	8	1−	1−	NUM
ap-1860	116	9	a−	a−	PROPN
ap-1860	116	10	b	b	PROPN
ap-1860	116	11	if	if	SCONJ
ap-1860	116	12	x	x	SYM
ap-1860	116	13	∈	∈	PROPN
ap-1860	116	14	[	[	X
ap-1860	116	15	a	a	DET
ap-1860	116	16	,	,	PUNCT
ap-1860	116	17	b	b	NOUN
ap-1860	116	18	)	)	PUNCT
ap-1860	116	19	,	,	PUNCT
ap-1860	117	1	x−	x−	PROPN
ap-1860	117	2	b	b	PROPN
ap-1860	117	3	if	if	SCONJ
ap-1860	117	4	x	x	SYM
ap-1860	117	5	∈	∈	PROPN
ap-1860	117	6	[	[	X
ap-1860	117	7	b	b	NOUN
ap-1860	117	8	,	,	PUNCT
ap-1860	117	9	1	1	NUM
ap-1860	117	10	)	)	PUNCT
ap-1860	117	11	,	,	PUNCT
ap-1860	117	12	(	(	PUNCT
ap-1860	117	13	4	4	X
ap-1860	117	14	)	)	PUNCT
ap-1860	117	15	is	be	AUX
ap-1860	117	16	a	a	DET
ap-1860	117	17	3	3	NUM
ap-1860	117	18	-	-	PUNCT
ap-1860	117	19	interval	interval	NOUN
ap-1860	117	20	exchange	exchange	NOUN
ap-1860	117	21	with	with	ADP
ap-1860	117	22	permutation	permutation	NOUN
ap-1860	117	23	π	π	X
ap-1860	117	24	=	=	PUNCT
ap-1860	117	25	(	(	PUNCT
ap-1860	117	26	321	321	NUM
ap-1860	117	27	)	)	PUNCT
ap-1860	117	28	,	,	PUNCT
ap-1860	117	29	see	see	VERB
ap-1860	117	30	figure	figure	NOUN
ap-1860	117	31	1	1	NUM
ap-1860	117	32	.	.	PUNCT
ap-1860	117	33	from	from	ADP
ap-1860	117	34	now	now	ADV
ap-1860	117	35	on	on	ADV
ap-1860	117	36	,	,	PUNCT
ap-1860	117	37	we	we	PRON
ap-1860	117	38	focus	focus	VERB
ap-1860	117	39	on	on	ADP
ap-1860	117	40	the	the	DET
ap-1860	117	41	exchange	exchange	NOUN
ap-1860	117	42	t	t	PROPN
ap-1860	117	43	of	of	ADP
ap-1860	117	44	two	two	NUM
ap-1860	117	45	intervals	interval	NOUN
ap-1860	117	46	given	give	VERB
ap-1860	117	47	by	by	ADP
ap-1860	117	48	the	the	DET
ap-1860	117	49	prescription	prescription	NOUN
ap-1860	117	50	(	(	PUNCT
ap-1860	117	51	1	1	NUM
ap-1860	117	52	)	)	PUNCT
ap-1860	117	53	with	with	ADP
ap-1860	117	54	an	an	DET
ap-1860	117	55	irrational	irrational	ADJ
ap-1860	117	56	slope	slope	NOUN
ap-1860	117	57	α	α	NOUN
ap-1860	117	58	.	.	PUNCT
ap-1860	118	1	we	we	PRON
ap-1860	118	2	will	will	AUX
ap-1860	118	3	study	study	VERB
ap-1860	118	4	the	the	DET
ap-1860	118	5	first	first	ADJ
ap-1860	118	6	return	return	NOUN
ap-1860	118	7	map	map	NOUN
ap-1860	118	8	ti	ti	NOUN
ap-1860	118	9	defined	define	VERB
ap-1860	118	10	by	by	ADP
ap-1860	118	11	(	(	PUNCT
ap-1860	118	12	3	3	X
ap-1860	118	13	)	)	PUNCT
ap-1860	118	14	to	to	ADP
ap-1860	118	15	the	the	DET
ap-1860	118	16	subinterval	subinterval	NOUN
ap-1860	119	1	i	i	PRON
ap-1860	119	2	⊂	⊂	PROPN
ap-1860	120	1	[	[	X
ap-1860	120	2	0	0	NUM
ap-1860	120	3	,	,	PUNCT
ap-1860	120	4	1	1	NUM
ap-1860	120	5	)	)	PUNCT
ap-1860	120	6	.	.	PUNCT
ap-1860	121	1	in	in	ADP
ap-1860	121	2	[	[	X
ap-1860	121	3	10	10	NUM
ap-1860	121	4	]	]	X
ap-1860	121	5	it	it	PRON
ap-1860	121	6	is	be	AUX
ap-1860	121	7	shown	show	VERB
ap-1860	121	8	how	how	SCONJ
ap-1860	121	9	ti	ti	NOUN
ap-1860	121	10	depends	depend	VERB
ap-1860	121	11	on	on	ADP
ap-1860	121	12	the	the	DET
ap-1860	121	13	length	length	NOUN
ap-1860	121	14	of	of	ADP
ap-1860	121	15	the	the	DET
ap-1860	121	16	interval	interval	NOUN
ap-1860	121	17	i.	i.	NOUN
ap-1860	121	18	for	for	ADP
ap-1860	121	19	an	an	DET
ap-1860	121	20	irrational	irrational	ADJ
ap-1860	121	21	α	α	NOUN
ap-1860	121	22	∈	∈	PROPN
ap-1860	121	23	(	(	PUNCT
ap-1860	121	24	0	0	NUM
ap-1860	121	25	,	,	PUNCT
ap-1860	121	26	1	1	NUM
ap-1860	121	27	)	)	PUNCT
ap-1860	121	28	with	with	ADP
ap-1860	121	29	the	the	DET
ap-1860	121	30	continued	continue	VERB
ap-1860	121	31	fraction	fraction	NOUN
ap-1860	122	1	α	α	NOUN
ap-1860	122	2	=	=	PUNCT
ap-1860	123	1	[	[	X
ap-1860	123	2	0	0	NUM
ap-1860	123	3	,	,	PUNCT
ap-1860	123	4	a1	a1	NOUN
ap-1860	123	5	,	,	PUNCT
ap-1860	123	6	a2	a2	PROPN
ap-1860	123	7	,	,	PUNCT
ap-1860	123	8	.	.	PUNCT
ap-1860	123	9	.	.	PUNCT
ap-1860	123	10	.	.	PUNCT
ap-1860	124	1	]	]	PUNCT
ap-1860	125	1	and	and	CCONJ
ap-1860	125	2	convergents	convergent	NOUN
ap-1860	125	3	pn	pn	PROPN
ap-1860	125	4	qn	qn	PROPN
ap-1860	125	5	set	set	PROPN
ap-1860	125	6	δk	δk	PROPN
ap-1860	125	7	,	,	PUNCT
ap-1860	125	8	s	s	PART
ap-1860	125	9	:	:	PUNCT
ap-1860	125	10	=	=	SYM
ap-1860	125	11	∣∣(s−	∣∣(s−	ADJ
ap-1860	125	12	1)(pk	1)(pk	NUM
ap-1860	125	13	−	−	PROPN
ap-1860	125	14	αqk	αqk	NOUN
ap-1860	125	15	)	)	PUNCT
ap-1860	125	16	+	+	NOUN
ap-1860	126	1	pk−1	pk−1	ADP
ap-1860	126	2	−	−	PROPN
ap-1860	126	3	αqk−1	αqk−1	PROPN
ap-1860	126	4	∣∣	∣∣	PROPN
ap-1860	126	5	,	,	PUNCT
ap-1860	126	6	for	for	ADP
ap-1860	126	7	k	k	PROPN
ap-1860	126	8	≥	≥	PROPN
ap-1860	126	9	0	0	NUM
ap-1860	126	10	,	,	PUNCT
ap-1860	126	11	1	1	NUM
ap-1860	126	12	≤	≤	NUM
ap-1860	126	13	s	s	PART
ap-1860	126	14	≤	≤	NUM
ap-1860	126	15	ak+1	ak+1	VERB
ap-1860	126	16	.	.	PUNCT
ap-1860	127	1	(	(	PUNCT
ap-1860	127	2	5	5	NUM
ap-1860	127	3	)	)	PUNCT
ap-1860	127	4	for	for	ADP
ap-1860	127	5	the	the	DET
ap-1860	127	6	numbers	number	NOUN
ap-1860	127	7	δk	δk	ADP
ap-1860	127	8	,	,	PUNCT
ap-1860	127	9	s	s	VERB
ap-1860	127	10	one	one	NUM
ap-1860	127	11	has	have	VERB
ap-1860	127	12	δk	δk	VERB
ap-1860	127	13	,	,	PUNCT
ap-1860	127	14	s	s	PART
ap-1860	127	15	>	>	X
ap-1860	127	16	δk′,s′	δk′,s′	PROPN
ap-1860	128	1	if	if	SCONJ
ap-1860	128	2	and	and	CCONJ
ap-1860	128	3	only	only	ADV
ap-1860	128	4	if	if	SCONJ
ap-1860	128	5	k′	k′	PROPN
ap-1860	128	6	>	>	X
ap-1860	128	7	k	k	PROPN
ap-1860	128	8	or	or	CCONJ
ap-1860	128	9	k′	k′	PROPN
ap-1860	128	10	=	=	SYM
ap-1860	128	11	k	k	PROPN
ap-1860	128	12	and	and	CCONJ
ap-1860	128	13	s′	s′	X
ap-1860	128	14	>	>	PUNCT
ap-1860	129	1	s.	s.	PROPN
ap-1860	129	2	445	445	PROPN
ap-1860	129	3	z.	z.	PROPN
ap-1860	129	4	masáková	masáková	PROPN
ap-1860	129	5	,	,	PUNCT
ap-1860	129	6	e.	e.	PROPN
ap-1860	129	7	pelantová	pelantová	PROPN
ap-1860	129	8	acta	acta	PROPN
ap-1860	129	9	polytechnica	polytechnica	PROPN
ap-1860	129	10	i0︷	i0︷	NOUN
ap-1860	129	11	︸︸	︸︸	CCONJ
ap-1860	129	12	︷	︷	PROPN
ap-1860	130	1	i1︷	i1︷	INTJ
ap-1860	130	2	︸︸	︸︸	PUNCT
ap-1860	130	3	︷	︷	PROPN
ap-1860	130	4	i2︷	i2︷	ADV
ap-1860	130	5	︸︸	︸︸	PUNCT
ap-1860	130	6	︷	︷	AUX
ap-1860	131	1	︸	︸	X
ap-1860	131	2	︷︷	︷︷	PROPN
ap-1860	131	3	︸	︸	X
ap-1860	131	4	t	t	PROPN
ap-1860	131	5	(	(	PUNCT
ap-1860	131	6	i2	i2	PROPN
ap-1860	131	7	)	)	PUNCT
ap-1860	131	8	︸	︸	X
ap-1860	132	1	︷︷	︷︷	PROPN
ap-1860	132	2	︸	︸	X
ap-1860	132	3	t	t	PROPN
ap-1860	132	4	(	(	PUNCT
ap-1860	132	5	i1	i1	PROPN
ap-1860	132	6	)	)	PUNCT
ap-1860	132	7	︸	︸	X
ap-1860	133	1	︷︷	︷︷	PROPN
ap-1860	133	2	︸	︸	X
ap-1860	133	3	t	t	PROPN
ap-1860	133	4	(	(	PUNCT
ap-1860	133	5	i0	i0	PROPN
ap-1860	133	6	)	)	PUNCT
ap-1860	133	7	figure	figure	NOUN
ap-1860	133	8	1	1	NUM
ap-1860	133	9	.	.	PUNCT
ap-1860	134	1	exchange	exchange	NOUN
ap-1860	134	2	of	of	ADP
ap-1860	134	3	three	three	NUM
ap-1860	134	4	intervals	interval	NOUN
ap-1860	134	5	.	.	PUNCT
ap-1860	135	1	in	in	ADP
ap-1860	135	2	[	[	X
ap-1860	135	3	10	10	NUM
ap-1860	135	4	]	]	PUNCT
ap-1860	135	5	,	,	PUNCT
ap-1860	135	6	we	we	PRON
ap-1860	135	7	study	study	VERB
ap-1860	135	8	infinite	infinite	ADJ
ap-1860	135	9	words	word	NOUN
ap-1860	135	10	associated	associate	VERB
ap-1860	135	11	to	to	ADP
ap-1860	135	12	cutand	cutand	NOUN
ap-1860	135	13	-	-	PUNCT
ap-1860	135	14	project	project	NOUN
ap-1860	135	15	sequences	sequence	NOUN
ap-1860	135	16	which	which	PRON
ap-1860	135	17	we	we	PRON
ap-1860	135	18	show	show	VERB
ap-1860	135	19	to	to	PART
ap-1860	135	20	be	be	AUX
ap-1860	135	21	exactly	exactly	ADV
ap-1860	135	22	codings	coding	NOUN
ap-1860	135	23	of	of	ADP
ap-1860	135	24	exchanges	exchange	NOUN
ap-1860	135	25	of	of	ADP
ap-1860	135	26	two	two	NUM
ap-1860	135	27	or	or	CCONJ
ap-1860	135	28	three	three	NUM
ap-1860	135	29	intervals	interval	NOUN
ap-1860	135	30	.	.	PUNCT
ap-1860	136	1	the	the	DET
ap-1860	136	2	following	follow	VERB
ap-1860	136	3	proposition	proposition	NOUN
ap-1860	136	4	is	be	AUX
ap-1860	136	5	a	a	DET
ap-1860	136	6	reformulation	reformulation	NOUN
ap-1860	136	7	of	of	ADP
ap-1860	136	8	statements	statement	NOUN
ap-1860	136	9	of	of	ADP
ap-1860	136	10	theorem	theorem	ADJ
ap-1860	136	11	4.1	4.1	NUM
ap-1860	136	12	and	and	CCONJ
ap-1860	136	13	proposition	proposition	NOUN
ap-1860	136	14	4.5	4.5	NUM
ap-1860	136	15	of	of	ADP
ap-1860	136	16	[	[	X
ap-1860	136	17	10	10	NUM
ap-1860	136	18	]	]	PUNCT
ap-1860	136	19	in	in	ADP
ap-1860	136	20	the	the	DET
ap-1860	136	21	framework	framework	NOUN
ap-1860	136	22	of	of	ADP
ap-1860	136	23	interval	interval	NOUN
ap-1860	136	24	exchanges	exchange	NOUN
ap-1860	136	25	.	.	PUNCT
ap-1860	137	1	proposition	proposition	NOUN
ap-1860	137	2	2.3	2.3	NUM
ap-1860	137	3	.	.	PUNCT
ap-1860	138	1	let	let	VERB
ap-1860	138	2	t	t	NOUN
ap-1860	138	3	:	:	PUNCT
ap-1860	139	1	[	[	X
ap-1860	139	2	0	0	NUM
ap-1860	139	3	,	,	PUNCT
ap-1860	139	4	1	1	NUM
ap-1860	139	5	)	)	PUNCT
ap-1860	139	6	→	→	PUNCT
ap-1860	139	7	[	[	X
ap-1860	139	8	0	0	NUM
ap-1860	139	9	,	,	PUNCT
ap-1860	139	10	1	1	NUM
ap-1860	139	11	)	)	PUNCT
ap-1860	139	12	be	be	AUX
ap-1860	139	13	an	an	DET
ap-1860	139	14	exchange	exchange	NOUN
ap-1860	139	15	of	of	ADP
ap-1860	139	16	two	two	NUM
ap-1860	139	17	intervals	interval	NOUN
ap-1860	139	18	with	with	ADP
ap-1860	139	19	irrational	irrational	ADJ
ap-1860	139	20	slope	slope	NOUN
ap-1860	139	21	α	α	NOUN
ap-1860	139	22	and	and	CCONJ
ap-1860	139	23	let	let	VERB
ap-1860	139	24	i	i	PRON
ap-1860	139	25	=	=	PUNCT
ap-1860	140	1	[	[	X
ap-1860	140	2	c	c	X
ap-1860	140	3	,	,	PUNCT
ap-1860	140	4	d	d	NOUN
ap-1860	140	5	)	)	PUNCT
ap-1860	140	6	⊂	⊂	PROPN
ap-1860	141	1	[	[	X
ap-1860	141	2	0	0	NUM
ap-1860	141	3	,	,	PUNCT
ap-1860	141	4	1	1	NUM
ap-1860	141	5	)	)	PUNCT
ap-1860	141	6	.	.	PUNCT
ap-1860	142	1	for	for	ADP
ap-1860	142	2	the	the	DET
ap-1860	142	3	induced	induced	ADJ
ap-1860	142	4	map	map	NOUN
ap-1860	142	5	ti	ti	NOUN
ap-1860	142	6	one	one	NUM
ap-1860	142	7	has	have	VERB
ap-1860	142	8	(	(	PUNCT
ap-1860	142	9	1	1	NUM
ap-1860	142	10	.	.	PUNCT
ap-1860	142	11	)	)	PUNCT
ap-1860	143	1	if	if	SCONJ
ap-1860	143	2	d−	d−	PROPN
ap-1860	143	3	c	c	PROPN
ap-1860	143	4	=	=	SYM
ap-1860	143	5	δk	δk	PROPN
ap-1860	143	6	,	,	PUNCT
ap-1860	143	7	s	s	X
ap-1860	143	8	for	for	ADP
ap-1860	143	9	some	some	DET
ap-1860	143	10	k	k	PROPN
ap-1860	143	11	,	,	PUNCT
ap-1860	143	12	s	s	PROPN
ap-1860	143	13	,	,	PUNCT
ap-1860	143	14	defined	define	VERB
ap-1860	143	15	in	in	ADP
ap-1860	143	16	(	(	PUNCT
ap-1860	143	17	5	5	NUM
ap-1860	143	18	)	)	PUNCT
ap-1860	143	19	,	,	PUNCT
ap-1860	143	20	then	then	ADV
ap-1860	143	21	ti	ti	PROPN
ap-1860	143	22	is	be	AUX
ap-1860	143	23	an	an	DET
ap-1860	143	24	exchange	exchange	NOUN
ap-1860	143	25	of	of	ADP
ap-1860	143	26	two	two	NUM
ap-1860	143	27	intervals	interval	NOUN
ap-1860	143	28	.	.	PUNCT
ap-1860	144	1	(	(	PUNCT
ap-1860	144	2	2	2	NUM
ap-1860	144	3	.	.	PUNCT
ap-1860	144	4	)	)	PUNCT
ap-1860	144	5	otherwise	otherwise	ADV
ap-1860	144	6	,	,	PUNCT
ap-1860	144	7	ti	ti	PROPN
ap-1860	144	8	is	be	AUX
ap-1860	144	9	an	an	DET
ap-1860	144	10	exchange	exchange	NOUN
ap-1860	144	11	of	of	ADP
ap-1860	144	12	three	three	NUM
ap-1860	144	13	intervals	interval	NOUN
ap-1860	144	14	with	with	ADP
ap-1860	144	15	permutation	permutation	NOUN
ap-1860	144	16	(	(	PUNCT
ap-1860	144	17	321	321	NUM
ap-1860	144	18	)	)	PUNCT
ap-1860	144	19	.	.	PUNCT
ap-1860	145	1	moreover	moreover	ADV
ap-1860	145	2	,	,	PUNCT
ap-1860	145	3	the	the	DET
ap-1860	145	4	lengths	length	NOUN
ap-1860	145	5	of	of	ADP
ap-1860	145	6	intervals	interval	NOUN
ap-1860	145	7	i0	i0	PROPN
ap-1860	145	8	,	,	PUNCT
ap-1860	145	9	i1	i1	PROPN
ap-1860	145	10	,	,	PUNCT
ap-1860	145	11	i2	i2	PROPN
ap-1860	145	12	forming	form	VERB
ap-1860	145	13	the	the	DET
ap-1860	145	14	partition	partition	NOUN
ap-1860	145	15	of	of	ADP
ap-1860	145	16	i	i	PRON
ap-1860	145	17	depend	depend	VERB
ap-1860	145	18	only	only	ADV
ap-1860	145	19	on	on	ADP
ap-1860	145	20	d−	d−	PROPN
ap-1860	145	21	c	c	PROPN
ap-1860	145	22	and	and	CCONJ
ap-1860	145	23	for	for	ADP
ap-1860	145	24	the	the	DET
ap-1860	145	25	return	return	NOUN
ap-1860	145	26	time	time	NOUN
ap-1860	145	27	r(x0	r(x0	NOUN
ap-1860	145	28	)	)	PUNCT
ap-1860	145	29	,	,	PUNCT
ap-1860	145	30	r(x1	r(x1	PROPN
ap-1860	145	31	)	)	PUNCT
ap-1860	145	32	,	,	PUNCT
ap-1860	145	33	r(x2	r(x2	NOUN
ap-1860	145	34	)	)	PUNCT
ap-1860	145	35	of	of	ADP
ap-1860	145	36	points	point	NOUN
ap-1860	145	37	x0	x0	PROPN
ap-1860	145	38	∈	∈	PROPN
ap-1860	145	39	i0	i0	PROPN
ap-1860	145	40	,	,	PUNCT
ap-1860	145	41	x1	x1	PROPN
ap-1860	145	42	∈	∈	PROPN
ap-1860	145	43	i1	i1	PROPN
ap-1860	145	44	,	,	PUNCT
ap-1860	145	45	x2	x2	PROPN
ap-1860	145	46	∈	∈	PROPN
ap-1860	145	47	i2	i2	PROPN
ap-1860	145	48	,	,	PUNCT
ap-1860	145	49	x0	x0	PROPN
ap-1860	145	50	<	<	X
ap-1860	146	1	x1	x1	X
ap-1860	146	2	<	<	X
ap-1860	146	3	x2	x2	PROPN
ap-1860	146	4	,	,	PUNCT
ap-1860	146	5	one	one	NUM
ap-1860	146	6	has	have	VERB
ap-1860	146	7	r(x1	r(x1	NOUN
ap-1860	146	8	)	)	PUNCT
ap-1860	146	9	=	=	SYM
ap-1860	146	10	r(x0	r(x0	NOUN
ap-1860	146	11	)	)	PUNCT
ap-1860	146	12	+	+	NUM
ap-1860	146	13	r(x2	r(x2	NOUN
ap-1860	146	14	)	)	PUNCT
ap-1860	146	15	.	.	PUNCT
ap-1860	147	1	remark	remark	VERB
ap-1860	147	2	2.4	2.4	NUM
ap-1860	147	3	.	.	PUNCT
ap-1860	148	1	proposition	proposition	NOUN
ap-1860	148	2	4.5	4.5	NUM
ap-1860	148	3	of	of	ADP
ap-1860	148	4	[	[	X
ap-1860	148	5	10	10	NUM
ap-1860	148	6	]	]	PUNCT
ap-1860	148	7	also	also	ADV
ap-1860	148	8	allows	allow	VERB
ap-1860	148	9	to	to	PART
ap-1860	148	10	determine	determine	VERB
ap-1860	148	11	the	the	DET
ap-1860	148	12	exact	exact	ADJ
ap-1860	148	13	two	two	NUM
ap-1860	148	14	or	or	CCONJ
ap-1860	148	15	three	three	NUM
ap-1860	148	16	values	value	NOUN
ap-1860	148	17	of	of	ADP
ap-1860	148	18	return	return	NOUN
ap-1860	148	19	time	time	NOUN
ap-1860	148	20	r(x	r(x	PROPN
ap-1860	148	21	)	)	PUNCT
ap-1860	148	22	to	to	PART
ap-1860	148	23	i.	i.	VERB
ap-1860	148	24	in	in	ADP
ap-1860	148	25	fact	fact	NOUN
ap-1860	148	26	,	,	PUNCT
ap-1860	148	27	if	if	SCONJ
ap-1860	148	28	d−	d−	PROPN
ap-1860	148	29	c	c	PROPN
ap-1860	148	30	=	=	SYM
ap-1860	148	31	δk	δk	PROPN
ap-1860	148	32	,	,	PUNCT
ap-1860	148	33	s	s	PART
ap-1860	148	34	,	,	PUNCT
ap-1860	148	35	then	then	ADV
ap-1860	148	36	—	—	PUNCT
ap-1860	148	37	keeping	keep	VERB
ap-1860	148	38	the	the	DET
ap-1860	148	39	notation	notation	NOUN
ap-1860	148	40	of	of	ADP
ap-1860	148	41	(	(	PUNCT
ap-1860	148	42	5	5	NUM
ap-1860	148	43	)	)	PUNCT
ap-1860	148	44	—	—	PUNCT
ap-1860	149	1	r(x	r(x	NOUN
ap-1860	149	2	)	)	PUNCT
ap-1860	149	3	takes	take	VERB
ap-1860	149	4	two	two	NUM
ap-1860	149	5	values	value	NOUN
ap-1860	149	6	{	{	PUNCT
ap-1860	149	7	r(x	r(x	NOUN
ap-1860	149	8	)	)	PUNCT
ap-1860	149	9	:	:	PUNCT
ap-1860	150	1	x	x	X
ap-1860	150	2	∈	∈	NOUN
ap-1860	150	3	i	i	NOUN
ap-1860	150	4	}	}	PUNCT
ap-1860	150	5	=	=	SYM
ap-1860	150	6	{	{	PUNCT
ap-1860	150	7	qk	qk	NOUN
ap-1860	150	8	,	,	PUNCT
ap-1860	150	9	sqk	sqk	VERB
ap-1860	150	10	+	+	CCONJ
ap-1860	150	11	qk−1	qk−1	NOUN
ap-1860	150	12	}	}	PUNCT
ap-1860	150	13	.	.	PUNCT
ap-1860	151	1	if	if	SCONJ
ap-1860	151	2	d	d	PROPN
ap-1860	151	3	−	−	PROPN
ap-1860	151	4	c	c	PROPN
ap-1860	151	5	is	be	AUX
ap-1860	151	6	between	between	ADP
ap-1860	151	7	δk	δk	NOUN
ap-1860	151	8	,	,	PUNCT
ap-1860	151	9	s	s	PART
ap-1860	151	10	and	and	CCONJ
ap-1860	151	11	its	its	PRON
ap-1860	151	12	successor	successor	NOUN
ap-1860	151	13	in	in	ADP
ap-1860	151	14	the	the	DET
ap-1860	151	15	decreasing	decrease	VERB
ap-1860	151	16	sequence	sequence	NOUN
ap-1860	151	17	(	(	PUNCT
ap-1860	151	18	δk	δk	PROPN
ap-1860	151	19	,	,	PUNCT
ap-1860	151	20	s	s	PART
ap-1860	151	21	)	)	PUNCT
ap-1860	151	22	,	,	PUNCT
ap-1860	151	23	then	then	ADV
ap-1860	151	24	r(x	r(x	PROPN
ap-1860	151	25	)	)	PUNCT
ap-1860	151	26	takes	take	VERB
ap-1860	151	27	three	three	NUM
ap-1860	151	28	values	value	NOUN
ap-1860	151	29	{	{	PUNCT
ap-1860	151	30	r(x	r(x	NOUN
ap-1860	151	31	)	)	PUNCT
ap-1860	151	32	:	:	PUNCT
ap-1860	152	1	x	x	X
ap-1860	152	2	∈	∈	NOUN
ap-1860	152	3	i	i	NOUN
ap-1860	152	4	}	}	PUNCT
ap-1860	152	5	=	=	SYM
ap-1860	152	6	{	{	PUNCT
ap-1860	152	7	qk	qk	NOUN
ap-1860	152	8	,	,	PUNCT
ap-1860	152	9	sqk	sqk	VERB
ap-1860	152	10	+	+	CCONJ
ap-1860	152	11	qk−1	qk−1	NOUN
ap-1860	152	12	,	,	PUNCT
ap-1860	152	13	(	(	PUNCT
ap-1860	152	14	s+	s+	ADV
ap-1860	152	15	1)qk	1)qk	PROPN
ap-1860	153	1	+	+	CCONJ
ap-1860	153	2	qk−1	qk−1	NOUN
ap-1860	153	3	}	}	PUNCT
ap-1860	153	4	.	.	PUNCT
ap-1860	154	1	the	the	DET
ap-1860	154	2	values	value	NOUN
ap-1860	154	3	of	of	ADP
ap-1860	154	4	return	return	NOUN
ap-1860	154	5	time	time	NOUN
ap-1860	154	6	are	be	AUX
ap-1860	154	7	connected	connect	VERB
ap-1860	154	8	to	to	ADP
ap-1860	154	9	the	the	DET
ap-1860	154	10	socalled	socalled	ADJ
ap-1860	154	11	three	three	NUM
ap-1860	154	12	-	-	PUNCT
ap-1860	154	13	distance	distance	NOUN
ap-1860	154	14	theorem	theorem	NOUN
ap-1860	154	15	[	[	X
ap-1860	154	16	11	11	NUM
ap-1860	154	17	,	,	PUNCT
ap-1860	154	18	12	12	NUM
ap-1860	154	19	]	]	PUNCT
ap-1860	154	20	.	.	PUNCT
ap-1860	155	1	another	another	DET
ap-1860	155	2	point	point	NOUN
ap-1860	155	3	of	of	ADP
ap-1860	155	4	view	view	NOUN
ap-1860	155	5	on	on	ADP
ap-1860	155	6	return	return	NOUN
ap-1860	155	7	time	time	NOUN
ap-1860	155	8	in	in	ADP
ap-1860	155	9	sturmian	sturmian	ADJ
ap-1860	155	10	words	word	NOUN
ap-1860	155	11	is	be	AUX
ap-1860	155	12	presented	present	VERB
ap-1860	155	13	in	in	ADP
ap-1860	155	14	[	[	X
ap-1860	155	15	13	13	NUM
ap-1860	155	16	]	]	PUNCT
ap-1860	155	17	.	.	PUNCT
ap-1860	156	1	although	although	SCONJ
ap-1860	156	2	the	the	DET
ap-1860	156	3	return	return	NOUN
ap-1860	156	4	time	time	NOUN
ap-1860	156	5	r(x	r(x	PROPN
ap-1860	156	6	)	)	PUNCT
ap-1860	156	7	to	to	ADP
ap-1860	156	8	a	a	DET
ap-1860	156	9	given	give	VERB
ap-1860	156	10	interval	interval	NOUN
ap-1860	156	11	i	i	PRON
ap-1860	156	12	can	can	AUX
ap-1860	156	13	take	take	VERB
ap-1860	156	14	only	only	ADV
ap-1860	156	15	three	three	NUM
ap-1860	156	16	values	value	NOUN
ap-1860	156	17	,	,	PUNCT
ap-1860	156	18	the	the	DET
ap-1860	156	19	set	set	ADJ
ap-1860	156	20	iti	iti	PROPN
ap-1860	156	21	of	of	ADP
ap-1860	156	22	i	i	PROPN
ap-1860	156	23	-	-	PUNCT
ap-1860	156	24	itineraries	itinerary	NOUN
ap-1860	156	25	can	can	AUX
ap-1860	156	26	have	have	VERB
ap-1860	156	27	more	more	ADJ
ap-1860	156	28	than	than	ADP
ap-1860	156	29	three	three	NUM
ap-1860	156	30	elements	element	NOUN
ap-1860	156	31	.	.	PUNCT
ap-1860	157	1	the	the	DET
ap-1860	157	2	following	follow	VERB
ap-1860	157	3	statement	statement	NOUN
ap-1860	157	4	can	can	AUX
ap-1860	157	5	be	be	AUX
ap-1860	157	6	extracted	extract	VERB
ap-1860	157	7	from	from	ADP
ap-1860	157	8	the	the	DET
ap-1860	157	9	proof	proof	NOUN
ap-1860	157	10	of	of	ADP
ap-1860	157	11	the	the	DET
ap-1860	157	12	theorem	theorem	NOUN
ap-1860	157	13	in	in	ADP
ap-1860	157	14	[	[	X
ap-1860	157	15	9	9	NUM
ap-1860	157	16	,	,	PUNCT
ap-1860	157	17	§	§	NOUN
ap-1860	157	18	2	2	NUM
ap-1860	157	19	]	]	PUNCT
ap-1860	157	20	.	.	PUNCT
ap-1860	158	1	it	it	PRON
ap-1860	158	2	is	be	AUX
ap-1860	158	3	convenient	convenient	ADJ
ap-1860	158	4	to	to	PART
ap-1860	158	5	provide	provide	VERB
ap-1860	158	6	the	the	DET
ap-1860	158	7	demonstration	demonstration	NOUN
ap-1860	158	8	here	here	ADV
ap-1860	158	9	.	.	PUNCT
ap-1860	159	1	proposition	proposition	NOUN
ap-1860	159	2	2.5	2.5	NUM
ap-1860	159	3	.	.	PUNCT
ap-1860	160	1	let	let	VERB
ap-1860	160	2	t	t	NOUN
ap-1860	160	3	:	:	PUNCT
ap-1860	161	1	[	[	X
ap-1860	161	2	0	0	NUM
ap-1860	161	3	,	,	PUNCT
ap-1860	161	4	1	1	NUM
ap-1860	161	5	)	)	PUNCT
ap-1860	161	6	→	→	PUNCT
ap-1860	161	7	[	[	X
ap-1860	161	8	0	0	NUM
ap-1860	161	9	,	,	PUNCT
ap-1860	161	10	1	1	NUM
ap-1860	161	11	)	)	PUNCT
ap-1860	161	12	be	be	AUX
ap-1860	161	13	an	an	DET
ap-1860	161	14	exchange	exchange	NOUN
ap-1860	161	15	of	of	ADP
ap-1860	161	16	two	two	NUM
ap-1860	161	17	intervals	interval	NOUN
ap-1860	161	18	with	with	ADP
ap-1860	161	19	irrational	irrational	ADJ
ap-1860	161	20	slope	slope	NOUN
ap-1860	161	21	α	α	NOUN
ap-1860	161	22	and	and	CCONJ
ap-1860	161	23	let	let	VERB
ap-1860	161	24	i	i	PRON
ap-1860	161	25	=	=	PUNCT
ap-1860	162	1	[	[	X
ap-1860	162	2	c	c	X
ap-1860	162	3	,	,	PUNCT
ap-1860	162	4	d	d	NOUN
ap-1860	162	5	)	)	PUNCT
ap-1860	162	6	⊂	⊂	PROPN
ap-1860	163	1	[	[	X
ap-1860	163	2	0	0	NUM
ap-1860	163	3	,	,	PUNCT
ap-1860	163	4	1	1	NUM
ap-1860	163	5	)	)	PUNCT
ap-1860	163	6	.	.	PUNCT
ap-1860	164	1	then	then	ADV
ap-1860	164	2	iti	iti	PROPN
ap-1860	164	3	has	have	VERB
ap-1860	164	4	at	at	ADP
ap-1860	164	5	most	most	ADJ
ap-1860	164	6	4	4	NUM
ap-1860	164	7	elements	element	NOUN
ap-1860	164	8	.	.	PUNCT
ap-1860	165	1	proof	proof	NOUN
ap-1860	165	2	.	.	PUNCT
ap-1860	166	1	choose	choose	VERB
ap-1860	166	2	x	x	SYM
ap-1860	166	3	∈	∈	PROPN
ap-1860	166	4	i.	i.	PROPN
ap-1860	166	5	denote	denote	PROPN
ap-1860	166	6	r(x	r(x	PROPN
ap-1860	166	7	)	)	PUNCT
ap-1860	166	8	its	its	PRON
ap-1860	166	9	i	i	NOUN
ap-1860	166	10	-	-	PUNCT
ap-1860	166	11	itinerary	itinerary	NOUN
ap-1860	166	12	and	and	CCONJ
ap-1860	166	13	r	r	NOUN
ap-1860	166	14	=	=	SYM
ap-1860	166	15	r(x	r(x	PROPN
ap-1860	166	16	)	)	PUNCT
ap-1860	166	17	its	its	PRON
ap-1860	166	18	return	return	NOUN
ap-1860	166	19	time	time	NOUN
ap-1860	166	20	.	.	PUNCT
ap-1860	167	1	let	let	VERB
ap-1860	167	2	h	h	PRON
ap-1860	167	3	⊂	⊂	PRON
ap-1860	167	4	i	i	PRON
ap-1860	167	5	be	be	VERB
ap-1860	167	6	the	the	DET
ap-1860	167	7	maximal	maximal	ADJ
ap-1860	167	8	interval	interval	NOUN
ap-1860	167	9	containing	contain	VERB
ap-1860	167	10	x	x	PUNCT
ap-1860	167	11	such	such	ADJ
ap-1860	167	12	that	that	PRON
ap-1860	167	13	for	for	ADP
ap-1860	167	14	every	every	DET
ap-1860	167	15	x′	x′	PROPN
ap-1860	167	16	∈	∈	PROPN
ap-1860	167	17	h	h	NOUN
ap-1860	167	18	one	one	NOUN
ap-1860	167	19	has	have	VERB
ap-1860	167	20	r(x	r(x	PROPN
ap-1860	167	21	)	)	PUNCT
ap-1860	168	1	=	=	SYM
ap-1860	168	2	r(x′	r(x′	PROPN
ap-1860	168	3	)	)	PUNCT
ap-1860	168	4	.	.	PUNCT
ap-1860	169	1	for	for	ADP
ap-1860	169	2	h	h	NOUN
ap-1860	169	3	,	,	PUNCT
ap-1860	169	4	it	it	PRON
ap-1860	169	5	holds	hold	VERB
ap-1860	169	6	that	that	SCONJ
ap-1860	169	7	(	(	PUNCT
ap-1860	169	8	1	1	NUM
ap-1860	169	9	.	.	PUNCT
ap-1860	169	10	)	)	PUNCT
ap-1860	170	1	t	t	PROPN
ap-1860	170	2	i(h	i(h	NOUN
ap-1860	170	3	)	)	PUNCT
ap-1860	171	1	⊂	⊂	PROPN
ap-1860	172	1	[	[	X
ap-1860	172	2	0	0	NUM
ap-1860	172	3	,	,	PUNCT
ap-1860	172	4	α	α	NOUN
ap-1860	172	5	)	)	PUNCT
ap-1860	172	6	or	or	CCONJ
ap-1860	172	7	t	t	PROPN
ap-1860	172	8	i(h	i(h	NOUN
ap-1860	172	9	)	)	PUNCT
ap-1860	172	10	⊂	⊂	PROPN
ap-1860	173	1	[	[	X
ap-1860	173	2	α	α	X
ap-1860	173	3	,	,	PUNCT
ap-1860	173	4	1	1	NUM
ap-1860	173	5	)	)	PUNCT
ap-1860	173	6	for	for	ADP
ap-1860	173	7	i	i	PROPN
ap-1860	173	8	=	=	SYM
ap-1860	173	9	0	0	NUM
ap-1860	173	10	,	,	PUNCT
ap-1860	173	11	1	1	NUM
ap-1860	173	12	,	,	PUNCT
ap-1860	173	13	.	.	PUNCT
ap-1860	173	14	.	.	PUNCT
ap-1860	174	1	.	.	PUNCT
ap-1860	175	1	,	,	PUNCT
ap-1860	175	2	r	r	NOUN
ap-1860	175	3	−	−	PROPN
ap-1860	175	4	1	1	NUM
ap-1860	175	5	;	;	PUNCT
ap-1860	175	6	(	(	PUNCT
ap-1860	175	7	2	2	NUM
ap-1860	175	8	.	.	PUNCT
ap-1860	175	9	)	)	PUNCT
ap-1860	176	1	t	t	PROPN
ap-1860	176	2	i(h	i(h	NOUN
ap-1860	176	3	)	)	PUNCT
ap-1860	176	4	∩	∩	NOUN
ap-1860	176	5	i	i	NOUN
ap-1860	176	6	=	=	NOUN
ap-1860	176	7	∅	∅	NOUN
ap-1860	176	8	for	for	ADP
ap-1860	176	9	i	i	PRON
ap-1860	176	10	=	=	NOUN
ap-1860	176	11	1	1	NUM
ap-1860	176	12	,	,	PUNCT
ap-1860	176	13	.	.	PUNCT
ap-1860	176	14	.	.	PUNCT
ap-1860	177	1	.	.	PUNCT
ap-1860	178	1	,	,	PUNCT
ap-1860	178	2	r	r	NOUN
ap-1860	178	3	−	−	PROPN
ap-1860	178	4	1	1	NUM
ap-1860	178	5	;	;	PUNCT
ap-1860	178	6	(	(	PUNCT
ap-1860	178	7	3	3	NUM
ap-1860	178	8	.	.	PUNCT
ap-1860	178	9	)	)	PUNCT
ap-1860	179	1	t	t	PROPN
ap-1860	179	2	r(h	r(h	PROPN
ap-1860	179	3	)	)	PUNCT
ap-1860	180	1	⊂	⊂	PROPN
ap-1860	180	2	i.	i.	PROPN
ap-1860	180	3	the	the	DET
ap-1860	180	4	theorem	theorem	NOUN
ap-1860	180	5	will	will	AUX
ap-1860	180	6	be	be	AUX
ap-1860	180	7	established	establish	VERB
ap-1860	180	8	by	by	ADP
ap-1860	180	9	showing	show	VERB
ap-1860	180	10	that	that	SCONJ
ap-1860	180	11	there	there	PRON
ap-1860	180	12	are	be	VERB
ap-1860	180	13	only	only	ADV
ap-1860	180	14	four	four	NUM
ap-1860	180	15	candidates	candidate	NOUN
ap-1860	180	16	for	for	ADP
ap-1860	180	17	the	the	DET
ap-1860	180	18	left	left	ADJ
ap-1860	180	19	end	end	NOUN
ap-1860	180	20	-	-	PUNCT
ap-1860	180	21	point	point	NOUN
ap-1860	180	22	of	of	ADP
ap-1860	180	23	the	the	DET
ap-1860	180	24	interval	interval	NOUN
ap-1860	180	25	h	h	NOUN
ap-1860	181	1	=	=	PUNCT
ap-1860	182	1	[	[	X
ap-1860	182	2	c̃	c̃	PROPN
ap-1860	182	3	,	,	PUNCT
ap-1860	182	4	d̃	d̃	PROPN
ap-1860	182	5	)	)	PUNCT
ap-1860	182	6	.	.	PUNCT
ap-1860	183	1	obviously	obviously	ADV
ap-1860	183	2	,	,	PUNCT
ap-1860	183	3	one	one	NUM
ap-1860	183	4	of	of	ADP
ap-1860	183	5	them	they	PRON
ap-1860	183	6	is	be	AUX
ap-1860	183	7	c̃	c̃	PROPN
ap-1860	183	8	=	=	SYM
ap-1860	183	9	c.	c.	PROPN
ap-1860	183	10	if	if	SCONJ
ap-1860	183	11	it	it	PRON
ap-1860	183	12	is	be	AUX
ap-1860	183	13	not	not	PART
ap-1860	183	14	the	the	DET
ap-1860	183	15	case	case	NOUN
ap-1860	183	16	,	,	PUNCT
ap-1860	183	17	maximality	maximality	NOUN
ap-1860	183	18	of	of	ADP
ap-1860	183	19	h	h	NOUN
ap-1860	183	20	and	and	CCONJ
ap-1860	183	21	properties	property	NOUN
ap-1860	183	22	(	(	PUNCT
ap-1860	183	23	1	1	NUM
ap-1860	183	24	.	.	NUM
ap-1860	183	25	)	)	PUNCT
ap-1860	183	26	,	,	PUNCT
ap-1860	183	27	(	(	PUNCT
ap-1860	183	28	2	2	NUM
ap-1860	183	29	.	.	NUM
ap-1860	183	30	)	)	PUNCT
ap-1860	183	31	,	,	PUNCT
ap-1860	183	32	and	and	CCONJ
ap-1860	183	33	(	(	PUNCT
ap-1860	183	34	3	3	NUM
ap-1860	183	35	.	.	PUNCT
ap-1860	183	36	)	)	PUNCT
ap-1860	183	37	imply	imply	VERB
ap-1860	183	38	that	that	SCONJ
ap-1860	183	39	c	c	VERB
ap-1860	183	40	<	<	X
ap-1860	183	41	c̃	c̃	PROPN
ap-1860	183	42	<	<	X
ap-1860	183	43	d̃	d̃	PROPN
ap-1860	183	44	≤	≤	PROPN
ap-1860	183	45	d	d	NOUN
ap-1860	183	46	and	and	CCONJ
ap-1860	183	47	there	there	PRON
ap-1860	183	48	exists	exist	VERB
ap-1860	183	49	(	(	PUNCT
ap-1860	183	50	a	a	X
ap-1860	183	51	)	)	PUNCT
ap-1860	183	52	l̃	l̃	PROPN
ap-1860	183	53	,	,	PUNCT
ap-1860	183	54	r	r	NOUN
ap-1860	183	55	−	−	PROPN
ap-1860	183	56	1	1	NUM
ap-1860	183	57	≥	≥	NOUN
ap-1860	184	1	l̃	l̃	PROPN
ap-1860	184	2	≥	≥	NUM
ap-1860	184	3	1	1	NUM
ap-1860	184	4	such	such	ADJ
ap-1860	184	5	that	that	SCONJ
ap-1860	184	6	t	t	NOUN
ap-1860	184	7	l̃(c̃	l̃(c̃	NOUN
ap-1860	184	8	)	)	PUNCT
ap-1860	184	9	=	=	SYM
ap-1860	185	1	d	d	NOUN
ap-1860	185	2	;	;	PUNCT
ap-1860	185	3	or	or	CCONJ
ap-1860	185	4	(	(	PUNCT
ap-1860	185	5	b	b	NOUN
ap-1860	185	6	)	)	PUNCT
ap-1860	185	7	ñ	ñ	PROPN
ap-1860	185	8	,	,	PUNCT
ap-1860	185	9	r	r	NOUN
ap-1860	185	10	−	−	PROPN
ap-1860	185	11	1	1	NUM
ap-1860	185	12	≥	≥	NOUN
ap-1860	185	13	ñ	ñ	VERB
ap-1860	185	14	≥	≥	NOUN
ap-1860	185	15	0	0	NUM
ap-1860	186	1	such	such	ADJ
ap-1860	186	2	that	that	SCONJ
ap-1860	186	3	t	t	PROPN
ap-1860	186	4	ñ(c̃	ñ(c̃	NUM
ap-1860	186	5	)	)	PUNCT
ap-1860	186	6	=	=	SYM
ap-1860	186	7	α	α	NOUN
ap-1860	186	8	;	;	PUNCT
ap-1860	186	9	or	or	CCONJ
ap-1860	186	10	(	(	PUNCT
ap-1860	186	11	c	c	X
ap-1860	186	12	)	)	PUNCT
ap-1860	186	13	m̃	m̃	PROPN
ap-1860	186	14	,	,	PUNCT
ap-1860	186	15	r	r	NOUN
ap-1860	186	16	−	−	PROPN
ap-1860	186	17	1	1	NUM
ap-1860	186	18	≥	≥	NOUN
ap-1860	186	19	m̃	m̃	PROPN
ap-1860	186	20	≥	≥	NUM
ap-1860	186	21	1	1	NUM
ap-1860	186	22	such	such	ADJ
ap-1860	186	23	that	that	SCONJ
ap-1860	186	24	t	t	NOUN
ap-1860	186	25	m̃(c̃	m̃(c̃	X
ap-1860	186	26	)	)	PUNCT
ap-1860	187	1	=	=	SYM
ap-1860	187	2	c.	c.	PROPN
ap-1860	187	3	suppose	suppose	VERB
ap-1860	187	4	that	that	SCONJ
ap-1860	187	5	possibility	possibility	NOUN
ap-1860	187	6	(	(	PUNCT
ap-1860	187	7	a	a	PRON
ap-1860	187	8	)	)	PUNCT
ap-1860	187	9	happened	happen	VERB
ap-1860	187	10	.	.	PUNCT
ap-1860	188	1	let	let	VERB
ap-1860	188	2	us	we	PRON
ap-1860	188	3	mention	mention	VERB
ap-1860	188	4	that	that	SCONJ
ap-1860	188	5	it	it	PRON
ap-1860	188	6	is	be	AUX
ap-1860	188	7	possible	possible	ADJ
ap-1860	188	8	only	only	ADV
ap-1860	188	9	if	if	SCONJ
ap-1860	188	10	d	d	PROPN
ap-1860	188	11	<	<	X
ap-1860	188	12	1	1	NUM
ap-1860	188	13	.	.	PUNCT
ap-1860	188	14	denote	denote	NOUN
ap-1860	189	1	l	l	NOUN
ap-1860	189	2	=	=	SYM
ap-1860	189	3	min	min	PROPN
ap-1860	189	4	{	{	PUNCT
ap-1860	189	5	k	k	PROPN
ap-1860	189	6	∈	∈	PROPN
ap-1860	189	7	z	z	PROPN
ap-1860	189	8	,	,	PUNCT
ap-1860	189	9	k	k	PROPN
ap-1860	189	10	≥	≥	NUM
ap-1860	189	11	1	1	NUM
ap-1860	189	12	:	:	PUNCT
ap-1860	189	13	t−k(d	t−k(d	ADJ
ap-1860	189	14	)	)	PUNCT
ap-1860	189	15	∈	∈	PROPN
ap-1860	189	16	i	i	PRON
ap-1860	189	17	}	}	PUNCT
ap-1860	189	18	.	.	PUNCT
ap-1860	190	1	(	(	PUNCT
ap-1860	190	2	6	6	NUM
ap-1860	190	3	)	)	PUNCT
ap-1860	190	4	since	since	SCONJ
ap-1860	190	5	t−l̃(d	t−l̃(d	NUM
ap-1860	190	6	)	)	PUNCT
ap-1860	190	7	=	=	PUNCT
ap-1860	191	1	c̃	c̃	PROPN
ap-1860	191	2	∈	∈	PROPN
ap-1860	191	3	h	h	NOUN
ap-1860	192	1	⊂	⊂	PROPN
ap-1860	192	2	i	i	PRON
ap-1860	192	3	,	,	PUNCT
ap-1860	192	4	we	we	PRON
ap-1860	192	5	have	have	VERB
ap-1860	192	6	by	by	ADP
ap-1860	192	7	definition	definition	NOUN
ap-1860	192	8	of	of	ADP
ap-1860	192	9	l	l	NOUN
ap-1860	193	1	that	that	PRON
ap-1860	193	2	l̃	l̃	PROPN
ap-1860	193	3	≥	≥	NOUN
ap-1860	193	4	l.	l.	NOUN
ap-1860	193	5	we	we	PRON
ap-1860	193	6	will	will	AUX
ap-1860	193	7	show	show	VERB
ap-1860	193	8	by	by	ADP
ap-1860	193	9	contradiction	contradiction	NOUN
ap-1860	193	10	that	that	PRON
ap-1860	193	11	l̃	l̃	PROPN
ap-1860	193	12	=	=	PUNCT
ap-1860	193	13	l.	l.	PROPN
ap-1860	193	14	if	if	SCONJ
ap-1860	193	15	l̃	l̃	PROPN
ap-1860	193	16	>	>	X
ap-1860	193	17	l	l	PROPN
ap-1860	193	18	,	,	PUNCT
ap-1860	193	19	then	then	ADV
ap-1860	193	20	t	t	PROPN
ap-1860	193	21	l̃−l(c̃	l̃−l(c̃	PROPN
ap-1860	193	22	)	)	PUNCT
ap-1860	193	23	=	=	SYM
ap-1860	194	1	t−l	t−l	PROPN
ap-1860	194	2	(	(	PUNCT
ap-1860	194	3	t	t	NOUN
ap-1860	194	4	l̃(c̃	l̃(c̃	NUM
ap-1860	194	5	)	)	PUNCT
ap-1860	194	6	)	)	PUNCT
ap-1860	195	1	=	=	SYM
ap-1860	195	2	t−l(d	t−l(d	ADJ
ap-1860	195	3	)	)	PUNCT
ap-1860	195	4	∈	∈	PROPN
ap-1860	195	5	i	i	PRON
ap-1860	195	6	,	,	PUNCT
ap-1860	195	7	and	and	CCONJ
ap-1860	195	8	by	by	ADP
ap-1860	195	9	definition	definition	NOUN
ap-1860	195	10	of	of	ADP
ap-1860	195	11	return	return	NOUN
ap-1860	195	12	time	time	NOUN
ap-1860	195	13	r	r	NOUN
ap-1860	195	14	=	=	SYM
ap-1860	195	15	r(c̃	r(c̃	PROPN
ap-1860	195	16	)	)	PUNCT
ap-1860	195	17	≤	≤	PUNCT
ap-1860	196	1	l̃	l̃	PROPN
ap-1860	196	2	−	−	PROPN
ap-1860	196	3	l.	l.	NOUN
ap-1860	196	4	this	this	PRON
ap-1860	196	5	contradicts	contradict	VERB
ap-1860	196	6	the	the	DET
ap-1860	196	7	fact	fact	NOUN
ap-1860	196	8	that	that	SCONJ
ap-1860	196	9	l̃	l̃	PROPN
ap-1860	196	10	≤	≤	PROPN
ap-1860	196	11	r	r	NOUN
ap-1860	196	12	−	−	NOUN
ap-1860	196	13	1	1	NUM
ap-1860	196	14	.	.	PUNCT
ap-1860	196	15	similar	similar	ADJ
ap-1860	196	16	discussion	discussion	NOUN
ap-1860	196	17	for	for	ADP
ap-1860	196	18	possibilities	possibility	NOUN
ap-1860	196	19	(	(	PUNCT
ap-1860	196	20	b	b	NOUN
ap-1860	196	21	)	)	PUNCT
ap-1860	196	22	and	and	CCONJ
ap-1860	196	23	(	(	PUNCT
ap-1860	196	24	c	c	X
ap-1860	196	25	)	)	PUNCT
ap-1860	196	26	shows	show	VERB
ap-1860	196	27	that	that	SCONJ
ap-1860	196	28	the	the	DET
ap-1860	196	29	left	left	ADJ
ap-1860	196	30	end	end	NOUN
ap-1860	196	31	-	-	PUNCT
ap-1860	196	32	point	point	NOUN
ap-1860	196	33	of	of	ADP
ap-1860	196	34	the	the	DET
ap-1860	196	35	interval	interval	NOUN
ap-1860	196	36	h	h	NOUN
ap-1860	196	37	is	be	AUX
ap-1860	196	38	equal	equal	ADJ
ap-1860	196	39	either	either	ADV
ap-1860	196	40	to	to	PART
ap-1860	196	41	t−l(d	t−l(d	VERB
ap-1860	196	42	)	)	PUNCT
ap-1860	196	43	where	where	SCONJ
ap-1860	196	44	l	l	NOUN
ap-1860	196	45	is	be	AUX
ap-1860	196	46	defined	define	VERB
ap-1860	196	47	by	by	ADP
ap-1860	196	48	(	(	PUNCT
ap-1860	196	49	6	6	NUM
ap-1860	196	50	)	)	PUNCT
ap-1860	196	51	,	,	PUNCT
ap-1860	196	52	or	or	CCONJ
ap-1860	196	53	t−n(α	t−n(α	PROPN
ap-1860	196	54	)	)	PUNCT
ap-1860	196	55	,	,	PUNCT
ap-1860	196	56	where	where	SCONJ
ap-1860	196	57	n	n	NOUN
ap-1860	196	58	=	=	SYM
ap-1860	196	59	min	min	PROPN
ap-1860	196	60	{	{	PUNCT
ap-1860	196	61	k	k	PROPN
ap-1860	196	62	∈	∈	PROPN
ap-1860	196	63	z	z	PROPN
ap-1860	196	64	,	,	PUNCT
ap-1860	196	65	k	k	X
ap-1860	196	66	≥	≥	X
ap-1860	196	67	0	0	NUM
ap-1860	196	68	:	:	PUNCT
ap-1860	196	69	t−k(α	t−k(α	NOUN
ap-1860	196	70	)	)	PUNCT
ap-1860	196	71	∈	∈	PROPN
ap-1860	197	1	i	i	PRON
ap-1860	197	2	}	}	PUNCT
ap-1860	197	3	,	,	PUNCT
ap-1860	197	4	(	(	PUNCT
ap-1860	197	5	7	7	NUM
ap-1860	197	6	)	)	PUNCT
ap-1860	197	7	or	or	CCONJ
ap-1860	197	8	t−m(c	t−m(c	NOUN
ap-1860	197	9	)	)	PUNCT
ap-1860	197	10	,	,	PUNCT
ap-1860	197	11	where	where	SCONJ
ap-1860	197	12	m	m	VERB
ap-1860	197	13	=	=	SYM
ap-1860	197	14	min	min	PROPN
ap-1860	197	15	{	{	PUNCT
ap-1860	197	16	k	k	PROPN
ap-1860	197	17	∈	∈	PROPN
ap-1860	197	18	z	z	PROPN
ap-1860	197	19	,	,	PUNCT
ap-1860	197	20	k	k	PROPN
ap-1860	197	21	≥	≥	NUM
ap-1860	197	22	1	1	NUM
ap-1860	197	23	:	:	PUNCT
ap-1860	197	24	t−k(c	t−k(c	X
ap-1860	197	25	)	)	PUNCT
ap-1860	197	26	∈	∈	PROPN
ap-1860	197	27	i	i	PRON
ap-1860	197	28	}	}	PUNCT
ap-1860	197	29	.	.	PUNCT
ap-1860	198	1	(	(	PUNCT
ap-1860	198	2	8)	8)	NUM
ap-1860	198	3	this	this	PRON
ap-1860	198	4	means	mean	VERB
ap-1860	198	5	that	that	SCONJ
ap-1860	198	6	i	i	PRON
ap-1860	198	7	is	be	AUX
ap-1860	198	8	divided	divide	VERB
ap-1860	198	9	by	by	ADP
ap-1860	198	10	the	the	DET
ap-1860	198	11	three	three	NUM
ap-1860	198	12	(	(	PUNCT
ap-1860	198	13	not	not	PART
ap-1860	198	14	necessarily	necessarily	ADV
ap-1860	198	15	distinct	distinct	ADJ
ap-1860	198	16	)	)	PUNCT
ap-1860	198	17	points	point	NOUN
ap-1860	198	18	t−l(d	t−l(d	NUM
ap-1860	198	19	)	)	PUNCT
ap-1860	198	20	,	,	PUNCT
ap-1860	198	21	t−n(α	t−n(α	PROPN
ap-1860	198	22	)	)	PUNCT
ap-1860	198	23	,	,	PUNCT
ap-1860	198	24	t−m(c	t−m(c	NOUN
ap-1860	198	25	)	)	PUNCT
ap-1860	198	26	into	into	ADP
ap-1860	198	27	at	at	ADP
ap-1860	198	28	most	most	ADV
ap-1860	198	29	4	4	NUM
ap-1860	198	30	subintervals	subinterval	NOUN
ap-1860	198	31	h	h	NOUN
ap-1860	198	32	on	on	ADP
ap-1860	198	33	which	which	PRON
ap-1860	198	34	the	the	DET
ap-1860	198	35	i	i	NOUN
ap-1860	198	36	-	-	PUNCT
ap-1860	198	37	itinerary	itinerary	NOUN
ap-1860	198	38	is	be	AUX
ap-1860	198	39	constant	constant	ADJ
ap-1860	198	40	.	.	PUNCT
ap-1860	199	1	proposition	proposition	NOUN
ap-1860	199	2	2.6	2.6	NUM
ap-1860	199	3	.	.	PUNCT
ap-1860	200	1	let	let	VERB
ap-1860	200	2	iti	iti	PROPN
ap-1860	200	3	be	be	AUX
ap-1860	200	4	the	the	DET
ap-1860	200	5	set	set	NOUN
ap-1860	200	6	of	of	ADP
ap-1860	200	7	i	i	PROPN
ap-1860	200	8	-	-	PUNCT
ap-1860	200	9	itineraries	itinerary	NOUN
ap-1860	200	10	for	for	ADP
ap-1860	200	11	the	the	DET
ap-1860	200	12	interval	interval	NOUN
ap-1860	201	1	i	i	PRON
ap-1860	201	2	=	=	PUNCT
ap-1860	202	1	[	[	X
ap-1860	202	2	c	c	X
ap-1860	202	3	,	,	PUNCT
ap-1860	202	4	d	d	NOUN
ap-1860	202	5	)	)	PUNCT
ap-1860	202	6	⊂	⊂	PROPN
ap-1860	203	1	[	[	X
ap-1860	203	2	0	0	NUM
ap-1860	203	3	,	,	PUNCT
ap-1860	203	4	1	1	NUM
ap-1860	203	5	)	)	PUNCT
ap-1860	203	6	under	under	ADP
ap-1860	203	7	an	an	DET
ap-1860	203	8	exchange	exchange	NOUN
ap-1860	203	9	of	of	ADP
ap-1860	203	10	two	two	NUM
ap-1860	203	11	intervals	interval	NOUN
ap-1860	203	12	with	with	ADP
ap-1860	203	13	irrational	irrational	ADJ
ap-1860	203	14	slope	slope	NOUN
ap-1860	203	15	α	α	NOUN
ap-1860	203	16	.	.	PUNCT
ap-1860	204	1	there	there	PRON
ap-1860	204	2	exist	exist	VERB
ap-1860	204	3	neighbourhoods	neighbourhood	NOUN
ap-1860	204	4	hc	hc	PROPN
ap-1860	204	5	and	and	CCONJ
ap-1860	204	6	hd	hd	PROPN
ap-1860	204	7	of	of	ADP
ap-1860	204	8	c	c	PROPN
ap-1860	204	9	,	,	PUNCT
ap-1860	204	10	d	d	NOUN
ap-1860	204	11	,	,	PUNCT
ap-1860	204	12	respectively	respectively	ADV
ap-1860	204	13	,	,	PUNCT
ap-1860	204	14	such	such	ADJ
ap-1860	204	15	that	that	PRON
ap-1860	204	16	for	for	SCONJ
ap-1860	204	17	every	every	DET
ap-1860	204	18	c̃	c̃	PROPN
ap-1860	204	19	∈	∈	PROPN
ap-1860	204	20	hc	hc	PROPN
ap-1860	204	21	and	and	CCONJ
ap-1860	204	22	d̃	d̃	PROPN
ap-1860	204	23	∈	∈	PROPN
ap-1860	204	24	hd	hd	PROPN
ap-1860	204	25	,	,	PUNCT
ap-1860	204	26	0	0	NUM
ap-1860	204	27	≤	≤	NUM
ap-1860	205	1	c̃	c̃	PROPN
ap-1860	205	2	<	<	X
ap-1860	205	3	d̃	d̃	PROPN
ap-1860	205	4	≤	≤	NUM
ap-1860	205	5	1	1	NUM
ap-1860	205	6	one	one	NUM
ap-1860	205	7	has	have	VERB
ap-1860	205	8	it	it	PRON
ap-1860	205	9	ĩ	ĩ	PROPN
ap-1860	205	10	⊇	⊇	NOUN
ap-1860	205	11	iti	iti	PROPN
ap-1860	205	12	,	,	PUNCT
ap-1860	205	13	where	where	SCONJ
ap-1860	205	14	ĩ	ĩ	PROPN
ap-1860	205	15	=	=	PUNCT
ap-1860	206	1	[	[	X
ap-1860	206	2	c̃	c̃	PROPN
ap-1860	206	3	,	,	PUNCT
ap-1860	206	4	d̃	d̃	PROPN
ap-1860	206	5	)	)	PUNCT
ap-1860	206	6	.	.	PUNCT
ap-1860	207	1	proof	proof	NOUN
ap-1860	207	2	.	.	PUNCT
ap-1860	208	1	let	let	VERB
ap-1860	208	2	iti	iti	PROPN
ap-1860	208	3	=	=	PUNCT
ap-1860	208	4	{	{	PUNCT
ap-1860	208	5	r1	r1	PROPN
ap-1860	208	6	,	,	PUNCT
ap-1860	208	7	.	.	PUNCT
ap-1860	208	8	.	.	PUNCT
ap-1860	209	1	.	.	PUNCT
ap-1860	210	1	,	,	PUNCT
ap-1860	210	2	rp	rp	NOUN
ap-1860	210	3	}	}	PUNCT
ap-1860	210	4	.	.	PUNCT
ap-1860	211	1	proposition	proposition	NOUN
ap-1860	211	2	2.5	2.5	NUM
ap-1860	211	3	implies	imply	VERB
ap-1860	211	4	that	that	SCONJ
ap-1860	211	5	p	p	PROPN
ap-1860	211	6	≤	≤	NOUN
ap-1860	211	7	4	4	NUM
ap-1860	211	8	and	and	CCONJ
ap-1860	211	9	for	for	ADP
ap-1860	211	10	every	every	DET
ap-1860	211	11	1	1	NUM
ap-1860	211	12	≤	≤	NUM
ap-1860	212	1	i	i	NOUN
ap-1860	212	2	≤	≤	NOUN
ap-1860	212	3	p	p	ADP
ap-1860	212	4	the	the	DET
ap-1860	212	5	elements	element	NOUN
ap-1860	212	6	x	x	PUNCT
ap-1860	212	7	such	such	ADJ
ap-1860	212	8	that	that	SCONJ
ap-1860	212	9	r(x	r(x	NOUN
ap-1860	212	10	)	)	PUNCT
ap-1860	212	11	=	=	SYM
ap-1860	212	12	ri	ri	NOUN
ap-1860	212	13	form	form	VERB
ap-1860	212	14	an	an	DET
ap-1860	212	15	interval	interval	NOUN
ap-1860	212	16	,	,	PUNCT
ap-1860	212	17	say	say	VERB
ap-1860	212	18	ii	ii	NOUN
ap-1860	212	19	.	.	PUNCT
ap-1860	212	20	choose	choose	VERB
ap-1860	212	21	xi	xi	NUM
ap-1860	212	22	∈	∈	PROPN
ap-1860	212	23	ii	ii	PROPN
ap-1860	212	24	such	such	ADJ
ap-1860	212	25	that	that	PRON
ap-1860	212	26	for	for	ADP
ap-1860	212	27	q	q	NOUN
ap-1860	212	28	with	with	ADP
ap-1860	212	29	0	0	NUM
ap-1860	212	30	≤	≤	NUM
ap-1860	212	31	q	q	PROPN
ap-1860	212	32	≤	≤	NUM
ap-1860	212	33	r(xi	r(xi	NUM
ap-1860	212	34	)	)	PUNCT
ap-1860	212	35	−	−	PROPN
ap-1860	213	1	1	1	NUM
ap-1860	213	2	=	=	SYM
ap-1860	213	3	|ri|	|ri|	NOUN
ap-1860	213	4	−	−	NOUN
ap-1860	213	5	1	1	NUM
ap-1860	213	6	one	one	NOUN
ap-1860	213	7	has	have	VERB
ap-1860	213	8	t	t	PROPN
ap-1860	213	9	q(xi	q(xi	PROPN
ap-1860	213	10	)	)	PUNCT
ap-1860	213	11	/∈	/∈	PUNCT
ap-1860	214	1	{	{	PUNCT
ap-1860	214	2	c	c	X
ap-1860	214	3	,	,	PUNCT
ap-1860	214	4	d	d	NOUN
ap-1860	214	5	,	,	PUNCT
ap-1860	214	6	α	α	NOUN
ap-1860	214	7	}	}	PUNCT
ap-1860	214	8	,	,	PUNCT
ap-1860	214	9	(	(	PUNCT
ap-1860	214	10	it	it	PRON
ap-1860	214	11	suffices	suffice	VERB
ap-1860	214	12	to	to	PART
ap-1860	214	13	choose	choose	VERB
ap-1860	214	14	xi	xi	PROPN
ap-1860	214	15	/∈	/∈	PUNCT
ap-1860	215	1	z[c	z[c	ADJ
ap-1860	215	2	,	,	PUNCT
ap-1860	215	3	d	d	PROPN
ap-1860	215	4	,	,	PUNCT
ap-1860	215	5	α	α	NOUN
ap-1860	215	6	]	]	X
ap-1860	215	7	)	)	PUNCT
ap-1860	215	8	.	.	PUNCT
ap-1860	216	1	denote	denote	VERB
ap-1860	216	2	m	m	NOUN
ap-1860	216	3	=	=	PUNCT
ap-1860	216	4	{	{	PUNCT
ap-1860	216	5	c	c	NOUN
ap-1860	216	6	,	,	PUNCT
ap-1860	216	7	d	d	PROPN
ap-1860	216	8	,	,	PUNCT
ap-1860	216	9	α	α	NOUN
ap-1860	216	10	}	}	PUNCT
ap-1860	216	11	and	and	CCONJ
ap-1860	216	12	n	n	NOUN
ap-1860	216	13	=	=	SYM
ap-1860	216	14	{	{	PUNCT
ap-1860	216	15	t	t	PROPN
ap-1860	216	16	q(xi	q(xi	PROPN
ap-1860	216	17	)	)	PUNCT
ap-1860	216	18	:	:	PUNCT
ap-1860	217	1	i	i	NOUN
ap-1860	217	2	=	=	NOUN
ap-1860	217	3	1	1	NUM
ap-1860	217	4	,	,	PUNCT
ap-1860	217	5	.	.	PUNCT
ap-1860	217	6	.	.	PUNCT
ap-1860	217	7	.	.	PUNCT
ap-1860	218	1	,	,	PUNCT
ap-1860	218	2	p	p	X
ap-1860	218	3	,	,	PUNCT
ap-1860	218	4	0	0	NUM
ap-1860	218	5	≤	≤	NUM
ap-1860	218	6	q	q	ADJ
ap-1860	218	7	≤	≤	NUM
ap-1860	218	8	r(xi)−	r(xi)−	NOUN
ap-1860	218	9	1	1	NUM
ap-1860	218	10	}	}	PUNCT
ap-1860	218	11	.	.	PUNCT
ap-1860	219	1	put	put	VERB
ap-1860	219	2	ε	ε	PROPN
ap-1860	220	1	:	:	PUNCT
ap-1860	220	2	=	=	SYM
ap-1860	220	3	min	min	PROPN
ap-1860	220	4	{	{	PUNCT
ap-1860	220	5	|a−	|a−	NOUN
ap-1860	220	6	b|	b|	PROPN
ap-1860	220	7	:	:	PUNCT
ap-1860	220	8	a	a	DET
ap-1860	220	9	∈m	∈m	NOUN
ap-1860	220	10	,	,	PUNCT
ap-1860	220	11	b	b	X
ap-1860	220	12	∈	∈	PROPN
ap-1860	220	13	n	n	X
ap-1860	220	14	}	}	PUNCT
ap-1860	220	15	.	.	PUNCT
ap-1860	221	1	446	446	NUM
ap-1860	221	2	vol	vol	NOUN
ap-1860	221	3	.	.	PUNCT
ap-1860	222	1	53	53	NUM
ap-1860	222	2	no	no	NOUN
ap-1860	222	3	.	.	PUNCT
ap-1860	223	1	5/2013	5/2013	NUM
ap-1860	223	2	itineraries	itinerary	NOUN
ap-1860	223	3	induced	induce	VERB
ap-1860	223	4	by	by	ADP
ap-1860	223	5	exchange	exchange	NOUN
ap-1860	223	6	of	of	ADP
ap-1860	223	7	two	two	NUM
ap-1860	223	8	intervals	interval	NOUN
ap-1860	223	9	then	then	ADV
ap-1860	223	10	for	for	ADP
ap-1860	223	11	every	every	DET
ap-1860	223	12	c̃	c̃	PROPN
ap-1860	223	13	∈	∈	PROPN
ap-1860	223	14	(	(	PUNCT
ap-1860	223	15	c−ε	c−ε	NOUN
ap-1860	223	16	,	,	PUNCT
ap-1860	223	17	c+ε	c+ε	NUM
ap-1860	223	18	)	)	PUNCT
ap-1860	223	19	and	and	CCONJ
ap-1860	223	20	d̃	d̃	PROPN
ap-1860	223	21	∈	∈	PROPN
ap-1860	223	22	(	(	PUNCT
ap-1860	223	23	d−ε	d−ε	PROPN
ap-1860	223	24	,	,	PUNCT
ap-1860	223	25	d+ε	d+ε	PROPN
ap-1860	223	26	)	)	PUNCT
ap-1860	223	27	,	,	PUNCT
ap-1860	223	28	the	the	DET
ap-1860	223	29	i	i	NOUN
ap-1860	223	30	-	-	PUNCT
ap-1860	223	31	itineraries	itinerary	NOUN
ap-1860	223	32	r(x1	r(x1	NOUN
ap-1860	223	33	)	)	PUNCT
ap-1860	223	34	,	,	PUNCT
ap-1860	223	35	.	.	PUNCT
ap-1860	223	36	.	.	PUNCT
ap-1860	223	37	.	.	PUNCT
ap-1860	224	1	,	,	PUNCT
ap-1860	224	2	r(xp	r(xp	NOUN
ap-1860	224	3	)	)	PUNCT
ap-1860	224	4	are	be	AUX
ap-1860	224	5	also	also	ADV
ap-1860	224	6	ĩ-itineraries	ĩ-itinerarie	NOUN
ap-1860	224	7	,	,	PUNCT
ap-1860	224	8	where	where	SCONJ
ap-1860	224	9	ĩ	ĩ	PROPN
ap-1860	224	10	=	=	PUNCT
ap-1860	224	11	[	[	X
ap-1860	224	12	c̃	c̃	PROPN
ap-1860	224	13	,	,	PUNCT
ap-1860	224	14	d̃	d̃	PROPN
ap-1860	224	15	)	)	PUNCT
ap-1860	224	16	.	.	PUNCT
ap-1860	225	1	proof	proof	NOUN
ap-1860	225	2	of	of	ADP
ap-1860	225	3	theorem	theorem	ADJ
ap-1860	225	4	1.1	1.1	NUM
ap-1860	225	5	.	.	PUNCT
ap-1860	226	1	if	if	SCONJ
ap-1860	226	2	i	i	PRON
ap-1860	226	3	=	=	PUNCT
ap-1860	227	1	[	[	X
ap-1860	227	2	c	c	X
ap-1860	227	3	,	,	PUNCT
ap-1860	227	4	d	d	NOUN
ap-1860	227	5	)	)	PUNCT
ap-1860	227	6	where	where	SCONJ
ap-1860	227	7	c	c	NOUN
ap-1860	227	8	=	=	SYM
ap-1860	227	9	0	0	PROPN
ap-1860	227	10	or	or	CCONJ
ap-1860	227	11	d	d	NOUN
ap-1860	227	12	=	=	SYM
ap-1860	227	13	1	1	NUM
ap-1860	227	14	,	,	PUNCT
ap-1860	227	15	then	then	ADV
ap-1860	227	16	by	by	ADP
ap-1860	227	17	theorem	theorem	NOUN
ap-1860	227	18	4.5	4.5	NUM
ap-1860	227	19	of	of	ADP
ap-1860	227	20	[	[	X
ap-1860	227	21	8	8	NUM
ap-1860	227	22	]	]	PUNCT
ap-1860	227	23	,	,	PUNCT
ap-1860	227	24	the	the	DET
ap-1860	227	25	set	set	PROPN
ap-1860	227	26	iti	iti	PROPN
ap-1860	227	27	of	of	ADP
ap-1860	227	28	iitineraries	iitinerarie	NOUN
ap-1860	227	29	is	be	AUX
ap-1860	227	30	of	of	ADP
ap-1860	227	31	the	the	DET
ap-1860	227	32	form	form	NOUN
ap-1860	227	33	iti	iti	PROPN
ap-1860	227	34	⊂	⊂	PROPN
ap-1860	227	35	{	{	PUNCT
ap-1860	227	36	r	r	NOUN
ap-1860	227	37	,	,	PUNCT
ap-1860	227	38	r′	r′	NUM
ap-1860	227	39	,	,	PUNCT
ap-1860	227	40	rr′	rr′	NOUN
ap-1860	227	41	}	}	PUNCT
ap-1860	227	42	.	.	PUNCT
ap-1860	228	1	without	without	ADP
ap-1860	228	2	loss	loss	NOUN
ap-1860	228	3	of	of	ADP
ap-1860	228	4	generality	generality	NOUN
ap-1860	228	5	,	,	PUNCT
ap-1860	228	6	we	we	PRON
ap-1860	228	7	can	can	AUX
ap-1860	228	8	therefore	therefore	ADV
ap-1860	228	9	assume	assume	VERB
ap-1860	228	10	that	that	SCONJ
ap-1860	228	11	c	c	PROPN
ap-1860	228	12	6=	6=	ADP
ap-1860	228	13	0	0	NUM
ap-1860	228	14	and	and	CCONJ
ap-1860	228	15	d	d	X
ap-1860	228	16	6=	6=	PROPN
ap-1860	228	17	1	1	NUM
ap-1860	228	18	.	.	PUNCT
ap-1860	229	1	if	if	SCONJ
ap-1860	229	2	c	c	X
ap-1860	229	3	,	,	PUNCT
ap-1860	229	4	d	d	NOUN
ap-1860	229	5	,	,	PUNCT
ap-1860	229	6	or	or	CCONJ
ap-1860	229	7	d−	d−	PROPN
ap-1860	229	8	c	c	PROPN
ap-1860	229	9	belongs	belong	VERB
ap-1860	229	10	to	to	ADP
ap-1860	229	11	z[α	z[α	NOUN
ap-1860	229	12	]	]	X
ap-1860	229	13	(	(	PUNCT
ap-1860	229	14	which	which	PRON
ap-1860	229	15	is	be	AUX
ap-1860	229	16	dense	dense	ADJ
ap-1860	229	17	in	in	ADP
ap-1860	229	18	r	r	NOUN
ap-1860	229	19	)	)	PUNCT
ap-1860	229	20	,	,	PUNCT
ap-1860	229	21	we	we	PRON
ap-1860	229	22	can	can	AUX
ap-1860	229	23	always	always	ADV
ap-1860	229	24	use	use	VERB
ap-1860	229	25	proposition	proposition	NOUN
ap-1860	229	26	2.6	2.6	NUM
ap-1860	229	27	to	to	PART
ap-1860	229	28	find	find	VERB
ap-1860	229	29	ĩ	ĩ	PROPN
ap-1860	229	30	=	=	PUNCT
ap-1860	230	1	[	[	X
ap-1860	230	2	c̃	c̃	PROPN
ap-1860	230	3	,	,	PUNCT
ap-1860	230	4	d̃	d̃	PROPN
ap-1860	230	5	)	)	PUNCT
ap-1860	230	6	such	such	ADJ
ap-1860	230	7	that	that	SCONJ
ap-1860	230	8	it	it	PRON
ap-1860	230	9	ĩ	ĩ	VERB
ap-1860	230	10	⊇	⊇	PROPN
ap-1860	230	11	iti	iti	PROPN
ap-1860	230	12	.	.	PUNCT
ap-1860	231	1	therefore	therefore	ADV
ap-1860	231	2	,	,	PUNCT
ap-1860	231	3	without	without	ADP
ap-1860	231	4	loss	loss	NOUN
ap-1860	231	5	of	of	ADP
ap-1860	231	6	generality	generality	NOUN
ap-1860	231	7	we	we	PRON
ap-1860	231	8	assume	assume	VERB
ap-1860	231	9	c	c	NOUN
ap-1860	231	10	,	,	PUNCT
ap-1860	231	11	d	d	NOUN
ap-1860	231	12	,	,	PUNCT
ap-1860	231	13	d−	d−	PROPN
ap-1860	231	14	c	c	PROPN
ap-1860	231	15	/∈	/∈	PUNCT
ap-1860	232	1	z[α	z[α	NUM
ap-1860	232	2	]	]	PUNCT
ap-1860	232	3	.	.	PUNCT
ap-1860	233	1	in	in	ADP
ap-1860	233	2	particular	particular	ADJ
ap-1860	233	3	,	,	PUNCT
ap-1860	233	4	d−	d−	PROPN
ap-1860	233	5	c	c	PROPN
ap-1860	233	6	6=	6=	PROPN
ap-1860	233	7	δk	δk	PROPN
ap-1860	233	8	,	,	PUNCT
ap-1860	233	9	s.	s.	PROPN
ap-1860	233	10	from	from	ADP
ap-1860	233	11	the	the	DET
ap-1860	233	12	proof	proof	NOUN
ap-1860	233	13	of	of	ADP
ap-1860	233	14	proposition	proposition	NOUN
ap-1860	233	15	2.5	2.5	NUM
ap-1860	233	16	,	,	PUNCT
ap-1860	233	17	the	the	DET
ap-1860	233	18	interval	interval	NOUN
ap-1860	233	19	i	i	PRON
ap-1860	233	20	is	be	AUX
ap-1860	233	21	divided	divide	VERB
ap-1860	233	22	into	into	ADP
ap-1860	233	23	at	at	ADP
ap-1860	233	24	most	most	ADJ
ap-1860	233	25	four	four	NUM
ap-1860	233	26	subintervals	subinterval	NOUN
ap-1860	233	27	with	with	ADP
ap-1860	233	28	constant	constant	ADJ
ap-1860	233	29	i	i	PROPN
ap-1860	233	30	-	-	PUNCT
ap-1860	233	31	itinerary	itinerary	NOUN
ap-1860	233	32	by	by	ADP
ap-1860	233	33	points	point	NOUN
ap-1860	233	34	λ	λ	X
ap-1860	233	35	=	=	PUNCT
ap-1860	233	36	t−l(d	t−l(d	NUM
ap-1860	233	37	)	)	PUNCT
ap-1860	233	38	,	,	PUNCT
ap-1860	233	39	l	l	NOUN
ap-1860	233	40	=	=	SYM
ap-1860	233	41	min	min	PROPN
ap-1860	233	42	{	{	PUNCT
ap-1860	233	43	k	k	X
ap-1860	233	44	≥	≥	NUM
ap-1860	233	45	1	1	NUM
ap-1860	233	46	:	:	PUNCT
ap-1860	233	47	t−k(d	t−k(d	ADJ
ap-1860	233	48	)	)	PUNCT
ap-1860	233	49	∈	∈	PROPN
ap-1860	234	1	i	i	PRON
ap-1860	234	2	}	}	PUNCT
ap-1860	234	3	,	,	PUNCT
ap-1860	234	4	µ	µ	X
ap-1860	234	5	=	=	SYM
ap-1860	234	6	t−m(c	t−m(c	NOUN
ap-1860	234	7	)	)	PUNCT
ap-1860	234	8	,	,	PUNCT
ap-1860	234	9	m	m	VERB
ap-1860	234	10	=	=	VERB
ap-1860	234	11	min	min	PROPN
ap-1860	234	12	{	{	PUNCT
ap-1860	234	13	k	k	X
ap-1860	234	14	≥	≥	NUM
ap-1860	234	15	1	1	NUM
ap-1860	234	16	:	:	PUNCT
ap-1860	234	17	t−k(c	t−k(c	X
ap-1860	234	18	)	)	PUNCT
ap-1860	234	19	∈	∈	PROPN
ap-1860	235	1	i	i	PRON
ap-1860	235	2	}	}	PUNCT
ap-1860	235	3	,	,	PUNCT
ap-1860	235	4	ν	ν	X
ap-1860	235	5	=	=	SYM
ap-1860	235	6	t−n(α	t−n(α	ADJ
ap-1860	235	7	)	)	PUNCT
ap-1860	235	8	,	,	PUNCT
ap-1860	235	9	n	n	PROPN
ap-1860	235	10	=	=	SYM
ap-1860	235	11	min	min	PROPN
ap-1860	235	12	{	{	PUNCT
ap-1860	235	13	k	k	X
ap-1860	235	14	≥	≥	NUM
ap-1860	235	15	0	0	NUM
ap-1860	235	16	:	:	PUNCT
ap-1860	235	17	t−k(α	t−k(α	NOUN
ap-1860	235	18	)	)	PUNCT
ap-1860	235	19	∈	∈	PROPN
ap-1860	235	20	i	i	PRON
ap-1860	235	21	}	}	PUNCT
ap-1860	235	22	.	.	PUNCT
ap-1860	236	1	moreover	moreover	ADV
ap-1860	236	2	,	,	PUNCT
ap-1860	236	3	λ	λ	PROPN
ap-1860	236	4	and	and	CCONJ
ap-1860	236	5	µ	µ	DET
ap-1860	236	6	separate	separate	ADJ
ap-1860	236	7	intervals	interval	NOUN
ap-1860	236	8	with	with	ADP
ap-1860	236	9	different	different	ADJ
ap-1860	236	10	return	return	NOUN
ap-1860	236	11	times	time	NOUN
ap-1860	236	12	.	.	PUNCT
ap-1860	237	1	in	in	ADP
ap-1860	237	2	particular	particular	ADJ
ap-1860	237	3	,	,	PUNCT
ap-1860	237	4	for	for	ADP
ap-1860	237	5	sufficiently	sufficiently	ADV
ap-1860	237	6	small	small	ADJ
ap-1860	237	7	ε	ε	PROPN
ap-1860	237	8	,	,	PUNCT
ap-1860	237	9	one	one	NUM
ap-1860	237	10	has	have	VERB
ap-1860	237	11	l	l	NOUN
ap-1860	237	12	=	=	SYM
ap-1860	237	13	r(λ−	r(λ−	PROPN
ap-1860	237	14	ε	ε	PROPN
ap-1860	237	15	)	)	PUNCT
ap-1860	237	16	<	<	X
ap-1860	237	17	r(λ+	r(λ+	X
ap-1860	237	18	ε	ε	PROPN
ap-1860	237	19	)	)	PUNCT
ap-1860	237	20	,	,	PUNCT
ap-1860	237	21	m	m	VERB
ap-1860	237	22	=	=	X
ap-1860	237	23	r(µ+	r(µ+	PROPN
ap-1860	237	24	ε	ε	PROPN
ap-1860	237	25	)	)	PUNCT
ap-1860	237	26	<	<	X
ap-1860	237	27	r(µ−	r(µ−	X
ap-1860	237	28	ε	ε	PROPN
ap-1860	237	29	)	)	PUNCT
ap-1860	237	30	.	.	PUNCT
ap-1860	238	1	(	(	PUNCT
ap-1860	238	2	9	9	X
ap-1860	238	3	)	)	PUNCT
ap-1860	238	4	by	by	ADP
ap-1860	238	5	proposition	proposition	NOUN
ap-1860	238	6	2.3	2.3	NUM
ap-1860	238	7	,	,	PUNCT
ap-1860	238	8	the	the	DET
ap-1860	238	9	induced	induced	ADJ
ap-1860	238	10	map	map	NOUN
ap-1860	238	11	ti	ti	NOUN
ap-1860	238	12	is	be	AUX
ap-1860	238	13	an	an	DET
ap-1860	238	14	exchange	exchange	NOUN
ap-1860	238	15	of	of	ADP
ap-1860	238	16	three	three	NUM
ap-1860	238	17	intervals	interval	NOUN
ap-1860	238	18	with	with	ADP
ap-1860	238	19	permutation	permutation	NOUN
ap-1860	238	20	(	(	PUNCT
ap-1860	238	21	321	321	NUM
ap-1860	238	22	)	)	PUNCT
ap-1860	238	23	.	.	PUNCT
ap-1860	239	1	let	let	VERB
ap-1860	239	2	i	i	PRON
ap-1860	239	3	=	=	PUNCT
ap-1860	239	4	i0∪	i0∪	PROPN
ap-1860	239	5	i1	i1	PROPN
ap-1860	239	6	∪	∪	PROPN
ap-1860	239	7	i2	i2	PROPN
ap-1860	239	8	be	be	VERB
ap-1860	239	9	the	the	DET
ap-1860	239	10	corresponding	corresponding	ADJ
ap-1860	239	11	partition	partition	NOUN
ap-1860	239	12	of	of	ADP
ap-1860	239	13	i	i	PRON
ap-1860	239	14	,	,	PUNCT
ap-1860	239	15	where	where	SCONJ
ap-1860	239	16	for	for	ADP
ap-1860	239	17	every	every	DET
ap-1860	239	18	x0	x0	PROPN
ap-1860	239	19	∈	∈	PROPN
ap-1860	239	20	i0	i0	PROPN
ap-1860	239	21	,	,	PUNCT
ap-1860	239	22	x1	x1	PROPN
ap-1860	239	23	∈	∈	PROPN
ap-1860	239	24	i1	i1	PROPN
ap-1860	239	25	,	,	PUNCT
ap-1860	239	26	x2	x2	PROPN
ap-1860	239	27	∈	∈	PROPN
ap-1860	239	28	i2	i2	PROPN
ap-1860	239	29	one	one	NUM
ap-1860	239	30	has	have	VERB
ap-1860	239	31	x0	x0	PROPN
ap-1860	239	32	<	<	X
ap-1860	240	1	x1	x1	X
ap-1860	240	2	<	<	X
ap-1860	240	3	x2	x2	PROPN
ap-1860	240	4	.	.	PUNCT
ap-1860	241	1	by	by	ADP
ap-1860	241	2	the	the	DET
ap-1860	241	3	same	same	ADJ
ap-1860	241	4	proposition	proposition	NOUN
ap-1860	241	5	r(x1	r(x1	NOUN
ap-1860	241	6	)	)	PUNCT
ap-1860	241	7	=	=	SYM
ap-1860	241	8	r(x0	r(x0	NOUN
ap-1860	241	9	)	)	PUNCT
ap-1860	241	10	+	+	NUM
ap-1860	241	11	r(x2	r(x2	NOUN
ap-1860	241	12	)	)	PUNCT
ap-1860	241	13	,	,	PUNCT
ap-1860	241	14	which	which	PRON
ap-1860	241	15	together	together	ADV
ap-1860	241	16	with	with	ADP
ap-1860	241	17	inequalities	inequality	NOUN
ap-1860	241	18	(	(	PUNCT
ap-1860	241	19	9	9	NUM
ap-1860	241	20	)	)	PUNCT
ap-1860	241	21	implies	imply	VERB
ap-1860	241	22	that	that	SCONJ
ap-1860	241	23	the	the	DET
ap-1860	241	24	right	right	ADJ
ap-1860	241	25	end	end	NOUN
ap-1860	241	26	-	-	PUNCT
ap-1860	241	27	point	point	NOUN
ap-1860	241	28	of	of	ADP
ap-1860	241	29	i0	i0	PROPN
ap-1860	241	30	is	be	AUX
ap-1860	241	31	equal	equal	ADJ
ap-1860	241	32	to	to	ADP
ap-1860	241	33	λ	λ	PROPN
ap-1860	241	34	,	,	PUNCT
ap-1860	241	35	the	the	DET
ap-1860	241	36	left	left	ADJ
ap-1860	241	37	end	end	NOUN
ap-1860	241	38	-	-	PUNCT
ap-1860	241	39	point	point	NOUN
ap-1860	241	40	of	of	ADP
ap-1860	241	41	i2	i2	PROPN
ap-1860	241	42	is	be	AUX
ap-1860	241	43	equal	equal	ADJ
ap-1860	241	44	to	to	ADP
ap-1860	241	45	µ	µ	NUM
ap-1860	241	46	,	,	PUNCT
ap-1860	241	47	and	and	CCONJ
ap-1860	241	48	r(x1	r(x1	ADJ
ap-1860	241	49	)	)	PUNCT
ap-1860	241	50	=	=	PUNCT
ap-1860	242	1	l	l	PUNCT
ap-1860	242	2	+	+	NOUN
ap-1860	242	3	m.	m.	NOUN
ap-1860	242	4	since	since	SCONJ
ap-1860	242	5	c	c	PROPN
ap-1860	242	6	,	,	PUNCT
ap-1860	242	7	d	d	NOUN
ap-1860	242	8	,	,	PUNCT
ap-1860	242	9	d−	d−	PROPN
ap-1860	242	10	c	c	PROPN
ap-1860	242	11	/∈	/∈	PUNCT
ap-1860	243	1	z[α	z[α	NUM
ap-1860	243	2	]	]	PUNCT
ap-1860	243	3	,	,	PUNCT
ap-1860	243	4	we	we	PRON
ap-1860	243	5	also	also	ADV
ap-1860	243	6	have	have	VERB
ap-1860	243	7	λ	λ	X
ap-1860	243	8	/∈	/∈	PUNCT
ap-1860	244	1	z[α	z[α	NUM
ap-1860	244	2	]	]	PUNCT
ap-1860	244	3	,	,	PUNCT
ap-1860	244	4	and	and	CCONJ
ap-1860	244	5	thus	thus	ADV
ap-1860	244	6	one	one	NUM
ap-1860	244	7	can	can	AUX
ap-1860	244	8	choose	choose	VERB
ap-1860	244	9	ε	ε	PROPN
ap-1860	244	10	sufficiently	sufficiently	ADV
ap-1860	244	11	small	small	ADJ
ap-1860	244	12	,	,	PUNCT
ap-1860	244	13	so	so	SCONJ
ap-1860	244	14	that	that	SCONJ
ap-1860	244	15	the	the	DET
ap-1860	244	16	interval	interval	NOUN
ap-1860	244	17	[	[	X
ap-1860	244	18	λ	λ	X
ap-1860	244	19	−	−	PROPN
ap-1860	244	20	ε	ε	PROPN
ap-1860	244	21	,	,	PUNCT
ap-1860	244	22	λ	λ	PROPN
ap-1860	244	23	+	+	CCONJ
ap-1860	244	24	ε	ε	PROPN
ap-1860	244	25	]	]	X
ap-1860	244	26	does	do	AUX
ap-1860	244	27	not	not	PART
ap-1860	244	28	contain	contain	VERB
ap-1860	244	29	any	any	PRON
ap-1860	244	30	of	of	ADP
ap-1860	244	31	the	the	DET
ap-1860	244	32	points	point	NOUN
ap-1860	244	33	t−j(α	t−j(α	NOUN
ap-1860	244	34	)	)	PUNCT
ap-1860	244	35	for	for	ADP
ap-1860	244	36	0	0	NUM
ap-1860	244	37	≤	≤	NUM
ap-1860	245	1	j	j	PROPN
ap-1860	245	2	≤	≤	NUM
ap-1860	245	3	l	l	NOUN
ap-1860	245	4	+	+	CCONJ
ap-1860	245	5	m.	m.	NOUN
ap-1860	245	6	this	this	PRON
ap-1860	245	7	implies	imply	VERB
ap-1860	245	8	that	that	SCONJ
ap-1860	245	9	t	t	PROPN
ap-1860	245	10	j	j	PROPN
ap-1860	245	11	(	(	PUNCT
ap-1860	245	12	[	[	X
ap-1860	245	13	λ−	λ−	PROPN
ap-1860	245	14	ε	ε	PROPN
ap-1860	245	15	,	,	PUNCT
ap-1860	245	16	λ+	λ+	ADP
ap-1860	245	17	ε	ε	X
ap-1860	245	18	]	]	PUNCT
ap-1860	245	19	)	)	PUNCT
ap-1860	245	20	is	be	AUX
ap-1860	245	21	an	an	DET
ap-1860	245	22	interval	interval	NOUN
ap-1860	245	23	not	not	PART
ap-1860	245	24	containing	contain	VERB
ap-1860	245	25	α	α	NOUN
ap-1860	245	26	for	for	ADP
ap-1860	245	27	any	any	DET
ap-1860	245	28	j	j	PROPN
ap-1860	245	29	=	=	SYM
ap-1860	245	30	0	0	NUM
ap-1860	245	31	,	,	PUNCT
ap-1860	245	32	1	1	NUM
ap-1860	245	33	,	,	PUNCT
ap-1860	245	34	.	.	PUNCT
ap-1860	245	35	.	.	PUNCT
ap-1860	245	36	.	.	PUNCT
ap-1860	246	1	,	,	PUNCT
ap-1860	246	2	l	l	PROPN
ap-1860	247	1	+	+	NOUN
ap-1860	247	2	m	m	VERB
ap-1860	247	3	−	−	NOUN
ap-1860	247	4	1	1	NUM
ap-1860	247	5	,	,	PUNCT
ap-1860	247	6	and	and	CCONJ
ap-1860	247	7	consequently	consequently	ADV
ap-1860	247	8	,	,	PUNCT
ap-1860	247	9	the	the	DET
ap-1860	247	10	prefix	prefix	NOUN
ap-1860	247	11	of	of	ADP
ap-1860	247	12	length	length	NOUN
ap-1860	247	13	l	l	NOUN
ap-1860	248	1	+	+	X
ap-1860	248	2	m	m	NOUN
ap-1860	248	3	of	of	ADP
ap-1860	248	4	the	the	DET
ap-1860	248	5	infinite	infinite	ADJ
ap-1860	248	6	word	word	NOUN
ap-1860	248	7	uρ	uρ	INTJ
ap-1860	248	8	is	be	AUX
ap-1860	248	9	the	the	DET
ap-1860	248	10	same	same	ADJ
ap-1860	248	11	for	for	ADP
ap-1860	248	12	any	any	DET
ap-1860	248	13	ρ	ρ	PROPN
ap-1860	248	14	∈	∈	PROPN
ap-1860	248	15	[	[	X
ap-1860	248	16	λ−	λ−	PROPN
ap-1860	248	17	ε	ε	PROPN
ap-1860	248	18	,	,	PUNCT
ap-1860	248	19	λ+	λ+	VERB
ap-1860	248	20	ε	ε	X
ap-1860	248	21	]	]	PUNCT
ap-1860	248	22	.	.	PUNCT
ap-1860	249	1	we	we	PRON
ap-1860	249	2	have	have	VERB
ap-1860	249	3	t	t	X
ap-1860	249	4	l(λ−	l(λ−	PROPN
ap-1860	249	5	ε	ε	PROPN
ap-1860	249	6	)	)	PUNCT
ap-1860	249	7	=	=	PUNCT
ap-1860	250	1	d−	d−	PROPN
ap-1860	250	2	ε	ε	PROPN
ap-1860	250	3	∈	∈	PROPN
ap-1860	250	4	i	i	PROPN
ap-1860	250	5	,	,	PUNCT
ap-1860	250	6	t	t	PROPN
ap-1860	250	7	l(λ+	l(λ+	NOUN
ap-1860	250	8	ε	ε	PROPN
ap-1860	250	9	)	)	PUNCT
ap-1860	250	10	=	=	SYM
ap-1860	250	11	d+	d+	PUNCT
ap-1860	250	12	ε	ε	PROPN
ap-1860	250	13	/∈	/∈	PUNCT
ap-1860	250	14	i.	i.	PROPN
ap-1860	250	15	for	for	ADP
ap-1860	250	16	the	the	DET
ap-1860	250	17	corresponding	correspond	VERB
ap-1860	250	18	i	i	PROPN
ap-1860	250	19	-	-	PUNCT
ap-1860	250	20	itineraries	itinerary	NOUN
ap-1860	250	21	,	,	PUNCT
ap-1860	250	22	we	we	PRON
ap-1860	250	23	thus	thus	ADV
ap-1860	250	24	have	have	VERB
ap-1860	250	25	r(λ+	r(λ+	NOUN
ap-1860	250	26	ε	ε	NOUN
ap-1860	250	27	)	)	PUNCT
ap-1860	250	28	=	=	SYM
ap-1860	250	29	r(λ−	r(λ−	PROPN
ap-1860	250	30	ε)r(d−	ε)r(d−	X
ap-1860	250	31	ε	ε	PROPN
ap-1860	250	32	)	)	PUNCT
ap-1860	250	33	.	.	PUNCT
ap-1860	251	1	we	we	PRON
ap-1860	251	2	can	can	AUX
ap-1860	251	3	set	set	VERB
ap-1860	251	4	r1	r1	PROPN
ap-1860	251	5	=	=	SYM
ap-1860	251	6	r(λ−	r(λ−	PROPN
ap-1860	251	7	ε	ε	PROPN
ap-1860	251	8	)	)	PUNCT
ap-1860	251	9	,	,	PUNCT
ap-1860	251	10	r2	r2	PROPN
ap-1860	251	11	=	=	SYM
ap-1860	251	12	r(d−	r(d−	PROPN
ap-1860	251	13	ε	ε	PROPN
ap-1860	251	14	)	)	PUNCT
ap-1860	251	15	,	,	PUNCT
ap-1860	251	16	to	to	PART
ap-1860	251	17	have	have	VERB
ap-1860	251	18	iti	iti	PROPN
ap-1860	251	19	⊃	⊃	PROPN
ap-1860	251	20	{	{	PUNCT
ap-1860	251	21	r1	r1	PROPN
ap-1860	251	22	,	,	PUNCT
ap-1860	251	23	r2	r2	PROPN
ap-1860	251	24	,	,	PUNCT
ap-1860	251	25	r1r2	r1r2	VERB
ap-1860	251	26	}	}	PUNCT
ap-1860	251	27	.	.	PUNCT
ap-1860	252	1	by	by	ADP
ap-1860	252	2	proposition	proposition	NOUN
ap-1860	252	3	2.5	2.5	NUM
ap-1860	252	4	,	,	PUNCT
ap-1860	252	5	the	the	DET
ap-1860	252	6	set	set	NOUN
ap-1860	252	7	iti	iti	PROPN
ap-1860	252	8	may	may	AUX
ap-1860	252	9	have	have	VERB
ap-1860	252	10	four	four	NUM
ap-1860	252	11	elements	element	NOUN
ap-1860	252	12	.	.	PUNCT
ap-1860	253	1	let	let	VERB
ap-1860	253	2	us	we	PRON
ap-1860	253	3	determine	determine	VERB
ap-1860	253	4	the	the	DET
ap-1860	253	5	fourth	fourth	ADJ
ap-1860	253	6	element	element	NOUN
ap-1860	253	7	q.	q.	PROPN
ap-1860	253	8	consider	consider	VERB
ap-1860	253	9	the	the	DET
ap-1860	253	10	point	point	NOUN
ap-1860	253	11	ν	ν	X
ap-1860	253	12	=	=	SYM
ap-1860	253	13	t−n(α	t−n(α	ADJ
ap-1860	253	14	)	)	PUNCT
ap-1860	253	15	,	,	PUNCT
ap-1860	253	16	n	n	NOUN
ap-1860	253	17	=	=	SYM
ap-1860	253	18	min{k	min{k	NOUN
ap-1860	253	19	≥	≥	NOUN
ap-1860	253	20	0	0	NUM
ap-1860	253	21	:	:	PUNCT
ap-1860	253	22	t−k(α	t−k(α	NOUN
ap-1860	253	23	)	)	PUNCT
ap-1860	253	24	∈	∈	PROPN
ap-1860	254	1	i	i	PRON
ap-1860	254	2	}	}	PUNCT
ap-1860	254	3	,	,	PUNCT
ap-1860	254	4	which	which	PRON
ap-1860	254	5	,	,	PUNCT
ap-1860	254	6	by	by	ADP
ap-1860	254	7	the	the	DET
ap-1860	254	8	proof	proof	NOUN
ap-1860	254	9	of	of	ADP
ap-1860	254	10	proposition	proposition	NOUN
ap-1860	254	11	2.5	2.5	NUM
ap-1860	254	12	splits	split	VERB
ap-1860	254	13	one	one	NUM
ap-1860	254	14	of	of	ADP
ap-1860	254	15	the	the	DET
ap-1860	254	16	intervals	interval	NOUN
ap-1860	254	17	i0	i0	PROPN
ap-1860	254	18	,	,	PUNCT
ap-1860	254	19	i1	i1	PROPN
ap-1860	254	20	,	,	PUNCT
ap-1860	254	21	i2	i2	PROPN
ap-1860	254	22	,	,	PUNCT
ap-1860	254	23	into	into	ADP
ap-1860	254	24	two	two	NUM
ap-1860	254	25	,	,	PUNCT
ap-1860	254	26	so	so	SCONJ
ap-1860	254	27	that	that	SCONJ
ap-1860	254	28	the	the	DET
ap-1860	254	29	i	i	NOUN
ap-1860	254	30	-	-	PUNCT
ap-1860	254	31	itinerary	itinerary	NOUN
ap-1860	254	32	on	on	ADP
ap-1860	254	33	the	the	DET
ap-1860	254	34	new	new	ADJ
ap-1860	254	35	partition	partition	NOUN
ap-1860	254	36	is	be	AUX
ap-1860	254	37	constant	constant	ADJ
ap-1860	254	38	.	.	PUNCT
ap-1860	255	1	by	by	ADP
ap-1860	255	2	the	the	DET
ap-1860	255	3	assumption	assumption	NOUN
ap-1860	255	4	that	that	SCONJ
ap-1860	255	5	c	c	X
ap-1860	255	6	,	,	PUNCT
ap-1860	255	7	d	d	PROPN
ap-1860	255	8	/∈	/∈	PUNCT
ap-1860	256	1	z[α	z[α	NUM
ap-1860	256	2	]	]	PUNCT
ap-1860	256	3	,	,	PUNCT
ap-1860	256	4	we	we	PRON
ap-1860	256	5	have	have	VERB
ap-1860	256	6	ν	ν	NOUN
ap-1860	256	7	6=	6=	ADP
ap-1860	256	8	λ	λ	PROPN
ap-1860	256	9	,	,	PUNCT
ap-1860	256	10	ν	ν	PROPN
ap-1860	256	11	6=	6=	SYM
ap-1860	256	12	µ.	µ.	NOUN
ap-1860	256	13	consider	consider	VERB
ap-1860	256	14	the	the	DET
ap-1860	256	15	points	point	NOUN
ap-1860	256	16	ν−ε	ν−ε	PROPN
ap-1860	256	17	,	,	PUNCT
ap-1860	256	18	ν+ε	ν+ε	NOUN
ap-1860	256	19	for	for	ADP
ap-1860	256	20	sufficiently	sufficiently	ADV
ap-1860	256	21	small	small	ADJ
ap-1860	256	22	ε	ε	PROPN
ap-1860	256	23	.	.	PUNCT
ap-1860	257	1	obviously	obviously	ADV
ap-1860	257	2	,	,	PUNCT
ap-1860	257	3	their	their	PRON
ap-1860	257	4	return	return	NOUN
ap-1860	257	5	time	time	NOUN
ap-1860	257	6	coincides	coincide	VERB
ap-1860	257	7	,	,	PUNCT
ap-1860	257	8	r(ν	r(ν	PROPN
ap-1860	257	9	−	−	PROPN
ap-1860	257	10	ε	ε	PROPN
ap-1860	257	11	)	)	PUNCT
ap-1860	257	12	=	=	SYM
ap-1860	257	13	r(ν+ε	r(ν+ε	NOUN
ap-1860	257	14	)	)	PUNCT
ap-1860	257	15	=	=	SYM
ap-1860	257	16	r(ν	r(ν	PROPN
ap-1860	257	17	)	)	PUNCT
ap-1860	257	18	,	,	PUNCT
ap-1860	257	19	thus	thus	ADV
ap-1860	257	20	the	the	DET
ap-1860	257	21	i	i	PROPN
ap-1860	257	22	-	-	PUNCT
ap-1860	257	23	itineraries	itinerary	NOUN
ap-1860	257	24	r(ν−ε	r(ν−ε	NOUN
ap-1860	257	25	)	)	PUNCT
ap-1860	257	26	,	,	PUNCT
ap-1860	257	27	r(ν+ε	r(ν+ε	PROPN
ap-1860	257	28	)	)	PUNCT
ap-1860	257	29	are	be	AUX
ap-1860	257	30	of	of	ADP
ap-1860	257	31	the	the	DET
ap-1860	257	32	same	same	ADJ
ap-1860	257	33	length	length	NOUN
ap-1860	257	34	r(ν	r(ν	PROPN
ap-1860	257	35	)	)	PUNCT
ap-1860	257	36	.	.	PUNCT
ap-1860	258	1	since	since	SCONJ
ap-1860	258	2	tn(ν	tn(ν	NOUN
ap-1860	258	3	)	)	PUNCT
ap-1860	258	4	=	=	SYM
ap-1860	258	5	α	α	X
ap-1860	258	6	,	,	PUNCT
ap-1860	258	7	we	we	PRON
ap-1860	258	8	have	have	VERB
ap-1860	258	9	tn+1(ν	tn+1(ν	PROPN
ap-1860	258	10	)	)	PUNCT
ap-1860	259	1	=	=	SYM
ap-1860	259	2	0	0	NUM
ap-1860	259	3	/∈	/∈	INTJ
ap-1860	260	1	i	i	PRON
ap-1860	260	2	,	,	PUNCT
ap-1860	260	3	and	and	CCONJ
ap-1860	260	4	thus	thus	ADV
ap-1860	260	5	r(ν	r(ν	PROPN
ap-1860	260	6	)	)	PUNCT
ap-1860	260	7	≥	≥	PRON
ap-1860	260	8	n+	n+	PUNCT
ap-1860	261	1	1	1	X
ap-1860	261	2	.	.	X
ap-1860	262	1	we	we	PRON
ap-1860	262	2	can	can	AUX
ap-1860	262	3	see	see	VERB
ap-1860	262	4	that	that	DET
ap-1860	262	5	tn+1(ν	tn+1(ν	PROPN
ap-1860	262	6	+	+	CCONJ
ap-1860	262	7	ε	ε	PROPN
ap-1860	262	8	)	)	PUNCT
ap-1860	262	9	=	=	SYM
ap-1860	262	10	ε	ε	PROPN
ap-1860	262	11	,	,	PUNCT
ap-1860	262	12	tn+2(ν	tn+2(ν	ADV
ap-1860	262	13	+	+	CCONJ
ap-1860	262	14	ε	ε	PROPN
ap-1860	262	15	)	)	PUNCT
ap-1860	262	16	=	=	SYM
ap-1860	262	17	1−	1−	NUM
ap-1860	262	18	α+	α+	PUNCT
ap-1860	262	19	ε	ε	PROPN
ap-1860	262	20	,	,	PUNCT
ap-1860	262	21	tn+1(ν	tn+1(ν	PROPN
ap-1860	262	22	−	−	PROPN
ap-1860	262	23	ε	ε	PROPN
ap-1860	262	24	)	)	PUNCT
ap-1860	262	25	=	=	SYM
ap-1860	262	26	1−	1−	NUM
ap-1860	262	27	ε	ε	PROPN
ap-1860	262	28	,	,	PUNCT
ap-1860	262	29	tn+2(ν	tn+2(ν	ADV
ap-1860	262	30	−	−	NOUN
ap-1860	262	31	ε	ε	PROPN
ap-1860	262	32	)	)	PUNCT
ap-1860	262	33	=	=	SYM
ap-1860	262	34	1−	1−	NUM
ap-1860	262	35	α−	α−	ADP
ap-1860	262	36	ε	ε	PROPN
ap-1860	262	37	,	,	PUNCT
ap-1860	262	38	which	which	PRON
ap-1860	262	39	implies	imply	VERB
ap-1860	262	40	that	that	SCONJ
ap-1860	262	41	r(ν	r(ν	PROPN
ap-1860	262	42	−	−	ADP
ap-1860	262	43	ε	ε	PROPN
ap-1860	262	44	)	)	PUNCT
ap-1860	262	45	=	=	VERB
ap-1860	262	46	u0	u0	ADJ
ap-1860	262	47	·	·	PUNCT
ap-1860	262	48	·	·	PUNCT
ap-1860	262	49	·	·	PUNCT
ap-1860	262	50	un−101un+2	un−101un+2	X
ap-1860	262	51	·	·	PUNCT
ap-1860	262	52	·	·	PUNCT
ap-1860	262	53	·	·	PUNCT
ap-1860	262	54	ur(ν)−1	ur(ν)−1	NOUN
ap-1860	262	55	,	,	PUNCT
ap-1860	262	56	r(ν	r(ν	PROPN
ap-1860	262	57	+	+	CCONJ
ap-1860	262	58	ε	ε	PROPN
ap-1860	262	59	)	)	PUNCT
ap-1860	262	60	=	=	VERB
ap-1860	262	61	u0	u0	ADJ
ap-1860	262	62	·	·	PUNCT
ap-1860	262	63	·	·	PUNCT
ap-1860	262	64	·	·	PUNCT
ap-1860	262	65	un−110un+2	un−110un+2	X
ap-1860	262	66	·	·	PUNCT
ap-1860	262	67	·	·	PUNCT
ap-1860	262	68	·	·	PUNCT
ap-1860	262	69	ur(ν)−1	ur(ν)−1	NOUN
ap-1860	262	70	.	.	PUNCT
ap-1860	263	1	necessarily	necessarily	ADV
ap-1860	263	2	,	,	PUNCT
ap-1860	263	3	r(ν−ε	r(ν−ε	ADJ
ap-1860	263	4	)	)	PUNCT
ap-1860	263	5	and	and	CCONJ
ap-1860	263	6	r(ν+ε	r(ν+ε	NOUN
ap-1860	263	7	)	)	PUNCT
ap-1860	263	8	are	be	AUX
ap-1860	263	9	amicable	amicable	ADJ
ap-1860	263	10	words	word	NOUN
ap-1860	263	11	.	.	PUNCT
ap-1860	264	1	one	one	NUM
ap-1860	264	2	of	of	ADP
ap-1860	264	3	them	they	PRON
ap-1860	264	4	is	be	AUX
ap-1860	264	5	q	q	ADJ
ap-1860	264	6	,	,	PUNCT
ap-1860	264	7	the	the	DET
ap-1860	264	8	other	other	ADJ
ap-1860	264	9	one	one	NOUN
ap-1860	264	10	is	be	AUX
ap-1860	264	11	equal	equal	ADJ
ap-1860	264	12	to	to	ADP
ap-1860	264	13	r1	r1	PROPN
ap-1860	264	14	,	,	PUNCT
ap-1860	264	15	r2	r2	PROPN
ap-1860	264	16	or	or	CCONJ
ap-1860	264	17	r1r2	r1r2	NOUN
ap-1860	264	18	,	,	PUNCT
ap-1860	264	19	according	accord	VERB
ap-1860	264	20	to	to	ADP
ap-1860	264	21	whether	whether	SCONJ
ap-1860	264	22	the	the	DET
ap-1860	264	23	point	point	NOUN
ap-1860	264	24	ν	ν	NOUN
ap-1860	264	25	belongs	belong	VERB
ap-1860	264	26	to	to	ADP
ap-1860	264	27	i0	i0	PROPN
ap-1860	264	28	,	,	PUNCT
ap-1860	264	29	i1	i1	PROPN
ap-1860	264	30	or	or	CCONJ
ap-1860	264	31	i2	i2	PROPN
ap-1860	264	32	.	.	PUNCT
ap-1860	265	1	3	3	X
ap-1860	265	2	.	.	X
ap-1860	265	3	case	case	NOUN
ap-1860	265	4	study	study	NOUN
ap-1860	265	5	let	let	VERB
ap-1860	265	6	us	we	PRON
ap-1860	265	7	give	give	VERB
ap-1860	265	8	several	several	ADJ
ap-1860	265	9	examples	example	NOUN
ap-1860	265	10	illustrating	illustrate	VERB
ap-1860	265	11	the	the	DET
ap-1860	265	12	possible	possible	ADJ
ap-1860	265	13	outcomes	outcome	NOUN
ap-1860	265	14	for	for	ADP
ap-1860	265	15	the	the	DET
ap-1860	265	16	set	set	ADJ
ap-1860	265	17	iti	iti	PROPN
ap-1860	265	18	of	of	ADP
ap-1860	265	19	i	i	PROPN
ap-1860	265	20	-	-	PUNCT
ap-1860	265	21	itineraries	itinerary	NOUN
ap-1860	265	22	for	for	ADP
ap-1860	265	23	general	general	ADJ
ap-1860	265	24	subinterval	subinterval	NOUN
ap-1860	266	1	i	i	PRON
ap-1860	266	2	=	=	PUNCT
ap-1860	267	1	[	[	X
ap-1860	267	2	c	c	X
ap-1860	267	3	,	,	PUNCT
ap-1860	267	4	d	d	NOUN
ap-1860	267	5	)	)	PUNCT
ap-1860	267	6	⊂	⊂	PROPN
ap-1860	268	1	[	[	X
ap-1860	268	2	0	0	NUM
ap-1860	268	3	,	,	PUNCT
ap-1860	268	4	1	1	NUM
ap-1860	268	5	)	)	PUNCT
ap-1860	268	6	.	.	PUNCT
ap-1860	269	1	according	accord	VERB
ap-1860	269	2	to	to	ADP
ap-1860	269	3	our	our	PRON
ap-1860	269	4	main	main	ADJ
ap-1860	269	5	theorem	theorem	NOUN
ap-1860	269	6	1.1	1.1	NUM
ap-1860	269	7	,	,	PUNCT
ap-1860	269	8	we	we	PRON
ap-1860	269	9	have	have	VERB
ap-1860	269	10	iti	iti	PROPN
ap-1860	269	11	⊂	⊂	PROPN
ap-1860	269	12	{	{	PUNCT
ap-1860	269	13	r1	r1	PROPN
ap-1860	269	14	,	,	PUNCT
ap-1860	269	15	r2	r2	PROPN
ap-1860	269	16	,	,	PUNCT
ap-1860	269	17	r1r2	r1r2	NOUN
ap-1860	269	18	,	,	PUNCT
ap-1860	269	19	q	q	NOUN
ap-1860	269	20	}	}	PUNCT
ap-1860	269	21	,	,	PUNCT
ap-1860	269	22	where	where	SCONJ
ap-1860	269	23	q	q	NOUN
ap-1860	269	24	is	be	AUX
ap-1860	269	25	a	a	DET
ap-1860	269	26	word	word	NOUN
ap-1860	269	27	amicable	amicable	ADJ
ap-1860	269	28	with	with	ADP
ap-1860	269	29	one	one	NUM
ap-1860	269	30	of	of	ADP
ap-1860	269	31	r1	r1	NOUN
ap-1860	269	32	,	,	PUNCT
ap-1860	269	33	r2	r2	PROPN
ap-1860	269	34	,	,	PUNCT
ap-1860	269	35	r1r2	r1r2	X
ap-1860	269	36	.	.	PUNCT
ap-1860	270	1	in	in	ADP
ap-1860	270	2	fact	fact	NOUN
ap-1860	270	3	,	,	PUNCT
ap-1860	270	4	as	as	SCONJ
ap-1860	270	5	we	we	PRON
ap-1860	270	6	see	see	VERB
ap-1860	270	7	in	in	ADP
ap-1860	270	8	the	the	DET
ap-1860	270	9	following	follow	VERB
ap-1860	270	10	examples	example	NOUN
ap-1860	270	11	,	,	PUNCT
ap-1860	270	12	we	we	PRON
ap-1860	270	13	can	can	AUX
ap-1860	270	14	have	have	VERB
ap-1860	270	15	all	all	DET
ap-1860	270	16	possibilities	possibility	NOUN
ap-1860	270	17	.	.	PUNCT
ap-1860	271	1	for	for	ADP
ap-1860	271	2	simplicity	simplicity	NOUN
ap-1860	271	3	in	in	ADP
ap-1860	271	4	the	the	DET
ap-1860	271	5	examples	example	NOUN
ap-1860	271	6	,	,	PUNCT
ap-1860	271	7	we	we	PRON
ap-1860	271	8	always	always	ADV
ap-1860	271	9	keep	keep	VERB
ap-1860	271	10	α	α	NOUN
ap-1860	271	11	=	=	SYM
ap-1860	271	12	σ	σ	PROPN
ap-1860	271	13	,	,	PUNCT
ap-1860	271	14	where	where	SCONJ
ap-1860	271	15	σ	σ	PROPN
ap-1860	271	16	=	=	SYM
ap-1860	271	17	1	1	NUM
ap-1860	271	18	2	2	NUM
ap-1860	271	19	(	(	PUNCT
ap-1860	271	20	√	√	NUM
ap-1860	271	21	5	5	NUM
ap-1860	271	22	−	−	NOUN
ap-1860	271	23	1	1	NUM
ap-1860	271	24	)	)	PUNCT
ap-1860	271	25	is	be	AUX
ap-1860	271	26	the	the	DET
ap-1860	271	27	reciprocal	reciprocal	NOUN
ap-1860	271	28	of	of	ADP
ap-1860	271	29	the	the	DET
ap-1860	271	30	golden	golden	ADJ
ap-1860	271	31	ratio	ratio	NOUN
ap-1860	271	32	.	.	PUNCT
ap-1860	272	1	in	in	ADP
ap-1860	272	2	calculations	calculation	NOUN
ap-1860	272	3	,	,	PUNCT
ap-1860	272	4	we	we	PRON
ap-1860	272	5	use	use	VERB
ap-1860	272	6	the	the	DET
ap-1860	272	7	relation	relation	NOUN
ap-1860	272	8	σ2	σ2	PROPN
ap-1860	272	9	=	=	PROPN
ap-1860	272	10	σ	σ	PROPN
ap-1860	273	1	+	+	NOUN
ap-1860	273	2	1	1	X
ap-1860	273	3	.	.	X
ap-1860	274	1	first	first	ADV
ap-1860	274	2	,	,	PUNCT
ap-1860	274	3	we	we	PRON
ap-1860	274	4	choose	choose	VERB
ap-1860	274	5	the	the	DET
ap-1860	274	6	most	most	ADV
ap-1860	274	7	generic	generic	ADJ
ap-1860	274	8	cases	case	NOUN
ap-1860	274	9	,	,	PUNCT
ap-1860	274	10	namely	namely	ADV
ap-1860	274	11	examples	example	NOUN
ap-1860	274	12	where	where	SCONJ
ap-1860	274	13	#	#	SYM
ap-1860	274	14	iti	iti	NOUN
ap-1860	274	15	=	=	NOUN
ap-1860	274	16	4	4	X
ap-1860	274	17	.	.	PUNCT
ap-1860	275	1	let	let	VERB
ap-1860	275	2	i	i	PRON
ap-1860	275	3	=	=	PUNCT
ap-1860	276	1	[	[	X
ap-1860	276	2	c	c	X
ap-1860	276	3	,	,	PUNCT
ap-1860	276	4	d	d	NOUN
ap-1860	276	5	)	)	PUNCT
ap-1860	276	6	where	where	SCONJ
ap-1860	276	7	d	d	NOUN
ap-1860	276	8	−	−	PROPN
ap-1860	276	9	c	c	NOUN
ap-1860	276	10	=	=	SYM
ap-1860	276	11	σ3	σ3	PROPN
ap-1860	276	12	+	+	X
ap-1860	276	13	σ6	σ6	NOUN
ap-1860	276	14	.	.	PUNCT
ap-1860	277	1	since	since	SCONJ
ap-1860	277	2	d	d	PROPN
ap-1860	277	3	−	−	PROPN
ap-1860	277	4	c	c	PROPN
ap-1860	277	5	6=	6=	PROPN
ap-1860	277	6	δk	δk	PROPN
ap-1860	277	7	,	,	PUNCT
ap-1860	277	8	s	s	X
ap-1860	277	9	for	for	ADP
ap-1860	277	10	any	any	DET
ap-1860	277	11	k	k	PROPN
ap-1860	277	12	,	,	PUNCT
ap-1860	277	13	s	s	AUX
ap-1860	277	14	,	,	PUNCT
ap-1860	277	15	by	by	ADP
ap-1860	277	16	proposition	proposition	NOUN
ap-1860	277	17	2.3	2.3	NUM
ap-1860	277	18	,	,	PUNCT
ap-1860	277	19	the	the	DET
ap-1860	277	20	induced	induced	ADJ
ap-1860	277	21	map	map	NOUN
ap-1860	277	22	ti	ti	NOUN
ap-1860	277	23	is	be	AUX
ap-1860	277	24	an	an	DET
ap-1860	277	25	exchange	exchange	NOUN
ap-1860	277	26	of	of	ADP
ap-1860	277	27	three	three	NUM
ap-1860	277	28	intervals	interval	NOUN
ap-1860	277	29	with	with	ADP
ap-1860	277	30	permutation	permutation	NOUN
ap-1860	277	31	(	(	PUNCT
ap-1860	277	32	321	321	NUM
ap-1860	277	33	)	)	PUNCT
ap-1860	277	34	,	,	PUNCT
ap-1860	277	35	and	and	CCONJ
ap-1860	277	36	,	,	PUNCT
ap-1860	277	37	moreover	moreover	ADV
ap-1860	277	38	,	,	PUNCT
ap-1860	277	39	the	the	DET
ap-1860	277	40	lengths	length	NOUN
ap-1860	277	41	of	of	ADP
ap-1860	277	42	exchanged	exchange	VERB
ap-1860	277	43	intervals	interval	NOUN
ap-1860	277	44	i0	i0	PROPN
ap-1860	277	45	,	,	PUNCT
ap-1860	277	46	i1	i1	PROPN
ap-1860	277	47	,	,	PUNCT
ap-1860	277	48	i2	i2	PROPN
ap-1860	277	49	do	do	AUX
ap-1860	277	50	not	not	PART
ap-1860	277	51	depend	depend	VERB
ap-1860	277	52	on	on	ADP
ap-1860	277	53	the	the	DET
ap-1860	277	54	position	position	NOUN
ap-1860	277	55	of	of	ADP
ap-1860	277	56	the	the	DET
ap-1860	277	57	interval	interval	NOUN
ap-1860	277	58	i.	i.	NOUN
ap-1860	277	59	in	in	ADP
ap-1860	277	60	the	the	DET
ap-1860	277	61	notation	notation	NOUN
ap-1860	277	62	introduced	introduce	VERB
ap-1860	277	63	in	in	ADP
ap-1860	277	64	the	the	DET
ap-1860	277	65	proof	proof	NOUN
ap-1860	277	66	of	of	ADP
ap-1860	277	67	theorem	theorem	ADJ
ap-1860	277	68	1.1	1.1	NUM
ap-1860	277	69	,	,	PUNCT
ap-1860	277	70	λ	λ	X
ap-1860	277	71	=	=	SYM
ap-1860	277	72	c+	c+	VERB
ap-1860	277	73	σ4	σ4	NOUN
ap-1860	277	74	,	,	PUNCT
ap-1860	277	75	µ	µ	X
ap-1860	277	76	=	=	SYM
ap-1860	277	77	c+	c+	NOUN
ap-1860	277	78	σ3	σ3	PROPN
ap-1860	277	79	.	.	PUNCT
ap-1860	278	1	hence	hence	ADV
ap-1860	278	2	,	,	PUNCT
ap-1860	278	3	in	in	ADP
ap-1860	278	4	particular	particular	ADJ
ap-1860	278	5	,	,	PUNCT
ap-1860	278	6	i0	i0	PROPN
ap-1860	278	7	=	=	PUNCT
ap-1860	279	1	[	[	X
ap-1860	279	2	c	c	X
ap-1860	279	3	,	,	PUNCT
ap-1860	279	4	c+	c+	X
ap-1860	279	5	σ4	σ4	NOUN
ap-1860	279	6	)	)	PUNCT
ap-1860	279	7	,	,	PUNCT
ap-1860	279	8	i1	i1	PROPN
ap-1860	279	9	=	=	PUNCT
ap-1860	280	1	[	[	X
ap-1860	280	2	c+	c+	NOUN
ap-1860	280	3	σ4	σ4	NOUN
ap-1860	280	4	,	,	PUNCT
ap-1860	280	5	c+	c+	NOUN
ap-1860	280	6	σ3	σ3	PROPN
ap-1860	280	7	)	)	PUNCT
ap-1860	280	8	,	,	PUNCT
ap-1860	280	9	i2	i2	PROPN
ap-1860	280	10	=	=	PUNCT
ap-1860	281	1	[	[	X
ap-1860	281	2	c+	c+	NOUN
ap-1860	281	3	σ3	σ3	NOUN
ap-1860	281	4	,	,	PUNCT
ap-1860	281	5	c+	c+	NOUN
ap-1860	281	6	σ3	σ3	NOUN
ap-1860	281	7	+	+	CCONJ
ap-1860	281	8	σ6	σ6	NOUN
ap-1860	281	9	)	)	PUNCT
ap-1860	281	10	.	.	PUNCT
ap-1860	282	1	independently	independently	ADV
ap-1860	282	2	on	on	ADP
ap-1860	282	3	c	c	NOUN
ap-1860	282	4	,	,	PUNCT
ap-1860	282	5	the	the	DET
ap-1860	282	6	return	return	NOUN
ap-1860	282	7	time	time	NOUN
ap-1860	282	8	r(x	r(x	PROPN
ap-1860	282	9	)	)	PUNCT
ap-1860	282	10	to	to	ADP
ap-1860	282	11	the	the	DET
ap-1860	282	12	interval	interval	NOUN
ap-1860	282	13	i	i	PRON
ap-1860	282	14	satisfies	satisfy	VERB
ap-1860	282	15	r(x	r(x	PROPN
ap-1860	282	16	)	)	PUNCT
ap-1860	283	1	=	=	PUNCT
ap-1860	283	2			NOUN
ap-1860	283	3	3	3	NUM
ap-1860	283	4	if	if	SCONJ
ap-1860	283	5	x	x	PROPN
ap-1860	283	6	∈	∈	PROPN
ap-1860	283	7	i0	i0	PROPN
ap-1860	283	8	,	,	PUNCT
ap-1860	283	9	5	5	NUM
ap-1860	283	10	if	if	SCONJ
ap-1860	283	11	x	x	PROPN
ap-1860	283	12	∈	∈	PROPN
ap-1860	283	13	i1	i1	PROPN
ap-1860	283	14	,	,	PUNCT
ap-1860	283	15	2	2	NUM
ap-1860	283	16	if	if	SCONJ
ap-1860	283	17	x	x	PROPN
ap-1860	283	18	∈	∈	PROPN
ap-1860	283	19	i2	i2	PROPN
ap-1860	283	20	.	.	PUNCT
ap-1860	284	1	447	447	NUM
ap-1860	284	2	z.	z.	PROPN
ap-1860	284	3	masáková	masáková	PROPN
ap-1860	284	4	,	,	PUNCT
ap-1860	284	5	e.	e.	PROPN
ap-1860	284	6	pelantová	pelantová	PROPN
ap-1860	284	7	acta	acta	PROPN
ap-1860	284	8	polytechnica	polytechnica	PROPN
ap-1860	284	9	(	(	PUNCT
ap-1860	284	10	in	in	ADP
ap-1860	284	11	fact	fact	NOUN
ap-1860	284	12	,	,	PUNCT
ap-1860	284	13	for	for	ADP
ap-1860	284	14	any	any	DET
ap-1860	284	15	subinterval	subinterval	NOUN
ap-1860	285	1	i	i	PRON
ap-1860	285	2	⊂	⊂	PROPN
ap-1860	286	1	[	[	X
ap-1860	286	2	0	0	NUM
ap-1860	286	3	,	,	PUNCT
ap-1860	286	4	1	1	NUM
ap-1860	286	5	)	)	PUNCT
ap-1860	286	6	the	the	DET
ap-1860	286	7	return	return	NOUN
ap-1860	286	8	time	time	NOUN
ap-1860	286	9	takes	take	VERB
ap-1860	286	10	two	two	NUM
ap-1860	286	11	or	or	CCONJ
ap-1860	286	12	three	three	NUM
ap-1860	286	13	values	value	NOUN
ap-1860	286	14	,	,	PUNCT
ap-1860	286	15	for	for	ADP
ap-1860	286	16	α	α	NOUN
ap-1860	286	17	=	=	SYM
ap-1860	286	18	σ	σ	NOUN
ap-1860	286	19	always	always	ADV
ap-1860	286	20	equal	equal	ADJ
ap-1860	286	21	to	to	ADP
ap-1860	286	22	two	two	NUM
ap-1860	286	23	or	or	CCONJ
ap-1860	286	24	three	three	NUM
ap-1860	286	25	consecutive	consecutive	ADJ
ap-1860	286	26	fibonacci	fibonacci	NOUN
ap-1860	286	27	numbers	number	NOUN
ap-1860	286	28	.	.	PUNCT
ap-1860	286	29	)	)	PUNCT
ap-1860	287	1	we	we	PRON
ap-1860	287	2	consider	consider	VERB
ap-1860	287	3	several	several	ADJ
ap-1860	287	4	examples	example	NOUN
ap-1860	287	5	of	of	ADP
ap-1860	287	6	positions	position	NOUN
ap-1860	287	7	of	of	ADP
ap-1860	287	8	the	the	DET
ap-1860	287	9	interval	interval	NOUN
ap-1860	287	10	i.	i.	PROPN
ap-1860	287	11	example	example	PROPN
ap-1860	287	12	3.1	3.1	NUM
ap-1860	287	13	.	.	PUNCT
ap-1860	288	1	let	let	VERB
ap-1860	288	2	c	c	NOUN
ap-1860	288	3	=	=	SYM
ap-1860	288	4	σ4	σ4	PROPN
ap-1860	288	5	.	.	PUNCT
ap-1860	289	1	then	then	ADV
ap-1860	289	2	ν	ν	X
ap-1860	289	3	=	=	SYM
ap-1860	289	4	t−1(α	t−1(α	PROPN
ap-1860	289	5	)	)	PUNCT
ap-1860	290	1	=	=	SYM
ap-1860	290	2	σ3	σ3	PROPN
ap-1860	290	3	∈	∈	PROPN
ap-1860	290	4	i0	i0	PROPN
ap-1860	290	5	splits	split	VERB
ap-1860	290	6	the	the	DET
ap-1860	290	7	interval	interval	NOUN
ap-1860	290	8	i0	i0	PROPN
ap-1860	290	9	into	into	ADP
ap-1860	290	10	i0	i0	PROPN
ap-1860	290	11	=	=	PUNCT
ap-1860	290	12	il0	il0	VERB
ap-1860	290	13	∪	∪	ADJ
ap-1860	290	14	ir0	ir0	NOUN
ap-1860	290	15	,	,	PUNCT
ap-1860	290	16	where	where	SCONJ
ap-1860	290	17	il0	il0	NOUN
ap-1860	290	18	=	=	SYM
ap-1860	291	1	[	[	X
ap-1860	291	2	σ4	σ4	NOUN
ap-1860	291	3	,	,	PUNCT
ap-1860	291	4	σ3	σ3	PROPN
ap-1860	291	5	)	)	PUNCT
ap-1860	291	6	,	,	PUNCT
ap-1860	291	7	ir0	ir0	NOUN
ap-1860	291	8	=	=	PUNCT
ap-1860	292	1	[	[	X
ap-1860	292	2	σ3	σ3	PROPN
ap-1860	292	3	,	,	PUNCT
ap-1860	292	4	σ3	σ3	NOUN
ap-1860	292	5	+	+	X
ap-1860	292	6	σ6	σ6	NOUN
ap-1860	292	7	)	)	PUNCT
ap-1860	292	8	.	.	PUNCT
ap-1860	293	1	the	the	DET
ap-1860	293	2	i	i	PROPN
ap-1860	293	3	-	-	PUNCT
ap-1860	293	4	itinerary	itinerary	ADJ
ap-1860	293	5	satisfies	satisfie	NOUN
ap-1860	293	6	r(x	r(x	PROPN
ap-1860	293	7	)	)	PUNCT
ap-1860	293	8	=	=	PUNCT
ap-1860	294	1			NUM
ap-1860	294	2	001	001	NUM
ap-1860	295	1	if	if	SCONJ
ap-1860	295	2	x	x	PROPN
ap-1860	295	3	∈	∈	PROPN
ap-1860	295	4	il0	il0	NOUN
ap-1860	295	5	,	,	PUNCT
ap-1860	295	6	010	010	NUM
ap-1860	295	7	if	if	SCONJ
ap-1860	295	8	x	x	PROPN
ap-1860	295	9	∈	∈	PROPN
ap-1860	295	10	ir0	ir0	NOUN
ap-1860	295	11	,	,	PUNCT
ap-1860	295	12	01001	01001	NUM
ap-1860	295	13	if	if	SCONJ
ap-1860	295	14	x	x	PROPN
ap-1860	295	15	∈	∈	PROPN
ap-1860	295	16	i1	i1	PROPN
ap-1860	295	17	,	,	PUNCT
ap-1860	295	18	01	01	NUM
ap-1860	295	19	if	if	SCONJ
ap-1860	295	20	x	x	X
ap-1860	295	21	∈	∈	PROPN
ap-1860	295	22	i2	i2	PROPN
ap-1860	295	23	.	.	PUNCT
ap-1860	296	1	we	we	PRON
ap-1860	296	2	put	put	VERB
ap-1860	296	3	r1	r1	PROPN
ap-1860	296	4	=	=	SYM
ap-1860	296	5	01	01	NUM
ap-1860	296	6	,	,	PUNCT
ap-1860	296	7	r2	r2	PROPN
ap-1860	296	8	=	=	SYM
ap-1860	296	9	001	001	NUM
ap-1860	296	10	,	,	PUNCT
ap-1860	296	11	r1r2	r1r2	ADJ
ap-1860	296	12	=	=	SYM
ap-1860	296	13	01001	01001	NUM
ap-1860	296	14	,	,	PUNCT
ap-1860	296	15	q	q	PROPN
ap-1860	296	16	=	=	SYM
ap-1860	296	17	010	010	NUM
ap-1860	296	18	,	,	PUNCT
ap-1860	296	19	where	where	SCONJ
ap-1860	296	20	q	q	NOUN
ap-1860	296	21	is	be	AUX
ap-1860	296	22	amicable	amicable	ADJ
ap-1860	296	23	with	with	ADP
ap-1860	296	24	r2	r2	PROPN
ap-1860	296	25	.	.	PUNCT
ap-1860	297	1	note	note	VERB
ap-1860	297	2	that	that	SCONJ
ap-1860	297	3	we	we	PRON
ap-1860	297	4	have	have	VERB
ap-1860	297	5	another	another	DET
ap-1860	297	6	choice	choice	NOUN
ap-1860	297	7	for	for	ADP
ap-1860	297	8	notation	notation	NOUN
ap-1860	297	9	,	,	PUNCT
ap-1860	297	10	r1	r1	PROPN
ap-1860	297	11	=	=	SYM
ap-1860	297	12	010	010	NUM
ap-1860	297	13	,	,	PUNCT
ap-1860	297	14	r2	r2	PROPN
ap-1860	297	15	=	=	SYM
ap-1860	297	16	01	01	NUM
ap-1860	297	17	,	,	PUNCT
ap-1860	297	18	r1r2	r1r2	ADJ
ap-1860	297	19	=	=	SYM
ap-1860	297	20	01001	01001	NUM
ap-1860	297	21	,	,	PUNCT
ap-1860	297	22	q	q	NOUN
ap-1860	297	23	=	=	NOUN
ap-1860	297	24	001	001	NUM
ap-1860	297	25	,	,	PUNCT
ap-1860	297	26	where	where	SCONJ
ap-1860	297	27	q	q	NOUN
ap-1860	297	28	is	be	AUX
ap-1860	297	29	amicable	amicable	ADJ
ap-1860	297	30	with	with	ADP
ap-1860	297	31	r1	r1	PROPN
ap-1860	297	32	.	.	PUNCT
ap-1860	297	33	example	example	NOUN
ap-1860	298	1	3.2	3.2	NUM
ap-1860	298	2	.	.	PUNCT
ap-1860	299	1	let	let	VERB
ap-1860	299	2	c	c	NOUN
ap-1860	299	3	=	=	SYM
ap-1860	299	4	σ6	σ6	PROPN
ap-1860	299	5	.	.	PUNCT
ap-1860	300	1	then	then	ADV
ap-1860	300	2	ν	ν	X
ap-1860	300	3	=	=	SYM
ap-1860	300	4	t−1(α	t−1(α	PROPN
ap-1860	300	5	)	)	PUNCT
ap-1860	301	1	=	=	SYM
ap-1860	301	2	σ3	σ3	PROPN
ap-1860	301	3	∈	∈	PROPN
ap-1860	301	4	i1	i1	PROPN
ap-1860	301	5	splits	split	VERB
ap-1860	301	6	the	the	DET
ap-1860	301	7	interval	interval	NOUN
ap-1860	301	8	i1	i1	PROPN
ap-1860	301	9	into	into	ADP
ap-1860	301	10	i1	i1	PROPN
ap-1860	301	11	=	=	PUNCT
ap-1860	302	1	il1	il1	PROPN
ap-1860	302	2	∪	∪	ADP
ap-1860	303	1	ir1	ir1	NOUN
ap-1860	303	2	,	,	PUNCT
ap-1860	303	3	where	where	SCONJ
ap-1860	303	4	il1	il1	NOUN
ap-1860	304	1	=	=	PUNCT
ap-1860	305	1	[	[	X
ap-1860	305	2	σ4	σ4	NOUN
ap-1860	305	3	+	+	X
ap-1860	305	4	σ6	σ6	PROPN
ap-1860	305	5	,	,	PUNCT
ap-1860	305	6	σ3	σ3	PROPN
ap-1860	305	7	)	)	PUNCT
ap-1860	305	8	,	,	PUNCT
ap-1860	305	9	ir1	ir1	VERB
ap-1860	305	10	=	=	PUNCT
ap-1860	306	1	[	[	X
ap-1860	306	2	σ3	σ3	PROPN
ap-1860	306	3	,	,	PUNCT
ap-1860	306	4	σ3	σ3	NOUN
ap-1860	306	5	+	+	X
ap-1860	306	6	σ6	σ6	NOUN
ap-1860	306	7	)	)	PUNCT
ap-1860	306	8	.	.	PUNCT
ap-1860	307	1	the	the	DET
ap-1860	307	2	i	i	PROPN
ap-1860	307	3	-	-	PUNCT
ap-1860	307	4	itinerary	itinerary	ADJ
ap-1860	307	5	satisfies	satisfie	NOUN
ap-1860	307	6	r(x	r(x	PROPN
ap-1860	307	7	)	)	PUNCT
ap-1860	307	8	=	=	PUNCT
ap-1860	308	1			NUM
ap-1860	308	2	001	001	NUM
ap-1860	309	1	if	if	SCONJ
ap-1860	309	2	x	x	PROPN
ap-1860	309	3	∈	∈	PROPN
ap-1860	309	4	i0	i0	PROPN
ap-1860	309	5	,	,	PUNCT
ap-1860	309	6	00101	00101	PUNCT
ap-1860	310	1	if	if	SCONJ
ap-1860	310	2	x	x	SYM
ap-1860	310	3	∈	∈	PROPN
ap-1860	310	4	il1	il1	X
ap-1860	310	5	,	,	PUNCT
ap-1860	310	6	01001	01001	NUM
ap-1860	310	7	if	if	SCONJ
ap-1860	310	8	x	x	SYM
ap-1860	310	9	∈	∈	PROPN
ap-1860	310	10	ir1	ir1	NOUN
ap-1860	310	11	,	,	PUNCT
ap-1860	310	12	01	01	NUM
ap-1860	310	13	if	if	SCONJ
ap-1860	310	14	x	x	X
ap-1860	310	15	∈	∈	PROPN
ap-1860	310	16	i2	i2	PROPN
ap-1860	310	17	.	.	PUNCT
ap-1860	311	1	we	we	PRON
ap-1860	311	2	put	put	VERB
ap-1860	311	3	r1	r1	NOUN
ap-1860	311	4	=	=	SYM
ap-1860	311	5	001	001	NUM
ap-1860	311	6	,	,	PUNCT
ap-1860	311	7	r2	r2	PROPN
ap-1860	311	8	=	=	SYM
ap-1860	311	9	01	01	NUM
ap-1860	311	10	,	,	PUNCT
ap-1860	311	11	r1r2	r1r2	ADJ
ap-1860	311	12	=	=	SYM
ap-1860	311	13	00101	00101	NUM
ap-1860	311	14	,	,	PUNCT
ap-1860	311	15	q	q	X
ap-1860	311	16	=	=	SYM
ap-1860	311	17	01001	01001	NUM
ap-1860	311	18	,	,	PUNCT
ap-1860	311	19	where	where	SCONJ
ap-1860	311	20	q	q	NOUN
ap-1860	311	21	=	=	PRON
ap-1860	311	22	r2r1	r2r1	NOUN
ap-1860	311	23	is	be	AUX
ap-1860	311	24	amicable	amicable	ADJ
ap-1860	311	25	with	with	ADP
ap-1860	311	26	r1r2	r1r2	PROPN
ap-1860	311	27	.	.	NOUN
ap-1860	311	28	example	example	NOUN
ap-1860	311	29	3.3	3.3	NUM
ap-1860	311	30	.	.	PUNCT
ap-1860	312	1	let	let	VERB
ap-1860	312	2	c	c	NOUN
ap-1860	312	3	=	=	SYM
ap-1860	312	4	σ3	σ3	PROPN
ap-1860	312	5	+	+	CCONJ
ap-1860	312	6	σ5	σ5	PROPN
ap-1860	312	7	+	+	CCONJ
ap-1860	312	8	σ7	σ7	VERB
ap-1860	312	9	.	.	PUNCT
ap-1860	313	1	then	then	ADV
ap-1860	313	2	ν	ν	X
ap-1860	313	3	=	=	SYM
ap-1860	313	4	t	t	PROPN
ap-1860	313	5	0(α	0(α	NUM
ap-1860	313	6	)	)	PUNCT
ap-1860	314	1	=	=	PUNCT
ap-1860	314	2	σ	σ	PROPN
ap-1860	314	3	∈	∈	PROPN
ap-1860	314	4	i2	i2	PROPN
ap-1860	314	5	splits	split	VERB
ap-1860	314	6	the	the	DET
ap-1860	314	7	interval	interval	NOUN
ap-1860	314	8	i2	i2	PROPN
ap-1860	314	9	into	into	ADP
ap-1860	314	10	i2	i2	PROPN
ap-1860	314	11	=	=	PUNCT
ap-1860	314	12	il2	il2	PROPN
ap-1860	314	13	∪ir2	∪ir2	NOUN
ap-1860	314	14	,	,	PUNCT
ap-1860	314	15	where	where	SCONJ
ap-1860	314	16	il2	il2	NOUN
ap-1860	314	17	=	=	PUNCT
ap-1860	315	1	[	[	X
ap-1860	315	2	σ2	σ2	NOUN
ap-1860	315	3	+	+	CCONJ
ap-1860	315	4	σ4	σ4	NOUN
ap-1860	315	5	+	+	CCONJ
ap-1860	315	6	σ6	σ6	NOUN
ap-1860	315	7	+	+	CCONJ
ap-1860	315	8	σ9	σ9	PROPN
ap-1860	315	9	,	,	PUNCT
ap-1860	315	10	σ	σ	PROPN
ap-1860	315	11	)	)	PUNCT
ap-1860	315	12	,	,	PUNCT
ap-1860	315	13	ir2	ir2	X
ap-1860	316	1	=	=	PUNCT
ap-1860	317	1	[	[	X
ap-1860	317	2	σ	σ	PROPN
ap-1860	317	3	,	,	PUNCT
ap-1860	317	4	σ	σ	PROPN
ap-1860	317	5	+	+	CCONJ
ap-1860	317	6	σ7	σ7	VERB
ap-1860	317	7	)	)	PUNCT
ap-1860	317	8	.	.	PUNCT
ap-1860	318	1	the	the	DET
ap-1860	318	2	i	i	PROPN
ap-1860	318	3	-	-	PUNCT
ap-1860	318	4	itinerary	itinerary	ADJ
ap-1860	318	5	satisfies	satisfie	NOUN
ap-1860	318	6	r(x	r(x	PROPN
ap-1860	318	7	)	)	PUNCT
ap-1860	318	8	=	=	PUNCT
ap-1860	319	1			PROPN
ap-1860	319	2	010	010	NUM
ap-1860	320	1	if	if	SCONJ
ap-1860	320	2	x	x	PROPN
ap-1860	320	3	∈	∈	PROPN
ap-1860	320	4	i0	i0	PROPN
ap-1860	320	5	,	,	PUNCT
ap-1860	320	6	01010	01010	NUM
ap-1860	321	1	if	if	SCONJ
ap-1860	321	2	x	x	PROPN
ap-1860	321	3	∈	∈	PROPN
ap-1860	321	4	i1	i1	PROPN
ap-1860	321	5	,	,	PUNCT
ap-1860	321	6	01	01	NUM
ap-1860	321	7	if	if	SCONJ
ap-1860	321	8	x	x	X
ap-1860	321	9	∈	∈	PROPN
ap-1860	321	10	il2	il2	NOUN
ap-1860	321	11	,	,	PUNCT
ap-1860	321	12	10	10	NUM
ap-1860	322	1	if	if	SCONJ
ap-1860	322	2	x	x	SYM
ap-1860	322	3	∈	∈	PROPN
ap-1860	322	4	ir2	ir2	INTJ
ap-1860	322	5	.	.	PUNCT
ap-1860	323	1	we	we	PRON
ap-1860	323	2	put	put	VERB
ap-1860	323	3	r1	r1	PROPN
ap-1860	323	4	=	=	SYM
ap-1860	323	5	01	01	NUM
ap-1860	323	6	,	,	PUNCT
ap-1860	323	7	r2	r2	PROPN
ap-1860	323	8	=	=	SYM
ap-1860	323	9	010	010	NUM
ap-1860	323	10	,	,	PUNCT
ap-1860	323	11	r1r2	r1r2	VERB
ap-1860	323	12	=	=	SYM
ap-1860	323	13	01010	01010	NUM
ap-1860	323	14	,	,	PUNCT
ap-1860	323	15	q	q	NOUN
ap-1860	324	1	=	=	NUM
ap-1860	324	2	10	10	NUM
ap-1860	324	3	,	,	PUNCT
ap-1860	324	4	where	where	SCONJ
ap-1860	324	5	q	q	NOUN
ap-1860	324	6	is	be	AUX
ap-1860	324	7	amicable	amicable	ADJ
ap-1860	324	8	with	with	ADP
ap-1860	324	9	r1	r1	PROPN
ap-1860	324	10	,	,	PUNCT
ap-1860	324	11	or	or	CCONJ
ap-1860	324	12	r1	r1	PROPN
ap-1860	324	13	=	=	SYM
ap-1860	324	14	010	010	NUM
ap-1860	324	15	,	,	PUNCT
ap-1860	324	16	r2	r2	PROPN
ap-1860	324	17	=	=	SYM
ap-1860	324	18	10	10	NUM
ap-1860	324	19	,	,	PUNCT
ap-1860	324	20	r1r2	r1r2	VERB
ap-1860	324	21	=	=	SYM
ap-1860	324	22	01010	01010	NUM
ap-1860	324	23	,	,	PUNCT
ap-1860	324	24	q	q	PROPN
ap-1860	325	1	=	=	SYM
ap-1860	325	2	01	01	NUM
ap-1860	325	3	,	,	PUNCT
ap-1860	325	4	where	where	SCONJ
ap-1860	325	5	q	q	NOUN
ap-1860	325	6	is	be	AUX
ap-1860	325	7	amicable	amicable	ADJ
ap-1860	325	8	with	with	ADP
ap-1860	325	9	r2	r2	PROPN
ap-1860	325	10	.	.	PUNCT
ap-1860	326	1	let	let	VERB
ap-1860	326	2	us	we	PRON
ap-1860	326	3	discuss	discuss	VERB
ap-1860	326	4	the	the	DET
ap-1860	326	5	cases	case	NOUN
ap-1860	326	6	for	for	ADP
ap-1860	326	7	which	which	PRON
ap-1860	326	8	#	#	SYM
ap-1860	326	9	iti	iti	NOUN
ap-1860	326	10	<	<	X
ap-1860	326	11	4	4	NUM
ap-1860	326	12	.	.	PUNCT
ap-1860	327	1	this	this	PRON
ap-1860	327	2	can	can	AUX
ap-1860	327	3	happen	happen	VERB
ap-1860	327	4	if	if	SCONJ
ap-1860	327	5	d−	d−	PROPN
ap-1860	327	6	c	c	PROPN
ap-1860	327	7	6=	6=	PROPN
ap-1860	327	8	δk	δk	PROPN
ap-1860	327	9	,	,	PUNCT
ap-1860	327	10	s	s	PART
ap-1860	327	11	,	,	PUNCT
ap-1860	327	12	(	(	PUNCT
ap-1860	327	13	i.e.	i.e.	X
ap-1860	327	14	,	,	PUNCT
ap-1860	327	15	ti	ti	PROPN
ap-1860	327	16	is	be	AUX
ap-1860	327	17	still	still	ADV
ap-1860	327	18	an	an	DET
ap-1860	327	19	exchange	exchange	NOUN
ap-1860	327	20	of	of	ADP
ap-1860	327	21	three	three	NUM
ap-1860	327	22	intervals	interval	NOUN
ap-1860	327	23	)	)	PUNCT
ap-1860	327	24	,	,	PUNCT
ap-1860	327	25	but	but	CCONJ
ap-1860	327	26	ν	ν	X
ap-1860	327	27	∈	∈	PROPN
ap-1860	327	28	{	{	PUNCT
ap-1860	327	29	c	c	NOUN
ap-1860	327	30	,	,	PUNCT
ap-1860	327	31	λ	λ	PROPN
ap-1860	327	32	,	,	PUNCT
ap-1860	327	33	µ	µ	NOUN
ap-1860	327	34	}	}	PUNCT
ap-1860	327	35	.	.	PUNCT
ap-1860	328	1	it	it	PRON
ap-1860	328	2	can	can	AUX
ap-1860	328	3	be	be	AUX
ap-1860	328	4	derived	derive	VERB
ap-1860	328	5	from	from	ADP
ap-1860	328	6	the	the	DET
ap-1860	328	7	proof	proof	NOUN
ap-1860	328	8	of	of	ADP
ap-1860	328	9	theorem	theorem	ADJ
ap-1860	328	10	1.1	1.1	NUM
ap-1860	328	11	,	,	PUNCT
ap-1860	328	12	that	that	SCONJ
ap-1860	328	13	,	,	PUNCT
ap-1860	328	14	in	in	ADP
ap-1860	328	15	this	this	DET
ap-1860	328	16	case	case	NOUN
ap-1860	328	17	,	,	PUNCT
ap-1860	328	18	the	the	DET
ap-1860	328	19	set	set	NOUN
ap-1860	328	20	of	of	ADP
ap-1860	328	21	i	i	PROPN
ap-1860	328	22	-	-	PUNCT
ap-1860	328	23	itineraries	itinerary	NOUN
ap-1860	328	24	is	be	AUX
ap-1860	328	25	of	of	ADP
ap-1860	328	26	the	the	DET
ap-1860	328	27	form	form	NOUN
ap-1860	328	28	iti	iti	NOUN
ap-1860	328	29	=	=	PUNCT
ap-1860	328	30	{	{	PUNCT
ap-1860	328	31	r1	r1	PROPN
ap-1860	328	32	,	,	PUNCT
ap-1860	328	33	r2	r2	PROPN
ap-1860	328	34	,	,	PUNCT
ap-1860	328	35	r1r2	r1r2	VERB
ap-1860	328	36	}	}	PUNCT
ap-1860	328	37	.	.	PUNCT
ap-1860	329	1	note	note	VERB
ap-1860	329	2	that	that	SCONJ
ap-1860	329	3	c	c	AUX
ap-1860	329	4	=	=	SYM
ap-1860	329	5	0	0	NUM
ap-1860	329	6	is	be	AUX
ap-1860	329	7	a	a	DET
ap-1860	329	8	special	special	ADJ
ap-1860	329	9	case	case	NOUN
ap-1860	329	10	of	of	ADP
ap-1860	329	11	such	such	ADJ
ap-1860	329	12	situation	situation	NOUN
ap-1860	329	13	.	.	PUNCT
ap-1860	330	1	for	for	ADP
ap-1860	330	2	,	,	PUNCT
ap-1860	330	3	we	we	PRON
ap-1860	330	4	have	have	VERB
ap-1860	330	5	c	c	NOUN
ap-1860	330	6	=	=	SYM
ap-1860	330	7	0	0	PUNCT
ap-1860	331	1	=	=	SYM
ap-1860	331	2	t	t	PROPN
ap-1860	331	3	(	(	PUNCT
ap-1860	331	4	α	α	NOUN
ap-1860	331	5	)	)	PUNCT
ap-1860	331	6	,	,	PUNCT
ap-1860	331	7	whence	whence	ADP
ap-1860	331	8	µ	µ	NOUN
ap-1860	331	9	=	=	SYM
ap-1860	331	10	t−m(0	t−m(0	NOUN
ap-1860	331	11	)	)	PUNCT
ap-1860	331	12	=	=	SYM
ap-1860	331	13	t−m+1(α	t−m+1(α	X
ap-1860	331	14	)	)	PUNCT
ap-1860	331	15	=	=	SYM
ap-1860	332	1	ν	ν	X
ap-1860	332	2	.	.	PUNCT
ap-1860	332	3	similarly	similarly	ADV
ap-1860	332	4	,	,	PUNCT
ap-1860	332	5	the	the	DET
ap-1860	332	6	case	case	NOUN
ap-1860	332	7	d	d	X
ap-1860	332	8	=	=	SYM
ap-1860	332	9	1	1	NUM
ap-1860	332	10	corresponds	correspond	VERB
ap-1860	332	11	to	to	ADP
ap-1860	332	12	λ	λ	PROPN
ap-1860	332	13	=	=	SYM
ap-1860	332	14	ν	ν	PROPN
ap-1860	332	15	.	.	PROPN
ap-1860	332	16	example	example	NOUN
ap-1860	332	17	3.4	3.4	NUM
ap-1860	332	18	.	.	PUNCT
ap-1860	333	1	let	let	VERB
ap-1860	333	2	c	c	NOUN
ap-1860	333	3	=	=	SYM
ap-1860	333	4	σ2	σ2	PROPN
ap-1860	333	5	,	,	PUNCT
ap-1860	333	6	d	d	NOUN
ap-1860	333	7	−	−	PROPN
ap-1860	333	8	c	c	NOUN
ap-1860	333	9	=	=	SYM
ap-1860	333	10	σ3	σ3	PROPN
ap-1860	333	11	+	+	X
ap-1860	333	12	σ6	σ6	NOUN
ap-1860	333	13	.	.	PUNCT
ap-1860	334	1	then	then	ADV
ap-1860	334	2	ν	ν	X
ap-1860	334	3	=	=	SYM
ap-1860	334	4	t	t	PROPN
ap-1860	334	5	0(α	0(α	NUM
ap-1860	334	6	)	)	PUNCT
ap-1860	335	1	=	=	SYM
ap-1860	335	2	σ	σ	NOUN
ap-1860	335	3	=	=	PUNCT
ap-1860	335	4	µ.	µ.	NOUN
ap-1860	335	5	the	the	DET
ap-1860	335	6	i	i	NOUN
ap-1860	335	7	-	-	PUNCT
ap-1860	335	8	itinerary	itinerary	ADJ
ap-1860	335	9	satisfies	satisfie	NOUN
ap-1860	335	10	r(x	r(x	PROPN
ap-1860	335	11	)	)	PUNCT
ap-1860	336	1	=	=	PUNCT
ap-1860	337	1			NOUN
ap-1860	337	2	010	010	NUM
ap-1860	338	1	if	if	SCONJ
ap-1860	338	2	x	x	PROPN
ap-1860	338	3	∈	∈	PROPN
ap-1860	338	4	i0	i0	PROPN
ap-1860	338	5	,	,	PUNCT
ap-1860	338	6	01010	01010	NUM
ap-1860	339	1	if	if	SCONJ
ap-1860	339	2	x	x	PROPN
ap-1860	339	3	∈	∈	PROPN
ap-1860	339	4	i1	i1	PROPN
ap-1860	339	5	,	,	PUNCT
ap-1860	339	6	10	10	NUM
ap-1860	339	7	if	if	SCONJ
ap-1860	339	8	x	x	X
ap-1860	339	9	∈	∈	PROPN
ap-1860	339	10	i2	i2	PROPN
ap-1860	339	11	.	.	PUNCT
ap-1860	339	12	with	with	ADP
ap-1860	339	13	r1	r1	PROPN
ap-1860	339	14	=	=	SYM
ap-1860	339	15	010	010	NUM
ap-1860	339	16	,	,	PUNCT
ap-1860	339	17	r2	r2	PROPN
ap-1860	339	18	=	=	SYM
ap-1860	339	19	10	10	NUM
ap-1860	339	20	,	,	PUNCT
ap-1860	339	21	we	we	PRON
ap-1860	339	22	have	have	VERB
ap-1860	339	23	iti	iti	NOUN
ap-1860	339	24	=	=	SYM
ap-1860	339	25	{	{	PUNCT
ap-1860	339	26	r1	r1	PROPN
ap-1860	339	27	,	,	PUNCT
ap-1860	339	28	r2	r2	PROPN
ap-1860	339	29	,	,	PUNCT
ap-1860	339	30	r1r2	r1r2	VERB
ap-1860	339	31	}	}	PUNCT
ap-1860	339	32	.	.	PUNCT
ap-1860	340	1	consider	consider	VERB
ap-1860	340	2	the	the	DET
ap-1860	340	3	situation	situation	NOUN
ap-1860	340	4	that	that	SCONJ
ap-1860	341	1	d	d	ADP
ap-1860	341	2	−	−	X
ap-1860	341	3	c	c	NOUN
ap-1860	341	4	=	=	SYM
ap-1860	341	5	δk	δk	PROPN
ap-1860	341	6	,	,	PUNCT
ap-1860	341	7	s	s	X
ap-1860	341	8	for	for	ADP
ap-1860	341	9	some	some	DET
ap-1860	341	10	k	k	NOUN
ap-1860	341	11	,	,	PUNCT
ap-1860	341	12	s	s	X
ap-1860	341	13	as	as	SCONJ
ap-1860	341	14	defined	define	VERB
ap-1860	341	15	in	in	ADP
ap-1860	341	16	(	(	PUNCT
ap-1860	341	17	5	5	NUM
ap-1860	341	18	)	)	PUNCT
ap-1860	341	19	.	.	PUNCT
ap-1860	342	1	by	by	ADP
ap-1860	342	2	proposition	proposition	NOUN
ap-1860	342	3	2.3	2.3	NUM
ap-1860	342	4	,	,	PUNCT
ap-1860	342	5	the	the	DET
ap-1860	342	6	induced	induced	ADJ
ap-1860	342	7	map	map	NOUN
ap-1860	342	8	ti	ti	NOUN
ap-1860	342	9	is	be	AUX
ap-1860	342	10	an	an	DET
ap-1860	342	11	exchange	exchange	NOUN
ap-1860	342	12	of	of	ADP
ap-1860	342	13	two	two	NUM
ap-1860	342	14	intervals	interval	NOUN
ap-1860	342	15	,	,	PUNCT
ap-1860	342	16	since	since	SCONJ
ap-1860	342	17	λ	λ	X
ap-1860	342	18	=	=	PRON
ap-1860	342	19	µ.	µ.	NOUN
ap-1860	342	20	the	the	DET
ap-1860	342	21	set	set	NOUN
ap-1860	342	22	of	of	ADP
ap-1860	342	23	i	i	PROPN
ap-1860	342	24	-	-	PUNCT
ap-1860	342	25	itineraries	itinerary	NOUN
ap-1860	342	26	is	be	AUX
ap-1860	342	27	then	then	ADV
ap-1860	342	28	either	either	CCONJ
ap-1860	342	29	iti	iti	PROPN
ap-1860	343	1	=	=	PUNCT
ap-1860	343	2	{	{	PUNCT
ap-1860	343	3	r1	r1	PROPN
ap-1860	343	4	,	,	PUNCT
ap-1860	343	5	r2	r2	PROPN
ap-1860	343	6	}	}	PUNCT
ap-1860	343	7	,	,	PUNCT
ap-1860	343	8	which	which	PRON
ap-1860	343	9	happens	happen	VERB
ap-1860	343	10	if	if	SCONJ
ap-1860	343	11	ν	ν	PROPN
ap-1860	343	12	∈	∈	PROPN
ap-1860	343	13	{	{	PUNCT
ap-1860	343	14	c	c	NOUN
ap-1860	343	15	,	,	PUNCT
ap-1860	343	16	λ	λ	NOUN
ap-1860	343	17	}	}	PUNCT
ap-1860	343	18	,	,	PUNCT
ap-1860	343	19	or	or	CCONJ
ap-1860	343	20	iti	iti	NOUN
ap-1860	343	21	=	=	SYM
ap-1860	343	22	{	{	PUNCT
ap-1860	343	23	r1	r1	PROPN
ap-1860	343	24	,	,	PUNCT
ap-1860	343	25	r2	r2	PROPN
ap-1860	343	26	,	,	PUNCT
ap-1860	343	27	q	q	NOUN
ap-1860	343	28	}	}	PUNCT
ap-1860	343	29	,	,	PUNCT
ap-1860	343	30	where	where	SCONJ
ap-1860	343	31	q	q	NOUN
ap-1860	343	32	is	be	AUX
ap-1860	343	33	amicable	amicable	ADJ
ap-1860	343	34	with	with	ADP
ap-1860	343	35	r1	r1	PROPN
ap-1860	343	36	or	or	CCONJ
ap-1860	343	37	with	with	ADP
ap-1860	343	38	r2	r2	NOUN
ap-1860	343	39	,	,	PUNCT
ap-1860	343	40	according	accord	VERB
ap-1860	343	41	to	to	ADP
ap-1860	343	42	the	the	DET
ap-1860	343	43	position	position	NOUN
ap-1860	343	44	of	of	ADP
ap-1860	343	45	ν	ν	NOUN
ap-1860	343	46	in	in	ADP
ap-1860	343	47	the	the	DET
ap-1860	343	48	interval	interval	NOUN
ap-1860	343	49	i.	i.	NOUN
ap-1860	343	50	4	4	NUM
ap-1860	343	51	.	.	PUNCT
ap-1860	344	1	conclusions	conclusion	NOUN
ap-1860	344	2	notions	notion	NOUN
ap-1860	344	3	such	such	ADJ
ap-1860	344	4	as	as	ADP
ap-1860	344	5	return	return	NOUN
ap-1860	344	6	time	time	NOUN
ap-1860	344	7	,	,	PUNCT
ap-1860	344	8	return	return	VERB
ap-1860	344	9	itinerary	itinerary	NOUN
ap-1860	344	10	,	,	PUNCT
ap-1860	344	11	first	first	ADJ
ap-1860	344	12	return	return	NOUN
ap-1860	344	13	map	map	NOUN
ap-1860	344	14	,	,	PUNCT
ap-1860	344	15	etc	etc	X
ap-1860	344	16	.	.	X
ap-1860	344	17	for	for	ADP
ap-1860	344	18	the	the	DET
ap-1860	344	19	exchange	exchange	NOUN
ap-1860	344	20	of	of	ADP
ap-1860	344	21	two	two	NUM
ap-1860	344	22	intervals	interval	NOUN
ap-1860	344	23	have	have	AUX
ap-1860	344	24	been	be	AUX
ap-1860	344	25	studied	study	VERB
ap-1860	344	26	by	by	ADP
ap-1860	344	27	many	many	ADJ
ap-1860	344	28	authors	author	NOUN
ap-1860	344	29	.	.	PUNCT
ap-1860	345	1	for	for	ADP
ap-1860	345	2	an	an	DET
ap-1860	345	3	overview	overview	NOUN
ap-1860	345	4	,	,	PUNCT
ap-1860	345	5	see	see	VERB
ap-1860	345	6	for	for	ADP
ap-1860	345	7	example	example	NOUN
ap-1860	345	8	[	[	X
ap-1860	345	9	14	14	NUM
ap-1860	345	10	]	]	PUNCT
ap-1860	345	11	.	.	PUNCT
ap-1860	346	1	this	this	DET
ap-1860	346	2	notion	notion	NOUN
ap-1860	346	3	occurs	occur	VERB
ap-1860	346	4	in	in	ADP
ap-1860	346	5	various	various	ADJ
ap-1860	346	6	contexts	context	NOUN
ap-1860	346	7	such	such	ADJ
ap-1860	346	8	as	as	ADP
ap-1860	346	9	return	return	NOUN
ap-1860	346	10	words	word	NOUN
ap-1860	346	11	,	,	PUNCT
ap-1860	346	12	abelian	abelian	PROPN
ap-1860	346	13	return	return	NOUN
ap-1860	346	14	words	word	NOUN
ap-1860	346	15	,	,	PUNCT
ap-1860	346	16	or	or	CCONJ
ap-1860	346	17	substitution	substitution	NOUN
ap-1860	346	18	invariance	invariance	NOUN
ap-1860	346	19	of	of	ADP
ap-1860	346	20	the	the	DET
ap-1860	346	21	corresponding	corresponding	ADJ
ap-1860	346	22	codings	coding	NOUN
ap-1860	346	23	,	,	PUNCT
ap-1860	346	24	i.e.	i.e.	X
ap-1860	346	25	,	,	PUNCT
ap-1860	346	26	sturmian	sturmian	NOUN
ap-1860	346	27	words	word	NOUN
ap-1860	346	28	.	.	PUNCT
ap-1860	347	1	the	the	DET
ap-1860	347	2	many	many	ADJ
ap-1860	347	3	equivalent	equivalent	ADJ
ap-1860	347	4	definitions	definition	NOUN
ap-1860	347	5	of	of	ADP
ap-1860	347	6	sturmian	sturmian	ADJ
ap-1860	347	7	words	word	NOUN
ap-1860	347	8	allow	allow	VERB
ap-1860	347	9	one	one	PRON
ap-1860	347	10	to	to	PART
ap-1860	347	11	combine	combine	VERB
ap-1860	347	12	different	different	ADJ
ap-1860	347	13	points	point	NOUN
ap-1860	347	14	of	of	ADP
ap-1860	347	15	view	view	NOUN
ap-1860	347	16	which	which	PRON
ap-1860	347	17	contributes	contribute	VERB
ap-1860	347	18	substantially	substantially	ADV
ap-1860	347	19	to	to	ADP
ap-1860	347	20	the	the	DET
ap-1860	347	21	solution	solution	NOUN
ap-1860	347	22	of	of	ADP
ap-1860	347	23	such	such	ADJ
ap-1860	347	24	problems	problem	NOUN
ap-1860	347	25	.	.	PUNCT
ap-1860	348	1	a	a	DET
ap-1860	348	2	detailed	detailed	ADJ
ap-1860	348	3	solution	solution	NOUN
ap-1860	348	4	of	of	ADP
ap-1860	348	5	analogous	analogous	ADJ
ap-1860	348	6	questions	question	NOUN
ap-1860	348	7	for	for	ADP
ap-1860	348	8	exchanges	exchange	NOUN
ap-1860	348	9	of	of	ADP
ap-1860	348	10	more	more	ADJ
ap-1860	348	11	than	than	ADP
ap-1860	348	12	two	two	NUM
ap-1860	348	13	intervals	interval	NOUN
ap-1860	348	14	is	be	AUX
ap-1860	348	15	still	still	ADV
ap-1860	348	16	unknown	unknown	ADJ
ap-1860	348	17	.	.	PUNCT
ap-1860	349	1	we	we	PRON
ap-1860	349	2	believe	believe	VERB
ap-1860	349	3	that	that	SCONJ
ap-1860	349	4	at	at	ADP
ap-1860	349	5	least	least	ADJ
ap-1860	349	6	for	for	ADP
ap-1860	349	7	exchanges	exchange	NOUN
ap-1860	349	8	of	of	ADP
ap-1860	349	9	three	three	NUM
ap-1860	349	10	intervals	interval	NOUN
ap-1860	349	11	one	one	PRON
ap-1860	349	12	can	can	AUX
ap-1860	349	13	obtain	obtain	VERB
ap-1860	349	14	an	an	DET
ap-1860	349	15	explicit	explicit	ADJ
ap-1860	349	16	description	description	NOUN
ap-1860	349	17	of	of	ADP
ap-1860	349	18	return	return	NOUN
ap-1860	349	19	times	time	NOUN
ap-1860	349	20	and	and	CCONJ
ap-1860	349	21	return	return	VERB
ap-1860	349	22	itineraries	itinerary	NOUN
ap-1860	349	23	,	,	PUNCT
ap-1860	349	24	since	since	SCONJ
ap-1860	349	25	the	the	DET
ap-1860	349	26	corresponding	corresponding	ADJ
ap-1860	349	27	codings	coding	NOUN
ap-1860	349	28	are	be	AUX
ap-1860	349	29	geometrically	geometrically	ADV
ap-1860	349	30	representable	representable	ADJ
ap-1860	349	31	by	by	ADP
ap-1860	349	32	cut	cut	VERB
ap-1860	349	33	-	-	PUNCT
ap-1860	349	34	andproject	andproject	NOUN
ap-1860	349	35	sequences	sequence	NOUN
ap-1860	349	36	,	,	PUNCT
ap-1860	349	37	in	in	ADP
ap-1860	349	38	a	a	DET
ap-1860	349	39	similar	similar	ADJ
ap-1860	349	40	way	way	NOUN
ap-1860	349	41	that	that	PRON
ap-1860	349	42	sturmian	sturmian	NOUN
ap-1860	349	43	words	word	NOUN
ap-1860	349	44	are	be	AUX
ap-1860	349	45	identified	identify	VERB
ap-1860	349	46	with	with	ADP
ap-1860	349	47	mechanical	mechanical	ADJ
ap-1860	349	48	words	word	NOUN
ap-1860	349	49	.	.	PUNCT
ap-1860	350	1	acknowledgements	acknowledgement	NOUN
ap-1860	350	2	the	the	DET
ap-1860	350	3	results	result	NOUN
ap-1860	350	4	presented	present	VERB
ap-1860	350	5	in	in	ADP
ap-1860	350	6	this	this	DET
ap-1860	350	7	paper	paper	NOUN
ap-1860	350	8	,	,	PUNCT
ap-1860	350	9	as	as	ADV
ap-1860	350	10	well	well	ADV
ap-1860	350	11	as	as	ADP
ap-1860	350	12	other	other	ADJ
ap-1860	350	13	results	result	NOUN
ap-1860	350	14	about	about	ADP
ap-1860	350	15	exchange	exchange	NOUN
ap-1860	350	16	of	of	ADP
ap-1860	350	17	intervals	interval	NOUN
ap-1860	350	18	,	,	PUNCT
ap-1860	350	19	have	have	AUX
ap-1860	350	20	been	be	AUX
ap-1860	350	21	obtained	obtain	VERB
ap-1860	350	22	with	with	ADP
ap-1860	350	23	the	the	DET
ap-1860	350	24	use	use	NOUN
ap-1860	350	25	of	of	ADP
ap-1860	350	26	geometric	geometric	ADJ
ap-1860	350	27	representation	representation	NOUN
ap-1860	350	28	of	of	ADP
ap-1860	350	29	the	the	DET
ap-1860	350	30	associated	associated	ADJ
ap-1860	350	31	codings	coding	NOUN
ap-1860	350	32	in	in	ADP
ap-1860	350	33	the	the	DET
ap-1860	350	34	frame	frame	NOUN
ap-1860	350	35	of	of	ADP
ap-1860	350	36	cut	cut	NOUN
ap-1860	350	37	-	-	PUNCT
ap-1860	350	38	and	and	CCONJ
ap-1860	350	39	-	-	PUNCT
ap-1860	350	40	project	project	NOUN
ap-1860	350	41	scheme	scheme	NOUN
ap-1860	350	42	,	,	PUNCT
ap-1860	350	43	see	see	VERB
ap-1860	350	44	[	[	X
ap-1860	350	45	10	10	NUM
ap-1860	350	46	]	]	PUNCT
ap-1860	350	47	.	.	PUNCT
ap-1860	351	1	we	we	PRON
ap-1860	351	2	were	be	AUX
ap-1860	351	3	lead	lead	ADJ
ap-1860	351	4	to	to	ADP
ap-1860	351	5	this	this	DET
ap-1860	351	6	topic	topic	NOUN
ap-1860	351	7	from	from	ADP
ap-1860	351	8	the	the	DET
ap-1860	351	9	study	study	NOUN
ap-1860	351	10	of	of	ADP
ap-1860	351	11	mathematical	mathematical	ADJ
ap-1860	351	12	models	model	NOUN
ap-1860	351	13	of	of	ADP
ap-1860	351	14	quasicrystals	quasicrystal	NOUN
ap-1860	351	15	,	,	PUNCT
ap-1860	351	16	based	base	VERB
ap-1860	351	17	on	on	ADP
ap-1860	351	18	the	the	DET
ap-1860	351	19	rich	rich	ADJ
ap-1860	351	20	collaboration	collaboration	NOUN
ap-1860	351	21	of	of	ADP
ap-1860	351	22	our	our	PRON
ap-1860	351	23	institute	institute	NOUN
ap-1860	351	24	with	with	ADP
ap-1860	351	25	jiří	jiří	NOUN
ap-1860	351	26	patera	patera	NOUN
ap-1860	351	27	from	from	ADP
ap-1860	351	28	centre	centre	NOUN
ap-1860	351	29	de	de	X
ap-1860	351	30	recherches	recherche	NOUN
ap-1860	351	31	mathématiques	mathématique	NOUN
ap-1860	351	32	in	in	ADP
ap-1860	351	33	montréal	montréal	PROPN
ap-1860	351	34	.	.	PUNCT
ap-1860	352	1	this	this	DET
ap-1860	352	2	collaboration	collaboration	NOUN
ap-1860	352	3	was	be	AUX
ap-1860	352	4	initiated	initiate	VERB
ap-1860	352	5	,	,	PUNCT
ap-1860	352	6	encouraged	encourage	VERB
ap-1860	352	7	and	and	CCONJ
ap-1860	352	8	is	be	AUX
ap-1860	352	9	still	still	ADV
ap-1860	352	10	maintained	maintain	VERB
ap-1860	352	11	by	by	ADP
ap-1860	352	12	prof	prof	PROPN
ap-1860	352	13	.	.	PUNCT
ap-1860	353	1	miloslav	miloslav	PROPN
ap-1860	353	2	448	448	NUM
ap-1860	353	3	vol	vol	NOUN
ap-1860	353	4	.	.	PUNCT
ap-1860	354	1	53	53	NUM
ap-1860	354	2	no	no	NOUN
ap-1860	354	3	.	.	PUNCT
ap-1860	355	1	5/2013	5/2013	NUM
ap-1860	355	2	itineraries	itinerary	NOUN
ap-1860	355	3	induced	induce	VERB
ap-1860	355	4	by	by	ADP
ap-1860	355	5	exchange	exchange	NOUN
ap-1860	355	6	of	of	ADP
ap-1860	355	7	two	two	NUM
ap-1860	355	8	intervals	interval	NOUN
ap-1860	355	9	havlíček	havlíček	VERB
ap-1860	355	10	,	,	PUNCT
ap-1860	355	11	to	to	ADP
ap-1860	355	12	whose	whose	DET
ap-1860	355	13	honour	honour	VERB
ap-1860	355	14	this	this	DET
ap-1860	355	15	issue	issue	NOUN
ap-1860	355	16	of	of	ADP
ap-1860	355	17	acta	acta	PROPN
ap-1860	355	18	polytechnica	polytechnica	PROPN
ap-1860	355	19	is	be	AUX
ap-1860	355	20	published	publish	VERB
ap-1860	355	21	.	.	PUNCT
ap-1860	356	1	we	we	PRON
ap-1860	356	2	acknowledge	acknowledge	VERB
ap-1860	356	3	financial	financial	ADJ
ap-1860	356	4	support	support	NOUN
ap-1860	356	5	from	from	ADP
ap-1860	356	6	czech	czech	PROPN
ap-1860	356	7	science	science	NOUN
ap-1860	356	8	foundation	foundation	NOUN
ap-1860	356	9	grant	grant	VERB
ap-1860	356	10	13	13	NUM
ap-1860	356	11	-	-	PUNCT
ap-1860	356	12	03538s	03538s	NUM
ap-1860	356	13	.	.	PUNCT
ap-1860	357	1	references	reference	NOUN
ap-1860	357	2	[	[	X
ap-1860	357	3	1	1	NUM
ap-1860	357	4	]	]	X
ap-1860	357	5	o.	o.	PROPN
ap-1860	357	6	m.	m.	PROPN
ap-1860	357	7	sharkovskii	sharkovskii	PROPN
ap-1860	357	8	.	.	PUNCT
ap-1860	358	1	co	co	NOUN
ap-1860	359	1	-	-	NOUN
ap-1860	359	2	existence	existence	NOUN
ap-1860	359	3	of	of	ADP
ap-1860	359	4	cycles	cycle	NOUN
ap-1860	359	5	of	of	ADP
ap-1860	359	6	a	a	DET
ap-1860	359	7	continuous	continuous	ADJ
ap-1860	359	8	mapping	mapping	NOUN
ap-1860	359	9	of	of	ADP
ap-1860	359	10	the	the	DET
ap-1860	359	11	line	line	NOUN
ap-1860	359	12	into	into	ADP
ap-1860	359	13	itself	itself	PRON
ap-1860	359	14	.	.	PUNCT
ap-1860	360	1	ukrain	ukrain	PROPN
ap-1860	360	2	mat	mat	PROPN
ap-1860	360	3	z̆	z̆	PROPN
ap-1860	360	4	16:61–71	16:61–71	NUM
ap-1860	360	5	,	,	PUNCT
ap-1860	360	6	1964	1964	NUM
ap-1860	360	7	.	.	PUNCT
ap-1860	361	1	[	[	X
ap-1860	361	2	2	2	X
ap-1860	361	3	]	]	PUNCT
ap-1860	361	4	j.	j.	PROPN
ap-1860	361	5	berstel	berstel	PROPN
ap-1860	361	6	.	.	PUNCT
ap-1860	362	1	sturmian	sturmian	PROPN
ap-1860	362	2	and	and	CCONJ
ap-1860	362	3	episturmian	episturmian	ADJ
ap-1860	362	4	words	word	NOUN
ap-1860	362	5	(	(	PUNCT
ap-1860	362	6	a	a	DET
ap-1860	362	7	survey	survey	NOUN
ap-1860	362	8	of	of	ADP
ap-1860	362	9	some	some	DET
ap-1860	362	10	recent	recent	ADJ
ap-1860	362	11	results	result	NOUN
ap-1860	362	12	)	)	PUNCT
ap-1860	362	13	.	.	PUNCT
ap-1860	363	1	in	in	ADP
ap-1860	363	2	algebraic	algebraic	PROPN
ap-1860	363	3	informatics	informatic	NOUN
ap-1860	363	4	,	,	PUNCT
ap-1860	363	5	vol	vol	NOUN
ap-1860	363	6	.	.	PUNCT
ap-1860	363	7	4728	4728	NUM
ap-1860	363	8	of	of	ADP
ap-1860	363	9	lecture	lecture	NOUN
ap-1860	363	10	notes	note	NOUN
ap-1860	363	11	in	in	ADP
ap-1860	363	12	comput	comput	NOUN
ap-1860	363	13	.	.	PUNCT
ap-1860	364	1	sci	sci	PROPN
ap-1860	364	2	.	.	PROPN
ap-1860	364	3	,	,	PUNCT
ap-1860	364	4	pp	pp	PROPN
ap-1860	364	5	.	.	PUNCT
ap-1860	365	1	23–47	23–47	NUM
ap-1860	365	2	.	.	PUNCT
ap-1860	365	3	springer	springer	NOUN
ap-1860	365	4	,	,	PUNCT
ap-1860	365	5	berlin	berlin	PROPN
ap-1860	365	6	,	,	PUNCT
ap-1860	365	7	2007	2007	NUM
ap-1860	365	8	.	.	PUNCT
ap-1860	366	1	[	[	X
ap-1860	366	2	3	3	X
ap-1860	366	3	]	]	X
ap-1860	366	4	l.	l.	PROPN
ap-1860	366	5	balková	balková	PROPN
ap-1860	366	6	,	,	PUNCT
ap-1860	366	7	e.	e.	PROPN
ap-1860	366	8	pelantová	pelantová	PROPN
ap-1860	366	9	,	,	PUNCT
ap-1860	366	10	v.	v.	PROPN
ap-1860	366	11	starosta	starosta	PROPN
ap-1860	366	12	.	.	PUNCT
ap-1860	367	1	sturmian	sturmian	NOUN
ap-1860	367	2	jungle	jungle	NOUN
ap-1860	367	3	(	(	PUNCT
ap-1860	367	4	or	or	CCONJ
ap-1860	367	5	garden	garden	NOUN
ap-1860	367	6	?	?	PUNCT
ap-1860	367	7	)	)	PUNCT
ap-1860	367	8	on	on	ADP
ap-1860	367	9	multiliteral	multiliteral	ADJ
ap-1860	367	10	alphabets	alphabet	NOUN
ap-1860	367	11	.	.	PUNCT
ap-1860	368	1	rairo	rairo	PROPN
ap-1860	368	2	theor	theor	PROPN
ap-1860	368	3	inform	inform	VERB
ap-1860	368	4	appl	appl	PROPN
ap-1860	368	5	44(4):443–470	44(4):443–470	PROPN
ap-1860	368	6	,	,	PUNCT
ap-1860	368	7	2010	2010	NUM
ap-1860	368	8	.	.	PUNCT
ap-1860	369	1	[	[	X
ap-1860	369	2	4	4	NUM
ap-1860	369	3	]	]	X
ap-1860	369	4	l.	l.	PROPN
ap-1860	369	5	vuillon	vuillon	PROPN
ap-1860	369	6	.	.	PUNCT
ap-1860	370	1	a	a	DET
ap-1860	370	2	characterization	characterization	NOUN
ap-1860	370	3	of	of	ADP
ap-1860	370	4	sturmian	sturmian	ADJ
ap-1860	370	5	words	word	NOUN
ap-1860	370	6	by	by	ADP
ap-1860	370	7	return	return	NOUN
ap-1860	370	8	words	word	NOUN
ap-1860	370	9	.	.	PUNCT
ap-1860	371	1	european	european	PROPN
ap-1860	371	2	j	j	PROPN
ap-1860	371	3	combin	combin	PROPN
ap-1860	371	4	22(2):263–275	22(2):263–275	PROPN
ap-1860	371	5	,	,	PUNCT
ap-1860	371	6	2001	2001	NUM
ap-1860	371	7	.	.	PUNCT
ap-1860	372	1	[	[	X
ap-1860	372	2	5	5	X
ap-1860	372	3	]	]	PUNCT
ap-1860	372	4	s.	s.	PROPN
ap-1860	372	5	i.	i.	PROPN
ap-1860	372	6	yasutomi	yasutomi	PROPN
ap-1860	372	7	.	.	PUNCT
ap-1860	373	1	on	on	ADP
ap-1860	373	2	sturmian	sturmian	ADJ
ap-1860	373	3	sequences	sequence	NOUN
ap-1860	373	4	which	which	PRON
ap-1860	373	5	are	be	AUX
ap-1860	373	6	invariant	invariant	ADJ
ap-1860	373	7	under	under	ADP
ap-1860	373	8	some	some	DET
ap-1860	373	9	substitutions	substitution	NOUN
ap-1860	373	10	.	.	PUNCT
ap-1860	374	1	in	in	ADP
ap-1860	374	2	number	number	NOUN
ap-1860	374	3	theory	theory	NOUN
ap-1860	374	4	and	and	CCONJ
ap-1860	374	5	its	its	PRON
ap-1860	374	6	applications	application	NOUN
ap-1860	374	7	(	(	PUNCT
ap-1860	374	8	kyoto	kyoto	NOUN
ap-1860	374	9	,	,	PUNCT
ap-1860	374	10	1997	1997	NUM
ap-1860	374	11	)	)	PUNCT
ap-1860	374	12	,	,	PUNCT
ap-1860	374	13	vol	vol	NOUN
ap-1860	374	14	.	.	PROPN
ap-1860	374	15	2	2	NUM
ap-1860	374	16	of	of	ADP
ap-1860	374	17	dev	dev	PROPN
ap-1860	374	18	.	.	PUNCT
ap-1860	374	19	math	math	PROPN
ap-1860	374	20	.	.	PUNCT
ap-1860	374	21	,	,	PUNCT
ap-1860	374	22	pp	pp	ADJ
ap-1860	374	23	.	.	PUNCT
ap-1860	375	1	347–373	347–373	NUM
ap-1860	375	2	.	.	PUNCT
ap-1860	376	1	kluwer	kluwer	PROPN
ap-1860	376	2	acad	acad	PROPN
ap-1860	376	3	.	.	PUNCT
ap-1860	377	1	publ	publ	PROPN
ap-1860	377	2	.	.	PUNCT
ap-1860	377	3	,	,	PUNCT
ap-1860	377	4	dordrecht	dordrecht	PROPN
ap-1860	377	5	,	,	PUNCT
ap-1860	377	6	1999	1999	NUM
ap-1860	377	7	.	.	PUNCT
ap-1860	378	1	[	[	X
ap-1860	378	2	6	6	NUM
ap-1860	378	3	]	]	PUNCT
ap-1860	378	4	m.	m.	NOUN
ap-1860	378	5	rigo	rigo	PROPN
ap-1860	378	6	,	,	PUNCT
ap-1860	378	7	p.	p.	PROPN
ap-1860	378	8	salimov	salimov	PROPN
ap-1860	378	9	,	,	PUNCT
ap-1860	378	10	e.	e.	PROPN
ap-1860	378	11	vandomme	vandomme	PROPN
ap-1860	378	12	.	.	PUNCT
ap-1860	379	1	some	some	DET
ap-1860	379	2	properties	property	NOUN
ap-1860	379	3	of	of	ADP
ap-1860	379	4	abelian	abelian	ADJ
ap-1860	379	5	return	return	NOUN
ap-1860	379	6	words	word	NOUN
ap-1860	379	7	.	.	PUNCT
ap-1860	380	1	j	j	PROPN
ap-1860	380	2	int	int	NOUN
ap-1860	380	3	sequences	sequence	NOUN
ap-1860	380	4	16:13.2.5	16:13.2.5	NOUN
ap-1860	380	5	,	,	PUNCT
ap-1860	380	6	2013	2013	NUM
ap-1860	380	7	.	.	PUNCT
ap-1860	381	1	[	[	X
ap-1860	381	2	7	7	X
ap-1860	381	3	]	]	X
ap-1860	381	4	s.	s.	PROPN
ap-1860	381	5	puzynina	puzynina	PROPN
ap-1860	381	6	,	,	PUNCT
ap-1860	381	7	l.	l.	PROPN
ap-1860	381	8	q.	q.	PROPN
ap-1860	381	9	zamboni	zamboni	PROPN
ap-1860	381	10	.	.	PUNCT
ap-1860	382	1	abelian	abelian	PROPN
ap-1860	382	2	returns	return	NOUN
ap-1860	382	3	in	in	ADP
ap-1860	382	4	sturmian	sturmian	ADJ
ap-1860	382	5	words	word	NOUN
ap-1860	382	6	.	.	PUNCT
ap-1860	383	1	j	j	PROPN
ap-1860	383	2	combin	combin	PROPN
ap-1860	383	3	theory	theory	NOUN
ap-1860	383	4	ser	ser	NOUN
ap-1860	383	5	a	a	DET
ap-1860	383	6	120(2):390–408	120(2):390–408	NUM
ap-1860	383	7	,	,	PUNCT
ap-1860	383	8	2013	2013	NUM
ap-1860	383	9	.	.	PUNCT
ap-1860	384	1	[	[	X
ap-1860	384	2	8	8	NUM
ap-1860	384	3	]	]	PUNCT
ap-1860	384	4	z.	z.	PROPN
ap-1860	384	5	masáková	masáková	PROPN
ap-1860	384	6	,	,	PUNCT
ap-1860	384	7	e.	e.	PROPN
ap-1860	384	8	pelantová	pelantová	PROPN
ap-1860	384	9	.	.	PUNCT
ap-1860	385	1	enumerating	enumerate	VERB
ap-1860	385	2	abelian	abelian	ADJ
ap-1860	385	3	returns	return	NOUN
ap-1860	385	4	to	to	ADP
ap-1860	385	5	prefixes	prefix	NOUN
ap-1860	385	6	of	of	ADP
ap-1860	385	7	sturmian	sturmian	NOUN
ap-1860	385	8	words	word	NOUN
ap-1860	385	9	.	.	PUNCT
ap-1860	386	1	in	in	ADP
ap-1860	386	2	words	word	NOUN
ap-1860	386	3	2013	2013	NUM
ap-1860	386	4	,	,	PUNCT
ap-1860	386	5	vol	vol	NOUN
ap-1860	386	6	.	.	PROPN
ap-1860	386	7	8079	8079	NUM
ap-1860	386	8	of	of	ADP
ap-1860	386	9	lecture	lecture	NOUN
ap-1860	386	10	notes	note	NOUN
ap-1860	386	11	in	in	ADP
ap-1860	386	12	comput	comput	NOUN
ap-1860	386	13	.	.	PUNCT
ap-1860	387	1	sci	sci	PROPN
ap-1860	387	2	.	.	PROPN
ap-1860	387	3	,	,	PUNCT
ap-1860	387	4	pp	pp	ADP
ap-1860	387	5	.	.	PUNCT
ap-1860	388	1	193–204	193–204	NUM
ap-1860	388	2	.	.	PUNCT
ap-1860	388	3	springer	springer	NOUN
ap-1860	388	4	,	,	PUNCT
ap-1860	388	5	heidelberg	heidelberg	PROPN
ap-1860	388	6	,	,	PUNCT
ap-1860	388	7	2013	2013	NUM
ap-1860	388	8	.	.	PUNCT
ap-1860	389	1	[	[	X
ap-1860	389	2	9	9	NUM
ap-1860	389	3	]	]	PUNCT
ap-1860	389	4	m.	m.	NOUN
ap-1860	389	5	keane	keane	PROPN
ap-1860	389	6	.	.	PUNCT
ap-1860	390	1	interval	interval	NOUN
ap-1860	390	2	exchange	exchange	NOUN
ap-1860	390	3	transformations	transformation	NOUN
ap-1860	390	4	.	.	PUNCT
ap-1860	391	1	math	math	NOUN
ap-1860	391	2	z	z	PROPN
ap-1860	391	3	141:25–31	141:25–31	PROPN
ap-1860	391	4	,	,	PUNCT
ap-1860	391	5	1975	1975	NUM
ap-1860	391	6	.	.	PUNCT
ap-1860	392	1	[	[	X
ap-1860	392	2	10	10	NUM
ap-1860	392	3	]	]	PUNCT
ap-1860	392	4	l.-s	l.-	NOUN
ap-1860	392	5	.	.	PUNCT
ap-1860	393	1	guimond	guimond	NOUN
ap-1860	393	2	,	,	PUNCT
ap-1860	393	3	z.	z.	PROPN
ap-1860	393	4	masáková	masáková	PROPN
ap-1860	393	5	,	,	PUNCT
ap-1860	393	6	e.	e.	PROPN
ap-1860	393	7	pelantová	pelantová	PROPN
ap-1860	393	8	.	.	PUNCT
ap-1860	394	1	combinatorial	combinatorial	ADJ
ap-1860	394	2	properties	property	NOUN
ap-1860	394	3	of	of	ADP
ap-1860	394	4	infinite	infinite	ADJ
ap-1860	394	5	words	word	NOUN
ap-1860	394	6	associated	associate	VERB
ap-1860	394	7	with	with	ADP
ap-1860	394	8	cut	cut	VERB
ap-1860	394	9	-	-	PUNCT
ap-1860	394	10	and	and	CCONJ
ap-1860	394	11	-	-	PUNCT
ap-1860	394	12	project	project	NOUN
ap-1860	394	13	sequences	sequence	NOUN
ap-1860	394	14	.	.	PUNCT
ap-1860	395	1	j	j	PROPN
ap-1860	395	2	théor	théor	PROPN
ap-1860	395	3	nombres	nombre	NOUN
ap-1860	395	4	bordeaux	bordeaux	PROPN
ap-1860	395	5	15(3):697–725	15(3):697–725	PROPN
ap-1860	395	6	,	,	PUNCT
ap-1860	395	7	2003	2003	NUM
ap-1860	395	8	.	.	PUNCT
ap-1860	396	1	[	[	X
ap-1860	396	2	11	11	NUM
ap-1860	396	3	]	]	X
ap-1860	396	4	v.	v.	X
ap-1860	396	5	sós	sós	PROPN
ap-1860	396	6	.	.	PUNCT
ap-1860	397	1	on	on	ADP
ap-1860	397	2	the	the	DET
ap-1860	397	3	distribution	distribution	NOUN
ap-1860	397	4	mod	mod	NOUN
ap-1860	397	5	1	1	NUM
ap-1860	397	6	of	of	ADP
ap-1860	397	7	the	the	DET
ap-1860	397	8	sequence	sequence	NOUN
ap-1860	397	9	nα	nα	NOUN
ap-1860	397	10	.	.	PUNCT
ap-1860	398	1	ann	ann	PROPN
ap-1860	398	2	univ	univ	PROPN
ap-1860	398	3	sci	sci	PROPN
ap-1860	398	4	budapest	budapest	PROPN
ap-1860	398	5	eötös	eötös	PROPN
ap-1860	398	6	sect	sect	PROPN
ap-1860	398	7	math	math	PROPN
ap-1860	398	8	1:127–134	1:127–134	PROPN
ap-1860	398	9	,	,	PUNCT
ap-1860	398	10	1958	1958	NUM
ap-1860	398	11	.	.	PUNCT
ap-1860	399	1	[	[	X
ap-1860	399	2	12	12	NUM
ap-1860	399	3	]	]	X
ap-1860	399	4	n.	n.	PROPN
ap-1860	399	5	b.	b.	PROPN
ap-1860	399	6	slater	slater	PROPN
ap-1860	399	7	.	.	PUNCT
ap-1860	400	1	gaps	gap	NOUN
ap-1860	400	2	and	and	CCONJ
ap-1860	400	3	steps	step	NOUN
ap-1860	400	4	for	for	ADP
ap-1860	400	5	the	the	DET
ap-1860	400	6	sequence	sequence	NOUN
ap-1860	400	7	nθ	nθ	NOUN
ap-1860	400	8	mod	mod	PROPN
ap-1860	400	9	1	1	NUM
ap-1860	400	10	.	.	PUNCT
ap-1860	401	1	proc	proc	PROPN
ap-1860	401	2	cambridge	cambridge	PROPN
ap-1860	401	3	philos	philos	PROPN
ap-1860	401	4	soc	soc	PROPN
ap-1860	401	5	63:1115–1123	63:1115–1123	NUM
ap-1860	401	6	,	,	PUNCT
ap-1860	401	7	1967	1967	NUM
ap-1860	401	8	.	.	PUNCT
ap-1860	402	1	[	[	X
ap-1860	402	2	13	13	NUM
ap-1860	402	3	]	]	PUNCT
ap-1860	402	4	m.	m.	NOUN
ap-1860	402	5	kupsa	kupsa	PROPN
ap-1860	402	6	.	.	PUNCT
ap-1860	403	1	local	local	ADJ
ap-1860	403	2	return	return	NOUN
ap-1860	403	3	rates	rate	NOUN
ap-1860	403	4	in	in	ADP
ap-1860	403	5	sturmian	sturmian	ADJ
ap-1860	403	6	subshifts	subshift	NOUN
ap-1860	403	7	.	.	PUNCT
ap-1860	404	1	acta	acta	PROPN
ap-1860	404	2	univ	univ	PROPN
ap-1860	404	3	carolin	carolin	PROPN
ap-1860	404	4	math	math	PROPN
ap-1860	404	5	phys	phy	NOUN
ap-1860	404	6	44(2):17–28	44(2):17–28	NUM
ap-1860	404	7	,	,	PUNCT
ap-1860	404	8	2003	2003	NUM
ap-1860	404	9	.	.	PUNCT
ap-1860	405	1	[	[	X
ap-1860	405	2	14	14	NUM
ap-1860	405	3	]	]	PUNCT
ap-1860	405	4	p.	p.	PROPN
ap-1860	405	5	kůrka	kůrka	PROPN
ap-1860	405	6	.	.	PUNCT
ap-1860	406	1	topological	topological	ADJ
ap-1860	406	2	and	and	CCONJ
ap-1860	406	3	symbolic	symbolic	ADJ
ap-1860	406	4	dynamics	dynamic	NOUN
ap-1860	406	5	,	,	PUNCT
ap-1860	406	6	vol	vol	NOUN
ap-1860	406	7	.	.	PROPN
ap-1860	406	8	11	11	NUM
ap-1860	406	9	of	of	ADP
ap-1860	406	10	cours	cours	PROPN
ap-1860	406	11	spécialisés	spécialisés	PROPN
ap-1860	407	1	[	[	PUNCT
ap-1860	407	2	specialized	specialized	ADJ
ap-1860	407	3	courses	course	NOUN
ap-1860	407	4	]	]	PUNCT
ap-1860	407	5	.	.	PUNCT
ap-1860	408	1	société	société	PROPN
ap-1860	408	2	mathématique	mathématique	PROPN
ap-1860	408	3	de	de	PROPN
ap-1860	408	4	france	france	PROPN
ap-1860	408	5	,	,	PUNCT
ap-1860	408	6	paris	paris	PROPN
ap-1860	408	7	,	,	PUNCT
ap-1860	408	8	2003	2003	NUM
ap-1860	408	9	.	.	PUNCT
ap-1860	409	1	449	449	NUM
ap-1860	409	2	acta	acta	PROPN
ap-1860	409	3	polytechnica	polytechnica	PROPN
ap-1860	409	4	53(5):444–449	53(5):444–449	PROPN
ap-1860	409	5	,	,	PUNCT
ap-1860	409	6	2013	2013	NUM
ap-1860	409	7	1	1	NUM
ap-1860	409	8	introduction	introduction	NOUN
ap-1860	409	9	2	2	NUM
ap-1860	409	10	interval	interval	NOUN
ap-1860	409	11	exchange	exchange	NOUN
ap-1860	409	12	transformations	transformation	NOUN
ap-1860	409	13	3	3	NUM
ap-1860	409	14	case	case	NOUN
ap-1860	409	15	study	study	NOUN
ap-1860	409	16	4	4	NUM
ap-1860	409	17	conclusions	conclusion	NOUN
ap-1860	409	18	acknowledgements	acknowledgement	NOUN
ap-1860	409	19	references	reference	NOUN