id	sid	tid	token	lemma	pos
ap-5897	1	1	acta	acta	PROPN
ap-5897	1	2	polytechnica	polytechnica	PROPN
ap-5897	1	3	doi:10.14311	doi:10.14311	PROPN
ap-5897	1	4	/	/	SYM
ap-5897	1	5	ap.2020.60.0214	ap.2020.60.0214	PROPN
ap-5897	1	6	acta	acta	PROPN
ap-5897	1	7	polytechnica	polytechnica	PROPN
ap-5897	1	8	60(3):214–224	60(3):214–224	PROPN
ap-5897	1	9	,	,	PUNCT
ap-5897	1	10	2020	2020	NUM
ap-5897	1	11	©	©	PROPN
ap-5897	1	12	czech	czech	PROPN
ap-5897	1	13	technical	technical	PROPN
ap-5897	1	14	university	university	PROPN
ap-5897	1	15	in	in	ADP
ap-5897	1	16	prague	prague	PROPN
ap-5897	1	17	,	,	PUNCT
ap-5897	1	18	2020	2020	NUM
ap-5897	1	19	available	available	ADJ
ap-5897	1	20	online	online	ADV
ap-5897	1	21	at	at	ADP
ap-5897	1	22	https://ojs.cvut.cz/ojs/index.php/ap	https://ojs.cvut.cz/ojs/index.php/ap	PROPN
ap-5897	1	23	beta	beta	ADJ
ap-5897	1	24	cantor	cantor	NOUN
ap-5897	1	25	series	series	NOUN
ap-5897	1	26	expansion	expansion	NOUN
ap-5897	1	27	and	and	CCONJ
ap-5897	1	28	admissible	admissible	ADJ
ap-5897	1	29	sequences	sequence	NOUN
ap-5897	1	30	jonathan	jonathan	PROPN
ap-5897	1	31	caalima	caalima	PROPN
ap-5897	1	32	,	,	PUNCT
ap-5897	1	33	shiela	shiela	PROPN
ap-5897	1	34	demegilloa	demegilloa	NOUN
ap-5897	1	35	,	,	PUNCT
ap-5897	1	36	b,∗	b,∗	PROPN
ap-5897	1	37	a	a	DET
ap-5897	1	38	university	university	NOUN
ap-5897	1	39	of	of	ADP
ap-5897	1	40	the	the	DET
ap-5897	1	41	philippines	philippines	PROPN
ap-5897	1	42	diliman	diliman	PROPN
ap-5897	1	43	,	,	PUNCT
ap-5897	1	44	institute	institute	PROPN
ap-5897	1	45	of	of	ADP
ap-5897	1	46	mathematics	mathematics	PROPN
ap-5897	1	47	,	,	PUNCT
ap-5897	1	48	c.p	c.p	PROPN
ap-5897	1	49	.	.	PROPN
ap-5897	1	50	garcia	garcia	PROPN
ap-5897	1	51	,	,	PUNCT
ap-5897	1	52	1101	1101	NUM
ap-5897	1	53	quezon	quezon	PROPN
ap-5897	1	54	city	city	PROPN
ap-5897	1	55	,	,	PUNCT
ap-5897	1	56	philippines	philippines	PROPN
ap-5897	1	57	b	b	PROPN
ap-5897	1	58	adamson	adamson	PROPN
ap-5897	1	59	university	university	PROPN
ap-5897	1	60	,	,	PUNCT
ap-5897	1	61	mathematics	mathematics	PROPN
ap-5897	1	62	and	and	CCONJ
ap-5897	1	63	physics	physics	PROPN
ap-5897	1	64	department	department	PROPN
ap-5897	1	65	,	,	PUNCT
ap-5897	2	1	san	san	PROPN
ap-5897	2	2	marcelino	marcelino	PROPN
ap-5897	2	3	st	st	PROPN
ap-5897	2	4	.	.	PROPN
ap-5897	2	5	,	,	PUNCT
ap-5897	2	6	1000	1000	NUM
ap-5897	2	7	manila	manila	NOUN
ap-5897	2	8	,	,	PUNCT
ap-5897	2	9	philippines	philippine	NOUN
ap-5897	2	10	∗	∗	NOUN
ap-5897	2	11	corresponding	correspond	VERB
ap-5897	2	12	author	author	NOUN
ap-5897	2	13	:	:	PUNCT
ap-5897	2	14	ssdemegillo@upd.edu.ph	ssdemegillo@upd.edu.ph	PROPN
ap-5897	2	15	abstract	abstract	NOUN
ap-5897	2	16	.	.	PUNCT
ap-5897	3	1	we	we	PRON
ap-5897	3	2	introduce	introduce	VERB
ap-5897	3	3	a	a	DET
ap-5897	3	4	numeration	numeration	NOUN
ap-5897	3	5	system	system	NOUN
ap-5897	3	6	,	,	PUNCT
ap-5897	3	7	called	call	VERB
ap-5897	3	8	the	the	DET
ap-5897	3	9	beta	beta	PROPN
ap-5897	3	10	cantor	cantor	PROPN
ap-5897	3	11	series	series	PROPN
ap-5897	3	12	expansion	expansion	NOUN
ap-5897	3	13	,	,	PUNCT
ap-5897	3	14	that	that	PRON
ap-5897	3	15	generalizes	generalize	VERB
ap-5897	3	16	the	the	DET
ap-5897	3	17	classical	classical	ADJ
ap-5897	3	18	positive	positive	ADJ
ap-5897	3	19	and	and	CCONJ
ap-5897	3	20	negative	negative	ADJ
ap-5897	3	21	beta	beta	ADJ
ap-5897	3	22	expansions	expansion	NOUN
ap-5897	3	23	by	by	ADP
ap-5897	3	24	allowing	allow	VERB
ap-5897	3	25	non	non	ADJ
ap-5897	3	26	-	-	ADJ
ap-5897	3	27	integer	integer	ADJ
ap-5897	3	28	bases	basis	NOUN
ap-5897	3	29	in	in	ADP
ap-5897	3	30	the	the	DET
ap-5897	3	31	q	q	ADJ
ap-5897	3	32	-	-	PUNCT
ap-5897	3	33	cantor	cantor	PROPN
ap-5897	3	34	series	series	NOUN
ap-5897	3	35	expansion	expansion	NOUN
ap-5897	3	36	.	.	PUNCT
ap-5897	4	1	in	in	ADP
ap-5897	4	2	particular	particular	ADJ
ap-5897	4	3	,	,	PUNCT
ap-5897	4	4	we	we	PRON
ap-5897	4	5	show	show	VERB
ap-5897	4	6	that	that	SCONJ
ap-5897	4	7	for	for	ADP
ap-5897	4	8	a	a	DET
ap-5897	4	9	fix	fix	NOUN
ap-5897	4	10	γ	γ	X
ap-5897	4	11	∈	∈	NOUN
ap-5897	4	12	r	r	NOUN
ap-5897	4	13	and	and	CCONJ
ap-5897	4	14	a	a	DET
ap-5897	4	15	sequence	sequence	NOUN
ap-5897	4	16	b	b	NOUN
ap-5897	4	17	of	of	ADP
ap-5897	4	18	real	real	ADJ
ap-5897	4	19	number	number	NOUN
ap-5897	4	20	bases	basis	NOUN
ap-5897	4	21	,	,	PUNCT
ap-5897	4	22	every	every	DET
ap-5897	4	23	element	element	NOUN
ap-5897	4	24	of	of	ADP
ap-5897	4	25	the	the	DET
ap-5897	4	26	interval	interval	NOUN
ap-5897	4	27	[	[	X
ap-5897	4	28	γ	γ	X
ap-5897	4	29	,	,	PUNCT
ap-5897	4	30	γ	γ	X
ap-5897	4	31	+	+	NOUN
ap-5897	4	32	1	1	NUM
ap-5897	4	33	)	)	PUNCT
ap-5897	4	34	has	have	VERB
ap-5897	4	35	a	a	DET
ap-5897	4	36	beta	beta	ADJ
ap-5897	4	37	cantor	cantor	NOUN
ap-5897	4	38	series	series	NOUN
ap-5897	4	39	expansion	expansion	NOUN
ap-5897	4	40	with	with	ADP
ap-5897	4	41	respect	respect	NOUN
ap-5897	4	42	to	to	ADP
ap-5897	4	43	b	b	NOUN
ap-5897	4	44	where	where	SCONJ
ap-5897	4	45	the	the	DET
ap-5897	4	46	digits	digit	NOUN
ap-5897	4	47	are	be	AUX
ap-5897	4	48	integers	integer	NOUN
ap-5897	4	49	in	in	ADP
ap-5897	4	50	some	some	DET
ap-5897	4	51	alphabet	alphabet	NOUN
ap-5897	4	52	a(b	a(b	PROPN
ap-5897	4	53	)	)	PUNCT
ap-5897	4	54	.	.	PUNCT
ap-5897	5	1	we	we	PRON
ap-5897	5	2	give	give	VERB
ap-5897	5	3	a	a	DET
ap-5897	5	4	criterion	criterion	NOUN
ap-5897	5	5	in	in	ADP
ap-5897	5	6	determining	determine	VERB
ap-5897	5	7	whether	whether	SCONJ
ap-5897	5	8	an	an	DET
ap-5897	5	9	integer	integer	NOUN
ap-5897	5	10	sequence	sequence	NOUN
ap-5897	5	11	is	be	AUX
ap-5897	5	12	admissible	admissible	ADJ
ap-5897	5	13	when	when	SCONJ
ap-5897	5	14	b	b	NOUN
ap-5897	5	15	satisfies	satisfy	VERB
ap-5897	5	16	some	some	DET
ap-5897	5	17	condition	condition	NOUN
ap-5897	5	18	.	.	PUNCT
ap-5897	6	1	we	we	PRON
ap-5897	6	2	provide	provide	VERB
ap-5897	6	3	a	a	DET
ap-5897	6	4	description	description	NOUN
ap-5897	6	5	of	of	ADP
ap-5897	6	6	the	the	DET
ap-5897	6	7	reference	reference	NOUN
ap-5897	6	8	strings	string	NOUN
ap-5897	6	9	,	,	PUNCT
ap-5897	6	10	namely	namely	ADV
ap-5897	6	11	the	the	DET
ap-5897	6	12	expansion	expansion	NOUN
ap-5897	6	13	of	of	ADP
ap-5897	6	14	γ	γ	PROPN
ap-5897	6	15	and	and	CCONJ
ap-5897	6	16	γ	γ	X
ap-5897	6	17	+	+	ADP
ap-5897	6	18	1	1	NUM
ap-5897	6	19	,	,	PUNCT
ap-5897	6	20	used	use	VERB
ap-5897	6	21	in	in	ADP
ap-5897	6	22	the	the	DET
ap-5897	6	23	admissibility	admissibility	NOUN
ap-5897	6	24	criterion	criterion	NOUN
ap-5897	6	25	.	.	PUNCT
ap-5897	7	1	keywords	keyword	NOUN
ap-5897	7	2	:	:	PUNCT
ap-5897	7	3	beta	beta	ADJ
ap-5897	7	4	expansion	expansion	NOUN
ap-5897	7	5	,	,	PUNCT
ap-5897	7	6	q	q	ADJ
ap-5897	7	7	-	-	PUNCT
ap-5897	7	8	cantor	cantor	PROPN
ap-5897	7	9	series	series	NOUN
ap-5897	7	10	expansion	expansion	NOUN
ap-5897	7	11	,	,	PUNCT
ap-5897	7	12	numeration	numeration	NOUN
ap-5897	7	13	system	system	NOUN
ap-5897	7	14	,	,	PUNCT
ap-5897	7	15	admissibility	admissibility	NOUN
ap-5897	7	16	.	.	PUNCT
ap-5897	8	1	1	1	X
ap-5897	8	2	.	.	X
ap-5897	8	3	introduction	introduction	NOUN
ap-5897	8	4	the	the	DET
ap-5897	8	5	subject	subject	NOUN
ap-5897	8	6	of	of	ADP
ap-5897	8	7	representations	representation	NOUN
ap-5897	8	8	of	of	ADP
ap-5897	8	9	real	real	ADJ
ap-5897	8	10	numbers	number	NOUN
ap-5897	8	11	is	be	AUX
ap-5897	8	12	an	an	DET
ap-5897	8	13	extensively	extensively	ADV
ap-5897	8	14	studied	study	VERB
ap-5897	8	15	research	research	NOUN
ap-5897	8	16	field	field	NOUN
ap-5897	8	17	.	.	PUNCT
ap-5897	9	1	in	in	ADP
ap-5897	9	2	the	the	DET
ap-5897	9	3	seminal	seminal	ADJ
ap-5897	9	4	work	work	NOUN
ap-5897	9	5	[	[	X
ap-5897	9	6	1	1	NUM
ap-5897	9	7	]	]	PUNCT
ap-5897	9	8	,	,	PUNCT
ap-5897	9	9	renyi	renyi	PROPN
ap-5897	9	10	introduced	introduce	VERB
ap-5897	9	11	the	the	DET
ap-5897	9	12	now	now	ADV
ap-5897	9	13	well	well	ADV
ap-5897	9	14	-	-	PUNCT
ap-5897	9	15	known	know	VERB
ap-5897	9	16	concept	concept	NOUN
ap-5897	9	17	of	of	ADP
ap-5897	9	18	beta	beta	ADJ
ap-5897	9	19	expansions	expansion	NOUN
ap-5897	9	20	.	.	PUNCT
ap-5897	10	1	beta	beta	ADJ
ap-5897	10	2	expansions	expansion	NOUN
ap-5897	10	3	are	be	AUX
ap-5897	10	4	representations	representation	NOUN
ap-5897	10	5	of	of	ADP
ap-5897	10	6	real	real	ADJ
ap-5897	10	7	numbers	number	NOUN
ap-5897	10	8	using	use	VERB
ap-5897	10	9	an	an	DET
ap-5897	10	10	arbitrary	arbitrary	ADJ
ap-5897	10	11	positive	positive	ADJ
ap-5897	10	12	real	real	ADJ
ap-5897	10	13	base	base	NOUN
ap-5897	10	14	β	β	X
ap-5897	10	15	>	>	X
ap-5897	10	16	1	1	NUM
ap-5897	10	17	obtained	obtain	VERB
ap-5897	10	18	via	via	ADP
ap-5897	10	19	the	the	DET
ap-5897	10	20	beta	beta	ADJ
ap-5897	10	21	transformation	transformation	NOUN
ap-5897	11	1	tβ	tβ	NOUN
ap-5897	11	2	:	:	PUNCT
ap-5897	12	1	[	[	X
ap-5897	12	2	0	0	NUM
ap-5897	12	3	,	,	PUNCT
ap-5897	12	4	1	1	X
ap-5897	12	5	)	)	PUNCT
ap-5897	12	6	−→	−→	NOUN
ap-5897	12	7	[	[	X
ap-5897	12	8	0	0	NUM
ap-5897	12	9	,	,	PUNCT
ap-5897	12	10	1	1	NUM
ap-5897	12	11	)	)	PUNCT
ap-5897	12	12	given	give	VERB
ap-5897	12	13	by	by	ADP
ap-5897	12	14	tβ(x	tβ(x	NOUN
ap-5897	12	15	)	)	PUNCT
ap-5897	12	16	=	=	SYM
ap-5897	12	17	βx−	βx−	PUNCT
ap-5897	12	18	bβxc	bβxc	NOUN
ap-5897	12	19	.	.	PUNCT
ap-5897	13	1	the	the	DET
ap-5897	13	2	iterates	iterate	NOUN
ap-5897	13	3	of	of	ADP
ap-5897	13	4	t	t	NOUN
ap-5897	13	5	induce	induce	VERB
ap-5897	13	6	a	a	DET
ap-5897	13	7	numeration	numeration	NOUN
ap-5897	13	8	system	system	NOUN
ap-5897	13	9	on	on	ADP
ap-5897	13	10	[	[	X
ap-5897	13	11	0	0	NUM
ap-5897	13	12	,	,	PUNCT
ap-5897	13	13	1	1	NUM
ap-5897	13	14	)	)	PUNCT
ap-5897	13	15	wherein	wherein	SCONJ
ap-5897	13	16	the	the	DET
ap-5897	13	17	expansion	expansion	NOUN
ap-5897	13	18	of	of	ADP
ap-5897	13	19	an	an	DET
ap-5897	13	20	element	element	NOUN
ap-5897	13	21	x	x	SYM
ap-5897	13	22	∈	∈	PROPN
ap-5897	14	1	[	[	X
ap-5897	14	2	0	0	NUM
ap-5897	14	3	,	,	PUNCT
ap-5897	14	4	1	1	NUM
ap-5897	14	5	)	)	PUNCT
ap-5897	14	6	is	be	AUX
ap-5897	14	7	given	give	VERB
ap-5897	14	8	by	by	ADP
ap-5897	14	9	the	the	DET
ap-5897	14	10	sequence	sequence	NOUN
ap-5897	14	11	d(β;x	d(β;x	PROPN
ap-5897	14	12	)	)	PUNCT
ap-5897	14	13	=	=	PRON
ap-5897	14	14	(	(	PUNCT
ap-5897	14	15	d1	d1	PROPN
ap-5897	14	16	,	,	PUNCT
ap-5897	14	17	d2	d2	PROPN
ap-5897	14	18	,	,	PUNCT
ap-5897	14	19	.	.	PUNCT
ap-5897	14	20	.	.	PUNCT
ap-5897	14	21	.	.	PUNCT
ap-5897	14	22	)	)	PUNCT
ap-5897	15	1	with	with	ADP
ap-5897	15	2	di	di	NOUN
ap-5897	15	3	=	=	PUNCT
ap-5897	15	4	bβt	bβt	PROPN
ap-5897	15	5	i−1(x)c	i−1(x)c	PROPN
ap-5897	15	6	.	.	PUNCT
ap-5897	15	7	thus	thus	ADV
ap-5897	15	8	,	,	PUNCT
ap-5897	15	9	the	the	DET
ap-5897	15	10	digits	digit	NOUN
ap-5897	15	11	di	di	AUX
ap-5897	15	12	belong	belong	VERB
ap-5897	15	13	to	to	ADP
ap-5897	15	14	the	the	DET
ap-5897	15	15	alphabet	alphabet	NOUN
ap-5897	15	16	a	a	PROPN
ap-5897	15	17	=	=	X
ap-5897	15	18	{	{	PUNCT
ap-5897	15	19	0	0	NUM
ap-5897	15	20	,	,	PUNCT
ap-5897	15	21	1	1	NUM
ap-5897	15	22	,	,	PUNCT
ap-5897	15	23	.	.	PUNCT
ap-5897	15	24	.	.	PUNCT
ap-5897	16	1	.	.	PUNCT
ap-5897	17	1	,	,	PUNCT
ap-5897	17	2	bβc	bβc	NOUN
ap-5897	17	3	}	}	PUNCT
ap-5897	17	4	if	if	SCONJ
ap-5897	17	5	β	β	NOUN
ap-5897	17	6	/∈	/∈	PUNCT
ap-5897	18	1	n	n	CCONJ
ap-5897	18	2	or	or	CCONJ
ap-5897	18	3	a	a	DET
ap-5897	18	4	=	=	X
ap-5897	18	5	{	{	PUNCT
ap-5897	18	6	0	0	NUM
ap-5897	18	7	,	,	PUNCT
ap-5897	18	8	1	1	NUM
ap-5897	18	9	,	,	PUNCT
ap-5897	18	10	.	.	PUNCT
ap-5897	18	11	.	.	PUNCT
ap-5897	19	1	.	.	PUNCT
ap-5897	20	1	,	,	PUNCT
ap-5897	20	2	β	β	X
ap-5897	20	3	−	−	NOUN
ap-5897	20	4	1	1	X
ap-5897	20	5	}	}	PUNCT
ap-5897	20	6	if	if	SCONJ
ap-5897	20	7	β	β	PROPN
ap-5897	20	8	∈	∈	PROPN
ap-5897	20	9	n.	n.	PROPN
ap-5897	20	10	parry	parry	PROPN
ap-5897	20	11	,	,	PUNCT
ap-5897	20	12	in	in	ADP
ap-5897	20	13	[	[	PUNCT
ap-5897	20	14	2	2	NUM
ap-5897	20	15	]	]	PUNCT
ap-5897	20	16	,	,	PUNCT
ap-5897	20	17	considered	consider	VERB
ap-5897	20	18	the	the	DET
ap-5897	20	19	admissibility	admissibility	NOUN
ap-5897	20	20	problem	problem	NOUN
ap-5897	20	21	of	of	ADP
ap-5897	20	22	determining	determine	VERB
ap-5897	20	23	the	the	DET
ap-5897	20	24	integer	integer	NOUN
ap-5897	20	25	sequences	sequence	NOUN
ap-5897	20	26	over	over	ADP
ap-5897	20	27	the	the	DET
ap-5897	20	28	alphabet	alphabet	NOUN
ap-5897	20	29	a	a	PRON
ap-5897	20	30	that	that	PRON
ap-5897	20	31	appear	appear	VERB
ap-5897	20	32	as	as	ADP
ap-5897	20	33	the	the	DET
ap-5897	20	34	beta	beta	ADJ
ap-5897	20	35	expansion	expansion	NOUN
ap-5897	20	36	of	of	ADP
ap-5897	20	37	a	a	DET
ap-5897	20	38	real	real	ADJ
ap-5897	20	39	number	number	NOUN
ap-5897	20	40	in	in	ADP
ap-5897	20	41	the	the	DET
ap-5897	20	42	domain	domain	NOUN
ap-5897	20	43	[	[	X
ap-5897	20	44	0	0	NUM
ap-5897	20	45	,	,	PUNCT
ap-5897	20	46	1	1	NUM
ap-5897	20	47	)	)	PUNCT
ap-5897	20	48	.	.	PUNCT
ap-5897	21	1	parry	parry	PROPN
ap-5897	21	2	provided	provide	VERB
ap-5897	21	3	a	a	DET
ap-5897	21	4	necessary	necessary	ADJ
ap-5897	21	5	and	and	CCONJ
ap-5897	21	6	sufficient	sufficient	ADJ
ap-5897	21	7	condition	condition	NOUN
ap-5897	21	8	(	(	PUNCT
ap-5897	21	9	formulated	formulate	VERB
ap-5897	21	10	in	in	ADP
ap-5897	21	11	terms	term	NOUN
ap-5897	21	12	of	of	ADP
ap-5897	21	13	the	the	DET
ap-5897	21	14	beta	beta	ADJ
ap-5897	21	15	expansion	expansion	NOUN
ap-5897	21	16	of	of	ADP
ap-5897	21	17	1	1	NUM
ap-5897	21	18	)	)	PUNCT
ap-5897	21	19	for	for	ADP
ap-5897	21	20	a	a	DET
ap-5897	21	21	sequence	sequence	NOUN
ap-5897	21	22	of	of	ADP
ap-5897	21	23	integers	integer	NOUN
ap-5897	21	24	to	to	PART
ap-5897	21	25	be	be	AUX
ap-5897	21	26	beta	beta	ADJ
ap-5897	21	27	admissible	admissible	ADJ
ap-5897	21	28	.	.	PUNCT
ap-5897	22	1	in	in	ADP
ap-5897	22	2	the	the	DET
ap-5897	22	3	subsequent	subsequent	ADJ
ap-5897	22	4	paper	paper	NOUN
ap-5897	22	5	[	[	X
ap-5897	22	6	3	3	NUM
ap-5897	22	7	]	]	PUNCT
ap-5897	22	8	,	,	PUNCT
ap-5897	22	9	parry	parry	PROPN
ap-5897	22	10	extended	extend	VERB
ap-5897	22	11	the	the	DET
ap-5897	22	12	definition	definition	NOUN
ap-5897	22	13	of	of	ADP
ap-5897	22	14	the	the	DET
ap-5897	22	15	beta	beta	ADJ
ap-5897	22	16	transformation	transformation	NOUN
ap-5897	22	17	to	to	ADP
ap-5897	22	18	t	t	NOUN
ap-5897	22	19	:	:	PUNCT
ap-5897	23	1	[	[	X
ap-5897	23	2	0	0	NUM
ap-5897	23	3	,	,	PUNCT
ap-5897	23	4	1	1	X
ap-5897	23	5	)	)	PUNCT
ap-5897	23	6	−→	−→	NOUN
ap-5897	24	1	[	[	X
ap-5897	24	2	0	0	NUM
ap-5897	24	3	,	,	PUNCT
ap-5897	24	4	1	1	NUM
ap-5897	24	5	)	)	PUNCT
ap-5897	25	1	where	where	SCONJ
ap-5897	25	2	t	t	PROPN
ap-5897	25	3	(	(	PUNCT
ap-5897	25	4	x	x	X
ap-5897	25	5	)	)	PUNCT
ap-5897	25	6	=	=	SYM
ap-5897	25	7	βx+	βx+	NOUN
ap-5897	25	8	α+	α+	X
ap-5897	25	9	bβx+	bβx+	NOUN
ap-5897	25	10	αc	αc	ADV
ap-5897	25	11	with	with	ADP
ap-5897	25	12	β	β	PROPN
ap-5897	25	13	>	>	X
ap-5897	25	14	1	1	NUM
ap-5897	25	15	and	and	CCONJ
ap-5897	25	16	0	0	NUM
ap-5897	25	17	≤	≤	NUM
ap-5897	25	18	α	α	NOUN
ap-5897	25	19	<	<	X
ap-5897	25	20	1	1	NUM
ap-5897	25	21	and	and	CCONJ
ap-5897	25	22	he	he	PRON
ap-5897	25	23	also	also	ADV
ap-5897	25	24	tackled	tackle	VERB
ap-5897	25	25	the	the	DET
ap-5897	25	26	admissibility	admissibility	NOUN
ap-5897	25	27	problem	problem	NOUN
ap-5897	25	28	in	in	ADP
ap-5897	25	29	this	this	DET
ap-5897	25	30	setting	setting	NOUN
ap-5897	25	31	.	.	PUNCT
ap-5897	26	1	an	an	DET
ap-5897	26	2	important	important	ADJ
ap-5897	26	3	generalization	generalization	NOUN
ap-5897	26	4	of	of	ADP
ap-5897	26	5	beta	beta	ADJ
ap-5897	26	6	expansion	expansion	NOUN
ap-5897	26	7	is	be	AUX
ap-5897	26	8	a	a	DET
ap-5897	26	9	positional	positional	ADJ
ap-5897	26	10	numeration	numeration	NOUN
ap-5897	26	11	system	system	NOUN
ap-5897	26	12	that	that	PRON
ap-5897	26	13	uses	use	VERB
ap-5897	26	14	negative	negative	ADJ
ap-5897	26	15	bases	basis	NOUN
ap-5897	26	16	.	.	PUNCT
ap-5897	27	1	as	as	SCONJ
ap-5897	27	2	remarked	remark	VERB
ap-5897	27	3	by	by	ADP
ap-5897	27	4	frougny	frougny	NOUN
ap-5897	27	5	and	and	CCONJ
ap-5897	27	6	lai	lai	X
ap-5897	27	7	in	in	ADP
ap-5897	27	8	[	[	X
ap-5897	27	9	4	4	NUM
ap-5897	27	10	]	]	PUNCT
ap-5897	27	11	,	,	PUNCT
ap-5897	27	12	it	it	PRON
ap-5897	27	13	appears	appear	VERB
ap-5897	27	14	that	that	SCONJ
ap-5897	27	15	grünwald	grünwald	NOUN
ap-5897	27	16	was	be	AUX
ap-5897	27	17	the	the	DET
ap-5897	27	18	first	first	ADJ
ap-5897	27	19	to	to	PART
ap-5897	27	20	introduce	introduce	VERB
ap-5897	27	21	this	this	DET
ap-5897	27	22	idea	idea	NOUN
ap-5897	27	23	in	in	ADP
ap-5897	27	24	[	[	X
ap-5897	27	25	5	5	NUM
ap-5897	27	26	]	]	PUNCT
ap-5897	27	27	.	.	PUNCT
ap-5897	28	1	here	here	ADV
ap-5897	28	2	,	,	PUNCT
ap-5897	28	3	we	we	PRON
ap-5897	28	4	present	present	VERB
ap-5897	28	5	a	a	DET
ap-5897	28	6	general	general	ADJ
ap-5897	28	7	formulation	formulation	NOUN
ap-5897	28	8	considered	consider	VERB
ap-5897	28	9	by	by	ADP
ap-5897	28	10	ito	ito	PROPN
ap-5897	28	11	and	and	CCONJ
ap-5897	28	12	sadahiro	sadahiro	PROPN
ap-5897	28	13	in	in	ADP
ap-5897	28	14	[	[	X
ap-5897	28	15	6	6	NUM
ap-5897	28	16	]	]	PUNCT
ap-5897	28	17	.	.	PUNCT
ap-5897	29	1	let	let	VERB
ap-5897	29	2	1	1	NUM
ap-5897	29	3	<	<	X
ap-5897	29	4	β	β	X
ap-5897	29	5	∈	∈	NOUN
ap-5897	29	6	r	r	NOUN
ap-5897	29	7	and	and	CCONJ
ap-5897	29	8	define	define	VERB
ap-5897	29	9	lβ	lβ	NOUN
ap-5897	29	10	:	:	PUNCT
ap-5897	29	11	=	=	PUNCT
ap-5897	29	12	−β/(β	−β/(β	VERB
ap-5897	29	13	+	+	CCONJ
ap-5897	29	14	1	1	NUM
ap-5897	29	15	)	)	PUNCT
ap-5897	29	16	and	and	CCONJ
ap-5897	29	17	rβ	rβ	VERB
ap-5897	29	18	:	:	PUNCT
ap-5897	30	1	=	=	SYM
ap-5897	30	2	1/(β	1/(β	NUM
ap-5897	31	1	+	+	CCONJ
ap-5897	31	2	1	1	NUM
ap-5897	31	3	)	)	PUNCT
ap-5897	31	4	.	.	PUNCT
ap-5897	32	1	the	the	DET
ap-5897	32	2	negative	negative	ADJ
ap-5897	32	3	beta	beta	ADJ
ap-5897	32	4	transformation	transformation	NOUN
ap-5897	32	5	is	be	AUX
ap-5897	32	6	the	the	DET
ap-5897	32	7	map	map	NOUN
ap-5897	32	8	t−β	t−β	NOUN
ap-5897	32	9	:	:	PUNCT
ap-5897	33	1	[	[	X
ap-5897	33	2	lβ	lβ	INTJ
ap-5897	33	3	,	,	PUNCT
ap-5897	33	4	rβ	rβ	X
ap-5897	33	5	)	)	PUNCT
ap-5897	33	6	−→	−→	NOUN
ap-5897	33	7	[	[	X
ap-5897	33	8	lβ	lβ	INTJ
ap-5897	33	9	,	,	PUNCT
ap-5897	33	10	rβ	rβ	NOUN
ap-5897	33	11	)	)	PUNCT
ap-5897	33	12	given	give	VERB
ap-5897	33	13	by	by	ADP
ap-5897	33	14	t−β(x	t−β(x	NOUN
ap-5897	33	15	)	)	PUNCT
ap-5897	33	16	=	=	PUNCT
ap-5897	34	1	−βx−	−βx−	X
ap-5897	34	2	b−βx−	b−βx−	X
ap-5897	34	3	lβc	lβc	PROPN
ap-5897	34	4	.	.	PUNCT
ap-5897	35	1	the	the	DET
ap-5897	35	2	map	map	NOUN
ap-5897	35	3	t−β	t−β	NOUN
ap-5897	35	4	also	also	ADV
ap-5897	35	5	induces	induce	VERB
ap-5897	35	6	an	an	DET
ap-5897	35	7	expansion	expansion	NOUN
ap-5897	35	8	on	on	ADP
ap-5897	35	9	the	the	DET
ap-5897	35	10	domain	domain	NOUN
ap-5897	35	11	[	[	X
ap-5897	35	12	lβ	lβ	INTJ
ap-5897	35	13	,	,	PUNCT
ap-5897	35	14	rβ	rβ	PROPN
ap-5897	35	15	)	)	PUNCT
ap-5897	35	16	,	,	PUNCT
ap-5897	35	17	where	where	SCONJ
ap-5897	35	18	the	the	DET
ap-5897	35	19	digits	digit	NOUN
ap-5897	35	20	are	be	AUX
ap-5897	35	21	given	give	VERB
ap-5897	35	22	by	by	ADP
ap-5897	35	23	b−βt	b−βt	NOUN
ap-5897	35	24	i(x	i(x	NOUN
ap-5897	35	25	)	)	PUNCT
ap-5897	35	26	−	−	PROPN
ap-5897	35	27	lβc	lβc	PROPN
ap-5897	35	28	.	.	PUNCT
ap-5897	36	1	an	an	DET
ap-5897	36	2	admissibility	admissibility	NOUN
ap-5897	36	3	criterion	criterion	NOUN
ap-5897	36	4	was	be	AUX
ap-5897	36	5	also	also	ADV
ap-5897	36	6	given	give	VERB
ap-5897	36	7	in	in	ADP
ap-5897	36	8	[	[	PUNCT
ap-5897	36	9	6	6	NUM
ap-5897	36	10	,	,	PUNCT
ap-5897	36	11	theorem	theorem	VERB
ap-5897	36	12	10	10	NUM
ap-5897	36	13	]	]	PUNCT
ap-5897	36	14	.	.	PUNCT
ap-5897	37	1	(	(	PUNCT
ap-5897	37	2	in	in	ADP
ap-5897	37	3	[	[	X
ap-5897	37	4	7	7	NUM
ap-5897	37	5	]	]	PUNCT
ap-5897	37	6	,	,	PUNCT
ap-5897	37	7	liao	liao	PROPN
ap-5897	37	8	and	and	CCONJ
ap-5897	37	9	steiner	steiner	PROPN
ap-5897	37	10	introduced	introduce	VERB
ap-5897	37	11	the	the	DET
ap-5897	37	12	self	self	NOUN
ap-5897	37	13	-	-	PUNCT
ap-5897	37	14	map	map	NOUN
ap-5897	37	15	t̂	t̂	X
ap-5897	37	16	:	:	PUNCT
ap-5897	37	17	(	(	PUNCT
ap-5897	37	18	0	0	NUM
ap-5897	37	19	,	,	PUNCT
ap-5897	37	20	1	1	NUM
ap-5897	37	21	]	]	X
ap-5897	37	22	−→	−→	NOUN
ap-5897	37	23	(	(	PUNCT
ap-5897	37	24	0	0	NUM
ap-5897	37	25	,	,	PUNCT
ap-5897	37	26	1	1	NUM
ap-5897	37	27	]	]	PUNCT
ap-5897	37	28	given	give	VERB
ap-5897	37	29	by	by	ADP
ap-5897	37	30	t̂	t̂	NUM
ap-5897	37	31	(	(	PUNCT
ap-5897	37	32	x	x	NOUN
ap-5897	37	33	)	)	PUNCT
ap-5897	37	34	=	=	PUNCT
ap-5897	38	1	−βx+	−βx+	NOUN
ap-5897	38	2	bβxc+	bβxc+	NUM
ap-5897	38	3	1	1	NUM
ap-5897	38	4	.	.	PUNCT
ap-5897	39	1	this	this	DET
ap-5897	39	2	transformation	transformation	NOUN
ap-5897	39	3	is	be	AUX
ap-5897	39	4	conjugate	conjugate	ADJ
ap-5897	39	5	to	to	ADP
ap-5897	39	6	the	the	DET
ap-5897	39	7	one	one	NOUN
ap-5897	39	8	defined	define	VERB
ap-5897	39	9	by	by	ADP
ap-5897	39	10	ito	ito	PROPN
ap-5897	39	11	and	and	CCONJ
ap-5897	39	12	sadahiro	sadahiro	PROPN
ap-5897	39	13	and	and	CCONJ
ap-5897	39	14	,	,	PUNCT
ap-5897	39	15	as	as	ADP
ap-5897	39	16	such	such	ADJ
ap-5897	39	17	,	,	PUNCT
ap-5897	39	18	the	the	DET
ap-5897	39	19	results	result	NOUN
ap-5897	39	20	for	for	ADP
ap-5897	39	21	the	the	DET
ap-5897	39	22	negative	negative	ADJ
ap-5897	39	23	beta	beta	NOUN
ap-5897	39	24	expansion	expansion	NOUN
ap-5897	39	25	can	can	AUX
ap-5897	39	26	be	be	AUX
ap-5897	39	27	restated	restate	VERB
ap-5897	39	28	using	use	VERB
ap-5897	39	29	the	the	DET
ap-5897	39	30	map	map	NOUN
ap-5897	39	31	t̂	t̂	PUNCT
ap-5897	39	32	.	.	PUNCT
ap-5897	39	33	)	)	PUNCT
ap-5897	40	1	as	as	ADP
ap-5897	40	2	with	with	ADP
ap-5897	40	3	the	the	DET
ap-5897	40	4	positive	positive	ADJ
ap-5897	40	5	beta	beta	ADJ
ap-5897	40	6	transformations	transformation	NOUN
ap-5897	40	7	,	,	PUNCT
ap-5897	40	8	dombek	dombek	NOUN
ap-5897	40	9	,	,	PUNCT
ap-5897	40	10	et.al	et.al	PROPN
ap-5897	40	11	,	,	PUNCT
ap-5897	40	12	in	in	ADP
ap-5897	40	13	[	[	X
ap-5897	40	14	8	8	NUM
ap-5897	40	15	]	]	PUNCT
ap-5897	40	16	introduced	introduce	VERB
ap-5897	40	17	a	a	DET
ap-5897	40	18	parameter	parameter	NOUN
ap-5897	40	19	α	α	NOUN
ap-5897	40	20	to	to	PART
ap-5897	40	21	generalize	generalize	VERB
ap-5897	40	22	the	the	DET
ap-5897	40	23	negative	negative	ADJ
ap-5897	40	24	beta	beta	NOUN
ap-5897	40	25	transformation	transformation	NOUN
ap-5897	40	26	defined	define	VERB
ap-5897	40	27	by	by	ADP
ap-5897	40	28	ito	ito	PROPN
ap-5897	40	29	and	and	CCONJ
ap-5897	40	30	sadahiro	sadahiro	PROPN
ap-5897	40	31	.	.	PUNCT
ap-5897	41	1	they	they	PRON
ap-5897	41	2	considered	consider	VERB
ap-5897	41	3	the	the	DET
ap-5897	41	4	map	map	NOUN
ap-5897	41	5	t	t	NOUN
ap-5897	41	6	:	:	PUNCT
ap-5897	42	1	[	[	X
ap-5897	42	2	α	α	X
ap-5897	42	3	,	,	PUNCT
ap-5897	42	4	α+	α+	NOUN
ap-5897	42	5	1	1	X
ap-5897	42	6	)	)	PUNCT
ap-5897	42	7	−→	−→	NOUN
ap-5897	43	1	[	[	X
ap-5897	43	2	α	α	NOUN
ap-5897	43	3	,	,	PUNCT
ap-5897	43	4	α	α	NOUN
ap-5897	43	5	+	+	NOUN
ap-5897	43	6	1	1	NUM
ap-5897	43	7	)	)	PUNCT
ap-5897	43	8	given	give	VERB
ap-5897	43	9	by	by	ADP
ap-5897	43	10	t	t	PROPN
ap-5897	43	11	(	(	PUNCT
ap-5897	43	12	x	x	NOUN
ap-5897	43	13	)	)	PUNCT
ap-5897	43	14	=	=	PUNCT
ap-5897	43	15	−βx−	−βx−	X
ap-5897	43	16	b−βx−	b−βx−	NOUN
ap-5897	43	17	αc	αc	ADP
ap-5897	43	18	where	where	SCONJ
ap-5897	43	19	β	β	X
ap-5897	43	20	>	>	X
ap-5897	43	21	1	1	NUM
ap-5897	43	22	and	and	CCONJ
ap-5897	43	23	α	α	PRON
ap-5897	43	24	∈	∈	PROPN
ap-5897	43	25	(	(	PUNCT
ap-5897	43	26	−1	−1	NOUN
ap-5897	43	27	,	,	PUNCT
ap-5897	43	28	0	0	NUM
ap-5897	43	29	]	]	PUNCT
ap-5897	43	30	.	.	PUNCT
ap-5897	44	1	(	(	PUNCT
ap-5897	44	2	see	see	VERB
ap-5897	44	3	also	also	ADV
ap-5897	44	4	[	[	X
ap-5897	44	5	9	9	NUM
ap-5897	44	6	,	,	PUNCT
ap-5897	44	7	10	10	NUM
ap-5897	44	8	]	]	PUNCT
ap-5897	44	9	for	for	ADP
ap-5897	44	10	other	other	ADJ
ap-5897	44	11	transformations	transformation	NOUN
ap-5897	44	12	inducing	induce	VERB
ap-5897	44	13	an	an	DET
ap-5897	44	14	expansion	expansion	NOUN
ap-5897	44	15	in	in	ADP
ap-5897	44	16	a	a	DET
ap-5897	44	17	negative	negative	ADJ
ap-5897	44	18	base	base	NOUN
ap-5897	44	19	.	.	PUNCT
ap-5897	44	20	)	)	PUNCT
ap-5897	45	1	the	the	DET
ap-5897	45	2	motivation	motivation	NOUN
ap-5897	45	3	of	of	ADP
ap-5897	45	4	the	the	DET
ap-5897	45	5	current	current	ADJ
ap-5897	45	6	study	study	NOUN
ap-5897	45	7	originates	originate	NOUN
ap-5897	45	8	from	from	ADP
ap-5897	45	9	a	a	DET
ap-5897	45	10	certain	certain	ADJ
ap-5897	45	11	class	class	NOUN
ap-5897	45	12	of	of	ADP
ap-5897	45	13	rotational	rotational	ADJ
ap-5897	45	14	beta	beta	ADJ
ap-5897	45	15	expansions	expansion	NOUN
ap-5897	45	16	in	in	ADP
ap-5897	45	17	dimension	dimension	NOUN
ap-5897	45	18	two	two	NUM
ap-5897	45	19	(	(	PUNCT
ap-5897	45	20	see	see	VERB
ap-5897	45	21	[	[	X
ap-5897	45	22	11	11	NUM
ap-5897	45	23	,	,	PUNCT
ap-5897	45	24	12	12	NUM
ap-5897	45	25	]	]	PUNCT
ap-5897	45	26	)	)	PUNCT
ap-5897	45	27	.	.	PUNCT
ap-5897	46	1	rotational	rotational	ADJ
ap-5897	46	2	beta	beta	ADJ
ap-5897	46	3	expansions	expansion	NOUN
ap-5897	46	4	generalize	generalize	VERB
ap-5897	46	5	the	the	DET
ap-5897	46	6	notion	notion	NOUN
ap-5897	46	7	of	of	ADP
ap-5897	46	8	beta	beta	ADJ
ap-5897	46	9	expansions	expansion	NOUN
ap-5897	46	10	in	in	ADP
ap-5897	46	11	higher	high	ADJ
ap-5897	46	12	dimensions	dimension	NOUN
ap-5897	46	13	.	.	PUNCT
ap-5897	47	1	let	let	VERB
ap-5897	47	2	z	z	NOUN
ap-5897	47	3	=	=	PUNCT
ap-5897	48	1	[	[	X
ap-5897	48	2	0	0	NUM
ap-5897	48	3	,	,	PUNCT
ap-5897	48	4	1	1	NUM
ap-5897	48	5	)	)	PUNCT
ap-5897	48	6	×	×	NOUN
ap-5897	49	1	[	[	X
ap-5897	49	2	0	0	NUM
ap-5897	49	3	,	,	PUNCT
ap-5897	49	4	1	1	NUM
ap-5897	49	5	)	)	PUNCT
ap-5897	49	6	and	and	CCONJ
ap-5897	49	7	1	1	NUM
ap-5897	49	8	<	<	X
ap-5897	49	9	β	β	X
ap-5897	49	10	∈	∈	PROPN
ap-5897	49	11	r.	r.	PROPN
ap-5897	49	12	define	define	VERB
ap-5897	49	13	the	the	DET
ap-5897	49	14	four	four	NUM
ap-5897	49	15	-	-	ADJ
ap-5897	49	16	fold	fold	ADJ
ap-5897	49	17	rotational	rotational	ADJ
ap-5897	49	18	beta	beta	NOUN
ap-5897	49	19	transformation	transformation	NOUN
ap-5897	49	20	t	t	NOUN
ap-5897	49	21	:	:	PUNCT
ap-5897	49	22	z	z	NOUN
ap-5897	50	1	−→	−→	NOUN
ap-5897	50	2	z	z	PROPN
ap-5897	50	3	by	by	ADP
ap-5897	50	4	t	t	PROPN
ap-5897	50	5	(	(	PUNCT
ap-5897	50	6	[	[	PUNCT
ap-5897	50	7	x	x	X
ap-5897	50	8	y	y	NOUN
ap-5897	50	9	]	]	PUNCT
ap-5897	50	10	)	)	PUNCT
ap-5897	51	1	=	=	PUNCT
ap-5897	51	2	[	[	PUNCT
ap-5897	51	3	−βy	−βy	X
ap-5897	51	4	−	−	NOUN
ap-5897	51	5	b−βyc	b−βyc	NOUN
ap-5897	51	6	βx−	βx−	PUNCT
ap-5897	51	7	bβxc	bβxc	NOUN
ap-5897	51	8	]	]	PUNCT
ap-5897	51	9	.	.	PUNCT
ap-5897	52	1	it	it	PRON
ap-5897	52	2	is	be	AUX
ap-5897	52	3	easy	easy	ADJ
ap-5897	52	4	to	to	PART
ap-5897	52	5	see	see	VERB
ap-5897	52	6	that	that	SCONJ
ap-5897	52	7	if	if	SCONJ
ap-5897	52	8	we	we	PRON
ap-5897	52	9	wish	wish	VERB
ap-5897	52	10	to	to	PART
ap-5897	52	11	keep	keep	VERB
ap-5897	52	12	track	track	NOUN
ap-5897	52	13	of	of	ADP
ap-5897	52	14	the	the	DET
ap-5897	52	15	itinerary	itinerary	NOUN
ap-5897	52	16	of	of	ADP
ap-5897	52	17	a	a	DET
ap-5897	52	18	point	point	NOUN
ap-5897	52	19	z	z	NOUN
ap-5897	52	20	∈	∈	PROPN
ap-5897	52	21	z	z	PROPN
ap-5897	52	22	under	under	ADP
ap-5897	52	23	t	t	PROPN
ap-5897	52	24	,	,	PUNCT
ap-5897	52	25	then	then	ADV
ap-5897	52	26	we	we	PRON
ap-5897	52	27	need	need	VERB
ap-5897	52	28	to	to	PART
ap-5897	52	29	alternatingly	alternatingly	ADV
ap-5897	52	30	apply	apply	VERB
ap-5897	52	31	the	the	DET
ap-5897	52	32	functions	function	NOUN
ap-5897	52	33	f1(x	f1(x	NOUN
ap-5897	52	34	)	)	PUNCT
ap-5897	52	35	=	=	PUNCT
ap-5897	52	36	−βx−	−βx−	PUNCT
ap-5897	52	37	b−βxc	b−βxc	NOUN
ap-5897	52	38	and	and	CCONJ
ap-5897	52	39	f2(x	f2(x	NUM
ap-5897	52	40	)	)	PUNCT
ap-5897	52	41	=	=	SYM
ap-5897	52	42	βx−	βx−	PUNCT
ap-5897	52	43	bβxc	bβxc	NOUN
ap-5897	52	44	to	to	ADP
ap-5897	52	45	an	an	DET
ap-5897	52	46	element	element	NOUN
ap-5897	52	47	x	x	SYM
ap-5897	52	48	∈	∈	PROPN
ap-5897	53	1	[	[	X
ap-5897	53	2	0	0	NUM
ap-5897	53	3	,	,	PUNCT
ap-5897	53	4	1	1	NUM
ap-5897	53	5	)	)	PUNCT
ap-5897	53	6	.	.	PUNCT
ap-5897	54	1	this	this	DET
ap-5897	54	2	series	series	NOUN
ap-5897	54	3	of	of	ADP
ap-5897	54	4	applications	application	NOUN
ap-5897	54	5	of	of	ADP
ap-5897	54	6	the	the	DET
ap-5897	54	7	maps	map	NOUN
ap-5897	54	8	f1	f1	NOUN
ap-5897	54	9	and	and	CCONJ
ap-5897	54	10	f2	f2	PROPN
ap-5897	54	11	yields	yield	VERB
ap-5897	54	12	a	a	DET
ap-5897	54	13	numeration	numeration	NOUN
ap-5897	54	14	system	system	NOUN
ap-5897	54	15	in	in	ADP
ap-5897	54	16	[	[	X
ap-5897	54	17	0	0	NUM
ap-5897	54	18	,	,	PUNCT
ap-5897	54	19	1	1	NUM
ap-5897	54	20	)	)	PUNCT
ap-5897	54	21	in	in	ADP
ap-5897	54	22	two	two	NUM
ap-5897	54	23	bases	basis	NOUN
ap-5897	54	24	−β	−β	NOUN
ap-5897	54	25	and	and	CCONJ
ap-5897	54	26	β	β	PROPN
ap-5897	54	27	as	as	SCONJ
ap-5897	54	28	discussed	discuss	VERB
ap-5897	54	29	in	in	ADP
ap-5897	54	30	section	section	NOUN
ap-5897	54	31	2	2	NUM
ap-5897	54	32	below	below	ADV
ap-5897	54	33	.	.	PUNCT
ap-5897	55	1	this	this	DET
ap-5897	55	2	numeration	numeration	NOUN
ap-5897	55	3	system	system	NOUN
ap-5897	55	4	is	be	AUX
ap-5897	55	5	akin	akin	ADJ
ap-5897	55	6	to	to	ADP
ap-5897	55	7	the	the	DET
ap-5897	55	8	q	q	ADJ
ap-5897	55	9	-	-	PUNCT
ap-5897	55	10	cantor	cantor	PROPN
ap-5897	55	11	series	series	NOUN
ap-5897	55	12	expansion	expansion	NOUN
ap-5897	55	13	[	[	X
ap-5897	55	14	13	13	NUM
ap-5897	55	15	]	]	PUNCT
ap-5897	55	16	.	.	PUNCT
ap-5897	56	1	given	give	VERB
ap-5897	56	2	a	a	DET
ap-5897	56	3	sequence	sequence	NOUN
ap-5897	56	4	q	q	NOUN
ap-5897	56	5	=	=	PUNCT
ap-5897	56	6	(	(	PUNCT
ap-5897	56	7	qn)n≥1	qn)n≥1	NOUN
ap-5897	56	8	of	of	ADP
ap-5897	56	9	integers	integer	NOUN
ap-5897	56	10	qn	qn	VERB
ap-5897	56	11	≥	≥	NOUN
ap-5897	56	12	2	2	NUM
ap-5897	56	13	,	,	PUNCT
ap-5897	56	14	the	the	DET
ap-5897	56	15	q	q	ADJ
ap-5897	56	16	-	-	PUNCT
ap-5897	56	17	cantor	cantor	NOUN
ap-5897	56	18	expansion	expansion	NOUN
ap-5897	56	19	of	of	ADP
ap-5897	56	20	a	a	DET
ap-5897	56	21	real	real	ADJ
ap-5897	56	22	214	214	NUM
ap-5897	56	23	https://doi.org/10.14311/ap.2020.60.0214	https://doi.org/10.14311/ap.2020.60.0214	ADV
ap-5897	56	24	https://ojs.cvut.cz/ojs/index.php/ap	https://ojs.cvut.cz/ojs/index.php/ap	NUM
ap-5897	56	25	vol	vol	NOUN
ap-5897	56	26	.	.	PUNCT
ap-5897	57	1	60	60	NUM
ap-5897	57	2	no	no	NOUN
ap-5897	57	3	.	.	PUNCT
ap-5897	58	1	3/2020	3/2020	NUM
ap-5897	58	2	beta	beta	PROPN
ap-5897	58	3	cantor	cantor	PROPN
ap-5897	58	4	series	series	NOUN
ap-5897	58	5	expansion	expansion	NOUN
ap-5897	58	6	and	and	CCONJ
ap-5897	58	7	admissible	admissible	ADJ
ap-5897	58	8	sequences	sequence	NOUN
ap-5897	58	9	number	number	NOUN
ap-5897	58	10	x	x	PUNCT
ap-5897	58	11	is	be	AUX
ap-5897	58	12	the	the	DET
ap-5897	58	13	unique	unique	ADJ
ap-5897	58	14	expansion	expansion	NOUN
ap-5897	58	15	of	of	ADP
ap-5897	58	16	the	the	DET
ap-5897	58	17	form	form	NOUN
ap-5897	58	18	x	x	NOUN
ap-5897	58	19	=	=	SYM
ap-5897	58	20	e0	e0	PROPN
ap-5897	58	21	+	+	CCONJ
ap-5897	58	22	∑	∑	ADP
ap-5897	58	23	n≥1	n≥1	PROPN
ap-5897	58	24	en	en	X
ap-5897	58	25	πn	πn	INTJ
ap-5897	58	26	j=1qj	j=1qj	NOUN
ap-5897	58	27	where	where	SCONJ
ap-5897	58	28	e0	e0	PROPN
ap-5897	58	29	=	=	SYM
ap-5897	58	30	bxc	bxc	NOUN
ap-5897	58	31	and	and	CCONJ
ap-5897	58	32	en	en	ADP
ap-5897	58	33	∈	∈	PROPN
ap-5897	58	34	{	{	PUNCT
ap-5897	58	35	0	0	NUM
ap-5897	58	36	,	,	PUNCT
ap-5897	58	37	1	1	NUM
ap-5897	58	38	,	,	PUNCT
ap-5897	58	39	.	.	PUNCT
ap-5897	58	40	.	.	PUNCT
ap-5897	59	1	.	.	PUNCT
ap-5897	60	1	,	,	PUNCT
ap-5897	60	2	qn	qn	INTJ
ap-5897	60	3	−	−	NOUN
ap-5897	60	4	1	1	NUM
ap-5897	60	5	}	}	PUNCT
ap-5897	60	6	for	for	ADP
ap-5897	60	7	all	all	DET
ap-5897	60	8	n	n	PRON
ap-5897	60	9	≥	≥	NOUN
ap-5897	60	10	1	1	NUM
ap-5897	60	11	such	such	ADJ
ap-5897	60	12	that	that	SCONJ
ap-5897	60	13	en	en	PROPN
ap-5897	60	14	6=	6=	ADP
ap-5897	60	15	qn	qn	NOUN
ap-5897	60	16	−	−	PROPN
ap-5897	60	17	1	1	NUM
ap-5897	60	18	infinitely	infinitely	ADV
ap-5897	60	19	many	many	ADJ
ap-5897	60	20	number	number	NOUN
ap-5897	60	21	of	of	ADP
ap-5897	60	22	times	time	NOUN
ap-5897	60	23	.	.	PUNCT
ap-5897	61	1	we	we	PRON
ap-5897	61	2	call	call	VERB
ap-5897	61	3	the	the	DET
ap-5897	61	4	numeration	numeration	NOUN
ap-5897	61	5	system	system	NOUN
ap-5897	61	6	considered	consider	VERB
ap-5897	61	7	in	in	ADP
ap-5897	61	8	this	this	DET
ap-5897	61	9	paper	paper	NOUN
ap-5897	61	10	the	the	DET
ap-5897	61	11	beta	beta	PROPN
ap-5897	61	12	cantor	cantor	PROPN
ap-5897	61	13	series	series	NOUN
ap-5897	61	14	expansion	expansion	NOUN
ap-5897	61	15	as	as	SCONJ
ap-5897	61	16	it	it	PRON
ap-5897	61	17	marries	marry	VERB
ap-5897	61	18	the	the	DET
ap-5897	61	19	notions	notion	NOUN
ap-5897	61	20	of	of	ADP
ap-5897	61	21	beta	beta	ADJ
ap-5897	61	22	expansion	expansion	NOUN
ap-5897	61	23	and	and	CCONJ
ap-5897	61	24	q	q	ADJ
ap-5897	61	25	-	-	PUNCT
ap-5897	61	26	cantor	cantor	PROPN
ap-5897	61	27	series	series	NOUN
ap-5897	61	28	expansion	expansion	NOUN
ap-5897	61	29	.	.	PUNCT
ap-5897	62	1	as	as	SCONJ
ap-5897	62	2	mentioned	mention	VERB
ap-5897	62	3	in	in	ADP
ap-5897	62	4	section	section	NOUN
ap-5897	62	5	2	2	NUM
ap-5897	62	6	,	,	PUNCT
ap-5897	62	7	the	the	DET
ap-5897	62	8	beta	beta	ADJ
ap-5897	62	9	expansion	expansion	NOUN
ap-5897	62	10	of	of	ADP
ap-5897	62	11	parry	parry	PROPN
ap-5897	62	12	and	and	CCONJ
ap-5897	62	13	the	the	DET
ap-5897	62	14	negative	negative	ADJ
ap-5897	62	15	beta	beta	ADJ
ap-5897	62	16	expansion	expansion	NOUN
ap-5897	62	17	of	of	ADP
ap-5897	62	18	ito	ito	PROPN
ap-5897	62	19	and	and	CCONJ
ap-5897	62	20	sadahiro	sadahiro	PROPN
ap-5897	62	21	are	be	AUX
ap-5897	62	22	examples	example	NOUN
ap-5897	62	23	of	of	ADP
ap-5897	62	24	beta	beta	ADJ
ap-5897	62	25	cantor	cantor	PROPN
ap-5897	62	26	series	series	PROPN
ap-5897	62	27	expansion	expansion	NOUN
ap-5897	62	28	.	.	PUNCT
ap-5897	63	1	it	it	PRON
ap-5897	63	2	is	be	AUX
ap-5897	63	3	the	the	DET
ap-5897	63	4	hope	hope	NOUN
ap-5897	63	5	of	of	ADP
ap-5897	63	6	the	the	DET
ap-5897	63	7	authors	author	NOUN
ap-5897	63	8	that	that	PRON
ap-5897	63	9	the	the	DET
ap-5897	63	10	beta	beta	PROPN
ap-5897	63	11	cantor	cantor	PROPN
ap-5897	63	12	series	series	PROPN
ap-5897	63	13	expansion	expansion	NOUN
ap-5897	63	14	provides	provide	VERB
ap-5897	63	15	a	a	DET
ap-5897	63	16	unified	unified	ADJ
ap-5897	63	17	formulation	formulation	NOUN
ap-5897	63	18	for	for	ADP
ap-5897	63	19	the	the	DET
ap-5897	63	20	positive	positive	ADJ
ap-5897	63	21	and	and	CCONJ
ap-5897	63	22	negative	negative	ADJ
ap-5897	63	23	beta	beta	NOUN
ap-5897	63	24	numeration	numeration	NOUN
ap-5897	63	25	systems	system	NOUN
ap-5897	63	26	to	to	PART
ap-5897	63	27	further	far	ADV
ap-5897	63	28	highlight	highlight	VERB
ap-5897	63	29	their	their	PRON
ap-5897	63	30	similarities	similarity	NOUN
ap-5897	63	31	.	.	PUNCT
ap-5897	64	1	after	after	ADV
ap-5897	64	2	all	all	ADV
ap-5897	64	3	,	,	PUNCT
ap-5897	64	4	the	the	DET
ap-5897	64	5	positive	positive	ADJ
ap-5897	64	6	and	and	CCONJ
ap-5897	64	7	negative	negative	ADJ
ap-5897	64	8	beta	beta	ADJ
ap-5897	64	9	expansions	expansion	NOUN
ap-5897	64	10	share	share	VERB
ap-5897	64	11	many	many	ADJ
ap-5897	64	12	similar	similar	ADJ
ap-5897	64	13	properties	property	NOUN
ap-5897	64	14	(	(	PUNCT
ap-5897	64	15	see	see	VERB
ap-5897	65	1	e.g.	e.g.	ADV
ap-5897	65	2	[	[	X
ap-5897	65	3	9	9	NUM
ap-5897	65	4	,	,	PUNCT
ap-5897	65	5	14	14	NUM
ap-5897	65	6	,	,	PUNCT
ap-5897	65	7	15	15	NUM
ap-5897	65	8	]	]	NUM
ap-5897	65	9	)	)	PUNCT
ap-5897	65	10	.	.	PUNCT
ap-5897	66	1	our	our	PRON
ap-5897	66	2	goal	goal	NOUN
ap-5897	66	3	is	be	AUX
ap-5897	66	4	to	to	PART
ap-5897	66	5	extend	extend	VERB
ap-5897	66	6	the	the	DET
ap-5897	66	7	work	work	NOUN
ap-5897	66	8	of	of	ADP
ap-5897	66	9	parry	parry	NOUN
ap-5897	66	10	on	on	ADP
ap-5897	66	11	admissibility	admissibility	NOUN
ap-5897	66	12	to	to	ADP
ap-5897	66	13	beta	beta	PROPN
ap-5897	66	14	cantor	cantor	PROPN
ap-5897	66	15	series	series	NOUN
ap-5897	66	16	expansions	expansion	NOUN
ap-5897	66	17	.	.	PUNCT
ap-5897	67	1	in	in	ADP
ap-5897	67	2	section	section	NOUN
ap-5897	67	3	2	2	NUM
ap-5897	67	4	,	,	PUNCT
ap-5897	67	5	we	we	PRON
ap-5897	67	6	define	define	VERB
ap-5897	67	7	the	the	DET
ap-5897	67	8	transformations	transformation	NOUN
ap-5897	67	9	that	that	PRON
ap-5897	67	10	induce	induce	VERB
ap-5897	67	11	the	the	DET
ap-5897	67	12	beta	beta	ADJ
ap-5897	67	13	cantor	cantor	PROPN
ap-5897	67	14	series	series	PROPN
ap-5897	67	15	expansion	expansion	NOUN
ap-5897	67	16	.	.	PUNCT
ap-5897	68	1	in	in	ADP
ap-5897	68	2	section	section	NOUN
ap-5897	68	3	3	3	NUM
ap-5897	68	4	,	,	PUNCT
ap-5897	68	5	we	we	PRON
ap-5897	68	6	provide	provide	VERB
ap-5897	68	7	a	a	DET
ap-5897	68	8	discussion	discussion	NOUN
ap-5897	68	9	on	on	ADP
ap-5897	68	10	the	the	DET
ap-5897	68	11	relationship	relationship	NOUN
ap-5897	68	12	between	between	ADP
ap-5897	68	13	two	two	NUM
ap-5897	68	14	different	different	ADJ
ap-5897	68	15	definitions	definition	NOUN
ap-5897	68	16	of	of	ADP
ap-5897	68	17	the	the	DET
ap-5897	68	18	expansion	expansion	NOUN
ap-5897	68	19	of	of	ADP
ap-5897	68	20	γ	γ	PROPN
ap-5897	68	21	+	+	PROPN
ap-5897	68	22	1	1	NUM
ap-5897	68	23	(	(	PUNCT
ap-5897	68	24	similar	similar	ADJ
ap-5897	68	25	to	to	ADP
ap-5897	68	26	the	the	DET
ap-5897	68	27	expansion	expansion	NOUN
ap-5897	68	28	of	of	ADP
ap-5897	68	29	1	1	NUM
ap-5897	68	30	in	in	ADP
ap-5897	68	31	[	[	X
ap-5897	68	32	2	2	NUM
ap-5897	68	33	]	]	PUNCT
ap-5897	68	34	and	and	CCONJ
ap-5897	68	35	the	the	DET
ap-5897	68	36	expansion	expansion	NOUN
ap-5897	68	37	of	of	ADP
ap-5897	68	38	rβ	rβ	NOUN
ap-5897	68	39	in	in	ADP
ap-5897	68	40	[	[	X
ap-5897	68	41	6	6	NUM
ap-5897	68	42	]	]	PUNCT
ap-5897	68	43	)	)	PUNCT
ap-5897	68	44	.	.	PUNCT
ap-5897	69	1	in	in	ADP
ap-5897	69	2	section	section	NOUN
ap-5897	69	3	4	4	NUM
ap-5897	69	4	,	,	PUNCT
ap-5897	69	5	we	we	PRON
ap-5897	69	6	tackle	tackle	VERB
ap-5897	69	7	the	the	DET
ap-5897	69	8	problem	problem	NOUN
ap-5897	69	9	of	of	ADP
ap-5897	69	10	finding	find	VERB
ap-5897	69	11	a	a	DET
ap-5897	69	12	necessary	necessary	ADJ
ap-5897	69	13	and	and	CCONJ
ap-5897	69	14	sufficient	sufficient	ADJ
ap-5897	69	15	condition	condition	NOUN
ap-5897	69	16	for	for	ADP
ap-5897	69	17	a	a	DET
ap-5897	69	18	sequence	sequence	NOUN
ap-5897	69	19	to	to	PART
ap-5897	69	20	be	be	AUX
ap-5897	69	21	admissible	admissible	ADJ
ap-5897	69	22	with	with	ADP
ap-5897	69	23	respect	respect	NOUN
ap-5897	69	24	to	to	ADP
ap-5897	69	25	the	the	DET
ap-5897	69	26	beta	beta	PROPN
ap-5897	69	27	cantor	cantor	PROPN
ap-5897	69	28	series	series	PROPN
ap-5897	69	29	expansion	expansion	NOUN
ap-5897	69	30	.	.	PUNCT
ap-5897	70	1	2	2	NUM
ap-5897	70	2	.	.	X
ap-5897	70	3	b	b	X
ap-5897	70	4	-	-	PUNCT
ap-5897	70	5	expansion	expansion	NOUN
ap-5897	70	6	maps	map	NOUN
ap-5897	70	7	fix	fix	VERB
ap-5897	70	8	γ	γ	NOUN
ap-5897	70	9	∈	∈	NOUN
ap-5897	70	10	r	r	NOUN
ap-5897	70	11	and	and	CCONJ
ap-5897	70	12	let	let	VERB
ap-5897	70	13	b	b	NOUN
ap-5897	70	14	=	=	SYM
ap-5897	70	15	(	(	PUNCT
ap-5897	70	16	β1	β1	PROPN
ap-5897	70	17	,	,	PUNCT
ap-5897	70	18	β2	β2	NOUN
ap-5897	70	19	,	,	PUNCT
ap-5897	70	20	.	.	PUNCT
ap-5897	70	21	.	.	PUNCT
ap-5897	70	22	.	.	PUNCT
ap-5897	70	23	)	)	PUNCT
ap-5897	71	1	where	where	SCONJ
ap-5897	71	2	βi	βi	PRON
ap-5897	71	3	∈	∈	NOUN
ap-5897	71	4	r	r	NOUN
ap-5897	71	5	for	for	ADP
ap-5897	71	6	all	all	DET
ap-5897	71	7	i	i	PRON
ap-5897	71	8	∈	∈	PROPN
ap-5897	71	9	n.	n.	NOUN
ap-5897	71	10	for	for	ADP
ap-5897	71	11	j	j	PROPN
ap-5897	71	12	∈	∈	PROPN
ap-5897	71	13	n	n	CCONJ
ap-5897	71	14	,	,	PUNCT
ap-5897	71	15	we	we	PRON
ap-5897	71	16	define	define	VERB
ap-5897	71	17	fj	fj	X
ap-5897	71	18	:	:	PUNCT
ap-5897	71	19	[	[	X
ap-5897	71	20	γ	γ	X
ap-5897	71	21	,	,	PUNCT
ap-5897	71	22	γ	γ	X
ap-5897	71	23	+	+	NOUN
ap-5897	71	24	1	1	NUM
ap-5897	71	25	)	)	PUNCT
ap-5897	71	26	−→	−→	NOUN
ap-5897	71	27	[	[	X
ap-5897	71	28	γ	γ	X
ap-5897	71	29	,	,	PUNCT
ap-5897	71	30	γ	γ	X
ap-5897	71	31	+	+	NOUN
ap-5897	71	32	1	1	NUM
ap-5897	71	33	)	)	PUNCT
ap-5897	71	34	by	by	ADP
ap-5897	71	35	fj(x	fj(x	NOUN
ap-5897	71	36	)	)	PUNCT
ap-5897	71	37	=	=	SYM
ap-5897	71	38	βjx	βjx	PROPN
ap-5897	71	39	−	−	PROPN
ap-5897	71	40	bβjx−	bβjx−	PROPN
ap-5897	71	41	γc	γc	PROPN
ap-5897	71	42	.	.	PROPN
ap-5897	72	1	for	for	ADP
ap-5897	72	2	m	m	PROPN
ap-5897	72	3	∈	∈	PROPN
ap-5897	72	4	n	n	CCONJ
ap-5897	72	5	,	,	PUNCT
ap-5897	72	6	consider	consider	VERB
ap-5897	72	7	the	the	DET
ap-5897	72	8	transformation	transformation	NOUN
ap-5897	72	9	tm	tm	NOUN
ap-5897	72	10	=	=	PROPN
ap-5897	72	11	tmb	tmb	PROPN
ap-5897	72	12	=	=	SYM
ap-5897	72	13	tmb	tmb	PROPN
ap-5897	72	14	,	,	PUNCT
ap-5897	72	15	γ	γ	X
ap-5897	72	16	on	on	ADP
ap-5897	72	17	[	[	X
ap-5897	72	18	γ	γ	X
ap-5897	72	19	,	,	PUNCT
ap-5897	72	20	γ	γ	X
ap-5897	72	21	+	+	NOUN
ap-5897	72	22	1	1	NUM
ap-5897	72	23	)	)	PUNCT
ap-5897	72	24	given	give	VERB
ap-5897	72	25	by	by	ADP
ap-5897	72	26	tm(x	tm(x	PROPN
ap-5897	72	27	)	)	PUNCT
ap-5897	72	28	=	=	SYM
ap-5897	72	29	fm	fm	PROPN
ap-5897	72	30	(	(	PUNCT
ap-5897	72	31	.	.	PUNCT
ap-5897	72	32	.	.	PUNCT
ap-5897	72	33	.	.	PUNCT
ap-5897	73	1	f3(f2(f1(x	f3(f2(f1(x	NOUN
ap-5897	73	2	)	)	PUNCT
ap-5897	73	3	)	)	PUNCT
ap-5897	73	4	)	)	PUNCT
ap-5897	73	5	.	.	PUNCT
ap-5897	73	6	.	.	PUNCT
ap-5897	73	7	.	.	PUNCT
ap-5897	73	8	)	)	PUNCT
ap-5897	73	9	.	.	PUNCT
ap-5897	74	1	hence	hence	ADV
ap-5897	74	2	,	,	PUNCT
ap-5897	74	3	tm(x	tm(x	PUNCT
ap-5897	74	4	)	)	PUNCT
ap-5897	74	5	=	=	SYM
ap-5897	74	6	βmt	βmt	NOUN
ap-5897	74	7	m−1(x)−	m−1(x)−	PROPN
ap-5897	74	8	am(x	am(x	PUNCT
ap-5897	74	9	)	)	PUNCT
ap-5897	74	10	where	where	SCONJ
ap-5897	74	11	am(x	am(x	NOUN
ap-5897	74	12	)	)	PUNCT
ap-5897	74	13	=	=	SYM
ap-5897	75	1	⌊	⌊	VERB
ap-5897	75	2	βmt	βmt	VERB
ap-5897	75	3	m−1(x)−	m−1(x)−	PROPN
ap-5897	75	4	γ	γ	PROPN
ap-5897	75	5	⌋	⌋	PROPN
ap-5897	75	6	.	.	PUNCT
ap-5897	76	1	for	for	ADP
ap-5897	76	2	β	β	X
ap-5897	76	3	=	=	SYM
ap-5897	76	4	βm	βm	PROPN
ap-5897	76	5	,	,	PUNCT
ap-5897	76	6	we	we	PRON
ap-5897	76	7	also	also	ADV
ap-5897	76	8	define	define	VERB
ap-5897	76	9	uβ	uβ	NOUN
ap-5897	76	10	:	:	PUNCT
ap-5897	76	11	=	=	SYM
ap-5897	76	12	min{bβγ	min{bβγ	VERB
ap-5897	76	13	−	−	PROPN
ap-5897	76	14	γc	γc	PROPN
ap-5897	76	15	,	,	PUNCT
ap-5897	76	16	bβ(γ	bβ(γ	X
ap-5897	76	17	+	+	PROPN
ap-5897	77	1	1)−	1)−	NUM
ap-5897	77	2	γc	γc	NOUN
ap-5897	77	3	}	}	PUNCT
ap-5897	77	4	,	,	PUNCT
ap-5897	77	5	vβ	vβ	ADP
ap-5897	77	6	:	:	PUNCT
ap-5897	77	7	=	=	PUNCT
ap-5897	77	8	max{bβγ	max{bβγ	NOUN
ap-5897	77	9	−	−	PROPN
ap-5897	77	10	γc	γc	PROPN
ap-5897	77	11	,	,	PUNCT
ap-5897	77	12	bβ(γ	bβ(γ	X
ap-5897	77	13	+	+	PUNCT
ap-5897	77	14	1)−	1)−	NUM
ap-5897	77	15	γc	γc	NOUN
ap-5897	77	16	}	}	PUNCT
ap-5897	77	17	and	and	CCONJ
ap-5897	77	18	a(β	a(β	NOUN
ap-5897	77	19	)	)	PUNCT
ap-5897	77	20	:	:	PUNCT
ap-5897	78	1	=	=	X
ap-5897	78	2	{	{	PUNCT
ap-5897	79	1	[	[	X
ap-5897	79	2	uβ	uβ	INTJ
ap-5897	79	3	,	,	PUNCT
ap-5897	79	4	vβ	vβ	NOUN
ap-5897	79	5	)	)	PUNCT
ap-5897	79	6	∩	∩	NOUN
ap-5897	79	7	z	z	NOUN
ap-5897	79	8	if	if	SCONJ
ap-5897	79	9	β	β	PROPN
ap-5897	79	10	>	>	X
ap-5897	79	11	0	0	PUNCT
ap-5897	79	12	and	and	CCONJ
ap-5897	79	13	β	β	X
ap-5897	79	14	+	+	CCONJ
ap-5897	79	15	γ(β	γ(β	PROPN
ap-5897	79	16	−	−	NOUN
ap-5897	79	17	1	1	X
ap-5897	79	18	)	)	PUNCT
ap-5897	79	19	∈	∈	PROPN
ap-5897	79	20	z	z	NOUN
ap-5897	80	1	[	[	X
ap-5897	80	2	uβ	uβ	X
ap-5897	80	3	,	,	PUNCT
ap-5897	80	4	vβ	vβ	X
ap-5897	80	5	]	]	PUNCT
ap-5897	80	6	∩	∩	PROPN
ap-5897	80	7	z	z	NOUN
ap-5897	80	8	otherwise	otherwise	ADV
ap-5897	80	9	.	.	PUNCT
ap-5897	81	1	then	then	ADV
ap-5897	81	2	am(x	am(x	NOUN
ap-5897	81	3	)	)	PUNCT
ap-5897	81	4	∈	∈	PROPN
ap-5897	81	5	a(βm	a(βm	PROPN
ap-5897	81	6	)	)	PUNCT
ap-5897	81	7	.	.	PUNCT
ap-5897	82	1	define	define	VERB
ap-5897	82	2	a(b	a(b	NOUN
ap-5897	82	3	)	)	PUNCT
ap-5897	82	4	:	:	PUNCT
ap-5897	83	1	=	=	SYM
ap-5897	83	2	π∞m=1a(βm	π∞m=1a(βm	PROPN
ap-5897	83	3	)	)	PUNCT
ap-5897	83	4	,	,	PUNCT
ap-5897	83	5	which	which	PRON
ap-5897	83	6	is	be	AUX
ap-5897	83	7	the	the	DET
ap-5897	83	8	set	set	NOUN
ap-5897	83	9	of	of	ADP
ap-5897	83	10	all	all	DET
ap-5897	83	11	sequences	sequence	NOUN
ap-5897	83	12	(	(	PUNCT
ap-5897	83	13	d1	d1	NOUN
ap-5897	83	14	,	,	PUNCT
ap-5897	83	15	d2	d2	PROPN
ap-5897	83	16	,	,	PUNCT
ap-5897	83	17	.	.	PUNCT
ap-5897	83	18	.	.	PUNCT
ap-5897	83	19	.	.	PUNCT
ap-5897	83	20	)	)	PUNCT
ap-5897	84	1	where	where	SCONJ
ap-5897	84	2	dm	dm	PROPN
ap-5897	84	3	∈	∈	PROPN
ap-5897	84	4	a(βm	a(βm	PROPN
ap-5897	84	5	)	)	PUNCT
ap-5897	84	6	.	.	PUNCT
ap-5897	85	1	for	for	ADP
ap-5897	85	2	ease	ease	NOUN
ap-5897	85	3	of	of	ADP
ap-5897	85	4	notation	notation	NOUN
ap-5897	85	5	,	,	PUNCT
ap-5897	85	6	we	we	PRON
ap-5897	85	7	define	define	VERB
ap-5897	85	8	b[i	b[i	NUM
ap-5897	85	9	,	,	PUNCT
ap-5897	85	10	j	j	NOUN
ap-5897	85	11	]	]	X
ap-5897	85	12	:	:	PUNCT
ap-5897	85	13	=	=	PUNCT
ap-5897	85	14	πj	πj	VERB
ap-5897	85	15	m	m	NOUN
ap-5897	85	16	=	=	NOUN
ap-5897	85	17	iβm	iβm	NOUN
ap-5897	85	18	.	.	PUNCT
ap-5897	86	1	when	when	SCONJ
ap-5897	86	2	i	i	PRON
ap-5897	86	3	=	=	NOUN
ap-5897	86	4	1	1	NUM
ap-5897	86	5	,	,	PUNCT
ap-5897	86	6	we	we	PRON
ap-5897	86	7	write	write	VERB
ap-5897	86	8	b[j	b[j	NOUN
ap-5897	86	9	]	]	PUNCT
ap-5897	86	10	instead	instead	ADV
ap-5897	86	11	of	of	ADP
ap-5897	86	12	b[1	b[1	PROPN
ap-5897	86	13	,	,	PUNCT
ap-5897	86	14	j	j	PROPN
ap-5897	86	15	]	]	X
ap-5897	86	16	with	with	ADP
ap-5897	86	17	the	the	DET
ap-5897	86	18	convention	convention	NOUN
ap-5897	86	19	that	that	DET
ap-5897	86	20	b[0	b[0	VERB
ap-5897	86	21	]	]	X
ap-5897	86	22	:	:	PUNCT
ap-5897	86	23	=	=	SYM
ap-5897	86	24	1	1	X
ap-5897	86	25	.	.	X
ap-5897	86	26	observe	observe	VERB
ap-5897	86	27	that	that	SCONJ
ap-5897	86	28	b[m+	b[m+	ADJ
ap-5897	86	29	i	i	X
ap-5897	86	30	]	]	X
ap-5897	86	31	=	=	SYM
ap-5897	86	32	b[m]b[m+	b[m]b[m+	NOUN
ap-5897	86	33	1,m+	1,m+	NUM
ap-5897	87	1	i	i	PRON
ap-5897	87	2	]	]	X
ap-5897	87	3	.	.	PUNCT
ap-5897	88	1	the	the	DET
ap-5897	88	2	transformations	transformation	NOUN
ap-5897	88	3	tm	tm	NOUN
ap-5897	88	4	induce	induce	VERB
ap-5897	88	5	a	a	DET
ap-5897	88	6	numeration	numeration	NOUN
ap-5897	88	7	system	system	NOUN
ap-5897	88	8	on	on	ADP
ap-5897	88	9	the	the	DET
ap-5897	88	10	interval	interval	NOUN
ap-5897	88	11	[	[	X
ap-5897	88	12	γ	γ	X
ap-5897	88	13	,	,	PUNCT
ap-5897	88	14	γ	γ	X
ap-5897	88	15	+	+	NOUN
ap-5897	88	16	1	1	NUM
ap-5897	88	17	)	)	PUNCT
ap-5897	88	18	over	over	ADP
ap-5897	88	19	the	the	DET
ap-5897	88	20	alphabet	alphabet	NOUN
ap-5897	88	21	a(b	a(b	PROPN
ap-5897	88	22	)	)	PUNCT
ap-5897	88	23	if	if	SCONJ
ap-5897	88	24	limm→∞	limm→∞	PROPN
ap-5897	88	25	|b[m]|	|b[m]|	NOUN
ap-5897	88	26	=	=	NOUN
ap-5897	88	27	∞.	∞.	PROPN
ap-5897	88	28	proposition	proposition	NOUN
ap-5897	88	29	2.1	2.1	NUM
ap-5897	88	30	.	.	PUNCT
ap-5897	89	1	let	let	VERB
ap-5897	89	2	b	b	NOUN
ap-5897	89	3	=	=	SYM
ap-5897	89	4	(	(	PUNCT
ap-5897	89	5	β1	β1	PROPN
ap-5897	89	6	,	,	PUNCT
ap-5897	89	7	β2	β2	NOUN
ap-5897	89	8	,	,	PUNCT
ap-5897	89	9	.	.	PUNCT
ap-5897	89	10	.	.	PUNCT
ap-5897	89	11	.	.	PUNCT
ap-5897	89	12	)	)	PUNCT
ap-5897	90	1	∈	∈	PROPN
ap-5897	90	2	rn	rn	PROPN
ap-5897	90	3	and	and	CCONJ
ap-5897	90	4	x	x	PROPN
ap-5897	90	5	∈	∈	PROPN
ap-5897	91	1	[	[	X
ap-5897	91	2	γ	γ	X
ap-5897	91	3	,	,	PUNCT
ap-5897	91	4	γ	γ	X
ap-5897	91	5	+	+	NOUN
ap-5897	91	6	1	1	NUM
ap-5897	91	7	)	)	PUNCT
ap-5897	91	8	.	.	PUNCT
ap-5897	92	1	if	if	SCONJ
ap-5897	92	2	limm→∞	limm→∞	PROPN
ap-5897	92	3	|b[m]|	|b[m]|	NOUN
ap-5897	92	4	=	=	NOUN
ap-5897	92	5	∞	∞	PROPN
ap-5897	92	6	,	,	PUNCT
ap-5897	92	7	then	then	ADV
ap-5897	92	8	x	x	X
ap-5897	92	9	=	=	PUNCT
ap-5897	92	10	∞∑	∞∑	NUM
ap-5897	92	11	i=1	i=1	PRON
ap-5897	92	12	ai(x	ai(x	CCONJ
ap-5897	92	13	)	)	PUNCT
ap-5897	92	14	b[i	b[i	ADV
ap-5897	92	15	]	]	PUNCT
ap-5897	92	16	.	.	PUNCT
ap-5897	93	1	proof	proof	NOUN
ap-5897	93	2	.	.	PUNCT
ap-5897	94	1	for	for	ADP
ap-5897	94	2	simplicity	simplicity	NOUN
ap-5897	94	3	,	,	PUNCT
ap-5897	94	4	let	let	VERB
ap-5897	94	5	aj	aj	PROPN
ap-5897	94	6	=	=	PRON
ap-5897	94	7	aj(x	aj(x	PUNCT
ap-5897	94	8	)	)	PUNCT
ap-5897	94	9	.	.	PUNCT
ap-5897	95	1	note	note	VERB
ap-5897	95	2	that	that	SCONJ
ap-5897	95	3	t	t	PROPN
ap-5897	95	4	j−1(x	j−1(x	PROPN
ap-5897	95	5	)	)	PUNCT
ap-5897	96	1	=	=	SYM
ap-5897	96	2	t	t	PROPN
ap-5897	96	3	j(x	j(x	PROPN
ap-5897	96	4	)	)	PUNCT
ap-5897	97	1	+	+	NUM
ap-5897	97	2	aj	aj	PROPN
ap-5897	97	3	βj	βj	PROPN
ap-5897	97	4	.	.	PUNCT
ap-5897	98	1	hence	hence	ADV
ap-5897	98	2	,	,	PUNCT
ap-5897	98	3	x	x	PUNCT
ap-5897	98	4	=	=	SYM
ap-5897	98	5	a1	a1	NOUN
ap-5897	98	6	β1	β1	NOUN
ap-5897	98	7	+	+	PROPN
ap-5897	98	8	t	t	PROPN
ap-5897	98	9	(	(	PUNCT
ap-5897	98	10	x	x	X
ap-5897	98	11	)	)	PUNCT
ap-5897	98	12	β1	β1	NOUN
ap-5897	98	13	=	=	SYM
ap-5897	98	14	a1	a1	NOUN
ap-5897	98	15	β1	β1	NOUN
ap-5897	98	16	+	+	NUM
ap-5897	98	17	a2	a2	PROPN
ap-5897	98	18	β1β2	β1β2	PUNCT
ap-5897	98	19	+	+	CCONJ
ap-5897	98	20	t	t	NOUN
ap-5897	98	21	2(x	2(x	NUM
ap-5897	98	22	)	)	PUNCT
ap-5897	98	23	β1β2	β1β2	X
ap-5897	98	24	.	.	PUNCT
ap-5897	99	1	in	in	ADP
ap-5897	99	2	general	general	ADJ
ap-5897	99	3	,	,	PUNCT
ap-5897	99	4	x	x	SYM
ap-5897	99	5	=	=	PUNCT
ap-5897	99	6	m∑	m∑	INTJ
ap-5897	99	7	i=1	i=1	PROPN
ap-5897	99	8	ai	ai	AUX
ap-5897	99	9	b[i	b[i	VERB
ap-5897	99	10	]	]	PUNCT
ap-5897	99	11	+	+	NUM
ap-5897	99	12	tm(x	tm(x	X
ap-5897	99	13	)	)	PUNCT
ap-5897	99	14	b[m	b[m	NOUN
ap-5897	99	15	]	]	PUNCT
ap-5897	99	16	.	.	PUNCT
ap-5897	100	1	this	this	PRON
ap-5897	100	2	implies	imply	VERB
ap-5897	100	3	that	that	SCONJ
ap-5897	100	4	as	as	SCONJ
ap-5897	100	5	m→∞,∣∣∣∣∣x−	m→∞,∣∣∣∣∣x−	PROPN
ap-5897	100	6	m∑	m∑	CCONJ
ap-5897	100	7	i=1	i=1	PROPN
ap-5897	100	8	ai	ai	AUX
ap-5897	100	9	b[i	b[i	VERB
ap-5897	100	10	]	]	PUNCT
ap-5897	100	11	∣∣∣∣∣	∣∣∣∣∣	PROPN
ap-5897	100	12	=	=	SYM
ap-5897	100	13	∣∣∣∣tm(x	∣∣∣∣tm(x	PROPN
ap-5897	100	14	)	)	PUNCT
ap-5897	100	15	b[m	b[m	NOUN
ap-5897	100	16	]	]	PUNCT
ap-5897	100	17	∣∣∣∣	∣∣∣∣	PROPN
ap-5897	100	18	≤	≤	NUM
ap-5897	100	19	max	max	NOUN
ap-5897	100	20	{	{	PUNCT
ap-5897	100	21	|γ|	|γ|	PROPN
ap-5897	100	22	,	,	PUNCT
ap-5897	100	23	|γ	|γ	ADV
ap-5897	100	24	+	+	CCONJ
ap-5897	100	25	1|	1|	NUM
ap-5897	100	26	}	}	PUNCT
ap-5897	100	27	|b[m]|	|b[m]|	PROPN
ap-5897	100	28	→	→	SYM
ap-5897	100	29	0	0	NUM
ap-5897	100	30	.	.	PUNCT
ap-5897	101	1	we	we	PRON
ap-5897	101	2	write	write	VERB
ap-5897	101	3	x	x	PUNCT
ap-5897	101	4	=	=	SYM
ap-5897	102	1	∞∑	∞∑	NOUN
ap-5897	102	2	i=1	i=1	AUX
ap-5897	102	3	ai	ai	AUX
ap-5897	102	4	b[i	b[i	NOUN
ap-5897	102	5	]	]	PUNCT
ap-5897	102	6	as	as	ADP
ap-5897	102	7	(	(	PUNCT
ap-5897	102	8	a1	a1	NOUN
ap-5897	102	9	,	,	PUNCT
ap-5897	102	10	a2	a2	PROPN
ap-5897	102	11	,	,	PUNCT
ap-5897	102	12	.	.	PUNCT
ap-5897	102	13	.	.	PUNCT
ap-5897	102	14	.	.	PUNCT
ap-5897	102	15	)	)	PUNCT
ap-5897	103	1	b.	b.	PROPN
ap-5897	104	1	we	we	PRON
ap-5897	104	2	call	call	VERB
ap-5897	104	3	the	the	DET
ap-5897	104	4	sequence	sequence	NOUN
ap-5897	104	5	d(b;x	d(b;x	NOUN
ap-5897	104	6	)	)	PUNCT
ap-5897	104	7	:	:	PUNCT
ap-5897	105	1	=	=	SYM
ap-5897	105	2	(	(	PUNCT
ap-5897	105	3	a1	a1	PROPN
ap-5897	105	4	,	,	PUNCT
ap-5897	105	5	a2	a2	PROPN
ap-5897	105	6	,	,	PUNCT
ap-5897	105	7	.	.	PUNCT
ap-5897	105	8	.	.	PUNCT
ap-5897	105	9	.	.	PUNCT
ap-5897	105	10	)	)	PUNCT
ap-5897	106	1	the	the	DET
ap-5897	106	2	b	b	NOUN
ap-5897	106	3	-	-	PUNCT
ap-5897	106	4	expansion	expansion	NOUN
ap-5897	106	5	of	of	ADP
ap-5897	106	6	x.	x.	NOUN
ap-5897	106	7	let	let	VERB
ap-5897	106	8	1	1	NUM
ap-5897	106	9	<	<	X
ap-5897	106	10	β	β	X
ap-5897	106	11	∈	∈	PROPN
ap-5897	106	12	r.	r.	PROPN
ap-5897	106	13	note	note	VERB
ap-5897	106	14	that	that	SCONJ
ap-5897	106	15	if	if	SCONJ
ap-5897	106	16	γ	γ	X
ap-5897	106	17	=	=	SYM
ap-5897	106	18	0	0	NUM
ap-5897	106	19	and	and	CCONJ
ap-5897	106	20	b	b	X
ap-5897	106	21	=	=	SYM
ap-5897	106	22	(	(	PUNCT
ap-5897	106	23	β	β	NOUN
ap-5897	106	24	)	)	PUNCT
ap-5897	106	25	,	,	PUNCT
ap-5897	106	26	then	then	ADV
ap-5897	106	27	the	the	DET
ap-5897	106	28	b	b	NOUN
ap-5897	106	29	-	-	PUNCT
ap-5897	106	30	expansion	expansion	NOUN
ap-5897	106	31	of	of	ADP
ap-5897	106	32	x	x	SYM
ap-5897	106	33	coincides	coincide	NOUN
ap-5897	106	34	with	with	ADP
ap-5897	106	35	the	the	DET
ap-5897	106	36	classical	classical	ADJ
ap-5897	106	37	β	β	NOUN
ap-5897	106	38	-	-	NOUN
ap-5897	106	39	expansion	expansion	NOUN
ap-5897	106	40	.	.	PUNCT
ap-5897	107	1	(	(	PUNCT
ap-5897	107	2	here	here	ADV
ap-5897	107	3	,	,	PUNCT
ap-5897	107	4	v	v	NOUN
ap-5897	107	5	stands	stand	VERB
ap-5897	107	6	for	for	ADP
ap-5897	107	7	the	the	DET
ap-5897	107	8	periodic	periodic	ADJ
ap-5897	107	9	repetition	repetition	NOUN
ap-5897	107	10	of	of	ADP
ap-5897	107	11	a	a	DET
ap-5897	107	12	word	word	NOUN
ap-5897	107	13	v.	v.	ADP
ap-5897	107	14	)	)	PUNCT
ap-5897	107	15	when	when	SCONJ
ap-5897	107	16	γ	γ	X
ap-5897	107	17	=	=	PRON
ap-5897	107	18	−β/(β	−β/(β	VERB
ap-5897	107	19	+	+	CCONJ
ap-5897	107	20	1	1	NUM
ap-5897	107	21	)	)	PUNCT
ap-5897	107	22	and	and	CCONJ
ap-5897	107	23	b	b	X
ap-5897	107	24	=	=	SYM
ap-5897	107	25	(	(	PUNCT
ap-5897	107	26	−β	−β	PROPN
ap-5897	107	27	)	)	PUNCT
ap-5897	107	28	,	,	PUNCT
ap-5897	107	29	then	then	ADV
ap-5897	107	30	the	the	DET
ap-5897	107	31	b	b	NOUN
ap-5897	107	32	-	-	PUNCT
ap-5897	107	33	expansion	expansion	NOUN
ap-5897	107	34	coincides	coincide	VERB
ap-5897	107	35	with	with	ADP
ap-5897	107	36	the	the	DET
ap-5897	107	37	(	(	PUNCT
ap-5897	107	38	−β)-expansion	−β)-expansion	NOUN
ap-5897	107	39	.	.	PUNCT
ap-5897	108	1	if	if	SCONJ
ap-5897	108	2	b	b	PROPN
ap-5897	108	3	is	be	AUX
ap-5897	108	4	periodic	periodic	ADJ
ap-5897	108	5	,	,	PUNCT
ap-5897	108	6	say	say	VERB
ap-5897	108	7	b	b	X
ap-5897	108	8	=	=	SYM
ap-5897	108	9	(	(	PUNCT
ap-5897	108	10	β1	β1	PROPN
ap-5897	108	11	,	,	PUNCT
ap-5897	108	12	β2	β2	NOUN
ap-5897	108	13	,	,	PUNCT
ap-5897	108	14	.	.	PUNCT
ap-5897	108	15	.	.	PUNCT
ap-5897	108	16	.	.	PUNCT
ap-5897	109	1	,	,	PUNCT
ap-5897	109	2	βn	βn	PROPN
ap-5897	109	3	)	)	PUNCT
ap-5897	109	4	for	for	ADP
ap-5897	109	5	some	some	DET
ap-5897	109	6	n	n	PRON
ap-5897	109	7	∈	∈	PROPN
ap-5897	109	8	n	n	CCONJ
ap-5897	109	9	,	,	PUNCT
ap-5897	109	10	we	we	PRON
ap-5897	109	11	also	also	ADV
ap-5897	109	12	call	call	VERB
ap-5897	109	13	the	the	DET
ap-5897	109	14	bexpansion	bexpansion	NOUN
ap-5897	109	15	as	as	ADP
ap-5897	109	16	the	the	DET
ap-5897	109	17	{	{	PUNCT
ap-5897	109	18	β1	β1	NOUN
ap-5897	109	19	,	,	PUNCT
ap-5897	109	20	.	.	PUNCT
ap-5897	109	21	.	.	PUNCT
ap-5897	110	1	.	.	PUNCT
ap-5897	111	1	,	,	PUNCT
ap-5897	111	2	βn}-expansion	βn}-expansion	NOUN
ap-5897	111	3	of	of	ADP
ap-5897	111	4	x.	x.	NOUN
ap-5897	111	5	in	in	ADP
ap-5897	111	6	this	this	DET
ap-5897	111	7	case	case	NOUN
ap-5897	111	8	,	,	PUNCT
ap-5897	111	9	we	we	PRON
ap-5897	111	10	may	may	AUX
ap-5897	111	11	write	write	VERB
ap-5897	111	12	d(b;x	d(b;x	NOUN
ap-5897	111	13	)	)	PUNCT
ap-5897	111	14	as	as	ADP
ap-5897	111	15	d(β1	d(β1	NOUN
ap-5897	111	16	,	,	PUNCT
ap-5897	111	17	.	.	PUNCT
ap-5897	111	18	.	.	PUNCT
ap-5897	112	1	.	.	PUNCT
ap-5897	113	1	,	,	PUNCT
ap-5897	113	2	βn;x	βn;x	PUNCT
ap-5897	113	3	)	)	PUNCT
ap-5897	113	4	.	.	PUNCT
ap-5897	114	1	we	we	PRON
ap-5897	114	2	may	may	AUX
ap-5897	114	3	extend	extend	VERB
ap-5897	114	4	the	the	DET
ap-5897	114	5	definition	definition	NOUN
ap-5897	114	6	of	of	ADP
ap-5897	114	7	t	t	PROPN
ap-5897	114	8	j	j	PROPN
ap-5897	114	9	to	to	ADP
ap-5897	114	10	γ	γ	PROPN
ap-5897	114	11	+	+	ADP
ap-5897	114	12	1	1	NUM
ap-5897	114	13	as	as	SCONJ
ap-5897	114	14	has	have	AUX
ap-5897	114	15	been	be	AUX
ap-5897	114	16	done	do	VERB
ap-5897	114	17	in	in	ADP
ap-5897	114	18	[	[	X
ap-5897	114	19	2	2	NUM
ap-5897	114	20	]	]	PUNCT
ap-5897	114	21	and	and	CCONJ
ap-5897	114	22	[	[	X
ap-5897	114	23	6	6	NUM
ap-5897	114	24	]	]	PUNCT
ap-5897	114	25	.	.	PUNCT
ap-5897	115	1	for	for	ADP
ap-5897	115	2	all	all	DET
ap-5897	115	3	j	j	PROPN
ap-5897	115	4	∈	∈	PROPN
ap-5897	115	5	n	n	CCONJ
ap-5897	115	6	,	,	PUNCT
ap-5897	115	7	define	define	VERB
ap-5897	115	8	t	t	PROPN
ap-5897	115	9	j(γ	j(γ	PROPN
ap-5897	116	1	+	+	CCONJ
ap-5897	116	2	1	1	NUM
ap-5897	116	3	)	)	PUNCT
ap-5897	116	4	:	:	PUNCT
ap-5897	116	5	=	=	PUNCT
ap-5897	116	6	βjt	βjt	X
ap-5897	116	7	j−1(γ	j−1(γ	PROPN
ap-5897	117	1	+	+	CCONJ
ap-5897	118	1	1)−	1)−	NUM
ap-5897	118	2	⌊	⌊	PROPN
ap-5897	118	3	βjt	βjt	PRON
ap-5897	118	4	j−1(γ	j−1(γ	NOUN
ap-5897	119	1	+	+	CCONJ
ap-5897	120	1	1)−	1)−	PROPN
ap-5897	120	2	γ	γ	NOUN
ap-5897	120	3	⌋	⌋	NOUN
ap-5897	120	4	.	.	PUNCT
ap-5897	121	1	as	as	ADP
ap-5897	121	2	in	in	ADP
ap-5897	121	3	proposition	proposition	NOUN
ap-5897	121	4	2.1	2.1	NUM
ap-5897	121	5	,	,	PUNCT
ap-5897	121	6	we	we	PRON
ap-5897	121	7	have	have	VERB
ap-5897	121	8	γ	γ	NOUN
ap-5897	121	9	+	+	CCONJ
ap-5897	121	10	1	1	NUM
ap-5897	121	11	=	=	SYM
ap-5897	121	12	∞∑	∞∑	NUM
ap-5897	121	13	i=1	i=1	PROPN
ap-5897	121	14	ci	ci	NOUN
ap-5897	121	15	b[i	b[i	NOUN
ap-5897	121	16	]	]	PUNCT
ap-5897	121	17	where	where	SCONJ
ap-5897	121	18	ci	ci	NOUN
ap-5897	121	19	:	:	PUNCT
ap-5897	121	20	=	=	PUNCT
ap-5897	121	21	⌊	⌊	PART
ap-5897	121	22	βit	βit	NOUN
ap-5897	122	1	i−1(γ	i−1(γ	ADV
ap-5897	122	2	+	+	X
ap-5897	123	1	1)−	1)−	PROPN
ap-5897	123	2	γ	γ	NOUN
ap-5897	123	3	⌋	⌋	NOUN
ap-5897	123	4	.	.	PUNCT
ap-5897	124	1	we	we	PRON
ap-5897	124	2	also	also	ADV
ap-5897	124	3	write	write	VERB
ap-5897	124	4	d(b	d(b	PRON
ap-5897	124	5	;	;	PUNCT
ap-5897	124	6	γ+1	γ+1	NUM
ap-5897	124	7	)	)	PUNCT
ap-5897	124	8	=	=	SYM
ap-5897	124	9	(	(	PUNCT
ap-5897	124	10	c1	c1	PROPN
ap-5897	124	11	,	,	PUNCT
ap-5897	124	12	c2	c2	PROPN
ap-5897	124	13	,	,	PUNCT
ap-5897	124	14	.	.	PUNCT
ap-5897	124	15	.	.	PUNCT
ap-5897	124	16	.	.	PUNCT
ap-5897	124	17	)	)	PUNCT
ap-5897	124	18	.	.	PUNCT
ap-5897	125	1	note	note	VERB
ap-5897	125	2	that	that	SCONJ
ap-5897	125	3	c1	c1	PROPN
ap-5897	125	4	∈	∈	PROPN
ap-5897	126	1	[	[	X
ap-5897	126	2	uβ1	uβ1	NOUN
ap-5897	126	3	,	,	PUNCT
ap-5897	126	4	vβ1	vβ1	NOUN
ap-5897	126	5	]	]	X
ap-5897	126	6	∩z	∩z	ADJ
ap-5897	126	7	and	and	CCONJ
ap-5897	126	8	for	for	ADP
ap-5897	126	9	j	j	PROPN
ap-5897	126	10	>	>	X
ap-5897	126	11	1	1	NUM
ap-5897	126	12	,	,	PUNCT
ap-5897	126	13	cj	cj	PROPN
ap-5897	126	14	∈	∈	PROPN
ap-5897	126	15	a(βj	a(βj	PROPN
ap-5897	126	16	)	)	PUNCT
ap-5897	126	17	since	since	SCONJ
ap-5897	126	18	t	t	PROPN
ap-5897	126	19	j−1(γ	j−1(γ	PROPN
ap-5897	126	20	+	+	CCONJ
ap-5897	126	21	1	1	X
ap-5897	126	22	)	)	PUNCT
ap-5897	126	23	<	<	X
ap-5897	126	24	γ	γ	X
ap-5897	126	25	+	+	PROPN
ap-5897	126	26	1	1	NUM
ap-5897	126	27	.	.	PUNCT
ap-5897	126	28	example	example	NOUN
ap-5897	126	29	.	.	PUNCT
ap-5897	127	1	let	let	VERB
ap-5897	127	2	α	α	NOUN
ap-5897	127	3	=	=	X
ap-5897	127	4	−β	−β	NOUN
ap-5897	127	5	=	=	PUNCT
ap-5897	127	6	(	(	PUNCT
ap-5897	127	7	1	1	NUM
ap-5897	127	8	+	+	CCONJ
ap-5897	127	9	√	√	PROPN
ap-5897	127	10	5)/2	5)/2	NUM
ap-5897	127	11	be	be	AUX
ap-5897	127	12	the	the	DET
ap-5897	127	13	golden	golden	ADJ
ap-5897	127	14	mean	mean	NOUN
ap-5897	127	15	.	.	PUNCT
ap-5897	128	1	table	table	NOUN
ap-5897	128	2	1	1	NUM
ap-5897	128	3	gives	give	VERB
ap-5897	128	4	some	some	DET
ap-5897	128	5	information	information	NOUN
ap-5897	128	6	on	on	ADP
ap-5897	128	7	the	the	DET
ap-5897	128	8	{	{	PUNCT
ap-5897	128	9	α	α	NOUN
ap-5897	128	10	,	,	PUNCT
ap-5897	128	11	β}transformations	β}transformation	NOUN
ap-5897	128	12	for	for	ADP
ap-5897	128	13	various	various	ADJ
ap-5897	128	14	values	value	NOUN
ap-5897	128	15	of	of	ADP
ap-5897	128	16	γ	γ	PROPN
ap-5897	128	17	.	.	PROPN
ap-5897	128	18	215	215	NUM
ap-5897	128	19	jonathan	jonathan	PROPN
ap-5897	128	20	caalim	caalim	PROPN
ap-5897	128	21	,	,	PUNCT
ap-5897	128	22	shiela	shiela	PROPN
ap-5897	128	23	demegillo	demegillo	PROPN
ap-5897	128	24	acta	acta	PROPN
ap-5897	128	25	polytechnica	polytechnica	PROPN
ap-5897	128	26	γ	γ	PROPN
ap-5897	128	27	a(α	a(α	ADV
ap-5897	128	28	)	)	PUNCT
ap-5897	128	29	a(β	a(β	NOUN
ap-5897	128	30	)	)	PUNCT
ap-5897	128	31	d(α	d(α	NOUN
ap-5897	128	32	,	,	PUNCT
ap-5897	128	33	β	β	X
ap-5897	128	34	;	;	PUNCT
ap-5897	128	35	γ	γ	X
ap-5897	128	36	)	)	PUNCT
ap-5897	128	37	d(α	d(α	PROPN
ap-5897	128	38	,	,	PUNCT
ap-5897	128	39	β	β	X
ap-5897	128	40	;	;	PUNCT
ap-5897	128	41	γ	γ	X
ap-5897	128	42	+	+	PROPN
ap-5897	128	43	1	1	NUM
ap-5897	128	44	)	)	PUNCT
ap-5897	128	45	0	0	NUM
ap-5897	129	1	{	{	PUNCT
ap-5897	129	2	0	0	NUM
ap-5897	129	3	,	,	PUNCT
ap-5897	129	4	1	1	NUM
ap-5897	129	5	}	}	PUNCT
ap-5897	129	6	{	{	PUNCT
ap-5897	129	7	−2,−1	−2,−1	ADJ
ap-5897	129	8	,	,	PUNCT
ap-5897	129	9	0	0	NUM
ap-5897	129	10	}	}	PUNCT
ap-5897	129	11	(	(	PUNCT
ap-5897	129	12	0	0	NUM
ap-5897	129	13	)	)	PUNCT
ap-5897	129	14	(	(	PUNCT
ap-5897	129	15	1,−1	1,−1	NUM
ap-5897	129	16	,	,	PUNCT
ap-5897	129	17	0	0	NUM
ap-5897	129	18	)	)	PUNCT
ap-5897	129	19	1	1	NUM
ap-5897	129	20	/	/	SYM
ap-5897	129	21	α	α	PRON
ap-5897	129	22	{	{	PUNCT
ap-5897	129	23	0	0	NUM
ap-5897	129	24	,	,	PUNCT
ap-5897	129	25	1	1	NUM
ap-5897	129	26	}	}	PUNCT
ap-5897	129	27	{	{	PUNCT
ap-5897	129	28	−4,−3,−2	−4,−3,−2	NUM
ap-5897	129	29	}	}	PUNCT
ap-5897	129	30	(	(	PUNCT
ap-5897	129	31	0,−3	0,−3	PROPN
ap-5897	129	32	,	,	PUNCT
ap-5897	129	33	1,−3	1,−3	NUM
ap-5897	129	34	,	,	PUNCT
ap-5897	129	35	1,−2	1,−2	NUM
ap-5897	129	36	)	)	PUNCT
ap-5897	129	37	(	(	PUNCT
ap-5897	129	38	2,−2	2,−2	NUM
ap-5897	129	39	,	,	PUNCT
ap-5897	129	40	1	1	NUM
ap-5897	129	41	)	)	SYM
ap-5897	129	42	2	2	NUM
ap-5897	129	43	{	{	PUNCT
ap-5897	129	44	1	1	NUM
ap-5897	129	45	,	,	PUNCT
ap-5897	129	46	2	2	NUM
ap-5897	129	47	}	}	PUNCT
ap-5897	129	48	{	{	PUNCT
ap-5897	129	49	−7,−6	−7,−6	NOUN
ap-5897	129	50	}	}	PUNCT
ap-5897	129	51	(	(	PUNCT
ap-5897	129	52	1,−6	1,−6	PROPN
ap-5897	129	53	,	,	PUNCT
ap-5897	129	54	1,−7	1,−7	NUM
ap-5897	129	55	)	)	PUNCT
ap-5897	129	56	(	(	PUNCT
ap-5897	129	57	2,−7	2,−7	PROPN
ap-5897	129	58	,	,	PUNCT
ap-5897	129	59	1	1	NUM
ap-5897	129	60	)	)	PUNCT
ap-5897	129	61	table	table	NOUN
ap-5897	129	62	1	1	NUM
ap-5897	129	63	.	.	PUNCT
ap-5897	130	1	the	the	DET
ap-5897	130	2	expansion	expansion	NOUN
ap-5897	130	3	of	of	ADP
ap-5897	130	4	γ	γ	PROPN
ap-5897	130	5	and	and	CCONJ
ap-5897	130	6	γ	γ	X
ap-5897	130	7	+	+	ADP
ap-5897	130	8	1	1	NUM
ap-5897	130	9	under	under	ADP
ap-5897	130	10	various	various	ADJ
ap-5897	130	11	values	value	NOUN
ap-5897	130	12	of	of	ADP
ap-5897	130	13	γ	γ	NOUN
ap-5897	130	14	when	when	SCONJ
ap-5897	130	15	α	α	NOUN
ap-5897	130	16	=	=	X
ap-5897	130	17	−β	−β	NOUN
ap-5897	130	18	=	=	PUNCT
ap-5897	130	19	(	(	PUNCT
ap-5897	130	20	1	1	NUM
ap-5897	130	21	+	+	CCONJ
ap-5897	130	22	√	√	ADJ
ap-5897	130	23	5)/2	5)/2	NUM
ap-5897	130	24	figure	figure	NOUN
ap-5897	130	25	1	1	NUM
ap-5897	130	26	.	.	PUNCT
ap-5897	131	1	the	the	DET
ap-5897	131	2	maps	maps	PROPN
ap-5897	131	3	t	t	PROPN
ap-5897	131	4	and	and	CCONJ
ap-5897	131	5	t	t	PROPN
ap-5897	131	6	2	2	NUM
ap-5897	132	1	when	when	SCONJ
ap-5897	132	2	γ	γ	X
ap-5897	132	3	=	=	SYM
ap-5897	132	4	0	0	PROPN
ap-5897	132	5	,	,	PUNCT
ap-5897	132	6	α	α	NOUN
ap-5897	132	7	=	=	X
ap-5897	132	8	−β	−β	NOUN
ap-5897	132	9	=	=	PUNCT
ap-5897	132	10	(	(	PUNCT
ap-5897	132	11	1	1	NUM
ap-5897	132	12	+	+	CCONJ
ap-5897	132	13	√	√	PROPN
ap-5897	132	14	5)/2	5)/2	NUM
ap-5897	132	15	let	let	VERB
ap-5897	132	16	us	we	PRON
ap-5897	132	17	consider	consider	VERB
ap-5897	132	18	the	the	DET
ap-5897	132	19	particular	particular	ADJ
ap-5897	132	20	case	case	NOUN
ap-5897	132	21	of	of	ADP
ap-5897	132	22	γ	γ	X
ap-5897	132	23	=	=	SYM
ap-5897	132	24	0	0	PROPN
ap-5897	132	25	.	.	PUNCT
ap-5897	133	1	then	then	ADV
ap-5897	133	2	fα(x	fα(x	NOUN
ap-5897	133	3	)	)	PUNCT
ap-5897	134	1	=	=	PRON
ap-5897	134	2	{	{	PUNCT
ap-5897	134	3	αx	αx	ADV
ap-5897	134	4	if	if	SCONJ
ap-5897	134	5	x	x	X
ap-5897	134	6	∈	∈	PROPN
ap-5897	134	7	[	[	X
ap-5897	134	8	0	0	NUM
ap-5897	134	9	,	,	PUNCT
ap-5897	134	10	1	1	NUM
ap-5897	134	11	/	/	SYM
ap-5897	134	12	α	α	NOUN
ap-5897	134	13	)	)	PUNCT
ap-5897	134	14	αx−	αx−	NUM
ap-5897	134	15	1	1	NUM
ap-5897	134	16	if	if	SCONJ
ap-5897	134	17	x	x	X
ap-5897	134	18	∈	∈	PROPN
ap-5897	135	1	[	[	X
ap-5897	135	2	1	1	NUM
ap-5897	135	3	/	/	SYM
ap-5897	135	4	α	α	NOUN
ap-5897	135	5	,	,	PUNCT
ap-5897	135	6	1	1	NUM
ap-5897	135	7	)	)	PUNCT
ap-5897	135	8	and	and	CCONJ
ap-5897	135	9	fβ(x	fβ(x	NOUN
ap-5897	135	10	)	)	PUNCT
ap-5897	135	11	=	=	SYM
ap-5897	136	1			NOUN
ap-5897	136	2	−αx	−αx	VERB
ap-5897	136	3	if	if	SCONJ
ap-5897	136	4	x	x	PROPN
ap-5897	136	5	=	=	NOUN
ap-5897	136	6	0	0	SYM
ap-5897	137	1	−αx+	−αx+	NUM
ap-5897	137	2	1	1	NUM
ap-5897	137	3	if	if	SCONJ
ap-5897	137	4	x	x	X
ap-5897	137	5	∈	∈	PROPN
ap-5897	137	6	(	(	PUNCT
ap-5897	137	7	0	0	NUM
ap-5897	137	8	,	,	PUNCT
ap-5897	137	9	1	1	NUM
ap-5897	137	10	/	/	SYM
ap-5897	137	11	α	α	NOUN
ap-5897	137	12	]	]	X
ap-5897	137	13	−αx+	−αx+	NUM
ap-5897	137	14	2	2	NUM
ap-5897	137	15	if	if	SCONJ
ap-5897	137	16	x	x	X
ap-5897	137	17	∈	∈	PROPN
ap-5897	137	18	(	(	PUNCT
ap-5897	137	19	1	1	NUM
ap-5897	137	20	/	/	SYM
ap-5897	137	21	α	α	NOUN
ap-5897	137	22	,	,	PUNCT
ap-5897	137	23	1	1	NUM
ap-5897	137	24	)	)	PUNCT
ap-5897	137	25	.	.	PUNCT
ap-5897	138	1	figure	figure	VERB
ap-5897	138	2	1	1	NUM
ap-5897	138	3	depicts	depict	VERB
ap-5897	138	4	the	the	DET
ap-5897	138	5	shape	shape	NOUN
ap-5897	138	6	of	of	ADP
ap-5897	138	7	the	the	DET
ap-5897	138	8	{	{	PUNCT
ap-5897	138	9	α	α	NOUN
ap-5897	138	10	,	,	PUNCT
ap-5897	138	11	β}transformations	β}transformation	NOUN
ap-5897	138	12	t	t	NOUN
ap-5897	138	13	and	and	CCONJ
ap-5897	138	14	t	t	PROPN
ap-5897	138	15	2	2	NUM
ap-5897	138	16	.	.	PUNCT
ap-5897	139	1	from	from	ADP
ap-5897	139	2	these	these	PRON
ap-5897	139	3	,	,	PUNCT
ap-5897	139	4	we	we	PRON
ap-5897	139	5	obtain	obtain	VERB
ap-5897	139	6	a	a	DET
ap-5897	139	7	graph	graph	NOUN
ap-5897	139	8	g	g	NOUN
ap-5897	139	9	(	(	PUNCT
ap-5897	139	10	figure	figure	NOUN
ap-5897	139	11	2	2	NUM
ap-5897	139	12	)	)	PUNCT
ap-5897	139	13	describing	describe	VERB
ap-5897	139	14	the	the	DET
ap-5897	139	15	dynamics	dynamic	NOUN
ap-5897	139	16	of	of	ADP
ap-5897	139	17	the	the	DET
ap-5897	139	18	map	map	NOUN
ap-5897	139	19	tm	tm	PROPN
ap-5897	139	20	.	.	PROPN
ap-5897	140	1	in	in	ADP
ap-5897	140	2	this	this	DET
ap-5897	140	3	graph	graph	NOUN
ap-5897	140	4	,	,	PUNCT
ap-5897	140	5	the	the	DET
ap-5897	140	6	vertices	vertex	NOUN
ap-5897	140	7	are	be	AUX
ap-5897	140	8	subintervals	subinterval	NOUN
ap-5897	140	9	of	of	ADP
ap-5897	140	10	[	[	X
ap-5897	140	11	γ	γ	X
ap-5897	140	12	,	,	PUNCT
ap-5897	140	13	γ	γ	X
ap-5897	140	14	+	+	NOUN
ap-5897	140	15	1	1	NUM
ap-5897	140	16	)	)	PUNCT
ap-5897	140	17	that	that	PRON
ap-5897	140	18	form	form	VERB
ap-5897	140	19	its	its	PRON
ap-5897	140	20	partition	partition	NOUN
ap-5897	140	21	and	and	CCONJ
ap-5897	140	22	there	there	PRON
ap-5897	140	23	is	be	VERB
ap-5897	140	24	a	a	DET
ap-5897	140	25	directed	direct	VERB
ap-5897	140	26	edge	edge	NOUN
ap-5897	140	27	(	(	PUNCT
ap-5897	140	28	dashed	dash	VERB
ap-5897	140	29	,	,	PUNCT
ap-5897	140	30	if	if	SCONJ
ap-5897	140	31	τ	τ	PROPN
ap-5897	140	32	=	=	SYM
ap-5897	140	33	α	α	PROPN
ap-5897	140	34	;	;	PUNCT
ap-5897	140	35	and	and	CCONJ
ap-5897	140	36	solid	solid	ADJ
ap-5897	140	37	,	,	PUNCT
ap-5897	140	38	otherwise	otherwise	ADV
ap-5897	140	39	)	)	PUNCT
ap-5897	140	40	from	from	ADP
ap-5897	140	41	vertex	vertex	NOUN
ap-5897	140	42	v1	v1	NOUN
ap-5897	140	43	to	to	PART
ap-5897	140	44	vertex	vertex	VERB
ap-5897	140	45	v2	v2	NOUN
ap-5897	140	46	labelled	label	VERB
ap-5897	140	47	d	d	NOUN
ap-5897	140	48	if	if	SCONJ
ap-5897	141	1	and	and	CCONJ
ap-5897	141	2	only	only	ADV
ap-5897	141	3	if	if	SCONJ
ap-5897	141	4	v2	v2	PROPN
ap-5897	141	5	⊂	⊂	PRON
ap-5897	141	6	fτ	fτ	X
ap-5897	141	7	(	(	PUNCT
ap-5897	141	8	v1	v1	NOUN
ap-5897	141	9	)	)	PUNCT
ap-5897	141	10	and	and	CCONJ
ap-5897	141	11	the	the	DET
ap-5897	141	12	corresponding	corresponding	ADJ
ap-5897	141	13	digit	digit	NOUN
ap-5897	141	14	is	be	AUX
ap-5897	141	15	d.	d.	PROPN
ap-5897	141	16	now	now	ADV
ap-5897	141	17	,	,	PUNCT
ap-5897	141	18	let	let	VERB
ap-5897	141	19	γ	γ	X
ap-5897	141	20	=	=	SYM
ap-5897	141	21	1	1	NUM
ap-5897	141	22	/	/	SYM
ap-5897	141	23	α	α	NOUN
ap-5897	141	24	.	.	PUNCT
ap-5897	142	1	we	we	PRON
ap-5897	142	2	have	have	VERB
ap-5897	142	3	fα(x	fα(x	NOUN
ap-5897	142	4	)	)	PUNCT
ap-5897	143	1	=	=	PRON
ap-5897	143	2	{	{	PUNCT
ap-5897	143	3	αx	αx	ADV
ap-5897	143	4	if	if	SCONJ
ap-5897	143	5	x	x	X
ap-5897	143	6	∈	∈	PROPN
ap-5897	144	1	[	[	X
ap-5897	144	2	1	1	NUM
ap-5897	144	3	/	/	SYM
ap-5897	144	4	α	α	NOUN
ap-5897	144	5	,	,	PUNCT
ap-5897	144	6	1	1	NUM
ap-5897	144	7	)	)	PUNCT
ap-5897	144	8	αx−	αx−	NUM
ap-5897	144	9	1	1	NUM
ap-5897	144	10	if	if	SCONJ
ap-5897	144	11	x	x	X
ap-5897	144	12	∈	∈	PROPN
ap-5897	145	1	[	[	X
ap-5897	145	2	1	1	NUM
ap-5897	145	3	,	,	PUNCT
ap-5897	145	4	α	α	NOUN
ap-5897	145	5	)	)	PUNCT
ap-5897	145	6	and	and	CCONJ
ap-5897	145	7	fβ(x	fβ(x	NOUN
ap-5897	145	8	)	)	PUNCT
ap-5897	146	1	=	=	PUNCT
ap-5897	147	1			NOUN
ap-5897	147	2	−αx+	−αx+	PRON
ap-5897	147	3	2	2	NUM
ap-5897	147	4	if	if	SCONJ
ap-5897	147	5	x	x	X
ap-5897	147	6	∈	∈	PROPN
ap-5897	147	7	[	[	X
ap-5897	147	8	1	1	NUM
ap-5897	147	9	/	/	SYM
ap-5897	147	10	α	α	NOUN
ap-5897	147	11	,	,	PUNCT
ap-5897	147	12	3	3	NUM
ap-5897	147	13	/	/	SYM
ap-5897	147	14	α−	α−	ADP
ap-5897	147	15	1	1	NUM
ap-5897	147	16	]	]	PUNCT
ap-5897	147	17	−αx+	−αx+	PRON
ap-5897	147	18	3	3	NUM
ap-5897	147	19	if	if	SCONJ
ap-5897	147	20	x	x	X
ap-5897	147	21	∈	∈	PROPN
ap-5897	147	22	(	(	PUNCT
ap-5897	147	23	3	3	NUM
ap-5897	147	24	/	/	SYM
ap-5897	147	25	α−	α−	ADP
ap-5897	147	26	1	1	NUM
ap-5897	147	27	,	,	PUNCT
ap-5897	147	28	4	4	NUM
ap-5897	147	29	/	/	SYM
ap-5897	147	30	α−	α−	ADP
ap-5897	147	31	1	1	NUM
ap-5897	147	32	]	]	PUNCT
ap-5897	147	33	−αx+	−αx+	PRON
ap-5897	147	34	4	4	NUM
ap-5897	147	35	if	if	SCONJ
ap-5897	147	36	x	x	X
ap-5897	147	37	∈	∈	NOUN
ap-5897	147	38	(	(	PUNCT
ap-5897	147	39	4	4	NUM
ap-5897	147	40	/	/	SYM
ap-5897	147	41	α−	α−	ADP
ap-5897	147	42	1	1	NUM
ap-5897	147	43	,	,	PUNCT
ap-5897	147	44	α	α	NOUN
ap-5897	147	45	)	)	PUNCT
ap-5897	147	46	.	.	PUNCT
ap-5897	148	1	figure	figure	NOUN
ap-5897	148	2	3	3	NUM
ap-5897	148	3	gives	give	VERB
ap-5897	148	4	a	a	DET
ap-5897	148	5	graph	graph	NOUN
ap-5897	148	6	corresponding	correspond	VERB
ap-5897	148	7	to	to	ADP
ap-5897	148	8	γ	γ	PROPN
ap-5897	148	9	=	=	SYM
ap-5897	148	10	1	1	NUM
ap-5897	148	11	/	/	SYM
ap-5897	148	12	α	α	NOUN
ap-5897	148	13	where	where	SCONJ
ap-5897	148	14	j	j	NOUN
ap-5897	148	15	:	:	PUNCT
ap-5897	148	16	=	=	SYM
ap-5897	148	17	(	(	PUNCT
ap-5897	148	18	1	1	NUM
ap-5897	148	19	/	/	SYM
ap-5897	148	20	α	α	NOUN
ap-5897	148	21	,	,	PUNCT
ap-5897	148	22	4	4	NUM
ap-5897	148	23	−	−	NOUN
ap-5897	148	24	2α	2α	NOUN
ap-5897	148	25	)	)	PUNCT
ap-5897	148	26	,	,	PUNCT
ap-5897	148	27	k	k	X
ap-5897	149	1	:	:	PUNCT
ap-5897	149	2	=	=	SYM
ap-5897	149	3	(	(	PUNCT
ap-5897	149	4	4	4	NUM
ap-5897	149	5	−	−	NOUN
ap-5897	149	6	2α	2α	NOUN
ap-5897	149	7	,	,	PUNCT
ap-5897	149	8	3	3	NUM
ap-5897	149	9	/	/	SYM
ap-5897	149	10	α	α	NOUN
ap-5897	149	11	−	−	NOUN
ap-5897	149	12	1	1	NUM
ap-5897	149	13	)	)	PUNCT
ap-5897	149	14	,	,	PUNCT
ap-5897	149	15	l	l	NOUN
ap-5897	149	16	:	:	PUNCT
ap-5897	149	17	=	=	SYM
ap-5897	149	18	(	(	PUNCT
ap-5897	149	19	3	3	NUM
ap-5897	149	20	/	/	SYM
ap-5897	149	21	α−1	α−1	PROPN
ap-5897	149	22	,	,	PUNCT
ap-5897	149	23	1),m	1),m	NUM
ap-5897	149	24	:	:	PUNCT
ap-5897	149	25	=	=	SYM
ap-5897	149	26	(	(	PUNCT
ap-5897	149	27	1	1	NUM
ap-5897	149	28	,	,	PUNCT
ap-5897	149	29	2α−2	2α−2	NUM
ap-5897	149	30	)	)	PUNCT
ap-5897	149	31	,	,	PUNCT
ap-5897	149	32	p	p	X
ap-5897	149	33	:	:	PUNCT
ap-5897	149	34	=	=	SYM
ap-5897	149	35	(	(	PUNCT
ap-5897	149	36	2α−2	2α−2	NUM
ap-5897	149	37	,	,	PUNCT
ap-5897	149	38	3−α	3−α	NUM
ap-5897	149	39	)	)	PUNCT
ap-5897	149	40	,	,	PUNCT
ap-5897	149	41	q	q	NOUN
ap-5897	149	42	:	:	PUNCT
ap-5897	149	43	=	=	SYM
ap-5897	149	44	(	(	PUNCT
ap-5897	149	45	3−	3−	NUM
ap-5897	149	46	α	α	NOUN
ap-5897	149	47	,	,	PUNCT
ap-5897	149	48	4	4	NUM
ap-5897	149	49	/	/	SYM
ap-5897	149	50	α−	α−	ADP
ap-5897	149	51	1	1	NUM
ap-5897	149	52	)	)	PUNCT
ap-5897	149	53	and	and	CCONJ
ap-5897	149	54	r	r	NOUN
ap-5897	149	55	:	:	PUNCT
ap-5897	149	56	=	=	SYM
ap-5897	149	57	(	(	PUNCT
ap-5897	149	58	4	4	NUM
ap-5897	149	59	/	/	SYM
ap-5897	149	60	α−	α−	ADP
ap-5897	149	61	1	1	NUM
ap-5897	149	62	,	,	PUNCT
ap-5897	149	63	α	α	NOUN
ap-5897	149	64	)	)	PUNCT
ap-5897	149	65	.	.	PUNCT
ap-5897	150	1	3	3	X
ap-5897	150	2	.	.	X
ap-5897	150	3	expansion	expansion	NOUN
ap-5897	150	4	of	of	ADP
ap-5897	150	5	γ	γ	PROPN
ap-5897	150	6	+	+	PROPN
ap-5897	150	7	1	1	NUM
ap-5897	150	8	the	the	DET
ap-5897	150	9	expansion	expansion	NOUN
ap-5897	150	10	of	of	ADP
ap-5897	150	11	γ+	γ+	NUM
ap-5897	150	12	1	1	NUM
ap-5897	150	13	defined	define	VERB
ap-5897	150	14	in	in	ADP
ap-5897	150	15	the	the	DET
ap-5897	150	16	previous	previous	ADJ
ap-5897	150	17	section	section	NOUN
ap-5897	150	18	proves	prove	VERB
ap-5897	150	19	to	to	PART
ap-5897	150	20	be	be	AUX
ap-5897	150	21	insufficient	insufficient	ADJ
ap-5897	150	22	for	for	ADP
ap-5897	150	23	our	our	PRON
ap-5897	150	24	purposes	purpose	NOUN
ap-5897	150	25	and	and	CCONJ
ap-5897	150	26	hence	hence	ADV
ap-5897	150	27	,	,	PUNCT
ap-5897	150	28	the	the	DET
ap-5897	150	29	definition	definition	NOUN
ap-5897	150	30	needs	need	VERB
ap-5897	150	31	to	to	PART
ap-5897	150	32	be	be	AUX
ap-5897	150	33	modified	modify	VERB
ap-5897	150	34	.	.	PUNCT
ap-5897	151	1	in	in	ADP
ap-5897	151	2	this	this	DET
ap-5897	151	3	section	section	NOUN
ap-5897	151	4	,	,	PUNCT
ap-5897	151	5	we	we	PRON
ap-5897	151	6	present	present	VERB
ap-5897	151	7	another	another	DET
ap-5897	151	8	definition	definition	NOUN
ap-5897	151	9	of	of	ADP
ap-5897	151	10	the	the	DET
ap-5897	151	11	expansion	expansion	NOUN
ap-5897	151	12	of	of	ADP
ap-5897	151	13	γ+	γ+	X
ap-5897	151	14	1	1	NUM
ap-5897	151	15	analogous	analogous	ADJ
ap-5897	151	16	to	to	ADP
ap-5897	151	17	those	those	PRON
ap-5897	151	18	defined	define	VERB
ap-5897	151	19	in	in	ADP
ap-5897	151	20	[	[	X
ap-5897	151	21	2	2	NUM
ap-5897	151	22	]	]	PUNCT
ap-5897	151	23	and	and	CCONJ
ap-5897	151	24	[	[	X
ap-5897	151	25	6	6	NUM
ap-5897	151	26	,	,	PUNCT
ap-5897	151	27	lemma	lemma	PROPN
ap-5897	151	28	6	6	NUM
ap-5897	151	29	]	]	PUNCT
ap-5897	151	30	.	.	PUNCT
ap-5897	152	1	hereafter	hereafter	ADV
ap-5897	152	2	,	,	PUNCT
ap-5897	152	3	we	we	PRON
ap-5897	152	4	assume	assume	VERB
ap-5897	152	5	b	b	X
ap-5897	152	6	=	=	SYM
ap-5897	152	7	(	(	PUNCT
ap-5897	152	8	β1	β1	PROPN
ap-5897	152	9	,	,	PUNCT
ap-5897	152	10	β2	β2	NOUN
ap-5897	152	11	,	,	PUNCT
ap-5897	152	12	.	.	PUNCT
ap-5897	152	13	.	.	PUNCT
ap-5897	152	14	.	.	PUNCT
ap-5897	152	15	)	)	PUNCT
ap-5897	153	1	∈	∈	PROPN
ap-5897	153	2	rn	rn	PROPN
ap-5897	153	3	with	with	ADP
ap-5897	153	4	limm→∞	limm→∞	PROPN
ap-5897	153	5	|b[m]|	|b[m]|	NOUN
ap-5897	154	1	=	=	SYM
ap-5897	154	2	∞.	∞.	PROPN
ap-5897	154	3	definition	definition	NOUN
ap-5897	154	4	1	1	NUM
ap-5897	154	5	.	.	PUNCT
ap-5897	155	1	we	we	PRON
ap-5897	155	2	define	define	VERB
ap-5897	155	3	d∗(b	d∗(b	PROPN
ap-5897	155	4	;	;	PUNCT
ap-5897	155	5	γ	γ	X
ap-5897	155	6	+	+	NOUN
ap-5897	155	7	1	1	NUM
ap-5897	155	8	)	)	PUNCT
ap-5897	155	9	=	=	NOUN
ap-5897	155	10	(	(	PUNCT
ap-5897	155	11	c∗1	c∗1	ADJ
ap-5897	155	12	,	,	PUNCT
ap-5897	155	13	c∗2	c∗2	NOUN
ap-5897	155	14	,	,	PUNCT
ap-5897	155	15	.	.	PUNCT
ap-5897	155	16	.	.	PUNCT
ap-5897	155	17	.	.	PUNCT
ap-5897	155	18	)	)	PUNCT
ap-5897	156	1	as	as	ADP
ap-5897	156	2	the	the	DET
ap-5897	156	3	limit	limit	NOUN
ap-5897	156	4	lim	lim	NOUN
ap-5897	156	5	x→(γ+1)−	x→(γ+1)−	PUNCT
ap-5897	156	6	d(b;x	d(b;x	PROPN
ap-5897	156	7	)	)	PUNCT
ap-5897	156	8	.	.	PUNCT
ap-5897	157	1	that	that	PRON
ap-5897	157	2	is	be	AUX
ap-5897	157	3	,	,	PUNCT
ap-5897	157	4	for	for	ADP
ap-5897	157	5	any	any	DET
ap-5897	157	6	n	n	PRON
ap-5897	157	7	∈	∈	PROPN
ap-5897	157	8	n	n	CCONJ
ap-5897	157	9	,	,	PUNCT
ap-5897	157	10	there	there	PRON
ap-5897	157	11	exists	exist	VERB
ap-5897	157	12	an	an	DET
ap-5897	157	13	εn	εn	ADJ
ap-5897	157	14	>	>	X
ap-5897	157	15	0	0	NUM
ap-5897	157	16	such	such	ADJ
ap-5897	157	17	that	that	PRON
ap-5897	157	18	for	for	SCONJ
ap-5897	157	19	all	all	DET
ap-5897	157	20	x	x	SYM
ap-5897	157	21	∈	∈	PROPN
ap-5897	157	22	(	(	PUNCT
ap-5897	157	23	γ	γ	X
ap-5897	157	24	+	+	NOUN
ap-5897	157	25	1	1	NUM
ap-5897	157	26	−	−	NOUN
ap-5897	157	27	εn	εn	ADJ
ap-5897	157	28	,	,	PUNCT
ap-5897	157	29	γ	γ	X
ap-5897	157	30	+	+	NOUN
ap-5897	157	31	1	1	NUM
ap-5897	157	32	)	)	PUNCT
ap-5897	157	33	and	and	CCONJ
ap-5897	157	34	for	for	ADP
ap-5897	157	35	all	all	DET
ap-5897	157	36	i	i	PRON
ap-5897	157	37	<	<	X
ap-5897	157	38	n	n	CCONJ
ap-5897	157	39	,	,	PUNCT
ap-5897	157	40	the	the	DET
ap-5897	157	41	i	i	NOUN
ap-5897	157	42	-	-	PUNCT
ap-5897	157	43	th	th	X
ap-5897	157	44	digit	digit	NOUN
ap-5897	157	45	of	of	ADP
ap-5897	157	46	d(b;x	d(b;x	NOUN
ap-5897	157	47	)	)	PUNCT
ap-5897	157	48	is	be	AUX
ap-5897	157	49	c∗i	c∗i	NUM
ap-5897	157	50	where	where	SCONJ
ap-5897	157	51	εn+1	εn+1	ADJ
ap-5897	157	52	<	<	X
ap-5897	157	53	εn	εn	ADJ
ap-5897	157	54	and	and	CCONJ
ap-5897	157	55	εn	εn	ADJ
ap-5897	157	56	→	→	SYM
ap-5897	157	57	0	0	NUM
ap-5897	157	58	as	as	ADP
ap-5897	157	59	n→∞.	n→∞.	PROPN
ap-5897	157	60	example	example	NOUN
ap-5897	157	61	.	.	PUNCT
ap-5897	158	1	let	let	VERB
ap-5897	158	2	β	β	PRON
ap-5897	158	3	be	be	AUX
ap-5897	158	4	a	a	DET
ap-5897	158	5	quadratic	quadratic	ADJ
ap-5897	158	6	pisot	pisot	ADJ
ap-5897	158	7	number	number	NOUN
ap-5897	158	8	.	.	PUNCT
ap-5897	159	1	then	then	ADV
ap-5897	159	2	β	β	X
ap-5897	159	3	satisfies	satisfy	VERB
ap-5897	159	4	the	the	DET
ap-5897	159	5	minimal	minimal	ADJ
ap-5897	159	6	polynomial	polynomial	ADJ
ap-5897	159	7	x2	x2	ADJ
ap-5897	159	8	−	−	PROPN
ap-5897	159	9	bx−	bx−	PROPN
ap-5897	159	10	c	c	NOUN
ap-5897	159	11	where	where	SCONJ
ap-5897	159	12	b	b	X
ap-5897	159	13	∈	∈	PROPN
ap-5897	159	14	n	n	NOUN
ap-5897	159	15	and	and	CCONJ
ap-5897	159	16	1	1	NUM
ap-5897	159	17	≤	≤	NUM
ap-5897	159	18	c	c	NOUN
ap-5897	159	19	≤	≤	NUM
ap-5897	159	20	b	b	NOUN
ap-5897	159	21	;	;	PUNCT
ap-5897	159	22	or	or	CCONJ
ap-5897	159	23	b	b	X
ap-5897	159	24	∈	∈	PROPN
ap-5897	159	25	n−{1	n−{1	NOUN
ap-5897	159	26	,	,	PUNCT
ap-5897	159	27	2	2	NUM
ap-5897	159	28	}	}	PUNCT
ap-5897	159	29	and	and	CCONJ
ap-5897	159	30	2−b	2−b	NUM
ap-5897	159	31	≤	≤	NUM
ap-5897	160	1	c	c	NOUN
ap-5897	160	2	≤	≤	NUM
ap-5897	160	3	−1	−1	NOUN
ap-5897	160	4	.	.	PUNCT
ap-5897	161	1	let	let	VERB
ap-5897	161	2	γ	γ	X
ap-5897	161	3	=	=	SYM
ap-5897	161	4	0	0	NUM
ap-5897	161	5	.	.	PUNCT
ap-5897	162	1	we	we	PRON
ap-5897	162	2	compute	compute	VERB
ap-5897	162	3	for	for	ADP
ap-5897	162	4	d∗(β,−β	d∗(β,−β	NOUN
ap-5897	162	5	;	;	PUNCT
ap-5897	162	6	γ	γ	X
ap-5897	162	7	+	+	NOUN
ap-5897	162	8	1	1	NUM
ap-5897	162	9	)	)	PUNCT
ap-5897	162	10	=	=	PUNCT
ap-5897	162	11	d∗(β,−β	d∗(β,−β	NOUN
ap-5897	162	12	;	;	PUNCT
ap-5897	162	13	1	1	NUM
ap-5897	162	14	)	)	PUNCT
ap-5897	162	15	.	.	PUNCT
ap-5897	163	1	let	let	VERB
ap-5897	163	2	ε	ε	PROPN
ap-5897	163	3	>	>	X
ap-5897	163	4	0	0	PUNCT
ap-5897	163	5	be	be	AUX
ap-5897	163	6	arbitrarily	arbitrarily	ADV
ap-5897	163	7	small	small	ADJ
ap-5897	163	8	.	.	PUNCT
ap-5897	164	1	case	case	NOUN
ap-5897	164	2	1	1	X
ap-5897	164	3	.	.	PUNCT
ap-5897	165	1	let	let	VERB
ap-5897	165	2	1	1	NUM
ap-5897	165	3	≤	≤	NUM
ap-5897	165	4	c	c	PROPN
ap-5897	165	5	≤	≤	PROPN
ap-5897	165	6	b.	b.	PROPN
ap-5897	166	1	then	then	ADV
ap-5897	166	2	b	b	X
ap-5897	166	3	<	<	X
ap-5897	166	4	β	β	X
ap-5897	166	5	<	<	X
ap-5897	166	6	b+	b+	X
ap-5897	166	7	1	1	NUM
ap-5897	166	8	.	.	PUNCT
ap-5897	167	1	we	we	PRON
ap-5897	167	2	have	have	VERB
ap-5897	167	3	t	t	X
ap-5897	167	4	(	(	PUNCT
ap-5897	167	5	1−	1−	NUM
ap-5897	167	6	ε	ε	PROPN
ap-5897	167	7	)	)	PUNCT
ap-5897	167	8	=	=	SYM
ap-5897	168	1	β	β	X
ap-5897	168	2	(	(	PUNCT
ap-5897	168	3	1−	1−	NUM
ap-5897	168	4	ε)−	ε)−	PROPN
ap-5897	168	5	bβ	bβ	NOUN
ap-5897	168	6	(	(	PUNCT
ap-5897	168	7	1−	1−	NUM
ap-5897	168	8	ε)c	ε)c	X
ap-5897	168	9	=	=	X
ap-5897	168	10	β	β	X
ap-5897	168	11	−	−	X
ap-5897	168	12	ε−	ε−	PROPN
ap-5897	168	13	b	b	PROPN
ap-5897	168	14	t	t	NOUN
ap-5897	168	15	2	2	NUM
ap-5897	168	16	(	(	PUNCT
ap-5897	168	17	1−	1−	NUM
ap-5897	168	18	ε	ε	PROPN
ap-5897	168	19	)	)	PUNCT
ap-5897	168	20	=	=	SYM
ap-5897	168	21	−β	−β	NOUN
ap-5897	168	22	(	(	PUNCT
ap-5897	168	23	β	β	NOUN
ap-5897	168	24	−	−	NOUN
ap-5897	168	25	b−	b−	PROPN
ap-5897	168	26	ε)−	ε)−	PROPN
ap-5897	168	27	b−β	b−β	PROPN
ap-5897	168	28	(	(	PUNCT
ap-5897	168	29	β	β	NOUN
ap-5897	168	30	−	−	NOUN
ap-5897	168	31	b−	b−	NOUN
ap-5897	168	32	ε)c	ε)c	ADV
ap-5897	168	33	=	=	PUNCT
ap-5897	168	34	ε	ε	PROPN
ap-5897	168	35	t	t	PROPN
ap-5897	168	36	3	3	NUM
ap-5897	168	37	(	(	PUNCT
ap-5897	168	38	1−	1−	NUM
ap-5897	168	39	ε	ε	PROPN
ap-5897	168	40	)	)	PUNCT
ap-5897	168	41	=	=	SYM
ap-5897	168	42	βε−	βε−	PUNCT
ap-5897	168	43	bβεc	bβεc	NOUN
ap-5897	168	44	=	=	SYM
ap-5897	168	45	ε	ε	PROPN
ap-5897	168	46	t	t	PROPN
ap-5897	168	47	4	4	NUM
ap-5897	168	48	(	(	PUNCT
ap-5897	168	49	1−	1−	NUM
ap-5897	168	50	ε	ε	PROPN
ap-5897	168	51	)	)	PUNCT
ap-5897	168	52	=	=	PUNCT
ap-5897	169	1	−βε−	−βε−	AUX
ap-5897	169	2	b−βεc	b−βεc	NOUN
ap-5897	169	3	=	=	NOUN
ap-5897	169	4	−ε+	−ε+	ADP
ap-5897	169	5	1	1	NUM
ap-5897	169	6	.	.	PUNCT
ap-5897	169	7	hence	hence	ADV
ap-5897	169	8	,	,	PUNCT
ap-5897	169	9	d∗(β,−β	d∗(β,−β	PROPN
ap-5897	169	10	;	;	PUNCT
ap-5897	169	11	γ	γ	X
ap-5897	169	12	+	+	NOUN
ap-5897	169	13	1	1	NUM
ap-5897	169	14	)	)	PUNCT
ap-5897	169	15	=	=	SYM
ap-5897	169	16	(	(	PUNCT
ap-5897	169	17	b,−c	b,−c	PROPN
ap-5897	169	18	,	,	PUNCT
ap-5897	169	19	0,−1	0,−1	NUM
ap-5897	169	20	)	)	PUNCT
ap-5897	169	21	.	.	PUNCT
ap-5897	170	1	it	it	PRON
ap-5897	170	2	can	can	AUX
ap-5897	170	3	be	be	AUX
ap-5897	170	4	shown	show	VERB
ap-5897	170	5	that	that	SCONJ
ap-5897	170	6	d(β,−β	d(β,−β	NOUN
ap-5897	170	7	;	;	PUNCT
ap-5897	170	8	γ	γ	X
ap-5897	170	9	+	+	NOUN
ap-5897	170	10	1	1	NUM
ap-5897	170	11	)	)	PUNCT
ap-5897	170	12	=	=	SYM
ap-5897	170	13	(	(	PUNCT
ap-5897	170	14	b,−c	b,−c	PROPN
ap-5897	170	15	,	,	PUNCT
ap-5897	170	16	0	0	NUM
ap-5897	170	17	)	)	PUNCT
ap-5897	170	18	.	.	PUNCT
ap-5897	171	1	216	216	NUM
ap-5897	171	2	vol	vol	NOUN
ap-5897	171	3	.	.	PUNCT
ap-5897	171	4	60	60	NUM
ap-5897	171	5	no	no	NOUN
ap-5897	171	6	.	.	PUNCT
ap-5897	172	1	3/2020	3/2020	NUM
ap-5897	172	2	beta	beta	PROPN
ap-5897	172	3	cantor	cantor	PROPN
ap-5897	172	4	series	series	NOUN
ap-5897	172	5	expansion	expansion	NOUN
ap-5897	172	6	and	and	CCONJ
ap-5897	172	7	admissible	admissible	ADJ
ap-5897	172	8	sequences	sequence	NOUN
ap-5897	172	9	(	(	PUNCT
ap-5897	172	10	0	0	NUM
ap-5897	172	11	,	,	PUNCT
ap-5897	172	12	1	1	NUM
ap-5897	172	13	α2	α2	ADJ
ap-5897	172	14	)	)	PUNCT
ap-5897	172	15	(	(	PUNCT
ap-5897	172	16	1	1	NUM
ap-5897	172	17	α2	α2	ADJ
ap-5897	172	18	,	,	PUNCT
ap-5897	172	19	1	1	NUM
ap-5897	172	20	α	α	NOUN
ap-5897	172	21	)	)	PUNCT
ap-5897	173	1	(	(	PUNCT
ap-5897	173	2	1	1	NUM
ap-5897	173	3	α	α	NOUN
ap-5897	173	4	,	,	PUNCT
ap-5897	173	5	1	1	NUM
ap-5897	173	6	)	)	PUNCT
ap-5897	173	7	0	0	NUM
ap-5897	173	8	1	1	NUM
ap-5897	173	9	α2	α2	ADJ
ap-5897	173	10	1	1	NUM
ap-5897	173	11	α	α	NOUN
ap-5897	173	12	0	0	NUM
ap-5897	173	13	0	0	NUM
ap-5897	173	14	0	0	NUM
ap-5897	173	15	1	1	NUM
ap-5897	173	16	1	1	NUM
ap-5897	173	17	−1	−1	NOUN
ap-5897	173	18	−1	−1	NOUN
ap-5897	173	19	−1	−1	NOUN
ap-5897	173	20	−2	−2	PROPN
ap-5897	173	21	−2	−2	NOUN
ap-5897	173	22	−1	−1	NOUN
ap-5897	173	23	1	1	NUM
ap-5897	173	24	0	0	NUM
ap-5897	173	25	0	0	NUM
ap-5897	173	26	0	0	NUM
ap-5897	173	27	−1	−1	NOUN
ap-5897	173	28	figure	figure	NOUN
ap-5897	173	29	2	2	NUM
ap-5897	173	30	.	.	PUNCT
ap-5897	174	1	tα	tα	PROPN
ap-5897	174	2	,	,	PUNCT
ap-5897	174	3	β	β	X
ap-5897	174	4	:	:	PUNCT
ap-5897	175	1	[	[	X
ap-5897	175	2	0	0	NUM
ap-5897	175	3	,	,	PUNCT
ap-5897	175	4	1	1	X
ap-5897	175	5	)	)	PUNCT
ap-5897	175	6	−→	−→	NOUN
ap-5897	175	7	[	[	X
ap-5897	175	8	0	0	NUM
ap-5897	175	9	,	,	PUNCT
ap-5897	175	10	1	1	NUM
ap-5897	175	11	)	)	PUNCT
ap-5897	175	12	with	with	ADP
ap-5897	175	13	α	α	NOUN
ap-5897	175	14	=	=	SYM
ap-5897	175	15	−β	−β	NOUN
ap-5897	175	16	=	=	PUNCT
ap-5897	175	17	(	(	PUNCT
ap-5897	175	18	1	1	NUM
ap-5897	175	19	+	+	CCONJ
ap-5897	175	20	√	√	PROPN
ap-5897	175	21	5)/2	5)/2	NUM
ap-5897	175	22	j	j	PROPN
ap-5897	175	23	k	k	PROPN
ap-5897	176	1	l	l	NOUN
ap-5897	176	2	m	m	VERB
ap-5897	176	3	p	p	X
ap-5897	176	4	q	q	NOUN
ap-5897	176	5	r	r	NOUN
ap-5897	176	6	0	0	NUM
ap-5897	176	7	0	0	NUM
ap-5897	176	8	0	0	NUM
ap-5897	176	9	01	01	NUM
ap-5897	176	10	1	1	NUM
ap-5897	176	11	1	1	NUM
ap-5897	176	12	1	1	NUM
ap-5897	176	13	1	1	NUM
ap-5897	176	14	1	1	NUM
ap-5897	176	15	1	1	NUM
ap-5897	176	16	−2	−2	NOUN
ap-5897	176	17	−2	−2	NOUN
ap-5897	176	18	−2	−2	NOUN
ap-5897	176	19	−	−	PROPN
ap-5897	176	20	3	3	NUM
ap-5897	176	21	−3	−3	NOUN
ap-5897	176	22	−3−3	−3−3	ADJ
ap-5897	177	1	−3	−3	PROPN
ap-5897	177	2	−3−3	−3−3	ADJ
ap-5897	177	3	−4	−4	X
ap-5897	177	4	−4	−4	X
ap-5897	177	5	figure	figure	NOUN
ap-5897	177	6	3	3	X
ap-5897	177	7	.	.	PUNCT
ap-5897	178	1	tα	tα	PROPN
ap-5897	178	2	,	,	PUNCT
ap-5897	178	3	β	β	X
ap-5897	178	4	:	:	PUNCT
ap-5897	179	1	[	[	X
ap-5897	179	2	1	1	NUM
ap-5897	179	3	/	/	SYM
ap-5897	179	4	α	α	NOUN
ap-5897	179	5	,	,	PUNCT
ap-5897	179	6	α	α	NOUN
ap-5897	179	7	)	)	PUNCT
ap-5897	179	8	−→	−→	NOUN
ap-5897	179	9	[	[	X
ap-5897	179	10	1	1	NUM
ap-5897	179	11	/	/	SYM
ap-5897	179	12	α	α	NOUN
ap-5897	179	13	,	,	PUNCT
ap-5897	179	14	α	α	NOUN
ap-5897	179	15	)	)	PUNCT
ap-5897	179	16	with	with	ADP
ap-5897	179	17	α	α	NOUN
ap-5897	179	18	=	=	SYM
ap-5897	179	19	−β	−β	NOUN
ap-5897	179	20	=	=	PUNCT
ap-5897	179	21	(	(	PUNCT
ap-5897	179	22	1	1	NUM
ap-5897	179	23	+	+	CCONJ
ap-5897	180	1	√	√	NUM
ap-5897	180	2	5)/2	5)/2	NUM
ap-5897	180	3	case	case	NOUN
ap-5897	180	4	2	2	NUM
ap-5897	180	5	.	.	PUNCT
ap-5897	180	6	let	let	VERB
ap-5897	180	7	2−	2−	NUM
ap-5897	180	8	b	b	NOUN
ap-5897	180	9	≤	≤	NUM
ap-5897	180	10	c	c	NOUN
ap-5897	180	11	≤	≤	NUM
ap-5897	180	12	−1	−1	NOUN
ap-5897	180	13	.	.	PUNCT
ap-5897	181	1	then	then	ADV
ap-5897	181	2	b−	b−	PROPN
ap-5897	181	3	1	1	NUM
ap-5897	181	4	<	<	X
ap-5897	181	5	β	β	X
ap-5897	181	6	<	<	X
ap-5897	181	7	b.	b.	PROPN
ap-5897	182	1	so	so	ADV
ap-5897	182	2	t	t	PROPN
ap-5897	182	3	(	(	PUNCT
ap-5897	182	4	1−	1−	NUM
ap-5897	182	5	ε	ε	PROPN
ap-5897	182	6	)	)	PUNCT
ap-5897	182	7	=	=	SYM
ap-5897	183	1	β	β	X
ap-5897	183	2	(	(	PUNCT
ap-5897	183	3	1−	1−	NUM
ap-5897	183	4	ε)−	ε)−	PROPN
ap-5897	183	5	bβ	bβ	NOUN
ap-5897	183	6	(	(	PUNCT
ap-5897	183	7	1−	1−	NUM
ap-5897	183	8	ε)c	ε)c	X
ap-5897	183	9	=	=	X
ap-5897	183	10	β	β	X
ap-5897	183	11	−	−	X
ap-5897	183	12	ε−	ε−	ADV
ap-5897	183	13	b+	b+	ADJ
ap-5897	183	14	1	1	NUM
ap-5897	183	15	t	t	NOUN
ap-5897	183	16	2	2	NUM
ap-5897	183	17	(	(	PUNCT
ap-5897	183	18	1−	1−	NUM
ap-5897	183	19	ε	ε	PROPN
ap-5897	183	20	)	)	PUNCT
ap-5897	183	21	=	=	SYM
ap-5897	184	1	−β	−β	NOUN
ap-5897	184	2	(	(	PUNCT
ap-5897	184	3	β	β	X
ap-5897	184	4	−	−	NOUN
ap-5897	184	5	b+	b+	X
ap-5897	184	6	1−	1−	NUM
ap-5897	184	7	ε	ε	PROPN
ap-5897	184	8	)	)	PUNCT
ap-5897	184	9	−b−β	−b−β	PROPN
ap-5897	184	10	(	(	PUNCT
ap-5897	184	11	β	β	X
ap-5897	184	12	−	−	NOUN
ap-5897	184	13	b+	b+	X
ap-5897	184	14	1−	1−	NUM
ap-5897	184	15	ε)c	ε)c	X
ap-5897	184	16	=	=	X
ap-5897	184	17	−β	−β	NOUN
ap-5897	184	18	+	+	CCONJ
ap-5897	184	19	b+	b+	X
ap-5897	184	20	ε	ε	PROPN
ap-5897	184	21	t	t	PROPN
ap-5897	184	22	3	3	NUM
ap-5897	184	23	(	(	PUNCT
ap-5897	184	24	1−	1−	NUM
ap-5897	184	25	ε	ε	PROPN
ap-5897	184	26	)	)	PUNCT
ap-5897	184	27	=	=	PUNCT
ap-5897	184	28	β(−β	β(−β	PUNCT
ap-5897	184	29	+	+	NUM
ap-5897	184	30	b+	b+	X
ap-5897	184	31	ε)−	ε)−	PROPN
ap-5897	184	32	bβ(−β	bβ(−β	PROPN
ap-5897	184	33	+	+	CCONJ
ap-5897	184	34	b+	b+	X
ap-5897	184	35	ε)c	ε)c	X
ap-5897	184	36	=	=	PUNCT
ap-5897	184	37	ε	ε	PROPN
ap-5897	184	38	t	t	PROPN
ap-5897	184	39	4	4	NUM
ap-5897	184	40	(	(	PUNCT
ap-5897	184	41	1−	1−	NUM
ap-5897	184	42	ε	ε	PROPN
ap-5897	184	43	)	)	PUNCT
ap-5897	184	44	=	=	PUNCT
ap-5897	185	1	−βε−	−βε−	AUX
ap-5897	185	2	b−βεc	b−βεc	NOUN
ap-5897	185	3	=	=	NOUN
ap-5897	185	4	−ε+	−ε+	ADP
ap-5897	185	5	1	1	NUM
ap-5897	185	6	.	.	PUNCT
ap-5897	185	7	hence	hence	ADV
ap-5897	185	8	,	,	PUNCT
ap-5897	185	9	d∗(β,−β	d∗(β,−β	PROPN
ap-5897	185	10	;	;	PUNCT
ap-5897	185	11	γ	γ	X
ap-5897	185	12	+	+	NOUN
ap-5897	185	13	1	1	NUM
ap-5897	185	14	)	)	PUNCT
ap-5897	185	15	=	=	SYM
ap-5897	185	16	(	(	PUNCT
ap-5897	185	17	b−	b−	PROPN
ap-5897	185	18	1,−b−	1,−b−	NUM
ap-5897	185	19	c,−c,−1	c,−c,−1	NOUN
ap-5897	185	20	)	)	PUNCT
ap-5897	185	21	.	.	PUNCT
ap-5897	186	1	also	also	ADV
ap-5897	186	2	,	,	PUNCT
ap-5897	186	3	we	we	PRON
ap-5897	186	4	have	have	VERB
ap-5897	186	5	d(β,−β	d(β,−β	NOUN
ap-5897	186	6	;	;	PUNCT
ap-5897	186	7	γ	γ	X
ap-5897	186	8	+	+	NOUN
ap-5897	186	9	1	1	NUM
ap-5897	186	10	)	)	PUNCT
ap-5897	186	11	=	=	SYM
ap-5897	186	12	(	(	PUNCT
ap-5897	186	13	b−	b−	PROPN
ap-5897	186	14	1,−b−	1,−b−	NUM
ap-5897	186	15	c,−c	c,−c	PROPN
ap-5897	186	16	,	,	PUNCT
ap-5897	186	17	0	0	NUM
ap-5897	186	18	)	)	PUNCT
ap-5897	186	19	.	.	PUNCT
ap-5897	187	1	example	example	NOUN
ap-5897	187	2	.	.	PUNCT
ap-5897	188	1	let	let	VERB
ap-5897	188	2	b	b	NOUN
ap-5897	188	3	=	=	SYM
ap-5897	188	4	(	(	PUNCT
ap-5897	188	5	α	α	X
ap-5897	188	6	,	,	PUNCT
ap-5897	188	7	β	β	NOUN
ap-5897	188	8	)	)	PUNCT
ap-5897	188	9	where	where	SCONJ
ap-5897	188	10	α	α	X
ap-5897	188	11	∈	∈	PROPN
ap-5897	188	12	z	z	X
ap-5897	188	13	<	<	X
ap-5897	188	14	0	0	NUM
ap-5897	188	15	and	and	CCONJ
ap-5897	188	16	0	0	NUM
ap-5897	188	17	>	>	X
ap-5897	188	18	β	β	PUNCT
ap-5897	188	19	∈	∈	PROPN
ap-5897	188	20	r.	r.	PROPN
ap-5897	188	21	suppose	suppose	VERB
ap-5897	188	22	α(γ	α(γ	PROPN
ap-5897	188	23	+	+	CCONJ
ap-5897	188	24	1	1	X
ap-5897	188	25	)	)	PUNCT
ap-5897	188	26	−	−	PROPN
ap-5897	188	27	γ	γ	PROPN
ap-5897	188	28	∈	∈	PROPN
ap-5897	188	29	z	z	PROPN
ap-5897	188	30	and	and	CCONJ
ap-5897	188	31	t	t	PROPN
ap-5897	188	32	2n(γ	2n(γ	NUM
ap-5897	189	1	+	+	CCONJ
ap-5897	189	2	1	1	X
ap-5897	189	3	)	)	PUNCT
ap-5897	189	4	=	=	PUNCT
ap-5897	189	5	fn(β	fn(β	X
ap-5897	189	6	)	)	PUNCT
ap-5897	189	7	−	−	PROPN
ap-5897	190	1	bfn(β)−	bfn(β)−	PROPN
ap-5897	190	2	γc	γc	PROPN
ap-5897	190	3	and	and	CCONJ
ap-5897	190	4	t	t	PROPN
ap-5897	191	1	2n+1(γ	2n+1(γ	NUM
ap-5897	192	1	+	+	CCONJ
ap-5897	192	2	1	1	X
ap-5897	192	3	)	)	PUNCT
ap-5897	192	4	=	=	NOUN
ap-5897	192	5	gn(β	gn(β	X
ap-5897	192	6	)	)	PUNCT
ap-5897	192	7	−	−	PROPN
ap-5897	193	1	bgn(β)−	bgn(β)−	PROPN
ap-5897	193	2	γc	γc	VERB
ap-5897	193	3	for	for	ADP
ap-5897	193	4	some	some	DET
ap-5897	193	5	polynomials	polynomial	NOUN
ap-5897	193	6	fn	fn	ADJ
ap-5897	193	7	and	and	CCONJ
ap-5897	193	8	gn	gn	PROPN
ap-5897	193	9	of	of	ADP
ap-5897	193	10	degree	degree	NOUN
ap-5897	193	11	n	n	PROPN
ap-5897	193	12	in	in	ADP
ap-5897	193	13	z[x	z[x	NOUN
ap-5897	193	14	]	]	PUNCT
ap-5897	193	15	where	where	SCONJ
ap-5897	193	16	fn(β	fn(β	NUM
ap-5897	193	17	)	)	PUNCT
ap-5897	193	18	−	−	PROPN
ap-5897	193	19	γ	γ	X
ap-5897	193	20	/∈	/∈	PUNCT
ap-5897	193	21	z	z	PROPN
ap-5897	193	22	and	and	CCONJ
ap-5897	193	23	gn(β	gn(β	NUM
ap-5897	193	24	)	)	PUNCT
ap-5897	193	25	−	−	PROPN
ap-5897	193	26	γ	γ	X
ap-5897	193	27	/∈	/∈	PUNCT
ap-5897	193	28	z	z	NOUN
ap-5897	193	29	for	for	ADP
ap-5897	193	30	all	all	PRON
ap-5897	193	31	n	n	PRON
ap-5897	193	32	∈	∈	PROPN
ap-5897	193	33	n	n	CCONJ
ap-5897	193	34	(	(	PUNCT
ap-5897	193	35	e.g.	e.g.	ADV
ap-5897	193	36	β	β	X
ap-5897	193	37	may	may	AUX
ap-5897	193	38	be	be	AUX
ap-5897	193	39	taken	take	VERB
ap-5897	193	40	to	to	PART
ap-5897	193	41	be	be	AUX
ap-5897	193	42	transcendental	transcendental	ADJ
ap-5897	193	43	over	over	ADP
ap-5897	193	44	q	q	NOUN
ap-5897	193	45	and	and	CCONJ
ap-5897	193	46	γ	γ	X
ap-5897	193	47	∈	∈	PROPN
ap-5897	193	48	q	q	NOUN
ap-5897	193	49	)	)	PUNCT
ap-5897	193	50	.	.	PUNCT
ap-5897	194	1	let	let	VERB
ap-5897	194	2	d(α	d(α	NOUN
ap-5897	194	3	,	,	PUNCT
ap-5897	194	4	β	β	X
ap-5897	194	5	;	;	PUNCT
ap-5897	194	6	γ	γ	X
ap-5897	194	7	+	+	NOUN
ap-5897	194	8	1	1	NUM
ap-5897	194	9	)	)	PUNCT
ap-5897	194	10	=	=	SYM
ap-5897	194	11	(	(	PUNCT
ap-5897	194	12	c1	c1	PROPN
ap-5897	194	13	,	,	PUNCT
ap-5897	194	14	c2	c2	PROPN
ap-5897	194	15	,	,	PUNCT
ap-5897	194	16	.	.	PUNCT
ap-5897	194	17	.	.	PUNCT
ap-5897	194	18	.	.	PUNCT
ap-5897	194	19	)	)	PUNCT
ap-5897	195	1	and	and	CCONJ
ap-5897	195	2	d∗(α	d∗(α	PROPN
ap-5897	195	3	,	,	PUNCT
ap-5897	195	4	β	β	X
ap-5897	195	5	;	;	PUNCT
ap-5897	195	6	γ	γ	X
ap-5897	195	7	+	+	NOUN
ap-5897	195	8	1	1	NUM
ap-5897	195	9	)	)	PUNCT
ap-5897	195	10	=	=	NOUN
ap-5897	195	11	(	(	PUNCT
ap-5897	195	12	c∗1	c∗1	ADJ
ap-5897	195	13	,	,	PUNCT
ap-5897	195	14	c∗2	c∗2	NOUN
ap-5897	195	15	,	,	PUNCT
ap-5897	195	16	.	.	PUNCT
ap-5897	195	17	.	.	PUNCT
ap-5897	195	18	.	.	PUNCT
ap-5897	195	19	)	)	PUNCT
ap-5897	195	20	.	.	PUNCT
ap-5897	196	1	then	then	ADV
ap-5897	196	2	,	,	PUNCT
ap-5897	196	3	for	for	ADP
ap-5897	196	4	small	small	ADJ
ap-5897	196	5	ε	ε	PROPN
ap-5897	196	6	>	>	X
ap-5897	196	7	0	0	PROPN
ap-5897	196	8	,	,	PUNCT
ap-5897	196	9	c∗1	c∗1	NOUN
ap-5897	196	10	=	=	SYM
ap-5897	196	11	bα(γ	bα(γ	X
ap-5897	196	12	+	+	PROPN
ap-5897	196	13	1)−	1)−	NUM
ap-5897	196	14	γ	γ	NOUN
ap-5897	196	15	+	+	X
ap-5897	196	16	εc	εc	NOUN
ap-5897	196	17	=	=	SYM
ap-5897	196	18	α(γ	α(γ	NOUN
ap-5897	196	19	+	+	CCONJ
ap-5897	196	20	1	1	X
ap-5897	196	21	)	)	PUNCT
ap-5897	196	22	−	−	PROPN
ap-5897	196	23	γ	γ	PROPN
ap-5897	196	24	=	=	PROPN
ap-5897	196	25	c1	c1	PROPN
ap-5897	196	26	.	.	PUNCT
ap-5897	197	1	since	since	SCONJ
ap-5897	197	2	fn(β)−γ	fn(β)−γ	PROPN
ap-5897	197	3	and	and	CCONJ
ap-5897	197	4	gn(β)−γ	gn(β)−γ	PROPN
ap-5897	197	5	are	be	AUX
ap-5897	197	6	not	not	PART
ap-5897	197	7	integers	integer	NOUN
ap-5897	197	8	,	,	PUNCT
ap-5897	197	9	we	we	PRON
ap-5897	197	10	can	can	AUX
ap-5897	197	11	show	show	VERB
ap-5897	197	12	that	that	SCONJ
ap-5897	197	13	tn(γ+	tn(γ+	PROPN
ap-5897	197	14	1−	1−	NUM
ap-5897	197	15	ε	ε	PROPN
ap-5897	197	16	)	)	PUNCT
ap-5897	197	17	=	=	PRON
ap-5897	197	18	tn(γ+	tn(γ+	PROPN
ap-5897	197	19	1	1	X
ap-5897	197	20	)	)	PUNCT
ap-5897	197	21	+	+	CCONJ
ap-5897	197	22	(	(	PUNCT
ap-5897	197	23	−1)n+1ε	−1)n+1ε	NOUN
ap-5897	197	24	.	.	PUNCT
ap-5897	198	1	moreover	moreover	ADV
ap-5897	198	2	,	,	PUNCT
ap-5897	198	3	c∗2n	c∗2n	PROPN
ap-5897	198	4	=	=	PUNCT
ap-5897	198	5	bfn(β)−	bfn(β)−	PROPN
ap-5897	198	6	γ	γ	NOUN
ap-5897	198	7	−	−	PROPN
ap-5897	198	8	εc	εc	NOUN
ap-5897	198	9	=	=	PUNCT
ap-5897	198	10	bfn(β)−	bfn(β)−	PROPN
ap-5897	198	11	γc	γc	PROPN
ap-5897	198	12	=	=	SYM
ap-5897	198	13	c2n	c2n	NOUN
ap-5897	198	14	and	and	CCONJ
ap-5897	198	15	c∗2n+1	c∗2n+1	PROPN
ap-5897	198	16	=	=	SYM
ap-5897	198	17	bgn(β)−	bgn(β)−	PROPN
ap-5897	198	18	γ	γ	X
ap-5897	198	19	+	+	PROPN
ap-5897	198	20	εc	εc	PROPN
ap-5897	198	21	=	=	PUNCT
ap-5897	198	22	bgn(β)−	bgn(β)−	PROPN
ap-5897	198	23	γc	γc	PROPN
ap-5897	199	1	=	=	SYM
ap-5897	199	2	c2n+1	c2n+1	PROPN
ap-5897	199	3	.	.	PUNCT
ap-5897	200	1	hence	hence	ADV
ap-5897	200	2	,	,	PUNCT
ap-5897	200	3	d(α	d(α	PROPN
ap-5897	200	4	,	,	PUNCT
ap-5897	200	5	β	β	X
ap-5897	200	6	;	;	PUNCT
ap-5897	200	7	γ	γ	X
ap-5897	200	8	+	+	NOUN
ap-5897	200	9	1	1	NUM
ap-5897	200	10	)	)	PUNCT
ap-5897	200	11	=	=	SYM
ap-5897	200	12	d∗(α	d∗(α	PROPN
ap-5897	200	13	,	,	PUNCT
ap-5897	200	14	β	β	X
ap-5897	200	15	;	;	PUNCT
ap-5897	200	16	γ	γ	X
ap-5897	200	17	+	+	NOUN
ap-5897	200	18	1	1	NUM
ap-5897	200	19	)	)	PUNCT
ap-5897	200	20	.	.	PUNCT
ap-5897	201	1	from	from	ADP
ap-5897	201	2	the	the	DET
ap-5897	201	3	examples	example	NOUN
ap-5897	201	4	above	above	ADV
ap-5897	201	5	,	,	PUNCT
ap-5897	201	6	we	we	PRON
ap-5897	201	7	see	see	VERB
ap-5897	201	8	that	that	SCONJ
ap-5897	201	9	d(b	d(b	PRON
ap-5897	201	10	;	;	PUNCT
ap-5897	201	11	γ	γ	X
ap-5897	201	12	+	+	NOUN
ap-5897	201	13	1	1	NUM
ap-5897	201	14	)	)	PUNCT
ap-5897	201	15	may	may	AUX
ap-5897	201	16	or	or	CCONJ
ap-5897	201	17	may	may	AUX
ap-5897	201	18	not	not	PART
ap-5897	201	19	be	be	AUX
ap-5897	201	20	equal	equal	ADJ
ap-5897	201	21	to	to	ADP
ap-5897	201	22	d∗(b	d∗(b	PROPN
ap-5897	201	23	;	;	PUNCT
ap-5897	201	24	γ	γ	X
ap-5897	201	25	+	+	NOUN
ap-5897	201	26	1	1	NUM
ap-5897	201	27	)	)	PUNCT
ap-5897	201	28	.	.	PUNCT
ap-5897	202	1	in	in	ADP
ap-5897	202	2	what	what	PRON
ap-5897	202	3	follows	follow	VERB
ap-5897	202	4	,	,	PUNCT
ap-5897	202	5	we	we	PRON
ap-5897	202	6	characterize	characterize	VERB
ap-5897	202	7	the	the	DET
ap-5897	202	8	b	b	NOUN
ap-5897	202	9	-	-	PUNCT
ap-5897	202	10	expansions	expansion	NOUN
ap-5897	202	11	such	such	ADJ
ap-5897	202	12	that	that	SCONJ
ap-5897	202	13	d(b	d(b	PROPN
ap-5897	202	14	;	;	PUNCT
ap-5897	202	15	γ	γ	X
ap-5897	202	16	+	+	NOUN
ap-5897	202	17	1	1	NUM
ap-5897	202	18	)	)	PUNCT
ap-5897	202	19	=	=	PUNCT
ap-5897	202	20	d∗(b	d∗(b	PROPN
ap-5897	202	21	;	;	PUNCT
ap-5897	202	22	γ	γ	X
ap-5897	202	23	+	+	NOUN
ap-5897	202	24	1	1	NUM
ap-5897	202	25	)	)	PUNCT
ap-5897	202	26	.	.	PUNCT
ap-5897	203	1	let	let	VERB
ap-5897	203	2	sgn	sgn	NOUN
ap-5897	203	3	denote	denote	VERB
ap-5897	203	4	the	the	DET
ap-5897	203	5	signum	signum	PROPN
ap-5897	203	6	function	function	NOUN
ap-5897	203	7	.	.	PUNCT
ap-5897	204	1	define	define	VERB
ap-5897	204	2	ib	ib	NOUN
ap-5897	204	3	:	:	PUNCT
ap-5897	204	4	=	=	SYM
ap-5897	204	5	{	{	PUNCT
ap-5897	204	6	n	n	CCONJ
ap-5897	204	7	∈	∈	PROPN
ap-5897	204	8	n	n	NOUN
ap-5897	204	9	∪	∪	X
ap-5897	204	10	{	{	PUNCT
ap-5897	204	11	0	0	NUM
ap-5897	204	12	}	}	PUNCT
ap-5897	204	13	|	|	ADV
ap-5897	204	14	sgn(b[n+	sgn(b[n+	ADP
ap-5897	204	15	1	1	NUM
ap-5897	204	16	]	]	PUNCT
ap-5897	204	17	)	)	PUNCT
ap-5897	204	18	>	>	X
ap-5897	204	19	0	0	X
ap-5897	204	20	}	}	PUNCT
ap-5897	204	21	and	and	CCONJ
ap-5897	204	22	cb	cb	NOUN
ap-5897	205	1	:	:	PUNCT
ap-5897	205	2	=	=	X
ap-5897	205	3	{	{	PUNCT
ap-5897	205	4	γ	γ	X
ap-5897	205	5	∈	∈	PROPN
ap-5897	205	6	r	r	NOUN
ap-5897	205	7	|	|	NOUN
ap-5897	205	8	βn+1	βn+1	NUM
ap-5897	205	9	t	t	NOUN
ap-5897	205	10	n(γ	n(γ	NOUN
ap-5897	205	11	+	+	CCONJ
ap-5897	205	12	1)−	1)−	NUM
ap-5897	205	13	γ	γ	X
ap-5897	205	14	/∈	/∈	PUNCT
ap-5897	205	15	z	z	NOUN
ap-5897	205	16	for	for	ADP
ap-5897	205	17	n	n	PRON
ap-5897	205	18	∈	∈	PROPN
ap-5897	205	19	ib	ib	NOUN
ap-5897	205	20	}	}	PUNCT
ap-5897	205	21	.	.	PUNCT
ap-5897	206	1	proposition	proposition	NOUN
ap-5897	206	2	3.1	3.1	NUM
ap-5897	206	3	.	.	PUNCT
ap-5897	207	1	if	if	SCONJ
ap-5897	207	2	γ	γ	PROPN
ap-5897	207	3	∈	∈	PROPN
ap-5897	207	4	cb	cb	PROPN
ap-5897	207	5	,	,	PUNCT
ap-5897	207	6	then	then	ADV
ap-5897	207	7	d(b	d(b	PROPN
ap-5897	207	8	;	;	PUNCT
ap-5897	207	9	γ	γ	X
ap-5897	207	10	+	+	NOUN
ap-5897	207	11	1	1	NUM
ap-5897	207	12	)	)	PUNCT
ap-5897	207	13	=	=	PUNCT
ap-5897	207	14	d∗(b	d∗(b	PROPN
ap-5897	207	15	;	;	PUNCT
ap-5897	207	16	γ	γ	X
ap-5897	207	17	+	+	NOUN
ap-5897	207	18	1	1	NUM
ap-5897	207	19	)	)	PUNCT
ap-5897	207	20	.	.	PUNCT
ap-5897	208	1	proof	proof	NOUN
ap-5897	208	2	.	.	PUNCT
ap-5897	209	1	we	we	PRON
ap-5897	209	2	show	show	VERB
ap-5897	209	3	by	by	ADP
ap-5897	209	4	induction	induction	NOUN
ap-5897	209	5	on	on	ADP
ap-5897	209	6	n	n	DET
ap-5897	209	7	∈	∈	PROPN
ap-5897	209	8	n∪{0	n∪{0	NOUN
ap-5897	209	9	}	}	PUNCT
ap-5897	209	10	that	that	PRON
ap-5897	209	11	,	,	PUNCT
ap-5897	209	12	for	for	ADP
ap-5897	209	13	arbitrarily	arbitrarily	ADV
ap-5897	209	14	small	small	ADJ
ap-5897	209	15	constant	constant	ADJ
ap-5897	209	16	,	,	PUNCT
ap-5897	209	17	ε	ε	PROPN
ap-5897	209	18	>	>	X
ap-5897	209	19	0	0	PROPN
ap-5897	209	20	,	,	PUNCT
ap-5897	209	21	tn(γ	tn(γ	PUNCT
ap-5897	210	1	+	+	CCONJ
ap-5897	210	2	1−	1−	NUM
ap-5897	210	3	ε	ε	PROPN
ap-5897	210	4	)	)	PUNCT
ap-5897	210	5	=	=	SYM
ap-5897	210	6	tn(γ	tn(γ	X
ap-5897	211	1	+	+	CCONJ
ap-5897	211	2	1)−b[n]ε	1)−b[n]ε	NUM
ap-5897	211	3	.	.	PUNCT
ap-5897	212	1	(	(	PUNCT
ap-5897	212	2	?	?	PUNCT
ap-5897	212	3	)	)	PUNCT
ap-5897	213	1	the	the	DET
ap-5897	213	2	case	case	NOUN
ap-5897	213	3	where	where	SCONJ
ap-5897	213	4	n	n	PROPN
ap-5897	213	5	=	=	SYM
ap-5897	213	6	0	0	NUM
ap-5897	213	7	is	be	AUX
ap-5897	213	8	clear	clear	ADJ
ap-5897	213	9	.	.	PUNCT
ap-5897	214	1	suppose	suppose	VERB
ap-5897	214	2	(	(	PUNCT
ap-5897	214	3	?	?	PUNCT
ap-5897	214	4	)	)	PUNCT
ap-5897	214	5	holds	hold	VERB
ap-5897	214	6	for	for	ADP
ap-5897	214	7	some	some	DET
ap-5897	214	8	n	n	PRON
ap-5897	214	9	∈	∈	PROPN
ap-5897	214	10	n	n	NOUN
ap-5897	214	11	∪	∪	X
ap-5897	214	12	{	{	PUNCT
ap-5897	214	13	0	0	NUM
ap-5897	214	14	}	}	PUNCT
ap-5897	214	15	.	.	PUNCT
ap-5897	215	1	then	then	ADV
ap-5897	215	2	tn+1(γ	tn+1(γ	PROPN
ap-5897	215	3	+	+	CCONJ
ap-5897	215	4	1−	1−	NUM
ap-5897	215	5	ε	ε	PROPN
ap-5897	215	6	)	)	PUNCT
ap-5897	215	7	=	=	PUNCT
ap-5897	216	1	βn+1	βn+1	SYM
ap-5897	216	2	t	t	NOUN
ap-5897	216	3	n(γ	n(γ	X
ap-5897	216	4	+	+	CCONJ
ap-5897	216	5	1−	1−	NUM
ap-5897	216	6	ε)−	ε)−	PROPN
ap-5897	216	7	bβn+1	bβn+1	PROPN
ap-5897	216	8	t	t	NOUN
ap-5897	216	9	n(γ	n(γ	NOUN
ap-5897	216	10	+	+	CCONJ
ap-5897	216	11	1−	1−	NUM
ap-5897	216	12	ε)−	ε)−	PROPN
ap-5897	216	13	γc	γc	PROPN
ap-5897	216	14	=	=	SYM
ap-5897	216	15	βn+1	βn+1	SYM
ap-5897	216	16	t	t	NOUN
ap-5897	216	17	n(γ	n(γ	NOUN
ap-5897	216	18	+	+	CCONJ
ap-5897	216	19	1)−b[n+	1)−b[n+	NUM
ap-5897	216	20	1]ε	1]ε	NUM
ap-5897	216	21	−bβn+1	−bβn+1	NOUN
ap-5897	216	22	t	t	NOUN
ap-5897	216	23	n(γ	n(γ	NOUN
ap-5897	216	24	+	+	CCONJ
ap-5897	216	25	1)−b[n+	1)−b[n+	NUM
ap-5897	216	26	1]ε−	1]ε−	NUM
ap-5897	216	27	γc	γc	X
ap-5897	216	28	.	.	PUNCT
ap-5897	217	1	since	since	SCONJ
ap-5897	217	2	γ	γ	PROPN
ap-5897	217	3	∈	∈	PROPN
ap-5897	217	4	cb	cb	PROPN
ap-5897	217	5	and	and	CCONJ
ap-5897	217	6	ε	ε	PROPN
ap-5897	217	7	is	be	AUX
ap-5897	217	8	arbitrarily	arbitrarily	ADV
ap-5897	217	9	small	small	ADJ
ap-5897	217	10	,	,	PUNCT
ap-5897	217	11	then	then	ADV
ap-5897	217	12	bβn+1	bβn+1	NOUN
ap-5897	217	13	t	t	NOUN
ap-5897	217	14	n(γ	n(γ	NOUN
ap-5897	217	15	+	+	CCONJ
ap-5897	217	16	1)−b[n+	1)−b[n+	NUM
ap-5897	217	17	1]ε−	1]ε−	ADJ
ap-5897	217	18	γc	γc	PROPN
ap-5897	217	19	=	=	SYM
ap-5897	217	20	bβn+1	bβn+1	PROPN
ap-5897	217	21	t	t	NOUN
ap-5897	217	22	n(γ	n(γ	NOUN
ap-5897	217	23	+	+	PROPN
ap-5897	218	1	1)−	1)−	NUM
ap-5897	218	2	γc	γc	PROPN
ap-5897	218	3	.	.	PUNCT
ap-5897	218	4	therefore	therefore	ADV
ap-5897	218	5	,	,	PUNCT
ap-5897	218	6	tn+1(γ	tn+1(γ	PROPN
ap-5897	218	7	+	+	CCONJ
ap-5897	218	8	1−	1−	NUM
ap-5897	218	9	ε	ε	PROPN
ap-5897	218	10	)	)	PUNCT
ap-5897	218	11	=	=	PUNCT
ap-5897	219	1	βn+1	βn+1	SYM
ap-5897	219	2	t	t	NOUN
ap-5897	219	3	n(γ	n(γ	NOUN
ap-5897	219	4	+	+	CCONJ
ap-5897	219	5	1)−b[n+	1)−b[n+	NUM
ap-5897	219	6	1]ε	1]ε	NUM
ap-5897	219	7	−bβn+1	−bβn+1	NOUN
ap-5897	219	8	t	t	NOUN
ap-5897	219	9	n(γ	n(γ	NOUN
ap-5897	219	10	+	+	PROPN
ap-5897	219	11	1)−	1)−	NUM
ap-5897	219	12	γc	γc	NOUN
ap-5897	219	13	=	=	PUNCT
ap-5897	219	14	tn+1(γ	tn+1(γ	PROPN
ap-5897	219	15	+	+	CCONJ
ap-5897	219	16	1)−b[n+	1)−b[n+	NUM
ap-5897	219	17	1]ε	1]ε	NOUN
ap-5897	219	18	.	.	PUNCT
ap-5897	220	1	thus	thus	ADV
ap-5897	220	2	,	,	PUNCT
ap-5897	220	3	we	we	PRON
ap-5897	220	4	get	get	VERB
ap-5897	220	5	d(b	d(b	PROPN
ap-5897	220	6	;	;	PUNCT
ap-5897	220	7	γ	γ	X
ap-5897	220	8	+	+	NOUN
ap-5897	220	9	1	1	NUM
ap-5897	220	10	)	)	PUNCT
ap-5897	220	11	=	=	PUNCT
ap-5897	220	12	d∗(b	d∗(b	PROPN
ap-5897	220	13	;	;	PUNCT
ap-5897	220	14	γ	γ	X
ap-5897	220	15	+	+	NOUN
ap-5897	220	16	1	1	NUM
ap-5897	220	17	)	)	PUNCT
ap-5897	220	18	.	.	PUNCT
ap-5897	221	1	proposition	proposition	NOUN
ap-5897	221	2	3.2	3.2	NUM
ap-5897	221	3	.	.	PUNCT
ap-5897	222	1	if	if	SCONJ
ap-5897	222	2	d(b	d(b	X
ap-5897	222	3	;	;	PUNCT
ap-5897	222	4	γ+	γ+	NUM
ap-5897	222	5	1	1	X
ap-5897	222	6	)	)	PUNCT
ap-5897	222	7	=	=	PUNCT
ap-5897	222	8	d∗(b	d∗(b	PROPN
ap-5897	222	9	;	;	PUNCT
ap-5897	222	10	γ+	γ+	NUM
ap-5897	222	11	1	1	NUM
ap-5897	222	12	)	)	PUNCT
ap-5897	222	13	,	,	PUNCT
ap-5897	222	14	then	then	ADV
ap-5897	222	15	γ	γ	PROPN
ap-5897	222	16	∈	∈	PROPN
ap-5897	222	17	cb	cb	PROPN
ap-5897	222	18	.	.	PUNCT
ap-5897	223	1	proof	proof	NOUN
ap-5897	223	2	.	.	PUNCT
ap-5897	224	1	suppose	suppose	VERB
ap-5897	224	2	d(b	d(b	NOUN
ap-5897	224	3	;	;	PUNCT
ap-5897	224	4	γ	γ	X
ap-5897	224	5	+	+	NOUN
ap-5897	224	6	1	1	NUM
ap-5897	224	7	)	)	PUNCT
ap-5897	224	8	=	=	SYM
ap-5897	224	9	(	(	PUNCT
ap-5897	224	10	c1	c1	PROPN
ap-5897	224	11	,	,	PUNCT
ap-5897	224	12	c2	c2	PROPN
ap-5897	224	13	,	,	PUNCT
ap-5897	224	14	.	.	PUNCT
ap-5897	224	15	.	.	PUNCT
ap-5897	224	16	.	.	PUNCT
ap-5897	224	17	)	)	PUNCT
ap-5897	225	1	and	and	CCONJ
ap-5897	225	2	d∗(b	d∗(b	PROPN
ap-5897	225	3	;	;	PUNCT
ap-5897	225	4	γ	γ	X
ap-5897	225	5	+	+	NOUN
ap-5897	225	6	1	1	NUM
ap-5897	225	7	)	)	PUNCT
ap-5897	225	8	=	=	NOUN
ap-5897	225	9	(	(	PUNCT
ap-5897	225	10	c∗1	c∗1	ADJ
ap-5897	225	11	,	,	PUNCT
ap-5897	225	12	c∗2	c∗2	NOUN
ap-5897	225	13	,	,	PUNCT
ap-5897	225	14	.	.	PUNCT
ap-5897	225	15	.	.	PUNCT
ap-5897	225	16	.	.	PUNCT
ap-5897	225	17	)	)	PUNCT
ap-5897	225	18	are	be	AUX
ap-5897	225	19	equal	equal	ADJ
ap-5897	225	20	.	.	PUNCT
ap-5897	226	1	we	we	PRON
ap-5897	226	2	show	show	VERB
ap-5897	226	3	,	,	PUNCT
ap-5897	226	4	by	by	ADP
ap-5897	226	5	induction	induction	NOUN
ap-5897	226	6	on	on	ADP
ap-5897	226	7	n	n	DET
ap-5897	226	8	∈	∈	PROPN
ap-5897	226	9	n	n	NOUN
ap-5897	226	10	∪	∪	X
ap-5897	226	11	{	{	PUNCT
ap-5897	226	12	0	0	NUM
ap-5897	226	13	}	}	PUNCT
ap-5897	226	14	,	,	PUNCT
ap-5897	226	15	that	that	SCONJ
ap-5897	226	16	tn(γ	tn(γ	NUM
ap-5897	226	17	+	+	PROPN
ap-5897	226	18	1−	1−	NUM
ap-5897	226	19	ε	ε	PROPN
ap-5897	226	20	)	)	PUNCT
ap-5897	226	21	=	=	SYM
ap-5897	226	22	tn(γ	tn(γ	X
ap-5897	227	1	+	+	CCONJ
ap-5897	227	2	1)−b[n]ε	1)−b[n]ε	NUM
ap-5897	227	3	.	.	PUNCT
ap-5897	228	1	(	(	PUNCT
ap-5897	228	2	?	?	PUNCT
ap-5897	228	3	)	)	PUNCT
ap-5897	228	4	217	217	NUM
ap-5897	228	5	jonathan	jonathan	PROPN
ap-5897	228	6	caalim	caalim	PROPN
ap-5897	228	7	,	,	PUNCT
ap-5897	228	8	shiela	shiela	PROPN
ap-5897	228	9	demegillo	demegillo	PROPN
ap-5897	228	10	acta	acta	PROPN
ap-5897	228	11	polytechnica	polytechnica	PROPN
ap-5897	228	12	the	the	DET
ap-5897	228	13	base	base	NOUN
ap-5897	228	14	case	case	NOUN
ap-5897	228	15	n	n	NOUN
ap-5897	228	16	=	=	SYM
ap-5897	228	17	0	0	NUM
ap-5897	228	18	is	be	AUX
ap-5897	228	19	clear	clear	ADJ
ap-5897	228	20	.	.	PUNCT
ap-5897	229	1	suppose	suppose	VERB
ap-5897	229	2	(	(	PUNCT
ap-5897	229	3	?	?	PUNCT
ap-5897	229	4	)	)	PUNCT
ap-5897	229	5	holds	hold	VERB
ap-5897	229	6	for	for	ADP
ap-5897	229	7	some	some	DET
ap-5897	229	8	n	n	PRON
ap-5897	229	9	∈	∈	PROPN
ap-5897	229	10	n	n	NOUN
ap-5897	229	11	∪	∪	X
ap-5897	229	12	{	{	PUNCT
ap-5897	229	13	0	0	NUM
ap-5897	229	14	}	}	PUNCT
ap-5897	229	15	.	.	PUNCT
ap-5897	230	1	then	then	ADV
ap-5897	230	2	tn+1(γ	tn+1(γ	PROPN
ap-5897	230	3	+	+	CCONJ
ap-5897	230	4	1−	1−	NUM
ap-5897	230	5	ε	ε	PROPN
ap-5897	230	6	)	)	PUNCT
ap-5897	230	7	=	=	SYM
ap-5897	231	1	βn+1(tn(γ	βn+1(tn(γ	PROPN
ap-5897	232	1	+	+	CCONJ
ap-5897	232	2	1)−b[n]ε)−	1)−b[n]ε)−	NUM
ap-5897	232	3	c∗n+1	c∗n+1	NOUN
ap-5897	232	4	=	=	SYM
ap-5897	232	5	βn+1	βn+1	SYM
ap-5897	232	6	t	t	NOUN
ap-5897	232	7	n(γ	n(γ	NOUN
ap-5897	232	8	+	+	CCONJ
ap-5897	232	9	1)−b[n+	1)−b[n+	NUM
ap-5897	232	10	1]ε−	1]ε−	ADJ
ap-5897	232	11	cn+1	cn+1	NOUN
ap-5897	232	12	=	=	SYM
ap-5897	232	13	tn+1(γ	tn+1(γ	PROPN
ap-5897	232	14	+	+	CCONJ
ap-5897	232	15	1)−b[n+	1)−b[n+	NUM
ap-5897	232	16	1]ε	1]ε	NOUN
ap-5897	232	17	.	.	PUNCT
ap-5897	233	1	thus	thus	ADV
ap-5897	233	2	,	,	PUNCT
ap-5897	233	3	for	for	ADP
ap-5897	233	4	all	all	DET
ap-5897	233	5	n	n	PRON
ap-5897	233	6	∈	∈	NOUN
ap-5897	233	7	n	n	NOUN
ap-5897	233	8	∪	∪	X
ap-5897	233	9	{	{	PUNCT
ap-5897	233	10	0	0	NUM
ap-5897	233	11	}	}	PUNCT
ap-5897	233	12	and	and	CCONJ
ap-5897	233	13	ε	ε	PROPN
ap-5897	233	14	>	>	X
ap-5897	233	15	0	0	PUNCT
ap-5897	233	16	sufficiently	sufficiently	ADV
ap-5897	233	17	small	small	ADJ
ap-5897	233	18	,	,	PUNCT
ap-5897	233	19	c∗n+1	c∗n+1	ADJ
ap-5897	233	20	=	=	SYM
ap-5897	233	21	bβn+1	bβn+1	PROPN
ap-5897	233	22	t	t	NOUN
ap-5897	233	23	n(γ	n(γ	NOUN
ap-5897	233	24	+	+	CCONJ
ap-5897	233	25	1)−b[n+	1)−b[n+	NUM
ap-5897	233	26	1]ε−	1]ε−	ADJ
ap-5897	233	27	γc	γc	PROPN
ap-5897	233	28	=	=	SYM
ap-5897	233	29	bβn+1	bβn+1	PROPN
ap-5897	233	30	t	t	NOUN
ap-5897	233	31	n(γ	n(γ	NOUN
ap-5897	233	32	+	+	PROPN
ap-5897	234	1	1)−	1)−	NUM
ap-5897	234	2	γc	γc	NOUN
ap-5897	234	3	=	=	SYM
ap-5897	234	4	cn+1	cn+1	PROPN
ap-5897	234	5	.	.	PUNCT
ap-5897	235	1	if	if	SCONJ
ap-5897	235	2	n	n	NUM
ap-5897	235	3	∈	∈	NOUN
ap-5897	235	4	ib	ib	NOUN
ap-5897	235	5	,	,	PUNCT
ap-5897	235	6	then	then	ADV
ap-5897	235	7	βn+1	βn+1	NUM
ap-5897	235	8	t	t	NOUN
ap-5897	235	9	n(γ	n(γ	NOUN
ap-5897	235	10	+	+	CCONJ
ap-5897	235	11	1	1	X
ap-5897	235	12	)	)	PUNCT
ap-5897	235	13	−	−	PROPN
ap-5897	235	14	γ	γ	PROPN
ap-5897	235	15	/∈	/∈	PUNCT
ap-5897	235	16	z.	z.	PROPN
ap-5897	236	1	thus	thus	ADV
ap-5897	236	2	,	,	PUNCT
ap-5897	236	3	γ	γ	PROPN
ap-5897	236	4	∈	∈	PROPN
ap-5897	236	5	cb	cb	X
ap-5897	236	6	.	.	PUNCT
ap-5897	237	1	combining	combine	VERB
ap-5897	237	2	prop	prop	NOUN
ap-5897	237	3	.	.	PUNCT
ap-5897	237	4	3.1	3.1	NUM
ap-5897	237	5	and	and	CCONJ
ap-5897	237	6	prop	prop	NOUN
ap-5897	237	7	.	.	PUNCT
ap-5897	238	1	3.2	3.2	NUM
ap-5897	238	2	,	,	PUNCT
ap-5897	238	3	we	we	PRON
ap-5897	238	4	have	have	VERB
ap-5897	238	5	the	the	DET
ap-5897	238	6	following	follow	VERB
ap-5897	238	7	theorem	theorem	VERB
ap-5897	238	8	.	.	PUNCT
ap-5897	238	9	theorem	theorem	VERB
ap-5897	238	10	3.3	3.3	NUM
ap-5897	238	11	.	.	PUNCT
ap-5897	239	1	γ	γ	PROPN
ap-5897	239	2	∈	∈	PROPN
ap-5897	239	3	cb	cb	PROPN
ap-5897	239	4	if	if	SCONJ
ap-5897	239	5	and	and	CCONJ
ap-5897	239	6	only	only	ADV
ap-5897	239	7	if	if	SCONJ
ap-5897	239	8	d(b	d(b	NOUN
ap-5897	239	9	;	;	PUNCT
ap-5897	239	10	γ	γ	X
ap-5897	239	11	+	+	NOUN
ap-5897	239	12	1	1	NUM
ap-5897	239	13	)	)	PUNCT
ap-5897	239	14	=	=	PUNCT
ap-5897	239	15	d∗(b	d∗(b	PROPN
ap-5897	239	16	;	;	PUNCT
ap-5897	239	17	γ	γ	X
ap-5897	239	18	+	+	NOUN
ap-5897	239	19	1	1	NUM
ap-5897	239	20	)	)	PUNCT
ap-5897	239	21	.	.	PUNCT
ap-5897	240	1	theorem	theorem	VERB
ap-5897	240	2	3.3	3.3	NUM
ap-5897	240	3	and	and	CCONJ
ap-5897	240	4	[	[	X
ap-5897	240	5	2	2	NUM
ap-5897	240	6	,	,	PUNCT
ap-5897	240	7	theorem	theorem	VERB
ap-5897	240	8	3	3	NUM
ap-5897	240	9	]	]	PUNCT
ap-5897	240	10	imply	imply	VERB
ap-5897	240	11	corollary	corollary	ADJ
ap-5897	240	12	3.3.1	3.3.1	NUM
ap-5897	240	13	while	while	SCONJ
ap-5897	240	14	theorem	theorem	VERB
ap-5897	240	15	3.3	3.3	NUM
ap-5897	240	16	and	and	CCONJ
ap-5897	240	17	[	[	X
ap-5897	240	18	6	6	NUM
ap-5897	240	19	,	,	PUNCT
ap-5897	240	20	lemma	lemma	PROPN
ap-5897	240	21	6	6	NUM
ap-5897	240	22	]	]	PUNCT
ap-5897	240	23	imply	imply	VERB
ap-5897	240	24	corollary	corollary	ADJ
ap-5897	240	25	3.3.2	3.3.2	NUM
ap-5897	240	26	.	.	PUNCT
ap-5897	241	1	corollary	corollary	ADJ
ap-5897	241	2	3.3.1	3.3.1	PROPN
ap-5897	241	3	.	.	PUNCT
ap-5897	242	1	let	let	VERB
ap-5897	242	2	1	1	NUM
ap-5897	242	3	<	<	X
ap-5897	242	4	β	β	X
ap-5897	242	5	∈	∈	PROPN
ap-5897	242	6	r.	r.	PROPN
ap-5897	242	7	let	let	VERB
ap-5897	242	8	t	t	NOUN
ap-5897	242	9	:	:	PUNCT
ap-5897	243	1	[	[	X
ap-5897	243	2	0	0	NUM
ap-5897	243	3	,	,	PUNCT
ap-5897	243	4	1	1	X
ap-5897	243	5	)	)	PUNCT
ap-5897	243	6	−→	−→	NOUN
ap-5897	243	7	[	[	X
ap-5897	243	8	0	0	NUM
ap-5897	243	9	,	,	PUNCT
ap-5897	243	10	1	1	NUM
ap-5897	243	11	)	)	PUNCT
ap-5897	243	12	be	be	AUX
ap-5897	243	13	the	the	DET
ap-5897	243	14	beta	beta	ADJ
ap-5897	243	15	transformation	transformation	NOUN
ap-5897	243	16	given	give	VERB
ap-5897	243	17	by	by	ADP
ap-5897	243	18	t	t	PROPN
ap-5897	243	19	(	(	PUNCT
ap-5897	243	20	x	x	NOUN
ap-5897	243	21	)	)	PUNCT
ap-5897	243	22	=	=	SYM
ap-5897	243	23	βx−	βx−	PUNCT
ap-5897	243	24	bβxc	bβxc	NOUN
ap-5897	243	25	.	.	PUNCT
ap-5897	244	1	then	then	ADV
ap-5897	244	2	the	the	DET
ap-5897	244	3	following	follow	VERB
ap-5897	244	4	are	be	AUX
ap-5897	244	5	equivalent	equivalent	ADJ
ap-5897	244	6	:	:	PUNCT
ap-5897	244	7	(	(	PUNCT
ap-5897	244	8	1	1	NUM
ap-5897	244	9	.	.	PUNCT
ap-5897	244	10	)	)	PUNCT
ap-5897	245	1	d(b	d(b	PROPN
ap-5897	245	2	;	;	PUNCT
ap-5897	245	3	1	1	X
ap-5897	245	4	)	)	PUNCT
ap-5897	245	5	=	=	PUNCT
ap-5897	245	6	d∗(b	d∗(b	PROPN
ap-5897	245	7	;	;	PUNCT
ap-5897	245	8	1	1	NUM
ap-5897	245	9	)	)	PUNCT
ap-5897	245	10	;	;	PUNCT
ap-5897	245	11	(	(	PUNCT
ap-5897	245	12	2	2	NUM
ap-5897	245	13	.	.	PUNCT
ap-5897	245	14	)	)	PUNCT
ap-5897	245	15	βt	βt	PROPN
ap-5897	245	16	j(1	j(1	NOUN
ap-5897	245	17	)	)	PUNCT
ap-5897	245	18	/∈	/∈	PUNCT
ap-5897	246	1	z	z	NOUN
ap-5897	246	2	for	for	ADP
ap-5897	246	3	all	all	DET
ap-5897	246	4	j	j	PROPN
ap-5897	246	5	∈	∈	PROPN
ap-5897	246	6	n	n	PART
ap-5897	246	7	∪	∪	X
ap-5897	246	8	{	{	PUNCT
ap-5897	246	9	0	0	NUM
ap-5897	246	10	}	}	PUNCT
ap-5897	246	11	;	;	PUNCT
ap-5897	246	12	(	(	PUNCT
ap-5897	246	13	3	3	NUM
ap-5897	246	14	.	.	PUNCT
ap-5897	246	15	)	)	PUNCT
ap-5897	247	1	d(b	d(b	PROPN
ap-5897	247	2	;	;	PUNCT
ap-5897	247	3	1	1	X
ap-5897	247	4	)	)	PUNCT
ap-5897	247	5	is	be	AUX
ap-5897	247	6	infinite	infinite	ADJ
ap-5897	247	7	.	.	PUNCT
ap-5897	247	8	corollary	corollary	ADJ
ap-5897	247	9	3.3.2	3.3.2	PROPN
ap-5897	247	10	.	.	PUNCT
ap-5897	248	1	let	let	VERB
ap-5897	248	2	1	1	NUM
ap-5897	248	3	<	<	X
ap-5897	248	4	β	β	X
ap-5897	248	5	∈	∈	PROPN
ap-5897	248	6	r.	r.	PROPN
ap-5897	248	7	let	let	VERB
ap-5897	248	8	t−β	t−β	NOUN
ap-5897	248	9	be	be	AUX
ap-5897	248	10	the	the	DET
ap-5897	248	11	negative	negative	ADJ
ap-5897	248	12	beta	beta	ADJ
ap-5897	248	13	transformation	transformation	NOUN
ap-5897	248	14	on	on	ADP
ap-5897	248	15	[	[	X
ap-5897	248	16	lβ	lβ	INTJ
ap-5897	248	17	,	,	PUNCT
ap-5897	248	18	rβ	rβ	NOUN
ap-5897	248	19	)	)	PUNCT
ap-5897	248	20	given	give	VERB
ap-5897	248	21	by	by	ADP
ap-5897	248	22	t−β(x	t−β(x	NOUN
ap-5897	248	23	)	)	PUNCT
ap-5897	248	24	=	=	PUNCT
ap-5897	248	25	−βx−	−βx−	X
ap-5897	248	26	b−βx−	b−βx−	X
ap-5897	248	27	lβc	lβc	PROPN
ap-5897	248	28	.	.	PUNCT
ap-5897	249	1	then	then	ADV
ap-5897	249	2	the	the	DET
ap-5897	249	3	following	follow	VERB
ap-5897	249	4	are	be	AUX
ap-5897	249	5	equivalent	equivalent	ADJ
ap-5897	249	6	:	:	PUNCT
ap-5897	249	7	(	(	PUNCT
ap-5897	249	8	1	1	NUM
ap-5897	249	9	.	.	PUNCT
ap-5897	249	10	)	)	PUNCT
ap-5897	250	1	d(b	d(b	PROPN
ap-5897	250	2	;	;	PUNCT
ap-5897	250	3	rβ	rβ	X
ap-5897	250	4	)	)	PUNCT
ap-5897	250	5	=	=	PUNCT
ap-5897	250	6	d∗(b	d∗(b	PROPN
ap-5897	250	7	;	;	PUNCT
ap-5897	250	8	rβ	rβ	NUM
ap-5897	250	9	)	)	PUNCT
ap-5897	250	10	;	;	PUNCT
ap-5897	250	11	(	(	PUNCT
ap-5897	250	12	2	2	NUM
ap-5897	250	13	.	.	PUNCT
ap-5897	250	14	)	)	PUNCT
ap-5897	251	1	−βt	−βt	NOUN
ap-5897	252	1	2j+1	2j+1	PROPN
ap-5897	252	2	−β	−β	NOUN
ap-5897	252	3	(	(	PUNCT
ap-5897	252	4	rβ)−	rβ)−	VERB
ap-5897	252	5	lβ	lβ	X
ap-5897	252	6	/∈	/∈	PUNCT
ap-5897	252	7	z	z	NOUN
ap-5897	252	8	for	for	ADP
ap-5897	252	9	all	all	DET
ap-5897	252	10	j	j	PROPN
ap-5897	252	11	∈	∈	PROPN
ap-5897	252	12	n	n	PART
ap-5897	252	13	∪	∪	X
ap-5897	252	14	{	{	PUNCT
ap-5897	252	15	0	0	NUM
ap-5897	252	16	}	}	PUNCT
ap-5897	252	17	;	;	PUNCT
ap-5897	252	18	(	(	PUNCT
ap-5897	252	19	3	3	NUM
ap-5897	252	20	.	.	PUNCT
ap-5897	252	21	)	)	PUNCT
ap-5897	253	1	d(b	d(b	PROPN
ap-5897	253	2	;	;	PUNCT
ap-5897	253	3	rβ	rβ	X
ap-5897	253	4	)	)	PUNCT
ap-5897	253	5	is	be	AUX
ap-5897	253	6	not	not	PART
ap-5897	253	7	purely	purely	ADV
ap-5897	253	8	periodic	periodic	ADJ
ap-5897	253	9	of	of	ADP
ap-5897	253	10	odd	odd	ADJ
ap-5897	253	11	period	period	NOUN
ap-5897	253	12	.	.	PUNCT
ap-5897	254	1	next	next	ADV
ap-5897	254	2	,	,	PUNCT
ap-5897	254	3	we	we	PRON
ap-5897	254	4	determine	determine	VERB
ap-5897	254	5	the	the	DET
ap-5897	254	6	relation	relation	NOUN
ap-5897	254	7	between	between	ADP
ap-5897	254	8	d∗(b	d∗(b	PROPN
ap-5897	254	9	;	;	PUNCT
ap-5897	254	10	γ+1	γ+1	NUM
ap-5897	254	11	)	)	PUNCT
ap-5897	254	12	and	and	CCONJ
ap-5897	254	13	d(b	d(b	PROPN
ap-5897	254	14	;	;	PUNCT
ap-5897	254	15	γ+	γ+	NUM
ap-5897	254	16	1	1	X
ap-5897	254	17	)	)	PUNCT
ap-5897	254	18	when	when	SCONJ
ap-5897	254	19	they	they	PRON
ap-5897	254	20	are	be	AUX
ap-5897	254	21	not	not	PART
ap-5897	254	22	equal	equal	ADJ
ap-5897	254	23	(	(	PUNCT
ap-5897	254	24	i.e.	i.e.	X
ap-5897	254	25	,	,	PUNCT
ap-5897	254	26	γ	γ	PROPN
ap-5897	254	27	/∈	/∈	PUNCT
ap-5897	254	28	cb	cb	PROPN
ap-5897	254	29	)	)	PUNCT
ap-5897	254	30	.	.	PUNCT
ap-5897	255	1	define	define	VERB
ap-5897	255	2	the	the	DET
ap-5897	255	3	propositional	propositional	ADJ
ap-5897	255	4	statement	statement	NOUN
ap-5897	255	5	e(b	e(b	ADJ
ap-5897	255	6	;	;	PUNCT
ap-5897	255	7	k	k	X
ap-5897	255	8	)	)	PUNCT
ap-5897	255	9	to	to	PART
ap-5897	255	10	mean	mean	VERB
ap-5897	255	11	βk+1	βk+1	NOUN
ap-5897	255	12	t	t	NOUN
ap-5897	255	13	k(γ	k(γ	PROPN
ap-5897	255	14	+	+	PROPN
ap-5897	256	1	1)−	1)−	PROPN
ap-5897	256	2	γ	γ	PROPN
ap-5897	256	3	∈	∈	PROPN
ap-5897	256	4	z	z	NOUN
ap-5897	256	5	and	and	CCONJ
ap-5897	256	6	sgn(b[k	sgn(b[k	NUM
ap-5897	256	7	+	+	CCONJ
ap-5897	256	8	1	1	NUM
ap-5897	256	9	]	]	PUNCT
ap-5897	256	10	)	)	PUNCT
ap-5897	256	11	>	>	X
ap-5897	257	1	0	0	X
ap-5897	257	2	.	.	PUNCT
ap-5897	257	3	suppose	suppose	VERB
ap-5897	257	4	e(b	e(b	PROPN
ap-5897	257	5	;	;	PUNCT
ap-5897	257	6	k	k	X
ap-5897	257	7	)	)	PUNCT
ap-5897	257	8	holds	hold	VERB
ap-5897	257	9	and	and	CCONJ
ap-5897	257	10	k	k	PROPN
ap-5897	257	11	is	be	AUX
ap-5897	257	12	minimal	minimal	ADJ
ap-5897	257	13	with	with	ADP
ap-5897	257	14	such	such	ADJ
ap-5897	257	15	property	property	NOUN
ap-5897	257	16	.	.	PUNCT
ap-5897	258	1	then	then	ADV
ap-5897	258	2	t	t	PROPN
ap-5897	258	3	k+1(γ+1	k+1(γ+1	PROPN
ap-5897	258	4	)	)	PUNCT
ap-5897	258	5	=	=	SYM
ap-5897	258	6	βk+1	βk+1	PROPN
ap-5897	258	7	t	t	PROPN
ap-5897	258	8	k(γ+1)−bβk+1	k(γ+1)−bβk+1	PROPN
ap-5897	258	9	t	t	NOUN
ap-5897	258	10	k(γ+1)−γc	k(γ+1)−γc	NOUN
ap-5897	259	1	=	=	SYM
ap-5897	259	2	γ	γ	X
ap-5897	259	3	.	.	PUNCT
ap-5897	260	1	thus	thus	ADV
ap-5897	260	2	,	,	PUNCT
ap-5897	260	3	if	if	SCONJ
ap-5897	260	4	d(b	d(b	X
ap-5897	260	5	;	;	PUNCT
ap-5897	260	6	γ	γ	X
ap-5897	260	7	+	+	NOUN
ap-5897	260	8	1	1	NUM
ap-5897	260	9	)	)	PUNCT
ap-5897	260	10	=	=	SYM
ap-5897	260	11	(	(	PUNCT
ap-5897	260	12	c1	c1	PROPN
ap-5897	260	13	,	,	PUNCT
ap-5897	260	14	c2	c2	PROPN
ap-5897	260	15	,	,	PUNCT
ap-5897	260	16	.	.	PUNCT
ap-5897	260	17	.	.	PUNCT
ap-5897	260	18	.	.	PUNCT
ap-5897	260	19	)	)	PUNCT
ap-5897	261	1	,	,	PUNCT
ap-5897	261	2	then	then	ADV
ap-5897	261	3	d(b	d(b	NUM
ap-5897	261	4	;	;	PUNCT
ap-5897	261	5	γ	γ	X
ap-5897	261	6	+	+	NOUN
ap-5897	261	7	1	1	NUM
ap-5897	261	8	)	)	PUNCT
ap-5897	261	9	=	=	SYM
ap-5897	261	10	(	(	PUNCT
ap-5897	261	11	c1	c1	PROPN
ap-5897	261	12	,	,	PUNCT
ap-5897	261	13	c2	c2	PROPN
ap-5897	261	14	,	,	PUNCT
ap-5897	261	15	.	.	PUNCT
ap-5897	261	16	.	.	PUNCT
ap-5897	261	17	.	.	PUNCT
ap-5897	262	1	,	,	PUNCT
ap-5897	262	2	ck+1	ck+1	X
ap-5897	262	3	)	)	PUNCT
ap-5897	262	4	◦	◦	NOUN
ap-5897	262	5	d(σk+1(b	d(σk+1(b	ADJ
ap-5897	262	6	)	)	PUNCT
ap-5897	262	7	;	;	PUNCT
ap-5897	262	8	γ	γ	X
ap-5897	262	9	)	)	PUNCT
ap-5897	262	10	,	,	PUNCT
ap-5897	262	11	where	where	SCONJ
ap-5897	262	12	◦	◦	NOUN
ap-5897	262	13	denotes	denote	VERB
ap-5897	262	14	the	the	DET
ap-5897	262	15	usual	usual	ADJ
ap-5897	262	16	word	word	NOUN
ap-5897	262	17	concatenation	concatenation	NOUN
ap-5897	262	18	and	and	CCONJ
ap-5897	262	19	σj	σj	ADJ
ap-5897	262	20	(	(	PUNCT
ap-5897	262	21	j	j	PROPN
ap-5897	262	22	∈	∈	PROPN
ap-5897	262	23	n	n	CCONJ
ap-5897	262	24	)	)	PUNCT
ap-5897	262	25	is	be	AUX
ap-5897	262	26	the	the	DET
ap-5897	262	27	shift	shift	NOUN
ap-5897	262	28	operator	operator	NOUN
ap-5897	262	29	in	in	ADP
ap-5897	262	30	rn	rn	PROPN
ap-5897	262	31	given	give	VERB
ap-5897	262	32	by	by	ADP
ap-5897	262	33	σj(r1	σj(r1	NUM
ap-5897	262	34	,	,	PUNCT
ap-5897	262	35	r2	r2	PROPN
ap-5897	262	36	,	,	PUNCT
ap-5897	262	37	.	.	PUNCT
ap-5897	262	38	.	.	PUNCT
ap-5897	262	39	.	.	PUNCT
ap-5897	262	40	)	)	PUNCT
ap-5897	263	1	=	=	PRON
ap-5897	263	2	(	(	PUNCT
ap-5897	263	3	rj+1	rj+1	X
ap-5897	263	4	,	,	PUNCT
ap-5897	263	5	rj+2	rj+2	NOUN
ap-5897	263	6	,	,	PUNCT
ap-5897	263	7	.	.	PUNCT
ap-5897	263	8	.	.	PUNCT
ap-5897	263	9	.	.	PUNCT
ap-5897	263	10	)	)	PUNCT
ap-5897	263	11	.	.	PUNCT
ap-5897	264	1	moreover	moreover	ADV
ap-5897	264	2	,	,	PUNCT
ap-5897	264	3	from	from	ADP
ap-5897	264	4	the	the	DET
ap-5897	264	5	proof	proof	NOUN
ap-5897	264	6	of	of	ADP
ap-5897	264	7	proposition	proposition	NOUN
ap-5897	264	8	3.1	3.1	NUM
ap-5897	264	9	,	,	PUNCT
ap-5897	264	10	we	we	PRON
ap-5897	264	11	see	see	VERB
ap-5897	264	12	that	that	DET
ap-5897	264	13	t	t	PROPN
ap-5897	264	14	k+1(γ	k+1(γ	PROPN
ap-5897	265	1	+	+	CCONJ
ap-5897	265	2	1−	1−	NUM
ap-5897	265	3	ε	ε	PROPN
ap-5897	265	4	)	)	PUNCT
ap-5897	265	5	=	=	SYM
ap-5897	266	1	βk+1	βk+1	PROPN
ap-5897	266	2	t	t	NOUN
ap-5897	266	3	k(γ	k(γ	PROPN
ap-5897	266	4	+	+	CCONJ
ap-5897	266	5	1)−b[k	1)−b[k	NUM
ap-5897	266	6	+	+	NUM
ap-5897	266	7	1]ε	1]ε	NUM
ap-5897	266	8	−bβk+1	−bβk+1	NOUN
ap-5897	266	9	t	t	NOUN
ap-5897	266	10	k(γ	k(γ	PROPN
ap-5897	266	11	+	+	CCONJ
ap-5897	266	12	1)−b[k	1)−b[k	NUM
ap-5897	267	1	+	+	CCONJ
ap-5897	267	2	1]ε−	1]ε−	NUM
ap-5897	267	3	γc	γc	X
ap-5897	267	4	=	=	SYM
ap-5897	267	5	γ	γ	X
ap-5897	267	6	+	+	X
ap-5897	267	7	1−	1−	NUM
ap-5897	267	8	ε	ε	PROPN
ap-5897	267	9	.	.	PUNCT
ap-5897	268	1	therefore	therefore	ADV
ap-5897	268	2	,	,	PUNCT
ap-5897	268	3	d∗(b	d∗(b	PROPN
ap-5897	268	4	;	;	PUNCT
ap-5897	268	5	γ+1	γ+1	NUM
ap-5897	268	6	)	)	PUNCT
ap-5897	268	7	=	=	SYM
ap-5897	268	8	(	(	PUNCT
ap-5897	268	9	c1	c1	PROPN
ap-5897	268	10	,	,	PUNCT
ap-5897	268	11	c2	c2	PROPN
ap-5897	268	12	,	,	PUNCT
ap-5897	268	13	.	.	PUNCT
ap-5897	268	14	.	.	PUNCT
ap-5897	269	1	.	.	PUNCT
ap-5897	270	1	,	,	PUNCT
ap-5897	270	2	ck+1−1)	ck+1−1)	PROPN
ap-5897	270	3	◦	◦	NOUN
ap-5897	270	4	d∗(σk+1(b	d∗(σk+1(b	NOUN
ap-5897	270	5	)	)	PUNCT
ap-5897	270	6	;	;	PUNCT
ap-5897	270	7	γ+1	γ+1	PROPN
ap-5897	270	8	)	)	PUNCT
ap-5897	270	9	.	.	PUNCT
ap-5897	271	1	from	from	ADP
ap-5897	271	2	the	the	DET
ap-5897	271	3	computation	computation	NOUN
ap-5897	271	4	above	above	ADV
ap-5897	271	5	,	,	PUNCT
ap-5897	271	6	we	we	PRON
ap-5897	271	7	see	see	VERB
ap-5897	271	8	that	that	SCONJ
ap-5897	271	9	the	the	DET
ap-5897	271	10	process	process	NOUN
ap-5897	271	11	of	of	ADP
ap-5897	271	12	determining	determine	VERB
ap-5897	271	13	d∗(b	d∗(b	PROPN
ap-5897	271	14	;	;	PUNCT
ap-5897	271	15	γ	γ	X
ap-5897	271	16	+	+	NOUN
ap-5897	271	17	1	1	NUM
ap-5897	271	18	)	)	PUNCT
ap-5897	271	19	depends	depend	VERB
ap-5897	271	20	on	on	ADP
ap-5897	271	21	the	the	DET
ap-5897	271	22	other	other	ADJ
ap-5897	271	23	sequences	sequence	NOUN
ap-5897	271	24	d∗(σi(b	d∗(σi(b	PROPN
ap-5897	271	25	)	)	PUNCT
ap-5897	271	26	;	;	PUNCT
ap-5897	271	27	γ	γ	X
ap-5897	271	28	+	+	NOUN
ap-5897	271	29	1	1	NUM
ap-5897	271	30	)	)	PUNCT
ap-5897	271	31	,	,	PUNCT
ap-5897	271	32	i	i	PRON
ap-5897	271	33	∈	∈	PROPN
ap-5897	271	34	n.	n.	NOUN
ap-5897	271	35	to	to	PART
ap-5897	271	36	illustrate	illustrate	VERB
ap-5897	271	37	this	this	DET
ap-5897	271	38	process	process	NOUN
ap-5897	271	39	,	,	PUNCT
ap-5897	271	40	we	we	PRON
ap-5897	271	41	present	present	VERB
ap-5897	271	42	the	the	DET
ap-5897	271	43	two	two	NUM
ap-5897	271	44	-	-	PUNCT
ap-5897	271	45	base	base	NOUN
ap-5897	271	46	expansion	expansion	NOUN
ap-5897	271	47	case	case	NOUN
ap-5897	271	48	where	where	SCONJ
ap-5897	271	49	we	we	PRON
ap-5897	271	50	set	set	VERB
ap-5897	271	51	α	α	NOUN
ap-5897	271	52	:	:	PUNCT
ap-5897	271	53	=	=	SYM
ap-5897	271	54	β1	β1	X
ap-5897	271	55	>	>	X
ap-5897	271	56	0	0	PUNCT
ap-5897	271	57	and	and	CCONJ
ap-5897	271	58	β	β	X
ap-5897	271	59	:	:	PUNCT
ap-5897	271	60	=	=	SYM
ap-5897	271	61	β2	β2	VERB
ap-5897	271	62	>	>	X
ap-5897	271	63	0	0	X
ap-5897	271	64	.	.	PUNCT
ap-5897	272	1	we	we	PRON
ap-5897	272	2	easily	easily	ADV
ap-5897	272	3	compute	compute	VERB
ap-5897	272	4	ib	ib	NOUN
ap-5897	272	5	to	to	PART
ap-5897	272	6	be	be	AUX
ap-5897	272	7	n∪{0	n∪{0	NOUN
ap-5897	272	8	}	}	PUNCT
ap-5897	272	9	.	.	PUNCT
ap-5897	273	1	suppose	suppose	VERB
ap-5897	273	2	e(b	e(b	PROPN
ap-5897	273	3	;	;	PUNCT
ap-5897	273	4	k	k	X
ap-5897	273	5	)	)	PUNCT
ap-5897	273	6	is	be	AUX
ap-5897	273	7	satisfied	satisfied	ADJ
ap-5897	273	8	and	and	CCONJ
ap-5897	273	9	k	k	PROPN
ap-5897	273	10	is	be	AUX
ap-5897	273	11	minimal	minimal	ADJ
ap-5897	273	12	.	.	PUNCT
ap-5897	274	1	on	on	ADP
ap-5897	274	2	the	the	DET
ap-5897	274	3	one	one	NUM
ap-5897	274	4	hand	hand	NOUN
ap-5897	274	5	,	,	PUNCT
ap-5897	274	6	suppose	suppose	VERB
ap-5897	274	7	k	k	PROPN
ap-5897	274	8	is	be	AUX
ap-5897	274	9	odd	odd	ADJ
ap-5897	274	10	.	.	PUNCT
ap-5897	275	1	then	then	ADV
ap-5897	275	2	d(α	d(α	PROPN
ap-5897	275	3	,	,	PUNCT
ap-5897	275	4	β	β	X
ap-5897	275	5	;	;	PUNCT
ap-5897	275	6	γ	γ	X
ap-5897	275	7	+	+	NOUN
ap-5897	275	8	1	1	NUM
ap-5897	275	9	)	)	PUNCT
ap-5897	275	10	=	=	SYM
ap-5897	275	11	(	(	PUNCT
ap-5897	275	12	c1	c1	PROPN
ap-5897	275	13	,	,	PUNCT
ap-5897	275	14	c2	c2	PROPN
ap-5897	275	15	,	,	PUNCT
ap-5897	275	16	.	.	PUNCT
ap-5897	275	17	.	.	PUNCT
ap-5897	275	18	.	.	PUNCT
ap-5897	276	1	,	,	PUNCT
ap-5897	276	2	ck+1	ck+1	X
ap-5897	276	3	)	)	PUNCT
ap-5897	276	4	◦	◦	NOUN
ap-5897	276	5	d(α	d(α	PROPN
ap-5897	276	6	,	,	PUNCT
ap-5897	276	7	β	β	X
ap-5897	276	8	;	;	PUNCT
ap-5897	276	9	γ	γ	X
ap-5897	276	10	)	)	PUNCT
ap-5897	276	11	and	and	CCONJ
ap-5897	276	12	d∗(α	d∗(α	PROPN
ap-5897	276	13	,	,	PUNCT
ap-5897	276	14	β	β	X
ap-5897	276	15	;	;	PUNCT
ap-5897	276	16	γ+1	γ+1	NUM
ap-5897	276	17	)	)	PUNCT
ap-5897	276	18	=	=	SYM
ap-5897	276	19	(	(	PUNCT
ap-5897	276	20	c1	c1	PROPN
ap-5897	276	21	,	,	PUNCT
ap-5897	276	22	c2	c2	PROPN
ap-5897	276	23	,	,	PUNCT
ap-5897	276	24	.	.	PUNCT
ap-5897	276	25	.	.	PUNCT
ap-5897	277	1	.	.	PUNCT
ap-5897	278	1	,	,	PUNCT
ap-5897	278	2	ck+1−1)	ck+1−1)	PROPN
ap-5897	278	3	◦	◦	NOUN
ap-5897	278	4	d∗(α	d∗(α	PROPN
ap-5897	278	5	,	,	PUNCT
ap-5897	278	6	β	β	X
ap-5897	278	7	;	;	PUNCT
ap-5897	278	8	γ+1	γ+1	NUM
ap-5897	278	9	)	)	PUNCT
ap-5897	278	10	.	.	PUNCT
ap-5897	279	1	this	this	PRON
ap-5897	279	2	implies	imply	VERB
ap-5897	279	3	that	that	SCONJ
ap-5897	279	4	d∗(α	d∗(α	PROPN
ap-5897	279	5	,	,	PUNCT
ap-5897	279	6	β	β	X
ap-5897	279	7	;	;	PUNCT
ap-5897	279	8	γ	γ	X
ap-5897	279	9	+	+	NOUN
ap-5897	279	10	1	1	NUM
ap-5897	279	11	)	)	PUNCT
ap-5897	279	12	=	=	SYM
ap-5897	279	13	(	(	PUNCT
ap-5897	279	14	c1	c1	PROPN
ap-5897	279	15	,	,	PUNCT
ap-5897	279	16	c2	c2	PROPN
ap-5897	279	17	,	,	PUNCT
ap-5897	279	18	.	.	PUNCT
ap-5897	279	19	.	.	PUNCT
ap-5897	279	20	.	.	PUNCT
ap-5897	280	1	,	,	PUNCT
ap-5897	280	2	ck+1	ck+1	VERB
ap-5897	280	3	−	−	PROPN
ap-5897	280	4	1	1	NUM
ap-5897	280	5	)	)	PUNCT
ap-5897	280	6	.	.	PUNCT
ap-5897	281	1	on	on	ADP
ap-5897	281	2	the	the	DET
ap-5897	281	3	other	other	ADJ
ap-5897	281	4	hand	hand	NOUN
ap-5897	281	5	,	,	PUNCT
ap-5897	281	6	suppose	suppose	VERB
ap-5897	281	7	k	k	PROPN
ap-5897	281	8	is	be	AUX
ap-5897	281	9	even	even	ADV
ap-5897	281	10	.	.	PUNCT
ap-5897	282	1	then	then	ADV
ap-5897	282	2	d(α	d(α	PROPN
ap-5897	282	3	,	,	PUNCT
ap-5897	282	4	β	β	X
ap-5897	282	5	;	;	PUNCT
ap-5897	282	6	γ	γ	X
ap-5897	282	7	+	+	NOUN
ap-5897	282	8	1	1	NUM
ap-5897	282	9	)	)	PUNCT
ap-5897	282	10	=	=	SYM
ap-5897	282	11	(	(	PUNCT
ap-5897	282	12	c1	c1	PROPN
ap-5897	282	13	,	,	PUNCT
ap-5897	282	14	c2	c2	PROPN
ap-5897	282	15	,	,	PUNCT
ap-5897	282	16	.	.	PUNCT
ap-5897	282	17	.	.	PUNCT
ap-5897	282	18	.	.	PUNCT
ap-5897	283	1	,	,	PUNCT
ap-5897	283	2	ck+1	ck+1	X
ap-5897	283	3	)	)	PUNCT
ap-5897	283	4	◦	◦	NOUN
ap-5897	283	5	d(β	d(β	PROPN
ap-5897	283	6	,	,	PUNCT
ap-5897	283	7	α	α	NOUN
ap-5897	283	8	;	;	PUNCT
ap-5897	283	9	γ	γ	X
ap-5897	283	10	)	)	PUNCT
ap-5897	283	11	and	and	CCONJ
ap-5897	283	12	d∗(α	d∗(α	PROPN
ap-5897	283	13	,	,	PUNCT
ap-5897	283	14	β	β	X
ap-5897	283	15	;	;	PUNCT
ap-5897	283	16	γ+1	γ+1	NUM
ap-5897	283	17	)	)	PUNCT
ap-5897	283	18	=	=	SYM
ap-5897	283	19	(	(	PUNCT
ap-5897	283	20	c1	c1	PROPN
ap-5897	283	21	,	,	PUNCT
ap-5897	283	22	c2	c2	PROPN
ap-5897	283	23	,	,	PUNCT
ap-5897	283	24	.	.	PUNCT
ap-5897	283	25	.	.	PUNCT
ap-5897	284	1	.	.	PUNCT
ap-5897	285	1	,	,	PUNCT
ap-5897	285	2	ck+1−1)	ck+1−1)	PROPN
ap-5897	285	3	◦	◦	PROPN
ap-5897	285	4	d∗(β	d∗(β	PROPN
ap-5897	285	5	,	,	PUNCT
ap-5897	285	6	α	α	NOUN
ap-5897	285	7	;	;	PUNCT
ap-5897	285	8	γ+1	γ+1	NUM
ap-5897	285	9	)	)	PUNCT
ap-5897	285	10	.	.	PUNCT
ap-5897	286	1	note	note	VERB
ap-5897	286	2	that	that	SCONJ
ap-5897	286	3	iσ(b	iσ(b	NOUN
ap-5897	286	4	)	)	PUNCT
ap-5897	286	5	=	=	SYM
ap-5897	287	1	n	n	NOUN
ap-5897	287	2	∪	∪	X
ap-5897	287	3	{	{	PUNCT
ap-5897	287	4	0	0	NUM
ap-5897	287	5	}	}	PUNCT
ap-5897	287	6	.	.	PUNCT
ap-5897	288	1	suppose	suppose	VERB
ap-5897	288	2	that	that	SCONJ
ap-5897	288	3	there	there	PRON
ap-5897	288	4	is	be	VERB
ap-5897	288	5	no	no	DET
ap-5897	288	6	m	m	NOUN
ap-5897	288	7	∈	∈	NOUN
ap-5897	288	8	n	n	PRON
ap-5897	288	9	such	such	ADJ
ap-5897	288	10	that	that	SCONJ
ap-5897	288	11	e(σ(b);m	e(σ(b);m	PROPN
ap-5897	288	12	)	)	PUNCT
ap-5897	288	13	holds	hold	VERB
ap-5897	288	14	.	.	PUNCT
ap-5897	289	1	then	then	ADV
ap-5897	289	2	d∗(β	d∗(β	PROPN
ap-5897	289	3	,	,	PUNCT
ap-5897	289	4	α	α	X
ap-5897	289	5	;	;	PUNCT
ap-5897	289	6	γ	γ	X
ap-5897	289	7	+	+	NOUN
ap-5897	289	8	1	1	NUM
ap-5897	289	9	)	)	PUNCT
ap-5897	289	10	=	=	PUNCT
ap-5897	289	11	d(β	d(β	PROPN
ap-5897	289	12	,	,	PUNCT
ap-5897	289	13	α	α	X
ap-5897	289	14	;	;	PUNCT
ap-5897	289	15	γ	γ	X
ap-5897	289	16	+	+	NOUN
ap-5897	289	17	1	1	NUM
ap-5897	289	18	)	)	PUNCT
ap-5897	289	19	and	and	CCONJ
ap-5897	289	20	so	so	ADV
ap-5897	289	21	,	,	PUNCT
ap-5897	289	22	d∗(α	d∗(α	PROPN
ap-5897	289	23	,	,	PUNCT
ap-5897	289	24	β	β	X
ap-5897	289	25	;	;	PUNCT
ap-5897	289	26	γ	γ	X
ap-5897	289	27	+	+	NOUN
ap-5897	289	28	1	1	NUM
ap-5897	289	29	)	)	PUNCT
ap-5897	289	30	=	=	SYM
ap-5897	289	31	(	(	PUNCT
ap-5897	289	32	c1	c1	PROPN
ap-5897	289	33	,	,	PUNCT
ap-5897	289	34	c2	c2	PROPN
ap-5897	289	35	,	,	PUNCT
ap-5897	289	36	.	.	PUNCT
ap-5897	289	37	.	.	PUNCT
ap-5897	289	38	.	.	PUNCT
ap-5897	290	1	,	,	PUNCT
ap-5897	290	2	ck+1−	ck+1−	PROPN
ap-5897	290	3	1	1	NUM
ap-5897	290	4	)	)	PUNCT
ap-5897	290	5	◦	◦	PROPN
ap-5897	290	6	d(β	d(β	PROPN
ap-5897	290	7	,	,	PUNCT
ap-5897	290	8	α	α	X
ap-5897	290	9	;	;	PUNCT
ap-5897	290	10	γ	γ	X
ap-5897	290	11	+	+	NOUN
ap-5897	290	12	1	1	NUM
ap-5897	290	13	)	)	PUNCT
ap-5897	290	14	.	.	PUNCT
ap-5897	291	1	let	let	VERB
ap-5897	291	2	d(β	d(β	PROPN
ap-5897	291	3	,	,	PUNCT
ap-5897	291	4	α	α	X
ap-5897	291	5	;	;	PUNCT
ap-5897	291	6	γ	γ	X
ap-5897	291	7	+	+	NOUN
ap-5897	291	8	1	1	NUM
ap-5897	291	9	)	)	PUNCT
ap-5897	291	10	=	=	SYM
ap-5897	291	11	(	(	PUNCT
ap-5897	291	12	q1	q1	PROPN
ap-5897	291	13	,	,	PUNCT
ap-5897	291	14	q2	q2	NOUN
ap-5897	291	15	,	,	PUNCT
ap-5897	291	16	.	.	PUNCT
ap-5897	291	17	.	.	PUNCT
ap-5897	291	18	.	.	PUNCT
ap-5897	291	19	)	)	PUNCT
ap-5897	291	20	.	.	PUNCT
ap-5897	292	1	if	if	SCONJ
ap-5897	292	2	there	there	PRON
ap-5897	292	3	exists	exist	VERB
ap-5897	292	4	m	m	VERB
ap-5897	292	5	∈	∈	NOUN
ap-5897	292	6	n∪{0	n∪{0	NOUN
ap-5897	292	7	}	}	PUNCT
ap-5897	292	8	such	such	ADJ
ap-5897	292	9	that	that	SCONJ
ap-5897	292	10	e(σ(b);m	e(σ(b);m	NOUN
ap-5897	292	11	)	)	PUNCT
ap-5897	292	12	holds	hold	VERB
ap-5897	292	13	and	and	CCONJ
ap-5897	292	14	m	m	VERB
ap-5897	292	15	is	be	AUX
ap-5897	292	16	minimal	minimal	ADJ
ap-5897	292	17	and	and	CCONJ
ap-5897	292	18	odd	odd	ADJ
ap-5897	292	19	,	,	PUNCT
ap-5897	292	20	then	then	ADV
ap-5897	292	21	d∗(β	d∗(β	PROPN
ap-5897	292	22	,	,	PUNCT
ap-5897	292	23	α	α	X
ap-5897	292	24	;	;	PUNCT
ap-5897	292	25	γ	γ	X
ap-5897	292	26	+	+	NOUN
ap-5897	292	27	1	1	NUM
ap-5897	292	28	)	)	PUNCT
ap-5897	292	29	=	=	SYM
ap-5897	292	30	(	(	PUNCT
ap-5897	292	31	q1	q1	PROPN
ap-5897	292	32	,	,	PUNCT
ap-5897	292	33	q2	q2	NOUN
ap-5897	292	34	,	,	PUNCT
ap-5897	292	35	.	.	PUNCT
ap-5897	292	36	.	.	PUNCT
ap-5897	293	1	.	.	PUNCT
ap-5897	294	1	,	,	PUNCT
ap-5897	294	2	qm+1	qm+1	PROPN
ap-5897	295	1	−	−	PROPN
ap-5897	295	2	1	1	NUM
ap-5897	295	3	)	)	PUNCT
ap-5897	295	4	.	.	PUNCT
ap-5897	296	1	therefore	therefore	ADV
ap-5897	296	2	,	,	PUNCT
ap-5897	296	3	d∗(α	d∗(α	PROPN
ap-5897	296	4	,	,	PUNCT
ap-5897	296	5	β	β	X
ap-5897	296	6	;	;	PUNCT
ap-5897	296	7	γ	γ	X
ap-5897	296	8	+	+	NOUN
ap-5897	296	9	1	1	NUM
ap-5897	296	10	)	)	PUNCT
ap-5897	296	11	=	=	SYM
ap-5897	296	12	(	(	PUNCT
ap-5897	296	13	c1	c1	PROPN
ap-5897	296	14	,	,	PUNCT
ap-5897	296	15	c2	c2	PROPN
ap-5897	296	16	,	,	PUNCT
ap-5897	296	17	.	.	PUNCT
ap-5897	296	18	.	.	PUNCT
ap-5897	296	19	.	.	PUNCT
ap-5897	297	1	,	,	PUNCT
ap-5897	297	2	ck+1	ck+1	VERB
ap-5897	297	3	−	−	PROPN
ap-5897	297	4	1	1	X
ap-5897	297	5	)	)	PUNCT
ap-5897	297	6	◦	◦	NOUN
ap-5897	297	7	(	(	PUNCT
ap-5897	297	8	q1	q1	PROPN
ap-5897	297	9	,	,	PUNCT
ap-5897	297	10	q2	q2	NOUN
ap-5897	297	11	,	,	PUNCT
ap-5897	297	12	.	.	PUNCT
ap-5897	297	13	.	.	PUNCT
ap-5897	298	1	.	.	PUNCT
ap-5897	299	1	,	,	PUNCT
ap-5897	299	2	qm+1	qm+1	PROPN
ap-5897	300	1	−	−	PROPN
ap-5897	300	2	1	1	NUM
ap-5897	300	3	)	)	PUNCT
ap-5897	300	4	.	.	PUNCT
ap-5897	301	1	finally	finally	ADV
ap-5897	301	2	,	,	PUNCT
ap-5897	301	3	if	if	SCONJ
ap-5897	301	4	m	m	NOUN
ap-5897	301	5	is	be	AUX
ap-5897	301	6	even	even	ADV
ap-5897	301	7	,	,	PUNCT
ap-5897	301	8	we	we	PRON
ap-5897	301	9	have	have	VERB
ap-5897	301	10	d∗(β	d∗(β	PROPN
ap-5897	301	11	,	,	PUNCT
ap-5897	301	12	α	α	NOUN
ap-5897	301	13	;	;	PUNCT
ap-5897	301	14	γ+1	γ+1	NUM
ap-5897	301	15	)	)	PUNCT
ap-5897	301	16	=	=	SYM
ap-5897	301	17	(	(	PUNCT
ap-5897	301	18	q1	q1	PROPN
ap-5897	301	19	,	,	PUNCT
ap-5897	301	20	q2	q2	NOUN
ap-5897	301	21	,	,	PUNCT
ap-5897	301	22	.	.	PUNCT
ap-5897	301	23	.	.	PUNCT
ap-5897	302	1	.	.	PUNCT
ap-5897	303	1	,	,	PUNCT
ap-5897	303	2	qm+1−1)	qm+1−1)	X
ap-5897	303	3	◦	◦	NOUN
ap-5897	303	4	d∗(α	d∗(α	PROPN
ap-5897	303	5	,	,	PUNCT
ap-5897	303	6	β	β	X
ap-5897	303	7	;	;	PUNCT
ap-5897	303	8	γ+1	γ+1	NUM
ap-5897	303	9	)	)	PUNCT
ap-5897	303	10	.	.	PUNCT
ap-5897	304	1	218	218	NUM
ap-5897	304	2	vol	vol	NOUN
ap-5897	304	3	.	.	PUNCT
ap-5897	305	1	60	60	NUM
ap-5897	305	2	no	no	NOUN
ap-5897	305	3	.	.	PUNCT
ap-5897	306	1	3/2020	3/2020	NUM
ap-5897	306	2	beta	beta	PROPN
ap-5897	306	3	cantor	cantor	PROPN
ap-5897	306	4	series	series	NOUN
ap-5897	306	5	expansion	expansion	NOUN
ap-5897	306	6	and	and	CCONJ
ap-5897	306	7	admissible	admissible	ADJ
ap-5897	306	8	sequences	sequence	NOUN
ap-5897	306	9	hence	hence	ADV
ap-5897	306	10	,	,	PUNCT
ap-5897	306	11	d∗(α	d∗(α	PROPN
ap-5897	306	12	,	,	PUNCT
ap-5897	306	13	β	β	X
ap-5897	306	14	;	;	PUNCT
ap-5897	306	15	γ	γ	X
ap-5897	306	16	+	+	NOUN
ap-5897	306	17	1	1	NUM
ap-5897	306	18	)	)	PUNCT
ap-5897	306	19	=	=	SYM
ap-5897	306	20	(	(	PUNCT
ap-5897	306	21	c1	c1	PROPN
ap-5897	306	22	,	,	PUNCT
ap-5897	306	23	c2	c2	PROPN
ap-5897	306	24	,	,	PUNCT
ap-5897	306	25	.	.	PUNCT
ap-5897	306	26	.	.	PUNCT
ap-5897	306	27	.	.	PUNCT
ap-5897	307	1	,	,	PUNCT
ap-5897	307	2	ck+1	ck+1	VERB
ap-5897	307	3	−	−	PROPN
ap-5897	307	4	1	1	X
ap-5897	307	5	)	)	PUNCT
ap-5897	307	6	◦	◦	NOUN
ap-5897	307	7	(	(	PUNCT
ap-5897	307	8	q1	q1	PROPN
ap-5897	307	9	,	,	PUNCT
ap-5897	307	10	q2	q2	NOUN
ap-5897	307	11	,	,	PUNCT
ap-5897	307	12	.	.	PUNCT
ap-5897	307	13	.	.	PUNCT
ap-5897	308	1	.	.	PUNCT
ap-5897	309	1	,	,	PUNCT
ap-5897	309	2	qm+1	qm+1	PROPN
ap-5897	310	1	−	−	PROPN
ap-5897	310	2	1	1	X
ap-5897	310	3	)	)	PUNCT
ap-5897	310	4	◦	◦	PROPN
ap-5897	310	5	d∗(α	d∗(α	PROPN
ap-5897	310	6	,	,	PUNCT
ap-5897	310	7	β	β	X
ap-5897	310	8	;	;	PUNCT
ap-5897	310	9	γ	γ	X
ap-5897	310	10	+	+	NOUN
ap-5897	310	11	1	1	NUM
ap-5897	310	12	)	)	PUNCT
ap-5897	310	13	=	=	SYM
ap-5897	310	14	(	(	PUNCT
ap-5897	310	15	c1	c1	PROPN
ap-5897	310	16	,	,	PUNCT
ap-5897	310	17	c2	c2	PROPN
ap-5897	310	18	,	,	PUNCT
ap-5897	310	19	.	.	PUNCT
ap-5897	310	20	.	.	PUNCT
ap-5897	310	21	.	.	PUNCT
ap-5897	311	1	,	,	PUNCT
ap-5897	311	2	ck+1	ck+1	VERB
ap-5897	311	3	−	−	PROPN
ap-5897	311	4	1	1	NUM
ap-5897	311	5	,	,	PUNCT
ap-5897	311	6	q1	q1	PROPN
ap-5897	311	7	,	,	PUNCT
ap-5897	311	8	q2	q2	NOUN
ap-5897	311	9	,	,	PUNCT
ap-5897	311	10	.	.	PUNCT
ap-5897	311	11	.	.	PUNCT
ap-5897	312	1	.	.	PUNCT
ap-5897	313	1	,	,	PUNCT
ap-5897	313	2	qm+1	qm+1	PROPN
ap-5897	314	1	−	−	PROPN
ap-5897	314	2	1	1	NUM
ap-5897	314	3	)	)	PUNCT
ap-5897	314	4	.	.	PUNCT
ap-5897	315	1	to	to	PART
ap-5897	315	2	sum	sum	VERB
ap-5897	315	3	up	up	ADP
ap-5897	315	4	,	,	PUNCT
ap-5897	315	5	we	we	PRON
ap-5897	315	6	have	have	VERB
ap-5897	315	7	the	the	DET
ap-5897	315	8	following	follow	VERB
ap-5897	315	9	proposition	proposition	NOUN
ap-5897	315	10	.	.	PUNCT
ap-5897	316	1	proposition	proposition	NOUN
ap-5897	316	2	3.4	3.4	NUM
ap-5897	316	3	.	.	PUNCT
ap-5897	317	1	let	let	VERB
ap-5897	317	2	b	b	NOUN
ap-5897	317	3	=	=	SYM
ap-5897	317	4	(	(	PUNCT
ap-5897	317	5	α	α	X
ap-5897	317	6	,	,	PUNCT
ap-5897	317	7	β	β	NOUN
ap-5897	317	8	)	)	PUNCT
ap-5897	317	9	where	where	SCONJ
ap-5897	317	10	α	α	X
ap-5897	317	11	,	,	PUNCT
ap-5897	317	12	β	β	X
ap-5897	317	13	∈	∈	NOUN
ap-5897	317	14	r	r	NOUN
ap-5897	317	15	>	>	X
ap-5897	317	16	0	0	PUNCT
ap-5897	318	1	and	and	CCONJ
ap-5897	318	2	αβ	αβ	INTJ
ap-5897	318	3	>	>	X
ap-5897	318	4	1	1	X
ap-5897	318	5	.	.	PUNCT
ap-5897	319	1	let	let	VERB
ap-5897	319	2	d(α	d(α	NOUN
ap-5897	319	3	,	,	PUNCT
ap-5897	319	4	β	β	X
ap-5897	319	5	;	;	PUNCT
ap-5897	319	6	γ	γ	X
ap-5897	319	7	+	+	NOUN
ap-5897	319	8	1	1	NUM
ap-5897	319	9	)	)	PUNCT
ap-5897	319	10	=	=	SYM
ap-5897	319	11	(	(	PUNCT
ap-5897	319	12	c1	c1	PROPN
ap-5897	319	13	,	,	PUNCT
ap-5897	319	14	c2	c2	PROPN
ap-5897	319	15	,	,	PUNCT
ap-5897	319	16	.	.	PUNCT
ap-5897	319	17	.	.	PUNCT
ap-5897	319	18	.	.	PUNCT
ap-5897	319	19	)	)	PUNCT
ap-5897	320	1	and	and	CCONJ
ap-5897	320	2	d(β	d(β	PROPN
ap-5897	320	3	,	,	PUNCT
ap-5897	320	4	α	α	X
ap-5897	320	5	;	;	PUNCT
ap-5897	320	6	γ	γ	X
ap-5897	320	7	+	+	NOUN
ap-5897	320	8	1	1	NUM
ap-5897	320	9	)	)	PUNCT
ap-5897	320	10	=	=	SYM
ap-5897	320	11	(	(	PUNCT
ap-5897	320	12	q1	q1	PROPN
ap-5897	320	13	,	,	PUNCT
ap-5897	320	14	q2	q2	NOUN
ap-5897	320	15	,	,	PUNCT
ap-5897	320	16	.	.	PUNCT
ap-5897	320	17	.	.	PUNCT
ap-5897	320	18	.	.	PUNCT
ap-5897	320	19	)	)	PUNCT
ap-5897	320	20	.	.	PUNCT
ap-5897	321	1	then	then	ADV
ap-5897	321	2	d∗(α	d∗(α	PROPN
ap-5897	321	3	,	,	PUNCT
ap-5897	321	4	β	β	X
ap-5897	321	5	;	;	PUNCT
ap-5897	321	6	γ	γ	X
ap-5897	321	7	+	+	NOUN
ap-5897	321	8	1	1	NUM
ap-5897	321	9	)	)	PUNCT
ap-5897	321	10	can	can	AUX
ap-5897	321	11	only	only	ADV
ap-5897	321	12	assume	assume	VERB
ap-5897	321	13	one	one	NUM
ap-5897	321	14	of	of	ADP
ap-5897	321	15	the	the	DET
ap-5897	321	16	following	follow	VERB
ap-5897	321	17	forms	form	NOUN
ap-5897	321	18	:	:	PUNCT
ap-5897	321	19	(	(	PUNCT
ap-5897	321	20	1	1	NUM
ap-5897	321	21	.	.	NUM
ap-5897	321	22	)	)	PUNCT
ap-5897	322	1	(	(	PUNCT
ap-5897	322	2	c1	c1	PROPN
ap-5897	322	3	,	,	PUNCT
ap-5897	322	4	c2	c2	PROPN
ap-5897	322	5	,	,	PUNCT
ap-5897	322	6	.	.	PUNCT
ap-5897	322	7	.	.	PUNCT
ap-5897	322	8	.	.	PUNCT
ap-5897	323	1	,	,	PUNCT
ap-5897	324	1	c2k	c2k	INTJ
ap-5897	324	2	−	−	NOUN
ap-5897	325	1	1	1	NUM
ap-5897	325	2	)	)	PUNCT
ap-5897	325	3	(	(	PUNCT
ap-5897	325	4	2	2	NUM
ap-5897	325	5	.	.	NUM
ap-5897	325	6	)	)	PUNCT
ap-5897	326	1	(	(	PUNCT
ap-5897	326	2	c1	c1	PROPN
ap-5897	326	3	,	,	PUNCT
ap-5897	326	4	c2	c2	PROPN
ap-5897	326	5	,	,	PUNCT
ap-5897	326	6	.	.	PUNCT
ap-5897	326	7	.	.	PUNCT
ap-5897	326	8	.	.	PUNCT
ap-5897	327	1	,	,	PUNCT
ap-5897	327	2	c2k+1	c2k+1	NOUN
ap-5897	327	3	−	−	NOUN
ap-5897	327	4	1	1	NUM
ap-5897	327	5	,	,	PUNCT
ap-5897	327	6	q1	q1	PROPN
ap-5897	327	7	,	,	PUNCT
ap-5897	327	8	q2	q2	NOUN
ap-5897	327	9	,	,	PUNCT
ap-5897	327	10	.	.	PUNCT
ap-5897	327	11	.	.	PUNCT
ap-5897	327	12	.	.	PUNCT
ap-5897	327	13	)	)	PUNCT
ap-5897	328	1	(	(	PUNCT
ap-5897	328	2	3	3	NUM
ap-5897	328	3	.	.	NUM
ap-5897	328	4	)	)	PUNCT
ap-5897	329	1	(	(	PUNCT
ap-5897	329	2	c1	c1	PROPN
ap-5897	329	3	,	,	PUNCT
ap-5897	329	4	c2	c2	PROPN
ap-5897	329	5	,	,	PUNCT
ap-5897	329	6	.	.	PUNCT
ap-5897	329	7	.	.	PUNCT
ap-5897	329	8	.	.	PUNCT
ap-5897	330	1	,	,	PUNCT
ap-5897	330	2	c2k+1	c2k+1	NOUN
ap-5897	330	3	−	−	NOUN
ap-5897	330	4	1	1	NUM
ap-5897	330	5	,	,	PUNCT
ap-5897	330	6	q1	q1	PROPN
ap-5897	330	7	,	,	PUNCT
ap-5897	330	8	q2	q2	NOUN
ap-5897	330	9	,	,	PUNCT
ap-5897	330	10	.	.	PUNCT
ap-5897	330	11	.	.	PUNCT
ap-5897	331	1	.	.	PUNCT
ap-5897	332	1	,	,	PUNCT
ap-5897	332	2	q2	q2	PROPN
ap-5897	332	3	m	m	PROPN
ap-5897	332	4	−	−	PROPN
ap-5897	332	5	1	1	NUM
ap-5897	332	6	)	)	PUNCT
ap-5897	332	7	(	(	PUNCT
ap-5897	332	8	4	4	NUM
ap-5897	332	9	.	.	NUM
ap-5897	332	10	)	)	PUNCT
ap-5897	333	1	(	(	PUNCT
ap-5897	333	2	c1	c1	PROPN
ap-5897	333	3	,	,	PUNCT
ap-5897	333	4	c2	c2	PROPN
ap-5897	333	5	,	,	PUNCT
ap-5897	333	6	.	.	PUNCT
ap-5897	333	7	.	.	PUNCT
ap-5897	333	8	.	.	PUNCT
ap-5897	334	1	,	,	PUNCT
ap-5897	334	2	c2k+1	c2k+1	NOUN
ap-5897	334	3	−	−	NOUN
ap-5897	334	4	1	1	NUM
ap-5897	334	5	,	,	PUNCT
ap-5897	334	6	q1	q1	PROPN
ap-5897	334	7	,	,	PUNCT
ap-5897	334	8	q2	q2	NOUN
ap-5897	334	9	,	,	PUNCT
ap-5897	334	10	.	.	PUNCT
ap-5897	334	11	.	.	PUNCT
ap-5897	335	1	.	.	PUNCT
ap-5897	336	1	,	,	PUNCT
ap-5897	336	2	q2m+1	q2m+1	NOUN
ap-5897	336	3	−	−	ADP
ap-5897	336	4	1	1	NUM
ap-5897	336	5	)	)	PUNCT
ap-5897	336	6	(	(	PUNCT
ap-5897	336	7	5	5	NUM
ap-5897	336	8	.	.	NUM
ap-5897	336	9	)	)	PUNCT
ap-5897	337	1	(	(	PUNCT
ap-5897	337	2	c1	c1	PROPN
ap-5897	337	3	,	,	PUNCT
ap-5897	337	4	c2	c2	PROPN
ap-5897	337	5	,	,	PUNCT
ap-5897	337	6	.	.	PUNCT
ap-5897	337	7	.	.	PUNCT
ap-5897	337	8	.	.	PUNCT
ap-5897	337	9	)	)	PUNCT
ap-5897	338	1	examples	example	NOUN
ap-5897	338	2	.	.	PUNCT
ap-5897	339	1	we	we	PRON
ap-5897	339	2	now	now	ADV
ap-5897	339	3	give	give	VERB
ap-5897	339	4	examples	example	NOUN
ap-5897	339	5	to	to	PART
ap-5897	339	6	illustrate	illustrate	VERB
ap-5897	339	7	prop	prop	NOUN
ap-5897	339	8	.	.	PUNCT
ap-5897	340	1	3.4	3.4	NUM
ap-5897	340	2	(	(	PUNCT
ap-5897	340	3	1–5	1–5	NUM
ap-5897	340	4	)	)	PUNCT
ap-5897	340	5	by	by	ADP
ap-5897	340	6	providing	provide	VERB
ap-5897	340	7	values	value	NOUN
ap-5897	340	8	of	of	ADP
ap-5897	340	9	α	α	PROPN
ap-5897	340	10	>	>	X
ap-5897	340	11	0	0	PROPN
ap-5897	340	12	and	and	CCONJ
ap-5897	340	13	β	β	X
ap-5897	340	14	>	>	X
ap-5897	340	15	0	0	PUNCT
ap-5897	341	1	with	with	ADP
ap-5897	341	2	αβ	αβ	INTJ
ap-5897	341	3	>	>	SYM
ap-5897	341	4	1	1	NUM
ap-5897	341	5	and	and	CCONJ
ap-5897	341	6	γ	γ	X
ap-5897	341	7	=	=	SYM
ap-5897	341	8	0	0	NUM
ap-5897	341	9	for	for	ADP
ap-5897	341	10	each	each	DET
ap-5897	341	11	case	case	NOUN
ap-5897	341	12	.	.	PUNCT
ap-5897	342	1	let	let	VERB
ap-5897	343	1	r	r	NOUN
ap-5897	343	2	,	,	PUNCT
ap-5897	343	3	s	s	NOUN
ap-5897	343	4	∈	∈	PROPN
ap-5897	343	5	n.	n.	NOUN
ap-5897	343	6	(	(	PUNCT
ap-5897	343	7	1	1	NUM
ap-5897	343	8	.	.	PUNCT
ap-5897	343	9	)	)	PUNCT
ap-5897	343	10	let	let	VERB
ap-5897	343	11	α	α	NOUN
ap-5897	343	12	=	=	SYM
ap-5897	343	13	r	r	X
ap-5897	343	14	/	/	SYM
ap-5897	343	15	s	s	NOUN
ap-5897	343	16	/∈	/∈	NOUN
ap-5897	343	17	z	z	PROPN
ap-5897	343	18	and	and	CCONJ
ap-5897	343	19	β	β	X
ap-5897	343	20	=	=	PUNCT
ap-5897	343	21	s.	s.	PROPN
ap-5897	343	22	d(α	d(α	PROPN
ap-5897	343	23	,	,	PUNCT
ap-5897	343	24	β	β	X
ap-5897	343	25	;	;	PUNCT
ap-5897	343	26	γ	γ	X
ap-5897	343	27	+	+	NOUN
ap-5897	343	28	1	1	NUM
ap-5897	343	29	)	)	PUNCT
ap-5897	343	30	=	=	SYM
ap-5897	343	31	(	(	PUNCT
ap-5897	343	32	br	br	PROPN
ap-5897	343	33	/	/	SYM
ap-5897	343	34	sc	sc	PROPN
ap-5897	343	35	,	,	PUNCT
ap-5897	343	36	r	r	NOUN
ap-5897	343	37	−	−	PROPN
ap-5897	343	38	sbr	sbr	PROPN
ap-5897	343	39	/	/	SYM
ap-5897	343	40	sc	sc	PROPN
ap-5897	343	41	,	,	PUNCT
ap-5897	343	42	0	0	NUM
ap-5897	343	43	)	)	PUNCT
ap-5897	343	44	d∗(α	d∗(α	PROPN
ap-5897	343	45	,	,	PUNCT
ap-5897	343	46	β	β	X
ap-5897	343	47	;	;	PUNCT
ap-5897	343	48	γ	γ	X
ap-5897	343	49	+	+	NOUN
ap-5897	343	50	1	1	NUM
ap-5897	343	51	)	)	PUNCT
ap-5897	343	52	=	=	SYM
ap-5897	343	53	(	(	PUNCT
ap-5897	343	54	br	br	PROPN
ap-5897	343	55	/	/	SYM
ap-5897	343	56	sc	sc	PROPN
ap-5897	343	57	,	,	PUNCT
ap-5897	343	58	r	r	NOUN
ap-5897	343	59	−	−	PROPN
ap-5897	343	60	sbr	sbr	PROPN
ap-5897	343	61	/	/	SYM
ap-5897	343	62	sc	sc	PROPN
ap-5897	343	63	−	−	NOUN
ap-5897	343	64	1	1	NUM
ap-5897	343	65	)	)	PUNCT
ap-5897	343	66	(	(	PUNCT
ap-5897	343	67	2	2	NUM
ap-5897	343	68	.	.	PUNCT
ap-5897	343	69	)	)	PUNCT
ap-5897	343	70	let	let	VERB
ap-5897	343	71	α	α	NOUN
ap-5897	343	72	=	=	SYM
ap-5897	343	73	r	r	NOUN
ap-5897	343	74	and	and	CCONJ
ap-5897	343	75	β	β	X
ap-5897	343	76	be	be	AUX
ap-5897	343	77	transcendental	transcendental	ADJ
ap-5897	343	78	over	over	ADP
ap-5897	343	79	q.	q.	PROPN
ap-5897	343	80	d(α	d(α	PROPN
ap-5897	343	81	,	,	PUNCT
ap-5897	343	82	β	β	X
ap-5897	343	83	;	;	PUNCT
ap-5897	343	84	γ	γ	X
ap-5897	343	85	+	+	NOUN
ap-5897	343	86	1	1	NUM
ap-5897	343	87	)	)	PUNCT
ap-5897	343	88	=	=	SYM
ap-5897	344	1	(	(	PUNCT
ap-5897	344	2	r	r	NOUN
ap-5897	344	3	,	,	PUNCT
ap-5897	344	4	0	0	NUM
ap-5897	344	5	)	)	PUNCT
ap-5897	344	6	d(β	d(β	PROPN
ap-5897	344	7	,	,	PUNCT
ap-5897	344	8	α	α	X
ap-5897	344	9	;	;	PUNCT
ap-5897	344	10	γ	γ	X
ap-5897	344	11	+	+	NOUN
ap-5897	344	12	1	1	NUM
ap-5897	344	13	)	)	PUNCT
ap-5897	344	14	=	=	SYM
ap-5897	345	1	d∗(β	d∗(β	PROPN
ap-5897	345	2	,	,	PUNCT
ap-5897	345	3	α	α	X
ap-5897	345	4	;	;	PUNCT
ap-5897	345	5	γ	γ	X
ap-5897	345	6	+	+	NOUN
ap-5897	345	7	1	1	NUM
ap-5897	345	8	)	)	PUNCT
ap-5897	345	9	=	=	SYM
ap-5897	345	10	(	(	PUNCT
ap-5897	345	11	q1	q1	PROPN
ap-5897	345	12	,	,	PUNCT
ap-5897	345	13	q2	q2	NOUN
ap-5897	345	14	,	,	PUNCT
ap-5897	345	15	.	.	PUNCT
ap-5897	345	16	.	.	PUNCT
ap-5897	345	17	.	.	PUNCT
ap-5897	345	18	)	)	PUNCT
ap-5897	346	1	d∗(α	d∗(α	PROPN
ap-5897	346	2	,	,	PUNCT
ap-5897	346	3	β	β	X
ap-5897	346	4	;	;	PUNCT
ap-5897	346	5	γ	γ	X
ap-5897	346	6	+	+	NOUN
ap-5897	346	7	1	1	NUM
ap-5897	346	8	)	)	PUNCT
ap-5897	346	9	=	=	SYM
ap-5897	346	10	(	(	PUNCT
ap-5897	346	11	r	r	NOUN
ap-5897	346	12	−	−	PROPN
ap-5897	346	13	1	1	NUM
ap-5897	346	14	,	,	PUNCT
ap-5897	346	15	q1	q1	PROPN
ap-5897	346	16	,	,	PUNCT
ap-5897	346	17	q2	q2	NOUN
ap-5897	346	18	,	,	PUNCT
ap-5897	346	19	.	.	PUNCT
ap-5897	346	20	.	.	PUNCT
ap-5897	346	21	.	.	PUNCT
ap-5897	346	22	)	)	PUNCT
ap-5897	347	1	(	(	PUNCT
ap-5897	347	2	3	3	X
ap-5897	347	3	.	.	PUNCT
ap-5897	347	4	)	)	PUNCT
ap-5897	347	5	let	let	VERB
ap-5897	347	6	α	α	NOUN
ap-5897	347	7	=	=	PUNCT
ap-5897	347	8	(	(	PUNCT
ap-5897	347	9	1	1	NUM
ap-5897	347	10	+	+	CCONJ
ap-5897	347	11	√	√	PROPN
ap-5897	347	12	5)/2	5)/2	NUM
ap-5897	347	13	and	and	CCONJ
ap-5897	347	14	β	β	X
ap-5897	347	15	=	=	PUNCT
ap-5897	347	16	α2	α2	PROPN
ap-5897	347	17	.	.	PUNCT
ap-5897	348	1	d(α	d(α	NOUN
ap-5897	348	2	,	,	PUNCT
ap-5897	348	3	β	β	X
ap-5897	348	4	;	;	PUNCT
ap-5897	348	5	γ	γ	X
ap-5897	348	6	+	+	NOUN
ap-5897	348	7	1	1	NUM
ap-5897	348	8	)	)	PUNCT
ap-5897	348	9	=	=	NOUN
ap-5897	348	10	(	(	PUNCT
ap-5897	348	11	1	1	NUM
ap-5897	348	12	,	,	PUNCT
ap-5897	348	13	1	1	NUM
ap-5897	348	14	,	,	PUNCT
ap-5897	348	15	1	1	NUM
ap-5897	348	16	,	,	PUNCT
ap-5897	348	17	0	0	NUM
ap-5897	348	18	)	)	PUNCT
ap-5897	348	19	d(β	d(β	PROPN
ap-5897	348	20	,	,	PUNCT
ap-5897	348	21	α	α	X
ap-5897	348	22	;	;	PUNCT
ap-5897	348	23	γ	γ	X
ap-5897	348	24	+	+	NOUN
ap-5897	348	25	1	1	NUM
ap-5897	348	26	)	)	PUNCT
ap-5897	348	27	=	=	NOUN
ap-5897	348	28	(	(	PUNCT
ap-5897	348	29	2	2	NUM
ap-5897	348	30	,	,	PUNCT
ap-5897	348	31	1	1	NUM
ap-5897	348	32	,	,	PUNCT
ap-5897	348	33	0	0	NUM
ap-5897	348	34	)	)	PUNCT
ap-5897	348	35	d∗(α	d∗(α	PROPN
ap-5897	348	36	,	,	PUNCT
ap-5897	348	37	β	β	X
ap-5897	348	38	;	;	PUNCT
ap-5897	348	39	γ	γ	X
ap-5897	348	40	+	+	NOUN
ap-5897	348	41	1	1	NUM
ap-5897	348	42	)	)	PUNCT
ap-5897	348	43	=	=	NOUN
ap-5897	348	44	(	(	PUNCT
ap-5897	348	45	1	1	NUM
ap-5897	348	46	,	,	PUNCT
ap-5897	348	47	1	1	NUM
ap-5897	348	48	,	,	PUNCT
ap-5897	348	49	0	0	NUM
ap-5897	348	50	,	,	PUNCT
ap-5897	348	51	2	2	NUM
ap-5897	348	52	,	,	PUNCT
ap-5897	348	53	0	0	NUM
ap-5897	348	54	)	)	PUNCT
ap-5897	348	55	(	(	PUNCT
ap-5897	348	56	4	4	NUM
ap-5897	348	57	.	.	PUNCT
ap-5897	348	58	)	)	PUNCT
ap-5897	348	59	let	let	VERB
ap-5897	348	60	β	β	X
ap-5897	348	61	=	=	PUNCT
ap-5897	348	62	α	α	PROPN
ap-5897	349	1	+	+	NOUN
ap-5897	349	2	1	1	NUM
ap-5897	349	3	where	where	SCONJ
ap-5897	349	4	α	α	NOUN
ap-5897	349	5	is	be	AUX
ap-5897	349	6	the	the	DET
ap-5897	349	7	(	(	PUNCT
ap-5897	349	8	smallest	small	ADJ
ap-5897	349	9	)	)	PUNCT
ap-5897	349	10	pisot	pisot	ADJ
ap-5897	349	11	number	number	NOUN
ap-5897	349	12	which	which	PRON
ap-5897	349	13	satisfies	satisfy	VERB
ap-5897	349	14	α3	α3	ADP
ap-5897	349	15	−	−	PROPN
ap-5897	349	16	α−	α−	ADP
ap-5897	349	17	1	1	NUM
ap-5897	349	18	=	=	SYM
ap-5897	349	19	0	0	NUM
ap-5897	349	20	.	.	PUNCT
ap-5897	350	1	d(α	d(α	NOUN
ap-5897	350	2	,	,	PUNCT
ap-5897	350	3	β	β	X
ap-5897	350	4	;	;	PUNCT
ap-5897	350	5	γ	γ	X
ap-5897	350	6	+	+	NOUN
ap-5897	350	7	1	1	NUM
ap-5897	350	8	)	)	PUNCT
ap-5897	350	9	=	=	NOUN
ap-5897	350	10	(	(	PUNCT
ap-5897	350	11	1	1	NUM
ap-5897	350	12	,	,	PUNCT
ap-5897	350	13	0	0	NUM
ap-5897	350	14	,	,	PUNCT
ap-5897	350	15	1	1	NUM
ap-5897	350	16	,	,	PUNCT
ap-5897	350	17	0	0	NUM
ap-5897	350	18	)	)	PUNCT
ap-5897	350	19	d(β	d(β	PROPN
ap-5897	350	20	,	,	PUNCT
ap-5897	350	21	α	α	X
ap-5897	350	22	;	;	PUNCT
ap-5897	350	23	γ	γ	X
ap-5897	350	24	+	+	NOUN
ap-5897	350	25	1	1	NUM
ap-5897	350	26	)	)	PUNCT
ap-5897	350	27	=	=	NOUN
ap-5897	350	28	(	(	PUNCT
ap-5897	350	29	2	2	NUM
ap-5897	350	30	,	,	PUNCT
ap-5897	350	31	0	0	NUM
ap-5897	350	32	,	,	PUNCT
ap-5897	350	33	1	1	NUM
ap-5897	350	34	,	,	PUNCT
ap-5897	350	35	0	0	NUM
ap-5897	350	36	)	)	PUNCT
ap-5897	350	37	d∗(α	d∗(α	PROPN
ap-5897	350	38	,	,	PUNCT
ap-5897	350	39	β	β	X
ap-5897	350	40	;	;	PUNCT
ap-5897	350	41	γ	γ	X
ap-5897	350	42	+	+	NOUN
ap-5897	350	43	1	1	NUM
ap-5897	350	44	)	)	PUNCT
ap-5897	350	45	=	=	NOUN
ap-5897	350	46	(	(	PUNCT
ap-5897	350	47	1	1	NUM
ap-5897	350	48	,	,	PUNCT
ap-5897	350	49	0	0	NUM
ap-5897	350	50	,	,	PUNCT
ap-5897	350	51	0	0	NUM
ap-5897	350	52	,	,	PUNCT
ap-5897	350	53	2	2	NUM
ap-5897	350	54	,	,	PUNCT
ap-5897	350	55	0	0	NUM
ap-5897	350	56	,	,	PUNCT
ap-5897	350	57	0	0	NUM
ap-5897	350	58	)	)	PUNCT
ap-5897	350	59	(	(	PUNCT
ap-5897	350	60	5	5	NUM
ap-5897	350	61	.	.	PUNCT
ap-5897	350	62	)	)	PUNCT
ap-5897	350	63	let	let	VERB
ap-5897	350	64	α	α	PRON
ap-5897	350	65	be	be	AUX
ap-5897	350	66	transcendental	transcendental	ADJ
ap-5897	350	67	over	over	ADP
ap-5897	350	68	q	q	NOUN
ap-5897	350	69	and	and	CCONJ
ap-5897	350	70	β	β	X
ap-5897	350	71	=	=	SYM
ap-5897	350	72	r.	r.	PROPN
ap-5897	350	73	then	then	ADV
ap-5897	350	74	d(α	d(α	PROPN
ap-5897	350	75	,	,	PUNCT
ap-5897	350	76	β	β	X
ap-5897	350	77	;	;	PUNCT
ap-5897	350	78	γ	γ	X
ap-5897	350	79	+	+	NOUN
ap-5897	350	80	1	1	NUM
ap-5897	350	81	)	)	PUNCT
ap-5897	350	82	=	=	SYM
ap-5897	350	83	d∗(α	d∗(α	PROPN
ap-5897	350	84	,	,	PUNCT
ap-5897	350	85	β	β	X
ap-5897	350	86	;	;	PUNCT
ap-5897	350	87	γ	γ	X
ap-5897	350	88	+	+	NOUN
ap-5897	350	89	1	1	NUM
ap-5897	350	90	)	)	PUNCT
ap-5897	350	91	.	.	PUNCT
ap-5897	351	1	to	to	PART
ap-5897	351	2	end	end	VERB
ap-5897	351	3	this	this	DET
ap-5897	351	4	section	section	NOUN
ap-5897	351	5	,	,	PUNCT
ap-5897	351	6	we	we	PRON
ap-5897	351	7	recover	recover	VERB
ap-5897	351	8	the	the	DET
ap-5897	351	9	classical	classical	ADJ
ap-5897	351	10	results	result	NOUN
ap-5897	351	11	for	for	ADP
ap-5897	351	12	beta	beta	ADJ
ap-5897	351	13	and	and	CCONJ
ap-5897	351	14	negative	negative	ADJ
ap-5897	351	15	beta	beta	ADJ
ap-5897	351	16	expansions	expansion	NOUN
ap-5897	351	17	.	.	PUNCT
ap-5897	352	1	let	let	VERB
ap-5897	352	2	1	1	NUM
ap-5897	352	3	<	<	X
ap-5897	352	4	β	β	X
ap-5897	352	5	∈	∈	PROPN
ap-5897	352	6	r.	r.	PROPN
ap-5897	352	7	for	for	ADP
ap-5897	352	8	the	the	DET
ap-5897	352	9	positive	positive	ADJ
ap-5897	352	10	beta	beta	NOUN
ap-5897	352	11	expansion	expansion	NOUN
ap-5897	352	12	,	,	PUNCT
ap-5897	352	13	we	we	PRON
ap-5897	352	14	see	see	VERB
ap-5897	352	15	that	that	DET
ap-5897	352	16	ib	ib	NOUN
ap-5897	352	17	=	=	SYM
ap-5897	352	18	n∪	n∪	PROPN
ap-5897	352	19	{	{	PUNCT
ap-5897	352	20	0	0	NUM
ap-5897	352	21	}	}	PUNCT
ap-5897	352	22	and	and	CCONJ
ap-5897	352	23	cb	cb	PROPN
ap-5897	352	24	=	=	PRON
ap-5897	352	25	{	{	PUNCT
ap-5897	352	26	γ	γ	X
ap-5897	352	27	∈	∈	NOUN
ap-5897	352	28	r	r	NOUN
ap-5897	352	29	|	|	ADV
ap-5897	352	30	βtn(γ+1)−γ	βtn(γ+1)−γ	NOUN
ap-5897	352	31	/∈	/∈	PUNCT
ap-5897	353	1	z	z	NOUN
ap-5897	353	2	for	for	ADP
ap-5897	353	3	all	all	DET
ap-5897	353	4	n	n	PRON
ap-5897	353	5	∈	∈	NOUN
ap-5897	353	6	ib	ib	NOUN
ap-5897	353	7	}	}	PUNCT
ap-5897	353	8	.	.	PUNCT
ap-5897	354	1	suppose	suppose	VERB
ap-5897	354	2	0	0	NUM
ap-5897	355	1	/∈	/∈	PUNCT
ap-5897	356	1	cb	cb	PROPN
ap-5897	356	2	.	.	PUNCT
ap-5897	357	1	then	then	ADV
ap-5897	357	2	there	there	PRON
ap-5897	357	3	exists	exist	VERB
ap-5897	357	4	a	a	DET
ap-5897	357	5	minimal	minimal	ADJ
ap-5897	357	6	k	k	PROPN
ap-5897	357	7	∈	∈	PROPN
ap-5897	357	8	n	n	PART
ap-5897	357	9	∪	∪	X
ap-5897	357	10	{	{	PUNCT
ap-5897	357	11	0	0	NUM
ap-5897	357	12	}	}	PUNCT
ap-5897	357	13	such	such	ADJ
ap-5897	357	14	that	that	SCONJ
ap-5897	357	15	e(b	e(b	PROPN
ap-5897	357	16	;	;	PUNCT
ap-5897	357	17	k	k	X
ap-5897	357	18	)	)	PUNCT
ap-5897	357	19	holds	hold	VERB
ap-5897	357	20	.	.	PUNCT
ap-5897	358	1	we	we	PRON
ap-5897	358	2	have	have	VERB
ap-5897	358	3	d(b	d(b	NOUN
ap-5897	358	4	;	;	PUNCT
ap-5897	358	5	1	1	X
ap-5897	358	6	)	)	PUNCT
ap-5897	358	7	=	=	SYM
ap-5897	358	8	(	(	PUNCT
ap-5897	358	9	c1	c1	PROPN
ap-5897	358	10	,	,	PUNCT
ap-5897	358	11	.	.	PUNCT
ap-5897	358	12	.	.	PUNCT
ap-5897	358	13	.	.	PUNCT
ap-5897	359	1	,	,	PUNCT
ap-5897	359	2	ck+1	ck+1	X
ap-5897	359	3	)	)	PUNCT
ap-5897	359	4	◦	◦	NOUN
ap-5897	359	5	d(b	d(b	PROPN
ap-5897	359	6	;	;	PUNCT
ap-5897	359	7	0	0	NUM
ap-5897	359	8	)	)	PUNCT
ap-5897	359	9	=	=	SYM
ap-5897	359	10	(	(	PUNCT
ap-5897	359	11	c1	c1	PROPN
ap-5897	359	12	,	,	PUNCT
ap-5897	359	13	.	.	PUNCT
ap-5897	359	14	.	.	PUNCT
ap-5897	360	1	.	.	PUNCT
ap-5897	361	1	,	,	PUNCT
ap-5897	361	2	ck+1	ck+1	NOUN
ap-5897	361	3	,	,	PUNCT
ap-5897	361	4	0	0	NUM
ap-5897	361	5	)	)	PUNCT
ap-5897	361	6	and	and	CCONJ
ap-5897	361	7	d∗(b	d∗(b	PROPN
ap-5897	361	8	;	;	PUNCT
ap-5897	361	9	1	1	X
ap-5897	361	10	)	)	PUNCT
ap-5897	361	11	=	=	SYM
ap-5897	361	12	(	(	PUNCT
ap-5897	361	13	c1	c1	PROPN
ap-5897	361	14	,	,	PUNCT
ap-5897	361	15	.	.	PUNCT
ap-5897	361	16	.	.	PUNCT
ap-5897	361	17	.	.	PUNCT
ap-5897	362	1	,	,	PUNCT
ap-5897	362	2	ck+1	ck+1	VERB
ap-5897	362	3	−	−	PROPN
ap-5897	362	4	1	1	NUM
ap-5897	362	5	)	)	PUNCT
ap-5897	362	6	.	.	PUNCT
ap-5897	363	1	in	in	ADP
ap-5897	363	2	other	other	ADJ
ap-5897	363	3	words	word	NOUN
ap-5897	363	4	,	,	PUNCT
ap-5897	363	5	d∗(b	d∗(b	PROPN
ap-5897	363	6	;	;	PUNCT
ap-5897	363	7	1	1	X
ap-5897	363	8	)	)	PUNCT
ap-5897	363	9	=	=	NOUN
ap-5897	363	10	{	{	PUNCT
ap-5897	363	11	d(b	d(b	X
ap-5897	363	12	;	;	PUNCT
ap-5897	363	13	1	1	X
ap-5897	363	14	)	)	PUNCT
ap-5897	363	15	if	if	SCONJ
ap-5897	363	16	d(b	d(b	NOUN
ap-5897	363	17	;	;	PUNCT
ap-5897	363	18	1	1	X
ap-5897	363	19	)	)	PUNCT
ap-5897	363	20	is	be	AUX
ap-5897	363	21	infinite	infinite	ADJ
ap-5897	363	22	(	(	PUNCT
ap-5897	363	23	c1	c1	NOUN
ap-5897	363	24	,	,	PUNCT
ap-5897	363	25	c2	c2	PROPN
ap-5897	363	26	,	,	PUNCT
ap-5897	363	27	.	.	PUNCT
ap-5897	363	28	.	.	PUNCT
ap-5897	363	29	.	.	PUNCT
ap-5897	364	1	,	,	PUNCT
ap-5897	364	2	cn	cn	INTJ
ap-5897	364	3	−	−	PROPN
ap-5897	364	4	1	1	NUM
ap-5897	364	5	)	)	PUNCT
ap-5897	364	6	if	if	SCONJ
ap-5897	364	7	d(b	d(b	NOUN
ap-5897	364	8	;	;	PUNCT
ap-5897	364	9	1	1	X
ap-5897	364	10	)	)	PUNCT
ap-5897	364	11	=	=	SYM
ap-5897	364	12	(	(	PUNCT
ap-5897	364	13	c1	c1	PROPN
ap-5897	364	14	,	,	PUNCT
ap-5897	364	15	.	.	PUNCT
ap-5897	364	16	.	.	PUNCT
ap-5897	365	1	.	.	PUNCT
ap-5897	366	1	,	,	PUNCT
ap-5897	366	2	cn	cn	INTJ
ap-5897	366	3	,	,	PUNCT
ap-5897	366	4	0	0	NUM
ap-5897	366	5	)	)	PUNCT
ap-5897	366	6	.	.	PUNCT
ap-5897	367	1	for	for	ADP
ap-5897	367	2	the	the	DET
ap-5897	367	3	negative	negative	ADJ
ap-5897	367	4	beta	beta	NOUN
ap-5897	367	5	expansion	expansion	NOUN
ap-5897	367	6	,	,	PUNCT
ap-5897	367	7	we	we	PRON
ap-5897	367	8	have	have	VERB
ap-5897	367	9	ib	ib	NOUN
ap-5897	367	10	=	=	SYM
ap-5897	367	11	2n	2n	NUM
ap-5897	368	1	−	−	NOUN
ap-5897	368	2	1	1	NUM
ap-5897	368	3	and	and	CCONJ
ap-5897	368	4	cb	cb	PROPN
ap-5897	368	5	=	=	PRON
ap-5897	368	6	{	{	PUNCT
ap-5897	368	7	γ	γ	X
ap-5897	368	8	∈	∈	NOUN
ap-5897	368	9	r	r	NOUN
ap-5897	368	10	|	|	NOUN
ap-5897	368	11	−βt	−βt	NOUN
ap-5897	368	12	2n−1(γ	2n−1(γ	NUM
ap-5897	368	13	+	+	CCONJ
ap-5897	368	14	1	1	NUM
ap-5897	368	15	)	)	PUNCT
ap-5897	368	16	−	−	PROPN
ap-5897	368	17	γ	γ	X
ap-5897	368	18	/∈	/∈	PUNCT
ap-5897	368	19	z	z	NOUN
ap-5897	368	20	for	for	ADP
ap-5897	368	21	all	all	PRON
ap-5897	368	22	n	n	PRON
ap-5897	368	23	∈	∈	NOUN
ap-5897	368	24	n	n	CCONJ
ap-5897	368	25	}	}	PUNCT
ap-5897	368	26	.	.	PUNCT
ap-5897	369	1	suppose	suppose	VERB
ap-5897	369	2	lβ	lβ	PROPN
ap-5897	369	3	/∈	/∈	PUNCT
ap-5897	370	1	cb	cb	PROPN
ap-5897	370	2	.	.	PUNCT
ap-5897	371	1	then	then	ADV
ap-5897	371	2	there	there	PRON
ap-5897	371	3	exists	exist	VERB
ap-5897	371	4	minimal	minimal	ADJ
ap-5897	371	5	k	k	PROPN
ap-5897	371	6	∈	∈	PROPN
ap-5897	371	7	n	n	PRON
ap-5897	371	8	such	such	ADJ
ap-5897	371	9	that	that	DET
ap-5897	371	10	e(b	e(b	PROPN
ap-5897	371	11	;	;	PUNCT
ap-5897	371	12	2k	2k	NUM
ap-5897	371	13	−	−	NOUN
ap-5897	371	14	1	1	NUM
ap-5897	371	15	)	)	PUNCT
ap-5897	371	16	holds	hold	VERB
ap-5897	371	17	.	.	PUNCT
ap-5897	372	1	let	let	VERB
ap-5897	372	2	d(b	d(b	PRON
ap-5897	372	3	;	;	PUNCT
ap-5897	372	4	lβ	lβ	PROPN
ap-5897	372	5	)	)	PUNCT
ap-5897	372	6	=	=	SYM
ap-5897	372	7	(	(	PUNCT
ap-5897	372	8	a1	a1	PROPN
ap-5897	372	9	,	,	PUNCT
ap-5897	372	10	a2	a2	PROPN
ap-5897	372	11	,	,	PUNCT
ap-5897	372	12	.	.	PUNCT
ap-5897	372	13	.	.	PUNCT
ap-5897	372	14	.	.	PUNCT
ap-5897	372	15	)	)	PUNCT
ap-5897	372	16	.	.	PUNCT
ap-5897	373	1	it	it	PRON
ap-5897	373	2	is	be	AUX
ap-5897	373	3	easy	easy	ADJ
ap-5897	373	4	to	to	PART
ap-5897	373	5	see	see	VERB
ap-5897	373	6	that	that	SCONJ
ap-5897	373	7	d(b	d(b	PRON
ap-5897	373	8	;	;	PUNCT
ap-5897	373	9	rβ	rβ	X
ap-5897	373	10	)	)	PUNCT
ap-5897	373	11	=	=	SYM
ap-5897	373	12	(	(	PUNCT
ap-5897	373	13	0	0	NUM
ap-5897	373	14	,	,	PUNCT
ap-5897	373	15	a1	a1	NOUN
ap-5897	373	16	,	,	PUNCT
ap-5897	373	17	a2	a2	PROPN
ap-5897	373	18	,	,	PUNCT
ap-5897	373	19	.	.	PUNCT
ap-5897	373	20	.	.	PUNCT
ap-5897	373	21	.	.	PUNCT
ap-5897	373	22	)	)	PUNCT
ap-5897	373	23	.	.	PUNCT
ap-5897	374	1	from	from	ADP
ap-5897	374	2	e(b	e(b	PROPN
ap-5897	374	3	;	;	PUNCT
ap-5897	374	4	2k	2k	NUM
ap-5897	374	5	−	−	PROPN
ap-5897	374	6	1	1	NUM
ap-5897	374	7	)	)	PUNCT
ap-5897	374	8	,	,	PUNCT
ap-5897	374	9	it	it	PRON
ap-5897	374	10	follows	follow	VERB
ap-5897	374	11	that	that	SCONJ
ap-5897	374	12	d(b	d(b	PRON
ap-5897	374	13	;	;	PUNCT
ap-5897	374	14	rβ	rβ	X
ap-5897	374	15	)	)	PUNCT
ap-5897	375	1	=	=	SYM
ap-5897	375	2	(	(	PUNCT
ap-5897	375	3	c1	c1	PROPN
ap-5897	375	4	,	,	PUNCT
ap-5897	375	5	.	.	PUNCT
ap-5897	375	6	.	.	PUNCT
ap-5897	375	7	.	.	PUNCT
ap-5897	376	1	,	,	PUNCT
ap-5897	376	2	c2k	c2k	NOUN
ap-5897	376	3	,	,	PUNCT
ap-5897	376	4	a1	a1	NOUN
ap-5897	376	5	,	,	PUNCT
ap-5897	376	6	a2	a2	PROPN
ap-5897	376	7	,	,	PUNCT
ap-5897	376	8	.	.	PUNCT
ap-5897	376	9	.	.	PUNCT
ap-5897	376	10	.	.	PUNCT
ap-5897	376	11	)	)	PUNCT
ap-5897	376	12	.	.	PUNCT
ap-5897	377	1	this	this	PRON
ap-5897	377	2	means	mean	VERB
ap-5897	377	3	that	that	SCONJ
ap-5897	377	4	c1	c1	NOUN
ap-5897	377	5	=	=	PROPN
ap-5897	377	6	0	0	NUM
ap-5897	377	7	;	;	PUNCT
ap-5897	377	8	ci	ci	NOUN
ap-5897	377	9	=	=	SYM
ap-5897	377	10	ai−1	ai−1	PROPN
ap-5897	377	11	for	for	ADP
ap-5897	377	12	all	all	DET
ap-5897	377	13	i	i	PRON
ap-5897	377	14	=	=	NOUN
ap-5897	377	15	2	2	NUM
ap-5897	377	16	,	,	PUNCT
ap-5897	377	17	.	.	PUNCT
ap-5897	377	18	.	.	PUNCT
ap-5897	378	1	.	.	PUNCT
ap-5897	379	1	,	,	PUNCT
ap-5897	379	2	2k	2k	NUM
ap-5897	379	3	;	;	PUNCT
ap-5897	379	4	and	and	CCONJ
ap-5897	379	5	ai	ai	VERB
ap-5897	379	6	=	=	PUNCT
ap-5897	379	7	a2k−1+i	a2k−1+i	NOUN
ap-5897	379	8	for	for	ADP
ap-5897	379	9	all	all	PRON
ap-5897	379	10	i	i	PRON
ap-5897	379	11	∈	∈	PROPN
ap-5897	379	12	n.	n.	NOUN
ap-5897	379	13	therefore	therefore	ADV
ap-5897	379	14	,	,	PUNCT
ap-5897	379	15	d(b	d(b	PROPN
ap-5897	379	16	;	;	PUNCT
ap-5897	379	17	lβ	lβ	PROPN
ap-5897	379	18	)	)	PUNCT
ap-5897	379	19	=	=	SYM
ap-5897	379	20	(	(	PUNCT
ap-5897	379	21	a1	a1	PROPN
ap-5897	379	22	,	,	PUNCT
ap-5897	379	23	.	.	PUNCT
ap-5897	379	24	.	.	PUNCT
ap-5897	379	25	.	.	PUNCT
ap-5897	380	1	,	,	PUNCT
ap-5897	380	2	a2k−1	a2k−1	PROPN
ap-5897	380	3	)	)	PUNCT
ap-5897	380	4	and	and	CCONJ
ap-5897	380	5	d∗(b	d∗(b	PROPN
ap-5897	380	6	;	;	PUNCT
ap-5897	380	7	rβ	rβ	X
ap-5897	380	8	)	)	PUNCT
ap-5897	380	9	=	=	SYM
ap-5897	380	10	(	(	PUNCT
ap-5897	380	11	c1	c1	PROPN
ap-5897	380	12	,	,	PUNCT
ap-5897	380	13	.	.	PUNCT
ap-5897	380	14	.	.	PUNCT
ap-5897	380	15	.	.	PUNCT
ap-5897	381	1	,	,	PUNCT
ap-5897	382	1	c2k	c2k	INTJ
ap-5897	382	2	−	−	NOUN
ap-5897	382	3	1	1	X
ap-5897	382	4	)	)	PUNCT
ap-5897	382	5	◦	◦	NOUN
ap-5897	382	6	d∗(b	d∗(b	PROPN
ap-5897	382	7	;	;	PUNCT
ap-5897	382	8	rβ	rβ	X
ap-5897	382	9	)	)	PUNCT
ap-5897	383	1	=	=	SYM
ap-5897	383	2	(	(	PUNCT
ap-5897	383	3	c1	c1	PROPN
ap-5897	383	4	,	,	PUNCT
ap-5897	383	5	.	.	PUNCT
ap-5897	383	6	.	.	PUNCT
ap-5897	383	7	.	.	PUNCT
ap-5897	384	1	,	,	PUNCT
ap-5897	385	1	c2k	c2k	INTJ
ap-5897	385	2	−	−	NOUN
ap-5897	386	1	1	1	X
ap-5897	386	2	)	)	PUNCT
ap-5897	386	3	=	=	SYM
ap-5897	386	4	(	(	PUNCT
ap-5897	386	5	0	0	NUM
ap-5897	386	6	,	,	PUNCT
ap-5897	386	7	a1	a1	NOUN
ap-5897	386	8	,	,	PUNCT
ap-5897	386	9	.	.	PUNCT
ap-5897	386	10	.	.	PUNCT
ap-5897	387	1	.	.	PUNCT
ap-5897	388	1	,	,	PUNCT
ap-5897	388	2	a2k−1	a2k−1	PROPN
ap-5897	388	3	−	−	PROPN
ap-5897	388	4	1	1	NUM
ap-5897	388	5	)	)	PUNCT
ap-5897	388	6	.	.	PUNCT
ap-5897	389	1	this	this	PRON
ap-5897	389	2	is	be	AUX
ap-5897	389	3	equivalent	equivalent	ADJ
ap-5897	389	4	to	to	ADP
ap-5897	389	5	d∗(b	d∗(b	PROPN
ap-5897	389	6	;	;	PUNCT
ap-5897	389	7	rβ	rβ	X
ap-5897	389	8	)	)	PUNCT
ap-5897	390	1	=	=	SYM
ap-5897	390	2			PROPN
ap-5897	390	3	(	(	PUNCT
ap-5897	390	4	0	0	NUM
ap-5897	390	5	,	,	PUNCT
ap-5897	390	6	a1	a1	NOUN
ap-5897	390	7	,	,	PUNCT
ap-5897	390	8	a2	a2	PROPN
ap-5897	390	9	,	,	PUNCT
ap-5897	390	10	.	.	PUNCT
ap-5897	390	11	.	.	PUNCT
ap-5897	390	12	.	.	PUNCT
ap-5897	391	1	,	,	PUNCT
ap-5897	391	2	a2n−1	a2n−1	ADJ
ap-5897	391	3	−	−	NOUN
ap-5897	391	4	1	1	NUM
ap-5897	391	5	)	)	PUNCT
ap-5897	391	6	if	if	SCONJ
ap-5897	391	7	d(b	d(b	X
ap-5897	391	8	;	;	PUNCT
ap-5897	391	9	lβ	lβ	PROPN
ap-5897	391	10	)	)	PUNCT
ap-5897	391	11	=	=	SYM
ap-5897	391	12	(	(	PUNCT
ap-5897	391	13	a1	a1	PROPN
ap-5897	391	14	,	,	PUNCT
ap-5897	391	15	a2	a2	PROPN
ap-5897	391	16	,	,	PUNCT
ap-5897	391	17	.	.	PUNCT
ap-5897	391	18	.	.	PUNCT
ap-5897	392	1	.	.	PUNCT
ap-5897	393	1	,	,	PUNCT
ap-5897	393	2	a2n−1	a2n−1	PROPN
ap-5897	393	3	)	)	PUNCT
ap-5897	393	4	d(b	d(b	PROPN
ap-5897	393	5	;	;	PUNCT
ap-5897	393	6	rβ	rβ	X
ap-5897	393	7	)	)	PUNCT
ap-5897	393	8	otherwise	otherwise	ADV
ap-5897	393	9	.	.	PUNCT
ap-5897	394	1	4	4	X
ap-5897	394	2	.	.	NOUN
ap-5897	394	3	admissible	admissible	ADJ
ap-5897	394	4	sequences	sequence	NOUN
ap-5897	394	5	throughout	throughout	ADP
ap-5897	394	6	this	this	DET
ap-5897	394	7	section	section	NOUN
ap-5897	394	8	,	,	PUNCT
ap-5897	394	9	we	we	PRON
ap-5897	394	10	let	let	VERB
ap-5897	394	11	b	b	NOUN
ap-5897	394	12	=	=	SYM
ap-5897	394	13	(	(	PUNCT
ap-5897	394	14	β1	β1	PROPN
ap-5897	394	15	,	,	PUNCT
ap-5897	394	16	β2	β2	NOUN
ap-5897	394	17	,	,	PUNCT
ap-5897	394	18	.	.	PUNCT
ap-5897	394	19	.	.	PUNCT
ap-5897	394	20	.	.	PUNCT
ap-5897	394	21	)	)	PUNCT
ap-5897	395	1	∈	∈	PROPN
ap-5897	395	2	rn	rn	PROPN
ap-5897	395	3	with	with	ADP
ap-5897	395	4	limm→∞	limm→∞	PROPN
ap-5897	395	5	|b[m]|	|b[m]|	NOUN
ap-5897	396	1	=	=	SYM
ap-5897	396	2	∞.	∞.	PROPN
ap-5897	396	3	a	a	DET
ap-5897	396	4	b	b	NOUN
ap-5897	396	5	-	-	PUNCT
ap-5897	396	6	representation	representation	NOUN
ap-5897	396	7	of	of	ADP
ap-5897	396	8	a	a	DET
ap-5897	396	9	real	real	ADJ
ap-5897	396	10	number	number	NOUN
ap-5897	396	11	x	x	SYM
ap-5897	396	12	∈	∈	PROPN
ap-5897	396	13	[	[	X
ap-5897	396	14	γ	γ	X
ap-5897	396	15	,	,	PUNCT
ap-5897	396	16	γ	γ	X
ap-5897	396	17	+	+	NOUN
ap-5897	396	18	1	1	NUM
ap-5897	396	19	)	)	PUNCT
ap-5897	396	20	is	be	AUX
ap-5897	396	21	an	an	DET
ap-5897	396	22	expansion	expansion	NOUN
ap-5897	396	23	of	of	ADP
ap-5897	396	24	the	the	DET
ap-5897	396	25	form	form	NOUN
ap-5897	396	26	x	x	PUNCT
ap-5897	397	1	=	=	SYM
ap-5897	397	2	∞∑	∞∑	NUM
ap-5897	397	3	i=1	i=1	PROPN
ap-5897	397	4	di	di	NOUN
ap-5897	397	5	b[i	b[i	NOUN
ap-5897	397	6	]	]	PUNCT
ap-5897	397	7	with	with	ADP
ap-5897	397	8	(	(	PUNCT
ap-5897	397	9	d1	d1	PROPN
ap-5897	397	10	,	,	PUNCT
ap-5897	397	11	d2	d2	PROPN
ap-5897	397	12	,	,	PUNCT
ap-5897	397	13	.	.	PUNCT
ap-5897	397	14	.	.	PUNCT
ap-5897	397	15	.	.	PUNCT
ap-5897	397	16	)	)	PUNCT
ap-5897	398	1	∈	∈	PROPN
ap-5897	398	2	a(b	a(b	NOUN
ap-5897	398	3	)	)	PUNCT
ap-5897	398	4	.	.	PUNCT
ap-5897	399	1	note	note	VERB
ap-5897	399	2	that	that	SCONJ
ap-5897	399	3	the	the	DET
ap-5897	399	4	condition	condition	NOUN
ap-5897	399	5	lim	lim	NOUN
ap-5897	399	6	|b[m]|	|b[m]|	PROPN
ap-5897	400	1	=	=	NOUN
ap-5897	400	2	∞	∞	PROPN
ap-5897	400	3	does	do	AUX
ap-5897	400	4	not	not	PART
ap-5897	400	5	guarantee	guarantee	VERB
ap-5897	400	6	that	that	SCONJ
ap-5897	400	7	any	any	DET
ap-5897	400	8	sequence	sequence	NOUN
ap-5897	400	9	(	(	PUNCT
ap-5897	400	10	d1	d1	PROPN
ap-5897	400	11	,	,	PUNCT
ap-5897	400	12	d2	d2	PROPN
ap-5897	400	13	,	,	PUNCT
ap-5897	400	14	.	.	PUNCT
ap-5897	400	15	.	.	PUNCT
ap-5897	400	16	.	.	PUNCT
ap-5897	400	17	)	)	PUNCT
ap-5897	400	18	in	in	ADP
ap-5897	400	19	a(b	a(b	NOUN
ap-5897	400	20	)	)	PUNCT
ap-5897	400	21	is	be	AUX
ap-5897	400	22	a	a	DET
ap-5897	400	23	b	b	NOUN
ap-5897	400	24	-	-	PUNCT
ap-5897	400	25	representation	representation	NOUN
ap-5897	400	26	of	of	ADP
ap-5897	400	27	a	a	DET
ap-5897	400	28	real	real	ADJ
ap-5897	400	29	number	number	NOUN
ap-5897	400	30	x	x	PUNCT
ap-5897	400	31	since	since	SCONJ
ap-5897	400	32	the	the	DET
ap-5897	400	33	series	series	PROPN
ap-5897	400	34	∑∞	∑∞	X
ap-5897	400	35	i=1	i=1	PROPN
ap-5897	400	36	di	di	PROPN
ap-5897	400	37	/	/	SYM
ap-5897	400	38	b[i	b[i	NOUN
ap-5897	400	39	]	]	PUNCT
ap-5897	400	40	may	may	AUX
ap-5897	400	41	not	not	PART
ap-5897	400	42	converge	converge	VERB
ap-5897	400	43	.	.	PUNCT
ap-5897	401	1	if	if	SCONJ
ap-5897	401	2	the	the	DET
ap-5897	401	3	sum	sum	NOUN
ap-5897	401	4	converges	converge	VERB
ap-5897	401	5	,	,	PUNCT
ap-5897	401	6	we	we	PRON
ap-5897	401	7	adopt	adopt	VERB
ap-5897	401	8	the	the	DET
ap-5897	401	9	notation	notation	NOUN
ap-5897	401	10	(	(	PUNCT
ap-5897	401	11	d1	d1	PROPN
ap-5897	401	12	,	,	PUNCT
ap-5897	401	13	d2	d2	PROPN
ap-5897	401	14	,	,	PUNCT
ap-5897	401	15	.	.	PUNCT
ap-5897	401	16	.	.	PUNCT
ap-5897	401	17	.	.	PUNCT
ap-5897	402	1	)	)	PUNCT
ap-5897	403	1	b	b	X
ap-5897	403	2	=	=	PUNCT
ap-5897	403	3	∑∞	∑∞	X
ap-5897	403	4	i=1	i=1	X
ap-5897	403	5	di	di	NOUN
ap-5897	403	6	/	/	SYM
ap-5897	403	7	b[i	b[i	NOUN
ap-5897	403	8	]	]	PUNCT
ap-5897	403	9	.	.	PUNCT
ap-5897	404	1	now	now	ADV
ap-5897	404	2	,	,	PUNCT
ap-5897	404	3	the	the	DET
ap-5897	404	4	b	b	NOUN
ap-5897	404	5	-	-	PUNCT
ap-5897	404	6	expansion	expansion	NOUN
ap-5897	404	7	of	of	ADP
ap-5897	404	8	x	x	SYM
ap-5897	404	9	is	be	AUX
ap-5897	404	10	a	a	DET
ap-5897	404	11	particular	particular	ADJ
ap-5897	404	12	brepresentation	brepresentation	NOUN
ap-5897	404	13	of	of	ADP
ap-5897	404	14	x.	x.	NOUN
ap-5897	404	15	deciding	decide	VERB
ap-5897	404	16	whether	whether	SCONJ
ap-5897	404	17	a	a	DET
ap-5897	404	18	sequence	sequence	NOUN
ap-5897	404	19	(	(	PUNCT
ap-5897	404	20	d1	d1	PROPN
ap-5897	404	21	,	,	PUNCT
ap-5897	404	22	d2	d2	PROPN
ap-5897	404	23	,	,	PUNCT
ap-5897	404	24	.	.	PUNCT
ap-5897	404	25	.	.	PUNCT
ap-5897	404	26	.	.	PUNCT
ap-5897	404	27	)	)	PUNCT
ap-5897	405	1	in	in	ADP
ap-5897	405	2	a(b	a(b	NOUN
ap-5897	405	3	)	)	PUNCT
ap-5897	405	4	is	be	AUX
ap-5897	405	5	the	the	DET
ap-5897	405	6	b	b	NOUN
ap-5897	405	7	-	-	PUNCT
ap-5897	405	8	expansion	expansion	NOUN
ap-5897	405	9	of	of	ADP
ap-5897	405	10	an	an	DET
ap-5897	405	11	element	element	NOUN
ap-5897	405	12	of	of	ADP
ap-5897	405	13	[	[	X
ap-5897	405	14	γ	γ	X
ap-5897	405	15	,	,	PUNCT
ap-5897	405	16	γ	γ	X
ap-5897	405	17	+	+	NOUN
ap-5897	405	18	1	1	NUM
ap-5897	405	19	)	)	PUNCT
ap-5897	405	20	,	,	PUNCT
ap-5897	405	21	thus	thus	ADV
ap-5897	405	22	,	,	PUNCT
ap-5897	405	23	entails	entail	VERB
ap-5897	405	24	showing	show	VERB
ap-5897	405	25	that	that	SCONJ
ap-5897	405	26	the	the	DET
ap-5897	405	27	series	series	NOUN
ap-5897	405	28	converges	converge	VERB
ap-5897	405	29	.	.	PUNCT
ap-5897	406	1	definition	definition	NOUN
ap-5897	406	2	2	2	NUM
ap-5897	406	3	.	.	PUNCT
ap-5897	407	1	an	an	DET
ap-5897	407	2	integer	integer	NOUN
ap-5897	407	3	sequence	sequence	NOUN
ap-5897	407	4	(	(	PUNCT
ap-5897	407	5	d1	d1	PROPN
ap-5897	407	6	,	,	PUNCT
ap-5897	407	7	d2	d2	PROPN
ap-5897	407	8	,	,	PUNCT
ap-5897	407	9	.	.	PUNCT
ap-5897	407	10	.	.	PUNCT
ap-5897	407	11	.	.	PUNCT
ap-5897	407	12	)	)	PUNCT
ap-5897	408	1	∈	∈	PROPN
ap-5897	408	2	a(b	a(b	NOUN
ap-5897	408	3	)	)	PUNCT
ap-5897	408	4	is	be	AUX
ap-5897	408	5	b	b	NOUN
ap-5897	408	6	-	-	PUNCT
ap-5897	408	7	admissible	admissible	ADJ
ap-5897	408	8	if	if	SCONJ
ap-5897	408	9	there	there	PRON
ap-5897	408	10	is	be	VERB
ap-5897	408	11	an	an	DET
ap-5897	408	12	x	x	SYM
ap-5897	408	13	∈	∈	PROPN
ap-5897	408	14	[	[	X
ap-5897	408	15	γ	γ	X
ap-5897	408	16	,	,	PUNCT
ap-5897	408	17	γ	γ	X
ap-5897	408	18	+	+	NOUN
ap-5897	408	19	1	1	NUM
ap-5897	408	20	)	)	PUNCT
ap-5897	408	21	such	such	ADJ
ap-5897	408	22	that	that	DET
ap-5897	408	23	d(b;x	d(b;x	NOUN
ap-5897	408	24	)	)	PUNCT
ap-5897	409	1	=	=	SYM
ap-5897	409	2	(	(	PUNCT
ap-5897	409	3	d1	d1	PROPN
ap-5897	409	4	,	,	PUNCT
ap-5897	409	5	d2	d2	PROPN
ap-5897	409	6	,	,	PUNCT
ap-5897	409	7	.	.	PUNCT
ap-5897	409	8	.	.	PUNCT
ap-5897	409	9	.	.	PUNCT
ap-5897	409	10	)	)	PUNCT
ap-5897	409	11	.	.	PUNCT
ap-5897	410	1	219	219	NUM
ap-5897	410	2	jonathan	jonathan	PROPN
ap-5897	410	3	caalim	caalim	PROPN
ap-5897	410	4	,	,	PUNCT
ap-5897	410	5	shiela	shiela	PROPN
ap-5897	410	6	demegillo	demegillo	PROPN
ap-5897	410	7	acta	acta	PROPN
ap-5897	410	8	polytechnica	polytechnica	PROPN
ap-5897	410	9	the	the	DET
ap-5897	410	10	admissibility	admissibility	NOUN
ap-5897	410	11	of	of	ADP
ap-5897	410	12	sequences	sequence	NOUN
ap-5897	410	13	with	with	ADP
ap-5897	410	14	respect	respect	NOUN
ap-5897	410	15	to	to	ADP
ap-5897	410	16	the	the	DET
ap-5897	410	17	b	b	NOUN
ap-5897	410	18	-	-	PUNCT
ap-5897	410	19	expansion	expansion	NOUN
ap-5897	410	20	map	map	NOUN
ap-5897	410	21	is	be	AUX
ap-5897	410	22	related	relate	VERB
ap-5897	410	23	to	to	ADP
ap-5897	410	24	the	the	DET
ap-5897	410	25	admissibility	admissibility	NOUN
ap-5897	410	26	of	of	ADP
ap-5897	410	27	sequences	sequence	NOUN
ap-5897	410	28	for	for	ADP
ap-5897	410	29	a	a	DET
ap-5897	410	30	special	special	ADJ
ap-5897	410	31	class	class	NOUN
ap-5897	410	32	of	of	ADP
ap-5897	410	33	rotational	rotational	ADJ
ap-5897	410	34	beta	beta	NOUN
ap-5897	410	35	expansion	expansion	NOUN
ap-5897	410	36	map	map	NOUN
ap-5897	410	37	.	.	PUNCT
ap-5897	411	1	let	let	VERB
ap-5897	411	2	z	z	NOUN
ap-5897	411	3	=	=	PUNCT
ap-5897	412	1	[	[	X
ap-5897	412	2	0	0	NUM
ap-5897	412	3	,	,	PUNCT
ap-5897	412	4	1	1	NUM
ap-5897	412	5	)	)	PUNCT
ap-5897	412	6	×	×	NOUN
ap-5897	413	1	[	[	X
ap-5897	413	2	0	0	NUM
ap-5897	413	3	,	,	PUNCT
ap-5897	413	4	1	1	NUM
ap-5897	413	5	)	)	PUNCT
ap-5897	413	6	and	and	CCONJ
ap-5897	413	7	1	1	NUM
ap-5897	413	8	<	<	X
ap-5897	413	9	β	β	X
ap-5897	413	10	∈	∈	PROPN
ap-5897	413	11	r.	r.	PROPN
ap-5897	413	12	define	define	VERB
ap-5897	413	13	the	the	DET
ap-5897	413	14	map	map	NOUN
ap-5897	413	15	t	t	NOUN
ap-5897	413	16	:	:	PUNCT
ap-5897	413	17	z	z	NOUN
ap-5897	414	1	−→	−→	NOUN
ap-5897	414	2	z	z	PROPN
ap-5897	414	3	by	by	ADP
ap-5897	414	4	t	t	PROPN
ap-5897	414	5	(	(	PUNCT
ap-5897	414	6	(	(	PUNCT
ap-5897	414	7	x	x	NOUN
ap-5897	414	8	,	,	PUNCT
ap-5897	414	9	y	y	NOUN
ap-5897	414	10	)	)	PUNCT
ap-5897	414	11	)	)	PUNCT
ap-5897	415	1	=	=	SYM
ap-5897	415	2	(	(	PUNCT
ap-5897	415	3	−βy	−βy	X
ap-5897	415	4	−	−	NOUN
ap-5897	415	5	b−βyc	b−βyc	NOUN
ap-5897	415	6	,	,	PUNCT
ap-5897	415	7	βx−	βx−	PUNCT
ap-5897	415	8	bβxc	bβxc	NOUN
ap-5897	415	9	)	)	PUNCT
ap-5897	415	10	.	.	PUNCT
ap-5897	416	1	let	let	VERB
ap-5897	416	2	t	t	NOUN
ap-5897	416	3	be	be	AUX
ap-5897	416	4	the	the	DET
ap-5897	416	5	b	b	NOUN
ap-5897	416	6	-	-	PUNCT
ap-5897	416	7	expansion	expansion	NOUN
ap-5897	416	8	map	map	NOUN
ap-5897	416	9	on	on	ADP
ap-5897	416	10	[	[	X
ap-5897	416	11	0	0	NUM
ap-5897	416	12	,	,	PUNCT
ap-5897	416	13	1	1	NUM
ap-5897	416	14	)	)	PUNCT
ap-5897	416	15	with	with	ADP
ap-5897	416	16	b	b	NOUN
ap-5897	416	17	=	=	SYM
ap-5897	416	18	(	(	PUNCT
ap-5897	416	19	−β	−β	PROPN
ap-5897	416	20	,	,	PUNCT
ap-5897	416	21	β	β	NOUN
ap-5897	416	22	)	)	PUNCT
ap-5897	416	23	.	.	PUNCT
ap-5897	417	1	it	it	PRON
ap-5897	417	2	follows	follow	VERB
ap-5897	417	3	that	that	SCONJ
ap-5897	417	4	for	for	ADP
ap-5897	417	5	all	all	DET
ap-5897	417	6	n	n	DET
ap-5897	417	7	∈	∈	PROPN
ap-5897	417	8	n	n	CCONJ
ap-5897	417	9	,	,	PUNCT
ap-5897	417	10	we	we	PRON
ap-5897	417	11	have	have	VERB
ap-5897	417	12	t	t	PROPN
ap-5897	417	13	2n−1(x	2n−1(x	NOUN
ap-5897	417	14	,	,	PUNCT
ap-5897	417	15	y	y	NOUN
ap-5897	417	16	)	)	PUNCT
ap-5897	417	17	=	=	PUNCT
ap-5897	418	1	(	(	PUNCT
ap-5897	418	2	t	t	PROPN
ap-5897	418	3	2n−1	2n−1	NUM
ap-5897	418	4	b	b	PROPN
ap-5897	418	5	(	(	PUNCT
ap-5897	418	6	y	y	PROPN
ap-5897	418	7	)	)	PUNCT
ap-5897	418	8	,	,	PUNCT
ap-5897	418	9	t	t	PROPN
ap-5897	418	10	2n−1	2n−1	NUM
ap-5897	418	11	σ(b	σ(b	PROPN
ap-5897	418	12	)	)	PUNCT
ap-5897	418	13	(	(	PUNCT
ap-5897	418	14	x	x	NOUN
ap-5897	418	15	)	)	PUNCT
ap-5897	418	16	)	)	PUNCT
ap-5897	418	17	and	and	CCONJ
ap-5897	418	18	t	t	PROPN
ap-5897	418	19	2n(x	2n(x	NUM
ap-5897	418	20	,	,	PUNCT
ap-5897	418	21	y	y	NOUN
ap-5897	418	22	)	)	PUNCT
ap-5897	418	23	=	=	PUNCT
ap-5897	418	24	(	(	PUNCT
ap-5897	418	25	t	t	PROPN
ap-5897	418	26	2n	2n	NUM
ap-5897	418	27	σ(b)(x	σ(b)(x	NOUN
ap-5897	418	28	)	)	PUNCT
ap-5897	418	29	,	,	PUNCT
ap-5897	418	30	t	t	PROPN
ap-5897	418	31	2n	2n	NUM
ap-5897	418	32	b	b	PROPN
ap-5897	418	33	(	(	PUNCT
ap-5897	418	34	y	y	NOUN
ap-5897	418	35	)	)	PUNCT
ap-5897	418	36	)	)	PUNCT
ap-5897	418	37	.	.	PUNCT
ap-5897	419	1	so	so	ADV
ap-5897	419	2	,	,	PUNCT
ap-5897	419	3	if	if	SCONJ
ap-5897	419	4	d(b	d(b	X
ap-5897	419	5	;	;	PUNCT
ap-5897	419	6	y	y	X
ap-5897	419	7	)	)	PUNCT
ap-5897	419	8	=	=	SYM
ap-5897	419	9	(	(	PUNCT
ap-5897	419	10	a1	a1	PROPN
ap-5897	419	11	,	,	PUNCT
ap-5897	419	12	a2	a2	PROPN
ap-5897	419	13	,	,	PUNCT
ap-5897	419	14	.	.	PUNCT
ap-5897	419	15	.	.	PUNCT
ap-5897	419	16	.	.	PUNCT
ap-5897	419	17	)	)	PUNCT
ap-5897	420	1	and	and	CCONJ
ap-5897	420	2	d(σ(b);x	d(σ(b);x	PROPN
ap-5897	420	3	)	)	PUNCT
ap-5897	421	1	=	=	PUNCT
ap-5897	421	2	(	(	PUNCT
ap-5897	421	3	b1	b1	NOUN
ap-5897	421	4	,	,	PUNCT
ap-5897	421	5	b2	b2	NOUN
ap-5897	421	6	,	,	PUNCT
ap-5897	421	7	.	.	PUNCT
ap-5897	421	8	.	.	PUNCT
ap-5897	421	9	.	.	PUNCT
ap-5897	421	10	)	)	PUNCT
ap-5897	422	1	,	,	PUNCT
ap-5897	422	2	then	then	ADV
ap-5897	422	3	the	the	DET
ap-5897	422	4	expansion	expansion	NOUN
ap-5897	422	5	of	of	ADP
ap-5897	422	6	(	(	PUNCT
ap-5897	422	7	x	x	NOUN
ap-5897	422	8	,	,	PUNCT
ap-5897	422	9	y	y	PROPN
ap-5897	422	10	)	)	PUNCT
ap-5897	422	11	with	with	ADP
ap-5897	422	12	respect	respect	NOUN
ap-5897	422	13	to	to	ADP
ap-5897	422	14	t	t	PROPN
ap-5897	422	15	is	be	AUX
ap-5897	422	16	(	(	PUNCT
ap-5897	422	17	(	(	PUNCT
ap-5897	422	18	a1	a1	NOUN
ap-5897	422	19	,	,	PUNCT
ap-5897	422	20	b1	b1	NOUN
ap-5897	422	21	)	)	PUNCT
ap-5897	422	22	,	,	PUNCT
ap-5897	422	23	(	(	PUNCT
ap-5897	422	24	b2	b2	NOUN
ap-5897	422	25	,	,	PUNCT
ap-5897	422	26	a2	a2	PROPN
ap-5897	422	27	)	)	PUNCT
ap-5897	422	28	,	,	PUNCT
ap-5897	422	29	(	(	PUNCT
ap-5897	422	30	a3	a3	NOUN
ap-5897	422	31	,	,	PUNCT
ap-5897	422	32	b3	b3	PROPN
ap-5897	422	33	)	)	PUNCT
ap-5897	422	34	,	,	PUNCT
ap-5897	422	35	.	.	PUNCT
ap-5897	422	36	.	.	PUNCT
ap-5897	422	37	.	.	PUNCT
ap-5897	422	38	)	)	PUNCT
ap-5897	422	39	.	.	PUNCT
ap-5897	423	1	proposition	proposition	NOUN
ap-5897	423	2	4.1	4.1	NUM
ap-5897	423	3	.	.	PUNCT
ap-5897	424	1	let	let	VERB
ap-5897	424	2	b	b	NOUN
ap-5897	424	3	=	=	SYM
ap-5897	424	4	(	(	PUNCT
ap-5897	424	5	−β	−β	PROPN
ap-5897	424	6	,	,	PUNCT
ap-5897	424	7	β	β	NOUN
ap-5897	424	8	)	)	PUNCT
ap-5897	424	9	with	with	ADP
ap-5897	424	10	β	β	X
ap-5897	424	11	>	>	X
ap-5897	424	12	1	1	NUM
ap-5897	424	13	.	.	PUNCT
ap-5897	425	1	then	then	ADV
ap-5897	425	2	(	(	PUNCT
ap-5897	425	3	a1	a1	PROPN
ap-5897	425	4	,	,	PUNCT
ap-5897	425	5	a2	a2	PROPN
ap-5897	425	6	,	,	PUNCT
ap-5897	425	7	.	.	PUNCT
ap-5897	425	8	.	.	PUNCT
ap-5897	425	9	.	.	PUNCT
ap-5897	425	10	)	)	PUNCT
ap-5897	426	1	∈	∈	PROPN
ap-5897	426	2	a(b	a(b	NOUN
ap-5897	426	3	)	)	PUNCT
ap-5897	426	4	is	be	AUX
ap-5897	426	5	b	b	NOUN
ap-5897	426	6	-	-	PUNCT
ap-5897	426	7	admissible	admissible	ADJ
ap-5897	426	8	and	and	CCONJ
ap-5897	426	9	(	(	PUNCT
ap-5897	426	10	b1	b1	NOUN
ap-5897	426	11	,	,	PUNCT
ap-5897	426	12	b2	b2	NOUN
ap-5897	426	13	,	,	PUNCT
ap-5897	426	14	.	.	PUNCT
ap-5897	426	15	.	.	PUNCT
ap-5897	426	16	.	.	PUNCT
ap-5897	426	17	)	)	PUNCT
ap-5897	427	1	∈	∈	PROPN
ap-5897	427	2	a(σ(b	a(σ(b	PROPN
ap-5897	427	3	)	)	PUNCT
ap-5897	427	4	)	)	PUNCT
ap-5897	427	5	is	be	AUX
ap-5897	427	6	σ(b)-admissible	σ(b)-admissible	ADJ
ap-5897	427	7	if	if	SCONJ
ap-5897	427	8	and	and	CCONJ
ap-5897	427	9	only	only	ADV
ap-5897	427	10	if	if	SCONJ
ap-5897	427	11	(	(	PUNCT
ap-5897	427	12	(	(	PUNCT
ap-5897	427	13	a1	a1	NOUN
ap-5897	427	14	,	,	PUNCT
ap-5897	427	15	b1	b1	NOUN
ap-5897	427	16	)	)	PUNCT
ap-5897	427	17	,	,	PUNCT
ap-5897	427	18	(	(	PUNCT
ap-5897	427	19	b2	b2	NOUN
ap-5897	427	20	,	,	PUNCT
ap-5897	427	21	a2	a2	PROPN
ap-5897	427	22	)	)	PUNCT
ap-5897	427	23	,	,	PUNCT
ap-5897	427	24	(	(	PUNCT
ap-5897	427	25	a3	a3	NOUN
ap-5897	427	26	,	,	PUNCT
ap-5897	427	27	b3	b3	PROPN
ap-5897	427	28	)	)	PUNCT
ap-5897	427	29	,	,	PUNCT
ap-5897	427	30	.	.	PUNCT
ap-5897	427	31	.	.	PUNCT
ap-5897	427	32	.	.	PUNCT
ap-5897	427	33	)	)	PUNCT
ap-5897	428	1	is	be	AUX
ap-5897	428	2	admissible	admissible	ADJ
ap-5897	428	3	with	with	ADP
ap-5897	428	4	respect	respect	NOUN
ap-5897	428	5	to	to	ADP
ap-5897	428	6	t	t	PROPN
ap-5897	428	7	.	.	PUNCT
ap-5897	429	1	in	in	ADP
ap-5897	429	2	this	this	DET
ap-5897	429	3	section	section	NOUN
ap-5897	429	4	,	,	PUNCT
ap-5897	429	5	our	our	PRON
ap-5897	429	6	goal	goal	NOUN
ap-5897	429	7	is	be	AUX
ap-5897	429	8	to	to	PART
ap-5897	429	9	provide	provide	VERB
ap-5897	429	10	an	an	DET
ap-5897	429	11	admissibility	admissibility	NOUN
ap-5897	429	12	criterion	criterion	NOUN
ap-5897	429	13	for	for	ADP
ap-5897	429	14	sequences	sequence	NOUN
ap-5897	429	15	in	in	ADP
ap-5897	429	16	a(b	a(b	NOUN
ap-5897	429	17	)	)	PUNCT
ap-5897	429	18	.	.	PUNCT
ap-5897	430	1	we	we	PRON
ap-5897	430	2	first	first	ADV
ap-5897	430	3	mention	mention	VERB
ap-5897	430	4	few	few	ADJ
ap-5897	430	5	results	result	NOUN
ap-5897	430	6	.	.	PUNCT
ap-5897	431	1	lemma	lemma	PROPN
ap-5897	431	2	4.2	4.2	NUM
ap-5897	431	3	.	.	PUNCT
ap-5897	432	1	let	let	VERB
ap-5897	432	2	x	x	X
ap-5897	432	3	∈	∈	PROPN
ap-5897	432	4	[	[	X
ap-5897	432	5	γ	γ	X
ap-5897	432	6	,	,	PUNCT
ap-5897	432	7	γ	γ	X
ap-5897	432	8	+	+	NOUN
ap-5897	432	9	1	1	NUM
ap-5897	432	10	)	)	PUNCT
ap-5897	432	11	such	such	ADJ
ap-5897	432	12	that	that	DET
ap-5897	432	13	d(b;x	d(b;x	NOUN
ap-5897	432	14	)	)	PUNCT
ap-5897	432	15	=	=	SYM
ap-5897	432	16	(	(	PUNCT
ap-5897	432	17	a1	a1	PROPN
ap-5897	432	18	,	,	PUNCT
ap-5897	432	19	a2	a2	PROPN
ap-5897	432	20	,	,	PUNCT
ap-5897	432	21	.	.	PUNCT
ap-5897	432	22	.	.	PUNCT
ap-5897	432	23	.	.	PUNCT
ap-5897	432	24	)	)	PUNCT
ap-5897	432	25	.	.	PUNCT
ap-5897	433	1	for	for	ADP
ap-5897	433	2	n	n	PRON
ap-5897	433	3	∈	∈	PROPN
ap-5897	433	4	n	n	CCONJ
ap-5897	433	5	,	,	PUNCT
ap-5897	433	6	tn(x	tn(x	PUNCT
ap-5897	433	7	)	)	PUNCT
ap-5897	433	8	=	=	SYM
ap-5897	433	9	b[n]x−	b[n]x−	NUM
ap-5897	433	10	n∑	n∑	PROPN
ap-5897	433	11	i=1	i=1	PROPN
ap-5897	434	1	aib[n	aib[n	X
ap-5897	434	2	]	]	PUNCT
ap-5897	434	3	b[i	b[i	ADV
ap-5897	434	4	]	]	PUNCT
ap-5897	434	5	.	.	PUNCT
ap-5897	435	1	proof	proof	NOUN
ap-5897	435	2	.	.	PUNCT
ap-5897	436	1	we	we	PRON
ap-5897	436	2	prove	prove	VERB
ap-5897	436	3	this	this	DET
ap-5897	436	4	lemma	lemma	PROPN
ap-5897	436	5	by	by	ADP
ap-5897	436	6	induction	induction	NOUN
ap-5897	436	7	.	.	PUNCT
ap-5897	437	1	let	let	VERB
ap-5897	437	2	x	x	X
ap-5897	437	3	∈	∈	PROPN
ap-5897	437	4	[	[	X
ap-5897	437	5	γ	γ	X
ap-5897	437	6	,	,	PUNCT
ap-5897	437	7	γ	γ	X
ap-5897	437	8	+	+	NOUN
ap-5897	437	9	1	1	NUM
ap-5897	437	10	)	)	PUNCT
ap-5897	437	11	.	.	PUNCT
ap-5897	438	1	then	then	ADV
ap-5897	438	2	t	t	PROPN
ap-5897	438	3	(	(	PUNCT
ap-5897	438	4	x	x	NOUN
ap-5897	438	5	)	)	PUNCT
ap-5897	438	6	=	=	SYM
ap-5897	438	7	b[1]x−	b[1]x−	PROPN
ap-5897	438	8	a1	a1	NOUN
ap-5897	438	9	.	.	PUNCT
ap-5897	438	10	suppose	suppose	VERB
ap-5897	438	11	that	that	SCONJ
ap-5897	438	12	for	for	ADP
ap-5897	438	13	some	some	DET
ap-5897	438	14	k	k	PROPN
ap-5897	438	15	∈	∈	PROPN
ap-5897	438	16	n	n	CCONJ
ap-5897	438	17	,	,	PUNCT
ap-5897	438	18	t	t	PROPN
ap-5897	438	19	k(x	k(x	PROPN
ap-5897	438	20	)	)	PUNCT
ap-5897	438	21	=	=	SYM
ap-5897	438	22	b[k]x−	b[k]x−	PROPN
ap-5897	438	23	k∑	k∑	VERB
ap-5897	438	24	i=1	i=1	VERB
ap-5897	438	25	aib[k]/b[i	aib[k]/b[i	PROPN
ap-5897	438	26	]	]	PUNCT
ap-5897	438	27	.	.	PUNCT
ap-5897	439	1	thus	thus	ADV
ap-5897	439	2	,	,	PUNCT
ap-5897	439	3	t	t	PROPN
ap-5897	439	4	k+1(x	k+1(x	PROPN
ap-5897	439	5	)	)	PUNCT
ap-5897	439	6	=	=	PUNCT
ap-5897	439	7	βk+1	βk+1	PROPN
ap-5897	439	8	t	t	PROPN
ap-5897	439	9	k(x)−	k(x)−	NOUN
ap-5897	439	10	ak+1	ak+1	VERB
ap-5897	439	11	=	=	SYM
ap-5897	439	12	b[k	b[k	NOUN
ap-5897	439	13	+	+	CCONJ
ap-5897	439	14	1]x−	1]x−	NUM
ap-5897	439	15	k∑	k∑	NOUN
ap-5897	439	16	i=1	i=1	PROPN
ap-5897	439	17	aib[k	aib[k	PROPN
ap-5897	439	18	+	+	CCONJ
ap-5897	440	1	1	1	NUM
ap-5897	440	2	]	]	PUNCT
ap-5897	440	3	b[i	b[i	NOUN
ap-5897	440	4	]	]	PUNCT
ap-5897	440	5	−ak+1	−ak+1	PROPN
ap-5897	440	6	b[k	b[k	NOUN
ap-5897	440	7	+	+	CCONJ
ap-5897	440	8	1	1	NUM
ap-5897	440	9	]	]	PUNCT
ap-5897	440	10	b[k	b[k	NOUN
ap-5897	440	11	+	+	CCONJ
ap-5897	440	12	1	1	NUM
ap-5897	440	13	]	]	X
ap-5897	440	14	=	=	PUNCT
ap-5897	440	15	b[k	b[k	NOUN
ap-5897	440	16	+	+	CCONJ
ap-5897	440	17	1]x−	1]x−	NUM
ap-5897	440	18	k+1∑	k+1∑	PROPN
ap-5897	440	19	i=1	i=1	PROPN
ap-5897	441	1	aib[k	aib[k	PROPN
ap-5897	441	2	+	+	CCONJ
ap-5897	441	3	1	1	NUM
ap-5897	441	4	]	]	PUNCT
ap-5897	441	5	b[i	b[i	NOUN
ap-5897	441	6	]	]	PUNCT
ap-5897	441	7	.	.	PUNCT
ap-5897	442	1	in	in	ADP
ap-5897	442	2	the	the	DET
ap-5897	442	3	following	follow	VERB
ap-5897	442	4	lemma	lemma	PROPN
ap-5897	442	5	,	,	PUNCT
ap-5897	442	6	we	we	PRON
ap-5897	442	7	give	give	VERB
ap-5897	442	8	certain	certain	ADJ
ap-5897	442	9	conditions	condition	NOUN
ap-5897	442	10	for	for	ADP
ap-5897	442	11	a	a	DET
ap-5897	442	12	b	b	NOUN
ap-5897	442	13	-	-	PUNCT
ap-5897	442	14	representation	representation	NOUN
ap-5897	442	15	(	(	PUNCT
ap-5897	442	16	d1	d1	PROPN
ap-5897	442	17	,	,	PUNCT
ap-5897	442	18	d2	d2	PROPN
ap-5897	442	19	,	,	PUNCT
ap-5897	442	20	.	.	PUNCT
ap-5897	442	21	.	.	PUNCT
ap-5897	442	22	.	.	PUNCT
ap-5897	442	23	)	)	PUNCT
ap-5897	443	1	to	to	PART
ap-5897	443	2	be	be	AUX
ap-5897	443	3	a	a	DET
ap-5897	443	4	b	b	NOUN
ap-5897	443	5	-	-	PUNCT
ap-5897	443	6	expansion	expansion	NOUN
ap-5897	443	7	.	.	PUNCT
ap-5897	444	1	note	note	VERB
ap-5897	444	2	that	that	SCONJ
ap-5897	444	3	the	the	DET
ap-5897	444	4	convergence	convergence	NOUN
ap-5897	444	5	of	of	ADP
ap-5897	444	6	the	the	DET
ap-5897	444	7	sum	sum	NOUN
ap-5897	444	8	(	(	PUNCT
ap-5897	444	9	d1	d1	PROPN
ap-5897	444	10	,	,	PUNCT
ap-5897	444	11	d2	d2	PROPN
ap-5897	444	12	,	,	PUNCT
ap-5897	444	13	.	.	PUNCT
ap-5897	444	14	.	.	PUNCT
ap-5897	444	15	.	.	PUNCT
ap-5897	445	1	)	)	PUNCT
ap-5897	446	1	b	b	NOUN
ap-5897	446	2	implies	imply	VERB
ap-5897	446	3	the	the	DET
ap-5897	446	4	convergence	convergence	NOUN
ap-5897	446	5	of	of	ADP
ap-5897	446	6	(	(	PUNCT
ap-5897	446	7	dk+1	dk+1	PROPN
ap-5897	446	8	,	,	PUNCT
ap-5897	446	9	dk+2	dk+2	NOUN
ap-5897	446	10	,	,	PUNCT
ap-5897	446	11	.	.	PUNCT
ap-5897	446	12	.	.	PUNCT
ap-5897	446	13	.	.	PUNCT
ap-5897	446	14	)	)	PUNCT
ap-5897	447	1	σk(b	σk(b	X
ap-5897	447	2	)	)	PUNCT
ap-5897	448	1	for	for	ADP
ap-5897	448	2	all	all	DET
ap-5897	448	3	k	k	PROPN
ap-5897	448	4	∈	∈	PROPN
ap-5897	448	5	n	n	PART
ap-5897	448	6	∪	∪	X
ap-5897	448	7	{	{	PUNCT
ap-5897	448	8	0	0	NUM
ap-5897	448	9	}	}	PUNCT
ap-5897	448	10	.	.	PUNCT
ap-5897	449	1	lemma	lemma	PROPN
ap-5897	449	2	4.3	4.3	NUM
ap-5897	449	3	.	.	PUNCT
ap-5897	450	1	let	let	VERB
ap-5897	450	2	(	(	PUNCT
ap-5897	450	3	d1	d1	NOUN
ap-5897	450	4	,	,	PUNCT
ap-5897	450	5	d2	d2	PROPN
ap-5897	450	6	,	,	PUNCT
ap-5897	450	7	.	.	PUNCT
ap-5897	450	8	.	.	PUNCT
ap-5897	450	9	.	.	PUNCT
ap-5897	450	10	)	)	PUNCT
ap-5897	451	1	be	be	AUX
ap-5897	451	2	a	a	DET
ap-5897	451	3	b	b	NOUN
ap-5897	451	4	-	-	PUNCT
ap-5897	451	5	representation	representation	NOUN
ap-5897	451	6	of	of	ADP
ap-5897	451	7	x	x	X
ap-5897	451	8	∈	∈	PROPN
ap-5897	451	9	[	[	X
ap-5897	451	10	γ	γ	X
ap-5897	451	11	,	,	PUNCT
ap-5897	451	12	γ	γ	X
ap-5897	451	13	+	+	NOUN
ap-5897	451	14	1	1	NUM
ap-5897	451	15	)	)	PUNCT
ap-5897	451	16	.	.	PUNCT
ap-5897	452	1	if	if	SCONJ
ap-5897	452	2	(	(	PUNCT
ap-5897	452	3	dk+1	dk+1	NOUN
ap-5897	452	4	,	,	PUNCT
ap-5897	452	5	dk+2	dk+2	NOUN
ap-5897	452	6	,	,	PUNCT
ap-5897	452	7	.	.	PUNCT
ap-5897	452	8	.	.	PUNCT
ap-5897	452	9	.	.	PUNCT
ap-5897	452	10	)	)	PUNCT
ap-5897	453	1	σk(b	σk(b	X
ap-5897	453	2	)	)	PUNCT
ap-5897	453	3	∈	∈	NOUN
ap-5897	454	1	[	[	X
ap-5897	454	2	γ	γ	X
ap-5897	454	3	,	,	PUNCT
ap-5897	454	4	γ	γ	X
ap-5897	454	5	+	+	NOUN
ap-5897	454	6	1	1	NUM
ap-5897	454	7	)	)	PUNCT
ap-5897	454	8	for	for	ADP
ap-5897	454	9	all	all	DET
ap-5897	454	10	k	k	PROPN
ap-5897	454	11	∈	∈	PROPN
ap-5897	454	12	n	n	PART
ap-5897	454	13	∪	∪	X
ap-5897	454	14	{	{	PUNCT
ap-5897	454	15	0	0	NUM
ap-5897	454	16	}	}	PUNCT
ap-5897	454	17	,	,	PUNCT
ap-5897	454	18	then	then	ADV
ap-5897	454	19	d(b;x	d(b;x	PROPN
ap-5897	454	20	)	)	PUNCT
ap-5897	455	1	=	=	SYM
ap-5897	455	2	(	(	PUNCT
ap-5897	455	3	d1	d1	PROPN
ap-5897	455	4	,	,	PUNCT
ap-5897	455	5	d2	d2	PROPN
ap-5897	455	6	,	,	PUNCT
ap-5897	455	7	.	.	PUNCT
ap-5897	455	8	.	.	PUNCT
ap-5897	455	9	.	.	PUNCT
ap-5897	455	10	)	)	PUNCT
ap-5897	455	11	.	.	PUNCT
ap-5897	456	1	proof	proof	NOUN
ap-5897	456	2	.	.	PUNCT
ap-5897	457	1	by	by	ADP
ap-5897	457	2	induction	induction	NOUN
ap-5897	457	3	on	on	ADP
ap-5897	457	4	n	n	PRON
ap-5897	457	5	∈	∈	PROPN
ap-5897	457	6	n	n	CCONJ
ap-5897	457	7	,	,	PUNCT
ap-5897	457	8	we	we	PRON
ap-5897	457	9	prove	prove	VERB
ap-5897	457	10	that	that	SCONJ
ap-5897	457	11	dn	dn	NOUN
ap-5897	457	12	=	=	VERB
ap-5897	457	13	⌊	⌊	PROPN
ap-5897	457	14	βnt	βnt	ADV
ap-5897	457	15	n−1(x)−	n−1(x)−	PROPN
ap-5897	457	16	γ	γ	PROPN
ap-5897	457	17	⌋	⌋	PROPN
ap-5897	457	18	and	and	CCONJ
ap-5897	457	19	tn(x	tn(x	PUNCT
ap-5897	457	20	)	)	PUNCT
ap-5897	457	21	=	=	SYM
ap-5897	457	22	(	(	PUNCT
ap-5897	457	23	dn+1	dn+1	PROPN
ap-5897	457	24	,	,	PUNCT
ap-5897	457	25	dn+2	dn+2	ADV
ap-5897	457	26	,	,	PUNCT
ap-5897	457	27	.	.	PUNCT
ap-5897	457	28	.	.	PUNCT
ap-5897	457	29	.	.	PUNCT
ap-5897	458	1	)	)	PUNCT
ap-5897	459	1	σn(b	σn(b	NUM
ap-5897	459	2	)	)	PUNCT
ap-5897	459	3	.	.	PUNCT
ap-5897	460	1	note	note	VERB
ap-5897	460	2	that	that	SCONJ
ap-5897	460	3	β1x	β1x	NOUN
ap-5897	460	4	−	−	NOUN
ap-5897	460	5	d1	d1	NOUN
ap-5897	460	6	=	=	SYM
ap-5897	460	7	(	(	PUNCT
ap-5897	460	8	d2	d2	PROPN
ap-5897	460	9	,	,	PUNCT
ap-5897	460	10	d3	d3	PROPN
ap-5897	460	11	,	,	PUNCT
ap-5897	460	12	.	.	PUNCT
ap-5897	460	13	.	.	PUNCT
ap-5897	460	14	.	.	PUNCT
ap-5897	460	15	)	)	PUNCT
ap-5897	461	1	σ(b	σ(b	PROPN
ap-5897	461	2	)	)	PUNCT
ap-5897	461	3	∈	∈	PROPN
ap-5897	462	1	[	[	X
ap-5897	462	2	γ	γ	X
ap-5897	462	3	,	,	PUNCT
ap-5897	462	4	γ	γ	X
ap-5897	462	5	+	+	NOUN
ap-5897	462	6	1	1	NUM
ap-5897	462	7	)	)	PUNCT
ap-5897	462	8	.	.	PUNCT
ap-5897	463	1	hence	hence	ADV
ap-5897	463	2	,	,	PUNCT
ap-5897	463	3	d1	d1	PROPN
ap-5897	463	4	=	=	PUNCT
ap-5897	463	5	⌊	⌊	AUX
ap-5897	463	6	β1	β1	VERB
ap-5897	463	7	t	t	NOUN
ap-5897	463	8	0(x)−	0(x)−	NUM
ap-5897	464	1	γ	γ	PROPN
ap-5897	464	2	⌋	⌋	PROPN
ap-5897	464	3	and	and	CCONJ
ap-5897	464	4	t	t	PROPN
ap-5897	464	5	(	(	PUNCT
ap-5897	464	6	x	x	X
ap-5897	464	7	)	)	PUNCT
ap-5897	464	8	=	=	SYM
ap-5897	464	9	(	(	PUNCT
ap-5897	464	10	d2	d2	PROPN
ap-5897	464	11	,	,	PUNCT
ap-5897	464	12	d3	d3	PROPN
ap-5897	464	13	,	,	PUNCT
ap-5897	464	14	.	.	PUNCT
ap-5897	464	15	.	.	PUNCT
ap-5897	464	16	.	.	PUNCT
ap-5897	465	1	)	)	PUNCT
ap-5897	465	2	σ(b	σ(b	PROPN
ap-5897	465	3	)	)	PUNCT
ap-5897	465	4	.	.	PUNCT
ap-5897	466	1	suppose	suppose	VERB
ap-5897	466	2	the	the	DET
ap-5897	466	3	claim	claim	NOUN
ap-5897	466	4	holds	hold	VERB
ap-5897	466	5	for	for	ADP
ap-5897	466	6	n	n	DET
ap-5897	466	7	≤	≤	NOUN
ap-5897	466	8	k	k	NOUN
ap-5897	466	9	where	where	SCONJ
ap-5897	466	10	k	k	PROPN
ap-5897	466	11	∈	∈	PROPN
ap-5897	466	12	n.	n.	NOUN
ap-5897	466	13	then	then	ADV
ap-5897	466	14	βk+1	βk+1	NOUN
ap-5897	466	15	t	t	PROPN
ap-5897	466	16	k(x)−	k(x)−	PROPN
ap-5897	466	17	dk+1	dk+1	PROPN
ap-5897	466	18	=	=	SYM
ap-5897	466	19	βk+1	βk+1	PROPN
ap-5897	466	20	(	(	PUNCT
ap-5897	466	21	dk+1	dk+1	X
ap-5897	466	22	βk+1	βk+1	X
ap-5897	466	23	+	+	CCONJ
ap-5897	466	24	dk+2	dk+2	PRON
ap-5897	466	25	βk+1βk+2	βk+1βk+2	NOUN
ap-5897	466	26	+	+	X
ap-5897	466	27	.	.	PUNCT
ap-5897	466	28	.	.	PUNCT
ap-5897	466	29	.	.	PUNCT
ap-5897	466	30	)	)	PUNCT
ap-5897	467	1	−	−	NOUN
ap-5897	467	2	dk+1	dk+1	NOUN
ap-5897	467	3	=	=	SYM
ap-5897	467	4	dk+2	dk+2	X
ap-5897	467	5	βk+2	βk+2	NUM
ap-5897	468	1	+	+	CCONJ
ap-5897	468	2	dk+3	dk+3	NOUN
ap-5897	468	3	βk+2βk+3	βk+2βk+3	NOUN
ap-5897	469	1	+	+	CCONJ
ap-5897	469	2	.	.	PUNCT
ap-5897	469	3	.	.	PUNCT
ap-5897	469	4	.	.	PUNCT
ap-5897	470	1	=	=	PUNCT
ap-5897	470	2	(	(	PUNCT
ap-5897	470	3	dk+2	dk+2	NOUN
ap-5897	470	4	,	,	PUNCT
ap-5897	470	5	dk+3	dk+3	NOUN
ap-5897	470	6	,	,	PUNCT
ap-5897	470	7	.	.	PUNCT
ap-5897	470	8	.	.	PUNCT
ap-5897	470	9	.	.	PUNCT
ap-5897	471	1	)	)	PUNCT
ap-5897	471	2	σk+1(b	σk+1(b	X
ap-5897	471	3	)	)	PUNCT
ap-5897	471	4	∈	∈	PROPN
ap-5897	472	1	[	[	X
ap-5897	472	2	γ	γ	X
ap-5897	472	3	,	,	PUNCT
ap-5897	472	4	γ	γ	X
ap-5897	472	5	+	+	NOUN
ap-5897	472	6	1	1	NUM
ap-5897	472	7	)	)	PUNCT
ap-5897	472	8	.	.	PUNCT
ap-5897	473	1	hence	hence	ADV
ap-5897	473	2	,	,	PUNCT
ap-5897	473	3	dk+1	dk+1	X
ap-5897	473	4	=	=	SYM
ap-5897	473	5	⌊	⌊	PROPN
ap-5897	473	6	βk+1	βk+1	NUM
ap-5897	473	7	t	t	PROPN
ap-5897	473	8	k+1(x)−	k+1(x)−	PROPN
ap-5897	473	9	γ	γ	PROPN
ap-5897	473	10	⌋	⌋	PROPN
ap-5897	473	11	and	and	CCONJ
ap-5897	473	12	t	t	PROPN
ap-5897	473	13	k+1(x	k+1(x	PROPN
ap-5897	473	14	)	)	PUNCT
ap-5897	474	1	=	=	PUNCT
ap-5897	474	2	(	(	PUNCT
ap-5897	474	3	dk+2	dk+2	NOUN
ap-5897	474	4	,	,	PUNCT
ap-5897	474	5	dk+3	dk+3	NOUN
ap-5897	474	6	,	,	PUNCT
ap-5897	474	7	.	.	PUNCT
ap-5897	474	8	.	.	PUNCT
ap-5897	474	9	.	.	PUNCT
ap-5897	474	10	)	)	PUNCT
ap-5897	475	1	σk+1(b	σk+1(b	PROPN
ap-5897	475	2	)	)	PUNCT
ap-5897	475	3	.	.	PUNCT
ap-5897	476	1	corollary	corollary	NOUN
ap-5897	476	2	4.3.1	4.3.1	X
ap-5897	476	3	.	.	PUNCT
ap-5897	477	1	let	let	VERB
ap-5897	477	2	x	x	X
ap-5897	477	3	∈	∈	PROPN
ap-5897	477	4	[	[	X
ap-5897	477	5	γ	γ	X
ap-5897	477	6	,	,	PUNCT
ap-5897	477	7	γ	γ	X
ap-5897	477	8	+	+	NOUN
ap-5897	477	9	1	1	NUM
ap-5897	477	10	)	)	PUNCT
ap-5897	477	11	such	such	ADJ
ap-5897	477	12	that	that	DET
ap-5897	477	13	d(b;x	d(b;x	NOUN
ap-5897	477	14	)	)	PUNCT
ap-5897	477	15	=	=	SYM
ap-5897	477	16	(	(	PUNCT
ap-5897	477	17	a1	a1	PROPN
ap-5897	477	18	,	,	PUNCT
ap-5897	477	19	a2	a2	PROPN
ap-5897	477	20	,	,	PUNCT
ap-5897	477	21	.	.	PUNCT
ap-5897	477	22	.	.	PUNCT
ap-5897	477	23	.	.	PUNCT
ap-5897	477	24	)	)	PUNCT
ap-5897	477	25	.	.	PUNCT
ap-5897	478	1	then	then	ADV
ap-5897	478	2	d(σn(b);tn(x	d(σn(b);tn(x	NOUN
ap-5897	478	3	)	)	PUNCT
ap-5897	478	4	)	)	PUNCT
ap-5897	479	1	=	=	SYM
ap-5897	479	2	(	(	PUNCT
ap-5897	479	3	an+1	an+1	NOUN
ap-5897	479	4	,	,	PUNCT
ap-5897	479	5	an+2	an+2	ADV
ap-5897	479	6	,	,	PUNCT
ap-5897	479	7	.	.	PUNCT
ap-5897	479	8	.	.	PUNCT
ap-5897	479	9	.	.	PUNCT
ap-5897	479	10	)	)	PUNCT
ap-5897	479	11	.	.	PUNCT
ap-5897	480	1	proof	proof	NOUN
ap-5897	480	2	.	.	PUNCT
ap-5897	481	1	by	by	ADP
ap-5897	481	2	lemma	lemma	PROPN
ap-5897	481	3	4.2	4.2	NUM
ap-5897	481	4	,	,	PUNCT
ap-5897	481	5	tn(x	tn(x	NUM
ap-5897	481	6	)	)	PUNCT
ap-5897	481	7	=	=	SYM
ap-5897	481	8	b[n	b[n	PROPN
ap-5897	481	9	]	]	X
ap-5897	481	10	(	(	PUNCT
ap-5897	481	11	x−	x−	PROPN
ap-5897	481	12	n∑	n∑	PROPN
ap-5897	481	13	i=1	i=1	PROPN
ap-5897	482	1	ai	ai	VERB
ap-5897	482	2	b[i	b[i	NOUN
ap-5897	482	3	]	]	PUNCT
ap-5897	482	4	)	)	PUNCT
ap-5897	482	5	=	=	SYM
ap-5897	482	6	b[n	b[n	PROPN
ap-5897	482	7	]	]	PUNCT
ap-5897	482	8	∑	∑	PUNCT
ap-5897	482	9	i≥n+1	i≥n+1	PROPN
ap-5897	482	10	ai	ai	AUX
ap-5897	482	11	b[i	b[i	NOUN
ap-5897	482	12	]	]	PUNCT
ap-5897	482	13	=	=	SYM
ap-5897	482	14	b[n	b[n	PROPN
ap-5897	482	15	]	]	PUNCT
ap-5897	482	16	∑	∑	PUNCT
ap-5897	482	17	i≥1	i≥1	ADJ
ap-5897	482	18	an+i	an+i	VERB
ap-5897	482	19	b[n+	b[n+	NOUN
ap-5897	482	20	i	i	PRON
ap-5897	482	21	]	]	X
ap-5897	482	22	=	=	PUNCT
ap-5897	482	23	∑	∑	PUNCT
ap-5897	482	24	i≥1	i≥1	ADJ
ap-5897	482	25	an+i	an+i	ADJ
ap-5897	482	26	b[n+	b[n+	NOUN
ap-5897	482	27	1	1	NUM
ap-5897	482	28	,	,	PUNCT
ap-5897	482	29	n+	n+	PUNCT
ap-5897	482	30	i	i	NOUN
ap-5897	482	31	]	]	PUNCT
ap-5897	482	32	.	.	PUNCT
ap-5897	483	1	thus	thus	ADV
ap-5897	483	2	,	,	PUNCT
ap-5897	483	3	(	(	PUNCT
ap-5897	483	4	an+1	an+1	NOUN
ap-5897	483	5	,	,	PUNCT
ap-5897	483	6	an+2	an+2	ADV
ap-5897	483	7	,	,	PUNCT
ap-5897	483	8	.	.	PUNCT
ap-5897	483	9	.	.	PUNCT
ap-5897	483	10	.	.	PUNCT
ap-5897	483	11	)	)	PUNCT
ap-5897	484	1	is	be	AUX
ap-5897	484	2	a	a	DET
ap-5897	484	3	σn(b)-representation	σn(b)-representation	NOUN
ap-5897	484	4	of	of	ADP
ap-5897	484	5	tn(x	tn(x	NOUN
ap-5897	484	6	)	)	PUNCT
ap-5897	484	7	.	.	PUNCT
ap-5897	485	1	for	for	ADP
ap-5897	485	2	all	all	DET
ap-5897	485	3	k	k	PROPN
ap-5897	485	4	∈	∈	PROPN
ap-5897	485	5	n	n	CCONJ
ap-5897	485	6	,	,	PUNCT
ap-5897	485	7	we	we	PRON
ap-5897	485	8	have	have	VERB
ap-5897	485	9	σk(an+1	σk(an+1	PROPN
ap-5897	485	10	,	,	PUNCT
ap-5897	485	11	an+2	an+2	ADV
ap-5897	485	12	,	,	PUNCT
ap-5897	485	13	.	.	PUNCT
ap-5897	485	14	.	.	PUNCT
ap-5897	485	15	.	.	PUNCT
ap-5897	485	16	)	)	PUNCT
ap-5897	485	17	σk(σn(b	σk(σn(b	X
ap-5897	485	18	)	)	PUNCT
ap-5897	485	19	)	)	PUNCT
ap-5897	486	1	=	=	SYM
ap-5897	486	2	σn+k(a1	σn+k(a1	X
ap-5897	486	3	,	,	PUNCT
ap-5897	486	4	a2	a2	PROPN
ap-5897	486	5	,	,	PUNCT
ap-5897	486	6	.	.	PUNCT
ap-5897	486	7	.	.	PUNCT
ap-5897	486	8	.	.	PUNCT
ap-5897	486	9	)	)	PUNCT
ap-5897	487	1	σn+k(b	σn+k(b	X
ap-5897	487	2	)	)	PUNCT
ap-5897	487	3	=	=	SYM
ap-5897	487	4	tn+k(x	tn+k(x	X
ap-5897	487	5	)	)	PUNCT
ap-5897	487	6	∈	∈	NOUN
ap-5897	488	1	[	[	X
ap-5897	488	2	γ	γ	X
ap-5897	488	3	,	,	PUNCT
ap-5897	488	4	γ	γ	X
ap-5897	488	5	+	+	NOUN
ap-5897	488	6	1	1	NUM
ap-5897	488	7	)	)	PUNCT
ap-5897	488	8	.	.	PUNCT
ap-5897	489	1	the	the	DET
ap-5897	489	2	conclusion	conclusion	NOUN
ap-5897	489	3	then	then	ADV
ap-5897	489	4	follows	follow	VERB
ap-5897	489	5	from	from	ADP
ap-5897	489	6	lemma	lemma	PROPN
ap-5897	489	7	4.3	4.3	NUM
ap-5897	489	8	.	.	PUNCT
ap-5897	489	9	remark	remark	PROPN
ap-5897	489	10	.	.	PUNCT
ap-5897	490	1	proposition	proposition	NOUN
ap-5897	490	2	2.1	2.1	NUM
ap-5897	490	3	,	,	PUNCT
ap-5897	490	4	lemma	lemma	PROPN
ap-5897	490	5	4.2	4.2	NUM
ap-5897	490	6	,	,	PUNCT
ap-5897	490	7	and	and	CCONJ
ap-5897	490	8	corollary	corollary	ADJ
ap-5897	490	9	4.3.1	4.3.1	PRON
ap-5897	490	10	also	also	ADV
ap-5897	490	11	hold	hold	VERB
ap-5897	490	12	when	when	SCONJ
ap-5897	490	13	x	x	PROPN
ap-5897	490	14	=	=	SYM
ap-5897	490	15	γ	γ	X
ap-5897	490	16	+	+	NOUN
ap-5897	490	17	1	1	NUM
ap-5897	490	18	.	.	X
ap-5897	490	19	from	from	ADP
ap-5897	490	20	lemma	lemma	PROPN
ap-5897	490	21	4.3	4.3	NUM
ap-5897	490	22	and	and	CCONJ
ap-5897	490	23	corollary	corollary	ADJ
ap-5897	490	24	4.3.1	4.3.1	NUM
ap-5897	490	25	,	,	PUNCT
ap-5897	490	26	we	we	PRON
ap-5897	490	27	obtain	obtain	VERB
ap-5897	490	28	the	the	DET
ap-5897	490	29	following	follow	VERB
ap-5897	490	30	proposition	proposition	NOUN
ap-5897	490	31	,	,	PUNCT
ap-5897	490	32	which	which	PRON
ap-5897	490	33	gives	give	VERB
ap-5897	490	34	an	an	DET
ap-5897	490	35	admissibility	admissibility	NOUN
ap-5897	490	36	criterion	criterion	NOUN
ap-5897	490	37	for	for	ADP
ap-5897	490	38	a	a	DET
ap-5897	490	39	sequence	sequence	NOUN
ap-5897	490	40	(	(	PUNCT
ap-5897	490	41	d1	d1	PROPN
ap-5897	490	42	,	,	PUNCT
ap-5897	490	43	d2	d2	PROPN
ap-5897	490	44	,	,	PUNCT
ap-5897	490	45	.	.	PUNCT
ap-5897	490	46	.	.	PUNCT
ap-5897	490	47	.	.	PUNCT
ap-5897	490	48	)	)	PUNCT
ap-5897	491	1	∈	∈	PROPN
ap-5897	491	2	a(b	a(b	NOUN
ap-5897	491	3	)	)	PUNCT
ap-5897	491	4	in	in	ADP
ap-5897	491	5	terms	term	NOUN
ap-5897	491	6	of	of	ADP
ap-5897	491	7	σk(d1	σk(d1	PROPN
ap-5897	491	8	,	,	PUNCT
ap-5897	491	9	d2	d2	PROPN
ap-5897	491	10	,	,	PUNCT
ap-5897	491	11	.	.	PUNCT
ap-5897	491	12	.	.	PUNCT
ap-5897	491	13	.	.	PUNCT
ap-5897	491	14	)	)	PUNCT
ap-5897	492	1	σk(b	σk(b	X
ap-5897	492	2	)	)	PUNCT
ap-5897	492	3	.	.	PUNCT
ap-5897	493	1	proposition	proposition	NOUN
ap-5897	493	2	4.4	4.4	NUM
ap-5897	493	3	.	.	PUNCT
ap-5897	494	1	a	a	DET
ap-5897	494	2	sequence	sequence	NOUN
ap-5897	494	3	(	(	PUNCT
ap-5897	494	4	d1	d1	PROPN
ap-5897	494	5	,	,	PUNCT
ap-5897	494	6	d2	d2	PROPN
ap-5897	494	7	,	,	PUNCT
ap-5897	494	8	.	.	PUNCT
ap-5897	494	9	.	.	PUNCT
ap-5897	494	10	.	.	PUNCT
ap-5897	494	11	)	)	PUNCT
ap-5897	495	1	∈	∈	PROPN
ap-5897	495	2	a(b	a(b	NOUN
ap-5897	495	3	)	)	PUNCT
ap-5897	495	4	is	be	AUX
ap-5897	495	5	b	b	NOUN
ap-5897	495	6	-	-	PUNCT
ap-5897	495	7	admissible	admissible	ADJ
ap-5897	495	8	if	if	SCONJ
ap-5897	496	1	and	and	CCONJ
ap-5897	496	2	only	only	ADV
ap-5897	496	3	if	if	SCONJ
ap-5897	496	4	σk(d1	σk(d1	X
ap-5897	496	5	,	,	PUNCT
ap-5897	496	6	d2	d2	PROPN
ap-5897	496	7	,	,	PUNCT
ap-5897	496	8	.	.	PUNCT
ap-5897	496	9	.	.	PUNCT
ap-5897	496	10	.	.	PUNCT
ap-5897	497	1	)	)	PUNCT
ap-5897	498	1	σk(b	σk(b	X
ap-5897	498	2	)	)	PUNCT
ap-5897	498	3	∈	∈	NOUN
ap-5897	499	1	[	[	X
ap-5897	499	2	γ	γ	X
ap-5897	499	3	,	,	PUNCT
ap-5897	499	4	γ	γ	X
ap-5897	499	5	+	+	NOUN
ap-5897	499	6	1	1	NUM
ap-5897	499	7	)	)	PUNCT
ap-5897	499	8	for	for	ADP
ap-5897	499	9	all	all	DET
ap-5897	499	10	k	k	PROPN
ap-5897	499	11	∈	∈	PROPN
ap-5897	499	12	n	n	PART
ap-5897	499	13	∪	∪	X
ap-5897	499	14	{	{	PUNCT
ap-5897	499	15	0	0	NUM
ap-5897	499	16	}	}	PUNCT
ap-5897	499	17	.	.	PUNCT
ap-5897	500	1	now	now	ADV
ap-5897	500	2	,	,	PUNCT
ap-5897	500	3	we	we	PRON
ap-5897	500	4	provide	provide	VERB
ap-5897	500	5	another	another	DET
ap-5897	500	6	admissibility	admissibility	NOUN
ap-5897	500	7	criterion	criterion	NOUN
ap-5897	500	8	–	–	PUNCT
ap-5897	500	9	this	this	DET
ap-5897	500	10	time	time	NOUN
ap-5897	500	11	,	,	PUNCT
ap-5897	500	12	in	in	ADP
ap-5897	500	13	terms	term	NOUN
ap-5897	500	14	of	of	ADP
ap-5897	500	15	the	the	DET
ap-5897	500	16	shifts	shift	NOUN
ap-5897	500	17	of	of	ADP
ap-5897	500	18	a	a	DET
ap-5897	500	19	sequence	sequence	NOUN
ap-5897	500	20	(	(	PUNCT
ap-5897	500	21	x1	x1	PROPN
ap-5897	500	22	,	,	PUNCT
ap-5897	500	23	x2	x2	PROPN
ap-5897	500	24	,	,	PUNCT
ap-5897	500	25	.	.	PUNCT
ap-5897	500	26	.	.	PUNCT
ap-5897	500	27	.	.	PUNCT
ap-5897	500	28	)	)	PUNCT
ap-5897	501	1	∈	∈	PROPN
ap-5897	501	2	a(b	a(b	NOUN
ap-5897	501	3	)	)	PUNCT
ap-5897	501	4	.	.	PUNCT
ap-5897	502	1	to	to	ADP
ap-5897	502	2	this	this	DET
ap-5897	502	3	end	end	NOUN
ap-5897	502	4	,	,	PUNCT
ap-5897	502	5	we	we	PRON
ap-5897	502	6	need	need	VERB
ap-5897	502	7	to	to	PART
ap-5897	502	8	introduce	introduce	VERB
ap-5897	502	9	an	an	DET
ap-5897	502	10	order	order	NOUN
ap-5897	502	11	≺b	≺b	NOUN
ap-5897	502	12	on	on	ADP
ap-5897	502	13	a(b	a(b	PROPN
ap-5897	502	14	)	)	PUNCT
ap-5897	502	15	.	.	PUNCT
ap-5897	503	1	220	220	NUM
ap-5897	503	2	vol	vol	NOUN
ap-5897	503	3	.	.	PUNCT
ap-5897	504	1	60	60	NUM
ap-5897	504	2	no	no	NOUN
ap-5897	504	3	.	.	PUNCT
ap-5897	505	1	3/2020	3/2020	NUM
ap-5897	505	2	beta	beta	PROPN
ap-5897	505	3	cantor	cantor	PROPN
ap-5897	505	4	series	series	NOUN
ap-5897	505	5	expansion	expansion	NOUN
ap-5897	505	6	and	and	CCONJ
ap-5897	505	7	admissible	admissible	ADJ
ap-5897	505	8	sequences	sequence	NOUN
ap-5897	505	9	definition	definition	NOUN
ap-5897	505	10	3	3	X
ap-5897	505	11	.	.	PUNCT
ap-5897	506	1	let	let	VERB
ap-5897	506	2	(	(	PUNCT
ap-5897	506	3	a1	a1	NOUN
ap-5897	506	4	,	,	PUNCT
ap-5897	506	5	a2	a2	PROPN
ap-5897	506	6	,	,	PUNCT
ap-5897	506	7	.	.	PUNCT
ap-5897	506	8	.	.	PUNCT
ap-5897	506	9	.	.	PUNCT
ap-5897	506	10	)	)	PUNCT
ap-5897	507	1	and	and	CCONJ
ap-5897	507	2	(	(	PUNCT
ap-5897	507	3	b1	b1	NOUN
ap-5897	507	4	,	,	PUNCT
ap-5897	507	5	b2	b2	NOUN
ap-5897	507	6	,	,	PUNCT
ap-5897	507	7	.	.	PUNCT
ap-5897	507	8	.	.	PUNCT
ap-5897	507	9	.	.	PUNCT
ap-5897	507	10	)	)	PUNCT
ap-5897	507	11	be	be	AUX
ap-5897	507	12	in	in	ADP
ap-5897	507	13	a(b	a(b	NOUN
ap-5897	507	14	)	)	PUNCT
ap-5897	507	15	.	.	PUNCT
ap-5897	508	1	we	we	PRON
ap-5897	508	2	say	say	VERB
ap-5897	508	3	(	(	PUNCT
ap-5897	508	4	a1	a1	NOUN
ap-5897	508	5	,	,	PUNCT
ap-5897	508	6	a2	a2	PROPN
ap-5897	508	7	,	,	PUNCT
ap-5897	508	8	.	.	PUNCT
ap-5897	508	9	.	.	PUNCT
ap-5897	508	10	.	.	PUNCT
ap-5897	508	11	)	)	PUNCT
ap-5897	509	1	≺b	≺b	NOUN
ap-5897	509	2	(	(	PUNCT
ap-5897	509	3	b1	b1	NOUN
ap-5897	509	4	,	,	PUNCT
ap-5897	509	5	b2	b2	NOUN
ap-5897	509	6	,	,	PUNCT
ap-5897	509	7	.	.	PUNCT
ap-5897	509	8	.	.	PUNCT
ap-5897	509	9	.	.	PUNCT
ap-5897	509	10	)	)	PUNCT
ap-5897	510	1	if	if	SCONJ
ap-5897	510	2	and	and	CCONJ
ap-5897	510	3	only	only	ADV
ap-5897	510	4	if	if	SCONJ
ap-5897	510	5	there	there	PRON
ap-5897	510	6	exists	exist	VERB
ap-5897	510	7	k	k	PROPN
ap-5897	510	8	∈	∈	PROPN
ap-5897	510	9	n	n	PRON
ap-5897	510	10	such	such	ADJ
ap-5897	510	11	that	that	SCONJ
ap-5897	510	12	bi	bi	NOUN
ap-5897	510	13	=	=	NOUN
ap-5897	510	14	ai	ai	VERB
ap-5897	510	15	for	for	ADP
ap-5897	510	16	all	all	PRON
ap-5897	510	17	i	i	PRON
ap-5897	510	18	=	=	NOUN
ap-5897	510	19	1	1	NUM
ap-5897	510	20	,	,	PUNCT
ap-5897	510	21	2	2	NUM
ap-5897	510	22	,	,	PUNCT
ap-5897	510	23	.	.	PUNCT
ap-5897	510	24	.	.	PUNCT
ap-5897	511	1	.	.	PUNCT
ap-5897	512	1	,	,	PUNCT
ap-5897	513	1	k	k	PROPN
ap-5897	514	1	−	−	PROPN
ap-5897	514	2	1	1	NUM
ap-5897	514	3	and	and	CCONJ
ap-5897	514	4	bk	bk	PROPN
ap-5897	514	5	6=	6=	PROPN
ap-5897	514	6	ak	ak	PROPN
ap-5897	514	7	where	where	SCONJ
ap-5897	514	8	(	(	PUNCT
ap-5897	514	9	bk	bk	PROPN
ap-5897	514	10	−	−	PROPN
ap-5897	514	11	ak	ak	PROPN
ap-5897	514	12	)	)	PUNCT
ap-5897	514	13	sgn(b[k	sgn(b[k	NOUN
ap-5897	514	14	]	]	PUNCT
ap-5897	514	15	)	)	PUNCT
ap-5897	514	16	≥	≥	NOUN
ap-5897	514	17	1	1	NUM
ap-5897	514	18	.	.	PUNCT
ap-5897	515	1	if	if	SCONJ
ap-5897	515	2	(	(	PUNCT
ap-5897	515	3	a1	a1	NOUN
ap-5897	515	4	,	,	PUNCT
ap-5897	515	5	a2	a2	PROPN
ap-5897	515	6	,	,	PUNCT
ap-5897	515	7	.	.	PUNCT
ap-5897	515	8	.	.	PUNCT
ap-5897	515	9	.	.	PUNCT
ap-5897	515	10	)	)	PUNCT
ap-5897	516	1	≺b	≺b	NOUN
ap-5897	516	2	(	(	PUNCT
ap-5897	516	3	b1	b1	NOUN
ap-5897	516	4	,	,	PUNCT
ap-5897	516	5	b2	b2	NOUN
ap-5897	516	6	,	,	PUNCT
ap-5897	516	7	.	.	PUNCT
ap-5897	516	8	.	.	PUNCT
ap-5897	516	9	.	.	PUNCT
ap-5897	516	10	)	)	PUNCT
ap-5897	517	1	or	or	CCONJ
ap-5897	517	2	(	(	PUNCT
ap-5897	517	3	a1	a1	PROPN
ap-5897	517	4	,	,	PUNCT
ap-5897	517	5	a2	a2	PROPN
ap-5897	517	6	,	,	PUNCT
ap-5897	517	7	.	.	PUNCT
ap-5897	517	8	.	.	PUNCT
ap-5897	517	9	.	.	PUNCT
ap-5897	517	10	)	)	PUNCT
ap-5897	518	1	=	=	PRON
ap-5897	518	2	(	(	PUNCT
ap-5897	518	3	b1	b1	NOUN
ap-5897	518	4	,	,	PUNCT
ap-5897	518	5	b2	b2	NOUN
ap-5897	518	6	,	,	PUNCT
ap-5897	518	7	.	.	PUNCT
ap-5897	518	8	.	.	PUNCT
ap-5897	518	9	.	.	PUNCT
ap-5897	518	10	)	)	PUNCT
ap-5897	519	1	,	,	PUNCT
ap-5897	519	2	we	we	PRON
ap-5897	519	3	write	write	VERB
ap-5897	519	4	(	(	PUNCT
ap-5897	519	5	a1	a1	PROPN
ap-5897	519	6	,	,	PUNCT
ap-5897	519	7	a2	a2	PROPN
ap-5897	519	8	,	,	PUNCT
ap-5897	519	9	.	.	PUNCT
ap-5897	519	10	.	.	PUNCT
ap-5897	519	11	.	.	PUNCT
ap-5897	519	12	)	)	PUNCT
ap-5897	520	1	�	�	PROPN
ap-5897	520	2	b	b	PROPN
ap-5897	520	3	(	(	PUNCT
ap-5897	520	4	b1	b1	NOUN
ap-5897	520	5	,	,	PUNCT
ap-5897	520	6	b2	b2	NOUN
ap-5897	520	7	,	,	PUNCT
ap-5897	520	8	.	.	PUNCT
ap-5897	520	9	.	.	PUNCT
ap-5897	520	10	.	.	PUNCT
ap-5897	520	11	)	)	PUNCT
ap-5897	520	12	.	.	PUNCT
ap-5897	521	1	remark	remark	PROPN
ap-5897	521	2	.	.	PUNCT
ap-5897	522	1	for	for	ADP
ap-5897	522	2	the	the	DET
ap-5897	522	3	classical	classical	ADJ
ap-5897	522	4	positive	positive	ADJ
ap-5897	522	5	and	and	CCONJ
ap-5897	522	6	negative	negative	ADJ
ap-5897	522	7	beta	beta	ADJ
ap-5897	522	8	expansions	expansion	NOUN
ap-5897	522	9	,	,	PUNCT
ap-5897	522	10	the	the	DET
ap-5897	522	11	order	order	NOUN
ap-5897	522	12	≺b	≺b	PROPN
ap-5897	522	13	coincides	coincide	VERB
ap-5897	522	14	with	with	ADP
ap-5897	522	15	the	the	DET
ap-5897	522	16	orders	order	NOUN
ap-5897	522	17	defined	define	VERB
ap-5897	522	18	in	in	ADP
ap-5897	522	19	[	[	X
ap-5897	522	20	2	2	NUM
ap-5897	522	21	]	]	PUNCT
ap-5897	522	22	and	and	CCONJ
ap-5897	522	23	[	[	X
ap-5897	522	24	6	6	NUM
ap-5897	522	25	]	]	PUNCT
ap-5897	522	26	,	,	PUNCT
ap-5897	522	27	respectively	respectively	ADV
ap-5897	522	28	.	.	PUNCT
ap-5897	523	1	the	the	DET
ap-5897	523	2	following	follow	VERB
ap-5897	523	3	proposition	proposition	NOUN
ap-5897	523	4	tells	tell	VERB
ap-5897	523	5	us	we	PRON
ap-5897	523	6	that	that	SCONJ
ap-5897	523	7	the	the	DET
ap-5897	523	8	monotonicity	monotonicity	NOUN
ap-5897	523	9	of	of	ADP
ap-5897	523	10	points	point	NOUN
ap-5897	523	11	in	in	ADP
ap-5897	523	12	[	[	X
ap-5897	523	13	γ	γ	X
ap-5897	523	14	,	,	PUNCT
ap-5897	523	15	γ	γ	X
ap-5897	523	16	+	+	NOUN
ap-5897	523	17	1	1	NUM
ap-5897	523	18	)	)	PUNCT
ap-5897	523	19	is	be	AUX
ap-5897	523	20	carried	carry	VERB
ap-5897	523	21	over	over	ADP
ap-5897	523	22	to	to	ADP
ap-5897	523	23	the	the	DET
ap-5897	523	24	ordering	ordering	NOUN
ap-5897	523	25	of	of	ADP
ap-5897	523	26	words	word	NOUN
ap-5897	523	27	with	with	ADP
ap-5897	523	28	respect	respect	NOUN
ap-5897	523	29	to	to	ADP
ap-5897	523	30	≺b	≺b	PROPN
ap-5897	523	31	.	.	PUNCT
ap-5897	524	1	proposition	proposition	NOUN
ap-5897	524	2	4.5	4.5	NUM
ap-5897	524	3	.	.	PUNCT
ap-5897	525	1	let	let	VERB
ap-5897	525	2	x	x	PRON
ap-5897	525	3	,	,	PUNCT
ap-5897	525	4	y	y	PROPN
ap-5897	525	5	∈	∈	PROPN
ap-5897	526	1	[	[	X
ap-5897	526	2	γ	γ	X
ap-5897	526	3	,	,	PUNCT
ap-5897	526	4	γ	γ	X
ap-5897	526	5	+	+	NOUN
ap-5897	526	6	1	1	NUM
ap-5897	526	7	)	)	PUNCT
ap-5897	526	8	.	.	PUNCT
ap-5897	527	1	then	then	ADV
ap-5897	527	2	d(b;x	d(b;x	NOUN
ap-5897	527	3	)	)	PUNCT
ap-5897	527	4	≺b	≺b	PROPN
ap-5897	527	5	d(b	d(b	PROPN
ap-5897	527	6	;	;	PUNCT
ap-5897	527	7	y	y	X
ap-5897	527	8	)	)	PUNCT
ap-5897	528	1	if	if	SCONJ
ap-5897	528	2	and	and	CCONJ
ap-5897	528	3	only	only	ADV
ap-5897	528	4	if	if	SCONJ
ap-5897	528	5	x	x	X
ap-5897	528	6	<	<	X
ap-5897	528	7	y.	y.	PROPN
ap-5897	528	8	proof	proof	NOUN
ap-5897	528	9	.	.	PUNCT
ap-5897	529	1	let	let	VERB
ap-5897	529	2	d(b;x	d(b;x	NOUN
ap-5897	529	3	)	)	PUNCT
ap-5897	529	4	=	=	PRON
ap-5897	530	1	(	(	PUNCT
ap-5897	530	2	x1	x1	PROPN
ap-5897	530	3	,	,	PUNCT
ap-5897	530	4	x2	x2	PROPN
ap-5897	530	5	,	,	PUNCT
ap-5897	530	6	.	.	PUNCT
ap-5897	530	7	.	.	PUNCT
ap-5897	530	8	.	.	PUNCT
ap-5897	530	9	)	)	PUNCT
ap-5897	531	1	and	and	CCONJ
ap-5897	531	2	d(b	d(b	PROPN
ap-5897	531	3	;	;	PUNCT
ap-5897	531	4	y	y	X
ap-5897	531	5	)	)	PUNCT
ap-5897	531	6	=	=	SYM
ap-5897	531	7	(	(	PUNCT
ap-5897	531	8	y1	y1	INTJ
ap-5897	531	9	,	,	PUNCT
ap-5897	531	10	y2	y2	PROPN
ap-5897	531	11	,	,	PUNCT
ap-5897	531	12	.	.	PUNCT
ap-5897	531	13	.	.	PUNCT
ap-5897	531	14	.	.	PUNCT
ap-5897	531	15	)	)	PUNCT
ap-5897	531	16	.	.	PUNCT
ap-5897	532	1	let	let	VERB
ap-5897	532	2	k	k	PROPN
ap-5897	532	3	∈	∈	PROPN
ap-5897	532	4	n	n	AUX
ap-5897	532	5	be	be	AUX
ap-5897	532	6	the	the	DET
ap-5897	532	7	least	least	ADJ
ap-5897	532	8	integer	integer	NOUN
ap-5897	532	9	such	such	ADJ
ap-5897	532	10	that	that	SCONJ
ap-5897	532	11	xk	xk	PROPN
ap-5897	532	12	6=	6=	PROPN
ap-5897	532	13	yk	yk	PROPN
ap-5897	532	14	.	.	PUNCT
ap-5897	532	15	suppose	suppose	VERB
ap-5897	532	16	d(b;x	d(b;x	NOUN
ap-5897	532	17	)	)	PUNCT
ap-5897	532	18	≺b	≺b	PROPN
ap-5897	532	19	d(b	d(b	PROPN
ap-5897	532	20	;	;	PUNCT
ap-5897	532	21	y	y	X
ap-5897	532	22	)	)	PUNCT
ap-5897	532	23	.	.	PUNCT
ap-5897	533	1	then	then	ADV
ap-5897	533	2	y	y	PROPN
ap-5897	533	3	−	−	NOUN
ap-5897	533	4	x	x	PUNCT
ap-5897	533	5	=	=	SYM
ap-5897	533	6	yk	yk	PROPN
ap-5897	533	7	−	−	PROPN
ap-5897	533	8	xk	xk	PROPN
ap-5897	533	9	b[k	b[k	PROPN
ap-5897	533	10	]	]	PUNCT
ap-5897	534	1	+	+	CCONJ
ap-5897	534	2	∑	∑	ADP
ap-5897	534	3	i≥k+1	i≥k+1	ADV
ap-5897	534	4	yi	yi	NOUN
ap-5897	534	5	−	−	NOUN
ap-5897	534	6	xi	xi	ADP
ap-5897	534	7	b[i	b[i	PROPN
ap-5897	534	8	]	]	PUNCT
ap-5897	534	9	.	.	PUNCT
ap-5897	535	1	we	we	PRON
ap-5897	535	2	have∑	have∑	VERB
ap-5897	535	3	i≥k+1	i≥k+1	ADV
ap-5897	535	4	yi	yi	PROPN
ap-5897	536	1	−	−	NOUN
ap-5897	536	2	xi	xi	ADP
ap-5897	536	3	b[i	b[i	VERB
ap-5897	536	4	]	]	PUNCT
ap-5897	536	5	=	=	PUNCT
ap-5897	536	6	∑	∑	PUNCT
ap-5897	536	7	i≥1	i≥1	PROPN
ap-5897	536	8	yk+i	yk+i	PROPN
ap-5897	536	9	−	−	PROPN
ap-5897	536	10	xk+i	xk+i	PROPN
ap-5897	536	11	b[k	b[k	NOUN
ap-5897	537	1	+	+	CCONJ
ap-5897	538	1	i	i	PRON
ap-5897	538	2	]	]	X
ap-5897	538	3	=	=	SYM
ap-5897	538	4	1	1	NUM
ap-5897	538	5	b[k	b[k	NOUN
ap-5897	538	6	]	]	PUNCT
ap-5897	538	7	∑	∑	PUNCT
ap-5897	538	8	i≥1	i≥1	PROPN
ap-5897	538	9	yk+i	yk+i	PROPN
ap-5897	538	10	−	−	PROPN
ap-5897	538	11	xk+i	xk+i	PROPN
ap-5897	538	12	b[k	b[k	NOUN
ap-5897	538	13	+	+	CCONJ
ap-5897	538	14	1	1	NUM
ap-5897	538	15	,	,	PUNCT
ap-5897	538	16	k	k	PROPN
ap-5897	539	1	+	+	CCONJ
ap-5897	539	2	i	i	X
ap-5897	539	3	]	]	X
ap-5897	540	1	=	=	PUNCT
ap-5897	540	2	t	t	PROPN
ap-5897	540	3	k(y)−	k(y)−	PROPN
ap-5897	540	4	t	t	PROPN
ap-5897	540	5	k(x	k(x	PROPN
ap-5897	540	6	)	)	PUNCT
ap-5897	540	7	b[k	b[k	NOUN
ap-5897	540	8	]	]	X
ap-5897	541	1	=	=	SYM
ap-5897	542	1	(	(	PUNCT
ap-5897	542	2	t	t	PROPN
ap-5897	542	3	k(y)−	k(y)−	PROPN
ap-5897	542	4	t	t	PROPN
ap-5897	542	5	k(x	k(x	PROPN
ap-5897	542	6	)	)	PUNCT
ap-5897	542	7	)	)	PUNCT
ap-5897	543	1	sgn(b[k	sgn(b[k	NUM
ap-5897	543	2	]	]	PUNCT
ap-5897	543	3	)	)	PUNCT
ap-5897	543	4	|b[k]|	|b[k]|	X
ap-5897	543	5	>	>	X
ap-5897	543	6	−1	−1	NOUN
ap-5897	543	7	|b[k]|	|b[k]|	NUM
ap-5897	543	8	.	.	PUNCT
ap-5897	544	1	thus	thus	ADV
ap-5897	544	2	,	,	PUNCT
ap-5897	544	3	y	y	PROPN
ap-5897	544	4	−	−	PROPN
ap-5897	544	5	x	x	SYM
ap-5897	544	6	>	>	X
ap-5897	544	7	(	(	PUNCT
ap-5897	544	8	yk	yk	PROPN
ap-5897	544	9	−	−	PROPN
ap-5897	544	10	xk	xk	PROPN
ap-5897	544	11	)	)	PUNCT
ap-5897	544	12	sgn(b[k])−	sgn(b[k])−	PROPN
ap-5897	544	13	1	1	NUM
ap-5897	544	14	|b[k]|	|b[k]|	NUM
ap-5897	544	15	≥	≥	NOUN
ap-5897	544	16	0	0	NUM
ap-5897	544	17	.	.	PUNCT
ap-5897	545	1	for	for	ADP
ap-5897	545	2	the	the	DET
ap-5897	545	3	reverse	reverse	ADJ
ap-5897	545	4	implication	implication	NOUN
ap-5897	545	5	,	,	PUNCT
ap-5897	545	6	suppose	suppose	VERB
ap-5897	545	7	0	0	PUNCT
ap-5897	545	8	<	<	X
ap-5897	545	9	y	y	NOUN
ap-5897	546	1	−	−	NOUN
ap-5897	546	2	x	x	PUNCT
ap-5897	546	3	=	=	SYM
ap-5897	546	4	yk	yk	PROPN
ap-5897	546	5	−	−	PROPN
ap-5897	546	6	xk	xk	PROPN
ap-5897	547	1	+	+	CCONJ
ap-5897	547	2	t	t	PROPN
ap-5897	547	3	k(y)−	k(y)−	PROPN
ap-5897	547	4	t	t	PROPN
ap-5897	547	5	k(x	k(x	PROPN
ap-5897	547	6	)	)	PUNCT
ap-5897	548	1	b[k	b[k	NOUN
ap-5897	548	2	]	]	PUNCT
ap-5897	548	3	.	.	PUNCT
ap-5897	549	1	note	note	VERB
ap-5897	549	2	that	that	SCONJ
ap-5897	549	3	−1	−1	NOUN
ap-5897	549	4	<	<	X
ap-5897	549	5	t	t	PROPN
ap-5897	549	6	k(y	k(y	PROPN
ap-5897	549	7	)	)	PUNCT
ap-5897	549	8	−	−	PROPN
ap-5897	549	9	t	t	PROPN
ap-5897	549	10	k(x	k(x	PROPN
ap-5897	549	11	)	)	PUNCT
ap-5897	549	12	<	<	X
ap-5897	550	1	1	1	X
ap-5897	550	2	.	.	PUNCT
ap-5897	550	3	when	when	SCONJ
ap-5897	550	4	sgn(b[k	sgn(b[k	NOUN
ap-5897	550	5	]	]	PUNCT
ap-5897	550	6	)	)	PUNCT
ap-5897	550	7	>	>	X
ap-5897	550	8	0	0	NUM
ap-5897	550	9	,	,	PUNCT
ap-5897	550	10	then	then	ADV
ap-5897	550	11	yk	yk	PROPN
ap-5897	550	12	−	−	PROPN
ap-5897	550	13	xk	xk	PROPN
ap-5897	551	1	+	+	CCONJ
ap-5897	551	2	1	1	NUM
ap-5897	551	3	>	>	SYM
ap-5897	551	4	0	0	NUM
ap-5897	551	5	.	.	PUNCT
ap-5897	552	1	this	this	PRON
ap-5897	552	2	implies	imply	VERB
ap-5897	552	3	that	that	SCONJ
ap-5897	552	4	yk	yk	PROPN
ap-5897	552	5	−	−	PROPN
ap-5897	552	6	xk	xk	PROPN
ap-5897	552	7	≥	≥	PROPN
ap-5897	552	8	0	0	NUM
ap-5897	552	9	since	since	SCONJ
ap-5897	552	10	both	both	CCONJ
ap-5897	552	11	yk	yk	PROPN
ap-5897	552	12	and	and	CCONJ
ap-5897	552	13	xk	xk	PROPN
ap-5897	552	14	are	be	AUX
ap-5897	552	15	integers	integer	NOUN
ap-5897	552	16	.	.	PUNCT
ap-5897	553	1	but	but	CCONJ
ap-5897	553	2	since	since	SCONJ
ap-5897	553	3	yk	yk	PROPN
ap-5897	553	4	6=	6=	PROPN
ap-5897	553	5	xk	xk	PROPN
ap-5897	553	6	,	,	PUNCT
ap-5897	553	7	then	then	ADV
ap-5897	553	8	yk	yk	PROPN
ap-5897	553	9	−	−	PROPN
ap-5897	553	10	xk	xk	PROPN
ap-5897	553	11	≥	≥	PROPN
ap-5897	553	12	1	1	NUM
ap-5897	553	13	.	.	PUNCT
ap-5897	554	1	however	however	ADV
ap-5897	554	2	,	,	PUNCT
ap-5897	554	3	when	when	SCONJ
ap-5897	554	4	sgn(b[k	sgn(b[k	NOUN
ap-5897	554	5	]	]	PUNCT
ap-5897	554	6	)	)	PUNCT
ap-5897	554	7	<	<	X
ap-5897	554	8	0	0	NUM
ap-5897	554	9	,	,	PUNCT
ap-5897	554	10	then	then	ADV
ap-5897	554	11	0	0	NUM
ap-5897	554	12	>	>	X
ap-5897	554	13	yk−xk−1	yk−xk−1	NOUN
ap-5897	554	14	.	.	PUNCT
ap-5897	555	1	thus	thus	ADV
ap-5897	555	2	,	,	PUNCT
ap-5897	555	3	yk−xk	yk−xk	PROPN
ap-5897	555	4	≤	≤	ADV
ap-5897	555	5	0	0	NUM
ap-5897	555	6	.	.	PUNCT
ap-5897	556	1	but	but	CCONJ
ap-5897	556	2	since	since	SCONJ
ap-5897	556	3	yk	yk	PROPN
ap-5897	556	4	6=	6=	PROPN
ap-5897	556	5	xk	xk	PROPN
ap-5897	556	6	,	,	PUNCT
ap-5897	556	7	then	then	ADV
ap-5897	556	8	yk	yk	PROPN
ap-5897	556	9	−	−	PROPN
ap-5897	556	10	xk	xk	PROPN
ap-5897	556	11	≤	≤	PROPN
ap-5897	556	12	−1	−1	NOUN
ap-5897	556	13	.	.	PUNCT
ap-5897	557	1	in	in	ADP
ap-5897	557	2	both	both	DET
ap-5897	557	3	cases	case	NOUN
ap-5897	557	4	,	,	PUNCT
ap-5897	557	5	(	(	PUNCT
ap-5897	557	6	yk	yk	PROPN
ap-5897	557	7	−	−	PROPN
ap-5897	557	8	xk	xk	PROPN
ap-5897	557	9	)	)	PUNCT
ap-5897	557	10	sgn(b[k	sgn(b[k	NOUN
ap-5897	557	11	]	]	PUNCT
ap-5897	557	12	)	)	PUNCT
ap-5897	557	13	≥	≥	NOUN
ap-5897	557	14	1	1	NUM
ap-5897	557	15	.	.	PUNCT
ap-5897	557	16	proposition	proposition	NOUN
ap-5897	557	17	4.5	4.5	NUM
ap-5897	557	18	,	,	PUNCT
ap-5897	557	19	together	together	ADV
ap-5897	557	20	with	with	ADP
ap-5897	557	21	corollary	corollary	ADJ
ap-5897	557	22	4.3.1	4.3.1	NUM
ap-5897	557	23	,	,	PUNCT
ap-5897	557	24	implies	imply	VERB
ap-5897	557	25	the	the	DET
ap-5897	557	26	following	follow	VERB
ap-5897	557	27	result	result	NOUN
ap-5897	557	28	.	.	PUNCT
ap-5897	558	1	corollary	corollary	ADJ
ap-5897	558	2	4.5.1	4.5.1	X
ap-5897	558	3	.	.	PUNCT
ap-5897	559	1	if	if	SCONJ
ap-5897	559	2	(	(	PUNCT
ap-5897	559	3	d1	d1	NOUN
ap-5897	559	4	,	,	PUNCT
ap-5897	559	5	d2	d2	PROPN
ap-5897	559	6	,	,	PUNCT
ap-5897	559	7	.	.	PUNCT
ap-5897	559	8	.	.	PUNCT
ap-5897	559	9	.	.	PUNCT
ap-5897	559	10	)	)	PUNCT
ap-5897	560	1	∈	∈	PROPN
ap-5897	560	2	a(b	a(b	NOUN
ap-5897	560	3	)	)	PUNCT
ap-5897	560	4	is	be	AUX
ap-5897	560	5	badmissible	badmissible	ADJ
ap-5897	560	6	,	,	PUNCT
ap-5897	560	7	then	then	ADV
ap-5897	560	8	,	,	PUNCT
ap-5897	560	9	for	for	ADP
ap-5897	560	10	all	all	DET
ap-5897	560	11	n	n	PRON
ap-5897	560	12	∈	∈	PROPN
ap-5897	560	13	n	n	CCONJ
ap-5897	560	14	,	,	PUNCT
ap-5897	560	15	d(σn(b	d(σn(b	PROPN
ap-5897	560	16	)	)	PUNCT
ap-5897	560	17	;	;	PUNCT
ap-5897	560	18	γ	γ	X
ap-5897	560	19	)	)	PUNCT
ap-5897	560	20	�	�	NOUN
ap-5897	560	21	σn(b	σn(b	NOUN
ap-5897	560	22	)	)	PUNCT
ap-5897	560	23	(	(	PUNCT
ap-5897	560	24	dn+1	dn+1	PROPN
ap-5897	560	25	,	,	PUNCT
ap-5897	560	26	dn+2	dn+2	ADV
ap-5897	560	27	,	,	PUNCT
ap-5897	560	28	.	.	PUNCT
ap-5897	560	29	.	.	PUNCT
ap-5897	560	30	.	.	PUNCT
ap-5897	560	31	)	)	PUNCT
ap-5897	560	32	.	.	PUNCT
ap-5897	561	1	analogous	analogous	ADJ
ap-5897	561	2	to	to	PART
ap-5897	561	3	proposition	proposition	VERB
ap-5897	561	4	2.1	2.1	NUM
ap-5897	561	5	and	and	CCONJ
ap-5897	561	6	lemma	lemma	PROPN
ap-5897	561	7	4.3	4.3	NUM
ap-5897	561	8	,	,	PUNCT
ap-5897	561	9	we	we	PRON
ap-5897	561	10	provide	provide	VERB
ap-5897	561	11	a	a	DET
ap-5897	561	12	relation	relation	NOUN
ap-5897	561	13	between	between	ADP
ap-5897	561	14	d∗(b	d∗(b	PROPN
ap-5897	561	15	;	;	PUNCT
ap-5897	561	16	γ	γ	X
ap-5897	561	17	+	+	NOUN
ap-5897	561	18	1	1	NUM
ap-5897	561	19	)	)	PUNCT
ap-5897	561	20	and	and	CCONJ
ap-5897	561	21	γ	γ	X
ap-5897	561	22	+	+	NOUN
ap-5897	561	23	1	1	NUM
ap-5897	561	24	.	.	X
ap-5897	561	25	proposition	proposition	NOUN
ap-5897	561	26	4.6	4.6	NUM
ap-5897	561	27	.	.	PUNCT
ap-5897	562	1	if	if	SCONJ
ap-5897	562	2	d∗(b	d∗(b	PROPN
ap-5897	562	3	;	;	PUNCT
ap-5897	562	4	γ+1	γ+1	NUM
ap-5897	562	5	)	)	PUNCT
ap-5897	562	6	=	=	SYM
ap-5897	562	7	(	(	PUNCT
ap-5897	562	8	c∗1	c∗1	ADJ
ap-5897	562	9	,	,	PUNCT
ap-5897	562	10	c∗2	c∗2	NOUN
ap-5897	562	11	,	,	PUNCT
ap-5897	562	12	.	.	PUNCT
ap-5897	562	13	.	.	PUNCT
ap-5897	562	14	.	.	PUNCT
ap-5897	562	15	)	)	PUNCT
ap-5897	563	1	,	,	PUNCT
ap-5897	563	2	then	then	ADV
ap-5897	563	3	γ	γ	PROPN
ap-5897	563	4	+	+	PROPN
ap-5897	563	5	1	1	NUM
ap-5897	563	6	=	=	SYM
ap-5897	563	7	(	(	PUNCT
ap-5897	563	8	c∗1	c∗1	ADJ
ap-5897	563	9	,	,	PUNCT
ap-5897	563	10	c∗2	c∗2	NOUN
ap-5897	563	11	,	,	PUNCT
ap-5897	563	12	.	.	PUNCT
ap-5897	563	13	.	.	PUNCT
ap-5897	563	14	.	.	PUNCT
ap-5897	563	15	)	)	PUNCT
ap-5897	564	1	b	b	X
ap-5897	564	2	and	and	CCONJ
ap-5897	564	3	(	(	PUNCT
ap-5897	564	4	c∗k+1	c∗k+1	X
ap-5897	564	5	,	,	PUNCT
ap-5897	564	6	c	c	PROPN
ap-5897	564	7	∗	∗	NOUN
ap-5897	564	8	k+2	k+2	NUM
ap-5897	564	9	,	,	PUNCT
ap-5897	564	10	.	.	PUNCT
ap-5897	564	11	.	.	PUNCT
ap-5897	564	12	.	.	PUNCT
ap-5897	564	13	)	)	PUNCT
ap-5897	565	1	σk(b	σk(b	X
ap-5897	565	2	)	)	PUNCT
ap-5897	565	3	∈	∈	NOUN
ap-5897	566	1	[	[	X
ap-5897	566	2	γ	γ	X
ap-5897	566	3	,	,	PUNCT
ap-5897	566	4	γ	γ	X
ap-5897	566	5	+	+	NOUN
ap-5897	566	6	1	1	NUM
ap-5897	566	7	]	]	PUNCT
ap-5897	566	8	for	for	ADP
ap-5897	566	9	all	all	DET
ap-5897	566	10	k	k	PROPN
ap-5897	566	11	∈	∈	PROPN
ap-5897	566	12	n.	n.	NOUN
ap-5897	566	13	proof	proof	NOUN
ap-5897	566	14	.	.	PUNCT
ap-5897	567	1	suppose	suppose	VERB
ap-5897	567	2	d∗(b	d∗(b	PROPN
ap-5897	567	3	;	;	PUNCT
ap-5897	567	4	γ	γ	X
ap-5897	567	5	+	+	NOUN
ap-5897	567	6	1	1	NUM
ap-5897	567	7	)	)	PUNCT
ap-5897	567	8	=	=	NOUN
ap-5897	567	9	(	(	PUNCT
ap-5897	567	10	c∗1	c∗1	ADJ
ap-5897	567	11	,	,	PUNCT
ap-5897	567	12	c∗2	c∗2	NOUN
ap-5897	567	13	,	,	PUNCT
ap-5897	567	14	.	.	PUNCT
ap-5897	567	15	.	.	PUNCT
ap-5897	567	16	.	.	PUNCT
ap-5897	567	17	)	)	PUNCT
ap-5897	567	18	.	.	PUNCT
ap-5897	568	1	then	then	ADV
ap-5897	568	2	there	there	PRON
ap-5897	568	3	exist	exist	VERB
ap-5897	568	4	a	a	DET
ap-5897	568	5	sequence	sequence	NOUN
ap-5897	568	6	{	{	PUNCT
ap-5897	568	7	εn	εn	ADJ
ap-5897	568	8	}	}	PUNCT
ap-5897	568	9	converging	converge	VERB
ap-5897	568	10	to	to	ADP
ap-5897	568	11	0	0	NUM
ap-5897	568	12	and	and	CCONJ
ap-5897	568	13	a	a	DET
ap-5897	568	14	sequence	sequence	NOUN
ap-5897	568	15	{	{	PUNCT
ap-5897	568	16	yn	yn	NOUN
ap-5897	568	17	}	}	PUNCT
ap-5897	568	18	such	such	ADJ
ap-5897	568	19	that	that	SCONJ
ap-5897	568	20	yn	yn	PROPN
ap-5897	568	21	∈	∈	PROPN
ap-5897	568	22	(	(	PUNCT
ap-5897	568	23	γ	γ	X
ap-5897	568	24	+	+	X
ap-5897	568	25	1−	1−	NUM
ap-5897	568	26	εn	εn	ADJ
ap-5897	568	27	,	,	PUNCT
ap-5897	568	28	γ	γ	X
ap-5897	568	29	+	+	NOUN
ap-5897	568	30	1	1	NUM
ap-5897	568	31	)	)	PUNCT
ap-5897	568	32	and	and	CCONJ
ap-5897	568	33	d(b;yn	d(b;yn	NOUN
ap-5897	568	34	)	)	PUNCT
ap-5897	568	35	=	=	SYM
ap-5897	568	36	(	(	PUNCT
ap-5897	568	37	c∗1	c∗1	NOUN
ap-5897	568	38	,	,	PUNCT
ap-5897	568	39	.	.	PUNCT
ap-5897	568	40	.	.	PUNCT
ap-5897	569	1	.	.	PUNCT
ap-5897	570	1	,	,	PUNCT
ap-5897	570	2	c∗n	c∗n	PROPN
ap-5897	570	3	,	,	PUNCT
ap-5897	570	4	yn,1	yn,1	NOUN
ap-5897	570	5	,	,	PUNCT
ap-5897	570	6	yn,2	yn,2	PROPN
ap-5897	570	7	,	,	PUNCT
ap-5897	570	8	.	.	PUNCT
ap-5897	570	9	.	.	PUNCT
ap-5897	570	10	.	.	PUNCT
ap-5897	570	11	)	)	PUNCT
ap-5897	570	12	.	.	PUNCT
ap-5897	571	1	thus	thus	ADV
ap-5897	571	2	,	,	PUNCT
ap-5897	571	3	yn	yn	PROPN
ap-5897	571	4	=	=	PUNCT
ap-5897	571	5	n∑	n∑	PROPN
ap-5897	571	6	i=1	i=1	PROPN
ap-5897	571	7	c∗i	c∗i	VERB
ap-5897	571	8	b[i	b[i	NUM
ap-5897	571	9	]	]	PUNCT
ap-5897	572	1	+	+	CCONJ
ap-5897	572	2	∑	∑	ADV
ap-5897	572	3	i≥1	i≥1	PROPN
ap-5897	572	4	yn	yn	PROPN
ap-5897	572	5	,	,	PUNCT
ap-5897	572	6	i	i	PRON
ap-5897	572	7	b[n+	b[n+	VERB
ap-5897	572	8	i	i	PRON
ap-5897	572	9	]	]	PUNCT
ap-5897	572	10	.	.	PUNCT
ap-5897	573	1	since∑	since∑	VERB
ap-5897	573	2	i≥1	i≥1	ADJ
ap-5897	573	3	yn	yn	PROPN
ap-5897	573	4	,	,	PUNCT
ap-5897	573	5	i	i	PRON
ap-5897	573	6	b[n+	b[n+	VERB
ap-5897	574	1	i	i	PRON
ap-5897	574	2	]	]	X
ap-5897	574	3	=	=	SYM
ap-5897	574	4	1	1	NUM
ap-5897	574	5	b[n	b[n	NOUN
ap-5897	574	6	]	]	PUNCT
ap-5897	574	7	∑	∑	PUNCT
ap-5897	574	8	i≥1	i≥1	PROPN
ap-5897	574	9	yn	yn	PROPN
ap-5897	574	10	,	,	PUNCT
ap-5897	574	11	i	i	PRON
ap-5897	574	12	σn(b)[i	σn(b)[i	VERB
ap-5897	574	13	]	]	PUNCT
ap-5897	574	14	∈	∈	PROPN
ap-5897	574	15	1	1	NUM
ap-5897	574	16	b[n	b[n	NOUN
ap-5897	574	17	]	]	X
ap-5897	575	1	[	[	X
ap-5897	575	2	γ	γ	X
ap-5897	575	3	,	,	PUNCT
ap-5897	575	4	γ	γ	X
ap-5897	575	5	+	+	NOUN
ap-5897	575	6	1	1	NUM
ap-5897	575	7	)	)	PUNCT
ap-5897	575	8	,	,	PUNCT
ap-5897	575	9	then	then	ADV
ap-5897	575	10	lim	lim	PROPN
ap-5897	575	11	n→∞	n→∞	PROPN
ap-5897	575	12	∑	∑	PUNCT
ap-5897	575	13	i≥1	i≥1	PROPN
ap-5897	575	14	yn	yn	PROPN
ap-5897	575	15	,	,	PUNCT
ap-5897	575	16	i	i	PRON
ap-5897	575	17	b[n+	b[n+	VERB
ap-5897	575	18	i	i	PRON
ap-5897	575	19	]	]	X
ap-5897	575	20	=	=	PUNCT
ap-5897	576	1	0	0	X
ap-5897	576	2	.	.	PUNCT
ap-5897	577	1	hence	hence	ADV
ap-5897	577	2	,	,	PUNCT
ap-5897	577	3	γ	γ	X
ap-5897	577	4	+	+	NOUN
ap-5897	577	5	1	1	NUM
ap-5897	577	6	=	=	SYM
ap-5897	577	7	lim	lim	PROPN
ap-5897	577	8	n→∞	n→∞	X
ap-5897	578	1	yn	yn	PROPN
ap-5897	578	2	=	=	PROPN
ap-5897	578	3	lim	lim	PROPN
ap-5897	578	4	n→∞	n→∞	X
ap-5897	579	1			PROPN
ap-5897	579	2	n∑	n∑	PROPN
ap-5897	579	3	i=1	i=1	PROPN
ap-5897	579	4	c∗i	c∗i	VERB
ap-5897	579	5	b[i	b[i	NUM
ap-5897	579	6	]	]	PUNCT
ap-5897	580	1	+	+	CCONJ
ap-5897	580	2	∑	∑	ADV
ap-5897	580	3	i≥1	i≥1	PROPN
ap-5897	580	4	yn	yn	PROPN
ap-5897	580	5	,	,	PUNCT
ap-5897	580	6	i	i	PROPN
ap-5897	580	7	b[i	b[i	VERB
ap-5897	580	8	]	]	PUNCT
ap-5897	581	1			PROPN
ap-5897	581	2	=	=	SYM
ap-5897	581	3	∑	∑	PUNCT
ap-5897	581	4	i≥1	i≥1	ADJ
ap-5897	581	5	c∗i	c∗i	X
ap-5897	581	6	b[i	b[i	NOUN
ap-5897	581	7	]	]	PUNCT
ap-5897	581	8	.	.	PUNCT
ap-5897	582	1	now	now	ADV
ap-5897	582	2	,	,	PUNCT
ap-5897	582	3	for	for	ADP
ap-5897	582	4	j	j	PROPN
ap-5897	582	5	,	,	PUNCT
ap-5897	582	6	k	k	PROPN
ap-5897	582	7	∈	∈	PROPN
ap-5897	582	8	n	n	CCONJ
ap-5897	582	9	,	,	PUNCT
ap-5897	582	10	let	let	VERB
ap-5897	582	11	us	we	PRON
ap-5897	582	12	consider	consider	VERB
ap-5897	582	13	d(b;yk+j	d(b;yk+j	PRON
ap-5897	582	14	)	)	PUNCT
ap-5897	583	1	=	=	SYM
ap-5897	583	2	(	(	PUNCT
ap-5897	583	3	c∗1	c∗1	ADJ
ap-5897	583	4	,	,	PUNCT
ap-5897	583	5	c∗2	c∗2	NOUN
ap-5897	583	6	,	,	PUNCT
ap-5897	583	7	.	.	PUNCT
ap-5897	583	8	.	.	PUNCT
ap-5897	584	1	.	.	PUNCT
ap-5897	585	1	,	,	PUNCT
ap-5897	585	2	c∗k+j	c∗k+j	PROPN
ap-5897	585	3	,	,	PUNCT
ap-5897	585	4	yk+j,1	yk+j,1	PROPN
ap-5897	585	5	,	,	PUNCT
ap-5897	585	6	yk+j,2	yk+j,2	PROPN
ap-5897	585	7	,	,	PUNCT
ap-5897	585	8	.	.	PUNCT
ap-5897	585	9	.	.	PUNCT
ap-5897	585	10	.	.	PUNCT
ap-5897	585	11	)	)	PUNCT
ap-5897	585	12	.	.	PUNCT
ap-5897	586	1	by	by	ADP
ap-5897	586	2	corollary	corollary	ADJ
ap-5897	586	3	4.3.1	4.3.1	NUM
ap-5897	586	4	,	,	PUNCT
ap-5897	586	5	d(σk(b);t	d(σk(b);t	ADJ
ap-5897	586	6	k(yk+j	k(yk+j	NOUN
ap-5897	586	7	)	)	PUNCT
ap-5897	586	8	)	)	PUNCT
ap-5897	587	1	=	=	PUNCT
ap-5897	587	2	(	(	PUNCT
ap-5897	587	3	c∗k+1	c∗k+1	X
ap-5897	587	4	,	,	PUNCT
ap-5897	587	5	.	.	PUNCT
ap-5897	587	6	.	.	PUNCT
ap-5897	588	1	.	.	PUNCT
ap-5897	589	1	,	,	PUNCT
ap-5897	589	2	c	c	NOUN
ap-5897	589	3	∗	∗	X
ap-5897	589	4	k+j	k+j	PROPN
ap-5897	589	5	,	,	PUNCT
ap-5897	589	6	yk+j,1	yk+j,1	PROPN
ap-5897	589	7	,	,	PUNCT
ap-5897	589	8	yk+j,2	yk+j,2	PROPN
ap-5897	589	9	,	,	PUNCT
ap-5897	589	10	.	.	PUNCT
ap-5897	589	11	.	.	PUNCT
ap-5897	589	12	.	.	PUNCT
ap-5897	589	13	)	)	PUNCT
ap-5897	589	14	.	.	PUNCT
ap-5897	590	1	hence	hence	ADV
ap-5897	590	2	,	,	PUNCT
ap-5897	590	3	(	(	PUNCT
ap-5897	590	4	c∗k+1	c∗k+1	X
ap-5897	590	5	,	,	PUNCT
ap-5897	590	6	.	.	PUNCT
ap-5897	590	7	.	.	PUNCT
ap-5897	590	8	.	.	PUNCT
ap-5897	591	1	,	,	PUNCT
ap-5897	591	2	c	c	NOUN
ap-5897	591	3	∗	∗	X
ap-5897	591	4	k+j	k+j	PROPN
ap-5897	591	5	,	,	PUNCT
ap-5897	591	6	yk+j,1	yk+j,1	PROPN
ap-5897	591	7	,	,	PUNCT
ap-5897	591	8	yk+j,2	yk+j,2	PROPN
ap-5897	591	9	,	,	PUNCT
ap-5897	591	10	.	.	PUNCT
ap-5897	591	11	.	.	PUNCT
ap-5897	591	12	.	.	PUNCT
ap-5897	591	13	)	)	PUNCT
ap-5897	592	1	σk(b	σk(b	X
ap-5897	592	2	)	)	PUNCT
ap-5897	592	3	∈	∈	NOUN
ap-5897	593	1	[	[	X
ap-5897	593	2	γ	γ	X
ap-5897	593	3	,	,	PUNCT
ap-5897	593	4	γ	γ	X
ap-5897	593	5	+	+	NOUN
ap-5897	593	6	1	1	NUM
ap-5897	593	7	)	)	PUNCT
ap-5897	593	8	.	.	PUNCT
ap-5897	594	1	that	that	PRON
ap-5897	594	2	is	be	AUX
ap-5897	594	3	,	,	PUNCT
ap-5897	594	4	γ	γ	PROPN
ap-5897	594	5	≤	≤	ADJ
ap-5897	594	6	wj	wj	NOUN
ap-5897	594	7	:	:	PUNCT
ap-5897	594	8	=	=	NOUN
ap-5897	594	9	j∑	j∑	PROPN
ap-5897	594	10	i=1	i=1	X
ap-5897	594	11	c∗k+i	c∗k+i	NOUN
ap-5897	594	12	σk(b)[i	σk(b)[i	NOUN
ap-5897	594	13	]	]	X
ap-5897	594	14	+	+	CCONJ
ap-5897	594	15	∑	∑	ADV
ap-5897	594	16	i≥1	i≥1	PROPN
ap-5897	594	17	yk+j	yk+j	PROPN
ap-5897	594	18	,	,	PUNCT
ap-5897	594	19	i	i	PRON
ap-5897	594	20	σk(b)[j	σk(b)[j	VERB
ap-5897	595	1	+	+	CCONJ
ap-5897	595	2	i	i	X
ap-5897	595	3	]	]	X
ap-5897	595	4	<	<	X
ap-5897	595	5	γ	γ	X
ap-5897	595	6	+	+	PROPN
ap-5897	595	7	1	1	NUM
ap-5897	595	8	.	.	PUNCT
ap-5897	595	9	since	since	SCONJ
ap-5897	595	10	{	{	PUNCT
ap-5897	595	11	wj	wj	NOUN
ap-5897	595	12	}	}	PUNCT
ap-5897	595	13	tends	tend	VERB
ap-5897	595	14	to	to	PART
ap-5897	595	15	(	(	PUNCT
ap-5897	595	16	c∗k+1	c∗k+1	X
ap-5897	595	17	,	,	PUNCT
ap-5897	595	18	c	c	NOUN
ap-5897	595	19	∗	∗	NOUN
ap-5897	595	20	k+2	k+2	NUM
ap-5897	595	21	,	,	PUNCT
ap-5897	595	22	.	.	PUNCT
ap-5897	595	23	.	.	PUNCT
ap-5897	595	24	.	.	PUNCT
ap-5897	595	25	)	)	PUNCT
ap-5897	596	1	σk(b	σk(b	X
ap-5897	596	2	)	)	PUNCT
ap-5897	597	1	,	,	PUNCT
ap-5897	597	2	then	then	ADV
ap-5897	597	3	γ	γ	PROPN
ap-5897	597	4	≤	≤	X
ap-5897	597	5	(	(	PUNCT
ap-5897	597	6	c∗k+1	c∗k+1	X
ap-5897	597	7	,	,	PUNCT
ap-5897	597	8	c	c	NOUN
ap-5897	597	9	∗	∗	NOUN
ap-5897	597	10	k+2	k+2	NUM
ap-5897	597	11	,	,	PUNCT
ap-5897	597	12	.	.	PUNCT
ap-5897	597	13	.	.	PUNCT
ap-5897	597	14	.	.	PUNCT
ap-5897	597	15	)	)	PUNCT
ap-5897	598	1	≤	≤	NUM
ap-5897	598	2	γ	γ	X
ap-5897	598	3	+	+	NOUN
ap-5897	598	4	1	1	X
ap-5897	598	5	.	.	X
ap-5897	598	6	proposition	proposition	NOUN
ap-5897	598	7	4.7	4.7	NUM
ap-5897	598	8	.	.	PUNCT
ap-5897	599	1	if	if	SCONJ
ap-5897	599	2	x	x	SYM
ap-5897	599	3	∈	∈	PROPN
ap-5897	599	4	[	[	X
ap-5897	599	5	γ	γ	X
ap-5897	599	6	,	,	PUNCT
ap-5897	599	7	γ	γ	X
ap-5897	599	8	+	+	NOUN
ap-5897	599	9	1	1	NUM
ap-5897	599	10	)	)	PUNCT
ap-5897	599	11	,	,	PUNCT
ap-5897	599	12	then	then	ADV
ap-5897	599	13	d(b;x	d(b;x	PROPN
ap-5897	599	14	)	)	PUNCT
ap-5897	599	15	≺b	≺b	PROPN
ap-5897	599	16	d∗(b	d∗(b	PROPN
ap-5897	599	17	;	;	PUNCT
ap-5897	599	18	γ	γ	X
ap-5897	599	19	+	+	NOUN
ap-5897	599	20	1	1	NUM
ap-5897	599	21	)	)	PUNCT
ap-5897	599	22	.	.	PUNCT
ap-5897	600	1	221	221	NUM
ap-5897	600	2	jonathan	jonathan	PROPN
ap-5897	600	3	caalim	caalim	PROPN
ap-5897	600	4	,	,	PUNCT
ap-5897	600	5	shiela	shiela	PROPN
ap-5897	600	6	demegillo	demegillo	PROPN
ap-5897	600	7	acta	acta	PROPN
ap-5897	600	8	polytechnica	polytechnica	PROPN
ap-5897	600	9	proof	proof	NOUN
ap-5897	600	10	.	.	PUNCT
ap-5897	601	1	let	let	VERB
ap-5897	601	2	d∗(b	d∗(b	PROPN
ap-5897	601	3	;	;	PUNCT
ap-5897	601	4	γ	γ	X
ap-5897	601	5	+	+	NOUN
ap-5897	601	6	1	1	NUM
ap-5897	601	7	)	)	PUNCT
ap-5897	601	8	=	=	NOUN
ap-5897	601	9	(	(	PUNCT
ap-5897	601	10	c∗1	c∗1	ADJ
ap-5897	601	11	,	,	PUNCT
ap-5897	601	12	c∗2	c∗2	NOUN
ap-5897	601	13	,	,	PUNCT
ap-5897	601	14	.	.	PUNCT
ap-5897	601	15	.	.	PUNCT
ap-5897	601	16	.	.	PUNCT
ap-5897	601	17	)	)	PUNCT
ap-5897	601	18	.	.	PUNCT
ap-5897	602	1	then	then	ADV
ap-5897	602	2	there	there	PRON
ap-5897	602	3	exist	exist	VERB
ap-5897	602	4	a	a	DET
ap-5897	602	5	sequence	sequence	NOUN
ap-5897	602	6	εn	εn	ADP
ap-5897	602	7	tending	tend	VERB
ap-5897	602	8	to	to	ADP
ap-5897	602	9	zero	zero	NUM
ap-5897	602	10	and	and	CCONJ
ap-5897	602	11	yn	yn	PROPN
ap-5897	602	12	∈	∈	PROPN
ap-5897	602	13	(	(	PUNCT
ap-5897	602	14	γ	γ	X
ap-5897	602	15	+	+	NOUN
ap-5897	602	16	1	1	NUM
ap-5897	602	17	−	−	NOUN
ap-5897	602	18	εn	εn	ADJ
ap-5897	602	19	,	,	PUNCT
ap-5897	602	20	γ	γ	X
ap-5897	602	21	+	+	NOUN
ap-5897	602	22	1	1	NUM
ap-5897	602	23	)	)	PUNCT
ap-5897	602	24	such	such	ADJ
ap-5897	602	25	that	that	DET
ap-5897	602	26	d(b;yn	d(b;yn	NOUN
ap-5897	602	27	)	)	PUNCT
ap-5897	603	1	=	=	SYM
ap-5897	603	2	(	(	PUNCT
ap-5897	603	3	c∗1	c∗1	NOUN
ap-5897	603	4	,	,	PUNCT
ap-5897	603	5	.	.	PUNCT
ap-5897	603	6	.	.	PUNCT
ap-5897	604	1	.	.	PUNCT
ap-5897	605	1	,	,	PUNCT
ap-5897	605	2	c∗n	c∗n	PROPN
ap-5897	605	3	,	,	PUNCT
ap-5897	605	4	yn,1	yn,1	NOUN
ap-5897	605	5	,	,	PUNCT
ap-5897	605	6	yn,2	yn,2	PROPN
ap-5897	605	7	,	,	PUNCT
ap-5897	605	8	.	.	PUNCT
ap-5897	605	9	.	.	PUNCT
ap-5897	605	10	.	.	PUNCT
ap-5897	605	11	)	)	PUNCT
ap-5897	606	1	with	with	ADP
ap-5897	606	2	c∗n+1	c∗n+1	PROPN
ap-5897	606	3	6=	6=	ADP
ap-5897	606	4	yn,1	yn,1	PROPN
ap-5897	606	5	,	,	PUNCT
ap-5897	606	6	so	so	SCONJ
ap-5897	606	7	that	that	SCONJ
ap-5897	606	8	d(b;yn	d(b;yn	NOUN
ap-5897	606	9	)	)	PUNCT
ap-5897	606	10	6=	6=	ADP
ap-5897	607	1	d∗(b	d∗(b	PROPN
ap-5897	607	2	;	;	PUNCT
ap-5897	607	3	γ	γ	X
ap-5897	607	4	+	+	NOUN
ap-5897	607	5	1	1	NUM
ap-5897	607	6	)	)	PUNCT
ap-5897	607	7	.	.	PUNCT
ap-5897	608	1	suppose	suppose	VERB
ap-5897	608	2	d(b;yn	d(b;yn	PROPN
ap-5897	608	3	)	)	PUNCT
ap-5897	608	4	�	�	PROPN
ap-5897	608	5	b	b	PROPN
ap-5897	608	6	d∗(b	d∗(b	PROPN
ap-5897	608	7	;	;	PUNCT
ap-5897	608	8	γ	γ	X
ap-5897	608	9	+	+	NOUN
ap-5897	608	10	1	1	NUM
ap-5897	608	11	)	)	PUNCT
ap-5897	608	12	.	.	PUNCT
ap-5897	609	1	there	there	PRON
ap-5897	609	2	exists	exist	VERB
ap-5897	609	3	yn+m	yn+m	PROPN
ap-5897	609	4	∈	∈	PROPN
ap-5897	609	5	(	(	PUNCT
ap-5897	609	6	yn	yn	PROPN
ap-5897	609	7	,	,	PUNCT
ap-5897	609	8	γ	γ	X
ap-5897	609	9	+	+	NOUN
ap-5897	609	10	1	1	NUM
ap-5897	609	11	)	)	PUNCT
ap-5897	609	12	where	where	SCONJ
ap-5897	609	13	m	m	PROPN
ap-5897	609	14	≥	≥	VERB
ap-5897	609	15	1	1	NUM
ap-5897	609	16	such	such	ADJ
ap-5897	609	17	that	that	DET
ap-5897	609	18	d(b;yn+m	d(b;yn+m	PROPN
ap-5897	609	19	)	)	PUNCT
ap-5897	609	20	=	=	PUNCT
ap-5897	609	21	(	(	PUNCT
ap-5897	609	22	c∗1	c∗1	NOUN
ap-5897	609	23	,	,	PUNCT
ap-5897	609	24	.	.	PUNCT
ap-5897	609	25	.	.	PUNCT
ap-5897	610	1	.	.	PUNCT
ap-5897	611	1	,	,	PUNCT
ap-5897	611	2	c∗n+m	c∗n+m	PROPN
ap-5897	611	3	,	,	PUNCT
ap-5897	611	4	yn+m,1	yn+m,1	PROPN
ap-5897	611	5	,	,	PUNCT
ap-5897	611	6	yn+m,2	yn+m,2	PROPN
ap-5897	611	7	,	,	PUNCT
ap-5897	611	8	.	.	PUNCT
ap-5897	611	9	.	.	PUNCT
ap-5897	611	10	.	.	PUNCT
ap-5897	611	11	)	)	PUNCT
ap-5897	611	12	.	.	PUNCT
ap-5897	612	1	since	since	SCONJ
ap-5897	612	2	d(b;yn	d(b;yn	PROPN
ap-5897	612	3	)	)	PUNCT
ap-5897	612	4	�	�	PROPN
ap-5897	612	5	b	b	PROPN
ap-5897	612	6	d∗(b	d∗(b	PROPN
ap-5897	612	7	,	,	PUNCT
ap-5897	612	8	γ	γ	X
ap-5897	612	9	+	+	NOUN
ap-5897	612	10	1	1	NUM
ap-5897	612	11	)	)	PUNCT
ap-5897	612	12	,	,	PUNCT
ap-5897	612	13	then	then	ADV
ap-5897	612	14	(	(	PUNCT
ap-5897	612	15	yn,1	yn,1	PROPN
ap-5897	612	16	−	−	PROPN
ap-5897	612	17	c∗n+1	c∗n+1	NOUN
ap-5897	612	18	)	)	PUNCT
ap-5897	612	19	sgn(b[n+	sgn(b[n+	NOUN
ap-5897	612	20	1	1	NUM
ap-5897	612	21	]	]	PUNCT
ap-5897	612	22	)	)	PUNCT
ap-5897	612	23	≥	≥	NOUN
ap-5897	612	24	1	1	NUM
ap-5897	612	25	,	,	PUNCT
ap-5897	612	26	implying	imply	VERB
ap-5897	612	27	that	that	SCONJ
ap-5897	612	28	d(b;yn	d(b;yn	NOUN
ap-5897	612	29	)	)	PUNCT
ap-5897	612	30	�	�	PROPN
ap-5897	612	31	b	b	PROPN
ap-5897	612	32	d(b;yn+m	d(b;yn+m	PROPN
ap-5897	612	33	)	)	PUNCT
ap-5897	612	34	.	.	PUNCT
ap-5897	613	1	by	by	ADP
ap-5897	613	2	proposition	proposition	NOUN
ap-5897	613	3	4.5	4.5	NUM
ap-5897	613	4	,	,	PUNCT
ap-5897	613	5	yn	yn	PROPN
ap-5897	613	6	>	>	X
ap-5897	613	7	yn+m	yn+m	PROPN
ap-5897	614	1	which	which	PRON
ap-5897	614	2	is	be	AUX
ap-5897	614	3	a	a	DET
ap-5897	614	4	contradiction	contradiction	NOUN
ap-5897	614	5	since	since	SCONJ
ap-5897	614	6	yn+m	yn+m	PROPN
ap-5897	614	7	∈	∈	PROPN
ap-5897	614	8	(	(	PUNCT
ap-5897	614	9	yn	yn	PROPN
ap-5897	614	10	,	,	PUNCT
ap-5897	614	11	γ	γ	X
ap-5897	614	12	+	+	NOUN
ap-5897	614	13	1	1	NUM
ap-5897	614	14	)	)	PUNCT
ap-5897	614	15	.	.	PUNCT
ap-5897	615	1	hence	hence	ADV
ap-5897	615	2	,	,	PUNCT
ap-5897	615	3	if	if	SCONJ
ap-5897	615	4	x	x	ADP
ap-5897	615	5	<	<	X
ap-5897	615	6	yn	yn	PROPN
ap-5897	615	7	,	,	PUNCT
ap-5897	615	8	then	then	ADV
ap-5897	615	9	d(b;x	d(b;x	PROPN
ap-5897	615	10	)	)	PUNCT
ap-5897	615	11	≺b	≺b	PROPN
ap-5897	615	12	d(b;yn	d(b;yn	NOUN
ap-5897	615	13	)	)	PUNCT
ap-5897	615	14	≺b	≺b	PROPN
ap-5897	615	15	d∗(b	d∗(b	PROPN
ap-5897	615	16	;	;	PUNCT
ap-5897	615	17	γ+1	γ+1	NUM
ap-5897	615	18	)	)	PUNCT
ap-5897	615	19	.	.	PUNCT
ap-5897	616	1	definition	definition	NOUN
ap-5897	616	2	4	4	NUM
ap-5897	616	3	.	.	PUNCT
ap-5897	617	1	a	a	DET
ap-5897	617	2	sequence	sequence	NOUN
ap-5897	617	3	(	(	PUNCT
ap-5897	617	4	d1	d1	PROPN
ap-5897	617	5	,	,	PUNCT
ap-5897	617	6	d2	d2	PROPN
ap-5897	617	7	,	,	PUNCT
ap-5897	617	8	.	.	PUNCT
ap-5897	617	9	.	.	PUNCT
ap-5897	617	10	.	.	PUNCT
ap-5897	617	11	)	)	PUNCT
ap-5897	618	1	∈	∈	PROPN
ap-5897	618	2	a(b	a(b	NOUN
ap-5897	618	3	)	)	PUNCT
ap-5897	618	4	satisfies	satisfy	VERB
ap-5897	618	5	the	the	DET
ap-5897	618	6	lexicographic	lexicographic	ADJ
ap-5897	618	7	restriction	restriction	NOUN
ap-5897	618	8	if	if	SCONJ
ap-5897	618	9	,	,	PUNCT
ap-5897	618	10	for	for	ADP
ap-5897	618	11	all	all	DET
ap-5897	618	12	k	k	PROPN
ap-5897	618	13	∈	∈	PROPN
ap-5897	618	14	n∪{0	n∪{0	NOUN
ap-5897	618	15	}	}	PUNCT
ap-5897	618	16	,	,	PUNCT
ap-5897	618	17	d(σk(b	d(σk(b	PROPN
ap-5897	618	18	)	)	PUNCT
ap-5897	618	19	;	;	PUNCT
ap-5897	618	20	γ	γ	X
ap-5897	618	21	)	)	PUNCT
ap-5897	618	22	�	�	PROPN
ap-5897	618	23	σk(b	σk(b	X
ap-5897	618	24	)	)	PUNCT
ap-5897	618	25	σk(d1	σk(d1	PROPN
ap-5897	618	26	,	,	PUNCT
ap-5897	618	27	d2	d2	PROPN
ap-5897	618	28	,	,	PUNCT
ap-5897	618	29	.	.	PUNCT
ap-5897	618	30	.	.	PUNCT
ap-5897	618	31	.	.	PUNCT
ap-5897	618	32	)	)	PUNCT
ap-5897	619	1	≺σk(b	≺σk(b	NOUN
ap-5897	619	2	)	)	PUNCT
ap-5897	619	3	d∗(σk(b	d∗(σk(b	PROPN
ap-5897	619	4	)	)	PUNCT
ap-5897	619	5	;	;	PUNCT
ap-5897	619	6	γ	γ	X
ap-5897	619	7	+	+	NOUN
ap-5897	619	8	1	1	NUM
ap-5897	619	9	)	)	PUNCT
ap-5897	619	10	.	.	PUNCT
ap-5897	620	1	combining	combine	VERB
ap-5897	620	2	corollary	corollary	ADJ
ap-5897	620	3	4.5.1	4.5.1	NOUN
ap-5897	620	4	and	and	CCONJ
ap-5897	620	5	proposition	proposition	NOUN
ap-5897	620	6	4.7	4.7	NUM
ap-5897	620	7	yields	yield	NOUN
ap-5897	620	8	the	the	DET
ap-5897	620	9	following	follow	VERB
ap-5897	620	10	proposition	proposition	NOUN
ap-5897	620	11	.	.	PUNCT
ap-5897	621	1	proposition	proposition	NOUN
ap-5897	621	2	4.8	4.8	NUM
ap-5897	621	3	.	.	PUNCT
ap-5897	622	1	let	let	VERB
ap-5897	622	2	x	x	X
ap-5897	622	3	∈	∈	PROPN
ap-5897	622	4	[	[	X
ap-5897	622	5	γ	γ	X
ap-5897	622	6	,	,	PUNCT
ap-5897	622	7	γ	γ	X
ap-5897	622	8	+	+	NOUN
ap-5897	622	9	1	1	NUM
ap-5897	622	10	)	)	PUNCT
ap-5897	622	11	.	.	PUNCT
ap-5897	623	1	then	then	ADV
ap-5897	623	2	d(b;x	d(b;x	NOUN
ap-5897	623	3	)	)	PUNCT
ap-5897	623	4	satisfies	satisfy	VERB
ap-5897	623	5	the	the	DET
ap-5897	623	6	lexicographic	lexicographic	ADJ
ap-5897	623	7	restriction	restriction	NOUN
ap-5897	623	8	.	.	PUNCT
ap-5897	624	1	we	we	PRON
ap-5897	624	2	now	now	ADV
ap-5897	624	3	show	show	VERB
ap-5897	624	4	that	that	SCONJ
ap-5897	624	5	the	the	DET
ap-5897	624	6	converse	converse	NOUN
ap-5897	624	7	of	of	ADP
ap-5897	624	8	prop	prop	PROPN
ap-5897	624	9	.	.	PUNCT
ap-5897	624	10	4.8	4.8	NUM
ap-5897	624	11	holds	hold	VERB
ap-5897	624	12	under	under	ADP
ap-5897	624	13	some	some	DET
ap-5897	624	14	condition	condition	NOUN
ap-5897	624	15	.	.	PUNCT
ap-5897	625	1	to	to	PART
ap-5897	625	2	proceed	proceed	VERB
ap-5897	625	3	,	,	PUNCT
ap-5897	625	4	consider	consider	VERB
ap-5897	625	5	a	a	DET
ap-5897	625	6	sequence	sequence	NOUN
ap-5897	625	7	z	z	NOUN
ap-5897	625	8	=	=	SYM
ap-5897	625	9	(	(	PUNCT
ap-5897	625	10	z1	z1	PROPN
ap-5897	625	11	,	,	PUNCT
ap-5897	625	12	z2	z2	PROPN
ap-5897	625	13	,	,	PUNCT
ap-5897	625	14	.	.	PUNCT
ap-5897	625	15	.	.	PUNCT
ap-5897	625	16	.	.	PUNCT
ap-5897	625	17	)	)	PUNCT
ap-5897	626	1	∈	∈	PROPN
ap-5897	626	2	a(b	a(b	NOUN
ap-5897	626	3	)	)	PUNCT
ap-5897	626	4	.	.	PUNCT
ap-5897	627	1	for	for	ADP
ap-5897	627	2	i	i	PROPN
ap-5897	627	3	∈	∈	PROPN
ap-5897	627	4	n	n	CCONJ
ap-5897	627	5	,	,	PUNCT
ap-5897	627	6	we	we	PRON
ap-5897	627	7	define	define	VERB
ap-5897	627	8	z(i	z(i	PROPN
ap-5897	627	9	,	,	PUNCT
ap-5897	627	10	j	j	NOUN
ap-5897	627	11	)	)	PUNCT
ap-5897	627	12	=	=	PRON
ap-5897	627	13	{	{	PUNCT
ap-5897	627	14	(	(	PUNCT
ap-5897	627	15	zi	zi	NOUN
ap-5897	627	16	,	,	PUNCT
ap-5897	627	17	zi+1	zi+1	X
ap-5897	627	18	,	,	PUNCT
ap-5897	627	19	.	.	PUNCT
ap-5897	627	20	.	.	PUNCT
ap-5897	627	21	.	.	PUNCT
ap-5897	628	1	,	,	PUNCT
ap-5897	628	2	zi+j	zi+j	NUM
ap-5897	628	3	)	)	PUNCT
ap-5897	628	4	if	if	SCONJ
ap-5897	628	5	j	j	PROPN
ap-5897	628	6	∈	∈	PROPN
ap-5897	628	7	n	n	PART
ap-5897	628	8	∪	∪	X
ap-5897	628	9	{	{	PUNCT
ap-5897	628	10	0	0	NUM
ap-5897	628	11	}	}	PUNCT
ap-5897	628	12	(	(	PUNCT
ap-5897	628	13	zi	zi	NOUN
ap-5897	628	14	,	,	PUNCT
ap-5897	628	15	zi+1	zi+1	X
ap-5897	628	16	,	,	PUNCT
ap-5897	628	17	.	.	PUNCT
ap-5897	628	18	.	.	PUNCT
ap-5897	628	19	.	.	PUNCT
ap-5897	628	20	)	)	PUNCT
ap-5897	629	1	if	if	SCONJ
ap-5897	629	2	j	j	PROPN
ap-5897	629	3	=	=	SYM
ap-5897	629	4	∞	∞	PROPN
ap-5897	629	5	and	and	CCONJ
ap-5897	629	6	set	set	VERB
ap-5897	629	7	z(i	z(i	PROPN
ap-5897	629	8	,	,	PUNCT
ap-5897	629	9	j)σi−1(b	j)σi−1(b	PROPN
ap-5897	629	10	)	)	PUNCT
ap-5897	629	11	=	=	SYM
ap-5897	629	12	j∑	j∑	PROPN
ap-5897	629	13	k=0	k=0	PUNCT
ap-5897	629	14	zi+k	zi+k	PROPN
ap-5897	629	15	b[i	b[i	NOUN
ap-5897	629	16	,	,	PUNCT
ap-5897	629	17	i+	i+	X
ap-5897	629	18	k	k	X
ap-5897	629	19	]	]	PUNCT
ap-5897	629	20	,	,	PUNCT
ap-5897	629	21	provided	provide	VERB
ap-5897	629	22	that	that	SCONJ
ap-5897	629	23	the	the	DET
ap-5897	629	24	sum	sum	NOUN
ap-5897	629	25	converges	converge	VERB
ap-5897	629	26	if	if	SCONJ
ap-5897	629	27	j	j	PROPN
ap-5897	629	28	=	=	AUX
ap-5897	629	29	∞.	∞.	PROPN
ap-5897	629	30	for	for	ADP
ap-5897	629	31	n	n	PRON
ap-5897	629	32	∈	∈	PROPN
ap-5897	629	33	n∪	n∪	PROPN
ap-5897	629	34	{	{	PUNCT
ap-5897	629	35	0	0	NUM
ap-5897	629	36	}	}	PUNCT
ap-5897	629	37	,	,	PUNCT
ap-5897	629	38	let	let	VERB
ap-5897	629	39	u(n	u(n	PRON
ap-5897	629	40	)	)	PUNCT
ap-5897	629	41	=	=	SYM
ap-5897	630	1	d(σn(b	d(σn(b	PROPN
ap-5897	630	2	)	)	PUNCT
ap-5897	630	3	;	;	PUNCT
ap-5897	630	4	γ	γ	X
ap-5897	630	5	)	)	PUNCT
ap-5897	630	6	and	and	CCONJ
ap-5897	630	7	v(n	v(n	NOUN
ap-5897	630	8	)	)	PUNCT
ap-5897	630	9	=	=	SYM
ap-5897	630	10	d∗(σn(b	d∗(σn(b	NUM
ap-5897	630	11	)	)	PUNCT
ap-5897	630	12	;	;	PUNCT
ap-5897	630	13	γ	γ	PROPN
ap-5897	630	14	+	+	NOUN
ap-5897	630	15	1	1	NUM
ap-5897	630	16	)	)	PUNCT
ap-5897	630	17	.	.	PUNCT
ap-5897	631	1	lemma	lemma	PROPN
ap-5897	631	2	4.9	4.9	NUM
ap-5897	631	3	.	.	PUNCT
ap-5897	632	1	let	let	VERB
ap-5897	632	2	w	w	NOUN
ap-5897	632	3	=	=	SYM
ap-5897	632	4	(	(	PUNCT
ap-5897	632	5	w1	w1	NOUN
ap-5897	632	6	,	,	PUNCT
ap-5897	632	7	w2	w2	NOUN
ap-5897	632	8	,	,	PUNCT
ap-5897	632	9	.	.	PUNCT
ap-5897	632	10	.	.	PUNCT
ap-5897	632	11	.	.	PUNCT
ap-5897	632	12	)	)	PUNCT
ap-5897	633	1	∈	∈	PROPN
ap-5897	633	2	a(b	a(b	NOUN
ap-5897	633	3	)	)	PUNCT
ap-5897	633	4	.	.	PUNCT
ap-5897	634	1	if	if	SCONJ
ap-5897	634	2	w	w	NOUN
ap-5897	634	3	satisfies	satisfy	VERB
ap-5897	634	4	the	the	DET
ap-5897	634	5	lexicographic	lexicographic	ADJ
ap-5897	634	6	restriction	restriction	NOUN
ap-5897	634	7	,	,	PUNCT
ap-5897	634	8	then	then	ADV
ap-5897	634	9	there	there	PRON
ap-5897	634	10	are	be	VERB
ap-5897	634	11	infinitely	infinitely	ADV
ap-5897	634	12	many	many	ADJ
ap-5897	634	13	n	n	ADP
ap-5897	634	14	such	such	ADJ
ap-5897	634	15	that	that	SCONJ
ap-5897	634	16	at	at	ADV
ap-5897	634	17	least	least	ADJ
ap-5897	634	18	one	one	NUM
ap-5897	634	19	of	of	ADP
ap-5897	634	20	the	the	DET
ap-5897	634	21	two	two	NUM
ap-5897	634	22	holds	hold	NOUN
ap-5897	634	23	:	:	PUNCT
ap-5897	634	24	(	(	PUNCT
ap-5897	634	25	1	1	NUM
ap-5897	634	26	.	.	PUNCT
ap-5897	634	27	)	)	PUNCT
ap-5897	635	1	b[n	b[n	VERB
ap-5897	635	2	]	]	PUNCT
ap-5897	635	3	>	>	X
ap-5897	635	4	0	0	PUNCT
ap-5897	635	5	and	and	CCONJ
ap-5897	635	6	(	(	PUNCT
ap-5897	635	7	w(1	w(1	INTJ
ap-5897	635	8	,	,	PUNCT
ap-5897	635	9	n−	n−	NOUN
ap-5897	635	10	1	1	NUM
ap-5897	635	11	)	)	PUNCT
ap-5897	635	12	◦	◦	NOUN
ap-5897	635	13	u(n))b	u(n))b	PROPN
ap-5897	635	14	≥	≥	PROPN
ap-5897	635	15	γ	γ	NOUN
ap-5897	635	16	;	;	PUNCT
ap-5897	635	17	(	(	PUNCT
ap-5897	635	18	2	2	NUM
ap-5897	635	19	.	.	PUNCT
ap-5897	635	20	)	)	PUNCT
ap-5897	636	1	b[n	b[n	VERB
ap-5897	636	2	]	]	PUNCT
ap-5897	636	3	<	<	X
ap-5897	636	4	0	0	PUNCT
ap-5897	636	5	and	and	CCONJ
ap-5897	636	6	(	(	PUNCT
ap-5897	636	7	w(1	w(1	INTJ
ap-5897	636	8	,	,	PUNCT
ap-5897	636	9	n−	n−	NOUN
ap-5897	636	10	1	1	NUM
ap-5897	636	11	)	)	PUNCT
ap-5897	636	12	◦	◦	NOUN
ap-5897	636	13	v(n))b	v(n))b	PROPN
ap-5897	636	14	≥	≥	NUM
ap-5897	636	15	γ	γ	X
ap-5897	636	16	.	.	PROPN
ap-5897	636	17	proof	proof	NOUN
ap-5897	636	18	.	.	PUNCT
ap-5897	637	1	suppose	suppose	VERB
ap-5897	637	2	w	w	ADP
ap-5897	637	3	6=	6=	ADP
ap-5897	637	4	d(b	d(b	PROPN
ap-5897	637	5	;	;	PUNCT
ap-5897	637	6	γ	γ	X
ap-5897	637	7	)	)	PUNCT
ap-5897	637	8	.	.	PUNCT
ap-5897	638	1	for	for	ADP
ap-5897	638	2	the	the	DET
ap-5897	638	3	base	base	NOUN
ap-5897	638	4	of	of	ADP
ap-5897	638	5	the	the	DET
ap-5897	638	6	induction	induction	NOUN
ap-5897	638	7	,	,	PUNCT
ap-5897	638	8	we	we	PRON
ap-5897	638	9	set	set	VERB
ap-5897	638	10	n	n	NOUN
ap-5897	638	11	=	=	SYM
ap-5897	638	12	0	0	PUNCT
ap-5897	638	13	and	and	CCONJ
ap-5897	638	14	define	define	VERB
ap-5897	638	15	w(1,−1	w(1,−1	PROPN
ap-5897	638	16	)	)	PUNCT
ap-5897	638	17	as	as	ADP
ap-5897	638	18	the	the	DET
ap-5897	638	19	empty	empty	ADJ
ap-5897	638	20	word	word	NOUN
ap-5897	638	21	.	.	PUNCT
ap-5897	639	1	then	then	ADV
ap-5897	639	2	(	(	PUNCT
ap-5897	639	3	w(1,−1	w(1,−1	PROPN
ap-5897	639	4	)	)	PUNCT
ap-5897	639	5	◦	◦	NOUN
ap-5897	639	6	u(0))b	u(0))b	NOUN
ap-5897	639	7	=	=	SYM
ap-5897	639	8	γ	γ	X
ap-5897	639	9	.	.	PROPN
ap-5897	640	1	likewise	likewise	ADV
ap-5897	640	2	(	(	PUNCT
ap-5897	640	3	w(1,−1	w(1,−1	PROPN
ap-5897	640	4	)	)	PUNCT
ap-5897	640	5	◦	◦	NOUN
ap-5897	640	6	v(0))b	v(0))b	NOUN
ap-5897	640	7	=	=	PUNCT
ap-5897	640	8	γ+	γ+	PUNCT
ap-5897	640	9	1	1	NUM
ap-5897	640	10	≥	≥	NUM
ap-5897	640	11	γ	γ	X
ap-5897	640	12	.	.	PROPN
ap-5897	641	1	now	now	ADV
ap-5897	641	2	,	,	PUNCT
ap-5897	641	3	let	let	VERB
ap-5897	641	4	m	m	PRON
ap-5897	641	5	∈	∈	NOUN
ap-5897	641	6	n∪{0	n∪{0	NOUN
ap-5897	641	7	}	}	PUNCT
ap-5897	641	8	.	.	PUNCT
ap-5897	642	1	case	case	NOUN
ap-5897	642	2	1	1	NUM
ap-5897	642	3	suppose	suppose	VERB
ap-5897	642	4	b[m	b[m	PROPN
ap-5897	642	5	]	]	PUNCT
ap-5897	642	6	>	>	X
ap-5897	642	7	0	0	PUNCT
ap-5897	643	1	and	and	CCONJ
ap-5897	643	2	(	(	PUNCT
ap-5897	643	3	w(1,m	w(1,m	NOUN
ap-5897	643	4	−	−	PROPN
ap-5897	643	5	1	1	NUM
ap-5897	643	6	)	)	PUNCT
ap-5897	643	7	◦	◦	NOUN
ap-5897	643	8	u(m))b	u(m))b	PROPN
ap-5897	643	9	≥	≥	NOUN
ap-5897	643	10	γ	γ	NOUN
ap-5897	643	11	hold	hold	VERB
ap-5897	643	12	.	.	PUNCT
ap-5897	644	1	by	by	ADP
ap-5897	644	2	the	the	DET
ap-5897	644	3	lexicographic	lexicographic	ADJ
ap-5897	644	4	restriction	restriction	NOUN
ap-5897	644	5	,	,	PUNCT
ap-5897	644	6	u(m	u(m	ADJ
ap-5897	644	7	)	)	PUNCT
ap-5897	644	8	=	=	SYM
ap-5897	644	9	d(σm(b	d(σm(b	NOUN
ap-5897	644	10	)	)	PUNCT
ap-5897	644	11	;	;	PUNCT
ap-5897	644	12	γ	γ	X
ap-5897	644	13	)	)	PUNCT
ap-5897	644	14	≺σm(b	≺σm(b	NOUN
ap-5897	644	15	)	)	PUNCT
ap-5897	644	16	σ	σ	NOUN
ap-5897	644	17	m(w	m(w	NOUN
ap-5897	644	18	)	)	PUNCT
ap-5897	644	19	=	=	SYM
ap-5897	644	20	w(m+	w(m+	X
ap-5897	644	21	1,∞	1,∞	NUM
ap-5897	644	22	)	)	PUNCT
ap-5897	644	23	.	.	PUNCT
ap-5897	645	1	thus	thus	ADV
ap-5897	645	2	,	,	PUNCT
ap-5897	645	3	there	there	PRON
ap-5897	645	4	exists	exist	VERB
ap-5897	645	5	a	a	DET
ap-5897	645	6	least	least	ADV
ap-5897	645	7	positive	positive	ADJ
ap-5897	645	8	integer	integer	NOUN
ap-5897	645	9	l	l	NOUN
ap-5897	645	10	such	such	ADJ
ap-5897	645	11	that	that	SCONJ
ap-5897	645	12	wm+i	wm+i	PROPN
ap-5897	645	13	=	=	SYM
ap-5897	645	14	u(m)(i	u(m)(i	PROPN
ap-5897	645	15	,	,	PUNCT
ap-5897	645	16	0	0	NUM
ap-5897	645	17	)	)	PUNCT
ap-5897	645	18	for	for	ADP
ap-5897	645	19	all	all	DET
ap-5897	645	20	i	i	PRON
ap-5897	645	21	<	<	X
ap-5897	645	22	l	l	NOUN
ap-5897	645	23	and	and	CCONJ
ap-5897	645	24	(	(	PUNCT
ap-5897	645	25	wm+l	wm+l	PROPN
ap-5897	645	26	−	−	PROPN
ap-5897	645	27	u(m)(l	u(m)(l	NUM
ap-5897	645	28	,	,	PUNCT
ap-5897	645	29	0	0	NUM
ap-5897	645	30	)	)	PUNCT
ap-5897	645	31	)	)	PUNCT
ap-5897	646	1	sgn(σm(b)[l	sgn(σm(b)[l	PROPN
ap-5897	646	2	]	]	PUNCT
ap-5897	646	3	)	)	PUNCT
ap-5897	646	4	≥	≥	NOUN
ap-5897	646	5	1	1	NUM
ap-5897	646	6	.	.	PUNCT
ap-5897	646	7	since	since	SCONJ
ap-5897	646	8	b[m	b[m	NUM
ap-5897	646	9	]	]	PUNCT
ap-5897	646	10	>	>	X
ap-5897	646	11	0	0	NUM
ap-5897	646	12	,	,	PUNCT
ap-5897	646	13	then	then	ADV
ap-5897	646	14	sgn(σm(b)[l	sgn(σm(b)[l	PROPN
ap-5897	646	15	]	]	PUNCT
ap-5897	646	16	)	)	PUNCT
ap-5897	646	17	=	=	SYM
ap-5897	646	18	sgn(b[m+	sgn(b[m+	NOUN
ap-5897	646	19	l	l	NOUN
ap-5897	646	20	]	]	X
ap-5897	646	21	)	)	PUNCT
ap-5897	646	22	.	.	PUNCT
ap-5897	647	1	case	case	NOUN
ap-5897	647	2	1.1	1.1	NUM
ap-5897	647	3	suppose	suppose	VERB
ap-5897	647	4	b[m	b[m	ADJ
ap-5897	648	1	+	+	NOUN
ap-5897	648	2	l	l	X
ap-5897	648	3	]	]	X
ap-5897	648	4	>	>	X
ap-5897	648	5	0	0	PUNCT
ap-5897	649	1	so	so	SCONJ
ap-5897	649	2	that	that	SCONJ
ap-5897	649	3	(	(	PUNCT
ap-5897	649	4	wm+l	wm+l	PROPN
ap-5897	649	5	−	−	PROPN
ap-5897	649	6	u(m)(l	u(m)(l	NUM
ap-5897	649	7	,	,	PUNCT
ap-5897	649	8	0	0	NUM
ap-5897	649	9	)	)	PUNCT
ap-5897	649	10	)	)	PUNCT
ap-5897	649	11	≥	≥	NOUN
ap-5897	649	12	1	1	X
ap-5897	649	13	.	.	PUNCT
ap-5897	649	14	we	we	PRON
ap-5897	649	15	have	have	VERB
ap-5897	649	16	(	(	PUNCT
ap-5897	649	17	w(1,m+	w(1,m+	NOUN
ap-5897	649	18	l	l	NOUN
ap-5897	649	19	−	−	NOUN
ap-5897	649	20	1	1	X
ap-5897	649	21	)	)	PUNCT
ap-5897	649	22	◦	◦	NOUN
ap-5897	650	1	u(m+l))b	u(m+l))b	PROPN
ap-5897	650	2	−	−	PROPN
ap-5897	650	3	(	(	PUNCT
ap-5897	650	4	w(1,m−	w(1,m−	NUM
ap-5897	650	5	1	1	NUM
ap-5897	650	6	)	)	PUNCT
ap-5897	650	7	◦	◦	NOUN
ap-5897	650	8	u(m))b	u(m))b	NOUN
ap-5897	650	9	=	=	SYM
ap-5897	650	10	wm+l	wm+l	PROPN
ap-5897	650	11	−	−	PROPN
ap-5897	650	12	u(m)(l	u(m)(l	NUM
ap-5897	650	13	,	,	PUNCT
ap-5897	650	14	0	0	NUM
ap-5897	650	15	)	)	PUNCT
ap-5897	650	16	b[m+	b[m+	NOUN
ap-5897	650	17	l	l	X
ap-5897	650	18	]	]	X
ap-5897	651	1	+	+	CCONJ
ap-5897	651	2	(	(	PUNCT
ap-5897	651	3	u(m+l))σm+l(b	u(m+l))σm+l(b	ADJ
ap-5897	651	4	)	)	PUNCT
ap-5897	651	5	−	−	PROPN
ap-5897	651	6	(	(	PUNCT
ap-5897	652	1	u(m)(l	u(m)(l	PROPN
ap-5897	652	2	+	+	NUM
ap-5897	652	3	1,∞))σm+l(b	1,∞))σm+l(b	NUM
ap-5897	652	4	)	)	PUNCT
ap-5897	652	5	b[m+	b[m+	NOUN
ap-5897	652	6	l	l	NOUN
ap-5897	652	7	]	]	PUNCT
ap-5897	652	8	.	.	PUNCT
ap-5897	653	1	now	now	ADV
ap-5897	653	2	,	,	PUNCT
ap-5897	653	3	wm+l	wm+l	PROPN
ap-5897	653	4	−	−	PROPN
ap-5897	653	5	u(m)(l	u(m)(l	NUM
ap-5897	653	6	,	,	PUNCT
ap-5897	653	7	0	0	NUM
ap-5897	653	8	)	)	PUNCT
ap-5897	653	9	b[m+	b[m+	ADP
ap-5897	653	10	l	l	X
ap-5897	653	11	]	]	X
ap-5897	653	12	≥	≥	PROPN
ap-5897	653	13	1	1	NUM
ap-5897	653	14	b[m+	b[m+	NOUN
ap-5897	653	15	l	l	NOUN
ap-5897	653	16	]	]	PUNCT
ap-5897	653	17	.	.	PUNCT
ap-5897	654	1	meanwhile	meanwhile	ADV
ap-5897	654	2	,	,	PUNCT
ap-5897	654	3	(	(	PUNCT
ap-5897	654	4	u(m+l))σm+l(b	u(m+l))σm+l(b	NOUN
ap-5897	654	5	)	)	PUNCT
ap-5897	654	6	−	−	PROPN
ap-5897	654	7	(	(	PUNCT
ap-5897	654	8	u(m)(l	u(m)(l	PROPN
ap-5897	654	9	+	+	NUM
ap-5897	654	10	1,∞))σm+l(b	1,∞))σm+l(b	NUM
ap-5897	654	11	)	)	PUNCT
ap-5897	654	12	=	=	SYM
ap-5897	654	13	γ	γ	X
ap-5897	654	14	−	−	PROPN
ap-5897	654	15	t	t	PROPN
ap-5897	654	16	lσm(b)(γ	lσm(b)(γ	NUM
ap-5897	654	17	)	)	PUNCT
ap-5897	654	18	.	.	PUNCT
ap-5897	655	1	hence	hence	ADV
ap-5897	655	2	,	,	PUNCT
ap-5897	655	3	(	(	PUNCT
ap-5897	655	4	u(m+l))σm+l(b	u(m+l))σm+l(b	NOUN
ap-5897	655	5	)	)	PUNCT
ap-5897	655	6	−	−	PROPN
ap-5897	655	7	(	(	PUNCT
ap-5897	655	8	u(m)(l	u(m)(l	PROPN
ap-5897	655	9	+	+	NUM
ap-5897	655	10	1,∞))σm+l(b	1,∞))σm+l(b	NUM
ap-5897	655	11	)	)	PUNCT
ap-5897	655	12	b[m+	b[m+	NOUN
ap-5897	655	13	l	l	NOUN
ap-5897	655	14	]	]	PUNCT
ap-5897	655	15	is	be	AUX
ap-5897	655	16	greater	great	ADJ
ap-5897	655	17	than	than	ADP
ap-5897	655	18	−1	−1	NOUN
ap-5897	655	19	/	/	SYM
ap-5897	655	20	b[m+	b[m+	NOUN
ap-5897	655	21	l	l	NOUN
ap-5897	655	22	]	]	PUNCT
ap-5897	655	23	and	and	CCONJ
ap-5897	655	24	less	less	ADJ
ap-5897	655	25	than	than	ADP
ap-5897	655	26	or	or	CCONJ
ap-5897	655	27	equal	equal	ADJ
ap-5897	655	28	to	to	ADP
ap-5897	655	29	0	0	NUM
ap-5897	655	30	.	.	PUNCT
ap-5897	656	1	therefore	therefore	ADV
ap-5897	656	2	,	,	PUNCT
ap-5897	656	3	(	(	PUNCT
ap-5897	656	4	w(1,m+l−1)	w(1,m+l−1)	PROPN
ap-5897	656	5	◦	◦	NOUN
ap-5897	656	6	u(m+l))b−(w(1,m−1)	u(m+l))b−(w(1,m−1)	PROPN
ap-5897	656	7	◦	◦	NOUN
ap-5897	656	8	u(m))b	u(m))b	PROPN
ap-5897	656	9	≥	≥	NOUN
ap-5897	656	10	0	0	NUM
ap-5897	656	11	,	,	PUNCT
ap-5897	656	12	implying	imply	VERB
ap-5897	656	13	that	that	SCONJ
ap-5897	656	14	(	(	PUNCT
ap-5897	656	15	w(1,m+l−1)	w(1,m+l−1)	X
ap-5897	656	16	◦	◦	NOUN
ap-5897	656	17	u(m+l))b	u(m+l))b	PROPN
ap-5897	656	18	>	>	X
ap-5897	656	19	(	(	PUNCT
ap-5897	656	20	w(1,m−1)	w(1,m−1)	PROPN
ap-5897	656	21	◦	◦	NOUN
ap-5897	656	22	u(m))b	u(m))b	PROPN
ap-5897	656	23	≥	≥	NOUN
ap-5897	656	24	γ	γ	PROPN
ap-5897	656	25	.	.	PROPN
ap-5897	656	26	case	case	NOUN
ap-5897	656	27	1.2	1.2	NUM
ap-5897	656	28	suppose	suppose	VERB
ap-5897	656	29	b[m+	b[m+	PROPN
ap-5897	656	30	l	l	PROPN
ap-5897	656	31	]	]	X
ap-5897	656	32	<	<	X
ap-5897	656	33	0	0	X
ap-5897	656	34	.	.	PUNCT
ap-5897	657	1	then	then	ADV
ap-5897	657	2	(	(	PUNCT
ap-5897	657	3	w(1,m+	w(1,m+	NOUN
ap-5897	657	4	l	l	NOUN
ap-5897	657	5	−	−	NOUN
ap-5897	657	6	1	1	X
ap-5897	657	7	)	)	PUNCT
ap-5897	657	8	◦	◦	NOUN
ap-5897	657	9	v(m+l))b	v(m+l))b	PROPN
ap-5897	657	10	−	−	PROPN
ap-5897	657	11	(	(	PUNCT
ap-5897	657	12	w(1,m−	w(1,m−	NUM
ap-5897	657	13	1	1	NUM
ap-5897	657	14	)	)	PUNCT
ap-5897	657	15	◦	◦	NOUN
ap-5897	657	16	u(m))b	u(m))b	NOUN
ap-5897	657	17	=	=	SYM
ap-5897	657	18	wm+l	wm+l	PROPN
ap-5897	657	19	−	−	PROPN
ap-5897	657	20	u(m)(l	u(m)(l	NUM
ap-5897	657	21	,	,	PUNCT
ap-5897	657	22	0	0	NUM
ap-5897	657	23	)	)	PUNCT
ap-5897	657	24	b[m+	b[m+	NOUN
ap-5897	657	25	l	l	X
ap-5897	657	26	]	]	X
ap-5897	658	1	+	+	CCONJ
ap-5897	658	2	(	(	PUNCT
ap-5897	658	3	v(m+l))σm+l(b	v(m+l))σm+l(b	NOUN
ap-5897	658	4	)	)	PUNCT
ap-5897	658	5	−	−	PROPN
ap-5897	658	6	(	(	PUNCT
ap-5897	658	7	u(m)(l	u(m)(l	PROPN
ap-5897	658	8	+	+	NUM
ap-5897	658	9	1,∞))σm+l(b	1,∞))σm+l(b	NUM
ap-5897	658	10	)	)	PUNCT
ap-5897	658	11	b[m+	b[m+	NOUN
ap-5897	658	12	l	l	NOUN
ap-5897	658	13	]	]	PUNCT
ap-5897	658	14	.	.	PUNCT
ap-5897	659	1	since	since	SCONJ
ap-5897	659	2	b[m+	b[m+	PROPN
ap-5897	659	3	l	l	PROPN
ap-5897	659	4	]	]	X
ap-5897	659	5	<	<	X
ap-5897	659	6	0	0	NUM
ap-5897	659	7	,	,	PUNCT
ap-5897	659	8	we	we	PRON
ap-5897	659	9	have	have	VERB
ap-5897	659	10	wm+l	wm+l	VERB
ap-5897	659	11	−	−	PROPN
ap-5897	659	12	u(m)(l	u(m)(l	NUM
ap-5897	659	13	,	,	PUNCT
ap-5897	659	14	0	0	NUM
ap-5897	659	15	)	)	PUNCT
ap-5897	659	16	b[m+	b[m+	ADP
ap-5897	659	17	l	l	X
ap-5897	659	18	]	]	X
ap-5897	659	19	≥	≥	X
ap-5897	659	20	−1	−1	NOUN
ap-5897	659	21	b[m+	b[m+	PROPN
ap-5897	659	22	l	l	X
ap-5897	659	23	]	]	PUNCT
ap-5897	659	24	.	.	PUNCT
ap-5897	660	1	moreover	moreover	ADV
ap-5897	660	2	,	,	PUNCT
ap-5897	660	3	(	(	PUNCT
ap-5897	660	4	v(m+l))σm+l(b	v(m+l))σm+l(b	NOUN
ap-5897	660	5	)	)	PUNCT
ap-5897	660	6	−	−	PROPN
ap-5897	660	7	(	(	PUNCT
ap-5897	661	1	u(m)(l	u(m)(l	PROPN
ap-5897	661	2	+	+	NUM
ap-5897	661	3	1,∞))σm+l(b	1,∞))σm+l(b	NUM
ap-5897	661	4	)	)	PUNCT
ap-5897	661	5	=	=	SYM
ap-5897	662	1	γ	γ	X
ap-5897	662	2	+	+	SYM
ap-5897	662	3	1−	1−	NUM
ap-5897	662	4	t	t	NOUN
ap-5897	662	5	lσm(b)(γ	lσm(b)(γ	NUM
ap-5897	662	6	)	)	PUNCT
ap-5897	662	7	.	.	PUNCT
ap-5897	663	1	it	it	PRON
ap-5897	663	2	follows	follow	VERB
ap-5897	663	3	that	that	SCONJ
ap-5897	663	4	(	(	PUNCT
ap-5897	663	5	v(m+l))σm+l(b	v(m+l))σm+l(b	NOUN
ap-5897	663	6	)	)	PUNCT
ap-5897	663	7	−	−	PROPN
ap-5897	663	8	(	(	PUNCT
ap-5897	663	9	u(m)(l	u(m)(l	PROPN
ap-5897	663	10	+	+	NUM
ap-5897	663	11	1,∞))σm+l(b	1,∞))σm+l(b	NUM
ap-5897	663	12	)	)	PUNCT
ap-5897	663	13	b[m+	b[m+	ADP
ap-5897	663	14	l	l	X
ap-5897	663	15	]	]	PUNCT
ap-5897	663	16	222	222	NUM
ap-5897	663	17	vol	vol	NOUN
ap-5897	663	18	.	.	PUNCT
ap-5897	664	1	60	60	NUM
ap-5897	664	2	no	no	NOUN
ap-5897	664	3	.	.	PUNCT
ap-5897	665	1	3/2020	3/2020	NUM
ap-5897	665	2	beta	beta	PROPN
ap-5897	665	3	cantor	cantor	PROPN
ap-5897	665	4	series	series	NOUN
ap-5897	665	5	expansion	expansion	NOUN
ap-5897	665	6	and	and	CCONJ
ap-5897	665	7	admissible	admissible	ADJ
ap-5897	665	8	sequences	sequence	NOUN
ap-5897	665	9	is	be	AUX
ap-5897	665	10	less	less	ADJ
ap-5897	665	11	than	than	ADP
ap-5897	665	12	0	0	NUM
ap-5897	665	13	but	but	CCONJ
ap-5897	665	14	greater	great	ADJ
ap-5897	665	15	than	than	ADP
ap-5897	665	16	or	or	CCONJ
ap-5897	665	17	equal	equal	ADJ
ap-5897	665	18	to	to	ADP
ap-5897	665	19	1	1	NUM
ap-5897	665	20	/	/	SYM
ap-5897	665	21	b[m+	b[m+	NOUN
ap-5897	665	22	l	l	NOUN
ap-5897	665	23	]	]	PUNCT
ap-5897	665	24	.	.	PUNCT
ap-5897	666	1	therefore	therefore	ADV
ap-5897	666	2	,	,	PUNCT
ap-5897	666	3	(	(	PUNCT
ap-5897	666	4	w(1,m+	w(1,m+	NOUN
ap-5897	666	5	l−1)	l−1)	NOUN
ap-5897	666	6	◦	◦	NOUN
ap-5897	666	7	v(m+l))b−(w(1,m−1)	v(m+l))b−(w(1,m−1)	NOUN
ap-5897	666	8	◦	◦	NOUN
ap-5897	666	9	u(m))b	u(m))b	NOUN
ap-5897	666	10	≥	≥	NOUN
ap-5897	666	11	0	0	NUM
ap-5897	666	12	,	,	PUNCT
ap-5897	666	13	and	and	CCONJ
ap-5897	666	14	consequently	consequently	ADV
ap-5897	666	15	,	,	PUNCT
ap-5897	666	16	(	(	PUNCT
ap-5897	666	17	w(1,m+l−1)	w(1,m+l−1)	X
ap-5897	666	18	◦	◦	NOUN
ap-5897	666	19	v(m+l))b	v(m+l))b	PROPN
ap-5897	666	20	>	>	X
ap-5897	666	21	(	(	PUNCT
ap-5897	666	22	w(1,m−1)	w(1,m−1)	PROPN
ap-5897	666	23	◦	◦	NOUN
ap-5897	666	24	u(m))b	u(m))b	PROPN
ap-5897	666	25	≥	≥	NOUN
ap-5897	666	26	γ	γ	PROPN
ap-5897	666	27	.	.	PROPN
ap-5897	666	28	case	case	NOUN
ap-5897	666	29	2	2	NUM
ap-5897	666	30	suppose	suppose	VERB
ap-5897	666	31	b[m	b[m	PROPN
ap-5897	666	32	]	]	PUNCT
ap-5897	666	33	<	<	X
ap-5897	666	34	0	0	PUNCT
ap-5897	667	1	and	and	CCONJ
ap-5897	667	2	(	(	PUNCT
ap-5897	667	3	w(1,m	w(1,m	NOUN
ap-5897	667	4	−	−	PROPN
ap-5897	667	5	1	1	NUM
ap-5897	667	6	)	)	PUNCT
ap-5897	667	7	◦	◦	NOUN
ap-5897	667	8	v(m))b	v(m))b	PROPN
ap-5897	667	9	≥	≥	NOUN
ap-5897	667	10	γ	γ	NOUN
ap-5897	667	11	hold	hold	VERB
ap-5897	667	12	.	.	PUNCT
ap-5897	668	1	by	by	ADP
ap-5897	668	2	the	the	DET
ap-5897	668	3	lexicographic	lexicographic	ADJ
ap-5897	668	4	restriction	restriction	NOUN
ap-5897	668	5	,	,	PUNCT
ap-5897	668	6	σm(w	σm(w	PUNCT
ap-5897	668	7	)	)	PUNCT
ap-5897	668	8	=	=	SYM
ap-5897	668	9	w(m+1,∞	w(m+1,∞	NOUN
ap-5897	668	10	)	)	PUNCT
ap-5897	668	11	≺σm(b	≺σm(b	NOUN
ap-5897	668	12	)	)	PUNCT
ap-5897	668	13	v	v	NOUN
ap-5897	668	14	(	(	PUNCT
ap-5897	668	15	m	m	NOUN
ap-5897	668	16	)	)	PUNCT
ap-5897	668	17	=	=	SYM
ap-5897	668	18	d∗(σm(b	d∗(σm(b	NUM
ap-5897	668	19	)	)	PUNCT
ap-5897	668	20	;	;	PUNCT
ap-5897	668	21	γ+1	γ+1	PROPN
ap-5897	668	22	)	)	PUNCT
ap-5897	668	23	.	.	PUNCT
ap-5897	669	1	thus	thus	ADV
ap-5897	669	2	,	,	PUNCT
ap-5897	669	3	there	there	PRON
ap-5897	669	4	exists	exist	VERB
ap-5897	669	5	a	a	DET
ap-5897	669	6	least	least	ADV
ap-5897	669	7	positive	positive	ADJ
ap-5897	669	8	integer	integer	NOUN
ap-5897	669	9	l	l	NOUN
ap-5897	670	1	such	such	ADJ
ap-5897	670	2	that	that	SCONJ
ap-5897	670	3	wm+i	wm+i	PROPN
ap-5897	670	4	=	=	SYM
ap-5897	670	5	v(m)(i	v(m)(i	NUM
ap-5897	670	6	,	,	PUNCT
ap-5897	670	7	0	0	NUM
ap-5897	670	8	)	)	PUNCT
ap-5897	670	9	for	for	ADP
ap-5897	670	10	all	all	DET
ap-5897	670	11	i	i	PRON
ap-5897	670	12	<	<	X
ap-5897	670	13	l	l	NOUN
ap-5897	670	14	and	and	CCONJ
ap-5897	670	15	(	(	PUNCT
ap-5897	670	16	v(m)(l	v(m)(l	NUM
ap-5897	670	17	,	,	PUNCT
ap-5897	670	18	0)−	0)−	NOUN
ap-5897	670	19	wm+l	wm+l	NUM
ap-5897	670	20	)	)	PUNCT
ap-5897	670	21	sgn(σm(b)[l	sgn(σm(b)[l	PROPN
ap-5897	670	22	]	]	PUNCT
ap-5897	670	23	)	)	PUNCT
ap-5897	670	24	≥	≥	NOUN
ap-5897	670	25	1	1	NUM
ap-5897	670	26	.	.	PUNCT
ap-5897	670	27	since	since	SCONJ
ap-5897	670	28	b[m	b[m	NUM
ap-5897	670	29	]	]	PUNCT
ap-5897	670	30	<	<	X
ap-5897	670	31	0	0	NUM
ap-5897	670	32	,	,	PUNCT
ap-5897	670	33	we	we	PRON
ap-5897	670	34	have	have	VERB
ap-5897	670	35	sgn(σm(b)[l	sgn(σm(b)[l	NOUN
ap-5897	670	36	]	]	X
ap-5897	670	37	)	)	PUNCT
ap-5897	671	1	=	=	SYM
ap-5897	672	1	−	−	PROPN
ap-5897	672	2	sgn(b[m	sgn(b[m	NUM
ap-5897	672	3	+	+	CCONJ
ap-5897	672	4	l	l	NOUN
ap-5897	672	5	]	]	X
ap-5897	672	6	)	)	PUNCT
ap-5897	672	7	.	.	PUNCT
ap-5897	673	1	as	as	ADP
ap-5897	673	2	before	before	ADV
ap-5897	673	3	,	,	PUNCT
ap-5897	673	4	we	we	PRON
ap-5897	673	5	have	have	VERB
ap-5897	673	6	two	two	NUM
ap-5897	673	7	subcases	subcase	NOUN
ap-5897	673	8	:	:	PUNCT
ap-5897	673	9	sgn(b[m+	sgn(b[m+	NOUN
ap-5897	673	10	l	l	NOUN
ap-5897	673	11	]	]	X
ap-5897	673	12	)	)	PUNCT
ap-5897	673	13	>	>	X
ap-5897	673	14	0	0	PUNCT
ap-5897	674	1	and	and	CCONJ
ap-5897	674	2	sgn(b[m+	sgn(b[m+	X
ap-5897	674	3	l	l	NOUN
ap-5897	674	4	]	]	X
ap-5897	674	5	)	)	PUNCT
ap-5897	674	6	<	<	X
ap-5897	674	7	0	0	X
ap-5897	674	8	.	.	PUNCT
ap-5897	675	1	the	the	DET
ap-5897	675	2	proofs	proof	NOUN
ap-5897	675	3	are	be	AUX
ap-5897	675	4	similar	similar	ADJ
ap-5897	675	5	.	.	PUNCT
ap-5897	676	1	analogous	analogous	ADJ
ap-5897	676	2	to	to	PART
ap-5897	676	3	lemma	lemma	PROPN
ap-5897	676	4	4.9	4.9	NUM
ap-5897	676	5	,	,	PUNCT
ap-5897	676	6	we	we	PRON
ap-5897	676	7	have	have	VERB
ap-5897	676	8	the	the	DET
ap-5897	676	9	following	follow	VERB
ap-5897	676	10	result	result	NOUN
ap-5897	676	11	.	.	PUNCT
ap-5897	677	1	lemma	lemma	PROPN
ap-5897	677	2	4.10	4.10	NUM
ap-5897	677	3	.	.	PUNCT
ap-5897	678	1	let	let	VERB
ap-5897	678	2	w	w	NOUN
ap-5897	678	3	∈	∈	PROPN
ap-5897	678	4	a(b	a(b	PROPN
ap-5897	678	5	)	)	PUNCT
ap-5897	678	6	.	.	PUNCT
ap-5897	679	1	if	if	SCONJ
ap-5897	679	2	w	w	NOUN
ap-5897	679	3	satisfies	satisfy	VERB
ap-5897	679	4	the	the	DET
ap-5897	679	5	lexicographic	lexicographic	ADJ
ap-5897	679	6	restriction	restriction	NOUN
ap-5897	679	7	,	,	PUNCT
ap-5897	679	8	then	then	ADV
ap-5897	679	9	there	there	PRON
ap-5897	679	10	are	be	VERB
ap-5897	679	11	infinitely	infinitely	ADV
ap-5897	679	12	many	many	ADJ
ap-5897	679	13	n	n	ADP
ap-5897	679	14	such	such	ADJ
ap-5897	679	15	that	that	SCONJ
ap-5897	679	16	at	at	ADV
ap-5897	679	17	least	least	ADJ
ap-5897	679	18	one	one	NUM
ap-5897	679	19	of	of	ADP
ap-5897	679	20	the	the	DET
ap-5897	679	21	two	two	NUM
ap-5897	679	22	holds	hold	NOUN
ap-5897	679	23	:	:	PUNCT
ap-5897	679	24	(	(	PUNCT
ap-5897	679	25	1	1	NUM
ap-5897	679	26	.	.	PUNCT
ap-5897	679	27	)	)	PUNCT
ap-5897	680	1	b[n	b[n	VERB
ap-5897	680	2	]	]	PUNCT
ap-5897	680	3	<	<	X
ap-5897	680	4	0	0	PUNCT
ap-5897	680	5	and	and	CCONJ
ap-5897	680	6	(	(	PUNCT
ap-5897	680	7	w(1	w(1	INTJ
ap-5897	680	8	,	,	PUNCT
ap-5897	680	9	n−	n−	NOUN
ap-5897	680	10	1	1	NUM
ap-5897	680	11	)	)	PUNCT
ap-5897	680	12	◦	◦	VERB
ap-5897	680	13	u(n))b	u(n))b	ADP
ap-5897	680	14	≤	≤	NOUN
ap-5897	680	15	γ	γ	X
ap-5897	680	16	+	+	ADP
ap-5897	680	17	1	1	NUM
ap-5897	680	18	;	;	PUNCT
ap-5897	680	19	(	(	PUNCT
ap-5897	680	20	2	2	NUM
ap-5897	680	21	.	.	PUNCT
ap-5897	680	22	)	)	PUNCT
ap-5897	681	1	b[n	b[n	VERB
ap-5897	681	2	]	]	PUNCT
ap-5897	681	3	>	>	X
ap-5897	681	4	0	0	PUNCT
ap-5897	681	5	and	and	CCONJ
ap-5897	681	6	(	(	PUNCT
ap-5897	681	7	w(1	w(1	INTJ
ap-5897	681	8	,	,	PUNCT
ap-5897	681	9	n−	n−	NOUN
ap-5897	681	10	1	1	NUM
ap-5897	681	11	)	)	PUNCT
ap-5897	681	12	◦	◦	NOUN
ap-5897	681	13	v(n))b	v(n))b	NOUN
ap-5897	681	14	≤	≤	ADJ
ap-5897	681	15	γ	γ	X
ap-5897	681	16	+	+	NOUN
ap-5897	681	17	1	1	X
ap-5897	681	18	.	.	PUNCT
ap-5897	681	19	we	we	PRON
ap-5897	681	20	now	now	ADV
ap-5897	681	21	apply	apply	VERB
ap-5897	681	22	lemmas	lemmas	PROPN
ap-5897	681	23	4.9	4.9	NUM
ap-5897	681	24	and	and	CCONJ
ap-5897	681	25	4.10	4.10	NUM
ap-5897	681	26	to	to	PART
ap-5897	681	27	prove	prove	VERB
ap-5897	681	28	the	the	DET
ap-5897	681	29	next	next	ADJ
ap-5897	681	30	proposition	proposition	NOUN
ap-5897	681	31	.	.	PUNCT
ap-5897	682	1	proposition	proposition	NOUN
ap-5897	682	2	4.11	4.11	NUM
ap-5897	682	3	.	.	PUNCT
ap-5897	683	1	let	let	VERB
ap-5897	683	2	w	w	NOUN
ap-5897	683	3	=	=	SYM
ap-5897	683	4	(	(	PUNCT
ap-5897	683	5	w1	w1	NOUN
ap-5897	683	6	,	,	PUNCT
ap-5897	683	7	w2	w2	NOUN
ap-5897	683	8	,	,	PUNCT
ap-5897	683	9	.	.	PUNCT
ap-5897	683	10	.	.	PUNCT
ap-5897	683	11	.	.	PUNCT
ap-5897	683	12	)	)	PUNCT
ap-5897	684	1	∈	∈	PROPN
ap-5897	684	2	a(b	a(b	NOUN
ap-5897	684	3	)	)	PUNCT
ap-5897	684	4	such	such	ADJ
ap-5897	685	1	that	that	SCONJ
ap-5897	685	2	the	the	DET
ap-5897	685	3	sum	sum	NOUN
ap-5897	685	4	σk(w)σk(b	σk(w)σk(b	PROPN
ap-5897	685	5	)	)	PUNCT
ap-5897	685	6	converges	converge	VERB
ap-5897	685	7	for	for	ADP
ap-5897	685	8	all	all	DET
ap-5897	685	9	k	k	PROPN
ap-5897	685	10	∈	∈	PROPN
ap-5897	685	11	n	n	PART
ap-5897	685	12	∪	∪	X
ap-5897	685	13	{	{	PUNCT
ap-5897	685	14	0	0	NUM
ap-5897	685	15	}	}	PUNCT
ap-5897	685	16	.	.	PUNCT
ap-5897	686	1	if	if	SCONJ
ap-5897	686	2	w	w	NOUN
ap-5897	686	3	satisfies	satisfy	VERB
ap-5897	686	4	the	the	DET
ap-5897	686	5	lexicographic	lexicographic	ADJ
ap-5897	686	6	restriction	restriction	NOUN
ap-5897	686	7	,	,	PUNCT
ap-5897	686	8	then	then	ADV
ap-5897	686	9	σk(w)σk(b	σk(w)σk(b	NOUN
ap-5897	686	10	)	)	PUNCT
ap-5897	686	11	∈	∈	PROPN
ap-5897	687	1	[	[	X
ap-5897	687	2	γ	γ	X
ap-5897	687	3	,	,	PUNCT
ap-5897	687	4	γ	γ	X
ap-5897	687	5	+	+	NOUN
ap-5897	687	6	1	1	NUM
ap-5897	687	7	)	)	PUNCT
ap-5897	687	8	.	.	PUNCT
ap-5897	688	1	proof	proof	NOUN
ap-5897	688	2	.	.	PUNCT
ap-5897	689	1	we	we	PRON
ap-5897	689	2	show	show	VERB
ap-5897	689	3	that	that	SCONJ
ap-5897	689	4	γ	γ	PROPN
ap-5897	689	5	≤	≤	X
ap-5897	689	6	wb	wb	PROPN
ap-5897	689	7	≤	≤	PROPN
ap-5897	689	8	γ	γ	X
ap-5897	689	9	+	+	NOUN
ap-5897	689	10	1	1	X
ap-5897	689	11	.	.	X
ap-5897	689	12	let	let	VERB
ap-5897	689	13	en(w	en(w	NUM
ap-5897	689	14	)	)	PUNCT
ap-5897	689	15	:	:	PUNCT
ap-5897	689	16	=	=	SYM
ap-5897	689	17	σn(w)σn(b	σn(w)σn(b	NUM
ap-5897	689	18	)	)	PUNCT
ap-5897	689	19	,	,	PUNCT
ap-5897	689	20	which	which	PRON
ap-5897	689	21	by	by	ADP
ap-5897	689	22	assumption	assumption	NOUN
ap-5897	689	23	converges	converge	NOUN
ap-5897	689	24	.	.	PUNCT
ap-5897	690	1	then	then	ADV
ap-5897	690	2	,	,	PUNCT
ap-5897	690	3	for	for	ADP
ap-5897	690	4	all	all	DET
ap-5897	690	5	n	n	DET
ap-5897	690	6	∈	∈	PROPN
ap-5897	690	7	n	n	CCONJ
ap-5897	690	8	,	,	PUNCT
ap-5897	690	9	wb	wb	PROPN
ap-5897	690	10	=	=	SYM
ap-5897	690	11	n∑	n∑	NOUN
ap-5897	690	12	k=1	k=1	PROPN
ap-5897	690	13	wk	wk	PUNCT
ap-5897	690	14	b[k	b[k	NOUN
ap-5897	690	15	]	]	PUNCT
ap-5897	690	16	+	+	CCONJ
ap-5897	690	17	en(w	en(w	X
ap-5897	690	18	)	)	PUNCT
ap-5897	690	19	b[n	b[n	NOUN
ap-5897	690	20	]	]	X
ap-5897	690	21	=	=	SYM
ap-5897	690	22	w(1	w(1	X
ap-5897	690	23	,	,	PUNCT
ap-5897	690	24	n−	n−	VERB
ap-5897	690	25	1)b	1)b	NUM
ap-5897	690	26	+	+	CCONJ
ap-5897	690	27	en(w	en(w	NOUN
ap-5897	690	28	)	)	PUNCT
ap-5897	690	29	b[n	b[n	NOUN
ap-5897	690	30	]	]	PUNCT
ap-5897	690	31	.	.	PUNCT
ap-5897	691	1	thus	thus	ADV
ap-5897	691	2	,	,	PUNCT
ap-5897	691	3	as	as	SCONJ
ap-5897	691	4	n	n	PRON
ap-5897	691	5	tends	tend	VERB
ap-5897	691	6	to∞	to∞	PROPN
ap-5897	691	7	,	,	PUNCT
ap-5897	691	8	the	the	DET
ap-5897	691	9	quotient	quotient	NOUN
ap-5897	691	10	en(w)/b[n	en(w)/b[n	X
ap-5897	691	11	]	]	PUNCT
ap-5897	691	12	tends	tend	VERB
ap-5897	691	13	to	to	ADP
ap-5897	691	14	0	0	NUM
ap-5897	691	15	.	.	PUNCT
ap-5897	692	1	for	for	ADP
ap-5897	692	2	sufficiently	sufficiently	ADV
ap-5897	692	3	large	large	ADJ
ap-5897	692	4	n	n	CCONJ
ap-5897	692	5	,	,	PUNCT
ap-5897	692	6	(	(	PUNCT
ap-5897	692	7	w(1	w(1	INTJ
ap-5897	692	8	,	,	PUNCT
ap-5897	692	9	n−	n−	NOUN
ap-5897	692	10	1	1	NUM
ap-5897	692	11	)	)	PUNCT
ap-5897	692	12	◦	◦	NOUN
ap-5897	692	13	u(n))b	u(n))b	VERB
ap-5897	692	14	≥	≥	NOUN
ap-5897	692	15	γ	γ	PROPN
ap-5897	692	16	or	or	CCONJ
ap-5897	692	17	(	(	PUNCT
ap-5897	692	18	w(1	w(1	INTJ
ap-5897	692	19	,	,	PUNCT
ap-5897	692	20	n−	n−	NOUN
ap-5897	692	21	1	1	NUM
ap-5897	692	22	)	)	PUNCT
ap-5897	692	23	◦	◦	NOUN
ap-5897	692	24	v(n))b	v(n))b	PROPN
ap-5897	692	25	≥	≥	NOUN
ap-5897	692	26	γ	γ	X
ap-5897	692	27	by	by	ADP
ap-5897	692	28	lemma	lemma	PROPN
ap-5897	692	29	4.9	4.9	NUM
ap-5897	692	30	.	.	PUNCT
ap-5897	693	1	so	so	ADV
ap-5897	693	2	,	,	PUNCT
ap-5897	693	3	wb	wb	PROPN
ap-5897	693	4	−	−	PROPN
ap-5897	693	5	(	(	PUNCT
ap-5897	693	6	w(1	w(1	INTJ
ap-5897	693	7	,	,	PUNCT
ap-5897	693	8	n−	n−	NOUN
ap-5897	693	9	1	1	NUM
ap-5897	693	10	)	)	PUNCT
ap-5897	693	11	◦	◦	NOUN
ap-5897	693	12	t(n))b	t(n))b	NOUN
ap-5897	693	13	=	=	SYM
ap-5897	693	14	en(w	en(w	X
ap-5897	693	15	)	)	PUNCT
ap-5897	693	16	b[n	b[n	NOUN
ap-5897	693	17	]	]	X
ap-5897	693	18	−	−	PROPN
ap-5897	693	19	(	(	PUNCT
ap-5897	693	20	t(n))σn(b	t(n))σn(b	PROPN
ap-5897	693	21	)	)	PUNCT
ap-5897	693	22	b[n	b[n	NOUN
ap-5897	693	23	]	]	PUNCT
ap-5897	693	24	=	=	SYM
ap-5897	693	25	en(w	en(w	X
ap-5897	693	26	)	)	PUNCT
ap-5897	693	27	b[n	b[n	NOUN
ap-5897	693	28	]	]	PUNCT
ap-5897	693	29	−	−	PROPN
ap-5897	693	30	c	c	NOUN
ap-5897	693	31	b[n	b[n	NOUN
ap-5897	693	32	]	]	X
ap-5897	693	33	−→	−→	NOUN
ap-5897	693	34	0	0	NUM
ap-5897	694	1	where	where	SCONJ
ap-5897	694	2	(	(	PUNCT
ap-5897	694	3	t(n	t(n	PROPN
ap-5897	694	4	)	)	PUNCT
ap-5897	694	5	,	,	PUNCT
ap-5897	694	6	c	c	X
ap-5897	694	7	)	)	PUNCT
ap-5897	694	8	is	be	AUX
ap-5897	694	9	either	either	PRON
ap-5897	694	10	(	(	PUNCT
ap-5897	694	11	u(n	u(n	PROPN
ap-5897	694	12	)	)	PUNCT
ap-5897	694	13	,	,	PUNCT
ap-5897	694	14	γ	γ	NOUN
ap-5897	694	15	)	)	PUNCT
ap-5897	694	16	or	or	CCONJ
ap-5897	694	17	(	(	PUNCT
ap-5897	694	18	v(n	v(n	PROPN
ap-5897	694	19	)	)	PUNCT
ap-5897	694	20	,	,	PUNCT
ap-5897	694	21	γ+1	γ+1	PROPN
ap-5897	694	22	)	)	PUNCT
ap-5897	694	23	.	.	PUNCT
ap-5897	695	1	therefore	therefore	ADV
ap-5897	695	2	,	,	PUNCT
ap-5897	695	3	wb	wb	PROPN
ap-5897	695	4	≥	≥	PROPN
ap-5897	695	5	γ	γ	PROPN
ap-5897	695	6	.	.	PROPN
ap-5897	695	7	in	in	ADP
ap-5897	695	8	general	general	ADJ
ap-5897	695	9	,	,	PUNCT
ap-5897	695	10	observe	observe	VERB
ap-5897	695	11	that	that	SCONJ
ap-5897	695	12	as	as	SCONJ
ap-5897	695	13	w	w	PROPN
ap-5897	695	14	satisfies	satisfie	NOUN
ap-5897	695	15	the	the	DET
ap-5897	695	16	lexicographic	lexicographic	ADJ
ap-5897	695	17	restriction	restriction	NOUN
ap-5897	695	18	with	with	ADP
ap-5897	695	19	respect	respect	NOUN
ap-5897	695	20	to	to	ADP
ap-5897	695	21	b	b	NOUN
ap-5897	695	22	,	,	PUNCT
ap-5897	695	23	then	then	ADV
ap-5897	695	24	σm(w	σm(w	PUNCT
ap-5897	695	25	)	)	PUNCT
ap-5897	695	26	also	also	ADV
ap-5897	695	27	satisfies	satisfy	VERB
ap-5897	695	28	the	the	DET
ap-5897	695	29	lexicographic	lexicographic	ADJ
ap-5897	695	30	restriction	restriction	NOUN
ap-5897	695	31	with	with	ADP
ap-5897	695	32	respect	respect	NOUN
ap-5897	695	33	to	to	ADP
ap-5897	695	34	σm(b	σm(b	NUM
ap-5897	695	35	)	)	PUNCT
ap-5897	695	36	.	.	PUNCT
ap-5897	696	1	consequently	consequently	ADV
ap-5897	696	2	,	,	PUNCT
ap-5897	696	3	lemmas	lemma	VERB
ap-5897	696	4	4.9	4.9	NUM
ap-5897	696	5	and	and	CCONJ
ap-5897	696	6	4.10	4.10	NUM
ap-5897	696	7	apply	apply	NOUN
ap-5897	696	8	.	.	PUNCT
ap-5897	697	1	in	in	ADP
ap-5897	697	2	other	other	ADJ
ap-5897	697	3	words	word	NOUN
ap-5897	697	4	,	,	PUNCT
ap-5897	697	5	letting	let	VERB
ap-5897	697	6	σm(w	σm(w	NOUN
ap-5897	697	7	)	)	PUNCT
ap-5897	697	8	,	,	PUNCT
ap-5897	697	9	σm(b	σm(b	NUM
ap-5897	697	10	)	)	PUNCT
ap-5897	697	11	,	,	PUNCT
ap-5897	697	12	(	(	PUNCT
ap-5897	697	13	u(m))σm(b	u(m))σm(b	PROPN
ap-5897	697	14	)	)	PUNCT
ap-5897	697	15	and	and	CCONJ
ap-5897	697	16	(	(	PUNCT
ap-5897	697	17	v(m))σm(b	v(m))σm(b	NOUN
ap-5897	697	18	)	)	PUNCT
ap-5897	697	19	take	take	VERB
ap-5897	697	20	the	the	DET
ap-5897	697	21	role	role	NOUN
ap-5897	697	22	of	of	ADP
ap-5897	697	23	w	w	PROPN
ap-5897	697	24	,	,	PUNCT
ap-5897	697	25	b	b	PROPN
ap-5897	697	26	,	,	PUNCT
ap-5897	697	27	γ	γ	NOUN
ap-5897	697	28	and	and	CCONJ
ap-5897	697	29	γ	γ	X
ap-5897	697	30	+	+	NOUN
ap-5897	697	31	1	1	NUM
ap-5897	697	32	,	,	PUNCT
ap-5897	697	33	respectively	respectively	ADV
ap-5897	697	34	,	,	PUNCT
ap-5897	697	35	in	in	ADP
ap-5897	697	36	lemma	lemma	PROPN
ap-5897	697	37	4.9	4.9	NUM
ap-5897	697	38	,	,	PUNCT
ap-5897	697	39	we	we	PRON
ap-5897	697	40	obtain	obtain	VERB
ap-5897	697	41	the	the	DET
ap-5897	697	42	conclusion	conclusion	NOUN
ap-5897	697	43	that	that	SCONJ
ap-5897	697	44	σm(w)σm(b	σm(w)σm(b	PROPN
ap-5897	697	45	)	)	PUNCT
ap-5897	697	46	≥	≥	PROPN
ap-5897	697	47	γ	γ	PROPN
ap-5897	697	48	.	.	PROPN
ap-5897	698	1	likewise	likewise	ADV
ap-5897	698	2	,	,	PUNCT
ap-5897	698	3	we	we	PRON
ap-5897	698	4	have	have	VERB
ap-5897	698	5	σm(w)σm(b	σm(w)σm(b	NOUN
ap-5897	698	6	)	)	PUNCT
ap-5897	698	7	≤	≤	NOUN
ap-5897	698	8	γ	γ	X
ap-5897	698	9	+	+	ADP
ap-5897	698	10	1	1	NUM
ap-5897	698	11	for	for	ADP
ap-5897	698	12	all	all	DET
ap-5897	698	13	m	m	NOUN
ap-5897	698	14	∈	∈	NOUN
ap-5897	698	15	n	n	ADV
ap-5897	698	16	∪	∪	X
ap-5897	698	17	{	{	PUNCT
ap-5897	698	18	0	0	NUM
ap-5897	698	19	}	}	PUNCT
ap-5897	698	20	by	by	ADP
ap-5897	698	21	lemma	lemma	PROPN
ap-5897	698	22	4.10	4.10	NUM
ap-5897	698	23	.	.	PUNCT
ap-5897	699	1	the	the	DET
ap-5897	699	2	only	only	ADJ
ap-5897	699	3	thing	thing	NOUN
ap-5897	699	4	we	we	PRON
ap-5897	699	5	are	be	AUX
ap-5897	699	6	left	leave	VERB
ap-5897	699	7	to	to	PART
ap-5897	699	8	do	do	VERB
ap-5897	699	9	is	be	AUX
ap-5897	699	10	to	to	PART
ap-5897	699	11	show	show	VERB
ap-5897	699	12	that	that	SCONJ
ap-5897	699	13	σm(w)σm(b	σm(w)σm(b	NOUN
ap-5897	699	14	)	)	PUNCT
ap-5897	699	15	6=	6=	ADP
ap-5897	699	16	γ	γ	PROPN
ap-5897	699	17	+	+	NOUN
ap-5897	699	18	1	1	X
ap-5897	699	19	.	.	PUNCT
ap-5897	700	1	let	let	VERB
ap-5897	700	2	z	z	NOUN
ap-5897	700	3	=	=	SYM
ap-5897	700	4	(	(	PUNCT
ap-5897	700	5	z1	z1	PROPN
ap-5897	700	6	,	,	PUNCT
ap-5897	700	7	z2	z2	PROPN
ap-5897	700	8	,	,	PUNCT
ap-5897	700	9	.	.	PUNCT
ap-5897	700	10	.	.	PUNCT
ap-5897	700	11	.	.	PUNCT
ap-5897	700	12	)	)	PUNCT
ap-5897	701	1	denote	denote	VERB
ap-5897	701	2	the	the	DET
ap-5897	701	3	sequence	sequence	NOUN
ap-5897	701	4	d∗(σm−1(b	d∗(σm−1(b	PROPN
ap-5897	701	5	)	)	PUNCT
ap-5897	701	6	;	;	PUNCT
ap-5897	701	7	γ	γ	X
ap-5897	701	8	+	+	NOUN
ap-5897	701	9	1	1	NUM
ap-5897	701	10	)	)	PUNCT
ap-5897	701	11	.	.	PUNCT
ap-5897	702	1	let	let	VERB
ap-5897	702	2	s	s	PRON
ap-5897	702	3	be	be	AUX
ap-5897	702	4	the	the	DET
ap-5897	702	5	least	least	ADV
ap-5897	702	6	positive	positive	ADJ
ap-5897	702	7	integer	integer	NOUN
ap-5897	702	8	such	such	ADJ
ap-5897	702	9	that	that	DET
ap-5897	702	10	wm+i−1	wm+i−1	PROPN
ap-5897	702	11	=	=	SYM
ap-5897	702	12	zi	zi	PROPN
ap-5897	702	13	for	for	ADP
ap-5897	702	14	1	1	NUM
ap-5897	702	15	≤	≤	NUM
ap-5897	703	1	i	i	PRON
ap-5897	703	2	<	<	X
ap-5897	703	3	s	s	X
ap-5897	703	4	and	and	CCONJ
ap-5897	703	5	(	(	PUNCT
ap-5897	703	6	zs	zs	NOUN
ap-5897	703	7	−	−	PROPN
ap-5897	703	8	wm+s−1	wm+s−1	PROPN
ap-5897	703	9	)	)	PUNCT
ap-5897	703	10	sgn(σm−1(b)[s	sgn(σm−1(b)[s	NOUN
ap-5897	703	11	]	]	PUNCT
ap-5897	703	12	)	)	PUNCT
ap-5897	703	13	≥	≥	NOUN
ap-5897	703	14	1	1	X
ap-5897	703	15	.	.	X
ap-5897	703	16	note	note	VERB
ap-5897	703	17	that	that	SCONJ
ap-5897	703	18	there	there	PRON
ap-5897	703	19	exists	exist	VERB
ap-5897	703	20	y	y	PROPN
ap-5897	703	21	∈	∈	PROPN
ap-5897	703	22	[	[	X
ap-5897	703	23	γ	γ	X
ap-5897	703	24	,	,	PUNCT
ap-5897	703	25	γ	γ	X
ap-5897	703	26	+	+	NOUN
ap-5897	703	27	1	1	NUM
ap-5897	703	28	)	)	PUNCT
ap-5897	703	29	such	such	ADJ
ap-5897	703	30	that	that	DET
ap-5897	703	31	d(σm−1(b);y	d(σm−1(b);y	NOUN
ap-5897	703	32	)	)	PUNCT
ap-5897	703	33	=	=	SYM
ap-5897	703	34	(	(	PUNCT
ap-5897	703	35	z1	z1	PROPN
ap-5897	703	36	,	,	PUNCT
ap-5897	703	37	.	.	PUNCT
ap-5897	703	38	.	.	PUNCT
ap-5897	703	39	.	.	PUNCT
ap-5897	704	1	,	,	PUNCT
ap-5897	704	2	zs	zs	PROPN
ap-5897	704	3	,	,	PUNCT
ap-5897	704	4	ys+1	ys+1	PROPN
ap-5897	704	5	,	,	PUNCT
ap-5897	704	6	ys+2	ys+2	PROPN
ap-5897	704	7	.	.	PUNCT
ap-5897	704	8	.	.	PUNCT
ap-5897	704	9	.	.	PUNCT
ap-5897	704	10	)	)	PUNCT
ap-5897	704	11	.	.	PUNCT
ap-5897	705	1	then	then	ADV
ap-5897	705	2	,	,	PUNCT
ap-5897	705	3	|y	|y	NOUN
ap-5897	705	4	(	(	PUNCT
ap-5897	705	5	s+	s+	NUM
ap-5897	705	6	1,∞)σm+s−1(b	1,∞)σm+s−1(b	NUM
ap-5897	705	7	)	)	PUNCT
ap-5897	705	8	−	−	PROPN
ap-5897	705	9	w(m	w(m	PROPN
ap-5897	705	10	+	+	CCONJ
ap-5897	705	11	s,∞)σm+s−1(b)|	s,∞)σm+s−1(b)|	PROPN
ap-5897	705	12	≤	≤	NUM
ap-5897	705	13	(	(	PUNCT
ap-5897	705	14	γ	γ	X
ap-5897	705	15	+	+	PROPN
ap-5897	705	16	1)−	1)−	PROPN
ap-5897	705	17	γ	γ	X
ap-5897	705	18	=	=	SYM
ap-5897	705	19	1	1	NUM
ap-5897	705	20	.	.	PUNCT
ap-5897	705	21	therefore	therefore	ADV
ap-5897	705	22	,	,	PUNCT
ap-5897	705	23	y	y	PROPN
ap-5897	705	24	−	−	PROPN
ap-5897	705	25	w(m,∞)σm−1(b	w(m,∞)σm−1(b	NOUN
ap-5897	705	26	)	)	PUNCT
ap-5897	705	27	=	=	SYM
ap-5897	706	1	(	(	PUNCT
ap-5897	706	2	zs	zs	NOUN
ap-5897	706	3	−	−	PROPN
ap-5897	706	4	wm+s−1	wm+s−1	PROPN
ap-5897	706	5	)	)	PUNCT
ap-5897	707	1	+	+	CCONJ
ap-5897	707	2	y	y	PROPN
ap-5897	707	3	(	(	PUNCT
ap-5897	707	4	s+	s+	NOUN
ap-5897	707	5	1,∞)σm+s−1(b	1,∞)σm+s−1(b	NUM
ap-5897	707	6	)	)	PUNCT
ap-5897	707	7	σm−1(b)[s	σm−1(b)[s	NOUN
ap-5897	707	8	]	]	PUNCT
ap-5897	707	9	−	−	PROPN
ap-5897	707	10	w(m	w(m	PROPN
ap-5897	707	11	+	+	CCONJ
ap-5897	707	12	s,∞)σm+s−1(b	s,∞)σm+s−1(b	ADJ
ap-5897	707	13	)	)	PUNCT
ap-5897	707	14	σm−1(b)[s	σm−1(b)[s	NOUN
ap-5897	707	15	]	]	X
ap-5897	707	16	≥	≥	NOUN
ap-5897	707	17	1	1	NUM
ap-5897	707	18	+	+	CCONJ
ap-5897	707	19	sgn(σm−1(b)[s])(y	sgn(σm−1(b)[s])(y	NUM
ap-5897	707	20	(	(	PUNCT
ap-5897	707	21	s+	s+	NOUN
ap-5897	707	22	1,∞)σm+s−1(b	1,∞)σm+s−1(b	NUM
ap-5897	707	23	)	)	PUNCT
ap-5897	707	24	|σm−1(b)[s]|	|σm−1(b)[s]|	ADJ
ap-5897	707	25	−	−	PROPN
ap-5897	707	26	w(m	w(m	PROPN
ap-5897	707	27	+	+	CCONJ
ap-5897	707	28	s,∞)σm+s−1(b	s,∞)σm+s−1(b	ADJ
ap-5897	707	29	)	)	PUNCT
ap-5897	707	30	)	)	PUNCT
ap-5897	708	1	|σm−1(b)[s]|	|σm−1(b)[s]|	ADJ
ap-5897	708	2	≥	≥	NOUN
ap-5897	708	3	1−	1−	NUM
ap-5897	708	4	1	1	NUM
ap-5897	708	5	|σm−1(b)[s]|	|σm−1(b)[s]|	NOUN
ap-5897	708	6	=	=	X
ap-5897	709	1	0	0	X
ap-5897	709	2	.	.	PUNCT
ap-5897	710	1	since	since	SCONJ
ap-5897	710	2	γ	γ	PROPN
ap-5897	710	3	+	+	CCONJ
ap-5897	710	4	1	1	NUM
ap-5897	710	5	>	>	SYM
ap-5897	710	6	y	y	PROPN
ap-5897	710	7	≥	≥	PROPN
ap-5897	710	8	w(m,∞)σm−1(b	w(m,∞)σm−1(b	PROPN
ap-5897	710	9	)	)	PUNCT
ap-5897	710	10	,	,	PUNCT
ap-5897	710	11	then	then	ADV
ap-5897	710	12	w(m,∞)σm−1(b	w(m,∞)σm−1(b	NOUN
ap-5897	710	13	)	)	PUNCT
ap-5897	711	1	6=	6=	ADP
ap-5897	711	2	γ	γ	X
ap-5897	711	3	+	+	NOUN
ap-5897	711	4	1	1	X
ap-5897	711	5	.	.	PUNCT
ap-5897	712	1	in	in	ADP
ap-5897	712	2	the	the	DET
ap-5897	712	3	previous	previous	ADJ
ap-5897	712	4	proposition	proposition	NOUN
ap-5897	712	5	,	,	PUNCT
ap-5897	712	6	an	an	DET
ap-5897	712	7	important	important	ADJ
ap-5897	712	8	part	part	NOUN
ap-5897	712	9	of	of	ADP
ap-5897	712	10	the	the	DET
ap-5897	712	11	proof	proof	NOUN
ap-5897	712	12	is	be	AUX
ap-5897	712	13	the	the	DET
ap-5897	712	14	assumption	assumption	NOUN
ap-5897	712	15	that	that	SCONJ
ap-5897	712	16	the	the	DET
ap-5897	712	17	sequence	sequence	NOUN
ap-5897	712	18	w	w	NOUN
ap-5897	712	19	=	=	SYM
ap-5897	712	20	(	(	PUNCT
ap-5897	712	21	w1	w1	NOUN
ap-5897	712	22	,	,	PUNCT
ap-5897	712	23	w2	w2	NOUN
ap-5897	712	24	,	,	PUNCT
ap-5897	712	25	.	.	PUNCT
ap-5897	712	26	.	.	PUNCT
ap-5897	712	27	.	.	PUNCT
ap-5897	712	28	)	)	PUNCT
ap-5897	713	1	∈	∈	PROPN
ap-5897	713	2	a(b	a(b	NOUN
ap-5897	713	3	)	)	PUNCT
ap-5897	713	4	has	have	VERB
ap-5897	713	5	the	the	DET
ap-5897	713	6	property	property	NOUN
ap-5897	713	7	that	that	PRON
ap-5897	713	8	the	the	DET
ap-5897	713	9	series	series	PROPN
ap-5897	713	10	σk(w)σk(b	σk(w)σk(b	PROPN
ap-5897	713	11	)	)	PUNCT
ap-5897	713	12	converges	converge	VERB
ap-5897	713	13	for	for	ADP
ap-5897	713	14	all	all	DET
ap-5897	713	15	k	k	PROPN
ap-5897	713	16	∈	∈	PROPN
ap-5897	713	17	n	n	PART
ap-5897	713	18	∪	∪	X
ap-5897	713	19	{	{	PUNCT
ap-5897	713	20	0	0	NUM
ap-5897	713	21	}	}	PUNCT
ap-5897	713	22	.	.	PUNCT
ap-5897	714	1	it	it	PRON
ap-5897	714	2	is	be	AUX
ap-5897	714	3	clear	clear	ADJ
ap-5897	714	4	that	that	SCONJ
ap-5897	714	5	if	if	SCONJ
ap-5897	714	6	the	the	DET
ap-5897	714	7	base	base	NOUN
ap-5897	714	8	b	b	PROPN
ap-5897	714	9	=	=	PUNCT
ap-5897	714	10	(	(	PUNCT
ap-5897	714	11	β1	β1	PROPN
ap-5897	714	12	,	,	PUNCT
ap-5897	714	13	β2	β2	NOUN
ap-5897	714	14	,	,	PUNCT
ap-5897	714	15	.	.	PUNCT
ap-5897	714	16	.	.	PUNCT
ap-5897	714	17	.	.	PUNCT
ap-5897	714	18	)	)	PUNCT
ap-5897	715	1	is	be	AUX
ap-5897	715	2	eventually	eventually	ADV
ap-5897	715	3	periodic	periodic	ADJ
ap-5897	715	4	,	,	PUNCT
ap-5897	715	5	then	then	ADV
ap-5897	715	6	this	this	DET
ap-5897	715	7	property	property	NOUN
ap-5897	715	8	holds	hold	VERB
ap-5897	715	9	for	for	ADP
ap-5897	715	10	w.	w.	NOUN
ap-5897	715	11	we	we	PRON
ap-5897	715	12	can	can	AUX
ap-5897	715	13	say	say	VERB
ap-5897	715	14	more	more	ADJ
ap-5897	715	15	.	.	PUNCT
ap-5897	716	1	first	first	ADV
ap-5897	716	2	,	,	PUNCT
ap-5897	716	3	note	note	VERB
ap-5897	716	4	that	that	SCONJ
ap-5897	716	5	the	the	DET
ap-5897	716	6	digits	digit	NOUN
ap-5897	716	7	are	be	AUX
ap-5897	716	8	bounded	bound	VERB
ap-5897	716	9	by	by	ADP
ap-5897	716	10	uβi	uβi	NOUN
ap-5897	716	11	and	and	CCONJ
ap-5897	716	12	vβi	vβi	PROPN
ap-5897	716	13	(	(	PUNCT
ap-5897	716	14	see	see	VERB
ap-5897	716	15	section	section	NOUN
ap-5897	716	16	2	2	NUM
ap-5897	716	17	)	)	PUNCT
ap-5897	716	18	,	,	PUNCT
ap-5897	716	19	which	which	PRON
ap-5897	716	20	,	,	PUNCT
ap-5897	716	21	in	in	ADP
ap-5897	716	22	turn	turn	NOUN
ap-5897	716	23	,	,	PUNCT
ap-5897	716	24	satisfy	satisfy	VERB
ap-5897	716	25	max(|uβi	max(|uβi	ADV
ap-5897	716	26	|	|	ADV
ap-5897	716	27	,	,	PUNCT
ap-5897	716	28	|vβi	|vβi	NOUN
ap-5897	716	29	|	|	ADV
ap-5897	716	30	)	)	PUNCT
ap-5897	716	31	≤	≤	NOUN
ap-5897	716	32	(	(	PUNCT
ap-5897	716	33	|βi|+	|βi|+	NOUN
ap-5897	716	34	1)(|γ|+	1)(|γ|+	NUM
ap-5897	716	35	1	1	NUM
ap-5897	716	36	)	)	PUNCT
ap-5897	716	37	.	.	PUNCT
ap-5897	717	1	now	now	ADV
ap-5897	717	2	,	,	PUNCT
ap-5897	717	3	let	let	VERB
ap-5897	717	4	us	we	PRON
ap-5897	717	5	consider	consider	VERB
ap-5897	717	6	the	the	DET
ap-5897	717	7	following	following	NOUN
ap-5897	717	8	.	.	PUNCT
ap-5897	718	1	for	for	ADP
ap-5897	718	2	the	the	DET
ap-5897	718	3	base	base	PROPN
ap-5897	718	4	b	b	NOUN
ap-5897	718	5	,	,	PUNCT
ap-5897	718	6	let	let	VERB
ap-5897	718	7	|b|	|b|	PRON
ap-5897	718	8	be	be	AUX
ap-5897	718	9	the	the	DET
ap-5897	718	10	sequence	sequence	NOUN
ap-5897	718	11	(	(	PUNCT
ap-5897	718	12	|β1|	|β1|	ADJ
ap-5897	718	13	,	,	PUNCT
ap-5897	718	14	|β2|	|β2|	NOUN
ap-5897	718	15	,	,	PUNCT
ap-5897	718	16	.	.	PUNCT
ap-5897	718	17	.	.	PUNCT
ap-5897	718	18	.	.	PUNCT
ap-5897	718	19	)	)	PUNCT
ap-5897	718	20	.	.	PUNCT
ap-5897	719	1	suppose	suppose	VERB
ap-5897	719	2	that	that	SCONJ
ap-5897	719	3	(	(	PUNCT
ap-5897	719	4	|β1|+	|β1|+	NOUN
ap-5897	719	5	1	1	NUM
ap-5897	719	6	,	,	PUNCT
ap-5897	719	7	|β2|+	|β2|+	X
ap-5897	719	8	1	1	NUM
ap-5897	719	9	,	,	PUNCT
ap-5897	719	10	.	.	PUNCT
ap-5897	719	11	.	.	PUNCT
ap-5897	719	12	.	.	PUNCT
ap-5897	719	13	)	)	PUNCT
ap-5897	720	1	|b|	|b|	X
ap-5897	721	1	=	=	PUNCT
ap-5897	722	1	∞∑	∞∑	NUM
ap-5897	722	2	n=1	n=1	NUM
ap-5897	722	3	|βn|+	|βn|+	SYM
ap-5897	722	4	1	1	NUM
ap-5897	722	5	|b[n]|	|b[n]|	NUM
ap-5897	722	6	<	<	X
ap-5897	722	7	∞.	∞.	PROPN
ap-5897	722	8	(	(	PUNCT
ap-5897	722	9	?	?	PUNCT
ap-5897	722	10	?	?	PUNCT
ap-5897	722	11	)	)	PUNCT
ap-5897	723	1	223	223	NUM
ap-5897	724	1	jonathan	jonathan	PROPN
ap-5897	724	2	caalim	caalim	PROPN
ap-5897	724	3	,	,	PUNCT
ap-5897	724	4	shiela	shiela	PROPN
ap-5897	724	5	demegillo	demegillo	PROPN
ap-5897	724	6	acta	acta	PROPN
ap-5897	724	7	polytechnica	polytechnica	PROPN
ap-5897	724	8	then	then	ADV
ap-5897	724	9	for	for	ADP
ap-5897	724	10	every	every	DET
ap-5897	724	11	sequence	sequence	NOUN
ap-5897	724	12	w	w	NOUN
ap-5897	724	13	∈	∈	PROPN
ap-5897	724	14	a(b	a(b	PROPN
ap-5897	724	15	)	)	PUNCT
ap-5897	724	16	,	,	PUNCT
ap-5897	724	17	the	the	DET
ap-5897	724	18	sum	sum	NOUN
ap-5897	724	19	wb	wb	PROPN
ap-5897	724	20	is	be	AUX
ap-5897	724	21	convergent	convergent	ADJ
ap-5897	724	22	.	.	PUNCT
ap-5897	725	1	indeed,∣∣∣∣∣	indeed,∣∣∣∣∣	PROPN
ap-5897	725	2	∞∑	∞∑	NUM
ap-5897	725	3	n=1	n=1	PROPN
ap-5897	725	4	wn	wn	PROPN
ap-5897	725	5	b[n	b[n	PROPN
ap-5897	725	6	]	]	PUNCT
ap-5897	726	1	∣∣∣∣∣	∣∣∣∣∣	ADP
ap-5897	726	2	≤	≤	NOUN
ap-5897	726	3	∞∑	∞∑	NUM
ap-5897	726	4	n=1	n=1	PROPN
ap-5897	726	5	∣∣∣∣	∣∣∣∣	PROPN
ap-5897	726	6	wnb[n	wnb[n	PROPN
ap-5897	726	7	]	]	PUNCT
ap-5897	726	8	∣∣∣∣	∣∣∣∣	NOUN
ap-5897	726	9	≤	≤	NOUN
ap-5897	727	1	∞∑	∞∑	NUM
ap-5897	727	2	n=1	n=1	PROPN
ap-5897	727	3	(	(	PUNCT
ap-5897	727	4	|βn|+	|βn|+	NUM
ap-5897	727	5	1)(|γ|+	1)(|γ|+	NUM
ap-5897	727	6	1	1	NUM
ap-5897	727	7	)	)	PUNCT
ap-5897	727	8	|b[n]|	|b[n]|	ADV
ap-5897	727	9	≤	≤	NOUN
ap-5897	727	10	(	(	PUNCT
ap-5897	727	11	|γ|+	|γ|+	VERB
ap-5897	727	12	1	1	NUM
ap-5897	727	13	)	)	PUNCT
ap-5897	727	14	∞∑	∞∑	NUM
ap-5897	727	15	n=1	n=1	PART
ap-5897	727	16	|βn|+	|βn|+	SYM
ap-5897	727	17	1	1	NUM
ap-5897	727	18	|b[n]|	|b[n]|	NUM
ap-5897	727	19	<	<	X
ap-5897	727	20	∞.	∞.	PROPN
ap-5897	727	21	note	note	VERB
ap-5897	727	22	that	that	SCONJ
ap-5897	727	23	if	if	SCONJ
ap-5897	727	24	b	b	NOUN
ap-5897	727	25	is	be	AUX
ap-5897	727	26	eventually	eventually	ADV
ap-5897	727	27	periodic	periodic	ADJ
ap-5897	727	28	,	,	PUNCT
ap-5897	727	29	then	then	ADV
ap-5897	727	30	(	(	PUNCT
ap-5897	727	31	?	?	PUNCT
ap-5897	727	32	?	?	PUNCT
ap-5897	727	33	)	)	PUNCT
ap-5897	728	1	holds	hold	VERB
ap-5897	728	2	.	.	PUNCT
ap-5897	729	1	however	however	ADV
ap-5897	729	2	,	,	PUNCT
ap-5897	729	3	if	if	SCONJ
ap-5897	729	4	b	b	X
ap-5897	729	5	=	=	SYM
ap-5897	729	6	(	(	PUNCT
ap-5897	729	7	b1	b1	PROPN
ap-5897	729	8	,	,	PUNCT
ap-5897	729	9	b2	b2	NOUN
ap-5897	729	10	,	,	PUNCT
ap-5897	729	11	.	.	PUNCT
ap-5897	729	12	.	.	PUNCT
ap-5897	729	13	.	.	PUNCT
ap-5897	729	14	)	)	PUNCT
ap-5897	730	1	with	with	ADP
ap-5897	730	2	bn	bn	NOUN
ap-5897	730	3	=	=	SYM
ap-5897	730	4	(	(	PUNCT
ap-5897	730	5	n+1)/n	n+1)/n	NOUN
ap-5897	730	6	,	,	PUNCT
ap-5897	730	7	then	then	ADV
ap-5897	730	8	(	(	PUNCT
ap-5897	730	9	?	?	PUNCT
ap-5897	730	10	?	?	PUNCT
ap-5897	730	11	)	)	PUNCT
ap-5897	730	12	does	do	AUX
ap-5897	730	13	not	not	PART
ap-5897	730	14	hold	hold	VERB
ap-5897	730	15	.	.	PUNCT
ap-5897	731	1	we	we	PRON
ap-5897	731	2	now	now	ADV
ap-5897	731	3	state	state	VERB
ap-5897	731	4	the	the	DET
ap-5897	731	5	main	main	ADJ
ap-5897	731	6	result	result	NOUN
ap-5897	731	7	of	of	ADP
ap-5897	731	8	this	this	DET
ap-5897	731	9	article	article	NOUN
ap-5897	731	10	,	,	PUNCT
ap-5897	731	11	which	which	PRON
ap-5897	731	12	provides	provide	VERB
ap-5897	731	13	a	a	DET
ap-5897	731	14	sufficient	sufficient	ADJ
ap-5897	731	15	and	and	CCONJ
ap-5897	731	16	necessary	necessary	ADJ
ap-5897	731	17	condition	condition	NOUN
ap-5897	731	18	for	for	ADP
ap-5897	731	19	admissibility	admissibility	NOUN
ap-5897	731	20	of	of	ADP
ap-5897	731	21	integer	integer	NOUN
ap-5897	731	22	sequence	sequence	NOUN
ap-5897	731	23	in	in	ADP
ap-5897	731	24	a(b	a(b	NOUN
ap-5897	731	25	)	)	PUNCT
ap-5897	731	26	with	with	ADP
ap-5897	731	27	respect	respect	NOUN
ap-5897	731	28	to	to	ADP
ap-5897	731	29	the	the	DET
ap-5897	731	30	beta	beta	PROPN
ap-5897	731	31	cantor	cantor	PROPN
ap-5897	731	32	series	series	NOUN
ap-5897	731	33	expansion	expansion	NOUN
ap-5897	731	34	for	for	ADP
ap-5897	731	35	a	a	DET
ap-5897	731	36	base	base	NOUN
ap-5897	731	37	sequence	sequence	NOUN
ap-5897	731	38	b	b	NOUN
ap-5897	731	39	satisfying	satisfying	NOUN
ap-5897	731	40	(	(	PUNCT
ap-5897	731	41	?	?	PUNCT
ap-5897	731	42	?	?	PUNCT
ap-5897	731	43	)	)	PUNCT
ap-5897	731	44	.	.	PUNCT
ap-5897	732	1	it	it	PRON
ap-5897	732	2	would	would	AUX
ap-5897	732	3	be	be	AUX
ap-5897	732	4	interesting	interesting	ADJ
ap-5897	732	5	to	to	PART
ap-5897	732	6	know	know	VERB
ap-5897	732	7	how	how	SCONJ
ap-5897	732	8	the	the	DET
ap-5897	732	9	result	result	NOUN
ap-5897	732	10	can	can	AUX
ap-5897	732	11	be	be	AUX
ap-5897	732	12	extended	extend	VERB
ap-5897	732	13	beyond	beyond	ADP
ap-5897	732	14	property	property	NOUN
ap-5897	732	15	(	(	PUNCT
ap-5897	732	16	?	?	PUNCT
ap-5897	732	17	?	?	PUNCT
ap-5897	732	18	)	)	PUNCT
ap-5897	732	19	.	.	PUNCT
ap-5897	733	1	theorem	theorem	VERB
ap-5897	733	2	4.12	4.12	NUM
ap-5897	733	3	.	.	PUNCT
ap-5897	734	1	let	let	VERB
ap-5897	734	2	b	b	X
ap-5897	734	3	∈	∈	PROPN
ap-5897	734	4	rn	rn	PROPN
ap-5897	734	5	such	such	ADJ
ap-5897	734	6	that	that	SCONJ
ap-5897	734	7	limm→∞	limm→∞	PROPN
ap-5897	734	8	|b[m]|	|b[m]|	NOUN
ap-5897	734	9	=	=	SYM
ap-5897	734	10	∞	∞	PROPN
ap-5897	734	11	and	and	CCONJ
ap-5897	734	12	(	(	PUNCT
ap-5897	734	13	?	?	PUNCT
ap-5897	734	14	?	?	PUNCT
ap-5897	734	15	)	)	PUNCT
ap-5897	735	1	holds	hold	VERB
ap-5897	735	2	.	.	PUNCT
ap-5897	736	1	let	let	VERB
ap-5897	736	2	(	(	PUNCT
ap-5897	736	3	d1	d1	NOUN
ap-5897	736	4	,	,	PUNCT
ap-5897	736	5	d2	d2	PROPN
ap-5897	736	6	,	,	PUNCT
ap-5897	736	7	.	.	PUNCT
ap-5897	736	8	.	.	PUNCT
ap-5897	736	9	.	.	PUNCT
ap-5897	736	10	)	)	PUNCT
ap-5897	737	1	∈	∈	PROPN
ap-5897	737	2	a(b	a(b	NOUN
ap-5897	737	3	)	)	PUNCT
ap-5897	737	4	.	.	PUNCT
ap-5897	738	1	then	then	ADV
ap-5897	738	2	(	(	PUNCT
ap-5897	738	3	d1	d1	PROPN
ap-5897	738	4	,	,	PUNCT
ap-5897	738	5	d2	d2	PROPN
ap-5897	738	6	,	,	PUNCT
ap-5897	738	7	.	.	PUNCT
ap-5897	738	8	.	.	PUNCT
ap-5897	738	9	.	.	PUNCT
ap-5897	738	10	)	)	PUNCT
ap-5897	738	11	is	be	AUX
ap-5897	738	12	badmissible	badmissible	ADJ
ap-5897	738	13	if	if	SCONJ
ap-5897	739	1	and	and	CCONJ
ap-5897	739	2	only	only	ADV
ap-5897	739	3	if	if	SCONJ
ap-5897	739	4	(	(	PUNCT
ap-5897	739	5	d1	d1	NOUN
ap-5897	739	6	,	,	PUNCT
ap-5897	739	7	d2	d2	PROPN
ap-5897	739	8	,	,	PUNCT
ap-5897	739	9	.	.	PUNCT
ap-5897	739	10	.	.	PUNCT
ap-5897	739	11	.	.	PUNCT
ap-5897	739	12	)	)	PUNCT
ap-5897	739	13	satisfies	satisfy	VERB
ap-5897	739	14	the	the	DET
ap-5897	739	15	lexicographic	lexicographic	ADJ
ap-5897	739	16	restriction	restriction	NOUN
ap-5897	739	17	.	.	PUNCT
ap-5897	740	1	acknowledgements	acknowledgement	NOUN
ap-5897	740	2	the	the	DET
ap-5897	740	3	authors	author	NOUN
ap-5897	740	4	would	would	AUX
ap-5897	740	5	like	like	VERB
ap-5897	740	6	to	to	PART
ap-5897	740	7	thank	thank	VERB
ap-5897	740	8	the	the	DET
ap-5897	740	9	anonymous	anonymous	ADJ
ap-5897	740	10	reviewer	reviewer	NOUN
ap-5897	740	11	for	for	ADP
ap-5897	740	12	valuable	valuable	ADJ
ap-5897	740	13	remarks	remark	NOUN
ap-5897	740	14	that	that	PRON
ap-5897	740	15	improved	improve	VERB
ap-5897	740	16	the	the	DET
ap-5897	740	17	quality	quality	NOUN
ap-5897	740	18	of	of	ADP
ap-5897	740	19	the	the	DET
ap-5897	740	20	paper	paper	NOUN
ap-5897	740	21	.	.	PUNCT
ap-5897	741	1	j.	j.	PROPN
ap-5897	741	2	caalim	caalim	PROPN
ap-5897	741	3	is	be	AUX
ap-5897	741	4	grateful	grateful	ADJ
ap-5897	741	5	to	to	ADP
ap-5897	741	6	the	the	DET
ap-5897	741	7	university	university	NOUN
ap-5897	741	8	of	of	ADP
ap-5897	741	9	the	the	DET
ap-5897	741	10	philippines	philippine	NOUN
ap-5897	741	11	for	for	ADP
ap-5897	741	12	the	the	DET
ap-5897	741	13	financial	financial	ADJ
ap-5897	741	14	support	support	NOUN
ap-5897	741	15	through	through	ADP
ap-5897	741	16	its	its	PRON
ap-5897	741	17	phd	phd	NOUN
ap-5897	741	18	incentive	incentive	NOUN
ap-5897	741	19	award	award	PROPN
ap-5897	741	20	under	under	ADP
ap-5897	741	21	the	the	DET
ap-5897	741	22	office	office	NOUN
ap-5897	741	23	of	of	ADP
ap-5897	741	24	the	the	DET
ap-5897	741	25	vice	vice	NOUN
ap-5897	741	26	chancellor	chancellor	NOUN
ap-5897	741	27	for	for	ADP
ap-5897	741	28	research	research	NOUN
ap-5897	741	29	and	and	CCONJ
ap-5897	741	30	development	development	NOUN
ap-5897	741	31	.	.	PUNCT
ap-5897	742	1	s.	s.	PROPN
ap-5897	742	2	demegillo	demegillo	PROPN
ap-5897	742	3	is	be	AUX
ap-5897	742	4	grateful	grateful	ADJ
ap-5897	742	5	to	to	ADP
ap-5897	742	6	the	the	DET
ap-5897	742	7	department	department	NOUN
ap-5897	742	8	of	of	ADP
ap-5897	742	9	science	science	NOUN
ap-5897	742	10	and	and	CCONJ
ap-5897	742	11	technology	technology	NOUN
ap-5897	742	12	of	of	ADP
ap-5897	742	13	the	the	DET
ap-5897	742	14	philippine	philippine	ADJ
ap-5897	742	15	government	government	NOUN
ap-5897	742	16	for	for	ADP
ap-5897	742	17	the	the	DET
ap-5897	742	18	financial	financial	ADJ
ap-5897	742	19	support	support	NOUN
ap-5897	742	20	under	under	ADP
ap-5897	742	21	the	the	DET
ap-5897	742	22	dost	dost	NOUN
ap-5897	742	23	-	-	PUNCT
ap-5897	742	24	asthrdp	asthrdp	PROPN
ap-5897	742	25	scholarship	scholarship	NOUN
ap-5897	742	26	grant	grant	NOUN
ap-5897	742	27	.	.	PUNCT
ap-5897	743	1	references	reference	NOUN
ap-5897	743	2	[	[	X
ap-5897	743	3	1	1	NUM
ap-5897	743	4	]	]	PUNCT
ap-5897	743	5	a.	a.	NOUN
ap-5897	743	6	rényi	rényi	PROPN
ap-5897	743	7	.	.	PUNCT
ap-5897	744	1	representations	representation	NOUN
ap-5897	744	2	for	for	ADP
ap-5897	744	3	real	real	ADJ
ap-5897	744	4	numbers	number	NOUN
ap-5897	744	5	and	and	CCONJ
ap-5897	744	6	their	their	PRON
ap-5897	744	7	ergodic	ergodic	ADJ
ap-5897	744	8	properties	property	NOUN
ap-5897	744	9	.	.	PUNCT
ap-5897	745	1	acta	acta	PROPN
ap-5897	745	2	math	math	PROPN
ap-5897	745	3	acad	acad	PROPN
ap-5897	745	4	sci	sci	PROPN
ap-5897	745	5	hungar	hungar	NOUN
ap-5897	745	6	8:477–493	8:477–493	NUM
ap-5897	745	7	,	,	PUNCT
ap-5897	745	8	1957	1957	NUM
ap-5897	745	9	.	.	PUNCT
ap-5897	746	1	doi:10.1007	doi:10.1007	PROPN
ap-5897	746	2	/	/	SYM
ap-5897	746	3	bf02020331	bf02020331	PROPN
ap-5897	746	4	.	.	PUNCT
ap-5897	747	1	[	[	X
ap-5897	747	2	2	2	X
ap-5897	747	3	]	]	PUNCT
ap-5897	747	4	w.	w.	PROPN
ap-5897	747	5	parry	parry	PROPN
ap-5897	747	6	.	.	PUNCT
ap-5897	748	1	on	on	ADP
ap-5897	748	2	the	the	DET
ap-5897	748	3	β	β	NOUN
ap-5897	748	4	-	-	NOUN
ap-5897	748	5	expansions	expansion	NOUN
ap-5897	748	6	of	of	ADP
ap-5897	748	7	real	real	ADJ
ap-5897	748	8	numbers	number	NOUN
ap-5897	748	9	.	.	PUNCT
ap-5897	749	1	acta	acta	PROPN
ap-5897	749	2	math	math	PROPN
ap-5897	749	3	acad	acad	PROPN
ap-5897	749	4	sci	sci	PROPN
ap-5897	749	5	hungar	hungar	PROPN
ap-5897	749	6	11:401–426	11:401–426	PROPN
ap-5897	749	7	,	,	PUNCT
ap-5897	749	8	1960	1960	NUM
ap-5897	749	9	.	.	PUNCT
ap-5897	750	1	doi:10.1007	doi:10.1007	NOUN
ap-5897	750	2	/	/	SYM
ap-5897	750	3	bf02020954	bf02020954	PROPN
ap-5897	750	4	.	.	PUNCT
ap-5897	751	1	[	[	X
ap-5897	751	2	3	3	X
ap-5897	751	3	]	]	X
ap-5897	751	4	w.	w.	PROPN
ap-5897	751	5	parry	parry	PROPN
ap-5897	751	6	.	.	PUNCT
ap-5897	752	1	representations	representation	NOUN
ap-5897	752	2	for	for	ADP
ap-5897	752	3	real	real	ADJ
ap-5897	752	4	numbers	number	NOUN
ap-5897	752	5	.	.	PUNCT
ap-5897	753	1	acta	acta	PROPN
ap-5897	753	2	math	math	PROPN
ap-5897	753	3	acad	acad	PROPN
ap-5897	753	4	sci	sci	PROPN
ap-5897	753	5	hungar	hungar	PROPN
ap-5897	753	6	15:95–105	15:95–105	NUM
ap-5897	753	7	,	,	PUNCT
ap-5897	753	8	1964	1964	NUM
ap-5897	753	9	.	.	PUNCT
ap-5897	754	1	doi:10.1007	doi:10.1007	VERB
ap-5897	754	2	/	/	SYM
ap-5897	754	3	bf01897025	bf01897025	NOUN
ap-5897	754	4	.	.	PUNCT
ap-5897	755	1	[	[	X
ap-5897	755	2	4	4	NUM
ap-5897	755	3	]	]	X
ap-5897	755	4	c.	c.	PROPN
ap-5897	755	5	frougny	frougny	PROPN
ap-5897	755	6	,	,	PUNCT
ap-5897	755	7	a.	a.	PROPN
ap-5897	755	8	lai	lai	PROPN
ap-5897	755	9	.	.	PUNCT
ap-5897	756	1	on	on	ADP
ap-5897	756	2	negative	negative	ADJ
ap-5897	756	3	bases	basis	NOUN
ap-5897	756	4	.	.	PUNCT
ap-5897	757	1	development	development	NOUN
ap-5897	757	2	in	in	ADP
ap-5897	757	3	language	language	NOUN
ap-5897	757	4	theory	theory	NOUN
ap-5897	757	5	pp	pp	X
ap-5897	757	6	.	.	PUNCT
ap-5897	758	1	252–263	252–263	NUM
ap-5897	758	2	,	,	PUNCT
ap-5897	758	3	2009	2009	NUM
ap-5897	758	4	.	.	PUNCT
ap-5897	759	1	doi:10.1007/978	doi:10.1007/978	ADJ
ap-5897	759	2	-	-	PUNCT
ap-5897	759	3	3	3	NUM
ap-5897	759	4	-	-	PUNCT
ap-5897	759	5	642	642	NUM
ap-5897	759	6	-	-	PUNCT
ap-5897	759	7	02737	02737	NUM
ap-5897	759	8	-	-	PUNCT
ap-5897	759	9	6_20	6_20	NOUN
ap-5897	759	10	.	.	PUNCT
ap-5897	760	1	[	[	X
ap-5897	760	2	5	5	X
ap-5897	760	3	]	]	PUNCT
ap-5897	760	4	v.	v.	ADP
ap-5897	760	5	grünwald	grünwald	PROPN
ap-5897	760	6	.	.	PUNCT
ap-5897	761	1	intorno	intorno	PROPN
ap-5897	761	2	all’aritmetica	all’aritmetica	PROPN
ap-5897	761	3	dei	dei	PROPN
ap-5897	761	4	sistemi	sistemi	PROPN
ap-5897	761	5	numerici	numerici	PROPN
ap-5897	761	6	a	a	DET
ap-5897	761	7	base	base	PROPN
ap-5897	761	8	negativa	negativa	PROPN
ap-5897	761	9	con	con	PROPN
ap-5897	761	10	particolare	particolare	PROPN
ap-5897	761	11	riguardo	riguardo	PROPN
ap-5897	761	12	al	al	PROPN
ap-5897	761	13	sistema	sistema	PROPN
ap-5897	761	14	numerico	numerico	PROPN
ap-5897	761	15	a	a	DET
ap-5897	761	16	base	base	NOUN
ap-5897	761	17	negativo	negativo	ADJ
ap-5897	761	18	-	-	PUNCT
ap-5897	761	19	decimale	decimale	NOUN
ap-5897	761	20	per	per	ADP
ap-5897	761	21	lo	lo	PROPN
ap-5897	761	22	studio	studio	NOUN
ap-5897	761	23	delle	delle	NOUN
ap-5897	761	24	sue	sue	PROPN
ap-5897	761	25	analogie	analogie	PROPN
ap-5897	761	26	coll’aritmetica	coll’aritmetica	PROPN
ap-5897	761	27	ordinaria	ordinaria	PROPN
ap-5897	761	28	(	(	PUNCT
ap-5897	761	29	decimale	decimale	NOUN
ap-5897	761	30	)	)	PUNCT
ap-5897	761	31	.	.	PUNCT
ap-5897	762	1	giornale	giornale	PROPN
ap-5897	762	2	di	di	PROPN
ap-5897	762	3	matematiche	matematiche	PROPN
ap-5897	762	4	di	di	PROPN
ap-5897	762	5	battaglini	battaglini	PROPN
ap-5897	762	6	pp	pp	ADV
ap-5897	762	7	.	.	PUNCT
ap-5897	763	1	203–221	203–221	NUM
ap-5897	763	2	,	,	PUNCT
ap-5897	763	3	1885	1885	NUM
ap-5897	763	4	.	.	PUNCT
ap-5897	764	1	[	[	X
ap-5897	764	2	6	6	NUM
ap-5897	764	3	]	]	PUNCT
ap-5897	764	4	s.	s.	PROPN
ap-5897	764	5	ito	ito	PROPN
ap-5897	764	6	,	,	PUNCT
ap-5897	764	7	t.	t.	PROPN
ap-5897	764	8	sadahiro	sadahiro	PROPN
ap-5897	764	9	.	.	PUNCT
ap-5897	765	1	beta	beta	NOUN
ap-5897	765	2	-	-	PUNCT
ap-5897	765	3	expansions	expansion	NOUN
ap-5897	765	4	with	with	ADP
ap-5897	765	5	negative	negative	ADJ
ap-5897	765	6	bases	basis	NOUN
ap-5897	765	7	.	.	PUNCT
ap-5897	766	1	integers	integer	NOUN
ap-5897	766	2	9:239–259	9:239–259	NOUN
ap-5897	766	3	,	,	PUNCT
ap-5897	766	4	2009	2009	NUM
ap-5897	766	5	.	.	PUNCT
ap-5897	767	1	doi:10.1515	doi:10.1515	NOUN
ap-5897	767	2	/	/	SYM
ap-5897	767	3	integ.2009.023	integ.2009.023	NOUN
ap-5897	767	4	.	.	PUNCT
ap-5897	768	1	[	[	X
ap-5897	768	2	7	7	X
ap-5897	768	3	]	]	X
ap-5897	768	4	l.	l.	PROPN
ap-5897	768	5	liao	liao	PROPN
ap-5897	768	6	,	,	PUNCT
ap-5897	768	7	w.	w.	PROPN
ap-5897	768	8	steiner	steiner	PROPN
ap-5897	768	9	.	.	PUNCT
ap-5897	769	1	dynamical	dynamical	ADJ
ap-5897	769	2	properties	property	NOUN
ap-5897	769	3	of	of	ADP
ap-5897	769	4	the	the	DET
ap-5897	769	5	negative	negative	ADJ
ap-5897	769	6	beta	beta	ADJ
ap-5897	769	7	transformation	transformation	NOUN
ap-5897	769	8	.	.	PUNCT
ap-5897	770	1	ergod	ergod	PROPN
ap-5897	770	2	th	th	PROPN
ap-5897	770	3	&	&	CCONJ
ap-5897	770	4	dynam	dynam	PROPN
ap-5897	770	5	sys	sys	VERB
ap-5897	770	6	32(5):1673–1690	32(5):1673–1690	NUM
ap-5897	770	7	,	,	PUNCT
ap-5897	770	8	2012	2012	NUM
ap-5897	770	9	.	.	PUNCT
ap-5897	771	1	doi:10.1017	doi:10.1017	NOUN
ap-5897	771	2	/	/	SYM
ap-5897	771	3	s0143385711000514	s0143385711000514	NOUN
ap-5897	771	4	.	.	PUNCT
ap-5897	772	1	[	[	X
ap-5897	772	2	8	8	NUM
ap-5897	772	3	]	]	X
ap-5897	772	4	d.	d.	PROPN
ap-5897	772	5	dombek	dombek	PROPN
ap-5897	772	6	,	,	PUNCT
ap-5897	772	7	z.	z.	PROPN
ap-5897	772	8	masáková	masáková	PROPN
ap-5897	772	9	,	,	PUNCT
ap-5897	772	10	e.	e.	PROPN
ap-5897	772	11	pelantová	pelantová	PROPN
ap-5897	772	12	.	.	PUNCT
ap-5897	773	1	number	number	NOUN
ap-5897	773	2	representation	representation	NOUN
ap-5897	773	3	using	use	VERB
ap-5897	773	4	the	the	DET
ap-5897	773	5	generalized	generalized	ADJ
ap-5897	773	6	(	(	PUNCT
ap-5897	773	7	−β)-transformation	−β)-transformation	NOUN
ap-5897	773	8	.	.	PUNCT
ap-5897	774	1	theoretical	theoretical	ADJ
ap-5897	774	2	computer	computer	NOUN
ap-5897	774	3	science	science	NOUN
ap-5897	774	4	412(48):6653–6665	412(48):6653–6665	NUM
ap-5897	774	5	,	,	PUNCT
ap-5897	774	6	2011	2011	NUM
ap-5897	774	7	.	.	PUNCT
ap-5897	775	1	doi:10.1016	doi:10.1016	PROPN
ap-5897	775	2	/	/	SYM
ap-5897	775	3	j.tcs.2011.08.028	j.tcs.2011.08.028	PROPN
ap-5897	775	4	.	.	PUNCT
ap-5897	776	1	[	[	X
ap-5897	776	2	9	9	NUM
ap-5897	776	3	]	]	PUNCT
ap-5897	776	4	c.	c.	PROPN
ap-5897	776	5	kalle	kalle	PROPN
ap-5897	776	6	.	.	PUNCT
ap-5897	777	1	isomorphisms	isomorphism	NOUN
ap-5897	777	2	between	between	ADP
ap-5897	777	3	positive	positive	ADJ
ap-5897	777	4	and	and	CCONJ
ap-5897	777	5	negative	negative	ADJ
ap-5897	777	6	(	(	PUNCT
ap-5897	777	7	β)-transformations	β)-transformation	NOUN
ap-5897	777	8	.	.	PUNCT
ap-5897	777	9	ergod	ergod	PROPN
ap-5897	777	10	th	th	PROPN
ap-5897	777	11	&	&	CCONJ
ap-5897	777	12	dynam	dynam	PROPN
ap-5897	777	13	sys	sys	VERB
ap-5897	777	14	34(1):153–170	34(1):153–170	PROPN
ap-5897	777	15	,	,	PUNCT
ap-5897	777	16	2014	2014	NUM
ap-5897	777	17	.	.	PUNCT
ap-5897	778	1	doi:10.1017	doi:10.1017	NOUN
ap-5897	778	2	/	/	SYM
ap-5897	778	3	etds.2012.127	etds.2012.127	PROPN
ap-5897	778	4	.	.	PUNCT
ap-5897	779	1	[	[	X
ap-5897	779	2	10	10	NUM
ap-5897	779	3	]	]	PUNCT
ap-5897	779	4	k.	k.	PROPN
ap-5897	779	5	dajani	dajani	PROPN
ap-5897	779	6	,	,	PUNCT
ap-5897	779	7	c.	c.	PROPN
ap-5897	779	8	kalle	kalle	PROPN
ap-5897	779	9	.	.	PUNCT
ap-5897	780	1	transformations	transformation	NOUN
ap-5897	780	2	generating	generate	VERB
ap-5897	780	3	negative	negative	ADJ
ap-5897	780	4	(	(	PUNCT
ap-5897	780	5	−β)-expansions	−β)-expansion	NOUN
ap-5897	780	6	.	.	PUNCT
ap-5897	781	1	integers	integer	NOUN
ap-5897	781	2	11(b):1–18	11(b):1–18	NUM
ap-5897	781	3	,	,	PUNCT
ap-5897	781	4	2011	2011	NUM
ap-5897	781	5	.	.	PUNCT
ap-5897	782	1	[	[	X
ap-5897	782	2	11	11	NUM
ap-5897	782	3	]	]	PUNCT
ap-5897	782	4	s.	s.	PROPN
ap-5897	782	5	akiyama	akiyama	PROPN
ap-5897	782	6	,	,	PUNCT
ap-5897	782	7	j.	j.	PROPN
ap-5897	782	8	caalim	caalim	PROPN
ap-5897	782	9	.	.	PUNCT
ap-5897	783	1	rotational	rotational	ADJ
ap-5897	783	2	beta	beta	ADJ
ap-5897	783	3	expansion	expansion	NOUN
ap-5897	783	4	:	:	PUNCT
ap-5897	783	5	ergodicity	ergodicity	NOUN
ap-5897	783	6	and	and	CCONJ
ap-5897	783	7	soficness	soficness	NOUN
ap-5897	783	8	.	.	PUNCT
ap-5897	784	1	j	j	PROPN
ap-5897	784	2	math	math	PROPN
ap-5897	784	3	soc	soc	PROPN
ap-5897	784	4	japan	japan	PROPN
ap-5897	784	5	69(1):397–415	69(1):397–415	PROPN
ap-5897	784	6	,	,	PUNCT
ap-5897	784	7	2017	2017	NUM
ap-5897	784	8	.	.	PUNCT
ap-5897	785	1	doi:10.2969	doi:10.2969	NOUN
ap-5897	785	2	/	/	SYM
ap-5897	785	3	jmsj/06910397	jmsj/06910397	PROPN
ap-5897	785	4	.	.	PUNCT
ap-5897	786	1	[	[	X
ap-5897	786	2	12	12	NUM
ap-5897	786	3	]	]	X
ap-5897	786	4	s.	s.	PROPN
ap-5897	786	5	akiyama	akiyama	PROPN
ap-5897	786	6	,	,	PUNCT
ap-5897	786	7	j.	j.	PROPN
ap-5897	786	8	caalim	caalim	PROPN
ap-5897	786	9	.	.	PUNCT
ap-5897	787	1	invariant	invariant	ADJ
ap-5897	787	2	measure	measure	NOUN
ap-5897	787	3	of	of	ADP
ap-5897	787	4	rotational	rotational	ADJ
ap-5897	787	5	beta	beta	NOUN
ap-5897	787	6	expansion	expansion	NOUN
ap-5897	787	7	and	and	CCONJ
ap-5897	787	8	tarski	tarski	NOUN
ap-5897	787	9	’s	’s	PART
ap-5897	787	10	plank	plank	NOUN
ap-5897	787	11	problem	problem	NOUN
ap-5897	787	12	.	.	PUNCT
ap-5897	788	1	discrete	discrete	ADJ
ap-5897	788	2	comput	comput	ADJ
ap-5897	788	3	geom	geom	PROPN
ap-5897	788	4	57(2):357–370	57(2):357–370	PROPN
ap-5897	788	5	,	,	PUNCT
ap-5897	788	6	2017	2017	NUM
ap-5897	788	7	.	.	PUNCT
ap-5897	789	1	doi:10.1007	doi:10.1007	NOUN
ap-5897	789	2	/	/	SYM
ap-5897	789	3	s00454	s00454	NOUN
ap-5897	789	4	-	-	PUNCT
ap-5897	789	5	016	016	NUM
ap-5897	789	6	-	-	PUNCT
ap-5897	789	7	9849	9849	NUM
ap-5897	789	8	-	-	SYM
ap-5897	789	9	4	4	NUM
ap-5897	789	10	.	.	PUNCT
ap-5897	790	1	[	[	X
ap-5897	790	2	13	13	NUM
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ap-5897	790	6	.	.	PUNCT
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ap-5897	791	4	zahlensysteme	zahlensysteme	PROPN
ap-5897	791	5	.	.	PUNCT
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ap-5897	792	7	,	,	PUNCT
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ap-5897	792	9	.	.	PUNCT
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ap-5897	793	6	,	,	PUNCT
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ap-5897	793	9	.	.	PUNCT
ap-5897	794	1	ito	ito	PROPN
ap-5897	794	2	-	-	PUNCT
ap-5897	794	3	sadahiro	sadahiro	PROPN
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ap-5897	794	5	vs.	vs.	ADP
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ap-5897	794	8	.	.	PUNCT
ap-5897	795	1	acta	acta	PROPN
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ap-5897	795	4	,	,	PUNCT
ap-5897	795	5	2011	2011	NUM
ap-5897	795	6	.	.	PUNCT
ap-5897	796	1	[	[	X
ap-5897	796	2	15	15	NUM
ap-5897	796	3	]	]	X
ap-5897	796	4	k.	k.	PROPN
ap-5897	796	5	dajani	dajani	PROPN
ap-5897	796	6	,	,	PUNCT
ap-5897	796	7	s.	s.	PROPN
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ap-5897	796	9	.	.	PUNCT
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ap-5897	797	4	(	(	PUNCT
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ap-5897	797	6	.	.	PUNCT
ap-5897	798	1	journal	journal	NOUN
ap-5897	798	2	of	of	ADP
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ap-5897	798	5	15:12.2.6/1	15:12.2.6/1	NUM
ap-5897	798	6	–	–	PUNCT
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ap-5897	798	8	,	,	PUNCT
ap-5897	798	9	2012	2012	NUM
ap-5897	798	10	.	.	PUNCT
ap-5897	799	1	224	224	NUM
ap-5897	799	2	http://dx.doi.org/10.1007/bf02020331	http://dx.doi.org/10.1007/bf02020331	NOUN
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ap-5897	799	5	http://dx.doi.org/10.1007/978-3-642-02737-6_20	http://dx.doi.org/10.1007/978-3-642-02737-6_20	NOUN
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ap-5897	799	7	http://dx.doi.org/10.1017/s0143385711000514	http://dx.doi.org/10.1017/s0143385711000514	NOUN
ap-5897	799	8	http://dx.doi.org/10.1016/j.tcs.2011.08.028	http://dx.doi.org/10.1016/j.tcs.2011.08.028	NOUN
ap-5897	799	9	http://dx.doi.org/10.1017/etds.2012.127	http://dx.doi.org/10.1017/etds.2012.127	NOUN
ap-5897	799	10	http://dx.doi.org/10.2969/jmsj/06910397	http://dx.doi.org/10.2969/jmsj/06910397	PROPN
ap-5897	799	11	http://dx.doi.org/10.1007/s00454-016-9849-4	http://dx.doi.org/10.1007/s00454-016-9849-4	PROPN
ap-5897	799	12	acta	acta	PROPN
ap-5897	799	13	polytechnica	polytechnica	PROPN
ap-5897	799	14	60(3):214–224	60(3):214–224	PROPN
ap-5897	799	15	,	,	PUNCT
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ap-5897	799	17	1	1	NUM
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ap-5897	799	19	2	2	NUM
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ap-5897	799	21	-	-	PUNCT
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ap-5897	799	23	maps	map	VERB
ap-5897	799	24	3	3	NUM
ap-5897	799	25	expansion	expansion	NOUN
ap-5897	799	26	of	of	ADP
ap-5897	799	27	+1	+1	ADJ
ap-5897	799	28	4	4	NUM
ap-5897	799	29	admissible	admissible	ADJ
ap-5897	799	30	sequences	sequence	NOUN
ap-5897	799	31	acknowledgements	acknowledgement	NOUN
ap-5897	799	32	references	reference	NOUN