id	sid	tid	token	lemma	pos
ap-8536	1	1	acta	acta	PROPN
ap-8536	1	2	polytechnica	polytechnica	PROPN
ap-8536	1	3	https://doi.org/10.14311/ap.2023.63.0188	https://doi.org/10.14311/ap.2023.63.0188	PROPN
ap-8536	1	4	acta	acta	PROPN
ap-8536	1	5	polytechnica	polytechnica	PROPN
ap-8536	1	6	63(3):188–198	63(3):188–198	PROPN
ap-8536	1	7	,	,	PUNCT
ap-8536	1	8	2023	2023	NUM
ap-8536	1	9	©	©	ADP
ap-8536	1	10	2023	2023	NUM
ap-8536	1	11	the	the	DET
ap-8536	1	12	author(s	author(s	NOUN
ap-8536	1	13	)	)	PUNCT
ap-8536	1	14	.	.	PUNCT
ap-8536	2	1	licensed	license	VERB
ap-8536	2	2	under	under	ADP
ap-8536	2	3	a	a	DET
ap-8536	2	4	cc	cc	NOUN
ap-8536	2	5	-	-	PUNCT
ap-8536	2	6	by	by	ADP
ap-8536	2	7	4.0	4.0	NUM
ap-8536	2	8	licence	licence	NOUN
ap-8536	2	9	published	publish	VERB
ap-8536	2	10	by	by	ADP
ap-8536	2	11	the	the	DET
ap-8536	2	12	czech	czech	PROPN
ap-8536	2	13	technical	technical	PROPN
ap-8536	2	14	university	university	PROPN
ap-8536	2	15	in	in	ADP
ap-8536	2	16	prague	prague	PROPN
ap-8536	2	17	from	from	ADP
ap-8536	2	18	positional	positional	ADJ
ap-8536	2	19	representation	representation	NOUN
ap-8536	2	20	of	of	ADP
ap-8536	2	21	numbers	number	NOUN
ap-8536	2	22	to	to	ADP
ap-8536	2	23	positional	positional	ADJ
ap-8536	2	24	representation	representation	NOUN
ap-8536	2	25	of	of	ADP
ap-8536	2	26	vectors	vector	NOUN
ap-8536	2	27	izabella	izabella	PROPN
ap-8536	2	28	ingrid	ingrid	PROPN
ap-8536	2	29	farkasa	farkasa	PROPN
ap-8536	2	30	,	,	PUNCT
ap-8536	2	31	edita	edita	PROPN
ap-8536	2	32	pelantováb,∗	pelantováb,∗	PROPN
ap-8536	2	33	,	,	PUNCT
ap-8536	2	34	milena	milena	NOUN
ap-8536	2	35	svobodováb	svobodováb	NOUN
ap-8536	2	36	a	a	DET
ap-8536	2	37	eötvös	eötvös	PROPN
ap-8536	2	38	loránd	loránd	PROPN
ap-8536	2	39	university	university	PROPN
ap-8536	2	40	,	,	PUNCT
ap-8536	2	41	doctoral	doctoral	ADJ
ap-8536	2	42	school	school	NOUN
ap-8536	2	43	of	of	ADP
ap-8536	2	44	informatics	informatic	NOUN
ap-8536	2	45	,	,	PUNCT
ap-8536	2	46	pázmány	pázmány	PROPN
ap-8536	2	47	p.	p.	PROPN
ap-8536	2	48	sétány	sétány	PROPN
ap-8536	3	1	1	1	NUM
ap-8536	3	2	/	/	SYM
ap-8536	3	3	c	c	PROPN
ap-8536	3	4	,	,	PUNCT
ap-8536	3	5	1117	1117	NUM
ap-8536	3	6	budapest	budapest	NOUN
ap-8536	3	7	,	,	PUNCT
ap-8536	3	8	hungary	hungary	PROPN
ap-8536	3	9	b	b	PROPN
ap-8536	3	10	czech	czech	PROPN
ap-8536	3	11	technical	technical	PROPN
ap-8536	3	12	university	university	PROPN
ap-8536	3	13	in	in	ADP
ap-8536	3	14	prague	prague	PROPN
ap-8536	3	15	,	,	PUNCT
ap-8536	3	16	faculty	faculty	NOUN
ap-8536	3	17	of	of	ADP
ap-8536	3	18	nuclear	nuclear	ADJ
ap-8536	3	19	sciences	science	NOUN
ap-8536	3	20	and	and	CCONJ
ap-8536	3	21	physical	physical	ADJ
ap-8536	3	22	engineering	engineering	NOUN
ap-8536	3	23	,	,	PUNCT
ap-8536	3	24	department	department	NOUN
ap-8536	3	25	of	of	ADP
ap-8536	3	26	mathematics	mathematic	NOUN
ap-8536	3	27	,	,	PUNCT
ap-8536	3	28	trojanova	trojanova	X
ap-8536	3	29	13	13	NUM
ap-8536	3	30	,	,	PUNCT
ap-8536	3	31	120	120	NUM
ap-8536	3	32	00	00	NUM
ap-8536	3	33	prague	prague	PROPN
ap-8536	3	34	,	,	PUNCT
ap-8536	3	35	czech	czech	PROPN
ap-8536	3	36	republic	republic	NOUN
ap-8536	3	37	∗	∗	NOUN
ap-8536	3	38	corresponding	correspond	VERB
ap-8536	3	39	author	author	NOUN
ap-8536	3	40	:	:	PUNCT
ap-8536	3	41	edita.pelantova@fjfi.cvut.cz	edita.pelantova@fjfi.cvut.cz	NOUN
ap-8536	3	42	abstract	abstract	NOUN
ap-8536	3	43	.	.	PUNCT
ap-8536	4	1	to	to	PART
ap-8536	4	2	represent	represent	VERB
ap-8536	4	3	real	real	ADJ
ap-8536	4	4	m	m	ADJ
ap-8536	4	5	-	-	ADJ
ap-8536	4	6	dimensional	dimensional	ADJ
ap-8536	4	7	vectors	vector	NOUN
ap-8536	4	8	,	,	PUNCT
ap-8536	4	9	a	a	DET
ap-8536	4	10	positional	positional	ADJ
ap-8536	4	11	vector	vector	NOUN
ap-8536	4	12	system	system	NOUN
ap-8536	4	13	given	give	VERB
ap-8536	4	14	by	by	ADP
ap-8536	4	15	a	a	DET
ap-8536	4	16	non	non	ADJ
ap-8536	4	17	-	-	ADJ
ap-8536	4	18	singular	singular	ADJ
ap-8536	4	19	matrix	matrix	NOUN
ap-8536	4	20	m	m	NOUN
ap-8536	4	21	∈	∈	PROPN
ap-8536	4	22	zm×m	zm×m	NOUN
ap-8536	4	23	and	and	CCONJ
ap-8536	4	24	a	a	DET
ap-8536	4	25	digit	digit	NOUN
ap-8536	4	26	set	set	VERB
ap-8536	4	27	d	d	PROPN
ap-8536	4	28	⊂	⊂	PROPN
ap-8536	4	29	zm	zm	PROPN
ap-8536	4	30	is	be	AUX
ap-8536	4	31	used	use	VERB
ap-8536	4	32	.	.	PUNCT
ap-8536	5	1	if	if	SCONJ
ap-8536	5	2	m	m	VERB
ap-8536	5	3	=	=	NOUN
ap-8536	5	4	1	1	NUM
ap-8536	5	5	,	,	PUNCT
ap-8536	5	6	the	the	DET
ap-8536	5	7	system	system	NOUN
ap-8536	5	8	coincides	coincide	VERB
ap-8536	5	9	with	with	ADP
ap-8536	5	10	the	the	DET
ap-8536	5	11	well	well	ADV
ap-8536	5	12	known	know	VERB
ap-8536	5	13	numeration	numeration	NOUN
ap-8536	5	14	system	system	NOUN
ap-8536	5	15	used	use	VERB
ap-8536	5	16	to	to	PART
ap-8536	5	17	represent	represent	VERB
ap-8536	5	18	real	real	ADJ
ap-8536	5	19	numbers	number	NOUN
ap-8536	5	20	.	.	PUNCT
ap-8536	6	1	we	we	PRON
ap-8536	6	2	study	study	VERB
ap-8536	6	3	some	some	DET
ap-8536	6	4	properties	property	NOUN
ap-8536	6	5	of	of	ADP
ap-8536	6	6	the	the	DET
ap-8536	6	7	vector	vector	NOUN
ap-8536	6	8	systems	system	NOUN
ap-8536	6	9	which	which	PRON
ap-8536	6	10	are	be	AUX
ap-8536	6	11	transformable	transformable	ADJ
ap-8536	6	12	from	from	ADP
ap-8536	6	13	the	the	DET
ap-8536	6	14	case	case	NOUN
ap-8536	6	15	m	m	NOUN
ap-8536	6	16	=	=	NOUN
ap-8536	6	17	1	1	NUM
ap-8536	6	18	to	to	ADP
ap-8536	6	19	higher	high	ADJ
ap-8536	6	20	dimensions	dimension	NOUN
ap-8536	6	21	.	.	PUNCT
ap-8536	7	1	we	we	PRON
ap-8536	7	2	focus	focus	VERB
ap-8536	7	3	on	on	ADP
ap-8536	7	4	an	an	DET
ap-8536	7	5	algorithm	algorithm	NOUN
ap-8536	7	6	for	for	ADP
ap-8536	7	7	parallel	parallel	ADJ
ap-8536	7	8	addition	addition	NOUN
ap-8536	7	9	and	and	CCONJ
ap-8536	7	10	on	on	ADP
ap-8536	7	11	systems	system	NOUN
ap-8536	7	12	allowing	allow	VERB
ap-8536	7	13	an	an	DET
ap-8536	7	14	eventually	eventually	ADV
ap-8536	7	15	periodic	periodic	ADJ
ap-8536	7	16	representation	representation	NOUN
ap-8536	7	17	of	of	ADP
ap-8536	7	18	vectors	vector	NOUN
ap-8536	7	19	with	with	ADP
ap-8536	7	20	rational	rational	ADJ
ap-8536	7	21	coordinates	coordinate	NOUN
ap-8536	7	22	.	.	PUNCT
ap-8536	8	1	keywords	keyword	NOUN
ap-8536	8	2	:	:	PUNCT
ap-8536	8	3	number	number	NOUN
ap-8536	8	4	system	system	NOUN
ap-8536	8	5	,	,	PUNCT
ap-8536	8	6	positional	positional	ADJ
ap-8536	8	7	representation	representation	NOUN
ap-8536	8	8	,	,	PUNCT
ap-8536	8	9	local	local	ADJ
ap-8536	8	10	function	function	NOUN
ap-8536	8	11	,	,	PUNCT
ap-8536	8	12	parallel	parallel	ADJ
ap-8536	8	13	addition	addition	NOUN
ap-8536	8	14	,	,	PUNCT
ap-8536	8	15	eventually	eventually	ADV
ap-8536	8	16	periodic	periodic	ADJ
ap-8536	8	17	representation	representation	NOUN
ap-8536	8	18	.	.	PUNCT
ap-8536	9	1	1	1	X
ap-8536	9	2	.	.	X
ap-8536	9	3	introduction	introduction	NOUN
ap-8536	9	4	expression	expression	NOUN
ap-8536	9	5	of	of	ADP
ap-8536	9	6	a	a	DET
ap-8536	9	7	number	number	NOUN
ap-8536	9	8	as	as	ADP
ap-8536	9	9	a	a	DET
ap-8536	9	10	linear	linear	ADJ
ap-8536	9	11	combination	combination	NOUN
ap-8536	9	12	of	of	ADP
ap-8536	9	13	elements	element	NOUN
ap-8536	9	14	of	of	ADP
ap-8536	9	15	the	the	DET
ap-8536	9	16	sequence	sequence	NOUN
ap-8536	9	17	(	(	PUNCT
ap-8536	9	18	βj)j∈z	βj)j∈z	NUM
ap-8536	9	19	with	with	ADP
ap-8536	9	20	coefficients	coefficient	NOUN
ap-8536	9	21	from	from	ADP
ap-8536	9	22	a	a	DET
ap-8536	9	23	finite	finite	NOUN
ap-8536	9	24	set	set	NOUN
ap-8536	9	25	d	d	X
ap-8536	9	26	is	be	AUX
ap-8536	9	27	nowadays	nowadays	ADV
ap-8536	9	28	the	the	DET
ap-8536	9	29	most	most	ADV
ap-8536	9	30	used	used	ADJ
ap-8536	9	31	way	way	NOUN
ap-8536	9	32	to	to	PART
ap-8536	9	33	represent	represent	VERB
ap-8536	9	34	numbers	number	NOUN
ap-8536	9	35	.	.	PUNCT
ap-8536	10	1	such	such	DET
ap-8536	10	2	a	a	DET
ap-8536	10	3	system	system	NOUN
ap-8536	10	4	is	be	AUX
ap-8536	10	5	called	call	VERB
ap-8536	10	6	a	a	DET
ap-8536	10	7	positional	positional	ADJ
ap-8536	10	8	number	number	NOUN
ap-8536	10	9	system	system	NOUN
ap-8536	10	10	with	with	ADP
ap-8536	10	11	the	the	DET
ap-8536	10	12	base	base	NOUN
ap-8536	10	13	β	β	X
ap-8536	10	14	and	and	CCONJ
ap-8536	10	15	the	the	DET
ap-8536	10	16	digit	digit	NOUN
ap-8536	10	17	set	set	VERB
ap-8536	10	18	d.	d.	PROPN
ap-8536	10	19	the	the	DET
ap-8536	10	20	decimal	decimal	ADJ
ap-8536	10	21	number	number	NOUN
ap-8536	10	22	system	system	NOUN
ap-8536	10	23	with	with	ADP
ap-8536	10	24	the	the	DET
ap-8536	10	25	base	base	NOUN
ap-8536	10	26	ten	ten	NUM
ap-8536	10	27	and	and	CCONJ
ap-8536	10	28	digits	digit	NOUN
ap-8536	10	29	0	0	NUM
ap-8536	10	30	,	,	PUNCT
ap-8536	10	31	1	1	NUM
ap-8536	10	32	,	,	PUNCT
ap-8536	10	33	.	.	PUNCT
ap-8536	10	34	.	.	PUNCT
ap-8536	11	1	.	.	PUNCT
ap-8536	12	1	,	,	PUNCT
ap-8536	12	2	9	9	NUM
ap-8536	12	3	prevails	prevail	NOUN
ap-8536	12	4	in	in	ADP
ap-8536	12	5	europe	europe	PROPN
ap-8536	12	6	for	for	ADP
ap-8536	12	7	several	several	ADJ
ap-8536	12	8	centuries	century	NOUN
ap-8536	12	9	.	.	PUNCT
ap-8536	13	1	in	in	ADP
ap-8536	13	2	the	the	DET
ap-8536	13	3	age	age	NOUN
ap-8536	13	4	of	of	ADP
ap-8536	13	5	computers	computer	NOUN
ap-8536	13	6	,	,	PUNCT
ap-8536	13	7	the	the	DET
ap-8536	13	8	binary	binary	ADJ
ap-8536	13	9	and	and	CCONJ
ap-8536	13	10	hexadecimal	hexadecimal	ADJ
ap-8536	13	11	number	number	NOUN
ap-8536	13	12	systems	system	NOUN
ap-8536	13	13	broke	break	VERB
ap-8536	13	14	the	the	DET
ap-8536	13	15	domination	domination	NOUN
ap-8536	13	16	of	of	ADP
ap-8536	13	17	the	the	DET
ap-8536	13	18	decimal	decimal	ADJ
ap-8536	13	19	number	number	NOUN
ap-8536	13	20	system	system	NOUN
ap-8536	13	21	.	.	PUNCT
ap-8536	14	1	but	but	CCONJ
ap-8536	14	2	advantages	advantage	NOUN
ap-8536	14	3	of	of	ADP
ap-8536	14	4	working	work	VERB
ap-8536	14	5	with	with	ADP
ap-8536	14	6	another	another	DET
ap-8536	14	7	number	number	NOUN
ap-8536	14	8	system	system	NOUN
ap-8536	14	9	were	be	AUX
ap-8536	14	10	observed	observe	VERB
ap-8536	14	11	before	before	SCONJ
ap-8536	14	12	computers	computer	NOUN
ap-8536	14	13	came	come	VERB
ap-8536	14	14	on	on	ADP
ap-8536	14	15	the	the	DET
ap-8536	14	16	scene	scene	NOUN
ap-8536	14	17	:	:	PUNCT
ap-8536	15	1	a.	a.	NOUN
ap-8536	15	2	cauchy	cauchy	PROPN
ap-8536	16	1	[	[	X
ap-8536	16	2	1	1	X
ap-8536	16	3	]	]	PUNCT
ap-8536	16	4	checked	check	VERB
ap-8536	16	5	correctness	correctness	NOUN
ap-8536	16	6	of	of	ADP
ap-8536	16	7	his	his	PRON
ap-8536	16	8	computations	computation	NOUN
ap-8536	16	9	using	use	VERB
ap-8536	16	10	simultaneously	simultaneously	ADV
ap-8536	16	11	the	the	DET
ap-8536	16	12	classical	classical	ADJ
ap-8536	16	13	decimal	decimal	ADJ
ap-8536	16	14	number	number	NOUN
ap-8536	16	15	system	system	NOUN
ap-8536	16	16	and	and	CCONJ
ap-8536	16	17	decimal	decimal	ADJ
ap-8536	16	18	number	number	NOUN
ap-8536	16	19	system	system	NOUN
ap-8536	16	20	with	with	ADP
ap-8536	16	21	the	the	DET
ap-8536	16	22	symmetric	symmetric	ADJ
ap-8536	16	23	set	set	NOUN
ap-8536	16	24	of	of	ADP
ap-8536	16	25	digits	digit	NOUN
ap-8536	16	26	{	{	PUNCT
ap-8536	16	27	−5	−5	NOUN
ap-8536	16	28	,	,	PUNCT
ap-8536	16	29	−4	−4	X
ap-8536	16	30	,	,	PUNCT
ap-8536	16	31	.	.	PUNCT
ap-8536	16	32	.	.	PUNCT
ap-8536	17	1	.	.	PUNCT
ap-8536	18	1	,	,	PUNCT
ap-8536	18	2	0	0	NUM
ap-8536	18	3	,	,	PUNCT
ap-8536	18	4	1	1	NUM
ap-8536	18	5	,	,	PUNCT
ap-8536	18	6	.	.	PUNCT
ap-8536	18	7	.	.	PUNCT
ap-8536	19	1	.	.	PUNCT
ap-8536	20	1	,	,	PUNCT
ap-8536	20	2	5	5	NUM
ap-8536	20	3	}	}	PUNCT
ap-8536	20	4	.	.	PUNCT
ap-8536	21	1	v.	v.	ADP
ap-8536	21	2	grünwald	grünwald	PROPN
ap-8536	22	1	[	[	X
ap-8536	22	2	2	2	NUM
ap-8536	22	3	]	]	PUNCT
ap-8536	22	4	considered	consider	VERB
ap-8536	22	5	a	a	DET
ap-8536	22	6	number	number	NOUN
ap-8536	22	7	system	system	NOUN
ap-8536	22	8	with	with	ADP
ap-8536	22	9	base	base	NOUN
ap-8536	22	10	β	β	NOUN
ap-8536	22	11	=	=	PUNCT
ap-8536	22	12	−2	−2	NOUN
ap-8536	22	13	and	and	CCONJ
ap-8536	22	14	digits	digit	NOUN
ap-8536	22	15	{	{	PUNCT
ap-8536	22	16	0	0	NUM
ap-8536	22	17	,	,	PUNCT
ap-8536	22	18	1	1	NUM
ap-8536	22	19	}	}	PUNCT
ap-8536	22	20	.	.	PUNCT
ap-8536	23	1	an	an	DET
ap-8536	23	2	important	important	ADJ
ap-8536	23	3	moment	moment	NOUN
ap-8536	23	4	for	for	ADP
ap-8536	23	5	the	the	DET
ap-8536	23	6	number	number	NOUN
ap-8536	23	7	systems	system	NOUN
ap-8536	23	8	,	,	PUNCT
ap-8536	23	9	making	make	VERB
ap-8536	23	10	them	they	PRON
ap-8536	23	11	a	a	DET
ap-8536	23	12	source	source	NOUN
ap-8536	23	13	of	of	ADP
ap-8536	23	14	interest	interest	NOUN
ap-8536	23	15	for	for	ADP
ap-8536	23	16	many	many	ADJ
ap-8536	23	17	areas	area	NOUN
ap-8536	23	18	of	of	ADP
ap-8536	23	19	mathematics	mathematic	NOUN
ap-8536	23	20	,	,	PUNCT
ap-8536	23	21	came	come	VERB
ap-8536	23	22	in	in	ADP
ap-8536	23	23	1957	1957	NUM
ap-8536	23	24	,	,	PUNCT
ap-8536	23	25	when	when	SCONJ
ap-8536	23	26	a.	a.	NOUN
ap-8536	23	27	rényi	rényi	PROPN
ap-8536	24	1	[	[	X
ap-8536	24	2	3	3	NUM
ap-8536	24	3	]	]	PUNCT
ap-8536	24	4	introduced	introduce	VERB
ap-8536	24	5	number	number	NOUN
ap-8536	24	6	systems	system	NOUN
ap-8536	24	7	with	with	ADP
ap-8536	24	8	an	an	DET
ap-8536	24	9	arbitrary	arbitrary	ADJ
ap-8536	24	10	real	real	ADJ
ap-8536	24	11	base	base	NOUN
ap-8536	24	12	β	β	X
ap-8536	24	13	>	>	X
ap-8536	24	14	1	1	X
ap-8536	24	15	.	.	PUNCT
ap-8536	25	1	algebraic	algebraic	ADJ
ap-8536	25	2	,	,	PUNCT
ap-8536	25	3	dynamical	dynamical	ADJ
ap-8536	25	4	,	,	PUNCT
ap-8536	25	5	topological	topological	ADJ
ap-8536	25	6	,	,	PUNCT
ap-8536	25	7	geometric	geometric	ADJ
ap-8536	25	8	and	and	CCONJ
ap-8536	25	9	algorithmic	algorithmic	ADJ
ap-8536	25	10	properties	property	NOUN
ap-8536	25	11	of	of	ADP
ap-8536	25	12	the	the	DET
ap-8536	25	13	rényi	rényi	NOUN
ap-8536	25	14	number	number	NOUN
ap-8536	25	15	systems	system	NOUN
ap-8536	25	16	have	have	AUX
ap-8536	25	17	been	be	AUX
ap-8536	25	18	very	very	ADV
ap-8536	25	19	intensively	intensively	ADV
ap-8536	25	20	studied	study	VERB
ap-8536	25	21	since	since	SCONJ
ap-8536	25	22	then	then	ADV
ap-8536	25	23	,	,	PUNCT
ap-8536	25	24	from	from	ADP
ap-8536	25	25	both	both	CCONJ
ap-8536	25	26	theoretical	theoretical	ADJ
ap-8536	25	27	and	and	CCONJ
ap-8536	25	28	practical	practical	ADJ
ap-8536	25	29	points	point	NOUN
ap-8536	25	30	of	of	ADP
ap-8536	25	31	view	view	NOUN
ap-8536	25	32	.	.	PUNCT
ap-8536	26	1	for	for	ADP
ap-8536	26	2	example	example	NOUN
ap-8536	26	3	,	,	PUNCT
ap-8536	26	4	a	a	DET
ap-8536	26	5	suitable	suitable	ADJ
ap-8536	26	6	choice	choice	NOUN
ap-8536	26	7	of	of	ADP
ap-8536	26	8	a	a	DET
ap-8536	26	9	base	base	NOUN
ap-8536	26	10	in	in	ADP
ap-8536	26	11	an	an	DET
ap-8536	26	12	algebraic	algebraic	ADJ
ap-8536	26	13	extension	extension	NOUN
ap-8536	26	14	of	of	ADP
ap-8536	26	15	rational	rational	ADJ
ap-8536	26	16	numbers	number	NOUN
ap-8536	26	17	enables	enable	VERB
ap-8536	26	18	to	to	PART
ap-8536	26	19	represent	represent	VERB
ap-8536	26	20	all	all	DET
ap-8536	26	21	elements	element	NOUN
ap-8536	26	22	of	of	ADP
ap-8536	26	23	the	the	DET
ap-8536	26	24	algebraic	algebraic	ADJ
ap-8536	26	25	field	field	NOUN
ap-8536	26	26	by	by	ADP
ap-8536	26	27	a	a	DET
ap-8536	26	28	finite	finite	NOUN
ap-8536	26	29	or	or	CCONJ
ap-8536	26	30	eventually	eventually	ADV
ap-8536	26	31	periodic	periodic	ADJ
ap-8536	26	32	string	string	NOUN
ap-8536	26	33	of	of	ADP
ap-8536	26	34	digits	digit	NOUN
ap-8536	26	35	,	,	PUNCT
ap-8536	26	36	see	see	VERB
ap-8536	26	37	[	[	X
ap-8536	26	38	4	4	X
ap-8536	26	39	]	]	PUNCT
ap-8536	26	40	and	and	CCONJ
ap-8536	26	41	[	[	X
ap-8536	26	42	5	5	NUM
ap-8536	26	43	]	]	PUNCT
ap-8536	26	44	.	.	PUNCT
ap-8536	27	1	further	further	ADJ
ap-8536	27	2	generalisations	generalisation	NOUN
ap-8536	27	3	of	of	ADP
ap-8536	27	4	numeration	numeration	NOUN
ap-8536	27	5	systems	system	NOUN
ap-8536	27	6	emerged	emerge	VERB
ap-8536	27	7	in	in	ADP
ap-8536	27	8	the	the	DET
ap-8536	27	9	following	following	ADJ
ap-8536	27	10	years	year	NOUN
ap-8536	27	11	.	.	PUNCT
ap-8536	28	1	knuth	knuth	PROPN
ap-8536	29	1	[	[	X
ap-8536	29	2	6	6	NUM
ap-8536	29	3	]	]	PUNCT
ap-8536	29	4	and	and	CCONJ
ap-8536	29	5	penney	penney	NOUN
ap-8536	29	6	[	[	X
ap-8536	29	7	7	7	X
ap-8536	29	8	]	]	PUNCT
ap-8536	29	9	came	come	VERB
ap-8536	29	10	with	with	ADP
ap-8536	29	11	positional	positional	ADJ
ap-8536	29	12	representations	representation	NOUN
ap-8536	29	13	of	of	ADP
ap-8536	29	14	complex	complex	ADJ
ap-8536	29	15	numbers	number	NOUN
ap-8536	29	16	,	,	PUNCT
ap-8536	29	17	wherein	wherein	ADJ
ap-8536	29	18	,	,	PUNCT
ap-8536	29	19	instead	instead	ADV
ap-8536	29	20	of	of	ADP
ap-8536	29	21	two	two	NUM
ap-8536	29	22	strings	string	NOUN
ap-8536	29	23	of	of	ADP
ap-8536	29	24	digits	digit	NOUN
ap-8536	29	25	representing	represent	VERB
ap-8536	29	26	the	the	DET
ap-8536	29	27	real	real	ADJ
ap-8536	29	28	and	and	CCONJ
ap-8536	29	29	the	the	DET
ap-8536	29	30	imaginary	imaginary	ADJ
ap-8536	29	31	part	part	NOUN
ap-8536	29	32	of	of	ADP
ap-8536	29	33	complex	complex	ADJ
ap-8536	29	34	numbers	number	NOUN
ap-8536	29	35	separately	separately	ADV
ap-8536	29	36	,	,	PUNCT
ap-8536	29	37	they	they	PRON
ap-8536	29	38	suggested	suggest	VERB
ap-8536	29	39	to	to	PART
ap-8536	29	40	use	use	VERB
ap-8536	29	41	a	a	DET
ap-8536	29	42	complex	complex	ADJ
ap-8536	29	43	base	base	NOUN
ap-8536	29	44	β	β	NOUN
ap-8536	29	45	,	,	PUNCT
ap-8536	29	46	in	in	ADP
ap-8536	29	47	order	order	NOUN
ap-8536	29	48	to	to	PART
ap-8536	29	49	represent	represent	VERB
ap-8536	29	50	the	the	DET
ap-8536	29	51	complex	complex	ADJ
ap-8536	29	52	number	number	NOUN
ap-8536	29	53	by	by	ADP
ap-8536	29	54	a	a	DET
ap-8536	29	55	single	single	ADJ
ap-8536	29	56	string	string	NOUN
ap-8536	29	57	of	of	ADP
ap-8536	29	58	digits	digit	NOUN
ap-8536	29	59	.	.	PUNCT
ap-8536	30	1	this	this	DET
ap-8536	30	2	type	type	NOUN
ap-8536	30	3	of	of	ADP
ap-8536	30	4	representation	representation	NOUN
ap-8536	30	5	was	be	AUX
ap-8536	30	6	then	then	ADV
ap-8536	30	7	further	far	ADV
ap-8536	30	8	developed	develop	VERB
ap-8536	30	9	in	in	ADP
ap-8536	30	10	the	the	DET
ap-8536	30	11	concept	concept	NOUN
ap-8536	30	12	of	of	ADP
ap-8536	30	13	canonical	canonical	ADJ
ap-8536	30	14	number	number	NOUN
ap-8536	30	15	systems	system	NOUN
ap-8536	30	16	[	[	X
ap-8536	30	17	8	8	NUM
ap-8536	30	18	]	]	PUNCT
ap-8536	30	19	and	and	CCONJ
ap-8536	30	20	(	(	PUNCT
ap-8536	30	21	even	even	ADV
ap-8536	30	22	more	more	ADV
ap-8536	30	23	general	general	ADJ
ap-8536	30	24	)	)	PUNCT
ap-8536	30	25	shift	shift	NOUN
ap-8536	30	26	radix	radix	PROPN
ap-8536	30	27	systems	system	NOUN
ap-8536	30	28	,	,	PUNCT
ap-8536	30	29	see	see	VERB
ap-8536	30	30	[	[	X
ap-8536	30	31	9	9	X
ap-8536	30	32	]	]	PUNCT
ap-8536	30	33	for	for	ADP
ap-8536	30	34	a	a	DET
ap-8536	30	35	survey	survey	NOUN
ap-8536	30	36	on	on	ADP
ap-8536	30	37	the	the	DET
ap-8536	30	38	topic	topic	NOUN
ap-8536	30	39	.	.	PUNCT
ap-8536	31	1	in	in	ADP
ap-8536	31	2	most	most	ADJ
ap-8536	31	3	of	of	ADP
ap-8536	31	4	the	the	DET
ap-8536	31	5	above	above	ADJ
ap-8536	31	6	mentioned	mention	VERB
ap-8536	31	7	generalisations	generalisation	NOUN
ap-8536	31	8	of	of	ADP
ap-8536	31	9	numeration	numeration	NOUN
ap-8536	31	10	systems	system	NOUN
ap-8536	31	11	,	,	PUNCT
ap-8536	31	12	every	every	DET
ap-8536	31	13	number	number	NOUN
ap-8536	31	14	has	have	VERB
ap-8536	31	15	a	a	DET
ap-8536	31	16	unique	unique	ADJ
ap-8536	31	17	representation	representation	NOUN
ap-8536	31	18	,	,	PUNCT
ap-8536	31	19	whose	whose	DET
ap-8536	31	20	digits	digit	NOUN
ap-8536	31	21	are	be	AUX
ap-8536	31	22	determined	determine	VERB
ap-8536	31	23	by	by	ADP
ap-8536	31	24	iterations	iteration	NOUN
ap-8536	31	25	of	of	ADP
ap-8536	31	26	some	some	DET
ap-8536	31	27	transformation	transformation	NOUN
ap-8536	31	28	function	function	NOUN
ap-8536	31	29	.	.	PUNCT
ap-8536	32	1	one	one	NUM
ap-8536	32	2	exception	exception	NOUN
ap-8536	32	3	is	be	AUX
ap-8536	32	4	the	the	DET
ap-8536	32	5	decimal	decimal	ADJ
ap-8536	32	6	system	system	NOUN
ap-8536	32	7	with	with	ADP
ap-8536	32	8	a	a	DET
ap-8536	32	9	symmetric	symmetric	ADJ
ap-8536	32	10	digit	digit	NOUN
ap-8536	32	11	set	set	VERB
ap-8536	32	12	used	use	VERB
ap-8536	32	13	by	by	ADP
ap-8536	32	14	cauchy	cauchy	PROPN
ap-8536	32	15	.	.	PUNCT
ap-8536	33	1	it	it	PRON
ap-8536	33	2	has	have	VERB
ap-8536	33	3	eleven	eleven	NUM
ap-8536	33	4	digits	digit	NOUN
ap-8536	33	5	–	–	PUNCT
ap-8536	33	6	more	more	ADJ
ap-8536	33	7	than	than	ADP
ap-8536	33	8	necessary	necessary	ADJ
ap-8536	33	9	for	for	ADP
ap-8536	33	10	representing	represent	VERB
ap-8536	33	11	all	all	DET
ap-8536	33	12	positive	positive	ADJ
ap-8536	33	13	integers	integer	NOUN
ap-8536	33	14	.	.	PUNCT
ap-8536	34	1	such	such	DET
ap-8536	34	2	a	a	DET
ap-8536	34	3	system	system	NOUN
ap-8536	34	4	is	be	AUX
ap-8536	34	5	called	call	VERB
ap-8536	34	6	redundant	redundant	ADJ
ap-8536	34	7	.	.	PUNCT
ap-8536	35	1	already	already	ADV
ap-8536	35	2	cauchy	cauchy	PROPN
ap-8536	35	3	noticed	notice	VERB
ap-8536	35	4	that	that	SCONJ
ap-8536	35	5	,	,	PUNCT
ap-8536	35	6	with	with	ADP
ap-8536	35	7	this	this	DET
ap-8536	35	8	redundant	redundant	ADJ
ap-8536	35	9	system	system	NOUN
ap-8536	35	10	,	,	PUNCT
ap-8536	35	11	the	the	DET
ap-8536	35	12	addition	addition	NOUN
ap-8536	35	13	of	of	ADP
ap-8536	35	14	two	two	NUM
ap-8536	35	15	numbers	number	NOUN
ap-8536	35	16	is	be	AUX
ap-8536	35	17	easier	easy	ADJ
ap-8536	35	18	than	than	ADP
ap-8536	35	19	in	in	ADP
ap-8536	35	20	the	the	DET
ap-8536	35	21	classical	classical	ADJ
ap-8536	35	22	system	system	NOUN
ap-8536	35	23	,	,	PUNCT
ap-8536	35	24	as	as	SCONJ
ap-8536	35	25	the	the	DET
ap-8536	35	26	carry	carry	NOUN
ap-8536	35	27	propagation	propagation	NOUN
ap-8536	35	28	is	be	AUX
ap-8536	35	29	limited	limited	ADJ
ap-8536	35	30	.	.	PUNCT
ap-8536	36	1	this	this	DET
ap-8536	36	2	property	property	NOUN
ap-8536	36	3	was	be	AUX
ap-8536	36	4	further	far	ADV
ap-8536	36	5	exploited	exploit	VERB
ap-8536	36	6	by	by	ADP
ap-8536	36	7	a.	a.	NOUN
ap-8536	36	8	avizienis	avizienis	PROPN
ap-8536	36	9	,	,	PUNCT
ap-8536	36	10	aiming	aim	VERB
ap-8536	36	11	to	to	PART
ap-8536	36	12	speed	speed	VERB
ap-8536	36	13	up	up	ADP
ap-8536	36	14	addition	addition	NOUN
ap-8536	36	15	.	.	PUNCT
ap-8536	37	1	in	in	ADP
ap-8536	37	2	the	the	DET
ap-8536	37	3	classical	classical	ADJ
ap-8536	37	4	b	b	NOUN
ap-8536	37	5	-	-	PUNCT
ap-8536	37	6	ary	ary	ADJ
ap-8536	37	7	numeration	numeration	NOUN
ap-8536	37	8	system	system	NOUN
ap-8536	37	9	,	,	PUNCT
ap-8536	37	10	where	where	SCONJ
ap-8536	37	11	the	the	DET
ap-8536	37	12	base	base	NOUN
ap-8536	37	13	is	be	AUX
ap-8536	37	14	an	an	DET
ap-8536	37	15	integer	integer	NOUN
ap-8536	37	16	β	β	X
ap-8536	37	17	=	=	SYM
ap-8536	37	18	b	b	PROPN
ap-8536	37	19	≥	≥	NUM
ap-8536	37	20	2	2	NUM
ap-8536	37	21	,	,	PUNCT
ap-8536	37	22	addition	addition	NOUN
ap-8536	37	23	has	have	VERB
ap-8536	37	24	a	a	DET
ap-8536	37	25	linear	linear	ADJ
ap-8536	37	26	time	time	NOUN
ap-8536	37	27	complexity	complexity	NOUN
ap-8536	37	28	with	with	ADP
ap-8536	37	29	respect	respect	NOUN
ap-8536	37	30	to	to	ADP
ap-8536	37	31	the	the	DET
ap-8536	37	32	length	length	NOUN
ap-8536	37	33	of	of	ADP
ap-8536	37	34	representations	representation	NOUN
ap-8536	37	35	of	of	ADP
ap-8536	37	36	the	the	DET
ap-8536	37	37	summands	summand	NOUN
ap-8536	37	38	.	.	PUNCT
ap-8536	38	1	avizienis	avizienis	PROPN
ap-8536	38	2	[	[	X
ap-8536	38	3	10	10	NUM
ap-8536	38	4	]	]	PUNCT
ap-8536	38	5	designed	design	VERB
ap-8536	38	6	an	an	DET
ap-8536	38	7	algorithm	algorithm	NOUN
ap-8536	38	8	with	with	ADP
ap-8536	38	9	a	a	DET
ap-8536	38	10	constant	constant	ADJ
ap-8536	38	11	time	time	NOUN
ap-8536	38	12	complexity	complexity	NOUN
ap-8536	38	13	for	for	ADP
ap-8536	38	14	addition	addition	NOUN
ap-8536	38	15	in	in	ADP
ap-8536	38	16	redundant	redundant	ADJ
ap-8536	38	17	number	number	NOUN
ap-8536	38	18	systems	system	NOUN
ap-8536	38	19	using	use	VERB
ap-8536	38	20	base	base	NOUN
ap-8536	38	21	β	β	NOUN
ap-8536	38	22	=	=	SYM
ap-8536	38	23	b	b	PROPN
ap-8536	38	24	≥	≥	NUM
ap-8536	38	25	3	3	NUM
ap-8536	38	26	and	and	CCONJ
ap-8536	38	27	a	a	DET
ap-8536	38	28	symmetric	symmetric	ADJ
ap-8536	38	29	integer	integer	NOUN
ap-8536	38	30	digit	digit	NOUN
ap-8536	38	31	set	set	NOUN
ap-8536	38	32	.	.	PUNCT
ap-8536	39	1	in	in	ADP
ap-8536	39	2	this	this	DET
ap-8536	39	3	paper	paper	NOUN
ap-8536	39	4	,	,	PUNCT
ap-8536	39	5	we	we	PRON
ap-8536	39	6	consider	consider	VERB
ap-8536	39	7	representations	representation	NOUN
ap-8536	39	8	of	of	ADP
ap-8536	39	9	mdimensional	mdimensional	ADJ
ap-8536	39	10	vectors	vector	NOUN
ap-8536	39	11	.	.	PUNCT
ap-8536	40	1	the	the	DET
ap-8536	40	2	numeration	numeration	NOUN
ap-8536	40	3	system	system	NOUN
ap-8536	40	4	is	be	AUX
ap-8536	40	5	given	give	VERB
ap-8536	40	6	by	by	ADP
ap-8536	40	7	a	a	DET
ap-8536	40	8	square	square	ADJ
ap-8536	40	9	matrix	matrix	NOUN
ap-8536	40	10	base	base	NOUN
ap-8536	40	11	m	m	NOUN
ap-8536	40	12	∈	∈	PROPN
ap-8536	40	13	zm×m	zm×m	NOUN
ap-8536	40	14	and	and	CCONJ
ap-8536	40	15	by	by	ADP
ap-8536	40	16	a	a	DET
ap-8536	40	17	finite	finite	ADJ
ap-8536	40	18	set	set	NOUN
ap-8536	40	19	of	of	ADP
ap-8536	40	20	digits	digit	NOUN
ap-8536	40	21	d	d	X
ap-8536	40	22	⊂	⊂	PROPN
ap-8536	40	23	zm	zm	PROPN
ap-8536	40	24	.	.	PUNCT
ap-8536	41	1	an	an	DET
ap-8536	41	2	origin	origin	NOUN
ap-8536	41	3	of	of	ADP
ap-8536	41	4	such	such	ADJ
ap-8536	41	5	numeration	numeration	NOUN
ap-8536	41	6	systems	system	NOUN
ap-8536	41	7	can	can	AUX
ap-8536	41	8	be	be	AUX
ap-8536	41	9	found	find	VERB
ap-8536	41	10	in	in	ADP
ap-8536	41	11	works	work	NOUN
ap-8536	41	12	[	[	X
ap-8536	41	13	11	11	NUM
ap-8536	41	14	]	]	PUNCT
ap-8536	41	15	and	and	CCONJ
ap-8536	41	16	[	[	X
ap-8536	41	17	12	12	NUM
ap-8536	41	18	]	]	PUNCT
ap-8536	41	19	of	of	ADP
ap-8536	41	20	a.	a.	NOUN
ap-8536	41	21	vince	vince	PROPN
ap-8536	41	22	,	,	PUNCT
ap-8536	41	23	showing	show	VERB
ap-8536	41	24	that	that	SCONJ
ap-8536	41	25	for	for	ADP
ap-8536	41	26	any	any	DET
ap-8536	41	27	expansive	expansive	ADJ
ap-8536	41	28	matrix	matrix	NOUN
ap-8536	41	29	m	m	NOUN
ap-8536	41	30	∈	∈	NOUN
ap-8536	41	31	zm×m	zm×m	NOUN
ap-8536	41	32	,	,	PUNCT
ap-8536	41	33	there	there	PRON
ap-8536	41	34	exists	exist	VERB
ap-8536	41	35	a	a	DET
ap-8536	41	36	digit	digit	NOUN
ap-8536	41	37	set	set	VERB
ap-8536	41	38	d	d	PROPN
ap-8536	41	39	⊂	⊂	PROPN
ap-8536	41	40	zm	zm	PROPN
ap-8536	41	41	such	such	ADJ
ap-8536	41	42	that	that	SCONJ
ap-8536	41	43	any	any	DET
ap-8536	41	44	element	element	NOUN
ap-8536	41	45	x	x	PUNCT
ap-8536	41	46	of	of	ADP
ap-8536	41	47	the	the	DET
ap-8536	41	48	lattice	lattice	PROPN
ap-8536	41	49	zm	zm	PROPN
ap-8536	41	50	can	can	AUX
ap-8536	41	51	be	be	AUX
ap-8536	41	52	written	write	VERB
ap-8536	41	53	in	in	ADP
ap-8536	41	54	the	the	DET
ap-8536	41	55	form	form	NOUN
ap-8536	41	56	x	x	PUNCT
ap-8536	42	1	=	=	SYM
ap-8536	42	2	∑n	∑n	PROPN
ap-8536	42	3	j=0	j=0	PROPN
ap-8536	42	4	m	m	PROPN
ap-8536	42	5	jdj	jdj	PROPN
ap-8536	42	6	,	,	PUNCT
ap-8536	42	7	where	where	SCONJ
ap-8536	42	8	dj	dj	ADP
ap-8536	42	9	∈	∈	PROPN
ap-8536	42	10	d.	d.	NOUN
ap-8536	42	11	in	in	ADP
ap-8536	42	12	other	other	ADJ
ap-8536	42	13	words	word	NOUN
ap-8536	42	14	,	,	PUNCT
ap-8536	42	15	the	the	DET
ap-8536	42	16	whole	whole	ADJ
ap-8536	42	17	lattice	lattice	PROPN
ap-8536	42	18	zm	zm	PROPN
ap-8536	42	19	is	be	AUX
ap-8536	42	20	representable	representable	ADJ
ap-8536	42	21	in	in	ADP
ap-8536	42	22	the	the	DET
ap-8536	42	23	matrix	matrix	NOUN
ap-8536	42	24	numeration	numeration	NOUN
ap-8536	42	25	system	system	NOUN
ap-8536	42	26	(	(	PUNCT
ap-8536	42	27	m	m	PROPN
ap-8536	42	28	,	,	PUNCT
ap-8536	42	29	d	d	NOUN
ap-8536	42	30	)	)	PUNCT
ap-8536	42	31	.	.	PUNCT
ap-8536	43	1	however	however	ADV
ap-8536	43	2	,	,	PUNCT
ap-8536	43	3	if	if	SCONJ
ap-8536	43	4	a	a	DET
ap-8536	43	5	matrix	matrix	NOUN
ap-8536	43	6	m	m	AUX
ap-8536	43	7	has	have	VERB
ap-8536	43	8	an	an	DET
ap-8536	43	9	eigenvalue	eigenvalue	NOUN
ap-8536	43	10	inside	inside	ADP
ap-8536	43	11	the	the	DET
ap-8536	43	12	unit	unit	NOUN
ap-8536	43	13	circle	circle	NOUN
ap-8536	43	14	,	,	PUNCT
ap-8536	43	15	then	then	ADV
ap-8536	43	16	no	no	DET
ap-8536	43	17	choice	choice	NOUN
ap-8536	43	18	of	of	ADP
ap-8536	43	19	the	the	DET
ap-8536	43	20	digit	digit	NOUN
ap-8536	43	21	set	set	VERB
ap-8536	43	22	d	d	PROPN
ap-8536	43	23	⊂	⊂	PROPN
ap-8536	43	24	zm	zm	PROPN
ap-8536	43	25	allows	allow	VERB
ap-8536	43	26	to	to	PART
ap-8536	43	27	represent	represent	VERB
ap-8536	43	28	all	all	DET
ap-8536	43	29	in188	in188	PROPN
ap-8536	43	30	https://doi.org/10.14311/ap.2023.63.0188	https://doi.org/10.14311/ap.2023.63.0188	DET
ap-8536	43	31	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
ap-8536	43	32	https://www.cvut.cz/en	https://www.cvut.cz/en	NOUN
ap-8536	43	33	vol	vol	NOUN
ap-8536	43	34	.	.	PROPN
ap-8536	44	1	63	63	NUM
ap-8536	45	1	no	no	NOUN
ap-8536	45	2	.	.	PUNCT
ap-8536	46	1	3/2023	3/2023	NUM
ap-8536	46	2	positional	positional	ADJ
ap-8536	46	3	representation	representation	NOUN
ap-8536	46	4	of	of	ADP
ap-8536	46	5	vectors	vector	NOUN
ap-8536	46	6	teger	teger	NOUN
ap-8536	46	7	vectors	vector	NOUN
ap-8536	46	8	as	as	ADP
ap-8536	46	9	a	a	DET
ap-8536	46	10	combination	combination	NOUN
ap-8536	46	11	of	of	ADP
ap-8536	46	12	only	only	ADJ
ap-8536	46	13	non	non	ADJ
ap-8536	46	14	-	-	ADJ
ap-8536	46	15	negative	negative	ADJ
ap-8536	46	16	powers	power	NOUN
ap-8536	46	17	of	of	ADP
ap-8536	46	18	m	m	PROPN
ap-8536	46	19	.	.	PUNCT
ap-8536	47	1	the	the	DET
ap-8536	47	2	matrix	matrix	NOUN
ap-8536	47	3	formalism	formalism	NOUN
ap-8536	47	4	for	for	ADP
ap-8536	47	5	numeration	numeration	NOUN
ap-8536	47	6	systems	system	NOUN
ap-8536	47	7	(	(	PUNCT
ap-8536	47	8	under	under	ADP
ap-8536	47	9	the	the	DET
ap-8536	47	10	name	name	NOUN
ap-8536	47	11	numeration	numeration	NOUN
ap-8536	47	12	systems	system	NOUN
ap-8536	47	13	in	in	ADP
ap-8536	47	14	lattices	lattice	NOUN
ap-8536	47	15	)	)	PUNCT
ap-8536	47	16	was	be	AUX
ap-8536	47	17	systematically	systematically	ADV
ap-8536	47	18	used	use	VERB
ap-8536	47	19	by	by	ADP
ap-8536	47	20	a.	a.	PROPN
ap-8536	47	21	kovács	kovács	PROPN
ap-8536	47	22	in	in	ADP
ap-8536	47	23	[	[	X
ap-8536	47	24	13	13	NUM
ap-8536	47	25	]	]	PUNCT
ap-8536	47	26	.	.	PUNCT
ap-8536	48	1	of	of	ADP
ap-8536	48	2	course	course	ADV
ap-8536	48	3	,	,	PUNCT
ap-8536	48	4	kovács	kovács	NOUN
ap-8536	48	5	,	,	PUNCT
ap-8536	48	6	just	just	ADV
ap-8536	48	7	like	like	ADP
ap-8536	48	8	vince	vince	NOUN
ap-8536	48	9	,	,	PUNCT
ap-8536	48	10	considers	consider	VERB
ap-8536	48	11	integer	integer	NOUN
ap-8536	48	12	matrices	matrix	NOUN
ap-8536	48	13	,	,	PUNCT
ap-8536	48	14	as	as	SCONJ
ap-8536	48	15	they	they	PRON
ap-8536	48	16	map	map	VERB
ap-8536	48	17	a	a	DET
ap-8536	48	18	lattice	lattice	NOUN
ap-8536	48	19	into	into	ADP
ap-8536	48	20	itself	itself	PRON
ap-8536	48	21	.	.	PUNCT
ap-8536	49	1	in	in	ADP
ap-8536	49	2	fact	fact	NOUN
ap-8536	49	3	,	,	PUNCT
ap-8536	49	4	already	already	ADV
ap-8536	49	5	positional	positional	ADJ
ap-8536	49	6	representations	representation	NOUN
ap-8536	49	7	of	of	ADP
ap-8536	49	8	gaussian	gaussian	ADJ
ap-8536	49	9	integers	integer	NOUN
ap-8536	49	10	or	or	CCONJ
ap-8536	49	11	algebraic	algebraic	ADJ
ap-8536	49	12	integers	integer	NOUN
ap-8536	49	13	from	from	ADP
ap-8536	49	14	an	an	DET
ap-8536	49	15	algebraic	algebraic	ADJ
ap-8536	49	16	extension	extension	NOUN
ap-8536	49	17	of	of	ADP
ap-8536	49	18	rational	rational	ADJ
ap-8536	49	19	numbers	number	NOUN
ap-8536	49	20	can	can	AUX
ap-8536	49	21	be	be	AUX
ap-8536	49	22	interpreted	interpret	VERB
ap-8536	49	23	as	as	ADP
ap-8536	49	24	special	special	ADJ
ap-8536	49	25	cases	case	NOUN
ap-8536	49	26	of	of	ADP
ap-8536	49	27	the	the	DET
ap-8536	49	28	matrix	matrix	NOUN
ap-8536	49	29	numeration	numeration	NOUN
ap-8536	49	30	systems	system	NOUN
ap-8536	49	31	.	.	PUNCT
ap-8536	50	1	j.	j.	PROPN
ap-8536	50	2	jankauskas	jankauskas	PROPN
ap-8536	50	3	and	and	CCONJ
ap-8536	50	4	j.	j.	PROPN
ap-8536	50	5	thuswaldner	thuswaldner	PROPN
ap-8536	51	1	[	[	X
ap-8536	51	2	14	14	NUM
ap-8536	51	3	]	]	PUNCT
ap-8536	51	4	generalised	generalise	VERB
ap-8536	51	5	the	the	DET
ap-8536	51	6	vince	vince	NOUN
ap-8536	51	7	’s	’s	PART
ap-8536	51	8	results	result	NOUN
ap-8536	51	9	to	to	PART
ap-8536	51	10	matrix	matrix	VERB
ap-8536	51	11	bases	basis	NOUN
ap-8536	51	12	m	m	NOUN
ap-8536	51	13	∈	∈	NOUN
ap-8536	51	14	qm×m	qm×m	NOUN
ap-8536	51	15	with	with	ADP
ap-8536	51	16	rational	rational	ADJ
ap-8536	51	17	entries	entry	NOUN
ap-8536	51	18	and	and	CCONJ
ap-8536	51	19	without	without	ADP
ap-8536	51	20	eigenvalues	eigenvalue	NOUN
ap-8536	51	21	in	in	ADP
ap-8536	51	22	modulus	modulus	NOUN
ap-8536	51	23	strictly	strictly	ADV
ap-8536	51	24	smaller	small	ADJ
ap-8536	51	25	than	than	ADP
ap-8536	51	26	1	1	NUM
ap-8536	51	27	.	.	PUNCT
ap-8536	52	1	another	another	DET
ap-8536	52	2	generalisation	generalisation	NOUN
ap-8536	52	3	introduced	introduce	VERB
ap-8536	52	4	recently	recently	ADV
ap-8536	52	5	allows	allow	VERB
ap-8536	52	6	to	to	PART
ap-8536	52	7	use	use	VERB
ap-8536	52	8	both	both	CCONJ
ap-8536	52	9	positive	positive	ADJ
ap-8536	52	10	and	and	CCONJ
ap-8536	52	11	negative	negative	ADJ
ap-8536	52	12	powers	power	NOUN
ap-8536	52	13	of	of	ADP
ap-8536	52	14	the	the	DET
ap-8536	52	15	matrix	matrix	NOUN
ap-8536	52	16	base	base	NOUN
ap-8536	52	17	for	for	ADP
ap-8536	52	18	representation	representation	NOUN
ap-8536	52	19	of	of	ADP
ap-8536	52	20	vectors	vector	NOUN
ap-8536	52	21	.	.	PUNCT
ap-8536	53	1	in	in	ADP
ap-8536	53	2	[	[	X
ap-8536	53	3	15	15	NUM
ap-8536	53	4	]	]	PUNCT
ap-8536	53	5	,	,	PUNCT
ap-8536	53	6	it	it	PRON
ap-8536	53	7	is	be	AUX
ap-8536	53	8	shown	show	VERB
ap-8536	53	9	that	that	SCONJ
ap-8536	53	10	for	for	ADP
ap-8536	53	11	m	m	PROPN
ap-8536	53	12	∈	∈	PROPN
ap-8536	53	13	zm×m	zm×m	PROPN
ap-8536	53	14	with	with	ADP
ap-8536	53	15	det	det	PROPN
ap-8536	53	16	m	m	PROPN
ap-8536	53	17	=	=	NOUN
ap-8536	53	18	∆	∆	PROPN
ap-8536	53	19	̸=	̸=	PROPN
ap-8536	53	20	0	0	NUM
ap-8536	53	21	,	,	PUNCT
ap-8536	53	22	there	there	PRON
ap-8536	53	23	exists	exist	VERB
ap-8536	53	24	a	a	DET
ap-8536	53	25	finite	finite	ADJ
ap-8536	53	26	digit	digit	NOUN
ap-8536	53	27	set	set	VERB
ap-8536	53	28	d	d	PROPN
ap-8536	53	29	⊂	⊂	PROPN
ap-8536	53	30	zm	zm	PROPN
ap-8536	53	31	such	such	ADJ
ap-8536	53	32	that	that	SCONJ
ap-8536	53	33	every	every	DET
ap-8536	53	34	integer	integer	NOUN
ap-8536	53	35	vector	vector	NOUN
ap-8536	53	36	from	from	ADP
ap-8536	53	37	zm	zm	PROPN
ap-8536	53	38	has	have	AUX
ap-8536	53	39	a	a	DET
ap-8536	53	40	finite	finite	NOUN
ap-8536	53	41	(	(	PUNCT
ap-8536	53	42	m	m	PROPN
ap-8536	53	43	,	,	PUNCT
ap-8536	53	44	d)-representation	d)-representation	NOUN
ap-8536	53	45	,	,	PUNCT
ap-8536	53	46	i.e.	i.e.	X
ap-8536	53	47	,	,	PUNCT
ap-8536	53	48	zm	zm	PROPN
ap-8536	53	49	⊂	⊂	PROPN
ap-8536	53	50	find(m	find(m	PROPN
ap-8536	53	51	)	)	PUNCT
ap-8536	53	52	,	,	PUNCT
ap-8536	53	53	where	where	SCONJ
ap-8536	53	54	(	(	PUNCT
ap-8536	53	55	1	1	X
ap-8536	53	56	)	)	PUNCT
ap-8536	53	57	find(m	find(m	NOUN
ap-8536	53	58	)	)	PUNCT
ap-8536	53	59	=	=	PRON
ap-8536	53	60	{	{	PUNCT
ap-8536	53	61	∑	∑	PROPN
ap-8536	53	62	j∈i	j∈i	PROPN
ap-8536	53	63	m	m	PROPN
ap-8536	53	64	jdj	jdj	PROPN
ap-8536	53	65	:	:	PUNCT
ap-8536	54	1	i	i	PRON
ap-8536	54	2	⊂	⊂	PROPN
ap-8536	55	1	z	z	X
ap-8536	55	2	,	,	PUNCT
ap-8536	55	3	i	i	PRON
ap-8536	55	4	finite	finite	VERB
ap-8536	55	5	,	,	PUNCT
ap-8536	55	6	dj	dj	ADP
ap-8536	55	7	∈	∈	PROPN
ap-8536	55	8	d	d	NOUN
ap-8536	55	9	}	}	PUNCT
ap-8536	55	10	.	.	PUNCT
ap-8536	56	1	we	we	PRON
ap-8536	56	2	show	show	VERB
ap-8536	56	3	(	(	PUNCT
ap-8536	56	4	in	in	ADP
ap-8536	56	5	theorem	theorem	NOUN
ap-8536	56	6	9	9	NUM
ap-8536	56	7	)	)	PUNCT
ap-8536	56	8	that	that	SCONJ
ap-8536	56	9	if	if	SCONJ
ap-8536	56	10	,	,	PUNCT
ap-8536	56	11	moreover	moreover	ADV
ap-8536	56	12	,	,	PUNCT
ap-8536	56	13	no	no	DET
ap-8536	56	14	eigenvalue	eigenvalue	NOUN
ap-8536	56	15	of	of	ADP
ap-8536	56	16	m	m	PROPN
ap-8536	56	17	lies	lie	VERB
ap-8536	56	18	on	on	ADP
ap-8536	56	19	the	the	DET
ap-8536	56	20	unit	unit	NOUN
ap-8536	56	21	circle	circle	NOUN
ap-8536	56	22	,	,	PUNCT
ap-8536	56	23	then	then	ADV
ap-8536	56	24	for	for	ADP
ap-8536	56	25	a	a	DET
ap-8536	56	26	suitable	suitable	ADJ
ap-8536	56	27	finite	finite	NOUN
ap-8536	56	28	digit	digit	NOUN
ap-8536	56	29	set	set	PROPN
ap-8536	56	30	d	d	PROPN
ap-8536	56	31	⊂	⊂	PROPN
ap-8536	56	32	zm	zm	PROPN
ap-8536	56	33	,	,	PUNCT
ap-8536	56	34	addition	addition	NOUN
ap-8536	56	35	and	and	CCONJ
ap-8536	56	36	subtraction	subtraction	NOUN
ap-8536	56	37	on	on	ADP
ap-8536	56	38	find(m	find(m	NOUN
ap-8536	56	39	)	)	PUNCT
ap-8536	56	40	can	can	AUX
ap-8536	56	41	be	be	AUX
ap-8536	56	42	performed	perform	VERB
ap-8536	56	43	by	by	ADP
ap-8536	56	44	a	a	DET
ap-8536	56	45	parallel	parallel	ADJ
ap-8536	56	46	algorithm	algorithm	NOUN
ap-8536	56	47	,	,	PUNCT
ap-8536	56	48	i.e.	i.e.	X
ap-8536	56	49	,	,	PUNCT
ap-8536	56	50	in	in	ADP
ap-8536	56	51	a	a	DET
ap-8536	56	52	constant	constant	ADJ
ap-8536	56	53	number	number	NOUN
ap-8536	56	54	of	of	ADP
ap-8536	56	55	steps	step	NOUN
ap-8536	56	56	independent	independent	ADJ
ap-8536	56	57	of	of	ADP
ap-8536	56	58	the	the	DET
ap-8536	56	59	length	length	NOUN
ap-8536	56	60	of	of	ADP
ap-8536	56	61	(	(	PUNCT
ap-8536	56	62	m	m	PROPN
ap-8536	56	63	,	,	PUNCT
ap-8536	56	64	d)-representation	d)-representation	NOUN
ap-8536	56	65	of	of	ADP
ap-8536	56	66	summands	summand	NOUN
ap-8536	56	67	.	.	PUNCT
ap-8536	57	1	according	accord	VERB
ap-8536	57	2	to	to	ADP
ap-8536	57	3	proposition	proposition	NOUN
ap-8536	57	4	15	15	NUM
ap-8536	57	5	,	,	PUNCT
ap-8536	57	6	the	the	DET
ap-8536	57	7	required	require	VERB
ap-8536	57	8	assumption	assumption	NOUN
ap-8536	57	9	on	on	ADP
ap-8536	57	10	eigenvalues	eigenvalue	NOUN
ap-8536	57	11	of	of	ADP
ap-8536	57	12	m	m	NOUN
ap-8536	57	13	is	be	AUX
ap-8536	57	14	in	in	ADP
ap-8536	57	15	fact	fact	NOUN
ap-8536	57	16	necessary	necessary	ADJ
ap-8536	57	17	for	for	ADP
ap-8536	57	18	the	the	DET
ap-8536	57	19	existence	existence	NOUN
ap-8536	57	20	of	of	ADP
ap-8536	57	21	a	a	DET
ap-8536	57	22	parallel	parallel	ADJ
ap-8536	57	23	addition	addition	NOUN
ap-8536	57	24	algorithm	algorithm	NOUN
ap-8536	57	25	on	on	ADP
ap-8536	57	26	(	(	PUNCT
ap-8536	57	27	m	m	PROPN
ap-8536	57	28	,	,	PUNCT
ap-8536	57	29	d	d	NOUN
ap-8536	57	30	)	)	PUNCT
ap-8536	57	31	.	.	PUNCT
ap-8536	58	1	then	then	ADV
ap-8536	58	2	,	,	PUNCT
ap-8536	58	3	we	we	PRON
ap-8536	58	4	restrict	restrict	VERB
ap-8536	58	5	our	our	PRON
ap-8536	58	6	study	study	NOUN
ap-8536	58	7	to	to	ADP
ap-8536	58	8	expansive	expansive	ADJ
ap-8536	58	9	matrices	matrix	NOUN
ap-8536	58	10	–	–	PUNCT
ap-8536	58	11	i.e.	i.e.	X
ap-8536	58	12	,	,	PUNCT
ap-8536	58	13	matrices	matrix	NOUN
ap-8536	58	14	with	with	ADP
ap-8536	58	15	all	all	PRON
ap-8536	58	16	eigenvalues	eigenvalue	NOUN
ap-8536	58	17	strictly	strictly	ADV
ap-8536	58	18	outside	outside	ADP
ap-8536	58	19	the	the	DET
ap-8536	58	20	unit	unit	NOUN
ap-8536	58	21	circle	circle	NOUN
ap-8536	58	22	.	.	PUNCT
ap-8536	59	1	in	in	ADP
ap-8536	59	2	theorem	theorem	NOUN
ap-8536	59	3	21	21	NUM
ap-8536	59	4	,	,	PUNCT
ap-8536	59	5	we	we	PRON
ap-8536	59	6	show	show	VERB
ap-8536	59	7	that	that	SCONJ
ap-8536	59	8	the	the	DET
ap-8536	59	9	digit	digit	NOUN
ap-8536	59	10	set	set	VERB
ap-8536	59	11	d	d	PROPN
ap-8536	59	12	⊂	⊂	PROPN
ap-8536	59	13	zm	zm	PROPN
ap-8536	59	14	allowing	allow	VERB
ap-8536	59	15	a	a	DET
ap-8536	59	16	parallel	parallel	ADJ
ap-8536	59	17	addition	addition	NOUN
ap-8536	59	18	enables	enable	VERB
ap-8536	59	19	(	(	PUNCT
ap-8536	59	20	for	for	ADP
ap-8536	59	21	an	an	DET
ap-8536	59	22	expansive	expansive	ADJ
ap-8536	59	23	matrix	matrix	NOUN
ap-8536	59	24	base	base	NOUN
ap-8536	59	25	m	m	PROPN
ap-8536	59	26	)	)	PUNCT
ap-8536	59	27	an	an	DET
ap-8536	59	28	eventually	eventually	ADV
ap-8536	59	29	periodic	periodic	ADJ
ap-8536	59	30	(	(	PUNCT
ap-8536	59	31	m	m	PROPN
ap-8536	59	32	,	,	PUNCT
ap-8536	59	33	d)-representation	d)-representation	NOUN
ap-8536	59	34	of	of	ADP
ap-8536	59	35	every	every	DET
ap-8536	59	36	element	element	NOUN
ap-8536	59	37	of	of	ADP
ap-8536	59	38	qm	qm	PROPN
ap-8536	59	39	,	,	PUNCT
ap-8536	59	40	i.e.	i.e.	X
ap-8536	59	41	qm	qm	PROPN
ap-8536	59	42	=	=	SYM
ap-8536	59	43	perd(m	perd(m	PROPN
ap-8536	59	44	)	)	PUNCT
ap-8536	59	45	,	,	PUNCT
ap-8536	59	46	where	where	SCONJ
ap-8536	59	47	perd(m	perd(m	NOUN
ap-8536	59	48	)	)	PUNCT
ap-8536	59	49	=	=	PRON
ap-8536	59	50	{	{	PUNCT
ap-8536	60	1	n∑	n∑	INTJ
ap-8536	60	2	j=−∞	j=−∞	PROPN
ap-8536	61	1	m	m	PROPN
ap-8536	61	2	jdj	jdj	PROPN
ap-8536	61	3	:	:	PUNCT
ap-8536	61	4	dj	dj	X
ap-8536	61	5	∈	∈	PROPN
ap-8536	61	6	d	d	NOUN
ap-8536	61	7	,	,	PUNCT
ap-8536	61	8	(	(	PUNCT
ap-8536	61	9	dj)n	dj)n	PROPN
ap-8536	61	10	−∞	−∞	PUNCT
ap-8536	61	11	eventually	eventually	ADV
ap-8536	61	12	periodic	periodic	ADJ
ap-8536	61	13	}	}	PUNCT
ap-8536	61	14	.	.	PUNCT
ap-8536	62	1	consequently	consequently	ADV
ap-8536	62	2	,	,	PUNCT
ap-8536	62	3	every	every	DET
ap-8536	62	4	element	element	NOUN
ap-8536	62	5	of	of	ADP
ap-8536	62	6	rm	rm	PROPN
ap-8536	62	7	has	have	VERB
ap-8536	62	8	an	an	DET
ap-8536	62	9	(	(	PUNCT
ap-8536	62	10	m	m	NOUN
ap-8536	62	11	,	,	PUNCT
ap-8536	62	12	d)representation	d)representation	NOUN
ap-8536	62	13	(	(	PUNCT
ap-8536	62	14	corollary	corollary	NOUN
ap-8536	62	15	22	22	NUM
ap-8536	62	16	)	)	PUNCT
ap-8536	62	17	.	.	PUNCT
ap-8536	63	1	the	the	DET
ap-8536	63	2	methods	method	NOUN
ap-8536	63	3	we	we	PRON
ap-8536	63	4	use	use	VERB
ap-8536	63	5	in	in	ADP
ap-8536	63	6	our	our	PRON
ap-8536	63	7	proofs	proof	NOUN
ap-8536	63	8	are	be	AUX
ap-8536	63	9	based	base	VERB
ap-8536	63	10	on	on	ADP
ap-8536	63	11	proofs	proof	NOUN
ap-8536	63	12	of	of	ADP
ap-8536	63	13	analogous	analogous	ADJ
ap-8536	63	14	results	result	NOUN
ap-8536	63	15	for	for	ADP
ap-8536	63	16	a	a	DET
ap-8536	63	17	positional	positional	ADJ
ap-8536	63	18	representation	representation	NOUN
ap-8536	63	19	of	of	ADP
ap-8536	63	20	real	real	ADJ
ap-8536	63	21	and	and	CCONJ
ap-8536	63	22	complex	complex	ADJ
ap-8536	63	23	numbers	number	NOUN
ap-8536	63	24	,	,	PUNCT
ap-8536	63	25	modified	modify	VERB
ap-8536	63	26	accordingly	accordingly	ADV
ap-8536	63	27	to	to	ADP
ap-8536	63	28	the	the	DET
ap-8536	63	29	formalism	formalism	NOUN
ap-8536	63	30	of	of	ADP
ap-8536	63	31	matrices	matrix	NOUN
ap-8536	63	32	and	and	CCONJ
ap-8536	63	33	vectors	vector	NOUN
ap-8536	63	34	.	.	PUNCT
ap-8536	64	1	2	2	X
ap-8536	64	2	.	.	NUM
ap-8536	64	3	preliminaries	preliminary	NOUN
ap-8536	64	4	a	a	DET
ap-8536	64	5	numeration	numeration	NOUN
ap-8536	64	6	system	system	NOUN
ap-8536	64	7	used	use	VERB
ap-8536	64	8	for	for	ADP
ap-8536	64	9	a	a	DET
ap-8536	64	10	positional	positional	ADJ
ap-8536	64	11	representation	representation	NOUN
ap-8536	64	12	of	of	ADP
ap-8536	64	13	complex	complex	ADJ
ap-8536	64	14	numbers	number	NOUN
ap-8536	64	15	is	be	AUX
ap-8536	64	16	given	give	VERB
ap-8536	64	17	by	by	ADP
ap-8536	64	18	a	a	DET
ap-8536	64	19	base	base	NOUN
ap-8536	64	20	β	β	X
ap-8536	64	21	∈	∈	PROPN
ap-8536	64	22	c	c	NOUN
ap-8536	64	23	with	with	ADP
ap-8536	64	24	|β|	|β|	PRON
ap-8536	64	25	>	>	SYM
ap-8536	64	26	1	1	NUM
ap-8536	64	27	and	and	CCONJ
ap-8536	64	28	a	a	DET
ap-8536	64	29	finite	finite	ADJ
ap-8536	64	30	digit	digit	NOUN
ap-8536	64	31	set	set	VERB
ap-8536	64	32	a	a	DET
ap-8536	64	33	⊂	⊂	PROPN
ap-8536	64	34	c.	c.	NOUN
ap-8536	64	35	if	if	SCONJ
ap-8536	64	36	x	x	SYM
ap-8536	64	37	∈	∈	PROPN
ap-8536	64	38	c	c	NOUN
ap-8536	64	39	can	can	AUX
ap-8536	64	40	be	be	AUX
ap-8536	64	41	written	write	VERB
ap-8536	64	42	in	in	ADP
ap-8536	64	43	the	the	DET
ap-8536	64	44	form	form	NOUN
ap-8536	64	45	x	x	PUNCT
ap-8536	64	46	=	=	SYM
ap-8536	64	47	∑n	∑n	PROPN
ap-8536	64	48	j=−∞	j=−∞	PROPN
ap-8536	64	49	ajβj	ajβj	PROPN
ap-8536	64	50	,	,	PUNCT
ap-8536	64	51	where	where	SCONJ
ap-8536	64	52	aj	aj	PROPN
ap-8536	64	53	∈	∈	PROPN
ap-8536	64	54	a	a	PRON
ap-8536	64	55	for	for	ADP
ap-8536	64	56	each	each	DET
ap-8536	64	57	j	j	PROPN
ap-8536	64	58	∈	∈	PROPN
ap-8536	64	59	z	z	PROPN
ap-8536	64	60	,	,	PUNCT
ap-8536	64	61	j	j	PROPN
ap-8536	64	62	≤	≤	PROPN
ap-8536	64	63	n	n	CCONJ
ap-8536	64	64	,	,	PUNCT
ap-8536	64	65	we	we	PRON
ap-8536	64	66	say	say	VERB
ap-8536	64	67	that	that	SCONJ
ap-8536	64	68	x	x	PRON
ap-8536	64	69	has	have	VERB
ap-8536	64	70	a	a	DET
ap-8536	64	71	(	(	PUNCT
ap-8536	64	72	β	β	NOUN
ap-8536	64	73	,	,	PUNCT
ap-8536	64	74	a)-representation	a)-representation	NOUN
ap-8536	64	75	.	.	PUNCT
ap-8536	65	1	the	the	DET
ap-8536	65	2	assumption	assumption	NOUN
ap-8536	65	3	|β|	|β|	ADP
ap-8536	65	4	>	>	SYM
ap-8536	65	5	1	1	NUM
ap-8536	65	6	guarantees	guarantee	VERB
ap-8536	65	7	that	that	SCONJ
ap-8536	65	8	the	the	DET
ap-8536	65	9	series	series	NOUN
ap-8536	65	10	∑n	∑n	PROPN
ap-8536	65	11	j=−∞	j=−∞	PROPN
ap-8536	65	12	ajβj	ajβj	PROPN
ap-8536	65	13	is	be	AUX
ap-8536	65	14	convergent	convergent	NOUN
ap-8536	65	15	for	for	ADP
ap-8536	65	16	any	any	DET
ap-8536	65	17	choice	choice	NOUN
ap-8536	65	18	of	of	ADP
ap-8536	65	19	digits	digit	NOUN
ap-8536	65	20	aj	aj	PROPN
ap-8536	65	21	∈	∈	PROPN
ap-8536	65	22	a.	a.	NOUN
ap-8536	65	23	w.	w.	PROPN
ap-8536	65	24	penney	penney	PROPN
ap-8536	65	25	in	in	ADP
ap-8536	65	26	[	[	X
ap-8536	65	27	7	7	NUM
ap-8536	65	28	]	]	PUNCT
ap-8536	65	29	introduced	introduce	VERB
ap-8536	65	30	the	the	DET
ap-8536	65	31	following	follow	VERB
ap-8536	65	32	numeration	numeration	NOUN
ap-8536	65	33	system	system	NOUN
ap-8536	65	34	,	,	PUNCT
ap-8536	65	35	which	which	PRON
ap-8536	65	36	we	we	PRON
ap-8536	65	37	use	use	VERB
ap-8536	65	38	to	to	PART
ap-8536	65	39	demonstrate	demonstrate	VERB
ap-8536	65	40	our	our	PRON
ap-8536	65	41	approach	approach	NOUN
ap-8536	65	42	.	.	PUNCT
ap-8536	66	1	example	example	NOUN
ap-8536	67	1	1	1	NUM
ap-8536	67	2	.	.	PUNCT
ap-8536	67	3	let	let	VERB
ap-8536	67	4	us	we	PRON
ap-8536	67	5	consider	consider	VERB
ap-8536	67	6	β	β	NOUN
ap-8536	67	7	=	=	PUNCT
ap-8536	68	1	i	i	PRON
ap-8536	68	2	−	−	PROPN
ap-8536	69	1	1	1	X
ap-8536	69	2	.	.	X
ap-8536	69	3	penney	penney	NOUN
ap-8536	69	4	in	in	ADP
ap-8536	69	5	[	[	X
ap-8536	69	6	7	7	NUM
ap-8536	69	7	]	]	PUNCT
ap-8536	69	8	showed	show	VERB
ap-8536	69	9	that	that	SCONJ
ap-8536	69	10	(	(	PUNCT
ap-8536	69	11	1	1	NUM
ap-8536	69	12	.	.	PUNCT
ap-8536	69	13	)	)	PUNCT
ap-8536	70	1	each	each	DET
ap-8536	70	2	x	x	PUNCT
ap-8536	70	3	∈	∈	PROPN
ap-8536	70	4	z[i	z[i	NUM
ap-8536	70	5	]	]	X
ap-8536	70	6	=	=	X
ap-8536	70	7	{	{	PUNCT
ap-8536	70	8	a	a	X
ap-8536	70	9	+	+	X
ap-8536	70	10	ıb	ıb	NOUN
ap-8536	70	11	:	:	PUNCT
ap-8536	70	12	a	a	DET
ap-8536	70	13	,	,	PUNCT
ap-8536	70	14	b	b	X
ap-8536	70	15	∈	∈	PROPN
ap-8536	70	16	z	z	NOUN
ap-8536	70	17	}	}	PUNCT
ap-8536	70	18	,	,	PUNCT
ap-8536	70	19	x	x	SYM
ap-8536	70	20	̸=	̸=	PROPN
ap-8536	70	21	0	0	NUM
ap-8536	70	22	can	can	AUX
ap-8536	70	23	be	be	AUX
ap-8536	70	24	expressed	express	VERB
ap-8536	70	25	uniquely	uniquely	ADV
ap-8536	70	26	as	as	ADP
ap-8536	70	27	x	x	PROPN
ap-8536	70	28	=	=	SYM
ap-8536	70	29	∑n	∑n	PROPN
ap-8536	70	30	j=0	j=0	PROPN
ap-8536	70	31	ajβj	ajβj	PROPN
ap-8536	70	32	,	,	PUNCT
ap-8536	70	33	where	where	SCONJ
ap-8536	70	34	a0	a0	PROPN
ap-8536	70	35	,	,	PUNCT
ap-8536	70	36	a1	a1	PROPN
ap-8536	70	37	,	,	PUNCT
ap-8536	70	38	.	.	PUNCT
ap-8536	70	39	.	.	PUNCT
ap-8536	71	1	.	.	PUNCT
ap-8536	72	1	,	,	PUNCT
ap-8536	72	2	an	an	DET
ap-8536	72	3	∈	∈	NOUN
ap-8536	72	4	{	{	PUNCT
ap-8536	72	5	0	0	NUM
ap-8536	72	6	,	,	PUNCT
ap-8536	72	7	1	1	NUM
ap-8536	72	8	}	}	PUNCT
ap-8536	72	9	and	and	CCONJ
ap-8536	72	10	an	an	DET
ap-8536	72	11	̸=	̸=	PROPN
ap-8536	72	12	0	0	NUM
ap-8536	72	13	;	;	PUNCT
ap-8536	72	14	(	(	PUNCT
ap-8536	72	15	2	2	NUM
ap-8536	72	16	.	.	PUNCT
ap-8536	72	17	)	)	PUNCT
ap-8536	73	1	each	each	DET
ap-8536	73	2	x	x	PUNCT
ap-8536	73	3	∈	∈	PROPN
ap-8536	73	4	c	c	NOUN
ap-8536	73	5	can	can	AUX
ap-8536	73	6	be	be	AUX
ap-8536	73	7	expressed	express	VERB
ap-8536	73	8	as	as	ADP
ap-8536	73	9	x	x	X
ap-8536	73	10	=	=	SYM
ap-8536	73	11	∑n	∑n	PROPN
ap-8536	73	12	j=−∞	j=−∞	PROPN
ap-8536	73	13	ajβj	ajβj	PROPN
ap-8536	73	14	,	,	PUNCT
ap-8536	73	15	where	where	SCONJ
ap-8536	73	16	aj	aj	PROPN
ap-8536	73	17	∈	∈	PROPN
ap-8536	73	18	{	{	PUNCT
ap-8536	73	19	0	0	NUM
ap-8536	73	20	,	,	PUNCT
ap-8536	73	21	1	1	NUM
ap-8536	73	22	}	}	PUNCT
ap-8536	73	23	for	for	ADP
ap-8536	73	24	every	every	DET
ap-8536	73	25	j	j	PROPN
ap-8536	73	26	∈	∈	PROPN
ap-8536	73	27	z	z	PROPN
ap-8536	73	28	,	,	PUNCT
ap-8536	73	29	j	j	PROPN
ap-8536	73	30	≤	≤	PROPN
ap-8536	73	31	n.	n.	NOUN
ap-8536	73	32	our	our	PRON
ap-8536	73	33	aim	aim	NOUN
ap-8536	73	34	is	be	AUX
ap-8536	73	35	to	to	PART
ap-8536	73	36	study	study	VERB
ap-8536	73	37	selected	select	VERB
ap-8536	73	38	properties	property	NOUN
ap-8536	73	39	of	of	ADP
ap-8536	73	40	matrix	matrix	NOUN
ap-8536	73	41	numeration	numeration	NOUN
ap-8536	73	42	systems	system	NOUN
ap-8536	73	43	used	use	VERB
ap-8536	73	44	to	to	PART
ap-8536	73	45	represent	represent	VERB
ap-8536	73	46	m	m	ADJ
ap-8536	73	47	-	-	ADJ
ap-8536	73	48	dimensional	dimensional	ADJ
ap-8536	73	49	vectors	vector	NOUN
ap-8536	73	50	.	.	PUNCT
ap-8536	74	1	any	any	DET
ap-8536	74	2	matrix	matrix	NOUN
ap-8536	74	3	numeration	numeration	NOUN
ap-8536	74	4	system	system	NOUN
ap-8536	74	5	used	use	VERB
ap-8536	74	6	in	in	ADP
ap-8536	74	7	this	this	DET
ap-8536	74	8	paper	paper	NOUN
ap-8536	74	9	is	be	AUX
ap-8536	74	10	given	give	VERB
ap-8536	74	11	by	by	ADP
ap-8536	74	12	a	a	DET
ap-8536	74	13	non	non	ADJ
ap-8536	74	14	-	-	ADJ
ap-8536	74	15	singular	singular	ADJ
ap-8536	74	16	matrix	matrix	NOUN
ap-8536	74	17	base	base	NOUN
ap-8536	74	18	m	m	NOUN
ap-8536	74	19	∈	∈	PROPN
ap-8536	74	20	zm×m	zm×m	NOUN
ap-8536	74	21	and	and	CCONJ
ap-8536	74	22	a	a	DET
ap-8536	74	23	finite	finite	NOUN
ap-8536	74	24	(	(	PUNCT
ap-8536	74	25	vector	vector	NOUN
ap-8536	74	26	)	)	PUNCT
ap-8536	74	27	digit	digit	NOUN
ap-8536	74	28	set	set	PROPN
ap-8536	75	1	d	d	PROPN
ap-8536	75	2	⊂	⊂	PROPN
ap-8536	75	3	zm	zm	PROPN
ap-8536	75	4	.	.	PUNCT
ap-8536	76	1	thanks	thank	NOUN
ap-8536	76	2	to	to	ADP
ap-8536	76	3	the	the	DET
ap-8536	76	4	result	result	NOUN
ap-8536	76	5	of	of	ADP
ap-8536	76	6	[	[	X
ap-8536	76	7	15	15	NUM
ap-8536	76	8	]	]	PUNCT
ap-8536	76	9	mentioned	mention	VERB
ap-8536	76	10	earlier	early	ADV
ap-8536	76	11	,	,	PUNCT
ap-8536	76	12	we	we	PRON
ap-8536	76	13	always	always	ADV
ap-8536	76	14	assume	assume	VERB
ap-8536	76	15	that	that	SCONJ
ap-8536	76	16	zm	zm	PROPN
ap-8536	76	17	is	be	AUX
ap-8536	76	18	a	a	DET
ap-8536	76	19	subset	subset	NOUN
ap-8536	76	20	of	of	ADP
ap-8536	76	21	find(m	find(m	NOUN
ap-8536	76	22	)	)	PUNCT
ap-8536	76	23	and	and	CCONJ
ap-8536	76	24	,	,	PUNCT
ap-8536	76	25	(	(	PUNCT
ap-8536	76	26	2	2	X
ap-8536	76	27	)	)	PUNCT
ap-8536	76	28	moreover	moreover	ADV
ap-8536	76	29	,	,	PUNCT
ap-8536	76	30	d	d	PROPN
ap-8536	76	31	contains	contain	VERB
ap-8536	76	32	the	the	DET
ap-8536	76	33	zero	zero	NUM
ap-8536	76	34	vector	vector	NOUN
ap-8536	76	35	.	.	PUNCT
ap-8536	77	1	(	(	PUNCT
ap-8536	77	2	3	3	X
ap-8536	77	3	)	)	PUNCT
ap-8536	77	4	in	in	ADP
ap-8536	77	5	the	the	DET
ap-8536	77	6	first	first	ADJ
ap-8536	77	7	part	part	NOUN
ap-8536	77	8	of	of	ADP
ap-8536	77	9	this	this	DET
ap-8536	77	10	paper	paper	NOUN
ap-8536	77	11	,	,	PUNCT
ap-8536	77	12	we	we	PRON
ap-8536	77	13	work	work	VERB
ap-8536	77	14	only	only	ADV
ap-8536	77	15	with	with	ADP
ap-8536	77	16	vectors	vector	NOUN
ap-8536	77	17	from	from	ADP
ap-8536	77	18	find(m	find(m	NOUN
ap-8536	77	19	)	)	PUNCT
ap-8536	77	20	.	.	PUNCT
ap-8536	78	1	therefore	therefore	ADV
ap-8536	78	2	,	,	PUNCT
ap-8536	78	3	we	we	PRON
ap-8536	78	4	do	do	AUX
ap-8536	78	5	not	not	PART
ap-8536	78	6	yet	yet	ADV
ap-8536	78	7	impose	impose	VERB
ap-8536	78	8	additional	additional	ADJ
ap-8536	78	9	assumptions	assumption	NOUN
ap-8536	78	10	on	on	ADP
ap-8536	78	11	the	the	DET
ap-8536	78	12	matrix	matrix	NOUN
ap-8536	78	13	base	base	NOUN
ap-8536	78	14	m	m	ADP
ap-8536	78	15	,	,	PUNCT
ap-8536	78	16	analogous	analogous	ADJ
ap-8536	78	17	to	to	ADP
ap-8536	78	18	the	the	DET
ap-8536	78	19	assumption	assumption	NOUN
ap-8536	78	20	|β|	|β|	ADP
ap-8536	78	21	>	>	X
ap-8536	78	22	1	1	NUM
ap-8536	78	23	required	require	VERB
ap-8536	78	24	for	for	ADP
ap-8536	78	25	number	number	NOUN
ap-8536	78	26	bases	basis	NOUN
ap-8536	78	27	(	(	PUNCT
ap-8536	78	28	which	which	PRON
ap-8536	78	29	is	be	AUX
ap-8536	78	30	important	important	ADJ
ap-8536	78	31	to	to	PART
ap-8536	78	32	ensure	ensure	VERB
ap-8536	78	33	the	the	DET
ap-8536	78	34	convergence	convergence	NOUN
ap-8536	78	35	of	of	ADP
ap-8536	78	36	the	the	DET
ap-8536	78	37	infinite	infinite	ADJ
ap-8536	78	38	series	series	NOUN
ap-8536	78	39	∑n	∑n	PROPN
ap-8536	78	40	j=−∞	j=−∞	PROPN
ap-8536	78	41	ajβj	ajβj	PROPN
ap-8536	78	42	)	)	PUNCT
ap-8536	78	43	.	.	PUNCT
ap-8536	79	1	only	only	ADV
ap-8536	79	2	in	in	ADP
ap-8536	79	3	the	the	DET
ap-8536	79	4	second	second	ADJ
ap-8536	79	5	part	part	NOUN
ap-8536	79	6	of	of	ADP
ap-8536	79	7	the	the	DET
ap-8536	79	8	paper	paper	NOUN
ap-8536	79	9	,	,	PUNCT
ap-8536	79	10	we	we	PRON
ap-8536	79	11	revisit	revisit	VERB
ap-8536	79	12	the	the	DET
ap-8536	79	13	question	question	NOUN
ap-8536	79	14	of	of	ADP
ap-8536	79	15	the	the	DET
ap-8536	79	16	infinite	infinite	ADJ
ap-8536	79	17	representations	representation	NOUN
ap-8536	79	18	convergence	convergence	NOUN
ap-8536	79	19	for	for	ADP
ap-8536	79	20	matrix	matrix	NOUN
ap-8536	79	21	numeration	numeration	NOUN
ap-8536	79	22	systems	system	NOUN
ap-8536	79	23	as	as	ADV
ap-8536	79	24	well	well	ADV
ap-8536	79	25	.	.	PUNCT
ap-8536	80	1	let	let	VERB
ap-8536	80	2	us	we	PRON
ap-8536	80	3	list	list	VERB
ap-8536	80	4	some	some	DET
ap-8536	80	5	obvious	obvious	ADJ
ap-8536	80	6	properties	property	NOUN
ap-8536	80	7	of	of	ADP
ap-8536	80	8	the	the	DET
ap-8536	80	9	set	set	NOUN
ap-8536	80	10	find(m	find(m	NOUN
ap-8536	80	11	):	):	PUNCT
ap-8536	80	12	•	•	NUM
ap-8536	80	13	mpfind(m	mpfind(m	NOUN
ap-8536	80	14	)	)	PUNCT
ap-8536	81	1	=	=	SYM
ap-8536	81	2	find(m	find(m	NOUN
ap-8536	81	3	)	)	PUNCT
ap-8536	81	4	for	for	ADP
ap-8536	81	5	every	every	DET
ap-8536	81	6	p	p	PROPN
ap-8536	81	7	∈	∈	PROPN
ap-8536	81	8	z.	z.	PROPN
ap-8536	81	9	•	•	NUM
ap-8536	81	10	find(m	find(m	PROPN
ap-8536	81	11	)	)	PUNCT
ap-8536	82	1	⊂	⊂	PROPN
ap-8536	82	2	qm	qm	PROPN
ap-8536	82	3	,	,	PUNCT
ap-8536	82	4	more	more	ADV
ap-8536	82	5	precisely	precisely	ADV
ap-8536	82	6	,	,	PUNCT
ap-8536	82	7	find(m	find(m	NOUN
ap-8536	82	8	)	)	PUNCT
ap-8536	82	9	⊂	⊂	PROPN
ap-8536	82	10	⋃	⋃	PROPN
ap-8536	82	11	k∈n	k∈n	PROPN
ap-8536	82	12	1	1	NUM
ap-8536	82	13	∆k	∆k	PROPN
ap-8536	82	14	zm	zm	PROPN
ap-8536	82	15	,	,	PUNCT
ap-8536	82	16	where	where	SCONJ
ap-8536	82	17	∆	∆	PROPN
ap-8536	82	18	=	=	SYM
ap-8536	82	19	det	det	PROPN
ap-8536	82	20	m	m	PROPN
ap-8536	82	21	.	.	PUNCT
ap-8536	83	1	•	•	INTJ
ap-8536	83	2	if	if	SCONJ
ap-8536	83	3	x	x	PRON
ap-8536	83	4	=	=	SYM
ap-8536	83	5	∑n	∑n	PROPN
ap-8536	83	6	j=0	j=0	PROPN
ap-8536	83	7	m	m	VERB
ap-8536	83	8	jdj	jdj	PROPN
ap-8536	83	9	for	for	ADP
ap-8536	83	10	some	some	DET
ap-8536	83	11	n	n	PRON
ap-8536	83	12	∈	∈	PROPN
ap-8536	83	13	n	n	CCONJ
ap-8536	83	14	,	,	PUNCT
ap-8536	83	15	then	then	ADV
ap-8536	83	16	x	x	PROPN
ap-8536	83	17	∈	∈	PROPN
ap-8536	83	18	zm	zm	PROPN
ap-8536	83	19	.	.	PROPN
ap-8536	83	20	•	•	NUM
ap-8536	83	21	find(m	find(m	NOUN
ap-8536	83	22	)	)	PUNCT
ap-8536	83	23	is	be	AUX
ap-8536	83	24	closed	close	VERB
ap-8536	83	25	under	under	ADP
ap-8536	83	26	addition	addition	NOUN
ap-8536	83	27	and	and	CCONJ
ap-8536	83	28	subtraction	subtraction	NOUN
ap-8536	83	29	:	:	PUNCT
ap-8536	83	30	indeed	indeed	ADV
ap-8536	83	31	,	,	PUNCT
ap-8536	83	32	if	if	SCONJ
ap-8536	83	33	x	x	X
ap-8536	83	34	,	,	PUNCT
ap-8536	83	35	y	y	PROPN
ap-8536	83	36	∈	∈	PROPN
ap-8536	83	37	find(m	find(m	PROPN
ap-8536	83	38	)	)	PUNCT
ap-8536	83	39	,	,	PUNCT
ap-8536	83	40	then	then	ADV
ap-8536	83	41	there	there	PRON
ap-8536	83	42	exists	exist	VERB
ap-8536	83	43	p	p	PROPN
ap-8536	83	44	∈	∈	PROPN
ap-8536	83	45	n	n	CCONJ
ap-8536	83	46	such	such	ADJ
ap-8536	83	47	that	that	SCONJ
ap-8536	83	48	mpx	mpx	VERB
ap-8536	83	49	∈	∈	PROPN
ap-8536	83	50	zm	zm	PROPN
ap-8536	83	51	and	and	CCONJ
ap-8536	83	52	mpy	mpy	PROPN
ap-8536	83	53	∈	∈	PROPN
ap-8536	83	54	zm	zm	PROPN
ap-8536	83	55	,	,	PUNCT
ap-8536	83	56	and	and	CCONJ
ap-8536	83	57	hence	hence	ADV
ap-8536	83	58	mp(x	mp(x	PUNCT
ap-8536	83	59	±	±	PROPN
ap-8536	83	60	y	y	NOUN
ap-8536	83	61	)	)	PUNCT
ap-8536	83	62	∈	∈	PROPN
ap-8536	84	1	zm	zm	PROPN
ap-8536	84	2	.	.	PUNCT
ap-8536	85	1	by	by	ADP
ap-8536	85	2	assumption	assumption	NOUN
ap-8536	85	3	(	(	PUNCT
ap-8536	85	4	2	2	NUM
ap-8536	85	5	)	)	PUNCT
ap-8536	85	6	,	,	PUNCT
ap-8536	85	7	mp(x	mp(x	PUNCT
ap-8536	85	8	±	±	PROPN
ap-8536	85	9	y	y	NOUN
ap-8536	85	10	)	)	PUNCT
ap-8536	85	11	∈	∈	PROPN
ap-8536	85	12	find(m	find(m	NOUN
ap-8536	85	13	)	)	PUNCT
ap-8536	85	14	=	=	SYM
ap-8536	85	15	mpfind(m	mpfind(m	NOUN
ap-8536	85	16	)	)	PUNCT
ap-8536	85	17	,	,	PUNCT
ap-8536	85	18	and	and	CCONJ
ap-8536	85	19	thus	thus	ADV
ap-8536	85	20	x	x	X
ap-8536	85	21	±	±	NUM
ap-8536	85	22	y	y	PROPN
ap-8536	85	23	∈	∈	PROPN
ap-8536	85	24	find(m	find(m	PROPN
ap-8536	85	25	)	)	PUNCT
ap-8536	85	26	.	.	PUNCT
ap-8536	86	1	if	if	SCONJ
ap-8536	86	2	a	a	DET
ap-8536	86	3	vector	vector	NOUN
ap-8536	86	4	x	x	X
ap-8536	86	5	∈	∈	PROPN
ap-8536	86	6	qm	qm	PROPN
ap-8536	86	7	is	be	AUX
ap-8536	86	8	expressed	express	VERB
ap-8536	86	9	as	as	ADP
ap-8536	86	10	x	x	X
ap-8536	86	11	=	=	SYM
ap-8536	86	12	∑	∑	PROPN
ap-8536	86	13	j∈i	j∈i	PROPN
ap-8536	86	14	m	m	PROPN
ap-8536	86	15	jdj	jdj	PROPN
ap-8536	86	16	for	for	ADP
ap-8536	86	17	a	a	DET
ap-8536	86	18	finite	finite	NOUN
ap-8536	86	19	i	i	PRON
ap-8536	86	20	⊂	⊂	PROPN
ap-8536	86	21	z	z	NOUN
ap-8536	86	22	and	and	CCONJ
ap-8536	86	23	dj	dj	X
ap-8536	86	24	∈	∈	PROPN
ap-8536	87	1	d	d	NOUN
ap-8536	87	2	,	,	PUNCT
ap-8536	87	3	we	we	PRON
ap-8536	87	4	can	can	AUX
ap-8536	87	5	,	,	PUNCT
ap-8536	87	6	for	for	ADP
ap-8536	87	7	some	some	DET
ap-8536	87	8	integer	integer	NOUN
ap-8536	87	9	numbers	number	NOUN
ap-8536	87	10	s	s	PART
ap-8536	87	11	≤	≤	NUM
ap-8536	87	12	0	0	NUM
ap-8536	87	13	≤	≤	NUM
ap-8536	87	14	n	n	CCONJ
ap-8536	87	15	,	,	PUNCT
ap-8536	87	16	write	write	VERB
ap-8536	87	17	x	x	PUNCT
ap-8536	87	18	=	=	SYM
ap-8536	88	1	∑n	∑n	PROPN
ap-8536	88	2	j	j	PROPN
ap-8536	88	3	=	=	PROPN
ap-8536	88	4	s	s	PART
ap-8536	88	5	m	m	VERB
ap-8536	88	6	jdj	jdj	PROPN
ap-8536	88	7	,	,	PUNCT
ap-8536	88	8	because	because	SCONJ
ap-8536	88	9	the	the	DET
ap-8536	88	10	zero	zero	NUM
ap-8536	88	11	vector	vector	NOUN
ap-8536	88	12	belongs	belong	VERB
ap-8536	88	13	to	to	ADP
ap-8536	88	14	d.	d.	PROPN
ap-8536	88	15	hence	hence	ADV
ap-8536	88	16	x	x	PUNCT
ap-8536	88	17	can	can	AUX
ap-8536	88	18	be	be	AUX
ap-8536	88	19	identified	identify	VERB
ap-8536	88	20	with	with	ADP
ap-8536	88	21	a	a	DET
ap-8536	88	22	bi	bi	ADJ
ap-8536	88	23	-	-	ADJ
ap-8536	88	24	infinite	infinite	ADJ
ap-8536	88	25	string	string	NOUN
ap-8536	88	26	(	(	PUNCT
ap-8536	88	27	dj)j∈z	dj)j∈z	PROPN
ap-8536	88	28	∈	∈	PROPN
ap-8536	88	29	dz	dz	NOUN
ap-8536	88	30	usually	usually	ADV
ap-8536	88	31	referred	refer	VERB
ap-8536	88	32	to	to	ADP
ap-8536	88	33	as	as	ADP
ap-8536	88	34	(	(	PUNCT
ap-8536	88	35	m	m	PROPN
ap-8536	88	36	,	,	PUNCT
ap-8536	88	37	d)-representation	d)-representation	NOUN
ap-8536	88	38	of	of	ADP
ap-8536	88	39	x	x	X
ap-8536	88	40	:	:	PUNCT
ap-8536	88	41	(	(	PUNCT
ap-8536	88	42	x)m	x)m	X
ap-8536	88	43	,	,	PUNCT
ap-8536	88	44	d	d	X
ap-8536	88	45	=	=	SYM
ap-8536	88	46	ω0dndn−1	ω0dndn−1	NUM
ap-8536	88	47	·	·	PUNCT
ap-8536	88	48	·	·	PUNCT
ap-8536	88	49	·	·	PUNCT
ap-8536	89	1	d1d0	d1d0	X
ap-8536	89	2	•	•	NOUN
ap-8536	89	3	d−1d−2	d−1d−2	ADP
ap-8536	89	4	·	·	PUNCT
ap-8536	89	5	·	·	PUNCT
ap-8536	89	6	·	·	PUNCT
ap-8536	90	1	ds0ω	ds0ω	NOUN
ap-8536	90	2	,	,	PUNCT
ap-8536	90	3	where	where	SCONJ
ap-8536	90	4	the	the	DET
ap-8536	90	5	zero	zero	NUM
ap-8536	90	6	index	index	NOUN
ap-8536	90	7	in	in	ADP
ap-8536	90	8	the	the	DET
ap-8536	90	9	bi	bi	ADJ
ap-8536	90	10	-	-	ADJ
ap-8536	90	11	infinite	infinite	ADJ
ap-8536	90	12	string	string	NOUN
ap-8536	90	13	is	be	AUX
ap-8536	90	14	indicated	indicate	VERB
ap-8536	90	15	by	by	ADP
ap-8536	90	16	•.	•.	NOUN
ap-8536	90	17	189	189	NUM
ap-8536	90	18	i.	i.	NOUN
ap-8536	90	19	farkas	farkas	PROPN
ap-8536	90	20	,	,	PUNCT
ap-8536	90	21	e.	e.	PROPN
ap-8536	90	22	pelantová	pelantová	PROPN
ap-8536	90	23	,	,	PUNCT
ap-8536	90	24	m.	m.	NOUN
ap-8536	90	25	svobodová	svobodová	PROPN
ap-8536	90	26	acta	acta	PROPN
ap-8536	90	27	polytechnica	polytechnica	PROPN
ap-8536	90	28	as	as	SCONJ
ap-8536	90	29	mentioned	mention	VERB
ap-8536	90	30	in	in	ADP
ap-8536	90	31	the	the	DET
ap-8536	90	32	introduction	introduction	NOUN
ap-8536	90	33	,	,	PUNCT
ap-8536	90	34	some	some	DET
ap-8536	90	35	numeration	numeration	NOUN
ap-8536	90	36	systems	system	NOUN
ap-8536	90	37	used	use	VERB
ap-8536	90	38	for	for	ADP
ap-8536	90	39	representation	representation	NOUN
ap-8536	90	40	of	of	ADP
ap-8536	90	41	numbers	number	NOUN
ap-8536	90	42	can	can	AUX
ap-8536	90	43	also	also	ADV
ap-8536	90	44	be	be	AUX
ap-8536	90	45	interpreted	interpret	VERB
ap-8536	90	46	as	as	ADP
ap-8536	90	47	matrix	matrix	NOUN
ap-8536	90	48	numeration	numeration	NOUN
ap-8536	90	49	systems	system	NOUN
ap-8536	90	50	.	.	PUNCT
ap-8536	91	1	let	let	VERB
ap-8536	91	2	us	we	PRON
ap-8536	91	3	illustrate	illustrate	VERB
ap-8536	91	4	this	this	DET
ap-8536	91	5	concept	concept	NOUN
ap-8536	91	6	on	on	ADP
ap-8536	91	7	the	the	DET
ap-8536	91	8	penney	penney	PROPN
ap-8536	91	9	numeration	numeration	NOUN
ap-8536	91	10	system	system	NOUN
ap-8536	91	11	introduced	introduce	VERB
ap-8536	91	12	in	in	ADP
ap-8536	91	13	example	example	NOUN
ap-8536	91	14	1	1	NUM
ap-8536	91	15	.	.	PUNCT
ap-8536	91	16	example	example	NOUN
ap-8536	92	1	2	2	NUM
ap-8536	92	2	.	.	PUNCT
ap-8536	92	3	let	let	VERB
ap-8536	92	4	us	we	PRON
ap-8536	92	5	transform	transform	VERB
ap-8536	92	6	the	the	DET
ap-8536	92	7	number	number	NOUN
ap-8536	92	8	numeration	numeration	NOUN
ap-8536	92	9	system	system	NOUN
ap-8536	92	10	from	from	ADP
ap-8536	92	11	example	example	NOUN
ap-8536	92	12	1	1	NUM
ap-8536	92	13	into	into	ADP
ap-8536	92	14	a	a	DET
ap-8536	92	15	matrix	matrix	NOUN
ap-8536	92	16	numeration	numeration	NOUN
ap-8536	92	17	system	system	NOUN
ap-8536	92	18	on	on	ADP
ap-8536	92	19	z2	z2	PROPN
ap-8536	92	20	.	.	PUNCT
ap-8536	93	1	in	in	ADP
ap-8536	93	2	place	place	NOUN
ap-8536	93	3	of	of	ADP
ap-8536	93	4	the	the	DET
ap-8536	93	5	number	number	NOUN
ap-8536	93	6	base	base	NOUN
ap-8536	93	7	β	β	X
ap-8536	93	8	=	=	PUNCT
ap-8536	93	9	ı	ı	PROPN
ap-8536	93	10	−	−	NOUN
ap-8536	93	11	1	1	NUM
ap-8536	93	12	,	,	PUNCT
ap-8536	93	13	we	we	PRON
ap-8536	93	14	use	use	VERB
ap-8536	93	15	the	the	DET
ap-8536	93	16	matrix	matrix	NOUN
ap-8536	93	17	base	base	NOUN
ap-8536	93	18	m	m	NOUN
ap-8536	93	19	=	=	PUNCT
ap-8536	93	20	(	(	PUNCT
ap-8536	93	21	−1	−1	NOUN
ap-8536	93	22	−1	−1	NOUN
ap-8536	93	23	+1	+1	PROPN
ap-8536	93	24	−1	−1	NOUN
ap-8536	93	25	)	)	PUNCT
ap-8536	93	26	∈	∈	PROPN
ap-8536	93	27	z2×2	z2×2	NOUN
ap-8536	93	28	.	.	PUNCT
ap-8536	94	1	it	it	PRON
ap-8536	94	2	is	be	AUX
ap-8536	94	3	easily	easily	ADV
ap-8536	94	4	seen	see	VERB
ap-8536	94	5	that	that	DET
ap-8536	94	6	multiplication	multiplication	NOUN
ap-8536	94	7	of	of	ADP
ap-8536	94	8	a	a	DET
ap-8536	94	9	gaussian	gaussian	ADJ
ap-8536	94	10	integer	integer	NOUN
ap-8536	94	11	–	–	PUNCT
ap-8536	94	12	complex	complex	ADJ
ap-8536	94	13	number	number	NOUN
ap-8536	94	14	x	x	X
ap-8536	94	15	=	=	SYM
ap-8536	94	16	b+	b+	X
ap-8536	94	17	ıc	ıc	ADP
ap-8536	94	18	∈	∈	PROPN
ap-8536	94	19	z[ı	z[ı	PROPN
ap-8536	94	20	]	]	X
ap-8536	94	21	,	,	PUNCT
ap-8536	94	22	with	with	ADP
ap-8536	94	23	b	b	NOUN
ap-8536	94	24	,	,	PUNCT
ap-8536	94	25	c	c	PROPN
ap-8536	94	26	∈	∈	PROPN
ap-8536	94	27	z	z	X
ap-8536	94	28	,	,	PUNCT
ap-8536	94	29	by	by	ADP
ap-8536	94	30	the	the	DET
ap-8536	94	31	number	number	NOUN
ap-8536	94	32	base	base	NOUN
ap-8536	94	33	β	β	X
ap-8536	94	34	corresponds	correspond	VERB
ap-8536	94	35	to	to	ADP
ap-8536	94	36	multiplication	multiplication	NOUN
ap-8536	94	37	of	of	ADP
ap-8536	94	38	an	an	DET
ap-8536	94	39	integer	integer	NOUN
ap-8536	94	40	vector	vector	NOUN
ap-8536	94	41	v	v	NOUN
ap-8536	94	42	=	=	SYM
ap-8536	94	43	(	(	PUNCT
ap-8536	94	44	b	b	NOUN
ap-8536	94	45	,	,	PUNCT
ap-8536	94	46	c)⊤	c)⊤	PROPN
ap-8536	94	47	∈	∈	PROPN
ap-8536	94	48	z2	z2	PROPN
ap-8536	94	49	by	by	ADP
ap-8536	94	50	the	the	DET
ap-8536	94	51	matrix	matrix	NOUN
ap-8536	94	52	base	base	NOUN
ap-8536	94	53	m	m	VERB
ap-8536	94	54	,	,	PUNCT
ap-8536	94	55	as	as	SCONJ
ap-8536	94	56	follows	follow	VERB
ap-8536	94	57	:	:	PUNCT
ap-8536	94	58	βx	βx	PROPN
ap-8536	94	59	=	=	PUNCT
ap-8536	94	60	(	(	PUNCT
ap-8536	94	61	ı	ı	NOUN
ap-8536	94	62	−	−	PROPN
ap-8536	94	63	1	1	NUM
ap-8536	94	64	)	)	PUNCT
ap-8536	94	65	·	·	PUNCT
ap-8536	95	1	(	(	PUNCT
ap-8536	95	2	b	b	X
ap-8536	95	3	+	+	CCONJ
ap-8536	95	4	ıc	ıc	PROPN
ap-8536	95	5	)	)	PUNCT
ap-8536	95	6	=	=	SYM
ap-8536	95	7	(	(	PUNCT
ap-8536	95	8	−b	−b	INTJ
ap-8536	95	9	−	−	NOUN
ap-8536	95	10	c	c	X
ap-8536	95	11	)	)	PUNCT
ap-8536	95	12	+	+	NUM
ap-8536	95	13	ı(b	ı(b	NOUN
ap-8536	95	14	−	−	PROPN
ap-8536	95	15	c	c	X
ap-8536	95	16	)	)	PUNCT
ap-8536	95	17	,	,	PUNCT
ap-8536	95	18	mv	mv	PROPN
ap-8536	95	19	=	=	PUNCT
ap-8536	95	20	(	(	PUNCT
ap-8536	95	21	−1	−1	NOUN
ap-8536	95	22	−1	−1	NOUN
ap-8536	95	23	+1	+1	PROPN
ap-8536	95	24	−1	−1	NOUN
ap-8536	95	25	)	)	PUNCT
ap-8536	95	26	·	·	PUNCT
ap-8536	96	1	(	(	PUNCT
ap-8536	96	2	b	b	X
ap-8536	96	3	c	c	NOUN
ap-8536	96	4	)	)	PUNCT
ap-8536	96	5	=	=	SYM
ap-8536	97	1	(	(	PUNCT
ap-8536	97	2	−b	−b	INTJ
ap-8536	97	3	−	−	PROPN
ap-8536	97	4	c	c	PROPN
ap-8536	97	5	b	b	PROPN
ap-8536	97	6	−	−	PROPN
ap-8536	97	7	c	c	NOUN
ap-8536	97	8	)	)	PUNCT
ap-8536	97	9	.	.	PUNCT
ap-8536	98	1	let	let	VERB
ap-8536	98	2	us	we	PRON
ap-8536	98	3	define	define	VERB
ap-8536	98	4	a	a	DET
ap-8536	98	5	mapping	mapping	NOUN
ap-8536	98	6	ξ	ξ	NOUN
ap-8536	98	7	:	:	PUNCT
ap-8536	98	8	z[ı	z[ı	NUM
ap-8536	98	9	]	]	X
ap-8536	98	10	7→	7→	NUM
ap-8536	98	11	z2	z2	NOUN
ap-8536	98	12	by	by	ADP
ap-8536	98	13	the	the	DET
ap-8536	98	14	formula	formula	NOUN
ap-8536	98	15	ξ(b	ξ(b	VERB
ap-8536	98	16	+	+	CCONJ
ap-8536	98	17	ıc	ıc	PROPN
ap-8536	98	18	)	)	PUNCT
ap-8536	98	19	:	:	PUNCT
ap-8536	99	1	=	=	SYM
ap-8536	99	2	(	(	PUNCT
ap-8536	99	3	b	b	NOUN
ap-8536	99	4	,	,	PUNCT
ap-8536	99	5	c)⊤	c)⊤	NOUN
ap-8536	99	6	for	for	ADP
ap-8536	99	7	any	any	DET
ap-8536	99	8	b	b	NOUN
ap-8536	99	9	,	,	PUNCT
ap-8536	99	10	c	c	PROPN
ap-8536	99	11	∈	∈	PROPN
ap-8536	99	12	z	z	NOUN
ap-8536	99	13	.	.	PUNCT
ap-8536	100	1	(	(	PUNCT
ap-8536	100	2	4	4	X
ap-8536	100	3	)	)	PUNCT
ap-8536	100	4	obviously	obviously	ADV
ap-8536	100	5	,	,	PUNCT
ap-8536	100	6	ξ(x	ξ(x	PROPN
ap-8536	100	7	+	+	CCONJ
ap-8536	100	8	y	y	NOUN
ap-8536	100	9	)	)	PUNCT
ap-8536	100	10	=	=	SYM
ap-8536	100	11	ξ(x	ξ(x	NOUN
ap-8536	100	12	)	)	PUNCT
ap-8536	101	1	+	+	CCONJ
ap-8536	101	2	ξ(y	ξ(y	PROPN
ap-8536	101	3	)	)	PUNCT
ap-8536	101	4	for	for	ADP
ap-8536	101	5	every	every	DET
ap-8536	101	6	x	x	PROPN
ap-8536	101	7	,	,	PUNCT
ap-8536	101	8	y	y	PROPN
ap-8536	101	9	∈	∈	PROPN
ap-8536	101	10	z[ı	z[ı	PROPN
ap-8536	101	11	]	]	PUNCT
ap-8536	101	12	.	.	PUNCT
ap-8536	102	1	thus	thus	ADV
ap-8536	102	2	,	,	PUNCT
ap-8536	102	3	the	the	DET
ap-8536	102	4	mapping	mapping	NOUN
ap-8536	102	5	ξ	ξ	PROPN
ap-8536	102	6	is	be	AUX
ap-8536	102	7	an	an	DET
ap-8536	102	8	isomorphism	isomorphism	NOUN
ap-8536	102	9	between	between	ADP
ap-8536	102	10	the	the	DET
ap-8536	102	11	lattices	lattice	NOUN
ap-8536	102	12	z[ı	z[ı	X
ap-8536	102	13	]	]	X
ap-8536	102	14	and	and	CCONJ
ap-8536	102	15	z2	z2	PROPN
ap-8536	102	16	.	.	PUNCT
ap-8536	103	1	moreover	moreover	ADV
ap-8536	103	2	,	,	PUNCT
ap-8536	103	3	it	it	PRON
ap-8536	103	4	fulfils	fulfil	VERB
ap-8536	103	5	the	the	DET
ap-8536	103	6	equality	equality	NOUN
ap-8536	103	7	ξ(β	ξ(β	PROPN
ap-8536	103	8	·	·	PUNCT
ap-8536	104	1	x	x	X
ap-8536	104	2	)	)	PUNCT
ap-8536	104	3	=	=	SYM
ap-8536	104	4	m	m	PROPN
ap-8536	104	5	·	·	PUNCT
ap-8536	104	6	ξ(x	ξ(x	NOUN
ap-8536	104	7	)	)	PUNCT
ap-8536	104	8	for	for	ADP
ap-8536	104	9	every	every	DET
ap-8536	104	10	x	x	PROPN
ap-8536	104	11	∈	∈	PROPN
ap-8536	104	12	z[ı	z[ı	PROPN
ap-8536	104	13	]	]	X
ap-8536	104	14	.	.	PUNCT
ap-8536	105	1	(	(	PUNCT
ap-8536	105	2	5	5	NUM
ap-8536	105	3	)	)	PUNCT
ap-8536	105	4	hence	hence	ADV
ap-8536	105	5	,	,	PUNCT
ap-8536	105	6	if	if	SCONJ
ap-8536	105	7	b	b	X
ap-8536	105	8	+	+	CCONJ
ap-8536	105	9	ıc	ıc	PROPN
ap-8536	105	10	∈	∈	PROPN
ap-8536	105	11	z[ı	z[ı	PROPN
ap-8536	105	12	]	]	X
ap-8536	105	13	is	be	AUX
ap-8536	105	14	written	write	VERB
ap-8536	105	15	in	in	ADP
ap-8536	105	16	the	the	DET
ap-8536	105	17	form	form	NOUN
ap-8536	105	18	b	b	NOUN
ap-8536	105	19	+	+	CCONJ
ap-8536	105	20	ıc	ıc	PROPN
ap-8536	105	21	=	=	ADJ
ap-8536	105	22	∑n	∑n	PROPN
ap-8536	105	23	j=0	j=0	PROPN
ap-8536	105	24	ajβj	ajβj	PROPN
ap-8536	105	25	with	with	ADP
ap-8536	105	26	aj	aj	PROPN
ap-8536	105	27	∈	∈	PROPN
ap-8536	105	28	{	{	PUNCT
ap-8536	105	29	0	0	NUM
ap-8536	105	30	,	,	PUNCT
ap-8536	105	31	1	1	NUM
ap-8536	105	32	}	}	PUNCT
ap-8536	105	33	,	,	PUNCT
ap-8536	105	34	then	then	ADV
ap-8536	105	35	(	(	PUNCT
ap-8536	105	36	b	b	NOUN
ap-8536	105	37	c	c	NOUN
ap-8536	105	38	)	)	PUNCT
ap-8536	105	39	=	=	SYM
ap-8536	106	1	ξ(b+ıc	ξ(b+ıc	NOUN
ap-8536	106	2	)	)	PUNCT
ap-8536	107	1	=	=	SYM
ap-8536	107	2	ξ	ξ	PROPN
ap-8536	107	3	(	(	PUNCT
ap-8536	107	4	n∑	n∑	NOUN
ap-8536	107	5	j=0	j=0	PROPN
ap-8536	107	6	βjaj	βjaj	PROPN
ap-8536	107	7	)	)	PUNCT
ap-8536	108	1	=	=	PUNCT
ap-8536	108	2	n∑	n∑	PROPN
ap-8536	108	3	j=0	j=0	PROPN
ap-8536	108	4	m	m	PROPN
ap-8536	108	5	jdj	jdj	PROPN
ap-8536	108	6	,	,	PUNCT
ap-8536	108	7	where	where	SCONJ
ap-8536	108	8	dj	dj	NOUN
ap-8536	108	9	=	=	SYM
ap-8536	108	10	ξ(aj	ξ(aj	NUM
ap-8536	108	11	)	)	PUNCT
ap-8536	108	12	∈	∈	NOUN
ap-8536	108	13	{	{	PUNCT
ap-8536	108	14	(	(	PUNCT
ap-8536	108	15	0	0	NUM
ap-8536	108	16	0	0	NUM
ap-8536	108	17	)	)	PUNCT
ap-8536	108	18	,	,	PUNCT
ap-8536	108	19	(	(	PUNCT
ap-8536	108	20	1	1	NUM
ap-8536	108	21	0	0	NUM
ap-8536	108	22	)	)	PUNCT
ap-8536	108	23	}	}	PUNCT
ap-8536	108	24	=	=	SYM
ap-8536	108	25	{	{	PUNCT
ap-8536	108	26	ξ(0	ξ(0	NOUN
ap-8536	108	27	)	)	PUNCT
ap-8536	108	28	,	,	PUNCT
ap-8536	108	29	ξ(1	ξ(1	PROPN
ap-8536	108	30	)	)	PUNCT
ap-8536	108	31	}	}	PUNCT
ap-8536	108	32	.	.	PUNCT
ap-8536	109	1	using	use	VERB
ap-8536	109	2	the	the	DET
ap-8536	109	3	properties	property	NOUN
ap-8536	109	4	of	of	ADP
ap-8536	109	5	the	the	DET
ap-8536	109	6	penney	penney	PROPN
ap-8536	109	7	numeration	numeration	NOUN
ap-8536	109	8	system	system	NOUN
ap-8536	109	9	from	from	ADP
ap-8536	109	10	example	example	NOUN
ap-8536	109	11	1	1	NUM
ap-8536	109	12	,	,	PUNCT
ap-8536	109	13	we	we	PRON
ap-8536	109	14	conclude	conclude	VERB
ap-8536	109	15	that	that	SCONJ
ap-8536	109	16	the	the	DET
ap-8536	109	17	matrix	matrix	NOUN
ap-8536	109	18	numeration	numeration	NOUN
ap-8536	109	19	system	system	NOUN
ap-8536	109	20	given	give	VERB
ap-8536	109	21	by	by	ADP
ap-8536	109	22	the	the	DET
ap-8536	109	23	base	base	NOUN
ap-8536	109	24	m	m	NOUN
ap-8536	109	25	=	=	PUNCT
ap-8536	109	26	(	(	PUNCT
ap-8536	109	27	−1	−1	NOUN
ap-8536	109	28	−1	−1	NOUN
ap-8536	109	29	+1	+1	ADJ
ap-8536	109	30	−1	−1	NOUN
ap-8536	109	31	)	)	PUNCT
ap-8536	109	32	and	and	CCONJ
ap-8536	109	33	the	the	DET
ap-8536	109	34	digit	digit	NOUN
ap-8536	109	35	set	set	VERB
ap-8536	109	36	d	d	NOUN
ap-8536	109	37	=	=	SYM
ap-8536	109	38	{	{	PUNCT
ap-8536	109	39	(	(	PUNCT
ap-8536	109	40	0	0	NUM
ap-8536	109	41	,	,	PUNCT
ap-8536	109	42	0)⊤	0)⊤	PROPN
ap-8536	109	43	,	,	PUNCT
ap-8536	109	44	(	(	PUNCT
ap-8536	109	45	1	1	NUM
ap-8536	109	46	,	,	PUNCT
ap-8536	109	47	0)⊤	0)⊤	PROPN
ap-8536	109	48	}	}	PUNCT
ap-8536	109	49	provides	provide	VERB
ap-8536	109	50	for	for	ADP
ap-8536	109	51	any	any	DET
ap-8536	109	52	vector	vector	NOUN
ap-8536	109	53	v	v	ADP
ap-8536	109	54	∈	∈	PROPN
ap-8536	109	55	z2	z2	NOUN
ap-8536	109	56	a	a	DET
ap-8536	109	57	unique	unique	ADJ
ap-8536	109	58	(	(	PUNCT
ap-8536	109	59	m	m	NOUN
ap-8536	109	60	,	,	PUNCT
ap-8536	109	61	d)-representation	d)-representation	NOUN
ap-8536	109	62	in	in	ADP
ap-8536	109	63	the	the	DET
ap-8536	109	64	form	form	NOUN
ap-8536	109	65	v	v	ADP
ap-8536	109	66	=	=	SYM
ap-8536	109	67	∑n	∑n	PROPN
ap-8536	109	68	j=0	j=0	PROPN
ap-8536	109	69	m	m	PROPN
ap-8536	109	70	jdj	jdj	PROPN
ap-8536	109	71	,	,	PUNCT
ap-8536	109	72	dj	dj	NOUN
ap-8536	109	73	∈	∈	PROPN
ap-8536	109	74	d	d	NOUN
ap-8536	109	75	and	and	CCONJ
ap-8536	109	76	dn	dn	ADP
ap-8536	109	77	̸=	̸=	PROPN
ap-8536	109	78	0	0	NUM
ap-8536	110	1	(	(	PUNCT
ap-8536	110	2	if	if	SCONJ
ap-8536	110	3	v	v	ADP
ap-8536	110	4	̸=	̸=	PROPN
ap-8536	110	5	0	0	NUM
ap-8536	110	6	)	)	PUNCT
ap-8536	110	7	.	.	PUNCT
ap-8536	111	1	3	3	X
ap-8536	111	2	.	.	X
ap-8536	111	3	parallel	parallel	ADJ
ap-8536	111	4	addition	addition	NOUN
ap-8536	111	5	in	in	ADP
ap-8536	111	6	matrix	matrix	NOUN
ap-8536	111	7	numeration	numeration	NOUN
ap-8536	111	8	systems	system	NOUN
ap-8536	111	9	let	let	VERB
ap-8536	111	10	us	we	PRON
ap-8536	111	11	consider	consider	VERB
ap-8536	111	12	the	the	DET
ap-8536	111	13	operations	operation	NOUN
ap-8536	111	14	of	of	ADP
ap-8536	111	15	addition	addition	NOUN
ap-8536	111	16	and	and	CCONJ
ap-8536	111	17	subtraction	subtraction	NOUN
ap-8536	111	18	on	on	ADP
ap-8536	111	19	the	the	DET
ap-8536	111	20	set	set	NOUN
ap-8536	111	21	of	of	ADP
ap-8536	111	22	m	m	ADJ
ap-8536	111	23	-	-	ADJ
ap-8536	111	24	dimensional	dimensional	ADJ
ap-8536	111	25	vectors	vector	NOUN
ap-8536	111	26	from	from	ADP
ap-8536	111	27	an	an	DET
ap-8536	111	28	algorithmic	algorithmic	ADJ
ap-8536	111	29	point	point	NOUN
ap-8536	111	30	of	of	ADP
ap-8536	111	31	view	view	NOUN
ap-8536	111	32	.	.	PUNCT
ap-8536	112	1	similarly	similarly	ADV
ap-8536	112	2	to	to	ADP
ap-8536	112	3	the	the	DET
ap-8536	112	4	classical	classical	ADJ
ap-8536	112	5	algorithms	algorithm	NOUN
ap-8536	112	6	for	for	ADP
ap-8536	112	7	arithmetic	arithmetic	ADJ
ap-8536	112	8	operations	operation	NOUN
ap-8536	112	9	,	,	PUNCT
ap-8536	112	10	we	we	PRON
ap-8536	112	11	work	work	VERB
ap-8536	112	12	only	only	ADV
ap-8536	112	13	with	with	ADP
ap-8536	112	14	finite	finite	ADJ
ap-8536	112	15	representations	representation	NOUN
ap-8536	112	16	–	–	PUNCT
ap-8536	112	17	i.e.	i.e.	X
ap-8536	112	18	,	,	PUNCT
ap-8536	112	19	on	on	ADP
ap-8536	112	20	the	the	DET
ap-8536	112	21	set	set	NOUN
ap-8536	112	22	find(m	find(m	NOUN
ap-8536	112	23	)	)	PUNCT
ap-8536	112	24	.	.	PUNCT
ap-8536	113	1	let	let	VERB
ap-8536	113	2	x	x	PRON
ap-8536	113	3	,	,	PUNCT
ap-8536	113	4	y	y	PROPN
ap-8536	113	5	∈	∈	PROPN
ap-8536	113	6	find(m	find(m	PROPN
ap-8536	113	7	)	)	PUNCT
ap-8536	113	8	,	,	PUNCT
ap-8536	113	9	with	with	ADP
ap-8536	113	10	(	(	PUNCT
ap-8536	113	11	x)m	x)m	NOUN
ap-8536	113	12	,	,	PUNCT
ap-8536	113	13	d	d	PROPN
ap-8536	113	14	=	=	SYM
ap-8536	113	15	ω0xnxn−1	ω0xnxn−1	PROPN
ap-8536	113	16	·	·	PUNCT
ap-8536	113	17	·	·	PUNCT
ap-8536	113	18	·	·	PUNCT
ap-8536	114	1	x1x0•x−1x−2	x1x0•x−1x−2	X
ap-8536	114	2	·	·	PUNCT
ap-8536	114	3	·	·	PUNCT
ap-8536	114	4	·	·	PUNCT
ap-8536	114	5	xs0ω	xs0ω	PROPN
ap-8536	114	6	and	and	CCONJ
ap-8536	114	7	(	(	PUNCT
ap-8536	114	8	y)m	y)m	NOUN
ap-8536	114	9	,	,	PUNCT
ap-8536	114	10	d	d	X
ap-8536	114	11	=	=	X
ap-8536	114	12	ω0ynyn−1	ω0ynyn−1	PROPN
ap-8536	114	13	·	·	PUNCT
ap-8536	114	14	·	·	PUNCT
ap-8536	114	15	·	·	PUNCT
ap-8536	114	16	y1y0	y1y0	X
ap-8536	114	17	•	•	NOUN
ap-8536	114	18	y−1y−2	y−1y−2	PROPN
ap-8536	114	19	·	·	PUNCT
ap-8536	114	20	·	·	PUNCT
ap-8536	114	21	·	·	PUNCT
ap-8536	114	22	ys0ω	ys0ω	X
ap-8536	114	23	.	.	PUNCT
ap-8536	115	1	adding	add	VERB
ap-8536	115	2	x	x	PUNCT
ap-8536	115	3	and	and	CCONJ
ap-8536	115	4	y	y	PROPN
ap-8536	115	5	means	mean	VERB
ap-8536	115	6	to	to	PART
ap-8536	115	7	rewrite	rewrite	VERB
ap-8536	115	8	the	the	DET
ap-8536	115	9	(	(	PUNCT
ap-8536	115	10	m	m	PROPN
ap-8536	115	11	,	,	PUNCT
ap-8536	115	12	d	d	X
ap-8536	115	13	+	+	NUM
ap-8536	115	14	d)representation	d)representation	NOUN
ap-8536	115	15	ω0(xn	ω0(xn	PUNCT
ap-8536	115	16	+	+	CCONJ
ap-8536	115	17	yn	yn	PROPN
ap-8536	115	18	)	)	PUNCT
ap-8536	115	19	·	·	PUNCT
ap-8536	115	20	·	·	PUNCT
ap-8536	115	21	·	·	PUNCT
ap-8536	116	1	(	(	PUNCT
ap-8536	116	2	x0	x0	PROPN
ap-8536	116	3	+	+	CCONJ
ap-8536	116	4	y0	y0	NOUN
ap-8536	116	5	)	)	PUNCT
ap-8536	116	6	•	•	NOUN
ap-8536	116	7	(	(	PUNCT
ap-8536	116	8	x−1	x−1	PROPN
ap-8536	116	9	+	+	PROPN
ap-8536	116	10	y−1	y−1	PROPN
ap-8536	116	11	)	)	PUNCT
ap-8536	116	12	·	·	PUNCT
ap-8536	116	13	·	·	PUNCT
ap-8536	116	14	·	·	PUNCT
ap-8536	116	15	(	(	PUNCT
ap-8536	116	16	xs	xs	PROPN
ap-8536	116	17	+	+	NUM
ap-8536	116	18	ys)0ω	ys)0ω	NOUN
ap-8536	116	19	of	of	ADP
ap-8536	116	20	the	the	DET
ap-8536	116	21	number	number	NOUN
ap-8536	116	22	x	x	PUNCT
ap-8536	117	1	+	+	CCONJ
ap-8536	117	2	y	y	PROPN
ap-8536	117	3	into	into	ADP
ap-8536	117	4	an	an	DET
ap-8536	117	5	(	(	PUNCT
ap-8536	117	6	m	m	PROPN
ap-8536	117	7	,	,	PUNCT
ap-8536	117	8	d)-representation	d)-representation	NOUN
ap-8536	117	9	of	of	ADP
ap-8536	117	10	x	x	PROPN
ap-8536	117	11	+	+	X
ap-8536	117	12	y.	y.	NOUN
ap-8536	117	13	as	as	SCONJ
ap-8536	117	14	already	already	ADV
ap-8536	117	15	announced	announce	VERB
ap-8536	117	16	,	,	PUNCT
ap-8536	117	17	we	we	PRON
ap-8536	117	18	are	be	AUX
ap-8536	117	19	interested	interested	ADJ
ap-8536	117	20	in	in	ADP
ap-8536	117	21	parallel	parallel	ADJ
ap-8536	117	22	algorithms	algorithm	NOUN
ap-8536	117	23	for	for	ADP
ap-8536	117	24	addition	addition	NOUN
ap-8536	117	25	.	.	PUNCT
ap-8536	118	1	let	let	VERB
ap-8536	118	2	us	we	PRON
ap-8536	118	3	mathematically	mathematically	ADV
ap-8536	118	4	formalise	formalise	VERB
ap-8536	118	5	the	the	DET
ap-8536	118	6	parallelism	parallelism	NOUN
ap-8536	118	7	.	.	PUNCT
ap-8536	119	1	firstly	firstly	ADV
ap-8536	119	2	,	,	PUNCT
ap-8536	119	3	we	we	PRON
ap-8536	119	4	recall	recall	VERB
ap-8536	119	5	the	the	DET
ap-8536	119	6	notion	notion	NOUN
ap-8536	119	7	of	of	ADP
ap-8536	119	8	a	a	DET
ap-8536	119	9	local	local	ADJ
ap-8536	119	10	function	function	NOUN
ap-8536	119	11	,	,	PUNCT
ap-8536	119	12	which	which	PRON
ap-8536	119	13	comes	come	VERB
ap-8536	119	14	from	from	ADP
ap-8536	119	15	symbolic	symbolic	ADJ
ap-8536	119	16	dynamics	dynamic	NOUN
ap-8536	119	17	,	,	PUNCT
ap-8536	119	18	see	see	VERB
ap-8536	119	19	[	[	X
ap-8536	119	20	16	16	NUM
ap-8536	119	21	]	]	PUNCT
ap-8536	119	22	.	.	PUNCT
ap-8536	120	1	definition	definition	NOUN
ap-8536	120	2	3	3	X
ap-8536	120	3	.	.	PUNCT
ap-8536	121	1	let	let	VERB
ap-8536	121	2	a	a	PRON
ap-8536	121	3	and	and	CCONJ
ap-8536	121	4	b	b	NOUN
ap-8536	121	5	be	be	AUX
ap-8536	121	6	finite	finite	ADJ
ap-8536	121	7	sets	set	NOUN
ap-8536	121	8	.	.	PUNCT
ap-8536	122	1	a	a	DET
ap-8536	122	2	function	function	NOUN
ap-8536	122	3	φ	φ	NOUN
ap-8536	122	4	:	:	PUNCT
ap-8536	122	5	az	az	PROPN
ap-8536	122	6	→	→	SYM
ap-8536	122	7	bz	bz	PROPN
ap-8536	122	8	is	be	AUX
ap-8536	122	9	said	say	VERB
ap-8536	122	10	to	to	PART
ap-8536	122	11	be	be	AUX
ap-8536	122	12	p	p	NOUN
ap-8536	122	13	-	-	ADJ
ap-8536	122	14	local	local	ADJ
ap-8536	122	15	if	if	SCONJ
ap-8536	122	16	there	there	PRON
ap-8536	122	17	exist	exist	VERB
ap-8536	122	18	nonnegative	nonnegative	ADJ
ap-8536	122	19	integers	integer	NOUN
ap-8536	122	20	r	r	NOUN
ap-8536	122	21	and	and	CCONJ
ap-8536	122	22	t	t	NOUN
ap-8536	122	23	satisfying	satisfy	VERB
ap-8536	122	24	p	p	X
ap-8536	122	25	=	=	PUNCT
ap-8536	122	26	r	r	NOUN
ap-8536	123	1	+	+	NOUN
ap-8536	123	2	t+1	t+1	PROPN
ap-8536	123	3	,	,	PUNCT
ap-8536	123	4	and	and	CCONJ
ap-8536	123	5	a	a	DET
ap-8536	123	6	function	function	NOUN
ap-8536	123	7	φ	φ	NOUN
ap-8536	123	8	:	:	PUNCT
ap-8536	124	1	ap	ap	PROPN
ap-8536	124	2	→	→	SYM
ap-8536	124	3	b	b	X
ap-8536	125	1	such	such	ADJ
ap-8536	125	2	that	that	PRON
ap-8536	125	3	,	,	PUNCT
ap-8536	125	4	for	for	ADP
ap-8536	125	5	any	any	DET
ap-8536	125	6	u	u	NOUN
ap-8536	125	7	=	=	PUNCT
ap-8536	125	8	(	(	PUNCT
ap-8536	125	9	uj)j∈z	uj)j∈z	PROPN
ap-8536	125	10	∈	∈	PROPN
ap-8536	125	11	az	az	PROPN
ap-8536	125	12	and	and	CCONJ
ap-8536	125	13	its	its	PRON
ap-8536	125	14	image	image	NOUN
ap-8536	125	15	v	v	NOUN
ap-8536	125	16	=	=	PUNCT
ap-8536	125	17	φ(u	φ(u	PROPN
ap-8536	125	18	)	)	PUNCT
ap-8536	125	19	=	=	SYM
ap-8536	125	20	(	(	PUNCT
ap-8536	125	21	vj)j∈z	vj)j∈z	NOUN
ap-8536	125	22	∈	∈	PROPN
ap-8536	125	23	bz	bz	PROPN
ap-8536	125	24	,	,	PUNCT
ap-8536	125	25	we	we	PRON
ap-8536	125	26	have	have	VERB
ap-8536	125	27	vj	vj	PROPN
ap-8536	125	28	=	=	PROPN
ap-8536	125	29	φ(uj+t	φ(uj+t	PROPN
ap-8536	125	30	·	·	PUNCT
ap-8536	125	31	·	·	PUNCT
ap-8536	125	32	·	·	PUNCT
ap-8536	125	33	uj−r	uj−r	NOUN
ap-8536	125	34	)	)	PUNCT
ap-8536	125	35	for	for	ADP
ap-8536	125	36	every	every	DET
ap-8536	125	37	j	j	PROPN
ap-8536	125	38	∈	∈	PROPN
ap-8536	125	39	z.	z.	PROPN
ap-8536	126	1	this	this	PRON
ap-8536	126	2	means	mean	VERB
ap-8536	126	3	that	that	SCONJ
ap-8536	126	4	the	the	DET
ap-8536	126	5	image	image	NOUN
ap-8536	126	6	of	of	ADP
ap-8536	126	7	u	u	NOUN
ap-8536	126	8	by	by	ADP
ap-8536	126	9	φ	φ	PROPN
ap-8536	126	10	is	be	AUX
ap-8536	126	11	obtained	obtain	VERB
ap-8536	126	12	through	through	ADP
ap-8536	126	13	a	a	DET
ap-8536	126	14	window	window	NOUN
ap-8536	126	15	of	of	ADP
ap-8536	126	16	limited	limited	ADJ
ap-8536	126	17	length	length	NOUN
ap-8536	126	18	p.	p.	NOUN
ap-8536	127	1	the	the	DET
ap-8536	127	2	parameter	parameter	NOUN
ap-8536	127	3	r	r	NOUN
ap-8536	127	4	is	be	AUX
ap-8536	127	5	called	call	VERB
ap-8536	127	6	memory	memory	NOUN
ap-8536	127	7	and	and	CCONJ
ap-8536	127	8	the	the	DET
ap-8536	127	9	parameter	parameter	NOUN
ap-8536	127	10	t	t	PROPN
ap-8536	127	11	is	be	AUX
ap-8536	127	12	called	call	VERB
ap-8536	127	13	anticipation	anticipation	NOUN
ap-8536	127	14	of	of	ADP
ap-8536	127	15	the	the	DET
ap-8536	127	16	function	function	NOUN
ap-8536	127	17	φ	φ	PROPN
ap-8536	127	18	.	.	PUNCT
ap-8536	128	1	such	such	ADJ
ap-8536	128	2	functions	function	NOUN
ap-8536	128	3	,	,	PUNCT
ap-8536	128	4	restricted	restrict	VERB
ap-8536	128	5	to	to	PART
ap-8536	128	6	finite	finite	VERB
ap-8536	128	7	sequences	sequence	NOUN
ap-8536	128	8	,	,	PUNCT
ap-8536	128	9	are	be	AUX
ap-8536	128	10	computable	computable	ADJ
ap-8536	128	11	by	by	ADP
ap-8536	128	12	a	a	DET
ap-8536	128	13	parallel	parallel	ADJ
ap-8536	128	14	algorithm	algorithm	NOUN
ap-8536	128	15	in	in	ADP
ap-8536	128	16	constant	constant	ADJ
ap-8536	128	17	time	time	NOUN
ap-8536	128	18	,	,	PUNCT
ap-8536	128	19	irrespective	irrespective	ADV
ap-8536	128	20	of	of	ADP
ap-8536	128	21	the	the	DET
ap-8536	128	22	length	length	NOUN
ap-8536	128	23	of	of	ADP
ap-8536	128	24	the	the	DET
ap-8536	128	25	operands	operand	NOUN
ap-8536	128	26	’	'	PUNCT
ap-8536	128	27	representations	representation	NOUN
ap-8536	128	28	.	.	PUNCT
ap-8536	129	1	definition	definition	NOUN
ap-8536	129	2	4	4	NUM
ap-8536	129	3	.	.	PUNCT
ap-8536	129	4	given	give	VERB
ap-8536	129	5	a	a	DET
ap-8536	129	6	(	(	PUNCT
ap-8536	129	7	matrix	matrix	NOUN
ap-8536	129	8	)	)	PUNCT
ap-8536	129	9	base	base	NOUN
ap-8536	129	10	m	m	PROPN
ap-8536	129	11	∈	∈	PROPN
ap-8536	129	12	zm×m	zm×m	NOUN
ap-8536	129	13	with	with	ADP
ap-8536	129	14	det	det	PROPN
ap-8536	129	15	m	m	PROPN
ap-8536	129	16	̸=	̸=	PROPN
ap-8536	129	17	0	0	NUM
ap-8536	129	18	and	and	CCONJ
ap-8536	129	19	(	(	PUNCT
ap-8536	129	20	vector	vector	NOUN
ap-8536	129	21	)	)	PUNCT
ap-8536	129	22	digit	digit	NOUN
ap-8536	129	23	sets	set	VERB
ap-8536	129	24	a	a	DET
ap-8536	129	25	,	,	PUNCT
ap-8536	129	26	b	b	PROPN
ap-8536	129	27	⊂	⊂	PROPN
ap-8536	129	28	zm	zm	PROPN
ap-8536	129	29	containing	contain	VERB
ap-8536	129	30	0	0	NUM
ap-8536	129	31	,	,	PUNCT
ap-8536	129	32	a	a	DET
ap-8536	129	33	digit	digit	NOUN
ap-8536	129	34	set	set	VERB
ap-8536	129	35	conversion	conversion	NOUN
ap-8536	129	36	in	in	ADP
ap-8536	129	37	base	base	NOUN
ap-8536	129	38	m	m	PROPN
ap-8536	129	39	from	from	ADP
ap-8536	129	40	a	a	DET
ap-8536	129	41	to	to	PART
ap-8536	129	42	b	b	NOUN
ap-8536	129	43	is	be	AUX
ap-8536	129	44	a	a	DET
ap-8536	129	45	function	function	NOUN
ap-8536	129	46	φ	φ	NOUN
ap-8536	129	47	:	:	PUNCT
ap-8536	129	48	az	az	PROPN
ap-8536	129	49	→	→	PUNCT
ap-8536	129	50	bz	bz	PROPN
ap-8536	129	51	such	such	ADJ
ap-8536	129	52	that	that	SCONJ
ap-8536	129	53	(	(	PUNCT
ap-8536	129	54	1	1	NUM
ap-8536	129	55	.	.	PUNCT
ap-8536	129	56	)	)	PUNCT
ap-8536	130	1	for	for	ADP
ap-8536	130	2	any	any	DET
ap-8536	130	3	u	u	NOUN
ap-8536	130	4	=	=	PUNCT
ap-8536	130	5	(	(	PUNCT
ap-8536	130	6	uj)j∈z	uj)j∈z	PROPN
ap-8536	130	7	∈	∈	PROPN
ap-8536	130	8	az	az	PROPN
ap-8536	130	9	with	with	ADP
ap-8536	130	10	a	a	DET
ap-8536	130	11	finite	finite	ADJ
ap-8536	130	12	number	number	NOUN
ap-8536	130	13	of	of	ADP
ap-8536	130	14	non	non	ADJ
ap-8536	130	15	-	-	ADJ
ap-8536	130	16	zero	zero	NUM
ap-8536	130	17	digits	digit	NOUN
ap-8536	130	18	,	,	PUNCT
ap-8536	130	19	v	v	NOUN
ap-8536	130	20	=	=	SYM
ap-8536	130	21	(	(	PUNCT
ap-8536	130	22	vj)j∈z	vj)j∈z	NOUN
ap-8536	130	23	=	=	SYM
ap-8536	130	24	φ(u	φ(u	NOUN
ap-8536	130	25	)	)	PUNCT
ap-8536	130	26	∈	∈	PROPN
ap-8536	130	27	bz	bz	PROPN
ap-8536	130	28	has	have	VERB
ap-8536	130	29	only	only	ADV
ap-8536	130	30	a	a	DET
ap-8536	130	31	finite	finite	ADJ
ap-8536	130	32	number	number	NOUN
ap-8536	130	33	of	of	ADP
ap-8536	130	34	non	non	ADJ
ap-8536	130	35	-	-	ADJ
ap-8536	130	36	zero	zero	NUM
ap-8536	130	37	digits	digit	NOUN
ap-8536	130	38	,	,	PUNCT
ap-8536	130	39	and	and	CCONJ
ap-8536	130	40	(	(	PUNCT
ap-8536	130	41	2	2	NUM
ap-8536	130	42	.	.	PUNCT
ap-8536	130	43	)	)	PUNCT
ap-8536	130	44	∑	∑	PUNCT
ap-8536	131	1	j∈z	j∈z	PROPN
ap-8536	131	2	m	m	PROPN
ap-8536	131	3	jvj	jvj	PROPN
ap-8536	132	1	=	=	PUNCT
ap-8536	132	2	∑	∑	PUNCT
ap-8536	132	3	j∈z	j∈z	PROPN
ap-8536	132	4	m	m	PROPN
ap-8536	132	5	juj	juj	ADJ
ap-8536	132	6	.	.	PUNCT
ap-8536	133	1	such	such	DET
ap-8536	133	2	a	a	DET
ap-8536	133	3	conversion	conversion	NOUN
ap-8536	133	4	is	be	AUX
ap-8536	133	5	said	say	VERB
ap-8536	133	6	to	to	PART
ap-8536	133	7	be	be	AUX
ap-8536	133	8	computable	computable	ADJ
ap-8536	133	9	in	in	ADP
ap-8536	133	10	parallel	parallel	NOUN
ap-8536	133	11	if	if	SCONJ
ap-8536	133	12	it	it	PRON
ap-8536	133	13	is	be	AUX
ap-8536	133	14	a	a	DET
ap-8536	133	15	p	p	ADJ
ap-8536	133	16	-	-	PUNCT
ap-8536	133	17	local	local	ADJ
ap-8536	133	18	function	function	NOUN
ap-8536	133	19	for	for	ADP
ap-8536	133	20	some	some	DET
ap-8536	133	21	p	p	PROPN
ap-8536	133	22	∈	∈	PROPN
ap-8536	133	23	n.	n.	NOUN
ap-8536	133	24	thus	thus	ADV
ap-8536	133	25	,	,	PUNCT
ap-8536	133	26	the	the	DET
ap-8536	133	27	operation	operation	NOUN
ap-8536	133	28	of	of	ADP
ap-8536	133	29	addition	addition	NOUN
ap-8536	133	30	on	on	ADP
ap-8536	133	31	find(m	find(m	NOUN
ap-8536	133	32	)	)	PUNCT
ap-8536	133	33	is	be	AUX
ap-8536	133	34	computable	computable	ADJ
ap-8536	133	35	in	in	ADP
ap-8536	133	36	parallel	parallel	NOUN
ap-8536	133	37	if	if	SCONJ
ap-8536	133	38	there	there	PRON
ap-8536	133	39	exists	exist	VERB
ap-8536	133	40	a	a	DET
ap-8536	133	41	digit	digit	NOUN
ap-8536	133	42	set	set	VERB
ap-8536	133	43	conversion	conversion	NOUN
ap-8536	133	44	in	in	ADP
ap-8536	133	45	base	base	NOUN
ap-8536	133	46	m	m	VERB
ap-8536	133	47	from	from	ADP
ap-8536	133	48	d	d	PROPN
ap-8536	133	49	+	+	CCONJ
ap-8536	133	50	d	d	NOUN
ap-8536	133	51	to	to	ADP
ap-8536	133	52	d	d	PRON
ap-8536	133	53	which	which	PRON
ap-8536	133	54	is	be	AUX
ap-8536	133	55	computable	computable	ADJ
ap-8536	133	56	in	in	ADP
ap-8536	133	57	parallel	parallel	NOUN
ap-8536	133	58	.	.	PUNCT
ap-8536	134	1	two	two	NUM
ap-8536	134	2	useful	useful	ADJ
ap-8536	134	3	lemmas	lemma	NOUN
ap-8536	134	4	precede	precede	VERB
ap-8536	134	5	the	the	DET
ap-8536	134	6	statement	statement	NOUN
ap-8536	134	7	about	about	ADP
ap-8536	134	8	the	the	DET
ap-8536	134	9	parallel	parallel	ADJ
ap-8536	134	10	addition	addition	NOUN
ap-8536	134	11	on	on	ADP
ap-8536	134	12	matrix	matrix	NOUN
ap-8536	134	13	numeration	numeration	NOUN
ap-8536	134	14	systems	system	NOUN
ap-8536	134	15	:	:	PUNCT
ap-8536	134	16	lemma	lemma	PROPN
ap-8536	134	17	5	5	X
ap-8536	134	18	.	.	PUNCT
ap-8536	135	1	let	let	VERB
ap-8536	135	2	m	m	PRON
ap-8536	135	3	∈	∈	PROPN
ap-8536	135	4	zm×m	zm×m	PROPN
ap-8536	135	5	be	be	AUX
ap-8536	135	6	a	a	DET
ap-8536	135	7	non	non	ADJ
ap-8536	135	8	-	-	ADJ
ap-8536	135	9	singular	singular	ADJ
ap-8536	135	10	matrix	matrix	NOUN
ap-8536	135	11	and	and	CCONJ
ap-8536	135	12	d	d	PROPN
ap-8536	135	13	⊂	⊂	PROPN
ap-8536	135	14	zm	zm	PROPN
ap-8536	135	15	be	be	AUX
ap-8536	135	16	a	a	DET
ap-8536	135	17	finite	finite	ADJ
ap-8536	135	18	digit	digit	NOUN
ap-8536	135	19	set	set	VERB
ap-8536	135	20	such	such	ADJ
ap-8536	135	21	that	that	SCONJ
ap-8536	135	22	every	every	DET
ap-8536	135	23	x	x	PROPN
ap-8536	135	24	∈	∈	PROPN
ap-8536	135	25	zm	zm	PROPN
ap-8536	135	26	is	be	AUX
ap-8536	135	27	representable	representable	ADJ
ap-8536	135	28	in	in	ADP
ap-8536	135	29	the	the	DET
ap-8536	135	30	numeration	numeration	NOUN
ap-8536	135	31	system	system	NOUN
ap-8536	135	32	(	(	PUNCT
ap-8536	135	33	m	m	PROPN
ap-8536	135	34	,	,	PUNCT
ap-8536	135	35	d	d	NOUN
ap-8536	135	36	)	)	PUNCT
ap-8536	135	37	.	.	PUNCT
ap-8536	136	1	if	if	SCONJ
ap-8536	136	2	addition	addition	NOUN
ap-8536	136	3	is	be	AUX
ap-8536	136	4	computable	computable	ADJ
ap-8536	136	5	in	in	ADP
ap-8536	136	6	parallel	parallel	NOUN
ap-8536	136	7	in	in	ADP
ap-8536	136	8	(	(	PUNCT
ap-8536	136	9	m	m	PROPN
ap-8536	136	10	,	,	PUNCT
ap-8536	136	11	d	d	PROPN
ap-8536	136	12	)	)	PUNCT
ap-8536	136	13	,	,	PUNCT
ap-8536	136	14	then	then	ADV
ap-8536	136	15	it	it	PRON
ap-8536	136	16	is	be	AUX
ap-8536	136	17	computable	computable	ADJ
ap-8536	136	18	in	in	ADP
ap-8536	136	19	parallel	parallel	NOUN
ap-8536	136	20	also	also	ADV
ap-8536	136	21	in	in	ADP
ap-8536	136	22	(	(	PUNCT
ap-8536	136	23	m	m	PROPN
ap-8536	136	24	,	,	PUNCT
ap-8536	136	25	d′	d′	NUM
ap-8536	136	26	)	)	PUNCT
ap-8536	136	27	for	for	SCONJ
ap-8536	136	28	each	each	DET
ap-8536	136	29	finite	finite	ADJ
ap-8536	136	30	digit	digit	NOUN
ap-8536	136	31	set	set	VERB
ap-8536	136	32	d′	d′	PROPN
ap-8536	136	33	⊂	⊂	PROPN
ap-8536	136	34	zm	zm	PROPN
ap-8536	136	35	containing	contain	VERB
ap-8536	136	36	d.	d.	PROPN
ap-8536	136	37	proof	proof	NOUN
ap-8536	136	38	.	.	PUNCT
ap-8536	137	1	each	each	PRON
ap-8536	137	2	digit	digit	VERB
ap-8536	137	3	d′	d′	NUM
ap-8536	137	4	∈	∈	PROPN
ap-8536	137	5	d′	d′	PRON
ap-8536	137	6	can	can	AUX
ap-8536	137	7	be	be	AUX
ap-8536	137	8	written	write	VERB
ap-8536	137	9	in	in	ADP
ap-8536	137	10	the	the	DET
ap-8536	137	11	form	form	NOUN
ap-8536	137	12	d′	d′	X
ap-8536	137	13	=	=	SYM
ap-8536	137	14	∑	∑	PUNCT
ap-8536	137	15	j∈i(d′	j∈i(d′	PROPN
ap-8536	137	16	)	)	PUNCT
ap-8536	138	1	m	m	PROPN
ap-8536	138	2	jdj	jdj	PROPN
ap-8536	138	3	,	,	PUNCT
ap-8536	138	4	where	where	SCONJ
ap-8536	138	5	i(d′	i(d′	PROPN
ap-8536	138	6	)	)	PUNCT
ap-8536	138	7	is	be	AUX
ap-8536	138	8	a	a	DET
ap-8536	138	9	finite	finite	NOUN
ap-8536	138	10	subset	subset	NOUN
ap-8536	138	11	of	of	ADP
ap-8536	138	12	z	z	NOUN
ap-8536	138	13	and	and	CCONJ
ap-8536	138	14	dj	dj	X
ap-8536	138	15	∈	∈	PROPN
ap-8536	138	16	d	d	NOUN
ap-8536	138	17	for	for	ADP
ap-8536	138	18	each	each	DET
ap-8536	138	19	j	j	PROPN
ap-8536	138	20	∈	∈	PROPN
ap-8536	138	21	i(d′	i(d′	PROPN
ap-8536	138	22	)	)	PUNCT
ap-8536	138	23	.	.	PUNCT
ap-8536	139	1	denote	denote	VERB
ap-8536	139	2	q	q	NOUN
ap-8536	140	1	=	=	SYM
ap-8536	140	2	max{|x|	max{|x|	X
ap-8536	140	3	:	:	PUNCT
ap-8536	140	4	d′	d′	X
ap-8536	140	5	∈	∈	PROPN
ap-8536	140	6	d′	d′	PROPN
ap-8536	140	7	,	,	PUNCT
ap-8536	140	8	x	x	PROPN
ap-8536	140	9	∈	∈	PROPN
ap-8536	140	10	i(d′	i(d′	PROPN
ap-8536	140	11	)	)	PUNCT
ap-8536	140	12	}	}	PUNCT
ap-8536	140	13	.	.	PUNCT
ap-8536	141	1	190	190	NUM
ap-8536	141	2	vol	vol	NOUN
ap-8536	141	3	.	.	PUNCT
ap-8536	141	4	63	63	NUM
ap-8536	141	5	no	no	NOUN
ap-8536	141	6	.	.	PUNCT
ap-8536	142	1	3/2023	3/2023	NUM
ap-8536	142	2	positional	positional	ADJ
ap-8536	142	3	representation	representation	NOUN
ap-8536	142	4	of	of	ADP
ap-8536	142	5	vectors	vector	NOUN
ap-8536	142	6	the	the	DET
ap-8536	142	7	string	string	NOUN
ap-8536	142	8	0ωd′	0ωd′	NOUN
ap-8536	142	9	nd′	nd′	X
ap-8536	142	10	n−1	n−1	PROPN
ap-8536	142	11	·	·	PUNCT
ap-8536	142	12	·	·	PUNCT
ap-8536	142	13	·	·	PUNCT
ap-8536	142	14	d′	d′	NOUN
ap-8536	142	15	0	0	NUM
ap-8536	142	16	•	•	NOUN
ap-8536	142	17	d′	d′	NUM
ap-8536	142	18	−1d′	−1d′	PROPN
ap-8536	142	19	−2	−2	X
ap-8536	142	20	·	·	PUNCT
ap-8536	142	21	·	·	PUNCT
ap-8536	142	22	·	·	PUNCT
ap-8536	142	23	d′	d′	NUM
ap-8536	142	24	−n	−n	ADJ
ap-8536	142	25	0ω	0ω	NOUN
ap-8536	142	26	can	can	AUX
ap-8536	142	27	be	be	AUX
ap-8536	142	28	transformed	transform	VERB
ap-8536	142	29	by	by	ADP
ap-8536	142	30	a	a	DET
ap-8536	142	31	(	(	PUNCT
ap-8536	142	32	2q	2q	NOUN
ap-8536	142	33	+	+	SYM
ap-8536	142	34	1)-local	1)-local	NUM
ap-8536	142	35	function	function	NOUN
ap-8536	142	36	into	into	ADP
ap-8536	142	37	a	a	DET
ap-8536	142	38	string	string	NOUN
ap-8536	142	39	having	have	VERB
ap-8536	142	40	the	the	DET
ap-8536	142	41	form	form	NOUN
ap-8536	142	42	0ωen+qen+q−1	0ωen+qen+q−1	PROPN
ap-8536	142	43	·	·	PUNCT
ap-8536	142	44	·	·	PUNCT
ap-8536	142	45	·	·	PUNCT
ap-8536	143	1	e0	e0	PROPN
ap-8536	143	2	•	•	ADV
ap-8536	143	3	e−1e−2	e−1e−2	PROPN
ap-8536	143	4	·	·	PUNCT
ap-8536	143	5	·	·	PUNCT
ap-8536	143	6	·	·	PUNCT
ap-8536	143	7	e−n−q0ω	e−n−q0ω	NOUN
ap-8536	143	8	,	,	PUNCT
ap-8536	143	9	where	where	SCONJ
ap-8536	143	10	ed	ed	NOUN
ap-8536	143	11	∈	∈	PROPN
ap-8536	144	1	d	d	PROPN
ap-8536	144	2	+	+	CCONJ
ap-8536	144	3	d	d	PROPN
ap-8536	144	4	+	+	CCONJ
ap-8536	144	5	·	·	PUNCT
ap-8536	144	6	·	·	PUNCT
ap-8536	144	7	·	·	PUNCT
ap-8536	145	1	+	+	CCONJ
ap-8536	145	2	d︸	d︸	PROPN
ap-8536	145	3	︷︷	︷︷	PROPN
ap-8536	145	4	︸	︸	X
ap-8536	145	5	(	(	PUNCT
ap-8536	145	6	2q+1)−times	2q+1)−times	NUM
ap-8536	145	7	.	.	PUNCT
ap-8536	146	1	in	in	ADP
ap-8536	146	2	other	other	ADJ
ap-8536	146	3	words	word	NOUN
ap-8536	146	4	,	,	PUNCT
ap-8536	146	5	a	a	DET
ap-8536	146	6	sum	sum	NOUN
ap-8536	146	7	of	of	ADP
ap-8536	146	8	two	two	NUM
ap-8536	146	9	finite	finite	NOUN
ap-8536	146	10	(	(	PUNCT
ap-8536	146	11	m	m	PROPN
ap-8536	146	12	,	,	PUNCT
ap-8536	146	13	d′)representations	d′)representation	NOUN
ap-8536	146	14	can	can	AUX
ap-8536	146	15	be	be	AUX
ap-8536	146	16	rewritten	rewrite	VERB
ap-8536	146	17	as	as	ADP
ap-8536	146	18	sum	sum	NOUN
ap-8536	146	19	of	of	ADP
ap-8536	146	20	2(2q	2(2q	NUM
ap-8536	146	21	+	+	CCONJ
ap-8536	146	22	1	1	X
ap-8536	146	23	)	)	PUNCT
ap-8536	146	24	finite	finite	NOUN
ap-8536	146	25	(	(	PUNCT
ap-8536	146	26	m	m	NOUN
ap-8536	146	27	,	,	PUNCT
ap-8536	146	28	d)-representations	d)-representation	NOUN
ap-8536	146	29	.	.	PUNCT
ap-8536	147	1	since	since	SCONJ
ap-8536	147	2	addition	addition	NOUN
ap-8536	147	3	of	of	ADP
ap-8536	147	4	two	two	NUM
ap-8536	147	5	strings	string	NOUN
ap-8536	147	6	is	be	AUX
ap-8536	147	7	doable	doable	ADJ
ap-8536	147	8	in	in	ADP
ap-8536	147	9	parallel	parallel	NOUN
ap-8536	147	10	in	in	ADP
ap-8536	147	11	(	(	PUNCT
ap-8536	147	12	m	m	PROPN
ap-8536	147	13	,	,	PUNCT
ap-8536	147	14	d	d	NOUN
ap-8536	147	15	)	)	PUNCT
ap-8536	147	16	,	,	PUNCT
ap-8536	147	17	addition	addition	NOUN
ap-8536	147	18	of	of	ADP
ap-8536	147	19	2(2q	2(2q	NUM
ap-8536	147	20	+	+	CCONJ
ap-8536	147	21	1	1	NUM
ap-8536	147	22	)	)	PUNCT
ap-8536	147	23	strings	string	NOUN
ap-8536	147	24	with	with	ADP
ap-8536	147	25	fixed	fix	VERB
ap-8536	147	26	q	q	NOUN
ap-8536	147	27	is	be	AUX
ap-8536	147	28	possible	possible	ADJ
ap-8536	147	29	in	in	ADP
ap-8536	147	30	parallel	parallel	ADJ
ap-8536	147	31	(	(	PUNCT
ap-8536	147	32	m	m	PROPN
ap-8536	147	33	,	,	PUNCT
ap-8536	147	34	d	d	NOUN
ap-8536	147	35	)	)	PUNCT
ap-8536	147	36	as	as	ADV
ap-8536	147	37	well	well	ADV
ap-8536	147	38	,	,	PUNCT
ap-8536	147	39	and	and	CCONJ
ap-8536	147	40	the	the	DET
ap-8536	147	41	resulting	result	VERB
ap-8536	147	42	(	(	PUNCT
ap-8536	147	43	m	m	PROPN
ap-8536	147	44	,	,	PUNCT
ap-8536	147	45	d)representation	d)representation	NOUN
ap-8536	147	46	is	be	AUX
ap-8536	147	47	also	also	ADV
ap-8536	147	48	an	an	DET
ap-8536	147	49	(	(	PUNCT
ap-8536	147	50	m	m	NOUN
ap-8536	147	51	,	,	PUNCT
ap-8536	147	52	d′)-representation	d′)-representation	NOUN
ap-8536	147	53	,	,	PUNCT
ap-8536	147	54	due	due	ADP
ap-8536	147	55	to	to	ADP
ap-8536	147	56	d	d	PROPN
ap-8536	147	57	⊂	⊂	PROPN
ap-8536	147	58	d′.	d′.	VERB
ap-8536	147	59	the	the	DET
ap-8536	147	60	following	follow	VERB
ap-8536	147	61	lemma	lemma	PROPN
ap-8536	147	62	is	be	AUX
ap-8536	147	63	stated	state	VERB
ap-8536	147	64	without	without	ADP
ap-8536	147	65	a	a	DET
ap-8536	147	66	proof	proof	NOUN
ap-8536	147	67	here	here	ADV
ap-8536	147	68	,	,	PUNCT
ap-8536	147	69	as	as	SCONJ
ap-8536	147	70	the	the	DET
ap-8536	147	71	course	course	NOUN
ap-8536	147	72	of	of	ADP
ap-8536	147	73	the	the	DET
ap-8536	147	74	proof	proof	NOUN
ap-8536	147	75	would	would	AUX
ap-8536	147	76	be	be	AUX
ap-8536	147	77	identical	identical	ADJ
ap-8536	147	78	to	to	ADP
ap-8536	147	79	that	that	PRON
ap-8536	147	80	of	of	ADP
ap-8536	147	81	proposition	proposition	NOUN
ap-8536	147	82	5.1	5.1	NUM
ap-8536	147	83	in	in	ADP
ap-8536	147	84	[	[	X
ap-8536	147	85	17	17	NUM
ap-8536	147	86	]	]	PUNCT
ap-8536	147	87	.	.	PUNCT
ap-8536	148	1	although	although	SCONJ
ap-8536	148	2	that	that	DET
ap-8536	148	3	proposition	proposition	NOUN
ap-8536	148	4	works	work	VERB
ap-8536	148	5	with	with	ADP
ap-8536	148	6	roots	root	NOUN
ap-8536	148	7	of	of	ADP
ap-8536	148	8	the	the	DET
ap-8536	148	9	minimal	minimal	ADJ
ap-8536	148	10	polynomial	polynomial	NOUN
ap-8536	148	11	of	of	ADP
ap-8536	148	12	an	an	DET
ap-8536	148	13	algebraic	algebraic	ADJ
ap-8536	148	14	number	number	NOUN
ap-8536	148	15	,	,	PUNCT
ap-8536	148	16	the	the	DET
ap-8536	148	17	minimality	minimality	NOUN
ap-8536	148	18	of	of	ADP
ap-8536	148	19	the	the	DET
ap-8536	148	20	polynomial	polynomial	NOUN
ap-8536	148	21	is	be	AUX
ap-8536	148	22	not	not	PART
ap-8536	148	23	used	use	VERB
ap-8536	148	24	for	for	ADP
ap-8536	148	25	the	the	DET
ap-8536	148	26	proof	proof	NOUN
ap-8536	148	27	itself	itself	PRON
ap-8536	148	28	.	.	PUNCT
ap-8536	149	1	in	in	ADP
ap-8536	149	2	fact	fact	NOUN
ap-8536	149	3	,	,	PUNCT
ap-8536	149	4	the	the	DET
ap-8536	149	5	idea	idea	NOUN
ap-8536	149	6	of	of	ADP
ap-8536	149	7	the	the	DET
ap-8536	149	8	proof	proof	NOUN
ap-8536	149	9	comes	come	VERB
ap-8536	149	10	from	from	ADP
ap-8536	149	11	[	[	X
ap-8536	149	12	18	18	NUM
ap-8536	149	13	]	]	PUNCT
ap-8536	149	14	,	,	PUNCT
ap-8536	149	15	where	where	SCONJ
ap-8536	149	16	expansive	expansive	ADJ
ap-8536	149	17	polynomials	polynomial	NOUN
ap-8536	149	18	are	be	AUX
ap-8536	149	19	considered	consider	VERB
ap-8536	149	20	.	.	PUNCT
ap-8536	150	1	in	in	ADP
ap-8536	150	2	our	our	PRON
ap-8536	150	3	case	case	NOUN
ap-8536	150	4	,	,	PUNCT
ap-8536	150	5	we	we	PRON
ap-8536	150	6	extend	extend	VERB
ap-8536	150	7	the	the	DET
ap-8536	150	8	considerations	consideration	NOUN
ap-8536	150	9	to	to	PART
ap-8536	150	10	polynomials	polynomial	VERB
ap-8536	150	11	with	with	ADP
ap-8536	150	12	no	no	DET
ap-8536	150	13	roots	root	NOUN
ap-8536	150	14	on	on	ADP
ap-8536	150	15	the	the	DET
ap-8536	150	16	unit	unit	NOUN
ap-8536	150	17	circle	circle	NOUN
ap-8536	150	18	.	.	PUNCT
ap-8536	151	1	lemma	lemma	PROPN
ap-8536	151	2	6	6	NUM
ap-8536	151	3	.	.	PUNCT
ap-8536	152	1	let	let	VERB
ap-8536	152	2	α1	α1	PROPN
ap-8536	152	3	,	,	PUNCT
ap-8536	152	4	.	.	PUNCT
ap-8536	152	5	.	.	PUNCT
ap-8536	153	1	.	.	PUNCT
ap-8536	154	1	,	,	PUNCT
ap-8536	154	2	αn	αn	X
ap-8536	154	3	∈	∈	PROPN
ap-8536	154	4	c	c	AUX
ap-8536	154	5	be	be	AUX
ap-8536	154	6	the	the	DET
ap-8536	154	7	roots	root	NOUN
ap-8536	154	8	of	of	ADP
ap-8536	154	9	a	a	DET
ap-8536	154	10	polynomial	polynomial	ADJ
ap-8536	154	11	f	f	PROPN
ap-8536	154	12	∈	∈	PROPN
ap-8536	154	13	z[x	z[x	NOUN
ap-8536	154	14	]	]	PUNCT
ap-8536	154	15	satisfying	satisfy	VERB
ap-8536	154	16	|αk|	|αk|	PROPN
ap-8536	154	17	̸=	̸=	PROPN
ap-8536	154	18	1	1	NUM
ap-8536	154	19	for	for	ADP
ap-8536	154	20	all	all	PRON
ap-8536	154	21	k	k	NOUN
ap-8536	154	22	=	=	SYM
ap-8536	154	23	1	1	NUM
ap-8536	154	24	,	,	PUNCT
ap-8536	154	25	.	.	PUNCT
ap-8536	154	26	.	.	PUNCT
ap-8536	155	1	.	.	PUNCT
ap-8536	156	1	,	,	PUNCT
ap-8536	156	2	n.	n.	PROPN
ap-8536	156	3	then	then	ADV
ap-8536	156	4	,	,	PUNCT
ap-8536	156	5	for	for	ADP
ap-8536	156	6	any	any	DET
ap-8536	156	7	t	t	PROPN
ap-8536	156	8	≥	≥	NOUN
ap-8536	156	9	1	1	NUM
ap-8536	156	10	,	,	PUNCT
ap-8536	156	11	there	there	PRON
ap-8536	156	12	exists	exist	VERB
ap-8536	156	13	a	a	DET
ap-8536	156	14	non	non	ADJ
ap-8536	156	15	-	-	ADJ
ap-8536	156	16	zero	zero	ADJ
ap-8536	156	17	polynomial	polynomial	ADJ
ap-8536	156	18	g	g	PROPN
ap-8536	156	19	∈	∈	PROPN
ap-8536	156	20	z[x	z[x	NOUN
ap-8536	156	21	]	]	X
ap-8536	156	22	,	,	PUNCT
ap-8536	156	23	g(x	g(x	NOUN
ap-8536	156	24	)	)	PUNCT
ap-8536	157	1	=	=	PRON
ap-8536	157	2	∑p−1	∑p−1	NOUN
ap-8536	158	1	l=0	l=0	PROPN
ap-8536	158	2	clx	clx	PROPN
ap-8536	158	3	l	l	NOUN
ap-8536	158	4	such	such	ADJ
ap-8536	158	5	that	that	SCONJ
ap-8536	158	6	g	g	PROPN
ap-8536	158	7	is	be	AUX
ap-8536	158	8	divisible	divisible	ADJ
ap-8536	158	9	by	by	ADP
ap-8536	158	10	f	f	PROPN
ap-8536	158	11	and	and	CCONJ
ap-8536	158	12	for	for	ADP
ap-8536	158	13	one	one	NUM
ap-8536	158	14	coefficient	coefficient	NOUN
ap-8536	158	15	cl	cl	NOUN
ap-8536	158	16	we	we	PRON
ap-8536	158	17	have	have	VERB
ap-8536	158	18	1	1	NUM
ap-8536	158	19	t	t	NOUN
ap-8536	158	20	cl	cl	NOUN
ap-8536	158	21	>	>	X
ap-8536	158	22	p−1∑	p−1∑	PROPN
ap-8536	158	23	l=0,l	l=0,l	ADP
ap-8536	158	24	̸=l	̸=l	PROPN
ap-8536	158	25	|cl|	|cl|	PROPN
ap-8536	158	26	.	.	PUNCT
ap-8536	159	1	(	(	PUNCT
ap-8536	159	2	6	6	NUM
ap-8536	159	3	)	)	PUNCT
ap-8536	159	4	in	in	ADP
ap-8536	159	5	particular	particular	ADJ
ap-8536	159	6	,	,	PUNCT
ap-8536	159	7	if	if	SCONJ
ap-8536	159	8	|αk|	|αk|	PROPN
ap-8536	159	9	>	>	X
ap-8536	159	10	1	1	NUM
ap-8536	159	11	for	for	ADP
ap-8536	159	12	all	all	PRON
ap-8536	159	13	k	k	NOUN
ap-8536	159	14	=	=	SYM
ap-8536	159	15	1	1	NUM
ap-8536	159	16	,	,	PUNCT
ap-8536	159	17	.	.	PUNCT
ap-8536	159	18	.	.	PUNCT
ap-8536	160	1	.	.	PUNCT
ap-8536	161	1	,	,	PUNCT
ap-8536	162	1	n	n	CCONJ
ap-8536	162	2	,	,	PUNCT
ap-8536	162	3	then	then	ADV
ap-8536	162	4	l	l	NOUN
ap-8536	162	5	=	=	SYM
ap-8536	162	6	0	0	X
ap-8536	162	7	.	.	PUNCT
ap-8536	162	8	corollary	corollary	ADJ
ap-8536	162	9	7	7	NUM
ap-8536	162	10	.	.	PUNCT
ap-8536	163	1	let	let	VERB
ap-8536	163	2	m	m	PRON
ap-8536	163	3	∈	∈	PROPN
ap-8536	163	4	zm×m	zm×m	NOUN
ap-8536	163	5	,	,	PUNCT
ap-8536	163	6	with	with	ADP
ap-8536	163	7	det	det	PROPN
ap-8536	163	8	m	m	PROPN
ap-8536	163	9	̸=	̸=	PROPN
ap-8536	163	10	0	0	NUM
ap-8536	163	11	and	and	CCONJ
ap-8536	163	12	no	no	DET
ap-8536	163	13	eigenvalue	eigenvalue	NOUN
ap-8536	163	14	of	of	ADP
ap-8536	163	15	m	m	PROPN
ap-8536	163	16	equals	equal	VERB
ap-8536	163	17	1	1	NUM
ap-8536	163	18	in	in	ADP
ap-8536	163	19	modulus	modulus	NOUN
ap-8536	163	20	.	.	PUNCT
ap-8536	164	1	then	then	ADV
ap-8536	164	2	,	,	PUNCT
ap-8536	164	3	there	there	PRON
ap-8536	164	4	exists	exist	VERB
ap-8536	164	5	a	a	DET
ap-8536	164	6	polynomial	polynomial	ADJ
ap-8536	164	7	g	g	PROPN
ap-8536	164	8	∈	∈	PROPN
ap-8536	164	9	z[x	z[x	NOUN
ap-8536	164	10	]	]	X
ap-8536	164	11	,	,	PUNCT
ap-8536	164	12	g(x	g(x	NOUN
ap-8536	164	13	)	)	PUNCT
ap-8536	165	1	=	=	PRON
ap-8536	165	2	∑p−1	∑p−1	NOUN
ap-8536	166	1	l=0	l=0	PROPN
ap-8536	166	2	clx	clx	PROPN
ap-8536	166	3	l	l	NOUN
ap-8536	166	4	such	such	ADJ
ap-8536	166	5	that	that	DET
ap-8536	166	6	g(m	g(m	VERB
ap-8536	166	7	)	)	PUNCT
ap-8536	166	8	=	=	SYM
ap-8536	166	9	θ	θ	PROPN
ap-8536	166	10	and	and	CCONJ
ap-8536	166	11	for	for	ADP
ap-8536	166	12	one	one	NUM
ap-8536	166	13	coefficient	coefficient	NOUN
ap-8536	166	14	cl	cl	NOUN
ap-8536	166	15	we	we	PRON
ap-8536	166	16	have	have	VERB
ap-8536	166	17	cl	cl	NOUN
ap-8536	166	18	≥	≥	NUM
ap-8536	166	19	4	4	NUM
ap-8536	166	20	·	·	PUNCT
ap-8536	166	21	(	(	PUNCT
ap-8536	166	22	c	c	X
ap-8536	166	23	+	+	CCONJ
ap-8536	166	24	p−1∑	p−1∑	NOUN
ap-8536	166	25	l=0,l	l=0,l	ADP
ap-8536	166	26	̸=l	̸=l	PROPN
ap-8536	166	27	|cl|	|cl|	PROPN
ap-8536	166	28	)	)	PUNCT
ap-8536	166	29	,	,	PUNCT
ap-8536	166	30	(	(	PUNCT
ap-8536	166	31	7	7	X
ap-8536	166	32	)	)	PUNCT
ap-8536	166	33	where	where	SCONJ
ap-8536	166	34	c	c	NOUN
ap-8536	166	35	=	=	SYM
ap-8536	166	36	cl	cl	NOUN
ap-8536	166	37	−	−	NOUN
ap-8536	166	38	4	4	NUM
ap-8536	166	39	⌊	⌊	NOUN
ap-8536	166	40	cl	cl	NOUN
ap-8536	166	41	4	4	NUM
ap-8536	166	42	⌋	⌋	NOUN
ap-8536	166	43	∈	∈	PROPN
ap-8536	166	44	{	{	PUNCT
ap-8536	166	45	0	0	NUM
ap-8536	166	46	,	,	PUNCT
ap-8536	166	47	1	1	NUM
ap-8536	166	48	,	,	PUNCT
ap-8536	166	49	2	2	NUM
ap-8536	166	50	,	,	PUNCT
ap-8536	166	51	3	3	NUM
ap-8536	166	52	}	}	PUNCT
ap-8536	166	53	.	.	PUNCT
ap-8536	167	1	proof	proof	NOUN
ap-8536	167	2	.	.	PUNCT
ap-8536	168	1	let	let	VERB
ap-8536	168	2	f	f	PRON
ap-8536	168	3	be	be	AUX
ap-8536	168	4	the	the	DET
ap-8536	168	5	characteristic	characteristic	ADJ
ap-8536	168	6	polynomial	polynomial	NOUN
ap-8536	168	7	of	of	ADP
ap-8536	168	8	the	the	DET
ap-8536	168	9	matrix	matrix	NOUN
ap-8536	168	10	m	m	NOUN
ap-8536	168	11	.	.	PUNCT
ap-8536	169	1	the	the	DET
ap-8536	169	2	hamilton	hamilton	PROPN
ap-8536	169	3	–	–	PUNCT
ap-8536	169	4	cayley	cayley	PROPN
ap-8536	169	5	theorem	theorem	NOUN
ap-8536	169	6	says	say	VERB
ap-8536	169	7	that	that	SCONJ
ap-8536	169	8	f(m	f(m	PROPN
ap-8536	169	9	)	)	PUNCT
ap-8536	169	10	=	=	SYM
ap-8536	169	11	θ	θ	X
ap-8536	169	12	.	.	PUNCT
ap-8536	169	13	by	by	ADP
ap-8536	169	14	lemma	lemma	PROPN
ap-8536	169	15	6	6	NUM
ap-8536	169	16	applied	apply	VERB
ap-8536	169	17	on	on	ADP
ap-8536	169	18	f	f	PROPN
ap-8536	169	19	with	with	ADP
ap-8536	169	20	t	t	PROPN
ap-8536	169	21	=	=	SYM
ap-8536	169	22	16	16	NUM
ap-8536	169	23	,	,	PUNCT
ap-8536	169	24	we	we	PRON
ap-8536	169	25	find	find	VERB
ap-8536	169	26	a	a	DET
ap-8536	169	27	polynomial	polynomial	ADJ
ap-8536	169	28	g(x	g(x	NOUN
ap-8536	169	29	)	)	PUNCT
ap-8536	170	1	=	=	PRON
ap-8536	170	2	∑p−1	∑p−1	NOUN
ap-8536	171	1	l=0	l=0	PROPN
ap-8536	171	2	clx	clx	PROPN
ap-8536	171	3	l	l	PROPN
ap-8536	171	4	∈	∈	PROPN
ap-8536	171	5	z[x	z[x	NOUN
ap-8536	171	6	]	]	PUNCT
ap-8536	171	7	such	such	ADJ
ap-8536	171	8	that	that	DET
ap-8536	171	9	cl	cl	NOUN
ap-8536	171	10	>	>	X
ap-8536	171	11	16	16	NUM
ap-8536	171	12	·	·	PUNCT
ap-8536	171	13	p−1∑	p−1∑	NOUN
ap-8536	171	14	l=0,l	l=0,l	ADP
ap-8536	171	15	̸=l	̸=l	PROPN
ap-8536	171	16	|cl|	|cl|	PROPN
ap-8536	171	17	,	,	PUNCT
ap-8536	171	18	for	for	ADP
ap-8536	171	19	some	some	DET
ap-8536	171	20	l	l	NOUN
ap-8536	171	21	∈	∈	PROPN
ap-8536	171	22	{	{	PUNCT
ap-8536	171	23	0	0	NUM
ap-8536	171	24	,	,	PUNCT
ap-8536	171	25	.	.	PUNCT
ap-8536	171	26	.	.	PUNCT
ap-8536	171	27	.	.	PUNCT
ap-8536	172	1	,	,	PUNCT
ap-8536	172	2	p	p	NOUN
ap-8536	172	3	−	−	PROPN
ap-8536	172	4	1	1	NUM
ap-8536	172	5	}	}	PUNCT
ap-8536	172	6	.	.	PUNCT
ap-8536	173	1	using	use	VERB
ap-8536	173	2	the	the	DET
ap-8536	173	3	fact	fact	NOUN
ap-8536	173	4	that	that	SCONJ
ap-8536	173	5	∑p−1	∑p−1	VERB
ap-8536	174	1	l=0,l	l=0,l	ADV
ap-8536	174	2	̸=l	̸=l	PROPN
ap-8536	174	3	|cl|	|cl|	PROPN
ap-8536	174	4	≥	≥	NOUN
ap-8536	174	5	1	1	NUM
ap-8536	174	6	and	and	CCONJ
ap-8536	174	7	denoting	denote	VERB
ap-8536	174	8	c	c	NOUN
ap-8536	174	9	:	:	PUNCT
ap-8536	174	10	=	=	NOUN
ap-8536	174	11	cl	cl	NOUN
ap-8536	174	12	−4	−4	X
ap-8536	174	13	⌊	⌊	NOUN
ap-8536	174	14	cl	cl	NOUN
ap-8536	174	15	4	4	NUM
ap-8536	174	16	⌋	⌋	NOUN
ap-8536	174	17	∈	∈	PROPN
ap-8536	174	18	{	{	PUNCT
ap-8536	174	19	0	0	NUM
ap-8536	174	20	,	,	PUNCT
ap-8536	174	21	1	1	NUM
ap-8536	174	22	,	,	PUNCT
ap-8536	174	23	2	2	NUM
ap-8536	174	24	,	,	PUNCT
ap-8536	174	25	3	3	NUM
ap-8536	174	26	}	}	PUNCT
ap-8536	174	27	,	,	PUNCT
ap-8536	174	28	we	we	PRON
ap-8536	174	29	obtain	obtain	VERB
ap-8536	174	30	the	the	DET
ap-8536	174	31	following	follow	VERB
ap-8536	174	32	estimate	estimate	NOUN
ap-8536	174	33	:	:	PUNCT
ap-8536	174	34	cl	cl	NOUN
ap-8536	174	35	>	>	X
ap-8536	174	36	16	16	NUM
ap-8536	174	37	p−1∑	p−1∑	NOUN
ap-8536	174	38	l=0,l	l=0,l	ADP
ap-8536	174	39	̸=l	̸=l	PROPN
ap-8536	174	40	|cl|	|cl|	NOUN
ap-8536	174	41	=	=	SYM
ap-8536	174	42	4	4	NUM
ap-8536	174	43	(	(	PUNCT
ap-8536	174	44	3	3	NUM
ap-8536	174	45	p−1∑	p−1∑	NOUN
ap-8536	174	46	l=0,l	l=0,l	ADP
ap-8536	174	47	̸=l	̸=l	PROPN
ap-8536	174	48	|cl|	|cl|	NOUN
ap-8536	174	49	+	+	CCONJ
ap-8536	174	50	p−1∑	p−1∑	PROPN
ap-8536	174	51	l=0,l	l=0,l	ADP
ap-8536	174	52	̸=l	̸=l	PROPN
ap-8536	174	53	|cl|	|cl|	PROPN
ap-8536	174	54	)	)	PUNCT
ap-8536	174	55	≥	≥	NOUN
ap-8536	174	56	≥	≥	NUM
ap-8536	174	57	4	4	NUM
ap-8536	174	58	(	(	PUNCT
ap-8536	174	59	3	3	NUM
ap-8536	174	60	+	+	NUM
ap-8536	174	61	p−1∑	p−1∑	NOUN
ap-8536	174	62	l=0,l	l=0,l	ADP
ap-8536	174	63	̸=l	̸=l	PROPN
ap-8536	174	64	|cl|	|cl|	PROPN
ap-8536	174	65	)	)	PUNCT
ap-8536	174	66	≥	≥	NOUN
ap-8536	174	67	4	4	NUM
ap-8536	174	68	(	(	PUNCT
ap-8536	174	69	c	c	NOUN
ap-8536	174	70	+	+	CCONJ
ap-8536	174	71	p−1∑	p−1∑	NOUN
ap-8536	174	72	l=0,j	l=0,j	PROPN
ap-8536	174	73	̸=l	̸=l	PROPN
ap-8536	174	74	|cl|	|cl|	PROPN
ap-8536	174	75	)	)	PUNCT
ap-8536	174	76	.	.	PUNCT
ap-8536	175	1	since	since	SCONJ
ap-8536	175	2	the	the	DET
ap-8536	175	3	characteristic	characteristic	ADJ
ap-8536	175	4	polynomial	polynomial	NOUN
ap-8536	175	5	f	f	PROPN
ap-8536	175	6	divides	divide	VERB
ap-8536	175	7	g	g	NOUN
ap-8536	175	8	,	,	PUNCT
ap-8536	175	9	we	we	PRON
ap-8536	175	10	have	have	AUX
ap-8536	175	11	g(m	g(m	VERB
ap-8536	175	12	)	)	PUNCT
ap-8536	175	13	=	=	SYM
ap-8536	175	14	θ	θ	PROPN
ap-8536	175	15	.	.	PROPN
ap-8536	175	16	example	example	NOUN
ap-8536	175	17	8	8	NUM
ap-8536	175	18	.	.	PUNCT
ap-8536	176	1	the	the	DET
ap-8536	176	2	minimal	minimal	ADJ
ap-8536	176	3	polynomial	polynomial	NOUN
ap-8536	176	4	of	of	ADP
ap-8536	176	5	the	the	DET
ap-8536	176	6	complex	complex	ADJ
ap-8536	176	7	number	number	NOUN
ap-8536	176	8	β	β	X
ap-8536	176	9	=	=	PUNCT
ap-8536	176	10	ı	ı	NOUN
ap-8536	176	11	−	−	NOUN
ap-8536	176	12	1	1	NUM
ap-8536	176	13	and	and	CCONJ
ap-8536	176	14	the	the	DET
ap-8536	176	15	characteristic	characteristic	ADJ
ap-8536	176	16	polynomial	polynomial	NOUN
ap-8536	176	17	of	of	ADP
ap-8536	176	18	the	the	DET
ap-8536	176	19	matrix	matrix	NOUN
ap-8536	176	20	m	m	AUX
ap-8536	176	21	defined	define	VERB
ap-8536	176	22	in	in	ADP
ap-8536	176	23	example	example	NOUN
ap-8536	176	24	2	2	NUM
ap-8536	176	25	are	be	AUX
ap-8536	176	26	both	both	ADV
ap-8536	176	27	equal	equal	ADJ
ap-8536	176	28	to	to	ADP
ap-8536	176	29	f(x	f(x	PROPN
ap-8536	176	30	)	)	PUNCT
ap-8536	177	1	=	=	SYM
ap-8536	178	1	x2	x2	PROPN
ap-8536	179	1	+	+	NUM
ap-8536	179	2	2x	2x	NUM
ap-8536	179	3	+	+	X
ap-8536	179	4	2	2	X
ap-8536	179	5	.	.	PUNCT
ap-8536	179	6	the	the	DET
ap-8536	179	7	polynomial	polynomial	ADJ
ap-8536	179	8	g(x	g(x	NOUN
ap-8536	179	9	)	)	PUNCT
ap-8536	180	1	=	=	PUNCT
ap-8536	180	2	x4	x4	PROPN
ap-8536	181	1	+	+	CCONJ
ap-8536	182	1	4	4	NUM
ap-8536	182	2	=	=	SYM
ap-8536	182	3	(	(	PUNCT
ap-8536	182	4	x2	x2	NOUN
ap-8536	182	5	+	+	PUNCT
ap-8536	182	6	2x	2x	NUM
ap-8536	182	7	+	+	CCONJ
ap-8536	182	8	2)(x2	2)(x2	NOUN
ap-8536	182	9	−	−	NOUN
ap-8536	182	10	2x	2x	NUM
ap-8536	182	11	+	+	CCONJ
ap-8536	182	12	2	2	X
ap-8536	182	13	)	)	PUNCT
ap-8536	182	14	satisfies	satisfie	NOUN
ap-8536	182	15	g(m	g(m	NOUN
ap-8536	182	16	)	)	PUNCT
ap-8536	182	17	=	=	SYM
ap-8536	182	18	θ	θ	PROPN
ap-8536	182	19	and	and	CCONJ
ap-8536	182	20	(	(	PUNCT
ap-8536	182	21	7	7	NUM
ap-8536	182	22	)	)	PUNCT
ap-8536	182	23	with	with	ADP
ap-8536	182	24	l	l	NOUN
ap-8536	182	25	=	=	SYM
ap-8536	182	26	0	0	X
ap-8536	182	27	.	.	PUNCT
ap-8536	182	28	theorem	theorem	NOUN
ap-8536	182	29	9	9	NUM
ap-8536	182	30	.	.	PUNCT
ap-8536	183	1	let	let	VERB
ap-8536	183	2	m	m	PRON
ap-8536	183	3	∈	∈	PROPN
ap-8536	183	4	zm×m	zm×m	NOUN
ap-8536	183	5	,	,	PUNCT
ap-8536	183	6	with	with	ADP
ap-8536	183	7	det	det	PROPN
ap-8536	183	8	m	m	PROPN
ap-8536	183	9	=	=	NOUN
ap-8536	183	10	̸	̸	NUM
ap-8536	183	11	0	0	PUNCT
ap-8536	184	1	and	and	CCONJ
ap-8536	184	2	no	no	DET
ap-8536	184	3	eigenvalue	eigenvalue	NOUN
ap-8536	184	4	of	of	ADP
ap-8536	184	5	m	m	PROPN
ap-8536	184	6	equals	equal	VERB
ap-8536	184	7	1	1	NUM
ap-8536	184	8	in	in	ADP
ap-8536	184	9	modulus	modulus	NOUN
ap-8536	184	10	.	.	PUNCT
ap-8536	185	1	then	then	ADV
ap-8536	185	2	there	there	PRON
ap-8536	185	3	exists	exist	VERB
ap-8536	185	4	a	a	DET
ap-8536	185	5	finite	finite	NOUN
ap-8536	185	6	(	(	PUNCT
ap-8536	185	7	vector	vector	NOUN
ap-8536	185	8	)	)	PUNCT
ap-8536	185	9	digit	digit	NOUN
ap-8536	185	10	set	set	VERB
ap-8536	185	11	d	d	PROPN
ap-8536	185	12	⊂	⊂	PROPN
ap-8536	185	13	zm	zm	PROPN
ap-8536	185	14	such	such	ADJ
ap-8536	185	15	that	that	SCONJ
ap-8536	185	16	zm	zm	PROPN
ap-8536	185	17	⊂	⊂	PROPN
ap-8536	185	18	find(m	find(m	PROPN
ap-8536	185	19	)	)	PUNCT
ap-8536	185	20	and	and	CCONJ
ap-8536	185	21	both	both	DET
ap-8536	185	22	addition	addition	NOUN
ap-8536	185	23	and	and	CCONJ
ap-8536	185	24	subtraction	subtraction	NOUN
ap-8536	185	25	on	on	ADP
ap-8536	185	26	find(m	find(m	NOUN
ap-8536	185	27	)	)	PUNCT
ap-8536	185	28	are	be	AUX
ap-8536	185	29	computable	computable	ADJ
ap-8536	185	30	in	in	ADP
ap-8536	185	31	parallel	parallel	NOUN
ap-8536	185	32	.	.	PUNCT
ap-8536	186	1	proof	proof	NOUN
ap-8536	186	2	.	.	PUNCT
ap-8536	187	1	let	let	VERB
ap-8536	187	2	g(x	g(x	NOUN
ap-8536	187	3	)	)	PUNCT
ap-8536	188	1	=	=	PRON
ap-8536	188	2	∑p−1	∑p−1	NOUN
ap-8536	189	1	l=0	l=0	PROPN
ap-8536	189	2	clx	clx	PROPN
ap-8536	189	3	l	l	NOUN
ap-8536	189	4	be	be	VERB
ap-8536	189	5	the	the	DET
ap-8536	189	6	polynomial	polynomial	NOUN
ap-8536	189	7	from	from	ADP
ap-8536	189	8	corollary	corollary	ADJ
ap-8536	189	9	7	7	NUM
ap-8536	189	10	.	.	PUNCT
ap-8536	189	11	denote	denote	NOUN
ap-8536	189	12	k	k	NOUN
ap-8536	190	1	=	=	PUNCT
ap-8536	190	2	⌊	⌊	PART
ap-8536	190	3	cl	cl	NOUN
ap-8536	190	4	4	4	NUM
ap-8536	190	5	⌋	⌋	NOUN
ap-8536	190	6	≥	≥	NOUN
ap-8536	190	7	1	1	NUM
ap-8536	190	8	and	and	CCONJ
ap-8536	190	9	define	define	VERB
ap-8536	190	10	d	d	NOUN
ap-8536	190	11	=	=	PUNCT
ap-8536	191	1	[	[	X
ap-8536	191	2	−3k	−3k	PROPN
ap-8536	191	3	,	,	PUNCT
ap-8536	191	4	3k)m	3k)m	NUM
ap-8536	191	5	∩	∩	PROPN
ap-8536	191	6	zm	zm	PROPN
ap-8536	191	7	.	.	PUNCT
ap-8536	192	1	in	in	ADP
ap-8536	192	2	order	order	NOUN
ap-8536	192	3	to	to	PART
ap-8536	192	4	show	show	VERB
ap-8536	192	5	that	that	SCONJ
ap-8536	192	6	the	the	DET
ap-8536	192	7	digit	digit	NOUN
ap-8536	192	8	set	set	VERB
ap-8536	192	9	d	d	ADP
ap-8536	192	10	enables	enable	VERB
ap-8536	192	11	parallel	parallel	ADJ
ap-8536	192	12	addition	addition	NOUN
ap-8536	192	13	,	,	PUNCT
ap-8536	192	14	we	we	PRON
ap-8536	192	15	introduce	introduce	VERB
ap-8536	192	16	two	two	NUM
ap-8536	192	17	auxiliary	auxiliary	ADJ
ap-8536	192	18	sets	set	NOUN
ap-8536	192	19	d′	d′	X
ap-8536	192	20	=	=	PUNCT
ap-8536	193	1	[	[	X
ap-8536	193	2	−2k	−2k	PROPN
ap-8536	193	3	,	,	PUNCT
ap-8536	193	4	2k)m	2k)m	NUM
ap-8536	193	5	∩	∩	NOUN
ap-8536	193	6	zm	zm	PROPN
ap-8536	193	7	and	and	CCONJ
ap-8536	193	8	q	q	NOUN
ap-8536	193	9	=	=	PUNCT
ap-8536	194	1	[	[	X
ap-8536	194	2	−1	−1	NOUN
ap-8536	194	3	,	,	PUNCT
ap-8536	194	4	1]m	1]m	NUM
ap-8536	194	5	∩	∩	X
ap-8536	194	6	zm	zm	NOUN
ap-8536	194	7	,	,	PUNCT
ap-8536	194	8	and	and	CCONJ
ap-8536	194	9	then	then	ADV
ap-8536	194	10	exploit	exploit	VERB
ap-8536	194	11	the	the	DET
ap-8536	194	12	obvious	obvious	ADJ
ap-8536	194	13	fact	fact	NOUN
ap-8536	194	14	that	that	SCONJ
ap-8536	194	15	d	d	X
ap-8536	194	16	+	+	X
ap-8536	195	1	d	d	X
ap-8536	195	2	⊂	⊂	X
ap-8536	195	3	d′	d′	X
ap-8536	196	1	+	+	X
ap-8536	197	1	4kq	4kq	ADJ
ap-8536	197	2	.	.	PUNCT
ap-8536	198	1	(	(	PUNCT
ap-8536	198	2	8)	8)	NUM
ap-8536	198	3	let	let	VERB
ap-8536	198	4	x	x	PUNCT
ap-8536	198	5	=	=	PUNCT
ap-8536	198	6	∑	∑	PUNCT
ap-8536	198	7	j∈z	j∈z	PROPN
ap-8536	198	8	m	m	PROPN
ap-8536	198	9	jaj	jaj	PROPN
ap-8536	198	10	and	and	CCONJ
ap-8536	198	11	y	y	PROPN
ap-8536	198	12	=	=	PUNCT
ap-8536	198	13	∑	∑	PUNCT
ap-8536	198	14	j∈z	j∈z	PROPN
ap-8536	198	15	m	m	PROPN
ap-8536	198	16	jbj	jbj	PROPN
ap-8536	198	17	,	,	PUNCT
ap-8536	198	18	where	where	SCONJ
ap-8536	198	19	aj	aj	PROPN
ap-8536	198	20	,	,	PUNCT
ap-8536	198	21	bj	bj	ADP
ap-8536	198	22	∈	∈	PROPN
ap-8536	198	23	d.	d.	NOUN
ap-8536	198	24	moreover	moreover	ADV
ap-8536	198	25	,	,	PUNCT
ap-8536	198	26	we	we	PRON
ap-8536	198	27	assume	assume	VERB
ap-8536	198	28	that	that	SCONJ
ap-8536	198	29	aj	aj	PROPN
ap-8536	198	30	,	,	PUNCT
ap-8536	198	31	bj	bj	ADP
ap-8536	198	32	̸=	̸=	PROPN
ap-8536	198	33	0	0	NUM
ap-8536	199	1	for	for	ADP
ap-8536	199	2	just	just	ADV
ap-8536	199	3	a	a	DET
ap-8536	199	4	finite	finite	ADJ
ap-8536	199	5	number	number	NOUN
ap-8536	199	6	of	of	ADP
ap-8536	199	7	indices	index	NOUN
ap-8536	199	8	j	j	PROPN
ap-8536	199	9	∈	∈	PROPN
ap-8536	199	10	z.	z.	PROPN
ap-8536	199	11	clearly	clearly	ADV
ap-8536	199	12	,	,	PUNCT
ap-8536	199	13	aj	aj	PROPN
ap-8536	199	14	+	+	X
ap-8536	199	15	bj	bj	VERB
ap-8536	199	16	∈	∈	PROPN
ap-8536	199	17	d	d	PROPN
ap-8536	199	18	+	+	CCONJ
ap-8536	199	19	d.	d.	NOUN
ap-8536	199	20	due	due	ADP
ap-8536	199	21	to	to	ADP
ap-8536	199	22	(	(	PUNCT
ap-8536	199	23	8)	8)	NUM
ap-8536	199	24	,	,	PUNCT
ap-8536	199	25	we	we	PRON
ap-8536	199	26	find	find	VERB
ap-8536	199	27	for	for	ADP
ap-8536	199	28	each	each	DET
ap-8536	199	29	j	j	PROPN
ap-8536	199	30	a	a	DET
ap-8536	199	31	vector	vector	NOUN
ap-8536	199	32	qj	qj	PROPN
ap-8536	199	33	∈	∈	PROPN
ap-8536	199	34	q	q	NOUN
ap-8536	199	35	such	such	ADJ
ap-8536	199	36	that	that	SCONJ
ap-8536	199	37	aj	aj	PROPN
ap-8536	199	38	+	+	X
ap-8536	199	39	bj	bj	VERB
ap-8536	199	40	−	−	PROPN
ap-8536	199	41	4kqj	4kqj	PROPN
ap-8536	199	42	∈	∈	PROPN
ap-8536	199	43	d′.	d′.	VERB
ap-8536	199	44	then	then	ADV
ap-8536	199	45	x	x	PUNCT
ap-8536	200	1	+	+	CCONJ
ap-8536	200	2	y	y	PROPN
ap-8536	200	3	=	=	PUNCT
ap-8536	200	4	∑	∑	PUNCT
ap-8536	200	5	j∈z	j∈z	NOUN
ap-8536	200	6	m	m	PROPN
ap-8536	200	7	j(aj	j(aj	PROPN
ap-8536	200	8	+	+	CCONJ
ap-8536	200	9	bj)=	bj)=	X
ap-8536	200	10	=	=	PUNCT
ap-8536	200	11	∑	∑	PUNCT
ap-8536	200	12	j∈z	j∈z	NOUN
ap-8536	200	13	(	(	PUNCT
ap-8536	200	14	m	m	VERB
ap-8536	200	15	j(aj	j(aj	PROPN
ap-8536	200	16	+	+	CCONJ
ap-8536	200	17	bj	bj	VERB
ap-8536	200	18	)	)	PUNCT
ap-8536	200	19	−	−	PROPN
ap-8536	200	20	m	m	VERB
ap-8536	200	21	j−lg(m)︸	j−lg(m)︸	AUX
ap-8536	200	22	︷︷	︷︷	VERB
ap-8536	200	23	︸	︸	PUNCT
ap-8536	201	1	=	=	NOUN
ap-8536	201	2	θ	θ	PROPN
ap-8536	201	3	qj	qj	PROPN
ap-8536	201	4	)	)	PUNCT
ap-8536	201	5	.	.	PUNCT
ap-8536	202	1	we	we	PRON
ap-8536	202	2	express	express	VERB
ap-8536	202	3	g(m	g(m	NOUN
ap-8536	202	4	)	)	PUNCT
ap-8536	202	5	in	in	ADP
ap-8536	202	6	the	the	DET
ap-8536	202	7	explicit	explicit	ADJ
ap-8536	202	8	polynomial	polynomial	ADJ
ap-8536	202	9	form	form	NOUN
ap-8536	202	10	g(m	g(m	VERB
ap-8536	202	11	)	)	PUNCT
ap-8536	202	12	=	=	PUNCT
ap-8536	202	13	∑p−1	∑p−1	X
ap-8536	203	1	l=0	l=0	PROPN
ap-8536	203	2	clm	clm	X
ap-8536	203	3	l	l	NOUN
ap-8536	203	4	in	in	ADP
ap-8536	203	5	the	the	DET
ap-8536	203	6	rightmost	rightmost	ADJ
ap-8536	203	7	sum	sum	NOUN
ap-8536	203	8	:	:	PUNCT
ap-8536	203	9	∑	∑	PUNCT
ap-8536	204	1	j∈z	j∈z	PROPN
ap-8536	204	2	m	m	PROPN
ap-8536	204	3	j−lg(m)qj	j−lg(m)qj	PROPN
ap-8536	204	4	=	=	PUNCT
ap-8536	204	5	∑	∑	PUNCT
ap-8536	204	6	j∈z	j∈z	PROPN
ap-8536	204	7	p−1∑	p−1∑	VERB
ap-8536	204	8	l=0	l=0	PROPN
ap-8536	204	9	m	m	VERB
ap-8536	204	10	j−l+lclqj	j−l+lclqj	NOUN
ap-8536	204	11	=	=	PUNCT
ap-8536	204	12	=	=	PUNCT
ap-8536	204	13	∑	∑	PUNCT
ap-8536	204	14	j∈z	j∈z	PROPN
ap-8536	204	15	m	m	PROPN
ap-8536	204	16	j	j	NOUN
ap-8536	204	17	(	(	PUNCT
ap-8536	204	18	p−1∑	p−1∑	NOUN
ap-8536	204	19	l=0	l=0	PROPN
ap-8536	204	20	clqj+l−l	clqj+l−l	NOUN
ap-8536	204	21	)	)	PUNCT
ap-8536	204	22	.	.	PUNCT
ap-8536	205	1	therefore	therefore	ADV
ap-8536	205	2	,	,	PUNCT
ap-8536	205	3	x	x	PUNCT
ap-8536	205	4	+	+	PUNCT
ap-8536	205	5	y	y	NOUN
ap-8536	205	6	=	=	PUNCT
ap-8536	205	7	∑	∑	PUNCT
ap-8536	206	1	j∈z	j∈z	PROPN
ap-8536	207	1	m	m	VERB
ap-8536	207	2	jzj	jzj	PROPN
ap-8536	207	3	,	,	PUNCT
ap-8536	207	4	with	with	ADP
ap-8536	207	5	zj	zj	PROPN
ap-8536	207	6	=	=	SYM
ap-8536	207	7	aj	aj	PROPN
ap-8536	207	8	+	+	X
ap-8536	207	9	bj	bj	VERB
ap-8536	207	10	−	−	PROPN
ap-8536	207	11	(	(	PUNCT
ap-8536	207	12	p−1∑	p−1∑	NOUN
ap-8536	207	13	l=0	l=0	PROPN
ap-8536	207	14	clqj+l−l	clqj+l−l	NOUN
ap-8536	207	15	)	)	PUNCT
ap-8536	207	16	.	.	PUNCT
ap-8536	208	1	(	(	PUNCT
ap-8536	208	2	9	9	X
ap-8536	208	3	)	)	PUNCT
ap-8536	208	4	using	use	VERB
ap-8536	208	5	cl	cl	NOUN
ap-8536	208	6	=	=	PUNCT
ap-8536	208	7	4k	4k	PRON
ap-8536	209	1	+	+	CCONJ
ap-8536	209	2	c	c	X
ap-8536	209	3	,	,	PUNCT
ap-8536	209	4	we	we	PRON
ap-8536	209	5	get	get	VERB
ap-8536	209	6	zj	zj	X
ap-8536	209	7	=	=	SYM
ap-8536	209	8	aj	aj	PROPN
ap-8536	209	9	+	+	X
ap-8536	209	10	bj	bj	ADP
ap-8536	209	11	−	−	PROPN
ap-8536	209	12	4kqj︸	4kqj︸	PROPN
ap-8536	209	13	︷︷	︷︷	PROPN
ap-8536	209	14	︸	︸	ADP
ap-8536	209	15	∈d′	∈d′	NOUN
ap-8536	209	16	−	−	PROPN
ap-8536	210	1	(	(	PUNCT
ap-8536	210	2	cqj	cqj	PROPN
ap-8536	210	3	+	+	CCONJ
ap-8536	210	4	p−1∑	p−1∑	NOUN
ap-8536	210	5	l=0,l	l=0,l	ADP
ap-8536	210	6	̸=l	̸=l	PROPN
ap-8536	210	7	clqj+l−l︸	clqj+l−l︸	PROPN
ap-8536	210	8	︷︷	︷︷	PROPN
ap-8536	210	9	︸	︸	PRON
ap-8536	211	1	=	=	ADJ
ap-8536	211	2	:	:	PUNCT
ap-8536	211	3	u	u	NOUN
ap-8536	211	4	)	)	PUNCT
ap-8536	211	5	.	.	PUNCT
ap-8536	212	1	191	191	NUM
ap-8536	212	2	i.	i.	PROPN
ap-8536	212	3	farkas	farkas	PROPN
ap-8536	212	4	,	,	PUNCT
ap-8536	212	5	e.	e.	PROPN
ap-8536	212	6	pelantová	pelantová	PROPN
ap-8536	212	7	,	,	PUNCT
ap-8536	212	8	m.	m.	NOUN
ap-8536	212	9	svobodová	svobodová	PROPN
ap-8536	212	10	acta	acta	PROPN
ap-8536	212	11	polytechnica	polytechnica	PROPN
ap-8536	212	12	all	all	DET
ap-8536	212	13	entries	entry	NOUN
ap-8536	212	14	of	of	ADP
ap-8536	212	15	all	all	DET
ap-8536	212	16	vectors	vector	NOUN
ap-8536	212	17	qj	qj	X
ap-8536	212	18	belong	belong	VERB
ap-8536	212	19	to	to	ADP
ap-8536	212	20	{	{	PUNCT
ap-8536	212	21	0	0	NUM
ap-8536	212	22	,	,	PUNCT
ap-8536	212	23	1	1	NUM
ap-8536	212	24	,	,	PUNCT
ap-8536	212	25	−1	−1	NOUN
ap-8536	212	26	}	}	PUNCT
ap-8536	212	27	,	,	PUNCT
ap-8536	212	28	and	and	CCONJ
ap-8536	212	29	thus	thus	ADV
ap-8536	212	30	any	any	DET
ap-8536	212	31	component	component	NOUN
ap-8536	212	32	of	of	ADP
ap-8536	212	33	the	the	DET
ap-8536	212	34	vector	vector	NOUN
ap-8536	212	35	u	u	NOUN
ap-8536	212	36	in	in	ADP
ap-8536	212	37	modulus	modulus	NOUN
ap-8536	212	38	is	be	AUX
ap-8536	212	39	at	at	ADP
ap-8536	212	40	most	most	ADJ
ap-8536	212	41	c	c	NOUN
ap-8536	213	1	+	+	CCONJ
ap-8536	213	2	∑	∑	PUNCT
ap-8536	213	3	l=0,l	l=0,l	PROPN
ap-8536	213	4	̸=l	̸=l	PROPN
ap-8536	213	5	|cl|	|cl|	PROPN
ap-8536	213	6	.	.	PUNCT
ap-8536	214	1	equation	equation	NOUN
ap-8536	214	2	(	(	PUNCT
ap-8536	214	3	7	7	X
ap-8536	214	4	)	)	PUNCT
ap-8536	214	5	guarantees	guarantee	VERB
ap-8536	214	6	that	that	SCONJ
ap-8536	214	7	the	the	DET
ap-8536	214	8	components	component	NOUN
ap-8536	214	9	of	of	ADP
ap-8536	214	10	u	u	NOUN
ap-8536	214	11	are	be	AUX
ap-8536	214	12	not	not	PART
ap-8536	214	13	greater	great	ADJ
ap-8536	214	14	than	than	ADP
ap-8536	214	15	cl	cl	NOUN
ap-8536	214	16	4	4	NUM
ap-8536	214	17	.	.	PUNCT
ap-8536	215	1	since	since	SCONJ
ap-8536	215	2	the	the	DET
ap-8536	215	3	components	component	NOUN
ap-8536	215	4	are	be	AUX
ap-8536	215	5	integer	integer	ADJ
ap-8536	215	6	,	,	PUNCT
ap-8536	215	7	they	they	PRON
ap-8536	215	8	are	be	AUX
ap-8536	215	9	at	at	ADP
ap-8536	215	10	most	most	ADJ
ap-8536	215	11	k	k	NOUN
ap-8536	215	12	=	=	PUNCT
ap-8536	215	13	⌊	⌊	VERB
ap-8536	215	14	cl	cl	NOUN
ap-8536	215	15	4	4	NUM
ap-8536	215	16	⌋	⌋	NOUN
ap-8536	215	17	.	.	PUNCT
ap-8536	216	1	as	as	ADP
ap-8536	216	2	d′	d′	PRON
ap-8536	216	3	+	+	CCONJ
ap-8536	216	4	[	[	X
ap-8536	216	5	−k	−k	ADJ
ap-8536	216	6	,	,	PUNCT
ap-8536	216	7	k]m	k]m	NOUN
ap-8536	216	8	⊂	⊂	PROPN
ap-8536	217	1	[	[	X
ap-8536	217	2	−3k	−3k	PROPN
ap-8536	217	3	,	,	PUNCT
ap-8536	217	4	3k)m	3k)m	NUM
ap-8536	217	5	,	,	PUNCT
ap-8536	217	6	so	so	SCONJ
ap-8536	217	7	we	we	PRON
ap-8536	217	8	can	can	AUX
ap-8536	217	9	conclude	conclude	VERB
ap-8536	217	10	that	that	DET
ap-8536	217	11	zj	zj	PROPN
ap-8536	217	12	∈	∈	PROPN
ap-8536	217	13	d.	d.	PROPN
ap-8536	217	14	in	in	ADP
ap-8536	217	15	order	order	NOUN
ap-8536	217	16	to	to	PART
ap-8536	217	17	compute	compute	VERB
ap-8536	217	18	zj	zj	PROPN
ap-8536	217	19	,	,	PUNCT
ap-8536	217	20	we	we	PRON
ap-8536	217	21	needed	need	VERB
ap-8536	217	22	to	to	PART
ap-8536	217	23	know	know	VERB
ap-8536	217	24	,	,	PUNCT
ap-8536	217	25	besides	besides	SCONJ
ap-8536	217	26	aj	aj	PROPN
ap-8536	217	27	and	and	CCONJ
ap-8536	217	28	bj	bj	VERB
ap-8536	217	29	,	,	PUNCT
ap-8536	217	30	also	also	ADV
ap-8536	217	31	qj+l	qj+l	NOUN
ap-8536	217	32	,	,	PUNCT
ap-8536	217	33	qj+l−1	qj+l−1	NOUN
ap-8536	217	34	,	,	PUNCT
ap-8536	217	35	.	.	PUNCT
ap-8536	217	36	.	.	PUNCT
ap-8536	218	1	.	.	PUNCT
ap-8536	219	1	,	,	PUNCT
ap-8536	219	2	qj+l−p+1	qj+l−p+1	X
ap-8536	219	3	.	.	PUNCT
ap-8536	220	1	let	let	VERB
ap-8536	220	2	us	we	PRON
ap-8536	220	3	stress	stress	VERB
ap-8536	220	4	that	that	SCONJ
ap-8536	220	5	qj	qj	PROPN
ap-8536	220	6	depends	depend	VERB
ap-8536	220	7	only	only	ADV
ap-8536	220	8	on	on	ADP
ap-8536	220	9	aj	aj	PROPN
ap-8536	220	10	+	+	CCONJ
ap-8536	220	11	bj	bj	VERB
ap-8536	220	12	.	.	PUNCT
ap-8536	221	1	hence	hence	ADV
ap-8536	221	2	,	,	PUNCT
ap-8536	221	3	zj	zj	PROPN
ap-8536	221	4	is	be	AUX
ap-8536	221	5	determined	determine	VERB
ap-8536	221	6	by	by	ADP
ap-8536	221	7	digits	digit	NOUN
ap-8536	221	8	on	on	ADP
ap-8536	221	9	p	p	NOUN
ap-8536	221	10	positions	position	NOUN
ap-8536	221	11	,	,	PUNCT
ap-8536	221	12	i.e.	i.e.	X
ap-8536	221	13	,	,	PUNCT
ap-8536	221	14	the	the	DET
ap-8536	221	15	addition	addition	NOUN
ap-8536	221	16	is	be	AUX
ap-8536	221	17	performed	perform	VERB
ap-8536	221	18	by	by	ADP
ap-8536	221	19	a	a	DET
ap-8536	221	20	p	p	ADJ
ap-8536	221	21	-	-	PUNCT
ap-8536	221	22	local	local	ADJ
ap-8536	221	23	function	function	NOUN
ap-8536	221	24	.	.	PUNCT
ap-8536	222	1	to	to	PART
ap-8536	222	2	demonstrate	demonstrate	VERB
ap-8536	222	3	the	the	DET
ap-8536	222	4	point	point	NOUN
ap-8536	222	5	(	(	PUNCT
ap-8536	222	6	1	1	NUM
ap-8536	222	7	)	)	PUNCT
ap-8536	222	8	of	of	ADP
ap-8536	222	9	definition	definition	NOUN
ap-8536	222	10	4	4	NUM
ap-8536	222	11	,	,	PUNCT
ap-8536	222	12	we	we	PRON
ap-8536	222	13	have	have	VERB
ap-8536	222	14	to	to	PART
ap-8536	222	15	show	show	VERB
ap-8536	222	16	that	that	SCONJ
ap-8536	222	17	zj	zj	PROPN
ap-8536	222	18	̸=	̸=	PROPN
ap-8536	222	19	0	0	NUM
ap-8536	222	20	for	for	ADP
ap-8536	222	21	only	only	ADV
ap-8536	222	22	finitely	finitely	ADV
ap-8536	222	23	many	many	ADJ
ap-8536	222	24	indices	index	NOUN
ap-8536	222	25	j	j	PROPN
ap-8536	222	26	∈	∈	PROPN
ap-8536	222	27	z.	z.	PROPN
ap-8536	222	28	the	the	DET
ap-8536	222	29	form	form	NOUN
ap-8536	222	30	of	of	ADP
ap-8536	222	31	d	d	PROPN
ap-8536	222	32	,	,	PUNCT
ap-8536	222	33	d′	d′	PRON
ap-8536	222	34	and	and	CCONJ
ap-8536	222	35	q	q	NOUN
ap-8536	222	36	guarantees	guarantee	NOUN
ap-8536	222	37	that	that	SCONJ
ap-8536	222	38	there	there	PRON
ap-8536	222	39	exists	exist	VERB
ap-8536	222	40	a	a	DET
ap-8536	222	41	unique	unique	ADJ
ap-8536	222	42	qj	qj	NOUN
ap-8536	222	43	satisfying	satisfy	VERB
ap-8536	222	44	aj	aj	PROPN
ap-8536	223	1	+	+	X
ap-8536	223	2	bj	bj	VERB
ap-8536	223	3	−	−	PROPN
ap-8536	223	4	4kqj	4kqj	PROPN
ap-8536	223	5	∈	∈	PROPN
ap-8536	223	6	d′.	d′.	VERB
ap-8536	223	7	in	in	ADP
ap-8536	223	8	particular	particular	ADJ
ap-8536	223	9	,	,	PUNCT
ap-8536	223	10	if	if	SCONJ
ap-8536	223	11	aj	aj	PROPN
ap-8536	223	12	=	=	VERB
ap-8536	223	13	bj	bj	VERB
ap-8536	223	14	=	=	SYM
ap-8536	223	15	0	0	NUM
ap-8536	223	16	,	,	PUNCT
ap-8536	223	17	then	then	ADV
ap-8536	223	18	qj	qj	PROPN
ap-8536	223	19	=	=	SYM
ap-8536	223	20	0	0	PROPN
ap-8536	223	21	.	.	PUNCT
ap-8536	224	1	the	the	DET
ap-8536	224	2	formula	formula	NOUN
ap-8536	224	3	(	(	PUNCT
ap-8536	224	4	9	9	NUM
ap-8536	224	5	)	)	PUNCT
ap-8536	224	6	implies	imply	VERB
ap-8536	224	7	that	that	SCONJ
ap-8536	224	8	zj	zj	PROPN
ap-8536	224	9	is	be	AUX
ap-8536	224	10	non	non	ADJ
ap-8536	224	11	-	-	ADJ
ap-8536	224	12	zero	zero	NUM
ap-8536	224	13	for	for	ADP
ap-8536	224	14	only	only	ADV
ap-8536	224	15	finitely	finitely	ADV
ap-8536	224	16	many	many	ADJ
ap-8536	224	17	indices	index	NOUN
ap-8536	224	18	j	j	PROPN
ap-8536	224	19	∈	∈	PROPN
ap-8536	224	20	z.	z.	PROPN
ap-8536	224	21	let	let	VERB
ap-8536	224	22	us	we	PRON
ap-8536	224	23	note	note	VERB
ap-8536	224	24	that	that	SCONJ
ap-8536	224	25	the	the	DET
ap-8536	224	26	digit	digit	NOUN
ap-8536	224	27	set	set	NOUN
ap-8536	224	28	d	d	NOUN
ap-8536	224	29	is	be	AUX
ap-8536	224	30	not	not	PART
ap-8536	224	31	closed	close	VERB
ap-8536	224	32	under	under	ADP
ap-8536	224	33	multiplication	multiplication	NOUN
ap-8536	224	34	by	by	ADP
ap-8536	224	35	−1	−1	NOUN
ap-8536	224	36	.	.	PUNCT
ap-8536	225	1	but	but	CCONJ
ap-8536	225	2	for	for	ADP
ap-8536	225	3	each	each	DET
ap-8536	225	4	b	b	PROPN
ap-8536	225	5	∈	∈	PROPN
ap-8536	225	6	d	d	NOUN
ap-8536	225	7	,	,	PUNCT
ap-8536	225	8	we	we	PRON
ap-8536	225	9	can	can	AUX
ap-8536	225	10	find	find	VERB
ap-8536	225	11	c	c	NOUN
ap-8536	225	12	,	,	PUNCT
ap-8536	225	13	d	d	PROPN
ap-8536	225	14	∈	∈	PROPN
ap-8536	226	1	d	d	ADP
ap-8536	226	2	such	such	ADJ
ap-8536	226	3	that	that	DET
ap-8536	226	4	−b	−b	NOUN
ap-8536	226	5	=	=	SYM
ap-8536	226	6	c	c	PROPN
ap-8536	226	7	+	+	CCONJ
ap-8536	226	8	d.	d.	PROPN
ap-8536	226	9	hence	hence	ADV
ap-8536	226	10	,	,	PUNCT
ap-8536	226	11	a	a	DET
ap-8536	226	12	subtraction	subtraction	NOUN
ap-8536	226	13	of	of	ADP
ap-8536	226	14	two	two	NUM
ap-8536	226	15	vectors	vector	NOUN
ap-8536	226	16	x	x	PUNCT
ap-8536	226	17	=	=	PUNCT
ap-8536	226	18	∑	∑	PUNCT
ap-8536	226	19	z	z	PROPN
ap-8536	226	20	m	m	VERB
ap-8536	226	21	jaj	jaj	PROPN
ap-8536	226	22	and	and	CCONJ
ap-8536	226	23	y	y	PROPN
ap-8536	226	24	=	=	PUNCT
ap-8536	227	1	∑	∑	PUNCT
ap-8536	227	2	z	z	PROPN
ap-8536	227	3	m	m	VERB
ap-8536	227	4	jbj	jbj	PROPN
ap-8536	227	5	can	can	AUX
ap-8536	227	6	be	be	AUX
ap-8536	227	7	viewed	view	VERB
ap-8536	227	8	as	as	ADP
ap-8536	227	9	addition	addition	NOUN
ap-8536	227	10	of	of	ADP
ap-8536	227	11	three	three	NUM
ap-8536	227	12	vectors	vector	NOUN
ap-8536	227	13	,	,	PUNCT
ap-8536	227	14	and	and	CCONJ
ap-8536	227	15	therefore	therefore	ADV
ap-8536	227	16	it	it	PRON
ap-8536	227	17	is	be	AUX
ap-8536	227	18	computable	computable	ADJ
ap-8536	227	19	in	in	ADP
ap-8536	227	20	parallel	parallel	NOUN
ap-8536	227	21	as	as	ADV
ap-8536	227	22	well	well	ADV
ap-8536	227	23	.	.	PUNCT
ap-8536	228	1	it	it	PRON
ap-8536	228	2	remains	remain	VERB
ap-8536	228	3	to	to	PART
ap-8536	228	4	prove	prove	VERB
ap-8536	228	5	that	that	SCONJ
ap-8536	228	6	zm	zm	PROPN
ap-8536	228	7	⊂	⊂	PROPN
ap-8536	228	8	find(m	find(m	PROPN
ap-8536	228	9	)	)	PUNCT
ap-8536	228	10	.	.	PUNCT
ap-8536	229	1	but	but	CCONJ
ap-8536	229	2	this	this	PRON
ap-8536	229	3	is	be	AUX
ap-8536	229	4	clear	clear	ADJ
ap-8536	229	5	,	,	PUNCT
ap-8536	229	6	since	since	SCONJ
ap-8536	229	7	d	d	PROPN
ap-8536	229	8	⊂	⊂	PROPN
ap-8536	229	9	find(m	find(m	PROPN
ap-8536	229	10	)	)	PUNCT
ap-8536	229	11	,	,	PUNCT
ap-8536	229	12	find(m	find(m	NOUN
ap-8536	229	13	)	)	PUNCT
ap-8536	229	14	is	be	AUX
ap-8536	229	15	closed	close	VERB
ap-8536	229	16	under	under	ADP
ap-8536	229	17	addition	addition	NOUN
ap-8536	229	18	and	and	CCONJ
ap-8536	229	19	each	each	PRON
ap-8536	229	20	x	x	PROPN
ap-8536	229	21	∈	∈	PROPN
ap-8536	229	22	zm	zm	PROPN
ap-8536	229	23	can	can	AUX
ap-8536	229	24	be	be	AUX
ap-8536	229	25	expressed	express	VERB
ap-8536	229	26	as	as	ADP
ap-8536	229	27	a	a	DET
ap-8536	229	28	finite	finite	ADJ
ap-8536	229	29	sum	sum	NOUN
ap-8536	229	30	of	of	ADP
ap-8536	229	31	digits	digit	NOUN
ap-8536	229	32	from	from	ADP
ap-8536	229	33	d.	d.	PROPN
ap-8536	229	34	remark	remark	PROPN
ap-8536	229	35	10	10	NUM
ap-8536	229	36	.	.	PUNCT
ap-8536	230	1	the	the	DET
ap-8536	230	2	vectors	vector	NOUN
ap-8536	230	3	we	we	PRON
ap-8536	230	4	add	add	VERB
ap-8536	230	5	by	by	ADP
ap-8536	230	6	the	the	DET
ap-8536	230	7	parallel	parallel	ADJ
ap-8536	230	8	algorithm	algorithm	NOUN
ap-8536	230	9	as	as	SCONJ
ap-8536	230	10	described	describe	VERB
ap-8536	230	11	in	in	ADP
ap-8536	230	12	the	the	DET
ap-8536	230	13	previous	previous	ADJ
ap-8536	230	14	proof	proof	NOUN
ap-8536	230	15	are	be	AUX
ap-8536	230	16	represented	represent	VERB
ap-8536	230	17	by	by	ADP
ap-8536	230	18	both	both	ADV
ap-8536	230	19	-	-	PUNCT
ap-8536	230	20	sided	sided	ADJ
ap-8536	230	21	infinite	infinite	ADJ
ap-8536	230	22	strings	string	NOUN
ap-8536	230	23	.	.	PUNCT
ap-8536	231	1	but	but	CCONJ
ap-8536	231	2	only	only	ADV
ap-8536	231	3	finitely	finitely	ADV
ap-8536	231	4	many	many	ADJ
ap-8536	231	5	entries	entry	NOUN
ap-8536	231	6	of	of	ADP
ap-8536	231	7	the	the	DET
ap-8536	231	8	strings	string	NOUN
ap-8536	231	9	are	be	AUX
ap-8536	231	10	occupied	occupy	VERB
ap-8536	231	11	by	by	ADP
ap-8536	231	12	non	non	ADJ
ap-8536	231	13	-	-	ADJ
ap-8536	231	14	zero	zero	NUM
ap-8536	231	15	digits	digit	NOUN
ap-8536	231	16	.	.	PUNCT
ap-8536	232	1	assume	assume	VERB
ap-8536	232	2	that	that	SCONJ
ap-8536	232	3	x	x	X
ap-8536	233	1	+	+	NUM
ap-8536	233	2	y	y	NOUN
ap-8536	233	3	=	=	SYM
ap-8536	234	1	∑n	∑n	PROPN
ap-8536	234	2	j	j	PROPN
ap-8536	234	3	=	=	PROPN
ap-8536	234	4	n	n	PROPN
ap-8536	234	5	m	m	VERB
ap-8536	234	6	j(aj	j(aj	PROPN
ap-8536	234	7	+	+	CCONJ
ap-8536	234	8	bj	bj	VERB
ap-8536	234	9	)	)	PUNCT
ap-8536	234	10	,	,	PUNCT
ap-8536	234	11	for	for	ADP
ap-8536	234	12	some	some	DET
ap-8536	234	13	integers	integer	NOUN
ap-8536	234	14	n	n	PRON
ap-8536	234	15	≤	≤	NOUN
ap-8536	234	16	n	n	ADV
ap-8536	234	17	.	.	PUNCT
ap-8536	235	1	as	as	SCONJ
ap-8536	235	2	stated	state	VERB
ap-8536	235	3	in	in	ADP
ap-8536	235	4	the	the	DET
ap-8536	235	5	previous	previous	ADJ
ap-8536	235	6	proof	proof	NOUN
ap-8536	235	7	,	,	PUNCT
ap-8536	235	8	if	if	SCONJ
ap-8536	235	9	both	both	DET
ap-8536	235	10	digits	digit	NOUN
ap-8536	235	11	aj	aj	VERB
ap-8536	235	12	and	and	CCONJ
ap-8536	235	13	bj	bj	VERB
ap-8536	235	14	are	be	AUX
ap-8536	235	15	zero	zero	NUM
ap-8536	235	16	,	,	PUNCT
ap-8536	235	17	then	then	ADV
ap-8536	235	18	the	the	DET
ap-8536	235	19	algorithm	algorithm	NOUN
ap-8536	235	20	puts	put	VERB
ap-8536	235	21	qj	qj	PROPN
ap-8536	235	22	=	=	NOUN
ap-8536	235	23	0	0	PROPN
ap-8536	235	24	.	.	PUNCT
ap-8536	236	1	the	the	DET
ap-8536	236	2	formula	formula	NOUN
ap-8536	236	3	(	(	PUNCT
ap-8536	236	4	9	9	NUM
ap-8536	236	5	)	)	PUNCT
ap-8536	236	6	for	for	ADP
ap-8536	236	7	zj	zj	PROPN
ap-8536	236	8	implies	imply	VERB
ap-8536	236	9	that	that	SCONJ
ap-8536	236	10	zj	zj	PROPN
ap-8536	236	11	is	be	AUX
ap-8536	236	12	zero	zero	NUM
ap-8536	236	13	for	for	ADP
ap-8536	236	14	all	all	DET
ap-8536	236	15	j	j	PROPN
ap-8536	236	16	≤	≤	NUM
ap-8536	236	17	n	n	CCONJ
ap-8536	236	18	−	−	PROPN
ap-8536	236	19	l	l	NOUN
ap-8536	236	20	−	−	NOUN
ap-8536	236	21	1	1	NUM
ap-8536	236	22	and	and	CCONJ
ap-8536	236	23	for	for	ADP
ap-8536	236	24	all	all	DET
ap-8536	236	25	j	j	PROPN
ap-8536	236	26	≥	≥	NOUN
ap-8536	236	27	n	n	NOUN
ap-8536	236	28	+	+	CCONJ
ap-8536	236	29	p	p	X
ap-8536	236	30	−	−	PROPN
ap-8536	236	31	l.	l.	NOUN
ap-8536	236	32	hence	hence	ADV
ap-8536	236	33	,	,	PUNCT
ap-8536	236	34	∑n	∑n	PROPN
ap-8536	236	35	j	j	PROPN
ap-8536	236	36	=	=	PROPN
ap-8536	236	37	n	n	PROPN
ap-8536	236	38	m	m	VERB
ap-8536	236	39	j(aj	j(aj	PROPN
ap-8536	236	40	+	+	CCONJ
ap-8536	236	41	bj	bj	VERB
ap-8536	236	42	)	)	PUNCT
ap-8536	236	43	=	=	SYM
ap-8536	237	1	∑n	∑n	PROPN
ap-8536	237	2	′	′	NUM
ap-8536	238	1	j	j	PROPN
ap-8536	239	1	=	=	NOUN
ap-8536	239	2	n′	n′	ADV
ap-8536	239	3	m	m	VERB
ap-8536	239	4	jzj	jzj	NOUN
ap-8536	239	5	,	,	PUNCT
ap-8536	239	6	where	where	SCONJ
ap-8536	239	7	n′	n′	PROPN
ap-8536	239	8	=	=	SYM
ap-8536	239	9	n	n	CCONJ
ap-8536	239	10	−	−	PROPN
ap-8536	239	11	l	l	NOUN
ap-8536	239	12	and	and	CCONJ
ap-8536	239	13	n	n	NOUN
ap-8536	239	14	′	′	NOUN
ap-8536	239	15	=	=	PUNCT
ap-8536	239	16	n	n	PROPN
ap-8536	239	17	+	+	CCONJ
ap-8536	239	18	p	p	NOUN
ap-8536	239	19	−	−	PROPN
ap-8536	239	20	l	l	NOUN
ap-8536	239	21	−	−	NOUN
ap-8536	239	22	1	1	NUM
ap-8536	239	23	.	.	PUNCT
ap-8536	239	24	example	example	NOUN
ap-8536	239	25	11	11	NUM
ap-8536	239	26	.	.	PUNCT
ap-8536	239	27	consider	consider	VERB
ap-8536	239	28	the	the	DET
ap-8536	239	29	matrix	matrix	NOUN
ap-8536	239	30	numeration	numeration	NOUN
ap-8536	239	31	system	system	NOUN
ap-8536	239	32	with	with	ADP
ap-8536	239	33	base	base	NOUN
ap-8536	239	34	matrix	matrix	NOUN
ap-8536	239	35	m	m	NOUN
ap-8536	239	36	=	=	PUNCT
ap-8536	239	37	(	(	PUNCT
ap-8536	239	38	−1	−1	NOUN
ap-8536	239	39	−1	−1	NOUN
ap-8536	239	40	+1	+1	PROPN
ap-8536	239	41	−1	−1	NOUN
ap-8536	239	42	)	)	PUNCT
ap-8536	239	43	∈	∈	PROPN
ap-8536	239	44	z2×2	z2×2	NOUN
ap-8536	239	45	.	.	PUNCT
ap-8536	240	1	by	by	ADP
ap-8536	240	2	example	example	NOUN
ap-8536	240	3	8	8	NUM
ap-8536	240	4	,	,	PUNCT
ap-8536	240	5	the	the	DET
ap-8536	240	6	polynomial	polynomial	ADJ
ap-8536	240	7	g(x	g(x	NOUN
ap-8536	240	8	)	)	PUNCT
ap-8536	241	1	=	=	PUNCT
ap-8536	241	2	x4	x4	PROPN
ap-8536	242	1	+	+	CCONJ
ap-8536	242	2	4	4	NUM
ap-8536	242	3	with	with	ADP
ap-8536	242	4	cl	cl	NOUN
ap-8536	242	5	=	=	SYM
ap-8536	242	6	c0	c0	NOUN
ap-8536	242	7	=	=	SYM
ap-8536	242	8	4	4	NUM
ap-8536	242	9	is	be	AUX
ap-8536	242	10	suitable	suitable	ADJ
ap-8536	242	11	for	for	ADP
ap-8536	242	12	the	the	DET
ap-8536	242	13	parallel	parallel	ADJ
ap-8536	242	14	addition	addition	NOUN
ap-8536	242	15	algorithm	algorithm	NOUN
ap-8536	242	16	as	as	SCONJ
ap-8536	242	17	described	describe	VERB
ap-8536	242	18	in	in	ADP
ap-8536	242	19	the	the	DET
ap-8536	242	20	proof	proof	NOUN
ap-8536	242	21	of	of	ADP
ap-8536	242	22	theorem	theorem	NOUN
ap-8536	242	23	9	9	NUM
ap-8536	242	24	.	.	PUNCT
ap-8536	243	1	following	follow	VERB
ap-8536	243	2	the	the	DET
ap-8536	243	3	proof	proof	NOUN
ap-8536	243	4	,	,	PUNCT
ap-8536	243	5	we	we	PRON
ap-8536	243	6	put	put	VERB
ap-8536	243	7	k	k	X
ap-8536	243	8	=	=	PUNCT
ap-8536	243	9	⌊	⌊	NOUN
ap-8536	243	10	cl	cl	NOUN
ap-8536	243	11	4	4	NUM
ap-8536	243	12	⌋	⌋	NOUN
ap-8536	243	13	=	=	SYM
ap-8536	243	14	1	1	NUM
ap-8536	243	15	and	and	CCONJ
ap-8536	243	16	define	define	VERB
ap-8536	243	17	d	d	NOUN
ap-8536	243	18	=	=	PUNCT
ap-8536	244	1	[	[	X
ap-8536	244	2	−3	−3	ADJ
ap-8536	244	3	,	,	PUNCT
ap-8536	244	4	3)2	3)2	NUM
ap-8536	244	5	∩	∩	ADJ
ap-8536	244	6	z2	z2	NOUN
ap-8536	244	7	,	,	PUNCT
ap-8536	244	8	i.e.	i.e.	X
ap-8536	244	9	,	,	PUNCT
ap-8536	244	10	the	the	DET
ap-8536	244	11	digit	digit	NOUN
ap-8536	244	12	set	set	NOUN
ap-8536	244	13	has	have	VERB
ap-8536	244	14	36	36	NUM
ap-8536	244	15	elements	element	NOUN
ap-8536	244	16	.	.	PUNCT
ap-8536	245	1	with	with	ADP
ap-8536	245	2	such	such	DET
ap-8536	245	3	a	a	DET
ap-8536	245	4	choice	choice	NOUN
ap-8536	245	5	of	of	ADP
ap-8536	245	6	the	the	DET
ap-8536	245	7	digit	digit	NOUN
ap-8536	245	8	set	set	VERB
ap-8536	245	9	d	d	PROPN
ap-8536	245	10	,	,	PUNCT
ap-8536	245	11	the	the	DET
ap-8536	245	12	addition	addition	NOUN
ap-8536	245	13	in	in	ADP
ap-8536	245	14	(	(	PUNCT
ap-8536	245	15	m	m	PROPN
ap-8536	245	16	,	,	PUNCT
ap-8536	245	17	d	d	X
ap-8536	245	18	)	)	PUNCT
ap-8536	245	19	is	be	AUX
ap-8536	245	20	computable	computable	ADJ
ap-8536	245	21	in	in	ADP
ap-8536	245	22	parallel	parallel	NOUN
ap-8536	245	23	.	.	PUNCT
ap-8536	246	1	the	the	DET
ap-8536	246	2	algorithm	algorithm	NOUN
ap-8536	246	3	for	for	ADP
ap-8536	246	4	parallel	parallel	ADJ
ap-8536	246	5	addition	addition	NOUN
ap-8536	246	6	constructed	construct	VERB
ap-8536	246	7	in	in	ADP
ap-8536	246	8	the	the	DET
ap-8536	246	9	proof	proof	NOUN
ap-8536	246	10	of	of	ADP
ap-8536	246	11	theorem	theorem	ADJ
ap-8536	246	12	9	9	NUM
ap-8536	246	13	is	be	AUX
ap-8536	246	14	very	very	ADV
ap-8536	246	15	simple	simple	ADJ
ap-8536	246	16	,	,	PUNCT
ap-8536	246	17	as	as	SCONJ
ap-8536	246	18	the	the	DET
ap-8536	246	19	value	value	NOUN
ap-8536	246	20	qj	qj	PROPN
ap-8536	246	21	depends	depend	VERB
ap-8536	246	22	only	only	ADV
ap-8536	246	23	on	on	ADP
ap-8536	246	24	the	the	DET
ap-8536	246	25	digits	digit	NOUN
ap-8536	246	26	aj	aj	PROPN
ap-8536	246	27	and	and	CCONJ
ap-8536	246	28	bj	bj	VERB
ap-8536	246	29	having	have	VERB
ap-8536	246	30	the	the	DET
ap-8536	246	31	same	same	ADJ
ap-8536	246	32	index	index	NOUN
ap-8536	246	33	j.	j.	PROPN
ap-8536	246	34	an	an	DET
ap-8536	246	35	algorithm	algorithm	NOUN
ap-8536	246	36	with	with	ADP
ap-8536	246	37	such	such	DET
ap-8536	246	38	a	a	DET
ap-8536	246	39	property	property	NOUN
ap-8536	246	40	is	be	AUX
ap-8536	246	41	usually	usually	ADV
ap-8536	246	42	called	call	VERB
ap-8536	246	43	neighbour	neighbour	NOUN
ap-8536	246	44	free	free	ADJ
ap-8536	246	45	.	.	PUNCT
ap-8536	247	1	however	however	ADV
ap-8536	247	2	,	,	PUNCT
ap-8536	247	3	we	we	PRON
ap-8536	247	4	pay	pay	VERB
ap-8536	247	5	a	a	DET
ap-8536	247	6	large	large	ADJ
ap-8536	247	7	price	price	NOUN
ap-8536	247	8	for	for	ADP
ap-8536	247	9	the	the	DET
ap-8536	247	10	simplicity	simplicity	NOUN
ap-8536	247	11	of	of	ADP
ap-8536	247	12	the	the	DET
ap-8536	247	13	algorithm	algorithm	NOUN
ap-8536	247	14	–	–	PUNCT
ap-8536	247	15	the	the	DET
ap-8536	247	16	digit	digit	NOUN
ap-8536	247	17	set	set	NOUN
ap-8536	247	18	is	be	AUX
ap-8536	247	19	huge	huge	ADJ
ap-8536	247	20	.	.	PUNCT
ap-8536	248	1	with	with	ADP
ap-8536	248	2	another	another	DET
ap-8536	248	3	choice	choice	NOUN
ap-8536	248	4	of	of	ADP
ap-8536	248	5	the	the	DET
ap-8536	248	6	algorithm	algorithm	NOUN
ap-8536	248	7	,	,	PUNCT
ap-8536	248	8	the	the	DET
ap-8536	248	9	digit	digit	NOUN
ap-8536	248	10	set	set	NOUN
ap-8536	248	11	could	could	AUX
ap-8536	248	12	be	be	AUX
ap-8536	248	13	substantially	substantially	ADV
ap-8536	248	14	smaller	small	ADJ
ap-8536	248	15	,	,	PUNCT
ap-8536	248	16	and	and	CCONJ
ap-8536	248	17	still	still	ADV
ap-8536	248	18	sufficient	sufficient	ADJ
ap-8536	248	19	to	to	PART
ap-8536	248	20	perform	perform	VERB
ap-8536	248	21	the	the	DET
ap-8536	248	22	addition	addition	NOUN
ap-8536	248	23	in	in	ADP
ap-8536	248	24	parallel	parallel	NOUN
ap-8536	248	25	by	by	ADP
ap-8536	248	26	means	mean	NOUN
ap-8536	248	27	of	of	ADP
ap-8536	248	28	a	a	DET
ap-8536	248	29	p	p	ADJ
ap-8536	248	30	-	-	PUNCT
ap-8536	248	31	local	local	ADJ
ap-8536	248	32	function	function	NOUN
ap-8536	248	33	(	(	PUNCT
ap-8536	248	34	with	with	ADP
ap-8536	248	35	a	a	DET
ap-8536	248	36	larger	large	ADJ
ap-8536	248	37	parameter	parameter	NOUN
ap-8536	248	38	p	p	NOUN
ap-8536	248	39	,	,	PUNCT
ap-8536	248	40	though	though	ADV
ap-8536	248	41	)	)	PUNCT
ap-8536	248	42	.	.	PUNCT
ap-8536	249	1	example	example	NOUN
ap-8536	249	2	12	12	NUM
ap-8536	249	3	.	.	PUNCT
ap-8536	250	1	consider	consider	VERB
ap-8536	250	2	the	the	DET
ap-8536	250	3	numeration	numeration	NOUN
ap-8536	250	4	system	system	NOUN
ap-8536	250	5	in	in	ADP
ap-8536	250	6	c	c	PROPN
ap-8536	250	7	with	with	ADP
ap-8536	250	8	base	base	NOUN
ap-8536	250	9	β	β	X
ap-8536	250	10	=	=	PUNCT
ap-8536	250	11	ı	ı	PROPN
ap-8536	250	12	−	−	NOUN
ap-8536	250	13	1	1	NUM
ap-8536	250	14	.	.	PUNCT
ap-8536	251	1	in	in	ADP
ap-8536	251	2	[	[	X
ap-8536	251	3	19	19	NUM
ap-8536	251	4	]	]	PUNCT
ap-8536	251	5	,	,	PUNCT
ap-8536	251	6	a	a	DET
ap-8536	251	7	7	7	NUM
ap-8536	251	8	-	-	PUNCT
ap-8536	251	9	local	local	ADJ
ap-8536	251	10	function	function	NOUN
ap-8536	251	11	of	of	ADP
ap-8536	251	12	parallel	parallel	ADJ
ap-8536	251	13	addition	addition	NOUN
ap-8536	251	14	in	in	ADP
ap-8536	251	15	system	system	NOUN
ap-8536	251	16	(	(	PUNCT
ap-8536	251	17	β	β	X
ap-8536	251	18	,	,	PUNCT
ap-8536	251	19	a	a	PRON
ap-8536	251	20	)	)	PUNCT
ap-8536	251	21	is	be	AUX
ap-8536	251	22	found	find	VERB
ap-8536	251	23	for	for	ADP
ap-8536	251	24	the	the	DET
ap-8536	251	25	digit	digit	NOUN
ap-8536	251	26	set	set	VERB
ap-8536	251	27	a	a	PRON
ap-8536	251	28	=	=	PUNCT
ap-8536	251	29	{	{	PUNCT
ap-8536	251	30	0	0	NUM
ap-8536	251	31	,	,	PUNCT
ap-8536	251	32	±1	±1	VERB
ap-8536	251	33	,	,	PUNCT
ap-8536	251	34	±ı	±ı	PROPN
ap-8536	251	35	}	}	PUNCT
ap-8536	251	36	.	.	PUNCT
ap-8536	252	1	let	let	VERB
ap-8536	252	2	us	we	PRON
ap-8536	252	3	denote	denote	VERB
ap-8536	252	4	the	the	DET
ap-8536	252	5	7	7	NUM
ap-8536	252	6	-	-	PUNCT
ap-8536	252	7	local	local	ADJ
ap-8536	252	8	function	function	NOUN
ap-8536	252	9	as	as	ADP
ap-8536	252	10	φ	φ	PROPN
ap-8536	252	11	:	:	PUNCT
ap-8536	252	12	(	(	PUNCT
ap-8536	252	13	a	a	DET
ap-8536	252	14	+	+	X
ap-8536	252	15	a)7	a)7	PROPN
ap-8536	252	16	7→	7→	NUM
ap-8536	252	17	a	a	PRON
ap-8536	252	18	,	,	PUNCT
ap-8536	252	19	acting	act	VERB
ap-8536	252	20	on	on	ADP
ap-8536	252	21	a	a	DET
ap-8536	252	22	7	7	NUM
ap-8536	252	23	-	-	PUNCT
ap-8536	252	24	tuple	tuple	NOUN
ap-8536	252	25	(	(	PUNCT
ap-8536	252	26	wj	wj	PROPN
ap-8536	252	27	,	,	PUNCT
ap-8536	252	28	.	.	PUNCT
ap-8536	252	29	.	.	PUNCT
ap-8536	253	1	.	.	PUNCT
ap-8536	254	1	,	,	PUNCT
ap-8536	254	2	wj−6	wj−6	VERB
ap-8536	254	3	)	)	PUNCT
ap-8536	254	4	∈	∈	PROPN
ap-8536	254	5	(	(	PUNCT
ap-8536	254	6	a	a	DET
ap-8536	254	7	+	+	X
ap-8536	254	8	a)7	a)7	ADJ
ap-8536	254	9	by	by	ADP
ap-8536	254	10	means	mean	NOUN
ap-8536	254	11	of	of	ADP
ap-8536	254	12	an	an	DET
ap-8536	254	13	auxiliary	auxiliary	ADJ
ap-8536	254	14	quotient	quotient	NOUN
ap-8536	254	15	function	function	NOUN
ap-8536	254	16	q	q	NOUN
ap-8536	254	17	:	:	PUNCT
ap-8536	254	18	(	(	PUNCT
ap-8536	254	19	a	a	DET
ap-8536	254	20	+	+	X
ap-8536	254	21	a)6	a)6	PROPN
ap-8536	254	22	7→	7→	NUM
ap-8536	254	23	q	q	NOUN
ap-8536	254	24	⊂	⊂	PROPN
ap-8536	254	25	z[ı	z[ı	X
ap-8536	254	26	]	]	X
ap-8536	254	27	as	as	SCONJ
ap-8536	254	28	follows	follow	VERB
ap-8536	254	29	:	:	PUNCT
ap-8536	254	30	qj	qj	NOUN
ap-8536	254	31	:	:	PUNCT
ap-8536	254	32	=	=	SYM
ap-8536	254	33	q(wj	q(wj	PRON
ap-8536	254	34	,	,	PUNCT
ap-8536	254	35	.	.	PUNCT
ap-8536	254	36	.	.	PUNCT
ap-8536	254	37	.	.	PUNCT
ap-8536	255	1	,	,	PUNCT
ap-8536	255	2	wj−5	wj−5	PROPN
ap-8536	255	3	)	)	PUNCT
ap-8536	255	4	∈	∈	PROPN
ap-8536	255	5	q	q	NOUN
ap-8536	256	1	and	and	CCONJ
ap-8536	256	2	,	,	PUNCT
ap-8536	256	3	consequently	consequently	ADV
ap-8536	256	4	,	,	PUNCT
ap-8536	256	5	zj	zj	PROPN
ap-8536	256	6	:	:	PUNCT
ap-8536	256	7	=	=	SYM
ap-8536	256	8	wj	wj	X
ap-8536	256	9	+	+	CCONJ
ap-8536	256	10	qj−1	qj−1	ADV
ap-8536	256	11	−	−	PROPN
ap-8536	256	12	βqj	βqj	NOUN
ap-8536	256	13	=	=	PUNCT
ap-8536	257	1	φ(wj	φ(wj	PROPN
ap-8536	257	2	,	,	PUNCT
ap-8536	257	3	.	.	PUNCT
ap-8536	257	4	.	.	PUNCT
ap-8536	258	1	.	.	PUNCT
ap-8536	259	1	,	,	PUNCT
ap-8536	259	2	wj−6	wj−6	VERB
ap-8536	259	3	)	)	PUNCT
ap-8536	259	4	∈	∈	PROPN
ap-8536	259	5	a	a	PRON
ap-8536	259	6	.	.	PUNCT
ap-8536	260	1	the	the	DET
ap-8536	260	2	local	local	ADJ
ap-8536	260	3	functions	function	NOUN
ap-8536	260	4	φ	φ	PROPN
ap-8536	260	5	and	and	CCONJ
ap-8536	260	6	q	q	NOUN
ap-8536	260	7	acting	act	VERB
ap-8536	260	8	on	on	ADP
ap-8536	260	9	numbers	number	NOUN
ap-8536	260	10	can	can	AUX
ap-8536	260	11	be	be	AUX
ap-8536	260	12	transformed	transform	VERB
ap-8536	260	13	to	to	ADP
ap-8536	260	14	local	local	ADJ
ap-8536	260	15	functions	function	NOUN
ap-8536	260	16	φ′	φ′	NUM
ap-8536	260	17	:	:	PUNCT
ap-8536	260	18	(	(	PUNCT
ap-8536	260	19	d	d	X
ap-8536	260	20	+	+	CCONJ
ap-8536	260	21	d)7	d)7	PROPN
ap-8536	260	22	7→	7→	PROPN
ap-8536	261	1	d	d	NOUN
ap-8536	261	2	=	=	PUNCT
ap-8536	261	3	ξ(a	ξ(a	VERB
ap-8536	261	4	)	)	PUNCT
ap-8536	261	5	and	and	CCONJ
ap-8536	261	6	q′	q′	NOUN
ap-8536	261	7	:	:	PUNCT
ap-8536	262	1	(	(	PUNCT
ap-8536	262	2	d	d	X
ap-8536	262	3	+	+	CCONJ
ap-8536	262	4	d)6	d)6	NOUN
ap-8536	262	5	7→	7→	NUM
ap-8536	262	6	q′	q′	NOUN
ap-8536	262	7	=	=	SYM
ap-8536	262	8	ξ(q	ξ(q	PROPN
ap-8536	262	9	)	)	PUNCT
ap-8536	262	10	acting	act	VERB
ap-8536	262	11	on	on	ADP
ap-8536	262	12	vectors	vector	NOUN
ap-8536	262	13	,	,	PUNCT
ap-8536	262	14	by	by	ADP
ap-8536	262	15	means	mean	NOUN
ap-8536	262	16	of	of	ADP
ap-8536	262	17	the	the	DET
ap-8536	262	18	isomorphism	isomorphism	NOUN
ap-8536	262	19	ξ	ξ	X
ap-8536	262	20	:	:	PUNCT
ap-8536	262	21	z[ı	z[ı	NUM
ap-8536	262	22	]	]	X
ap-8536	262	23	7→	7→	NUM
ap-8536	262	24	z2	z2	NOUN
ap-8536	262	25	defined	define	VERB
ap-8536	262	26	in	in	ADP
ap-8536	262	27	example	example	NOUN
ap-8536	262	28	2	2	NUM
ap-8536	262	29	.	.	PUNCT
ap-8536	262	30	thereby	thereby	ADV
ap-8536	262	31	,	,	PUNCT
ap-8536	262	32	we	we	PRON
ap-8536	262	33	can	can	AUX
ap-8536	262	34	define	define	VERB
ap-8536	262	35	q′	q′	NOUN
ap-8536	262	36	=	=	SYM
ap-8536	262	37	ξ	ξ	PROPN
ap-8536	262	38	◦	◦	NOUN
ap-8536	262	39	q	q	PUNCT
ap-8536	262	40	◦	◦	NOUN
ap-8536	262	41	ξ−1	ξ−1	NOUN
ap-8536	262	42	and	and	CCONJ
ap-8536	262	43	φ′	φ′	NUM
ap-8536	262	44	=	=	SYM
ap-8536	263	1	ξ	ξ	PROPN
ap-8536	263	2	◦	◦	NOUN
ap-8536	263	3	φ	φ	NUM
ap-8536	263	4	◦	◦	NOUN
ap-8536	263	5	ξ−1	ξ−1	PROPN
ap-8536	263	6	.	.	PUNCT
ap-8536	264	1	in	in	ADP
ap-8536	264	2	other	other	ADJ
ap-8536	264	3	words	word	NOUN
ap-8536	264	4	,	,	PUNCT
ap-8536	264	5	with	with	ADP
ap-8536	264	6	the	the	DET
ap-8536	264	7	help	help	NOUN
ap-8536	264	8	of	of	ADP
ap-8536	264	9	the	the	DET
ap-8536	264	10	formulas	formula	NOUN
ap-8536	264	11	from	from	ADP
ap-8536	264	12	7	7	NUM
ap-8536	264	13	-	-	PUNCT
ap-8536	264	14	local	local	ADJ
ap-8536	264	15	parallel	parallel	ADJ
ap-8536	264	16	addition	addition	NOUN
ap-8536	264	17	on	on	ADP
ap-8536	264	18	the	the	DET
ap-8536	264	19	number	number	NOUN
ap-8536	264	20	system	system	NOUN
ap-8536	264	21	(	(	PUNCT
ap-8536	264	22	β	β	X
ap-8536	264	23	,	,	PUNCT
ap-8536	264	24	a	a	X
ap-8536	264	25	)	)	PUNCT
ap-8536	264	26	,	,	PUNCT
ap-8536	264	27	we	we	PRON
ap-8536	264	28	obtain	obtain	VERB
ap-8536	264	29	a	a	DET
ap-8536	264	30	7	7	NUM
ap-8536	264	31	-	-	PUNCT
ap-8536	264	32	local	local	ADJ
ap-8536	264	33	parallel	parallel	ADJ
ap-8536	264	34	addition	addition	NOUN
ap-8536	264	35	on	on	ADP
ap-8536	264	36	the	the	DET
ap-8536	264	37	matrix	matrix	NOUN
ap-8536	264	38	system	system	NOUN
ap-8536	264	39	(	(	PUNCT
ap-8536	264	40	m	m	PROPN
ap-8536	264	41	,	,	PUNCT
ap-8536	264	42	d	d	NOUN
ap-8536	264	43	)	)	PUNCT
ap-8536	264	44	with	with	ADP
ap-8536	264	45	digit	digit	NOUN
ap-8536	264	46	set	set	VERB
ap-8536	264	47	size	size	NOUN
ap-8536	264	48	#	#	NOUN
ap-8536	264	49	d	d	NOUN
ap-8536	264	50	=	=	PUNCT
ap-8536	264	51	#	#	NOUN
ap-8536	264	52	(	(	PUNCT
ap-8536	264	53	ξ(a	ξ(a	ADJ
ap-8536	264	54	)	)	PUNCT
ap-8536	264	55	)	)	PUNCT
ap-8536	265	1	=	=	PUNCT
ap-8536	265	2	#	#	SYM
ap-8536	265	3	a	a	PRON
ap-8536	265	4	=	=	SYM
ap-8536	265	5	5	5	NUM
ap-8536	265	6	.	.	PUNCT
ap-8536	266	1	the	the	DET
ap-8536	266	2	vector	vector	NOUN
ap-8536	266	3	digit	digit	NOUN
ap-8536	266	4	set	set	NOUN
ap-8536	266	5	of	of	ADP
ap-8536	266	6	size	size	NOUN
ap-8536	266	7	5	5	NUM
ap-8536	266	8	has	have	VERB
ap-8536	266	9	elements	element	NOUN
ap-8536	266	10	{	{	PUNCT
ap-8536	266	11	(	(	PUNCT
ap-8536	266	12	0	0	NUM
ap-8536	266	13	,	,	PUNCT
ap-8536	266	14	0)⊤	0)⊤	PROPN
ap-8536	266	15	,	,	PUNCT
ap-8536	266	16	(	(	PUNCT
ap-8536	266	17	1	1	NUM
ap-8536	266	18	,	,	PUNCT
ap-8536	266	19	0)⊤	0)⊤	PROPN
ap-8536	266	20	,	,	PUNCT
ap-8536	266	21	(	(	PUNCT
ap-8536	266	22	−1	−1	NOUN
ap-8536	266	23	,	,	PUNCT
ap-8536	266	24	0)⊤	0)⊤	PROPN
ap-8536	266	25	,	,	PUNCT
ap-8536	266	26	(	(	PUNCT
ap-8536	266	27	0	0	NUM
ap-8536	266	28	,	,	PUNCT
ap-8536	266	29	1)⊤	1)⊤	NUM
ap-8536	266	30	,	,	PUNCT
ap-8536	266	31	(	(	PUNCT
ap-8536	266	32	0	0	NUM
ap-8536	266	33	,	,	PUNCT
ap-8536	266	34	−1)⊤	−1)⊤	NOUN
ap-8536	266	35	}	}	PUNCT
ap-8536	266	36	=	=	SYM
ap-8536	266	37	d.	d.	NOUN
ap-8536	266	38	as	as	SCONJ
ap-8536	266	39	proved	prove	VERB
ap-8536	266	40	in	in	ADP
ap-8536	266	41	[	[	X
ap-8536	266	42	20	20	NUM
ap-8536	266	43	]	]	PUNCT
ap-8536	266	44	,	,	PUNCT
ap-8536	266	45	the	the	DET
ap-8536	266	46	size	size	NOUN
ap-8536	266	47	of	of	ADP
ap-8536	266	48	5	5	NUM
ap-8536	266	49	is	be	AUX
ap-8536	266	50	minimal	minimal	ADJ
ap-8536	266	51	for	for	ADP
ap-8536	266	52	a	a	DET
ap-8536	266	53	digit	digit	NOUN
ap-8536	266	54	set	set	NOUN
ap-8536	266	55	allowing	allow	VERB
ap-8536	266	56	parallel	parallel	ADJ
ap-8536	266	57	addition	addition	NOUN
ap-8536	266	58	on	on	ADP
ap-8536	266	59	the	the	DET
ap-8536	266	60	number	number	NOUN
ap-8536	266	61	system	system	NOUN
ap-8536	266	62	with	with	ADP
ap-8536	266	63	base	base	NOUN
ap-8536	266	64	β	β	X
ap-8536	266	65	=	=	PUNCT
ap-8536	266	66	ı	ı	PROPN
ap-8536	266	67	−	−	NOUN
ap-8536	266	68	1	1	NUM
ap-8536	266	69	.	.	PUNCT
ap-8536	267	1	consequently	consequently	ADV
ap-8536	267	2	,	,	PUNCT
ap-8536	267	3	the	the	DET
ap-8536	267	4	digit	digit	NOUN
ap-8536	267	5	set	set	VERB
ap-8536	267	6	size	size	NOUN
ap-8536	267	7	#	#	NOUN
ap-8536	267	8	d	d	NOUN
ap-8536	267	9	=	=	SYM
ap-8536	267	10	5	5	NUM
ap-8536	267	11	must	must	AUX
ap-8536	267	12	be	be	AUX
ap-8536	267	13	minimal	minimal	ADJ
ap-8536	267	14	for	for	ADP
ap-8536	267	15	parallel	parallel	ADJ
ap-8536	267	16	addition	addition	NOUN
ap-8536	267	17	on	on	ADP
ap-8536	267	18	the	the	DET
ap-8536	267	19	matrix	matrix	NOUN
ap-8536	267	20	system	system	NOUN
ap-8536	267	21	(	(	PUNCT
ap-8536	267	22	m	m	PROPN
ap-8536	267	23	,	,	PUNCT
ap-8536	267	24	d	d	NOUN
ap-8536	267	25	)	)	PUNCT
ap-8536	267	26	as	as	ADV
ap-8536	267	27	well	well	ADV
ap-8536	267	28	,	,	PUNCT
ap-8536	267	29	due	due	ADP
ap-8536	267	30	to	to	ADP
ap-8536	267	31	the	the	DET
ap-8536	267	32	isomorphism	isomorphism	NOUN
ap-8536	267	33	ξ	ξ	PROPN
ap-8536	267	34	.	.	PUNCT
ap-8536	268	1	the	the	DET
ap-8536	268	2	algorithm	algorithm	NOUN
ap-8536	268	3	for	for	ADP
ap-8536	268	4	parallel	parallel	ADJ
ap-8536	268	5	addition	addition	NOUN
ap-8536	268	6	of	of	ADP
ap-8536	268	7	vectors	vector	NOUN
ap-8536	268	8	in	in	ADP
ap-8536	268	9	z2	z2	PROPN
ap-8536	268	10	presented	present	VERB
ap-8536	268	11	in	in	ADP
ap-8536	268	12	example	example	NOUN
ap-8536	268	13	12	12	NUM
ap-8536	268	14	uses	use	NOUN
ap-8536	268	15	,	,	PUNCT
ap-8536	268	16	for	for	ADP
ap-8536	268	17	the	the	DET
ap-8536	268	18	given	give	VERB
ap-8536	268	19	matrix	matrix	NOUN
ap-8536	268	20	base	base	NOUN
ap-8536	268	21	m	m	PROPN
ap-8536	268	22	,	,	PUNCT
ap-8536	268	23	a	a	DET
ap-8536	268	24	digit	digit	NOUN
ap-8536	268	25	set	set	NOUN
ap-8536	268	26	of	of	ADP
ap-8536	268	27	the	the	DET
ap-8536	268	28	minimal	minimal	ADJ
ap-8536	268	29	possible	possible	ADJ
ap-8536	268	30	size	size	NOUN
ap-8536	268	31	for	for	ADP
ap-8536	268	32	parallel	parallel	ADJ
ap-8536	268	33	addition	addition	NOUN
ap-8536	268	34	.	.	PUNCT
ap-8536	269	1	however	however	ADV
ap-8536	269	2	,	,	PUNCT
ap-8536	269	3	the	the	DET
ap-8536	269	4	way	way	NOUN
ap-8536	269	5	to	to	PART
ap-8536	269	6	determine	determine	VERB
ap-8536	269	7	the	the	DET
ap-8536	269	8	coefficients	coefficient	NOUN
ap-8536	269	9	qj	qj	PROPN
ap-8536	270	1	=	=	SYM
ap-8536	271	1	q(wj	q(wj	PROPN
ap-8536	271	2	,	,	PUNCT
ap-8536	271	3	.	.	PUNCT
ap-8536	271	4	.	.	PUNCT
ap-8536	271	5	.	.	PUNCT
ap-8536	272	1	,	,	PUNCT
ap-8536	272	2	wj−5	wj−5	PROPN
ap-8536	272	3	)	)	PUNCT
ap-8536	272	4	is	be	AUX
ap-8536	272	5	very	very	ADV
ap-8536	272	6	laborious	laborious	ADJ
ap-8536	272	7	,	,	PUNCT
ap-8536	272	8	as	as	SCONJ
ap-8536	272	9	the	the	DET
ap-8536	272	10	formula	formula	NOUN
ap-8536	272	11	for	for	ADP
ap-8536	272	12	q	q	NOUN
ap-8536	272	13	is	be	AUX
ap-8536	272	14	in	in	ADP
ap-8536	272	15	fact	fact	NOUN
ap-8536	272	16	a	a	DET
ap-8536	272	17	look	look	VERB
ap-8536	272	18	up	up	ADP
ap-8536	272	19	table	table	NOUN
ap-8536	272	20	with	with	ADP
ap-8536	272	21	136	136	NUM
ap-8536	272	22	rows	row	NOUN
ap-8536	272	23	.	.	PUNCT
ap-8536	273	1	with	with	SCONJ
ap-8536	273	2	the	the	DET
ap-8536	273	3	digit	digit	NOUN
ap-8536	273	4	set	set	VERB
ap-8536	273	5	size	size	NOUN
ap-8536	273	6	increased	increase	VERB
ap-8536	273	7	from	from	ADP
ap-8536	273	8	5	5	NUM
ap-8536	273	9	to	to	PART
ap-8536	273	10	9	9	NUM
ap-8536	273	11	elements	element	NOUN
ap-8536	273	12	,	,	PUNCT
ap-8536	273	13	a	a	DET
ap-8536	273	14	lot	lot	NOUN
ap-8536	273	15	simpler	simple	ADJ
ap-8536	273	16	algorithm	algorithm	NOUN
ap-8536	273	17	for	for	ADP
ap-8536	273	18	parallel	parallel	ADJ
ap-8536	273	19	addition	addition	NOUN
ap-8536	273	20	can	can	AUX
ap-8536	273	21	be	be	AUX
ap-8536	273	22	obtained	obtain	VERB
ap-8536	273	23	,	,	PUNCT
ap-8536	273	24	as	as	SCONJ
ap-8536	273	25	presented	present	VERB
ap-8536	273	26	in	in	ADP
ap-8536	273	27	the	the	DET
ap-8536	273	28	following	follow	VERB
ap-8536	273	29	example	example	NOUN
ap-8536	273	30	.	.	PUNCT
ap-8536	274	1	example	example	NOUN
ap-8536	274	2	13	13	NUM
ap-8536	274	3	.	.	PUNCT
ap-8536	275	1	let	let	VERB
ap-8536	275	2	us	we	PRON
ap-8536	275	3	consider	consider	VERB
ap-8536	275	4	m	m	VERB
ap-8536	275	5	=	=	PRON
ap-8536	275	6	(	(	PUNCT
ap-8536	275	7	−1	−1	NOUN
ap-8536	275	8	−1	−1	NOUN
ap-8536	275	9	+1	+1	ADJ
ap-8536	275	10	−1	−1	NOUN
ap-8536	275	11	)	)	PUNCT
ap-8536	275	12	and	and	CCONJ
ap-8536	275	13	the	the	DET
ap-8536	275	14	digit	digit	NOUN
ap-8536	275	15	set	set	NOUN
ap-8536	275	16	of	of	ADP
ap-8536	275	17	size	size	NOUN
ap-8536	275	18	9	9	NUM
ap-8536	275	19	d̃	d̃	PROPN
ap-8536	275	20	=	=	SYM
ap-8536	275	21	{	{	PUNCT
ap-8536	275	22	(	(	PUNCT
ap-8536	275	23	b	b	NOUN
ap-8536	275	24	c	c	NOUN
ap-8536	275	25	)	)	PUNCT
ap-8536	275	26	:	:	PUNCT
ap-8536	276	1	b	b	X
ap-8536	276	2	,	,	PUNCT
ap-8536	276	3	c	c	PROPN
ap-8536	276	4	∈	∈	PROPN
ap-8536	276	5	{	{	PUNCT
ap-8536	276	6	0	0	NUM
ap-8536	276	7	,	,	PUNCT
ap-8536	276	8	±1	±1	VERB
ap-8536	276	9	}	}	PUNCT
ap-8536	276	10	}	}	PUNCT
ap-8536	276	11	⊂	⊂	PROPN
ap-8536	276	12	z2	z2	PROPN
ap-8536	276	13	.	.	PUNCT
ap-8536	277	1	again	again	ADV
ap-8536	277	2	,	,	PUNCT
ap-8536	277	3	we	we	PRON
ap-8536	277	4	construct	construct	VERB
ap-8536	277	5	an	an	DET
ap-8536	277	6	auxiliary	auxiliary	ADJ
ap-8536	277	7	coefficient	coefficient	NOUN
ap-8536	277	8	function	function	NOUN
ap-8536	277	9	q̃	q̃	PROPN
ap-8536	277	10	:	:	PUNCT
ap-8536	277	11	(	(	PUNCT
ap-8536	277	12	d̃	d̃	PROPN
ap-8536	277	13	+	+	CCONJ
ap-8536	277	14	d̃)2	d̃)2	PROPN
ap-8536	277	15	7→	7→	NUM
ap-8536	277	16	q̃	q̃	PROPN
ap-8536	277	17	⊂	⊂	PROPN
ap-8536	277	18	z2	z2	PROPN
ap-8536	277	19	.	.	PUNCT
ap-8536	278	1	the	the	DET
ap-8536	278	2	coefficients	coefficient	NOUN
ap-8536	278	3	q̃j	q̃j	VERB
ap-8536	278	4	∈	∈	PROPN
ap-8536	278	5	q̃	q̃	PROPN
ap-8536	278	6	produced	produce	VERB
ap-8536	278	7	by	by	ADP
ap-8536	278	8	q̃	q̃	PROPN
ap-8536	278	9	then	then	ADV
ap-8536	278	10	provide	provide	VERB
ap-8536	278	11	the	the	DET
ap-8536	278	12	result	result	NOUN
ap-8536	278	13	sum	sum	NOUN
ap-8536	278	14	digits	digit	NOUN
ap-8536	278	15	z̃j	z̃j	NOUN
ap-8536	278	16	∈	∈	PROPN
ap-8536	278	17	d̃	d̃	PROPN
ap-8536	278	18	via	via	ADP
ap-8536	278	19	local	local	ADJ
ap-8536	278	20	function	function	NOUN
ap-8536	278	21	φ̃	φ̃	PROPN
ap-8536	278	22	:	:	PUNCT
ap-8536	278	23	(	(	PUNCT
ap-8536	278	24	d̃	d̃	PROPN
ap-8536	278	25	+	+	CCONJ
ap-8536	278	26	d̃)3	d̃)3	PROPN
ap-8536	278	27	7→	7→	PROPN
ap-8536	278	28	d̃	d̃	PROPN
ap-8536	278	29	,	,	PUNCT
ap-8536	278	30	as	as	SCONJ
ap-8536	278	31	follows	follow	VERB
ap-8536	278	32	:	:	PUNCT
ap-8536	278	33	q̃j	q̃j	NOUN
ap-8536	278	34	:	:	PUNCT
ap-8536	278	35	=	=	SYM
ap-8536	278	36	q̃(w̃j	q̃(w̃j	NOUN
ap-8536	278	37	,	,	PUNCT
ap-8536	278	38	w̃j−2	w̃j−2	PROPN
ap-8536	278	39	)	)	PUNCT
ap-8536	278	40	∈	∈	PROPN
ap-8536	278	41	q̃	q̃	PROPN
ap-8536	278	42	and	and	CCONJ
ap-8536	278	43	,	,	PUNCT
ap-8536	278	44	consequently	consequently	ADV
ap-8536	278	45	z̃j	z̃j	NOUN
ap-8536	278	46	:	:	PUNCT
ap-8536	278	47	=	=	SYM
ap-8536	278	48	w̃j	w̃j	NOUN
ap-8536	278	49	+	+	CCONJ
ap-8536	278	50	q̃j−2	q̃j−2	NOUN
ap-8536	278	51	−	−	NOUN
ap-8536	278	52	m2q̃j	m2q̃j	NOUN
ap-8536	278	53	=	=	PUNCT
ap-8536	278	54	φ̃(w̃j	φ̃(w̃j	NOUN
ap-8536	278	55	,	,	PUNCT
ap-8536	278	56	w̃j−2	w̃j−2	PROPN
ap-8536	278	57	,	,	PUNCT
ap-8536	278	58	w̃j−4	w̃j−4	NUM
ap-8536	278	59	)	)	PUNCT
ap-8536	278	60	∈	∈	PROPN
ap-8536	278	61	d̃	d̃	PROPN
ap-8536	278	62	.	.	PUNCT
ap-8536	279	1	192	192	NUM
ap-8536	279	2	vol	vol	NOUN
ap-8536	279	3	.	.	PUNCT
ap-8536	279	4	63	63	NUM
ap-8536	279	5	no	no	NOUN
ap-8536	279	6	.	.	PUNCT
ap-8536	280	1	3/2023	3/2023	NUM
ap-8536	280	2	positional	positional	ADJ
ap-8536	280	3	representation	representation	NOUN
ap-8536	280	4	of	of	ADP
ap-8536	280	5	vectors	vector	NOUN
ap-8536	280	6	the	the	DET
ap-8536	280	7	coefficient	coefficient	NOUN
ap-8536	280	8	set	set	VERB
ap-8536	280	9	q̃	q̃	PROPN
ap-8536	280	10	is	be	AUX
ap-8536	280	11	,	,	PUNCT
ap-8536	280	12	just	just	ADV
ap-8536	280	13	by	by	ADP
ap-8536	280	14	coincidence	coincidence	NOUN
ap-8536	280	15	,	,	PUNCT
ap-8536	280	16	equal	equal	ADJ
ap-8536	280	17	to	to	ADP
ap-8536	280	18	the	the	DET
ap-8536	280	19	digit	digit	NOUN
ap-8536	280	20	set	set	VERB
ap-8536	280	21	d̃	d̃	PROPN
ap-8536	280	22	:	:	PUNCT
ap-8536	280	23	q̃	q̃	PROPN
ap-8536	280	24	=	=	PRON
ap-8536	280	25	{	{	PUNCT
ap-8536	280	26	(	(	PUNCT
ap-8536	280	27	0	0	NUM
ap-8536	280	28	0	0	NUM
ap-8536	280	29	)	)	PUNCT
ap-8536	280	30	,	,	PUNCT
ap-8536	280	31	±	±	NUM
ap-8536	280	32	(	(	PUNCT
ap-8536	280	33	1	1	NUM
ap-8536	280	34	0	0	NUM
ap-8536	280	35	)	)	PUNCT
ap-8536	280	36	,	,	PUNCT
ap-8536	280	37	±	±	NUM
ap-8536	280	38	(	(	PUNCT
ap-8536	280	39	0	0	NUM
ap-8536	280	40	1	1	NUM
ap-8536	280	41	)	)	PUNCT
ap-8536	280	42	,	,	PUNCT
ap-8536	280	43	±	±	NUM
ap-8536	280	44	(	(	PUNCT
ap-8536	280	45	1	1	NUM
ap-8536	280	46	1	1	NUM
ap-8536	280	47	)	)	PUNCT
ap-8536	280	48	,	,	PUNCT
ap-8536	280	49	±	±	NUM
ap-8536	280	50	(	(	PUNCT
ap-8536	280	51	1	1	NUM
ap-8536	280	52	−1	−1	NOUN
ap-8536	280	53	)	)	PUNCT
ap-8536	280	54	}	}	PUNCT
ap-8536	280	55	.	.	PUNCT
ap-8536	281	1	the	the	DET
ap-8536	281	2	interim	interim	ADJ
ap-8536	281	3	sum	sum	NOUN
ap-8536	281	4	digit	digit	NOUN
ap-8536	281	5	set	set	VERB
ap-8536	281	6	w̃	w̃	PROPN
ap-8536	281	7	=	=	SYM
ap-8536	281	8	d̃	d̃	PROPN
ap-8536	281	9	+	+	CCONJ
ap-8536	281	10	d̃	d̃	PROPN
ap-8536	281	11	=	=	SYM
ap-8536	281	12	{	{	PUNCT
ap-8536	281	13	(	(	PUNCT
ap-8536	281	14	b	b	NOUN
ap-8536	281	15	c	c	NOUN
ap-8536	281	16	)	)	PUNCT
ap-8536	281	17	:	:	PUNCT
ap-8536	282	1	b	b	X
ap-8536	282	2	,	,	PUNCT
ap-8536	282	3	c	c	PROPN
ap-8536	282	4	∈	∈	PROPN
ap-8536	282	5	{	{	PUNCT
ap-8536	282	6	0	0	NUM
ap-8536	282	7	,	,	PUNCT
ap-8536	282	8	±1	±1	VERB
ap-8536	282	9	,	,	PUNCT
ap-8536	282	10	±2	±2	NOUN
ap-8536	282	11	}	}	PUNCT
ap-8536	282	12	}	}	PUNCT
ap-8536	282	13	has	have	VERB
ap-8536	282	14	25	25	NUM
ap-8536	282	15	elements	element	NOUN
ap-8536	282	16	and	and	CCONJ
ap-8536	282	17	can	can	AUX
ap-8536	282	18	be	be	AUX
ap-8536	282	19	rewritten	rewrite	VERB
ap-8536	282	20	using	use	VERB
ap-8536	282	21	the	the	DET
ap-8536	282	22	rotation	rotation	NOUN
ap-8536	282	23	symmetry	symmetry	NOUN
ap-8536	282	24	by	by	ADP
ap-8536	282	25	the	the	DET
ap-8536	282	26	angle	angle	NOUN
ap-8536	282	27	π	π	PROPN
ap-8536	282	28	2	2	NUM
ap-8536	282	29	as	as	SCONJ
ap-8536	282	30	follows	follow	VERB
ap-8536	282	31	w̃	w̃	PROPN
ap-8536	282	32	=	=	SYM
ap-8536	282	33	w̃0	w̃0	PROPN
ap-8536	282	34	∪	∪	VERB
ap-8536	282	35	w̃1	w̃1	PROPN
ap-8536	282	36	∪	∪	ADP
ap-8536	282	37	w̃2	w̃2	PROPN
ap-8536	282	38	∪	∪	PROPN
ap-8536	282	39	w̃3	w̃3	PROPN
ap-8536	282	40	,	,	PUNCT
ap-8536	282	41	where	where	SCONJ
ap-8536	282	42	w̃0	w̃0	PROPN
ap-8536	282	43	=	=	SYM
ap-8536	282	44	{	{	PUNCT
ap-8536	282	45	(	(	PUNCT
ap-8536	282	46	0	0	NUM
ap-8536	282	47	0	0	NUM
ap-8536	282	48	)	)	PUNCT
ap-8536	282	49	,	,	PUNCT
ap-8536	282	50	(	(	PUNCT
ap-8536	282	51	1	1	NUM
ap-8536	282	52	0	0	NUM
ap-8536	282	53	)	)	PUNCT
ap-8536	282	54	,	,	PUNCT
ap-8536	282	55	(	(	PUNCT
ap-8536	282	56	2	2	NUM
ap-8536	282	57	0	0	NUM
ap-8536	282	58	)	)	PUNCT
ap-8536	282	59	,	,	PUNCT
ap-8536	282	60	(	(	PUNCT
ap-8536	282	61	1	1	NUM
ap-8536	282	62	1	1	NUM
ap-8536	282	63	)	)	PUNCT
ap-8536	282	64	,	,	PUNCT
ap-8536	282	65	(	(	PUNCT
ap-8536	282	66	2	2	NUM
ap-8536	282	67	1	1	NUM
ap-8536	282	68	)	)	PUNCT
ap-8536	282	69	,	,	PUNCT
ap-8536	282	70	(	(	PUNCT
ap-8536	282	71	1	1	NUM
ap-8536	282	72	2	2	NUM
ap-8536	282	73	)	)	PUNCT
ap-8536	282	74	,	,	PUNCT
ap-8536	282	75	(	(	PUNCT
ap-8536	282	76	2	2	NUM
ap-8536	282	77	2	2	NUM
ap-8536	282	78	)	)	PUNCT
ap-8536	282	79	}	}	PUNCT
ap-8536	282	80	and	and	CCONJ
ap-8536	282	81	w̃k	w̃k	ADV
ap-8536	282	82	=	=	VERB
ap-8536	282	83	rk	rk	PROPN
ap-8536	282	84	w̃0	w̃0	PROPN
ap-8536	282	85	for	for	ADP
ap-8536	282	86	k	k	PROPN
ap-8536	282	87	=	=	SYM
ap-8536	282	88	1	1	NUM
ap-8536	282	89	,	,	PUNCT
ap-8536	282	90	2	2	NUM
ap-8536	282	91	,	,	PUNCT
ap-8536	282	92	3	3	NUM
ap-8536	282	93	,	,	PUNCT
ap-8536	282	94	with	with	ADP
ap-8536	282	95	r	r	NOUN
ap-8536	282	96	=	=	SYM
ap-8536	282	97	(	(	PUNCT
ap-8536	282	98	0	0	NUM
ap-8536	282	99	−1	−1	NOUN
ap-8536	282	100	+1	+1	ADJ
ap-8536	282	101	0	0	NUM
ap-8536	282	102	)	)	PUNCT
ap-8536	282	103	.	.	PUNCT
ap-8536	283	1	thanks	thank	NOUN
ap-8536	283	2	to	to	ADP
ap-8536	283	3	the	the	DET
ap-8536	283	4	π	π	PROPN
ap-8536	283	5	2	2	NUM
ap-8536	283	6	-rotation	-rotation	NOUN
ap-8536	283	7	symmetry	symmetry	NOUN
ap-8536	283	8	of	of	ADP
ap-8536	283	9	all	all	DET
ap-8536	283	10	the	the	DET
ap-8536	283	11	sets	set	NOUN
ap-8536	283	12	in	in	ADP
ap-8536	283	13	question	question	NOUN
ap-8536	283	14	,	,	PUNCT
ap-8536	283	15	i.e.	i.e.	X
ap-8536	283	16	,	,	PUNCT
ap-8536	283	17	d̃	d̃	PROPN
ap-8536	283	18	,	,	PUNCT
ap-8536	283	19	q̃	q̃	PROPN
ap-8536	283	20	,	,	PUNCT
ap-8536	283	21	and	and	CCONJ
ap-8536	283	22	w̃	w̃	PROPN
ap-8536	283	23	=	=	SYM
ap-8536	283	24	d̃	d̃	PROPN
ap-8536	283	25	+	+	CCONJ
ap-8536	283	26	d̃	d̃	PROPN
ap-8536	283	27	,	,	PUNCT
ap-8536	283	28	it	it	PRON
ap-8536	283	29	is	be	AUX
ap-8536	283	30	enough	enough	ADJ
ap-8536	283	31	to	to	PART
ap-8536	283	32	specify	specify	VERB
ap-8536	283	33	the	the	DET
ap-8536	283	34	coefficient	coefficient	NOUN
ap-8536	283	35	function	function	NOUN
ap-8536	283	36	by	by	ADP
ap-8536	283	37	listing	list	VERB
ap-8536	283	38	its	its	PRON
ap-8536	283	39	values	value	NOUN
ap-8536	283	40	q̃(w̃j	q̃(w̃j	NOUN
ap-8536	283	41	,	,	PUNCT
ap-8536	283	42	w̃j−2	w̃j−2	PROPN
ap-8536	283	43	)	)	PUNCT
ap-8536	283	44	just	just	ADV
ap-8536	283	45	for	for	ADP
ap-8536	283	46	w̃j	w̃j	NOUN
ap-8536	283	47	∈	∈	PROPN
ap-8536	283	48	w̃0	w̃0	PROPN
ap-8536	283	49	.	.	PUNCT
ap-8536	284	1	all	all	DET
ap-8536	284	2	the	the	DET
ap-8536	284	3	rest	rest	NOUN
ap-8536	284	4	can	can	AUX
ap-8536	284	5	then	then	ADV
ap-8536	284	6	be	be	AUX
ap-8536	284	7	obtained	obtain	VERB
ap-8536	284	8	by	by	ADP
ap-8536	284	9	rotation	rotation	NOUN
ap-8536	284	10	:	:	PUNCT
ap-8536	284	11	q̃	q̃	PROPN
ap-8536	284	12	(	(	PUNCT
ap-8536	284	13	rkw̃j	rkw̃j	PROPN
ap-8536	284	14	,	,	PUNCT
ap-8536	284	15	rkw̃j−2	rkw̃j−2	X
ap-8536	284	16	)	)	PUNCT
ap-8536	285	1	=	=	VERB
ap-8536	285	2	rk	rk	PROPN
ap-8536	285	3	·	·	PUNCT
ap-8536	285	4	q̃(w̃j	q̃(w̃j	NOUN
ap-8536	285	5	,	,	PUNCT
ap-8536	285	6	w̃j−2	w̃j−2	PROPN
ap-8536	285	7	)	)	PUNCT
ap-8536	285	8	.	.	PUNCT
ap-8536	286	1	for	for	ADP
ap-8536	286	2	all	all	DET
ap-8536	286	3	w̃j	w̃j	NOUN
ap-8536	286	4	∈	∈	PROPN
ap-8536	286	5	w̃0	w̃0	PROPN
ap-8536	286	6	and	and	CCONJ
ap-8536	286	7	w̃j−2	w̃j−2	PROPN
ap-8536	286	8	=	=	SYM
ap-8536	287	1	(	(	PUNCT
ap-8536	287	2	b	b	X
ap-8536	287	3	c	c	NOUN
ap-8536	287	4	)	)	PUNCT
ap-8536	287	5	∈	∈	PROPN
ap-8536	287	6	w̃	w̃	PROPN
ap-8536	287	7	,	,	PUNCT
ap-8536	287	8	the	the	DET
ap-8536	287	9	coefficients	coefficient	NOUN
ap-8536	287	10	q̃j	q̃j	VERB
ap-8536	287	11	assigned	assign	VERB
ap-8536	287	12	by	by	ADP
ap-8536	287	13	q̃	q̃	PROPN
ap-8536	287	14	are	be	AUX
ap-8536	287	15	listed	list	VERB
ap-8536	287	16	below	below	ADP
ap-8536	287	17	•	•	NOUN
ap-8536	287	18	q̃	q̃	PROPN
ap-8536	287	19	(	(	PUNCT
ap-8536	287	20	(	(	PUNCT
ap-8536	287	21	0	0	NUM
ap-8536	287	22	0	0	NUM
ap-8536	287	23	)	)	PUNCT
ap-8536	287	24	,	,	PUNCT
ap-8536	287	25	(	(	PUNCT
ap-8536	287	26	b	b	X
ap-8536	287	27	c	c	NOUN
ap-8536	287	28	)	)	PUNCT
ap-8536	287	29	)	)	PUNCT
ap-8536	288	1	:	:	PUNCT
ap-8536	288	2	=	=	PUNCT
ap-8536	288	3	(	(	PUNCT
ap-8536	288	4	0	0	NUM
ap-8536	288	5	0	0	NUM
ap-8536	288	6	)	)	PUNCT
ap-8536	288	7	;	;	PUNCT
ap-8536	288	8	•	•	NUM
ap-8536	288	9	q̃	q̃	PROPN
ap-8536	288	10	(	(	PUNCT
ap-8536	288	11	(	(	PUNCT
ap-8536	288	12	2	2	NUM
ap-8536	288	13	0	0	NUM
ap-8536	288	14	)	)	PUNCT
ap-8536	288	15	,	,	PUNCT
ap-8536	288	16	(	(	PUNCT
ap-8536	288	17	b	b	X
ap-8536	288	18	c	c	NOUN
ap-8536	288	19	)	)	PUNCT
ap-8536	288	20	)	)	PUNCT
ap-8536	289	1	:	:	PUNCT
ap-8536	289	2	=	=	PUNCT
ap-8536	289	3	(	(	PUNCT
ap-8536	289	4	0	0	NUM
ap-8536	289	5	1	1	NUM
ap-8536	289	6	)	)	PUNCT
ap-8536	289	7	;	;	PUNCT
ap-8536	289	8	•	•	NUM
ap-8536	289	9	q̃	q̃	PROPN
ap-8536	289	10	(	(	PUNCT
ap-8536	289	11	(	(	PUNCT
ap-8536	289	12	2	2	NUM
ap-8536	289	13	2	2	NUM
ap-8536	289	14	)	)	PUNCT
ap-8536	289	15	,	,	PUNCT
ap-8536	289	16	(	(	PUNCT
ap-8536	289	17	b	b	X
ap-8536	289	18	c	c	NOUN
ap-8536	289	19	)	)	PUNCT
ap-8536	289	20	)	)	PUNCT
ap-8536	289	21	:	:	PUNCT
ap-8536	289	22	=	=	SYM
ap-8536	289	23	(	(	PUNCT
ap-8536	289	24	−1	−1	NOUN
ap-8536	289	25	1	1	NUM
ap-8536	289	26	)	)	PUNCT
ap-8536	289	27	;	;	PUNCT
ap-8536	289	28	•	•	NUM
ap-8536	289	29	q̃	q̃	PROPN
ap-8536	289	30	(	(	PUNCT
ap-8536	289	31	(	(	PUNCT
ap-8536	289	32	1	1	NUM
ap-8536	289	33	0	0	NUM
ap-8536	289	34	)	)	PUNCT
ap-8536	289	35	,	,	PUNCT
ap-8536	289	36	(	(	PUNCT
ap-8536	289	37	b	b	X
ap-8536	289	38	c	c	NOUN
ap-8536	289	39	)	)	PUNCT
ap-8536	289	40	)	)	PUNCT
ap-8536	289	41	:	:	PUNCT
ap-8536	289	42	=	=	PUNCT
ap-8536	289	43	(	(	PUNCT
ap-8536	289	44	0	0	NUM
ap-8536	289	45	0	0	NUM
ap-8536	289	46	)	)	PUNCT
ap-8536	289	47	,	,	PUNCT
ap-8536	289	48	if	if	SCONJ
ap-8536	289	49	c	c	PROPN
ap-8536	289	50	≥	≥	NOUN
ap-8536	289	51	0	0	NUM
ap-8536	289	52	;	;	PUNCT
ap-8536	289	53	•	•	NUM
ap-8536	289	54	q̃	q̃	PROPN
ap-8536	289	55	(	(	PUNCT
ap-8536	289	56	(	(	PUNCT
ap-8536	289	57	1	1	NUM
ap-8536	289	58	0	0	NUM
ap-8536	289	59	)	)	PUNCT
ap-8536	289	60	,	,	PUNCT
ap-8536	289	61	(	(	PUNCT
ap-8536	289	62	b	b	X
ap-8536	289	63	c	c	NOUN
ap-8536	289	64	)	)	PUNCT
ap-8536	289	65	)	)	PUNCT
ap-8536	290	1	:	:	PUNCT
ap-8536	290	2	=	=	PUNCT
ap-8536	290	3	(	(	PUNCT
ap-8536	290	4	0	0	NUM
ap-8536	290	5	1	1	NUM
ap-8536	290	6	)	)	PUNCT
ap-8536	290	7	,	,	PUNCT
ap-8536	290	8	if	if	SCONJ
ap-8536	290	9	c	c	PROPN
ap-8536	290	10	<	<	X
ap-8536	290	11	0	0	NUM
ap-8536	290	12	;	;	PUNCT
ap-8536	290	13	•	•	NUM
ap-8536	290	14	q̃	q̃	PROPN
ap-8536	290	15	(	(	PUNCT
ap-8536	290	16	(	(	PUNCT
ap-8536	290	17	1	1	NUM
ap-8536	290	18	2	2	NUM
ap-8536	290	19	)	)	PUNCT
ap-8536	290	20	,	,	PUNCT
ap-8536	290	21	(	(	PUNCT
ap-8536	290	22	b	b	X
ap-8536	290	23	c	c	NOUN
ap-8536	290	24	)	)	PUNCT
ap-8536	290	25	)	)	PUNCT
ap-8536	290	26	:	:	PUNCT
ap-8536	290	27	=	=	PUNCT
ap-8536	290	28	(	(	PUNCT
ap-8536	290	29	−1	−1	NOUN
ap-8536	290	30	0	0	NUM
ap-8536	290	31	)	)	PUNCT
ap-8536	290	32	,	,	PUNCT
ap-8536	290	33	if	if	SCONJ
ap-8536	290	34	c	c	PROPN
ap-8536	290	35	≥	≥	NOUN
ap-8536	290	36	0	0	NUM
ap-8536	290	37	;	;	PUNCT
ap-8536	290	38	•	•	NUM
ap-8536	290	39	q̃	q̃	PROPN
ap-8536	290	40	(	(	PUNCT
ap-8536	290	41	(	(	PUNCT
ap-8536	290	42	1	1	NUM
ap-8536	290	43	2	2	NUM
ap-8536	290	44	)	)	PUNCT
ap-8536	290	45	,	,	PUNCT
ap-8536	290	46	(	(	PUNCT
ap-8536	290	47	b	b	X
ap-8536	290	48	c	c	NOUN
ap-8536	290	49	)	)	PUNCT
ap-8536	290	50	)	)	PUNCT
ap-8536	291	1	:	:	PUNCT
ap-8536	291	2	=	=	SYM
ap-8536	291	3	(	(	PUNCT
ap-8536	291	4	−1	−1	NOUN
ap-8536	291	5	1	1	NUM
ap-8536	291	6	)	)	PUNCT
ap-8536	291	7	,	,	PUNCT
ap-8536	291	8	if	if	SCONJ
ap-8536	291	9	c	c	PROPN
ap-8536	291	10	<	<	X
ap-8536	291	11	0	0	NUM
ap-8536	291	12	;	;	PUNCT
ap-8536	291	13	•	•	NUM
ap-8536	291	14	q̃	q̃	PROPN
ap-8536	291	15	(	(	PUNCT
ap-8536	291	16	(	(	PUNCT
ap-8536	291	17	2	2	NUM
ap-8536	291	18	1	1	NUM
ap-8536	291	19	)	)	PUNCT
ap-8536	291	20	,	,	PUNCT
ap-8536	291	21	(	(	PUNCT
ap-8536	291	22	b	b	X
ap-8536	291	23	c	c	NOUN
ap-8536	291	24	)	)	PUNCT
ap-8536	291	25	)	)	PUNCT
ap-8536	292	1	:	:	PUNCT
ap-8536	292	2	=	=	PUNCT
ap-8536	292	3	(	(	PUNCT
ap-8536	292	4	0	0	NUM
ap-8536	292	5	1	1	NUM
ap-8536	292	6	)	)	PUNCT
ap-8536	292	7	,	,	PUNCT
ap-8536	292	8	if	if	SCONJ
ap-8536	292	9	b	b	PROPN
ap-8536	292	10	≤	≤	ADV
ap-8536	292	11	0	0	NUM
ap-8536	292	12	;	;	PUNCT
ap-8536	292	13	•	•	NUM
ap-8536	292	14	q̃	q̃	PROPN
ap-8536	292	15	(	(	PUNCT
ap-8536	292	16	(	(	PUNCT
ap-8536	292	17	2	2	NUM
ap-8536	292	18	1	1	NUM
ap-8536	292	19	)	)	PUNCT
ap-8536	292	20	,	,	PUNCT
ap-8536	292	21	(	(	PUNCT
ap-8536	292	22	b	b	X
ap-8536	292	23	c	c	NOUN
ap-8536	292	24	)	)	PUNCT
ap-8536	292	25	)	)	PUNCT
ap-8536	292	26	:	:	PUNCT
ap-8536	292	27	=	=	SYM
ap-8536	292	28	(	(	PUNCT
ap-8536	292	29	−1	−1	NOUN
ap-8536	292	30	1	1	NUM
ap-8536	292	31	)	)	PUNCT
ap-8536	292	32	,	,	PUNCT
ap-8536	292	33	if	if	SCONJ
ap-8536	292	34	b	b	X
ap-8536	292	35	>	>	X
ap-8536	292	36	0	0	NUM
ap-8536	292	37	;	;	PUNCT
ap-8536	292	38	•	•	NUM
ap-8536	292	39	q̃	q̃	PROPN
ap-8536	292	40	(	(	PUNCT
ap-8536	292	41	(	(	PUNCT
ap-8536	292	42	1	1	NUM
ap-8536	292	43	1	1	NUM
ap-8536	292	44	)	)	PUNCT
ap-8536	292	45	,	,	PUNCT
ap-8536	292	46	(	(	PUNCT
ap-8536	292	47	b	b	X
ap-8536	292	48	c	c	NOUN
ap-8536	292	49	)	)	PUNCT
ap-8536	292	50	)	)	PUNCT
ap-8536	292	51	:	:	PUNCT
ap-8536	292	52	=	=	PUNCT
ap-8536	292	53	(	(	PUNCT
ap-8536	292	54	0	0	NUM
ap-8536	292	55	0	0	NUM
ap-8536	292	56	)	)	PUNCT
ap-8536	292	57	,	,	PUNCT
ap-8536	292	58	if	if	SCONJ
ap-8536	292	59	b	b	PROPN
ap-8536	292	60	≤	≤	X
ap-8536	292	61	0	0	NUM
ap-8536	292	62	≤	≤	NUM
ap-8536	292	63	c	c	NOUN
ap-8536	292	64	;	;	PUNCT
ap-8536	292	65	•	•	NUM
ap-8536	292	66	q̃	q̃	PROPN
ap-8536	292	67	(	(	PUNCT
ap-8536	292	68	(	(	PUNCT
ap-8536	292	69	1	1	NUM
ap-8536	292	70	1	1	NUM
ap-8536	292	71	)	)	PUNCT
ap-8536	292	72	,	,	PUNCT
ap-8536	292	73	(	(	PUNCT
ap-8536	292	74	b	b	X
ap-8536	292	75	c	c	NOUN
ap-8536	292	76	)	)	PUNCT
ap-8536	292	77	)	)	PUNCT
ap-8536	292	78	:	:	PUNCT
ap-8536	292	79	=	=	PUNCT
ap-8536	292	80	(	(	PUNCT
ap-8536	292	81	−1	−1	NOUN
ap-8536	292	82	0	0	NUM
ap-8536	292	83	)	)	PUNCT
ap-8536	292	84	,	,	PUNCT
ap-8536	292	85	if	if	SCONJ
ap-8536	292	86	b	b	X
ap-8536	292	87	>	>	X
ap-8536	292	88	0	0	PUNCT
ap-8536	292	89	and	and	CCONJ
ap-8536	292	90	c	c	X
ap-8536	292	91	>	>	X
ap-8536	292	92	0	0	NUM
ap-8536	292	93	;	;	PUNCT
ap-8536	292	94	•	•	NUM
ap-8536	292	95	q̃	q̃	PROPN
ap-8536	292	96	(	(	PUNCT
ap-8536	292	97	(	(	PUNCT
ap-8536	292	98	1	1	NUM
ap-8536	292	99	1	1	NUM
ap-8536	292	100	)	)	PUNCT
ap-8536	292	101	,	,	PUNCT
ap-8536	292	102	(	(	PUNCT
ap-8536	292	103	b	b	X
ap-8536	292	104	c	c	NOUN
ap-8536	292	105	)	)	PUNCT
ap-8536	292	106	)	)	PUNCT
ap-8536	292	107	:	:	PUNCT
ap-8536	292	108	=	=	PUNCT
ap-8536	292	109	(	(	PUNCT
ap-8536	292	110	0	0	NUM
ap-8536	292	111	1	1	NUM
ap-8536	292	112	)	)	PUNCT
ap-8536	292	113	,	,	PUNCT
ap-8536	292	114	if	if	SCONJ
ap-8536	292	115	b	b	PROPN
ap-8536	292	116	<	<	X
ap-8536	292	117	0	0	NUM
ap-8536	292	118	and	and	CCONJ
ap-8536	292	119	c	c	X
ap-8536	292	120	<	<	X
ap-8536	292	121	0	0	NUM
ap-8536	292	122	;	;	PUNCT
ap-8536	292	123	•	•	NUM
ap-8536	292	124	q̃	q̃	PROPN
ap-8536	292	125	(	(	PUNCT
ap-8536	292	126	(	(	PUNCT
ap-8536	292	127	1	1	NUM
ap-8536	292	128	1	1	NUM
ap-8536	292	129	)	)	PUNCT
ap-8536	292	130	,	,	PUNCT
ap-8536	292	131	(	(	PUNCT
ap-8536	292	132	b	b	X
ap-8536	292	133	c	c	NOUN
ap-8536	292	134	)	)	PUNCT
ap-8536	292	135	)	)	PUNCT
ap-8536	292	136	:	:	PUNCT
ap-8536	292	137	=	=	SYM
ap-8536	292	138	(	(	PUNCT
ap-8536	292	139	−1	−1	NOUN
ap-8536	292	140	1	1	NUM
ap-8536	292	141	)	)	PUNCT
ap-8536	292	142	,	,	PUNCT
ap-8536	292	143	if	if	SCONJ
ap-8536	292	144	c	c	PROPN
ap-8536	292	145	≤	≤	X
ap-8536	292	146	0	0	NUM
ap-8536	292	147	≤	≤	NUM
ap-8536	292	148	b	b	NOUN
ap-8536	292	149	and	and	CCONJ
ap-8536	292	150	(	(	PUNCT
ap-8536	292	151	b	b	PROPN
ap-8536	292	152	c	c	NOUN
ap-8536	292	153	)	)	PUNCT
ap-8536	292	154	̸=	̸=	PROPN
ap-8536	292	155	(	(	PUNCT
ap-8536	292	156	0	0	NUM
ap-8536	292	157	0	0	NUM
ap-8536	292	158	)	)	PUNCT
ap-8536	292	159	.	.	PUNCT
ap-8536	293	1	by	by	ADP
ap-8536	293	2	checking	check	VERB
ap-8536	293	3	all	all	DET
ap-8536	293	4	the	the	DET
ap-8536	293	5	possible	possible	ADJ
ap-8536	293	6	variants	variant	NOUN
ap-8536	293	7	(	(	PUNCT
ap-8536	293	8	w̃j	w̃j	NOUN
ap-8536	293	9	,	,	PUNCT
ap-8536	294	1	w̃j−2	w̃j−2	PROPN
ap-8536	294	2	,	,	PUNCT
ap-8536	294	3	w̃j−4	w̃j−4	NUM
ap-8536	294	4	)	)	PUNCT
ap-8536	294	5	∈	∈	PROPN
ap-8536	294	6	w̃3	w̃3	PROPN
ap-8536	294	7	,	,	PUNCT
ap-8536	294	8	it	it	PRON
ap-8536	294	9	can	can	AUX
ap-8536	294	10	be	be	AUX
ap-8536	294	11	verified	verify	VERB
ap-8536	294	12	that	that	SCONJ
ap-8536	294	13	the	the	DET
ap-8536	294	14	final	final	ADJ
ap-8536	294	15	digit	digit	NOUN
ap-8536	294	16	z̃j	z̃j	NOUN
ap-8536	294	17	calculated	calculate	VERB
ap-8536	294	18	by	by	ADP
ap-8536	294	19	z̃j	z̃j	NOUN
ap-8536	294	20	:	:	PUNCT
ap-8536	294	21	=	=	SYM
ap-8536	294	22	w̃j	w̃j	X
ap-8536	294	23	+	+	CCONJ
ap-8536	294	24	q̃j−2	q̃j−2	PROPN
ap-8536	294	25	−m2q̃j	−m2q̃j	PROPN
ap-8536	294	26	is	be	AUX
ap-8536	294	27	always	always	ADV
ap-8536	294	28	an	an	DET
ap-8536	294	29	element	element	NOUN
ap-8536	294	30	of	of	ADP
ap-8536	294	31	the	the	DET
ap-8536	294	32	desired	desire	VERB
ap-8536	294	33	digit	digit	NOUN
ap-8536	294	34	set	set	VERB
ap-8536	294	35	d̃.	d̃.	VERB
ap-8536	294	36	correct	correct	ADJ
ap-8536	294	37	value	value	NOUN
ap-8536	294	38	of	of	ADP
ap-8536	294	39	the	the	DET
ap-8536	294	40	final	final	ADJ
ap-8536	294	41	sum	sum	NOUN
ap-8536	294	42	z̃	z̃	PROPN
ap-8536	294	43	=	=	PUNCT
ap-8536	294	44	∑	∑	PUNCT
ap-8536	294	45	j∈z	j∈z	PROPN
ap-8536	294	46	m	m	PROPN
ap-8536	294	47	j	j	PROPN
ap-8536	294	48	z̃j	z̃j	PROPN
ap-8536	294	49	is	be	AUX
ap-8536	294	50	guaranteed	guarantee	VERB
ap-8536	294	51	by	by	ADP
ap-8536	294	52	z̃	z̃	PROPN
ap-8536	294	53	=	=	PUNCT
ap-8536	294	54	∑	∑	PUNCT
ap-8536	294	55	j∈z	j∈z	PROPN
ap-8536	294	56	m	m	PROPN
ap-8536	294	57	j	j	PROPN
ap-8536	294	58	z̃j	z̃j	NOUN
ap-8536	294	59	=	=	PUNCT
ap-8536	294	60	∑	∑	PUNCT
ap-8536	294	61	j∈z	j∈z	PROPN
ap-8536	294	62	m	m	PROPN
ap-8536	294	63	j(w̃j	j(w̃j	PROPN
ap-8536	294	64	+	+	CCONJ
ap-8536	294	65	q̃j−2	q̃j−2	PROPN
ap-8536	294	66	−	−	NOUN
ap-8536	294	67	m2q̃j	m2q̃j	NOUN
ap-8536	294	68	)	)	PUNCT
ap-8536	294	69	=	=	PUNCT
ap-8536	295	1	∑	∑	PUNCT
ap-8536	295	2	j∈z	j∈z	PROPN
ap-8536	295	3	m	m	PROPN
ap-8536	295	4	jw̃j	jw̃j	PROPN
ap-8536	296	1	+	+	CCONJ
ap-8536	296	2	∑	∑	PUNCT
ap-8536	296	3	j∈z	j∈z	PROPN
ap-8536	296	4	m	m	PROPN
ap-8536	296	5	j	j	PROPN
ap-8536	296	6	q̃j−2	q̃j−2	PROPN
ap-8536	296	7	−	−	PROPN
ap-8536	296	8	∑	∑	PROPN
ap-8536	296	9	j∈z	j∈z	PROPN
ap-8536	296	10	m	m	PROPN
ap-8536	296	11	j+2q̃j	j+2q̃j	NOUN
ap-8536	296	12	=	=	PUNCT
ap-8536	297	1	=	=	PUNCT
ap-8536	297	2	w̃	w̃	PROPN
ap-8536	298	1	+	+	CCONJ
ap-8536	298	2	∑	∑	PROPN
ap-8536	298	3	j∈z	j∈z	PROPN
ap-8536	298	4	m	m	PROPN
ap-8536	298	5	j	j	PROPN
ap-8536	298	6	q̃j−2	q̃j−2	PROPN
ap-8536	298	7	−	−	PROPN
ap-8536	298	8	∑	∑	PROPN
ap-8536	298	9	l∈z	l∈z	NOUN
ap-8536	298	10	m	m	VERB
ap-8536	299	1	lq̃l−2	lq̃l−2	NOUN
ap-8536	299	2	=	=	SYM
ap-8536	299	3	w̃	w̃	PROPN
ap-8536	299	4	+	+	CCONJ
ap-8536	299	5	0	0	NUM
ap-8536	299	6	=	=	SYM
ap-8536	299	7	w̃	w̃	PROPN
ap-8536	299	8	.	.	PUNCT
ap-8536	300	1	remark	remark	PROPN
ap-8536	300	2	14	14	NUM
ap-8536	300	3	.	.	PUNCT
ap-8536	301	1	consider	consider	VERB
ap-8536	301	2	m	m	NOUN
ap-8536	301	3	∈	∈	NOUN
ap-8536	301	4	zm×m	zm×m	NOUN
ap-8536	301	5	with	with	ADP
ap-8536	301	6	no	no	DET
ap-8536	301	7	eigenvalue	eigenvalue	NOUN
ap-8536	301	8	on	on	ADP
ap-8536	301	9	the	the	DET
ap-8536	301	10	unit	unit	NOUN
ap-8536	301	11	circle	circle	NOUN
ap-8536	301	12	.	.	PUNCT
ap-8536	302	1	then	then	ADV
ap-8536	302	2	,	,	PUNCT
ap-8536	302	3	at	at	ADV
ap-8536	302	4	least	least	ADV
ap-8536	302	5	one	one	NUM
ap-8536	302	6	eigenvalue	eigenvalue	NOUN
ap-8536	302	7	λ	λ	PROPN
ap-8536	302	8	of	of	ADP
ap-8536	302	9	m	m	PROPN
ap-8536	302	10	satisfies	satisfie	NOUN
ap-8536	302	11	|λ|	|λ|	PROPN
ap-8536	302	12	>	>	X
ap-8536	302	13	1	1	X
ap-8536	302	14	.	.	PUNCT
ap-8536	303	1	the	the	DET
ap-8536	303	2	eigenvalue	eigenvalue	PROPN
ap-8536	303	3	λ	λ	PROPN
ap-8536	303	4	is	be	AUX
ap-8536	303	5	an	an	DET
ap-8536	303	6	algebraic	algebraic	ADJ
ap-8536	303	7	integer	integer	NOUN
ap-8536	303	8	,	,	PUNCT
ap-8536	303	9	as	as	SCONJ
ap-8536	303	10	it	it	PRON
ap-8536	303	11	is	be	AUX
ap-8536	303	12	a	a	DET
ap-8536	303	13	root	root	NOUN
ap-8536	303	14	of	of	ADP
ap-8536	303	15	the	the	DET
ap-8536	303	16	characteristic	characteristic	ADJ
ap-8536	303	17	polynomial	polynomial	ADJ
ap-8536	303	18	f	f	PROPN
ap-8536	303	19	∈	∈	PROPN
ap-8536	303	20	z[x	z[x	NOUN
ap-8536	303	21	]	]	PUNCT
ap-8536	303	22	of	of	ADP
ap-8536	303	23	the	the	DET
ap-8536	303	24	matrix	matrix	NOUN
ap-8536	303	25	m	m	NOUN
ap-8536	303	26	,	,	PUNCT
ap-8536	303	27	and	and	CCONJ
ap-8536	303	28	,	,	PUNCT
ap-8536	303	29	obviously	obviously	ADV
ap-8536	303	30	,	,	PUNCT
ap-8536	303	31	f	f	PROPN
ap-8536	303	32	is	be	AUX
ap-8536	303	33	monic	monic	ADJ
ap-8536	303	34	.	.	PUNCT
ap-8536	304	1	if	if	SCONJ
ap-8536	304	2	,	,	PUNCT
ap-8536	304	3	moreover	moreover	ADV
ap-8536	304	4	,	,	PUNCT
ap-8536	304	5	f	f	PROPN
ap-8536	304	6	is	be	AUX
ap-8536	304	7	irreducible	irreducible	ADJ
ap-8536	304	8	over	over	ADP
ap-8536	304	9	q	q	NOUN
ap-8536	304	10	,	,	PUNCT
ap-8536	304	11	then	then	ADV
ap-8536	304	12	z[λ	z[λ	NUM
ap-8536	304	13	]	]	X
ap-8536	304	14	=	=	SYM
ap-8536	304	15	{	{	PUNCT
ap-8536	304	16	a0+a1λ+	a0+a1λ+	PROPN
ap-8536	304	17	·	·	PUNCT
ap-8536	304	18	·	·	PUNCT
ap-8536	304	19	·	·	PUNCT
ap-8536	305	1	+	+	ADJ
ap-8536	305	2	am−1λm−1	am−1λm−1	NUM
ap-8536	305	3	:	:	PUNCT
ap-8536	305	4	a0	a0	NOUN
ap-8536	305	5	,	,	PUNCT
ap-8536	305	6	.	.	PUNCT
ap-8536	305	7	.	.	PUNCT
ap-8536	306	1	.	.	PUNCT
ap-8536	307	1	,	,	PUNCT
ap-8536	307	2	am−1	am−1	PROPN
ap-8536	307	3	∈	∈	PROPN
ap-8536	307	4	z	z	PROPN
ap-8536	307	5	}	}	PUNCT
ap-8536	307	6	and	and	CCONJ
ap-8536	307	7	ξ	ξ	X
ap-8536	307	8	:	:	PUNCT
ap-8536	307	9	z[λ	z[λ	NUM
ap-8536	307	10	]	]	X
ap-8536	307	11	7→	7→	NUM
ap-8536	307	12	zm	zm	NOUN
ap-8536	307	13	defined	define	VERB
ap-8536	307	14	by	by	ADP
ap-8536	307	15	ξ(a0	ξ(a0	VERB
ap-8536	307	16	+	+	ADJ
ap-8536	307	17	a1λ+	a1λ+	NOUN
ap-8536	307	18	·	·	PUNCT
ap-8536	307	19	·	·	PUNCT
ap-8536	307	20	·	·	PUNCT
ap-8536	307	21	+	+	NUM
ap-8536	307	22	am−1	am−1	PROPN
ap-8536	307	23	)	)	PUNCT
ap-8536	307	24	=	=	PUNCT
ap-8536	307	25	(	(	PUNCT
ap-8536	307	26	a0	a0	PROPN
ap-8536	307	27	,	,	PUNCT
ap-8536	307	28	a1	a1	NOUN
ap-8536	307	29	,	,	PUNCT
ap-8536	307	30	.	.	PUNCT
ap-8536	307	31	.	.	PUNCT
ap-8536	308	1	.	.	PUNCT
ap-8536	309	1	,	,	PUNCT
ap-8536	309	2	am−1)⊤	am−1)⊤	PROPN
ap-8536	309	3	is	be	AUX
ap-8536	309	4	an	an	DET
ap-8536	309	5	isomorphism	isomorphism	NOUN
ap-8536	309	6	.	.	PUNCT
ap-8536	310	1	consequently	consequently	ADV
ap-8536	310	2	,	,	PUNCT
ap-8536	310	3	any	any	DET
ap-8536	310	4	algorithm	algorithm	NOUN
ap-8536	310	5	for	for	ADP
ap-8536	310	6	parallel	parallel	ADJ
ap-8536	310	7	addition	addition	NOUN
ap-8536	310	8	in	in	ADP
ap-8536	310	9	the	the	DET
ap-8536	310	10	number	number	NOUN
ap-8536	310	11	system	system	NOUN
ap-8536	310	12	(	(	PUNCT
ap-8536	310	13	λ	λ	NOUN
ap-8536	310	14	,	,	PUNCT
ap-8536	310	15	a	a	PRON
ap-8536	310	16	)	)	PUNCT
ap-8536	310	17	with	with	ADP
ap-8536	310	18	a	a	DET
ap-8536	310	19	⊂	⊂	PROPN
ap-8536	310	20	z[λ	z[λ	NUM
ap-8536	310	21	]	]	X
ap-8536	310	22	can	can	AUX
ap-8536	310	23	be	be	AUX
ap-8536	310	24	transformed	transform	VERB
ap-8536	310	25	by	by	ADP
ap-8536	310	26	the	the	DET
ap-8536	310	27	isomorphism	isomorphism	NOUN
ap-8536	310	28	ξ	ξ	X
ap-8536	310	29	to	to	ADP
ap-8536	310	30	the	the	DET
ap-8536	310	31	matrix	matrix	NOUN
ap-8536	310	32	numeration	numeration	NOUN
ap-8536	310	33	system	system	NOUN
ap-8536	310	34	(	(	PUNCT
ap-8536	310	35	m	m	PROPN
ap-8536	310	36	,	,	PUNCT
ap-8536	310	37	ξ(a	ξ(a	NOUN
ap-8536	310	38	)	)	PUNCT
ap-8536	310	39	)	)	PUNCT
ap-8536	310	40	.	.	PUNCT
ap-8536	311	1	in	in	ADP
ap-8536	311	2	particular	particular	ADJ
ap-8536	311	3	,	,	PUNCT
ap-8536	311	4	if	if	SCONJ
ap-8536	311	5	a	a	DET
ap-8536	311	6	⊂	⊂	PROPN
ap-8536	311	7	z	z	PROPN
ap-8536	311	8	,	,	PUNCT
ap-8536	311	9	then	then	ADV
ap-8536	311	10	the	the	DET
ap-8536	311	11	digit	digit	NOUN
ap-8536	311	12	set	set	VERB
ap-8536	311	13	ξ(a	ξ(a	PROPN
ap-8536	311	14	)	)	PUNCT
ap-8536	311	15	⊂	⊂	PROPN
ap-8536	311	16	{	{	PUNCT
ap-8536	311	17	ce1	ce1	NOUN
ap-8536	311	18	:	:	PUNCT
ap-8536	312	1	c	c	PROPN
ap-8536	312	2	∈	∈	PROPN
ap-8536	313	1	z	z	X
ap-8536	313	2	}	}	PUNCT
ap-8536	313	3	,	,	PUNCT
ap-8536	313	4	where	where	SCONJ
ap-8536	313	5	e1	e1	NOUN
ap-8536	313	6	=	=	SYM
ap-8536	313	7	(	(	PUNCT
ap-8536	313	8	1	1	NUM
ap-8536	313	9	,	,	PUNCT
ap-8536	313	10	0	0	NUM
ap-8536	313	11	,	,	PUNCT
ap-8536	313	12	.	.	PUNCT
ap-8536	313	13	.	.	PUNCT
ap-8536	313	14	.	.	PUNCT
ap-8536	314	1	,	,	PUNCT
ap-8536	314	2	0)⊤.	0)⊤.	NUM
ap-8536	314	3	any	any	DET
ap-8536	314	4	known	know	VERB
ap-8536	314	5	result	result	NOUN
ap-8536	314	6	on	on	ADP
ap-8536	314	7	the	the	DET
ap-8536	314	8	minimal	minimal	ADJ
ap-8536	314	9	size	size	NOUN
ap-8536	314	10	of	of	ADP
ap-8536	314	11	the	the	DET
ap-8536	314	12	digit	digit	NOUN
ap-8536	314	13	set	set	VERB
ap-8536	314	14	allowing	allow	VERB
ap-8536	314	15	parallel	parallel	ADJ
ap-8536	314	16	addition	addition	NOUN
ap-8536	314	17	in	in	ADP
ap-8536	314	18	the	the	DET
ap-8536	314	19	number	number	NOUN
ap-8536	314	20	system	system	NOUN
ap-8536	314	21	with	with	ADP
ap-8536	314	22	base	base	NOUN
ap-8536	314	23	λ	λ	PROPN
ap-8536	314	24	can	can	AUX
ap-8536	314	25	be	be	AUX
ap-8536	314	26	applied	apply	VERB
ap-8536	314	27	to	to	ADP
ap-8536	314	28	the	the	DET
ap-8536	314	29	matrix	matrix	NOUN
ap-8536	314	30	numeration	numeration	NOUN
ap-8536	314	31	system	system	NOUN
ap-8536	314	32	with	with	ADP
ap-8536	314	33	base	base	NOUN
ap-8536	314	34	m	m	PROPN
ap-8536	314	35	,	,	PUNCT
ap-8536	314	36	thanks	thank	NOUN
ap-8536	314	37	to	to	ADP
ap-8536	314	38	irreducibility	irreducibility	NOUN
ap-8536	314	39	of	of	ADP
ap-8536	314	40	f	f	PROPN
ap-8536	314	41	over	over	ADP
ap-8536	314	42	q.	q.	PROPN
ap-8536	314	43	some	some	DET
ap-8536	314	44	results	result	NOUN
ap-8536	314	45	on	on	ADP
ap-8536	314	46	the	the	DET
ap-8536	314	47	minimal	minimal	ADJ
ap-8536	314	48	size	size	NOUN
ap-8536	314	49	of	of	ADP
ap-8536	314	50	digit	digit	NOUN
ap-8536	314	51	sets	set	NOUN
ap-8536	314	52	for	for	ADP
ap-8536	314	53	parallel	parallel	ADJ
ap-8536	314	54	addition	addition	NOUN
ap-8536	314	55	in	in	ADP
ap-8536	314	56	number	number	NOUN
ap-8536	314	57	systems	system	NOUN
ap-8536	314	58	with	with	ADP
ap-8536	314	59	base	base	NOUN
ap-8536	314	60	β	β	X
ap-8536	314	61	∈	∈	PROPN
ap-8536	314	62	c	c	NOUN
ap-8536	314	63	can	can	AUX
ap-8536	314	64	be	be	AUX
ap-8536	314	65	found	find	VERB
ap-8536	314	66	e.g.	e.g.	ADV
ap-8536	314	67	in	in	ADP
ap-8536	314	68	[	[	X
ap-8536	314	69	21	21	NUM
ap-8536	314	70	]	]	PUNCT
ap-8536	314	71	.	.	PUNCT
ap-8536	315	1	theorem	theorem	ADJ
ap-8536	315	2	9	9	NUM
ap-8536	315	3	states	state	NOUN
ap-8536	315	4	that	that	SCONJ
ap-8536	315	5	each	each	DET
ap-8536	315	6	non	non	ADJ
ap-8536	315	7	-	-	ADJ
ap-8536	315	8	singular	singular	ADJ
ap-8536	315	9	matrix	matrix	NOUN
ap-8536	315	10	m	m	NOUN
ap-8536	315	11	∈	∈	PROPN
ap-8536	315	12	zm×m	zm×m	NOUN
ap-8536	315	13	with	with	ADP
ap-8536	315	14	no	no	DET
ap-8536	315	15	eigenvalue	eigenvalue	NOUN
ap-8536	315	16	on	on	ADP
ap-8536	315	17	the	the	DET
ap-8536	315	18	unit	unit	NOUN
ap-8536	315	19	circle	circle	NOUN
ap-8536	315	20	can	can	AUX
ap-8536	315	21	be	be	AUX
ap-8536	315	22	equipped	equip	VERB
ap-8536	315	23	with	with	ADP
ap-8536	315	24	a	a	DET
ap-8536	315	25	suitable	suitable	ADJ
ap-8536	315	26	finite	finite	NOUN
ap-8536	315	27	digit	digit	NOUN
ap-8536	315	28	set	set	VERB
ap-8536	315	29	d	d	PROPN
ap-8536	315	30	⊂	⊂	PROPN
ap-8536	315	31	zm	zm	PROPN
ap-8536	315	32	such	such	ADJ
ap-8536	315	33	that	that	SCONJ
ap-8536	315	34	any	any	DET
ap-8536	315	35	integer	integer	NOUN
ap-8536	315	36	vector	vector	NOUN
ap-8536	315	37	x	x	PROPN
ap-8536	315	38	∈	∈	PROPN
ap-8536	315	39	zm	zm	PROPN
ap-8536	315	40	is	be	AUX
ap-8536	315	41	representable	representable	ADJ
ap-8536	315	42	in	in	ADP
ap-8536	315	43	the	the	DET
ap-8536	315	44	system	system	NOUN
ap-8536	315	45	(	(	PUNCT
ap-8536	315	46	m	m	PROPN
ap-8536	315	47	,	,	PUNCT
ap-8536	315	48	d	d	NOUN
ap-8536	315	49	)	)	PUNCT
ap-8536	315	50	.	.	PUNCT
ap-8536	316	1	as	as	SCONJ
ap-8536	316	2	shown	show	VERB
ap-8536	316	3	in	in	ADP
ap-8536	316	4	[	[	X
ap-8536	316	5	15	15	NUM
ap-8536	316	6	]	]	PUNCT
ap-8536	316	7	,	,	PUNCT
ap-8536	316	8	the	the	DET
ap-8536	316	9	assumption	assumption	NOUN
ap-8536	316	10	|λ|	|λ|	NOUN
ap-8536	316	11	=	=	SYM
ap-8536	316	12	̸	̸	ADV
ap-8536	316	13	1	1	NUM
ap-8536	316	14	for	for	ADP
ap-8536	316	15	each	each	DET
ap-8536	316	16	eigenvalue	eigenvalue	PROPN
ap-8536	316	17	λ	λ	PROPN
ap-8536	316	18	of	of	ADP
ap-8536	316	19	m	m	PROPN
ap-8536	316	20	is	be	AUX
ap-8536	316	21	not	not	PART
ap-8536	316	22	necessary	necessary	ADJ
ap-8536	316	23	for	for	ADP
ap-8536	316	24	the	the	DET
ap-8536	316	25	representability	representability	NOUN
ap-8536	316	26	of	of	ADP
ap-8536	316	27	zm	zm	PROPN
ap-8536	316	28	in	in	ADP
ap-8536	316	29	(	(	PUNCT
ap-8536	316	30	m	m	PROPN
ap-8536	316	31	,	,	PUNCT
ap-8536	316	32	d	d	NOUN
ap-8536	316	33	)	)	PUNCT
ap-8536	316	34	.	.	PUNCT
ap-8536	317	1	nevertheless	nevertheless	ADV
ap-8536	317	2	,	,	PUNCT
ap-8536	317	3	this	this	DET
ap-8536	317	4	assumption	assumption	NOUN
ap-8536	317	5	is	be	AUX
ap-8536	317	6	necessary	necessary	ADJ
ap-8536	317	7	for	for	ADP
ap-8536	317	8	parallel	parallel	ADJ
ap-8536	317	9	addition	addition	NOUN
ap-8536	317	10	in	in	ADP
ap-8536	317	11	(	(	PUNCT
ap-8536	317	12	m	m	PROPN
ap-8536	317	13	,	,	PUNCT
ap-8536	317	14	d	d	NOUN
ap-8536	317	15	)	)	PUNCT
ap-8536	317	16	,	,	PUNCT
ap-8536	317	17	as	as	SCONJ
ap-8536	317	18	shown	show	VERB
ap-8536	317	19	below	below	ADV
ap-8536	317	20	.	.	PUNCT
ap-8536	318	1	proposition	proposition	NOUN
ap-8536	318	2	15	15	NUM
ap-8536	318	3	.	.	PUNCT
ap-8536	319	1	if	if	SCONJ
ap-8536	319	2	addition	addition	NOUN
ap-8536	319	3	in	in	ADP
ap-8536	319	4	a	a	DET
ap-8536	319	5	matrix	matrix	NOUN
ap-8536	319	6	numeration	numeration	NOUN
ap-8536	319	7	system	system	NOUN
ap-8536	319	8	(	(	PUNCT
ap-8536	319	9	m	m	PROPN
ap-8536	319	10	,	,	PUNCT
ap-8536	319	11	d	d	NOUN
ap-8536	319	12	)	)	PUNCT
ap-8536	319	13	with	with	ADP
ap-8536	319	14	base	base	NOUN
ap-8536	319	15	m	m	PROPN
ap-8536	319	16	∈	∈	PROPN
ap-8536	319	17	zm×m	zm×m	NOUN
ap-8536	319	18	and	and	CCONJ
ap-8536	319	19	a	a	DET
ap-8536	319	20	finite	finite	ADJ
ap-8536	319	21	digit	digit	NOUN
ap-8536	319	22	set	set	PROPN
ap-8536	319	23	d	d	PROPN
ap-8536	319	24	⊂	⊂	PROPN
ap-8536	319	25	zm	zm	PROPN
ap-8536	319	26	is	be	AUX
ap-8536	319	27	doable	doable	ADJ
ap-8536	319	28	in	in	ADP
ap-8536	319	29	parallel	parallel	NOUN
ap-8536	319	30	,	,	PUNCT
ap-8536	319	31	then	then	ADV
ap-8536	319	32	no	no	DET
ap-8536	319	33	eigenvalue	eigenvalue	NOUN
ap-8536	319	34	of	of	ADP
ap-8536	319	35	m	m	PROPN
ap-8536	319	36	lies	lie	VERB
ap-8536	319	37	on	on	ADP
ap-8536	319	38	the	the	DET
ap-8536	319	39	unit	unit	NOUN
ap-8536	319	40	circle	circle	NOUN
ap-8536	319	41	.	.	PUNCT
ap-8536	320	1	proof	proof	NOUN
ap-8536	320	2	.	.	PUNCT
ap-8536	321	1	by	by	ADP
ap-8536	321	2	lemma	lemma	PROPN
ap-8536	321	3	5	5	NUM
ap-8536	321	4	,	,	PUNCT
ap-8536	321	5	we	we	PRON
ap-8536	321	6	consider	consider	VERB
ap-8536	321	7	,	,	PUNCT
ap-8536	321	8	without	without	ADP
ap-8536	321	9	a	a	DET
ap-8536	321	10	loss	loss	NOUN
ap-8536	321	11	of	of	ADP
ap-8536	321	12	generality	generality	NOUN
ap-8536	321	13	,	,	PUNCT
ap-8536	321	14	that	that	SCONJ
ap-8536	321	15	the	the	DET
ap-8536	321	16	digit	digit	NOUN
ap-8536	321	17	set	set	VERB
ap-8536	321	18	d	d	ADP
ap-8536	321	19	generates	generate	VERB
ap-8536	321	20	rm	rm	NOUN
ap-8536	321	21	–	–	PUNCT
ap-8536	321	22	i.e.	i.e.	X
ap-8536	321	23	,	,	PUNCT
ap-8536	321	24	that	that	DET
ap-8536	321	25	rm	rm	PROPN
ap-8536	321	26	is	be	AUX
ap-8536	321	27	the	the	DET
ap-8536	321	28	linear	linear	PROPN
ap-8536	321	29	hull	hull	NOUN
ap-8536	321	30	of	of	ADP
ap-8536	321	31	d.	d.	PROPN
ap-8536	321	32	assume	assume	PROPN
ap-8536	321	33	,	,	PUNCT
ap-8536	321	34	for	for	ADP
ap-8536	321	35	a	a	DET
ap-8536	321	36	contradiction	contradiction	NOUN
ap-8536	321	37	,	,	PUNCT
ap-8536	321	38	that	that	PRON
ap-8536	321	39	m	m	VERB
ap-8536	321	40	has	have	VERB
ap-8536	321	41	an	an	DET
ap-8536	321	42	eigenvalue	eigenvalue	PROPN
ap-8536	321	43	λ	λ	SYM
ap-8536	321	44	∈	∈	PROPN
ap-8536	321	45	c	c	NOUN
ap-8536	321	46	on	on	ADP
ap-8536	321	47	the	the	DET
ap-8536	321	48	unit	unit	NOUN
ap-8536	321	49	circle	circle	NOUN
ap-8536	321	50	,	,	PUNCT
ap-8536	321	51	|λ|	|λ|	NOUN
ap-8536	321	52	=	=	SYM
ap-8536	322	1	1	1	X
ap-8536	322	2	.	.	PUNCT
ap-8536	322	3	let	let	VERB
ap-8536	322	4	u	u	PRON
ap-8536	322	5	∈	∈	PROPN
ap-8536	322	6	cm	cm	NOUN
ap-8536	322	7	be	be	AUX
ap-8536	322	8	an	an	DET
ap-8536	322	9	eigenvector	eigenvector	NOUN
ap-8536	322	10	of	of	ADP
ap-8536	322	11	the	the	DET
ap-8536	322	12	matrix	matrix	NOUN
ap-8536	322	13	m⊤	m⊤	NOUN
ap-8536	322	14	to	to	ADP
ap-8536	322	15	the	the	DET
ap-8536	322	16	eigenvalue	eigenvalue	PROPN
ap-8536	322	17	λ	λ	PROPN
ap-8536	322	18	–	–	PUNCT
ap-8536	322	19	i.e.	i.e.	X
ap-8536	322	20	,	,	PUNCT
ap-8536	323	1	m⊤u	m⊤u	ADJ
ap-8536	323	2	=	=	SYM
ap-8536	323	3	λu	λu	X
ap-8536	323	4	.	.	PUNCT
ap-8536	324	1	as	as	SCONJ
ap-8536	324	2	d	d	PROPN
ap-8536	324	3	generates	generate	VERB
ap-8536	324	4	rm	rm	PROPN
ap-8536	324	5	,	,	PUNCT
ap-8536	324	6	the	the	DET
ap-8536	324	7	vector	vector	NOUN
ap-8536	324	8	u	u	NOUN
ap-8536	324	9	can	can	AUX
ap-8536	324	10	not	not	PART
ap-8536	324	11	be	be	AUX
ap-8536	324	12	orthogonal	orthogonal	ADJ
ap-8536	324	13	to	to	ADP
ap-8536	324	14	all	all	DET
ap-8536	324	15	digits	digit	NOUN
ap-8536	324	16	,	,	PUNCT
ap-8536	324	17	hence	hence	ADV
ap-8536	324	18	there	there	PRON
ap-8536	324	19	exists	exist	VERB
ap-8536	324	20	a	a	DET
ap-8536	324	21	digit	digit	NOUN
ap-8536	324	22	d	d	X
ap-8536	324	23	∈	∈	PROPN
ap-8536	324	24	d	d	ADP
ap-8536	324	25	such	such	ADJ
ap-8536	324	26	that	that	DET
ap-8536	324	27	αu⊤d	αu⊤d	NUM
ap-8536	324	28	=	=	SYM
ap-8536	324	29	1	1	NUM
ap-8536	324	30	for	for	ADP
ap-8536	324	31	some	some	DET
ap-8536	324	32	α	α	PROPN
ap-8536	324	33	∈	∈	PROPN
ap-8536	324	34	c.	c.	NOUN
ap-8536	324	35	let	let	VERB
ap-8536	324	36	v	v	PART
ap-8536	324	37	be	be	AUX
ap-8536	324	38	the	the	DET
ap-8536	324	39	eigenvector	eigenvector	NOUN
ap-8536	324	40	v	v	NOUN
ap-8536	324	41	=	=	SYM
ap-8536	324	42	αu	αu	NOUN
ap-8536	324	43	,	,	PUNCT
ap-8536	324	44	so	so	SCONJ
ap-8536	324	45	that	that	SCONJ
ap-8536	324	46	v⊤d	v⊤d	X
ap-8536	324	47	=	=	SYM
ap-8536	324	48	1	1	NUM
ap-8536	324	49	and	and	CCONJ
ap-8536	324	50	v⊤m	v⊤m	NOUN
ap-8536	324	51	=	=	PUNCT
ap-8536	325	1	λv⊤.	λv⊤.	PUNCT
ap-8536	325	2	let	let	VERB
ap-8536	325	3	parallel	parallel	ADJ
ap-8536	325	4	addition	addition	NOUN
ap-8536	325	5	be	be	AUX
ap-8536	325	6	performed	perform	VERB
ap-8536	325	7	by	by	ADP
ap-8536	325	8	a	a	DET
ap-8536	325	9	p	p	ADJ
ap-8536	325	10	-	-	PUNCT
ap-8536	325	11	local	local	ADJ
ap-8536	325	12	function	function	NOUN
ap-8536	325	13	.	.	PUNCT
ap-8536	326	1	denote	denote	PROPN
ap-8536	326	2	s	s	PART
ap-8536	326	3	=	=	X
ap-8536	326	4	max	max	PROPN
ap-8536	326	5	{	{	PUNCT
ap-8536	326	6	∣∣p−1∑	∣∣p−1∑	ADP
ap-8536	326	7	j=0	j=0	PROPN
ap-8536	326	8	λj	λj	PROPN
ap-8536	326	9	v⊤dj	v⊤dj	PROPN
ap-8536	326	10	∣∣	∣∣	NUM
ap-8536	326	11	:	:	PUNCT
ap-8536	326	12	dj	dj	X
ap-8536	326	13	∈	∈	PROPN
ap-8536	326	14	d	d	NOUN
ap-8536	326	15	}	}	PUNCT
ap-8536	326	16	.	.	PUNCT
ap-8536	327	1	as	as	ADP
ap-8536	327	2	|λ|	|λ|	PROPN
ap-8536	327	3	=	=	SYM
ap-8536	327	4	1	1	NUM
ap-8536	327	5	,	,	PUNCT
ap-8536	327	6	there	there	PRON
ap-8536	327	7	exist	exist	VERB
ap-8536	327	8	infinitely	infinitely	ADV
ap-8536	327	9	many	many	ADJ
ap-8536	327	10	j	j	NOUN
ap-8536	327	11	∈	∈	PROPN
ap-8536	327	12	n	n	PRON
ap-8536	327	13	such	such	ADJ
ap-8536	327	14	that	that	DET
ap-8536	327	15	ℜ(λj	ℜ(λj	NOUN
ap-8536	327	16	)	)	PUNCT
ap-8536	327	17	>	>	SYM
ap-8536	327	18	1	1	NUM
ap-8536	327	19	2	2	NUM
ap-8536	327	20	.	.	PUNCT
ap-8536	328	1	hence	hence	ADV
ap-8536	328	2	one	one	PRON
ap-8536	328	3	can	can	AUX
ap-8536	328	4	find	find	VERB
ap-8536	328	5	n	n	PRON
ap-8536	328	6	∈	∈	PROPN
ap-8536	328	7	n	n	NOUN
ap-8536	328	8	and	and	CCONJ
ap-8536	328	9	coefficients	coefficient	NOUN
ap-8536	328	10	ε0	ε0	PROPN
ap-8536	328	11	,	,	PUNCT
ap-8536	328	12	ε1	ε1	PROPN
ap-8536	328	13	,	,	PUNCT
ap-8536	328	14	.	.	PUNCT
ap-8536	328	15	.	.	PUNCT
ap-8536	329	1	.	.	PUNCT
ap-8536	330	1	,	,	PUNCT
ap-8536	330	2	εn−1	εn−1	PROPN
ap-8536	330	3	∈	∈	PROPN
ap-8536	330	4	{	{	PUNCT
ap-8536	330	5	0	0	NUM
ap-8536	330	6	,	,	PUNCT
ap-8536	330	7	1	1	NUM
ap-8536	330	8	}	}	PUNCT
ap-8536	330	9	such	such	ADJ
ap-8536	330	10	that	that	PRON
ap-8536	330	11	ℜ	ℜ	PROPN
ap-8536	330	12	(	(	PUNCT
ap-8536	330	13	∑n−1	∑n−1	ADP
ap-8536	330	14	j=0	j=0	PROPN
ap-8536	330	15	εjλj	εjλj	PROPN
ap-8536	330	16	)	)	PUNCT
ap-8536	330	17	>	>	X
ap-8536	330	18	193	193	NUM
ap-8536	330	19	i.	i.	NOUN
ap-8536	330	20	farkas	farkas	PROPN
ap-8536	330	21	,	,	PUNCT
ap-8536	330	22	e.	e.	PROPN
ap-8536	330	23	pelantová	pelantová	PROPN
ap-8536	330	24	,	,	PUNCT
ap-8536	330	25	m.	m.	NOUN
ap-8536	330	26	svobodová	svobodová	PROPN
ap-8536	330	27	acta	acta	PROPN
ap-8536	330	28	polytechnica	polytechnica	PROPN
ap-8536	330	29	2s	2s	PROPN
ap-8536	330	30	.	.	PUNCT
ap-8536	331	1	let	let	VERB
ap-8536	331	2	x	x	SYM
ap-8536	331	3	=	=	PUNCT
ap-8536	331	4	∑n−1	∑n−1	X
ap-8536	331	5	j=0	j=0	PROPN
ap-8536	331	6	m	m	PROPN
ap-8536	331	7	jxj	jxj	ADJ
ap-8536	331	8	with	with	ADP
ap-8536	331	9	xj	xj	PROPN
ap-8536	331	10	∈	∈	PROPN
ap-8536	331	11	d	d	ADP
ap-8536	331	12	such	such	ADJ
ap-8536	331	13	that	that	PRON
ap-8536	331	14	|ℜ(v⊤x)|	|ℜ(v⊤x)|	PUNCT
ap-8536	331	15	=	=	SYM
ap-8536	331	16	max	max	NOUN
ap-8536	331	17	{	{	PUNCT
ap-8536	331	18	∣∣ℜ	∣∣ℜ	NOUN
ap-8536	331	19	(	(	PUNCT
ap-8536	331	20	v⊤	v⊤	ADJ
ap-8536	331	21	(	(	PUNCT
ap-8536	331	22	n−1∑	n−1∑	PROPN
ap-8536	331	23	j=0	j=0	PROPN
ap-8536	331	24	m	m	PROPN
ap-8536	331	25	jdj	jdj	PROPN
ap-8536	331	26	)	)	PUNCT
ap-8536	331	27	)	)	PUNCT
ap-8536	332	1	∣∣	∣∣	NUM
ap-8536	332	2	:	:	PUNCT
ap-8536	332	3	dj	dj	X
ap-8536	332	4	∈	∈	ADJ
ap-8536	332	5	d	d	NOUN
ap-8536	332	6	}	}	PUNCT
ap-8536	332	7	=	=	SYM
ap-8536	332	8	max	max	PROPN
ap-8536	332	9	{	{	PUNCT
ap-8536	332	10	∣∣ℜ(n−1∑	∣∣ℜ(n−1∑	PROPN
ap-8536	332	11	j=0	j=0	PROPN
ap-8536	332	12	λj	λj	PROPN
ap-8536	332	13	v⊤dj	v⊤dj	PROPN
ap-8536	332	14	)	)	PUNCT
ap-8536	332	15	∣∣	∣∣	X
ap-8536	332	16	:	:	PUNCT
ap-8536	332	17	dj	dj	X
ap-8536	332	18	∈	∈	PROPN
ap-8536	332	19	d	d	NOUN
ap-8536	332	20	}	}	PUNCT
ap-8536	332	21	.	.	PUNCT
ap-8536	333	1	since	since	SCONJ
ap-8536	333	2	1	1	NUM
ap-8536	333	3	and	and	CCONJ
ap-8536	333	4	0	0	NUM
ap-8536	333	5	belong	belong	VERB
ap-8536	333	6	to	to	ADP
ap-8536	333	7	{	{	PUNCT
ap-8536	333	8	v⊤d	v⊤d	NOUN
ap-8536	333	9	:	:	PUNCT
ap-8536	333	10	d	d	X
ap-8536	333	11	∈	∈	PROPN
ap-8536	333	12	d	d	X
ap-8536	333	13	}	}	PUNCT
ap-8536	333	14	,	,	PUNCT
ap-8536	333	15	we	we	PRON
ap-8536	333	16	have	have	VERB
ap-8536	333	17	|ℜ(v⊤x)|	|ℜ(v⊤x)|	NOUN
ap-8536	333	18	≥	≥	X
ap-8536	333	19	ℜ	ℜ	PROPN
ap-8536	333	20	(	(	PUNCT
ap-8536	333	21	n−1∑	n−1∑	NUM
ap-8536	333	22	j=0	j=0	PROPN
ap-8536	333	23	εjλj	εjλj	PROPN
ap-8536	333	24	)	)	PUNCT
ap-8536	333	25	>	>	X
ap-8536	334	1	2s	2s	X
ap-8536	334	2	.	.	PUNCT
ap-8536	335	1	(	(	PUNCT
ap-8536	335	2	10	10	NUM
ap-8536	335	3	)	)	PUNCT
ap-8536	335	4	the	the	DET
ap-8536	335	5	p	p	ADJ
ap-8536	335	6	-	-	PUNCT
ap-8536	335	7	local	local	ADJ
ap-8536	335	8	function	function	NOUN
ap-8536	335	9	used	use	VERB
ap-8536	335	10	to	to	PART
ap-8536	335	11	add	add	VERB
ap-8536	335	12	x	x	PUNCT
ap-8536	335	13	+	+	CCONJ
ap-8536	335	14	x	x	PRON
ap-8536	335	15	produces	produce	VERB
ap-8536	335	16	digits	digit	NOUN
ap-8536	335	17	zj	zj	X
ap-8536	335	18	∈	∈	PROPN
ap-8536	336	1	d	d	ADP
ap-8536	336	2	such	such	ADJ
ap-8536	336	3	that	that	PRON
ap-8536	336	4	x	x	PUNCT
ap-8536	337	1	+	+	NUM
ap-8536	337	2	x	x	SYM
ap-8536	337	3	=	=	SYM
ap-8536	337	4	n+p−2∑	n+p−2∑	PROPN
ap-8536	337	5	j	j	PROPN
ap-8536	337	6	=	=	PROPN
ap-8536	337	7	n	n	PROPN
ap-8536	337	8	m	m	VERB
ap-8536	337	9	jzj	jzj	NOUN
ap-8536	338	1	+	+	CCONJ
ap-8536	338	2	n−1∑	n−1∑	PROPN
ap-8536	338	3	j=0	j=0	PROPN
ap-8536	338	4	m	m	VERB
ap-8536	338	5	jzj	jzj	NOUN
ap-8536	339	1	+	+	CCONJ
ap-8536	340	1	−1∑	−1∑	PROPN
ap-8536	340	2	j=−p+1	j=−p+1	NOUN
ap-8536	340	3	m	m	VERB
ap-8536	340	4	jzj	jzj	PROPN
ap-8536	340	5	.	.	PUNCT
ap-8536	341	1	after	after	ADP
ap-8536	341	2	multiplication	multiplication	NOUN
ap-8536	341	3	of	of	ADP
ap-8536	341	4	x	x	PROPN
ap-8536	341	5	+	+	CCONJ
ap-8536	341	6	x	x	SYM
ap-8536	341	7	by	by	ADP
ap-8536	341	8	the	the	DET
ap-8536	341	9	vector	vector	NOUN
ap-8536	341	10	v⊤	v⊤	INTJ
ap-8536	341	11	from	from	ADP
ap-8536	341	12	the	the	DET
ap-8536	341	13	left	left	NOUN
ap-8536	341	14	,	,	PUNCT
ap-8536	341	15	we	we	PRON
ap-8536	341	16	have	have	VERB
ap-8536	341	17	v⊤(x	v⊤(x	ADJ
ap-8536	341	18	+	+	NOUN
ap-8536	341	19	x	x	X
ap-8536	341	20	)	)	PUNCT
ap-8536	342	1	=	=	SYM
ap-8536	342	2	λn	λn	PROPN
ap-8536	342	3	p−2∑	p−2∑	ADV
ap-8536	342	4	j=0	j=0	PROPN
ap-8536	342	5	λj	λj	PROPN
ap-8536	342	6	v⊤zn+j+v⊤	v⊤zn+j+v⊤	PROPN
ap-8536	342	7	n−1∑	n−1∑	PROPN
ap-8536	342	8	j=0	j=0	PROPN
ap-8536	342	9	m	m	PROPN
ap-8536	342	10	j	j	NOUN
ap-8536	342	11	zj+λ−p	zj+λ−p	PROPN
ap-8536	342	12	p−1∑	p−1∑	X
ap-8536	342	13	j=1	j=1	NOUN
ap-8536	342	14	λj	λj	X
ap-8536	342	15	v⊤zj−p	v⊤zj−p	PROPN
ap-8536	342	16	.	.	PUNCT
ap-8536	343	1	(	(	PUNCT
ap-8536	343	2	11	11	NUM
ap-8536	343	3	)	)	PUNCT
ap-8536	343	4	the	the	DET
ap-8536	343	5	definitions	definition	NOUN
ap-8536	343	6	of	of	ADP
ap-8536	343	7	s	s	PRON
ap-8536	343	8	and	and	CCONJ
ap-8536	343	9	x	x	PUNCT
ap-8536	343	10	guarantee	guarantee	VERB
ap-8536	343	11	that	that	SCONJ
ap-8536	343	12	∣∣p−2∑	∣∣p−2∑	ADV
ap-8536	343	13	j=0	j=0	VERB
ap-8536	343	14	λj	λj	PROPN
ap-8536	343	15	v⊤zn+j	v⊤zn+j	PROPN
ap-8536	343	16	∣∣	∣∣	X
ap-8536	343	17	≤	≤	NUM
ap-8536	343	18	s	s	PART
ap-8536	343	19	,	,	PUNCT
ap-8536	343	20	∣∣p−1∑	∣∣p−1∑	VERB
ap-8536	343	21	j=1	j=1	PROPN
ap-8536	343	22	λj	λj	PROPN
ap-8536	343	23	v⊤zj−p	v⊤zj−p	X
ap-8536	343	24	∣∣	∣∣	X
ap-8536	343	25	≤	≤	PROPN
ap-8536	343	26	s	s	PART
ap-8536	343	27	,	,	PUNCT
ap-8536	343	28	and	and	CCONJ
ap-8536	343	29	∣∣ℜ	∣∣ℜ	NOUN
ap-8536	343	30	(	(	PUNCT
ap-8536	343	31	v⊤(n−1∑	v⊤(n−1∑	PROPN
ap-8536	343	32	j=0	j=0	PROPN
ap-8536	343	33	m	m	PROPN
ap-8536	343	34	j	j	PROPN
ap-8536	343	35	zj	zj	PROPN
ap-8536	343	36	)	)	PUNCT
ap-8536	343	37	)	)	PUNCT
ap-8536	343	38	∣∣	∣∣	PROPN
ap-8536	343	39	≤	≤	NUM
ap-8536	343	40	|ℜ(v⊤x)|	|ℜ(v⊤x)|	PUNCT
ap-8536	343	41	.	.	PUNCT
ap-8536	344	1	using	use	VERB
ap-8536	344	2	these	these	DET
ap-8536	344	3	inequalities	inequality	NOUN
ap-8536	344	4	and	and	CCONJ
ap-8536	344	5	the	the	DET
ap-8536	344	6	triangle	triangle	NOUN
ap-8536	344	7	inequality	inequality	NOUN
ap-8536	344	8	,	,	PUNCT
ap-8536	344	9	together	together	ADV
ap-8536	344	10	with	with	ADP
ap-8536	344	11	(	(	PUNCT
ap-8536	344	12	11	11	NUM
ap-8536	344	13	)	)	PUNCT
ap-8536	344	14	and	and	CCONJ
ap-8536	344	15	the	the	DET
ap-8536	344	16	fact	fact	NOUN
ap-8536	344	17	that	that	SCONJ
ap-8536	344	18	|ℜ(y)|	|ℜ(y)|	PROPN
ap-8536	344	19	≤	≤	PROPN
ap-8536	344	20	|y|	|y|	PROPN
ap-8536	344	21	for	for	ADP
ap-8536	344	22	every	every	DET
ap-8536	344	23	y	y	PROPN
ap-8536	344	24	∈	∈	PROPN
ap-8536	344	25	c	c	NOUN
ap-8536	344	26	,	,	PUNCT
ap-8536	344	27	and	and	CCONJ
ap-8536	344	28	with	with	ADP
ap-8536	344	29	|λ|	|λ|	NOUN
ap-8536	344	30	=	=	SYM
ap-8536	344	31	1	1	NUM
ap-8536	344	32	,	,	PUNCT
ap-8536	344	33	we	we	PRON
ap-8536	344	34	get	get	VERB
ap-8536	344	35	2|ℜ(v⊤x)|	2|ℜ(v⊤x)|	NOUN
ap-8536	344	36	≤	≤	NOUN
ap-8536	344	37	|2v⊤x|	|2v⊤x|	PUNCT
ap-8536	344	38	≤	≤	ADJ
ap-8536	344	39	2s	2s	NUM
ap-8536	344	40	+	+	CCONJ
ap-8536	344	41	|ℜ(v⊤x)|	|ℜ(v⊤x)|	NUM
ap-8536	344	42	which	which	PRON
ap-8536	344	43	contradicts	contradict	VERB
ap-8536	344	44	(	(	PUNCT
ap-8536	344	45	10	10	NUM
ap-8536	344	46	)	)	PUNCT
ap-8536	344	47	.	.	PUNCT
ap-8536	345	1	4	4	X
ap-8536	345	2	.	.	X
ap-8536	345	3	eventually	eventually	ADV
ap-8536	345	4	periodic	periodic	ADJ
ap-8536	345	5	representations	representation	NOUN
ap-8536	345	6	with	with	ADP
ap-8536	345	7	expansive	expansive	ADJ
ap-8536	345	8	matrix	matrix	NOUN
ap-8536	345	9	base	base	NOUN
ap-8536	345	10	t.	t.	NOUN
ap-8536	345	11	vávra	vávra	NOUN
ap-8536	345	12	in	in	ADP
ap-8536	345	13	[	[	X
ap-8536	345	14	22	22	NUM
ap-8536	345	15	]	]	PUNCT
ap-8536	345	16	shows	show	NOUN
ap-8536	345	17	,	,	PUNCT
ap-8536	345	18	for	for	ADP
ap-8536	345	19	any	any	DET
ap-8536	345	20	algebraic	algebraic	ADJ
ap-8536	345	21	complex	complex	ADJ
ap-8536	345	22	base	base	NOUN
ap-8536	345	23	β	β	X
ap-8536	345	24	with	with	ADP
ap-8536	345	25	|β|	|β|	PROPN
ap-8536	345	26	>	>	X
ap-8536	345	27	1	1	NUM
ap-8536	345	28	,	,	PUNCT
ap-8536	345	29	that	that	SCONJ
ap-8536	345	30	there	there	PRON
ap-8536	345	31	exists	exist	VERB
ap-8536	345	32	a	a	DET
ap-8536	345	33	suitable	suitable	ADJ
ap-8536	345	34	(	(	PUNCT
ap-8536	345	35	finite	finite	NOUN
ap-8536	345	36	)	)	PUNCT
ap-8536	345	37	digit	digit	NOUN
ap-8536	345	38	set	set	VERB
ap-8536	345	39	a	a	DET
ap-8536	345	40	⊂	⊂	PROPN
ap-8536	345	41	z	z	NOUN
ap-8536	345	42	such	such	ADJ
ap-8536	345	43	that	that	SCONJ
ap-8536	345	44	any	any	DET
ap-8536	345	45	x	x	SYM
ap-8536	345	46	∈	∈	PROPN
ap-8536	345	47	q(β	q(β	PROPN
ap-8536	345	48	)	)	PUNCT
ap-8536	345	49	has	have	VERB
ap-8536	345	50	an	an	DET
ap-8536	345	51	eventually	eventually	ADV
ap-8536	345	52	periodic	periodic	ADJ
ap-8536	345	53	expansion	expansion	NOUN
ap-8536	345	54	in	in	ADP
ap-8536	345	55	this	this	DET
ap-8536	345	56	base	base	NOUN
ap-8536	345	57	,	,	PUNCT
ap-8536	345	58	i.e.	i.e.	X
ap-8536	345	59	,	,	PUNCT
ap-8536	345	60	x	x	SYM
ap-8536	345	61	=	=	SYM
ap-8536	345	62	∑n	∑n	PROPN
ap-8536	345	63	j=−∞	j=−∞	PROPN
ap-8536	345	64	ajβj	ajβj	PROPN
ap-8536	345	65	,	,	PUNCT
ap-8536	345	66	where	where	SCONJ
ap-8536	345	67	the	the	DET
ap-8536	345	68	sequence	sequence	NOUN
ap-8536	345	69	(	(	PUNCT
ap-8536	345	70	aj)j≤n	aj)j≤n	NUM
ap-8536	345	71	of	of	ADP
ap-8536	345	72	digits	digit	NOUN
ap-8536	345	73	from	from	ADP
ap-8536	345	74	a	a	PRON
ap-8536	345	75	is	be	AUX
ap-8536	345	76	eventually	eventually	ADV
ap-8536	345	77	periodic	periodic	ADJ
ap-8536	345	78	.	.	PUNCT
ap-8536	346	1	looking	look	VERB
ap-8536	346	2	for	for	ADP
ap-8536	346	3	an	an	DET
ap-8536	346	4	analogy	analogy	NOUN
ap-8536	346	5	to	to	ADP
ap-8536	346	6	this	this	DET
ap-8536	346	7	result	result	NOUN
ap-8536	346	8	in	in	ADP
ap-8536	346	9	the	the	DET
ap-8536	346	10	matrix	matrix	NOUN
ap-8536	346	11	systems	system	NOUN
ap-8536	346	12	,	,	PUNCT
ap-8536	346	13	we	we	PRON
ap-8536	346	14	first	first	ADV
ap-8536	346	15	have	have	VERB
ap-8536	346	16	to	to	PART
ap-8536	346	17	give	give	VERB
ap-8536	346	18	a	a	DET
ap-8536	346	19	meaning	meaning	NOUN
ap-8536	346	20	of	of	ADP
ap-8536	346	21	the	the	DET
ap-8536	346	22	previous	previous	ADJ
ap-8536	346	23	sum	sum	NOUN
ap-8536	346	24	in	in	ADP
ap-8536	346	25	the	the	DET
ap-8536	346	26	case	case	NOUN
ap-8536	346	27	when	when	SCONJ
ap-8536	346	28	the	the	DET
ap-8536	346	29	number	number	NOUN
ap-8536	346	30	base	base	NOUN
ap-8536	346	31	β	β	X
ap-8536	346	32	is	be	AUX
ap-8536	346	33	replaced	replace	VERB
ap-8536	346	34	with	with	ADP
ap-8536	346	35	a	a	DET
ap-8536	346	36	matrix	matrix	NOUN
ap-8536	346	37	base	base	NOUN
ap-8536	346	38	m	m	PROPN
ap-8536	346	39	and	and	CCONJ
ap-8536	346	40	(	(	PUNCT
ap-8536	346	41	integer	integer	NOUN
ap-8536	346	42	)	)	PUNCT
ap-8536	346	43	number	number	NOUN
ap-8536	346	44	digits	digit	NOUN
ap-8536	346	45	aj	aj	PROPN
ap-8536	346	46	by	by	ADP
ap-8536	346	47	(	(	PUNCT
ap-8536	346	48	integer	integer	NOUN
ap-8536	346	49	)	)	PUNCT
ap-8536	346	50	vector	vector	NOUN
ap-8536	346	51	digits	digit	NOUN
ap-8536	346	52	dj	dj	VERB
ap-8536	346	53	.	.	PUNCT
ap-8536	347	1	if	if	SCONJ
ap-8536	347	2	a	a	DET
ap-8536	347	3	matrix	matrix	NOUN
ap-8536	347	4	m	m	VERB
ap-8536	347	5	∈	∈	NOUN
ap-8536	347	6	zm×m	zm×m	NOUN
ap-8536	347	7	is	be	AUX
ap-8536	347	8	expansive	expansive	ADJ
ap-8536	347	9	,	,	PUNCT
ap-8536	347	10	then	then	ADV
ap-8536	347	11	m−1	m−1	PROPN
ap-8536	347	12	is	be	AUX
ap-8536	347	13	contractive	contractive	ADJ
ap-8536	347	14	and	and	CCONJ
ap-8536	347	15	there	there	PRON
ap-8536	347	16	exists	exist	VERB
ap-8536	347	17	a	a	DET
ap-8536	347	18	vector	vector	NOUN
ap-8536	347	19	norm	norm	NOUN
ap-8536	347	20	∥·∥c	∥·∥c	ADJ
ap-8536	347	21	in	in	ADP
ap-8536	347	22	rm	rm	PROPN
ap-8536	347	23	such	such	ADJ
ap-8536	347	24	that	that	DET
ap-8536	347	25	∥∥m−1	∥∥m−1	PROPN
ap-8536	348	1	∥∥	∥∥	X
ap-8536	348	2	<	<	X
ap-8536	348	3	1	1	NUM
ap-8536	348	4	,	,	PUNCT
ap-8536	348	5	where	where	SCONJ
ap-8536	348	6	∥·∥	∥·∥	PROPN
ap-8536	348	7	is	be	AUX
ap-8536	348	8	the	the	DET
ap-8536	348	9	matrix	matrix	NOUN
ap-8536	348	10	norm	norm	NOUN
ap-8536	348	11	induced	induce	VERB
ap-8536	348	12	by	by	ADP
ap-8536	348	13	the	the	DET
ap-8536	348	14	vector	vector	NOUN
ap-8536	348	15	norm	norm	NOUN
ap-8536	348	16	∥·∥c	∥·∥c	ADJ
ap-8536	348	17	,	,	PUNCT
ap-8536	348	18	see	see	VERB
ap-8536	348	19	[	[	X
ap-8536	348	20	23	23	NUM
ap-8536	348	21	]	]	PUNCT
ap-8536	348	22	.	.	PUNCT
ap-8536	349	1	let	let	VERB
ap-8536	349	2	us	we	PRON
ap-8536	349	3	recall	recall	VERB
ap-8536	349	4	that	that	PRON
ap-8536	349	5	for	for	ADP
ap-8536	349	6	these	these	DET
ap-8536	349	7	two	two	NUM
ap-8536	349	8	norms	norm	NOUN
ap-8536	349	9	,	,	PUNCT
ap-8536	349	10	the	the	DET
ap-8536	349	11	following	follow	VERB
ap-8536	349	12	inequalities	inequality	NOUN
ap-8536	349	13	hold	hold	VERB
ap-8536	349	14	:	:	PUNCT
ap-8536	349	15	∥ax∥c	∥ax∥c	NOUN
ap-8536	349	16	≤	≤	NUM
ap-8536	349	17	∥a∥	∥a∥	NOUN
ap-8536	349	18	.	.	PUNCT
ap-8536	350	1	∥x∥c	∥x∥c	NOUN
ap-8536	350	2	and	and	CCONJ
ap-8536	350	3	∥ab∥	∥ab∥	ADJ
ap-8536	350	4	≤	≤	NUM
ap-8536	350	5	∥a∥	∥a∥	NOUN
ap-8536	350	6	.	.	PUNCT
ap-8536	351	1	∥b∥	∥b∥	VERB
ap-8536	351	2	for	for	ADP
ap-8536	351	3	every	every	DET
ap-8536	351	4	x	x	PROPN
ap-8536	351	5	∈	∈	PROPN
ap-8536	351	6	rm	rm	NOUN
ap-8536	351	7	and	and	CCONJ
ap-8536	351	8	a	a	DET
ap-8536	351	9	,	,	PUNCT
ap-8536	351	10	b	b	PROPN
ap-8536	351	11	∈	∈	PROPN
ap-8536	351	12	rm×m	rm×m	NOUN
ap-8536	351	13	.	.	PUNCT
ap-8536	352	1	if	if	SCONJ
ap-8536	352	2	(	(	PUNCT
ap-8536	352	3	dj)n	dj)n	PROPN
ap-8536	352	4	−∞	−∞	PUNCT
ap-8536	352	5	is	be	AUX
ap-8536	352	6	a	a	DET
ap-8536	352	7	sequence	sequence	NOUN
ap-8536	352	8	of	of	ADP
ap-8536	352	9	(	(	PUNCT
ap-8536	352	10	vector	vector	NOUN
ap-8536	352	11	)	)	PUNCT
ap-8536	352	12	digits	digit	NOUN
ap-8536	352	13	from	from	ADP
ap-8536	352	14	a	a	DET
ap-8536	352	15	finite	finite	NOUN
ap-8536	352	16	set	set	VERB
ap-8536	352	17	d	d	PROPN
ap-8536	352	18	⊂	⊂	PROPN
ap-8536	352	19	rm	rm	PROPN
ap-8536	352	20	,	,	PUNCT
ap-8536	352	21	then	then	ADV
ap-8536	352	22	the	the	DET
ap-8536	352	23	vector	vector	NOUN
ap-8536	352	24	∑n	∑n	PROPN
ap-8536	352	25	j=−∞	j=−∞	NOUN
ap-8536	352	26	m	m	VERB
ap-8536	352	27	j	j	PROPN
ap-8536	352	28	dj	dj	NOUN
ap-8536	352	29	is	be	AUX
ap-8536	352	30	well	well	ADV
ap-8536	352	31	defined	define	VERB
ap-8536	352	32	,	,	PUNCT
ap-8536	352	33	as	as	ADP
ap-8536	352	34	the	the	DET
ap-8536	352	35	sequence	sequence	NOUN
ap-8536	352	36	(	(	PUNCT
ap-8536	352	37	n∑	n∑	NOUN
ap-8536	352	38	j=−n	j=−n	PROPN
ap-8536	352	39	m	m	VERB
ap-8536	352	40	j	j	NOUN
ap-8536	352	41	dj	dj	NOUN
ap-8536	352	42	)	)	PUNCT
ap-8536	353	1	+	+	NOUN
ap-8536	353	2	∞	∞	PROPN
ap-8536	353	3	n=0	n=0	NUM
ap-8536	353	4	is	be	AUX
ap-8536	353	5	a	a	DET
ap-8536	353	6	cauchy	cauchy	ADJ
ap-8536	353	7	sequence	sequence	NOUN
ap-8536	353	8	.	.	PUNCT
ap-8536	354	1	indeed	indeed	ADV
ap-8536	354	2	,	,	PUNCT
ap-8536	354	3	let	let	VERB
ap-8536	354	4	us	we	PRON
ap-8536	354	5	denote	denote	VERB
ap-8536	354	6	r	r	NOUN
ap-8536	354	7	:	:	PUNCT
ap-8536	354	8	=	=	ADJ
ap-8536	354	9	∥∥m−1	∥∥m−1	X
ap-8536	354	10	∥∥	∥∥	X
ap-8536	354	11	<	<	X
ap-8536	354	12	1	1	NUM
ap-8536	354	13	and	and	CCONJ
ap-8536	354	14	d	d	NOUN
ap-8536	354	15	:	:	PUNCT
ap-8536	354	16	=	=	NOUN
ap-8536	354	17	max{∥d∥c	max{∥d∥c	NOUN
ap-8536	354	18	:	:	PUNCT
ap-8536	355	1	d	d	X
ap-8536	355	2	∈	∈	PROPN
ap-8536	355	3	d	d	NOUN
ap-8536	355	4	}	}	PUNCT
ap-8536	355	5	.	.	PUNCT
ap-8536	356	1	the	the	DET
ap-8536	356	2	triangle	triangle	NOUN
ap-8536	356	3	inequality	inequality	NOUN
ap-8536	356	4	implies	imply	VERB
ap-8536	356	5	for	for	ADP
ap-8536	356	6	every	every	DET
ap-8536	356	7	n	n	NOUN
ap-8536	356	8	,	,	PUNCT
ap-8536	356	9	q	q	PUNCT
ap-8536	356	10	∈	∈	PROPN
ap-8536	356	11	n	n	CCONJ
ap-8536	356	12	that∥∥∥∥∥∥	that∥∥∥∥∥∥	ADV
ap-8536	356	13	n∑	n∑	PROPN
ap-8536	357	1	j=−n	j=−n	PROPN
ap-8536	357	2	m	m	VERB
ap-8536	357	3	j	j	NOUN
ap-8536	357	4	dj	dj	X
ap-8536	357	5	−	−	PROPN
ap-8536	358	1	n∑	n∑	INTJ
ap-8536	359	1	j=−n−q	j=−n−q	PROPN
ap-8536	359	2	m	m	VERB
ap-8536	359	3	j	j	NOUN
ap-8536	359	4	dj	dj	X
ap-8536	359	5	∥∥∥∥∥∥	∥∥∥∥∥∥	NUM
ap-8536	359	6	c	c	NOUN
ap-8536	359	7	=	=	SYM
ap-8536	359	8	∥∥∥∥∥∥	∥∥∥∥∥∥	PROPN
ap-8536	360	1	−n−1∑	−n−1∑	PROPN
ap-8536	361	1	j=−n−q	j=−n−q	PROPN
ap-8536	361	2	m	m	VERB
ap-8536	361	3	j	j	NOUN
ap-8536	362	1	dj	dj	X
ap-8536	362	2	∥∥∥∥∥∥	∥∥∥∥∥∥	NUM
ap-8536	362	3	c	c	NOUN
ap-8536	362	4	=	=	SYM
ap-8536	362	5	∥∥∥∥∥∥	∥∥∥∥∥∥	PROPN
ap-8536	363	1	n+q∑	n+q∑	PROPN
ap-8536	363	2	j	j	X
ap-8536	364	1	=	=	SYM
ap-8536	364	2	n+1	n+1	PROPN
ap-8536	364	3	(	(	PUNCT
ap-8536	364	4	m−1)j	m−1)j	NOUN
ap-8536	364	5	d−j	d−j	NOUN
ap-8536	364	6	∥∥∥∥∥∥	∥∥∥∥∥∥	NUM
ap-8536	364	7	c	c	PART
ap-8536	364	8	≤	≤	NOUN
ap-8536	364	9	n+q∑	n+q∑	PROPN
ap-8536	364	10	j	j	PROPN
ap-8536	364	11	=	=	NOUN
ap-8536	364	12	n+1	n+1	PROPN
ap-8536	364	13	rjd	rjd	PROPN
ap-8536	364	14	≤	≤	PROPN
ap-8536	364	15	drn+1	drn+1	PROPN
ap-8536	364	16	1	1	NUM
ap-8536	364	17	−	−	NOUN
ap-8536	364	18	r	r	NOUN
ap-8536	364	19	,	,	PUNCT
ap-8536	364	20	so	so	CCONJ
ap-8536	364	21	the	the	DET
ap-8536	364	22	value	value	NOUN
ap-8536	364	23	drn+1	drn+1	PROPN
ap-8536	364	24	1−r	1−r	PROPN
ap-8536	364	25	can	can	AUX
ap-8536	364	26	be	be	AUX
ap-8536	364	27	made	make	VERB
ap-8536	364	28	arbitrarily	arbitrarily	ADV
ap-8536	364	29	small	small	ADJ
ap-8536	364	30	for	for	ADP
ap-8536	364	31	all	all	PRON
ap-8536	364	32	n	n	PRON
ap-8536	364	33	>	>	X
ap-8536	364	34	n0	n0	PROPN
ap-8536	364	35	,	,	PUNCT
ap-8536	364	36	with	with	ADP
ap-8536	364	37	sufficiently	sufficiently	ADV
ap-8536	364	38	large	large	ADJ
ap-8536	364	39	n0	n0	X
ap-8536	364	40	∈	∈	PROPN
ap-8536	364	41	n.	n.	NOUN
ap-8536	364	42	consequently	consequently	ADV
ap-8536	364	43	,	,	PUNCT
ap-8536	364	44	if	if	SCONJ
ap-8536	364	45	m	m	NOUN
ap-8536	364	46	is	be	AUX
ap-8536	364	47	expansive	expansive	ADJ
ap-8536	364	48	,	,	PUNCT
ap-8536	364	49	then	then	ADV
ap-8536	364	50	there	there	PRON
ap-8536	364	51	exists	exist	VERB
ap-8536	364	52	lim	lim	PROPN
ap-8536	364	53	n→+∞	n→+∞	VERB
ap-8536	364	54	∑n	∑n	PROPN
ap-8536	364	55	j=−n	j=−n	PROPN
ap-8536	364	56	m	m	VERB
ap-8536	364	57	jdj	jdj	PROPN
ap-8536	364	58	,	,	PUNCT
ap-8536	364	59	which	which	PRON
ap-8536	364	60	may	may	AUX
ap-8536	364	61	be	be	AUX
ap-8536	364	62	denoted	denote	VERB
ap-8536	364	63	as∑n	as∑n	PROPN
ap-8536	364	64	j=−∞	j=−∞	PROPN
ap-8536	364	65	m	m	PROPN
ap-8536	364	66	jdj	jdj	PROPN
ap-8536	364	67	.	.	PUNCT
ap-8536	365	1	in	in	ADP
ap-8536	365	2	the	the	DET
ap-8536	365	3	remaining	remain	VERB
ap-8536	365	4	part	part	NOUN
ap-8536	365	5	of	of	ADP
ap-8536	365	6	this	this	DET
ap-8536	365	7	chapter	chapter	NOUN
ap-8536	365	8	,	,	PUNCT
ap-8536	365	9	we	we	PRON
ap-8536	365	10	focus	focus	VERB
ap-8536	365	11	on	on	ADP
ap-8536	365	12	matrix	matrix	NOUN
ap-8536	365	13	numeration	numeration	NOUN
ap-8536	365	14	systems	system	NOUN
ap-8536	365	15	with	with	ADP
ap-8536	365	16	m	m	AUX
ap-8536	365	17	being	be	AUX
ap-8536	365	18	an	an	DET
ap-8536	365	19	expansive	expansive	ADJ
ap-8536	365	20	matrix	matrix	NOUN
ap-8536	365	21	.	.	PUNCT
ap-8536	366	1	matrix	matrix	NOUN
ap-8536	366	2	numeration	numeration	NOUN
ap-8536	366	3	systems	system	NOUN
ap-8536	366	4	with	with	ADP
ap-8536	366	5	base	base	NOUN
ap-8536	366	6	m	m	VERB
ap-8536	366	7	being	be	AUX
ap-8536	366	8	an	an	DET
ap-8536	366	9	expansive	expansive	ADJ
ap-8536	366	10	matrix	matrix	NOUN
ap-8536	366	11	have	have	AUX
ap-8536	366	12	been	be	AUX
ap-8536	366	13	intensively	intensively	ADV
ap-8536	366	14	studied	study	VERB
ap-8536	366	15	since	since	SCONJ
ap-8536	366	16	the	the	DET
ap-8536	366	17	work	work	NOUN
ap-8536	366	18	of	of	ADP
ap-8536	366	19	vince	vince	NOUN
ap-8536	366	20	.	.	PUNCT
ap-8536	367	1	the	the	DET
ap-8536	367	2	main	main	ADJ
ap-8536	367	3	focus	focus	NOUN
ap-8536	367	4	of	of	ADP
ap-8536	367	5	the	the	DET
ap-8536	367	6	research	research	NOUN
ap-8536	367	7	in	in	ADP
ap-8536	367	8	this	this	DET
ap-8536	367	9	area	area	NOUN
ap-8536	367	10	is	be	AUX
ap-8536	367	11	on	on	ADP
ap-8536	367	12	the	the	DET
ap-8536	367	13	systems	system	NOUN
ap-8536	367	14	where	where	SCONJ
ap-8536	367	15	each	each	DET
ap-8536	367	16	lattice	lattice	NOUN
ap-8536	367	17	point	point	NOUN
ap-8536	367	18	has	have	VERB
ap-8536	367	19	a	a	DET
ap-8536	367	20	unique	unique	ADJ
ap-8536	367	21	representation	representation	NOUN
ap-8536	367	22	.	.	PUNCT
ap-8536	368	1	recall	recall	VERB
ap-8536	368	2	that	that	SCONJ
ap-8536	368	3	a	a	DET
ap-8536	368	4	lattice	lattice	NOUN
ap-8536	368	5	in	in	ADP
ap-8536	368	6	rm	rm	PROPN
ap-8536	368	7	is	be	AUX
ap-8536	368	8	the	the	DET
ap-8536	368	9	set	set	NOUN
ap-8536	368	10	of	of	ADP
ap-8536	368	11	all	all	DET
ap-8536	368	12	integer	integer	NOUN
ap-8536	368	13	combinations	combination	NOUN
ap-8536	368	14	of	of	ADP
ap-8536	368	15	m	m	VERB
ap-8536	368	16	linearly	linearly	ADV
ap-8536	368	17	independent	independent	ADJ
ap-8536	368	18	vectors	vector	NOUN
ap-8536	368	19	.	.	PUNCT
ap-8536	369	1	a	a	DET
ap-8536	369	2	lattice	lattice	PROPN
ap-8536	369	3	numeration	numeration	NOUN
ap-8536	369	4	system	system	NOUN
ap-8536	369	5	can	can	AUX
ap-8536	369	6	be	be	AUX
ap-8536	369	7	formalised	formalise	VERB
ap-8536	369	8	as	as	SCONJ
ap-8536	369	9	follows	follow	VERB
ap-8536	369	10	(	(	PUNCT
ap-8536	369	11	see	see	VERB
ap-8536	369	12	[	[	X
ap-8536	369	13	24	24	NUM
ap-8536	369	14	]	]	PUNCT
ap-8536	369	15	):	):	PUNCT
ap-8536	369	16	definition	definition	NOUN
ap-8536	369	17	16	16	NUM
ap-8536	369	18	.	.	PUNCT
ap-8536	370	1	let	let	VERB
ap-8536	370	2	λ	λ	PRON
ap-8536	370	3	be	be	AUX
ap-8536	370	4	a	a	DET
ap-8536	370	5	lattice	lattice	NOUN
ap-8536	370	6	,	,	PUNCT
ap-8536	370	7	m	m	VERB
ap-8536	370	8	:	:	PUNCT
ap-8536	370	9	λ	λ	X
ap-8536	370	10	→	→	SYM
ap-8536	370	11	λ	λ	X
ap-8536	370	12	be	be	AUX
ap-8536	370	13	a	a	DET
ap-8536	370	14	linear	linear	ADJ
ap-8536	370	15	operator	operator	NOUN
ap-8536	370	16	(	(	PUNCT
ap-8536	370	17	also	also	ADV
ap-8536	370	18	called	call	VERB
ap-8536	370	19	the	the	DET
ap-8536	370	20	base	base	NOUN
ap-8536	370	21	or	or	CCONJ
ap-8536	370	22	radix	radix	NOUN
ap-8536	370	23	)	)	PUNCT
ap-8536	370	24	and	and	CCONJ
ap-8536	370	25	let	let	VERB
ap-8536	370	26	d	d	PRON
ap-8536	370	27	be	be	AUX
ap-8536	370	28	a	a	DET
ap-8536	370	29	finite	finite	NOUN
ap-8536	370	30	subset	subset	NOUN
ap-8536	370	31	of	of	ADP
ap-8536	370	32	λ	λ	PROPN
ap-8536	370	33	containing	contain	VERB
ap-8536	370	34	0	0	NUM
ap-8536	370	35	(	(	PUNCT
ap-8536	370	36	called	call	VERB
ap-8536	370	37	the	the	DET
ap-8536	370	38	digit	digit	NOUN
ap-8536	370	39	set	set	NOUN
ap-8536	370	40	)	)	PUNCT
ap-8536	370	41	.	.	PUNCT
ap-8536	371	1	the	the	DET
ap-8536	371	2	triplet	triplet	NOUN
ap-8536	371	3	(	(	PUNCT
ap-8536	371	4	λ	λ	PROPN
ap-8536	371	5	,	,	PUNCT
ap-8536	371	6	m	m	PROPN
ap-8536	371	7	,	,	PUNCT
ap-8536	371	8	d	d	X
ap-8536	371	9	)	)	PUNCT
ap-8536	371	10	is	be	AUX
ap-8536	371	11	called	call	VERB
ap-8536	371	12	a	a	DET
ap-8536	371	13	generalised	generalise	VERB
ap-8536	371	14	number	number	NOUN
ap-8536	371	15	system	system	NOUN
ap-8536	371	16	(	(	PUNCT
ap-8536	371	17	gns	gns	NOUN
ap-8536	371	18	)	)	PUNCT
ap-8536	371	19	if	if	SCONJ
ap-8536	371	20	every	every	DET
ap-8536	371	21	element	element	NOUN
ap-8536	371	22	x	x	SYM
ap-8536	371	23	∈	∈	NOUN
ap-8536	371	24	λ	λ	NOUN
ap-8536	371	25	has	have	VERB
ap-8536	371	26	a	a	DET
ap-8536	371	27	unique	unique	ADJ
ap-8536	371	28	finite	finite	ADJ
ap-8536	371	29	representation	representation	NOUN
ap-8536	371	30	of	of	ADP
ap-8536	371	31	the	the	DET
ap-8536	371	32	form	form	NOUN
ap-8536	371	33	x	x	PUNCT
ap-8536	371	34	=	=	SYM
ap-8536	371	35	n∑	n∑	PROPN
ap-8536	371	36	j=0	j=0	PROPN
ap-8536	371	37	m	m	PROPN
ap-8536	371	38	jdj	jdj	PROPN
ap-8536	371	39	,	,	PUNCT
ap-8536	371	40	where	where	SCONJ
ap-8536	371	41	n	n	X
ap-8536	371	42	∈	∈	PROPN
ap-8536	371	43	n	n	CCONJ
ap-8536	371	44	,	,	PUNCT
ap-8536	371	45	dj	dj	NOUN
ap-8536	371	46	∈	∈	PROPN
ap-8536	371	47	d	d	NOUN
ap-8536	371	48	for	for	ADP
ap-8536	371	49	every	every	DET
ap-8536	371	50	j	j	NOUN
ap-8536	371	51	=	=	SYM
ap-8536	371	52	0	0	NUM
ap-8536	371	53	,	,	PUNCT
ap-8536	371	54	1	1	NUM
ap-8536	371	55	,	,	PUNCT
ap-8536	371	56	.	.	PUNCT
ap-8536	371	57	.	.	PUNCT
ap-8536	372	1	.	.	PUNCT
ap-8536	373	1	,	,	PUNCT
ap-8536	373	2	n	n	PROPN
ap-8536	373	3	and	and	CCONJ
ap-8536	373	4	dn	dn	PROPN
ap-8536	373	5	is	be	AUX
ap-8536	373	6	a	a	DET
ap-8536	373	7	non	non	ADJ
ap-8536	373	8	-	-	ADJ
ap-8536	373	9	zero	zero	NUM
ap-8536	373	10	digit	digit	NOUN
ap-8536	373	11	(	(	PUNCT
ap-8536	373	12	if	if	SCONJ
ap-8536	373	13	x	x	NUM
ap-8536	373	14	̸=	̸=	PROPN
ap-8536	373	15	0	0	NUM
ap-8536	373	16	)	)	PUNCT
ap-8536	373	17	.	.	PUNCT
ap-8536	374	1	characterisation	characterisation	NOUN
ap-8536	374	2	of	of	ADP
ap-8536	374	3	triplets	triplet	NOUN
ap-8536	374	4	(	(	PUNCT
ap-8536	374	5	λ	λ	X
ap-8536	374	6	,	,	PUNCT
ap-8536	374	7	m	m	PROPN
ap-8536	374	8	,	,	PUNCT
ap-8536	374	9	d	d	NOUN
ap-8536	374	10	)	)	PUNCT
ap-8536	374	11	forming	form	VERB
ap-8536	374	12	gns	gns	NOUN
ap-8536	374	13	seems	seem	VERB
ap-8536	374	14	to	to	PART
ap-8536	374	15	be	be	AUX
ap-8536	374	16	a	a	DET
ap-8536	374	17	difficult	difficult	ADJ
ap-8536	374	18	task	task	NOUN
ap-8536	374	19	.	.	PUNCT
ap-8536	375	1	to	to	PART
ap-8536	375	2	quote	quote	VERB
ap-8536	375	3	a	a	DET
ap-8536	375	4	necessary	necessary	ADJ
ap-8536	375	5	194	194	NUM
ap-8536	375	6	vol	vol	NOUN
ap-8536	375	7	.	.	PUNCT
ap-8536	376	1	63	63	NUM
ap-8536	376	2	no	no	NOUN
ap-8536	376	3	.	.	PUNCT
ap-8536	377	1	3/2023	3/2023	NUM
ap-8536	377	2	positional	positional	ADJ
ap-8536	377	3	representation	representation	NOUN
ap-8536	377	4	of	of	ADP
ap-8536	377	5	vectors	vector	NOUN
ap-8536	377	6	condition	condition	NOUN
ap-8536	377	7	,	,	PUNCT
ap-8536	377	8	we	we	PRON
ap-8536	377	9	recall	recall	VERB
ap-8536	377	10	that	that	SCONJ
ap-8536	377	11	two	two	NUM
ap-8536	377	12	elements	element	NOUN
ap-8536	377	13	(	(	PUNCT
ap-8536	377	14	lattice	lattice	NOUN
ap-8536	377	15	points	point	NOUN
ap-8536	377	16	)	)	PUNCT
ap-8536	377	17	x	x	X
ap-8536	377	18	,	,	PUNCT
ap-8536	377	19	y	y	PROPN
ap-8536	377	20	∈	∈	PROPN
ap-8536	377	21	λ	λ	NOUN
ap-8536	377	22	are	be	AUX
ap-8536	377	23	congruent	congruent	ADJ
ap-8536	377	24	modulo	modulo	NOUN
ap-8536	377	25	m	m	VERB
ap-8536	377	26	if	if	SCONJ
ap-8536	377	27	they	they	PRON
ap-8536	377	28	belong	belong	VERB
ap-8536	377	29	to	to	ADP
ap-8536	377	30	the	the	DET
ap-8536	377	31	same	same	ADJ
ap-8536	377	32	residue	residue	NOUN
ap-8536	377	33	class	class	NOUN
ap-8536	377	34	,	,	PUNCT
ap-8536	377	35	i.	i.	PROPN
ap-8536	377	36	e.	e.	PROPN
ap-8536	377	37	,	,	PUNCT
ap-8536	377	38	if	if	SCONJ
ap-8536	377	39	(	(	PUNCT
ap-8536	377	40	x	x	SYM
ap-8536	377	41	−	−	PROPN
ap-8536	377	42	y	y	X
ap-8536	377	43	)	)	PUNCT
ap-8536	377	44	∈	∈	PROPN
ap-8536	377	45	mλ	mλ	NOUN
ap-8536	377	46	.	.	PUNCT
ap-8536	378	1	we	we	PRON
ap-8536	378	2	denote	denote	VERB
ap-8536	378	3	this	this	DET
ap-8536	378	4	fact	fact	NOUN
ap-8536	378	5	by	by	ADP
ap-8536	378	6	x	x	PUNCT
ap-8536	378	7	≡λ	≡λ	NOUN
ap-8536	378	8	y	y	PROPN
ap-8536	378	9	(	(	PUNCT
ap-8536	378	10	mod	mod	PROPN
ap-8536	378	11	m	m	PROPN
ap-8536	378	12	)	)	PUNCT
ap-8536	378	13	.	.	PUNCT
ap-8536	379	1	theorem	theorem	VERB
ap-8536	379	2	17	17	NUM
ap-8536	379	3	.	.	PUNCT
ap-8536	380	1	[	[	X
ap-8536	380	2	13	13	NUM
ap-8536	380	3	]	]	X
ap-8536	380	4	if	if	SCONJ
ap-8536	380	5	(	(	PUNCT
ap-8536	380	6	λ	λ	X
ap-8536	380	7	,	,	PUNCT
ap-8536	380	8	m	m	PROPN
ap-8536	380	9	,	,	PUNCT
ap-8536	380	10	d	d	X
ap-8536	380	11	)	)	PUNCT
ap-8536	380	12	is	be	AUX
ap-8536	380	13	a	a	DET
ap-8536	380	14	gns	gns	NOUN
ap-8536	380	15	,	,	PUNCT
ap-8536	380	16	then	then	ADV
ap-8536	380	17	the	the	DET
ap-8536	380	18	following	follow	VERB
ap-8536	380	19	holds	hold	VERB
ap-8536	380	20	:	:	PUNCT
ap-8536	380	21	(	(	PUNCT
ap-8536	380	22	1	1	NUM
ap-8536	380	23	.	.	PUNCT
ap-8536	380	24	)	)	PUNCT
ap-8536	381	1	the	the	DET
ap-8536	381	2	operator	operator	NOUN
ap-8536	381	3	m	m	VERB
ap-8536	381	4	is	be	AUX
ap-8536	381	5	expansive	expansive	ADJ
ap-8536	381	6	.	.	PUNCT
ap-8536	382	1	(	(	PUNCT
ap-8536	382	2	2	2	NUM
ap-8536	382	3	.	.	PUNCT
ap-8536	382	4	)	)	PUNCT
ap-8536	383	1	the	the	DET
ap-8536	383	2	digit	digit	NOUN
ap-8536	383	3	set	set	NOUN
ap-8536	383	4	d	d	PUNCT
ap-8536	383	5	is	be	AUX
ap-8536	383	6	a	a	DET
ap-8536	383	7	complete	complete	ADJ
ap-8536	383	8	residue	residue	NOUN
ap-8536	383	9	system	system	NOUN
ap-8536	383	10	modulo	modulo	NOUN
ap-8536	383	11	m.	m.	NOUN
ap-8536	383	12	(	(	PUNCT
ap-8536	383	13	3	3	NUM
ap-8536	383	14	.	.	PUNCT
ap-8536	383	15	)	)	PUNCT
ap-8536	384	1	det(m	det(m	PROPN
ap-8536	384	2	−	−	PROPN
ap-8536	385	1	i	i	NOUN
ap-8536	385	2	)	)	PUNCT
ap-8536	385	3	̸=	̸=	PROPN
ap-8536	385	4	±1	±1	VERB
ap-8536	385	5	.	.	PUNCT
ap-8536	386	1	note	note	VERB
ap-8536	386	2	that	that	SCONJ
ap-8536	386	3	the	the	DET
ap-8536	386	4	number	number	NOUN
ap-8536	386	5	of	of	ADP
ap-8536	386	6	residue	residue	NOUN
ap-8536	386	7	classes	class	NOUN
ap-8536	386	8	modulo	modulo	VERB
ap-8536	386	9	m	m	AUX
ap-8536	386	10	equals	equal	VERB
ap-8536	386	11	|	|	ADV
ap-8536	386	12	det(m)|	det(m)|	PROPN
ap-8536	386	13	,	,	PUNCT
ap-8536	386	14	hence	hence	ADV
ap-8536	386	15	the	the	DET
ap-8536	386	16	gns	gns	NOUN
ap-8536	386	17	has	have	VERB
ap-8536	386	18	exactly	exactly	ADV
ap-8536	386	19	#	#	SYM
ap-8536	386	20	d	d	NOUN
ap-8536	386	21	=	=	SYM
ap-8536	387	1	|	|	NOUN
ap-8536	387	2	det(m)|	det(m)|	PROPN
ap-8536	387	3	digits	digit	NOUN
ap-8536	387	4	.	.	PUNCT
ap-8536	388	1	a	a	DET
ap-8536	388	2	sufficient	sufficient	ADJ
ap-8536	388	3	condition	condition	NOUN
ap-8536	388	4	for	for	ADP
ap-8536	388	5	gns	gns	NOUN
ap-8536	388	6	was	be	AUX
ap-8536	388	7	provided	provide	VERB
ap-8536	388	8	by	by	ADP
ap-8536	388	9	l.	l.	PROPN
ap-8536	388	10	germán	germán	PROPN
ap-8536	388	11	and	and	CCONJ
ap-8536	388	12	a.	a.	NOUN
ap-8536	388	13	kovács	kovács	NOUN
ap-8536	388	14	in	in	ADP
ap-8536	388	15	[	[	X
ap-8536	388	16	24	24	NUM
ap-8536	388	17	]	]	PUNCT
ap-8536	388	18	.	.	PUNCT
ap-8536	389	1	theorem	theorem	PROPN
ap-8536	389	2	18	18	NUM
ap-8536	389	3	.	.	PUNCT
ap-8536	390	1	[	[	X
ap-8536	390	2	24	24	NUM
ap-8536	390	3	]	]	PUNCT
ap-8536	390	4	let	let	VERB
ap-8536	390	5	m	m	PRON
ap-8536	390	6	:	:	PUNCT
ap-8536	391	1	λ	λ	X
ap-8536	391	2	7→	7→	NUM
ap-8536	391	3	λ	λ	NOUN
ap-8536	391	4	be	be	AUX
ap-8536	391	5	a	a	DET
ap-8536	391	6	non	non	ADJ
ap-8536	391	7	-	-	ADJ
ap-8536	391	8	singular	singular	ADJ
ap-8536	391	9	linear	linear	ADJ
ap-8536	391	10	operator	operator	NOUN
ap-8536	391	11	.	.	PUNCT
ap-8536	392	1	if	if	SCONJ
ap-8536	392	2	the	the	DET
ap-8536	392	3	spectral	spectral	ADJ
ap-8536	392	4	radius	radius	NOUN
ap-8536	392	5	of	of	ADP
ap-8536	392	6	the	the	DET
ap-8536	392	7	inverse	inverse	NOUN
ap-8536	392	8	operator	operator	NOUN
ap-8536	392	9	m−1	m−1	PROPN
ap-8536	392	10	is	be	AUX
ap-8536	392	11	less	less	ADJ
ap-8536	392	12	than	than	ADP
ap-8536	392	13	1	1	NUM
ap-8536	392	14	2	2	NUM
ap-8536	392	15	,	,	PUNCT
ap-8536	392	16	then	then	ADV
ap-8536	392	17	there	there	PRON
ap-8536	392	18	exists	exist	VERB
ap-8536	392	19	a	a	DET
ap-8536	392	20	digit	digit	NOUN
ap-8536	392	21	set	set	VERB
ap-8536	392	22	d	d	X
ap-8536	392	23	⊂	⊂	PROPN
ap-8536	392	24	λ	λ	NOUN
ap-8536	392	25	such	such	ADJ
ap-8536	392	26	that	that	SCONJ
ap-8536	392	27	(	(	PUNCT
ap-8536	392	28	λ	λ	PROPN
ap-8536	392	29	,	,	PUNCT
ap-8536	392	30	m	m	PROPN
ap-8536	392	31	,	,	PUNCT
ap-8536	392	32	d	d	X
ap-8536	392	33	)	)	PUNCT
ap-8536	392	34	is	be	AUX
ap-8536	392	35	a	a	DET
ap-8536	392	36	gns	gns	NOUN
ap-8536	392	37	.	.	PUNCT
ap-8536	393	1	any	any	DET
ap-8536	393	2	lattice	lattice	NOUN
ap-8536	393	3	λ	λ	PROPN
ap-8536	393	4	⊂	⊂	PROPN
ap-8536	393	5	rm	rm	PROPN
ap-8536	393	6	is	be	AUX
ap-8536	393	7	just	just	ADV
ap-8536	393	8	an	an	DET
ap-8536	393	9	image	image	NOUN
ap-8536	393	10	of	of	ADP
ap-8536	393	11	zm	zm	PROPN
ap-8536	393	12	under	under	ADP
ap-8536	393	13	a	a	DET
ap-8536	393	14	non	non	ADJ
ap-8536	393	15	-	-	ADJ
ap-8536	393	16	singular	singular	ADJ
ap-8536	393	17	linear	linear	PROPN
ap-8536	393	18	map	map	NOUN
ap-8536	393	19	.	.	PUNCT
ap-8536	394	1	since	since	SCONJ
ap-8536	394	2	a	a	DET
ap-8536	394	3	linear	linear	ADJ
ap-8536	394	4	map	map	NOUN
ap-8536	394	5	does	do	AUX
ap-8536	394	6	not	not	PART
ap-8536	394	7	change	change	VERB
ap-8536	394	8	the	the	DET
ap-8536	394	9	gns	gns	NOUN
ap-8536	394	10	property	property	NOUN
ap-8536	394	11	,	,	PUNCT
ap-8536	394	12	we	we	PRON
ap-8536	394	13	can	can	AUX
ap-8536	394	14	consider	consider	VERB
ap-8536	394	15	,	,	PUNCT
ap-8536	394	16	without	without	ADP
ap-8536	394	17	a	a	DET
ap-8536	394	18	loss	loss	NOUN
ap-8536	394	19	of	of	ADP
ap-8536	394	20	generality	generality	NOUN
ap-8536	394	21	,	,	PUNCT
ap-8536	394	22	that	that	SCONJ
ap-8536	394	23	λ	λ	X
ap-8536	394	24	=	=	SYM
ap-8536	394	25	zm	zm	PROPN
ap-8536	394	26	.	.	PUNCT
ap-8536	395	1	a	a	DET
ap-8536	395	2	linear	linear	ADJ
ap-8536	395	3	operator	operator	NOUN
ap-8536	395	4	mapping	mapping	NOUN
ap-8536	395	5	zm	zm	PROPN
ap-8536	395	6	to	to	ADP
ap-8536	395	7	zm	zm	PROPN
ap-8536	395	8	corresponds	correspond	VERB
ap-8536	395	9	to	to	ADP
ap-8536	395	10	multiplication	multiplication	NOUN
ap-8536	395	11	of	of	ADP
ap-8536	395	12	integer	integer	NOUN
ap-8536	395	13	vectors	vector	NOUN
ap-8536	395	14	by	by	ADP
ap-8536	395	15	a	a	DET
ap-8536	395	16	matrix	matrix	NOUN
ap-8536	395	17	m	m	NOUN
ap-8536	395	18	∈	∈	PROPN
ap-8536	395	19	zm×m	zm×m	NOUN
ap-8536	395	20	.	.	PUNCT
ap-8536	396	1	further	further	ADJ
ap-8536	396	2	results	result	NOUN
ap-8536	396	3	are	be	AUX
ap-8536	396	4	known	know	VERB
ap-8536	396	5	regarding	regard	VERB
ap-8536	396	6	the	the	DET
ap-8536	396	7	existence	existence	NOUN
ap-8536	396	8	of	of	ADP
ap-8536	396	9	gns	gns	NOUN
ap-8536	396	10	with	with	ADP
ap-8536	396	11	special	special	ADJ
ap-8536	396	12	types	type	NOUN
ap-8536	396	13	of	of	ADP
ap-8536	396	14	the	the	DET
ap-8536	396	15	digits	digit	NOUN
ap-8536	396	16	sets	set	VERB
ap-8536	396	17	:	:	PUNCT
ap-8536	396	18	•	•	ADP
ap-8536	396	19	if	if	SCONJ
ap-8536	396	20	d	d	NOUN
ap-8536	396	21	=	=	SYM
ap-8536	396	22	{	{	PUNCT
ap-8536	396	23	k(1	k(1	PROPN
ap-8536	396	24	,	,	PUNCT
ap-8536	396	25	0	0	NUM
ap-8536	396	26	,	,	PUNCT
ap-8536	396	27	.	.	PUNCT
ap-8536	396	28	.	.	PUNCT
ap-8536	397	1	.	.	PUNCT
ap-8536	398	1	,	,	PUNCT
ap-8536	398	2	0)⊤	0)⊤	PROPN
ap-8536	398	3	)	)	PUNCT
ap-8536	398	4	:	:	PUNCT
ap-8536	399	1	k	k	PROPN
ap-8536	399	2	∈	∈	PROPN
ap-8536	399	3	z	z	PROPN
ap-8536	399	4	,	,	PUNCT
ap-8536	399	5	0	0	NUM
ap-8536	399	6	≤	≤	NOUN
ap-8536	400	1	k	k	X
ap-8536	400	2	<	<	X
ap-8536	400	3	|	|	PROPN
ap-8536	400	4	det(m)|	det(m)|	PROPN
ap-8536	400	5	}	}	PUNCT
ap-8536	400	6	,	,	PUNCT
ap-8536	400	7	we	we	PRON
ap-8536	400	8	speak	speak	VERB
ap-8536	400	9	about	about	ADP
ap-8536	400	10	canonical	canonical	ADJ
ap-8536	400	11	systems	system	NOUN
ap-8536	400	12	.	.	PUNCT
ap-8536	401	1	•	•	INTJ
ap-8536	401	2	if	if	SCONJ
ap-8536	401	3	d	d	NOUN
ap-8536	401	4	=	=	SYM
ap-8536	401	5	{	{	PUNCT
ap-8536	401	6	k(1	k(1	PROPN
ap-8536	401	7	,	,	PUNCT
ap-8536	401	8	0	0	NUM
ap-8536	401	9	,	,	PUNCT
ap-8536	401	10	.	.	PUNCT
ap-8536	401	11	.	.	PUNCT
ap-8536	402	1	.	.	PUNCT
ap-8536	403	1	,	,	PUNCT
ap-8536	403	2	0)⊤	0)⊤	PROPN
ap-8536	403	3	)	)	PUNCT
ap-8536	403	4	:	:	PUNCT
ap-8536	404	1	k	k	PROPN
ap-8536	404	2	∈	∈	PROPN
ap-8536	404	3	z	z	PROPN
ap-8536	404	4	,	,	PUNCT
ap-8536	404	5	−	−	PROPN
ap-8536	404	6	1	1	NUM
ap-8536	404	7	2	2	NUM
ap-8536	404	8	|	|	NOUN
ap-8536	404	9	det(m)|	det(m)|	VERB
ap-8536	404	10	<	<	X
ap-8536	404	11	k	k	X
ap-8536	404	12	≤	≤	NUM
ap-8536	404	13	1	1	NUM
ap-8536	404	14	2	2	NUM
ap-8536	404	15	|	|	ADV
ap-8536	404	16	det(m)|	det(m)|	PROPN
ap-8536	404	17	}	}	PUNCT
ap-8536	404	18	,	,	PUNCT
ap-8536	404	19	the	the	DET
ap-8536	404	20	digit	digit	NOUN
ap-8536	404	21	set	set	NOUN
ap-8536	404	22	is	be	AUX
ap-8536	404	23	called	call	VERB
ap-8536	404	24	symmetric	symmetric	ADJ
ap-8536	404	25	.	.	PUNCT
ap-8536	405	1	•	•	INTJ
ap-8536	405	2	if	if	SCONJ
ap-8536	405	3	d	d	PROPN
ap-8536	405	4	contains	contain	VERB
ap-8536	405	5	the	the	DET
ap-8536	405	6	lattice	lattice	NOUN
ap-8536	405	7	point	point	NOUN
ap-8536	405	8	of	of	ADP
ap-8536	405	9	the	the	DET
ap-8536	405	10	smallest	small	ADJ
ap-8536	405	11	norm	norm	NOUN
ap-8536	405	12	from	from	ADP
ap-8536	405	13	each	each	DET
ap-8536	405	14	residue	residue	NOUN
ap-8536	405	15	class	class	NOUN
ap-8536	405	16	(	(	PUNCT
ap-8536	405	17	modulo	modulo	PROPN
ap-8536	405	18	m	m	PROPN
ap-8536	405	19	)	)	PUNCT
ap-8536	405	20	,	,	PUNCT
ap-8536	405	21	then	then	ADV
ap-8536	405	22	the	the	DET
ap-8536	405	23	digits	digit	NOUN
ap-8536	405	24	set	set	VERB
ap-8536	405	25	d	d	NOUN
ap-8536	405	26	is	be	AUX
ap-8536	405	27	said	say	VERB
ap-8536	405	28	to	to	PART
ap-8536	405	29	be	be	AUX
ap-8536	405	30	dense	dense	ADJ
ap-8536	405	31	.	.	PUNCT
ap-8536	406	1	the	the	DET
ap-8536	406	2	choice	choice	NOUN
ap-8536	406	3	of	of	ADP
ap-8536	406	4	the	the	DET
ap-8536	406	5	smallest	small	ADJ
ap-8536	406	6	lattice	lattice	NOUN
ap-8536	406	7	point	point	NOUN
ap-8536	406	8	is	be	AUX
ap-8536	406	9	not	not	PART
ap-8536	406	10	necessarily	necessarily	ADV
ap-8536	406	11	unique	unique	ADJ
ap-8536	406	12	.	.	PUNCT
ap-8536	407	1	•	•	ADP
ap-8536	407	2	the	the	DET
ap-8536	407	3	adjoint	adjoint	NOUN
ap-8536	407	4	digit	digit	NOUN
ap-8536	407	5	set	set	NOUN
ap-8536	407	6	consists	consist	VERB
ap-8536	407	7	of	of	ADP
ap-8536	407	8	those	those	DET
ap-8536	407	9	lattice	lattice	NOUN
ap-8536	407	10	points	point	NOUN
ap-8536	407	11	which	which	PRON
ap-8536	407	12	belong	belong	VERB
ap-8536	407	13	to	to	ADP
ap-8536	407	14	|	|	ADV
ap-8536	407	15	det(m)|	det(m)|	ADJ
ap-8536	407	16	·	·	PUNCT
ap-8536	408	1	[	[	X
ap-8536	408	2	−	−	NOUN
ap-8536	408	3	1	1	NUM
ap-8536	408	4	2	2	NUM
ap-8536	408	5	,	,	PUNCT
ap-8536	408	6	1	1	NUM
ap-8536	408	7	2	2	NUM
ap-8536	408	8	)	)	PUNCT
ap-8536	408	9	m.	m.	NOUN
ap-8536	408	10	the	the	DET
ap-8536	408	11	results	result	NOUN
ap-8536	408	12	on	on	ADP
ap-8536	408	13	gns	gns	NOUN
ap-8536	408	14	with	with	ADP
ap-8536	408	15	the	the	DET
ap-8536	408	16	special	special	ADJ
ap-8536	408	17	digits	digits	NOUN
ap-8536	408	18	sets	set	NOUN
ap-8536	408	19	can	can	AUX
ap-8536	408	20	be	be	AUX
ap-8536	408	21	consulted	consult	VERB
ap-8536	408	22	in	in	ADP
ap-8536	408	23	[	[	X
ap-8536	408	24	25	25	NUM
ap-8536	408	25	]	]	PUNCT
ap-8536	408	26	.	.	PUNCT
ap-8536	409	1	if	if	SCONJ
ap-8536	409	2	we	we	PRON
ap-8536	409	3	abandon	abandon	VERB
ap-8536	409	4	the	the	DET
ap-8536	409	5	requirement	requirement	NOUN
ap-8536	409	6	of	of	ADP
ap-8536	409	7	uniqueness	uniqueness	NOUN
ap-8536	409	8	for	for	ADP
ap-8536	409	9	the	the	DET
ap-8536	409	10	representation	representation	NOUN
ap-8536	409	11	of	of	ADP
ap-8536	409	12	vectors	vector	NOUN
ap-8536	409	13	from	from	ADP
ap-8536	409	14	zm	zm	PROPN
ap-8536	409	15	,	,	PUNCT
ap-8536	409	16	then	then	ADV
ap-8536	409	17	the	the	DET
ap-8536	409	18	question	question	NOUN
ap-8536	409	19	on	on	ADP
ap-8536	409	20	existence	existence	NOUN
ap-8536	409	21	of	of	ADP
ap-8536	409	22	a	a	DET
ap-8536	409	23	suitable	suitable	ADJ
ap-8536	409	24	digit	digit	NOUN
ap-8536	409	25	set	set	VERB
ap-8536	409	26	for	for	ADP
ap-8536	409	27	a	a	DET
ap-8536	409	28	given	give	VERB
ap-8536	409	29	expansive	expansive	ADJ
ap-8536	409	30	base	base	NOUN
ap-8536	409	31	m	m	VERB
ap-8536	409	32	is	be	AUX
ap-8536	409	33	much	much	ADV
ap-8536	409	34	more	more	ADV
ap-8536	409	35	simpler	simple	ADJ
ap-8536	409	36	.	.	PUNCT
ap-8536	410	1	in	in	ADP
ap-8536	410	2	fact	fact	NOUN
ap-8536	410	3	,	,	PUNCT
ap-8536	410	4	using	use	VERB
ap-8536	410	5	a	a	DET
ap-8536	410	6	sufficiently	sufficiently	ADV
ap-8536	410	7	redundant	redundant	ADJ
ap-8536	410	8	digit	digit	NOUN
ap-8536	410	9	set	set	VERB
ap-8536	410	10	d	d	ADP
ap-8536	410	11	allows	allow	VERB
ap-8536	410	12	to	to	PART
ap-8536	410	13	represent	represent	VERB
ap-8536	410	14	all	all	DET
ap-8536	410	15	vectors	vector	NOUN
ap-8536	410	16	in	in	ADP
ap-8536	410	17	rm	rm	PROPN
ap-8536	410	18	.	.	PUNCT
ap-8536	411	1	first	first	ADV
ap-8536	411	2	,	,	PUNCT
ap-8536	411	3	we	we	PRON
ap-8536	411	4	focus	focus	VERB
ap-8536	411	5	on	on	ADP
ap-8536	411	6	vectors	vector	NOUN
ap-8536	411	7	from	from	ADP
ap-8536	411	8	rm	rm	PROPN
ap-8536	411	9	which	which	PRON
ap-8536	411	10	have	have	VERB
ap-8536	411	11	eventually	eventually	ADV
ap-8536	411	12	periodic	periodic	ADJ
ap-8536	411	13	representation	representation	NOUN
ap-8536	411	14	in	in	ADP
ap-8536	411	15	the	the	DET
ap-8536	411	16	numeration	numeration	NOUN
ap-8536	411	17	system	system	NOUN
ap-8536	411	18	(	(	PUNCT
ap-8536	411	19	m	m	PROPN
ap-8536	411	20	,	,	PUNCT
ap-8536	411	21	d	d	NOUN
ap-8536	411	22	)	)	PUNCT
ap-8536	411	23	,	,	PUNCT
ap-8536	411	24	i.e.	i.e.	X
ap-8536	411	25	,	,	PUNCT
ap-8536	411	26	we	we	PRON
ap-8536	411	27	focus	focus	VERB
ap-8536	411	28	on	on	ADP
ap-8536	411	29	the	the	DET
ap-8536	411	30	set	set	NOUN
ap-8536	411	31	perd(m	perd(m	NOUN
ap-8536	411	32	)	)	PUNCT
ap-8536	411	33	=	=	PRON
ap-8536	411	34	{	{	PUNCT
ap-8536	412	1	n∑	n∑	INTJ
ap-8536	412	2	j=−∞	j=−∞	PROPN
ap-8536	413	1	m	m	PROPN
ap-8536	413	2	jdj	jdj	PROPN
ap-8536	413	3	:	:	PUNCT
ap-8536	413	4	dj	dj	X
ap-8536	413	5	∈	∈	PROPN
ap-8536	413	6	d	d	NOUN
ap-8536	413	7	,	,	PUNCT
ap-8536	413	8	(	(	PUNCT
ap-8536	413	9	dj)n	dj)n	PROPN
ap-8536	413	10	−∞	−∞	PUNCT
ap-8536	413	11	eventually	eventually	ADV
ap-8536	413	12	periodic	periodic	ADJ
ap-8536	413	13	}	}	PUNCT
ap-8536	413	14	.	.	PUNCT
ap-8536	414	1	to	to	PART
ap-8536	414	2	work	work	VERB
ap-8536	414	3	with	with	ADP
ap-8536	414	4	eventually	eventually	ADV
ap-8536	414	5	periodic	periodic	ADJ
ap-8536	414	6	representations	representation	NOUN
ap-8536	414	7	,	,	PUNCT
ap-8536	414	8	we	we	PRON
ap-8536	414	9	exploit	exploit	VERB
ap-8536	414	10	a	a	DET
ap-8536	414	11	well	well	ADV
ap-8536	414	12	known	know	VERB
ap-8536	414	13	fact	fact	NOUN
ap-8536	414	14	about	about	ADP
ap-8536	414	15	contractive	contractive	ADJ
ap-8536	414	16	matrices	matrix	NOUN
ap-8536	414	17	,	,	PUNCT
ap-8536	414	18	namely	namely	ADV
ap-8536	414	19	that	that	SCONJ
ap-8536	414	20	(	(	PUNCT
ap-8536	414	21	i	i	PRON
ap-8536	414	22	−	−	PROPN
ap-8536	414	23	a)−1	a)−1	NOUN
ap-8536	414	24	=	=	PUNCT
ap-8536	414	25	∑+∞	∑+∞	NUM
ap-8536	414	26	j=0	j=0	PROPN
ap-8536	414	27	aj	aj	PROPN
ap-8536	414	28	for	for	ADP
ap-8536	414	29	any	any	DET
ap-8536	414	30	contractive	contractive	ADJ
ap-8536	414	31	matrix	matrix	NOUN
ap-8536	414	32	a	a	DET
ap-8536	414	33	∈	∈	PROPN
ap-8536	414	34	cm×m	cm×m	NOUN
ap-8536	414	35	.	.	PUNCT
ap-8536	415	1	lemma	lemma	PROPN
ap-8536	415	2	19	19	NUM
ap-8536	415	3	.	.	PUNCT
ap-8536	416	1	let	let	VERB
ap-8536	416	2	m	m	PRON
ap-8536	416	3	∈	∈	PROPN
ap-8536	416	4	zm×m	zm×m	PROPN
ap-8536	416	5	be	be	AUX
ap-8536	416	6	an	an	DET
ap-8536	416	7	expansive	expansive	ADJ
ap-8536	416	8	matrix	matrix	NOUN
ap-8536	416	9	and	and	CCONJ
ap-8536	417	1	d	d	PROPN
ap-8536	417	2	⊂	⊂	PROPN
ap-8536	417	3	zm	zm	PROPN
ap-8536	417	4	.	.	PUNCT
ap-8536	418	1	if	if	SCONJ
ap-8536	418	2	x	x	SYM
ap-8536	418	3	∈	∈	PROPN
ap-8536	418	4	perd(m	perd(m	NOUN
ap-8536	418	5	)	)	PUNCT
ap-8536	418	6	,	,	PUNCT
ap-8536	418	7	then	then	ADV
ap-8536	418	8	x	x	PROPN
ap-8536	418	9	∈	∈	PROPN
ap-8536	418	10	qm	qm	PROPN
ap-8536	418	11	.	.	PUNCT
ap-8536	418	12	proof	proof	NOUN
ap-8536	418	13	.	.	PUNCT
ap-8536	419	1	first	first	ADV
ap-8536	419	2	,	,	PUNCT
ap-8536	419	3	assume	assume	VERB
ap-8536	419	4	that	that	SCONJ
ap-8536	419	5	an	an	DET
ap-8536	419	6	(	(	PUNCT
ap-8536	419	7	m	m	NOUN
ap-8536	419	8	,	,	PUNCT
ap-8536	419	9	d)-representation	d)-representation	PUNCT
ap-8536	419	10	of	of	ADP
ap-8536	419	11	x	x	PUNCT
ap-8536	419	12	has	have	VERB
ap-8536	419	13	the	the	DET
ap-8536	419	14	form	form	NOUN
ap-8536	419	15	0	0	NUM
ap-8536	419	16	•	•	NOUN
ap-8536	419	17	(	(	PUNCT
ap-8536	419	18	d−1d−2	d−1d−2	PROPN
ap-8536	419	19	·	·	PUNCT
ap-8536	419	20	·	·	PUNCT
ap-8536	419	21	·	·	PUNCT
ap-8536	420	1	d−p)ω	d−p)ω	NOUN
ap-8536	420	2	.	.	PUNCT
ap-8536	421	1	it	it	PRON
ap-8536	421	2	means	mean	VERB
ap-8536	421	3	that	that	SCONJ
ap-8536	421	4	x	x	PROPN
ap-8536	421	5	=	=	PUNCT
ap-8536	421	6	m−1d−1	m−1d−1	PROPN
ap-8536	421	7	+	+	CCONJ
ap-8536	421	8	m−2d−2	m−2d−2	PROPN
ap-8536	421	9	+	+	CCONJ
ap-8536	421	10	·	·	PUNCT
ap-8536	421	11	·	·	PUNCT
ap-8536	421	12	·	·	PUNCT
ap-8536	422	1	+	+	CCONJ
ap-8536	423	1	m−pd−p	m−pd−p	PUNCT
ap-8536	423	2	+	+	CCONJ
ap-8536	423	3	m−px	m−px	PROPN
ap-8536	423	4	,	,	PUNCT
ap-8536	423	5	which	which	PRON
ap-8536	423	6	implies	imply	VERB
ap-8536	423	7	that	that	SCONJ
ap-8536	424	1	x	x	SYM
ap-8536	424	2	=	=	PRON
ap-8536	424	3	(	(	PUNCT
ap-8536	424	4	i−m−p)−1(m−1d−1+m−2d−2	i−m−p)−1(m−1d−1+m−2d−2	PROPN
ap-8536	424	5	+	+	PROPN
ap-8536	424	6	·	·	PUNCT
ap-8536	424	7	·	·	PUNCT
ap-8536	424	8	·	·	PUNCT
ap-8536	424	9	+	+	NOUN
ap-8536	424	10	m−pd−p	m−pd−p	X
ap-8536	424	11	)	)	PUNCT
ap-8536	424	12	.	.	PUNCT
ap-8536	425	1	since	since	SCONJ
ap-8536	425	2	all	all	DET
ap-8536	425	3	elements	element	NOUN
ap-8536	425	4	of	of	ADP
ap-8536	425	5	m	m	NOUN
ap-8536	425	6	are	be	AUX
ap-8536	425	7	integer	integer	ADJ
ap-8536	425	8	,	,	PUNCT
ap-8536	425	9	it	it	PRON
ap-8536	425	10	follows	follow	VERB
ap-8536	425	11	that	that	SCONJ
ap-8536	425	12	all	all	DET
ap-8536	425	13	the	the	DET
ap-8536	425	14	matrix	matrix	NOUN
ap-8536	425	15	powers	power	NOUN
ap-8536	425	16	m−j	m−j	PROPN
ap-8536	425	17	with	with	ADP
ap-8536	425	18	j	j	PROPN
ap-8536	425	19	∈	∈	PROPN
ap-8536	425	20	n	n	PROPN
ap-8536	426	1	and	and	CCONJ
ap-8536	426	2	(	(	PUNCT
ap-8536	426	3	i	i	NOUN
ap-8536	426	4	m	m	VERB
ap-8536	426	5	−m−p)−1	−m−p)−1	VERB
ap-8536	426	6	belong	belong	VERB
ap-8536	426	7	to	to	ADP
ap-8536	426	8	qm×m	qm×m	PROPN
ap-8536	426	9	.	.	PUNCT
ap-8536	427	1	therefore	therefore	ADV
ap-8536	427	2	,	,	PUNCT
ap-8536	427	3	x	x	PROPN
ap-8536	427	4	∈	∈	PROPN
ap-8536	427	5	qm	qm	PROPN
ap-8536	427	6	.	.	PUNCT
ap-8536	428	1	now	now	ADV
ap-8536	428	2	,	,	PUNCT
ap-8536	428	3	let	let	VERB
ap-8536	428	4	us	we	PRON
ap-8536	428	5	assume	assume	VERB
ap-8536	428	6	that	that	SCONJ
ap-8536	428	7	a	a	DET
ap-8536	428	8	representation	representation	NOUN
ap-8536	428	9	of	of	ADP
ap-8536	428	10	x	x	SYM
ap-8536	428	11	is	be	AUX
ap-8536	428	12	eventually	eventually	ADV
ap-8536	428	13	periodic	periodic	ADJ
ap-8536	428	14	and	and	CCONJ
ap-8536	428	15	the	the	DET
ap-8536	428	16	preperiodic	preperiodic	ADJ
ap-8536	428	17	part	part	NOUN
ap-8536	428	18	ends	end	VERB
ap-8536	428	19	at	at	ADP
ap-8536	428	20	an	an	DET
ap-8536	428	21	index	index	NOUN
ap-8536	428	22	−k	−k	NOUN
ap-8536	428	23	∈	∈	PROPN
ap-8536	429	1	z	z	X
ap-8536	429	2	,	,	PUNCT
ap-8536	429	3	−k	−k	ADJ
ap-8536	429	4	≤	≤	NUM
ap-8536	429	5	0	0	NUM
ap-8536	429	6	.	.	PUNCT
ap-8536	430	1	then	then	ADV
ap-8536	430	2	mkx	mkx	PROPN
ap-8536	430	3	equals	equal	VERB
ap-8536	430	4	z	z	PROPN
ap-8536	430	5	+	+	NUM
ap-8536	430	6	y	y	PROPN
ap-8536	430	7	,	,	PUNCT
ap-8536	430	8	where	where	SCONJ
ap-8536	430	9	z	z	NOUN
ap-8536	430	10	=	=	SYM
ap-8536	431	1	∑n	∑n	PROPN
ap-8536	431	2	j=0	j=0	PROPN
ap-8536	431	3	m	m	PROPN
ap-8536	431	4	jdj	jdj	PROPN
ap-8536	431	5	and	and	CCONJ
ap-8536	431	6	y	y	PROPN
ap-8536	431	7	has	have	VERB
ap-8536	431	8	the	the	DET
ap-8536	431	9	purely	purely	ADV
ap-8536	431	10	periodic	periodic	ADJ
ap-8536	431	11	form	form	NOUN
ap-8536	432	1	y	y	PROPN
ap-8536	432	2	=	=	SYM
ap-8536	432	3	0	0	NUM
ap-8536	432	4	•	•	NOUN
ap-8536	432	5	(	(	PUNCT
ap-8536	432	6	d−1d−2	d−1d−2	NOUN
ap-8536	432	7	·	·	PUNCT
ap-8536	432	8	·	·	PUNCT
ap-8536	432	9	·	·	PUNCT
ap-8536	433	1	d−p)ω	d−p)ω	PROPN
ap-8536	433	2	.	.	PUNCT
ap-8536	434	1	obviously	obviously	ADV
ap-8536	434	2	,	,	PUNCT
ap-8536	434	3	z	z	PROPN
ap-8536	434	4	∈	∈	PROPN
ap-8536	434	5	zm	zm	PROPN
ap-8536	434	6	and	and	CCONJ
ap-8536	434	7	,	,	PUNCT
ap-8536	434	8	by	by	ADP
ap-8536	434	9	the	the	DET
ap-8536	434	10	previous	previous	ADJ
ap-8536	434	11	argumentation	argumentation	NOUN
ap-8536	434	12	,	,	PUNCT
ap-8536	434	13	y	y	PROPN
ap-8536	434	14	∈	∈	PROPN
ap-8536	434	15	qm	qm	PROPN
ap-8536	434	16	.	.	PUNCT
ap-8536	435	1	hence	hence	ADV
ap-8536	435	2	x	x	X
ap-8536	435	3	=	=	PUNCT
ap-8536	435	4	m−k(z	m−k(z	PROPN
ap-8536	435	5	+	+	CCONJ
ap-8536	435	6	y	y	X
ap-8536	435	7	)	)	PUNCT
ap-8536	435	8	belongs	belong	VERB
ap-8536	435	9	to	to	ADP
ap-8536	435	10	qm	qm	PROPN
ap-8536	435	11	as	as	ADV
ap-8536	435	12	well	well	ADV
ap-8536	435	13	.	.	PUNCT
ap-8536	436	1	to	to	PART
ap-8536	436	2	study	study	VERB
ap-8536	436	3	the	the	DET
ap-8536	436	4	implication	implication	NOUN
ap-8536	436	5	opposite	opposite	NOUN
ap-8536	436	6	to	to	ADP
ap-8536	436	7	the	the	DET
ap-8536	436	8	previous	previous	ADJ
ap-8536	436	9	lemma	lemma	PROPN
ap-8536	436	10	19	19	NUM
ap-8536	436	11	,	,	PUNCT
ap-8536	436	12	we	we	PRON
ap-8536	436	13	use	use	VERB
ap-8536	436	14	the	the	DET
ap-8536	436	15	concept	concept	NOUN
ap-8536	436	16	of	of	ADP
ap-8536	436	17	parallel	parallel	ADJ
ap-8536	436	18	addition	addition	NOUN
ap-8536	436	19	.	.	PUNCT
ap-8536	437	1	lemma	lemma	PROPN
ap-8536	437	2	20	20	NUM
ap-8536	437	3	.	.	PUNCT
ap-8536	438	1	if	if	SCONJ
ap-8536	438	2	the	the	DET
ap-8536	438	3	digit	digit	NOUN
ap-8536	438	4	set	set	VERB
ap-8536	438	5	d	d	ADP
ap-8536	438	6	allows	allow	VERB
ap-8536	438	7	parallel	parallel	ADJ
ap-8536	438	8	addition	addition	NOUN
ap-8536	438	9	on	on	ADP
ap-8536	438	10	find(m	find(m	NOUN
ap-8536	438	11	)	)	PUNCT
ap-8536	438	12	,	,	PUNCT
ap-8536	438	13	then	then	ADV
ap-8536	438	14	it	it	PRON
ap-8536	438	15	allows	allow	VERB
ap-8536	438	16	parallel	parallel	ADJ
ap-8536	438	17	addition	addition	NOUN
ap-8536	438	18	also	also	ADV
ap-8536	438	19	on	on	ADP
ap-8536	438	20	perd(m	perd(m	NOUN
ap-8536	438	21	)	)	PUNCT
ap-8536	438	22	,	,	PUNCT
ap-8536	438	23	and	and	CCONJ
ap-8536	438	24	the	the	DET
ap-8536	438	25	result	result	NOUN
ap-8536	438	26	of	of	ADP
ap-8536	438	27	addition	addition	NOUN
ap-8536	438	28	is	be	AUX
ap-8536	438	29	again	again	ADV
ap-8536	438	30	eventually	eventually	ADV
ap-8536	438	31	periodic	periodic	ADJ
ap-8536	438	32	.	.	PUNCT
ap-8536	439	1	it	it	PRON
ap-8536	439	2	means	mean	VERB
ap-8536	439	3	that	that	SCONJ
ap-8536	439	4	,	,	PUNCT
ap-8536	439	5	for	for	ADP
ap-8536	439	6	such	such	DET
ap-8536	439	7	a	a	DET
ap-8536	439	8	digit	digit	NOUN
ap-8536	439	9	set	set	VERB
ap-8536	439	10	d	d	PROPN
ap-8536	439	11	,	,	PUNCT
ap-8536	439	12	the	the	DET
ap-8536	439	13	set	set	NOUN
ap-8536	439	14	perd(m	perd(m	NOUN
ap-8536	439	15	)	)	PUNCT
ap-8536	439	16	is	be	AUX
ap-8536	439	17	closed	close	VERB
ap-8536	439	18	under	under	ADP
ap-8536	439	19	addition	addition	NOUN
ap-8536	439	20	.	.	PUNCT
ap-8536	440	1	proof	proof	NOUN
ap-8536	440	2	.	.	PUNCT
ap-8536	441	1	let	let	VERB
ap-8536	441	2	us	we	PRON
ap-8536	441	3	explain	explain	VERB
ap-8536	441	4	this	this	DET
ap-8536	441	5	statement	statement	NOUN
ap-8536	441	6	,	,	PUNCT
ap-8536	441	7	assuming	assume	VERB
ap-8536	441	8	the	the	DET
ap-8536	441	9	parallel	parallel	ADJ
ap-8536	441	10	addition	addition	NOUN
ap-8536	441	11	on	on	ADP
ap-8536	441	12	find(m	find(m	NOUN
ap-8536	441	13	)	)	PUNCT
ap-8536	441	14	is	be	AUX
ap-8536	441	15	a	a	DET
ap-8536	441	16	p	p	ADJ
ap-8536	441	17	-	-	PUNCT
ap-8536	441	18	local	local	ADJ
ap-8536	441	19	function	function	NOUN
ap-8536	441	20	,	,	PUNCT
ap-8536	441	21	with	with	ADP
ap-8536	441	22	p	p	NOUN
ap-8536	441	23	=	=	PUNCT
ap-8536	441	24	r	r	NOUN
ap-8536	441	25	+	+	NUM
ap-8536	441	26	1	1	NUM
ap-8536	441	27	+	+	NUM
ap-8536	441	28	t	t	NOUN
ap-8536	441	29	,	,	PUNCT
ap-8536	441	30	memory	memory	NOUN
ap-8536	441	31	r	r	NOUN
ap-8536	441	32	and	and	CCONJ
ap-8536	441	33	anticipation	anticipation	NOUN
ap-8536	441	34	t.	t.	PROPN
ap-8536	441	35	let	let	VERB
ap-8536	441	36	ℓx	ℓx	INTJ
ap-8536	441	37	,	,	PUNCT
ap-8536	441	38	ℓy	ℓy	PROPN
ap-8536	441	39	be	be	VERB
ap-8536	441	40	the	the	DET
ap-8536	441	41	lengths	length	NOUN
ap-8536	441	42	of	of	ADP
ap-8536	441	43	the	the	DET
ap-8536	441	44	periods	period	NOUN
ap-8536	441	45	of	of	ADP
ap-8536	441	46	(	(	PUNCT
ap-8536	441	47	m	m	PROPN
ap-8536	441	48	,	,	PUNCT
ap-8536	441	49	d)representations	d)representation	NOUN
ap-8536	441	50	of	of	ADP
ap-8536	441	51	the	the	DET
ap-8536	441	52	summands	summand	NOUN
ap-8536	441	53	x	x	NOUN
ap-8536	441	54	,	,	PUNCT
ap-8536	441	55	y	y	PROPN
ap-8536	441	56	,	,	PUNCT
ap-8536	441	57	and	and	CCONJ
ap-8536	441	58	denote	denote	VERB
ap-8536	441	59	ℓxy	ℓxy	ADJ
ap-8536	441	60	:	:	PUNCT
ap-8536	442	1	=	=	SYM
ap-8536	442	2	ℓxℓy	ℓxℓy	PROPN
ap-8536	442	3	.	.	PUNCT
ap-8536	443	1	clearly	clearly	ADV
ap-8536	443	2	,	,	PUNCT
ap-8536	443	3	both	both	CCONJ
ap-8536	443	4	x	x	X
ap-8536	443	5	and	and	CCONJ
ap-8536	443	6	y	y	PROPN
ap-8536	443	7	have	have	VERB
ap-8536	443	8	(	(	PUNCT
ap-8536	443	9	m	m	PROPN
ap-8536	443	10	,	,	PUNCT
ap-8536	443	11	d)representations	d)representation	NOUN
ap-8536	443	12	with	with	ADP
ap-8536	443	13	the	the	DET
ap-8536	443	14	same	same	ADJ
ap-8536	443	15	period	period	NOUN
ap-8536	443	16	ℓxy	ℓxy	ADJ
ap-8536	443	17	,	,	PUNCT
ap-8536	443	18	and	and	CCONJ
ap-8536	443	19	the	the	DET
ap-8536	443	20	same	same	ADJ
ap-8536	443	21	holds	hold	VERB
ap-8536	443	22	for	for	ADP
ap-8536	443	23	the	the	DET
ap-8536	443	24	(	(	PUNCT
ap-8536	443	25	m	m	PROPN
ap-8536	443	26	,	,	PUNCT
ap-8536	443	27	d	d	PROPN
ap-8536	443	28	+	+	CCONJ
ap-8536	443	29	d)-representation	d)-representation	NOUN
ap-8536	443	30	of	of	ADP
ap-8536	443	31	their	their	PRON
ap-8536	443	32	(	(	PUNCT
ap-8536	443	33	eventually	eventually	ADV
ap-8536	443	34	periodic	periodic	ADJ
ap-8536	443	35	)	)	PUNCT
ap-8536	443	36	sum	sum	NOUN
ap-8536	443	37	w	w	NOUN
ap-8536	444	1	=	=	PUNCT
ap-8536	445	1	x	x	NOUN
ap-8536	446	1	+	+	CCONJ
ap-8536	446	2	y	y	NOUN
ap-8536	446	3	calculated	calculate	VERB
ap-8536	446	4	by	by	ADP
ap-8536	446	5	a	a	DET
ap-8536	446	6	pure	pure	ADJ
ap-8536	446	7	summation	summation	NOUN
ap-8536	446	8	of	of	ADP
ap-8536	446	9	digits	digit	NOUN
ap-8536	446	10	on	on	ADP
ap-8536	446	11	each	each	DET
ap-8536	446	12	position	position	NOUN
ap-8536	446	13	separately	separately	ADV
ap-8536	446	14	.	.	PUNCT
ap-8536	447	1	it	it	PRON
ap-8536	447	2	is	be	AUX
ap-8536	447	3	clear	clear	ADJ
ap-8536	447	4	that	that	SCONJ
ap-8536	447	5	,	,	PUNCT
ap-8536	447	6	by	by	ADP
ap-8536	447	7	applying	apply	VERB
ap-8536	447	8	the	the	DET
ap-8536	447	9	p	p	ADJ
ap-8536	447	10	-	-	PUNCT
ap-8536	447	11	local	local	ADJ
ap-8536	447	12	function	function	NOUN
ap-8536	447	13	φ	φ	NOUN
ap-8536	447	14	:	:	PUNCT
ap-8536	447	15	(	(	PUNCT
ap-8536	447	16	d	d	X
ap-8536	447	17	+	+	PROPN
ap-8536	447	18	d)p	d)p	NOUN
ap-8536	447	19	7→	7→	X
ap-8536	447	20	d	d	NOUN
ap-8536	447	21	onto	onto	ADP
ap-8536	447	22	the	the	DET
ap-8536	447	23	(	(	PUNCT
ap-8536	447	24	m	m	PROPN
ap-8536	447	25	,	,	PUNCT
ap-8536	447	26	d	d	PROPN
ap-8536	447	27	+	+	NOUN
ap-8536	447	28	d)-representation	d)-representation	NOUN
ap-8536	447	29	of	of	ADP
ap-8536	447	30	the	the	DET
ap-8536	447	31	interim	interim	ADJ
ap-8536	447	32	sum	sum	NOUN
ap-8536	447	33	w	w	PROPN
ap-8536	447	34	,	,	PUNCT
ap-8536	447	35	we	we	PRON
ap-8536	447	36	obtain	obtain	VERB
ap-8536	447	37	an	an	DET
ap-8536	447	38	eventually	eventually	ADV
ap-8536	447	39	periodic	periodic	ADJ
ap-8536	447	40	string	string	NOUN
ap-8536	447	41	over	over	ADP
ap-8536	447	42	the	the	DET
ap-8536	447	43	alphabet	alphabet	PROPN
ap-8536	447	44	d.	d.	PROPN
ap-8536	447	45	in	in	ADP
ap-8536	447	46	other	other	ADJ
ap-8536	447	47	words	word	NOUN
ap-8536	447	48	,	,	PUNCT
ap-8536	447	49	x	x	PUNCT
ap-8536	448	1	+	+	CCONJ
ap-8536	448	2	y	y	PROPN
ap-8536	448	3	has	have	VERB
ap-8536	448	4	an	an	DET
ap-8536	448	5	eventualy	eventualy	ADJ
ap-8536	448	6	periodic	periodic	ADJ
ap-8536	448	7	(	(	PUNCT
ap-8536	448	8	m	m	PROPN
ap-8536	448	9	,	,	PUNCT
ap-8536	448	10	d)-representation	d)-representation	PROPN
ap-8536	448	11	.	.	PUNCT
ap-8536	449	1	theorem	theorem	PROPN
ap-8536	449	2	21	21	NUM
ap-8536	449	3	.	.	PUNCT
ap-8536	450	1	let	let	VERB
ap-8536	450	2	m	m	PRON
ap-8536	450	3	∈	∈	PROPN
ap-8536	450	4	zm×m	zm×m	PROPN
ap-8536	450	5	be	be	AUX
ap-8536	450	6	an	an	DET
ap-8536	450	7	expansive	expansive	ADJ
ap-8536	450	8	matrix	matrix	NOUN
ap-8536	450	9	.	.	PUNCT
ap-8536	451	1	then	then	ADV
ap-8536	451	2	there	there	PRON
ap-8536	451	3	exists	exist	VERB
ap-8536	451	4	a	a	DET
ap-8536	451	5	finite	finite	ADJ
ap-8536	451	6	digit	digit	NOUN
ap-8536	451	7	set	set	VERB
ap-8536	451	8	d	d	PROPN
ap-8536	451	9	⊂	⊂	PROPN
ap-8536	451	10	zm	zm	PROPN
ap-8536	451	11	such	such	ADJ
ap-8536	451	12	that	that	DET
ap-8536	451	13	perd(m	perd(m	NOUN
ap-8536	451	14	)	)	PUNCT
ap-8536	451	15	=	=	SYM
ap-8536	451	16	qm	qm	PROPN
ap-8536	451	17	.	.	PUNCT
ap-8536	451	18	proof	proof	NOUN
ap-8536	451	19	.	.	PUNCT
ap-8536	452	1	by	by	ADP
ap-8536	452	2	theorem	theorem	NOUN
ap-8536	452	3	9	9	NUM
ap-8536	452	4	and	and	CCONJ
ap-8536	452	5	lemma	lemma	PROPN
ap-8536	452	6	5	5	NUM
ap-8536	452	7	,	,	PUNCT
ap-8536	452	8	there	there	PRON
ap-8536	452	9	exists	exist	VERB
ap-8536	452	10	a	a	DET
ap-8536	452	11	finite	finite	ADJ
ap-8536	452	12	digit	digit	NOUN
ap-8536	452	13	set	set	VERB
ap-8536	452	14	d	d	ADP
ap-8536	452	15	such	such	ADJ
ap-8536	452	16	that	that	DET
ap-8536	452	17	addition	addition	NOUN
ap-8536	452	18	in	in	ADP
ap-8536	452	19	(	(	PUNCT
ap-8536	452	20	m	m	PROPN
ap-8536	452	21	,	,	PUNCT
ap-8536	452	22	d	d	NOUN
ap-8536	452	23	)	)	PUNCT
ap-8536	452	24	can	can	AUX
ap-8536	452	25	be	be	AUX
ap-8536	452	26	performed	perform	VERB
ap-8536	452	27	in	in	ADP
ap-8536	452	28	parallel	parallel	NOUN
ap-8536	452	29	.	.	PUNCT
ap-8536	453	1	due	due	ADP
ap-8536	453	2	to	to	ADP
ap-8536	453	3	lemma	lemma	PROPN
ap-8536	453	4	19	19	NUM
ap-8536	453	5	,	,	PUNCT
ap-8536	453	6	it	it	PRON
ap-8536	453	7	remains	remain	VERB
ap-8536	453	8	to	to	PART
ap-8536	453	9	prove	prove	VERB
ap-8536	453	10	only	only	ADV
ap-8536	453	11	the	the	DET
ap-8536	453	12	inclusion	inclusion	NOUN
ap-8536	453	13	qm	qm	PROPN
ap-8536	453	14	⊂	⊂	PROPN
ap-8536	453	15	perd(m	perd(m	PROPN
ap-8536	453	16	)	)	PUNCT
ap-8536	453	17	.	.	PUNCT
ap-8536	454	1	we	we	PRON
ap-8536	454	2	denote	denote	VERB
ap-8536	454	3	by	by	ADP
ap-8536	454	4	es	es	ADP
ap-8536	454	5	the	the	DET
ap-8536	454	6	vector	vector	NOUN
ap-8536	454	7	from	from	ADP
ap-8536	454	8	zm	zm	PROPN
ap-8536	454	9	whose	whose	DET
ap-8536	454	10	sth	sth	NOUN
ap-8536	454	11	coordinate	coordinate	NOUN
ap-8536	454	12	equals	equal	VERB
ap-8536	454	13	1	1	NUM
ap-8536	454	14	and	and	CCONJ
ap-8536	454	15	all	all	DET
ap-8536	454	16	other	other	ADJ
ap-8536	454	17	coordinates	coordinate	NOUN
ap-8536	454	18	are	be	AUX
ap-8536	454	19	zero	zero	NUM
ap-8536	454	20	.	.	PUNCT
ap-8536	455	1	in	in	ADP
ap-8536	455	2	the	the	DET
ap-8536	455	3	195	195	NUM
ap-8536	455	4	i.	i.	NOUN
ap-8536	455	5	farkas	farkas	PROPN
ap-8536	455	6	,	,	PUNCT
ap-8536	455	7	e.	e.	PROPN
ap-8536	455	8	pelantová	pelantová	PROPN
ap-8536	455	9	,	,	PUNCT
ap-8536	455	10	m.	m.	NOUN
ap-8536	455	11	svobodová	svobodová	PROPN
ap-8536	455	12	acta	acta	PROPN
ap-8536	455	13	polytechnica	polytechnica	PROPN
ap-8536	455	14	first	first	ADJ
ap-8536	455	15	step	step	NOUN
ap-8536	455	16	,	,	PUNCT
ap-8536	455	17	we	we	PRON
ap-8536	455	18	show	show	VERB
ap-8536	455	19	that	that	SCONJ
ap-8536	455	20	for	for	ADP
ap-8536	455	21	each	each	DET
ap-8536	455	22	q	q	PROPN
ap-8536	455	23	∈	∈	PROPN
ap-8536	455	24	n	n	CCONJ
ap-8536	455	25	and	and	CCONJ
ap-8536	455	26	each	each	PRON
ap-8536	455	27	s	s	PART
ap-8536	455	28	=	=	SYM
ap-8536	455	29	1	1	NUM
ap-8536	455	30	,	,	PUNCT
ap-8536	455	31	2	2	NUM
ap-8536	455	32	,	,	PUNCT
ap-8536	455	33	.	.	PUNCT
ap-8536	455	34	.	.	PUNCT
ap-8536	456	1	.	.	PUNCT
ap-8536	457	1	,	,	PUNCT
ap-8536	457	2	m	m	X
ap-8536	457	3	,	,	PUNCT
ap-8536	457	4	the	the	DET
ap-8536	457	5	vector	vector	NOUN
ap-8536	457	6	1	1	NUM
ap-8536	457	7	q	q	NOUN
ap-8536	457	8	es	es	PROPN
ap-8536	457	9	has	have	VERB
ap-8536	457	10	an	an	DET
ap-8536	457	11	eventually	eventually	ADV
ap-8536	457	12	periodic	periodic	ADJ
ap-8536	457	13	(	(	PUNCT
ap-8536	457	14	m	m	PROPN
ap-8536	457	15	,	,	PUNCT
ap-8536	457	16	d)-representation	d)-representation	PROPN
ap-8536	457	17	.	.	PUNCT
ap-8536	458	1	for	for	ADP
ap-8536	458	2	this	this	DET
ap-8536	458	3	purpose	purpose	NOUN
ap-8536	458	4	,	,	PUNCT
ap-8536	458	5	we	we	PRON
ap-8536	458	6	define	define	VERB
ap-8536	458	7	a	a	DET
ap-8536	458	8	congruence	congruence	NOUN
ap-8536	458	9	relation	relation	NOUN
ap-8536	458	10	on	on	ADP
ap-8536	458	11	the	the	DET
ap-8536	458	12	matrices	matrix	NOUN
ap-8536	458	13	from	from	ADP
ap-8536	458	14	zm×m	zm×m	NOUN
ap-8536	458	15	.	.	PUNCT
ap-8536	459	1	we	we	PRON
ap-8536	459	2	say	say	VERB
ap-8536	459	3	that	that	SCONJ
ap-8536	459	4	a	a	DET
ap-8536	459	5	∈	∈	PROPN
ap-8536	459	6	zm×m	zm×m	NOUN
ap-8536	459	7	is	be	AUX
ap-8536	459	8	congruent	congruent	ADJ
ap-8536	459	9	modulo	modulo	NOUN
ap-8536	459	10	q	q	X
ap-8536	459	11	to	to	ADP
ap-8536	459	12	b	b	PROPN
ap-8536	459	13	∈	∈	PROPN
ap-8536	459	14	zm×m	zm×m	NOUN
ap-8536	459	15	,	,	PUNCT
ap-8536	459	16	if	if	SCONJ
ap-8536	459	17	(	(	PUNCT
ap-8536	459	18	a	a	DET
ap-8536	459	19	−	−	PROPN
ap-8536	459	20	b	b	NOUN
ap-8536	459	21	)	)	PUNCT
ap-8536	459	22	∈	∈	PROPN
ap-8536	459	23	q	q	NOUN
ap-8536	459	24	zm×m	zm×m	NOUN
ap-8536	459	25	.	.	PUNCT
ap-8536	460	1	as	as	SCONJ
ap-8536	460	2	the	the	DET
ap-8536	460	3	number	number	NOUN
ap-8536	460	4	of	of	ADP
ap-8536	460	5	congruence	congruence	PROPN
ap-8536	460	6	classes	class	NOUN
ap-8536	460	7	is	be	AUX
ap-8536	460	8	finite	finite	ADJ
ap-8536	460	9	(	(	PUNCT
ap-8536	460	10	for	for	ADP
ap-8536	460	11	a	a	DET
ap-8536	460	12	fixed	fix	VERB
ap-8536	460	13	q	q	NOUN
ap-8536	460	14	∈	∈	PROPN
ap-8536	460	15	n	n	CCONJ
ap-8536	460	16	)	)	PUNCT
ap-8536	460	17	,	,	PUNCT
ap-8536	460	18	we	we	PRON
ap-8536	460	19	find	find	VERB
ap-8536	460	20	in	in	ADP
ap-8536	460	21	the	the	DET
ap-8536	460	22	list	list	NOUN
ap-8536	460	23	i	i	PRON
ap-8536	460	24	,	,	PUNCT
ap-8536	460	25	m	m	PROPN
ap-8536	460	26	,	,	PUNCT
ap-8536	460	27	m2	m2	PROPN
ap-8536	460	28	,	,	PUNCT
ap-8536	460	29	m3	m3	PROPN
ap-8536	460	30	,	,	PUNCT
ap-8536	460	31	.	.	PUNCT
ap-8536	460	32	.	.	PUNCT
ap-8536	460	33	.	.	PUNCT
ap-8536	461	1	two	two	NUM
ap-8536	461	2	matrices	matrix	NOUN
ap-8536	461	3	from	from	ADP
ap-8536	461	4	the	the	DET
ap-8536	461	5	same	same	ADJ
ap-8536	461	6	congruence	congruence	NOUN
ap-8536	461	7	class	class	NOUN
ap-8536	461	8	.	.	PUNCT
ap-8536	462	1	in	in	ADP
ap-8536	462	2	other	other	ADJ
ap-8536	462	3	words	word	NOUN
ap-8536	462	4	,	,	PUNCT
ap-8536	462	5	there	there	PRON
ap-8536	462	6	exist	exist	VERB
ap-8536	462	7	k	k	PROPN
ap-8536	462	8	,	,	PUNCT
ap-8536	462	9	ℓ	ℓ	PROPN
ap-8536	462	10	∈	∈	PROPN
ap-8536	462	11	n	n	CCONJ
ap-8536	462	12	,	,	PUNCT
ap-8536	462	13	ℓ	ℓ	PROPN
ap-8536	462	14	>	>	X
ap-8536	462	15	0	0	NUM
ap-8536	463	1	such	such	ADJ
ap-8536	463	2	that	that	SCONJ
ap-8536	463	3	mk+ℓ	mk+ℓ	PROPN
ap-8536	463	4	−	−	PROPN
ap-8536	463	5	mk	mk	PROPN
ap-8536	463	6	=	=	NOUN
ap-8536	463	7	qc	qc	PROPN
ap-8536	463	8	for	for	ADP
ap-8536	463	9	some	some	DET
ap-8536	463	10	c	c	PROPN
ap-8536	463	11	∈	∈	PROPN
ap-8536	463	12	zm×m	zm×m	NOUN
ap-8536	463	13	,	,	PUNCT
ap-8536	463	14	or	or	CCONJ
ap-8536	463	15	,	,	PUNCT
ap-8536	463	16	equivalently	equivalently	ADV
ap-8536	463	17	1	1	NUM
ap-8536	463	18	q	q	NOUN
ap-8536	464	1	i	i	PRON
ap-8536	464	2	=	=	PUNCT
ap-8536	464	3	m−ℓ−k	m−ℓ−k	PROPN
ap-8536	464	4	(	(	PUNCT
ap-8536	464	5	i	i	PRON
ap-8536	464	6	−	−	PROPN
ap-8536	464	7	m−ℓ)−1	m−ℓ)−1	PROPN
ap-8536	465	1	c	c	NOUN
ap-8536	465	2	=	=	PUNCT
ap-8536	465	3	m−ℓ−k	m−ℓ−k	PROPN
ap-8536	465	4	(	(	PUNCT
ap-8536	465	5	+	+	ADJ
ap-8536	465	6	∞∑	∞∑	NUM
ap-8536	465	7	j=0	j=0	ADJ
ap-8536	465	8	m−ℓj	m−ℓj	PROPN
ap-8536	465	9	)	)	PUNCT
ap-8536	466	1	c	c	NOUN
ap-8536	466	2	.	.	PUNCT
ap-8536	467	1	(	(	PUNCT
ap-8536	467	2	12	12	NUM
ap-8536	467	3	)	)	PUNCT
ap-8536	467	4	let	let	VERB
ap-8536	467	5	c1	c1	PROPN
ap-8536	467	6	denote	denote	VERB
ap-8536	467	7	the	the	DET
ap-8536	467	8	first	first	ADJ
ap-8536	467	9	column	column	NOUN
ap-8536	467	10	of	of	ADP
ap-8536	467	11	the	the	DET
ap-8536	467	12	matrix	matrix	NOUN
ap-8536	467	13	c.	c.	NOUN
ap-8536	467	14	as	as	ADP
ap-8536	467	15	zm	zm	PROPN
ap-8536	467	16	⊂	⊂	PROPN
ap-8536	467	17	find(m	find(m	PROPN
ap-8536	467	18	)	)	PUNCT
ap-8536	467	19	,	,	PUNCT
ap-8536	467	20	we	we	PRON
ap-8536	467	21	can	can	AUX
ap-8536	467	22	write	write	VERB
ap-8536	467	23	c1	c1	PROPN
ap-8536	467	24	=	=	PUNCT
ap-8536	468	1	∑n	∑n	PROPN
ap-8536	468	2	t=0	t=0	ADJ
ap-8536	468	3	m	m	VERB
ap-8536	468	4	tft	tft	ADJ
ap-8536	468	5	with	with	ADP
ap-8536	468	6	ft	ft	PROPN
ap-8536	468	7	∈	∈	PROPN
ap-8536	468	8	d.	d.	NOUN
ap-8536	468	9	it	it	PRON
ap-8536	468	10	is	be	AUX
ap-8536	468	11	due	due	ADJ
ap-8536	468	12	to	to	PART
ap-8536	468	13	lemma	lemma	PROPN
ap-8536	468	14	6	6	NUM
ap-8536	468	15	and	and	CCONJ
ap-8536	468	16	remark	remark	VERB
ap-8536	468	17	10	10	NUM
ap-8536	468	18	that	that	SCONJ
ap-8536	468	19	the	the	DET
ap-8536	468	20	bottom	bottom	ADJ
ap-8536	468	21	index	index	NOUN
ap-8536	468	22	of	of	ADP
ap-8536	468	23	the	the	DET
ap-8536	468	24	sum	sum	NOUN
ap-8536	468	25	equals	equal	VERB
ap-8536	468	26	zero	zero	NUM
ap-8536	468	27	,	,	PUNCT
ap-8536	468	28	as	as	SCONJ
ap-8536	468	29	m	m	PROPN
ap-8536	468	30	is	be	AUX
ap-8536	468	31	expansive	expansive	ADJ
ap-8536	468	32	,	,	PUNCT
ap-8536	468	33	so	so	ADV
ap-8536	468	34	all	all	PRON
ap-8536	468	35	of	of	ADP
ap-8536	468	36	its	its	PRON
ap-8536	468	37	eigenvalues	eigenvalue	NOUN
ap-8536	468	38	are	be	AUX
ap-8536	468	39	>	>	X
ap-8536	468	40	1	1	NUM
ap-8536	468	41	in	in	ADP
ap-8536	468	42	modulus	modulus	NOUN
ap-8536	468	43	.	.	PUNCT
ap-8536	469	1	the	the	DET
ap-8536	469	2	first	first	ADJ
ap-8536	469	3	column	column	NOUN
ap-8536	469	4	of	of	ADP
ap-8536	469	5	the	the	DET
ap-8536	469	6	matrix	matrix	NOUN
ap-8536	469	7	equality	equality	NOUN
ap-8536	469	8	(	(	PUNCT
ap-8536	469	9	12	12	NUM
ap-8536	469	10	)	)	PUNCT
ap-8536	469	11	then	then	ADV
ap-8536	469	12	equals	equal	VERB
ap-8536	469	13	1	1	NUM
ap-8536	469	14	q	q	NOUN
ap-8536	469	15	e1	e1	NOUN
ap-8536	469	16	=	=	NOUN
ap-8536	470	1	m−ℓ−k	m−ℓ−k	NOUN
ap-8536	471	1	+	+	ADJ
ap-8536	471	2	∞∑	∞∑	ADJ
ap-8536	471	3	j=0	j=0	ADJ
ap-8536	471	4	m−ℓjc1	m−ℓjc1	NOUN
ap-8536	471	5	=	=	SYM
ap-8536	471	6	n∑	n∑	NOUN
ap-8536	471	7	t=0	t=0	PUNCT
ap-8536	472	1	+	+	ADJ
ap-8536	472	2	∞∑	∞∑	NUM
ap-8536	472	3	j=0	j=0	ADJ
ap-8536	472	4	m−ℓj−ℓ−k+tft	m−ℓj−ℓ−k+tft	PROPN
ap-8536	472	5	.	.	PUNCT
ap-8536	473	1	thus	thus	ADV
ap-8536	473	2	,	,	PUNCT
ap-8536	473	3	we	we	PRON
ap-8536	473	4	have	have	AUX
ap-8536	473	5	expressed	express	VERB
ap-8536	473	6	1	1	NUM
ap-8536	473	7	q	q	NOUN
ap-8536	473	8	e1	e1	PROPN
ap-8536	473	9	as	as	ADP
ap-8536	473	10	a	a	DET
ap-8536	473	11	sum	sum	NOUN
ap-8536	473	12	of	of	ADP
ap-8536	473	13	n+1	n+1	PROPN
ap-8536	473	14	vectors	vector	NOUN
ap-8536	473	15	with	with	ADP
ap-8536	473	16	eventually	eventually	ADV
ap-8536	473	17	periodic	periodic	ADJ
ap-8536	473	18	(	(	PUNCT
ap-8536	473	19	m	m	PROPN
ap-8536	473	20	,	,	PUNCT
ap-8536	473	21	d)-representation	d)-representation	PROPN
ap-8536	473	22	.	.	PUNCT
ap-8536	474	1	since	since	SCONJ
ap-8536	474	2	perd(m	perd(m	NOUN
ap-8536	474	3	)	)	PUNCT
ap-8536	474	4	is	be	AUX
ap-8536	474	5	closed	close	VERB
ap-8536	474	6	under	under	ADP
ap-8536	474	7	addition	addition	NOUN
ap-8536	474	8	,	,	PUNCT
ap-8536	474	9	due	due	ADP
ap-8536	474	10	to	to	ADP
ap-8536	474	11	lemma	lemma	PROPN
ap-8536	474	12	20	20	NUM
ap-8536	474	13	,	,	PUNCT
ap-8536	474	14	the	the	DET
ap-8536	474	15	vector	vector	NOUN
ap-8536	474	16	1	1	NUM
ap-8536	474	17	q	q	PROPN
ap-8536	474	18	e1	e1	PROPN
ap-8536	474	19	belongs	belong	VERB
ap-8536	474	20	to	to	PART
ap-8536	474	21	perd(m	perd(m	VERB
ap-8536	474	22	)	)	PUNCT
ap-8536	474	23	as	as	ADV
ap-8536	474	24	well	well	ADV
ap-8536	474	25	.	.	PUNCT
ap-8536	475	1	analogously	analogously	ADV
ap-8536	475	2	,	,	PUNCT
ap-8536	475	3	the	the	DET
ap-8536	475	4	same	same	ADJ
ap-8536	475	5	holds	hold	VERB
ap-8536	475	6	for	for	ADP
ap-8536	475	7	1	1	NUM
ap-8536	475	8	q	q	PROPN
ap-8536	475	9	e2	e2	PROPN
ap-8536	475	10	,	,	PUNCT
ap-8536	475	11	.	.	PUNCT
ap-8536	475	12	.	.	PUNCT
ap-8536	476	1	.	.	PUNCT
ap-8536	477	1	,	,	PUNCT
ap-8536	477	2	1	1	NUM
ap-8536	477	3	q	q	NOUN
ap-8536	477	4	em	em	PRON
ap-8536	477	5	and	and	CCONJ
ap-8536	477	6	for	for	ADP
ap-8536	477	7	−	−	PROPN
ap-8536	477	8	1	1	NUM
ap-8536	477	9	q	q	PROPN
ap-8536	477	10	e1	e1	PROPN
ap-8536	477	11	,	,	PUNCT
ap-8536	477	12	.	.	PUNCT
ap-8536	477	13	.	.	PUNCT
ap-8536	478	1	.	.	PUNCT
ap-8536	479	1	,	,	PUNCT
ap-8536	479	2	−	−	PROPN
ap-8536	479	3	1	1	NUM
ap-8536	479	4	q	q	NOUN
ap-8536	479	5	em	em	PRON
ap-8536	479	6	,	,	PUNCT
ap-8536	479	7	therefore	therefore	ADV
ap-8536	479	8	,	,	PUNCT
ap-8536	479	9	they	they	PRON
ap-8536	479	10	are	be	AUX
ap-8536	479	11	also	also	ADV
ap-8536	479	12	elements	element	NOUN
ap-8536	479	13	of	of	ADP
ap-8536	479	14	perd(m	perd(m	NOUN
ap-8536	479	15	)	)	PUNCT
ap-8536	479	16	.	.	PUNCT
ap-8536	480	1	in	in	ADP
ap-8536	480	2	the	the	DET
ap-8536	480	3	second	second	ADJ
ap-8536	480	4	step	step	NOUN
ap-8536	480	5	,	,	PUNCT
ap-8536	480	6	we	we	PRON
ap-8536	480	7	consider	consider	VERB
ap-8536	480	8	an	an	DET
ap-8536	480	9	arbitrary	arbitrary	ADJ
ap-8536	480	10	vector	vector	NOUN
ap-8536	480	11	x	x	SYM
ap-8536	480	12	∈	∈	PROPN
ap-8536	480	13	qm	qm	PROPN
ap-8536	480	14	.	.	PUNCT
ap-8536	481	1	we	we	PRON
ap-8536	481	2	find	find	VERB
ap-8536	481	3	q	q	X
ap-8536	481	4	∈	∈	PROPN
ap-8536	481	5	n	n	PRON
ap-8536	481	6	and	and	CCONJ
ap-8536	481	7	integer	integer	PROPN
ap-8536	481	8	nubers	nuber	NOUN
ap-8536	481	9	p1	p1	PROPN
ap-8536	481	10	,	,	PUNCT
ap-8536	481	11	.	.	PUNCT
ap-8536	481	12	.	.	PUNCT
ap-8536	481	13	.	.	PUNCT
ap-8536	482	1	,	,	PUNCT
ap-8536	482	2	pm	pm	VERB
ap-8536	482	3	such	such	ADJ
ap-8536	482	4	that	that	SCONJ
ap-8536	482	5	x	x	SYM
ap-8536	482	6	=	=	SYM
ap-8536	482	7	p1	p1	NOUN
ap-8536	482	8	(	(	PUNCT
ap-8536	482	9	1	1	NUM
ap-8536	482	10	q	q	NOUN
ap-8536	482	11	e1	e1	PROPN
ap-8536	482	12	)	)	PUNCT
ap-8536	483	1	+	+	CCONJ
ap-8536	483	2	·	·	PUNCT
ap-8536	483	3	·	·	PUNCT
ap-8536	483	4	·	·	PUNCT
ap-8536	484	1	+	+	CCONJ
ap-8536	484	2	pm	pm	NOUN
ap-8536	484	3	(	(	PUNCT
ap-8536	484	4	1	1	NUM
ap-8536	484	5	q	q	NOUN
ap-8536	484	6	em	em	PROPN
ap-8536	484	7	)	)	PUNCT
ap-8536	484	8	.	.	PUNCT
ap-8536	485	1	it	it	PRON
ap-8536	485	2	means	mean	VERB
ap-8536	485	3	that	that	SCONJ
ap-8536	485	4	x	x	PRON
ap-8536	485	5	is	be	AUX
ap-8536	485	6	a	a	DET
ap-8536	485	7	sum	sum	NOUN
ap-8536	485	8	of	of	ADP
ap-8536	485	9	(	(	PUNCT
ap-8536	485	10	|p1|	|p1|	PROPN
ap-8536	485	11	+	+	CCONJ
ap-8536	485	12	·	·	PUNCT
ap-8536	485	13	·	·	PUNCT
ap-8536	485	14	·	·	PUNCT
ap-8536	486	1	+	+	PUNCT
ap-8536	486	2	|pm|	|pm|	NUM
ap-8536	486	3	)	)	PUNCT
ap-8536	486	4	vectors	vector	NOUN
ap-8536	486	5	from	from	ADP
ap-8536	486	6	the	the	DET
ap-8536	486	7	set	set	NOUN
ap-8536	486	8	perd(m	perd(m	NOUN
ap-8536	486	9	)	)	PUNCT
ap-8536	486	10	,	,	PUNCT
ap-8536	486	11	which	which	PRON
ap-8536	486	12	is	be	AUX
ap-8536	486	13	closed	close	VERB
ap-8536	486	14	under	under	ADP
ap-8536	486	15	addition	addition	NOUN
ap-8536	486	16	.	.	PUNCT
ap-8536	487	1	hence	hence	ADV
ap-8536	487	2	x	x	X
ap-8536	487	3	∈	∈	PROPN
ap-8536	487	4	perd(m	perd(m	NOUN
ap-8536	487	5	)	)	PUNCT
ap-8536	487	6	.	.	PUNCT
ap-8536	488	1	corollary	corollary	ADJ
ap-8536	488	2	22	22	NUM
ap-8536	488	3	.	.	PUNCT
ap-8536	489	1	let	let	VERB
ap-8536	489	2	m	m	PRON
ap-8536	489	3	∈	∈	PROPN
ap-8536	489	4	zm×m	zm×m	PROPN
ap-8536	489	5	be	be	AUX
ap-8536	489	6	an	an	DET
ap-8536	489	7	expansive	expansive	ADJ
ap-8536	489	8	matrix	matrix	NOUN
ap-8536	489	9	.	.	PUNCT
ap-8536	490	1	then	then	ADV
ap-8536	490	2	there	there	PRON
ap-8536	490	3	exists	exist	VERB
ap-8536	490	4	a	a	DET
ap-8536	490	5	finite	finite	ADJ
ap-8536	490	6	digit	digit	NOUN
ap-8536	490	7	set	set	VERB
ap-8536	490	8	d	d	PROPN
ap-8536	490	9	⊂	⊂	PROPN
ap-8536	490	10	z	z	NOUN
ap-8536	490	11	such	such	ADJ
ap-8536	490	12	that	that	SCONJ
ap-8536	490	13	every	every	DET
ap-8536	490	14	x	x	PROPN
ap-8536	490	15	∈	∈	PROPN
ap-8536	490	16	rm	rm	NOUN
ap-8536	490	17	has	have	VERB
ap-8536	490	18	an	an	DET
ap-8536	490	19	(	(	PUNCT
ap-8536	490	20	m	m	NOUN
ap-8536	490	21	,	,	PUNCT
ap-8536	490	22	d)-representation	d)-representation	PROPN
ap-8536	490	23	.	.	PUNCT
ap-8536	491	1	proof	proof	NOUN
ap-8536	491	2	.	.	PUNCT
ap-8536	492	1	let	let	VERB
ap-8536	492	2	∥·∥c	∥·∥c	ADJ
ap-8536	492	3	be	be	AUX
ap-8536	492	4	the	the	DET
ap-8536	492	5	vector	vector	NOUN
ap-8536	492	6	norm	norm	NOUN
ap-8536	492	7	of	of	ADP
ap-8536	492	8	rm	rm	PROPN
ap-8536	492	9	,	,	PUNCT
ap-8536	492	10	for	for	ADP
ap-8536	492	11	which	which	PRON
ap-8536	492	12	the	the	DET
ap-8536	492	13	induced	induced	ADJ
ap-8536	492	14	matrix	matrix	NOUN
ap-8536	492	15	norm	norm	NOUN
ap-8536	492	16	of	of	ADP
ap-8536	492	17	the	the	DET
ap-8536	492	18	contractive	contractive	ADJ
ap-8536	492	19	matrix	matrix	NOUN
ap-8536	492	20	m−1	m−1	PROPN
ap-8536	492	21	is	be	AUX
ap-8536	492	22	∥∥m−1	∥∥m−1	ADJ
ap-8536	492	23	∥∥	∥∥	X
ap-8536	493	1	=	=	SYM
ap-8536	493	2	r	r	NOUN
ap-8536	493	3	<	<	X
ap-8536	493	4	1	1	NUM
ap-8536	493	5	.	.	PUNCT
ap-8536	494	1	since	since	SCONJ
ap-8536	494	2	m	m	PROPN
ap-8536	494	3	is	be	AUX
ap-8536	494	4	expansive	expansive	ADJ
ap-8536	494	5	,	,	PUNCT
ap-8536	494	6	any	any	DET
ap-8536	494	7	vector	vector	NOUN
ap-8536	494	8	y	y	PROPN
ap-8536	494	9	∈	∈	PROPN
ap-8536	494	10	zm	zm	PROPN
ap-8536	494	11	has	have	VERB
ap-8536	494	12	an	an	DET
ap-8536	494	13	(	(	PUNCT
ap-8536	494	14	m	m	PROPN
ap-8536	494	15	,	,	PUNCT
ap-8536	494	16	d)-representation	d)-representation	NOUN
ap-8536	494	17	in	in	ADP
ap-8536	494	18	the	the	DET
ap-8536	494	19	form	form	NOUN
ap-8536	494	20	y	y	PROPN
ap-8536	494	21	=	=	SYM
ap-8536	494	22	∑n	∑n	PROPN
ap-8536	494	23	j=0	j=0	PROPN
ap-8536	494	24	m	m	VERB
ap-8536	494	25	jdj	jdj	PROPN
ap-8536	494	26	for	for	ADP
ap-8536	494	27	some	some	DET
ap-8536	494	28	n	n	PRON
ap-8536	494	29	∈	∈	PROPN
ap-8536	494	30	n	n	NOUN
ap-8536	494	31	and	and	CCONJ
ap-8536	494	32	dj	dj	X
ap-8536	494	33	∈	∈	PROPN
ap-8536	494	34	d.	d.	PROPN
ap-8536	494	35	in	in	ADP
ap-8536	494	36	general	general	ADJ
ap-8536	494	37	,	,	PUNCT
ap-8536	494	38	the	the	DET
ap-8536	494	39	representation	representation	NOUN
ap-8536	494	40	is	be	AUX
ap-8536	494	41	not	not	PART
ap-8536	494	42	unique	unique	ADJ
ap-8536	494	43	.	.	PUNCT
ap-8536	495	1	we	we	PRON
ap-8536	495	2	denote	denote	VERB
ap-8536	495	3	by	by	ADP
ap-8536	495	4	height(y	height(y	PROPN
ap-8536	495	5	)	)	PUNCT
ap-8536	495	6	the	the	DET
ap-8536	495	7	minimal	minimal	ADJ
ap-8536	495	8	n	n	PROPN
ap-8536	495	9	∈	∈	PROPN
ap-8536	495	10	n	n	CCONJ
ap-8536	495	11	among	among	ADP
ap-8536	495	12	all	all	DET
ap-8536	495	13	the	the	DET
ap-8536	495	14	(	(	PUNCT
ap-8536	495	15	m	m	PROPN
ap-8536	495	16	,	,	PUNCT
ap-8536	495	17	d)-representations	d)-representation	NOUN
ap-8536	495	18	of	of	ADP
ap-8536	495	19	y.	y.	NOUN
ap-8536	495	20	let	let	VERB
ap-8536	495	21	x	x	SYM
ap-8536	495	22	∈	∈	PROPN
ap-8536	495	23	rm	rm	PROPN
ap-8536	495	24	and	and	CCONJ
ap-8536	495	25	(	(	PUNCT
ap-8536	495	26	x(n	x(n	NOUN
ap-8536	495	27	)	)	PUNCT
ap-8536	495	28	)	)	PUNCT
ap-8536	496	1	n∈n	n∈n	NOUN
ap-8536	496	2	be	be	AUX
ap-8536	496	3	a	a	DET
ap-8536	496	4	sequence	sequence	NOUN
ap-8536	496	5	of	of	ADP
ap-8536	496	6	vectors	vector	NOUN
ap-8536	496	7	from	from	ADP
ap-8536	496	8	qm	qm	PROPN
ap-8536	496	9	such	such	ADJ
ap-8536	496	10	that	that	SCONJ
ap-8536	496	11	lim	lim	PROPN
ap-8536	496	12	n→∞	n→∞	X
ap-8536	496	13	xn	xn	PROPN
ap-8536	497	1	=	=	PUNCT
ap-8536	497	2	x.	x.	NOUN
ap-8536	497	3	by	by	ADP
ap-8536	497	4	the	the	DET
ap-8536	497	5	previous	previous	ADJ
ap-8536	497	6	theorem	theorem	NOUN
ap-8536	497	7	,	,	PUNCT
ap-8536	497	8	we	we	PRON
ap-8536	497	9	have	have	VERB
ap-8536	497	10	x(n	x(n	NOUN
ap-8536	497	11	)	)	PUNCT
ap-8536	498	1	=	=	SYM
ap-8536	499	1	∑nn	∑nn	X
ap-8536	500	1	j=−∞	j=−∞	PROPN
ap-8536	501	1	m	m	PROPN
ap-8536	501	2	jd	jd	PROPN
ap-8536	501	3	(	(	PUNCT
ap-8536	501	4	n	n	CCONJ
ap-8536	501	5	)	)	PUNCT
ap-8536	501	6	j	j	PROPN
ap-8536	501	7	.	.	PUNCT
ap-8536	502	1	denote	denote	VERB
ap-8536	502	2	the	the	DET
ap-8536	502	3	integer	integer	NOUN
ap-8536	502	4	and	and	CCONJ
ap-8536	502	5	fractional	fractional	ADJ
ap-8536	502	6	parts	part	NOUN
ap-8536	502	7	of	of	ADP
ap-8536	502	8	x(n	x(n	NOUN
ap-8536	502	9	)	)	PUNCT
ap-8536	502	10	by	by	ADP
ap-8536	502	11	y(n	y(n	PROPN
ap-8536	502	12	)	)	PUNCT
ap-8536	503	1	:	:	PUNCT
ap-8536	503	2	=	=	SYM
ap-8536	503	3	∑nn	∑nn	PROPN
ap-8536	503	4	j=0	j=0	PROPN
ap-8536	503	5	m	m	PROPN
ap-8536	503	6	jd	jd	PROPN
ap-8536	503	7	(	(	PUNCT
ap-8536	503	8	n	n	CCONJ
ap-8536	503	9	)	)	PUNCT
ap-8536	503	10	j	j	PROPN
ap-8536	503	11	and	and	CCONJ
ap-8536	503	12	z(n	z(n	NOUN
ap-8536	503	13	)	)	PUNCT
ap-8536	503	14	:	:	PUNCT
ap-8536	504	1	=	=	PUNCT
ap-8536	504	2	∑−1	∑−1	NOUN
ap-8536	504	3	j=−∞	j=−∞	NOUN
ap-8536	505	1	m	m	PROPN
ap-8536	505	2	jd	jd	PROPN
ap-8536	505	3	(	(	PUNCT
ap-8536	505	4	n	n	CCONJ
ap-8536	505	5	)	)	PUNCT
ap-8536	505	6	j	j	PROPN
ap-8536	505	7	,	,	PUNCT
ap-8536	505	8	respectively	respectively	ADV
ap-8536	505	9	.	.	PUNCT
ap-8536	506	1	obviously	obviously	ADV
ap-8536	506	2	,	,	PUNCT
ap-8536	506	3	y(n	y(n	PROPN
ap-8536	506	4	)	)	PUNCT
ap-8536	506	5	∈	∈	PROPN
ap-8536	506	6	zm	zm	PROPN
ap-8536	506	7	.	.	PROPN
ap-8536	506	8	assume	assume	VERB
ap-8536	506	9	that	that	SCONJ
ap-8536	506	10	(	(	PUNCT
ap-8536	506	11	m	m	NOUN
ap-8536	506	12	,	,	PUNCT
ap-8536	506	13	d)-representations	d)-representation	NOUN
ap-8536	506	14	of	of	ADP
ap-8536	506	15	the	the	DET
ap-8536	506	16	integer	integer	NOUN
ap-8536	506	17	parts	part	NOUN
ap-8536	506	18	y(n	y(n	PROPN
ap-8536	506	19	)	)	PUNCT
ap-8536	506	20	satisfy	satisfy	NOUN
ap-8536	506	21	nn	nn	PROPN
ap-8536	506	22	=	=	SYM
ap-8536	506	23	height(y(n	height(y(n	PROPN
ap-8536	506	24	)	)	PUNCT
ap-8536	506	25	)	)	PUNCT
ap-8536	506	26	for	for	ADP
ap-8536	506	27	every	every	DET
ap-8536	506	28	n	n	PRON
ap-8536	506	29	∈	∈	PROPN
ap-8536	506	30	n.	n.	NOUN
ap-8536	506	31	the	the	DET
ap-8536	506	32	size	size	NOUN
ap-8536	506	33	of	of	ADP
ap-8536	506	34	the	the	DET
ap-8536	506	35	fractional	fractional	ADJ
ap-8536	506	36	parts	part	NOUN
ap-8536	506	37	z(n	z(n	NOUN
ap-8536	506	38	)	)	PUNCT
ap-8536	506	39	is	be	AUX
ap-8536	506	40	bounded	bound	VERB
ap-8536	506	41	.	.	PUNCT
ap-8536	507	1	indeed,∥∥z(n	indeed,∥∥z(n	NOUN
ap-8536	507	2	)	)	PUNCT
ap-8536	508	1	∥∥	∥∥	X
ap-8536	509	1	c	c	NOUN
ap-8536	509	2	=	=	PUNCT
ap-8536	510	1	∥∥∥∑−1	∥∥∥∑−1	PROPN
ap-8536	511	1	j=−∞	j=−∞	PROPN
ap-8536	512	1	m	m	PROPN
ap-8536	512	2	jd	jd	PROPN
ap-8536	512	3	(	(	PUNCT
ap-8536	512	4	n	n	CCONJ
ap-8536	512	5	)	)	PUNCT
ap-8536	512	6	j	j	PROPN
ap-8536	512	7	∥∥∥	∥∥∥	PROPN
ap-8536	512	8	c	c	PROPN
ap-8536	512	9	≤	≤	PROPN
ap-8536	512	10	∑−1	∑−1	NOUN
ap-8536	512	11	j=−∞	j=−∞	NOUN
ap-8536	513	1	r−jd	r−jd	PROPN
ap-8536	513	2	=	=	SYM
ap-8536	513	3	rd	rd	PROPN
ap-8536	513	4	1−r	1−r	NUM
ap-8536	513	5	.	.	PUNCT
ap-8536	514	1	since	since	SCONJ
ap-8536	514	2	any	any	DET
ap-8536	514	3	convergent	convergent	NOUN
ap-8536	514	4	sequence	sequence	NOUN
ap-8536	514	5	is	be	AUX
ap-8536	514	6	bounded	bound	VERB
ap-8536	514	7	,	,	PUNCT
ap-8536	514	8	the	the	DET
ap-8536	514	9	integer	integer	NOUN
ap-8536	514	10	parts	part	NOUN
ap-8536	514	11	y(n	y(n	PROPN
ap-8536	514	12	)	)	PUNCT
ap-8536	515	1	∈	∈	PROPN
ap-8536	515	2	zm	zm	PROPN
ap-8536	515	3	are	be	AUX
ap-8536	515	4	bounded	bound	VERB
ap-8536	515	5	as	as	ADV
ap-8536	515	6	well	well	ADV
ap-8536	515	7	,	,	PUNCT
ap-8536	515	8	as∥∥y(n	as∥∥y(n	PROPN
ap-8536	515	9	)	)	PUNCT
ap-8536	516	1	∥∥	∥∥	PROPN
ap-8536	516	2	c	c	NOUN
ap-8536	516	3	≤	≤	NUM
ap-8536	516	4	∥∥x(n	∥∥x(n	NOUN
ap-8536	516	5	)	)	PUNCT
ap-8536	516	6	−	−	PROPN
ap-8536	516	7	z(n	z(n	NOUN
ap-8536	516	8	)	)	PUNCT
ap-8536	516	9	∥∥	∥∥	PROPN
ap-8536	516	10	c	c	NOUN
ap-8536	516	11	≤	≤	NUM
ap-8536	516	12	∥∥z(n	∥∥z(n	NOUN
ap-8536	516	13	)	)	PUNCT
ap-8536	517	1	∥∥	∥∥	X
ap-8536	517	2	c	c	NOUN
ap-8536	518	1	+	+	NUM
ap-8536	519	1	∥∥x(n	∥∥x(n	NOUN
ap-8536	519	2	)	)	PUNCT
ap-8536	520	1	∥∥	∥∥	PROPN
ap-8536	520	2	c	c	NOUN
ap-8536	520	3	.	.	PUNCT
ap-8536	521	1	it	it	PRON
ap-8536	521	2	means	mean	VERB
ap-8536	521	3	that	that	SCONJ
ap-8536	521	4	y(n	y(n	PROPN
ap-8536	521	5	)	)	PUNCT
ap-8536	521	6	can	can	AUX
ap-8536	521	7	take	take	VERB
ap-8536	521	8	only	only	ADV
ap-8536	521	9	finitely	finitely	ADV
ap-8536	521	10	many	many	ADJ
ap-8536	521	11	values	value	NOUN
ap-8536	521	12	in	in	ADP
ap-8536	521	13	zm	zm	PROPN
ap-8536	521	14	,	,	PUNCT
ap-8536	521	15	and	and	CCONJ
ap-8536	521	16	thus	thus	ADV
ap-8536	521	17	their	their	PRON
ap-8536	521	18	heights	height	NOUN
ap-8536	521	19	nn	nn	PROPN
ap-8536	521	20	are	be	AUX
ap-8536	521	21	bounded	bound	VERB
ap-8536	521	22	as	as	ADV
ap-8536	521	23	well	well	ADV
ap-8536	521	24	,	,	PUNCT
ap-8536	521	25	say	say	VERB
ap-8536	521	26	by	by	ADP
ap-8536	521	27	n	n	X
ap-8536	521	28	.	.	PUNCT
ap-8536	522	1	hence	hence	ADV
ap-8536	522	2	,	,	PUNCT
ap-8536	522	3	we	we	PRON
ap-8536	522	4	can	can	AUX
ap-8536	522	5	rewrite	rewrite	VERB
ap-8536	522	6	the	the	DET
ap-8536	522	7	representation	representation	NOUN
ap-8536	522	8	of	of	ADP
ap-8536	522	9	x(n	x(n	NOUN
ap-8536	522	10	)	)	PUNCT
ap-8536	522	11	for	for	ADP
ap-8536	522	12	each	each	DET
ap-8536	522	13	n	n	PRON
ap-8536	522	14	∈	∈	PROPN
ap-8536	522	15	n	n	NOUN
ap-8536	522	16	into	into	ADP
ap-8536	522	17	the	the	DET
ap-8536	522	18	form	form	NOUN
ap-8536	522	19	x(n	x(n	NOUN
ap-8536	522	20	)	)	PUNCT
ap-8536	522	21	=	=	SYM
ap-8536	523	1	∑n	∑n	PROPN
ap-8536	523	2	j=−∞	j=−∞	NOUN
ap-8536	523	3	m	m	PROPN
ap-8536	523	4	jd	jd	PROPN
ap-8536	523	5	(	(	PUNCT
ap-8536	523	6	n	n	CCONJ
ap-8536	523	7	)	)	PUNCT
ap-8536	523	8	j	j	NOUN
ap-8536	523	9	with	with	ADP
ap-8536	523	10	the	the	DET
ap-8536	523	11	uniform	uniform	ADJ
ap-8536	523	12	upper	upper	ADJ
ap-8536	523	13	index	index	NOUN
ap-8536	523	14	n	n	PROPN
ap-8536	523	15	of	of	ADP
ap-8536	523	16	the	the	DET
ap-8536	523	17	sums	sum	NOUN
ap-8536	523	18	(	(	PUNCT
ap-8536	523	19	if	if	SCONJ
ap-8536	523	20	necessary	necessary	ADJ
ap-8536	523	21	,	,	PUNCT
ap-8536	523	22	we	we	PRON
ap-8536	523	23	add	add	VERB
ap-8536	523	24	leading	lead	VERB
ap-8536	523	25	zero	zero	NUM
ap-8536	523	26	coefficients	coefficient	NOUN
ap-8536	523	27	to	to	ADP
ap-8536	523	28	sums	sum	NOUN
ap-8536	523	29	)	)	PUNCT
ap-8536	523	30	.	.	PUNCT
ap-8536	524	1	now	now	ADV
ap-8536	524	2	,	,	PUNCT
ap-8536	524	3	we	we	PRON
ap-8536	524	4	are	be	AUX
ap-8536	524	5	ready	ready	ADJ
ap-8536	524	6	to	to	PART
ap-8536	524	7	find	find	VERB
ap-8536	524	8	an	an	DET
ap-8536	524	9	(	(	PUNCT
ap-8536	524	10	m	m	PROPN
ap-8536	524	11	,	,	PUNCT
ap-8536	524	12	d)-representation	d)-representation	NOUN
ap-8536	524	13	of	of	ADP
ap-8536	524	14	x	x	X
ap-8536	524	15	=	=	SYM
ap-8536	524	16	lim	lim	PROPN
ap-8536	524	17	n→∞	n→∞	NUM
ap-8536	524	18	x(n	x(n	NOUN
ap-8536	524	19	)	)	PUNCT
ap-8536	524	20	.	.	PUNCT
ap-8536	525	1	we	we	PRON
ap-8536	525	2	construct	construct	VERB
ap-8536	525	3	the	the	DET
ap-8536	525	4	sequence	sequence	NOUN
ap-8536	525	5	dn	dn	PROPN
ap-8536	525	6	dn−1	dn−1	PROPN
ap-8536	525	7	·	·	PUNCT
ap-8536	525	8	·	·	PUNCT
ap-8536	525	9	·	·	PUNCT
ap-8536	525	10	d0	d0	NOUN
ap-8536	525	11	•	•	NOUN
ap-8536	525	12	d−1d−2	d−1d−2	PROPN
ap-8536	525	13	·	·	PUNCT
ap-8536	525	14	·	·	PUNCT
ap-8536	525	15	·	·	PUNCT
ap-8536	525	16	of	of	ADP
ap-8536	525	17	digits	digit	NOUN
ap-8536	525	18	from	from	ADP
ap-8536	525	19	d.	d.	PROPN
ap-8536	525	20	a	a	DET
ap-8536	525	21	digit	digit	NOUN
ap-8536	525	22	which	which	PRON
ap-8536	525	23	appears	appear	VERB
ap-8536	525	24	infinitely	infinitely	ADV
ap-8536	525	25	times	time	NOUN
ap-8536	525	26	among	among	ADP
ap-8536	525	27	x	x	X
ap-8536	525	28	(	(	PUNCT
ap-8536	525	29	n	n	CCONJ
ap-8536	525	30	)	)	PUNCT
ap-8536	525	31	n	n	X
ap-8536	525	32	will	will	AUX
ap-8536	525	33	be	be	AUX
ap-8536	525	34	chosen	choose	VERB
ap-8536	525	35	as	as	ADP
ap-8536	525	36	dn	dn	PROPN
ap-8536	525	37	.	.	PUNCT
ap-8536	526	1	then	then	ADV
ap-8536	526	2	,	,	PUNCT
ap-8536	526	3	we	we	PRON
ap-8536	526	4	chose	choose	VERB
ap-8536	526	5	dn−1	dn−1	PROPN
ap-8536	526	6	as	as	ADP
ap-8536	526	7	such	such	DET
ap-8536	526	8	a	a	DET
ap-8536	526	9	digit	digit	NOUN
ap-8536	526	10	that	that	SCONJ
ap-8536	526	11	the	the	DET
ap-8536	526	12	pair	pair	NOUN
ap-8536	526	13	of	of	ADP
ap-8536	526	14	digits	digit	NOUN
ap-8536	526	15	dn	dn	PROPN
ap-8536	526	16	dn−1	dn−1	PROPN
ap-8536	526	17	appears	appear	VERB
ap-8536	526	18	infinitely	infinitely	ADV
ap-8536	526	19	many	many	ADJ
ap-8536	526	20	times	time	NOUN
ap-8536	526	21	among	among	ADP
ap-8536	526	22	x	x	X
ap-8536	526	23	(	(	PUNCT
ap-8536	526	24	n	n	CCONJ
ap-8536	526	25	)	)	PUNCT
ap-8536	526	26	n	n	NOUN
ap-8536	526	27	x	x	SYM
ap-8536	526	28	(	(	PUNCT
ap-8536	526	29	n	n	CCONJ
ap-8536	526	30	)	)	PUNCT
ap-8536	526	31	n−1	n−1	PROPN
ap-8536	526	32	.	.	PROPN
ap-8536	527	1	similarly	similarly	ADV
ap-8536	527	2	,	,	PUNCT
ap-8536	527	3	dn−2	dn−2	PROPN
ap-8536	527	4	is	be	AUX
ap-8536	527	5	chosen	choose	VERB
ap-8536	527	6	as	as	ADP
ap-8536	527	7	such	such	DET
ap-8536	527	8	a	a	DET
ap-8536	527	9	digit	digit	NOUN
ap-8536	527	10	that	that	SCONJ
ap-8536	527	11	the	the	DET
ap-8536	527	12	triplet	triplet	NOUN
ap-8536	527	13	dn	dn	NOUN
ap-8536	527	14	dn−1dn−2	dn−1dn−2	PROPN
ap-8536	527	15	appears	appear	VERB
ap-8536	527	16	infinitely	infinitely	ADV
ap-8536	527	17	many	many	ADJ
ap-8536	527	18	times	time	NOUN
ap-8536	527	19	among	among	ADP
ap-8536	527	20	x	x	X
ap-8536	527	21	(	(	PUNCT
ap-8536	527	22	n	n	CCONJ
ap-8536	527	23	)	)	PUNCT
ap-8536	527	24	n	n	NOUN
ap-8536	527	25	x	x	SYM
ap-8536	527	26	(	(	PUNCT
ap-8536	527	27	n	n	CCONJ
ap-8536	527	28	)	)	PUNCT
ap-8536	527	29	n−1x	n−1x	X
ap-8536	527	30	(	(	PUNCT
ap-8536	527	31	n	n	CCONJ
ap-8536	527	32	)	)	PUNCT
ap-8536	527	33	n−2	n−2	PROPN
ap-8536	527	34	,	,	PUNCT
ap-8536	527	35	and	and	CCONJ
ap-8536	527	36	so	so	ADV
ap-8536	527	37	on	on	ADV
ap-8536	527	38	.	.	PUNCT
ap-8536	528	1	by	by	ADP
ap-8536	528	2	this	this	DET
ap-8536	528	3	construction	construction	NOUN
ap-8536	528	4	,	,	PUNCT
ap-8536	528	5	for	for	ADP
ap-8536	528	6	every	every	DET
ap-8536	528	7	step	step	NOUN
ap-8536	528	8	l	l	NOUN
ap-8536	528	9	∈	∈	PROPN
ap-8536	528	10	n	n	CCONJ
ap-8536	528	11	,	,	PUNCT
ap-8536	528	12	l	l	X
ap-8536	528	13	≥	≥	NUM
ap-8536	528	14	1	1	NUM
ap-8536	528	15	,	,	PUNCT
ap-8536	528	16	there	there	PRON
ap-8536	528	17	exists	exist	VERB
ap-8536	528	18	kl	kl	PROPN
ap-8536	528	19	∈	∈	PROPN
ap-8536	528	20	z	z	NOUN
ap-8536	528	21	such	such	ADJ
ap-8536	528	22	that	that	SCONJ
ap-8536	528	23	the	the	DET
ap-8536	528	24	strings	string	NOUN
ap-8536	528	25	dn	dn	ADP
ap-8536	528	26	dn−1	dn−1	PROPN
ap-8536	528	27	·	·	PUNCT
ap-8536	528	28	·	·	PUNCT
ap-8536	528	29	·	·	PUNCT
ap-8536	528	30	d0•d−1d−2	d0•d−1d−2	PROPN
ap-8536	528	31	·	·	PUNCT
ap-8536	528	32	·	·	PUNCT
ap-8536	528	33	·	·	PUNCT
ap-8536	529	1	and	and	CCONJ
ap-8536	529	2	x	x	X
ap-8536	529	3	(	(	PUNCT
ap-8536	529	4	kl	kl	NOUN
ap-8536	529	5	)	)	PUNCT
ap-8536	529	6	n	n	NOUN
ap-8536	529	7	x	x	SYM
ap-8536	529	8	(	(	PUNCT
ap-8536	529	9	kl	kl	NOUN
ap-8536	529	10	)	)	PUNCT
ap-8536	529	11	n−1	n−1	PROPN
ap-8536	529	12	·	·	PUNCT
ap-8536	529	13	·	·	PUNCT
ap-8536	529	14	·	·	PUNCT
ap-8536	529	15	x	x	X
ap-8536	529	16	(	(	PUNCT
ap-8536	529	17	kl	kl	NOUN
ap-8536	529	18	)	)	PUNCT
ap-8536	529	19	0	0	NUM
ap-8536	529	20	•	•	NOUN
ap-8536	529	21	x	x	SYM
ap-8536	529	22	(	(	PUNCT
ap-8536	529	23	kl	kl	NOUN
ap-8536	529	24	)	)	PUNCT
ap-8536	529	25	−1	−1	NOUN
ap-8536	529	26	x	x	SYM
ap-8536	529	27	(	(	PUNCT
ap-8536	529	28	kl	kl	NOUN
ap-8536	529	29	)	)	PUNCT
ap-8536	529	30	−2	−2	NOUN
ap-8536	529	31	·	·	PUNCT
ap-8536	529	32	·	·	PUNCT
ap-8536	529	33	·	·	PUNCT
ap-8536	529	34	have	have	VERB
ap-8536	529	35	a	a	DET
ap-8536	529	36	common	common	ADJ
ap-8536	529	37	prefix	prefix	NOUN
ap-8536	529	38	of	of	ADP
ap-8536	529	39	length	length	NOUN
ap-8536	529	40	at	at	ADP
ap-8536	529	41	least	least	ADJ
ap-8536	529	42	l.	l.	PROPN
ap-8536	529	43	therefore,∥∥∥∥∥∥	therefore,∥∥∥∥∥∥	PROPN
ap-8536	530	1	n∑	n∑	PROPN
ap-8536	531	1	j=−∞	j=−∞	PROPN
ap-8536	532	1	m	m	PROPN
ap-8536	532	2	jdj	jdj	PROPN
ap-8536	532	3	−	−	PROPN
ap-8536	533	1	n∑	n∑	PROPN
ap-8536	533	2	j=−∞	j=−∞	PROPN
ap-8536	534	1	m	m	PROPN
ap-8536	534	2	jx	jx	PROPN
ap-8536	534	3	(	(	PUNCT
ap-8536	534	4	kl	kl	PROPN
ap-8536	534	5	)	)	PUNCT
ap-8536	534	6	j	j	PROPN
ap-8536	534	7	∥∥∥∥∥∥	∥∥∥∥∥∥	PROPN
ap-8536	535	1	c	c	PROPN
ap-8536	535	2	=	=	SYM
ap-8536	535	3	∥∥∥∥∥∥	∥∥∥∥∥∥	PROPN
ap-8536	535	4	n−l∑	n−l∑	PROPN
ap-8536	535	5	j=−∞	j=−∞	PROPN
ap-8536	536	1	m	m	PROPN
ap-8536	536	2	j	j	PROPN
ap-8536	536	3	(	(	PUNCT
ap-8536	536	4	dj	dj	NOUN
ap-8536	536	5	−	−	NOUN
ap-8536	536	6	x	x	SYM
ap-8536	536	7	(	(	PUNCT
ap-8536	536	8	kl	kl	PROPN
ap-8536	536	9	)	)	PUNCT
ap-8536	536	10	j	j	PROPN
ap-8536	536	11	)	)	PUNCT
ap-8536	536	12	∥∥∥∥∥∥	∥∥∥∥∥∥	PROPN
ap-8536	537	1	c	c	NOUN
ap-8536	537	2	≤	≤	NOUN
ap-8536	537	3	2d	2d	NOUN
ap-8536	537	4	+	+	PUNCT
ap-8536	537	5	∞∑	∞∑	PROPN
ap-8536	537	6	j	j	NOUN
ap-8536	537	7	=	=	X
ap-8536	537	8	l−n	l−n	PROPN
ap-8536	537	9	rj	rj	X
ap-8536	537	10	=	=	SYM
ap-8536	537	11	2d	2d	PROPN
ap-8536	537	12	1	1	NUM
ap-8536	537	13	−	−	NOUN
ap-8536	537	14	r	r	NOUN
ap-8536	537	15	rl−n	rl−n	NOUN
ap-8536	537	16	,	,	PUNCT
ap-8536	537	17	where	where	SCONJ
ap-8536	537	18	d	d	PROPN
ap-8536	537	19	denotes	denote	VERB
ap-8536	537	20	max{||d||c	max{||d||c	NOUN
ap-8536	537	21	:	:	PUNCT
ap-8536	537	22	d	d	X
ap-8536	537	23	∈	∈	PROPN
ap-8536	537	24	d	d	NOUN
ap-8536	537	25	}	}	PUNCT
ap-8536	537	26	.	.	PUNCT
ap-8536	538	1	since	since	SCONJ
ap-8536	538	2	2d	2d	PROPN
ap-8536	538	3	1−r	1−r	NUM
ap-8536	538	4	rl−n	rl−n	PROPN
ap-8536	538	5	tends	tend	VERB
ap-8536	538	6	to	to	ADP
ap-8536	538	7	0	0	NUM
ap-8536	538	8	with	with	ADP
ap-8536	538	9	l	l	NOUN
ap-8536	538	10	→	→	SYM
ap-8536	538	11	+	+	PROPN
ap-8536	538	12	∞	∞	PROPN
ap-8536	538	13	,	,	PUNCT
ap-8536	538	14	we	we	PRON
ap-8536	538	15	obtain	obtain	VERB
ap-8536	538	16	n∑	n∑	PROPN
ap-8536	538	17	j=−∞	j=−∞	NOUN
ap-8536	539	1	m	m	PROPN
ap-8536	539	2	jdj	jdj	PROPN
ap-8536	539	3	=	=	SYM
ap-8536	539	4	lim	lim	PROPN
ap-8536	539	5	l→+∞	l→+∞	PROPN
ap-8536	539	6	x(kl	x(kl	PROPN
ap-8536	539	7	)	)	PUNCT
ap-8536	539	8	=	=	SYM
ap-8536	539	9	lim	lim	PROPN
ap-8536	539	10	n→+∞	n→+∞	VERB
ap-8536	539	11	x(n	x(n	NOUN
ap-8536	539	12	)	)	PUNCT
ap-8536	540	1	=	=	SYM
ap-8536	540	2	x	x	X
ap-8536	540	3	,	,	PUNCT
ap-8536	540	4	and	and	CCONJ
ap-8536	540	5	thus	thus	ADV
ap-8536	540	6	∑n	∑n	PROPN
ap-8536	540	7	j=−∞	j=−∞	PROPN
ap-8536	541	1	m	m	PROPN
ap-8536	541	2	jdj	jdj	PROPN
ap-8536	541	3	is	be	AUX
ap-8536	541	4	an	an	DET
ap-8536	541	5	(	(	PUNCT
ap-8536	541	6	m	m	PROPN
ap-8536	541	7	,	,	PUNCT
ap-8536	541	8	d)-representation	d)-representation	NOUN
ap-8536	541	9	of	of	ADP
ap-8536	541	10	x.	x.	NOUN
ap-8536	541	11	5	5	X
ap-8536	541	12	.	.	PUNCT
ap-8536	542	1	open	open	ADJ
ap-8536	542	2	questions	question	NOUN
ap-8536	542	3	we	we	PRON
ap-8536	542	4	have	have	AUX
ap-8536	542	5	focused	focus	VERB
ap-8536	542	6	only	only	ADV
ap-8536	542	7	on	on	ADP
ap-8536	542	8	two	two	NUM
ap-8536	542	9	questions	question	NOUN
ap-8536	542	10	connected	connect	VERB
ap-8536	542	11	with	with	ADP
ap-8536	542	12	(	(	PUNCT
ap-8536	542	13	m	m	PROPN
ap-8536	542	14	,	,	PUNCT
ap-8536	542	15	d)-representation	d)-representation	NOUN
ap-8536	542	16	of	of	ADP
ap-8536	542	17	vectors	vector	NOUN
ap-8536	542	18	:	:	PUNCT
ap-8536	542	19	computability	computability	NOUN
ap-8536	542	20	of	of	ADP
ap-8536	542	21	addition	addition	NOUN
ap-8536	542	22	in	in	ADP
ap-8536	542	23	parallel	parallel	ADJ
ap-8536	542	24	and	and	CCONJ
ap-8536	542	25	eventually	eventually	ADV
ap-8536	542	26	periodic	periodic	ADJ
ap-8536	542	27	representations	representation	NOUN
ap-8536	542	28	.	.	PUNCT
ap-8536	543	1	many	many	ADJ
ap-8536	543	2	open	open	ADJ
ap-8536	543	3	problems	problem	NOUN
ap-8536	543	4	still	still	ADV
ap-8536	543	5	remain	remain	VERB
ap-8536	543	6	unresolved	unresolved	ADJ
ap-8536	543	7	.	.	PUNCT
ap-8536	544	1	let	let	VERB
ap-8536	544	2	us	we	PRON
ap-8536	544	3	list	list	VERB
ap-8536	544	4	some	some	PRON
ap-8536	544	5	of	of	ADP
ap-8536	544	6	them	they	PRON
ap-8536	544	7	:	:	PUNCT
ap-8536	544	8	(	(	PUNCT
ap-8536	544	9	1	1	X
ap-8536	544	10	.	.	PUNCT
ap-8536	544	11	)	)	PUNCT
ap-8536	545	1	what	what	PRON
ap-8536	545	2	is	be	AUX
ap-8536	545	3	the	the	DET
ap-8536	545	4	minimal	minimal	ADJ
ap-8536	545	5	size	size	NOUN
ap-8536	545	6	of	of	ADP
ap-8536	545	7	a	a	DET
ap-8536	545	8	digit	digit	NOUN
ap-8536	545	9	set	set	VERB
ap-8536	545	10	d	d	PROPN
ap-8536	545	11	⊂	⊂	PROPN
ap-8536	545	12	zm	zm	PROPN
ap-8536	545	13	with	with	ADP
ap-8536	545	14	the	the	DET
ap-8536	545	15	property	property	NOUN
ap-8536	545	16	zm	zm	PROPN
ap-8536	545	17	⊂	⊂	PROPN
ap-8536	545	18	find(m	find(m	PROPN
ap-8536	545	19	)	)	PUNCT
ap-8536	545	20	?	?	PUNCT
ap-8536	546	1	this	this	DET
ap-8536	546	2	question	question	NOUN
ap-8536	546	3	196	196	NUM
ap-8536	546	4	vol	vol	NOUN
ap-8536	546	5	.	.	PUNCT
ap-8536	547	1	63	63	NUM
ap-8536	547	2	no	no	NOUN
ap-8536	547	3	.	.	PUNCT
ap-8536	548	1	3/2023	3/2023	NUM
ap-8536	548	2	positional	positional	ADJ
ap-8536	548	3	representation	representation	NOUN
ap-8536	548	4	of	of	ADP
ap-8536	548	5	vectors	vector	NOUN
ap-8536	548	6	was	be	AUX
ap-8536	548	7	already	already	ADV
ap-8536	548	8	tackled	tackle	VERB
ap-8536	548	9	in	in	ADP
ap-8536	548	10	[	[	X
ap-8536	548	11	26	26	NUM
ap-8536	548	12	]	]	PUNCT
ap-8536	548	13	for	for	ADP
ap-8536	548	14	very	very	ADV
ap-8536	548	15	special	special	ADJ
ap-8536	548	16	matrices	matrix	NOUN
ap-8536	548	17	,	,	PUNCT
ap-8536	548	18	namely	namely	ADV
ap-8536	548	19	for	for	ADP
ap-8536	548	20	jordan	jordan	PROPN
ap-8536	548	21	blocks	blocks	PROPN
ap-8536	548	22	jm(1	jm(1	PROPN
ap-8536	548	23	)	)	PUNCT
ap-8536	548	24	corresponding	correspond	VERB
ap-8536	548	25	to	to	ADP
ap-8536	548	26	the	the	DET
ap-8536	548	27	eigenvalue	eigenvalue	PROPN
ap-8536	548	28	1	1	NUM
ap-8536	548	29	.	.	PUNCT
ap-8536	549	1	(	(	PUNCT
ap-8536	549	2	2	2	NUM
ap-8536	549	3	.	.	PUNCT
ap-8536	549	4	)	)	PUNCT
ap-8536	549	5	is	be	AUX
ap-8536	549	6	it	it	PRON
ap-8536	549	7	possible	possible	ADJ
ap-8536	549	8	for	for	SCONJ
ap-8536	549	9	a	a	DET
ap-8536	549	10	given	give	VERB
ap-8536	549	11	non	non	ADJ
ap-8536	549	12	-	-	ADJ
ap-8536	549	13	singular	singular	ADJ
ap-8536	549	14	matrix	matrix	NOUN
ap-8536	549	15	m	m	VERB
ap-8536	549	16	∈	∈	PROPN
ap-8536	549	17	zm×m	zm×m	NOUN
ap-8536	549	18	to	to	PART
ap-8536	549	19	find	find	VERB
ap-8536	549	20	a	a	DET
ap-8536	549	21	(	(	PUNCT
ap-8536	549	22	finite	finite	NOUN
ap-8536	549	23	)	)	PUNCT
ap-8536	549	24	digit	digit	NOUN
ap-8536	549	25	set	set	VERB
ap-8536	549	26	d	d	PROPN
ap-8536	549	27	⊂	⊂	PROPN
ap-8536	549	28	zm	zm	PROPN
ap-8536	549	29	such	such	ADJ
ap-8536	549	30	that	that	SCONJ
ap-8536	549	31	every	every	DET
ap-8536	549	32	element	element	NOUN
ap-8536	549	33	in	in	ADP
ap-8536	549	34	find(m	find(m	NOUN
ap-8536	549	35	)	)	PUNCT
ap-8536	549	36	has	have	VERB
ap-8536	549	37	a	a	DET
ap-8536	549	38	unique	unique	ADJ
ap-8536	549	39	representation	representation	NOUN
ap-8536	549	40	in	in	ADP
ap-8536	549	41	this	this	DET
ap-8536	549	42	numeration	numeration	NOUN
ap-8536	549	43	system	system	NOUN
ap-8536	549	44	?	?	PUNCT
ap-8536	550	1	if	if	SCONJ
ap-8536	550	2	m	m	NOUN
ap-8536	550	3	is	be	AUX
ap-8536	550	4	expansive	expansive	ADJ
ap-8536	550	5	,	,	PUNCT
ap-8536	550	6	then	then	ADV
ap-8536	550	7	the	the	DET
ap-8536	550	8	size	size	NOUN
ap-8536	550	9	of	of	ADP
ap-8536	550	10	the	the	DET
ap-8536	550	11	suitable	suitable	ADJ
ap-8536	550	12	digit	digit	NOUN
ap-8536	550	13	set	set	NOUN
ap-8536	550	14	(	(	PUNCT
ap-8536	550	15	if	if	SCONJ
ap-8536	550	16	it	it	PRON
ap-8536	550	17	exists	exist	VERB
ap-8536	550	18	)	)	PUNCT
ap-8536	550	19	is	be	AUX
ap-8536	550	20	|	|	ADV
ap-8536	550	21	det(m)|	det(m)|	PROPN
ap-8536	550	22	,	,	PUNCT
ap-8536	550	23	see	see	VERB
ap-8536	550	24	[	[	X
ap-8536	550	25	13	13	NUM
ap-8536	550	26	]	]	PUNCT
ap-8536	550	27	.	.	PUNCT
ap-8536	551	1	what	what	PRON
ap-8536	551	2	is	be	AUX
ap-8536	551	3	an	an	DET
ap-8536	551	4	analogy	analogy	NOUN
ap-8536	551	5	of	of	ADP
ap-8536	551	6	this	this	DET
ap-8536	551	7	result	result	NOUN
ap-8536	551	8	for	for	ADP
ap-8536	551	9	a	a	DET
ap-8536	551	10	non	non	ADJ
ap-8536	551	11	-	-	ADJ
ap-8536	551	12	expansive	expansive	ADJ
ap-8536	551	13	matrix	matrix	NOUN
ap-8536	551	14	?	?	PUNCT
ap-8536	552	1	(	(	PUNCT
ap-8536	552	2	3	3	X
ap-8536	552	3	.	.	PUNCT
ap-8536	552	4	)	)	PUNCT
ap-8536	552	5	does	do	AUX
ap-8536	552	6	there	there	PRON
ap-8536	552	7	exist	exist	VERB
ap-8536	552	8	any	any	DET
ap-8536	552	9	(	(	PUNCT
ap-8536	552	10	finite	finite	NOUN
ap-8536	552	11	)	)	PUNCT
ap-8536	552	12	digit	digit	NOUN
ap-8536	552	13	set	set	NOUN
ap-8536	552	14	such	such	ADJ
ap-8536	552	15	that	that	DET
ap-8536	552	16	multiplication	multiplication	NOUN
ap-8536	552	17	of	of	ADP
ap-8536	552	18	vectors	vector	NOUN
ap-8536	552	19	from	from	ADP
ap-8536	552	20	rm	rm	NOUN
ap-8536	552	21	by	by	ADP
ap-8536	552	22	a	a	DET
ap-8536	552	23	scalar	scalar	ADJ
ap-8536	552	24	x	x	SYM
ap-8536	552	25	∈	∈	NOUN
ap-8536	552	26	r	r	NOUN
ap-8536	552	27	can	can	AUX
ap-8536	552	28	be	be	AUX
ap-8536	552	29	performed	perform	VERB
ap-8536	552	30	by	by	ADP
ap-8536	552	31	an	an	DET
ap-8536	552	32	on	on	ADP
ap-8536	552	33	-	-	PUNCT
ap-8536	552	34	line	line	NOUN
ap-8536	552	35	algorithm	algorithm	NOUN
ap-8536	552	36	?	?	PUNCT
ap-8536	553	1	note	note	VERB
ap-8536	553	2	that	that	SCONJ
ap-8536	553	3	k.	k.	PROPN
ap-8536	553	4	trivedi	trivedi	PROPN
ap-8536	553	5	and	and	CCONJ
ap-8536	553	6	m.	m.	NOUN
ap-8536	553	7	ercegovac	ercegovac	PROPN
ap-8536	553	8	in	in	ADP
ap-8536	553	9	[	[	X
ap-8536	553	10	27	27	NUM
ap-8536	553	11	]	]	PUNCT
ap-8536	553	12	designed	design	VERB
ap-8536	553	13	on	on	ADP
ap-8536	553	14	-	-	PUNCT
ap-8536	553	15	line	line	NOUN
ap-8536	553	16	algorithms	algorithm	NOUN
ap-8536	553	17	for	for	ADP
ap-8536	553	18	the	the	DET
ap-8536	553	19	multiplication	multiplication	NOUN
ap-8536	553	20	and	and	CCONJ
ap-8536	553	21	division	division	NOUN
ap-8536	553	22	of	of	ADP
ap-8536	553	23	two	two	NUM
ap-8536	553	24	numbers	number	NOUN
ap-8536	553	25	represented	represent	VERB
ap-8536	553	26	in	in	ADP
ap-8536	553	27	a	a	DET
ap-8536	553	28	numeration	numeration	NOUN
ap-8536	553	29	system	system	NOUN
ap-8536	553	30	(	(	PUNCT
ap-8536	553	31	β	β	X
ap-8536	553	32	,	,	PUNCT
ap-8536	553	33	a	a	PRON
ap-8536	553	34	)	)	PUNCT
ap-8536	553	35	with	with	ADP
ap-8536	553	36	β	β	X
ap-8536	553	37	∈	∈	PROPN
ap-8536	553	38	n.	n.	NOUN
ap-8536	553	39	their	their	PRON
ap-8536	553	40	algorithms	algorithm	NOUN
ap-8536	553	41	were	be	AUX
ap-8536	553	42	later	later	ADV
ap-8536	553	43	generalised	generalise	VERB
ap-8536	553	44	to	to	ADP
ap-8536	553	45	numeration	numeration	NOUN
ap-8536	553	46	systems	system	NOUN
ap-8536	553	47	(	(	PUNCT
ap-8536	553	48	β	β	X
ap-8536	553	49	,	,	PUNCT
ap-8536	553	50	a	a	PRON
ap-8536	553	51	)	)	PUNCT
ap-8536	553	52	with	with	ADP
ap-8536	553	53	base	base	NOUN
ap-8536	553	54	β	β	X
ap-8536	553	55	being	be	AUX
ap-8536	553	56	a	a	DET
ap-8536	553	57	(	(	PUNCT
ap-8536	553	58	real	real	ADJ
ap-8536	553	59	or	or	CCONJ
ap-8536	553	60	complex	complex	ADJ
ap-8536	553	61	)	)	PUNCT
ap-8536	553	62	pisot	pisot	ADJ
ap-8536	553	63	number	number	NOUN
ap-8536	553	64	[	[	X
ap-8536	553	65	28	28	NUM
ap-8536	553	66	]	]	PUNCT
ap-8536	553	67	.	.	PUNCT
ap-8536	554	1	acknowledgements	acknowledgement	NOUN
ap-8536	554	2	edita	edita	PROPN
ap-8536	554	3	pelantová	pelantová	PROPN
ap-8536	554	4	acknowledges	acknowledge	VERB
ap-8536	554	5	financial	financial	ADJ
ap-8536	554	6	support	support	NOUN
ap-8536	554	7	by	by	ADP
ap-8536	554	8	the	the	DET
ap-8536	554	9	ministry	ministry	PROPN
ap-8536	554	10	of	of	ADP
ap-8536	554	11	education	education	PROPN
ap-8536	554	12	,	,	PUNCT
ap-8536	554	13	youth	youth	NOUN
ap-8536	554	14	and	and	CCONJ
ap-8536	554	15	sports	sport	NOUN
ap-8536	554	16	of	of	ADP
ap-8536	554	17	the	the	DET
ap-8536	554	18	czech	czech	PROPN
ap-8536	554	19	republic	republic	NOUN
ap-8536	554	20	,	,	PUNCT
ap-8536	554	21	project	project	NOUN
ap-8536	554	22	no	no	INTJ
ap-8536	554	23	.	.	PUNCT
ap-8536	554	24	cz.02.1.01/0.0/0.0/16_019/0000778	cz.02.1.01/0.0/0.0/16_019/0000778	PROPN
ap-8536	554	25	.	.	PUNCT
ap-8536	555	1	references	reference	NOUN
ap-8536	555	2	[	[	X
ap-8536	555	3	1	1	NUM
ap-8536	555	4	]	]	PUNCT
ap-8536	555	5	a.	a.	NOUN
ap-8536	555	6	cauchy	cauchy	PROPN
ap-8536	555	7	.	.	PUNCT
ap-8536	556	1	sur	sur	PROPN
ap-8536	556	2	les	les	PROPN
ap-8536	556	3	moyens	moyens	PROPN
ap-8536	556	4	d’éviter	d’éviter	PROPN
ap-8536	556	5	les	les	PROPN
ap-8536	556	6	erreurs	erreurs	X
ap-8536	556	7	dans	dans	PROPN
ap-8536	556	8	les	les	X
ap-8536	556	9	calculs	calculs	PROPN
ap-8536	556	10	numériques	numériques	X
ap-8536	556	11	.	.	PUNCT
ap-8536	557	1	no	no	INTJ
ap-8536	557	2	.	.	NOUN
ap-8536	557	3	11	11	NUM
ap-8536	557	4	in	in	ADP
ap-8536	557	5	série	série	PROPN
ap-8536	557	6	i.	i.	PROPN
ap-8536	557	7	c.r	c.r	PROPN
ap-8536	557	8	.	.	PROPN
ap-8536	557	9	acad	acad	PROPN
ap-8536	557	10	.	.	PUNCT
ap-8536	558	1	sc	sc	PROPN
ap-8536	558	2	.	.	PROPN
ap-8536	558	3	paris	paris	PROPN
ap-8536	558	4	,	,	PUNCT
ap-8536	558	5	france	france	PROPN
ap-8536	558	6	,	,	PUNCT
ap-8536	558	7	1840	1840	NUM
ap-8536	558	8	.	.	PUNCT
ap-8536	559	1	[	[	X
ap-8536	559	2	2	2	X
ap-8536	559	3	]	]	PUNCT
ap-8536	559	4	v.	v.	ADP
ap-8536	559	5	grünwald	grünwald	PROPN
ap-8536	559	6	.	.	PUNCT
ap-8536	560	1	intorno	intorno	PROPN
ap-8536	560	2	all’aritmetica	all’aritmetica	PROPN
ap-8536	560	3	dei	dei	PROPN
ap-8536	560	4	sistemi	sistemi	PROPN
ap-8536	560	5	numerici	numerici	PROPN
ap-8536	560	6	a	a	DET
ap-8536	560	7	base	base	PROPN
ap-8536	560	8	negativa	negativa	PROPN
ap-8536	560	9	con	con	PROPN
ap-8536	560	10	particolare	particolare	PROPN
ap-8536	560	11	riguardo	riguardo	PROPN
ap-8536	560	12	al	al	PROPN
ap-8536	560	13	sistema	sistema	PROPN
ap-8536	560	14	numerico	numerico	PROPN
ap-8536	560	15	a	a	DET
ap-8536	560	16	base	base	NOUN
ap-8536	560	17	negativo	negativo	ADJ
ap-8536	560	18	-	-	PUNCT
ap-8536	560	19	decimale	decimale	NOUN
ap-8536	560	20	per	per	ADP
ap-8536	560	21	lo	lo	PROPN
ap-8536	560	22	studio	studio	NOUN
ap-8536	560	23	delle	delle	NOUN
ap-8536	560	24	sue	sue	PROPN
ap-8536	560	25	analogie	analogie	PROPN
ap-8536	560	26	coll’aritmetica	coll’aritmetica	PROPN
ap-8536	560	27	ordinaria	ordinaria	PROPN
ap-8536	560	28	(	(	PUNCT
ap-8536	560	29	decimale	decimale	NOUN
ap-8536	560	30	)	)	PUNCT
ap-8536	560	31	.	.	PUNCT
ap-8536	561	1	23	23	NUM
ap-8536	561	2	.	.	PUNCT
ap-8536	561	3	giornale	giornale	PROPN
ap-8536	561	4	di	di	PROPN
ap-8536	561	5	matematiche	matematiche	PROPN
ap-8536	561	6	di	di	PROPN
ap-8536	561	7	battaglini	battaglini	PROPN
ap-8536	561	8	,	,	PUNCT
ap-8536	561	9	italy	italy	PROPN
ap-8536	561	10	,	,	PUNCT
ap-8536	561	11	1885	1885	NUM
ap-8536	561	12	.	.	PUNCT
ap-8536	562	1	[	[	X
ap-8536	562	2	3	3	NUM
ap-8536	562	3	]	]	PUNCT
ap-8536	562	4	a.	a.	NOUN
ap-8536	562	5	rényi	rényi	PROPN
ap-8536	562	6	.	.	PUNCT
ap-8536	563	1	representations	representation	NOUN
ap-8536	563	2	for	for	ADP
ap-8536	563	3	real	real	ADJ
ap-8536	563	4	numbers	number	NOUN
ap-8536	563	5	and	and	CCONJ
ap-8536	563	6	their	their	PRON
ap-8536	563	7	ergodic	ergodic	ADJ
ap-8536	563	8	properties	property	NOUN
ap-8536	563	9	.	.	PUNCT
ap-8536	564	1	acta	acta	PROPN
ap-8536	564	2	mathematica	mathematica	PROPN
ap-8536	564	3	academiae	academiae	PROPN
ap-8536	564	4	scientiarum	scientiarum	PROPN
ap-8536	564	5	hungaricae	hungaricae	PROPN
ap-8536	564	6	8:477–493	8:477–493	PROPN
ap-8536	564	7	,	,	PUNCT
ap-8536	564	8	1957	1957	NUM
ap-8536	564	9	.	.	PUNCT
ap-8536	565	1	https://doi.org/10.1007/bf02020331	https://doi.org/10.1007/bf02020331	X
ap-8536	566	1	[	[	X
ap-8536	566	2	4	4	NUM
ap-8536	566	3	]	]	PUNCT
ap-8536	566	4	k.	k.	PROPN
ap-8536	566	5	schmidt	schmidt	PROPN
ap-8536	566	6	.	.	PUNCT
ap-8536	567	1	on	on	ADP
ap-8536	567	2	periodic	periodic	ADJ
ap-8536	567	3	expansions	expansion	NOUN
ap-8536	567	4	of	of	ADP
ap-8536	567	5	pisot	pisot	ADJ
ap-8536	567	6	numbers	number	NOUN
ap-8536	567	7	and	and	CCONJ
ap-8536	567	8	salem	salem	NOUN
ap-8536	567	9	numbers	number	NOUN
ap-8536	567	10	.	.	PUNCT
ap-8536	568	1	bulletin	bulletin	NOUN
ap-8536	568	2	of	of	ADP
ap-8536	568	3	the	the	DET
ap-8536	568	4	london	london	PROPN
ap-8536	568	5	mathematical	mathematical	ADJ
ap-8536	568	6	society	society	PROPN
ap-8536	568	7	12(4):269–278	12(4):269–278	PROPN
ap-8536	568	8	,	,	PUNCT
ap-8536	568	9	1980	1980	NUM
ap-8536	568	10	.	.	PUNCT
ap-8536	569	1	https://doi.org/10.1112/blms/12.4.269	https://doi.org/10.1112/blms/12.4.269	PUNCT
ap-8536	569	2	[	[	X
ap-8536	569	3	5	5	X
ap-8536	569	4	]	]	PUNCT
ap-8536	569	5	t.	t.	NOUN
ap-8536	569	6	vávra	vávra	PROPN
ap-8536	569	7	,	,	PUNCT
ap-8536	569	8	f.	f.	PROPN
ap-8536	569	9	veneziano	veneziano	PROPN
ap-8536	569	10	.	.	PROPN
ap-8536	569	11	pisot	pisot	ADJ
ap-8536	569	12	unit	unit	NOUN
ap-8536	569	13	generators	generator	NOUN
ap-8536	569	14	in	in	ADP
ap-8536	569	15	number	number	NOUN
ap-8536	569	16	fields	field	NOUN
ap-8536	569	17	.	.	PUNCT
ap-8536	570	1	journal	journal	NOUN
ap-8536	570	2	of	of	ADP
ap-8536	570	3	symbolic	symbolic	ADJ
ap-8536	570	4	computation	computation	NOUN
ap-8536	570	5	89:94–108	89:94–108	NUM
ap-8536	570	6	,	,	PUNCT
ap-8536	570	7	2018	2018	NUM
ap-8536	570	8	.	.	PUNCT
ap-8536	571	1	https://doi.org/10.1016/j.jsc.2017.11.005	https://doi.org/10.1016/j.jsc.2017.11.005	PROPN
ap-8536	572	1	[	[	X
ap-8536	572	2	6	6	NUM
ap-8536	572	3	]	]	PUNCT
ap-8536	572	4	d.	d.	PROPN
ap-8536	572	5	e.	e.	PROPN
ap-8536	572	6	knuth	knuth	PROPN
ap-8536	572	7	.	.	PUNCT
ap-8536	573	1	a	a	DET
ap-8536	573	2	imaginary	imaginary	ADJ
ap-8536	573	3	number	number	NOUN
ap-8536	573	4	system	system	NOUN
ap-8536	573	5	.	.	PUNCT
ap-8536	574	1	communications	communication	NOUN
ap-8536	574	2	of	of	ADP
ap-8536	574	3	the	the	DET
ap-8536	574	4	acm	acm	PROPN
ap-8536	574	5	3(4):245–247	3(4):245–247	NUM
ap-8536	574	6	,	,	PUNCT
ap-8536	574	7	1960	1960	NUM
ap-8536	574	8	.	.	PUNCT
ap-8536	575	1	https://doi.org/10.1145/367177.367233	https://doi.org/10.1145/367177.367233	NOUN
ap-8536	576	1	[	[	X
ap-8536	576	2	7	7	X
ap-8536	576	3	]	]	X
ap-8536	576	4	w.	w.	PROPN
ap-8536	576	5	penney	penney	PROPN
ap-8536	576	6	.	.	PUNCT
ap-8536	577	1	a	a	DET
ap-8536	577	2	“	"	PUNCT
ap-8536	577	3	binary	binary	ADJ
ap-8536	577	4	”	"	PUNCT
ap-8536	577	5	system	system	NOUN
ap-8536	577	6	for	for	ADP
ap-8536	577	7	complex	complex	ADJ
ap-8536	577	8	numbers	number	NOUN
ap-8536	577	9	.	.	PUNCT
ap-8536	578	1	journal	journal	NOUN
ap-8536	578	2	of	of	ADP
ap-8536	578	3	the	the	DET
ap-8536	578	4	acm	acm	PROPN
ap-8536	578	5	12(2):247–248	12(2):247–248	PROPN
ap-8536	578	6	,	,	PUNCT
ap-8536	578	7	1965	1965	NUM
ap-8536	578	8	.	.	PUNCT
ap-8536	579	1	https://doi.org/10.1145/321264.321274	https://doi.org/10.1145/321264.321274	NOUN
ap-8536	580	1	[	[	X
ap-8536	580	2	8	8	NUM
ap-8536	580	3	]	]	X
ap-8536	580	4	b.	b.	PROPN
ap-8536	580	5	kovács	kovács	PROPN
ap-8536	580	6	,	,	PUNCT
ap-8536	580	7	a.	a.	NOUN
ap-8536	580	8	pethö	pethö	PROPN
ap-8536	580	9	.	.	PUNCT
ap-8536	581	1	number	number	NOUN
ap-8536	581	2	systems	system	NOUN
ap-8536	581	3	in	in	ADP
ap-8536	581	4	integral	integral	ADJ
ap-8536	581	5	domains	domain	NOUN
ap-8536	581	6	,	,	PUNCT
ap-8536	581	7	especially	especially	ADV
ap-8536	581	8	in	in	ADP
ap-8536	581	9	orders	order	NOUN
ap-8536	581	10	of	of	ADP
ap-8536	581	11	algebraic	algebraic	ADJ
ap-8536	581	12	number	number	NOUN
ap-8536	581	13	fields	field	NOUN
ap-8536	581	14	.	.	PUNCT
ap-8536	582	1	acta	acta	PROPN
ap-8536	582	2	scientiarum	scientiarum	PROPN
ap-8536	582	3	mathematicarum	mathematicarum	PROPN
ap-8536	582	4	55:287–299	55:287–299	PROPN
ap-8536	582	5	,	,	PUNCT
ap-8536	582	6	1991	1991	NUM
ap-8536	582	7	.	.	PUNCT
ap-8536	583	1	http://acta.bibl.u-szeged.hu/15313/1/math_055	http://acta.bibl.u-szeged.hu/15313/1/math_055	X
ap-8536	583	2	_	_	PUNCT
ap-8536	584	1	fasc_003_004_287-299.pdf	fasc_003_004_287-299.pdf	NOUN
ap-8536	584	2	.	.	PUNCT
ap-8536	585	1	[	[	X
ap-8536	585	2	9	9	NUM
ap-8536	585	3	]	]	PUNCT
ap-8536	585	4	p.	p.	NOUN
ap-8536	585	5	kirschenhofer	kirschenhofer	NOUN
ap-8536	585	6	,	,	PUNCT
ap-8536	585	7	j.	j.	PROPN
ap-8536	585	8	thuswaldner	thuswaldner	PROPN
ap-8536	585	9	.	.	PUNCT
ap-8536	586	1	shift	shift	NOUN
ap-8536	586	2	radix	radix	PROPN
ap-8536	586	3	systems	system	NOUN
ap-8536	586	4	:	:	PUNCT
ap-8536	586	5	a	a	DET
ap-8536	586	6	survey	survey	NOUN
ap-8536	586	7	.	.	PUNCT
ap-8536	587	1	numeration	numeration	NOUN
ap-8536	587	2	and	and	CCONJ
ap-8536	587	3	substitution	substitution	NOUN
ap-8536	587	4	b46:1–59	b46:1–59	NOUN
ap-8536	587	5	,	,	PUNCT
ap-8536	587	6	2014	2014	NUM
ap-8536	587	7	.	.	PUNCT
ap-8536	588	1	http://hdl.handle.net/2433/226207	http://hdl.handle.net/2433/226207	NOUN
ap-8536	588	2	.	.	PUNCT
ap-8536	589	1	[	[	X
ap-8536	589	2	10	10	NUM
ap-8536	589	3	]	]	PUNCT
ap-8536	589	4	a.	a.	NOUN
ap-8536	589	5	avizienis	avizienis	PROPN
ap-8536	589	6	.	.	PUNCT
ap-8536	590	1	signed	sign	VERB
ap-8536	590	2	-	-	PUNCT
ap-8536	590	3	digit	digit	NOUN
ap-8536	590	4	numbe	numbe	NOUN
ap-8536	590	5	representations	representation	NOUN
ap-8536	590	6	for	for	ADP
ap-8536	590	7	fast	fast	ADJ
ap-8536	590	8	parallel	parallel	ADJ
ap-8536	590	9	arithmetic	arithmetic	ADJ
ap-8536	590	10	.	.	PUNCT
ap-8536	591	1	ire	ire	ADJ
ap-8536	591	2	transactions	transaction	NOUN
ap-8536	591	3	on	on	ADP
ap-8536	591	4	electronic	electronic	ADJ
ap-8536	591	5	computers	computer	NOUN
ap-8536	591	6	ec-10(3):389–400	ec-10(3):389–400	NOUN
ap-8536	591	7	,	,	PUNCT
ap-8536	591	8	1961	1961	NUM
ap-8536	591	9	.	.	PUNCT
ap-8536	592	1	https://doi.org/10.1109/tec.1961.5219227	https://doi.org/10.1109/tec.1961.5219227	PROPN
ap-8536	593	1	[	[	X
ap-8536	593	2	11	11	NUM
ap-8536	593	3	]	]	PUNCT
ap-8536	593	4	a.	a.	NOUN
ap-8536	593	5	vince	vince	PROPN
ap-8536	593	6	.	.	PUNCT
ap-8536	594	1	radix	radix	PROPN
ap-8536	594	2	representation	representation	PROPN
ap-8536	594	3	and	and	CCONJ
ap-8536	594	4	rep	rep	NOUN
ap-8536	594	5	-	-	ADJ
ap-8536	594	6	tiling	tiling	NOUN
ap-8536	594	7	.	.	PUNCT
ap-8536	595	1	in	in	ADP
ap-8536	595	2	proceedings	proceeding	NOUN
ap-8536	595	3	of	of	ADP
ap-8536	595	4	the	the	DET
ap-8536	595	5	24	24	NUM
ap-8536	595	6	-	-	PUNCT
ap-8536	595	7	th	th	X
ap-8536	595	8	southeastern	southeastern	ADJ
ap-8536	595	9	international	international	ADJ
ap-8536	595	10	conference	conference	NOUN
ap-8536	595	11	on	on	ADP
ap-8536	595	12	combinatorics	combinatoric	NOUN
ap-8536	595	13	,	,	PUNCT
ap-8536	595	14	graph	graph	NOUN
ap-8536	595	15	theory	theory	NOUN
ap-8536	595	16	,	,	PUNCT
ap-8536	595	17	and	and	CCONJ
ap-8536	595	18	computing	computing	NOUN
ap-8536	595	19	,	,	PUNCT
ap-8536	595	20	vol	vol	NOUN
ap-8536	595	21	.	.	PROPN
ap-8536	595	22	98	98	NUM
ap-8536	595	23	,	,	PUNCT
ap-8536	595	24	pp	pp	ADJ
ap-8536	595	25	.	.	PUNCT
ap-8536	596	1	199–212	199–212	NUM
ap-8536	596	2	.	.	PUNCT
ap-8536	596	3	1993	1993	NUM
ap-8536	596	4	.	.	PUNCT
ap-8536	597	1	[	[	X
ap-8536	597	2	12	12	NUM
ap-8536	597	3	]	]	PUNCT
ap-8536	597	4	a.	a.	NOUN
ap-8536	597	5	vince	vince	NOUN
ap-8536	597	6	.	.	PUNCT
ap-8536	598	1	replicating	replicate	VERB
ap-8536	598	2	tessellations	tessellation	NOUN
ap-8536	598	3	.	.	PUNCT
ap-8536	599	1	siam	siam	PROPN
ap-8536	599	2	journal	journal	PROPN
ap-8536	599	3	on	on	ADP
ap-8536	599	4	discrete	discrete	ADJ
ap-8536	599	5	mathematics	mathematic	NOUN
ap-8536	599	6	6(3):501–521	6(3):501–521	NOUN
ap-8536	599	7	,	,	PUNCT
ap-8536	599	8	1993	1993	NUM
ap-8536	599	9	.	.	PUNCT
ap-8536	600	1	https://doi.org/10.1137/0406040	https://doi.org/10.1137/0406040	PUNCT
ap-8536	601	1	[	[	X
ap-8536	601	2	13	13	NUM
ap-8536	601	3	]	]	PUNCT
ap-8536	601	4	a.	a.	NOUN
ap-8536	601	5	kovács	kovács	PROPN
ap-8536	601	6	.	.	PUNCT
ap-8536	602	1	number	number	NOUN
ap-8536	602	2	expansions	expansion	NOUN
ap-8536	602	3	in	in	ADP
ap-8536	602	4	lattices	lattice	NOUN
ap-8536	602	5	.	.	PUNCT
ap-8536	603	1	mathematical	mathematical	ADJ
ap-8536	603	2	and	and	CCONJ
ap-8536	603	3	computer	computer	NOUN
ap-8536	603	4	modelling	model	VERB
ap-8536	603	5	38(7):909–915	38(7):909–915	PROPN
ap-8536	603	6	,	,	PUNCT
ap-8536	603	7	2003	2003	NUM
ap-8536	603	8	.	.	PUNCT
ap-8536	604	1	https://doi.org/10.1016/s0895-7177(03)90076-8	https://doi.org/10.1016/s0895-7177(03)90076-8	PROPN
ap-8536	605	1	[	[	X
ap-8536	605	2	14	14	NUM
ap-8536	605	3	]	]	X
ap-8536	605	4	j.	j.	PROPN
ap-8536	605	5	jankauskas	jankauskas	PROPN
ap-8536	605	6	,	,	PUNCT
ap-8536	605	7	j.	j.	PROPN
ap-8536	605	8	thuswaldner	thuswaldner	PROPN
ap-8536	605	9	.	.	PUNCT
ap-8536	606	1	characterization	characterization	NOUN
ap-8536	606	2	of	of	ADP
ap-8536	606	3	rational	rational	ADJ
ap-8536	606	4	matrices	matrix	NOUN
ap-8536	606	5	that	that	PRON
ap-8536	606	6	admit	admit	VERB
ap-8536	606	7	finite	finite	ADJ
ap-8536	606	8	digit	digit	NOUN
ap-8536	606	9	representations	representation	NOUN
ap-8536	606	10	.	.	PUNCT
ap-8536	607	1	linear	linear	ADJ
ap-8536	607	2	algebra	algebra	NOUN
ap-8536	607	3	and	and	CCONJ
ap-8536	607	4	its	its	PRON
ap-8536	607	5	applications	application	NOUN
ap-8536	607	6	557:350–358	557:350–358	NUM
ap-8536	607	7	,	,	PUNCT
ap-8536	607	8	2018	2018	NUM
ap-8536	607	9	.	.	PUNCT
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ap-8536	609	2	15	15	NUM
ap-8536	609	3	]	]	X
ap-8536	609	4	e.	e.	PROPN
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ap-8536	609	9	.	.	PUNCT
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ap-8536	611	7	,	,	PUNCT
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ap-8536	611	9	.	.	PUNCT
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ap-8536	613	1	an	an	DET
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ap-8536	613	4	symbolic	symbolic	ADJ
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ap-8536	613	8	.	.	PUNCT
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ap-8536	614	6	,	,	PUNCT
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ap-8536	614	8	.	.	PUNCT
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ap-8536	618	5	,	,	PUNCT
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ap-8536	618	7	.	.	PUNCT
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ap-8536	620	2	18	18	NUM
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ap-8536	620	9	,	,	PUNCT
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ap-8536	621	7	.	.	PUNCT
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ap-8536	622	7	,	,	PUNCT
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ap-8536	622	9	.	.	PUNCT
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ap-8536	623	4	]	]	PUNCT
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ap-8536	625	5	,	,	PUNCT
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ap-8536	625	7	.	.	PUNCT
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ap-8536	632	6	,	,	PUNCT
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ap-8536	632	8	.	.	PUNCT
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ap-8536	633	7	/	/	SYM
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ap-8536	636	8	.	.	PUNCT
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ap-8536	643	5	,	,	PUNCT
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ap-8536	643	7	.	.	PUNCT
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