id	sid	tid	token	lemma	pos
ejpam-4669	1	1	european	european	PROPN
ejpam-4669	1	2	journal	journal	PROPN
ejpam-4669	1	3	of	of	ADP
ejpam-4669	1	4	pure	pure	ADJ
ejpam-4669	1	5	and	and	CCONJ
ejpam-4669	1	6	applied	apply	VERB
ejpam-4669	1	7	mathematics	mathematic	NOUN
ejpam-4669	1	8	vol	vol	NOUN
ejpam-4669	1	9	.	.	PUNCT
ejpam-4669	2	1	16	16	NUM
ejpam-4669	2	2	,	,	PUNCT
ejpam-4669	2	3	no	no	INTJ
ejpam-4669	2	4	.	.	NOUN
ejpam-4669	2	5	1	1	NUM
ejpam-4669	2	6	,	,	PUNCT
ejpam-4669	2	7	2023	2023	NUM
ejpam-4669	2	8	,	,	PUNCT
ejpam-4669	2	9	587	587	NUM
ejpam-4669	2	10	-	-	SYM
ejpam-4669	2	11	594	594	NUM
ejpam-4669	2	12	issn	issn	PROPN
ejpam-4669	2	13	1307	1307	NUM
ejpam-4669	2	14	-	-	SYM
ejpam-4669	2	15	5543	5543	NUM
ejpam-4669	2	16	–	–	PUNCT
ejpam-4669	3	1	ejpam.com	ejpam.com	X
ejpam-4669	3	2	published	publish	VERB
ejpam-4669	3	3	by	by	ADP
ejpam-4669	3	4	new	new	PROPN
ejpam-4669	3	5	york	york	PROPN
ejpam-4669	3	6	business	business	PROPN
ejpam-4669	3	7	global	global	PROPN
ejpam-4669	3	8	on	on	ADP
ejpam-4669	3	9	gaussian	gaussian	PROPN
ejpam-4669	3	10	fibonacci	fibonacci	NOUN
ejpam-4669	3	11	functions	function	NOUN
ejpam-4669	3	12	with	with	ADP
ejpam-4669	3	13	periodicity	periodicity	NOUN
ejpam-4669	3	14	hariwan	hariwan	NOUN
ejpam-4669	3	15	fadhil	fadhil	PROPN
ejpam-4669	3	16	m.salih	m.salih	NOUN
ejpam-4669	3	17	department	department	PROPN
ejpam-4669	3	18	of	of	ADP
ejpam-4669	3	19	mathematics	mathematics	PROPN
ejpam-4669	3	20	,	,	PUNCT
ejpam-4669	3	21	college	college	NOUN
ejpam-4669	3	22	of	of	ADP
ejpam-4669	3	23	science	science	NOUN
ejpam-4669	3	24	,	,	PUNCT
ejpam-4669	3	25	university	university	NOUN
ejpam-4669	3	26	of	of	ADP
ejpam-4669	3	27	duhok	duhok	PROPN
ejpam-4669	3	28	,	,	PUNCT
ejpam-4669	3	29	iraq	iraq	PROPN
ejpam-4669	3	30	abstract	abstract	NOUN
ejpam-4669	3	31	.	.	PUNCT
ejpam-4669	4	1	in	in	ADP
ejpam-4669	4	2	this	this	DET
ejpam-4669	4	3	work	work	NOUN
ejpam-4669	4	4	,	,	PUNCT
ejpam-4669	4	5	gaussian	gaussian	ADJ
ejpam-4669	4	6	fibonacci	fibonacci	NOUN
ejpam-4669	4	7	functions	function	NOUN
ejpam-4669	4	8	with	with	ADP
ejpam-4669	4	9	the	the	DET
ejpam-4669	4	10	use	use	NOUN
ejpam-4669	4	11	of	of	ADP
ejpam-4669	4	12	the	the	DET
ejpam-4669	4	13	(	(	PUNCT
ejpam-4669	4	14	ultimately	ultimately	ADV
ejpam-4669	4	15	)	)	PUNCT
ejpam-4669	4	16	periodicity	periodicity	NOUN
ejpam-4669	4	17	and	and	CCONJ
ejpam-4669	4	18	exponential	exponential	ADJ
ejpam-4669	4	19	gaussian	gaussian	NOUN
ejpam-4669	4	20	fibonacci	fibonacci	NOUN
ejpam-4669	4	21	functions	function	NOUN
ejpam-4669	4	22	are	be	AUX
ejpam-4669	4	23	also	also	ADV
ejpam-4669	4	24	discussed	discuss	VERB
ejpam-4669	4	25	.	.	PUNCT
ejpam-4669	5	1	especially	especially	ADV
ejpam-4669	5	2	,	,	PUNCT
ejpam-4669	5	3	by	by	ADP
ejpam-4669	5	4	giving	give	VERB
ejpam-4669	5	5	a	a	DET
ejpam-4669	5	6	nonnegative	nonnegative	ADJ
ejpam-4669	5	7	real	real	ADJ
ejpam-4669	5	8	valued	value	VERB
ejpam-4669	5	9	function	function	NOUN
ejpam-4669	5	10	,	,	PUNCT
ejpam-4669	5	11	several	several	ADJ
ejpam-4669	5	12	exponential	exponential	ADJ
ejpam-4669	5	13	gaussian	gaussian	NOUN
ejpam-4669	5	14	fibonacci	fibonacci	NOUN
ejpam-4669	5	15	functions	function	NOUN
ejpam-4669	5	16	are	be	AUX
ejpam-4669	5	17	attained	attain	VERB
ejpam-4669	5	18	.	.	PUNCT
ejpam-4669	6	1	2020	2020	NUM
ejpam-4669	6	2	mathematics	mathematic	NOUN
ejpam-4669	6	3	subject	subject	NOUN
ejpam-4669	6	4	classifications	classification	NOUN
ejpam-4669	6	5	:	:	PUNCT
ejpam-4669	6	6	11b39	11b39	NUM
ejpam-4669	6	7	,	,	PUNCT
ejpam-4669	6	8	11b99	11b99	NUM
ejpam-4669	6	9	key	key	ADJ
ejpam-4669	6	10	words	word	NOUN
ejpam-4669	6	11	and	and	CCONJ
ejpam-4669	6	12	phrases	phrase	NOUN
ejpam-4669	6	13	:	:	PUNCT
ejpam-4669	6	14	gaussian	gaussian	ADJ
ejpam-4669	6	15	fibonacci	fibonacci	NOUN
ejpam-4669	6	16	functions	function	NOUN
ejpam-4669	6	17	,	,	PUNCT
ejpam-4669	6	18	gaussian	gaussian	NOUN
ejpam-4669	6	19	(	(	PUNCT
ejpam-4669	6	20	ultimately	ultimately	ADV
ejpam-4669	6	21	)	)	PUNCT
ejpam-4669	6	22	periodic	periodic	ADJ
ejpam-4669	6	23	,	,	PUNCT
ejpam-4669	6	24	fibonacci	fibonacci	NOUN
ejpam-4669	6	25	functions	function	NOUN
ejpam-4669	6	26	1	1	NUM
ejpam-4669	6	27	.	.	PUNCT
ejpam-4669	7	1	introduction	introduction	NOUN
ejpam-4669	7	2	fibonacci	fibonacci	NOUN
ejpam-4669	7	3	numbers	number	NOUN
ejpam-4669	7	4	have	have	VERB
ejpam-4669	7	5	many	many	ADJ
ejpam-4669	7	6	applications	application	NOUN
ejpam-4669	7	7	in	in	ADP
ejpam-4669	7	8	different	different	ADJ
ejpam-4669	7	9	disciplines	discipline	NOUN
ejpam-4669	7	10	such	such	ADJ
ejpam-4669	7	11	as	as	ADP
ejpam-4669	7	12	in	in	ADP
ejpam-4669	7	13	mathematics	mathematic	NOUN
ejpam-4669	7	14	,	,	PUNCT
ejpam-4669	7	15	philosophy	philosophy	NOUN
ejpam-4669	7	16	,	,	PUNCT
ejpam-4669	7	17	physics	physics	NOUN
ejpam-4669	7	18	,	,	PUNCT
ejpam-4669	7	19	art	art	NOUN
ejpam-4669	7	20	,	,	PUNCT
ejpam-4669	7	21	architecture	architecture	NOUN
ejpam-4669	7	22	etc	etc	NOUN
ejpam-4669	7	23	,	,	PUNCT
ejpam-4669	7	24	where	where	SCONJ
ejpam-4669	7	25	can	can	AUX
ejpam-4669	7	26	be	be	AUX
ejpam-4669	7	27	found	find	VERB
ejpam-4669	7	28	in	in	ADP
ejpam-4669	7	29	[	[	X
ejpam-4669	7	30	2	2	NUM
ejpam-4669	7	31	,	,	PUNCT
ejpam-4669	7	32	3	3	NUM
ejpam-4669	7	33	,	,	PUNCT
ejpam-4669	7	34	7	7	NUM
ejpam-4669	7	35	]	]	PUNCT
ejpam-4669	7	36	.	.	PUNCT
ejpam-4669	8	1	a	a	DET
ejpam-4669	8	2	series	series	NOUN
ejpam-4669	8	3	of	of	ADP
ejpam-4669	8	4	the	the	DET
ejpam-4669	8	5	fibonacci	fibonacci	NOUN
ejpam-4669	8	6	numbers	number	NOUN
ejpam-4669	8	7	is	be	AUX
ejpam-4669	8	8	1	1	NUM
ejpam-4669	8	9	,	,	PUNCT
ejpam-4669	8	10	1	1	NUM
ejpam-4669	8	11	,	,	PUNCT
ejpam-4669	8	12	2	2	NUM
ejpam-4669	8	13	,	,	PUNCT
ejpam-4669	8	14	3	3	NUM
ejpam-4669	8	15	,	,	PUNCT
ejpam-4669	8	16	5	5	NUM
ejpam-4669	8	17	,	,	PUNCT
ejpam-4669	8	18	8	8	NUM
ejpam-4669	8	19	,	,	PUNCT
ejpam-4669	8	20	.	.	PUNCT
ejpam-4669	8	21	.	.	PUNCT
ejpam-4669	9	1	.	.	PUNCT
ejpam-4669	10	1	,	,	PUNCT
ejpam-4669	10	2	where	where	SCONJ
ejpam-4669	10	3	the	the	DET
ejpam-4669	10	4	first	first	ADJ
ejpam-4669	10	5	two	two	NUM
ejpam-4669	10	6	initiated	initiate	VERB
ejpam-4669	10	7	numbers	number	NOUN
ejpam-4669	10	8	are	be	AUX
ejpam-4669	10	9	1	1	NUM
ejpam-4669	10	10	and	and	CCONJ
ejpam-4669	10	11	every	every	DET
ejpam-4669	10	12	other	other	ADJ
ejpam-4669	10	13	number	number	NOUN
ejpam-4669	10	14	comes	come	VERB
ejpam-4669	10	15	from	from	ADP
ejpam-4669	10	16	the	the	DET
ejpam-4669	10	17	sum	sum	NOUN
ejpam-4669	10	18	of	of	ADP
ejpam-4669	10	19	the	the	DET
ejpam-4669	10	20	two	two	NUM
ejpam-4669	10	21	preceding	precede	VERB
ejpam-4669	10	22	numbers	number	NOUN
ejpam-4669	10	23	.	.	PUNCT
ejpam-4669	11	1	in	in	ADP
ejpam-4669	11	2	1963	1963	NUM
ejpam-4669	11	3	,	,	PUNCT
ejpam-4669	11	4	fibonacci	fibonacci	NOUN
ejpam-4669	11	5	numbers	number	NOUN
ejpam-4669	11	6	were	be	AUX
ejpam-4669	11	7	examined	examine	VERB
ejpam-4669	11	8	on	on	ADP
ejpam-4669	11	9	the	the	DET
ejpam-4669	11	10	complex	complex	ADJ
ejpam-4669	11	11	plane	plane	NOUN
ejpam-4669	11	12	and	and	CCONJ
ejpam-4669	11	13	some	some	DET
ejpam-4669	11	14	interesting	interesting	ADJ
ejpam-4669	11	15	properties	property	NOUN
ejpam-4669	11	16	about	about	ADP
ejpam-4669	11	17	them	they	PRON
ejpam-4669	11	18	are	be	AUX
ejpam-4669	11	19	established	establish	VERB
ejpam-4669	11	20	[	[	PUNCT
ejpam-4669	11	21	1	1	NUM
ejpam-4669	11	22	]	]	PUNCT
ejpam-4669	11	23	.	.	PUNCT
ejpam-4669	12	1	by	by	ADP
ejpam-4669	12	2	the	the	DET
ejpam-4669	12	3	same	same	ADJ
ejpam-4669	12	4	strategy	strategy	NOUN
ejpam-4669	12	5	of	of	ADP
ejpam-4669	12	6	finding	find	VERB
ejpam-4669	12	7	the	the	DET
ejpam-4669	12	8	fibonacci	fibonacci	NOUN
ejpam-4669	12	9	numbers	number	NOUN
ejpam-4669	12	10	,	,	PUNCT
ejpam-4669	12	11	gaussian	gaussian	ADJ
ejpam-4669	12	12	fibonacci	fibonacci	NOUN
ejpam-4669	12	13	numbers	number	NOUN
ejpam-4669	12	14	gfn	gfn	VERB
ejpam-4669	12	15	are	be	AUX
ejpam-4669	12	16	defined	define	VERB
ejpam-4669	12	17	recursively	recursively	ADV
ejpam-4669	12	18	by	by	ADP
ejpam-4669	12	19	gfn	gfn	NOUN
ejpam-4669	12	20	=	=	SYM
ejpam-4669	12	21	gfn−1	gfn−1	PROPN
ejpam-4669	12	22	+	+	CCONJ
ejpam-4669	12	23	gfn−2	gfn−2	PROPN
ejpam-4669	12	24	,	,	PUNCT
ejpam-4669	12	25	where	where	SCONJ
ejpam-4669	12	26	gf0	gf0	NOUN
ejpam-4669	12	27	=	=	SYM
ejpam-4669	12	28	i	i	PROPN
ejpam-4669	12	29	,	,	PUNCT
ejpam-4669	12	30	gf1	gf1	NOUN
ejpam-4669	12	31	=	=	NOUN
ejpam-4669	12	32	1	1	NUM
ejpam-4669	12	33	,	,	PUNCT
ejpam-4669	12	34	and	and	CCONJ
ejpam-4669	12	35	n	n	PRON
ejpam-4669	12	36	≥	≥	NOUN
ejpam-4669	12	37	2	2	NUM
ejpam-4669	13	1	[	[	X
ejpam-4669	13	2	6	6	NUM
ejpam-4669	13	3	]	]	PUNCT
ejpam-4669	13	4	.	.	PUNCT
ejpam-4669	14	1	in	in	ADP
ejpam-4669	14	2	[	[	X
ejpam-4669	14	3	4	4	NUM
ejpam-4669	14	4	]	]	PUNCT
ejpam-4669	14	5	,	,	PUNCT
ejpam-4669	14	6	it	it	PRON
ejpam-4669	14	7	is	be	AUX
ejpam-4669	14	8	showed	show	VERB
ejpam-4669	14	9	that	that	SCONJ
ejpam-4669	14	10	if	if	SCONJ
ejpam-4669	14	11	fg	fg	PROPN
ejpam-4669	14	12	is	be	AUX
ejpam-4669	14	13	a	a	DET
ejpam-4669	14	14	gaussian	gaussian	ADJ
ejpam-4669	14	15	fibonacci	fibonacci	NOUN
ejpam-4669	14	16	function	function	NOUN
ejpam-4669	14	17	,	,	PUNCT
ejpam-4669	14	18	we	we	PRON
ejpam-4669	14	19	have	have	VERB
ejpam-4669	14	20	that	that	PRON
ejpam-4669	14	21	limx→∞	limx→∞	PROPN
ejpam-4669	14	22	fg(x+1	fg(x+1	ADJ
ejpam-4669	14	23	)	)	PUNCT
ejpam-4669	14	24	fg(x	fg(x	NUM
ejpam-4669	14	25	)	)	PUNCT
ejpam-4669	15	1	=	=	SYM
ejpam-4669	15	2	ϕ	ϕ	NOUN
ejpam-4669	15	3	,	,	PUNCT
ejpam-4669	15	4	where	where	SCONJ
ejpam-4669	15	5	ϕ	ϕ	NOUN
ejpam-4669	15	6	=	=	NOUN
ejpam-4669	15	7	1	1	NUM
ejpam-4669	15	8	+	+	NUM
ejpam-4669	15	9	√	√	NUM
ejpam-4669	15	10	5	5	NUM
ejpam-4669	15	11	2	2	NUM
ejpam-4669	15	12	.	.	PUNCT
ejpam-4669	16	1	similarly	similarly	ADV
ejpam-4669	16	2	,	,	PUNCT
ejpam-4669	16	3	it	it	PRON
ejpam-4669	16	4	is	be	AUX
ejpam-4669	16	5	showed	show	VERB
ejpam-4669	16	6	that	that	SCONJ
ejpam-4669	16	7	if	if	SCONJ
ejpam-4669	16	8	fg	fg	PROPN
ejpam-4669	16	9	is	be	AUX
ejpam-4669	16	10	a	a	DET
ejpam-4669	16	11	gaussian	gaussian	ADJ
ejpam-4669	16	12	fibonacci	fibonacci	NOUN
ejpam-4669	16	13	function	function	NOUN
ejpam-4669	16	14	and	and	CCONJ
ejpam-4669	16	15	f	f	PROPN
ejpam-4669	16	16	is	be	AUX
ejpam-4669	16	17	a	a	DET
ejpam-4669	16	18	fibonacci	fibonacci	NOUN
ejpam-4669	16	19	function	function	NOUN
ejpam-4669	16	20	,	,	PUNCT
ejpam-4669	16	21	then	then	ADV
ejpam-4669	16	22	limx→∞	limx→∞	PROPN
ejpam-4669	16	23	fg(x+1	fg(x+1	ADJ
ejpam-4669	16	24	)	)	PUNCT
ejpam-4669	16	25	f(x	f(x	PROPN
ejpam-4669	16	26	)	)	PUNCT
ejpam-4669	17	1	=	=	PUNCT
ejpam-4669	17	2	ϕ+	ϕ+	PUNCT
ejpam-4669	18	1	i	i	PRON
ejpam-4669	18	2	,	,	PUNCT
ejpam-4669	18	3	where	where	SCONJ
ejpam-4669	18	4	ϕ	ϕ	NOUN
ejpam-4669	18	5	=	=	NOUN
ejpam-4669	18	6	1	1	NUM
ejpam-4669	18	7	+	+	NUM
ejpam-4669	18	8	√	√	NUM
ejpam-4669	18	9	5	5	NUM
ejpam-4669	18	10	2	2	NUM
ejpam-4669	18	11	.	.	PUNCT
ejpam-4669	19	1	the	the	DET
ejpam-4669	19	2	fibonacci	fibonacci	NOUN
ejpam-4669	19	3	functions	function	NOUN
ejpam-4669	19	4	with	with	ADP
ejpam-4669	19	5	periodicity	periodicity	NOUN
ejpam-4669	19	6	is	be	AUX
ejpam-4669	19	7	studied	study	VERB
ejpam-4669	19	8	in	in	ADP
ejpam-4669	19	9	[	[	X
ejpam-4669	19	10	5	5	NUM
ejpam-4669	19	11	]	]	PUNCT
ejpam-4669	19	12	.	.	PUNCT
ejpam-4669	20	1	in	in	ADP
ejpam-4669	20	2	this	this	DET
ejpam-4669	20	3	paper	paper	NOUN
ejpam-4669	20	4	,	,	PUNCT
ejpam-4669	20	5	gaussian	gaussian	ADJ
ejpam-4669	20	6	fibonacci	fibonacci	NOUN
ejpam-4669	20	7	functions	function	NOUN
ejpam-4669	20	8	with	with	ADP
ejpam-4669	20	9	periodicity	periodicity	NOUN
ejpam-4669	20	10	are	be	AUX
ejpam-4669	20	11	discussed	discuss	VERB
ejpam-4669	20	12	and	and	CCONJ
ejpam-4669	20	13	studied	study	VERB
ejpam-4669	20	14	as	as	ADV
ejpam-4669	20	15	well	well	ADV
ejpam-4669	20	16	as	as	ADP
ejpam-4669	20	17	discussing	discuss	VERB
ejpam-4669	20	18	the	the	DET
ejpam-4669	20	19	exponential	exponential	ADJ
ejpam-4669	20	20	gaussian	gaussian	NOUN
ejpam-4669	20	21	fibonacci	fibonacci	NOUN
ejpam-4669	20	22	functions	function	NOUN
ejpam-4669	20	23	,	,	PUNCT
ejpam-4669	20	24	especially	especially	ADV
ejpam-4669	20	25	,	,	PUNCT
ejpam-4669	20	26	by	by	ADP
ejpam-4669	20	27	giving	give	VERB
ejpam-4669	20	28	a	a	DET
ejpam-4669	20	29	non	non	ADJ
ejpam-4669	20	30	-	-	ADJ
ejpam-4669	20	31	negative	negative	ADJ
ejpam-4669	20	32	real	real	ADJ
ejpam-4669	20	33	valued	value	VERB
ejpam-4669	20	34	function	function	NOUN
ejpam-4669	20	35	,	,	PUNCT
ejpam-4669	20	36	several	several	ADJ
ejpam-4669	20	37	exponential	exponential	ADJ
ejpam-4669	20	38	gaussian	gaussian	NOUN
ejpam-4669	20	39	fibonacci	fibonacci	NOUN
ejpam-4669	20	40	functions	function	NOUN
ejpam-4669	20	41	are	be	AUX
ejpam-4669	20	42	obtained	obtain	VERB
ejpam-4669	20	43	.	.	PUNCT
ejpam-4669	21	1	doi	doi	NOUN
ejpam-4669	21	2	:	:	PUNCT
ejpam-4669	21	3	https://doi.org/10.29020/nybg.ejpam.v16i1.4669	https://doi.org/10.29020/nybg.ejpam.v16i1.4669	PROPN
ejpam-4669	21	4	email	email	NOUN
ejpam-4669	21	5	address	address	NOUN
ejpam-4669	21	6	:	:	PUNCT
ejpam-4669	21	7	hariwan.msalih@uod.ac	hariwan.msalih@uod.ac	PROPN
ejpam-4669	21	8	(	(	PUNCT
ejpam-4669	21	9	h.	h.	PROPN
ejpam-4669	21	10	f.	f.	PROPN
ejpam-4669	21	11	m.salih	m.salih	PROPN
ejpam-4669	21	12	)	)	PUNCT
ejpam-4669	21	13	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4669	22	1	587	587	NUM
ejpam-4669	23	1	©	©	ADP
ejpam-4669	23	2	2023	2023	NUM
ejpam-4669	23	3	ejpam	ejpam	NOUN
ejpam-4669	23	4	all	all	DET
ejpam-4669	23	5	rights	right	NOUN
ejpam-4669	23	6	reserved	reserve	VERB
ejpam-4669	23	7	.	.	PUNCT
ejpam-4669	24	1	h.	h.	PROPN
ejpam-4669	24	2	f.	f.	PROPN
ejpam-4669	24	3	m.salih	m.salih	PROPN
ejpam-4669	24	4	/	/	SYM
ejpam-4669	24	5	eur	eur	PROPN
ejpam-4669	24	6	.	.	PUNCT
ejpam-4669	25	1	j.	j.	PROPN
ejpam-4669	25	2	pure	pure	PROPN
ejpam-4669	25	3	appl	appl	PROPN
ejpam-4669	25	4	.	.	PROPN
ejpam-4669	25	5	math	math	PROPN
ejpam-4669	25	6	,	,	PUNCT
ejpam-4669	25	7	16	16	NUM
ejpam-4669	25	8	(	(	PUNCT
ejpam-4669	25	9	1	1	NUM
ejpam-4669	25	10	)	)	PUNCT
ejpam-4669	25	11	(	(	PUNCT
ejpam-4669	25	12	2023	2023	NUM
ejpam-4669	25	13	)	)	PUNCT
ejpam-4669	25	14	,	,	PUNCT
ejpam-4669	25	15	587	587	NUM
ejpam-4669	25	16	-	-	SYM
ejpam-4669	25	17	594	594	NUM
ejpam-4669	25	18	588	588	NUM
ejpam-4669	25	19	2	2	NUM
ejpam-4669	25	20	.	.	PUNCT
ejpam-4669	26	1	preliminaries	preliminary	NOUN
ejpam-4669	26	2	definition	definition	NOUN
ejpam-4669	26	3	1	1	NUM
ejpam-4669	26	4	.	.	PUNCT
ejpam-4669	27	1	[	[	X
ejpam-4669	27	2	4	4	X
ejpam-4669	27	3	]	]	PUNCT
ejpam-4669	27	4	a	a	DET
ejpam-4669	27	5	gaussian	gaussian	ADJ
ejpam-4669	27	6	function	function	NOUN
ejpam-4669	27	7	fg	fg	NOUN
ejpam-4669	27	8	on	on	ADP
ejpam-4669	27	9	the	the	DET
ejpam-4669	27	10	real	real	ADJ
ejpam-4669	27	11	numbers	number	NOUN
ejpam-4669	27	12	r	r	NOUN
ejpam-4669	27	13	is	be	AUX
ejpam-4669	27	14	said	say	VERB
ejpam-4669	27	15	to	to	PART
ejpam-4669	27	16	be	be	AUX
ejpam-4669	27	17	a	a	DET
ejpam-4669	27	18	gaussian	gaussian	ADJ
ejpam-4669	27	19	fibonacci	fibonacci	NOUN
ejpam-4669	27	20	function	function	NOUN
ejpam-4669	27	21	if	if	SCONJ
ejpam-4669	27	22	it	it	PRON
ejpam-4669	27	23	satisfies	satisfy	VERB
ejpam-4669	27	24	the	the	DET
ejpam-4669	27	25	formula	formula	NOUN
ejpam-4669	27	26	fg(x+	fg(x+	ADV
ejpam-4669	27	27	2	2	NUM
ejpam-4669	27	28	)	)	PUNCT
ejpam-4669	27	29	=	=	PUNCT
ejpam-4669	27	30	f(x+	f(x+	NUM
ejpam-4669	27	31	2	2	NUM
ejpam-4669	27	32	)	)	PUNCT
ejpam-4669	27	33	+	+	CCONJ
ejpam-4669	27	34	f(x+	f(x+	ADV
ejpam-4669	27	35	1)i	1)i	NOUN
ejpam-4669	27	36	,	,	PUNCT
ejpam-4669	27	37	where	where	SCONJ
ejpam-4669	27	38	f	f	PROPN
ejpam-4669	27	39	is	be	AUX
ejpam-4669	27	40	a	a	DET
ejpam-4669	27	41	fibonacci	fibonacci	NOUN
ejpam-4669	27	42	function	function	NOUN
ejpam-4669	27	43	and	and	CCONJ
ejpam-4669	27	44	for	for	ADP
ejpam-4669	27	45	any	any	DET
ejpam-4669	27	46	x	x	SYM
ejpam-4669	27	47	∈	∈	PROPN
ejpam-4669	27	48	r.	r.	PROPN
ejpam-4669	27	49	remark	remark	NOUN
ejpam-4669	27	50	1	1	NUM
ejpam-4669	27	51	.	.	PUNCT
ejpam-4669	28	1	[	[	X
ejpam-4669	28	2	4	4	X
ejpam-4669	28	3	]	]	PUNCT
ejpam-4669	28	4	for	for	ADP
ejpam-4669	28	5	all	all	DET
ejpam-4669	28	6	n	n	PRON
ejpam-4669	28	7	≥	≥	NOUN
ejpam-4669	28	8	0	0	NUM
ejpam-4669	28	9	.	.	PUNCT
ejpam-4669	29	1	the	the	DET
ejpam-4669	29	2	full	full	ADJ
ejpam-4669	29	3	gaussian	gaussian	PROPN
ejpam-4669	29	4	fibonacci	fibonacci	NOUN
ejpam-4669	29	5	sequence	sequence	NOUN
ejpam-4669	29	6	,	,	PUNCT
ejpam-4669	29	7	where	where	SCONJ
ejpam-4669	29	8	gu0	gu0	NOUN
ejpam-4669	29	9	=	=	PUNCT
ejpam-4669	29	10	i	i	PRON
ejpam-4669	29	11	and	and	CCONJ
ejpam-4669	29	12	gu1	gu1	VERB
ejpam-4669	29	13	=	=	SYM
ejpam-4669	29	14	1	1	NUM
ejpam-4669	29	15	,	,	PUNCT
ejpam-4669	29	16	are	be	AUX
ejpam-4669	29	17	formed	form	VERB
ejpam-4669	29	18	by	by	ADP
ejpam-4669	29	19	the	the	DET
ejpam-4669	29	20	following	follow	VERB
ejpam-4669	29	21	formula	formula	NOUN
ejpam-4669	29	22	gu−n	gu−n	PROPN
ejpam-4669	29	23	=	=	SYM
ejpam-4669	29	24	(	(	PUNCT
ejpam-4669	29	25	−1)n	−1)n	PROPN
ejpam-4669	29	26	∗	∗	X
ejpam-4669	29	27	i	i	PRON
ejpam-4669	29	28	∗gn+1	∗gn+1	PROPN
ejpam-4669	29	29	.	.	PUNCT
ejpam-4669	30	1	then	then	ADV
ejpam-4669	30	2	the	the	DET
ejpam-4669	30	3	full	full	ADJ
ejpam-4669	30	4	gaussian	gaussian	ADJ
ejpam-4669	30	5	fibonacci	fibonacci	NOUN
ejpam-4669	30	6	sequence	sequence	NOUN
ejpam-4669	30	7	,	,	PUNCT
ejpam-4669	30	8	where	where	SCONJ
ejpam-4669	30	9	gun	gun	NOUN
ejpam-4669	30	10	=	=	PUNCT
ejpam-4669	30	11	gfn	gfn	VERB
ejpam-4669	30	12	the	the	DET
ejpam-4669	30	13	nth	nth	PROPN
ejpam-4669	30	14	gaussian	gaussian	PROPN
ejpam-4669	30	15	fibonacci	fibonacci	NOUN
ejpam-4669	30	16	numbers	number	NOUN
ejpam-4669	30	17	,	,	PUNCT
ejpam-4669	30	18	are	be	AUX
ejpam-4669	30	19	:	:	PUNCT
ejpam-4669	30	20	.	.	PUNCT
ejpam-4669	30	21	.	.	PUNCT
ejpam-4669	31	1	.	.	PUNCT
ejpam-4669	32	1	,	,	PUNCT
ejpam-4669	32	2	−3	−3	PROPN
ejpam-4669	33	1	+	+	NUM
ejpam-4669	33	2	5i	5i	NUM
ejpam-4669	33	3	,	,	PUNCT
ejpam-4669	33	4	2−	2−	NUM
ejpam-4669	33	5	3i,−1	3i,−1	NOUN
ejpam-4669	33	6	+	+	NUM
ejpam-4669	33	7	2i	2i	NUM
ejpam-4669	33	8	,	,	PUNCT
ejpam-4669	33	9	1−	1−	NUM
ejpam-4669	34	1	i	i	PRON
ejpam-4669	34	2	,	,	PUNCT
ejpam-4669	34	3	i	i	PRON
ejpam-4669	34	4	,	,	PUNCT
ejpam-4669	34	5	1	1	NUM
ejpam-4669	34	6	,	,	PUNCT
ejpam-4669	34	7	1	1	NUM
ejpam-4669	34	8	+	+	CCONJ
ejpam-4669	34	9	i	i	PRON
ejpam-4669	34	10	,	,	PUNCT
ejpam-4669	34	11	2	2	NUM
ejpam-4669	34	12	+	+	CCONJ
ejpam-4669	34	13	i	i	PRON
ejpam-4669	34	14	,	,	PUNCT
ejpam-4669	34	15	3	3	NUM
ejpam-4669	34	16	+	+	NUM
ejpam-4669	34	17	2i	2i	NUM
ejpam-4669	34	18	,	,	PUNCT
ejpam-4669	34	19	5	5	NUM
ejpam-4669	34	20	+	+	NUM
ejpam-4669	34	21	3i	3i	NOUN
ejpam-4669	34	22	,	,	PUNCT
ejpam-4669	34	23	.	.	PUNCT
ejpam-4669	34	24	.	.	PUNCT
ejpam-4669	34	25	.	.	PUNCT
ejpam-4669	34	26	.	.	PUNCT
ejpam-4669	35	1	example	example	NOUN
ejpam-4669	36	1	1	1	NUM
ejpam-4669	36	2	.	.	PUNCT
ejpam-4669	37	1	[	[	X
ejpam-4669	37	2	4	4	X
ejpam-4669	37	3	]	]	X
ejpam-4669	37	4	let	let	VERB
ejpam-4669	37	5	{	{	PUNCT
ejpam-4669	37	6	gun}∞n=−∞	gun}∞n=−∞	ADJ
ejpam-4669	37	7	and	and	CCONJ
ejpam-4669	37	8	{	{	PUNCT
ejpam-4669	37	9	gvn}∞n=−∞	gvn}∞n=−∞	NOUN
ejpam-4669	37	10	be	be	AUX
ejpam-4669	37	11	full	full	ADJ
ejpam-4669	37	12	gaussian	gaussian	ADJ
ejpam-4669	37	13	fibonacci	fibonacci	NOUN
ejpam-4669	37	14	sequences	sequence	NOUN
ejpam-4669	37	15	.	.	PUNCT
ejpam-4669	38	1	we	we	PRON
ejpam-4669	38	2	define	define	VERB
ejpam-4669	38	3	a	a	DET
ejpam-4669	38	4	function	function	NOUN
ejpam-4669	38	5	fg	fg	ADV
ejpam-4669	38	6	by	by	ADP
ejpam-4669	38	7	fg(x	fg(x	NUM
ejpam-4669	38	8	)	)	PUNCT
ejpam-4669	38	9	:	:	PUNCT
ejpam-4669	38	10	=	=	NOUN
ejpam-4669	38	11	gu⌊x⌋	gu⌊x⌋	X
ejpam-4669	38	12	+	+	CCONJ
ejpam-4669	38	13	gv⌊x⌋t	gv⌊x⌋t	NOUN
ejpam-4669	38	14	and	and	CCONJ
ejpam-4669	38	15	f(x	f(x	PROPN
ejpam-4669	38	16	)	)	PUNCT
ejpam-4669	38	17	:	:	PUNCT
ejpam-4669	39	1	=	=	SYM
ejpam-4669	39	2	u⌊x⌋	u⌊x⌋	X
ejpam-4669	39	3	+	+	CCONJ
ejpam-4669	39	4	v⌊x⌋t	v⌊x⌋t	PROPN
ejpam-4669	39	5	,	,	PUNCT
ejpam-4669	39	6	where	where	SCONJ
ejpam-4669	39	7	t	t	PROPN
ejpam-4669	39	8	=	=	SYM
ejpam-4669	39	9	x−	x−	PROPN
ejpam-4669	39	10	⌊x⌋	⌊x⌋	X
ejpam-4669	39	11	∈	∈	PROPN
ejpam-4669	39	12	(	(	PUNCT
ejpam-4669	39	13	0	0	NUM
ejpam-4669	39	14	,	,	PUNCT
ejpam-4669	39	15	1	1	NUM
ejpam-4669	39	16	)	)	PUNCT
ejpam-4669	39	17	and	and	CCONJ
ejpam-4669	39	18	x	x	SYM
ejpam-4669	39	19	∈	∈	PROPN
ejpam-4669	39	20	r.	r.	NOUN
ejpam-4669	39	21	then	then	ADV
ejpam-4669	39	22	fg(x+	fg(x+	ADV
ejpam-4669	39	23	2	2	NUM
ejpam-4669	39	24	)	)	PUNCT
ejpam-4669	39	25	=	=	PUNCT
ejpam-4669	40	1	gu⌊x+2⌋	gu⌊x+2⌋	PROPN
ejpam-4669	40	2	+	+	PROPN
ejpam-4669	40	3	gv⌊x+2⌋t	gv⌊x+2⌋t	NOUN
ejpam-4669	40	4	=	=	X
ejpam-4669	40	5	gu⌊x⌋+2	gu⌊x⌋+2	PROPN
ejpam-4669	41	1	+	+	ADP
ejpam-4669	41	2	gv⌊x⌋+2	gv⌊x⌋+2	X
ejpam-4669	41	3	t	t	X
ejpam-4669	41	4	by	by	ADP
ejpam-4669	41	5	the	the	DET
ejpam-4669	41	6	fact	fact	NOUN
ejpam-4669	41	7	that	that	SCONJ
ejpam-4669	41	8	gu⌊x⌋+2	gu⌊x⌋+2	PROPN
ejpam-4669	41	9	=	=	PUNCT
ejpam-4669	42	1	u⌊x⌋+2	u⌊x⌋+2	ADP
ejpam-4669	42	2	+	+	NOUN
ejpam-4669	42	3	iu⌊x⌋+1	iu⌊x⌋+1	NOUN
ejpam-4669	42	4	and	and	CCONJ
ejpam-4669	42	5	gv⌊x⌋+2	gv⌊x⌋+2	VERB
ejpam-4669	42	6	=	=	PUNCT
ejpam-4669	43	1	v⌊x⌋+2	v⌊x⌋+2	PROPN
ejpam-4669	43	2	+	+	CCONJ
ejpam-4669	43	3	iv⌊x⌋+1	iv⌊x⌋+1	NOUN
ejpam-4669	43	4	,	,	PUNCT
ejpam-4669	43	5	we	we	PRON
ejpam-4669	43	6	obtain	obtain	VERB
ejpam-4669	43	7	that	that	SCONJ
ejpam-4669	43	8	fg(x+	fg(x+	ADV
ejpam-4669	43	9	2	2	NUM
ejpam-4669	43	10	)	)	PUNCT
ejpam-4669	43	11	=	=	NOUN
ejpam-4669	44	1	(	(	PUNCT
ejpam-4669	44	2	u⌊x⌋+2	u⌊x⌋+2	X
ejpam-4669	44	3	+	+	NUM
ejpam-4669	44	4	iu⌊x⌋+1	iu⌊x⌋+1	NOUN
ejpam-4669	44	5	)	)	PUNCT
ejpam-4669	45	1	+	+	CCONJ
ejpam-4669	45	2	(	(	PUNCT
ejpam-4669	45	3	v⌊x⌋+2	v⌊x⌋+2	X
ejpam-4669	45	4	+	+	NUM
ejpam-4669	45	5	iv⌊x⌋+1)t	iv⌊x⌋+1)t	NOUN
ejpam-4669	45	6	=	=	SYM
ejpam-4669	45	7	(	(	PUNCT
ejpam-4669	45	8	u⌊x+2⌋	u⌊x+2⌋	X
ejpam-4669	45	9	+	+	NUM
ejpam-4669	45	10	v⌊x+2⌋t	v⌊x+2⌋t	NOUN
ejpam-4669	45	11	)	)	PUNCT
ejpam-4669	45	12	+	+	CCONJ
ejpam-4669	45	13	i(u⌊x+1⌋	i(u⌊x+1⌋	NOUN
ejpam-4669	45	14	+	+	CCONJ
ejpam-4669	45	15	v⌊x+1⌋t	v⌊x+1⌋t	NOUN
ejpam-4669	45	16	)	)	PUNCT
ejpam-4669	45	17	=	=	PUNCT
ejpam-4669	45	18	f(x+	f(x+	NUM
ejpam-4669	45	19	2	2	NUM
ejpam-4669	45	20	)	)	PUNCT
ejpam-4669	45	21	+	+	CCONJ
ejpam-4669	45	22	f(x+	f(x+	ADV
ejpam-4669	45	23	1)i	1)i	NOUN
ejpam-4669	45	24	.	.	PUNCT
ejpam-4669	46	1	therefore	therefore	ADV
ejpam-4669	46	2	,	,	PUNCT
ejpam-4669	46	3	fg	fg	PROPN
ejpam-4669	46	4	is	be	AUX
ejpam-4669	46	5	a	a	DET
ejpam-4669	46	6	gaussian	gaussian	ADJ
ejpam-4669	46	7	fibonacci	fibonacci	NOUN
ejpam-4669	46	8	function	function	NOUN
ejpam-4669	46	9	.	.	PUNCT
ejpam-4669	47	1	example	example	NOUN
ejpam-4669	48	1	2	2	NUM
ejpam-4669	48	2	.	.	PUNCT
ejpam-4669	48	3	let	let	VERB
ejpam-4669	48	4	ϕ(t	ϕ(t	NUM
ejpam-4669	48	5	)	)	PUNCT
ejpam-4669	48	6	,	,	PUNCT
ejpam-4669	48	7	ψ(t	ψ(t	PROPN
ejpam-4669	48	8	)	)	PUNCT
ejpam-4669	48	9	be	be	VERB
ejpam-4669	48	10	any	any	DET
ejpam-4669	48	11	real	real	ADV
ejpam-4669	48	12	valued	value	VERB
ejpam-4669	48	13	functions	function	NOUN
ejpam-4669	48	14	which	which	PRON
ejpam-4669	48	15	are	be	AUX
ejpam-4669	48	16	defined	define	VERB
ejpam-4669	48	17	on	on	ADP
ejpam-4669	48	18	[	[	X
ejpam-4669	48	19	0	0	NUM
ejpam-4669	48	20	,	,	PUNCT
ejpam-4669	48	21	1	1	NUM
ejpam-4669	48	22	)	)	PUNCT
ejpam-4669	48	23	and	and	CCONJ
ejpam-4669	48	24	let	let	VERB
ejpam-4669	48	25	{	{	PUNCT
ejpam-4669	48	26	gu−n	gu−n	ADV
ejpam-4669	48	27	}	}	PUNCT
ejpam-4669	48	28	and	and	CCONJ
ejpam-4669	48	29	{	{	PUNCT
ejpam-4669	48	30	gv−n	gv−n	PROPN
ejpam-4669	48	31	}	}	PUNCT
ejpam-4669	48	32	be	be	AUX
ejpam-4669	48	33	gaussian	gaussian	ADJ
ejpam-4669	48	34	fibonacci	fibonacci	NOUN
ejpam-4669	48	35	sequences	sequence	NOUN
ejpam-4669	48	36	.	.	PUNCT
ejpam-4669	49	1	define	define	VERB
ejpam-4669	49	2	a	a	DET
ejpam-4669	49	3	map	map	NOUN
ejpam-4669	49	4	fg(x	fg(x	NUM
ejpam-4669	49	5	)	)	PUNCT
ejpam-4669	49	6	:	:	PUNCT
ejpam-4669	50	1	=	=	PUNCT
ejpam-4669	50	2	gu⌊x⌋ϕ(t)+	gu⌊x⌋ϕ(t)+	VERB
ejpam-4669	50	3	gv⌊x⌋ψ(t	gv⌊x⌋ψ(t	PROPN
ejpam-4669	50	4	)	)	PUNCT
ejpam-4669	50	5	,	,	PUNCT
ejpam-4669	50	6	and	and	CCONJ
ejpam-4669	50	7	f(x	f(x	PROPN
ejpam-4669	50	8	)	)	PUNCT
ejpam-4669	50	9	=	=	SYM
ejpam-4669	50	10	u⌊x⌋ϕ(t	u⌊x⌋ϕ(t	X
ejpam-4669	50	11	)	)	PUNCT
ejpam-4669	50	12	+	+	CCONJ
ejpam-4669	50	13	v⌊x⌋ψ(t	v⌊x⌋ψ(t	VERB
ejpam-4669	50	14	)	)	PUNCT
ejpam-4669	50	15	where	where	SCONJ
ejpam-4669	50	16	t	t	NOUN
ejpam-4669	50	17	=	=	SYM
ejpam-4669	50	18	x−	x−	PROPN
ejpam-4669	50	19	⌊x⌋	⌊x⌋	PUNCT
ejpam-4669	50	20	∈	∈	PROPN
ejpam-4669	51	1	[	[	X
ejpam-4669	51	2	0	0	NUM
ejpam-4669	51	3	,	,	PUNCT
ejpam-4669	51	4	1	1	NUM
ejpam-4669	51	5	)	)	PUNCT
ejpam-4669	51	6	.	.	PUNCT
ejpam-4669	52	1	then	then	ADV
ejpam-4669	52	2	fg(x+	fg(x+	ADV
ejpam-4669	52	3	2	2	NUM
ejpam-4669	52	4	)	)	PUNCT
ejpam-4669	52	5	:	:	PUNCT
ejpam-4669	52	6	=	=	SYM
ejpam-4669	52	7	gu⌊x+2⌋ϕ(t	gu⌊x+2⌋ϕ(t	PROPN
ejpam-4669	52	8	)	)	PUNCT
ejpam-4669	53	1	+	+	PROPN
ejpam-4669	53	2	gv⌊x+2⌋ψ(t	gv⌊x+2⌋ψ(t	PROPN
ejpam-4669	53	3	)	)	PUNCT
ejpam-4669	53	4	.	.	PUNCT
ejpam-4669	54	1	since	since	SCONJ
ejpam-4669	54	2	⌊x+	⌊x+	PUNCT
ejpam-4669	55	1	2⌋	2⌋	X
ejpam-4669	55	2	=	=	SYM
ejpam-4669	55	3	⌊x⌋+2	⌊x⌋+2	PROPN
ejpam-4669	55	4	and	and	CCONJ
ejpam-4669	55	5	hence	hence	ADV
ejpam-4669	55	6	x+2−⌊x+	x+2−⌊x+	PUNCT
ejpam-4669	55	7	2⌋	2⌋	NOUN
ejpam-4669	56	1	=	=	SYM
ejpam-4669	56	2	x−⌊x⌋	x−⌊x⌋	PROPN
ejpam-4669	56	3	,	,	PUNCT
ejpam-4669	56	4	we	we	PRON
ejpam-4669	56	5	obtain	obtain	VERB
ejpam-4669	56	6	fg(x+	fg(x+	ADV
ejpam-4669	56	7	2	2	NUM
ejpam-4669	56	8	)	)	PUNCT
ejpam-4669	56	9	=	=	SYM
ejpam-4669	56	10	gu⌊x⌋+2ϕ(t	gu⌊x⌋+2ϕ(t	PROPN
ejpam-4669	56	11	)	)	PUNCT
ejpam-4669	57	1	+	+	NOUN
ejpam-4669	57	2	gv⌊x⌋+2ψ(t	gv⌊x⌋+2ψ(t	X
ejpam-4669	57	3	)	)	PUNCT
ejpam-4669	57	4	=	=	SYM
ejpam-4669	58	1	(	(	PUNCT
ejpam-4669	58	2	u⌊x⌋+2	u⌊x⌋+2	X
ejpam-4669	58	3	+	+	ADJ
ejpam-4669	58	4	iu⌊x⌋+1)ϕ(t	iu⌊x⌋+1)ϕ(t	NOUN
ejpam-4669	58	5	)	)	PUNCT
ejpam-4669	58	6	+	+	CCONJ
ejpam-4669	58	7	(	(	PUNCT
ejpam-4669	58	8	v⌊x⌋+2	v⌊x⌋+2	PROPN
ejpam-4669	58	9	+	+	NUM
ejpam-4669	58	10	iv⌊x⌋+1)ψ(t	iv⌊x⌋+1)ψ(t	PROPN
ejpam-4669	58	11	)	)	PUNCT
ejpam-4669	58	12	=	=	PUNCT
ejpam-4669	58	13	(	(	PUNCT
ejpam-4669	58	14	u⌊x⌋+2ϕ(t	u⌊x⌋+2ϕ(t	NOUN
ejpam-4669	58	15	)	)	PUNCT
ejpam-4669	58	16	+	+	X
ejpam-4669	58	17	v⌊x⌋+2ψ(t	v⌊x⌋+2ψ(t	X
ejpam-4669	58	18	)	)	PUNCT
ejpam-4669	58	19	)	)	PUNCT
ejpam-4669	59	1	+	+	CCONJ
ejpam-4669	59	2	i(u⌊x⌋+1ϕ(t	i(u⌊x⌋+1ϕ(t	NOUN
ejpam-4669	59	3	)	)	PUNCT
ejpam-4669	59	4	+	+	NUM
ejpam-4669	59	5	v⌊x⌋+1ψ(t	v⌊x⌋+1ψ(t	NOUN
ejpam-4669	59	6	)	)	PUNCT
ejpam-4669	59	7	)	)	PUNCT
ejpam-4669	60	1	=	=	PUNCT
ejpam-4669	60	2	f(x+	f(x+	NUM
ejpam-4669	60	3	2	2	NUM
ejpam-4669	60	4	)	)	PUNCT
ejpam-4669	60	5	+	+	CCONJ
ejpam-4669	60	6	if(x+	if(x+	ADV
ejpam-4669	60	7	1	1	NUM
ejpam-4669	60	8	)	)	PUNCT
ejpam-4669	60	9	.	.	PUNCT
ejpam-4669	61	1	therefore	therefore	ADV
ejpam-4669	61	2	,	,	PUNCT
ejpam-4669	61	3	fg(x	fg(x	NUM
ejpam-4669	61	4	)	)	PUNCT
ejpam-4669	61	5	is	be	AUX
ejpam-4669	61	6	a	a	DET
ejpam-4669	61	7	gaussian	gaussian	ADJ
ejpam-4669	61	8	fibonacci	fibonacci	NOUN
ejpam-4669	61	9	function	function	NOUN
ejpam-4669	61	10	.	.	PUNCT
ejpam-4669	62	1	by	by	ADP
ejpam-4669	62	2	using	use	VERB
ejpam-4669	62	3	the	the	DET
ejpam-4669	62	4	concept	concept	NOUN
ejpam-4669	62	5	of	of	ADP
ejpam-4669	62	6	an	an	DET
ejpam-4669	62	7	fg	fg	NOUN
ejpam-4669	62	8	-	-	ADJ
ejpam-4669	62	9	even	even	ADV
ejpam-4669	62	10	and	and	CCONJ
ejpam-4669	62	11	fg	fg	PROPN
ejpam-4669	62	12	-	-	PUNCT
ejpam-4669	62	13	odd	odd	ADJ
ejpam-4669	62	14	functions	function	NOUN
ejpam-4669	62	15	,	,	PUNCT
ejpam-4669	62	16	we	we	PRON
ejpam-4669	62	17	attain	attain	VERB
ejpam-4669	62	18	some	some	DET
ejpam-4669	62	19	gaussian	gaussian	ADJ
ejpam-4669	62	20	fibonacci	fibonacci	NOUN
ejpam-4669	62	21	functions	function	NOUN
ejpam-4669	62	22	which	which	PRON
ejpam-4669	62	23	are	be	AUX
ejpam-4669	62	24	discussed	discuss	VERB
ejpam-4669	62	25	in	in	ADP
ejpam-4669	62	26	[	[	X
ejpam-4669	62	27	1	1	NUM
ejpam-4669	62	28	]	]	PUNCT
ejpam-4669	62	29	h.	h.	PROPN
ejpam-4669	62	30	f.	f.	PROPN
ejpam-4669	62	31	m.salih	m.salih	PROPN
ejpam-4669	62	32	/	/	SYM
ejpam-4669	62	33	eur	eur	PROPN
ejpam-4669	62	34	.	.	PUNCT
ejpam-4669	63	1	j.	j.	PROPN
ejpam-4669	63	2	pure	pure	PROPN
ejpam-4669	63	3	appl	appl	PROPN
ejpam-4669	63	4	.	.	PROPN
ejpam-4669	63	5	math	math	PROPN
ejpam-4669	63	6	,	,	PUNCT
ejpam-4669	63	7	16	16	NUM
ejpam-4669	63	8	(	(	PUNCT
ejpam-4669	63	9	1	1	NUM
ejpam-4669	63	10	)	)	PUNCT
ejpam-4669	63	11	(	(	PUNCT
ejpam-4669	63	12	2023	2023	NUM
ejpam-4669	63	13	)	)	PUNCT
ejpam-4669	63	14	,	,	PUNCT
ejpam-4669	63	15	587	587	NUM
ejpam-4669	63	16	-	-	SYM
ejpam-4669	63	17	594	594	NUM
ejpam-4669	63	18	589	589	NUM
ejpam-4669	63	19	definition	definition	NOUN
ejpam-4669	63	20	2	2	NUM
ejpam-4669	63	21	.	.	PUNCT
ejpam-4669	64	1	[	[	X
ejpam-4669	64	2	4	4	X
ejpam-4669	64	3	]	]	X
ejpam-4669	64	4	let	let	AUX
ejpam-4669	64	5	c(x	c(x	NOUN
ejpam-4669	64	6	)	)	PUNCT
ejpam-4669	64	7	be	be	AUX
ejpam-4669	64	8	real	real	ADV
ejpam-4669	64	9	-	-	PUNCT
ejpam-4669	64	10	valued	value	VERB
ejpam-4669	64	11	function	function	NOUN
ejpam-4669	64	12	of	of	ADP
ejpam-4669	64	13	a	a	DET
ejpam-4669	64	14	real	real	ADJ
ejpam-4669	64	15	variable	variable	NOUN
ejpam-4669	64	16	such	such	ADJ
ejpam-4669	64	17	that	that	DET
ejpam-4669	64	18	c(x)h(x	c(x)h(x	NOUN
ejpam-4669	64	19	)	)	PUNCT
ejpam-4669	64	20	≡	≡	PROPN
ejpam-4669	64	21	0	0	NUM
ejpam-4669	64	22	and	and	CCONJ
ejpam-4669	64	23	h(x	h(x	PROPN
ejpam-4669	64	24	)	)	PUNCT
ejpam-4669	64	25	is	be	AUX
ejpam-4669	64	26	continuous	continuous	ADJ
ejpam-4669	64	27	,	,	PUNCT
ejpam-4669	64	28	then	then	ADV
ejpam-4669	64	29	h(x	h(x	PROPN
ejpam-4669	64	30	)	)	PUNCT
ejpam-4669	65	1	=	=	PUNCT
ejpam-4669	66	1	0	0	X
ejpam-4669	66	2	.	.	PUNCT
ejpam-4669	67	1	the	the	DET
ejpam-4669	67	2	function	function	NOUN
ejpam-4669	67	3	c(x	c(x	NOUN
ejpam-4669	67	4	)	)	PUNCT
ejpam-4669	67	5	is	be	AUX
ejpam-4669	67	6	said	say	VERB
ejpam-4669	67	7	to	to	PART
ejpam-4669	67	8	be	be	AUX
ejpam-4669	67	9	fg	fg	VERB
ejpam-4669	67	10	-	-	ADJ
ejpam-4669	67	11	even	even	ADV
ejpam-4669	67	12	function	function	NOUN
ejpam-4669	67	13	(	(	PUNCT
ejpam-4669	67	14	resp	resp	NOUN
ejpam-4669	67	15	.	.	PUNCT
ejpam-4669	67	16	,	,	PUNCT
ejpam-4669	67	17	fg	fg	PROPN
ejpam-4669	67	18	-	-	PUNCT
ejpam-4669	67	19	odd	odd	ADJ
ejpam-4669	67	20	function	function	NOUN
ejpam-4669	67	21	)	)	PUNCT
ejpam-4669	67	22	if	if	SCONJ
ejpam-4669	67	23	c(x+	c(x+	PROPN
ejpam-4669	67	24	1	1	NUM
ejpam-4669	67	25	)	)	PUNCT
ejpam-4669	67	26	=	=	SYM
ejpam-4669	67	27	c(x	c(x	NOUN
ejpam-4669	67	28	)	)	PUNCT
ejpam-4669	67	29	(	(	PUNCT
ejpam-4669	67	30	resp	resp	NOUN
ejpam-4669	67	31	.	.	PUNCT
ejpam-4669	67	32	,	,	PUNCT
ejpam-4669	67	33	c(x+	c(x+	VERB
ejpam-4669	67	34	1	1	NUM
ejpam-4669	67	35	)	)	PUNCT
ejpam-4669	67	36	=	=	SYM
ejpam-4669	67	37	−c(x	−c(x	NOUN
ejpam-4669	67	38	)	)	PUNCT
ejpam-4669	67	39	)	)	PUNCT
ejpam-4669	67	40	for	for	ADP
ejpam-4669	67	41	any	any	DET
ejpam-4669	67	42	x	x	SYM
ejpam-4669	67	43	∈	∈	PROPN
ejpam-4669	67	44	r.	r.	PROPN
ejpam-4669	67	45	theorem	theorem	NOUN
ejpam-4669	67	46	1	1	NUM
ejpam-4669	67	47	.	.	PUNCT
ejpam-4669	68	1	[	[	X
ejpam-4669	68	2	4	4	X
ejpam-4669	68	3	]	]	PUNCT
ejpam-4669	68	4	let	let	NOUN
ejpam-4669	68	5	fg(x	fg(x	PUNCT
ejpam-4669	68	6	)	)	PUNCT
ejpam-4669	68	7	=	=	PUNCT
ejpam-4669	69	1	c(x)gg(x	c(x)gg(x	X
ejpam-4669	69	2	)	)	PUNCT
ejpam-4669	69	3	be	be	VERB
ejpam-4669	69	4	a	a	DET
ejpam-4669	69	5	function	function	NOUN
ejpam-4669	69	6	and	and	CCONJ
ejpam-4669	69	7	f(x	f(x	NOUN
ejpam-4669	69	8	)	)	PUNCT
ejpam-4669	70	1	=	=	PUNCT
ejpam-4669	70	2	c(x)g(x	c(x)g(x	ADJ
ejpam-4669	70	3	)	)	PUNCT
ejpam-4669	70	4	be	be	VERB
ejpam-4669	70	5	a	a	DET
ejpam-4669	70	6	fibonacci	fibonacci	NOUN
ejpam-4669	70	7	function	function	NOUN
ejpam-4669	70	8	,	,	PUNCT
ejpam-4669	70	9	where	where	SCONJ
ejpam-4669	70	10	c(x	c(x	NOUN
ejpam-4669	70	11	)	)	PUNCT
ejpam-4669	70	12	is	be	AUX
ejpam-4669	70	13	an	an	DET
ejpam-4669	70	14	fg	fg	VERB
ejpam-4669	70	15	-	-	ADJ
ejpam-4669	70	16	even	even	ADV
ejpam-4669	70	17	function	function	NOUN
ejpam-4669	70	18	and	and	CCONJ
ejpam-4669	70	19	gg(x	gg(x	PUNCT
ejpam-4669	70	20	)	)	PUNCT
ejpam-4669	70	21	and	and	CCONJ
ejpam-4669	70	22	g(x	g(x	NOUN
ejpam-4669	70	23	)	)	PUNCT
ejpam-4669	70	24	are	be	AUX
ejpam-4669	70	25	continuous	continuous	ADJ
ejpam-4669	70	26	functions	function	NOUN
ejpam-4669	70	27	.	.	PUNCT
ejpam-4669	71	1	then	then	ADV
ejpam-4669	71	2	fg(x	fg(x	VERB
ejpam-4669	71	3	)	)	PUNCT
ejpam-4669	71	4	is	be	AUX
ejpam-4669	71	5	a	a	DET
ejpam-4669	71	6	gaussian	gaussian	ADJ
ejpam-4669	71	7	fibonacci	fibonacci	NOUN
ejpam-4669	71	8	function	function	VERB
ejpam-4669	71	9	if	if	SCONJ
ejpam-4669	71	10	and	and	CCONJ
ejpam-4669	71	11	only	only	ADV
ejpam-4669	71	12	if	if	SCONJ
ejpam-4669	71	13	gg(x	gg(x	NOUN
ejpam-4669	71	14	)	)	PUNCT
ejpam-4669	71	15	is	be	AUX
ejpam-4669	71	16	a	a	DET
ejpam-4669	71	17	gaussian	gaussian	ADJ
ejpam-4669	71	18	fibonacci	fibonacci	NOUN
ejpam-4669	71	19	function	function	NOUN
ejpam-4669	71	20	.	.	PUNCT
ejpam-4669	72	1	note	note	VERB
ejpam-4669	72	2	that	that	SCONJ
ejpam-4669	72	3	if	if	SCONJ
ejpam-4669	72	4	a	a	DET
ejpam-4669	72	5	gaussian	gaussian	ADJ
ejpam-4669	72	6	fibonacci	fibonacci	NOUN
ejpam-4669	72	7	function	function	NOUN
ejpam-4669	72	8	is	be	AUX
ejpam-4669	72	9	differentiable	differentiable	ADJ
ejpam-4669	72	10	on	on	ADP
ejpam-4669	72	11	r	r	NOUN
ejpam-4669	72	12	,	,	PUNCT
ejpam-4669	72	13	then	then	ADV
ejpam-4669	72	14	its	its	PRON
ejpam-4669	72	15	derivative	derivative	NOUN
ejpam-4669	72	16	is	be	AUX
ejpam-4669	72	17	also	also	ADV
ejpam-4669	72	18	a	a	DET
ejpam-4669	72	19	gaussian	gaussian	ADJ
ejpam-4669	72	20	fibonacci	fibonacci	NOUN
ejpam-4669	72	21	function	function	NOUN
ejpam-4669	72	22	.	.	PUNCT
ejpam-4669	73	1	proposition	proposition	NOUN
ejpam-4669	73	2	1	1	NUM
ejpam-4669	73	3	.	.	PUNCT
ejpam-4669	74	1	let	let	VERB
ejpam-4669	74	2	fg	fg	PRON
ejpam-4669	74	3	be	be	AUX
ejpam-4669	74	4	a	a	DET
ejpam-4669	74	5	gaussian	gaussian	ADJ
ejpam-4669	74	6	fibonacci	fibonacci	NOUN
ejpam-4669	74	7	function	function	NOUN
ejpam-4669	74	8	.	.	PUNCT
ejpam-4669	75	1	if	if	SCONJ
ejpam-4669	75	2	we	we	PRON
ejpam-4669	75	3	define	define	VERB
ejpam-4669	75	4	gg(x	gg(x	PUNCT
ejpam-4669	75	5	)	)	PUNCT
ejpam-4669	75	6	:	:	PUNCT
ejpam-4669	75	7	=	=	SYM
ejpam-4669	75	8	fg(x+	fg(x+	NOUN
ejpam-4669	75	9	t	t	PROPN
ejpam-4669	75	10	)	)	PUNCT
ejpam-4669	75	11	and	and	CCONJ
ejpam-4669	75	12	g(x	g(x	NOUN
ejpam-4669	75	13	)	)	PUNCT
ejpam-4669	75	14	:	:	PUNCT
ejpam-4669	76	1	=	=	PUNCT
ejpam-4669	76	2	f(x	f(x	PROPN
ejpam-4669	76	3	+	+	CCONJ
ejpam-4669	76	4	t	t	PROPN
ejpam-4669	76	5	)	)	PUNCT
ejpam-4669	76	6	where	where	SCONJ
ejpam-4669	76	7	t	t	PROPN
ejpam-4669	76	8	∈	∈	PROPN
ejpam-4669	76	9	r	r	PROPN
ejpam-4669	76	10	,	,	PUNCT
ejpam-4669	76	11	for	for	ADP
ejpam-4669	76	12	any	any	DET
ejpam-4669	76	13	x	x	SYM
ejpam-4669	76	14	∈	∈	PROPN
ejpam-4669	76	15	r.	r.	NOUN
ejpam-4669	76	16	if	if	SCONJ
ejpam-4669	76	17	g	g	PROPN
ejpam-4669	76	18	is	be	AUX
ejpam-4669	76	19	a	a	DET
ejpam-4669	76	20	gaussian	gaussian	ADJ
ejpam-4669	76	21	fibonacci	fibonacci	NOUN
ejpam-4669	76	22	function	function	NOUN
ejpam-4669	76	23	,	,	PUNCT
ejpam-4669	76	24	then	then	ADV
ejpam-4669	76	25	gg	gg	PROPN
ejpam-4669	76	26	is	be	AUX
ejpam-4669	76	27	also	also	ADV
ejpam-4669	76	28	a	a	DET
ejpam-4669	76	29	gaussian	gaussian	ADJ
ejpam-4669	76	30	fibonacci	fibonacci	NOUN
ejpam-4669	76	31	function	function	NOUN
ejpam-4669	76	32	.	.	PUNCT
ejpam-4669	77	1	theorem	theorem	NOUN
ejpam-4669	77	2	2	2	NUM
ejpam-4669	77	3	.	.	PUNCT
ejpam-4669	78	1	[	[	X
ejpam-4669	78	2	4	4	X
ejpam-4669	78	3	]	]	X
ejpam-4669	78	4	if	if	SCONJ
ejpam-4669	78	5	fg(x	fg(x	NUM
ejpam-4669	78	6	)	)	PUNCT
ejpam-4669	78	7	is	be	AUX
ejpam-4669	78	8	a	a	DET
ejpam-4669	78	9	gaussian	gaussian	ADJ
ejpam-4669	78	10	fibonacci	fibonacci	NOUN
ejpam-4669	78	11	function	function	NOUN
ejpam-4669	78	12	,	,	PUNCT
ejpam-4669	78	13	then	then	ADV
ejpam-4669	78	14	the	the	DET
ejpam-4669	78	15	limit	limit	NOUN
ejpam-4669	78	16	of	of	ADP
ejpam-4669	78	17	quotient	quotient	NOUN
ejpam-4669	78	18	fg(x+1	fg(x+1	ADJ
ejpam-4669	78	19	)	)	PUNCT
ejpam-4669	78	20	fg(x	fg(x	NUM
ejpam-4669	78	21	)	)	PUNCT
ejpam-4669	78	22	exists	exist	VERB
ejpam-4669	78	23	.	.	PUNCT
ejpam-4669	79	1	corollary	corollary	ADJ
ejpam-4669	79	2	1	1	NUM
ejpam-4669	79	3	.	.	PUNCT
ejpam-4669	80	1	[	[	X
ejpam-4669	80	2	4	4	X
ejpam-4669	80	3	]	]	X
ejpam-4669	80	4	if	if	SCONJ
ejpam-4669	80	5	fg(x	fg(x	NUM
ejpam-4669	80	6	)	)	PUNCT
ejpam-4669	80	7	is	be	AUX
ejpam-4669	80	8	a	a	DET
ejpam-4669	80	9	gaussian	gaussian	ADJ
ejpam-4669	80	10	fibonacci	fibonacci	NOUN
ejpam-4669	80	11	function	function	NOUN
ejpam-4669	80	12	,	,	PUNCT
ejpam-4669	80	13	then	then	ADV
ejpam-4669	80	14	lim	lim	PROPN
ejpam-4669	80	15	x→∞	x→∞	NUM
ejpam-4669	81	1	fg(x+	fg(x+	ADV
ejpam-4669	81	2	1	1	NUM
ejpam-4669	81	3	)	)	PUNCT
ejpam-4669	81	4	fg(x	fg(x	NUM
ejpam-4669	81	5	)	)	PUNCT
ejpam-4669	81	6	=	=	SYM
ejpam-4669	81	7	1	1	NUM
ejpam-4669	81	8	+	+	CCONJ
ejpam-4669	81	9	√	√	NUM
ejpam-4669	81	10	5	5	NUM
ejpam-4669	81	11	2	2	NUM
ejpam-4669	81	12	=	=	SYM
ejpam-4669	81	13	ϕ.	ϕ.	NOUN
ejpam-4669	81	14	3	3	X
ejpam-4669	81	15	.	.	PUNCT
ejpam-4669	82	1	gaussian	gaussian	ADJ
ejpam-4669	82	2	fibonacci	fibonacci	NOUN
ejpam-4669	82	3	functions	function	NOUN
ejpam-4669	82	4	with	with	ADP
ejpam-4669	82	5	periodicity	periodicity	NOUN
ejpam-4669	82	6	in	in	ADP
ejpam-4669	82	7	this	this	DET
ejpam-4669	82	8	section	section	NOUN
ejpam-4669	82	9	,	,	PUNCT
ejpam-4669	82	10	several	several	ADJ
ejpam-4669	82	11	results	result	NOUN
ejpam-4669	82	12	of	of	ADP
ejpam-4669	82	13	gaussian	gaussian	ADJ
ejpam-4669	82	14	fibonacci	fibonacci	NOUN
ejpam-4669	82	15	functions	function	NOUN
ejpam-4669	82	16	with	with	ADP
ejpam-4669	82	17	periodicity	periodicity	NOUN
ejpam-4669	82	18	is	be	AUX
ejpam-4669	82	19	obtained	obtain	VERB
ejpam-4669	82	20	.	.	PUNCT
ejpam-4669	83	1	theorem	theorem	NOUN
ejpam-4669	83	2	3	3	X
ejpam-4669	83	3	.	.	PUNCT
ejpam-4669	83	4	let	let	VERB
ejpam-4669	83	5	fg(x	fg(x	NUM
ejpam-4669	83	6	)	)	PUNCT
ejpam-4669	83	7	,	,	PUNCT
ejpam-4669	83	8	gg(x	gg(x	PUNCT
ejpam-4669	83	9	)	)	PUNCT
ejpam-4669	83	10	be	be	AUX
ejpam-4669	83	11	gaussian	gaussian	ADJ
ejpam-4669	83	12	fibonacci	fibonacci	NOUN
ejpam-4669	83	13	functions	function	NOUN
ejpam-4669	83	14	with	with	ADP
ejpam-4669	83	15	gg(x	gg(x	NOUN
ejpam-4669	83	16	)	)	PUNCT
ejpam-4669	83	17	=	=	PUNCT
ejpam-4669	83	18	ag(x)fg(x	ag(x)fg(x	X
ejpam-4669	83	19	)	)	PUNCT
ejpam-4669	83	20	.	.	PUNCT
ejpam-4669	84	1	if	if	SCONJ
ejpam-4669	84	2	ag(x+	ag(x+	ADJ
ejpam-4669	84	3	1	1	X
ejpam-4669	84	4	)	)	PUNCT
ejpam-4669	84	5	̸=	̸=	PROPN
ejpam-4669	84	6	ag(x	ag(x	ADV
ejpam-4669	84	7	)	)	PUNCT
ejpam-4669	84	8	for	for	ADP
ejpam-4669	84	9	all	all	DET
ejpam-4669	84	10	x	x	SYM
ejpam-4669	84	11	∈	∈	PROPN
ejpam-4669	84	12	r	r	NOUN
ejpam-4669	84	13	,	,	PUNCT
ejpam-4669	84	14	then	then	ADV
ejpam-4669	84	15	lim	lim	PROPN
ejpam-4669	84	16	x→∞	x→∞	X
ejpam-4669	84	17	ag(x+	ag(x+	PROPN
ejpam-4669	84	18	1	1	NUM
ejpam-4669	84	19	)	)	PUNCT
ejpam-4669	84	20	ag(x	ag(x	PUNCT
ejpam-4669	84	21	)	)	PUNCT
ejpam-4669	84	22	=	=	SYM
ejpam-4669	84	23	1	1	X
ejpam-4669	84	24	.	.	PUNCT
ejpam-4669	84	25	proof	proof	NOUN
ejpam-4669	84	26	.	.	PUNCT
ejpam-4669	85	1	since	since	SCONJ
ejpam-4669	85	2	ag(x+	ag(x+	ADV
ejpam-4669	85	3	1	1	NUM
ejpam-4669	85	4	)	)	PUNCT
ejpam-4669	85	5	̸=	̸=	PROPN
ejpam-4669	85	6	ag(x	ag(x	ADV
ejpam-4669	85	7	)	)	PUNCT
ejpam-4669	85	8	for	for	ADP
ejpam-4669	85	9	all	all	DET
ejpam-4669	85	10	x	x	SYM
ejpam-4669	85	11	∈	∈	PROPN
ejpam-4669	85	12	r	r	NOUN
ejpam-4669	85	13	,	,	PUNCT
ejpam-4669	85	14	we	we	PRON
ejpam-4669	85	15	have	have	VERB
ejpam-4669	85	16	ag(x+	ag(x+	ADV
ejpam-4669	85	17	1)[f(x+	1)[f(x+	NUM
ejpam-4669	85	18	1	1	NUM
ejpam-4669	85	19	)	)	PUNCT
ejpam-4669	85	20	+	+	NUM
ejpam-4669	85	21	if(x	if(x	NOUN
ejpam-4669	85	22	)	)	PUNCT
ejpam-4669	85	23	]	]	PUNCT
ejpam-4669	86	1	=	=	SYM
ejpam-4669	86	2	ag(x+	ag(x+	ADV
ejpam-4669	86	3	1)fg(x+	1)fg(x+	ADJ
ejpam-4669	86	4	1	1	X
ejpam-4669	86	5	)	)	PUNCT
ejpam-4669	86	6	=	=	PUNCT
ejpam-4669	86	7	gg(x+	gg(x+	ADV
ejpam-4669	86	8	1	1	NUM
ejpam-4669	86	9	)	)	PUNCT
ejpam-4669	86	10	=	=	PUNCT
ejpam-4669	87	1	g(x+	g(x+	ADJ
ejpam-4669	87	2	1	1	X
ejpam-4669	87	3	)	)	PUNCT
ejpam-4669	87	4	+	+	CCONJ
ejpam-4669	87	5	ig(x	ig(x	X
ejpam-4669	87	6	)	)	PUNCT
ejpam-4669	88	1	=	=	SYM
ejpam-4669	88	2	ag(x+	ag(x+	PROPN
ejpam-4669	89	1	1)f(x+	1)f(x+	NUM
ejpam-4669	89	2	1	1	NUM
ejpam-4669	89	3	)	)	PUNCT
ejpam-4669	89	4	+	+	CCONJ
ejpam-4669	89	5	iag(x)f(x	iag(x)f(x	NOUN
ejpam-4669	89	6	)	)	PUNCT
ejpam-4669	89	7	comparing	compare	VERB
ejpam-4669	89	8	the	the	DET
ejpam-4669	89	9	two	two	NUM
ejpam-4669	89	10	sides	side	NOUN
ejpam-4669	89	11	,	,	PUNCT
ejpam-4669	89	12	we	we	PRON
ejpam-4669	89	13	obtain	obtain	VERB
ejpam-4669	89	14	lim	lim	PROPN
ejpam-4669	89	15	x→∞	x→∞	X
ejpam-4669	89	16	ag(x+	ag(x+	PROPN
ejpam-4669	89	17	1	1	NUM
ejpam-4669	89	18	)	)	PUNCT
ejpam-4669	89	19	ag(x	ag(x	PUNCT
ejpam-4669	89	20	)	)	PUNCT
ejpam-4669	90	1	=	=	SYM
ejpam-4669	90	2	1	1	X
ejpam-4669	90	3	.	.	PUNCT
ejpam-4669	90	4	h.	h.	PROPN
ejpam-4669	90	5	f.	f.	PROPN
ejpam-4669	90	6	m.salih	m.salih	PROPN
ejpam-4669	90	7	/	/	SYM
ejpam-4669	90	8	eur	eur	PROPN
ejpam-4669	90	9	.	.	PUNCT
ejpam-4669	91	1	j.	j.	PROPN
ejpam-4669	91	2	pure	pure	PROPN
ejpam-4669	91	3	appl	appl	PROPN
ejpam-4669	91	4	.	.	PROPN
ejpam-4669	91	5	math	math	PROPN
ejpam-4669	91	6	,	,	PUNCT
ejpam-4669	91	7	16	16	NUM
ejpam-4669	91	8	(	(	PUNCT
ejpam-4669	91	9	1	1	NUM
ejpam-4669	91	10	)	)	PUNCT
ejpam-4669	91	11	(	(	PUNCT
ejpam-4669	91	12	2023	2023	NUM
ejpam-4669	91	13	)	)	PUNCT
ejpam-4669	91	14	,	,	PUNCT
ejpam-4669	91	15	587	587	NUM
ejpam-4669	91	16	-	-	SYM
ejpam-4669	91	17	594	594	NUM
ejpam-4669	91	18	590	590	NUM
ejpam-4669	91	19	corollary	corollary	ADJ
ejpam-4669	91	20	2	2	NUM
ejpam-4669	91	21	.	.	PUNCT
ejpam-4669	92	1	let	let	AUX
ejpam-4669	92	2	fg(x	fg(x	NUM
ejpam-4669	92	3	)	)	PUNCT
ejpam-4669	92	4	,	,	PUNCT
ejpam-4669	92	5	gg(x	gg(x	PUNCT
ejpam-4669	92	6	)	)	PUNCT
ejpam-4669	92	7	be	be	AUX
ejpam-4669	92	8	gaussian	gaussian	ADJ
ejpam-4669	92	9	fibonacci	fibonacci	NOUN
ejpam-4669	92	10	functions	function	NOUN
ejpam-4669	92	11	with	with	ADP
ejpam-4669	92	12	gg(x	gg(x	NOUN
ejpam-4669	92	13	)	)	PUNCT
ejpam-4669	92	14	=	=	PUNCT
ejpam-4669	92	15	ag(x)fg(x	ag(x)fg(x	X
ejpam-4669	92	16	)	)	PUNCT
ejpam-4669	92	17	.	.	PUNCT
ejpam-4669	93	1	if	if	SCONJ
ejpam-4669	93	2	ag(x+	ag(x+	ADV
ejpam-4669	93	3	p	p	X
ejpam-4669	93	4	)	)	PUNCT
ejpam-4669	93	5	̸=	̸=	PROPN
ejpam-4669	93	6	ag(x	ag(x	ADV
ejpam-4669	93	7	)	)	PUNCT
ejpam-4669	93	8	for	for	ADP
ejpam-4669	93	9	all	all	DET
ejpam-4669	93	10	x	x	SYM
ejpam-4669	93	11	∈	∈	PROPN
ejpam-4669	93	12	r	r	NOUN
ejpam-4669	93	13	,	,	PUNCT
ejpam-4669	93	14	thwn	thwn	ADJ
ejpam-4669	93	15	lim	lim	NOUN
ejpam-4669	93	16	x→∞	x→∞	PUNCT
ejpam-4669	93	17	ag(x+	ag(x+	PROPN
ejpam-4669	93	18	p	p	X
ejpam-4669	93	19	)	)	PUNCT
ejpam-4669	93	20	ag(x	ag(x	PUNCT
ejpam-4669	93	21	)	)	PUNCT
ejpam-4669	93	22	=	=	SYM
ejpam-4669	94	1	1	1	X
ejpam-4669	94	2	.	.	PUNCT
ejpam-4669	94	3	proof	proof	NOUN
ejpam-4669	94	4	.	.	PUNCT
ejpam-4669	95	1	the	the	DET
ejpam-4669	95	2	proof	proof	NOUN
ejpam-4669	95	3	is	be	AUX
ejpam-4669	95	4	similar	similar	ADJ
ejpam-4669	95	5	to	to	ADP
ejpam-4669	95	6	the	the	DET
ejpam-4669	95	7	proof	proof	NOUN
ejpam-4669	95	8	of	of	ADP
ejpam-4669	95	9	the	the	DET
ejpam-4669	95	10	proposition	proposition	NOUN
ejpam-4669	95	11	3	3	NUM
ejpam-4669	95	12	.	.	PUNCT
ejpam-4669	95	13	corollary	corollary	ADJ
ejpam-4669	95	14	3	3	NUM
ejpam-4669	95	15	.	.	PUNCT
ejpam-4669	96	1	let	let	AUX
ejpam-4669	96	2	fg(x	fg(x	NUM
ejpam-4669	96	3	)	)	PUNCT
ejpam-4669	96	4	and	and	CCONJ
ejpam-4669	96	5	gg(x	gg(x	PUNCT
ejpam-4669	96	6	)	)	PUNCT
ejpam-4669	96	7	be	be	AUX
ejpam-4669	96	8	gaussian	gaussian	ADJ
ejpam-4669	96	9	fibonacci	fibonacci	NOUN
ejpam-4669	96	10	functions	function	NOUN
ejpam-4669	96	11	with	with	ADP
ejpam-4669	96	12	gg(x	gg(x	NOUN
ejpam-4669	96	13	)	)	PUNCT
ejpam-4669	96	14	=	=	PUNCT
ejpam-4669	97	1	a(x)fg(x	a(x)fg(x	X
ejpam-4669	97	2	)	)	PUNCT
ejpam-4669	97	3	for	for	ADP
ejpam-4669	97	4	some	some	DET
ejpam-4669	97	5	a(x	a(x	NOUN
ejpam-4669	97	6	)	)	PUNCT
ejpam-4669	97	7	.	.	PUNCT
ejpam-4669	98	1	if	if	SCONJ
ejpam-4669	98	2	y	y	PROPN
ejpam-4669	98	3	>	>	X
ejpam-4669	98	4	0	0	PROPN
ejpam-4669	98	5	,	,	PUNCT
ejpam-4669	98	6	then	then	ADV
ejpam-4669	98	7	lim	lim	PROPN
ejpam-4669	98	8	x→∞	x→∞	PUNCT
ejpam-4669	99	1	a(x+	a(x+	CCONJ
ejpam-4669	99	2	y	y	PROPN
ejpam-4669	99	3	)	)	PUNCT
ejpam-4669	99	4	a(x	a(x	PROPN
ejpam-4669	99	5	)	)	PUNCT
ejpam-4669	99	6	=	=	SYM
ejpam-4669	100	1	lim	lim	PROPN
ejpam-4669	100	2	x→∞	x→∞	PUNCT
ejpam-4669	101	1	a(x+	a(x+	PROPN
ejpam-4669	101	2	y	y	PROPN
ejpam-4669	101	3	)	)	PUNCT
ejpam-4669	101	4	a(x+	a(x+	ADP
ejpam-4669	101	5	y	y	PROPN
ejpam-4669	101	6	−	−	PROPN
ejpam-4669	101	7	⌊y⌋	⌊y⌋	NUM
ejpam-4669	101	8	)	)	PUNCT
ejpam-4669	101	9	proof	proof	NOUN
ejpam-4669	101	10	.	.	PUNCT
ejpam-4669	102	1	lim	lim	NOUN
ejpam-4669	102	2	x→∞	x→∞	PUNCT
ejpam-4669	102	3	a(x+	a(x+	CCONJ
ejpam-4669	102	4	y	y	PROPN
ejpam-4669	102	5	)	)	PUNCT
ejpam-4669	102	6	a(x	a(x	PROPN
ejpam-4669	102	7	)	)	PUNCT
ejpam-4669	102	8	=	=	SYM
ejpam-4669	102	9	lim	lim	PROPN
ejpam-4669	102	10	x→∞	x→∞	PUNCT
ejpam-4669	103	1	a(x+	a(x+	ADP
ejpam-4669	103	2	y)a(x+	y)a(x+	NUM
ejpam-4669	103	3	y	y	PROPN
ejpam-4669	103	4	−	−	PROPN
ejpam-4669	103	5	⌊y⌋	⌊y⌋	NUM
ejpam-4669	103	6	)	)	PUNCT
ejpam-4669	104	1	a(x+	a(x+	ADP
ejpam-4669	104	2	y	y	PROPN
ejpam-4669	104	3	−	−	PROPN
ejpam-4669	104	4	⌊y⌋)a(x	⌊y⌋)a(x	PROPN
ejpam-4669	104	5	)	)	PUNCT
ejpam-4669	104	6	=	=	SYM
ejpam-4669	105	1	lim	lim	PROPN
ejpam-4669	105	2	x→∞	x→∞	PUNCT
ejpam-4669	106	1	a(x+	a(x+	PROPN
ejpam-4669	106	2	y	y	PROPN
ejpam-4669	106	3	)	)	PUNCT
ejpam-4669	106	4	a(x+	a(x+	ADP
ejpam-4669	106	5	y	y	PROPN
ejpam-4669	106	6	−	−	PROPN
ejpam-4669	106	7	⌊y⌋	⌊y⌋	NUM
ejpam-4669	106	8	)	)	PUNCT
ejpam-4669	107	1	lim	lim	NOUN
ejpam-4669	107	2	x→∞	x→∞	NUM
ejpam-4669	108	1	a(x+	a(x+	PROPN
ejpam-4669	108	2	y	y	PROPN
ejpam-4669	108	3	−	−	PROPN
ejpam-4669	108	4	⌊y⌋	⌊y⌋	NUM
ejpam-4669	108	5	)	)	PUNCT
ejpam-4669	108	6	a(x	a(x	PROPN
ejpam-4669	108	7	)	)	PUNCT
ejpam-4669	108	8	=	=	SYM
ejpam-4669	109	1	lim	lim	PROPN
ejpam-4669	109	2	x→∞	x→∞	NUM
ejpam-4669	110	1	a(x+	a(x+	CCONJ
ejpam-4669	110	2	y	y	PROPN
ejpam-4669	110	3	−	−	PROPN
ejpam-4669	110	4	⌊y⌋+	⌊y⌋+	NOUN
ejpam-4669	110	5	⌊y⌋	⌊y⌋	PROPN
ejpam-4669	110	6	)	)	PUNCT
ejpam-4669	111	1	a(x+	a(x+	ADP
ejpam-4669	111	2	y	y	PROPN
ejpam-4669	111	3	−	−	PROPN
ejpam-4669	111	4	⌊y⌋	⌊y⌋	NUM
ejpam-4669	111	5	)	)	PUNCT
ejpam-4669	111	6	lim	lim	NOUN
ejpam-4669	111	7	x→∞	x→∞	NUM
ejpam-4669	112	1	a(x+	a(x+	PROPN
ejpam-4669	112	2	y	y	PROPN
ejpam-4669	112	3	−	−	PROPN
ejpam-4669	112	4	⌊y⌋	⌊y⌋	NUM
ejpam-4669	112	5	)	)	PUNCT
ejpam-4669	112	6	a(x	a(x	PROPN
ejpam-4669	112	7	)	)	PUNCT
ejpam-4669	112	8	=	=	SYM
ejpam-4669	113	1	lim	lim	PROPN
ejpam-4669	113	2	x→∞	x→∞	PUNCT
ejpam-4669	114	1	a(x+	a(x+	PROPN
ejpam-4669	114	2	y	y	PROPN
ejpam-4669	114	3	)	)	PUNCT
ejpam-4669	114	4	a(x+	a(x+	ADP
ejpam-4669	114	5	y	y	PROPN
ejpam-4669	114	6	−	−	PROPN
ejpam-4669	114	7	⌊y⌋	⌊y⌋	NUM
ejpam-4669	114	8	)	)	PUNCT
ejpam-4669	114	9	definition	definition	NOUN
ejpam-4669	114	10	3	3	NUM
ejpam-4669	114	11	.	.	PUNCT
ejpam-4669	115	1	a	a	DET
ejpam-4669	115	2	map	map	NOUN
ejpam-4669	115	3	tg(x	tg(x	ADV
ejpam-4669	115	4	)	)	PUNCT
ejpam-4669	115	5	is	be	AUX
ejpam-4669	115	6	said	say	VERB
ejpam-4669	115	7	to	to	PART
ejpam-4669	115	8	be	be	AUX
ejpam-4669	115	9	gaussian	gaussian	ADJ
ejpam-4669	115	10	ultimately	ultimately	ADV
ejpam-4669	115	11	periodic	periodic	ADJ
ejpam-4669	115	12	of	of	ADP
ejpam-4669	115	13	period	period	NOUN
ejpam-4669	115	14	p	p	X
ejpam-4669	115	15	>	>	X
ejpam-4669	115	16	0	0	PUNCT
ejpam-4669	116	1	if	if	SCONJ
ejpam-4669	116	2	lim	lim	PROPN
ejpam-4669	116	3	x→∞	x→∞	NUM
ejpam-4669	116	4	tg(x+	tg(x+	PROPN
ejpam-4669	116	5	p	p	NOUN
ejpam-4669	116	6	)	)	PUNCT
ejpam-4669	116	7	tg(x	tg(x	PUNCT
ejpam-4669	116	8	)	)	PUNCT
ejpam-4669	116	9	=	=	SYM
ejpam-4669	116	10	1	1	X
ejpam-4669	116	11	.	.	X
ejpam-4669	116	12	note	note	VERB
ejpam-4669	116	13	that	that	SCONJ
ejpam-4669	116	14	a(x	a(x	NOUN
ejpam-4669	116	15	)	)	PUNCT
ejpam-4669	116	16	discussed	discuss	VERB
ejpam-4669	116	17	in	in	ADP
ejpam-4669	116	18	proposition	proposition	NOUN
ejpam-4669	116	19	3	3	NUM
ejpam-4669	116	20	is	be	AUX
ejpam-4669	116	21	gaussian	gaussian	ADJ
ejpam-4669	116	22	ultimately	ultimately	ADV
ejpam-4669	116	23	periodic	periodic	ADJ
ejpam-4669	116	24	of	of	ADP
ejpam-4669	116	25	period	period	NOUN
ejpam-4669	116	26	1	1	NUM
ejpam-4669	116	27	.	.	PUNCT
ejpam-4669	116	28	example	example	NOUN
ejpam-4669	117	1	3	3	X
ejpam-4669	117	2	.	.	PUNCT
ejpam-4669	117	3	let	let	VERB
ejpam-4669	117	4	tg(x	tg(x	PUNCT
ejpam-4669	117	5	)	)	PUNCT
ejpam-4669	117	6	:	:	PUNCT
ejpam-4669	117	7	=	=	SYM
ejpam-4669	117	8	mx+	mx+	PROPN
ejpam-4669	117	9	b.	b.	NOUN
ejpam-4669	118	1	if	if	SCONJ
ejpam-4669	118	2	m	m	VERB
ejpam-4669	118	3	̸=	̸=	PROPN
ejpam-4669	118	4	0,then	0,then	NOUN
ejpam-4669	118	5	lim	lim	PROPN
ejpam-4669	118	6	x→∞	x→∞	NUM
ejpam-4669	119	1	tg(x+	tg(x+	PROPN
ejpam-4669	119	2	p	p	NOUN
ejpam-4669	119	3	)	)	PUNCT
ejpam-4669	119	4	tg(x	tg(x	PUNCT
ejpam-4669	119	5	)	)	PUNCT
ejpam-4669	120	1	=	=	VERB
ejpam-4669	120	2	lim	lim	PROPN
ejpam-4669	120	3	x→∞	x→∞	PROPN
ejpam-4669	121	1	m(x+	m(x+	NUM
ejpam-4669	121	2	p	p	X
ejpam-4669	121	3	)	)	PUNCT
ejpam-4669	122	1	+	+	PUNCT
ejpam-4669	122	2	b	b	X
ejpam-4669	122	3	mx+	mx+	NOUN
ejpam-4669	122	4	b	b	NOUN
ejpam-4669	122	5	=	=	SYM
ejpam-4669	122	6	1	1	NUM
ejpam-4669	122	7	,	,	PUNCT
ejpam-4669	122	8	showing	show	VERB
ejpam-4669	122	9	that	that	PRON
ejpam-4669	122	10	tg(x	tg(x	PUNCT
ejpam-4669	122	11	)	)	PUNCT
ejpam-4669	122	12	is	be	AUX
ejpam-4669	122	13	a	a	DET
ejpam-4669	122	14	gaussian	gaussian	NOUN
ejpam-4669	122	15	ultimately	ultimately	ADV
ejpam-4669	122	16	periodic	periodic	ADJ
ejpam-4669	122	17	of	of	ADP
ejpam-4669	122	18	period	period	NOUN
ejpam-4669	122	19	p	p	NOUN
ejpam-4669	122	20	for	for	ADP
ejpam-4669	122	21	all	all	DET
ejpam-4669	122	22	p	p	X
ejpam-4669	122	23	>	>	X
ejpam-4669	122	24	0	0	X
ejpam-4669	122	25	.	.	PUNCT
ejpam-4669	123	1	using	use	VERB
ejpam-4669	123	2	example	example	NOUN
ejpam-4669	123	3	3	3	NUM
ejpam-4669	123	4	,	,	PUNCT
ejpam-4669	123	5	we	we	PRON
ejpam-4669	123	6	obtain	obtain	VERB
ejpam-4669	123	7	the	the	DET
ejpam-4669	123	8	following	follow	VERB
ejpam-4669	123	9	example	example	NOUN
ejpam-4669	123	10	.	.	PUNCT
ejpam-4669	124	1	example	example	NOUN
ejpam-4669	125	1	4	4	NUM
ejpam-4669	125	2	.	.	PUNCT
ejpam-4669	126	1	if	if	SCONJ
ejpam-4669	126	2	tg(x	tg(x	NUM
ejpam-4669	126	3	)	)	PUNCT
ejpam-4669	127	1	:	:	PUNCT
ejpam-4669	127	2	=	=	SYM
ejpam-4669	127	3	anx	anx	ADJ
ejpam-4669	127	4	n	n	PROPN
ejpam-4669	127	5	+	+	CCONJ
ejpam-4669	127	6	an−1x	an−1x	PROPN
ejpam-4669	127	7	n−1	n−1	PROPN
ejpam-4669	127	8	+	+	CCONJ
ejpam-4669	127	9	·	·	PUNCT
ejpam-4669	127	10	·	·	PUNCT
ejpam-4669	127	11	·	·	PUNCT
ejpam-4669	127	12	+	+	NUM
ejpam-4669	127	13	a0	a0	NOUN
ejpam-4669	127	14	,	,	PUNCT
ejpam-4669	127	15	then	then	ADV
ejpam-4669	127	16	tg(x	tg(x	PUNCT
ejpam-4669	127	17	)	)	PUNCT
ejpam-4669	127	18	is	be	AUX
ejpam-4669	127	19	a	a	DET
ejpam-4669	127	20	gaussian	gaussian	NOUN
ejpam-4669	127	21	ultimately	ultimately	ADV
ejpam-4669	127	22	periodic	periodic	ADJ
ejpam-4669	127	23	of	of	ADP
ejpam-4669	127	24	period	period	NOUN
ejpam-4669	127	25	p	p	NOUN
ejpam-4669	127	26	for	for	ADP
ejpam-4669	127	27	all	all	DET
ejpam-4669	127	28	p	p	X
ejpam-4669	127	29	>	>	X
ejpam-4669	127	30	0	0	NUM
ejpam-4669	127	31	.	.	PUNCT
ejpam-4669	128	1	h.	h.	PROPN
ejpam-4669	128	2	f.	f.	PROPN
ejpam-4669	128	3	m.salih	m.salih	PROPN
ejpam-4669	128	4	/	/	SYM
ejpam-4669	128	5	eur	eur	PROPN
ejpam-4669	128	6	.	.	PUNCT
ejpam-4669	129	1	j.	j.	PROPN
ejpam-4669	129	2	pure	pure	PROPN
ejpam-4669	129	3	appl	appl	PROPN
ejpam-4669	129	4	.	.	PROPN
ejpam-4669	129	5	math	math	PROPN
ejpam-4669	129	6	,	,	PUNCT
ejpam-4669	129	7	16	16	NUM
ejpam-4669	129	8	(	(	PUNCT
ejpam-4669	129	9	1	1	NUM
ejpam-4669	129	10	)	)	PUNCT
ejpam-4669	129	11	(	(	PUNCT
ejpam-4669	129	12	2023	2023	NUM
ejpam-4669	129	13	)	)	PUNCT
ejpam-4669	129	14	,	,	PUNCT
ejpam-4669	129	15	587	587	NUM
ejpam-4669	129	16	-	-	SYM
ejpam-4669	129	17	594	594	NUM
ejpam-4669	129	18	591	591	NUM
ejpam-4669	129	19	example	example	NOUN
ejpam-4669	129	20	5	5	NUM
ejpam-4669	129	21	.	.	PUNCT
ejpam-4669	130	1	if	if	SCONJ
ejpam-4669	130	2	tg(x	tg(x	PRON
ejpam-4669	131	1	)	)	PUNCT
ejpam-4669	131	2	=	=	SYM
ejpam-4669	131	3	cos(x	cos(x	PROPN
ejpam-4669	131	4	)	)	PUNCT
ejpam-4669	131	5	,	,	PUNCT
ejpam-4669	131	6	then	then	ADV
ejpam-4669	131	7	lim	lim	PROPN
ejpam-4669	131	8	x→∞	x→∞	PROPN
ejpam-4669	132	1	tg(x+	tg(x+	PROPN
ejpam-4669	132	2	p	p	NOUN
ejpam-4669	132	3	)	)	PUNCT
ejpam-4669	132	4	tg(x	tg(x	PUNCT
ejpam-4669	132	5	)	)	PUNCT
ejpam-4669	133	1	=	=	SYM
ejpam-4669	133	2	lim	lim	PROPN
ejpam-4669	133	3	x→∞	x→∞	NUM
ejpam-4669	134	1	cos(x+	cos(x+	PROPN
ejpam-4669	134	2	p	p	X
ejpam-4669	134	3	)	)	PUNCT
ejpam-4669	134	4	cos(x	cos(x	PROPN
ejpam-4669	134	5	)	)	PUNCT
ejpam-4669	135	1	=	=	SYM
ejpam-4669	135	2	lim	lim	PROPN
ejpam-4669	135	3	x→∞	x→∞	NUM
ejpam-4669	135	4	cos(x	cos(x	PROPN
ejpam-4669	135	5	)	)	PUNCT
ejpam-4669	135	6	cos(p	cos(p	PROPN
ejpam-4669	135	7	)	)	PUNCT
ejpam-4669	136	1	+	+	CCONJ
ejpam-4669	136	2	sin(x	sin(x	PROPN
ejpam-4669	136	3	)	)	PUNCT
ejpam-4669	136	4	sin(p	sin(p	PROPN
ejpam-4669	136	5	)	)	PUNCT
ejpam-4669	136	6	cos(x	cos(x	PROPN
ejpam-4669	136	7	)	)	PUNCT
ejpam-4669	137	1	=	=	SYM
ejpam-4669	137	2	cos(p	cos(p	PROPN
ejpam-4669	137	3	)	)	PUNCT
ejpam-4669	137	4	+	+	X
ejpam-4669	137	5	sin(p	sin(p	ADJ
ejpam-4669	137	6	)	)	PUNCT
ejpam-4669	137	7	lim	lim	NOUN
ejpam-4669	137	8	x→∞	x→∞	NUM
ejpam-4669	137	9	tan(x	tan(x	ADP
ejpam-4669	137	10	)	)	PUNCT
ejpam-4669	137	11	.	.	PUNCT
ejpam-4669	138	1	since	since	SCONJ
ejpam-4669	138	2	limx→∞	limx→∞	PROPN
ejpam-4669	138	3	tan(x	tan(x	X
ejpam-4669	138	4	)	)	PUNCT
ejpam-4669	138	5	does	do	AUX
ejpam-4669	138	6	not	not	PART
ejpam-4669	138	7	exist	exist	VERB
ejpam-4669	138	8	,	,	PUNCT
ejpam-4669	138	9	tg(x	tg(x	ADV
ejpam-4669	138	10	)	)	PUNCT
ejpam-4669	138	11	is	be	AUX
ejpam-4669	138	12	not	not	PART
ejpam-4669	138	13	a	a	DET
ejpam-4669	138	14	gaussian	gaussian	NOUN
ejpam-4669	138	15	ultimately	ultimately	ADV
ejpam-4669	138	16	periodic	periodic	ADJ
ejpam-4669	138	17	of	of	ADP
ejpam-4669	138	18	period	period	NOUN
ejpam-4669	138	19	p	p	X
ejpam-4669	138	20	>	>	X
ejpam-4669	138	21	0	0	PUNCT
ejpam-4669	139	1	unless	unless	SCONJ
ejpam-4669	139	2	sin(p	sin(p	ADJ
ejpam-4669	139	3	)	)	PUNCT
ejpam-4669	139	4	=	=	SYM
ejpam-4669	139	5	0	0	NUM
ejpam-4669	139	6	and	and	CCONJ
ejpam-4669	139	7	cos(p	cos(p	PROPN
ejpam-4669	139	8	)	)	PUNCT
ejpam-4669	139	9	=	=	SYM
ejpam-4669	140	1	1	1	X
ejpam-4669	140	2	.	.	X
ejpam-4669	140	3	proposition	proposition	NOUN
ejpam-4669	140	4	2	2	NUM
ejpam-4669	140	5	.	.	PUNCT
ejpam-4669	141	1	if	if	SCONJ
ejpam-4669	141	2	ag(x	ag(x	NUM
ejpam-4669	141	3	)	)	PUNCT
ejpam-4669	141	4	and	and	CCONJ
ejpam-4669	141	5	bg(x	bg(x	VERB
ejpam-4669	141	6	)	)	PUNCT
ejpam-4669	141	7	are	be	AUX
ejpam-4669	141	8	gaussian	gaussian	ADJ
ejpam-4669	141	9	ultimately	ultimately	ADV
ejpam-4669	141	10	periodic	periodic	ADJ
ejpam-4669	141	11	of	of	ADP
ejpam-4669	141	12	period	period	NOUN
ejpam-4669	141	13	p	p	X
ejpam-4669	141	14	>	>	X
ejpam-4669	141	15	0	0	NUM
ejpam-4669	141	16	,	,	PUNCT
ejpam-4669	141	17	then	then	ADV
ejpam-4669	141	18	αag(x	αag(x	X
ejpam-4669	141	19	)	)	PUNCT
ejpam-4669	142	1	+	+	CCONJ
ejpam-4669	142	2	βbg(x	βbg(x	X
ejpam-4669	142	3	)	)	PUNCT
ejpam-4669	142	4	is	be	AUX
ejpam-4669	142	5	also	also	ADV
ejpam-4669	142	6	a	a	DET
ejpam-4669	142	7	gaussian	gaussian	NOUN
ejpam-4669	142	8	ultimately	ultimately	ADV
ejpam-4669	142	9	periodic	periodic	ADJ
ejpam-4669	142	10	of	of	ADP
ejpam-4669	142	11	period	period	NOUN
ejpam-4669	142	12	p	p	X
ejpam-4669	142	13	>	>	X
ejpam-4669	142	14	0	0	NUM
ejpam-4669	142	15	,	,	PUNCT
ejpam-4669	142	16	for	for	ADP
ejpam-4669	142	17	all	all	DET
ejpam-4669	142	18	α	α	NOUN
ejpam-4669	142	19	,	,	PUNCT
ejpam-4669	142	20	β	β	X
ejpam-4669	142	21	>	>	X
ejpam-4669	142	22	0	0	NUM
ejpam-4669	143	1	proof	proof	NOUN
ejpam-4669	143	2	.	.	PUNCT
ejpam-4669	144	1	since	since	SCONJ
ejpam-4669	144	2	ag(x	ag(x	NUM
ejpam-4669	144	3	)	)	PUNCT
ejpam-4669	144	4	and	and	CCONJ
ejpam-4669	144	5	bg(x	bg(x	VERB
ejpam-4669	144	6	)	)	PUNCT
ejpam-4669	144	7	are	be	AUX
ejpam-4669	144	8	gaussian	gaussian	ADJ
ejpam-4669	144	9	ultimately	ultimately	ADV
ejpam-4669	144	10	periodic	periodic	ADJ
ejpam-4669	144	11	of	of	ADP
ejpam-4669	144	12	period	period	NOUN
ejpam-4669	144	13	p	p	X
ejpam-4669	144	14	>	>	X
ejpam-4669	144	15	0	0	NUM
ejpam-4669	144	16	,	,	PUNCT
ejpam-4669	144	17	there	there	PRON
ejpam-4669	144	18	exist	exist	VERB
ejpam-4669	144	19	ϵ1(x	ϵ1(x	NOUN
ejpam-4669	144	20	)	)	PUNCT
ejpam-4669	144	21	,	,	PUNCT
ejpam-4669	144	22	ϵ2(x	ϵ2(x	X
ejpam-4669	144	23	)	)	PUNCT
ejpam-4669	144	24	>	>	X
ejpam-4669	144	25	0	0	NUM
ejpam-4669	145	1	such	such	ADJ
ejpam-4669	145	2	that	that	PRON
ejpam-4669	145	3	ag(x+p	ag(x+p	NOUN
ejpam-4669	145	4	)	)	PUNCT
ejpam-4669	145	5	ag(p	ag(p	NOUN
ejpam-4669	145	6	)	)	PUNCT
ejpam-4669	146	1	=	=	SYM
ejpam-4669	146	2	1	1	NUM
ejpam-4669	146	3	+	+	NUM
ejpam-4669	146	4	ϵ1(x	ϵ1(x	NOUN
ejpam-4669	146	5	)	)	PUNCT
ejpam-4669	146	6	and	and	CCONJ
ejpam-4669	146	7	bg(x+p	bg(x+p	PROPN
ejpam-4669	146	8	)	)	PUNCT
ejpam-4669	146	9	bg(p	bg(p	X
ejpam-4669	146	10	)	)	PUNCT
ejpam-4669	147	1	=	=	SYM
ejpam-4669	147	2	1	1	NUM
ejpam-4669	147	3	+	+	NUM
ejpam-4669	147	4	ϵ2(x	ϵ2(x	NOUN
ejpam-4669	147	5	)	)	PUNCT
ejpam-4669	147	6	where	where	SCONJ
ejpam-4669	147	7	ϵ1(x	ϵ1(x	NOUN
ejpam-4669	147	8	)	)	PUNCT
ejpam-4669	147	9	,	,	PUNCT
ejpam-4669	147	10	ϵ2(x	ϵ2(x	X
ejpam-4669	147	11	)	)	PUNCT
ejpam-4669	147	12	→	→	SYM
ejpam-4669	147	13	0	0	X
ejpam-4669	147	14	.	.	PUNCT
ejpam-4669	148	1	we	we	PRON
ejpam-4669	148	2	know	know	VERB
ejpam-4669	148	3	that	that	SCONJ
ejpam-4669	148	4	1+ϵ1(x	1+ϵ1(x	NOUN
ejpam-4669	148	5	)	)	PUNCT
ejpam-4669	148	6	1+ϵ2(x	1+ϵ2(x	NUM
ejpam-4669	148	7	)	)	PUNCT
ejpam-4669	148	8	=	=	SYM
ejpam-4669	148	9	1	1	NUM
ejpam-4669	148	10	+	+	CCONJ
ejpam-4669	148	11	ϵ(x	ϵ(x	NOUN
ejpam-4669	148	12	)	)	PUNCT
ejpam-4669	148	13	.	.	PUNCT
ejpam-4669	149	1	in	in	ADP
ejpam-4669	149	2	fact	fact	NOUN
ejpam-4669	149	3	,	,	PUNCT
ejpam-4669	149	4	ϵ(x	ϵ(x	PROPN
ejpam-4669	149	5	)	)	PUNCT
ejpam-4669	149	6	=	=	SYM
ejpam-4669	149	7	ϵ1(x)−ϵ2(x	ϵ1(x)−ϵ2(x	PROPN
ejpam-4669	149	8	)	)	PUNCT
ejpam-4669	149	9	1+ϵ1(x	1+ϵ1(x	NOUN
ejpam-4669	149	10	)	)	PUNCT
ejpam-4669	149	11	→	→	SYM
ejpam-4669	150	1	0	0	X
ejpam-4669	150	2	.	.	PUNCT
ejpam-4669	151	1	this	this	PRON
ejpam-4669	151	2	shows	show	VERB
ejpam-4669	151	3	that	that	SCONJ
ejpam-4669	151	4	αag(x+	αag(x+	VERB
ejpam-4669	151	5	p	p	X
ejpam-4669	151	6	)	)	PUNCT
ejpam-4669	151	7	+	+	CCONJ
ejpam-4669	151	8	βbg(x+	βbg(x+	ADV
ejpam-4669	151	9	p	p	X
ejpam-4669	151	10	)	)	PUNCT
ejpam-4669	151	11	αag(x	αag(x	NUM
ejpam-4669	151	12	)	)	PUNCT
ejpam-4669	152	1	+	+	CCONJ
ejpam-4669	152	2	βbg(x	βbg(x	X
ejpam-4669	152	3	)	)	PUNCT
ejpam-4669	152	4	=	=	SYM
ejpam-4669	152	5	1	1	NUM
ejpam-4669	152	6	+	+	NUM
ejpam-4669	152	7	βbg(x+p	βbg(x+p	SYM
ejpam-4669	152	8	)	)	PUNCT
ejpam-4669	152	9	αag(x+p	αag(x+p	PROPN
ejpam-4669	152	10	)	)	PUNCT
ejpam-4669	152	11	1	1	NUM
ejpam-4669	153	1	+	+	CCONJ
ejpam-4669	153	2	βbg(x	βbg(x	X
ejpam-4669	153	3	)	)	PUNCT
ejpam-4669	153	4	αag(x	αag(x	X
ejpam-4669	153	5	)	)	PUNCT
ejpam-4669	153	6	αag(x+	αag(x+	VERB
ejpam-4669	153	7	p	p	X
ejpam-4669	153	8	)	)	PUNCT
ejpam-4669	153	9	αag(x	αag(x	PROPN
ejpam-4669	153	10	)	)	PUNCT
ejpam-4669	153	11	=	=	SYM
ejpam-4669	154	1	1	1	NUM
ejpam-4669	154	2	+	+	CCONJ
ejpam-4669	154	3	β(1+ϵ2(x))bg(x	β(1+ϵ2(x))bg(x	NOUN
ejpam-4669	154	4	)	)	PUNCT
ejpam-4669	154	5	α(1+ϵ1(x))ag(x	α(1+ϵ1(x))ag(x	NOUN
ejpam-4669	154	6	)	)	PUNCT
ejpam-4669	154	7	1	1	NUM
ejpam-4669	155	1	+	+	CCONJ
ejpam-4669	155	2	βbg(x	βbg(x	X
ejpam-4669	155	3	)	)	PUNCT
ejpam-4669	155	4	αag(x	αag(x	X
ejpam-4669	155	5	)	)	PUNCT
ejpam-4669	155	6	ag(x+	ag(x+	ADV
ejpam-4669	156	1	p	p	X
ejpam-4669	156	2	)	)	PUNCT
ejpam-4669	156	3	ag(x	ag(x	ADV
ejpam-4669	156	4	)	)	PUNCT
ejpam-4669	156	5	→	→	SYM
ejpam-4669	156	6	ag(x+	ag(x+	ADV
ejpam-4669	156	7	p	p	X
ejpam-4669	156	8	)	)	PUNCT
ejpam-4669	156	9	ag(x	ag(x	ADV
ejpam-4669	156	10	)	)	PUNCT
ejpam-4669	157	1	→	→	SYM
ejpam-4669	157	2	1	1	X
ejpam-4669	157	3	.	.	X
ejpam-4669	158	1	hence	hence	ADV
ejpam-4669	158	2	,	,	PUNCT
ejpam-4669	158	3	the	the	DET
ejpam-4669	158	4	proposition	proposition	NOUN
ejpam-4669	158	5	is	be	AUX
ejpam-4669	158	6	proved	prove	VERB
ejpam-4669	158	7	.	.	PUNCT
ejpam-4669	159	1	proposition	proposition	NOUN
ejpam-4669	159	2	3	3	NUM
ejpam-4669	159	3	.	.	PUNCT
ejpam-4669	160	1	if	if	SCONJ
ejpam-4669	160	2	ag(x	ag(x	NUM
ejpam-4669	160	3	)	)	PUNCT
ejpam-4669	160	4	and	and	CCONJ
ejpam-4669	160	5	bg(x	bg(x	VERB
ejpam-4669	160	6	)	)	PUNCT
ejpam-4669	160	7	are	be	AUX
ejpam-4669	160	8	gaussian	gaussian	ADJ
ejpam-4669	160	9	ultimately	ultimately	ADV
ejpam-4669	160	10	periodic	periodic	ADJ
ejpam-4669	160	11	of	of	ADP
ejpam-4669	160	12	period	period	NOUN
ejpam-4669	160	13	p	p	X
ejpam-4669	160	14	>	>	X
ejpam-4669	160	15	0	0	NUM
ejpam-4669	160	16	,	,	PUNCT
ejpam-4669	160	17	then	then	ADV
ejpam-4669	160	18	ag(x)bg(x	ag(x)bg(x	NOUN
ejpam-4669	160	19	)	)	PUNCT
ejpam-4669	161	1	is	be	AUX
ejpam-4669	161	2	also	also	ADV
ejpam-4669	161	3	a	a	DET
ejpam-4669	161	4	gaussian	gaussian	NOUN
ejpam-4669	161	5	ultimately	ultimately	ADV
ejpam-4669	161	6	periodic	periodic	ADJ
ejpam-4669	161	7	of	of	ADP
ejpam-4669	161	8	period	period	NOUN
ejpam-4669	161	9	p	p	X
ejpam-4669	161	10	>	>	X
ejpam-4669	161	11	0	0	X
ejpam-4669	161	12	.	.	PUNCT
ejpam-4669	162	1	proof	proof	NOUN
ejpam-4669	162	2	.	.	PUNCT
ejpam-4669	163	1	this	this	PRON
ejpam-4669	163	2	can	can	AUX
ejpam-4669	163	3	be	be	AUX
ejpam-4669	163	4	prove	prove	VERB
ejpam-4669	163	5	by	by	ADP
ejpam-4669	163	6	the	the	DET
ejpam-4669	163	7	following	follow	VERB
ejpam-4669	163	8	equation	equation	NOUN
ejpam-4669	163	9	:	:	PUNCT
ejpam-4669	163	10	lim	lim	PROPN
ejpam-4669	163	11	x→∞	x→∞	PROPN
ejpam-4669	163	12	ag(x+	ag(x+	PROPN
ejpam-4669	163	13	p)bg(x+	p)bg(x+	VERB
ejpam-4669	163	14	p	p	NOUN
ejpam-4669	163	15	)	)	PUNCT
ejpam-4669	163	16	ag(x)bg(x	ag(x)bg(x	NOUN
ejpam-4669	163	17	)	)	PUNCT
ejpam-4669	164	1	=	=	SYM
ejpam-4669	164	2	lim	lim	PROPN
ejpam-4669	164	3	x→∞	x→∞	NUM
ejpam-4669	164	4	ag(x+	ag(x+	PROPN
ejpam-4669	164	5	p	p	X
ejpam-4669	164	6	)	)	PUNCT
ejpam-4669	164	7	ag(x	ag(x	ADV
ejpam-4669	164	8	)	)	PUNCT
ejpam-4669	165	1	lim	lim	PROPN
ejpam-4669	165	2	x→∞	x→∞	NUM
ejpam-4669	166	1	bg(x+	bg(x+	PROPN
ejpam-4669	166	2	p	p	X
ejpam-4669	166	3	)	)	PUNCT
ejpam-4669	166	4	bg(x	bg(x	PUNCT
ejpam-4669	166	5	)	)	PUNCT
ejpam-4669	167	1	=	=	SYM
ejpam-4669	167	2	1	1	X
ejpam-4669	167	3	.	.	X
ejpam-4669	167	4	note	note	VERB
ejpam-4669	167	5	that	that	SCONJ
ejpam-4669	167	6	gup	gup	PROPN
ejpam-4669	167	7	is	be	AUX
ejpam-4669	167	8	the	the	DET
ejpam-4669	167	9	collection	collection	NOUN
ejpam-4669	167	10	of	of	ADP
ejpam-4669	167	11	all	all	DET
ejpam-4669	167	12	functions	function	NOUN
ejpam-4669	167	13	which	which	PRON
ejpam-4669	167	14	are	be	AUX
ejpam-4669	167	15	gaussian	gaussian	ADJ
ejpam-4669	167	16	ultimately	ultimately	ADV
ejpam-4669	167	17	periodic	periodic	ADJ
ejpam-4669	167	18	of	of	ADP
ejpam-4669	167	19	period	period	NOUN
ejpam-4669	167	20	p	p	X
ejpam-4669	167	21	>	>	X
ejpam-4669	167	22	0	0	X
ejpam-4669	167	23	.	.	PUNCT
ejpam-4669	168	1	proposition	proposition	NOUN
ejpam-4669	168	2	4	4	NUM
ejpam-4669	168	3	.	.	PUNCT
ejpam-4669	169	1	if	if	SCONJ
ejpam-4669	169	2	ag(x	ag(x	NOUN
ejpam-4669	169	3	)	)	PUNCT
ejpam-4669	169	4	∈	∈	PROPN
ejpam-4669	169	5	gup	gup	NOUN
ejpam-4669	169	6	and	and	CCONJ
ejpam-4669	169	7	ag(x	ag(x	ADV
ejpam-4669	169	8	)	)	PUNCT
ejpam-4669	170	1	̸=	̸=	NOUN
ejpam-4669	170	2	0	0	NUM
ejpam-4669	170	3	for	for	ADP
ejpam-4669	170	4	all	all	DET
ejpam-4669	170	5	x	x	SYM
ejpam-4669	170	6	∈	∈	PROPN
ejpam-4669	170	7	[	[	X
ejpam-4669	170	8	λ,∞	λ,∞	X
ejpam-4669	170	9	)	)	PUNCT
ejpam-4669	170	10	,	,	PUNCT
ejpam-4669	170	11	then	then	ADV
ejpam-4669	170	12	1	1	NUM
ejpam-4669	170	13	ag(x	ag(x	ADV
ejpam-4669	170	14	)	)	PUNCT
ejpam-4669	170	15	∈	∈	PROPN
ejpam-4669	170	16	gup	gup	NOUN
ejpam-4669	170	17	.	.	PUNCT
ejpam-4669	171	1	proof	proof	NOUN
ejpam-4669	171	2	.	.	PUNCT
ejpam-4669	172	1	lim	lim	NOUN
ejpam-4669	172	2	x→∞	x→∞	NUM
ejpam-4669	172	3	1	1	NUM
ejpam-4669	172	4	ag(x+p	ag(x+p	NOUN
ejpam-4669	172	5	)	)	PUNCT
ejpam-4669	172	6	1	1	NUM
ejpam-4669	172	7	ag(x	ag(x	PUNCT
ejpam-4669	172	8	)	)	PUNCT
ejpam-4669	173	1	=	=	SYM
ejpam-4669	173	2	lim	lim	PROPN
ejpam-4669	173	3	x→∞	x→∞	NUM
ejpam-4669	173	4	ag(x	ag(x	CCONJ
ejpam-4669	173	5	)	)	PUNCT
ejpam-4669	173	6	ag(x+	ag(x+	ADV
ejpam-4669	174	1	p	p	X
ejpam-4669	174	2	)	)	PUNCT
ejpam-4669	174	3	=	=	SYM
ejpam-4669	174	4	1	1	X
ejpam-4669	174	5	.	.	PUNCT
ejpam-4669	174	6	h.	h.	PROPN
ejpam-4669	174	7	f.	f.	PROPN
ejpam-4669	174	8	m.salih	m.salih	PROPN
ejpam-4669	174	9	/	/	SYM
ejpam-4669	174	10	eur	eur	PROPN
ejpam-4669	174	11	.	.	PUNCT
ejpam-4669	175	1	j.	j.	PROPN
ejpam-4669	175	2	pure	pure	PROPN
ejpam-4669	175	3	appl	appl	PROPN
ejpam-4669	175	4	.	.	PROPN
ejpam-4669	175	5	math	math	PROPN
ejpam-4669	175	6	,	,	PUNCT
ejpam-4669	175	7	16	16	NUM
ejpam-4669	175	8	(	(	PUNCT
ejpam-4669	175	9	1	1	NUM
ejpam-4669	175	10	)	)	PUNCT
ejpam-4669	175	11	(	(	PUNCT
ejpam-4669	175	12	2023	2023	NUM
ejpam-4669	175	13	)	)	PUNCT
ejpam-4669	175	14	,	,	PUNCT
ejpam-4669	175	15	587	587	NUM
ejpam-4669	175	16	-	-	SYM
ejpam-4669	175	17	594	594	NUM
ejpam-4669	175	18	592	592	NUM
ejpam-4669	175	19	a	a	DET
ejpam-4669	175	20	map	map	NOUN
ejpam-4669	175	21	fg	fg	PROPN
ejpam-4669	175	22	which	which	PRON
ejpam-4669	175	23	is	be	AUX
ejpam-4669	175	24	defined	define	VERB
ejpam-4669	175	25	on	on	ADP
ejpam-4669	175	26	the	the	DET
ejpam-4669	175	27	set	set	NOUN
ejpam-4669	175	28	of	of	ADP
ejpam-4669	175	29	all	all	DET
ejpam-4669	175	30	real	real	ADJ
ejpam-4669	175	31	numbers	number	NOUN
ejpam-4669	175	32	r	r	NOUN
ejpam-4669	175	33	is	be	AUX
ejpam-4669	175	34	said	say	VERB
ejpam-4669	175	35	to	to	PART
ejpam-4669	175	36	be	be	AUX
ejpam-4669	175	37	gaussian	gaussian	ADJ
ejpam-4669	175	38	periodic	periodic	NOUN
ejpam-4669	175	39	of	of	ADP
ejpam-4669	175	40	period	period	NOUN
ejpam-4669	175	41	p	p	X
ejpam-4669	175	42	>	>	X
ejpam-4669	175	43	0	0	PUNCT
ejpam-4669	176	1	if	if	SCONJ
ejpam-4669	176	2	fg(x	fg(x	NUM
ejpam-4669	176	3	+	+	CCONJ
ejpam-4669	176	4	p	p	X
ejpam-4669	176	5	)	)	PUNCT
ejpam-4669	176	6	=	=	PUNCT
ejpam-4669	176	7	fg(x	fg(x	X
ejpam-4669	176	8	)	)	PUNCT
ejpam-4669	176	9	for	for	ADP
ejpam-4669	176	10	all	all	DET
ejpam-4669	176	11	x	x	SYM
ejpam-4669	176	12	∈	∈	PROPN
ejpam-4669	176	13	r.	r.	NOUN
ejpam-4669	176	14	it	it	PRON
ejpam-4669	176	15	is	be	AUX
ejpam-4669	176	16	obvious	obvious	ADJ
ejpam-4669	176	17	that	that	SCONJ
ejpam-4669	176	18	every	every	DET
ejpam-4669	176	19	gaussian	gaussian	ADJ
ejpam-4669	176	20	map	map	NOUN
ejpam-4669	176	21	of	of	ADP
ejpam-4669	176	22	period	period	NOUN
ejpam-4669	176	23	of	of	ADP
ejpam-4669	176	24	periodic	periodic	ADJ
ejpam-4669	176	25	1	1	NUM
ejpam-4669	176	26	is	be	AUX
ejpam-4669	176	27	gaussian	gaussian	NOUN
ejpam-4669	176	28	ultimately	ultimately	ADV
ejpam-4669	176	29	periodic	periodic	ADJ
ejpam-4669	176	30	of	of	ADP
ejpam-4669	176	31	period	period	NOUN
ejpam-4669	176	32	p.	p.	NOUN
ejpam-4669	176	33	proposition	proposition	NOUN
ejpam-4669	176	34	5	5	NUM
ejpam-4669	176	35	.	.	PUNCT
ejpam-4669	177	1	let	let	AUX
ejpam-4669	177	2	fg(x	fg(x	NUM
ejpam-4669	177	3	)	)	PUNCT
ejpam-4669	177	4	be	be	AUX
ejpam-4669	177	5	a	a	DET
ejpam-4669	177	6	gaussian	gaussian	ADJ
ejpam-4669	177	7	fibonacci	fibonacci	NOUN
ejpam-4669	177	8	function	function	NOUN
ejpam-4669	177	9	and	and	CCONJ
ejpam-4669	177	10	f(x	f(x	PROPN
ejpam-4669	177	11	)	)	PUNCT
ejpam-4669	177	12	be	be	VERB
ejpam-4669	177	13	a	a	DET
ejpam-4669	177	14	fibonacci	fibonacci	NOUN
ejpam-4669	177	15	function	function	NOUN
ejpam-4669	177	16	and	and	CCONJ
ejpam-4669	177	17	let	let	VERB
ejpam-4669	177	18	ag(x	ag(x	PUNCT
ejpam-4669	177	19	)	)	PUNCT
ejpam-4669	177	20	be	be	AUX
ejpam-4669	177	21	a	a	DET
ejpam-4669	177	22	gaussian	gaussian	ADJ
ejpam-4669	177	23	periodic	periodic	NOUN
ejpam-4669	177	24	of	of	ADP
ejpam-4669	177	25	period	period	NOUN
ejpam-4669	177	26	1	1	NUM
ejpam-4669	177	27	.	.	PUNCT
ejpam-4669	178	1	if	if	SCONJ
ejpam-4669	178	2	gg(x	gg(x	NUM
ejpam-4669	178	3	)	)	PUNCT
ejpam-4669	179	1	:	:	PUNCT
ejpam-4669	179	2	=	=	PUNCT
ejpam-4669	179	3	ag(x)fg(x	ag(x)fg(x	X
ejpam-4669	179	4	)	)	PUNCT
ejpam-4669	179	5	and	and	CCONJ
ejpam-4669	179	6	g(x	g(x	NOUN
ejpam-4669	179	7	)	)	PUNCT
ejpam-4669	179	8	:	:	PUNCT
ejpam-4669	179	9	=	=	SYM
ejpam-4669	179	10	ag(x)f(x	ag(x)f(x	PROPN
ejpam-4669	179	11	)	)	PUNCT
ejpam-4669	179	12	,	,	PUNCT
ejpam-4669	179	13	then	then	ADV
ejpam-4669	179	14	gg(x	gg(x	PUNCT
ejpam-4669	179	15	)	)	PUNCT
ejpam-4669	179	16	is	be	AUX
ejpam-4669	179	17	a	a	DET
ejpam-4669	179	18	gaussian	gaussian	ADJ
ejpam-4669	179	19	fibonacci	fibonacci	NOUN
ejpam-4669	179	20	function	function	NOUN
ejpam-4669	179	21	.	.	PUNCT
ejpam-4669	180	1	proof	proof	NOUN
ejpam-4669	180	2	.	.	PUNCT
ejpam-4669	181	1	given	give	VERB
ejpam-4669	181	2	x	x	PROPN
ejpam-4669	181	3	∈	∈	PROPN
ejpam-4669	181	4	r.	r.	NOUN
ejpam-4669	181	5	since	since	SCONJ
ejpam-4669	181	6	ag(x	ag(x	PROPN
ejpam-4669	181	7	)	)	PUNCT
ejpam-4669	181	8	is	be	AUX
ejpam-4669	181	9	a	a	DET
ejpam-4669	181	10	gaussian	gaussian	ADJ
ejpam-4669	181	11	periodic	periodic	NOUN
ejpam-4669	181	12	of	of	ADP
ejpam-4669	181	13	period	period	NOUN
ejpam-4669	181	14	1	1	NUM
ejpam-4669	181	15	,	,	PUNCT
ejpam-4669	181	16	we	we	PRON
ejpam-4669	181	17	have	have	VERB
ejpam-4669	181	18	gg(x+	gg(x+	ADV
ejpam-4669	181	19	2	2	NUM
ejpam-4669	181	20	)	)	PUNCT
ejpam-4669	181	21	=	=	SYM
ejpam-4669	181	22	ag(x+	ag(x+	ADV
ejpam-4669	181	23	2)fg(x+	2)fg(x+	CCONJ
ejpam-4669	181	24	2	2	X
ejpam-4669	181	25	)	)	PUNCT
ejpam-4669	181	26	=	=	PUNCT
ejpam-4669	182	1	ag(x)[f(x+	ag(x)[f(x+	ADV
ejpam-4669	182	2	2	2	NUM
ejpam-4669	182	3	)	)	PUNCT
ejpam-4669	182	4	+	+	CCONJ
ejpam-4669	182	5	if(x+	if(x+	ADV
ejpam-4669	182	6	1	1	NUM
ejpam-4669	182	7	)	)	PUNCT
ejpam-4669	182	8	]	]	PUNCT
ejpam-4669	183	1	=	=	PUNCT
ejpam-4669	183	2	ag(x)f(x+	ag(x)f(x+	ADP
ejpam-4669	183	3	2	2	NUM
ejpam-4669	183	4	)	)	PUNCT
ejpam-4669	183	5	+	+	CCONJ
ejpam-4669	183	6	iag(x)f(x+	iag(x)f(x+	ADV
ejpam-4669	183	7	1	1	NUM
ejpam-4669	183	8	)	)	PUNCT
ejpam-4669	183	9	=	=	SYM
ejpam-4669	184	1	ag(x+	ag(x+	PROPN
ejpam-4669	184	2	2)f(x+	2)f(x+	NOUN
ejpam-4669	184	3	2	2	NUM
ejpam-4669	184	4	)	)	PUNCT
ejpam-4669	184	5	+	+	CCONJ
ejpam-4669	185	1	iag(x+	iag(x+	PROPN
ejpam-4669	185	2	1)f(x+	1)f(x+	NUM
ejpam-4669	185	3	1	1	NUM
ejpam-4669	185	4	)	)	PUNCT
ejpam-4669	185	5	=	=	PUNCT
ejpam-4669	185	6	g(x+	g(x+	ADJ
ejpam-4669	185	7	2	2	X
ejpam-4669	185	8	)	)	PUNCT
ejpam-4669	185	9	+	+	CCONJ
ejpam-4669	185	10	ig(x+	ig(x+	ADP
ejpam-4669	185	11	1	1	NUM
ejpam-4669	185	12	)	)	PUNCT
ejpam-4669	185	13	.	.	PUNCT
ejpam-4669	186	1	hence	hence	ADV
ejpam-4669	186	2	,	,	PUNCT
ejpam-4669	186	3	gg(x	gg(x	PUNCT
ejpam-4669	186	4	)	)	PUNCT
ejpam-4669	186	5	is	be	AUX
ejpam-4669	186	6	a	a	DET
ejpam-4669	186	7	gaussian	gaussian	ADJ
ejpam-4669	186	8	fibonacci	fibonacci	NOUN
ejpam-4669	186	9	function	function	NOUN
ejpam-4669	186	10	.	.	PUNCT
ejpam-4669	187	1	we	we	PRON
ejpam-4669	187	2	ask	ask	VERB
ejpam-4669	187	3	the	the	DET
ejpam-4669	187	4	following	follow	VERB
ejpam-4669	187	5	question	question	NOUN
ejpam-4669	187	6	:	:	PUNCT
ejpam-4669	187	7	are	be	AUX
ejpam-4669	187	8	there	there	PRON
ejpam-4669	187	9	a	a	DET
ejpam-4669	187	10	gaussian	gaussian	ADJ
ejpam-4669	187	11	fibonacci	fibonacci	NOUN
ejpam-4669	187	12	function	function	NOUN
ejpam-4669	187	13	fg(x	fg(x	PUNCT
ejpam-4669	187	14	)	)	PUNCT
ejpam-4669	187	15	and	and	CCONJ
ejpam-4669	187	16	a	a	DET
ejpam-4669	187	17	function	function	NOUN
ejpam-4669	187	18	ag(x	ag(x	PUNCT
ejpam-4669	187	19	)	)	PUNCT
ejpam-4669	187	20	which	which	PRON
ejpam-4669	187	21	is	be	AUX
ejpam-4669	187	22	a	a	DET
ejpam-4669	187	23	gaussian	gaussian	NOUN
ejpam-4669	187	24	ultimately	ultimately	ADV
ejpam-4669	187	25	periodic	periodic	ADJ
ejpam-4669	187	26	of	of	ADP
ejpam-4669	187	27	period	period	NOUN
ejpam-4669	187	28	1	1	NUM
ejpam-4669	187	29	but	but	CCONJ
ejpam-4669	187	30	not	not	PART
ejpam-4669	187	31	periodic	periodic	NOUN
ejpam-4669	187	32	of	of	ADP
ejpam-4669	187	33	period	period	NOUN
ejpam-4669	187	34	1	1	NUM
ejpam-4669	187	35	such	such	ADJ
ejpam-4669	187	36	that	that	PRON
ejpam-4669	187	37	gg(x	gg(x	PUNCT
ejpam-4669	187	38	)	)	PUNCT
ejpam-4669	187	39	=	=	PUNCT
ejpam-4669	188	1	ag(x)fg(x	ag(x)fg(x	X
ejpam-4669	188	2	)	)	PUNCT
ejpam-4669	188	3	is	be	AUX
ejpam-4669	188	4	also	also	ADV
ejpam-4669	188	5	a	a	DET
ejpam-4669	188	6	gaussian	gaussian	ADJ
ejpam-4669	188	7	fibonacci	fibonacci	NOUN
ejpam-4669	188	8	function	function	NOUN
ejpam-4669	188	9	?	?	PUNCT
ejpam-4669	189	1	4	4	X
ejpam-4669	189	2	.	.	X
ejpam-4669	189	3	exponential	exponential	ADJ
ejpam-4669	189	4	gaussian	gaussian	PROPN
ejpam-4669	189	5	fibonacci	fibonacci	NOUN
ejpam-4669	189	6	functions	function	NOUN
ejpam-4669	189	7	consider	consider	VERB
ejpam-4669	189	8	a	a	DET
ejpam-4669	189	9	gaussian	gaussian	ADJ
ejpam-4669	189	10	map	map	NOUN
ejpam-4669	189	11	tg(x	tg(x	PUNCT
ejpam-4669	189	12	)	)	PUNCT
ejpam-4669	189	13	=	=	SYM
ejpam-4669	189	14	ln(x+i	ln(x+i	NOUN
ejpam-4669	189	15	)	)	PUNCT
ejpam-4669	189	16	ln(x	ln(x	X
ejpam-4669	189	17	)	)	PUNCT
ejpam-4669	189	18	with	with	ADP
ejpam-4669	189	19	domain	domain	NOUN
ejpam-4669	189	20	d	d	X
ejpam-4669	189	21	=	=	SYM
ejpam-4669	189	22	(	(	PUNCT
ejpam-4669	189	23	0,∞	0,∞	NUM
ejpam-4669	189	24	)	)	PUNCT
ejpam-4669	189	25	\	\	NOUN
ejpam-4669	189	26	{	{	PUNCT
ejpam-4669	189	27	1	1	NUM
ejpam-4669	189	28	}	}	PUNCT
ejpam-4669	189	29	.	.	PUNCT
ejpam-4669	190	1	if	if	SCONJ
ejpam-4669	190	2	we	we	PRON
ejpam-4669	190	3	let	let	VERB
ejpam-4669	190	4	c	c	NOUN
ejpam-4669	190	5	:	:	PUNCT
ejpam-4669	191	1	=	=	PUNCT
ejpam-4669	191	2	c	c	X
ejpam-4669	191	3	\	\	PUNCT
ejpam-4669	192	1	[	[	X
ejpam-4669	192	2	0	0	NUM
ejpam-4669	192	3	,	,	PUNCT
ejpam-4669	192	4	1	1	NUM
ejpam-4669	192	5	]	]	PUNCT
ejpam-4669	192	6	,	,	PUNCT
ejpam-4669	192	7	then	then	ADV
ejpam-4669	192	8	tg	tg	INTJ
ejpam-4669	192	9	:	:	PUNCT
ejpam-4669	192	10	d	d	X
ejpam-4669	192	11	→	→	SYM
ejpam-4669	192	12	c	c	NOUN
ejpam-4669	192	13	is	be	AUX
ejpam-4669	192	14	a	a	DET
ejpam-4669	192	15	bijective	bijective	ADJ
ejpam-4669	192	16	function	function	NOUN
ejpam-4669	192	17	.	.	PUNCT
ejpam-4669	193	1	proposition	proposition	NOUN
ejpam-4669	193	2	6	6	NUM
ejpam-4669	193	3	.	.	PUNCT
ejpam-4669	194	1	if	if	SCONJ
ejpam-4669	194	2	gg(x	gg(x	NOUN
ejpam-4669	194	3	)	)	PUNCT
ejpam-4669	194	4	=	=	SYM
ejpam-4669	194	5	a(x)f(x	a(x)f(x	NOUN
ejpam-4669	194	6	)	)	PUNCT
ejpam-4669	194	7	is	be	AUX
ejpam-4669	194	8	a	a	DET
ejpam-4669	194	9	gaussian	gaussian	ADJ
ejpam-4669	194	10	fibonacci	fibonacci	NOUN
ejpam-4669	194	11	function	function	NOUN
ejpam-4669	194	12	where	where	SCONJ
ejpam-4669	194	13	a(x	a(x	NOUN
ejpam-4669	194	14	)	)	PUNCT
ejpam-4669	194	15	>	>	X
ejpam-4669	194	16	0	0	NUM
ejpam-4669	194	17	,	,	PUNCT
ejpam-4669	194	18	then	then	ADV
ejpam-4669	194	19	there	there	PRON
ejpam-4669	194	20	exists	exist	VERB
ejpam-4669	194	21	γ(x	γ(x	NOUN
ejpam-4669	194	22	)	)	PUNCT
ejpam-4669	194	23	∈	∈	PROPN
ejpam-4669	194	24	c	c	NOUN
ejpam-4669	194	25	such	such	ADJ
ejpam-4669	194	26	that	that	SCONJ
ejpam-4669	194	27	gg(x+	gg(x+	PROPN
ejpam-4669	194	28	2	2	NUM
ejpam-4669	194	29	)	)	PUNCT
ejpam-4669	194	30	g(x+	g(x+	NOUN
ejpam-4669	194	31	1	1	X
ejpam-4669	194	32	)	)	PUNCT
ejpam-4669	195	1	=	=	NOUN
ejpam-4669	196	1	[	[	PUNCT
ejpam-4669	196	2	g(x+	g(x+	ADP
ejpam-4669	196	3	2	2	NUM
ejpam-4669	196	4	)	)	PUNCT
ejpam-4669	196	5	g(x+	g(x+	ADV
ejpam-4669	196	6	1	1	NUM
ejpam-4669	196	7	)	)	PUNCT
ejpam-4669	196	8	]	]	PUNCT
ejpam-4669	196	9	γ(x	γ(x	NOUN
ejpam-4669	196	10	)	)	PUNCT
ejpam-4669	196	11	.	.	PUNCT
ejpam-4669	197	1	proof	proof	NOUN
ejpam-4669	197	2	.	.	PUNCT
ejpam-4669	198	1	if	if	SCONJ
ejpam-4669	198	2	gg(x	gg(x	NOUN
ejpam-4669	198	3	)	)	PUNCT
ejpam-4669	198	4	=	=	SYM
ejpam-4669	198	5	a(x)f(x	a(x)f(x	NOUN
ejpam-4669	198	6	)	)	PUNCT
ejpam-4669	198	7	,	,	PUNCT
ejpam-4669	198	8	a(x	a(x	PROPN
ejpam-4669	198	9	)	)	PUNCT
ejpam-4669	198	10	>	>	X
ejpam-4669	198	11	0	0	NUM
ejpam-4669	198	12	,	,	PUNCT
ejpam-4669	198	13	then	then	ADV
ejpam-4669	198	14	gg(x	gg(x	PUNCT
ejpam-4669	198	15	)	)	PUNCT
ejpam-4669	198	16	>	>	X
ejpam-4669	199	1	0	0	X
ejpam-4669	199	2	.	.	PUNCT
ejpam-4669	199	3	assume	assume	VERB
ejpam-4669	199	4	gg(x+	gg(x+	ADV
ejpam-4669	199	5	2	2	NUM
ejpam-4669	199	6	)	)	PUNCT
ejpam-4669	199	7	g(x+	g(x+	NOUN
ejpam-4669	199	8	1	1	X
ejpam-4669	199	9	)	)	PUNCT
ejpam-4669	200	1	=	=	NOUN
ejpam-4669	201	1	[	[	PUNCT
ejpam-4669	201	2	g(x+	g(x+	ADP
ejpam-4669	201	3	2	2	NUM
ejpam-4669	201	4	)	)	PUNCT
ejpam-4669	201	5	g(x+	g(x+	ADV
ejpam-4669	201	6	1	1	NUM
ejpam-4669	201	7	)	)	PUNCT
ejpam-4669	201	8	]	]	PUNCT
ejpam-4669	201	9	γ(x	γ(x	NOUN
ejpam-4669	201	10	)	)	PUNCT
ejpam-4669	201	11	for	for	ADP
ejpam-4669	201	12	some	some	DET
ejpam-4669	201	13	γ(x	γ(x	NOUN
ejpam-4669	201	14	)	)	PUNCT
ejpam-4669	201	15	.	.	PUNCT
ejpam-4669	202	1	if	if	SCONJ
ejpam-4669	202	2	we	we	PRON
ejpam-4669	202	3	let	let	VERB
ejpam-4669	202	4	b(x	b(x	NOUN
ejpam-4669	202	5	)	)	PUNCT
ejpam-4669	202	6	:	:	PUNCT
ejpam-4669	202	7	=	=	PUNCT
ejpam-4669	202	8	g(x+2	g(x+2	PROPN
ejpam-4669	202	9	)	)	PUNCT
ejpam-4669	202	10	g(x+1	g(x+1	PROPN
ejpam-4669	202	11	)	)	PUNCT
ejpam-4669	202	12	,	,	PUNCT
ejpam-4669	202	13	then	then	ADV
ejpam-4669	202	14	b(x)γ(x	b(x)γ(x	VERB
ejpam-4669	202	15	)	)	PUNCT
ejpam-4669	202	16	=	=	PUNCT
ejpam-4669	203	1	gg(x+	gg(x+	ADV
ejpam-4669	203	2	2	2	NUM
ejpam-4669	203	3	)	)	PUNCT
ejpam-4669	203	4	g(x+	g(x+	NOUN
ejpam-4669	203	5	1	1	X
ejpam-4669	203	6	)	)	PUNCT
ejpam-4669	203	7	=	=	SYM
ejpam-4669	204	1	g(x+	g(x+	ADJ
ejpam-4669	204	2	2	2	X
ejpam-4669	204	3	)	)	PUNCT
ejpam-4669	204	4	+	+	CCONJ
ejpam-4669	204	5	ig(x+	ig(x+	ADP
ejpam-4669	204	6	1	1	NUM
ejpam-4669	204	7	)	)	PUNCT
ejpam-4669	204	8	g(x+	g(x+	ADV
ejpam-4669	204	9	1	1	X
ejpam-4669	204	10	)	)	PUNCT
ejpam-4669	204	11	=	=	SYM
ejpam-4669	204	12	b(x	b(x	NOUN
ejpam-4669	204	13	)	)	PUNCT
ejpam-4669	205	1	+	+	NUM
ejpam-4669	205	2	i.	i.	NOUN
ejpam-4669	205	3	it	it	PRON
ejpam-4669	205	4	follows	follow	VERB
ejpam-4669	205	5	that	that	SCONJ
ejpam-4669	205	6	γ(x	γ(x	VERB
ejpam-4669	205	7	)	)	PUNCT
ejpam-4669	205	8	=	=	PUNCT
ejpam-4669	205	9	ln(b(x	ln(b(x	ADJ
ejpam-4669	205	10	)	)	PUNCT
ejpam-4669	206	1	+	+	CCONJ
ejpam-4669	206	2	i	i	NOUN
ejpam-4669	206	3	)	)	PUNCT
ejpam-4669	206	4	lnb(x	lnb(x	PROPN
ejpam-4669	206	5	)	)	PUNCT
ejpam-4669	207	1	=	=	SYM
ejpam-4669	207	2	ln	ln	ADJ
ejpam-4669	207	3	(	(	PUNCT
ejpam-4669	207	4	g(x+2	g(x+2	PROPN
ejpam-4669	207	5	)	)	PUNCT
ejpam-4669	207	6	g(x+1	g(x+1	PROPN
ejpam-4669	207	7	)	)	PUNCT
ejpam-4669	208	1	+	+	CCONJ
ejpam-4669	208	2	i	i	NOUN
ejpam-4669	208	3	)	)	PUNCT
ejpam-4669	209	1	ln	ln	INTJ
ejpam-4669	209	2	(	(	PUNCT
ejpam-4669	209	3	g(x+2	g(x+2	PROPN
ejpam-4669	209	4	)	)	PUNCT
ejpam-4669	209	5	g(x+1	g(x+1	PROPN
ejpam-4669	209	6	)	)	PUNCT
ejpam-4669	209	7	)	)	PUNCT
ejpam-4669	210	1	=	=	PUNCT
ejpam-4669	210	2	tg(b(x	tg(b(x	NOUN
ejpam-4669	210	3	)	)	PUNCT
ejpam-4669	210	4	)	)	PUNCT
ejpam-4669	211	1	∈	∈	PROPN
ejpam-4669	211	2	c.	c.	PROPN
ejpam-4669	211	3	h.	h.	PROPN
ejpam-4669	211	4	f.	f.	PROPN
ejpam-4669	211	5	m.salih	m.salih	PROPN
ejpam-4669	211	6	/	/	SYM
ejpam-4669	211	7	eur	eur	PROPN
ejpam-4669	211	8	.	.	PUNCT
ejpam-4669	212	1	j.	j.	PROPN
ejpam-4669	212	2	pure	pure	PROPN
ejpam-4669	212	3	appl	appl	PROPN
ejpam-4669	212	4	.	.	PROPN
ejpam-4669	212	5	math	math	PROPN
ejpam-4669	212	6	,	,	PUNCT
ejpam-4669	212	7	16	16	NUM
ejpam-4669	212	8	(	(	PUNCT
ejpam-4669	212	9	1	1	NUM
ejpam-4669	212	10	)	)	PUNCT
ejpam-4669	212	11	(	(	PUNCT
ejpam-4669	212	12	2023	2023	NUM
ejpam-4669	212	13	)	)	PUNCT
ejpam-4669	212	14	,	,	PUNCT
ejpam-4669	212	15	587	587	NUM
ejpam-4669	212	16	-	-	SYM
ejpam-4669	212	17	594	594	NUM
ejpam-4669	212	18	593	593	NUM
ejpam-4669	212	19	this	this	PRON
ejpam-4669	212	20	has	have	AUX
ejpam-4669	212	21	proved	prove	VERB
ejpam-4669	212	22	the	the	DET
ejpam-4669	212	23	proposition	proposition	NOUN
ejpam-4669	212	24	.	.	PUNCT
ejpam-4669	213	1	proposition	proposition	NOUN
ejpam-4669	213	2	7	7	NUM
ejpam-4669	213	3	.	.	PUNCT
ejpam-4669	214	1	there	there	PRON
ejpam-4669	214	2	is	be	VERB
ejpam-4669	214	3	no	no	DET
ejpam-4669	214	4	gaussian	gaussian	ADJ
ejpam-4669	214	5	fibonacci	fibonacci	NOUN
ejpam-4669	214	6	function	function	VERB
ejpam-4669	214	7	fg(x	fg(x	PUNCT
ejpam-4669	214	8	)	)	PUNCT
ejpam-4669	214	9	such	such	ADJ
ejpam-4669	214	10	that	that	PRON
ejpam-4669	214	11	gg(x	gg(x	PUNCT
ejpam-4669	214	12	)	)	PUNCT
ejpam-4669	214	13	=	=	SYM
ejpam-4669	214	14	afg(x	afg(x	X
ejpam-4669	214	15	)	)	PUNCT
ejpam-4669	214	16	and	and	CCONJ
ejpam-4669	214	17	g(x	g(x	NOUN
ejpam-4669	214	18	)	)	PUNCT
ejpam-4669	214	19	=	=	PUNCT
ejpam-4669	214	20	af(x	af(x	NOUN
ejpam-4669	214	21	)	)	PUNCT
ejpam-4669	214	22	,	,	PUNCT
ejpam-4669	214	23	a	a	PRON
ejpam-4669	214	24	>	>	X
ejpam-4669	214	25	0	0	NUM
ejpam-4669	214	26	where	where	SCONJ
ejpam-4669	214	27	fg(x	fg(x	NUM
ejpam-4669	214	28	)	)	PUNCT
ejpam-4669	214	29	and	and	CCONJ
ejpam-4669	214	30	f(x	f(x	PROPN
ejpam-4669	214	31	)	)	PUNCT
ejpam-4669	214	32	are	be	AUX
ejpam-4669	214	33	differentiable	differentiable	ADJ
ejpam-4669	214	34	and	and	CCONJ
ejpam-4669	214	35	gg(x	gg(x	X
ejpam-4669	214	36	)	)	PUNCT
ejpam-4669	214	37	and	and	CCONJ
ejpam-4669	214	38	g(x	g(x	NOUN
ejpam-4669	214	39	)	)	PUNCT
ejpam-4669	214	40	are	be	AUX
ejpam-4669	214	41	gaussian	gaussian	ADJ
ejpam-4669	214	42	fibonacci	fibonacci	NOUN
ejpam-4669	214	43	function	function	PROPN
ejpam-4669	214	44	and	and	CCONJ
ejpam-4669	214	45	fibonacci	fibonacci	NOUN
ejpam-4669	214	46	function	function	NOUN
ejpam-4669	214	47	,	,	PUNCT
ejpam-4669	214	48	respectively	respectively	ADV
ejpam-4669	214	49	.	.	PUNCT
ejpam-4669	215	1	proof	proof	NOUN
ejpam-4669	215	2	.	.	PUNCT
ejpam-4669	216	1	suppose	suppose	VERB
ejpam-4669	216	2	that	that	SCONJ
ejpam-4669	216	3	fg(x	fg(x	NUM
ejpam-4669	216	4	)	)	PUNCT
ejpam-4669	216	5	is	be	AUX
ejpam-4669	216	6	a	a	DET
ejpam-4669	216	7	gaussian	gaussian	ADJ
ejpam-4669	216	8	fibonacci	fibonacci	NOUN
ejpam-4669	216	9	function	function	NOUN
ejpam-4669	216	10	.	.	PUNCT
ejpam-4669	217	1	since	since	SCONJ
ejpam-4669	217	2	fg(x	fg(x	NUM
ejpam-4669	217	3	)	)	PUNCT
ejpam-4669	217	4	is	be	AUX
ejpam-4669	217	5	differentiable	differentiable	ADJ
ejpam-4669	217	6	,	,	PUNCT
ejpam-4669	217	7	we	we	PRON
ejpam-4669	217	8	have	have	VERB
ejpam-4669	217	9	f	f	PROPN
ejpam-4669	217	10	′g(x+	′g(x+	PROPN
ejpam-4669	217	11	2	2	NUM
ejpam-4669	217	12	)	)	PUNCT
ejpam-4669	217	13	=	=	SYM
ejpam-4669	217	14	f	f	PROPN
ejpam-4669	217	15	′(x+	′(x+	ADP
ejpam-4669	217	16	2	2	NUM
ejpam-4669	217	17	)	)	PUNCT
ejpam-4669	218	1	+	+	CCONJ
ejpam-4669	218	2	if	if	SCONJ
ejpam-4669	218	3	′(x+	′(x+	ADP
ejpam-4669	218	4	2	2	NUM
ejpam-4669	218	5	)	)	PUNCT
ejpam-4669	218	6	(	(	PUNCT
ejpam-4669	218	7	1	1	X
ejpam-4669	218	8	)	)	PUNCT
ejpam-4669	218	9	since	since	SCONJ
ejpam-4669	218	10	gg(x	gg(x	NOUN
ejpam-4669	218	11	)	)	PUNCT
ejpam-4669	218	12	is	be	AUX
ejpam-4669	218	13	a	a	DET
ejpam-4669	218	14	gaussian	gaussian	ADJ
ejpam-4669	218	15	fibonacci	fibonacci	NOUN
ejpam-4669	218	16	function	function	NOUN
ejpam-4669	218	17	,	,	PUNCT
ejpam-4669	218	18	then	then	ADV
ejpam-4669	218	19	gg(x+2	gg(x+2	NOUN
ejpam-4669	218	20	)	)	PUNCT
ejpam-4669	218	21	=	=	SYM
ejpam-4669	218	22	g(x+2)+	g(x+2)+	NOUN
ejpam-4669	218	23	ig(x+1	ig(x+1	NOUN
ejpam-4669	218	24	)	)	PUNCT
ejpam-4669	218	25	.	.	PUNCT
ejpam-4669	219	1	since	since	SCONJ
ejpam-4669	219	2	gg(x	gg(x	ADV
ejpam-4669	219	3	+	+	CCONJ
ejpam-4669	219	4	2	2	X
ejpam-4669	219	5	)	)	PUNCT
ejpam-4669	219	6	=	=	SYM
ejpam-4669	219	7	afg(x	afg(x	X
ejpam-4669	219	8	)	)	PUNCT
ejpam-4669	219	9	and	and	CCONJ
ejpam-4669	219	10	g(x	g(x	NOUN
ejpam-4669	219	11	)	)	PUNCT
ejpam-4669	219	12	=	=	PUNCT
ejpam-4669	219	13	af(x	af(x	X
ejpam-4669	219	14	)	)	PUNCT
ejpam-4669	219	15	and	and	CCONJ
ejpam-4669	219	16	fg(x	fg(x	NUM
ejpam-4669	219	17	)	)	PUNCT
ejpam-4669	219	18	and	and	CCONJ
ejpam-4669	219	19	fg(x	fg(x	NUM
ejpam-4669	219	20	)	)	PUNCT
ejpam-4669	219	21	are	be	AUX
ejpam-4669	219	22	differential	differential	ADJ
ejpam-4669	219	23	,	,	PUNCT
ejpam-4669	219	24	g′g(x	g′g(x	NOUN
ejpam-4669	219	25	+	+	NOUN
ejpam-4669	219	26	2	2	X
ejpam-4669	219	27	)	)	PUNCT
ejpam-4669	219	28	=	=	PUNCT
ejpam-4669	220	1	g′(x	g′(x	PROPN
ejpam-4669	221	1	+	+	CCONJ
ejpam-4669	221	2	2	2	X
ejpam-4669	221	3	)	)	PUNCT
ejpam-4669	221	4	+	+	CCONJ
ejpam-4669	221	5	ig′(x	ig′(x	NOUN
ejpam-4669	221	6	+	+	ADJ
ejpam-4669	221	7	2	2	NUM
ejpam-4669	221	8	)	)	PUNCT
ejpam-4669	221	9	,	,	PUNCT
ejpam-4669	221	10	i.e.	i.e.	X
ejpam-4669	221	11	,	,	PUNCT
ejpam-4669	221	12	g′g(x	g′g(x	NOUN
ejpam-4669	221	13	)	)	PUNCT
ejpam-4669	221	14	is	be	AUX
ejpam-4669	221	15	also	also	ADV
ejpam-4669	221	16	a	a	DET
ejpam-4669	221	17	gaussian	gaussian	ADJ
ejpam-4669	221	18	fibonacci	fibonacci	NOUN
ejpam-4669	221	19	function	function	NOUN
ejpam-4669	221	20	.	.	PUNCT
ejpam-4669	222	1	it	it	PRON
ejpam-4669	222	2	follows	follow	VERB
ejpam-4669	222	3	from	from	ADP
ejpam-4669	222	4	g′g(x	g′g(x	NOUN
ejpam-4669	222	5	)	)	PUNCT
ejpam-4669	223	1	=	=	SYM
ejpam-4669	223	2	gg(x	gg(x	X
ejpam-4669	223	3	)	)	PUNCT
ejpam-4669	223	4	lnaf	lnaf	NOUN
ejpam-4669	223	5	′	′	NUM
ejpam-4669	223	6	g(x	g(x	NOUN
ejpam-4669	223	7	)	)	PUNCT
ejpam-4669	223	8	and	and	CCONJ
ejpam-4669	223	9	g	g	NOUN
ejpam-4669	223	10	′(x	′(x	NOUN
ejpam-4669	223	11	)	)	PUNCT
ejpam-4669	223	12	=	=	SYM
ejpam-4669	223	13	g(x	g(x	NOUN
ejpam-4669	223	14	)	)	PUNCT
ejpam-4669	223	15	lnaf	lnaf	NOUN
ejpam-4669	223	16	′(x	′(x	NOUN
ejpam-4669	223	17	)	)	PUNCT
ejpam-4669	223	18	that	that	SCONJ
ejpam-4669	223	19	gg(x+	gg(x+	ADV
ejpam-4669	223	20	2	2	NUM
ejpam-4669	223	21	)	)	PUNCT
ejpam-4669	223	22	lnaf	lnaf	NOUN
ejpam-4669	224	1	′g(x+	′g(x+	ADP
ejpam-4669	224	2	2	2	NUM
ejpam-4669	224	3	)	)	PUNCT
ejpam-4669	224	4	=	=	NOUN
ejpam-4669	224	5	g′g(x+	g′g(x+	X
ejpam-4669	224	6	2	2	X
ejpam-4669	224	7	)	)	PUNCT
ejpam-4669	224	8	=	=	NOUN
ejpam-4669	224	9	g′(x+	g′(x+	ADP
ejpam-4669	224	10	2	2	NUM
ejpam-4669	224	11	)	)	PUNCT
ejpam-4669	224	12	+	+	CCONJ
ejpam-4669	224	13	ig′(x+	ig′(x+	ADP
ejpam-4669	224	14	1	1	NUM
ejpam-4669	224	15	)	)	PUNCT
ejpam-4669	224	16	=	=	PUNCT
ejpam-4669	224	17	g(x+	g(x+	ADJ
ejpam-4669	224	18	2	2	X
ejpam-4669	224	19	)	)	PUNCT
ejpam-4669	224	20	lnaf	lnaf	NOUN
ejpam-4669	224	21	′(x+	′(x+	ADP
ejpam-4669	224	22	2	2	NUM
ejpam-4669	224	23	)	)	PUNCT
ejpam-4669	224	24	+	+	CCONJ
ejpam-4669	224	25	ig(x+	ig(x+	ADP
ejpam-4669	224	26	1	1	NUM
ejpam-4669	224	27	)	)	PUNCT
ejpam-4669	224	28	lnaf	lnaf	NOUN
ejpam-4669	224	29	′(x+	′(x+	ADP
ejpam-4669	224	30	1	1	NUM
ejpam-4669	224	31	)	)	PUNCT
ejpam-4669	224	32	comparing	compare	VERB
ejpam-4669	224	33	the	the	DET
ejpam-4669	224	34	two	two	NUM
ejpam-4669	224	35	sides	side	NOUN
ejpam-4669	224	36	,	,	PUNCT
ejpam-4669	224	37	we	we	PRON
ejpam-4669	224	38	obtain	obtain	VERB
ejpam-4669	224	39	f	f	PROPN
ejpam-4669	225	1	′g(x+	′g(x+	PROPN
ejpam-4669	225	2	2	2	NUM
ejpam-4669	225	3	)	)	PUNCT
ejpam-4669	225	4	=	=	SYM
ejpam-4669	225	5	g(x+	g(x+	ADJ
ejpam-4669	225	6	2	2	X
ejpam-4669	225	7	)	)	PUNCT
ejpam-4669	225	8	gg(x+	gg(x+	ADV
ejpam-4669	225	9	2	2	NUM
ejpam-4669	225	10	)	)	PUNCT
ejpam-4669	225	11	f	f	NOUN
ejpam-4669	225	12	′(x+	′(x+	ADP
ejpam-4669	225	13	2	2	NUM
ejpam-4669	225	14	)	)	PUNCT
ejpam-4669	226	1	+	+	CCONJ
ejpam-4669	226	2	i	i	PRON
ejpam-4669	226	3	g(x+	g(x+	NOUN
ejpam-4669	226	4	1	1	X
ejpam-4669	226	5	)	)	PUNCT
ejpam-4669	226	6	gg(x+	gg(x+	ADV
ejpam-4669	226	7	2	2	NUM
ejpam-4669	226	8	)	)	PUNCT
ejpam-4669	226	9	f	f	NOUN
ejpam-4669	226	10	′(x+	′(x+	ADP
ejpam-4669	226	11	1	1	NUM
ejpam-4669	226	12	)	)	PUNCT
ejpam-4669	226	13	(	(	PUNCT
ejpam-4669	226	14	2	2	NUM
ejpam-4669	226	15	)	)	PUNCT
ejpam-4669	226	16	from	from	ADP
ejpam-4669	226	17	equations	equation	NOUN
ejpam-4669	226	18	(	(	PUNCT
ejpam-4669	226	19	1	1	NUM
ejpam-4669	226	20	)	)	PUNCT
ejpam-4669	226	21	and	and	CCONJ
ejpam-4669	226	22	(	(	PUNCT
ejpam-4669	226	23	2	2	NUM
ejpam-4669	226	24	)	)	PUNCT
ejpam-4669	226	25	,	,	PUNCT
ejpam-4669	226	26	we	we	PRON
ejpam-4669	226	27	obtain	obtain	VERB
ejpam-4669	226	28	[	[	PUNCT
ejpam-4669	226	29	g(x+	g(x+	NOUN
ejpam-4669	226	30	2	2	NUM
ejpam-4669	226	31	)	)	PUNCT
ejpam-4669	226	32	gg(x+	gg(x+	ADV
ejpam-4669	226	33	2	2	NUM
ejpam-4669	226	34	)	)	PUNCT
ejpam-4669	226	35	−	−	NOUN
ejpam-4669	226	36	1	1	NUM
ejpam-4669	226	37	]	]	SYM
ejpam-4669	226	38	f	f	PROPN
ejpam-4669	226	39	′(x+	′(x+	ADP
ejpam-4669	226	40	2	2	NUM
ejpam-4669	226	41	)	)	PUNCT
ejpam-4669	227	1	+	+	CCONJ
ejpam-4669	227	2	i	i	PRON
ejpam-4669	227	3	[	[	PUNCT
ejpam-4669	227	4	g(x+	g(x+	NOUN
ejpam-4669	227	5	1	1	X
ejpam-4669	227	6	)	)	PUNCT
ejpam-4669	227	7	gg(x+	gg(x+	ADV
ejpam-4669	227	8	2	2	NUM
ejpam-4669	227	9	)	)	PUNCT
ejpam-4669	227	10	−	−	NOUN
ejpam-4669	227	11	1	1	NUM
ejpam-4669	227	12	]	]	SYM
ejpam-4669	227	13	f	f	PROPN
ejpam-4669	227	14	′(x+	′(x+	ADP
ejpam-4669	227	15	1	1	NUM
ejpam-4669	227	16	)	)	PUNCT
ejpam-4669	227	17	=	=	SYM
ejpam-4669	227	18	0	0	PUNCT
ejpam-4669	228	1	this	this	PRON
ejpam-4669	228	2	implies	imply	VERB
ejpam-4669	228	3	that	that	SCONJ
ejpam-4669	228	4	f	f	PROPN
ejpam-4669	228	5	′(x+	′(x+	ADP
ejpam-4669	228	6	2	2	X
ejpam-4669	228	7	)	)	PUNCT
ejpam-4669	228	8	if	if	SCONJ
ejpam-4669	228	9	′(x+	′(x+	ADP
ejpam-4669	228	10	1	1	X
ejpam-4669	228	11	)	)	PUNCT
ejpam-4669	228	12	=	=	VERB
ejpam-4669	228	13	g(x+	g(x+	PROPN
ejpam-4669	229	1	1)−	1)−	NUM
ejpam-4669	229	2	gg(x+	gg(x+	ADV
ejpam-4669	229	3	2	2	NUM
ejpam-4669	229	4	)	)	PUNCT
ejpam-4669	229	5	g(x+	g(x+	X
ejpam-4669	230	1	2)−	2)−	NUM
ejpam-4669	230	2	gg(x+	gg(x+	ADP
ejpam-4669	230	3	2	2	NUM
ejpam-4669	230	4	)	)	PUNCT
ejpam-4669	230	5	=	=	PUNCT
ejpam-4669	230	6	g(x+	g(x+	PROPN
ejpam-4669	231	1	1)−	1)−	NUM
ejpam-4669	231	2	gg(x+	gg(x+	ADP
ejpam-4669	231	3	2	2	NUM
ejpam-4669	231	4	)	)	PUNCT
ejpam-4669	231	5	ig(x+	ig(x+	ADP
ejpam-4669	231	6	1	1	NUM
ejpam-4669	231	7	)	)	PUNCT
ejpam-4669	231	8	=	=	SYM
ejpam-4669	232	1	1	1	NUM
ejpam-4669	232	2	i	i	PRON
ejpam-4669	232	3	−	−	VERB
ejpam-4669	233	1	g(x+	g(x+	CCONJ
ejpam-4669	233	2	2	2	X
ejpam-4669	233	3	)	)	PUNCT
ejpam-4669	233	4	ig(x+	ig(x+	ADP
ejpam-4669	233	5	1	1	NUM
ejpam-4669	233	6	)	)	PUNCT
ejpam-4669	233	7	−	−	PROPN
ejpam-4669	233	8	1	1	NUM
ejpam-4669	233	9	this	this	PRON
ejpam-4669	233	10	follows	follow	VERB
ejpam-4669	233	11	that	that	PRON
ejpam-4669	233	12	ϕ	ϕ	PROPN
ejpam-4669	233	13	=	=	PROPN
ejpam-4669	233	14	lim	lim	PROPN
ejpam-4669	234	1	x→∞	x→∞	PROPN
ejpam-4669	234	2	f	f	PROPN
ejpam-4669	234	3	′(x+	′(x+	ADP
ejpam-4669	234	4	2	2	NUM
ejpam-4669	234	5	)	)	PUNCT
ejpam-4669	234	6	f	f	NOUN
ejpam-4669	234	7	′(x+	′(x+	ADP
ejpam-4669	234	8	1	1	NUM
ejpam-4669	234	9	)	)	PUNCT
ejpam-4669	234	10	=	=	SYM
ejpam-4669	235	1	1−	1−	NUM
ejpam-4669	235	2	lim	lim	NOUN
ejpam-4669	235	3	x→∞	x→∞	X
ejpam-4669	236	1	g(x+	g(x+	CCONJ
ejpam-4669	236	2	2	2	X
ejpam-4669	236	3	)	)	PUNCT
ejpam-4669	236	4	g(x+	g(x+	ADV
ejpam-4669	236	5	1	1	X
ejpam-4669	236	6	)	)	PUNCT
ejpam-4669	236	7	−	−	NOUN
ejpam-4669	237	1	i	i	PRON
ejpam-4669	237	2	=	=	SYM
ejpam-4669	237	3	1−	1−	NUM
ejpam-4669	237	4	ϕ−	ϕ−	NUM
ejpam-4669	237	5	i	i	PRON
ejpam-4669	237	6	which	which	PRON
ejpam-4669	237	7	is	be	AUX
ejpam-4669	237	8	contradiction	contradiction	NOUN
ejpam-4669	237	9	because	because	SCONJ
ejpam-4669	237	10	ϕ	ϕ	NOUN
ejpam-4669	237	11	=	=	SYM
ejpam-4669	237	12	1	1	NUM
ejpam-4669	237	13	+	+	NUM
ejpam-4669	237	14	√	√	NUM
ejpam-4669	237	15	5	5	NUM
ejpam-4669	237	16	2	2	NUM
ejpam-4669	237	17	but	but	CCONJ
ejpam-4669	237	18	we	we	PRON
ejpam-4669	237	19	obtain	obtain	VERB
ejpam-4669	237	20	that	that	DET
ejpam-4669	237	21	ϕ	ϕ	NOUN
ejpam-4669	237	22	=	=	SYM
ejpam-4669	237	23	1−i	1−i	NUM
ejpam-4669	237	24	2	2	NUM
ejpam-4669	237	25	.	.	PUNCT
ejpam-4669	238	1	hence	hence	ADV
ejpam-4669	238	2	,	,	PUNCT
ejpam-4669	238	3	fg(x	fg(x	NUM
ejpam-4669	238	4	)	)	PUNCT
ejpam-4669	238	5	is	be	AUX
ejpam-4669	238	6	a	a	DET
ejpam-4669	238	7	gaussian	gaussian	ADJ
ejpam-4669	238	8	fibonacci	fibonacci	NOUN
ejpam-4669	238	9	function	function	NOUN
ejpam-4669	238	10	.	.	PUNCT
ejpam-4669	239	1	references	reference	NOUN
ejpam-4669	239	2	594	594	NUM
ejpam-4669	239	3	references	reference	NOUN
ejpam-4669	239	4	[	[	X
ejpam-4669	239	5	1	1	NUM
ejpam-4669	239	6	]	]	PUNCT
ejpam-4669	239	7	a.	a.	NOUN
ejpam-4669	239	8	f.	f.	PROPN
ejpam-4669	239	9	horadam	horadam	PROPN
ejpam-4669	239	10	,	,	PUNCT
ejpam-4669	239	11	further	further	ADJ
ejpam-4669	239	12	appearance	appearance	NOUN
ejpam-4669	239	13	of	of	ADP
ejpam-4669	239	14	the	the	DET
ejpam-4669	239	15	fibonacci	fibonacci	NOUN
ejpam-4669	239	16	sequence	sequence	NOUN
ejpam-4669	239	17	·	·	PUNCT
ejpam-4669	239	18	the	the	DET
ejpam-4669	239	19	fibonacci	fibonacci	NOUN
ejpam-4669	239	20	quarterly	quarterly	ADV
ejpam-4669	239	21	,	,	PUNCT
ejpam-4669	239	22	1:4	1:4	NUM
ejpam-4669	239	23	(	(	PUNCT
ejpam-4669	239	24	dec	dec	PROPN
ejpam-4669	239	25	)	)	PUNCT
ejpam-4669	239	26	,	,	PUNCT
ejpam-4669	239	27	41	41	NUM
ejpam-4669	239	28	-	-	SYM
ejpam-4669	239	29	42	42	NUM
ejpam-4669	239	30	,	,	PUNCT
ejpam-4669	239	31	46	46	NUM
ejpam-4669	239	32	1963	1963	NUM
ejpam-4669	239	33	.	.	PUNCT
ejpam-4669	240	1	[	[	X
ejpam-4669	240	2	2	2	NUM
ejpam-4669	240	3	]	]	PUNCT
ejpam-4669	240	4	f.	f.	PROPN
ejpam-4669	240	5	t.	t.	PROPN
ejpam-4669	240	6	howard	howard	PROPN
ejpam-4669	240	7	,	,	PUNCT
ejpam-4669	240	8	applications	application	NOUN
ejpam-4669	240	9	of	of	ADP
ejpam-4669	240	10	fibonacci	fibonacci	NOUN
ejpam-4669	240	11	numbers	number	NOUN
ejpam-4669	240	12	:	:	PUNCT
ejpam-4669	240	13	volume	volume	NOUN
ejpam-4669	240	14	9	9	NUM
ejpam-4669	240	15	:	:	PUNCT
ejpam-4669	240	16	proceedings	proceeding	NOUN
ejpam-4669	240	17	of	of	ADP
ejpam-4669	240	18	the	the	DET
ejpam-4669	240	19	tenth	tenth	ADJ
ejpam-4669	240	20	international	international	ADJ
ejpam-4669	240	21	research	research	NOUN
ejpam-4669	240	22	conference	conference	NOUN
ejpam-4669	240	23	on	on	ADP
ejpam-4669	240	24	fibonacci	fibonacci	NOUN
ejpam-4669	240	25	numbers	number	NOUN
ejpam-4669	240	26	and	and	CCONJ
ejpam-4669	240	27	their	their	PRON
ejpam-4669	240	28	applications	application	NOUN
ejpam-4669	240	29	.	.	PUNCT
ejpam-4669	241	1	dordrecht	dordrecht	PROPN
ejpam-4669	241	2	,	,	PUNCT
ejpam-4669	241	3	springer	springer	NOUN
ejpam-4669	241	4	netherlands	netherlands	PROPN
ejpam-4669	241	5	,	,	PUNCT
ejpam-4669	241	6	2004	2004	NUM
ejpam-4669	241	7	.	.	PUNCT
ejpam-4669	242	1	[	[	X
ejpam-4669	242	2	3	3	X
ejpam-4669	242	3	]	]	X
ejpam-4669	242	4	h.	h.	PROPN
ejpam-4669	242	5	s.	s.	PROPN
ejpam-4669	242	6	jeong	jeong	PROPN
ejpam-4669	242	7	,	,	PUNCT
ejpam-4669	242	8	k.	k.	PROPN
ejpam-4669	242	9	s.	s.	PROPN
ejpam-4669	242	10	hee	hee	PROPN
ejpam-4669	242	11	,	,	PUNCT
ejpam-4669	242	12	&	&	CCONJ
ejpam-4669	242	13	n.	n.	PROPN
ejpam-4669	242	14	joseph	joseph	PROPN
ejpam-4669	242	15	,	,	PUNCT
ejpam-4669	242	16	on	on	ADP
ejpam-4669	242	17	fibonacci	fibonacci	NOUN
ejpam-4669	242	18	functions	function	NOUN
ejpam-4669	242	19	with	with	ADP
ejpam-4669	242	20	fibonacci	fibonacci	NOUN
ejpam-4669	242	21	numbers	number	NOUN
ejpam-4669	242	22	.	.	PUNCT
ejpam-4669	243	1	advances	advance	NOUN
ejpam-4669	243	2	in	in	ADP
ejpam-4669	243	3	difference	difference	NOUN
ejpam-4669	243	4	equations	equation	NOUN
ejpam-4669	243	5	.	.	PUNCT
ejpam-4669	244	1	126	126	NUM
ejpam-4669	244	2	,	,	PUNCT
ejpam-4669	244	3	2012	2012	NUM
ejpam-4669	244	4	.	.	PUNCT
ejpam-4669	245	1	[	[	X
ejpam-4669	245	2	4	4	NUM
ejpam-4669	245	3	]	]	X
ejpam-4669	245	4	m.s	m.s	PROPN
ejpam-4669	245	5	.	.	PUNCT
ejpam-4669	245	6	f.	f.	PROPN
ejpam-4669	245	7	hariwan	hariwan	PROPN
ejpam-4669	245	8	,	,	PUNCT
ejpam-4669	245	9	on	on	ADP
ejpam-4669	245	10	gaussian	gaussian	ADJ
ejpam-4669	245	11	fibonacci	fibonacci	NOUN
ejpam-4669	245	12	functions	function	NOUN
ejpam-4669	245	13	with	with	ADP
ejpam-4669	245	14	gaussian	gaussian	ADJ
ejpam-4669	245	15	fibonacci	fibonacci	NOUN
ejpam-4669	245	16	and	and	CCONJ
ejpam-4669	245	17	fibonacci	fibonacci	NOUN
ejpam-4669	245	18	numbers	number	NOUN
ejpam-4669	245	19	.	.	PUNCT
ejpam-4669	246	1	journal	journal	PROPN
ejpam-4669	246	2	of	of	ADP
ejpam-4669	246	3	xi′an	xi′an	PROPN
ejpam-4669	246	4	university	university	PROPN
ejpam-4669	246	5	of	of	ADP
ejpam-4669	246	6	architecture	architecture	NOUN
ejpam-4669	246	7	&	&	CCONJ
ejpam-4669	246	8	technology	technology	NOUN
ejpam-4669	246	9	.	.	PUNCT
ejpam-4669	247	1	126	126	NUM
ejpam-4669	247	2	,	,	PUNCT
ejpam-4669	247	3	2020	2020	NUM
ejpam-4669	247	4	.	.	PUNCT
ejpam-4669	248	1	[	[	X
ejpam-4669	248	2	5	5	X
ejpam-4669	248	3	]	]	PUNCT
ejpam-4669	248	4	k.	k.	PROPN
ejpam-4669	248	5	s.	s.	PROPN
ejpam-4669	248	6	hee	hee	PROPN
ejpam-4669	248	7	,	,	PUNCT
ejpam-4669	248	8	n.	n.	PROPN
ejpam-4669	248	9	joseph	joseph	PROPN
ejpam-4669	248	10	,	,	PUNCT
ejpam-4669	248	11	on	on	ADP
ejpam-4669	248	12	fibonacci	fibonacci	NOUN
ejpam-4669	248	13	functions	function	NOUN
ejpam-4669	248	14	with	with	ADP
ejpam-4669	248	15	periodicity	periodicity	NOUN
ejpam-4669	248	16	.	.	PUNCT
ejpam-4669	249	1	advances	advance	NOUN
ejpam-4669	249	2	in	in	ADP
ejpam-4669	249	3	difference	difference	NOUN
ejpam-4669	249	4	equations	equation	NOUN
ejpam-4669	249	5	,	,	PUNCT
ejpam-4669	249	6	(	(	PUNCT
ejpam-4669	249	7	2014	2014	NUM
ejpam-4669	249	8	)	)	PUNCT
ejpam-4669	249	9	,	,	PUNCT
ejpam-4669	249	10	2014:293	2014:293	NOUN
ejpam-4669	249	11	.	.	PUNCT
ejpam-4669	250	1	[	[	X
ejpam-4669	250	2	6	6	NUM
ejpam-4669	250	3	]	]	PUNCT
ejpam-4669	250	4	t.	t.	PROPN
ejpam-4669	250	5	koshy	koshy	PROPN
ejpam-4669	250	6	,	,	PUNCT
ejpam-4669	250	7	fibonacci	fibonacci	NOUN
ejpam-4669	250	8	and	and	CCONJ
ejpam-4669	250	9	lucas	lucas	PROPN
ejpam-4669	250	10	numbers	number	NOUN
ejpam-4669	250	11	with	with	ADP
ejpam-4669	250	12	applications	application	NOUN
ejpam-4669	250	13	,	,	PUNCT
ejpam-4669	250	14	volume	volume	NOUN
ejpam-4669	250	15	1	1	NUM
ejpam-4669	250	16	,	,	PUNCT
ejpam-4669	250	17	2nd	2nd	ADJ
ejpam-4669	250	18	edition	edition	NOUN
ejpam-4669	250	19	,	,	PUNCT
ejpam-4669	250	20	(	(	PUNCT
ejpam-4669	250	21	2017	2017	NUM
ejpam-4669	250	22	)	)	PUNCT
ejpam-4669	250	23	.	.	PUNCT
ejpam-4669	251	1	[	[	X
ejpam-4669	251	2	7	7	X
ejpam-4669	251	3	]	]	PUNCT
ejpam-4669	251	4	a.	a.	NOUN
ejpam-4669	251	5	s.	s.	PROPN
ejpam-4669	251	6	posamentier	posamentier	PROPN
ejpam-4669	251	7	,	,	PUNCT
ejpam-4669	251	8	&	&	CCONJ
ejpam-4669	251	9	i.	i.	PROPN
ejpam-4669	251	10	lehmann	lehmann	PROPN
ejpam-4669	251	11	,	,	PUNCT
ejpam-4669	251	12	the	the	DET
ejpam-4669	251	13	fabulous	fabulous	ADJ
ejpam-4669	251	14	)	)	PUNCT
ejpam-4669	251	15	fibonacci	fibonacci	NOUN
ejpam-4669	251	16	numbers	number	NOUN
ejpam-4669	251	17	.	.	PUNCT
ejpam-4669	252	1	amherst	amherst	PROPN
ejpam-4669	252	2	,	,	PUNCT
ejpam-4669	252	3	n.y	n.y	PROPN
ejpam-4669	252	4	,	,	PUNCT
ejpam-4669	252	5	prometheus	prometheus	NOUN
ejpam-4669	252	6	books	book	NOUN
ejpam-4669	252	7	,	,	PUNCT
ejpam-4669	252	8	(	(	PUNCT
ejpam-4669	252	9	2007	2007	NUM
ejpam-4669	252	10	)	)	PUNCT
ejpam-4669	252	11	.	.	PUNCT