id	sid	tid	token	lemma	pos
ejpam-5572	1	1	european	european	PROPN
ejpam-5572	1	2	journal	journal	PROPN
ejpam-5572	1	3	of	of	ADP
ejpam-5572	1	4	pure	pure	ADJ
ejpam-5572	1	5	and	and	CCONJ
ejpam-5572	1	6	applied	applied	ADJ
ejpam-5572	1	7	mathematics	mathematic	NOUN
ejpam-5572	1	8	2025	2025	NUM
ejpam-5572	1	9	,	,	PUNCT
ejpam-5572	1	10	vol	vol	NOUN
ejpam-5572	1	11	.	.	PROPN
ejpam-5572	1	12	18	18	NUM
ejpam-5572	1	13	,	,	PUNCT
ejpam-5572	1	14	issue	issue	NOUN
ejpam-5572	1	15	1	1	NUM
ejpam-5572	1	16	,	,	PUNCT
ejpam-5572	1	17	article	article	NOUN
ejpam-5572	1	18	number	number	NOUN
ejpam-5572	1	19	5572	5572	NUM
ejpam-5572	1	20	issn	issn	PROPN
ejpam-5572	1	21	1307	1307	NUM
ejpam-5572	1	22	-	-	SYM
ejpam-5572	1	23	5543	5543	NUM
ejpam-5572	1	24	–	–	PUNCT
ejpam-5572	1	25	ejpam.com	ejpam.com	X
ejpam-5572	1	26	published	publish	VERB
ejpam-5572	1	27	by	by	ADP
ejpam-5572	1	28	new	new	PROPN
ejpam-5572	1	29	york	york	PROPN
ejpam-5572	1	30	business	business	PROPN
ejpam-5572	1	31	global	global	PROPN
ejpam-5572	1	32	some	some	DET
ejpam-5572	1	33	algebraic	algebraic	ADJ
ejpam-5572	1	34	structures	structure	NOUN
ejpam-5572	1	35	of	of	ADP
ejpam-5572	1	36	complex	complex	ADJ
ejpam-5572	1	37	fermatean	fermatean	ADJ
ejpam-5572	1	38	fuzzy	fuzzy	NOUN
ejpam-5572	1	39	subgroups	subgroup	NOUN
ejpam-5572	1	40	eman	eman	NOUN
ejpam-5572	1	41	a.	a.	PROPN
ejpam-5572	1	42	abuhijleh1,∗	abuhijleh1,∗	PROPN
ejpam-5572	1	43	,	,	PUNCT
ejpam-5572	1	44	abd	abd	PROPN
ejpam-5572	1	45	ulazeez	ulazeez	PROPN
ejpam-5572	1	46	alkouri2	alkouri2	PROPN
ejpam-5572	1	47	1	1	NUM
ejpam-5572	1	48	department	department	NOUN
ejpam-5572	1	49	of	of	ADP
ejpam-5572	1	50	basic	basic	ADJ
ejpam-5572	1	51	sciences	sciences	PROPN
ejpam-5572	1	52	,	,	PUNCT
ejpam-5572	1	53	al	al	PROPN
ejpam-5572	1	54	-	-	PUNCT
ejpam-5572	1	55	zarqa	zarqa	PROPN
ejpam-5572	1	56	university	university	PROPN
ejpam-5572	1	57	college	college	PROPN
ejpam-5572	1	58	,	,	PUNCT
ejpam-5572	1	59	al	al	PROPN
ejpam-5572	1	60	-	-	PUNCT
ejpam-5572	1	61	balqa	balqa	NOUN
ejpam-5572	1	62	applied	apply	VERB
ejpam-5572	1	63	university	university	NOUN
ejpam-5572	1	64	,	,	PUNCT
ejpam-5572	1	65	jordan	jordan	PROPN
ejpam-5572	1	66	2	2	NUM
ejpam-5572	1	67	mathematics	mathematics	PROPN
ejpam-5572	1	68	department	department	NOUN
ejpam-5572	1	69	,	,	PUNCT
ejpam-5572	1	70	science	science	NOUN
ejpam-5572	1	71	college	college	PROPN
ejpam-5572	1	72	,	,	PUNCT
ejpam-5572	1	73	ajloun	ajloun	ADJ
ejpam-5572	1	74	national	national	PROPN
ejpam-5572	1	75	university	university	PROPN
ejpam-5572	1	76	,	,	PUNCT
ejpam-5572	1	77	jordan	jordan	PROPN
ejpam-5572	1	78	abstract	abstract	PROPN
ejpam-5572	1	79	.	.	PUNCT
ejpam-5572	2	1	a	a	DET
ejpam-5572	2	2	complex	complex	ADJ
ejpam-5572	2	3	fermatean	fermatean	ADJ
ejpam-5572	2	4	fuzzy	fuzzy	ADJ
ejpam-5572	2	5	set	set	NOUN
ejpam-5572	2	6	provides	provide	VERB
ejpam-5572	2	7	a	a	DET
ejpam-5572	2	8	detailed	detailed	ADJ
ejpam-5572	2	9	framework	framework	NOUN
ejpam-5572	2	10	for	for	ADP
ejpam-5572	2	11	representing	represent	VERB
ejpam-5572	2	12	a	a	DET
ejpam-5572	2	13	specific	specific	ADJ
ejpam-5572	2	14	type	type	NOUN
ejpam-5572	2	15	of	of	ADP
ejpam-5572	2	16	information	information	NOUN
ejpam-5572	2	17	and	and	CCONJ
ejpam-5572	2	18	has	have	AUX
ejpam-5572	2	19	been	be	AUX
ejpam-5572	2	20	effectively	effectively	ADV
ejpam-5572	2	21	applied	apply	VERB
ejpam-5572	2	22	to	to	ADP
ejpam-5572	2	23	decision	decision	NOUN
ejpam-5572	2	24	-	-	PUNCT
ejpam-5572	2	25	making	make	VERB
ejpam-5572	2	26	problems	problem	NOUN
ejpam-5572	2	27	.	.	PUNCT
ejpam-5572	3	1	this	this	DET
ejpam-5572	3	2	study	study	NOUN
ejpam-5572	3	3	introduces	introduce	VERB
ejpam-5572	3	4	complex	complex	ADJ
ejpam-5572	3	5	fermatean	fermatean	ADJ
ejpam-5572	3	6	fuzzy	fuzzy	ADJ
ejpam-5572	3	7	subgroups	subgroup	NOUN
ejpam-5572	3	8	(	(	PUNCT
ejpam-5572	3	9	cffsgs	cffsg	NOUN
ejpam-5572	3	10	)	)	PUNCT
ejpam-5572	3	11	,	,	PUNCT
ejpam-5572	3	12	an	an	DET
ejpam-5572	3	13	extension	extension	NOUN
ejpam-5572	3	14	of	of	ADP
ejpam-5572	3	15	fermatean	fermatean	ADJ
ejpam-5572	3	16	fuzzy	fuzzy	ADJ
ejpam-5572	3	17	subgroups	subgroup	NOUN
ejpam-5572	3	18	,	,	PUNCT
ejpam-5572	3	19	and	and	CCONJ
ejpam-5572	3	20	complex	complex	ADJ
ejpam-5572	3	21	pythagorean	pythagorean	ADJ
ejpam-5572	3	22	fuzzy	fuzzy	ADJ
ejpam-5572	3	23	subgroups	subgroup	NOUN
ejpam-5572	3	24	.	.	PUNCT
ejpam-5572	4	1	the	the	DET
ejpam-5572	4	2	key	key	ADJ
ejpam-5572	4	3	innovation	innovation	NOUN
ejpam-5572	4	4	of	of	ADP
ejpam-5572	4	5	cffsgs	cffsg	NOUN
ejpam-5572	4	6	lies	lie	VERB
ejpam-5572	4	7	in	in	ADP
ejpam-5572	4	8	their	their	PRON
ejpam-5572	4	9	capacity	capacity	NOUN
ejpam-5572	4	10	to	to	PART
ejpam-5572	4	11	represent	represent	VERB
ejpam-5572	4	12	two	two	NUM
ejpam-5572	4	13	variables	variable	NOUN
ejpam-5572	4	14	within	within	ADP
ejpam-5572	4	15	their	their	PRON
ejpam-5572	4	16	algebraic	algebraic	ADJ
ejpam-5572	4	17	structure	structure	NOUN
ejpam-5572	4	18	,	,	PUNCT
ejpam-5572	4	19	surpassing	surpass	VERB
ejpam-5572	4	20	the	the	DET
ejpam-5572	4	21	capabilities	capability	NOUN
ejpam-5572	4	22	of	of	ADP
ejpam-5572	4	23	traditional	traditional	ADJ
ejpam-5572	4	24	fermatean	fermatean	ADJ
ejpam-5572	4	25	fuzzy	fuzzy	ADJ
ejpam-5572	4	26	subgroups	subgroup	NOUN
ejpam-5572	4	27	.	.	PUNCT
ejpam-5572	5	1	the	the	DET
ejpam-5572	5	2	research	research	NOUN
ejpam-5572	5	3	establishes	establish	VERB
ejpam-5572	5	4	the	the	DET
ejpam-5572	5	5	formal	formal	ADJ
ejpam-5572	5	6	definition	definition	NOUN
ejpam-5572	5	7	and	and	CCONJ
ejpam-5572	5	8	properties	property	NOUN
ejpam-5572	5	9	of	of	ADP
ejpam-5572	5	10	cffsgs	cffsg	NOUN
ejpam-5572	5	11	,	,	PUNCT
ejpam-5572	5	12	adapting	adapt	VERB
ejpam-5572	5	13	them	they	PRON
ejpam-5572	5	14	to	to	ADP
ejpam-5572	5	15	a	a	DET
ejpam-5572	5	16	complex	complex	ADJ
ejpam-5572	5	17	framework	framework	NOUN
ejpam-5572	5	18	that	that	PRON
ejpam-5572	5	19	incorporates	incorporate	VERB
ejpam-5572	5	20	amplitude	amplitude	NOUN
ejpam-5572	5	21	and	and	CCONJ
ejpam-5572	5	22	phase	phase	NOUN
ejpam-5572	5	23	components	component	NOUN
ejpam-5572	5	24	.	.	PUNCT
ejpam-5572	6	1	additionally	additionally	ADV
ejpam-5572	6	2	,	,	PUNCT
ejpam-5572	6	3	the	the	DET
ejpam-5572	6	4	concepts	concept	NOUN
ejpam-5572	6	5	of	of	ADP
ejpam-5572	6	6	complex	complex	ADJ
ejpam-5572	6	7	fermatean	fermatean	ADJ
ejpam-5572	6	8	fuzzy	fuzzy	ADJ
ejpam-5572	6	9	cosets	coset	NOUN
ejpam-5572	6	10	and	and	CCONJ
ejpam-5572	6	11	complex	complex	ADJ
ejpam-5572	6	12	fermatean	fermatean	NOUN
ejpam-5572	6	13	fuzzy	fuzzy	ADJ
ejpam-5572	6	14	normal	normal	ADJ
ejpam-5572	6	15	subgroups	subgroup	NOUN
ejpam-5572	6	16	are	be	AUX
ejpam-5572	6	17	introduced	introduce	VERB
ejpam-5572	6	18	.	.	PUNCT
ejpam-5572	7	1	the	the	DET
ejpam-5572	7	2	study	study	NOUN
ejpam-5572	7	3	also	also	ADV
ejpam-5572	7	4	investigates	investigate	VERB
ejpam-5572	7	5	and	and	CCONJ
ejpam-5572	7	6	examines	examine	VERB
ejpam-5572	7	7	the	the	DET
ejpam-5572	7	8	characteristics	characteristic	NOUN
ejpam-5572	7	9	of	of	ADP
ejpam-5572	7	10	homomorphisms	homomorphism	NOUN
ejpam-5572	7	11	between	between	ADP
ejpam-5572	7	12	complex	complex	ADJ
ejpam-5572	7	13	fermatean	fermatean	ADJ
ejpam-5572	7	14	fuzzy	fuzzy	ADJ
ejpam-5572	7	15	subgroups	subgroup	NOUN
ejpam-5572	7	16	.	.	PUNCT
ejpam-5572	8	1	2020	2020	NUM
ejpam-5572	8	2	mathematics	mathematic	NOUN
ejpam-5572	8	3	subject	subject	NOUN
ejpam-5572	8	4	classifications	classification	NOUN
ejpam-5572	8	5	:	:	PUNCT
ejpam-5572	8	6	08a35	08a35	NUM
ejpam-5572	8	7	,	,	PUNCT
ejpam-5572	8	8	08a72	08a72	NUM
ejpam-5572	8	9	,	,	PUNCT
ejpam-5572	8	10	20n25	20n25	NOUN
ejpam-5572	8	11	key	key	ADJ
ejpam-5572	8	12	words	word	NOUN
ejpam-5572	8	13	and	and	CCONJ
ejpam-5572	8	14	phrases	phrase	NOUN
ejpam-5572	8	15	:	:	PUNCT
ejpam-5572	8	16	complex	complex	ADJ
ejpam-5572	8	17	fermatean	fermatean	ADJ
ejpam-5572	8	18	fuzzy	fuzzy	ADJ
ejpam-5572	8	19	subgroup	subgroup	NOUN
ejpam-5572	8	20	,	,	PUNCT
ejpam-5572	8	21	complex	complex	ADJ
ejpam-5572	8	22	fermatean	fermatean	NOUN
ejpam-5572	8	23	fuzzy	fuzzy	ADJ
ejpam-5572	8	24	normal	normal	ADJ
ejpam-5572	8	25	subgroup	subgroup	NOUN
ejpam-5572	8	26	,	,	PUNCT
ejpam-5572	8	27	homomorphism	homomorphism	NOUN
ejpam-5572	8	28	of	of	ADP
ejpam-5572	8	29	fermatean	fermatean	ADJ
ejpam-5572	8	30	fuzzy	fuzzy	ADJ
ejpam-5572	8	31	subgroup	subgroup	PROPN
ejpam-5572	8	32	1	1	NUM
ejpam-5572	8	33	.	.	PUNCT
ejpam-5572	9	1	introduction	introduction	NOUN
ejpam-5572	9	2	zadeh	zadeh	PROPN
ejpam-5572	10	1	[	[	X
ejpam-5572	10	2	40	40	NUM
ejpam-5572	10	3	]	]	PUNCT
ejpam-5572	10	4	introduced	introduce	VERB
ejpam-5572	10	5	the	the	DET
ejpam-5572	10	6	fuzzy	fuzzy	ADJ
ejpam-5572	10	7	set	set	NOUN
ejpam-5572	10	8	(	(	PUNCT
ejpam-5572	10	9	fs	fs	NOUN
ejpam-5572	10	10	)	)	PUNCT
ejpam-5572	10	11	as	as	ADP
ejpam-5572	10	12	a	a	DET
ejpam-5572	10	13	novel	novel	ADJ
ejpam-5572	10	14	concept	concept	NOUN
ejpam-5572	10	15	to	to	PART
ejpam-5572	10	16	address	address	VERB
ejpam-5572	10	17	uncertainty	uncertainty	NOUN
ejpam-5572	10	18	and	and	CCONJ
ejpam-5572	10	19	vagueness	vagueness	NOUN
ejpam-5572	10	20	in	in	ADP
ejpam-5572	10	21	decision	decision	NOUN
ejpam-5572	10	22	-	-	PUNCT
ejpam-5572	10	23	making	making	NOUN
ejpam-5572	10	24	(	(	PUNCT
ejpam-5572	10	25	dm	dm	NOUN
ejpam-5572	10	26	)	)	PUNCT
ejpam-5572	10	27	problems	problem	NOUN
ejpam-5572	10	28	.	.	PUNCT
ejpam-5572	11	1	this	this	DET
ejpam-5572	11	2	foundational	foundational	ADJ
ejpam-5572	11	3	work	work	NOUN
ejpam-5572	11	4	sparked	spark	VERB
ejpam-5572	11	5	the	the	DET
ejpam-5572	11	6	publication	publication	NOUN
ejpam-5572	11	7	of	of	ADP
ejpam-5572	11	8	hundreds	hundred	NOUN
ejpam-5572	11	9	of	of	ADP
ejpam-5572	11	10	studies	study	NOUN
ejpam-5572	11	11	that	that	PRON
ejpam-5572	11	12	extended	extend	VERB
ejpam-5572	11	13	and	and	CCONJ
ejpam-5572	11	14	applied	apply	VERB
ejpam-5572	11	15	fs	fs	ADP
ejpam-5572	11	16	concepts	concept	NOUN
ejpam-5572	11	17	across	across	ADP
ejpam-5572	11	18	various	various	ADJ
ejpam-5572	11	19	fields	field	NOUN
ejpam-5572	11	20	.	.	PUNCT
ejpam-5572	12	1	a	a	DET
ejpam-5572	12	2	significant	significant	ADJ
ejpam-5572	12	3	advancement	advancement	NOUN
ejpam-5572	12	4	was	be	AUX
ejpam-5572	12	5	made	make	VERB
ejpam-5572	12	6	by	by	ADP
ejpam-5572	12	7	atanassov	atanassov	NOUN
ejpam-5572	12	8	in	in	ADP
ejpam-5572	12	9	1986	1986	NUM
ejpam-5572	12	10	[	[	X
ejpam-5572	12	11	12	12	NUM
ejpam-5572	12	12	]	]	PUNCT
ejpam-5572	12	13	,	,	PUNCT
ejpam-5572	12	14	who	who	PRON
ejpam-5572	12	15	introduced	introduce	VERB
ejpam-5572	12	16	the	the	DET
ejpam-5572	12	17	intuitionistic	intuitionistic	ADJ
ejpam-5572	12	18	fuzzy	fuzzy	ADJ
ejpam-5572	12	19	set	set	NOUN
ejpam-5572	12	20	(	(	PUNCT
ejpam-5572	12	21	ifs	ifs	PROPN
ejpam-5572	12	22	)	)	PUNCT
ejpam-5572	12	23	.	.	PUNCT
ejpam-5572	13	1	later	later	ADV
ejpam-5572	13	2	,	,	PUNCT
ejpam-5572	13	3	yager	yager	NOUN
ejpam-5572	13	4	[	[	X
ejpam-5572	13	5	38	38	NUM
ejpam-5572	13	6	]	]	PUNCT
ejpam-5572	13	7	expanded	expand	VERB
ejpam-5572	13	8	this	this	DET
ejpam-5572	13	9	framework	framework	NOUN
ejpam-5572	13	10	in	in	ADP
ejpam-5572	13	11	2013	2013	NUM
ejpam-5572	13	12	by	by	ADP
ejpam-5572	13	13	defining	define	VERB
ejpam-5572	13	14	the	the	DET
ejpam-5572	13	15	pythagorean	pythagorean	PROPN
ejpam-5572	13	16	fuzzy	fuzzy	ADJ
ejpam-5572	13	17	set	set	NOUN
ejpam-5572	13	18	(	(	PUNCT
ejpam-5572	13	19	pfs	pfs	PROPN
ejpam-5572	13	20	)	)	PUNCT
ejpam-5572	13	21	,	,	PUNCT
ejpam-5572	13	22	positioning	position	VERB
ejpam-5572	13	23	fs	f	NOUN
ejpam-5572	13	24	as	as	ADP
ejpam-5572	13	25	a	a	DET
ejpam-5572	13	26	subset	subset	NOUN
ejpam-5572	13	27	of	of	ADP
ejpam-5572	13	28	ifs	ifs	PROPN
ejpam-5572	13	29	,	,	PUNCT
ejpam-5572	13	30	which	which	PRON
ejpam-5572	13	31	in	in	ADP
ejpam-5572	13	32	turn	turn	NOUN
ejpam-5572	13	33	is	be	AUX
ejpam-5572	13	34	a	a	DET
ejpam-5572	13	35	subset	subset	NOUN
ejpam-5572	13	36	of	of	ADP
ejpam-5572	13	37	pfs	pfs	PROPN
ejpam-5572	13	38	.	.	PUNCT
ejpam-5572	14	1	in	in	ADP
ejpam-5572	14	2	2017	2017	NUM
ejpam-5572	14	3	,	,	PUNCT
ejpam-5572	14	4	yager	yager	NOUN
ejpam-5572	14	5	further	far	ADV
ejpam-5572	14	6	generalized	generalize	VERB
ejpam-5572	14	7	these	these	DET
ejpam-5572	14	8	concepts	concept	NOUN
ejpam-5572	14	9	by	by	ADP
ejpam-5572	14	10	introducing	introduce	VERB
ejpam-5572	14	11	q	q	ADJ
ejpam-5572	14	12	-	-	PUNCT
ejpam-5572	14	13	rung	rung	ADJ
ejpam-5572	14	14	orthopair	orthopair	NOUN
ejpam-5572	14	15	fuzzy	fuzzy	ADJ
ejpam-5572	14	16	sets	set	NOUN
ejpam-5572	14	17	[	[	X
ejpam-5572	14	18	39	39	NUM
ejpam-5572	14	19	]	]	PUNCT
ejpam-5572	14	20	.	.	PUNCT
ejpam-5572	15	1	∗corresponding	∗corresponde	VERB
ejpam-5572	15	2	author	author	NOUN
ejpam-5572	15	3	.	.	PUNCT
ejpam-5572	16	1	doi	doi	NOUN
ejpam-5572	16	2	:	:	PUNCT
ejpam-5572	16	3	https://doi.org/10.29020/nybg.ejpam.v18i1.5572	https://doi.org/10.29020/nybg.ejpam.v18i1.5572	PRON
ejpam-5572	16	4	email	email	NOUN
ejpam-5572	16	5	addresses	address	NOUN
ejpam-5572	16	6	:	:	PUNCT
ejpam-5572	16	7	emanhijleh@bau.edu.jo	emanhijleh@bau.edu.jo	PROPN
ejpam-5572	16	8	(	(	PUNCT
ejpam-5572	16	9	e.a	e.a	PROPN
ejpam-5572	16	10	.	.	PROPN
ejpam-5572	16	11	abuhijleh	abuhijleh	PROPN
ejpam-5572	16	12	)	)	PUNCT
ejpam-5572	16	13	,	,	PUNCT
ejpam-5572	16	14	alkouriabdulazeez@anu.edu.jo	alkouriabdulazeez@anu.edu.jo	NOUN
ejpam-5572	16	15	(	(	PUNCT
ejpam-5572	16	16	a.	a.	NOUN
ejpam-5572	16	17	alkouri	alkouri	PROPN
ejpam-5572	16	18	)	)	PUNCT
ejpam-5572	16	19	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5572	16	20	1	1	NUM
ejpam-5572	16	21	copyright	copyright	NOUN
ejpam-5572	16	22	:	:	PUNCT
ejpam-5572	17	1	©	©	PROPN
ejpam-5572	17	2	2025	2025	NUM
ejpam-5572	17	3	the	the	DET
ejpam-5572	17	4	author(s	author(s	NOUN
ejpam-5572	17	5	)	)	PUNCT
ejpam-5572	17	6	.	.	PUNCT
ejpam-5572	18	1	(	(	PUNCT
ejpam-5572	18	2	cc	cc	NOUN
ejpam-5572	18	3	by	by	ADP
ejpam-5572	18	4	-	-	PUNCT
ejpam-5572	18	5	nc	nc	PROPN
ejpam-5572	18	6	4.0	4.0	NUM
ejpam-5572	18	7	)	)	PUNCT
ejpam-5572	18	8	e.a	e.a	PROPN
ejpam-5572	18	9	.	.	PROPN
ejpam-5572	18	10	abuhijleh	abuhijleh	PROPN
ejpam-5572	18	11	,	,	PUNCT
ejpam-5572	18	12	a.	a.	PROPN
ejpam-5572	18	13	alkouri	alkouri	PROPN
ejpam-5572	18	14	/	/	PROPN
ejpam-5572	18	15	eur	eur	PROPN
ejpam-5572	18	16	.	.	PUNCT
ejpam-5572	19	1	j.	j.	PROPN
ejpam-5572	19	2	pure	pure	PROPN
ejpam-5572	19	3	appl	appl	PROPN
ejpam-5572	19	4	.	.	PROPN
ejpam-5572	19	5	math	math	PROPN
ejpam-5572	19	6	,	,	PUNCT
ejpam-5572	19	7	18	18	NUM
ejpam-5572	19	8	(	(	PUNCT
ejpam-5572	19	9	1	1	NUM
ejpam-5572	19	10	)	)	PUNCT
ejpam-5572	19	11	(	(	PUNCT
ejpam-5572	19	12	2025	2025	NUM
ejpam-5572	19	13	)	)	PUNCT
ejpam-5572	19	14	,	,	PUNCT
ejpam-5572	19	15	5572	5572	NUM
ejpam-5572	19	16	2	2	NUM
ejpam-5572	19	17	of	of	ADP
ejpam-5572	19	18	19	19	NUM
ejpam-5572	19	19	this	this	DET
ejpam-5572	19	20	generalization	generalization	NOUN
ejpam-5572	19	21	was	be	AUX
ejpam-5572	19	22	followed	follow	VERB
ejpam-5572	19	23	by	by	ADP
ejpam-5572	19	24	specific	specific	ADJ
ejpam-5572	19	25	cases	case	NOUN
ejpam-5572	19	26	,	,	PUNCT
ejpam-5572	19	27	such	such	ADJ
ejpam-5572	19	28	as	as	ADP
ejpam-5572	19	29	the	the	DET
ejpam-5572	19	30	fermatean	fermatean	ADJ
ejpam-5572	19	31	fuzzy	fuzzy	ADJ
ejpam-5572	19	32	set	set	NOUN
ejpam-5572	19	33	(	(	PUNCT
ejpam-5572	19	34	ffs	ff	NOUN
ejpam-5572	19	35	)	)	PUNCT
ejpam-5572	19	36	defined	define	VERB
ejpam-5572	19	37	by	by	ADP
ejpam-5572	19	38	senapati	senapati	PROPN
ejpam-5572	19	39	et	et	PROPN
ejpam-5572	19	40	al	al	PROPN
ejpam-5572	19	41	.	.	PROPN
ejpam-5572	20	1	in	in	ADP
ejpam-5572	20	2	2020	2020	NUM
ejpam-5572	20	3	[	[	X
ejpam-5572	20	4	33	33	NUM
ejpam-5572	20	5	]	]	PUNCT
ejpam-5572	20	6	.	.	PUNCT
ejpam-5572	21	1	these	these	DET
ejpam-5572	21	2	sets	set	NOUN
ejpam-5572	21	3	differ	differ	VERB
ejpam-5572	21	4	in	in	ADP
ejpam-5572	21	5	their	their	PRON
ejpam-5572	21	6	constraints	constraint	NOUN
ejpam-5572	21	7	;	;	PUNCT
ejpam-5572	21	8	for	for	ADP
ejpam-5572	21	9	ifs	ifs	PROPN
ejpam-5572	21	10	,	,	PUNCT
ejpam-5572	21	11	the	the	DET
ejpam-5572	21	12	sum	sum	NOUN
ejpam-5572	21	13	of	of	ADP
ejpam-5572	21	14	membership	membership	NOUN
ejpam-5572	21	15	and	and	CCONJ
ejpam-5572	21	16	non	non	ADJ
ejpam-5572	21	17	-	-	ADJ
ejpam-5572	21	18	membership	membership	ADJ
ejpam-5572	21	19	degrees	degree	NOUN
ejpam-5572	21	20	must	must	AUX
ejpam-5572	21	21	lie	lie	VERB
ejpam-5572	21	22	between	between	ADP
ejpam-5572	21	23	zero	zero	NUM
ejpam-5572	21	24	and	and	CCONJ
ejpam-5572	21	25	one	one	NUM
ejpam-5572	21	26	;	;	PUNCT
ejpam-5572	21	27	for	for	ADP
ejpam-5572	21	28	pfs	pfs	PROPN
ejpam-5572	21	29	,	,	PUNCT
ejpam-5572	21	30	the	the	DET
ejpam-5572	21	31	sum	sum	NOUN
ejpam-5572	21	32	of	of	ADP
ejpam-5572	21	33	their	their	PRON
ejpam-5572	21	34	squares	square	NOUN
ejpam-5572	21	35	must	must	AUX
ejpam-5572	21	36	meet	meet	VERB
ejpam-5572	21	37	this	this	DET
ejpam-5572	21	38	condition	condition	NOUN
ejpam-5572	21	39	;	;	PUNCT
ejpam-5572	21	40	for	for	ADP
ejpam-5572	21	41	ffs	ff	NOUN
ejpam-5572	21	42	,	,	PUNCT
ejpam-5572	21	43	it	it	PRON
ejpam-5572	21	44	is	be	AUX
ejpam-5572	21	45	the	the	DET
ejpam-5572	21	46	sum	sum	NOUN
ejpam-5572	21	47	of	of	ADP
ejpam-5572	21	48	their	their	PRON
ejpam-5572	21	49	cubes	cube	NOUN
ejpam-5572	21	50	;	;	PUNCT
ejpam-5572	21	51	and	and	CCONJ
ejpam-5572	21	52	for	for	ADP
ejpam-5572	21	53	q	q	ADJ
ejpam-5572	21	54	-	-	PUNCT
ejpam-5572	21	55	rung	rung	ADJ
ejpam-5572	21	56	orthopair	orthopair	ADJ
ejpam-5572	21	57	fuzzy	fuzzy	ADJ
ejpam-5572	21	58	sets	set	NOUN
ejpam-5572	21	59	,	,	PUNCT
ejpam-5572	21	60	the	the	DET
ejpam-5572	21	61	sum	sum	NOUN
ejpam-5572	21	62	of	of	ADP
ejpam-5572	21	63	the	the	DET
ejpam-5572	21	64	kth	kth	PROPN
ejpam-5572	21	65	powers	power	NOUN
ejpam-5572	21	66	of	of	ADP
ejpam-5572	21	67	membership	membership	NOUN
ejpam-5572	21	68	and	and	CCONJ
ejpam-5572	21	69	non	non	ADJ
ejpam-5572	21	70	-	-	ADJ
ejpam-5572	21	71	membership	membership	ADJ
ejpam-5572	21	72	degrees	degree	NOUN
ejpam-5572	21	73	must	must	AUX
ejpam-5572	21	74	satisfy	satisfy	VERB
ejpam-5572	21	75	this	this	DET
ejpam-5572	21	76	constraint	constraint	NOUN
ejpam-5572	21	77	.	.	PUNCT
ejpam-5572	22	1	researchers	researcher	NOUN
ejpam-5572	22	2	have	have	AUX
ejpam-5572	22	3	explored	explore	VERB
ejpam-5572	22	4	the	the	DET
ejpam-5572	22	5	properties	property	NOUN
ejpam-5572	22	6	and	and	CCONJ
ejpam-5572	22	7	operations	operation	NOUN
ejpam-5572	22	8	of	of	ADP
ejpam-5572	22	9	these	these	DET
ejpam-5572	22	10	sets	set	NOUN
ejpam-5572	22	11	,	,	PUNCT
ejpam-5572	22	12	developed	develop	VERB
ejpam-5572	22	13	new	new	ADJ
ejpam-5572	22	14	measures	measure	NOUN
ejpam-5572	22	15	,	,	PUNCT
ejpam-5572	22	16	and	and	CCONJ
ejpam-5572	22	17	applied	apply	VERB
ejpam-5572	22	18	them	they	PRON
ejpam-5572	22	19	to	to	ADP
ejpam-5572	22	20	dm	dm	PROPN
ejpam-5572	22	21	problems	problem	NOUN
ejpam-5572	22	22	,	,	PUNCT
ejpam-5572	22	23	as	as	SCONJ
ejpam-5572	22	24	demonstrated	demonstrate	VERB
ejpam-5572	22	25	in	in	ADP
ejpam-5572	22	26	studies	study	NOUN
ejpam-5572	22	27	like	like	ADP
ejpam-5572	22	28	[	[	X
ejpam-5572	22	29	18	18	NUM
ejpam-5572	22	30	]	]	PUNCT
ejpam-5572	22	31	,	,	PUNCT
ejpam-5572	22	32	[	[	X
ejpam-5572	22	33	23	23	NUM
ejpam-5572	22	34	]	]	PUNCT
ejpam-5572	22	35	,	,	PUNCT
ejpam-5572	22	36	and	and	CCONJ
ejpam-5572	22	37	[	[	X
ejpam-5572	22	38	27	27	NUM
ejpam-5572	22	39	]	]	PUNCT
ejpam-5572	22	40	.	.	PUNCT
ejpam-5572	23	1	more	more	ADV
ejpam-5572	23	2	recently	recently	ADV
ejpam-5572	23	3	,	,	PUNCT
ejpam-5572	23	4	in	in	ADP
ejpam-5572	23	5	2022	2022	NUM
ejpam-5572	23	6	,	,	PUNCT
ejpam-5572	23	7	m.	m.	NOUN
ejpam-5572	23	8	akram	akram	PROPN
ejpam-5572	23	9	et	et	PROPN
ejpam-5572	23	10	al	al	PROPN
ejpam-5572	23	11	.	.	PUNCT
ejpam-5572	24	1	[	[	X
ejpam-5572	24	2	4	4	X
ejpam-5572	24	3	]	]	PUNCT
ejpam-5572	24	4	proposed	propose	VERB
ejpam-5572	24	5	a	a	DET
ejpam-5572	24	6	dm	dm	PROPN
ejpam-5572	24	7	approach	approach	NOUN
ejpam-5572	24	8	integrating	integrate	VERB
ejpam-5572	24	9	the	the	DET
ejpam-5572	24	10	attributes	attribute	NOUN
ejpam-5572	24	11	of	of	ADP
ejpam-5572	24	12	the	the	DET
ejpam-5572	24	13	traditional	traditional	ADJ
ejpam-5572	24	14	vikor	vikor	ADJ
ejpam-5572	24	15	method	method	NOUN
ejpam-5572	24	16	within	within	ADP
ejpam-5572	24	17	the	the	DET
ejpam-5572	24	18	framework	framework	NOUN
ejpam-5572	24	19	of	of	ADP
ejpam-5572	24	20	a	a	DET
ejpam-5572	24	21	multidimensional	multidimensional	ADJ
ejpam-5572	24	22	complex	complex	ADJ
ejpam-5572	24	23	fermatean	fermatean	ADJ
ejpam-5572	24	24	fuzzy	fuzzy	ADJ
ejpam-5572	24	25	n	n	CCONJ
ejpam-5572	24	26	-	-	PUNCT
ejpam-5572	24	27	soft	soft	ADJ
ejpam-5572	24	28	set	set	NOUN
ejpam-5572	24	29	.	.	PUNCT
ejpam-5572	25	1	real	real	ADJ
ejpam-5572	25	2	-	-	PUNCT
ejpam-5572	25	3	world	world	NOUN
ejpam-5572	25	4	problems	problem	NOUN
ejpam-5572	25	5	often	often	ADV
ejpam-5572	25	6	involve	involve	VERB
ejpam-5572	25	7	multiple	multiple	ADJ
ejpam-5572	25	8	variables	variable	NOUN
ejpam-5572	25	9	,	,	PUNCT
ejpam-5572	25	10	necessitating	necessitate	VERB
ejpam-5572	25	11	more	more	ADV
ejpam-5572	25	12	advanced	advanced	ADJ
ejpam-5572	25	13	conceptual	conceptual	ADJ
ejpam-5572	25	14	frameworks	framework	NOUN
ejpam-5572	25	15	.	.	PUNCT
ejpam-5572	26	1	this	this	DET
ejpam-5572	26	2	need	need	NOUN
ejpam-5572	26	3	led	lead	VERB
ejpam-5572	26	4	to	to	ADP
ejpam-5572	26	5	the	the	DET
ejpam-5572	26	6	intellectual	intellectual	ADJ
ejpam-5572	26	7	introduction	introduction	NOUN
ejpam-5572	26	8	of	of	ADP
ejpam-5572	26	9	the	the	DET
ejpam-5572	26	10	complex	complex	ADJ
ejpam-5572	26	11	fuzzy	fuzzy	ADJ
ejpam-5572	26	12	set	set	NOUN
ejpam-5572	26	13	(	(	PUNCT
ejpam-5572	26	14	cfs	cfs	PROPN
ejpam-5572	26	15	)	)	PUNCT
ejpam-5572	26	16	by	by	ADP
ejpam-5572	26	17	ramot	ramot	NOUN
ejpam-5572	26	18	et	et	PROPN
ejpam-5572	26	19	al	al	PROPN
ejpam-5572	26	20	.	.	PROPN
ejpam-5572	27	1	(	(	PUNCT
ejpam-5572	27	2	2002	2002	NUM
ejpam-5572	27	3	)	)	PUNCT
ejpam-5572	28	1	[	[	X
ejpam-5572	28	2	30	30	NUM
ejpam-5572	28	3	]	]	PUNCT
ejpam-5572	28	4	,	,	PUNCT
ejpam-5572	28	5	which	which	PRON
ejpam-5572	28	6	incorporates	incorporate	VERB
ejpam-5572	28	7	two	two	NUM
ejpam-5572	28	8	key	key	ADJ
ejpam-5572	28	9	variables	variable	NOUN
ejpam-5572	28	10	:	:	PUNCT
ejpam-5572	28	11	amplitude	amplitude	NOUN
ejpam-5572	28	12	and	and	CCONJ
ejpam-5572	28	13	phase	phase	NOUN
ejpam-5572	28	14	.	.	PUNCT
ejpam-5572	29	1	building	build	VERB
ejpam-5572	29	2	on	on	ADP
ejpam-5572	29	3	this	this	DET
ejpam-5572	29	4	concept	concept	NOUN
ejpam-5572	29	5	,	,	PUNCT
ejpam-5572	29	6	alkouri	alkouri	PROPN
ejpam-5572	29	7	and	and	CCONJ
ejpam-5572	29	8	salleh	salleh	PROPN
ejpam-5572	29	9	(	(	PUNCT
ejpam-5572	29	10	2012	2012	NUM
ejpam-5572	29	11	)	)	PUNCT
ejpam-5572	30	1	[	[	X
ejpam-5572	30	2	8	8	NUM
ejpam-5572	30	3	]	]	PUNCT
ejpam-5572	30	4	introduced	introduce	VERB
ejpam-5572	30	5	the	the	DET
ejpam-5572	30	6	complex	complex	ADJ
ejpam-5572	30	7	intuitionistic	intuitionistic	ADJ
ejpam-5572	30	8	fuzzy	fuzzy	ADJ
ejpam-5572	30	9	set	set	NOUN
ejpam-5572	30	10	(	(	PUNCT
ejpam-5572	30	11	cifs	cifs	PROPN
ejpam-5572	30	12	)	)	PUNCT
ejpam-5572	30	13	,	,	PUNCT
ejpam-5572	30	14	enhancing	enhance	VERB
ejpam-5572	30	15	the	the	DET
ejpam-5572	30	16	properties	property	NOUN
ejpam-5572	30	17	of	of	ADP
ejpam-5572	30	18	cfs	cfs	PROPN
ejpam-5572	30	19	.	.	PUNCT
ejpam-5572	31	1	subsequent	subsequent	ADJ
ejpam-5572	31	2	works	work	NOUN
ejpam-5572	31	3	have	have	AUX
ejpam-5572	31	4	extended	extend	VERB
ejpam-5572	31	5	and	and	CCONJ
ejpam-5572	31	6	refined	refine	VERB
ejpam-5572	31	7	these	these	DET
ejpam-5572	31	8	ideas	idea	NOUN
ejpam-5572	31	9	,	,	PUNCT
ejpam-5572	31	10	including	include	VERB
ejpam-5572	31	11	contributions	contribution	NOUN
ejpam-5572	31	12	from	from	ADP
ejpam-5572	31	13	alkouri	alkouri	PROPN
ejpam-5572	31	14	(	(	PUNCT
ejpam-5572	31	15	2013	2013	NUM
ejpam-5572	31	16	)	)	PUNCT
ejpam-5572	32	1	[	[	X
ejpam-5572	32	2	9	9	NUM
ejpam-5572	32	3	]	]	PUNCT
ejpam-5572	32	4	and	and	CCONJ
ejpam-5572	32	5	ramot	ramot	NOUN
ejpam-5572	32	6	(	(	PUNCT
ejpam-5572	32	7	2003	2003	NUM
ejpam-5572	32	8	)	)	PUNCT
ejpam-5572	33	1	[	[	X
ejpam-5572	33	2	29	29	NUM
ejpam-5572	33	3	]	]	PUNCT
ejpam-5572	33	4	.	.	PUNCT
ejpam-5572	34	1	in	in	ADP
ejpam-5572	34	2	2019	2019	NUM
ejpam-5572	34	3	,	,	PUNCT
ejpam-5572	34	4	ullah	ullah	PROPN
ejpam-5572	34	5	et	et	PROPN
ejpam-5572	34	6	al	al	PROPN
ejpam-5572	34	7	.	.	PUNCT
ejpam-5572	35	1	[	[	X
ejpam-5572	35	2	36	36	NUM
ejpam-5572	35	3	]	]	PUNCT
ejpam-5572	35	4	developed	develop	VERB
ejpam-5572	35	5	the	the	DET
ejpam-5572	35	6	complex	complex	ADJ
ejpam-5572	35	7	pythagorean	pythagorean	ADJ
ejpam-5572	35	8	fuzzy	fuzzy	ADJ
ejpam-5572	35	9	set	set	NOUN
ejpam-5572	35	10	(	(	PUNCT
ejpam-5572	35	11	cpfs	cpfs	PROPN
ejpam-5572	35	12	)	)	PUNCT
ejpam-5572	35	13	concept	concept	NOUN
ejpam-5572	35	14	,	,	PUNCT
ejpam-5572	35	15	introducing	introduce	VERB
ejpam-5572	35	16	various	various	ADJ
ejpam-5572	35	17	associated	associated	ADJ
ejpam-5572	35	18	measures	measure	NOUN
ejpam-5572	35	19	.	.	PUNCT
ejpam-5572	36	1	this	this	PRON
ejpam-5572	36	2	was	be	AUX
ejpam-5572	36	3	followed	follow	VERB
ejpam-5572	36	4	in	in	ADP
ejpam-5572	36	5	2020	2020	NUM
ejpam-5572	36	6	by	by	ADP
ejpam-5572	36	7	liu	liu	PROPN
ejpam-5572	36	8	et	et	PROPN
ejpam-5572	36	9	al	al	PROPN
ejpam-5572	36	10	.	.	PUNCT
ejpam-5572	37	1	[	[	X
ejpam-5572	37	2	25	25	NUM
ejpam-5572	37	3	]	]	PUNCT
ejpam-5572	37	4	,	,	PUNCT
ejpam-5572	37	5	who	who	PRON
ejpam-5572	37	6	proposed	propose	VERB
ejpam-5572	37	7	complex	complex	ADJ
ejpam-5572	37	8	q	q	ADJ
ejpam-5572	37	9	-	-	PUNCT
ejpam-5572	37	10	rung	rung	ADJ
ejpam-5572	37	11	orthopair	orthopair	ADJ
ejpam-5572	37	12	fuzzy	fuzzy	ADJ
ejpam-5572	37	13	sets	set	NOUN
ejpam-5572	37	14	,	,	PUNCT
ejpam-5572	37	15	a	a	DET
ejpam-5572	37	16	generalization	generalization	NOUN
ejpam-5572	37	17	encompassing	encompass	VERB
ejpam-5572	37	18	cifs	cif	NOUN
ejpam-5572	37	19	,	,	PUNCT
ejpam-5572	37	20	cpfs	cpfs	PROPN
ejpam-5572	37	21	,	,	PUNCT
ejpam-5572	37	22	cffs	cff	NOUN
ejpam-5572	37	23	,	,	PUNCT
ejpam-5572	37	24	and	and	CCONJ
ejpam-5572	37	25	orthopair	orthopair	ADJ
ejpam-5572	37	26	fuzzy	fuzzy	ADJ
ejpam-5572	37	27	sets	set	NOUN
ejpam-5572	37	28	.	.	PUNCT
ejpam-5572	38	1	further	further	ADJ
ejpam-5572	38	2	advancements	advancement	NOUN
ejpam-5572	38	3	were	be	AUX
ejpam-5572	38	4	made	make	VERB
ejpam-5572	38	5	in	in	ADP
ejpam-5572	38	6	2021	2021	NUM
ejpam-5572	38	7	when	when	SCONJ
ejpam-5572	38	8	chinnadurai	chinnadurai	PROPN
ejpam-5572	38	9	et	et	PROPN
ejpam-5572	38	10	al	al	PROPN
ejpam-5572	38	11	.	.	PUNCT
ejpam-5572	39	1	[	[	X
ejpam-5572	39	2	17	17	NUM
ejpam-5572	39	3	]	]	PUNCT
ejpam-5572	39	4	defined	define	VERB
ejpam-5572	39	5	the	the	DET
ejpam-5572	39	6	complex	complex	ADJ
ejpam-5572	39	7	fermatean	fermatean	ADJ
ejpam-5572	39	8	fuzzy	fuzzy	ADJ
ejpam-5572	39	9	set	set	NOUN
ejpam-5572	39	10	(	(	PUNCT
ejpam-5572	39	11	cffs	cff	NOUN
ejpam-5572	39	12	)	)	PUNCT
ejpam-5572	39	13	and	and	CCONJ
ejpam-5572	39	14	explored	explore	VERB
ejpam-5572	39	15	its	its	PRON
ejpam-5572	39	16	applications	application	NOUN
ejpam-5572	39	17	in	in	ADP
ejpam-5572	39	18	decision	decision	NOUN
ejpam-5572	39	19	-	-	PUNCT
ejpam-5572	39	20	making	making	NOUN
ejpam-5572	39	21	(	(	PUNCT
ejpam-5572	39	22	dm	dm	NOUN
ejpam-5572	39	23	)	)	PUNCT
ejpam-5572	39	24	problems	problem	NOUN
ejpam-5572	39	25	.	.	PUNCT
ejpam-5572	40	1	the	the	DET
ejpam-5572	40	2	application	application	NOUN
ejpam-5572	40	3	of	of	ADP
ejpam-5572	40	4	complex	complex	ADJ
ejpam-5572	40	5	fuzzy	fuzzy	ADJ
ejpam-5572	40	6	theories	theory	NOUN
ejpam-5572	40	7	in	in	ADP
ejpam-5572	40	8	dm	dm	PROPN
ejpam-5572	40	9	has	have	AUX
ejpam-5572	40	10	since	since	ADV
ejpam-5572	40	11	expanded	expand	VERB
ejpam-5572	40	12	,	,	PUNCT
ejpam-5572	40	13	as	as	SCONJ
ejpam-5572	40	14	demonstrated	demonstrate	VERB
ejpam-5572	40	15	by	by	ADP
ejpam-5572	40	16	recent	recent	ADJ
ejpam-5572	40	17	works	work	NOUN
ejpam-5572	40	18	such	such	ADJ
ejpam-5572	40	19	as	as	ADP
ejpam-5572	40	20	chen	chen	PROPN
ejpam-5572	40	21	(	(	PUNCT
ejpam-5572	40	22	2023	2023	NUM
ejpam-5572	40	23	)	)	PUNCT
ejpam-5572	41	1	[	[	X
ejpam-5572	41	2	16	16	NUM
ejpam-5572	41	3	]	]	PUNCT
ejpam-5572	41	4	and	and	CCONJ
ejpam-5572	41	5	wang	wang	PROPN
ejpam-5572	41	6	(	(	PUNCT
ejpam-5572	41	7	2023	2023	NUM
ejpam-5572	41	8	)	)	PUNCT
ejpam-5572	42	1	[	[	X
ejpam-5572	42	2	37	37	NUM
ejpam-5572	42	3	]	]	PUNCT
ejpam-5572	42	4	.	.	PUNCT
ejpam-5572	43	1	other	other	ADJ
ejpam-5572	43	2	notable	notable	ADJ
ejpam-5572	43	3	generalizations	generalization	NOUN
ejpam-5572	43	4	include	include	VERB
ejpam-5572	43	5	the	the	DET
ejpam-5572	43	6	use	use	NOUN
ejpam-5572	43	7	of	of	ADP
ejpam-5572	43	8	complex	complex	ADJ
ejpam-5572	43	9	neutrosophic	neutrosophic	ADJ
ejpam-5572	43	10	graphs	graph	NOUN
ejpam-5572	43	11	for	for	ADP
ejpam-5572	43	12	hospital	hospital	NOUN
ejpam-5572	43	13	infrastructure	infrastructure	NOUN
ejpam-5572	43	14	design	design	NOUN
ejpam-5572	43	15	by	by	ADP
ejpam-5572	43	16	alqahtani	alqahtani	PROPN
ejpam-5572	43	17	et	et	PROPN
ejpam-5572	43	18	al	al	PROPN
ejpam-5572	43	19	.	.	PROPN
ejpam-5572	44	1	(	(	PUNCT
ejpam-5572	44	2	2024	2024	NUM
ejpam-5572	44	3	)	)	PUNCT
ejpam-5572	45	1	[	[	X
ejpam-5572	45	2	10	10	NUM
ejpam-5572	45	3	]	]	PUNCT
ejpam-5572	45	4	and	and	CCONJ
ejpam-5572	45	5	the	the	DET
ejpam-5572	45	6	application	application	NOUN
ejpam-5572	45	7	of	of	ADP
ejpam-5572	45	8	complex	complex	ADJ
ejpam-5572	45	9	hesitant	hesitant	ADJ
ejpam-5572	45	10	fuzzy	fuzzy	ADJ
ejpam-5572	45	11	graphs	graph	NOUN
ejpam-5572	45	12	by	by	ADP
ejpam-5572	45	13	abuhijleh	abuhijleh	PROPN
ejpam-5572	45	14	et	et	PROPN
ejpam-5572	45	15	al	al	PROPN
ejpam-5572	45	16	.	.	PROPN
ejpam-5572	45	17	(	(	PUNCT
ejpam-5572	45	18	2023	2023	NUM
ejpam-5572	45	19	)	)	PUNCT
ejpam-5572	46	1	[	[	X
ejpam-5572	46	2	2	2	X
ejpam-5572	46	3	]	]	PUNCT
ejpam-5572	46	4	and	and	CCONJ
ejpam-5572	46	5	alkouri	alkouri	PROPN
ejpam-5572	46	6	(	(	PUNCT
ejpam-5572	46	7	2023	2023	NUM
ejpam-5572	46	8	)	)	PUNCT
ejpam-5572	47	1	[	[	X
ejpam-5572	47	2	6	6	NUM
ejpam-5572	47	3	]	]	PUNCT
ejpam-5572	47	4	.	.	PUNCT
ejpam-5572	48	1	additionally	additionally	ADV
ejpam-5572	48	2	,	,	PUNCT
ejpam-5572	48	3	alqaraleh	alqaraleh	PROPN
ejpam-5572	48	4	et	et	PROPN
ejpam-5572	48	5	al	al	PROPN
ejpam-5572	48	6	.	.	PROPN
ejpam-5572	48	7	(	(	PUNCT
ejpam-5572	48	8	2022	2022	NUM
ejpam-5572	48	9	)	)	PUNCT
ejpam-5572	49	1	[	[	X
ejpam-5572	49	2	11	11	NUM
ejpam-5572	49	3	]	]	PUNCT
ejpam-5572	49	4	introduced	introduce	VERB
ejpam-5572	49	5	bipolar	bipolar	ADJ
ejpam-5572	49	6	complex	complex	ADJ
ejpam-5572	49	7	fuzzy	fuzzy	ADJ
ejpam-5572	49	8	soft	soft	ADJ
ejpam-5572	49	9	sets	set	NOUN
ejpam-5572	49	10	with	with	ADP
ejpam-5572	49	11	practical	practical	ADJ
ejpam-5572	49	12	applications	application	NOUN
ejpam-5572	49	13	.	.	PUNCT
ejpam-5572	50	1	for	for	ADP
ejpam-5572	50	2	further	further	ADJ
ejpam-5572	50	3	exploration	exploration	NOUN
ejpam-5572	50	4	of	of	ADP
ejpam-5572	50	5	generalizations	generalization	NOUN
ejpam-5572	50	6	and	and	CCONJ
ejpam-5572	50	7	applications	application	NOUN
ejpam-5572	50	8	within	within	ADP
ejpam-5572	50	9	this	this	DET
ejpam-5572	50	10	domain	domain	NOUN
ejpam-5572	50	11	,	,	PUNCT
ejpam-5572	50	12	refer	refer	VERB
ejpam-5572	50	13	to	to	ADP
ejpam-5572	50	14	recent	recent	ADJ
ejpam-5572	50	15	works	work	NOUN
ejpam-5572	50	16	by	by	ADP
ejpam-5572	50	17	al	al	PROPN
ejpam-5572	50	18	-	-	PROPN
ejpam-5572	50	19	masarwah	masarwah	PROPN
ejpam-5572	50	20	et	et	PROPN
ejpam-5572	50	21	al	al	PROPN
ejpam-5572	50	22	.	.	PROPN
ejpam-5572	51	1	(	(	PUNCT
ejpam-5572	51	2	2023	2023	NUM
ejpam-5572	51	3	)	)	PUNCT
ejpam-5572	52	1	[	[	X
ejpam-5572	52	2	5	5	NUM
ejpam-5572	52	3	]	]	PUNCT
ejpam-5572	52	4	,	,	PUNCT
ejpam-5572	52	5	fallat	fallat	PROPN
ejpam-5572	52	6	et	et	PROPN
ejpam-5572	52	7	al	al	PROPN
ejpam-5572	52	8	.	.	PROPN
ejpam-5572	52	9	(	(	PUNCT
ejpam-5572	52	10	2022	2022	NUM
ejpam-5572	52	11	)	)	PUNCT
ejpam-5572	53	1	[	[	X
ejpam-5572	53	2	19	19	NUM
ejpam-5572	53	3	]	]	PUNCT
ejpam-5572	53	4	,	,	PUNCT
ejpam-5572	53	5	and	and	CCONJ
ejpam-5572	53	6	hazaymeh	hazaymeh	NOUN
ejpam-5572	53	7	et	et	PROPN
ejpam-5572	53	8	al	al	PROPN
ejpam-5572	53	9	.	.	PROPN
ejpam-5572	54	1	(	(	PUNCT
ejpam-5572	54	2	2024–2025	2024–2025	NUM
ejpam-5572	54	3	)	)	PUNCT
ejpam-5572	55	1	[	[	X
ejpam-5572	55	2	21	21	NUM
ejpam-5572	55	3	,	,	PUNCT
ejpam-5572	55	4	22	22	NUM
ejpam-5572	55	5	]	]	PUNCT
ejpam-5572	55	6	.	.	PUNCT
ejpam-5572	56	1	in	in	ADP
ejpam-5572	56	2	parallel	parallel	NOUN
ejpam-5572	56	3	with	with	ADP
ejpam-5572	56	4	the	the	DET
ejpam-5572	56	5	advancements	advancement	NOUN
ejpam-5572	56	6	in	in	ADP
ejpam-5572	56	7	fuzzy	fuzzy	ADJ
ejpam-5572	56	8	set	set	NOUN
ejpam-5572	56	9	theory	theory	NOUN
ejpam-5572	56	10	,	,	PUNCT
ejpam-5572	56	11	significant	significant	ADJ
ejpam-5572	56	12	progress	progress	NOUN
ejpam-5572	56	13	has	have	AUX
ejpam-5572	56	14	also	also	ADV
ejpam-5572	56	15	been	be	AUX
ejpam-5572	56	16	made	make	VERB
ejpam-5572	56	17	in	in	ADP
ejpam-5572	56	18	fuzzy	fuzzy	ADJ
ejpam-5572	56	19	group	group	NOUN
ejpam-5572	56	20	theory	theory	NOUN
ejpam-5572	56	21	.	.	PUNCT
ejpam-5572	57	1	rosenfeld	rosenfeld	PROPN
ejpam-5572	57	2	(	(	PUNCT
ejpam-5572	57	3	1971	1971	NUM
ejpam-5572	57	4	)	)	PUNCT
ejpam-5572	58	1	[	[	X
ejpam-5572	58	2	32	32	NUM
ejpam-5572	58	3	]	]	PUNCT
ejpam-5572	58	4	introduced	introduce	VERB
ejpam-5572	58	5	the	the	DET
ejpam-5572	58	6	concept	concept	NOUN
ejpam-5572	58	7	of	of	ADP
ejpam-5572	58	8	a	a	DET
ejpam-5572	58	9	fuzzy	fuzzy	ADJ
ejpam-5572	58	10	subgroup	subgroup	NOUN
ejpam-5572	58	11	(	(	PUNCT
ejpam-5572	58	12	fsg	fsg	PROPN
ejpam-5572	58	13	)	)	PUNCT
ejpam-5572	58	14	as	as	ADP
ejpam-5572	58	15	a	a	DET
ejpam-5572	58	16	generalization	generalization	NOUN
ejpam-5572	58	17	of	of	ADP
ejpam-5572	58	18	the	the	DET
ejpam-5572	58	19	classical	classical	ADJ
ejpam-5572	58	20	group	group	NOUN
ejpam-5572	58	21	.	.	PUNCT
ejpam-5572	59	1	this	this	DET
ejpam-5572	59	2	foundational	foundational	ADJ
ejpam-5572	59	3	work	work	NOUN
ejpam-5572	59	4	inspired	inspire	VERB
ejpam-5572	59	5	numerous	numerous	ADJ
ejpam-5572	59	6	mathematicians	mathematician	NOUN
ejpam-5572	59	7	to	to	PART
ejpam-5572	59	8	explore	explore	VERB
ejpam-5572	59	9	group	group	NOUN
ejpam-5572	59	10	theory	theory	NOUN
ejpam-5572	59	11	through	through	ADP
ejpam-5572	59	12	fuzzy	fuzzy	ADJ
ejpam-5572	59	13	sets	set	NOUN
ejpam-5572	59	14	.	.	PUNCT
ejpam-5572	60	1	in	in	ADP
ejpam-5572	60	2	1989	1989	NUM
ejpam-5572	60	3	,	,	PUNCT
ejpam-5572	60	4	biswas	biswa	VERB
ejpam-5572	61	1	[	[	X
ejpam-5572	61	2	15	15	NUM
ejpam-5572	61	3	]	]	PUNCT
ejpam-5572	61	4	expanded	expand	VERB
ejpam-5572	61	5	on	on	ADP
ejpam-5572	61	6	this	this	DET
ejpam-5572	61	7	idea	idea	NOUN
ejpam-5572	61	8	by	by	ADP
ejpam-5572	61	9	defining	define	VERB
ejpam-5572	61	10	the	the	DET
ejpam-5572	61	11	intuitionistic	intuitionistic	ADJ
ejpam-5572	61	12	fuzzy	fuzzy	ADJ
ejpam-5572	61	13	subgroup	subgroup	NOUN
ejpam-5572	61	14	(	(	PUNCT
ejpam-5572	61	15	ifsg	ifsg	NOUN
ejpam-5572	61	16	)	)	PUNCT
ejpam-5572	61	17	and	and	CCONJ
ejpam-5572	61	18	analyzing	analyze	VERB
ejpam-5572	61	19	its	its	PRON
ejpam-5572	61	20	algebraic	algebraic	ADJ
ejpam-5572	61	21	properties	property	NOUN
ejpam-5572	61	22	.	.	PUNCT
ejpam-5572	62	1	in	in	ADP
ejpam-5572	62	2	2020	2020	NUM
ejpam-5572	62	3	,	,	PUNCT
ejpam-5572	62	4	bhunia	bhunia	NOUN
ejpam-5572	62	5	et	et	PROPN
ejpam-5572	62	6	al	al	PROPN
ejpam-5572	62	7	.	.	PUNCT
ejpam-5572	63	1	[	[	X
ejpam-5572	63	2	14	14	NUM
ejpam-5572	63	3	]	]	PUNCT
ejpam-5572	63	4	introduced	introduce	VERB
ejpam-5572	63	5	the	the	DET
ejpam-5572	63	6	pythagorean	pythagorean	PROPN
ejpam-5572	63	7	fuzzy	fuzzy	ADJ
ejpam-5572	63	8	subgroup	subgroup	NOUN
ejpam-5572	63	9	(	(	PUNCT
ejpam-5572	63	10	pfsg	pfsg	NOUN
ejpam-5572	63	11	)	)	PUNCT
ejpam-5572	63	12	,	,	PUNCT
ejpam-5572	63	13	further	far	ADV
ejpam-5572	63	14	examining	examine	VERB
ejpam-5572	63	15	its	its	PRON
ejpam-5572	63	16	algebraic	algebraic	ADJ
ejpam-5572	63	17	structure	structure	NOUN
ejpam-5572	63	18	.	.	PUNCT
ejpam-5572	64	1	the	the	DET
ejpam-5572	64	2	same	same	ADJ
ejpam-5572	64	3	year	year	NOUN
ejpam-5572	64	4	,	,	PUNCT
ejpam-5572	64	5	manuscripts	manuscript	NOUN
ejpam-5572	64	6	on	on	ADP
ejpam-5572	64	7	the	the	DET
ejpam-5572	64	8	complex	complex	ADJ
ejpam-5572	64	9	intuitionistic	intuitionistic	ADJ
ejpam-5572	64	10	fuzzy	fuzzy	ADJ
ejpam-5572	64	11	subgroup	subgroup	NOUN
ejpam-5572	64	12	(	(	PUNCT
ejpam-5572	64	13	cifsg	cifsg	PROPN
ejpam-5572	64	14	)	)	PUNCT
ejpam-5572	65	1	[	[	X
ejpam-5572	65	2	20	20	NUM
ejpam-5572	65	3	]	]	PUNCT
ejpam-5572	65	4	and	and	CCONJ
ejpam-5572	65	5	the	the	DET
ejpam-5572	65	6	complex	complex	ADJ
ejpam-5572	65	7	fuzzy	fuzzy	ADJ
ejpam-5572	65	8	subgroup	subgroup	NOUN
ejpam-5572	65	9	(	(	PUNCT
ejpam-5572	65	10	cfsg	cfsg	PROPN
ejpam-5572	65	11	)	)	PUNCT
ejpam-5572	66	1	[	[	X
ejpam-5572	66	2	3	3	X
ejpam-5572	66	3	]	]	PUNCT
ejpam-5572	66	4	were	be	AUX
ejpam-5572	66	5	published	publish	VERB
ejpam-5572	66	6	,	,	PUNCT
ejpam-5572	66	7	adding	add	VERB
ejpam-5572	66	8	new	new	ADJ
ejpam-5572	66	9	dimensions	dimension	NOUN
ejpam-5572	66	10	to	to	ADP
ejpam-5572	66	11	the	the	DET
ejpam-5572	66	12	field	field	NOUN
ejpam-5572	66	13	.	.	PUNCT
ejpam-5572	67	1	most	most	ADV
ejpam-5572	67	2	recently	recently	ADV
ejpam-5572	67	3	,	,	PUNCT
ejpam-5572	67	4	a	a	DET
ejpam-5572	67	5	study	study	NOUN
ejpam-5572	67	6	on	on	ADP
ejpam-5572	67	7	the	the	DET
ejpam-5572	67	8	complex	complex	ADJ
ejpam-5572	67	9	e.a	e.a	PROPN
ejpam-5572	67	10	.	.	PROPN
ejpam-5572	67	11	abuhijleh	abuhijleh	PROPN
ejpam-5572	67	12	,	,	PUNCT
ejpam-5572	67	13	a.	a.	PROPN
ejpam-5572	67	14	alkouri	alkouri	PROPN
ejpam-5572	67	15	/	/	PROPN
ejpam-5572	67	16	eur	eur	PROPN
ejpam-5572	67	17	.	.	PUNCT
ejpam-5572	68	1	j.	j.	PROPN
ejpam-5572	68	2	pure	pure	PROPN
ejpam-5572	68	3	appl	appl	PROPN
ejpam-5572	68	4	.	.	PROPN
ejpam-5572	68	5	math	math	PROPN
ejpam-5572	68	6	,	,	PUNCT
ejpam-5572	68	7	18	18	NUM
ejpam-5572	68	8	(	(	PUNCT
ejpam-5572	68	9	1	1	NUM
ejpam-5572	68	10	)	)	PUNCT
ejpam-5572	68	11	(	(	PUNCT
ejpam-5572	68	12	2025	2025	NUM
ejpam-5572	68	13	)	)	PUNCT
ejpam-5572	68	14	,	,	PUNCT
ejpam-5572	68	15	5572	5572	NUM
ejpam-5572	68	16	3	3	NUM
ejpam-5572	68	17	of	of	ADP
ejpam-5572	68	18	19	19	NUM
ejpam-5572	68	19	pythagorean	pythagorean	ADJ
ejpam-5572	68	20	fuzzy	fuzzy	ADJ
ejpam-5572	68	21	subgroup	subgroup	NOUN
ejpam-5572	68	22	(	(	PUNCT
ejpam-5572	68	23	cpfsg	cpfsg	PROPN
ejpam-5572	68	24	)	)	PUNCT
ejpam-5572	68	25	was	be	AUX
ejpam-5572	68	26	published	publish	VERB
ejpam-5572	68	27	for	for	ADP
ejpam-5572	68	28	publication	publication	NOUN
ejpam-5572	68	29	[	[	X
ejpam-5572	68	30	7	7	NUM
ejpam-5572	68	31	]	]	PUNCT
ejpam-5572	68	32	.	.	PUNCT
ejpam-5572	69	1	building	build	VERB
ejpam-5572	69	2	on	on	ADP
ejpam-5572	69	3	these	these	DET
ejpam-5572	69	4	developments	development	NOUN
ejpam-5572	69	5	,	,	PUNCT
ejpam-5572	69	6	silambarasan	silambarasan	NOUN
ejpam-5572	69	7	(	(	PUNCT
ejpam-5572	69	8	2021	2021	NUM
ejpam-5572	69	9	)	)	PUNCT
ejpam-5572	70	1	[	[	X
ejpam-5572	70	2	35	35	NUM
ejpam-5572	70	3	]	]	PUNCT
ejpam-5572	70	4	introduced	introduce	VERB
ejpam-5572	70	5	the	the	DET
ejpam-5572	70	6	fermatean	fermatean	ADJ
ejpam-5572	70	7	fuzzy	fuzzy	ADJ
ejpam-5572	70	8	subgroup	subgroup	NOUN
ejpam-5572	70	9	(	(	PUNCT
ejpam-5572	70	10	ffsg	ffsg	NOUN
ejpam-5572	70	11	)	)	PUNCT
ejpam-5572	70	12	,	,	PUNCT
ejpam-5572	70	13	analyzing	analyze	VERB
ejpam-5572	70	14	its	its	PRON
ejpam-5572	70	15	properties	property	NOUN
ejpam-5572	70	16	and	and	CCONJ
ejpam-5572	70	17	its	its	PRON
ejpam-5572	70	18	relationships	relationship	NOUN
ejpam-5572	70	19	with	with	ADP
ejpam-5572	70	20	ifsg	ifsg	NOUN
ejpam-5572	70	21	and	and	CCONJ
ejpam-5572	70	22	pfsg	pfsg	NOUN
ejpam-5572	70	23	.	.	PUNCT
ejpam-5572	71	1	this	this	DET
ejpam-5572	71	2	work	work	NOUN
ejpam-5572	71	3	has	have	AUX
ejpam-5572	71	4	inspired	inspire	VERB
ejpam-5572	71	5	further	further	ADJ
ejpam-5572	71	6	research	research	NOUN
ejpam-5572	71	7	into	into	ADP
ejpam-5572	71	8	ffsg	ffsg	NOUN
ejpam-5572	71	9	,	,	PUNCT
ejpam-5572	71	10	including	include	VERB
ejpam-5572	71	11	studies	study	NOUN
ejpam-5572	71	12	by	by	ADP
ejpam-5572	71	13	balamurugan	balamurugan	NOUN
ejpam-5572	71	14	(	(	PUNCT
ejpam-5572	71	15	2022	2022	NUM
ejpam-5572	71	16	)	)	PUNCT
ejpam-5572	72	1	[	[	X
ejpam-5572	72	2	13	13	NUM
ejpam-5572	72	3	]	]	PUNCT
ejpam-5572	72	4	,	,	PUNCT
ejpam-5572	72	5	kalaichelvan	kalaichelvan	NOUN
ejpam-5572	72	6	(	(	PUNCT
ejpam-5572	72	7	2022	2022	NUM
ejpam-5572	72	8	)	)	PUNCT
ejpam-5572	73	1	[	[	X
ejpam-5572	73	2	24	24	NUM
ejpam-5572	73	3	]	]	PUNCT
ejpam-5572	73	4	,	,	PUNCT
ejpam-5572	73	5	nagarajan	nagarajan	NOUN
ejpam-5572	73	6	(	(	PUNCT
ejpam-5572	73	7	2021	2021	NUM
ejpam-5572	73	8	)	)	PUNCT
ejpam-5572	74	1	[	[	X
ejpam-5572	74	2	26	26	NUM
ejpam-5572	74	3	]	]	PUNCT
ejpam-5572	74	4	,	,	PUNCT
ejpam-5572	74	5	onasanya	onasanya	NOUN
ejpam-5572	74	6	(	(	PUNCT
ejpam-5572	74	7	2022	2022	NUM
ejpam-5572	74	8	)	)	PUNCT
ejpam-5572	75	1	[	[	X
ejpam-5572	75	2	28	28	NUM
ejpam-5572	75	3	]	]	PUNCT
ejpam-5572	75	4	,	,	PUNCT
ejpam-5572	75	5	and	and	CCONJ
ejpam-5572	75	6	muhammad	muhammad	X
ejpam-5572	75	7	(	(	PUNCT
ejpam-5572	75	8	2022	2022	NUM
ejpam-5572	75	9	)	)	PUNCT
ejpam-5572	76	1	[	[	X
ejpam-5572	76	2	31	31	NUM
ejpam-5572	76	3	]	]	PUNCT
ejpam-5572	76	4	.	.	PUNCT
ejpam-5572	77	1	onasanya	onasanya	INTJ
ejpam-5572	77	2	et	et	PROPN
ejpam-5572	77	3	al	al	PROPN
ejpam-5572	77	4	.	.	PROPN
ejpam-5572	78	1	(	(	PUNCT
ejpam-5572	78	2	2022	2022	NUM
ejpam-5572	78	3	)	)	PUNCT
ejpam-5572	79	1	[	[	X
ejpam-5572	79	2	28	28	NUM
ejpam-5572	79	3	]	]	PUNCT
ejpam-5572	79	4	introduced	introduce	VERB
ejpam-5572	79	5	fermatean	fermatean	NOUN
ejpam-5572	79	6	fuzzy	fuzzy	ADJ
ejpam-5572	79	7	subgroups	subgroup	NOUN
ejpam-5572	79	8	within	within	ADP
ejpam-5572	79	9	the	the	DET
ejpam-5572	79	10	qrung	qrung	PROPN
ejpam-5572	79	11	orthopair	orthopair	NOUN
ejpam-5572	79	12	fuzzy	fuzzy	ADJ
ejpam-5572	79	13	sets	set	NOUN
ejpam-5572	79	14	framework	framework	NOUN
ejpam-5572	79	15	in	in	ADP
ejpam-5572	79	16	group	group	NOUN
ejpam-5572	79	17	theory	theory	NOUN
ejpam-5572	79	18	,	,	PUNCT
ejpam-5572	79	19	coining	coin	VERB
ejpam-5572	79	20	the	the	DET
ejpam-5572	79	21	term	term	NOUN
ejpam-5572	79	22	”	"	PUNCT
ejpam-5572	79	23	harmonized	harmonized	ADJ
ejpam-5572	79	24	fuzzy	fuzzy	ADJ
ejpam-5572	79	25	groups	group	NOUN
ejpam-5572	79	26	”	"	PUNCT
ejpam-5572	79	27	.	.	PUNCT
ejpam-5572	80	1	these	these	DET
ejpam-5572	80	2	groups	group	NOUN
ejpam-5572	80	3	unify	unify	VERB
ejpam-5572	80	4	various	various	ADJ
ejpam-5572	80	5	subgroup	subgroup	NOUN
ejpam-5572	80	6	types	type	NOUN
ejpam-5572	80	7	as	as	ADP
ejpam-5572	80	8	special	special	ADJ
ejpam-5572	80	9	cases	case	NOUN
ejpam-5572	80	10	and	and	CCONJ
ejpam-5572	80	11	provide	provide	VERB
ejpam-5572	80	12	a	a	DET
ejpam-5572	80	13	versatile	versatile	ADJ
ejpam-5572	80	14	foundation	foundation	NOUN
ejpam-5572	80	15	for	for	ADP
ejpam-5572	80	16	further	further	ADJ
ejpam-5572	80	17	exploration	exploration	NOUN
ejpam-5572	80	18	.	.	PUNCT
ejpam-5572	81	1	while	while	SCONJ
ejpam-5572	81	2	their	their	PRON
ejpam-5572	81	3	study	study	NOUN
ejpam-5572	81	4	addressed	address	VERB
ejpam-5572	81	5	several	several	ADJ
ejpam-5572	81	6	properties	property	NOUN
ejpam-5572	81	7	of	of	ADP
ejpam-5572	81	8	harmonized	harmonized	ADJ
ejpam-5572	81	9	fuzzy	fuzzy	ADJ
ejpam-5572	81	10	groups	group	NOUN
ejpam-5572	81	11	,	,	PUNCT
ejpam-5572	81	12	the	the	DET
ejpam-5572	81	13	extension	extension	NOUN
ejpam-5572	81	14	to	to	ADP
ejpam-5572	81	15	complex	complex	ADJ
ejpam-5572	81	16	harmonized	harmonized	ADJ
ejpam-5572	81	17	fuzzy	fuzzy	ADJ
ejpam-5572	81	18	groups	group	NOUN
ejpam-5572	81	19	remains	remain	VERB
ejpam-5572	81	20	an	an	DET
ejpam-5572	81	21	open	open	ADJ
ejpam-5572	81	22	area	area	NOUN
ejpam-5572	81	23	of	of	ADP
ejpam-5572	81	24	research	research	NOUN
ejpam-5572	81	25	.	.	PUNCT
ejpam-5572	82	1	the	the	DET
ejpam-5572	82	2	motivation	motivation	NOUN
ejpam-5572	82	3	for	for	ADP
ejpam-5572	82	4	constructing	construct	VERB
ejpam-5572	82	5	complex	complex	ADJ
ejpam-5572	82	6	fermatean	fermatean	ADJ
ejpam-5572	82	7	fuzzy	fuzzy	ADJ
ejpam-5572	82	8	subgroups	subgroup	NOUN
ejpam-5572	82	9	(	(	PUNCT
ejpam-5572	82	10	cffsg	cffsg	ADJ
ejpam-5572	82	11	)	)	PUNCT
ejpam-5572	82	12	lies	lie	VERB
ejpam-5572	82	13	in	in	ADP
ejpam-5572	82	14	advancing	advance	VERB
ejpam-5572	82	15	the	the	DET
ejpam-5572	82	16	mathematical	mathematical	ADJ
ejpam-5572	82	17	framework	framework	NOUN
ejpam-5572	82	18	of	of	ADP
ejpam-5572	82	19	fuzzy	fuzzy	ADJ
ejpam-5572	82	20	group	group	NOUN
ejpam-5572	82	21	theory	theory	NOUN
ejpam-5572	82	22	.	.	PUNCT
ejpam-5572	83	1	specifically	specifically	ADV
ejpam-5572	83	2	,	,	PUNCT
ejpam-5572	83	3	it	it	PRON
ejpam-5572	83	4	seeks	seek	VERB
ejpam-5572	83	5	to	to	PART
ejpam-5572	83	6	incorporate	incorporate	VERB
ejpam-5572	83	7	periodic	periodic	ADJ
ejpam-5572	83	8	information	information	NOUN
ejpam-5572	83	9	inherent	inherent	ADJ
ejpam-5572	83	10	in	in	ADP
ejpam-5572	83	11	complex	complex	ADJ
ejpam-5572	83	12	fermatean	fermatean	ADJ
ejpam-5572	83	13	fuzzy	fuzzy	ADJ
ejpam-5572	83	14	sets	set	NOUN
ejpam-5572	83	15	(	(	PUNCT
ejpam-5572	83	16	cffs	cff	NOUN
ejpam-5572	83	17	)	)	PUNCT
ejpam-5572	83	18	and	and	CCONJ
ejpam-5572	83	19	leverage	leverage	VERB
ejpam-5572	83	20	their	their	PRON
ejpam-5572	83	21	capacity	capacity	NOUN
ejpam-5572	83	22	to	to	PART
ejpam-5572	83	23	represent	represent	VERB
ejpam-5572	83	24	larger	large	ADJ
ejpam-5572	83	25	values	value	NOUN
ejpam-5572	83	26	compared	compare	VERB
ejpam-5572	83	27	to	to	ADP
ejpam-5572	83	28	complex	complex	ADJ
ejpam-5572	83	29	pythagorean	pythagorean	ADJ
ejpam-5572	83	30	fuzzy	fuzzy	ADJ
ejpam-5572	83	31	sets	set	NOUN
ejpam-5572	83	32	(	(	PUNCT
ejpam-5572	83	33	cpfs	cpfs	PROPN
ejpam-5572	83	34	)	)	PUNCT
ejpam-5572	83	35	and	and	CCONJ
ejpam-5572	83	36	complex	complex	ADJ
ejpam-5572	83	37	intuitionistic	intuitionistic	ADJ
ejpam-5572	83	38	fuzzy	fuzzy	ADJ
ejpam-5572	83	39	sets	set	NOUN
ejpam-5572	83	40	(	(	PUNCT
ejpam-5572	83	41	cifs	cif	NOUN
ejpam-5572	83	42	)	)	PUNCT
ejpam-5572	83	43	.	.	PUNCT
ejpam-5572	84	1	this	this	DET
ejpam-5572	84	2	development	development	NOUN
ejpam-5572	84	3	not	not	PART
ejpam-5572	84	4	only	only	ADV
ejpam-5572	84	5	enriches	enrich	VERB
ejpam-5572	84	6	the	the	DET
ejpam-5572	84	7	theoretical	theoretical	ADJ
ejpam-5572	84	8	landscape	landscape	NOUN
ejpam-5572	84	9	but	but	CCONJ
ejpam-5572	84	10	also	also	ADV
ejpam-5572	84	11	opens	open	VERB
ejpam-5572	84	12	pathways	pathway	NOUN
ejpam-5572	84	13	for	for	ADP
ejpam-5572	84	14	practical	practical	ADJ
ejpam-5572	84	15	applications	application	NOUN
ejpam-5572	84	16	,	,	PUNCT
ejpam-5572	84	17	such	such	ADJ
ejpam-5572	84	18	as	as	ADP
ejpam-5572	84	19	cryptographic	cryptographic	ADJ
ejpam-5572	84	20	primitives	primitive	NOUN
ejpam-5572	84	21	and	and	CCONJ
ejpam-5572	84	22	generalized	generalize	VERB
ejpam-5572	84	23	periodic	periodic	ADJ
ejpam-5572	84	24	algorithms	algorithm	NOUN
ejpam-5572	84	25	.	.	PUNCT
ejpam-5572	85	1	future	future	ADJ
ejpam-5572	85	2	work	work	NOUN
ejpam-5572	85	3	includes	include	VERB
ejpam-5572	85	4	the	the	DET
ejpam-5572	85	5	construction	construction	NOUN
ejpam-5572	85	6	of	of	ADP
ejpam-5572	85	7	cffsg	cffsg	NOUN
ejpam-5572	85	8	and	and	CCONJ
ejpam-5572	85	9	the	the	DET
ejpam-5572	85	10	development	development	NOUN
ejpam-5572	85	11	of	of	ADP
ejpam-5572	85	12	cyclic	cyclic	ADJ
ejpam-5572	85	13	cffsg	cffsg	NOUN
ejpam-5572	85	14	as	as	ADP
ejpam-5572	85	15	a	a	DET
ejpam-5572	85	16	specialized	specialized	ADJ
ejpam-5572	85	17	extension	extension	NOUN
ejpam-5572	85	18	.	.	PUNCT
ejpam-5572	86	1	additionally	additionally	ADV
ejpam-5572	86	2	,	,	PUNCT
ejpam-5572	86	3	integrating	integrate	VERB
ejpam-5572	86	4	results	result	NOUN
ejpam-5572	86	5	from	from	ADP
ejpam-5572	86	6	fixed	fix	VERB
ejpam-5572	86	7	-	-	PUNCT
ejpam-5572	86	8	point	point	NOUN
ejpam-5572	86	9	theory	theory	NOUN
ejpam-5572	86	10	(	(	PUNCT
ejpam-5572	86	11	e.g.	e.g.	ADV
ejpam-5572	86	12	,	,	PUNCT
ejpam-5572	86	13	[	[	X
ejpam-5572	86	14	1	1	NUM
ejpam-5572	86	15	]	]	PUNCT
ejpam-5572	86	16	and	and	CCONJ
ejpam-5572	86	17	[	[	X
ejpam-5572	86	18	34	34	NUM
ejpam-5572	86	19	]	]	PUNCT
ejpam-5572	86	20	)	)	PUNCT
ejpam-5572	86	21	with	with	ADP
ejpam-5572	86	22	complex	complex	ADJ
ejpam-5572	86	23	fermatean	fermatean	ADJ
ejpam-5572	86	24	fuzzy	fuzzy	ADJ
ejpam-5572	86	25	algebra	algebra	NOUN
ejpam-5572	86	26	presents	present	VERB
ejpam-5572	86	27	a	a	DET
ejpam-5572	86	28	promising	promising	ADJ
ejpam-5572	86	29	avenue	avenue	NOUN
ejpam-5572	86	30	for	for	ADP
ejpam-5572	86	31	novel	novel	ADJ
ejpam-5572	86	32	applications	application	NOUN
ejpam-5572	86	33	.	.	PUNCT
ejpam-5572	87	1	such	such	ADJ
ejpam-5572	87	2	integration	integration	NOUN
ejpam-5572	87	3	could	could	AUX
ejpam-5572	87	4	address	address	VERB
ejpam-5572	87	5	real	real	ADJ
ejpam-5572	87	6	-	-	PUNCT
ejpam-5572	87	7	world	world	NOUN
ejpam-5572	87	8	problems	problem	NOUN
ejpam-5572	87	9	by	by	ADP
ejpam-5572	87	10	employing	employ	VERB
ejpam-5572	87	11	metric	metric	ADJ
ejpam-5572	87	12	space	space	NOUN
ejpam-5572	87	13	frameworks	framework	NOUN
ejpam-5572	87	14	to	to	PART
ejpam-5572	87	15	create	create	VERB
ejpam-5572	87	16	innovative	innovative	ADJ
ejpam-5572	87	17	solutions	solution	NOUN
ejpam-5572	87	18	.	.	PUNCT
ejpam-5572	88	1	this	this	DET
ejpam-5572	88	2	paper	paper	NOUN
ejpam-5572	88	3	explores	explore	VERB
ejpam-5572	88	4	the	the	DET
ejpam-5572	88	5	concept	concept	NOUN
ejpam-5572	88	6	of	of	ADP
ejpam-5572	88	7	the	the	DET
ejpam-5572	88	8	complex	complex	ADJ
ejpam-5572	88	9	fermatean	fermatean	ADJ
ejpam-5572	88	10	fuzzy	fuzzy	ADJ
ejpam-5572	88	11	subgroup	subgroup	NOUN
ejpam-5572	88	12	(	(	PUNCT
ejpam-5572	88	13	cffsg	cffsg	ADJ
ejpam-5572	88	14	)	)	PUNCT
ejpam-5572	88	15	as	as	ADP
ejpam-5572	88	16	an	an	DET
ejpam-5572	88	17	enhancement	enhancement	NOUN
ejpam-5572	88	18	of	of	ADP
ejpam-5572	88	19	both	both	CCONJ
ejpam-5572	88	20	the	the	DET
ejpam-5572	88	21	complex	complex	ADJ
ejpam-5572	88	22	pythagorean	pythagorean	ADJ
ejpam-5572	88	23	fuzzy	fuzzy	ADJ
ejpam-5572	88	24	subgroup	subgroup	NOUN
ejpam-5572	88	25	(	(	PUNCT
ejpam-5572	88	26	cpfsg	cpfsg	NOUN
ejpam-5572	88	27	)	)	PUNCT
ejpam-5572	88	28	and	and	CCONJ
ejpam-5572	88	29	the	the	DET
ejpam-5572	88	30	fermatean	fermatean	ADJ
ejpam-5572	88	31	fuzzy	fuzzy	ADJ
ejpam-5572	88	32	subgroup	subgroup	NOUN
ejpam-5572	88	33	(	(	PUNCT
ejpam-5572	88	34	ffsg	ffsg	NOUN
ejpam-5572	88	35	)	)	PUNCT
ejpam-5572	88	36	.	.	PUNCT
ejpam-5572	89	1	section	section	NOUN
ejpam-5572	89	2	2	2	NUM
ejpam-5572	89	3	provides	provide	VERB
ejpam-5572	89	4	an	an	DET
ejpam-5572	89	5	overview	overview	NOUN
ejpam-5572	89	6	of	of	ADP
ejpam-5572	89	7	key	key	ADJ
ejpam-5572	89	8	definitions	definition	NOUN
ejpam-5572	89	9	from	from	ADP
ejpam-5572	89	10	relevant	relevant	ADJ
ejpam-5572	89	11	literature	literature	NOUN
ejpam-5572	89	12	,	,	PUNCT
ejpam-5572	89	13	establishing	establish	VERB
ejpam-5572	89	14	the	the	DET
ejpam-5572	89	15	foundational	foundational	ADJ
ejpam-5572	89	16	concepts	concept	NOUN
ejpam-5572	89	17	.	.	PUNCT
ejpam-5572	90	1	section	section	NOUN
ejpam-5572	90	2	3	3	NUM
ejpam-5572	90	3	introduces	introduce	NOUN
ejpam-5572	90	4	the	the	DET
ejpam-5572	90	5	formal	formal	ADJ
ejpam-5572	90	6	definition	definition	NOUN
ejpam-5572	90	7	of	of	ADP
ejpam-5572	90	8	cffsg	cffsg	NOUN
ejpam-5572	90	9	and	and	CCONJ
ejpam-5572	90	10	examines	examine	VERB
ejpam-5572	90	11	its	its	PRON
ejpam-5572	90	12	fundamental	fundamental	ADJ
ejpam-5572	90	13	properties	property	NOUN
ejpam-5572	90	14	.	.	PUNCT
ejpam-5572	91	1	section	section	NOUN
ejpam-5572	91	2	4	4	NUM
ejpam-5572	91	3	extends	extend	VERB
ejpam-5572	91	4	the	the	DET
ejpam-5572	91	5	discussion	discussion	NOUN
ejpam-5572	91	6	to	to	ADP
ejpam-5572	91	7	complex	complex	ADJ
ejpam-5572	91	8	fermatean	fermatean	ADJ
ejpam-5572	91	9	fuzzy	fuzzy	ADJ
ejpam-5572	91	10	normal	normal	ADJ
ejpam-5572	91	11	subgroups	subgroup	NOUN
ejpam-5572	91	12	,	,	PUNCT
ejpam-5572	91	13	detailing	detail	VERB
ejpam-5572	91	14	their	their	PRON
ejpam-5572	91	15	characteristics	characteristic	NOUN
ejpam-5572	91	16	.	.	PUNCT
ejpam-5572	92	1	section	section	NOUN
ejpam-5572	92	2	5	5	NUM
ejpam-5572	92	3	analyzes	analyze	VERB
ejpam-5572	92	4	homomorphisms	homomorphism	NOUN
ejpam-5572	92	5	within	within	ADP
ejpam-5572	92	6	the	the	DET
ejpam-5572	92	7	context	context	NOUN
ejpam-5572	92	8	of	of	ADP
ejpam-5572	92	9	cffsg	cffsg	ADJ
ejpam-5572	92	10	,	,	PUNCT
ejpam-5572	92	11	highlighting	highlight	VERB
ejpam-5572	92	12	their	their	PRON
ejpam-5572	92	13	properties	property	NOUN
ejpam-5572	92	14	and	and	CCONJ
ejpam-5572	92	15	implications	implication	NOUN
ejpam-5572	92	16	.	.	PUNCT
ejpam-5572	93	1	finally	finally	ADV
ejpam-5572	93	2	,	,	PUNCT
ejpam-5572	93	3	section	section	NOUN
ejpam-5572	93	4	6	6	NUM
ejpam-5572	93	5	summarizes	summarize	NOUN
ejpam-5572	93	6	the	the	DET
ejpam-5572	93	7	findings	finding	NOUN
ejpam-5572	93	8	of	of	ADP
ejpam-5572	93	9	this	this	DET
ejpam-5572	93	10	study	study	NOUN
ejpam-5572	93	11	and	and	CCONJ
ejpam-5572	93	12	proposes	propose	VERB
ejpam-5572	93	13	potential	potential	ADJ
ejpam-5572	93	14	directions	direction	NOUN
ejpam-5572	93	15	for	for	ADP
ejpam-5572	93	16	future	future	ADJ
ejpam-5572	93	17	research	research	NOUN
ejpam-5572	93	18	.	.	PUNCT
ejpam-5572	94	1	2	2	X
ejpam-5572	94	2	.	.	X
ejpam-5572	94	3	preliminaries	preliminary	NOUN
ejpam-5572	94	4	zadeh	zadeh	PROPN
ejpam-5572	94	5	defined	define	VERB
ejpam-5572	94	6	fuzzy	fuzzy	ADJ
ejpam-5572	94	7	set	set	VERB
ejpam-5572	94	8	in	in	ADP
ejpam-5572	94	9	1965	1965	NUM
ejpam-5572	94	10	[	[	X
ejpam-5572	94	11	40	40	NUM
ejpam-5572	94	12	]	]	PUNCT
ejpam-5572	94	13	.	.	PUNCT
ejpam-5572	95	1	definition	definition	NOUN
ejpam-5572	95	2	1	1	NUM
ejpam-5572	95	3	.	.	PUNCT
ejpam-5572	96	1	[	[	X
ejpam-5572	96	2	40	40	NUM
ejpam-5572	96	3	]	]	PUNCT
ejpam-5572	96	4	a	a	DET
ejpam-5572	96	5	fuzzy	fuzzy	ADJ
ejpam-5572	96	6	set	set	NOUN
ejpam-5572	96	7	(	(	PUNCT
ejpam-5572	96	8	fs	fs	PROPN
ejpam-5572	96	9	)	)	PUNCT
ejpam-5572	96	10	k	k	NOUN
ejpam-5572	96	11	of	of	ADP
ejpam-5572	96	12	the	the	DET
ejpam-5572	96	13	universe	universe	NOUN
ejpam-5572	96	14	of	of	ADP
ejpam-5572	96	15	discourse	discourse	NOUN
ejpam-5572	96	16	x	x	PUNCT
ejpam-5572	96	17	is	be	AUX
ejpam-5572	96	18	defined	define	VERB
ejpam-5572	96	19	by	by	ADP
ejpam-5572	96	20	membership	membership	NOUN
ejpam-5572	96	21	function	function	NOUN
ejpam-5572	96	22	;	;	PUNCT
ejpam-5572	96	23	k	k	X
ejpam-5572	96	24	:	:	PUNCT
ejpam-5572	96	25	x	x	X
ejpam-5572	96	26	→	→	PUNCT
ejpam-5572	97	1	[	[	X
ejpam-5572	97	2	0	0	NUM
ejpam-5572	97	3	,	,	PUNCT
ejpam-5572	97	4	1	1	NUM
ejpam-5572	97	5	]	]	PUNCT
ejpam-5572	97	6	,	,	PUNCT
ejpam-5572	97	7	whereas	whereas	SCONJ
ejpam-5572	97	8	k(x	k(x	NOUN
ejpam-5572	97	9	)	)	PUNCT
ejpam-5572	97	10	is	be	AUX
ejpam-5572	97	11	a	a	DET
ejpam-5572	97	12	degree	degree	NOUN
ejpam-5572	97	13	of	of	ADP
ejpam-5572	97	14	membership	membership	NOUN
ejpam-5572	97	15	for	for	ADP
ejpam-5572	97	16	any	any	DET
ejpam-5572	97	17	x	x	NOUN
ejpam-5572	97	18	in	in	ADP
ejpam-5572	97	19	x.	x.	PROPN
ejpam-5572	97	20	ramot	ramot	PROPN
ejpam-5572	97	21	et	et	PROPN
ejpam-5572	97	22	al	al	PROPN
ejpam-5572	97	23	.	.	PROPN
ejpam-5572	97	24	defined	define	VERB
ejpam-5572	97	25	complex	complex	ADJ
ejpam-5572	97	26	fuzzy	fuzzy	ADJ
ejpam-5572	97	27	set	set	NOUN
ejpam-5572	97	28	(	(	PUNCT
ejpam-5572	97	29	cfs	cfs	PROPN
ejpam-5572	97	30	)	)	PUNCT
ejpam-5572	97	31	on	on	ADP
ejpam-5572	97	32	a	a	DET
ejpam-5572	97	33	crisp	crisp	ADJ
ejpam-5572	97	34	set	set	NOUN
ejpam-5572	97	35	in	in	ADP
ejpam-5572	97	36	2002	2002	NUM
ejpam-5572	97	37	[	[	X
ejpam-5572	97	38	30	30	NUM
ejpam-5572	97	39	]	]	PUNCT
ejpam-5572	97	40	.	.	PUNCT
ejpam-5572	98	1	e.a	e.a	PROPN
ejpam-5572	98	2	.	.	PROPN
ejpam-5572	98	3	abuhijleh	abuhijleh	PROPN
ejpam-5572	98	4	,	,	PUNCT
ejpam-5572	98	5	a.	a.	PROPN
ejpam-5572	98	6	alkouri	alkouri	PROPN
ejpam-5572	98	7	/	/	PROPN
ejpam-5572	98	8	eur	eur	PROPN
ejpam-5572	98	9	.	.	PUNCT
ejpam-5572	99	1	j.	j.	PROPN
ejpam-5572	99	2	pure	pure	PROPN
ejpam-5572	99	3	appl	appl	PROPN
ejpam-5572	99	4	.	.	PROPN
ejpam-5572	99	5	math	math	PROPN
ejpam-5572	99	6	,	,	PUNCT
ejpam-5572	99	7	18	18	NUM
ejpam-5572	99	8	(	(	PUNCT
ejpam-5572	99	9	1	1	NUM
ejpam-5572	99	10	)	)	PUNCT
ejpam-5572	99	11	(	(	PUNCT
ejpam-5572	99	12	2025	2025	NUM
ejpam-5572	99	13	)	)	PUNCT
ejpam-5572	99	14	,	,	PUNCT
ejpam-5572	99	15	5572	5572	NUM
ejpam-5572	99	16	4	4	NUM
ejpam-5572	99	17	of	of	ADP
ejpam-5572	99	18	19	19	NUM
ejpam-5572	99	19	definition	definition	NOUN
ejpam-5572	99	20	2	2	NUM
ejpam-5572	99	21	.	.	PUNCT
ejpam-5572	100	1	[	[	X
ejpam-5572	100	2	30	30	NUM
ejpam-5572	100	3	]	]	X
ejpam-5572	100	4	a	a	DET
ejpam-5572	100	5	complex	complex	ADJ
ejpam-5572	100	6	fuzzy	fuzzy	ADJ
ejpam-5572	100	7	set	set	NOUN
ejpam-5572	100	8	(	(	PUNCT
ejpam-5572	100	9	cfs	cfs	PROPN
ejpam-5572	100	10	)	)	PUNCT
ejpam-5572	100	11	k	k	PROPN
ejpam-5572	100	12	of	of	ADP
ejpam-5572	100	13	the	the	DET
ejpam-5572	100	14	universe	universe	NOUN
ejpam-5572	100	15	of	of	ADP
ejpam-5572	100	16	discourse	discourse	NOUN
ejpam-5572	100	17	x	x	PUNCT
ejpam-5572	100	18	is	be	AUX
ejpam-5572	100	19	defined	define	VERB
ejpam-5572	100	20	by	by	ADP
ejpam-5572	100	21	membership	membership	NOUN
ejpam-5572	100	22	function	function	NOUN
ejpam-5572	100	23	;	;	PUNCT
ejpam-5572	100	24	k(x	k(x	PROPN
ejpam-5572	100	25	)	)	PUNCT
ejpam-5572	100	26	:	:	PUNCT
ejpam-5572	101	1	x	x	X
ejpam-5572	101	2	→	→	PUNCT
ejpam-5572	101	3	{	{	PUNCT
ejpam-5572	101	4	z	z	NOUN
ejpam-5572	101	5	:	:	PUNCT
ejpam-5572	101	6	z	z	X
ejpam-5572	101	7	∈	∈	PROPN
ejpam-5572	101	8	c	c	NOUN
ejpam-5572	101	9	,	,	PUNCT
ejpam-5572	101	10	|z|	|z|	VERB
ejpam-5572	101	11	≤	≤	NUM
ejpam-5572	101	12	1	1	NUM
ejpam-5572	101	13	}	}	PUNCT
ejpam-5572	101	14	,	,	PUNCT
ejpam-5572	101	15	that	that	PRON
ejpam-5572	101	16	assigns	assign	VERB
ejpam-5572	101	17	a	a	DET
ejpam-5572	101	18	degree	degree	NOUN
ejpam-5572	101	19	of	of	ADP
ejpam-5572	101	20	membership	membership	NOUN
ejpam-5572	101	21	k(x	k(x	PROPN
ejpam-5572	101	22	)	)	PUNCT
ejpam-5572	102	1	=	=	SYM
ejpam-5572	102	2	p(x)e2πiω(x	p(x)e2πiω(x	NOUN
ejpam-5572	102	3	)	)	PUNCT
ejpam-5572	102	4	for	for	ADP
ejpam-5572	102	5	any	any	DET
ejpam-5572	102	6	x	x	SYM
ejpam-5572	102	7	in	in	ADP
ejpam-5572	102	8	x	x	NOUN
ejpam-5572	102	9	,	,	PUNCT
ejpam-5572	102	10	where	where	SCONJ
ejpam-5572	102	11	the	the	DET
ejpam-5572	102	12	value	value	NOUN
ejpam-5572	102	13	of	of	ADP
ejpam-5572	102	14	k(x	k(x	PROPN
ejpam-5572	102	15	)	)	PUNCT
ejpam-5572	102	16	is	be	AUX
ejpam-5572	102	17	defined	define	VERB
ejpam-5572	102	18	by	by	ADP
ejpam-5572	102	19	two	two	NUM
ejpam-5572	102	20	variables	variable	NOUN
ejpam-5572	102	21	p(x	p(x	PROPN
ejpam-5572	102	22	)	)	PUNCT
ejpam-5572	102	23	and	and	CCONJ
ejpam-5572	102	24	ω(x	ω(x	NOUN
ejpam-5572	102	25	)	)	PUNCT
ejpam-5572	102	26	and	and	CCONJ
ejpam-5572	102	27	both	both	PRON
ejpam-5572	102	28	are	be	AUX
ejpam-5572	102	29	located	locate	VERB
ejpam-5572	102	30	within	within	ADP
ejpam-5572	102	31	zero	zero	NUM
ejpam-5572	102	32	and	and	CCONJ
ejpam-5572	102	33	one	one	NUM
ejpam-5572	102	34	.	.	PUNCT
ejpam-5572	103	1	on	on	ADP
ejpam-5572	103	2	the	the	DET
ejpam-5572	103	3	other	other	ADJ
ejpam-5572	103	4	hand	hand	NOUN
ejpam-5572	103	5	,	,	PUNCT
ejpam-5572	103	6	atanassov	atanassov	PROPN
ejpam-5572	103	7	(	(	PUNCT
ejpam-5572	103	8	1986	1986	NUM
ejpam-5572	103	9	)	)	PUNCT
ejpam-5572	104	1	[	[	X
ejpam-5572	104	2	12	12	NUM
ejpam-5572	104	3	]	]	PUNCT
ejpam-5572	104	4	introduced	introduce	VERB
ejpam-5572	104	5	the	the	DET
ejpam-5572	104	6	concept	concept	NOUN
ejpam-5572	104	7	of	of	ADP
ejpam-5572	104	8	an	an	DET
ejpam-5572	104	9	intuitionistic	intuitionistic	ADJ
ejpam-5572	104	10	fuzzy	fuzzy	ADJ
ejpam-5572	104	11	set	set	NOUN
ejpam-5572	104	12	(	(	PUNCT
ejpam-5572	104	13	ifs	ifs	PROPN
ejpam-5572	104	14	)	)	PUNCT
ejpam-5572	104	15	by	by	ADP
ejpam-5572	104	16	incorporating	incorporate	VERB
ejpam-5572	104	17	a	a	DET
ejpam-5572	104	18	non	non	ADJ
ejpam-5572	104	19	-	-	ADJ
ejpam-5572	104	20	membership	membership	ADJ
ejpam-5572	104	21	degree	degree	NOUN
ejpam-5572	104	22	,	,	PUNCT
ejpam-5572	104	23	where	where	SCONJ
ejpam-5572	104	24	the	the	DET
ejpam-5572	104	25	sum	sum	NOUN
ejpam-5572	104	26	of	of	ADP
ejpam-5572	104	27	the	the	DET
ejpam-5572	104	28	membership	membership	NOUN
ejpam-5572	104	29	degree	degree	NOUN
ejpam-5572	104	30	and	and	CCONJ
ejpam-5572	104	31	the	the	DET
ejpam-5572	104	32	non	non	ADJ
ejpam-5572	104	33	-	-	ADJ
ejpam-5572	104	34	membership	membership	ADJ
ejpam-5572	104	35	degree	degree	NOUN
ejpam-5572	104	36	lies	lie	VERB
ejpam-5572	104	37	between	between	ADP
ejpam-5572	104	38	zero	zero	NUM
ejpam-5572	104	39	and	and	CCONJ
ejpam-5572	104	40	one	one	NUM
ejpam-5572	104	41	.	.	PUNCT
ejpam-5572	105	1	this	this	DET
ejpam-5572	105	2	framework	framework	NOUN
ejpam-5572	105	3	was	be	AUX
ejpam-5572	105	4	later	later	ADV
ejpam-5572	105	5	expanded	expand	VERB
ejpam-5572	105	6	in	in	ADP
ejpam-5572	105	7	various	various	ADJ
ejpam-5572	105	8	ways	way	NOUN
ejpam-5572	105	9	,	,	PUNCT
ejpam-5572	105	10	one	one	NUM
ejpam-5572	105	11	of	of	ADP
ejpam-5572	105	12	which	which	PRON
ejpam-5572	105	13	is	be	AUX
ejpam-5572	105	14	the	the	DET
ejpam-5572	105	15	pythagorean	pythagorean	PROPN
ejpam-5572	105	16	fuzzy	fuzzy	ADJ
ejpam-5572	105	17	set	set	NOUN
ejpam-5572	105	18	(	(	PUNCT
ejpam-5572	105	19	pfs	pfs	PROPN
ejpam-5572	105	20	)	)	PUNCT
ejpam-5572	106	1	[	[	X
ejpam-5572	106	2	38	38	NUM
ejpam-5572	106	3	]	]	PUNCT
ejpam-5572	106	4	,	,	PUNCT
ejpam-5572	106	5	introduced	introduce	VERB
ejpam-5572	106	6	by	by	ADP
ejpam-5572	106	7	yager	yager	NOUN
ejpam-5572	106	8	in	in	ADP
ejpam-5572	106	9	2013	2013	NUM
ejpam-5572	106	10	,	,	PUNCT
ejpam-5572	106	11	where	where	SCONJ
ejpam-5572	106	12	the	the	DET
ejpam-5572	106	13	sum	sum	NOUN
ejpam-5572	106	14	of	of	ADP
ejpam-5572	106	15	the	the	DET
ejpam-5572	106	16	squares	square	NOUN
ejpam-5572	106	17	of	of	ADP
ejpam-5572	106	18	the	the	DET
ejpam-5572	106	19	membership	membership	NOUN
ejpam-5572	106	20	and	and	CCONJ
ejpam-5572	106	21	non	non	ADJ
ejpam-5572	106	22	-	-	ADJ
ejpam-5572	106	23	membership	membership	ADJ
ejpam-5572	106	24	degrees	degree	NOUN
ejpam-5572	106	25	is	be	AUX
ejpam-5572	106	26	constrained	constrain	VERB
ejpam-5572	106	27	between	between	ADP
ejpam-5572	106	28	zero	zero	NUM
ejpam-5572	106	29	and	and	CCONJ
ejpam-5572	106	30	one	one	NUM
ejpam-5572	106	31	.	.	PUNCT
ejpam-5572	107	1	subsequently	subsequently	ADV
ejpam-5572	107	2	,	,	PUNCT
ejpam-5572	107	3	senapti	senapti	NOUN
ejpam-5572	107	4	and	and	CCONJ
ejpam-5572	107	5	yager	yager	NOUN
ejpam-5572	107	6	(	(	PUNCT
ejpam-5572	107	7	2020	2020	NUM
ejpam-5572	107	8	)	)	PUNCT
ejpam-5572	108	1	[	[	X
ejpam-5572	108	2	33	33	NUM
ejpam-5572	108	3	]	]	PUNCT
ejpam-5572	108	4	defined	define	VERB
ejpam-5572	108	5	the	the	DET
ejpam-5572	108	6	fermatean	fermatean	ADJ
ejpam-5572	108	7	fuzzy	fuzzy	ADJ
ejpam-5572	108	8	set	set	NOUN
ejpam-5572	108	9	(	(	PUNCT
ejpam-5572	108	10	ffs	ff	NOUN
ejpam-5572	108	11	)	)	PUNCT
ejpam-5572	108	12	,	,	PUNCT
ejpam-5572	108	13	as	as	SCONJ
ejpam-5572	108	14	described	describe	VERB
ejpam-5572	108	15	below	below	ADV
ejpam-5572	108	16	.	.	PUNCT
ejpam-5572	109	1	definition	definition	NOUN
ejpam-5572	109	2	3	3	NUM
ejpam-5572	109	3	.	.	PUNCT
ejpam-5572	110	1	[	[	X
ejpam-5572	110	2	33	33	NUM
ejpam-5572	110	3	]	]	PUNCT
ejpam-5572	110	4	let	let	VERB
ejpam-5572	110	5	x	x	PRON
ejpam-5572	110	6	be	be	AUX
ejpam-5572	110	7	a	a	DET
ejpam-5572	110	8	universe	universe	NOUN
ejpam-5572	110	9	of	of	ADP
ejpam-5572	110	10	discourse	discourse	NOUN
ejpam-5572	110	11	,	,	PUNCT
ejpam-5572	110	12	then	then	ADV
ejpam-5572	110	13	a	a	DET
ejpam-5572	110	14	fermatean	fermatean	ADJ
ejpam-5572	110	15	fuzzy	fuzzy	NOUN
ejpam-5572	110	16	set	set	VERB
ejpam-5572	110	17	f	f	PROPN
ejpam-5572	110	18	on	on	ADP
ejpam-5572	110	19	x	x	SYM
ejpam-5572	110	20	defined	define	VERB
ejpam-5572	110	21	by	by	ADP
ejpam-5572	110	22	f	f	X
ejpam-5572	110	23	=	=	PRON
ejpam-5572	110	24	{	{	PUNCT
ejpam-5572	110	25	(	(	PUNCT
ejpam-5572	110	26	x	x	X
ejpam-5572	110	27	,	,	PUNCT
ejpam-5572	110	28	k(x),l(x	k(x),l(x	PROPN
ejpam-5572	110	29	)	)	PUNCT
ejpam-5572	110	30	)	)	PUNCT
ejpam-5572	110	31	:	:	PUNCT
ejpam-5572	111	1	x	x	X
ejpam-5572	111	2	∈	∈	NOUN
ejpam-5572	111	3	x	x	X
ejpam-5572	111	4	}	}	PUNCT
ejpam-5572	111	5	.	.	PUNCT
ejpam-5572	112	1	such	such	ADJ
ejpam-5572	112	2	that	that	SCONJ
ejpam-5572	112	3	k(x	k(x	PROPN
ejpam-5572	112	4	)	)	PUNCT
ejpam-5572	112	5	∈	∈	PROPN
ejpam-5572	113	1	[	[	X
ejpam-5572	113	2	0	0	NUM
ejpam-5572	113	3	,	,	PUNCT
ejpam-5572	113	4	1	1	NUM
ejpam-5572	113	5	]	]	PUNCT
ejpam-5572	113	6	and	and	CCONJ
ejpam-5572	113	7	l(x	l(x	PROPN
ejpam-5572	113	8	)	)	PUNCT
ejpam-5572	113	9	∈	∈	PROPN
ejpam-5572	114	1	[	[	X
ejpam-5572	114	2	0	0	NUM
ejpam-5572	114	3	,	,	PUNCT
ejpam-5572	114	4	1	1	NUM
ejpam-5572	114	5	]	]	PUNCT
ejpam-5572	114	6	are	be	AUX
ejpam-5572	114	7	the	the	DET
ejpam-5572	114	8	degree	degree	NOUN
ejpam-5572	114	9	of	of	ADP
ejpam-5572	114	10	membership	membership	NOUN
ejpam-5572	114	11	and	and	CCONJ
ejpam-5572	114	12	the	the	DET
ejpam-5572	114	13	degree	degree	NOUN
ejpam-5572	114	14	of	of	ADP
ejpam-5572	114	15	non	non	ADJ
ejpam-5572	114	16	-	-	NOUN
ejpam-5572	114	17	membership	membership	NOUN
ejpam-5572	114	18	for	for	ADP
ejpam-5572	114	19	any	any	DET
ejpam-5572	114	20	x	x	SYM
ejpam-5572	114	21	∈	∈	PROPN
ejpam-5572	114	22	x	x	NOUN
ejpam-5572	114	23	,	,	PUNCT
ejpam-5572	114	24	respectively	respectively	ADV
ejpam-5572	114	25	,	,	PUNCT
ejpam-5572	114	26	and	and	CCONJ
ejpam-5572	114	27	0	0	NUM
ejpam-5572	114	28	≤	≤	NUM
ejpam-5572	114	29	k3(x	k3(x	PROPN
ejpam-5572	114	30	)	)	PUNCT
ejpam-5572	114	31	+	+	NUM
ejpam-5572	114	32	l3(x	l3(x	NOUN
ejpam-5572	114	33	)	)	PUNCT
ejpam-5572	114	34	≤	≤	NUM
ejpam-5572	114	35	1	1	NUM
ejpam-5572	114	36	,	,	PUNCT
ejpam-5572	114	37	for	for	ADP
ejpam-5572	114	38	all	all	PRON
ejpam-5572	114	39	x	x	SYM
ejpam-5572	114	40	∈	∈	PROPN
ejpam-5572	114	41	x.	x.	NOUN
ejpam-5572	114	42	additionally	additionally	ADV
ejpam-5572	114	43	,	,	PUNCT
ejpam-5572	114	44	yager	yager	NOUN
ejpam-5572	114	45	(	(	PUNCT
ejpam-5572	114	46	2017	2017	NUM
ejpam-5572	114	47	)	)	PUNCT
ejpam-5572	115	1	[	[	X
ejpam-5572	115	2	39	39	NUM
ejpam-5572	115	3	]	]	PUNCT
ejpam-5572	115	4	introduced	introduce	VERB
ejpam-5572	115	5	a	a	DET
ejpam-5572	115	6	broader	broad	ADJ
ejpam-5572	115	7	class	class	NOUN
ejpam-5572	115	8	of	of	ADP
ejpam-5572	115	9	fuzzy	fuzzy	ADJ
ejpam-5572	115	10	sets	set	NOUN
ejpam-5572	115	11	known	know	VERB
ejpam-5572	115	12	as	as	ADP
ejpam-5572	115	13	q	q	NOUN
ejpam-5572	115	14	-	-	PUNCT
ejpam-5572	115	15	rung	rung	ADJ
ejpam-5572	115	16	orthopair	orthopair	ADJ
ejpam-5572	115	17	fuzzy	fuzzy	ADJ
ejpam-5572	115	18	sets	set	NOUN
ejpam-5572	115	19	,	,	PUNCT
ejpam-5572	115	20	in	in	ADP
ejpam-5572	115	21	which	which	PRON
ejpam-5572	115	22	the	the	DET
ejpam-5572	115	23	sum	sum	NOUN
ejpam-5572	115	24	of	of	ADP
ejpam-5572	115	25	the	the	DET
ejpam-5572	115	26	qth	qth	NOUN
ejpam-5572	115	27	powers	power	NOUN
ejpam-5572	115	28	of	of	ADP
ejpam-5572	115	29	the	the	DET
ejpam-5572	115	30	membership	membership	NOUN
ejpam-5572	115	31	and	and	CCONJ
ejpam-5572	115	32	nonmembership	nonmembership	NOUN
ejpam-5572	115	33	degrees	degree	NOUN
ejpam-5572	115	34	is	be	AUX
ejpam-5572	115	35	constrained	constrain	VERB
ejpam-5572	115	36	between	between	ADP
ejpam-5572	115	37	zero	zero	NUM
ejpam-5572	115	38	and	and	CCONJ
ejpam-5572	115	39	one	one	NUM
ejpam-5572	115	40	.	.	PUNCT
ejpam-5572	116	1	he	he	PRON
ejpam-5572	116	2	demonstrated	demonstrate	VERB
ejpam-5572	116	3	that	that	SCONJ
ejpam-5572	116	4	as	as	ADP
ejpam-5572	116	5	the	the	DET
ejpam-5572	116	6	value	value	NOUN
ejpam-5572	116	7	of	of	ADP
ejpam-5572	116	8	q	q	NOUN
ejpam-5572	116	9	increases	increase	NOUN
ejpam-5572	116	10	,	,	PUNCT
ejpam-5572	116	11	the	the	DET
ejpam-5572	116	12	space	space	NOUN
ejpam-5572	116	13	of	of	ADP
ejpam-5572	116	14	acceptable	acceptable	ADJ
ejpam-5572	116	15	orthopair	orthopair	NOUN
ejpam-5572	116	16	sets	set	NOUN
ejpam-5572	116	17	expands	expand	VERB
ejpam-5572	116	18	,	,	PUNCT
ejpam-5572	116	19	thereby	thereby	ADV
ejpam-5572	116	20	providing	provide	VERB
ejpam-5572	116	21	users	user	NOUN
ejpam-5572	116	22	with	with	ADP
ejpam-5572	116	23	greater	great	ADJ
ejpam-5572	116	24	flexibility	flexibility	NOUN
ejpam-5572	116	25	in	in	ADP
ejpam-5572	116	26	expressing	express	VERB
ejpam-5572	116	27	their	their	PRON
ejpam-5572	116	28	beliefs	belief	NOUN
ejpam-5572	116	29	regarding	regard	VERB
ejpam-5572	116	30	the	the	DET
ejpam-5572	116	31	degree	degree	NOUN
ejpam-5572	116	32	of	of	ADP
ejpam-5572	116	33	membership	membership	NOUN
ejpam-5572	116	34	.	.	PUNCT
ejpam-5572	117	1	on	on	ADP
ejpam-5572	117	2	the	the	DET
ejpam-5572	117	3	other	other	ADJ
ejpam-5572	117	4	hand	hand	NOUN
ejpam-5572	117	5	,	,	PUNCT
ejpam-5572	117	6	the	the	DET
ejpam-5572	117	7	concept	concept	NOUN
ejpam-5572	117	8	of	of	ADP
ejpam-5572	117	9	the	the	DET
ejpam-5572	117	10	complex	complex	ADJ
ejpam-5572	117	11	intuitionistic	intuitionistic	ADJ
ejpam-5572	117	12	fuzzy	fuzzy	ADJ
ejpam-5572	117	13	set	set	NOUN
ejpam-5572	117	14	(	(	PUNCT
ejpam-5572	117	15	cifs	cifs	PROPN
ejpam-5572	117	16	)	)	PUNCT
ejpam-5572	117	17	was	be	AUX
ejpam-5572	117	18	introduced	introduce	VERB
ejpam-5572	117	19	in	in	ADP
ejpam-5572	117	20	2012	2012	NUM
ejpam-5572	117	21	[	[	X
ejpam-5572	117	22	8	8	NUM
ejpam-5572	117	23	]	]	PUNCT
ejpam-5572	117	24	,	,	PUNCT
ejpam-5572	117	25	followed	follow	VERB
ejpam-5572	117	26	by	by	ADP
ejpam-5572	117	27	the	the	DET
ejpam-5572	117	28	complex	complex	ADJ
ejpam-5572	117	29	pythagorean	pythagorean	ADJ
ejpam-5572	117	30	fuzzy	fuzzy	ADJ
ejpam-5572	117	31	set	set	NOUN
ejpam-5572	117	32	(	(	PUNCT
ejpam-5572	117	33	cpfs	cpfs	PROPN
ejpam-5572	117	34	)	)	PUNCT
ejpam-5572	117	35	in	in	ADP
ejpam-5572	117	36	2019	2019	NUM
ejpam-5572	117	37	[	[	X
ejpam-5572	117	38	36	36	NUM
ejpam-5572	117	39	]	]	PUNCT
ejpam-5572	117	40	,	,	PUNCT
ejpam-5572	117	41	and	and	CCONJ
ejpam-5572	117	42	more	more	ADV
ejpam-5572	117	43	recently	recently	ADV
ejpam-5572	117	44	,	,	PUNCT
ejpam-5572	117	45	the	the	DET
ejpam-5572	117	46	complex	complex	ADJ
ejpam-5572	117	47	fermatean	fermatean	ADJ
ejpam-5572	117	48	fuzzy	fuzzy	ADJ
ejpam-5572	117	49	set	set	NOUN
ejpam-5572	117	50	(	(	PUNCT
ejpam-5572	117	51	cffs	cff	NOUN
ejpam-5572	117	52	)	)	PUNCT
ejpam-5572	117	53	was	be	AUX
ejpam-5572	117	54	presented	present	VERB
ejpam-5572	117	55	in	in	ADP
ejpam-5572	117	56	2021	2021	NUM
ejpam-5572	118	1	[	[	X
ejpam-5572	118	2	17	17	NUM
ejpam-5572	118	3	]	]	PUNCT
ejpam-5572	118	4	.	.	PUNCT
ejpam-5572	119	1	these	these	DET
ejpam-5572	119	2	advancements	advancement	NOUN
ejpam-5572	119	3	involve	involve	VERB
ejpam-5572	119	4	extending	extend	VERB
ejpam-5572	119	5	traditional	traditional	ADJ
ejpam-5572	119	6	fuzzy	fuzzy	ADJ
ejpam-5572	119	7	sets	set	NOUN
ejpam-5572	119	8	to	to	PART
ejpam-5572	119	9	complex	complex	ADJ
ejpam-5572	119	10	fuzzy	fuzzy	ADJ
ejpam-5572	119	11	sets	set	NOUN
ejpam-5572	119	12	for	for	ADP
ejpam-5572	119	13	both	both	CCONJ
ejpam-5572	119	14	membership	membership	NOUN
ejpam-5572	119	15	and	and	CCONJ
ejpam-5572	119	16	non	non	ADJ
ejpam-5572	119	17	-	-	ADJ
ejpam-5572	119	18	membership	membership	ADJ
ejpam-5572	119	19	degrees	degree	NOUN
ejpam-5572	119	20	.	.	PUNCT
ejpam-5572	120	1	the	the	DET
ejpam-5572	120	2	definition	definition	NOUN
ejpam-5572	120	3	of	of	ADP
ejpam-5572	120	4	the	the	DET
ejpam-5572	120	5	complex	complex	ADJ
ejpam-5572	120	6	fermatean	fermatean	ADJ
ejpam-5572	120	7	fuzzy	fuzzy	ADJ
ejpam-5572	120	8	set	set	NOUN
ejpam-5572	120	9	(	(	PUNCT
ejpam-5572	120	10	cffs	cff	NOUN
ejpam-5572	120	11	)	)	PUNCT
ejpam-5572	120	12	is	be	AUX
ejpam-5572	120	13	provided	provide	VERB
ejpam-5572	120	14	below	below	ADV
ejpam-5572	120	15	.	.	PUNCT
ejpam-5572	121	1	definition	definition	NOUN
ejpam-5572	121	2	4	4	NUM
ejpam-5572	121	3	.	.	PUNCT
ejpam-5572	122	1	[	[	X
ejpam-5572	122	2	17	17	NUM
ejpam-5572	122	3	]	]	PUNCT
ejpam-5572	122	4	let	let	VERB
ejpam-5572	122	5	x	x	PRON
ejpam-5572	122	6	be	be	AUX
ejpam-5572	122	7	a	a	DET
ejpam-5572	122	8	universe	universe	NOUN
ejpam-5572	122	9	of	of	ADP
ejpam-5572	122	10	discourse	discourse	NOUN
ejpam-5572	122	11	and	and	CCONJ
ejpam-5572	122	12	defined	define	VERB
ejpam-5572	122	13	a	a	DET
ejpam-5572	122	14	complex	complex	ADJ
ejpam-5572	122	15	fermatean	fermatean	ADJ
ejpam-5572	122	16	fuzzy	fuzzy	NOUN
ejpam-5572	122	17	set	set	VERB
ejpam-5572	122	18	ϕ	ϕ	NOUN
ejpam-5572	122	19	on	on	ADP
ejpam-5572	122	20	x	x	SYM
ejpam-5572	122	21	,	,	PUNCT
ejpam-5572	122	22	where	where	SCONJ
ejpam-5572	122	23	ϕ	ϕ	NOUN
ejpam-5572	122	24	=	=	PRON
ejpam-5572	122	25	{	{	PUNCT
ejpam-5572	122	26	(	(	PUNCT
ejpam-5572	122	27	x	x	NOUN
ejpam-5572	122	28	,	,	PUNCT
ejpam-5572	122	29	k(x),l(x	k(x),l(x	PROPN
ejpam-5572	122	30	)	)	PUNCT
ejpam-5572	122	31	)	)	PUNCT
ejpam-5572	122	32	:	:	PUNCT
ejpam-5572	123	1	x	x	X
ejpam-5572	123	2	∈	∈	NOUN
ejpam-5572	123	3	x	x	X
ejpam-5572	123	4	}	}	PUNCT
ejpam-5572	123	5	.	.	PUNCT
ejpam-5572	124	1	such	such	ADJ
ejpam-5572	124	2	that	that	SCONJ
ejpam-5572	124	3	k(x	k(x	PROPN
ejpam-5572	124	4	)	)	PUNCT
ejpam-5572	124	5	:	:	PUNCT
ejpam-5572	124	6	x	x	X
ejpam-5572	124	7	→	→	PUNCT
ejpam-5572	124	8	{	{	PUNCT
ejpam-5572	124	9	z	z	NOUN
ejpam-5572	124	10	:	:	PUNCT
ejpam-5572	124	11	z	z	X
ejpam-5572	124	12	∈	∈	PROPN
ejpam-5572	124	13	c	c	NOUN
ejpam-5572	124	14	,	,	PUNCT
ejpam-5572	124	15	|z|	|z|	VERB
ejpam-5572	124	16	≤	≤	NUM
ejpam-5572	124	17	1	1	NUM
ejpam-5572	124	18	}	}	PUNCT
ejpam-5572	124	19	and	and	CCONJ
ejpam-5572	124	20	l(x	l(x	PROPN
ejpam-5572	124	21	)	)	PUNCT
ejpam-5572	124	22	:	:	PUNCT
ejpam-5572	125	1	x	x	X
ejpam-5572	125	2	→	→	PUNCT
ejpam-5572	125	3	{	{	PUNCT
ejpam-5572	125	4	z	z	NOUN
ejpam-5572	125	5	:	:	PUNCT
ejpam-5572	125	6	z	z	X
ejpam-5572	125	7	∈	∈	PROPN
ejpam-5572	125	8	c	c	NOUN
ejpam-5572	125	9	,	,	PUNCT
ejpam-5572	125	10	|z|	|z|	VERB
ejpam-5572	125	11	≤	≤	NUM
ejpam-5572	125	12	1	1	NUM
ejpam-5572	125	13	}	}	PUNCT
ejpam-5572	125	14	,	,	PUNCT
ejpam-5572	125	15	are	be	AUX
ejpam-5572	125	16	the	the	DET
ejpam-5572	125	17	degree	degree	NOUN
ejpam-5572	125	18	of	of	ADP
ejpam-5572	125	19	membership	membership	NOUN
ejpam-5572	125	20	and	and	CCONJ
ejpam-5572	125	21	nonmembership	nonmembership	NOUN
ejpam-5572	125	22	of	of	ADP
ejpam-5572	125	23	x	x	X
ejpam-5572	125	24	∈	∈	PROPN
ejpam-5572	125	25	x	x	NOUN
ejpam-5572	125	26	,	,	PUNCT
ejpam-5572	125	27	respectively	respectively	ADV
ejpam-5572	125	28	.	.	PUNCT
ejpam-5572	126	1	moreover	moreover	ADV
ejpam-5572	126	2	,	,	PUNCT
ejpam-5572	126	3	k(x	k(x	PROPN
ejpam-5572	126	4	)	)	PUNCT
ejpam-5572	126	5	=	=	SYM
ejpam-5572	126	6	p(x)e2πiω(x	p(x)e2πiω(x	NOUN
ejpam-5572	126	7	)	)	PUNCT
ejpam-5572	126	8	and	and	CCONJ
ejpam-5572	126	9	l(x	l(x	PROPN
ejpam-5572	126	10	)	)	PUNCT
ejpam-5572	126	11	=	=	SYM
ejpam-5572	126	12	q(x)e2πiν(x	q(x)e2πiν(x	NOUN
ejpam-5572	126	13	)	)	PUNCT
ejpam-5572	126	14	are	be	AUX
ejpam-5572	126	15	satisfying	satisfy	VERB
ejpam-5572	126	16	the	the	DET
ejpam-5572	126	17	conditions	condition	NOUN
ejpam-5572	126	18	;	;	PUNCT
ejpam-5572	126	19	0	0	NUM
ejpam-5572	126	20	≤	≤	NUM
ejpam-5572	126	21	p3(x	p3(x	X
ejpam-5572	126	22	)	)	PUNCT
ejpam-5572	126	23	+	+	NUM
ejpam-5572	126	24	q3(x	q3(x	X
ejpam-5572	126	25	)	)	PUNCT
ejpam-5572	126	26	≤	≤	NOUN
ejpam-5572	126	27	1	1	NUM
ejpam-5572	126	28	and	and	CCONJ
ejpam-5572	126	29	0	0	NUM
ejpam-5572	126	30	≤	≤	NUM
ejpam-5572	126	31	ω3(x	ω3(x	PROPN
ejpam-5572	126	32	)	)	PUNCT
ejpam-5572	126	33	+	+	CCONJ
ejpam-5572	126	34	ν3(x	ν3(x	SYM
ejpam-5572	126	35	)	)	PUNCT
ejpam-5572	126	36	≤	≤	NOUN
ejpam-5572	126	37	1	1	NUM
ejpam-5572	126	38	.	.	PUNCT
ejpam-5572	127	1	in	in	ADP
ejpam-5572	127	2	the	the	DET
ejpam-5572	127	3	previous	previous	ADJ
ejpam-5572	127	4	definition	definition	NOUN
ejpam-5572	127	5	,	,	PUNCT
ejpam-5572	127	6	if	if	SCONJ
ejpam-5572	127	7	conditions	condition	NOUN
ejpam-5572	127	8	become	become	VERB
ejpam-5572	127	9	0	0	NUM
ejpam-5572	127	10	≤	≤	NOUN
ejpam-5572	127	11	pk(x	pk(x	NUM
ejpam-5572	127	12	)	)	PUNCT
ejpam-5572	128	1	+	+	NUM
ejpam-5572	129	1	qk(x	qk(x	X
ejpam-5572	129	2	)	)	PUNCT
ejpam-5572	129	3	≤	≤	NUM
ejpam-5572	129	4	1	1	NUM
ejpam-5572	129	5	and	and	CCONJ
ejpam-5572	129	6	0	0	NUM
ejpam-5572	129	7	≤	≤	NOUN
ejpam-5572	129	8	ωk(x	ωk(x	PUNCT
ejpam-5572	129	9	)	)	PUNCT
ejpam-5572	129	10	+	+	CCONJ
ejpam-5572	129	11	νk(x	νk(x	X
ejpam-5572	129	12	)	)	PUNCT
ejpam-5572	129	13	≤	≤	NUM
ejpam-5572	129	14	1	1	NUM
ejpam-5572	129	15	,	,	PUNCT
ejpam-5572	129	16	for	for	ADP
ejpam-5572	129	17	all	all	DET
ejpam-5572	129	18	x	x	SYM
ejpam-5572	129	19	∈	∈	PROPN
ejpam-5572	129	20	x	x	PUNCT
ejpam-5572	129	21	with	with	ADP
ejpam-5572	129	22	k	k	PROPN
ejpam-5572	129	23	≥	≥	NUM
ejpam-5572	129	24	1	1	NUM
ejpam-5572	129	25	.	.	PUNCT
ejpam-5572	130	1	then	then	ADV
ejpam-5572	130	2	ϕ	ϕ	NOUN
ejpam-5572	130	3	define	define	VERB
ejpam-5572	130	4	a	a	DET
ejpam-5572	130	5	complex	complex	ADJ
ejpam-5572	130	6	q	q	ADJ
ejpam-5572	130	7	-	-	PUNCT
ejpam-5572	130	8	rung	rung	ADJ
ejpam-5572	130	9	orthopair	orthopair	ADJ
ejpam-5572	130	10	fuzzy	fuzzy	ADJ
ejpam-5572	130	11	sets	set	NOUN
ejpam-5572	130	12	(	(	PUNCT
ejpam-5572	130	13	2020	2020	NUM
ejpam-5572	130	14	)	)	PUNCT
ejpam-5572	131	1	[	[	X
ejpam-5572	131	2	25	25	NUM
ejpam-5572	131	3	]	]	PUNCT
ejpam-5572	131	4	.	.	PUNCT
ejpam-5572	132	1	another	another	DET
ejpam-5572	132	2	approach	approach	NOUN
ejpam-5572	132	3	to	to	ADP
ejpam-5572	132	4	fuzzy	fuzzy	ADJ
ejpam-5572	132	5	sets	set	NOUN
ejpam-5572	132	6	is	be	AUX
ejpam-5572	132	7	the	the	DET
ejpam-5572	132	8	concept	concept	NOUN
ejpam-5572	132	9	of	of	ADP
ejpam-5572	132	10	fuzzy	fuzzy	ADJ
ejpam-5572	132	11	subgroups	subgroup	NOUN
ejpam-5572	132	12	,	,	PUNCT
ejpam-5572	132	13	first	first	ADV
ejpam-5572	132	14	introduced	introduce	VERB
ejpam-5572	132	15	by	by	ADP
ejpam-5572	132	16	rosenfeld	rosenfeld	PROPN
ejpam-5572	132	17	in	in	ADP
ejpam-5572	132	18	1971	1971	NUM
ejpam-5572	132	19	[	[	X
ejpam-5572	132	20	32	32	NUM
ejpam-5572	132	21	]	]	PUNCT
ejpam-5572	132	22	.	.	PUNCT
ejpam-5572	133	1	this	this	DET
ejpam-5572	133	2	concept	concept	NOUN
ejpam-5572	133	3	was	be	AUX
ejpam-5572	133	4	subsequently	subsequently	ADV
ejpam-5572	133	5	enhanced	enhance	VERB
ejpam-5572	133	6	with	with	ADP
ejpam-5572	133	7	the	the	DET
ejpam-5572	133	8	introduction	introduction	NOUN
ejpam-5572	133	9	of	of	ADP
ejpam-5572	133	10	e.a	e.a	PROPN
ejpam-5572	133	11	.	.	PROPN
ejpam-5572	133	12	abuhijleh	abuhijleh	PROPN
ejpam-5572	133	13	,	,	PUNCT
ejpam-5572	133	14	a.	a.	PROPN
ejpam-5572	133	15	alkouri	alkouri	PROPN
ejpam-5572	133	16	/	/	PROPN
ejpam-5572	133	17	eur	eur	PROPN
ejpam-5572	133	18	.	.	PUNCT
ejpam-5572	134	1	j.	j.	PROPN
ejpam-5572	134	2	pure	pure	PROPN
ejpam-5572	134	3	appl	appl	PROPN
ejpam-5572	134	4	.	.	PROPN
ejpam-5572	134	5	math	math	PROPN
ejpam-5572	134	6	,	,	PUNCT
ejpam-5572	134	7	18	18	NUM
ejpam-5572	134	8	(	(	PUNCT
ejpam-5572	134	9	1	1	NUM
ejpam-5572	134	10	)	)	PUNCT
ejpam-5572	134	11	(	(	PUNCT
ejpam-5572	134	12	2025	2025	NUM
ejpam-5572	134	13	)	)	PUNCT
ejpam-5572	134	14	,	,	PUNCT
ejpam-5572	134	15	5572	5572	NUM
ejpam-5572	134	16	5	5	NUM
ejpam-5572	134	17	of	of	ADP
ejpam-5572	134	18	19	19	NUM
ejpam-5572	134	19	the	the	DET
ejpam-5572	134	20	intuitionistic	intuitionistic	ADJ
ejpam-5572	134	21	fuzzy	fuzzy	ADJ
ejpam-5572	134	22	subgroup	subgroup	NOUN
ejpam-5572	134	23	(	(	PUNCT
ejpam-5572	134	24	ifsg	ifsg	NOUN
ejpam-5572	134	25	)	)	PUNCT
ejpam-5572	134	26	in	in	ADP
ejpam-5572	134	27	1989	1989	NUM
ejpam-5572	134	28	[	[	X
ejpam-5572	134	29	15	15	NUM
ejpam-5572	134	30	]	]	PUNCT
ejpam-5572	134	31	.	.	PUNCT
ejpam-5572	135	1	in	in	ADP
ejpam-5572	135	2	2021	2021	NUM
ejpam-5572	135	3	,	,	PUNCT
ejpam-5572	135	4	e.a	e.a	PROPN
ejpam-5572	135	5	.	.	PROPN
ejpam-5572	135	6	abuhijleh	abuhijleh	PROPN
ejpam-5572	135	7	et	et	PROPN
ejpam-5572	135	8	al	al	PROPN
ejpam-5572	135	9	.	.	PUNCT
ejpam-5572	136	1	[	[	X
ejpam-5572	136	2	3	3	X
ejpam-5572	136	3	]	]	PUNCT
ejpam-5572	136	4	defined	define	VERB
ejpam-5572	136	5	the	the	DET
ejpam-5572	136	6	complex	complex	ADJ
ejpam-5572	136	7	fuzzy	fuzzy	ADJ
ejpam-5572	136	8	subgroup	subgroup	NOUN
ejpam-5572	136	9	(	(	PUNCT
ejpam-5572	136	10	cfsg	cfsg	PROPN
ejpam-5572	136	11	)	)	PUNCT
ejpam-5572	136	12	,	,	PUNCT
ejpam-5572	136	13	while	while	SCONJ
ejpam-5572	136	14	the	the	DET
ejpam-5572	136	15	complex	complex	ADJ
ejpam-5572	136	16	intuitionistic	intuitionistic	ADJ
ejpam-5572	136	17	fuzzy	fuzzy	ADJ
ejpam-5572	136	18	subgroup	subgroup	NOUN
ejpam-5572	136	19	(	(	PUNCT
ejpam-5572	136	20	cifsg	cifsg	PROPN
ejpam-5572	136	21	)	)	PUNCT
ejpam-5572	136	22	was	be	AUX
ejpam-5572	136	23	introduced	introduce	VERB
ejpam-5572	136	24	in	in	ADP
ejpam-5572	136	25	2020	2020	NUM
ejpam-5572	137	1	[	[	X
ejpam-5572	137	2	20	20	NUM
ejpam-5572	137	3	]	]	PUNCT
ejpam-5572	137	4	.	.	PUNCT
ejpam-5572	138	1	the	the	DET
ejpam-5572	138	2	pythagorean	pythagorean	PROPN
ejpam-5572	138	3	fuzzy	fuzzy	PROPN
ejpam-5572	138	4	subgroup	subgroup	NOUN
ejpam-5572	138	5	(	(	PUNCT
ejpam-5572	138	6	pfsg	pfsg	NOUN
ejpam-5572	138	7	)	)	PUNCT
ejpam-5572	138	8	was	be	AUX
ejpam-5572	138	9	presented	present	VERB
ejpam-5572	138	10	in	in	ADP
ejpam-5572	138	11	2020	2020	NUM
ejpam-5572	138	12	[	[	X
ejpam-5572	138	13	14	14	NUM
ejpam-5572	138	14	]	]	PUNCT
ejpam-5572	138	15	,	,	PUNCT
ejpam-5572	138	16	followed	follow	VERB
ejpam-5572	138	17	by	by	ADP
ejpam-5572	138	18	the	the	DET
ejpam-5572	138	19	complex	complex	ADJ
ejpam-5572	138	20	pythagorean	pythagorean	ADJ
ejpam-5572	138	21	fuzzy	fuzzy	ADJ
ejpam-5572	138	22	subgroup	subgroup	NOUN
ejpam-5572	138	23	(	(	PUNCT
ejpam-5572	138	24	cpfsg	cpfsg	PROPN
ejpam-5572	138	25	)	)	PUNCT
ejpam-5572	138	26	in	in	ADP
ejpam-5572	138	27	2023	2023	NUM
ejpam-5572	138	28	[	[	X
ejpam-5572	138	29	7	7	NUM
ejpam-5572	138	30	]	]	PUNCT
ejpam-5572	138	31	.	.	PUNCT
ejpam-5572	139	1	additionally	additionally	ADV
ejpam-5572	139	2	,	,	PUNCT
ejpam-5572	139	3	the	the	DET
ejpam-5572	139	4	fermatean	fermatean	ADJ
ejpam-5572	139	5	fuzzy	fuzzy	ADJ
ejpam-5572	139	6	subgroup	subgroup	NOUN
ejpam-5572	139	7	(	(	PUNCT
ejpam-5572	139	8	ffsg	ffsg	NOUN
ejpam-5572	139	9	)	)	PUNCT
ejpam-5572	139	10	was	be	AUX
ejpam-5572	139	11	defined	define	VERB
ejpam-5572	139	12	in	in	ADP
ejpam-5572	139	13	2021	2021	NUM
ejpam-5572	139	14	[	[	X
ejpam-5572	139	15	35	35	NUM
ejpam-5572	139	16	]	]	PUNCT
ejpam-5572	139	17	,	,	PUNCT
ejpam-5572	139	18	with	with	ADP
ejpam-5572	139	19	onasanya	onasanya	PROPN
ejpam-5572	139	20	et	et	PROPN
ejpam-5572	139	21	al	al	PROPN
ejpam-5572	139	22	.	.	PROPN
ejpam-5572	140	1	(	(	PUNCT
ejpam-5572	140	2	2022	2022	NUM
ejpam-5572	140	3	)	)	PUNCT
ejpam-5572	141	1	[	[	X
ejpam-5572	141	2	28	28	NUM
ejpam-5572	141	3	]	]	PUNCT
ejpam-5572	141	4	also	also	ADV
ejpam-5572	141	5	contributing	contribute	VERB
ejpam-5572	141	6	to	to	ADP
ejpam-5572	141	7	the	the	DET
ejpam-5572	141	8	development	development	NOUN
ejpam-5572	141	9	of	of	ADP
ejpam-5572	141	10	the	the	DET
ejpam-5572	141	11	fermatean	fermatean	ADJ
ejpam-5572	141	12	fuzzy	fuzzy	ADJ
ejpam-5572	141	13	subgroup	subgroup	NOUN
ejpam-5572	141	14	.	.	PUNCT
ejpam-5572	142	1	here	here	ADV
ejpam-5572	142	2	,	,	PUNCT
ejpam-5572	142	3	we	we	PRON
ejpam-5572	142	4	produce	produce	VERB
ejpam-5572	142	5	definitions	definition	NOUN
ejpam-5572	142	6	of	of	ADP
ejpam-5572	142	7	fsg	fsg	PROPN
ejpam-5572	142	8	,	,	PUNCT
ejpam-5572	142	9	cfsg	cfsg	NOUN
ejpam-5572	142	10	,	,	PUNCT
ejpam-5572	142	11	and	and	CCONJ
ejpam-5572	142	12	ffsg	ffsg	NOUN
ejpam-5572	142	13	,	,	PUNCT
ejpam-5572	142	14	respectively	respectively	ADV
ejpam-5572	142	15	.	.	PUNCT
ejpam-5572	143	1	definition	definition	NOUN
ejpam-5572	143	2	5	5	NUM
ejpam-5572	143	3	.	.	PUNCT
ejpam-5572	144	1	[	[	X
ejpam-5572	144	2	32	32	NUM
ejpam-5572	144	3	]	]	PUNCT
ejpam-5572	144	4	let	let	VERB
ejpam-5572	144	5	k	k	NOUN
ejpam-5572	144	6	:	:	PUNCT
ejpam-5572	144	7	x	x	X
ejpam-5572	144	8	→	→	PUNCT
ejpam-5572	145	1	[	[	X
ejpam-5572	145	2	0	0	NUM
ejpam-5572	145	3	,	,	PUNCT
ejpam-5572	145	4	1	1	NUM
ejpam-5572	145	5	]	]	PUNCT
ejpam-5572	145	6	defined	define	VERB
ejpam-5572	145	7	a	a	DET
ejpam-5572	145	8	fuzzy	fuzzy	ADJ
ejpam-5572	145	9	subset	subset	NOUN
ejpam-5572	145	10	of	of	ADP
ejpam-5572	145	11	a	a	DET
ejpam-5572	145	12	group	group	NOUN
ejpam-5572	145	13	(	(	PUNCT
ejpam-5572	145	14	x	x	X
ejpam-5572	145	15	,	,	PUNCT
ejpam-5572	145	16	∗	∗	NOUN
ejpam-5572	145	17	)	)	PUNCT
ejpam-5572	145	18	.	.	PUNCT
ejpam-5572	146	1	then	then	ADV
ejpam-5572	146	2	k	k	PROPN
ejpam-5572	146	3	presented	present	VERB
ejpam-5572	146	4	a	a	DET
ejpam-5572	146	5	fuzzy	fuzzy	ADJ
ejpam-5572	146	6	subgroup	subgroup	NOUN
ejpam-5572	146	7	(	(	PUNCT
ejpam-5572	146	8	fsg	fsg	PROPN
ejpam-5572	146	9	)	)	PUNCT
ejpam-5572	146	10	of	of	ADP
ejpam-5572	146	11	(	(	PUNCT
ejpam-5572	146	12	x	x	NOUN
ejpam-5572	146	13	,	,	PUNCT
ejpam-5572	146	14	∗	∗	NOUN
ejpam-5572	146	15	)	)	PUNCT
ejpam-5572	146	16	,	,	PUNCT
ejpam-5572	146	17	if	if	SCONJ
ejpam-5572	146	18	the	the	DET
ejpam-5572	146	19	following	follow	VERB
ejpam-5572	146	20	conditions	condition	NOUN
ejpam-5572	146	21	hold	hold	VERB
ejpam-5572	146	22	:	:	PUNCT
ejpam-5572	146	23	i	i	X
ejpam-5572	146	24	)	)	PUNCT
ejpam-5572	146	25	k(x	k(x	PROPN
ejpam-5572	146	26	∗	∗	PROPN
ejpam-5572	146	27	y	y	PROPN
ejpam-5572	146	28	)	)	PUNCT
ejpam-5572	146	29	≥	≥	NOUN
ejpam-5572	146	30	k(x	k(x	PROPN
ejpam-5572	146	31	)	)	PUNCT
ejpam-5572	146	32	∧k(y	∧k(y	NUM
ejpam-5572	146	33	)	)	PUNCT
ejpam-5572	146	34	.	.	PUNCT
ejpam-5572	147	1	ii	ii	X
ejpam-5572	147	2	)	)	PUNCT
ejpam-5572	147	3	k(x−1	k(x−1	NOUN
ejpam-5572	147	4	)	)	PUNCT
ejpam-5572	147	5	≥	≥	NOUN
ejpam-5572	147	6	k(x	k(x	PROPN
ejpam-5572	147	7	)	)	PUNCT
ejpam-5572	147	8	,	,	PUNCT
ejpam-5572	147	9	for	for	ADP
ejpam-5572	147	10	all	all	DET
ejpam-5572	147	11	x	x	NOUN
ejpam-5572	147	12	,	,	PUNCT
ejpam-5572	147	13	y	y	PROPN
ejpam-5572	147	14	∈	∈	PROPN
ejpam-5572	147	15	x	x	NOUN
ejpam-5572	147	16	definition	definition	NOUN
ejpam-5572	147	17	6	6	NUM
ejpam-5572	147	18	.	.	PUNCT
ejpam-5572	148	1	[	[	X
ejpam-5572	148	2	3	3	X
ejpam-5572	148	3	]	]	PUNCT
ejpam-5572	148	4	let	let	VERB
ejpam-5572	148	5	k(x	k(x	NOUN
ejpam-5572	148	6	)	)	PUNCT
ejpam-5572	148	7	:	:	PUNCT
ejpam-5572	149	1	x	x	X
ejpam-5572	149	2	→	→	PUNCT
ejpam-5572	149	3	{	{	PUNCT
ejpam-5572	149	4	z	z	NOUN
ejpam-5572	149	5	:	:	PUNCT
ejpam-5572	149	6	z	z	X
ejpam-5572	149	7	∈	∈	PROPN
ejpam-5572	149	8	c	c	NOUN
ejpam-5572	149	9	,	,	PUNCT
ejpam-5572	149	10	|z|	|z|	VERB
ejpam-5572	149	11	≤	≤	NOUN
ejpam-5572	149	12	1	1	NUM
ejpam-5572	149	13	}	}	PUNCT
ejpam-5572	149	14	be	be	AUX
ejpam-5572	149	15	a	a	DET
ejpam-5572	149	16	complex	complex	ADJ
ejpam-5572	149	17	fuzzy	fuzzy	ADJ
ejpam-5572	149	18	subset	subset	NOUN
ejpam-5572	149	19	of	of	ADP
ejpam-5572	149	20	a	a	DET
ejpam-5572	149	21	group	group	NOUN
ejpam-5572	149	22	(	(	PUNCT
ejpam-5572	149	23	x	x	X
ejpam-5572	149	24	,	,	PUNCT
ejpam-5572	149	25	∗	∗	NOUN
ejpam-5572	149	26	)	)	PUNCT
ejpam-5572	149	27	.	.	PUNCT
ejpam-5572	150	1	then	then	ADV
ejpam-5572	150	2	k	k	PROPN
ejpam-5572	150	3	presented	present	VERB
ejpam-5572	150	4	a	a	DET
ejpam-5572	150	5	complex	complex	ADJ
ejpam-5572	150	6	fuzzy	fuzzy	ADJ
ejpam-5572	150	7	subgroup	subgroup	NOUN
ejpam-5572	150	8	,	,	PUNCT
ejpam-5572	150	9	of	of	ADP
ejpam-5572	150	10	(	(	PUNCT
ejpam-5572	150	11	x	x	NOUN
ejpam-5572	150	12	,	,	PUNCT
ejpam-5572	150	13	∗	∗	NOUN
ejpam-5572	150	14	)	)	PUNCT
ejpam-5572	150	15	,	,	PUNCT
ejpam-5572	150	16	if	if	SCONJ
ejpam-5572	150	17	the	the	DET
ejpam-5572	150	18	following	follow	VERB
ejpam-5572	150	19	conditions	condition	NOUN
ejpam-5572	150	20	hold	hold	VERB
ejpam-5572	150	21	:	:	PUNCT
ejpam-5572	150	22	i	i	X
ejpam-5572	150	23	)	)	PUNCT
ejpam-5572	150	24	k(x	k(x	PROPN
ejpam-5572	150	25	∗	∗	PROPN
ejpam-5572	150	26	y	y	PROPN
ejpam-5572	150	27	)	)	PUNCT
ejpam-5572	150	28	≥	≥	NOUN
ejpam-5572	150	29	k(x	k(x	PROPN
ejpam-5572	150	30	)	)	PUNCT
ejpam-5572	150	31	∧k(y	∧k(y	NUM
ejpam-5572	150	32	)	)	PUNCT
ejpam-5572	150	33	.	.	PUNCT
ejpam-5572	151	1	ii	ii	X
ejpam-5572	151	2	)	)	PUNCT
ejpam-5572	151	3	k(x−1	k(x−1	NOUN
ejpam-5572	151	4	)	)	PUNCT
ejpam-5572	151	5	≥	≥	NOUN
ejpam-5572	151	6	k(x	k(x	PROPN
ejpam-5572	151	7	)	)	PUNCT
ejpam-5572	151	8	,	,	PUNCT
ejpam-5572	151	9	for	for	ADP
ejpam-5572	151	10	all	all	DET
ejpam-5572	151	11	x	x	NOUN
ejpam-5572	151	12	,	,	PUNCT
ejpam-5572	151	13	y	y	PROPN
ejpam-5572	151	14	∈	∈	PROPN
ejpam-5572	151	15	x	x	PUNCT
ejpam-5572	151	16	equivalently	equivalently	ADV
ejpam-5572	151	17	,	,	PUNCT
ejpam-5572	151	18	for	for	ADP
ejpam-5572	151	19	any	any	DET
ejpam-5572	151	20	x	x	NOUN
ejpam-5572	151	21	,	,	PUNCT
ejpam-5572	151	22	y	y	PROPN
ejpam-5572	151	23	∈	∈	PROPN
ejpam-5572	151	24	x	x	X
ejpam-5572	151	25	and	and	CCONJ
ejpam-5572	151	26	k(x	k(x	PROPN
ejpam-5572	151	27	)	)	PUNCT
ejpam-5572	151	28	=	=	SYM
ejpam-5572	151	29	p(x)e2πiω(x	p(x)e2πiω(x	NOUN
ejpam-5572	151	30	)	)	PUNCT
ejpam-5572	151	31	,	,	PUNCT
ejpam-5572	151	32	we	we	PRON
ejpam-5572	151	33	have	have	AUX
ejpam-5572	151	34	:	:	PUNCT
ejpam-5572	151	35	i	i	NOUN
ejpam-5572	151	36	)	)	PUNCT
ejpam-5572	151	37	p(x	p(x	PROPN
ejpam-5572	151	38	∗	∗	NOUN
ejpam-5572	151	39	y	y	PROPN
ejpam-5572	151	40	)	)	PUNCT
ejpam-5572	151	41	≥	≥	NOUN
ejpam-5572	151	42	p(x	p(x	NOUN
ejpam-5572	151	43	)	)	PUNCT
ejpam-5572	151	44	∧	∧	PROPN
ejpam-5572	151	45	p(y	p(y	PROPN
ejpam-5572	151	46	)	)	PUNCT
ejpam-5572	151	47	and	and	CCONJ
ejpam-5572	151	48	ω(x	ω(x	X
ejpam-5572	151	49	∗	∗	NOUN
ejpam-5572	151	50	y	y	PROPN
ejpam-5572	151	51	)	)	PUNCT
ejpam-5572	151	52	≥	≥	NOUN
ejpam-5572	151	53	ω(x	ω(x	NOUN
ejpam-5572	151	54	)	)	PUNCT
ejpam-5572	151	55	∧	∧	PROPN
ejpam-5572	151	56	ω(y	ω(y	PROPN
ejpam-5572	151	57	)	)	PUNCT
ejpam-5572	151	58	.	.	PUNCT
ejpam-5572	152	1	ii	ii	X
ejpam-5572	152	2	)	)	PUNCT
ejpam-5572	152	3	p(x−1	p(x−1	PROPN
ejpam-5572	152	4	)	)	PUNCT
ejpam-5572	152	5	≥	≥	NOUN
ejpam-5572	152	6	p(x	p(x	PROPN
ejpam-5572	152	7	)	)	PUNCT
ejpam-5572	152	8	and	and	CCONJ
ejpam-5572	152	9	ω(x−1	ω(x−1	NOUN
ejpam-5572	152	10	)	)	PUNCT
ejpam-5572	152	11	≥	≥	NOUN
ejpam-5572	152	12	ω(x	ω(x	NOUN
ejpam-5572	152	13	)	)	PUNCT
ejpam-5572	152	14	.	.	PUNCT
ejpam-5572	153	1	definition	definition	NOUN
ejpam-5572	153	2	7	7	NUM
ejpam-5572	153	3	.	.	PUNCT
ejpam-5572	154	1	[	[	X
ejpam-5572	154	2	35	35	NUM
ejpam-5572	154	3	]	]	X
ejpam-5572	154	4	let	let	VERB
ejpam-5572	154	5	(	(	PUNCT
ejpam-5572	154	6	x	x	NOUN
ejpam-5572	154	7	,	,	PUNCT
ejpam-5572	154	8	∗	∗	NOUN
ejpam-5572	154	9	)	)	PUNCT
ejpam-5572	154	10	be	be	VERB
ejpam-5572	154	11	a	a	DET
ejpam-5572	154	12	group	group	NOUN
ejpam-5572	154	13	and	and	CCONJ
ejpam-5572	154	14	f	f	NOUN
ejpam-5572	154	15	=	=	SYM
ejpam-5572	154	16	(	(	PUNCT
ejpam-5572	154	17	k	k	X
ejpam-5572	154	18	,	,	PUNCT
ejpam-5572	154	19	l	l	NOUN
ejpam-5572	154	20	)	)	PUNCT
ejpam-5572	154	21	be	be	AUX
ejpam-5572	154	22	a	a	DET
ejpam-5572	154	23	fermatean	fermatean	ADJ
ejpam-5572	154	24	fuzzy	fuzzy	ADJ
ejpam-5572	154	25	set	set	NOUN
ejpam-5572	154	26	of	of	ADP
ejpam-5572	154	27	x.	x.	NOUN
ejpam-5572	154	28	then	then	ADV
ejpam-5572	154	29	f	f	PROPN
ejpam-5572	154	30	is	be	AUX
ejpam-5572	154	31	a	a	DET
ejpam-5572	154	32	fermatean	fermatean	ADJ
ejpam-5572	154	33	fuzzy	fuzzy	ADJ
ejpam-5572	154	34	subgroup	subgroup	NOUN
ejpam-5572	154	35	of	of	ADP
ejpam-5572	154	36	x	x	SYM
ejpam-5572	154	37	if	if	SCONJ
ejpam-5572	154	38	the	the	DET
ejpam-5572	154	39	following	follow	VERB
ejpam-5572	154	40	conditions	condition	NOUN
ejpam-5572	154	41	hold	hold	VERB
ejpam-5572	154	42	:	:	PUNCT
ejpam-5572	154	43	(	(	PUNCT
ejpam-5572	154	44	i	i	NOUN
ejpam-5572	154	45	)	)	PUNCT
ejpam-5572	155	1	k3(x	k3(x	PROPN
ejpam-5572	155	2	∗	∗	PROPN
ejpam-5572	155	3	y	y	PROPN
ejpam-5572	155	4	)	)	PUNCT
ejpam-5572	155	5	≥	≥	NOUN
ejpam-5572	155	6	k3(x	k3(x	PROPN
ejpam-5572	155	7	)	)	PUNCT
ejpam-5572	155	8	∧k3(y	∧k3(y	PROPN
ejpam-5572	155	9	)	)	PUNCT
ejpam-5572	155	10	and	and	CCONJ
ejpam-5572	155	11	l3(x	l3(x	PROPN
ejpam-5572	155	12	∗	∗	PROPN
ejpam-5572	155	13	y	y	NOUN
ejpam-5572	155	14	)	)	PUNCT
ejpam-5572	155	15	≤	≤	NUM
ejpam-5572	155	16	l3(x	l3(x	PROPN
ejpam-5572	155	17	)	)	PUNCT
ejpam-5572	155	18	∨	∨	PROPN
ejpam-5572	155	19	l3(y	l3(y	PROPN
ejpam-5572	155	20	)	)	PUNCT
ejpam-5572	155	21	.	.	PUNCT
ejpam-5572	156	1	(	(	PUNCT
ejpam-5572	156	2	ii	ii	X
ejpam-5572	156	3	)	)	PUNCT
ejpam-5572	156	4	k3(x−1	k3(x−1	PROPN
ejpam-5572	156	5	)	)	PUNCT
ejpam-5572	156	6	≥	≥	NOUN
ejpam-5572	156	7	k3(x	k3(x	PROPN
ejpam-5572	156	8	)	)	PUNCT
ejpam-5572	156	9	and	and	CCONJ
ejpam-5572	156	10	l3(x−1	l3(x−1	NOUN
ejpam-5572	156	11	)	)	PUNCT
ejpam-5572	156	12	≤	≤	NOUN
ejpam-5572	156	13	l3(x	l3(x	PROPN
ejpam-5572	156	14	)	)	PUNCT
ejpam-5572	156	15	,	,	PUNCT
ejpam-5572	156	16	∀	∀	X
ejpam-5572	156	17	x	x	NOUN
ejpam-5572	156	18	,	,	PUNCT
ejpam-5572	156	19	y	y	PROPN
ejpam-5572	156	20	∈	∈	PROPN
ejpam-5572	156	21	x	x	PUNCT
ejpam-5572	156	22	in	in	ADP
ejpam-5572	156	23	the	the	DET
ejpam-5572	156	24	previous	previous	ADJ
ejpam-5572	156	25	definition	definition	NOUN
ejpam-5572	156	26	,	,	PUNCT
ejpam-5572	156	27	if	if	SCONJ
ejpam-5572	156	28	the	the	DET
ejpam-5572	156	29	power	power	NOUN
ejpam-5572	156	30	k	k	NOUN
ejpam-5572	156	31	=	=	SYM
ejpam-5572	156	32	1	1	NUM
ejpam-5572	156	33	,	,	PUNCT
ejpam-5572	156	34	2	2	NUM
ejpam-5572	156	35	we	we	PRON
ejpam-5572	156	36	get	get	VERB
ejpam-5572	156	37	intuitionistic	intuitionistic	ADJ
ejpam-5572	156	38	fuzzy	fuzzy	ADJ
ejpam-5572	156	39	subgroup	subgroup	NOUN
ejpam-5572	156	40	(	(	PUNCT
ejpam-5572	156	41	ifsg	ifsg	NOUN
ejpam-5572	156	42	)	)	PUNCT
ejpam-5572	157	1	[	[	X
ejpam-5572	157	2	15	15	NUM
ejpam-5572	157	3	]	]	PUNCT
ejpam-5572	157	4	and	and	CCONJ
ejpam-5572	157	5	pythagorean	pythagorean	PROPN
ejpam-5572	157	6	fuzzy	fuzzy	ADJ
ejpam-5572	157	7	subgroup	subgroup	NOUN
ejpam-5572	157	8	(	(	PUNCT
ejpam-5572	157	9	pfsg	pfsg	NOUN
ejpam-5572	157	10	)	)	PUNCT
ejpam-5572	158	1	[	[	X
ejpam-5572	158	2	14	14	NUM
ejpam-5572	158	3	]	]	X
ejpam-5572	158	4	,	,	PUNCT
ejpam-5572	158	5	respectively	respectively	ADV
ejpam-5572	158	6	.	.	PUNCT
ejpam-5572	159	1	the	the	DET
ejpam-5572	159	2	union	union	NOUN
ejpam-5572	159	3	,	,	PUNCT
ejpam-5572	159	4	intersection	intersection	NOUN
ejpam-5572	159	5	,	,	PUNCT
ejpam-5572	159	6	and	and	CCONJ
ejpam-5572	159	7	complement	complement	NOUN
ejpam-5572	159	8	of	of	ADP
ejpam-5572	159	9	cffs	cff	NOUN
ejpam-5572	159	10	defined	define	VERB
ejpam-5572	159	11	in	in	ADP
ejpam-5572	159	12	2021	2021	NUM
ejpam-5572	159	13	[	[	X
ejpam-5572	159	14	17	17	NUM
ejpam-5572	159	15	]	]	PUNCT
ejpam-5572	159	16	,	,	PUNCT
ejpam-5572	159	17	as	as	SCONJ
ejpam-5572	159	18	follows	follow	VERB
ejpam-5572	159	19	.	.	PUNCT
ejpam-5572	160	1	definition	definition	NOUN
ejpam-5572	160	2	8	8	NUM
ejpam-5572	160	3	.	.	PUNCT
ejpam-5572	161	1	let	let	VERB
ejpam-5572	161	2	ϕ1	ϕ1	NOUN
ejpam-5572	161	3	=	=	SYM
ejpam-5572	161	4	(	(	PUNCT
ejpam-5572	161	5	k1,l1	k1,l1	PROPN
ejpam-5572	161	6	)	)	PUNCT
ejpam-5572	161	7	and	and	CCONJ
ejpam-5572	161	8	ϕ2	ϕ2	ADV
ejpam-5572	161	9	=	=	SYM
ejpam-5572	161	10	(	(	PUNCT
ejpam-5572	161	11	k2,l2	k2,l2	PROPN
ejpam-5572	161	12	)	)	PUNCT
ejpam-5572	161	13	be	be	AUX
ejpam-5572	161	14	two	two	NUM
ejpam-5572	161	15	cffss	cffss	NOUN
ejpam-5572	161	16	on	on	ADP
ejpam-5572	161	17	x	x	NOUN
ejpam-5572	161	18	,	,	PUNCT
ejpam-5572	161	19	where	where	SCONJ
ejpam-5572	161	20	:	:	PUNCT
ejpam-5572	161	21	kj(x	kj(x	X
ejpam-5572	161	22	)	)	PUNCT
ejpam-5572	161	23	:	:	PUNCT
ejpam-5572	162	1	x	x	X
ejpam-5572	162	2	→	→	X
ejpam-5572	162	3	{	{	PUNCT
ejpam-5572	162	4	pj(x)e2πiωj(x	pj(x)e2πiωj(x	PROPN
ejpam-5572	162	5	)	)	PUNCT
ejpam-5572	162	6	:	:	PUNCT
ejpam-5572	162	7	0	0	NUM
ejpam-5572	162	8	≤	≤	NOUN
ejpam-5572	162	9	pj(x	pj(x	PRON
ejpam-5572	162	10	)	)	PUNCT
ejpam-5572	162	11	,	,	PUNCT
ejpam-5572	162	12	ωj(x	ωj(x	NUM
ejpam-5572	162	13	)	)	PUNCT
ejpam-5572	162	14	≤	≤	NUM
ejpam-5572	162	15	1	1	NUM
ejpam-5572	162	16	}	}	PUNCT
ejpam-5572	162	17	,	,	PUNCT
ejpam-5572	162	18	and	and	CCONJ
ejpam-5572	162	19	lj(x	lj(x	NUM
ejpam-5572	162	20	)	)	PUNCT
ejpam-5572	162	21	:	:	PUNCT
ejpam-5572	162	22	x	x	X
ejpam-5572	162	23	→	→	PUNCT
ejpam-5572	162	24	{	{	PUNCT
ejpam-5572	162	25	qj(x)e2πiνj(x	qj(x)e2πiνj(x	PROPN
ejpam-5572	162	26	)	)	PUNCT
ejpam-5572	162	27	:	:	PUNCT
ejpam-5572	162	28	0	0	NUM
ejpam-5572	162	29	≤	≤	NUM
ejpam-5572	162	30	qj(x	qj(x	NUM
ejpam-5572	162	31	)	)	PUNCT
ejpam-5572	162	32	,	,	PUNCT
ejpam-5572	162	33	νj(x	νj(x	NOUN
ejpam-5572	162	34	)	)	PUNCT
ejpam-5572	162	35	≤	≤	NUM
ejpam-5572	162	36	1	1	NUM
ejpam-5572	162	37	}	}	PUNCT
ejpam-5572	162	38	,	,	PUNCT
ejpam-5572	162	39	for	for	ADP
ejpam-5572	162	40	j	j	PROPN
ejpam-5572	162	41	=	=	SYM
ejpam-5572	162	42	1	1	NUM
ejpam-5572	162	43	,	,	PUNCT
ejpam-5572	162	44	2	2	NUM
ejpam-5572	162	45	,	,	PUNCT
ejpam-5572	162	46	then	then	ADV
ejpam-5572	162	47	:	:	PUNCT
ejpam-5572	162	48	1	1	X
ejpam-5572	162	49	.	.	X
ejpam-5572	163	1	ϕ1	ϕ1	NOUN
ejpam-5572	163	2	∩	∩	NOUN
ejpam-5572	163	3	ϕ2	ϕ2	ADV
ejpam-5572	163	4	=	=	SYM
ejpam-5572	163	5	(	(	PUNCT
ejpam-5572	163	6	k1	k1	ADJ
ejpam-5572	163	7	∩k2,l1	∩k2,l1	ADJ
ejpam-5572	163	8	∩	∩	ADJ
ejpam-5572	163	9	l2	l2	NOUN
ejpam-5572	163	10	)	)	PUNCT
ejpam-5572	163	11	,	,	PUNCT
ejpam-5572	163	12	where	where	SCONJ
ejpam-5572	163	13	(	(	PUNCT
ejpam-5572	163	14	k1	k1	PROPN
ejpam-5572	163	15	∩k2)(x	∩k2)(x	PROPN
ejpam-5572	163	16	)	)	PUNCT
ejpam-5572	163	17	=	=	PUNCT
ejpam-5572	163	18	(	(	PUNCT
ejpam-5572	163	19	p1(x	p1(x	NOUN
ejpam-5572	163	20	)	)	PUNCT
ejpam-5572	163	21	∧	∧	PROPN
ejpam-5572	163	22	p2(x))e	p2(x))e	PROPN
ejpam-5572	163	23	2πi(ω1(x)∧ω2(x	2πi(ω1(x)∧ω2(x	NUM
ejpam-5572	163	24	)	)	PUNCT
ejpam-5572	163	25	)	)	PUNCT
ejpam-5572	164	1	and	and	CCONJ
ejpam-5572	164	2	(	(	PUNCT
ejpam-5572	164	3	l1	l1	PROPN
ejpam-5572	164	4	∩	∩	ADJ
ejpam-5572	164	5	l2)(x	l2)(x	PROPN
ejpam-5572	164	6	)	)	PUNCT
ejpam-5572	164	7	=	=	PUNCT
ejpam-5572	164	8	(	(	PUNCT
ejpam-5572	164	9	q1(x	q1(x	NOUN
ejpam-5572	164	10	)	)	PUNCT
ejpam-5572	164	11	∨	∨	NOUN
ejpam-5572	164	12	q2(x))e	q2(x))e	NOUN
ejpam-5572	164	13	2πi(ν1(x)∨ν2(x	2πi(ν1(x)∨ν2(x	NUM
ejpam-5572	164	14	)	)	PUNCT
ejpam-5572	164	15	)	)	PUNCT
ejpam-5572	164	16	.	.	PUNCT
ejpam-5572	165	1	e.a	e.a	PROPN
ejpam-5572	165	2	.	.	PROPN
ejpam-5572	165	3	abuhijleh	abuhijleh	PROPN
ejpam-5572	165	4	,	,	PUNCT
ejpam-5572	165	5	a.	a.	PROPN
ejpam-5572	165	6	alkouri	alkouri	PROPN
ejpam-5572	165	7	/	/	PROPN
ejpam-5572	165	8	eur	eur	PROPN
ejpam-5572	165	9	.	.	PUNCT
ejpam-5572	166	1	j.	j.	PROPN
ejpam-5572	166	2	pure	pure	PROPN
ejpam-5572	166	3	appl	appl	PROPN
ejpam-5572	166	4	.	.	PROPN
ejpam-5572	166	5	math	math	PROPN
ejpam-5572	166	6	,	,	PUNCT
ejpam-5572	166	7	18	18	NUM
ejpam-5572	166	8	(	(	PUNCT
ejpam-5572	166	9	1	1	NUM
ejpam-5572	166	10	)	)	PUNCT
ejpam-5572	166	11	(	(	PUNCT
ejpam-5572	166	12	2025	2025	NUM
ejpam-5572	166	13	)	)	PUNCT
ejpam-5572	166	14	,	,	PUNCT
ejpam-5572	166	15	5572	5572	NUM
ejpam-5572	166	16	6	6	NUM
ejpam-5572	166	17	of	of	ADP
ejpam-5572	166	18	19	19	NUM
ejpam-5572	166	19	2	2	NUM
ejpam-5572	166	20	.	.	PUNCT
ejpam-5572	167	1	ϕ1	ϕ1	NOUN
ejpam-5572	167	2	∪	∪	VERB
ejpam-5572	167	3	ϕ2	ϕ2	ADV
ejpam-5572	167	4	=	=	SYM
ejpam-5572	167	5	(	(	PUNCT
ejpam-5572	167	6	k1	k1	NOUN
ejpam-5572	167	7	∪k2,l1	∪k2,l1	ADJ
ejpam-5572	167	8	∪	∪	NOUN
ejpam-5572	167	9	l2	l2	NOUN
ejpam-5572	167	10	)	)	PUNCT
ejpam-5572	167	11	,	,	PUNCT
ejpam-5572	167	12	where	where	SCONJ
ejpam-5572	167	13	(	(	PUNCT
ejpam-5572	167	14	k1	k1	NOUN
ejpam-5572	167	15	∪k2)(x	∪k2)(x	NUM
ejpam-5572	167	16	)	)	PUNCT
ejpam-5572	168	1	=	=	PRON
ejpam-5572	168	2	(	(	PUNCT
ejpam-5572	168	3	p1(x	p1(x	NOUN
ejpam-5572	168	4	)	)	PUNCT
ejpam-5572	168	5	∨	∨	NUM
ejpam-5572	168	6	p2(x))e	p2(x))e	PROPN
ejpam-5572	168	7	2πi(ω1(x)∨ω2(x	2πi(ω1(x)∨ω2(x	NUM
ejpam-5572	168	8	)	)	PUNCT
ejpam-5572	168	9	)	)	PUNCT
ejpam-5572	169	1	and	and	CCONJ
ejpam-5572	169	2	(	(	PUNCT
ejpam-5572	169	3	l1	l1	PROPN
ejpam-5572	169	4	∪	∪	PROPN
ejpam-5572	169	5	l2)(x	l2)(x	PROPN
ejpam-5572	169	6	)	)	PUNCT
ejpam-5572	169	7	=	=	PUNCT
ejpam-5572	169	8	(	(	PUNCT
ejpam-5572	169	9	q1(x	q1(x	NOUN
ejpam-5572	169	10	)	)	PUNCT
ejpam-5572	169	11	∧	∧	NOUN
ejpam-5572	169	12	q2(x))e	q2(x))e	PROPN
ejpam-5572	169	13	2πi(ν1(x)∧ν2(x	2πi(ν1(x)∧ν2(x	NUM
ejpam-5572	169	14	)	)	PUNCT
ejpam-5572	169	15	)	)	PUNCT
ejpam-5572	169	16	.	.	PUNCT
ejpam-5572	170	1	3	3	X
ejpam-5572	170	2	.	.	X
ejpam-5572	171	1	ϕc	ϕc	NOUN
ejpam-5572	171	2	=	=	SYM
ejpam-5572	172	1	ϕ̄	ϕ̄	PROPN
ejpam-5572	172	2	=	=	PUNCT
ejpam-5572	172	3	(	(	PUNCT
ejpam-5572	172	4	l	l	NOUN
ejpam-5572	172	5	,	,	PUNCT
ejpam-5572	172	6	k	k	NOUN
ejpam-5572	172	7	)	)	PUNCT
ejpam-5572	172	8	=	=	SYM
ejpam-5572	172	9	(	(	PUNCT
ejpam-5572	172	10	q(x)e2πiν(x	q(x)e2πiν(x	PROPN
ejpam-5572	172	11	)	)	PUNCT
ejpam-5572	172	12	,	,	PUNCT
ejpam-5572	172	13	p(x)e2πiω(x	p(x)e2πiω(x	NOUN
ejpam-5572	172	14	)	)	PUNCT
ejpam-5572	172	15	)	)	PUNCT
ejpam-5572	172	16	.	.	PUNCT
ejpam-5572	173	1	3	3	X
ejpam-5572	173	2	.	.	X
ejpam-5572	173	3	complex	complex	ADJ
ejpam-5572	173	4	fermatean	fermatean	ADJ
ejpam-5572	173	5	fuzzy	fuzzy	NOUN
ejpam-5572	173	6	subgroups	subgroup	VERB
ejpam-5572	173	7	a	a	DET
ejpam-5572	173	8	generalization	generalization	NOUN
ejpam-5572	173	9	of	of	ADP
ejpam-5572	173	10	fermatean	fermatean	ADJ
ejpam-5572	173	11	fuzzy	fuzzy	ADJ
ejpam-5572	173	12	subgroups	subgroup	NOUN
ejpam-5572	173	13	[	[	X
ejpam-5572	173	14	35	35	NUM
ejpam-5572	173	15	]	]	PUNCT
ejpam-5572	173	16	and	and	CCONJ
ejpam-5572	173	17	complex	complex	ADJ
ejpam-5572	173	18	fermatean	fermatean	ADJ
ejpam-5572	173	19	fuzzy	fuzzy	ADJ
ejpam-5572	173	20	sets	set	NOUN
ejpam-5572	173	21	[	[	X
ejpam-5572	173	22	17	17	NUM
ejpam-5572	173	23	]	]	PUNCT
ejpam-5572	173	24	is	be	AUX
ejpam-5572	173	25	presented	present	VERB
ejpam-5572	173	26	in	in	ADP
ejpam-5572	173	27	the	the	DET
ejpam-5572	173	28	following	follow	VERB
ejpam-5572	173	29	definition	definition	NOUN
ejpam-5572	173	30	,	,	PUNCT
ejpam-5572	173	31	which	which	PRON
ejpam-5572	173	32	also	also	ADV
ejpam-5572	173	33	serves	serve	VERB
ejpam-5572	173	34	as	as	ADP
ejpam-5572	173	35	a	a	DET
ejpam-5572	173	36	generalization	generalization	NOUN
ejpam-5572	173	37	of	of	ADP
ejpam-5572	173	38	complex	complex	ADJ
ejpam-5572	173	39	pythagorean	pythagorean	ADJ
ejpam-5572	173	40	fuzzy	fuzzy	ADJ
ejpam-5572	173	41	subgroups	subgroup	NOUN
ejpam-5572	173	42	[	[	X
ejpam-5572	173	43	7	7	NUM
ejpam-5572	173	44	]	]	PUNCT
ejpam-5572	173	45	.	.	PUNCT
ejpam-5572	174	1	definition	definition	NOUN
ejpam-5572	174	2	9	9	NUM
ejpam-5572	174	3	.	.	PUNCT
ejpam-5572	175	1	let	let	AUX
ejpam-5572	175	2	(	(	PUNCT
ejpam-5572	175	3	x	x	NOUN
ejpam-5572	175	4	,	,	PUNCT
ejpam-5572	175	5	∗	∗	NOUN
ejpam-5572	175	6	)	)	PUNCT
ejpam-5572	175	7	be	be	VERB
ejpam-5572	175	8	a	a	DET
ejpam-5572	175	9	group	group	NOUN
ejpam-5572	175	10	and	and	CCONJ
ejpam-5572	175	11	ϕ	ϕ	NOUN
ejpam-5572	175	12	=	=	PUNCT
ejpam-5572	175	13	(	(	PUNCT
ejpam-5572	175	14	p	p	NOUN
ejpam-5572	175	15	e2πiω	e2πiω	PROPN
ejpam-5572	175	16	,	,	PUNCT
ejpam-5572	175	17	q	q	PROPN
ejpam-5572	175	18	e2πiν	e2πiν	PROPN
ejpam-5572	175	19	)	)	PUNCT
ejpam-5572	175	20	be	be	VERB
ejpam-5572	175	21	a	a	DET
ejpam-5572	175	22	cffs	cff	NOUN
ejpam-5572	175	23	of	of	ADP
ejpam-5572	175	24	x.	x.	NOUN
ejpam-5572	175	25	then	then	ADV
ejpam-5572	175	26	ϕ	ϕ	PROPN
ejpam-5572	175	27	is	be	AUX
ejpam-5572	175	28	complex	complex	ADJ
ejpam-5572	175	29	fermatean	fermatean	ADJ
ejpam-5572	175	30	fuzzy	fuzzy	ADJ
ejpam-5572	175	31	subgroup	subgroup	NOUN
ejpam-5572	175	32	(	(	PUNCT
ejpam-5572	175	33	cffsg	cffsg	ADJ
ejpam-5572	175	34	)	)	PUNCT
ejpam-5572	175	35	of	of	ADP
ejpam-5572	175	36	x	x	PRON
ejpam-5572	175	37	,	,	PUNCT
ejpam-5572	175	38	where	where	SCONJ
ejpam-5572	175	39	p3	p3	PROPN
ejpam-5572	175	40	+	+	CCONJ
ejpam-5572	175	41	q3	q3	PROPN
ejpam-5572	175	42	≤	≤	NUM
ejpam-5572	175	43	1	1	NUM
ejpam-5572	175	44	and	and	CCONJ
ejpam-5572	175	45	ω3	ω3	NOUN
ejpam-5572	175	46	+	+	CCONJ
ejpam-5572	175	47	ν3	ν3	ADJ
ejpam-5572	175	48	≤	≤	NUM
ejpam-5572	175	49	1	1	NUM
ejpam-5572	175	50	,	,	PUNCT
ejpam-5572	175	51	if	if	SCONJ
ejpam-5572	175	52	the	the	DET
ejpam-5572	175	53	following	follow	VERB
ejpam-5572	175	54	holds	hold	VERB
ejpam-5572	175	55	:	:	PUNCT
ejpam-5572	175	56	1a	1a	NUM
ejpam-5572	175	57	.	.	PUNCT
ejpam-5572	176	1	p3(x	p3(x	X
ejpam-5572	176	2	∗	∗	NOUN
ejpam-5572	176	3	y)e2πiω	y)e2πiω	PROPN
ejpam-5572	176	4	3(x	3(x	NUM
ejpam-5572	176	5	∗	∗	NOUN
ejpam-5572	176	6	y	y	PROPN
ejpam-5572	176	7	)	)	PUNCT
ejpam-5572	176	8	≥	≥	NOUN
ejpam-5572	176	9	p3(x)e2πiω	p3(x)e2πiω	PROPN
ejpam-5572	176	10	3(x	3(x	NUM
ejpam-5572	176	11	)	)	PUNCT
ejpam-5572	176	12	∧	∧	PROPN
ejpam-5572	176	13	p3(y)e2πiω	p3(y)e2πiω	NOUN
ejpam-5572	176	14	3(y	3(y	NUM
ejpam-5572	176	15	)	)	PUNCT
ejpam-5572	176	16	.	.	PUNCT
ejpam-5572	177	1	where	where	SCONJ
ejpam-5572	177	2	,	,	PUNCT
ejpam-5572	177	3	p3(x	p3(x	PROPN
ejpam-5572	177	4	∗	∗	NOUN
ejpam-5572	177	5	y	y	PROPN
ejpam-5572	177	6	)	)	PUNCT
ejpam-5572	177	7	≥	≥	NOUN
ejpam-5572	177	8	p3(x	p3(x	NOUN
ejpam-5572	177	9	)	)	PUNCT
ejpam-5572	177	10	∧	∧	PROPN
ejpam-5572	177	11	p3(y	p3(y	PROPN
ejpam-5572	177	12	)	)	PUNCT
ejpam-5572	177	13	and	and	CCONJ
ejpam-5572	177	14	ω3(x	ω3(x	PROPN
ejpam-5572	177	15	∗	∗	PROPN
ejpam-5572	177	16	y	y	PROPN
ejpam-5572	177	17	)	)	PUNCT
ejpam-5572	177	18	≥	≥	NOUN
ejpam-5572	177	19	ω3(x	ω3(x	PROPN
ejpam-5572	177	20	)	)	PUNCT
ejpam-5572	177	21	∧	∧	PROPN
ejpam-5572	177	22	ω3(y	ω3(y	PROPN
ejpam-5572	177	23	)	)	PUNCT
ejpam-5572	177	24	.	.	PUNCT
ejpam-5572	178	1	1b	1b	NUM
ejpam-5572	178	2	.	.	PUNCT
ejpam-5572	179	1	q3(x	q3(x	PRON
ejpam-5572	179	2	∗	∗	NOUN
ejpam-5572	179	3	y)e2πiν	y)e2πiν	PROPN
ejpam-5572	179	4	3(x	3(x	NUM
ejpam-5572	179	5	∗	∗	X
ejpam-5572	179	6	y	y	NOUN
ejpam-5572	179	7	)	)	PUNCT
ejpam-5572	179	8	≤	≤	NUM
ejpam-5572	180	1	q3(x)e2πiν	q3(x)e2πiν	ADP
ejpam-5572	180	2	3(x	3(x	NUM
ejpam-5572	180	3	)	)	PUNCT
ejpam-5572	180	4	∨	∨	NUM
ejpam-5572	180	5	q3(y)e2πiν	q3(y)e2πiν	NOUN
ejpam-5572	180	6	3(y	3(y	NUM
ejpam-5572	180	7	)	)	PUNCT
ejpam-5572	180	8	where	where	SCONJ
ejpam-5572	180	9	,	,	PUNCT
ejpam-5572	180	10	q3(x	q3(x	PROPN
ejpam-5572	180	11	∗	∗	X
ejpam-5572	180	12	y	y	NOUN
ejpam-5572	180	13	)	)	PUNCT
ejpam-5572	180	14	≤	≤	NUM
ejpam-5572	180	15	q3(x	q3(x	PROPN
ejpam-5572	180	16	)	)	PUNCT
ejpam-5572	180	17	∨	∨	NUM
ejpam-5572	180	18	q3(y	q3(y	PROPN
ejpam-5572	180	19	)	)	PUNCT
ejpam-5572	180	20	and	and	CCONJ
ejpam-5572	180	21	ν3(x	ν3(x	PROPN
ejpam-5572	180	22	∗	∗	PROPN
ejpam-5572	180	23	y	y	PROPN
ejpam-5572	180	24	)	)	PUNCT
ejpam-5572	180	25	≤	≤	NOUN
ejpam-5572	180	26	ν3(x	ν3(x	PROPN
ejpam-5572	180	27	)	)	PUNCT
ejpam-5572	180	28	∨	∨	NUM
ejpam-5572	180	29	ν3(y	ν3(y	NUM
ejpam-5572	180	30	)	)	PUNCT
ejpam-5572	180	31	2a	2a	NUM
ejpam-5572	180	32	.	.	PUNCT
ejpam-5572	181	1	p3(x−1)e2πiω	p3(x−1)e2πiω	PROPN
ejpam-5572	181	2	3(x−1	3(x−1	NUM
ejpam-5572	181	3	)	)	PUNCT
ejpam-5572	181	4	≥	≥	PROPN
ejpam-5572	181	5	p3(x)e2πiω	p3(x)e2πiω	PROPN
ejpam-5572	181	6	3(x	3(x	NUM
ejpam-5572	181	7	)	)	PUNCT
ejpam-5572	181	8	where	where	SCONJ
ejpam-5572	181	9	,	,	PUNCT
ejpam-5572	181	10	p3(x−1	p3(x−1	NOUN
ejpam-5572	181	11	)	)	PUNCT
ejpam-5572	181	12	≥	≥	NOUN
ejpam-5572	181	13	p3(x	p3(x	NOUN
ejpam-5572	181	14	)	)	PUNCT
ejpam-5572	181	15	and	and	CCONJ
ejpam-5572	181	16	ω3(x−1	ω3(x−1	NUM
ejpam-5572	181	17	)	)	PUNCT
ejpam-5572	181	18	≥	≥	NOUN
ejpam-5572	181	19	ω3(x	ω3(x	NUM
ejpam-5572	181	20	)	)	PUNCT
ejpam-5572	181	21	.	.	PUNCT
ejpam-5572	182	1	2b	2b	NOUN
ejpam-5572	182	2	.	.	PUNCT
ejpam-5572	183	1	q3(x−1)e2πiν	q3(x−1)e2πiν	PROPN
ejpam-5572	183	2	3(x−1	3(x−1	NUM
ejpam-5572	183	3	)	)	PUNCT
ejpam-5572	183	4	≤	≤	NUM
ejpam-5572	183	5	q3(x)e2πiν	q3(x)e2πiν	PROPN
ejpam-5572	183	6	3(x	3(x	NUM
ejpam-5572	183	7	)	)	PUNCT
ejpam-5572	183	8	where	where	SCONJ
ejpam-5572	183	9	,	,	PUNCT
ejpam-5572	183	10	q3(x−1	q3(x−1	NOUN
ejpam-5572	183	11	)	)	PUNCT
ejpam-5572	183	12	≤	≤	NOUN
ejpam-5572	183	13	q3(x	q3(x	PROPN
ejpam-5572	183	14	)	)	PUNCT
ejpam-5572	183	15	and	and	CCONJ
ejpam-5572	183	16	ν3(x−1	ν3(x−1	NUM
ejpam-5572	183	17	)	)	PUNCT
ejpam-5572	183	18	≤	≤	NOUN
ejpam-5572	183	19	ν3(x	ν3(x	NOUN
ejpam-5572	183	20	)	)	PUNCT
ejpam-5572	183	21	in	in	ADP
ejpam-5572	183	22	the	the	DET
ejpam-5572	183	23	previous	previous	ADJ
ejpam-5572	183	24	definition	definition	NOUN
ejpam-5572	183	25	,	,	PUNCT
ejpam-5572	183	26	if	if	SCONJ
ejpam-5572	183	27	the	the	DET
ejpam-5572	183	28	power	power	NOUN
ejpam-5572	183	29	k	k	NOUN
ejpam-5572	183	30	=	=	SYM
ejpam-5572	183	31	1	1	NUM
ejpam-5572	183	32	,	,	PUNCT
ejpam-5572	183	33	2	2	NUM
ejpam-5572	183	34	we	we	PRON
ejpam-5572	183	35	get	get	VERB
ejpam-5572	183	36	a	a	DET
ejpam-5572	183	37	complex	complex	ADJ
ejpam-5572	183	38	intuitionistic	intuitionistic	ADJ
ejpam-5572	183	39	fuzzy	fuzzy	ADJ
ejpam-5572	183	40	subgroups	subgroup	NOUN
ejpam-5572	183	41	(	(	PUNCT
ejpam-5572	183	42	cifsgs	cifsg	NOUN
ejpam-5572	183	43	)	)	PUNCT
ejpam-5572	184	1	[	[	X
ejpam-5572	184	2	20	20	NUM
ejpam-5572	184	3	]	]	PUNCT
ejpam-5572	184	4	,	,	PUNCT
ejpam-5572	184	5	and	and	CCONJ
ejpam-5572	184	6	a	a	DET
ejpam-5572	184	7	complex	complex	ADJ
ejpam-5572	184	8	pythagorean	pythagorean	ADJ
ejpam-5572	184	9	fuzzy	fuzzy	ADJ
ejpam-5572	184	10	subgroups	subgroup	NOUN
ejpam-5572	184	11	(	(	PUNCT
ejpam-5572	184	12	cpfsgs	cpfsg	NOUN
ejpam-5572	184	13	)	)	PUNCT
ejpam-5572	185	1	[	[	X
ejpam-5572	185	2	7	7	NUM
ejpam-5572	185	3	]	]	NUM
ejpam-5572	185	4	,	,	PUNCT
ejpam-5572	185	5	respectively	respectively	ADV
ejpam-5572	185	6	.	.	PUNCT
ejpam-5572	186	1	proposition	proposition	NOUN
ejpam-5572	186	2	1	1	NUM
ejpam-5572	186	3	.	.	PUNCT
ejpam-5572	187	1	let	let	VERB
ejpam-5572	187	2	ϕ	ϕ	NOUN
ejpam-5572	187	3	=	=	PUNCT
ejpam-5572	187	4	(	(	PUNCT
ejpam-5572	187	5	p	p	NOUN
ejpam-5572	187	6	e2πiω	e2πiω	PROPN
ejpam-5572	187	7	,	,	PUNCT
ejpam-5572	187	8	q	q	PROPN
ejpam-5572	187	9	e2πiν	e2πiν	PROPN
ejpam-5572	187	10	)	)	PUNCT
ejpam-5572	187	11	be	be	VERB
ejpam-5572	187	12	a	a	DET
ejpam-5572	187	13	cffsg	cffsg	NOUN
ejpam-5572	187	14	of	of	ADP
ejpam-5572	187	15	a	a	DET
ejpam-5572	187	16	group	group	NOUN
ejpam-5572	187	17	(	(	PUNCT
ejpam-5572	187	18	x	x	X
ejpam-5572	187	19	,	,	PUNCT
ejpam-5572	187	20	∗	∗	NOUN
ejpam-5572	187	21	)	)	PUNCT
ejpam-5572	187	22	,	,	PUNCT
ejpam-5572	187	23	then	then	ADV
ejpam-5572	187	24	the	the	DET
ejpam-5572	187	25	following	follow	VERB
ejpam-5572	187	26	holds	hold	VERB
ejpam-5572	187	27	:	:	PUNCT
ejpam-5572	187	28	(	(	PUNCT
ejpam-5572	187	29	i	i	NOUN
ejpam-5572	187	30	)	)	PUNCT
ejpam-5572	187	31	p3(id)e2πiω	p3(id)e2πiω	NUM
ejpam-5572	187	32	3(id	3(id	NUM
ejpam-5572	187	33	)	)	PUNCT
ejpam-5572	187	34	≥	≥	NOUN
ejpam-5572	187	35	p3(x)e2πiω	p3(x)e2πiω	NOUN
ejpam-5572	187	36	3(x	3(x	NUM
ejpam-5572	187	37	)	)	PUNCT
ejpam-5572	187	38	,	,	PUNCT
ejpam-5572	187	39	where	where	SCONJ
ejpam-5572	187	40	p3(id	p3(id	NUM
ejpam-5572	187	41	)	)	PUNCT
ejpam-5572	187	42	≥	≥	NOUN
ejpam-5572	187	43	p3(x	p3(x	SYM
ejpam-5572	187	44	)	)	PUNCT
ejpam-5572	187	45	and	and	CCONJ
ejpam-5572	187	46	ω3(id	ω3(id	PROPN
ejpam-5572	187	47	)	)	PUNCT
ejpam-5572	187	48	≥	≥	NOUN
ejpam-5572	187	49	ω3(x	ω3(x	PROPN
ejpam-5572	187	50	)	)	PUNCT
ejpam-5572	187	51	.	.	PUNCT
ejpam-5572	188	1	(	(	PUNCT
ejpam-5572	188	2	ii	ii	NOUN
ejpam-5572	188	3	)	)	PUNCT
ejpam-5572	188	4	q3(id)e2πiν	q3(id)e2πiν	NUM
ejpam-5572	188	5	3(id	3(id	NUM
ejpam-5572	188	6	)	)	PUNCT
ejpam-5572	188	7	≤	≤	NUM
ejpam-5572	188	8	q3(x)e2πiν	q3(x)e2πiν	PROPN
ejpam-5572	188	9	3(x	3(x	NUM
ejpam-5572	188	10	)	)	PUNCT
ejpam-5572	188	11	,	,	PUNCT
ejpam-5572	188	12	where	where	SCONJ
ejpam-5572	188	13	q3(id	q3(id	NUM
ejpam-5572	188	14	)	)	PUNCT
ejpam-5572	188	15	≤	≤	NUM
ejpam-5572	188	16	q3(x	q3(x	PROPN
ejpam-5572	188	17	)	)	PUNCT
ejpam-5572	188	18	and	and	CCONJ
ejpam-5572	188	19	ν3(id	ν3(id	PROPN
ejpam-5572	188	20	)	)	PUNCT
ejpam-5572	188	21	≤	≤	NOUN
ejpam-5572	188	22	ν3(x	ν3(x	NOUN
ejpam-5572	188	23	)	)	PUNCT
ejpam-5572	188	24	.	.	PUNCT
ejpam-5572	189	1	(	(	PUNCT
ejpam-5572	189	2	iii	iii	X
ejpam-5572	189	3	)	)	PUNCT
ejpam-5572	189	4	p3(x−1)e2πiω	p3(x−1)e2πiω	NOUN
ejpam-5572	189	5	3(x−1	3(x−1	NUM
ejpam-5572	189	6	)	)	PUNCT
ejpam-5572	190	1	=	=	SYM
ejpam-5572	190	2	p3(x)e2πiω	p3(x)e2πiω	PROPN
ejpam-5572	190	3	3(x	3(x	NUM
ejpam-5572	190	4	)	)	PUNCT
ejpam-5572	190	5	,	,	PUNCT
ejpam-5572	190	6	where	where	SCONJ
ejpam-5572	190	7	p3(x−1	p3(x−1	NOUN
ejpam-5572	190	8	)	)	PUNCT
ejpam-5572	190	9	=	=	SYM
ejpam-5572	190	10	p3(x	p3(x	PROPN
ejpam-5572	190	11	)	)	PUNCT
ejpam-5572	190	12	and	and	CCONJ
ejpam-5572	190	13	ω3(x−1	ω3(x−1	NUM
ejpam-5572	190	14	)	)	PUNCT
ejpam-5572	190	15	=	=	SYM
ejpam-5572	190	16	ω3(x	ω3(x	PROPN
ejpam-5572	190	17	)	)	PUNCT
ejpam-5572	190	18	.	.	PUNCT
ejpam-5572	191	1	(	(	PUNCT
ejpam-5572	191	2	iv	iv	X
ejpam-5572	191	3	)	)	PUNCT
ejpam-5572	191	4	q3(x−1)e2πiν	q3(x−1)e2πiν	NOUN
ejpam-5572	191	5	3(x−1	3(x−1	NUM
ejpam-5572	191	6	)	)	PUNCT
ejpam-5572	192	1	=	=	PUNCT
ejpam-5572	192	2	q3(x)e2πiν	q3(x)e2πiν	ADP
ejpam-5572	192	3	3(x	3(x	NUM
ejpam-5572	192	4	)	)	PUNCT
ejpam-5572	192	5	,	,	PUNCT
ejpam-5572	192	6	where	where	SCONJ
ejpam-5572	192	7	q3(x−1	q3(x−1	NOUN
ejpam-5572	192	8	)	)	PUNCT
ejpam-5572	192	9	=	=	PUNCT
ejpam-5572	192	10	q3(x	q3(x	PROPN
ejpam-5572	192	11	)	)	PUNCT
ejpam-5572	192	12	and	and	CCONJ
ejpam-5572	192	13	ν3(x−1	ν3(x−1	NUM
ejpam-5572	192	14	)	)	PUNCT
ejpam-5572	192	15	=	=	SYM
ejpam-5572	192	16	ν3(x	ν3(x	PROPN
ejpam-5572	192	17	)	)	PUNCT
ejpam-5572	192	18	.	.	PUNCT
ejpam-5572	193	1	for	for	ADP
ejpam-5572	193	2	all	all	DET
ejpam-5572	193	3	x	x	SYM
ejpam-5572	193	4	∈	∈	PROPN
ejpam-5572	193	5	x	x	NOUN
ejpam-5572	193	6	,	,	PUNCT
ejpam-5572	193	7	where	where	SCONJ
ejpam-5572	193	8	i	i	PRON
ejpam-5572	193	9	d	d	PROPN
ejpam-5572	193	10	is	be	AUX
ejpam-5572	193	11	the	the	DET
ejpam-5572	193	12	identity	identity	NOUN
ejpam-5572	193	13	of	of	ADP
ejpam-5572	193	14	all	all	DET
ejpam-5572	193	15	elements	element	NOUN
ejpam-5572	193	16	.	.	PUNCT
ejpam-5572	194	1	proof	proof	NOUN
ejpam-5572	194	2	.	.	PUNCT
ejpam-5572	195	1	since	since	SCONJ
ejpam-5572	195	2	ϕ	ϕ	NOUN
ejpam-5572	195	3	is	be	AUX
ejpam-5572	195	4	cffsg	cffsg	ADJ
ejpam-5572	195	5	then	then	ADV
ejpam-5572	195	6	by	by	ADP
ejpam-5572	195	7	definition	definition	NOUN
ejpam-5572	195	8	9	9	NUM
ejpam-5572	195	9	:	:	PUNCT
ejpam-5572	195	10	”	"	PUNCT
ejpam-5572	195	11	1	1	NUM
ejpam-5572	195	12	”	"	PUNCT
ejpam-5572	195	13	and	and	CCONJ
ejpam-5572	195	14	”	"	PUNCT
ejpam-5572	195	15	2	2	NUM
ejpam-5572	195	16	”	"	PUNCT
ejpam-5572	195	17	can	can	AUX
ejpam-5572	195	18	be	be	AUX
ejpam-5572	195	19	proved	prove	VERB
ejpam-5572	195	20	as	as	ADP
ejpam-5572	195	21	follow	follow	NOUN
ejpam-5572	195	22	,	,	PUNCT
ejpam-5572	195	23	p3(id)e2πiω	p3(id)e2πiω	X
ejpam-5572	195	24	3(id	3(id	NUM
ejpam-5572	195	25	)	)	PUNCT
ejpam-5572	195	26	=	=	PUNCT
ejpam-5572	196	1	p3(x	p3(x	X
ejpam-5572	196	2	o	o	X
ejpam-5572	196	3	x−1)e2πiω	x−1)e2πiω	PROPN
ejpam-5572	197	1	3(x	3(x	NUM
ejpam-5572	197	2	o	o	NOUN
ejpam-5572	197	3	x−1	x−1	PROPN
ejpam-5572	197	4	)	)	PUNCT
ejpam-5572	197	5	≥	≥	NOUN
ejpam-5572	197	6	min{p3(x)e2πiω3(x	min{p3(x)e2πiω3(x	NOUN
ejpam-5572	197	7	)	)	PUNCT
ejpam-5572	197	8	,	,	PUNCT
ejpam-5572	197	9	p3(x−1)e2πiω	p3(x−1)e2πiω	PROPN
ejpam-5572	197	10	3(x−1)}=	3(x−1)}=	NUM
ejpam-5572	197	11	min{p3(x	min{p3(x	PROPN
ejpam-5572	197	12	)	)	PUNCT
ejpam-5572	197	13	,	,	PUNCT
ejpam-5572	197	14	p3(x−1)}e2πimin{ω3(x),ω3(x−1	p3(x−1)}e2πimin{ω3(x),ω3(x−1	NOUN
ejpam-5572	197	15	)	)	PUNCT
ejpam-5572	197	16	}	}	PUNCT
ejpam-5572	198	1	=	=	SYM
ejpam-5572	198	2	p3(x)e2πiω	p3(x)e2πiω	PROPN
ejpam-5572	198	3	3(x	3(x	NUM
ejpam-5572	198	4	)	)	PUNCT
ejpam-5572	198	5	.	.	PUNCT
ejpam-5572	199	1	e.a	e.a	PROPN
ejpam-5572	199	2	.	.	PROPN
ejpam-5572	199	3	abuhijleh	abuhijleh	PROPN
ejpam-5572	199	4	,	,	PUNCT
ejpam-5572	199	5	a.	a.	PROPN
ejpam-5572	199	6	alkouri	alkouri	PROPN
ejpam-5572	199	7	/	/	PROPN
ejpam-5572	199	8	eur	eur	PROPN
ejpam-5572	199	9	.	.	PUNCT
ejpam-5572	200	1	j.	j.	PROPN
ejpam-5572	200	2	pure	pure	PROPN
ejpam-5572	200	3	appl	appl	PROPN
ejpam-5572	200	4	.	.	PROPN
ejpam-5572	200	5	math	math	PROPN
ejpam-5572	200	6	,	,	PUNCT
ejpam-5572	200	7	18	18	NUM
ejpam-5572	200	8	(	(	PUNCT
ejpam-5572	200	9	1	1	NUM
ejpam-5572	200	10	)	)	PUNCT
ejpam-5572	200	11	(	(	PUNCT
ejpam-5572	200	12	2025	2025	NUM
ejpam-5572	200	13	)	)	PUNCT
ejpam-5572	200	14	,	,	PUNCT
ejpam-5572	200	15	5572	5572	NUM
ejpam-5572	200	16	7	7	NUM
ejpam-5572	200	17	of	of	ADP
ejpam-5572	200	18	19	19	NUM
ejpam-5572	200	19	in	in	ADP
ejpam-5572	200	20	addition	addition	NOUN
ejpam-5572	200	21	,	,	PUNCT
ejpam-5572	200	22	q3(id)e2πiν	q3(id)e2πiν	NOUN
ejpam-5572	200	23	3(id	3(id	NUM
ejpam-5572	200	24	)	)	PUNCT
ejpam-5572	201	1	=	=	PUNCT
ejpam-5572	202	1	q3(x	q3(x	PUNCT
ejpam-5572	202	2	o	o	X
ejpam-5572	202	3	x−1)e2πiν	x−1)e2πiν	PUNCT
ejpam-5572	203	1	3(x	3(x	NUM
ejpam-5572	203	2	o	o	NOUN
ejpam-5572	203	3	x−1	x−1	PROPN
ejpam-5572	203	4	)	)	PUNCT
ejpam-5572	203	5	≤	≤	NOUN
ejpam-5572	203	6	max{q3(x)e2πiω3	max{q3(x)e2πiω3	PROPN
ejpam-5572	203	7	q	q	PROPN
ejpam-5572	203	8	(	(	PUNCT
ejpam-5572	203	9	x	x	NOUN
ejpam-5572	203	10	)	)	PUNCT
ejpam-5572	203	11	,	,	PUNCT
ejpam-5572	203	12	q3(x−1)e2πiω	q3(x−1)e2πiω	NOUN
ejpam-5572	203	13	3	3	NUM
ejpam-5572	203	14	q	q	NOUN
ejpam-5572	203	15	(	(	PUNCT
ejpam-5572	203	16	x	x	NOUN
ejpam-5572	203	17	−1	−1	NOUN
ejpam-5572	203	18	)	)	PUNCT
ejpam-5572	203	19	}	}	PUNCT
ejpam-5572	203	20	=	=	SYM
ejpam-5572	203	21	max{q3(x	max{q3(x	PROPN
ejpam-5572	203	22	)	)	PUNCT
ejpam-5572	203	23	,	,	PUNCT
ejpam-5572	203	24	q3(x−1)}e2πimax{ν3(x),ν3(x−1	q3(x−1)}e2πimax{ν3(x),ν3(x−1	NOUN
ejpam-5572	203	25	)	)	PUNCT
ejpam-5572	203	26	}	}	PUNCT
ejpam-5572	203	27	=	=	PUNCT
ejpam-5572	203	28	q3(x)e2πiν	q3(x)e2πiν	ADP
ejpam-5572	203	29	3(x	3(x	NUM
ejpam-5572	203	30	)	)	PUNCT
ejpam-5572	203	31	.	.	PUNCT
ejpam-5572	204	1	”	"	PUNCT
ejpam-5572	204	2	3	3	X
ejpam-5572	204	3	”	"	PUNCT
ejpam-5572	204	4	and	and	CCONJ
ejpam-5572	204	5	”	"	PUNCT
ejpam-5572	204	6	4	4	NUM
ejpam-5572	204	7	”	"	PUNCT
ejpam-5572	204	8	can	can	AUX
ejpam-5572	204	9	be	be	AUX
ejpam-5572	204	10	prove	prove	VERB
ejpam-5572	204	11	in	in	ADP
ejpam-5572	204	12	the	the	DET
ejpam-5572	204	13	same	same	ADJ
ejpam-5572	204	14	manner	manner	NOUN
ejpam-5572	204	15	of	of	ADP
ejpam-5572	204	16	”	"	PUNCT
ejpam-5572	204	17	1	1	NUM
ejpam-5572	204	18	”	"	PUNCT
ejpam-5572	204	19	and	and	CCONJ
ejpam-5572	204	20	”	"	PUNCT
ejpam-5572	204	21	2	2	NUM
ejpam-5572	204	22	”	"	PUNCT
ejpam-5572	204	23	.	.	PUNCT
ejpam-5572	205	1	the	the	DET
ejpam-5572	205	2	following	follow	VERB
ejpam-5572	205	3	is	be	AUX
ejpam-5572	205	4	an	an	DET
ejpam-5572	205	5	example	example	NOUN
ejpam-5572	205	6	of	of	ADP
ejpam-5572	205	7	complex	complex	ADJ
ejpam-5572	205	8	fermatean	fermatean	ADJ
ejpam-5572	205	9	fuzzy	fuzzy	ADJ
ejpam-5572	205	10	subgroup	subgroup	NOUN
ejpam-5572	205	11	(	(	PUNCT
ejpam-5572	205	12	cffsg	cffsg	ADJ
ejpam-5572	205	13	)	)	PUNCT
ejpam-5572	205	14	.	.	PUNCT
ejpam-5572	205	15	example	example	NOUN
ejpam-5572	206	1	1	1	NUM
ejpam-5572	206	2	.	.	X
ejpam-5572	206	3	for	for	ADP
ejpam-5572	206	4	the	the	DET
ejpam-5572	206	5	set	set	VERB
ejpam-5572	206	6	d2	d2	PROPN
ejpam-5572	206	7	which	which	PRON
ejpam-5572	206	8	define	define	VERB
ejpam-5572	206	9	a	a	DET
ejpam-5572	206	10	dihedral	dihedral	ADJ
ejpam-5572	206	11	group	group	NOUN
ejpam-5572	206	12	of	of	ADP
ejpam-5572	206	13	order	order	NOUN
ejpam-5572	206	14	four	four	NUM
ejpam-5572	206	15	and	and	CCONJ
ejpam-5572	206	16	isomorphic	isomorphic	ADJ
ejpam-5572	206	17	to	to	ADP
ejpam-5572	206	18	the	the	DET
ejpam-5572	206	19	direct	direct	ADJ
ejpam-5572	206	20	sum	sum	NOUN
ejpam-5572	206	21	of	of	ADP
ejpam-5572	206	22	two	two	NUM
ejpam-5572	206	23	cyclic	cyclic	ADJ
ejpam-5572	206	24	group	group	NOUN
ejpam-5572	206	25	z2	z2	PROPN
ejpam-5572	206	26	.	.	PUNCT
ejpam-5572	207	1	hence	hence	ADV
ejpam-5572	207	2	,	,	PUNCT
ejpam-5572	207	3	d2	d2	PROPN
ejpam-5572	207	4	=	=	SYM
ejpam-5572	207	5	{	{	PUNCT
ejpam-5572	207	6	(	(	PUNCT
ejpam-5572	207	7	0	0	NUM
ejpam-5572	207	8	,	,	PUNCT
ejpam-5572	207	9	0	0	NUM
ejpam-5572	207	10	)	)	PUNCT
ejpam-5572	207	11	,	,	PUNCT
ejpam-5572	207	12	(	(	PUNCT
ejpam-5572	207	13	1	1	NUM
ejpam-5572	207	14	,	,	PUNCT
ejpam-5572	207	15	0	0	NUM
ejpam-5572	207	16	)	)	PUNCT
ejpam-5572	207	17	,	,	PUNCT
ejpam-5572	207	18	(	(	PUNCT
ejpam-5572	207	19	0	0	NUM
ejpam-5572	207	20	,	,	PUNCT
ejpam-5572	207	21	1	1	NUM
ejpam-5572	207	22	)	)	PUNCT
ejpam-5572	207	23	,	,	PUNCT
ejpam-5572	207	24	(	(	PUNCT
ejpam-5572	207	25	1	1	NUM
ejpam-5572	207	26	,	,	PUNCT
ejpam-5572	207	27	1	1	NUM
ejpam-5572	207	28	)	)	PUNCT
ejpam-5572	207	29	}	}	PUNCT
ejpam-5572	207	30	.	.	PUNCT
ejpam-5572	208	1	also	also	ADV
ejpam-5572	208	2	,	,	PUNCT
ejpam-5572	208	3	let	let	VERB
ejpam-5572	208	4	ϕ	ϕ	X
ejpam-5572	208	5	=	=	X
ejpam-5572	208	6	(	(	PUNCT
ejpam-5572	208	7	k	k	X
ejpam-5572	208	8	,	,	PUNCT
ejpam-5572	208	9	l	l	NOUN
ejpam-5572	208	10	)	)	PUNCT
ejpam-5572	208	11	be	be	AUX
ejpam-5572	208	12	a	a	DET
ejpam-5572	208	13	cffs	cff	NOUN
ejpam-5572	208	14	on	on	ADP
ejpam-5572	208	15	d2	d2	PROPN
ejpam-5572	208	16	,	,	PUNCT
ejpam-5572	208	17	such	such	ADJ
ejpam-5572	208	18	that	that	SCONJ
ejpam-5572	208	19	:	:	PUNCT
ejpam-5572	208	20	ϕ((0	ϕ((0	PROPN
ejpam-5572	208	21	,	,	PUNCT
ejpam-5572	208	22	0	0	NUM
ejpam-5572	208	23	)	)	PUNCT
ejpam-5572	208	24	)	)	PUNCT
ejpam-5572	209	1	=	=	PRON
ejpam-5572	209	2	(	(	PUNCT
ejpam-5572	209	3	0.8e2πi(0.85	0.8e2πi(0.85	NUM
ejpam-5572	209	4	)	)	PUNCT
ejpam-5572	209	5	,	,	PUNCT
ejpam-5572	209	6	0.5e2πi(0.6	0.5e2πi(0.6	NUM
ejpam-5572	209	7	)	)	PUNCT
ejpam-5572	209	8	)	)	PUNCT
ejpam-5572	209	9	,	,	PUNCT
ejpam-5572	209	10	ϕ((1	ϕ((1	PROPN
ejpam-5572	209	11	,	,	PUNCT
ejpam-5572	209	12	0	0	NUM
ejpam-5572	209	13	)	)	PUNCT
ejpam-5572	209	14	)	)	PUNCT
ejpam-5572	210	1	=	=	PRON
ejpam-5572	210	2	(	(	PUNCT
ejpam-5572	210	3	0.78e2πi(0.83	0.78e2πi(0.83	NOUN
ejpam-5572	210	4	)	)	PUNCT
ejpam-5572	210	5	,	,	PUNCT
ejpam-5572	210	6	0.6e2πi(0.65	0.6e2πi(0.65	NUM
ejpam-5572	210	7	)	)	PUNCT
ejpam-5572	210	8	)	)	PUNCT
ejpam-5572	210	9	,	,	PUNCT
ejpam-5572	210	10	ϕ((0	ϕ((0	PROPN
ejpam-5572	210	11	,	,	PUNCT
ejpam-5572	210	12	1	1	NUM
ejpam-5572	210	13	)	)	PUNCT
ejpam-5572	210	14	)	)	PUNCT
ejpam-5572	211	1	=	=	SYM
ejpam-5572	211	2	(	(	PUNCT
ejpam-5572	211	3	0.75e2πi(0.8	0.75e2πi(0.8	NUM
ejpam-5572	211	4	)	)	PUNCT
ejpam-5572	211	5	,	,	PUNCT
ejpam-5572	211	6	0.65e2πi(0.68	0.65e2πi(0.68	NOUN
ejpam-5572	211	7	)	)	PUNCT
ejpam-5572	211	8	)	)	PUNCT
ejpam-5572	211	9	=	=	SYM
ejpam-5572	212	1	ϕ(xy	ϕ(xy	NUM
ejpam-5572	212	2	)	)	PUNCT
ejpam-5572	212	3	.	.	PUNCT
ejpam-5572	213	1	at	at	ADP
ejpam-5572	213	2	first	first	ADV
ejpam-5572	213	3	,	,	PUNCT
ejpam-5572	213	4	by	by	ADP
ejpam-5572	213	5	definition	definition	NOUN
ejpam-5572	213	6	4	4	NUM
ejpam-5572	213	7	,	,	PUNCT
ejpam-5572	213	8	it	it	PRON
ejpam-5572	213	9	is	be	AUX
ejpam-5572	213	10	easy	easy	ADJ
ejpam-5572	213	11	to	to	PART
ejpam-5572	213	12	check	check	VERB
ejpam-5572	213	13	that	that	PRON
ejpam-5572	213	14	ϕ	ϕ	NOUN
ejpam-5572	213	15	is	be	AUX
ejpam-5572	213	16	cffs	cff	NOUN
ejpam-5572	213	17	,	,	PUNCT
ejpam-5572	213	18	but	but	CCONJ
ejpam-5572	213	19	it	it	PRON
ejpam-5572	213	20	is	be	AUX
ejpam-5572	213	21	not	not	PART
ejpam-5572	213	22	cpfs	cpfs	ADJ
ejpam-5572	213	23	.	.	PUNCT
ejpam-5572	214	1	for	for	ADP
ejpam-5572	214	2	example	example	NOUN
ejpam-5572	214	3	in	in	ADP
ejpam-5572	214	4	ϕ((0	ϕ((0	PROPN
ejpam-5572	214	5	,	,	PUNCT
ejpam-5572	214	6	0	0	NUM
ejpam-5572	214	7	)	)	PUNCT
ejpam-5572	214	8	)	)	PUNCT
ejpam-5572	214	9	,	,	PUNCT
ejpam-5572	214	10	we	we	PRON
ejpam-5572	214	11	have	have	VERB
ejpam-5572	214	12	ω3+ν3	ω3+ν3	PROPN
ejpam-5572	214	13	=	=	SYM
ejpam-5572	214	14	(	(	PUNCT
ejpam-5572	214	15	0.614	0.614	NUM
ejpam-5572	214	16	+	+	NUM
ejpam-5572	214	17	0.216	0.216	NUM
ejpam-5572	214	18	)	)	PUNCT
ejpam-5572	214	19	=	=	SYM
ejpam-5572	214	20	0.83	0.83	NUM
ejpam-5572	214	21	≤	≤	NUM
ejpam-5572	214	22	1	1	NUM
ejpam-5572	214	23	and	and	CCONJ
ejpam-5572	214	24	ω2+ν2	ω2+ν2	NOUN
ejpam-5572	214	25	=	=	PUNCT
ejpam-5572	215	1	0.723	0.723	NUM
ejpam-5572	215	2	+	+	SYM
ejpam-5572	215	3	0.36	0.36	NUM
ejpam-5572	215	4	≰	≰	NOUN
ejpam-5572	215	5	1	1	NUM
ejpam-5572	215	6	.	.	PUNCT
ejpam-5572	216	1	in	in	ADP
ejpam-5572	216	2	the	the	DET
ejpam-5572	216	3	second	second	ADJ
ejpam-5572	216	4	part	part	NOUN
ejpam-5572	216	5	,	,	PUNCT
ejpam-5572	216	6	it	it	PRON
ejpam-5572	216	7	suffices	suffice	VERB
ejpam-5572	216	8	to	to	PART
ejpam-5572	216	9	prove	prove	VERB
ejpam-5572	216	10	that	that	SCONJ
ejpam-5572	216	11	the	the	DET
ejpam-5572	216	12	set	set	NOUN
ejpam-5572	216	13	ϕ(ℓ	ϕ(ℓ	PROPN
ejpam-5572	216	14	)	)	PUNCT
ejpam-5572	217	1	=	=	PRON
ejpam-5572	217	2	(	(	PUNCT
ejpam-5572	217	3	p(ℓ)e2πiω(ℓ	p(ℓ)e2πiω(ℓ	NOUN
ejpam-5572	217	4	)	)	PUNCT
ejpam-5572	217	5	,	,	PUNCT
ejpam-5572	217	6	q(ℓ)e2πiν(ℓ	q(ℓ)e2πiν(ℓ	NOUN
ejpam-5572	217	7	)	)	PUNCT
ejpam-5572	217	8	)	)	PUNCT
ejpam-5572	217	9	is	be	AUX
ejpam-5572	217	10	cffsg	cffsg	ADJ
ejpam-5572	217	11	on	on	ADP
ejpam-5572	217	12	d2	d2	PROPN
ejpam-5572	217	13	,	,	PUNCT
ejpam-5572	217	14	for	for	ADP
ejpam-5572	217	15	any	any	DET
ejpam-5572	217	16	ℓ	ℓ	PROPN
ejpam-5572	217	17	∈	∈	PROPN
ejpam-5572	217	18	d2	d2	PROPN
ejpam-5572	217	19	.	.	PUNCT
ejpam-5572	218	1	so	so	SCONJ
ejpam-5572	218	2	that	that	PRON
ejpam-5572	218	3	check	check	VERB
ejpam-5572	218	4	conditions	condition	NOUN
ejpam-5572	218	5	(	(	PUNCT
ejpam-5572	218	6	1a	1a	X
ejpam-5572	218	7	)	)	PUNCT
ejpam-5572	218	8	and	and	CCONJ
ejpam-5572	218	9	(	(	PUNCT
ejpam-5572	218	10	1b	1b	NUM
ejpam-5572	218	11	)	)	PUNCT
ejpam-5572	218	12	in	in	ADP
ejpam-5572	218	13	the	the	DET
ejpam-5572	218	14	definition	definition	NOUN
ejpam-5572	218	15	9	9	NUM
ejpam-5572	218	16	,	,	PUNCT
ejpam-5572	218	17	as	as	SCONJ
ejpam-5572	218	18	follows	follow	VERB
ejpam-5572	218	19	:	:	PUNCT
ejpam-5572	218	20	consider	consider	VERB
ejpam-5572	218	21	ℓ1	ℓ1	VERB
ejpam-5572	218	22	=	=	SYM
ejpam-5572	218	23	(	(	PUNCT
ejpam-5572	218	24	1	1	NUM
ejpam-5572	218	25	,	,	PUNCT
ejpam-5572	218	26	0	0	NUM
ejpam-5572	218	27	)	)	PUNCT
ejpam-5572	218	28	,	,	PUNCT
ejpam-5572	218	29	ℓ2	ℓ2	NOUN
ejpam-5572	218	30	=	=	SYM
ejpam-5572	218	31	(	(	PUNCT
ejpam-5572	218	32	0	0	NUM
ejpam-5572	218	33	,	,	PUNCT
ejpam-5572	218	34	1	1	NUM
ejpam-5572	218	35	)	)	PUNCT
ejpam-5572	218	36	,	,	PUNCT
ejpam-5572	218	37	then	then	ADV
ejpam-5572	218	38	(	(	PUNCT
ejpam-5572	218	39	1	1	NUM
ejpam-5572	218	40	,	,	PUNCT
ejpam-5572	218	41	0	0	NUM
ejpam-5572	218	42	)	)	PUNCT
ejpam-5572	218	43	⊕	⊕	PROPN
ejpam-5572	218	44	(	(	PUNCT
ejpam-5572	218	45	0	0	NUM
ejpam-5572	218	46	,	,	PUNCT
ejpam-5572	218	47	1	1	NUM
ejpam-5572	218	48	)	)	PUNCT
ejpam-5572	218	49	=	=	NOUN
ejpam-5572	218	50	(	(	PUNCT
ejpam-5572	218	51	1	1	NUM
ejpam-5572	218	52	,	,	PUNCT
ejpam-5572	218	53	1	1	NUM
ejpam-5572	218	54	)	)	PUNCT
ejpam-5572	218	55	1a	1a	NOUN
ejpam-5572	218	56	.	.	PUNCT
ejpam-5572	219	1	p3((1	p3((1	PROPN
ejpam-5572	219	2	,	,	PUNCT
ejpam-5572	219	3	0	0	NUM
ejpam-5572	219	4	)	)	PUNCT
ejpam-5572	219	5	⊕	⊕	PROPN
ejpam-5572	219	6	(	(	PUNCT
ejpam-5572	219	7	0	0	NUM
ejpam-5572	219	8	,	,	PUNCT
ejpam-5572	219	9	1	1	NUM
ejpam-5572	219	10	)	)	PUNCT
ejpam-5572	219	11	)	)	PUNCT
ejpam-5572	220	1	=	=	SYM
ejpam-5572	220	2	p3((1	p3((1	PROPN
ejpam-5572	220	3	,	,	PUNCT
ejpam-5572	220	4	1	1	NUM
ejpam-5572	220	5	)	)	PUNCT
ejpam-5572	220	6	)	)	PUNCT
ejpam-5572	221	1	=	=	PUNCT
ejpam-5572	222	1	0.422	0.422	NUM
ejpam-5572	222	2	≥	≥	NOUN
ejpam-5572	222	3	min{p3((1	min{p3((1	PROPN
ejpam-5572	222	4	,	,	PUNCT
ejpam-5572	222	5	0	0	NUM
ejpam-5572	222	6	)	)	PUNCT
ejpam-5572	222	7	)	)	PUNCT
ejpam-5572	222	8	,	,	PUNCT
ejpam-5572	222	9	p3((0	p3((0	PROPN
ejpam-5572	222	10	,	,	PUNCT
ejpam-5572	222	11	1	1	NUM
ejpam-5572	222	12	)	)	PUNCT
ejpam-5572	222	13	)	)	PUNCT
ejpam-5572	222	14	}	}	PUNCT
ejpam-5572	223	1	=	=	PUNCT
ejpam-5572	223	2	0.422	0.422	NUM
ejpam-5572	223	3	and	and	CCONJ
ejpam-5572	223	4	ω3((1	ω3((1	NOUN
ejpam-5572	223	5	,	,	PUNCT
ejpam-5572	223	6	0)⊕(0	0)⊕(0	NUM
ejpam-5572	223	7	,	,	PUNCT
ejpam-5572	223	8	1	1	NUM
ejpam-5572	223	9	)	)	PUNCT
ejpam-5572	223	10	)	)	PUNCT
ejpam-5572	224	1	=	=	SYM
ejpam-5572	224	2	ω3((1	ω3((1	NOUN
ejpam-5572	224	3	,	,	PUNCT
ejpam-5572	224	4	1	1	NUM
ejpam-5572	224	5	)	)	PUNCT
ejpam-5572	224	6	)	)	PUNCT
ejpam-5572	225	1	=	=	NOUN
ejpam-5572	225	2	0.512	0.512	NUM
ejpam-5572	225	3	≥	≥	NOUN
ejpam-5572	225	4	min{ω3((1	min{ω3((1	PROPN
ejpam-5572	225	5	,	,	PUNCT
ejpam-5572	225	6	0	0	NUM
ejpam-5572	225	7	)	)	PUNCT
ejpam-5572	225	8	)	)	PUNCT
ejpam-5572	225	9	,	,	PUNCT
ejpam-5572	225	10	ω3((0	ω3((0	NOUN
ejpam-5572	225	11	,	,	PUNCT
ejpam-5572	225	12	1	1	NUM
ejpam-5572	225	13	)	)	PUNCT
ejpam-5572	225	14	)	)	PUNCT
ejpam-5572	225	15	}	}	PUNCT
ejpam-5572	226	1	=	=	PUNCT
ejpam-5572	226	2	0.512	0.512	NUM
ejpam-5572	226	3	.	.	PUNCT
ejpam-5572	227	1	1b	1b	NUM
ejpam-5572	227	2	.	.	PUNCT
ejpam-5572	228	1	q3((1	q3((1	PROPN
ejpam-5572	228	2	,	,	PUNCT
ejpam-5572	228	3	0	0	X
ejpam-5572	228	4	)	)	PUNCT
ejpam-5572	228	5	⊕	⊕	PROPN
ejpam-5572	228	6	(	(	PUNCT
ejpam-5572	228	7	0	0	NUM
ejpam-5572	228	8	,	,	PUNCT
ejpam-5572	228	9	1	1	NUM
ejpam-5572	228	10	)	)	PUNCT
ejpam-5572	228	11	)	)	PUNCT
ejpam-5572	229	1	=	=	SYM
ejpam-5572	229	2	q3((1	q3((1	PROPN
ejpam-5572	229	3	,	,	PUNCT
ejpam-5572	229	4	1	1	NUM
ejpam-5572	229	5	)	)	PUNCT
ejpam-5572	229	6	)	)	PUNCT
ejpam-5572	229	7	=	=	PUNCT
ejpam-5572	230	1	0.275	0.275	NUM
ejpam-5572	230	2	≤	≤	NUM
ejpam-5572	230	3	max{q3((1	max{q3((1	PROPN
ejpam-5572	230	4	,	,	PUNCT
ejpam-5572	230	5	0	0	NUM
ejpam-5572	230	6	)	)	PUNCT
ejpam-5572	230	7	)	)	PUNCT
ejpam-5572	230	8	,	,	PUNCT
ejpam-5572	230	9	q3((0	q3((0	PROPN
ejpam-5572	230	10	,	,	PUNCT
ejpam-5572	230	11	1	1	NUM
ejpam-5572	230	12	)	)	PUNCT
ejpam-5572	230	13	)	)	PUNCT
ejpam-5572	230	14	}	}	PUNCT
ejpam-5572	231	1	=	=	PUNCT
ejpam-5572	231	2	0.275	0.275	NUM
ejpam-5572	231	3	and	and	CCONJ
ejpam-5572	231	4	ν3((1	ν3((1	PROPN
ejpam-5572	231	5	,	,	PUNCT
ejpam-5572	231	6	0)⊕	0)⊕	NUM
ejpam-5572	231	7	(	(	PUNCT
ejpam-5572	231	8	0	0	NUM
ejpam-5572	231	9	,	,	PUNCT
ejpam-5572	231	10	1	1	NUM
ejpam-5572	231	11	)	)	PUNCT
ejpam-5572	231	12	)	)	PUNCT
ejpam-5572	232	1	=	=	SYM
ejpam-5572	232	2	ν3((1	ν3((1	PROPN
ejpam-5572	232	3	,	,	PUNCT
ejpam-5572	232	4	1	1	NUM
ejpam-5572	232	5	)	)	PUNCT
ejpam-5572	232	6	)	)	PUNCT
ejpam-5572	233	1	=	=	SYM
ejpam-5572	233	2	0.314	0.314	NUM
ejpam-5572	233	3	≤	≤	ADJ
ejpam-5572	233	4	max{ν3((1	max{ν3((1	PROPN
ejpam-5572	233	5	,	,	PUNCT
ejpam-5572	233	6	0	0	NUM
ejpam-5572	233	7	)	)	PUNCT
ejpam-5572	233	8	)	)	PUNCT
ejpam-5572	233	9	,	,	PUNCT
ejpam-5572	233	10	ν3((0	ν3((0	ADJ
ejpam-5572	233	11	,	,	PUNCT
ejpam-5572	233	12	1	1	NUM
ejpam-5572	233	13	)	)	PUNCT
ejpam-5572	233	14	)	)	PUNCT
ejpam-5572	233	15	}	}	PUNCT
ejpam-5572	234	1	=	=	PUNCT
ejpam-5572	234	2	0.314	0.314	NUM
ejpam-5572	234	3	.	.	PUNCT
ejpam-5572	235	1	then	then	ADV
ejpam-5572	235	2	the	the	DET
ejpam-5572	235	3	property	property	NOUN
ejpam-5572	235	4	satisfied	satisfied	ADJ
ejpam-5572	235	5	at	at	ADP
ejpam-5572	235	6	ℓ1	ℓ1	NOUN
ejpam-5572	235	7	=	=	SYM
ejpam-5572	235	8	(	(	PUNCT
ejpam-5572	235	9	1	1	NUM
ejpam-5572	235	10	,	,	PUNCT
ejpam-5572	235	11	0	0	NUM
ejpam-5572	235	12	)	)	PUNCT
ejpam-5572	235	13	,	,	PUNCT
ejpam-5572	235	14	ℓ2	ℓ2	NOUN
ejpam-5572	235	15	=	=	SYM
ejpam-5572	235	16	(	(	PUNCT
ejpam-5572	235	17	0	0	NUM
ejpam-5572	235	18	,	,	PUNCT
ejpam-5572	235	19	1	1	NUM
ejpam-5572	235	20	)	)	PUNCT
ejpam-5572	235	21	,	,	PUNCT
ejpam-5572	235	22	and	and	CCONJ
ejpam-5572	235	23	one	one	PRON
ejpam-5572	235	24	can	can	AUX
ejpam-5572	235	25	go	go	VERB
ejpam-5572	235	26	through	through	ADP
ejpam-5572	235	27	all	all	DET
ejpam-5572	235	28	ℓi	ℓi	NOUN
ejpam-5572	235	29	and	and	CCONJ
ejpam-5572	235	30	check	check	VERB
ejpam-5572	235	31	conditions	condition	NOUN
ejpam-5572	235	32	.	.	PUNCT
ejpam-5572	236	1	in	in	ADP
ejpam-5572	236	2	addition	addition	NOUN
ejpam-5572	236	3	,	,	PUNCT
ejpam-5572	236	4	for	for	ADP
ejpam-5572	236	5	conditions	condition	NOUN
ejpam-5572	236	6	(	(	PUNCT
ejpam-5572	236	7	2a	2a	NUM
ejpam-5572	236	8	)	)	PUNCT
ejpam-5572	236	9	and	and	CCONJ
ejpam-5572	236	10	(	(	PUNCT
ejpam-5572	236	11	2b	2b	NOUN
ejpam-5572	236	12	)	)	PUNCT
ejpam-5572	236	13	in	in	ADP
ejpam-5572	236	14	the	the	DET
ejpam-5572	236	15	definition	definition	NOUN
ejpam-5572	236	16	9	9	NUM
ejpam-5572	236	17	,	,	PUNCT
ejpam-5572	236	18	they	they	PRON
ejpam-5572	236	19	satisfied	satisfy	VERB
ejpam-5572	236	20	too	too	ADV
ejpam-5572	236	21	,	,	PUNCT
ejpam-5572	236	22	since	since	SCONJ
ejpam-5572	236	23	ℓ	ℓ	NOUN
ejpam-5572	236	24	=	=	SYM
ejpam-5572	237	1	ℓ−1	ℓ−1	PROPN
ejpam-5572	237	2	for	for	ADP
ejpam-5572	237	3	any	any	DET
ejpam-5572	237	4	ℓ	ℓ	PROPN
ejpam-5572	237	5	∈	∈	PROPN
ejpam-5572	237	6	d2	d2	PROPN
ejpam-5572	237	7	.	.	PUNCT
ejpam-5572	238	1	note	note	VERB
ejpam-5572	238	2	that	that	SCONJ
ejpam-5572	238	3	,	,	PUNCT
ejpam-5572	238	4	it	it	PRON
ejpam-5572	238	5	is	be	AUX
ejpam-5572	238	6	easy	easy	ADJ
ejpam-5572	238	7	to	to	PART
ejpam-5572	238	8	see	see	VERB
ejpam-5572	238	9	that	that	PRON
ejpam-5572	238	10	ϕ	ϕ	NOUN
ejpam-5572	238	11	is	be	AUX
ejpam-5572	238	12	satisfied	satisfied	ADJ
ejpam-5572	238	13	the	the	DET
ejpam-5572	238	14	previous	previous	ADJ
ejpam-5572	238	15	proposition	proposition	NOUN
ejpam-5572	238	16	.	.	PUNCT
ejpam-5572	239	1	the	the	DET
ejpam-5572	239	2	following	follow	VERB
ejpam-5572	239	3	theorem	theorem	NOUN
ejpam-5572	239	4	prove	prove	VERB
ejpam-5572	239	5	that	that	SCONJ
ejpam-5572	239	6	any	any	DET
ejpam-5572	239	7	cpfsg	cpfsg	NOUN
ejpam-5572	239	8	is	be	AUX
ejpam-5572	239	9	a	a	DET
ejpam-5572	239	10	cffsg	cffsg	ADJ
ejpam-5572	239	11	.	.	PUNCT
ejpam-5572	240	1	theorem	theorem	NOUN
ejpam-5572	240	2	1	1	NUM
ejpam-5572	240	3	.	.	PUNCT
ejpam-5572	241	1	if	if	SCONJ
ejpam-5572	241	2	ϕ	ϕ	NOUN
ejpam-5572	241	3	is	be	AUX
ejpam-5572	241	4	a	a	DET
ejpam-5572	241	5	cpfsg	cpfsg	NOUN
ejpam-5572	241	6	of	of	ADP
ejpam-5572	241	7	the	the	DET
ejpam-5572	241	8	group	group	NOUN
ejpam-5572	241	9	(	(	PUNCT
ejpam-5572	241	10	x	x	X
ejpam-5572	241	11	,	,	PUNCT
ejpam-5572	241	12	∗	∗	NOUN
ejpam-5572	241	13	)	)	PUNCT
ejpam-5572	241	14	,	,	PUNCT
ejpam-5572	241	15	then	then	ADV
ejpam-5572	241	16	ϕ	ϕ	PROPN
ejpam-5572	241	17	is	be	AUX
ejpam-5572	241	18	a	a	DET
ejpam-5572	241	19	cffsg	cffsg	NOUN
ejpam-5572	241	20	of	of	ADP
ejpam-5572	241	21	the	the	DET
ejpam-5572	241	22	group	group	NOUN
ejpam-5572	241	23	(	(	PUNCT
ejpam-5572	241	24	x	x	X
ejpam-5572	241	25	,	,	PUNCT
ejpam-5572	241	26	∗	∗	NOUN
ejpam-5572	241	27	)	)	PUNCT
ejpam-5572	241	28	.	.	PUNCT
ejpam-5572	242	1	proof	proof	NOUN
ejpam-5572	242	2	.	.	PUNCT
ejpam-5572	243	1	at	at	ADP
ejpam-5572	243	2	first	first	ADV
ejpam-5572	243	3	,	,	PUNCT
ejpam-5572	243	4	to	to	PART
ejpam-5572	243	5	show	show	VERB
ejpam-5572	243	6	that	that	SCONJ
ejpam-5572	243	7	p3(x	p3(x	PROPN
ejpam-5572	243	8	∗	∗	NOUN
ejpam-5572	243	9	y)e2πiω	y)e2πiω	PROPN
ejpam-5572	243	10	3(x	3(x	NUM
ejpam-5572	243	11	∗	∗	NOUN
ejpam-5572	243	12	y	y	PROPN
ejpam-5572	243	13	)	)	PUNCT
ejpam-5572	243	14	≥	≥	NOUN
ejpam-5572	243	15	p3(x)e2πiω	p3(x)e2πiω	PROPN
ejpam-5572	243	16	3(x	3(x	NUM
ejpam-5572	243	17	)	)	PUNCT
ejpam-5572	243	18	∧	∧	PROPN
ejpam-5572	243	19	p3(y)e2πiω	p3(y)e2πiω	NOUN
ejpam-5572	243	20	3(y	3(y	NUM
ejpam-5572	243	21	)	)	PUNCT
ejpam-5572	243	22	and	and	CCONJ
ejpam-5572	243	23	q3(x	q3(x	PROPN
ejpam-5572	243	24	∗	∗	NOUN
ejpam-5572	243	25	y)e2πiν	y)e2πiν	PROPN
ejpam-5572	243	26	3(x	3(x	NUM
ejpam-5572	243	27	∗	∗	X
ejpam-5572	243	28	y	y	NOUN
ejpam-5572	243	29	)	)	PUNCT
ejpam-5572	243	30	≤	≤	NUM
ejpam-5572	243	31	q3(x)e2πiν	q3(x)e2πiν	ADP
ejpam-5572	243	32	3(x	3(x	NUM
ejpam-5572	243	33	)	)	PUNCT
ejpam-5572	243	34	∨	∨	NUM
ejpam-5572	243	35	q3(y)e2πiν	q3(y)e2πiν	NOUN
ejpam-5572	243	36	3(y	3(y	NUM
ejpam-5572	243	37	)	)	PUNCT
ejpam-5572	243	38	.	.	PUNCT
ejpam-5572	244	1	we	we	PRON
ejpam-5572	244	2	know	know	VERB
ejpam-5572	244	3	that	that	SCONJ
ejpam-5572	244	4	ϕ	ϕ	NOUN
ejpam-5572	244	5	is	be	AUX
ejpam-5572	244	6	a	a	DET
ejpam-5572	244	7	cpfsg	cpfsg	NOUN
ejpam-5572	244	8	,	,	PUNCT
ejpam-5572	244	9	then	then	ADV
ejpam-5572	244	10	p2(x	p2(x	PRON
ejpam-5572	244	11	∗	∗	NOUN
ejpam-5572	244	12	y)e2πiω	y)e2πiω	PROPN
ejpam-5572	244	13	2(x	2(x	NUM
ejpam-5572	244	14	∗	∗	PROPN
ejpam-5572	244	15	y	y	PROPN
ejpam-5572	244	16	)	)	PUNCT
ejpam-5572	244	17	≥	≥	NOUN
ejpam-5572	244	18	p2(x)e2πiω	p2(x)e2πiω	PROPN
ejpam-5572	244	19	2(x	2(x	NUM
ejpam-5572	244	20	)	)	PUNCT
ejpam-5572	245	1	∧	∧	NOUN
ejpam-5572	245	2	p2(y)e2πiω	p2(y)e2πiω	NOUN
ejpam-5572	245	3	2(y	2(y	NUM
ejpam-5572	245	4	)	)	PUNCT
ejpam-5572	245	5	and	and	CCONJ
ejpam-5572	245	6	q2(x	q2(x	PROPN
ejpam-5572	245	7	∗	∗	NOUN
ejpam-5572	245	8	y)e2πiν	y)e2πiν	PROPN
ejpam-5572	245	9	2(x	2(x	NUM
ejpam-5572	245	10	∗	∗	PROPN
ejpam-5572	245	11	y	y	NOUN
ejpam-5572	245	12	)	)	PUNCT
ejpam-5572	245	13	≤	≤	NUM
ejpam-5572	245	14	q2(x)e2πiν	q2(x)e2πiν	PROPN
ejpam-5572	245	15	2(x	2(x	NUM
ejpam-5572	245	16	)	)	PUNCT
ejpam-5572	245	17	∨	∨	NUM
ejpam-5572	245	18	q2(y)e2πiν	q2(y)e2πiν	NOUN
ejpam-5572	245	19	2(y	2(y	NUM
ejpam-5572	245	20	)	)	PUNCT
ejpam-5572	245	21	,	,	PUNCT
ejpam-5572	245	22	where	where	SCONJ
ejpam-5572	245	23	p2	p2	PROPN
ejpam-5572	245	24	+	+	CCONJ
ejpam-5572	245	25	q2	q2	NOUN
ejpam-5572	245	26	≤	≤	NUM
ejpam-5572	245	27	1	1	NUM
ejpam-5572	245	28	and	and	CCONJ
ejpam-5572	245	29	ω2	ω2	ADJ
ejpam-5572	245	30	+	+	CCONJ
ejpam-5572	245	31	ν2	ν2	ADV
ejpam-5572	245	32	≤	≤	NUM
ejpam-5572	245	33	1	1	NUM
ejpam-5572	245	34	.	.	PUNCT
ejpam-5572	246	1	then	then	ADV
ejpam-5572	246	2	,	,	PUNCT
ejpam-5572	246	3	we	we	PRON
ejpam-5572	246	4	have	have	VERB
ejpam-5572	246	5	four	four	NUM
ejpam-5572	246	6	cases	case	NOUN
ejpam-5572	246	7	to	to	PART
ejpam-5572	246	8	consider	consider	VERB
ejpam-5572	246	9	:	:	PUNCT
ejpam-5572	246	10	a	a	X
ejpam-5572	246	11	)	)	PUNCT
ejpam-5572	246	12	let	let	VERB
ejpam-5572	246	13	p2(x)e2πiω	p2(x)e2πiω	PROPN
ejpam-5572	246	14	2(x	2(x	NUM
ejpam-5572	246	15	)	)	PUNCT
ejpam-5572	246	16	≥	≥	NOUN
ejpam-5572	246	17	p2(y)e2πiω	p2(y)e2πiω	NOUN
ejpam-5572	246	18	2(y	2(y	NUM
ejpam-5572	246	19	)	)	PUNCT
ejpam-5572	246	20	and	and	CCONJ
ejpam-5572	246	21	q2(x)e2πiν	q2(x)e2πiν	PROPN
ejpam-5572	246	22	2(x	2(x	NUM
ejpam-5572	246	23	)	)	PUNCT
ejpam-5572	246	24	≥	≥	NOUN
ejpam-5572	246	25	q2(y)e2πiν	q2(y)e2πiν	NOUN
ejpam-5572	246	26	2(y	2(y	NUM
ejpam-5572	246	27	)	)	PUNCT
ejpam-5572	246	28	,	,	PUNCT
ejpam-5572	246	29	then	then	ADV
ejpam-5572	246	30	p2(x	p2(x	PRON
ejpam-5572	246	31	∗	∗	NOUN
ejpam-5572	246	32	y)e2πiω	y)e2πiω	PROPN
ejpam-5572	246	33	2(x	2(x	NUM
ejpam-5572	246	34	∗	∗	PROPN
ejpam-5572	246	35	y	y	PROPN
ejpam-5572	246	36	)	)	PUNCT
ejpam-5572	246	37	≥	≥	NOUN
ejpam-5572	246	38	p2(y)e2πiω	p2(y)e2πiω	NOUN
ejpam-5572	246	39	2(y	2(y	NUM
ejpam-5572	246	40	)	)	PUNCT
ejpam-5572	246	41	.	.	PUNCT
ejpam-5572	247	1	now	now	ADV
ejpam-5572	247	2	consider	consider	VERB
ejpam-5572	247	3	p3(x	p3(x	PRON
ejpam-5572	247	4	∗	∗	NOUN
ejpam-5572	247	5	y)e2πiω	y)e2πiω	PROPN
ejpam-5572	247	6	3(x	3(x	NUM
ejpam-5572	247	7	∗	∗	NOUN
ejpam-5572	247	8	y	y	PROPN
ejpam-5572	247	9	)	)	PUNCT
ejpam-5572	247	10	≥	≥	PROPN
ejpam-5572	247	11	p3(y)e2πiω	p3(y)e2πiω	NOUN
ejpam-5572	247	12	3(y	3(y	NUM
ejpam-5572	247	13	)	)	PUNCT
ejpam-5572	247	14	=	=	SYM
ejpam-5572	247	15	p3(x)e2πiω	p3(x)e2πiω	PROPN
ejpam-5572	247	16	3(x	3(x	NUM
ejpam-5572	247	17	)	)	PUNCT
ejpam-5572	247	18	∧	∧	PROPN
ejpam-5572	247	19	p3(y)e2πiω	p3(y)e2πiω	NOUN
ejpam-5572	247	20	3(y	3(y	NUM
ejpam-5572	247	21	)	)	PUNCT
ejpam-5572	247	22	.	.	PUNCT
ejpam-5572	248	1	moreover	moreover	ADV
ejpam-5572	248	2	,	,	PUNCT
ejpam-5572	248	3	q2(x	q2(x	PROPN
ejpam-5572	248	4	∗	∗	NOUN
ejpam-5572	248	5	y)e2πiν	y)e2πiν	PROPN
ejpam-5572	248	6	2(x	2(x	NUM
ejpam-5572	248	7	∗	∗	PROPN
ejpam-5572	248	8	y	y	NOUN
ejpam-5572	248	9	)	)	PUNCT
ejpam-5572	248	10	≤	≤	NUM
ejpam-5572	248	11	q2(x)e2πiν	q2(x)e2πiν	PROPN
ejpam-5572	248	12	2(x	2(x	NUM
ejpam-5572	248	13	)	)	PUNCT
ejpam-5572	248	14	.	.	PUNCT
ejpam-5572	249	1	now	now	ADV
ejpam-5572	249	2	consider	consider	VERB
ejpam-5572	249	3	q3(x	q3(x	PRON
ejpam-5572	249	4	∗	∗	NOUN
ejpam-5572	249	5	y)e2πiν	y)e2πiν	PROPN
ejpam-5572	249	6	3(x	3(x	NUM
ejpam-5572	249	7	∗	∗	X
ejpam-5572	249	8	y	y	NOUN
ejpam-5572	249	9	)	)	PUNCT
ejpam-5572	249	10	≤	≤	NUM
ejpam-5572	249	11	q3(x)e2πiν	q3(x)e2πiν	ADP
ejpam-5572	249	12	3(x	3(x	NUM
ejpam-5572	249	13	)	)	PUNCT
ejpam-5572	249	14	=	=	SYM
ejpam-5572	249	15	q3(x)e2πiν	q3(x)e2πiν	ADP
ejpam-5572	249	16	3(x	3(x	NUM
ejpam-5572	249	17	)	)	PUNCT
ejpam-5572	249	18	∨q3(y)e2πiωq	∨q3(y)e2πiωq	NOUN
ejpam-5572	249	19	3(y	3(y	NUM
ejpam-5572	249	20	)	)	PUNCT
ejpam-5572	249	21	.	.	PUNCT
ejpam-5572	250	1	e.a	e.a	PROPN
ejpam-5572	250	2	.	.	PROPN
ejpam-5572	250	3	abuhijleh	abuhijleh	PROPN
ejpam-5572	250	4	,	,	PUNCT
ejpam-5572	250	5	a.	a.	PROPN
ejpam-5572	250	6	alkouri	alkouri	PROPN
ejpam-5572	250	7	/	/	PROPN
ejpam-5572	250	8	eur	eur	PROPN
ejpam-5572	250	9	.	.	PUNCT
ejpam-5572	251	1	j.	j.	PROPN
ejpam-5572	251	2	pure	pure	PROPN
ejpam-5572	251	3	appl	appl	PROPN
ejpam-5572	251	4	.	.	PROPN
ejpam-5572	251	5	math	math	PROPN
ejpam-5572	251	6	,	,	PUNCT
ejpam-5572	251	7	18	18	NUM
ejpam-5572	251	8	(	(	PUNCT
ejpam-5572	251	9	1	1	NUM
ejpam-5572	251	10	)	)	PUNCT
ejpam-5572	251	11	(	(	PUNCT
ejpam-5572	251	12	2025	2025	NUM
ejpam-5572	251	13	)	)	PUNCT
ejpam-5572	251	14	,	,	PUNCT
ejpam-5572	251	15	5572	5572	NUM
ejpam-5572	251	16	8	8	NUM
ejpam-5572	251	17	of	of	ADP
ejpam-5572	251	18	19	19	NUM
ejpam-5572	251	19	b	b	NOUN
ejpam-5572	251	20	)	)	PUNCT
ejpam-5572	251	21	let	let	VERB
ejpam-5572	251	22	p2(x)e2πiω	p2(x)e2πiω	PROPN
ejpam-5572	251	23	2(x	2(x	NUM
ejpam-5572	251	24	)	)	PUNCT
ejpam-5572	251	25	≤	≤	NOUN
ejpam-5572	251	26	p2(y)e2πiω	p2(y)e2πiω	NOUN
ejpam-5572	251	27	2(y	2(y	NUM
ejpam-5572	251	28	)	)	PUNCT
ejpam-5572	251	29	and	and	CCONJ
ejpam-5572	251	30	q2(x)e2πiν	q2(x)e2πiν	PROPN
ejpam-5572	251	31	2(x	2(x	NUM
ejpam-5572	251	32	)	)	PUNCT
ejpam-5572	251	33	≤	≤	NUM
ejpam-5572	251	34	q2(y)e2πiν	q2(y)e2πiν	NOUN
ejpam-5572	251	35	2(y	2(y	NUM
ejpam-5572	251	36	)	)	PUNCT
ejpam-5572	251	37	,	,	PUNCT
ejpam-5572	251	38	then	then	ADV
ejpam-5572	251	39	with	with	ADP
ejpam-5572	251	40	same	same	ADJ
ejpam-5572	251	41	argument	argument	NOUN
ejpam-5572	251	42	of	of	ADP
ejpam-5572	251	43	case	case	NOUN
ejpam-5572	251	44	a	a	X
ejpam-5572	251	45	,	,	PUNCT
ejpam-5572	251	46	we	we	PRON
ejpam-5572	251	47	get	get	VERB
ejpam-5572	251	48	the	the	DET
ejpam-5572	251	49	result	result	NOUN
ejpam-5572	251	50	.	.	PUNCT
ejpam-5572	252	1	c	c	X
ejpam-5572	252	2	)	)	PUNCT
ejpam-5572	252	3	let	let	VERB
ejpam-5572	252	4	p2(x)e2πiω	p2(x)e2πiω	PROPN
ejpam-5572	252	5	2(x	2(x	NUM
ejpam-5572	252	6	)	)	PUNCT
ejpam-5572	252	7	≤	≤	NOUN
ejpam-5572	252	8	p2(y)e2πiω	p2(y)e2πiω	NOUN
ejpam-5572	252	9	2(y	2(y	NUM
ejpam-5572	252	10	)	)	PUNCT
ejpam-5572	252	11	and	and	CCONJ
ejpam-5572	252	12	q2(x)e2πiν	q2(x)e2πiν	PROPN
ejpam-5572	252	13	2(x	2(x	NUM
ejpam-5572	252	14	)	)	PUNCT
ejpam-5572	252	15	≥	≥	NOUN
ejpam-5572	252	16	q2(y)e2πiν	q2(y)e2πiν	NOUN
ejpam-5572	252	17	2(y	2(y	NUM
ejpam-5572	252	18	)	)	PUNCT
ejpam-5572	252	19	,	,	PUNCT
ejpam-5572	252	20	then	then	ADV
ejpam-5572	252	21	p2(x	p2(x	PRON
ejpam-5572	252	22	∗	∗	NOUN
ejpam-5572	252	23	y)e2πiω	y)e2πiω	PROPN
ejpam-5572	252	24	2(x	2(x	NUM
ejpam-5572	252	25	∗	∗	PROPN
ejpam-5572	252	26	y	y	PROPN
ejpam-5572	252	27	)	)	PUNCT
ejpam-5572	252	28	≥	≥	NOUN
ejpam-5572	252	29	p2(x)e2πiω	p2(x)e2πiω	PROPN
ejpam-5572	252	30	2(x	2(x	NUM
ejpam-5572	252	31	)	)	PUNCT
ejpam-5572	252	32	.	.	PUNCT
ejpam-5572	253	1	now	now	ADV
ejpam-5572	253	2	consider	consider	VERB
ejpam-5572	253	3	p3(x	p3(x	PRON
ejpam-5572	253	4	∗	∗	NOUN
ejpam-5572	253	5	y)e2πiω	y)e2πiω	PROPN
ejpam-5572	253	6	3(x	3(x	NUM
ejpam-5572	253	7	∗	∗	NOUN
ejpam-5572	253	8	y	y	PROPN
ejpam-5572	253	9	)	)	PUNCT
ejpam-5572	253	10	≥	≥	NOUN
ejpam-5572	253	11	p3(x)e2πiω	p3(x)e2πiω	NOUN
ejpam-5572	253	12	3(x	3(x	NUM
ejpam-5572	253	13	)	)	PUNCT
ejpam-5572	253	14	=	=	X
ejpam-5572	253	15	p3(x)e2πiω	p3(x)e2πiω	PROPN
ejpam-5572	253	16	3(x	3(x	NUM
ejpam-5572	253	17	)	)	PUNCT
ejpam-5572	253	18	∧	∧	PROPN
ejpam-5572	253	19	p3(y)e2πiω	p3(y)e2πiω	NOUN
ejpam-5572	253	20	3(y	3(y	NUM
ejpam-5572	253	21	)	)	PUNCT
ejpam-5572	253	22	.	.	PUNCT
ejpam-5572	254	1	moreover	moreover	ADV
ejpam-5572	254	2	,	,	PUNCT
ejpam-5572	254	3	q2(x	q2(x	PROPN
ejpam-5572	254	4	∗	∗	NOUN
ejpam-5572	254	5	y)e2πiν	y)e2πiν	PROPN
ejpam-5572	254	6	2(x	2(x	NUM
ejpam-5572	254	7	∗	∗	PROPN
ejpam-5572	254	8	y	y	NOUN
ejpam-5572	254	9	)	)	PUNCT
ejpam-5572	254	10	≤	≤	NUM
ejpam-5572	254	11	q2(x)e2πiν	q2(x)e2πiν	PROPN
ejpam-5572	254	12	2(x	2(x	NUM
ejpam-5572	254	13	)	)	PUNCT
ejpam-5572	254	14	.	.	PUNCT
ejpam-5572	255	1	now	now	ADV
ejpam-5572	255	2	consider	consider	VERB
ejpam-5572	255	3	q3(x	q3(x	PRON
ejpam-5572	255	4	∗	∗	NOUN
ejpam-5572	255	5	y)e2πiν	y)e2πiν	PROPN
ejpam-5572	255	6	3(x	3(x	NUM
ejpam-5572	255	7	∗	∗	X
ejpam-5572	255	8	y	y	NOUN
ejpam-5572	255	9	)	)	PUNCT
ejpam-5572	255	10	≤	≤	NUM
ejpam-5572	255	11	q3(x)e2πiν	q3(x)e2πiν	ADP
ejpam-5572	255	12	3(x	3(x	NUM
ejpam-5572	255	13	)	)	PUNCT
ejpam-5572	255	14	=	=	SYM
ejpam-5572	255	15	q3(x)e2πiν	q3(x)e2πiν	ADP
ejpam-5572	255	16	3(x	3(x	NUM
ejpam-5572	255	17	)	)	PUNCT
ejpam-5572	255	18	∨q3(y)e2πiν3(y	∨q3(y)e2πiν3(y	PROPN
ejpam-5572	255	19	)	)	PUNCT
ejpam-5572	255	20	.	.	PUNCT
ejpam-5572	256	1	d	d	X
ejpam-5572	256	2	)	)	PUNCT
ejpam-5572	256	3	let	let	VERB
ejpam-5572	256	4	p2(x)e2πiω	p2(x)e2πiω	PROPN
ejpam-5572	256	5	2(x	2(x	NUM
ejpam-5572	256	6	)	)	PUNCT
ejpam-5572	256	7	≥	≥	NOUN
ejpam-5572	256	8	p2(y)e2πiω	p2(y)e2πiω	NOUN
ejpam-5572	256	9	2(y	2(y	NUM
ejpam-5572	256	10	)	)	PUNCT
ejpam-5572	256	11	and	and	CCONJ
ejpam-5572	256	12	q2(x)e2πiν	q2(x)e2πiν	PROPN
ejpam-5572	256	13	2(x	2(x	NUM
ejpam-5572	256	14	)	)	PUNCT
ejpam-5572	256	15	≤	≤	NUM
ejpam-5572	256	16	q2(y)e2πiν	q2(y)e2πiν	NOUN
ejpam-5572	256	17	2(y	2(y	NUM
ejpam-5572	256	18	)	)	PUNCT
ejpam-5572	256	19	,	,	PUNCT
ejpam-5572	256	20	then	then	ADV
ejpam-5572	256	21	with	with	ADP
ejpam-5572	256	22	same	same	ADJ
ejpam-5572	256	23	argument	argument	NOUN
ejpam-5572	256	24	of	of	ADP
ejpam-5572	256	25	case	case	NOUN
ejpam-5572	256	26	c	c	X
ejpam-5572	256	27	,	,	PUNCT
ejpam-5572	256	28	we	we	PRON
ejpam-5572	256	29	get	get	VERB
ejpam-5572	256	30	the	the	DET
ejpam-5572	256	31	result	result	NOUN
ejpam-5572	256	32	.	.	PUNCT
ejpam-5572	257	1	secondly	secondly	ADV
ejpam-5572	257	2	,	,	PUNCT
ejpam-5572	257	3	since	since	SCONJ
ejpam-5572	257	4	p2(x−1)e2πiω	p2(x−1)e2πiω	NOUN
ejpam-5572	257	5	2(x−1	2(x−1	NOUN
ejpam-5572	257	6	)	)	PUNCT
ejpam-5572	257	7	≥	≥	NOUN
ejpam-5572	257	8	p2(x)e2πiω	p2(x)e2πiω	PROPN
ejpam-5572	257	9	2(x	2(x	NUM
ejpam-5572	257	10	)	)	PUNCT
ejpam-5572	257	11	and	and	CCONJ
ejpam-5572	257	12	q2(x−1)e2πiν	q2(x−1)e2πiν	NOUN
ejpam-5572	257	13	2(x−1	2(x−1	NOUN
ejpam-5572	257	14	)	)	PUNCT
ejpam-5572	257	15	≤	≤	NUM
ejpam-5572	257	16	q2(x)e2πiν	q2(x)e2πiν	PROPN
ejpam-5572	257	17	2(x	2(x	NUM
ejpam-5572	257	18	)	)	PUNCT
ejpam-5572	257	19	,	,	PUNCT
ejpam-5572	257	20	then	then	ADV
ejpam-5572	257	21	p3(x−1)e2πiω	p3(x−1)e2πiω	PROPN
ejpam-5572	257	22	3(x−1	3(x−1	NUM
ejpam-5572	257	23	)	)	PUNCT
ejpam-5572	257	24	≥	≥	PROPN
ejpam-5572	257	25	p3(x)e2πiω	p3(x)e2πiω	PROPN
ejpam-5572	257	26	3(x	3(x	NUM
ejpam-5572	257	27	)	)	PUNCT
ejpam-5572	257	28	and	and	CCONJ
ejpam-5572	257	29	q3(x−1)e2πiν	q3(x−1)e2πiν	PROPN
ejpam-5572	257	30	3(x−1	3(x−1	NUM
ejpam-5572	257	31	)	)	PUNCT
ejpam-5572	257	32	≤	≤	NUM
ejpam-5572	257	33	q3(x)e2πiν	q3(x)e2πiν	PROPN
ejpam-5572	257	34	3(x	3(x	NUM
ejpam-5572	257	35	)	)	PUNCT
ejpam-5572	257	36	too	too	ADV
ejpam-5572	257	37	.	.	PUNCT
ejpam-5572	258	1	the	the	DET
ejpam-5572	258	2	converse	converse	NOUN
ejpam-5572	258	3	of	of	ADP
ejpam-5572	258	4	theorem	theorem	NOUN
ejpam-5572	258	5	1	1	NUM
ejpam-5572	258	6	is	be	AUX
ejpam-5572	258	7	not	not	PART
ejpam-5572	258	8	always	always	ADV
ejpam-5572	258	9	true	true	ADJ
ejpam-5572	258	10	,	,	PUNCT
ejpam-5572	258	11	please	please	INTJ
ejpam-5572	258	12	see	see	VERB
ejpam-5572	258	13	the	the	DET
ejpam-5572	258	14	following	follow	VERB
ejpam-5572	258	15	example	example	NOUN
ejpam-5572	258	16	.	.	PUNCT
ejpam-5572	259	1	example	example	NOUN
ejpam-5572	260	1	2	2	NUM
ejpam-5572	260	2	.	.	X
ejpam-5572	260	3	for	for	ADP
ejpam-5572	260	4	the	the	DET
ejpam-5572	260	5	set	set	NOUN
ejpam-5572	260	6	x	x	X
ejpam-5572	260	7	=	=	PUNCT
ejpam-5572	260	8	{	{	PUNCT
ejpam-5572	260	9	1,−1	1,−1	PROPN
ejpam-5572	260	10	,	,	PUNCT
ejpam-5572	260	11	i,−i	i,−i	NOUN
ejpam-5572	260	12	}	}	PUNCT
ejpam-5572	260	13	,	,	PUNCT
ejpam-5572	260	14	define	define	VERB
ejpam-5572	260	15	a	a	DET
ejpam-5572	260	16	group	group	NOUN
ejpam-5572	260	17	(	(	PUNCT
ejpam-5572	260	18	x	x	X
ejpam-5572	260	19	,	,	PUNCT
ejpam-5572	260	20	.	.	PUNCT
ejpam-5572	260	21	)	)	PUNCT
ejpam-5572	261	1	,	,	PUNCT
ejpam-5572	261	2	where	where	SCONJ
ejpam-5572	261	3	′.′	′.′	NOUN
ejpam-5572	261	4	is	be	AUX
ejpam-5572	261	5	the	the	DET
ejpam-5572	261	6	known	know	VERB
ejpam-5572	261	7	multiplication	multiplication	NOUN
ejpam-5572	261	8	.	.	PUNCT
ejpam-5572	262	1	also	also	ADV
ejpam-5572	262	2	define	define	VERB
ejpam-5572	262	3	ϕ	ϕ	NOUN
ejpam-5572	262	4	=	=	SYM
ejpam-5572	262	5	(	(	PUNCT
ejpam-5572	262	6	k	k	X
ejpam-5572	262	7	,	,	PUNCT
ejpam-5572	262	8	l	l	NOUN
ejpam-5572	262	9	)	)	PUNCT
ejpam-5572	262	10	be	be	AUX
ejpam-5572	262	11	a	a	DET
ejpam-5572	262	12	cffs	cff	NOUN
ejpam-5572	262	13	on	on	ADP
ejpam-5572	262	14	x	x	NOUN
ejpam-5572	262	15	,	,	PUNCT
ejpam-5572	262	16	where	where	SCONJ
ejpam-5572	262	17	:	:	PUNCT
ejpam-5572	262	18	ϕ(1	ϕ(1	X
ejpam-5572	262	19	)	)	PUNCT
ejpam-5572	263	1	=	=	PRON
ejpam-5572	263	2	(	(	PUNCT
ejpam-5572	263	3	0.8e2πi(0.75	0.8e2πi(0.75	NOUN
ejpam-5572	263	4	)	)	PUNCT
ejpam-5572	263	5	,	,	PUNCT
ejpam-5572	263	6	0.3e2πi(0.3	0.3e2πi(0.3	NUM
ejpam-5572	263	7	)	)	PUNCT
ejpam-5572	263	8	)	)	PUNCT
ejpam-5572	263	9	,	,	PUNCT
ejpam-5572	263	10	ϕ(−1	ϕ(−1	PROPN
ejpam-5572	263	11	)	)	PUNCT
ejpam-5572	263	12	=	=	PUNCT
ejpam-5572	263	13	(	(	PUNCT
ejpam-5572	263	14	0.8e2πi(0.5	0.8e2πi(0.5	NUM
ejpam-5572	263	15	)	)	PUNCT
ejpam-5572	263	16	,	,	PUNCT
ejpam-5572	263	17	0.65e2πi(0.45	0.65e2πi(0.45	NOUN
ejpam-5572	263	18	)	)	PUNCT
ejpam-5572	263	19	)	)	PUNCT
ejpam-5572	263	20	,	,	PUNCT
ejpam-5572	263	21	ϕ(i	ϕ(i	PROPN
ejpam-5572	263	22	)	)	PUNCT
ejpam-5572	264	1	=	=	PRON
ejpam-5572	264	2	(	(	PUNCT
ejpam-5572	264	3	0.7e2πi(0.45	0.7e2πi(0.45	NOUN
ejpam-5572	264	4	)	)	PUNCT
ejpam-5572	264	5	,	,	PUNCT
ejpam-5572	264	6	0.9e2πi(0.6	0.9e2πi(0.6	NUM
ejpam-5572	264	7	)	)	PUNCT
ejpam-5572	264	8	=	=	PUNCT
ejpam-5572	264	9	ϕ(−i	ϕ(−i	NOUN
ejpam-5572	264	10	)	)	PUNCT
ejpam-5572	264	11	.	.	PUNCT
ejpam-5572	265	1	now	now	ADV
ejpam-5572	265	2	,	,	PUNCT
ejpam-5572	265	3	it	it	PRON
ejpam-5572	265	4	is	be	AUX
ejpam-5572	265	5	easy	easy	ADJ
ejpam-5572	265	6	to	to	PART
ejpam-5572	265	7	check	check	VERB
ejpam-5572	265	8	that	that	PRON
ejpam-5572	265	9	for	for	ADP
ejpam-5572	265	10	all	all	PRON
ejpam-5572	265	11	x	x	SYM
ejpam-5572	265	12	∈	∈	PROPN
ejpam-5572	265	13	x	x	NOUN
ejpam-5572	265	14	,	,	PUNCT
ejpam-5572	265	15	that	that	SCONJ
ejpam-5572	265	16	ϕ	ϕ	NOUN
ejpam-5572	265	17	is	be	AUX
ejpam-5572	265	18	cffs	cff	NOUN
ejpam-5572	265	19	,	,	PUNCT
ejpam-5572	265	20	c.f	c.f	PROPN
ejpam-5572	265	21	.	.	PROPN
ejpam-5572	265	22	definition	definition	NOUN
ejpam-5572	265	23	4	4	NUM
ejpam-5572	265	24	.	.	PUNCT
ejpam-5572	266	1	but	but	CCONJ
ejpam-5572	266	2	,	,	PUNCT
ejpam-5572	266	3	ϕ	ϕ	NOUN
ejpam-5572	266	4	is	be	AUX
ejpam-5572	266	5	not	not	PART
ejpam-5572	266	6	cpfs	cpfs	ADJ
ejpam-5572	266	7	,	,	PUNCT
ejpam-5572	266	8	for	for	ADP
ejpam-5572	266	9	example	example	NOUN
ejpam-5572	266	10	at	at	ADP
ejpam-5572	266	11	x	x	X
ejpam-5572	266	12	=	=	VERB
ejpam-5572	266	13	−1	−1	NOUN
ejpam-5572	266	14	we	we	PRON
ejpam-5572	266	15	have	have	VERB
ejpam-5572	266	16	p2	p2	VERB
ejpam-5572	266	17	+	+	X
ejpam-5572	266	18	q2	q2	NOUN
ejpam-5572	266	19	=	=	NOUN
ejpam-5572	267	1	0.64	0.64	NUM
ejpam-5572	267	2	+	+	NUM
ejpam-5572	267	3	0.4225	0.4225	NUM
ejpam-5572	267	4	≰	≰	NOUN
ejpam-5572	267	5	1	1	NUM
ejpam-5572	267	6	.	.	PUNCT
ejpam-5572	268	1	now	now	ADV
ejpam-5572	268	2	,	,	PUNCT
ejpam-5572	268	3	it	it	PRON
ejpam-5572	268	4	suffices	suffice	VERB
ejpam-5572	268	5	to	to	PART
ejpam-5572	268	6	prove	prove	VERB
ejpam-5572	268	7	that	that	SCONJ
ejpam-5572	268	8	the	the	DET
ejpam-5572	268	9	set	set	NOUN
ejpam-5572	268	10	ϕ(x	ϕ(x	NOUN
ejpam-5572	268	11	)	)	PUNCT
ejpam-5572	268	12	=	=	SYM
ejpam-5572	268	13	(	(	PUNCT
ejpam-5572	268	14	p(x	p(x	PROPN
ejpam-5572	268	15	)	)	PUNCT
ejpam-5572	268	16	e2πiω(x	e2πiω(x	NUM
ejpam-5572	268	17	)	)	PUNCT
ejpam-5572	268	18	,	,	PUNCT
ejpam-5572	268	19	q(x	q(x	PROPN
ejpam-5572	268	20	)	)	PUNCT
ejpam-5572	268	21	e2πiν(x	e2πiν(x	NOUN
ejpam-5572	268	22	)	)	PUNCT
ejpam-5572	268	23	)	)	PUNCT
ejpam-5572	268	24	is	be	AUX
ejpam-5572	268	25	cffsg	cffsg	ADJ
ejpam-5572	268	26	:	:	PUNCT
ejpam-5572	268	27	i	i	NOUN
ejpam-5572	268	28	)	)	PUNCT
ejpam-5572	268	29	first	first	ADV
ejpam-5572	268	30	,	,	PUNCT
ejpam-5572	268	31	for	for	ADP
ejpam-5572	268	32	any	any	DET
ejpam-5572	268	33	x	x	NOUN
ejpam-5572	268	34	,	,	PUNCT
ejpam-5572	268	35	y	y	PROPN
ejpam-5572	268	36	∈	∈	PROPN
ejpam-5572	268	37	x	x	X
ejpam-5572	268	38	,	,	PUNCT
ejpam-5572	268	39	we	we	PRON
ejpam-5572	268	40	check	check	VERB
ejpam-5572	268	41	that	that	PRON
ejpam-5572	268	42	:	:	PUNCT
ejpam-5572	268	43	a.	a.	PROPN
ejpam-5572	268	44	p3(x	p3(x	PROPN
ejpam-5572	268	45	∗	∗	X
ejpam-5572	268	46	y	y	PROPN
ejpam-5572	268	47	)	)	PUNCT
ejpam-5572	268	48	≥	≥	PROPN
ejpam-5572	268	49	min{p3(x	min{p3(x	PROPN
ejpam-5572	268	50	)	)	PUNCT
ejpam-5572	268	51	,	,	PUNCT
ejpam-5572	268	52	p3(y	p3(y	PROPN
ejpam-5572	268	53	)	)	PUNCT
ejpam-5572	268	54	}	}	PUNCT
ejpam-5572	268	55	and	and	CCONJ
ejpam-5572	268	56	ω3(x	ω3(x	PROPN
ejpam-5572	268	57	∗	∗	PROPN
ejpam-5572	268	58	y	y	PROPN
ejpam-5572	268	59	)	)	PUNCT
ejpam-5572	268	60	≥	≥	PROPN
ejpam-5572	268	61	min{ω3(x	min{ω3(x	PROPN
ejpam-5572	268	62	)	)	PUNCT
ejpam-5572	268	63	,	,	PUNCT
ejpam-5572	268	64	ω3(y	ω3(y	PROPN
ejpam-5572	268	65	)	)	PUNCT
ejpam-5572	268	66	}	}	PUNCT
ejpam-5572	268	67	.	.	PUNCT
ejpam-5572	269	1	b.	b.	PROPN
ejpam-5572	270	1	q3(x	q3(x	PROPN
ejpam-5572	270	2	∗	∗	PROPN
ejpam-5572	270	3	y	y	NOUN
ejpam-5572	270	4	)	)	PUNCT
ejpam-5572	270	5	≤	≤	PROPN
ejpam-5572	270	6	max{q3(x	max{q3(x	PROPN
ejpam-5572	270	7	)	)	PUNCT
ejpam-5572	270	8	,	,	PUNCT
ejpam-5572	270	9	q3(y	q3(y	NOUN
ejpam-5572	270	10	)	)	PUNCT
ejpam-5572	270	11	}	}	PUNCT
ejpam-5572	270	12	and	and	CCONJ
ejpam-5572	270	13	ν3(x	ν3(x	PROPN
ejpam-5572	270	14	∗	∗	PROPN
ejpam-5572	270	15	y	y	NOUN
ejpam-5572	270	16	)	)	PUNCT
ejpam-5572	270	17	≤	≤	NUM
ejpam-5572	270	18	max{ν3(x	max{ν3(x	PROPN
ejpam-5572	270	19	)	)	PUNCT
ejpam-5572	270	20	,	,	PUNCT
ejpam-5572	270	21	ν3(y	ν3(y	NUM
ejpam-5572	270	22	)	)	PUNCT
ejpam-5572	270	23	}	}	PUNCT
ejpam-5572	270	24	.	.	PUNCT
ejpam-5572	271	1	hence	hence	ADV
ejpam-5572	271	2	,	,	PUNCT
ejpam-5572	271	3	consider	consider	VERB
ejpam-5572	271	4	x	x	X
ejpam-5572	271	5	=	=	SYM
ejpam-5572	271	6	i	i	PROPN
ejpam-5572	271	7	,	,	PUNCT
ejpam-5572	271	8	y	y	PROPN
ejpam-5572	271	9	=	=	SYM
ejpam-5572	271	10	−i	−i	PROPN
ejpam-5572	271	11	,	,	PUNCT
ejpam-5572	271	12	then	then	ADV
ejpam-5572	271	13	i.−	i.−	ADV
ejpam-5572	272	1	i	i	NOUN
ejpam-5572	272	2	=	=	NOUN
ejpam-5572	272	3	1	1	NUM
ejpam-5572	272	4	:	:	PUNCT
ejpam-5572	272	5	a.	a.	NOUN
ejpam-5572	272	6	p3(i	p3(i	PROPN
ejpam-5572	272	7	∗	∗	NOUN
ejpam-5572	272	8	−i	−i	NOUN
ejpam-5572	272	9	)	)	PUNCT
ejpam-5572	273	1	=	=	SYM
ejpam-5572	273	2	p3(1	p3(1	PROPN
ejpam-5572	273	3	)	)	PUNCT
ejpam-5572	273	4	=	=	PUNCT
ejpam-5572	273	5	0.512	0.512	NUM
ejpam-5572	273	6	≥	≥	NUM
ejpam-5572	273	7	min{p3(i	min{p3(i	NOUN
ejpam-5572	273	8	)	)	PUNCT
ejpam-5572	273	9	,	,	PUNCT
ejpam-5572	273	10	p3(−i	p3(−i	NOUN
ejpam-5572	273	11	)	)	PUNCT
ejpam-5572	273	12	}	}	PUNCT
ejpam-5572	274	1	=	=	SYM
ejpam-5572	274	2	0.343	0.343	NUM
ejpam-5572	274	3	and	and	CCONJ
ejpam-5572	274	4	ω3(i	ω3(i	PROPN
ejpam-5572	274	5	∗	∗	NOUN
ejpam-5572	274	6	−i	−i	NOUN
ejpam-5572	274	7	)	)	PUNCT
ejpam-5572	274	8	=	=	SYM
ejpam-5572	274	9	ω3(1	ω3(1	X
ejpam-5572	274	10	)	)	PUNCT
ejpam-5572	274	11	=	=	PUNCT
ejpam-5572	275	1	0.42188	0.42188	NUM
ejpam-5572	275	2	≥	≥	NUM
ejpam-5572	275	3	min{ω3(i	min{ω3(i	PROPN
ejpam-5572	275	4	)	)	PUNCT
ejpam-5572	275	5	,	,	PUNCT
ejpam-5572	275	6	ω3(−i	ω3(−i	NOUN
ejpam-5572	275	7	)	)	PUNCT
ejpam-5572	275	8	}	}	PUNCT
ejpam-5572	276	1	=	=	SYM
ejpam-5572	276	2	0.09112	0.09112	NUM
ejpam-5572	276	3	.	.	PUNCT
ejpam-5572	277	1	b.	b.	PROPN
ejpam-5572	278	1	q3(i	q3(i	PROPN
ejpam-5572	278	2	∗	∗	NOUN
ejpam-5572	278	3	−i	−i	PROPN
ejpam-5572	278	4	)	)	PUNCT
ejpam-5572	279	1	=	=	PUNCT
ejpam-5572	279	2	q3(1	q3(1	PROPN
ejpam-5572	279	3	)	)	PUNCT
ejpam-5572	279	4	=	=	PUNCT
ejpam-5572	280	1	0.027	0.027	NUM
ejpam-5572	280	2	≤	≤	NUM
ejpam-5572	280	3	max{q3(i	max{q3(i	NOUN
ejpam-5572	280	4	)	)	PUNCT
ejpam-5572	280	5	,	,	PUNCT
ejpam-5572	280	6	q3(−i	q3(−i	PROPN
ejpam-5572	280	7	)	)	PUNCT
ejpam-5572	280	8	}	}	PUNCT
ejpam-5572	281	1	=	=	PUNCT
ejpam-5572	281	2	0.729	0.729	NUM
ejpam-5572	281	3	and	and	CCONJ
ejpam-5572	281	4	ν3(i	ν3(i	PROPN
ejpam-5572	281	5	∗	∗	NOUN
ejpam-5572	281	6	−i	−i	NOUN
ejpam-5572	281	7	)	)	PUNCT
ejpam-5572	281	8	=	=	SYM
ejpam-5572	281	9	ν3(1	ν3(1	ADJ
ejpam-5572	281	10	)	)	PUNCT
ejpam-5572	281	11	=	=	PUNCT
ejpam-5572	281	12	0.027	0.027	NUM
ejpam-5572	281	13	≤	≤	NUM
ejpam-5572	281	14	max{ν3(i	max{ν3(i	PROPN
ejpam-5572	281	15	)	)	PUNCT
ejpam-5572	281	16	,	,	PUNCT
ejpam-5572	281	17	ν3(−i	ν3(−i	NOUN
ejpam-5572	281	18	)	)	PUNCT
ejpam-5572	281	19	}	}	PUNCT
ejpam-5572	281	20	=	=	PUNCT
ejpam-5572	281	21	0.216	0.216	NUM
ejpam-5572	281	22	.	.	PUNCT
ejpam-5572	282	1	,	,	PUNCT
ejpam-5572	282	2	then	then	ADV
ejpam-5572	282	3	the	the	DET
ejpam-5572	282	4	property	property	NOUN
ejpam-5572	282	5	satisfied	satisfied	ADJ
ejpam-5572	282	6	at	at	ADP
ejpam-5572	282	7	x	x	X
ejpam-5572	282	8	=	=	SYM
ejpam-5572	282	9	i	i	PROPN
ejpam-5572	282	10	,	,	PUNCT
ejpam-5572	282	11	y	y	PROPN
ejpam-5572	282	12	=	=	PUNCT
ejpam-5572	282	13	−i	−i	PROPN
ejpam-5572	282	14	;	;	PUNCT
ejpam-5572	282	15	one	one	PRON
ejpam-5572	282	16	can	can	AUX
ejpam-5572	282	17	go	go	VERB
ejpam-5572	282	18	through	through	ADP
ejpam-5572	282	19	all	all	DET
ejpam-5572	282	20	x	x	PUNCT
ejpam-5572	282	21	and	and	CCONJ
ejpam-5572	282	22	see	see	VERB
ejpam-5572	282	23	that	that	PRON
ejpam-5572	282	24	property	property	NOUN
ejpam-5572	282	25	satisfied	satisfied	ADJ
ejpam-5572	282	26	.	.	PUNCT
ejpam-5572	283	1	ii	ii	X
ejpam-5572	283	2	)	)	PUNCT
ejpam-5572	283	3	second	second	ADV
ejpam-5572	283	4	,	,	PUNCT
ejpam-5572	283	5	since	since	SCONJ
ejpam-5572	283	6	1	1	NUM
ejpam-5572	283	7	=	=	SYM
ejpam-5572	283	8	1−1	1−1	NUM
ejpam-5572	283	9	,	,	PUNCT
ejpam-5572	283	10	−1	−1	NOUN
ejpam-5572	283	11	=	=	SYM
ejpam-5572	283	12	−1−1	−1−1	PROPN
ejpam-5572	283	13	and	and	CCONJ
ejpam-5572	283	14	i	i	PRON
ejpam-5572	283	15	=	=	SYM
ejpam-5572	283	16	−i−1	−i−1	NUM
ejpam-5572	283	17	,	,	PUNCT
ejpam-5572	283	18	where	where	SCONJ
ejpam-5572	283	19	ϕ(i	ϕ(i	X
ejpam-5572	283	20	)	)	PUNCT
ejpam-5572	283	21	=	=	PUNCT
ejpam-5572	284	1	ϕ(−i	ϕ(−i	NOUN
ejpam-5572	284	2	)	)	PUNCT
ejpam-5572	284	3	,	,	PUNCT
ejpam-5572	284	4	then	then	ADV
ejpam-5572	284	5	property	property	NOUN
ejpam-5572	284	6	2	2	NUM
ejpam-5572	284	7	in	in	ADP
ejpam-5572	284	8	of	of	ADP
ejpam-5572	284	9	cffsg	cffsg	NOUN
ejpam-5572	284	10	is	be	AUX
ejpam-5572	284	11	satisfied	satisfied	ADJ
ejpam-5572	284	12	too	too	ADV
ejpam-5572	284	13	.	.	PUNCT
ejpam-5572	285	1	note	note	VERB
ejpam-5572	285	2	that	that	SCONJ
ejpam-5572	285	3	,	,	PUNCT
ejpam-5572	285	4	since	since	SCONJ
ejpam-5572	285	5	cifsg	cifsg	NOUN
ejpam-5572	285	6	is	be	AUX
ejpam-5572	285	7	subclass	subclass	NOUN
ejpam-5572	285	8	of	of	ADP
ejpam-5572	285	9	cpfsg	cpfsg	NOUN
ejpam-5572	286	1	[	[	X
ejpam-5572	286	2	7	7	NUM
ejpam-5572	286	3	]	]	PUNCT
ejpam-5572	286	4	,	,	PUNCT
ejpam-5572	286	5	then	then	ADV
ejpam-5572	286	6	cifsg	cifsg	NOUN
ejpam-5572	286	7	is	be	AUX
ejpam-5572	286	8	subclass	subclass	NOUN
ejpam-5572	286	9	of	of	ADP
ejpam-5572	286	10	cffsg	cffsg	ADJ
ejpam-5572	286	11	.	.	PUNCT
ejpam-5572	287	1	proposition	proposition	NOUN
ejpam-5572	287	2	2	2	NUM
ejpam-5572	287	3	.	.	X
ejpam-5572	288	1	for	for	ADP
ejpam-5572	288	2	a	a	DET
ejpam-5572	288	3	cffs	cff	NOUN
ejpam-5572	288	4	ϕ	ϕ	NOUN
ejpam-5572	288	5	=	=	PUNCT
ejpam-5572	288	6	(	(	PUNCT
ejpam-5572	288	7	p	p	NOUN
ejpam-5572	288	8	e2πiω	e2πiω	PROPN
ejpam-5572	288	9	,	,	PUNCT
ejpam-5572	288	10	q	q	PROPN
ejpam-5572	288	11	e2πiν	e2πiν	PROPN
ejpam-5572	288	12	)	)	PUNCT
ejpam-5572	288	13	of	of	ADP
ejpam-5572	288	14	a	a	DET
ejpam-5572	288	15	group	group	NOUN
ejpam-5572	288	16	(	(	PUNCT
ejpam-5572	288	17	x	x	X
ejpam-5572	288	18	,	,	PUNCT
ejpam-5572	288	19	∗	∗	NOUN
ejpam-5572	288	20	)	)	PUNCT
ejpam-5572	288	21	,	,	PUNCT
ejpam-5572	288	22	it	it	PRON
ejpam-5572	288	23	is	be	AUX
ejpam-5572	288	24	a	a	DET
ejpam-5572	288	25	cffsg	cffsg	ADJ
ejpam-5572	288	26	if	if	SCONJ
ejpam-5572	289	1	and	and	CCONJ
ejpam-5572	289	2	only	only	ADV
ejpam-5572	289	3	if	if	SCONJ
ejpam-5572	289	4	:	:	PUNCT
ejpam-5572	289	5	1	1	X
ejpam-5572	289	6	.	.	X
ejpam-5572	289	7	p3(x	p3(x	NOUN
ejpam-5572	289	8	∗	∗	NOUN
ejpam-5572	289	9	y−1)e2πiω	y−1)e2πiω	NOUN
ejpam-5572	289	10	3(x	3(x	NUM
ejpam-5572	289	11	∗	∗	NOUN
ejpam-5572	289	12	y−1	y−1	PROPN
ejpam-5572	289	13	)	)	PUNCT
ejpam-5572	289	14	≥	≥	NOUN
ejpam-5572	289	15	p3(x)e2πiω	p3(x)e2πiω	PROPN
ejpam-5572	289	16	3(x	3(x	NUM
ejpam-5572	289	17	)	)	PUNCT
ejpam-5572	289	18	∧	∧	PROPN
ejpam-5572	289	19	p3(y)e2πiω	p3(y)e2πiω	NOUN
ejpam-5572	289	20	3(y	3(y	NUM
ejpam-5572	289	21	)	)	PUNCT
ejpam-5572	289	22	,	,	PUNCT
ejpam-5572	289	23	where	where	SCONJ
ejpam-5572	289	24	p3(x	p3(x	PROPN
ejpam-5572	289	25	∗	∗	NOUN
ejpam-5572	289	26	y−1	y−1	PROPN
ejpam-5572	289	27	)	)	PUNCT
ejpam-5572	289	28	≥	≥	PART
ejpam-5572	289	29	p3(x	p3(x	NOUN
ejpam-5572	289	30	)	)	PUNCT
ejpam-5572	289	31	∧	∧	PROPN
ejpam-5572	289	32	p3(y	p3(y	PROPN
ejpam-5572	289	33	)	)	PUNCT
ejpam-5572	289	34	and	and	CCONJ
ejpam-5572	289	35	ω3(x	ω3(x	PROPN
ejpam-5572	289	36	∗	∗	NOUN
ejpam-5572	289	37	y−1	y−1	PROPN
ejpam-5572	289	38	)	)	PUNCT
ejpam-5572	289	39	≥	≥	PART
ejpam-5572	289	40	ω3(x	ω3(x	PROPN
ejpam-5572	289	41	)	)	PUNCT
ejpam-5572	289	42	∧	∧	PROPN
ejpam-5572	289	43	ω3(y	ω3(y	PROPN
ejpam-5572	289	44	)	)	PUNCT
ejpam-5572	289	45	e.a	e.a	PROPN
ejpam-5572	289	46	.	.	PROPN
ejpam-5572	289	47	abuhijleh	abuhijleh	PROPN
ejpam-5572	289	48	,	,	PUNCT
ejpam-5572	289	49	a.	a.	PROPN
ejpam-5572	289	50	alkouri	alkouri	PROPN
ejpam-5572	289	51	/	/	PROPN
ejpam-5572	289	52	eur	eur	PROPN
ejpam-5572	289	53	.	.	PUNCT
ejpam-5572	290	1	j.	j.	PROPN
ejpam-5572	290	2	pure	pure	PROPN
ejpam-5572	290	3	appl	appl	PROPN
ejpam-5572	290	4	.	.	PROPN
ejpam-5572	290	5	math	math	PROPN
ejpam-5572	290	6	,	,	PUNCT
ejpam-5572	290	7	18	18	NUM
ejpam-5572	290	8	(	(	PUNCT
ejpam-5572	290	9	1	1	NUM
ejpam-5572	290	10	)	)	PUNCT
ejpam-5572	290	11	(	(	PUNCT
ejpam-5572	290	12	2025	2025	NUM
ejpam-5572	290	13	)	)	PUNCT
ejpam-5572	290	14	,	,	PUNCT
ejpam-5572	290	15	5572	5572	NUM
ejpam-5572	290	16	9	9	NUM
ejpam-5572	290	17	of	of	ADP
ejpam-5572	290	18	19	19	NUM
ejpam-5572	290	19	2	2	NUM
ejpam-5572	290	20	.	.	PUNCT
ejpam-5572	291	1	q3(x	q3(x	PROPN
ejpam-5572	291	2	∗	∗	NOUN
ejpam-5572	291	3	y−1)e2πiν	y−1)e2πiν	PROPN
ejpam-5572	291	4	3(x	3(x	NUM
ejpam-5572	291	5	∗	∗	NOUN
ejpam-5572	291	6	y−1	y−1	PROPN
ejpam-5572	291	7	)	)	PUNCT
ejpam-5572	291	8	≤	≤	NUM
ejpam-5572	291	9	q3(x)e2πiν	q3(x)e2πiν	ADP
ejpam-5572	291	10	3(x	3(x	NUM
ejpam-5572	291	11	)	)	PUNCT
ejpam-5572	291	12	∨	∨	NUM
ejpam-5572	291	13	q3(y)e2πiν	q3(y)e2πiν	NOUN
ejpam-5572	291	14	3(y	3(y	NUM
ejpam-5572	291	15	)	)	PUNCT
ejpam-5572	291	16	,	,	PUNCT
ejpam-5572	291	17	where	where	SCONJ
ejpam-5572	291	18	q3(x	q3(x	PROPN
ejpam-5572	291	19	∗	∗	X
ejpam-5572	291	20	y−1	y−1	PROPN
ejpam-5572	291	21	)	)	PUNCT
ejpam-5572	291	22	≤	≤	PROPN
ejpam-5572	291	23	q3(x	q3(x	PROPN
ejpam-5572	291	24	)	)	PUNCT
ejpam-5572	291	25	∨	∨	NUM
ejpam-5572	291	26	q3(y	q3(y	PROPN
ejpam-5572	291	27	)	)	PUNCT
ejpam-5572	291	28	and	and	CCONJ
ejpam-5572	291	29	ν3(x	ν3(x	PROPN
ejpam-5572	291	30	∗	∗	NOUN
ejpam-5572	291	31	y−1	y−1	PROPN
ejpam-5572	291	32	)	)	PUNCT
ejpam-5572	291	33	≤	≤	PUNCT
ejpam-5572	291	34	ν3(x	ν3(x	PROPN
ejpam-5572	291	35	)	)	PUNCT
ejpam-5572	291	36	∨	∨	NUM
ejpam-5572	291	37	ν3(y	ν3(y	NUM
ejpam-5572	291	38	)	)	PUNCT
ejpam-5572	291	39	proof	proof	NOUN
ejpam-5572	291	40	.	.	PUNCT
ejpam-5572	292	1	(=	(=	X
ejpam-5572	292	2	⇒	⇒	NOUN
ejpam-5572	292	3	)	)	PUNCT
ejpam-5572	292	4	according	accord	VERB
ejpam-5572	292	5	to	to	ADP
ejpam-5572	292	6	proposition	proposition	NOUN
ejpam-5572	292	7	1	1	NUM
ejpam-5572	292	8	,	,	PUNCT
ejpam-5572	292	9	we	we	PRON
ejpam-5572	292	10	have	have	VERB
ejpam-5572	292	11	p3(x−1)e2πiω	p3(x−1)e2πiω	PROPN
ejpam-5572	292	12	3(x−1	3(x−1	NUM
ejpam-5572	292	13	)	)	PUNCT
ejpam-5572	293	1	=	=	SYM
ejpam-5572	293	2	p3(x)e2πiω	p3(x)e2πiω	PROPN
ejpam-5572	293	3	3(x	3(x	NUM
ejpam-5572	293	4	)	)	PUNCT
ejpam-5572	293	5	and	and	CCONJ
ejpam-5572	293	6	q3(x−1)e2πiν	q3(x−1)e2πiν	PROPN
ejpam-5572	293	7	3(x−1	3(x−1	NUM
ejpam-5572	293	8	)	)	PUNCT
ejpam-5572	293	9	=	=	PUNCT
ejpam-5572	293	10	q3(x)e2πiν	q3(x)e2πiν	ADP
ejpam-5572	293	11	3(x	3(x	NUM
ejpam-5572	293	12	)	)	PUNCT
ejpam-5572	293	13	for	for	ADP
ejpam-5572	293	14	all	all	DET
ejpam-5572	293	15	x	x	SYM
ejpam-5572	293	16	∈	∈	PROPN
ejpam-5572	293	17	x	x	NOUN
ejpam-5572	293	18	,	,	PUNCT
ejpam-5572	293	19	then	then	ADV
ejpam-5572	293	20	results	result	NOUN
ejpam-5572	293	21	follow	follow	VERB
ejpam-5572	293	22	by	by	ADP
ejpam-5572	293	23	definition	definition	NOUN
ejpam-5572	293	24	9	9	NUM
ejpam-5572	293	25	.	.	PUNCT
ejpam-5572	294	1	(	(	PUNCT
ejpam-5572	294	2	⇐	⇐	ADJ
ejpam-5572	294	3	=)	=)	PROPN
ejpam-5572	294	4	first	first	ADJ
ejpam-5572	294	5	,	,	PUNCT
ejpam-5572	294	6	ϕ	ϕ	PROPN
ejpam-5572	294	7	is	be	AUX
ejpam-5572	294	8	cffs	cff	NOUN
ejpam-5572	294	9	and	and	CCONJ
ejpam-5572	294	10	is	be	AUX
ejpam-5572	294	11	defined	define	VERB
ejpam-5572	294	12	on	on	ADP
ejpam-5572	294	13	group	group	NOUN
ejpam-5572	294	14	(	(	PUNCT
ejpam-5572	294	15	x	x	X
ejpam-5572	294	16	,	,	PUNCT
ejpam-5572	294	17	∗	∗	NOUN
ejpam-5572	294	18	)	)	PUNCT
ejpam-5572	294	19	,	,	PUNCT
ejpam-5572	294	20	then	then	ADV
ejpam-5572	294	21	:	:	PUNCT
ejpam-5572	294	22	(	(	PUNCT
ejpam-5572	294	23	i	i	NOUN
ejpam-5572	294	24	)	)	PUNCT
ejpam-5572	294	25	p3(id)e2πiω	p3(id)e2πiω	NUM
ejpam-5572	294	26	3(id	3(id	NUM
ejpam-5572	294	27	)	)	PUNCT
ejpam-5572	295	1	=	=	SYM
ejpam-5572	295	2	p3(x	p3(x	PROPN
ejpam-5572	295	3	∗	∗	NOUN
ejpam-5572	295	4	x−1)e2πiω	x−1)e2πiω	PROPN
ejpam-5572	296	1	3(x	3(x	NUM
ejpam-5572	296	2	∗	∗	NOUN
ejpam-5572	296	3	x−1	x−1	PROPN
ejpam-5572	296	4	)	)	PUNCT
ejpam-5572	296	5	≥	≥	NOUN
ejpam-5572	296	6	p3(x)e2πiω	p3(x)e2πiω	PROPN
ejpam-5572	296	7	3(x	3(x	NUM
ejpam-5572	296	8	)	)	PUNCT
ejpam-5572	296	9	,	,	PUNCT
ejpam-5572	296	10	where	where	SCONJ
ejpam-5572	296	11	p3(x	p3(x	PROPN
ejpam-5572	296	12	∗	∗	X
ejpam-5572	296	13	x−1	x−1	PROPN
ejpam-5572	296	14	)	)	PUNCT
ejpam-5572	296	15	≥	≥	NOUN
ejpam-5572	296	16	p3(x	p3(x	SYM
ejpam-5572	296	17	)	)	PUNCT
ejpam-5572	296	18	and	and	CCONJ
ejpam-5572	296	19	ω3(x	ω3(x	PROPN
ejpam-5572	296	20	∗	∗	PROPN
ejpam-5572	296	21	x−1	x−1	PROPN
ejpam-5572	296	22	)	)	PUNCT
ejpam-5572	296	23	≥	≥	NOUN
ejpam-5572	296	24	ω3(x	ω3(x	NUM
ejpam-5572	296	25	)	)	PUNCT
ejpam-5572	296	26	.	.	PUNCT
ejpam-5572	297	1	(	(	PUNCT
ejpam-5572	297	2	ii	ii	X
ejpam-5572	297	3	)	)	PUNCT
ejpam-5572	297	4	p3(x−1)e2πiω	p3(x−1)e2πiω	NOUN
ejpam-5572	297	5	3(x−1	3(x−1	NUM
ejpam-5572	297	6	)	)	PUNCT
ejpam-5572	297	7	=	=	PUNCT
ejpam-5572	298	1	p3(id	p3(id	PROPN
ejpam-5572	298	2	∗	∗	NOUN
ejpam-5572	298	3	x−1)e2πiω	x−1)e2πiω	PUNCT
ejpam-5572	299	1	3(id	3(id	NUM
ejpam-5572	299	2	∗	∗	NOUN
ejpam-5572	299	3	x−1	x−1	PROPN
ejpam-5572	299	4	)	)	PUNCT
ejpam-5572	299	5	≥	≥	NOUN
ejpam-5572	299	6	min{p3(id)e2πiω3(di	min{p3(id)e2πiω3(di	NOUN
ejpam-5572	299	7	)	)	PUNCT
ejpam-5572	299	8	,	,	PUNCT
ejpam-5572	299	9	p3(x)e2πiω	p3(x)e2πiω	PROPN
ejpam-5572	299	10	3(x	3(x	NUM
ejpam-5572	299	11	)	)	PUNCT
ejpam-5572	299	12	}	}	PUNCT
ejpam-5572	299	13	=	=	SYM
ejpam-5572	299	14	min{p3(id	min{p3(id	PROPN
ejpam-5572	299	15	)	)	PUNCT
ejpam-5572	299	16	,	,	PUNCT
ejpam-5572	299	17	p3(x)}e2πimin{ω3(di),ω3(x	p3(x)}e2πimin{ω3(di),ω3(x	PROPN
ejpam-5572	299	18	)	)	PUNCT
ejpam-5572	299	19	}	}	PUNCT
ejpam-5572	299	20	=	=	SYM
ejpam-5572	299	21	p3(x)e2πiω	p3(x)e2πiω	PROPN
ejpam-5572	299	22	3(x	3(x	NUM
ejpam-5572	299	23	)	)	PUNCT
ejpam-5572	299	24	,	,	PUNCT
ejpam-5572	299	25	by	by	ADP
ejpam-5572	299	26	(	(	PUNCT
ejpam-5572	299	27	i	i	NOUN
ejpam-5572	299	28	)	)	PUNCT
ejpam-5572	299	29	.	.	PUNCT
ejpam-5572	300	1	(	(	PUNCT
ejpam-5572	300	2	ii	ii	X
ejpam-5572	300	3	)	)	PUNCT
ejpam-5572	300	4	p3(x	p3(x	PROPN
ejpam-5572	300	5	∗	∗	NOUN
ejpam-5572	300	6	y)e2πiω	y)e2πiω	PROPN
ejpam-5572	300	7	3(x	3(x	NUM
ejpam-5572	300	8	∗	∗	NOUN
ejpam-5572	300	9	y	y	NOUN
ejpam-5572	300	10	)	)	PUNCT
ejpam-5572	301	1	=	=	SYM
ejpam-5572	301	2	p3(x	p3(x	PROPN
ejpam-5572	301	3	∗	∗	NOUN
ejpam-5572	301	4	(	(	PUNCT
ejpam-5572	301	5	y−1)−1)e2πiω	y−1)−1)e2πiω	X
ejpam-5572	301	6	3(x	3(x	NUM
ejpam-5572	301	7	∗	∗	NOUN
ejpam-5572	301	8	(	(	PUNCT
ejpam-5572	301	9	y−1)−1	y−1)−1	NOUN
ejpam-5572	301	10	)	)	PUNCT
ejpam-5572	301	11	≥	≥	NOUN
ejpam-5572	301	12	p3(x)e2πiω	p3(x)e2πiω	PROPN
ejpam-5572	301	13	3(x	3(x	NUM
ejpam-5572	301	14	)	)	PUNCT
ejpam-5572	301	15	∧	∧	PROPN
ejpam-5572	301	16	p3(y−1)e2πiω	p3(y−1)e2πiω	PROPN
ejpam-5572	301	17	3(y−1	3(y−1	NOUN
ejpam-5572	301	18	)	)	PUNCT
ejpam-5572	301	19	≥	≥	NOUN
ejpam-5572	301	20	p3(x)e2πiω	p3(x)e2πiω	PROPN
ejpam-5572	301	21	3(x	3(x	NUM
ejpam-5572	301	22	)	)	PUNCT
ejpam-5572	301	23	∧	∧	PROPN
ejpam-5572	301	24	p3(y)e2πiω	p3(y)e2πiω	NOUN
ejpam-5572	301	25	3(y	3(y	NUM
ejpam-5572	301	26	)	)	PUNCT
ejpam-5572	301	27	,	,	PUNCT
ejpam-5572	301	28	by	by	ADP
ejpam-5572	301	29	(	(	PUNCT
ejpam-5572	301	30	ii	ii	NOUN
ejpam-5572	301	31	)	)	PUNCT
ejpam-5572	301	32	.	.	PUNCT
ejpam-5572	302	1	similarly	similarly	ADV
ejpam-5572	302	2	,	,	PUNCT
ejpam-5572	302	3	we	we	PRON
ejpam-5572	302	4	have	have	VERB
ejpam-5572	302	5	:	:	PUNCT
ejpam-5572	302	6	(	(	PUNCT
ejpam-5572	302	7	iv	iv	X
ejpam-5572	302	8	)	)	PUNCT
ejpam-5572	302	9	q3(id)e2πiν	q3(id)e2πiν	NUM
ejpam-5572	302	10	3(id	3(id	NUM
ejpam-5572	302	11	)	)	PUNCT
ejpam-5572	302	12	=	=	PUNCT
ejpam-5572	303	1	q3(x	q3(x	PROPN
ejpam-5572	303	2	∗	∗	NOUN
ejpam-5572	303	3	x−1)e2πiν	x−1)e2πiν	X
ejpam-5572	304	1	3(x	3(x	NUM
ejpam-5572	304	2	∗	∗	NOUN
ejpam-5572	304	3	x−1	x−1	PROPN
ejpam-5572	304	4	)	)	PUNCT
ejpam-5572	304	5	≤	≤	NUM
ejpam-5572	304	6	q3(x)e2πiν	q3(x)e2πiν	PROPN
ejpam-5572	304	7	3(x	3(x	NUM
ejpam-5572	304	8	)	)	PUNCT
ejpam-5572	304	9	,	,	PUNCT
ejpam-5572	304	10	where	where	SCONJ
ejpam-5572	304	11	q3(x	q3(x	PROPN
ejpam-5572	304	12	∗	∗	NOUN
ejpam-5572	304	13	x−1	x−1	NOUN
ejpam-5572	304	14	)	)	PUNCT
ejpam-5572	304	15	≤	≤	NUM
ejpam-5572	304	16	q3(x	q3(x	PROPN
ejpam-5572	304	17	)	)	PUNCT
ejpam-5572	304	18	and	and	CCONJ
ejpam-5572	304	19	ν3(x	ν3(x	PROPN
ejpam-5572	304	20	∗	∗	NOUN
ejpam-5572	304	21	x−1	x−1	PROPN
ejpam-5572	304	22	)	)	PUNCT
ejpam-5572	304	23	≤	≤	NOUN
ejpam-5572	304	24	ν3(x	ν3(x	NOUN
ejpam-5572	304	25	)	)	PUNCT
ejpam-5572	304	26	.	.	PUNCT
ejpam-5572	305	1	(	(	PUNCT
ejpam-5572	305	2	v	v	X
ejpam-5572	305	3	)	)	PUNCT
ejpam-5572	305	4	q3(x−1)e2πiν	q3(x−1)e2πiν	NOUN
ejpam-5572	305	5	3(x−1	3(x−1	NUM
ejpam-5572	305	6	)	)	PUNCT
ejpam-5572	305	7	=	=	SYM
ejpam-5572	306	1	q3(id	q3(id	VERB
ejpam-5572	306	2	∗	∗	NOUN
ejpam-5572	306	3	x−1)e2πiν	x−1)e2πiν	X
ejpam-5572	307	1	3(id	3(id	NUM
ejpam-5572	307	2	∗	∗	NOUN
ejpam-5572	307	3	x−1	x−1	NOUN
ejpam-5572	307	4	)	)	PUNCT
ejpam-5572	307	5	≤	≤	NUM
ejpam-5572	307	6	max{q3(id)e2πiν3(di	max{q3(id)e2πiν3(di	NOUN
ejpam-5572	307	7	)	)	PUNCT
ejpam-5572	307	8	,	,	PUNCT
ejpam-5572	307	9	q3(x)e2πiν3(x	q3(x)e2πiν3(x	PROPN
ejpam-5572	307	10	)	)	PUNCT
ejpam-5572	307	11	}	}	PUNCT
ejpam-5572	307	12	=	=	SYM
ejpam-5572	307	13	max{q3(id	max{q3(id	NOUN
ejpam-5572	307	14	)	)	PUNCT
ejpam-5572	307	15	,	,	PUNCT
ejpam-5572	307	16	q3(x)}e2πimax{ν3(di),ν3(x	q3(x)}e2πimax{ν3(di),ν3(x	NOUN
ejpam-5572	307	17	)	)	PUNCT
ejpam-5572	307	18	}	}	PUNCT
ejpam-5572	307	19	=	=	PUNCT
ejpam-5572	307	20	q3(x)e2πiν	q3(x)e2πiν	PRON
ejpam-5572	307	21	3(x	3(x	NUM
ejpam-5572	307	22	)	)	PUNCT
ejpam-5572	307	23	,	,	PUNCT
ejpam-5572	307	24	by	by	ADP
ejpam-5572	307	25	(	(	PUNCT
ejpam-5572	307	26	iv	iv	X
ejpam-5572	307	27	)	)	PUNCT
ejpam-5572	307	28	.	.	PUNCT
ejpam-5572	308	1	(	(	PUNCT
ejpam-5572	308	2	vi	vi	X
ejpam-5572	308	3	)	)	PUNCT
ejpam-5572	308	4	q3(x	q3(x	PROPN
ejpam-5572	308	5	∗	∗	NOUN
ejpam-5572	308	6	y)e2πiν	y)e2πiν	PROPN
ejpam-5572	308	7	3(x	3(x	NUM
ejpam-5572	308	8	∗	∗	NOUN
ejpam-5572	308	9	y	y	NOUN
ejpam-5572	308	10	)	)	PUNCT
ejpam-5572	308	11	=	=	PUNCT
ejpam-5572	309	1	q3(x	q3(x	PROPN
ejpam-5572	309	2	∗	∗	NOUN
ejpam-5572	309	3	(	(	PUNCT
ejpam-5572	309	4	y−1)−1)e2πiν	y−1)−1)e2πiν	NOUN
ejpam-5572	309	5	3(x	3(x	NUM
ejpam-5572	309	6	∗	∗	NOUN
ejpam-5572	309	7	(	(	PUNCT
ejpam-5572	309	8	y−1)−1	y−1)−1	NOUN
ejpam-5572	309	9	)	)	PUNCT
ejpam-5572	309	10	≤	≤	NUM
ejpam-5572	309	11	q3(x)e2πiν	q3(x)e2πiν	ADP
ejpam-5572	309	12	3(x	3(x	NUM
ejpam-5572	309	13	)	)	PUNCT
ejpam-5572	309	14	∨	∨	PROPN
ejpam-5572	309	15	q3(y−1)e2πiν	q3(y−1)e2πiν	PROPN
ejpam-5572	309	16	3(y−1	3(y−1	NUM
ejpam-5572	309	17	)	)	PUNCT
ejpam-5572	309	18	≤	≤	NUM
ejpam-5572	309	19	q3(x)e2πiν	q3(x)e2πiν	ADP
ejpam-5572	309	20	3(x	3(x	NUM
ejpam-5572	309	21	)	)	PUNCT
ejpam-5572	309	22	∨	∨	NUM
ejpam-5572	309	23	q3(y)e2πiν	q3(y)e2πiν	NOUN
ejpam-5572	309	24	3(y	3(y	NUM
ejpam-5572	309	25	)	)	PUNCT
ejpam-5572	309	26	,	,	PUNCT
ejpam-5572	309	27	by	by	ADP
ejpam-5572	309	28	(	(	PUNCT
ejpam-5572	309	29	v	v	NOUN
ejpam-5572	309	30	)	)	PUNCT
ejpam-5572	309	31	.	.	PUNCT
ejpam-5572	310	1	finally	finally	ADV
ejpam-5572	310	2	,	,	PUNCT
ejpam-5572	310	3	by	by	ADP
ejpam-5572	310	4	(	(	PUNCT
ejpam-5572	310	5	iii	iii	NOUN
ejpam-5572	310	6	)	)	PUNCT
ejpam-5572	310	7	and	and	CCONJ
ejpam-5572	310	8	(	(	PUNCT
ejpam-5572	310	9	vi	vi	X
ejpam-5572	310	10	)	)	PUNCT
ejpam-5572	310	11	the	the	DET
ejpam-5572	310	12	first	first	ADJ
ejpam-5572	310	13	condition	condition	NOUN
ejpam-5572	310	14	was	be	AUX
ejpam-5572	310	15	satisfied	satisfied	ADJ
ejpam-5572	310	16	,	,	PUNCT
ejpam-5572	310	17	and	and	CCONJ
ejpam-5572	310	18	by	by	ADP
ejpam-5572	310	19	(	(	PUNCT
ejpam-5572	310	20	ii	ii	NOUN
ejpam-5572	310	21	)	)	PUNCT
ejpam-5572	310	22	and	and	CCONJ
ejpam-5572	310	23	(	(	PUNCT
ejpam-5572	310	24	v	v	NOUN
ejpam-5572	310	25	)	)	PUNCT
ejpam-5572	310	26	the	the	DET
ejpam-5572	310	27	second	second	ADJ
ejpam-5572	310	28	condition	condition	NOUN
ejpam-5572	310	29	was	be	AUX
ejpam-5572	310	30	satisfied	satisfied	ADJ
ejpam-5572	310	31	in	in	ADP
ejpam-5572	310	32	the	the	DET
ejpam-5572	310	33	definition	definition	NOUN
ejpam-5572	310	34	9	9	NUM
ejpam-5572	310	35	,	,	PUNCT
ejpam-5572	310	36	hence	hence	ADV
ejpam-5572	310	37	ϕ	ϕ	NOUN
ejpam-5572	310	38	is	be	AUX
ejpam-5572	310	39	cffsg	cffsg	ADJ
ejpam-5572	310	40	of	of	ADP
ejpam-5572	310	41	a	a	DET
ejpam-5572	310	42	group	group	NOUN
ejpam-5572	310	43	(	(	PUNCT
ejpam-5572	310	44	x	x	X
ejpam-5572	310	45	,	,	PUNCT
ejpam-5572	310	46	∗	∗	NOUN
ejpam-5572	310	47	)	)	PUNCT
ejpam-5572	310	48	.	.	PUNCT
ejpam-5572	311	1	proposition	proposition	NOUN
ejpam-5572	311	2	3	3	NUM
ejpam-5572	311	3	.	.	PUNCT
ejpam-5572	312	1	the	the	DET
ejpam-5572	312	2	intersection	intersection	NOUN
ejpam-5572	312	3	of	of	ADP
ejpam-5572	312	4	two	two	NUM
ejpam-5572	312	5	cffsgs	cffsg	NOUN
ejpam-5572	312	6	of	of	ADP
ejpam-5572	312	7	a	a	DET
ejpam-5572	312	8	group	group	NOUN
ejpam-5572	312	9	(	(	PUNCT
ejpam-5572	312	10	x	x	X
ejpam-5572	312	11	,	,	PUNCT
ejpam-5572	312	12	∗	∗	NOUN
ejpam-5572	312	13	)	)	PUNCT
ejpam-5572	312	14	is	be	AUX
ejpam-5572	312	15	a	a	DET
ejpam-5572	312	16	cffsg	cffsg	ADJ
ejpam-5572	312	17	.	.	PUNCT
ejpam-5572	313	1	proof	proof	NOUN
ejpam-5572	313	2	.	.	PUNCT
ejpam-5572	314	1	let	let	VERB
ejpam-5572	314	2	s1	s1	NOUN
ejpam-5572	314	3	,	,	PUNCT
ejpam-5572	314	4	s2	s2	NOUN
ejpam-5572	314	5	be	be	VERB
ejpam-5572	314	6	two	two	NUM
ejpam-5572	314	7	cffsgs	cffsg	NOUN
ejpam-5572	314	8	of	of	ADP
ejpam-5572	314	9	x	x	PUNCT
ejpam-5572	314	10	and	and	CCONJ
ejpam-5572	314	11	using	use	VERB
ejpam-5572	314	12	previous	previous	ADJ
ejpam-5572	314	13	proposition	proposition	NOUN
ejpam-5572	314	14	,	,	PUNCT
ejpam-5572	314	15	then	then	ADV
ejpam-5572	314	16	:	:	PUNCT
ejpam-5572	314	17	i	i	NOUN
ejpam-5572	314	18	)	)	PUNCT
ejpam-5572	314	19	p3s1∩s2	p3s1∩s2	PROPN
ejpam-5572	314	20	(	(	PUNCT
ejpam-5572	314	21	x	x	SYM
ejpam-5572	314	22	∗	∗	NOUN
ejpam-5572	314	23	y−1)e	y−1)e	NUM
ejpam-5572	314	24	2πiω3	2πiω3	NUM
ejpam-5572	314	25	s1∩s2	s1∩s2	NOUN
ejpam-5572	314	26	(	(	PUNCT
ejpam-5572	314	27	x	x	NOUN
ejpam-5572	314	28	∗	∗	NOUN
ejpam-5572	314	29	y−1	y−1	PROPN
ejpam-5572	314	30	)	)	PUNCT
ejpam-5572	315	1	=	=	PRON
ejpam-5572	315	2	(	(	PUNCT
ejpam-5572	315	3	p3s1	p3s1	X
ejpam-5572	315	4	(	(	PUNCT
ejpam-5572	315	5	x	x	NOUN
ejpam-5572	315	6	∗	∗	PROPN
ejpam-5572	315	7	y−1	y−1	NOUN
ejpam-5572	315	8	)	)	PUNCT
ejpam-5572	315	9	∧	∧	PROPN
ejpam-5572	315	10	p3s2	p3s2	PROPN
ejpam-5572	315	11	(	(	PUNCT
ejpam-5572	315	12	x	x	SYM
ejpam-5572	315	13	∗	∗	NUM
ejpam-5572	315	14	y−1))e	y−1))e	NOUN
ejpam-5572	315	15	2πi(ω3	2πi(ω3	PROPN
ejpam-5572	315	16	s1	s1	NOUN
ejpam-5572	315	17	(	(	PUNCT
ejpam-5572	315	18	x	x	NOUN
ejpam-5572	315	19	∗	∗	NOUN
ejpam-5572	315	20	y−1)∧	y−1)∧	PROPN
ejpam-5572	315	21	ω3	ω3	PROPN
ejpam-5572	315	22	s2	s2	NOUN
ejpam-5572	315	23	(	(	PUNCT
ejpam-5572	315	24	x	x	NOUN
ejpam-5572	315	25	∗	∗	NOUN
ejpam-5572	315	26	y−1	y−1	PROPN
ejpam-5572	315	27	)	)	PUNCT
ejpam-5572	315	28	)	)	PUNCT
ejpam-5572	315	29	≥	≥	X
ejpam-5572	315	30	(	(	PUNCT
ejpam-5572	315	31	min{p3s1	min{p3s1	NOUN
ejpam-5572	315	32	(	(	PUNCT
ejpam-5572	315	33	x	x	NOUN
ejpam-5572	315	34	)	)	PUNCT
ejpam-5572	315	35	,	,	PUNCT
ejpam-5572	315	36	p3s1	p3s1	X
ejpam-5572	315	37	(	(	PUNCT
ejpam-5572	315	38	y	y	NOUN
ejpam-5572	315	39	)	)	PUNCT
ejpam-5572	315	40	}	}	PUNCT
ejpam-5572	315	41	∧	∧	NOUN
ejpam-5572	315	42	min{p3s2	min{p3s2	NOUN
ejpam-5572	315	43	(	(	PUNCT
ejpam-5572	315	44	x	x	NOUN
ejpam-5572	315	45	)	)	PUNCT
ejpam-5572	315	46	,	,	PUNCT
ejpam-5572	315	47	p3s2	p3s2	PROPN
ejpam-5572	315	48	(	(	PUNCT
ejpam-5572	315	49	y)})e2πi(min{ω3	y)})e2πi(min{ω3	PROPN
ejpam-5572	315	50	s1	s1	PROPN
ejpam-5572	315	51	(	(	PUNCT
ejpam-5572	315	52	x),ω3	x),ω3	PROPN
ejpam-5572	315	53	s1	s1	PROPN
ejpam-5572	315	54	(	(	PUNCT
ejpam-5572	315	55	y)}∧	y)}∧	PROPN
ejpam-5572	315	56	min{ω3	min{ω3	PROPN
ejpam-5572	315	57	s2	s2	PROPN
ejpam-5572	315	58	(	(	PUNCT
ejpam-5572	315	59	x),ω3	x),ω3	PROPN
ejpam-5572	315	60	s2	s2	PROPN
ejpam-5572	315	61	(	(	PUNCT
ejpam-5572	315	62	y	y	NOUN
ejpam-5572	315	63	)	)	PUNCT
ejpam-5572	315	64	}	}	PUNCT
ejpam-5572	315	65	)	)	PUNCT
ejpam-5572	315	66	=	=	SYM
ejpam-5572	315	67	(	(	PUNCT
ejpam-5572	315	68	min{p3s1	min{p3s1	NOUN
ejpam-5572	315	69	(	(	PUNCT
ejpam-5572	315	70	x	x	NOUN
ejpam-5572	315	71	)	)	PUNCT
ejpam-5572	315	72	,	,	PUNCT
ejpam-5572	315	73	p3s2	p3s2	PROPN
ejpam-5572	315	74	(	(	PUNCT
ejpam-5572	315	75	x	x	X
ejpam-5572	315	76	)	)	PUNCT
ejpam-5572	315	77	}	}	PUNCT
ejpam-5572	315	78	∧	∧	NOUN
ejpam-5572	315	79	min{p3s1	min{p3s1	NOUN
ejpam-5572	315	80	(	(	PUNCT
ejpam-5572	315	81	y	y	NOUN
ejpam-5572	315	82	)	)	PUNCT
ejpam-5572	315	83	,	,	PUNCT
ejpam-5572	315	84	p3s2	p3s2	PROPN
ejpam-5572	315	85	(	(	PUNCT
ejpam-5572	315	86	y)})e2πi(min{ω3	y)})e2πi(min{ω3	PROPN
ejpam-5572	315	87	s1	s1	PROPN
ejpam-5572	315	88	(	(	PUNCT
ejpam-5572	315	89	x),ω3	x),ω3	PROPN
ejpam-5572	315	90	s2	s2	PROPN
ejpam-5572	315	91	(	(	PUNCT
ejpam-5572	315	92	x)}∧	x)}∧	PROPN
ejpam-5572	315	93	min{ω3	min{ω3	NOUN
ejpam-5572	315	94	s1	s1	PROPN
ejpam-5572	315	95	(	(	PUNCT
ejpam-5572	315	96	y),ω3	y),ω3	NOUN
ejpam-5572	315	97	s2	s2	NOUN
ejpam-5572	315	98	(	(	PUNCT
ejpam-5572	315	99	y	y	NOUN
ejpam-5572	315	100	)	)	PUNCT
ejpam-5572	315	101	}	}	PUNCT
ejpam-5572	315	102	)	)	PUNCT
ejpam-5572	315	103	=	=	SYM
ejpam-5572	315	104	(	(	PUNCT
ejpam-5572	315	105	p3s1∩s2	p3s1∩s2	PROPN
ejpam-5572	315	106	(	(	PUNCT
ejpam-5572	315	107	x	x	NOUN
ejpam-5572	315	108	)	)	PUNCT
ejpam-5572	315	109	∧	∧	PROPN
ejpam-5572	315	110	p3s1∩s2	p3s1∩s2	PROPN
ejpam-5572	315	111	(	(	PUNCT
ejpam-5572	315	112	y))e	y))e	PROPN
ejpam-5572	315	113	2πi(ω3	2πi(ω3	NUM
ejpam-5572	315	114	s1∩s2	s1∩s2	NOUN
ejpam-5572	315	115	(	(	PUNCT
ejpam-5572	315	116	x)∧	x)∧	X
ejpam-5572	315	117	ω3	ω3	ADJ
ejpam-5572	315	118	s1∩s2	s1∩s2	PROPN
ejpam-5572	315	119	(	(	PUNCT
ejpam-5572	315	120	y	y	NOUN
ejpam-5572	315	121	)	)	PUNCT
ejpam-5572	315	122	)	)	PUNCT
ejpam-5572	316	1	=	=	PUNCT
ejpam-5572	316	2	p3s1∩s2	p3s1∩s2	PROPN
ejpam-5572	316	3	(	(	PUNCT
ejpam-5572	316	4	x)e	x)e	X
ejpam-5572	316	5	2πiω3	2πiω3	NUM
ejpam-5572	316	6	s1∩s2	s1∩s2	NOUN
ejpam-5572	316	7	(	(	PUNCT
ejpam-5572	316	8	x	x	NOUN
ejpam-5572	316	9	)	)	PUNCT
ejpam-5572	316	10	∧	∧	PROPN
ejpam-5572	316	11	p3s1∩s2	p3s1∩s2	PROPN
ejpam-5572	316	12	(	(	PUNCT
ejpam-5572	316	13	y)e	y)e	NOUN
ejpam-5572	316	14	2πiω3	2πiω3	NUM
ejpam-5572	316	15	s1∩s2	s1∩s2	NOUN
ejpam-5572	316	16	(	(	PUNCT
ejpam-5572	316	17	y	y	NOUN
ejpam-5572	316	18	)	)	PUNCT
ejpam-5572	316	19	.	.	PUNCT
ejpam-5572	317	1	ii	ii	X
ejpam-5572	317	2	)	)	PUNCT
ejpam-5572	317	3	q3s1∩s2	q3s1∩s2	PROPN
ejpam-5572	317	4	(	(	PUNCT
ejpam-5572	317	5	x	x	X
ejpam-5572	317	6	∗	∗	VERB
ejpam-5572	317	7	y−1)e	y−1)e	NOUN
ejpam-5572	317	8	2πiν3s1∩s2	2πiν3s1∩s2	NUM
ejpam-5572	317	9	(	(	PUNCT
ejpam-5572	317	10	x	x	NOUN
ejpam-5572	317	11	∗	∗	NOUN
ejpam-5572	317	12	y−1	y−1	PROPN
ejpam-5572	317	13	)	)	PUNCT
ejpam-5572	318	1	=	=	PRON
ejpam-5572	318	2	(	(	PUNCT
ejpam-5572	318	3	q3s1	q3s1	INTJ
ejpam-5572	318	4	(	(	PUNCT
ejpam-5572	318	5	x	x	NOUN
ejpam-5572	318	6	∗	∗	PROPN
ejpam-5572	318	7	y−1	y−1	PROPN
ejpam-5572	318	8	)	)	PUNCT
ejpam-5572	318	9	∨	∨	NUM
ejpam-5572	319	1	q3s2	q3s2	X
ejpam-5572	319	2	(	(	PUNCT
ejpam-5572	319	3	x	x	SYM
ejpam-5572	319	4	∗	∗	NOUN
ejpam-5572	319	5	y−1))e	y−1))e	NOUN
ejpam-5572	319	6	2πi(ν3s1	2πi(ν3s1	NUM
ejpam-5572	319	7	(	(	PUNCT
ejpam-5572	319	8	x	x	SYM
ejpam-5572	319	9	∗	∗	VERB
ejpam-5572	319	10	y−1)∨	y−1)∨	NOUN
ejpam-5572	319	11	ν3s2	ν3s2	NOUN
ejpam-5572	319	12	(	(	PUNCT
ejpam-5572	319	13	x	x	NOUN
ejpam-5572	319	14	∗	∗	NOUN
ejpam-5572	319	15	y−1	y−1	PROPN
ejpam-5572	319	16	)	)	PUNCT
ejpam-5572	319	17	)	)	PUNCT
ejpam-5572	320	1	≤	≤	NOUN
ejpam-5572	320	2	(	(	PUNCT
ejpam-5572	320	3	max{q3s1	max{q3s1	PROPN
ejpam-5572	320	4	(	(	PUNCT
ejpam-5572	320	5	x	x	NOUN
ejpam-5572	320	6	)	)	PUNCT
ejpam-5572	320	7	,	,	PUNCT
ejpam-5572	320	8	q3s1	q3s1	X
ejpam-5572	320	9	(	(	PUNCT
ejpam-5572	320	10	y	y	NOUN
ejpam-5572	320	11	)	)	PUNCT
ejpam-5572	320	12	}	}	PUNCT
ejpam-5572	320	13	∨	∨	NUM
ejpam-5572	320	14	max{q3s2	max{q3s2	NOUN
ejpam-5572	320	15	(	(	PUNCT
ejpam-5572	320	16	x	x	NOUN
ejpam-5572	320	17	)	)	PUNCT
ejpam-5572	320	18	,	,	PUNCT
ejpam-5572	320	19	q3s2	q3s2	PROPN
ejpam-5572	320	20	(	(	PUNCT
ejpam-5572	320	21	y)})e2πi(max{ν3s1	y)})e2πi(max{ν3s1	PROPN
ejpam-5572	320	22	(	(	PUNCT
ejpam-5572	320	23	x),ν3s1	x),ν3s1	PROPN
ejpam-5572	320	24	(	(	PUNCT
ejpam-5572	320	25	y)}∨	y)}∨	NOUN
ejpam-5572	320	26	max{ν3s2	max{ν3s2	PROPN
ejpam-5572	320	27	(	(	PUNCT
ejpam-5572	320	28	x),ν3s2	x),ν3s2	PROPN
ejpam-5572	320	29	(	(	PUNCT
ejpam-5572	320	30	y	y	NOUN
ejpam-5572	320	31	)	)	PUNCT
ejpam-5572	320	32	}	}	PUNCT
ejpam-5572	320	33	)	)	PUNCT
ejpam-5572	321	1	=	=	SYM
ejpam-5572	321	2	(	(	PUNCT
ejpam-5572	321	3	max{q3s1	max{q3s1	PROPN
ejpam-5572	321	4	(	(	PUNCT
ejpam-5572	321	5	x	x	NOUN
ejpam-5572	321	6	)	)	PUNCT
ejpam-5572	321	7	,	,	PUNCT
ejpam-5572	321	8	q3s2	q3s2	PROPN
ejpam-5572	321	9	(	(	PUNCT
ejpam-5572	321	10	x	x	NOUN
ejpam-5572	321	11	)	)	PUNCT
ejpam-5572	321	12	}	}	PUNCT
ejpam-5572	321	13	∨	∨	NUM
ejpam-5572	321	14	max{q3s1	max{q3s1	PROPN
ejpam-5572	321	15	(	(	PUNCT
ejpam-5572	321	16	y	y	NOUN
ejpam-5572	321	17	)	)	PUNCT
ejpam-5572	321	18	,	,	PUNCT
ejpam-5572	321	19	q3s2	q3s2	PROPN
ejpam-5572	321	20	(	(	PUNCT
ejpam-5572	321	21	y)})e2πi(max{ν3s1	y)})e2πi(max{ν3s1	PROPN
ejpam-5572	321	22	(	(	PUNCT
ejpam-5572	321	23	x),ν3s2	x),ν3s2	PROPN
ejpam-5572	321	24	(	(	PUNCT
ejpam-5572	321	25	x)}∨	x)}∨	X
ejpam-5572	321	26	max{ν3s1	max{ν3s1	NOUN
ejpam-5572	321	27	(	(	PUNCT
ejpam-5572	321	28	y),ν3s2	y),ν3s2	PROPN
ejpam-5572	321	29	(	(	PUNCT
ejpam-5572	321	30	y	y	NOUN
ejpam-5572	321	31	)	)	PUNCT
ejpam-5572	321	32	}	}	PUNCT
ejpam-5572	321	33	)	)	PUNCT
ejpam-5572	321	34	=	=	SYM
ejpam-5572	321	35	(	(	PUNCT
ejpam-5572	321	36	q3s1∩s2	q3s1∩s2	PROPN
ejpam-5572	321	37	(	(	PUNCT
ejpam-5572	321	38	x	x	NOUN
ejpam-5572	321	39	)	)	PUNCT
ejpam-5572	321	40	∨	∨	NUM
ejpam-5572	321	41	q3s1∩s2	q3s1∩s2	PROPN
ejpam-5572	321	42	(	(	PUNCT
ejpam-5572	321	43	y))e	y))e	PROPN
ejpam-5572	321	44	2πi(ν3s1∩s2	2πi(ν3s1∩s2	NUM
ejpam-5572	321	45	(	(	PUNCT
ejpam-5572	321	46	x)∨ν3s1∩s2	x)∨ν3s1∩s2	X
ejpam-5572	321	47	(	(	PUNCT
ejpam-5572	321	48	y	y	NOUN
ejpam-5572	321	49	)	)	PUNCT
ejpam-5572	321	50	)	)	PUNCT
ejpam-5572	322	1	e.a	e.a	PROPN
ejpam-5572	322	2	.	.	PROPN
ejpam-5572	322	3	abuhijleh	abuhijleh	PROPN
ejpam-5572	322	4	,	,	PUNCT
ejpam-5572	322	5	a.	a.	PROPN
ejpam-5572	322	6	alkouri	alkouri	PROPN
ejpam-5572	322	7	/	/	PROPN
ejpam-5572	322	8	eur	eur	PROPN
ejpam-5572	322	9	.	.	PUNCT
ejpam-5572	323	1	j.	j.	PROPN
ejpam-5572	323	2	pure	pure	PROPN
ejpam-5572	323	3	appl	appl	PROPN
ejpam-5572	323	4	.	.	PROPN
ejpam-5572	323	5	math	math	PROPN
ejpam-5572	323	6	,	,	PUNCT
ejpam-5572	323	7	18	18	NUM
ejpam-5572	323	8	(	(	PUNCT
ejpam-5572	323	9	1	1	NUM
ejpam-5572	323	10	)	)	PUNCT
ejpam-5572	323	11	(	(	PUNCT
ejpam-5572	323	12	2025	2025	NUM
ejpam-5572	323	13	)	)	PUNCT
ejpam-5572	323	14	,	,	PUNCT
ejpam-5572	323	15	5572	5572	NUM
ejpam-5572	323	16	10	10	NUM
ejpam-5572	323	17	of	of	ADP
ejpam-5572	323	18	19	19	NUM
ejpam-5572	323	19	=	=	NUM
ejpam-5572	323	20	q2a∩b(x)e	q2a∩b(x)e	PROPN
ejpam-5572	323	21	2πiν2a∩b(x	2πiν2a∩b(x	NUM
ejpam-5572	323	22	)	)	PUNCT
ejpam-5572	323	23	∨	∨	NUM
ejpam-5572	323	24	q2a∩b(y)e	q2a∩b(y)e	PROPN
ejpam-5572	323	25	2πiν2a∩b(y	2πiν2a∩b(y	NUM
ejpam-5572	323	26	)	)	PUNCT
ejpam-5572	323	27	.	.	PUNCT
ejpam-5572	324	1	the	the	DET
ejpam-5572	324	2	union	union	NOUN
ejpam-5572	324	3	of	of	ADP
ejpam-5572	324	4	two	two	NUM
ejpam-5572	324	5	cffsg	cffsg	NOUN
ejpam-5572	324	6	is	be	AUX
ejpam-5572	324	7	not	not	PART
ejpam-5572	324	8	necessary	necessary	ADJ
ejpam-5572	324	9	a	a	DET
ejpam-5572	324	10	cffsg	cffsg	ADJ
ejpam-5572	324	11	,	,	PUNCT
ejpam-5572	324	12	see	see	VERB
ejpam-5572	324	13	the	the	DET
ejpam-5572	324	14	following	follow	VERB
ejpam-5572	324	15	example	example	NOUN
ejpam-5572	324	16	.	.	PUNCT
ejpam-5572	325	1	example	example	NOUN
ejpam-5572	326	1	3	3	X
ejpam-5572	326	2	.	.	PUNCT
ejpam-5572	327	1	let	let	VERB
ejpam-5572	327	2	(	(	PUNCT
ejpam-5572	327	3	x	x	NOUN
ejpam-5572	327	4	,	,	PUNCT
ejpam-5572	327	5	∗	∗	NOUN
ejpam-5572	327	6	)	)	PUNCT
ejpam-5572	327	7	=	=	SYM
ejpam-5572	327	8	(	(	PUNCT
ejpam-5572	327	9	z,+	z,+	NUM
ejpam-5572	327	10	)	)	PUNCT
ejpam-5572	327	11	be	be	VERB
ejpam-5572	327	12	a	a	DET
ejpam-5572	327	13	group	group	NOUN
ejpam-5572	327	14	,	,	PUNCT
ejpam-5572	327	15	also	also	ADV
ejpam-5572	327	16	ϕ1	ϕ1	VERB
ejpam-5572	327	17	=	=	SYM
ejpam-5572	327	18	5z	5z	NOUN
ejpam-5572	327	19	and	and	CCONJ
ejpam-5572	327	20	ϕ2	ϕ2	ADV
ejpam-5572	327	21	=	=	PRON
ejpam-5572	327	22	2z	2z	NOUN
ejpam-5572	327	23	be	be	AUX
ejpam-5572	327	24	two	two	NUM
ejpam-5572	327	25	cffsg	cffsg	NOUN
ejpam-5572	327	26	of	of	ADP
ejpam-5572	327	27	z.	z.	PROPN
ejpam-5572	328	1	where	where	SCONJ
ejpam-5572	328	2	,	,	PUNCT
ejpam-5572	328	3	ϕj	ϕj	PROPN
ejpam-5572	328	4	=	=	SYM
ejpam-5572	328	5	(	(	PUNCT
ejpam-5572	328	6	kϕj	kϕj	NOUN
ejpam-5572	328	7	=	=	SYM
ejpam-5572	328	8	pϕj	pϕj	PROPN
ejpam-5572	328	9	(	(	PUNCT
ejpam-5572	328	10	x)e	x)e	X
ejpam-5572	328	11	2πiωϕj	2πiωϕj	NUM
ejpam-5572	328	12	(	(	PUNCT
ejpam-5572	328	13	x	x	X
ejpam-5572	328	14	)	)	PUNCT
ejpam-5572	328	15	,	,	PUNCT
ejpam-5572	328	16	lϕj	lϕj	NOUN
ejpam-5572	328	17	=	=	PUNCT
ejpam-5572	328	18	qϕj	qϕj	INTJ
ejpam-5572	328	19	(	(	PUNCT
ejpam-5572	328	20	x)e	x)e	X
ejpam-5572	328	21	2πiνϕj	2πiνϕj	NUM
ejpam-5572	328	22	(	(	PUNCT
ejpam-5572	328	23	x	x	NOUN
ejpam-5572	328	24	)	)	PUNCT
ejpam-5572	328	25	)	)	PUNCT
ejpam-5572	328	26	;	;	PUNCT
ejpam-5572	329	1	j	j	PROPN
ejpam-5572	329	2	=	=	SYM
ejpam-5572	329	3	1	1	NUM
ejpam-5572	329	4	,	,	PUNCT
ejpam-5572	329	5	2	2	NUM
ejpam-5572	329	6	.	.	X
ejpam-5572	330	1	they	they	PRON
ejpam-5572	330	2	defined	define	VERB
ejpam-5572	330	3	by	by	ADP
ejpam-5572	330	4	:	:	PUNCT
ejpam-5572	330	5	ϕ1(x	ϕ1(x	NUM
ejpam-5572	330	6	)	)	PUNCT
ejpam-5572	330	7	=	=	SYM
ejpam-5572	330	8	(	(	PUNCT
ejpam-5572	330	9	kϕ1	kϕ1	PROPN
ejpam-5572	330	10	,	,	PUNCT
ejpam-5572	330	11	lϕ1	lϕ1	PROPN
ejpam-5572	330	12	)	)	PUNCT
ejpam-5572	331	1	=	=	PRON
ejpam-5572	331	2	{	{	PUNCT
ejpam-5572	331	3	(	(	PUNCT
ejpam-5572	331	4	0.8e2πi	0.8e2πi	NOUN
ejpam-5572	331	5	0.6	0.6	NUM
ejpam-5572	331	6	,	,	PUNCT
ejpam-5572	331	7	0.7e2πi	0.7e2πi	ADP
ejpam-5572	331	8	0.3	0.3	NUM
ejpam-5572	331	9	)	)	PUNCT
ejpam-5572	331	10	:	:	PUNCT
ejpam-5572	332	1	x	x	PUNCT
ejpam-5572	332	2	∈	∈	NOUN
ejpam-5572	332	3	5z	5z	NOUN
ejpam-5572	332	4	(	(	PUNCT
ejpam-5572	332	5	0.0e2πi	0.0e2πi	NOUN
ejpam-5572	332	6	0.0	0.0	NUM
ejpam-5572	332	7	,	,	PUNCT
ejpam-5572	332	8	0.5e2πi	0.5e2πi	NOUN
ejpam-5572	332	9	0.4	0.4	NUM
ejpam-5572	332	10	)	)	PUNCT
ejpam-5572	332	11	:	:	PUNCT
ejpam-5572	332	12	elsewhere	elsewhere	ADV
ejpam-5572	332	13	ϕ2(x	ϕ2(x	PROPN
ejpam-5572	332	14	)	)	PUNCT
ejpam-5572	332	15	=	=	PRON
ejpam-5572	332	16	(	(	PUNCT
ejpam-5572	332	17	kϕ2	kϕ2	NOUN
ejpam-5572	332	18	,	,	PUNCT
ejpam-5572	332	19	lϕ2	lϕ2	X
ejpam-5572	332	20	)	)	PUNCT
ejpam-5572	332	21	=	=	SYM
ejpam-5572	332	22	{	{	PUNCT
ejpam-5572	332	23	(	(	PUNCT
ejpam-5572	332	24	0.8e2πi	0.8e2πi	NOUN
ejpam-5572	332	25	0.7	0.7	NUM
ejpam-5572	332	26	,	,	PUNCT
ejpam-5572	332	27	0.4e2πi	0.4e2πi	NOUN
ejpam-5572	332	28	0.2	0.2	NUM
ejpam-5572	332	29	)	)	PUNCT
ejpam-5572	332	30	:	:	PUNCT
ejpam-5572	333	1	x	x	PUNCT
ejpam-5572	333	2	∈	∈	X
ejpam-5572	333	3	7z	7z	NOUN
ejpam-5572	333	4	(	(	PUNCT
ejpam-5572	333	5	0.1e2πi	0.1e2πi	NUM
ejpam-5572	333	6	0.5	0.5	NUM
ejpam-5572	333	7	,	,	PUNCT
ejpam-5572	333	8	0.6e2πi	0.6e2πi	NOUN
ejpam-5572	333	9	0.9	0.9	NUM
ejpam-5572	333	10	)	)	PUNCT
ejpam-5572	333	11	:	:	PUNCT
ejpam-5572	333	12	elsewhere	elsewhere	ADV
ejpam-5572	333	13	then	then	ADV
ejpam-5572	333	14	,	,	PUNCT
ejpam-5572	333	15	we	we	PRON
ejpam-5572	333	16	get	get	VERB
ejpam-5572	333	17	:	:	PUNCT
ejpam-5572	333	18	ϕ	ϕ	NOUN
ejpam-5572	333	19	=	=	PUNCT
ejpam-5572	333	20	ϕ1	ϕ1	NOUN
ejpam-5572	333	21	∪	∪	ADJ
ejpam-5572	333	22	ϕ2	ϕ2	ADV
ejpam-5572	333	23	=	=	PUNCT
ejpam-5572	333	24			PUNCT
ejpam-5572	333	25	(	(	PUNCT
ejpam-5572	333	26	0.8e2πi	0.8e2πi	NOUN
ejpam-5572	333	27	0.7	0.7	NUM
ejpam-5572	333	28	,	,	PUNCT
ejpam-5572	333	29	0.4e2πi	0.4e2πi	NOUN
ejpam-5572	333	30	0.2	0.2	NUM
ejpam-5572	333	31	)	)	PUNCT
ejpam-5572	333	32	:	:	PUNCT
ejpam-5572	334	1	x	x	PUNCT
ejpam-5572	334	2	∈	∈	X
ejpam-5572	334	3	7z	7z	PROPN
ejpam-5572	334	4	(	(	PUNCT
ejpam-5572	334	5	0.8e2πi	0.8e2πi	NOUN
ejpam-5572	334	6	0.6	0.6	NUM
ejpam-5572	334	7	,	,	PUNCT
ejpam-5572	334	8	0.7e2πi	0.7e2πi	ADP
ejpam-5572	334	9	0.3	0.3	NUM
ejpam-5572	334	10	)	)	PUNCT
ejpam-5572	334	11	:	:	PUNCT
ejpam-5572	335	1	x	x	PUNCT
ejpam-5572	335	2	∈	∈	PROPN
ejpam-5572	336	1	5z−	5z−	NUM
ejpam-5572	336	2	7z	7z	PROPN
ejpam-5572	336	3	(	(	PUNCT
ejpam-5572	336	4	0.0e2πi	0.0e2πi	NOUN
ejpam-5572	336	5	0.0	0.0	NUM
ejpam-5572	336	6	,	,	PUNCT
ejpam-5572	336	7	0.5e2πi	0.5e2πi	NOUN
ejpam-5572	336	8	0.4	0.4	NUM
ejpam-5572	336	9	)	)	PUNCT
ejpam-5572	336	10	:	:	PUNCT
ejpam-5572	336	11	elsewhere	elsewhere	ADV
ejpam-5572	336	12	now	now	ADV
ejpam-5572	336	13	,	,	PUNCT
ejpam-5572	336	14	it	it	PRON
ejpam-5572	336	15	is	be	AUX
ejpam-5572	336	16	easy	easy	ADJ
ejpam-5572	336	17	to	to	PART
ejpam-5572	336	18	check	check	VERB
ejpam-5572	336	19	that	that	PRON
ejpam-5572	336	20	ϕ	ϕ	NOUN
ejpam-5572	336	21	is	be	AUX
ejpam-5572	336	22	cffs	cff	NOUN
ejpam-5572	336	23	.	.	PUNCT
ejpam-5572	337	1	then	then	ADV
ejpam-5572	337	2	,	,	PUNCT
ejpam-5572	337	3	consider	consider	VERB
ejpam-5572	337	4	definition	definition	NOUN
ejpam-5572	337	5	9	9	NUM
ejpam-5572	337	6	to	to	PART
ejpam-5572	337	7	check	check	VERB
ejpam-5572	337	8	if	if	SCONJ
ejpam-5572	337	9	ϕ	ϕ	NOUN
ejpam-5572	337	10	is	be	AUX
ejpam-5572	337	11	cffsg	cffsg	ADJ
ejpam-5572	337	12	:	:	PUNCT
ejpam-5572	337	13	for	for	ADP
ejpam-5572	337	14	x1	x1	PROPN
ejpam-5572	337	15	=	=	SYM
ejpam-5572	337	16	15	15	NUM
ejpam-5572	337	17	and	and	CCONJ
ejpam-5572	337	18	x2	x2	NOUN
ejpam-5572	337	19	=	=	PUNCT
ejpam-5572	337	20	−7	−7	PROPN
ejpam-5572	337	21	,	,	PUNCT
ejpam-5572	337	22	then	then	ADV
ejpam-5572	337	23	:	:	PUNCT
ejpam-5572	337	24	<	<	X
ejpam-5572	337	25	k3	k3	X
ejpam-5572	337	26	ϕ(15	ϕ(15	PROPN
ejpam-5572	338	1	+	+	CCONJ
ejpam-5572	338	2	−7	−7	NOUN
ejpam-5572	338	3	)	)	PUNCT
ejpam-5572	338	4	,	,	PUNCT
ejpam-5572	338	5	l3	l3	PROPN
ejpam-5572	338	6	ϕ(15	ϕ(15	PUNCT
ejpam-5572	339	1	+	+	CCONJ
ejpam-5572	339	2	−7	−7	NOUN
ejpam-5572	339	3	)	)	PUNCT
ejpam-5572	339	4	>	>	X
ejpam-5572	340	1	=	=	PUNCT
ejpam-5572	340	2	<	<	X
ejpam-5572	340	3	k3	k3	PROPN
ejpam-5572	340	4	ϕ(8	ϕ(8	PROPN
ejpam-5572	340	5	)	)	PUNCT
ejpam-5572	340	6	,	,	PUNCT
ejpam-5572	340	7	l3	l3	PROPN
ejpam-5572	340	8	ϕ(8	ϕ(8	PROPN
ejpam-5572	340	9	)	)	PUNCT
ejpam-5572	340	10	>	>	X
ejpam-5572	341	1	=	=	PUNCT
ejpam-5572	341	2	<	<	X
ejpam-5572	341	3	0.0e2πi	0.0e2πi	X
ejpam-5572	341	4	0.0	0.0	NUM
ejpam-5572	341	5	,	,	PUNCT
ejpam-5572	341	6	0.125e2πi	0.125e2πi	NOUN
ejpam-5572	341	7	0.064	0.064	NUM
ejpam-5572	341	8	>	>	PUNCT
ejpam-5572	341	9	and	and	CCONJ
ejpam-5572	341	10	,	,	PUNCT
ejpam-5572	341	11	<	<	X
ejpam-5572	341	12	k3	k3	PROPN
ejpam-5572	341	13	ϕ(15	ϕ(15	PROPN
ejpam-5572	341	14	)	)	PUNCT
ejpam-5572	341	15	∧k3	∧k3	NOUN
ejpam-5572	341	16	ϕ(−7	ϕ(−7	NOUN
ejpam-5572	341	17	)	)	PUNCT
ejpam-5572	341	18	,	,	PUNCT
ejpam-5572	341	19	l3	l3	PROPN
ejpam-5572	341	20	ϕ(15	ϕ(15	PROPN
ejpam-5572	341	21	)	)	PUNCT
ejpam-5572	341	22	∨	∨	NUM
ejpam-5572	341	23	l3	l3	PROPN
ejpam-5572	341	24	ϕ(−7	ϕ(−7	PROPN
ejpam-5572	341	25	)	)	PUNCT
ejpam-5572	341	26	>	>	X
ejpam-5572	342	1	=	=	PUNCT
ejpam-5572	342	2	<	<	X
ejpam-5572	342	3	0.512e2πi	0.512e2πi	X
ejpam-5572	342	4	0.216	0.216	NUM
ejpam-5572	342	5	∧	∧	PROPN
ejpam-5572	342	6	0.512e2πi	0.512e2πi	X
ejpam-5572	342	7	0.343	0.343	NUM
ejpam-5572	342	8	,	,	PUNCT
ejpam-5572	342	9	0.343e2πi	0.343e2πi	NUM
ejpam-5572	342	10	0.027	0.027	NUM
ejpam-5572	342	11	∨	∨	NUM
ejpam-5572	342	12	0.064e2πi	0.064e2πi	NOUN
ejpam-5572	342	13	0.008	0.008	NUM
ejpam-5572	342	14	>	>	X
ejpam-5572	342	15	=	=	PUNCT
ejpam-5572	342	16	<	<	X
ejpam-5572	342	17	0.512e2πi	0.512e2πi	X
ejpam-5572	342	18	0.216	0.216	NUM
ejpam-5572	342	19	,	,	PUNCT
ejpam-5572	342	20	0.343e2πi	0.343e2πi	NOUN
ejpam-5572	342	21	0.027	0.027	NUM
ejpam-5572	342	22	>	>	X
ejpam-5572	343	1	but	but	CCONJ
ejpam-5572	343	2	,	,	PUNCT
ejpam-5572	343	3	k3	k3	VERB
ejpam-5572	343	4	ϕ(15	ϕ(15	PROPN
ejpam-5572	344	1	+	+	CCONJ
ejpam-5572	344	2	−7	−7	NOUN
ejpam-5572	344	3	)	)	PUNCT
ejpam-5572	344	4	≱	≱	PROPN
ejpam-5572	344	5	k3	k3	VERB
ejpam-5572	344	6	ϕ(15	ϕ(15	PROPN
ejpam-5572	344	7	)	)	PUNCT
ejpam-5572	344	8	∧k3	∧k3	NOUN
ejpam-5572	344	9	ϕ(−7	ϕ(−7	NOUN
ejpam-5572	344	10	)	)	PUNCT
ejpam-5572	344	11	;	;	PUNCT
ejpam-5572	344	12	0.0e2πi	0.0e2πi	PROPN
ejpam-5572	344	13	0.0	0.0	NUM
ejpam-5572	344	14	≱	≱	PROPN
ejpam-5572	344	15	0.512e2πi	0.512e2πi	NOUN
ejpam-5572	344	16	0.216	0.216	NUM
ejpam-5572	344	17	and	and	CCONJ
ejpam-5572	344	18	l3	l3	NOUN
ejpam-5572	344	19	ϕ(15	ϕ(15	PUNCT
ejpam-5572	345	1	+	+	CCONJ
ejpam-5572	345	2	−7	−7	NOUN
ejpam-5572	345	3	)	)	PUNCT
ejpam-5572	345	4	≰	≰	PROPN
ejpam-5572	345	5	l3	l3	PROPN
ejpam-5572	345	6	ϕ(15	ϕ(15	PUNCT
ejpam-5572	345	7	)	)	PUNCT
ejpam-5572	346	1	∨	∨	NUM
ejpam-5572	346	2	l3	l3	PROPN
ejpam-5572	346	3	ϕ(−7	ϕ(−7	PROPN
ejpam-5572	346	4	)	)	PUNCT
ejpam-5572	346	5	;	;	PUNCT
ejpam-5572	347	1	0.125e2πi	0.125e2πi	PROPN
ejpam-5572	347	2	0.064	0.064	NUM
ejpam-5572	347	3	≰	≰	PROPN
ejpam-5572	347	4	0.343e2πi	0.343e2πi	X
ejpam-5572	347	5	0.027	0.027	NUM
ejpam-5572	347	6	.	.	PUNCT
ejpam-5572	348	1	therefore	therefore	ADV
ejpam-5572	348	2	,	,	PUNCT
ejpam-5572	348	3	ϕ	ϕ	X
ejpam-5572	348	4	=	=	SYM
ejpam-5572	348	5	φ1	φ1	PROPN
ejpam-5572	348	6	∪	∪	ADP
ejpam-5572	348	7	φ2	φ2	PROPN
ejpam-5572	348	8	is	be	AUX
ejpam-5572	348	9	not	not	PART
ejpam-5572	348	10	a	a	DET
ejpam-5572	348	11	cffsg	cffsg	NOUN
ejpam-5572	348	12	of	of	ADP
ejpam-5572	348	13	(	(	PUNCT
ejpam-5572	348	14	z,+	z,+	NUM
ejpam-5572	348	15	)	)	PUNCT
ejpam-5572	348	16	.	.	PUNCT
ejpam-5572	349	1	proposition	proposition	NOUN
ejpam-5572	349	2	4	4	NUM
ejpam-5572	349	3	.	.	X
ejpam-5572	350	1	for	for	ADP
ejpam-5572	350	2	a	a	DET
ejpam-5572	350	3	cffs	cff	NOUN
ejpam-5572	350	4	ϕ	ϕ	NOUN
ejpam-5572	350	5	=	=	PUNCT
ejpam-5572	350	6	(	(	PUNCT
ejpam-5572	350	7	pe2πiω	pe2πiω	NOUN
ejpam-5572	350	8	,	,	PUNCT
ejpam-5572	350	9	qe2πiν	qe2πiν	NOUN
ejpam-5572	350	10	)	)	PUNCT
ejpam-5572	350	11	of	of	ADP
ejpam-5572	350	12	a	a	DET
ejpam-5572	350	13	group	group	NOUN
ejpam-5572	350	14	(	(	PUNCT
ejpam-5572	350	15	x	x	X
ejpam-5572	350	16	,	,	PUNCT
ejpam-5572	350	17	∗	∗	NOUN
ejpam-5572	350	18	)	)	PUNCT
ejpam-5572	350	19	.	.	PUNCT
ejpam-5572	351	1	then	then	ADV
ejpam-5572	351	2	p3(x	p3(x	PRON
ejpam-5572	351	3	∗	∗	NOUN
ejpam-5572	351	4	x	x	SYM
ejpam-5572	351	5	∗	∗	NOUN
ejpam-5572	351	6	·	·	PUNCT
ejpam-5572	351	7	·	·	PUNCT
ejpam-5572	351	8	·	·	PUNCT
ejpam-5572	351	9	∗	∗	NOUN
ejpam-5572	351	10	x)e2πiω	x)e2πiω	PROPN
ejpam-5572	351	11	3(x∗x∗···∗x	3(x∗x∗···∗x	NUM
ejpam-5572	351	12	)	)	PUNCT
ejpam-5572	351	13	≥	≥	NOUN
ejpam-5572	351	14	p3(x)e2πiω	p3(x)e2πiω	NOUN
ejpam-5572	351	15	3(x	3(x	NUM
ejpam-5572	351	16	)	)	PUNCT
ejpam-5572	351	17	,	,	PUNCT
ejpam-5572	351	18	where	where	SCONJ
ejpam-5572	351	19	p3(x	p3(x	X
ejpam-5572	351	20	∗x	∗x	NOUN
ejpam-5572	351	21	∗	∗	NOUN
ejpam-5572	351	22	·	·	PUNCT
ejpam-5572	351	23	·	·	PUNCT
ejpam-5572	351	24	·	·	PUNCT
ejpam-5572	351	25	∗x	∗x	X
ejpam-5572	351	26	)	)	PUNCT
ejpam-5572	351	27	≥	≥	NOUN
ejpam-5572	351	28	p3(x	p3(x	SYM
ejpam-5572	351	29	)	)	PUNCT
ejpam-5572	351	30	and	and	CCONJ
ejpam-5572	351	31	ω3(x	ω3(x	NUM
ejpam-5572	351	32	∗x	∗x	NOUN
ejpam-5572	351	33	∗	∗	NOUN
ejpam-5572	351	34	·	·	PUNCT
ejpam-5572	351	35	·	·	PUNCT
ejpam-5572	351	36	·	·	PUNCT
ejpam-5572	351	37	∗x	∗x	X
ejpam-5572	351	38	)	)	PUNCT
ejpam-5572	351	39	≥	≥	NOUN
ejpam-5572	351	40	ω3(x	ω3(x	PROPN
ejpam-5572	351	41	)	)	PUNCT
ejpam-5572	351	42	.	.	PUNCT
ejpam-5572	352	1	also	also	ADV
ejpam-5572	352	2	,	,	PUNCT
ejpam-5572	352	3	q3(x∗x∗	q3(x∗x∗	PROPN
ejpam-5572	352	4	·	·	SYM
ejpam-5572	352	5	·	·	PUNCT
ejpam-5572	352	6	·	·	PUNCT
ejpam-5572	352	7	∗x)e2πiν3(x∗x∗···∗x	∗x)e2πiν3(x∗x∗···∗x	NOUN
ejpam-5572	352	8	)	)	PUNCT
ejpam-5572	352	9	≤	≤	NUM
ejpam-5572	352	10	q3(x)e2πiν	q3(x)e2πiν	PROPN
ejpam-5572	352	11	3(x	3(x	NUM
ejpam-5572	352	12	)	)	PUNCT
ejpam-5572	352	13	,	,	PUNCT
ejpam-5572	352	14	where	where	SCONJ
ejpam-5572	352	15	q3(x∗x∗	q3(x∗x∗	NOUN
ejpam-5572	352	16	·	·	SYM
ejpam-5572	352	17	·	·	PUNCT
ejpam-5572	352	18	·	·	PUNCT
ejpam-5572	352	19	∗x	∗x	X
ejpam-5572	352	20	)	)	PUNCT
ejpam-5572	352	21	≤	≤	NUM
ejpam-5572	352	22	q3(x	q3(x	PROPN
ejpam-5572	352	23	)	)	PUNCT
ejpam-5572	352	24	and	and	CCONJ
ejpam-5572	352	25	ν3(x	ν3(x	PROPN
ejpam-5572	352	26	∗	∗	NOUN
ejpam-5572	352	27	x	x	X
ejpam-5572	352	28	∗	∗	NOUN
ejpam-5572	352	29	·	·	PUNCT
ejpam-5572	352	30	·	·	PUNCT
ejpam-5572	352	31	·	·	PUNCT
ejpam-5572	353	1	∗	∗	NOUN
ejpam-5572	353	2	x	x	SYM
ejpam-5572	353	3	)	)	PUNCT
ejpam-5572	353	4	≤	≤	NUM
ejpam-5572	353	5	ν3(x	ν3(x	NOUN
ejpam-5572	353	6	)	)	PUNCT
ejpam-5572	353	7	.	.	PUNCT
ejpam-5572	354	1	proof	proof	NOUN
ejpam-5572	354	2	.	.	PUNCT
ejpam-5572	355	1	by	by	ADP
ejpam-5572	355	2	induction	induction	NOUN
ejpam-5572	355	3	the	the	DET
ejpam-5572	355	4	results	result	NOUN
ejpam-5572	355	5	will	will	AUX
ejpam-5572	355	6	follow	follow	VERB
ejpam-5572	355	7	,	,	PUNCT
ejpam-5572	355	8	such	such	ADJ
ejpam-5572	355	9	that	that	PRON
ejpam-5572	355	10	p3(x∗x)e2πiω3(x∗x	p3(x∗x)e2πiω3(x∗x	NOUN
ejpam-5572	355	11	)	)	PUNCT
ejpam-5572	355	12	≥	≥	NOUN
ejpam-5572	355	13	p3(x)e2πiω	p3(x)e2πiω	NOUN
ejpam-5572	355	14	3(x	3(x	NUM
ejpam-5572	355	15	)	)	PUNCT
ejpam-5572	355	16	,	,	PUNCT
ejpam-5572	355	17	where	where	SCONJ
ejpam-5572	355	18	p3(x	p3(x	PROPN
ejpam-5572	355	19	∗	∗	NOUN
ejpam-5572	355	20	x	x	NOUN
ejpam-5572	355	21	)	)	PUNCT
ejpam-5572	355	22	≥	≥	NOUN
ejpam-5572	355	23	p3(x	p3(x	SYM
ejpam-5572	355	24	)	)	PUNCT
ejpam-5572	355	25	and	and	CCONJ
ejpam-5572	355	26	ω3(x	ω3(x	PROPN
ejpam-5572	355	27	∗	∗	NOUN
ejpam-5572	355	28	x	x	NOUN
ejpam-5572	355	29	)	)	PUNCT
ejpam-5572	355	30	≥	≥	NOUN
ejpam-5572	355	31	ω3(x	ω3(x	PROPN
ejpam-5572	355	32	)	)	PUNCT
ejpam-5572	355	33	.	.	PUNCT
ejpam-5572	356	1	also	also	ADV
ejpam-5572	356	2	,	,	PUNCT
ejpam-5572	356	3	q3(x	q3(x	PROPN
ejpam-5572	356	4	∗	∗	NOUN
ejpam-5572	356	5	x)e2πiν3(x∗x	x)e2πiν3(x∗x	PUNCT
ejpam-5572	356	6	)	)	PUNCT
ejpam-5572	357	1	≤	≤	NUM
ejpam-5572	357	2	q3(x)e2πiν	q3(x)e2πiν	ADP
ejpam-5572	357	3	3(x	3(x	NUM
ejpam-5572	357	4	)	)	PUNCT
ejpam-5572	357	5	,	,	PUNCT
ejpam-5572	357	6	where	where	SCONJ
ejpam-5572	357	7	q3(x	q3(x	PROPN
ejpam-5572	357	8	∗	∗	X
ejpam-5572	357	9	x	x	NOUN
ejpam-5572	357	10	)	)	PUNCT
ejpam-5572	357	11	≤	≤	NUM
ejpam-5572	357	12	q3(x	q3(x	PROPN
ejpam-5572	357	13	)	)	PUNCT
ejpam-5572	357	14	and	and	CCONJ
ejpam-5572	357	15	ν3(x	ν3(x	PROPN
ejpam-5572	357	16	∗	∗	NOUN
ejpam-5572	357	17	x	x	NOUN
ejpam-5572	357	18	)	)	PUNCT
ejpam-5572	357	19	≤	≤	NUM
ejpam-5572	357	20	ν3(x	ν3(x	NOUN
ejpam-5572	357	21	)	)	PUNCT
ejpam-5572	357	22	.	.	PUNCT
ejpam-5572	358	1	theorem	theorem	NOUN
ejpam-5572	358	2	2	2	NUM
ejpam-5572	358	3	.	.	X
ejpam-5572	358	4	for	for	ADP
ejpam-5572	358	5	a	a	DET
ejpam-5572	358	6	cffs	cff	NOUN
ejpam-5572	358	7	ϕ	ϕ	NOUN
ejpam-5572	358	8	=	=	PUNCT
ejpam-5572	358	9	(	(	PUNCT
ejpam-5572	358	10	pe2πiω	pe2πiω	NOUN
ejpam-5572	358	11	,	,	PUNCT
ejpam-5572	358	12	qe2πiν	qe2πiν	NOUN
ejpam-5572	358	13	)	)	PUNCT
ejpam-5572	358	14	of	of	ADP
ejpam-5572	358	15	a	a	DET
ejpam-5572	358	16	group	group	NOUN
ejpam-5572	358	17	(	(	PUNCT
ejpam-5572	358	18	x	x	X
ejpam-5572	358	19	,	,	PUNCT
ejpam-5572	358	20	∗	∗	NOUN
ejpam-5572	358	21	)	)	PUNCT
ejpam-5572	358	22	.	.	PUNCT
ejpam-5572	359	1	the	the	DET
ejpam-5572	359	2	set	set	NOUN
ejpam-5572	359	3	m	m	NOUN
ejpam-5572	359	4	=	=	SYM
ejpam-5572	359	5	{	{	PUNCT
ejpam-5572	359	6	x	x	PUNCT
ejpam-5572	359	7	∈	∈	PROPN
ejpam-5572	359	8	x	x	X
ejpam-5572	359	9	:	:	PUNCT
ejpam-5572	359	10	p3(id)e2πiω	p3(id)e2πiω	X
ejpam-5572	359	11	3(id	3(id	NUM
ejpam-5572	359	12	)	)	PUNCT
ejpam-5572	359	13	=	=	X
ejpam-5572	359	14	p3(x)e2πiω	p3(x)e2πiω	PROPN
ejpam-5572	359	15	3(x	3(x	NUM
ejpam-5572	359	16	)	)	PUNCT
ejpam-5572	359	17	and	and	CCONJ
ejpam-5572	359	18	q3(id)e2πiν	q3(id)e2πiν	NUM
ejpam-5572	359	19	3(id	3(id	NUM
ejpam-5572	359	20	)	)	PUNCT
ejpam-5572	359	21	=	=	PUNCT
ejpam-5572	359	22	q3(x)e2πiν	q3(x)e2πiν	ADP
ejpam-5572	359	23	3(x	3(x	NUM
ejpam-5572	359	24	)	)	PUNCT
ejpam-5572	359	25	}	}	PUNCT
ejpam-5572	359	26	,	,	PUNCT
ejpam-5572	359	27	is	be	AUX
ejpam-5572	359	28	a	a	DET
ejpam-5572	359	29	subgroup	subgroup	NOUN
ejpam-5572	359	30	of	of	ADP
ejpam-5572	359	31	x	x	NOUN
ejpam-5572	359	32	,	,	PUNCT
ejpam-5572	359	33	where	where	SCONJ
ejpam-5572	359	34	i	i	PRON
ejpam-5572	359	35	d	d	PROPN
ejpam-5572	359	36	is	be	AUX
ejpam-5572	359	37	the	the	DET
ejpam-5572	359	38	identity	identity	NOUN
ejpam-5572	359	39	of	of	ADP
ejpam-5572	359	40	it	it	PRON
ejpam-5572	359	41	.	.	PUNCT
ejpam-5572	360	1	e.a	e.a	PROPN
ejpam-5572	360	2	.	.	PROPN
ejpam-5572	360	3	abuhijleh	abuhijleh	PROPN
ejpam-5572	360	4	,	,	PUNCT
ejpam-5572	360	5	a.	a.	PROPN
ejpam-5572	360	6	alkouri	alkouri	PROPN
ejpam-5572	360	7	/	/	PROPN
ejpam-5572	360	8	eur	eur	PROPN
ejpam-5572	360	9	.	.	PUNCT
ejpam-5572	361	1	j.	j.	PROPN
ejpam-5572	361	2	pure	pure	PROPN
ejpam-5572	361	3	appl	appl	PROPN
ejpam-5572	361	4	.	.	PROPN
ejpam-5572	361	5	math	math	PROPN
ejpam-5572	361	6	,	,	PUNCT
ejpam-5572	361	7	18	18	NUM
ejpam-5572	361	8	(	(	PUNCT
ejpam-5572	361	9	1	1	NUM
ejpam-5572	361	10	)	)	PUNCT
ejpam-5572	361	11	(	(	PUNCT
ejpam-5572	361	12	2025	2025	NUM
ejpam-5572	361	13	)	)	PUNCT
ejpam-5572	361	14	,	,	PUNCT
ejpam-5572	361	15	5572	5572	NUM
ejpam-5572	361	16	11	11	NUM
ejpam-5572	361	17	of	of	ADP
ejpam-5572	361	18	19	19	NUM
ejpam-5572	361	19	proof	proof	NOUN
ejpam-5572	361	20	.	.	PUNCT
ejpam-5572	362	1	at	at	ADP
ejpam-5572	362	2	first	first	ADV
ejpam-5572	362	3	,	,	PUNCT
ejpam-5572	362	4	we	we	PRON
ejpam-5572	362	5	have	have	AUX
ejpam-5572	362	6	i	i	NOUN
ejpam-5572	362	7	d	d	PROPN
ejpam-5572	362	8	∈	∈	PROPN
ejpam-5572	362	9	m	m	PROPN
ejpam-5572	362	10	,	,	PUNCT
ejpam-5572	362	11	hence	hence	ADV
ejpam-5572	362	12	m	m	VERB
ejpam-5572	362	13	is	be	AUX
ejpam-5572	362	14	not	not	PART
ejpam-5572	362	15	empty	empty	ADJ
ejpam-5572	362	16	.	.	PUNCT
ejpam-5572	363	1	moreover	moreover	ADV
ejpam-5572	363	2	,	,	PUNCT
ejpam-5572	363	3	we	we	PRON
ejpam-5572	363	4	need	need	VERB
ejpam-5572	363	5	to	to	PART
ejpam-5572	363	6	show	show	VERB
ejpam-5572	363	7	that	that	SCONJ
ejpam-5572	363	8	x	x	PUNCT
ejpam-5572	363	9	∗	∗	PUNCT
ejpam-5572	363	10	y−1	y−1	NOUN
ejpam-5572	363	11	∈	∈	PROPN
ejpam-5572	363	12	m	m	VERB
ejpam-5572	363	13	for	for	ADP
ejpam-5572	363	14	all	all	DET
ejpam-5572	363	15	x	x	NOUN
ejpam-5572	363	16	,	,	PUNCT
ejpam-5572	363	17	y	y	PROPN
ejpam-5572	363	18	∈	∈	PROPN
ejpam-5572	363	19	x.	x.	NOUN
ejpam-5572	363	20	assume	assume	VERB
ejpam-5572	363	21	that	that	SCONJ
ejpam-5572	363	22	x	x	X
ejpam-5572	363	23	,	,	PUNCT
ejpam-5572	363	24	y	y	PROPN
ejpam-5572	363	25	∈	∈	PROPN
ejpam-5572	363	26	m	m	PROPN
ejpam-5572	363	27	,	,	PUNCT
ejpam-5572	363	28	where	where	SCONJ
ejpam-5572	363	29	ϕ	ϕ	NOUN
ejpam-5572	363	30	is	be	AUX
ejpam-5572	363	31	cffsg	cffsg	ADJ
ejpam-5572	363	32	of	of	ADP
ejpam-5572	363	33	x	x	NOUN
ejpam-5572	363	34	,	,	PUNCT
ejpam-5572	363	35	then	then	ADV
ejpam-5572	363	36	,	,	PUNCT
ejpam-5572	363	37	by	by	ADP
ejpam-5572	363	38	proposition	proposition	NOUN
ejpam-5572	363	39	2	2	NUM
ejpam-5572	363	40	,	,	PUNCT
ejpam-5572	363	41	p3(x∗y−1)e2πiω	p3(x∗y−1)e2πiω	NOUN
ejpam-5572	363	42	3(x∗y−1	3(x∗y−1	NUM
ejpam-5572	363	43	)	)	PUNCT
ejpam-5572	363	44	≥	≥	NOUN
ejpam-5572	363	45	p3(x)e2πiω	p3(x)e2πiω	PROPN
ejpam-5572	363	46	3(x	3(x	NUM
ejpam-5572	363	47	)	)	PUNCT
ejpam-5572	363	48	∧	∧	PROPN
ejpam-5572	363	49	p3(y)e2πiω	p3(y)e2πiω	NOUN
ejpam-5572	363	50	3(y	3(y	NUM
ejpam-5572	363	51	)	)	PUNCT
ejpam-5572	363	52	=	=	SYM
ejpam-5572	363	53	p3(id)e2πiω	p3(id)e2πiω	X
ejpam-5572	363	54	3(id	3(id	NUM
ejpam-5572	363	55	)	)	PUNCT
ejpam-5572	363	56	,	,	PUNCT
ejpam-5572	363	57	according	accord	VERB
ejpam-5572	363	58	to	to	ADP
ejpam-5572	363	59	definition	definition	NOUN
ejpam-5572	363	60	of	of	ADP
ejpam-5572	363	61	m.	m.	NOUN
ejpam-5572	363	62	but	but	CCONJ
ejpam-5572	363	63	,	,	PUNCT
ejpam-5572	363	64	x	x	PROPN
ejpam-5572	363	65	∗	∗	X
ejpam-5572	363	66	y−1	y−1	PROPN
ejpam-5572	364	1	∈	∈	PROPN
ejpam-5572	365	1	ϕ	ϕ	NOUN
ejpam-5572	365	2	,	,	PUNCT
ejpam-5572	365	3	hence	hence	ADV
ejpam-5572	365	4	p3(id)e2πiω	p3(id)e2πiω	X
ejpam-5572	365	5	3(id	3(id	NUM
ejpam-5572	365	6	)	)	PUNCT
ejpam-5572	365	7	≥	≥	PART
ejpam-5572	365	8	p3(x	p3(x	NOUN
ejpam-5572	365	9	∗	∗	NOUN
ejpam-5572	365	10	y−1)e2πiω	y−1)e2πiω	NOUN
ejpam-5572	365	11	3(x∗y−1	3(x∗y−1	NUM
ejpam-5572	365	12	)	)	PUNCT
ejpam-5572	365	13	)	)	PUNCT
ejpam-5572	365	14	,	,	PUNCT
ejpam-5572	365	15	by	by	ADP
ejpam-5572	365	16	proposition	proposition	NOUN
ejpam-5572	365	17	1	1	NUM
ejpam-5572	365	18	.	.	PUNCT
ejpam-5572	366	1	so	so	SCONJ
ejpam-5572	366	2	that	that	SCONJ
ejpam-5572	366	3	equality	equality	NOUN
ejpam-5572	366	4	holds	hold	VERB
ejpam-5572	366	5	and	and	CCONJ
ejpam-5572	366	6	p3(id)e2πiω	p3(id)e2πiω	NUM
ejpam-5572	366	7	3(id	3(id	NUM
ejpam-5572	366	8	)	)	PUNCT
ejpam-5572	367	1	=	=	X
ejpam-5572	367	2	p3(x∗y−1)e2πiω	p3(x∗y−1)e2πiω	NOUN
ejpam-5572	367	3	3(x∗y−1	3(x∗y−1	NUM
ejpam-5572	367	4	)	)	PUNCT
ejpam-5572	367	5	)	)	PUNCT
ejpam-5572	367	6	.	.	PUNCT
ejpam-5572	368	1	similarly	similarly	ADV
ejpam-5572	368	2	,	,	PUNCT
ejpam-5572	368	3	we	we	PRON
ejpam-5572	368	4	can	can	AUX
ejpam-5572	368	5	prove	prove	VERB
ejpam-5572	368	6	that	that	SCONJ
ejpam-5572	368	7	q3(id)e2πiν	q3(id)e2πiν	NUM
ejpam-5572	368	8	3(id	3(id	NUM
ejpam-5572	368	9	)	)	PUNCT
ejpam-5572	369	1	=	=	PUNCT
ejpam-5572	370	1	q3(x	q3(x	PROPN
ejpam-5572	370	2	∗	∗	NOUN
ejpam-5572	370	3	y−1)e2πiν	y−1)e2πiν	PROPN
ejpam-5572	371	1	3(x∗y−1	3(x∗y−1	NUM
ejpam-5572	371	2	)	)	PUNCT
ejpam-5572	371	3	,	,	PUNCT
ejpam-5572	371	4	by	by	ADP
ejpam-5572	371	5	proposition	proposition	NOUN
ejpam-5572	371	6	1	1	NUM
ejpam-5572	371	7	and	and	CCONJ
ejpam-5572	371	8	proposition	proposition	NOUN
ejpam-5572	371	9	2	2	NUM
ejpam-5572	371	10	.	.	PUNCT
ejpam-5572	372	1	so	so	ADV
ejpam-5572	372	2	that	that	SCONJ
ejpam-5572	372	3	x	x	PUNCT
ejpam-5572	372	4	∗	∗	PUNCT
ejpam-5572	372	5	y−1	y−1	NOUN
ejpam-5572	372	6	∈	∈	PROPN
ejpam-5572	372	7	m	m	NOUN
ejpam-5572	372	8	and	and	CCONJ
ejpam-5572	372	9	m	m	PROPN
ejpam-5572	372	10	is	be	AUX
ejpam-5572	372	11	subgroup	subgroup	NOUN
ejpam-5572	372	12	of	of	ADP
ejpam-5572	372	13	x.	x.	PROPN
ejpam-5572	372	14	4	4	NUM
ejpam-5572	372	15	.	.	PUNCT
ejpam-5572	372	16	complex	complex	ADJ
ejpam-5572	372	17	fermatean	fermatean	ADJ
ejpam-5572	372	18	fuzzy	fuzzy	ADJ
ejpam-5572	372	19	normal	normal	ADJ
ejpam-5572	372	20	subgroup	subgroup	NOUN
ejpam-5572	372	21	in	in	ADP
ejpam-5572	372	22	this	this	DET
ejpam-5572	372	23	section	section	NOUN
ejpam-5572	372	24	,	,	PUNCT
ejpam-5572	372	25	we	we	PRON
ejpam-5572	372	26	define	define	VERB
ejpam-5572	372	27	complex	complex	ADJ
ejpam-5572	372	28	fermatean	fermatean	ADJ
ejpam-5572	372	29	fuzzy	fuzzy	ADJ
ejpam-5572	372	30	normal	normal	ADJ
ejpam-5572	372	31	subgroup	subgroup	NOUN
ejpam-5572	372	32	(	(	PUNCT
ejpam-5572	372	33	cffnsg	cffnsg	ADJ
ejpam-5572	372	34	)	)	PUNCT
ejpam-5572	372	35	and	and	CCONJ
ejpam-5572	372	36	give	give	VERB
ejpam-5572	372	37	equivalent	equivalent	ADJ
ejpam-5572	372	38	conditions	condition	NOUN
ejpam-5572	372	39	and	and	CCONJ
ejpam-5572	372	40	some	some	DET
ejpam-5572	372	41	properties	property	NOUN
ejpam-5572	372	42	for	for	ADP
ejpam-5572	372	43	it	it	PRON
ejpam-5572	372	44	.	.	PUNCT
ejpam-5572	373	1	definition	definition	NOUN
ejpam-5572	373	2	10	10	NUM
ejpam-5572	373	3	.	.	PUNCT
ejpam-5572	374	1	let	let	VERB
ejpam-5572	374	2	ϕ	ϕ	X
ejpam-5572	374	3	=	=	X
ejpam-5572	374	4	(	(	PUNCT
ejpam-5572	374	5	pe2πiω	pe2πiω	NOUN
ejpam-5572	374	6	,	,	PUNCT
ejpam-5572	374	7	qe2πiν	qe2πiν	NOUN
ejpam-5572	374	8	)	)	PUNCT
ejpam-5572	374	9	be	be	VERB
ejpam-5572	374	10	a	a	DET
ejpam-5572	374	11	cffsg	cffsg	NOUN
ejpam-5572	374	12	of	of	ADP
ejpam-5572	374	13	a	a	DET
ejpam-5572	374	14	group	group	NOUN
ejpam-5572	374	15	(	(	PUNCT
ejpam-5572	374	16	x	x	X
ejpam-5572	374	17	,	,	PUNCT
ejpam-5572	374	18	∗	∗	NOUN
ejpam-5572	374	19	)	)	PUNCT
ejpam-5572	374	20	.	.	PUNCT
ejpam-5572	375	1	then	then	ADV
ejpam-5572	375	2	for	for	ADP
ejpam-5572	375	3	z	z	PROPN
ejpam-5572	375	4	∈	∈	PROPN
ejpam-5572	375	5	x	x	NOUN
ejpam-5572	375	6	,	,	PUNCT
ejpam-5572	375	7	the	the	DET
ejpam-5572	375	8	complex	complex	ADJ
ejpam-5572	375	9	fermatean	fermatean	ADJ
ejpam-5572	375	10	fuzzy	fuzzy	NOUN
ejpam-5572	375	11	left	leave	VERB
ejpam-5572	375	12	coset	coset	NOUN
ejpam-5572	375	13	of	of	ADP
ejpam-5572	375	14	ϕ	ϕ	NOUN
ejpam-5572	375	15	is	be	AUX
ejpam-5572	375	16	the	the	DET
ejpam-5572	375	17	cffs	cff	NOUN
ejpam-5572	375	18	zϕ	zϕ	NOUN
ejpam-5572	376	1	=	=	SYM
ejpam-5572	377	1	(	(	PUNCT
ejpam-5572	377	2	(	(	PUNCT
ejpam-5572	377	3	zp)e2πi(zω	zp)e2πi(zω	NOUN
ejpam-5572	377	4	)	)	PUNCT
ejpam-5572	377	5	,	,	PUNCT
ejpam-5572	377	6	(	(	PUNCT
ejpam-5572	377	7	zq)e2πi(zν	zq)e2πi(zν	NOUN
ejpam-5572	377	8	)	)	PUNCT
ejpam-5572	377	9	)	)	PUNCT
ejpam-5572	377	10	,	,	PUNCT
ejpam-5572	377	11	which	which	PRON
ejpam-5572	377	12	defined	define	VERB
ejpam-5572	377	13	for	for	ADP
ejpam-5572	377	14	membership	membership	NOUN
ejpam-5572	377	15	by	by	ADP
ejpam-5572	377	16	,	,	PUNCT
ejpam-5572	377	17	(	(	PUNCT
ejpam-5572	377	18	zp3(x))e2πi(zω	zp3(x))e2πi(zω	NOUN
ejpam-5572	377	19	3(x	3(x	NUM
ejpam-5572	377	20	)	)	PUNCT
ejpam-5572	377	21	)	)	PUNCT
ejpam-5572	378	1	=	=	SYM
ejpam-5572	378	2	p3(z−1	p3(z−1	PROPN
ejpam-5572	378	3	∗	∗	NOUN
ejpam-5572	378	4	x)e2πiω	x)e2πiω	PROPN
ejpam-5572	378	5	3(z−1	3(z−1	PROPN
ejpam-5572	378	6	∗x	∗x	NOUN
ejpam-5572	378	7	)	)	PUNCT
ejpam-5572	378	8	.	.	PUNCT
ejpam-5572	379	1	also	also	ADV
ejpam-5572	379	2	,	,	PUNCT
ejpam-5572	379	3	for	for	ADP
ejpam-5572	379	4	nonmembership	nonmembership	NOUN
ejpam-5572	379	5	it	it	PRON
ejpam-5572	379	6	is	be	AUX
ejpam-5572	379	7	defined	define	VERB
ejpam-5572	379	8	by	by	ADP
ejpam-5572	379	9	,	,	PUNCT
ejpam-5572	379	10	(	(	PUNCT
ejpam-5572	379	11	zq3(x))e2πi(zν	zq3(x))e2πi(zν	NOUN
ejpam-5572	379	12	3(x	3(x	NUM
ejpam-5572	379	13	)	)	PUNCT
ejpam-5572	379	14	)	)	PUNCT
ejpam-5572	380	1	=	=	PUNCT
ejpam-5572	380	2	q3(z−1	q3(z−1	NOUN
ejpam-5572	380	3	∗	∗	NOUN
ejpam-5572	380	4	x)e2πiν3(z−1	x)e2πiν3(z−1	NUM
ejpam-5572	381	1	∗x	∗x	NOUN
ejpam-5572	381	2	)	)	PUNCT
ejpam-5572	381	3	.	.	PUNCT
ejpam-5572	382	1	in	in	ADP
ejpam-5572	382	2	the	the	DET
ejpam-5572	382	3	same	same	ADJ
ejpam-5572	382	4	manner	manner	NOUN
ejpam-5572	382	5	,	,	PUNCT
ejpam-5572	382	6	the	the	DET
ejpam-5572	382	7	complex	complex	ADJ
ejpam-5572	382	8	fermatean	fermatean	NOUN
ejpam-5572	382	9	fuzzy	fuzzy	ADJ
ejpam-5572	382	10	right	right	ADJ
ejpam-5572	382	11	coset	coset	NOUN
ejpam-5572	382	12	of	of	ADP
ejpam-5572	382	13	ϕ	ϕ	NOUN
ejpam-5572	382	14	is	be	AUX
ejpam-5572	382	15	the	the	DET
ejpam-5572	382	16	cffs	cff	NOUN
ejpam-5572	382	17	ϕz	ϕz	NOUN
ejpam-5572	382	18	=	=	SYM
ejpam-5572	382	19	(	(	PUNCT
ejpam-5572	382	20	(	(	PUNCT
ejpam-5572	382	21	pz)e2πi(ωz	pz)e2πi(ωz	NOUN
ejpam-5572	382	22	)	)	PUNCT
ejpam-5572	382	23	,	,	PUNCT
ejpam-5572	382	24	(	(	PUNCT
ejpam-5572	382	25	qz)e2πi(νz	qz)e2πi(νz	PROPN
ejpam-5572	382	26	)	)	PUNCT
ejpam-5572	382	27	)	)	PUNCT
ejpam-5572	382	28	and	and	CCONJ
ejpam-5572	382	29	is	be	AUX
ejpam-5572	382	30	defined	define	VERB
ejpam-5572	382	31	by	by	ADP
ejpam-5572	382	32	(	(	PUNCT
ejpam-5572	382	33	p3(x)z)e2πi(ω	p3(x)z)e2πi(ω	NUM
ejpam-5572	382	34	3(x)z	3(x)z	NUM
ejpam-5572	382	35	)	)	PUNCT
ejpam-5572	383	1	=	=	SYM
ejpam-5572	383	2	p3(x	p3(x	PROPN
ejpam-5572	383	3	∗	∗	X
ejpam-5572	383	4	z−1)e2πiω	z−1)e2πiω	X
ejpam-5572	383	5	3(x	3(x	NUM
ejpam-5572	383	6	∗	∗	NOUN
ejpam-5572	383	7	z−1	z−1	NUM
ejpam-5572	383	8	)	)	PUNCT
ejpam-5572	383	9	and	and	CCONJ
ejpam-5572	383	10	(	(	PUNCT
ejpam-5572	383	11	q3(x)z)e2πi(ν	q3(x)z)e2πi(ν	NOUN
ejpam-5572	383	12	3(x)z	3(x)z	NUM
ejpam-5572	383	13	)	)	PUNCT
ejpam-5572	383	14	=	=	PUNCT
ejpam-5572	384	1	q3(x	q3(x	PROPN
ejpam-5572	384	2	∗	∗	NOUN
ejpam-5572	384	3	z−1)e2πiν	z−1)e2πiν	NOUN
ejpam-5572	384	4	3(x	3(x	NUM
ejpam-5572	384	5	∗	∗	NOUN
ejpam-5572	384	6	z−1	z−1	NUM
ejpam-5572	384	7	)	)	PUNCT
ejpam-5572	384	8	,	,	PUNCT
ejpam-5572	384	9	for	for	ADP
ejpam-5572	384	10	membership	membership	NOUN
ejpam-5572	384	11	and	and	CCONJ
ejpam-5572	384	12	nonmembership	nonmembership	NOUN
ejpam-5572	384	13	,	,	PUNCT
ejpam-5572	384	14	respectively	respectively	ADV
ejpam-5572	384	15	.	.	PUNCT
ejpam-5572	385	1	definition	definition	NOUN
ejpam-5572	385	2	11	11	NUM
ejpam-5572	385	3	.	.	PUNCT
ejpam-5572	386	1	let	let	VERB
ejpam-5572	386	2	ϕ	ϕ	X
ejpam-5572	386	3	=	=	X
ejpam-5572	386	4	(	(	PUNCT
ejpam-5572	386	5	p(x)e2πiω(x	p(x)e2πiω(x	NOUN
ejpam-5572	386	6	)	)	PUNCT
ejpam-5572	386	7	,	,	PUNCT
ejpam-5572	386	8	q(x)e2πiν(x	q(x)e2πiν(x	NOUN
ejpam-5572	386	9	)	)	PUNCT
ejpam-5572	386	10	)	)	PUNCT
ejpam-5572	386	11	be	be	AUX
ejpam-5572	386	12	a	a	DET
ejpam-5572	386	13	cffsg	cffsg	NOUN
ejpam-5572	386	14	of	of	ADP
ejpam-5572	386	15	a	a	DET
ejpam-5572	386	16	group	group	NOUN
ejpam-5572	386	17	(	(	PUNCT
ejpam-5572	386	18	x	x	X
ejpam-5572	386	19	,	,	PUNCT
ejpam-5572	386	20	∗	∗	NOUN
ejpam-5572	386	21	)	)	PUNCT
ejpam-5572	386	22	.	.	PUNCT
ejpam-5572	387	1	then	then	ADV
ejpam-5572	387	2	ϕ	ϕ	PROPN
ejpam-5572	387	3	is	be	AUX
ejpam-5572	387	4	a	a	DET
ejpam-5572	387	5	complex	complex	ADJ
ejpam-5572	387	6	fermatean	fermatean	ADJ
ejpam-5572	387	7	fuzzy	fuzzy	ADJ
ejpam-5572	387	8	normal	normal	ADJ
ejpam-5572	387	9	subgroup	subgroup	NOUN
ejpam-5572	387	10	,	,	PUNCT
ejpam-5572	387	11	of	of	ADP
ejpam-5572	387	12	the	the	DET
ejpam-5572	387	13	group	group	NOUN
ejpam-5572	387	14	(	(	PUNCT
ejpam-5572	387	15	x	x	X
ejpam-5572	387	16	,	,	PUNCT
ejpam-5572	387	17	∗	∗	NOUN
ejpam-5572	387	18	)	)	PUNCT
ejpam-5572	387	19	if	if	SCONJ
ejpam-5572	387	20	every	every	DET
ejpam-5572	387	21	complex	complex	ADJ
ejpam-5572	387	22	fermatean	fermatean	ADJ
ejpam-5572	387	23	fuzzy	fuzzy	NOUN
ejpam-5572	387	24	left	leave	VERB
ejpam-5572	387	25	coset	coset	NOUN
ejpam-5572	387	26	is	be	AUX
ejpam-5572	387	27	complex	complex	ADJ
ejpam-5572	387	28	fermatean	fermatean	ADJ
ejpam-5572	387	29	fuzzy	fuzzy	ADJ
ejpam-5572	387	30	right	right	ADJ
ejpam-5572	387	31	coset	coset	NOUN
ejpam-5572	387	32	of	of	ADP
ejpam-5572	387	33	ϕ	ϕ	NOUN
ejpam-5572	387	34	in	in	ADP
ejpam-5572	387	35	x	x	PROPN
ejpam-5572	387	36	,	,	PUNCT
ejpam-5572	387	37	equivalently	equivalently	ADV
ejpam-5572	387	38	,	,	PUNCT
ejpam-5572	387	39	zϕ	zϕ	PROPN
ejpam-5572	387	40	=	=	SYM
ejpam-5572	387	41	ϕz	ϕz	PROPN
ejpam-5572	387	42	.	.	PUNCT
ejpam-5572	387	43	example	example	NOUN
ejpam-5572	388	1	4	4	NUM
ejpam-5572	388	2	.	.	PUNCT
ejpam-5572	389	1	let	let	AUX
ejpam-5572	389	2	(	(	PUNCT
ejpam-5572	389	3	x	x	NOUN
ejpam-5572	389	4	,	,	PUNCT
ejpam-5572	389	5	∗	∗	NOUN
ejpam-5572	389	6	)	)	PUNCT
ejpam-5572	389	7	=	=	SYM
ejpam-5572	389	8	(	(	PUNCT
ejpam-5572	389	9	z3,+3	z3,+3	X
ejpam-5572	389	10	)	)	PUNCT
ejpam-5572	389	11	be	be	VERB
ejpam-5572	389	12	a	a	DET
ejpam-5572	389	13	group	group	NOUN
ejpam-5572	389	14	with	with	ADP
ejpam-5572	389	15	addition	addition	NOUN
ejpam-5572	389	16	integer	integer	NOUN
ejpam-5572	389	17	modulo	modulo	PROPN
ejpam-5572	389	18	3	3	NUM
ejpam-5572	389	19	.	.	PUNCT
ejpam-5572	389	20	define	define	VERB
ejpam-5572	389	21	a	a	DET
ejpam-5572	389	22	cffs	cff	NOUN
ejpam-5572	389	23	ϕ	ϕ	NOUN
ejpam-5572	389	24	,	,	PUNCT
ejpam-5572	389	25	as	as	SCONJ
ejpam-5572	389	26	follows	follow	VERB
ejpam-5572	389	27	:	:	PUNCT
ejpam-5572	389	28	ϕ	ϕ	NOUN
ejpam-5572	389	29	=	=	SYM
ejpam-5572	389	30	(	(	PUNCT
ejpam-5572	389	31	k(x	k(x	PROPN
ejpam-5572	389	32	)	)	PUNCT
ejpam-5572	389	33	,	,	PUNCT
ejpam-5572	389	34	l(x	l(x	PROPN
ejpam-5572	389	35	)	)	PUNCT
ejpam-5572	389	36	)	)	PUNCT
ejpam-5572	390	1	=	=	PUNCT
ejpam-5572	390	2			PUNCT
ejpam-5572	390	3	(	(	PUNCT
ejpam-5572	390	4	0.9e2πi	0.9e2πi	NOUN
ejpam-5572	390	5	0.7	0.7	NUM
ejpam-5572	390	6	,	,	PUNCT
ejpam-5572	390	7	0.8e2πi	0.8e2πi	NOUN
ejpam-5572	390	8	0.8	0.8	NUM
ejpam-5572	390	9	)	)	PUNCT
ejpam-5572	390	10	:	:	PUNCT
ejpam-5572	391	1	x	x	X
ejpam-5572	391	2	=	=	SYM
ejpam-5572	391	3	0	0	NUM
ejpam-5572	391	4	(	(	PUNCT
ejpam-5572	391	5	0.8e2πi	0.8e2πi	NOUN
ejpam-5572	391	6	0.8	0.8	NUM
ejpam-5572	391	7	,	,	PUNCT
ejpam-5572	391	8	0.7e2πi	0.7e2πi	ADP
ejpam-5572	391	9	0.6	0.6	NUM
ejpam-5572	391	10	)	)	PUNCT
ejpam-5572	391	11	:	:	PUNCT
ejpam-5572	392	1	x	x	X
ejpam-5572	392	2	=	=	SYM
ejpam-5572	392	3	1	1	NUM
ejpam-5572	392	4	(	(	PUNCT
ejpam-5572	392	5	0.3e2πi	0.3e2πi	NOUN
ejpam-5572	392	6	0.6	0.6	NUM
ejpam-5572	392	7	,	,	PUNCT
ejpam-5572	392	8	0.5e2πi	0.5e2πi	NOUN
ejpam-5572	392	9	0.6	0.6	NUM
ejpam-5572	392	10	)	)	PUNCT
ejpam-5572	392	11	:	:	PUNCT
ejpam-5572	393	1	x	x	X
ejpam-5572	393	2	=	=	SYM
ejpam-5572	393	3	2	2	NUM
ejpam-5572	393	4	first	first	ADV
ejpam-5572	393	5	,	,	PUNCT
ejpam-5572	393	6	it	it	PRON
ejpam-5572	393	7	is	be	AUX
ejpam-5572	393	8	easy	easy	ADJ
ejpam-5572	393	9	to	to	PART
ejpam-5572	393	10	see	see	VERB
ejpam-5572	393	11	at	at	ADP
ejpam-5572	393	12	x	x	X
ejpam-5572	393	13	=	=	SYM
ejpam-5572	393	14	0	0	NUM
ejpam-5572	393	15	,	,	PUNCT
ejpam-5572	393	16	ϕ	ϕ	PROPN
ejpam-5572	393	17	is	be	AUX
ejpam-5572	393	18	cffs	cff	NOUN
ejpam-5572	393	19	and	and	CCONJ
ejpam-5572	393	20	not	not	PART
ejpam-5572	393	21	cpfs	cpf	NOUN
ejpam-5572	393	22	(	(	PUNCT
ejpam-5572	393	23	so	so	ADV
ejpam-5572	393	24	,	,	PUNCT
ejpam-5572	393	25	not	not	PART
ejpam-5572	393	26	cifs	cif	VERB
ejpam-5572	393	27	too	too	ADV
ejpam-5572	393	28	)	)	PUNCT
ejpam-5572	393	29	.	.	PUNCT
ejpam-5572	394	1	second	second	ADJ
ejpam-5572	394	2	,	,	PUNCT
ejpam-5572	394	3	to	to	PART
ejpam-5572	394	4	prove	prove	VERB
ejpam-5572	394	5	that	that	SCONJ
ejpam-5572	394	6	ϕ	ϕ	NOUN
ejpam-5572	394	7	is	be	AUX
ejpam-5572	394	8	cffnsg	cffnsg	ADJ
ejpam-5572	394	9	:	:	PUNCT
ejpam-5572	394	10	assume	assume	VERB
ejpam-5572	394	11	that	that	SCONJ
ejpam-5572	394	12	z	z	NOUN
ejpam-5572	394	13	=	=	SYM
ejpam-5572	394	14	2	2	NUM
ejpam-5572	394	15	and	and	CCONJ
ejpam-5572	394	16	x	x	SYM
ejpam-5572	394	17	=	=	SYM
ejpam-5572	394	18	0	0	NUM
ejpam-5572	394	19	and	and	CCONJ
ejpam-5572	394	20	take	take	VERB
ejpam-5572	394	21	2ϕ	2ϕ	NUM
ejpam-5572	394	22	,	,	PUNCT
ejpam-5572	394	23	then	then	ADV
ejpam-5572	394	24	<	<	X
ejpam-5572	394	25	(	(	PUNCT
ejpam-5572	394	26	2k)3(0	2k)3(0	NUM
ejpam-5572	394	27	)	)	PUNCT
ejpam-5572	394	28	,	,	PUNCT
ejpam-5572	394	29	(	(	PUNCT
ejpam-5572	394	30	2l)3(0	2l)3(0	NUM
ejpam-5572	394	31	)	)	PUNCT
ejpam-5572	394	32	>	>	X
ejpam-5572	395	1	=	=	X
ejpam-5572	395	2	<	<	X
ejpam-5572	395	3	k3(2−1	k3(2−1	PROPN
ejpam-5572	395	4	+	+	NOUN
ejpam-5572	395	5	30	30	NUM
ejpam-5572	395	6	)	)	PUNCT
ejpam-5572	395	7	,	,	PUNCT
ejpam-5572	395	8	l3(2−1	l3(2−1	PROPN
ejpam-5572	395	9	+	+	NOUN
ejpam-5572	395	10	30	30	NUM
ejpam-5572	395	11	)	)	PUNCT
ejpam-5572	395	12	>	>	X
ejpam-5572	396	1	=	=	X
ejpam-5572	396	2	<	<	X
ejpam-5572	396	3	p3(2−1	p3(2−1	PROPN
ejpam-5572	396	4	+	+	PROPN
ejpam-5572	396	5	30)e	30)e	NUM
ejpam-5572	396	6	2πiω3(2−1	2πiω3(2−1	NUM
ejpam-5572	396	7	+	+	NOUN
ejpam-5572	396	8	30	30	NUM
ejpam-5572	396	9	)	)	PUNCT
ejpam-5572	396	10	,	,	PUNCT
ejpam-5572	396	11	q3(2−1	q3(2−1	ADP
ejpam-5572	396	12	+	+	NOUN
ejpam-5572	396	13	30)e	30)e	NUM
ejpam-5572	396	14	2πiν3(2−1	2πiν3(2−1	ADP
ejpam-5572	396	15	+	+	NOUN
ejpam-5572	396	16	30	30	NUM
ejpam-5572	396	17	)	)	PUNCT
ejpam-5572	396	18	>	>	X
ejpam-5572	397	1	=	=	X
ejpam-5572	397	2	<	<	X
ejpam-5572	397	3	p3(1	p3(1	X
ejpam-5572	397	4	+	+	PROPN
ejpam-5572	397	5	30)e	30)e	PROPN
ejpam-5572	397	6	2πiω3(1	2πiω3(1	NUM
ejpam-5572	397	7	+	+	NOUN
ejpam-5572	397	8	30	30	NUM
ejpam-5572	397	9	)	)	PUNCT
ejpam-5572	397	10	,	,	PUNCT
ejpam-5572	397	11	q3(1	q3(1	PROPN
ejpam-5572	397	12	+	+	PROPN
ejpam-5572	397	13	30)e	30)e	NUM
ejpam-5572	397	14	2πiν3(1	2πiν3(1	NUM
ejpam-5572	397	15	+	+	NOUN
ejpam-5572	397	16	30	30	NUM
ejpam-5572	397	17	)	)	PUNCT
ejpam-5572	397	18	>	>	X
ejpam-5572	398	1	=	=	PUNCT
ejpam-5572	398	2	<	<	X
ejpam-5572	398	3	p3(1)e2πiω	p3(1)e2πiω	NOUN
ejpam-5572	398	4	3(1	3(1	NUM
ejpam-5572	398	5	)	)	PUNCT
ejpam-5572	398	6	,	,	PUNCT
ejpam-5572	398	7	q3(1)e2πiν	q3(1)e2πiν	PROPN
ejpam-5572	398	8	3(1	3(1	NUM
ejpam-5572	398	9	)	)	PUNCT
ejpam-5572	398	10	>	>	X
ejpam-5572	398	11	e.a	e.a	PROPN
ejpam-5572	398	12	.	.	PROPN
ejpam-5572	398	13	abuhijleh	abuhijleh	PROPN
ejpam-5572	398	14	,	,	PUNCT
ejpam-5572	398	15	a.	a.	PROPN
ejpam-5572	398	16	alkouri	alkouri	PROPN
ejpam-5572	398	17	/	/	PROPN
ejpam-5572	398	18	eur	eur	PROPN
ejpam-5572	398	19	.	.	PUNCT
ejpam-5572	399	1	j.	j.	PROPN
ejpam-5572	399	2	pure	pure	PROPN
ejpam-5572	399	3	appl	appl	PROPN
ejpam-5572	399	4	.	.	PROPN
ejpam-5572	399	5	math	math	PROPN
ejpam-5572	399	6	,	,	PUNCT
ejpam-5572	399	7	18	18	NUM
ejpam-5572	399	8	(	(	PUNCT
ejpam-5572	399	9	1	1	NUM
ejpam-5572	399	10	)	)	PUNCT
ejpam-5572	399	11	(	(	PUNCT
ejpam-5572	399	12	2025	2025	NUM
ejpam-5572	399	13	)	)	PUNCT
ejpam-5572	399	14	,	,	PUNCT
ejpam-5572	399	15	5572	5572	NUM
ejpam-5572	399	16	12	12	NUM
ejpam-5572	399	17	of	of	ADP
ejpam-5572	399	18	19	19	NUM
ejpam-5572	399	19	=	=	NOUN
ejpam-5572	399	20	<	<	X
ejpam-5572	399	21	p3(0	p3(0	PROPN
ejpam-5572	399	22	+	+	PROPN
ejpam-5572	399	23	31)e	31)e	PROPN
ejpam-5572	399	24	2πiω3(0	2πiω3(0	NUM
ejpam-5572	399	25	+	+	NOUN
ejpam-5572	399	26	31	31	NUM
ejpam-5572	399	27	)	)	PUNCT
ejpam-5572	399	28	,	,	PUNCT
ejpam-5572	399	29	q3(0	q3(0	PROPN
ejpam-5572	399	30	+	+	PROPN
ejpam-5572	399	31	31)e	31)e	NUM
ejpam-5572	399	32	2πiν3(0	2πiν3(0	NUM
ejpam-5572	399	33	+	+	NOUN
ejpam-5572	399	34	31	31	NUM
ejpam-5572	399	35	)	)	PUNCT
ejpam-5572	399	36	>	>	X
ejpam-5572	400	1	=	=	PUNCT
ejpam-5572	400	2	<	<	X
ejpam-5572	400	3	p3(0	p3(0	PROPN
ejpam-5572	400	4	+	+	PROPN
ejpam-5572	400	5	32	32	NUM
ejpam-5572	400	6	−1)e2πiω	−1)e2πiω	NOUN
ejpam-5572	400	7	3(0	3(0	NUM
ejpam-5572	401	1	+	+	ADJ
ejpam-5572	401	2	32	32	NUM
ejpam-5572	401	3	−1	−1	NOUN
ejpam-5572	401	4	)	)	PUNCT
ejpam-5572	402	1	,	,	PUNCT
ejpam-5572	402	2	q3(0	q3(0	PROPN
ejpam-5572	402	3	+	+	PROPN
ejpam-5572	402	4	32	32	NUM
ejpam-5572	402	5	−1)e2πiν	−1)e2πiν	NOUN
ejpam-5572	402	6	3(0	3(0	NUM
ejpam-5572	402	7	+	+	ADJ
ejpam-5572	402	8	32	32	NUM
ejpam-5572	402	9	−1	−1	NOUN
ejpam-5572	402	10	)	)	PUNCT
ejpam-5572	402	11	>	>	X
ejpam-5572	403	1	=	=	PUNCT
ejpam-5572	403	2	<	<	X
ejpam-5572	403	3	k3(0	k3(0	PROPN
ejpam-5572	403	4	+	+	PROPN
ejpam-5572	403	5	32	32	NUM
ejpam-5572	403	6	−1),l3(0	−1),l3(0	NOUN
ejpam-5572	403	7	+	+	NOUN
ejpam-5572	403	8	32	32	NUM
ejpam-5572	403	9	−1	−1	NOUN
ejpam-5572	403	10	)	)	PUNCT
ejpam-5572	403	11	>	>	X
ejpam-5572	404	1	=	=	X
ejpam-5572	404	2	<	<	X
ejpam-5572	404	3	(	(	PUNCT
ejpam-5572	404	4	k2)3(0	k2)3(0	PROPN
ejpam-5572	404	5	)	)	PUNCT
ejpam-5572	404	6	,	,	PUNCT
ejpam-5572	404	7	(	(	PUNCT
ejpam-5572	404	8	l2)3(0	l2)3(0	PROPN
ejpam-5572	404	9	)	)	PUNCT
ejpam-5572	404	10	>	>	X
ejpam-5572	404	11	so	so	SCONJ
ejpam-5572	404	12	that	that	SCONJ
ejpam-5572	404	13	,	,	PUNCT
ejpam-5572	404	14	2ϕ	2ϕ	NUM
ejpam-5572	404	15	=	=	SYM
ejpam-5572	404	16	ϕ2	ϕ2	ADV
ejpam-5572	404	17	.	.	PUNCT
ejpam-5572	405	1	hence	hence	ADV
ejpam-5572	405	2	,	,	PUNCT
ejpam-5572	405	3	for	for	ADP
ejpam-5572	405	4	all	all	PRON
ejpam-5572	405	5	x	x	SYM
ejpam-5572	405	6	=	=	SYM
ejpam-5572	405	7	0	0	NUM
ejpam-5572	405	8	,	,	PUNCT
ejpam-5572	405	9	1	1	NUM
ejpam-5572	405	10	,	,	PUNCT
ejpam-5572	405	11	2	2	NUM
ejpam-5572	405	12	we	we	PRON
ejpam-5572	405	13	can	can	AUX
ejpam-5572	405	14	verify	verify	VERB
ejpam-5572	405	15	that	that	PRON
ejpam-5572	405	16	zϕ	zϕ	PROPN
ejpam-5572	405	17	=	=	SYM
ejpam-5572	405	18	ϕz	ϕz	PROPN
ejpam-5572	405	19	,	,	PUNCT
ejpam-5572	405	20	where	where	SCONJ
ejpam-5572	405	21	z	z	PROPN
ejpam-5572	405	22	∈	∈	PROPN
ejpam-5572	405	23	x	x	X
ejpam-5572	405	24	,	,	PUNCT
ejpam-5572	405	25	i.e.	i.e.	X
ejpam-5572	405	26	ϕ	ϕ	NOUN
ejpam-5572	405	27	is	be	AUX
ejpam-5572	405	28	cffnsg	cffnsg	ADJ
ejpam-5572	405	29	of	of	ADP
ejpam-5572	405	30	the	the	DET
ejpam-5572	405	31	group	group	NOUN
ejpam-5572	405	32	(	(	PUNCT
ejpam-5572	405	33	z3,+3	z3,+3	PROPN
ejpam-5572	405	34	)	)	PUNCT
ejpam-5572	405	35	.	.	PUNCT
ejpam-5572	406	1	proposition	proposition	NOUN
ejpam-5572	406	2	5	5	NUM
ejpam-5572	406	3	.	.	PUNCT
ejpam-5572	407	1	let	let	VERB
ejpam-5572	407	2	ϕ	ϕ	NOUN
ejpam-5572	407	3	=	=	X
ejpam-5572	407	4	(	(	PUNCT
ejpam-5572	407	5	pe2πiω	pe2πiω	NOUN
ejpam-5572	407	6	,	,	PUNCT
ejpam-5572	407	7	qe2πiν	qe2πiν	NOUN
ejpam-5572	407	8	)	)	PUNCT
ejpam-5572	407	9	be	be	VERB
ejpam-5572	407	10	a	a	DET
ejpam-5572	407	11	cffsg	cffsg	NOUN
ejpam-5572	407	12	of	of	ADP
ejpam-5572	407	13	a	a	DET
ejpam-5572	407	14	group	group	NOUN
ejpam-5572	407	15	(	(	PUNCT
ejpam-5572	407	16	x	x	X
ejpam-5572	407	17	,	,	PUNCT
ejpam-5572	407	18	∗	∗	NOUN
ejpam-5572	407	19	)	)	PUNCT
ejpam-5572	407	20	.	.	PUNCT
ejpam-5572	408	1	then	then	ADV
ejpam-5572	408	2	ϕ	ϕ	PROPN
ejpam-5572	408	3	is	be	AUX
ejpam-5572	408	4	a	a	DET
ejpam-5572	408	5	cffnsg	cffnsg	NOUN
ejpam-5572	408	6	of	of	ADP
ejpam-5572	408	7	x	x	SYM
ejpam-5572	408	8	if	if	SCONJ
ejpam-5572	409	1	and	and	CCONJ
ejpam-5572	409	2	only	only	ADV
ejpam-5572	409	3	if	if	SCONJ
ejpam-5572	409	4	p3(z1	p3(z1	NOUN
ejpam-5572	409	5	∗	∗	NOUN
ejpam-5572	409	6	z2)e	z2)e	PROPN
ejpam-5572	409	7	2πiω3(z1	2πiω3(z1	PROPN
ejpam-5572	409	8	∗	∗	NOUN
ejpam-5572	409	9	z2	z2	NUM
ejpam-5572	409	10	)	)	PUNCT
ejpam-5572	409	11	=	=	SYM
ejpam-5572	409	12	p3(z2	p3(z2	PROPN
ejpam-5572	409	13	∗	∗	NOUN
ejpam-5572	409	14	z1)e	z1)e	NOUN
ejpam-5572	409	15	2πiω3(z2	2πiω3(z2	NUM
ejpam-5572	409	16	∗	∗	NOUN
ejpam-5572	409	17	z1	z1	PROPN
ejpam-5572	409	18	)	)	PUNCT
ejpam-5572	409	19	and	and	CCONJ
ejpam-5572	409	20	q3(z1	q3(z1	ADJ
ejpam-5572	409	21	∗	∗	NOUN
ejpam-5572	409	22	z2)e	z2)e	PROPN
ejpam-5572	409	23	2πiν3(z1	2πiν3(z1	NUM
ejpam-5572	409	24	∗	∗	NOUN
ejpam-5572	409	25	z2	z2	NUM
ejpam-5572	409	26	)	)	PUNCT
ejpam-5572	409	27	=	=	PUNCT
ejpam-5572	410	1	q3(z2	q3(z2	NOUN
ejpam-5572	410	2	∗	∗	NOUN
ejpam-5572	410	3	z1)e	z1)e	NOUN
ejpam-5572	410	4	2πiν3(z2	2πiν3(z2	NUM
ejpam-5572	410	5	∗	∗	NOUN
ejpam-5572	410	6	z1	z1	NUM
ejpam-5572	410	7	)	)	PUNCT
ejpam-5572	410	8	.	.	PUNCT
ejpam-5572	411	1	proof	proof	NOUN
ejpam-5572	411	2	.	.	PUNCT
ejpam-5572	412	1	⇒assume	⇒assume	ADJ
ejpam-5572	412	2	that	that	SCONJ
ejpam-5572	412	3	ϕ	ϕ	NOUN
ejpam-5572	412	4	is	be	AUX
ejpam-5572	412	5	a	a	DET
ejpam-5572	412	6	cffnsg	cffnsg	NOUN
ejpam-5572	412	7	of	of	ADP
ejpam-5572	412	8	x	x	NOUN
ejpam-5572	412	9	,	,	PUNCT
ejpam-5572	412	10	then	then	ADV
ejpam-5572	412	11	(	(	PUNCT
ejpam-5572	412	12	z2p	z2p	X
ejpam-5572	412	13	3(z1))e	3(z1))e	NOUN
ejpam-5572	412	14	2πi(z2ω3(z1	2πi(z2ω3(z1	NUM
ejpam-5572	412	15	)	)	PUNCT
ejpam-5572	412	16	)	)	PUNCT
ejpam-5572	413	1	=	=	PRON
ejpam-5572	413	2	(	(	PUNCT
ejpam-5572	413	3	p3(z1)z2)e	p3(z1)z2)e	NOUN
ejpam-5572	413	4	2πi(ω3(z1)z2	2πi(ω3(z1)z2	NUM
ejpam-5572	413	5	)	)	PUNCT
ejpam-5572	413	6	,	,	PUNCT
ejpam-5572	413	7	for	for	ADP
ejpam-5572	413	8	all	all	DET
ejpam-5572	413	9	z1	z1	VERB
ejpam-5572	413	10	,	,	PUNCT
ejpam-5572	413	11	z2	z2	PROPN
ejpam-5572	413	12	∈	∈	PROPN
ejpam-5572	413	13	x.	x.	NOUN
ejpam-5572	413	14	equivalently	equivalently	PROPN
ejpam-5572	413	15	,	,	PUNCT
ejpam-5572	413	16	p3(z−1	p3(z−1	PROPN
ejpam-5572	413	17	2	2	NUM
ejpam-5572	413	18	∗	∗	NOUN
ejpam-5572	413	19	z1)e	z1)e	NOUN
ejpam-5572	413	20	2πiω3(z−1	2πiω3(z−1	NUM
ejpam-5572	413	21	2	2	NUM
ejpam-5572	413	22	∗	∗	NOUN
ejpam-5572	413	23	z1	z1	NUM
ejpam-5572	413	24	)	)	PUNCT
ejpam-5572	413	25	=	=	SYM
ejpam-5572	414	1	p3(z1	p3(z1	NOUN
ejpam-5572	414	2	∗	∗	NOUN
ejpam-5572	414	3	z−1	z−1	ADJ
ejpam-5572	414	4	2	2	NUM
ejpam-5572	414	5	)	)	PUNCT
ejpam-5572	414	6	e2πiω	e2πiω	NOUN
ejpam-5572	415	1	3(z1	3(z1	NUM
ejpam-5572	415	2	∗	∗	NOUN
ejpam-5572	415	3	z−1	z−1	ADJ
ejpam-5572	415	4	2	2	NUM
ejpam-5572	415	5	)	)	PUNCT
ejpam-5572	415	6	.	.	PUNCT
ejpam-5572	416	1	hence	hence	ADV
ejpam-5572	416	2	,	,	PUNCT
ejpam-5572	416	3	p3(z2	p3(z2	NOUN
ejpam-5572	416	4	∗	∗	NOUN
ejpam-5572	416	5	z1)e2πiω	z1)e2πiω	PROPN
ejpam-5572	416	6	3(z2	3(z2	NUM
ejpam-5572	416	7	∗	∗	NOUN
ejpam-5572	416	8	z1	z1	NUM
ejpam-5572	416	9	)	)	PUNCT
ejpam-5572	416	10	=	=	SYM
ejpam-5572	416	11	p3((z−1	p3((z−1	ADJ
ejpam-5572	416	12	2	2	NUM
ejpam-5572	416	13	)	)	PUNCT
ejpam-5572	416	14	−1	−1	NOUN
ejpam-5572	416	15	∗	∗	NOUN
ejpam-5572	416	16	z1)e2πiω	z1)e2πiω	NUM
ejpam-5572	416	17	3((z−1	3((z−1	NUM
ejpam-5572	416	18	2	2	NUM
ejpam-5572	416	19	)	)	PUNCT
ejpam-5572	416	20	−1	−1	NOUN
ejpam-5572	416	21	∗	∗	NOUN
ejpam-5572	416	22	z1	z1	NUM
ejpam-5572	416	23	)	)	PUNCT
ejpam-5572	416	24	=	=	SYM
ejpam-5572	417	1	p3(z1	p3(z1	NOUN
ejpam-5572	417	2	∗	∗	NOUN
ejpam-5572	417	3	(	(	PUNCT
ejpam-5572	417	4	z−1	z−1	PROPN
ejpam-5572	417	5	2	2	NUM
ejpam-5572	417	6	)	)	PUNCT
ejpam-5572	417	7	−1)e2πiω	−1)e2πiω	NOUN
ejpam-5572	418	1	3(z1	3(z1	NUM
ejpam-5572	418	2	∗	∗	NOUN
ejpam-5572	418	3	(	(	PUNCT
ejpam-5572	418	4	z−1	z−1	ADJ
ejpam-5572	418	5	2	2	NUM
ejpam-5572	418	6	)	)	PUNCT
ejpam-5572	418	7	−1	−1	NOUN
ejpam-5572	418	8	)	)	PUNCT
ejpam-5572	418	9	=	=	SYM
ejpam-5572	419	1	p3(z1	p3(z1	NOUN
ejpam-5572	419	2	∗	∗	NOUN
ejpam-5572	419	3	z2)e	z2)e	PROPN
ejpam-5572	419	4	2πiω3(z1	2πiω3(z1	PROPN
ejpam-5572	419	5	∗	∗	NOUN
ejpam-5572	419	6	z2	z2	NUM
ejpam-5572	419	7	)	)	PUNCT
ejpam-5572	419	8	.	.	PUNCT
ejpam-5572	420	1	similarly	similarly	ADV
ejpam-5572	420	2	,	,	PUNCT
ejpam-5572	420	3	we	we	PRON
ejpam-5572	420	4	can	can	AUX
ejpam-5572	420	5	verify	verify	VERB
ejpam-5572	420	6	that	that	SCONJ
ejpam-5572	420	7	q3(z1	q3(z1	ADJ
ejpam-5572	420	8	∗	∗	NOUN
ejpam-5572	420	9	z2)e	z2)e	PROPN
ejpam-5572	420	10	2πiν3(z1	2πiν3(z1	NUM
ejpam-5572	420	11	∗	∗	NOUN
ejpam-5572	420	12	z2	z2	NUM
ejpam-5572	420	13	)	)	PUNCT
ejpam-5572	420	14	=	=	PUNCT
ejpam-5572	421	1	q3(z2	q3(z2	NOUN
ejpam-5572	421	2	∗	∗	NOUN
ejpam-5572	421	3	z1)e	z1)e	NOUN
ejpam-5572	421	4	2πiν3(z2	2πiν3(z2	NUM
ejpam-5572	421	5	∗	∗	NOUN
ejpam-5572	421	6	z1	z1	NUM
ejpam-5572	421	7	)	)	PUNCT
ejpam-5572	421	8	.	.	PUNCT
ejpam-5572	422	1	⇐	⇐	PROPN
ejpam-5572	422	2	assume	assume	VERB
ejpam-5572	422	3	that	that	SCONJ
ejpam-5572	422	4	z3	z3	NOUN
ejpam-5572	422	5	=	=	PUNCT
ejpam-5572	422	6	z−1	z−1	NUM
ejpam-5572	422	7	1	1	NUM
ejpam-5572	422	8	,	,	PUNCT
ejpam-5572	422	9	then	then	ADV
ejpam-5572	422	10	for	for	ADP
ejpam-5572	422	11	arbitrary	arbitrary	ADJ
ejpam-5572	422	12	z1	z1	ADJ
ejpam-5572	422	13	,	,	PUNCT
ejpam-5572	422	14	z2	z2	PROPN
ejpam-5572	422	15	∈	∈	PROPN
ejpam-5572	422	16	x.	x.	NOUN
ejpam-5572	423	1	we	we	PRON
ejpam-5572	423	2	have	have	VERB
ejpam-5572	423	3	p3(z1	p3(z1	NOUN
ejpam-5572	423	4	∗	∗	NOUN
ejpam-5572	423	5	z2)e2πiω	z2)e2πiω	NUM
ejpam-5572	423	6	3(z1	3(z1	NUM
ejpam-5572	423	7	∗	∗	NOUN
ejpam-5572	423	8	z2	z2	NUM
ejpam-5572	423	9	)	)	PUNCT
ejpam-5572	423	10	=	=	SYM
ejpam-5572	423	11	p3(z2	p3(z2	PROPN
ejpam-5572	423	12	∗	∗	NOUN
ejpam-5572	423	13	z1)e	z1)e	NOUN
ejpam-5572	423	14	2πiω3(z2	2πiω3(z2	NUM
ejpam-5572	423	15	∗	∗	NOUN
ejpam-5572	423	16	z1	z1	NUM
ejpam-5572	423	17	)	)	PUNCT
ejpam-5572	423	18	,	,	PUNCT
ejpam-5572	423	19	hence	hence	ADV
ejpam-5572	423	20	p3(z−1	p3(z−1	PROPN
ejpam-5572	423	21	3	3	NUM
ejpam-5572	423	22	∗	∗	NOUN
ejpam-5572	423	23	z2)e	z2)e	PROPN
ejpam-5572	423	24	2πiω3(z−1	2πiω3(z−1	NUM
ejpam-5572	423	25	3	3	NUM
ejpam-5572	423	26	∗	∗	NOUN
ejpam-5572	423	27	z2	z2	NUM
ejpam-5572	423	28	)	)	PUNCT
ejpam-5572	423	29	=	=	SYM
ejpam-5572	423	30	p3(z2	p3(z2	PROPN
ejpam-5572	423	31	∗	∗	NOUN
ejpam-5572	423	32	z−1	z−1	NUM
ejpam-5572	423	33	3	3	NUM
ejpam-5572	423	34	)	)	PUNCT
ejpam-5572	423	35	e2πiω	e2πiω	NOUN
ejpam-5572	424	1	3(z2	3(z2	NUM
ejpam-5572	424	2	∗	∗	NOUN
ejpam-5572	424	3	z−1	z−1	NUM
ejpam-5572	424	4	3	3	NUM
ejpam-5572	424	5	)	)	PUNCT
ejpam-5572	424	6	for	for	ADP
ejpam-5572	424	7	any	any	DET
ejpam-5572	424	8	z3	z3	NOUN
ejpam-5572	424	9	,	,	PUNCT
ejpam-5572	424	10	z2	z2	PROPN
ejpam-5572	424	11	∈	∈	PROPN
ejpam-5572	424	12	x.	x.	NOUN
ejpam-5572	425	1	so	so	SCONJ
ejpam-5572	425	2	that	that	SCONJ
ejpam-5572	425	3	,	,	PUNCT
ejpam-5572	425	4	(	(	PUNCT
ejpam-5572	425	5	z3p	z3p	PROPN
ejpam-5572	425	6	3(z2))e	3(z2))e	NUM
ejpam-5572	425	7	2πi(z3ω3(z2	2πi(z3ω3(z2	NUM
ejpam-5572	425	8	)	)	PUNCT
ejpam-5572	425	9	)	)	PUNCT
ejpam-5572	426	1	=	=	SYM
ejpam-5572	426	2	(	(	PUNCT
ejpam-5572	426	3	p3(z2)z3)e	p3(z2)z3)e	NUM
ejpam-5572	426	4	2πi(ω3(z2)z3	2πi(ω3(z2)z3	NUM
ejpam-5572	426	5	)	)	PUNCT
ejpam-5572	426	6	.	.	PUNCT
ejpam-5572	427	1	similarly	similarly	ADV
ejpam-5572	427	2	,	,	PUNCT
ejpam-5572	427	3	we	we	PRON
ejpam-5572	427	4	can	can	AUX
ejpam-5572	427	5	prove	prove	VERB
ejpam-5572	427	6	that	that	SCONJ
ejpam-5572	427	7	(	(	PUNCT
ejpam-5572	427	8	z3q	z3q	PROPN
ejpam-5572	427	9	3(z2))e	3(z2))e	NUM
ejpam-5572	427	10	2πi(z3ν3(z2	2πi(z3ν3(z2	NUM
ejpam-5572	427	11	)	)	PUNCT
ejpam-5572	427	12	)	)	PUNCT
ejpam-5572	428	1	=	=	PUNCT
ejpam-5572	428	2	(	(	PUNCT
ejpam-5572	428	3	q3(z2)z3)e	q3(z2)z3)e	NOUN
ejpam-5572	428	4	2πi(ν3(z2)z3	2πi(ν3(z2)z3	NUM
ejpam-5572	428	5	)	)	PUNCT
ejpam-5572	428	6	,	,	PUNCT
ejpam-5572	428	7	then	then	ADV
ejpam-5572	428	8	z3ϕ	z3ϕ	X
ejpam-5572	428	9	=	=	PUNCT
ejpam-5572	428	10	ϕz3	ϕz3	CCONJ
ejpam-5572	428	11	for	for	ADP
ejpam-5572	428	12	any	any	DET
ejpam-5572	428	13	z3	z3	PROPN
ejpam-5572	428	14	∈	∈	PROPN
ejpam-5572	428	15	x	x	NOUN
ejpam-5572	428	16	,	,	PUNCT
ejpam-5572	428	17	which	which	PRON
ejpam-5572	428	18	implies	imply	VERB
ejpam-5572	428	19	that	that	SCONJ
ejpam-5572	428	20	ϕ	ϕ	PROPN
ejpam-5572	428	21	is	be	AUX
ejpam-5572	428	22	cffnsg	cffnsg	ADJ
ejpam-5572	428	23	of	of	ADP
ejpam-5572	428	24	a	a	DET
ejpam-5572	428	25	group	group	NOUN
ejpam-5572	428	26	(	(	PUNCT
ejpam-5572	428	27	x	x	X
ejpam-5572	428	28	,	,	PUNCT
ejpam-5572	428	29	∗	∗	NOUN
ejpam-5572	428	30	)	)	PUNCT
ejpam-5572	428	31	.	.	PUNCT
ejpam-5572	429	1	proposition	proposition	NOUN
ejpam-5572	429	2	6	6	NUM
ejpam-5572	429	3	.	.	PUNCT
ejpam-5572	430	1	for	for	ADP
ejpam-5572	430	2	a	a	DET
ejpam-5572	430	3	group	group	NOUN
ejpam-5572	430	4	(	(	PUNCT
ejpam-5572	430	5	x	x	X
ejpam-5572	430	6	,	,	PUNCT
ejpam-5572	430	7	∗	∗	NOUN
ejpam-5572	430	8	)	)	PUNCT
ejpam-5572	430	9	that	that	PRON
ejpam-5572	430	10	was	be	AUX
ejpam-5572	430	11	defined	define	VERB
ejpam-5572	430	12	on	on	ADP
ejpam-5572	430	13	cffsg	cffsg	ADJ
ejpam-5572	430	14	,	,	PUNCT
ejpam-5572	430	15	ϕ	ϕ	X
ejpam-5572	430	16	=	=	PUNCT
ejpam-5572	430	17	(	(	PUNCT
ejpam-5572	430	18	pe2πiω	pe2πiω	NOUN
ejpam-5572	430	19	,	,	PUNCT
ejpam-5572	430	20	qe2πiν	qe2πiν	NOUN
ejpam-5572	430	21	)	)	PUNCT
ejpam-5572	430	22	.	.	PUNCT
ejpam-5572	431	1	then	then	ADV
ejpam-5572	431	2	ϕ	ϕ	PROPN
ejpam-5572	431	3	is	be	AUX
ejpam-5572	431	4	a	a	DET
ejpam-5572	431	5	cffnsg	cffnsg	NOUN
ejpam-5572	431	6	of	of	ADP
ejpam-5572	431	7	x	x	SYM
ejpam-5572	431	8	if	if	SCONJ
ejpam-5572	432	1	and	and	CCONJ
ejpam-5572	432	2	only	only	ADV
ejpam-5572	432	3	if	if	SCONJ
ejpam-5572	432	4	p3(x)e2πiω	p3(x)e2πiω	PROPN
ejpam-5572	432	5	3(x	3(x	NUM
ejpam-5572	432	6	)	)	PUNCT
ejpam-5572	432	7	=	=	SYM
ejpam-5572	432	8	p3(z	p3(z	PROPN
ejpam-5572	432	9	∗	∗	NOUN
ejpam-5572	432	10	x	x	SYM
ejpam-5572	432	11	∗	∗	NOUN
ejpam-5572	432	12	z−1)e2πiω	z−1)e2πiω	NOUN
ejpam-5572	432	13	3(z∗x∗z−1	3(z∗x∗z−1	NUM
ejpam-5572	432	14	)	)	PUNCT
ejpam-5572	432	15	,	,	PUNCT
ejpam-5572	432	16	and	and	CCONJ
ejpam-5572	432	17	q3(x)e2πiν	q3(x)e2πiν	ADP
ejpam-5572	432	18	3(x	3(x	NUM
ejpam-5572	432	19	)	)	PUNCT
ejpam-5572	433	1	=	=	SYM
ejpam-5572	433	2	q3(z	q3(z	X
ejpam-5572	433	3	∗	∗	NOUN
ejpam-5572	433	4	x	x	SYM
ejpam-5572	433	5	∗	∗	NOUN
ejpam-5572	433	6	z−1)e2πiν	z−1)e2πiν	NOUN
ejpam-5572	433	7	3(z∗x∗z−1	3(z∗x∗z−1	NUM
ejpam-5572	433	8	)	)	PUNCT
ejpam-5572	433	9	,	,	PUNCT
ejpam-5572	433	10	for	for	ADP
ejpam-5572	433	11	all	all	DET
ejpam-5572	433	12	z	z	PROPN
ejpam-5572	433	13	,	,	PUNCT
ejpam-5572	433	14	x	x	SYM
ejpam-5572	433	15	∈	∈	NOUN
ejpam-5572	433	16	x	x	NOUN
ejpam-5572	433	17	proof	proof	NOUN
ejpam-5572	433	18	.	.	PUNCT
ejpam-5572	434	1	first	first	ADV
ejpam-5572	434	2	consider	consider	VERB
ejpam-5572	434	3	,	,	PUNCT
ejpam-5572	434	4	p3(x)e2πiω	p3(x)e2πiω	NOUN
ejpam-5572	434	5	3(x	3(x	NUM
ejpam-5572	434	6	)	)	PUNCT
ejpam-5572	434	7	=	=	SYM
ejpam-5572	434	8	p3(x∗id)e2πiω3(x∗id	p3(x∗id)e2πiω3(x∗id	NOUN
ejpam-5572	434	9	)	)	PUNCT
ejpam-5572	434	10	=	=	PUNCT
ejpam-5572	435	1	p3(x∗z∗z−1)e2πiω	p3(x∗z∗z−1)e2πiω	PRON
ejpam-5572	435	2	3(x∗z∗z−1	3(x∗z∗z−1	NUM
ejpam-5572	435	3	)	)	PUNCT
ejpam-5572	435	4	=	=	SYM
ejpam-5572	435	5	p3(x	p3(x	PROPN
ejpam-5572	435	6	∗	∗	NOUN
ejpam-5572	435	7	(	(	PUNCT
ejpam-5572	435	8	z	z	NOUN
ejpam-5572	435	9	∗	∗	NOUN
ejpam-5572	435	10	z−1))e2πiω	z−1))e2πiω	PROPN
ejpam-5572	435	11	3(x∗(z∗z−1	3(x∗(z∗z−1	NUM
ejpam-5572	435	12	)	)	PUNCT
ejpam-5572	435	13	)	)	PUNCT
ejpam-5572	436	1	=	=	PUNCT
ejpam-5572	436	2	p3((x	p3((x	INTJ
ejpam-5572	436	3	∗	∗	PROPN
ejpam-5572	436	4	z	z	NOUN
ejpam-5572	436	5	)	)	PUNCT
ejpam-5572	436	6	∗	∗	NOUN
ejpam-5572	436	7	z−1)e2πiω	z−1)e2πiω	NOUN
ejpam-5572	436	8	3((x∗z)∗z−1	3((x∗z)∗z−1	NOUN
ejpam-5572	436	9	)	)	PUNCT
ejpam-5572	436	10	=	=	SYM
ejpam-5572	436	11	p3(z−1	p3(z−1	PROPN
ejpam-5572	436	12	∗	∗	NOUN
ejpam-5572	436	13	(	(	PUNCT
ejpam-5572	436	14	x	x	X
ejpam-5572	436	15	∗	∗	PUNCT
ejpam-5572	436	16	z))e2πiω	z))e2πiω	PROPN
ejpam-5572	436	17	3(z−1∗(x∗z	3(z−1∗(x∗z	NUM
ejpam-5572	436	18	)	)	PUNCT
ejpam-5572	436	19	)	)	PUNCT
ejpam-5572	436	20	,	,	PUNCT
ejpam-5572	436	21	whereas	whereas	SCONJ
ejpam-5572	436	22	ϕ	ϕ	NOUN
ejpam-5572	436	23	is	be	AUX
ejpam-5572	436	24	cffnsg	cffnsg	ADJ
ejpam-5572	436	25	of	of	ADP
ejpam-5572	436	26	x.	x.	PROPN
ejpam-5572	436	27	but	but	CCONJ
ejpam-5572	436	28	z	z	NOUN
ejpam-5572	436	29	=	=	SYM
ejpam-5572	436	30	(	(	PUNCT
ejpam-5572	436	31	z−1)−1	z−1)−1	NOUN
ejpam-5572	436	32	and	and	CCONJ
ejpam-5572	436	33	by	by	ADP
ejpam-5572	436	34	similarity	similarity	NOUN
ejpam-5572	436	35	p3(x)e2πiω	p3(x)e2πiω	NOUN
ejpam-5572	436	36	3(x	3(x	NUM
ejpam-5572	436	37	)	)	PUNCT
ejpam-5572	436	38	=	=	SYM
ejpam-5572	437	1	p3(z	p3(z	X
ejpam-5572	437	2	∗x∗	∗x∗	NUM
ejpam-5572	437	3	z−1)e2πiω	z−1)e2πiω	NOUN
ejpam-5572	437	4	3(z∗x∗z−1	3(z∗x∗z−1	NUM
ejpam-5572	437	5	)	)	PUNCT
ejpam-5572	437	6	.	.	PUNCT
ejpam-5572	438	1	also	also	ADV
ejpam-5572	438	2	,	,	PUNCT
ejpam-5572	438	3	it	it	PRON
ejpam-5572	438	4	is	be	AUX
ejpam-5572	438	5	easy	easy	ADJ
ejpam-5572	438	6	to	to	PART
ejpam-5572	438	7	show	show	VERB
ejpam-5572	438	8	that	that	SCONJ
ejpam-5572	438	9	q3(x)e2πiν	q3(x)e2πiν	NOUN
ejpam-5572	438	10	3(x	3(x	NUM
ejpam-5572	438	11	)	)	PUNCT
ejpam-5572	438	12	=	=	SYM
ejpam-5572	439	1	q3(z	q3(z	X
ejpam-5572	439	2	∗	∗	NOUN
ejpam-5572	439	3	x	x	SYM
ejpam-5572	439	4	∗	∗	NOUN
ejpam-5572	439	5	z−1)e2πiν	z−1)e2πiν	NOUN
ejpam-5572	439	6	3(z∗x∗z−1	3(z∗x∗z−1	NUM
ejpam-5572	439	7	)	)	PUNCT
ejpam-5572	439	8	too	too	ADV
ejpam-5572	439	9	.	.	PUNCT
ejpam-5572	440	1	conversely	conversely	ADV
ejpam-5572	440	2	,	,	PUNCT
ejpam-5572	440	3	p3(z∗x)e2πiω3(z∗x	p3(z∗x)e2πiω3(z∗x	NOUN
ejpam-5572	440	4	)	)	PUNCT
ejpam-5572	440	5	=	=	SYM
ejpam-5572	440	6	p3(z∗x∗id)e2πiω3(z∗x∗id	p3(z∗x∗id)e2πiω3(z∗x∗id	NOUN
ejpam-5572	440	7	)	)	PUNCT
ejpam-5572	440	8	=	=	SYM
ejpam-5572	440	9	p3(z∗(x∗z)∗z−1	p3(z∗(x∗z)∗z−1	PROPN
ejpam-5572	440	10	)	)	PUNCT
ejpam-5572	440	11	e2πiω	e2πiω	NUM
ejpam-5572	441	1	3(z∗(x∗z)∗z−1	3(z∗(x∗z)∗z−1	NUM
ejpam-5572	441	2	)	)	PUNCT
ejpam-5572	441	3	=	=	SYM
ejpam-5572	441	4	p3(x∗z)e2πiω3(x∗z	p3(x∗z)e2πiω3(x∗z	NOUN
ejpam-5572	441	5	)	)	PUNCT
ejpam-5572	441	6	.	.	PUNCT
ejpam-5572	442	1	also	also	ADV
ejpam-5572	442	2	,	,	PUNCT
ejpam-5572	442	3	it	it	PRON
ejpam-5572	442	4	is	be	AUX
ejpam-5572	442	5	easy	easy	ADJ
ejpam-5572	442	6	to	to	PART
ejpam-5572	442	7	show	show	VERB
ejpam-5572	442	8	that	that	SCONJ
ejpam-5572	442	9	q3(z	q3(z	PROPN
ejpam-5572	442	10	∗x)e2πiν3(z∗x	∗x)e2πiν3(z∗x	NOUN
ejpam-5572	442	11	)	)	PUNCT
ejpam-5572	442	12	=	=	SYM
ejpam-5572	442	13	q3(x∗z)e2πiν3(x∗z	q3(x∗z)e2πiν3(x∗z	NOUN
ejpam-5572	442	14	)	)	PUNCT
ejpam-5572	442	15	.	.	PUNCT
ejpam-5572	443	1	then	then	ADV
ejpam-5572	443	2	by	by	ADP
ejpam-5572	443	3	previous	previous	ADJ
ejpam-5572	443	4	proposition	proposition	NOUN
ejpam-5572	443	5	,	,	PUNCT
ejpam-5572	443	6	ϕ	ϕ	PROPN
ejpam-5572	443	7	is	be	AUX
ejpam-5572	443	8	cffnsg	cffnsg	ADJ
ejpam-5572	443	9	of	of	ADP
ejpam-5572	443	10	x.	x.	PROPN
ejpam-5572	443	11	theorem	theorem	NOUN
ejpam-5572	443	12	3	3	X
ejpam-5572	443	13	.	.	PUNCT
ejpam-5572	444	1	let	let	VERB
ejpam-5572	444	2	ϕ	ϕ	NOUN
ejpam-5572	444	3	be	be	AUX
ejpam-5572	444	4	a	a	DET
ejpam-5572	444	5	cffnsg	cffnsg	NOUN
ejpam-5572	444	6	of	of	ADP
ejpam-5572	444	7	a	a	DET
ejpam-5572	444	8	group	group	NOUN
ejpam-5572	444	9	(	(	PUNCT
ejpam-5572	444	10	x	x	X
ejpam-5572	444	11	,	,	PUNCT
ejpam-5572	444	12	∗	∗	NOUN
ejpam-5572	444	13	)	)	PUNCT
ejpam-5572	444	14	.	.	PUNCT
ejpam-5572	445	1	then	then	ADV
ejpam-5572	445	2	the	the	DET
ejpam-5572	445	3	set	set	NOUN
ejpam-5572	445	4	m	m	NOUN
ejpam-5572	445	5	=	=	PUNCT
ejpam-5572	445	6	{	{	PUNCT
ejpam-5572	445	7	y	y	PROPN
ejpam-5572	445	8	∈	∈	PROPN
ejpam-5572	445	9	x	x	X
ejpam-5572	445	10	:	:	PUNCT
ejpam-5572	445	11	p3(id)e2πiω	p3(id)e2πiω	X
ejpam-5572	445	12	3(id	3(id	NUM
ejpam-5572	445	13	)	)	PUNCT
ejpam-5572	445	14	=	=	SYM
ejpam-5572	445	15	p3(y)e2πiω	p3(y)e2πiω	NOUN
ejpam-5572	445	16	3(y	3(y	NUM
ejpam-5572	445	17	)	)	PUNCT
ejpam-5572	445	18	and	and	CCONJ
ejpam-5572	445	19	q3(id)e2πiν	q3(id)e2πiν	NUM
ejpam-5572	445	20	3(id	3(id	NUM
ejpam-5572	445	21	)	)	PUNCT
ejpam-5572	445	22	=	=	SYM
ejpam-5572	445	23	q3(y)e2πiν	q3(y)e2πiν	NOUN
ejpam-5572	445	24	3(y	3(y	NUM
ejpam-5572	445	25	)	)	PUNCT
ejpam-5572	445	26	}	}	PUNCT
ejpam-5572	445	27	,	,	PUNCT
ejpam-5572	445	28	is	be	AUX
ejpam-5572	445	29	a	a	DET
ejpam-5572	445	30	normal	normal	ADJ
ejpam-5572	445	31	subgroup	subgroup	NOUN
ejpam-5572	445	32	of	of	ADP
ejpam-5572	445	33	x	x	PROPN
ejpam-5572	445	34	,	,	PUNCT
ejpam-5572	445	35	where	where	SCONJ
ejpam-5572	445	36	i	i	PRON
ejpam-5572	445	37	d	d	PROPN
ejpam-5572	445	38	is	be	AUX
ejpam-5572	445	39	the	the	DET
ejpam-5572	445	40	identity	identity	NOUN
ejpam-5572	445	41	of	of	ADP
ejpam-5572	445	42	it	it	PRON
ejpam-5572	445	43	.	.	PUNCT
ejpam-5572	446	1	e.a	e.a	PROPN
ejpam-5572	446	2	.	.	PROPN
ejpam-5572	446	3	abuhijleh	abuhijleh	PROPN
ejpam-5572	446	4	,	,	PUNCT
ejpam-5572	446	5	a.	a.	PROPN
ejpam-5572	446	6	alkouri	alkouri	PROPN
ejpam-5572	446	7	/	/	PROPN
ejpam-5572	446	8	eur	eur	PROPN
ejpam-5572	446	9	.	.	PUNCT
ejpam-5572	447	1	j.	j.	PROPN
ejpam-5572	447	2	pure	pure	PROPN
ejpam-5572	447	3	appl	appl	PROPN
ejpam-5572	447	4	.	.	PROPN
ejpam-5572	447	5	math	math	PROPN
ejpam-5572	447	6	,	,	PUNCT
ejpam-5572	447	7	18	18	NUM
ejpam-5572	447	8	(	(	PUNCT
ejpam-5572	447	9	1	1	NUM
ejpam-5572	447	10	)	)	PUNCT
ejpam-5572	447	11	(	(	PUNCT
ejpam-5572	447	12	2025	2025	NUM
ejpam-5572	447	13	)	)	PUNCT
ejpam-5572	447	14	,	,	PUNCT
ejpam-5572	447	15	5572	5572	NUM
ejpam-5572	447	16	13	13	NUM
ejpam-5572	447	17	of	of	ADP
ejpam-5572	447	18	19	19	NUM
ejpam-5572	447	19	proof	proof	NOUN
ejpam-5572	447	20	.	.	PUNCT
ejpam-5572	448	1	at	at	ADP
ejpam-5572	448	2	first	first	ADV
ejpam-5572	448	3	i	i	PROPN
ejpam-5572	448	4	d	d	PROPN
ejpam-5572	448	5	∈	∈	PROPN
ejpam-5572	448	6	m	m	PROPN
ejpam-5572	448	7	,	,	PUNCT
ejpam-5572	448	8	i.e.	i.e.	X
ejpam-5572	448	9	m	m	VERB
ejpam-5572	448	10	is	be	AUX
ejpam-5572	448	11	not	not	PART
ejpam-5572	448	12	empty	empty	ADJ
ejpam-5572	448	13	.	.	PUNCT
ejpam-5572	449	1	moreover	moreover	ADV
ejpam-5572	449	2	,	,	PUNCT
ejpam-5572	449	3	it	it	PRON
ejpam-5572	449	4	is	be	AUX
ejpam-5572	449	5	subgroup	subgroup	NOUN
ejpam-5572	449	6	of	of	ADP
ejpam-5572	449	7	x	x	PRON
ejpam-5572	449	8	,	,	PUNCT
ejpam-5572	449	9	by	by	ADP
ejpam-5572	449	10	theorem	theorem	NOUN
ejpam-5572	449	11	2	2	NUM
ejpam-5572	449	12	.	.	PUNCT
ejpam-5572	450	1	so	so	ADV
ejpam-5572	450	2	that	that	SCONJ
ejpam-5572	450	3	,	,	PUNCT
ejpam-5572	450	4	p3(id)e2πiω	p3(id)e2πiω	X
ejpam-5572	450	5	3(id	3(id	NUM
ejpam-5572	450	6	)	)	PUNCT
ejpam-5572	451	1	=	=	SYM
ejpam-5572	451	2	p3(y	p3(y	PROPN
ejpam-5572	451	3	∗	∗	NOUN
ejpam-5572	451	4	z−1)e2πiω	z−1)e2πiω	X
ejpam-5572	451	5	3(y	3(y	NUM
ejpam-5572	451	6	∗	∗	NOUN
ejpam-5572	451	7	z−1	z−1	NUM
ejpam-5572	451	8	)	)	PUNCT
ejpam-5572	451	9	and	and	CCONJ
ejpam-5572	451	10	q3(id)e2πiν	q3(id)e2πiν	NUM
ejpam-5572	451	11	3(id	3(id	NUM
ejpam-5572	451	12	)	)	PUNCT
ejpam-5572	451	13	=	=	PUNCT
ejpam-5572	452	1	q3(y	q3(y	PROPN
ejpam-5572	452	2	∗	∗	NOUN
ejpam-5572	452	3	z−1)e2πiν	z−1)e2πiν	PROPN
ejpam-5572	452	4	3(y	3(y	NUM
ejpam-5572	452	5	∗	∗	NOUN
ejpam-5572	452	6	z−1	z−1	NUM
ejpam-5572	452	7	)	)	PUNCT
ejpam-5572	452	8	.	.	PUNCT
ejpam-5572	453	1	but	but	CCONJ
ejpam-5572	453	2	,	,	PUNCT
ejpam-5572	453	3	ϕ	ϕ	PROPN
ejpam-5572	453	4	is	be	AUX
ejpam-5572	453	5	a	a	DET
ejpam-5572	453	6	cffnsg	cffnsg	NOUN
ejpam-5572	453	7	of	of	ADP
ejpam-5572	453	8	(	(	PUNCT
ejpam-5572	453	9	x	x	NOUN
ejpam-5572	453	10	,	,	PUNCT
ejpam-5572	453	11	∗	∗	NOUN
ejpam-5572	453	12	)	)	PUNCT
ejpam-5572	453	13	.	.	PUNCT
ejpam-5572	454	1	then	then	ADV
ejpam-5572	454	2	p3(y	p3(y	PROPN
ejpam-5572	454	3	∗	∗	NOUN
ejpam-5572	454	4	z−1)e2πiω	z−1)e2πiω	X
ejpam-5572	455	1	3(y	3(y	NUM
ejpam-5572	455	2	∗	∗	NOUN
ejpam-5572	455	3	z−1	z−1	NUM
ejpam-5572	455	4	)	)	PUNCT
ejpam-5572	455	5	=	=	SYM
ejpam-5572	455	6	p3(z−1	p3(z−1	PROPN
ejpam-5572	455	7	∗	∗	NOUN
ejpam-5572	455	8	y)e2πiω	y)e2πiω	PROPN
ejpam-5572	455	9	3(z−1	3(z−1	PROPN
ejpam-5572	455	10	∗	∗	X
ejpam-5572	455	11	y	y	PROPN
ejpam-5572	455	12	)	)	PUNCT
ejpam-5572	455	13	and	and	CCONJ
ejpam-5572	455	14	q3(y	q3(y	PROPN
ejpam-5572	455	15	∗	∗	NOUN
ejpam-5572	455	16	z−1)e2πiν	z−1)e2πiν	PROPN
ejpam-5572	455	17	3(y	3(y	NUM
ejpam-5572	455	18	∗	∗	NOUN
ejpam-5572	455	19	z−1	z−1	NUM
ejpam-5572	455	20	)	)	PUNCT
ejpam-5572	455	21	=	=	SYM
ejpam-5572	455	22	q3(z−1	q3(z−1	NOUN
ejpam-5572	455	23	∗	∗	NOUN
ejpam-5572	455	24	y)e2πiν	y)e2πiν	PROPN
ejpam-5572	455	25	3(z−1	3(z−1	PROPN
ejpam-5572	455	26	∗	∗	PROPN
ejpam-5572	455	27	y	y	PROPN
ejpam-5572	455	28	)	)	PUNCT
ejpam-5572	455	29	.	.	PUNCT
ejpam-5572	456	1	hence	hence	ADV
ejpam-5572	456	2	,	,	PUNCT
ejpam-5572	456	3	(	(	PUNCT
ejpam-5572	456	4	z−1	z−1	PROPN
ejpam-5572	456	5	∗	∗	NOUN
ejpam-5572	456	6	y	y	NOUN
ejpam-5572	456	7	)	)	PUNCT
ejpam-5572	456	8	∈	∈	PROPN
ejpam-5572	456	9	m	m	NOUN
ejpam-5572	456	10	and	and	CCONJ
ejpam-5572	456	11	m	m	PROPN
ejpam-5572	456	12	is	be	AUX
ejpam-5572	456	13	a	a	DET
ejpam-5572	456	14	normal	normal	ADJ
ejpam-5572	456	15	subgroup	subgroup	NOUN
ejpam-5572	456	16	of	of	ADP
ejpam-5572	456	17	m.	m.	NOUN
ejpam-5572	456	18	5	5	NUM
ejpam-5572	456	19	.	.	PUNCT
ejpam-5572	457	1	homomorphism	homomorphism	NOUN
ejpam-5572	457	2	on	on	ADP
ejpam-5572	457	3	complex	complex	ADJ
ejpam-5572	457	4	fermatean	fermatean	ADJ
ejpam-5572	457	5	fuzzy	fuzzy	ADJ
ejpam-5572	457	6	subgroup	subgroup	NOUN
ejpam-5572	457	7	in	in	ADP
ejpam-5572	457	8	this	this	DET
ejpam-5572	457	9	section	section	NOUN
ejpam-5572	457	10	,	,	PUNCT
ejpam-5572	457	11	we	we	PRON
ejpam-5572	457	12	discuss	discuss	VERB
ejpam-5572	457	13	the	the	DET
ejpam-5572	457	14	effect	effect	NOUN
ejpam-5572	457	15	of	of	ADP
ejpam-5572	457	16	homomorphism	homomorphism	PROPN
ejpam-5572	457	17	on	on	ADP
ejpam-5572	457	18	cffsg	cffsg	ADJ
ejpam-5572	457	19	.	.	PUNCT
ejpam-5572	458	1	definition	definition	NOUN
ejpam-5572	458	2	12	12	NUM
ejpam-5572	458	3	.	.	PUNCT
ejpam-5572	459	1	a	a	DET
ejpam-5572	459	2	homomorphism	homomorphism	PROPN
ejpam-5572	459	3	function	function	NOUN
ejpam-5572	459	4	h	h	NOUN
ejpam-5572	459	5	:	:	PUNCT
ejpam-5572	459	6	x	x	X
ejpam-5572	459	7	→	→	SYM
ejpam-5572	459	8	u	u	NOUN
ejpam-5572	459	9	from	from	ADP
ejpam-5572	459	10	group	group	NOUN
ejpam-5572	459	11	x	x	PROPN
ejpam-5572	459	12	to	to	ADP
ejpam-5572	459	13	group	group	NOUN
ejpam-5572	459	14	u.	u.	PROPN
ejpam-5572	459	15	let	let	VERB
ejpam-5572	459	16	a	a	DET
ejpam-5572	459	17	be	be	AUX
ejpam-5572	459	18	cffsg	cffsg	ADJ
ejpam-5572	459	19	of	of	ADP
ejpam-5572	459	20	x	x	X
ejpam-5572	459	21	and	and	CCONJ
ejpam-5572	459	22	b	b	NOUN
ejpam-5572	459	23	be	be	AUX
ejpam-5572	459	24	cffsg	cffsg	ADJ
ejpam-5572	459	25	of	of	ADP
ejpam-5572	459	26	u.	u.	NOUN
ejpam-5572	459	27	let	let	VERB
ejpam-5572	459	28	x	x	X
ejpam-5572	459	29	∈	∈	PROPN
ejpam-5572	459	30	x	x	X
ejpam-5572	459	31	and	and	CCONJ
ejpam-5572	459	32	y	y	PROPN
ejpam-5572	459	33	∈	∈	PROPN
ejpam-5572	459	34	u	u	PROPN
ejpam-5572	459	35	,	,	PUNCT
ejpam-5572	459	36	then	then	ADV
ejpam-5572	459	37	we	we	PRON
ejpam-5572	459	38	have	have	VERB
ejpam-5572	459	39	:	:	PUNCT
ejpam-5572	459	40	h(a)(y	h(a)(y	X
ejpam-5572	459	41	)	)	PUNCT
ejpam-5572	459	42	=	=	SYM
ejpam-5572	459	43	{	{	PUNCT
ejpam-5572	459	44	(	(	PUNCT
ejpam-5572	459	45	y	y	PROPN
ejpam-5572	459	46	,	,	PUNCT
ejpam-5572	459	47	h(ka)(y	h(ka)(y	PROPN
ejpam-5572	459	48	)	)	PUNCT
ejpam-5572	459	49	,	,	PUNCT
ejpam-5572	459	50	h(la)(y	h(la)(y	PROPN
ejpam-5572	459	51	)	)	PUNCT
ejpam-5572	459	52	)	)	PUNCT
ejpam-5572	459	53	}	}	PUNCT
ejpam-5572	459	54	,	,	PUNCT
ejpam-5572	459	55	is	be	AUX
ejpam-5572	459	56	the	the	DET
ejpam-5572	459	57	image	image	NOUN
ejpam-5572	459	58	of	of	ADP
ejpam-5572	459	59	a	a	DET
ejpam-5572	459	60	,	,	PUNCT
ejpam-5572	459	61	where	where	SCONJ
ejpam-5572	459	62	:	:	PUNCT
ejpam-5572	459	63	h(k3	h(k3	NOUN
ejpam-5572	459	64	a	a	NOUN
ejpam-5572	459	65	)	)	PUNCT
ejpam-5572	459	66	=	=	PRON
ejpam-5572	459	67	{	{	PUNCT
ejpam-5572	459	68	sup	sup	NOUN
ejpam-5572	459	69	x∈h−1(y	x∈h−1(y	PROPN
ejpam-5572	459	70	)	)	PUNCT
ejpam-5572	459	71	k3	k3	VERB
ejpam-5572	459	72	a(x	a(x	NOUN
ejpam-5572	459	73	)	)	PUNCT
ejpam-5572	459	74	,	,	PUNCT
ejpam-5572	459	75	h(x	h(x	PROPN
ejpam-5572	459	76	)	)	PUNCT
ejpam-5572	460	1	=	=	PUNCT
ejpam-5572	460	2	y	y	PROPN
ejpam-5572	460	3	0	0	NUM
ejpam-5572	460	4	,	,	PUNCT
ejpam-5572	460	5	otherwise	otherwise	ADV
ejpam-5572	460	6	.	.	PUNCT
ejpam-5572	461	1	=	=	PUNCT
ejpam-5572	462	1			NOUN
ejpam-5572	462	2	(	(	PUNCT
ejpam-5572	462	3	sup	sup	PROPN
ejpam-5572	462	4	x∈h−1(y	x∈h−1(y	PROPN
ejpam-5572	462	5	)	)	PUNCT
ejpam-5572	462	6	p3a(x))e	p3a(x))e	PROPN
ejpam-5572	462	7	2πi	2πi	NOUN
ejpam-5572	462	8	(	(	PUNCT
ejpam-5572	462	9	sup	sup	NOUN
ejpam-5572	462	10	x∈h−1(y	x∈h−1(y	PROPN
ejpam-5572	462	11	)	)	PUNCT
ejpam-5572	462	12	ω3	ω3	PROPN
ejpam-5572	462	13	a(x	a(x	NOUN
ejpam-5572	462	14	)	)	PUNCT
ejpam-5572	462	15	)	)	PUNCT
ejpam-5572	462	16	,	,	PUNCT
ejpam-5572	462	17	h(x	h(x	PROPN
ejpam-5572	462	18	)	)	PUNCT
ejpam-5572	463	1	=	=	PUNCT
ejpam-5572	463	2	y	y	PROPN
ejpam-5572	463	3	0	0	NUM
ejpam-5572	463	4	e2πi	e2πi	X
ejpam-5572	463	5	0	0	NUM
ejpam-5572	463	6	,	,	PUNCT
ejpam-5572	463	7	otherwise	otherwise	ADV
ejpam-5572	463	8	.	.	PUNCT
ejpam-5572	463	9	h(l3	h(l3	PRON
ejpam-5572	464	1	a	a	X
ejpam-5572	464	2	)	)	PUNCT
ejpam-5572	464	3	=	=	PRON
ejpam-5572	464	4	{	{	PUNCT
ejpam-5572	464	5	inf	inf	PROPN
ejpam-5572	464	6	x∈h−1(y	x∈h−1(y	PROPN
ejpam-5572	464	7	)	)	PUNCT
ejpam-5572	464	8	l3	l3	PROPN
ejpam-5572	464	9	a(x	a(x	NOUN
ejpam-5572	464	10	)	)	PUNCT
ejpam-5572	464	11	,	,	PUNCT
ejpam-5572	464	12	h(x	h(x	PROPN
ejpam-5572	464	13	)	)	PUNCT
ejpam-5572	464	14	=	=	SYM
ejpam-5572	465	1	y	y	PROPN
ejpam-5572	465	2	1	1	NUM
ejpam-5572	465	3	,	,	PUNCT
ejpam-5572	465	4	otherwise	otherwise	ADV
ejpam-5572	465	5	.	.	PUNCT
ejpam-5572	466	1	=	=	PUNCT
ejpam-5572	466	2			PROPN
ejpam-5572	466	3	(	(	PUNCT
ejpam-5572	466	4	inf	inf	NOUN
ejpam-5572	466	5	x∈h−1(x	x∈h−1(x	NOUN
ejpam-5572	466	6	)	)	PUNCT
ejpam-5572	466	7	q3a(x))e	q3a(x))e	NOUN
ejpam-5572	466	8	2πi	2πi	NOUN
ejpam-5572	466	9	(	(	PUNCT
ejpam-5572	466	10	inf	inf	PROPN
ejpam-5572	466	11	x∈h−1(y	x∈h−1(y	PROPN
ejpam-5572	466	12	)	)	PUNCT
ejpam-5572	466	13	ν3a(x	ν3a(x	NOUN
ejpam-5572	466	14	)	)	PUNCT
ejpam-5572	466	15	)	)	PUNCT
ejpam-5572	466	16	,	,	PUNCT
ejpam-5572	466	17	h(x	h(x	PROPN
ejpam-5572	466	18	)	)	PUNCT
ejpam-5572	467	1	=	=	SYM
ejpam-5572	467	2	y	y	PROPN
ejpam-5572	467	3	1	1	NUM
ejpam-5572	467	4	,	,	PUNCT
ejpam-5572	467	5	otherwise	otherwise	ADV
ejpam-5572	467	6	.	.	PUNCT
ejpam-5572	468	1	and	and	CCONJ
ejpam-5572	468	2	the	the	DET
ejpam-5572	468	3	set	set	NOUN
ejpam-5572	468	4	of	of	ADP
ejpam-5572	468	5	pre	pre	NOUN
ejpam-5572	468	6	-	-	NOUN
ejpam-5572	468	7	image	image	NOUN
ejpam-5572	468	8	of	of	ADP
ejpam-5572	468	9	b	b	NOUN
ejpam-5572	468	10	is	be	AUX
ejpam-5572	468	11	h−1(b)(x	h−1(b)(x	PROPN
ejpam-5572	468	12	)	)	PUNCT
ejpam-5572	469	1	=	=	PRON
ejpam-5572	469	2	{	{	PUNCT
ejpam-5572	469	3	(	(	PUNCT
ejpam-5572	469	4	x	x	NOUN
ejpam-5572	469	5	,	,	PUNCT
ejpam-5572	469	6	h−1(kb)(x	h−1(kb)(x	NOUN
ejpam-5572	469	7	)	)	PUNCT
ejpam-5572	469	8	,	,	PUNCT
ejpam-5572	469	9	h	h	NOUN
ejpam-5572	469	10	−1(lb)(x	−1(lb)(x	PROPN
ejpam-5572	469	11	)	)	PUNCT
ejpam-5572	469	12	)	)	PUNCT
ejpam-5572	469	13	}	}	PUNCT
ejpam-5572	469	14	,	,	PUNCT
ejpam-5572	469	15	where	where	SCONJ
ejpam-5572	469	16	:	:	PUNCT
ejpam-5572	469	17	h−1(k3	h−1(k3	ADP
ejpam-5572	469	18	b)(x	b)(x	PROPN
ejpam-5572	469	19	)	)	PUNCT
ejpam-5572	469	20	=(	=(	PROPN
ejpam-5572	469	21	kb	kb	PROPN
ejpam-5572	469	22	)	)	PUNCT
ejpam-5572	469	23	3(h(x	3(h(x	NUM
ejpam-5572	469	24	)	)	PUNCT
ejpam-5572	469	25	)	)	PUNCT
ejpam-5572	470	1	=	=	SYM
ejpam-5572	470	2	p3b(h(x))e	p3b(h(x))e	NOUN
ejpam-5572	470	3	2πiω3	2πiω3	NUM
ejpam-5572	470	4	b(h(x	b(h(x	NOUN
ejpam-5572	470	5	)	)	PUNCT
ejpam-5572	470	6	)	)	PUNCT
ejpam-5572	470	7	h−1(l3	h−1(l3	PRON
ejpam-5572	470	8	b)(x	b)(x	PROPN
ejpam-5572	470	9	)	)	PUNCT
ejpam-5572	470	10	=(	=(	NOUN
ejpam-5572	470	11	lb	lb	NOUN
ejpam-5572	470	12	)	)	PUNCT
ejpam-5572	470	13	3(h(x	3(h(x	NUM
ejpam-5572	470	14	)	)	PUNCT
ejpam-5572	470	15	)	)	PUNCT
ejpam-5572	471	1	=	=	PROPN
ejpam-5572	471	2	q3b(h(x))e	q3b(h(x))e	PROPN
ejpam-5572	471	3	2πiν3b(h(x	2πiν3b(h(x	NUM
ejpam-5572	471	4	)	)	PUNCT
ejpam-5572	471	5	)	)	PUNCT
ejpam-5572	471	6	,	,	PUNCT
ejpam-5572	471	7	∀	∀	PUNCT
ejpam-5572	471	8	x	x	SYM
ejpam-5572	471	9	∈	∈	NOUN
ejpam-5572	471	10	x.	x.	NOUN
ejpam-5572	471	11	lemma	lemma	PROPN
ejpam-5572	472	1	1	1	X
ejpam-5572	472	2	.	.	PUNCT
ejpam-5572	473	1	let	let	VERB
ejpam-5572	473	2	h	h	NOUN
ejpam-5572	473	3	:	:	PUNCT
ejpam-5572	473	4	x	x	X
ejpam-5572	473	5	→	→	SYM
ejpam-5572	473	6	u	u	NOUN
ejpam-5572	473	7	be	be	VERB
ejpam-5572	473	8	a	a	DET
ejpam-5572	473	9	homomorphism	homomorphism	NOUN
ejpam-5572	473	10	from	from	ADP
ejpam-5572	473	11	group	group	NOUN
ejpam-5572	473	12	x	x	NUM
ejpam-5572	473	13	to	to	ADP
ejpam-5572	473	14	group	group	NOUN
ejpam-5572	473	15	u	u	PROPN
ejpam-5572	473	16	,	,	PUNCT
ejpam-5572	473	17	and	and	CCONJ
ejpam-5572	473	18	let	let	VERB
ejpam-5572	473	19	a	a	DET
ejpam-5572	473	20	be	be	AUX
ejpam-5572	473	21	cffsg	cffsg	ADJ
ejpam-5572	473	22	of	of	ADP
ejpam-5572	473	23	x	x	PROPN
ejpam-5572	473	24	,	,	PUNCT
ejpam-5572	473	25	b	b	X
ejpam-5572	473	26	be	be	AUX
ejpam-5572	473	27	cffsg	cffsg	ADJ
ejpam-5572	473	28	of	of	ADP
ejpam-5572	473	29	u.	u.	NOUN
ejpam-5572	473	30	then	then	ADV
ejpam-5572	473	31	:	:	PUNCT
ejpam-5572	473	32	1	1	X
ejpam-5572	473	33	)	)	PUNCT
ejpam-5572	473	34	h(k3	h(k3	NOUN
ejpam-5572	473	35	a)(y	a)(y	PROPN
ejpam-5572	473	36	)	)	PUNCT
ejpam-5572	474	1	=	=	SYM
ejpam-5572	474	2	h(p3a)(y)e	h(p3a)(y)e	X
ejpam-5572	474	3	2πih(ω3	2πih(ω3	NUM
ejpam-5572	474	4	a)(y	a)(y	PROPN
ejpam-5572	474	5	)	)	PUNCT
ejpam-5572	474	6	∀	∀	PUNCT
ejpam-5572	475	1	y	y	PROPN
ejpam-5572	475	2	∈	∈	PROPN
ejpam-5572	475	3	u.	u.	NOUN
ejpam-5572	475	4	2	2	NUM
ejpam-5572	475	5	)	)	PUNCT
ejpam-5572	475	6	h(l3	h(l3	ADJ
ejpam-5572	475	7	a)(y	a)(y	X
ejpam-5572	475	8	)	)	PUNCT
ejpam-5572	475	9	=	=	SYM
ejpam-5572	475	10	h(q3a)(y)e	h(q3a)(y)e	PROPN
ejpam-5572	475	11	2πif(ν3a)(y	2πif(ν3a)(y	NUM
ejpam-5572	475	12	)	)	PUNCT
ejpam-5572	475	13	∀	∀	X
ejpam-5572	476	1	y	y	PROPN
ejpam-5572	476	2	∈	∈	PROPN
ejpam-5572	476	3	u.	u.	PROPN
ejpam-5572	476	4	3	3	NUM
ejpam-5572	476	5	)	)	PUNCT
ejpam-5572	476	6	h−1(k3	h−1(k3	ADP
ejpam-5572	476	7	b)(x	b)(x	PROPN
ejpam-5572	476	8	)	)	PUNCT
ejpam-5572	476	9	=	=	SYM
ejpam-5572	476	10	h−1(p3b)(x)e	h−1(p3b)(x)e	PROPN
ejpam-5572	477	1	2πih−1(ω3	2πih−1(ω3	NUM
ejpam-5572	477	2	b)(x	b)(x	NOUN
ejpam-5572	477	3	)	)	PUNCT
ejpam-5572	477	4	∀	∀	X
ejpam-5572	478	1	x	x	SYM
ejpam-5572	478	2	∈	∈	NOUN
ejpam-5572	478	3	x.	x.	NOUN
ejpam-5572	478	4	4	4	X
ejpam-5572	478	5	)	)	PUNCT
ejpam-5572	478	6	h−1(l3	h−1(l3	PRON
ejpam-5572	478	7	b)(x	b)(x	PROPN
ejpam-5572	478	8	)	)	PUNCT
ejpam-5572	478	9	=	=	PUNCT
ejpam-5572	478	10	h−1(q3b)(x)e	h−1(q3b)(x)e	PROPN
ejpam-5572	478	11	2πih−1(ν3b)(x	2πih−1(ν3b)(x	PROPN
ejpam-5572	478	12	)	)	PUNCT
ejpam-5572	478	13	∀	∀	X
ejpam-5572	478	14	x	x	SYM
ejpam-5572	478	15	∈	∈	NOUN
ejpam-5572	478	16	x.	x.	NOUN
ejpam-5572	478	17	proof	proof	NOUN
ejpam-5572	478	18	.	.	PUNCT
ejpam-5572	479	1	1	1	X
ejpam-5572	479	2	)	)	PUNCT
ejpam-5572	479	3	h(k3	h(k3	NOUN
ejpam-5572	479	4	a)(y	a)(y	PROPN
ejpam-5572	479	5	)	)	PUNCT
ejpam-5572	480	1	=	=	SYM
ejpam-5572	480	2	sup	sup	NOUN
ejpam-5572	480	3	x∈h−1(y	x∈h−1(y	PROPN
ejpam-5572	480	4	)	)	PUNCT
ejpam-5572	480	5	{	{	PUNCT
ejpam-5572	480	6	k3	k3	VERB
ejpam-5572	480	7	a(x	a(x	NOUN
ejpam-5572	480	8	)	)	PUNCT
ejpam-5572	480	9	;	;	PUNCT
ejpam-5572	480	10	h(x	h(x	PROPN
ejpam-5572	480	11	)	)	PUNCT
ejpam-5572	480	12	=	=	PUNCT
ejpam-5572	481	1	y	y	X
ejpam-5572	481	2	}	}	PUNCT
ejpam-5572	481	3	=	=	SYM
ejpam-5572	481	4	sup	sup	NOUN
ejpam-5572	481	5	x∈h−1(y	x∈h−1(y	PROPN
ejpam-5572	481	6	)	)	PUNCT
ejpam-5572	481	7	{	{	PUNCT
ejpam-5572	481	8	p3a(x)e2πiω	p3a(x)e2πiω	PROPN
ejpam-5572	481	9	3	3	NUM
ejpam-5572	481	10	a(x	a(x	NOUN
ejpam-5572	481	11	)	)	PUNCT
ejpam-5572	481	12	;	;	PUNCT
ejpam-5572	481	13	h(x	h(x	PROPN
ejpam-5572	481	14	)	)	PUNCT
ejpam-5572	481	15	=	=	SYM
ejpam-5572	481	16	y	y	PROPN
ejpam-5572	481	17	}	}	PUNCT
ejpam-5572	481	18	e.a	e.a	PROPN
ejpam-5572	481	19	.	.	PROPN
ejpam-5572	481	20	abuhijleh	abuhijleh	PROPN
ejpam-5572	481	21	,	,	PUNCT
ejpam-5572	481	22	a.	a.	PROPN
ejpam-5572	481	23	alkouri	alkouri	PROPN
ejpam-5572	481	24	/	/	PROPN
ejpam-5572	481	25	eur	eur	PROPN
ejpam-5572	481	26	.	.	PUNCT
ejpam-5572	482	1	j.	j.	PROPN
ejpam-5572	482	2	pure	pure	PROPN
ejpam-5572	482	3	appl	appl	PROPN
ejpam-5572	482	4	.	.	PROPN
ejpam-5572	482	5	math	math	PROPN
ejpam-5572	482	6	,	,	PUNCT
ejpam-5572	482	7	18	18	NUM
ejpam-5572	482	8	(	(	PUNCT
ejpam-5572	482	9	1	1	NUM
ejpam-5572	482	10	)	)	PUNCT
ejpam-5572	482	11	(	(	PUNCT
ejpam-5572	482	12	2025	2025	NUM
ejpam-5572	482	13	)	)	PUNCT
ejpam-5572	482	14	,	,	PUNCT
ejpam-5572	482	15	5572	5572	NUM
ejpam-5572	482	16	14	14	NUM
ejpam-5572	482	17	of	of	ADP
ejpam-5572	482	18	19	19	NUM
ejpam-5572	482	19	=	=	NOUN
ejpam-5572	482	20	sup	sup	NOUN
ejpam-5572	482	21	x∈h−1(y	x∈h−1(y	PROPN
ejpam-5572	482	22	)	)	PUNCT
ejpam-5572	482	23	{	{	PUNCT
ejpam-5572	482	24	p3a(x)}e	p3a(x)}e	DET
ejpam-5572	482	25	2πi	2πi	ADJ
ejpam-5572	482	26	sup	sup	NOUN
ejpam-5572	482	27	x∈h−1(y	x∈h−1(y	PROPN
ejpam-5572	482	28	)	)	PUNCT
ejpam-5572	482	29	{	{	PUNCT
ejpam-5572	482	30	ω3	ω3	NOUN
ejpam-5572	482	31	a(x	a(x	NOUN
ejpam-5572	482	32	)	)	PUNCT
ejpam-5572	482	33	}	}	PUNCT
ejpam-5572	482	34	=	=	SYM
ejpam-5572	482	35	h(p3a)(y)e	h(p3a)(y)e	PROPN
ejpam-5572	482	36	2πih(ω3	2πih(ω3	NUM
ejpam-5572	482	37	a)(y	a)(y	NUM
ejpam-5572	482	38	)	)	PUNCT
ejpam-5572	482	39	.	.	PUNCT
ejpam-5572	483	1	2	2	X
ejpam-5572	483	2	)	)	PUNCT
ejpam-5572	483	3	h(l3	h(l3	ADJ
ejpam-5572	483	4	a)(y	a)(y	X
ejpam-5572	483	5	)	)	PUNCT
ejpam-5572	483	6	=	=	PROPN
ejpam-5572	483	7	inf	inf	PROPN
ejpam-5572	483	8	x∈h−1(y	x∈h−1(y	PROPN
ejpam-5572	483	9	)	)	PUNCT
ejpam-5572	483	10	{	{	PUNCT
ejpam-5572	483	11	l3	l3	PROPN
ejpam-5572	483	12	a(x	a(x	NOUN
ejpam-5572	483	13	)	)	PUNCT
ejpam-5572	483	14	;	;	PUNCT
ejpam-5572	483	15	h(x	h(x	PROPN
ejpam-5572	483	16	)	)	PUNCT
ejpam-5572	483	17	=	=	PUNCT
ejpam-5572	483	18	y	y	X
ejpam-5572	483	19	}	}	PUNCT
ejpam-5572	483	20	=	=	SYM
ejpam-5572	483	21	inf	inf	PROPN
ejpam-5572	483	22	x∈h−1(y	x∈h−1(y	PROPN
ejpam-5572	483	23	)	)	PUNCT
ejpam-5572	483	24	{	{	PUNCT
ejpam-5572	483	25	q3a(x)e2πiν	q3a(x)e2πiν	NOUN
ejpam-5572	483	26	3	3	NUM
ejpam-5572	483	27	a(x	a(x	NOUN
ejpam-5572	483	28	)	)	PUNCT
ejpam-5572	483	29	;	;	PUNCT
ejpam-5572	483	30	h(x	h(x	PROPN
ejpam-5572	483	31	)	)	PUNCT
ejpam-5572	483	32	=	=	PUNCT
ejpam-5572	483	33	y	y	X
ejpam-5572	483	34	}	}	PUNCT
ejpam-5572	483	35	=	=	SYM
ejpam-5572	483	36	inf	inf	PROPN
ejpam-5572	483	37	x∈h−1(y	x∈h−1(y	PROPN
ejpam-5572	483	38	)	)	PUNCT
ejpam-5572	483	39	{	{	PUNCT
ejpam-5572	483	40	q3a(x)}e	q3a(x)}e	NOUN
ejpam-5572	483	41	2πi	2πi	PROPN
ejpam-5572	483	42	inf	inf	PROPN
ejpam-5572	483	43	x∈h−1(y	x∈h−1(y	PROPN
ejpam-5572	483	44	)	)	PUNCT
ejpam-5572	483	45	{	{	PUNCT
ejpam-5572	483	46	ν3a(x	ν3a(x	NOUN
ejpam-5572	483	47	)	)	PUNCT
ejpam-5572	483	48	}	}	PUNCT
ejpam-5572	483	49	=	=	SYM
ejpam-5572	483	50	h(q3a)(y)e	h(q3a)(y)e	PROPN
ejpam-5572	483	51	2πih(ν3a)(y	2πih(ν3a)(y	NUM
ejpam-5572	483	52	)	)	PUNCT
ejpam-5572	483	53	.	.	PUNCT
ejpam-5572	484	1	3	3	X
ejpam-5572	484	2	)	)	PUNCT
ejpam-5572	484	3	h−1(k3	h−1(k3	ADP
ejpam-5572	484	4	b)(x	b)(x	PROPN
ejpam-5572	484	5	)	)	PUNCT
ejpam-5572	485	1	=	=	SYM
ejpam-5572	485	2	(	(	PUNCT
ejpam-5572	485	3	kb	kb	PROPN
ejpam-5572	485	4	)	)	PUNCT
ejpam-5572	485	5	3(h(x	3(h(x	NUM
ejpam-5572	485	6	)	)	PUNCT
ejpam-5572	485	7	)	)	PUNCT
ejpam-5572	486	1	=	=	SYM
ejpam-5572	486	2	p3b(h(x))e	p3b(h(x))e	NOUN
ejpam-5572	486	3	2πω3	2πω3	NUM
ejpam-5572	486	4	b(h(x	b(h(x	NOUN
ejpam-5572	486	5	)	)	PUNCT
ejpam-5572	486	6	)	)	PUNCT
ejpam-5572	487	1	=	=	SYM
ejpam-5572	487	2	h−1(p3b)(x)e	h−1(p3b)(x)e	PROPN
ejpam-5572	488	1	2πih−1(ω3	2πih−1(ω3	NUM
ejpam-5572	488	2	b)(x	b)(x	NOUN
ejpam-5572	488	3	)	)	PUNCT
ejpam-5572	488	4	4	4	NUM
ejpam-5572	488	5	)	)	PUNCT
ejpam-5572	488	6	h−1(l3	h−1(l3	PRON
ejpam-5572	488	7	b)(x	b)(x	PROPN
ejpam-5572	488	8	)	)	PUNCT
ejpam-5572	488	9	=	=	PUNCT
ejpam-5572	488	10	(	(	PUNCT
ejpam-5572	488	11	lb	lb	NOUN
ejpam-5572	488	12	)	)	PUNCT
ejpam-5572	488	13	3(h(x	3(h(x	NUM
ejpam-5572	488	14	)	)	PUNCT
ejpam-5572	488	15	)	)	PUNCT
ejpam-5572	489	1	=	=	PRON
ejpam-5572	489	2	q3b(h(x))e	q3b(h(x))e	PROPN
ejpam-5572	489	3	2πν3b(h(x	2πν3b(h(x	NUM
ejpam-5572	489	4	)	)	PUNCT
ejpam-5572	489	5	)	)	PUNCT
ejpam-5572	490	1	=	=	PUNCT
ejpam-5572	490	2	h−1(q3b)(x)e	h−1(q3b)(x)e	PROPN
ejpam-5572	490	3	2πih−1(ν3b)(x	2πih−1(ν3b)(x	PROPN
ejpam-5572	490	4	)	)	PUNCT
ejpam-5572	490	5	example	example	NOUN
ejpam-5572	490	6	5	5	NUM
ejpam-5572	490	7	.	.	PUNCT
ejpam-5572	491	1	let	let	VERB
ejpam-5572	491	2	(	(	PUNCT
ejpam-5572	491	3	z3,+3	z3,+3	NUM
ejpam-5572	491	4	)	)	PUNCT
ejpam-5572	491	5	and	and	CCONJ
ejpam-5572	491	6	(	(	PUNCT
ejpam-5572	491	7	z,+	z,+	NUM
ejpam-5572	491	8	)	)	PUNCT
ejpam-5572	491	9	be	be	AUX
ejpam-5572	491	10	complex	complex	ADJ
ejpam-5572	491	11	fermatean	fermatean	ADJ
ejpam-5572	491	12	fuzzy	fuzzy	ADJ
ejpam-5572	491	13	group	group	NOUN
ejpam-5572	491	14	(	(	PUNCT
ejpam-5572	491	15	cffg	cffg	PROPN
ejpam-5572	491	16	)	)	PUNCT
ejpam-5572	491	17	,	,	PUNCT
ejpam-5572	491	18	where	where	SCONJ
ejpam-5572	491	19	we	we	PRON
ejpam-5572	491	20	define	define	VERB
ejpam-5572	491	21	(	(	PUNCT
ejpam-5572	491	22	z3,+3	z3,+3	NUM
ejpam-5572	491	23	)	)	PUNCT
ejpam-5572	491	24	as	as	ADP
ejpam-5572	491	25	in	in	ADP
ejpam-5572	491	26	example	example	NOUN
ejpam-5572	491	27	4	4	NUM
ejpam-5572	491	28	.	.	PUNCT
ejpam-5572	492	1	the	the	DET
ejpam-5572	492	2	map	map	NOUN
ejpam-5572	492	3	h	h	NOUN
ejpam-5572	492	4	:	:	PUNCT
ejpam-5572	492	5	(	(	PUNCT
ejpam-5572	492	6	z,+	z,+	NUM
ejpam-5572	492	7	)	)	PUNCT
ejpam-5572	492	8	→	→	SYM
ejpam-5572	492	9	(	(	PUNCT
ejpam-5572	492	10	z3,+3	z3,+3	NUM
ejpam-5572	492	11	)	)	PUNCT
ejpam-5572	492	12	is	be	AUX
ejpam-5572	492	13	complex	complex	ADJ
ejpam-5572	492	14	fermatean	fermatean	ADJ
ejpam-5572	492	15	fuzzy	fuzzy	ADJ
ejpam-5572	492	16	homomorphism	homomorphism	NOUN
ejpam-5572	492	17	.	.	PUNCT
ejpam-5572	493	1	consider	consider	VERB
ejpam-5572	493	2	a	a	DET
ejpam-5572	493	3	=	=	SYM
ejpam-5572	493	4	{	{	PUNCT
ejpam-5572	493	5	1	1	NUM
ejpam-5572	493	6	,	,	PUNCT
ejpam-5572	493	7	4	4	NUM
ejpam-5572	493	8	,	,	PUNCT
ejpam-5572	493	9	5	5	NUM
ejpam-5572	493	10	,	,	PUNCT
ejpam-5572	493	11	8	8	NUM
ejpam-5572	493	12	,	,	PUNCT
ejpam-5572	493	13	11	11	NUM
ejpam-5572	493	14	,	,	PUNCT
ejpam-5572	493	15	12	12	NUM
ejpam-5572	493	16	}	}	SYM
ejpam-5572	493	17	⊆	⊆	NUM
ejpam-5572	493	18	z	z	PROPN
ejpam-5572	493	19	,	,	PUNCT
ejpam-5572	493	20	then	then	ADV
ejpam-5572	493	21	h(a	h(a	PROPN
ejpam-5572	493	22	)	)	PUNCT
ejpam-5572	494	1	=	=	PRON
ejpam-5572	494	2	(	(	PUNCT
ejpam-5572	494	3	x	x	X
ejpam-5572	494	4	,	,	PUNCT
ejpam-5572	494	5	h(ka)(x	h(ka)(x	PROPN
ejpam-5572	494	6	)	)	PUNCT
ejpam-5572	494	7	,	,	PUNCT
ejpam-5572	494	8	h(la)(x	h(la)(x	PROPN
ejpam-5572	494	9	)	)	PUNCT
ejpam-5572	494	10	)	)	PUNCT
ejpam-5572	494	11	.	.	PUNCT
ejpam-5572	495	1	then	then	ADV
ejpam-5572	495	2	:	:	PUNCT
ejpam-5572	495	3	1	1	X
ejpam-5572	495	4	)	)	PUNCT
ejpam-5572	495	5	h(k3	h(k3	NOUN
ejpam-5572	495	6	a)(z	a)(z	NOUN
ejpam-5572	495	7	)	)	PUNCT
ejpam-5572	496	1	=	=	SYM
ejpam-5572	496	2	sup	sup	NOUN
ejpam-5572	496	3	x∈h−1(z	x∈h−1(z	NOUN
ejpam-5572	496	4	)	)	PUNCT
ejpam-5572	496	5	{	{	PUNCT
ejpam-5572	496	6	k3	k3	VERB
ejpam-5572	496	7	a(x	a(x	NOUN
ejpam-5572	496	8	)	)	PUNCT
ejpam-5572	496	9	;	;	PUNCT
ejpam-5572	496	10	h(x	h(x	PROPN
ejpam-5572	496	11	)	)	PUNCT
ejpam-5572	497	1	=	=	PUNCT
ejpam-5572	498	1	z	z	NOUN
ejpam-5572	498	2	(	(	PUNCT
ejpam-5572	498	3	mod	mod	NOUN
ejpam-5572	498	4	3	3	NUM
ejpam-5572	498	5	)	)	PUNCT
ejpam-5572	498	6	}	}	PUNCT
ejpam-5572	498	7	=	=	PUNCT
ejpam-5572	498	8	sup{k3	sup{k3	NOUN
ejpam-5572	498	9	a(1),k3	a(1),k3	PROPN
ejpam-5572	498	10	a(4),k3	a(4),k3	PROPN
ejpam-5572	498	11	a(5),k3	a(5),k3	PROPN
ejpam-5572	498	12	a(8),k3	a(8),k3	PROPN
ejpam-5572	498	13	a(11),k3	a(11),k3	PROPN
ejpam-5572	498	14	a(12	a(12	ADP
ejpam-5572	498	15	)	)	PUNCT
ejpam-5572	498	16	}	}	PUNCT
ejpam-5572	499	1	=	=	SYM
ejpam-5572	499	2	sup{p3a(1)e2πiω	sup{p3a(1)e2πiω	NOUN
ejpam-5572	499	3	3	3	NUM
ejpam-5572	499	4	a(1	a(1	NUM
ejpam-5572	499	5	)	)	PUNCT
ejpam-5572	499	6	,	,	PUNCT
ejpam-5572	499	7	p3a(4)e	p3a(4)e	PROPN
ejpam-5572	499	8	2πiω3	2πiω3	PROPN
ejpam-5572	499	9	a(4	a(4	PROPN
ejpam-5572	499	10	)	)	PUNCT
ejpam-5572	499	11	,	,	PUNCT
ejpam-5572	499	12	p3a(5)e	p3a(5)e	NOUN
ejpam-5572	499	13	2πiω3	2πiω3	NUM
ejpam-5572	499	14	a(5	a(5	PROPN
ejpam-5572	499	15	)	)	PUNCT
ejpam-5572	499	16	,	,	PUNCT
ejpam-5572	499	17	,	,	PUNCT
ejpam-5572	499	18	p3a(8)e	p3a(8)e	X
ejpam-5572	499	19	2πiω3	2πiω3	NUM
ejpam-5572	499	20	a(8	a(8	NOUN
ejpam-5572	499	21	)	)	PUNCT
ejpam-5572	499	22	,	,	PUNCT
ejpam-5572	499	23	p3a(11)e	p3a(11)e	PROPN
ejpam-5572	499	24	2πiω3	2πiω3	NUM
ejpam-5572	499	25	a(11	a(11	NUM
ejpam-5572	499	26	)	)	PUNCT
ejpam-5572	499	27	,	,	PUNCT
ejpam-5572	499	28	p3a(12)e	p3a(12)e	NOUN
ejpam-5572	499	29	2πiω3	2πiω3	NUM
ejpam-5572	499	30	a(12	a(12	ADV
ejpam-5572	499	31	)	)	PUNCT
ejpam-5572	499	32	}	}	PUNCT
ejpam-5572	499	33	=	=	SYM
ejpam-5572	499	34	sup{p3a(1	sup{p3a(1	NUM
ejpam-5572	499	35	)	)	PUNCT
ejpam-5572	499	36	,	,	PUNCT
ejpam-5572	499	37	.	.	PUNCT
ejpam-5572	499	38	.	.	PUNCT
ejpam-5572	499	39	.	.	PUNCT
ejpam-5572	500	1	,	,	PUNCT
ejpam-5572	500	2	p3a(12)}e2πi	p3a(12)}e2πi	NOUN
ejpam-5572	500	3	sup{ω	sup{ω	PROPN
ejpam-5572	500	4	3	3	NUM
ejpam-5572	500	5	a(1),	a(1),	PROPN
ejpam-5572	500	6	...	...	PUNCT
ejpam-5572	500	7	,ω3	,ω3	PUNCT
ejpam-5572	500	8	a(12	a(12	ADP
ejpam-5572	500	9	)	)	PUNCT
ejpam-5572	500	10	}	}	PUNCT
ejpam-5572	500	11	=	=	SYM
ejpam-5572	500	12	sup{0.512	sup{0.512	NOUN
ejpam-5572	500	13	,	,	PUNCT
ejpam-5572	500	14	0.027	0.027	NUM
ejpam-5572	500	15	,	,	PUNCT
ejpam-5572	500	16	0.729}e2πi	0.729}e2πi	PROPN
ejpam-5572	500	17	sup{0.512,0.216,0.343	sup{0.512,0.216,0.343	NOUN
ejpam-5572	500	18	}	}	PUNCT
ejpam-5572	500	19	=	=	PUNCT
ejpam-5572	500	20	0.729e2πi	0.729e2πi	NOUN
ejpam-5572	500	21	0.512	0.512	NUM
ejpam-5572	500	22	2	2	NUM
ejpam-5572	500	23	)	)	PUNCT
ejpam-5572	500	24	h(l3	h(l3	PRON
ejpam-5572	500	25	a)(z	a)(z	NOUN
ejpam-5572	500	26	)	)	PUNCT
ejpam-5572	501	1	=	=	SYM
ejpam-5572	501	2	inf	inf	PROPN
ejpam-5572	501	3	x∈h−1(z	x∈h−1(z	PROPN
ejpam-5572	501	4	)	)	PUNCT
ejpam-5572	501	5	{	{	PUNCT
ejpam-5572	501	6	l3	l3	PROPN
ejpam-5572	501	7	a(x	a(x	NOUN
ejpam-5572	501	8	)	)	PUNCT
ejpam-5572	501	9	;	;	PUNCT
ejpam-5572	501	10	h(x	h(x	PROPN
ejpam-5572	501	11	)	)	PUNCT
ejpam-5572	501	12	=	=	PUNCT
ejpam-5572	502	1	z	z	NOUN
ejpam-5572	502	2	(	(	PUNCT
ejpam-5572	502	3	mod	mod	NOUN
ejpam-5572	502	4	3	3	NUM
ejpam-5572	502	5	)	)	PUNCT
ejpam-5572	502	6	}	}	PUNCT
ejpam-5572	502	7	e.a	e.a	PROPN
ejpam-5572	502	8	.	.	PROPN
ejpam-5572	502	9	abuhijleh	abuhijleh	PROPN
ejpam-5572	502	10	,	,	PUNCT
ejpam-5572	502	11	a.	a.	PROPN
ejpam-5572	502	12	alkouri	alkouri	PROPN
ejpam-5572	502	13	/	/	PROPN
ejpam-5572	502	14	eur	eur	PROPN
ejpam-5572	502	15	.	.	PUNCT
ejpam-5572	503	1	j.	j.	PROPN
ejpam-5572	503	2	pure	pure	PROPN
ejpam-5572	503	3	appl	appl	PROPN
ejpam-5572	503	4	.	.	PROPN
ejpam-5572	503	5	math	math	PROPN
ejpam-5572	503	6	,	,	PUNCT
ejpam-5572	503	7	18	18	NUM
ejpam-5572	503	8	(	(	PUNCT
ejpam-5572	503	9	1	1	NUM
ejpam-5572	503	10	)	)	PUNCT
ejpam-5572	503	11	(	(	PUNCT
ejpam-5572	503	12	2025	2025	NUM
ejpam-5572	503	13	)	)	PUNCT
ejpam-5572	503	14	,	,	PUNCT
ejpam-5572	503	15	5572	5572	NUM
ejpam-5572	503	16	15	15	NUM
ejpam-5572	503	17	of	of	ADP
ejpam-5572	503	18	19	19	NUM
ejpam-5572	503	19	=	=	NOUN
ejpam-5572	503	20	inf{l3	inf{l3	VERB
ejpam-5572	503	21	a(1),l3	a(1),l3	NUM
ejpam-5572	503	22	a(4),l3	a(4),l3	PROPN
ejpam-5572	503	23	a(5),l3	a(5),l3	PROPN
ejpam-5572	503	24	a(8),l3	a(8),l3	PROPN
ejpam-5572	503	25	a(11),l3	a(11),l3	PROPN
ejpam-5572	503	26	a(12	a(12	ADP
ejpam-5572	503	27	)	)	PUNCT
ejpam-5572	503	28	}	}	PUNCT
ejpam-5572	503	29	=	=	SYM
ejpam-5572	503	30	inf{q3a(1)e2πiν	inf{q3a(1)e2πiν	NOUN
ejpam-5572	503	31	3	3	NUM
ejpam-5572	503	32	a(1	a(1	NOUN
ejpam-5572	503	33	)	)	PUNCT
ejpam-5572	503	34	,	,	PUNCT
ejpam-5572	503	35	q3a(4)e	q3a(4)e	VERB
ejpam-5572	503	36	2πiν3a(4	2πiν3a(4	NUM
ejpam-5572	503	37	)	)	PUNCT
ejpam-5572	503	38	,	,	PUNCT
ejpam-5572	503	39	q3a(5)e	q3a(5)e	NOUN
ejpam-5572	503	40	2πiν3a(5	2πiν3a(5	PROPN
ejpam-5572	503	41	)	)	PUNCT
ejpam-5572	503	42	,	,	PUNCT
ejpam-5572	503	43	,	,	PUNCT
ejpam-5572	503	44	q3a(8)e	q3a(8)e	NOUN
ejpam-5572	503	45	2πiν3a(8	2πiν3a(8	NUM
ejpam-5572	503	46	)	)	PUNCT
ejpam-5572	503	47	,	,	PUNCT
ejpam-5572	503	48	q3a(11)e	q3a(11)e	PROPN
ejpam-5572	503	49	2πiν3a(11	2πiν3a(11	NUM
ejpam-5572	503	50	)	)	PUNCT
ejpam-5572	503	51	,	,	PUNCT
ejpam-5572	503	52	q3a(12)e	q3a(12)e	NOUN
ejpam-5572	503	53	2πiν3a(12	2πiν3a(12	NUM
ejpam-5572	503	54	)	)	PUNCT
ejpam-5572	503	55	}	}	PUNCT
ejpam-5572	503	56	=	=	SYM
ejpam-5572	503	57	inf{q3a(1	inf{q3a(1	X
ejpam-5572	503	58	)	)	PUNCT
ejpam-5572	503	59	,	,	PUNCT
ejpam-5572	503	60	.	.	PUNCT
ejpam-5572	503	61	.	.	PUNCT
ejpam-5572	504	1	.	.	PUNCT
ejpam-5572	505	1	,	,	PUNCT
ejpam-5572	505	2	q3a(12)}e2πi	q3a(12)}e2πi	PROPN
ejpam-5572	505	3	inf{ν	inf{ν	PROPN
ejpam-5572	505	4	3	3	NUM
ejpam-5572	505	5	a(1),	a(1),	AUX
ejpam-5572	505	6	...	...	PUNCT
ejpam-5572	505	7	,ν3a(12	,ν3a(12	PROPN
ejpam-5572	505	8	)	)	PUNCT
ejpam-5572	505	9	}	}	PUNCT
ejpam-5572	505	10	=	=	SYM
ejpam-5572	505	11	inf{0.343	inf{0.343	PROPN
ejpam-5572	505	12	,	,	PUNCT
ejpam-5572	505	13	0.125	0.125	NUM
ejpam-5572	505	14	,	,	PUNCT
ejpam-5572	505	15	0.512}e2πi	0.512}e2πi	NOUN
ejpam-5572	505	16	inf{0.216,0.512	inf{0.216,0.512	NOUN
ejpam-5572	505	17	}	}	PUNCT
ejpam-5572	505	18	=	=	SYM
ejpam-5572	505	19	0.125e2πi	0.125e2πi	NOUN
ejpam-5572	505	20	0.216	0.216	NUM
ejpam-5572	505	21	theorem	theorem	NOUN
ejpam-5572	505	22	4	4	NUM
ejpam-5572	505	23	.	.	PUNCT
ejpam-5572	506	1	let	let	VERB
ejpam-5572	506	2	h	h	NOUN
ejpam-5572	506	3	:	:	PUNCT
ejpam-5572	506	4	x	x	X
ejpam-5572	506	5	epimorphism−−−−−−−−→	epimorphism−−−−−−−−→	NOUN
ejpam-5572	506	6	u	u	NOUN
ejpam-5572	506	7	,	,	PUNCT
ejpam-5572	506	8	from	from	ADP
ejpam-5572	506	9	(	(	PUNCT
ejpam-5572	506	10	x	x	NOUN
ejpam-5572	506	11	,	,	PUNCT
ejpam-5572	506	12	∗1	∗1	PROPN
ejpam-5572	506	13	)	)	PUNCT
ejpam-5572	506	14	to	to	ADP
ejpam-5572	506	15	(	(	PUNCT
ejpam-5572	506	16	u	u	NOUN
ejpam-5572	506	17	,	,	PUNCT
ejpam-5572	506	18	∗2	∗2	PROPN
ejpam-5572	506	19	)	)	PUNCT
ejpam-5572	506	20	,	,	PUNCT
ejpam-5572	506	21	and	and	CCONJ
ejpam-5572	506	22	let	let	VERB
ejpam-5572	506	23	a	a	DET
ejpam-5572	506	24	be	be	AUX
ejpam-5572	506	25	cffsg	cffsg	ADJ
ejpam-5572	506	26	of	of	ADP
ejpam-5572	506	27	x.	x.	NOUN
ejpam-5572	506	28	then	then	ADV
ejpam-5572	506	29	h(a	h(a	PROPN
ejpam-5572	506	30	)	)	PUNCT
ejpam-5572	506	31	is	be	AUX
ejpam-5572	506	32	cffsg	cffsg	ADJ
ejpam-5572	506	33	of	of	ADP
ejpam-5572	506	34	u.	u.	NOUN
ejpam-5572	506	35	proof	proof	NOUN
ejpam-5572	506	36	.	.	PUNCT
ejpam-5572	507	1	consider	consider	VERB
ejpam-5572	507	2	a	a	DET
ejpam-5572	507	3	two	two	NUM
ejpam-5572	507	4	groups	group	NOUN
ejpam-5572	507	5	(	(	PUNCT
ejpam-5572	507	6	x	x	X
ejpam-5572	507	7	,	,	PUNCT
ejpam-5572	507	8	∗1	∗1	PROPN
ejpam-5572	507	9	)	)	PUNCT
ejpam-5572	507	10	and	and	CCONJ
ejpam-5572	507	11	(	(	PUNCT
ejpam-5572	507	12	u	u	NOUN
ejpam-5572	507	13	,	,	PUNCT
ejpam-5572	507	14	∗2	∗2	PROPN
ejpam-5572	507	15	)	)	PUNCT
ejpam-5572	507	16	,	,	PUNCT
ejpam-5572	507	17	with	with	ADP
ejpam-5572	507	18	a	a	DET
ejpam-5572	507	19	=	=	SYM
ejpam-5572	507	20	(	(	PUNCT
ejpam-5572	507	21	ka	ka	PROPN
ejpam-5572	507	22	,	,	PUNCT
ejpam-5572	507	23	la	la	NOUN
ejpam-5572	507	24	)	)	PUNCT
ejpam-5572	507	25	is	be	AUX
ejpam-5572	507	26	cffsg	cffsg	ADJ
ejpam-5572	507	27	,	,	PUNCT
ejpam-5572	507	28	and	and	CCONJ
ejpam-5572	507	29	want	want	VERB
ejpam-5572	507	30	to	to	PART
ejpam-5572	507	31	show	show	VERB
ejpam-5572	507	32	that	that	PRON
ejpam-5572	507	33	h(a	h(a	PROPN
ejpam-5572	507	34	)	)	PUNCT
ejpam-5572	508	1	=	=	PRON
ejpam-5572	508	2	(	(	PUNCT
ejpam-5572	508	3	h(ka	h(ka	NOUN
ejpam-5572	508	4	)	)	PUNCT
ejpam-5572	508	5	,	,	PUNCT
ejpam-5572	508	6	h(la	h(la	NUM
ejpam-5572	508	7	)	)	PUNCT
ejpam-5572	508	8	)	)	PUNCT
ejpam-5572	509	1	=	=	SYM
ejpam-5572	509	2	(	(	PUNCT
ejpam-5572	509	3	h(pa)(y)e	h(pa)(y)e	X
ejpam-5572	509	4	2πih(ωa)(y	2πih(ωa)(y	NUM
ejpam-5572	509	5	)	)	PUNCT
ejpam-5572	509	6	,	,	PUNCT
ejpam-5572	509	7	h(qa)(y)e	h(qa)(y)e	NOUN
ejpam-5572	509	8	2πih(νa)(y	2πih(νa)(y	NUM
ejpam-5572	509	9	)	)	PUNCT
ejpam-5572	509	10	)	)	PUNCT
ejpam-5572	509	11	is	be	AUX
ejpam-5572	509	12	cffsg	cffsg	ADJ
ejpam-5572	509	13	.	.	PUNCT
ejpam-5572	510	1	at	at	ADP
ejpam-5572	510	2	first	first	ADV
ejpam-5572	510	3	,	,	PUNCT
ejpam-5572	510	4	the	the	DET
ejpam-5572	510	5	set	set	NOUN
ejpam-5572	510	6	s1	s1	NOUN
ejpam-5572	510	7	=	=	SYM
ejpam-5572	510	8	{	{	PUNCT
ejpam-5572	510	9	(	(	PUNCT
ejpam-5572	510	10	x	x	NOUN
ejpam-5572	510	11	,	,	PUNCT
ejpam-5572	510	12	pa(x	pa(x	NOUN
ejpam-5572	510	13	)	)	PUNCT
ejpam-5572	510	14	,	,	PUNCT
ejpam-5572	510	15	qa(x	qa(x	NOUN
ejpam-5572	510	16	)	)	PUNCT
ejpam-5572	510	17	)	)	PUNCT
ejpam-5572	510	18	:	:	PUNCT
ejpam-5572	510	19	x	x	X
ejpam-5572	510	20	∈	∈	NOUN
ejpam-5572	510	21	x	x	X
ejpam-5572	510	22	,	,	PUNCT
ejpam-5572	510	23	0	0	NUM
ejpam-5572	510	24	≤	≤	NUM
ejpam-5572	510	25	p3a(x	p3a(x	NOUN
ejpam-5572	510	26	)	)	PUNCT
ejpam-5572	511	1	+	+	NUM
ejpam-5572	511	2	q3a(x	q3a(x	NOUN
ejpam-5572	511	3	)	)	PUNCT
ejpam-5572	511	4	≤	≤	NOUN
ejpam-5572	511	5	1	1	NUM
ejpam-5572	511	6	}	}	PUNCT
ejpam-5572	511	7	and	and	CCONJ
ejpam-5572	511	8	s2	s2	VERB
ejpam-5572	511	9	=	=	SYM
ejpam-5572	511	10	{	{	PUNCT
ejpam-5572	511	11	(	(	PUNCT
ejpam-5572	511	12	x	x	NOUN
ejpam-5572	511	13	,	,	PUNCT
ejpam-5572	511	14	ωa(x	ωa(x	NOUN
ejpam-5572	511	15	)	)	PUNCT
ejpam-5572	511	16	,	,	PUNCT
ejpam-5572	511	17	νa(x	νa(x	NOUN
ejpam-5572	511	18	)	)	PUNCT
ejpam-5572	511	19	)	)	PUNCT
ejpam-5572	511	20	:	:	PUNCT
ejpam-5572	512	1	x	x	X
ejpam-5572	512	2	∈	∈	NOUN
ejpam-5572	512	3	x	x	X
ejpam-5572	512	4	,	,	PUNCT
ejpam-5572	512	5	0	0	NUM
ejpam-5572	512	6	≤	≤	NUM
ejpam-5572	512	7	ω3	ω3	NOUN
ejpam-5572	512	8	a(x	a(x	NOUN
ejpam-5572	512	9	)	)	PUNCT
ejpam-5572	512	10	+	+	NUM
ejpam-5572	512	11	ν3a(x	ν3a(x	NOUN
ejpam-5572	512	12	)	)	PUNCT
ejpam-5572	512	13	≤	≤	NOUN
ejpam-5572	512	14	1	1	NUM
ejpam-5572	512	15	}	}	PUNCT
ejpam-5572	512	16	are	be	AUX
ejpam-5572	512	17	the	the	DET
ejpam-5572	512	18	amplitude	amplitude	NOUN
ejpam-5572	512	19	and	and	CCONJ
ejpam-5572	512	20	phase	phase	NOUN
ejpam-5572	512	21	terms	term	NOUN
ejpam-5572	512	22	of	of	ADP
ejpam-5572	512	23	cffsg	cffsg	ADJ
ejpam-5572	512	24	,	,	PUNCT
ejpam-5572	512	25	since	since	SCONJ
ejpam-5572	512	26	a	a	PRON
ejpam-5572	512	27	is	be	AUX
ejpam-5572	512	28	cffsg	cffsg	ADJ
ejpam-5572	512	29	and	and	CCONJ
ejpam-5572	512	30	using	use	VERB
ejpam-5572	512	31	lemma	lemma	PROPN
ejpam-5572	512	32	1	1	NUM
ejpam-5572	512	33	.	.	PUNCT
ejpam-5572	512	34	then	then	ADV
ejpam-5572	512	35	by	by	ADP
ejpam-5572	512	36	theorem[6.1	theorem[6.1	VERB
ejpam-5572	512	37	]	]	X
ejpam-5572	513	1	[	[	X
ejpam-5572	513	2	35	35	NUM
ejpam-5572	513	3	]	]	PUNCT
ejpam-5572	513	4	and	and	CCONJ
ejpam-5572	513	5	h	h	NOUN
ejpam-5572	513	6	is	be	AUX
ejpam-5572	513	7	homomorphism	homomorphism	NOUN
ejpam-5572	513	8	,	,	PUNCT
ejpam-5572	513	9	we	we	PRON
ejpam-5572	513	10	have	have	VERB
ejpam-5572	513	11	:	:	PUNCT
ejpam-5572	513	12	i	i	NOUN
ejpam-5572	513	13	)	)	PUNCT
ejpam-5572	513	14	a	a	X
ejpam-5572	513	15	)	)	PUNCT
ejpam-5572	513	16	h(p3a)(x1	h(p3a)(x1	PROPN
ejpam-5572	513	17	∗2	∗2	PROPN
ejpam-5572	513	18	x2	x2	NUM
ejpam-5572	513	19	)	)	PUNCT
ejpam-5572	514	1	=	=	SYM
ejpam-5572	514	2	(	(	PUNCT
ejpam-5572	514	3	h(pa	h(pa	NOUN
ejpam-5572	514	4	)	)	PUNCT
ejpam-5572	514	5	)	)	PUNCT
ejpam-5572	515	1	3(x1	3(x1	NUM
ejpam-5572	515	2	∗2	∗2	NOUN
ejpam-5572	515	3	x2	x2	NUM
ejpam-5572	515	4	)	)	PUNCT
ejpam-5572	515	5	≥	≥	NOUN
ejpam-5572	515	6	(	(	PUNCT
ejpam-5572	515	7	h(pa	h(pa	NOUN
ejpam-5572	515	8	)	)	PUNCT
ejpam-5572	515	9	)	)	PUNCT
ejpam-5572	516	1	3(x1	3(x1	X
ejpam-5572	516	2	)	)	PUNCT
ejpam-5572	516	3	∧	∧	PROPN
ejpam-5572	516	4	(	(	PUNCT
ejpam-5572	516	5	h(pa	h(pa	NOUN
ejpam-5572	516	6	)	)	PUNCT
ejpam-5572	516	7	)	)	PUNCT
ejpam-5572	516	8	3(x2	3(x2	NOUN
ejpam-5572	516	9	)	)	PUNCT
ejpam-5572	516	10	,	,	PUNCT
ejpam-5572	516	11	b	b	X
ejpam-5572	516	12	)	)	PUNCT
ejpam-5572	516	13	h(q3a)(x1	h(q3a)(x1	PROPN
ejpam-5572	516	14	∗2	∗2	NOUN
ejpam-5572	516	15	x2	x2	NOUN
ejpam-5572	516	16	)	)	PUNCT
ejpam-5572	516	17	=	=	PUNCT
ejpam-5572	516	18	(	(	PUNCT
ejpam-5572	516	19	h(qa	h(qa	NOUN
ejpam-5572	516	20	)	)	PUNCT
ejpam-5572	516	21	)	)	PUNCT
ejpam-5572	516	22	3(x1	3(x1	NUM
ejpam-5572	516	23	∗2	∗2	NOUN
ejpam-5572	516	24	x2	x2	NUM
ejpam-5572	516	25	)	)	PUNCT
ejpam-5572	516	26	≤	≤	NOUN
ejpam-5572	516	27	(	(	PUNCT
ejpam-5572	516	28	h(qa	h(qa	NOUN
ejpam-5572	516	29	)	)	PUNCT
ejpam-5572	516	30	)	)	PUNCT
ejpam-5572	516	31	3(x1	3(x1	NUM
ejpam-5572	516	32	)	)	PUNCT
ejpam-5572	516	33	∨	∨	NOUN
ejpam-5572	516	34	(	(	PUNCT
ejpam-5572	516	35	h(qa	h(qa	NOUN
ejpam-5572	516	36	)	)	PUNCT
ejpam-5572	516	37	)	)	PUNCT
ejpam-5572	516	38	3(x2	3(x2	NOUN
ejpam-5572	516	39	)	)	PUNCT
ejpam-5572	516	40	,	,	PUNCT
ejpam-5572	516	41	c	c	X
ejpam-5572	516	42	)	)	PUNCT
ejpam-5572	516	43	h(ω3	h(ω3	NOUN
ejpam-5572	516	44	a)(x1	a)(x1	NOUN
ejpam-5572	516	45	∗2	∗2	NOUN
ejpam-5572	516	46	x2	x2	NUM
ejpam-5572	516	47	)	)	PUNCT
ejpam-5572	516	48	=	=	SYM
ejpam-5572	516	49	(	(	PUNCT
ejpam-5572	516	50	h(ωa	h(ωa	NOUN
ejpam-5572	516	51	)	)	PUNCT
ejpam-5572	516	52	)	)	PUNCT
ejpam-5572	517	1	3(x1	3(x1	NUM
ejpam-5572	517	2	∗2	∗2	NOUN
ejpam-5572	517	3	x2	x2	NUM
ejpam-5572	517	4	)	)	PUNCT
ejpam-5572	517	5	≥	≥	NOUN
ejpam-5572	517	6	(	(	PUNCT
ejpam-5572	517	7	h(ωa	h(ωa	NOUN
ejpam-5572	517	8	)	)	PUNCT
ejpam-5572	517	9	)	)	PUNCT
ejpam-5572	518	1	3(x1	3(x1	X
ejpam-5572	518	2	)	)	PUNCT
ejpam-5572	518	3	∧	∧	PROPN
ejpam-5572	518	4	(	(	PUNCT
ejpam-5572	518	5	h(ωa	h(ωa	NOUN
ejpam-5572	518	6	)	)	PUNCT
ejpam-5572	518	7	)	)	PUNCT
ejpam-5572	518	8	3(x2	3(x2	NOUN
ejpam-5572	518	9	)	)	PUNCT
ejpam-5572	518	10	,	,	PUNCT
ejpam-5572	518	11	d	d	X
ejpam-5572	518	12	)	)	PUNCT
ejpam-5572	518	13	h(ν3a)(x1	h(ν3a)(x1	PROPN
ejpam-5572	518	14	∗2	∗2	PROPN
ejpam-5572	518	15	x2	x2	NUM
ejpam-5572	518	16	)	)	PUNCT
ejpam-5572	518	17	=	=	SYM
ejpam-5572	518	18	(	(	PUNCT
ejpam-5572	518	19	h(νa	h(νa	NOUN
ejpam-5572	518	20	)	)	PUNCT
ejpam-5572	518	21	)	)	PUNCT
ejpam-5572	519	1	3(x1	3(x1	NUM
ejpam-5572	519	2	∗2	∗2	NOUN
ejpam-5572	519	3	x2	x2	NUM
ejpam-5572	519	4	)	)	PUNCT
ejpam-5572	519	5	≤	≤	NOUN
ejpam-5572	519	6	(	(	PUNCT
ejpam-5572	519	7	h(νa	h(νa	NOUN
ejpam-5572	519	8	)	)	PUNCT
ejpam-5572	519	9	)	)	PUNCT
ejpam-5572	520	1	3(x1	3(x1	NUM
ejpam-5572	520	2	)	)	PUNCT
ejpam-5572	520	3	∨	∨	NOUN
ejpam-5572	520	4	(	(	PUNCT
ejpam-5572	520	5	h(νa	h(νa	NOUN
ejpam-5572	520	6	)	)	PUNCT
ejpam-5572	520	7	)	)	PUNCT
ejpam-5572	520	8	3(x2	3(x2	NOUN
ejpam-5572	520	9	)	)	PUNCT
ejpam-5572	520	10	.	.	PUNCT
ejpam-5572	521	1	ii	ii	X
ejpam-5572	521	2	)	)	PUNCT
ejpam-5572	521	3	a	a	X
ejpam-5572	521	4	)	)	PUNCT
ejpam-5572	521	5	h(p3a)(x	h(p3a)(x	PROPN
ejpam-5572	521	6	−1	−1	NOUN
ejpam-5572	521	7	)	)	PUNCT
ejpam-5572	521	8	=	=	PUNCT
ejpam-5572	521	9	(	(	PUNCT
ejpam-5572	521	10	h(pa	h(pa	NOUN
ejpam-5572	521	11	)	)	PUNCT
ejpam-5572	521	12	)	)	PUNCT
ejpam-5572	522	1	3(x−1	3(x−1	NUM
ejpam-5572	522	2	)	)	PUNCT
ejpam-5572	523	1	=	=	SYM
ejpam-5572	523	2	(	(	PUNCT
ejpam-5572	523	3	h(pa	h(pa	NOUN
ejpam-5572	523	4	)	)	PUNCT
ejpam-5572	523	5	)	)	PUNCT
ejpam-5572	523	6	3(x−1	3(x−1	NUM
ejpam-5572	523	7	)	)	PUNCT
ejpam-5572	523	8	,	,	PUNCT
ejpam-5572	523	9	b	b	X
ejpam-5572	523	10	)	)	PUNCT
ejpam-5572	523	11	h(q3a)(x	h(q3a)(x	NOUN
ejpam-5572	523	12	−1	−1	NOUN
ejpam-5572	523	13	)	)	PUNCT
ejpam-5572	523	14	=	=	PUNCT
ejpam-5572	523	15	(	(	PUNCT
ejpam-5572	523	16	h(qa	h(qa	NOUN
ejpam-5572	523	17	)	)	PUNCT
ejpam-5572	523	18	)	)	PUNCT
ejpam-5572	523	19	3(x−1	3(x−1	NUM
ejpam-5572	523	20	)	)	PUNCT
ejpam-5572	523	21	=	=	PUNCT
ejpam-5572	523	22	(	(	PUNCT
ejpam-5572	523	23	h(qa	h(qa	NOUN
ejpam-5572	523	24	)	)	PUNCT
ejpam-5572	523	25	)	)	PUNCT
ejpam-5572	523	26	3(x−1	3(x−1	NUM
ejpam-5572	523	27	)	)	PUNCT
ejpam-5572	523	28	,	,	PUNCT
ejpam-5572	523	29	c	c	X
ejpam-5572	523	30	)	)	PUNCT
ejpam-5572	523	31	h(ω3	h(ω3	NOUN
ejpam-5572	523	32	a)(x	a)(x	NOUN
ejpam-5572	523	33	−1	−1	NOUN
ejpam-5572	523	34	)	)	PUNCT
ejpam-5572	523	35	=	=	PUNCT
ejpam-5572	523	36	(	(	PUNCT
ejpam-5572	523	37	h(ωa	h(ωa	NOUN
ejpam-5572	523	38	)	)	PUNCT
ejpam-5572	523	39	)	)	PUNCT
ejpam-5572	523	40	3(x−1	3(x−1	NUM
ejpam-5572	523	41	)	)	PUNCT
ejpam-5572	523	42	=	=	SYM
ejpam-5572	523	43	(	(	PUNCT
ejpam-5572	523	44	h(ωa	h(ωa	NOUN
ejpam-5572	523	45	)	)	PUNCT
ejpam-5572	523	46	)	)	PUNCT
ejpam-5572	523	47	3(x−1	3(x−1	NUM
ejpam-5572	523	48	)	)	PUNCT
ejpam-5572	523	49	,	,	PUNCT
ejpam-5572	523	50	d	d	X
ejpam-5572	523	51	)	)	PUNCT
ejpam-5572	523	52	h(ω3	h(ω3	NOUN
ejpam-5572	523	53	a)(x	a)(x	NOUN
ejpam-5572	523	54	−1	−1	NOUN
ejpam-5572	523	55	)	)	PUNCT
ejpam-5572	523	56	=	=	SYM
ejpam-5572	523	57	(	(	PUNCT
ejpam-5572	523	58	h(νa	h(νa	NOUN
ejpam-5572	523	59	)	)	PUNCT
ejpam-5572	523	60	)	)	PUNCT
ejpam-5572	523	61	3(x−1	3(x−1	NUM
ejpam-5572	523	62	)	)	PUNCT
ejpam-5572	523	63	=	=	SYM
ejpam-5572	523	64	(	(	PUNCT
ejpam-5572	523	65	h(νa	h(νa	NOUN
ejpam-5572	523	66	)	)	PUNCT
ejpam-5572	523	67	)	)	PUNCT
ejpam-5572	523	68	3(x−1	3(x−1	NUM
ejpam-5572	523	69	)	)	PUNCT
ejpam-5572	523	70	.	.	PUNCT
ejpam-5572	524	1	consequently	consequently	ADV
ejpam-5572	524	2	and	and	CCONJ
ejpam-5572	524	3	by	by	ADP
ejpam-5572	524	4	lemma	lemma	PROPN
ejpam-5572	524	5	1	1	NUM
ejpam-5572	524	6	,	,	PUNCT
ejpam-5572	524	7	we	we	PRON
ejpam-5572	524	8	have	have	VERB
ejpam-5572	524	9	:	:	PUNCT
ejpam-5572	524	10	1	1	NUM
ejpam-5572	524	11	)	)	PUNCT
ejpam-5572	524	12	(	(	PUNCT
ejpam-5572	524	13	h(ka	h(ka	NOUN
ejpam-5572	524	14	)	)	PUNCT
ejpam-5572	524	15	)	)	PUNCT
ejpam-5572	525	1	3(x1	3(x1	NUM
ejpam-5572	525	2	∗2	∗2	NOUN
ejpam-5572	525	3	x2	x2	NOUN
ejpam-5572	525	4	)	)	PUNCT
ejpam-5572	526	1	=	=	SYM
ejpam-5572	526	2	h(k3	h(k3	NOUN
ejpam-5572	526	3	a)(x1	a)(x1	PROPN
ejpam-5572	526	4	∗2	∗2	NOUN
ejpam-5572	526	5	x2	x2	NUM
ejpam-5572	526	6	)	)	PUNCT
ejpam-5572	527	1	=	=	PUNCT
ejpam-5572	527	2	h(p3a)(x1	h(p3a)(x1	PROPN
ejpam-5572	527	3	∗2	∗2	PROPN
ejpam-5572	527	4	x2)e2πi	x2)e2πi	CCONJ
ejpam-5572	527	5	h(ω3	h(ω3	PROPN
ejpam-5572	527	6	a)(x1∗2x2	a)(x1∗2x2	ADJ
ejpam-5572	527	7	)	)	PUNCT
ejpam-5572	527	8	≥	≥	PROPN
ejpam-5572	527	9	(	(	PUNCT
ejpam-5572	527	10	h(p3a)(x1	h(p3a)(x1	PROPN
ejpam-5572	527	11	)	)	PUNCT
ejpam-5572	527	12	∧	∧	PROPN
ejpam-5572	527	13	h(p3a)(x2	h(p3a)(x2	NOUN
ejpam-5572	527	14	)	)	PUNCT
ejpam-5572	527	15	)	)	PUNCT
ejpam-5572	528	1	∗	∗	NOUN
ejpam-5572	528	2	e2πi(h(ω	e2πi(h(ω	NOUN
ejpam-5572	528	3	3	3	NUM
ejpam-5572	528	4	a)(x1)∧h(ω3	a)(x1)∧h(ω3	NUM
ejpam-5572	528	5	a)(x2	a)(x2	NOUN
ejpam-5572	528	6	)	)	PUNCT
ejpam-5572	528	7	)	)	PUNCT
ejpam-5572	529	1	=	=	PRON
ejpam-5572	529	2	{	{	PUNCT
ejpam-5572	529	3	h(p3a)(x1)e2πi	h(p3a)(x1)e2πi	PROPN
ejpam-5572	529	4	h(ω3	h(ω3	NOUN
ejpam-5572	529	5	a)(x1	a)(x1	NOUN
ejpam-5572	529	6	)	)	PUNCT
ejpam-5572	529	7	∧	∧	PROPN
ejpam-5572	529	8	h(p3a)(x2)e	h(p3a)(x2)e	PROPN
ejpam-5572	529	9	2πi	2πi	NOUN
ejpam-5572	529	10	f(γ3	f(γ3	NOUN
ejpam-5572	529	11	a)(x2	a)(x2	ADJ
ejpam-5572	529	12	)	)	PUNCT
ejpam-5572	529	13	}	}	PUNCT
ejpam-5572	530	1	=	=	PUNCT
ejpam-5572	530	2	h(k3	h(k3	NOUN
ejpam-5572	530	3	a)(x1	a)(x1	NOUN
ejpam-5572	530	4	)	)	PUNCT
ejpam-5572	530	5	∧	∧	PROPN
ejpam-5572	530	6	h(k3	h(k3	NOUN
ejpam-5572	530	7	a)(x2	a)(x2	PROPN
ejpam-5572	530	8	)	)	PUNCT
ejpam-5572	530	9	=	=	PUNCT
ejpam-5572	530	10	(	(	PUNCT
ejpam-5572	530	11	h(ka	h(ka	NOUN
ejpam-5572	530	12	)	)	PUNCT
ejpam-5572	530	13	)	)	PUNCT
ejpam-5572	531	1	3(x1	3(x1	X
ejpam-5572	531	2	)	)	PUNCT
ejpam-5572	531	3	∧	∧	NOUN
ejpam-5572	531	4	(	(	PUNCT
ejpam-5572	531	5	h(ka	h(ka	NOUN
ejpam-5572	531	6	)	)	PUNCT
ejpam-5572	531	7	)	)	PUNCT
ejpam-5572	531	8	3(x2	3(x2	NOUN
ejpam-5572	531	9	)	)	PUNCT
ejpam-5572	531	10	.	.	PUNCT
ejpam-5572	532	1	2	2	X
ejpam-5572	532	2	)	)	PUNCT
ejpam-5572	532	3	(	(	PUNCT
ejpam-5572	532	4	h(la	h(la	NUM
ejpam-5572	532	5	)	)	PUNCT
ejpam-5572	532	6	)	)	PUNCT
ejpam-5572	532	7	3(x1	3(x1	NUM
ejpam-5572	532	8	∗2	∗2	NOUN
ejpam-5572	532	9	x2	x2	NUM
ejpam-5572	532	10	)	)	PUNCT
ejpam-5572	532	11	=	=	PUNCT
ejpam-5572	532	12	h(l3	h(l3	ADJ
ejpam-5572	532	13	a)(x1	a)(x1	NOUN
ejpam-5572	532	14	∗2	∗2	NOUN
ejpam-5572	532	15	x2	x2	NUM
ejpam-5572	532	16	)	)	PUNCT
ejpam-5572	533	1	=	=	PUNCT
ejpam-5572	534	1	h(q3a)(x1	h(q3a)(x1	NUM
ejpam-5572	534	2	∗2	∗2	PROPN
ejpam-5572	534	3	x2)e2πi	x2)e2πi	ADV
ejpam-5572	534	4	f(ν3a)(x1∗2x2	f(ν3a)(x1∗2x2	NOUN
ejpam-5572	534	5	)	)	PUNCT
ejpam-5572	534	6	≤	≤	NOUN
ejpam-5572	534	7	(	(	PUNCT
ejpam-5572	534	8	h(q3a)(x1	h(q3a)(x1	ADJ
ejpam-5572	534	9	)	)	PUNCT
ejpam-5572	534	10	∨	∨	NUM
ejpam-5572	534	11	h(q3a)(x2	h(q3a)(x2	NOUN
ejpam-5572	534	12	)	)	PUNCT
ejpam-5572	534	13	)	)	PUNCT
ejpam-5572	534	14	∗	∗	NOUN
ejpam-5572	534	15	e2πi(h(ν	e2πi(h(ν	NOUN
ejpam-5572	534	16	3	3	NUM
ejpam-5572	534	17	a)(x1)∨h(ν3a)(x2	a)(x1)∨h(ν3a)(x2	NOUN
ejpam-5572	534	18	)	)	PUNCT
ejpam-5572	534	19	)	)	PUNCT
ejpam-5572	535	1	=	=	PRON
ejpam-5572	535	2	{	{	PUNCT
ejpam-5572	535	3	h(q3a)(x1)e2πi	h(q3a)(x1)e2πi	PROPN
ejpam-5572	535	4	h(ν3a)(x1	h(ν3a)(x1	PROPN
ejpam-5572	535	5	)	)	PUNCT
ejpam-5572	535	6	∨	∨	NUM
ejpam-5572	535	7	h(q3a)(x2)e	h(q3a)(x2)e	NUM
ejpam-5572	535	8	2πi	2πi	ADJ
ejpam-5572	535	9	h(ν3a)(x2	h(ν3a)(x2	NOUN
ejpam-5572	535	10	)	)	PUNCT
ejpam-5572	535	11	}	}	PUNCT
ejpam-5572	535	12	=	=	PUNCT
ejpam-5572	535	13	h(l3	h(l3	ADJ
ejpam-5572	535	14	a)(x1	a)(x1	NOUN
ejpam-5572	535	15	)	)	PUNCT
ejpam-5572	535	16	∨	∨	NUM
ejpam-5572	535	17	h(l3	h(l3	PRON
ejpam-5572	535	18	a)(x2	a)(x2	ADJ
ejpam-5572	535	19	)	)	PUNCT
ejpam-5572	535	20	=	=	PUNCT
ejpam-5572	535	21	(	(	PUNCT
ejpam-5572	535	22	h(la	h(la	PROPN
ejpam-5572	535	23	)	)	PUNCT
ejpam-5572	535	24	)	)	PUNCT
ejpam-5572	536	1	3(x1	3(x1	NUM
ejpam-5572	536	2	)	)	PUNCT
ejpam-5572	536	3	∨	∨	NUM
ejpam-5572	536	4	(	(	PUNCT
ejpam-5572	536	5	h(la	h(la	NUM
ejpam-5572	536	6	)	)	PUNCT
ejpam-5572	536	7	)	)	PUNCT
ejpam-5572	536	8	3(x2	3(x2	NOUN
ejpam-5572	536	9	)	)	PUNCT
ejpam-5572	536	10	.	.	PUNCT
ejpam-5572	537	1	3	3	X
ejpam-5572	537	2	)	)	PUNCT
ejpam-5572	537	3	(	(	PUNCT
ejpam-5572	537	4	h(ka	h(ka	NOUN
ejpam-5572	537	5	)	)	PUNCT
ejpam-5572	537	6	)	)	PUNCT
ejpam-5572	538	1	3(x−1	3(x−1	NUM
ejpam-5572	538	2	)	)	PUNCT
ejpam-5572	539	1	=	=	SYM
ejpam-5572	539	2	h(k3	h(k3	NOUN
ejpam-5572	539	3	a)(x	a)(x	NOUN
ejpam-5572	539	4	−1	−1	NOUN
ejpam-5572	539	5	)	)	PUNCT
ejpam-5572	540	1	=	=	SYM
ejpam-5572	540	2	h(p3a)(x	h(p3a)(x	PROPN
ejpam-5572	540	3	−1)e2πi	−1)e2πi	ADV
ejpam-5572	540	4	h(ω3	h(ω3	NOUN
ejpam-5572	540	5	a)(x−1	a)(x−1	X
ejpam-5572	540	6	)	)	PUNCT
ejpam-5572	540	7	=	=	SYM
ejpam-5572	541	1	h(p3a)(x)e	h(p3a)(x)e	ADJ
ejpam-5572	541	2	2πi	2πi	NOUN
ejpam-5572	541	3	h(ω3	h(ω3	NOUN
ejpam-5572	541	4	a)(x	a)(x	NOUN
ejpam-5572	541	5	)	)	PUNCT
ejpam-5572	541	6	=	=	SYM
ejpam-5572	541	7	h(k3	h(k3	NOUN
ejpam-5572	541	8	a)(x	a)(x	PROPN
ejpam-5572	541	9	)	)	PUNCT
ejpam-5572	542	1	=	=	PUNCT
ejpam-5572	542	2	(	(	PUNCT
ejpam-5572	542	3	h(ka	h(ka	NOUN
ejpam-5572	542	4	)	)	PUNCT
ejpam-5572	542	5	)	)	PUNCT
ejpam-5572	542	6	3(x	3(x	NUM
ejpam-5572	542	7	)	)	PUNCT
ejpam-5572	542	8	.	.	PUNCT
ejpam-5572	543	1	4	4	X
ejpam-5572	543	2	)	)	PUNCT
ejpam-5572	543	3	(	(	PUNCT
ejpam-5572	543	4	h(la	h(la	NUM
ejpam-5572	543	5	)	)	PUNCT
ejpam-5572	543	6	)	)	PUNCT
ejpam-5572	543	7	3(x−1	3(x−1	NUM
ejpam-5572	543	8	)	)	PUNCT
ejpam-5572	544	1	=	=	PRON
ejpam-5572	544	2	h(l3	h(l3	PRON
ejpam-5572	544	3	a)(x	a)(x	ADP
ejpam-5572	544	4	−1	−1	NOUN
ejpam-5572	544	5	)	)	PUNCT
ejpam-5572	544	6	=	=	SYM
ejpam-5572	544	7	e.a	e.a	PROPN
ejpam-5572	544	8	.	.	PROPN
ejpam-5572	544	9	abuhijleh	abuhijleh	PROPN
ejpam-5572	544	10	,	,	PUNCT
ejpam-5572	544	11	a.	a.	PROPN
ejpam-5572	544	12	alkouri	alkouri	PROPN
ejpam-5572	544	13	/	/	PROPN
ejpam-5572	544	14	eur	eur	PROPN
ejpam-5572	544	15	.	.	PUNCT
ejpam-5572	545	1	j.	j.	PROPN
ejpam-5572	545	2	pure	pure	PROPN
ejpam-5572	545	3	appl	appl	PROPN
ejpam-5572	545	4	.	.	PROPN
ejpam-5572	545	5	math	math	PROPN
ejpam-5572	545	6	,	,	PUNCT
ejpam-5572	545	7	18	18	NUM
ejpam-5572	545	8	(	(	PUNCT
ejpam-5572	545	9	1	1	NUM
ejpam-5572	545	10	)	)	PUNCT
ejpam-5572	545	11	(	(	PUNCT
ejpam-5572	545	12	2025	2025	NUM
ejpam-5572	545	13	)	)	PUNCT
ejpam-5572	545	14	,	,	PUNCT
ejpam-5572	545	15	5572	5572	NUM
ejpam-5572	545	16	16	16	NUM
ejpam-5572	545	17	of	of	ADP
ejpam-5572	545	18	19	19	NUM
ejpam-5572	545	19	h(q3a)(x	h(q3a)(x	NOUN
ejpam-5572	545	20	−1)e2πi	−1)e2πi	ADV
ejpam-5572	545	21	h(ν3a)(x−1	h(ν3a)(x−1	PROPN
ejpam-5572	545	22	)	)	PUNCT
ejpam-5572	545	23	=	=	SYM
ejpam-5572	546	1	h(q3a)(x)e	h(q3a)(x)e	NUM
ejpam-5572	546	2	2πi	2πi	NOUN
ejpam-5572	546	3	f(ν3a)(x	f(ν3a)(x	NOUN
ejpam-5572	546	4	)	)	PUNCT
ejpam-5572	546	5	=	=	PRON
ejpam-5572	546	6	h(l3	h(l3	PRON
ejpam-5572	546	7	a)(x	a)(x	NOUN
ejpam-5572	546	8	)	)	PUNCT
ejpam-5572	546	9	=	=	SYM
ejpam-5572	546	10	(	(	PUNCT
ejpam-5572	546	11	h(la	h(la	PROPN
ejpam-5572	546	12	)	)	PUNCT
ejpam-5572	546	13	)	)	PUNCT
ejpam-5572	546	14	3(x	3(x	NUM
ejpam-5572	546	15	)	)	PUNCT
ejpam-5572	546	16	.	.	PUNCT
ejpam-5572	547	1	hence	hence	ADV
ejpam-5572	547	2	result	result	NOUN
ejpam-5572	547	3	follows	follow	VERB
ejpam-5572	547	4	.	.	PUNCT
ejpam-5572	548	1	theorem	theorem	ADJ
ejpam-5572	548	2	5	5	NUM
ejpam-5572	548	3	.	.	PUNCT
ejpam-5572	549	1	let	let	VERB
ejpam-5572	549	2	f	f	NOUN
ejpam-5572	549	3	:	:	PUNCT
ejpam-5572	549	4	x	x	PUNCT
ejpam-5572	549	5	isomorphism−−−−−−−−→	isomorphism−−−−−−−−→	PROPN
ejpam-5572	549	6	u	u	PROPN
ejpam-5572	549	7	,	,	PUNCT
ejpam-5572	549	8	from	from	ADP
ejpam-5572	549	9	(	(	PUNCT
ejpam-5572	549	10	x	x	NOUN
ejpam-5572	549	11	,	,	PUNCT
ejpam-5572	549	12	∗1	∗1	PROPN
ejpam-5572	549	13	)	)	PUNCT
ejpam-5572	549	14	to	to	ADP
ejpam-5572	549	15	(	(	PUNCT
ejpam-5572	549	16	u	u	NOUN
ejpam-5572	549	17	,	,	PUNCT
ejpam-5572	549	18	∗2	∗2	PROPN
ejpam-5572	549	19	)	)	PUNCT
ejpam-5572	549	20	,	,	PUNCT
ejpam-5572	549	21	and	and	CCONJ
ejpam-5572	549	22	let	let	VERB
ejpam-5572	549	23	b	b	X
ejpam-5572	549	24	be	be	AUX
ejpam-5572	549	25	cffsg	cffsg	ADJ
ejpam-5572	549	26	of	of	ADP
ejpam-5572	549	27	u.	u.	PROPN
ejpam-5572	549	28	then	then	ADV
ejpam-5572	549	29	h−1(b	h−1(b	PROPN
ejpam-5572	549	30	)	)	PUNCT
ejpam-5572	549	31	is	be	AUX
ejpam-5572	549	32	cffsg	cffsg	ADJ
ejpam-5572	549	33	of	of	ADP
ejpam-5572	549	34	x.	x.	NOUN
ejpam-5572	549	35	proof	proof	NOUN
ejpam-5572	549	36	.	.	PUNCT
ejpam-5572	550	1	the	the	DET
ejpam-5572	550	2	proof	proof	NOUN
ejpam-5572	550	3	will	will	AUX
ejpam-5572	550	4	be	be	AUX
ejpam-5572	550	5	similar	similar	ADJ
ejpam-5572	550	6	to	to	ADP
ejpam-5572	550	7	previous	previous	ADJ
ejpam-5572	550	8	theorem	theorem	VERB
ejpam-5572	550	9	,	,	PUNCT
ejpam-5572	550	10	and	and	CCONJ
ejpam-5572	550	11	that	that	SCONJ
ejpam-5572	550	12	by	by	ADP
ejpam-5572	550	13	using	use	VERB
ejpam-5572	550	14	lemma	lemma	PROPN
ejpam-5572	550	15	1	1	NUM
ejpam-5572	550	16	with	with	ADP
ejpam-5572	550	17	theorem[6.2	theorem[6.2	NOUN
ejpam-5572	550	18	]	]	X
ejpam-5572	551	1	[	[	X
ejpam-5572	551	2	35	35	NUM
ejpam-5572	551	3	]	]	PUNCT
ejpam-5572	551	4	.	.	PUNCT
ejpam-5572	552	1	theorem	theorem	ADJ
ejpam-5572	552	2	6	6	NUM
ejpam-5572	552	3	.	.	PUNCT
ejpam-5572	553	1	let	let	VERB
ejpam-5572	553	2	h	h	NOUN
ejpam-5572	553	3	:	:	PUNCT
ejpam-5572	553	4	x	x	X
ejpam-5572	553	5	epimorphism−−−−−−−−→	epimorphism−−−−−−−−→	NOUN
ejpam-5572	553	6	u	u	NOUN
ejpam-5572	553	7	,	,	PUNCT
ejpam-5572	553	8	from	from	ADP
ejpam-5572	553	9	(	(	PUNCT
ejpam-5572	553	10	x	x	NOUN
ejpam-5572	553	11	,	,	PUNCT
ejpam-5572	553	12	∗1	∗1	PROPN
ejpam-5572	553	13	)	)	PUNCT
ejpam-5572	553	14	to	to	ADP
ejpam-5572	553	15	(	(	PUNCT
ejpam-5572	553	16	u	u	NOUN
ejpam-5572	553	17	,	,	PUNCT
ejpam-5572	553	18	∗2	∗2	PROPN
ejpam-5572	553	19	)	)	PUNCT
ejpam-5572	553	20	,	,	PUNCT
ejpam-5572	553	21	and	and	CCONJ
ejpam-5572	553	22	let	let	VERB
ejpam-5572	553	23	a	a	PRON
ejpam-5572	553	24	be	be	AUX
ejpam-5572	553	25	cffnsg	cffnsg	ADJ
ejpam-5572	553	26	of	of	ADP
ejpam-5572	553	27	x.	x.	PROPN
ejpam-5572	553	28	then	then	ADV
ejpam-5572	553	29	h(a	h(a	PROPN
ejpam-5572	553	30	)	)	PUNCT
ejpam-5572	553	31	is	be	AUX
ejpam-5572	553	32	cffnsg	cffnsg	ADJ
ejpam-5572	553	33	of	of	ADP
ejpam-5572	553	34	u.	u.	PROPN
ejpam-5572	553	35	proof	proof	NOUN
ejpam-5572	553	36	.	.	PUNCT
ejpam-5572	554	1	according	accord	VERB
ejpam-5572	554	2	to	to	ADP
ejpam-5572	554	3	proposition	proposition	NOUN
ejpam-5572	554	4	5	5	NUM
ejpam-5572	554	5	with	with	ADP
ejpam-5572	554	6	theorem[6.3	theorem[6.3	X
ejpam-5572	554	7	]	]	X
ejpam-5572	554	8	[	[	X
ejpam-5572	554	9	35	35	NUM
ejpam-5572	554	10	]	]	PUNCT
ejpam-5572	554	11	,	,	PUNCT
ejpam-5572	554	12	and	and	CCONJ
ejpam-5572	554	13	inspiring	inspire	VERB
ejpam-5572	554	14	the	the	DET
ejpam-5572	554	15	proof	proof	NOUN
ejpam-5572	554	16	of	of	ADP
ejpam-5572	554	17	theorem	theorem	ADJ
ejpam-5572	554	18	4	4	NUM
ejpam-5572	554	19	,	,	PUNCT
ejpam-5572	554	20	result	result	NOUN
ejpam-5572	554	21	will	will	AUX
ejpam-5572	554	22	follows	follow	VERB
ejpam-5572	554	23	.	.	PUNCT
ejpam-5572	555	1	theorem	theorem	ADJ
ejpam-5572	555	2	7	7	NUM
ejpam-5572	555	3	.	.	PUNCT
ejpam-5572	556	1	let	let	VERB
ejpam-5572	556	2	h	h	NOUN
ejpam-5572	556	3	:	:	PUNCT
ejpam-5572	556	4	x	x	X
ejpam-5572	556	5	isomorphism−−−−−−−−→	isomorphism−−−−−−−−→	PROPN
ejpam-5572	556	6	u	u	PROPN
ejpam-5572	556	7	,	,	PUNCT
ejpam-5572	556	8	from	from	ADP
ejpam-5572	556	9	(	(	PUNCT
ejpam-5572	556	10	x	x	NOUN
ejpam-5572	556	11	,	,	PUNCT
ejpam-5572	556	12	∗1	∗1	PROPN
ejpam-5572	556	13	)	)	PUNCT
ejpam-5572	556	14	to	to	ADP
ejpam-5572	556	15	(	(	PUNCT
ejpam-5572	556	16	u	u	NOUN
ejpam-5572	556	17	,	,	PUNCT
ejpam-5572	556	18	∗2	∗2	PROPN
ejpam-5572	556	19	)	)	PUNCT
ejpam-5572	556	20	,	,	PUNCT
ejpam-5572	556	21	and	and	CCONJ
ejpam-5572	556	22	let	let	VERB
ejpam-5572	556	23	b	b	X
ejpam-5572	556	24	be	be	AUX
ejpam-5572	556	25	cffnsg	cffnsg	ADJ
ejpam-5572	556	26	of	of	ADP
ejpam-5572	556	27	u.	u.	PROPN
ejpam-5572	556	28	then	then	ADV
ejpam-5572	556	29	h−1(b	h−1(b	PROPN
ejpam-5572	556	30	)	)	PUNCT
ejpam-5572	556	31	is	be	AUX
ejpam-5572	556	32	cffnsg	cffnsg	ADJ
ejpam-5572	556	33	of	of	ADP
ejpam-5572	556	34	x.	x.	NOUN
ejpam-5572	556	35	proof	proof	NOUN
ejpam-5572	556	36	.	.	PUNCT
ejpam-5572	557	1	according	accord	VERB
ejpam-5572	557	2	to	to	ADP
ejpam-5572	557	3	proposition	proposition	NOUN
ejpam-5572	557	4	5	5	NUM
ejpam-5572	557	5	with	with	ADP
ejpam-5572	557	6	theorem[6.4	theorem[6.4	PROPN
ejpam-5572	557	7	]	]	X
ejpam-5572	558	1	[	[	X
ejpam-5572	558	2	35	35	NUM
ejpam-5572	558	3	]	]	PUNCT
ejpam-5572	558	4	,	,	PUNCT
ejpam-5572	558	5	and	and	CCONJ
ejpam-5572	558	6	inspiring	inspire	VERB
ejpam-5572	558	7	the	the	DET
ejpam-5572	558	8	proof	proof	NOUN
ejpam-5572	558	9	of	of	ADP
ejpam-5572	558	10	theorem	theorem	ADJ
ejpam-5572	558	11	4	4	NUM
ejpam-5572	558	12	,	,	PUNCT
ejpam-5572	558	13	result	result	NOUN
ejpam-5572	558	14	will	will	AUX
ejpam-5572	558	15	follows	follow	VERB
ejpam-5572	558	16	.	.	PUNCT
ejpam-5572	559	1	example	example	NOUN
ejpam-5572	559	2	6	6	NUM
ejpam-5572	559	3	.	.	PUNCT
ejpam-5572	560	1	let	let	VERB
ejpam-5572	560	2	(	(	PUNCT
ejpam-5572	560	3	x	x	NOUN
ejpam-5572	560	4	,	,	PUNCT
ejpam-5572	560	5	∗1	∗1	PROPN
ejpam-5572	560	6	)	)	PUNCT
ejpam-5572	560	7	=	=	SYM
ejpam-5572	561	1	k4	k4	NOUN
ejpam-5572	561	2	=	=	X
ejpam-5572	561	3	<	<	X
ejpam-5572	561	4	a	a	DET
ejpam-5572	561	5	,	,	PUNCT
ejpam-5572	561	6	b|a2	b|a2	NOUN
ejpam-5572	561	7	=	=	NOUN
ejpam-5572	561	8	b2	b2	NOUN
ejpam-5572	561	9	=	=	SYM
ejpam-5572	561	10	(	(	PUNCT
ejpam-5572	561	11	ab)2	ab)2	PROPN
ejpam-5572	561	12	=	=	SYM
ejpam-5572	561	13	1	1	NUM
ejpam-5572	561	14	>	>	PUNCT
ejpam-5572	561	15	,	,	PUNCT
ejpam-5572	561	16	i.e.	i.e.	X
ejpam-5572	561	17	the	the	DET
ejpam-5572	561	18	klein	klein	PROPN
ejpam-5572	561	19	fourgroup	fourgroup	NOUN
ejpam-5572	561	20	,	,	PUNCT
ejpam-5572	561	21	and	and	CCONJ
ejpam-5572	561	22	(	(	PUNCT
ejpam-5572	561	23	u	u	NOUN
ejpam-5572	561	24	,	,	PUNCT
ejpam-5572	561	25	∗2	∗2	NOUN
ejpam-5572	561	26	)	)	PUNCT
ejpam-5572	561	27	=	=	SYM
ejpam-5572	561	28	d2	d2	PROPN
ejpam-5572	561	29	∼=	∼=	PROPN
ejpam-5572	561	30	z2	z2	PROPN
ejpam-5572	561	31	⊕	⊕	PROPN
ejpam-5572	561	32	z2	z2	PROPN
ejpam-5572	561	33	=	=	SYM
ejpam-5572	561	34	{	{	PUNCT
ejpam-5572	561	35	(	(	PUNCT
ejpam-5572	561	36	0	0	NUM
ejpam-5572	561	37	,	,	PUNCT
ejpam-5572	561	38	0	0	NUM
ejpam-5572	561	39	)	)	PUNCT
ejpam-5572	561	40	,	,	PUNCT
ejpam-5572	561	41	(	(	PUNCT
ejpam-5572	561	42	1	1	NUM
ejpam-5572	561	43	,	,	PUNCT
ejpam-5572	561	44	0	0	NUM
ejpam-5572	561	45	)	)	PUNCT
ejpam-5572	561	46	,	,	PUNCT
ejpam-5572	561	47	(	(	PUNCT
ejpam-5572	561	48	0	0	NUM
ejpam-5572	561	49	,	,	PUNCT
ejpam-5572	561	50	1	1	NUM
ejpam-5572	561	51	)	)	PUNCT
ejpam-5572	561	52	,	,	PUNCT
ejpam-5572	561	53	(	(	PUNCT
ejpam-5572	561	54	1	1	NUM
ejpam-5572	561	55	,	,	PUNCT
ejpam-5572	561	56	1	1	NUM
ejpam-5572	561	57	)	)	PUNCT
ejpam-5572	561	58	}	}	PUNCT
ejpam-5572	561	59	.	.	PUNCT
ejpam-5572	562	1	in	in	ADP
ejpam-5572	562	2	addition	addition	NOUN
ejpam-5572	562	3	,	,	PUNCT
ejpam-5572	562	4	for	for	ADP
ejpam-5572	562	5	b	b	NOUN
ejpam-5572	562	6	=	=	SYM
ejpam-5572	562	7	<	<	X
ejpam-5572	562	8	ℓ	ℓ	PROPN
ejpam-5572	562	9	,	,	PUNCT
ejpam-5572	562	10	p(ℓ)e2πiω(ℓ	p(ℓ)e2πiω(ℓ	NOUN
ejpam-5572	562	11	)	)	PUNCT
ejpam-5572	562	12	,	,	PUNCT
ejpam-5572	562	13	q(ℓ)e2πiν(ℓ	q(ℓ)e2πiν(ℓ	PROPN
ejpam-5572	562	14	)	)	PUNCT
ejpam-5572	562	15	>	>	PUNCT
ejpam-5572	563	1	that	that	PRON
ejpam-5572	563	2	defined	define	VERB
ejpam-5572	563	3	a	a	DET
ejpam-5572	563	4	cffsg	cffsg	NOUN
ejpam-5572	563	5	on	on	ADP
ejpam-5572	563	6	d2	d2	PROPN
ejpam-5572	563	7	,	,	PUNCT
ejpam-5572	563	8	see	see	VERB
ejpam-5572	563	9	example	example	NOUN
ejpam-5572	564	1	1	1	X
ejpam-5572	564	2	.	.	PUNCT
ejpam-5572	564	3	then	then	ADV
ejpam-5572	564	4	b	b	NUM
ejpam-5572	564	5	defined	define	VERB
ejpam-5572	564	6	cffnsg	cffnsg	ADJ
ejpam-5572	564	7	of	of	ADP
ejpam-5572	564	8	d2	d2	PROPN
ejpam-5572	564	9	,	,	PUNCT
ejpam-5572	564	10	where	where	SCONJ
ejpam-5572	564	11	the	the	DET
ejpam-5572	564	12	dihedral	dihedral	ADJ
ejpam-5572	564	13	group	group	NOUN
ejpam-5572	564	14	d2	d2	PROPN
ejpam-5572	564	15	is	be	AUX
ejpam-5572	564	16	an	an	DET
ejpam-5572	564	17	abelian	abelian	ADJ
ejpam-5572	564	18	group	group	NOUN
ejpam-5572	564	19	.	.	PUNCT
ejpam-5572	565	1	hence	hence	ADV
ejpam-5572	565	2	,	,	PUNCT
ejpam-5572	565	3	we	we	PRON
ejpam-5572	565	4	can	can	AUX
ejpam-5572	565	5	find	find	VERB
ejpam-5572	565	6	homomorphism	homomorphism	NOUN
ejpam-5572	565	7	function	function	NOUN
ejpam-5572	565	8	that	that	PRON
ejpam-5572	565	9	is	be	AUX
ejpam-5572	565	10	bijection	bijection	ADJ
ejpam-5572	565	11	;	;	PUNCT
ejpam-5572	565	12	h	h	NOUN
ejpam-5572	565	13	:	:	PUNCT
ejpam-5572	565	14	k4	k4	PROPN
ejpam-5572	565	15	isomorphism−−−−−−−−→	isomorphism−−−−−−−−→	PROPN
ejpam-5572	565	16	d2	d2	PROPN
ejpam-5572	565	17	,	,	PUNCT
ejpam-5572	565	18	with	with	ADP
ejpam-5572	565	19	h(1	h(1	PROPN
ejpam-5572	565	20	)	)	PUNCT
ejpam-5572	565	21	=	=	PUNCT
ejpam-5572	565	22	(	(	PUNCT
ejpam-5572	565	23	0	0	NUM
ejpam-5572	565	24	,	,	PUNCT
ejpam-5572	565	25	0	0	NUM
ejpam-5572	565	26	)	)	PUNCT
ejpam-5572	565	27	,	,	PUNCT
ejpam-5572	565	28	h(a	h(a	PROPN
ejpam-5572	565	29	)	)	PUNCT
ejpam-5572	566	1	=	=	PUNCT
ejpam-5572	566	2	(	(	PUNCT
ejpam-5572	566	3	1	1	NUM
ejpam-5572	566	4	,	,	PUNCT
ejpam-5572	566	5	0	0	NUM
ejpam-5572	566	6	)	)	PUNCT
ejpam-5572	566	7	,	,	PUNCT
ejpam-5572	566	8	h(b	h(b	PROPN
ejpam-5572	566	9	)	)	PUNCT
ejpam-5572	566	10	=	=	PUNCT
ejpam-5572	567	1	(	(	PUNCT
ejpam-5572	567	2	0	0	NUM
ejpam-5572	567	3	,	,	PUNCT
ejpam-5572	567	4	1	1	NUM
ejpam-5572	567	5	)	)	PUNCT
ejpam-5572	567	6	,	,	PUNCT
ejpam-5572	567	7	h(ab	h(ab	PROPN
ejpam-5572	567	8	)	)	PUNCT
ejpam-5572	567	9	=	=	PUNCT
ejpam-5572	567	10	(	(	PUNCT
ejpam-5572	567	11	1	1	NUM
ejpam-5572	567	12	,	,	PUNCT
ejpam-5572	567	13	1	1	NUM
ejpam-5572	567	14	)	)	PUNCT
ejpam-5572	567	15	.	.	PUNCT
ejpam-5572	568	1	now	now	ADV
ejpam-5572	568	2	,	,	PUNCT
ejpam-5572	568	3	according	accord	VERB
ejpam-5572	568	4	to	to	ADP
ejpam-5572	568	5	definition	definition	NOUN
ejpam-5572	568	6	12	12	NUM
ejpam-5572	568	7	h−1(k3	h−1(k3	NOUN
ejpam-5572	568	8	b)(u	b)(u	ADJ
ejpam-5572	568	9	)	)	PUNCT
ejpam-5572	568	10	=	=	SYM
ejpam-5572	568	11	p3b(h(u))e	p3b(h(u))e	PROPN
ejpam-5572	568	12	2πiω3	2πiω3	NUM
ejpam-5572	568	13	b(h(u	b(h(u	NOUN
ejpam-5572	568	14	)	)	PUNCT
ejpam-5572	568	15	)	)	PUNCT
ejpam-5572	568	16	and	and	CCONJ
ejpam-5572	568	17	h−1(l3	h−1(l3	NOUN
ejpam-5572	568	18	b)(u	b)(u	ADJ
ejpam-5572	568	19	)	)	PUNCT
ejpam-5572	568	20	=	=	SYM
ejpam-5572	568	21	q3b(h(u))e	q3b(h(u))e	ADJ
ejpam-5572	568	22	2πiν3b(h(u	2πiν3b(h(u	NUM
ejpam-5572	568	23	)	)	PUNCT
ejpam-5572	568	24	)	)	PUNCT
ejpam-5572	568	25	,	,	PUNCT
ejpam-5572	568	26	∀	∀	X
ejpam-5572	568	27	u	u	NOUN
ejpam-5572	568	28	∈	∈	PROPN
ejpam-5572	568	29	k4	k4	NOUN
ejpam-5572	568	30	.	.	PUNCT
ejpam-5572	569	1	hence	hence	ADV
ejpam-5572	569	2	,	,	PUNCT
ejpam-5572	569	3	by	by	ADP
ejpam-5572	569	4	this	this	DET
ejpam-5572	569	5	definition	definition	NOUN
ejpam-5572	569	6	we	we	PRON
ejpam-5572	569	7	get	get	VERB
ejpam-5572	569	8	that	that	DET
ejpam-5572	569	9	h−1(b	h−1(b	PROPN
ejpam-5572	569	10	)	)	PUNCT
ejpam-5572	569	11	is	be	AUX
ejpam-5572	569	12	cffnsg	cffnsg	ADJ
ejpam-5572	569	13	of	of	ADP
ejpam-5572	569	14	k4	k4	PROPN
ejpam-5572	569	15	.	.	PUNCT
ejpam-5572	570	1	6	6	NUM
ejpam-5572	570	2	.	.	X
ejpam-5572	570	3	conclusion	conclusion	NOUN
ejpam-5572	570	4	this	this	DET
ejpam-5572	570	5	research	research	NOUN
ejpam-5572	570	6	provides	provide	VERB
ejpam-5572	570	7	a	a	DET
ejpam-5572	570	8	theoretical	theoretical	ADJ
ejpam-5572	570	9	foundation	foundation	NOUN
ejpam-5572	570	10	for	for	ADP
ejpam-5572	570	11	the	the	DET
ejpam-5572	570	12	complex	complex	ADJ
ejpam-5572	570	13	fermatean	fermatean	ADJ
ejpam-5572	570	14	fuzzy	fuzzy	ADJ
ejpam-5572	570	15	subgroup	subgroup	NOUN
ejpam-5572	570	16	(	(	PUNCT
ejpam-5572	570	17	cffsg	cffsg	ADJ
ejpam-5572	570	18	)	)	PUNCT
ejpam-5572	570	19	and	and	CCONJ
ejpam-5572	570	20	examines	examine	VERB
ejpam-5572	570	21	its	its	PRON
ejpam-5572	570	22	algebraic	algebraic	ADJ
ejpam-5572	570	23	properties	property	NOUN
ejpam-5572	570	24	.	.	PUNCT
ejpam-5572	571	1	the	the	DET
ejpam-5572	571	2	concepts	concept	NOUN
ejpam-5572	571	3	of	of	ADP
ejpam-5572	571	4	complex	complex	ADJ
ejpam-5572	571	5	fermatean	fermatean	ADJ
ejpam-5572	571	6	fuzzy	fuzzy	ADJ
ejpam-5572	571	7	normal	normal	ADJ
ejpam-5572	571	8	subgroups	subgroup	NOUN
ejpam-5572	571	9	and	and	CCONJ
ejpam-5572	571	10	complex	complex	ADJ
ejpam-5572	571	11	fermatean	fermatean	ADJ
ejpam-5572	571	12	fuzzy	fuzzy	ADJ
ejpam-5572	571	13	cosets	coset	NOUN
ejpam-5572	571	14	were	be	AUX
ejpam-5572	571	15	introduced	introduce	VERB
ejpam-5572	571	16	.	.	PUNCT
ejpam-5572	572	1	additionally	additionally	ADV
ejpam-5572	572	2	,	,	PUNCT
ejpam-5572	572	3	the	the	DET
ejpam-5572	572	4	conditions	condition	NOUN
ejpam-5572	572	5	under	under	ADP
ejpam-5572	572	6	which	which	PRON
ejpam-5572	572	7	a	a	DET
ejpam-5572	572	8	complex	complex	ADJ
ejpam-5572	572	9	fermatean	fermatean	ADJ
ejpam-5572	572	10	fuzzy	fuzzy	ADJ
ejpam-5572	572	11	subgroup	subgroup	NOUN
ejpam-5572	572	12	can	can	AUX
ejpam-5572	572	13	be	be	AUX
ejpam-5572	572	14	a	a	DET
ejpam-5572	572	15	complex	complex	ADJ
ejpam-5572	572	16	fermatean	fermatean	NOUN
ejpam-5572	572	17	fuzzy	fuzzy	ADJ
ejpam-5572	572	18	normal	normal	ADJ
ejpam-5572	572	19	subgroup	subgroup	NOUN
ejpam-5572	572	20	were	be	AUX
ejpam-5572	572	21	explored	explore	VERB
ejpam-5572	572	22	.	.	PUNCT
ejpam-5572	573	1	a	a	DET
ejpam-5572	573	2	homomorphism	homomorphism	NOUN
ejpam-5572	573	3	between	between	ADP
ejpam-5572	573	4	two	two	NUM
ejpam-5572	573	5	complex	complex	ADJ
ejpam-5572	573	6	fermatean	fermatean	ADJ
ejpam-5572	573	7	fuzzy	fuzzy	ADJ
ejpam-5572	573	8	subgroups	subgroup	NOUN
ejpam-5572	573	9	and	and	CCONJ
ejpam-5572	573	10	its	its	PRON
ejpam-5572	573	11	properties	property	NOUN
ejpam-5572	573	12	were	be	AUX
ejpam-5572	573	13	also	also	ADV
ejpam-5572	573	14	discussed	discuss	VERB
ejpam-5572	573	15	.	.	PUNCT
ejpam-5572	574	1	as	as	ADP
ejpam-5572	574	2	a	a	DET
ejpam-5572	574	3	direction	direction	NOUN
ejpam-5572	574	4	for	for	ADP
ejpam-5572	574	5	future	future	ADJ
ejpam-5572	574	6	research	research	NOUN
ejpam-5572	574	7	,	,	PUNCT
ejpam-5572	574	8	we	we	PRON
ejpam-5572	574	9	plan	plan	VERB
ejpam-5572	574	10	to	to	PART
ejpam-5572	574	11	refine	refine	VERB
ejpam-5572	574	12	the	the	DET
ejpam-5572	574	13	definition	definition	NOUN
ejpam-5572	574	14	of	of	ADP
ejpam-5572	574	15	cffsg	cffsg	ADJ
ejpam-5572	574	16	by	by	ADP
ejpam-5572	574	17	replacing	replace	VERB
ejpam-5572	574	18	the	the	DET
ejpam-5572	574	19	minimum	minimum	ADJ
ejpam-5572	574	20	and	and	CCONJ
ejpam-5572	574	21	maximum	maximum	ADJ
ejpam-5572	574	22	operations	operation	NOUN
ejpam-5572	574	23	with	with	ADP
ejpam-5572	574	24	t	t	NOUN
ejpam-5572	574	25	-	-	PUNCT
ejpam-5572	574	26	norm	norm	NOUN
ejpam-5572	574	27	and	and	CCONJ
ejpam-5572	574	28	s	s	NOUN
ejpam-5572	574	29	-	-	PUNCT
ejpam-5572	574	30	norm	norm	NOUN
ejpam-5572	574	31	functions	function	NOUN
ejpam-5572	574	32	,	,	PUNCT
ejpam-5572	574	33	respectively	respectively	ADV
ejpam-5572	574	34	.	.	PUNCT
ejpam-5572	575	1	furthermore	furthermore	ADV
ejpam-5572	575	2	,	,	PUNCT
ejpam-5572	575	3	fixed	fix	VERB
ejpam-5572	575	4	-	-	PUNCT
ejpam-5572	575	5	point	point	NOUN
ejpam-5572	575	6	theory	theory	NOUN
ejpam-5572	575	7	could	could	AUX
ejpam-5572	575	8	be	be	AUX
ejpam-5572	575	9	integrated	integrate	VERB
ejpam-5572	575	10	and	and	CCONJ
ejpam-5572	575	11	extended	extend	VERB
ejpam-5572	575	12	within	within	ADP
ejpam-5572	575	13	the	the	DET
ejpam-5572	575	14	context	context	NOUN
ejpam-5572	575	15	of	of	ADP
ejpam-5572	575	16	cffsg	cffsg	ADJ
ejpam-5572	575	17	.	.	PUNCT
ejpam-5572	576	1	the	the	DET
ejpam-5572	576	2	approach	approach	NOUN
ejpam-5572	576	3	presented	present	VERB
ejpam-5572	576	4	here	here	ADV
ejpam-5572	576	5	can	can	AUX
ejpam-5572	576	6	be	be	AUX
ejpam-5572	576	7	progressively	progressively	ADV
ejpam-5572	576	8	applied	apply	VERB
ejpam-5572	576	9	to	to	ADP
ejpam-5572	576	10	other	other	ADJ
ejpam-5572	576	11	algebraic	algebraic	ADJ
ejpam-5572	576	12	structures	structure	NOUN
ejpam-5572	576	13	,	,	PUNCT
ejpam-5572	576	14	such	such	ADJ
ejpam-5572	576	15	e.a	e.a	PROPN
ejpam-5572	576	16	.	.	PROPN
ejpam-5572	576	17	abuhijleh	abuhijleh	PROPN
ejpam-5572	576	18	,	,	PUNCT
ejpam-5572	576	19	a.	a.	PROPN
ejpam-5572	576	20	alkouri	alkouri	PROPN
ejpam-5572	576	21	/	/	PROPN
ejpam-5572	576	22	eur	eur	PROPN
ejpam-5572	576	23	.	.	PUNCT
ejpam-5572	577	1	j.	j.	PROPN
ejpam-5572	577	2	pure	pure	PROPN
ejpam-5572	577	3	appl	appl	PROPN
ejpam-5572	577	4	.	.	PROPN
ejpam-5572	577	5	math	math	PROPN
ejpam-5572	577	6	,	,	PUNCT
ejpam-5572	577	7	18	18	NUM
ejpam-5572	577	8	(	(	PUNCT
ejpam-5572	577	9	1	1	NUM
ejpam-5572	577	10	)	)	PUNCT
ejpam-5572	577	11	(	(	PUNCT
ejpam-5572	577	12	2025	2025	NUM
ejpam-5572	577	13	)	)	PUNCT
ejpam-5572	577	14	,	,	PUNCT
ejpam-5572	577	15	5572	5572	NUM
ejpam-5572	577	16	17	17	NUM
ejpam-5572	577	17	of	of	ADP
ejpam-5572	577	18	19	19	NUM
ejpam-5572	577	19	as	as	ADP
ejpam-5572	577	20	integral	integral	ADJ
ejpam-5572	577	21	domains	domain	NOUN
ejpam-5572	577	22	,	,	PUNCT
ejpam-5572	577	23	fields	field	NOUN
ejpam-5572	577	24	,	,	PUNCT
ejpam-5572	577	25	rings	ring	NOUN
ejpam-5572	577	26	,	,	PUNCT
ejpam-5572	577	27	and	and	CCONJ
ejpam-5572	577	28	factor	factor	NOUN
ejpam-5572	577	29	groups	group	NOUN
ejpam-5572	577	30	.	.	PUNCT
ejpam-5572	578	1	by	by	ADP
ejpam-5572	578	2	incorporating	incorporate	VERB
ejpam-5572	578	3	periodic	periodic	ADJ
ejpam-5572	578	4	information	information	NOUN
ejpam-5572	578	5	into	into	ADP
ejpam-5572	578	6	the	the	DET
ejpam-5572	578	7	cffsg	cffsg	ADJ
ejpam-5572	578	8	framework	framework	NOUN
ejpam-5572	578	9	,	,	PUNCT
ejpam-5572	578	10	the	the	DET
ejpam-5572	578	11	current	current	ADJ
ejpam-5572	578	12	structure	structure	NOUN
ejpam-5572	578	13	could	could	AUX
ejpam-5572	578	14	facilitate	facilitate	VERB
ejpam-5572	578	15	the	the	DET
ejpam-5572	578	16	development	development	NOUN
ejpam-5572	578	17	of	of	ADP
ejpam-5572	578	18	cryptographic	cryptographic	ADJ
ejpam-5572	578	19	primitives	primitive	NOUN
ejpam-5572	578	20	and	and	CCONJ
ejpam-5572	578	21	be	be	AUX
ejpam-5572	578	22	applied	apply	VERB
ejpam-5572	578	23	to	to	ADP
ejpam-5572	578	24	the	the	DET
ejpam-5572	578	25	generalization	generalization	NOUN
ejpam-5572	578	26	of	of	ADP
ejpam-5572	578	27	new	new	ADJ
ejpam-5572	578	28	algorithms	algorithm	NOUN
ejpam-5572	578	29	.	.	PUNCT
ejpam-5572	579	1	another	another	DET
ejpam-5572	579	2	avenue	avenue	NOUN
ejpam-5572	579	3	for	for	ADP
ejpam-5572	579	4	future	future	ADJ
ejpam-5572	579	5	work	work	NOUN
ejpam-5572	579	6	is	be	AUX
ejpam-5572	579	7	upgrading	upgrade	VERB
ejpam-5572	579	8	cffsg	cffsg	ADJ
ejpam-5572	579	9	to	to	ADP
ejpam-5572	579	10	the	the	DET
ejpam-5572	579	11	complex	complex	ADJ
ejpam-5572	579	12	q	q	ADJ
ejpam-5572	579	13	-	-	PUNCT
ejpam-5572	579	14	rung	rung	ADJ
ejpam-5572	579	15	orthopair	orthopair	ADJ
ejpam-5572	579	16	fuzzy	fuzzy	ADJ
ejpam-5572	579	17	subgroup	subgroup	NOUN
ejpam-5572	579	18	.	.	PUNCT
ejpam-5572	580	1	references	reference	NOUN
ejpam-5572	580	2	[	[	X
ejpam-5572	580	3	1	1	NUM
ejpam-5572	580	4	]	]	X
ejpam-5572	580	5	i.	i.	PROPN
ejpam-5572	580	6	abu	abu	PROPN
ejpam-5572	580	7	-	-	PUNCT
ejpam-5572	580	8	irwaq	irwaq	PROPN
ejpam-5572	580	9	,	,	PUNCT
ejpam-5572	580	10	w.	w.	PROPN
ejpam-5572	580	11	shatanawi	shatanawi	PROPN
ejpam-5572	580	12	,	,	PUNCT
ejpam-5572	580	13	a.	a.	NOUN
ejpam-5572	580	14	bataihah	bataihah	PROPN
ejpam-5572	580	15	,	,	PUNCT
ejpam-5572	580	16	and	and	CCONJ
ejpam-5572	580	17	i.	i.	PROPN
ejpam-5572	580	18	nuseir	nuseir	PROPN
ejpam-5572	580	19	.	.	PUNCT
ejpam-5572	581	1	fixed	fix	VERB
ejpam-5572	581	2	point	point	NOUN
ejpam-5572	581	3	results	result	NOUN
ejpam-5572	581	4	for	for	ADP
ejpam-5572	581	5	nonlinear	nonlinear	ADJ
ejpam-5572	581	6	contractions	contraction	NOUN
ejpam-5572	581	7	with	with	ADP
ejpam-5572	581	8	generalized	generalized	ADJ
ejpam-5572	581	9	ω	ω	NUM
ejpam-5572	581	10	-	-	PUNCT
ejpam-5572	581	11	distance	distance	NOUN
ejpam-5572	581	12	mappings	mapping	NOUN
ejpam-5572	581	13	.	.	PUNCT
ejpam-5572	582	1	upb	upb	PROPN
ejpam-5572	582	2	sci	sci	PROPN
ejpam-5572	582	3	.	.	PUNCT
ejpam-5572	582	4	bull	bull	PROPN
ejpam-5572	582	5	.	.	PUNCT
ejpam-5572	583	1	ser	ser	PROPN
ejpam-5572	583	2	.	.	PUNCT
ejpam-5572	584	1	a	a	DET
ejpam-5572	584	2	,	,	PUNCT
ejpam-5572	584	3	81(1):57–64	81(1):57–64	NUM
ejpam-5572	584	4	,	,	PUNCT
ejpam-5572	584	5	2019	2019	NUM
ejpam-5572	584	6	.	.	PUNCT
ejpam-5572	585	1	[	[	X
ejpam-5572	585	2	2	2	X
ejpam-5572	585	3	]	]	PUNCT
ejpam-5572	585	4	e.	e.	PROPN
ejpam-5572	585	5	abuhijleh	abuhijleh	PROPN
ejpam-5572	585	6	.	.	PUNCT
ejpam-5572	586	1	complex	complex	ADJ
ejpam-5572	586	2	hesitant	hesitant	ADJ
ejpam-5572	586	3	fuzzy	fuzzy	ADJ
ejpam-5572	586	4	graph	graph	NOUN
ejpam-5572	586	5	.	.	PUNCT
ejpam-5572	587	1	fuzzy	fuzzy	ADJ
ejpam-5572	587	2	information	information	NOUN
ejpam-5572	587	3	and	and	CCONJ
ejpam-5572	587	4	engineering	engineering	NOUN
ejpam-5572	587	5	,	,	PUNCT
ejpam-5572	587	6	15(2):149–161	15(2):149–161	NUM
ejpam-5572	587	7	,	,	PUNCT
ejpam-5572	587	8	2023	2023	NUM
ejpam-5572	587	9	.	.	PUNCT
ejpam-5572	588	1	[	[	X
ejpam-5572	588	2	3	3	X
ejpam-5572	588	3	]	]	PUNCT
ejpam-5572	588	4	e.	e.	PROPN
ejpam-5572	588	5	abuhijleh	abuhijleh	PROPN
ejpam-5572	588	6	,	,	PUNCT
ejpam-5572	588	7	m.	m.	NOUN
ejpam-5572	588	8	massa’deh	massa’deh	PROPN
ejpam-5572	588	9	,	,	PUNCT
ejpam-5572	588	10	a.	a.	NOUN
ejpam-5572	588	11	sheimat	sheimat	NOUN
ejpam-5572	588	12	,	,	PUNCT
ejpam-5572	588	13	and	and	CCONJ
ejpam-5572	588	14	a.	a.	PROPN
ejpam-5572	588	15	alkouri	alkouri	PROPN
ejpam-5572	588	16	.	.	PUNCT
ejpam-5572	589	1	complex	complex	ADJ
ejpam-5572	589	2	fuzzy	fuzzy	ADJ
ejpam-5572	589	3	groups	group	NOUN
ejpam-5572	589	4	based	base	VERB
ejpam-5572	589	5	on	on	ADP
ejpam-5572	589	6	rosenfeld	rosenfeld	PROPN
ejpam-5572	589	7	’s	’s	PART
ejpam-5572	589	8	approach	approach	NOUN
ejpam-5572	589	9	.	.	PUNCT
ejpam-5572	590	1	wseas	wseas	PROPN
ejpam-5572	590	2	trans	trans	PROPN
ejpam-5572	590	3	.	.	PROPN
ejpam-5572	590	4	math	math	PROPN
ejpam-5572	590	5	.	.	PUNCT
ejpam-5572	591	1	,	,	PUNCT
ejpam-5572	592	1	20:368–377	20:368–377	NUM
ejpam-5572	592	2	,	,	PUNCT
ejpam-5572	592	3	2021	2021	NUM
ejpam-5572	592	4	.	.	PUNCT
ejpam-5572	593	1	[	[	X
ejpam-5572	593	2	4	4	NUM
ejpam-5572	593	3	]	]	PUNCT
ejpam-5572	593	4	m.	m.	NOUN
ejpam-5572	593	5	akram	akram	PROPN
ejpam-5572	593	6	,	,	PUNCT
ejpam-5572	593	7	g.	g.	PROPN
ejpam-5572	593	8	muhiuddin	muhiuddin	PROPN
ejpam-5572	593	9	,	,	PUNCT
ejpam-5572	593	10	and	and	CCONJ
ejpam-5572	593	11	g.	g.	PROPN
ejpam-5572	593	12	santos	santos	PROPN
ejpam-5572	593	13	-	-	PUNCT
ejpam-5572	593	14	garćıa	garćıa	NOUN
ejpam-5572	593	15	.	.	PUNCT
ejpam-5572	594	1	an	an	DET
ejpam-5572	594	2	enhanced	enhance	VERB
ejpam-5572	594	3	vikor	vikor	ADJ
ejpam-5572	594	4	method	method	NOUN
ejpam-5572	594	5	for	for	ADP
ejpam-5572	594	6	multi	multi	ADJ
ejpam-5572	594	7	-	-	ADJ
ejpam-5572	594	8	criteria	criterion	NOUN
ejpam-5572	594	9	group	group	NOUN
ejpam-5572	594	10	decision	decision	NOUN
ejpam-5572	594	11	-	-	PUNCT
ejpam-5572	594	12	making	making	NOUN
ejpam-5572	594	13	with	with	ADP
ejpam-5572	594	14	complex	complex	ADJ
ejpam-5572	594	15	fermatean	fermatean	ADJ
ejpam-5572	594	16	fuzzy	fuzzy	ADJ
ejpam-5572	594	17	sets	set	NOUN
ejpam-5572	594	18	.	.	PUNCT
ejpam-5572	595	1	math	math	NOUN
ejpam-5572	595	2	.	.	PUNCT
ejpam-5572	596	1	biosci	biosci	PROPN
ejpam-5572	596	2	.	.	PUNCT
ejpam-5572	597	1	eng	eng	PROPN
ejpam-5572	597	2	.	.	PROPN
ejpam-5572	597	3	,	,	PUNCT
ejpam-5572	597	4	19(7):7201–7231	19(7):7201–7231	NUM
ejpam-5572	597	5	,	,	PUNCT
ejpam-5572	597	6	2022	2022	NUM
ejpam-5572	597	7	.	.	PUNCT
ejpam-5572	598	1	[	[	X
ejpam-5572	598	2	5	5	NUM
ejpam-5572	598	3	]	]	PUNCT
ejpam-5572	598	4	a.	a.	PROPN
ejpam-5572	598	5	al	al	PROPN
ejpam-5572	598	6	-	-	PROPN
ejpam-5572	598	7	masarwah	masarwah	PROPN
ejpam-5572	598	8	and	and	CCONJ
ejpam-5572	598	9	m.	m.	NOUN
ejpam-5572	598	10	alqahtani	alqahtani	PROPN
ejpam-5572	598	11	.	.	PUNCT
ejpam-5572	599	1	operational	operational	ADJ
ejpam-5572	599	2	algebraic	algebraic	ADJ
ejpam-5572	599	3	properties	property	NOUN
ejpam-5572	599	4	and	and	CCONJ
ejpam-5572	599	5	subsemigroups	subsemigroup	NOUN
ejpam-5572	599	6	of	of	ADP
ejpam-5572	599	7	semigroups	semigroup	NOUN
ejpam-5572	599	8	in	in	ADP
ejpam-5572	599	9	view	view	NOUN
ejpam-5572	599	10	of	of	ADP
ejpam-5572	599	11	k	k	NOUN
ejpam-5572	599	12	-	-	PUNCT
ejpam-5572	599	13	folded	fold	VERB
ejpam-5572	599	14	n	n	CCONJ
ejpam-5572	599	15	-	-	PUNCT
ejpam-5572	599	16	structures	structure	NOUN
ejpam-5572	599	17	.	.	PUNCT
ejpam-5572	600	1	aims	aim	VERB
ejpam-5572	600	2	mathematics	mathematic	NOUN
ejpam-5572	600	3	,	,	PUNCT
ejpam-5572	600	4	8(9):22081–22096	8(9):22081–22096	NUM
ejpam-5572	600	5	,	,	PUNCT
ejpam-5572	600	6	2023	2023	NUM
ejpam-5572	600	7	.	.	PUNCT
ejpam-5572	601	1	[	[	X
ejpam-5572	601	2	6	6	NUM
ejpam-5572	601	3	]	]	PUNCT
ejpam-5572	601	4	a.	a.	NOUN
ejpam-5572	601	5	alkouri	alkouri	PROPN
ejpam-5572	601	6	,	,	PUNCT
ejpam-5572	601	7	e.	e.	PROPN
ejpam-5572	601	8	abuhijleh	abuhijleh	PROPN
ejpam-5572	601	9	,	,	PUNCT
ejpam-5572	601	10	g.	g.	PROPN
ejpam-5572	601	11	alafifi	alafifi	PROPN
ejpam-5572	601	12	,	,	PUNCT
ejpam-5572	601	13	e.	e.	PROPN
ejpam-5572	601	14	almuhur	almuhur	PROPN
ejpam-5572	601	15	,	,	PUNCT
ejpam-5572	601	16	and	and	CCONJ
ejpam-5572	601	17	f.	f.	PROPN
ejpam-5572	601	18	m.	m.	PROPN
ejpam-5572	601	19	al	al	PROPN
ejpam-5572	601	20	-	-	PUNCT
ejpam-5572	601	21	zubi	zubi	PROPN
ejpam-5572	601	22	.	.	PUNCT
ejpam-5572	602	1	more	more	ADJ
ejpam-5572	602	2	on	on	ADP
ejpam-5572	602	3	complex	complex	ADJ
ejpam-5572	602	4	hesitant	hesitant	ADJ
ejpam-5572	602	5	fuzzy	fuzzy	ADJ
ejpam-5572	602	6	graphs	graph	NOUN
ejpam-5572	602	7	.	.	PUNCT
ejpam-5572	603	1	aims	aim	VERB
ejpam-5572	603	2	mathematics	mathematic	NOUN
ejpam-5572	603	3	,	,	PUNCT
ejpam-5572	603	4	8(12):30429–30444	8(12):30429–30444	NUM
ejpam-5572	603	5	,	,	PUNCT
ejpam-5572	603	6	2023	2023	NUM
ejpam-5572	603	7	.	.	PUNCT
ejpam-5572	604	1	[	[	X
ejpam-5572	604	2	7	7	NUM
ejpam-5572	604	3	]	]	PUNCT
ejpam-5572	604	4	a.	a.	NOUN
ejpam-5572	604	5	alkouri	alkouri	PROPN
ejpam-5572	604	6	,	,	PUNCT
ejpam-5572	604	7	e.	e.	PROPN
ejpam-5572	604	8	abuhijleh	abuhijleh	PROPN
ejpam-5572	604	9	,	,	PUNCT
ejpam-5572	604	10	e.	e.	PROPN
ejpam-5572	604	11	almuhur	almuhur	PROPN
ejpam-5572	604	12	,	,	PUNCT
ejpam-5572	604	13	and	and	CCONJ
ejpam-5572	604	14	g.	g.	PROPN
ejpam-5572	604	15	alafifi	alafifi	PROPN
ejpam-5572	604	16	.	.	PUNCT
ejpam-5572	605	1	subgroups	subgroup	NOUN
ejpam-5572	605	2	and	and	CCONJ
ejpam-5572	605	3	homomorphism	homomorphism	PROPN
ejpam-5572	605	4	structures	structure	NOUN
ejpam-5572	605	5	of	of	ADP
ejpam-5572	605	6	complex	complex	ADJ
ejpam-5572	605	7	pythagorean	pythagorean	ADJ
ejpam-5572	605	8	fuzzy	fuzzy	ADJ
ejpam-5572	605	9	sets	set	NOUN
ejpam-5572	605	10	.	.	PUNCT
ejpam-5572	606	1	wseas	wseas	PROPN
ejpam-5572	606	2	trans	trans	PROPN
ejpam-5572	606	3	.	.	PROPN
ejpam-5572	606	4	math	math	PROPN
ejpam-5572	606	5	.	.	PUNCT
ejpam-5572	606	6	,	,	PUNCT
ejpam-5572	606	7	23:614–626	23:614–626	NUM
ejpam-5572	606	8	,	,	PUNCT
ejpam-5572	606	9	2024	2024	NUM
ejpam-5572	606	10	.	.	PUNCT
ejpam-5572	607	1	[	[	X
ejpam-5572	607	2	8	8	NUM
ejpam-5572	607	3	]	]	X
ejpam-5572	607	4	a.	a.	NOUN
ejpam-5572	607	5	alkouri	alkouri	PROPN
ejpam-5572	607	6	and	and	CCONJ
ejpam-5572	607	7	a.	a.	PROPN
ejpam-5572	607	8	salleh	salleh	PROPN
ejpam-5572	607	9	.	.	PUNCT
ejpam-5572	608	1	complex	complex	ADJ
ejpam-5572	608	2	intuitionistic	intuitionistic	ADJ
ejpam-5572	608	3	fuzzy	fuzzy	ADJ
ejpam-5572	608	4	sets	set	NOUN
ejpam-5572	608	5	.	.	PUNCT
ejpam-5572	609	1	in	in	ADP
ejpam-5572	609	2	aip	aip	PROPN
ejpam-5572	609	3	conf	conf	PROPN
ejpam-5572	609	4	.	.	PUNCT
ejpam-5572	610	1	proc	proc	PROPN
ejpam-5572	610	2	.	.	PROPN
ejpam-5572	610	3	,	,	PUNCT
ejpam-5572	610	4	pages	page	NOUN
ejpam-5572	610	5	464–470	464–470	NUM
ejpam-5572	610	6	,	,	PUNCT
ejpam-5572	610	7	kuala	kuala	PROPN
ejpam-5572	610	8	lumpur	lumpur	PROPN
ejpam-5572	610	9	,	,	PUNCT
ejpam-5572	610	10	2012	2012	NUM
ejpam-5572	610	11	.	.	PUNCT
ejpam-5572	611	1	icfas2012	icfas2012	PROPN
ejpam-5572	611	2	.	.	PUNCT
ejpam-5572	612	1	[	[	X
ejpam-5572	612	2	9	9	NUM
ejpam-5572	612	3	]	]	SYM
ejpam-5572	612	4	a.	a.	NOUN
ejpam-5572	612	5	alkouri	alkouri	PROPN
ejpam-5572	612	6	and	and	CCONJ
ejpam-5572	612	7	a.	a.	PROPN
ejpam-5572	612	8	salleh	salleh	PROPN
ejpam-5572	612	9	.	.	PUNCT
ejpam-5572	613	1	complex	complex	ADJ
ejpam-5572	613	2	atanassov	atanassov	NOUN
ejpam-5572	613	3	’s	’s	PART
ejpam-5572	613	4	intuitionistic	intuitionistic	ADJ
ejpam-5572	613	5	fuzzy	fuzzy	ADJ
ejpam-5572	613	6	relation	relation	NOUN
ejpam-5572	613	7	.	.	PUNCT
ejpam-5572	614	1	abstract	abstract	ADJ
ejpam-5572	614	2	and	and	CCONJ
ejpam-5572	614	3	applied	apply	VERB
ejpam-5572	614	4	analysis	analysis	NOUN
ejpam-5572	614	5	,	,	PUNCT
ejpam-5572	614	6	2013(1	2013(1	NUM
ejpam-5572	614	7	)	)	PUNCT
ejpam-5572	614	8	,	,	PUNCT
ejpam-5572	614	9	2013	2013	NUM
ejpam-5572	614	10	.	.	PUNCT
ejpam-5572	615	1	[	[	X
ejpam-5572	615	2	10	10	NUM
ejpam-5572	615	3	]	]	X
ejpam-5572	615	4	m.	m.	NOUN
ejpam-5572	615	5	alqahtani	alqahtani	PROPN
ejpam-5572	615	6	,	,	PUNCT
ejpam-5572	615	7	m.	m.	NOUN
ejpam-5572	615	8	kaviyarasu	kaviyarasu	PROPN
ejpam-5572	615	9	,	,	PUNCT
ejpam-5572	615	10	a.	a.	PROPN
ejpam-5572	615	11	al	al	PROPN
ejpam-5572	615	12	-	-	PROPN
ejpam-5572	615	13	masarwah	masarwah	NOUN
ejpam-5572	615	14	,	,	PUNCT
ejpam-5572	615	15	and	and	CCONJ
ejpam-5572	615	16	m.	m.	NOUN
ejpam-5572	615	17	rajeshwari	rajeshwari	PROPN
ejpam-5572	615	18	.	.	PUNCT
ejpam-5572	616	1	application	application	NOUN
ejpam-5572	616	2	of	of	ADP
ejpam-5572	616	3	complex	complex	ADJ
ejpam-5572	616	4	neutrosophic	neutrosophic	ADJ
ejpam-5572	616	5	graphs	graph	NOUN
ejpam-5572	616	6	in	in	ADP
ejpam-5572	616	7	hospital	hospital	NOUN
ejpam-5572	616	8	infrastructure	infrastructure	NOUN
ejpam-5572	616	9	design	design	NOUN
ejpam-5572	616	10	.	.	PUNCT
ejpam-5572	617	1	mathematics	mathematic	NOUN
ejpam-5572	617	2	,	,	PUNCT
ejpam-5572	617	3	12(5	12(5	NUM
ejpam-5572	617	4	)	)	PUNCT
ejpam-5572	617	5	,	,	PUNCT
ejpam-5572	617	6	2024	2024	NUM
ejpam-5572	617	7	.	.	PUNCT
ejpam-5572	618	1	[	[	X
ejpam-5572	618	2	11	11	NUM
ejpam-5572	618	3	]	]	PUNCT
ejpam-5572	618	4	s.	s.	PROPN
ejpam-5572	618	5	m.	m.	PROPN
ejpam-5572	618	6	alqaraleh	alqaraleh	PROPN
ejpam-5572	618	7	,	,	PUNCT
ejpam-5572	618	8	a.	a.	PROPN
ejpam-5572	618	9	alkouri	alkouri	PROPN
ejpam-5572	618	10	,	,	PUNCT
ejpam-5572	618	11	m.	m.	NOUN
ejpam-5572	618	12	massa’deh	massa’deh	PROPN
ejpam-5572	618	13	,	,	PUNCT
ejpam-5572	618	14	a.	a.	NOUN
ejpam-5572	618	15	talafha	talafha	NOUN
ejpam-5572	618	16	,	,	PUNCT
ejpam-5572	618	17	and	and	CCONJ
ejpam-5572	618	18	a.	a.	NOUN
ejpam-5572	618	19	bataihah	bataihah	PROPN
ejpam-5572	618	20	.	.	PUNCT
ejpam-5572	619	1	bipolar	bipolar	ADJ
ejpam-5572	619	2	complex	complex	ADJ
ejpam-5572	619	3	fuzzy	fuzzy	ADJ
ejpam-5572	619	4	soft	soft	ADJ
ejpam-5572	619	5	sets	set	NOUN
ejpam-5572	619	6	and	and	CCONJ
ejpam-5572	619	7	their	their	PRON
ejpam-5572	619	8	application	application	NOUN
ejpam-5572	619	9	.	.	PUNCT
ejpam-5572	620	1	international	international	ADJ
ejpam-5572	620	2	journal	journal	NOUN
ejpam-5572	620	3	of	of	ADP
ejpam-5572	620	4	fuzzy	fuzzy	ADJ
ejpam-5572	620	5	system	system	NOUN
ejpam-5572	620	6	applications	application	NOUN
ejpam-5572	620	7	,	,	PUNCT
ejpam-5572	620	8	11(1):1–23	11(1):1–23	NUM
ejpam-5572	620	9	,	,	PUNCT
ejpam-5572	620	10	2022	2022	NUM
ejpam-5572	620	11	.	.	PUNCT
ejpam-5572	621	1	[	[	X
ejpam-5572	621	2	12	12	NUM
ejpam-5572	621	3	]	]	PUNCT
ejpam-5572	621	4	k.	k.	PROPN
ejpam-5572	621	5	atanassov	atanassov	PROPN
ejpam-5572	621	6	.	.	PUNCT
ejpam-5572	622	1	intuitionistic	intuitionistic	ADJ
ejpam-5572	622	2	fuzzy	fuzzy	ADJ
ejpam-5572	622	3	sets	set	NOUN
ejpam-5572	622	4	.	.	PUNCT
ejpam-5572	623	1	fuzzy	fuzzy	ADJ
ejpam-5572	623	2	sets	set	NOUN
ejpam-5572	623	3	and	and	CCONJ
ejpam-5572	623	4	systems	system	NOUN
ejpam-5572	623	5	,	,	PUNCT
ejpam-5572	623	6	20(1):87–96	20(1):87–96	NUM
ejpam-5572	623	7	,	,	PUNCT
ejpam-5572	623	8	1986	1986	NUM
ejpam-5572	623	9	.	.	PUNCT
ejpam-5572	624	1	[	[	X
ejpam-5572	624	2	13	13	NUM
ejpam-5572	624	3	]	]	PUNCT
ejpam-5572	624	4	k.	k.	PROPN
ejpam-5572	624	5	balamurugan	balamurugan	PROPN
ejpam-5572	624	6	and	and	CCONJ
ejpam-5572	624	7	r.	r.	PROPN
ejpam-5572	624	8	nagarajan	nagarajan	PROPN
ejpam-5572	624	9	.	.	PUNCT
ejpam-5572	625	1	fermatean	fermatean	PROPN
ejpam-5572	625	2	fuzzy	fuzzy	ADJ
ejpam-5572	625	3	soft	soft	ADJ
ejpam-5572	625	4	covered	cover	VERB
ejpam-5572	625	5	congruence	congruence	NOUN
ejpam-5572	625	6	relations	relation	NOUN
ejpam-5572	625	7	acting	act	VERB
ejpam-5572	625	8	on	on	ADP
ejpam-5572	625	9	a	a	DET
ejpam-5572	625	10	semi	semi	NOUN
ejpam-5572	625	11	-	-	NOUN
ejpam-5572	625	12	group	group	NOUN
ejpam-5572	625	13	.	.	PUNCT
ejpam-5572	626	1	journal	journal	PROPN
ejpam-5572	626	2	for	for	ADP
ejpam-5572	626	3	basic	basic	ADJ
ejpam-5572	626	4	sciences	science	NOUN
ejpam-5572	626	5	,	,	PUNCT
ejpam-5572	626	6	22(11):85–96	22(11):85–96	NUM
ejpam-5572	626	7	,	,	PUNCT
ejpam-5572	626	8	2022	2022	NUM
ejpam-5572	626	9	.	.	PUNCT
ejpam-5572	627	1	[	[	X
ejpam-5572	627	2	14	14	NUM
ejpam-5572	627	3	]	]	X
ejpam-5572	627	4	s.	s.	PROPN
ejpam-5572	627	5	bhunia	bhunia	PROPN
ejpam-5572	627	6	,	,	PUNCT
ejpam-5572	627	7	g.	g.	PROPN
ejpam-5572	627	8	ghorai	ghorai	PROPN
ejpam-5572	627	9	,	,	PUNCT
ejpam-5572	627	10	and	and	CCONJ
ejpam-5572	627	11	q.	q.	PROPN
ejpam-5572	627	12	xin	xin	PROPN
ejpam-5572	627	13	.	.	PUNCT
ejpam-5572	628	1	on	on	ADP
ejpam-5572	628	2	the	the	DET
ejpam-5572	628	3	characterization	characterization	NOUN
ejpam-5572	628	4	of	of	ADP
ejpam-5572	628	5	pythagorean	pythagorean	PROPN
ejpam-5572	628	6	fuzzy	fuzzy	ADJ
ejpam-5572	628	7	subgroups	subgroup	NOUN
ejpam-5572	628	8	.	.	PUNCT
ejpam-5572	629	1	aims	aim	VERB
ejpam-5572	629	2	mathematics	mathematic	NOUN
ejpam-5572	629	3	,	,	PUNCT
ejpam-5572	629	4	6(1):962–978	6(1):962–978	NOUN
ejpam-5572	629	5	,	,	PUNCT
ejpam-5572	629	6	2020	2020	NUM
ejpam-5572	629	7	.	.	PUNCT
ejpam-5572	630	1	[	[	X
ejpam-5572	630	2	15	15	NUM
ejpam-5572	630	3	]	]	X
ejpam-5572	630	4	r.	r.	PROPN
ejpam-5572	630	5	biswas	biswas	PROPN
ejpam-5572	630	6	.	.	PUNCT
ejpam-5572	631	1	intuitionistic	intuitionistic	ADJ
ejpam-5572	631	2	fuzzy	fuzzy	ADJ
ejpam-5572	631	3	subgroup	subgroup	NOUN
ejpam-5572	631	4	.	.	PUNCT
ejpam-5572	632	1	mathematical	mathematical	PROPN
ejpam-5572	632	2	forum	forum	PROPN
ejpam-5572	632	3	,	,	PUNCT
ejpam-5572	632	4	10:39–44	10:39–44	NUM
ejpam-5572	632	5	,	,	PUNCT
ejpam-5572	632	6	1989	1989	NUM
ejpam-5572	632	7	.	.	PUNCT
ejpam-5572	633	1	[	[	X
ejpam-5572	633	2	16	16	NUM
ejpam-5572	633	3	]	]	PUNCT
ejpam-5572	633	4	l.	l.	PROPN
ejpam-5572	633	5	chen	chen	PROPN
ejpam-5572	633	6	,	,	PUNCT
ejpam-5572	633	7	x.	x.	PROPN
ejpam-5572	633	8	zhou	zhou	PROPN
ejpam-5572	633	9	,	,	PUNCT
ejpam-5572	633	10	m.	m.	PROPN
ejpam-5572	633	11	wu	wu	PROPN
ejpam-5572	633	12	,	,	PUNCT
ejpam-5572	633	13	y.	y.	PROPN
ejpam-5572	633	14	shi	shi	PROPN
ejpam-5572	633	15	,	,	PUNCT
ejpam-5572	633	16	and	and	CCONJ
ejpam-5572	633	17	y.	y.	PROPN
ejpam-5572	633	18	wang	wang	PROPN
ejpam-5572	633	19	.	.	PUNCT
ejpam-5572	634	1	aczel	aczel	PROPN
ejpam-5572	634	2	-	-	PUNCT
ejpam-5572	634	3	alsina	alsina	PROPN
ejpam-5572	634	4	aggregation	aggregation	PROPN
ejpam-5572	634	5	operators	operators	PROPN
ejpam-5572	634	6	e.a	e.a	PROPN
ejpam-5572	634	7	.	.	PROPN
ejpam-5572	634	8	abuhijleh	abuhijleh	PROPN
ejpam-5572	634	9	,	,	PUNCT
ejpam-5572	634	10	a.	a.	PROPN
ejpam-5572	634	11	alkouri	alkouri	PROPN
ejpam-5572	634	12	/	/	PROPN
ejpam-5572	634	13	eur	eur	PROPN
ejpam-5572	634	14	.	.	PUNCT
ejpam-5572	635	1	j.	j.	PROPN
ejpam-5572	635	2	pure	pure	PROPN
ejpam-5572	635	3	appl	appl	PROPN
ejpam-5572	635	4	.	.	PROPN
ejpam-5572	635	5	math	math	PROPN
ejpam-5572	635	6	,	,	PUNCT
ejpam-5572	635	7	18	18	NUM
ejpam-5572	635	8	(	(	PUNCT
ejpam-5572	635	9	1	1	NUM
ejpam-5572	635	10	)	)	PUNCT
ejpam-5572	635	11	(	(	PUNCT
ejpam-5572	635	12	2025	2025	NUM
ejpam-5572	635	13	)	)	PUNCT
ejpam-5572	635	14	,	,	PUNCT
ejpam-5572	635	15	5572	5572	NUM
ejpam-5572	635	16	18	18	NUM
ejpam-5572	635	17	of	of	ADP
ejpam-5572	635	18	19	19	NUM
ejpam-5572	635	19	on	on	ADP
ejpam-5572	635	20	complex	complex	ADJ
ejpam-5572	635	21	fermatean	fermatean	ADJ
ejpam-5572	635	22	fuzzy	fuzzy	ADJ
ejpam-5572	635	23	information	information	NOUN
ejpam-5572	635	24	with	with	ADP
ejpam-5572	635	25	application	application	NOUN
ejpam-5572	635	26	to	to	ADP
ejpam-5572	635	27	multi	multi	ADJ
ejpam-5572	635	28	-	-	ADJ
ejpam-5572	635	29	attribute	attribute	NOUN
ejpam-5572	635	30	decision	decision	NOUN
ejpam-5572	635	31	making	making	NOUN
ejpam-5572	635	32	.	.	PUNCT
ejpam-5572	636	1	ieee	ieee	NOUN
ejpam-5572	636	2	access	access	NOUN
ejpam-5572	636	3	,	,	PUNCT
ejpam-5572	636	4	11:141703–141722	11:141703–141722	NUM
ejpam-5572	636	5	,	,	PUNCT
ejpam-5572	636	6	2023	2023	NUM
ejpam-5572	636	7	.	.	PUNCT
ejpam-5572	637	1	[	[	X
ejpam-5572	637	2	17	17	NUM
ejpam-5572	637	3	]	]	X
ejpam-5572	637	4	v.	v.	ADP
ejpam-5572	637	5	chinnadurai	chinnadurai	PROPN
ejpam-5572	637	6	,	,	PUNCT
ejpam-5572	637	7	s.	s.	PROPN
ejpam-5572	637	8	thayalan	thayalan	PROPN
ejpam-5572	637	9	,	,	PUNCT
ejpam-5572	637	10	and	and	CCONJ
ejpam-5572	637	11	a.	a.	NOUN
ejpam-5572	637	12	bobin	bobin	NOUN
ejpam-5572	637	13	.	.	PUNCT
ejpam-5572	638	1	multi	multi	ADJ
ejpam-5572	638	2	-	-	ADJ
ejpam-5572	638	3	criteria	criterion	NOUN
ejpam-5572	638	4	decision	decision	NOUN
ejpam-5572	638	5	-	-	PUNCT
ejpam-5572	638	6	making	making	NOUN
ejpam-5572	638	7	in	in	ADP
ejpam-5572	638	8	complex	complex	ADJ
ejpam-5572	638	9	fermatean	fermatean	ADJ
ejpam-5572	638	10	fuzzy	fuzzy	ADJ
ejpam-5572	638	11	environment	environment	NOUN
ejpam-5572	638	12	.	.	PUNCT
ejpam-5572	639	1	journal	journal	PROPN
ejpam-5572	639	2	of	of	ADP
ejpam-5572	639	3	mathematics	mathematics	PROPN
ejpam-5572	639	4	and	and	CCONJ
ejpam-5572	639	5	computer	computer	NOUN
ejpam-5572	639	6	science	science	NOUN
ejpam-5572	639	7	,	,	PUNCT
ejpam-5572	639	8	11:7209–7227	11:7209–7227	NUM
ejpam-5572	639	9	,	,	PUNCT
ejpam-5572	639	10	2021	2021	NUM
ejpam-5572	639	11	.	.	PUNCT
ejpam-5572	640	1	[	[	X
ejpam-5572	640	2	18	18	NUM
ejpam-5572	640	3	]	]	PUNCT
ejpam-5572	640	4	p.	p.	NOUN
ejpam-5572	640	5	ejegwa	ejegwa	NOUN
ejpam-5572	640	6	.	.	PUNCT
ejpam-5572	641	1	pythagorean	pythagorean	PROPN
ejpam-5572	641	2	fuzzy	fuzzy	PROPN
ejpam-5572	641	3	set	set	NOUN
ejpam-5572	641	4	and	and	CCONJ
ejpam-5572	641	5	its	its	PRON
ejpam-5572	641	6	applications	application	NOUN
ejpam-5572	641	7	in	in	ADP
ejpam-5572	641	8	career	career	NOUN
ejpam-5572	641	9	placements	placement	NOUN
ejpam-5572	641	10	based	base	VERB
ejpam-5572	641	11	on	on	ADP
ejpam-5572	641	12	academics	academic	NOUN
ejpam-5572	641	13	performance	performance	NOUN
ejpam-5572	641	14	using	use	VERB
ejpam-5572	641	15	max	max	PROPN
ejpam-5572	641	16	-	-	PUNCT
ejpam-5572	641	17	min	min	PROPN
ejpam-5572	641	18	-	-	ADJ
ejpam-5572	641	19	max	max	PROPN
ejpam-5572	641	20	composition	composition	NOUN
ejpam-5572	641	21	.	.	PUNCT
ejpam-5572	642	1	complex	complex	ADJ
ejpam-5572	642	2	intell	intell	PROPN
ejpam-5572	642	3	.	.	PUNCT
ejpam-5572	643	1	syst	syst	PROPN
ejpam-5572	643	2	.	.	PROPN
ejpam-5572	643	3	,	,	PUNCT
ejpam-5572	643	4	5:165–175	5:165–175	PROPN
ejpam-5572	643	5	,	,	PUNCT
ejpam-5572	643	6	2019	2019	NUM
ejpam-5572	643	7	.	.	PUNCT
ejpam-5572	644	1	[	[	X
ejpam-5572	644	2	19	19	NUM
ejpam-5572	644	3	]	]	PUNCT
ejpam-5572	644	4	a.	a.	NOUN
ejpam-5572	644	5	fallatah	fallatah	PROPN
ejpam-5572	644	6	,	,	PUNCT
ejpam-5572	644	7	m.	m.	NOUN
ejpam-5572	644	8	o.	o.	PROPN
ejpam-5572	644	9	massa’deh	massa’deh	PROPN
ejpam-5572	644	10	,	,	PUNCT
ejpam-5572	644	11	and	and	CCONJ
ejpam-5572	644	12	a.	a.	PROPN
ejpam-5572	644	13	alkouri	alkouri	PROPN
ejpam-5572	644	14	.	.	PUNCT
ejpam-5572	645	1	normal	normal	ADJ
ejpam-5572	645	2	and	and	CCONJ
ejpam-5572	645	3	cosets	coset	NOUN
ejpam-5572	645	4	of	of	ADP
ejpam-5572	645	5	(	(	PUNCT
ejpam-5572	645	6	γ	γ	X
ejpam-5572	645	7	,	,	PUNCT
ejpam-5572	645	8	∂)-fuzzy	∂)-fuzzy	PROPN
ejpam-5572	645	9	hxsubgroups	hxsubgroup	NOUN
ejpam-5572	645	10	.	.	PUNCT
ejpam-5572	646	1	journal	journal	PROPN
ejpam-5572	646	2	of	of	ADP
ejpam-5572	646	3	applied	apply	VERB
ejpam-5572	646	4	mathematics	mathematics	PROPN
ejpam-5572	646	5	&	&	CCONJ
ejpam-5572	646	6	informatics	informatics	PROPN
ejpam-5572	646	7	,	,	PUNCT
ejpam-5572	646	8	40(3	40(3	NUM
ejpam-5572	646	9	4):719–727	4):719–727	NUM
ejpam-5572	646	10	,	,	PUNCT
ejpam-5572	646	11	2022	2022	NUM
ejpam-5572	646	12	.	.	PUNCT
ejpam-5572	647	1	[	[	X
ejpam-5572	647	2	20	20	NUM
ejpam-5572	647	3	]	]	PUNCT
ejpam-5572	647	4	m.	m.	NOUN
ejpam-5572	647	5	gulzar	gulzar	PROPN
ejpam-5572	647	6	,	,	PUNCT
ejpam-5572	647	7	m.	m.	PROPN
ejpam-5572	647	8	h.	h.	PROPN
ejpam-5572	647	9	mateen	mateen	PROPN
ejpam-5572	647	10	,	,	PUNCT
ejpam-5572	647	11	d.	d.	PROPN
ejpam-5572	647	12	algazzawi	algazzawi	PROPN
ejpam-5572	647	13	,	,	PUNCT
ejpam-5572	647	14	and	and	CCONJ
ejpam-5572	647	15	n.	n.	PROPN
ejpam-5572	647	16	kausar	kausar	PROPN
ejpam-5572	647	17	.	.	PUNCT
ejpam-5572	648	1	a	a	DET
ejpam-5572	648	2	novel	novel	ADJ
ejpam-5572	648	3	applications	application	NOUN
ejpam-5572	648	4	of	of	ADP
ejpam-5572	648	5	complex	complex	ADJ
ejpam-5572	648	6	intuitionistic	intuitionistic	ADJ
ejpam-5572	648	7	fuzzy	fuzzy	ADJ
ejpam-5572	648	8	sets	set	NOUN
ejpam-5572	648	9	in	in	ADP
ejpam-5572	648	10	group	group	NOUN
ejpam-5572	648	11	theory	theory	NOUN
ejpam-5572	648	12	.	.	PUNCT
ejpam-5572	649	1	ieee	ieee	NOUN
ejpam-5572	649	2	access	access	NOUN
ejpam-5572	649	3	,	,	PUNCT
ejpam-5572	649	4	8:196075–196085	8:196075–196085	NUM
ejpam-5572	649	5	,	,	PUNCT
ejpam-5572	649	6	2020	2020	NUM
ejpam-5572	649	7	.	.	PUNCT
ejpam-5572	650	1	[	[	X
ejpam-5572	650	2	21	21	NUM
ejpam-5572	650	3	]	]	PUNCT
ejpam-5572	650	4	a.	a.	NOUN
ejpam-5572	650	5	hazaymeh	hazaymeh	NOUN
ejpam-5572	650	6	.	.	PUNCT
ejpam-5572	651	1	time	time	NOUN
ejpam-5572	651	2	effective	effective	ADJ
ejpam-5572	651	3	fuzzy	fuzzy	ADJ
ejpam-5572	651	4	soft	soft	ADJ
ejpam-5572	651	5	set	set	NOUN
ejpam-5572	651	6	and	and	CCONJ
ejpam-5572	651	7	its	its	PRON
ejpam-5572	651	8	some	some	DET
ejpam-5572	651	9	applications	application	NOUN
ejpam-5572	651	10	with	with	ADP
ejpam-5572	651	11	and	and	CCONJ
ejpam-5572	651	12	without	without	ADP
ejpam-5572	651	13	a	a	DET
ejpam-5572	651	14	neutrosophic	neutrosophic	ADJ
ejpam-5572	651	15	.	.	PUNCT
ejpam-5572	652	1	international	international	ADJ
ejpam-5572	652	2	journal	journal	PROPN
ejpam-5572	652	3	of	of	ADP
ejpam-5572	652	4	neutrosophic	neutrosophic	ADJ
ejpam-5572	652	5	science	science	NOUN
ejpam-5572	652	6	,	,	PUNCT
ejpam-5572	652	7	23(2):129–149	23(2):129–149	PROPN
ejpam-5572	652	8	,	,	PUNCT
ejpam-5572	652	9	2024	2024	NUM
ejpam-5572	652	10	.	.	PUNCT
ejpam-5572	653	1	[	[	X
ejpam-5572	653	2	22	22	NUM
ejpam-5572	653	3	]	]	PUNCT
ejpam-5572	653	4	a.	a.	NOUN
ejpam-5572	653	5	hazaymeh	hazaymeh	NOUN
ejpam-5572	653	6	.	.	PUNCT
ejpam-5572	654	1	time	time	NOUN
ejpam-5572	654	2	fuzzy	fuzzy	ADJ
ejpam-5572	654	3	soft	soft	ADJ
ejpam-5572	654	4	sets	set	NOUN
ejpam-5572	654	5	and	and	CCONJ
ejpam-5572	654	6	its	its	PRON
ejpam-5572	654	7	application	application	NOUN
ejpam-5572	654	8	in	in	ADP
ejpam-5572	654	9	design	design	NOUN
ejpam-5572	654	10	-	-	PUNCT
ejpam-5572	654	11	making	making	NOUN
ejpam-5572	654	12	.	.	PUNCT
ejpam-5572	655	1	international	international	ADJ
ejpam-5572	655	2	journal	journal	PROPN
ejpam-5572	655	3	of	of	ADP
ejpam-5572	655	4	neutrosophic	neutrosophic	ADJ
ejpam-5572	655	5	science	science	NOUN
ejpam-5572	655	6	,	,	PUNCT
ejpam-5572	655	7	25(3):37–50	25(3):37–50	NUM
ejpam-5572	655	8	,	,	PUNCT
ejpam-5572	655	9	2025	2025	NUM
ejpam-5572	655	10	.	.	PUNCT
ejpam-5572	656	1	[	[	X
ejpam-5572	656	2	23	23	NUM
ejpam-5572	656	3	]	]	X
ejpam-5572	656	4	h.	h.	PROPN
ejpam-5572	656	5	ibrahim	ibrahim	PROPN
ejpam-5572	656	6	.	.	PUNCT
ejpam-5572	657	1	new	new	ADJ
ejpam-5572	657	2	extensions	extension	NOUN
ejpam-5572	657	3	of	of	ADP
ejpam-5572	657	4	fuzzy	fuzzy	ADJ
ejpam-5572	657	5	sets	set	NOUN
ejpam-5572	657	6	with	with	ADP
ejpam-5572	657	7	applications	application	NOUN
ejpam-5572	657	8	to	to	ADP
ejpam-5572	657	9	rough	rough	ADJ
ejpam-5572	657	10	topology	topology	NOUN
ejpam-5572	657	11	and	and	CCONJ
ejpam-5572	657	12	medical	medical	ADJ
ejpam-5572	657	13	diagnosis	diagnosis	NOUN
ejpam-5572	657	14	.	.	PUNCT
ejpam-5572	658	1	soft	soft	ADJ
ejpam-5572	658	2	computing	computing	NOUN
ejpam-5572	658	3	,	,	PUNCT
ejpam-5572	658	4	27(2):821–835	27(2):821–835	PROPN
ejpam-5572	658	5	,	,	PUNCT
ejpam-5572	658	6	2023	2023	NUM
ejpam-5572	658	7	.	.	PUNCT
ejpam-5572	659	1	[	[	X
ejpam-5572	659	2	24	24	NUM
ejpam-5572	659	3	]	]	PUNCT
ejpam-5572	659	4	k.	k.	PROPN
ejpam-5572	659	5	kalaiarasi	kalaiarasi	PROPN
ejpam-5572	659	6	,	,	PUNCT
ejpam-5572	659	7	p.	p.	PROPN
ejpam-5572	659	8	sudha	sudha	PROPN
ejpam-5572	659	9	,	,	PUNCT
ejpam-5572	659	10	n.	n.	PROPN
ejpam-5572	659	11	kausar	kausar	PROPN
ejpam-5572	659	12	,	,	PUNCT
ejpam-5572	659	13	s.	s.	PROPN
ejpam-5572	659	14	kousar	kousar	PROPN
ejpam-5572	659	15	,	,	PUNCT
ejpam-5572	659	16	d.	d.	PROPN
ejpam-5572	659	17	pamucar	pamucar	PROPN
ejpam-5572	659	18	,	,	PUNCT
ejpam-5572	659	19	and	and	CCONJ
ejpam-5572	659	20	nasr	nasr	PROPN
ejpam-5572	659	21	al	al	PROPN
ejpam-5572	659	22	din	din	PROPN
ejpam-5572	659	23	ide	ide	PROPN
ejpam-5572	659	24	.	.	PUNCT
ejpam-5572	660	1	the	the	DET
ejpam-5572	660	2	characterization	characterization	NOUN
ejpam-5572	660	3	of	of	ADP
ejpam-5572	660	4	substructures	substructure	NOUN
ejpam-5572	660	5	of	of	ADP
ejpam-5572	660	6	γ	γ	PROPN
ejpam-5572	660	7	-	-	ADJ
ejpam-5572	660	8	anti	anti	ADJ
ejpam-5572	660	9	fuzzy	fuzzy	ADJ
ejpam-5572	660	10	subgroups	subgroup	NOUN
ejpam-5572	660	11	with	with	ADP
ejpam-5572	660	12	application	application	NOUN
ejpam-5572	660	13	in	in	ADP
ejpam-5572	660	14	genetics	genetic	NOUN
ejpam-5572	660	15	.	.	PUNCT
ejpam-5572	661	1	discrete	discrete	ADJ
ejpam-5572	661	2	dynamics	dynamic	NOUN
ejpam-5572	661	3	in	in	ADP
ejpam-5572	661	4	nature	nature	NOUN
ejpam-5572	661	5	and	and	CCONJ
ejpam-5572	661	6	society	society	NOUN
ejpam-5572	661	7	,	,	PUNCT
ejpam-5572	661	8	2022(1	2022(1	NUM
ejpam-5572	661	9	)	)	PUNCT
ejpam-5572	661	10	,	,	PUNCT
ejpam-5572	661	11	2022	2022	NUM
ejpam-5572	661	12	.	.	PUNCT
ejpam-5572	662	1	[	[	X
ejpam-5572	662	2	25	25	NUM
ejpam-5572	662	3	]	]	PUNCT
ejpam-5572	662	4	p.	p.	PROPN
ejpam-5572	662	5	liu	liu	PROPN
ejpam-5572	662	6	,	,	PUNCT
ejpam-5572	662	7	t.	t.	PROPN
ejpam-5572	662	8	mahmood	mahmood	PROPN
ejpam-5572	662	9	,	,	PUNCT
ejpam-5572	662	10	and	and	CCONJ
ejpam-5572	662	11	z.	z.	PROPN
ejpam-5572	662	12	ali	ali	PROPN
ejpam-5572	662	13	.	.	PUNCT
ejpam-5572	663	1	complex	complex	ADJ
ejpam-5572	663	2	q	q	ADJ
ejpam-5572	663	3	-	-	PUNCT
ejpam-5572	663	4	rung	rung	ADJ
ejpam-5572	663	5	orthopair	orthopair	ADJ
ejpam-5572	663	6	fuzzy	fuzzy	ADJ
ejpam-5572	663	7	aggregation	aggregation	NOUN
ejpam-5572	663	8	operators	operator	NOUN
ejpam-5572	663	9	and	and	CCONJ
ejpam-5572	663	10	their	their	PRON
ejpam-5572	663	11	applications	application	NOUN
ejpam-5572	663	12	in	in	ADP
ejpam-5572	663	13	multi	multi	ADJ
ejpam-5572	663	14	-	-	ADJ
ejpam-5572	663	15	attribute	attribute	NOUN
ejpam-5572	663	16	group	group	NOUN
ejpam-5572	663	17	decision	decision	NOUN
ejpam-5572	663	18	making	making	NOUN
ejpam-5572	663	19	.	.	PUNCT
ejpam-5572	664	1	information	information	NOUN
ejpam-5572	664	2	,	,	PUNCT
ejpam-5572	664	3	11(1	11(1	NUM
ejpam-5572	664	4	)	)	PUNCT
ejpam-5572	664	5	,	,	PUNCT
ejpam-5572	664	6	2020	2020	NUM
ejpam-5572	664	7	.	.	PUNCT
ejpam-5572	665	1	[	[	X
ejpam-5572	665	2	26	26	NUM
ejpam-5572	665	3	]	]	X
ejpam-5572	665	4	r.	r.	PROPN
ejpam-5572	665	5	nagarajan	nagarajan	PROPN
ejpam-5572	665	6	.	.	PUNCT
ejpam-5572	666	1	fermatean	fermatean	PROPN
ejpam-5572	666	2	fuzzy	fuzzy	ADJ
ejpam-5572	666	3	multi	multi	PROPN
ejpam-5572	666	4	group	group	NOUN
ejpam-5572	666	5	over	over	ADP
ejpam-5572	666	6	multi	multi	ADJ
ejpam-5572	666	7	-	-	ADJ
ejpam-5572	666	8	homomorphisms	homomorphism	NOUN
ejpam-5572	666	9	.	.	PUNCT
ejpam-5572	667	1	ijcset	ijcset	NOUN
ejpam-5572	667	2	,	,	PUNCT
ejpam-5572	667	3	11:18–22	11:18–22	NUM
ejpam-5572	667	4	,	,	PUNCT
ejpam-5572	667	5	2021	2021	NUM
ejpam-5572	667	6	.	.	PUNCT
ejpam-5572	668	1	[	[	X
ejpam-5572	668	2	27	27	NUM
ejpam-5572	668	3	]	]	PUNCT
ejpam-5572	668	4	s.	s.	PROPN
ejpam-5572	668	5	naz	naz	PROPN
ejpam-5572	668	6	,	,	PUNCT
ejpam-5572	668	7	s.	s.	PROPN
ejpam-5572	668	8	ashraf	ashraf	PROPN
ejpam-5572	668	9	,	,	PUNCT
ejpam-5572	668	10	and	and	CCONJ
ejpam-5572	668	11	m.	m.	PROPN
ejpam-5572	668	12	akram	akram	PROPN
ejpam-5572	668	13	.	.	PUNCT
ejpam-5572	669	1	a	a	DET
ejpam-5572	669	2	novel	novel	ADJ
ejpam-5572	669	3	approach	approach	NOUN
ejpam-5572	669	4	to	to	ADP
ejpam-5572	669	5	decision	decision	NOUN
ejpam-5572	669	6	-	-	PUNCT
ejpam-5572	669	7	making	making	NOUN
ejpam-5572	669	8	with	with	ADP
ejpam-5572	669	9	pythagorean	pythagorean	PROPN
ejpam-5572	669	10	fuzzy	fuzzy	ADJ
ejpam-5572	669	11	information	information	NOUN
ejpam-5572	669	12	.	.	PUNCT
ejpam-5572	670	1	mathematics	mathematic	NOUN
ejpam-5572	670	2	,	,	PUNCT
ejpam-5572	670	3	6(6	6(6	ADJ
ejpam-5572	670	4	)	)	PUNCT
ejpam-5572	670	5	,	,	PUNCT
ejpam-5572	670	6	2018	2018	NUM
ejpam-5572	670	7	.	.	PUNCT
ejpam-5572	671	1	[	[	X
ejpam-5572	671	2	28	28	NUM
ejpam-5572	671	3	]	]	X
ejpam-5572	671	4	b.	b.	PROPN
ejpam-5572	671	5	onasanya	onasanya	PROPN
ejpam-5572	671	6	,	,	PUNCT
ejpam-5572	671	7	x.	x.	PROPN
ejpam-5572	671	8	ming	ming	PROPN
ejpam-5572	671	9	,	,	PUNCT
ejpam-5572	671	10	y.	y.	PROPN
ejpam-5572	671	11	feng	feng	PROPN
ejpam-5572	671	12	,	,	PUNCT
ejpam-5572	671	13	and	and	CCONJ
ejpam-5572	671	14	w.	w.	PROPN
ejpam-5572	671	15	zhang	zhang	PROPN
ejpam-5572	671	16	.	.	PUNCT
ejpam-5572	672	1	harmonization	harmonization	NOUN
ejpam-5572	672	2	of	of	ADP
ejpam-5572	672	3	some	some	DET
ejpam-5572	672	4	fuzzy	fuzzy	ADJ
ejpam-5572	672	5	subgroups	subgroup	NOUN
ejpam-5572	672	6	.	.	PUNCT
ejpam-5572	673	1	italian	italian	ADJ
ejpam-5572	673	2	journal	journal	NOUN
ejpam-5572	673	3	of	of	ADP
ejpam-5572	673	4	pure	pure	ADJ
ejpam-5572	673	5	and	and	CCONJ
ejpam-5572	673	6	applied	applied	ADJ
ejpam-5572	673	7	mathematics	mathematic	NOUN
ejpam-5572	673	8	,	,	PUNCT
ejpam-5572	673	9	48:863–876	48:863–876	NOUN
ejpam-5572	673	10	,	,	PUNCT
ejpam-5572	673	11	2022	2022	NUM
ejpam-5572	673	12	.	.	PUNCT
ejpam-5572	674	1	[	[	X
ejpam-5572	674	2	29	29	NUM
ejpam-5572	674	3	]	]	X
ejpam-5572	674	4	d.	d.	PROPN
ejpam-5572	674	5	ramot	ramot	PROPN
ejpam-5572	674	6	,	,	PUNCT
ejpam-5572	674	7	m.	m.	NOUN
ejpam-5572	674	8	friedman	friedman	PROPN
ejpam-5572	674	9	,	,	PUNCT
ejpam-5572	674	10	g.	g.	PROPN
ejpam-5572	674	11	langholz	langholz	PROPN
ejpam-5572	674	12	,	,	PUNCT
ejpam-5572	674	13	and	and	CCONJ
ejpam-5572	674	14	a.	a.	NOUN
ejpam-5572	674	15	kandel	kandel	PROPN
ejpam-5572	674	16	.	.	PUNCT
ejpam-5572	675	1	complex	complex	ADJ
ejpam-5572	675	2	fuzzy	fuzzy	ADJ
ejpam-5572	675	3	logic	logic	NOUN
ejpam-5572	675	4	.	.	PUNCT
ejpam-5572	676	1	ieee	ieee	NOUN
ejpam-5572	676	2	transaction	transaction	NOUN
ejpam-5572	676	3	on	on	ADP
ejpam-5572	676	4	fuzzy	fuzzy	ADJ
ejpam-5572	676	5	systems	system	NOUN
ejpam-5572	676	6	,	,	PUNCT
ejpam-5572	676	7	11(4):450–461	11(4):450–461	NUM
ejpam-5572	676	8	,	,	PUNCT
ejpam-5572	676	9	2003	2003	NUM
ejpam-5572	676	10	.	.	PUNCT
ejpam-5572	677	1	[	[	X
ejpam-5572	677	2	30	30	NUM
ejpam-5572	677	3	]	]	X
ejpam-5572	677	4	d.	d.	PROPN
ejpam-5572	677	5	ramot	ramot	PROPN
ejpam-5572	677	6	,	,	PUNCT
ejpam-5572	677	7	r.	r.	PROPN
ejpam-5572	677	8	milo	milo	PROPN
ejpam-5572	677	9	,	,	PUNCT
ejpam-5572	677	10	m.	m.	NOUN
ejpam-5572	677	11	friedman	friedman	PROPN
ejpam-5572	677	12	,	,	PUNCT
ejpam-5572	677	13	and	and	CCONJ
ejpam-5572	677	14	a.	a.	NOUN
ejpam-5572	677	15	kandel	kandel	PROPN
ejpam-5572	677	16	.	.	PUNCT
ejpam-5572	678	1	complex	complex	ADJ
ejpam-5572	678	2	fuzzy	fuzzy	ADJ
ejpam-5572	678	3	sets	set	NOUN
ejpam-5572	678	4	.	.	PUNCT
ejpam-5572	679	1	ieee	ieee	NOUN
ejpam-5572	679	2	transaction	transaction	NOUN
ejpam-5572	679	3	on	on	ADP
ejpam-5572	679	4	fuzzy	fuzzy	ADJ
ejpam-5572	679	5	systems	system	NOUN
ejpam-5572	679	6	,	,	PUNCT
ejpam-5572	679	7	10(2):171–186	10(2):171–186	NUM
ejpam-5572	679	8	,	,	PUNCT
ejpam-5572	679	9	2002	2002	NUM
ejpam-5572	679	10	.	.	PUNCT
ejpam-5572	680	1	[	[	X
ejpam-5572	680	2	31	31	NUM
ejpam-5572	680	3	]	]	PUNCT
ejpam-5572	680	4	m.	m.	NOUN
ejpam-5572	680	5	riaz	riaz	PROPN
ejpam-5572	680	6	,	,	PUNCT
ejpam-5572	680	7	a.	a.	PROPN
ejpam-5572	680	8	zeb	zeb	PROPN
ejpam-5572	680	9	,	,	PUNCT
ejpam-5572	680	10	f.	f.	PROPN
ejpam-5572	680	11	ali	ali	PROPN
ejpam-5572	680	12	,	,	PUNCT
ejpam-5572	680	13	m.	m.	NOUN
ejpam-5572	680	14	naeem	naeem	PROPN
ejpam-5572	680	15	,	,	PUNCT
ejpam-5572	680	16	and	and	CCONJ
ejpam-5572	680	17	s.	s.	PROPN
ejpam-5572	680	18	arjika	arjika	PROPN
ejpam-5572	680	19	.	.	PUNCT
ejpam-5572	681	1	fermatean	fermatean	PROPN
ejpam-5572	681	2	cubic	cubic	ADJ
ejpam-5572	681	3	fuzzy	fuzzy	ADJ
ejpam-5572	681	4	aggregation	aggregation	NOUN
ejpam-5572	681	5	operators	operator	NOUN
ejpam-5572	681	6	and	and	CCONJ
ejpam-5572	681	7	their	their	PRON
ejpam-5572	681	8	application	application	NOUN
ejpam-5572	681	9	in	in	ADP
ejpam-5572	681	10	multi	multi	ADJ
ejpam-5572	681	11	-	-	ADJ
ejpam-5572	681	12	attribute	attribute	NOUN
ejpam-5572	681	13	decision	decision	NOUN
ejpam-5572	681	14	-	-	PUNCT
ejpam-5572	681	15	making	make	VERB
ejpam-5572	681	16	problems	problem	NOUN
ejpam-5572	681	17	.	.	PUNCT
ejpam-5572	682	1	journal	journal	NOUN
ejpam-5572	682	2	of	of	ADP
ejpam-5572	682	3	function	function	NOUN
ejpam-5572	682	4	spaces	space	NOUN
ejpam-5572	682	5	,	,	PUNCT
ejpam-5572	682	6	2022(1	2022(1	NUM
ejpam-5572	682	7	)	)	PUNCT
ejpam-5572	682	8	,	,	PUNCT
ejpam-5572	682	9	2022	2022	NUM
ejpam-5572	682	10	.	.	PUNCT
ejpam-5572	683	1	[	[	X
ejpam-5572	683	2	32	32	NUM
ejpam-5572	683	3	]	]	PUNCT
ejpam-5572	683	4	a.	a.	NOUN
ejpam-5572	683	5	rosenfeld	rosenfeld	PROPN
ejpam-5572	683	6	.	.	PUNCT
ejpam-5572	684	1	fuzzy	fuzzy	ADJ
ejpam-5572	684	2	groups	group	NOUN
ejpam-5572	684	3	.	.	PUNCT
ejpam-5572	685	1	j.	j.	PROPN
ejpam-5572	685	2	math	math	PROPN
ejpam-5572	685	3	.	.	PUNCT
ejpam-5572	686	1	anal	anal	PROPN
ejpam-5572	686	2	.	.	PUNCT
ejpam-5572	687	1	appl	appl	PROPN
ejpam-5572	687	2	.	.	PROPN
ejpam-5572	687	3	,	,	PUNCT
ejpam-5572	687	4	35:512–517	35:512–517	PROPN
ejpam-5572	687	5	,	,	PUNCT
ejpam-5572	687	6	1971	1971	NUM
ejpam-5572	687	7	.	.	PUNCT
ejpam-5572	688	1	[	[	X
ejpam-5572	688	2	33	33	NUM
ejpam-5572	688	3	]	]	PUNCT
ejpam-5572	688	4	t.	t.	NOUN
ejpam-5572	688	5	senapti	senapti	PROPN
ejpam-5572	688	6	and	and	CCONJ
ejpam-5572	688	7	r.	r.	PROPN
ejpam-5572	688	8	yager	yager	PROPN
ejpam-5572	688	9	.	.	PUNCT
ejpam-5572	689	1	fermatean	fermatean	ADJ
ejpam-5572	689	2	fuzzy	fuzzy	ADJ
ejpam-5572	689	3	sets	set	NOUN
ejpam-5572	689	4	.	.	PUNCT
ejpam-5572	690	1	journal	journal	PROPN
ejpam-5572	690	2	of	of	ADP
ejpam-5572	690	3	ambient	ambient	ADJ
ejpam-5572	690	4	intelligence	intelligence	NOUN
ejpam-5572	690	5	and	and	CCONJ
ejpam-5572	690	6	humanized	humanize	VERB
ejpam-5572	690	7	computing	computing	NOUN
ejpam-5572	690	8	,	,	PUNCT
ejpam-5572	690	9	11(2):663–674	11(2):663–674	NUM
ejpam-5572	690	10	,	,	PUNCT
ejpam-5572	690	11	2020	2020	NUM
ejpam-5572	690	12	.	.	PUNCT
ejpam-5572	691	1	[	[	X
ejpam-5572	691	2	34	34	NUM
ejpam-5572	691	3	]	]	X
ejpam-5572	691	4	w.	w.	PROPN
ejpam-5572	691	5	shatanawi	shatanawi	PROPN
ejpam-5572	691	6	,	,	PUNCT
ejpam-5572	691	7	t.	t.	NOUN
ejpam-5572	691	8	qawasmeh	qawasmeh	NOUN
ejpam-5572	691	9	,	,	PUNCT
ejpam-5572	691	10	a.	a.	NOUN
ejpam-5572	691	11	bataihah	bataihah	PROPN
ejpam-5572	691	12	,	,	PUNCT
ejpam-5572	691	13	and	and	CCONJ
ejpam-5572	691	14	a.	a.	NOUN
ejpam-5572	691	15	tallafha	tallafha	NOUN
ejpam-5572	691	16	.	.	PUNCT
ejpam-5572	692	1	new	new	ADJ
ejpam-5572	692	2	contractions	contraction	NOUN
ejpam-5572	692	3	and	and	CCONJ
ejpam-5572	692	4	some	some	DET
ejpam-5572	692	5	fixed	fix	VERB
ejpam-5572	692	6	point	point	NOUN
ejpam-5572	692	7	results	result	NOUN
ejpam-5572	692	8	with	with	ADP
ejpam-5572	692	9	application	application	NOUN
ejpam-5572	692	10	based	base	VERB
ejpam-5572	692	11	on	on	ADP
ejpam-5572	692	12	extended	extended	ADJ
ejpam-5572	692	13	quasi	quasi	ADJ
ejpam-5572	692	14	b	b	NOUN
ejpam-5572	692	15	-	-	ADJ
ejpam-5572	692	16	metric	metric	ADJ
ejpam-5572	692	17	spaces	space	NOUN
ejpam-5572	692	18	.	.	PUNCT
ejpam-5572	693	1	e.a	e.a	PROPN
ejpam-5572	693	2	.	.	PROPN
ejpam-5572	693	3	abuhijleh	abuhijleh	PROPN
ejpam-5572	693	4	,	,	PUNCT
ejpam-5572	693	5	a.	a.	PROPN
ejpam-5572	693	6	alkouri	alkouri	PROPN
ejpam-5572	693	7	/	/	PROPN
ejpam-5572	693	8	eur	eur	PROPN
ejpam-5572	693	9	.	.	PUNCT
ejpam-5572	694	1	j.	j.	PROPN
ejpam-5572	694	2	pure	pure	PROPN
ejpam-5572	694	3	appl	appl	PROPN
ejpam-5572	694	4	.	.	PROPN
ejpam-5572	694	5	math	math	PROPN
ejpam-5572	694	6	,	,	PUNCT
ejpam-5572	694	7	18	18	NUM
ejpam-5572	694	8	(	(	PUNCT
ejpam-5572	694	9	1	1	NUM
ejpam-5572	694	10	)	)	PUNCT
ejpam-5572	694	11	(	(	PUNCT
ejpam-5572	694	12	2025	2025	NUM
ejpam-5572	694	13	)	)	PUNCT
ejpam-5572	694	14	,	,	PUNCT
ejpam-5572	694	15	5572	5572	NUM
ejpam-5572	694	16	19	19	NUM
ejpam-5572	694	17	of	of	ADP
ejpam-5572	694	18	19	19	NUM
ejpam-5572	694	19	upb	upb	ADJ
ejpam-5572	694	20	scientific	scientific	ADJ
ejpam-5572	694	21	bulletin	bulletin	NOUN
ejpam-5572	694	22	,	,	PUNCT
ejpam-5572	694	23	series	series	NOUN
ejpam-5572	694	24	a	a	NOUN
ejpam-5572	694	25	,	,	PUNCT
ejpam-5572	694	26	83(2):39–48	83(2):39–48	NUM
ejpam-5572	694	27	,	,	PUNCT
ejpam-5572	694	28	2021	2021	NUM
ejpam-5572	694	29	.	.	PUNCT
ejpam-5572	695	1	[	[	X
ejpam-5572	695	2	35	35	NUM
ejpam-5572	695	3	]	]	PUNCT
ejpam-5572	695	4	i.	i.	PROPN
ejpam-5572	695	5	silambarasan	silambarasan	PROPN
ejpam-5572	695	6	.	.	PUNCT
ejpam-5572	696	1	fermatean	fermatean	PROPN
ejpam-5572	696	2	fuzzy	fuzzy	ADJ
ejpam-5572	696	3	subgroups	subgroup	NOUN
ejpam-5572	696	4	.	.	PUNCT
ejpam-5572	697	1	j.	j.	PROPN
ejpam-5572	697	2	int	int	PROPN
ejpam-5572	697	3	.	.	PUNCT
ejpam-5572	698	1	math	math	NOUN
ejpam-5572	698	2	.	.	PUNCT
ejpam-5572	699	1	virtual	virtual	ADJ
ejpam-5572	699	2	inst	inst	PROPN
ejpam-5572	699	3	.	.	PROPN
ejpam-5572	699	4	,	,	PUNCT
ejpam-5572	699	5	11(1):1–16	11(1):1–16	PROPN
ejpam-5572	699	6	,	,	PUNCT
ejpam-5572	699	7	2021	2021	NUM
ejpam-5572	699	8	.	.	PUNCT
ejpam-5572	700	1	[	[	X
ejpam-5572	700	2	36	36	NUM
ejpam-5572	700	3	]	]	PUNCT
ejpam-5572	700	4	k.	k.	PROPN
ejpam-5572	700	5	ullah	ullah	PROPN
ejpam-5572	700	6	,	,	PUNCT
ejpam-5572	700	7	t.	t.	PROPN
ejpam-5572	700	8	mahmood	mahmood	PROPN
ejpam-5572	700	9	,	,	PUNCT
ejpam-5572	700	10	z.	z.	PROPN
ejpam-5572	700	11	ali	ali	PROPN
ejpam-5572	700	12	,	,	PUNCT
ejpam-5572	700	13	and	and	CCONJ
ejpam-5572	700	14	n.	n.	PROPN
ejpam-5572	700	15	jan	jan	PROPN
ejpam-5572	700	16	.	.	PROPN
ejpam-5572	701	1	on	on	ADP
ejpam-5572	701	2	some	some	DET
ejpam-5572	701	3	distance	distance	NOUN
ejpam-5572	701	4	measures	measure	NOUN
ejpam-5572	701	5	of	of	ADP
ejpam-5572	701	6	complex	complex	ADJ
ejpam-5572	701	7	pythagorean	pythagorean	ADJ
ejpam-5572	701	8	fuzzy	fuzzy	ADJ
ejpam-5572	701	9	sets	set	NOUN
ejpam-5572	701	10	and	and	CCONJ
ejpam-5572	701	11	their	their	PRON
ejpam-5572	701	12	applications	application	NOUN
ejpam-5572	701	13	in	in	ADP
ejpam-5572	701	14	pattern	pattern	NOUN
ejpam-5572	701	15	recognition	recognition	NOUN
ejpam-5572	701	16	.	.	PUNCT
ejpam-5572	702	1	complex	complex	ADJ
ejpam-5572	702	2	and	and	CCONJ
ejpam-5572	702	3	intelligent	intelligent	ADJ
ejpam-5572	702	4	systems	system	NOUN
ejpam-5572	702	5	,	,	PUNCT
ejpam-5572	702	6	6:15–27	6:15–27	NUM
ejpam-5572	702	7	,	,	PUNCT
ejpam-5572	702	8	2020	2020	NUM
ejpam-5572	702	9	.	.	PUNCT
ejpam-5572	703	1	[	[	X
ejpam-5572	703	2	37	37	NUM
ejpam-5572	703	3	]	]	X
ejpam-5572	703	4	y.	y.	PROPN
ejpam-5572	703	5	wang	wang	PROPN
ejpam-5572	703	6	,	,	PUNCT
ejpam-5572	703	7	x.	x.	PROPN
ejpam-5572	703	8	ma	ma	PROPN
ejpam-5572	703	9	,	,	PUNCT
ejpam-5572	703	10	h.	h.	PROPN
ejpam-5572	703	11	qin	qin	PROPN
ejpam-5572	703	12	,	,	PUNCT
ejpam-5572	703	13	h.	h.	PROPN
ejpam-5572	703	14	sun	sun	PROPN
ejpam-5572	703	15	,	,	PUNCT
ejpam-5572	703	16	and	and	CCONJ
ejpam-5572	703	17	w.	w.	PROPN
ejpam-5572	703	18	wei	wei	PROPN
ejpam-5572	703	19	.	.	PUNCT
ejpam-5572	704	1	hesitant	hesitant	ADJ
ejpam-5572	704	2	fermatean	fermatean	PROPN
ejpam-5572	704	3	fuzzy	fuzzy	ADJ
ejpam-5572	704	4	bonferroni	bonferroni	NOUN
ejpam-5572	704	5	mean	mean	VERB
ejpam-5572	704	6	operators	operator	NOUN
ejpam-5572	704	7	for	for	ADP
ejpam-5572	704	8	multi	multi	ADJ
ejpam-5572	704	9	-	-	ADJ
ejpam-5572	704	10	attribute	attribute	NOUN
ejpam-5572	704	11	decision	decision	NOUN
ejpam-5572	704	12	-	-	PUNCT
ejpam-5572	704	13	making	making	NOUN
ejpam-5572	704	14	.	.	PUNCT
ejpam-5572	705	1	complex	complex	ADJ
ejpam-5572	705	2	and	and	CCONJ
ejpam-5572	705	3	intelligent	intelligent	ADJ
ejpam-5572	705	4	systems	system	NOUN
ejpam-5572	705	5	,	,	PUNCT
ejpam-5572	705	6	10:1425–1457	10:1425–1457	NUM
ejpam-5572	705	7	,	,	PUNCT
ejpam-5572	705	8	2024	2024	NUM
ejpam-5572	705	9	.	.	PUNCT
ejpam-5572	706	1	[	[	X
ejpam-5572	706	2	38	38	NUM
ejpam-5572	706	3	]	]	PUNCT
ejpam-5572	706	4	r.	r.	PROPN
ejpam-5572	706	5	yager	yager	PROPN
ejpam-5572	706	6	.	.	PUNCT
ejpam-5572	707	1	pythagorean	pythagorean	PROPN
ejpam-5572	707	2	fuzzy	fuzzy	ADJ
ejpam-5572	707	3	subsets	subset	NOUN
ejpam-5572	707	4	.	.	PUNCT
ejpam-5572	708	1	in	in	ADP
ejpam-5572	708	2	2013	2013	NUM
ejpam-5572	708	3	joint	joint	ADJ
ejpam-5572	708	4	ifsa	ifsa	PROPN
ejpam-5572	708	5	world	world	PROPN
ejpam-5572	708	6	congress	congress	PROPN
ejpam-5572	708	7	and	and	CCONJ
ejpam-5572	708	8	nafips	nafip	NOUN
ejpam-5572	708	9	annual	annual	ADJ
ejpam-5572	708	10	meeting	meeting	NOUN
ejpam-5572	708	11	(	(	PUNCT
ejpam-5572	708	12	ifsa	ifsa	PROPN
ejpam-5572	708	13	/	/	SYM
ejpam-5572	708	14	nafips	nafip	NOUN
ejpam-5572	708	15	)	)	PUNCT
ejpam-5572	708	16	,	,	PUNCT
ejpam-5572	708	17	pages	page	NOUN
ejpam-5572	708	18	57–61	57–61	NUM
ejpam-5572	708	19	,	,	PUNCT
ejpam-5572	708	20	2013	2013	NUM
ejpam-5572	708	21	.	.	PUNCT
ejpam-5572	709	1	[	[	X
ejpam-5572	709	2	39	39	NUM
ejpam-5572	709	3	]	]	PUNCT
ejpam-5572	709	4	r.	r.	PROPN
ejpam-5572	709	5	yager	yager	PROPN
ejpam-5572	709	6	.	.	PUNCT
ejpam-5572	710	1	generalized	generalized	ADJ
ejpam-5572	710	2	orthopair	orthopair	ADJ
ejpam-5572	710	3	fuzzy	fuzzy	ADJ
ejpam-5572	710	4	sets	set	NOUN
ejpam-5572	710	5	.	.	PUNCT
ejpam-5572	711	1	ieee	ieee	PROPN
ejpam-5572	711	2	trans	trans	PROPN
ejpam-5572	711	3	.	.	PUNCT
ejpam-5572	711	4	fuzzy	fuzzy	ADJ
ejpam-5572	711	5	syst	syst	PROPN
ejpam-5572	711	6	.	.	PUNCT
ejpam-5572	711	7	,	,	PUNCT
ejpam-5572	711	8	25(5):1222	25(5):1222	NUM
ejpam-5572	711	9	–	–	PUNCT
ejpam-5572	711	10	1230	1230	NUM
ejpam-5572	711	11	,	,	PUNCT
ejpam-5572	711	12	2017	2017	NUM
ejpam-5572	711	13	.	.	PUNCT
ejpam-5572	712	1	[	[	X
ejpam-5572	712	2	40	40	NUM
ejpam-5572	712	3	]	]	PUNCT
ejpam-5572	712	4	l.	l.	PROPN
ejpam-5572	712	5	zadeh	zadeh	PROPN
ejpam-5572	712	6	.	.	PUNCT
ejpam-5572	712	7	fuzzy	fuzzy	ADJ
ejpam-5572	712	8	sets	set	NOUN
ejpam-5572	712	9	.	.	PUNCT
ejpam-5572	713	1	inform	inform	NOUN
ejpam-5572	713	2	.	.	PUNCT
ejpam-5572	714	1	and	and	CCONJ
ejpam-5572	714	2	control	control	NOUN
ejpam-5572	714	3	,	,	PUNCT
ejpam-5572	714	4	8:338–353	8:338–353	NUM
ejpam-5572	714	5	,	,	PUNCT
ejpam-5572	714	6	1965	1965	NUM
ejpam-5572	714	7	.	.	PUNCT