id	sid	tid	token	lemma	pos
ejpam-6261	1	1	european	european	PROPN
ejpam-6261	1	2	journal	journal	PROPN
ejpam-6261	1	3	of	of	ADP
ejpam-6261	1	4	pure	pure	ADJ
ejpam-6261	1	5	and	and	CCONJ
ejpam-6261	1	6	applied	applied	ADJ
ejpam-6261	1	7	mathematics	mathematic	NOUN
ejpam-6261	1	8	2025	2025	NUM
ejpam-6261	1	9	,	,	PUNCT
ejpam-6261	1	10	vol	vol	NOUN
ejpam-6261	1	11	.	.	PROPN
ejpam-6261	1	12	18	18	NUM
ejpam-6261	1	13	,	,	PUNCT
ejpam-6261	1	14	issue	issue	NOUN
ejpam-6261	1	15	4	4	NUM
ejpam-6261	1	16	,	,	PUNCT
ejpam-6261	1	17	article	article	NOUN
ejpam-6261	1	18	number	number	NOUN
ejpam-6261	1	19	6261	6261	NUM
ejpam-6261	1	20	issn	issn	PROPN
ejpam-6261	1	21	1307	1307	NUM
ejpam-6261	1	22	-	-	SYM
ejpam-6261	1	23	5543	5543	NUM
ejpam-6261	1	24	–	–	PUNCT
ejpam-6261	2	1	ejpam.com	ejpam.com	X
ejpam-6261	2	2	published	publish	VERB
ejpam-6261	2	3	by	by	ADP
ejpam-6261	2	4	new	new	PROPN
ejpam-6261	2	5	york	york	PROPN
ejpam-6261	2	6	business	business	PROPN
ejpam-6261	2	7	global	global	ADJ
ejpam-6261	2	8	direct	direct	ADJ
ejpam-6261	2	9	product	product	NOUN
ejpam-6261	2	10	of	of	ADP
ejpam-6261	2	11	complex	complex	ADJ
ejpam-6261	2	12	neutrosophic	neutrosophic	ADJ
ejpam-6261	2	13	subrings	subring	NOUN
ejpam-6261	2	14	muhammad	muhammad	PROPN
ejpam-6261	2	15	haris	haris	PROPN
ejpam-6261	2	16	mateen1	mateen1	PROPN
ejpam-6261	2	17	,	,	PUNCT
ejpam-6261	2	18	sarka	sarka	PROPN
ejpam-6261	2	19	hoskova	hoskova	PROPN
ejpam-6261	2	20	-	-	PUNCT
ejpam-6261	2	21	mayerova2,∗	mayerova2,∗	PROPN
ejpam-6261	2	22	,	,	PUNCT
ejpam-6261	2	23	kholood	kholood	NOUN
ejpam-6261	2	24	alnefaie3	alnefaie3	NOUN
ejpam-6261	2	25	,	,	PUNCT
ejpam-6261	2	26	florentin	florentin	NOUN
ejpam-6261	2	27	smarandache4	smarandache4	NOUN
ejpam-6261	2	28	1	1	NUM
ejpam-6261	2	29	school	school	NOUN
ejpam-6261	2	30	of	of	ADP
ejpam-6261	2	31	mathematics	mathematic	NOUN
ejpam-6261	2	32	,	,	PUNCT
ejpam-6261	2	33	minhaj	minhaj	PROPN
ejpam-6261	2	34	university	university	PROPN
ejpam-6261	2	35	lahore	lahore	NOUN
ejpam-6261	2	36	,	,	PUNCT
ejpam-6261	2	37	pakistan	pakistan	PROPN
ejpam-6261	2	38	2	2	NUM
ejpam-6261	2	39	department	department	NOUN
ejpam-6261	2	40	of	of	ADP
ejpam-6261	2	41	mathematics	mathematics	PROPN
ejpam-6261	2	42	and	and	CCONJ
ejpam-6261	2	43	physics	physics	PROPN
ejpam-6261	2	44	,	,	PUNCT
ejpam-6261	2	45	university	university	NOUN
ejpam-6261	2	46	of	of	ADP
ejpam-6261	2	47	defence	defence	NOUN
ejpam-6261	2	48	,	,	PUNCT
ejpam-6261	2	49	66210	66210	NUM
ejpam-6261	2	50	brno	brno	NOUN
ejpam-6261	2	51	,	,	PUNCT
ejpam-6261	2	52	czech	czech	PROPN
ejpam-6261	2	53	republic	republic	NOUN
ejpam-6261	2	54	3	3	NUM
ejpam-6261	2	55	department	department	NOUN
ejpam-6261	2	56	of	of	ADP
ejpam-6261	2	57	mathematics	mathematics	PROPN
ejpam-6261	2	58	,	,	PUNCT
ejpam-6261	2	59	college	college	NOUN
ejpam-6261	2	60	of	of	ADP
ejpam-6261	2	61	science	science	PROPN
ejpam-6261	2	62	,	,	PUNCT
ejpam-6261	2	63	taibah	taibah	PROPN
ejpam-6261	2	64	university	university	PROPN
ejpam-6261	2	65	,	,	PUNCT
ejpam-6261	2	66	madinah	madinah	PROPN
ejpam-6261	2	67	42353	42353	NUM
ejpam-6261	2	68	,	,	PUNCT
ejpam-6261	2	69	saudi	saudi	PROPN
ejpam-6261	2	70	arabia	arabia	PROPN
ejpam-6261	2	71	4	4	NUM
ejpam-6261	2	72	university	university	NOUN
ejpam-6261	2	73	of	of	ADP
ejpam-6261	2	74	new	new	PROPN
ejpam-6261	2	75	mexico	mexico	PROPN
ejpam-6261	2	76	,	,	PUNCT
ejpam-6261	2	77	705	705	NUM
ejpam-6261	2	78	gurley	gurley	PROPN
ejpam-6261	2	79	ave	ave	PROPN
ejpam-6261	2	80	.	.	PROPN
ejpam-6261	2	81	,	,	PUNCT
ejpam-6261	2	82	gallup	gallup	PROPN
ejpam-6261	2	83	,	,	PUNCT
ejpam-6261	2	84	nm	nm	PROPN
ejpam-6261	2	85	87301	87301	NUM
ejpam-6261	2	86	,	,	PUNCT
ejpam-6261	2	87	usa	usa	PROPN
ejpam-6261	2	88	abstract	abstract	PROPN
ejpam-6261	2	89	.	.	PUNCT
ejpam-6261	3	1	the	the	DET
ejpam-6261	3	2	complex	complex	ADJ
ejpam-6261	3	3	neutrosophic	neutrosophic	ADJ
ejpam-6261	3	4	set	set	NOUN
ejpam-6261	3	5	is	be	AUX
ejpam-6261	3	6	a	a	DET
ejpam-6261	3	7	generalization	generalization	NOUN
ejpam-6261	3	8	of	of	ADP
ejpam-6261	3	9	the	the	DET
ejpam-6261	3	10	neutrosophic	neutrosophic	ADJ
ejpam-6261	3	11	set	set	NOUN
ejpam-6261	3	12	with	with	ADP
ejpam-6261	3	13	the	the	DET
ejpam-6261	3	14	addition	addition	NOUN
ejpam-6261	3	15	of	of	ADP
ejpam-6261	3	16	three	three	NUM
ejpam-6261	3	17	phase	phase	NOUN
ejpam-6261	3	18	terms	term	NOUN
ejpam-6261	3	19	.	.	PUNCT
ejpam-6261	4	1	the	the	DET
ejpam-6261	4	2	complex	complex	ADJ
ejpam-6261	4	3	neutrosophic	neutrosophic	ADJ
ejpam-6261	4	4	set	set	NOUN
ejpam-6261	4	5	deals	deal	NOUN
ejpam-6261	4	6	with	with	ADP
ejpam-6261	4	7	periodic	periodic	ADJ
ejpam-6261	4	8	data	datum	NOUN
ejpam-6261	4	9	that	that	PRON
ejpam-6261	4	10	contains	contain	VERB
ejpam-6261	4	11	uncertainty	uncertainty	NOUN
ejpam-6261	4	12	,	,	PUNCT
ejpam-6261	4	13	indeterminacy	indeterminacy	NOUN
ejpam-6261	4	14	,	,	PUNCT
ejpam-6261	4	15	and	and	CCONJ
ejpam-6261	4	16	falsity	falsity	NOUN
ejpam-6261	4	17	.	.	PUNCT
ejpam-6261	5	1	the	the	DET
ejpam-6261	5	2	complex	complex	ADJ
ejpam-6261	5	3	neutrosophic	neutrosophic	ADJ
ejpam-6261	5	4	set	set	NOUN
ejpam-6261	5	5	has	have	VERB
ejpam-6261	5	6	a	a	DET
ejpam-6261	5	7	variety	variety	NOUN
ejpam-6261	5	8	of	of	ADP
ejpam-6261	5	9	applications	application	NOUN
ejpam-6261	5	10	,	,	PUNCT
ejpam-6261	5	11	such	such	ADJ
ejpam-6261	5	12	as	as	ADP
ejpam-6261	5	13	signal	signal	ADJ
ejpam-6261	5	14	processing	processing	NOUN
ejpam-6261	5	15	,	,	PUNCT
ejpam-6261	5	16	hospital	hospital	NOUN
ejpam-6261	5	17	infrastructure	infrastructure	NOUN
ejpam-6261	5	18	design	design	NOUN
ejpam-6261	5	19	,	,	PUNCT
ejpam-6261	5	20	medical	medical	ADJ
ejpam-6261	5	21	image	image	NOUN
ejpam-6261	5	22	denoising	denoising	NOUN
ejpam-6261	5	23	,	,	PUNCT
ejpam-6261	5	24	segmentation	segmentation	NOUN
ejpam-6261	5	25	distance	distance	NOUN
ejpam-6261	5	26	measurement	measurement	NOUN
ejpam-6261	5	27	,	,	PUNCT
ejpam-6261	5	28	and	and	CCONJ
ejpam-6261	5	29	the	the	DET
ejpam-6261	5	30	game	game	NOUN
ejpam-6261	5	31	of	of	ADP
ejpam-6261	5	32	loser	loser	NOUN
ejpam-6261	5	33	,	,	PUNCT
ejpam-6261	5	34	neutral	neutral	ADJ
ejpam-6261	5	35	,	,	PUNCT
ejpam-6261	5	36	and	and	CCONJ
ejpam-6261	5	37	winner	winner	NOUN
ejpam-6261	5	38	.	.	PUNCT
ejpam-6261	6	1	this	this	DET
ejpam-6261	6	2	article	article	NOUN
ejpam-6261	6	3	presents	present	VERB
ejpam-6261	6	4	a	a	DET
ejpam-6261	6	5	novel	novel	ADJ
ejpam-6261	6	6	concept	concept	NOUN
ejpam-6261	6	7	for	for	ADP
ejpam-6261	6	8	complex	complex	ADJ
ejpam-6261	6	9	neutrosophic	neutrosophic	ADJ
ejpam-6261	6	10	subrings	subring	NOUN
ejpam-6261	6	11	and	and	CCONJ
ejpam-6261	6	12	illustrates	illustrate	VERB
ejpam-6261	6	13	how	how	SCONJ
ejpam-6261	6	14	these	these	DET
ejpam-6261	6	15	subrings	subring	NOUN
ejpam-6261	6	16	can	can	AUX
ejpam-6261	6	17	generate	generate	VERB
ejpam-6261	6	18	two	two	NUM
ejpam-6261	6	19	other	other	ADJ
ejpam-6261	6	20	neutrosophic	neutrosophic	ADJ
ejpam-6261	6	21	subrings	subring	NOUN
ejpam-6261	6	22	.	.	PUNCT
ejpam-6261	7	1	additionally	additionally	ADV
ejpam-6261	7	2	,	,	PUNCT
ejpam-6261	7	3	we	we	PRON
ejpam-6261	7	4	prove	prove	VERB
ejpam-6261	7	5	that	that	SCONJ
ejpam-6261	7	6	the	the	DET
ejpam-6261	7	7	intersection	intersection	NOUN
ejpam-6261	7	8	of	of	ADP
ejpam-6261	7	9	two	two	NUM
ejpam-6261	7	10	neutrosophic	neutrosophic	ADJ
ejpam-6261	7	11	subrings	subring	NOUN
ejpam-6261	7	12	is	be	AUX
ejpam-6261	7	13	a	a	DET
ejpam-6261	7	14	neutrosophic	neutrosophic	ADJ
ejpam-6261	7	15	subring	subring	NOUN
ejpam-6261	7	16	.	.	PUNCT
ejpam-6261	8	1	we	we	PRON
ejpam-6261	8	2	expand	expand	VERB
ejpam-6261	8	3	this	this	DET
ejpam-6261	8	4	idea	idea	NOUN
ejpam-6261	8	5	to	to	PART
ejpam-6261	8	6	talk	talk	VERB
ejpam-6261	8	7	about	about	ADP
ejpam-6261	8	8	the	the	DET
ejpam-6261	8	9	abstraction	abstraction	NOUN
ejpam-6261	8	10	of	of	ADP
ejpam-6261	8	11	level	level	NOUN
ejpam-6261	8	12	subsets	subset	NOUN
ejpam-6261	8	13	of	of	ADP
ejpam-6261	8	14	complex	complex	ADJ
ejpam-6261	8	15	neutrosophic	neutrosophic	ADJ
ejpam-6261	8	16	sets	set	NOUN
ejpam-6261	8	17	and	and	CCONJ
ejpam-6261	8	18	look	look	VERB
ejpam-6261	8	19	into	into	ADP
ejpam-6261	8	20	the	the	DET
ejpam-6261	8	21	basic	basic	ADJ
ejpam-6261	8	22	algebraic	algebraic	ADJ
ejpam-6261	8	23	properties	property	NOUN
ejpam-6261	8	24	of	of	ADP
ejpam-6261	8	25	this	this	DET
ejpam-6261	8	26	event	event	NOUN
ejpam-6261	8	27	.	.	PUNCT
ejpam-6261	9	1	we	we	PRON
ejpam-6261	9	2	prove	prove	VERB
ejpam-6261	9	3	that	that	SCONJ
ejpam-6261	9	4	the	the	DET
ejpam-6261	9	5	level	level	NOUN
ejpam-6261	9	6	subset	subset	NOUN
ejpam-6261	9	7	of	of	ADP
ejpam-6261	9	8	the	the	DET
ejpam-6261	9	9	complex	complex	ADJ
ejpam-6261	9	10	neutrosophic	neutrosophic	ADJ
ejpam-6261	9	11	subring	subring	NOUN
ejpam-6261	9	12	is	be	AUX
ejpam-6261	9	13	a	a	DET
ejpam-6261	9	14	subring	subring	NOUN
ejpam-6261	9	15	.	.	PUNCT
ejpam-6261	10	1	moreover	moreover	ADV
ejpam-6261	10	2	,	,	PUNCT
ejpam-6261	10	3	we	we	PRON
ejpam-6261	10	4	demonstrate	demonstrate	VERB
ejpam-6261	10	5	that	that	SCONJ
ejpam-6261	10	6	the	the	DET
ejpam-6261	10	7	product	product	NOUN
ejpam-6261	10	8	of	of	ADP
ejpam-6261	10	9	two	two	NUM
ejpam-6261	10	10	complex	complex	ADJ
ejpam-6261	10	11	neutrosophic	neutrosophic	ADJ
ejpam-6261	10	12	subrings	subring	NOUN
ejpam-6261	10	13	is	be	AUX
ejpam-6261	10	14	also	also	ADV
ejpam-6261	10	15	a	a	DET
ejpam-6261	10	16	complex	complex	ADJ
ejpam-6261	10	17	neutrosophic	neutrosophic	ADJ
ejpam-6261	10	18	subring	subring	NOUN
ejpam-6261	10	19	and	and	CCONJ
ejpam-6261	10	20	explore	explore	VERB
ejpam-6261	10	21	some	some	DET
ejpam-6261	10	22	novel	novel	ADJ
ejpam-6261	10	23	consequences	consequence	NOUN
ejpam-6261	10	24	about	about	ADP
ejpam-6261	10	25	the	the	DET
ejpam-6261	10	26	direct	direct	ADJ
ejpam-6261	10	27	product	product	NOUN
ejpam-6261	10	28	of	of	ADP
ejpam-6261	10	29	complex	complex	ADJ
ejpam-6261	10	30	neutrosophic	neutrosophic	ADJ
ejpam-6261	10	31	subrings	subring	NOUN
ejpam-6261	10	32	.	.	PUNCT
ejpam-6261	11	1	our	our	PRON
ejpam-6261	11	2	findings	finding	NOUN
ejpam-6261	11	3	generalize	generalize	VERB
ejpam-6261	11	4	and	and	CCONJ
ejpam-6261	11	5	extend	extend	VERB
ejpam-6261	11	6	the	the	DET
ejpam-6261	11	7	existing	exist	VERB
ejpam-6261	11	8	ring	ring	NOUN
ejpam-6261	11	9	theory	theory	NOUN
ejpam-6261	11	10	results	result	VERB
ejpam-6261	11	11	within	within	ADP
ejpam-6261	11	12	a	a	DET
ejpam-6261	11	13	complex	complex	ADJ
ejpam-6261	11	14	neutrosophic	neutrosophic	ADJ
ejpam-6261	11	15	framework	framework	NOUN
ejpam-6261	11	16	.	.	PUNCT
ejpam-6261	12	1	2020	2020	NUM
ejpam-6261	12	2	mathematics	mathematic	NOUN
ejpam-6261	12	3	subject	subject	NOUN
ejpam-6261	12	4	classifications	classification	NOUN
ejpam-6261	12	5	:	:	PUNCT
ejpam-6261	12	6	13e15	13e15	NUM
ejpam-6261	12	7	,	,	PUNCT
ejpam-6261	12	8	08a72	08a72	NUM
ejpam-6261	12	9	,	,	PUNCT
ejpam-6261	12	10	03e72	03e72	X
ejpam-6261	12	11	key	key	ADJ
ejpam-6261	12	12	words	word	NOUN
ejpam-6261	12	13	and	and	CCONJ
ejpam-6261	12	14	phrases	phrase	NOUN
ejpam-6261	12	15	:	:	PUNCT
ejpam-6261	12	16	complex	complex	ADJ
ejpam-6261	12	17	neutrosophic	neutrosophic	ADJ
ejpam-6261	12	18	subring	subring	NOUN
ejpam-6261	12	19	,	,	PUNCT
ejpam-6261	12	20	level	level	NOUN
ejpam-6261	12	21	subsets	subset	NOUN
ejpam-6261	12	22	of	of	ADP
ejpam-6261	12	23	complex	complex	ADJ
ejpam-6261	12	24	neutrosophic	neutrosophic	ADJ
ejpam-6261	12	25	subrings	subring	NOUN
ejpam-6261	12	26	,	,	PUNCT
ejpam-6261	12	27	product	product	NOUN
ejpam-6261	12	28	of	of	ADP
ejpam-6261	12	29	complex	complex	ADJ
ejpam-6261	12	30	neutrosophic	neutrosophic	ADJ
ejpam-6261	12	31	subrings	subring	NOUN
ejpam-6261	12	32	1	1	NUM
ejpam-6261	12	33	.	.	PUNCT
ejpam-6261	12	34	introduction	introduction	NOUN
ejpam-6261	12	35	in	in	ADP
ejpam-6261	12	36	the	the	DET
ejpam-6261	12	37	beginning	beginning	NOUN
ejpam-6261	12	38	,	,	PUNCT
ejpam-6261	12	39	initiatives	initiative	NOUN
ejpam-6261	12	40	to	to	PART
ejpam-6261	12	41	demonstrate	demonstrate	VERB
ejpam-6261	12	42	fermat	fermat	PROPN
ejpam-6261	12	43	’s	’s	PART
ejpam-6261	12	44	last	last	ADJ
ejpam-6261	12	45	theorem	theorem	NOUN
ejpam-6261	12	46	gave	give	VERB
ejpam-6261	12	47	rise	rise	NOUN
ejpam-6261	12	48	to	to	ADP
ejpam-6261	12	49	the	the	DET
ejpam-6261	12	50	idea	idea	NOUN
ejpam-6261	12	51	of	of	ADP
ejpam-6261	12	52	a	a	DET
ejpam-6261	12	53	ring	ring	NOUN
ejpam-6261	12	54	in	in	ADP
ejpam-6261	12	55	the	the	DET
ejpam-6261	12	56	1880s	1880	NOUN
ejpam-6261	12	57	,	,	PUNCT
ejpam-6261	12	58	commencing	commence	VERB
ejpam-6261	12	59	with	with	ADP
ejpam-6261	12	60	dedekind	dedekind	NOUN
ejpam-6261	12	61	[	[	X
ejpam-6261	12	62	1	1	NUM
ejpam-6261	12	63	]	]	PUNCT
ejpam-6261	12	64	.	.	PUNCT
ejpam-6261	13	1	in	in	ADP
ejpam-6261	13	2	the	the	DET
ejpam-6261	13	3	1920s	1920s	NUM
ejpam-6261	13	4	,	,	PUNCT
ejpam-6261	13	5	noether	noether	ADJ
ejpam-6261	13	6	and	and	CCONJ
ejpam-6261	13	7	krull	krull	PROPN
ejpam-6261	13	8	[	[	X
ejpam-6261	13	9	2	2	NUM
ejpam-6261	13	10	]	]	PUNCT
ejpam-6261	13	11	∗corresponding	∗corresponde	VERB
ejpam-6261	13	12	author	author	NOUN
ejpam-6261	13	13	.	.	PUNCT
ejpam-6261	14	1	doi	doi	NOUN
ejpam-6261	14	2	:	:	PUNCT
ejpam-6261	14	3	https://doi.org/10.29020/nybg.ejpam.v18i4.6261	https://doi.org/10.29020/nybg.ejpam.v18i4.6261	NOUN
ejpam-6261	14	4	email	email	NOUN
ejpam-6261	14	5	addresses	address	NOUN
ejpam-6261	14	6	:	:	PUNCT
ejpam-6261	14	7	harism.math@gmail.com	harism.math@gmail.com	X
ejpam-6261	14	8	(	(	PUNCT
ejpam-6261	14	9	m.	m.	PROPN
ejpam-6261	14	10	h.	h.	PROPN
ejpam-6261	14	11	mateen	mateen	PROPN
ejpam-6261	14	12	)	)	PUNCT
ejpam-6261	14	13	,	,	PUNCT
ejpam-6261	14	14	sarka.mayerova@unob.cz	sarka.mayerova@unob.cz	NOUN
ejpam-6261	14	15	(	(	PUNCT
ejpam-6261	14	16	s.	s.	PROPN
ejpam-6261	14	17	hoskova	hoskova	PROPN
ejpam-6261	14	18	-	-	PUNCT
ejpam-6261	14	19	mayerova	mayerova	X
ejpam-6261	14	20	)	)	PUNCT
ejpam-6261	14	21	,	,	PUNCT
ejpam-6261	14	22	knefaie@taibahu.edu.sa	knefaie@taibahu.edu.sa	PROPN
ejpam-6261	14	23	(	(	PUNCT
ejpam-6261	14	24	k.	k.	PROPN
ejpam-6261	14	25	alnefaie	alnefaie	PROPN
ejpam-6261	14	26	)	)	PUNCT
ejpam-6261	14	27	,	,	PUNCT
ejpam-6261	14	28	smarand@unm.edu	smarand@unm.edu	PROPN
ejpam-6261	14	29	(	(	PUNCT
ejpam-6261	14	30	f.	f.	PROPN
ejpam-6261	14	31	smarandache	smarandache	PROPN
ejpam-6261	14	32	)	)	PUNCT
ejpam-6261	14	33	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6261	15	1	1	1	NUM
ejpam-6261	15	2	copyright	copyright	NOUN
ejpam-6261	15	3	:	:	PUNCT
ejpam-6261	15	4	©	©	PROPN
ejpam-6261	15	5	2025	2025	NUM
ejpam-6261	15	6	the	the	DET
ejpam-6261	15	7	author(s	author(s	NOUN
ejpam-6261	15	8	)	)	PUNCT
ejpam-6261	15	9	.	.	PUNCT
ejpam-6261	16	1	(	(	PUNCT
ejpam-6261	16	2	cc	cc	NOUN
ejpam-6261	16	3	by	by	ADP
ejpam-6261	16	4	-	-	PUNCT
ejpam-6261	16	5	nc	nc	PROPN
ejpam-6261	16	6	4.0	4.0	NUM
ejpam-6261	16	7	)	)	PUNCT
ejpam-6261	16	8	m.	m.	NOUN
ejpam-6261	16	9	h.	h.	PROPN
ejpam-6261	16	10	mateen	mateen	PROPN
ejpam-6261	16	11	et	et	PROPN
ejpam-6261	16	12	al	al	PROPN
ejpam-6261	16	13	.	.	PUNCT
ejpam-6261	16	14	/	/	SYM
ejpam-6261	16	15	eur	eur	PROPN
ejpam-6261	16	16	.	.	PUNCT
ejpam-6261	17	1	j.	j.	PROPN
ejpam-6261	17	2	pure	pure	PROPN
ejpam-6261	17	3	appl	appl	PROPN
ejpam-6261	17	4	.	.	PROPN
ejpam-6261	17	5	math	math	PROPN
ejpam-6261	17	6	,	,	PUNCT
ejpam-6261	17	7	18	18	NUM
ejpam-6261	17	8	(	(	PUNCT
ejpam-6261	17	9	4	4	NUM
ejpam-6261	17	10	)	)	PUNCT
ejpam-6261	17	11	(	(	PUNCT
ejpam-6261	17	12	2025	2025	NUM
ejpam-6261	17	13	)	)	PUNCT
ejpam-6261	17	14	,	,	PUNCT
ejpam-6261	17	15	6261	6261	NUM
ejpam-6261	17	16	2	2	NUM
ejpam-6261	17	17	of	of	ADP
ejpam-6261	17	18	23	23	NUM
ejpam-6261	17	19	generalized	generalized	ADJ
ejpam-6261	17	20	and	and	CCONJ
ejpam-6261	17	21	firmly	firmly	ADV
ejpam-6261	17	22	established	establish	VERB
ejpam-6261	17	23	the	the	DET
ejpam-6261	17	24	idea	idea	NOUN
ejpam-6261	17	25	of	of	ADP
ejpam-6261	17	26	a	a	DET
ejpam-6261	17	27	ring	ring	NOUN
ejpam-6261	17	28	after	after	ADP
ejpam-6261	17	29	contributions	contribution	NOUN
ejpam-6261	17	30	from	from	ADP
ejpam-6261	17	31	other	other	ADJ
ejpam-6261	17	32	domains	domain	NOUN
ejpam-6261	17	33	,	,	PUNCT
ejpam-6261	17	34	especially	especially	ADV
ejpam-6261	17	35	number	number	NOUN
ejpam-6261	17	36	theory	theory	NOUN
ejpam-6261	17	37	.	.	PUNCT
ejpam-6261	18	1	modern	modern	ADJ
ejpam-6261	18	2	ring	ring	NOUN
ejpam-6261	18	3	theory	theory	NOUN
ejpam-6261	18	4	,	,	PUNCT
ejpam-6261	18	5	a	a	DET
ejpam-6261	18	6	relatively	relatively	ADV
ejpam-6261	18	7	active	active	ADJ
ejpam-6261	18	8	mathematical	mathematical	ADJ
ejpam-6261	18	9	field	field	NOUN
ejpam-6261	18	10	,	,	PUNCT
ejpam-6261	18	11	investigates	investigate	VERB
ejpam-6261	18	12	rings	ring	NOUN
ejpam-6261	18	13	independently	independently	ADV
ejpam-6261	18	14	.	.	PUNCT
ejpam-6261	19	1	fuzzy	fuzzy	ADJ
ejpam-6261	19	2	set	set	PROPN
ejpam-6261	19	3	theory	theory	NOUN
ejpam-6261	19	4	conceptualized	conceptualize	VERB
ejpam-6261	19	5	by	by	ADP
ejpam-6261	19	6	zadeh	zadeh	PROPN
ejpam-6261	20	1	[	[	X
ejpam-6261	20	2	3	3	NUM
ejpam-6261	20	3	]	]	PUNCT
ejpam-6261	20	4	as	as	ADP
ejpam-6261	20	5	an	an	DET
ejpam-6261	20	6	extension	extension	NOUN
ejpam-6261	20	7	of	of	ADP
ejpam-6261	20	8	the	the	DET
ejpam-6261	20	9	idea	idea	NOUN
ejpam-6261	20	10	of	of	ADP
ejpam-6261	20	11	classical	classical	ADJ
ejpam-6261	20	12	set	set	NOUN
ejpam-6261	20	13	theory	theory	NOUN
ejpam-6261	20	14	,	,	PUNCT
ejpam-6261	20	15	it	it	PRON
ejpam-6261	20	16	is	be	AUX
ejpam-6261	20	17	extremely	extremely	ADV
ejpam-6261	20	18	important	important	ADJ
ejpam-6261	20	19	for	for	ADP
ejpam-6261	20	20	managing	manage	VERB
ejpam-6261	20	21	uncertainty	uncertainty	NOUN
ejpam-6261	20	22	in	in	ADP
ejpam-6261	20	23	practical	practical	ADJ
ejpam-6261	20	24	applications	application	NOUN
ejpam-6261	20	25	and	and	CCONJ
ejpam-6261	20	26	described	describe	VERB
ejpam-6261	20	27	as	as	ADP
ejpam-6261	20	28	κ	κ	PROPN
ejpam-6261	20	29	:	:	PUNCT
ejpam-6261	20	30	→	→	PUNCT
ejpam-6261	20	31	q	q	X
ejpam-6261	20	32	such	such	ADJ
ejpam-6261	20	33	that	that	SCONJ
ejpam-6261	20	34	{	{	PUNCT
ejpam-6261	20	35	y	y	NOUN
ejpam-6261	20	36	:	:	PUNCT
ejpam-6261	20	37	(	(	PUNCT
ejpam-6261	20	38	y	y	NOUN
ejpam-6261	20	39	,	,	PUNCT
ejpam-6261	20	40	κq(y	κq(y	ADJ
ejpam-6261	20	41	)	)	PUNCT
ejpam-6261	20	42	)	)	PUNCT
ejpam-6261	20	43	∀	∀	PUNCT
ejpam-6261	21	1	y	y	NOUN
ejpam-6261	21	2	,	,	PUNCT
ejpam-6261	21	3	κq(y	κq(y	ADJ
ejpam-6261	21	4	)	)	PUNCT
ejpam-6261	21	5	∈	∈	PROPN
ejpam-6261	22	1	[	[	X
ejpam-6261	22	2	01	01	NUM
ejpam-6261	22	3	]	]	PUNCT
ejpam-6261	22	4	}	}	PUNCT
ejpam-6261	22	5	.	.	PUNCT
ejpam-6261	23	1	as	as	ADP
ejpam-6261	23	2	an	an	DET
ejpam-6261	23	3	extension	extension	NOUN
ejpam-6261	23	4	of	of	ADP
ejpam-6261	23	5	the	the	DET
ejpam-6261	23	6	fuzzy	fuzzy	ADJ
ejpam-6261	23	7	set	set	NOUN
ejpam-6261	23	8	,	,	PUNCT
ejpam-6261	23	9	atanassov	atanassov	VERB
ejpam-6261	24	1	[	[	X
ejpam-6261	24	2	4	4	NUM
ejpam-6261	24	3	]	]	PUNCT
ejpam-6261	24	4	presented	present	VERB
ejpam-6261	24	5	an	an	DET
ejpam-6261	24	6	intuitionistic	intuitionistic	ADJ
ejpam-6261	24	7	fuzzy	fuzzy	ADJ
ejpam-6261	24	8	(	(	PUNCT
ejpam-6261	24	9	if	if	SCONJ
ejpam-6261	24	10	)	)	PUNCT
ejpam-6261	24	11	set	set	VERB
ejpam-6261	24	12	and	and	CCONJ
ejpam-6261	24	13	characterize	characterize	VERB
ejpam-6261	24	14	as	as	ADP
ejpam-6261	24	15	χ	χ	X
ejpam-6261	24	16	:	:	PUNCT
ejpam-6261	24	17	t	t	PROPN
ejpam-6261	24	18	→	→	PUNCT
ejpam-6261	24	19	{	{	PUNCT
ejpam-6261	24	20	(	(	PUNCT
ejpam-6261	24	21	x	x	X
ejpam-6261	24	22	,	,	PUNCT
ejpam-6261	24	23	µ(y	µ(y	PROPN
ejpam-6261	24	24	)	)	PUNCT
ejpam-6261	24	25	,	,	PUNCT
ejpam-6261	24	26	µ(y	µ(y	PROPN
ejpam-6261	24	27	)	)	PUNCT
ejpam-6261	24	28	)	)	PUNCT
ejpam-6261	24	29	:	:	PUNCT
ejpam-6261	25	1	y	y	PROPN
ejpam-6261	25	2	∈	∈	PROPN
ejpam-6261	25	3	t	t	PROPN
ejpam-6261	25	4	}	}	PUNCT
ejpam-6261	25	5	where	where	SCONJ
ejpam-6261	25	6	,	,	PUNCT
ejpam-6261	25	7	µ(y	µ(y	PROPN
ejpam-6261	25	8	)	)	PUNCT
ejpam-6261	25	9	,	,	PUNCT
ejpam-6261	25	10	µ(y	µ(y	PROPN
ejpam-6261	25	11	)	)	PUNCT
ejpam-6261	25	12	∈	∈	NOUN
ejpam-6261	26	1	[	[	X
ejpam-6261	26	2	0	0	NUM
ejpam-6261	26	3	1	1	NUM
ejpam-6261	26	4	]	]	PUNCT
ejpam-6261	26	5	such	such	ADJ
ejpam-6261	26	6	that	that	SCONJ
ejpam-6261	26	7	0	0	NUM
ejpam-6261	26	8	≤	≤	NUM
ejpam-6261	26	9	µ(y	µ(y	PROPN
ejpam-6261	26	10	)	)	PUNCT
ejpam-6261	26	11	+	+	SYM
ejpam-6261	26	12	µ(y	µ(y	NOUN
ejpam-6261	26	13	)	)	PUNCT
ejpam-6261	26	14	≤	≤	NUM
ejpam-6261	26	15	1	1	NUM
ejpam-6261	26	16	.	.	PUNCT
ejpam-6261	27	1	in	in	ADP
ejpam-6261	27	2	decision	decision	NOUN
ejpam-6261	27	3	-	-	PUNCT
ejpam-6261	27	4	making	make	VERB
ejpam-6261	27	5	challenges	challenge	NOUN
ejpam-6261	27	6	,	,	PUNCT
ejpam-6261	27	7	the	the	DET
ejpam-6261	27	8	positive	positive	ADJ
ejpam-6261	27	9	and	and	CCONJ
ejpam-6261	27	10	negative	negative	ADJ
ejpam-6261	27	11	membership	membership	NOUN
ejpam-6261	27	12	functions	function	NOUN
ejpam-6261	27	13	of	of	ADP
ejpam-6261	27	14	if	if	SCONJ
ejpam-6261	27	15	sets	set	NOUN
ejpam-6261	27	16	in	in	ADP
ejpam-6261	27	17	contrast	contrast	NOUN
ejpam-6261	27	18	to	to	ADP
ejpam-6261	27	19	classical	classical	ADJ
ejpam-6261	27	20	fuzzy	fuzzy	ADJ
ejpam-6261	27	21	sets	set	NOUN
ejpam-6261	27	22	,	,	PUNCT
ejpam-6261	27	23	provide	provide	VERB
ejpam-6261	27	24	that	that	SCONJ
ejpam-6261	27	25	both	both	PRON
ejpam-6261	27	26	are	be	AUX
ejpam-6261	27	27	able	able	ADJ
ejpam-6261	27	28	to	to	PART
ejpam-6261	27	29	manage	manage	VERB
ejpam-6261	27	30	situations	situation	NOUN
ejpam-6261	27	31	that	that	PRON
ejpam-6261	27	32	are	be	AUX
ejpam-6261	27	33	unclear	unclear	ADJ
ejpam-6261	27	34	and	and	CCONJ
ejpam-6261	27	35	uncertain	uncertain	ADJ
ejpam-6261	27	36	in	in	ADP
ejpam-6261	27	37	physical	physical	ADJ
ejpam-6261	27	38	issues	issue	NOUN
ejpam-6261	27	39	.	.	PUNCT
ejpam-6261	28	1	alolaiyan	alolaiyan	VERB
ejpam-6261	29	1	[	[	X
ejpam-6261	29	2	5	5	NUM
ejpam-6261	29	3	]	]	PUNCT
ejpam-6261	29	4	et	et	PROPN
ejpam-6261	29	5	al	al	PROPN
ejpam-6261	29	6	.	.	PROPN
ejpam-6261	29	7	discussed	discuss	VERB
ejpam-6261	29	8	decision	decision	NOUN
ejpam-6261	29	9	making	make	VERB
ejpam-6261	29	10	problems	problem	NOUN
ejpam-6261	29	11	by	by	ADP
ejpam-6261	29	12	employing	employ	VERB
ejpam-6261	29	13	linguistic	linguistic	ADJ
ejpam-6261	29	14	intuitionistic	intuitionistic	ADJ
ejpam-6261	29	15	fuzzy	fuzzy	ADJ
ejpam-6261	29	16	set	set	VERB
ejpam-6261	29	17	with	with	ADP
ejpam-6261	29	18	a	a	DET
ejpam-6261	29	19	fuzzy	fuzzy	ADJ
ejpam-6261	29	20	dombi	dombi	NOUN
ejpam-6261	29	21	weighted	weight	VERB
ejpam-6261	29	22	geometric	geometric	ADJ
ejpam-6261	29	23	operator	operator	NOUN
ejpam-6261	29	24	.	.	PUNCT
ejpam-6261	30	1	rosenfeld	rosenfeld	PROPN
ejpam-6261	31	1	[	[	X
ejpam-6261	31	2	6	6	NUM
ejpam-6261	31	3	]	]	PUNCT
ejpam-6261	31	4	proposed	propose	VERB
ejpam-6261	31	5	the	the	DET
ejpam-6261	31	6	fuzzy	fuzzy	ADJ
ejpam-6261	31	7	subgroup	subgroup	NOUN
ejpam-6261	31	8	in	in	ADP
ejpam-6261	31	9	1971	1971	NUM
ejpam-6261	31	10	by	by	ADP
ejpam-6261	31	11	applying	apply	VERB
ejpam-6261	31	12	the	the	DET
ejpam-6261	31	13	fuzzy	fuzzy	ADJ
ejpam-6261	31	14	set	set	NOUN
ejpam-6261	31	15	theory	theory	NOUN
ejpam-6261	31	16	on	on	ADP
ejpam-6261	31	17	algebra	algebra	NOUN
ejpam-6261	31	18	.	.	PUNCT
ejpam-6261	32	1	if	if	SCONJ
ejpam-6261	32	2	subgroups	subgroup	NOUN
ejpam-6261	32	3	and	and	CCONJ
ejpam-6261	32	4	the	the	DET
ejpam-6261	32	5	algebraic	algebraic	ADJ
ejpam-6261	32	6	structure	structure	NOUN
ejpam-6261	32	7	of	of	ADP
ejpam-6261	32	8	intuitionistic	intuitionistic	ADJ
ejpam-6261	32	9	fuzzification	fuzzification	NOUN
ejpam-6261	32	10	introduced	introduce	VERB
ejpam-6261	32	11	by	by	ADP
ejpam-6261	32	12	biswas	biswas	PROPN
ejpam-6261	32	13	[	[	X
ejpam-6261	32	14	7	7	NUM
ejpam-6261	32	15	]	]	PUNCT
ejpam-6261	32	16	.	.	PUNCT
ejpam-6261	33	1	smarandache	smarandache	NOUN
ejpam-6261	34	1	[	[	X
ejpam-6261	34	2	8	8	NUM
ejpam-6261	34	3	]	]	PUNCT
ejpam-6261	34	4	was	be	AUX
ejpam-6261	34	5	the	the	DET
ejpam-6261	34	6	first	first	ADJ
ejpam-6261	34	7	to	to	PART
ejpam-6261	34	8	introduce	introduce	VERB
ejpam-6261	34	9	neutrosophy	neutrosophy	NOUN
ejpam-6261	34	10	as	as	ADP
ejpam-6261	34	11	a	a	DET
ejpam-6261	34	12	discipline	discipline	NOUN
ejpam-6261	34	13	of	of	ADP
ejpam-6261	34	14	philosophy	philosophy	NOUN
ejpam-6261	34	15	that	that	PRON
ejpam-6261	34	16	investigated	investigate	VERB
ejpam-6261	34	17	the	the	DET
ejpam-6261	34	18	origin	origin	NOUN
ejpam-6261	34	19	,	,	PUNCT
ejpam-6261	34	20	nature	nature	NOUN
ejpam-6261	34	21	,	,	PUNCT
ejpam-6261	34	22	and	and	CCONJ
ejpam-6261	34	23	scope	scope	NOUN
ejpam-6261	34	24	of	of	ADP
ejpam-6261	34	25	neutralities	neutrality	NOUN
ejpam-6261	34	26	as	as	ADV
ejpam-6261	34	27	well	well	ADV
ejpam-6261	34	28	as	as	ADP
ejpam-6261	34	29	how	how	SCONJ
ejpam-6261	34	30	they	they	PRON
ejpam-6261	34	31	interacted	interact	VERB
ejpam-6261	34	32	with	with	ADP
ejpam-6261	34	33	various	various	ADJ
ejpam-6261	34	34	ideational	ideational	ADJ
ejpam-6261	34	35	spectra	spectra	NOUN
ejpam-6261	34	36	.	.	PUNCT
ejpam-6261	35	1	a	a	DET
ejpam-6261	35	2	belonging	belong	VERB
ejpam-6261	35	3	membership	membership	NOUN
ejpam-6261	35	4	function	function	NOUN
ejpam-6261	35	5	,	,	PUNCT
ejpam-6261	35	6	a	a	DET
ejpam-6261	35	7	not	not	PART
ejpam-6261	35	8	belonging	belong	VERB
ejpam-6261	35	9	membership	membership	NOUN
ejpam-6261	35	10	function	function	NOUN
ejpam-6261	35	11	,	,	PUNCT
ejpam-6261	35	12	and	and	CCONJ
ejpam-6261	35	13	an	an	DET
ejpam-6261	35	14	indeterminacy	indeterminacy	NOUN
ejpam-6261	35	15	membership	membership	NOUN
ejpam-6261	35	16	function	function	NOUN
ejpam-6261	35	17	define	define	VERB
ejpam-6261	35	18	a	a	DET
ejpam-6261	35	19	neutrosophic	neutrosophic	ADJ
ejpam-6261	35	20	set	set	NOUN
ejpam-6261	35	21	(	(	PUNCT
ejpam-6261	35	22	ns	ns	NUM
ejpam-6261	35	23	)	)	PUNCT
ejpam-6261	35	24	.	.	PUNCT
ejpam-6261	36	1	agboola	agboola	PROPN
ejpam-6261	36	2	et	et	PROPN
ejpam-6261	36	3	al	al	PROPN
ejpam-6261	36	4	.	.	PUNCT
ejpam-6261	37	1	[	[	X
ejpam-6261	37	2	9	9	NUM
ejpam-6261	37	3	]	]	PUNCT
ejpam-6261	37	4	provided	provide	VERB
ejpam-6261	37	5	the	the	DET
ejpam-6261	37	6	idea	idea	NOUN
ejpam-6261	37	7	of	of	ADP
ejpam-6261	37	8	neutrosophic	neutrosophic	ADJ
ejpam-6261	37	9	bci	bci	PROPN
ejpam-6261	37	10	/	/	SYM
ejpam-6261	37	11	bck	bck	PROPN
ejpam-6261	37	12	algebras	algebra	NOUN
ejpam-6261	37	13	and	and	CCONJ
ejpam-6261	37	14	discussed	discuss	VERB
ejpam-6261	37	15	some	some	DET
ejpam-6261	37	16	fundamental	fundamental	ADJ
ejpam-6261	37	17	characteristics	characteristic	NOUN
ejpam-6261	37	18	of	of	ADP
ejpam-6261	37	19	neutrosophic	neutrosophic	ADJ
ejpam-6261	37	20	bci	bci	PROPN
ejpam-6261	37	21	/	/	SYM
ejpam-6261	37	22	bck	bck	PROPN
ejpam-6261	37	23	algebras	algebra	NOUN
ejpam-6261	37	24	.	.	PUNCT
ejpam-6261	38	1	the	the	DET
ejpam-6261	38	2	group	group	NOUN
ejpam-6261	38	3	structure	structure	NOUN
ejpam-6261	38	4	of	of	ADP
ejpam-6261	38	5	single	single	ADJ
ejpam-6261	38	6	valued	value	VERB
ejpam-6261	38	7	nss	nss	NOUN
ejpam-6261	38	8	was	be	AUX
ejpam-6261	38	9	investigated	investigate	VERB
ejpam-6261	38	10	by	by	ADP
ejpam-6261	38	11	cetkin	cetkin	NOUN
ejpam-6261	38	12	and	and	CCONJ
ejpam-6261	38	13	aygun	aygun	VERB
ejpam-6261	38	14	[	[	X
ejpam-6261	38	15	10	10	NUM
ejpam-6261	38	16	]	]	PUNCT
ejpam-6261	38	17	.	.	PUNCT
ejpam-6261	39	1	additionally	additionally	ADV
ejpam-6261	39	2	,	,	PUNCT
ejpam-6261	39	3	they	they	PRON
ejpam-6261	39	4	scrutinized	scrutinize	VERB
ejpam-6261	39	5	the	the	DET
ejpam-6261	39	6	essential	essential	ADJ
ejpam-6261	39	7	features	feature	NOUN
ejpam-6261	39	8	of	of	ADP
ejpam-6261	39	9	the	the	DET
ejpam-6261	39	10	neutrosophic	neutrosophic	ADJ
ejpam-6261	39	11	subgroup	subgroup	NOUN
ejpam-6261	39	12	and	and	CCONJ
ejpam-6261	39	13	showcased	showcase	VERB
ejpam-6261	39	14	the	the	DET
ejpam-6261	39	15	homomorphic	homomorphic	ADJ
ejpam-6261	39	16	image	image	NOUN
ejpam-6261	39	17	and	and	CCONJ
ejpam-6261	39	18	pre	pre	NOUN
ejpam-6261	39	19	-	-	NOUN
ejpam-6261	39	20	image	image	NOUN
ejpam-6261	39	21	of	of	ADP
ejpam-6261	39	22	a	a	DET
ejpam-6261	39	23	neutrosophic	neutrosophic	ADJ
ejpam-6261	39	24	(	(	PUNCT
ejpam-6261	39	25	normal	normal	ADJ
ejpam-6261	39	26	)	)	PUNCT
ejpam-6261	39	27	subgroup	subgroup	NOUN
ejpam-6261	39	28	.	.	PUNCT
ejpam-6261	40	1	song	song	PROPN
ejpam-6261	40	2	et	et	PROPN
ejpam-6261	40	3	al	al	PROPN
ejpam-6261	40	4	.	.	PUNCT
ejpam-6261	41	1	[	[	X
ejpam-6261	41	2	11	11	NUM
ejpam-6261	41	3	]	]	PUNCT
ejpam-6261	41	4	offered	offer	VERB
ejpam-6261	41	5	the	the	DET
ejpam-6261	41	6	idea	idea	NOUN
ejpam-6261	41	7	of	of	ADP
ejpam-6261	41	8	a	a	DET
ejpam-6261	41	9	neutrosophic	neutrosophic	ADJ
ejpam-6261	41	10	distributive	distributive	ADJ
ejpam-6261	41	11	n	n	PRON
ejpam-6261	41	12	-ideal	-ideal	NOUN
ejpam-6261	41	13	in	in	ADP
ejpam-6261	41	14	bck	bck	NOUN
ejpam-6261	41	15	-	-	PUNCT
ejpam-6261	41	16	algebras	algebras	PROPN
ejpam-6261	41	17	and	and	CCONJ
ejpam-6261	41	18	explored	explore	VERB
ejpam-6261	41	19	a	a	DET
ejpam-6261	41	20	number	number	NOUN
ejpam-6261	41	21	of	of	ADP
ejpam-6261	41	22	its	its	PRON
ejpam-6261	41	23	features	feature	NOUN
ejpam-6261	41	24	.	.	PUNCT
ejpam-6261	42	1	additionally	additionally	ADV
ejpam-6261	42	2	,	,	PUNCT
ejpam-6261	42	3	they	they	PRON
ejpam-6261	42	4	engaged	engage	VERB
ejpam-6261	42	5	in	in	ADP
ejpam-6261	42	6	a	a	DET
ejpam-6261	42	7	discussion	discussion	NOUN
ejpam-6261	42	8	about	about	ADP
ejpam-6261	42	9	the	the	DET
ejpam-6261	42	10	connections	connection	NOUN
ejpam-6261	42	11	between	between	ADP
ejpam-6261	42	12	a	a	DET
ejpam-6261	42	13	neutrosophic	neutrosophic	ADJ
ejpam-6261	42	14	commutative	commutative	ADJ
ejpam-6261	42	15	n	n	PRON
ejpam-6261	42	16	-ideal	-ideal	ADJ
ejpam-6261	42	17	and	and	CCONJ
ejpam-6261	42	18	a	a	DET
ejpam-6261	42	19	neutrosophic	neutrosophic	ADJ
ejpam-6261	42	20	n	n	PRON
ejpam-6261	42	21	-ideal	-ideal	NOUN
ejpam-6261	42	22	.	.	PUNCT
ejpam-6261	43	1	chalapathi	chalapathi	PROPN
ejpam-6261	43	2	and	and	CCONJ
ejpam-6261	43	3	kumar	kumar	PROPN
ejpam-6261	44	1	[	[	X
ejpam-6261	44	2	12	12	NUM
ejpam-6261	44	3	]	]	PUNCT
ejpam-6261	44	4	discussed	discuss	VERB
ejpam-6261	44	5	the	the	DET
ejpam-6261	44	6	finite	finite	ADJ
ejpam-6261	44	7	groups	group	NOUN
ejpam-6261	44	8	through	through	ADP
ejpam-6261	44	9	graphs	graph	NOUN
ejpam-6261	44	10	under	under	ADP
ejpam-6261	44	11	the	the	DET
ejpam-6261	44	12	framework	framework	NOUN
ejpam-6261	44	13	of	of	ADP
ejpam-6261	44	14	ns	ns	NOUN
ejpam-6261	44	15	.	.	PUNCT
ejpam-6261	45	1	the	the	DET
ejpam-6261	45	2	idea	idea	NOUN
ejpam-6261	45	3	of	of	ADP
ejpam-6261	45	4	the	the	DET
ejpam-6261	45	5	neutrosophic	neutrosophic	ADJ
ejpam-6261	45	6	triplet	triplet	NOUN
ejpam-6261	45	7	group	group	NOUN
ejpam-6261	45	8	,	,	PUNCT
ejpam-6261	45	9	it	it	PRON
ejpam-6261	45	10	includes	include	VERB
ejpam-6261	45	11	a	a	DET
ejpam-6261	45	12	novel	novel	ADJ
ejpam-6261	45	13	extension	extension	NOUN
ejpam-6261	45	14	of	of	ADP
ejpam-6261	45	15	the	the	DET
ejpam-6261	45	16	traditional	traditional	ADJ
ejpam-6261	45	17	group	group	NOUN
ejpam-6261	45	18	idea	idea	NOUN
ejpam-6261	45	19	,	,	PUNCT
ejpam-6261	45	20	was	be	AUX
ejpam-6261	45	21	derived	derive	VERB
ejpam-6261	45	22	from	from	ADP
ejpam-6261	45	23	the	the	DET
ejpam-6261	45	24	fundamental	fundamental	ADJ
ejpam-6261	45	25	idea	idea	NOUN
ejpam-6261	45	26	of	of	ADP
ejpam-6261	45	27	the	the	DET
ejpam-6261	45	28	ns	ns	NOUN
ejpam-6261	45	29	and	and	CCONJ
ejpam-6261	45	30	the	the	DET
ejpam-6261	45	31	structural	structural	ADJ
ejpam-6261	45	32	characteristics	characteristic	NOUN
ejpam-6261	45	33	of	of	ADP
ejpam-6261	45	34	the	the	DET
ejpam-6261	45	35	neutrosophic	neutrosophic	ADJ
ejpam-6261	45	36	triplet	triplet	NOUN
ejpam-6261	45	37	group	group	NOUN
ejpam-6261	45	38	are	be	AUX
ejpam-6261	45	39	examined	examine	VERB
ejpam-6261	45	40	in	in	ADP
ejpam-6261	45	41	detail	detail	NOUN
ejpam-6261	45	42	[	[	X
ejpam-6261	45	43	13	13	NUM
ejpam-6261	45	44	]	]	PUNCT
ejpam-6261	45	45	.	.	PUNCT
ejpam-6261	46	1	in	in	ADP
ejpam-6261	46	2	a	a	DET
ejpam-6261	46	3	bck	bck	NOUN
ejpam-6261	46	4	-	-	PUNCT
ejpam-6261	46	5	algebra	algebra	NOUN
ejpam-6261	46	6	,	,	PUNCT
ejpam-6261	46	7	borzooei	borzooei	PROPN
ejpam-6261	46	8	et	et	PROPN
ejpam-6261	46	9	al	al	PROPN
ejpam-6261	46	10	.	.	PUNCT
ejpam-6261	47	1	[	[	X
ejpam-6261	47	2	14	14	NUM
ejpam-6261	47	3	]	]	PUNCT
ejpam-6261	47	4	developed	develop	VERB
ejpam-6261	47	5	the	the	DET
ejpam-6261	47	6	idea	idea	NOUN
ejpam-6261	47	7	of	of	ADP
ejpam-6261	47	8	an	an	DET
ejpam-6261	47	9	extended	extend	VERB
ejpam-6261	47	10	neutrosophic	neutrosophic	ADJ
ejpam-6261	47	11	commutative	commutative	ADJ
ejpam-6261	47	12	ideal	ideal	NOUN
ejpam-6261	47	13	,	,	PUNCT
ejpam-6261	47	14	and	and	CCONJ
ejpam-6261	47	15	associated	associated	ADJ
ejpam-6261	47	16	features	feature	NOUN
ejpam-6261	47	17	were	be	AUX
ejpam-6261	47	18	demonstrated	demonstrate	VERB
ejpam-6261	47	19	.	.	PUNCT
ejpam-6261	48	1	moreover	moreover	ADV
ejpam-6261	48	2	,	,	PUNCT
ejpam-6261	48	3	some	some	DET
ejpam-6261	48	4	equivalence	equivalence	NOUN
ejpam-6261	48	5	relations	relation	NOUN
ejpam-6261	48	6	have	have	AUX
ejpam-6261	48	7	been	be	AUX
ejpam-6261	48	8	introduced	introduce	VERB
ejpam-6261	48	9	,	,	PUNCT
ejpam-6261	48	10	and	and	CCONJ
ejpam-6261	48	11	several	several	ADJ
ejpam-6261	48	12	features	feature	NOUN
ejpam-6261	48	13	of	of	ADP
ejpam-6261	48	14	the	the	DET
ejpam-6261	48	15	family	family	NOUN
ejpam-6261	48	16	of	of	ADP
ejpam-6261	48	17	all	all	DET
ejpam-6261	48	18	commutative	commutative	ADJ
ejpam-6261	48	19	modified	modify	VERB
ejpam-6261	48	20	neutrosophic	neutrosophic	ADJ
ejpam-6261	48	21	ideals	ideal	NOUN
ejpam-6261	48	22	in	in	ADP
ejpam-6261	48	23	bck	bck	NOUN
ejpam-6261	48	24	-	-	PUNCT
ejpam-6261	48	25	algebras	algebra	NOUN
ejpam-6261	48	26	are	be	AUX
ejpam-6261	48	27	investigated	investigate	VERB
ejpam-6261	48	28	.	.	PUNCT
ejpam-6261	49	1	interval	interval	NOUN
ejpam-6261	49	2	neutrosophic	neutrosophic	ADJ
ejpam-6261	49	3	subalgebra	subalgebra	NOUN
ejpam-6261	49	4	was	be	AUX
ejpam-6261	49	5	introduced	introduce	VERB
ejpam-6261	49	6	by	by	ADP
ejpam-6261	49	7	jun	jun	PROPN
ejpam-6261	49	8	et	et	PROPN
ejpam-6261	49	9	al	al	PROPN
ejpam-6261	49	10	.	.	PUNCT
ejpam-6261	50	1	[	[	X
ejpam-6261	50	2	15	15	NUM
ejpam-6261	50	3	]	]	PUNCT
ejpam-6261	50	4	.	.	PUNCT
ejpam-6261	51	1	in	in	ADP
ejpam-6261	51	2	bck	bck	PROPN
ejpam-6261	51	3	/	/	SYM
ejpam-6261	51	4	bci	bci	NOUN
ejpam-6261	51	5	-	-	NOUN
ejpam-6261	51	6	algebra	algebra	NOUN
ejpam-6261	51	7	,	,	PUNCT
ejpam-6261	51	8	their	their	PRON
ejpam-6261	51	9	characteristics	characteristic	NOUN
ejpam-6261	51	10	and	and	CCONJ
ejpam-6261	51	11	relationships	relationship	NOUN
ejpam-6261	51	12	are	be	AUX
ejpam-6261	51	13	studied	study	VERB
ejpam-6261	51	14	.	.	PUNCT
ejpam-6261	52	1	additionally	additionally	ADV
ejpam-6261	52	2	,	,	PUNCT
ejpam-6261	52	3	we	we	PRON
ejpam-6261	52	4	introduce	introduce	VERB
ejpam-6261	52	5	the	the	DET
ejpam-6261	52	6	concept	concept	NOUN
ejpam-6261	52	7	of	of	ADP
ejpam-6261	52	8	an	an	DET
ejpam-6261	52	9	interval	interval	NOUN
ejpam-6261	52	10	neutrosophic	neutrosophic	ADJ
ejpam-6261	52	11	length	length	NOUN
ejpam-6261	52	12	and	and	CCONJ
ejpam-6261	52	13	review	review	VERB
ejpam-6261	52	14	its	its	PRON
ejpam-6261	52	15	associated	associated	ADJ
ejpam-6261	52	16	features	feature	NOUN
ejpam-6261	52	17	.	.	PUNCT
ejpam-6261	53	1	the	the	DET
ejpam-6261	53	2	idea	idea	NOUN
ejpam-6261	53	3	of	of	ADP
ejpam-6261	53	4	a	a	DET
ejpam-6261	53	5	neutrosophic	neutrosophic	ADJ
ejpam-6261	53	6	positive	positive	ADJ
ejpam-6261	53	7	implicative	implicative	NOUN
ejpam-6261	53	8	n	n	PRON
ejpam-6261	53	9	-ideal	-ideal	NOUN
ejpam-6261	53	10	in	in	ADP
ejpam-6261	53	11	bckalgebras	bckalgebras	PROPN
ejpam-6261	53	12	was	be	AUX
ejpam-6261	53	13	suggested	suggest	VERB
ejpam-6261	53	14	by	by	ADP
ejpam-6261	53	15	jun	jun	PROPN
ejpam-6261	53	16	et	et	PROPN
ejpam-6261	53	17	al	al	PROPN
ejpam-6261	53	18	.	.	PUNCT
ejpam-6261	54	1	[	[	X
ejpam-6261	54	2	16	16	NUM
ejpam-6261	54	3	]	]	PUNCT
ejpam-6261	54	4	,	,	PUNCT
ejpam-6261	54	5	and	and	CCONJ
ejpam-6261	54	6	numerous	numerous	ADJ
ejpam-6261	54	7	features	feature	NOUN
ejpam-6261	54	8	were	be	AUX
ejpam-6261	54	9	studied	study	VERB
ejpam-6261	54	10	.	.	PUNCT
ejpam-6261	55	1	bender	bender	NOUN
ejpam-6261	55	2	et	et	PROPN
ejpam-6261	55	3	al	al	PROPN
ejpam-6261	55	4	.	.	PUNCT
ejpam-6261	56	1	[	[	X
ejpam-6261	56	2	17	17	NUM
ejpam-6261	56	3	]	]	PUNCT
ejpam-6261	56	4	investigated	investigate	VERB
ejpam-6261	56	5	the	the	DET
ejpam-6261	56	6	complex	complex	ADJ
ejpam-6261	56	7	anti	anti	ADJ
ejpam-6261	56	8	-	-	ADJ
ejpam-6261	56	9	fuzzy	fuzzy	ADJ
ejpam-6261	56	10	subgroups	subgroup	NOUN
ejpam-6261	56	11	and	and	CCONJ
ejpam-6261	56	12	proved	prove	VERB
ejpam-6261	56	13	some	some	DET
ejpam-6261	56	14	important	important	ADJ
ejpam-6261	56	15	results	result	NOUN
ejpam-6261	56	16	.	.	PUNCT
ejpam-6261	57	1	gulzar	gulzar	PROPN
ejpam-6261	57	2	et	et	PROPN
ejpam-6261	57	3	al	al	PROPN
ejpam-6261	57	4	.	.	PUNCT
ejpam-6261	58	1	[	[	X
ejpam-6261	58	2	18	18	NUM
ejpam-6261	58	3	]	]	PUNCT
ejpam-6261	58	4	discussed	discuss	VERB
ejpam-6261	58	5	the	the	DET
ejpam-6261	58	6	q	q	ADJ
ejpam-6261	58	7	-	-	PUNCT
ejpam-6261	58	8	fuzzy	fuzzy	ADJ
ejpam-6261	58	9	subrings	subring	NOUN
ejpam-6261	58	10	under	under	ADP
ejpam-6261	58	11	the	the	DET
ejpam-6261	58	12	framework	framework	NOUN
ejpam-6261	58	13	of	of	ADP
ejpam-6261	58	14	complex	complex	ADJ
ejpam-6261	58	15	fuzzy	fuzzy	ADJ
ejpam-6261	58	16	(	(	PUNCT
ejpam-6261	58	17	cf	cf	NOUN
ejpam-6261	58	18	)	)	PUNCT
ejpam-6261	58	19	sets	set	NOUN
ejpam-6261	58	20	.	.	PUNCT
ejpam-6261	59	1	alghazzawi	alghazzawi	PROPN
ejpam-6261	59	2	et	et	PROPN
ejpam-6261	59	3	al	al	PROPN
ejpam-6261	59	4	.	.	PUNCT
ejpam-6261	60	1	[	[	X
ejpam-6261	60	2	19	19	NUM
ejpam-6261	60	3	]	]	PUNCT
ejpam-6261	60	4	purposed	purpose	VERB
ejpam-6261	60	5	the	the	DET
ejpam-6261	60	6	optimal	optimal	ADJ
ejpam-6261	60	7	solution	solution	NOUN
ejpam-6261	60	8	for	for	ADP
ejpam-6261	60	9	energy	energy	NOUN
ejpam-6261	60	10	crises	crisis	NOUN
ejpam-6261	60	11	by	by	ADP
ejpam-6261	60	12	employing	employ	VERB
ejpam-6261	60	13	the	the	DET
ejpam-6261	60	14	interval	interval	NOUN
ejpam-6261	60	15	-	-	PUNCT
ejpam-6261	60	16	valued	value	VERB
ejpam-6261	60	17	intuitionistic	intuitionistic	ADJ
ejpam-6261	60	18	fuzzy	fuzzy	ADJ
ejpam-6261	60	19	sets	set	NOUN
ejpam-6261	60	20	.	.	PUNCT
ejpam-6261	61	1	alolaiyan	alolaiyan	VERB
ejpam-6261	61	2	et	et	PROPN
ejpam-6261	61	3	al	al	PROPN
ejpam-6261	61	4	.	.	PUNCT
ejpam-6261	62	1	[	[	X
ejpam-6261	62	2	20	20	NUM
ejpam-6261	62	3	]	]	PUNCT
ejpam-6261	62	4	studied	study	VERB
ejpam-6261	62	5	the	the	DET
ejpam-6261	62	6	algebraic	algebraic	PROPN
ejpam-6261	62	7	m.	m.	NOUN
ejpam-6261	62	8	h.	h.	PROPN
ejpam-6261	62	9	mateen	mateen	PROPN
ejpam-6261	62	10	et	et	PROPN
ejpam-6261	62	11	al	al	PROPN
ejpam-6261	62	12	.	.	PUNCT
ejpam-6261	62	13	/	/	SYM
ejpam-6261	62	14	eur	eur	PROPN
ejpam-6261	62	15	.	.	PUNCT
ejpam-6261	63	1	j.	j.	PROPN
ejpam-6261	63	2	pure	pure	PROPN
ejpam-6261	63	3	appl	appl	PROPN
ejpam-6261	63	4	.	.	PROPN
ejpam-6261	63	5	math	math	PROPN
ejpam-6261	63	6	,	,	PUNCT
ejpam-6261	63	7	18	18	NUM
ejpam-6261	63	8	(	(	PUNCT
ejpam-6261	63	9	4	4	NUM
ejpam-6261	63	10	)	)	PUNCT
ejpam-6261	63	11	(	(	PUNCT
ejpam-6261	63	12	2025	2025	NUM
ejpam-6261	63	13	)	)	PUNCT
ejpam-6261	63	14	,	,	PUNCT
ejpam-6261	63	15	6261	6261	NUM
ejpam-6261	63	16	3	3	NUM
ejpam-6261	63	17	of	of	ADP
ejpam-6261	63	18	23	23	NUM
ejpam-6261	63	19	structure	structure	NOUN
ejpam-6261	63	20	of	of	ADP
ejpam-6261	63	21	bipolar	bipolar	ADJ
ejpam-6261	63	22	fuzzy	fuzzy	ADJ
ejpam-6261	63	23	subrings	subring	NOUN
ejpam-6261	63	24	.	.	PUNCT
ejpam-6261	64	1	additionally	additionally	ADV
ejpam-6261	64	2	,	,	PUNCT
ejpam-6261	64	3	a	a	DET
ejpam-6261	64	4	detailed	detailed	ADJ
ejpam-6261	64	5	discussion	discussion	NOUN
ejpam-6261	64	6	of	of	ADP
ejpam-6261	64	7	the	the	DET
ejpam-6261	64	8	algebraic	algebraic	ADJ
ejpam-6261	64	9	attributes	attribute	NOUN
ejpam-6261	64	10	took	take	VERB
ejpam-6261	64	11	place	place	NOUN
ejpam-6261	64	12	.	.	PUNCT
ejpam-6261	65	1	altassan	altassan	ADV
ejpam-6261	65	2	et	et	PROPN
ejpam-6261	65	3	al	al	PROPN
ejpam-6261	65	4	.	.	PUNCT
ejpam-6261	66	1	[	[	X
ejpam-6261	66	2	21	21	NUM
ejpam-6261	66	3	]	]	PUNCT
ejpam-6261	66	4	discussed	discuss	VERB
ejpam-6261	66	5	the	the	DET
ejpam-6261	66	6	algebraic	algebraic	ADJ
ejpam-6261	66	7	product	product	NOUN
ejpam-6261	66	8	of	of	ADP
ejpam-6261	66	9	fuzzy	fuzzy	ADJ
ejpam-6261	66	10	subrings	subring	NOUN
ejpam-6261	66	11	.	.	PUNCT
ejpam-6261	67	1	furthermore	furthermore	ADV
ejpam-6261	67	2	,	,	PUNCT
ejpam-6261	67	3	the	the	DET
ejpam-6261	67	4	fundamental	fundamental	ADJ
ejpam-6261	67	5	theorems	theorem	NOUN
ejpam-6261	67	6	of	of	ADP
ejpam-6261	67	7	the	the	DET
ejpam-6261	67	8	fuzzy	fuzzy	ADJ
ejpam-6261	67	9	isomorphism	isomorphism	NOUN
ejpam-6261	67	10	subring	subre	VERB
ejpam-6261	67	11	extend	extend	NOUN
ejpam-6261	67	12	to	to	ADP
ejpam-6261	67	13	algebraic	algebraic	ADJ
ejpam-6261	67	14	products	product	NOUN
ejpam-6261	67	15	.	.	PUNCT
ejpam-6261	68	1	dilshad	dilshad	VERB
ejpam-6261	68	2	et	et	PROPN
ejpam-6261	68	3	al	al	PROPN
ejpam-6261	68	4	.	.	PUNCT
ejpam-6261	69	1	[	[	X
ejpam-6261	69	2	22	22	NUM
ejpam-6261	69	3	]	]	PUNCT
ejpam-6261	69	4	introduced	introduce	VERB
ejpam-6261	69	5	the	the	DET
ejpam-6261	69	6	novel	novel	ADJ
ejpam-6261	69	7	idea	idea	NOUN
ejpam-6261	69	8	of	of	ADP
ejpam-6261	69	9	q	q	NOUN
ejpam-6261	69	10	-	-	PUNCT
ejpam-6261	69	11	rung	rung	ADJ
ejpam-6261	69	12	orthopair	orthopair	NOUN
ejpam-6261	69	13	fuzzy	fuzzy	ADJ
ejpam-6261	69	14	(	(	PUNCT
ejpam-6261	69	15	rof	rof	NOUN
ejpam-6261	69	16	)	)	PUNCT
ejpam-6261	69	17	subrings	subring	NOUN
ejpam-6261	69	18	.	.	PUNCT
ejpam-6261	70	1	under	under	ADP
ejpam-6261	70	2	the	the	DET
ejpam-6261	70	3	influence	influence	NOUN
ejpam-6261	70	4	of	of	ADP
ejpam-6261	70	5	ring	ring	NOUN
ejpam-6261	70	6	theory	theory	NOUN
ejpam-6261	70	7	,	,	PUNCT
ejpam-6261	70	8	they	they	PRON
ejpam-6261	70	9	also	also	ADV
ejpam-6261	70	10	established	establish	VERB
ejpam-6261	70	11	various	various	ADJ
ejpam-6261	70	12	algebraic	algebraic	ADJ
ejpam-6261	70	13	operations	operation	NOUN
ejpam-6261	70	14	,	,	PUNCT
ejpam-6261	70	15	ideals	ideal	NOUN
ejpam-6261	70	16	,	,	PUNCT
ejpam-6261	70	17	homomorphic	homomorphic	ADJ
ejpam-6261	70	18	images	image	NOUN
ejpam-6261	70	19	,	,	PUNCT
ejpam-6261	70	20	and	and	CCONJ
ejpam-6261	70	21	pre	pre	NOUN
ejpam-6261	70	22	-	-	NOUN
ejpam-6261	70	23	images	image	NOUN
ejpam-6261	70	24	.	.	PUNCT
ejpam-6261	71	1	mateen	mateen	PROPN
ejpam-6261	71	2	et	et	PROPN
ejpam-6261	71	3	al	al	PROPN
ejpam-6261	71	4	.	.	PROPN
ejpam-6261	71	5	initiated	initiate	VERB
ejpam-6261	71	6	the	the	DET
ejpam-6261	71	7	novel	novel	ADJ
ejpam-6261	71	8	algebraic	algebraic	ADJ
ejpam-6261	71	9	structure	structure	NOUN
ejpam-6261	71	10	of	of	ADP
ejpam-6261	71	11	complex	complex	ADJ
ejpam-6261	71	12	pythagorean	pythagorean	ADJ
ejpam-6261	71	13	fuzzy	fuzzy	ADJ
ejpam-6261	71	14	subfield	subfield	PROPN
ejpam-6261	71	15	.	.	PUNCT
ejpam-6261	72	1	they	they	PRON
ejpam-6261	72	2	also	also	ADV
ejpam-6261	72	3	explored	explore	VERB
ejpam-6261	72	4	the	the	DET
ejpam-6261	72	5	direct	direct	ADJ
ejpam-6261	72	6	product	product	NOUN
ejpam-6261	72	7	and	and	CCONJ
ejpam-6261	72	8	homomorphism	homomorphism	NOUN
ejpam-6261	72	9	within	within	ADP
ejpam-6261	72	10	the	the	DET
ejpam-6261	72	11	context	context	NOUN
ejpam-6261	72	12	of	of	ADP
ejpam-6261	72	13	a	a	DET
ejpam-6261	72	14	complex	complex	ADJ
ejpam-6261	72	15	pythagorean	pythagorean	ADJ
ejpam-6261	72	16	fuzzy	fuzzy	ADJ
ejpam-6261	72	17	set	set	NOUN
ejpam-6261	72	18	.	.	PUNCT
ejpam-6261	73	1	bal	bal	PROPN
ejpam-6261	73	2	et	et	PROPN
ejpam-6261	73	3	al	al	PROPN
ejpam-6261	73	4	.	.	PUNCT
ejpam-6261	74	1	[	[	X
ejpam-6261	74	2	23	23	NUM
ejpam-6261	74	3	]	]	PUNCT
ejpam-6261	74	4	defined	define	VERB
ejpam-6261	74	5	the	the	DET
ejpam-6261	74	6	neutrosophic	neutrosophic	ADJ
ejpam-6261	74	7	extended	extended	ADJ
ejpam-6261	74	8	triplet	triplet	NOUN
ejpam-6261	74	9	subgroup	subgroup	NOUN
ejpam-6261	74	10	,	,	PUNCT
ejpam-6261	74	11	neutrosophic	neutrosophic	ADJ
ejpam-6261	74	12	kernel	kernel	NOUN
ejpam-6261	74	13	,	,	PUNCT
ejpam-6261	74	14	neutrosophic	neutrosophic	ADJ
ejpam-6261	74	15	inverse	inverse	NOUN
ejpam-6261	74	16	-	-	PUNCT
ejpam-6261	74	17	image	image	NOUN
ejpam-6261	74	18	,	,	PUNCT
ejpam-6261	74	19	and	and	CCONJ
ejpam-6261	74	20	neutrosophic	neutrosophic	ADJ
ejpam-6261	74	21	image	image	NOUN
ejpam-6261	74	22	in	in	ADP
ejpam-6261	74	23	this	this	DET
ejpam-6261	74	24	study	study	NOUN
ejpam-6261	74	25	using	use	VERB
ejpam-6261	74	26	the	the	DET
ejpam-6261	74	27	idea	idea	NOUN
ejpam-6261	74	28	of	of	ADP
ejpam-6261	74	29	a	a	DET
ejpam-6261	74	30	neutrosophic	neutrosophic	ADJ
ejpam-6261	74	31	extended	extend	VERB
ejpam-6261	74	32	triplet	triplet	NOUN
ejpam-6261	74	33	.	.	PUNCT
ejpam-6261	75	1	smarandache	smarandache	PROPN
ejpam-6261	75	2	et	et	PROPN
ejpam-6261	75	3	al	al	PROPN
ejpam-6261	75	4	.	.	PUNCT
ejpam-6261	76	1	[	[	X
ejpam-6261	76	2	24	24	NUM
ejpam-6261	76	3	]	]	PUNCT
ejpam-6261	76	4	examined	examine	VERB
ejpam-6261	76	5	the	the	DET
ejpam-6261	76	6	neutrosophic	neutrosophic	ADJ
ejpam-6261	76	7	triplet	triplet	NOUN
ejpam-6261	76	8	g	g	NOUN
ejpam-6261	76	9	-	-	PUNCT
ejpam-6261	76	10	module	module	NOUN
ejpam-6261	76	11	and	and	CCONJ
ejpam-6261	76	12	described	describe	VERB
ejpam-6261	76	13	its	its	PRON
ejpam-6261	76	14	characteristics	characteristic	NOUN
ejpam-6261	76	15	.	.	PUNCT
ejpam-6261	77	1	also	also	ADV
ejpam-6261	77	2	,	,	PUNCT
ejpam-6261	77	3	the	the	DET
ejpam-6261	77	4	definitions	definition	NOUN
ejpam-6261	77	5	of	of	ADP
ejpam-6261	77	6	neutrosophic	neutrosophic	ADJ
ejpam-6261	77	7	triplet	triplet	NOUN
ejpam-6261	77	8	g	g	NOUN
ejpam-6261	77	9	-	-	PUNCT
ejpam-6261	77	10	modules	module	NOUN
ejpam-6261	77	11	that	that	PRON
ejpam-6261	77	12	are	be	AUX
ejpam-6261	77	13	reducible	reducible	ADJ
ejpam-6261	77	14	,	,	PUNCT
ejpam-6261	77	15	irreducible	irreducible	ADJ
ejpam-6261	77	16	,	,	PUNCT
ejpam-6261	77	17	and	and	CCONJ
ejpam-6261	77	18	completely	completely	ADV
ejpam-6261	77	19	reducible	reducible	ADJ
ejpam-6261	77	20	were	be	AUX
ejpam-6261	77	21	given	give	VERB
ejpam-6261	77	22	,	,	PUNCT
ejpam-6261	77	23	along	along	ADP
ejpam-6261	77	24	with	with	ADP
ejpam-6261	77	25	an	an	DET
ejpam-6261	77	26	analysis	analysis	NOUN
ejpam-6261	77	27	of	of	ADP
ejpam-6261	77	28	the	the	DET
ejpam-6261	77	29	connections	connection	NOUN
ejpam-6261	77	30	between	between	ADP
ejpam-6261	77	31	these	these	DET
ejpam-6261	77	32	structures	structure	NOUN
ejpam-6261	77	33	.	.	PUNCT
ejpam-6261	78	1	ali	ali	PROPN
ejpam-6261	78	2	et	et	PROPN
ejpam-6261	78	3	al	al	PROPN
ejpam-6261	78	4	.	.	PROPN
ejpam-6261	78	5	established	establish	VERB
ejpam-6261	78	6	a	a	DET
ejpam-6261	78	7	neutrosophic	neutrosophic	ADJ
ejpam-6261	78	8	triplet	triplet	NOUN
ejpam-6261	78	9	subring	subring	NOUN
ejpam-6261	78	10	and	and	CCONJ
ejpam-6261	78	11	neutrosophic	neutrosophic	ADJ
ejpam-6261	78	12	triplet	triplet	NOUN
ejpam-6261	78	13	subfield	subfield	NOUN
ejpam-6261	78	14	,	,	PUNCT
ejpam-6261	78	15	as	as	ADV
ejpam-6261	78	16	well	well	ADV
ejpam-6261	78	17	as	as	ADP
ejpam-6261	78	18	some	some	PRON
ejpam-6261	78	19	of	of	ADP
ejpam-6261	78	20	its	its	PRON
ejpam-6261	78	21	fundamental	fundamental	ADJ
ejpam-6261	78	22	characteristics	characteristic	NOUN
ejpam-6261	78	23	.	.	PUNCT
ejpam-6261	79	1	in	in	ADP
ejpam-6261	79	2	a	a	DET
ejpam-6261	79	3	neutrosophic	neutrosophic	ADJ
ejpam-6261	79	4	group	group	NOUN
ejpam-6261	79	5	,	,	PUNCT
ejpam-6261	79	6	abobala	abobala	NOUN
ejpam-6261	79	7	et	et	PROPN
ejpam-6261	79	8	al	al	PROPN
ejpam-6261	79	9	.	.	PUNCT
ejpam-6261	80	1	[	[	X
ejpam-6261	80	2	25	25	NUM
ejpam-6261	80	3	]	]	PUNCT
ejpam-6261	80	4	defined	define	VERB
ejpam-6261	80	5	certain	certain	ADJ
ejpam-6261	80	6	novel	novel	NOUN
ejpam-6261	80	7	substructures	substructure	NOUN
ejpam-6261	80	8	(	(	PUNCT
ejpam-6261	80	9	ah	ah	INTJ
ejpam-6261	80	10	-	-	PUNCT
ejpam-6261	80	11	substructures	substructure	NOUN
ejpam-6261	80	12	)	)	PUNCT
ejpam-6261	80	13	.	.	PUNCT
ejpam-6261	81	1	it	it	PRON
ejpam-6261	81	2	also	also	ADV
ejpam-6261	81	3	covers	cover	VERB
ejpam-6261	81	4	some	some	DET
ejpam-6261	81	5	basic	basic	ADJ
ejpam-6261	81	6	ah	ah	INTJ
ejpam-6261	81	7	-	-	PUNCT
ejpam-6261	81	8	subgroup	subgroup	NOUN
ejpam-6261	81	9	characteristics	characteristic	NOUN
ejpam-6261	81	10	,	,	PUNCT
ejpam-6261	81	11	ah	ah	INTJ
ejpam-6261	81	12	-	-	PUNCT
ejpam-6261	81	13	normality	normality	NOUN
ejpam-6261	81	14	,	,	PUNCT
ejpam-6261	81	15	ah	ah	INTJ
ejpam-6261	81	16	-	-	PUNCT
ejpam-6261	81	17	homomorphisms	homomorphism	NOUN
ejpam-6261	81	18	,	,	PUNCT
ejpam-6261	81	19	ah	ah	INTJ
ejpam-6261	81	20	-	-	PUNCT
ejpam-6261	81	21	quotients	quotient	NOUN
ejpam-6261	81	22	,	,	PUNCT
ejpam-6261	81	23	and	and	CCONJ
ejpam-6261	81	24	ah	ah	INTJ
ejpam-6261	81	25	-	-	PUNCT
ejpam-6261	81	26	direct	direct	ADJ
ejpam-6261	81	27	products	product	NOUN
ejpam-6261	81	28	.	.	PUNCT
ejpam-6261	82	1	kandasamy	kandasamy	NOUN
ejpam-6261	82	2	et	et	PROPN
ejpam-6261	82	3	al	al	PROPN
ejpam-6261	82	4	.	.	PUNCT
ejpam-6261	83	1	[	[	X
ejpam-6261	83	2	26	26	NUM
ejpam-6261	83	3	]	]	PUNCT
ejpam-6261	83	4	discussed	discuss	VERB
ejpam-6261	83	5	the	the	DET
ejpam-6261	83	6	neutrosophic	neutrosophic	ADJ
ejpam-6261	83	7	triplets	triplet	NOUN
ejpam-6261	83	8	in	in	ADP
ejpam-6261	83	9	neutrosophic	neutrosophic	ADJ
ejpam-6261	83	10	rings	ring	NOUN
ejpam-6261	83	11	<	<	X
ejpam-6261	83	12	z	z	NOUN
ejpam-6261	83	13	∪	∪	VERB
ejpam-6261	83	14	i	i	PRON
ejpam-6261	83	15	>	>	X
ejpam-6261	83	16	<	<	X
ejpam-6261	83	17	q	q	X
ejpam-6261	83	18	∪	∪	PROPN
ejpam-6261	83	19	i	i	PRON
ejpam-6261	83	20	>	>	X
ejpam-6261	83	21	or	or	CCONJ
ejpam-6261	83	22	<	<	X
ejpam-6261	83	23	r	r	NOUN
ejpam-6261	83	24	∪	∪	NOUN
ejpam-6261	83	25	i	i	PRON
ejpam-6261	83	26	>	>	X
ejpam-6261	83	27	.	.	PUNCT
ejpam-6261	84	1	furthermore	furthermore	ADV
ejpam-6261	84	2	,	,	PUNCT
ejpam-6261	84	3	the	the	DET
ejpam-6261	84	4	study	study	NOUN
ejpam-6261	84	5	revealed	reveal	VERB
ejpam-6261	84	6	the	the	DET
ejpam-6261	84	7	existence	existence	NOUN
ejpam-6261	84	8	of	of	ADP
ejpam-6261	84	9	three	three	NUM
ejpam-6261	84	10	distinct	distinct	ADJ
ejpam-6261	84	11	types	type	NOUN
ejpam-6261	84	12	of	of	ADP
ejpam-6261	84	13	neutrosophic	neutrosophic	ADJ
ejpam-6261	84	14	triplets	triplet	NOUN
ejpam-6261	84	15	in	in	ADP
ejpam-6261	84	16	these	these	DET
ejpam-6261	84	17	neutrosophic	neutrosophic	ADJ
ejpam-6261	84	18	subrings	subring	NOUN
ejpam-6261	84	19	,	,	PUNCT
ejpam-6261	84	20	all	all	PRON
ejpam-6261	84	21	of	of	ADP
ejpam-6261	84	22	which	which	PRON
ejpam-6261	84	23	generate	generate	VERB
ejpam-6261	84	24	torsion	torsion	NOUN
ejpam-6261	84	25	-	-	PUNCT
ejpam-6261	84	26	free	free	ADJ
ejpam-6261	84	27	component	component	NOUN
ejpam-6261	84	28	-	-	PUNCT
ejpam-6261	84	29	wise	wise	ADJ
ejpam-6261	84	30	product	product	NOUN
ejpam-6261	84	31	abelian	abelian	NOUN
ejpam-6261	84	32	groups	group	NOUN
ejpam-6261	84	33	.	.	PUNCT
ejpam-6261	85	1	the	the	DET
ejpam-6261	85	2	idea	idea	NOUN
ejpam-6261	85	3	of	of	ADP
ejpam-6261	85	4	a	a	DET
ejpam-6261	85	5	generalized	generalize	VERB
ejpam-6261	85	6	neutrosophic	neutrosophic	ADJ
ejpam-6261	85	7	extended	extended	ADJ
ejpam-6261	85	8	triplet	triplet	NOUN
ejpam-6261	85	9	group	group	NOUN
ejpam-6261	85	10	is	be	AUX
ejpam-6261	85	11	introduced	introduce	VERB
ejpam-6261	85	12	by	by	ADP
ejpam-6261	85	13	ma	ma	PROPN
ejpam-6261	85	14	et	et	PROPN
ejpam-6261	85	15	al	al	PROPN
ejpam-6261	85	16	.	.	PUNCT
ejpam-6261	86	1	[	[	X
ejpam-6261	86	2	27	27	NUM
ejpam-6261	86	3	]	]	PUNCT
ejpam-6261	86	4	,	,	PUNCT
ejpam-6261	86	5	and	and	CCONJ
ejpam-6261	86	6	some	some	PRON
ejpam-6261	86	7	of	of	ADP
ejpam-6261	86	8	its	its	PRON
ejpam-6261	86	9	features	feature	NOUN
ejpam-6261	86	10	are	be	AUX
ejpam-6261	86	11	addressed	address	VERB
ejpam-6261	86	12	.	.	PUNCT
ejpam-6261	87	1	researchers	researcher	NOUN
ejpam-6261	87	2	have	have	AUX
ejpam-6261	87	3	demonstrated	demonstrate	VERB
ejpam-6261	87	4	that	that	SCONJ
ejpam-6261	87	5	the	the	DET
ejpam-6261	87	6	generalized	generalize	VERB
ejpam-6261	87	7	neutrosophic	neutrosophic	ADJ
ejpam-6261	87	8	extended	extended	ADJ
ejpam-6261	87	9	triplet	triplet	NOUN
ejpam-6261	87	10	group	group	NOUN
ejpam-6261	87	11	and	and	CCONJ
ejpam-6261	87	12	the	the	DET
ejpam-6261	87	13	weak	weak	ADJ
ejpam-6261	87	14	commutative	commutative	ADJ
ejpam-6261	87	15	generalized	generalized	ADJ
ejpam-6261	87	16	neutrosophic	neutrosophic	ADJ
ejpam-6261	87	17	extended	extended	ADJ
ejpam-6261	87	18	triplet	triplet	NOUN
ejpam-6261	87	19	group	group	NOUN
ejpam-6261	87	20	are	be	AUX
ejpam-6261	87	21	equivalent	equivalent	ADJ
ejpam-6261	87	22	to	to	ADP
ejpam-6261	87	23	the	the	DET
ejpam-6261	87	24	quasi	quasi	ADJ
ejpam-6261	87	25	-	-	ADJ
ejpam-6261	87	26	completely	completely	ADV
ejpam-6261	87	27	regular	regular	ADJ
ejpam-6261	87	28	semigroup	semigroup	NOUN
ejpam-6261	87	29	and	and	CCONJ
ejpam-6261	87	30	the	the	DET
ejpam-6261	87	31	quasi	quasi	ADJ
ejpam-6261	87	32	-	-	PROPN
ejpam-6261	87	33	clifford	clifford	PROPN
ejpam-6261	87	34	semigroup	semigroup	PROPN
ejpam-6261	87	35	,	,	PUNCT
ejpam-6261	87	36	respectively	respectively	ADV
ejpam-6261	87	37	.	.	PUNCT
ejpam-6261	88	1	numerous	numerous	ADJ
ejpam-6261	88	2	characteristics	characteristic	NOUN
ejpam-6261	88	3	of	of	ADP
ejpam-6261	88	4	implicative	implicative	PROPN
ejpam-6261	88	5	neutrosophic	neutrosophic	PROPN
ejpam-6261	88	6	quadruple	quadruple	NOUN
ejpam-6261	88	7	bck	bck	PROPN
ejpam-6261	88	8	-	-	PUNCT
ejpam-6261	88	9	algebras	algebras	PROPN
ejpam-6261	88	10	were	be	AUX
ejpam-6261	88	11	investigated	investigate	VERB
ejpam-6261	88	12	by	by	ADP
ejpam-6261	88	13	muhiuddin	muhiuddin	PROPN
ejpam-6261	88	14	et	et	PROPN
ejpam-6261	88	15	al	al	PROPN
ejpam-6261	88	16	.	.	PUNCT
ejpam-6261	89	1	[	[	X
ejpam-6261	89	2	28	28	NUM
ejpam-6261	89	3	]	]	PUNCT
ejpam-6261	89	4	,	,	PUNCT
ejpam-6261	89	5	and	and	CCONJ
ejpam-6261	89	6	developed	develop	VERB
ejpam-6261	89	7	criteria	criterion	NOUN
ejpam-6261	89	8	for	for	ADP
ejpam-6261	89	9	the	the	DET
ejpam-6261	89	10	neutrosophic	neutrosophic	ADJ
ejpam-6261	89	11	quadruple	quadruple	NOUN
ejpam-6261	89	12	set	set	VERB
ejpam-6261	89	13	to	to	PART
ejpam-6261	89	14	be	be	AUX
ejpam-6261	89	15	a	a	DET
ejpam-6261	89	16	neutrosophic	neutrosophic	ADJ
ejpam-6261	89	17	quadruple	quadruple	NOUN
ejpam-6261	89	18	bci	bci	NOUN
ejpam-6261	89	19	-	-	NOUN
ejpam-6261	89	20	algebra	algebra	NOUN
ejpam-6261	89	21	.	.	PUNCT
ejpam-6261	90	1	kandasamy	kandasamy	PROPN
ejpam-6261	90	2	et	et	PROPN
ejpam-6261	90	3	al	al	PROPN
ejpam-6261	90	4	.	.	PUNCT
ejpam-6261	91	1	[	[	X
ejpam-6261	91	2	26	26	NUM
ejpam-6261	91	3	]	]	PUNCT
ejpam-6261	91	4	examined	examine	VERB
ejpam-6261	91	5	the	the	DET
ejpam-6261	91	6	semi	semi	NOUN
ejpam-6261	91	7	-	-	NOUN
ejpam-6261	91	8	idempotents	idempotent	NOUN
ejpam-6261	91	9	in	in	ADP
ejpam-6261	91	10	neutrosophic	neutrosophic	ADJ
ejpam-6261	91	11	subrings	subring	NOUN
ejpam-6261	91	12	and	and	CCONJ
ejpam-6261	91	13	studied	study	VERB
ejpam-6261	91	14	the	the	DET
ejpam-6261	91	15	vital	vital	ADJ
ejpam-6261	91	16	characteristics	characteristic	NOUN
ejpam-6261	91	17	in	in	ADP
ejpam-6261	91	18	details	detail	NOUN
ejpam-6261	91	19	.	.	PUNCT
ejpam-6261	92	1	bashir	bashir	PROPN
ejpam-6261	92	2	et	et	PROPN
ejpam-6261	92	3	al	al	PROPN
ejpam-6261	92	4	.	.	PUNCT
ejpam-6261	93	1	[	[	X
ejpam-6261	93	2	29	29	NUM
ejpam-6261	93	3	]	]	PUNCT
ejpam-6261	93	4	introduced	introduce	VERB
ejpam-6261	93	5	the	the	DET
ejpam-6261	93	6	subsemirings	subsemiring	NOUN
ejpam-6261	93	7	,	,	PUNCT
ejpam-6261	93	8	ideals	ideal	NOUN
ejpam-6261	93	9	,	,	PUNCT
ejpam-6261	93	10	generalized	generalized	ADJ
ejpam-6261	93	11	bi	bi	NOUN
ejpam-6261	93	12	-	-	NOUN
ejpam-6261	93	13	ideals	ideal	NOUN
ejpam-6261	93	14	,	,	PUNCT
ejpam-6261	93	15	and	and	CCONJ
ejpam-6261	93	16	quasi	quasi	NOUN
ejpam-6261	93	17	-	-	NOUN
ejpam-6261	93	18	ideals	ideal	NOUN
ejpam-6261	93	19	under	under	ADP
ejpam-6261	93	20	the	the	DET
ejpam-6261	93	21	framework	framework	NOUN
ejpam-6261	93	22	of	of	ADP
ejpam-6261	93	23	a	a	DET
ejpam-6261	93	24	m	m	ADJ
ejpam-6261	93	25	-	-	ADJ
ejpam-6261	93	26	polar	polar	ADJ
ejpam-6261	93	27	fuzzy	fuzzy	ADJ
ejpam-6261	93	28	set	set	NOUN
ejpam-6261	93	29	in	in	ADP
ejpam-6261	93	30	semirings	semiring	NOUN
ejpam-6261	93	31	.	.	PUNCT
ejpam-6261	94	1	bashir	bashir	PROPN
ejpam-6261	94	2	et	et	PROPN
ejpam-6261	94	3	al	al	PROPN
ejpam-6261	94	4	.	.	PUNCT
ejpam-6261	95	1	[	[	X
ejpam-6261	95	2	30	30	NUM
ejpam-6261	95	3	]	]	PUNCT
ejpam-6261	95	4	purposed	purpose	VERB
ejpam-6261	95	5	the	the	DET
ejpam-6261	95	6	ternary	ternary	ADJ
ejpam-6261	95	7	multiplication	multiplication	NOUN
ejpam-6261	95	8	to	to	PART
ejpam-6261	95	9	extend	extend	VERB
ejpam-6261	95	10	the	the	DET
ejpam-6261	95	11	roughness	roughness	NOUN
ejpam-6261	95	12	of	of	ADP
ejpam-6261	95	13	a	a	DET
ejpam-6261	95	14	fuzzy	fuzzy	ADJ
ejpam-6261	95	15	set	set	NOUN
ejpam-6261	95	16	in	in	ADP
ejpam-6261	95	17	three	three	NUM
ejpam-6261	95	18	dimensions	dimension	NOUN
ejpam-6261	95	19	.	.	PUNCT
ejpam-6261	96	1	a	a	DET
ejpam-6261	96	2	number	number	NOUN
ejpam-6261	96	3	of	of	ADP
ejpam-6261	96	4	vital	vital	ADJ
ejpam-6261	96	5	characteristics	characteristic	NOUN
ejpam-6261	96	6	were	be	AUX
ejpam-6261	96	7	discussed	discuss	VERB
ejpam-6261	96	8	by	by	ADP
ejpam-6261	96	9	using	use	VERB
ejpam-6261	96	10	the	the	DET
ejpam-6261	96	11	idea	idea	NOUN
ejpam-6261	96	12	of	of	ADP
ejpam-6261	96	13	set	set	NOUN
ejpam-6261	96	14	-	-	PUNCT
ejpam-6261	96	15	valued	value	VERB
ejpam-6261	96	16	and	and	CCONJ
ejpam-6261	96	17	strong	strong	ADJ
ejpam-6261	96	18	set	set	NOUN
ejpam-6261	96	19	-	-	PUNCT
ejpam-6261	96	20	valued	value	VERB
ejpam-6261	96	21	homomorphism	homomorphism	NOUN
ejpam-6261	96	22	.	.	PUNCT
ejpam-6261	97	1	the	the	DET
ejpam-6261	97	2	notion	notion	NOUN
ejpam-6261	97	3	of	of	ADP
ejpam-6261	97	4	cf	cf	NOUN
ejpam-6261	97	5	sets	set	NOUN
ejpam-6261	97	6	and	and	CCONJ
ejpam-6261	97	7	their	their	PRON
ejpam-6261	97	8	fundamental	fundamental	ADJ
ejpam-6261	97	9	algebraic	algebraic	ADJ
ejpam-6261	97	10	operations	operation	NOUN
ejpam-6261	97	11	were	be	AUX
ejpam-6261	97	12	investigated	investigate	VERB
ejpam-6261	97	13	in	in	ADP
ejpam-6261	97	14	[	[	X
ejpam-6261	97	15	31	31	NUM
ejpam-6261	97	16	,	,	PUNCT
ejpam-6261	97	17	32	32	NUM
ejpam-6261	97	18	]	]	PUNCT
ejpam-6261	97	19	,	,	PUNCT
ejpam-6261	97	20	extending	extend	VERB
ejpam-6261	97	21	the	the	DET
ejpam-6261	97	22	range	range	NOUN
ejpam-6261	97	23	of	of	ADP
ejpam-6261	97	24	the	the	DET
ejpam-6261	97	25	belonging	belong	VERB
ejpam-6261	97	26	function	function	NOUN
ejpam-6261	97	27	from	from	ADP
ejpam-6261	97	28	real	real	ADJ
ejpam-6261	97	29	numbers	number	NOUN
ejpam-6261	97	30	to	to	ADP
ejpam-6261	97	31	complex	complex	ADJ
ejpam-6261	97	32	numbers	number	NOUN
ejpam-6261	97	33	with	with	ADP
ejpam-6261	97	34	the	the	DET
ejpam-6261	97	35	unit	unit	NOUN
ejpam-6261	97	36	disc	disc	NOUN
ejpam-6261	97	37	.	.	PUNCT
ejpam-6261	98	1	due	due	ADP
ejpam-6261	98	2	to	to	ADP
ejpam-6261	98	3	the	the	DET
ejpam-6261	98	4	cf	cf	NOUN
ejpam-6261	98	5	set	set	NOUN
ejpam-6261	98	6	only	only	ADV
ejpam-6261	98	7	considering	consider	VERB
ejpam-6261	98	8	the	the	DET
ejpam-6261	98	9	degree	degree	NOUN
ejpam-6261	98	10	of	of	ADP
ejpam-6261	98	11	belonging	belong	VERB
ejpam-6261	98	12	and	and	CCONJ
ejpam-6261	98	13	giving	give	VERB
ejpam-6261	98	14	no	no	DET
ejpam-6261	98	15	consideration	consideration	NOUN
ejpam-6261	98	16	to	to	ADP
ejpam-6261	98	17	the	the	DET
ejpam-6261	98	18	data	data	NOUN
ejpam-6261	98	19	entities	entity	NOUN
ejpam-6261	98	20	that	that	PRON
ejpam-6261	98	21	are	be	AUX
ejpam-6261	98	22	not	not	PART
ejpam-6261	98	23	members	member	NOUN
ejpam-6261	98	24	,	,	PUNCT
ejpam-6261	98	25	which	which	PRON
ejpam-6261	98	26	are	be	AUX
ejpam-6261	98	27	equally	equally	ADV
ejpam-6261	98	28	important	important	ADJ
ejpam-6261	98	29	in	in	ADP
ejpam-6261	98	30	the	the	DET
ejpam-6261	98	31	process	process	NOUN
ejpam-6261	98	32	of	of	ADP
ejpam-6261	98	33	making	make	VERB
ejpam-6261	98	34	decisions	decision	NOUN
ejpam-6261	98	35	on	on	ADP
ejpam-6261	98	36	system	system	NOUN
ejpam-6261	98	37	evaluation	evaluation	NOUN
ejpam-6261	98	38	.	.	PUNCT
ejpam-6261	99	1	however	however	ADV
ejpam-6261	99	2	,	,	PUNCT
ejpam-6261	99	3	it	it	PRON
ejpam-6261	99	4	is	be	AUX
ejpam-6261	99	5	usually	usually	ADV
ejpam-6261	99	6	difficult	difficult	ADJ
ejpam-6261	99	7	to	to	PART
ejpam-6261	99	8	measure	measure	VERB
ejpam-6261	99	9	the	the	DET
ejpam-6261	99	10	true	true	ADJ
ejpam-6261	99	11	value	value	NOUN
ejpam-6261	99	12	of	of	ADP
ejpam-6261	99	13	a	a	DET
ejpam-6261	99	14	truth	truth	NOUN
ejpam-6261	99	15	estimate	estimate	NOUN
ejpam-6261	99	16	by	by	ADP
ejpam-6261	99	17	the	the	DET
ejpam-6261	99	18	exact	exact	ADJ
ejpam-6261	99	19	value	value	NOUN
ejpam-6261	99	20	of	of	ADP
ejpam-6261	99	21	a	a	DET
ejpam-6261	99	22	fuzzy	fuzzy	ADJ
ejpam-6261	99	23	set	set	NOUN
ejpam-6261	99	24	in	in	ADP
ejpam-6261	99	25	real	real	ADJ
ejpam-6261	99	26	life	life	NOUN
ejpam-6261	99	27	.	.	PUNCT
ejpam-6261	100	1	in	in	ADP
ejpam-6261	100	2	some	some	DET
ejpam-6261	100	3	circumstances	circumstance	NOUN
ejpam-6261	100	4	,	,	PUNCT
ejpam-6261	100	5	it	it	PRON
ejpam-6261	100	6	could	could	AUX
ejpam-6261	100	7	be	be	AUX
ejpam-6261	100	8	simpler	simple	ADJ
ejpam-6261	100	9	to	to	PART
ejpam-6261	100	10	express	express	VERB
ejpam-6261	100	11	the	the	DET
ejpam-6261	100	12	ambiguity	ambiguity	NOUN
ejpam-6261	100	13	and	and	CCONJ
ejpam-6261	100	14	vagueness	vagueness	NOUN
ejpam-6261	100	15	m.	m.	PROPN
ejpam-6261	100	16	h.	h.	PROPN
ejpam-6261	100	17	mateen	mateen	PROPN
ejpam-6261	100	18	et	et	PROPN
ejpam-6261	100	19	al	al	PROPN
ejpam-6261	100	20	.	.	PUNCT
ejpam-6261	100	21	/	/	SYM
ejpam-6261	100	22	eur	eur	PROPN
ejpam-6261	100	23	.	.	PUNCT
ejpam-6261	101	1	j.	j.	PROPN
ejpam-6261	101	2	pure	pure	PROPN
ejpam-6261	101	3	appl	appl	PROPN
ejpam-6261	101	4	.	.	PROPN
ejpam-6261	101	5	math	math	PROPN
ejpam-6261	101	6	,	,	PUNCT
ejpam-6261	101	7	18	18	NUM
ejpam-6261	101	8	(	(	PUNCT
ejpam-6261	101	9	4	4	NUM
ejpam-6261	101	10	)	)	PUNCT
ejpam-6261	101	11	(	(	PUNCT
ejpam-6261	101	12	2025	2025	NUM
ejpam-6261	101	13	)	)	PUNCT
ejpam-6261	101	14	,	,	PUNCT
ejpam-6261	101	15	6261	6261	NUM
ejpam-6261	101	16	4	4	NUM
ejpam-6261	101	17	of	of	ADP
ejpam-6261	101	18	23	23	NUM
ejpam-6261	101	19	that	that	PRON
ejpam-6261	101	20	characterize	characterize	VERB
ejpam-6261	101	21	real	real	ADJ
ejpam-6261	101	22	-	-	PUNCT
ejpam-6261	101	23	world	world	NOUN
ejpam-6261	101	24	situations	situation	NOUN
ejpam-6261	101	25	using	use	VERB
ejpam-6261	101	26	two	two	NUM
ejpam-6261	101	27	-	-	PUNCT
ejpam-6261	101	28	dimensional	dimensional	ADJ
ejpam-6261	101	29	data	datum	NOUN
ejpam-6261	101	30	rather	rather	ADV
ejpam-6261	101	31	than	than	ADP
ejpam-6261	101	32	one	one	NUM
ejpam-6261	101	33	.	.	PUNCT
ejpam-6261	102	1	the	the	DET
ejpam-6261	102	2	complex	complex	ADJ
ejpam-6261	102	3	intuitionistic	intuitionistic	ADJ
ejpam-6261	102	4	fuzzy	fuzzy	ADJ
ejpam-6261	102	5	(	(	PUNCT
ejpam-6261	102	6	cif	cif	PROPN
ejpam-6261	102	7	)	)	PUNCT
ejpam-6261	102	8	sets	set	NOUN
ejpam-6261	102	9	and	and	CCONJ
ejpam-6261	102	10	their	their	PRON
ejpam-6261	102	11	set	set	ADJ
ejpam-6261	102	12	operations	operation	NOUN
ejpam-6261	102	13	were	be	AUX
ejpam-6261	102	14	presented	present	VERB
ejpam-6261	102	15	by	by	ADP
ejpam-6261	102	16	alkouri	alkouri	PROPN
ejpam-6261	102	17	and	and	CCONJ
ejpam-6261	102	18	salleh	salleh	PROPN
ejpam-6261	102	19	[	[	X
ejpam-6261	102	20	33	33	NUM
ejpam-6261	102	21	,	,	PUNCT
ejpam-6261	102	22	34	34	NUM
ejpam-6261	102	23	]	]	PUNCT
ejpam-6261	102	24	.	.	PUNCT
ejpam-6261	103	1	also	also	ADV
ejpam-6261	103	2	,	,	PUNCT
ejpam-6261	103	3	it	it	PRON
ejpam-6261	103	4	indicated	indicate	VERB
ejpam-6261	103	5	the	the	DET
ejpam-6261	103	6	unpredictability	unpredictability	NOUN
ejpam-6261	103	7	of	of	ADP
ejpam-6261	103	8	complex	complex	ADV
ejpam-6261	103	9	-	-	PUNCT
ejpam-6261	103	10	valued	value	VERB
ejpam-6261	103	11	functions	function	NOUN
ejpam-6261	103	12	in	in	ADP
ejpam-6261	103	13	a	a	DET
ejpam-6261	103	14	variety	variety	NOUN
ejpam-6261	103	15	of	of	ADP
ejpam-6261	103	16	physical	physical	ADJ
ejpam-6261	103	17	measurements	measurement	NOUN
ejpam-6261	103	18	.	.	PUNCT
ejpam-6261	104	1	as	as	ADP
ejpam-6261	104	2	an	an	DET
ejpam-6261	104	3	example	example	NOUN
ejpam-6261	104	4	,	,	PUNCT
ejpam-6261	104	5	complex	complex	ADJ
ejpam-6261	104	6	intensity	intensity	NOUN
ejpam-6261	104	7	,	,	PUNCT
ejpam-6261	104	8	impedance	impedance	NOUN
ejpam-6261	104	9	and	and	CCONJ
ejpam-6261	104	10	wave	wave	NOUN
ejpam-6261	104	11	function	function	NOUN
ejpam-6261	104	12	in	in	ADP
ejpam-6261	104	13	the	the	DET
ejpam-6261	104	14	fields	field	NOUN
ejpam-6261	104	15	of	of	ADP
ejpam-6261	104	16	quantum	quantum	ADJ
ejpam-6261	104	17	physics	physics	NOUN
ejpam-6261	104	18	and	and	CCONJ
ejpam-6261	104	19	electronics	electronic	NOUN
ejpam-6261	104	20	.	.	PUNCT
ejpam-6261	105	1	the	the	DET
ejpam-6261	105	2	following	follow	VERB
ejpam-6261	105	3	are	be	AUX
ejpam-6261	105	4	the	the	DET
ejpam-6261	105	5	motivations	motivation	NOUN
ejpam-6261	105	6	for	for	ADP
ejpam-6261	105	7	the	the	DET
ejpam-6261	105	8	new	new	ADJ
ejpam-6261	105	9	work	work	NOUN
ejpam-6261	105	10	.	.	PUNCT
ejpam-6261	106	1	(	(	PUNCT
ejpam-6261	106	2	i	i	NOUN
ejpam-6261	106	3	)	)	PUNCT
ejpam-6261	106	4	the	the	DET
ejpam-6261	106	5	concept	concept	NOUN
ejpam-6261	106	6	of	of	ADP
ejpam-6261	106	7	a	a	DET
ejpam-6261	106	8	cif	cif	PROPN
ejpam-6261	106	9	subgroup	subgroup	PROPN
ejpam-6261	106	10	and	and	CCONJ
ejpam-6261	106	11	cif	cif	PROPN
ejpam-6261	106	12	level	level	NOUN
ejpam-6261	106	13	subsets	subset	NOUN
ejpam-6261	106	14	were	be	AUX
ejpam-6261	106	15	introduced	introduce	VERB
ejpam-6261	106	16	by	by	ADP
ejpam-6261	106	17	gulzar	gulzar	PROPN
ejpam-6261	106	18	et	et	PROPN
ejpam-6261	106	19	al	al	PROPN
ejpam-6261	106	20	.	.	PUNCT
ejpam-6261	107	1	[	[	X
ejpam-6261	107	2	35	35	NUM
ejpam-6261	107	3	]	]	PUNCT
ejpam-6261	107	4	.	.	PUNCT
ejpam-6261	108	1	additionally	additionally	ADV
ejpam-6261	108	2	,	,	PUNCT
ejpam-6261	108	3	the	the	DET
ejpam-6261	108	4	cartesian	cartesian	ADJ
ejpam-6261	108	5	product	product	NOUN
ejpam-6261	108	6	of	of	ADP
ejpam-6261	108	7	two	two	NUM
ejpam-6261	108	8	cif	cif	PROPN
ejpam-6261	108	9	subgroups	subgroup	NOUN
ejpam-6261	108	10	was	be	AUX
ejpam-6261	108	11	introduced	introduce	VERB
ejpam-6261	108	12	,	,	PUNCT
ejpam-6261	108	13	along	along	ADP
ejpam-6261	108	14	with	with	ADP
ejpam-6261	108	15	the	the	DET
ejpam-6261	108	16	homomorphic	homomorphic	ADJ
ejpam-6261	108	17	image	image	NOUN
ejpam-6261	108	18	and	and	CCONJ
ejpam-6261	108	19	inverse	inverse	NOUN
ejpam-6261	108	20	image	image	NOUN
ejpam-6261	108	21	of	of	ADP
ejpam-6261	108	22	the	the	DET
ejpam-6261	108	23	cif	cif	PROPN
ejpam-6261	108	24	subgroup	subgroup	PROPN
ejpam-6261	108	25	under	under	ADP
ejpam-6261	108	26	group	group	PROPN
ejpam-6261	108	27	homomorphism	homomorphism	PROPN
ejpam-6261	108	28	.	.	PUNCT
ejpam-6261	109	1	(	(	PUNCT
ejpam-6261	109	2	ii	ii	X
ejpam-6261	109	3	)	)	PUNCT
ejpam-6261	109	4	gulzar	gulzar	PROPN
ejpam-6261	109	5	et	et	PROPN
ejpam-6261	109	6	al	al	PROPN
ejpam-6261	109	7	.	.	PUNCT
ejpam-6261	110	1	[	[	X
ejpam-6261	110	2	36	36	NUM
ejpam-6261	110	3	]	]	PUNCT
ejpam-6261	110	4	introduced	introduce	VERB
ejpam-6261	110	5	the	the	DET
ejpam-6261	110	6	idea	idea	NOUN
ejpam-6261	110	7	of	of	ADP
ejpam-6261	110	8	the	the	DET
ejpam-6261	110	9	direct	direct	ADJ
ejpam-6261	110	10	product	product	NOUN
ejpam-6261	110	11	of	of	ADP
ejpam-6261	110	12	two	two	NUM
ejpam-6261	110	13	cif	cif	PROPN
ejpam-6261	110	14	subrings	subring	NOUN
ejpam-6261	110	15	,	,	PUNCT
ejpam-6261	110	16	demonstrated	demonstrate	VERB
ejpam-6261	110	17	that	that	SCONJ
ejpam-6261	110	18	it	it	PRON
ejpam-6261	110	19	is	be	AUX
ejpam-6261	110	20	also	also	ADV
ejpam-6261	110	21	a	a	DET
ejpam-6261	110	22	cif	cif	PROPN
ejpam-6261	110	23	subring	subring	NOUN
ejpam-6261	110	24	,	,	PUNCT
ejpam-6261	110	25	and	and	CCONJ
ejpam-6261	110	26	discussed	discuss	VERB
ejpam-6261	110	27	about	about	ADP
ejpam-6261	110	28	its	its	PRON
ejpam-6261	110	29	different	different	ADJ
ejpam-6261	110	30	algebraic	algebraic	ADJ
ejpam-6261	110	31	characteristics	characteristic	NOUN
ejpam-6261	110	32	.	.	PUNCT
ejpam-6261	111	1	hameed	hameed	PROPN
ejpam-6261	111	2	et	et	PROPN
ejpam-6261	111	3	al	al	PROPN
ejpam-6261	111	4	.	.	PUNCT
ejpam-6261	112	1	[	[	X
ejpam-6261	112	2	37	37	NUM
ejpam-6261	112	3	]	]	PUNCT
ejpam-6261	112	4	discussed	discuss	VERB
ejpam-6261	112	5	the	the	DET
ejpam-6261	112	6	(	(	PUNCT
ejpam-6261	112	7	α	α	NOUN
ejpam-6261	112	8	,	,	PUNCT
ejpam-6261	112	9	β	β	X
ejpam-6261	112	10	,	,	PUNCT
ejpam-6261	112	11	γ	γ	NOUN
ejpam-6261	112	12	)	)	PUNCT
ejpam-6261	112	13	neutrosophic	neutrosophic	ADJ
ejpam-6261	112	14	submodules	submodule	NOUN
ejpam-6261	112	15	and	and	CCONJ
ejpam-6261	112	16	some	some	DET
ejpam-6261	112	17	fundamental	fundamental	ADJ
ejpam-6261	112	18	algebraic	algebraic	ADJ
ejpam-6261	112	19	properties	property	NOUN
ejpam-6261	112	20	were	be	AUX
ejpam-6261	112	21	investigated	investigate	VERB
ejpam-6261	112	22	.	.	PUNCT
ejpam-6261	113	1	(	(	PUNCT
ejpam-6261	113	2	iii	iii	X
ejpam-6261	113	3	)	)	PUNCT
ejpam-6261	113	4	elrawy	elrawy	NOUN
ejpam-6261	113	5	and	and	CCONJ
ejpam-6261	113	6	abdalla	abdalla	PROPN
ejpam-6261	114	1	[	[	X
ejpam-6261	114	2	38	38	NUM
ejpam-6261	114	3	]	]	PUNCT
ejpam-6261	114	4	defined	define	VERB
ejpam-6261	114	5	the	the	DET
ejpam-6261	114	6	algebraic	algebraic	ADJ
ejpam-6261	114	7	structure	structure	NOUN
ejpam-6261	114	8	based	base	VERB
ejpam-6261	114	9	on	on	ADP
ejpam-6261	114	10	single	single	ADJ
ejpam-6261	114	11	-	-	PUNCT
ejpam-6261	114	12	valued	value	VERB
ejpam-6261	114	13	nss	nss	NOUN
ejpam-6261	114	14	and	and	CCONJ
ejpam-6261	114	15	proposed	propose	VERB
ejpam-6261	114	16	a	a	DET
ejpam-6261	114	17	novel	novel	ADJ
ejpam-6261	114	18	method	method	NOUN
ejpam-6261	114	19	for	for	ADP
ejpam-6261	114	20	constructing	construct	VERB
ejpam-6261	114	21	the	the	DET
ejpam-6261	114	22	neutrosophic	neutrosophic	ADJ
ejpam-6261	114	23	subring	subring	NOUN
ejpam-6261	114	24	and	and	CCONJ
ejpam-6261	114	25	ideal	ideal	ADJ
ejpam-6261	114	26	by	by	ADP
ejpam-6261	114	27	combining	combine	VERB
ejpam-6261	114	28	nss	nss	NOUN
ejpam-6261	114	29	with	with	ADP
ejpam-6261	114	30	the	the	DET
ejpam-6261	114	31	classical	classical	ADJ
ejpam-6261	114	32	ring	ring	NOUN
ejpam-6261	114	33	.	.	PUNCT
ejpam-6261	115	1	(	(	PUNCT
ejpam-6261	115	2	iv	iv	X
ejpam-6261	115	3	)	)	PUNCT
ejpam-6261	115	4	ali	ali	PROPN
ejpam-6261	115	5	and	and	CCONJ
ejpam-6261	115	6	smarandache	smarandache	NOUN
ejpam-6261	115	7	established	establish	VERB
ejpam-6261	115	8	innovative	innovative	ADJ
ejpam-6261	115	9	complex	complex	ADJ
ejpam-6261	115	10	neutrosophic	neutrosophic	ADJ
ejpam-6261	115	11	sets	set	NOUN
ejpam-6261	115	12	(	(	PUNCT
ejpam-6261	115	13	cns	cns	NOUN
ejpam-6261	115	14	)	)	PUNCT
ejpam-6261	115	15	,	,	PUNCT
ejpam-6261	115	16	which	which	PRON
ejpam-6261	115	17	extend	extend	VERB
ejpam-6261	115	18	the	the	DET
ejpam-6261	115	19	range	range	NOUN
ejpam-6261	115	20	of	of	ADP
ejpam-6261	115	21	components	component	NOUN
ejpam-6261	115	22	in	in	ADP
ejpam-6261	115	23	the	the	DET
ejpam-6261	115	24	complex	complex	ADJ
ejpam-6261	115	25	plane	plane	NOUN
ejpam-6261	115	26	from	from	ADP
ejpam-6261	115	27	the	the	DET
ejpam-6261	115	28	unit	unit	NOUN
ejpam-6261	115	29	interval	interval	NOUN
ejpam-6261	115	30	to	to	ADP
ejpam-6261	115	31	the	the	DET
ejpam-6261	115	32	unit	unit	NOUN
ejpam-6261	115	33	disc	disc	NOUN
ejpam-6261	115	34	.	.	PUNCT
ejpam-6261	115	35	amplitude	amplitude	NOUN
ejpam-6261	115	36	and	and	CCONJ
ejpam-6261	115	37	phase	phase	NOUN
ejpam-6261	115	38	values	value	NOUN
ejpam-6261	115	39	are	be	AUX
ejpam-6261	115	40	assigned	assign	VERB
ejpam-6261	115	41	to	to	ADP
ejpam-6261	115	42	each	each	PRON
ejpam-6261	115	43	of	of	ADP
ejpam-6261	115	44	its	its	PRON
ejpam-6261	115	45	components	component	NOUN
ejpam-6261	115	46	.	.	PUNCT
ejpam-6261	116	1	furthermore	furthermore	ADV
ejpam-6261	116	2	,	,	PUNCT
ejpam-6261	116	3	cnss	cns	NOUN
ejpam-6261	116	4	have	have	AUX
ejpam-6261	116	5	been	be	AUX
ejpam-6261	116	6	used	use	VERB
ejpam-6261	116	7	in	in	ADP
ejpam-6261	116	8	the	the	DET
ejpam-6261	116	9	fields	field	NOUN
ejpam-6261	116	10	of	of	ADP
ejpam-6261	116	11	science	science	NOUN
ejpam-6261	116	12	and	and	CCONJ
ejpam-6261	116	13	engineering	engineering	NOUN
ejpam-6261	116	14	.	.	PUNCT
ejpam-6261	117	1	(	(	PUNCT
ejpam-6261	117	2	v	v	NOUN
ejpam-6261	117	3	)	)	PUNCT
ejpam-6261	117	4	gulistana	gulistana	PROPN
ejpam-6261	117	5	et	et	PROPN
ejpam-6261	117	6	al	al	PROPN
ejpam-6261	117	7	.	.	PUNCT
ejpam-6261	118	1	[	[	X
ejpam-6261	118	2	39	39	NUM
ejpam-6261	118	3	]	]	PUNCT
ejpam-6261	118	4	initiated	initiate	VERB
ejpam-6261	118	5	the	the	DET
ejpam-6261	118	6	idea	idea	NOUN
ejpam-6261	118	7	of	of	ADP
ejpam-6261	118	8	complex	complex	ADJ
ejpam-6261	118	9	neutrosophic	neutrosophic	ADJ
ejpam-6261	118	10	subgroups	subgroup	NOUN
ejpam-6261	118	11	and	and	CCONJ
ejpam-6261	118	12	defined	define	VERB
ejpam-6261	118	13	the	the	DET
ejpam-6261	118	14	term	term	NOUN
ejpam-6261	118	15	alpha	alpha	NOUN
ejpam-6261	118	16	-	-	PUNCT
ejpam-6261	118	17	cut	cut	NOUN
ejpam-6261	118	18	of	of	ADP
ejpam-6261	118	19	cns	cns	NOUN
ejpam-6261	118	20	.	.	PUNCT
ejpam-6261	119	1	the	the	DET
ejpam-6261	119	2	cartesian	cartesian	ADJ
ejpam-6261	119	3	product	product	NOUN
ejpam-6261	119	4	of	of	ADP
ejpam-6261	119	5	complex	complex	ADJ
ejpam-6261	119	6	neutrosophic	neutrosophic	ADJ
ejpam-6261	119	7	subgroups	subgroup	NOUN
ejpam-6261	119	8	is	be	AUX
ejpam-6261	119	9	also	also	ADV
ejpam-6261	119	10	defined	define	VERB
ejpam-6261	119	11	.	.	PUNCT
ejpam-6261	120	1	rahoumah	rahoumah	PROPN
ejpam-6261	120	2	et	et	PROPN
ejpam-6261	120	3	al	al	PROPN
ejpam-6261	120	4	.	.	PUNCT
ejpam-6261	121	1	[	[	X
ejpam-6261	121	2	40	40	NUM
ejpam-6261	121	3	]	]	PUNCT
ejpam-6261	121	4	introduced	introduce	VERB
ejpam-6261	121	5	the	the	DET
ejpam-6261	121	6	complex	complex	ADJ
ejpam-6261	121	7	neutrosophic	neutrosophic	ADJ
ejpam-6261	121	8	soft	soft	ADJ
ejpam-6261	121	9	subgroups	subgroup	NOUN
ejpam-6261	121	10	,	,	PUNCT
ejpam-6261	121	11	and	and	CCONJ
ejpam-6261	121	12	the	the	DET
ejpam-6261	121	13	fundamental	fundamental	ADJ
ejpam-6261	121	14	results	result	NOUN
ejpam-6261	121	15	related	relate	VERB
ejpam-6261	121	16	to	to	ADP
ejpam-6261	121	17	this	this	DET
ejpam-6261	121	18	phenomenon	phenomenon	NOUN
ejpam-6261	121	19	were	be	AUX
ejpam-6261	121	20	discussed	discuss	VERB
ejpam-6261	121	21	.	.	PUNCT
ejpam-6261	122	1	(	(	PUNCT
ejpam-6261	122	2	vi	vi	X
ejpam-6261	122	3	)	)	PUNCT
ejpam-6261	122	4	the	the	DET
ejpam-6261	122	5	complex	complex	ADJ
ejpam-6261	122	6	neutrosophic	neutrosophic	ADJ
ejpam-6261	122	7	set	set	NOUN
ejpam-6261	122	8	is	be	AUX
ejpam-6261	122	9	the	the	DET
ejpam-6261	122	10	generalization	generalization	NOUN
ejpam-6261	122	11	of	of	ADP
ejpam-6261	122	12	existing	exist	VERB
ejpam-6261	122	13	theorizes	theorize	NOUN
ejpam-6261	122	14	i.e.	i.e.	X
ejpam-6261	122	15	,	,	PUNCT
ejpam-6261	122	16	complex	complex	ADJ
ejpam-6261	122	17	fuzzy	fuzzy	ADJ
ejpam-6261	122	18	sets	set	NOUN
ejpam-6261	122	19	and	and	CCONJ
ejpam-6261	122	20	complex	complex	ADJ
ejpam-6261	122	21	intuitionistic	intuitionistic	ADJ
ejpam-6261	122	22	fuzzy	fuzzy	ADJ
ejpam-6261	122	23	sets	set	NOUN
ejpam-6261	122	24	.	.	PUNCT
ejpam-6261	123	1	the	the	DET
ejpam-6261	123	2	concept	concept	NOUN
ejpam-6261	123	3	of	of	ADP
ejpam-6261	123	4	complex	complex	ADJ
ejpam-6261	123	5	neutrosophic	neutrosophic	ADJ
ejpam-6261	123	6	set	set	NOUN
ejpam-6261	123	7	is	be	AUX
ejpam-6261	123	8	not	not	PART
ejpam-6261	123	9	yet	yet	ADV
ejpam-6261	123	10	applied	apply	VERB
ejpam-6261	123	11	to	to	ADP
ejpam-6261	123	12	submodules	submodule	NOUN
ejpam-6261	123	13	.	.	PUNCT
ejpam-6261	124	1	our	our	PRON
ejpam-6261	124	2	proposed	propose	VERB
ejpam-6261	124	3	model	model	NOUN
ejpam-6261	124	4	is	be	AUX
ejpam-6261	124	5	given	give	VERB
ejpam-6261	124	6	in	in	ADP
ejpam-6261	124	7	figure	figure	NOUN
ejpam-6261	124	8	1	1	NUM
ejpam-6261	124	9	.	.	PUNCT
ejpam-6261	125	1	in	in	ADP
ejpam-6261	125	2	this	this	DET
ejpam-6261	125	3	paper	paper	NOUN
ejpam-6261	125	4	,	,	PUNCT
ejpam-6261	125	5	we	we	PRON
ejpam-6261	125	6	initiate	initiate	VERB
ejpam-6261	125	7	the	the	DET
ejpam-6261	125	8	work	work	NOUN
ejpam-6261	125	9	on	on	ADP
ejpam-6261	125	10	complex	complex	ADJ
ejpam-6261	125	11	neutrosophic	neutrosophic	ADJ
ejpam-6261	125	12	subrings	subring	NOUN
ejpam-6261	125	13	(	(	PUNCT
ejpam-6261	125	14	cnsrs	cnsrs	NOUN
ejpam-6261	125	15	)	)	PUNCT
ejpam-6261	125	16	.	.	PUNCT
ejpam-6261	126	1	the	the	DET
ejpam-6261	126	2	paper	paper	NOUN
ejpam-6261	126	3	is	be	AUX
ejpam-6261	126	4	shaped	shape	VERB
ejpam-6261	126	5	as	as	ADP
ejpam-6261	126	6	follow	follow	NOUN
ejpam-6261	126	7	:	:	PUNCT
ejpam-6261	126	8	the	the	DET
ejpam-6261	126	9	concept	concept	NOUN
ejpam-6261	126	10	of	of	ADP
ejpam-6261	126	11	cns	cns	PROPN
ejpam-6261	126	12	is	be	AUX
ejpam-6261	126	13	defined	define	VERB
ejpam-6261	126	14	in	in	ADP
ejpam-6261	126	15	section	section	NOUN
ejpam-6261	126	16	2	2	NUM
ejpam-6261	126	17	.	.	PUNCT
ejpam-6261	127	1	some	some	DET
ejpam-6261	127	2	important	important	ADJ
ejpam-6261	127	3	algebraic	algebraic	ADJ
ejpam-6261	127	4	properties	property	NOUN
ejpam-6261	127	5	of	of	ADP
ejpam-6261	127	6	this	this	DET
ejpam-6261	127	7	abstraction	abstraction	NOUN
ejpam-6261	127	8	are	be	AUX
ejpam-6261	127	9	mentioned	mention	VERB
ejpam-6261	127	10	in	in	ADP
ejpam-6261	127	11	this	this	DET
ejpam-6261	127	12	section	section	NOUN
ejpam-6261	127	13	.	.	PUNCT
ejpam-6261	128	1	in	in	ADP
ejpam-6261	128	2	section	section	NOUN
ejpam-6261	128	3	3	3	NUM
ejpam-6261	128	4	the	the	DET
ejpam-6261	128	5	novel	novel	ADJ
ejpam-6261	128	6	concept	concept	NOUN
ejpam-6261	128	7	of	of	ADP
ejpam-6261	128	8	cnsr	cnsr	PROPN
ejpam-6261	128	9	is	be	AUX
ejpam-6261	128	10	present	present	ADJ
ejpam-6261	128	11	together	together	ADV
ejpam-6261	128	12	with	with	ADP
ejpam-6261	128	13	their	their	PRON
ejpam-6261	128	14	fundamental	fundamental	ADJ
ejpam-6261	128	15	results	result	NOUN
ejpam-6261	128	16	.	.	PUNCT
ejpam-6261	129	1	we	we	PRON
ejpam-6261	129	2	show	show	VERB
ejpam-6261	129	3	that	that	SCONJ
ejpam-6261	129	4	the	the	DET
ejpam-6261	129	5	intersection	intersection	NOUN
ejpam-6261	129	6	of	of	ADP
ejpam-6261	129	7	two	two	NUM
ejpam-6261	129	8	cnsrs	cnsrs	NOUN
ejpam-6261	129	9	is	be	AUX
ejpam-6261	129	10	cnsr	cnsr	VERB
ejpam-6261	129	11	.	.	PUNCT
ejpam-6261	130	1	additionally	additionally	ADV
ejpam-6261	130	2	,	,	PUNCT
ejpam-6261	130	3	we	we	PRON
ejpam-6261	130	4	depict	depict	VERB
ejpam-6261	130	5	level	level	NOUN
ejpam-6261	130	6	subset	subset	NOUN
ejpam-6261	130	7	of	of	ADP
ejpam-6261	130	8	complex	complex	ADJ
ejpam-6261	130	9	neutrosophic(cn	neutrosophic(cn	ADJ
ejpam-6261	130	10	)	)	PUNCT
ejpam-6261	130	11	subset	subset	NOUN
ejpam-6261	130	12	and	and	CCONJ
ejpam-6261	130	13	prove	prove	VERB
ejpam-6261	130	14	that	that	SCONJ
ejpam-6261	130	15	the	the	DET
ejpam-6261	130	16	level	level	NOUN
ejpam-6261	130	17	subsets	subset	NOUN
ejpam-6261	130	18	of	of	ADP
ejpam-6261	130	19	cnsr	cnsr	PROPN
ejpam-6261	130	20	is	be	AUX
ejpam-6261	130	21	cnr	cnr	PROPN
ejpam-6261	130	22	.	.	PUNCT
ejpam-6261	131	1	the	the	DET
ejpam-6261	131	2	4th	4th	ADJ
ejpam-6261	131	3	section	section	NOUN
ejpam-6261	131	4	reveals	reveal	VERB
ejpam-6261	131	5	the	the	DET
ejpam-6261	131	6	idea	idea	NOUN
ejpam-6261	131	7	about	about	ADP
ejpam-6261	131	8	direct	direct	ADJ
ejpam-6261	131	9	product	product	NOUN
ejpam-6261	131	10	of	of	ADP
ejpam-6261	131	11	cnsrs	cnsrs	NOUN
ejpam-6261	131	12	and	and	CCONJ
ejpam-6261	131	13	explore	explore	VERB
ejpam-6261	131	14	algebraic	algebraic	ADJ
ejpam-6261	131	15	characteristics	characteristic	NOUN
ejpam-6261	131	16	.	.	PUNCT
ejpam-6261	132	1	we	we	PRON
ejpam-6261	132	2	shown	show	VERB
ejpam-6261	132	3	that	that	SCONJ
ejpam-6261	132	4	direct	direct	ADJ
ejpam-6261	132	5	product	product	NOUN
ejpam-6261	132	6	of	of	ADP
ejpam-6261	132	7	cnsrs	cnsrs	NOUN
ejpam-6261	132	8	is	be	AUX
ejpam-6261	132	9	cnsr	cnsr	ADJ
ejpam-6261	132	10	.	.	PUNCT
ejpam-6261	133	1	m.	m.	PROPN
ejpam-6261	133	2	h.	h.	PROPN
ejpam-6261	133	3	mateen	mateen	PROPN
ejpam-6261	133	4	et	et	PROPN
ejpam-6261	133	5	al	al	PROPN
ejpam-6261	133	6	.	.	PUNCT
ejpam-6261	133	7	/	/	SYM
ejpam-6261	133	8	eur	eur	PROPN
ejpam-6261	133	9	.	.	PUNCT
ejpam-6261	134	1	j.	j.	PROPN
ejpam-6261	134	2	pure	pure	PROPN
ejpam-6261	134	3	appl	appl	PROPN
ejpam-6261	134	4	.	.	PROPN
ejpam-6261	134	5	math	math	PROPN
ejpam-6261	134	6	,	,	PUNCT
ejpam-6261	134	7	18	18	NUM
ejpam-6261	134	8	(	(	PUNCT
ejpam-6261	134	9	4	4	NUM
ejpam-6261	134	10	)	)	PUNCT
ejpam-6261	134	11	(	(	PUNCT
ejpam-6261	134	12	2025	2025	NUM
ejpam-6261	134	13	)	)	PUNCT
ejpam-6261	134	14	,	,	PUNCT
ejpam-6261	134	15	6261	6261	NUM
ejpam-6261	134	16	5	5	NUM
ejpam-6261	134	17	of	of	ADP
ejpam-6261	134	18	23	23	NUM
ejpam-6261	134	19	figure	figure	NOUN
ejpam-6261	134	20	1	1	NUM
ejpam-6261	134	21	:	:	PUNCT
ejpam-6261	134	22	flowchart	flowchart	NOUN
ejpam-6261	134	23	of	of	ADP
ejpam-6261	134	24	proposed	propose	VERB
ejpam-6261	134	25	model	model	NOUN
ejpam-6261	134	26	2	2	NUM
ejpam-6261	134	27	.	.	PUNCT
ejpam-6261	134	28	preliminaries	preliminary	NOUN
ejpam-6261	134	29	the	the	DET
ejpam-6261	134	30	following	follow	VERB
ejpam-6261	134	31	portion	portion	NOUN
ejpam-6261	134	32	recalls	recall	VERB
ejpam-6261	134	33	some	some	DET
ejpam-6261	134	34	essential	essential	ADJ
ejpam-6261	134	35	concepts	concept	NOUN
ejpam-6261	134	36	of	of	ADP
ejpam-6261	134	37	cns	cns	NOUN
ejpam-6261	134	38	and	and	CCONJ
ejpam-6261	134	39	cnsr	cnsr	NOUN
ejpam-6261	134	40	which	which	PRON
ejpam-6261	134	41	are	be	AUX
ejpam-6261	134	42	necessary	necessary	ADJ
ejpam-6261	134	43	for	for	ADP
ejpam-6261	134	44	our	our	PRON
ejpam-6261	134	45	further	further	ADJ
ejpam-6261	134	46	discussion	discussion	NOUN
ejpam-6261	134	47	.	.	PUNCT
ejpam-6261	135	1	definition	definition	NOUN
ejpam-6261	135	2	1	1	NUM
ejpam-6261	135	3	.	.	PUNCT
ejpam-6261	136	1	[	[	X
ejpam-6261	136	2	8	8	NUM
ejpam-6261	136	3	]	]	X
ejpam-6261	136	4	ns	ns	NUM
ejpam-6261	136	5	w	w	NOUN
ejpam-6261	136	6	of	of	ADP
ejpam-6261	136	7	universal	universal	ADJ
ejpam-6261	136	8	set	set	NOUN
ejpam-6261	136	9	p	p	NOUN
ejpam-6261	136	10	is	be	AUX
ejpam-6261	136	11	of	of	ADP
ejpam-6261	136	12	the	the	DET
ejpam-6261	136	13	form	form	NOUN
ejpam-6261	137	1	w	w	NOUN
ejpam-6261	137	2	=	=	PUNCT
ejpam-6261	137	3	{	{	PUNCT
ejpam-6261	137	4	<	<	X
ejpam-6261	137	5	c	c	PROPN
ejpam-6261	137	6	,	,	PUNCT
ejpam-6261	137	7	pw	pw	PROPN
ejpam-6261	137	8	(	(	PUNCT
ejpam-6261	137	9	c	c	NOUN
ejpam-6261	137	10	)	)	PUNCT
ejpam-6261	137	11	,	,	PUNCT
ejpam-6261	137	12	qw	qw	X
ejpam-6261	137	13	(	(	PUNCT
ejpam-6261	137	14	c	c	NOUN
ejpam-6261	137	15	)	)	PUNCT
ejpam-6261	137	16	,	,	PUNCT
ejpam-6261	137	17	rw	rw	PROPN
ejpam-6261	137	18	(	(	PUNCT
ejpam-6261	137	19	c	c	NOUN
ejpam-6261	137	20	)	)	PUNCT
ejpam-6261	137	21	>	>	PUNCT
ejpam-6261	137	22	:	:	PUNCT
ejpam-6261	137	23	c	c	X
ejpam-6261	137	24	∈	∈	PROPN
ejpam-6261	138	1	p	p	X
ejpam-6261	138	2	}	}	PUNCT
ejpam-6261	138	3	,	,	PUNCT
ejpam-6261	138	4	where	where	SCONJ
ejpam-6261	138	5	pw	pw	X
ejpam-6261	138	6	(	(	PUNCT
ejpam-6261	138	7	c	c	NOUN
ejpam-6261	138	8	)	)	PUNCT
ejpam-6261	138	9	,	,	PUNCT
ejpam-6261	138	10	qw	qw	X
ejpam-6261	138	11	(	(	PUNCT
ejpam-6261	138	12	c	c	NOUN
ejpam-6261	138	13	)	)	PUNCT
ejpam-6261	138	14	and	and	CCONJ
ejpam-6261	138	15	rw	rw	NOUN
ejpam-6261	138	16	(	(	PUNCT
ejpam-6261	138	17	c	c	NOUN
ejpam-6261	138	18	)	)	PUNCT
ejpam-6261	138	19	give	give	VERB
ejpam-6261	138	20	the	the	DET
ejpam-6261	138	21	degree	degree	NOUN
ejpam-6261	138	22	of	of	ADP
ejpam-6261	138	23	accuracy	accuracy	NOUN
ejpam-6261	138	24	,	,	PUNCT
ejpam-6261	138	25	level	level	NOUN
ejpam-6261	138	26	of	of	ADP
ejpam-6261	138	27	indeterminacy	indeterminacy	NOUN
ejpam-6261	138	28	and	and	CCONJ
ejpam-6261	138	29	degree	degree	NOUN
ejpam-6261	138	30	of	of	ADP
ejpam-6261	138	31	falsehood	falsehood	NOUN
ejpam-6261	138	32	of	of	ADP
ejpam-6261	138	33	c	c	PROPN
ejpam-6261	138	34	from	from	ADP
ejpam-6261	138	35	unit	unit	NOUN
ejpam-6261	138	36	interval	interval	NOUN
ejpam-6261	138	37	,	,	PUNCT
ejpam-6261	138	38	respectively	respectively	ADV
ejpam-6261	138	39	such	such	ADJ
ejpam-6261	138	40	that	that	SCONJ
ejpam-6261	138	41	0	0	NUM
ejpam-6261	138	42	≤	≤	NUM
ejpam-6261	138	43	pw	pw	X
ejpam-6261	138	44	(	(	PUNCT
ejpam-6261	138	45	c	c	NOUN
ejpam-6261	138	46	)	)	PUNCT
ejpam-6261	138	47	+	+	NUM
ejpam-6261	138	48	qw	qw	X
ejpam-6261	138	49	(	(	PUNCT
ejpam-6261	138	50	c	c	NOUN
ejpam-6261	138	51	)	)	PUNCT
ejpam-6261	138	52	+	+	CCONJ
ejpam-6261	138	53	rw	rw	NOUN
ejpam-6261	138	54	(	(	PUNCT
ejpam-6261	138	55	c	c	NOUN
ejpam-6261	138	56	)	)	PUNCT
ejpam-6261	138	57	≤	≤	NOUN
ejpam-6261	138	58	3	3	NUM
ejpam-6261	138	59	,	,	PUNCT
ejpam-6261	138	60	for	for	ADP
ejpam-6261	138	61	any	any	DET
ejpam-6261	138	62	c	c	PROPN
ejpam-6261	138	63	∈	∈	PROPN
ejpam-6261	138	64	p	p	PROPN
ejpam-6261	138	65	.	.	PUNCT
ejpam-6261	139	1	definition	definition	NOUN
ejpam-6261	139	2	2	2	NUM
ejpam-6261	139	3	.	.	PUNCT
ejpam-6261	140	1	[	[	X
ejpam-6261	140	2	38	38	NUM
ejpam-6261	140	3	]	]	PUNCT
ejpam-6261	140	4	a	a	DET
ejpam-6261	140	5	ns	ns	INTJ
ejpam-6261	140	6	w	w	NOUN
ejpam-6261	140	7	of	of	ADP
ejpam-6261	140	8	a	a	DET
ejpam-6261	140	9	ring	ring	NOUN
ejpam-6261	140	10	r	r	NOUN
ejpam-6261	140	11	is	be	AUX
ejpam-6261	140	12	known	know	VERB
ejpam-6261	140	13	a	a	DET
ejpam-6261	140	14	nsr	nsr	NOUN
ejpam-6261	140	15	of	of	ADP
ejpam-6261	140	16	a	a	DET
ejpam-6261	140	17	r	r	NOUN
ejpam-6261	140	18	,	,	PUNCT
ejpam-6261	140	19	if	if	SCONJ
ejpam-6261	140	20	these	these	DET
ejpam-6261	140	21	conditions	condition	NOUN
ejpam-6261	140	22	are	be	AUX
ejpam-6261	140	23	valid	valid	ADJ
ejpam-6261	140	24	:	:	PUNCT
ejpam-6261	140	25	(	(	PUNCT
ejpam-6261	140	26	i	i	NOUN
ejpam-6261	140	27	)	)	PUNCT
ejpam-6261	140	28	pw	pw	PROPN
ejpam-6261	141	1	(	(	PUNCT
ejpam-6261	141	2	m−	m−	PROPN
ejpam-6261	141	3	d	d	PROPN
ejpam-6261	141	4	)	)	PUNCT
ejpam-6261	141	5	≥	≥	NOUN
ejpam-6261	141	6	min{pw	min{pw	ADV
ejpam-6261	141	7	(	(	PUNCT
ejpam-6261	141	8	m	m	NOUN
ejpam-6261	141	9	)	)	PUNCT
ejpam-6261	141	10	,	,	PUNCT
ejpam-6261	141	11	pw	pw	PROPN
ejpam-6261	142	1	(	(	PUNCT
ejpam-6261	142	2	d	d	NOUN
ejpam-6261	142	3	)	)	PUNCT
ejpam-6261	142	4	}	}	PUNCT
ejpam-6261	142	5	,	,	PUNCT
ejpam-6261	142	6	for	for	ADP
ejpam-6261	142	7	all	all	DET
ejpam-6261	142	8	m	m	PROPN
ejpam-6261	142	9	,	,	PUNCT
ejpam-6261	142	10	d	d	PROPN
ejpam-6261	142	11	∈	∈	PROPN
ejpam-6261	142	12	r.	r.	PROPN
ejpam-6261	142	13	(	(	PUNCT
ejpam-6261	142	14	ii	ii	PROPN
ejpam-6261	142	15	)	)	PUNCT
ejpam-6261	142	16	pw	pw	PROPN
ejpam-6261	143	1	(	(	PUNCT
ejpam-6261	143	2	md	md	PROPN
ejpam-6261	143	3	)	)	PUNCT
ejpam-6261	143	4	≥	≥	PRON
ejpam-6261	143	5	min{pw	min{pw	ADV
ejpam-6261	143	6	(	(	PUNCT
ejpam-6261	143	7	m	m	NOUN
ejpam-6261	143	8	)	)	PUNCT
ejpam-6261	143	9	,	,	PUNCT
ejpam-6261	143	10	pw	pw	PROPN
ejpam-6261	144	1	(	(	PUNCT
ejpam-6261	144	2	d	d	NOUN
ejpam-6261	144	3	)	)	PUNCT
ejpam-6261	144	4	}	}	PUNCT
ejpam-6261	144	5	,	,	PUNCT
ejpam-6261	144	6	(	(	PUNCT
ejpam-6261	144	7	iii	iii	X
ejpam-6261	144	8	)	)	PUNCT
ejpam-6261	144	9	qw	qw	PROPN
ejpam-6261	144	10	(	(	PUNCT
ejpam-6261	144	11	md	md	PROPN
ejpam-6261	144	12	)	)	PUNCT
ejpam-6261	144	13	≤	≤	NOUN
ejpam-6261	144	14	max{qw	max{qw	CCONJ
ejpam-6261	144	15	(	(	PUNCT
ejpam-6261	144	16	m	m	NOUN
ejpam-6261	144	17	)	)	PUNCT
ejpam-6261	144	18	,	,	PUNCT
ejpam-6261	144	19	qw	qw	X
ejpam-6261	144	20	(	(	PUNCT
ejpam-6261	144	21	d	d	NOUN
ejpam-6261	144	22	)	)	PUNCT
ejpam-6261	144	23	}	}	PUNCT
ejpam-6261	144	24	,	,	PUNCT
ejpam-6261	144	25	(	(	PUNCT
ejpam-6261	144	26	iv	iv	X
ejpam-6261	144	27	)	)	PUNCT
ejpam-6261	144	28	qw	qw	X
ejpam-6261	144	29	(	(	PUNCT
ejpam-6261	144	30	m−	m−	PROPN
ejpam-6261	144	31	d	d	PROPN
ejpam-6261	144	32	)	)	PUNCT
ejpam-6261	144	33	≤	≤	NOUN
ejpam-6261	144	34	max{qw	max{qw	CCONJ
ejpam-6261	144	35	(	(	PUNCT
ejpam-6261	144	36	m	m	NOUN
ejpam-6261	144	37	)	)	PUNCT
ejpam-6261	144	38	,	,	PUNCT
ejpam-6261	144	39	qw	qw	X
ejpam-6261	144	40	(	(	PUNCT
ejpam-6261	144	41	d	d	NOUN
ejpam-6261	144	42	)	)	PUNCT
ejpam-6261	144	43	}	}	PUNCT
ejpam-6261	144	44	,	,	PUNCT
ejpam-6261	144	45	(	(	PUNCT
ejpam-6261	144	46	v	v	NOUN
ejpam-6261	144	47	)	)	PUNCT
ejpam-6261	144	48	rw	rw	NOUN
ejpam-6261	144	49	(	(	PUNCT
ejpam-6261	144	50	m−	m−	PROPN
ejpam-6261	144	51	d	d	PROPN
ejpam-6261	144	52	)	)	PUNCT
ejpam-6261	144	53	≤	≤	NOUN
ejpam-6261	144	54	max{rw	max{rw	NUM
ejpam-6261	144	55	(	(	PUNCT
ejpam-6261	144	56	m	m	NOUN
ejpam-6261	144	57	)	)	PUNCT
ejpam-6261	144	58	,	,	PUNCT
ejpam-6261	144	59	rw	rw	PROPN
ejpam-6261	144	60	(	(	PUNCT
ejpam-6261	144	61	d	d	NOUN
ejpam-6261	144	62	)	)	PUNCT
ejpam-6261	144	63	}	}	PUNCT
ejpam-6261	144	64	,	,	PUNCT
ejpam-6261	144	65	m.	m.	PROPN
ejpam-6261	144	66	h.	h.	PROPN
ejpam-6261	144	67	mateen	mateen	PROPN
ejpam-6261	144	68	et	et	PROPN
ejpam-6261	144	69	al	al	PROPN
ejpam-6261	144	70	.	.	PUNCT
ejpam-6261	144	71	/	/	SYM
ejpam-6261	144	72	eur	eur	PROPN
ejpam-6261	144	73	.	.	PUNCT
ejpam-6261	145	1	j.	j.	PROPN
ejpam-6261	145	2	pure	pure	PROPN
ejpam-6261	145	3	appl	appl	PROPN
ejpam-6261	145	4	.	.	PROPN
ejpam-6261	145	5	math	math	PROPN
ejpam-6261	145	6	,	,	PUNCT
ejpam-6261	145	7	18	18	NUM
ejpam-6261	145	8	(	(	PUNCT
ejpam-6261	145	9	4	4	NUM
ejpam-6261	145	10	)	)	PUNCT
ejpam-6261	145	11	(	(	PUNCT
ejpam-6261	145	12	2025	2025	NUM
ejpam-6261	145	13	)	)	PUNCT
ejpam-6261	145	14	,	,	PUNCT
ejpam-6261	145	15	6261	6261	NUM
ejpam-6261	145	16	6	6	NUM
ejpam-6261	145	17	of	of	ADP
ejpam-6261	145	18	23	23	NUM
ejpam-6261	145	19	table	table	NOUN
ejpam-6261	145	20	1	1	NUM
ejpam-6261	145	21	:	:	PUNCT
ejpam-6261	145	22	comparison	comparison	NOUN
ejpam-6261	145	23	of	of	ADP
ejpam-6261	145	24	complex	complex	ADJ
ejpam-6261	145	25	neutrosophic	neutrosophic	ADJ
ejpam-6261	145	26	sets	set	NOUN
ejpam-6261	145	27	to	to	ADP
ejpam-6261	145	28	the	the	DET
ejpam-6261	145	29	existing	exist	VERB
ejpam-6261	145	30	approaches	approach	NOUN
ejpam-6261	145	31	sets	set	VERB
ejpam-6261	145	32	domain	domain	NOUN
ejpam-6261	145	33	co	co	ADJ
ejpam-6261	145	34	-	-	NOUN
ejpam-6261	145	35	domain	domain	ADJ
ejpam-6261	145	36	truth	truth	NOUN
ejpam-6261	145	37	falsity	falsity	NOUN
ejpam-6261	145	38	indeterminacy	indeterminacy	NOUN
ejpam-6261	145	39	truth	truth	NOUN
ejpam-6261	145	40	falsity	falsity	NOUN
ejpam-6261	145	41	indeterminacy	indeterminacy	NOUN
ejpam-6261	145	42	with	with	ADP
ejpam-6261	145	43	periodicity	periodicity	NOUN
ejpam-6261	145	44	with	with	ADP
ejpam-6261	145	45	periodicity	periodicity	NOUN
ejpam-6261	145	46	with	with	ADP
ejpam-6261	145	47	periodicity	periodicity	NOUN
ejpam-6261	145	48	fuzzy	fuzzy	ADJ
ejpam-6261	145	49	sets	set	NOUN
ejpam-6261	145	50	universel	universel	ADJ
ejpam-6261	145	51	set	set	VERB
ejpam-6261	145	52	unit	unit	NOUN
ejpam-6261	145	53	interval	interval	NOUN
ejpam-6261	145	54	x	x	X
ejpam-6261	145	55	×	×	NOUN
ejpam-6261	145	56	×	×	NOUN
ejpam-6261	145	57	×	×	NOUN
ejpam-6261	145	58	×	×	NOUN
ejpam-6261	145	59	×	×	NOUN
ejpam-6261	145	60	intuitionistic	intuitionistic	ADJ
ejpam-6261	145	61	fuzzy	fuzzy	ADJ
ejpam-6261	145	62	sets	set	NOUN
ejpam-6261	145	63	universel	universel	ADJ
ejpam-6261	145	64	set	set	VERB
ejpam-6261	145	65	unit	unit	NOUN
ejpam-6261	145	66	interval	interval	NOUN
ejpam-6261	145	67	x	x	NOUN
ejpam-6261	146	1	x	x	SYM
ejpam-6261	146	2	×	×	NOUN
ejpam-6261	146	3	×	×	NOUN
ejpam-6261	146	4	×	×	NOUN
ejpam-6261	146	5	×	×	NOUN
ejpam-6261	146	6	neutrosophic	neutrosophic	ADJ
ejpam-6261	146	7	sets	set	NOUN
ejpam-6261	146	8	universel	universel	ADJ
ejpam-6261	146	9	set	set	VERB
ejpam-6261	146	10	unit	unit	NOUN
ejpam-6261	146	11	interval	interval	NOUN
ejpam-6261	146	12	x	x	NOUN
ejpam-6261	147	1	x	x	PUNCT
ejpam-6261	147	2	x	x	SYM
ejpam-6261	147	3	×	×	NOUN
ejpam-6261	147	4	×	×	NOUN
ejpam-6261	147	5	×	×	NOUN
ejpam-6261	147	6	complex	complex	ADJ
ejpam-6261	147	7	fuzzy	fuzzy	ADJ
ejpam-6261	147	8	sets	set	NOUN
ejpam-6261	147	9	universel	universel	ADJ
ejpam-6261	147	10	set	set	NOUN
ejpam-6261	147	11	unit	unit	NOUN
ejpam-6261	147	12	disc	disc	NOUN
ejpam-6261	147	13	x	x	SYM
ejpam-6261	147	14	×	×	NOUN
ejpam-6261	147	15	×	×	NOUN
ejpam-6261	147	16	x	x	SYM
ejpam-6261	147	17	×	×	NOUN
ejpam-6261	147	18	×	×	NOUN
ejpam-6261	147	19	complex	complex	ADJ
ejpam-6261	147	20	intuitionistic	intuitionistic	ADJ
ejpam-6261	147	21	fuzzy	fuzzy	ADJ
ejpam-6261	147	22	sets	set	NOUN
ejpam-6261	147	23	universel	universel	ADJ
ejpam-6261	147	24	set	set	VERB
ejpam-6261	147	25	unit	unit	NOUN
ejpam-6261	147	26	disc	disc	VERB
ejpam-6261	147	27	x	x	PUNCT
ejpam-6261	147	28	x	x	SYM
ejpam-6261	147	29	×	×	NOUN
ejpam-6261	147	30	x	x	PUNCT
ejpam-6261	147	31	x	x	SYM
ejpam-6261	147	32	×	×	PROPN
ejpam-6261	147	33	complex	complex	ADJ
ejpam-6261	147	34	neutrosophic	neutrosophic	ADJ
ejpam-6261	147	35	fuzzy	fuzzy	ADJ
ejpam-6261	147	36	sets	set	NOUN
ejpam-6261	147	37	universel	universel	ADJ
ejpam-6261	147	38	set	set	VERB
ejpam-6261	147	39	unit	unit	NOUN
ejpam-6261	147	40	disc	disc	VERB
ejpam-6261	147	41	x	x	PUNCT
ejpam-6261	147	42	x	x	PUNCT
ejpam-6261	147	43	x	x	PUNCT
ejpam-6261	147	44	x	x	PUNCT
ejpam-6261	147	45	x	x	SYM
ejpam-6261	147	46	x	x	X
ejpam-6261	147	47	(	(	PUNCT
ejpam-6261	147	48	vi	vi	NOUN
ejpam-6261	147	49	)	)	PUNCT
ejpam-6261	147	50	rw	rw	NOUN
ejpam-6261	147	51	(	(	PUNCT
ejpam-6261	147	52	md	md	PROPN
ejpam-6261	147	53	)	)	PUNCT
ejpam-6261	147	54	≤	≤	NOUN
ejpam-6261	148	1	max{rw	max{rw	NUM
ejpam-6261	148	2	(	(	PUNCT
ejpam-6261	148	3	m	m	NOUN
ejpam-6261	148	4	)	)	PUNCT
ejpam-6261	148	5	,	,	PUNCT
ejpam-6261	148	6	rw	rw	PROPN
ejpam-6261	148	7	(	(	PUNCT
ejpam-6261	148	8	d	d	NOUN
ejpam-6261	148	9	)	)	PUNCT
ejpam-6261	148	10	}	}	PUNCT
ejpam-6261	148	11	.	.	PUNCT
ejpam-6261	149	1	example	example	NOUN
ejpam-6261	150	1	1	1	NUM
ejpam-6261	150	2	.	.	X
ejpam-6261	150	3	consider	consider	VERB
ejpam-6261	150	4	the	the	DET
ejpam-6261	150	5	ring	ring	NOUN
ejpam-6261	150	6	r	r	NOUN
ejpam-6261	150	7	=	=	SYM
ejpam-6261	150	8	z6	z6	PROPN
ejpam-6261	150	9	=	=	SYM
ejpam-6261	150	10	{	{	PUNCT
ejpam-6261	150	11	0	0	NUM
ejpam-6261	150	12	,	,	PUNCT
ejpam-6261	150	13	1	1	NUM
ejpam-6261	150	14	,	,	PUNCT
ejpam-6261	150	15	2	2	NUM
ejpam-6261	150	16	,	,	PUNCT
ejpam-6261	150	17	3	3	NUM
ejpam-6261	150	18	,	,	PUNCT
ejpam-6261	150	19	4	4	NUM
ejpam-6261	150	20	,	,	PUNCT
ejpam-6261	150	21	5	5	NUM
ejpam-6261	150	22	}	}	PUNCT
ejpam-6261	150	23	under	under	ADP
ejpam-6261	150	24	addition	addition	NOUN
ejpam-6261	150	25	and	and	CCONJ
ejpam-6261	150	26	multiplication	multiplication	NOUN
ejpam-6261	150	27	modulo	modulo	VERB
ejpam-6261	150	28	6	6	NUM
ejpam-6261	150	29	.	.	PUNCT
ejpam-6261	150	30	define	define	VERB
ejpam-6261	150	31	a	a	DET
ejpam-6261	150	32	neutrosophic	neutrosophic	ADJ
ejpam-6261	150	33	subring	subre	VERB
ejpam-6261	150	34	n	n	NOUN
ejpam-6261	150	35	of	of	ADP
ejpam-6261	150	36	r	r	NOUN
ejpam-6261	150	37	as	as	SCONJ
ejpam-6261	150	38	follows	follow	VERB
ejpam-6261	150	39	:	:	PUNCT
ejpam-6261	150	40	pw	pw	X
ejpam-6261	150	41	(	(	PUNCT
ejpam-6261	150	42	x	x	X
ejpam-6261	150	43	)	)	PUNCT
ejpam-6261	150	44	=	=	SYM
ejpam-6261	151	1			PROPN
ejpam-6261	151	2	1	1	NUM
ejpam-6261	151	3	,	,	PUNCT
ejpam-6261	151	4	if	if	SCONJ
ejpam-6261	151	5	x	x	ADP
ejpam-6261	151	6	=	=	SYM
ejpam-6261	151	7	0	0	NUM
ejpam-6261	151	8	,	,	PUNCT
ejpam-6261	151	9	0.88	0.88	NUM
ejpam-6261	151	10	,	,	PUNCT
ejpam-6261	151	11	if	if	SCONJ
ejpam-6261	151	12	x	x	SYM
ejpam-6261	151	13	∈	∈	NOUN
ejpam-6261	151	14	{	{	PUNCT
ejpam-6261	151	15	2	2	NUM
ejpam-6261	151	16	,	,	PUNCT
ejpam-6261	151	17	4	4	NUM
ejpam-6261	151	18	}	}	PUNCT
ejpam-6261	151	19	,	,	PUNCT
ejpam-6261	151	20	0	0	NUM
ejpam-6261	151	21	,	,	PUNCT
ejpam-6261	151	22	otherwise	otherwise	ADV
ejpam-6261	151	23	.	.	PUNCT
ejpam-6261	152	1	qw	qw	X
ejpam-6261	152	2	(	(	PUNCT
ejpam-6261	152	3	x	x	NOUN
ejpam-6261	152	4	)	)	PUNCT
ejpam-6261	152	5	=	=	SYM
ejpam-6261	153	1			PROPN
ejpam-6261	153	2	0	0	NUM
ejpam-6261	153	3	,	,	PUNCT
ejpam-6261	153	4	if	if	SCONJ
ejpam-6261	153	5	x	x	ADP
ejpam-6261	153	6	=	=	SYM
ejpam-6261	153	7	0	0	NUM
ejpam-6261	153	8	,	,	PUNCT
ejpam-6261	153	9	0.11	0.11	NUM
ejpam-6261	153	10	,	,	PUNCT
ejpam-6261	153	11	if	if	SCONJ
ejpam-6261	153	12	x	x	SYM
ejpam-6261	153	13	∈	∈	NOUN
ejpam-6261	153	14	{	{	PUNCT
ejpam-6261	153	15	2	2	NUM
ejpam-6261	153	16	,	,	PUNCT
ejpam-6261	153	17	4	4	NUM
ejpam-6261	153	18	}	}	PUNCT
ejpam-6261	153	19	,	,	PUNCT
ejpam-6261	153	20	0.33	0.33	NUM
ejpam-6261	153	21	,	,	PUNCT
ejpam-6261	153	22	otherwise	otherwise	ADV
ejpam-6261	153	23	.	.	PUNCT
ejpam-6261	154	1	rw	rw	PROPN
ejpam-6261	154	2	(	(	PUNCT
ejpam-6261	154	3	x	x	NOUN
ejpam-6261	154	4	)	)	PUNCT
ejpam-6261	154	5	=	=	SYM
ejpam-6261	155	1			PROPN
ejpam-6261	155	2	0	0	NUM
ejpam-6261	155	3	,	,	PUNCT
ejpam-6261	155	4	if	if	SCONJ
ejpam-6261	155	5	x	x	ADP
ejpam-6261	155	6	=	=	SYM
ejpam-6261	155	7	0	0	NUM
ejpam-6261	155	8	,	,	PUNCT
ejpam-6261	155	9	0.11	0.11	NUM
ejpam-6261	155	10	,	,	PUNCT
ejpam-6261	155	11	if	if	SCONJ
ejpam-6261	155	12	x	x	SYM
ejpam-6261	155	13	∈	∈	NOUN
ejpam-6261	155	14	{	{	PUNCT
ejpam-6261	155	15	2	2	NUM
ejpam-6261	155	16	,	,	PUNCT
ejpam-6261	155	17	4	4	NUM
ejpam-6261	155	18	}	}	PUNCT
ejpam-6261	155	19	,	,	PUNCT
ejpam-6261	155	20	0.77	0.77	NUM
ejpam-6261	155	21	,	,	PUNCT
ejpam-6261	155	22	otherwise	otherwise	ADV
ejpam-6261	155	23	.	.	PUNCT
ejpam-6261	156	1	clearly	clearly	ADV
ejpam-6261	156	2	,	,	PUNCT
ejpam-6261	156	3	n	n	PRON
ejpam-6261	156	4	is	be	AUX
ejpam-6261	156	5	nsr	nsr	NOUN
ejpam-6261	156	6	of	of	ADP
ejpam-6261	156	7	subring	subre	VERB
ejpam-6261	156	8	r.	r.	PROPN
ejpam-6261	156	9	theorem	theorem	NOUN
ejpam-6261	156	10	1	1	NUM
ejpam-6261	156	11	.	.	PUNCT
ejpam-6261	157	1	[	[	X
ejpam-6261	157	2	38	38	NUM
ejpam-6261	157	3	]	]	PUNCT
ejpam-6261	157	4	intersection	intersection	NOUN
ejpam-6261	157	5	of	of	ADP
ejpam-6261	157	6	two	two	NUM
ejpam-6261	157	7	nsrs	nsrs	NOUN
ejpam-6261	157	8	of	of	ADP
ejpam-6261	157	9	ring	ring	NOUN
ejpam-6261	157	10	r	r	NOUN
ejpam-6261	157	11	is	be	AUX
ejpam-6261	157	12	nsr	nsr	PROPN
ejpam-6261	157	13	.	.	PROPN
ejpam-6261	157	14	definition	definition	NOUN
ejpam-6261	157	15	3	3	NUM
ejpam-6261	157	16	.	.	PUNCT
ejpam-6261	158	1	[	[	X
ejpam-6261	158	2	31	31	NUM
ejpam-6261	158	3	,	,	PUNCT
ejpam-6261	158	4	32	32	NUM
ejpam-6261	158	5	]	]	PUNCT
ejpam-6261	158	6	a	a	DET
ejpam-6261	158	7	cns	cns	PROPN
ejpam-6261	158	8	w	w	NOUN
ejpam-6261	158	9	of	of	ADP
ejpam-6261	158	10	universal	universal	ADJ
ejpam-6261	158	11	set	set	NOUN
ejpam-6261	158	12	p	p	NOUN
ejpam-6261	158	13	is	be	AUX
ejpam-6261	158	14	an	an	DET
ejpam-6261	158	15	object	object	NOUN
ejpam-6261	158	16	of	of	ADP
ejpam-6261	158	17	the	the	DET
ejpam-6261	158	18	form	form	NOUN
ejpam-6261	158	19	w	w	NOUN
ejpam-6261	158	20	=	=	PUNCT
ejpam-6261	158	21	{	{	PUNCT
ejpam-6261	158	22	<	<	X
ejpam-6261	158	23	f	f	PROPN
ejpam-6261	158	24	,	,	PUNCT
ejpam-6261	158	25	tw(f	tw(f	NUM
ejpam-6261	158	26	)	)	PUNCT
ejpam-6261	158	27	,	,	PUNCT
ejpam-6261	158	28	iw(f),fw(f	iw(f),fw(f	NOUN
ejpam-6261	158	29	)	)	PUNCT
ejpam-6261	158	30	>	>	PUNCT
ejpam-6261	158	31	:	:	PUNCT
ejpam-6261	159	1	f	f	PROPN
ejpam-6261	159	2	∈	∈	PROPN
ejpam-6261	159	3	p	p	X
ejpam-6261	159	4	}	}	PUNCT
ejpam-6261	159	5	,	,	PUNCT
ejpam-6261	159	6	where	where	SCONJ
ejpam-6261	159	7	the	the	DET
ejpam-6261	159	8	degree	degree	NOUN
ejpam-6261	159	9	of	of	ADP
ejpam-6261	159	10	truth	truth	NOUN
ejpam-6261	159	11	tw(f	tw(f	PUNCT
ejpam-6261	159	12	)	)	PUNCT
ejpam-6261	159	13	=	=	SYM
ejpam-6261	159	14	pw(f)eiθw(f	pw(f)eiθw(f	PROPN
ejpam-6261	159	15	)	)	PUNCT
ejpam-6261	159	16	,	,	PUNCT
ejpam-6261	159	17	and	and	CCONJ
ejpam-6261	159	18	is	be	AUX
ejpam-6261	159	19	expressed	express	VERB
ejpam-6261	159	20	as	as	ADP
ejpam-6261	159	21	tw	tw	NOUN
ejpam-6261	159	22	:	:	PUNCT
ejpam-6261	159	23	p	p	X
ejpam-6261	159	24	→	→	PUNCT
ejpam-6261	159	25	{	{	PUNCT
ejpam-6261	159	26	τ	τ	PROPN
ejpam-6261	159	27	∈	∈	PROPN
ejpam-6261	159	28	c	c	NOUN
ejpam-6261	159	29	:	:	PUNCT
ejpam-6261	159	30	|τ	|τ	PROPN
ejpam-6261	159	31	|	|	ADV
ejpam-6261	159	32	≤	≤	NUM
ejpam-6261	159	33	1	1	NUM
ejpam-6261	159	34	}	}	PUNCT
ejpam-6261	159	35	,	,	PUNCT
ejpam-6261	159	36	degree	degree	NOUN
ejpam-6261	159	37	of	of	ADP
ejpam-6261	159	38	indeterminacy	indeterminacy	NOUN
ejpam-6261	159	39	iw(f	iw(f	NOUN
ejpam-6261	159	40	)	)	PUNCT
ejpam-6261	159	41	=	=	SYM
ejpam-6261	160	1	qw(f)eiφw(f	qw(f)eiφw(f	PROPN
ejpam-6261	160	2	)	)	PUNCT
ejpam-6261	160	3	is	be	AUX
ejpam-6261	160	4	expressed	express	VERB
ejpam-6261	160	5	as	as	ADP
ejpam-6261	160	6	iw	iw	NOUN
ejpam-6261	160	7	:	:	PUNCT
ejpam-6261	160	8	p	p	X
ejpam-6261	160	9	→	→	PUNCT
ejpam-6261	160	10	{	{	PUNCT
ejpam-6261	160	11	τ	τ	PROPN
ejpam-6261	160	12	∈	∈	PROPN
ejpam-6261	160	13	c	c	NOUN
ejpam-6261	160	14	:	:	PUNCT
ejpam-6261	160	15	|τ	|τ	PROPN
ejpam-6261	160	16	|	|	ADV
ejpam-6261	160	17	≤	≤	NUM
ejpam-6261	160	18	1	1	NUM
ejpam-6261	160	19	}	}	PUNCT
ejpam-6261	160	20	and	and	CCONJ
ejpam-6261	160	21	degree	degree	NOUN
ejpam-6261	160	22	of	of	ADP
ejpam-6261	160	23	falsity	falsity	NOUN
ejpam-6261	160	24	fw(f	fw(f	NOUN
ejpam-6261	160	25	)	)	PUNCT
ejpam-6261	160	26	=	=	SYM
ejpam-6261	160	27	rw(f)eiωw(f	rw(f)eiωw(f	NOUN
ejpam-6261	160	28	)	)	PUNCT
ejpam-6261	160	29	and	and	CCONJ
ejpam-6261	160	30	is	be	AUX
ejpam-6261	160	31	defined	define	VERB
ejpam-6261	160	32	as	as	ADP
ejpam-6261	160	33	fw	fw	ADJ
ejpam-6261	160	34	:	:	PUNCT
ejpam-6261	160	35	p	p	X
ejpam-6261	160	36	→	→	PUNCT
ejpam-6261	160	37	{	{	PUNCT
ejpam-6261	160	38	τ	τ	PROPN
ejpam-6261	160	39	∈	∈	PROPN
ejpam-6261	160	40	c	c	NOUN
ejpam-6261	160	41	:	:	PUNCT
ejpam-6261	160	42	|τ	|τ	PROPN
ejpam-6261	160	43	|	|	ADV
ejpam-6261	160	44	≤	≤	NUM
ejpam-6261	160	45	1	1	NUM
ejpam-6261	160	46	}	}	PUNCT
ejpam-6261	160	47	,	,	PUNCT
ejpam-6261	160	48	where	where	SCONJ
ejpam-6261	160	49	|tw(f	|tw(f	NOUN
ejpam-6261	160	50	)	)	PUNCT
ejpam-6261	160	51	+	+	NUM
ejpam-6261	160	52	iw(f	iw(f	NUM
ejpam-6261	160	53	)	)	PUNCT
ejpam-6261	160	54	+	+	CCONJ
ejpam-6261	160	55	fw(f)|	fw(f)|	VERB
ejpam-6261	160	56	≤	≤	NUM
ejpam-6261	160	57	3	3	NUM
ejpam-6261	160	58	and	and	CCONJ
ejpam-6261	160	59	c	c	PROPN
ejpam-6261	160	60	is	be	AUX
ejpam-6261	160	61	the	the	DET
ejpam-6261	160	62	set	set	NOUN
ejpam-6261	160	63	of	of	ADP
ejpam-6261	160	64	complex	complex	ADJ
ejpam-6261	160	65	numbers	number	NOUN
ejpam-6261	160	66	.	.	PUNCT
ejpam-6261	161	1	the	the	DET
ejpam-6261	161	2	degree	degree	NOUN
ejpam-6261	161	3	of	of	ADP
ejpam-6261	161	4	accuracy	accuracy	NOUN
ejpam-6261	161	5	,	,	PUNCT
ejpam-6261	161	6	level	level	NOUN
ejpam-6261	161	7	of	of	ADP
ejpam-6261	161	8	indeterminacy	indeterminacy	NOUN
ejpam-6261	161	9	and	and	CCONJ
ejpam-6261	161	10	degree	degree	NOUN
ejpam-6261	161	11	of	of	ADP
ejpam-6261	161	12	falsehood	falsehood	NOUN
ejpam-6261	161	13	receive	receive	VERB
ejpam-6261	161	14	all	all	DET
ejpam-6261	161	15	complex	complex	ADJ
ejpam-6261	161	16	valued	value	VERB
ejpam-6261	161	17	grade	grade	NOUN
ejpam-6261	161	18	from	from	ADP
ejpam-6261	161	19	within	within	ADP
ejpam-6261	161	20	in	in	ADP
ejpam-6261	161	21	the	the	DET
ejpam-6261	161	22	unit	unit	NOUN
ejpam-6261	161	23	circle	circle	NOUN
ejpam-6261	161	24	of	of	ADP
ejpam-6261	161	25	complex	complex	ADJ
ejpam-6261	161	26	plane	plane	NOUN
ejpam-6261	161	27	,	,	PUNCT
ejpam-6261	161	28	respectively	respectively	ADV
ejpam-6261	161	29	.	.	PUNCT
ejpam-6261	162	1	where	where	SCONJ
ejpam-6261	162	2	i	i	PRON
ejpam-6261	162	3	=	=	PUNCT
ejpam-6261	162	4	√	√	NUM
ejpam-6261	162	5	−1	−1	NOUN
ejpam-6261	162	6	,	,	PUNCT
ejpam-6261	162	7	pw(f	pw(f	NUM
ejpam-6261	162	8	)	)	PUNCT
ejpam-6261	162	9	,	,	PUNCT
ejpam-6261	162	10	qw(f	qw(f	NUM
ejpam-6261	162	11	)	)	PUNCT
ejpam-6261	162	12	,	,	PUNCT
ejpam-6261	162	13	rw(f	rw(f	PUNCT
ejpam-6261	162	14	)	)	PUNCT
ejpam-6261	162	15	∈	∈	PROPN
ejpam-6261	163	1	[	[	X
ejpam-6261	163	2	0	0	NUM
ejpam-6261	163	3	,	,	PUNCT
ejpam-6261	163	4	1	1	NUM
ejpam-6261	163	5	]	]	PUNCT
ejpam-6261	163	6	,	,	PUNCT
ejpam-6261	163	7	and	and	CCONJ
ejpam-6261	163	8	θw(f	θw(f	NUM
ejpam-6261	163	9	)	)	PUNCT
ejpam-6261	163	10	,	,	PUNCT
ejpam-6261	163	11	φw(f	φw(f	NOUN
ejpam-6261	163	12	)	)	PUNCT
ejpam-6261	163	13	,	,	PUNCT
ejpam-6261	163	14	ωw(f	ωw(f	NUM
ejpam-6261	163	15	)	)	PUNCT
ejpam-6261	163	16	∈	∈	PROPN
ejpam-6261	164	1	[	[	X
ejpam-6261	164	2	0	0	NUM
ejpam-6261	164	3	,	,	PUNCT
ejpam-6261	164	4	2π	2π	NOUN
ejpam-6261	164	5	]	]	PUNCT
ejpam-6261	164	6	are	be	AUX
ejpam-6261	164	7	real	real	ADV
ejpam-6261	164	8	valued	value	VERB
ejpam-6261	164	9	such	such	ADJ
ejpam-6261	164	10	that	that	SCONJ
ejpam-6261	164	11	0−	0−	NUM
ejpam-6261	164	12	≤	≤	NUM
ejpam-6261	164	13	pw(f	pw(f	NUM
ejpam-6261	164	14	)	)	PUNCT
ejpam-6261	165	1	+	+	CCONJ
ejpam-6261	165	2	qw(f	qw(f	NUM
ejpam-6261	165	3	)	)	PUNCT
ejpam-6261	166	1	+	+	CCONJ
ejpam-6261	166	2	rw(f	rw(f	X
ejpam-6261	166	3	)	)	PUNCT
ejpam-6261	166	4	≤	≤	NUM
ejpam-6261	166	5	3	3	NUM
ejpam-6261	166	6	+	+	NUM
ejpam-6261	166	7	and	and	CCONJ
ejpam-6261	166	8	0	0	NUM
ejpam-6261	166	9	≤	≤	NOUN
ejpam-6261	166	10	θw(f	θw(f	NUM
ejpam-6261	166	11	)	)	PUNCT
ejpam-6261	167	1	+	+	CCONJ
ejpam-6261	167	2	φw(f	φw(f	NOUN
ejpam-6261	167	3	)	)	PUNCT
ejpam-6261	167	4	+	+	NUM
ejpam-6261	167	5	ωw(f	ωw(f	X
ejpam-6261	167	6	)	)	PUNCT
ejpam-6261	167	7	≤	≤	NOUN
ejpam-6261	167	8	6π	6π	NOUN
ejpam-6261	167	9	.	.	PUNCT
ejpam-6261	168	1	for	for	ADP
ejpam-6261	168	2	convenience	convenience	NOUN
ejpam-6261	168	3	we	we	PRON
ejpam-6261	168	4	shall	shall	AUX
ejpam-6261	168	5	use	use	VERB
ejpam-6261	168	6	tw(f	tw(f	PUNCT
ejpam-6261	168	7	)	)	PUNCT
ejpam-6261	168	8	=	=	SYM
ejpam-6261	168	9	pw(f)eiθw(f	pw(f)eiθw(f	PROPN
ejpam-6261	168	10	)	)	PUNCT
ejpam-6261	168	11	,	,	PUNCT
ejpam-6261	168	12	tx(f	tx(f	NOUN
ejpam-6261	168	13	)	)	PUNCT
ejpam-6261	168	14	=	=	SYM
ejpam-6261	168	15	px(f)e	px(f)e	PROPN
ejpam-6261	168	16	iθx(f	iθx(f	PROPN
ejpam-6261	168	17	)	)	PUNCT
ejpam-6261	168	18	as	as	ADP
ejpam-6261	168	19	degree	degree	NOUN
ejpam-6261	168	20	of	of	ADP
ejpam-6261	168	21	truth	truth	NOUN
ejpam-6261	168	22	,	,	PUNCT
ejpam-6261	168	23	iw(f	iw(f	NOUN
ejpam-6261	168	24	)	)	PUNCT
ejpam-6261	168	25	=	=	SYM
ejpam-6261	169	1	qw(f)eiφw(f	qw(f)eiφw(f	PROPN
ejpam-6261	169	2	)	)	PUNCT
ejpam-6261	169	3	,	,	PUNCT
ejpam-6261	169	4	ix(f	ix(f	NOUN
ejpam-6261	169	5	)	)	PUNCT
ejpam-6261	169	6	=	=	PUNCT
ejpam-6261	169	7	qx(f)e	qx(f)e	PROPN
ejpam-6261	169	8	iφx(f	iφx(f	PROPN
ejpam-6261	169	9	)	)	PUNCT
ejpam-6261	169	10	as	as	ADP
ejpam-6261	169	11	degree	degree	NOUN
ejpam-6261	169	12	of	of	ADP
ejpam-6261	169	13	indeterminacy	indeterminacy	NOUN
ejpam-6261	169	14	and	and	CCONJ
ejpam-6261	169	15	fw(f	fw(f	NOUN
ejpam-6261	169	16	)	)	PUNCT
ejpam-6261	169	17	=	=	SYM
ejpam-6261	170	1	rw(f)eiωw(f	rw(f)eiωw(f	NOUN
ejpam-6261	170	2	)	)	PUNCT
ejpam-6261	170	3	,	,	PUNCT
ejpam-6261	170	4	fx(f	fx(f	NOUN
ejpam-6261	170	5	)	)	PUNCT
ejpam-6261	170	6	=	=	SYM
ejpam-6261	171	1	rx(f)e	rx(f)e	NOUN
ejpam-6261	171	2	iω	iω	ADP
ejpam-6261	171	3	x	x	X
ejpam-6261	171	4	(	(	PUNCT
ejpam-6261	171	5	f	f	X
ejpam-6261	171	6	)	)	PUNCT
ejpam-6261	171	7	as	as	ADP
ejpam-6261	171	8	degree	degree	NOUN
ejpam-6261	171	9	of	of	ADP
ejpam-6261	171	10	falsity	falsity	NOUN
ejpam-6261	171	11	of	of	ADP
ejpam-6261	171	12	cnss	cnss	PROPN
ejpam-6261	171	13	w	w	PROPN
ejpam-6261	171	14	and	and	CCONJ
ejpam-6261	171	15	x.	x.	PROPN
ejpam-6261	171	16	m.	m.	PROPN
ejpam-6261	171	17	h.	h.	PROPN
ejpam-6261	171	18	mateen	mateen	PROPN
ejpam-6261	171	19	et	et	PROPN
ejpam-6261	171	20	al	al	PROPN
ejpam-6261	171	21	.	.	PUNCT
ejpam-6261	171	22	/	/	SYM
ejpam-6261	171	23	eur	eur	PROPN
ejpam-6261	171	24	.	.	PUNCT
ejpam-6261	172	1	j.	j.	PROPN
ejpam-6261	172	2	pure	pure	PROPN
ejpam-6261	172	3	appl	appl	PROPN
ejpam-6261	172	4	.	.	PROPN
ejpam-6261	172	5	math	math	PROPN
ejpam-6261	172	6	,	,	PUNCT
ejpam-6261	172	7	18	18	NUM
ejpam-6261	172	8	(	(	PUNCT
ejpam-6261	172	9	4	4	NUM
ejpam-6261	172	10	)	)	PUNCT
ejpam-6261	172	11	(	(	PUNCT
ejpam-6261	172	12	2025	2025	NUM
ejpam-6261	172	13	)	)	PUNCT
ejpam-6261	172	14	,	,	PUNCT
ejpam-6261	172	15	6261	6261	NUM
ejpam-6261	172	16	7	7	NUM
ejpam-6261	172	17	of	of	ADP
ejpam-6261	172	18	23	23	NUM
ejpam-6261	172	19	definition	definition	NOUN
ejpam-6261	172	20	4	4	NUM
ejpam-6261	172	21	.	.	PUNCT
ejpam-6261	173	1	[	[	X
ejpam-6261	173	2	41	41	NUM
ejpam-6261	173	3	]	]	PUNCT
ejpam-6261	173	4	assume	assume	VERB
ejpam-6261	173	5	that	that	SCONJ
ejpam-6261	173	6	w	w	PROPN
ejpam-6261	173	7	and	and	CCONJ
ejpam-6261	173	8	x	x	AUX
ejpam-6261	173	9	be	be	AUX
ejpam-6261	173	10	two	two	NUM
ejpam-6261	173	11	cnss	cns	NOUN
ejpam-6261	173	12	of	of	ADP
ejpam-6261	173	13	set	set	ADJ
ejpam-6261	173	14	p.	p.	NOUN
ejpam-6261	173	15	then	then	ADV
ejpam-6261	173	16	the	the	DET
ejpam-6261	173	17	intersection	intersection	NOUN
ejpam-6261	173	18	of	of	ADP
ejpam-6261	173	19	cnss	cnss	PROPN
ejpam-6261	173	20	w	w	PROPN
ejpam-6261	173	21	and	and	CCONJ
ejpam-6261	173	22	x	x	VERB
ejpam-6261	173	23	is	be	AUX
ejpam-6261	173	24	elaborated	elaborate	VERB
ejpam-6261	173	25	as	as	ADP
ejpam-6261	173	26	:	:	PUNCT
ejpam-6261	173	27	w	w	NOUN
ejpam-6261	173	28	∩	∩	NOUN
ejpam-6261	173	29	x	x	SYM
ejpam-6261	173	30	=	=	PRON
ejpam-6261	173	31	{	{	PUNCT
ejpam-6261	173	32	<	<	X
ejpam-6261	173	33	(	(	PUNCT
ejpam-6261	173	34	p),tw∩x(p	p),tw∩x(p	PROPN
ejpam-6261	173	35	)	)	PUNCT
ejpam-6261	173	36	,	,	PUNCT
ejpam-6261	173	37	iw∩x(p),fw∩x(p	iw∩x(p),fw∩x(p	PROPN
ejpam-6261	173	38	)	)	PUNCT
ejpam-6261	173	39	>	>	PUNCT
ejpam-6261	173	40	}	}	PUNCT
ejpam-6261	173	41	.	.	PUNCT
ejpam-6261	174	1	where	where	SCONJ
ejpam-6261	174	2	tw∩x(p	tw∩x(p	NOUN
ejpam-6261	174	3	)	)	PUNCT
ejpam-6261	174	4	=	=	SYM
ejpam-6261	174	5	pw∩x(p)e	pw∩x(p)e	NUM
ejpam-6261	174	6	iθw∩x(p	iθw∩x(p	NOUN
ejpam-6261	174	7	)	)	PUNCT
ejpam-6261	174	8	=	=	SYM
ejpam-6261	174	9	min{pw(p	min{pw(p	PROPN
ejpam-6261	174	10	)	)	PUNCT
ejpam-6261	174	11	,	,	PUNCT
ejpam-6261	174	12	px(p)}eimin{θw(p),θx(p	px(p)}eimin{θw(p),θx(p	NOUN
ejpam-6261	174	13	)	)	PUNCT
ejpam-6261	174	14	}	}	PUNCT
ejpam-6261	174	15	,	,	PUNCT
ejpam-6261	174	16	iw∩x(p	iw∩x(p	PROPN
ejpam-6261	174	17	)	)	PUNCT
ejpam-6261	174	18	=	=	SYM
ejpam-6261	174	19	qw∩x(p)e	qw∩x(p)e	PART
ejpam-6261	174	20	iφw∩x(p	iφw∩x(p	NOUN
ejpam-6261	174	21	)	)	PUNCT
ejpam-6261	174	22	=	=	SYM
ejpam-6261	174	23	max{qw(p	max{qw(p	NOUN
ejpam-6261	174	24	)	)	PUNCT
ejpam-6261	174	25	,	,	PUNCT
ejpam-6261	174	26	qx(p)}eimax{φw(p),φx(p	qx(p)}eimax{φw(p),φx(p	NOUN
ejpam-6261	174	27	)	)	PUNCT
ejpam-6261	174	28	}	}	PUNCT
ejpam-6261	174	29	,	,	PUNCT
ejpam-6261	174	30	fw∩x(p	fw∩x(p	PROPN
ejpam-6261	174	31	)	)	PUNCT
ejpam-6261	174	32	=	=	SYM
ejpam-6261	174	33	rw∩x(p)e	rw∩x(p)e	NUM
ejpam-6261	174	34	iωw∩x(p	iωw∩x(p	PROPN
ejpam-6261	174	35	)	)	PUNCT
ejpam-6261	174	36	=	=	SYM
ejpam-6261	174	37	max{rw(p	max{rw(p	PROPN
ejpam-6261	174	38	)	)	PUNCT
ejpam-6261	174	39	,	,	PUNCT
ejpam-6261	174	40	rx(p)}eimax{ωw(p),ωx(p	rx(p)}eimax{ωw(p),ωx(p	NOUN
ejpam-6261	174	41	)	)	PUNCT
ejpam-6261	174	42	}	}	PUNCT
ejpam-6261	174	43	.	.	PUNCT
ejpam-6261	175	1	definition	definition	NOUN
ejpam-6261	175	2	5	5	NUM
ejpam-6261	175	3	.	.	PUNCT
ejpam-6261	176	1	[	[	X
ejpam-6261	176	2	41	41	NUM
ejpam-6261	176	3	]	]	PUNCT
ejpam-6261	176	4	let	let	VERB
ejpam-6261	176	5	w	w	NOUN
ejpam-6261	176	6	and	and	CCONJ
ejpam-6261	176	7	x	x	PART
ejpam-6261	176	8	be	be	AUX
ejpam-6261	176	9	two	two	NUM
ejpam-6261	176	10	cnss	cns	NOUN
ejpam-6261	176	11	of	of	ADP
ejpam-6261	176	12	set	set	NOUN
ejpam-6261	176	13	p	p	PROPN
ejpam-6261	176	14	.	.	PUNCT
ejpam-6261	177	1	then	then	ADV
ejpam-6261	177	2	the	the	DET
ejpam-6261	177	3	union	union	NOUN
ejpam-6261	177	4	of	of	ADP
ejpam-6261	177	5	cnss	cnss	PROPN
ejpam-6261	177	6	w	w	PROPN
ejpam-6261	178	1	and	and	CCONJ
ejpam-6261	178	2	x	x	PRON
ejpam-6261	178	3	is	be	AUX
ejpam-6261	178	4	described	describe	VERB
ejpam-6261	178	5	as	as	ADP
ejpam-6261	178	6	:	:	PUNCT
ejpam-6261	178	7	w	w	NOUN
ejpam-6261	178	8	∩	∩	NOUN
ejpam-6261	178	9	x	x	SYM
ejpam-6261	178	10	=	=	PRON
ejpam-6261	178	11	{	{	PUNCT
ejpam-6261	178	12	<	<	X
ejpam-6261	178	13	(	(	PUNCT
ejpam-6261	178	14	p),tw∩x(p	p),tw∩x(p	PROPN
ejpam-6261	178	15	)	)	PUNCT
ejpam-6261	178	16	,	,	PUNCT
ejpam-6261	178	17	iw∩x(p),fw∩x(p	iw∩x(p),fw∩x(p	PROPN
ejpam-6261	178	18	)	)	PUNCT
ejpam-6261	178	19	>	>	PUNCT
ejpam-6261	178	20	}	}	PUNCT
ejpam-6261	178	21	.	.	PUNCT
ejpam-6261	179	1	where	where	SCONJ
ejpam-6261	179	2	tw∩x(p	tw∩x(p	NOUN
ejpam-6261	179	3	)	)	PUNCT
ejpam-6261	179	4	=	=	SYM
ejpam-6261	179	5	pw∩x(p)e	pw∩x(p)e	NUM
ejpam-6261	179	6	iθw∩x(p	iθw∩x(p	NOUN
ejpam-6261	179	7	)	)	PUNCT
ejpam-6261	179	8	=	=	SYM
ejpam-6261	179	9	max{pw(p	max{pw(p	ADJ
ejpam-6261	179	10	)	)	PUNCT
ejpam-6261	179	11	,	,	PUNCT
ejpam-6261	179	12	px(p)}eimax{θw(p),θx(p	px(p)}eimax{θw(p),θx(p	PROPN
ejpam-6261	179	13	)	)	PUNCT
ejpam-6261	179	14	}	}	PUNCT
ejpam-6261	179	15	,	,	PUNCT
ejpam-6261	179	16	iw∩x(p	iw∩x(p	PROPN
ejpam-6261	179	17	)	)	PUNCT
ejpam-6261	179	18	=	=	SYM
ejpam-6261	179	19	qw∩x(p)e	qw∩x(p)e	PART
ejpam-6261	179	20	iφw∩x(p	iφw∩x(p	NOUN
ejpam-6261	179	21	)	)	PUNCT
ejpam-6261	179	22	=	=	SYM
ejpam-6261	179	23	min{qw(p	min{qw(p	NOUN
ejpam-6261	179	24	)	)	PUNCT
ejpam-6261	179	25	,	,	PUNCT
ejpam-6261	179	26	qx(p)}eimin{φw(p),φx(p	qx(p)}eimin{φw(p),φx(p	NOUN
ejpam-6261	179	27	)	)	PUNCT
ejpam-6261	179	28	}	}	PUNCT
ejpam-6261	179	29	,	,	PUNCT
ejpam-6261	179	30	fw∩x(p	fw∩x(p	PROPN
ejpam-6261	179	31	)	)	PUNCT
ejpam-6261	179	32	=	=	SYM
ejpam-6261	179	33	rw∩x(p)e	rw∩x(p)e	NUM
ejpam-6261	179	34	iωw∩x(p	iωw∩x(p	PROPN
ejpam-6261	179	35	)	)	PUNCT
ejpam-6261	179	36	=	=	SYM
ejpam-6261	180	1	min{rw	min{rw	X
ejpam-6261	180	2	(	(	PUNCT
ejpam-6261	180	3	p	p	NOUN
ejpam-6261	180	4	)	)	PUNCT
ejpam-6261	180	5	,	,	PUNCT
ejpam-6261	180	6	rx(p)}eimin{ωw(p),ωx(p	rx(p)}eimin{ωw(p),ωx(p	PROPN
ejpam-6261	180	7	)	)	PUNCT
ejpam-6261	180	8	}	}	PUNCT
ejpam-6261	180	9	.	.	PUNCT
ejpam-6261	181	1	3	3	X
ejpam-6261	181	2	.	.	X
ejpam-6261	181	3	properties	property	NOUN
ejpam-6261	181	4	of	of	ADP
ejpam-6261	181	5	complex	complex	ADJ
ejpam-6261	181	6	neutrosophic	neutrosophic	ADJ
ejpam-6261	181	7	subrings	subring	NOUN
ejpam-6261	181	8	the	the	DET
ejpam-6261	181	9	investigation	investigation	NOUN
ejpam-6261	181	10	of	of	ADP
ejpam-6261	181	11	cnsrs	cnsrs	NOUN
ejpam-6261	181	12	and	and	CCONJ
ejpam-6261	181	13	level	level	NOUN
ejpam-6261	181	14	subsets	subset	NOUN
ejpam-6261	181	15	of	of	ADP
ejpam-6261	181	16	cnsrs	cnsrs	NOUN
ejpam-6261	181	17	is	be	AUX
ejpam-6261	181	18	the	the	DET
ejpam-6261	181	19	focus	focus	NOUN
ejpam-6261	181	20	of	of	ADP
ejpam-6261	181	21	this	this	DET
ejpam-6261	181	22	section	section	NOUN
ejpam-6261	181	23	.	.	PUNCT
ejpam-6261	182	1	we	we	PRON
ejpam-6261	182	2	investigate	investigate	VERB
ejpam-6261	182	3	that	that	SCONJ
ejpam-6261	182	4	a	a	DET
ejpam-6261	182	5	cnsrs	cnsrs	NOUN
ejpam-6261	182	6	produces	produce	VERB
ejpam-6261	182	7	two	two	NUM
ejpam-6261	182	8	neutrosophic	neutrosophic	ADJ
ejpam-6261	182	9	subrings	subring	NOUN
ejpam-6261	182	10	and	and	CCONJ
ejpam-6261	182	11	intersection	intersection	NOUN
ejpam-6261	182	12	of	of	ADP
ejpam-6261	182	13	two	two	NUM
ejpam-6261	182	14	cnsrs	cnsrs	NOUN
ejpam-6261	182	15	is	be	AUX
ejpam-6261	182	16	cnsr	cnsr	ADJ
ejpam-6261	182	17	.	.	PUNCT
ejpam-6261	183	1	we	we	PRON
ejpam-6261	183	2	describe	describe	VERB
ejpam-6261	183	3	the	the	DET
ejpam-6261	183	4	level	level	NOUN
ejpam-6261	183	5	-	-	PUNCT
ejpam-6261	183	6	subset	subset	NOUN
ejpam-6261	183	7	of	of	ADP
ejpam-6261	183	8	cns	cns	NOUN
ejpam-6261	183	9	and	and	CCONJ
ejpam-6261	183	10	prove	prove	VERB
ejpam-6261	183	11	that	that	DET
ejpam-6261	183	12	level	level	NOUN
ejpam-6261	183	13	-	-	PUNCT
ejpam-6261	183	14	subset	subset	NOUN
ejpam-6261	183	15	of	of	ADP
ejpam-6261	183	16	cnsr	cnsr	ADJ
ejpam-6261	183	17	form	form	NOUN
ejpam-6261	183	18	subring	subre	VERB
ejpam-6261	183	19	of	of	ADP
ejpam-6261	183	20	ring	ring	NOUN
ejpam-6261	183	21	.	.	PUNCT
ejpam-6261	184	1	definition	definition	NOUN
ejpam-6261	184	2	6	6	NUM
ejpam-6261	184	3	.	.	PUNCT
ejpam-6261	185	1	let	let	VERB
ejpam-6261	185	2	w	w	VERB
ejpam-6261	185	3	=	=	PRON
ejpam-6261	185	4	{	{	PUNCT
ejpam-6261	185	5	<	<	X
ejpam-6261	185	6	f	f	X
ejpam-6261	185	7	,	,	PUNCT
ejpam-6261	185	8	θw	θw	PROPN
ejpam-6261	185	9	(	(	PUNCT
ejpam-6261	185	10	f	f	X
ejpam-6261	185	11	)	)	PUNCT
ejpam-6261	185	12	,	,	PUNCT
ejpam-6261	185	13	φw	φw	PROPN
ejpam-6261	185	14	(	(	PUNCT
ejpam-6261	185	15	f	f	PROPN
ejpam-6261	185	16	)	)	PUNCT
ejpam-6261	185	17	,	,	PUNCT
ejpam-6261	185	18	ωw	ωw	PROPN
ejpam-6261	185	19	(	(	PUNCT
ejpam-6261	185	20	f	f	NOUN
ejpam-6261	185	21	)	)	PUNCT
ejpam-6261	185	22	,	,	PUNCT
ejpam-6261	185	23	>	>	X
ejpam-6261	185	24	:	:	PUNCT
ejpam-6261	185	25	f	f	PROPN
ejpam-6261	185	26	∈	∈	PROPN
ejpam-6261	185	27	k	k	AUX
ejpam-6261	185	28	}	}	PUNCT
ejpam-6261	185	29	be	be	AUX
ejpam-6261	185	30	a	a	DET
ejpam-6261	185	31	ns	ns	ADJ
ejpam-6261	185	32	,	,	PUNCT
ejpam-6261	185	33	where	where	SCONJ
ejpam-6261	185	34	k	k	PROPN
ejpam-6261	185	35	is	be	AUX
ejpam-6261	185	36	the	the	DET
ejpam-6261	185	37	ring	ring	NOUN
ejpam-6261	185	38	.	.	PUNCT
ejpam-6261	186	1	then	then	ADV
ejpam-6261	186	2	the	the	DET
ejpam-6261	186	3	π	π	PROPN
ejpam-6261	186	4	-	-	PROPN
ejpam-6261	186	5	ns	ns	ADJ
ejpam-6261	186	6	wπ	wπ	NOUN
ejpam-6261	186	7	is	be	AUX
ejpam-6261	186	8	described	describe	VERB
ejpam-6261	186	9	as	as	ADP
ejpam-6261	186	10	wπ	wπ	NOUN
ejpam-6261	186	11	=	=	PUNCT
ejpam-6261	186	12	{	{	PUNCT
ejpam-6261	186	13	<	<	X
ejpam-6261	186	14	f	f	PROPN
ejpam-6261	186	15	,	,	PUNCT
ejpam-6261	186	16	θwπ(f	θwπ(f	PROPN
ejpam-6261	186	17	)	)	PUNCT
ejpam-6261	186	18	,	,	PUNCT
ejpam-6261	186	19	φwπ(f	φwπ(f	PROPN
ejpam-6261	186	20	)	)	PUNCT
ejpam-6261	186	21	,	,	PUNCT
ejpam-6261	186	22	ωwπ(f	ωwπ(f	PROPN
ejpam-6261	186	23	)	)	PUNCT
ejpam-6261	186	24	>	>	PUNCT
ejpam-6261	186	25	:	:	PUNCT
ejpam-6261	186	26	f	f	PROPN
ejpam-6261	186	27	∈	∈	PROPN
ejpam-6261	186	28	k	k	NOUN
ejpam-6261	186	29	}	}	PUNCT
ejpam-6261	186	30	,	,	PUNCT
ejpam-6261	186	31	where	where	SCONJ
ejpam-6261	186	32	the	the	DET
ejpam-6261	186	33	function	function	NOUN
ejpam-6261	186	34	θwπ(f	θwπ(f	X
ejpam-6261	186	35	)	)	PUNCT
ejpam-6261	187	1	=	=	SYM
ejpam-6261	187	2	2πθw	2πθw	NUM
ejpam-6261	187	3	(	(	PUNCT
ejpam-6261	187	4	f	f	NOUN
ejpam-6261	187	5	)	)	PUNCT
ejpam-6261	187	6	,	,	PUNCT
ejpam-6261	187	7	φwπ(f	φwπ(f	PROPN
ejpam-6261	187	8	)	)	PUNCT
ejpam-6261	187	9	=	=	SYM
ejpam-6261	188	1	2πφw	2πφw	NUM
ejpam-6261	188	2	(	(	PUNCT
ejpam-6261	188	3	f	f	X
ejpam-6261	188	4	)	)	PUNCT
ejpam-6261	188	5	and	and	CCONJ
ejpam-6261	188	6	ωwπ(f	ωwπ(f	NUM
ejpam-6261	188	7	)	)	PUNCT
ejpam-6261	189	1	=	=	SYM
ejpam-6261	189	2	2πωw	2πωw	NUM
ejpam-6261	189	3	(	(	PUNCT
ejpam-6261	189	4	f	f	X
ejpam-6261	189	5	)	)	PUNCT
ejpam-6261	189	6	indicate	indicate	VERB
ejpam-6261	189	7	the	the	DET
ejpam-6261	189	8	measure	measure	NOUN
ejpam-6261	189	9	of	of	ADP
ejpam-6261	189	10	association	association	NOUN
ejpam-6261	189	11	,	,	PUNCT
ejpam-6261	189	12	measure	measure	NOUN
ejpam-6261	189	13	of	of	ADP
ejpam-6261	189	14	indeterminacy	indeterminacy	NOUN
ejpam-6261	189	15	and	and	CCONJ
ejpam-6261	189	16	measure	measure	NOUN
ejpam-6261	189	17	of	of	ADP
ejpam-6261	189	18	nonassociation	nonassociation	NOUN
ejpam-6261	189	19	of	of	ADP
ejpam-6261	189	20	an	an	DET
ejpam-6261	189	21	element	element	NOUN
ejpam-6261	189	22	f	f	PROPN
ejpam-6261	189	23	of	of	ADP
ejpam-6261	189	24	k	k	PROPN
ejpam-6261	189	25	,	,	PUNCT
ejpam-6261	189	26	consequently	consequently	ADV
ejpam-6261	189	27	.	.	PUNCT
ejpam-6261	190	1	and	and	CCONJ
ejpam-6261	190	2	fulfil	fulfil	VERB
ejpam-6261	190	3	the	the	DET
ejpam-6261	190	4	consequent	consequent	ADJ
ejpam-6261	190	5	criteria	criterion	NOUN
ejpam-6261	190	6	0	0	NUM
ejpam-6261	190	7	≤	≤	NUM
ejpam-6261	190	8	θwπ(f	θwπ(f	PROPN
ejpam-6261	190	9	)	)	PUNCT
ejpam-6261	191	1	+	+	SYM
ejpam-6261	191	2	φwπ(f	φwπ(f	X
ejpam-6261	191	3	)	)	PUNCT
ejpam-6261	191	4	+	+	NUM
ejpam-6261	191	5	ωwπ(f	ωwπ(f	PROPN
ejpam-6261	191	6	)	)	PUNCT
ejpam-6261	191	7	≤	≤	NUM
ejpam-6261	191	8	6π	6π	NOUN
ejpam-6261	191	9	.	.	PUNCT
ejpam-6261	192	1	definition	definition	NOUN
ejpam-6261	192	2	7	7	NUM
ejpam-6261	192	3	.	.	PUNCT
ejpam-6261	193	1	a	a	DET
ejpam-6261	193	2	π	π	PROPN
ejpam-6261	193	3	-	-	PROPN
ejpam-6261	193	4	ns	ns	INTJ
ejpam-6261	193	5	wπ	wπ	NOUN
ejpam-6261	193	6	of	of	ADP
ejpam-6261	193	7	ring	ring	PROPN
ejpam-6261	193	8	k	k	PROPN
ejpam-6261	193	9	is	be	AUX
ejpam-6261	193	10	known	know	VERB
ejpam-6261	193	11	as	as	ADP
ejpam-6261	193	12	π	π	PROPN
ejpam-6261	193	13	-	-	ADJ
ejpam-6261	193	14	neutrosophic	neutrosophic	ADJ
ejpam-6261	193	15	subring	subring	NOUN
ejpam-6261	193	16	of	of	ADP
ejpam-6261	193	17	k	k	NOUN
ejpam-6261	193	18	,	,	PUNCT
ejpam-6261	193	19	∀	∀	X
ejpam-6261	193	20	p	p	NOUN
ejpam-6261	193	21	,	,	PUNCT
ejpam-6261	194	1	u	u	NOUN
ejpam-6261	194	2	∈	∈	PROPN
ejpam-6261	194	3	k	k	NOUN
ejpam-6261	194	4	if	if	SCONJ
ejpam-6261	194	5	(	(	PUNCT
ejpam-6261	194	6	i	i	NOUN
ejpam-6261	194	7	)	)	PUNCT
ejpam-6261	194	8	θwπ(p−	θwπ(p−	PUNCT
ejpam-6261	194	9	u	u	NOUN
ejpam-6261	194	10	)	)	PUNCT
ejpam-6261	194	11	≥	≥	PROPN
ejpam-6261	194	12	min{θwπ(p	min{θwπ(p	PROPN
ejpam-6261	194	13	)	)	PUNCT
ejpam-6261	194	14	,	,	PUNCT
ejpam-6261	194	15	θwπ(u	θwπ(u	PROPN
ejpam-6261	194	16	)	)	PUNCT
ejpam-6261	194	17	}	}	PUNCT
ejpam-6261	194	18	,	,	PUNCT
ejpam-6261	194	19	(	(	PUNCT
ejpam-6261	194	20	ii	ii	NOUN
ejpam-6261	194	21	)	)	PUNCT
ejpam-6261	194	22	θwπ(pu	θwπ(pu	PROPN
ejpam-6261	194	23	)	)	PUNCT
ejpam-6261	194	24	≥	≥	PROPN
ejpam-6261	194	25	min{θwπ(p	min{θwπ(p	PROPN
ejpam-6261	194	26	)	)	PUNCT
ejpam-6261	194	27	,	,	PUNCT
ejpam-6261	194	28	θwπ(u	θwπ(u	PROPN
ejpam-6261	194	29	)	)	PUNCT
ejpam-6261	194	30	}	}	PUNCT
ejpam-6261	194	31	,	,	PUNCT
ejpam-6261	194	32	(	(	PUNCT
ejpam-6261	194	33	iii	iii	NOUN
ejpam-6261	194	34	)	)	PUNCT
ejpam-6261	194	35	φwπ(p−	φwπ(p−	VERB
ejpam-6261	194	36	u	u	NOUN
ejpam-6261	194	37	)	)	PUNCT
ejpam-6261	194	38	≤	≤	PROPN
ejpam-6261	194	39	max{φwπ(p	max{φwπ(p	PROPN
ejpam-6261	194	40	)	)	PUNCT
ejpam-6261	194	41	,	,	PUNCT
ejpam-6261	194	42	φwπ(u	φwπ(u	PROPN
ejpam-6261	194	43	)	)	PUNCT
ejpam-6261	194	44	}	}	PUNCT
ejpam-6261	194	45	,	,	PUNCT
ejpam-6261	194	46	(	(	PUNCT
ejpam-6261	194	47	iv	iv	X
ejpam-6261	194	48	)	)	PUNCT
ejpam-6261	194	49	φwπ(pu	φwπ(pu	NOUN
ejpam-6261	194	50	)	)	PUNCT
ejpam-6261	194	51	≤	≤	PROPN
ejpam-6261	194	52	max{φwπ(p	max{φwπ(p	PROPN
ejpam-6261	194	53	)	)	PUNCT
ejpam-6261	194	54	,	,	PUNCT
ejpam-6261	194	55	φwπ(u	φwπ(u	PROPN
ejpam-6261	194	56	)	)	PUNCT
ejpam-6261	194	57	}	}	PUNCT
ejpam-6261	194	58	,	,	PUNCT
ejpam-6261	194	59	(	(	PUNCT
ejpam-6261	194	60	v	v	NOUN
ejpam-6261	194	61	)	)	PUNCT
ejpam-6261	194	62	ωwπ(p−	ωwπ(p−	PART
ejpam-6261	194	63	u	u	NOUN
ejpam-6261	194	64	)	)	PUNCT
ejpam-6261	194	65	≤	≤	NOUN
ejpam-6261	195	1	max{ωwπ(p	max{ωwπ(p	PROPN
ejpam-6261	195	2	)	)	PUNCT
ejpam-6261	195	3	,	,	PUNCT
ejpam-6261	195	4	ωwπ(u	ωwπ(u	NOUN
ejpam-6261	195	5	)	)	PUNCT
ejpam-6261	195	6	}	}	PUNCT
ejpam-6261	195	7	,	,	PUNCT
ejpam-6261	195	8	(	(	PUNCT
ejpam-6261	195	9	vi	vi	NOUN
ejpam-6261	195	10	)	)	PUNCT
ejpam-6261	195	11	ωwπ(pu	ωwπ(pu	NOUN
ejpam-6261	195	12	)	)	PUNCT
ejpam-6261	195	13	≤	≤	NOUN
ejpam-6261	196	1	max{ωwπ(p	max{ωwπ(p	PROPN
ejpam-6261	196	2	)	)	PUNCT
ejpam-6261	196	3	,	,	PUNCT
ejpam-6261	196	4	ωwπ(u	ωwπ(u	NOUN
ejpam-6261	196	5	)	)	PUNCT
ejpam-6261	196	6	}	}	PUNCT
ejpam-6261	196	7	.	.	PUNCT
ejpam-6261	197	1	definition	definition	NOUN
ejpam-6261	197	2	8	8	NUM
ejpam-6261	197	3	.	.	PUNCT
ejpam-6261	198	1	let	let	VERB
ejpam-6261	198	2	w	w	NOUN
ejpam-6261	198	3	and	and	CCONJ
ejpam-6261	198	4	x	x	PART
ejpam-6261	198	5	be	be	AUX
ejpam-6261	198	6	two	two	NUM
ejpam-6261	198	7	cnss	cns	NOUN
ejpam-6261	198	8	of	of	ADP
ejpam-6261	198	9	k.	k.	PROPN
ejpam-6261	198	10	then	then	ADV
ejpam-6261	198	11	m.	m.	PROPN
ejpam-6261	198	12	h.	h.	PROPN
ejpam-6261	198	13	mateen	mateen	PROPN
ejpam-6261	198	14	et	et	PROPN
ejpam-6261	198	15	al	al	PROPN
ejpam-6261	198	16	.	.	PUNCT
ejpam-6261	198	17	/	/	SYM
ejpam-6261	198	18	eur	eur	PROPN
ejpam-6261	198	19	.	.	PUNCT
ejpam-6261	199	1	j.	j.	PROPN
ejpam-6261	199	2	pure	pure	PROPN
ejpam-6261	199	3	appl	appl	PROPN
ejpam-6261	199	4	.	.	PROPN
ejpam-6261	199	5	math	math	PROPN
ejpam-6261	199	6	,	,	PUNCT
ejpam-6261	199	7	18	18	NUM
ejpam-6261	199	8	(	(	PUNCT
ejpam-6261	199	9	4	4	NUM
ejpam-6261	199	10	)	)	PUNCT
ejpam-6261	199	11	(	(	PUNCT
ejpam-6261	199	12	2025	2025	NUM
ejpam-6261	199	13	)	)	PUNCT
ejpam-6261	199	14	,	,	PUNCT
ejpam-6261	199	15	6261	6261	NUM
ejpam-6261	199	16	8	8	NUM
ejpam-6261	199	17	of	of	ADP
ejpam-6261	199	18	23	23	NUM
ejpam-6261	199	19	(	(	PUNCT
ejpam-6261	199	20	i	i	NOUN
ejpam-6261	199	21	)	)	PUNCT
ejpam-6261	199	22	a	a	DET
ejpam-6261	199	23	cns	cns	NOUN
ejpam-6261	199	24	w	w	NOUN
ejpam-6261	199	25	is	be	AUX
ejpam-6261	199	26	homogeneous	homogeneous	ADJ
ejpam-6261	199	27	cns	cns	NOUN
ejpam-6261	199	28	,	,	PUNCT
ejpam-6261	199	29	if	if	SCONJ
ejpam-6261	199	30	for	for	ADP
ejpam-6261	199	31	all	all	DET
ejpam-6261	199	32	p	p	NOUN
ejpam-6261	199	33	,	,	PUNCT
ejpam-6261	199	34	a	a	DET
ejpam-6261	199	35	∈	∈	PROPN
ejpam-6261	199	36	k	k	NOUN
ejpam-6261	199	37	,	,	PUNCT
ejpam-6261	199	38	we	we	PRON
ejpam-6261	199	39	have	have	VERB
ejpam-6261	199	40	a	a	DET
ejpam-6261	199	41	pw	pw	X
ejpam-6261	199	42	(	(	PUNCT
ejpam-6261	199	43	p	p	NOUN
ejpam-6261	199	44	)	)	PUNCT
ejpam-6261	199	45	≤	≤	NUM
ejpam-6261	200	1	pw	pw	X
ejpam-6261	201	1	(	(	PUNCT
ejpam-6261	201	2	a	a	X
ejpam-6261	201	3	)	)	PUNCT
ejpam-6261	201	4	if	if	SCONJ
ejpam-6261	201	5	and	and	CCONJ
ejpam-6261	201	6	only	only	ADV
ejpam-6261	201	7	if	if	SCONJ
ejpam-6261	201	8	θw	θw	PROPN
ejpam-6261	201	9	(	(	PUNCT
ejpam-6261	201	10	p	p	NOUN
ejpam-6261	201	11	)	)	PUNCT
ejpam-6261	201	12	≤	≤	NOUN
ejpam-6261	201	13	θw	θw	X
ejpam-6261	201	14	(	(	PUNCT
ejpam-6261	201	15	a	a	NOUN
ejpam-6261	201	16	)	)	PUNCT
ejpam-6261	201	17	,	,	PUNCT
ejpam-6261	201	18	b	b	X
ejpam-6261	201	19	qw	qw	X
ejpam-6261	201	20	(	(	PUNCT
ejpam-6261	201	21	p	p	NOUN
ejpam-6261	201	22	)	)	PUNCT
ejpam-6261	201	23	≤	≤	NUM
ejpam-6261	201	24	qw	qw	X
ejpam-6261	201	25	(	(	PUNCT
ejpam-6261	201	26	a	a	X
ejpam-6261	201	27	)	)	PUNCT
ejpam-6261	201	28	if	if	SCONJ
ejpam-6261	202	1	and	and	CCONJ
ejpam-6261	202	2	only	only	ADV
ejpam-6261	202	3	if	if	SCONJ
ejpam-6261	202	4	φw	φw	PROPN
ejpam-6261	202	5	(	(	PUNCT
ejpam-6261	202	6	p	p	NOUN
ejpam-6261	202	7	)	)	PUNCT
ejpam-6261	202	8	≤	≤	NOUN
ejpam-6261	202	9	φw	φw	NOUN
ejpam-6261	202	10	(	(	PUNCT
ejpam-6261	202	11	a	a	NOUN
ejpam-6261	202	12	)	)	PUNCT
ejpam-6261	202	13	,	,	PUNCT
ejpam-6261	202	14	c	c	PROPN
ejpam-6261	202	15	rw	rw	PROPN
ejpam-6261	202	16	(	(	PUNCT
ejpam-6261	202	17	p	p	NOUN
ejpam-6261	202	18	)	)	PUNCT
ejpam-6261	202	19	≥	≥	NOUN
ejpam-6261	202	20	rw	rw	NOUN
ejpam-6261	202	21	(	(	PUNCT
ejpam-6261	202	22	a	a	NOUN
ejpam-6261	202	23	)	)	PUNCT
ejpam-6261	202	24	if	if	SCONJ
ejpam-6261	203	1	and	and	CCONJ
ejpam-6261	203	2	only	only	ADV
ejpam-6261	203	3	if	if	SCONJ
ejpam-6261	203	4	ωw	ωw	X
ejpam-6261	203	5	(	(	PUNCT
ejpam-6261	203	6	p	p	NOUN
ejpam-6261	203	7	)	)	PUNCT
ejpam-6261	203	8	≥	≥	X
ejpam-6261	203	9	ωw	ωw	X
ejpam-6261	203	10	(	(	PUNCT
ejpam-6261	203	11	a	a	NOUN
ejpam-6261	203	12	)	)	PUNCT
ejpam-6261	203	13	.	.	PUNCT
ejpam-6261	204	1	(	(	PUNCT
ejpam-6261	204	2	ii	ii	X
ejpam-6261	204	3	)	)	PUNCT
ejpam-6261	204	4	a	a	DET
ejpam-6261	204	5	cns	cns	NOUN
ejpam-6261	204	6	w	w	NOUN
ejpam-6261	204	7	is	be	AUX
ejpam-6261	204	8	homogeneous	homogeneous	ADJ
ejpam-6261	204	9	cns	cns	NOUN
ejpam-6261	204	10	with	with	ADP
ejpam-6261	204	11	x	x	SYM
ejpam-6261	204	12	,	,	PUNCT
ejpam-6261	204	13	if	if	SCONJ
ejpam-6261	204	14	for	for	ADP
ejpam-6261	204	15	all	all	DET
ejpam-6261	204	16	p	p	NOUN
ejpam-6261	204	17	∈	∈	PROPN
ejpam-6261	204	18	k	k	NOUN
ejpam-6261	204	19	,	,	PUNCT
ejpam-6261	204	20	we	we	PRON
ejpam-6261	204	21	have	have	VERB
ejpam-6261	204	22	a	a	DET
ejpam-6261	204	23	pw	pw	X
ejpam-6261	204	24	(	(	PUNCT
ejpam-6261	204	25	p	p	NOUN
ejpam-6261	204	26	)	)	PUNCT
ejpam-6261	204	27	≤	≤	NOUN
ejpam-6261	204	28	px(p	px(p	NUM
ejpam-6261	204	29	)	)	PUNCT
ejpam-6261	204	30	iff	iff	PROPN
ejpam-6261	204	31	θw	θw	PROPN
ejpam-6261	204	32	(	(	PUNCT
ejpam-6261	204	33	p	p	NOUN
ejpam-6261	204	34	)	)	PUNCT
ejpam-6261	204	35	≤	≤	NOUN
ejpam-6261	204	36	θx(p	θx(p	NUM
ejpam-6261	204	37	)	)	PUNCT
ejpam-6261	204	38	,	,	PUNCT
ejpam-6261	204	39	b	b	X
ejpam-6261	204	40	qw	qw	X
ejpam-6261	204	41	(	(	PUNCT
ejpam-6261	204	42	p	p	NOUN
ejpam-6261	204	43	)	)	PUNCT
ejpam-6261	204	44	≤	≤	NOUN
ejpam-6261	204	45	qx(p	qx(p	NUM
ejpam-6261	204	46	)	)	PUNCT
ejpam-6261	204	47	iff	iff	PROPN
ejpam-6261	204	48	φw	φw	PROPN
ejpam-6261	204	49	(	(	PUNCT
ejpam-6261	204	50	p	p	NOUN
ejpam-6261	204	51	)	)	PUNCT
ejpam-6261	204	52	≤	≤	NOUN
ejpam-6261	204	53	φx(p	φx(p	NUM
ejpam-6261	204	54	)	)	PUNCT
ejpam-6261	204	55	,	,	PUNCT
ejpam-6261	204	56	c	c	PROPN
ejpam-6261	204	57	rw	rw	PROPN
ejpam-6261	204	58	(	(	PUNCT
ejpam-6261	204	59	p	p	NOUN
ejpam-6261	204	60	)	)	PUNCT
ejpam-6261	204	61	≥	≥	NOUN
ejpam-6261	204	62	rx(p	rx(p	NUM
ejpam-6261	204	63	)	)	PUNCT
ejpam-6261	204	64	iff	iff	PROPN
ejpam-6261	204	65	ωw	ωw	ADP
ejpam-6261	204	66	(	(	PUNCT
ejpam-6261	204	67	p	p	NOUN
ejpam-6261	204	68	)	)	PUNCT
ejpam-6261	204	69	≥	≥	NOUN
ejpam-6261	204	70	ωx(p	ωx(p	PUNCT
ejpam-6261	204	71	)	)	PUNCT
ejpam-6261	204	72	.	.	PUNCT
ejpam-6261	205	1	in	in	ADP
ejpam-6261	205	2	this	this	DET
ejpam-6261	205	3	article	article	NOUN
ejpam-6261	205	4	,	,	PUNCT
ejpam-6261	205	5	we	we	PRON
ejpam-6261	205	6	shall	shall	AUX
ejpam-6261	205	7	take	take	VERB
ejpam-6261	205	8	cns	cns	NOUN
ejpam-6261	205	9	as	as	ADP
ejpam-6261	205	10	homogeneous	homogeneous	ADJ
ejpam-6261	205	11	cns	cns	NOUN
ejpam-6261	205	12	.	.	PUNCT
ejpam-6261	206	1	definition	definition	NOUN
ejpam-6261	206	2	9	9	NUM
ejpam-6261	206	3	.	.	PUNCT
ejpam-6261	207	1	a	a	DET
ejpam-6261	207	2	cns	cns	NOUN
ejpam-6261	207	3	w	w	NOUN
ejpam-6261	207	4	=	=	X
ejpam-6261	207	5	{	{	PUNCT
ejpam-6261	207	6	<	<	X
ejpam-6261	207	7	(	(	PUNCT
ejpam-6261	207	8	x	x	X
ejpam-6261	207	9	,	,	PUNCT
ejpam-6261	207	10	tw	tw	PROPN
ejpam-6261	207	11	(	(	PUNCT
ejpam-6261	207	12	p	p	NOUN
ejpam-6261	207	13	)	)	PUNCT
ejpam-6261	207	14	,	,	PUNCT
ejpam-6261	207	15	iw	iw	INTJ
ejpam-6261	207	16	(	(	PUNCT
ejpam-6261	207	17	p),fw	p),fw	X
ejpam-6261	207	18	(	(	PUNCT
ejpam-6261	207	19	p	p	NOUN
ejpam-6261	207	20	)	)	PUNCT
ejpam-6261	207	21	)	)	PUNCT
ejpam-6261	207	22	>	>	PUNCT
ejpam-6261	207	23	:	:	PUNCT
ejpam-6261	207	24	p	p	X
ejpam-6261	207	25	∈	∈	PROPN
ejpam-6261	207	26	k	k	X
ejpam-6261	207	27	}	}	PUNCT
ejpam-6261	207	28	of	of	ADP
ejpam-6261	207	29	ring	ring	NOUN
ejpam-6261	207	30	k	k	PROPN
ejpam-6261	207	31	is	be	AUX
ejpam-6261	207	32	called	call	VERB
ejpam-6261	207	33	a	a	DET
ejpam-6261	207	34	cnsr	cnsr	NOUN
ejpam-6261	207	35	,	,	PUNCT
ejpam-6261	207	36	∀	∀	PUNCT
ejpam-6261	207	37	p	p	NOUN
ejpam-6261	207	38	,	,	PUNCT
ejpam-6261	208	1	u	u	NOUN
ejpam-6261	208	2	∈	∈	PROPN
ejpam-6261	208	3	k	k	NOUN
ejpam-6261	209	1	if	if	SCONJ
ejpam-6261	209	2	(	(	PUNCT
ejpam-6261	209	3	i	i	NOUN
ejpam-6261	209	4	)	)	PUNCT
ejpam-6261	209	5	tw	tw	PROPN
ejpam-6261	209	6	(	(	PUNCT
ejpam-6261	209	7	p−	p−	NOUN
ejpam-6261	209	8	u	u	NOUN
ejpam-6261	209	9	)	)	PUNCT
ejpam-6261	209	10	≥	≥	NOUN
ejpam-6261	209	11	min{tw	min{tw	X
ejpam-6261	209	12	(	(	PUNCT
ejpam-6261	209	13	p),tw	p),tw	X
ejpam-6261	209	14	(	(	PUNCT
ejpam-6261	209	15	u	u	NOUN
ejpam-6261	209	16	)	)	PUNCT
ejpam-6261	209	17	}	}	PUNCT
ejpam-6261	209	18	,	,	PUNCT
ejpam-6261	209	19	(	(	PUNCT
ejpam-6261	209	20	ii	ii	NOUN
ejpam-6261	209	21	)	)	PUNCT
ejpam-6261	209	22	tw	tw	PROPN
ejpam-6261	209	23	(	(	PUNCT
ejpam-6261	209	24	pu	pu	PROPN
ejpam-6261	209	25	)	)	PUNCT
ejpam-6261	209	26	≥	≥	NOUN
ejpam-6261	209	27	min{tw	min{tw	X
ejpam-6261	209	28	(	(	PUNCT
ejpam-6261	209	29	p),tw	p),tw	X
ejpam-6261	209	30	(	(	PUNCT
ejpam-6261	209	31	u	u	NOUN
ejpam-6261	209	32	)	)	PUNCT
ejpam-6261	209	33	}	}	PUNCT
ejpam-6261	209	34	,	,	PUNCT
ejpam-6261	209	35	(	(	PUNCT
ejpam-6261	209	36	iii	iii	X
ejpam-6261	209	37	)	)	PUNCT
ejpam-6261	209	38	iw	iw	NOUN
ejpam-6261	209	39	(	(	PUNCT
ejpam-6261	209	40	p−	p−	NOUN
ejpam-6261	209	41	u	u	NOUN
ejpam-6261	209	42	)	)	PUNCT
ejpam-6261	209	43	≤	≤	NOUN
ejpam-6261	209	44	max{iw	max{iw	X
ejpam-6261	209	45	(	(	PUNCT
ejpam-6261	209	46	p	p	NOUN
ejpam-6261	209	47	)	)	PUNCT
ejpam-6261	209	48	,	,	PUNCT
ejpam-6261	209	49	iw	iw	PROPN
ejpam-6261	209	50	(	(	PUNCT
ejpam-6261	209	51	u	u	NOUN
ejpam-6261	209	52	)	)	PUNCT
ejpam-6261	209	53	}	}	PUNCT
ejpam-6261	209	54	,	,	PUNCT
ejpam-6261	209	55	(	(	PUNCT
ejpam-6261	209	56	iv	iv	X
ejpam-6261	209	57	)	)	PUNCT
ejpam-6261	209	58	iw	iw	PROPN
ejpam-6261	209	59	(	(	PUNCT
ejpam-6261	209	60	pu	pu	PROPN
ejpam-6261	209	61	)	)	PUNCT
ejpam-6261	209	62	≤	≤	NOUN
ejpam-6261	209	63	max{iw	max{iw	X
ejpam-6261	209	64	(	(	PUNCT
ejpam-6261	209	65	p	p	NOUN
ejpam-6261	209	66	)	)	PUNCT
ejpam-6261	209	67	,	,	PUNCT
ejpam-6261	209	68	iw	iw	PROPN
ejpam-6261	209	69	(	(	PUNCT
ejpam-6261	209	70	u	u	NOUN
ejpam-6261	209	71	)	)	PUNCT
ejpam-6261	209	72	}	}	PUNCT
ejpam-6261	209	73	,	,	PUNCT
ejpam-6261	209	74	(	(	PUNCT
ejpam-6261	209	75	v	v	NOUN
ejpam-6261	209	76	)	)	PUNCT
ejpam-6261	209	77	fw	fw	NOUN
ejpam-6261	209	78	(	(	PUNCT
ejpam-6261	209	79	p−	p−	NOUN
ejpam-6261	209	80	u	u	NOUN
ejpam-6261	209	81	)	)	PUNCT
ejpam-6261	209	82	≤	≤	ADJ
ejpam-6261	210	1	max{fw	max{fw	NOUN
ejpam-6261	210	2	(	(	PUNCT
ejpam-6261	210	3	p),fw	p),fw	X
ejpam-6261	210	4	(	(	PUNCT
ejpam-6261	210	5	u	u	NOUN
ejpam-6261	210	6	)	)	PUNCT
ejpam-6261	210	7	}	}	PUNCT
ejpam-6261	210	8	,	,	PUNCT
ejpam-6261	210	9	(	(	PUNCT
ejpam-6261	210	10	vi	vi	NOUN
ejpam-6261	210	11	)	)	PUNCT
ejpam-6261	210	12	fw	fw	PROPN
ejpam-6261	210	13	(	(	PUNCT
ejpam-6261	210	14	pu	pu	PROPN
ejpam-6261	210	15	)	)	PUNCT
ejpam-6261	210	16	≤	≤	NOUN
ejpam-6261	210	17	max{fw	max{fw	NOUN
ejpam-6261	210	18	(	(	PUNCT
ejpam-6261	210	19	p),fw	p),fw	X
ejpam-6261	210	20	(	(	PUNCT
ejpam-6261	210	21	u	u	NOUN
ejpam-6261	210	22	)	)	PUNCT
ejpam-6261	210	23	}	}	PUNCT
ejpam-6261	210	24	,	,	PUNCT
ejpam-6261	210	25	another	another	DET
ejpam-6261	210	26	way	way	NOUN
ejpam-6261	210	27	to	to	PART
ejpam-6261	210	28	describe	describe	VERB
ejpam-6261	210	29	the	the	DET
ejpam-6261	210	30	definition	definition	NOUN
ejpam-6261	210	31	cnsr	cnsr	NOUN
ejpam-6261	210	32	is	be	AUX
ejpam-6261	210	33	given	give	VERB
ejpam-6261	210	34	as	as	ADP
ejpam-6261	210	35	;	;	PUNCT
ejpam-6261	210	36	(	(	PUNCT
ejpam-6261	210	37	i	i	NOUN
ejpam-6261	210	38	)	)	PUNCT
ejpam-6261	210	39	pw	pw	PROPN
ejpam-6261	211	1	(	(	PUNCT
ejpam-6261	211	2	p−	p−	INTJ
ejpam-6261	211	3	u)eiθw	u)eiθw	X
ejpam-6261	211	4	(	(	PUNCT
ejpam-6261	211	5	p−u	p−u	NOUN
ejpam-6261	211	6	)	)	PUNCT
ejpam-6261	211	7	≥	≥	NOUN
ejpam-6261	211	8	min{pw	min{pw	ADV
ejpam-6261	211	9	(	(	PUNCT
ejpam-6261	211	10	p	p	NOUN
ejpam-6261	211	11	)	)	PUNCT
ejpam-6261	211	12	,	,	PUNCT
ejpam-6261	211	13	pw	pw	X
ejpam-6261	211	14	(	(	PUNCT
ejpam-6261	211	15	u)}eimin{θw	u)}eimin{θw	NOUN
ejpam-6261	211	16	(	(	PUNCT
ejpam-6261	211	17	p),θw	p),θw	NOUN
ejpam-6261	211	18	(	(	PUNCT
ejpam-6261	211	19	u	u	NOUN
ejpam-6261	211	20	)	)	PUNCT
ejpam-6261	211	21	}	}	PUNCT
ejpam-6261	211	22	,	,	PUNCT
ejpam-6261	211	23	(	(	PUNCT
ejpam-6261	211	24	ii	ii	NOUN
ejpam-6261	211	25	)	)	PUNCT
ejpam-6261	211	26	pw	pw	PROPN
ejpam-6261	211	27	(	(	PUNCT
ejpam-6261	211	28	pu)eiθw	pu)eiθw	NOUN
ejpam-6261	211	29	(	(	PUNCT
ejpam-6261	211	30	pu	pu	PROPN
ejpam-6261	211	31	)	)	PUNCT
ejpam-6261	211	32	≥	≥	NOUN
ejpam-6261	211	33	min{pw	min{pw	ADV
ejpam-6261	211	34	(	(	PUNCT
ejpam-6261	211	35	p	p	NOUN
ejpam-6261	211	36	)	)	PUNCT
ejpam-6261	211	37	,	,	PUNCT
ejpam-6261	211	38	pw	pw	X
ejpam-6261	211	39	(	(	PUNCT
ejpam-6261	211	40	u)}eimin{θw	u)}eimin{θw	NOUN
ejpam-6261	211	41	(	(	PUNCT
ejpam-6261	211	42	p),θw	p),θw	NOUN
ejpam-6261	211	43	(	(	PUNCT
ejpam-6261	211	44	u	u	NOUN
ejpam-6261	211	45	)	)	PUNCT
ejpam-6261	211	46	}	}	PUNCT
ejpam-6261	211	47	,	,	PUNCT
ejpam-6261	211	48	(	(	PUNCT
ejpam-6261	211	49	iii	iii	NOUN
ejpam-6261	211	50	)	)	PUNCT
ejpam-6261	211	51	qw	qw	NOUN
ejpam-6261	211	52	(	(	PUNCT
ejpam-6261	211	53	p−	p−	NOUN
ejpam-6261	211	54	u)eiφw	u)eiφw	PROPN
ejpam-6261	211	55	(	(	PUNCT
ejpam-6261	211	56	p−u	p−u	NOUN
ejpam-6261	211	57	)	)	PUNCT
ejpam-6261	211	58	≤	≤	NOUN
ejpam-6261	211	59	max{qw	max{qw	CCONJ
ejpam-6261	211	60	(	(	PUNCT
ejpam-6261	211	61	p	p	NOUN
ejpam-6261	211	62	)	)	PUNCT
ejpam-6261	211	63	,	,	PUNCT
ejpam-6261	211	64	qw	qw	X
ejpam-6261	211	65	(	(	PUNCT
ejpam-6261	211	66	u)}eimax{φw	u)}eimax{φw	PROPN
ejpam-6261	211	67	(	(	PUNCT
ejpam-6261	211	68	p),φw	p),φw	X
ejpam-6261	211	69	(	(	PUNCT
ejpam-6261	211	70	u	u	NOUN
ejpam-6261	211	71	)	)	PUNCT
ejpam-6261	211	72	}	}	PUNCT
ejpam-6261	211	73	,	,	PUNCT
ejpam-6261	211	74	(	(	PUNCT
ejpam-6261	211	75	iv	iv	X
ejpam-6261	211	76	)	)	PUNCT
ejpam-6261	211	77	qw	qw	X
ejpam-6261	211	78	(	(	PUNCT
ejpam-6261	211	79	pu)eiφw	pu)eiφw	PROPN
ejpam-6261	211	80	(	(	PUNCT
ejpam-6261	211	81	pu	pu	PROPN
ejpam-6261	211	82	)	)	PUNCT
ejpam-6261	211	83	≤	≤	NOUN
ejpam-6261	212	1	max{qw	max{qw	CCONJ
ejpam-6261	212	2	(	(	PUNCT
ejpam-6261	212	3	p	p	NOUN
ejpam-6261	212	4	)	)	PUNCT
ejpam-6261	212	5	,	,	PUNCT
ejpam-6261	212	6	qw	qw	X
ejpam-6261	212	7	(	(	PUNCT
ejpam-6261	212	8	u)}eimax{φw	u)}eimax{φw	PROPN
ejpam-6261	212	9	(	(	PUNCT
ejpam-6261	212	10	p),φw	p),φw	X
ejpam-6261	212	11	(	(	PUNCT
ejpam-6261	212	12	u	u	NOUN
ejpam-6261	212	13	)	)	PUNCT
ejpam-6261	212	14	}	}	PUNCT
ejpam-6261	212	15	,	,	PUNCT
ejpam-6261	212	16	(	(	PUNCT
ejpam-6261	212	17	v	v	NOUN
ejpam-6261	212	18	)	)	PUNCT
ejpam-6261	212	19	rw	rw	NOUN
ejpam-6261	212	20	(	(	PUNCT
ejpam-6261	212	21	p−	p−	X
ejpam-6261	212	22	u)eiωw	u)eiωw	X
ejpam-6261	212	23	(	(	PUNCT
ejpam-6261	212	24	p−u	p−u	NOUN
ejpam-6261	212	25	)	)	PUNCT
ejpam-6261	212	26	≤	≤	NOUN
ejpam-6261	213	1	max{rw	max{rw	NUM
ejpam-6261	213	2	(	(	PUNCT
ejpam-6261	213	3	p	p	NOUN
ejpam-6261	213	4	)	)	PUNCT
ejpam-6261	213	5	,	,	PUNCT
ejpam-6261	213	6	rw	rw	PROPN
ejpam-6261	213	7	(	(	PUNCT
ejpam-6261	213	8	u	u	NOUN
ejpam-6261	213	9	)	)	PUNCT
ejpam-6261	213	10	}	}	PUNCT
ejpam-6261	213	11	eimax{ωw	eimax{ωw	NOUN
ejpam-6261	213	12	(	(	PUNCT
ejpam-6261	213	13	p),ωw	p),ωw	X
ejpam-6261	213	14	(	(	PUNCT
ejpam-6261	213	15	u	u	NOUN
ejpam-6261	213	16	)	)	PUNCT
ejpam-6261	213	17	}	}	PUNCT
ejpam-6261	213	18	(	(	PUNCT
ejpam-6261	213	19	vi	vi	NOUN
ejpam-6261	213	20	)	)	PUNCT
ejpam-6261	213	21	rw	rw	NOUN
ejpam-6261	213	22	(	(	PUNCT
ejpam-6261	213	23	pu)eiωw	pu)eiωw	PROPN
ejpam-6261	213	24	(	(	PUNCT
ejpam-6261	213	25	pu	pu	PROPN
ejpam-6261	213	26	)	)	PUNCT
ejpam-6261	213	27	≤	≤	NOUN
ejpam-6261	213	28	max{rw	max{rw	NUM
ejpam-6261	213	29	(	(	PUNCT
ejpam-6261	213	30	p	p	NOUN
ejpam-6261	213	31	)	)	PUNCT
ejpam-6261	213	32	,	,	PUNCT
ejpam-6261	213	33	rw	rw	PROPN
ejpam-6261	213	34	(	(	PUNCT
ejpam-6261	213	35	u	u	NOUN
ejpam-6261	213	36	)	)	PUNCT
ejpam-6261	213	37	}	}	PUNCT
ejpam-6261	213	38	eimax{ωw	eimax{ωw	NOUN
ejpam-6261	213	39	(	(	PUNCT
ejpam-6261	213	40	p),ωw	p),ωw	X
ejpam-6261	213	41	(	(	PUNCT
ejpam-6261	213	42	u	u	NOUN
ejpam-6261	213	43	)	)	PUNCT
ejpam-6261	213	44	}	}	PUNCT
ejpam-6261	213	45	for	for	ADP
ejpam-6261	213	46	all	all	DET
ejpam-6261	213	47	p	p	NOUN
ejpam-6261	213	48	,	,	PUNCT
ejpam-6261	213	49	u	u	PROPN
ejpam-6261	213	50	∈	∈	PROPN
ejpam-6261	213	51	k.	k.	NOUN
ejpam-6261	213	52	in	in	ADP
ejpam-6261	213	53	the	the	DET
ejpam-6261	213	54	following	following	NOUN
ejpam-6261	213	55	theorem	theorem	NOUN
ejpam-6261	213	56	,	,	PUNCT
ejpam-6261	213	57	we	we	PRON
ejpam-6261	213	58	prove	prove	VERB
ejpam-6261	213	59	that	that	SCONJ
ejpam-6261	213	60	a	a	DET
ejpam-6261	213	61	cnsr	cnsr	NOUN
ejpam-6261	213	62	produce	produce	VERB
ejpam-6261	213	63	two	two	NUM
ejpam-6261	213	64	neutrosophic	neutrosophic	ADJ
ejpam-6261	213	65	subrings	subring	NOUN
ejpam-6261	213	66	,	,	PUNCT
ejpam-6261	213	67	namely	namely	ADV
ejpam-6261	213	68	neutrosophic	neutrosophic	ADJ
ejpam-6261	213	69	subring	subring	NOUN
ejpam-6261	213	70	and	and	CCONJ
ejpam-6261	213	71	π	π	PROPN
ejpam-6261	213	72	-	-	ADJ
ejpam-6261	213	73	neutrosophic	neutrosophic	ADJ
ejpam-6261	213	74	subring(πnsr	subring(πnsr	PROPN
ejpam-6261	213	75	)	)	PUNCT
ejpam-6261	213	76	.	.	PUNCT
ejpam-6261	214	1	theorem	theorem	NOUN
ejpam-6261	214	2	2	2	NUM
ejpam-6261	214	3	.	.	PUNCT
ejpam-6261	215	1	let	let	VERB
ejpam-6261	215	2	w	w	NOUN
ejpam-6261	215	3	be	be	AUX
ejpam-6261	215	4	a	a	DET
ejpam-6261	215	5	cns	cns	NOUN
ejpam-6261	215	6	of	of	ADP
ejpam-6261	215	7	ring	ring	PROPN
ejpam-6261	216	1	k.	k.	PROPN
ejpam-6261	216	2	then	then	ADV
ejpam-6261	216	3	w	w	PROPN
ejpam-6261	216	4	is	be	AUX
ejpam-6261	216	5	a	a	DET
ejpam-6261	216	6	cnsr	cnsr	NOUN
ejpam-6261	216	7	of	of	ADP
ejpam-6261	216	8	k	k	PROPN
ejpam-6261	216	9	if	if	SCONJ
ejpam-6261	217	1	and	and	CCONJ
ejpam-6261	217	2	only	only	ADV
ejpam-6261	217	3	if	if	SCONJ
ejpam-6261	217	4	(	(	PUNCT
ejpam-6261	217	5	i	i	NOUN
ejpam-6261	217	6	)	)	PUNCT
ejpam-6261	217	7	the	the	DET
ejpam-6261	217	8	fuzzy	fuzzy	ADJ
ejpam-6261	217	9	set	set	VERB
ejpam-6261	217	10	w	w	NOUN
ejpam-6261	217	11	=	=	PUNCT
ejpam-6261	217	12	{	{	PUNCT
ejpam-6261	217	13	<	<	X
ejpam-6261	217	14	k	k	X
ejpam-6261	217	15	,	,	PUNCT
ejpam-6261	217	16	pw	pw	PROPN
ejpam-6261	217	17	(	(	PUNCT
ejpam-6261	217	18	k	k	NOUN
ejpam-6261	217	19	)	)	PUNCT
ejpam-6261	217	20	,	,	PUNCT
ejpam-6261	217	21	qw	qw	X
ejpam-6261	217	22	(	(	PUNCT
ejpam-6261	217	23	k	k	NOUN
ejpam-6261	217	24	)	)	PUNCT
ejpam-6261	217	25	,	,	PUNCT
ejpam-6261	217	26	rw	rw	PROPN
ejpam-6261	217	27	(	(	PUNCT
ejpam-6261	217	28	k	k	NOUN
ejpam-6261	217	29	)	)	PUNCT
ejpam-6261	217	30	>	>	PUNCT
ejpam-6261	217	31	:	:	PUNCT
ejpam-6261	218	1	k	k	PROPN
ejpam-6261	218	2	∈	∈	PROPN
ejpam-6261	218	3	k	k	PROPN
ejpam-6261	218	4	,	,	PUNCT
ejpam-6261	218	5	pw	pw	PROPN
ejpam-6261	218	6	(	(	PUNCT
ejpam-6261	218	7	k	k	NOUN
ejpam-6261	218	8	)	)	PUNCT
ejpam-6261	218	9	,	,	PUNCT
ejpam-6261	218	10	qw	qw	X
ejpam-6261	218	11	(	(	PUNCT
ejpam-6261	218	12	k	k	NOUN
ejpam-6261	218	13	)	)	PUNCT
ejpam-6261	218	14	,	,	PUNCT
ejpam-6261	218	15	rw	rw	PROPN
ejpam-6261	218	16	(	(	PUNCT
ejpam-6261	218	17	k	k	NOUN
ejpam-6261	218	18	)	)	PUNCT
ejpam-6261	218	19	∈	∈	PROPN
ejpam-6261	219	1	[	[	X
ejpam-6261	219	2	0	0	NUM
ejpam-6261	219	3	,	,	PUNCT
ejpam-6261	219	4	1	1	NUM
ejpam-6261	219	5	]	]	PUNCT
ejpam-6261	219	6	and	and	CCONJ
ejpam-6261	219	7	0	0	NUM
ejpam-6261	219	8	≤	≤	NUM
ejpam-6261	219	9	pw	pw	X
ejpam-6261	219	10	(	(	PUNCT
ejpam-6261	219	11	k	k	NOUN
ejpam-6261	219	12	)	)	PUNCT
ejpam-6261	219	13	+	+	NUM
ejpam-6261	219	14	qw	qw	X
ejpam-6261	219	15	(	(	PUNCT
ejpam-6261	219	16	k	k	NOUN
ejpam-6261	219	17	)	)	PUNCT
ejpam-6261	219	18	+	+	NUM
ejpam-6261	219	19	rw	rw	NOUN
ejpam-6261	219	20	(	(	PUNCT
ejpam-6261	219	21	k)+	k)+	NOUN
ejpam-6261	219	22	≤	≤	NOUN
ejpam-6261	219	23	3	3	NUM
ejpam-6261	219	24	}	}	PUNCT
ejpam-6261	219	25	is	be	AUX
ejpam-6261	219	26	a	a	DET
ejpam-6261	219	27	nsr	nsr	PROPN
ejpam-6261	219	28	.	.	PUNCT
ejpam-6261	219	29	m.	m.	PROPN
ejpam-6261	219	30	h.	h.	PROPN
ejpam-6261	219	31	mateen	mateen	PROPN
ejpam-6261	219	32	et	et	PROPN
ejpam-6261	219	33	al	al	PROPN
ejpam-6261	219	34	.	.	PUNCT
ejpam-6261	219	35	/	/	SYM
ejpam-6261	219	36	eur	eur	PROPN
ejpam-6261	219	37	.	.	PUNCT
ejpam-6261	220	1	j.	j.	PROPN
ejpam-6261	220	2	pure	pure	PROPN
ejpam-6261	220	3	appl	appl	PROPN
ejpam-6261	220	4	.	.	PROPN
ejpam-6261	220	5	math	math	PROPN
ejpam-6261	220	6	,	,	PUNCT
ejpam-6261	220	7	18	18	NUM
ejpam-6261	220	8	(	(	PUNCT
ejpam-6261	220	9	4	4	NUM
ejpam-6261	220	10	)	)	PUNCT
ejpam-6261	220	11	(	(	PUNCT
ejpam-6261	220	12	2025	2025	NUM
ejpam-6261	220	13	)	)	PUNCT
ejpam-6261	220	14	,	,	PUNCT
ejpam-6261	220	15	6261	6261	NUM
ejpam-6261	220	16	9	9	NUM
ejpam-6261	220	17	of	of	ADP
ejpam-6261	220	18	23	23	NUM
ejpam-6261	220	19	(	(	PUNCT
ejpam-6261	220	20	ii	ii	NOUN
ejpam-6261	220	21	)	)	PUNCT
ejpam-6261	220	22	the	the	DET
ejpam-6261	220	23	π	π	PROPN
ejpam-6261	220	24	-	-	ADJ
ejpam-6261	220	25	fuzzy	fuzzy	ADJ
ejpam-6261	220	26	set	set	NOUN
ejpam-6261	220	27	w	w	NOUN
ejpam-6261	220	28	=	=	PUNCT
ejpam-6261	220	29	{	{	PUNCT
ejpam-6261	220	30	<	<	X
ejpam-6261	220	31	k	k	X
ejpam-6261	220	32	,	,	PUNCT
ejpam-6261	220	33	θw	θw	PROPN
ejpam-6261	220	34	(	(	PUNCT
ejpam-6261	220	35	k	k	NOUN
ejpam-6261	220	36	)	)	PUNCT
ejpam-6261	220	37	,	,	PUNCT
ejpam-6261	220	38	φw	φw	PROPN
ejpam-6261	220	39	(	(	PUNCT
ejpam-6261	220	40	k	k	NOUN
ejpam-6261	220	41	)	)	PUNCT
ejpam-6261	220	42	,	,	PUNCT
ejpam-6261	220	43	ωw	ωw	PROPN
ejpam-6261	220	44	(	(	PUNCT
ejpam-6261	220	45	k	k	NOUN
ejpam-6261	220	46	)	)	PUNCT
ejpam-6261	220	47	>	>	PUNCT
ejpam-6261	220	48	:	:	PUNCT
ejpam-6261	221	1	k	k	PROPN
ejpam-6261	221	2	∈	∈	PROPN
ejpam-6261	222	1	k	k	PROPN
ejpam-6261	222	2	,	,	PUNCT
ejpam-6261	222	3	θw	θw	PROPN
ejpam-6261	222	4	(	(	PUNCT
ejpam-6261	222	5	k	k	NOUN
ejpam-6261	222	6	)	)	PUNCT
ejpam-6261	222	7	,	,	PUNCT
ejpam-6261	222	8	φw	φw	PROPN
ejpam-6261	222	9	(	(	PUNCT
ejpam-6261	222	10	k	k	NOUN
ejpam-6261	222	11	)	)	PUNCT
ejpam-6261	222	12	,	,	PUNCT
ejpam-6261	222	13	ωw	ωw	PROPN
ejpam-6261	222	14	(	(	PUNCT
ejpam-6261	222	15	k	k	NOUN
ejpam-6261	222	16	)	)	PUNCT
ejpam-6261	222	17	∈	∈	PROPN
ejpam-6261	223	1	[	[	X
ejpam-6261	223	2	0	0	NUM
ejpam-6261	223	3	,	,	PUNCT
ejpam-6261	223	4	2π	2π	NOUN
ejpam-6261	223	5	]	]	PUNCT
ejpam-6261	223	6	}	}	PUNCT
ejpam-6261	223	7	is	be	AUX
ejpam-6261	223	8	a	a	DET
ejpam-6261	223	9	πnsr	πnsr	NOUN
ejpam-6261	223	10	.	.	PUNCT
ejpam-6261	224	1	proof	proof	NOUN
ejpam-6261	224	2	.	.	PUNCT
ejpam-6261	225	1	assume	assume	VERB
ejpam-6261	225	2	that	that	SCONJ
ejpam-6261	225	3	w	w	NOUN
ejpam-6261	225	4	is	be	AUX
ejpam-6261	225	5	a	a	DET
ejpam-6261	225	6	cnsr	cnsr	NOUN
ejpam-6261	225	7	and	and	CCONJ
ejpam-6261	225	8	k	k	NOUN
ejpam-6261	225	9	,	,	PUNCT
ejpam-6261	226	1	l	l	PROPN
ejpam-6261	226	2	∈	∈	PROPN
ejpam-6261	226	3	m	m	VERB
ejpam-6261	226	4	.	.	PUNCT
ejpam-6261	227	1	then	then	ADV
ejpam-6261	227	2	we	we	PRON
ejpam-6261	227	3	know	know	VERB
ejpam-6261	227	4	that	that	PRON
ejpam-6261	227	5	,	,	PUNCT
ejpam-6261	227	6	pw	pw	PROPN
ejpam-6261	227	7	(	(	PUNCT
ejpam-6261	227	8	k	k	X
ejpam-6261	227	9	−	−	PROPN
ejpam-6261	227	10	l)eiθw	l)eiθw	PROPN
ejpam-6261	227	11	(	(	PUNCT
ejpam-6261	227	12	k−l	k−l	NOUN
ejpam-6261	227	13	)	)	PUNCT
ejpam-6261	228	1	=	=	SYM
ejpam-6261	228	2	tw	tw	NOUN
ejpam-6261	228	3	(	(	PUNCT
ejpam-6261	228	4	k	k	NOUN
ejpam-6261	228	5	−	−	PROPN
ejpam-6261	228	6	l	l	NOUN
ejpam-6261	228	7	)	)	PUNCT
ejpam-6261	228	8	≥	≥	NOUN
ejpam-6261	228	9	min{tw	min{tw	X
ejpam-6261	228	10	(	(	PUNCT
ejpam-6261	228	11	k),tw	k),tw	X
ejpam-6261	228	12	(	(	PUNCT
ejpam-6261	228	13	l	l	NOUN
ejpam-6261	228	14	)	)	PUNCT
ejpam-6261	228	15	}	}	PUNCT
ejpam-6261	228	16	=	=	SYM
ejpam-6261	228	17	min{pw	min{pw	X
ejpam-6261	228	18	(	(	PUNCT
ejpam-6261	228	19	k)eiθw	k)eiθw	X
ejpam-6261	228	20	(	(	PUNCT
ejpam-6261	228	21	k	k	NOUN
ejpam-6261	228	22	)	)	PUNCT
ejpam-6261	228	23	,	,	PUNCT
ejpam-6261	228	24	pw	pw	PROPN
ejpam-6261	228	25	(	(	PUNCT
ejpam-6261	228	26	l)eiθw	l)eiθw	X
ejpam-6261	228	27	(	(	PUNCT
ejpam-6261	228	28	l	l	NOUN
ejpam-6261	228	29	)	)	PUNCT
ejpam-6261	228	30	}	}	PUNCT
ejpam-6261	228	31	=	=	PUNCT
ejpam-6261	228	32	min{pw	min{pw	X
ejpam-6261	228	33	(	(	PUNCT
ejpam-6261	228	34	k	k	NOUN
ejpam-6261	228	35	)	)	PUNCT
ejpam-6261	228	36	,	,	PUNCT
ejpam-6261	228	37	pw	pw	PROPN
ejpam-6261	228	38	(	(	PUNCT
ejpam-6261	228	39	l)}eimin{θw	l)}eimin{θw	NOUN
ejpam-6261	228	40	(	(	PUNCT
ejpam-6261	228	41	k),θw	k),θw	X
ejpam-6261	228	42	(	(	PUNCT
ejpam-6261	228	43	l	l	NOUN
ejpam-6261	228	44	)	)	PUNCT
ejpam-6261	228	45	}	}	PUNCT
ejpam-6261	228	46	.	.	PUNCT
ejpam-6261	229	1	as	as	SCONJ
ejpam-6261	229	2	w	w	PROPN
ejpam-6261	229	3	is	be	AUX
ejpam-6261	229	4	homogeneous	homogeneous	ADJ
ejpam-6261	229	5	,	,	PUNCT
ejpam-6261	229	6	so	so	ADV
ejpam-6261	229	7	pw	pw	PROPN
ejpam-6261	230	1	(	(	PUNCT
ejpam-6261	230	2	k	k	NOUN
ejpam-6261	230	3	−	−	PROPN
ejpam-6261	230	4	l	l	NOUN
ejpam-6261	230	5	)	)	PUNCT
ejpam-6261	230	6	≥	≥	NOUN
ejpam-6261	230	7	min{pw	min{pw	ADV
ejpam-6261	230	8	(	(	PUNCT
ejpam-6261	230	9	k	k	NOUN
ejpam-6261	230	10	)	)	PUNCT
ejpam-6261	230	11	,	,	PUNCT
ejpam-6261	230	12	pw	pw	PROPN
ejpam-6261	230	13	(	(	PUNCT
ejpam-6261	230	14	l	l	NOUN
ejpam-6261	230	15	)	)	PUNCT
ejpam-6261	230	16	}	}	PUNCT
ejpam-6261	230	17	and	and	CCONJ
ejpam-6261	230	18	θw	θw	PROPN
ejpam-6261	230	19	(	(	PUNCT
ejpam-6261	230	20	k	k	NOUN
ejpam-6261	230	21	−	−	PROPN
ejpam-6261	230	22	l	l	NOUN
ejpam-6261	230	23	)	)	PUNCT
ejpam-6261	230	24	≥	≥	NOUN
ejpam-6261	230	25	min{θw	min{θw	X
ejpam-6261	230	26	(	(	PUNCT
ejpam-6261	230	27	k	k	NOUN
ejpam-6261	230	28	)	)	PUNCT
ejpam-6261	230	29	,	,	PUNCT
ejpam-6261	230	30	θw	θw	X
ejpam-6261	230	31	(	(	PUNCT
ejpam-6261	230	32	l	l	NOUN
ejpam-6261	230	33	)	)	PUNCT
ejpam-6261	230	34	}	}	PUNCT
ejpam-6261	230	35	.	.	PUNCT
ejpam-6261	231	1	pw	pw	PROPN
ejpam-6261	231	2	(	(	PUNCT
ejpam-6261	231	3	kl)eiωw	kl)eiωw	PROPN
ejpam-6261	231	4	(	(	PUNCT
ejpam-6261	231	5	kl	kl	NOUN
ejpam-6261	231	6	)	)	PUNCT
ejpam-6261	231	7	=	=	SYM
ejpam-6261	231	8	tw	tw	NOUN
ejpam-6261	231	9	(	(	PUNCT
ejpam-6261	231	10	kl	kl	PROPN
ejpam-6261	231	11	)	)	PUNCT
ejpam-6261	231	12	≥	≥	NOUN
ejpam-6261	231	13	min{tw	min{tw	X
ejpam-6261	231	14	(	(	PUNCT
ejpam-6261	231	15	k),tw	k),tw	X
ejpam-6261	231	16	(	(	PUNCT
ejpam-6261	231	17	l	l	NOUN
ejpam-6261	231	18	)	)	PUNCT
ejpam-6261	231	19	}	}	PUNCT
ejpam-6261	231	20	=	=	SYM
ejpam-6261	231	21	min{pw	min{pw	X
ejpam-6261	231	22	(	(	PUNCT
ejpam-6261	231	23	k)eiθw	k)eiθw	X
ejpam-6261	231	24	(	(	PUNCT
ejpam-6261	231	25	k	k	NOUN
ejpam-6261	231	26	)	)	PUNCT
ejpam-6261	231	27	,	,	PUNCT
ejpam-6261	231	28	pw	pw	PROPN
ejpam-6261	231	29	(	(	PUNCT
ejpam-6261	231	30	l)eiθw	l)eiθw	X
ejpam-6261	231	31	(	(	PUNCT
ejpam-6261	231	32	l	l	NOUN
ejpam-6261	231	33	)	)	PUNCT
ejpam-6261	231	34	}	}	PUNCT
ejpam-6261	231	35	=	=	PUNCT
ejpam-6261	231	36	min{pw	min{pw	X
ejpam-6261	231	37	(	(	PUNCT
ejpam-6261	231	38	k	k	NOUN
ejpam-6261	231	39	)	)	PUNCT
ejpam-6261	231	40	,	,	PUNCT
ejpam-6261	231	41	pw	pw	PROPN
ejpam-6261	231	42	(	(	PUNCT
ejpam-6261	231	43	l)}eimin{θw	l)}eimin{θw	NOUN
ejpam-6261	231	44	(	(	PUNCT
ejpam-6261	231	45	k),θw	k),θw	X
ejpam-6261	231	46	(	(	PUNCT
ejpam-6261	231	47	l	l	NOUN
ejpam-6261	231	48	)	)	PUNCT
ejpam-6261	231	49	}	}	PUNCT
ejpam-6261	231	50	.	.	PUNCT
ejpam-6261	232	1	as	as	SCONJ
ejpam-6261	232	2	w	w	PROPN
ejpam-6261	232	3	is	be	AUX
ejpam-6261	232	4	homogeneous	homogeneous	ADJ
ejpam-6261	232	5	,	,	PUNCT
ejpam-6261	232	6	thus	thus	ADV
ejpam-6261	232	7	pw	pw	X
ejpam-6261	232	8	(	(	PUNCT
ejpam-6261	232	9	kl	kl	PROPN
ejpam-6261	232	10	)	)	PUNCT
ejpam-6261	232	11	≥	≥	NOUN
ejpam-6261	232	12	min{pw	min{pw	ADV
ejpam-6261	232	13	(	(	PUNCT
ejpam-6261	232	14	k	k	NOUN
ejpam-6261	232	15	)	)	PUNCT
ejpam-6261	232	16	,	,	PUNCT
ejpam-6261	232	17	pw	pw	PROPN
ejpam-6261	232	18	(	(	PUNCT
ejpam-6261	232	19	l	l	NOUN
ejpam-6261	232	20	)	)	PUNCT
ejpam-6261	232	21	}	}	PUNCT
ejpam-6261	232	22	and	and	CCONJ
ejpam-6261	232	23	θw	θw	PROPN
ejpam-6261	232	24	(	(	PUNCT
ejpam-6261	232	25	kl	kl	PROPN
ejpam-6261	232	26	)	)	PUNCT
ejpam-6261	232	27	≥	≥	NOUN
ejpam-6261	232	28	min{θw	min{θw	X
ejpam-6261	232	29	(	(	PUNCT
ejpam-6261	232	30	k	k	NOUN
ejpam-6261	232	31	)	)	PUNCT
ejpam-6261	232	32	,	,	PUNCT
ejpam-6261	232	33	θw	θw	X
ejpam-6261	232	34	(	(	PUNCT
ejpam-6261	232	35	l	l	NOUN
ejpam-6261	232	36	)	)	PUNCT
ejpam-6261	232	37	}	}	PUNCT
ejpam-6261	232	38	.	.	PUNCT
ejpam-6261	233	1	suppose	suppose	VERB
ejpam-6261	233	2	that	that	SCONJ
ejpam-6261	233	3	w	w	PROPN
ejpam-6261	233	4	is	be	AUX
ejpam-6261	233	5	a	a	DET
ejpam-6261	233	6	cnsr	cnsr	NOUN
ejpam-6261	233	7	and	and	CCONJ
ejpam-6261	233	8	k	k	NOUN
ejpam-6261	233	9	,	,	PUNCT
ejpam-6261	233	10	l	l	PROPN
ejpam-6261	233	11	∈	∈	PROPN
ejpam-6261	233	12	h.	h.	NOUN
ejpam-6261	233	13	then	then	ADV
ejpam-6261	233	14	we	we	PRON
ejpam-6261	233	15	have	have	VERB
ejpam-6261	233	16	,	,	PUNCT
ejpam-6261	233	17	qw	qw	X
ejpam-6261	233	18	(	(	PUNCT
ejpam-6261	233	19	k	k	PROPN
ejpam-6261	233	20	−	−	PROPN
ejpam-6261	233	21	l)eiφw	l)eiφw	PROPN
ejpam-6261	233	22	(	(	PUNCT
ejpam-6261	233	23	k−l	k−l	NOUN
ejpam-6261	233	24	)	)	PUNCT
ejpam-6261	234	1	=	=	SYM
ejpam-6261	234	2	iw	iw	INTJ
ejpam-6261	234	3	(	(	PUNCT
ejpam-6261	234	4	k	k	NOUN
ejpam-6261	234	5	−	−	PROPN
ejpam-6261	234	6	l	l	NOUN
ejpam-6261	234	7	)	)	PUNCT
ejpam-6261	234	8	≤	≤	NOUN
ejpam-6261	234	9	max{iw	max{iw	X
ejpam-6261	234	10	(	(	PUNCT
ejpam-6261	234	11	k	k	NOUN
ejpam-6261	234	12	)	)	PUNCT
ejpam-6261	234	13	,	,	PUNCT
ejpam-6261	234	14	iw	iw	PROPN
ejpam-6261	234	15	(	(	PUNCT
ejpam-6261	234	16	l	l	NOUN
ejpam-6261	234	17	)	)	PUNCT
ejpam-6261	234	18	}	}	PUNCT
ejpam-6261	234	19	=	=	SYM
ejpam-6261	234	20	max{qw	max{qw	NOUN
ejpam-6261	234	21	(	(	PUNCT
ejpam-6261	234	22	k)eiφw	k)eiφw	PROPN
ejpam-6261	234	23	(	(	PUNCT
ejpam-6261	234	24	k	k	NOUN
ejpam-6261	234	25	)	)	PUNCT
ejpam-6261	234	26	,	,	PUNCT
ejpam-6261	234	27	qw	qw	X
ejpam-6261	234	28	(	(	PUNCT
ejpam-6261	234	29	l)eiφw	l)eiφw	PROPN
ejpam-6261	234	30	(	(	PUNCT
ejpam-6261	234	31	l	l	NOUN
ejpam-6261	234	32	)	)	PUNCT
ejpam-6261	234	33	}	}	PUNCT
ejpam-6261	234	34	=	=	SYM
ejpam-6261	234	35	max{qw	max{qw	X
ejpam-6261	234	36	(	(	PUNCT
ejpam-6261	234	37	k	k	NOUN
ejpam-6261	234	38	)	)	PUNCT
ejpam-6261	234	39	,	,	PUNCT
ejpam-6261	234	40	qw	qw	X
ejpam-6261	234	41	(	(	PUNCT
ejpam-6261	234	42	l)}eimax{φw	l)}eimax{φw	PROPN
ejpam-6261	234	43	(	(	PUNCT
ejpam-6261	234	44	k),φw	k),φw	X
ejpam-6261	234	45	(	(	PUNCT
ejpam-6261	234	46	l	l	NOUN
ejpam-6261	234	47	)	)	PUNCT
ejpam-6261	234	48	}	}	PUNCT
ejpam-6261	234	49	.	.	PUNCT
ejpam-6261	235	1	as	as	SCONJ
ejpam-6261	235	2	w	w	PROPN
ejpam-6261	235	3	is	be	AUX
ejpam-6261	235	4	homogeneous	homogeneous	ADJ
ejpam-6261	235	5	,	,	PUNCT
ejpam-6261	235	6	so	so	ADV
ejpam-6261	235	7	qw	qw	INTJ
ejpam-6261	235	8	(	(	PUNCT
ejpam-6261	235	9	k	k	PROPN
ejpam-6261	235	10	−	−	PROPN
ejpam-6261	235	11	l	l	NOUN
ejpam-6261	235	12	)	)	PUNCT
ejpam-6261	235	13	≤	≤	NOUN
ejpam-6261	235	14	max{qw	max{qw	CCONJ
ejpam-6261	235	15	(	(	PUNCT
ejpam-6261	235	16	k	k	NOUN
ejpam-6261	235	17	)	)	PUNCT
ejpam-6261	235	18	,	,	PUNCT
ejpam-6261	235	19	qw	qw	X
ejpam-6261	235	20	(	(	PUNCT
ejpam-6261	235	21	l	l	NOUN
ejpam-6261	235	22	)	)	PUNCT
ejpam-6261	235	23	}	}	PUNCT
ejpam-6261	235	24	and	and	CCONJ
ejpam-6261	235	25	φw	φw	PROPN
ejpam-6261	235	26	(	(	PUNCT
ejpam-6261	235	27	k	k	PROPN
ejpam-6261	235	28	−	−	PROPN
ejpam-6261	235	29	l	l	NOUN
ejpam-6261	235	30	)	)	PUNCT
ejpam-6261	235	31	≤	≤	NUM
ejpam-6261	236	1	max{φw	max{φw	NOUN
ejpam-6261	236	2	(	(	PUNCT
ejpam-6261	236	3	k	k	NOUN
ejpam-6261	236	4	)	)	PUNCT
ejpam-6261	236	5	,	,	PUNCT
ejpam-6261	236	6	φw	φw	PROPN
ejpam-6261	236	7	(	(	PUNCT
ejpam-6261	236	8	l	l	NOUN
ejpam-6261	236	9	)	)	PUNCT
ejpam-6261	236	10	}	}	PUNCT
ejpam-6261	236	11	.	.	PUNCT
ejpam-6261	237	1	qw	qw	X
ejpam-6261	237	2	(	(	PUNCT
ejpam-6261	237	3	kl)eiφw	kl)eiφw	PROPN
ejpam-6261	237	4	(	(	PUNCT
ejpam-6261	237	5	kl	kl	NOUN
ejpam-6261	237	6	)	)	PUNCT
ejpam-6261	237	7	=	=	SYM
ejpam-6261	237	8	iw	iw	PROPN
ejpam-6261	237	9	(	(	PUNCT
ejpam-6261	237	10	kl	kl	NOUN
ejpam-6261	237	11	)	)	PUNCT
ejpam-6261	237	12	≤	≤	NOUN
ejpam-6261	237	13	max{iw	max{iw	X
ejpam-6261	237	14	(	(	PUNCT
ejpam-6261	237	15	k	k	NOUN
ejpam-6261	237	16	)	)	PUNCT
ejpam-6261	237	17	,	,	PUNCT
ejpam-6261	237	18	iw	iw	PROPN
ejpam-6261	237	19	(	(	PUNCT
ejpam-6261	237	20	l	l	NOUN
ejpam-6261	237	21	)	)	PUNCT
ejpam-6261	237	22	}	}	PUNCT
ejpam-6261	237	23	=	=	SYM
ejpam-6261	237	24	max{qw	max{qw	NOUN
ejpam-6261	237	25	(	(	PUNCT
ejpam-6261	237	26	k)eiφw	k)eiφw	PROPN
ejpam-6261	237	27	(	(	PUNCT
ejpam-6261	237	28	k	k	NOUN
ejpam-6261	237	29	)	)	PUNCT
ejpam-6261	237	30	,	,	PUNCT
ejpam-6261	237	31	qw	qw	X
ejpam-6261	237	32	(	(	PUNCT
ejpam-6261	237	33	l)eiφw	l)eiφw	PROPN
ejpam-6261	237	34	(	(	PUNCT
ejpam-6261	237	35	l	l	NOUN
ejpam-6261	237	36	)	)	PUNCT
ejpam-6261	237	37	}	}	PUNCT
ejpam-6261	237	38	=	=	SYM
ejpam-6261	237	39	max{qw	max{qw	X
ejpam-6261	237	40	(	(	PUNCT
ejpam-6261	237	41	k	k	NOUN
ejpam-6261	237	42	)	)	PUNCT
ejpam-6261	237	43	,	,	PUNCT
ejpam-6261	237	44	qw	qw	X
ejpam-6261	237	45	(	(	PUNCT
ejpam-6261	237	46	l)}eimax{φw	l)}eimax{φw	PROPN
ejpam-6261	237	47	(	(	PUNCT
ejpam-6261	237	48	k),φw	k),φw	X
ejpam-6261	237	49	(	(	PUNCT
ejpam-6261	237	50	l	l	NOUN
ejpam-6261	237	51	)	)	PUNCT
ejpam-6261	237	52	}	}	PUNCT
ejpam-6261	237	53	.	.	PUNCT
ejpam-6261	238	1	as	as	SCONJ
ejpam-6261	238	2	w	w	PROPN
ejpam-6261	238	3	is	be	AUX
ejpam-6261	238	4	homogeneous	homogeneous	ADJ
ejpam-6261	238	5	,	,	PUNCT
ejpam-6261	238	6	we	we	PRON
ejpam-6261	238	7	have	have	VERB
ejpam-6261	238	8	qw	qw	X
ejpam-6261	238	9	(	(	PUNCT
ejpam-6261	238	10	kl	kl	PROPN
ejpam-6261	238	11	)	)	PUNCT
ejpam-6261	238	12	≤	≤	NOUN
ejpam-6261	238	13	max{qw	max{qw	CCONJ
ejpam-6261	238	14	(	(	PUNCT
ejpam-6261	238	15	k	k	NOUN
ejpam-6261	238	16	)	)	PUNCT
ejpam-6261	238	17	,	,	PUNCT
ejpam-6261	238	18	qw	qw	X
ejpam-6261	238	19	(	(	PUNCT
ejpam-6261	238	20	l	l	NOUN
ejpam-6261	238	21	)	)	PUNCT
ejpam-6261	238	22	}	}	PUNCT
ejpam-6261	238	23	and	and	CCONJ
ejpam-6261	238	24	φw	φw	PROPN
ejpam-6261	238	25	(	(	PUNCT
ejpam-6261	238	26	kl	kl	PROPN
ejpam-6261	238	27	)	)	PUNCT
ejpam-6261	238	28	≤	≤	NOUN
ejpam-6261	238	29	max{φw	max{φw	NOUN
ejpam-6261	238	30	(	(	PUNCT
ejpam-6261	238	31	k	k	NOUN
ejpam-6261	238	32	)	)	PUNCT
ejpam-6261	238	33	,	,	PUNCT
ejpam-6261	238	34	φw	φw	PROPN
ejpam-6261	238	35	(	(	PUNCT
ejpam-6261	238	36	l	l	NOUN
ejpam-6261	238	37	)	)	PUNCT
ejpam-6261	238	38	}	}	PUNCT
ejpam-6261	238	39	.	.	PUNCT
ejpam-6261	239	1	rw	rw	PROPN
ejpam-6261	239	2	(	(	PUNCT
ejpam-6261	239	3	k	k	PROPN
ejpam-6261	239	4	−	−	PROPN
ejpam-6261	239	5	l)eiωw	l)eiωw	PROPN
ejpam-6261	239	6	(	(	PUNCT
ejpam-6261	239	7	k−l	k−l	NOUN
ejpam-6261	239	8	)	)	PUNCT
ejpam-6261	240	1	=	=	SYM
ejpam-6261	240	2	rw	rw	NOUN
ejpam-6261	240	3	(	(	PUNCT
ejpam-6261	240	4	k	k	NOUN
ejpam-6261	240	5	−	−	PROPN
ejpam-6261	240	6	l	l	NOUN
ejpam-6261	240	7	)	)	PUNCT
ejpam-6261	240	8	≤	≤	NOUN
ejpam-6261	241	1	max{fw	max{fw	NOUN
ejpam-6261	242	1	(	(	PUNCT
ejpam-6261	242	2	k),fw	k),fw	X
ejpam-6261	242	3	(	(	PUNCT
ejpam-6261	242	4	l	l	NOUN
ejpam-6261	242	5	)	)	PUNCT
ejpam-6261	242	6	}	}	PUNCT
ejpam-6261	242	7	=	=	SYM
ejpam-6261	243	1	max{rw	max{rw	X
ejpam-6261	243	2	(	(	PUNCT
ejpam-6261	243	3	k)eiωw	k)eiωw	PROPN
ejpam-6261	243	4	(	(	PUNCT
ejpam-6261	243	5	k	k	NOUN
ejpam-6261	243	6	)	)	PUNCT
ejpam-6261	243	7	,	,	PUNCT
ejpam-6261	243	8	rw	rw	PROPN
ejpam-6261	243	9	(	(	PUNCT
ejpam-6261	243	10	l)eiωw	l)eiωw	PROPN
ejpam-6261	243	11	(	(	PUNCT
ejpam-6261	243	12	l	l	NOUN
ejpam-6261	243	13	)	)	PUNCT
ejpam-6261	243	14	}	}	PUNCT
ejpam-6261	243	15	=	=	SYM
ejpam-6261	243	16	max{rw	max{rw	X
ejpam-6261	243	17	(	(	PUNCT
ejpam-6261	243	18	k	k	NOUN
ejpam-6261	243	19	)	)	PUNCT
ejpam-6261	243	20	,	,	PUNCT
ejpam-6261	243	21	rw	rw	NOUN
ejpam-6261	243	22	(	(	PUNCT
ejpam-6261	243	23	l)}eimax{ωw	l)}eimax{ωw	NOUN
ejpam-6261	243	24	(	(	PUNCT
ejpam-6261	243	25	k),ωw	k),ωw	X
ejpam-6261	243	26	(	(	PUNCT
ejpam-6261	243	27	l	l	NOUN
ejpam-6261	243	28	)	)	PUNCT
ejpam-6261	243	29	}	}	PUNCT
ejpam-6261	243	30	.	.	PUNCT
ejpam-6261	244	1	m.	m.	NOUN
ejpam-6261	244	2	h.	h.	PROPN
ejpam-6261	244	3	mateen	mateen	PROPN
ejpam-6261	244	4	et	et	PROPN
ejpam-6261	244	5	al	al	PROPN
ejpam-6261	244	6	.	.	PUNCT
ejpam-6261	244	7	/	/	SYM
ejpam-6261	244	8	eur	eur	PROPN
ejpam-6261	244	9	.	.	PUNCT
ejpam-6261	245	1	j.	j.	PROPN
ejpam-6261	245	2	pure	pure	PROPN
ejpam-6261	245	3	appl	appl	PROPN
ejpam-6261	245	4	.	.	PROPN
ejpam-6261	245	5	math	math	PROPN
ejpam-6261	245	6	,	,	PUNCT
ejpam-6261	245	7	18	18	NUM
ejpam-6261	245	8	(	(	PUNCT
ejpam-6261	245	9	4	4	NUM
ejpam-6261	245	10	)	)	PUNCT
ejpam-6261	245	11	(	(	PUNCT
ejpam-6261	245	12	2025	2025	NUM
ejpam-6261	245	13	)	)	PUNCT
ejpam-6261	245	14	,	,	PUNCT
ejpam-6261	245	15	6261	6261	NUM
ejpam-6261	245	16	10	10	NUM
ejpam-6261	245	17	of	of	ADP
ejpam-6261	245	18	23	23	NUM
ejpam-6261	245	19	as	as	SCONJ
ejpam-6261	245	20	w	w	NOUN
ejpam-6261	245	21	is	be	AUX
ejpam-6261	245	22	homogeneous	homogeneous	ADJ
ejpam-6261	245	23	,	,	PUNCT
ejpam-6261	245	24	so	so	ADV
ejpam-6261	245	25	rw	rw	PROPN
ejpam-6261	245	26	(	(	PUNCT
ejpam-6261	245	27	k	k	NOUN
ejpam-6261	245	28	−	−	PROPN
ejpam-6261	245	29	l	l	NOUN
ejpam-6261	245	30	)	)	PUNCT
ejpam-6261	245	31	≤	≤	NOUN
ejpam-6261	246	1	max{rw	max{rw	NUM
ejpam-6261	246	2	(	(	PUNCT
ejpam-6261	246	3	k	k	NOUN
ejpam-6261	246	4	)	)	PUNCT
ejpam-6261	246	5	,	,	PUNCT
ejpam-6261	246	6	rw	rw	PROPN
ejpam-6261	246	7	(	(	PUNCT
ejpam-6261	246	8	l	l	NOUN
ejpam-6261	246	9	)	)	PUNCT
ejpam-6261	246	10	}	}	PUNCT
ejpam-6261	246	11	and	and	CCONJ
ejpam-6261	246	12	ωw	ωw	X
ejpam-6261	246	13	(	(	PUNCT
ejpam-6261	246	14	k	k	NOUN
ejpam-6261	246	15	−	−	PROPN
ejpam-6261	246	16	l	l	NOUN
ejpam-6261	246	17	)	)	PUNCT
ejpam-6261	246	18	≤	≤	NOUN
ejpam-6261	246	19	max{ωw	max{ωw	VERB
ejpam-6261	246	20	(	(	PUNCT
ejpam-6261	246	21	k	k	NOUN
ejpam-6261	246	22	)	)	PUNCT
ejpam-6261	246	23	,	,	PUNCT
ejpam-6261	246	24	ωw	ωw	X
ejpam-6261	246	25	(	(	PUNCT
ejpam-6261	246	26	l	l	NOUN
ejpam-6261	246	27	)	)	PUNCT
ejpam-6261	246	28	}	}	PUNCT
ejpam-6261	246	29	.	.	PUNCT
ejpam-6261	247	1	rw	rw	PROPN
ejpam-6261	247	2	(	(	PUNCT
ejpam-6261	247	3	kl)eiωw	kl)eiωw	PROPN
ejpam-6261	247	4	(	(	PUNCT
ejpam-6261	247	5	kl	kl	NOUN
ejpam-6261	247	6	)	)	PUNCT
ejpam-6261	247	7	=	=	SYM
ejpam-6261	248	1	fw	fw	X
ejpam-6261	248	2	(	(	PUNCT
ejpam-6261	248	3	kl	kl	NOUN
ejpam-6261	248	4	)	)	PUNCT
ejpam-6261	248	5	≤	≤	NOUN
ejpam-6261	248	6	max{fw	max{fw	NOUN
ejpam-6261	248	7	(	(	PUNCT
ejpam-6261	248	8	k),fw	k),fw	X
ejpam-6261	248	9	(	(	PUNCT
ejpam-6261	248	10	l	l	NOUN
ejpam-6261	248	11	)	)	PUNCT
ejpam-6261	248	12	}	}	PUNCT
ejpam-6261	248	13	=	=	SYM
ejpam-6261	248	14	max{rw	max{rw	X
ejpam-6261	248	15	(	(	PUNCT
ejpam-6261	248	16	k)eiωw	k)eiωw	PROPN
ejpam-6261	248	17	(	(	PUNCT
ejpam-6261	248	18	k	k	NOUN
ejpam-6261	248	19	)	)	PUNCT
ejpam-6261	248	20	,	,	PUNCT
ejpam-6261	248	21	rw	rw	PROPN
ejpam-6261	248	22	(	(	PUNCT
ejpam-6261	248	23	l)eiωw	l)eiωw	PROPN
ejpam-6261	248	24	(	(	PUNCT
ejpam-6261	248	25	l	l	NOUN
ejpam-6261	248	26	)	)	PUNCT
ejpam-6261	248	27	}	}	PUNCT
ejpam-6261	248	28	=	=	SYM
ejpam-6261	248	29	max{rw	max{rw	X
ejpam-6261	248	30	(	(	PUNCT
ejpam-6261	248	31	k	k	NOUN
ejpam-6261	248	32	)	)	PUNCT
ejpam-6261	248	33	,	,	PUNCT
ejpam-6261	248	34	rw	rw	NOUN
ejpam-6261	248	35	(	(	PUNCT
ejpam-6261	248	36	l)}eimax{ωw	l)}eimax{ωw	NOUN
ejpam-6261	248	37	(	(	PUNCT
ejpam-6261	248	38	k),ωw	k),ωw	X
ejpam-6261	248	39	(	(	PUNCT
ejpam-6261	248	40	l	l	NOUN
ejpam-6261	248	41	)	)	PUNCT
ejpam-6261	248	42	}	}	PUNCT
ejpam-6261	248	43	.	.	PUNCT
ejpam-6261	249	1	as	as	SCONJ
ejpam-6261	249	2	w	w	PROPN
ejpam-6261	249	3	is	be	AUX
ejpam-6261	249	4	homogeneous	homogeneous	ADJ
ejpam-6261	249	5	,	,	PUNCT
ejpam-6261	249	6	so	so	ADV
ejpam-6261	249	7	rw	rw	PROPN
ejpam-6261	249	8	(	(	PUNCT
ejpam-6261	249	9	kl	kl	PROPN
ejpam-6261	249	10	)	)	PUNCT
ejpam-6261	249	11	≤	≤	NOUN
ejpam-6261	249	12	max{rw	max{rw	NUM
ejpam-6261	249	13	(	(	PUNCT
ejpam-6261	249	14	k	k	NOUN
ejpam-6261	249	15	)	)	PUNCT
ejpam-6261	249	16	,	,	PUNCT
ejpam-6261	249	17	rw	rw	PROPN
ejpam-6261	249	18	(	(	PUNCT
ejpam-6261	249	19	l	l	NOUN
ejpam-6261	249	20	)	)	PUNCT
ejpam-6261	249	21	}	}	PUNCT
ejpam-6261	249	22	and	and	CCONJ
ejpam-6261	249	23	ωw	ωw	ADJ
ejpam-6261	249	24	(	(	PUNCT
ejpam-6261	249	25	kl	kl	NOUN
ejpam-6261	249	26	)	)	PUNCT
ejpam-6261	249	27	≤	≤	NOUN
ejpam-6261	249	28	max{ωw	max{ωw	VERB
ejpam-6261	249	29	(	(	PUNCT
ejpam-6261	249	30	k	k	NOUN
ejpam-6261	249	31	)	)	PUNCT
ejpam-6261	249	32	,	,	PUNCT
ejpam-6261	249	33	ωw	ωw	X
ejpam-6261	249	34	(	(	PUNCT
ejpam-6261	249	35	l	l	NOUN
ejpam-6261	249	36	)	)	PUNCT
ejpam-6261	249	37	}	}	PUNCT
ejpam-6261	249	38	.	.	PUNCT
ejpam-6261	250	1	consequently	consequently	ADV
ejpam-6261	250	2	,	,	PUNCT
ejpam-6261	250	3	w	w	PROPN
ejpam-6261	250	4	is	be	AUX
ejpam-6261	250	5	nsr	nsr	PROPN
ejpam-6261	250	6	and	and	CCONJ
ejpam-6261	250	7	w	w	PROPN
ejpam-6261	250	8	is	be	AUX
ejpam-6261	250	9	π	π	PROPN
ejpam-6261	250	10	-	-	PUNCT
ejpam-6261	250	11	nsr	nsr	PROPN
ejpam-6261	250	12	.	.	PUNCT
ejpam-6261	251	1	conversely	conversely	ADV
ejpam-6261	251	2	,	,	PUNCT
ejpam-6261	251	3	assume	assume	VERB
ejpam-6261	251	4	that	that	SCONJ
ejpam-6261	251	5	w	w	PROPN
ejpam-6261	251	6	and	and	CCONJ
ejpam-6261	251	7	w	w	PROPN
ejpam-6261	251	8	is	be	AUX
ejpam-6261	251	9	nsr	nsr	PROPN
ejpam-6261	251	10	and	and	CCONJ
ejpam-6261	251	11	π	π	PROPN
ejpam-6261	251	12	-	-	PUNCT
ejpam-6261	251	13	nsr	nsr	PROPN
ejpam-6261	251	14	,	,	PUNCT
ejpam-6261	251	15	respectively	respectively	ADV
ejpam-6261	251	16	.	.	PUNCT
ejpam-6261	252	1	then	then	ADV
ejpam-6261	252	2	,	,	PUNCT
ejpam-6261	252	3	we	we	PRON
ejpam-6261	252	4	know	know	VERB
ejpam-6261	252	5	that	that	PRON
ejpam-6261	252	6	pw	pw	PROPN
ejpam-6261	252	7	(	(	PUNCT
ejpam-6261	252	8	k	k	NOUN
ejpam-6261	252	9	−	−	PROPN
ejpam-6261	252	10	l	l	NOUN
ejpam-6261	252	11	)	)	PUNCT
ejpam-6261	252	12	≥	≥	NOUN
ejpam-6261	252	13	min{tw	min{tw	X
ejpam-6261	252	14	(	(	PUNCT
ejpam-6261	252	15	k	k	NOUN
ejpam-6261	252	16	)	)	PUNCT
ejpam-6261	252	17	,	,	PUNCT
ejpam-6261	252	18	tw	tw	PROPN
ejpam-6261	252	19	(	(	PUNCT
ejpam-6261	252	20	l	l	NOUN
ejpam-6261	252	21	)	)	PUNCT
ejpam-6261	252	22	}	}	PUNCT
ejpam-6261	252	23	,	,	PUNCT
ejpam-6261	252	24	pw	pw	PROPN
ejpam-6261	252	25	(	(	PUNCT
ejpam-6261	252	26	kl	kl	PROPN
ejpam-6261	252	27	)	)	PUNCT
ejpam-6261	252	28	≥	≥	NOUN
ejpam-6261	252	29	min{pw	min{pw	ADV
ejpam-6261	252	30	(	(	PUNCT
ejpam-6261	252	31	k	k	NOUN
ejpam-6261	252	32	)	)	PUNCT
ejpam-6261	252	33	,	,	PUNCT
ejpam-6261	252	34	pw	pw	PROPN
ejpam-6261	252	35	(	(	PUNCT
ejpam-6261	252	36	l	l	NOUN
ejpam-6261	252	37	)	)	PUNCT
ejpam-6261	252	38	}	}	PUNCT
ejpam-6261	252	39	,	,	PUNCT
ejpam-6261	252	40	qw	qw	X
ejpam-6261	252	41	(	(	PUNCT
ejpam-6261	252	42	k	k	PROPN
ejpam-6261	252	43	−	−	PROPN
ejpam-6261	252	44	l	l	NOUN
ejpam-6261	252	45	)	)	PUNCT
ejpam-6261	252	46	≤	≤	NOUN
ejpam-6261	252	47	max{qw	max{qw	CCONJ
ejpam-6261	252	48	(	(	PUNCT
ejpam-6261	252	49	k	k	NOUN
ejpam-6261	252	50	)	)	PUNCT
ejpam-6261	252	51	,	,	PUNCT
ejpam-6261	252	52	qw	qw	X
ejpam-6261	252	53	(	(	PUNCT
ejpam-6261	252	54	l	l	NOUN
ejpam-6261	252	55	)	)	PUNCT
ejpam-6261	252	56	}	}	PUNCT
ejpam-6261	252	57	,	,	PUNCT
ejpam-6261	252	58	qw	qw	X
ejpam-6261	252	59	(	(	PUNCT
ejpam-6261	252	60	kl	kl	NOUN
ejpam-6261	252	61	)	)	PUNCT
ejpam-6261	252	62	≤	≤	NOUN
ejpam-6261	252	63	max{qw	max{qw	CCONJ
ejpam-6261	252	64	(	(	PUNCT
ejpam-6261	252	65	k	k	NOUN
ejpam-6261	252	66	)	)	PUNCT
ejpam-6261	252	67	,	,	PUNCT
ejpam-6261	252	68	qw	qw	X
ejpam-6261	252	69	(	(	PUNCT
ejpam-6261	252	70	l	l	NOUN
ejpam-6261	252	71	)	)	PUNCT
ejpam-6261	252	72	}	}	PUNCT
ejpam-6261	252	73	,	,	PUNCT
ejpam-6261	252	74	rw	rw	PROPN
ejpam-6261	252	75	(	(	PUNCT
ejpam-6261	252	76	k	k	NOUN
ejpam-6261	252	77	−	−	PROPN
ejpam-6261	252	78	l	l	NOUN
ejpam-6261	252	79	)	)	PUNCT
ejpam-6261	252	80	≤	≤	NOUN
ejpam-6261	253	1	max{rw	max{rw	NUM
ejpam-6261	253	2	(	(	PUNCT
ejpam-6261	253	3	k	k	NOUN
ejpam-6261	253	4	)	)	PUNCT
ejpam-6261	253	5	,	,	PUNCT
ejpam-6261	253	6	rw	rw	PROPN
ejpam-6261	253	7	(	(	PUNCT
ejpam-6261	253	8	l	l	NOUN
ejpam-6261	253	9	)	)	PUNCT
ejpam-6261	253	10	}	}	PUNCT
ejpam-6261	253	11	,	,	PUNCT
ejpam-6261	253	12	rw	rw	PROPN
ejpam-6261	253	13	(	(	PUNCT
ejpam-6261	253	14	kl	kl	NOUN
ejpam-6261	253	15	)	)	PUNCT
ejpam-6261	253	16	≤	≤	NOUN
ejpam-6261	253	17	max{rw	max{rw	NUM
ejpam-6261	253	18	(	(	PUNCT
ejpam-6261	253	19	k	k	NOUN
ejpam-6261	253	20	)	)	PUNCT
ejpam-6261	253	21	,	,	PUNCT
ejpam-6261	253	22	rw	rw	PROPN
ejpam-6261	253	23	(	(	PUNCT
ejpam-6261	253	24	l	l	NOUN
ejpam-6261	253	25	)	)	PUNCT
ejpam-6261	253	26	}	}	PUNCT
ejpam-6261	253	27	θw	θw	ADP
ejpam-6261	253	28	(	(	PUNCT
ejpam-6261	253	29	k	k	NOUN
ejpam-6261	253	30	−	−	PROPN
ejpam-6261	253	31	l	l	NOUN
ejpam-6261	253	32	)	)	PUNCT
ejpam-6261	253	33	≥	≥	NOUN
ejpam-6261	253	34	min{θw	min{θw	X
ejpam-6261	253	35	(	(	PUNCT
ejpam-6261	253	36	k	k	NOUN
ejpam-6261	253	37	)	)	PUNCT
ejpam-6261	253	38	,	,	PUNCT
ejpam-6261	253	39	θw	θw	X
ejpam-6261	253	40	(	(	PUNCT
ejpam-6261	253	41	l	l	NOUN
ejpam-6261	253	42	)	)	PUNCT
ejpam-6261	253	43	}	}	PUNCT
ejpam-6261	253	44	,	,	PUNCT
ejpam-6261	253	45	θw	θw	PROPN
ejpam-6261	253	46	(	(	PUNCT
ejpam-6261	253	47	kl	kl	PROPN
ejpam-6261	253	48	)	)	PUNCT
ejpam-6261	253	49	≥	≥	NOUN
ejpam-6261	253	50	min{θw	min{θw	X
ejpam-6261	253	51	(	(	PUNCT
ejpam-6261	253	52	k	k	NOUN
ejpam-6261	253	53	)	)	PUNCT
ejpam-6261	253	54	,	,	PUNCT
ejpam-6261	253	55	θw	θw	X
ejpam-6261	253	56	(	(	PUNCT
ejpam-6261	253	57	l	l	NOUN
ejpam-6261	253	58	)	)	PUNCT
ejpam-6261	253	59	}	}	PUNCT
ejpam-6261	253	60	,	,	PUNCT
ejpam-6261	253	61	φw	φw	PROPN
ejpam-6261	253	62	(	(	PUNCT
ejpam-6261	253	63	k	k	PROPN
ejpam-6261	253	64	−	−	PROPN
ejpam-6261	253	65	l	l	NOUN
ejpam-6261	253	66	)	)	PUNCT
ejpam-6261	253	67	≤	≤	NUM
ejpam-6261	253	68	max{φw	max{φw	NOUN
ejpam-6261	253	69	(	(	PUNCT
ejpam-6261	253	70	k	k	NOUN
ejpam-6261	253	71	)	)	PUNCT
ejpam-6261	253	72	,	,	PUNCT
ejpam-6261	253	73	φw	φw	PROPN
ejpam-6261	253	74	(	(	PUNCT
ejpam-6261	253	75	l	l	NOUN
ejpam-6261	253	76	)	)	PUNCT
ejpam-6261	253	77	}	}	PUNCT
ejpam-6261	253	78	,	,	PUNCT
ejpam-6261	253	79	φw	φw	PROPN
ejpam-6261	253	80	(	(	PUNCT
ejpam-6261	253	81	kl	kl	NOUN
ejpam-6261	253	82	)	)	PUNCT
ejpam-6261	253	83	≤	≤	NOUN
ejpam-6261	253	84	max{φw	max{φw	NOUN
ejpam-6261	254	1	(	(	PUNCT
ejpam-6261	254	2	k	k	NOUN
ejpam-6261	254	3	)	)	PUNCT
ejpam-6261	254	4	,	,	PUNCT
ejpam-6261	254	5	φw	φw	PROPN
ejpam-6261	254	6	(	(	PUNCT
ejpam-6261	254	7	l	l	NOUN
ejpam-6261	254	8	)	)	PUNCT
ejpam-6261	254	9	}	}	PUNCT
ejpam-6261	254	10	,	,	PUNCT
ejpam-6261	254	11	ωw	ωw	X
ejpam-6261	254	12	(	(	PUNCT
ejpam-6261	254	13	k	k	PROPN
ejpam-6261	254	14	−	−	PROPN
ejpam-6261	254	15	l	l	NOUN
ejpam-6261	254	16	)	)	PUNCT
ejpam-6261	254	17	≤	≤	NOUN
ejpam-6261	254	18	max{ωw	max{ωw	VERB
ejpam-6261	254	19	(	(	PUNCT
ejpam-6261	254	20	k	k	NOUN
ejpam-6261	254	21	)	)	PUNCT
ejpam-6261	254	22	,	,	PUNCT
ejpam-6261	254	23	ωw	ωw	X
ejpam-6261	254	24	(	(	PUNCT
ejpam-6261	254	25	l	l	NOUN
ejpam-6261	254	26	)	)	PUNCT
ejpam-6261	254	27	}	}	PUNCT
ejpam-6261	254	28	,	,	PUNCT
ejpam-6261	254	29	ωw	ωw	PROPN
ejpam-6261	254	30	(	(	PUNCT
ejpam-6261	254	31	kl	kl	NOUN
ejpam-6261	254	32	)	)	PUNCT
ejpam-6261	254	33	≤	≤	NOUN
ejpam-6261	254	34	max{ωw	max{ωw	VERB
ejpam-6261	254	35	(	(	PUNCT
ejpam-6261	254	36	k	k	NOUN
ejpam-6261	254	37	)	)	PUNCT
ejpam-6261	254	38	,	,	PUNCT
ejpam-6261	254	39	ωw	ωw	X
ejpam-6261	254	40	(	(	PUNCT
ejpam-6261	254	41	l	l	NOUN
ejpam-6261	254	42	)	)	PUNCT
ejpam-6261	254	43	}	}	PUNCT
ejpam-6261	254	44	,	,	PUNCT
ejpam-6261	254	45	for	for	ADP
ejpam-6261	254	46	this	this	PRON
ejpam-6261	254	47	,	,	PUNCT
ejpam-6261	254	48	we	we	PRON
ejpam-6261	254	49	consider	consider	VERB
ejpam-6261	254	50	tw	tw	PRON
ejpam-6261	254	51	(	(	PUNCT
ejpam-6261	254	52	k	k	NOUN
ejpam-6261	254	53	−	−	PROPN
ejpam-6261	254	54	l	l	NOUN
ejpam-6261	254	55	)	)	PUNCT
ejpam-6261	254	56	=	=	SYM
ejpam-6261	254	57	pw	pw	X
ejpam-6261	255	1	(	(	PUNCT
ejpam-6261	255	2	k	k	X
ejpam-6261	255	3	−	−	PROPN
ejpam-6261	255	4	l)eiθw	l)eiθw	PROPN
ejpam-6261	255	5	(	(	PUNCT
ejpam-6261	255	6	k−l	k−l	NOUN
ejpam-6261	255	7	)	)	PUNCT
ejpam-6261	255	8	≥	≥	NOUN
ejpam-6261	255	9	min{pw	min{pw	ADV
ejpam-6261	255	10	(	(	PUNCT
ejpam-6261	255	11	k	k	NOUN
ejpam-6261	255	12	)	)	PUNCT
ejpam-6261	255	13	,	,	PUNCT
ejpam-6261	255	14	pw	pw	PROPN
ejpam-6261	255	15	(	(	PUNCT
ejpam-6261	255	16	l)}eimin{θw	l)}eimin{θw	NOUN
ejpam-6261	255	17	(	(	PUNCT
ejpam-6261	255	18	k),θw	k),θw	X
ejpam-6261	255	19	(	(	PUNCT
ejpam-6261	255	20	l	l	NOUN
ejpam-6261	255	21	)	)	PUNCT
ejpam-6261	255	22	}	}	PUNCT
ejpam-6261	255	23	=	=	SYM
ejpam-6261	255	24	min{pw	min{pw	X
ejpam-6261	255	25	(	(	PUNCT
ejpam-6261	255	26	k)eiθw	k)eiθw	X
ejpam-6261	255	27	(	(	PUNCT
ejpam-6261	255	28	k	k	NOUN
ejpam-6261	255	29	)	)	PUNCT
ejpam-6261	255	30	,	,	PUNCT
ejpam-6261	255	31	pw	pw	PROPN
ejpam-6261	255	32	(	(	PUNCT
ejpam-6261	255	33	l)eiθw	l)eiθw	X
ejpam-6261	255	34	(	(	PUNCT
ejpam-6261	255	35	l	l	NOUN
ejpam-6261	255	36	)	)	PUNCT
ejpam-6261	255	37	}	}	PUNCT
ejpam-6261	255	38	=	=	SYM
ejpam-6261	255	39	min{tw	min{tw	X
ejpam-6261	255	40	(	(	PUNCT
ejpam-6261	255	41	k),tw	k),tw	X
ejpam-6261	255	42	(	(	PUNCT
ejpam-6261	255	43	l	l	NOUN
ejpam-6261	255	44	)	)	PUNCT
ejpam-6261	255	45	}	}	PUNCT
ejpam-6261	255	46	.	.	PUNCT
ejpam-6261	256	1	for	for	ADP
ejpam-6261	256	2	this	this	PRON
ejpam-6261	256	3	,	,	PUNCT
ejpam-6261	256	4	we	we	PRON
ejpam-6261	256	5	have	have	VERB
ejpam-6261	256	6	tw	tw	NOUN
ejpam-6261	256	7	(	(	PUNCT
ejpam-6261	256	8	kl	kl	NOUN
ejpam-6261	256	9	)	)	PUNCT
ejpam-6261	256	10	=	=	SYM
ejpam-6261	256	11	pw	pw	PROPN
ejpam-6261	256	12	(	(	PUNCT
ejpam-6261	256	13	kl)eiθw	kl)eiθw	PROPN
ejpam-6261	256	14	(	(	PUNCT
ejpam-6261	256	15	kl	kl	PROPN
ejpam-6261	256	16	)	)	PUNCT
ejpam-6261	256	17	≥	≥	NOUN
ejpam-6261	256	18	min{pw	min{pw	ADV
ejpam-6261	256	19	(	(	PUNCT
ejpam-6261	256	20	k	k	NOUN
ejpam-6261	256	21	)	)	PUNCT
ejpam-6261	256	22	,	,	PUNCT
ejpam-6261	256	23	pw	pw	PROPN
ejpam-6261	257	1	(	(	PUNCT
ejpam-6261	257	2	l)}eimin{θw	l)}eimin{θw	NOUN
ejpam-6261	257	3	(	(	PUNCT
ejpam-6261	257	4	k),θw	k),θw	X
ejpam-6261	257	5	(	(	PUNCT
ejpam-6261	257	6	l	l	NOUN
ejpam-6261	257	7	)	)	PUNCT
ejpam-6261	257	8	}	}	PUNCT
ejpam-6261	257	9	=	=	SYM
ejpam-6261	257	10	min{pw	min{pw	X
ejpam-6261	257	11	(	(	PUNCT
ejpam-6261	257	12	k)eiθw	k)eiθw	X
ejpam-6261	257	13	(	(	PUNCT
ejpam-6261	257	14	k	k	NOUN
ejpam-6261	257	15	)	)	PUNCT
ejpam-6261	257	16	,	,	PUNCT
ejpam-6261	257	17	pw	pw	PROPN
ejpam-6261	257	18	(	(	PUNCT
ejpam-6261	257	19	l)eiθw	l)eiθw	X
ejpam-6261	257	20	(	(	PUNCT
ejpam-6261	257	21	l	l	NOUN
ejpam-6261	257	22	)	)	PUNCT
ejpam-6261	257	23	}	}	PUNCT
ejpam-6261	257	24	=	=	SYM
ejpam-6261	257	25	min{tw	min{tw	X
ejpam-6261	257	26	(	(	PUNCT
ejpam-6261	257	27	k),tw	k),tw	X
ejpam-6261	257	28	(	(	PUNCT
ejpam-6261	257	29	l	l	NOUN
ejpam-6261	257	30	)	)	PUNCT
ejpam-6261	257	31	}	}	PUNCT
ejpam-6261	257	32	.	.	PUNCT
ejpam-6261	258	1	for	for	ADP
ejpam-6261	258	2	this	this	PRON
ejpam-6261	258	3	,	,	PUNCT
ejpam-6261	258	4	we	we	PRON
ejpam-6261	258	5	get	get	VERB
ejpam-6261	258	6	iw	iw	INTJ
ejpam-6261	258	7	(	(	PUNCT
ejpam-6261	258	8	k	k	NOUN
ejpam-6261	258	9	−	−	PROPN
ejpam-6261	258	10	l	l	NOUN
ejpam-6261	258	11	)	)	PUNCT
ejpam-6261	259	1	=	=	SYM
ejpam-6261	259	2	qw	qw	X
ejpam-6261	259	3	(	(	PUNCT
ejpam-6261	259	4	k	k	PROPN
ejpam-6261	259	5	−	−	PROPN
ejpam-6261	259	6	l)eiφw	l)eiφw	PROPN
ejpam-6261	259	7	(	(	PUNCT
ejpam-6261	259	8	k−l	k−l	NOUN
ejpam-6261	259	9	)	)	PUNCT
ejpam-6261	259	10	≤	≤	NOUN
ejpam-6261	259	11	max{qw	max{qw	CCONJ
ejpam-6261	259	12	(	(	PUNCT
ejpam-6261	259	13	k	k	NOUN
ejpam-6261	259	14	)	)	PUNCT
ejpam-6261	259	15	,	,	PUNCT
ejpam-6261	259	16	qw	qw	X
ejpam-6261	259	17	(	(	PUNCT
ejpam-6261	259	18	l)}eimax{φw	l)}eimax{φw	PROPN
ejpam-6261	259	19	(	(	PUNCT
ejpam-6261	259	20	k),φw	k),φw	X
ejpam-6261	259	21	(	(	PUNCT
ejpam-6261	259	22	l	l	NOUN
ejpam-6261	259	23	)	)	PUNCT
ejpam-6261	259	24	}	}	PUNCT
ejpam-6261	259	25	=	=	SYM
ejpam-6261	259	26	max{qw	max{qw	NOUN
ejpam-6261	259	27	(	(	PUNCT
ejpam-6261	259	28	k)eiφw	k)eiφw	PROPN
ejpam-6261	259	29	(	(	PUNCT
ejpam-6261	259	30	k	k	NOUN
ejpam-6261	259	31	)	)	PUNCT
ejpam-6261	259	32	,	,	PUNCT
ejpam-6261	259	33	qw	qw	X
ejpam-6261	259	34	(	(	PUNCT
ejpam-6261	259	35	l)eiφw	l)eiφw	PROPN
ejpam-6261	259	36	(	(	PUNCT
ejpam-6261	259	37	l	l	NOUN
ejpam-6261	259	38	)	)	PUNCT
ejpam-6261	259	39	}	}	PUNCT
ejpam-6261	259	40	=	=	SYM
ejpam-6261	259	41	max{iw	max{iw	X
ejpam-6261	259	42	(	(	PUNCT
ejpam-6261	259	43	k	k	NOUN
ejpam-6261	259	44	)	)	PUNCT
ejpam-6261	259	45	,	,	PUNCT
ejpam-6261	259	46	iw	iw	PROPN
ejpam-6261	259	47	(	(	PUNCT
ejpam-6261	259	48	l	l	NOUN
ejpam-6261	259	49	)	)	PUNCT
ejpam-6261	259	50	}	}	PUNCT
ejpam-6261	259	51	.	.	PUNCT
ejpam-6261	260	1	iw	iw	INTJ
ejpam-6261	260	2	(	(	PUNCT
ejpam-6261	260	3	kl	kl	NOUN
ejpam-6261	260	4	)	)	PUNCT
ejpam-6261	260	5	=	=	SYM
ejpam-6261	260	6	qw	qw	X
ejpam-6261	260	7	(	(	PUNCT
ejpam-6261	260	8	kl)eiφw	kl)eiφw	PROPN
ejpam-6261	260	9	(	(	PUNCT
ejpam-6261	260	10	kl	kl	NOUN
ejpam-6261	260	11	)	)	PUNCT
ejpam-6261	260	12	≤	≤	NOUN
ejpam-6261	260	13	max{qw	max{qw	CCONJ
ejpam-6261	260	14	(	(	PUNCT
ejpam-6261	260	15	k	k	NOUN
ejpam-6261	260	16	)	)	PUNCT
ejpam-6261	260	17	,	,	PUNCT
ejpam-6261	260	18	qw	qw	X
ejpam-6261	260	19	(	(	PUNCT
ejpam-6261	260	20	l)}eimax{φw	l)}eimax{φw	PROPN
ejpam-6261	260	21	(	(	PUNCT
ejpam-6261	260	22	k),φw	k),φw	X
ejpam-6261	260	23	(	(	PUNCT
ejpam-6261	260	24	l	l	NOUN
ejpam-6261	260	25	)	)	PUNCT
ejpam-6261	260	26	}	}	PUNCT
ejpam-6261	260	27	=	=	SYM
ejpam-6261	260	28	max{qw	max{qw	NOUN
ejpam-6261	260	29	(	(	PUNCT
ejpam-6261	260	30	k)eiφw	k)eiφw	PROPN
ejpam-6261	260	31	(	(	PUNCT
ejpam-6261	260	32	k	k	NOUN
ejpam-6261	260	33	)	)	PUNCT
ejpam-6261	260	34	,	,	PUNCT
ejpam-6261	260	35	qw	qw	X
ejpam-6261	260	36	(	(	PUNCT
ejpam-6261	260	37	l)eiφw	l)eiφw	PROPN
ejpam-6261	260	38	(	(	PUNCT
ejpam-6261	260	39	l	l	NOUN
ejpam-6261	260	40	)	)	PUNCT
ejpam-6261	260	41	}	}	PUNCT
ejpam-6261	261	1	=	=	SYM
ejpam-6261	261	2	max{iw	max{iw	X
ejpam-6261	261	3	(	(	PUNCT
ejpam-6261	261	4	k	k	NOUN
ejpam-6261	261	5	)	)	PUNCT
ejpam-6261	261	6	,	,	PUNCT
ejpam-6261	261	7	iw	iw	PROPN
ejpam-6261	261	8	(	(	PUNCT
ejpam-6261	261	9	l	l	NOUN
ejpam-6261	261	10	)	)	PUNCT
ejpam-6261	261	11	}	}	PUNCT
ejpam-6261	261	12	.	.	PUNCT
ejpam-6261	262	1	m.	m.	NOUN
ejpam-6261	262	2	h.	h.	PROPN
ejpam-6261	262	3	mateen	mateen	PROPN
ejpam-6261	262	4	et	et	PROPN
ejpam-6261	262	5	al	al	PROPN
ejpam-6261	262	6	.	.	PUNCT
ejpam-6261	262	7	/	/	SYM
ejpam-6261	262	8	eur	eur	PROPN
ejpam-6261	262	9	.	.	PUNCT
ejpam-6261	263	1	j.	j.	PROPN
ejpam-6261	263	2	pure	pure	PROPN
ejpam-6261	263	3	appl	appl	PROPN
ejpam-6261	263	4	.	.	PROPN
ejpam-6261	263	5	math	math	PROPN
ejpam-6261	263	6	,	,	PUNCT
ejpam-6261	263	7	18	18	NUM
ejpam-6261	263	8	(	(	PUNCT
ejpam-6261	263	9	4	4	NUM
ejpam-6261	263	10	)	)	PUNCT
ejpam-6261	263	11	(	(	PUNCT
ejpam-6261	263	12	2025	2025	NUM
ejpam-6261	263	13	)	)	PUNCT
ejpam-6261	263	14	,	,	PUNCT
ejpam-6261	263	15	6261	6261	NUM
ejpam-6261	263	16	11	11	NUM
ejpam-6261	263	17	of	of	ADP
ejpam-6261	263	18	23	23	NUM
ejpam-6261	263	19	consider	consider	VERB
ejpam-6261	263	20	,	,	PUNCT
ejpam-6261	263	21	fw	fw	PROPN
ejpam-6261	263	22	(	(	PUNCT
ejpam-6261	263	23	k	k	PROPN
ejpam-6261	263	24	−	−	PROPN
ejpam-6261	263	25	l	l	NOUN
ejpam-6261	263	26	)	)	PUNCT
ejpam-6261	264	1	=	=	SYM
ejpam-6261	264	2	rw	rw	NOUN
ejpam-6261	264	3	(	(	PUNCT
ejpam-6261	264	4	k	k	PROPN
ejpam-6261	264	5	−	−	PROPN
ejpam-6261	264	6	l)eiωw	l)eiωw	PROPN
ejpam-6261	264	7	(	(	PUNCT
ejpam-6261	264	8	k−l	k−l	NOUN
ejpam-6261	264	9	)	)	PUNCT
ejpam-6261	264	10	≤	≤	NOUN
ejpam-6261	265	1	max{rw	max{rw	NUM
ejpam-6261	265	2	(	(	PUNCT
ejpam-6261	265	3	k	k	NOUN
ejpam-6261	265	4	)	)	PUNCT
ejpam-6261	265	5	,	,	PUNCT
ejpam-6261	265	6	rw	rw	NOUN
ejpam-6261	265	7	(	(	PUNCT
ejpam-6261	265	8	l)}eimax{ωw	l)}eimax{ωw	NOUN
ejpam-6261	265	9	(	(	PUNCT
ejpam-6261	265	10	k),ωw	k),ωw	X
ejpam-6261	265	11	(	(	PUNCT
ejpam-6261	265	12	l	l	NOUN
ejpam-6261	265	13	)	)	PUNCT
ejpam-6261	265	14	}	}	PUNCT
ejpam-6261	265	15	=	=	SYM
ejpam-6261	265	16	max{rw	max{rw	X
ejpam-6261	265	17	(	(	PUNCT
ejpam-6261	265	18	k)eiωw	k)eiωw	PROPN
ejpam-6261	265	19	(	(	PUNCT
ejpam-6261	265	20	k	k	NOUN
ejpam-6261	265	21	)	)	PUNCT
ejpam-6261	265	22	,	,	PUNCT
ejpam-6261	265	23	rw	rw	PROPN
ejpam-6261	265	24	(	(	PUNCT
ejpam-6261	265	25	l)eiωw	l)eiωw	PROPN
ejpam-6261	265	26	(	(	PUNCT
ejpam-6261	265	27	l	l	NOUN
ejpam-6261	265	28	)	)	PUNCT
ejpam-6261	265	29	}	}	PUNCT
ejpam-6261	265	30	=	=	SYM
ejpam-6261	265	31	max{fw	max{fw	X
ejpam-6261	265	32	(	(	PUNCT
ejpam-6261	265	33	k),fw	k),fw	X
ejpam-6261	265	34	(	(	PUNCT
ejpam-6261	265	35	l	l	NOUN
ejpam-6261	265	36	)	)	PUNCT
ejpam-6261	265	37	}	}	PUNCT
ejpam-6261	265	38	.	.	PUNCT
ejpam-6261	266	1	fw	fw	INTJ
ejpam-6261	266	2	(	(	PUNCT
ejpam-6261	266	3	kl	kl	NOUN
ejpam-6261	266	4	)	)	PUNCT
ejpam-6261	266	5	=	=	SYM
ejpam-6261	266	6	rw	rw	NOUN
ejpam-6261	266	7	(	(	PUNCT
ejpam-6261	266	8	kl)eiωw	kl)eiωw	PROPN
ejpam-6261	266	9	(	(	PUNCT
ejpam-6261	266	10	kl	kl	NOUN
ejpam-6261	266	11	)	)	PUNCT
ejpam-6261	266	12	≤	≤	NOUN
ejpam-6261	266	13	max{rw	max{rw	NUM
ejpam-6261	266	14	(	(	PUNCT
ejpam-6261	266	15	k	k	NOUN
ejpam-6261	266	16	)	)	PUNCT
ejpam-6261	266	17	,	,	PUNCT
ejpam-6261	266	18	rw	rw	NOUN
ejpam-6261	266	19	(	(	PUNCT
ejpam-6261	266	20	l)}eimax{ωw	l)}eimax{ωw	NOUN
ejpam-6261	266	21	(	(	PUNCT
ejpam-6261	266	22	k),ωw	k),ωw	X
ejpam-6261	266	23	(	(	PUNCT
ejpam-6261	266	24	l	l	NOUN
ejpam-6261	266	25	)	)	PUNCT
ejpam-6261	266	26	}	}	PUNCT
ejpam-6261	266	27	=	=	SYM
ejpam-6261	266	28	max{rw	max{rw	X
ejpam-6261	266	29	(	(	PUNCT
ejpam-6261	266	30	k)eiωw	k)eiωw	PROPN
ejpam-6261	266	31	(	(	PUNCT
ejpam-6261	266	32	k	k	NOUN
ejpam-6261	266	33	)	)	PUNCT
ejpam-6261	266	34	,	,	PUNCT
ejpam-6261	266	35	rw	rw	PROPN
ejpam-6261	266	36	(	(	PUNCT
ejpam-6261	266	37	l)eiωw	l)eiωw	PROPN
ejpam-6261	266	38	(	(	PUNCT
ejpam-6261	266	39	l	l	NOUN
ejpam-6261	266	40	)	)	PUNCT
ejpam-6261	266	41	}	}	PUNCT
ejpam-6261	266	42	=	=	SYM
ejpam-6261	266	43	max{fw	max{fw	X
ejpam-6261	266	44	(	(	PUNCT
ejpam-6261	266	45	k),fw	k),fw	X
ejpam-6261	266	46	(	(	PUNCT
ejpam-6261	266	47	l	l	NOUN
ejpam-6261	266	48	)	)	PUNCT
ejpam-6261	266	49	}	}	PUNCT
ejpam-6261	266	50	.	.	PUNCT
ejpam-6261	267	1	hence	hence	ADV
ejpam-6261	267	2	,	,	PUNCT
ejpam-6261	267	3	w	w	PROPN
ejpam-6261	267	4	is	be	AUX
ejpam-6261	267	5	cnsr	cnsr	ADJ
ejpam-6261	267	6	.	.	PUNCT
ejpam-6261	268	1	in	in	ADP
ejpam-6261	268	2	the	the	DET
ejpam-6261	268	3	following	following	NOUN
ejpam-6261	268	4	theorem	theorem	NOUN
ejpam-6261	268	5	,	,	PUNCT
ejpam-6261	268	6	we	we	PRON
ejpam-6261	268	7	illustrate	illustrate	VERB
ejpam-6261	268	8	that	that	SCONJ
ejpam-6261	268	9	the	the	DET
ejpam-6261	268	10	intersection	intersection	NOUN
ejpam-6261	268	11	of	of	ADP
ejpam-6261	268	12	two	two	NUM
ejpam-6261	268	13	complex	complex	ADJ
ejpam-6261	268	14	neutrosophic	neutrosophic	ADJ
ejpam-6261	268	15	subrings	subring	NOUN
ejpam-6261	268	16	is	be	AUX
ejpam-6261	268	17	complex	complex	ADJ
ejpam-6261	268	18	neutrosophic	neutrosophic	ADJ
ejpam-6261	268	19	subring	subring	NOUN
ejpam-6261	268	20	.	.	PUNCT
ejpam-6261	269	1	theorem	theorem	ADJ
ejpam-6261	269	2	3	3	NUM
ejpam-6261	269	3	.	.	PUNCT
ejpam-6261	269	4	intersection	intersection	NOUN
ejpam-6261	269	5	of	of	ADP
ejpam-6261	269	6	two	two	NUM
ejpam-6261	269	7	cnsrs	cnsrs	NOUN
ejpam-6261	269	8	of	of	ADP
ejpam-6261	269	9	ring	ring	NOUN
ejpam-6261	269	10	r	r	NOUN
ejpam-6261	269	11	is	be	AUX
ejpam-6261	269	12	cnsr	cnsr	ADJ
ejpam-6261	269	13	.	.	PUNCT
ejpam-6261	270	1	proof	proof	NOUN
ejpam-6261	270	2	.	.	PUNCT
ejpam-6261	271	1	the	the	DET
ejpam-6261	271	2	proof	proof	NOUN
ejpam-6261	271	3	is	be	AUX
ejpam-6261	271	4	so	so	ADV
ejpam-6261	271	5	accomplished	accomplished	ADJ
ejpam-6261	271	6	.	.	PUNCT
ejpam-6261	272	1	remark	remark	PROPN
ejpam-6261	272	2	1	1	NUM
ejpam-6261	272	3	.	.	PUNCT
ejpam-6261	273	1	the	the	DET
ejpam-6261	273	2	union	union	NOUN
ejpam-6261	273	3	of	of	ADP
ejpam-6261	273	4	two	two	NUM
ejpam-6261	273	5	cnsrs	cnsrs	NOUN
ejpam-6261	273	6	of	of	ADP
ejpam-6261	273	7	ring	ring	NOUN
ejpam-6261	273	8	r	r	NOUN
ejpam-6261	273	9	may	may	AUX
ejpam-6261	273	10	not	not	PART
ejpam-6261	273	11	be	be	AUX
ejpam-6261	273	12	cnsr	cnsr	VERB
ejpam-6261	273	13	of	of	ADP
ejpam-6261	273	14	ring	ring	PROPN
ejpam-6261	273	15	r.	r.	PROPN
ejpam-6261	273	16	in	in	ADP
ejpam-6261	273	17	the	the	DET
ejpam-6261	273	18	upcoming	upcoming	ADJ
ejpam-6261	273	19	example	example	NOUN
ejpam-6261	273	20	,	,	PUNCT
ejpam-6261	273	21	we	we	PRON
ejpam-6261	273	22	describe	describe	VERB
ejpam-6261	273	23	that	that	SCONJ
ejpam-6261	273	24	the	the	DET
ejpam-6261	273	25	union	union	NOUN
ejpam-6261	273	26	of	of	ADP
ejpam-6261	273	27	two	two	NUM
ejpam-6261	273	28	cnsrs	cnsrs	NOUN
ejpam-6261	273	29	of	of	ADP
ejpam-6261	273	30	ring	ring	NOUN
ejpam-6261	273	31	may	may	AUX
ejpam-6261	273	32	not	not	PART
ejpam-6261	273	33	be	be	AUX
ejpam-6261	273	34	cnsr	cnsr	VERB
ejpam-6261	273	35	of	of	ADP
ejpam-6261	273	36	ring	ring	NOUN
ejpam-6261	273	37	.	.	PUNCT
ejpam-6261	274	1	example	example	NOUN
ejpam-6261	275	1	2	2	NUM
ejpam-6261	275	2	.	.	X
ejpam-6261	275	3	take	take	VERB
ejpam-6261	275	4	r	r	NOUN
ejpam-6261	275	5	=	=	PUNCT
ejpam-6261	275	6	z	z	NOUN
ejpam-6261	275	7	=	=	SYM
ejpam-6261	275	8	{	{	PUNCT
ejpam-6261	275	9	0,±1,±2,±3	0,±1,±2,±3	NOUN
ejpam-6261	275	10	,	,	PUNCT
ejpam-6261	275	11	...	...	PUNCT
ejpam-6261	275	12	}	}	PUNCT
ejpam-6261	275	13	is	be	AUX
ejpam-6261	275	14	a	a	DET
ejpam-6261	275	15	ring	ring	NOUN
ejpam-6261	275	16	of	of	ADP
ejpam-6261	275	17	integers	integer	NOUN
ejpam-6261	275	18	.	.	PUNCT
ejpam-6261	276	1	assume	assume	VERB
ejpam-6261	276	2	that	that	SCONJ
ejpam-6261	276	3	w	w	PROPN
ejpam-6261	276	4	and	and	CCONJ
ejpam-6261	276	5	x	x	NOUN
ejpam-6261	276	6	are	be	AUX
ejpam-6261	276	7	two	two	NUM
ejpam-6261	276	8	cnsrs	cnsrs	NOUN
ejpam-6261	276	9	of	of	ADP
ejpam-6261	276	10	ring	ring	NOUN
ejpam-6261	276	11	r	r	NOUN
ejpam-6261	276	12	and	and	CCONJ
ejpam-6261	276	13	described	describe	VERB
ejpam-6261	276	14	as	as	ADP
ejpam-6261	276	15	tw(q	tw(q	NUM
ejpam-6261	276	16	)	)	PUNCT
ejpam-6261	277	1	=	=	PRON
ejpam-6261	277	2	{	{	PUNCT
ejpam-6261	277	3	0.3e	0.3e	VERB
ejpam-6261	277	4	iπ	iπ	ADV
ejpam-6261	277	5	3	3	NUM
ejpam-6261	277	6	if	if	SCONJ
ejpam-6261	277	7	q	q	X
ejpam-6261	277	8	∈	∈	PROPN
ejpam-6261	277	9	3z	3z	NUM
ejpam-6261	277	10	0	0	NUM
ejpam-6261	277	11	otherwise	otherwise	ADV
ejpam-6261	277	12	.	.	PUNCT
ejpam-6261	278	1	iw(q	iw(q	NOUN
ejpam-6261	278	2	)	)	PUNCT
ejpam-6261	279	1	=	=	PRON
ejpam-6261	279	2	{	{	PUNCT
ejpam-6261	280	1	0.4e	0.4e	INTJ
ejpam-6261	280	2	iπ	iπ	ADV
ejpam-6261	280	3	3	3	NUM
ejpam-6261	280	4	if	if	SCONJ
ejpam-6261	280	5	q	q	X
ejpam-6261	280	6	∈	∈	PROPN
ejpam-6261	280	7	3z	3z	NUM
ejpam-6261	280	8	0	0	NUM
ejpam-6261	280	9	otherwise	otherwise	ADV
ejpam-6261	280	10	.	.	PUNCT
ejpam-6261	281	1	fw(q	fw(q	NOUN
ejpam-6261	281	2	)	)	PUNCT
ejpam-6261	282	1	=	=	PRON
ejpam-6261	282	2	{	{	PUNCT
ejpam-6261	282	3	0.2e	0.2e	VERB
ejpam-6261	282	4	iπ	iπ	ADV
ejpam-6261	282	5	9	9	NUM
ejpam-6261	282	6	if	if	SCONJ
ejpam-6261	282	7	q	q	PRON
ejpam-6261	282	8	∈	∈	PROPN
ejpam-6261	282	9	3z	3z	NUM
ejpam-6261	282	10	0.6e	0.6e	NOUN
ejpam-6261	282	11	iπ	iπ	ADV
ejpam-6261	282	12	4	4	NUM
ejpam-6261	282	13	else	else	ADV
ejpam-6261	282	14	.	.	PUNCT
ejpam-6261	283	1	tx(q	tx(q	PUNCT
ejpam-6261	283	2	)	)	PUNCT
ejpam-6261	284	1	=	=	PRON
ejpam-6261	284	2	{	{	PUNCT
ejpam-6261	284	3	0.2e	0.2e	VERB
ejpam-6261	284	4	iπ	iπ	ADV
ejpam-6261	284	5	3	3	NUM
ejpam-6261	284	6	if	if	SCONJ
ejpam-6261	284	7	q	q	X
ejpam-6261	284	8	∈	∈	PROPN
ejpam-6261	284	9	2z	2z	NOUN
ejpam-6261	285	1	0.01e	0.01e	NOUN
ejpam-6261	285	2	iπ	iπ	ADV
ejpam-6261	285	3	8	8	NUM
ejpam-6261	285	4	otherwise	otherwise	ADV
ejpam-6261	285	5	.	.	PUNCT
ejpam-6261	286	1	ix(q	ix(q	X
ejpam-6261	286	2	)	)	PUNCT
ejpam-6261	287	1	=	=	PRON
ejpam-6261	287	2	{	{	PUNCT
ejpam-6261	288	1	0.1e	0.1e	INTJ
ejpam-6261	288	2	iπ	iπ	ADV
ejpam-6261	288	3	4	4	NUM
ejpam-6261	288	4	if	if	SCONJ
ejpam-6261	288	5	q	q	PRON
ejpam-6261	288	6	∈	∈	NOUN
ejpam-6261	288	7	2z	2z	NOUN
ejpam-6261	289	1	0.01e	0.01e	NOUN
ejpam-6261	289	2	iπ	iπ	ADV
ejpam-6261	289	3	8	8	NUM
ejpam-6261	289	4	otherwise	otherwise	ADV
ejpam-6261	289	5	.	.	PUNCT
ejpam-6261	290	1	fx(q	fx(q	PUNCT
ejpam-6261	290	2	)	)	PUNCT
ejpam-6261	291	1	=	=	PRON
ejpam-6261	291	2	{	{	PUNCT
ejpam-6261	291	3	0.4e	0.4e	INTJ
ejpam-6261	291	4	iπ	iπ	ADV
ejpam-6261	291	5	8	8	NUM
ejpam-6261	291	6	if	if	SCONJ
ejpam-6261	291	7	q	q	PRON
ejpam-6261	291	8	∈	∈	PROPN
ejpam-6261	291	9	2z	2z	NOUN
ejpam-6261	291	10	0.2e	0.2e	VERB
ejpam-6261	291	11	iπ	iπ	ADV
ejpam-6261	291	12	9	9	NUM
ejpam-6261	291	13	otherwise	otherwise	ADV
ejpam-6261	291	14	.	.	PUNCT
ejpam-6261	292	1	m.	m.	PROPN
ejpam-6261	292	2	h.	h.	PROPN
ejpam-6261	292	3	mateen	mateen	PROPN
ejpam-6261	292	4	et	et	PROPN
ejpam-6261	292	5	al	al	PROPN
ejpam-6261	292	6	.	.	PUNCT
ejpam-6261	292	7	/	/	SYM
ejpam-6261	292	8	eur	eur	PROPN
ejpam-6261	292	9	.	.	PUNCT
ejpam-6261	293	1	j.	j.	PROPN
ejpam-6261	293	2	pure	pure	PROPN
ejpam-6261	293	3	appl	appl	PROPN
ejpam-6261	293	4	.	.	PROPN
ejpam-6261	293	5	math	math	PROPN
ejpam-6261	293	6	,	,	PUNCT
ejpam-6261	293	7	18	18	NUM
ejpam-6261	293	8	(	(	PUNCT
ejpam-6261	293	9	4	4	NUM
ejpam-6261	293	10	)	)	PUNCT
ejpam-6261	293	11	(	(	PUNCT
ejpam-6261	293	12	2025	2025	NUM
ejpam-6261	293	13	)	)	PUNCT
ejpam-6261	293	14	,	,	PUNCT
ejpam-6261	293	15	6261	6261	NUM
ejpam-6261	293	16	12	12	NUM
ejpam-6261	293	17	of	of	ADP
ejpam-6261	293	18	23	23	NUM
ejpam-6261	293	19	it	it	PRON
ejpam-6261	293	20	is	be	AUX
ejpam-6261	293	21	simple	simple	ADJ
ejpam-6261	293	22	to	to	PART
ejpam-6261	293	23	determine	determine	VERB
ejpam-6261	293	24	that	that	PRON
ejpam-6261	293	25	w	w	NOUN
ejpam-6261	293	26	and	and	CCONJ
ejpam-6261	293	27	x	x	NOUN
ejpam-6261	293	28	are	be	AUX
ejpam-6261	293	29	two	two	NUM
ejpam-6261	293	30	cnsrs	cnsrs	NOUN
ejpam-6261	293	31	of	of	ADP
ejpam-6261	293	32	ring	ring	PROPN
ejpam-6261	293	33	r.	r.	PROPN
ejpam-6261	293	34	using	use	VERB
ejpam-6261	293	35	definition	definition	NOUN
ejpam-6261	293	36	5	5	NUM
ejpam-6261	293	37	w	w	NOUN
ejpam-6261	293	38	∪	∪	NOUN
ejpam-6261	293	39	x	x	X
ejpam-6261	293	40	=	=	SYM
ejpam-6261	293	41	{	{	PUNCT
ejpam-6261	293	42	(	(	PUNCT
ejpam-6261	293	43	q	q	ADJ
ejpam-6261	293	44	,	,	PUNCT
ejpam-6261	293	45	tw∪x	tw∪x	ADJ
ejpam-6261	293	46	,	,	PUNCT
ejpam-6261	293	47	iw∪x	iw∪x	PROPN
ejpam-6261	293	48	,	,	PUNCT
ejpam-6261	293	49	fw∪x	fw∪x	PROPN
ejpam-6261	293	50	)	)	PUNCT
ejpam-6261	293	51	}	}	PUNCT
ejpam-6261	293	52	.	.	PUNCT
ejpam-6261	294	1	therefore	therefore	ADV
ejpam-6261	294	2	,	,	PUNCT
ejpam-6261	294	3	tw∪x(q	tw∪x(q	X
ejpam-6261	294	4	)	)	PUNCT
ejpam-6261	294	5	=	=	SYM
ejpam-6261	295	1			PRON
ejpam-6261	295	2	0.3e	0.3e	VERB
ejpam-6261	295	3	iπ	iπ	ADV
ejpam-6261	295	4	3	3	NUM
ejpam-6261	295	5	if	if	SCONJ
ejpam-6261	295	6	q	q	PROPN
ejpam-6261	295	7	∈	∈	PROPN
ejpam-6261	295	8	3z	3z	NUM
ejpam-6261	295	9	0.2e	0.2e	VERB
ejpam-6261	295	10	iπ	iπ	ADV
ejpam-6261	295	11	3	3	NUM
ejpam-6261	295	12	if	if	SCONJ
ejpam-6261	295	13	q	q	PROPN
ejpam-6261	295	14	∈	∈	PROPN
ejpam-6261	295	15	2z−	2z−	PROPN
ejpam-6261	295	16	3z	3z	ADJ
ejpam-6261	295	17	0.01e	0.01e	NOUN
ejpam-6261	295	18	iπ	iπ	ADV
ejpam-6261	295	19	8	8	NUM
ejpam-6261	295	20	otherwise	otherwise	ADV
ejpam-6261	295	21	.	.	PUNCT
ejpam-6261	296	1	iw∪x(q	iw∪x(q	X
ejpam-6261	296	2	)	)	PUNCT
ejpam-6261	297	1	=	=	PUNCT
ejpam-6261	298	1			PRON
ejpam-6261	298	2	0.4e	0.4e	ADV
ejpam-6261	299	1	iπ	iπ	ADV
ejpam-6261	299	2	3	3	NUM
ejpam-6261	299	3	if	if	SCONJ
ejpam-6261	299	4	q	q	PROPN
ejpam-6261	299	5	∈	∈	PROPN
ejpam-6261	299	6	3z	3z	NUM
ejpam-6261	299	7	0.1e	0.1e	PRON
ejpam-6261	299	8	iπ	iπ	ADV
ejpam-6261	299	9	4	4	NUM
ejpam-6261	299	10	if	if	SCONJ
ejpam-6261	299	11	q	q	PROPN
ejpam-6261	299	12	∈	∈	PROPN
ejpam-6261	299	13	2z−	2z−	PROPN
ejpam-6261	299	14	3z	3z	ADJ
ejpam-6261	299	15	0.01e	0.01e	NOUN
ejpam-6261	299	16	iπ	iπ	ADV
ejpam-6261	299	17	4	4	NUM
ejpam-6261	299	18	else	else	ADV
ejpam-6261	299	19	.	.	PUNCT
ejpam-6261	300	1	fw∪x(q	fw∪x(q	INTJ
ejpam-6261	300	2	)	)	PUNCT
ejpam-6261	300	3	=	=	SYM
ejpam-6261	301	1			PRON
ejpam-6261	301	2	0.2e	0.2e	VERB
ejpam-6261	301	3	iπ	iπ	ADV
ejpam-6261	301	4	9	9	NUM
ejpam-6261	301	5	if	if	SCONJ
ejpam-6261	301	6	q	q	PRON
ejpam-6261	301	7	∈	∈	PROPN
ejpam-6261	301	8	3z	3z	NUM
ejpam-6261	301	9	0.4e	0.4e	NOUN
ejpam-6261	302	1	iπ	iπ	ADV
ejpam-6261	302	2	9	9	NUM
ejpam-6261	302	3	if	if	SCONJ
ejpam-6261	302	4	q	q	PRON
ejpam-6261	302	5	∈	∈	PROPN
ejpam-6261	302	6	2z−	2z−	PROPN
ejpam-6261	302	7	3z	3z	NUM
ejpam-6261	302	8	0.2e	0.2e	VERB
ejpam-6261	302	9	iπ	iπ	ADV
ejpam-6261	302	10	9	9	NUM
ejpam-6261	302	11	else	else	ADV
ejpam-6261	302	12	.	.	PUNCT
ejpam-6261	303	1	assume	assume	VERB
ejpam-6261	303	2	s	s	X
ejpam-6261	303	3	=	=	SYM
ejpam-6261	303	4	15	15	NUM
ejpam-6261	303	5	and	and	CCONJ
ejpam-6261	303	6	t	t	NOUN
ejpam-6261	303	7	=	=	SYM
ejpam-6261	303	8	10	10	NUM
ejpam-6261	303	9	.	.	PUNCT
ejpam-6261	304	1	then	then	ADV
ejpam-6261	304	2	tw∪x(15	tw∪x(15	ADV
ejpam-6261	304	3	)	)	PUNCT
ejpam-6261	304	4	=	=	PUNCT
ejpam-6261	304	5	0.3e	0.3e	VERB
ejpam-6261	304	6	iπ	iπ	PRON
ejpam-6261	304	7	3	3	NUM
ejpam-6261	304	8	and	and	CCONJ
ejpam-6261	304	9	tw∪x(10	tw∪x(10	NUM
ejpam-6261	304	10	)	)	PUNCT
ejpam-6261	305	1	=	=	VERB
ejpam-6261	305	2	0.2e	0.2e	VERB
ejpam-6261	305	3	iπ	iπ	ADV
ejpam-6261	305	4	3	3	NUM
ejpam-6261	305	5	,	,	PUNCT
ejpam-6261	305	6	then	then	ADV
ejpam-6261	305	7	tw∪x(15−10	tw∪x(15−10	ADV
ejpam-6261	305	8	)	)	PUNCT
ejpam-6261	305	9	=	=	SYM
ejpam-6261	305	10	tw∪x(5	tw∪x(5	PROPN
ejpam-6261	305	11	)	)	PUNCT
ejpam-6261	305	12	=	=	SYM
ejpam-6261	306	1	0.01e	0.01e	NUM
ejpam-6261	306	2	iπ	iπ	DET
ejpam-6261	306	3	8	8	NUM
ejpam-6261	306	4	and	and	CCONJ
ejpam-6261	306	5	min{tw∪x(15),tw∪x(10	min{tw∪x(15),tw∪x(10	NOUN
ejpam-6261	306	6	)	)	PUNCT
ejpam-6261	306	7	}	}	PUNCT
ejpam-6261	306	8	=	=	PUNCT
ejpam-6261	306	9	min{0.3e	min{0.3e	VERB
ejpam-6261	306	10	iπ	iπ	ADV
ejpam-6261	306	11	3	3	NUM
ejpam-6261	306	12	,	,	PUNCT
ejpam-6261	306	13	0.2e	0.2e	VERB
ejpam-6261	306	14	iπ	iπ	PRON
ejpam-6261	306	15	3	3	X
ejpam-6261	306	16	}	}	PUNCT
ejpam-6261	306	17	=	=	SYM
ejpam-6261	306	18	0.2e	0.2e	VERB
ejpam-6261	306	19	iπ	iπ	ADV
ejpam-6261	306	20	3	3	NUM
ejpam-6261	306	21	.	.	PUNCT
ejpam-6261	307	1	clearly	clearly	ADV
ejpam-6261	307	2	,	,	PUNCT
ejpam-6261	307	3	tw∪x(15	tw∪x(15	VERB
ejpam-6261	307	4	−	−	PROPN
ejpam-6261	307	5	10	10	NUM
ejpam-6261	307	6	)	)	PUNCT
ejpam-6261	307	7	<	<	X
ejpam-6261	307	8	min{tw∪x(15),tw∪x(10	min{tw∪x(15),tw∪x(10	PROPN
ejpam-6261	307	9	)	)	PUNCT
ejpam-6261	307	10	}	}	PUNCT
ejpam-6261	307	11	.	.	PUNCT
ejpam-6261	308	1	this	this	DET
ejpam-6261	308	2	condition	condition	NOUN
ejpam-6261	308	3	does	do	AUX
ejpam-6261	308	4	not	not	PART
ejpam-6261	308	5	holds	hold	VERB
ejpam-6261	308	6	.	.	PUNCT
ejpam-6261	309	1	consequently	consequently	ADV
ejpam-6261	309	2	,	,	PUNCT
ejpam-6261	309	3	w	w	PROPN
ejpam-6261	309	4	∪	∪	NOUN
ejpam-6261	309	5	x	x	VERB
ejpam-6261	309	6	is	be	AUX
ejpam-6261	309	7	not	not	PART
ejpam-6261	309	8	cnsrs	cnsr	VERB
ejpam-6261	309	9	of	of	ADP
ejpam-6261	309	10	r.	r.	PROPN
ejpam-6261	309	11	now	now	ADV
ejpam-6261	309	12	,	,	PUNCT
ejpam-6261	309	13	we	we	PRON
ejpam-6261	309	14	define	define	VERB
ejpam-6261	309	15	the	the	DET
ejpam-6261	309	16	idea	idea	NOUN
ejpam-6261	309	17	of	of	ADP
ejpam-6261	309	18	complex	complex	ADJ
ejpam-6261	309	19	neutrosophic	neutrosophic	ADJ
ejpam-6261	309	20	level	level	NOUN
ejpam-6261	309	21	subset	subset	NOUN
ejpam-6261	309	22	of	of	ADP
ejpam-6261	309	23	cns	cns	NOUN
ejpam-6261	309	24	and	and	CCONJ
ejpam-6261	309	25	discuss	discuss	VERB
ejpam-6261	309	26	its	its	PRON
ejpam-6261	309	27	vital	vital	ADJ
ejpam-6261	309	28	results	result	NOUN
ejpam-6261	309	29	under	under	ADP
ejpam-6261	309	30	the	the	DET
ejpam-6261	309	31	framework	framework	NOUN
ejpam-6261	309	32	of	of	ADP
ejpam-6261	309	33	complex	complex	ADJ
ejpam-6261	309	34	neutrosophic	neutrosophic	ADJ
ejpam-6261	309	35	subring	subring	NOUN
ejpam-6261	309	36	.	.	PUNCT
ejpam-6261	310	1	definition	definition	NOUN
ejpam-6261	310	2	10	10	NUM
ejpam-6261	310	3	.	.	PUNCT
ejpam-6261	311	1	let	let	VERB
ejpam-6261	311	2	w	w	VERB
ejpam-6261	311	3	=	=	PRON
ejpam-6261	311	4	{	{	PUNCT
ejpam-6261	311	5	<	<	X
ejpam-6261	311	6	j	j	PROPN
ejpam-6261	311	7	,	,	PUNCT
ejpam-6261	311	8	tw(j	tw(j	NUM
ejpam-6261	311	9	)	)	PUNCT
ejpam-6261	311	10	,	,	PUNCT
ejpam-6261	311	11	iw(j),fw(j	iw(j),fw(j	PROPN
ejpam-6261	311	12	)	)	PUNCT
ejpam-6261	311	13	>	>	PUNCT
ejpam-6261	311	14	:	:	PUNCT
ejpam-6261	311	15	j	j	PROPN
ejpam-6261	311	16	∈	∈	PROPN
ejpam-6261	311	17	r	r	AUX
ejpam-6261	311	18	}	}	PUNCT
ejpam-6261	311	19	be	be	AUX
ejpam-6261	311	20	a	a	DET
ejpam-6261	311	21	cns	cns	NOUN
ejpam-6261	311	22	of	of	ADP
ejpam-6261	311	23	r	r	NOUN
ejpam-6261	311	24	,	,	PUNCT
ejpam-6261	311	25	for	for	ADP
ejpam-6261	311	26	all	all	DET
ejpam-6261	311	27	ν	ν	PROPN
ejpam-6261	311	28	,	,	PUNCT
ejpam-6261	311	29	η	η	PROPN
ejpam-6261	311	30	,	,	PUNCT
ejpam-6261	311	31	ρ	ρ	PROPN
ejpam-6261	311	32	∈	∈	PROPN
ejpam-6261	312	1	[	[	X
ejpam-6261	312	2	0	0	NUM
ejpam-6261	312	3	,	,	PUNCT
ejpam-6261	312	4	1	1	NUM
ejpam-6261	312	5	]	]	PUNCT
ejpam-6261	312	6	,	,	PUNCT
ejpam-6261	312	7	and	and	CCONJ
ejpam-6261	312	8	ν̂	ν̂	NUM
ejpam-6261	312	9	,	,	PUNCT
ejpam-6261	312	10	η̂	η̂	NUM
ejpam-6261	312	11	,	,	PUNCT
ejpam-6261	312	12	ρ̂	ρ̂	NUM
ejpam-6261	312	13	∈	∈	PROPN
ejpam-6261	313	1	[	[	X
ejpam-6261	313	2	0	0	NUM
ejpam-6261	313	3	,	,	PUNCT
ejpam-6261	313	4	2π	2π	NOUN
ejpam-6261	313	5	]	]	PUNCT
ejpam-6261	313	6	.	.	PUNCT
ejpam-6261	314	1	the	the	DET
ejpam-6261	314	2	level	level	NOUN
ejpam-6261	314	3	subset	subset	NOUN
ejpam-6261	314	4	of	of	ADP
ejpam-6261	314	5	cns	cns	PROPN
ejpam-6261	314	6	is	be	AUX
ejpam-6261	314	7	described	describe	VERB
ejpam-6261	314	8	as	as	ADP
ejpam-6261	314	9	;	;	PUNCT
ejpam-6261	314	10	w(ν	w(ν	PROPN
ejpam-6261	314	11	,	,	PUNCT
ejpam-6261	314	12	η	η	PROPN
ejpam-6261	314	13	,	,	PUNCT
ejpam-6261	314	14	ρ	ρ	PROPN
ejpam-6261	314	15	)	)	PUNCT
ejpam-6261	314	16	(	(	PUNCT
ejpam-6261	314	17	ν̂	ν̂	X
ejpam-6261	314	18	,	,	PUNCT
ejpam-6261	314	19	η̂	η̂	NUM
ejpam-6261	314	20	,	,	PUNCT
ejpam-6261	314	21	ρ̂	ρ̂	NUM
ejpam-6261	314	22	)	)	PUNCT
ejpam-6261	314	23	=	=	PRON
ejpam-6261	315	1	{	{	PUNCT
ejpam-6261	315	2	j	j	PROPN
ejpam-6261	315	3	∈	∈	PROPN
ejpam-6261	315	4	r	r	NOUN
ejpam-6261	315	5	:	:	PUNCT
ejpam-6261	315	6	pw(j	pw(j	X
ejpam-6261	315	7	)	)	PUNCT
ejpam-6261	315	8	≥	≥	NOUN
ejpam-6261	315	9	ν	ν	NOUN
ejpam-6261	315	10	,	,	PUNCT
ejpam-6261	315	11	θw(j	θw(j	NUM
ejpam-6261	315	12	)	)	PUNCT
ejpam-6261	315	13	≥	≥	NOUN
ejpam-6261	315	14	ν̂	ν̂	NUM
ejpam-6261	315	15	,	,	PUNCT
ejpam-6261	315	16	qw(j	qw(j	X
ejpam-6261	315	17	)	)	PUNCT
ejpam-6261	315	18	≥	≥	PROPN
ejpam-6261	315	19	η	η	PROPN
ejpam-6261	315	20	,	,	PUNCT
ejpam-6261	315	21	φw(j	φw(j	NOUN
ejpam-6261	315	22	)	)	PUNCT
ejpam-6261	315	23	≥	≥	X
ejpam-6261	315	24	η̂	η̂	VERB
ejpam-6261	315	25	,	,	PUNCT
ejpam-6261	315	26	rw(j	rw(j	NOUN
ejpam-6261	315	27	)	)	PUNCT
ejpam-6261	315	28	≤	≤	NUM
ejpam-6261	315	29	ρ	ρ	PROPN
ejpam-6261	315	30	,	,	PUNCT
ejpam-6261	315	31	ωw(j	ωw(j	NOUN
ejpam-6261	315	32	)	)	PUNCT
ejpam-6261	315	33	≤	≤	NOUN
ejpam-6261	315	34	ρ̂	ρ̂	NUM
ejpam-6261	315	35	}	}	PUNCT
ejpam-6261	315	36	.	.	PUNCT
ejpam-6261	316	1	for	for	ADP
ejpam-6261	316	2	η	η	PROPN
ejpam-6261	316	3	=	=	PROPN
ejpam-6261	316	4	ρ	ρ	PROPN
ejpam-6261	316	5	=	=	SYM
ejpam-6261	316	6	0	0	PUNCT
ejpam-6261	316	7	=	=	SYM
ejpam-6261	316	8	η̂	η̂	PROPN
ejpam-6261	316	9	=	=	SYM
ejpam-6261	316	10	ρ̂	ρ̂	NUM
ejpam-6261	316	11	,	,	PUNCT
ejpam-6261	316	12	we	we	PRON
ejpam-6261	316	13	get	get	VERB
ejpam-6261	316	14	wν	wν	NOUN
ejpam-6261	316	15	ν̂	ν̂	X
ejpam-6261	316	16	=	=	PUNCT
ejpam-6261	316	17	{	{	PUNCT
ejpam-6261	316	18	j	j	PROPN
ejpam-6261	316	19	∈	∈	PROPN
ejpam-6261	316	20	r	r	NOUN
ejpam-6261	316	21	:	:	PUNCT
ejpam-6261	316	22	pw(j	pw(j	X
ejpam-6261	316	23	)	)	PUNCT
ejpam-6261	316	24	≥	≥	NOUN
ejpam-6261	316	25	ν	ν	NOUN
ejpam-6261	316	26	,	,	PUNCT
ejpam-6261	316	27	θw(j	θw(j	NUM
ejpam-6261	316	28	)	)	PUNCT
ejpam-6261	316	29	≥	≥	NOUN
ejpam-6261	316	30	ν̂	ν̂	NUM
ejpam-6261	316	31	}	}	PUNCT
ejpam-6261	316	32	,	,	PUNCT
ejpam-6261	316	33	for	for	ADP
ejpam-6261	316	34	ν	ν	X
ejpam-6261	316	35	=	=	SYM
ejpam-6261	316	36	ρ	ρ	PROPN
ejpam-6261	316	37	=	=	SYM
ejpam-6261	316	38	0	0	PUNCT
ejpam-6261	316	39	=	=	SYM
ejpam-6261	316	40	ν̂	ν̂	X
ejpam-6261	316	41	=	=	SYM
ejpam-6261	316	42	ρ̂	ρ̂	NUM
ejpam-6261	316	43	,	,	PUNCT
ejpam-6261	316	44	we	we	PRON
ejpam-6261	316	45	get	get	VERB
ejpam-6261	316	46	,	,	PUNCT
ejpam-6261	316	47	wη	wη	X
ejpam-6261	317	1	η̂	η̂	PROPN
ejpam-6261	317	2	=	=	NOUN
ejpam-6261	317	3	{	{	PUNCT
ejpam-6261	317	4	j	j	PROPN
ejpam-6261	317	5	∈	∈	PROPN
ejpam-6261	317	6	r	r	NOUN
ejpam-6261	317	7	:	:	PUNCT
ejpam-6261	317	8	qw(j	qw(j	X
ejpam-6261	317	9	)	)	PUNCT
ejpam-6261	317	10	≥	≥	PROPN
ejpam-6261	317	11	η	η	PROPN
ejpam-6261	317	12	,	,	PUNCT
ejpam-6261	317	13	φw(j	φw(j	NOUN
ejpam-6261	317	14	)	)	PUNCT
ejpam-6261	317	15	≥	≥	X
ejpam-6261	317	16	η̂	η̂	X
ejpam-6261	317	17	}	}	PUNCT
ejpam-6261	317	18	and	and	CCONJ
ejpam-6261	317	19	for	for	ADP
ejpam-6261	317	20	ν	ν	X
ejpam-6261	317	21	=	=	SYM
ejpam-6261	317	22	η	η	PROPN
ejpam-6261	317	23	=	=	SYM
ejpam-6261	317	24	0	0	PUNCT
ejpam-6261	318	1	=	=	SYM
ejpam-6261	318	2	ν̂	ν̂	X
ejpam-6261	318	3	=	=	PUNCT
ejpam-6261	318	4	η̂	η̂	X
ejpam-6261	318	5	,	,	PUNCT
ejpam-6261	318	6	we	we	PRON
ejpam-6261	318	7	get	get	VERB
ejpam-6261	318	8	wρ	wρ	ADP
ejpam-6261	318	9	ρ̂	ρ̂	NUM
ejpam-6261	318	10	=	=	SYM
ejpam-6261	318	11	{	{	PUNCT
ejpam-6261	318	12	j	j	PROPN
ejpam-6261	318	13	∈	∈	PROPN
ejpam-6261	318	14	r	r	NOUN
ejpam-6261	318	15	:	:	PUNCT
ejpam-6261	318	16	rw(j	rw(j	X
ejpam-6261	318	17	)	)	PUNCT
ejpam-6261	318	18	≤	≤	NUM
ejpam-6261	318	19	ρ	ρ	PROPN
ejpam-6261	318	20	,	,	PUNCT
ejpam-6261	318	21	ωw(j	ωw(j	NOUN
ejpam-6261	318	22	)	)	PUNCT
ejpam-6261	318	23	≤	≤	NOUN
ejpam-6261	318	24	ρ̂	ρ̂	NUM
ejpam-6261	318	25	}	}	PUNCT
ejpam-6261	318	26	.	.	PUNCT
ejpam-6261	319	1	theorem	theorem	VERB
ejpam-6261	319	2	4	4	NUM
ejpam-6261	319	3	.	.	PUNCT
ejpam-6261	320	1	let	let	VERB
ejpam-6261	320	2	w	w	NOUN
ejpam-6261	320	3	be	be	AUX
ejpam-6261	320	4	cnnsr	cnnsr	NOUN
ejpam-6261	320	5	of	of	ADP
ejpam-6261	320	6	ring	ring	PROPN
ejpam-6261	320	7	r.	r.	PROPN
ejpam-6261	320	8	then	then	ADV
ejpam-6261	320	9	w(ν	w(ν	PROPN
ejpam-6261	320	10	,	,	PUNCT
ejpam-6261	320	11	η	η	PROPN
ejpam-6261	320	12	,	,	PUNCT
ejpam-6261	320	13	ρ	ρ	PROPN
ejpam-6261	320	14	)	)	PUNCT
ejpam-6261	320	15	(	(	PUNCT
ejpam-6261	320	16	ν̂	ν̂	X
ejpam-6261	320	17	,	,	PUNCT
ejpam-6261	320	18	η̂	η̂	NUM
ejpam-6261	320	19	,	,	PUNCT
ejpam-6261	320	20	ρ̂	ρ̂	NUM
ejpam-6261	320	21	)	)	PUNCT
ejpam-6261	320	22	is	be	AUX
ejpam-6261	320	23	a	a	DET
ejpam-6261	320	24	subring	subring	NOUN
ejpam-6261	320	25	of	of	ADP
ejpam-6261	320	26	ring	ring	NOUN
ejpam-6261	320	27	r	r	NOUN
ejpam-6261	320	28	,	,	PUNCT
ejpam-6261	320	29	for	for	ADP
ejpam-6261	320	30	all	all	DET
ejpam-6261	320	31	ν	ν	PROPN
ejpam-6261	320	32	,	,	PUNCT
ejpam-6261	320	33	η	η	PROPN
ejpam-6261	320	34	,	,	PUNCT
ejpam-6261	320	35	ρ	ρ	PROPN
ejpam-6261	320	36	∈	∈	PROPN
ejpam-6261	321	1	[	[	X
ejpam-6261	321	2	0	0	NUM
ejpam-6261	321	3	,	,	PUNCT
ejpam-6261	321	4	1	1	NUM
ejpam-6261	321	5	]	]	PUNCT
ejpam-6261	321	6	,	,	PUNCT
ejpam-6261	321	7	and	and	CCONJ
ejpam-6261	321	8	ν̂	ν̂	NUM
ejpam-6261	321	9	,	,	PUNCT
ejpam-6261	321	10	η̂	η̂	NUM
ejpam-6261	321	11	,	,	PUNCT
ejpam-6261	321	12	ρ̂	ρ̂	NUM
ejpam-6261	321	13	∈	∈	PROPN
ejpam-6261	322	1	[	[	X
ejpam-6261	322	2	0	0	NUM
ejpam-6261	322	3	,	,	PUNCT
ejpam-6261	322	4	2π	2π	NOUN
ejpam-6261	322	5	]	]	PUNCT
ejpam-6261	322	6	,	,	PUNCT
ejpam-6261	322	7	where	where	SCONJ
ejpam-6261	322	8	pw(j	pw(j	X
ejpam-6261	322	9	)	)	PUNCT
ejpam-6261	322	10	≥	≥	NOUN
ejpam-6261	322	11	ν	ν	NOUN
ejpam-6261	322	12	,	,	PUNCT
ejpam-6261	322	13	θw(j	θw(j	NUM
ejpam-6261	322	14	)	)	PUNCT
ejpam-6261	322	15	≥	≥	NOUN
ejpam-6261	322	16	ν̂	ν̂	NUM
ejpam-6261	322	17	,	,	PUNCT
ejpam-6261	322	18	qw(j	qw(j	X
ejpam-6261	322	19	)	)	PUNCT
ejpam-6261	322	20	≥	≥	PROPN
ejpam-6261	322	21	η	η	PROPN
ejpam-6261	322	22	,	,	PUNCT
ejpam-6261	322	23	φw(j	φw(j	NOUN
ejpam-6261	322	24	)	)	PUNCT
ejpam-6261	322	25	≥	≥	X
ejpam-6261	322	26	η̂	η̂	VERB
ejpam-6261	322	27	,	,	PUNCT
ejpam-6261	322	28	rw(j	rw(j	NOUN
ejpam-6261	322	29	)	)	PUNCT
ejpam-6261	322	30	≤	≤	NUM
ejpam-6261	322	31	ρ	ρ	PROPN
ejpam-6261	322	32	,	,	PUNCT
ejpam-6261	322	33	ωw(j	ωw(j	NOUN
ejpam-6261	322	34	)	)	PUNCT
ejpam-6261	322	35	≤	≤	NUM
ejpam-6261	322	36	ρ̂.	ρ̂.	NOUN
ejpam-6261	322	37	proof	proof	NOUN
ejpam-6261	322	38	.	.	PUNCT
ejpam-6261	323	1	we	we	PRON
ejpam-6261	323	2	know	know	VERB
ejpam-6261	323	3	that	that	SCONJ
ejpam-6261	323	4	w(ν	w(ν	PROPN
ejpam-6261	323	5	,	,	PUNCT
ejpam-6261	323	6	η	η	PROPN
ejpam-6261	323	7	,	,	PUNCT
ejpam-6261	323	8	ρ	ρ	PROPN
ejpam-6261	323	9	)	)	PUNCT
ejpam-6261	323	10	(	(	PUNCT
ejpam-6261	323	11	ν̂	ν̂	X
ejpam-6261	323	12	,	,	PUNCT
ejpam-6261	323	13	η̂	η̂	NUM
ejpam-6261	323	14	,	,	PUNCT
ejpam-6261	323	15	ρ̂	ρ̂	NUM
ejpam-6261	323	16	)	)	PUNCT
ejpam-6261	323	17	is	be	AUX
ejpam-6261	323	18	nonempty	nonempty	ADJ
ejpam-6261	323	19	,	,	PUNCT
ejpam-6261	323	20	as	as	ADP
ejpam-6261	323	21	e	e	PROPN
ejpam-6261	323	22	∈	∈	PROPN
ejpam-6261	323	23	a	a	DET
ejpam-6261	323	24	(	(	PUNCT
ejpam-6261	323	25	ν	ν	PROPN
ejpam-6261	323	26	,	,	PUNCT
ejpam-6261	323	27	η	η	PROPN
ejpam-6261	323	28	,	,	PUNCT
ejpam-6261	323	29	ρ	ρ	PROPN
ejpam-6261	323	30	)	)	PUNCT
ejpam-6261	323	31	(	(	PUNCT
ejpam-6261	323	32	ν̂	ν̂	X
ejpam-6261	323	33	,	,	PUNCT
ejpam-6261	323	34	η̂	η̂	NUM
ejpam-6261	323	35	,	,	PUNCT
ejpam-6261	323	36	ρ̂	ρ̂	NUM
ejpam-6261	323	37	)	)	PUNCT
ejpam-6261	323	38	.	.	PUNCT
ejpam-6261	324	1	let	let	VERB
ejpam-6261	324	2	f	f	X
ejpam-6261	324	3	,	,	PUNCT
ejpam-6261	324	4	y	y	PROPN
ejpam-6261	324	5	∈	∈	PROPN
ejpam-6261	324	6	a	a	DET
ejpam-6261	324	7	(	(	PUNCT
ejpam-6261	324	8	ν	ν	PROPN
ejpam-6261	324	9	,	,	PUNCT
ejpam-6261	324	10	η	η	PROPN
ejpam-6261	324	11	,	,	PUNCT
ejpam-6261	324	12	ρ	ρ	PROPN
ejpam-6261	324	13	)	)	PUNCT
ejpam-6261	324	14	(	(	PUNCT
ejpam-6261	324	15	ν̂	ν̂	X
ejpam-6261	324	16	,	,	PUNCT
ejpam-6261	324	17	η̂	η̂	NUM
ejpam-6261	324	18	,	,	PUNCT
ejpam-6261	324	19	ρ̂	ρ̂	NUM
ejpam-6261	324	20	)	)	PUNCT
ejpam-6261	324	21	be	be	VERB
ejpam-6261	324	22	any	any	DET
ejpam-6261	324	23	two	two	NUM
ejpam-6261	324	24	elements	element	NOUN
ejpam-6261	324	25	.	.	PUNCT
ejpam-6261	325	1	then	then	ADV
ejpam-6261	325	2	pw(j	pw(j	X
ejpam-6261	325	3	)	)	PUNCT
ejpam-6261	325	4	≥	≥	NOUN
ejpam-6261	325	5	ν	ν	NOUN
ejpam-6261	325	6	,	,	PUNCT
ejpam-6261	325	7	θw(j	θw(j	NUM
ejpam-6261	325	8	)	)	PUNCT
ejpam-6261	325	9	≥	≥	NOUN
ejpam-6261	325	10	ν̂	ν̂	NUM
ejpam-6261	325	11	,	,	PUNCT
ejpam-6261	325	12	qw(j	qw(j	X
ejpam-6261	325	13	)	)	PUNCT
ejpam-6261	325	14	≥	≥	PROPN
ejpam-6261	325	15	η	η	PROPN
ejpam-6261	325	16	,	,	PUNCT
ejpam-6261	325	17	φw(j	φw(j	NOUN
ejpam-6261	325	18	)	)	PUNCT
ejpam-6261	325	19	≥	≥	X
ejpam-6261	325	20	η̂	η̂	VERB
ejpam-6261	325	21	,	,	PUNCT
ejpam-6261	325	22	rw(j	rw(j	NOUN
ejpam-6261	325	23	)	)	PUNCT
ejpam-6261	325	24	≤	≤	NUM
ejpam-6261	325	25	ρ	ρ	PROPN
ejpam-6261	325	26	,	,	PUNCT
ejpam-6261	325	27	ωw(j	ωw(j	NOUN
ejpam-6261	325	28	)	)	PUNCT
ejpam-6261	325	29	≤	≤	NOUN
ejpam-6261	325	30	ρ̂.	ρ̂.	ADV
ejpam-6261	326	1	now	now	ADV
ejpam-6261	326	2	we	we	PRON
ejpam-6261	326	3	suppose	suppose	VERB
ejpam-6261	326	4	that	that	SCONJ
ejpam-6261	326	5	,	,	PUNCT
ejpam-6261	326	6	pw(j−	pw(j−	PROPN
ejpam-6261	326	7	g)eiθw(j−g	g)eiθw(j−g	ADJ
ejpam-6261	326	8	)	)	PUNCT
ejpam-6261	327	1	=	=	PRON
ejpam-6261	327	2	tw(j−	tw(j−	NUM
ejpam-6261	327	3	g	g	NOUN
ejpam-6261	327	4	)	)	PUNCT
ejpam-6261	327	5	≥	≥	NOUN
ejpam-6261	327	6	min{tw(j),tw(g	min{tw(j),tw(g	X
ejpam-6261	327	7	)	)	PUNCT
ejpam-6261	327	8	}	}	PUNCT
ejpam-6261	327	9	=	=	SYM
ejpam-6261	327	10	min{pw(j)eiθw(j	min{pw(j)eiθw(j	PROPN
ejpam-6261	327	11	)	)	PUNCT
ejpam-6261	327	12	,	,	PUNCT
ejpam-6261	327	13	pw(g)eiθw(g	pw(g)eiθw(g	PROPN
ejpam-6261	327	14	)	)	PUNCT
ejpam-6261	327	15	}	}	PUNCT
ejpam-6261	327	16	m.	m.	NOUN
ejpam-6261	327	17	h.	h.	PROPN
ejpam-6261	327	18	mateen	mateen	PROPN
ejpam-6261	327	19	et	et	PROPN
ejpam-6261	327	20	al	al	PROPN
ejpam-6261	327	21	.	.	PUNCT
ejpam-6261	327	22	/	/	SYM
ejpam-6261	327	23	eur	eur	PROPN
ejpam-6261	327	24	.	.	PUNCT
ejpam-6261	328	1	j.	j.	PROPN
ejpam-6261	328	2	pure	pure	PROPN
ejpam-6261	328	3	appl	appl	PROPN
ejpam-6261	328	4	.	.	PROPN
ejpam-6261	328	5	math	math	PROPN
ejpam-6261	328	6	,	,	PUNCT
ejpam-6261	328	7	18	18	NUM
ejpam-6261	328	8	(	(	PUNCT
ejpam-6261	328	9	4	4	NUM
ejpam-6261	328	10	)	)	PUNCT
ejpam-6261	328	11	(	(	PUNCT
ejpam-6261	328	12	2025	2025	NUM
ejpam-6261	328	13	)	)	PUNCT
ejpam-6261	328	14	,	,	PUNCT
ejpam-6261	328	15	6261	6261	NUM
ejpam-6261	328	16	13	13	NUM
ejpam-6261	328	17	of	of	ADP
ejpam-6261	328	18	23	23	NUM
ejpam-6261	328	19	=	=	SYM
ejpam-6261	328	20	min{pw(j	min{pw(j	PROPN
ejpam-6261	328	21	)	)	PUNCT
ejpam-6261	328	22	,	,	PUNCT
ejpam-6261	328	23	pw(g	pw(g	NOUN
ejpam-6261	328	24	)	)	PUNCT
ejpam-6261	328	25	}	}	PUNCT
ejpam-6261	328	26	eimin{θw(j),θw(g	eimin{θw(j),θw(g	NOUN
ejpam-6261	328	27	)	)	PUNCT
ejpam-6261	328	28	}	}	PUNCT
ejpam-6261	328	29	.	.	PUNCT
ejpam-6261	329	1	as	as	SCONJ
ejpam-6261	329	2	w	w	PROPN
ejpam-6261	329	3	is	be	AUX
ejpam-6261	329	4	homogeneous	homogeneous	ADJ
ejpam-6261	329	5	,	,	PUNCT
ejpam-6261	329	6	so	so	ADV
ejpam-6261	329	7	pw(j−	pw(j−	INTJ
ejpam-6261	329	8	g	g	PROPN
ejpam-6261	329	9	)	)	PUNCT
ejpam-6261	329	10	≥	≥	NOUN
ejpam-6261	329	11	min{pw(j	min{pw(j	PROPN
ejpam-6261	329	12	)	)	PUNCT
ejpam-6261	329	13	,	,	PUNCT
ejpam-6261	329	14	pw(g	pw(g	NOUN
ejpam-6261	329	15	)	)	PUNCT
ejpam-6261	329	16	}	}	PUNCT
ejpam-6261	329	17	=	=	SYM
ejpam-6261	329	18	min{ν	min{ν	NOUN
ejpam-6261	329	19	,	,	PUNCT
ejpam-6261	329	20	ν	ν	NOUN
ejpam-6261	329	21	}	}	PUNCT
ejpam-6261	329	22	=	=	SYM
ejpam-6261	329	23	ν	ν	NOUN
ejpam-6261	329	24	,	,	PUNCT
ejpam-6261	329	25	θw(j−	θw(j−	CCONJ
ejpam-6261	329	26	g	g	NOUN
ejpam-6261	329	27	)	)	PUNCT
ejpam-6261	329	28	≥	≥	NOUN
ejpam-6261	329	29	min{θw(j	min{θw(j	NOUN
ejpam-6261	329	30	)	)	PUNCT
ejpam-6261	329	31	,	,	PUNCT
ejpam-6261	329	32	θw(g	θw(g	NOUN
ejpam-6261	329	33	)	)	PUNCT
ejpam-6261	329	34	}	}	PUNCT
ejpam-6261	329	35	=	=	SYM
ejpam-6261	329	36	min{ν̂	min{ν̂	PROPN
ejpam-6261	329	37	,	,	PUNCT
ejpam-6261	329	38	ν̂	ν̂	NUM
ejpam-6261	329	39	}	}	PUNCT
ejpam-6261	329	40	=	=	SYM
ejpam-6261	329	41	ν̂.	ν̂.	ADJ
ejpam-6261	329	42	pw(jg)eiθw(jg	pw(jg)eiθw(jg	NOUN
ejpam-6261	329	43	)	)	PUNCT
ejpam-6261	329	44	=	=	SYM
ejpam-6261	329	45	tw(jg	tw(jg	PROPN
ejpam-6261	329	46	)	)	PUNCT
ejpam-6261	329	47	≥	≥	NOUN
ejpam-6261	329	48	min{tw(j),tw(g	min{tw(j),tw(g	X
ejpam-6261	329	49	)	)	PUNCT
ejpam-6261	329	50	}	}	PUNCT
ejpam-6261	329	51	=	=	SYM
ejpam-6261	329	52	min{pw(j)eiθw(j	min{pw(j)eiθw(j	PROPN
ejpam-6261	329	53	)	)	PUNCT
ejpam-6261	329	54	,	,	PUNCT
ejpam-6261	329	55	pw(g)eiθw(g	pw(g)eiθw(g	PROPN
ejpam-6261	329	56	)	)	PUNCT
ejpam-6261	329	57	}	}	PUNCT
ejpam-6261	329	58	=	=	SYM
ejpam-6261	329	59	min{pw(j	min{pw(j	X
ejpam-6261	329	60	)	)	PUNCT
ejpam-6261	329	61	,	,	PUNCT
ejpam-6261	329	62	pw(g	pw(g	NOUN
ejpam-6261	329	63	)	)	PUNCT
ejpam-6261	329	64	}	}	PUNCT
ejpam-6261	329	65	eimin{θw(j),θw(g	eimin{θw(j),θw(g	NOUN
ejpam-6261	329	66	)	)	PUNCT
ejpam-6261	329	67	}	}	PUNCT
ejpam-6261	329	68	.	.	PUNCT
ejpam-6261	330	1	as	as	SCONJ
ejpam-6261	330	2	w	w	PROPN
ejpam-6261	330	3	is	be	AUX
ejpam-6261	330	4	homogeneous	homogeneous	ADJ
ejpam-6261	330	5	,	,	PUNCT
ejpam-6261	330	6	so	so	ADV
ejpam-6261	330	7	pw(jg	pw(jg	PROPN
ejpam-6261	330	8	)	)	PUNCT
ejpam-6261	330	9	≥	≥	NOUN
ejpam-6261	330	10	min{pw(j	min{pw(j	PROPN
ejpam-6261	330	11	)	)	PUNCT
ejpam-6261	330	12	,	,	PUNCT
ejpam-6261	330	13	pw(g	pw(g	NOUN
ejpam-6261	330	14	)	)	PUNCT
ejpam-6261	330	15	}	}	PUNCT
ejpam-6261	330	16	=	=	SYM
ejpam-6261	330	17	min{ν	min{ν	NOUN
ejpam-6261	330	18	,	,	PUNCT
ejpam-6261	330	19	ν	ν	NOUN
ejpam-6261	330	20	}	}	PUNCT
ejpam-6261	330	21	=	=	SYM
ejpam-6261	330	22	ν	ν	PROPN
ejpam-6261	330	23	,	,	PUNCT
ejpam-6261	330	24	θw(jg	θw(jg	PROPN
ejpam-6261	330	25	)	)	PUNCT
ejpam-6261	330	26	≥	≥	PROPN
ejpam-6261	330	27	min{θw(j	min{θw(j	NOUN
ejpam-6261	330	28	)	)	PUNCT
ejpam-6261	330	29	,	,	PUNCT
ejpam-6261	330	30	θw(g	θw(g	NOUN
ejpam-6261	330	31	)	)	PUNCT
ejpam-6261	330	32	}	}	PUNCT
ejpam-6261	330	33	=	=	SYM
ejpam-6261	330	34	min{ν̂	min{ν̂	PROPN
ejpam-6261	330	35	,	,	PUNCT
ejpam-6261	330	36	ν̂	ν̂	NUM
ejpam-6261	330	37	}	}	PUNCT
ejpam-6261	330	38	=	=	PUNCT
ejpam-6261	330	39	ν̂.	ν̂.	ADJ
ejpam-6261	330	40	further	far	ADV
ejpam-6261	330	41	,	,	PUNCT
ejpam-6261	330	42	qw(j−	qw(j−	X
ejpam-6261	330	43	g)eiφw(j−g	g)eiφw(j−g	NUM
ejpam-6261	330	44	)	)	PUNCT
ejpam-6261	331	1	=	=	PRON
ejpam-6261	331	2	iw(j−	iw(j−	PROPN
ejpam-6261	331	3	g	g	PROPN
ejpam-6261	331	4	)	)	PUNCT
ejpam-6261	331	5	≥	≥	NOUN
ejpam-6261	331	6	min{iw(j	min{iw(j	NOUN
ejpam-6261	331	7	)	)	PUNCT
ejpam-6261	331	8	,	,	PUNCT
ejpam-6261	331	9	iw(g	iw(g	X
ejpam-6261	331	10	)	)	PUNCT
ejpam-6261	331	11	}	}	PUNCT
ejpam-6261	331	12	=	=	SYM
ejpam-6261	331	13	min{qw(j)eiφw(j	min{qw(j)eiφw(j	PROPN
ejpam-6261	331	14	)	)	PUNCT
ejpam-6261	331	15	,	,	PUNCT
ejpam-6261	331	16	qw(g)eiφw(g	qw(g)eiφw(g	PROPN
ejpam-6261	331	17	)	)	PUNCT
ejpam-6261	331	18	}	}	PUNCT
ejpam-6261	331	19	=	=	SYM
ejpam-6261	331	20	min{qw(j	min{qw(j	PROPN
ejpam-6261	331	21	)	)	PUNCT
ejpam-6261	331	22	,	,	PUNCT
ejpam-6261	331	23	qw(g	qw(g	NUM
ejpam-6261	331	24	)	)	PUNCT
ejpam-6261	331	25	}	}	PUNCT
ejpam-6261	331	26	eimin{φw(j),φw(g	eimin{φw(j),φw(g	ADV
ejpam-6261	331	27	)	)	PUNCT
ejpam-6261	331	28	}	}	PUNCT
ejpam-6261	331	29	.	.	PUNCT
ejpam-6261	332	1	as	as	SCONJ
ejpam-6261	332	2	w	w	PROPN
ejpam-6261	332	3	is	be	AUX
ejpam-6261	332	4	homogeneous	homogeneous	ADJ
ejpam-6261	332	5	,	,	PUNCT
ejpam-6261	332	6	so	so	CCONJ
ejpam-6261	332	7	qw(j−	qw(j−	PUNCT
ejpam-6261	332	8	g	g	NOUN
ejpam-6261	332	9	)	)	PUNCT
ejpam-6261	332	10	≥	≥	NOUN
ejpam-6261	332	11	min{qw(j	min{qw(j	PROPN
ejpam-6261	332	12	)	)	PUNCT
ejpam-6261	332	13	,	,	PUNCT
ejpam-6261	332	14	qw(g	qw(g	NUM
ejpam-6261	332	15	)	)	PUNCT
ejpam-6261	332	16	}	}	PUNCT
ejpam-6261	332	17	=	=	SYM
ejpam-6261	332	18	min{ν	min{ν	NOUN
ejpam-6261	332	19	,	,	PUNCT
ejpam-6261	332	20	ν	ν	NOUN
ejpam-6261	332	21	}	}	PUNCT
ejpam-6261	332	22	=	=	SYM
ejpam-6261	332	23	ν	ν	NOUN
ejpam-6261	332	24	,	,	PUNCT
ejpam-6261	332	25	φw(j−	φw(j−	PUNCT
ejpam-6261	332	26	g	g	NOUN
ejpam-6261	332	27	)	)	PUNCT
ejpam-6261	332	28	≥	≥	NOUN
ejpam-6261	332	29	min{φw(j	min{φw(j	X
ejpam-6261	332	30	)	)	PUNCT
ejpam-6261	332	31	,	,	PUNCT
ejpam-6261	332	32	φw(g	φw(g	X
ejpam-6261	332	33	)	)	PUNCT
ejpam-6261	332	34	}	}	PUNCT
ejpam-6261	332	35	=	=	SYM
ejpam-6261	332	36	min{ν̂	min{ν̂	PROPN
ejpam-6261	332	37	,	,	PUNCT
ejpam-6261	332	38	ν̂	ν̂	NUM
ejpam-6261	332	39	}	}	PUNCT
ejpam-6261	332	40	=	=	SYM
ejpam-6261	332	41	ν̂.	ν̂.	ADJ
ejpam-6261	332	42	qw(jg)eiφw(jg	qw(jg)eiφw(jg	NOUN
ejpam-6261	332	43	)	)	PUNCT
ejpam-6261	332	44	=	=	SYM
ejpam-6261	332	45	iw(jg	iw(jg	PROPN
ejpam-6261	332	46	)	)	PUNCT
ejpam-6261	332	47	≥	≥	PROPN
ejpam-6261	332	48	min{iw(j	min{iw(j	PROPN
ejpam-6261	332	49	)	)	PUNCT
ejpam-6261	332	50	,	,	PUNCT
ejpam-6261	332	51	iw(g	iw(g	X
ejpam-6261	332	52	)	)	PUNCT
ejpam-6261	332	53	}	}	PUNCT
ejpam-6261	332	54	=	=	SYM
ejpam-6261	332	55	min{qw(j)eiφw(j	min{qw(j)eiφw(j	PROPN
ejpam-6261	332	56	)	)	PUNCT
ejpam-6261	332	57	,	,	PUNCT
ejpam-6261	332	58	qw(g)eiφw(g	qw(g)eiφw(g	PROPN
ejpam-6261	332	59	)	)	PUNCT
ejpam-6261	332	60	}	}	PUNCT
ejpam-6261	332	61	=	=	SYM
ejpam-6261	332	62	min{qw(j	min{qw(j	PROPN
ejpam-6261	332	63	)	)	PUNCT
ejpam-6261	332	64	,	,	PUNCT
ejpam-6261	332	65	qw(g	qw(g	NUM
ejpam-6261	332	66	)	)	PUNCT
ejpam-6261	332	67	}	}	PUNCT
ejpam-6261	332	68	eimin{φw(j),φw(g	eimin{φw(j),φw(g	ADV
ejpam-6261	332	69	)	)	PUNCT
ejpam-6261	332	70	}	}	PUNCT
ejpam-6261	332	71	.	.	PUNCT
ejpam-6261	333	1	as	as	SCONJ
ejpam-6261	333	2	w	w	PROPN
ejpam-6261	333	3	is	be	AUX
ejpam-6261	333	4	homogeneous	homogeneous	ADJ
ejpam-6261	333	5	,	,	PUNCT
ejpam-6261	333	6	so	so	ADV
ejpam-6261	333	7	qw(jg	qw(jg	PROPN
ejpam-6261	333	8	)	)	PUNCT
ejpam-6261	333	9	≥	≥	PROPN
ejpam-6261	333	10	min{qw(j	min{qw(j	PROPN
ejpam-6261	333	11	)	)	PUNCT
ejpam-6261	333	12	,	,	PUNCT
ejpam-6261	333	13	qw(g	qw(g	NUM
ejpam-6261	333	14	)	)	PUNCT
ejpam-6261	333	15	}	}	PUNCT
ejpam-6261	333	16	=	=	SYM
ejpam-6261	333	17	min{ν	min{ν	NOUN
ejpam-6261	333	18	,	,	PUNCT
ejpam-6261	333	19	ν	ν	NOUN
ejpam-6261	333	20	}	}	PUNCT
ejpam-6261	333	21	=	=	SYM
ejpam-6261	333	22	ν	ν	PROPN
ejpam-6261	333	23	,	,	PUNCT
ejpam-6261	333	24	φw(jg	φw(jg	PROPN
ejpam-6261	333	25	)	)	PUNCT
ejpam-6261	333	26	≥	≥	NOUN
ejpam-6261	333	27	min{φw(j	min{φw(j	X
ejpam-6261	333	28	)	)	PUNCT
ejpam-6261	333	29	,	,	PUNCT
ejpam-6261	333	30	φw(g	φw(g	X
ejpam-6261	333	31	)	)	PUNCT
ejpam-6261	333	32	}	}	PUNCT
ejpam-6261	333	33	=	=	SYM
ejpam-6261	333	34	min{ν̂	min{ν̂	PROPN
ejpam-6261	333	35	,	,	PUNCT
ejpam-6261	333	36	ν̂	ν̂	NUM
ejpam-6261	333	37	}	}	PUNCT
ejpam-6261	333	38	=	=	PUNCT
ejpam-6261	333	39	ν̂.	ν̂.	ADJ
ejpam-6261	333	40	further	far	ADV
ejpam-6261	333	41	,	,	PUNCT
ejpam-6261	333	42	rw(j−	rw(j−	CCONJ
ejpam-6261	333	43	g)eiωw(j−g	g)eiωw(j−g	VERB
ejpam-6261	333	44	)	)	PUNCT
ejpam-6261	334	1	=	=	PRON
ejpam-6261	334	2	fw(j−	fw(j−	NUM
ejpam-6261	334	3	g	g	NOUN
ejpam-6261	334	4	)	)	PUNCT
ejpam-6261	334	5	≤	≤	NOUN
ejpam-6261	334	6	max{fw(j),fw(g	max{fw(j),fw(g	NOUN
ejpam-6261	334	7	)	)	PUNCT
ejpam-6261	334	8	}	}	PUNCT
ejpam-6261	334	9	=	=	SYM
ejpam-6261	334	10	max{rw(j)eiωw(j	max{rw(j)eiωw(j	PROPN
ejpam-6261	334	11	)	)	PUNCT
ejpam-6261	334	12	,	,	PUNCT
ejpam-6261	334	13	rw(g)eiωw(g	rw(g)eiωw(g	NOUN
ejpam-6261	334	14	)	)	PUNCT
ejpam-6261	334	15	}	}	PUNCT
ejpam-6261	334	16	=	=	SYM
ejpam-6261	334	17	max{rw(j	max{rw(j	X
ejpam-6261	334	18	)	)	PUNCT
ejpam-6261	334	19	,	,	PUNCT
ejpam-6261	334	20	rw(g	rw(g	X
ejpam-6261	334	21	)	)	PUNCT
ejpam-6261	334	22	}	}	PUNCT
ejpam-6261	334	23	eimax{ωw(j),ωw(g	eimax{ωw(j),ωw(g	NOUN
ejpam-6261	334	24	)	)	PUNCT
ejpam-6261	334	25	}	}	PUNCT
ejpam-6261	334	26	.	.	PUNCT
ejpam-6261	335	1	by	by	ADP
ejpam-6261	335	2	homogeneity	homogeneity	NOUN
ejpam-6261	335	3	,	,	PUNCT
ejpam-6261	335	4	so	so	CCONJ
ejpam-6261	335	5	rw(j−	rw(j−	CCONJ
ejpam-6261	335	6	g	g	NOUN
ejpam-6261	335	7	)	)	PUNCT
ejpam-6261	335	8	≤	≤	NUM
ejpam-6261	335	9	max{rw(j	max{rw(j	NOUN
ejpam-6261	335	10	)	)	PUNCT
ejpam-6261	335	11	,	,	PUNCT
ejpam-6261	335	12	rw(g	rw(g	NOUN
ejpam-6261	335	13	)	)	PUNCT
ejpam-6261	335	14	}	}	PUNCT
ejpam-6261	335	15	=	=	SYM
ejpam-6261	335	16	max{ρ	max{ρ	PROPN
ejpam-6261	335	17	,	,	PUNCT
ejpam-6261	335	18	ρ	ρ	NOUN
ejpam-6261	335	19	}	}	PUNCT
ejpam-6261	335	20	=	=	SYM
ejpam-6261	335	21	ρ	ρ	PROPN
ejpam-6261	335	22	,	,	PUNCT
ejpam-6261	335	23	ωw(j−	ωw(j−	PUNCT
ejpam-6261	335	24	g	g	NOUN
ejpam-6261	335	25	)	)	PUNCT
ejpam-6261	335	26	≤	≤	NUM
ejpam-6261	335	27	max{ωw(j	max{ωw(j	NOUN
ejpam-6261	335	28	)	)	PUNCT
ejpam-6261	335	29	,	,	PUNCT
ejpam-6261	335	30	ωw(g	ωw(g	NUM
ejpam-6261	335	31	)	)	PUNCT
ejpam-6261	335	32	}	}	PUNCT
ejpam-6261	335	33	=	=	SYM
ejpam-6261	335	34	max{ρ̂	max{ρ̂	PROPN
ejpam-6261	335	35	,	,	PUNCT
ejpam-6261	335	36	ρ̂	ρ̂	NUM
ejpam-6261	335	37	}	}	PUNCT
ejpam-6261	335	38	=	=	SYM
ejpam-6261	335	39	ρ̂.	ρ̂.	NOUN
ejpam-6261	335	40	m.	m.	NOUN
ejpam-6261	335	41	h.	h.	PROPN
ejpam-6261	335	42	mateen	mateen	PROPN
ejpam-6261	335	43	et	et	PROPN
ejpam-6261	335	44	al	al	PROPN
ejpam-6261	335	45	.	.	PUNCT
ejpam-6261	335	46	/	/	SYM
ejpam-6261	335	47	eur	eur	PROPN
ejpam-6261	335	48	.	.	PUNCT
ejpam-6261	336	1	j.	j.	PROPN
ejpam-6261	336	2	pure	pure	PROPN
ejpam-6261	336	3	appl	appl	PROPN
ejpam-6261	336	4	.	.	PROPN
ejpam-6261	336	5	math	math	PROPN
ejpam-6261	336	6	,	,	PUNCT
ejpam-6261	336	7	18	18	NUM
ejpam-6261	336	8	(	(	PUNCT
ejpam-6261	336	9	4	4	NUM
ejpam-6261	336	10	)	)	PUNCT
ejpam-6261	336	11	(	(	PUNCT
ejpam-6261	336	12	2025	2025	NUM
ejpam-6261	336	13	)	)	PUNCT
ejpam-6261	336	14	,	,	PUNCT
ejpam-6261	336	15	6261	6261	NUM
ejpam-6261	336	16	14	14	NUM
ejpam-6261	336	17	of	of	ADP
ejpam-6261	336	18	23	23	NUM
ejpam-6261	336	19	rw(jg)eiωw(jg	rw(jg)eiωw(jg	NOUN
ejpam-6261	336	20	)	)	PUNCT
ejpam-6261	337	1	=	=	SYM
ejpam-6261	337	2	fw(jg	fw(jg	ADJ
ejpam-6261	337	3	)	)	PUNCT
ejpam-6261	337	4	≤	≤	NOUN
ejpam-6261	337	5	max{fw(j),fw(g	max{fw(j),fw(g	NOUN
ejpam-6261	337	6	)	)	PUNCT
ejpam-6261	337	7	}	}	PUNCT
ejpam-6261	337	8	=	=	SYM
ejpam-6261	337	9	max{rw(j)eiωw(j	max{rw(j)eiωw(j	PROPN
ejpam-6261	337	10	)	)	PUNCT
ejpam-6261	337	11	,	,	PUNCT
ejpam-6261	337	12	rw(g)eiωw(g	rw(g)eiωw(g	NOUN
ejpam-6261	337	13	)	)	PUNCT
ejpam-6261	337	14	}	}	PUNCT
ejpam-6261	337	15	=	=	SYM
ejpam-6261	337	16	max{rw(j	max{rw(j	X
ejpam-6261	337	17	)	)	PUNCT
ejpam-6261	337	18	,	,	PUNCT
ejpam-6261	337	19	rw(g	rw(g	X
ejpam-6261	337	20	)	)	PUNCT
ejpam-6261	337	21	}	}	PUNCT
ejpam-6261	337	22	eimax{ωw(j),ωw(g	eimax{ωw(j),ωw(g	NOUN
ejpam-6261	337	23	)	)	PUNCT
ejpam-6261	337	24	}	}	PUNCT
ejpam-6261	337	25	.	.	PUNCT
ejpam-6261	338	1	by	by	ADP
ejpam-6261	338	2	homogeneity	homogeneity	NOUN
ejpam-6261	338	3	,	,	PUNCT
ejpam-6261	338	4	so	so	ADV
ejpam-6261	338	5	rw(jg	rw(jg	PROPN
ejpam-6261	338	6	)	)	PUNCT
ejpam-6261	338	7	≤	≤	NUM
ejpam-6261	338	8	max{rw(j	max{rw(j	PROPN
ejpam-6261	338	9	)	)	PUNCT
ejpam-6261	338	10	,	,	PUNCT
ejpam-6261	338	11	rw(g	rw(g	NOUN
ejpam-6261	338	12	)	)	PUNCT
ejpam-6261	338	13	}	}	PUNCT
ejpam-6261	338	14	=	=	SYM
ejpam-6261	338	15	max{ρ	max{ρ	PROPN
ejpam-6261	338	16	,	,	PUNCT
ejpam-6261	338	17	ρ	ρ	NOUN
ejpam-6261	338	18	}	}	PUNCT
ejpam-6261	338	19	=	=	SYM
ejpam-6261	338	20	ρ	ρ	PROPN
ejpam-6261	338	21	,	,	PUNCT
ejpam-6261	338	22	ωw(jg	ωw(jg	PROPN
ejpam-6261	338	23	)	)	PUNCT
ejpam-6261	338	24	≤	≤	NUM
ejpam-6261	338	25	max{ωw(j	max{ωw(j	NOUN
ejpam-6261	338	26	)	)	PUNCT
ejpam-6261	338	27	,	,	PUNCT
ejpam-6261	338	28	ωw(g	ωw(g	NUM
ejpam-6261	338	29	)	)	PUNCT
ejpam-6261	338	30	}	}	PUNCT
ejpam-6261	338	31	=	=	SYM
ejpam-6261	338	32	max{ρ̂	max{ρ̂	PROPN
ejpam-6261	338	33	,	,	PUNCT
ejpam-6261	338	34	ρ̂	ρ̂	NUM
ejpam-6261	338	35	}	}	PUNCT
ejpam-6261	338	36	=	=	PUNCT
ejpam-6261	338	37	ρ̂.	ρ̂.	ADP
ejpam-6261	338	38	this	this	PRON
ejpam-6261	338	39	implies	imply	VERB
ejpam-6261	338	40	that	that	SCONJ
ejpam-6261	338	41	mn	mn	PROPN
ejpam-6261	338	42	∈	∈	PROPN
ejpam-6261	338	43	w(ν	w(ν	PROPN
ejpam-6261	338	44	,	,	PUNCT
ejpam-6261	338	45	η	η	PROPN
ejpam-6261	338	46	,	,	PUNCT
ejpam-6261	338	47	ρ	ρ	PROPN
ejpam-6261	338	48	)	)	PUNCT
ejpam-6261	338	49	(	(	PUNCT
ejpam-6261	338	50	ν̂	ν̂	X
ejpam-6261	338	51	,	,	PUNCT
ejpam-6261	338	52	η̂	η̂	NUM
ejpam-6261	338	53	,	,	PUNCT
ejpam-6261	338	54	ρ̂	ρ̂	NUM
ejpam-6261	338	55	)	)	PUNCT
ejpam-6261	338	56	.	.	PUNCT
ejpam-6261	339	1	hence	hence	ADV
ejpam-6261	339	2	,	,	PUNCT
ejpam-6261	339	3	w(ν	w(ν	PROPN
ejpam-6261	339	4	,	,	PUNCT
ejpam-6261	339	5	η	η	PROPN
ejpam-6261	339	6	,	,	PUNCT
ejpam-6261	339	7	ρ	ρ	PROPN
ejpam-6261	339	8	)	)	PUNCT
ejpam-6261	339	9	(	(	PUNCT
ejpam-6261	339	10	ν̂	ν̂	X
ejpam-6261	339	11	,	,	PUNCT
ejpam-6261	339	12	η̂	η̂	NUM
ejpam-6261	339	13	,	,	PUNCT
ejpam-6261	339	14	ρ̂	ρ̂	NUM
ejpam-6261	339	15	)	)	PUNCT
ejpam-6261	339	16	is	be	AUX
ejpam-6261	339	17	subring	subre	VERB
ejpam-6261	339	18	.	.	PUNCT
ejpam-6261	340	1	theorem	theorem	ADJ
ejpam-6261	340	2	5	5	NUM
ejpam-6261	340	3	.	.	PUNCT
ejpam-6261	341	1	let	let	VERB
ejpam-6261	341	2	w(α	w(α	NOUN
ejpam-6261	341	3	,	,	PUNCT
ejpam-6261	341	4	β	β	X
ejpam-6261	341	5	,	,	PUNCT
ejpam-6261	341	6	γ	γ	NOUN
ejpam-6261	341	7	)	)	PUNCT
ejpam-6261	341	8	(	(	PUNCT
ejpam-6261	341	9	α̂	α̂	NOUN
ejpam-6261	341	10	,	,	PUNCT
ejpam-6261	341	11	β̂	β̂	ADV
ejpam-6261	341	12	,	,	PUNCT
ejpam-6261	341	13	γ̂	γ̂	PUNCT
ejpam-6261	341	14	)	)	PUNCT
ejpam-6261	341	15	be	be	AUX
ejpam-6261	341	16	a	a	DET
ejpam-6261	341	17	subring	subring	NOUN
ejpam-6261	341	18	of	of	ADP
ejpam-6261	341	19	ring	ring	NOUN
ejpam-6261	341	20	r	r	NOUN
ejpam-6261	341	21	,	,	PUNCT
ejpam-6261	341	22	then	then	ADV
ejpam-6261	341	23	w	w	PROPN
ejpam-6261	341	24	is	be	AUX
ejpam-6261	341	25	cnsr	cnsr	VERB
ejpam-6261	341	26	of	of	ADP
ejpam-6261	341	27	r	r	NOUN
ejpam-6261	341	28	if	if	SCONJ
ejpam-6261	341	29	pw(j	pw(j	NOUN
ejpam-6261	341	30	)	)	PUNCT
ejpam-6261	341	31	≥	≥	NOUN
ejpam-6261	341	32	α	α	NOUN
ejpam-6261	341	33	,	,	PUNCT
ejpam-6261	341	34	θw(j	θw(j	NUM
ejpam-6261	341	35	)	)	PUNCT
ejpam-6261	341	36	≥	≥	NOUN
ejpam-6261	341	37	α̂	α̂	NOUN
ejpam-6261	341	38	,	,	PUNCT
ejpam-6261	341	39	qw(j	qw(j	X
ejpam-6261	341	40	)	)	PUNCT
ejpam-6261	341	41	≥	≥	PROPN
ejpam-6261	341	42	β	β	X
ejpam-6261	341	43	,	,	PUNCT
ejpam-6261	341	44	φw(j	φw(j	NOUN
ejpam-6261	341	45	)	)	PUNCT
ejpam-6261	341	46	≥	≥	X
ejpam-6261	341	47	β̂	β̂	ADP
ejpam-6261	341	48	,	,	PUNCT
ejpam-6261	341	49	rw(j	rw(j	NOUN
ejpam-6261	341	50	)	)	PUNCT
ejpam-6261	341	51	≤	≤	NUM
ejpam-6261	341	52	γ	γ	NOUN
ejpam-6261	341	53	,	,	PUNCT
ejpam-6261	341	54	ωw(j	ωw(j	NOUN
ejpam-6261	341	55	)	)	PUNCT
ejpam-6261	341	56	≤	≤	NOUN
ejpam-6261	341	57	γ̂	γ̂	ADV
ejpam-6261	341	58	,	,	PUNCT
ejpam-6261	341	59	∀	∀	X
ejpam-6261	341	60	α	α	NOUN
ejpam-6261	341	61	,	,	PUNCT
ejpam-6261	341	62	β	β	X
ejpam-6261	341	63	,	,	PUNCT
ejpam-6261	341	64	γ	γ	PROPN
ejpam-6261	341	65	∈	∈	PROPN
ejpam-6261	342	1	[	[	X
ejpam-6261	342	2	0	0	NUM
ejpam-6261	342	3	,	,	PUNCT
ejpam-6261	342	4	1	1	NUM
ejpam-6261	342	5	]	]	PUNCT
ejpam-6261	342	6	,	,	PUNCT
ejpam-6261	342	7	and	and	CCONJ
ejpam-6261	342	8	α̂	α̂	NOUN
ejpam-6261	342	9	,	,	PUNCT
ejpam-6261	342	10	β̂	β̂	ADP
ejpam-6261	342	11	,	,	PUNCT
ejpam-6261	342	12	γ̂	γ̂	PRON
ejpam-6261	342	13	∈	∈	PROPN
ejpam-6261	343	1	[	[	X
ejpam-6261	343	2	0	0	NUM
ejpam-6261	343	3	,	,	PUNCT
ejpam-6261	343	4	2π	2π	NOUN
ejpam-6261	343	5	]	]	PUNCT
ejpam-6261	343	6	.	.	PUNCT
ejpam-6261	344	1	proof	proof	NOUN
ejpam-6261	344	2	.	.	PUNCT
ejpam-6261	345	1	suppose	suppose	VERB
ejpam-6261	345	2	that	that	SCONJ
ejpam-6261	345	3	min{pw(j	min{pw(j	NOUN
ejpam-6261	345	4	)	)	PUNCT
ejpam-6261	345	5	,	,	PUNCT
ejpam-6261	345	6	pw(g	pw(g	NOUN
ejpam-6261	345	7	)	)	PUNCT
ejpam-6261	345	8	}	}	PUNCT
ejpam-6261	345	9	=	=	SYM
ejpam-6261	345	10	α	α	PROPN
ejpam-6261	345	11	,	,	PUNCT
ejpam-6261	345	12	min{θw(j	min{θw(j	NOUN
ejpam-6261	345	13	)	)	PUNCT
ejpam-6261	345	14	,	,	PUNCT
ejpam-6261	345	15	θw(g	θw(g	NOUN
ejpam-6261	345	16	)	)	PUNCT
ejpam-6261	345	17	}	}	PUNCT
ejpam-6261	345	18	=	=	SYM
ejpam-6261	345	19	α̂,min{qw(j	α̂,min{qw(j	NOUN
ejpam-6261	345	20	)	)	PUNCT
ejpam-6261	345	21	,	,	PUNCT
ejpam-6261	345	22	qw(g	qw(g	NOUN
ejpam-6261	345	23	)	)	PUNCT
ejpam-6261	345	24	}	}	PUNCT
ejpam-6261	345	25	=	=	SYM
ejpam-6261	345	26	β	β	X
ejpam-6261	345	27	,	,	PUNCT
ejpam-6261	345	28	min{φw(j	min{φw(j	PROPN
ejpam-6261	345	29	)	)	PUNCT
ejpam-6261	345	30	,	,	PUNCT
ejpam-6261	345	31	φw(g	φw(g	X
ejpam-6261	345	32	)	)	PUNCT
ejpam-6261	345	33	}	}	PUNCT
ejpam-6261	345	34	=	=	SYM
ejpam-6261	345	35	β̂	β̂	ADP
ejpam-6261	345	36	and	and	CCONJ
ejpam-6261	345	37	max{rw(j	max{rw(j	PROPN
ejpam-6261	345	38	)	)	PUNCT
ejpam-6261	345	39	,	,	PUNCT
ejpam-6261	345	40	rw(g	rw(g	X
ejpam-6261	345	41	)	)	PUNCT
ejpam-6261	345	42	}	}	PUNCT
ejpam-6261	345	43	=	=	SYM
ejpam-6261	345	44	γ	γ	X
ejpam-6261	345	45	,	,	PUNCT
ejpam-6261	345	46	max{ωw(j	max{ωw(j	NOUN
ejpam-6261	345	47	)	)	PUNCT
ejpam-6261	345	48	,	,	PUNCT
ejpam-6261	345	49	ωw(g	ωw(g	NUM
ejpam-6261	345	50	)	)	PUNCT
ejpam-6261	345	51	}	}	PUNCT
ejpam-6261	345	52	=	=	SYM
ejpam-6261	345	53	γ̂.	γ̂.	NOUN
ejpam-6261	346	1	then	then	ADV
ejpam-6261	346	2	we	we	PRON
ejpam-6261	346	3	have	have	VERB
ejpam-6261	346	4	pw(j	pw(j	NOUN
ejpam-6261	346	5	)	)	PUNCT
ejpam-6261	346	6	≥	≥	NOUN
ejpam-6261	346	7	α	α	NOUN
ejpam-6261	346	8	,	,	PUNCT
ejpam-6261	346	9	qw(j	qw(j	X
ejpam-6261	346	10	)	)	PUNCT
ejpam-6261	346	11	≥	≥	PROPN
ejpam-6261	346	12	β	β	X
ejpam-6261	346	13	,	,	PUNCT
ejpam-6261	346	14	rw(j	rw(j	NOUN
ejpam-6261	346	15	)	)	PUNCT
ejpam-6261	346	16	≤	≤	NUM
ejpam-6261	346	17	γ	γ	X
ejpam-6261	346	18	,	,	PUNCT
ejpam-6261	346	19	θw(j	θw(j	NUM
ejpam-6261	346	20	)	)	PUNCT
ejpam-6261	346	21	≥	≥	NOUN
ejpam-6261	346	22	α̂	α̂	NOUN
ejpam-6261	346	23	,	,	PUNCT
ejpam-6261	346	24	φw(j	φw(j	NOUN
ejpam-6261	346	25	)	)	PUNCT
ejpam-6261	346	26	≥	≥	X
ejpam-6261	346	27	β̂	β̂	ADV
ejpam-6261	346	28	,	,	PUNCT
ejpam-6261	346	29	ωw(j	ωw(j	NOUN
ejpam-6261	346	30	)	)	PUNCT
ejpam-6261	346	31	≤	≤	NOUN
ejpam-6261	346	32	γ̂	γ̂	PUNCT
ejpam-6261	346	33	and	and	CCONJ
ejpam-6261	346	34	pw(g	pw(g	NUM
ejpam-6261	346	35	)	)	PUNCT
ejpam-6261	346	36	≥	≥	NOUN
ejpam-6261	346	37	α	α	NOUN
ejpam-6261	346	38	,	,	PUNCT
ejpam-6261	346	39	qw(g	qw(g	PRON
ejpam-6261	346	40	)	)	PUNCT
ejpam-6261	346	41	≥	≥	NOUN
ejpam-6261	346	42	β	β	X
ejpam-6261	346	43	,	,	PUNCT
ejpam-6261	346	44	rw(g	rw(g	NUM
ejpam-6261	346	45	)	)	PUNCT
ejpam-6261	346	46	≤	≤	NUM
ejpam-6261	346	47	γ	γ	X
ejpam-6261	346	48	,	,	PUNCT
ejpam-6261	346	49	θw(g	θw(g	NUM
ejpam-6261	346	50	)	)	PUNCT
ejpam-6261	346	51	≥	≥	NUM
ejpam-6261	346	52	α	α	NOUN
ejpam-6261	346	53	,	,	PUNCT
ejpam-6261	346	54	φw(g	φw(g	NUM
ejpam-6261	346	55	)	)	PUNCT
ejpam-6261	346	56	≥	≥	X
ejpam-6261	346	57	β	β	X
ejpam-6261	346	58	,	,	PUNCT
ejpam-6261	346	59	ωw(g	ωw(g	NUM
ejpam-6261	346	60	)	)	PUNCT
ejpam-6261	346	61	≤	≤	NUM
ejpam-6261	346	62	γ̂.	γ̂.	NOUN
ejpam-6261	346	63	this	this	PRON
ejpam-6261	346	64	means	mean	VERB
ejpam-6261	346	65	that	that	SCONJ
ejpam-6261	346	66	f	f	PROPN
ejpam-6261	346	67	∈	∈	PROPN
ejpam-6261	346	68	w(α	w(α	PROPN
ejpam-6261	346	69	,	,	PUNCT
ejpam-6261	346	70	β	β	X
ejpam-6261	346	71	,	,	PUNCT
ejpam-6261	346	72	γ	γ	NOUN
ejpam-6261	346	73	)	)	PUNCT
ejpam-6261	346	74	(	(	PUNCT
ejpam-6261	346	75	α̂	α̂	NOUN
ejpam-6261	346	76	,	,	PUNCT
ejpam-6261	346	77	β̂	β̂	ADP
ejpam-6261	346	78	,	,	PUNCT
ejpam-6261	346	79	γ̂	γ̂	PUNCT
ejpam-6261	346	80	)	)	PUNCT
ejpam-6261	346	81	and	and	CCONJ
ejpam-6261	346	82	n	n	PRON
ejpam-6261	346	83	∈	∈	PROPN
ejpam-6261	346	84	w(α	w(α	NOUN
ejpam-6261	346	85	,	,	PUNCT
ejpam-6261	346	86	β	β	X
ejpam-6261	346	87	,	,	PUNCT
ejpam-6261	346	88	γ	γ	NOUN
ejpam-6261	346	89	)	)	PUNCT
ejpam-6261	346	90	(	(	PUNCT
ejpam-6261	346	91	α̂	α̂	NOUN
ejpam-6261	346	92	,	,	PUNCT
ejpam-6261	346	93	β̂	β̂	ADP
ejpam-6261	346	94	,	,	PUNCT
ejpam-6261	346	95	γ̂	γ̂	PROPN
ejpam-6261	346	96	)	)	PUNCT
ejpam-6261	346	97	.	.	PUNCT
ejpam-6261	347	1	as	as	ADP
ejpam-6261	347	2	w(α	w(α	PROPN
ejpam-6261	347	3	,	,	PUNCT
ejpam-6261	347	4	β	β	X
ejpam-6261	347	5	,	,	PUNCT
ejpam-6261	347	6	γ	γ	NOUN
ejpam-6261	347	7	)	)	PUNCT
ejpam-6261	347	8	(	(	PUNCT
ejpam-6261	347	9	α̂	α̂	NOUN
ejpam-6261	347	10	,	,	PUNCT
ejpam-6261	347	11	β̂	β̂	ADV
ejpam-6261	347	12	,	,	PUNCT
ejpam-6261	347	13	γ̂	γ̂	PRON
ejpam-6261	347	14	)	)	PUNCT
ejpam-6261	347	15	is	be	AUX
ejpam-6261	347	16	subring	subre	VERB
ejpam-6261	347	17	,	,	PUNCT
ejpam-6261	347	18	so	so	SCONJ
ejpam-6261	347	19	mn	mn	PROPN
ejpam-6261	347	20	∈	∈	PROPN
ejpam-6261	347	21	w(α	w(α	PROPN
ejpam-6261	347	22	,	,	PUNCT
ejpam-6261	347	23	β	β	X
ejpam-6261	347	24	,	,	PUNCT
ejpam-6261	347	25	γ	γ	NOUN
ejpam-6261	347	26	)	)	PUNCT
ejpam-6261	347	27	(	(	PUNCT
ejpam-6261	347	28	α̂	α̂	NOUN
ejpam-6261	347	29	,	,	PUNCT
ejpam-6261	347	30	β̂	β̂	ADP
ejpam-6261	347	31	,	,	PUNCT
ejpam-6261	347	32	γ̂	γ̂	PROPN
ejpam-6261	347	33	)	)	PUNCT
ejpam-6261	347	34	.	.	PUNCT
ejpam-6261	348	1	then	then	ADV
ejpam-6261	348	2	we	we	PRON
ejpam-6261	348	3	have	have	VERB
ejpam-6261	348	4	pw(j−	pw(j−	PROPN
ejpam-6261	348	5	g	g	PROPN
ejpam-6261	348	6	)	)	PUNCT
ejpam-6261	348	7	≥	≥	NOUN
ejpam-6261	348	8	α	α	NOUN
ejpam-6261	348	9	and	and	CCONJ
ejpam-6261	348	10	θw(j−	θw(j−	CCONJ
ejpam-6261	348	11	g	g	NOUN
ejpam-6261	348	12	)	)	PUNCT
ejpam-6261	348	13	≥	≥	NOUN
ejpam-6261	348	14	α̂	α̂	NOUN
ejpam-6261	348	15	,	,	PUNCT
ejpam-6261	348	16	qw(j−	qw(j−	SYM
ejpam-6261	348	17	g	g	NOUN
ejpam-6261	348	18	)	)	PUNCT
ejpam-6261	348	19	≥	≥	NOUN
ejpam-6261	348	20	β	β	X
ejpam-6261	348	21	and	and	CCONJ
ejpam-6261	348	22	φw(j−	φw(j−	NUM
ejpam-6261	348	23	g	g	NOUN
ejpam-6261	348	24	)	)	PUNCT
ejpam-6261	348	25	≥	≥	NOUN
ejpam-6261	348	26	β̂	β̂	ADP
ejpam-6261	348	27	,	,	PUNCT
ejpam-6261	348	28	rw(j−	rw(j−	X
ejpam-6261	348	29	g	g	NOUN
ejpam-6261	348	30	)	)	PUNCT
ejpam-6261	348	31	≤	≤	NOUN
ejpam-6261	348	32	γ	γ	X
ejpam-6261	348	33	and	and	CCONJ
ejpam-6261	348	34	ωw(j−	ωw(j−	PUNCT
ejpam-6261	348	35	g	g	NOUN
ejpam-6261	348	36	)	)	PUNCT
ejpam-6261	348	37	≤	≤	NOUN
ejpam-6261	348	38	γ̂	γ̂	PUNCT
ejpam-6261	348	39	implies	imply	VERB
ejpam-6261	348	40	that	that	SCONJ
ejpam-6261	348	41	pw(j−	pw(j−	PROPN
ejpam-6261	348	42	g	g	NOUN
ejpam-6261	348	43	)	)	PUNCT
ejpam-6261	348	44	≥	≥	NOUN
ejpam-6261	348	45	min{pw(j	min{pw(j	PROPN
ejpam-6261	348	46	)	)	PUNCT
ejpam-6261	348	47	,	,	PUNCT
ejpam-6261	348	48	pw(g	pw(g	NOUN
ejpam-6261	348	49	)	)	PUNCT
ejpam-6261	348	50	}	}	PUNCT
ejpam-6261	348	51	and	and	CCONJ
ejpam-6261	348	52	θw(j−	θw(j−	CCONJ
ejpam-6261	348	53	g	g	NOUN
ejpam-6261	348	54	)	)	PUNCT
ejpam-6261	348	55	≥	≥	NOUN
ejpam-6261	348	56	min{θw(j	min{θw(j	NOUN
ejpam-6261	348	57	)	)	PUNCT
ejpam-6261	348	58	,	,	PUNCT
ejpam-6261	348	59	θw(g	θw(g	NOUN
ejpam-6261	348	60	)	)	PUNCT
ejpam-6261	348	61	}	}	PUNCT
ejpam-6261	348	62	,	,	PUNCT
ejpam-6261	348	63	qw(j−	qw(j−	CCONJ
ejpam-6261	348	64	g	g	NOUN
ejpam-6261	348	65	)	)	PUNCT
ejpam-6261	348	66	≥	≥	NOUN
ejpam-6261	348	67	min{qw(j	min{qw(j	PROPN
ejpam-6261	348	68	)	)	PUNCT
ejpam-6261	348	69	,	,	PUNCT
ejpam-6261	348	70	qw(g	qw(g	NOUN
ejpam-6261	348	71	)	)	PUNCT
ejpam-6261	348	72	}	}	PUNCT
ejpam-6261	348	73	and	and	CCONJ
ejpam-6261	348	74	φw(j−	φw(j−	NUM
ejpam-6261	348	75	g	g	NOUN
ejpam-6261	348	76	)	)	PUNCT
ejpam-6261	348	77	≥	≥	NOUN
ejpam-6261	348	78	min{φw(j	min{φw(j	X
ejpam-6261	348	79	)	)	PUNCT
ejpam-6261	348	80	,	,	PUNCT
ejpam-6261	348	81	φw(g	φw(g	X
ejpam-6261	348	82	)	)	PUNCT
ejpam-6261	348	83	}	}	PUNCT
ejpam-6261	348	84	,	,	PUNCT
ejpam-6261	348	85	rw(j−	rw(j−	CCONJ
ejpam-6261	348	86	g	g	X
ejpam-6261	348	87	)	)	PUNCT
ejpam-6261	348	88	≤	≤	NUM
ejpam-6261	348	89	max{rw(j	max{rw(j	NOUN
ejpam-6261	348	90	)	)	PUNCT
ejpam-6261	348	91	,	,	PUNCT
ejpam-6261	348	92	rw(g	rw(g	X
ejpam-6261	348	93	)	)	PUNCT
ejpam-6261	348	94	}	}	PUNCT
ejpam-6261	348	95	,	,	PUNCT
ejpam-6261	348	96	ωw(j−	ωw(j−	PUNCT
ejpam-6261	348	97	g	g	NOUN
ejpam-6261	348	98	)	)	PUNCT
ejpam-6261	348	99	≤	≤	NUM
ejpam-6261	348	100	max{ωw(j	max{ωw(j	NOUN
ejpam-6261	348	101	)	)	PUNCT
ejpam-6261	348	102	,	,	PUNCT
ejpam-6261	348	103	ωw(g	ωw(g	NUM
ejpam-6261	348	104	)	)	PUNCT
ejpam-6261	348	105	}	}	PUNCT
ejpam-6261	348	106	.	.	PUNCT
ejpam-6261	349	1	pw(jg	pw(jg	PROPN
ejpam-6261	349	2	)	)	PUNCT
ejpam-6261	349	3	≥	≥	PROPN
ejpam-6261	349	4	α	α	PROPN
ejpam-6261	349	5	and	and	CCONJ
ejpam-6261	349	6	θw(jg	θw(jg	PROPN
ejpam-6261	349	7	)	)	PUNCT
ejpam-6261	349	8	≥	≥	NOUN
ejpam-6261	349	9	α̂	α̂	PROPN
ejpam-6261	349	10	,	,	PUNCT
ejpam-6261	349	11	qw(jg	qw(jg	PROPN
ejpam-6261	349	12	)	)	PUNCT
ejpam-6261	349	13	≥	≥	PROPN
ejpam-6261	349	14	β	β	X
ejpam-6261	349	15	and	and	CCONJ
ejpam-6261	349	16	φw(jg	φw(jg	PROPN
ejpam-6261	349	17	)	)	PUNCT
ejpam-6261	349	18	≥	≥	NOUN
ejpam-6261	349	19	β̂	β̂	X
ejpam-6261	349	20	rw(jg	rw(jg	PROPN
ejpam-6261	349	21	)	)	PUNCT
ejpam-6261	349	22	≤	≤	NOUN
ejpam-6261	349	23	γ	γ	PROPN
ejpam-6261	349	24	and	and	CCONJ
ejpam-6261	349	25	ωw(jg	ωw(jg	PROPN
ejpam-6261	349	26	)	)	PUNCT
ejpam-6261	349	27	≤	≤	NOUN
ejpam-6261	349	28	γ̂	γ̂	PUNCT
ejpam-6261	349	29	implies	imply	VERB
ejpam-6261	349	30	that	that	SCONJ
ejpam-6261	349	31	pw(jg	pw(jg	PROPN
ejpam-6261	349	32	)	)	PUNCT
ejpam-6261	349	33	≥	≥	NOUN
ejpam-6261	349	34	min{pw(j	min{pw(j	PROPN
ejpam-6261	349	35	)	)	PUNCT
ejpam-6261	349	36	,	,	PUNCT
ejpam-6261	349	37	pw(g	pw(g	NOUN
ejpam-6261	349	38	)	)	PUNCT
ejpam-6261	349	39	}	}	PUNCT
ejpam-6261	349	40	and	and	CCONJ
ejpam-6261	349	41	θw(jg	θw(jg	PROPN
ejpam-6261	349	42	)	)	PUNCT
ejpam-6261	349	43	≥	≥	PROPN
ejpam-6261	349	44	min{θw(j	min{θw(j	NOUN
ejpam-6261	349	45	)	)	PUNCT
ejpam-6261	349	46	,	,	PUNCT
ejpam-6261	349	47	θw(g	θw(g	NOUN
ejpam-6261	349	48	)	)	PUNCT
ejpam-6261	349	49	}	}	PUNCT
ejpam-6261	349	50	qw(jg	qw(jg	PROPN
ejpam-6261	349	51	)	)	PUNCT
ejpam-6261	349	52	≥	≥	PROPN
ejpam-6261	349	53	min{qw(j	min{qw(j	PROPN
ejpam-6261	349	54	)	)	PUNCT
ejpam-6261	349	55	,	,	PUNCT
ejpam-6261	349	56	qw(g	qw(g	NUM
ejpam-6261	349	57	)	)	PUNCT
ejpam-6261	349	58	}	}	PUNCT
ejpam-6261	349	59	and	and	CCONJ
ejpam-6261	349	60	φw(jg	φw(jg	PROPN
ejpam-6261	349	61	)	)	PUNCT
ejpam-6261	349	62	≥	≥	NOUN
ejpam-6261	349	63	min{φw(j	min{φw(j	X
ejpam-6261	349	64	)	)	PUNCT
ejpam-6261	349	65	,	,	PUNCT
ejpam-6261	349	66	φw(g	φw(g	X
ejpam-6261	349	67	)	)	PUNCT
ejpam-6261	349	68	}	}	SYM
ejpam-6261	349	69	rw(jg	rw(jg	PROPN
ejpam-6261	349	70	)	)	PUNCT
ejpam-6261	349	71	≤	≤	NUM
ejpam-6261	349	72	max{rw(j	max{rw(j	NOUN
ejpam-6261	349	73	)	)	PUNCT
ejpam-6261	349	74	,	,	PUNCT
ejpam-6261	349	75	rw(g	rw(g	X
ejpam-6261	349	76	)	)	PUNCT
ejpam-6261	349	77	}	}	PUNCT
ejpam-6261	349	78	and	and	CCONJ
ejpam-6261	349	79	ωw(jg	ωw(jg	PROPN
ejpam-6261	349	80	)	)	PUNCT
ejpam-6261	349	81	≤	≤	NUM
ejpam-6261	349	82	max{ωw(j	max{ωw(j	NOUN
ejpam-6261	349	83	)	)	PUNCT
ejpam-6261	349	84	,	,	PUNCT
ejpam-6261	349	85	ωw(g	ωw(g	NUM
ejpam-6261	349	86	)	)	PUNCT
ejpam-6261	349	87	}	}	PUNCT
ejpam-6261	349	88	.	.	PUNCT
ejpam-6261	350	1	thus	thus	ADV
ejpam-6261	350	2	,	,	PUNCT
ejpam-6261	350	3	tw(j−	tw(j−	PUNCT
ejpam-6261	350	4	g	g	NOUN
ejpam-6261	350	5	)	)	PUNCT
ejpam-6261	350	6	=	=	PRON
ejpam-6261	351	1	pw(j−	pw(j−	NUM
ejpam-6261	351	2	g)eiθw(j−g	g)eiθw(j−g	ADJ
ejpam-6261	351	3	)	)	PUNCT
ejpam-6261	351	4	≥	≥	NOUN
ejpam-6261	351	5	min{pw(j	min{pw(j	PROPN
ejpam-6261	351	6	)	)	PUNCT
ejpam-6261	351	7	,	,	PUNCT
ejpam-6261	351	8	pw(g	pw(g	NOUN
ejpam-6261	351	9	)	)	PUNCT
ejpam-6261	351	10	}	}	PUNCT
ejpam-6261	351	11	eimin{θw(j),θw(g	eimin{θw(j),θw(g	NOUN
ejpam-6261	351	12	)	)	PUNCT
ejpam-6261	351	13	}	}	PUNCT
ejpam-6261	352	1	=	=	SYM
ejpam-6261	352	2	min{pw(j)eiθw(j	min{pw(j)eiθw(j	PROPN
ejpam-6261	352	3	)	)	PUNCT
ejpam-6261	352	4	,	,	PUNCT
ejpam-6261	352	5	pw(g)eiθw(g	pw(g)eiθw(g	PROPN
ejpam-6261	352	6	)	)	PUNCT
ejpam-6261	352	7	}	}	PUNCT
ejpam-6261	352	8	m.	m.	NOUN
ejpam-6261	352	9	h.	h.	PROPN
ejpam-6261	352	10	mateen	mateen	PROPN
ejpam-6261	352	11	et	et	PROPN
ejpam-6261	352	12	al	al	PROPN
ejpam-6261	352	13	.	.	PUNCT
ejpam-6261	352	14	/	/	SYM
ejpam-6261	352	15	eur	eur	PROPN
ejpam-6261	352	16	.	.	PUNCT
ejpam-6261	353	1	j.	j.	PROPN
ejpam-6261	353	2	pure	pure	PROPN
ejpam-6261	353	3	appl	appl	PROPN
ejpam-6261	353	4	.	.	PROPN
ejpam-6261	353	5	math	math	PROPN
ejpam-6261	353	6	,	,	PUNCT
ejpam-6261	353	7	18	18	NUM
ejpam-6261	353	8	(	(	PUNCT
ejpam-6261	353	9	4	4	NUM
ejpam-6261	353	10	)	)	PUNCT
ejpam-6261	353	11	(	(	PUNCT
ejpam-6261	353	12	2025	2025	NUM
ejpam-6261	353	13	)	)	PUNCT
ejpam-6261	353	14	,	,	PUNCT
ejpam-6261	353	15	6261	6261	NUM
ejpam-6261	353	16	15	15	NUM
ejpam-6261	353	17	of	of	ADP
ejpam-6261	353	18	23	23	NUM
ejpam-6261	353	19	tw(j−	tw(j−	NOUN
ejpam-6261	353	20	g	g	NOUN
ejpam-6261	353	21	)	)	PUNCT
ejpam-6261	353	22	≥	≥	NOUN
ejpam-6261	353	23	min{tw(j),tw(g	min{tw(j),tw(g	X
ejpam-6261	353	24	)	)	PUNCT
ejpam-6261	353	25	}	}	PUNCT
ejpam-6261	353	26	.	.	PUNCT
ejpam-6261	354	1	iw(j−	iw(j−	PROPN
ejpam-6261	354	2	g	g	NOUN
ejpam-6261	354	3	)	)	PUNCT
ejpam-6261	354	4	=	=	SYM
ejpam-6261	354	5	qw(j−	qw(j−	NOUN
ejpam-6261	354	6	g)eiφw(j−g	g)eiφw(j−g	NUM
ejpam-6261	354	7	)	)	PUNCT
ejpam-6261	354	8	≥	≥	PROPN
ejpam-6261	354	9	min{qw(j	min{qw(j	PROPN
ejpam-6261	354	10	)	)	PUNCT
ejpam-6261	354	11	,	,	PUNCT
ejpam-6261	354	12	qw(g	qw(g	NUM
ejpam-6261	354	13	)	)	PUNCT
ejpam-6261	354	14	}	}	PUNCT
ejpam-6261	354	15	eimin{φw(j),φw(g	eimin{φw(j),φw(g	ADV
ejpam-6261	354	16	)	)	PUNCT
ejpam-6261	354	17	}	}	PUNCT
ejpam-6261	354	18	=	=	SYM
ejpam-6261	354	19	min{qw(j)eiφw(j	min{qw(j)eiφw(j	PROPN
ejpam-6261	354	20	)	)	PUNCT
ejpam-6261	354	21	,	,	PUNCT
ejpam-6261	354	22	qw(g)eiφw(g	qw(g)eiφw(g	PROPN
ejpam-6261	354	23	)	)	PUNCT
ejpam-6261	354	24	}	}	PUNCT
ejpam-6261	354	25	iw(j−	iw(j−	PROPN
ejpam-6261	354	26	g	g	PROPN
ejpam-6261	354	27	)	)	PUNCT
ejpam-6261	354	28	≥	≥	NOUN
ejpam-6261	354	29	min{iw(j	min{iw(j	NOUN
ejpam-6261	354	30	)	)	PUNCT
ejpam-6261	354	31	,	,	PUNCT
ejpam-6261	354	32	iw(g	iw(g	NOUN
ejpam-6261	354	33	)	)	PUNCT
ejpam-6261	354	34	}	}	PUNCT
ejpam-6261	354	35	.	.	PUNCT
ejpam-6261	355	1	fw(j−	fw(j−	PRON
ejpam-6261	356	1	g	g	NOUN
ejpam-6261	356	2	)	)	PUNCT
ejpam-6261	356	3	=	=	PUNCT
ejpam-6261	356	4	rw(j−	rw(j−	NOUN
ejpam-6261	356	5	g)eiωw(j−g	g)eiωw(j−g	VERB
ejpam-6261	356	6	)	)	PUNCT
ejpam-6261	356	7	≤	≤	NUM
ejpam-6261	356	8	max{rw(j	max{rw(j	NOUN
ejpam-6261	356	9	)	)	PUNCT
ejpam-6261	356	10	,	,	PUNCT
ejpam-6261	356	11	rw(g	rw(g	X
ejpam-6261	356	12	)	)	PUNCT
ejpam-6261	356	13	}	}	PUNCT
ejpam-6261	356	14	eimax{ωw(j),ωw(g	eimax{ωw(j),ωw(g	NOUN
ejpam-6261	356	15	)	)	PUNCT
ejpam-6261	356	16	}	}	PUNCT
ejpam-6261	356	17	=	=	SYM
ejpam-6261	356	18	max{rw(j)eiωw(j	max{rw(j)eiωw(j	PROPN
ejpam-6261	356	19	)	)	PUNCT
ejpam-6261	356	20	,	,	PUNCT
ejpam-6261	356	21	rw(g)eiωw(g	rw(g)eiωw(g	NOUN
ejpam-6261	356	22	)	)	PUNCT
ejpam-6261	356	23	}	}	PUNCT
ejpam-6261	356	24	fw(j−	fw(j−	PROPN
ejpam-6261	357	1	g	g	NOUN
ejpam-6261	357	2	)	)	PUNCT
ejpam-6261	357	3	≤	≤	NUM
ejpam-6261	357	4	max{fw(j),fw(g	max{fw(j),fw(g	NOUN
ejpam-6261	357	5	)	)	PUNCT
ejpam-6261	357	6	}	}	PUNCT
ejpam-6261	357	7	tw(jg	tw(jg	PROPN
ejpam-6261	357	8	)	)	PUNCT
ejpam-6261	357	9	=	=	NOUN
ejpam-6261	357	10	pw(jg)eiθw(jg	pw(jg)eiθw(jg	NOUN
ejpam-6261	357	11	)	)	PUNCT
ejpam-6261	357	12	≥	≥	NOUN
ejpam-6261	357	13	min{pw(j	min{pw(j	PROPN
ejpam-6261	357	14	)	)	PUNCT
ejpam-6261	357	15	,	,	PUNCT
ejpam-6261	357	16	pw(g	pw(g	NOUN
ejpam-6261	357	17	)	)	PUNCT
ejpam-6261	357	18	}	}	PUNCT
ejpam-6261	357	19	eimin{θw(j),θw(g	eimin{θw(j),θw(g	NOUN
ejpam-6261	357	20	)	)	PUNCT
ejpam-6261	357	21	}	}	PUNCT
ejpam-6261	358	1	=	=	SYM
ejpam-6261	358	2	min{pw(j)eiθw(j	min{pw(j)eiθw(j	PROPN
ejpam-6261	358	3	)	)	PUNCT
ejpam-6261	358	4	,	,	PUNCT
ejpam-6261	358	5	pw(g)eiθw(g	pw(g)eiθw(g	PROPN
ejpam-6261	358	6	)	)	PUNCT
ejpam-6261	358	7	}	}	PUNCT
ejpam-6261	358	8	tw(jg	tw(jg	PROPN
ejpam-6261	358	9	)	)	PUNCT
ejpam-6261	358	10	≥	≥	NOUN
ejpam-6261	358	11	min{tw(j),tw(g	min{tw(j),tw(g	X
ejpam-6261	358	12	)	)	PUNCT
ejpam-6261	358	13	}	}	PUNCT
ejpam-6261	358	14	.	.	PUNCT
ejpam-6261	359	1	iw(jg	iw(jg	NOUN
ejpam-6261	359	2	)	)	PUNCT
ejpam-6261	359	3	=	=	SYM
ejpam-6261	359	4	qw(jg)eiφw(jg	qw(jg)eiφw(jg	PROPN
ejpam-6261	359	5	)	)	PUNCT
ejpam-6261	359	6	≥	≥	NOUN
ejpam-6261	359	7	min{qw(j	min{qw(j	PROPN
ejpam-6261	359	8	)	)	PUNCT
ejpam-6261	359	9	,	,	PUNCT
ejpam-6261	359	10	qw(g	qw(g	NUM
ejpam-6261	359	11	)	)	PUNCT
ejpam-6261	359	12	}	}	PUNCT
ejpam-6261	359	13	eimin{φw(j),φw(g	eimin{φw(j),φw(g	ADV
ejpam-6261	359	14	)	)	PUNCT
ejpam-6261	359	15	}	}	PUNCT
ejpam-6261	359	16	=	=	SYM
ejpam-6261	359	17	min{qw(j)eiφw(j	min{qw(j)eiφw(j	PROPN
ejpam-6261	359	18	)	)	PUNCT
ejpam-6261	359	19	,	,	PUNCT
ejpam-6261	359	20	qw(g)eiφw(g	qw(g)eiφw(g	PROPN
ejpam-6261	359	21	)	)	PUNCT
ejpam-6261	359	22	}	}	PUNCT
ejpam-6261	359	23	iw(jg	iw(jg	PROPN
ejpam-6261	359	24	)	)	PUNCT
ejpam-6261	359	25	≥	≥	PROPN
ejpam-6261	359	26	min{iw(j	min{iw(j	PROPN
ejpam-6261	359	27	)	)	PUNCT
ejpam-6261	359	28	,	,	PUNCT
ejpam-6261	359	29	iw(g	iw(g	NOUN
ejpam-6261	359	30	)	)	PUNCT
ejpam-6261	359	31	}	}	PUNCT
ejpam-6261	359	32	.	.	PUNCT
ejpam-6261	360	1	fw(jg	fw(jg	ADJ
ejpam-6261	360	2	)	)	PUNCT
ejpam-6261	360	3	=	=	SYM
ejpam-6261	360	4	rw(jg)eirw(jg	rw(jg)eirw(jg	ADJ
ejpam-6261	360	5	)	)	PUNCT
ejpam-6261	360	6	≤	≤	NUM
ejpam-6261	360	7	max{rw(j	max{rw(j	NOUN
ejpam-6261	360	8	)	)	PUNCT
ejpam-6261	360	9	,	,	PUNCT
ejpam-6261	360	10	rw(g	rw(g	X
ejpam-6261	360	11	)	)	PUNCT
ejpam-6261	360	12	}	}	PUNCT
ejpam-6261	360	13	eimax{ωw(j),ωw(g	eimax{ωw(j),ωw(g	NOUN
ejpam-6261	360	14	)	)	PUNCT
ejpam-6261	360	15	}	}	PUNCT
ejpam-6261	360	16	=	=	SYM
ejpam-6261	360	17	max{rw(j)eiωw(j	max{rw(j)eiωw(j	PROPN
ejpam-6261	360	18	)	)	PUNCT
ejpam-6261	360	19	,	,	PUNCT
ejpam-6261	360	20	rw(g)eiωw(g	rw(g)eiωw(g	NOUN
ejpam-6261	360	21	)	)	PUNCT
ejpam-6261	360	22	}	}	PUNCT
ejpam-6261	360	23	fw(jg	fw(jg	ADJ
ejpam-6261	360	24	)	)	PUNCT
ejpam-6261	360	25	≤	≤	NOUN
ejpam-6261	360	26	max{fw(j),fw(g	max{fw(j),fw(g	NOUN
ejpam-6261	360	27	)	)	PUNCT
ejpam-6261	360	28	}	}	PUNCT
ejpam-6261	360	29	.	.	PUNCT
ejpam-6261	361	1	further	far	ADV
ejpam-6261	361	2	,	,	PUNCT
ejpam-6261	361	3	let	let	VERB
ejpam-6261	361	4	f	f	PRON
ejpam-6261	361	5	∈	∈	PROPN
ejpam-6261	361	6	h	h	NOUN
ejpam-6261	361	7	be	be	AUX
ejpam-6261	361	8	any	any	DET
ejpam-6261	361	9	element	element	NOUN
ejpam-6261	361	10	.	.	PUNCT
ejpam-6261	362	1	let	let	VERB
ejpam-6261	362	2	pw(j	pw(j	PUNCT
ejpam-6261	362	3	)	)	PUNCT
ejpam-6261	363	1	=	=	SYM
ejpam-6261	363	2	α	α	X
ejpam-6261	363	3	,	,	PUNCT
ejpam-6261	363	4	θw(j	θw(j	NUM
ejpam-6261	363	5	)	)	PUNCT
ejpam-6261	363	6	=	=	SYM
ejpam-6261	363	7	α̂	α̂	NOUN
ejpam-6261	363	8	,	,	PUNCT
ejpam-6261	363	9	qw(j	qw(j	X
ejpam-6261	363	10	)	)	PUNCT
ejpam-6261	363	11	=	=	SYM
ejpam-6261	363	12	β	β	NOUN
ejpam-6261	363	13	,	,	PUNCT
ejpam-6261	363	14	φw(j	φw(j	NOUN
ejpam-6261	363	15	)	)	PUNCT
ejpam-6261	363	16	=	=	SYM
ejpam-6261	363	17	β̂	β̂	ADP
ejpam-6261	363	18	,	,	PUNCT
ejpam-6261	363	19	rw(j	rw(j	X
ejpam-6261	363	20	)	)	PUNCT
ejpam-6261	363	21	=	=	SYM
ejpam-6261	363	22	γ	γ	X
ejpam-6261	363	23	,	,	PUNCT
ejpam-6261	363	24	and	and	CCONJ
ejpam-6261	363	25	ωw(j	ωw(j	NOUN
ejpam-6261	363	26	)	)	PUNCT
ejpam-6261	363	27	=	=	SYM
ejpam-6261	363	28	γ̂.	γ̂.	NOUN
ejpam-6261	363	29	then	then	ADV
ejpam-6261	363	30	,	,	PUNCT
ejpam-6261	363	31	pw(j	pw(j	X
ejpam-6261	363	32	)	)	PUNCT
ejpam-6261	363	33	≥	≥	NOUN
ejpam-6261	363	34	α	α	NOUN
ejpam-6261	363	35	,	,	PUNCT
ejpam-6261	363	36	θw(j	θw(j	NUM
ejpam-6261	363	37	)	)	PUNCT
ejpam-6261	363	38	≥	≥	NOUN
ejpam-6261	363	39	α̂	α̂	NOUN
ejpam-6261	363	40	,	,	PUNCT
ejpam-6261	363	41	qw(j	qw(j	X
ejpam-6261	363	42	)	)	PUNCT
ejpam-6261	363	43	≥	≥	PROPN
ejpam-6261	363	44	β	β	X
ejpam-6261	363	45	,	,	PUNCT
ejpam-6261	363	46	φw(j	φw(j	NOUN
ejpam-6261	363	47	)	)	PUNCT
ejpam-6261	363	48	≥	≥	NOUN
ejpam-6261	363	49	β̂	β̂	ADP
ejpam-6261	363	50	,	,	PUNCT
ejpam-6261	363	51	and	and	CCONJ
ejpam-6261	363	52	rw(j	rw(j	NOUN
ejpam-6261	363	53	)	)	PUNCT
ejpam-6261	363	54	≤	≤	NOUN
ejpam-6261	363	55	γ̂	γ̂	ADV
ejpam-6261	363	56	,	,	PUNCT
ejpam-6261	363	57	ωw(j	ωw(j	NOUN
ejpam-6261	363	58	)	)	PUNCT
ejpam-6261	363	59	≤	≤	NOUN
ejpam-6261	363	60	γ̂	γ̂	VERB
ejpam-6261	363	61	is	be	AUX
ejpam-6261	363	62	true	true	ADJ
ejpam-6261	363	63	.	.	PUNCT
ejpam-6261	364	1	implies	imply	VERB
ejpam-6261	364	2	that	that	SCONJ
ejpam-6261	364	3	j	j	PROPN
ejpam-6261	364	4	∈	∈	PROPN
ejpam-6261	364	5	w	w	PROPN
ejpam-6261	364	6	(	(	PUNCT
ejpam-6261	364	7	α	α	X
ejpam-6261	364	8	,	,	PUNCT
ejpam-6261	364	9	β	β	X
ejpam-6261	364	10	,	,	PUNCT
ejpam-6261	364	11	γ	γ	NOUN
ejpam-6261	364	12	)	)	PUNCT
ejpam-6261	364	13	(	(	PUNCT
ejpam-6261	364	14	α̂	α̂	NOUN
ejpam-6261	364	15	,	,	PUNCT
ejpam-6261	364	16	β̂	β̂	ADP
ejpam-6261	364	17	,	,	PUNCT
ejpam-6261	364	18	γ̂	γ̂	PROPN
ejpam-6261	364	19	)	)	PUNCT
ejpam-6261	364	20	.	.	PUNCT
ejpam-6261	365	1	4	4	X
ejpam-6261	365	2	.	.	X
ejpam-6261	365	3	properties	property	NOUN
ejpam-6261	365	4	of	of	ADP
ejpam-6261	365	5	the	the	DET
ejpam-6261	365	6	direct	direct	ADJ
ejpam-6261	365	7	product	product	NOUN
ejpam-6261	365	8	of	of	ADP
ejpam-6261	365	9	complex	complex	ADJ
ejpam-6261	365	10	neutrosophic	neutrosophic	ADJ
ejpam-6261	365	11	subrings	subring	NOUN
ejpam-6261	365	12	in	in	ADP
ejpam-6261	365	13	this	this	DET
ejpam-6261	365	14	part	part	NOUN
ejpam-6261	365	15	,	,	PUNCT
ejpam-6261	365	16	we	we	PRON
ejpam-6261	365	17	describe	describe	VERB
ejpam-6261	365	18	the	the	DET
ejpam-6261	365	19	direct	direct	ADJ
ejpam-6261	365	20	product	product	NOUN
ejpam-6261	365	21	of	of	ADP
ejpam-6261	365	22	cnsrs	cnsrs	NOUN
ejpam-6261	365	23	.	.	PUNCT
ejpam-6261	366	1	we	we	PRON
ejpam-6261	366	2	use	use	VERB
ejpam-6261	366	3	the	the	DET
ejpam-6261	366	4	abstraction	abstraction	NOUN
ejpam-6261	366	5	of	of	ADP
ejpam-6261	366	6	cnss	cns	NOUN
ejpam-6261	366	7	to	to	PART
ejpam-6261	366	8	explore	explore	VERB
ejpam-6261	366	9	the	the	DET
ejpam-6261	366	10	fundamental	fundamental	ADJ
ejpam-6261	366	11	properties	property	NOUN
ejpam-6261	366	12	the	the	DET
ejpam-6261	366	13	direct	direct	ADJ
ejpam-6261	366	14	product	product	NOUN
ejpam-6261	366	15	of	of	ADP
ejpam-6261	366	16	cnsr	cnsr	PROPN
ejpam-6261	366	17	.	.	PUNCT
ejpam-6261	367	1	definition	definition	NOUN
ejpam-6261	367	2	11	11	NUM
ejpam-6261	367	3	.	.	PUNCT
ejpam-6261	368	1	assume	assume	VERB
ejpam-6261	368	2	that	that	SCONJ
ejpam-6261	368	3	w	w	PROPN
ejpam-6261	368	4	and	and	CCONJ
ejpam-6261	368	5	x	x	AUX
ejpam-6261	368	6	be	be	AUX
ejpam-6261	368	7	any	any	DET
ejpam-6261	368	8	two	two	NUM
ejpam-6261	368	9	π	π	PROPN
ejpam-6261	368	10	-	-	PROPN
ejpam-6261	368	11	ns	ns	NOUN
ejpam-6261	368	12	of	of	ADP
ejpam-6261	368	13	sets	set	NOUN
ejpam-6261	368	14	k1	k1	NOUN
ejpam-6261	368	15	and	and	CCONJ
ejpam-6261	368	16	k2	k2	NOUN
ejpam-6261	368	17	,	,	PUNCT
ejpam-6261	368	18	consequently	consequently	ADV
ejpam-6261	368	19	.	.	PUNCT
ejpam-6261	369	1	the	the	DET
ejpam-6261	369	2	cartesian	cartesian	ADJ
ejpam-6261	369	3	product	product	NOUN
ejpam-6261	369	4	of	of	ADP
ejpam-6261	369	5	π	π	PROPN
ejpam-6261	369	6	-	-	PROPN
ejpam-6261	369	7	ns	ns	INTJ
ejpam-6261	369	8	w	w	NOUN
ejpam-6261	369	9	and	and	CCONJ
ejpam-6261	369	10	x	x	PRON
ejpam-6261	369	11	is	be	AUX
ejpam-6261	369	12	expressed	express	VERB
ejpam-6261	369	13	as	as	ADP
ejpam-6261	369	14	(	(	PUNCT
ejpam-6261	369	15	wπ	wπ	INTJ
ejpam-6261	369	16	×	×	PROPN
ejpam-6261	369	17	xπ)(f	xπ)(f	PROPN
ejpam-6261	369	18	,	,	PUNCT
ejpam-6261	369	19	g	g	NOUN
ejpam-6261	369	20	)	)	PUNCT
ejpam-6261	369	21	=	=	PUNCT
ejpam-6261	370	1	{	{	PUNCT
ejpam-6261	370	2	<	<	X
ejpam-6261	370	3	(	(	PUNCT
ejpam-6261	370	4	f	f	NOUN
ejpam-6261	370	5	,	,	PUNCT
ejpam-6261	370	6	g	g	PROPN
ejpam-6261	370	7	)	)	PUNCT
ejpam-6261	370	8	,	,	PUNCT
ejpam-6261	370	9	twπ×xπ(f	twπ×xπ(f	PROPN
ejpam-6261	370	10	,	,	PUNCT
ejpam-6261	370	11	g	g	NOUN
ejpam-6261	370	12	)	)	PUNCT
ejpam-6261	370	13	,	,	PUNCT
ejpam-6261	370	14	iwπ×xπ(f	iwπ×xπ(f	PROPN
ejpam-6261	370	15	,	,	PUNCT
ejpam-6261	370	16	g),fwπ×xπ(f	g),fwπ×xπ(f	X
ejpam-6261	370	17	,	,	PUNCT
ejpam-6261	370	18	g	g	NOUN
ejpam-6261	370	19	)	)	PUNCT
ejpam-6261	370	20	>	>	PUNCT
ejpam-6261	370	21	}	}	PUNCT
ejpam-6261	370	22	,	,	PUNCT
ejpam-6261	370	23	∀	∀	PUNCT
ejpam-6261	370	24	f	f	PROPN
ejpam-6261	370	25	∈	∈	PROPN
ejpam-6261	370	26	k1	k1	PROPN
ejpam-6261	370	27	,	,	PUNCT
ejpam-6261	370	28	g	g	PROPN
ejpam-6261	370	29	∈	∈	PROPN
ejpam-6261	370	30	k2	k2	PROPN
ejpam-6261	370	31	.	.	PUNCT
ejpam-6261	370	32	remark	remark	PROPN
ejpam-6261	370	33	2	2	NUM
ejpam-6261	370	34	.	.	PUNCT
ejpam-6261	371	1	let	let	VERB
ejpam-6261	371	2	w	w	NOUN
ejpam-6261	371	3	and	and	CCONJ
ejpam-6261	371	4	x	x	PART
ejpam-6261	371	5	be	be	AUX
ejpam-6261	371	6	two	two	NUM
ejpam-6261	371	7	π	π	NOUN
ejpam-6261	371	8	-	-	NOUN
ejpam-6261	371	9	nsrs	nsrs	NOUN
ejpam-6261	371	10	of	of	ADP
ejpam-6261	371	11	k1	k1	NOUN
ejpam-6261	371	12	and	and	CCONJ
ejpam-6261	371	13	k2	k2	NOUN
ejpam-6261	371	14	,	,	PUNCT
ejpam-6261	371	15	respectively	respectively	ADV
ejpam-6261	371	16	.	.	PUNCT
ejpam-6261	372	1	then	then	ADV
ejpam-6261	372	2	wπ	wπ	VERB
ejpam-6261	372	3	×	×	NOUN
ejpam-6261	372	4	xπ	xπ	VERB
ejpam-6261	372	5	is	be	AUX
ejpam-6261	372	6	π	π	NOUN
ejpam-6261	372	7	-	-	ADJ
ejpam-6261	372	8	cnsr	cnsr	NOUN
ejpam-6261	372	9	of	of	ADP
ejpam-6261	372	10	k1	k1	PROPN
ejpam-6261	372	11	×k2	×k2	PROPN
ejpam-6261	372	12	.	.	PUNCT
ejpam-6261	373	1	remark	remark	NOUN
ejpam-6261	373	2	3	3	NUM
ejpam-6261	373	3	.	.	PUNCT
ejpam-6261	374	1	a	a	DET
ejpam-6261	374	2	π	π	NOUN
ejpam-6261	374	3	-	-	ADJ
ejpam-6261	374	4	cnsr	cnsr	ADJ
ejpam-6261	374	5	wπ	wπ	ADP
ejpam-6261	374	6	×xπ	×xπ	NOUN
ejpam-6261	374	7	of	of	ADP
ejpam-6261	374	8	ring	ring	NOUN
ejpam-6261	374	9	k1	k1	PROPN
ejpam-6261	374	10	×k2	×k2	PROPN
ejpam-6261	374	11	is	be	AUX
ejpam-6261	374	12	a	a	DET
ejpam-6261	374	13	π	π	PROPN
ejpam-6261	374	14	-	-	ADJ
ejpam-6261	374	15	cnsr	cnsr	NOUN
ejpam-6261	374	16	of	of	ADP
ejpam-6261	374	17	k1	k1	NOUN
ejpam-6261	374	18	×k2	×k2	PROPN
ejpam-6261	374	19	if	if	SCONJ
ejpam-6261	375	1	and	and	CCONJ
ejpam-6261	375	2	only	only	ADV
ejpam-6261	375	3	if	if	SCONJ
ejpam-6261	375	4	w×	w×	PROPN
ejpam-6261	375	5	x	x	PUNCT
ejpam-6261	375	6	is	be	AUX
ejpam-6261	375	7	cnsr	cnsr	VERB
ejpam-6261	375	8	of	of	ADP
ejpam-6261	375	9	k1	k1	PROPN
ejpam-6261	375	10	×k2	×k2	PROPN
ejpam-6261	375	11	m.	m.	NOUN
ejpam-6261	375	12	h.	h.	PROPN
ejpam-6261	375	13	mateen	mateen	PROPN
ejpam-6261	375	14	et	et	PROPN
ejpam-6261	375	15	al	al	PROPN
ejpam-6261	375	16	.	.	PUNCT
ejpam-6261	375	17	/	/	SYM
ejpam-6261	375	18	eur	eur	PROPN
ejpam-6261	375	19	.	.	PUNCT
ejpam-6261	376	1	j.	j.	PROPN
ejpam-6261	376	2	pure	pure	PROPN
ejpam-6261	376	3	appl	appl	PROPN
ejpam-6261	376	4	.	.	PROPN
ejpam-6261	376	5	math	math	PROPN
ejpam-6261	376	6	,	,	PUNCT
ejpam-6261	376	7	18	18	NUM
ejpam-6261	376	8	(	(	PUNCT
ejpam-6261	376	9	4	4	NUM
ejpam-6261	376	10	)	)	PUNCT
ejpam-6261	376	11	(	(	PUNCT
ejpam-6261	376	12	2025	2025	NUM
ejpam-6261	376	13	)	)	PUNCT
ejpam-6261	376	14	,	,	PUNCT
ejpam-6261	376	15	6261	6261	NUM
ejpam-6261	376	16	16	16	NUM
ejpam-6261	376	17	of	of	ADP
ejpam-6261	376	18	23	23	NUM
ejpam-6261	376	19	definition	definition	NOUN
ejpam-6261	376	20	12	12	NUM
ejpam-6261	376	21	.	.	PUNCT
ejpam-6261	377	1	let	let	VERB
ejpam-6261	377	2	w	w	NOUN
ejpam-6261	377	3	and	and	CCONJ
ejpam-6261	377	4	x	x	PART
ejpam-6261	377	5	be	be	AUX
ejpam-6261	377	6	two	two	NUM
ejpam-6261	377	7	cnss	cns	NOUN
ejpam-6261	377	8	of	of	ADP
ejpam-6261	377	9	set	set	NOUN
ejpam-6261	377	10	p	p	PROPN
ejpam-6261	377	11	.	.	PUNCT
ejpam-6261	378	1	the	the	DET
ejpam-6261	378	2	cartesian	cartesian	ADJ
ejpam-6261	378	3	product	product	NOUN
ejpam-6261	378	4	of	of	ADP
ejpam-6261	378	5	cnsss	cnsss	PROPN
ejpam-6261	378	6	w	w	PROPN
ejpam-6261	378	7	and	and	CCONJ
ejpam-6261	378	8	x	x	VERB
ejpam-6261	378	9	is	be	AUX
ejpam-6261	378	10	expressed	express	VERB
ejpam-6261	378	11	by	by	ADP
ejpam-6261	378	12	a	a	DET
ejpam-6261	378	13	function	function	NOUN
ejpam-6261	378	14	w×	w×	PROPN
ejpam-6261	378	15	x	x	PUNCT
ejpam-6261	379	1	=	=	PUNCT
ejpam-6261	379	2	{	{	PUNCT
ejpam-6261	379	3	<	<	X
ejpam-6261	379	4	(	(	PUNCT
ejpam-6261	379	5	ϑ	ϑ	X
ejpam-6261	379	6	,	,	PUNCT
ejpam-6261	379	7	),tw×x(ϑ	),tw×x(ϑ	NUM
ejpam-6261	379	8	,	,	PUNCT
ejpam-6261	379	9			PROPN
ejpam-6261	379	10	)	)	PUNCT
ejpam-6261	379	11	,	,	PUNCT
ejpam-6261	379	12	iw×x(ϑ	iw×x(ϑ	NOUN
ejpam-6261	379	13	,	,	PUNCT
ejpam-6261	379	14	),fw×x(ϑ	),fw×x(ϑ	NOUN
ejpam-6261	379	15	,	,	PUNCT
ejpam-6261	379	16			NOUN
ejpam-6261	379	17	)	)	PUNCT
ejpam-6261	379	18	>	>	PUNCT
ejpam-6261	379	19	}	}	PUNCT
ejpam-6261	379	20	,	,	PUNCT
ejpam-6261	379	21	tw×x(ϑ	tw×x(ϑ	NOUN
ejpam-6261	379	22	,	,	PUNCT
ejpam-6261	379	23			NOUN
ejpam-6261	379	24	)	)	PUNCT
ejpam-6261	379	25	=	=	SYM
ejpam-6261	379	26	pw×x(ϑ	pw×x(ϑ	PROPN
ejpam-6261	379	27	,	,	PUNCT
ejpam-6261	379	28	)e	)e	NOUN
ejpam-6261	379	29	iθw×x(ϑ,	iθw×x(ϑ,	NOUN
ejpam-6261	379	30	)	)	PUNCT
ejpam-6261	379	31	=	=	SYM
ejpam-6261	379	32	min{pw(ϑ	min{pw(ϑ	PROPN
ejpam-6261	379	33	)	)	PUNCT
ejpam-6261	379	34	,	,	PUNCT
ejpam-6261	379	35	px()}eimin{θw(ϑ),θx(	px()}eimin{θw(ϑ),θx(	NOUN
ejpam-6261	379	36	)	)	PUNCT
ejpam-6261	379	37	}	}	PUNCT
ejpam-6261	379	38	,	,	PUNCT
ejpam-6261	379	39	iw×x(ϑ	iw×x(ϑ	NOUN
ejpam-6261	379	40	,	,	PUNCT
ejpam-6261	379	41			NOUN
ejpam-6261	379	42	)	)	PUNCT
ejpam-6261	379	43	=	=	SYM
ejpam-6261	379	44	qw×x(ϑ	qw×x(ϑ	ADJ
ejpam-6261	379	45	,	,	PUNCT
ejpam-6261	379	46	)e	)e	NOUN
ejpam-6261	379	47	iϕw×x(ϑ,	iϕw×x(ϑ,	NOUN
ejpam-6261	379	48	)	)	PUNCT
ejpam-6261	379	49	=	=	SYM
ejpam-6261	379	50	min{qw(ϑ	min{qw(ϑ	NOUN
ejpam-6261	379	51	)	)	PUNCT
ejpam-6261	379	52	,	,	PUNCT
ejpam-6261	379	53	qx()}eimin{φw(ϑ),φx(	qx()}eimin{φw(ϑ),φx(	NOUN
ejpam-6261	379	54	)	)	PUNCT
ejpam-6261	379	55	}	}	PUNCT
ejpam-6261	379	56	,	,	PUNCT
ejpam-6261	379	57	fw×x(ϑ	fw×x(ϑ	NOUN
ejpam-6261	379	58	,	,	PUNCT
ejpam-6261	379	59			NOUN
ejpam-6261	379	60	)	)	PUNCT
ejpam-6261	379	61	=	=	SYM
ejpam-6261	379	62	rw×x(ϑ	rw×x(ϑ	NOUN
ejpam-6261	379	63	,	,	PUNCT
ejpam-6261	379	64	)e	)e	NOUN
ejpam-6261	379	65	irw×x(ϑ,	irw×x(ϑ,	NOUN
ejpam-6261	379	66	)	)	PUNCT
ejpam-6261	380	1	=	=	SYM
ejpam-6261	380	2	max{rw(ϑ	max{rw(ϑ	PROPN
ejpam-6261	380	3	)	)	PUNCT
ejpam-6261	380	4	,	,	PUNCT
ejpam-6261	380	5	rx()}eimax{ωw(ϑ),ωx(	rx()}eimax{ωw(ϑ),ωx(	NOUN
ejpam-6261	380	6	)	)	PUNCT
ejpam-6261	380	7	}	}	PUNCT
ejpam-6261	380	8	.	.	PUNCT
ejpam-6261	381	1	in	in	ADP
ejpam-6261	381	2	this	this	DET
ejpam-6261	381	3	paper	paper	NOUN
ejpam-6261	381	4	we	we	PRON
ejpam-6261	381	5	shall	shall	AUX
ejpam-6261	381	6	take	take	VERB
ejpam-6261	381	7	tw×x(ϑ	tw×x(ϑ	NOUN
ejpam-6261	381	8	,	,	PUNCT
ejpam-6261	381	9			NOUN
ejpam-6261	381	10	)	)	PUNCT
ejpam-6261	382	1	=	=	SYM
ejpam-6261	382	2	pw×x(ϑ	pw×x(ϑ	PROPN
ejpam-6261	382	3	,	,	PUNCT
ejpam-6261	382	4	)e	)e	NOUN
ejpam-6261	382	5	iθw×x(ϑ,	iθw×x(ϑ,	NOUN
ejpam-6261	382	6	)	)	PUNCT
ejpam-6261	382	7	,	,	PUNCT
ejpam-6261	382	8	iw×x(ϑ	iw×x(ϑ	NOUN
ejpam-6261	382	9	,	,	PUNCT
ejpam-6261	382	10			NOUN
ejpam-6261	382	11	)	)	PUNCT
ejpam-6261	382	12	=	=	SYM
ejpam-6261	382	13	qw×x(ϑ	qw×x(ϑ	ADJ
ejpam-6261	382	14	,	,	PUNCT
ejpam-6261	382	15	)e	)e	NOUN
ejpam-6261	382	16	iφw×x(ϑ,	iφw×x(ϑ,	NOUN
ejpam-6261	382	17	)	)	PUNCT
ejpam-6261	382	18	and	and	CCONJ
ejpam-6261	382	19	fw×x(ϑ	fw×x(ϑ	NOUN
ejpam-6261	382	20	,	,	PUNCT
ejpam-6261	382	21			NOUN
ejpam-6261	382	22	)	)	PUNCT
ejpam-6261	382	23	=	=	SYM
ejpam-6261	382	24	rw×x(ϑ	rw×x(ϑ	NOUN
ejpam-6261	382	25	,	,	PUNCT
ejpam-6261	382	26	)e	)e	NOUN
ejpam-6261	382	27	iωw×x(ϑ,	iωw×x(ϑ,	NOUN
ejpam-6261	382	28	)	)	PUNCT
ejpam-6261	382	29	for	for	ADP
ejpam-6261	382	30	the	the	DET
ejpam-6261	382	31	level	level	NOUN
ejpam-6261	382	32	of	of	ADP
ejpam-6261	382	33	truth	truth	NOUN
ejpam-6261	382	34	,	,	PUNCT
ejpam-6261	382	35	level	level	NOUN
ejpam-6261	382	36	of	of	ADP
ejpam-6261	382	37	neutral	neutral	ADJ
ejpam-6261	382	38	and	and	CCONJ
ejpam-6261	382	39	level	level	NOUN
ejpam-6261	382	40	of	of	ADP
ejpam-6261	382	41	falsehood	falsehood	NOUN
ejpam-6261	382	42	of	of	ADP
ejpam-6261	382	43	w×x	w×x	PROPN
ejpam-6261	382	44	.	.	PUNCT
ejpam-6261	383	1	the	the	DET
ejpam-6261	383	2	upcoming	upcoming	PROPN
ejpam-6261	383	3	theorem	theorem	NOUN
ejpam-6261	383	4	explain	explain	VERB
ejpam-6261	383	5	that	that	SCONJ
ejpam-6261	383	6	the	the	DET
ejpam-6261	383	7	cartesian	cartesian	ADJ
ejpam-6261	383	8	product	product	NOUN
ejpam-6261	383	9	of	of	ADP
ejpam-6261	383	10	two	two	NUM
ejpam-6261	383	11	cnsrs	cnsrs	NOUN
ejpam-6261	383	12	is	be	AUX
ejpam-6261	383	13	cnsr	cnsr	VERB
ejpam-6261	383	14	.	.	PUNCT
ejpam-6261	384	1	theorem	theorem	NOUN
ejpam-6261	384	2	6	6	NUM
ejpam-6261	384	3	.	.	PUNCT
ejpam-6261	385	1	let	let	VERB
ejpam-6261	385	2	w	w	NOUN
ejpam-6261	385	3	and	and	CCONJ
ejpam-6261	385	4	x	x	PART
ejpam-6261	385	5	be	be	AUX
ejpam-6261	385	6	two	two	NUM
ejpam-6261	385	7	cnsrs	cnsrs	NOUN
ejpam-6261	385	8	of	of	ADP
ejpam-6261	385	9	r1	r1	NOUN
ejpam-6261	385	10	and	and	CCONJ
ejpam-6261	385	11	r2	r2	PROPN
ejpam-6261	385	12	,	,	PUNCT
ejpam-6261	385	13	consequently	consequently	ADV
ejpam-6261	385	14	.	.	PUNCT
ejpam-6261	386	1	then	then	ADV
ejpam-6261	386	2	w	w	PROPN
ejpam-6261	386	3	×	×	NOUN
ejpam-6261	386	4	x	x	VERB
ejpam-6261	386	5	is	be	AUX
ejpam-6261	386	6	complex	complex	ADJ
ejpam-6261	386	7	neutrosophic	neutrosophic	ADJ
ejpam-6261	386	8	subrings	subring	NOUN
ejpam-6261	386	9	of	of	ADP
ejpam-6261	386	10	r1	r1	PROPN
ejpam-6261	386	11	×r2	×r2	PROPN
ejpam-6261	386	12	.	.	PUNCT
ejpam-6261	387	1	proof	proof	NOUN
ejpam-6261	387	2	.	.	PUNCT
ejpam-6261	388	1	let	let	VERB
ejpam-6261	388	2	ϑ	ϑ	X
ejpam-6261	388	3	,	,	PUNCT
ejpam-6261	388	4	k	k	PROPN
ejpam-6261	388	5	∈	∈	PROPN
ejpam-6261	388	6	r1	r1	NOUN
ejpam-6261	388	7	and	and	CCONJ
ejpam-6261	388	8			PROPN
ejpam-6261	388	9	,	,	PUNCT
ejpam-6261	388	10	l	l	PROPN
ejpam-6261	388	11	∈	∈	PROPN
ejpam-6261	388	12	r2	r2	NOUN
ejpam-6261	388	13	be	be	VERB
ejpam-6261	388	14	an	an	DET
ejpam-6261	388	15	elements	element	NOUN
ejpam-6261	388	16	.	.	PUNCT
ejpam-6261	389	1	then	then	ADV
ejpam-6261	389	2	(	(	PUNCT
ejpam-6261	389	3	ϑ	ϑ	X
ejpam-6261	389	4	,	,	PUNCT
ejpam-6261	389	5			NOUN
ejpam-6261	389	6	)	)	PUNCT
ejpam-6261	389	7	,	,	PUNCT
ejpam-6261	389	8	(	(	PUNCT
ejpam-6261	389	9	k	k	X
ejpam-6261	389	10	,	,	PUNCT
ejpam-6261	389	11	l	l	NOUN
ejpam-6261	389	12	)	)	PUNCT
ejpam-6261	389	13	∈	∈	PROPN
ejpam-6261	389	14	r1	r1	NOUN
ejpam-6261	389	15	×	×	NOUN
ejpam-6261	389	16	r2	r2	PROPN
ejpam-6261	389	17	.	.	PUNCT
ejpam-6261	390	1	consider	consider	VERB
ejpam-6261	390	2	tw×x((ϑ	tw×x((ϑ	PROPN
ejpam-6261	390	3	,	,	PUNCT
ejpam-6261	390	4	)−	)−	PROPN
ejpam-6261	390	5	(	(	PUNCT
ejpam-6261	390	6	k	k	NOUN
ejpam-6261	390	7	,	,	PUNCT
ejpam-6261	390	8	l	l	NOUN
ejpam-6261	390	9	)	)	PUNCT
ejpam-6261	390	10	)	)	PUNCT
ejpam-6261	391	1	=	=	PUNCT
ejpam-6261	392	1	tw×x(ϑ−	tw×x(ϑ−	NUM
ejpam-6261	392	2	k	k	NOUN
ejpam-6261	392	3	,	,	PUNCT
ejpam-6261	392	4	−	−	ADJ
ejpam-6261	392	5	l	l	NOUN
ejpam-6261	392	6	)	)	PUNCT
ejpam-6261	392	7	=	=	SYM
ejpam-6261	393	1	pw×x(ϑ−	pw×x(ϑ−	NUM
ejpam-6261	393	2	k	k	NOUN
ejpam-6261	393	3	,	,	PUNCT
ejpam-6261	393	4	−	−	ADJ
ejpam-6261	393	5	l)eiθw×x(x−k	l)eiθw×x(x−k	PROPN
ejpam-6261	393	6	,	,	PUNCT
ejpam-6261	393	7	y−l	y−l	X
ejpam-6261	393	8	)	)	PUNCT
ejpam-6261	393	9	=	=	SYM
ejpam-6261	393	10	min{pw(ϑ−	min{pw(ϑ−	NOUN
ejpam-6261	393	11	k	k	NOUN
ejpam-6261	393	12	)	)	PUNCT
ejpam-6261	393	13	,	,	PUNCT
ejpam-6261	393	14	px(−	px(−	ADJ
ejpam-6261	393	15	l	l	NOUN
ejpam-6261	393	16	)	)	PUNCT
ejpam-6261	393	17	}	}	PUNCT
ejpam-6261	393	18	eimin{θw(ϑ−k),θx(−l	eimin{θw(ϑ−k),θx(−l	NOUN
ejpam-6261	393	19	)	)	PUNCT
ejpam-6261	393	20	}	}	PUNCT
ejpam-6261	393	21	=	=	PUNCT
ejpam-6261	393	22	min{pw(ϑ−	min{pw(ϑ−	NOUN
ejpam-6261	393	23	k)eiθw(ϑ−k	k)eiθw(ϑ−k	NOUN
ejpam-6261	393	24	)	)	PUNCT
ejpam-6261	393	25	,	,	PUNCT
ejpam-6261	393	26	px(−	px(−	PRON
ejpam-6261	393	27	l)eiθw(−l	l)eiθw(−l	NOUN
ejpam-6261	393	28	)	)	PUNCT
ejpam-6261	393	29	}	}	PUNCT
ejpam-6261	393	30	=	=	PUNCT
ejpam-6261	394	1	min{tw(ϑ−	min{tw(ϑ−	PUNCT
ejpam-6261	394	2	k),tx(−	k),tx(−	ADJ
ejpam-6261	394	3	l	l	NOUN
ejpam-6261	394	4	)	)	PUNCT
ejpam-6261	394	5	}	}	PUNCT
ejpam-6261	394	6	≥	≥	NOUN
ejpam-6261	394	7	min{min{tw(ϑ),tw(k	min{min{tw(ϑ),tw(k	NOUN
ejpam-6261	394	8	)	)	PUNCT
ejpam-6261	394	9	}	}	PUNCT
ejpam-6261	394	10	,	,	PUNCT
ejpam-6261	394	11	min{tx(),tx(l	min{tx(),tx(l	PROPN
ejpam-6261	394	12	)	)	PUNCT
ejpam-6261	394	13	}	}	PUNCT
ejpam-6261	394	14	}	}	PUNCT
ejpam-6261	394	15	=	=	SYM
ejpam-6261	394	16	min{min{tw(ϑ),tx(	min{min{tw(ϑ),tx(	NOUN
ejpam-6261	394	17	)	)	PUNCT
ejpam-6261	394	18	}	}	PUNCT
ejpam-6261	394	19	,	,	PUNCT
ejpam-6261	394	20	min{tw(k),tx(l	min{tw(k),tx(l	NOUN
ejpam-6261	394	21	)	)	PUNCT
ejpam-6261	394	22	}	}	PUNCT
ejpam-6261	394	23	}	}	PUNCT
ejpam-6261	394	24	≥	≥	PROPN
ejpam-6261	394	25	min{tw×x(ϑ	min{tw×x(ϑ	PROPN
ejpam-6261	394	26	,	,	PUNCT
ejpam-6261	394	27	),tw×x(k	),tw×x(k	NOUN
ejpam-6261	394	28	,	,	PUNCT
ejpam-6261	394	29	l	l	NOUN
ejpam-6261	394	30	)	)	PUNCT
ejpam-6261	394	31	}	}	PUNCT
ejpam-6261	394	32	tw×x((ϑ	tw×x((ϑ	PROPN
ejpam-6261	394	33	,	,	PUNCT
ejpam-6261	394	34	)−	)−	PROPN
ejpam-6261	394	35	(	(	PUNCT
ejpam-6261	394	36	k	k	NOUN
ejpam-6261	394	37	,	,	PUNCT
ejpam-6261	394	38	l	l	NOUN
ejpam-6261	394	39	)	)	PUNCT
ejpam-6261	394	40	)	)	PUNCT
ejpam-6261	394	41	≥	≥	PROPN
ejpam-6261	395	1	min{tw×x(ϑ	min{tw×x(ϑ	PROPN
ejpam-6261	395	2	,	,	PUNCT
ejpam-6261	395	3	),tw×x(k	),tw×x(k	NOUN
ejpam-6261	395	4	,	,	PUNCT
ejpam-6261	395	5	l	l	NOUN
ejpam-6261	395	6	)	)	PUNCT
ejpam-6261	395	7	}	}	PUNCT
ejpam-6261	395	8	.	.	PUNCT
ejpam-6261	396	1	tw×x((ϑ	tw×x((ϑ	NOUN
ejpam-6261	396	2	,	,	PUNCT
ejpam-6261	396	3	)(k	)(k	PROPN
ejpam-6261	396	4	,	,	PUNCT
ejpam-6261	396	5	l	l	NOUN
ejpam-6261	396	6	)	)	PUNCT
ejpam-6261	396	7	)	)	PUNCT
ejpam-6261	397	1	=	=	SYM
ejpam-6261	397	2	tw×x((ϑ	tw×x((ϑ	PROPN
ejpam-6261	397	3	,	,	PUNCT
ejpam-6261	397	4	)(ϑ	)(ϑ	PROPN
ejpam-6261	397	5	,	,	PUNCT
ejpam-6261	397	6	))tw×x((ϑ	))tw×x((ϑ	PROPN
ejpam-6261	397	7	,	,	PUNCT
ejpam-6261	397	8	)(k	)(k	PROPN
ejpam-6261	397	9	,	,	PUNCT
ejpam-6261	397	10	l	l	NOUN
ejpam-6261	397	11	)	)	PUNCT
ejpam-6261	397	12	)	)	PUNCT
ejpam-6261	398	1	=	=	PUNCT
ejpam-6261	398	2	tw×x(ϑk	tw×x(ϑk	NOUN
ejpam-6261	398	3	,	,	PUNCT
ejpam-6261	398	4	l	l	NOUN
ejpam-6261	398	5	)	)	PUNCT
ejpam-6261	399	1	=	=	PUNCT
ejpam-6261	399	2	pw×x(ϑk	pw×x(ϑk	X
ejpam-6261	399	3	,	,	PUNCT
ejpam-6261	399	4	l)e	l)e	PROPN
ejpam-6261	399	5	iθw×x(ϑk,l	iθw×x(ϑk,l	NOUN
ejpam-6261	399	6	)	)	PUNCT
ejpam-6261	400	1	=	=	SYM
ejpam-6261	400	2	min{pw(ϑk	min{pw(ϑk	NOUN
ejpam-6261	400	3	)	)	PUNCT
ejpam-6261	400	4	,	,	PUNCT
ejpam-6261	400	5	px(l	px(l	NOUN
ejpam-6261	400	6	)	)	PUNCT
ejpam-6261	400	7	}	}	PUNCT
ejpam-6261	400	8	eimin{θw(ϑk),θx(l	eimin{θw(ϑk),θx(l	PROPN
ejpam-6261	400	9	)	)	PUNCT
ejpam-6261	400	10	}	}	PUNCT
ejpam-6261	401	1	=	=	SYM
ejpam-6261	401	2	min{pw(ϑk)eiθw(ϑk	min{pw(ϑk)eiθw(ϑk	NOUN
ejpam-6261	401	3	)	)	PUNCT
ejpam-6261	401	4	,	,	PUNCT
ejpam-6261	401	5	px(l)e	px(l)e	VERB
ejpam-6261	401	6	iθw	iθw	NOUN
ejpam-6261	401	7	(	(	PUNCT
ejpam-6261	401	8	l	l	PROPN
ejpam-6261	401	9	)	)	PUNCT
ejpam-6261	401	10	}	}	PUNCT
ejpam-6261	401	11	=	=	SYM
ejpam-6261	401	12	min{tw(ϑk),tx(l	min{tw(ϑk),tx(l	NOUN
ejpam-6261	401	13	)	)	PUNCT
ejpam-6261	401	14	}	}	PUNCT
ejpam-6261	401	15	≥	≥	NOUN
ejpam-6261	401	16	min{min{tw(ϑ),tw(k	min{min{tw(ϑ),tw(k	NOUN
ejpam-6261	401	17	)	)	PUNCT
ejpam-6261	401	18	}	}	PUNCT
ejpam-6261	401	19	,	,	PUNCT
ejpam-6261	401	20	min{tx(),tx(l	min{tx(),tx(l	PROPN
ejpam-6261	401	21	)	)	PUNCT
ejpam-6261	401	22	}	}	PUNCT
ejpam-6261	401	23	}	}	PUNCT
ejpam-6261	401	24	=	=	SYM
ejpam-6261	401	25	min{min{tw(ϑ),tx(	min{min{tw(ϑ),tx(	NOUN
ejpam-6261	401	26	)	)	PUNCT
ejpam-6261	401	27	}	}	PUNCT
ejpam-6261	401	28	,	,	PUNCT
ejpam-6261	401	29	min{tw(k),tx(l	min{tw(k),tx(l	NOUN
ejpam-6261	401	30	)	)	PUNCT
ejpam-6261	401	31	}	}	PUNCT
ejpam-6261	401	32	}	}	PUNCT
ejpam-6261	401	33	≥	≥	PROPN
ejpam-6261	401	34	min{tw×x(ϑ	min{tw×x(ϑ	PROPN
ejpam-6261	401	35	,	,	PUNCT
ejpam-6261	401	36	),tw×x(k	),tw×x(k	NOUN
ejpam-6261	401	37	,	,	PUNCT
ejpam-6261	401	38	l	l	NOUN
ejpam-6261	401	39	)	)	PUNCT
ejpam-6261	401	40	}	}	PUNCT
ejpam-6261	401	41	tw×x((ϑ	tw×x((ϑ	NOUN
ejpam-6261	401	42	,	,	PUNCT
ejpam-6261	401	43	)(k	)(k	PROPN
ejpam-6261	401	44	,	,	PUNCT
ejpam-6261	401	45	l	l	NOUN
ejpam-6261	401	46	)	)	PUNCT
ejpam-6261	401	47	)	)	PUNCT
ejpam-6261	402	1	≥	≥	PROPN
ejpam-6261	402	2	min{tw×x(ϑ	min{tw×x(ϑ	PROPN
ejpam-6261	402	3	,	,	PUNCT
ejpam-6261	402	4	),tw×x(k	),tw×x(k	NOUN
ejpam-6261	402	5	,	,	PUNCT
ejpam-6261	402	6	l	l	NOUN
ejpam-6261	402	7	)	)	PUNCT
ejpam-6261	402	8	}	}	PUNCT
ejpam-6261	402	9	.	.	PUNCT
ejpam-6261	403	1	consider	consider	VERB
ejpam-6261	403	2	iw×x((ϑ	iw×x((ϑ	ADJ
ejpam-6261	403	3	,	,	PUNCT
ejpam-6261	403	4	)−	)−	PROPN
ejpam-6261	403	5	(	(	PUNCT
ejpam-6261	403	6	k	k	NOUN
ejpam-6261	403	7	,	,	PUNCT
ejpam-6261	403	8	l	l	NOUN
ejpam-6261	403	9	)	)	PUNCT
ejpam-6261	403	10	)	)	PUNCT
ejpam-6261	404	1	=	=	SYM
ejpam-6261	404	2	χw×x((ϑ	χw×x((ϑ	PROPN
ejpam-6261	404	3	,	,	PUNCT
ejpam-6261	404	4	)(ϑ	)(ϑ	PROPN
ejpam-6261	404	5	,	,	PUNCT
ejpam-6261	404	6	))iw×x((ϑ	))iw×x((ϑ	PROPN
ejpam-6261	404	7	,	,	PUNCT
ejpam-6261	404	8	)−	)−	PROPN
ejpam-6261	404	9	(	(	PUNCT
ejpam-6261	404	10	k	k	NOUN
ejpam-6261	404	11	,	,	PUNCT
ejpam-6261	404	12	l	l	NOUN
ejpam-6261	404	13	)	)	PUNCT
ejpam-6261	404	14	)	)	PUNCT
ejpam-6261	405	1	=	=	PUNCT
ejpam-6261	406	1	iw×x(ϑ−	iw×x(ϑ−	NOUN
ejpam-6261	406	2	k	k	PROPN
ejpam-6261	406	3	,	,	PUNCT
ejpam-6261	406	4	−	−	ADJ
ejpam-6261	406	5	l	l	NOUN
ejpam-6261	406	6	)	)	PUNCT
ejpam-6261	406	7	=	=	SYM
ejpam-6261	407	1	qw×x(ϑ−	qw×x(ϑ−	NUM
ejpam-6261	407	2	k	k	NOUN
ejpam-6261	407	3	,	,	PUNCT
ejpam-6261	407	4	−	−	ADJ
ejpam-6261	407	5	l)eiφw×x(ϑ−k,−l	l)eiφw×x(ϑ−k,−l	PROPN
ejpam-6261	407	6	)	)	PUNCT
ejpam-6261	407	7	m.	m.	NOUN
ejpam-6261	407	8	h.	h.	PROPN
ejpam-6261	407	9	mateen	mateen	PROPN
ejpam-6261	407	10	et	et	PROPN
ejpam-6261	407	11	al	al	PROPN
ejpam-6261	407	12	.	.	PUNCT
ejpam-6261	407	13	/	/	SYM
ejpam-6261	407	14	eur	eur	PROPN
ejpam-6261	407	15	.	.	PUNCT
ejpam-6261	408	1	j.	j.	PROPN
ejpam-6261	408	2	pure	pure	PROPN
ejpam-6261	408	3	appl	appl	PROPN
ejpam-6261	408	4	.	.	PROPN
ejpam-6261	408	5	math	math	PROPN
ejpam-6261	408	6	,	,	PUNCT
ejpam-6261	408	7	18	18	NUM
ejpam-6261	408	8	(	(	PUNCT
ejpam-6261	408	9	4	4	NUM
ejpam-6261	408	10	)	)	PUNCT
ejpam-6261	408	11	(	(	PUNCT
ejpam-6261	408	12	2025	2025	NUM
ejpam-6261	408	13	)	)	PUNCT
ejpam-6261	408	14	,	,	PUNCT
ejpam-6261	408	15	6261	6261	NUM
ejpam-6261	408	16	17	17	NUM
ejpam-6261	408	17	of	of	ADP
ejpam-6261	408	18	23	23	NUM
ejpam-6261	408	19	=	=	SYM
ejpam-6261	408	20	min{qw(ϑ−	min{qw(ϑ−	PROPN
ejpam-6261	408	21	k	k	PROPN
ejpam-6261	408	22	)	)	PUNCT
ejpam-6261	408	23	,	,	PUNCT
ejpam-6261	408	24	qx(−	qx(−	PROPN
ejpam-6261	408	25	l	l	NOUN
ejpam-6261	408	26	)	)	PUNCT
ejpam-6261	408	27	}	}	PUNCT
ejpam-6261	408	28	eimin{φw(ϑ−k),φx(g−l	eimin{φw(ϑ−k),φx(g−l	NOUN
ejpam-6261	408	29	)	)	PUNCT
ejpam-6261	408	30	}	}	PUNCT
ejpam-6261	408	31	=	=	PUNCT
ejpam-6261	408	32	min{qw(ϑ−	min{qw(ϑ−	ADJ
ejpam-6261	408	33	k)eiφw(ϑ−k	k)eiφw(ϑ−k	NOUN
ejpam-6261	408	34	)	)	PUNCT
ejpam-6261	408	35	,	,	PUNCT
ejpam-6261	408	36	qx(−	qx(−	PROPN
ejpam-6261	408	37	l)eiφw	l)eiφw	PROPN
ejpam-6261	408	38	(	(	PUNCT
ejpam-6261	408	39	−l	−l	NOUN
ejpam-6261	408	40	)	)	PUNCT
ejpam-6261	408	41	}	}	PUNCT
ejpam-6261	408	42	=	=	PUNCT
ejpam-6261	408	43	min{iw(ϑ−	min{iw(ϑ−	NOUN
ejpam-6261	408	44	k	k	NOUN
ejpam-6261	408	45	)	)	PUNCT
ejpam-6261	408	46	,	,	PUNCT
ejpam-6261	408	47	ix(−	ix(−	ADJ
ejpam-6261	408	48	l	l	NOUN
ejpam-6261	408	49	)	)	PUNCT
ejpam-6261	408	50	}	}	PUNCT
ejpam-6261	408	51	≥	≥	NOUN
ejpam-6261	408	52	min{min{iw(ϑ	min{min{iw(ϑ	NOUN
ejpam-6261	408	53	)	)	PUNCT
ejpam-6261	408	54	,	,	PUNCT
ejpam-6261	408	55	iw(k	iw(k	NOUN
ejpam-6261	408	56	)	)	PUNCT
ejpam-6261	408	57	}	}	PUNCT
ejpam-6261	408	58	,	,	PUNCT
ejpam-6261	408	59	min{ix(	min{ix(	NOUN
ejpam-6261	408	60	)	)	PUNCT
ejpam-6261	408	61	,	,	PUNCT
ejpam-6261	408	62	ix(l	ix(l	ADV
ejpam-6261	408	63	)	)	PUNCT
ejpam-6261	408	64	}	}	PUNCT
ejpam-6261	408	65	}	}	PUNCT
ejpam-6261	408	66	=	=	SYM
ejpam-6261	408	67	min{min{iw(ϑ	min{min{iw(ϑ	PROPN
ejpam-6261	408	68	)	)	PUNCT
ejpam-6261	408	69	,	,	PUNCT
ejpam-6261	408	70	ix(	ix(	NOUN
ejpam-6261	408	71	)	)	PUNCT
ejpam-6261	408	72	}	}	PUNCT
ejpam-6261	408	73	,	,	PUNCT
ejpam-6261	408	74	min{iw(k	min{iw(k	PROPN
ejpam-6261	408	75	)	)	PUNCT
ejpam-6261	408	76	,	,	PUNCT
ejpam-6261	408	77	ix(l	ix(l	ADV
ejpam-6261	408	78	)	)	PUNCT
ejpam-6261	408	79	}	}	PUNCT
ejpam-6261	408	80	}	}	PUNCT
ejpam-6261	408	81	≥	≥	PROPN
ejpam-6261	408	82	min{iw×x(ϑ	min{iw×x(ϑ	PROPN
ejpam-6261	408	83	,	,	PUNCT
ejpam-6261	408	84			PROPN
ejpam-6261	408	85	)	)	PUNCT
ejpam-6261	408	86	,	,	PUNCT
ejpam-6261	408	87	iw×x(k	iw×x(k	NOUN
ejpam-6261	408	88	,	,	PUNCT
ejpam-6261	408	89	l	l	NOUN
ejpam-6261	408	90	)	)	PUNCT
ejpam-6261	408	91	}	}	PUNCT
ejpam-6261	408	92	iw×x((ϑ	iw×x((ϑ	PROPN
ejpam-6261	408	93	,	,	PUNCT
ejpam-6261	408	94	)−	)−	PROPN
ejpam-6261	408	95	(	(	PUNCT
ejpam-6261	408	96	k	k	NOUN
ejpam-6261	408	97	,	,	PUNCT
ejpam-6261	408	98	l	l	NOUN
ejpam-6261	408	99	)	)	PUNCT
ejpam-6261	408	100	)	)	PUNCT
ejpam-6261	408	101	≥	≥	PROPN
ejpam-6261	408	102	min{iw×x(ϑ	min{iw×x(ϑ	PROPN
ejpam-6261	408	103	,	,	PUNCT
ejpam-6261	408	104			PROPN
ejpam-6261	408	105	)	)	PUNCT
ejpam-6261	408	106	,	,	PUNCT
ejpam-6261	408	107	iw×x(k	iw×x(k	NOUN
ejpam-6261	408	108	,	,	PUNCT
ejpam-6261	408	109	l	l	NOUN
ejpam-6261	408	110	)	)	PUNCT
ejpam-6261	408	111	}	}	PUNCT
ejpam-6261	408	112	.	.	PUNCT
ejpam-6261	409	1	iw×x	iw×x	PROPN
ejpam-6261	409	2	(	(	PUNCT
ejpam-6261	409	3	(	(	PUNCT
ejpam-6261	409	4	ϑ	ϑ	X
ejpam-6261	409	5	,	,	PUNCT
ejpam-6261	409	6	)(k	)(k	PROPN
ejpam-6261	409	7	,	,	PUNCT
ejpam-6261	409	8	l	l	NOUN
ejpam-6261	409	9	)	)	PUNCT
ejpam-6261	409	10	)	)	PUNCT
ejpam-6261	410	1	=	=	PUNCT
ejpam-6261	410	2	iw×x(ϑk	iw×x(ϑk	ADJ
ejpam-6261	410	3	,	,	PUNCT
ejpam-6261	410	4	l	l	NOUN
ejpam-6261	410	5	)	)	PUNCT
ejpam-6261	411	1	=	=	SYM
ejpam-6261	411	2	qw×x(ϑk	qw×x(ϑk	PROPN
ejpam-6261	411	3	,	,	PUNCT
ejpam-6261	411	4	l)e	l)e	PROPN
ejpam-6261	411	5	iφw×x(ϑk,l	iφw×x(ϑk,l	NOUN
ejpam-6261	411	6	)	)	PUNCT
ejpam-6261	412	1	=	=	SYM
ejpam-6261	412	2	min	min	NOUN
ejpam-6261	412	3	{	{	PUNCT
ejpam-6261	412	4	qw(ϑk	qw(ϑk	PROPN
ejpam-6261	412	5	)	)	PUNCT
ejpam-6261	412	6	,	,	PUNCT
ejpam-6261	412	7	qx(l	qx(l	PROPN
ejpam-6261	412	8	)	)	PUNCT
ejpam-6261	412	9	}	}	PUNCT
ejpam-6261	412	10	eimin{φw(ϑk	eimin{φw(ϑk	NOUN
ejpam-6261	412	11	)	)	PUNCT
ejpam-6261	412	12	,	,	PUNCT
ejpam-6261	412	13	φx(l	φx(l	NOUN
ejpam-6261	412	14	)	)	PUNCT
ejpam-6261	412	15	}	}	PUNCT
ejpam-6261	412	16	=	=	SYM
ejpam-6261	412	17	min	min	NOUN
ejpam-6261	412	18	{	{	PUNCT
ejpam-6261	412	19	qw(ϑk)eiφw(ϑk	qw(ϑk)eiφw(ϑk	PROPN
ejpam-6261	412	20	)	)	PUNCT
ejpam-6261	412	21	,	,	PUNCT
ejpam-6261	412	22	qx(l)e	qx(l)e	X
ejpam-6261	412	23	iφx(l	iφx(l	NOUN
ejpam-6261	412	24	)	)	PUNCT
ejpam-6261	412	25	}	}	PUNCT
ejpam-6261	412	26	=	=	SYM
ejpam-6261	412	27	min	min	NOUN
ejpam-6261	412	28	{	{	PUNCT
ejpam-6261	412	29	iw(ϑk	iw(ϑk	PROPN
ejpam-6261	412	30	)	)	PUNCT
ejpam-6261	412	31	,	,	PUNCT
ejpam-6261	412	32	ix(l	ix(l	NOUN
ejpam-6261	412	33	)	)	PUNCT
ejpam-6261	412	34	}	}	PUNCT
ejpam-6261	412	35	≥	≥	PROPN
ejpam-6261	412	36	min	min	PROPN
ejpam-6261	412	37	{	{	PUNCT
ejpam-6261	412	38	min{iw(ϑ	min{iw(ϑ	PROPN
ejpam-6261	412	39	)	)	PUNCT
ejpam-6261	412	40	,	,	PUNCT
ejpam-6261	412	41	iw(k)},min{ix(	iw(k)},min{ix(	X
ejpam-6261	412	42	)	)	PUNCT
ejpam-6261	412	43	,	,	PUNCT
ejpam-6261	412	44	ix(l	ix(l	ADV
ejpam-6261	412	45	)	)	PUNCT
ejpam-6261	412	46	}	}	PUNCT
ejpam-6261	412	47	}	}	PUNCT
ejpam-6261	412	48	≥	≥	PROPN
ejpam-6261	412	49	min	min	PROPN
ejpam-6261	412	50	{	{	PUNCT
ejpam-6261	412	51	iw×x(ϑ	iw×x(ϑ	NOUN
ejpam-6261	412	52	,	,	PUNCT
ejpam-6261	412	53			PROPN
ejpam-6261	412	54	)	)	PUNCT
ejpam-6261	412	55	,	,	PUNCT
ejpam-6261	412	56	iw×x(k	iw×x(k	NOUN
ejpam-6261	412	57	,	,	PUNCT
ejpam-6261	412	58	l	l	NOUN
ejpam-6261	412	59	)	)	PUNCT
ejpam-6261	412	60	}	}	PUNCT
ejpam-6261	412	61	.	.	PUNCT
ejpam-6261	413	1	assume	assume	VERB
ejpam-6261	413	2	that	that	SCONJ
ejpam-6261	413	3	,	,	PUNCT
ejpam-6261	413	4	fw×x((ϑ	fw×x((ϑ	PROPN
ejpam-6261	413	5	,	,	PUNCT
ejpam-6261	413	6	)−	)−	NOUN
ejpam-6261	413	7	(	(	PUNCT
ejpam-6261	413	8	k	k	NOUN
ejpam-6261	413	9	,	,	PUNCT
ejpam-6261	413	10	l	l	NOUN
ejpam-6261	413	11	)	)	PUNCT
ejpam-6261	413	12	)	)	PUNCT
ejpam-6261	414	1	=	=	PUNCT
ejpam-6261	415	1	fw×x(ϑ−	fw×x(ϑ−	NUM
ejpam-6261	415	2	k	k	NOUN
ejpam-6261	415	3	,	,	PUNCT
ejpam-6261	415	4	−	−	ADJ
ejpam-6261	415	5	l	l	NOUN
ejpam-6261	415	6	)	)	PUNCT
ejpam-6261	415	7	=	=	SYM
ejpam-6261	415	8	rw×x(ϑ−	rw×x(ϑ−	PROPN
ejpam-6261	415	9	k	k	NOUN
ejpam-6261	415	10	,	,	PUNCT
ejpam-6261	415	11	−	−	ADJ
ejpam-6261	415	12	l)eiωw×x(ϑ−k,−l	l)eiωw×x(ϑ−k,−l	NOUN
ejpam-6261	415	13	)	)	PUNCT
ejpam-6261	415	14	=	=	PUNCT
ejpam-6261	416	1	max{rw(ϑ−	max{rw(ϑ−	X
ejpam-6261	416	2	k	k	NOUN
ejpam-6261	416	3	)	)	PUNCT
ejpam-6261	416	4	,	,	PUNCT
ejpam-6261	416	5	rx(−	rx(−	PROPN
ejpam-6261	416	6	l	l	NOUN
ejpam-6261	416	7	)	)	PUNCT
ejpam-6261	416	8	}	}	PUNCT
ejpam-6261	416	9	eimax{ωw(ϑ−k),ωx(−l	eimax{ωw(ϑ−k),ωx(−l	NOUN
ejpam-6261	416	10	)	)	PUNCT
ejpam-6261	416	11	}	}	PUNCT
ejpam-6261	417	1	=	=	PUNCT
ejpam-6261	417	2	max{rw(ϑ−	max{rw(ϑ−	PRON
ejpam-6261	417	3	k)eiωw(ϑ−k	k)eiωw(ϑ−k	NOUN
ejpam-6261	417	4	)	)	PUNCT
ejpam-6261	417	5	,	,	PUNCT
ejpam-6261	417	6	rx(−	rx(−	PROPN
ejpam-6261	417	7	l)eiωw(−l	l)eiωw(−l	NOUN
ejpam-6261	417	8	)	)	PUNCT
ejpam-6261	417	9	}	}	PUNCT
ejpam-6261	417	10	=	=	PUNCT
ejpam-6261	418	1	max{fw(ϑ−	max{fw(ϑ−	NOUN
ejpam-6261	418	2	k),fx(−	k),fx(−	PROPN
ejpam-6261	418	3	l	l	NOUN
ejpam-6261	418	4	)	)	PUNCT
ejpam-6261	418	5	}	}	PUNCT
ejpam-6261	418	6	≤	≤	ADV
ejpam-6261	418	7	max{max{fw(ϑ),fw(k	max{max{fw(ϑ),fw(k	NOUN
ejpam-6261	418	8	)	)	PUNCT
ejpam-6261	418	9	}	}	PUNCT
ejpam-6261	418	10	,	,	PUNCT
ejpam-6261	418	11	max{fx(),fx(l	max{fx(),fx(l	PROPN
ejpam-6261	418	12	)	)	PUNCT
ejpam-6261	418	13	}	}	PUNCT
ejpam-6261	418	14	}	}	PUNCT
ejpam-6261	419	1	=	=	SYM
ejpam-6261	419	2	max{max{fw(ϑ),fx(	max{max{fw(ϑ),fx(	X
ejpam-6261	419	3	)	)	PUNCT
ejpam-6261	419	4	}	}	PUNCT
ejpam-6261	419	5	,	,	PUNCT
ejpam-6261	419	6	max{fw(k),fx(l	max{fw(k),fx(l	NOUN
ejpam-6261	419	7	)	)	PUNCT
ejpam-6261	419	8	}	}	PUNCT
ejpam-6261	419	9	}	}	PUNCT
ejpam-6261	419	10	≤	≤	PROPN
ejpam-6261	419	11	max{fw×x(ϑ	max{fw×x(ϑ	PROPN
ejpam-6261	419	12	,	,	PUNCT
ejpam-6261	419	13	),fw×x(k	),fw×x(k	NOUN
ejpam-6261	419	14	,	,	PUNCT
ejpam-6261	419	15	l	l	NOUN
ejpam-6261	419	16	)	)	PUNCT
ejpam-6261	419	17	}	}	PUNCT
ejpam-6261	419	18	fw×x((ϑ	fw×x((ϑ	PROPN
ejpam-6261	419	19	,	,	PUNCT
ejpam-6261	419	20	)−	)−	NOUN
ejpam-6261	419	21	(	(	PUNCT
ejpam-6261	419	22	k	k	NOUN
ejpam-6261	419	23	,	,	PUNCT
ejpam-6261	419	24	l	l	NOUN
ejpam-6261	419	25	)	)	PUNCT
ejpam-6261	419	26	)	)	PUNCT
ejpam-6261	420	1	≤	≤	PROPN
ejpam-6261	420	2	max{fw×x(ϑ	max{fw×x(ϑ	PROPN
ejpam-6261	420	3	,	,	PUNCT
ejpam-6261	420	4	),fw×x(k	),fw×x(k	NOUN
ejpam-6261	420	5	,	,	PUNCT
ejpam-6261	420	6	l	l	NOUN
ejpam-6261	420	7	)	)	PUNCT
ejpam-6261	420	8	}	}	PUNCT
ejpam-6261	420	9	.	.	PUNCT
ejpam-6261	421	1	now	now	ADV
ejpam-6261	421	2	,	,	PUNCT
ejpam-6261	421	3	we	we	PRON
ejpam-6261	421	4	take	take	VERB
ejpam-6261	421	5	,	,	PUNCT
ejpam-6261	421	6	fw×x((ϑ	fw×x((ϑ	PROPN
ejpam-6261	421	7	,	,	PUNCT
ejpam-6261	421	8	)(k	)(k	PROPN
ejpam-6261	421	9	,	,	PUNCT
ejpam-6261	421	10	l	l	NOUN
ejpam-6261	421	11	)	)	PUNCT
ejpam-6261	421	12	)	)	PUNCT
ejpam-6261	422	1	=	=	PUNCT
ejpam-6261	423	1	fw×x(ϑk	fw×x(ϑk	ADJ
ejpam-6261	423	2	,	,	PUNCT
ejpam-6261	423	3	l	l	NOUN
ejpam-6261	423	4	)	)	PUNCT
ejpam-6261	424	1	=	=	SYM
ejpam-6261	424	2	rw×x(ϑk	rw×x(ϑk	PROPN
ejpam-6261	424	3	,	,	PUNCT
ejpam-6261	424	4	l)e	l)e	PROPN
ejpam-6261	424	5	iωw×x(ϑk,l	iωw×x(ϑk,l	NOUN
ejpam-6261	424	6	)	)	PUNCT
ejpam-6261	425	1	=	=	PUNCT
ejpam-6261	425	2	max{rw(ϑk	max{rw(ϑk	NOUN
ejpam-6261	425	3	)	)	PUNCT
ejpam-6261	425	4	,	,	PUNCT
ejpam-6261	425	5	rx(l	rx(l	NOUN
ejpam-6261	425	6	)	)	PUNCT
ejpam-6261	425	7	}	}	PUNCT
ejpam-6261	425	8	∗	∗	NOUN
ejpam-6261	425	9	eimax{ωw(ϑk),ωx(l	eimax{ωw(ϑk),ωx(l	PROPN
ejpam-6261	425	10	)	)	PUNCT
ejpam-6261	425	11	}	}	PUNCT
ejpam-6261	425	12	=	=	SYM
ejpam-6261	425	13	max{rw(ϑk)eiωw(ϑk	max{rw(ϑk)eiωw(ϑk	X
ejpam-6261	425	14	)	)	PUNCT
ejpam-6261	425	15	,	,	PUNCT
ejpam-6261	425	16	rx(l)e	rx(l)e	X
ejpam-6261	425	17	iωw(l	iωw(l	NOUN
ejpam-6261	425	18	)	)	PUNCT
ejpam-6261	425	19	}	}	PUNCT
ejpam-6261	425	20	=	=	SYM
ejpam-6261	425	21	max{fw(ϑk),fx(l	max{fw(ϑk),fx(l	NOUN
ejpam-6261	425	22	)	)	PUNCT
ejpam-6261	425	23	}	}	PUNCT
ejpam-6261	425	24	≤	≤	ADV
ejpam-6261	425	25	max{max{fw(ϑ),fw(k	max{max{fw(ϑ),fw(k	NOUN
ejpam-6261	425	26	)	)	PUNCT
ejpam-6261	425	27	}	}	PUNCT
ejpam-6261	425	28	,	,	PUNCT
ejpam-6261	425	29	max{fx(),fx(l	max{fx(),fx(l	PROPN
ejpam-6261	425	30	)	)	PUNCT
ejpam-6261	425	31	}	}	PUNCT
ejpam-6261	425	32	}	}	PUNCT
ejpam-6261	425	33	=	=	SYM
ejpam-6261	425	34	max{max{fw(ϑ),fx(	max{max{fw(ϑ),fx(	X
ejpam-6261	425	35	)	)	PUNCT
ejpam-6261	425	36	}	}	PUNCT
ejpam-6261	425	37	,	,	PUNCT
ejpam-6261	425	38	max{fw(k),fx(l	max{fw(k),fx(l	NOUN
ejpam-6261	425	39	)	)	PUNCT
ejpam-6261	425	40	}	}	PUNCT
ejpam-6261	425	41	}	}	PUNCT
ejpam-6261	425	42	≤	≤	PROPN
ejpam-6261	425	43	max{fw×x(ϑ	max{fw×x(ϑ	PROPN
ejpam-6261	425	44	,	,	PUNCT
ejpam-6261	425	45	),fw×x(k	),fw×x(k	NOUN
ejpam-6261	425	46	,	,	PUNCT
ejpam-6261	425	47	l	l	NOUN
ejpam-6261	425	48	)	)	PUNCT
ejpam-6261	425	49	}	}	PUNCT
ejpam-6261	425	50	fw×x((ϑ	fw×x((ϑ	PROPN
ejpam-6261	425	51	,	,	PUNCT
ejpam-6261	425	52	)(k	)(k	PROPN
ejpam-6261	425	53	,	,	PUNCT
ejpam-6261	425	54	l	l	NOUN
ejpam-6261	425	55	)	)	PUNCT
ejpam-6261	425	56	)	)	PUNCT
ejpam-6261	425	57	≤	≤	PROPN
ejpam-6261	426	1	max{fw×x(ϑ	max{fw×x(ϑ	PROPN
ejpam-6261	426	2	,	,	PUNCT
ejpam-6261	426	3	),fw×x(k	),fw×x(k	NOUN
ejpam-6261	426	4	,	,	PUNCT
ejpam-6261	426	5	l	l	NOUN
ejpam-6261	426	6	)	)	PUNCT
ejpam-6261	426	7	}	}	PUNCT
ejpam-6261	426	8	.	.	PUNCT
ejpam-6261	427	1	hence	hence	ADV
ejpam-6261	427	2	the	the	DET
ejpam-6261	427	3	desired	desire	VERB
ejpam-6261	427	4	result	result	NOUN
ejpam-6261	427	5	is	be	AUX
ejpam-6261	427	6	obtained	obtain	VERB
ejpam-6261	427	7	.	.	PUNCT
ejpam-6261	428	1	m.	m.	PROPN
ejpam-6261	428	2	h.	h.	PROPN
ejpam-6261	428	3	mateen	mateen	PROPN
ejpam-6261	428	4	et	et	PROPN
ejpam-6261	428	5	al	al	PROPN
ejpam-6261	428	6	.	.	PUNCT
ejpam-6261	428	7	/	/	SYM
ejpam-6261	428	8	eur	eur	PROPN
ejpam-6261	428	9	.	.	PUNCT
ejpam-6261	429	1	j.	j.	PROPN
ejpam-6261	429	2	pure	pure	PROPN
ejpam-6261	429	3	appl	appl	PROPN
ejpam-6261	429	4	.	.	PROPN
ejpam-6261	429	5	math	math	PROPN
ejpam-6261	429	6	,	,	PUNCT
ejpam-6261	429	7	18	18	NUM
ejpam-6261	429	8	(	(	PUNCT
ejpam-6261	429	9	4	4	NUM
ejpam-6261	429	10	)	)	PUNCT
ejpam-6261	429	11	(	(	PUNCT
ejpam-6261	429	12	2025	2025	NUM
ejpam-6261	429	13	)	)	PUNCT
ejpam-6261	429	14	,	,	PUNCT
ejpam-6261	429	15	6261	6261	NUM
ejpam-6261	429	16	18	18	NUM
ejpam-6261	429	17	of	of	ADP
ejpam-6261	429	18	23	23	NUM
ejpam-6261	429	19	corollary	corollary	ADJ
ejpam-6261	429	20	1	1	NUM
ejpam-6261	429	21	.	.	PUNCT
ejpam-6261	430	1	let	let	VERB
ejpam-6261	430	2	w1	w1	NOUN
ejpam-6261	430	3	,	,	PUNCT
ejpam-6261	430	4	w2	w2	NOUN
ejpam-6261	430	5	,	,	PUNCT
ejpam-6261	430	6	.	.	PUNCT
ejpam-6261	430	7	.	.	PUNCT
ejpam-6261	431	1	.	.	PUNCT
ejpam-6261	432	1	,	,	PUNCT
ejpam-6261	432	2	wn	wn	PROPN
ejpam-6261	432	3	be	be	AUX
ejpam-6261	432	4	cnsrs	cnsr	VERB
ejpam-6261	432	5	of	of	ADP
ejpam-6261	432	6	k1	k1	NOUN
ejpam-6261	432	7	,	,	PUNCT
ejpam-6261	432	8	k2	k2	NOUN
ejpam-6261	432	9	,	,	PUNCT
ejpam-6261	432	10	.	.	PUNCT
ejpam-6261	432	11	.	.	PUNCT
ejpam-6261	432	12	.	.	PUNCT
ejpam-6261	433	1	,	,	PUNCT
ejpam-6261	433	2	kn	kn	PROPN
ejpam-6261	433	3	,	,	PUNCT
ejpam-6261	433	4	respectively	respectively	ADV
ejpam-6261	433	5	.	.	PUNCT
ejpam-6261	434	1	then	then	ADV
ejpam-6261	434	2	w1	w1	PROPN
ejpam-6261	434	3	×w2×	×w2×	NUM
ejpam-6261	434	4	,	,	PUNCT
ejpam-6261	434	5	.	.	PUNCT
ejpam-6261	434	6	.	.	PUNCT
ejpam-6261	435	1	.	.	PUNCT
ejpam-6261	436	1	,	,	PUNCT
ejpam-6261	436	2	×wn	×wn	PROPN
ejpam-6261	436	3	is	be	AUX
ejpam-6261	436	4	cnsr	cnsr	VERB
ejpam-6261	436	5	of	of	ADP
ejpam-6261	436	6	k1	k1	PROPN
ejpam-6261	436	7	×k2	×k2	PROPN
ejpam-6261	436	8	×	×	NOUN
ejpam-6261	436	9	.	.	PUNCT
ejpam-6261	436	10	.	.	PUNCT
ejpam-6261	436	11	.	.	PUNCT
ejpam-6261	437	1	×kn	×kn	NOUN
ejpam-6261	437	2	.	.	PUNCT
ejpam-6261	438	1	remark	remark	PROPN
ejpam-6261	438	2	4	4	NUM
ejpam-6261	438	3	.	.	PUNCT
ejpam-6261	439	1	let	let	VERB
ejpam-6261	439	2	w	w	NOUN
ejpam-6261	439	3	and	and	CCONJ
ejpam-6261	439	4	x	x	PART
ejpam-6261	439	5	be	be	AUX
ejpam-6261	439	6	two	two	NUM
ejpam-6261	439	7	cnsrs	cnsrs	NOUN
ejpam-6261	439	8	of	of	ADP
ejpam-6261	439	9	k1	k1	NOUN
ejpam-6261	439	10	and	and	CCONJ
ejpam-6261	439	11	k2	k2	NOUN
ejpam-6261	439	12	,	,	PUNCT
ejpam-6261	439	13	consequently	consequently	ADV
ejpam-6261	439	14	and	and	CCONJ
ejpam-6261	439	15	w1	w1	NOUN
ejpam-6261	439	16	×w2	×w2	NOUN
ejpam-6261	439	17	be	be	AUX
ejpam-6261	439	18	cnsr	cnsr	VERB
ejpam-6261	439	19	of	of	ADP
ejpam-6261	439	20	k1	k1	PROPN
ejpam-6261	439	21	×k2	×k2	PROPN
ejpam-6261	439	22	.	.	PUNCT
ejpam-6261	440	1	then	then	ADV
ejpam-6261	440	2	it	it	PRON
ejpam-6261	440	3	is	be	AUX
ejpam-6261	440	4	not	not	PART
ejpam-6261	440	5	compulsory	compulsory	ADJ
ejpam-6261	440	6	both	both	CCONJ
ejpam-6261	440	7	w1	w1	NOUN
ejpam-6261	440	8	and	and	CCONJ
ejpam-6261	440	9	w2	w2	NOUN
ejpam-6261	440	10	should	should	AUX
ejpam-6261	440	11	be	be	AUX
ejpam-6261	440	12	cnsrs	cnsr	VERB
ejpam-6261	440	13	of	of	ADP
ejpam-6261	440	14	k1	k1	NOUN
ejpam-6261	440	15	and	and	CCONJ
ejpam-6261	440	16	k2	k2	NOUN
ejpam-6261	440	17	,	,	PUNCT
ejpam-6261	440	18	consequently	consequently	ADV
ejpam-6261	440	19	.	.	PUNCT
ejpam-6261	441	1	remark	remark	PROPN
ejpam-6261	441	2	5	5	NUM
ejpam-6261	441	3	.	.	PUNCT
ejpam-6261	442	1	let	let	VERB
ejpam-6261	442	2	w	w	PRON
ejpam-6261	442	3	×	×	VERB
ejpam-6261	442	4	x	x	VERB
ejpam-6261	442	5	be	be	AUX
ejpam-6261	442	6	cnsr	cnsr	VERB
ejpam-6261	442	7	of	of	ADP
ejpam-6261	442	8	ring	ring	NOUN
ejpam-6261	442	9	k1	k1	PROPN
ejpam-6261	442	10	×	×	PROPN
ejpam-6261	442	11	k2	k2	PROPN
ejpam-6261	442	12	.	.	PUNCT
ejpam-6261	443	1	then	then	ADV
ejpam-6261	443	2	pw×x(0	pw×x(0	PROPN
ejpam-6261	443	3	,	,	PUNCT
ejpam-6261	443	4	0	0	NUM
ejpam-6261	443	5	′	′	NUM
ejpam-6261	443	6	)	)	PUNCT
ejpam-6261	443	7	≥	≥	NOUN
ejpam-6261	443	8	pw×x(f	pw×x(f	NOUN
ejpam-6261	443	9	,	,	PUNCT
ejpam-6261	443	10	g	g	NOUN
ejpam-6261	443	11	)	)	PUNCT
ejpam-6261	443	12	,	,	PUNCT
ejpam-6261	443	13	θw×x(0	θw×x(0	PROPN
ejpam-6261	443	14	,	,	PUNCT
ejpam-6261	443	15	0	0	NUM
ejpam-6261	443	16	′	′	NUM
ejpam-6261	443	17	)	)	PUNCT
ejpam-6261	443	18	≥	≥	NOUN
ejpam-6261	443	19	θw×x(f	θw×x(f	ADV
ejpam-6261	443	20	,	,	PUNCT
ejpam-6261	443	21	g	g	NOUN
ejpam-6261	443	22	)	)	PUNCT
ejpam-6261	443	23	,	,	PUNCT
ejpam-6261	443	24	qw×x(0	qw×x(0	PROPN
ejpam-6261	443	25	,	,	PUNCT
ejpam-6261	443	26	0	0	NUM
ejpam-6261	443	27	′	′	NUM
ejpam-6261	443	28	)	)	PUNCT
ejpam-6261	443	29	≥	≥	NOUN
ejpam-6261	443	30	qw×x(f	qw×x(f	NOUN
ejpam-6261	443	31	,	,	PUNCT
ejpam-6261	443	32	g	g	NOUN
ejpam-6261	443	33	)	)	PUNCT
ejpam-6261	443	34	,	,	PUNCT
ejpam-6261	443	35	φw×x(0	φw×x(0	PROPN
ejpam-6261	443	36	,	,	PUNCT
ejpam-6261	443	37	0	0	NUM
ejpam-6261	443	38	′	′	NUM
ejpam-6261	443	39	)	)	PUNCT
ejpam-6261	443	40	≥	≥	NOUN
ejpam-6261	443	41	φw×x(f	φw×x(f	NOUN
ejpam-6261	443	42	,	,	PUNCT
ejpam-6261	443	43	g	g	NOUN
ejpam-6261	443	44	)	)	PUNCT
ejpam-6261	443	45	,	,	PUNCT
ejpam-6261	443	46	rw×x(0	rw×x(0	PROPN
ejpam-6261	443	47	,	,	PUNCT
ejpam-6261	443	48	0	0	NUM
ejpam-6261	443	49	′	′	NUM
ejpam-6261	443	50	)	)	PUNCT
ejpam-6261	444	1	≤	≤	NUM
ejpam-6261	444	2	rw×x(f	rw×x(f	NOUN
ejpam-6261	444	3	,	,	PUNCT
ejpam-6261	444	4	g	g	NOUN
ejpam-6261	444	5	)	)	PUNCT
ejpam-6261	444	6	,	,	PUNCT
ejpam-6261	444	7	and	and	CCONJ
ejpam-6261	444	8	ωw×x(0	ωw×x(0	NUM
ejpam-6261	444	9	,	,	PUNCT
ejpam-6261	444	10	0	0	NUM
ejpam-6261	444	11	′	′	NUM
ejpam-6261	444	12	)	)	PUNCT
ejpam-6261	444	13	≤	≤	NOUN
ejpam-6261	444	14	ωw×x(f	ωw×x(f	NOUN
ejpam-6261	444	15	,	,	PUNCT
ejpam-6261	444	16	g	g	NOUN
ejpam-6261	444	17	)	)	PUNCT
ejpam-6261	444	18	.	.	PUNCT
ejpam-6261	445	1	∀f	∀f	PROPN
ejpam-6261	445	2	∈	∈	PROPN
ejpam-6261	445	3	k1	k1	NOUN
ejpam-6261	445	4	,	,	PUNCT
ejpam-6261	445	5	g	g	PROPN
ejpam-6261	445	6	∈	∈	PROPN
ejpam-6261	445	7	k2	k2	NOUN
ejpam-6261	445	8	.	.	PUNCT
ejpam-6261	446	1	here	here	ADV
ejpam-6261	446	2	0	0	NUM
ejpam-6261	446	3	and	and	CCONJ
ejpam-6261	446	4	0	0	NUM
ejpam-6261	446	5	′	′	NUM
ejpam-6261	446	6	are	be	AUX
ejpam-6261	446	7	neutral	neutral	ADJ
ejpam-6261	446	8	elements	element	NOUN
ejpam-6261	446	9	of	of	ADP
ejpam-6261	446	10	k1	k1	NOUN
ejpam-6261	446	11	and	and	CCONJ
ejpam-6261	446	12	k2	k2	NOUN
ejpam-6261	446	13	,	,	PUNCT
ejpam-6261	446	14	respectively	respectively	ADV
ejpam-6261	446	15	.	.	PUNCT
ejpam-6261	447	1	theorem	theorem	VERB
ejpam-6261	447	2	7	7	NUM
ejpam-6261	447	3	.	.	PUNCT
ejpam-6261	448	1	let	let	VERB
ejpam-6261	448	2	w	w	NOUN
ejpam-6261	448	3	and	and	CCONJ
ejpam-6261	448	4	x	x	PART
ejpam-6261	448	5	be	be	AUX
ejpam-6261	448	6	two	two	NUM
ejpam-6261	448	7	cns	cns	NOUN
ejpam-6261	448	8	of	of	ADP
ejpam-6261	448	9	rings	ring	NOUN
ejpam-6261	448	10	k1	k1	NOUN
ejpam-6261	448	11	and	and	CCONJ
ejpam-6261	448	12	k2	k2	NOUN
ejpam-6261	448	13	.	.	PUNCT
ejpam-6261	449	1	if	if	SCONJ
ejpam-6261	449	2	w	w	NOUN
ejpam-6261	449	3	×	×	NOUN
ejpam-6261	449	4	x	x	VERB
ejpam-6261	449	5	is	be	AUX
ejpam-6261	449	6	a	a	DET
ejpam-6261	449	7	complex	complex	ADJ
ejpam-6261	449	8	neutrosophic	neutrosophic	ADJ
ejpam-6261	449	9	subring	subring	NOUN
ejpam-6261	449	10	of	of	ADP
ejpam-6261	449	11	k1	k1	PROPN
ejpam-6261	449	12	×k2	×k2	PROPN
ejpam-6261	449	13	,	,	PUNCT
ejpam-6261	449	14	then	then	ADV
ejpam-6261	449	15	one	one	NUM
ejpam-6261	449	16	of	of	ADP
ejpam-6261	449	17	the	the	DET
ejpam-6261	449	18	following	following	ADJ
ejpam-6261	449	19	statements	statement	NOUN
ejpam-6261	449	20	must	must	AUX
ejpam-6261	449	21	be	be	AUX
ejpam-6261	449	22	satisfied	satisfied	ADJ
ejpam-6261	449	23	.	.	PUNCT
ejpam-6261	450	1	(	(	PUNCT
ejpam-6261	450	2	i	i	NOUN
ejpam-6261	450	3	)	)	PUNCT
ejpam-6261	450	4	pw(0	pw(0	NOUN
ejpam-6261	450	5	)	)	PUNCT
ejpam-6261	450	6	≥	≥	NOUN
ejpam-6261	450	7	px(g	px(g	NOUN
ejpam-6261	450	8	)	)	PUNCT
ejpam-6261	450	9	,	,	PUNCT
ejpam-6261	450	10	θw(0	θw(0	PROPN
ejpam-6261	450	11	)	)	PUNCT
ejpam-6261	450	12	≥	≥	NOUN
ejpam-6261	450	13	θx(g	θx(g	NOUN
ejpam-6261	450	14	)	)	PUNCT
ejpam-6261	450	15	,	,	PUNCT
ejpam-6261	450	16	qw(0	qw(0	PROPN
ejpam-6261	450	17	)	)	PUNCT
ejpam-6261	450	18	≥	≥	NOUN
ejpam-6261	450	19	qx(g	qx(g	NOUN
ejpam-6261	450	20	)	)	PUNCT
ejpam-6261	450	21	,	,	PUNCT
ejpam-6261	450	22	φw(0	φw(0	PROPN
ejpam-6261	450	23	)	)	PUNCT
ejpam-6261	450	24	≥	≥	NOUN
ejpam-6261	450	25	φx(g	φx(g	NOUN
ejpam-6261	450	26	)	)	PUNCT
ejpam-6261	450	27	and	and	CCONJ
ejpam-6261	450	28	rw(0	rw(0	NOUN
ejpam-6261	450	29	)	)	PUNCT
ejpam-6261	450	30	≤	≤	NOUN
ejpam-6261	450	31	rx(g	rx(g	NOUN
ejpam-6261	450	32	)	)	PUNCT
ejpam-6261	450	33	,	,	PUNCT
ejpam-6261	450	34	ωw(0	ωw(0	NOUN
ejpam-6261	450	35	)	)	PUNCT
ejpam-6261	450	36	≤	≤	NOUN
ejpam-6261	450	37	ωx(g	ωx(g	NUM
ejpam-6261	450	38	)	)	PUNCT
ejpam-6261	450	39	,	,	PUNCT
ejpam-6261	450	40	∀g	∀g	X
ejpam-6261	450	41	∈	∈	PROPN
ejpam-6261	450	42	k2	k2	PROPN
ejpam-6261	450	43	.	.	PUNCT
ejpam-6261	451	1	(	(	PUNCT
ejpam-6261	451	2	ii	ii	X
ejpam-6261	451	3	)	)	PUNCT
ejpam-6261	452	1	px(0	px(0	PROPN
ejpam-6261	452	2	′	′	PROPN
ejpam-6261	452	3	)	)	PUNCT
ejpam-6261	452	4	≥	≥	NOUN
ejpam-6261	452	5	pw(f	pw(f	NUM
ejpam-6261	452	6	)	)	PUNCT
ejpam-6261	452	7	,	,	PUNCT
ejpam-6261	452	8	θx(0	θx(0	PROPN
ejpam-6261	452	9	′	′	NUM
ejpam-6261	452	10	)	)	PUNCT
ejpam-6261	452	11	≥	≥	NOUN
ejpam-6261	452	12	θw(f	θw(f	NUM
ejpam-6261	452	13	)	)	PUNCT
ejpam-6261	452	14	,	,	PUNCT
ejpam-6261	452	15	qx(0	qx(0	PROPN
ejpam-6261	452	16	′	′	NUM
ejpam-6261	452	17	)	)	PUNCT
ejpam-6261	452	18	≥	≥	NOUN
ejpam-6261	452	19	qw(f	qw(f	NUM
ejpam-6261	452	20	)	)	PUNCT
ejpam-6261	452	21	,	,	PUNCT
ejpam-6261	452	22	φx(0	φx(0	PROPN
ejpam-6261	452	23	′	′	NUM
ejpam-6261	452	24	)	)	PUNCT
ejpam-6261	452	25	≥	≥	NOUN
ejpam-6261	452	26	φw(f	φw(f	NOUN
ejpam-6261	452	27	)	)	PUNCT
ejpam-6261	452	28	,	,	PUNCT
ejpam-6261	452	29	rx(0	rx(0	PROPN
ejpam-6261	452	30	′	′	NOUN
ejpam-6261	452	31	)	)	PUNCT
ejpam-6261	452	32	≤	≤	NOUN
ejpam-6261	452	33	rw(f	rw(f	NUM
ejpam-6261	452	34	)	)	PUNCT
ejpam-6261	452	35	,	,	PUNCT
ejpam-6261	452	36	ωx(0	ωx(0	PROPN
ejpam-6261	452	37	′	′	NUM
ejpam-6261	452	38	)	)	PUNCT
ejpam-6261	452	39	≤	≤	NOUN
ejpam-6261	452	40	ωw(f	ωw(f	NUM
ejpam-6261	452	41	)	)	PUNCT
ejpam-6261	452	42	,	,	PUNCT
ejpam-6261	452	43	∀f	∀f	PROPN
ejpam-6261	452	44	∈	∈	PROPN
ejpam-6261	452	45	k1	k1	NOUN
ejpam-6261	452	46	.	.	PUNCT
ejpam-6261	453	1	here	here	ADV
ejpam-6261	453	2	0	0	NUM
ejpam-6261	453	3	and	and	CCONJ
ejpam-6261	453	4	0	0	NUM
ejpam-6261	453	5	′	′	NUM
ejpam-6261	453	6	are	be	AUX
ejpam-6261	453	7	neutral	neutral	ADJ
ejpam-6261	453	8	elements	element	NOUN
ejpam-6261	453	9	of	of	ADP
ejpam-6261	453	10	k1	k1	NOUN
ejpam-6261	453	11	and	and	CCONJ
ejpam-6261	453	12	k2	k2	NOUN
ejpam-6261	453	13	.	.	PUNCT
ejpam-6261	454	1	proof	proof	NOUN
ejpam-6261	454	2	.	.	PUNCT
ejpam-6261	455	1	let	let	VERB
ejpam-6261	455	2	w×	w×	VERB
ejpam-6261	455	3	x	x	PRON
ejpam-6261	455	4	be	be	AUX
ejpam-6261	455	5	a	a	DET
ejpam-6261	455	6	cnsr	cnsr	NOUN
ejpam-6261	455	7	of	of	ADP
ejpam-6261	455	8	k1	k1	PROPN
ejpam-6261	455	9	×k2	×k2	PROPN
ejpam-6261	455	10	.	.	PUNCT
ejpam-6261	456	1	on	on	ADP
ejpam-6261	456	2	contrary	contrary	ADV
ejpam-6261	456	3	,	,	PUNCT
ejpam-6261	456	4	assume	assume	VERB
ejpam-6261	456	5	that	that	SCONJ
ejpam-6261	456	6	the	the	DET
ejpam-6261	456	7	statements	statement	NOUN
ejpam-6261	456	8	[	[	X
ejpam-6261	456	9	1	1	X
ejpam-6261	456	10	]	]	PUNCT
ejpam-6261	456	11	and	and	CCONJ
ejpam-6261	456	12	[	[	X
ejpam-6261	456	13	2	2	X
ejpam-6261	456	14	]	]	PUNCT
ejpam-6261	456	15	do	do	AUX
ejpam-6261	456	16	not	not	PART
ejpam-6261	456	17	hold	hold	VERB
ejpam-6261	456	18	.	.	PUNCT
ejpam-6261	457	1	then	then	ADV
ejpam-6261	457	2	there	there	PRON
ejpam-6261	457	3	exist	exist	VERB
ejpam-6261	457	4	f	f	PROPN
ejpam-6261	457	5	∈	∈	PROPN
ejpam-6261	457	6	k1	k1	NOUN
ejpam-6261	457	7	and	and	CCONJ
ejpam-6261	457	8	g	g	PROPN
ejpam-6261	457	9	∈	∈	PROPN
ejpam-6261	457	10	k2	k2	PROPN
ejpam-6261	457	11	such	such	ADJ
ejpam-6261	457	12	that	that	SCONJ
ejpam-6261	457	13	(	(	PUNCT
ejpam-6261	457	14	i	i	NOUN
ejpam-6261	457	15	)	)	PUNCT
ejpam-6261	457	16	pw(0	pw(0	NOUN
ejpam-6261	457	17	)	)	PUNCT
ejpam-6261	457	18	≤	≤	NOUN
ejpam-6261	457	19	px(g	px(g	NOUN
ejpam-6261	457	20	)	)	PUNCT
ejpam-6261	457	21	,	,	PUNCT
ejpam-6261	457	22	θw(0	θw(0	PROPN
ejpam-6261	457	23	)	)	PUNCT
ejpam-6261	457	24	≤	≤	NOUN
ejpam-6261	457	25	θx(g	θx(g	NOUN
ejpam-6261	457	26	)	)	PUNCT
ejpam-6261	457	27	,	,	PUNCT
ejpam-6261	457	28	qw(0	qw(0	PROPN
ejpam-6261	457	29	)	)	PUNCT
ejpam-6261	457	30	≤	≤	NOUN
ejpam-6261	457	31	qx(g	qx(g	NOUN
ejpam-6261	457	32	)	)	PUNCT
ejpam-6261	457	33	,	,	PUNCT
ejpam-6261	457	34	φw(0	φw(0	NOUN
ejpam-6261	457	35	)	)	PUNCT
ejpam-6261	457	36	≤	≤	NOUN
ejpam-6261	457	37	φx(g	φx(g	NOUN
ejpam-6261	457	38	)	)	PUNCT
ejpam-6261	457	39	and	and	CCONJ
ejpam-6261	457	40	rw(0	rw(0	X
ejpam-6261	457	41	)	)	PUNCT
ejpam-6261	457	42	≥	≥	NOUN
ejpam-6261	457	43	rx(g	rx(g	NOUN
ejpam-6261	457	44	)	)	PUNCT
ejpam-6261	457	45	,	,	PUNCT
ejpam-6261	457	46	ωw(0	ωw(0	NOUN
ejpam-6261	457	47	)	)	PUNCT
ejpam-6261	457	48	≥	≥	NOUN
ejpam-6261	457	49	ωx(g	ωx(g	NUM
ejpam-6261	457	50	)	)	PUNCT
ejpam-6261	457	51	,	,	PUNCT
ejpam-6261	457	52	∀g	∀g	X
ejpam-6261	457	53	∈	∈	PROPN
ejpam-6261	457	54	k2	k2	PROPN
ejpam-6261	457	55	.	.	PUNCT
ejpam-6261	458	1	(	(	PUNCT
ejpam-6261	458	2	ii	ii	X
ejpam-6261	458	3	)	)	PUNCT
ejpam-6261	459	1	px(0	px(0	PROPN
ejpam-6261	459	2	′	′	NOUN
ejpam-6261	459	3	)	)	PUNCT
ejpam-6261	459	4	≤	≤	NOUN
ejpam-6261	459	5	pw(f	pw(f	NUM
ejpam-6261	459	6	)	)	PUNCT
ejpam-6261	459	7	,	,	PUNCT
ejpam-6261	459	8	θx(0	θx(0	PROPN
ejpam-6261	459	9	′	′	NUM
ejpam-6261	459	10	)	)	PUNCT
ejpam-6261	459	11	≤	≤	NOUN
ejpam-6261	459	12	θw(f	θw(f	NUM
ejpam-6261	459	13	)	)	PUNCT
ejpam-6261	459	14	,	,	PUNCT
ejpam-6261	459	15	qx(0	qx(0	PROPN
ejpam-6261	459	16	′	′	NUM
ejpam-6261	459	17	)	)	PUNCT
ejpam-6261	459	18	≤	≤	NOUN
ejpam-6261	459	19	qw(f	qw(f	NUM
ejpam-6261	459	20	)	)	PUNCT
ejpam-6261	459	21	,	,	PUNCT
ejpam-6261	459	22	φx(0	φx(0	PROPN
ejpam-6261	459	23	′	′	NUM
ejpam-6261	459	24	)	)	PUNCT
ejpam-6261	459	25	≤	≤	NOUN
ejpam-6261	459	26	φw(f	φw(f	NOUN
ejpam-6261	459	27	)	)	PUNCT
ejpam-6261	459	28	,	,	PUNCT
ejpam-6261	459	29	rx(0	rx(0	PROPN
ejpam-6261	459	30	′	′	NUM
ejpam-6261	459	31	)	)	PUNCT
ejpam-6261	459	32	≥	≥	NOUN
ejpam-6261	459	33	rw(f	rw(f	NUM
ejpam-6261	459	34	)	)	PUNCT
ejpam-6261	459	35	,	,	PUNCT
ejpam-6261	459	36	ωx(0	ωx(0	PROPN
ejpam-6261	459	37	′	′	NUM
ejpam-6261	459	38	)	)	PUNCT
ejpam-6261	459	39	≥	≥	NOUN
ejpam-6261	459	40	ωw(f	ωw(f	NUM
ejpam-6261	459	41	)	)	PUNCT
ejpam-6261	459	42	,	,	PUNCT
ejpam-6261	459	43	∀f	∀f	PROPN
ejpam-6261	459	44	∈	∈	PROPN
ejpam-6261	459	45	k1	k1	NOUN
ejpam-6261	459	46	.	.	PUNCT
ejpam-6261	460	1	consider	consider	VERB
ejpam-6261	460	2	,	,	PUNCT
ejpam-6261	460	3	tw×x(f	tw×x(f	NOUN
ejpam-6261	460	4	,	,	PUNCT
ejpam-6261	460	5	g	g	NOUN
ejpam-6261	460	6	)	)	PUNCT
ejpam-6261	460	7	=	=	SYM
ejpam-6261	460	8	min{pw(f	min{pw(f	PROPN
ejpam-6261	460	9	)	)	PUNCT
ejpam-6261	460	10	,	,	PUNCT
ejpam-6261	460	11	px(g	px(g	NOUN
ejpam-6261	460	12	)	)	PUNCT
ejpam-6261	460	13	}	}	PUNCT
ejpam-6261	460	14	eimin{θw(f),θx(g	eimin{θw(f),θx(g	NOUN
ejpam-6261	460	15	)	)	PUNCT
ejpam-6261	460	16	}	}	PUNCT
ejpam-6261	460	17	≥	≥	PROPN
ejpam-6261	460	18	min{pw(0	min{pw(0	PROPN
ejpam-6261	460	19	)	)	PUNCT
ejpam-6261	460	20	,	,	PUNCT
ejpam-6261	461	1	px(0	px(0	PROPN
ejpam-6261	461	2	′	′	NUM
ejpam-6261	461	3	)	)	PUNCT
ejpam-6261	461	4	}	}	PUNCT
ejpam-6261	461	5	eimin{θw(0),θx(0	eimin{θw(0),θx(0	PROPN
ejpam-6261	461	6	′	′	NUM
ejpam-6261	461	7	)	)	PUNCT
ejpam-6261	461	8	}	}	PUNCT
ejpam-6261	461	9	=	=	SYM
ejpam-6261	461	10	tw×x(0	tw×x(0	PROPN
ejpam-6261	461	11	,	,	PUNCT
ejpam-6261	461	12	0	0	NUM
ejpam-6261	461	13	′	′	NUM
ejpam-6261	461	14	)	)	PUNCT
ejpam-6261	461	15	.	.	PUNCT
ejpam-6261	462	1	iw×x(f	iw×x(f	PROPN
ejpam-6261	462	2	,	,	PUNCT
ejpam-6261	462	3	g	g	NOUN
ejpam-6261	462	4	)	)	PUNCT
ejpam-6261	462	5	=	=	SYM
ejpam-6261	462	6	min{qw(f	min{qw(f	PROPN
ejpam-6261	462	7	)	)	PUNCT
ejpam-6261	462	8	,	,	PUNCT
ejpam-6261	462	9	qx(g	qx(g	NOUN
ejpam-6261	462	10	)	)	PUNCT
ejpam-6261	462	11	}	}	PUNCT
ejpam-6261	462	12	eimin{φw(f),φx(g	eimin{φw(f),φx(g	NOUN
ejpam-6261	462	13	)	)	PUNCT
ejpam-6261	462	14	}	}	PUNCT
ejpam-6261	462	15	≥	≥	NUM
ejpam-6261	462	16	min{qw(0	min{qw(0	PROPN
ejpam-6261	462	17	)	)	PUNCT
ejpam-6261	462	18	,	,	PUNCT
ejpam-6261	462	19	qx(0	qx(0	PROPN
ejpam-6261	462	20	′	′	NUM
ejpam-6261	462	21	)	)	PUNCT
ejpam-6261	462	22	}	}	PUNCT
ejpam-6261	462	23	eimin{φw(0),φx(0	eimin{φw(0),φx(0	NUM
ejpam-6261	462	24	′	′	NUM
ejpam-6261	462	25	)	)	PUNCT
ejpam-6261	462	26	}	}	PUNCT
ejpam-6261	462	27	=	=	SYM
ejpam-6261	462	28	iw×x(0	iw×x(0	PROPN
ejpam-6261	462	29	,	,	PUNCT
ejpam-6261	462	30	0	0	NUM
ejpam-6261	462	31	′	′	NUM
ejpam-6261	462	32	)	)	PUNCT
ejpam-6261	462	33	and	and	CCONJ
ejpam-6261	462	34	fw×x(f	fw×x(f	ADJ
ejpam-6261	462	35	,	,	PUNCT
ejpam-6261	462	36	g	g	NOUN
ejpam-6261	462	37	)	)	PUNCT
ejpam-6261	462	38	=	=	SYM
ejpam-6261	463	1	max{rw	max{rw	X
ejpam-6261	463	2	(	(	PUNCT
ejpam-6261	463	3	f	f	NOUN
ejpam-6261	463	4	)	)	PUNCT
ejpam-6261	463	5	,	,	PUNCT
ejpam-6261	463	6	rx(g	rx(g	NOUN
ejpam-6261	463	7	)	)	PUNCT
ejpam-6261	463	8	}	}	PUNCT
ejpam-6261	463	9	eimax{ωw	eimax{ωw	NOUN
ejpam-6261	463	10	(	(	PUNCT
ejpam-6261	463	11	f),ωx(g	f),ωx(g	NOUN
ejpam-6261	463	12	)	)	PUNCT
ejpam-6261	463	13	}	}	PUNCT
ejpam-6261	463	14	≤	≤	PROPN
ejpam-6261	463	15	max{rw(0	max{rw(0	PROPN
ejpam-6261	463	16	)	)	PUNCT
ejpam-6261	463	17	,	,	PUNCT
ejpam-6261	463	18	rx(0	rx(0	PROPN
ejpam-6261	463	19	′	′	NUM
ejpam-6261	463	20	)	)	PUNCT
ejpam-6261	463	21	}	}	PUNCT
ejpam-6261	463	22	eimax{ωw(0),ωx(0	eimax{ωw(0),ωx(0	PROPN
ejpam-6261	463	23	′	′	NUM
ejpam-6261	463	24	)	)	PUNCT
ejpam-6261	463	25	}	}	PUNCT
ejpam-6261	463	26	=	=	SYM
ejpam-6261	463	27	fw×x(0	fw×x(0	PROPN
ejpam-6261	463	28	,	,	PUNCT
ejpam-6261	463	29	0	0	NUM
ejpam-6261	463	30	′	′	NUM
ejpam-6261	463	31	)	)	PUNCT
ejpam-6261	463	32	.	.	PUNCT
ejpam-6261	464	1	but	but	CCONJ
ejpam-6261	464	2	w	w	X
ejpam-6261	464	3	×	×	NOUN
ejpam-6261	464	4	x	x	VERB
ejpam-6261	464	5	is	be	AUX
ejpam-6261	464	6	cnsr	cnsr	ADJ
ejpam-6261	464	7	.	.	PUNCT
ejpam-6261	465	1	hence	hence	ADV
ejpam-6261	465	2	,	,	PUNCT
ejpam-6261	465	3	it	it	PRON
ejpam-6261	465	4	is	be	AUX
ejpam-6261	465	5	proved	prove	VERB
ejpam-6261	465	6	that	that	SCONJ
ejpam-6261	465	7	at	at	ADV
ejpam-6261	465	8	least	least	ADV
ejpam-6261	465	9	one	one	NUM
ejpam-6261	465	10	statements	statement	NOUN
ejpam-6261	465	11	must	must	AUX
ejpam-6261	465	12	be	be	AUX
ejpam-6261	465	13	satisfied	satisfied	ADJ
ejpam-6261	465	14	.	.	PUNCT
ejpam-6261	466	1	(	(	PUNCT
ejpam-6261	466	2	i	i	NOUN
ejpam-6261	466	3	)	)	PUNCT
ejpam-6261	466	4	pw(0	pw(0	NOUN
ejpam-6261	466	5	)	)	PUNCT
ejpam-6261	466	6	≥	≥	NOUN
ejpam-6261	466	7	px(g	px(g	NOUN
ejpam-6261	466	8	)	)	PUNCT
ejpam-6261	466	9	,	,	PUNCT
ejpam-6261	466	10	θw(0	θw(0	PROPN
ejpam-6261	466	11	)	)	PUNCT
ejpam-6261	466	12	≥	≥	NOUN
ejpam-6261	466	13	θx(g	θx(g	NOUN
ejpam-6261	466	14	)	)	PUNCT
ejpam-6261	466	15	,	,	PUNCT
ejpam-6261	466	16	qw(0	qw(0	PROPN
ejpam-6261	466	17	)	)	PUNCT
ejpam-6261	466	18	≥	≥	NOUN
ejpam-6261	466	19	qx(g	qx(g	NOUN
ejpam-6261	466	20	)	)	PUNCT
ejpam-6261	466	21	,	,	PUNCT
ejpam-6261	466	22	φw(0	φw(0	PROPN
ejpam-6261	466	23	)	)	PUNCT
ejpam-6261	466	24	≥	≥	NOUN
ejpam-6261	466	25	φx(g	φx(g	NOUN
ejpam-6261	466	26	)	)	PUNCT
ejpam-6261	466	27	and	and	CCONJ
ejpam-6261	466	28	rw(0	rw(0	NOUN
ejpam-6261	466	29	)	)	PUNCT
ejpam-6261	466	30	≤	≤	NOUN
ejpam-6261	466	31	rx(g	rx(g	NOUN
ejpam-6261	466	32	)	)	PUNCT
ejpam-6261	466	33	,	,	PUNCT
ejpam-6261	466	34	ωw(0	ωw(0	NOUN
ejpam-6261	466	35	)	)	PUNCT
ejpam-6261	466	36	≤	≤	NOUN
ejpam-6261	466	37	ωx(g	ωx(g	NUM
ejpam-6261	466	38	)	)	PUNCT
ejpam-6261	466	39	,	,	PUNCT
ejpam-6261	466	40	∀g	∀g	X
ejpam-6261	466	41	∈	∈	PROPN
ejpam-6261	466	42	k2	k2	PROPN
ejpam-6261	466	43	.	.	PUNCT
ejpam-6261	467	1	(	(	PUNCT
ejpam-6261	467	2	ii	ii	X
ejpam-6261	467	3	)	)	PUNCT
ejpam-6261	468	1	px(0	px(0	PROPN
ejpam-6261	468	2	′	′	PROPN
ejpam-6261	468	3	)	)	PUNCT
ejpam-6261	468	4	≥	≥	NOUN
ejpam-6261	468	5	pw(f	pw(f	NUM
ejpam-6261	468	6	)	)	PUNCT
ejpam-6261	468	7	,	,	PUNCT
ejpam-6261	468	8	θx(0	θx(0	PROPN
ejpam-6261	468	9	′	′	NUM
ejpam-6261	468	10	)	)	PUNCT
ejpam-6261	468	11	≥	≥	NOUN
ejpam-6261	468	12	θw(f	θw(f	NUM
ejpam-6261	468	13	)	)	PUNCT
ejpam-6261	468	14	,	,	PUNCT
ejpam-6261	468	15	qx(0	qx(0	PROPN
ejpam-6261	468	16	′	′	NUM
ejpam-6261	468	17	)	)	PUNCT
ejpam-6261	468	18	≥	≥	NOUN
ejpam-6261	468	19	qw(f	qw(f	NUM
ejpam-6261	468	20	)	)	PUNCT
ejpam-6261	468	21	,	,	PUNCT
ejpam-6261	468	22	φx(0	φx(0	PROPN
ejpam-6261	468	23	′	′	NUM
ejpam-6261	468	24	)	)	PUNCT
ejpam-6261	468	25	≥	≥	NOUN
ejpam-6261	468	26	φw(f	φw(f	NOUN
ejpam-6261	468	27	)	)	PUNCT
ejpam-6261	468	28	,	,	PUNCT
ejpam-6261	468	29	rx(0	rx(0	PROPN
ejpam-6261	468	30	′	′	NOUN
ejpam-6261	468	31	)	)	PUNCT
ejpam-6261	468	32	≤	≤	NOUN
ejpam-6261	468	33	rw(f	rw(f	NUM
ejpam-6261	468	34	)	)	PUNCT
ejpam-6261	468	35	,	,	PUNCT
ejpam-6261	468	36	ωx(0	ωx(0	PROPN
ejpam-6261	468	37	′	′	NUM
ejpam-6261	468	38	)	)	PUNCT
ejpam-6261	468	39	≤	≤	NOUN
ejpam-6261	468	40	ωw(f	ωw(f	NUM
ejpam-6261	468	41	)	)	PUNCT
ejpam-6261	468	42	,	,	PUNCT
ejpam-6261	468	43	∀f	∀f	PROPN
ejpam-6261	468	44	∈	∈	PROPN
ejpam-6261	468	45	k1	k1	NOUN
ejpam-6261	468	46	.	.	PUNCT
ejpam-6261	469	1	theorem	theorem	VERB
ejpam-6261	469	2	8	8	NUM
ejpam-6261	469	3	.	.	PUNCT
ejpam-6261	470	1	let	let	VERB
ejpam-6261	470	2	w	w	NOUN
ejpam-6261	470	3	and	and	CCONJ
ejpam-6261	470	4	x	x	PART
ejpam-6261	470	5	be	be	AUX
ejpam-6261	470	6	cnss	cns	NOUN
ejpam-6261	470	7	of	of	ADP
ejpam-6261	470	8	k1	k1	NOUN
ejpam-6261	470	9	,	,	PUNCT
ejpam-6261	470	10	k2	k2	NOUN
ejpam-6261	470	11	and	and	CCONJ
ejpam-6261	470	12	px(0	px(0	PROPN
ejpam-6261	470	13	′	′	PROPN
ejpam-6261	470	14	)	)	PUNCT
ejpam-6261	470	15	≥	≥	NOUN
ejpam-6261	470	16	pw(ϑ	pw(ϑ	NOUN
ejpam-6261	470	17	)	)	PUNCT
ejpam-6261	470	18	,	,	PUNCT
ejpam-6261	470	19	θx(0	θx(0	PROPN
ejpam-6261	470	20	′	′	NUM
ejpam-6261	470	21	)	)	PUNCT
ejpam-6261	470	22	≥	≥	NOUN
ejpam-6261	470	23	θw(ϑ	θw(ϑ	NUM
ejpam-6261	470	24	)	)	PUNCT
ejpam-6261	470	25	,	,	PUNCT
ejpam-6261	470	26	qx(0	qx(0	PROPN
ejpam-6261	470	27	′	′	NUM
ejpam-6261	470	28	)	)	PUNCT
ejpam-6261	470	29	≥	≥	NOUN
ejpam-6261	470	30	qw(ϑ	qw(ϑ	NUM
ejpam-6261	470	31	)	)	PUNCT
ejpam-6261	470	32	,	,	PUNCT
ejpam-6261	470	33	φx(0	φx(0	PROPN
ejpam-6261	470	34	′	′	NUM
ejpam-6261	470	35	)	)	PUNCT
ejpam-6261	470	36	≥	≥	NOUN
ejpam-6261	470	37	φw(ϑ	φw(ϑ	NUM
ejpam-6261	470	38	)	)	PUNCT
ejpam-6261	470	39	,	,	PUNCT
ejpam-6261	470	40	rx(0	rx(0	PROPN
ejpam-6261	470	41	′	′	NUM
ejpam-6261	470	42	)	)	PUNCT
ejpam-6261	471	1	≤	≤	NOUN
ejpam-6261	471	2	rw(ϑ	rw(ϑ	NOUN
ejpam-6261	471	3	)	)	PUNCT
ejpam-6261	471	4	,	,	PUNCT
ejpam-6261	471	5	ωx(0	ωx(0	PROPN
ejpam-6261	471	6	′	′	NUM
ejpam-6261	471	7	)	)	PUNCT
ejpam-6261	471	8	≤	≤	NOUN
ejpam-6261	471	9	ωw(ϑ	ωw(ϑ	NUM
ejpam-6261	471	10	)	)	PUNCT
ejpam-6261	471	11	,	,	PUNCT
ejpam-6261	471	12	∀	∀	X
ejpam-6261	471	13	ϑ	ϑ	X
ejpam-6261	471	14	∈	∈	PROPN
ejpam-6261	471	15	k1	k1	NOUN
ejpam-6261	471	16	,	,	PUNCT
ejpam-6261	471	17	0	0	NUM
ejpam-6261	471	18	′	′	NOUN
ejpam-6261	471	19	is	be	AUX
ejpam-6261	471	20	identity	identity	NOUN
ejpam-6261	471	21	of	of	ADP
ejpam-6261	471	22	k2	k2	NOUN
ejpam-6261	471	23	.	.	PUNCT
ejpam-6261	472	1	if	if	SCONJ
ejpam-6261	472	2	w×	w×	PROPN
ejpam-6261	472	3	x	x	PUNCT
ejpam-6261	472	4	is	be	AUX
ejpam-6261	472	5	cnsr	cnsr	VERB
ejpam-6261	472	6	of	of	ADP
ejpam-6261	472	7	k1	k1	PROPN
ejpam-6261	472	8	×k2	×k2	PROPN
ejpam-6261	472	9	,	,	PUNCT
ejpam-6261	472	10	then	then	ADV
ejpam-6261	472	11	w	w	PROPN
ejpam-6261	472	12	is	be	AUX
ejpam-6261	472	13	cnsr	cnsr	VERB
ejpam-6261	472	14	of	of	ADP
ejpam-6261	472	15	k1	k1	PROPN
ejpam-6261	472	16	.	.	PUNCT
ejpam-6261	473	1	m.	m.	PROPN
ejpam-6261	473	2	h.	h.	PROPN
ejpam-6261	473	3	mateen	mateen	PROPN
ejpam-6261	473	4	et	et	PROPN
ejpam-6261	473	5	al	al	PROPN
ejpam-6261	473	6	.	.	PUNCT
ejpam-6261	473	7	/	/	SYM
ejpam-6261	473	8	eur	eur	PROPN
ejpam-6261	473	9	.	.	PUNCT
ejpam-6261	474	1	j.	j.	PROPN
ejpam-6261	474	2	pure	pure	PROPN
ejpam-6261	474	3	appl	appl	PROPN
ejpam-6261	474	4	.	.	PROPN
ejpam-6261	474	5	math	math	PROPN
ejpam-6261	474	6	,	,	PUNCT
ejpam-6261	474	7	18	18	NUM
ejpam-6261	474	8	(	(	PUNCT
ejpam-6261	474	9	4	4	NUM
ejpam-6261	474	10	)	)	PUNCT
ejpam-6261	474	11	(	(	PUNCT
ejpam-6261	474	12	2025	2025	NUM
ejpam-6261	474	13	)	)	PUNCT
ejpam-6261	474	14	,	,	PUNCT
ejpam-6261	474	15	6261	6261	NUM
ejpam-6261	474	16	19	19	NUM
ejpam-6261	474	17	of	of	ADP
ejpam-6261	474	18	23	23	NUM
ejpam-6261	474	19	proof	proof	NOUN
ejpam-6261	474	20	.	.	PUNCT
ejpam-6261	475	1	let	let	VERB
ejpam-6261	475	2	(	(	PUNCT
ejpam-6261	475	3	ϑ	ϑ	X
ejpam-6261	475	4	,	,	PUNCT
ejpam-6261	475	5	0	0	NUM
ejpam-6261	475	6	′	′	NUM
ejpam-6261	475	7	)	)	PUNCT
ejpam-6261	475	8	,	,	PUNCT
ejpam-6261	475	9	(	(	PUNCT
ejpam-6261	475	10	a	a	X
ejpam-6261	475	11	,	,	PUNCT
ejpam-6261	475	12	0	0	NUM
ejpam-6261	475	13	′	′	NUM
ejpam-6261	475	14	)	)	PUNCT
ejpam-6261	475	15	be	be	AUX
ejpam-6261	475	16	elements	element	NOUN
ejpam-6261	475	17	of	of	ADP
ejpam-6261	475	18	k1	k1	PROPN
ejpam-6261	475	19	×k2	×k2	PROPN
ejpam-6261	475	20	.	.	PUNCT
ejpam-6261	476	1	by	by	ADP
ejpam-6261	476	2	given	give	VERB
ejpam-6261	476	3	condition	condition	NOUN
ejpam-6261	476	4	px(0	px(0	PROPN
ejpam-6261	476	5	′	′	PROPN
ejpam-6261	476	6	)	)	PUNCT
ejpam-6261	476	7	≥	≥	NOUN
ejpam-6261	476	8	pw(ϑ	pw(ϑ	NUM
ejpam-6261	476	9	)	)	PUNCT
ejpam-6261	476	10	and	and	CCONJ
ejpam-6261	476	11	θx(0	θx(0	PROPN
ejpam-6261	476	12	′	′	NUM
ejpam-6261	476	13	)	)	PUNCT
ejpam-6261	476	14	≥	≥	NOUN
ejpam-6261	476	15	θw(ϑ	θw(ϑ	NUM
ejpam-6261	476	16	)	)	PUNCT
ejpam-6261	476	17	,	,	PUNCT
ejpam-6261	476	18	for	for	ADP
ejpam-6261	476	19	all	all	DET
ejpam-6261	476	20	ϑ	ϑ	X
ejpam-6261	476	21	,	,	PUNCT
ejpam-6261	476	22	a	a	DET
ejpam-6261	476	23	∈	∈	NOUN
ejpam-6261	476	24	k1	k1	NOUN
ejpam-6261	476	25	and	and	CCONJ
ejpam-6261	476	26	0	0	NUM
ejpam-6261	476	27	′	′	NUM
ejpam-6261	476	28	∈	∈	PROPN
ejpam-6261	476	29	k2	k2	PROPN
ejpam-6261	476	30	.	.	PUNCT
ejpam-6261	477	1	consider	consider	VERB
ejpam-6261	477	2	,	,	PUNCT
ejpam-6261	477	3	tw(ϑ−	tw(ϑ−	ADV
ejpam-6261	477	4	a	a	X
ejpam-6261	477	5	)	)	PUNCT
ejpam-6261	478	1	=	=	SYM
ejpam-6261	478	2	pw(ϑ−	pw(ϑ−	NOUN
ejpam-6261	478	3	a)eiθw(ϑ−a	a)eiθw(ϑ−a	NUM
ejpam-6261	478	4	)	)	PUNCT
ejpam-6261	479	1	=	=	PUNCT
ejpam-6261	479	2	min{pw(ϑ−	min{pw(ϑ−	NOUN
ejpam-6261	479	3	a)eiθw(ϑ−a	a)eiθw(ϑ−a	NUM
ejpam-6261	479	4	)	)	PUNCT
ejpam-6261	479	5	,	,	PUNCT
ejpam-6261	480	1	px(0	px(0	PROPN
ejpam-6261	480	2	′	′	VERB
ejpam-6261	481	1	−	−	NOUN
ejpam-6261	481	2	0	0	NUM
ejpam-6261	482	1	′	′	NUM
ejpam-6261	482	2	)	)	PUNCT
ejpam-6261	483	1	eiθx(0	eiθx(0	PROPN
ejpam-6261	483	2	′−0	′−0	NOUN
ejpam-6261	483	3	′	′	NOUN
ejpam-6261	483	4	)	)	PUNCT
ejpam-6261	483	5	}	}	PUNCT
ejpam-6261	484	1	=	=	SYM
ejpam-6261	484	2	pw×x((ϑ	pw×x((ϑ	NOUN
ejpam-6261	484	3	,	,	PUNCT
ejpam-6261	484	4	0	0	NUM
ejpam-6261	484	5	′	′	NUM
ejpam-6261	484	6	)	)	PUNCT
ejpam-6261	484	7	(	(	PUNCT
ejpam-6261	484	8	ϑ	ϑ	X
ejpam-6261	484	9	,	,	PUNCT
ejpam-6261	484	10	0	0	NUM
ejpam-6261	484	11	′	′	NUM
ejpam-6261	484	12	)	)	PUNCT
ejpam-6261	484	13	)	)	PUNCT
ejpam-6261	485	1	eiθw×x((ϑ,0	eiθw×x((ϑ,0	ADJ
ejpam-6261	485	2	′	′	NUM
ejpam-6261	485	3	)	)	PUNCT
ejpam-6261	485	4	(	(	PUNCT
ejpam-6261	485	5	ϑ,0	ϑ,0	VERB
ejpam-6261	485	6	′	′	NUM
ejpam-6261	485	7	)	)	PUNCT
ejpam-6261	485	8	)	)	PUNCT
ejpam-6261	485	9	≥	≥	PROPN
ejpam-6261	486	1	min{pw×x(ϑ	min{pw×x(ϑ	PROPN
ejpam-6261	486	2	,	,	PUNCT
ejpam-6261	486	3	0	0	NUM
ejpam-6261	486	4	′	′	NUM
ejpam-6261	486	5	)	)	PUNCT
ejpam-6261	486	6	,	,	PUNCT
ejpam-6261	486	7	pw×x(ϑ	pw×x(ϑ	PROPN
ejpam-6261	486	8	,	,	PUNCT
ejpam-6261	486	9	0	0	NUM
ejpam-6261	486	10	′	′	NUM
ejpam-6261	486	11	)	)	PUNCT
ejpam-6261	486	12	}	}	PUNCT
ejpam-6261	486	13	eimin{θw×x(a,0	eimin{θw×x(a,0	ADJ
ejpam-6261	486	14	′	′	NOUN
ejpam-6261	486	15	)	)	PUNCT
ejpam-6261	486	16	,	,	PUNCT
ejpam-6261	486	17	θw×x(ϑ,0	θw×x(ϑ,0	VERB
ejpam-6261	486	18	′	′	NUM
ejpam-6261	486	19	)	)	PUNCT
ejpam-6261	486	20	}	}	PUNCT
ejpam-6261	486	21	=	=	SYM
ejpam-6261	486	22	min{min{pw(ϑ	min{min{pw(ϑ	NOUN
ejpam-6261	486	23	)	)	PUNCT
ejpam-6261	486	24	,	,	PUNCT
ejpam-6261	486	25	px(0	px(0	PROPN
ejpam-6261	486	26	′	′	NUM
ejpam-6261	486	27	)	)	PUNCT
ejpam-6261	486	28	}	}	PUNCT
ejpam-6261	486	29	,	,	PUNCT
ejpam-6261	486	30	min{pw(a	min{pw(a	PROPN
ejpam-6261	486	31	)	)	PUNCT
ejpam-6261	486	32	,	,	PUNCT
ejpam-6261	486	33	px(0	px(0	PROPN
ejpam-6261	486	34	′	′	NUM
ejpam-6261	486	35	)	)	PUNCT
ejpam-6261	486	36	}	}	PUNCT
ejpam-6261	486	37	}	}	PUNCT
ejpam-6261	486	38	∗	∗	NOUN
ejpam-6261	486	39	eimin{min{θx(0	eimin{min{θx(0	PROPN
ejpam-6261	486	40	′	′	NUM
ejpam-6261	486	41	)	)	PUNCT
ejpam-6261	486	42	}	}	PUNCT
ejpam-6261	486	43	,	,	PUNCT
ejpam-6261	486	44	min{θ	min{θ	PROPN
ejpam-6261	486	45	,	,	PUNCT
ejpam-6261	486	46	min{θw(a),θx(0	min{θw(a),θx(0	PROPN
ejpam-6261	486	47	′	′	NUM
ejpam-6261	486	48	)	)	PUNCT
ejpam-6261	486	49	}	}	PUNCT
ejpam-6261	486	50	}	}	PUNCT
ejpam-6261	486	51	=	=	SYM
ejpam-6261	486	52	min{tw(ϑ),tw(a	min{tw(ϑ),tw(a	X
ejpam-6261	486	53	)	)	PUNCT
ejpam-6261	486	54	}	}	PUNCT
ejpam-6261	486	55	.	.	PUNCT
ejpam-6261	487	1	thus	thus	ADV
ejpam-6261	487	2	,	,	PUNCT
ejpam-6261	487	3	tw(ϑ−	tw(ϑ−	ADP
ejpam-6261	487	4	a	a	X
ejpam-6261	487	5	)	)	PUNCT
ejpam-6261	487	6	≥	≥	NOUN
ejpam-6261	487	7	min{tw(ϑ),tw(a	min{tw(ϑ),tw(a	VERB
ejpam-6261	487	8	)	)	PUNCT
ejpam-6261	487	9	}	}	PUNCT
ejpam-6261	487	10	.	.	PUNCT
ejpam-6261	488	1	tw(ϑa	tw(ϑa	NOUN
ejpam-6261	488	2	)	)	PUNCT
ejpam-6261	489	1	=	=	SYM
ejpam-6261	489	2	pw(ϑa)eiθw(ϑa	pw(ϑa)eiθw(ϑa	X
ejpam-6261	489	3	)	)	PUNCT
ejpam-6261	489	4	=	=	SYM
ejpam-6261	489	5	min{pw(ϑa)eiθw(ϑa	min{pw(ϑa)eiθw(ϑa	PROPN
ejpam-6261	489	6	)	)	PUNCT
ejpam-6261	489	7	,	,	PUNCT
ejpam-6261	489	8	px(0	px(0	PROPN
ejpam-6261	489	9	′	′	NOUN
ejpam-6261	489	10	0	0	NUM
ejpam-6261	490	1	′	′	NUM
ejpam-6261	490	2	)	)	PUNCT
ejpam-6261	490	3	eiθx(0	eiθx(0	PROPN
ejpam-6261	491	1	′	′	NOUN
ejpam-6261	491	2	0	0	NUM
ejpam-6261	492	1	′	′	NUM
ejpam-6261	492	2	)	)	PUNCT
ejpam-6261	492	3	}	}	PUNCT
ejpam-6261	493	1	=	=	SYM
ejpam-6261	493	2	pw×x((ϑ	pw×x((ϑ	NOUN
ejpam-6261	493	3	,	,	PUNCT
ejpam-6261	493	4	0	0	NUM
ejpam-6261	493	5	′	′	NUM
ejpam-6261	493	6	)	)	PUNCT
ejpam-6261	493	7	(	(	PUNCT
ejpam-6261	493	8	ϑ	ϑ	X
ejpam-6261	493	9	,	,	PUNCT
ejpam-6261	493	10	0	0	NUM
ejpam-6261	493	11	′	′	NUM
ejpam-6261	493	12	)	)	PUNCT
ejpam-6261	493	13	)	)	PUNCT
ejpam-6261	494	1	eiθw×x((ϑ,0	eiθw×x((ϑ,0	ADJ
ejpam-6261	494	2	′	′	NUM
ejpam-6261	494	3	)	)	PUNCT
ejpam-6261	494	4	(	(	PUNCT
ejpam-6261	494	5	ϑ,0	ϑ,0	VERB
ejpam-6261	494	6	′	′	NUM
ejpam-6261	494	7	)	)	PUNCT
ejpam-6261	494	8	)	)	PUNCT
ejpam-6261	494	9	≥	≥	PROPN
ejpam-6261	495	1	min{pw×x(ϑ	min{pw×x(ϑ	PROPN
ejpam-6261	495	2	,	,	PUNCT
ejpam-6261	495	3	0	0	NUM
ejpam-6261	495	4	′	′	NUM
ejpam-6261	495	5	)	)	PUNCT
ejpam-6261	495	6	,	,	PUNCT
ejpam-6261	495	7	pw×x(ϑ	pw×x(ϑ	PROPN
ejpam-6261	495	8	,	,	PUNCT
ejpam-6261	495	9	0	0	NUM
ejpam-6261	495	10	′	′	NUM
ejpam-6261	495	11	)	)	PUNCT
ejpam-6261	495	12	}	}	PUNCT
ejpam-6261	495	13	eimin{θw×x(a,0	eimin{θw×x(a,0	ADJ
ejpam-6261	495	14	′	′	NOUN
ejpam-6261	495	15	)	)	PUNCT
ejpam-6261	495	16	,	,	PUNCT
ejpam-6261	495	17	θw×x(ϑ,0	θw×x(ϑ,0	VERB
ejpam-6261	495	18	′	′	NUM
ejpam-6261	495	19	)	)	PUNCT
ejpam-6261	495	20	}	}	PUNCT
ejpam-6261	495	21	=	=	SYM
ejpam-6261	495	22	min{min{pw(ϑ	min{min{pw(ϑ	NOUN
ejpam-6261	495	23	)	)	PUNCT
ejpam-6261	495	24	,	,	PUNCT
ejpam-6261	495	25	px(0	px(0	PROPN
ejpam-6261	495	26	′	′	NUM
ejpam-6261	495	27	)	)	PUNCT
ejpam-6261	495	28	}	}	PUNCT
ejpam-6261	495	29	,	,	PUNCT
ejpam-6261	495	30	min{pw(a	min{pw(a	PROPN
ejpam-6261	495	31	)	)	PUNCT
ejpam-6261	495	32	,	,	PUNCT
ejpam-6261	495	33	px(0	px(0	PROPN
ejpam-6261	495	34	′	′	NUM
ejpam-6261	495	35	)	)	PUNCT
ejpam-6261	495	36	}	}	PUNCT
ejpam-6261	495	37	}	}	PUNCT
ejpam-6261	495	38	∗	∗	NOUN
ejpam-6261	495	39	eimin{min{θx(0	eimin{min{θx(0	PROPN
ejpam-6261	495	40	′	′	NUM
ejpam-6261	495	41	)	)	PUNCT
ejpam-6261	495	42	}	}	PUNCT
ejpam-6261	495	43	,	,	PUNCT
ejpam-6261	495	44	min{θ	min{θ	PROPN
ejpam-6261	495	45	,	,	PUNCT
ejpam-6261	495	46	min{θw(a),θx(0	min{θw(a),θx(0	PROPN
ejpam-6261	495	47	′	′	NUM
ejpam-6261	495	48	)	)	PUNCT
ejpam-6261	495	49	}	}	PUNCT
ejpam-6261	495	50	}	}	PUNCT
ejpam-6261	495	51	=	=	SYM
ejpam-6261	495	52	min{tw(ϑ),tw(a	min{tw(ϑ),tw(a	X
ejpam-6261	495	53	)	)	PUNCT
ejpam-6261	495	54	}	}	PUNCT
ejpam-6261	495	55	.	.	PUNCT
ejpam-6261	496	1	thus	thus	ADV
ejpam-6261	496	2	,	,	PUNCT
ejpam-6261	496	3	tw(ϑa	tw(ϑa	PROPN
ejpam-6261	496	4	)	)	PUNCT
ejpam-6261	496	5	≥	≥	NOUN
ejpam-6261	496	6	min{tw(ϑ),tw(a	min{tw(ϑ),tw(a	VERB
ejpam-6261	496	7	)	)	PUNCT
ejpam-6261	496	8	}	}	PUNCT
ejpam-6261	496	9	.	.	PUNCT
ejpam-6261	497	1	consider	consider	VERB
ejpam-6261	497	2	,	,	PUNCT
ejpam-6261	497	3	iw(ϑ−	iw(ϑ−	ADP
ejpam-6261	497	4	a	a	X
ejpam-6261	497	5	)	)	PUNCT
ejpam-6261	497	6	=	=	SYM
ejpam-6261	497	7	qw(ϑ−	qw(ϑ−	PRON
ejpam-6261	497	8	a)eiθw(ϑ−a	a)eiθw(ϑ−a	NUM
ejpam-6261	497	9	)	)	PUNCT
ejpam-6261	498	1	=	=	SYM
ejpam-6261	499	1	min{qw(ϑ−	min{qw(ϑ−	ADJ
ejpam-6261	499	2	a)eiθw(ϑ−a	a)eiθw(ϑ−a	NOUN
ejpam-6261	499	3	)	)	PUNCT
ejpam-6261	499	4	,	,	PUNCT
ejpam-6261	500	1	qx(0	qx(0	PROPN
ejpam-6261	500	2	′	′	NUM
ejpam-6261	501	1	−	−	NOUN
ejpam-6261	501	2	0	0	NUM
ejpam-6261	502	1	′	′	NUM
ejpam-6261	502	2	)	)	PUNCT
ejpam-6261	503	1	eiθx(0	eiθx(0	PROPN
ejpam-6261	503	2	′−0	′−0	NOUN
ejpam-6261	503	3	′	′	NOUN
ejpam-6261	503	4	)	)	PUNCT
ejpam-6261	503	5	}	}	PUNCT
ejpam-6261	503	6	(	(	PUNCT
ejpam-6261	503	7	-6	-6	NOUN
ejpam-6261	503	8	)	)	PUNCT
ejpam-6261	503	9	=	=	SYM
ejpam-6261	503	10	qw×x((ϑ	qw×x((ϑ	PROPN
ejpam-6261	503	11	,	,	PUNCT
ejpam-6261	503	12	0	0	NUM
ejpam-6261	503	13	′	′	NUM
ejpam-6261	503	14	)	)	PUNCT
ejpam-6261	503	15	(	(	PUNCT
ejpam-6261	503	16	ϑ	ϑ	X
ejpam-6261	503	17	,	,	PUNCT
ejpam-6261	503	18	0	0	NUM
ejpam-6261	503	19	′	′	NUM
ejpam-6261	503	20	)	)	PUNCT
ejpam-6261	503	21	)	)	PUNCT
ejpam-6261	503	22	eiθw×x((ϑ,0	eiθw×x((ϑ,0	ADJ
ejpam-6261	503	23	′	′	NUM
ejpam-6261	503	24	)	)	PUNCT
ejpam-6261	503	25	(	(	PUNCT
ejpam-6261	503	26	ϑ,0	ϑ,0	VERB
ejpam-6261	503	27	′	′	NUM
ejpam-6261	503	28	)	)	PUNCT
ejpam-6261	503	29	)	)	PUNCT
ejpam-6261	503	30	≥	≥	PROPN
ejpam-6261	504	1	min{qw×x(ϑ	min{qw×x(ϑ	PROPN
ejpam-6261	504	2	,	,	PUNCT
ejpam-6261	504	3	0	0	NUM
ejpam-6261	504	4	′	′	NUM
ejpam-6261	504	5	)	)	PUNCT
ejpam-6261	504	6	,	,	PUNCT
ejpam-6261	504	7	qw×x(ϑ	qw×x(ϑ	INTJ
ejpam-6261	504	8	,	,	PUNCT
ejpam-6261	504	9	0	0	NUM
ejpam-6261	504	10	′	′	NUM
ejpam-6261	504	11	)	)	PUNCT
ejpam-6261	504	12	}	}	PUNCT
ejpam-6261	504	13	eimin{θw×x(a,0	eimin{θw×x(a,0	ADJ
ejpam-6261	504	14	′	′	NOUN
ejpam-6261	504	15	)	)	PUNCT
ejpam-6261	504	16	,	,	PUNCT
ejpam-6261	504	17	θw×x(ϑ,0	θw×x(ϑ,0	VERB
ejpam-6261	504	18	′	′	NUM
ejpam-6261	504	19	)	)	PUNCT
ejpam-6261	504	20	}	}	PUNCT
ejpam-6261	505	1	=	=	SYM
ejpam-6261	505	2	min{min{qw	min{min{qw	X
ejpam-6261	505	3	(	(	PUNCT
ejpam-6261	505	4	ϑ	ϑ	NOUN
ejpam-6261	505	5	)	)	PUNCT
ejpam-6261	505	6	,	,	PUNCT
ejpam-6261	505	7	qx(0	qx(0	PROPN
ejpam-6261	505	8	′	′	NUM
ejpam-6261	505	9	)	)	PUNCT
ejpam-6261	505	10	}	}	PUNCT
ejpam-6261	505	11	,	,	PUNCT
ejpam-6261	505	12	min{qw(a	min{qw(a	PROPN
ejpam-6261	505	13	)	)	PUNCT
ejpam-6261	505	14	,	,	PUNCT
ejpam-6261	505	15	px(0	px(0	PROPN
ejpam-6261	505	16	′	′	NUM
ejpam-6261	505	17	)	)	PUNCT
ejpam-6261	505	18	}	}	PUNCT
ejpam-6261	505	19	}	}	PUNCT
ejpam-6261	505	20	∗	∗	NOUN
ejpam-6261	505	21	eimin{min{θx(0	eimin{min{θx(0	PROPN
ejpam-6261	505	22	′	′	NUM
ejpam-6261	505	23	)	)	PUNCT
ejpam-6261	505	24	}	}	PUNCT
ejpam-6261	505	25	,	,	PUNCT
ejpam-6261	505	26	min{θ	min{θ	PROPN
ejpam-6261	505	27	,	,	PUNCT
ejpam-6261	505	28	min{θw(a),θx(0	min{θw(a),θx(0	PROPN
ejpam-6261	505	29	′	′	NUM
ejpam-6261	505	30	)	)	PUNCT
ejpam-6261	505	31	}	}	PUNCT
ejpam-6261	505	32	}	}	PUNCT
ejpam-6261	505	33	=	=	SYM
ejpam-6261	505	34	min{iw(ϑ	min{iw(ϑ	PROPN
ejpam-6261	505	35	)	)	PUNCT
ejpam-6261	505	36	,	,	PUNCT
ejpam-6261	505	37	iw(a	iw(a	PROPN
ejpam-6261	505	38	)	)	PUNCT
ejpam-6261	505	39	}	}	PUNCT
ejpam-6261	505	40	.	.	PUNCT
ejpam-6261	506	1	thus	thus	ADV
ejpam-6261	506	2	,	,	PUNCT
ejpam-6261	506	3	iw(ϑ−	iw(ϑ−	ADP
ejpam-6261	506	4	a	a	X
ejpam-6261	506	5	)	)	PUNCT
ejpam-6261	506	6	≥	≥	NOUN
ejpam-6261	506	7	min{iw(ϑ	min{iw(ϑ	PROPN
ejpam-6261	506	8	)	)	PUNCT
ejpam-6261	506	9	,	,	PUNCT
ejpam-6261	506	10	iw(a	iw(a	PROPN
ejpam-6261	506	11	)	)	PUNCT
ejpam-6261	506	12	}	}	PUNCT
ejpam-6261	506	13	.	.	PUNCT
ejpam-6261	507	1	iw(ϑa	iw(ϑa	PROPN
ejpam-6261	507	2	)	)	PUNCT
ejpam-6261	507	3	=	=	SYM
ejpam-6261	507	4	qw(ϑa)eiθw(ϑa	qw(ϑa)eiθw(ϑa	X
ejpam-6261	507	5	)	)	PUNCT
ejpam-6261	507	6	=	=	SYM
ejpam-6261	507	7	min{qw(ϑa)eiθw(ϑa	min{qw(ϑa)eiθw(ϑa	NOUN
ejpam-6261	507	8	)	)	PUNCT
ejpam-6261	507	9	,	,	PUNCT
ejpam-6261	507	10	qx(0	qx(0	PROPN
ejpam-6261	507	11	′	′	NOUN
ejpam-6261	507	12	0	0	NUM
ejpam-6261	507	13	′	′	NUM
ejpam-6261	508	1	)	)	PUNCT
ejpam-6261	508	2	eiθx(0	eiθx(0	PROPN
ejpam-6261	509	1	′	′	NOUN
ejpam-6261	509	2	0	0	NUM
ejpam-6261	510	1	′	′	NUM
ejpam-6261	510	2	)	)	PUNCT
ejpam-6261	510	3	}	}	PUNCT
ejpam-6261	511	1	=	=	SYM
ejpam-6261	511	2	qw×x((ϑ	qw×x((ϑ	PROPN
ejpam-6261	511	3	,	,	PUNCT
ejpam-6261	511	4	0	0	NUM
ejpam-6261	511	5	′	′	NUM
ejpam-6261	511	6	)	)	PUNCT
ejpam-6261	511	7	(	(	PUNCT
ejpam-6261	511	8	ϑ	ϑ	X
ejpam-6261	511	9	,	,	PUNCT
ejpam-6261	511	10	0	0	NUM
ejpam-6261	511	11	′	′	NUM
ejpam-6261	511	12	)	)	PUNCT
ejpam-6261	511	13	)	)	PUNCT
ejpam-6261	512	1	eiθw×x((ϑ,0	eiθw×x((ϑ,0	ADJ
ejpam-6261	512	2	′	′	NUM
ejpam-6261	512	3	)	)	PUNCT
ejpam-6261	512	4	(	(	PUNCT
ejpam-6261	512	5	ϑ,0	ϑ,0	VERB
ejpam-6261	512	6	′	′	NUM
ejpam-6261	512	7	)	)	PUNCT
ejpam-6261	512	8	)	)	PUNCT
ejpam-6261	512	9	≥	≥	PROPN
ejpam-6261	513	1	min{qw×x(ϑ	min{qw×x(ϑ	PROPN
ejpam-6261	513	2	,	,	PUNCT
ejpam-6261	513	3	0	0	NUM
ejpam-6261	513	4	′	′	NUM
ejpam-6261	513	5	)	)	PUNCT
ejpam-6261	513	6	,	,	PUNCT
ejpam-6261	513	7	qw×x(ϑ	qw×x(ϑ	INTJ
ejpam-6261	513	8	,	,	PUNCT
ejpam-6261	513	9	0	0	NUM
ejpam-6261	513	10	′	′	NUM
ejpam-6261	513	11	)	)	PUNCT
ejpam-6261	513	12	}	}	PUNCT
ejpam-6261	513	13	eimin{θw×x(a,0	eimin{θw×x(a,0	ADJ
ejpam-6261	513	14	′	′	NOUN
ejpam-6261	513	15	)	)	PUNCT
ejpam-6261	513	16	,	,	PUNCT
ejpam-6261	513	17	θw×x(ϑ,0	θw×x(ϑ,0	VERB
ejpam-6261	513	18	′	′	NUM
ejpam-6261	513	19	)	)	PUNCT
ejpam-6261	513	20	}	}	PUNCT
ejpam-6261	513	21	=	=	SYM
ejpam-6261	513	22	min{min{qw(ϑ	min{min{qw(ϑ	PROPN
ejpam-6261	513	23	)	)	PUNCT
ejpam-6261	513	24	,	,	PUNCT
ejpam-6261	513	25	qx(0	qx(0	PROPN
ejpam-6261	513	26	′	′	NUM
ejpam-6261	513	27	)	)	PUNCT
ejpam-6261	513	28	}	}	PUNCT
ejpam-6261	513	29	,	,	PUNCT
ejpam-6261	513	30	min{qw(a	min{qw(a	PROPN
ejpam-6261	513	31	)	)	PUNCT
ejpam-6261	513	32	,	,	PUNCT
ejpam-6261	513	33	px(0	px(0	PROPN
ejpam-6261	513	34	′	′	NUM
ejpam-6261	513	35	)	)	PUNCT
ejpam-6261	513	36	}	}	PUNCT
ejpam-6261	513	37	}	}	PUNCT
ejpam-6261	513	38	∗	∗	NOUN
ejpam-6261	513	39	eimin{min{θx(0	eimin{min{θx(0	PROPN
ejpam-6261	513	40	′	′	NUM
ejpam-6261	513	41	)	)	PUNCT
ejpam-6261	513	42	}	}	PUNCT
ejpam-6261	513	43	,	,	PUNCT
ejpam-6261	513	44	min{θ	min{θ	PROPN
ejpam-6261	513	45	,	,	PUNCT
ejpam-6261	513	46	min{θw(a),θx(0	min{θw(a),θx(0	PROPN
ejpam-6261	513	47	′	′	NUM
ejpam-6261	513	48	)	)	PUNCT
ejpam-6261	513	49	}	}	PUNCT
ejpam-6261	513	50	}	}	PUNCT
ejpam-6261	513	51	m.	m.	NOUN
ejpam-6261	513	52	h.	h.	PROPN
ejpam-6261	513	53	mateen	mateen	PROPN
ejpam-6261	513	54	et	et	PROPN
ejpam-6261	513	55	al	al	PROPN
ejpam-6261	513	56	.	.	PUNCT
ejpam-6261	513	57	/	/	SYM
ejpam-6261	513	58	eur	eur	PROPN
ejpam-6261	513	59	.	.	PUNCT
ejpam-6261	514	1	j.	j.	PROPN
ejpam-6261	514	2	pure	pure	PROPN
ejpam-6261	514	3	appl	appl	PROPN
ejpam-6261	514	4	.	.	PROPN
ejpam-6261	514	5	math	math	PROPN
ejpam-6261	514	6	,	,	PUNCT
ejpam-6261	514	7	18	18	NUM
ejpam-6261	514	8	(	(	PUNCT
ejpam-6261	514	9	4	4	NUM
ejpam-6261	514	10	)	)	PUNCT
ejpam-6261	514	11	(	(	PUNCT
ejpam-6261	514	12	2025	2025	NUM
ejpam-6261	514	13	)	)	PUNCT
ejpam-6261	514	14	,	,	PUNCT
ejpam-6261	514	15	6261	6261	NUM
ejpam-6261	514	16	20	20	NUM
ejpam-6261	514	17	of	of	ADP
ejpam-6261	514	18	23	23	NUM
ejpam-6261	514	19	=	=	SYM
ejpam-6261	514	20	min{iw(ϑ	min{iw(ϑ	PROPN
ejpam-6261	514	21	)	)	PUNCT
ejpam-6261	514	22	,	,	PUNCT
ejpam-6261	514	23	iw(a	iw(a	PROPN
ejpam-6261	514	24	)	)	PUNCT
ejpam-6261	514	25	}	}	PUNCT
ejpam-6261	514	26	.	.	PUNCT
ejpam-6261	515	1	thus	thus	ADV
ejpam-6261	515	2	,	,	PUNCT
ejpam-6261	515	3	iw(ϑa	iw(ϑa	PROPN
ejpam-6261	515	4	)	)	PUNCT
ejpam-6261	515	5	≥	≥	PROPN
ejpam-6261	515	6	min{iw(ϑ	min{iw(ϑ	PROPN
ejpam-6261	515	7	)	)	PUNCT
ejpam-6261	515	8	,	,	PUNCT
ejpam-6261	515	9	iw(a	iw(a	PROPN
ejpam-6261	515	10	)	)	PUNCT
ejpam-6261	515	11	}	}	PUNCT
ejpam-6261	515	12	.	.	PUNCT
ejpam-6261	516	1	further	far	ADV
ejpam-6261	516	2	,	,	PUNCT
ejpam-6261	516	3	fw(ϑ−	fw(ϑ−	ADP
ejpam-6261	516	4	a	a	X
ejpam-6261	516	5	)	)	PUNCT
ejpam-6261	516	6	=	=	SYM
ejpam-6261	516	7	rw(ϑ−	rw(ϑ−	NOUN
ejpam-6261	516	8	a)eiωw(ϑ−a	a)eiωw(ϑ−a	NOUN
ejpam-6261	516	9	)	)	PUNCT
ejpam-6261	517	1	=	=	PRON
ejpam-6261	517	2	{	{	PUNCT
ejpam-6261	517	3	max{rw(ϑ−	max{rw(ϑ−	ADV
ejpam-6261	517	4	a)eiωw(ϑ−a	a)eiωw(ϑ−a	NOUN
ejpam-6261	517	5	)	)	PUNCT
ejpam-6261	517	6	,	,	PUNCT
ejpam-6261	517	7	rx(0	rx(0	PROPN
ejpam-6261	517	8	′	′	NUM
ejpam-6261	518	1	−	−	NOUN
ejpam-6261	518	2	0	0	NUM
ejpam-6261	519	1	′	′	NUM
ejpam-6261	519	2	)	)	PUNCT
ejpam-6261	520	1	eiωx(0	eiωx(0	PROPN
ejpam-6261	520	2	′−0	′−0	NUM
ejpam-6261	520	3	′	′	NOUN
ejpam-6261	520	4	)	)	PUNCT
ejpam-6261	520	5	}	}	PUNCT
ejpam-6261	520	6	}	}	PUNCT
ejpam-6261	520	7	(	(	PUNCT
ejpam-6261	520	8	-17	-17	PUNCT
ejpam-6261	520	9	)	)	PUNCT
ejpam-6261	520	10	=	=	SYM
ejpam-6261	520	11	{	{	PUNCT
ejpam-6261	520	12	rw×x((ϑ	rw×x((ϑ	PROPN
ejpam-6261	520	13	,	,	PUNCT
ejpam-6261	520	14	0	0	NUM
ejpam-6261	520	15	′	′	NUM
ejpam-6261	520	16	)	)	PUNCT
ejpam-6261	520	17	(	(	PUNCT
ejpam-6261	520	18	a	a	PRON
ejpam-6261	520	19	,	,	PUNCT
ejpam-6261	520	20	0	0	NUM
ejpam-6261	520	21	′	′	NUM
ejpam-6261	520	22	)	)	PUNCT
ejpam-6261	520	23	)	)	PUNCT
ejpam-6261	520	24	}	}	PUNCT
ejpam-6261	520	25	ei{ωw×x((ϑ,0	ei{ωw×x((ϑ,0	VERB
ejpam-6261	520	26	′	′	NOUN
ejpam-6261	520	27	)	)	PUNCT
ejpam-6261	520	28	(	(	PUNCT
ejpam-6261	520	29	a,0	a,0	NUM
ejpam-6261	520	30	′	′	NUM
ejpam-6261	520	31	)	)	PUNCT
ejpam-6261	520	32	)	)	PUNCT
ejpam-6261	520	33	}	}	PUNCT
ejpam-6261	520	34	≤	≤	NUM
ejpam-6261	520	35	max{rw×x(ϑ	max{rw×x(ϑ	PROPN
ejpam-6261	520	36	,	,	PUNCT
ejpam-6261	520	37	0	0	NUM
ejpam-6261	520	38	′	′	NUM
ejpam-6261	520	39	)	)	PUNCT
ejpam-6261	520	40	,	,	PUNCT
ejpam-6261	520	41	rw×x(a	rw×x(a	NOUN
ejpam-6261	520	42	,	,	PUNCT
ejpam-6261	520	43	0	0	NUM
ejpam-6261	520	44	′	′	NUM
ejpam-6261	520	45	)	)	PUNCT
ejpam-6261	520	46	}	}	PUNCT
ejpam-6261	521	1	eimax{ωw×x(ϑ,0	eimax{ωw×x(ϑ,0	ADJ
ejpam-6261	521	2	′	′	NUM
ejpam-6261	521	3	)	)	PUNCT
ejpam-6261	521	4	,	,	PUNCT
ejpam-6261	521	5	ωw×x(a,0	ωw×x(a,0	PROPN
ejpam-6261	521	6	′	′	NUM
ejpam-6261	521	7	)	)	PUNCT
ejpam-6261	521	8	}	}	PUNCT
ejpam-6261	522	1	=	=	PUNCT
ejpam-6261	522	2	max{max{rw	max{max{rw	X
ejpam-6261	522	3	(	(	PUNCT
ejpam-6261	522	4	ϑ	ϑ	NOUN
ejpam-6261	522	5	)	)	PUNCT
ejpam-6261	522	6	,	,	PUNCT
ejpam-6261	522	7	rx(0	rx(0	PROPN
ejpam-6261	522	8	′	′	NUM
ejpam-6261	522	9	)	)	PUNCT
ejpam-6261	522	10	}	}	PUNCT
ejpam-6261	522	11	,	,	PUNCT
ejpam-6261	522	12	max{rw(a	max{rw(a	X
ejpam-6261	522	13	)	)	PUNCT
ejpam-6261	522	14	,	,	PUNCT
ejpam-6261	522	15	rx(0	rx(0	PROPN
ejpam-6261	522	16	′	′	NUM
ejpam-6261	522	17	)	)	PUNCT
ejpam-6261	522	18	}	}	PUNCT
ejpam-6261	522	19	}	}	PUNCT
ejpam-6261	522	20	∗	∗	NOUN
ejpam-6261	522	21	eimax{max{ωx(0	eimax{max{ωx(0	PROPN
ejpam-6261	522	22	′	′	NUM
ejpam-6261	522	23	)	)	PUNCT
ejpam-6261	522	24	}	}	PUNCT
ejpam-6261	522	25	,	,	PUNCT
ejpam-6261	522	26	max{ω	max{ω	PROPN
ejpam-6261	522	27	,	,	PUNCT
ejpam-6261	522	28	max{ωw(a),ωx(0	max{ωw(a),ωx(0	PROPN
ejpam-6261	522	29	′	′	NUM
ejpam-6261	522	30	)	)	PUNCT
ejpam-6261	522	31	}	}	PUNCT
ejpam-6261	522	32	}	}	PUNCT
ejpam-6261	522	33	=	=	SYM
ejpam-6261	522	34	max{fw(ϑ),fw(a	max{fw(ϑ),fw(a	X
ejpam-6261	522	35	)	)	PUNCT
ejpam-6261	522	36	}	}	PUNCT
ejpam-6261	522	37	.	.	PUNCT
ejpam-6261	523	1	thus	thus	ADV
ejpam-6261	523	2	,	,	PUNCT
ejpam-6261	523	3	fw(ϑ−	fw(ϑ−	ADP
ejpam-6261	523	4	a	a	PRON
ejpam-6261	523	5	)	)	PUNCT
ejpam-6261	523	6	≤	≤	NUM
ejpam-6261	523	7	max{fw(ϑ),fw(a	max{fw(ϑ),fw(a	NUM
ejpam-6261	523	8	)	)	PUNCT
ejpam-6261	523	9	}	}	PUNCT
ejpam-6261	523	10	.	.	PUNCT
ejpam-6261	524	1	further	far	ADV
ejpam-6261	524	2	,	,	PUNCT
ejpam-6261	524	3	fw(ϑa	fw(ϑa	PROPN
ejpam-6261	524	4	)	)	PUNCT
ejpam-6261	524	5	=	=	PUNCT
ejpam-6261	524	6	rw(ϑa)eiωw(ϑa	rw(ϑa)eiωw(ϑa	X
ejpam-6261	524	7	)	)	PUNCT
ejpam-6261	524	8	=	=	PRON
ejpam-6261	524	9	{	{	PUNCT
ejpam-6261	524	10	max{rw(ϑa)eiωw(ϑa	max{rw(ϑa)eiωw(ϑa	NOUN
ejpam-6261	524	11	)	)	PUNCT
ejpam-6261	524	12	,	,	PUNCT
ejpam-6261	524	13	rx(0	rx(0	VERB
ejpam-6261	524	14	′	′	NOUN
ejpam-6261	524	15	0	0	NUM
ejpam-6261	525	1	′	′	NUM
ejpam-6261	525	2	)	)	PUNCT
ejpam-6261	526	1	eiωx(0	eiωx(0	PROPN
ejpam-6261	526	2	′	′	NOUN
ejpam-6261	526	3	0	0	NUM
ejpam-6261	527	1	′	′	NUM
ejpam-6261	527	2	)	)	PUNCT
ejpam-6261	527	3	}	}	PUNCT
ejpam-6261	527	4	}	}	PUNCT
ejpam-6261	527	5	=	=	SYM
ejpam-6261	527	6	{	{	PUNCT
ejpam-6261	527	7	rw×x((ϑ	rw×x((ϑ	PROPN
ejpam-6261	527	8	,	,	PUNCT
ejpam-6261	527	9	0	0	NUM
ejpam-6261	527	10	′	′	NUM
ejpam-6261	527	11	)	)	PUNCT
ejpam-6261	528	1	(	(	PUNCT
ejpam-6261	528	2	a	a	PRON
ejpam-6261	528	3	,	,	PUNCT
ejpam-6261	528	4	0	0	NUM
ejpam-6261	528	5	′	′	NUM
ejpam-6261	528	6	)	)	PUNCT
ejpam-6261	528	7	)	)	PUNCT
ejpam-6261	528	8	}	}	PUNCT
ejpam-6261	528	9	ei{ωw×x((ϑ,0	ei{ωw×x((ϑ,0	VERB
ejpam-6261	528	10	′	′	NOUN
ejpam-6261	528	11	)	)	PUNCT
ejpam-6261	528	12	(	(	PUNCT
ejpam-6261	528	13	a,0	a,0	NUM
ejpam-6261	528	14	′	′	NUM
ejpam-6261	528	15	)	)	PUNCT
ejpam-6261	528	16	)	)	PUNCT
ejpam-6261	528	17	}	}	PUNCT
ejpam-6261	528	18	≤	≤	NUM
ejpam-6261	528	19	max{rw×x(ϑ	max{rw×x(ϑ	PROPN
ejpam-6261	528	20	,	,	PUNCT
ejpam-6261	528	21	0	0	NUM
ejpam-6261	528	22	′	′	NUM
ejpam-6261	528	23	)	)	PUNCT
ejpam-6261	528	24	,	,	PUNCT
ejpam-6261	528	25	rw×x(a	rw×x(a	NOUN
ejpam-6261	528	26	,	,	PUNCT
ejpam-6261	528	27	0	0	NUM
ejpam-6261	528	28	′	′	NUM
ejpam-6261	528	29	)	)	PUNCT
ejpam-6261	528	30	}	}	PUNCT
ejpam-6261	529	1	eimax{ωw×x(ϑ,0	eimax{ωw×x(ϑ,0	ADJ
ejpam-6261	529	2	′	′	NUM
ejpam-6261	529	3	)	)	PUNCT
ejpam-6261	529	4	,	,	PUNCT
ejpam-6261	529	5	ωw×x(a,0	ωw×x(a,0	PROPN
ejpam-6261	529	6	′	′	NUM
ejpam-6261	529	7	)	)	PUNCT
ejpam-6261	529	8	}	}	PUNCT
ejpam-6261	530	1	=	=	SYM
ejpam-6261	530	2	max{max{rw(ϑ	max{max{rw(ϑ	PROPN
ejpam-6261	530	3	)	)	PUNCT
ejpam-6261	530	4	,	,	PUNCT
ejpam-6261	530	5	rx(0	rx(0	PROPN
ejpam-6261	530	6	′	′	NUM
ejpam-6261	530	7	)	)	PUNCT
ejpam-6261	530	8	}	}	PUNCT
ejpam-6261	530	9	,	,	PUNCT
ejpam-6261	530	10	max{rw(a	max{rw(a	X
ejpam-6261	530	11	)	)	PUNCT
ejpam-6261	530	12	,	,	PUNCT
ejpam-6261	530	13	rx(0	rx(0	PROPN
ejpam-6261	530	14	′	′	NUM
ejpam-6261	530	15	)	)	PUNCT
ejpam-6261	530	16	}	}	PUNCT
ejpam-6261	530	17	}	}	PUNCT
ejpam-6261	530	18	∗	∗	NOUN
ejpam-6261	530	19	eimax{max{ωx(0	eimax{max{ωx(0	PROPN
ejpam-6261	530	20	′	′	NUM
ejpam-6261	530	21	)	)	PUNCT
ejpam-6261	530	22	}	}	PUNCT
ejpam-6261	530	23	,	,	PUNCT
ejpam-6261	530	24	max{ω	max{ω	PROPN
ejpam-6261	530	25	,	,	PUNCT
ejpam-6261	530	26	max{ωw(a),ωx(0	max{ωw(a),ωx(0	PROPN
ejpam-6261	530	27	′	′	NUM
ejpam-6261	530	28	)	)	PUNCT
ejpam-6261	530	29	}	}	PUNCT
ejpam-6261	530	30	}	}	PUNCT
ejpam-6261	530	31	=	=	SYM
ejpam-6261	530	32	max{fw(ϑ),fw(a	max{fw(ϑ),fw(a	X
ejpam-6261	530	33	)	)	PUNCT
ejpam-6261	530	34	}	}	PUNCT
ejpam-6261	530	35	.	.	PUNCT
ejpam-6261	531	1	thus	thus	ADV
ejpam-6261	531	2	,	,	PUNCT
ejpam-6261	531	3	fw(ϑa	fw(ϑa	PROPN
ejpam-6261	531	4	)	)	PUNCT
ejpam-6261	531	5	≤	≤	NUM
ejpam-6261	531	6	max{fw(ϑ),fw(a	max{fw(ϑ),fw(a	NOUN
ejpam-6261	531	7	)	)	PUNCT
ejpam-6261	531	8	}	}	PUNCT
ejpam-6261	531	9	.	.	PUNCT
ejpam-6261	532	1	hence	hence	ADV
ejpam-6261	532	2	,	,	PUNCT
ejpam-6261	532	3	we	we	PRON
ejpam-6261	532	4	obtained	obtain	VERB
ejpam-6261	532	5	the	the	DET
ejpam-6261	532	6	result	result	NOUN
ejpam-6261	532	7	.	.	PUNCT
ejpam-6261	533	1	theorem	theorem	ADJ
ejpam-6261	533	2	9	9	NUM
ejpam-6261	533	3	.	.	PUNCT
ejpam-6261	534	1	let	let	VERB
ejpam-6261	534	2	w	w	NOUN
ejpam-6261	534	3	and	and	CCONJ
ejpam-6261	534	4	x	x	SYM
ejpam-6261	534	5	two	two	NUM
ejpam-6261	534	6	cnsss	cnsss	NOUN
ejpam-6261	534	7	of	of	ADP
ejpam-6261	534	8	k1	k1	NOUN
ejpam-6261	534	9	and	and	CCONJ
ejpam-6261	534	10	k2	k2	ADJ
ejpam-6261	534	11	such	such	ADJ
ejpam-6261	534	12	that	that	DET
ejpam-6261	534	13	pw(0	pw(0	NOUN
ejpam-6261	534	14	)	)	PUNCT
ejpam-6261	534	15	≥	≥	NOUN
ejpam-6261	534	16	px(g	px(g	NOUN
ejpam-6261	534	17	)	)	PUNCT
ejpam-6261	534	18	,	,	PUNCT
ejpam-6261	534	19	qw(0	qw(0	PROPN
ejpam-6261	534	20	)	)	PUNCT
ejpam-6261	534	21	≥	≥	NOUN
ejpam-6261	534	22	qx(g	qx(g	NOUN
ejpam-6261	534	23	)	)	PUNCT
ejpam-6261	534	24	and	and	CCONJ
ejpam-6261	534	25	rw(0	rw(0	X
ejpam-6261	534	26	)	)	PUNCT
ejpam-6261	534	27	≥	≥	NOUN
ejpam-6261	534	28	rx(g	rx(g	NOUN
ejpam-6261	534	29	)	)	PUNCT
ejpam-6261	534	30	,	,	PUNCT
ejpam-6261	534	31	∀g	∀g	X
ejpam-6261	534	32	∈	∈	PROPN
ejpam-6261	534	33	k2	k2	NOUN
ejpam-6261	534	34	and	and	CCONJ
ejpam-6261	534	35	0	0	NUM
ejpam-6261	534	36	is	be	AUX
ejpam-6261	534	37	identity	identity	NOUN
ejpam-6261	534	38	of	of	ADP
ejpam-6261	534	39	k1	k1	NOUN
ejpam-6261	534	40	.	.	PUNCT
ejpam-6261	535	1	if	if	SCONJ
ejpam-6261	535	2	w×x	w×x	PROPN
ejpam-6261	535	3	is	be	AUX
ejpam-6261	535	4	cnsr	cnsr	VERB
ejpam-6261	535	5	of	of	ADP
ejpam-6261	535	6	k1×k2	k1×k2	PROPN
ejpam-6261	535	7	,	,	PUNCT
ejpam-6261	535	8	then	then	ADV
ejpam-6261	535	9	x	x	PUNCT
ejpam-6261	535	10	is	be	AUX
ejpam-6261	535	11	a	a	DET
ejpam-6261	535	12	cnsr	cnsr	NOUN
ejpam-6261	535	13	of	of	ADP
ejpam-6261	535	14	k2	k2	NOUN
ejpam-6261	535	15	.	.	PUNCT
ejpam-6261	536	1	proof	proof	NOUN
ejpam-6261	536	2	.	.	PUNCT
ejpam-6261	537	1	the	the	DET
ejpam-6261	537	2	proof	proof	NOUN
ejpam-6261	537	3	is	be	AUX
ejpam-6261	537	4	on	on	ADP
ejpam-6261	537	5	similar	similar	ADJ
ejpam-6261	537	6	lines	line	NOUN
ejpam-6261	537	7	as	as	SCONJ
ejpam-6261	537	8	theorem	theorem	ADJ
ejpam-6261	537	9	4.6	4.6	NUM
ejpam-6261	537	10	.	.	PUNCT
ejpam-6261	538	1	corollary	corollary	ADJ
ejpam-6261	538	2	2	2	NUM
ejpam-6261	538	3	.	.	PUNCT
ejpam-6261	539	1	let	let	VERB
ejpam-6261	539	2	w	w	NOUN
ejpam-6261	539	3	and	and	CCONJ
ejpam-6261	539	4	x	x	PART
ejpam-6261	539	5	be	be	AUX
ejpam-6261	539	6	two	two	NUM
ejpam-6261	539	7	cnsss	cnsss	NOUN
ejpam-6261	539	8	of	of	ADP
ejpam-6261	539	9	k1	k1	NOUN
ejpam-6261	539	10	and	and	CCONJ
ejpam-6261	539	11	k2	k2	NOUN
ejpam-6261	539	12	,	,	PUNCT
ejpam-6261	539	13	respectively	respectively	ADV
ejpam-6261	539	14	.	.	PUNCT
ejpam-6261	540	1	if	if	SCONJ
ejpam-6261	540	2	w×x	w×x	PROPN
ejpam-6261	540	3	is	be	AUX
ejpam-6261	540	4	cnsr	cnsr	VERB
ejpam-6261	540	5	of	of	ADP
ejpam-6261	540	6	k1	k1	PROPN
ejpam-6261	540	7	×k2	×k2	PROPN
ejpam-6261	540	8	,	,	PUNCT
ejpam-6261	540	9	then	then	ADV
ejpam-6261	540	10	w	w	PROPN
ejpam-6261	540	11	is	be	AUX
ejpam-6261	540	12	a	a	DET
ejpam-6261	540	13	cnsr	cnsr	NOUN
ejpam-6261	540	14	of	of	ADP
ejpam-6261	540	15	k1	k1	NOUN
ejpam-6261	540	16	or	or	CCONJ
ejpam-6261	540	17	x	x	NOUN
ejpam-6261	540	18	is	be	AUX
ejpam-6261	540	19	a	a	DET
ejpam-6261	540	20	cnsr	cnsr	NOUN
ejpam-6261	540	21	of	of	ADP
ejpam-6261	540	22	k2	k2	NOUN
ejpam-6261	540	23	.	.	PUNCT
ejpam-6261	541	1	5	5	NUM
ejpam-6261	541	2	.	.	X
ejpam-6261	541	3	conclusion	conclusion	NOUN
ejpam-6261	541	4	in	in	ADP
ejpam-6261	541	5	this	this	DET
ejpam-6261	541	6	manuscript	manuscript	NOUN
ejpam-6261	541	7	,	,	PUNCT
ejpam-6261	541	8	we	we	PRON
ejpam-6261	541	9	have	have	AUX
ejpam-6261	541	10	discussed	discuss	VERB
ejpam-6261	541	11	the	the	DET
ejpam-6261	541	12	complex	complex	ADJ
ejpam-6261	541	13	neutrosophic	neutrosophic	ADJ
ejpam-6261	541	14	subring	subring	NOUN
ejpam-6261	541	15	,	,	PUNCT
ejpam-6261	541	16	intersection	intersection	NOUN
ejpam-6261	541	17	,	,	PUNCT
ejpam-6261	541	18	and	and	CCONJ
ejpam-6261	541	19	level	level	NOUN
ejpam-6261	541	20	subset	subset	NOUN
ejpam-6261	541	21	of	of	ADP
ejpam-6261	541	22	complex	complex	ADJ
ejpam-6261	541	23	neutrosophic	neutrosophic	ADJ
ejpam-6261	541	24	subring	subring	NOUN
ejpam-6261	541	25	.	.	PUNCT
ejpam-6261	542	1	we	we	PRON
ejpam-6261	542	2	have	have	AUX
ejpam-6261	542	3	demonstrated	demonstrate	VERB
ejpam-6261	542	4	that	that	SCONJ
ejpam-6261	542	5	every	every	DET
ejpam-6261	542	6	complex	complex	ADJ
ejpam-6261	542	7	neutrosophic	neutrosophic	ADJ
ejpam-6261	542	8	subring	subring	NOUN
ejpam-6261	542	9	generates	generate	VERB
ejpam-6261	542	10	two	two	NUM
ejpam-6261	542	11	neutrosophic	neutrosophic	ADJ
ejpam-6261	542	12	subrings	subring	NOUN
ejpam-6261	542	13	and	and	CCONJ
ejpam-6261	542	14	examined	examine	VERB
ejpam-6261	542	15	important	important	ADJ
ejpam-6261	542	16	aspects	aspect	NOUN
ejpam-6261	542	17	of	of	ADP
ejpam-6261	542	18	this	this	DET
ejpam-6261	542	19	fact	fact	NOUN
ejpam-6261	542	20	.	.	PUNCT
ejpam-6261	543	1	we	we	PRON
ejpam-6261	543	2	have	have	AUX
ejpam-6261	543	3	shown	show	VERB
ejpam-6261	543	4	that	that	SCONJ
ejpam-6261	543	5	the	the	DET
ejpam-6261	543	6	level	level	NOUN
ejpam-6261	543	7	subset	subset	NOUN
ejpam-6261	543	8	of	of	ADP
ejpam-6261	543	9	the	the	DET
ejpam-6261	543	10	complex	complex	ADJ
ejpam-6261	543	11	neutrosophic	neutrosophic	ADJ
ejpam-6261	543	12	m.	m.	NOUN
ejpam-6261	543	13	h.	h.	PROPN
ejpam-6261	543	14	mateen	mateen	PROPN
ejpam-6261	543	15	et	et	PROPN
ejpam-6261	543	16	al	al	PROPN
ejpam-6261	543	17	.	.	PUNCT
ejpam-6261	543	18	/	/	SYM
ejpam-6261	543	19	eur	eur	PROPN
ejpam-6261	543	20	.	.	PUNCT
ejpam-6261	544	1	j.	j.	PROPN
ejpam-6261	544	2	pure	pure	PROPN
ejpam-6261	544	3	appl	appl	PROPN
ejpam-6261	544	4	.	.	PROPN
ejpam-6261	544	5	math	math	PROPN
ejpam-6261	544	6	,	,	PUNCT
ejpam-6261	544	7	18	18	NUM
ejpam-6261	544	8	(	(	PUNCT
ejpam-6261	544	9	4	4	NUM
ejpam-6261	544	10	)	)	PUNCT
ejpam-6261	544	11	(	(	PUNCT
ejpam-6261	544	12	2025	2025	NUM
ejpam-6261	544	13	)	)	PUNCT
ejpam-6261	544	14	,	,	PUNCT
ejpam-6261	544	15	6261	6261	NUM
ejpam-6261	544	16	21	21	NUM
ejpam-6261	544	17	of	of	ADP
ejpam-6261	544	18	23	23	NUM
ejpam-6261	544	19	subring	subring	NOUN
ejpam-6261	544	20	forms	form	VERB
ejpam-6261	544	21	the	the	DET
ejpam-6261	544	22	subring	subring	NOUN
ejpam-6261	544	23	of	of	ADP
ejpam-6261	544	24	the	the	DET
ejpam-6261	544	25	ring	ring	NOUN
ejpam-6261	544	26	and	and	CCONJ
ejpam-6261	544	27	we	we	PRON
ejpam-6261	544	28	have	have	AUX
ejpam-6261	544	29	talked	talk	VERB
ejpam-6261	544	30	about	about	ADP
ejpam-6261	544	31	some	some	PRON
ejpam-6261	544	32	of	of	ADP
ejpam-6261	544	33	the	the	DET
ejpam-6261	544	34	algebraic	algebraic	ADJ
ejpam-6261	544	35	characteristics	characteristic	NOUN
ejpam-6261	544	36	of	of	ADP
ejpam-6261	544	37	the	the	DET
ejpam-6261	544	38	level	level	NOUN
ejpam-6261	544	39	subset	subset	NOUN
ejpam-6261	544	40	.	.	PUNCT
ejpam-6261	545	1	in	in	ADP
ejpam-6261	545	2	future	future	NOUN
ejpam-6261	545	3	we	we	PRON
ejpam-6261	545	4	intend	intend	VERB
ejpam-6261	545	5	to	to	PART
ejpam-6261	545	6	extend	extend	VERB
ejpam-6261	545	7	this	this	DET
ejpam-6261	545	8	approch	approch	NOUN
ejpam-6261	545	9	to	to	PART
ejpam-6261	545	10	subfield	subfield	VERB
ejpam-6261	545	11	,	,	PUNCT
ejpam-6261	545	12	submodules	submodule	NOUN
ejpam-6261	545	13	and	and	CCONJ
ejpam-6261	545	14	bck	bck	VERB
ejpam-6261	545	15	/	/	SYM
ejpam-6261	545	16	bci	bci	PROPN
ejpam-6261	545	17	algebra	algebra	NOUN
ejpam-6261	545	18	.	.	PUNCT
ejpam-6261	546	1	also	also	ADV
ejpam-6261	546	2	we	we	PRON
ejpam-6261	546	3	intend	intend	VERB
ejpam-6261	546	4	to	to	PART
ejpam-6261	546	5	introduce	introduce	VERB
ejpam-6261	546	6	applications	application	NOUN
ejpam-6261	546	7	of	of	ADP
ejpam-6261	546	8	these	these	DET
ejpam-6261	546	9	defined	define	VERB
ejpam-6261	546	10	algebraic	algebraic	ADJ
ejpam-6261	546	11	structures	structure	NOUN
ejpam-6261	546	12	to	to	ADP
ejpam-6261	546	13	practical	practical	ADJ
ejpam-6261	546	14	and	and	CCONJ
ejpam-6261	546	15	theoretical	theoretical	ADJ
ejpam-6261	546	16	problems	problem	NOUN
ejpam-6261	546	17	.	.	PUNCT
ejpam-6261	547	1	acknowledgements	acknowledgement	NOUN
ejpam-6261	547	2	the	the	DET
ejpam-6261	547	3	second	second	ADJ
ejpam-6261	547	4	author	author	NOUN
ejpam-6261	547	5	thanks	thank	NOUN
ejpam-6261	547	6	the	the	DET
ejpam-6261	547	7	ministry	ministry	PROPN
ejpam-6261	547	8	of	of	ADP
ejpam-6261	547	9	defence	defence	PROPN
ejpam-6261	547	10	of	of	ADP
ejpam-6261	547	11	the	the	DET
ejpam-6261	547	12	czech	czech	PROPN
ejpam-6261	547	13	republic	republic	NOUN
ejpam-6261	547	14	for	for	ADP
ejpam-6261	547	15	the	the	DET
ejpam-6261	547	16	support	support	NOUN
ejpam-6261	547	17	via	via	ADP
ejpam-6261	547	18	grant	grant	NOUN
ejpam-6261	547	19	varops	varop	NOUN
ejpam-6261	547	20	.	.	PUNCT
ejpam-6261	548	1	data	datum	NOUN
ejpam-6261	548	2	availability	availability	NOUN
ejpam-6261	548	3	statement	statement	NOUN
ejpam-6261	548	4	:	:	PUNCT
ejpam-6261	548	5	no	no	DET
ejpam-6261	548	6	data	datum	NOUN
ejpam-6261	548	7	were	be	AUX
ejpam-6261	548	8	used	use	VERB
ejpam-6261	548	9	in	in	ADP
ejpam-6261	548	10	this	this	DET
ejpam-6261	548	11	study	study	NOUN
ejpam-6261	548	12	.	.	PUNCT
ejpam-6261	549	1	conflict	conflict	NOUN
ejpam-6261	549	2	of	of	ADP
ejpam-6261	549	3	interest	interest	NOUN
ejpam-6261	549	4	:	:	PUNCT
ejpam-6261	549	5	the	the	DET
ejpam-6261	549	6	authors	author	NOUN
ejpam-6261	549	7	state	state	VERB
ejpam-6261	549	8	that	that	SCONJ
ejpam-6261	549	9	they	they	PRON
ejpam-6261	549	10	do	do	AUX
ejpam-6261	549	11	not	not	PART
ejpam-6261	549	12	have	have	VERB
ejpam-6261	549	13	a	a	DET
ejpam-6261	549	14	conflict	conflict	NOUN
ejpam-6261	549	15	of	of	ADP
ejpam-6261	549	16	interest	interest	NOUN
ejpam-6261	549	17	in	in	ADP
ejpam-6261	549	18	relation	relation	NOUN
ejpam-6261	549	19	to	to	ADP
ejpam-6261	549	20	the	the	DET
ejpam-6261	549	21	publication	publication	NOUN
ejpam-6261	549	22	of	of	ADP
ejpam-6261	549	23	this	this	DET
ejpam-6261	549	24	research	research	NOUN
ejpam-6261	549	25	article	article	NOUN
ejpam-6261	549	26	.	.	PUNCT
ejpam-6261	550	1	references	reference	NOUN
ejpam-6261	550	2	[	[	X
ejpam-6261	550	3	1	1	NUM
ejpam-6261	550	4	]	]	PUNCT
ejpam-6261	550	5	i.	i.	PROPN
ejpam-6261	550	6	kleiner	kleiner	PROPN
ejpam-6261	550	7	.	.	PUNCT
ejpam-6261	551	1	from	from	ADP
ejpam-6261	551	2	numbers	number	NOUN
ejpam-6261	551	3	to	to	ADP
ejpam-6261	551	4	rings	ring	NOUN
ejpam-6261	551	5	:	:	PUNCT
ejpam-6261	551	6	the	the	DET
ejpam-6261	551	7	early	early	ADJ
ejpam-6261	551	8	history	history	NOUN
ejpam-6261	551	9	of	of	ADP
ejpam-6261	551	10	ring	ring	NOUN
ejpam-6261	551	11	theory	theory	NOUN
ejpam-6261	551	12	.	.	PUNCT
ejpam-6261	552	1	elemente	elemente	PROPN
ejpam-6261	552	2	der	der	PROPN
ejpam-6261	552	3	mathematik	mathematik	PROPN
ejpam-6261	552	4	,	,	PUNCT
ejpam-6261	552	5	53:1835	53:1835	NUM
ejpam-6261	552	6	,	,	PUNCT
ejpam-6261	552	7	1998	1998	NUM
ejpam-6261	552	8	.	.	PUNCT
ejpam-6261	553	1	[	[	X
ejpam-6261	553	2	2	2	NUM
ejpam-6261	553	3	]	]	X
ejpam-6261	553	4	technical	technical	ADJ
ejpam-6261	553	5	university	university	PROPN
ejpam-6261	553	6	of	of	ADP
ejpam-6261	553	7	catalonia	catalonia	PROPN
ejpam-6261	553	8	,	,	PUNCT
ejpam-6261	553	9	department	department	NOUN
ejpam-6261	553	10	of	of	ADP
ejpam-6261	553	11	mathematics	mathematic	NOUN
ejpam-6261	553	12	and	and	CCONJ
ejpam-6261	553	13	statistics	statistic	NOUN
ejpam-6261	553	14	.	.	PUNCT
ejpam-6261	554	1	conferences	conference	NOUN
ejpam-6261	554	2	of	of	ADP
ejpam-6261	554	3	the	the	DET
ejpam-6261	554	4	mathematics	mathematics	PROPN
ejpam-6261	554	5	and	and	CCONJ
ejpam-6261	554	6	statistics	statistics	PROPN
ejpam-6261	554	7	department	department	PROPN
ejpam-6261	554	8	of	of	ADP
ejpam-6261	554	9	the	the	DET
ejpam-6261	554	10	technical	technical	PROPN
ejpam-6261	554	11	university	university	PROPN
ejpam-6261	554	12	of	of	ADP
ejpam-6261	554	13	catalonia	catalonia	PROPN
ejpam-6261	554	14	.	.	PUNCT
ejpam-6261	555	1	emmy	emmy	PROPN
ejpam-6261	555	2	noether	noether	ADJ
ejpam-6261	555	3	course	course	NOUN
ejpam-6261	555	4	.	.	PUNCT
ejpam-6261	556	1	https://upcommons.upc.edu/bitstream/handle/	https://upcommons.upc.edu/bitstream/handle/	PROPN
ejpam-6261	556	2	2117/81399	2117/81399	NUM
ejpam-6261	556	3	/	/	SYM
ejpam-6261	556	4	cfme	cfme	NOUN
ejpam-6261	556	5	-	-	PUNCT
ejpam-6261	556	6	vol-6.pdf?sequence=1\&isallowed	vol-6.pdf?sequence=1\&isallowe	VERB
ejpam-6261	556	7	=	=	SYM
ejpam-6261	556	8	y.	y.	NOUN
ejpam-6261	556	9	accessed	access	VERB
ejpam-6261	556	10	on	on	ADP
ejpam-6261	556	11	21	21	NUM
ejpam-6261	556	12	march	march	NOUN
ejpam-6261	556	13	2018	2018	NUM
ejpam-6261	556	14	.	.	PUNCT
ejpam-6261	557	1	[	[	X
ejpam-6261	557	2	3	3	X
ejpam-6261	557	3	]	]	X
ejpam-6261	557	4	l.	l.	PROPN
ejpam-6261	557	5	a.	a.	PROPN
ejpam-6261	557	6	zadeh	zadeh	PROPN
ejpam-6261	557	7	.	.	PUNCT
ejpam-6261	557	8	fuzzy	fuzzy	ADJ
ejpam-6261	557	9	sets	set	NOUN
ejpam-6261	557	10	.	.	PUNCT
ejpam-6261	558	1	information	information	NOUN
ejpam-6261	558	2	and	and	CCONJ
ejpam-6261	558	3	control	control	NOUN
ejpam-6261	558	4	,	,	PUNCT
ejpam-6261	558	5	8(3):338–353	8(3):338–353	NUM
ejpam-6261	558	6	,	,	PUNCT
ejpam-6261	558	7	1965	1965	NUM
ejpam-6261	558	8	.	.	PUNCT
ejpam-6261	559	1	[	[	X
ejpam-6261	559	2	4	4	X
ejpam-6261	559	3	]	]	PUNCT
ejpam-6261	559	4	k.	k.	PROPN
ejpam-6261	559	5	t.	t.	PROPN
ejpam-6261	559	6	atanassov	atanassov	PROPN
ejpam-6261	559	7	and	and	CCONJ
ejpam-6261	559	8	s.	s.	PROPN
ejpam-6261	559	9	stoeva	stoeva	PROPN
ejpam-6261	559	10	.	.	PUNCT
ejpam-6261	560	1	intuitionistic	intuitionistic	ADJ
ejpam-6261	560	2	fuzzy	fuzzy	ADJ
ejpam-6261	560	3	sets	set	NOUN
ejpam-6261	560	4	.	.	PUNCT
ejpam-6261	561	1	fuzzy	fuzzy	ADJ
ejpam-6261	561	2	sets	set	NOUN
ejpam-6261	561	3	and	and	CCONJ
ejpam-6261	561	4	systems	system	NOUN
ejpam-6261	561	5	,	,	PUNCT
ejpam-6261	561	6	20(1):87–96	20(1):87–96	NUM
ejpam-6261	561	7	,	,	PUNCT
ejpam-6261	561	8	1986	1986	NUM
ejpam-6261	561	9	.	.	PUNCT
ejpam-6261	562	1	[	[	X
ejpam-6261	562	2	5	5	X
ejpam-6261	562	3	]	]	PUNCT
ejpam-6261	562	4	h.	h.	PROPN
ejpam-6261	562	5	alolaiyan	alolaiyan	PROPN
ejpam-6261	562	6	,	,	PUNCT
ejpam-6261	562	7	m.	m.	NOUN
ejpam-6261	562	8	hayat	hayat	PROPN
ejpam-6261	562	9	,	,	PUNCT
ejpam-6261	562	10	u.	u.	PROPN
ejpam-6261	562	11	shuaib	shuaib	PROPN
ejpam-6261	562	12	,	,	PUNCT
ejpam-6261	562	13	a.	a.	NOUN
ejpam-6261	562	14	razaq	razaq	NOUN
ejpam-6261	562	15	,	,	PUNCT
ejpam-6261	562	16	m.	m.	NOUN
ejpam-6261	562	17	a.	a.	PROPN
ejpam-6261	562	18	salman	salman	PROPN
ejpam-6261	562	19	,	,	PUNCT
ejpam-6261	562	20	and	and	CCONJ
ejpam-6261	562	21	q.	q.	PROPN
ejpam-6261	562	22	xin	xin	PROPN
ejpam-6261	562	23	.	.	PUNCT
ejpam-6261	563	1	optimizing	optimize	VERB
ejpam-6261	563	2	bioremediation	bioremediation	NOUN
ejpam-6261	563	3	techniques	technique	NOUN
ejpam-6261	563	4	for	for	ADP
ejpam-6261	563	5	soil	soil	NOUN
ejpam-6261	563	6	decontamination	decontamination	NOUN
ejpam-6261	563	7	in	in	ADP
ejpam-6261	563	8	a	a	DET
ejpam-6261	563	9	linguistic	linguistic	ADJ
ejpam-6261	563	10	intuitionistic	intuitionistic	ADJ
ejpam-6261	563	11	fuzzy	fuzzy	ADJ
ejpam-6261	563	12	framework	framework	NOUN
ejpam-6261	563	13	.	.	PUNCT
ejpam-6261	564	1	scientific	scientific	ADJ
ejpam-6261	564	2	reports	report	NOUN
ejpam-6261	564	3	,	,	PUNCT
ejpam-6261	564	4	14(1):15979	14(1):15979	NUM
ejpam-6261	564	5	,	,	PUNCT
ejpam-6261	564	6	2024	2024	NUM
ejpam-6261	564	7	.	.	PUNCT
ejpam-6261	565	1	[	[	X
ejpam-6261	565	2	6	6	NUM
ejpam-6261	565	3	]	]	PUNCT
ejpam-6261	565	4	a.	a.	NOUN
ejpam-6261	565	5	rosenfeld	rosenfeld	PROPN
ejpam-6261	565	6	.	.	PUNCT
ejpam-6261	566	1	fuzzy	fuzzy	ADJ
ejpam-6261	566	2	fields	field	NOUN
ejpam-6261	566	3	.	.	PUNCT
ejpam-6261	567	1	journal	journal	PROPN
ejpam-6261	567	2	of	of	ADP
ejpam-6261	567	3	mathematical	mathematical	ADJ
ejpam-6261	567	4	analysis	analysis	NOUN
ejpam-6261	567	5	and	and	CCONJ
ejpam-6261	567	6	applications	application	NOUN
ejpam-6261	567	7	,	,	PUNCT
ejpam-6261	567	8	35(3):512–517	35(3):512–517	PROPN
ejpam-6261	567	9	,	,	PUNCT
ejpam-6261	567	10	1971	1971	NUM
ejpam-6261	567	11	.	.	PUNCT
ejpam-6261	568	1	[	[	X
ejpam-6261	568	2	7	7	NUM
ejpam-6261	568	3	]	]	X
ejpam-6261	568	4	r.	r.	PROPN
ejpam-6261	568	5	biswas	biswas	PROPN
ejpam-6261	568	6	.	.	PUNCT
ejpam-6261	569	1	intuitionistic	intuitionistic	ADJ
ejpam-6261	569	2	fuzzy	fuzzy	ADJ
ejpam-6261	569	3	subgroups	subgroup	NOUN
ejpam-6261	569	4	.	.	PUNCT
ejpam-6261	570	1	mathematical	mathematical	ADJ
ejpam-6261	570	2	forum	forum	PROPN
ejpam-6261	570	3	,	,	PUNCT
ejpam-6261	570	4	10:37–46	10:37–46	NUM
ejpam-6261	570	5	,	,	PUNCT
ejpam-6261	570	6	1989	1989	NUM
ejpam-6261	570	7	.	.	PUNCT
ejpam-6261	571	1	[	[	X
ejpam-6261	571	2	8	8	NUM
ejpam-6261	571	3	]	]	X
ejpam-6261	571	4	f.	f.	PROPN
ejpam-6261	571	5	smarandache	smarandache	PROPN
ejpam-6261	571	6	.	.	PUNCT
ejpam-6261	571	7	neutrosophic	neutrosophic	PROPN
ejpam-6261	571	8	set	set	PROPN
ejpam-6261	571	9	,	,	PUNCT
ejpam-6261	571	10	a	a	DET
ejpam-6261	571	11	generalization	generalization	NOUN
ejpam-6261	571	12	of	of	ADP
ejpam-6261	571	13	the	the	DET
ejpam-6261	571	14	intuitionistic	intuitionistic	ADJ
ejpam-6261	571	15	fuzzy	fuzzy	ADJ
ejpam-6261	571	16	sets	set	NOUN
ejpam-6261	571	17	.	.	PUNCT
ejpam-6261	572	1	international	international	ADJ
ejpam-6261	572	2	journal	journal	NOUN
ejpam-6261	572	3	of	of	ADP
ejpam-6261	572	4	pure	pure	ADJ
ejpam-6261	572	5	and	and	CCONJ
ejpam-6261	572	6	applied	applied	ADJ
ejpam-6261	572	7	mathematics	mathematic	NOUN
ejpam-6261	572	8	,	,	PUNCT
ejpam-6261	572	9	24:287–297	24:287–297	PROPN
ejpam-6261	572	10	,	,	PUNCT
ejpam-6261	572	11	2005	2005	NUM
ejpam-6261	572	12	.	.	PUNCT
ejpam-6261	573	1	[	[	X
ejpam-6261	573	2	9	9	NUM
ejpam-6261	573	3	]	]	PUNCT
ejpam-6261	573	4	a.	a.	NOUN
ejpam-6261	573	5	a.	a.	PROPN
ejpam-6261	573	6	a.	a.	PROPN
ejpam-6261	573	7	agboola	agboola	PROPN
ejpam-6261	573	8	and	and	CCONJ
ejpam-6261	573	9	b.	b.	PROPN
ejpam-6261	573	10	davvaz	davvaz	PROPN
ejpam-6261	573	11	.	.	PUNCT
ejpam-6261	574	1	introduction	introduction	NOUN
ejpam-6261	574	2	to	to	ADP
ejpam-6261	574	3	neutrosophic	neutrosophic	ADJ
ejpam-6261	574	4	bci	bci	PROPN
ejpam-6261	574	5	/	/	SYM
ejpam-6261	574	6	bck	bck	NOUN
ejpam-6261	574	7	-	-	PUNCT
ejpam-6261	574	8	algebras	algebras	PROPN
ejpam-6261	574	9	.	.	PUNCT
ejpam-6261	575	1	international	international	ADJ
ejpam-6261	575	2	journal	journal	PROPN
ejpam-6261	575	3	of	of	ADP
ejpam-6261	575	4	mathematics	mathematics	PROPN
ejpam-6261	575	5	and	and	CCONJ
ejpam-6261	575	6	mathematical	mathematical	ADJ
ejpam-6261	575	7	sciences	science	NOUN
ejpam-6261	575	8	,	,	PUNCT
ejpam-6261	575	9	pages	page	NOUN
ejpam-6261	575	10	1–6	1–6	NUM
ejpam-6261	575	11	,	,	PUNCT
ejpam-6261	575	12	2015	2015	NUM
ejpam-6261	575	13	.	.	PUNCT
ejpam-6261	576	1	[	[	X
ejpam-6261	576	2	10	10	NUM
ejpam-6261	576	3	]	]	PUNCT
ejpam-6261	576	4	v.	v.	X
ejpam-6261	576	5	cetkin	cetkin	PROPN
ejpam-6261	576	6	and	and	CCONJ
ejpam-6261	576	7	h.	h.	PROPN
ejpam-6261	576	8	aygun	aygun	PROPN
ejpam-6261	576	9	.	.	PUNCT
ejpam-6261	577	1	an	an	DET
ejpam-6261	577	2	approach	approach	NOUN
ejpam-6261	577	3	to	to	ADP
ejpam-6261	577	4	neutrosophic	neutrosophic	ADJ
ejpam-6261	577	5	subgroup	subgroup	NOUN
ejpam-6261	577	6	and	and	CCONJ
ejpam-6261	577	7	its	its	PRON
ejpam-6261	577	8	fundamental	fundamental	ADJ
ejpam-6261	577	9	properties	property	NOUN
ejpam-6261	577	10	.	.	PUNCT
ejpam-6261	578	1	journal	journal	NOUN
ejpam-6261	578	2	of	of	ADP
ejpam-6261	578	3	intelligent	intelligent	ADJ
ejpam-6261	578	4	&	&	CCONJ
ejpam-6261	578	5	fuzzy	fuzzy	ADJ
ejpam-6261	578	6	systems	system	NOUN
ejpam-6261	578	7	,	,	PUNCT
ejpam-6261	578	8	29(5):1941–1947	29(5):1941–1947	NUM
ejpam-6261	578	9	,	,	PUNCT
ejpam-6261	578	10	2015	2015	NUM
ejpam-6261	578	11	.	.	PUNCT
ejpam-6261	579	1	[	[	X
ejpam-6261	579	2	11	11	NUM
ejpam-6261	579	3	]	]	PUNCT
ejpam-6261	579	4	s.	s.	PROPN
ejpam-6261	579	5	z.	z.	PROPN
ejpam-6261	579	6	song	song	PROPN
ejpam-6261	579	7	,	,	PUNCT
ejpam-6261	579	8	f.	f.	PROPN
ejpam-6261	579	9	smarandache	smarandache	PROPN
ejpam-6261	579	10	,	,	PUNCT
ejpam-6261	579	11	and	and	CCONJ
ejpam-6261	579	12	y.	y.	PROPN
ejpam-6261	579	13	b.	b.	PROPN
ejpam-6261	579	14	jun	jun	PROPN
ejpam-6261	579	15	.	.	PROPN
ejpam-6261	579	16	neutrosophic	neutrosophic	PROPN
ejpam-6261	579	17	commutative	commutative	ADJ
ejpam-6261	579	18	n	n	CCONJ
ejpam-6261	579	19	-	-	PUNCT
ejpam-6261	579	20	ideals	ideal	NOUN
ejpam-6261	579	21	in	in	ADP
ejpam-6261	579	22	bck	bck	NOUN
ejpam-6261	579	23	-	-	PUNCT
ejpam-6261	579	24	algebras	algebras	PROPN
ejpam-6261	579	25	.	.	PUNCT
ejpam-6261	580	1	information	information	NOUN
ejpam-6261	580	2	,	,	PUNCT
ejpam-6261	580	3	8(4):130	8(4):130	NUM
ejpam-6261	580	4	,	,	PUNCT
ejpam-6261	580	5	2017	2017	NUM
ejpam-6261	580	6	.	.	PUNCT
ejpam-6261	581	1	[	[	X
ejpam-6261	581	2	12	12	NUM
ejpam-6261	581	3	]	]	PUNCT
ejpam-6261	581	4	t.	t.	NOUN
ejpam-6261	581	5	chalapathi	chalapathi	NOUN
ejpam-6261	581	6	and	and	CCONJ
ejpam-6261	581	7	r.	r.	PROPN
ejpam-6261	581	8	v.	v.	PROPN
ejpam-6261	581	9	k.	k.	PROPN
ejpam-6261	581	10	kumar	kumar	PROPN
ejpam-6261	581	11	.	.	PROPN
ejpam-6261	582	1	neutrosophic	neutrosophic	ADJ
ejpam-6261	582	2	graphs	graph	NOUN
ejpam-6261	582	3	of	of	ADP
ejpam-6261	582	4	finite	finite	ADJ
ejpam-6261	582	5	groups	group	NOUN
ejpam-6261	582	6	.	.	PUNCT
ejpam-6261	583	1	neutrosophic	neutrosophic	ADJ
ejpam-6261	583	2	sets	set	NOUN
ejpam-6261	583	3	and	and	CCONJ
ejpam-6261	583	4	systems	system	NOUN
ejpam-6261	583	5	,	,	PUNCT
ejpam-6261	583	6	15:22–30	15:22–30	NUM
ejpam-6261	583	7	,	,	PUNCT
ejpam-6261	583	8	2017	2017	NUM
ejpam-6261	583	9	.	.	PUNCT
ejpam-6261	584	1	m.	m.	PROPN
ejpam-6261	584	2	h.	h.	PROPN
ejpam-6261	584	3	mateen	mateen	PROPN
ejpam-6261	584	4	et	et	PROPN
ejpam-6261	584	5	al	al	PROPN
ejpam-6261	584	6	.	.	PUNCT
ejpam-6261	584	7	/	/	SYM
ejpam-6261	584	8	eur	eur	PROPN
ejpam-6261	584	9	.	.	PUNCT
ejpam-6261	585	1	j.	j.	PROPN
ejpam-6261	585	2	pure	pure	PROPN
ejpam-6261	585	3	appl	appl	PROPN
ejpam-6261	585	4	.	.	PROPN
ejpam-6261	585	5	math	math	PROPN
ejpam-6261	585	6	,	,	PUNCT
ejpam-6261	585	7	18	18	NUM
ejpam-6261	585	8	(	(	PUNCT
ejpam-6261	585	9	4	4	NUM
ejpam-6261	585	10	)	)	PUNCT
ejpam-6261	585	11	(	(	PUNCT
ejpam-6261	585	12	2025	2025	NUM
ejpam-6261	585	13	)	)	PUNCT
ejpam-6261	585	14	,	,	PUNCT
ejpam-6261	585	15	6261	6261	NUM
ejpam-6261	585	16	22	22	NUM
ejpam-6261	585	17	of	of	ADP
ejpam-6261	585	18	23	23	NUM
ejpam-6261	586	1	[	[	SYM
ejpam-6261	586	2	13	13	NUM
ejpam-6261	586	3	]	]	PUNCT
ejpam-6261	586	4	x.	x.	PROPN
ejpam-6261	586	5	zhang	zhang	PROPN
ejpam-6261	586	6	,	,	PUNCT
ejpam-6261	586	7	f.	f.	PROPN
ejpam-6261	586	8	smarandache	smarandache	PROPN
ejpam-6261	586	9	,	,	PUNCT
ejpam-6261	586	10	and	and	CCONJ
ejpam-6261	586	11	x.	x.	PROPN
ejpam-6261	586	12	liang	liang	PROPN
ejpam-6261	586	13	.	.	PUNCT
ejpam-6261	587	1	neutrosophic	neutrosophic	PROPN
ejpam-6261	587	2	duplet	duplet	PROPN
ejpam-6261	587	3	semi	semi	NOUN
ejpam-6261	587	4	-	-	NOUN
ejpam-6261	587	5	group	group	NOUN
ejpam-6261	587	6	and	and	CCONJ
ejpam-6261	587	7	cancellable	cancellable	ADJ
ejpam-6261	587	8	neutrosophic	neutrosophic	ADJ
ejpam-6261	587	9	triplet	triplet	NOUN
ejpam-6261	587	10	groups	group	NOUN
ejpam-6261	587	11	.	.	PUNCT
ejpam-6261	588	1	symmetry	symmetry	NOUN
ejpam-6261	588	2	,	,	PUNCT
ejpam-6261	588	3	9(11):275	9(11):275	NUM
ejpam-6261	588	4	,	,	PUNCT
ejpam-6261	588	5	2017	2017	NUM
ejpam-6261	588	6	.	.	PUNCT
ejpam-6261	589	1	[	[	X
ejpam-6261	589	2	14	14	NUM
ejpam-6261	589	3	]	]	X
ejpam-6261	589	4	r.	r.	PROPN
ejpam-6261	589	5	a.	a.	PROPN
ejpam-6261	589	6	borzooei	borzooei	PROPN
ejpam-6261	589	7	,	,	PUNCT
ejpam-6261	589	8	x.	x.	PROPN
ejpam-6261	589	9	zhang	zhang	PROPN
ejpam-6261	589	10	,	,	PUNCT
ejpam-6261	589	11	f.	f.	PROPN
ejpam-6261	589	12	smarandache	smarandache	PROPN
ejpam-6261	589	13	,	,	PUNCT
ejpam-6261	589	14	and	and	CCONJ
ejpam-6261	589	15	y.	y.	PROPN
ejpam-6261	589	16	b.	b.	PROPN
ejpam-6261	589	17	jun	jun	PROPN
ejpam-6261	589	18	.	.	PROPN
ejpam-6261	590	1	commutative	commutative	PROPN
ejpam-6261	590	2	generalized	generalize	VERB
ejpam-6261	590	3	neutrosophic	neutrosophic	ADJ
ejpam-6261	590	4	ideals	ideal	NOUN
ejpam-6261	590	5	in	in	ADP
ejpam-6261	590	6	bck	bck	NOUN
ejpam-6261	590	7	-	-	PUNCT
ejpam-6261	590	8	algebras	algebras	PROPN
ejpam-6261	590	9	.	.	PUNCT
ejpam-6261	590	10	symmetry	symmetry	PROPN
ejpam-6261	590	11	,	,	PUNCT
ejpam-6261	590	12	10(8):350	10(8):350	NUM
ejpam-6261	590	13	,	,	PUNCT
ejpam-6261	590	14	2018	2018	NUM
ejpam-6261	590	15	.	.	PUNCT
ejpam-6261	591	1	[	[	X
ejpam-6261	591	2	15	15	NUM
ejpam-6261	591	3	]	]	X
ejpam-6261	591	4	y.	y.	PROPN
ejpam-6261	591	5	b.	b.	PROPN
ejpam-6261	591	6	jun	jun	PROPN
ejpam-6261	591	7	,	,	PUNCT
ejpam-6261	591	8	s.	s.	PROPN
ejpam-6261	591	9	j.	j.	PROPN
ejpam-6261	591	10	kim	kim	PROPN
ejpam-6261	591	11	,	,	PUNCT
ejpam-6261	591	12	and	and	CCONJ
ejpam-6261	591	13	f.	f.	PROPN
ejpam-6261	591	14	smarandache	smarandache	PROPN
ejpam-6261	591	15	.	.	PUNCT
ejpam-6261	592	1	interval	interval	NOUN
ejpam-6261	592	2	neutrosophic	neutrosophic	ADJ
ejpam-6261	592	3	sets	set	NOUN
ejpam-6261	592	4	with	with	ADP
ejpam-6261	592	5	applications	application	NOUN
ejpam-6261	592	6	in	in	ADP
ejpam-6261	592	7	bck	bck	PROPN
ejpam-6261	592	8	/	/	SYM
ejpam-6261	592	9	bci	bci	NOUN
ejpam-6261	592	10	-	-	NOUN
ejpam-6261	592	11	algebra	algebra	NOUN
ejpam-6261	592	12	.	.	PUNCT
ejpam-6261	593	1	axioms	axiom	NOUN
ejpam-6261	593	2	,	,	PUNCT
ejpam-6261	593	3	7(2):23	7(2):23	ADJ
ejpam-6261	593	4	,	,	PUNCT
ejpam-6261	593	5	2018	2018	NUM
ejpam-6261	593	6	.	.	PUNCT
ejpam-6261	594	1	[	[	X
ejpam-6261	594	2	16	16	NUM
ejpam-6261	594	3	]	]	X
ejpam-6261	594	4	y.	y.	PROPN
ejpam-6261	594	5	b.	b.	PROPN
ejpam-6261	594	6	jun	jun	PROPN
ejpam-6261	594	7	,	,	PUNCT
ejpam-6261	594	8	f.	f.	PROPN
ejpam-6261	594	9	smarandache	smarandache	PROPN
ejpam-6261	594	10	,	,	PUNCT
ejpam-6261	594	11	s.	s.	PROPN
ejpam-6261	594	12	z.	z.	PROPN
ejpam-6261	594	13	song	song	PROPN
ejpam-6261	594	14	,	,	PUNCT
ejpam-6261	594	15	and	and	CCONJ
ejpam-6261	594	16	m.	m.	PROPN
ejpam-6261	594	17	khan	khan	PROPN
ejpam-6261	594	18	.	.	PUNCT
ejpam-6261	595	1	neutrosophic	neutrosophic	ADJ
ejpam-6261	595	2	positive	positive	ADJ
ejpam-6261	595	3	implicative	implicative	ADJ
ejpam-6261	595	4	n	n	CCONJ
ejpam-6261	595	5	-	-	PUNCT
ejpam-6261	595	6	ideals	ideal	NOUN
ejpam-6261	595	7	in	in	ADP
ejpam-6261	595	8	bck	bck	NOUN
ejpam-6261	595	9	-	-	PUNCT
ejpam-6261	595	10	algebras	algebra	NOUN
ejpam-6261	595	11	.	.	PUNCT
ejpam-6261	596	1	axioms	axiom	NOUN
ejpam-6261	596	2	,	,	PUNCT
ejpam-6261	596	3	7(3):13	7(3):13	NUM
ejpam-6261	596	4	,	,	PUNCT
ejpam-6261	596	5	2018	2018	NUM
ejpam-6261	596	6	.	.	PUNCT
ejpam-6261	597	1	[	[	X
ejpam-6261	597	2	17	17	NUM
ejpam-6261	597	3	]	]	X
ejpam-6261	597	4	b.	b.	PROPN
ejpam-6261	597	5	almutairi	almutairi	PROPN
ejpam-6261	597	6	,	,	PUNCT
ejpam-6261	597	7	a.	a.	PROPN
ejpam-6261	597	8	ali	ali	PROPN
ejpam-6261	597	9	,	,	PUNCT
ejpam-6261	597	10	q.	q.	PROPN
ejpam-6261	597	11	xin	xin	PROPN
ejpam-6261	597	12	,	,	PUNCT
ejpam-6261	597	13	and	and	CCONJ
ejpam-6261	597	14	a.	a.	PROPN
ejpam-6261	597	15	khan	khan	PROPN
ejpam-6261	597	16	.	.	PUNCT
ejpam-6261	598	1	a	a	DET
ejpam-6261	598	2	novel	novel	ADJ
ejpam-6261	598	3	concept	concept	NOUN
ejpam-6261	598	4	of	of	ADP
ejpam-6261	598	5	complex	complex	ADJ
ejpam-6261	598	6	anti	anti	ADJ
ejpam-6261	598	7	-	-	ADJ
ejpam-6261	598	8	fuzzy	fuzzy	ADJ
ejpam-6261	598	9	isomorphism	isomorphism	NOUN
ejpam-6261	598	10	over	over	ADP
ejpam-6261	598	11	groups	group	NOUN
ejpam-6261	598	12	.	.	PUNCT
ejpam-6261	599	1	symmetry	symmetry	NOUN
ejpam-6261	599	2	,	,	PUNCT
ejpam-6261	599	3	15(9):1693	15(9):1693	NUM
ejpam-6261	599	4	,	,	PUNCT
ejpam-6261	599	5	2023	2023	NUM
ejpam-6261	599	6	.	.	PUNCT
ejpam-6261	600	1	[	[	X
ejpam-6261	600	2	18	18	NUM
ejpam-6261	600	3	]	]	PUNCT
ejpam-6261	600	4	m.	m.	NOUN
ejpam-6261	600	5	gulzar	gulzar	PROPN
ejpam-6261	600	6	,	,	PUNCT
ejpam-6261	600	7	d.	d.	PROPN
ejpam-6261	600	8	alghazzawi	alghazzawi	PROPN
ejpam-6261	600	9	,	,	PUNCT
ejpam-6261	600	10	m.	m.	PROPN
ejpam-6261	600	11	h.	h.	PROPN
ejpam-6261	600	12	mateen	mateen	PROPN
ejpam-6261	600	13	,	,	PUNCT
ejpam-6261	600	14	and	and	CCONJ
ejpam-6261	600	15	m.	m.	PROPN
ejpam-6261	600	16	premkumar	premkumar	PROPN
ejpam-6261	600	17	.	.	PUNCT
ejpam-6261	601	1	on	on	ADP
ejpam-6261	601	2	some	some	DET
ejpam-6261	601	3	characterization	characterization	NOUN
ejpam-6261	601	4	of	of	ADP
ejpam-6261	601	5	q	q	ADJ
ejpam-6261	601	6	-	-	PUNCT
ejpam-6261	601	7	complex	complex	ADJ
ejpam-6261	601	8	fuzzy	fuzzy	ADJ
ejpam-6261	601	9	sub	sub	NOUN
ejpam-6261	601	10	-	-	NOUN
ejpam-6261	601	11	rings	ring	NOUN
ejpam-6261	601	12	.	.	PUNCT
ejpam-6261	601	13	journal	journal	PROPN
ejpam-6261	601	14	of	of	ADP
ejpam-6261	601	15	mathematics	mathematics	PROPN
ejpam-6261	601	16	and	and	CCONJ
ejpam-6261	601	17	computer	computer	NOUN
ejpam-6261	601	18	science	science	NOUN
ejpam-6261	601	19	,	,	PUNCT
ejpam-6261	601	20	22:295–305	22:295–305	NUM
ejpam-6261	601	21	,	,	PUNCT
ejpam-6261	601	22	2021	2021	NUM
ejpam-6261	601	23	.	.	PUNCT
ejpam-6261	602	1	[	[	X
ejpam-6261	602	2	19	19	NUM
ejpam-6261	602	3	]	]	X
ejpam-6261	602	4	d.	d.	PROPN
ejpam-6261	602	5	alghazzawi	alghazzawi	PROPN
ejpam-6261	602	6	,	,	PUNCT
ejpam-6261	602	7	h.	h.	PROPN
ejpam-6261	602	8	alolaiyan	alolaiyan	PROPN
ejpam-6261	602	9	,	,	PUNCT
ejpam-6261	602	10	h.	h.	PROPN
ejpam-6261	602	11	ashfaq	ashfaq	PROPN
ejpam-6261	602	12	,	,	PUNCT
ejpam-6261	602	13	u.	u.	PROPN
ejpam-6261	602	14	shuaib	shuaib	PROPN
ejpam-6261	602	15	,	,	PUNCT
ejpam-6261	602	16	h.	h.	PROPN
ejpam-6261	602	17	a.	a.	PROPN
ejpam-6261	602	18	e.	e.	PROPN
ejpam-6261	602	19	w.	w.	PROPN
ejpam-6261	602	20	khalifa	khalifa	PROPN
ejpam-6261	602	21	,	,	PUNCT
ejpam-6261	602	22	h.	h.	PROPN
ejpam-6261	602	23	g.	g.	PROPN
ejpam-6261	602	24	gomaa	gomaa	PROPN
ejpam-6261	602	25	,	,	PUNCT
ejpam-6261	602	26	and	and	CCONJ
ejpam-6261	602	27	q.	q.	PROPN
ejpam-6261	602	28	xin	xin	PROPN
ejpam-6261	602	29	.	.	PUNCT
ejpam-6261	603	1	selecting	select	VERB
ejpam-6261	603	2	an	an	DET
ejpam-6261	603	3	optimal	optimal	ADJ
ejpam-6261	603	4	approach	approach	NOUN
ejpam-6261	603	5	to	to	PART
ejpam-6261	603	6	reduce	reduce	VERB
ejpam-6261	603	7	energy	energy	NOUN
ejpam-6261	603	8	crises	crisis	NOUN
ejpam-6261	603	9	under	under	ADP
ejpam-6261	603	10	interval	interval	NOUN
ejpam-6261	603	11	-	-	PUNCT
ejpam-6261	603	12	valued	value	VERB
ejpam-6261	603	13	intuitionistic	intuitionistic	ADJ
ejpam-6261	603	14	fuzzy	fuzzy	ADJ
ejpam-6261	603	15	environment	environment	NOUN
ejpam-6261	603	16	.	.	PUNCT
ejpam-6261	604	1	scientific	scientific	ADJ
ejpam-6261	604	2	reports	report	NOUN
ejpam-6261	604	3	,	,	PUNCT
ejpam-6261	604	4	14(1):8713	14(1):8713	NUM
ejpam-6261	604	5	,	,	PUNCT
ejpam-6261	604	6	2024	2024	NUM
ejpam-6261	604	7	.	.	PUNCT
ejpam-6261	605	1	[	[	X
ejpam-6261	605	2	20	20	NUM
ejpam-6261	605	3	]	]	PUNCT
ejpam-6261	605	4	h.	h.	PROPN
ejpam-6261	605	5	alolaiyan	alolaiyan	PROPN
ejpam-6261	605	6	,	,	PUNCT
ejpam-6261	605	7	m.	m.	PROPN
ejpam-6261	605	8	h.	h.	PROPN
ejpam-6261	605	9	mateen	mateen	PROPN
ejpam-6261	605	10	,	,	PUNCT
ejpam-6261	605	11	d.	d.	PROPN
ejpam-6261	605	12	pamucar	pamucar	PROPN
ejpam-6261	605	13	,	,	PUNCT
ejpam-6261	605	14	m.	m.	PROPN
ejpam-6261	605	15	k.	k.	PROPN
ejpam-6261	605	16	mahmmod	mahmmod	PROPN
ejpam-6261	605	17	,	,	PUNCT
ejpam-6261	605	18	and	and	CCONJ
ejpam-6261	605	19	f.	f.	PROPN
ejpam-6261	605	20	arslan	arslan	PROPN
ejpam-6261	605	21	.	.	PUNCT
ejpam-6261	606	1	a	a	DET
ejpam-6261	606	2	certain	certain	ADJ
ejpam-6261	606	3	structure	structure	NOUN
ejpam-6261	606	4	of	of	ADP
ejpam-6261	606	5	bipolar	bipolar	ADJ
ejpam-6261	606	6	fuzzy	fuzzy	ADJ
ejpam-6261	606	7	subrings	subring	NOUN
ejpam-6261	606	8	.	.	PUNCT
ejpam-6261	607	1	symmetry	symmetry	PROPN
ejpam-6261	607	2	,	,	PUNCT
ejpam-6261	607	3	13(8):1397	13(8):1397	NUM
ejpam-6261	607	4	,	,	PUNCT
ejpam-6261	607	5	2021	2021	NUM
ejpam-6261	607	6	.	.	PUNCT
ejpam-6261	608	1	[	[	X
ejpam-6261	608	2	21	21	NUM
ejpam-6261	608	3	]	]	PUNCT
ejpam-6261	608	4	a.	a.	NOUN
ejpam-6261	608	5	altassan	altassan	PROPN
ejpam-6261	608	6	,	,	PUNCT
ejpam-6261	608	7	m.	m.	NOUN
ejpam-6261	608	8	h.	h.	PROPN
ejpam-6261	608	9	mateen	mateen	PROPN
ejpam-6261	608	10	,	,	PUNCT
ejpam-6261	608	11	and	and	CCONJ
ejpam-6261	608	12	d.	d.	PROPN
ejpam-6261	608	13	pamucar	pamucar	PROPN
ejpam-6261	608	14	.	.	PUNCT
ejpam-6261	609	1	on	on	ADP
ejpam-6261	609	2	fundamental	fundamental	ADJ
ejpam-6261	609	3	theorems	theorem	NOUN
ejpam-6261	609	4	of	of	ADP
ejpam-6261	609	5	fuzzy	fuzzy	ADJ
ejpam-6261	609	6	isomorphism	isomorphism	NOUN
ejpam-6261	609	7	of	of	ADP
ejpam-6261	609	8	fuzzy	fuzzy	ADJ
ejpam-6261	609	9	subrings	subring	NOUN
ejpam-6261	609	10	over	over	ADP
ejpam-6261	609	11	a	a	DET
ejpam-6261	609	12	certain	certain	ADJ
ejpam-6261	609	13	algebraic	algebraic	ADJ
ejpam-6261	609	14	product	product	NOUN
ejpam-6261	609	15	.	.	PUNCT
ejpam-6261	610	1	symmetry	symmetry	NOUN
ejpam-6261	610	2	,	,	PUNCT
ejpam-6261	610	3	13(6):998	13(6):998	NUM
ejpam-6261	610	4	,	,	PUNCT
ejpam-6261	610	5	2021	2021	NUM
ejpam-6261	610	6	.	.	PUNCT
ejpam-6261	611	1	[	[	X
ejpam-6261	611	2	22	22	NUM
ejpam-6261	611	3	]	]	X
ejpam-6261	611	4	d.	d.	PROPN
ejpam-6261	611	5	alghazzwi	alghazzwi	PROPN
ejpam-6261	611	6	,	,	PUNCT
ejpam-6261	611	7	a.	a.	PROPN
ejpam-6261	611	8	ali	ali	PROPN
ejpam-6261	611	9	,	,	PUNCT
ejpam-6261	611	10	a.	a.	PROPN
ejpam-6261	611	11	almutlg	almutlg	PROPN
ejpam-6261	611	12	,	,	PUNCT
ejpam-6261	611	13	e.	e.	PROPN
ejpam-6261	611	14	a.	a.	PROPN
ejpam-6261	611	15	abo	abo	PROPN
ejpam-6261	611	16	-	-	PUNCT
ejpam-6261	611	17	tabl	tabl	NOUN
ejpam-6261	611	18	,	,	PUNCT
ejpam-6261	611	19	and	and	CCONJ
ejpam-6261	611	20	a.	a.	NOUN
ejpam-6261	611	21	a.	a.	PROPN
ejpam-6261	611	22	azzam	azzam	PROPN
ejpam-6261	611	23	.	.	PUNCT
ejpam-6261	612	1	a	a	DET
ejpam-6261	612	2	novel	novel	ADJ
ejpam-6261	612	3	structure	structure	NOUN
ejpam-6261	612	4	of	of	ADP
ejpam-6261	612	5	q	q	ADJ
ejpam-6261	612	6	-	-	PUNCT
ejpam-6261	612	7	rung	rung	ADJ
ejpam-6261	612	8	orthopair	orthopair	ADJ
ejpam-6261	612	9	fuzzy	fuzzy	ADJ
ejpam-6261	612	10	sets	set	NOUN
ejpam-6261	612	11	in	in	ADP
ejpam-6261	612	12	ring	ring	NOUN
ejpam-6261	612	13	theory	theory	NOUN
ejpam-6261	612	14	.	.	PUNCT
ejpam-6261	613	1	aims	aim	VERB
ejpam-6261	613	2	mathematics	mathematic	NOUN
ejpam-6261	613	3	,	,	PUNCT
ejpam-6261	613	4	8(4):8365–8385	8(4):8365–8385	PROPN
ejpam-6261	613	5	,	,	PUNCT
ejpam-6261	613	6	2023	2023	NUM
ejpam-6261	613	7	.	.	PUNCT
ejpam-6261	614	1	[	[	X
ejpam-6261	614	2	23	23	NUM
ejpam-6261	614	3	]	]	X
ejpam-6261	614	4	m.	m.	NOUN
ejpam-6261	614	5	bal	bal	PROPN
ejpam-6261	614	6	,	,	PUNCT
ejpam-6261	614	7	m.	m.	NOUN
ejpam-6261	614	8	m.	m.	NOUN
ejpam-6261	614	9	shalla	shalla	PROPN
ejpam-6261	614	10	,	,	PUNCT
ejpam-6261	614	11	and	and	CCONJ
ejpam-6261	614	12	n.	n.	PROPN
ejpam-6261	614	13	olgun	olgun	PROPN
ejpam-6261	614	14	.	.	PUNCT
ejpam-6261	615	1	neutrosophic	neutrosophic	ADJ
ejpam-6261	615	2	triplet	triplet	NOUN
ejpam-6261	615	3	cosets	coset	NOUN
ejpam-6261	615	4	and	and	CCONJ
ejpam-6261	615	5	quotient	quotient	NOUN
ejpam-6261	615	6	groups	group	NOUN
ejpam-6261	615	7	.	.	PUNCT
ejpam-6261	616	1	symmetry	symmetry	NOUN
ejpam-6261	616	2	,	,	PUNCT
ejpam-6261	616	3	10(4):126	10(4):126	PROPN
ejpam-6261	616	4	,	,	PUNCT
ejpam-6261	616	5	2018	2018	NUM
ejpam-6261	616	6	.	.	PUNCT
ejpam-6261	617	1	[	[	X
ejpam-6261	617	2	24	24	NUM
ejpam-6261	617	3	]	]	X
ejpam-6261	617	4	f.	f.	PROPN
ejpam-6261	617	5	smarandache	smarandache	PROPN
ejpam-6261	617	6	.	.	PUNCT
ejpam-6261	618	1	a	a	DET
ejpam-6261	618	2	unifying	unifying	ADJ
ejpam-6261	618	3	field	field	NOUN
ejpam-6261	618	4	in	in	ADP
ejpam-6261	618	5	logics	logic	NOUN
ejpam-6261	618	6	:	:	PUNCT
ejpam-6261	618	7	neutrosophic	neutrosophic	ADJ
ejpam-6261	618	8	logic	logic	NOUN
ejpam-6261	618	9	,	,	PUNCT
ejpam-6261	618	10	neutrosophy	neutrosophy	NOUN
ejpam-6261	618	11	,	,	PUNCT
ejpam-6261	618	12	neutrosophic	neutrosophic	ADJ
ejpam-6261	618	13	set	set	NOUN
ejpam-6261	618	14	,	,	PUNCT
ejpam-6261	618	15	neutrosophic	neutrosophic	ADJ
ejpam-6261	618	16	probability	probability	NOUN
ejpam-6261	618	17	and	and	CCONJ
ejpam-6261	618	18	statistics	statistic	NOUN
ejpam-6261	618	19	.	.	PUNCT
ejpam-6261	619	1	american	american	ADJ
ejpam-6261	619	2	research	research	PROPN
ejpam-6261	619	3	press	press	PROPN
ejpam-6261	619	4	,	,	PUNCT
ejpam-6261	619	5	5th	5th	ADJ
ejpam-6261	619	6	edition	edition	NOUN
ejpam-6261	619	7	,	,	PUNCT
ejpam-6261	619	8	2006	2006	NUM
ejpam-6261	619	9	.	.	PUNCT
ejpam-6261	620	1	[	[	X
ejpam-6261	620	2	25	25	NUM
ejpam-6261	620	3	]	]	PUNCT
ejpam-6261	620	4	a.	a.	NOUN
ejpam-6261	620	5	abobala	abobala	NOUN
ejpam-6261	620	6	,	,	PUNCT
ejpam-6261	620	7	m.	m.	NOUN
ejpam-6261	620	8	hatip	hatip	PROPN
ejpam-6261	620	9	,	,	PUNCT
ejpam-6261	620	10	and	and	CCONJ
ejpam-6261	620	11	r.	r.	PROPN
ejpam-6261	620	12	k.	k.	PROPN
ejpam-6261	620	13	alhamido	alhamido	PROPN
ejpam-6261	620	14	.	.	PUNCT
ejpam-6261	621	1	a	a	DET
ejpam-6261	621	2	contribution	contribution	NOUN
ejpam-6261	621	3	to	to	ADP
ejpam-6261	621	4	neutrosophic	neutrosophic	ADJ
ejpam-6261	621	5	groups	group	NOUN
ejpam-6261	621	6	.	.	PUNCT
ejpam-6261	622	1	infinite	infinite	ADJ
ejpam-6261	622	2	study	study	NOUN
ejpam-6261	622	3	,	,	PUNCT
ejpam-6261	622	4	2(1):67–76	2(1):67–76	NUM
ejpam-6261	622	5	,	,	PUNCT
ejpam-6261	622	6	2019	2019	NUM
ejpam-6261	622	7	.	.	PUNCT
ejpam-6261	623	1	[	[	X
ejpam-6261	623	2	26	26	NUM
ejpam-6261	623	3	]	]	X
ejpam-6261	623	4	w.	w.	PROPN
ejpam-6261	623	5	v.	v.	ADP
ejpam-6261	623	6	kandasamy	kandasamy	PROPN
ejpam-6261	623	7	and	and	CCONJ
ejpam-6261	623	8	f.	f.	PROPN
ejpam-6261	623	9	smarandache	smarandache	PROPN
ejpam-6261	623	10	.	.	PUNCT
ejpam-6261	624	1	some	some	DET
ejpam-6261	624	2	neutrosophic	neutrosophic	ADJ
ejpam-6261	624	3	algebraic	algebraic	ADJ
ejpam-6261	624	4	structures	structure	NOUN
ejpam-6261	624	5	and	and	CCONJ
ejpam-6261	624	6	neutrosophic	neutrosophic	ADJ
ejpam-6261	624	7	n	n	CCONJ
ejpam-6261	624	8	-algebraic	-algebraic	NOUN
ejpam-6261	624	9	structures	structure	NOUN
ejpam-6261	624	10	.	.	PUNCT
ejpam-6261	625	1	infinite	infinite	ADJ
ejpam-6261	625	2	study	study	NOUN
ejpam-6261	625	3	,	,	PUNCT
ejpam-6261	625	4	2006	2006	NUM
ejpam-6261	625	5	.	.	PUNCT
ejpam-6261	626	1	[	[	X
ejpam-6261	626	2	27	27	NUM
ejpam-6261	626	3	]	]	X
ejpam-6261	626	4	y.	y.	PROPN
ejpam-6261	626	5	ma	ma	PROPN
ejpam-6261	626	6	,	,	PUNCT
ejpam-6261	626	7	x.	x.	PROPN
ejpam-6261	626	8	zhang	zhang	PROPN
ejpam-6261	626	9	,	,	PUNCT
ejpam-6261	626	10	x.	x.	PROPN
ejpam-6261	626	11	yang	yang	PROPN
ejpam-6261	626	12	,	,	PUNCT
ejpam-6261	626	13	and	and	CCONJ
ejpam-6261	626	14	x.	x.	NOUN
ejpam-6261	626	15	zhou	zhou	PROPN
ejpam-6261	626	16	.	.	PUNCT
ejpam-6261	627	1	generalized	generalize	VERB
ejpam-6261	627	2	neutrosophic	neutrosophic	ADJ
ejpam-6261	627	3	extended	extended	ADJ
ejpam-6261	627	4	triplet	triplet	NOUN
ejpam-6261	627	5	group	group	NOUN
ejpam-6261	627	6	.	.	PUNCT
ejpam-6261	628	1	symmetry	symmetry	PROPN
ejpam-6261	628	2	,	,	PUNCT
ejpam-6261	628	3	11(3):327–342	11(3):327–342	NUM
ejpam-6261	628	4	,	,	PUNCT
ejpam-6261	628	5	2019	2019	NUM
ejpam-6261	628	6	.	.	PUNCT
ejpam-6261	629	1	[	[	X
ejpam-6261	629	2	28	28	NUM
ejpam-6261	629	3	]	]	X
ejpam-6261	629	4	g.	g.	PROPN
ejpam-6261	629	5	muhiuddin	muhiuddin	PROPN
ejpam-6261	629	6	,	,	PUNCT
ejpam-6261	629	7	a.	a.	PROPN
ejpam-6261	629	8	n.	n.	PROPN
ejpam-6261	629	9	al	al	PROPN
ejpam-6261	629	10	-	-	PUNCT
ejpam-6261	629	11	kenani	kenani	PROPN
ejpam-6261	629	12	,	,	PUNCT
ejpam-6261	629	13	e.	e.	PROPN
ejpam-6261	629	14	h.	h.	PROPN
ejpam-6261	629	15	roh	roh	PROPN
ejpam-6261	629	16	,	,	PUNCT
ejpam-6261	629	17	and	and	CCONJ
ejpam-6261	629	18	y.	y.	PROPN
ejpam-6261	629	19	b.	b.	PROPN
ejpam-6261	629	20	jun	jun	PROPN
ejpam-6261	629	21	.	.	PROPN
ejpam-6261	629	22	implicative	implicative	PROPN
ejpam-6261	629	23	neutrosophic	neutrosophic	PROPN
ejpam-6261	629	24	quadruple	quadruple	PROPN
ejpam-6261	629	25	bck	bck	PROPN
ejpam-6261	629	26	-	-	PUNCT
ejpam-6261	629	27	algebras	algebra	NOUN
ejpam-6261	629	28	and	and	CCONJ
ejpam-6261	629	29	ideals	ideal	NOUN
ejpam-6261	629	30	.	.	PUNCT
ejpam-6261	630	1	symmetry	symmetry	NOUN
ejpam-6261	630	2	,	,	PUNCT
ejpam-6261	630	3	11(2):277–286	11(2):277–286	PROPN
ejpam-6261	630	4	,	,	PUNCT
ejpam-6261	630	5	2019	2019	NUM
ejpam-6261	630	6	.	.	PUNCT
ejpam-6261	631	1	[	[	X
ejpam-6261	631	2	29	29	NUM
ejpam-6261	631	3	]	]	PUNCT
ejpam-6261	631	4	s.	s.	PROPN
ejpam-6261	631	5	bashir	bashir	PROPN
ejpam-6261	631	6	,	,	PUNCT
ejpam-6261	631	7	t.	t.	PROPN
ejpam-6261	631	8	alharbi	alharbi	PROPN
ejpam-6261	631	9	,	,	PUNCT
ejpam-6261	631	10	r.	r.	PROPN
ejpam-6261	631	11	mazhar	mazhar	PROPN
ejpam-6261	631	12	,	,	PUNCT
ejpam-6261	631	13	i.	i.	PROPN
ejpam-6261	631	14	khalid	khalid	PROPN
ejpam-6261	631	15	,	,	PUNCT
ejpam-6261	631	16	m.	m.	PROPN
ejpam-6261	631	17	ul	ul	PROPN
ejpam-6261	631	18	hassan	hassan	PROPN
ejpam-6261	631	19	afzal	afzal	PROPN
ejpam-6261	631	20	,	,	PUNCT
ejpam-6261	631	21	and	and	CCONJ
ejpam-6261	631	22	n.	n.	PROPN
ejpam-6261	631	23	riaz	riaz	PROPN
ejpam-6261	631	24	chaudhry	chaudhry	PROPN
ejpam-6261	631	25	.	.	PUNCT
ejpam-6261	632	1	an	an	DET
ejpam-6261	632	2	efficient	efficient	ADJ
ejpam-6261	632	3	approach	approach	NOUN
ejpam-6261	632	4	to	to	PART
ejpam-6261	632	5	study	study	VERB
ejpam-6261	632	6	multi	multi	ADJ
ejpam-6261	632	7	-	-	ADJ
ejpam-6261	632	8	polar	polar	ADJ
ejpam-6261	632	9	fuzzy	fuzzy	ADJ
ejpam-6261	632	10	ideals	ideal	NOUN
ejpam-6261	632	11	of	of	ADP
ejpam-6261	632	12	semirings	semiring	NOUN
ejpam-6261	632	13	.	.	PUNCT
ejpam-6261	633	1	scientific	scientific	ADJ
ejpam-6261	633	2	reports	report	NOUN
ejpam-6261	633	3	,	,	PUNCT
ejpam-6261	633	4	14(1):2446	14(1):2446	NUM
ejpam-6261	633	5	,	,	PUNCT
ejpam-6261	633	6	2024	2024	NUM
ejpam-6261	633	7	.	.	PUNCT
ejpam-6261	634	1	[	[	X
ejpam-6261	634	2	30	30	NUM
ejpam-6261	634	3	]	]	PUNCT
ejpam-6261	634	4	s.	s.	PROPN
ejpam-6261	634	5	bashir	bashir	PROPN
ejpam-6261	634	6	,	,	PUNCT
ejpam-6261	634	7	r.	r.	PROPN
ejpam-6261	634	8	mazhar	mazhar	PROPN
ejpam-6261	634	9	,	,	PUNCT
ejpam-6261	634	10	n.	n.	PROPN
ejpam-6261	634	11	kausar	kausar	PROPN
ejpam-6261	634	12	,	,	PUNCT
ejpam-6261	634	13	s.	s.	PROPN
ejpam-6261	634	14	yaman	yaman	PROPN
ejpam-6261	634	15	,	,	PUNCT
ejpam-6261	634	16	s.	s.	PROPN
ejpam-6261	634	17	s.	s.	PROPN
ejpam-6261	634	18	ali	ali	PROPN
ejpam-6261	634	19	,	,	PUNCT
ejpam-6261	634	20	and	and	CCONJ
ejpam-6261	634	21	m.	m.	NOUN
ejpam-6261	634	22	u.	u.	PROPN
ejpam-6261	634	23	h.	h.	PROPN
ejpam-6261	634	24	afzal	afzal	PROPN
ejpam-6261	634	25	.	.	PUNCT
ejpam-6261	635	1	generalized	generalized	ADJ
ejpam-6261	635	2	roughness	roughness	NOUN
ejpam-6261	635	3	of	of	ADP
ejpam-6261	635	4	three	three	NUM
ejpam-6261	635	5	dimensional	dimensional	ADJ
ejpam-6261	635	6	(	(	PUNCT
ejpam-6261	635	7	ε	ε	PROPN
ejpam-6261	635	8	,	,	PUNCT
ejpam-6261	635	9	ε,∨q)-fuzzy	ε,∨q)-fuzzy	PRON
ejpam-6261	635	10	ideals	ideal	NOUN
ejpam-6261	635	11	in	in	ADP
ejpam-6261	635	12	terms	term	NOUN
ejpam-6261	635	13	of	of	ADP
ejpam-6261	635	14	set	set	NOUN
ejpam-6261	635	15	-	-	PUNCT
ejpam-6261	635	16	valued	value	VERB
ejpam-6261	635	17	homomorphism	homomorphism	NOUN
ejpam-6261	635	18	.	.	PUNCT
ejpam-6261	636	1	scientific	scientific	ADJ
ejpam-6261	636	2	reports	report	NOUN
ejpam-6261	636	3	,	,	PUNCT
ejpam-6261	636	4	14(1):12301	14(1):12301	NUM
ejpam-6261	636	5	,	,	PUNCT
ejpam-6261	636	6	2024	2024	NUM
ejpam-6261	636	7	.	.	PUNCT
ejpam-6261	637	1	m.	m.	NOUN
ejpam-6261	637	2	h.	h.	PROPN
ejpam-6261	637	3	mateen	mateen	PROPN
ejpam-6261	637	4	et	et	PROPN
ejpam-6261	637	5	al	al	PROPN
ejpam-6261	637	6	.	.	PUNCT
ejpam-6261	637	7	/	/	SYM
ejpam-6261	637	8	eur	eur	PROPN
ejpam-6261	637	9	.	.	PUNCT
ejpam-6261	638	1	j.	j.	PROPN
ejpam-6261	638	2	pure	pure	PROPN
ejpam-6261	638	3	appl	appl	PROPN
ejpam-6261	638	4	.	.	PROPN
ejpam-6261	638	5	math	math	PROPN
ejpam-6261	638	6	,	,	PUNCT
ejpam-6261	638	7	18	18	NUM
ejpam-6261	638	8	(	(	PUNCT
ejpam-6261	638	9	4	4	NUM
ejpam-6261	638	10	)	)	PUNCT
ejpam-6261	638	11	(	(	PUNCT
ejpam-6261	638	12	2025	2025	NUM
ejpam-6261	638	13	)	)	PUNCT
ejpam-6261	638	14	,	,	PUNCT
ejpam-6261	638	15	6261	6261	NUM
ejpam-6261	638	16	23	23	NUM
ejpam-6261	638	17	of	of	ADP
ejpam-6261	638	18	23	23	NUM
ejpam-6261	639	1	[	[	X
ejpam-6261	639	2	31	31	NUM
ejpam-6261	639	3	]	]	X
ejpam-6261	639	4	d.	d.	PROPN
ejpam-6261	639	5	ramot	ramot	PROPN
ejpam-6261	639	6	,	,	PUNCT
ejpam-6261	639	7	r.	r.	PROPN
ejpam-6261	639	8	milo	milo	PROPN
ejpam-6261	639	9	,	,	PUNCT
ejpam-6261	639	10	m.	m.	NOUN
ejpam-6261	639	11	friedman	friedman	PROPN
ejpam-6261	639	12	,	,	PUNCT
ejpam-6261	639	13	and	and	CCONJ
ejpam-6261	639	14	a.	a.	NOUN
ejpam-6261	639	15	kandel	kandel	PROPN
ejpam-6261	639	16	.	.	PUNCT
ejpam-6261	640	1	complex	complex	ADJ
ejpam-6261	640	2	fuzzy	fuzzy	ADJ
ejpam-6261	640	3	sets	set	NOUN
ejpam-6261	640	4	.	.	PUNCT
ejpam-6261	641	1	ieee	ieee	NOUN
ejpam-6261	641	2	transactions	transaction	NOUN
ejpam-6261	641	3	on	on	ADP
ejpam-6261	641	4	fuzzy	fuzzy	ADJ
ejpam-6261	641	5	systems	system	NOUN
ejpam-6261	641	6	,	,	PUNCT
ejpam-6261	641	7	10:450–461	10:450–461	PROPN
ejpam-6261	641	8	,	,	PUNCT
ejpam-6261	641	9	2002	2002	NUM
ejpam-6261	641	10	.	.	PUNCT
ejpam-6261	642	1	[	[	X
ejpam-6261	642	2	32	32	NUM
ejpam-6261	642	3	]	]	X
ejpam-6261	642	4	d.	d.	PROPN
ejpam-6261	642	5	ramot	ramot	PROPN
ejpam-6261	642	6	,	,	PUNCT
ejpam-6261	642	7	m.	m.	NOUN
ejpam-6261	642	8	friedman	friedman	PROPN
ejpam-6261	642	9	,	,	PUNCT
ejpam-6261	642	10	g.	g.	PROPN
ejpam-6261	642	11	langholz	langholz	PROPN
ejpam-6261	642	12	,	,	PUNCT
ejpam-6261	642	13	and	and	CCONJ
ejpam-6261	642	14	a.	a.	NOUN
ejpam-6261	642	15	kandel	kandel	PROPN
ejpam-6261	642	16	.	.	PUNCT
ejpam-6261	643	1	complex	complex	ADJ
ejpam-6261	643	2	fuzzy	fuzzy	ADJ
ejpam-6261	643	3	logic	logic	NOUN
ejpam-6261	643	4	.	.	PUNCT
ejpam-6261	644	1	ieee	ieee	NOUN
ejpam-6261	644	2	transactions	transaction	NOUN
ejpam-6261	644	3	on	on	ADP
ejpam-6261	644	4	fuzzy	fuzzy	ADJ
ejpam-6261	644	5	systems	system	NOUN
ejpam-6261	644	6	,	,	PUNCT
ejpam-6261	644	7	11:171–186	11:171–186	PROPN
ejpam-6261	644	8	,	,	PUNCT
ejpam-6261	644	9	2003	2003	NUM
ejpam-6261	644	10	.	.	PUNCT
ejpam-6261	645	1	[	[	X
ejpam-6261	645	2	33	33	NUM
ejpam-6261	645	3	]	]	PUNCT
ejpam-6261	645	4	a.	a.	NOUN
ejpam-6261	645	5	alkouri	alkouri	PROPN
ejpam-6261	645	6	and	and	CCONJ
ejpam-6261	645	7	a.	a.	PROPN
ejpam-6261	645	8	r.	r.	PROPN
ejpam-6261	645	9	salleh	salleh	PROPN
ejpam-6261	645	10	.	.	PUNCT
ejpam-6261	646	1	complex	complex	ADJ
ejpam-6261	646	2	atanassov	atanassov	NOUN
ejpam-6261	646	3	’s	’s	PART
ejpam-6261	646	4	intuitionistic	intuitionistic	ADJ
ejpam-6261	646	5	fuzzy	fuzzy	ADJ
ejpam-6261	646	6	sets	set	NOUN
ejpam-6261	646	7	.	.	PUNCT
ejpam-6261	647	1	in	in	ADP
ejpam-6261	647	2	international	international	ADJ
ejpam-6261	647	3	conference	conference	NOUN
ejpam-6261	647	4	on	on	ADP
ejpam-6261	647	5	fundamental	fundamental	ADJ
ejpam-6261	647	6	and	and	CCONJ
ejpam-6261	647	7	applied	applied	ADJ
ejpam-6261	647	8	sciences	science	NOUN
ejpam-6261	647	9	,	,	PUNCT
ejpam-6261	647	10	aip	aip	PROPN
ejpam-6261	647	11	conference	conference	NOUN
ejpam-6261	647	12	proceedings	proceeding	NOUN
ejpam-6261	647	13	,	,	PUNCT
ejpam-6261	647	14	volume	volume	NOUN
ejpam-6261	647	15	1482	1482	NUM
ejpam-6261	647	16	,	,	PUNCT
ejpam-6261	647	17	pages	page	NOUN
ejpam-6261	647	18	464–470	464–470	NUM
ejpam-6261	647	19	,	,	PUNCT
ejpam-6261	647	20	2012	2012	NUM
ejpam-6261	647	21	.	.	PUNCT
ejpam-6261	648	1	[	[	X
ejpam-6261	648	2	34	34	NUM
ejpam-6261	648	3	]	]	X
ejpam-6261	648	4	a.	a.	NOUN
ejpam-6261	648	5	alkouri	alkouri	PROPN
ejpam-6261	648	6	and	and	CCONJ
ejpam-6261	648	7	a.	a.	PROPN
ejpam-6261	648	8	r.	r.	PROPN
ejpam-6261	648	9	salleh	salleh	PROPN
ejpam-6261	648	10	.	.	PUNCT
ejpam-6261	649	1	some	some	DET
ejpam-6261	649	2	operations	operation	NOUN
ejpam-6261	649	3	on	on	ADP
ejpam-6261	649	4	complex	complex	ADJ
ejpam-6261	649	5	atanassov	atanassov	NOUN
ejpam-6261	649	6	’s	’s	PART
ejpam-6261	649	7	intuitionistic	intuitionistic	ADJ
ejpam-6261	649	8	fuzzy	fuzzy	ADJ
ejpam-6261	649	9	sets	set	NOUN
ejpam-6261	649	10	.	.	PUNCT
ejpam-6261	650	1	in	in	ADP
ejpam-6261	650	2	aip	aip	PROPN
ejpam-6261	650	3	conference	conference	NOUN
ejpam-6261	650	4	proceedings	proceeding	NOUN
ejpam-6261	650	5	,	,	PUNCT
ejpam-6261	650	6	volume	volume	NOUN
ejpam-6261	650	7	1571	1571	NUM
ejpam-6261	650	8	,	,	PUNCT
ejpam-6261	650	9	pages	page	VERB
ejpam-6261	650	10	987–993	987–993	NUM
ejpam-6261	650	11	,	,	PUNCT
ejpam-6261	650	12	2013	2013	NUM
ejpam-6261	650	13	.	.	PUNCT
ejpam-6261	651	1	[	[	X
ejpam-6261	651	2	35	35	NUM
ejpam-6261	651	3	]	]	PUNCT
ejpam-6261	651	4	m.	m.	NOUN
ejpam-6261	651	5	gulzar	gulzar	PROPN
ejpam-6261	651	6	,	,	PUNCT
ejpam-6261	651	7	m.	m.	PROPN
ejpam-6261	651	8	h.	h.	PROPN
ejpam-6261	651	9	mateen	mateen	PROPN
ejpam-6261	651	10	,	,	PUNCT
ejpam-6261	651	11	d.	d.	PROPN
ejpam-6261	651	12	alghazzawi	alghazzawi	PROPN
ejpam-6261	651	13	,	,	PUNCT
ejpam-6261	651	14	and	and	CCONJ
ejpam-6261	651	15	n.	n.	PROPN
ejpam-6261	651	16	kausar	kausar	PROPN
ejpam-6261	651	17	.	.	PUNCT
ejpam-6261	652	1	a	a	DET
ejpam-6261	652	2	novel	novel	ADJ
ejpam-6261	652	3	applications	application	NOUN
ejpam-6261	652	4	of	of	ADP
ejpam-6261	652	5	complex	complex	ADJ
ejpam-6261	652	6	intuitionistic	intuitionistic	ADJ
ejpam-6261	652	7	fuzzy	fuzzy	ADJ
ejpam-6261	652	8	sets	set	NOUN
ejpam-6261	652	9	in	in	ADP
ejpam-6261	652	10	group	group	NOUN
ejpam-6261	652	11	theory	theory	NOUN
ejpam-6261	652	12	.	.	PUNCT
ejpam-6261	653	1	ieee	ieee	NOUN
ejpam-6261	653	2	access	access	NOUN
ejpam-6261	653	3	,	,	PUNCT
ejpam-6261	653	4	8:196075–196085	8:196075–196085	NUM
ejpam-6261	653	5	,	,	PUNCT
ejpam-6261	653	6	2020	2020	NUM
ejpam-6261	653	7	.	.	PUNCT
ejpam-6261	654	1	[	[	X
ejpam-6261	654	2	36	36	NUM
ejpam-6261	654	3	]	]	PUNCT
ejpam-6261	654	4	m.	m.	NOUN
ejpam-6261	654	5	gulzar	gulzar	PROPN
ejpam-6261	654	6	,	,	PUNCT
ejpam-6261	654	7	m.	m.	PROPN
ejpam-6261	654	8	h.	h.	PROPN
ejpam-6261	654	9	mateen	mateen	PROPN
ejpam-6261	654	10	,	,	PUNCT
ejpam-6261	654	11	y.	y.	PROPN
ejpam-6261	654	12	m.	m.	PROPN
ejpam-6261	654	13	chu	chu	PROPN
ejpam-6261	654	14	,	,	PUNCT
ejpam-6261	654	15	d.	d.	PROPN
ejpam-6261	654	16	alghazzawi	alghazzawi	PROPN
ejpam-6261	654	17	,	,	PUNCT
ejpam-6261	654	18	and	and	CCONJ
ejpam-6261	654	19	g.	g.	PROPN
ejpam-6261	654	20	abbas	abbas	PROPN
ejpam-6261	654	21	.	.	PUNCT
ejpam-6261	655	1	generalized	generalize	VERB
ejpam-6261	655	2	direct	direct	ADJ
ejpam-6261	655	3	product	product	NOUN
ejpam-6261	655	4	of	of	ADP
ejpam-6261	655	5	complex	complex	ADJ
ejpam-6261	655	6	intuitionistic	intuitionistic	ADJ
ejpam-6261	655	7	fuzzy	fuzzy	ADJ
ejpam-6261	655	8	subrings	subring	NOUN
ejpam-6261	655	9	.	.	PUNCT
ejpam-6261	656	1	international	international	ADJ
ejpam-6261	656	2	journal	journal	PROPN
ejpam-6261	656	3	of	of	ADP
ejpam-6261	656	4	computational	computational	ADJ
ejpam-6261	656	5	intelligence	intelligence	NOUN
ejpam-6261	656	6	systems	system	NOUN
ejpam-6261	656	7	,	,	PUNCT
ejpam-6261	656	8	14(1):582–593	14(1):582–593	NUM
ejpam-6261	656	9	,	,	PUNCT
ejpam-6261	656	10	2021	2021	NUM
ejpam-6261	656	11	.	.	PUNCT
ejpam-6261	657	1	[	[	X
ejpam-6261	657	2	37	37	NUM
ejpam-6261	657	3	]	]	PUNCT
ejpam-6261	657	4	m.	m.	NOUN
ejpam-6261	657	5	s.	s.	PROPN
ejpam-6261	657	6	hameed	hameed	PROPN
ejpam-6261	657	7	,	,	PUNCT
ejpam-6261	657	8	z.	z.	PROPN
ejpam-6261	657	9	ahmad	ahmad	PROPN
ejpam-6261	657	10	,	,	PUNCT
ejpam-6261	657	11	s.	s.	PROPN
ejpam-6261	657	12	ali	ali	PROPN
ejpam-6261	657	13	,	,	PUNCT
ejpam-6261	657	14	m.	m.	PROPN
ejpam-6261	657	15	kamran	kamran	PROPN
ejpam-6261	657	16	,	,	PUNCT
ejpam-6261	657	17	and	and	CCONJ
ejpam-6261	657	18	a.	a.	PROPN
ejpam-6261	657	19	r.	r.	PROPN
ejpam-6261	657	20	lula	lula	PROPN
ejpam-6261	657	21	babole	babole	PROPN
ejpam-6261	657	22	.	.	PUNCT
ejpam-6261	658	1	an	an	DET
ejpam-6261	658	2	approach	approach	NOUN
ejpam-6261	658	3	to	to	ADP
ejpam-6261	658	4	(	(	PUNCT
ejpam-6261	658	5	µ	µ	X
ejpam-6261	658	6	,	,	PUNCT
ejpam-6261	658	7	ν	ν	NOUN
ejpam-6261	658	8	,	,	PUNCT
ejpam-6261	658	9	ω)-single	ω)-single	NOUN
ejpam-6261	658	10	-	-	PUNCT
ejpam-6261	658	11	valued	value	VERB
ejpam-6261	658	12	neutrosophic	neutrosophic	ADJ
ejpam-6261	658	13	submodules	submodule	NOUN
ejpam-6261	658	14	.	.	PUNCT
ejpam-6261	659	1	scientific	scientific	ADJ
ejpam-6261	659	2	reports	report	NOUN
ejpam-6261	659	3	,	,	PUNCT
ejpam-6261	659	4	13(1):751	13(1):751	NUM
ejpam-6261	659	5	,	,	PUNCT
ejpam-6261	659	6	2023	2023	NUM
ejpam-6261	659	7	.	.	PUNCT
ejpam-6261	660	1	[	[	X
ejpam-6261	660	2	38	38	NUM
ejpam-6261	660	3	]	]	PUNCT
ejpam-6261	660	4	a.	a.	NOUN
ejpam-6261	660	5	elrawy	elrawy	NOUN
ejpam-6261	660	6	and	and	CCONJ
ejpam-6261	660	7	m.	m.	PROPN
ejpam-6261	660	8	abdalla	abdalla	PROPN
ejpam-6261	660	9	.	.	PUNCT
ejpam-6261	661	1	results	result	NOUN
ejpam-6261	661	2	on	on	ADP
ejpam-6261	661	3	a	a	DET
ejpam-6261	661	4	neutrosophic	neutrosophic	ADJ
ejpam-6261	661	5	subrings	subring	NOUN
ejpam-6261	661	6	.	.	PUNCT
ejpam-6261	662	1	aims	aim	VERB
ejpam-6261	662	2	mathematics	mathematic	NOUN
ejpam-6261	662	3	,	,	PUNCT
ejpam-6261	662	4	8(9):21393–21405	8(9):21393–21405	PROPN
ejpam-6261	662	5	,	,	PUNCT
ejpam-6261	662	6	2023	2023	NUM
ejpam-6261	662	7	.	.	PUNCT
ejpam-6261	663	1	[	[	X
ejpam-6261	663	2	39	39	NUM
ejpam-6261	663	3	]	]	PUNCT
ejpam-6261	663	4	m.	m.	NOUN
ejpam-6261	663	5	gulistana	gulistana	PROPN
ejpam-6261	663	6	,	,	PUNCT
ejpam-6261	663	7	f.	f.	PROPN
ejpam-6261	663	8	smarandache	smarandache	PROPN
ejpam-6261	663	9	,	,	PUNCT
ejpam-6261	663	10	and	and	CCONJ
ejpam-6261	663	11	a.	a.	NOUN
ejpam-6261	663	12	abdullaha	abdullaha	NOUN
ejpam-6261	663	13	.	.	PUNCT
ejpam-6261	664	1	an	an	DET
ejpam-6261	664	2	application	application	NOUN
ejpam-6261	664	3	of	of	ADP
ejpam-6261	664	4	complex	complex	ADJ
ejpam-6261	664	5	neutrosophic	neutrosophic	ADJ
ejpam-6261	664	6	sets	set	NOUN
ejpam-6261	664	7	to	to	ADP
ejpam-6261	664	8	the	the	DET
ejpam-6261	664	9	theory	theory	NOUN
ejpam-6261	664	10	of	of	ADP
ejpam-6261	664	11	groups	group	NOUN
ejpam-6261	664	12	.	.	PUNCT
ejpam-6261	665	1	in	in	ADP
ejpam-6261	665	2	collected	collect	VERB
ejpam-6261	665	3	papers	paper	NOUN
ejpam-6261	665	4	.	.	PUNCT
ejpam-6261	666	1	volume	volume	NOUN
ejpam-6261	666	2	vii	vii	PROPN
ejpam-6261	666	3	:	:	PUNCT
ejpam-6261	666	4	on	on	ADP
ejpam-6261	666	5	neutrosophic	neutrosophic	ADJ
ejpam-6261	666	6	theory	theory	NOUN
ejpam-6261	666	7	and	and	CCONJ
ejpam-6261	666	8	applications	application	NOUN
ejpam-6261	666	9	,	,	PUNCT
ejpam-6261	666	10	page	page	NOUN
ejpam-6261	666	11	341	341	NUM
ejpam-6261	666	12	.	.	PUNCT
ejpam-6261	666	13	2022	2022	NUM
ejpam-6261	666	14	.	.	PUNCT
ejpam-6261	667	1	[	[	X
ejpam-6261	667	2	40	40	NUM
ejpam-6261	667	3	]	]	PUNCT
ejpam-6261	667	4	f.	f.	PROPN
ejpam-6261	667	5	rahoumah	rahoumah	PROPN
ejpam-6261	667	6	,	,	PUNCT
ejpam-6261	667	7	k.	k.	PROPN
ejpam-6261	667	8	s.	s.	PROPN
ejpam-6261	667	9	yow	yow	PROPN
ejpam-6261	667	10	,	,	PUNCT
ejpam-6261	667	11	n.	n.	PROPN
ejpam-6261	667	12	m.	m.	PROPN
ejpam-6261	667	13	a.	a.	PROPN
ejpam-6261	667	14	nik	nik	PROPN
ejpam-6261	667	15	long	long	ADV
ejpam-6261	667	16	,	,	PUNCT
ejpam-6261	667	17	and	and	CCONJ
ejpam-6261	667	18	m.	m.	NOUN
ejpam-6261	667	19	gasim	gasim	PROPN
ejpam-6261	667	20	.	.	PUNCT
ejpam-6261	668	1	on	on	ADP
ejpam-6261	668	2	properties	property	NOUN
ejpam-6261	668	3	and	and	CCONJ
ejpam-6261	668	4	operations	operation	NOUN
ejpam-6261	668	5	of	of	ADP
ejpam-6261	668	6	complex	complex	ADJ
ejpam-6261	668	7	neutrosophic	neutrosophic	ADJ
ejpam-6261	668	8	soft	soft	ADJ
ejpam-6261	668	9	groups	group	NOUN
ejpam-6261	668	10	.	.	PUNCT
ejpam-6261	669	1	journal	journal	PROPN
ejpam-6261	669	2	of	of	ADP
ejpam-6261	669	3	inequalities	inequality	NOUN
ejpam-6261	669	4	and	and	CCONJ
ejpam-6261	669	5	applications	application	NOUN
ejpam-6261	669	6	,	,	PUNCT
ejpam-6261	669	7	2024(1):103	2024(1):103	NUM
ejpam-6261	669	8	,	,	PUNCT
ejpam-6261	669	9	2024	2024	NUM
ejpam-6261	669	10	.	.	PUNCT
ejpam-6261	670	1	[	[	X
ejpam-6261	670	2	41	41	NUM
ejpam-6261	670	3	]	]	X
ejpam-6261	670	4	f.	f.	PROPN
ejpam-6261	670	5	smarandache	smarandache	PROPN
ejpam-6261	670	6	.	.	PUNCT
ejpam-6261	671	1	neutrosophic	neutrosophic	PROPN
ejpam-6261	671	2	overset	overset	PROPN
ejpam-6261	671	3	,	,	PUNCT
ejpam-6261	671	4	neutrosophic	neutrosophic	ADJ
ejpam-6261	671	5	underset	underset	NOUN
ejpam-6261	671	6	,	,	PUNCT
ejpam-6261	671	7	and	and	CCONJ
ejpam-6261	671	8	neutrosophic	neutrosophic	ADJ
ejpam-6261	671	9	offset	offset	NOUN
ejpam-6261	671	10	:	:	PUNCT
ejpam-6261	671	11	similarly	similarly	ADV
ejpam-6261	671	12	for	for	ADP
ejpam-6261	671	13	neutrosophic	neutrosophic	ADJ
ejpam-6261	671	14	over-/under-/offlogic	over-/under-/offlogic	NOUN
ejpam-6261	671	15	,	,	PUNCT
ejpam-6261	671	16	probability	probability	NOUN
ejpam-6261	671	17	,	,	PUNCT
ejpam-6261	671	18	and	and	CCONJ
ejpam-6261	671	19	statistics	statistic	NOUN
ejpam-6261	671	20	.	.	PUNCT
ejpam-6261	672	1	pons	pon	NOUN
ejpam-6261	672	2	editions	edition	NOUN
ejpam-6261	672	3	,	,	PUNCT
ejpam-6261	672	4	bruxelles	bruxelle	NOUN
ejpam-6261	672	5	,	,	PUNCT
ejpam-6261	672	6	belgique	belgique	NOUN
ejpam-6261	672	7	,	,	PUNCT
ejpam-6261	672	8	2016	2016	NUM
ejpam-6261	672	9	.	.	PUNCT