id	sid	tid	token	lemma	pos
ejpam-5597	1	1	european	european	PROPN
ejpam-5597	1	2	journal	journal	PROPN
ejpam-5597	1	3	of	of	ADP
ejpam-5597	1	4	pure	pure	ADJ
ejpam-5597	1	5	and	and	CCONJ
ejpam-5597	1	6	applied	applied	ADJ
ejpam-5597	1	7	mathematics	mathematic	NOUN
ejpam-5597	1	8	2025	2025	NUM
ejpam-5597	1	9	,	,	PUNCT
ejpam-5597	1	10	vol	vol	NOUN
ejpam-5597	1	11	.	.	PROPN
ejpam-5597	1	12	18	18	NUM
ejpam-5597	1	13	,	,	PUNCT
ejpam-5597	1	14	issue	issue	NOUN
ejpam-5597	1	15	1	1	NUM
ejpam-5597	1	16	,	,	PUNCT
ejpam-5597	1	17	article	article	NOUN
ejpam-5597	1	18	number	number	NOUN
ejpam-5597	1	19	5597	5597	NUM
ejpam-5597	1	20	issn	issn	PROPN
ejpam-5597	1	21	1307	1307	NUM
ejpam-5597	1	22	-	-	SYM
ejpam-5597	1	23	5543	5543	NUM
ejpam-5597	1	24	–	–	PUNCT
ejpam-5597	1	25	ejpam.com	ejpam.com	X
ejpam-5597	1	26	published	publish	VERB
ejpam-5597	1	27	by	by	ADP
ejpam-5597	1	28	new	new	PROPN
ejpam-5597	1	29	york	york	PROPN
ejpam-5597	1	30	business	business	PROPN
ejpam-5597	1	31	global	global	PROPN
ejpam-5597	1	32	numerical	numerical	PROPN
ejpam-5597	1	33	radius	radius	PROPN
ejpam-5597	1	34	inequalities	inequality	NOUN
ejpam-5597	1	35	involving	involve	VERB
ejpam-5597	1	36	2×	2×	NUM
ejpam-5597	1	37	2	2	NUM
ejpam-5597	1	38	block	block	NOUN
ejpam-5597	1	39	matrices	matrix	NOUN
ejpam-5597	1	40	ahmad	ahmad	PROPN
ejpam-5597	1	41	al	al	PROPN
ejpam-5597	1	42	-	-	PUNCT
ejpam-5597	1	43	natoor1,∗	natoor1,∗	ADJ
ejpam-5597	1	44	,	,	PUNCT
ejpam-5597	1	45	fadi	fadi	NOUN
ejpam-5597	1	46	alrimawi2	alrimawi2	PROPN
ejpam-5597	2	1	1	1	NUM
ejpam-5597	2	2	department	department	NOUN
ejpam-5597	2	3	of	of	ADP
ejpam-5597	2	4	mathematics	mathematics	PROPN
ejpam-5597	2	5	,	,	PUNCT
ejpam-5597	2	6	isra	isra	PROPN
ejpam-5597	2	7	university	university	PROPN
ejpam-5597	2	8	,	,	PUNCT
ejpam-5597	2	9	amman	amman	PROPN
ejpam-5597	2	10	,	,	PUNCT
ejpam-5597	2	11	jordan	jordan	PROPN
ejpam-5597	2	12	2	2	NUM
ejpam-5597	2	13	department	department	NOUN
ejpam-5597	2	14	of	of	ADP
ejpam-5597	2	15	basic	basic	ADJ
ejpam-5597	2	16	sciences	sciences	PROPN
ejpam-5597	2	17	,	,	PUNCT
ejpam-5597	2	18	al	al	PROPN
ejpam-5597	2	19	-	-	PUNCT
ejpam-5597	2	20	ahliyya	ahliyya	PROPN
ejpam-5597	2	21	amman	amman	PROPN
ejpam-5597	2	22	university	university	PROPN
ejpam-5597	2	23	,	,	PUNCT
ejpam-5597	2	24	amman	amman	PROPN
ejpam-5597	2	25	,	,	PUNCT
ejpam-5597	2	26	jordan	jordan	PROPN
ejpam-5597	2	27	abstract	abstract	PROPN
ejpam-5597	2	28	.	.	PUNCT
ejpam-5597	3	1	in	in	ADP
ejpam-5597	3	2	this	this	DET
ejpam-5597	3	3	paper	paper	NOUN
ejpam-5597	3	4	,	,	PUNCT
ejpam-5597	3	5	we	we	PRON
ejpam-5597	3	6	give	give	VERB
ejpam-5597	3	7	several	several	ADJ
ejpam-5597	3	8	upper	upper	ADJ
ejpam-5597	3	9	and	and	CCONJ
ejpam-5597	3	10	lower	low	ADJ
ejpam-5597	3	11	bounds	bound	NOUN
ejpam-5597	3	12	for	for	ADP
ejpam-5597	3	13	the	the	DET
ejpam-5597	3	14	numerical	numerical	ADJ
ejpam-5597	3	15	radius	radius	NOUN
ejpam-5597	3	16	of	of	ADP
ejpam-5597	3	17	2×2	2×2	NUM
ejpam-5597	3	18	block	block	NOUN
ejpam-5597	3	19	matrices	matrix	NOUN
ejpam-5597	3	20	.	.	PUNCT
ejpam-5597	4	1	several	several	ADJ
ejpam-5597	4	2	special	special	ADJ
ejpam-5597	4	3	cases	case	NOUN
ejpam-5597	4	4	of	of	ADP
ejpam-5597	4	5	our	our	PRON
ejpam-5597	4	6	results	result	NOUN
ejpam-5597	4	7	are	be	AUX
ejpam-5597	4	8	given	give	VERB
ejpam-5597	4	9	.	.	PUNCT
ejpam-5597	5	1	2020	2020	NUM
ejpam-5597	5	2	mathematics	mathematic	NOUN
ejpam-5597	5	3	subject	subject	NOUN
ejpam-5597	5	4	classifications	classification	NOUN
ejpam-5597	5	5	:	:	PUNCT
ejpam-5597	5	6	47a12	47a12	NUM
ejpam-5597	5	7	,	,	PUNCT
ejpam-5597	5	8	47a30	47a30	NUM
ejpam-5597	5	9	,	,	PUNCT
ejpam-5597	5	10	15a60	15a60	NUM
ejpam-5597	5	11	key	key	ADJ
ejpam-5597	5	12	words	word	NOUN
ejpam-5597	5	13	and	and	CCONJ
ejpam-5597	5	14	phrases	phrase	NOUN
ejpam-5597	5	15	:	:	PUNCT
ejpam-5597	5	16	numerical	numerical	ADJ
ejpam-5597	5	17	radius	radius	NOUN
ejpam-5597	5	18	,	,	PUNCT
ejpam-5597	5	19	matrix	matrix	NOUN
ejpam-5597	5	20	,	,	PUNCT
ejpam-5597	5	21	inequality	inequality	NOUN
ejpam-5597	5	22	1	1	NUM
ejpam-5597	5	23	.	.	PUNCT
ejpam-5597	6	1	introduction	introduction	NOUN
ejpam-5597	6	2	let	let	AUX
ejpam-5597	6	3	mn(c	mn(c	X
ejpam-5597	6	4	)	)	PUNCT
ejpam-5597	6	5	denote	denote	VERB
ejpam-5597	6	6	the	the	DET
ejpam-5597	6	7	space	space	NOUN
ejpam-5597	6	8	of	of	ADP
ejpam-5597	6	9	all	all	DET
ejpam-5597	6	10	n	n	PRON
ejpam-5597	6	11	×	×	NOUN
ejpam-5597	6	12	n	n	CCONJ
ejpam-5597	6	13	complex	complex	ADJ
ejpam-5597	6	14	matrices	matrix	NOUN
ejpam-5597	6	15	.	.	PUNCT
ejpam-5597	7	1	the	the	DET
ejpam-5597	7	2	spectral	spectral	ADJ
ejpam-5597	7	3	norm	norm	NOUN
ejpam-5597	7	4	of	of	ADP
ejpam-5597	7	5	a	a	DET
ejpam-5597	7	6	matrix	matrix	NOUN
ejpam-5597	7	7	a	a	DET
ejpam-5597	7	8	∈	∈	NOUN
ejpam-5597	7	9	mn(c	mn(c	X
ejpam-5597	7	10	)	)	PUNCT
ejpam-5597	7	11	is	be	AUX
ejpam-5597	7	12	defined	define	VERB
ejpam-5597	7	13	by	by	ADP
ejpam-5597	7	14	∥a∥	∥a∥	NOUN
ejpam-5597	7	15	=	=	SYM
ejpam-5597	7	16	max	max	PROPN
ejpam-5597	7	17	∥x∥=1	∥x∥=1	PROPN
ejpam-5597	7	18	{	{	PUNCT
ejpam-5597	7	19	∥ax∥	∥ax∥	ADV
ejpam-5597	7	20	:	:	PUNCT
ejpam-5597	8	1	x	x	X
ejpam-5597	8	2	∈	∈	PROPN
ejpam-5597	8	3	cn	cn	PROPN
ejpam-5597	8	4	}	}	PUNCT
ejpam-5597	8	5	.	.	PUNCT
ejpam-5597	9	1	the	the	DET
ejpam-5597	9	2	numerical	numerical	ADJ
ejpam-5597	9	3	radius	radius	NOUN
ejpam-5597	9	4	of	of	ADP
ejpam-5597	9	5	a	a	DET
ejpam-5597	9	6	matrix	matrix	NOUN
ejpam-5597	9	7	a	a	DET
ejpam-5597	9	8	∈	∈	NOUN
ejpam-5597	9	9	mn(c	mn(c	X
ejpam-5597	9	10	)	)	PUNCT
ejpam-5597	9	11	is	be	AUX
ejpam-5597	9	12	defined	define	VERB
ejpam-5597	9	13	by	by	ADP
ejpam-5597	9	14	ω(a	ω(a	NOUN
ejpam-5597	9	15	)	)	PUNCT
ejpam-5597	9	16	=	=	SYM
ejpam-5597	9	17	max	max	PROPN
ejpam-5597	9	18	∥x∥=1	∥x∥=1	PROPN
ejpam-5597	9	19	{	{	PUNCT
ejpam-5597	9	20	|⟨ax	|⟨ax	VERB
ejpam-5597	9	21	,	,	PUNCT
ejpam-5597	9	22	x⟩|	x⟩|	ADP
ejpam-5597	9	23	:	:	PUNCT
ejpam-5597	9	24	x	x	PUNCT
ejpam-5597	9	25	∈	∈	PROPN
ejpam-5597	9	26	cn	cn	PROPN
ejpam-5597	9	27	}	}	PUNCT
ejpam-5597	9	28	.	.	PUNCT
ejpam-5597	10	1	in	in	ADP
ejpam-5597	10	2	[	[	X
ejpam-5597	10	3	22	22	NUM
ejpam-5597	10	4	]	]	PUNCT
ejpam-5597	10	5	,	,	PUNCT
ejpam-5597	10	6	the	the	DET
ejpam-5597	10	7	author	author	NOUN
ejpam-5597	10	8	proved	prove	VERB
ejpam-5597	10	9	that	that	SCONJ
ejpam-5597	10	10	the	the	DET
ejpam-5597	10	11	numerical	numerical	ADJ
ejpam-5597	10	12	radius	radius	NOUN
ejpam-5597	10	13	of	of	ADP
ejpam-5597	10	14	a	a	DET
ejpam-5597	10	15	matrix	matrix	NOUN
ejpam-5597	10	16	a	a	DET
ejpam-5597	10	17	∈	∈	NOUN
ejpam-5597	10	18	mn(c	mn(c	X
ejpam-5597	10	19	)	)	PUNCT
ejpam-5597	10	20	can	can	AUX
ejpam-5597	10	21	be	be	AUX
ejpam-5597	10	22	formulated	formulate	VERB
ejpam-5597	10	23	as	as	ADP
ejpam-5597	10	24	w	w	NOUN
ejpam-5597	10	25	(	(	PUNCT
ejpam-5597	10	26	a	a	NOUN
ejpam-5597	10	27	)	)	PUNCT
ejpam-5597	10	28	=	=	SYM
ejpam-5597	10	29	max	max	NOUN
ejpam-5597	10	30	θ∈r	θ∈r	NOUN
ejpam-5597	10	31	∥∥∥re(eiθa)∥∥∥	∥∥∥re(eiθa)∥∥∥	PROPN
ejpam-5597	10	32	,	,	PUNCT
ejpam-5597	10	33	where	where	SCONJ
ejpam-5597	10	34	re	re	X
ejpam-5597	10	35	(	(	PUNCT
ejpam-5597	10	36	eiθa	eiθa	PROPN
ejpam-5597	10	37	)	)	PUNCT
ejpam-5597	10	38	denotes	denote	VERB
ejpam-5597	10	39	the	the	DET
ejpam-5597	10	40	real	real	ADJ
ejpam-5597	10	41	part	part	NOUN
ejpam-5597	10	42	of	of	ADP
ejpam-5597	10	43	the	the	DET
ejpam-5597	10	44	matrix	matrix	NOUN
ejpam-5597	10	45	eiθa	eiθa	PROPN
ejpam-5597	10	46	.	.	PUNCT
ejpam-5597	11	1	clearly	clearly	ADV
ejpam-5597	11	2	,	,	PUNCT
ejpam-5597	11	3	we	we	PRON
ejpam-5597	11	4	always	always	ADV
ejpam-5597	11	5	have	have	AUX
ejpam-5597	11	6	w(a	w(a	VERB
ejpam-5597	11	7	)	)	PUNCT
ejpam-5597	11	8	≤	≤	NUM
ejpam-5597	11	9	∥a∥	∥a∥	NOUN
ejpam-5597	11	10	(	(	PUNCT
ejpam-5597	11	11	1	1	NUM
ejpam-5597	11	12	)	)	PUNCT
ejpam-5597	11	13	for	for	ADP
ejpam-5597	11	14	any	any	DET
ejpam-5597	11	15	a	a	DET
ejpam-5597	11	16	∈	∈	NOUN
ejpam-5597	11	17	mn(c	mn(c	X
ejpam-5597	11	18	)	)	PUNCT
ejpam-5597	11	19	.	.	PUNCT
ejpam-5597	12	1	many	many	ADJ
ejpam-5597	12	2	generalizations	generalization	NOUN
ejpam-5597	12	3	and	and	CCONJ
ejpam-5597	12	4	recent	recent	ADJ
ejpam-5597	12	5	related	related	ADJ
ejpam-5597	12	6	results	result	NOUN
ejpam-5597	12	7	of	of	ADP
ejpam-5597	12	8	the	the	DET
ejpam-5597	12	9	numerical	numerical	PROPN
ejpam-5597	12	10	radius	radius	PROPN
ejpam-5597	12	11	w	w	PROPN
ejpam-5597	12	12	(	(	PUNCT
ejpam-5597	12	13	·	·	PUNCT
ejpam-5597	12	14	)	)	PUNCT
ejpam-5597	12	15	were	be	AUX
ejpam-5597	12	16	discussed	discuss	VERB
ejpam-5597	12	17	by	by	ADP
ejpam-5597	12	18	many	many	ADJ
ejpam-5597	12	19	authors	author	NOUN
ejpam-5597	12	20	,	,	PUNCT
ejpam-5597	12	21	some	some	PRON
ejpam-5597	12	22	of	of	ADP
ejpam-5597	12	23	these	these	DET
ejpam-5597	12	24	results	result	NOUN
ejpam-5597	12	25	can	can	AUX
ejpam-5597	12	26	be	be	AUX
ejpam-5597	12	27	found	find	VERB
ejpam-5597	12	28	in	in	ADP
ejpam-5597	12	29	[	[	X
ejpam-5597	12	30	3	3	NUM
ejpam-5597	12	31	]	]	PUNCT
ejpam-5597	12	32	,	,	PUNCT
ejpam-5597	12	33	[	[	X
ejpam-5597	12	34	7	7	NUM
ejpam-5597	12	35	]	]	PUNCT
ejpam-5597	12	36	,	,	PUNCT
ejpam-5597	12	37	[	[	X
ejpam-5597	12	38	10	10	NUM
ejpam-5597	12	39	]	]	PUNCT
ejpam-5597	12	40	,	,	PUNCT
ejpam-5597	12	41	[	[	X
ejpam-5597	12	42	12	12	NUM
ejpam-5597	12	43	]	]	PUNCT
ejpam-5597	12	44	,	,	PUNCT
ejpam-5597	12	45	[	[	X
ejpam-5597	12	46	11	11	NUM
ejpam-5597	12	47	]	]	PUNCT
ejpam-5597	12	48	,	,	PUNCT
ejpam-5597	12	49	[	[	X
ejpam-5597	12	50	9	9	NUM
ejpam-5597	12	51	]	]	PUNCT
ejpam-5597	12	52	,	,	PUNCT
ejpam-5597	12	53	[	[	X
ejpam-5597	12	54	8	8	NUM
ejpam-5597	12	55	]	]	PUNCT
ejpam-5597	12	56	,	,	PUNCT
ejpam-5597	12	57	[	[	X
ejpam-5597	12	58	13	13	NUM
ejpam-5597	12	59	]	]	PUNCT
ejpam-5597	12	60	,	,	PUNCT
ejpam-5597	12	61	[	[	X
ejpam-5597	12	62	14	14	NUM
ejpam-5597	12	63	]	]	PUNCT
ejpam-5597	12	64	,	,	PUNCT
ejpam-5597	12	65	[	[	X
ejpam-5597	12	66	15	15	NUM
ejpam-5597	12	67	]	]	PUNCT
ejpam-5597	12	68	,	,	PUNCT
ejpam-5597	12	69	[	[	X
ejpam-5597	12	70	19	19	NUM
ejpam-5597	12	71	]	]	PUNCT
ejpam-5597	12	72	,	,	PUNCT
ejpam-5597	12	73	and	and	CCONJ
ejpam-5597	12	74	[	[	X
ejpam-5597	12	75	20	20	NUM
ejpam-5597	12	76	]	]	PUNCT
ejpam-5597	12	77	.	.	PUNCT
ejpam-5597	13	1	some	some	DET
ejpam-5597	13	2	basic	basic	ADJ
ejpam-5597	13	3	properties	property	NOUN
ejpam-5597	13	4	of	of	ADP
ejpam-5597	13	5	the	the	DET
ejpam-5597	13	6	numerical	numerical	ADJ
ejpam-5597	13	7	radii	radius	NOUN
ejpam-5597	13	8	and	and	CCONJ
ejpam-5597	13	9	the	the	DET
ejpam-5597	13	10	spectral	spectral	ADJ
ejpam-5597	13	11	norms	norm	NOUN
ejpam-5597	13	12	of	of	ADP
ejpam-5597	13	13	matrices	matrix	NOUN
ejpam-5597	13	14	that	that	PRON
ejpam-5597	13	15	we	we	PRON
ejpam-5597	13	16	need	need	VERB
ejpam-5597	13	17	in	in	ADP
ejpam-5597	13	18	our	our	PRON
ejpam-5597	13	19	paper	paper	NOUN
ejpam-5597	13	20	are	be	AUX
ejpam-5597	13	21	the	the	DET
ejpam-5597	13	22	following	following	NOUN
ejpam-5597	13	23	:	:	PUNCT
ejpam-5597	13	24	for	for	ADP
ejpam-5597	13	25	a	a	DET
ejpam-5597	13	26	,	,	PUNCT
ejpam-5597	13	27	b	b	PROPN
ejpam-5597	13	28	∈	∈	PROPN
ejpam-5597	13	29	mn(c	mn(c	X
ejpam-5597	13	30	)	)	PUNCT
ejpam-5597	13	31	,	,	PUNCT
ejpam-5597	13	32	we	we	PRON
ejpam-5597	13	33	have	have	VERB
ejpam-5597	13	34	the	the	DET
ejpam-5597	13	35	following	follow	VERB
ejpam-5597	13	36	relations	relation	NOUN
ejpam-5597	13	37	:	:	PUNCT
ejpam-5597	13	38	∗corresponding	∗corresponde	VERB
ejpam-5597	13	39	author	author	NOUN
ejpam-5597	13	40	.	.	PUNCT
ejpam-5597	14	1	doi	doi	NOUN
ejpam-5597	14	2	:	:	PUNCT
ejpam-5597	14	3	https://doi.org/10.29020/nybg.ejpam.v18i1.5597	https://doi.org/10.29020/nybg.ejpam.v18i1.5597	PRON
ejpam-5597	14	4	email	email	NOUN
ejpam-5597	14	5	addresses	address	NOUN
ejpam-5597	14	6	:	:	PUNCT
ejpam-5597	14	7	ahmad.alnatoor@iu.edu.jo	ahmad.alnatoor@iu.edu.jo	NOUN
ejpam-5597	14	8	(	(	PUNCT
ejpam-5597	14	9	a.	a.	NOUN
ejpam-5597	14	10	al	al	PROPN
ejpam-5597	14	11	-	-	PUNCT
ejpam-5597	14	12	natoor	natoor	NOUN
ejpam-5597	14	13	)	)	PUNCT
ejpam-5597	14	14	,	,	PUNCT
ejpam-5597	14	15	f.rimawi@ammanu.edu.jo	f.rimawi@ammanu.edu.jo	NOUN
ejpam-5597	14	16	(	(	PUNCT
ejpam-5597	14	17	f.	f.	PROPN
ejpam-5597	14	18	alrimawi	alrimawi	PROPN
ejpam-5597	14	19	)	)	PUNCT
ejpam-5597	14	20	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5597	15	1	1	1	NUM
ejpam-5597	15	2	copyright	copyright	NOUN
ejpam-5597	15	3	:	:	PUNCT
ejpam-5597	15	4	©	©	PROPN
ejpam-5597	15	5	2025	2025	NUM
ejpam-5597	15	6	the	the	DET
ejpam-5597	15	7	author(s	author(s	NOUN
ejpam-5597	15	8	)	)	PUNCT
ejpam-5597	15	9	.	.	PUNCT
ejpam-5597	16	1	(	(	PUNCT
ejpam-5597	16	2	cc	cc	NOUN
ejpam-5597	16	3	by	by	ADP
ejpam-5597	16	4	-	-	PUNCT
ejpam-5597	16	5	nc	nc	PROPN
ejpam-5597	16	6	4.0	4.0	NUM
ejpam-5597	16	7	)	)	PUNCT
ejpam-5597	16	8	a.	a.	NOUN
ejpam-5597	16	9	al	al	PROPN
ejpam-5597	16	10	-	-	PUNCT
ejpam-5597	16	11	natoor	natoor	NOUN
ejpam-5597	16	12	,	,	PUNCT
ejpam-5597	16	13	f.	f.	PROPN
ejpam-5597	16	14	alrimawi	alrimawi	PROPN
ejpam-5597	16	15	/	/	SYM
ejpam-5597	16	16	eur	eur	PROPN
ejpam-5597	16	17	.	.	PUNCT
ejpam-5597	17	1	j.	j.	PROPN
ejpam-5597	17	2	pure	pure	PROPN
ejpam-5597	17	3	appl	appl	PROPN
ejpam-5597	17	4	.	.	PROPN
ejpam-5597	17	5	math	math	PROPN
ejpam-5597	17	6	,	,	PUNCT
ejpam-5597	17	7	18	18	NUM
ejpam-5597	17	8	(	(	PUNCT
ejpam-5597	17	9	1	1	NUM
ejpam-5597	17	10	)	)	PUNCT
ejpam-5597	17	11	(	(	PUNCT
ejpam-5597	17	12	2025	2025	NUM
ejpam-5597	17	13	)	)	PUNCT
ejpam-5597	17	14	,	,	PUNCT
ejpam-5597	17	15	5597	5597	NUM
ejpam-5597	17	16	2	2	NUM
ejpam-5597	17	17	of	of	ADP
ejpam-5597	17	18	9	9	NUM
ejpam-5597	17	19	(	(	PUNCT
ejpam-5597	17	20	i	i	NOUN
ejpam-5597	17	21	)	)	PUNCT
ejpam-5597	17	22	w	w	PROPN
ejpam-5597	17	23	(	(	PUNCT
ejpam-5597	17	24	[	[	PUNCT
ejpam-5597	17	25	a	a	PRON
ejpam-5597	17	26	0	0	NUM
ejpam-5597	17	27	0	0	NUM
ejpam-5597	17	28	b	b	NOUN
ejpam-5597	17	29	]	]	X
ejpam-5597	17	30	)	)	PUNCT
ejpam-5597	17	31	=	=	SYM
ejpam-5597	18	1	max{w(a	max{w(a	NOUN
ejpam-5597	18	2	)	)	PUNCT
ejpam-5597	18	3	,	,	PUNCT
ejpam-5597	18	4	w(b	w(b	NOUN
ejpam-5597	18	5	)	)	PUNCT
ejpam-5597	18	6	)	)	PUNCT
ejpam-5597	18	7	}	}	PUNCT
ejpam-5597	18	8	(	(	PUNCT
ejpam-5597	18	9	ii	ii	NOUN
ejpam-5597	18	10	)	)	PUNCT
ejpam-5597	18	11	w(a∗	w(a∗	CCONJ
ejpam-5597	18	12	)	)	PUNCT
ejpam-5597	18	13	=	=	SYM
ejpam-5597	19	1	w(a	w(a	X
ejpam-5597	19	2	)	)	PUNCT
ejpam-5597	19	3	(	(	PUNCT
ejpam-5597	19	4	iii	iii	NOUN
ejpam-5597	19	5	)	)	PUNCT
ejpam-5597	19	6	∥a∗a∥	∥a∗a∥	NOUN
ejpam-5597	19	7	=	=	SYM
ejpam-5597	19	8	∥aa∗∥	∥aa∗∥	ADJ
ejpam-5597	19	9	=	=	SYM
ejpam-5597	19	10	∥a∥2	∥a∥2	PROPN
ejpam-5597	19	11	(	(	PUNCT
ejpam-5597	19	12	iv	iv	X
ejpam-5597	19	13	)	)	PUNCT
ejpam-5597	19	14	∥∥∥∥	∥∥∥∥	PROPN
ejpam-5597	19	15	[	[	PUNCT
ejpam-5597	19	16	a	a	PRON
ejpam-5597	19	17	0	0	NUM
ejpam-5597	19	18	0	0	NUM
ejpam-5597	19	19	b	b	NOUN
ejpam-5597	19	20	]	]	PUNCT
ejpam-5597	19	21	∥∥∥∥	∥∥∥∥	NUM
ejpam-5597	19	22	=	=	SYM
ejpam-5597	19	23	∥∥∥∥	∥∥∥∥	PROPN
ejpam-5597	19	24	[	[	PUNCT
ejpam-5597	19	25	0	0	NUM
ejpam-5597	19	26	a	a	DET
ejpam-5597	19	27	b	b	NOUN
ejpam-5597	19	28	0	0	NUM
ejpam-5597	19	29	]	]	SYM
ejpam-5597	19	30	∥∥∥∥	∥∥∥∥	NUM
ejpam-5597	19	31	=	=	SYM
ejpam-5597	19	32	max{∥a∥	max{∥a∥	VERB
ejpam-5597	19	33	,	,	PUNCT
ejpam-5597	19	34	∥b∥	∥b∥	NUM
ejpam-5597	19	35	}	}	PUNCT
ejpam-5597	19	36	.	.	PUNCT
ejpam-5597	20	1	recent	recent	ADJ
ejpam-5597	20	2	results	result	NOUN
ejpam-5597	20	3	concerning	concern	VERB
ejpam-5597	20	4	inequalities	inequality	NOUN
ejpam-5597	20	5	can	can	AUX
ejpam-5597	20	6	be	be	AUX
ejpam-5597	20	7	found	find	VERB
ejpam-5597	20	8	in	in	ADP
ejpam-5597	20	9	[	[	X
ejpam-5597	20	10	1	1	NUM
ejpam-5597	20	11	]	]	PUNCT
ejpam-5597	20	12	,	,	PUNCT
ejpam-5597	20	13	[	[	X
ejpam-5597	20	14	2	2	NUM
ejpam-5597	20	15	]	]	PUNCT
ejpam-5597	20	16	,	,	PUNCT
ejpam-5597	20	17	[	[	X
ejpam-5597	20	18	4	4	NUM
ejpam-5597	20	19	]	]	PUNCT
ejpam-5597	20	20	,	,	PUNCT
ejpam-5597	20	21	[	[	X
ejpam-5597	20	22	6	6	NUM
ejpam-5597	20	23	]	]	PUNCT
ejpam-5597	20	24	,	,	PUNCT
ejpam-5597	20	25	[	[	X
ejpam-5597	20	26	5	5	NUM
ejpam-5597	20	27	]	]	PUNCT
ejpam-5597	20	28	,	,	PUNCT
ejpam-5597	20	29	and	and	CCONJ
ejpam-5597	20	30	[	[	X
ejpam-5597	20	31	21	21	NUM
ejpam-5597	20	32	]	]	PUNCT
ejpam-5597	20	33	.	.	PUNCT
ejpam-5597	21	1	2	2	X
ejpam-5597	21	2	.	.	X
ejpam-5597	21	3	main	main	ADJ
ejpam-5597	21	4	results	result	NOUN
ejpam-5597	21	5	we	we	PRON
ejpam-5597	21	6	start	start	VERB
ejpam-5597	21	7	with	with	ADP
ejpam-5597	21	8	the	the	DET
ejpam-5597	21	9	following	follow	VERB
ejpam-5597	21	10	theorem	theorem	NOUN
ejpam-5597	21	11	.	.	PUNCT
ejpam-5597	21	12	theorem	theorem	NOUN
ejpam-5597	21	13	1	1	NUM
ejpam-5597	21	14	.	.	PUNCT
ejpam-5597	22	1	let	let	VERB
ejpam-5597	22	2	a	a	DET
ejpam-5597	22	3	,	,	PUNCT
ejpam-5597	22	4	b	b	NOUN
ejpam-5597	22	5	,	,	PUNCT
ejpam-5597	22	6	c	c	NOUN
ejpam-5597	22	7	,	,	PUNCT
ejpam-5597	22	8	d	d	PROPN
ejpam-5597	22	9	∈	∈	PROPN
ejpam-5597	22	10	mn(c	mn(c	X
ejpam-5597	22	11	)	)	PUNCT
ejpam-5597	22	12	.	.	PUNCT
ejpam-5597	23	1	then	then	ADV
ejpam-5597	23	2	w	w	X
ejpam-5597	23	3	(	(	PUNCT
ejpam-5597	23	4	[	[	PUNCT
ejpam-5597	23	5	a	a	PRON
ejpam-5597	23	6	b	b	NOUN
ejpam-5597	23	7	c	c	NOUN
ejpam-5597	23	8	d	d	X
ejpam-5597	23	9	]	]	X
ejpam-5597	23	10	)	)	PUNCT
ejpam-5597	23	11	≤	≤	NOUN
ejpam-5597	23	12	√	√	VERB
ejpam-5597	24	1	max{∥a∗a+	max{∥a∗a+	NOUN
ejpam-5597	24	2	c∗c∥	c∗c∥	NOUN
ejpam-5597	24	3	,	,	PUNCT
ejpam-5597	24	4	∥b∗b	∥b∗b	VERB
ejpam-5597	24	5	+	+	NOUN
ejpam-5597	24	6	d∗d∥}+	d∗d∥}+	NOUN
ejpam-5597	24	7	∥a∗b	∥a∗b	VERB
ejpam-5597	24	8	+	+	CCONJ
ejpam-5597	24	9	c∗d∥.	c∗d∥.	PROPN
ejpam-5597	24	10	(	(	PUNCT
ejpam-5597	24	11	2	2	NUM
ejpam-5597	24	12	)	)	PUNCT
ejpam-5597	24	13	proof	proof	NOUN
ejpam-5597	24	14	.	.	PUNCT
ejpam-5597	25	1	we	we	PRON
ejpam-5597	25	2	have	have	VERB
ejpam-5597	25	3	w	w	NOUN
ejpam-5597	25	4	(	(	PUNCT
ejpam-5597	25	5	[	[	PUNCT
ejpam-5597	25	6	a	a	PRON
ejpam-5597	25	7	b	b	NOUN
ejpam-5597	25	8	c	c	NOUN
ejpam-5597	25	9	d	d	X
ejpam-5597	25	10	]	]	X
ejpam-5597	25	11	)	)	PUNCT
ejpam-5597	25	12	≤	≤	NUM
ejpam-5597	25	13	∥∥∥∥	∥∥∥∥	PROPN
ejpam-5597	25	14	[	[	PUNCT
ejpam-5597	25	15	a	a	DET
ejpam-5597	25	16	b	b	NOUN
ejpam-5597	25	17	c	c	NOUN
ejpam-5597	25	18	d	d	X
ejpam-5597	25	19	]	]	X
ejpam-5597	25	20	∥∥∥∥	∥∥∥∥	NUM
ejpam-5597	25	21	(	(	PUNCT
ejpam-5597	25	22	by	by	ADP
ejpam-5597	25	23	inequality	inequality	NOUN
ejpam-5597	25	24	(	(	PUNCT
ejpam-5597	25	25	1	1	NUM
ejpam-5597	25	26	)	)	PUNCT
ejpam-5597	25	27	)	)	PUNCT
ejpam-5597	26	1	=	=	SYM
ejpam-5597	26	2	√∥∥∥∥	√∥∥∥∥	PROPN
ejpam-5597	26	3	[	[	PUNCT
ejpam-5597	26	4	a	a	PRON
ejpam-5597	26	5	b	b	NOUN
ejpam-5597	26	6	c	c	NOUN
ejpam-5597	26	7	d	d	X
ejpam-5597	26	8	]	]	X
ejpam-5597	26	9	∥∥∥∥2	∥∥∥∥2	NOUN
ejpam-5597	26	10	=	=	SYM
ejpam-5597	26	11	√∥∥∥∥	√∥∥∥∥	PROPN
ejpam-5597	26	12	[	[	PUNCT
ejpam-5597	26	13	a∗	a∗	PROPN
ejpam-5597	26	14	c∗	c∗	PROPN
ejpam-5597	26	15	b∗	b∗	ADJ
ejpam-5597	26	16	d∗	d∗	PROPN
ejpam-5597	26	17	]	]	PUNCT
ejpam-5597	26	18	[	[	PUNCT
ejpam-5597	26	19	a	a	PRON
ejpam-5597	26	20	b	b	NOUN
ejpam-5597	26	21	c	c	NOUN
ejpam-5597	26	22	d	d	X
ejpam-5597	26	23	]	]	X
ejpam-5597	26	24	∥∥∥∥	∥∥∥∥	PUNCT
ejpam-5597	26	25	=	=	SYM
ejpam-5597	26	26	√∥∥∥∥	√∥∥∥∥	PROPN
ejpam-5597	26	27	[	[	PUNCT
ejpam-5597	26	28	a∗a+	a∗a+	NOUN
ejpam-5597	26	29	c∗c	c∗c	NUM
ejpam-5597	26	30	a∗b	a∗b	NUM
ejpam-5597	26	31	+	+	NUM
ejpam-5597	26	32	c∗d	c∗d	NOUN
ejpam-5597	26	33	b∗a+d∗c	b∗a+d∗c	NOUN
ejpam-5597	26	34	b∗b	b∗b	NOUN
ejpam-5597	26	35	+	+	NOUN
ejpam-5597	26	36	d∗d	d∗d	PROPN
ejpam-5597	26	37	]	]	SYM
ejpam-5597	26	38	∥∥∥∥	∥∥∥∥	NUM
ejpam-5597	26	39	(	(	PUNCT
ejpam-5597	26	40	3	3	NUM
ejpam-5597	26	41	)	)	PUNCT
ejpam-5597	26	42	=	=	SYM
ejpam-5597	27	1	√∥∥∥∥	√∥∥∥∥	PROPN
ejpam-5597	27	2	[	[	PUNCT
ejpam-5597	27	3	a∗a+	a∗a+	X
ejpam-5597	27	4	c∗c	c∗c	X
ejpam-5597	27	5	0	0	NUM
ejpam-5597	27	6	0	0	NUM
ejpam-5597	27	7	b∗b	b∗b	NOUN
ejpam-5597	27	8	+	+	NOUN
ejpam-5597	27	9	d∗d	d∗d	PROPN
ejpam-5597	27	10	]	]	PUNCT
ejpam-5597	28	1	+	+	CCONJ
ejpam-5597	28	2	[	[	PUNCT
ejpam-5597	28	3	0	0	NUM
ejpam-5597	28	4	a∗b	a∗b	NUM
ejpam-5597	28	5	+	+	NUM
ejpam-5597	28	6	c∗d	c∗d	NOUN
ejpam-5597	28	7	b∗a+d∗c	b∗a+d∗c	NOUN
ejpam-5597	28	8	0	0	NUM
ejpam-5597	28	9	]	]	SYM
ejpam-5597	28	10	∥∥∥∥	∥∥∥∥	NUM
ejpam-5597	28	11	(	(	PUNCT
ejpam-5597	28	12	4	4	NUM
ejpam-5597	28	13	)	)	PUNCT
ejpam-5597	28	14	≤	≤	NOUN
ejpam-5597	29	1	√∥∥∥∥	√∥∥∥∥	PROPN
ejpam-5597	29	2	[	[	PUNCT
ejpam-5597	29	3	a∗a+	a∗a+	NOUN
ejpam-5597	29	4	c∗c	c∗c	X
ejpam-5597	29	5	0	0	NUM
ejpam-5597	29	6	0	0	NUM
ejpam-5597	29	7	b∗b	b∗b	NOUN
ejpam-5597	29	8	+	+	ADJ
ejpam-5597	29	9	d∗d	d∗d	NOUN
ejpam-5597	29	10	]	]	X
ejpam-5597	29	11	∥∥∥∥+	∥∥∥∥+	NOUN
ejpam-5597	29	12	∥∥∥∥	∥∥∥∥	PROPN
ejpam-5597	29	13	[	[	PUNCT
ejpam-5597	29	14	0	0	NUM
ejpam-5597	29	15	a∗b	a∗b	NUM
ejpam-5597	29	16	+	+	NUM
ejpam-5597	29	17	c∗d	c∗d	NOUN
ejpam-5597	29	18	b∗a+d∗c	b∗a+d∗c	NOUN
ejpam-5597	29	19	0	0	NUM
ejpam-5597	29	20	]	]	PUNCT
ejpam-5597	29	21	∥∥∥∥	∥∥∥∥	NUM
ejpam-5597	29	22	(	(	PUNCT
ejpam-5597	29	23	by	by	ADP
ejpam-5597	29	24	the	the	DET
ejpam-5597	29	25	triangle	triangle	NOUN
ejpam-5597	29	26	inequality	inequality	NOUN
ejpam-5597	29	27	)	)	PUNCT
ejpam-5597	29	28	=	=	SYM
ejpam-5597	30	1	√	√	NUM
ejpam-5597	30	2	max{∥a∗a+	max{∥a∗a+	NOUN
ejpam-5597	30	3	c∗c∥	c∗c∥	NOUN
ejpam-5597	30	4	,	,	PUNCT
ejpam-5597	30	5	∥b∗b	∥b∗b	VERB
ejpam-5597	30	6	+	+	NOUN
ejpam-5597	30	7	d∗d∥}+	d∗d∥}+	NOUN
ejpam-5597	30	8	∥a∗b	∥a∗b	VERB
ejpam-5597	30	9	+	+	CCONJ
ejpam-5597	31	1	c∗d∥	c∗d∥	PROPN
ejpam-5597	31	2	,	,	PUNCT
ejpam-5597	31	3	as	as	SCONJ
ejpam-5597	31	4	required	require	VERB
ejpam-5597	31	5	.	.	PUNCT
ejpam-5597	32	1	based	base	VERB
ejpam-5597	32	2	on	on	ADP
ejpam-5597	32	3	theorem	theorem	NOUN
ejpam-5597	32	4	1	1	NUM
ejpam-5597	32	5	and	and	CCONJ
ejpam-5597	32	6	its	its	PRON
ejpam-5597	32	7	proof	proof	NOUN
ejpam-5597	32	8	,	,	PUNCT
ejpam-5597	32	9	we	we	PRON
ejpam-5597	32	10	have	have	VERB
ejpam-5597	32	11	several	several	ADJ
ejpam-5597	32	12	corollaries	corollary	NOUN
ejpam-5597	32	13	.	.	PUNCT
ejpam-5597	33	1	a.	a.	PROPN
ejpam-5597	33	2	al	al	PROPN
ejpam-5597	33	3	-	-	PUNCT
ejpam-5597	33	4	natoor	natoor	NOUN
ejpam-5597	33	5	,	,	PUNCT
ejpam-5597	33	6	f.	f.	PROPN
ejpam-5597	33	7	alrimawi	alrimawi	PROPN
ejpam-5597	33	8	/	/	SYM
ejpam-5597	33	9	eur	eur	PROPN
ejpam-5597	33	10	.	.	PUNCT
ejpam-5597	34	1	j.	j.	PROPN
ejpam-5597	34	2	pure	pure	PROPN
ejpam-5597	34	3	appl	appl	PROPN
ejpam-5597	34	4	.	.	PROPN
ejpam-5597	34	5	math	math	PROPN
ejpam-5597	34	6	,	,	PUNCT
ejpam-5597	34	7	18	18	NUM
ejpam-5597	34	8	(	(	PUNCT
ejpam-5597	34	9	1	1	NUM
ejpam-5597	34	10	)	)	PUNCT
ejpam-5597	34	11	(	(	PUNCT
ejpam-5597	34	12	2025	2025	NUM
ejpam-5597	34	13	)	)	PUNCT
ejpam-5597	34	14	,	,	PUNCT
ejpam-5597	34	15	5597	5597	NUM
ejpam-5597	34	16	3	3	NUM
ejpam-5597	34	17	of	of	ADP
ejpam-5597	34	18	9	9	NUM
ejpam-5597	34	19	corollary	corollary	ADJ
ejpam-5597	34	20	1	1	NUM
ejpam-5597	34	21	.	.	PUNCT
ejpam-5597	35	1	let	let	VERB
ejpam-5597	35	2	a	a	DET
ejpam-5597	35	3	,	,	PUNCT
ejpam-5597	35	4	b	b	NOUN
ejpam-5597	35	5	∈	∈	NOUN
ejpam-5597	35	6	mn(c	mn(c	X
ejpam-5597	35	7	)	)	PUNCT
ejpam-5597	35	8	.	.	PUNCT
ejpam-5597	36	1	then	then	ADV
ejpam-5597	36	2	w	w	X
ejpam-5597	36	3	(	(	PUNCT
ejpam-5597	36	4	[	[	PUNCT
ejpam-5597	36	5	a	a	DET
ejpam-5597	36	6	b	b	NOUN
ejpam-5597	36	7	0	0	NUM
ejpam-5597	36	8	0	0	NUM
ejpam-5597	36	9	]	]	PUNCT
ejpam-5597	36	10	)	)	PUNCT
ejpam-5597	36	11	≤	≤	NUM
ejpam-5597	36	12	√	√	ADP
ejpam-5597	36	13	max{∥a∥2	max{∥a∥2	NOUN
ejpam-5597	36	14	,	,	PUNCT
ejpam-5597	36	15	∥b∥2}+	∥b∥2}+	NOUN
ejpam-5597	36	16	∥a∗b∥.	∥a∗b∥.	NOUN
ejpam-5597	36	17	(	(	PUNCT
ejpam-5597	36	18	5	5	X
ejpam-5597	36	19	)	)	PUNCT
ejpam-5597	36	20	proof	proof	NOUN
ejpam-5597	36	21	.	.	PUNCT
ejpam-5597	37	1	the	the	DET
ejpam-5597	37	2	result	result	NOUN
ejpam-5597	37	3	follows	follow	VERB
ejpam-5597	37	4	by	by	ADP
ejpam-5597	37	5	letting	let	VERB
ejpam-5597	37	6	c	c	NOUN
ejpam-5597	37	7	=	=	PUNCT
ejpam-5597	37	8	d	d	NOUN
ejpam-5597	37	9	=	=	SYM
ejpam-5597	37	10	0	0	NUM
ejpam-5597	37	11	in	in	ADP
ejpam-5597	37	12	inequality	inequality	NOUN
ejpam-5597	37	13	(	(	PUNCT
ejpam-5597	37	14	2	2	NUM
ejpam-5597	37	15	)	)	PUNCT
ejpam-5597	37	16	.	.	PUNCT
ejpam-5597	38	1	to	to	PART
ejpam-5597	38	2	state	state	VERB
ejpam-5597	38	3	our	our	PRON
ejpam-5597	38	4	next	next	ADJ
ejpam-5597	38	5	corollary	corollary	NOUN
ejpam-5597	38	6	,	,	PUNCT
ejpam-5597	38	7	we	we	PRON
ejpam-5597	38	8	need	need	VERB
ejpam-5597	38	9	the	the	DET
ejpam-5597	38	10	following	follow	VERB
ejpam-5597	38	11	lemma	lemma	PROPN
ejpam-5597	38	12	[	[	X
ejpam-5597	38	13	16	16	NUM
ejpam-5597	38	14	]	]	PUNCT
ejpam-5597	38	15	.	.	PUNCT
ejpam-5597	39	1	lemma	lemma	PROPN
ejpam-5597	39	2	1	1	X
ejpam-5597	39	3	.	.	PUNCT
ejpam-5597	40	1	let	let	VERB
ejpam-5597	40	2	a	a	DET
ejpam-5597	40	3	,	,	PUNCT
ejpam-5597	40	4	b	b	NOUN
ejpam-5597	40	5	∈	∈	PROPN
ejpam-5597	40	6	mn(c	mn(c	X
ejpam-5597	40	7	)	)	PUNCT
ejpam-5597	40	8	be	be	AUX
ejpam-5597	40	9	normal	normal	ADJ
ejpam-5597	40	10	.	.	PUNCT
ejpam-5597	41	1	then	then	ADV
ejpam-5597	41	2	∥a+b∥	∥a+b∥	PROPN
ejpam-5597	41	3	≤	≤	ADJ
ejpam-5597	41	4	∥|a|+	∥|a|+	NOUN
ejpam-5597	41	5	|b|∥	|b|∥	NOUN
ejpam-5597	41	6	,	,	PUNCT
ejpam-5597	41	7	where	where	SCONJ
ejpam-5597	41	8	|t	|t	PROPN
ejpam-5597	42	1	|	|	ADV
ejpam-5597	42	2	is	be	AUX
ejpam-5597	42	3	the	the	DET
ejpam-5597	42	4	absolute	absolute	ADJ
ejpam-5597	42	5	value	value	NOUN
ejpam-5597	42	6	of	of	ADP
ejpam-5597	42	7	t	t	PROPN
ejpam-5597	42	8	∈	∈	PROPN
ejpam-5597	42	9	mn(c	mn(c	X
ejpam-5597	42	10	)	)	PUNCT
ejpam-5597	42	11	which	which	PRON
ejpam-5597	42	12	is	be	AUX
ejpam-5597	42	13	defined	define	VERB
ejpam-5597	42	14	by	by	ADP
ejpam-5597	42	15	|t	|t	PROPN
ejpam-5597	43	1	|	|	ADV
ejpam-5597	43	2	=	=	PRON
ejpam-5597	43	3	(	(	PUNCT
ejpam-5597	43	4	t	t	X
ejpam-5597	43	5	∗t	∗t	PROPN
ejpam-5597	43	6	)	)	PUNCT
ejpam-5597	43	7	1/2	1/2	NUM
ejpam-5597	43	8	.	.	PUNCT
ejpam-5597	44	1	corollary	corollary	ADJ
ejpam-5597	44	2	2	2	NUM
ejpam-5597	44	3	.	.	PUNCT
ejpam-5597	45	1	let	let	VERB
ejpam-5597	45	2	a	a	DET
ejpam-5597	45	3	,	,	PUNCT
ejpam-5597	45	4	b	b	NOUN
ejpam-5597	45	5	,	,	PUNCT
ejpam-5597	45	6	c	c	NOUN
ejpam-5597	45	7	,	,	PUNCT
ejpam-5597	45	8	d	d	PROPN
ejpam-5597	45	9	∈	∈	PROPN
ejpam-5597	45	10	mn(c	mn(c	X
ejpam-5597	45	11	)	)	PUNCT
ejpam-5597	45	12	.	.	PUNCT
ejpam-5597	46	1	then	then	ADV
ejpam-5597	46	2	w	w	X
ejpam-5597	46	3	(	(	PUNCT
ejpam-5597	46	4	[	[	PUNCT
ejpam-5597	46	5	a	a	PRON
ejpam-5597	46	6	b	b	NOUN
ejpam-5597	46	7	c	c	NOUN
ejpam-5597	46	8	d	d	X
ejpam-5597	46	9	]	]	X
ejpam-5597	46	10	)	)	PUNCT
ejpam-5597	46	11	≤	≤	NOUN
ejpam-5597	46	12	√	√	VERB
ejpam-5597	46	13	max{∥a∗a+	max{∥a∗a+	PROPN
ejpam-5597	46	14	c∗c	c∗c	NOUN
ejpam-5597	46	15	+	+	CCONJ
ejpam-5597	46	16	|b∗a+d∗c|∥	|b∗a+d∗c|∥	PROPN
ejpam-5597	46	17	,	,	PUNCT
ejpam-5597	46	18	∥b∗b	∥b∗b	X
ejpam-5597	46	19	+	+	NOUN
ejpam-5597	46	20	d∗d	d∗d	PROPN
ejpam-5597	46	21	+	+	NUM
ejpam-5597	46	22	|a∗b	|a∗b	X
ejpam-5597	46	23	+	+	ADJ
ejpam-5597	46	24	c∗d|∥	c∗d|∥	PROPN
ejpam-5597	46	25	}	}	PUNCT
ejpam-5597	46	26	.	.	PUNCT
ejpam-5597	47	1	proof	proof	NOUN
ejpam-5597	47	2	.	.	PUNCT
ejpam-5597	48	1	by	by	ADP
ejpam-5597	48	2	inequality	inequality	NOUN
ejpam-5597	48	3	(	(	PUNCT
ejpam-5597	48	4	4	4	NUM
ejpam-5597	48	5	)	)	PUNCT
ejpam-5597	48	6	,	,	PUNCT
ejpam-5597	48	7	we	we	PRON
ejpam-5597	48	8	have	have	VERB
ejpam-5597	48	9	w	w	NOUN
ejpam-5597	48	10	(	(	PUNCT
ejpam-5597	48	11	[	[	PUNCT
ejpam-5597	48	12	a	a	PRON
ejpam-5597	48	13	b	b	NOUN
ejpam-5597	48	14	c	c	NOUN
ejpam-5597	48	15	d	d	X
ejpam-5597	48	16	]	]	X
ejpam-5597	48	17	)	)	PUNCT
ejpam-5597	48	18	≤	≤	PUNCT
ejpam-5597	49	1	√∥∥∥∥	√∥∥∥∥	PROPN
ejpam-5597	49	2	[	[	PUNCT
ejpam-5597	49	3	a∗a+	a∗a+	NOUN
ejpam-5597	49	4	c∗c	c∗c	X
ejpam-5597	49	5	0	0	NUM
ejpam-5597	49	6	0	0	NUM
ejpam-5597	49	7	b∗b	b∗b	NOUN
ejpam-5597	50	1	+	+	NOUN
ejpam-5597	50	2	d∗d	d∗d	PROPN
ejpam-5597	50	3	]	]	PUNCT
ejpam-5597	50	4	+	+	CCONJ
ejpam-5597	50	5	[	[	PUNCT
ejpam-5597	50	6	0	0	NUM
ejpam-5597	50	7	a∗b	a∗b	NUM
ejpam-5597	50	8	+	+	NUM
ejpam-5597	50	9	c∗d	c∗d	NOUN
ejpam-5597	50	10	b∗a+d∗c	b∗a+d∗c	NOUN
ejpam-5597	50	11	0	0	NUM
ejpam-5597	50	12	]	]	SYM
ejpam-5597	50	13	∥∥∥∥	∥∥∥∥	NUM
ejpam-5597	50	14	≤	≤	ADJ
ejpam-5597	50	15	√∥∥∥∥	√∥∥∥∥	PROPN
ejpam-5597	50	16	[	[	PUNCT
ejpam-5597	50	17	a∗a+	a∗a+	NOUN
ejpam-5597	50	18	c∗c	c∗c	X
ejpam-5597	50	19	0	0	NUM
ejpam-5597	50	20	0	0	NUM
ejpam-5597	50	21	b∗b	b∗b	NOUN
ejpam-5597	50	22	+	+	ADJ
ejpam-5597	50	23	d∗d	d∗d	PROPN
ejpam-5597	50	24	]	]	PUNCT
ejpam-5597	50	25	+	+	CCONJ
ejpam-5597	50	26	∣∣∣∣	∣∣∣∣	PROPN
ejpam-5597	50	27	[	[	PUNCT
ejpam-5597	50	28	0	0	NUM
ejpam-5597	50	29	a∗b	a∗b	NUM
ejpam-5597	50	30	+	+	NUM
ejpam-5597	50	31	c∗d	c∗d	NOUN
ejpam-5597	50	32	b∗a+d∗c	b∗a+d∗c	NOUN
ejpam-5597	50	33	0	0	NUM
ejpam-5597	50	34	]	]	X
ejpam-5597	50	35	∣∣∣∣∥∥∥∥	∣∣∣∣∥∥∥∥	NOUN
ejpam-5597	50	36	(	(	PUNCT
ejpam-5597	50	37	by	by	ADP
ejpam-5597	50	38	lemma	lemma	PROPN
ejpam-5597	50	39	1	1	NUM
ejpam-5597	50	40	)	)	PUNCT
ejpam-5597	50	41	=	=	SYM
ejpam-5597	50	42	√∥∥∥∥	√∥∥∥∥	PROPN
ejpam-5597	50	43	[	[	PUNCT
ejpam-5597	50	44	a∗a+	a∗a+	X
ejpam-5597	50	45	c∗c	c∗c	X
ejpam-5597	50	46	0	0	NUM
ejpam-5597	50	47	0	0	NUM
ejpam-5597	50	48	b∗b	b∗b	NOUN
ejpam-5597	50	49	+	+	NOUN
ejpam-5597	50	50	d∗d	d∗d	PROPN
ejpam-5597	50	51	]	]	PUNCT
ejpam-5597	50	52	+	+	CCONJ
ejpam-5597	50	53	[	[	PUNCT
ejpam-5597	50	54	|b∗a+d∗c|	|b∗a+d∗c|	NOUN
ejpam-5597	50	55	0	0	NUM
ejpam-5597	50	56	0	0	NUM
ejpam-5597	50	57	|a∗b	|a∗b	NOUN
ejpam-5597	50	58	+	+	X
ejpam-5597	50	59	c∗d|	c∗d|	X
ejpam-5597	50	60	]	]	PUNCT
ejpam-5597	50	61	∥∥∥∥	∥∥∥∥	PUNCT
ejpam-5597	50	62	=	=	SYM
ejpam-5597	50	63	√∥∥∥∥	√∥∥∥∥	PROPN
ejpam-5597	50	64	[	[	PUNCT
ejpam-5597	50	65	a∗a+	a∗a+	NOUN
ejpam-5597	50	66	c∗c	c∗c	NUM
ejpam-5597	50	67	+	+	NUM
ejpam-5597	50	68	|b∗a+d∗c|	|b∗a+d∗c|	NOUN
ejpam-5597	50	69	0	0	NUM
ejpam-5597	50	70	0	0	NUM
ejpam-5597	50	71	b∗b	b∗b	NOUN
ejpam-5597	50	72	+	+	NOUN
ejpam-5597	50	73	d∗d	d∗d	PROPN
ejpam-5597	50	74	+	+	NUM
ejpam-5597	50	75	|a∗b	|a∗b	X
ejpam-5597	50	76	+	+	X
ejpam-5597	50	77	c∗d|	c∗d|	X
ejpam-5597	50	78	]	]	PUNCT
ejpam-5597	50	79	∥∥∥∥	∥∥∥∥	PUNCT
ejpam-5597	50	80	=	=	SYM
ejpam-5597	50	81	√	√	NUM
ejpam-5597	50	82	max{∥a∗a+	max{∥a∗a+	NOUN
ejpam-5597	50	83	c∗c	c∗c	NUM
ejpam-5597	50	84	+	+	CCONJ
ejpam-5597	50	85	|b∗a+d∗c|∥	|b∗a+d∗c|∥	PROPN
ejpam-5597	50	86	,	,	PUNCT
ejpam-5597	50	87	∥b∗b	∥b∗b	X
ejpam-5597	50	88	+	+	NOUN
ejpam-5597	50	89	d∗d	d∗d	PROPN
ejpam-5597	50	90	+	+	NUM
ejpam-5597	50	91	|a∗b	|a∗b	X
ejpam-5597	50	92	+	+	ADJ
ejpam-5597	50	93	c∗d|∥	c∗d|∥	PROPN
ejpam-5597	50	94	}	}	PUNCT
ejpam-5597	50	95	,	,	PUNCT
ejpam-5597	50	96	as	as	SCONJ
ejpam-5597	50	97	required	require	VERB
ejpam-5597	50	98	.	.	PUNCT
ejpam-5597	51	1	letting	let	VERB
ejpam-5597	51	2	c	c	NOUN
ejpam-5597	51	3	=	=	PUNCT
ejpam-5597	51	4	d	d	NOUN
ejpam-5597	51	5	=	=	SYM
ejpam-5597	51	6	0	0	NUM
ejpam-5597	51	7	in	in	ADP
ejpam-5597	51	8	corollary	corollary	ADJ
ejpam-5597	51	9	2	2	NUM
ejpam-5597	51	10	,	,	PUNCT
ejpam-5597	51	11	we	we	PRON
ejpam-5597	51	12	have	have	VERB
ejpam-5597	51	13	the	the	DET
ejpam-5597	51	14	following	follow	VERB
ejpam-5597	51	15	result	result	NOUN
ejpam-5597	51	16	.	.	PUNCT
ejpam-5597	52	1	corollary	corollary	ADJ
ejpam-5597	52	2	3	3	X
ejpam-5597	52	3	.	.	PUNCT
ejpam-5597	53	1	let	let	VERB
ejpam-5597	53	2	a	a	DET
ejpam-5597	53	3	,	,	PUNCT
ejpam-5597	53	4	b	b	NOUN
ejpam-5597	53	5	∈	∈	NOUN
ejpam-5597	53	6	mn(c	mn(c	X
ejpam-5597	53	7	)	)	PUNCT
ejpam-5597	53	8	.	.	PUNCT
ejpam-5597	54	1	then	then	ADV
ejpam-5597	54	2	w2	w2	NOUN
ejpam-5597	54	3	(	(	PUNCT
ejpam-5597	54	4	[	[	PUNCT
ejpam-5597	54	5	a	a	DET
ejpam-5597	54	6	b	b	NOUN
ejpam-5597	54	7	0	0	NUM
ejpam-5597	54	8	0	0	NUM
ejpam-5597	54	9	]	]	PUNCT
ejpam-5597	54	10	)	)	PUNCT
ejpam-5597	54	11	≤	≤	NUM
ejpam-5597	54	12	max{∥a∗a+	max{∥a∗a+	PROPN
ejpam-5597	54	13	|b∗a|∥	|b∗a|∥	PROPN
ejpam-5597	54	14	,	,	PUNCT
ejpam-5597	54	15	∥b∗b	∥b∗b	NOUN
ejpam-5597	54	16	+	+	X
ejpam-5597	54	17	|a∗b|∥	|a∗b|∥	NOUN
ejpam-5597	54	18	}	}	PUNCT
ejpam-5597	54	19	.	.	PUNCT
ejpam-5597	55	1	a.	a.	PROPN
ejpam-5597	55	2	al	al	PROPN
ejpam-5597	55	3	-	-	PUNCT
ejpam-5597	55	4	natoor	natoor	NOUN
ejpam-5597	55	5	,	,	PUNCT
ejpam-5597	55	6	f.	f.	PROPN
ejpam-5597	55	7	alrimawi	alrimawi	PROPN
ejpam-5597	55	8	/	/	SYM
ejpam-5597	55	9	eur	eur	PROPN
ejpam-5597	55	10	.	.	PUNCT
ejpam-5597	56	1	j.	j.	PROPN
ejpam-5597	56	2	pure	pure	PROPN
ejpam-5597	56	3	appl	appl	PROPN
ejpam-5597	56	4	.	.	PROPN
ejpam-5597	56	5	math	math	PROPN
ejpam-5597	56	6	,	,	PUNCT
ejpam-5597	56	7	18	18	NUM
ejpam-5597	56	8	(	(	PUNCT
ejpam-5597	56	9	1	1	NUM
ejpam-5597	56	10	)	)	PUNCT
ejpam-5597	56	11	(	(	PUNCT
ejpam-5597	56	12	2025	2025	NUM
ejpam-5597	56	13	)	)	PUNCT
ejpam-5597	56	14	,	,	PUNCT
ejpam-5597	56	15	5597	5597	NUM
ejpam-5597	56	16	4	4	NUM
ejpam-5597	56	17	of	of	ADP
ejpam-5597	56	18	9	9	NUM
ejpam-5597	56	19	to	to	PART
ejpam-5597	56	20	state	state	VERB
ejpam-5597	56	21	the	the	DET
ejpam-5597	56	22	next	next	ADJ
ejpam-5597	56	23	corollary	corollary	NOUN
ejpam-5597	56	24	,	,	PUNCT
ejpam-5597	56	25	we	we	PRON
ejpam-5597	56	26	need	need	VERB
ejpam-5597	56	27	the	the	DET
ejpam-5597	56	28	following	follow	VERB
ejpam-5597	56	29	lemma	lemma	PROPN
ejpam-5597	56	30	[	[	X
ejpam-5597	56	31	18	18	NUM
ejpam-5597	56	32	]	]	PUNCT
ejpam-5597	56	33	.	.	PUNCT
ejpam-5597	57	1	lemma	lemma	PROPN
ejpam-5597	57	2	2	2	X
ejpam-5597	57	3	.	.	PUNCT
ejpam-5597	58	1	let	let	VERB
ejpam-5597	58	2	a	a	DET
ejpam-5597	58	3	,	,	PUNCT
ejpam-5597	58	4	b	b	NOUN
ejpam-5597	58	5	,	,	PUNCT
ejpam-5597	58	6	c	c	NOUN
ejpam-5597	58	7	,	,	PUNCT
ejpam-5597	58	8	d	d	PROPN
ejpam-5597	58	9	∈	∈	PROPN
ejpam-5597	58	10	mn(c	mn(c	X
ejpam-5597	58	11	)	)	PUNCT
ejpam-5597	58	12	.	.	PUNCT
ejpam-5597	59	1	then∥∥∥∥[a	then∥∥∥∥[a	PROPN
ejpam-5597	59	2	b	b	NOUN
ejpam-5597	59	3	c	c	NOUN
ejpam-5597	59	4	d	d	X
ejpam-5597	59	5	]	]	X
ejpam-5597	59	6	∥∥∥∥	∥∥∥∥	NOUN
ejpam-5597	59	7	≤	≤	X
ejpam-5597	60	1	∥∥∥∥[∥a∥	∥∥∥∥[∥a∥	PROPN
ejpam-5597	60	2	∥b∥	∥b∥	NUM
ejpam-5597	60	3	∥c∥	∥c∥	NOUN
ejpam-5597	60	4	∥d∥	∥d∥	ADJ
ejpam-5597	60	5	]	]	X
ejpam-5597	60	6	∥∥∥∥	∥∥∥∥	NUM
ejpam-5597	60	7	.	.	PUNCT
ejpam-5597	61	1	corollary	corollary	ADJ
ejpam-5597	61	2	4	4	NUM
ejpam-5597	61	3	.	.	PUNCT
ejpam-5597	62	1	let	let	VERB
ejpam-5597	62	2	a	a	DET
ejpam-5597	62	3	,	,	PUNCT
ejpam-5597	62	4	b	b	NOUN
ejpam-5597	62	5	,	,	PUNCT
ejpam-5597	62	6	c	c	NOUN
ejpam-5597	62	7	,	,	PUNCT
ejpam-5597	62	8	d	d	PROPN
ejpam-5597	62	9	∈	∈	PROPN
ejpam-5597	62	10	mn(c	mn(c	X
ejpam-5597	62	11	)	)	PUNCT
ejpam-5597	62	12	.	.	PUNCT
ejpam-5597	63	1	then	then	ADV
ejpam-5597	63	2	w	w	X
ejpam-5597	63	3	(	(	PUNCT
ejpam-5597	63	4	[	[	PUNCT
ejpam-5597	63	5	a	a	PRON
ejpam-5597	63	6	b	b	NOUN
ejpam-5597	63	7	c	c	NOUN
ejpam-5597	63	8	d	d	X
ejpam-5597	63	9	]	]	X
ejpam-5597	63	10	)	)	PUNCT
ejpam-5597	63	11	≤	≤	NOUN
ejpam-5597	63	12	√√√√	√√√√	ADP
ejpam-5597	63	13	1	1	NUM
ejpam-5597	63	14	2	2	NUM
ejpam-5597	63	15	∥a	∥a	PROPN
ejpam-5597	63	16	∗a+	∗a+	NOUN
ejpam-5597	63	17	c∗c∥+	c∗c∥+	NUM
ejpam-5597	63	18	1	1	NUM
ejpam-5597	63	19	2	2	NUM
ejpam-5597	63	20	∥b	∥b	PROPN
ejpam-5597	63	21	∗b	∗b	NOUN
ejpam-5597	63	22	+	+	PROPN
ejpam-5597	63	23	d∗d∥	d∗d∥	NOUN
ejpam-5597	63	24	+1	+1	ADJ
ejpam-5597	63	25	2	2	NUM
ejpam-5597	63	26	√	√	NUM
ejpam-5597	63	27	(	(	PUNCT
ejpam-5597	63	28	∥a∗a+	∥a∗a+	X
ejpam-5597	63	29	c∗c∥	c∗c∥	NOUN
ejpam-5597	63	30	−	−	NOUN
ejpam-5597	63	31	∥b∗b	∥b∗b	NOUN
ejpam-5597	63	32	+	+	NOUN
ejpam-5597	63	33	d∗d∥)2	d∗d∥)2	NOUN
ejpam-5597	63	34	+	+	CCONJ
ejpam-5597	63	35	4	4	NUM
ejpam-5597	63	36	∥b∗a+d∗c∥2	∥b∗a+d∗c∥2	NOUN
ejpam-5597	63	37	.	.	PUNCT
ejpam-5597	64	1	proof	proof	NOUN
ejpam-5597	64	2	.	.	PUNCT
ejpam-5597	65	1	by	by	ADP
ejpam-5597	65	2	inequality	inequality	NOUN
ejpam-5597	65	3	(	(	PUNCT
ejpam-5597	65	4	3	3	NUM
ejpam-5597	65	5	)	)	PUNCT
ejpam-5597	65	6	,	,	PUNCT
ejpam-5597	65	7	we	we	PRON
ejpam-5597	65	8	have	have	VERB
ejpam-5597	65	9	w	w	NOUN
ejpam-5597	65	10	(	(	PUNCT
ejpam-5597	65	11	[	[	PUNCT
ejpam-5597	65	12	a	a	PRON
ejpam-5597	65	13	b	b	NOUN
ejpam-5597	65	14	c	c	NOUN
ejpam-5597	65	15	d	d	X
ejpam-5597	65	16	]	]	X
ejpam-5597	65	17	)	)	PUNCT
ejpam-5597	65	18	≤	≤	PUNCT
ejpam-5597	66	1	√∥∥∥∥	√∥∥∥∥	PROPN
ejpam-5597	66	2	[	[	PUNCT
ejpam-5597	66	3	a∗a+	a∗a+	NOUN
ejpam-5597	66	4	c∗c	c∗c	NUM
ejpam-5597	66	5	a∗b	a∗b	NUM
ejpam-5597	66	6	+	+	NUM
ejpam-5597	66	7	c∗d	c∗d	NOUN
ejpam-5597	66	8	b∗a+d∗c	b∗a+d∗c	NOUN
ejpam-5597	66	9	b∗b	b∗b	NOUN
ejpam-5597	66	10	+	+	NOUN
ejpam-5597	66	11	d∗d	d∗d	PROPN
ejpam-5597	66	12	]	]	SYM
ejpam-5597	66	13	∥∥∥∥	∥∥∥∥	NUM
ejpam-5597	66	14	≤	≤	NOUN
ejpam-5597	67	1	√∥∥∥∥	√∥∥∥∥	PROPN
ejpam-5597	67	2	[	[	PUNCT
ejpam-5597	67	3	∥a∗a+	∥a∗a+	PROPN
ejpam-5597	67	4	c∗c∥	c∗c∥	PROPN
ejpam-5597	67	5	∥a∗b	∥a∗b	NOUN
ejpam-5597	67	6	+	+	CCONJ
ejpam-5597	67	7	c∗d∥	c∗d∥	PROPN
ejpam-5597	67	8	∥b∗a+d∗c∥	∥b∗a+d∗c∥	NUM
ejpam-5597	67	9	∥b∗b	∥b∗b	NOUN
ejpam-5597	67	10	+	+	NOUN
ejpam-5597	67	11	d∗d∥	d∗d∥	NOUN
ejpam-5597	67	12	]	]	PUNCT
ejpam-5597	67	13	∥∥∥∥	∥∥∥∥	PUNCT
ejpam-5597	67	14	=	=	SYM
ejpam-5597	67	15	√	√	ADP
ejpam-5597	67	16	r	r	NOUN
ejpam-5597	67	17	(	(	PUNCT
ejpam-5597	67	18	[	[	PUNCT
ejpam-5597	67	19	∥a∗a+	∥a∗a+	X
ejpam-5597	67	20	c∗c∥	c∗c∥	NOUN
ejpam-5597	67	21	∥a∗b	∥a∗b	NOUN
ejpam-5597	67	22	+	+	CCONJ
ejpam-5597	67	23	c∗d∥	c∗d∥	PROPN
ejpam-5597	67	24	∥b∗a+d∗c∥	∥b∗a+d∗c∥	NUM
ejpam-5597	67	25	∥b∗b	∥b∗b	NOUN
ejpam-5597	67	26	+	+	NOUN
ejpam-5597	67	27	d∗d∥	d∗d∥	NOUN
ejpam-5597	67	28	]	]	PUNCT
ejpam-5597	67	29	)	)	PUNCT
ejpam-5597	67	30	(	(	PUNCT
ejpam-5597	67	31	where	where	SCONJ
ejpam-5597	67	32	r	r	NOUN
ejpam-5597	67	33	denotes	denote	VERB
ejpam-5597	67	34	the	the	DET
ejpam-5597	67	35	spectral	spectral	ADJ
ejpam-5597	67	36	radius	radius	NOUN
ejpam-5597	67	37	of	of	ADP
ejpam-5597	67	38	matrices	matrix	NOUN
ejpam-5597	67	39	)	)	PUNCT
ejpam-5597	67	40	=	=	PUNCT
ejpam-5597	68	1	√√√√	√√√√	ADP
ejpam-5597	68	2	1	1	NUM
ejpam-5597	68	3	2	2	NUM
ejpam-5597	68	4	∥a	∥a	PROPN
ejpam-5597	68	5	∗a+	∗a+	NOUN
ejpam-5597	68	6	c∗c∥+	c∗c∥+	NUM
ejpam-5597	68	7	1	1	NUM
ejpam-5597	68	8	2	2	NUM
ejpam-5597	68	9	∥b	∥b	PROPN
ejpam-5597	68	10	∗b	∗b	NOUN
ejpam-5597	69	1	+	+	PROPN
ejpam-5597	69	2	d∗d∥	d∗d∥	NOUN
ejpam-5597	69	3	+1	+1	ADJ
ejpam-5597	69	4	2	2	NUM
ejpam-5597	69	5	√	√	NUM
ejpam-5597	69	6	(	(	PUNCT
ejpam-5597	69	7	∥a∗a+	∥a∗a+	X
ejpam-5597	69	8	c∗c∥	c∗c∥	NOUN
ejpam-5597	69	9	−	−	NOUN
ejpam-5597	69	10	∥b∗b	∥b∗b	NOUN
ejpam-5597	69	11	+	+	NOUN
ejpam-5597	69	12	d∗d∥)2	d∗d∥)2	NOUN
ejpam-5597	69	13	+	+	CCONJ
ejpam-5597	69	14	4	4	NUM
ejpam-5597	69	15	∥b∗a+d∗c∥2	∥b∗a+d∗c∥2	NOUN
ejpam-5597	69	16	,	,	PUNCT
ejpam-5597	69	17	as	as	SCONJ
ejpam-5597	69	18	required	require	VERB
ejpam-5597	69	19	.	.	PUNCT
ejpam-5597	70	1	corollary	corollary	ADJ
ejpam-5597	70	2	5	5	NUM
ejpam-5597	70	3	.	.	PUNCT
ejpam-5597	71	1	let	let	VERB
ejpam-5597	71	2	c	c	X
ejpam-5597	71	3	,	,	PUNCT
ejpam-5597	71	4	d	d	PROPN
ejpam-5597	71	5	∈	∈	PROPN
ejpam-5597	71	6	mn(c	mn(c	X
ejpam-5597	71	7	)	)	PUNCT
ejpam-5597	71	8	.	.	PUNCT
ejpam-5597	72	1	then	then	ADV
ejpam-5597	72	2	w2	w2	NOUN
ejpam-5597	72	3	(	(	PUNCT
ejpam-5597	72	4	[	[	PUNCT
ejpam-5597	72	5	i	i	PRON
ejpam-5597	72	6	i	i	NOUN
ejpam-5597	72	7	c	c	VERB
ejpam-5597	72	8	d	d	X
ejpam-5597	72	9	]	]	X
ejpam-5597	72	10	)	)	PUNCT
ejpam-5597	72	11	≤	≤	NUM
ejpam-5597	72	12	1	1	NUM
ejpam-5597	72	13	+	+	CCONJ
ejpam-5597	72	14	max	max	PROPN
ejpam-5597	72	15	{	{	PUNCT
ejpam-5597	72	16	∥c∥2	∥c∥2	PROPN
ejpam-5597	72	17	,	,	PUNCT
ejpam-5597	72	18	∥d∥2	∥d∥2	NOUN
ejpam-5597	72	19	}	}	PUNCT
ejpam-5597	72	20	+	+	CCONJ
ejpam-5597	72	21	∥i	∥i	PROPN
ejpam-5597	72	22	+	+	CCONJ
ejpam-5597	72	23	c∗d∥	c∗d∥	PROPN
ejpam-5597	72	24	.	.	PUNCT
ejpam-5597	73	1	proof	proof	NOUN
ejpam-5597	73	2	.	.	PUNCT
ejpam-5597	74	1	letting	let	VERB
ejpam-5597	74	2	a	a	DET
ejpam-5597	74	3	=	=	SYM
ejpam-5597	74	4	b	b	NOUN
ejpam-5597	74	5	=	=	SYM
ejpam-5597	74	6	i	i	PROPN
ejpam-5597	74	7	in	in	ADP
ejpam-5597	74	8	theorem	theorem	NOUN
ejpam-5597	74	9	1	1	NUM
ejpam-5597	74	10	,	,	PUNCT
ejpam-5597	74	11	we	we	PRON
ejpam-5597	74	12	have	have	VERB
ejpam-5597	74	13	w2	w2	NOUN
ejpam-5597	74	14	(	(	PUNCT
ejpam-5597	74	15	[	[	PUNCT
ejpam-5597	74	16	i	i	PRON
ejpam-5597	74	17	i	i	NOUN
ejpam-5597	74	18	c	c	VERB
ejpam-5597	75	1	d	d	X
ejpam-5597	75	2	]	]	X
ejpam-5597	75	3	)	)	PUNCT
ejpam-5597	75	4	≤	≤	NUM
ejpam-5597	75	5	max	max	PROPN
ejpam-5597	75	6	{	{	PUNCT
ejpam-5597	75	7	∥i	∥i	NOUN
ejpam-5597	75	8	+	+	CCONJ
ejpam-5597	75	9	c∗c∥	c∗c∥	PROPN
ejpam-5597	75	10	,	,	PUNCT
ejpam-5597	75	11	∥i	∥i	PROPN
ejpam-5597	75	12	+	+	NOUN
ejpam-5597	75	13	d∗d∥}+	d∗d∥}+	NOUN
ejpam-5597	75	14	∥i	∥i	PROPN
ejpam-5597	75	15	+	+	CCONJ
ejpam-5597	75	16	c∗d∥	c∗d∥	PROPN
ejpam-5597	75	17	≤	≤	NUM
ejpam-5597	75	18	1	1	NUM
ejpam-5597	75	19	+	+	CCONJ
ejpam-5597	75	20	max	max	PROPN
ejpam-5597	75	21	{	{	PUNCT
ejpam-5597	75	22	∥c∗c∥	∥c∗c∥	PROPN
ejpam-5597	75	23	,	,	PUNCT
ejpam-5597	75	24	∥d∗d∥}+	∥d∗d∥}+	PROPN
ejpam-5597	75	25	∥i	∥i	PROPN
ejpam-5597	75	26	+	+	CCONJ
ejpam-5597	75	27	c∗d∥	c∗d∥	PROPN
ejpam-5597	75	28	=	=	SYM
ejpam-5597	75	29	1	1	NUM
ejpam-5597	75	30	+	+	NOUN
ejpam-5597	75	31	max	max	PROPN
ejpam-5597	75	32	{	{	PUNCT
ejpam-5597	75	33	∥c∥2	∥c∥2	PROPN
ejpam-5597	75	34	,	,	PUNCT
ejpam-5597	75	35	∥d∥2	∥d∥2	NOUN
ejpam-5597	75	36	}	}	PUNCT
ejpam-5597	75	37	+	+	CCONJ
ejpam-5597	75	38	∥i	∥i	PROPN
ejpam-5597	75	39	+	+	CCONJ
ejpam-5597	75	40	c∗d∥	c∗d∥	PROPN
ejpam-5597	75	41	.	.	PUNCT
ejpam-5597	76	1	we	we	PRON
ejpam-5597	76	2	need	need	VERB
ejpam-5597	76	3	the	the	DET
ejpam-5597	76	4	following	follow	VERB
ejpam-5597	76	5	lemma	lemma	PROPN
ejpam-5597	76	6	[	[	X
ejpam-5597	76	7	17	17	NUM
ejpam-5597	76	8	]	]	PUNCT
ejpam-5597	76	9	to	to	PART
ejpam-5597	76	10	give	give	VERB
ejpam-5597	76	11	a	a	DET
ejpam-5597	76	12	lower	low	ADJ
ejpam-5597	76	13	bound	bind	VERB
ejpam-5597	76	14	for	for	ADP
ejpam-5597	76	15	inequality	inequality	NOUN
ejpam-5597	76	16	(	(	PUNCT
ejpam-5597	76	17	5	5	NUM
ejpam-5597	76	18	)	)	PUNCT
ejpam-5597	76	19	.	.	PUNCT
ejpam-5597	77	1	a.	a.	PROPN
ejpam-5597	77	2	al	al	PROPN
ejpam-5597	77	3	-	-	PUNCT
ejpam-5597	77	4	natoor	natoor	NOUN
ejpam-5597	77	5	,	,	PUNCT
ejpam-5597	77	6	f.	f.	PROPN
ejpam-5597	77	7	alrimawi	alrimawi	PROPN
ejpam-5597	77	8	/	/	SYM
ejpam-5597	77	9	eur	eur	PROPN
ejpam-5597	77	10	.	.	PUNCT
ejpam-5597	78	1	j.	j.	PROPN
ejpam-5597	78	2	pure	pure	PROPN
ejpam-5597	78	3	appl	appl	PROPN
ejpam-5597	78	4	.	.	PROPN
ejpam-5597	78	5	math	math	PROPN
ejpam-5597	78	6	,	,	PUNCT
ejpam-5597	78	7	18	18	NUM
ejpam-5597	78	8	(	(	PUNCT
ejpam-5597	78	9	1	1	NUM
ejpam-5597	78	10	)	)	PUNCT
ejpam-5597	78	11	(	(	PUNCT
ejpam-5597	78	12	2025	2025	NUM
ejpam-5597	78	13	)	)	PUNCT
ejpam-5597	78	14	,	,	PUNCT
ejpam-5597	78	15	5597	5597	NUM
ejpam-5597	78	16	5	5	NUM
ejpam-5597	78	17	of	of	ADP
ejpam-5597	78	18	9	9	NUM
ejpam-5597	78	19	lemma	lemma	PROPN
ejpam-5597	78	20	3	3	NUM
ejpam-5597	78	21	.	.	PUNCT
ejpam-5597	79	1	let	let	VERB
ejpam-5597	79	2	a	a	DET
ejpam-5597	79	3	,	,	PUNCT
ejpam-5597	79	4	b	b	NOUN
ejpam-5597	79	5	,	,	PUNCT
ejpam-5597	79	6	c	c	NOUN
ejpam-5597	79	7	,	,	PUNCT
ejpam-5597	79	8	d	d	PROPN
ejpam-5597	79	9	∈	∈	PROPN
ejpam-5597	79	10	mn(c	mn(c	X
ejpam-5597	79	11	)	)	PUNCT
ejpam-5597	79	12	.	.	PUNCT
ejpam-5597	80	1	then	then	ADV
ejpam-5597	80	2	w	w	X
ejpam-5597	80	3	(	(	PUNCT
ejpam-5597	80	4	[	[	PUNCT
ejpam-5597	80	5	a	a	PRON
ejpam-5597	80	6	b	b	NOUN
ejpam-5597	80	7	c	c	NOUN
ejpam-5597	80	8	d	d	NOUN
ejpam-5597	80	9	]	]	X
ejpam-5597	80	10	)	)	PUNCT
ejpam-5597	80	11	≥	≥	PROPN
ejpam-5597	80	12	max	max	PROPN
ejpam-5597	80	13	{	{	PUNCT
ejpam-5597	80	14	w(a	w(a	PROPN
ejpam-5597	80	15	)	)	PUNCT
ejpam-5597	80	16	,	,	PUNCT
ejpam-5597	80	17	w(d	w(d	PROPN
ejpam-5597	80	18	)	)	PUNCT
ejpam-5597	80	19	,	,	PUNCT
ejpam-5597	80	20	w(b	w(b	PROPN
ejpam-5597	80	21	+	+	PROPN
ejpam-5597	80	22	c	c	X
ejpam-5597	80	23	)	)	PUNCT
ejpam-5597	80	24	2	2	NUM
ejpam-5597	80	25	,	,	PUNCT
ejpam-5597	80	26	w(b	w(b	PROPN
ejpam-5597	80	27	−	−	PROPN
ejpam-5597	80	28	c	c	X
ejpam-5597	80	29	)	)	PUNCT
ejpam-5597	80	30	2	2	NUM
ejpam-5597	80	31	}	}	PUNCT
ejpam-5597	80	32	.	.	PUNCT
ejpam-5597	81	1	corollary	corollary	ADJ
ejpam-5597	81	2	6	6	NUM
ejpam-5597	81	3	.	.	PUNCT
ejpam-5597	82	1	let	let	VERB
ejpam-5597	82	2	a	a	DET
ejpam-5597	82	3	,	,	PUNCT
ejpam-5597	82	4	b	b	NOUN
ejpam-5597	82	5	∈	∈	PROPN
ejpam-5597	82	6	mn(c	mn(c	X
ejpam-5597	82	7	)	)	PUNCT
ejpam-5597	82	8	be	be	AUX
ejpam-5597	82	9	positive	positive	ADJ
ejpam-5597	82	10	semidefinite	semidefinite	NOUN
ejpam-5597	82	11	.	.	PUNCT
ejpam-5597	83	1	then	then	ADV
ejpam-5597	83	2	1	1	NUM
ejpam-5597	83	3	2	2	NUM
ejpam-5597	83	4	∥a+b∥	∥a+b∥	PROPN
ejpam-5597	83	5	≤	≤	NUM
ejpam-5597	83	6	w	w	NOUN
ejpam-5597	83	7	(	(	PUNCT
ejpam-5597	83	8	[	[	PUNCT
ejpam-5597	83	9	a	a	DET
ejpam-5597	83	10	b	b	NOUN
ejpam-5597	83	11	0	0	NUM
ejpam-5597	83	12	0	0	NUM
ejpam-5597	83	13	]	]	PUNCT
ejpam-5597	83	14	)	)	PUNCT
ejpam-5597	83	15	.	.	PUNCT
ejpam-5597	84	1	proof	proof	NOUN
ejpam-5597	84	2	.	.	PUNCT
ejpam-5597	85	1	let	let	VERB
ejpam-5597	85	2	u	u	PRON
ejpam-5597	85	3	=	=	PROPN
ejpam-5597	85	4	1√	1√	PROPN
ejpam-5597	85	5	2	2	NUM
ejpam-5597	85	6	[	[	PUNCT
ejpam-5597	85	7	i	i	PRON
ejpam-5597	85	8	i	i	PRON
ejpam-5597	85	9	−i	−i	VERB
ejpam-5597	86	1	i	i	PRON
ejpam-5597	86	2	]	]	PUNCT
ejpam-5597	86	3	,	,	PUNCT
ejpam-5597	86	4	where	where	SCONJ
ejpam-5597	86	5	i	i	PRON
ejpam-5597	86	6	is	be	AUX
ejpam-5597	86	7	the	the	DET
ejpam-5597	86	8	identity	identity	NOUN
ejpam-5597	86	9	matrix	matrix	NOUN
ejpam-5597	86	10	,	,	PUNCT
ejpam-5597	86	11	then	then	ADV
ejpam-5597	86	12	u	u	NOUN
ejpam-5597	86	13	is	be	AUX
ejpam-5597	86	14	unitary	unitary	ADJ
ejpam-5597	86	15	.	.	PUNCT
ejpam-5597	87	1	consequently	consequently	ADV
ejpam-5597	87	2	,	,	PUNCT
ejpam-5597	87	3	we	we	PRON
ejpam-5597	87	4	have	have	VERB
ejpam-5597	87	5	w	w	NOUN
ejpam-5597	87	6	(	(	PUNCT
ejpam-5597	87	7	[	[	PUNCT
ejpam-5597	87	8	a	a	DET
ejpam-5597	87	9	b	b	NOUN
ejpam-5597	87	10	0	0	NUM
ejpam-5597	87	11	0	0	NUM
ejpam-5597	87	12	]	]	PUNCT
ejpam-5597	87	13	)	)	PUNCT
ejpam-5597	88	1	=	=	SYM
ejpam-5597	88	2	w	w	PROPN
ejpam-5597	88	3	(	(	PUNCT
ejpam-5597	88	4	u∗	u∗	PROPN
ejpam-5597	88	5	[	[	PUNCT
ejpam-5597	88	6	a	a	DET
ejpam-5597	88	7	b	b	NOUN
ejpam-5597	88	8	0	0	NUM
ejpam-5597	88	9	0	0	NUM
ejpam-5597	88	10	]	]	PUNCT
ejpam-5597	88	11	u	u	NOUN
ejpam-5597	88	12	)	)	PUNCT
ejpam-5597	88	13	=	=	SYM
ejpam-5597	88	14	w	w	X
ejpam-5597	88	15	(	(	PUNCT
ejpam-5597	88	16	[	[	PUNCT
ejpam-5597	88	17	a−b	a−b	PROPN
ejpam-5597	88	18	2	2	NUM
ejpam-5597	88	19	a+b	a+b	NUM
ejpam-5597	88	20	2	2	NUM
ejpam-5597	88	21	a−b	a−b	NOUN
ejpam-5597	88	22	2	2	NUM
ejpam-5597	88	23	a+b	a+b	NUM
ejpam-5597	88	24	2	2	NUM
ejpam-5597	88	25	]	]	PUNCT
ejpam-5597	88	26	)	)	PUNCT
ejpam-5597	88	27	≥	≥	PROPN
ejpam-5597	89	1	max	max	PROPN
ejpam-5597	89	2	{	{	PUNCT
ejpam-5597	89	3	w	w	PROPN
ejpam-5597	89	4	(	(	PUNCT
ejpam-5597	89	5	a−b	a−b	PROPN
ejpam-5597	89	6	2	2	NUM
ejpam-5597	89	7	)	)	PUNCT
ejpam-5597	89	8	,	,	PUNCT
ejpam-5597	89	9	w	w	PROPN
ejpam-5597	89	10	(	(	PUNCT
ejpam-5597	89	11	a+b	a+b	NUM
ejpam-5597	89	12	2	2	NUM
ejpam-5597	89	13	)	)	PUNCT
ejpam-5597	89	14	,	,	PUNCT
ejpam-5597	89	15	w	w	PROPN
ejpam-5597	89	16	(	(	PUNCT
ejpam-5597	89	17	a	a	X
ejpam-5597	89	18	)	)	PUNCT
ejpam-5597	89	19	2	2	NUM
ejpam-5597	89	20	,	,	PUNCT
ejpam-5597	89	21	w	w	PROPN
ejpam-5597	89	22	(	(	PUNCT
ejpam-5597	89	23	b	b	NOUN
ejpam-5597	89	24	)	)	PUNCT
ejpam-5597	89	25	2	2	NUM
ejpam-5597	89	26	}	}	PUNCT
ejpam-5597	89	27	(	(	PUNCT
ejpam-5597	89	28	by	by	ADP
ejpam-5597	89	29	lemma	lemma	PROPN
ejpam-5597	89	30	3	3	NUM
ejpam-5597	89	31	)	)	PUNCT
ejpam-5597	89	32	=	=	SYM
ejpam-5597	89	33	1	1	NUM
ejpam-5597	89	34	2	2	NUM
ejpam-5597	89	35	∥a+b∥	∥a+b∥	NOUN
ejpam-5597	89	36	.	.	PUNCT
ejpam-5597	90	1	corollary	corollary	ADJ
ejpam-5597	90	2	7	7	NUM
ejpam-5597	90	3	.	.	PUNCT
ejpam-5597	91	1	let	let	VERB
ejpam-5597	91	2	a	a	DET
ejpam-5597	91	3	,	,	PUNCT
ejpam-5597	91	4	b	b	NOUN
ejpam-5597	91	5	∈	∈	PROPN
ejpam-5597	91	6	mn(c	mn(c	X
ejpam-5597	91	7	)	)	PUNCT
ejpam-5597	91	8	be	be	AUX
ejpam-5597	91	9	positive	positive	ADJ
ejpam-5597	91	10	semidefinite	semidefinite	NOUN
ejpam-5597	91	11	.	.	PUNCT
ejpam-5597	92	1	then	then	ADV
ejpam-5597	92	2	∥a∥+	∥a∥+	VERB
ejpam-5597	92	3	∥b∥	∥b∥	ADJ
ejpam-5597	92	4	≤	≤	ADJ
ejpam-5597	92	5	w2	w2	NOUN
ejpam-5597	92	6	(	(	PUNCT
ejpam-5597	92	7	[	[	PUNCT
ejpam-5597	92	8	a1/2	a1/2	VERB
ejpam-5597	92	9	b1/2	b1/2	NOUN
ejpam-5597	92	10	0	0	NUM
ejpam-5597	92	11	0	0	NUM
ejpam-5597	92	12	]	]	PUNCT
ejpam-5597	92	13	)	)	PUNCT
ejpam-5597	93	1	+	+	CCONJ
ejpam-5597	93	2	w2	w2	NOUN
ejpam-5597	93	3	(	(	PUNCT
ejpam-5597	93	4	[	[	PUNCT
ejpam-5597	93	5	b1/2	b1/2	NOUN
ejpam-5597	93	6	a1/2	a1/2	NOUN
ejpam-5597	93	7	0	0	NUM
ejpam-5597	93	8	0	0	NUM
ejpam-5597	93	9	]	]	PUNCT
ejpam-5597	93	10	)	)	PUNCT
ejpam-5597	93	11	.	.	PUNCT
ejpam-5597	94	1	(	(	PUNCT
ejpam-5597	94	2	6	6	X
ejpam-5597	94	3	)	)	PUNCT
ejpam-5597	94	4	proof	proof	NOUN
ejpam-5597	94	5	.	.	PUNCT
ejpam-5597	95	1	by	by	ADP
ejpam-5597	95	2	lemma	lemma	PROPN
ejpam-5597	95	3	3	3	NUM
ejpam-5597	95	4	,	,	PUNCT
ejpam-5597	95	5	we	we	PRON
ejpam-5597	95	6	have	have	VERB
ejpam-5597	95	7	w2	w2	NOUN
ejpam-5597	95	8	(	(	PUNCT
ejpam-5597	95	9	[	[	PUNCT
ejpam-5597	95	10	a1/2	a1/2	VERB
ejpam-5597	95	11	b1/2	b1/2	NOUN
ejpam-5597	95	12	0	0	NUM
ejpam-5597	95	13	0	0	NUM
ejpam-5597	95	14	]	]	PUNCT
ejpam-5597	95	15	)	)	PUNCT
ejpam-5597	95	16	≥	≥	PROPN
ejpam-5597	96	1	[	[	PUNCT
ejpam-5597	96	2	max	max	PROPN
ejpam-5597	96	3	{	{	PUNCT
ejpam-5597	96	4	w	w	PROPN
ejpam-5597	96	5	(	(	PUNCT
ejpam-5597	96	6	a1/2	a1/2	PROPN
ejpam-5597	96	7	)	)	PUNCT
ejpam-5597	96	8	,	,	PUNCT
ejpam-5597	96	9	w	w	PROPN
ejpam-5597	96	10	(	(	PUNCT
ejpam-5597	96	11	b1/2	b1/2	PROPN
ejpam-5597	96	12	)	)	PUNCT
ejpam-5597	96	13	2	2	NUM
ejpam-5597	96	14	}	}	PUNCT
ejpam-5597	96	15	]	]	SYM
ejpam-5597	96	16	2	2	X
ejpam-5597	96	17	=	=	SYM
ejpam-5597	96	18	max	max	X
ejpam-5597	96	19	{	{	PUNCT
ejpam-5597	96	20	∥a∥	∥a∥	NOUN
ejpam-5597	96	21	,	,	PUNCT
ejpam-5597	96	22	1	1	NUM
ejpam-5597	96	23	4	4	NUM
ejpam-5597	96	24	∥b∥	∥b∥	NUM
ejpam-5597	96	25	}	}	PUNCT
ejpam-5597	96	26	.	.	PUNCT
ejpam-5597	97	1	(	(	PUNCT
ejpam-5597	97	2	7	7	X
ejpam-5597	97	3	)	)	PUNCT
ejpam-5597	97	4	similarly	similarly	ADV
ejpam-5597	97	5	,	,	PUNCT
ejpam-5597	97	6	we	we	PRON
ejpam-5597	97	7	have	have	VERB
ejpam-5597	97	8	w2	w2	NOUN
ejpam-5597	97	9	(	(	PUNCT
ejpam-5597	97	10	[	[	PUNCT
ejpam-5597	97	11	b1/2	b1/2	NOUN
ejpam-5597	97	12	a1/2	a1/2	NOUN
ejpam-5597	97	13	0	0	NUM
ejpam-5597	97	14	0	0	NUM
ejpam-5597	97	15	]	]	PUNCT
ejpam-5597	97	16	)	)	PUNCT
ejpam-5597	97	17	≥	≥	PROPN
ejpam-5597	97	18	max	max	PROPN
ejpam-5597	97	19	{	{	PUNCT
ejpam-5597	97	20	∥b∥	∥b∥	NUM
ejpam-5597	97	21	,	,	PUNCT
ejpam-5597	97	22	1	1	NUM
ejpam-5597	97	23	4	4	NUM
ejpam-5597	97	24	∥a∥	∥a∥	NOUN
ejpam-5597	97	25	}	}	PUNCT
ejpam-5597	97	26	.	.	PUNCT
ejpam-5597	98	1	(	(	PUNCT
ejpam-5597	98	2	8)	8)	NUM
ejpam-5597	98	3	by	by	ADP
ejpam-5597	98	4	adding	add	VERB
ejpam-5597	98	5	inequalities	inequality	NOUN
ejpam-5597	98	6	(	(	PUNCT
ejpam-5597	98	7	7	7	NUM
ejpam-5597	98	8	)	)	PUNCT
ejpam-5597	98	9	and	and	CCONJ
ejpam-5597	98	10	(	(	PUNCT
ejpam-5597	98	11	8)	8)	NUM
ejpam-5597	98	12	and	and	CCONJ
ejpam-5597	98	13	then	then	ADV
ejpam-5597	98	14	using	use	VERB
ejpam-5597	98	15	the	the	DET
ejpam-5597	98	16	fact	fact	NOUN
ejpam-5597	98	17	that	that	SCONJ
ejpam-5597	98	18	max(a	max(a	PROPN
ejpam-5597	98	19	,	,	PUNCT
ejpam-5597	98	20	b	b	NOUN
ejpam-5597	98	21	)	)	PUNCT
ejpam-5597	98	22	=	=	VERB
ejpam-5597	98	23	a+b+|a−b|	a+b+|a−b|	VERB
ejpam-5597	98	24	2	2	NUM
ejpam-5597	98	25	,	,	PUNCT
ejpam-5597	98	26	we	we	PRON
ejpam-5597	98	27	have	have	VERB
ejpam-5597	98	28	w2	w2	NOUN
ejpam-5597	98	29	(	(	PUNCT
ejpam-5597	98	30	[	[	PUNCT
ejpam-5597	98	31	a1/2	a1/2	VERB
ejpam-5597	98	32	b1/2	b1/2	NOUN
ejpam-5597	98	33	0	0	NUM
ejpam-5597	98	34	0	0	NUM
ejpam-5597	98	35	]	]	PUNCT
ejpam-5597	98	36	)	)	PUNCT
ejpam-5597	99	1	+	+	CCONJ
ejpam-5597	99	2	w2	w2	NOUN
ejpam-5597	99	3	(	(	PUNCT
ejpam-5597	99	4	[	[	PUNCT
ejpam-5597	99	5	b1/2	b1/2	NOUN
ejpam-5597	99	6	a1/2	a1/2	NOUN
ejpam-5597	99	7	0	0	NUM
ejpam-5597	99	8	0	0	NUM
ejpam-5597	99	9	]	]	PUNCT
ejpam-5597	99	10	)	)	PUNCT
ejpam-5597	99	11	a.	a.	PROPN
ejpam-5597	99	12	al	al	PROPN
ejpam-5597	99	13	-	-	PUNCT
ejpam-5597	99	14	natoor	natoor	NOUN
ejpam-5597	99	15	,	,	PUNCT
ejpam-5597	99	16	f.	f.	PROPN
ejpam-5597	99	17	alrimawi	alrimawi	PROPN
ejpam-5597	99	18	/	/	SYM
ejpam-5597	99	19	eur	eur	PROPN
ejpam-5597	99	20	.	.	PUNCT
ejpam-5597	100	1	j.	j.	PROPN
ejpam-5597	100	2	pure	pure	PROPN
ejpam-5597	100	3	appl	appl	PROPN
ejpam-5597	100	4	.	.	PROPN
ejpam-5597	100	5	math	math	PROPN
ejpam-5597	100	6	,	,	PUNCT
ejpam-5597	100	7	18	18	NUM
ejpam-5597	100	8	(	(	PUNCT
ejpam-5597	100	9	1	1	NUM
ejpam-5597	100	10	)	)	PUNCT
ejpam-5597	100	11	(	(	PUNCT
ejpam-5597	100	12	2025	2025	NUM
ejpam-5597	100	13	)	)	PUNCT
ejpam-5597	100	14	,	,	PUNCT
ejpam-5597	100	15	5597	5597	NUM
ejpam-5597	100	16	6	6	NUM
ejpam-5597	100	17	of	of	ADP
ejpam-5597	100	18	9	9	NUM
ejpam-5597	100	19	≥	≥	NOUN
ejpam-5597	100	20	max	max	PROPN
ejpam-5597	100	21	{	{	PUNCT
ejpam-5597	100	22	∥a∥	∥a∥	PROPN
ejpam-5597	100	23	,	,	PUNCT
ejpam-5597	100	24	1	1	NUM
ejpam-5597	100	25	4	4	NUM
ejpam-5597	100	26	∥b∥	∥b∥	ADJ
ejpam-5597	100	27	}	}	PUNCT
ejpam-5597	101	1	+	+	PROPN
ejpam-5597	101	2	max	max	PROPN
ejpam-5597	101	3	{	{	PUNCT
ejpam-5597	101	4	∥b∥	∥b∥	NUM
ejpam-5597	101	5	,	,	PUNCT
ejpam-5597	101	6	1	1	NUM
ejpam-5597	101	7	4	4	NUM
ejpam-5597	101	8	∥a∥	∥a∥	NOUN
ejpam-5597	101	9	}	}	PUNCT
ejpam-5597	101	10	=	=	SYM
ejpam-5597	101	11	5	5	NUM
ejpam-5597	101	12	8	8	NUM
ejpam-5597	101	13	(	(	PUNCT
ejpam-5597	101	14	∥a∥+	∥a∥+	VERB
ejpam-5597	101	15	∥b∥	∥b∥	NUM
ejpam-5597	101	16	)	)	PUNCT
ejpam-5597	101	17	+	+	CCONJ
ejpam-5597	101	18	1	1	NUM
ejpam-5597	101	19	8	8	NUM
ejpam-5597	101	20	(	(	PUNCT
ejpam-5597	101	21	|4	|4	NUM
ejpam-5597	101	22	∥a∥	∥a∥	VERB
ejpam-5597	101	23	−	−	PUNCT
ejpam-5597	101	24	∥b∥|+	∥b∥|+	NOUN
ejpam-5597	101	25	|4	|4	NUM
ejpam-5597	101	26	∥b∥	∥b∥	NOUN
ejpam-5597	101	27	−	−	PROPN
ejpam-5597	101	28	∥a∥|	∥a∥|	PROPN
ejpam-5597	101	29	)	)	PUNCT
ejpam-5597	101	30	(	(	PUNCT
ejpam-5597	101	31	9	9	X
ejpam-5597	101	32	)	)	PUNCT
ejpam-5597	101	33	≥	≥	NOUN
ejpam-5597	101	34	∥a∥+	∥a∥+	VERB
ejpam-5597	101	35	∥b∥	∥b∥	NUM
ejpam-5597	101	36	.	.	PUNCT
ejpam-5597	102	1	in	in	ADP
ejpam-5597	102	2	inequality	inequality	NOUN
ejpam-5597	102	3	(	(	PUNCT
ejpam-5597	102	4	5	5	NUM
ejpam-5597	102	5	)	)	PUNCT
ejpam-5597	102	6	,	,	PUNCT
ejpam-5597	102	7	by	by	ADP
ejpam-5597	102	8	replacing	replace	VERB
ejpam-5597	102	9	a	a	PRON
ejpam-5597	102	10	and	and	CCONJ
ejpam-5597	102	11	b	b	NOUN
ejpam-5597	102	12	by	by	SCONJ
ejpam-5597	102	13	the	the	DET
ejpam-5597	102	14	positive	positive	ADJ
ejpam-5597	102	15	semidefinite	semidefinite	NOUN
ejpam-5597	102	16	matrices	matrix	NOUN
ejpam-5597	102	17	a1/2	a1/2	VERB
ejpam-5597	102	18	and	and	CCONJ
ejpam-5597	102	19	b1/2	b1/2	VERB
ejpam-5597	102	20	respectively	respectively	ADV
ejpam-5597	102	21	,	,	PUNCT
ejpam-5597	102	22	we	we	PRON
ejpam-5597	102	23	have	have	VERB
ejpam-5597	102	24	w	w	NOUN
ejpam-5597	102	25	(	(	PUNCT
ejpam-5597	102	26	[	[	PUNCT
ejpam-5597	102	27	a1/2	a1/2	VERB
ejpam-5597	102	28	b1/2	b1/2	NOUN
ejpam-5597	102	29	0	0	NUM
ejpam-5597	102	30	0	0	NUM
ejpam-5597	102	31	]	]	PUNCT
ejpam-5597	102	32	)	)	PUNCT
ejpam-5597	102	33	≤	≤	NUM
ejpam-5597	103	1	√	√	ADP
ejpam-5597	103	2	max{∥a∥	max{∥a∥	X
ejpam-5597	103	3	,	,	PUNCT
ejpam-5597	103	4	∥b∥}+	∥b∥}+	PROPN
ejpam-5597	103	5	∥∥a1/2b1/2	∥∥a1/2b1/2	NOUN
ejpam-5597	103	6	∥∥.	∥∥.	NUM
ejpam-5597	103	7	(	(	PUNCT
ejpam-5597	103	8	10	10	NUM
ejpam-5597	103	9	)	)	PUNCT
ejpam-5597	103	10	based	base	VERB
ejpam-5597	103	11	on	on	ADP
ejpam-5597	103	12	inequalities	inequality	NOUN
ejpam-5597	103	13	(	(	PUNCT
ejpam-5597	103	14	7	7	NUM
ejpam-5597	103	15	)	)	PUNCT
ejpam-5597	103	16	,	,	PUNCT
ejpam-5597	103	17	(	(	PUNCT
ejpam-5597	103	18	8)	8)	NUM
ejpam-5597	103	19	,	,	PUNCT
ejpam-5597	103	20	and	and	CCONJ
ejpam-5597	103	21	(	(	PUNCT
ejpam-5597	103	22	10	10	NUM
ejpam-5597	103	23	)	)	PUNCT
ejpam-5597	104	1	,	,	PUNCT
ejpam-5597	104	2	we	we	PRON
ejpam-5597	104	3	have	have	VERB
ejpam-5597	104	4	the	the	DET
ejpam-5597	104	5	following	follow	VERB
ejpam-5597	104	6	corollary	corollary	NOUN
ejpam-5597	104	7	.	.	PUNCT
ejpam-5597	105	1	corollary	corollary	ADJ
ejpam-5597	105	2	8	8	NUM
ejpam-5597	105	3	.	.	PUNCT
ejpam-5597	106	1	let	let	VERB
ejpam-5597	106	2	a	a	DET
ejpam-5597	106	3	,	,	PUNCT
ejpam-5597	106	4	b	b	NOUN
ejpam-5597	106	5	∈	∈	PROPN
ejpam-5597	106	6	mn(c	mn(c	X
ejpam-5597	106	7	)	)	PUNCT
ejpam-5597	106	8	be	be	AUX
ejpam-5597	106	9	positive	positive	ADJ
ejpam-5597	106	10	semidefinite	semidefinite	NOUN
ejpam-5597	106	11	.	.	PUNCT
ejpam-5597	107	1	then	then	ADV
ejpam-5597	107	2	max	max	PROPN
ejpam-5597	107	3	{	{	PUNCT
ejpam-5597	107	4	∥a∥	∥a∥	PROPN
ejpam-5597	107	5	,	,	PUNCT
ejpam-5597	107	6	∥b∥	∥b∥	NUM
ejpam-5597	107	7	}	}	PUNCT
ejpam-5597	107	8	≤	≤	ADJ
ejpam-5597	107	9	max	max	PROPN
ejpam-5597	107	10	{	{	PUNCT
ejpam-5597	107	11	w2	w2	NOUN
ejpam-5597	107	12	(	(	PUNCT
ejpam-5597	107	13	[	[	PUNCT
ejpam-5597	107	14	a1/2	a1/2	VERB
ejpam-5597	107	15	b1/2	b1/2	NOUN
ejpam-5597	107	16	0	0	NUM
ejpam-5597	107	17	0	0	NUM
ejpam-5597	107	18	]	]	PUNCT
ejpam-5597	107	19	)	)	PUNCT
ejpam-5597	107	20	,	,	PUNCT
ejpam-5597	107	21	w2	w2	NOUN
ejpam-5597	107	22	(	(	PUNCT
ejpam-5597	107	23	[	[	PUNCT
ejpam-5597	107	24	b1/2	b1/2	NOUN
ejpam-5597	107	25	a1/2	a1/2	NOUN
ejpam-5597	107	26	0	0	NUM
ejpam-5597	107	27	0	0	NUM
ejpam-5597	107	28	]	]	PUNCT
ejpam-5597	107	29	)	)	PUNCT
ejpam-5597	107	30	}	}	PUNCT
ejpam-5597	107	31	≤	≤	NUM
ejpam-5597	107	32	max	max	PROPN
ejpam-5597	107	33	{	{	PUNCT
ejpam-5597	107	34	∥a∥	∥a∥	PROPN
ejpam-5597	107	35	,	,	PUNCT
ejpam-5597	107	36	∥b∥}+	∥b∥}+	PROPN
ejpam-5597	107	37	∥∥∥a1/2b1/2	∥∥∥a1/2b1/2	NOUN
ejpam-5597	107	38	∥∥∥	∥∥∥	PROPN
ejpam-5597	107	39	.	.	PUNCT
ejpam-5597	108	1	in	in	ADP
ejpam-5597	108	2	particular	particular	ADJ
ejpam-5597	108	3	,	,	PUNCT
ejpam-5597	108	4	if	if	SCONJ
ejpam-5597	108	5	a1/2b1/2	a1/2b1/2	ADJ
ejpam-5597	108	6	=	=	SYM
ejpam-5597	108	7	0	0	NUM
ejpam-5597	108	8	,	,	PUNCT
ejpam-5597	108	9	then	then	ADV
ejpam-5597	108	10	max	max	PROPN
ejpam-5597	108	11	{	{	PUNCT
ejpam-5597	108	12	∥a∥	∥a∥	PROPN
ejpam-5597	108	13	,	,	PUNCT
ejpam-5597	108	14	∥b∥	∥b∥	NUM
ejpam-5597	108	15	}	}	PUNCT
ejpam-5597	108	16	=	=	SYM
ejpam-5597	108	17	max	max	PROPN
ejpam-5597	108	18	{	{	PUNCT
ejpam-5597	108	19	w2	w2	NOUN
ejpam-5597	108	20	(	(	PUNCT
ejpam-5597	108	21	[	[	PUNCT
ejpam-5597	108	22	a1/2	a1/2	VERB
ejpam-5597	108	23	b1/2	b1/2	NOUN
ejpam-5597	108	24	0	0	NUM
ejpam-5597	108	25	0	0	NUM
ejpam-5597	108	26	]	]	PUNCT
ejpam-5597	108	27	)	)	PUNCT
ejpam-5597	108	28	,	,	PUNCT
ejpam-5597	108	29	w2	w2	NOUN
ejpam-5597	108	30	(	(	PUNCT
ejpam-5597	108	31	[	[	PUNCT
ejpam-5597	108	32	b1/2	b1/2	NOUN
ejpam-5597	109	1	a1/2	a1/2	NOUN
ejpam-5597	109	2	0	0	NUM
ejpam-5597	109	3	0	0	NUM
ejpam-5597	109	4	]	]	PUNCT
ejpam-5597	109	5	)	)	PUNCT
ejpam-5597	109	6	}	}	PUNCT
ejpam-5597	109	7	.	.	PUNCT
ejpam-5597	110	1	proof	proof	NOUN
ejpam-5597	110	2	.	.	PUNCT
ejpam-5597	111	1	the	the	DET
ejpam-5597	111	2	first	first	ADJ
ejpam-5597	111	3	inequality	inequality	NOUN
ejpam-5597	111	4	follows	follow	VERB
ejpam-5597	111	5	from	from	ADP
ejpam-5597	111	6	inequalities	inequality	NOUN
ejpam-5597	111	7	(	(	PUNCT
ejpam-5597	111	8	7	7	NUM
ejpam-5597	111	9	)	)	PUNCT
ejpam-5597	111	10	and	and	CCONJ
ejpam-5597	111	11	(	(	PUNCT
ejpam-5597	111	12	8)	8)	NUM
ejpam-5597	111	13	.	.	PUNCT
ejpam-5597	112	1	in	in	ADP
ejpam-5597	112	2	inequality	inequality	NOUN
ejpam-5597	112	3	(	(	PUNCT
ejpam-5597	112	4	10	10	NUM
ejpam-5597	112	5	)	)	PUNCT
ejpam-5597	112	6	,	,	PUNCT
ejpam-5597	112	7	by	by	ADP
ejpam-5597	112	8	interchanging	interchange	VERB
ejpam-5597	112	9	the	the	DET
ejpam-5597	112	10	roles	role	NOUN
ejpam-5597	112	11	of	of	ADP
ejpam-5597	112	12	a1/2	a1/2	NOUN
ejpam-5597	112	13	and	and	CCONJ
ejpam-5597	112	14	b1/2	b1/2	ADJ
ejpam-5597	112	15	,	,	PUNCT
ejpam-5597	112	16	we	we	PRON
ejpam-5597	112	17	have	have	VERB
ejpam-5597	112	18	w2	w2	NOUN
ejpam-5597	112	19	(	(	PUNCT
ejpam-5597	112	20	[	[	PUNCT
ejpam-5597	112	21	b1/2	b1/2	NOUN
ejpam-5597	112	22	a1/2	a1/2	NOUN
ejpam-5597	112	23	0	0	NUM
ejpam-5597	112	24	0	0	NUM
ejpam-5597	112	25	]	]	PUNCT
ejpam-5597	112	26	)	)	PUNCT
ejpam-5597	112	27	≤	≤	NOUN
ejpam-5597	112	28	max{∥a∥	max{∥a∥	PUNCT
ejpam-5597	112	29	,	,	PUNCT
ejpam-5597	112	30	∥b∥}+	∥b∥}+	PROPN
ejpam-5597	112	31	∥∥∥a1/2b1/2	∥∥∥a1/2b1/2	NOUN
ejpam-5597	112	32	∥∥∥	∥∥∥	PROPN
ejpam-5597	112	33	.	.	PUNCT
ejpam-5597	113	1	(	(	PUNCT
ejpam-5597	113	2	11	11	NUM
ejpam-5597	113	3	)	)	PUNCT
ejpam-5597	113	4	so	so	ADV
ejpam-5597	113	5	,	,	PUNCT
ejpam-5597	113	6	we	we	PRON
ejpam-5597	113	7	can	can	AUX
ejpam-5597	113	8	obtain	obtain	VERB
ejpam-5597	113	9	the	the	DET
ejpam-5597	113	10	second	second	ADJ
ejpam-5597	113	11	inequality	inequality	NOUN
ejpam-5597	113	12	from	from	ADP
ejpam-5597	113	13	inequalities	inequality	NOUN
ejpam-5597	113	14	(	(	PUNCT
ejpam-5597	113	15	10	10	NUM
ejpam-5597	113	16	)	)	PUNCT
ejpam-5597	113	17	and	and	CCONJ
ejpam-5597	113	18	(	(	PUNCT
ejpam-5597	113	19	11	11	NUM
ejpam-5597	113	20	)	)	PUNCT
ejpam-5597	113	21	.	.	PUNCT
ejpam-5597	114	1	corollary	corollary	ADJ
ejpam-5597	114	2	9	9	NUM
ejpam-5597	114	3	.	.	PUNCT
ejpam-5597	115	1	let	let	VERB
ejpam-5597	115	2	a	a	DET
ejpam-5597	115	3	,	,	PUNCT
ejpam-5597	115	4	b	b	NOUN
ejpam-5597	115	5	∈	∈	PROPN
ejpam-5597	115	6	mn(c	mn(c	X
ejpam-5597	115	7	)	)	PUNCT
ejpam-5597	115	8	be	be	AUX
ejpam-5597	115	9	positive	positive	ADJ
ejpam-5597	115	10	semidefinite	semidefinite	NOUN
ejpam-5597	115	11	.	.	PUNCT
ejpam-5597	116	1	then	then	ADV
ejpam-5597	116	2	max	max	PROPN
ejpam-5597	116	3	{	{	PUNCT
ejpam-5597	116	4	∥a∥+	∥a∥+	VERB
ejpam-5597	116	5	∥b∥	∥b∥	NUM
ejpam-5597	116	6	2	2	NUM
ejpam-5597	116	7	,	,	PUNCT
ejpam-5597	116	8	5	5	NUM
ejpam-5597	116	9	8	8	NUM
ejpam-5597	116	10	∥a∥	∥a∥	NOUN
ejpam-5597	116	11	,	,	PUNCT
ejpam-5597	116	12	5	5	NUM
ejpam-5597	116	13	8	8	NUM
ejpam-5597	116	14	∥b∥	∥b∥	ADJ
ejpam-5597	116	15	}	}	PUNCT
ejpam-5597	116	16	≤	≤	NUM
ejpam-5597	116	17	1	1	NUM
ejpam-5597	116	18	2	2	NUM
ejpam-5597	116	19	[	[	PUNCT
ejpam-5597	116	20	w2	w2	NOUN
ejpam-5597	116	21	(	(	PUNCT
ejpam-5597	116	22	[	[	PUNCT
ejpam-5597	116	23	a1/2	a1/2	VERB
ejpam-5597	116	24	b1/2	b1/2	NOUN
ejpam-5597	116	25	0	0	NUM
ejpam-5597	116	26	0	0	NUM
ejpam-5597	116	27	]	]	PUNCT
ejpam-5597	116	28	)	)	PUNCT
ejpam-5597	117	1	+	+	CCONJ
ejpam-5597	117	2	w2	w2	NOUN
ejpam-5597	117	3	(	(	PUNCT
ejpam-5597	117	4	[	[	PUNCT
ejpam-5597	117	5	b1/2	b1/2	NOUN
ejpam-5597	117	6	a1/2	a1/2	NOUN
ejpam-5597	117	7	0	0	NUM
ejpam-5597	117	8	0	0	NUM
ejpam-5597	117	9	]	]	PUNCT
ejpam-5597	117	10	)	)	PUNCT
ejpam-5597	117	11	]	]	PUNCT
ejpam-5597	117	12	≤	≤	X
ejpam-5597	117	13	max{∥a∥	max{∥a∥	NUM
ejpam-5597	117	14	,	,	PUNCT
ejpam-5597	117	15	∥b∥}+	∥b∥}+	PROPN
ejpam-5597	117	16	∥∥∥a1/2b1/2	∥∥∥a1/2b1/2	NOUN
ejpam-5597	117	17	∥∥∥	∥∥∥	PROPN
ejpam-5597	117	18	.	.	PUNCT
ejpam-5597	118	1	a.	a.	PROPN
ejpam-5597	118	2	al	al	PROPN
ejpam-5597	118	3	-	-	PUNCT
ejpam-5597	118	4	natoor	natoor	NOUN
ejpam-5597	118	5	,	,	PUNCT
ejpam-5597	118	6	f.	f.	PROPN
ejpam-5597	118	7	alrimawi	alrimawi	PROPN
ejpam-5597	118	8	/	/	SYM
ejpam-5597	118	9	eur	eur	PROPN
ejpam-5597	118	10	.	.	PUNCT
ejpam-5597	119	1	j.	j.	PROPN
ejpam-5597	119	2	pure	pure	PROPN
ejpam-5597	119	3	appl	appl	PROPN
ejpam-5597	119	4	.	.	PROPN
ejpam-5597	119	5	math	math	PROPN
ejpam-5597	119	6	,	,	PUNCT
ejpam-5597	119	7	18	18	NUM
ejpam-5597	119	8	(	(	PUNCT
ejpam-5597	119	9	1	1	NUM
ejpam-5597	119	10	)	)	PUNCT
ejpam-5597	119	11	(	(	PUNCT
ejpam-5597	119	12	2025	2025	NUM
ejpam-5597	119	13	)	)	PUNCT
ejpam-5597	119	14	,	,	PUNCT
ejpam-5597	119	15	5597	5597	NUM
ejpam-5597	119	16	7	7	NUM
ejpam-5597	119	17	of	of	ADP
ejpam-5597	119	18	9	9	NUM
ejpam-5597	119	19	proof	proof	NOUN
ejpam-5597	119	20	.	.	PUNCT
ejpam-5597	120	1	using	use	VERB
ejpam-5597	120	2	inequality	inequality	NOUN
ejpam-5597	120	3	(	(	PUNCT
ejpam-5597	120	4	9	9	NUM
ejpam-5597	120	5	)	)	PUNCT
ejpam-5597	120	6	,	,	PUNCT
ejpam-5597	120	7	we	we	PRON
ejpam-5597	120	8	have	have	VERB
ejpam-5597	120	9	w2	w2	NOUN
ejpam-5597	120	10	(	(	PUNCT
ejpam-5597	120	11	[	[	PUNCT
ejpam-5597	120	12	a1/2	a1/2	VERB
ejpam-5597	120	13	b1/2	b1/2	NOUN
ejpam-5597	120	14	0	0	NUM
ejpam-5597	120	15	0	0	NUM
ejpam-5597	120	16	]	]	PUNCT
ejpam-5597	120	17	)	)	PUNCT
ejpam-5597	121	1	+	+	CCONJ
ejpam-5597	121	2	w2	w2	NOUN
ejpam-5597	121	3	(	(	PUNCT
ejpam-5597	121	4	[	[	PUNCT
ejpam-5597	121	5	b1/2	b1/2	NOUN
ejpam-5597	121	6	a1/2	a1/2	NOUN
ejpam-5597	121	7	0	0	NUM
ejpam-5597	121	8	0	0	NUM
ejpam-5597	121	9	]	]	PUNCT
ejpam-5597	121	10	)	)	PUNCT
ejpam-5597	121	11	≥	≥	NOUN
ejpam-5597	121	12	5	5	NUM
ejpam-5597	121	13	8	8	NUM
ejpam-5597	121	14	(	(	PUNCT
ejpam-5597	121	15	∥a∥+	∥a∥+	VERB
ejpam-5597	121	16	∥b∥	∥b∥	NUM
ejpam-5597	121	17	)	)	PUNCT
ejpam-5597	121	18	+	+	CCONJ
ejpam-5597	121	19	1	1	NUM
ejpam-5597	121	20	8	8	NUM
ejpam-5597	121	21	(	(	PUNCT
ejpam-5597	121	22	|4	|4	NUM
ejpam-5597	121	23	∥a∥	∥a∥	VERB
ejpam-5597	121	24	−	−	PUNCT
ejpam-5597	121	25	∥b∥|+	∥b∥|+	NOUN
ejpam-5597	121	26	|4	|4	NUM
ejpam-5597	121	27	∥b∥	∥b∥	NOUN
ejpam-5597	121	28	−	−	PROPN
ejpam-5597	121	29	∥a∥|	∥a∥|	NOUN
ejpam-5597	121	30	)	)	PUNCT
ejpam-5597	121	31	=	=	SYM
ejpam-5597	121	32	5	5	NUM
ejpam-5597	121	33	8	8	NUM
ejpam-5597	121	34	(	(	PUNCT
ejpam-5597	121	35	∥a∥+	∥a∥+	VERB
ejpam-5597	121	36	∥b∥	∥b∥	NUM
ejpam-5597	121	37	)	)	PUNCT
ejpam-5597	121	38	+	+	CCONJ
ejpam-5597	121	39	1	1	NUM
ejpam-5597	121	40	8	8	NUM
ejpam-5597	121	41	(	(	PUNCT
ejpam-5597	121	42	|4	|4	NUM
ejpam-5597	121	43	∥a∥	∥a∥	VERB
ejpam-5597	121	44	−	−	PUNCT
ejpam-5597	121	45	∥b∥|+	∥b∥|+	NOUN
ejpam-5597	121	46	|∥a∥	|∥a∥	VERB
ejpam-5597	121	47	−	−	NUM
ejpam-5597	121	48	4	4	NUM
ejpam-5597	121	49	∥b∥|	∥b∥|	NOUN
ejpam-5597	121	50	)	)	PUNCT
ejpam-5597	121	51	≥	≥	NOUN
ejpam-5597	121	52	5	5	NUM
ejpam-5597	121	53	8	8	NUM
ejpam-5597	121	54	(	(	PUNCT
ejpam-5597	121	55	∥a∥+	∥a∥+	VERB
ejpam-5597	121	56	∥b∥	∥b∥	NUM
ejpam-5597	121	57	)	)	PUNCT
ejpam-5597	121	58	+	+	CCONJ
ejpam-5597	121	59	5	5	NUM
ejpam-5597	121	60	8	8	NUM
ejpam-5597	121	61	|∥a∥	|∥a∥	VERB
ejpam-5597	121	62	−	−	NUM
ejpam-5597	121	63	∥b∥|	∥b∥|	NOUN
ejpam-5597	121	64	=	=	SYM
ejpam-5597	121	65	5	5	NUM
ejpam-5597	121	66	4	4	NUM
ejpam-5597	121	67	max	max	NOUN
ejpam-5597	121	68	{	{	PUNCT
ejpam-5597	121	69	∥a∥	∥a∥	PROPN
ejpam-5597	121	70	,	,	PUNCT
ejpam-5597	121	71	∥b∥	∥b∥	NUM
ejpam-5597	121	72	}	}	PUNCT
ejpam-5597	121	73	.	.	PUNCT
ejpam-5597	122	1	(	(	PUNCT
ejpam-5597	122	2	12	12	NUM
ejpam-5597	122	3	)	)	PUNCT
ejpam-5597	122	4	using	use	VERB
ejpam-5597	122	5	inequalities	inequality	NOUN
ejpam-5597	122	6	(	(	PUNCT
ejpam-5597	122	7	6	6	NUM
ejpam-5597	122	8	)	)	PUNCT
ejpam-5597	122	9	and	and	CCONJ
ejpam-5597	122	10	(	(	PUNCT
ejpam-5597	122	11	12	12	NUM
ejpam-5597	122	12	)	)	PUNCT
ejpam-5597	122	13	we	we	PRON
ejpam-5597	122	14	get	get	VERB
ejpam-5597	122	15	the	the	DET
ejpam-5597	122	16	first	first	ADJ
ejpam-5597	122	17	inequality	inequality	NOUN
ejpam-5597	122	18	.	.	PUNCT
ejpam-5597	123	1	also	also	ADV
ejpam-5597	123	2	,	,	PUNCT
ejpam-5597	123	3	the	the	DET
ejpam-5597	123	4	second	second	ADJ
ejpam-5597	123	5	inequality	inequality	NOUN
ejpam-5597	123	6	can	can	AUX
ejpam-5597	123	7	be	be	AUX
ejpam-5597	123	8	obtained	obtain	VERB
ejpam-5597	123	9	from	from	ADP
ejpam-5597	123	10	inequalities	inequality	NOUN
ejpam-5597	123	11	(	(	PUNCT
ejpam-5597	123	12	10	10	NUM
ejpam-5597	123	13	)	)	PUNCT
ejpam-5597	123	14	and	and	CCONJ
ejpam-5597	123	15	(	(	PUNCT
ejpam-5597	123	16	11	11	NUM
ejpam-5597	123	17	)	)	PUNCT
ejpam-5597	123	18	.	.	PUNCT
ejpam-5597	124	1	we	we	PRON
ejpam-5597	124	2	end	end	VERB
ejpam-5597	124	3	this	this	DET
ejpam-5597	124	4	paper	paper	NOUN
ejpam-5597	124	5	with	with	ADP
ejpam-5597	124	6	the	the	DET
ejpam-5597	124	7	following	follow	VERB
ejpam-5597	124	8	result	result	NOUN
ejpam-5597	124	9	.	.	PUNCT
ejpam-5597	125	1	theorem	theorem	NOUN
ejpam-5597	125	2	2	2	NUM
ejpam-5597	125	3	.	.	PUNCT
ejpam-5597	126	1	let	let	VERB
ejpam-5597	126	2	a	a	DET
ejpam-5597	126	3	,	,	PUNCT
ejpam-5597	126	4	b	b	NOUN
ejpam-5597	126	5	,	,	PUNCT
ejpam-5597	126	6	c	c	NOUN
ejpam-5597	126	7	,	,	PUNCT
ejpam-5597	126	8	d	d	PROPN
ejpam-5597	126	9	∈	∈	PROPN
ejpam-5597	126	10	mn(c	mn(c	X
ejpam-5597	126	11	)	)	PUNCT
ejpam-5597	126	12	.	.	PUNCT
ejpam-5597	127	1	then	then	ADV
ejpam-5597	127	2	w	w	X
ejpam-5597	127	3	(	(	PUNCT
ejpam-5597	127	4	[	[	PUNCT
ejpam-5597	127	5	a	a	PRON
ejpam-5597	127	6	b	b	NOUN
ejpam-5597	127	7	c	c	NOUN
ejpam-5597	127	8	d	d	NOUN
ejpam-5597	127	9	]	]	X
ejpam-5597	127	10	)	)	PUNCT
ejpam-5597	127	11	≥	≥	NOUN
ejpam-5597	127	12	∥∥∥∥	∥∥∥∥	ADP
ejpam-5597	127	13	[	[	PUNCT
ejpam-5597	127	14	re(a	re(a	NOUN
ejpam-5597	127	15	)	)	PUNCT
ejpam-5597	127	16	b+c∗	b+c∗	PROPN
ejpam-5597	127	17	2	2	NUM
ejpam-5597	127	18	c+b∗	c+b∗	NUM
ejpam-5597	127	19	2	2	NUM
ejpam-5597	127	20	re(d	re(d	NUM
ejpam-5597	127	21	)	)	PUNCT
ejpam-5597	127	22	]	]	PUNCT
ejpam-5597	127	23	∥∥∥∥	∥∥∥∥	NUM
ejpam-5597	127	24	.	.	PUNCT
ejpam-5597	128	1	(	(	PUNCT
ejpam-5597	128	2	13	13	NUM
ejpam-5597	128	3	)	)	PUNCT
ejpam-5597	128	4	proof	proof	NOUN
ejpam-5597	128	5	.	.	PUNCT
ejpam-5597	129	1	we	we	PRON
ejpam-5597	129	2	have	have	VERB
ejpam-5597	129	3	w	w	NOUN
ejpam-5597	129	4	(	(	PUNCT
ejpam-5597	129	5	[	[	PUNCT
ejpam-5597	129	6	a	a	PRON
ejpam-5597	129	7	b	b	NOUN
ejpam-5597	129	8	c	c	NOUN
ejpam-5597	129	9	d	d	NOUN
ejpam-5597	129	10	]	]	X
ejpam-5597	129	11	)	)	PUNCT
ejpam-5597	130	1	=	=	SYM
ejpam-5597	130	2	max	max	PROPN
ejpam-5597	130	3	θ∈r	θ∈r	NOUN
ejpam-5597	130	4	∥∥∥∥re	∥∥∥∥re	NOUN
ejpam-5597	130	5	(	(	PUNCT
ejpam-5597	130	6	eiθ	eiθ	PROPN
ejpam-5597	130	7	[	[	PUNCT
ejpam-5597	130	8	a	a	PRON
ejpam-5597	130	9	b	b	NOUN
ejpam-5597	130	10	c	c	NOUN
ejpam-5597	130	11	d	d	NOUN
ejpam-5597	130	12	]	]	X
ejpam-5597	130	13	)	)	PUNCT
ejpam-5597	130	14	∥∥∥∥	∥∥∥∥	NUM
ejpam-5597	130	15	≥	≥	NOUN
ejpam-5597	130	16	∥∥∥∥re	∥∥∥∥re	NUM
ejpam-5597	130	17	(	(	PUNCT
ejpam-5597	130	18	[	[	PUNCT
ejpam-5597	130	19	a	a	PRON
ejpam-5597	130	20	b	b	NOUN
ejpam-5597	130	21	c	c	NOUN
ejpam-5597	130	22	d	d	X
ejpam-5597	130	23	]	]	X
ejpam-5597	130	24	)	)	PUNCT
ejpam-5597	130	25	∥∥∥∥	∥∥∥∥	PUNCT
ejpam-5597	131	1	=	=	SYM
ejpam-5597	131	2	1	1	NUM
ejpam-5597	131	3	2	2	NUM
ejpam-5597	131	4	∥∥∥∥	∥∥∥∥	NUM
ejpam-5597	131	5	[	[	PUNCT
ejpam-5597	131	6	a	a	DET
ejpam-5597	131	7	b	b	NOUN
ejpam-5597	131	8	c	c	NOUN
ejpam-5597	131	9	d	d	X
ejpam-5597	131	10	]	]	X
ejpam-5597	132	1	+	+	CCONJ
ejpam-5597	132	2	[	[	PUNCT
ejpam-5597	132	3	a∗	a∗	ADJ
ejpam-5597	132	4	c∗	c∗	PROPN
ejpam-5597	132	5	b∗	b∗	ADJ
ejpam-5597	132	6	d∗	d∗	PROPN
ejpam-5597	132	7	]	]	PUNCT
ejpam-5597	132	8	∥∥∥∥	∥∥∥∥	PUNCT
ejpam-5597	132	9	=	=	SYM
ejpam-5597	132	10	1	1	NUM
ejpam-5597	132	11	2	2	NUM
ejpam-5597	132	12	∥∥∥∥	∥∥∥∥	NUM
ejpam-5597	132	13	[	[	PUNCT
ejpam-5597	132	14	a+a∗	a+a∗	NUM
ejpam-5597	132	15	b	b	NOUN
ejpam-5597	132	16	+	+	CCONJ
ejpam-5597	132	17	c∗	c∗	PROPN
ejpam-5597	132	18	c	c	PROPN
ejpam-5597	133	1	+	+	NOUN
ejpam-5597	133	2	b∗	b∗	ADJ
ejpam-5597	133	3	d	d	X
ejpam-5597	133	4	+	+	NOUN
ejpam-5597	133	5	d∗	d∗	NOUN
ejpam-5597	133	6	]	]	PUNCT
ejpam-5597	133	7	∥∥∥∥	∥∥∥∥	PUNCT
ejpam-5597	133	8	=	=	SYM
ejpam-5597	133	9	1	1	NUM
ejpam-5597	133	10	2	2	NUM
ejpam-5597	133	11	∥∥∥∥	∥∥∥∥	ADP
ejpam-5597	133	12	[	[	PUNCT
ejpam-5597	133	13	2re(a	2re(a	NUM
ejpam-5597	133	14	)	)	PUNCT
ejpam-5597	133	15	b	b	NOUN
ejpam-5597	133	16	+	+	CCONJ
ejpam-5597	133	17	c∗	c∗	PROPN
ejpam-5597	133	18	c	c	PROPN
ejpam-5597	134	1	+	+	NOUN
ejpam-5597	134	2	b∗	b∗	ADJ
ejpam-5597	134	3	2re(d	2re(d	NUM
ejpam-5597	134	4	)	)	PUNCT
ejpam-5597	134	5	]	]	PUNCT
ejpam-5597	134	6	∥∥∥∥	∥∥∥∥	PUNCT
ejpam-5597	134	7	=	=	SYM
ejpam-5597	134	8	∥∥∥∥	∥∥∥∥	PROPN
ejpam-5597	134	9	[	[	PUNCT
ejpam-5597	134	10	re(a	re(a	NOUN
ejpam-5597	134	11	)	)	PUNCT
ejpam-5597	134	12	b+c∗	b+c∗	PROPN
ejpam-5597	134	13	2	2	NUM
ejpam-5597	134	14	c+b∗	c+b∗	NUM
ejpam-5597	134	15	2	2	NUM
ejpam-5597	134	16	re(d	re(d	NUM
ejpam-5597	134	17	)	)	PUNCT
ejpam-5597	134	18	]	]	PUNCT
ejpam-5597	134	19	∥∥∥∥	∥∥∥∥	NUM
ejpam-5597	134	20	,	,	PUNCT
ejpam-5597	134	21	as	as	SCONJ
ejpam-5597	134	22	required	require	VERB
ejpam-5597	134	23	.	.	PUNCT
ejpam-5597	135	1	by	by	ADP
ejpam-5597	135	2	inequalities	inequality	NOUN
ejpam-5597	135	3	(	(	PUNCT
ejpam-5597	135	4	2	2	NUM
ejpam-5597	135	5	)	)	PUNCT
ejpam-5597	135	6	and	and	CCONJ
ejpam-5597	135	7	(	(	PUNCT
ejpam-5597	135	8	13	13	NUM
ejpam-5597	135	9	)	)	PUNCT
ejpam-5597	135	10	,	,	PUNCT
ejpam-5597	135	11	we	we	PRON
ejpam-5597	135	12	have∥∥∥∥	have∥∥∥∥	VERB
ejpam-5597	135	13	[	[	PUNCT
ejpam-5597	135	14	re(a	re(a	NOUN
ejpam-5597	135	15	)	)	PUNCT
ejpam-5597	135	16	b+c∗	b+c∗	PROPN
ejpam-5597	135	17	2	2	NUM
ejpam-5597	135	18	c+b∗	c+b∗	NUM
ejpam-5597	135	19	2	2	NUM
ejpam-5597	135	20	re(d	re(d	NUM
ejpam-5597	135	21	)	)	PUNCT
ejpam-5597	135	22	]	]	PUNCT
ejpam-5597	135	23	∥∥∥∥	∥∥∥∥	PUNCT
ejpam-5597	136	1	≤	≤	NUM
ejpam-5597	136	2	w	w	NOUN
ejpam-5597	136	3	(	(	PUNCT
ejpam-5597	136	4	[	[	PUNCT
ejpam-5597	136	5	a	a	PRON
ejpam-5597	136	6	b	b	NOUN
ejpam-5597	136	7	c	c	NOUN
ejpam-5597	136	8	d	d	X
ejpam-5597	136	9	]	]	X
ejpam-5597	136	10	)	)	PUNCT
ejpam-5597	136	11	≤	≤	NOUN
ejpam-5597	136	12	√	√	NUM
ejpam-5597	136	13	max	max	PROPN
ejpam-5597	136	14	{	{	PUNCT
ejpam-5597	136	15	∥a∗a+	∥a∗a+	PROPN
ejpam-5597	136	16	c∗c∥	c∗c∥	NOUN
ejpam-5597	136	17	,	,	PUNCT
ejpam-5597	136	18	∥b∗b	∥b∗b	VERB
ejpam-5597	136	19	+	+	NOUN
ejpam-5597	136	20	d∗d∥}+	d∗d∥}+	NOUN
ejpam-5597	136	21	∥a∗b	∥a∗b	VERB
ejpam-5597	136	22	+	+	CCONJ
ejpam-5597	136	23	c∗d∥.	c∗d∥.	PROPN
ejpam-5597	136	24	a.	a.	NOUN
ejpam-5597	136	25	al	al	PROPN
ejpam-5597	136	26	-	-	PUNCT
ejpam-5597	136	27	natoor	natoor	NOUN
ejpam-5597	136	28	,	,	PUNCT
ejpam-5597	136	29	f.	f.	PROPN
ejpam-5597	136	30	alrimawi	alrimawi	PROPN
ejpam-5597	136	31	/	/	SYM
ejpam-5597	136	32	eur	eur	PROPN
ejpam-5597	136	33	.	.	PUNCT
ejpam-5597	137	1	j.	j.	PROPN
ejpam-5597	137	2	pure	pure	PROPN
ejpam-5597	137	3	appl	appl	PROPN
ejpam-5597	137	4	.	.	PROPN
ejpam-5597	137	5	math	math	PROPN
ejpam-5597	137	6	,	,	PUNCT
ejpam-5597	137	7	18	18	NUM
ejpam-5597	137	8	(	(	PUNCT
ejpam-5597	137	9	1	1	NUM
ejpam-5597	137	10	)	)	PUNCT
ejpam-5597	137	11	(	(	PUNCT
ejpam-5597	137	12	2025	2025	NUM
ejpam-5597	137	13	)	)	PUNCT
ejpam-5597	137	14	,	,	PUNCT
ejpam-5597	137	15	5597	5597	NUM
ejpam-5597	137	16	8	8	NUM
ejpam-5597	137	17	of	of	ADP
ejpam-5597	137	18	9	9	NUM
ejpam-5597	137	19	3	3	NUM
ejpam-5597	137	20	.	.	PUNCT
ejpam-5597	137	21	conclusion	conclusion	NOUN
ejpam-5597	137	22	in	in	ADP
ejpam-5597	137	23	this	this	DET
ejpam-5597	137	24	paper	paper	NOUN
ejpam-5597	137	25	,	,	PUNCT
ejpam-5597	137	26	new	new	ADJ
ejpam-5597	137	27	results	result	NOUN
ejpam-5597	137	28	related	relate	VERB
ejpam-5597	137	29	to	to	ADP
ejpam-5597	137	30	numerical	numerical	ADJ
ejpam-5597	137	31	radii	radius	NOUN
ejpam-5597	137	32	of	of	ADP
ejpam-5597	137	33	block	block	NOUN
ejpam-5597	137	34	matrices	matrix	NOUN
ejpam-5597	137	35	were	be	AUX
ejpam-5597	137	36	given	give	VERB
ejpam-5597	137	37	.	.	PUNCT
ejpam-5597	138	1	several	several	ADJ
ejpam-5597	138	2	particular	particular	ADJ
ejpam-5597	138	3	cases	case	NOUN
ejpam-5597	138	4	were	be	AUX
ejpam-5597	138	5	also	also	ADV
ejpam-5597	138	6	given	give	VERB
ejpam-5597	138	7	.	.	PUNCT
ejpam-5597	139	1	at	at	ADP
ejpam-5597	139	2	the	the	DET
ejpam-5597	139	3	end	end	NOUN
ejpam-5597	139	4	of	of	ADP
ejpam-5597	139	5	the	the	DET
ejpam-5597	139	6	paper	paper	NOUN
ejpam-5597	139	7	,	,	PUNCT
ejpam-5597	139	8	an	an	DET
ejpam-5597	139	9	upper	upper	ADJ
ejpam-5597	139	10	bound	bind	VERB
ejpam-5597	139	11	and	and	CCONJ
ejpam-5597	139	12	a	a	DET
ejpam-5597	139	13	lower	low	ADJ
ejpam-5597	139	14	bound	bind	VERB
ejpam-5597	139	15	of	of	ADP
ejpam-5597	139	16	the	the	DET
ejpam-5597	139	17	numerical	numerical	ADJ
ejpam-5597	139	18	radius	radius	NOUN
ejpam-5597	139	19	of	of	ADP
ejpam-5597	139	20	the	the	DET
ejpam-5597	139	21	partitioned	partition	VERB
ejpam-5597	139	22	matrix	matrix	NOUN
ejpam-5597	139	23	[	[	PUNCT
ejpam-5597	139	24	a	a	PRON
ejpam-5597	139	25	b	b	NOUN
ejpam-5597	139	26	c	c	NOUN
ejpam-5597	139	27	d	d	X
ejpam-5597	139	28	]	]	X
ejpam-5597	139	29	were	be	AUX
ejpam-5597	139	30	established	establish	VERB
ejpam-5597	139	31	.	.	PUNCT
ejpam-5597	140	1	conflict	conflict	NOUN
ejpam-5597	140	2	of	of	ADP
ejpam-5597	140	3	interest	interest	NOUN
ejpam-5597	140	4	.	.	PUNCT
ejpam-5597	141	1	the	the	DET
ejpam-5597	141	2	authors	author	NOUN
ejpam-5597	141	3	declare	declare	VERB
ejpam-5597	141	4	that	that	SCONJ
ejpam-5597	141	5	they	they	PRON
ejpam-5597	141	6	have	have	VERB
ejpam-5597	141	7	no	no	DET
ejpam-5597	141	8	conflict	conflict	NOUN
ejpam-5597	141	9	of	of	ADP
ejpam-5597	141	10	interest	interest	NOUN
ejpam-5597	141	11	.	.	PUNCT
ejpam-5597	142	1	data	datum	NOUN
ejpam-5597	142	2	availability	availability	NOUN
ejpam-5597	142	3	.	.	PUNCT
ejpam-5597	143	1	not	not	PART
ejpam-5597	143	2	applicable	applicable	ADJ
ejpam-5597	143	3	.	.	PUNCT
ejpam-5597	144	1	author	author	NOUN
ejpam-5597	144	2	’s	’s	PART
ejpam-5597	144	3	contributions	contribution	NOUN
ejpam-5597	144	4	.	.	PUNCT
ejpam-5597	145	1	all	all	DET
ejpam-5597	145	2	authors	author	NOUN
ejpam-5597	145	3	contributed	contribute	VERB
ejpam-5597	145	4	to	to	ADP
ejpam-5597	145	5	each	each	DET
ejpam-5597	145	6	part	part	NOUN
ejpam-5597	145	7	of	of	ADP
ejpam-5597	145	8	this	this	DET
ejpam-5597	145	9	work	work	NOUN
ejpam-5597	145	10	,	,	PUNCT
ejpam-5597	145	11	and	and	CCONJ
ejpam-5597	145	12	they	they	PRON
ejpam-5597	145	13	all	all	PRON
ejpam-5597	145	14	read	read	VERB
ejpam-5597	145	15	and	and	CCONJ
ejpam-5597	145	16	approved	approve	VERB
ejpam-5597	145	17	the	the	DET
ejpam-5597	145	18	final	final	ADJ
ejpam-5597	145	19	manuscript	manuscript	NOUN
ejpam-5597	145	20	.	.	PUNCT
ejpam-5597	146	1	acknowledgements	acknowledgement	NOUN
ejpam-5597	146	2	the	the	DET
ejpam-5597	146	3	authors	author	NOUN
ejpam-5597	146	4	are	be	AUX
ejpam-5597	146	5	grateful	grateful	ADJ
ejpam-5597	146	6	to	to	ADP
ejpam-5597	146	7	the	the	DET
ejpam-5597	146	8	reviewers	reviewer	NOUN
ejpam-5597	146	9	for	for	ADP
ejpam-5597	146	10	their	their	PRON
ejpam-5597	146	11	careful	careful	ADJ
ejpam-5597	146	12	reading	reading	NOUN
ejpam-5597	146	13	and	and	CCONJ
ejpam-5597	146	14	valuable	valuable	ADJ
ejpam-5597	146	15	suggestions	suggestion	NOUN
ejpam-5597	146	16	.	.	PUNCT
ejpam-5597	147	1	references	reference	NOUN
ejpam-5597	147	2	[	[	X
ejpam-5597	147	3	1	1	NUM
ejpam-5597	147	4	]	]	X
ejpam-5597	147	5	al	al	PROPN
ejpam-5597	147	6	-	-	PUNCT
ejpam-5597	147	7	natoor	natoor	NOUN
ejpam-5597	147	8	.	.	PUNCT
ejpam-5597	148	1	norm	norm	NOUN
ejpam-5597	148	2	inequalities	inequality	NOUN
ejpam-5597	148	3	for	for	ADP
ejpam-5597	148	4	functions	function	NOUN
ejpam-5597	148	5	of	of	ADP
ejpam-5597	148	6	matrices	matrix	NOUN
ejpam-5597	148	7	.	.	PUNCT
ejpam-5597	149	1	heliyon	heliyon	NOUN
ejpam-5597	149	2	,	,	PUNCT
ejpam-5597	149	3	10	10	NUM
ejpam-5597	149	4	,	,	PUNCT
ejpam-5597	149	5	2024	2024	NUM
ejpam-5597	149	6	.	.	PUNCT
ejpam-5597	150	1	[	[	X
ejpam-5597	150	2	2	2	NUM
ejpam-5597	150	3	]	]	X
ejpam-5597	150	4	al	al	PROPN
ejpam-5597	150	5	-	-	PUNCT
ejpam-5597	150	6	natoor	natoor	NOUN
ejpam-5597	150	7	.	.	PUNCT
ejpam-5597	151	1	norm	norm	NOUN
ejpam-5597	151	2	inequalities	inequality	NOUN
ejpam-5597	151	3	for	for	ADP
ejpam-5597	151	4	product	product	NOUN
ejpam-5597	151	5	of	of	ADP
ejpam-5597	151	6	matrices	matrix	NOUN
ejpam-5597	151	7	.	.	PUNCT
ejpam-5597	152	1	acta	acta	PROPN
ejpam-5597	152	2	sci	sci	PROPN
ejpam-5597	152	3	.	.	PROPN
ejpam-5597	152	4	math	math	PROPN
ejpam-5597	152	5	.	.	PUNCT
ejpam-5597	153	1	(	(	PUNCT
ejpam-5597	153	2	szeged	szeged	PROPN
ejpam-5597	153	3	)	)	PUNCT
ejpam-5597	153	4	,	,	PUNCT
ejpam-5597	153	5	pages	page	NOUN
ejpam-5597	153	6	https://doi.org/10.1007/s44146–024–00121–1	https://doi.org/10.1007/s44146–024–00121–1	PROPN
ejpam-5597	153	7	,	,	PUNCT
ejpam-5597	153	8	2024	2024	NUM
ejpam-5597	153	9	.	.	PUNCT
ejpam-5597	154	1	[	[	X
ejpam-5597	154	2	3	3	NUM
ejpam-5597	154	3	]	]	PUNCT
ejpam-5597	154	4	a.	a.	NOUN
ejpam-5597	154	5	al	al	PROPN
ejpam-5597	154	6	-	-	PUNCT
ejpam-5597	154	7	natoor	natoor	NOUN
ejpam-5597	154	8	,	,	PUNCT
ejpam-5597	154	9	o.	o.	PROPN
ejpam-5597	154	10	hirzallah	hirzallah	PROPN
ejpam-5597	154	11	,	,	PUNCT
ejpam-5597	154	12	and	and	CCONJ
ejpam-5597	154	13	f.	f.	PROPN
ejpam-5597	154	14	kittaneh	kittaneh	PROPN
ejpam-5597	154	15	.	.	PUNCT
ejpam-5597	155	1	interpolating	interpolate	VERB
ejpam-5597	155	2	numerical	numerical	ADJ
ejpam-5597	155	3	radius	radius	PROPN
ejpam-5597	155	4	inequalities	inequality	NOUN
ejpam-5597	155	5	for	for	ADP
ejpam-5597	155	6	matrices	matrix	NOUN
ejpam-5597	155	7	.	.	PUNCT
ejpam-5597	156	1	adv	adv	PROPN
ejpam-5597	156	2	.	.	PUNCT
ejpam-5597	157	1	oper	oper	PROPN
ejpam-5597	157	2	.	.	PUNCT
ejpam-5597	157	3	theory	theory	NOUN
ejpam-5597	157	4	,	,	PUNCT
ejpam-5597	157	5	23	23	NUM
ejpam-5597	157	6	,	,	PUNCT
ejpam-5597	157	7	2024	2024	NUM
ejpam-5597	157	8	.	.	PUNCT
ejpam-5597	158	1	[	[	X
ejpam-5597	158	2	4	4	X
ejpam-5597	158	3	]	]	PUNCT
ejpam-5597	158	4	a.	a.	NOUN
ejpam-5597	158	5	al	al	PROPN
ejpam-5597	158	6	-	-	PUNCT
ejpam-5597	158	7	natoor	natoor	NOUN
ejpam-5597	158	8	,	,	PUNCT
ejpam-5597	158	9	o.	o.	PROPN
ejpam-5597	158	10	hirzallah	hirzallah	PROPN
ejpam-5597	158	11	,	,	PUNCT
ejpam-5597	158	12	and	and	CCONJ
ejpam-5597	158	13	f.	f.	PROPN
ejpam-5597	158	14	kittaneh	kittaneh	PROPN
ejpam-5597	158	15	.	.	PUNCT
ejpam-5597	159	1	singular	singular	PROPN
ejpam-5597	159	2	value	value	NOUN
ejpam-5597	159	3	and	and	CCONJ
ejpam-5597	159	4	norm	norm	NOUN
ejpam-5597	159	5	inequalities	inequality	NOUN
ejpam-5597	159	6	for	for	ADP
ejpam-5597	159	7	products	product	NOUN
ejpam-5597	159	8	and	and	CCONJ
ejpam-5597	159	9	sums	sum	NOUN
ejpam-5597	159	10	of	of	ADP
ejpam-5597	159	11	matrices	matrix	NOUN
ejpam-5597	159	12	.	.	PUNCT
ejpam-5597	160	1	period	period	NOUN
ejpam-5597	160	2	.	.	PUNCT
ejpam-5597	161	1	math	math	NOUN
ejpam-5597	161	2	.	.	PUNCT
ejpam-5597	162	1	hung	hung	PROPN
ejpam-5597	162	2	.	.	PROPN
ejpam-5597	162	3	,	,	PUNCT
ejpam-5597	163	1	88:204–217	88:204–217	NUM
ejpam-5597	163	2	,	,	PUNCT
ejpam-5597	163	3	2024	2024	NUM
ejpam-5597	163	4	.	.	PUNCT
ejpam-5597	164	1	[	[	X
ejpam-5597	164	2	5	5	NUM
ejpam-5597	164	3	]	]	PUNCT
ejpam-5597	164	4	a.	a.	NOUN
ejpam-5597	164	5	al	al	PROPN
ejpam-5597	164	6	-	-	PUNCT
ejpam-5597	164	7	natoor	natoor	PROPN
ejpam-5597	164	8	and	and	CCONJ
ejpam-5597	164	9	f.	f.	PROPN
ejpam-5597	164	10	kittaneh	kittaneh	PROPN
ejpam-5597	164	11	.	.	PUNCT
ejpam-5597	165	1	further	far	ADV
ejpam-5597	165	2	unitarily	unitarily	ADV
ejpam-5597	165	3	invariant	invariant	ADJ
ejpam-5597	165	4	norm	norm	NOUN
ejpam-5597	165	5	inequalities	inequality	NOUN
ejpam-5597	165	6	for	for	ADP
ejpam-5597	165	7	positive	positive	ADJ
ejpam-5597	165	8	semidefinite	semidefinite	NOUN
ejpam-5597	165	9	matrices	matrix	NOUN
ejpam-5597	165	10	.	.	PUNCT
ejpam-5597	166	1	positivity	positivity	NOUN
ejpam-5597	166	2	,	,	PUNCT
ejpam-5597	166	3	26	26	NUM
ejpam-5597	166	4	:	:	PUNCT
ejpam-5597	166	5	paper	paper	NOUN
ejpam-5597	166	6	no	no	NOUN
ejpam-5597	166	7	.	.	NOUN
ejpam-5597	166	8	8	8	NUM
ejpam-5597	166	9	,	,	PUNCT
ejpam-5597	166	10	11	11	NUM
ejpam-5597	166	11	pp	pp	NOUN
ejpam-5597	166	12	.	.	PUNCT
ejpam-5597	166	13	,	,	PUNCT
ejpam-5597	166	14	2022	2022	NUM
ejpam-5597	166	15	.	.	PUNCT
ejpam-5597	167	1	[	[	X
ejpam-5597	167	2	6	6	NUM
ejpam-5597	167	3	]	]	PUNCT
ejpam-5597	167	4	a.	a.	NOUN
ejpam-5597	167	5	al	al	PROPN
ejpam-5597	167	6	-	-	PUNCT
ejpam-5597	167	7	natoor	natoor	PROPN
ejpam-5597	167	8	and	and	CCONJ
ejpam-5597	167	9	f.	f.	PROPN
ejpam-5597	167	10	kittaneh	kittaneh	PROPN
ejpam-5597	167	11	.	.	PUNCT
ejpam-5597	168	1	singular	singular	PROPN
ejpam-5597	168	2	value	value	NOUN
ejpam-5597	168	3	and	and	CCONJ
ejpam-5597	168	4	norm	norm	NOUN
ejpam-5597	168	5	inequalities	inequality	NOUN
ejpam-5597	168	6	for	for	ADP
ejpam-5597	168	7	positive	positive	ADJ
ejpam-5597	168	8	semidefinite	semidefinite	NOUN
ejpam-5597	168	9	matrices	matrix	NOUN
ejpam-5597	168	10	.	.	PUNCT
ejpam-5597	169	1	linear	linear	PROPN
ejpam-5597	169	2	multilinear	multilinear	PROPN
ejpam-5597	169	3	algebra	algebra	PROPN
ejpam-5597	169	4	,	,	PUNCT
ejpam-5597	169	5	70:4498–4507	70:4498–4507	NOUN
ejpam-5597	169	6	,	,	PUNCT
ejpam-5597	169	7	2022	2022	NUM
ejpam-5597	169	8	.	.	PUNCT
ejpam-5597	170	1	[	[	X
ejpam-5597	170	2	7	7	X
ejpam-5597	170	3	]	]	X
ejpam-5597	170	4	m.w	m.w	PROPN
ejpam-5597	170	5	.	.	PROPN
ejpam-5597	170	6	alomari	alomari	PROPN
ejpam-5597	170	7	.	.	PUNCT
ejpam-5597	171	1	numerical	numerical	PROPN
ejpam-5597	171	2	radius	radius	PROPN
ejpam-5597	171	3	inequalities	inequality	NOUN
ejpam-5597	171	4	for	for	ADP
ejpam-5597	171	5	hilbert	hilbert	NOUN
ejpam-5597	171	6	space	space	NOUN
ejpam-5597	171	7	operators	operator	NOUN
ejpam-5597	171	8	.	.	PUNCT
ejpam-5597	172	1	complex	complex	ADJ
ejpam-5597	172	2	anal	anal	PROPN
ejpam-5597	172	3	.	.	PUNCT
ejpam-5597	173	1	oper	oper	PROPN
ejpam-5597	173	2	.	.	PROPN
ejpam-5597	173	3	theory	theory	NOUN
ejpam-5597	173	4	,	,	PUNCT
ejpam-5597	173	5	15:1–19	15:1–19	NUM
ejpam-5597	173	6	,	,	PUNCT
ejpam-5597	173	7	2021	2021	NUM
ejpam-5597	173	8	.	.	PUNCT
ejpam-5597	174	1	[	[	X
ejpam-5597	174	2	8	8	NUM
ejpam-5597	174	3	]	]	X
ejpam-5597	174	4	f.	f.	PROPN
ejpam-5597	174	5	alrimawi	alrimawi	PROPN
ejpam-5597	174	6	.	.	PUNCT
ejpam-5597	175	1	some	some	DET
ejpam-5597	175	2	inner	inner	ADJ
ejpam-5597	175	3	product	product	NOUN
ejpam-5597	175	4	inequalities	inequality	NOUN
ejpam-5597	175	5	with	with	ADP
ejpam-5597	175	6	applications	application	NOUN
ejpam-5597	175	7	to	to	ADP
ejpam-5597	175	8	numerical	numerical	ADJ
ejpam-5597	175	9	radius	radius	PROPN
ejpam-5597	175	10	inequalities	inequality	NOUN
ejpam-5597	175	11	.	.	PUNCT
ejpam-5597	176	1	j.	j.	PROPN
ejpam-5597	176	2	math	math	PROPN
ejpam-5597	176	3	.	.	PUNCT
ejpam-5597	177	1	computer	computer	NOUN
ejpam-5597	177	2	sci	sci	PROPN
ejpam-5597	177	3	.	.	PROPN
ejpam-5597	177	4	,	,	PUNCT
ejpam-5597	177	5	38:16–24	38:16–24	PROPN
ejpam-5597	177	6	,	,	PUNCT
ejpam-5597	177	7	2025	2025	NUM
ejpam-5597	177	8	.	.	PUNCT
ejpam-5597	178	1	[	[	X
ejpam-5597	178	2	9	9	NUM
ejpam-5597	178	3	]	]	X
ejpam-5597	178	4	f.	f.	PROPN
ejpam-5597	178	5	alrimawi	alrimawi	PROPN
ejpam-5597	178	6	,	,	PUNCT
ejpam-5597	178	7	f.a	f.a	PROPN
ejpam-5597	178	8	.	.	PROPN
ejpam-5597	178	9	abushaheen	abushaheen	PROPN
ejpam-5597	178	10	,	,	PUNCT
ejpam-5597	178	11	and	and	CCONJ
ejpam-5597	178	12	r.	r.	PROPN
ejpam-5597	178	13	alkhateeb	alkhateeb	PROPN
ejpam-5597	178	14	.	.	PUNCT
ejpam-5597	179	1	improved	improve	VERB
ejpam-5597	179	2	lower	low	ADJ
ejpam-5597	179	3	bounds	bound	NOUN
ejpam-5597	179	4	for	for	ADP
ejpam-5597	179	5	numerical	numerical	ADJ
ejpam-5597	179	6	radius	radius	NOUN
ejpam-5597	179	7	via	via	ADP
ejpam-5597	179	8	cartesian	cartesian	ADJ
ejpam-5597	179	9	decomposition	decomposition	NOUN
ejpam-5597	179	10	.	.	PUNCT
ejpam-5597	180	1	j.	j.	PROPN
ejpam-5597	180	2	math	math	PROPN
ejpam-5597	180	3	.	.	PUNCT
ejpam-5597	181	1	computer	computer	NOUN
ejpam-5597	181	2	sci	sci	PROPN
ejpam-5597	181	3	.	.	PROPN
ejpam-5597	181	4	,	,	PUNCT
ejpam-5597	181	5	33:169–175	33:169–175	PROPN
ejpam-5597	181	6	,	,	PUNCT
ejpam-5597	181	7	2024	2024	NUM
ejpam-5597	181	8	.	.	PUNCT
ejpam-5597	182	1	[	[	X
ejpam-5597	182	2	10	10	NUM
ejpam-5597	182	3	]	]	X
ejpam-5597	182	4	f.	f.	PROPN
ejpam-5597	182	5	alrimawi	alrimawi	PROPN
ejpam-5597	182	6	,	,	PUNCT
ejpam-5597	182	7	o.	o.	PROPN
ejpam-5597	182	8	hirzallah	hirzallah	PROPN
ejpam-5597	182	9	,	,	PUNCT
ejpam-5597	182	10	and	and	CCONJ
ejpam-5597	182	11	f.	f.	PROPN
ejpam-5597	182	12	kittaneh	kittaneh	PROPN
ejpam-5597	182	13	.	.	PUNCT
ejpam-5597	183	1	norm	norm	NOUN
ejpam-5597	183	2	inequalities	inequality	NOUN
ejpam-5597	183	3	involving	involve	VERB
ejpam-5597	183	4	the	the	DET
ejpam-5597	183	5	weighted	weight	VERB
ejpam-5597	183	6	numerical	numerical	ADJ
ejpam-5597	183	7	radii	radius	NOUN
ejpam-5597	183	8	of	of	ADP
ejpam-5597	183	9	operators	operator	NOUN
ejpam-5597	183	10	.	.	PUNCT
ejpam-5597	184	1	linear	linear	ADJ
ejpam-5597	184	2	algebra	algebra	PROPN
ejpam-5597	184	3	appl	appl	NOUN
ejpam-5597	184	4	.	.	PROPN
ejpam-5597	184	5	,	,	PUNCT
ejpam-5597	184	6	657:127–146	657:127–146	NUM
ejpam-5597	184	7	,	,	PUNCT
ejpam-5597	184	8	2023	2023	NUM
ejpam-5597	184	9	.	.	PUNCT
ejpam-5597	185	1	[	[	X
ejpam-5597	185	2	11	11	NUM
ejpam-5597	185	3	]	]	PUNCT
ejpam-5597	185	4	f.	f.	PROPN
ejpam-5597	185	5	alrimawi	alrimawi	PROPN
ejpam-5597	185	6	and	and	CCONJ
ejpam-5597	185	7	h.	h.	PROPN
ejpam-5597	185	8	kawariq	kawariq	PROPN
ejpam-5597	185	9	.	.	PUNCT
ejpam-5597	186	1	on	on	ADP
ejpam-5597	186	2	some	some	DET
ejpam-5597	186	3	generalized	generalize	VERB
ejpam-5597	186	4	numerical	numerical	ADJ
ejpam-5597	186	5	radius	radius	PROPN
ejpam-5597	186	6	inequalities	inequality	NOUN
ejpam-5597	186	7	for	for	ADP
ejpam-5597	186	8	hilbert	hilbert	NOUN
ejpam-5597	186	9	space	space	NOUN
ejpam-5597	186	10	operators	operator	NOUN
ejpam-5597	186	11	.	.	PUNCT
ejpam-5597	187	1	j.	j.	PROPN
ejpam-5597	187	2	math	math	PROPN
ejpam-5597	187	3	.	.	PUNCT
ejpam-5597	188	1	computer	computer	NOUN
ejpam-5597	188	2	sci	sci	PROPN
ejpam-5597	188	3	.	.	PROPN
ejpam-5597	188	4	,	,	PUNCT
ejpam-5597	188	5	32:257–262	32:257–262	NUM
ejpam-5597	188	6	,	,	PUNCT
ejpam-5597	188	7	2024	2024	NUM
ejpam-5597	188	8	.	.	PUNCT
ejpam-5597	189	1	[	[	X
ejpam-5597	189	2	12	12	NUM
ejpam-5597	189	3	]	]	X
ejpam-5597	189	4	f.	f.	PROPN
ejpam-5597	189	5	alrimawi	alrimawi	PROPN
ejpam-5597	189	6	,	,	PUNCT
ejpam-5597	189	7	h.	h.	PROPN
ejpam-5597	189	8	kawariq	kawariq	PROPN
ejpam-5597	189	9	,	,	PUNCT
ejpam-5597	189	10	and	and	CCONJ
ejpam-5597	189	11	f.a	f.a	PROPN
ejpam-5597	189	12	.	.	PROPN
ejpam-5597	189	13	abushaheen	abushaheen	PROPN
ejpam-5597	189	14	.	.	PUNCT
ejpam-5597	190	1	generalized	generalize	VERB
ejpam-5597	190	2	-	-	PUNCT
ejpam-5597	190	3	weighted	weight	VERB
ejpam-5597	190	4	numerical	numerical	ADJ
ejpam-5597	190	5	radius	radius	PROPN
ejpam-5597	190	6	inequalities	inequality	NOUN
ejpam-5597	190	7	for	for	ADP
ejpam-5597	190	8	schatten	schatten	ADJ
ejpam-5597	190	9	p	p	NOUN
ejpam-5597	190	10	-	-	PUNCT
ejpam-5597	190	11	norms	norm	NOUN
ejpam-5597	190	12	.	.	PUNCT
ejpam-5597	191	1	international	international	ADJ
ejpam-5597	191	2	journal	journal	NOUN
ejpam-5597	191	3	of	of	ADP
ejpam-5597	191	4	mathematics	mathematic	NOUN
ejpam-5597	191	5	and	and	CCONJ
ejpam-5597	191	6	computer	computer	NOUN
ejpam-5597	191	7	science	science	NOUN
ejpam-5597	191	8	,	,	PUNCT
ejpam-5597	191	9	17:1463–1473	17:1463–1473	NUM
ejpam-5597	191	10	,	,	PUNCT
ejpam-5597	191	11	2022	2022	NUM
ejpam-5597	191	12	.	.	PUNCT
ejpam-5597	192	1	a.	a.	PROPN
ejpam-5597	192	2	al	al	PROPN
ejpam-5597	192	3	-	-	PUNCT
ejpam-5597	192	4	natoor	natoor	NOUN
ejpam-5597	192	5	,	,	PUNCT
ejpam-5597	192	6	f.	f.	PROPN
ejpam-5597	192	7	alrimawi	alrimawi	PROPN
ejpam-5597	192	8	/	/	SYM
ejpam-5597	192	9	eur	eur	PROPN
ejpam-5597	192	10	.	.	PUNCT
ejpam-5597	193	1	j.	j.	PROPN
ejpam-5597	193	2	pure	pure	PROPN
ejpam-5597	193	3	appl	appl	PROPN
ejpam-5597	193	4	.	.	PROPN
ejpam-5597	193	5	math	math	PROPN
ejpam-5597	193	6	,	,	PUNCT
ejpam-5597	193	7	18	18	NUM
ejpam-5597	193	8	(	(	PUNCT
ejpam-5597	193	9	1	1	NUM
ejpam-5597	193	10	)	)	PUNCT
ejpam-5597	193	11	(	(	PUNCT
ejpam-5597	193	12	2025	2025	NUM
ejpam-5597	193	13	)	)	PUNCT
ejpam-5597	193	14	,	,	PUNCT
ejpam-5597	193	15	5597	5597	NUM
ejpam-5597	193	16	9	9	NUM
ejpam-5597	193	17	of	of	ADP
ejpam-5597	193	18	9	9	NUM
ejpam-5597	193	19	[	[	SYM
ejpam-5597	193	20	13	13	NUM
ejpam-5597	193	21	]	]	PUNCT
ejpam-5597	193	22	a.	a.	NOUN
ejpam-5597	193	23	benmakhlouf	benmakhlouf	PROPN
ejpam-5597	193	24	,	,	PUNCT
ejpam-5597	193	25	o.	o.	PROPN
ejpam-5597	193	26	hirzallah	hirzallah	PROPN
ejpam-5597	193	27	,	,	PUNCT
ejpam-5597	193	28	and	and	CCONJ
ejpam-5597	193	29	f.	f.	PROPN
ejpam-5597	193	30	kittaneh	kittaneh	PROPN
ejpam-5597	193	31	.	.	PUNCT
ejpam-5597	194	1	on	on	ADP
ejpam-5597	194	2	p	p	ADJ
ejpam-5597	194	3	-	-	PUNCT
ejpam-5597	194	4	numerical	numerical	ADJ
ejpam-5597	194	5	radii	radius	NOUN
ejpam-5597	194	6	of	of	ADP
ejpam-5597	194	7	hilbert	hilbert	NOUN
ejpam-5597	194	8	space	space	NOUN
ejpam-5597	194	9	operators	operator	NOUN
ejpam-5597	194	10	.	.	PUNCT
ejpam-5597	195	1	linear	linear	PROPN
ejpam-5597	195	2	multilinear	multilinear	PROPN
ejpam-5597	195	3	algebra	algebra	PROPN
ejpam-5597	195	4	,	,	PUNCT
ejpam-5597	195	5	69:2813–2829	69:2813–2829	NUM
ejpam-5597	195	6	,	,	PUNCT
ejpam-5597	195	7	2021	2021	NUM
ejpam-5597	195	8	.	.	PUNCT
ejpam-5597	196	1	[	[	X
ejpam-5597	196	2	14	14	NUM
ejpam-5597	196	3	]	]	X
ejpam-5597	196	4	p.	p.	NOUN
ejpam-5597	196	5	bhunia	bhunia	NOUN
ejpam-5597	196	6	and	and	CCONJ
ejpam-5597	196	7	k.	k.	PROPN
ejpam-5597	196	8	paul	paul	PROPN
ejpam-5597	196	9	.	.	PUNCT
ejpam-5597	197	1	some	some	DET
ejpam-5597	197	2	improvements	improvement	NOUN
ejpam-5597	197	3	of	of	ADP
ejpam-5597	197	4	numerical	numerical	ADJ
ejpam-5597	197	5	radius	radius	PROPN
ejpam-5597	197	6	inequalities	inequality	NOUN
ejpam-5597	197	7	of	of	ADP
ejpam-5597	197	8	operators	operator	NOUN
ejpam-5597	197	9	and	and	CCONJ
ejpam-5597	197	10	operator	operator	NOUN
ejpam-5597	197	11	matrices	matrix	NOUN
ejpam-5597	197	12	.	.	PUNCT
ejpam-5597	198	1	linear	linear	PROPN
ejpam-5597	198	2	multilinear	multilinear	PROPN
ejpam-5597	198	3	algebra	algebra	PROPN
ejpam-5597	198	4	,	,	PUNCT
ejpam-5597	198	5	70:1995–2013	70:1995–2013	NUM
ejpam-5597	198	6	,	,	PUNCT
ejpam-5597	198	7	2022	2022	NUM
ejpam-5597	198	8	.	.	PUNCT
ejpam-5597	199	1	[	[	X
ejpam-5597	199	2	15	15	NUM
ejpam-5597	199	3	]	]	X
ejpam-5597	199	4	t.	t.	NOUN
ejpam-5597	199	5	bottazzi	bottazzi	PROPN
ejpam-5597	199	6	and	and	CCONJ
ejpam-5597	199	7	c.	c.	PROPN
ejpam-5597	199	8	conde	conde	PROPN
ejpam-5597	199	9	.	.	PUNCT
ejpam-5597	200	1	generalized	generalize	VERB
ejpam-5597	200	2	numerical	numerical	ADJ
ejpam-5597	200	3	radius	radius	PROPN
ejpam-5597	200	4	and	and	CCONJ
ejpam-5597	200	5	related	related	ADJ
ejpam-5597	200	6	inequalities	inequality	NOUN
ejpam-5597	200	7	.	.	PUNCT
ejpam-5597	201	1	oper	oper	PROPN
ejpam-5597	201	2	.	.	PUNCT
ejpam-5597	201	3	matrices	matrix	NOUN
ejpam-5597	201	4	,	,	PUNCT
ejpam-5597	201	5	15:1289–1308	15:1289–1308	NUM
ejpam-5597	201	6	,	,	PUNCT
ejpam-5597	201	7	2021	2021	NUM
ejpam-5597	201	8	.	.	PUNCT
ejpam-5597	202	1	[	[	X
ejpam-5597	202	2	16	16	NUM
ejpam-5597	202	3	]	]	X
ejpam-5597	202	4	j.c	j.c	PROPN
ejpam-5597	202	5	.	.	PROPN
ejpam-5597	202	6	bourin	bourin	PROPN
ejpam-5597	202	7	.	.	PUNCT
ejpam-5597	203	1	a	a	DET
ejpam-5597	203	2	matrix	matrix	NOUN
ejpam-5597	203	3	subadditivity	subadditivity	NOUN
ejpam-5597	203	4	inequality	inequality	NOUN
ejpam-5597	203	5	for	for	ADP
ejpam-5597	203	6	symmetric	symmetric	ADJ
ejpam-5597	203	7	norms	norm	NOUN
ejpam-5597	203	8	.	.	PUNCT
ejpam-5597	204	1	proc	proc	NOUN
ejpam-5597	204	2	.	.	PUNCT
ejpam-5597	205	1	amer	amer	PROPN
ejpam-5597	205	2	.	.	PUNCT
ejpam-5597	205	3	math	math	PROPN
ejpam-5597	205	4	.	.	PUNCT
ejpam-5597	206	1	soc	soc	PROPN
ejpam-5597	206	2	.	.	PUNCT
ejpam-5597	206	3	,	,	PUNCT
ejpam-5597	206	4	138:495–504	138:495–504	NUM
ejpam-5597	206	5	,	,	PUNCT
ejpam-5597	206	6	2009	2009	NUM
ejpam-5597	206	7	.	.	PUNCT
ejpam-5597	207	1	[	[	X
ejpam-5597	207	2	17	17	NUM
ejpam-5597	207	3	]	]	X
ejpam-5597	207	4	o.	o.	PROPN
ejpam-5597	207	5	hirzallah	hirzallah	PROPN
ejpam-5597	207	6	,	,	PUNCT
ejpam-5597	207	7	f.	f.	PROPN
ejpam-5597	207	8	kittaneh	kittaneh	PROPN
ejpam-5597	207	9	,	,	PUNCT
ejpam-5597	207	10	and	and	CCONJ
ejpam-5597	207	11	k.	k.	PROPN
ejpam-5597	207	12	shebrawi	shebrawi	PROPN
ejpam-5597	207	13	.	.	PUNCT
ejpam-5597	208	1	numerical	numerical	PROPN
ejpam-5597	208	2	radius	radius	PROPN
ejpam-5597	208	3	inequalities	inequality	NOUN
ejpam-5597	208	4	for	for	ADP
ejpam-5597	208	5	certain	certain	ADJ
ejpam-5597	208	6	2×	2×	NUM
ejpam-5597	208	7	2	2	NUM
ejpam-5597	208	8	operator	operator	NOUN
ejpam-5597	208	9	matrices	matrix	NOUN
ejpam-5597	208	10	.	.	PUNCT
ejpam-5597	209	1	integr	integr	PROPN
ejpam-5597	209	2	.	.	PUNCT
ejpam-5597	210	1	equ	equ	PROPN
ejpam-5597	210	2	.	.	PUNCT
ejpam-5597	210	3	oper	oper	PROPN
ejpam-5597	210	4	.	.	PROPN
ejpam-5597	210	5	theory	theory	NOUN
ejpam-5597	210	6	,	,	PUNCT
ejpam-5597	210	7	71:129–147	71:129–147	PROPN
ejpam-5597	210	8	,	,	PUNCT
ejpam-5597	210	9	2011	2011	NUM
ejpam-5597	210	10	.	.	PUNCT
ejpam-5597	211	1	[	[	X
ejpam-5597	211	2	18	18	NUM
ejpam-5597	211	3	]	]	X
ejpam-5597	211	4	j.c	j.c	PROPN
ejpam-5597	211	5	.	.	PROPN
ejpam-5597	211	6	hou	hou	PROPN
ejpam-5597	211	7	and	and	CCONJ
ejpam-5597	211	8	h.k	h.k	PROPN
ejpam-5597	211	9	.	.	PROPN
ejpam-5597	211	10	du	du	PROPN
ejpam-5597	211	11	.	.	PROPN
ejpam-5597	211	12	norm	norm	PROPN
ejpam-5597	211	13	inequalities	inequality	NOUN
ejpam-5597	211	14	of	of	ADP
ejpam-5597	211	15	positive	positive	ADJ
ejpam-5597	211	16	operator	operator	NOUN
ejpam-5597	211	17	matrices	matrix	NOUN
ejpam-5597	211	18	.	.	PUNCT
ejpam-5597	212	1	integr	integr	PROPN
ejpam-5597	212	2	.	.	PUNCT
ejpam-5597	213	1	equ	equ	PROPN
ejpam-5597	213	2	.	.	PUNCT
ejpam-5597	213	3	oper	oper	PROPN
ejpam-5597	213	4	.	.	PROPN
ejpam-5597	213	5	theory	theory	NOUN
ejpam-5597	213	6	,	,	PUNCT
ejpam-5597	213	7	22:281–294	22:281–294	NUM
ejpam-5597	213	8	,	,	PUNCT
ejpam-5597	213	9	1995	1995	NUM
ejpam-5597	213	10	.	.	PUNCT
ejpam-5597	214	1	[	[	X
ejpam-5597	214	2	19	19	NUM
ejpam-5597	214	3	]	]	X
ejpam-5597	214	4	f.	f.	PROPN
ejpam-5597	214	5	kittaneh	kittaneh	PROPN
ejpam-5597	214	6	.	.	PUNCT
ejpam-5597	215	1	a	a	DET
ejpam-5597	215	2	numerical	numerical	ADJ
ejpam-5597	215	3	radius	radius	NOUN
ejpam-5597	215	4	inequality	inequality	NOUN
ejpam-5597	215	5	and	and	CCONJ
ejpam-5597	215	6	an	an	DET
ejpam-5597	215	7	estimate	estimate	NOUN
ejpam-5597	215	8	for	for	ADP
ejpam-5597	215	9	the	the	DET
ejpam-5597	215	10	numerical	numerical	ADJ
ejpam-5597	215	11	radius	radius	NOUN
ejpam-5597	215	12	of	of	ADP
ejpam-5597	215	13	the	the	DET
ejpam-5597	215	14	frobenius	frobenius	ADJ
ejpam-5597	215	15	companion	companion	NOUN
ejpam-5597	215	16	matrix	matrix	NOUN
ejpam-5597	215	17	.	.	PUNCT
ejpam-5597	216	1	studia	studia	PROPN
ejpam-5597	216	2	math	math	PROPN
ejpam-5597	216	3	.	.	PUNCT
ejpam-5597	216	4	,	,	PUNCT
ejpam-5597	217	1	158:11–17	158:11–17	NUM
ejpam-5597	217	2	,	,	PUNCT
ejpam-5597	217	3	2003	2003	NUM
ejpam-5597	217	4	.	.	PUNCT
ejpam-5597	218	1	[	[	X
ejpam-5597	218	2	20	20	NUM
ejpam-5597	218	3	]	]	PUNCT
ejpam-5597	218	4	m.	m.	NOUN
ejpam-5597	218	5	sattari	sattari	PROPN
ejpam-5597	218	6	,	,	PUNCT
ejpam-5597	218	7	m.s	m.s	PROPN
ejpam-5597	218	8	.	.	PROPN
ejpam-5597	218	9	moslehian	moslehian	PROPN
ejpam-5597	218	10	,	,	PUNCT
ejpam-5597	218	11	and	and	CCONJ
ejpam-5597	218	12	t.	t.	PROPN
ejpam-5597	218	13	yamazaki	yamazaki	PROPN
ejpam-5597	218	14	.	.	PUNCT
ejpam-5597	219	1	some	some	DET
ejpam-5597	219	2	generalized	generalized	ADJ
ejpam-5597	219	3	numerical	numerical	ADJ
ejpam-5597	219	4	radius	radius	PROPN
ejpam-5597	219	5	inequalities	inequality	NOUN
ejpam-5597	219	6	for	for	ADP
ejpam-5597	219	7	hilbert	hilbert	NOUN
ejpam-5597	219	8	space	space	NOUN
ejpam-5597	219	9	operators	operator	NOUN
ejpam-5597	219	10	.	.	PUNCT
ejpam-5597	220	1	linear	linear	ADJ
ejpam-5597	220	2	algebra	algebra	PROPN
ejpam-5597	220	3	appl	appl	NOUN
ejpam-5597	220	4	.	.	PROPN
ejpam-5597	220	5	,	,	PUNCT
ejpam-5597	220	6	470:216–227	470:216–227	PROPN
ejpam-5597	220	7	,	,	PUNCT
ejpam-5597	220	8	2015	2015	NUM
ejpam-5597	220	9	.	.	PUNCT
ejpam-5597	221	1	[	[	X
ejpam-5597	221	2	21	21	NUM
ejpam-5597	221	3	]	]	X
ejpam-5597	221	4	v.	v.	CCONJ
ejpam-5597	221	5	stojiljkovic	stojiljkovic	ADJ
ejpam-5597	221	6	.	.	PUNCT
ejpam-5597	222	1	hermite	hermite	ADJ
ejpam-5597	222	2	hadamard	hadamard	ADJ
ejpam-5597	222	3	type	type	NOUN
ejpam-5597	222	4	inequalities	inequality	NOUN
ejpam-5597	222	5	involving	involve	VERB
ejpam-5597	222	6	(	(	PUNCT
ejpam-5597	222	7	k	k	X
ejpam-5597	222	8	-	-	ADJ
ejpam-5597	222	9	p	p	ADJ
ejpam-5597	222	10	)	)	PUNCT
ejpam-5597	222	11	fractional	fractional	ADJ
ejpam-5597	222	12	operator	operator	NOUN
ejpam-5597	222	13	with	with	ADP
ejpam-5597	222	14	(	(	PUNCT
ejpam-5597	222	15	α	α	X
ejpam-5597	222	16	,	,	PUNCT
ejpam-5597	222	17	h	h	PROPN
ejpam-5597	222	18	m	m	NOUN
ejpam-5597	222	19	)	)	PUNCT
ejpam-5597	223	1	p	p	DET
ejpam-5597	223	2	convexity	convexity	NOUN
ejpam-5597	223	3	.	.	PUNCT
ejpam-5597	224	1	european	european	PROPN
ejpam-5597	224	2	journal	journal	PROPN
ejpam-5597	224	3	of	of	ADP
ejpam-5597	224	4	pure	pure	ADJ
ejpam-5597	224	5	and	and	CCONJ
ejpam-5597	224	6	applied	applied	ADJ
ejpam-5597	224	7	mathematics	mathematic	NOUN
ejpam-5597	224	8	,	,	PUNCT
ejpam-5597	224	9	16:503–522	16:503–522	PROPN
ejpam-5597	224	10	,	,	PUNCT
ejpam-5597	224	11	2023	2023	NUM
ejpam-5597	224	12	.	.	PUNCT
ejpam-5597	225	1	[	[	X
ejpam-5597	225	2	22	22	NUM
ejpam-5597	225	3	]	]	PUNCT
ejpam-5597	225	4	t.	t.	PROPN
ejpam-5597	225	5	yamazaki	yamazaki	PROPN
ejpam-5597	225	6	.	.	PUNCT
ejpam-5597	226	1	on	on	ADP
ejpam-5597	226	2	upper	upper	ADJ
ejpam-5597	226	3	and	and	CCONJ
ejpam-5597	226	4	lower	low	ADJ
ejpam-5597	226	5	bounds	bound	NOUN
ejpam-5597	226	6	of	of	ADP
ejpam-5597	226	7	the	the	DET
ejpam-5597	226	8	numerical	numerical	ADJ
ejpam-5597	226	9	radius	radius	NOUN
ejpam-5597	226	10	and	and	CCONJ
ejpam-5597	226	11	an	an	DET
ejpam-5597	226	12	equality	equality	NOUN
ejpam-5597	226	13	condition	condition	NOUN
ejpam-5597	226	14	.	.	PUNCT
ejpam-5597	227	1	studia	studia	PROPN
ejpam-5597	227	2	math	math	PROPN
ejpam-5597	227	3	.	.	PUNCT
ejpam-5597	227	4	,	,	PUNCT
ejpam-5597	227	5	178:83–89	178:83–89	NUM
ejpam-5597	227	6	,	,	PUNCT
ejpam-5597	227	7	2007	2007	NUM
ejpam-5597	227	8	.	.	PUNCT