id	sid	tid	token	lemma	pos
ejpam-5693	1	1	european	european	PROPN
ejpam-5693	1	2	journal	journal	PROPN
ejpam-5693	1	3	of	of	ADP
ejpam-5693	1	4	pure	pure	ADJ
ejpam-5693	1	5	and	and	CCONJ
ejpam-5693	1	6	applied	applied	ADJ
ejpam-5693	1	7	mathematics	mathematic	NOUN
ejpam-5693	1	8	2025	2025	NUM
ejpam-5693	1	9	,	,	PUNCT
ejpam-5693	1	10	vol	vol	NOUN
ejpam-5693	1	11	.	.	PROPN
ejpam-5693	1	12	18	18	NUM
ejpam-5693	1	13	,	,	PUNCT
ejpam-5693	1	14	issue	issue	NOUN
ejpam-5693	1	15	1	1	NUM
ejpam-5693	1	16	,	,	PUNCT
ejpam-5693	1	17	article	article	NOUN
ejpam-5693	1	18	number	number	NOUN
ejpam-5693	1	19	5693	5693	NUM
ejpam-5693	1	20	issn	issn	PROPN
ejpam-5693	1	21	1307	1307	NUM
ejpam-5693	1	22	-	-	SYM
ejpam-5693	1	23	5543	5543	NUM
ejpam-5693	1	24	–	–	PUNCT
ejpam-5693	1	25	ejpam.com	ejpam.com	X
ejpam-5693	1	26	published	publish	VERB
ejpam-5693	1	27	by	by	ADP
ejpam-5693	1	28	new	new	PROPN
ejpam-5693	1	29	york	york	PROPN
ejpam-5693	1	30	business	business	PROPN
ejpam-5693	1	31	global	global	ADJ
ejpam-5693	1	32	refinements	refinement	NOUN
ejpam-5693	1	33	of	of	ADP
ejpam-5693	1	34	reverse	reverse	ADJ
ejpam-5693	1	35	young	young	ADJ
ejpam-5693	1	36	inequality	inequality	NOUN
ejpam-5693	1	37	for	for	ADP
ejpam-5693	1	38	scalars	scalar	NOUN
ejpam-5693	1	39	and	and	CCONJ
ejpam-5693	1	40	matrices	matrix	NOUN
ejpam-5693	1	41	ola	ola	PROPN
ejpam-5693	1	42	ramadan1	ramadan1	PROPN
ejpam-5693	1	43	,	,	PUNCT
ejpam-5693	1	44	aliaa	aliaa	PROPN
ejpam-5693	1	45	burqan1,∗	burqan1,∗	NOUN
ejpam-5693	1	46	1	1	NUM
ejpam-5693	1	47	department	department	PROPN
ejpam-5693	1	48	of	of	ADP
ejpam-5693	1	49	mathematics	mathematic	NOUN
ejpam-5693	1	50	,	,	PUNCT
ejpam-5693	1	51	faculty	faculty	NOUN
ejpam-5693	1	52	of	of	ADP
ejpam-5693	1	53	science	science	NOUN
ejpam-5693	1	54	,	,	PUNCT
ejpam-5693	1	55	zarqa	zarqa	PROPN
ejpam-5693	1	56	university	university	PROPN
ejpam-5693	1	57	,	,	PUNCT
ejpam-5693	1	58	zarqa	zarqa	PROPN
ejpam-5693	1	59	13110	13110	NUM
ejpam-5693	1	60	,	,	PUNCT
ejpam-5693	1	61	jordan	jordan	PROPN
ejpam-5693	1	62	abstract	abstract	PROPN
ejpam-5693	1	63	.	.	PUNCT
ejpam-5693	2	1	in	in	ADP
ejpam-5693	2	2	this	this	DET
ejpam-5693	2	3	article	article	NOUN
ejpam-5693	2	4	,	,	PUNCT
ejpam-5693	2	5	we	we	PRON
ejpam-5693	2	6	introduce	introduce	VERB
ejpam-5693	2	7	some	some	DET
ejpam-5693	2	8	refinements	refinement	NOUN
ejpam-5693	2	9	of	of	ADP
ejpam-5693	2	10	the	the	DET
ejpam-5693	2	11	reverse	reverse	ADJ
ejpam-5693	2	12	young	young	ADJ
ejpam-5693	2	13	inequality	inequality	NOUN
ejpam-5693	2	14	for	for	ADP
ejpam-5693	2	15	scalars	scalar	NOUN
ejpam-5693	2	16	.	.	PUNCT
ejpam-5693	3	1	as	as	ADP
ejpam-5693	3	2	applications	application	NOUN
ejpam-5693	3	3	of	of	ADP
ejpam-5693	3	4	our	our	PRON
ejpam-5693	3	5	results	result	NOUN
ejpam-5693	3	6	,	,	PUNCT
ejpam-5693	3	7	we	we	PRON
ejpam-5693	3	8	establish	establish	VERB
ejpam-5693	3	9	corresponding	corresponding	ADJ
ejpam-5693	3	10	inequalities	inequality	NOUN
ejpam-5693	3	11	for	for	ADP
ejpam-5693	3	12	matrices	matrix	NOUN
ejpam-5693	3	13	.	.	PUNCT
ejpam-5693	4	1	the	the	DET
ejpam-5693	4	2	obtained	obtain	VERB
ejpam-5693	4	3	inequalities	inequality	NOUN
ejpam-5693	4	4	in	in	ADP
ejpam-5693	4	5	this	this	DET
ejpam-5693	4	6	article	article	NOUN
ejpam-5693	4	7	can	can	AUX
ejpam-5693	4	8	be	be	AUX
ejpam-5693	4	9	viewed	view	VERB
ejpam-5693	4	10	as	as	ADP
ejpam-5693	4	11	refinements	refinement	NOUN
ejpam-5693	4	12	of	of	ADP
ejpam-5693	4	13	the	the	DET
ejpam-5693	4	14	derived	derive	VERB
ejpam-5693	4	15	inequalities	inequality	NOUN
ejpam-5693	4	16	by	by	ADP
ejpam-5693	4	17	burqan	burqan	NOUN
ejpam-5693	4	18	and	and	CCONJ
ejpam-5693	4	19	khandaqji	khandaqji	ADJ
ejpam-5693	4	20	[	[	X
ejpam-5693	4	21	4	4	NUM
ejpam-5693	4	22	]	]	PUNCT
ejpam-5693	4	23	.	.	PUNCT
ejpam-5693	5	1	2020	2020	NUM
ejpam-5693	5	2	mathematics	mathematic	NOUN
ejpam-5693	5	3	subject	subject	NOUN
ejpam-5693	5	4	classifications	classification	NOUN
ejpam-5693	5	5	:	:	PUNCT
ejpam-5693	5	6	47a63	47a63	NUM
ejpam-5693	5	7	,	,	PUNCT
ejpam-5693	5	8	15a60	15a60	NUM
ejpam-5693	5	9	.	.	PUNCT
ejpam-5693	6	1	key	key	ADJ
ejpam-5693	6	2	words	word	NOUN
ejpam-5693	6	3	and	and	CCONJ
ejpam-5693	6	4	phrases	phrase	NOUN
ejpam-5693	6	5	:	:	PUNCT
ejpam-5693	6	6	young	young	ADJ
ejpam-5693	6	7	inequality	inequality	NOUN
ejpam-5693	6	8	,	,	PUNCT
ejpam-5693	6	9	positive	positive	ADJ
ejpam-5693	6	10	matrices	matrix	NOUN
ejpam-5693	6	11	,	,	PUNCT
ejpam-5693	6	12	unitarily	unitarily	ADV
ejpam-5693	6	13	invariant	invariant	ADJ
ejpam-5693	6	14	norms	norm	NOUN
ejpam-5693	6	15	,	,	PUNCT
ejpam-5693	6	16	singular	singular	ADJ
ejpam-5693	6	17	values	value	NOUN
ejpam-5693	6	18	.	.	PUNCT
ejpam-5693	7	1	1	1	X
ejpam-5693	7	2	.	.	X
ejpam-5693	7	3	introduction	introduction	NOUN
ejpam-5693	7	4	let	let	VERB
ejpam-5693	7	5	mn	mn	PROPN
ejpam-5693	7	6	be	be	AUX
ejpam-5693	7	7	the	the	DET
ejpam-5693	7	8	space	space	NOUN
ejpam-5693	7	9	of	of	ADP
ejpam-5693	7	10	n	n	NUM
ejpam-5693	7	11	×	×	NOUN
ejpam-5693	7	12	n	n	CCONJ
ejpam-5693	7	13	complex	complex	ADJ
ejpam-5693	7	14	matrices	matrix	NOUN
ejpam-5693	7	15	.	.	PUNCT
ejpam-5693	8	1	for	for	ADP
ejpam-5693	8	2	t	t	NOUN
ejpam-5693	8	3	=	=	PUNCT
ejpam-5693	9	1	[	[	X
ejpam-5693	9	2	tij	tij	X
ejpam-5693	9	3	]	]	PUNCT
ejpam-5693	9	4	∈	∈	PROPN
ejpam-5693	9	5	mn	mn	PROPN
ejpam-5693	9	6	,	,	PUNCT
ejpam-5693	9	7	the	the	DET
ejpam-5693	9	8	hilbertschmidt	hilbertschmidt	PROPN
ejpam-5693	9	9	norm	norm	NOUN
ejpam-5693	9	10	,	,	PUNCT
ejpam-5693	9	11	the	the	DET
ejpam-5693	9	12	trace	trace	NOUN
ejpam-5693	9	13	norm	norm	NOUN
ejpam-5693	9	14	,	,	PUNCT
ejpam-5693	9	15	and	and	CCONJ
ejpam-5693	9	16	the	the	DET
ejpam-5693	9	17	spectral	spectral	ADJ
ejpam-5693	9	18	norm	norm	NOUN
ejpam-5693	9	19	of	of	ADP
ejpam-5693	9	20	t	t	PROPN
ejpam-5693	9	21	are	be	AUX
ejpam-5693	9	22	defined	define	VERB
ejpam-5693	9	23	by	by	ADP
ejpam-5693	9	24	∥t∥2	∥t∥2	X
ejpam-5693	9	25	=	=	SYM
ejpam-5693	9	26	(	(	PUNCT
ejpam-5693	9	27	∑n	∑n	PROPN
ejpam-5693	9	28	j=1	j=1	PROPN
ejpam-5693	9	29	s	s	PART
ejpam-5693	9	30	2	2	NUM
ejpam-5693	9	31	j	j	PROPN
ejpam-5693	9	32	(	(	PUNCT
ejpam-5693	9	33	t	t	PROPN
ejpam-5693	9	34	)	)	PUNCT
ejpam-5693	9	35	)	)	PUNCT
ejpam-5693	9	36	1	1	NUM
ejpam-5693	9	37	2	2	NUM
ejpam-5693	9	38	,	,	PUNCT
ejpam-5693	9	39	∥t∥1	∥t∥1	X
ejpam-5693	9	40	=	=	SYM
ejpam-5693	10	1	∑n	∑n	PROPN
ejpam-5693	10	2	j=1	j=1	NOUN
ejpam-5693	10	3	sj(t	sj(t	PRON
ejpam-5693	10	4	)	)	PUNCT
ejpam-5693	10	5	and	and	CCONJ
ejpam-5693	10	6	∥t∥	∥t∥	CCONJ
ejpam-5693	10	7	=	=	SYM
ejpam-5693	10	8	s1(t	s1(t	X
ejpam-5693	10	9	)	)	PUNCT
ejpam-5693	10	10	,	,	PUNCT
ejpam-5693	10	11	respectively	respectively	ADV
ejpam-5693	10	12	,	,	PUNCT
ejpam-5693	10	13	where	where	SCONJ
ejpam-5693	10	14	s1(t	s1(t	X
ejpam-5693	10	15	)	)	PUNCT
ejpam-5693	10	16	≥	≥	NUM
ejpam-5693	10	17	...	...	PUNCT
ejpam-5693	10	18	≥	≥	X
ejpam-5693	10	19	sn(t	sn(t	NUM
ejpam-5693	10	20	)	)	PUNCT
ejpam-5693	10	21	are	be	AUX
ejpam-5693	10	22	the	the	DET
ejpam-5693	10	23	singular	singular	ADJ
ejpam-5693	10	24	values	value	NOUN
ejpam-5693	10	25	of	of	ADP
ejpam-5693	10	26	t	t	PROPN
ejpam-5693	10	27	,	,	PUNCT
ejpam-5693	10	28	that	that	ADV
ejpam-5693	10	29	is	is	ADV
ejpam-5693	10	30	,	,	PUNCT
ejpam-5693	10	31	the	the	DET
ejpam-5693	10	32	eigenvalues	eigenvalue	NOUN
ejpam-5693	10	33	of	of	ADP
ejpam-5693	10	34	the	the	DET
ejpam-5693	10	35	positive	positive	ADJ
ejpam-5693	10	36	semidefinite	semidefinite	NOUN
ejpam-5693	10	37	matrix	matrix	NOUN
ejpam-5693	10	38	|t	|t	VERB
ejpam-5693	11	1	|	|	ADV
ejpam-5693	11	2	=	=	SYM
ejpam-5693	11	3	(	(	PUNCT
ejpam-5693	11	4	t	t	PROPN
ejpam-5693	11	5	∗t	∗t	ADJ
ejpam-5693	11	6	)	)	PUNCT
ejpam-5693	11	7	1	1	NUM
ejpam-5693	11	8	2	2	NUM
ejpam-5693	11	9	,	,	PUNCT
ejpam-5693	11	10	arranged	arrange	VERB
ejpam-5693	11	11	in	in	ADP
ejpam-5693	11	12	decreasing	decrease	VERB
ejpam-5693	11	13	order	order	NOUN
ejpam-5693	11	14	and	and	CCONJ
ejpam-5693	11	15	repeated	repeat	VERB
ejpam-5693	11	16	according	accord	VERB
ejpam-5693	11	17	to	to	ADP
ejpam-5693	11	18	multiplicity	multiplicity	NOUN
ejpam-5693	11	19	.	.	PUNCT
ejpam-5693	12	1	the	the	DET
ejpam-5693	12	2	classical	classical	ADJ
ejpam-5693	12	3	young	young	ADJ
ejpam-5693	12	4	inequality	inequality	NOUN
ejpam-5693	12	5	says	say	VERB
ejpam-5693	12	6	that	that	SCONJ
ejpam-5693	12	7	,	,	PUNCT
ejpam-5693	12	8	if	if	SCONJ
ejpam-5693	12	9	a	a	DET
ejpam-5693	12	10	,	,	PUNCT
ejpam-5693	12	11	b	b	NOUN
ejpam-5693	12	12	≥	≥	NOUN
ejpam-5693	12	13	0	0	NUM
ejpam-5693	12	14	and	and	CCONJ
ejpam-5693	12	15	0	0	NUM
ejpam-5693	12	16	≤	≤	NUM
ejpam-5693	12	17	µ	µ	X
ejpam-5693	12	18	≤	≤	NUM
ejpam-5693	12	19	1	1	NUM
ejpam-5693	12	20	,	,	PUNCT
ejpam-5693	12	21	then	then	ADV
ejpam-5693	12	22	aµb1−µ	aµb1−µ	PUNCT
ejpam-5693	12	23	≤	≤	NUM
ejpam-5693	12	24	µa+	µa+	NOUN
ejpam-5693	12	25	(	(	PUNCT
ejpam-5693	12	26	1−	1−	NUM
ejpam-5693	12	27	µ)b	µ)b	NOUN
ejpam-5693	12	28	,	,	PUNCT
ejpam-5693	12	29	(	(	PUNCT
ejpam-5693	12	30	1.1	1.1	NUM
ejpam-5693	12	31	)	)	PUNCT
ejpam-5693	12	32	with	with	ADP
ejpam-5693	12	33	equality	equality	NOUN
ejpam-5693	12	34	if	if	SCONJ
ejpam-5693	12	35	and	and	CCONJ
ejpam-5693	12	36	only	only	ADV
ejpam-5693	12	37	if	if	SCONJ
ejpam-5693	12	38	a	a	DET
ejpam-5693	12	39	=	=	X
ejpam-5693	12	40	b.	b.	PROPN
ejpam-5693	12	41	kittaneh	kittaneh	PROPN
ejpam-5693	12	42	and	and	CCONJ
ejpam-5693	12	43	manasrah	manasrah	PROPN
ejpam-5693	13	1	[	[	X
ejpam-5693	13	2	5	5	NUM
ejpam-5693	13	3	]	]	PUNCT
ejpam-5693	13	4	obtained	obtain	VERB
ejpam-5693	13	5	a	a	DET
ejpam-5693	13	6	refinement	refinement	NOUN
ejpam-5693	13	7	of	of	ADP
ejpam-5693	13	8	inequality	inequality	NOUN
ejpam-5693	13	9	(	(	PUNCT
ejpam-5693	13	10	1.1	1.1	NUM
ejpam-5693	13	11	)	)	PUNCT
ejpam-5693	13	12	as	as	SCONJ
ejpam-5693	13	13	follows	follow	VERB
ejpam-5693	13	14	aµb1−µ	aµb1−µ	PUNCT
ejpam-5693	13	15	+	+	CCONJ
ejpam-5693	13	16	r0	r0	NOUN
ejpam-5693	13	17	(	(	PUNCT
ejpam-5693	13	18	√	√	NOUN
ejpam-5693	13	19	a−	a−	PROPN
ejpam-5693	13	20	√	√	NUM
ejpam-5693	13	21	b)2	b)2	PROPN
ejpam-5693	13	22	≤	≤	PROPN
ejpam-5693	13	23	µa+	µa+	NOUN
ejpam-5693	13	24	(	(	PUNCT
ejpam-5693	13	25	1−	1−	NUM
ejpam-5693	13	26	µ)b	µ)b	NOUN
ejpam-5693	13	27	,	,	PUNCT
ejpam-5693	13	28	(	(	PUNCT
ejpam-5693	13	29	1.2	1.2	NUM
ejpam-5693	13	30	)	)	PUNCT
ejpam-5693	13	31	where	where	SCONJ
ejpam-5693	13	32	r0	r0	NOUN
ejpam-5693	13	33	=	=	SYM
ejpam-5693	13	34	min{µ	min{µ	PROPN
ejpam-5693	13	35	,	,	PUNCT
ejpam-5693	13	36	1−	1−	NUM
ejpam-5693	13	37	µ	µ	NUM
ejpam-5693	13	38	}	}	PUNCT
ejpam-5693	13	39	.	.	PUNCT
ejpam-5693	14	1	kittaneh	kittaneh	PROPN
ejpam-5693	14	2	and	and	CCONJ
ejpam-5693	14	3	manasrah	manasrah	PROPN
ejpam-5693	15	1	[	[	X
ejpam-5693	15	2	6	6	NUM
ejpam-5693	15	3	]	]	PUNCT
ejpam-5693	15	4	gave	give	VERB
ejpam-5693	15	5	a	a	DET
ejpam-5693	15	6	reverse	reverse	NOUN
ejpam-5693	15	7	of	of	ADP
ejpam-5693	15	8	inequality	inequality	NOUN
ejpam-5693	15	9	(	(	PUNCT
ejpam-5693	15	10	1.2	1.2	NUM
ejpam-5693	15	11	)	)	PUNCT
ejpam-5693	15	12	as	as	SCONJ
ejpam-5693	15	13	follows	follow	VERB
ejpam-5693	15	14	µa+	µa+	VERB
ejpam-5693	15	15	(	(	PUNCT
ejpam-5693	15	16	1−	1−	NUM
ejpam-5693	15	17	µ)b	µ)b	NOUN
ejpam-5693	15	18	≤	≤	NOUN
ejpam-5693	15	19	aµb1−µ	aµb1−µ	PUNCT
ejpam-5693	15	20	+	+	NOUN
ejpam-5693	15	21	r0	r0	NOUN
ejpam-5693	15	22	(	(	PUNCT
ejpam-5693	15	23	√	√	PROPN
ejpam-5693	15	24	a−	a−	PROPN
ejpam-5693	15	25	√	√	NUM
ejpam-5693	15	26	b)2	b)2	PROPN
ejpam-5693	15	27	,	,	PUNCT
ejpam-5693	15	28	(	(	PUNCT
ejpam-5693	15	29	1.3	1.3	NUM
ejpam-5693	15	30	)	)	PUNCT
ejpam-5693	15	31	∗corresponding	∗corresponde	VERB
ejpam-5693	15	32	author	author	NOUN
ejpam-5693	15	33	.	.	PUNCT
ejpam-5693	16	1	doi	doi	NOUN
ejpam-5693	16	2	:	:	PUNCT
ejpam-5693	16	3	https://doi.org/10.29020/nybg.ejpam.v18i1.5693	https://doi.org/10.29020/nybg.ejpam.v18i1.5693	NOUN
ejpam-5693	16	4	email	email	NOUN
ejpam-5693	16	5	addresses	address	NOUN
ejpam-5693	16	6	:	:	PUNCT
ejpam-5693	17	1	ola	ola	PROPN
ejpam-5693	17	2	m	m	VERB
ejpam-5693	17	3	ramadan@yahoo.com	ramadan@yahoo.com	X
ejpam-5693	17	4	(	(	PUNCT
ejpam-5693	17	5	o.	o.	PROPN
ejpam-5693	17	6	ramadan	ramadan	PROPN
ejpam-5693	17	7	)	)	PUNCT
ejpam-5693	17	8	,	,	PUNCT
ejpam-5693	17	9	aliaaburqan@zu.edu.jo	aliaaburqan@zu.edu.jo	PROPN
ejpam-5693	17	10	(	(	PUNCT
ejpam-5693	17	11	a.	a.	NOUN
ejpam-5693	17	12	burqan	burqan	PROPN
ejpam-5693	17	13	)	)	PUNCT
ejpam-5693	17	14	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5693	17	15	1	1	NUM
ejpam-5693	17	16	copyright	copyright	NOUN
ejpam-5693	17	17	:	:	PUNCT
ejpam-5693	17	18	©	©	PROPN
ejpam-5693	17	19	2025	2025	NUM
ejpam-5693	17	20	the	the	DET
ejpam-5693	17	21	author(s	author(s	NOUN
ejpam-5693	17	22	)	)	PUNCT
ejpam-5693	17	23	.	.	PUNCT
ejpam-5693	18	1	(	(	PUNCT
ejpam-5693	18	2	cc	cc	NOUN
ejpam-5693	18	3	by	by	ADP
ejpam-5693	18	4	-	-	PUNCT
ejpam-5693	18	5	nc	nc	PROPN
ejpam-5693	18	6	4.0	4.0	NUM
ejpam-5693	18	7	)	)	PUNCT
ejpam-5693	18	8	o.	o.	PROPN
ejpam-5693	18	9	ramadan	ramadan	PROPN
ejpam-5693	18	10	,	,	PUNCT
ejpam-5693	18	11	a.	a.	PROPN
ejpam-5693	18	12	burqan	burqan	PROPN
ejpam-5693	18	13	/	/	SYM
ejpam-5693	18	14	eur	eur	PROPN
ejpam-5693	18	15	.	.	PUNCT
ejpam-5693	19	1	j.	j.	PROPN
ejpam-5693	19	2	pure	pure	PROPN
ejpam-5693	19	3	appl	appl	PROPN
ejpam-5693	19	4	.	.	PROPN
ejpam-5693	19	5	math	math	PROPN
ejpam-5693	19	6	,	,	PUNCT
ejpam-5693	19	7	18	18	NUM
ejpam-5693	19	8	(	(	PUNCT
ejpam-5693	19	9	1	1	NUM
ejpam-5693	19	10	)	)	PUNCT
ejpam-5693	19	11	(	(	PUNCT
ejpam-5693	19	12	2025	2025	NUM
ejpam-5693	19	13	)	)	PUNCT
ejpam-5693	19	14	,	,	PUNCT
ejpam-5693	19	15	5693	5693	NUM
ejpam-5693	19	16	2	2	NUM
ejpam-5693	19	17	of	of	ADP
ejpam-5693	19	18	8	8	NUM
ejpam-5693	19	19	where	where	SCONJ
ejpam-5693	19	20	r0	r0	NOUN
ejpam-5693	19	21	=	=	PUNCT
ejpam-5693	19	22	max{µ	max{µ	ADV
ejpam-5693	19	23	,	,	PUNCT
ejpam-5693	19	24	1−	1−	NUM
ejpam-5693	19	25	µ	µ	NUM
ejpam-5693	19	26	}	}	PUNCT
ejpam-5693	19	27	.	.	PUNCT
ejpam-5693	20	1	also	also	ADV
ejpam-5693	20	2	,	,	PUNCT
ejpam-5693	20	3	kai	kai	PROPN
ejpam-5693	21	1	[	[	X
ejpam-5693	21	2	9	9	NUM
ejpam-5693	21	3	]	]	PUNCT
ejpam-5693	21	4	gave	give	VERB
ejpam-5693	21	5	a	a	DET
ejpam-5693	21	6	refinement	refinement	NOUN
ejpam-5693	21	7	of	of	ADP
ejpam-5693	21	8	inequality	inequality	NOUN
ejpam-5693	21	9	(	(	PUNCT
ejpam-5693	21	10	1.1	1.1	NUM
ejpam-5693	21	11	)	)	PUNCT
ejpam-5693	21	12	as	as	SCONJ
ejpam-5693	21	13	follows	follow	VERB
ejpam-5693	21	14	if	if	SCONJ
ejpam-5693	21	15	0	0	NUM
ejpam-5693	21	16	≤	≤	NUM
ejpam-5693	21	17	µ	µ	X
ejpam-5693	21	18	≤	≤	NUM
ejpam-5693	21	19	1	1	NUM
ejpam-5693	21	20	2	2	NUM
ejpam-5693	21	21	,	,	PUNCT
ejpam-5693	21	22	then	then	ADV
ejpam-5693	21	23	[	[	PUNCT
ejpam-5693	21	24	(	(	PUNCT
ejpam-5693	21	25	µa)µb1−µ	µa)µb1−µ	NUM
ejpam-5693	21	26	]	]	SYM
ejpam-5693	21	27	2	2	NUM
ejpam-5693	21	28	+	+	NUM
ejpam-5693	21	29	µ2(a−	µ2(a−	DET
ejpam-5693	21	30	b)2	b)2	ADJ
ejpam-5693	21	31	≤	≤	NUM
ejpam-5693	21	32	µ2a2	µ2a2	PUNCT
ejpam-5693	22	1	+	+	CCONJ
ejpam-5693	22	2	(	(	PUNCT
ejpam-5693	22	3	1−	1−	NUM
ejpam-5693	22	4	µ)2b2	µ)2b2	NOUN
ejpam-5693	22	5	.	.	PUNCT
ejpam-5693	23	1	(	(	PUNCT
ejpam-5693	23	2	1.4	1.4	NUM
ejpam-5693	23	3	)	)	PUNCT
ejpam-5693	23	4	if	if	SCONJ
ejpam-5693	23	5	1	1	NUM
ejpam-5693	23	6	2	2	NUM
ejpam-5693	23	7	≤	≤	NOUN
ejpam-5693	23	8	µ	µ	PRON
ejpam-5693	23	9	≤	≤	NUM
ejpam-5693	23	10	1	1	NUM
ejpam-5693	23	11	,	,	PUNCT
ejpam-5693	23	12	then	then	ADV
ejpam-5693	23	13	[	[	PUNCT
ejpam-5693	23	14	aµ((1−	aµ((1−	PROPN
ejpam-5693	23	15	µ)b)1−µ	µ)b)1−µ	NOUN
ejpam-5693	23	16	]	]	X
ejpam-5693	23	17	2	2	NUM
ejpam-5693	23	18	+	+	CCONJ
ejpam-5693	23	19	(	(	PUNCT
ejpam-5693	23	20	1−	1−	NUM
ejpam-5693	23	21	µ)2(a−	µ)2(a−	PRON
ejpam-5693	23	22	b)2	b)2	ADJ
ejpam-5693	23	23	≤	≤	PUNCT
ejpam-5693	23	24	µ2a2	µ2a2	CCONJ
ejpam-5693	23	25	+	+	CCONJ
ejpam-5693	23	26	(	(	PUNCT
ejpam-5693	23	27	1−	1−	NUM
ejpam-5693	23	28	µ)2b2	µ)2b2	NOUN
ejpam-5693	23	29	.	.	PUNCT
ejpam-5693	24	1	(	(	PUNCT
ejpam-5693	24	2	1.5	1.5	NUM
ejpam-5693	24	3	)	)	PUNCT
ejpam-5693	24	4	reverses	reverse	NOUN
ejpam-5693	24	5	of	of	ADP
ejpam-5693	24	6	inequalities	inequality	NOUN
ejpam-5693	24	7	(	(	PUNCT
ejpam-5693	24	8	1.4	1.4	NUM
ejpam-5693	24	9	)	)	PUNCT
ejpam-5693	24	10	,	,	PUNCT
ejpam-5693	24	11	(	(	PUNCT
ejpam-5693	24	12	1.5	1.5	NUM
ejpam-5693	24	13	)	)	PUNCT
ejpam-5693	24	14	were	be	AUX
ejpam-5693	24	15	established	establish	VERB
ejpam-5693	24	16	by	by	ADP
ejpam-5693	24	17	burqan	burqan	NOUN
ejpam-5693	24	18	and	and	CCONJ
ejpam-5693	24	19	khandaqji	khandaqji	ADJ
ejpam-5693	24	20	[	[	X
ejpam-5693	24	21	4	4	NUM
ejpam-5693	24	22	]	]	PUNCT
ejpam-5693	24	23	as	as	SCONJ
ejpam-5693	24	24	follows	follow	VERB
ejpam-5693	24	25	if	if	SCONJ
ejpam-5693	24	26	0	0	NUM
ejpam-5693	24	27	≤	≤	NUM
ejpam-5693	25	1	µ	µ	X
ejpam-5693	25	2	≤	≤	NUM
ejpam-5693	25	3	1	1	NUM
ejpam-5693	25	4	2	2	NUM
ejpam-5693	25	5	,	,	PUNCT
ejpam-5693	25	6	then	then	ADV
ejpam-5693	25	7	µ2a2	µ2a2	CCONJ
ejpam-5693	25	8	+	+	CCONJ
ejpam-5693	25	9	(	(	PUNCT
ejpam-5693	25	10	1−	1−	NUM
ejpam-5693	25	11	µ)2b2	µ)2b2	NUM
ejpam-5693	25	12	≤	≤	NUM
ejpam-5693	25	13	(	(	PUNCT
ejpam-5693	25	14	1−	1−	NUM
ejpam-5693	25	15	µ)2(a−	µ)2(a−	X
ejpam-5693	25	16	b)2	b)2	PROPN
ejpam-5693	26	1	+	+	X
ejpam-5693	26	2	[	[	PUNCT
ejpam-5693	26	3	aµ((1−	aµ((1−	PROPN
ejpam-5693	26	4	µ)b)1−µ	µ)b)1−µ	NOUN
ejpam-5693	26	5	]	]	X
ejpam-5693	26	6	2	2	NUM
ejpam-5693	26	7	.	.	PUNCT
ejpam-5693	26	8	(	(	PUNCT
ejpam-5693	26	9	1.6	1.6	NUM
ejpam-5693	26	10	)	)	PUNCT
ejpam-5693	26	11	if	if	SCONJ
ejpam-5693	26	12	1	1	NUM
ejpam-5693	26	13	2	2	NUM
ejpam-5693	26	14	≤	≤	NOUN
ejpam-5693	26	15	µ	µ	PRON
ejpam-5693	26	16	≤	≤	NUM
ejpam-5693	26	17	1	1	NUM
ejpam-5693	26	18	,	,	PUNCT
ejpam-5693	26	19	then	then	ADV
ejpam-5693	26	20	µ2a2	µ2a2	CCONJ
ejpam-5693	26	21	+	+	CCONJ
ejpam-5693	26	22	(	(	PUNCT
ejpam-5693	26	23	1−	1−	NUM
ejpam-5693	26	24	µ)2b2	µ)2b2	ADJ
ejpam-5693	26	25	≤	≤	NOUN
ejpam-5693	26	26	µ2(a−	µ2(a−	PUNCT
ejpam-5693	26	27	b)2	b)2	PROPN
ejpam-5693	27	1	+	+	X
ejpam-5693	27	2	[	[	PUNCT
ejpam-5693	27	3	(	(	PUNCT
ejpam-5693	27	4	µa)µb1−µ	µa)µb1−µ	NUM
ejpam-5693	27	5	]	]	SYM
ejpam-5693	27	6	2	2	NUM
ejpam-5693	27	7	.	.	PUNCT
ejpam-5693	28	1	(	(	PUNCT
ejpam-5693	28	2	1.7	1.7	NUM
ejpam-5693	28	3	)	)	PUNCT
ejpam-5693	28	4	moreover	moreover	ADV
ejpam-5693	28	5	,	,	PUNCT
ejpam-5693	28	6	nasiri	nasiri	ADV
ejpam-5693	28	7	,	,	PUNCT
ejpam-5693	28	8	shokoori	shokoori	NOUN
ejpam-5693	28	9	,	,	PUNCT
ejpam-5693	28	10	and	and	CCONJ
ejpam-5693	28	11	liao	liao	NOUN
ejpam-5693	28	12	[	[	X
ejpam-5693	28	13	8	8	NUM
ejpam-5693	28	14	]	]	X
ejpam-5693	28	15	obtaind	obtaind	ADJ
ejpam-5693	28	16	refinements	refinement	NOUN
ejpam-5693	28	17	of	of	ADP
ejpam-5693	28	18	kai	kai	PROPN
ejpam-5693	28	19	results	result	NOUN
ejpam-5693	28	20	as	as	SCONJ
ejpam-5693	28	21	follows	follow	VERB
ejpam-5693	28	22	if	if	SCONJ
ejpam-5693	28	23	0	0	NUM
ejpam-5693	28	24	≤	≤	NUM
ejpam-5693	28	25	µ	µ	X
ejpam-5693	28	26	≤	≤	NUM
ejpam-5693	28	27	1	1	NUM
ejpam-5693	28	28	2	2	NUM
ejpam-5693	28	29	,	,	PUNCT
ejpam-5693	28	30	then	then	ADV
ejpam-5693	28	31	(	(	PUNCT
ejpam-5693	28	32	µa)2µb2−2µ	µa)2µb2−2µ	PROPN
ejpam-5693	28	33	+	+	NUM
ejpam-5693	28	34	µ2(a−	µ2(a−	PRON
ejpam-5693	28	35	b)2	b)2	ADJ
ejpam-5693	28	36	+	+	CCONJ
ejpam-5693	28	37	r0b	r0b	NOUN
ejpam-5693	28	38	(	(	PUNCT
ejpam-5693	28	39	√	√	ADP
ejpam-5693	28	40	µa−	µa−	NUM
ejpam-5693	28	41	√	√	NUM
ejpam-5693	28	42	b	b	NOUN
ejpam-5693	28	43	)	)	PUNCT
ejpam-5693	28	44	2	2	NUM
ejpam-5693	28	45	≤	≤	NOUN
ejpam-5693	28	46	µ2a2	µ2a2	ADV
ejpam-5693	28	47	+	+	CCONJ
ejpam-5693	28	48	(	(	PUNCT
ejpam-5693	28	49	1−	1−	NUM
ejpam-5693	28	50	µ)2b2	µ)2b2	NUM
ejpam-5693	28	51	,	,	PUNCT
ejpam-5693	28	52	(	(	PUNCT
ejpam-5693	28	53	1.8	1.8	NUM
ejpam-5693	28	54	)	)	PUNCT
ejpam-5693	28	55	where	where	SCONJ
ejpam-5693	28	56	r0	r0	NOUN
ejpam-5693	28	57	=	=	SYM
ejpam-5693	28	58	min{2µ	min{2µ	PROPN
ejpam-5693	28	59	,	,	PUNCT
ejpam-5693	28	60	1−	1−	NUM
ejpam-5693	28	61	2µ	2µ	NUM
ejpam-5693	28	62	}	}	PUNCT
ejpam-5693	28	63	.	.	PUNCT
ejpam-5693	29	1	if	if	SCONJ
ejpam-5693	29	2	1	1	NUM
ejpam-5693	29	3	2	2	NUM
ejpam-5693	29	4	≤	≤	NOUN
ejpam-5693	29	5	µ	µ	PRON
ejpam-5693	29	6	≤	≤	NUM
ejpam-5693	29	7	1	1	NUM
ejpam-5693	29	8	,	,	PUNCT
ejpam-5693	29	9	then	then	ADV
ejpam-5693	29	10	a2µ[(1−	a2µ[(1−	VERB
ejpam-5693	29	11	µ)b]2−2µ	µ)b]2−2µ	PROPN
ejpam-5693	29	12	+	+	CCONJ
ejpam-5693	29	13	(	(	PUNCT
ejpam-5693	29	14	1−	1−	NUM
ejpam-5693	29	15	µ)2(a−	µ)2(a−	X
ejpam-5693	29	16	b)2	b)2	PROPN
ejpam-5693	29	17	+	+	NUM
ejpam-5693	29	18	r0a	r0a	NOUN
ejpam-5693	29	19	(	(	PUNCT
ejpam-5693	29	20	√	√	X
ejpam-5693	29	21	a−	a−	PROPN
ejpam-5693	29	22	√	√	NUM
ejpam-5693	29	23	(	(	PUNCT
ejpam-5693	29	24	1−	1−	NUM
ejpam-5693	29	25	µ)b	µ)b	NOUN
ejpam-5693	29	26	)	)	PUNCT
ejpam-5693	29	27	2	2	NUM
ejpam-5693	29	28	≤	≤	NOUN
ejpam-5693	29	29	µ2a2	µ2a2	PUNCT
ejpam-5693	29	30	+	+	CCONJ
ejpam-5693	29	31	(	(	PUNCT
ejpam-5693	29	32	1−	1−	NUM
ejpam-5693	29	33	µ)2b2	µ)2b2	NUM
ejpam-5693	29	34	,	,	PUNCT
ejpam-5693	29	35	(	(	PUNCT
ejpam-5693	29	36	1.9	1.9	NUM
ejpam-5693	29	37	)	)	PUNCT
ejpam-5693	29	38	where	where	SCONJ
ejpam-5693	29	39	r0	r0	NOUN
ejpam-5693	29	40	=	=	SYM
ejpam-5693	29	41	min{2µ−	min{2µ−	PROPN
ejpam-5693	29	42	1	1	NUM
ejpam-5693	29	43	,	,	PUNCT
ejpam-5693	29	44	2−	2−	NUM
ejpam-5693	29	45	2µ	2µ	NUM
ejpam-5693	29	46	}	}	PUNCT
ejpam-5693	29	47	.	.	PUNCT
ejpam-5693	30	1	a	a	DET
ejpam-5693	30	2	matrix	matrix	NOUN
ejpam-5693	30	3	version	version	NOUN
ejpam-5693	30	4	of	of	ADP
ejpam-5693	30	5	(	(	PUNCT
ejpam-5693	30	6	1.1	1.1	NUM
ejpam-5693	30	7	)	)	PUNCT
ejpam-5693	30	8	proved	prove	VERB
ejpam-5693	30	9	in	in	ADP
ejpam-5693	30	10	[	[	X
ejpam-5693	30	11	1	1	NUM
ejpam-5693	30	12	]	]	PUNCT
ejpam-5693	30	13	says	say	VERB
ejpam-5693	30	14	that	that	SCONJ
ejpam-5693	30	15	if	if	SCONJ
ejpam-5693	30	16	a	a	PRON
ejpam-5693	30	17	,	,	PUNCT
ejpam-5693	30	18	b	b	PROPN
ejpam-5693	30	19	∈	∈	PROPN
ejpam-5693	30	20	mn	mn	PROPN
ejpam-5693	30	21	are	be	AUX
ejpam-5693	30	22	positive	positive	ADJ
ejpam-5693	30	23	semidefinit	semidefinit	ADV
ejpam-5693	30	24	,	,	PUNCT
ejpam-5693	30	25	then	then	ADV
ejpam-5693	30	26	∥aµb1−µ∥	∥aµb1−µ∥	NUM
ejpam-5693	30	27	≤	≤	NUM
ejpam-5693	31	1	∥µa+	∥µa+	NUM
ejpam-5693	31	2	(	(	PUNCT
ejpam-5693	31	3	1−	1−	NUM
ejpam-5693	31	4	µ)b∥	µ)b∥	ADV
ejpam-5693	31	5	,	,	PUNCT
ejpam-5693	31	6	for	for	ADP
ejpam-5693	31	7	0	0	NUM
ejpam-5693	31	8	≤	≤	NOUN
ejpam-5693	31	9	µ	µ	X
ejpam-5693	31	10	≤	≤	NUM
ejpam-5693	31	11	1	1	NUM
ejpam-5693	31	12	.	.	PUNCT
ejpam-5693	32	1	(	(	PUNCT
ejpam-5693	32	2	1.10	1.10	NUM
ejpam-5693	32	3	)	)	PUNCT
ejpam-5693	32	4	kosaki	kosaki	NOUN
ejpam-5693	33	1	[	[	X
ejpam-5693	33	2	7	7	NUM
ejpam-5693	33	3	]	]	PUNCT
ejpam-5693	33	4	,	,	PUNCT
ejpam-5693	33	5	bhatia	bhatia	PROPN
ejpam-5693	33	6	and	and	CCONJ
ejpam-5693	33	7	parthasarathy	parthasarathy	PROPN
ejpam-5693	33	8	[	[	X
ejpam-5693	33	9	3	3	NUM
ejpam-5693	33	10	]	]	PUNCT
ejpam-5693	33	11	proved	prove	VERB
ejpam-5693	33	12	that	that	SCONJ
ejpam-5693	33	13	if	if	SCONJ
ejpam-5693	33	14	a	a	DET
ejpam-5693	33	15	,	,	PUNCT
ejpam-5693	33	16	b	b	NOUN
ejpam-5693	33	17	,	,	PUNCT
ejpam-5693	33	18	x	x	PROPN
ejpam-5693	33	19	∈	∈	PROPN
ejpam-5693	33	20	mn	mn	PROPN
ejpam-5693	33	21	such	such	ADJ
ejpam-5693	33	22	that	that	SCONJ
ejpam-5693	33	23	a	a	PRON
ejpam-5693	33	24	and	and	CCONJ
ejpam-5693	33	25	b	b	NOUN
ejpam-5693	33	26	are	be	AUX
ejpam-5693	33	27	positive	positive	ADJ
ejpam-5693	33	28	semidefinite	semidefinite	NOUN
ejpam-5693	33	29	,	,	PUNCT
ejpam-5693	33	30	then	then	ADV
ejpam-5693	33	31	∥aµxb1−µ∥22	∥aµxb1−µ∥22	PROPN
ejpam-5693	33	32	≤	≤	PROPN
ejpam-5693	33	33	∥µax	∥µax	PROPN
ejpam-5693	33	34	+	+	CCONJ
ejpam-5693	34	1	(	(	PUNCT
ejpam-5693	34	2	1−	1−	NUM
ejpam-5693	34	3	µ)xb∥22	µ)xb∥22	PROPN
ejpam-5693	34	4	,	,	PUNCT
ejpam-5693	34	5	for	for	ADP
ejpam-5693	34	6	0	0	NUM
ejpam-5693	34	7	≤	≤	NOUN
ejpam-5693	34	8	µ	µ	X
ejpam-5693	34	9	≤	≤	NUM
ejpam-5693	34	10	1	1	NUM
ejpam-5693	34	11	.	.	PUNCT
ejpam-5693	35	1	(	(	PUNCT
ejpam-5693	35	2	1.11	1.11	NUM
ejpam-5693	35	3	)	)	PUNCT
ejpam-5693	35	4	based	base	VERB
ejpam-5693	35	5	on	on	ADP
ejpam-5693	35	6	the	the	DET
ejpam-5693	35	7	refined	refined	ADJ
ejpam-5693	35	8	young	young	ADJ
ejpam-5693	35	9	inequalities	inequality	NOUN
ejpam-5693	35	10	(	(	PUNCT
ejpam-5693	35	11	1.4	1.4	NUM
ejpam-5693	35	12	)	)	PUNCT
ejpam-5693	35	13	and	and	CCONJ
ejpam-5693	35	14	(	(	PUNCT
ejpam-5693	35	15	1.5	1.5	NUM
ejpam-5693	35	16	)	)	PUNCT
ejpam-5693	35	17	,	,	PUNCT
ejpam-5693	35	18	kai	kai	PROPN
ejpam-5693	36	1	[	[	X
ejpam-5693	36	2	9	9	NUM
ejpam-5693	36	3	]	]	PUNCT
ejpam-5693	36	4	have	have	AUX
ejpam-5693	36	5	showed	show	VERB
ejpam-5693	36	6	that	that	SCONJ
ejpam-5693	36	7	if	if	SCONJ
ejpam-5693	36	8	a	a	DET
ejpam-5693	36	9	,	,	PUNCT
ejpam-5693	36	10	b	b	NOUN
ejpam-5693	36	11	,	,	PUNCT
ejpam-5693	36	12	x	x	PROPN
ejpam-5693	36	13	∈	∈	PROPN
ejpam-5693	36	14	mn	mn	PROPN
ejpam-5693	36	15	such	such	ADJ
ejpam-5693	36	16	that	that	SCONJ
ejpam-5693	36	17	a	a	PRON
ejpam-5693	36	18	and	and	CCONJ
ejpam-5693	36	19	b	b	NOUN
ejpam-5693	36	20	are	be	AUX
ejpam-5693	36	21	positive	positive	ADJ
ejpam-5693	36	22	semidefinite	semidefinite	NOUN
ejpam-5693	36	23	,	,	PUNCT
ejpam-5693	36	24	then	then	ADV
ejpam-5693	36	25	µ2∥ax	µ2∥ax	ADJ
ejpam-5693	36	26	−xb∥22	−xb∥22	PUNCT
ejpam-5693	36	27	+	+	NUM
ejpam-5693	36	28	µ2µ∥aµxb1−µ∥22	µ2µ∥aµxb1−µ∥22	X
ejpam-5693	36	29	+	+	CCONJ
ejpam-5693	36	30	2µ(1−	2µ(1−	NUM
ejpam-5693	36	31	µ)∥a	µ)∥a	ADJ
ejpam-5693	36	32	1	1	NUM
ejpam-5693	36	33	2xb	2xb	ADJ
ejpam-5693	36	34	1	1	NUM
ejpam-5693	36	35	2	2	NUM
ejpam-5693	36	36	∥22	∥22	PROPN
ejpam-5693	36	37	≤	≤	ADJ
ejpam-5693	36	38	∥µax	∥µax	NOUN
ejpam-5693	36	39	+	+	CCONJ
ejpam-5693	36	40	(	(	PUNCT
ejpam-5693	36	41	1−	1−	NUM
ejpam-5693	36	42	µ)xb∥22	µ)xb∥22	PROPN
ejpam-5693	36	43	,	,	PUNCT
ejpam-5693	36	44	(	(	PUNCT
ejpam-5693	36	45	1.12	1.12	NUM
ejpam-5693	36	46	)	)	PUNCT
ejpam-5693	36	47	o.	o.	PROPN
ejpam-5693	36	48	ramadan	ramadan	PROPN
ejpam-5693	36	49	,	,	PUNCT
ejpam-5693	36	50	a.	a.	PROPN
ejpam-5693	36	51	burqan	burqan	PROPN
ejpam-5693	36	52	/	/	SYM
ejpam-5693	36	53	eur	eur	PROPN
ejpam-5693	36	54	.	.	PUNCT
ejpam-5693	37	1	j.	j.	PROPN
ejpam-5693	37	2	pure	pure	PROPN
ejpam-5693	37	3	appl	appl	PROPN
ejpam-5693	37	4	.	.	PROPN
ejpam-5693	37	5	math	math	PROPN
ejpam-5693	37	6	,	,	PUNCT
ejpam-5693	37	7	18	18	NUM
ejpam-5693	37	8	(	(	PUNCT
ejpam-5693	37	9	1	1	NUM
ejpam-5693	37	10	)	)	PUNCT
ejpam-5693	37	11	(	(	PUNCT
ejpam-5693	37	12	2025	2025	NUM
ejpam-5693	37	13	)	)	PUNCT
ejpam-5693	37	14	,	,	PUNCT
ejpam-5693	37	15	5693	5693	NUM
ejpam-5693	37	16	3	3	NUM
ejpam-5693	37	17	of	of	ADP
ejpam-5693	37	18	8	8	NUM
ejpam-5693	37	19	for	for	ADP
ejpam-5693	37	20	0	0	NUM
ejpam-5693	37	21	≤	≤	NOUN
ejpam-5693	37	22	µ	µ	PRON
ejpam-5693	37	23	≤	≤	NUM
ejpam-5693	37	24	1	1	NUM
ejpam-5693	37	25	2	2	NUM
ejpam-5693	37	26	.	.	PUNCT
ejpam-5693	38	1	(	(	PUNCT
ejpam-5693	38	2	1−	1−	NUM
ejpam-5693	38	3	µ)2∥ax	µ)2∥ax	ADJ
ejpam-5693	38	4	−xb∥22	−xb∥22	PUNCT
ejpam-5693	39	1	+	+	CCONJ
ejpam-5693	39	2	(	(	PUNCT
ejpam-5693	39	3	1−	1−	NUM
ejpam-5693	39	4	µ)2−2µ∥aµxb1−µ∥22	µ)2−2µ∥aµxb1−µ∥22	PUNCT
ejpam-5693	39	5	+	+	NUM
ejpam-5693	39	6	2µ(1−	2µ(1−	NUM
ejpam-5693	39	7	µ)∥a	µ)∥a	ADJ
ejpam-5693	39	8	1	1	NUM
ejpam-5693	39	9	2xb	2xb	ADJ
ejpam-5693	39	10	1	1	NUM
ejpam-5693	39	11	2	2	NUM
ejpam-5693	39	12	∥22	∥22	PROPN
ejpam-5693	39	13	≤	≤	ADJ
ejpam-5693	39	14	∥µax	∥µax	NOUN
ejpam-5693	39	15	+	+	CCONJ
ejpam-5693	39	16	(	(	PUNCT
ejpam-5693	39	17	1−	1−	NUM
ejpam-5693	39	18	µ)xb∥22	µ)xb∥22	PROPN
ejpam-5693	39	19	,	,	PUNCT
ejpam-5693	39	20	(	(	PUNCT
ejpam-5693	39	21	1.13	1.13	NUM
ejpam-5693	39	22	)	)	PUNCT
ejpam-5693	39	23	for	for	ADP
ejpam-5693	39	24	1	1	NUM
ejpam-5693	39	25	2	2	NUM
ejpam-5693	39	26	≤	≤	NOUN
ejpam-5693	39	27	µ	µ	PRON
ejpam-5693	39	28	≤	≤	NUM
ejpam-5693	39	29	1	1	NUM
ejpam-5693	39	30	.	.	PUNCT
ejpam-5693	40	1	burqan	burqan	NOUN
ejpam-5693	40	2	and	and	CCONJ
ejpam-5693	40	3	khandaqji	khandaqji	ADJ
ejpam-5693	40	4	[	[	X
ejpam-5693	40	5	4	4	NUM
ejpam-5693	40	6	]	]	PUNCT
ejpam-5693	40	7	gave	give	VERB
ejpam-5693	40	8	matrix	matrix	NOUN
ejpam-5693	40	9	versions	version	NOUN
ejpam-5693	40	10	of	of	ADP
ejpam-5693	40	11	the	the	DET
ejpam-5693	40	12	inequalities	inequality	NOUN
ejpam-5693	40	13	(	(	PUNCT
ejpam-5693	40	14	1.6	1.6	NUM
ejpam-5693	40	15	)	)	PUNCT
ejpam-5693	40	16	and	and	CCONJ
ejpam-5693	40	17	(	(	PUNCT
ejpam-5693	40	18	1.7	1.7	NUM
ejpam-5693	40	19	)	)	PUNCT
ejpam-5693	40	20	as	as	SCONJ
ejpam-5693	40	21	follows	follow	VERB
ejpam-5693	40	22	let	let	VERB
ejpam-5693	40	23	a	a	DET
ejpam-5693	40	24	,	,	PUNCT
ejpam-5693	40	25	b	b	NOUN
ejpam-5693	40	26	,	,	PUNCT
ejpam-5693	40	27	x	x	PROPN
ejpam-5693	40	28	∈	∈	PROPN
ejpam-5693	40	29	mn	mn	PROPN
ejpam-5693	40	30	such	such	ADJ
ejpam-5693	40	31	that	that	SCONJ
ejpam-5693	40	32	a	a	PRON
ejpam-5693	40	33	and	and	CCONJ
ejpam-5693	40	34	b	b	NOUN
ejpam-5693	40	35	are	be	AUX
ejpam-5693	40	36	positive	positive	ADJ
ejpam-5693	40	37	semidefinite	semidefinite	NOUN
ejpam-5693	40	38	.	.	PUNCT
ejpam-5693	41	1	if	if	SCONJ
ejpam-5693	41	2	0	0	NUM
ejpam-5693	41	3	≤	≤	NUM
ejpam-5693	41	4	µ	µ	X
ejpam-5693	41	5	≤	≤	NUM
ejpam-5693	41	6	1	1	NUM
ejpam-5693	41	7	2	2	NUM
ejpam-5693	41	8	,	,	PUNCT
ejpam-5693	41	9	then	then	ADV
ejpam-5693	41	10	∥µax	∥µax	PROPN
ejpam-5693	41	11	+	+	CCONJ
ejpam-5693	41	12	(	(	PUNCT
ejpam-5693	41	13	1−	1−	NUM
ejpam-5693	41	14	µ)xb∥22	µ)xb∥22	PROPN
ejpam-5693	41	15	≤	≤	PROPN
ejpam-5693	41	16	(	(	PUNCT
ejpam-5693	41	17	1−	1−	NUM
ejpam-5693	41	18	µ)2∥ax	µ)2∥ax	ADJ
ejpam-5693	41	19	−xb∥22	−xb∥22	PUNCT
ejpam-5693	42	1	+	+	SYM
ejpam-5693	42	2	2µ(1−	2µ(1−	NUM
ejpam-5693	42	3	µ)∥a	µ)∥a	ADJ
ejpam-5693	42	4	1	1	NUM
ejpam-5693	42	5	2xb	2xb	ADJ
ejpam-5693	42	6	1	1	NUM
ejpam-5693	42	7	2	2	NUM
ejpam-5693	42	8	∥22	∥22	ADJ
ejpam-5693	42	9	+	+	NUM
ejpam-5693	42	10	(	(	PUNCT
ejpam-5693	42	11	1−	1−	NUM
ejpam-5693	42	12	µ)2(1−µ)∥aµxb1−µ∥22	µ)2(1−µ)∥aµxb1−µ∥22	X
ejpam-5693	42	13	.	.	PUNCT
ejpam-5693	43	1	(	(	PUNCT
ejpam-5693	43	2	1.14	1.14	NUM
ejpam-5693	43	3	)	)	PUNCT
ejpam-5693	43	4	if	if	SCONJ
ejpam-5693	43	5	1	1	NUM
ejpam-5693	43	6	2	2	NUM
ejpam-5693	43	7	≤	≤	NOUN
ejpam-5693	43	8	µ	µ	PRON
ejpam-5693	43	9	≤	≤	NUM
ejpam-5693	43	10	1	1	NUM
ejpam-5693	43	11	,	,	PUNCT
ejpam-5693	43	12	then	then	ADV
ejpam-5693	43	13	∥µax	∥µax	PROPN
ejpam-5693	43	14	+	+	CCONJ
ejpam-5693	43	15	(	(	PUNCT
ejpam-5693	43	16	1−	1−	NUM
ejpam-5693	43	17	µ)xb∥22	µ)xb∥22	PROPN
ejpam-5693	43	18	≤	≤	ADV
ejpam-5693	43	19	µ2∥ax	µ2∥ax	ADJ
ejpam-5693	43	20	−xb∥22	−xb∥22	PUNCT
ejpam-5693	43	21	+	+	SYM
ejpam-5693	43	22	2µ(1−	2µ(1−	NUM
ejpam-5693	43	23	µ)∥a	µ)∥a	ADJ
ejpam-5693	43	24	1	1	NUM
ejpam-5693	43	25	2xb	2xb	ADJ
ejpam-5693	43	26	1	1	NUM
ejpam-5693	43	27	2	2	NUM
ejpam-5693	43	28	∥22	∥22	ADJ
ejpam-5693	43	29	+	+	CCONJ
ejpam-5693	43	30	µ2µ∥aµxb1−µ∥22	µ2µ∥aµxb1−µ∥22	NOUN
ejpam-5693	43	31	.	.	PUNCT
ejpam-5693	44	1	(	(	PUNCT
ejpam-5693	44	2	1.15	1.15	NUM
ejpam-5693	44	3	)	)	PUNCT
ejpam-5693	44	4	in	in	ADP
ejpam-5693	44	5	this	this	DET
ejpam-5693	44	6	paper	paper	NOUN
ejpam-5693	44	7	,	,	PUNCT
ejpam-5693	44	8	we	we	PRON
ejpam-5693	44	9	introduce	introduce	VERB
ejpam-5693	44	10	reverses	reverse	NOUN
ejpam-5693	44	11	of	of	ADP
ejpam-5693	44	12	the	the	DET
ejpam-5693	44	13	inequalities	inequality	NOUN
ejpam-5693	44	14	(	(	PUNCT
ejpam-5693	44	15	1.8	1.8	NUM
ejpam-5693	44	16	)	)	PUNCT
ejpam-5693	44	17	and	and	CCONJ
ejpam-5693	44	18	(	(	PUNCT
ejpam-5693	44	19	1.9	1.9	NUM
ejpam-5693	44	20	)	)	PUNCT
ejpam-5693	44	21	which	which	PRON
ejpam-5693	44	22	are	be	AUX
ejpam-5693	44	23	refinements	refinement	NOUN
ejpam-5693	44	24	of	of	ADP
ejpam-5693	44	25	the	the	DET
ejpam-5693	44	26	inequalities	inequality	NOUN
ejpam-5693	44	27	(	(	PUNCT
ejpam-5693	44	28	1.6	1.6	NUM
ejpam-5693	44	29	)	)	PUNCT
ejpam-5693	44	30	and	and	CCONJ
ejpam-5693	44	31	(	(	PUNCT
ejpam-5693	44	32	1.7	1.7	NUM
ejpam-5693	44	33	)	)	PUNCT
ejpam-5693	44	34	.	.	PUNCT
ejpam-5693	45	1	as	as	ADP
ejpam-5693	45	2	applications	application	NOUN
ejpam-5693	45	3	of	of	ADP
ejpam-5693	45	4	our	our	PRON
ejpam-5693	45	5	results	result	NOUN
ejpam-5693	45	6	,	,	PUNCT
ejpam-5693	45	7	we	we	PRON
ejpam-5693	45	8	obtain	obtain	VERB
ejpam-5693	45	9	corresponding	corresponding	ADJ
ejpam-5693	45	10	inequalities	inequality	NOUN
ejpam-5693	45	11	for	for	ADP
ejpam-5693	45	12	matrices	matrix	NOUN
ejpam-5693	45	13	.	.	PUNCT
ejpam-5693	46	1	2	2	X
ejpam-5693	46	2	.	.	X
ejpam-5693	46	3	main	main	ADJ
ejpam-5693	46	4	results	result	NOUN
ejpam-5693	46	5	we	we	PRON
ejpam-5693	46	6	will	will	AUX
ejpam-5693	46	7	divide	divide	VERB
ejpam-5693	46	8	our	our	PRON
ejpam-5693	46	9	main	main	ADJ
ejpam-5693	46	10	results	result	NOUN
ejpam-5693	46	11	into	into	ADP
ejpam-5693	46	12	two	two	NUM
ejpam-5693	46	13	categories	category	NOUN
ejpam-5693	46	14	,	,	PUNCT
ejpam-5693	46	15	the	the	DET
ejpam-5693	46	16	first	first	ADJ
ejpam-5693	46	17	is	be	AUX
ejpam-5693	46	18	about	about	ADP
ejpam-5693	46	19	scalars	scalar	NOUN
ejpam-5693	46	20	and	and	CCONJ
ejpam-5693	46	21	the	the	DET
ejpam-5693	46	22	other	other	ADJ
ejpam-5693	46	23	is	be	AUX
ejpam-5693	46	24	about	about	ADP
ejpam-5693	46	25	matrices	matrix	NOUN
ejpam-5693	46	26	.	.	PUNCT
ejpam-5693	47	1	2.1	2.1	NUM
ejpam-5693	47	2	.	.	PUNCT
ejpam-5693	47	3	inequalities	inequality	NOUN
ejpam-5693	47	4	for	for	ADP
ejpam-5693	47	5	scalars	scalar	NOUN
ejpam-5693	47	6	we	we	PRON
ejpam-5693	47	7	will	will	AUX
ejpam-5693	47	8	start	start	VERB
ejpam-5693	47	9	this	this	DET
ejpam-5693	47	10	section	section	NOUN
ejpam-5693	47	11	with	with	ADP
ejpam-5693	47	12	the	the	DET
ejpam-5693	47	13	following	follow	VERB
ejpam-5693	47	14	results	result	NOUN
ejpam-5693	47	15	for	for	ADP
ejpam-5693	47	16	scalars	scalar	NOUN
ejpam-5693	47	17	theorem	theorem	VERB
ejpam-5693	47	18	1	1	X
ejpam-5693	47	19	.	.	PUNCT
ejpam-5693	48	1	let	let	VERB
ejpam-5693	48	2	a	a	DET
ejpam-5693	48	3	,	,	PUNCT
ejpam-5693	48	4	b	b	PROPN
ejpam-5693	48	5	≥	≥	NOUN
ejpam-5693	48	6	0	0	NUM
ejpam-5693	48	7	.	.	PUNCT
ejpam-5693	49	1	if	if	SCONJ
ejpam-5693	49	2	0	0	NUM
ejpam-5693	49	3	≤	≤	NUM
ejpam-5693	49	4	µ	µ	X
ejpam-5693	49	5	≤	≤	NUM
ejpam-5693	49	6	1	1	NUM
ejpam-5693	49	7	2	2	NUM
ejpam-5693	49	8	,	,	PUNCT
ejpam-5693	49	9	then	then	ADV
ejpam-5693	49	10	µ2a2	µ2a2	CCONJ
ejpam-5693	49	11	+	+	CCONJ
ejpam-5693	49	12	(	(	PUNCT
ejpam-5693	49	13	1−	1−	NUM
ejpam-5693	49	14	µ)2b2	µ)2b2	NUM
ejpam-5693	49	15	≤	≤	NOUN
ejpam-5693	49	16	(	(	PUNCT
ejpam-5693	49	17	µa)2µb2−2µ	µa)2µb2−2µ	PROPN
ejpam-5693	49	18	+	+	NUM
ejpam-5693	49	19	µ2(a−	µ2(a−	PRON
ejpam-5693	49	20	b)2	b)2	ADJ
ejpam-5693	49	21	+	+	ADJ
ejpam-5693	49	22	r0b	r0b	NOUN
ejpam-5693	49	23	(	(	PUNCT
ejpam-5693	49	24	√	√	ADP
ejpam-5693	49	25	µa−	µa−	NUM
ejpam-5693	49	26	√	√	NUM
ejpam-5693	49	27	b	b	NOUN
ejpam-5693	49	28	)	)	PUNCT
ejpam-5693	49	29	2	2	NUM
ejpam-5693	49	30	,	,	PUNCT
ejpam-5693	49	31	(	(	PUNCT
ejpam-5693	49	32	2.1	2.1	NUM
ejpam-5693	49	33	)	)	PUNCT
ejpam-5693	49	34	where	where	SCONJ
ejpam-5693	49	35	r0	r0	NOUN
ejpam-5693	49	36	=	=	SYM
ejpam-5693	49	37	max{2µ	max{2µ	PROPN
ejpam-5693	49	38	,	,	PUNCT
ejpam-5693	49	39	1−	1−	NUM
ejpam-5693	49	40	2µ	2µ	NUM
ejpam-5693	49	41	}	}	PUNCT
ejpam-5693	49	42	.	.	PUNCT
ejpam-5693	50	1	if	if	SCONJ
ejpam-5693	50	2	1	1	NUM
ejpam-5693	50	3	2	2	NUM
ejpam-5693	50	4	≤	≤	NOUN
ejpam-5693	50	5	µ	µ	PRON
ejpam-5693	50	6	≤	≤	NUM
ejpam-5693	50	7	1	1	NUM
ejpam-5693	50	8	,	,	PUNCT
ejpam-5693	50	9	then	then	ADV
ejpam-5693	50	10	µ2a2	µ2a2	CCONJ
ejpam-5693	50	11	+	+	CCONJ
ejpam-5693	50	12	(	(	PUNCT
ejpam-5693	50	13	1−	1−	NUM
ejpam-5693	50	14	µ)2b2	µ)2b2	PUNCT
ejpam-5693	50	15	≤	≤	NUM
ejpam-5693	50	16	a2µ[(1−	a2µ[(1−	PROPN
ejpam-5693	50	17	µ)b]2−2µ	µ)b]2−2µ	NOUN
ejpam-5693	50	18	+	+	CCONJ
ejpam-5693	50	19	(	(	PUNCT
ejpam-5693	50	20	1−	1−	NUM
ejpam-5693	50	21	µ)2(a−	µ)2(a−	X
ejpam-5693	50	22	b)2	b)2	ADJ
ejpam-5693	50	23	+	+	ADJ
ejpam-5693	50	24	r0a	r0a	NOUN
ejpam-5693	50	25	(	(	PUNCT
ejpam-5693	50	26	√	√	X
ejpam-5693	50	27	a−	a−	PROPN
ejpam-5693	50	28	√	√	NUM
ejpam-5693	50	29	(	(	PUNCT
ejpam-5693	50	30	1−	1−	NUM
ejpam-5693	50	31	µ)b	µ)b	NOUN
ejpam-5693	50	32	)	)	PUNCT
ejpam-5693	50	33	2	2	NUM
ejpam-5693	50	34	,	,	PUNCT
ejpam-5693	50	35	(	(	PUNCT
ejpam-5693	50	36	2.2	2.2	NUM
ejpam-5693	50	37	)	)	PUNCT
ejpam-5693	50	38	where	where	SCONJ
ejpam-5693	50	39	r0	r0	NOUN
ejpam-5693	50	40	=	=	SYM
ejpam-5693	50	41	max{2µ−	max{2µ−	PROPN
ejpam-5693	50	42	1	1	NUM
ejpam-5693	50	43	,	,	PUNCT
ejpam-5693	50	44	2−	2−	NUM
ejpam-5693	50	45	2µ	2µ	NUM
ejpam-5693	50	46	}	}	PUNCT
ejpam-5693	50	47	.	.	PUNCT
ejpam-5693	51	1	o.	o.	PROPN
ejpam-5693	51	2	ramadan	ramadan	PROPN
ejpam-5693	51	3	,	,	PUNCT
ejpam-5693	51	4	a.	a.	PROPN
ejpam-5693	51	5	burqan	burqan	PROPN
ejpam-5693	51	6	/	/	SYM
ejpam-5693	51	7	eur	eur	PROPN
ejpam-5693	51	8	.	.	PUNCT
ejpam-5693	52	1	j.	j.	PROPN
ejpam-5693	52	2	pure	pure	PROPN
ejpam-5693	52	3	appl	appl	PROPN
ejpam-5693	52	4	.	.	PROPN
ejpam-5693	52	5	math	math	PROPN
ejpam-5693	52	6	,	,	PUNCT
ejpam-5693	52	7	18	18	NUM
ejpam-5693	52	8	(	(	PUNCT
ejpam-5693	52	9	1	1	NUM
ejpam-5693	52	10	)	)	PUNCT
ejpam-5693	52	11	(	(	PUNCT
ejpam-5693	52	12	2025	2025	NUM
ejpam-5693	52	13	)	)	PUNCT
ejpam-5693	52	14	,	,	PUNCT
ejpam-5693	52	15	5693	5693	NUM
ejpam-5693	52	16	4	4	NUM
ejpam-5693	52	17	of	of	ADP
ejpam-5693	52	18	8	8	NUM
ejpam-5693	52	19	proof	proof	NOUN
ejpam-5693	52	20	.	.	PUNCT
ejpam-5693	53	1	if	if	SCONJ
ejpam-5693	53	2	0	0	NUM
ejpam-5693	53	3	≤	≤	NUM
ejpam-5693	53	4	µ	µ	X
ejpam-5693	53	5	≤	≤	NUM
ejpam-5693	53	6	1	1	NUM
ejpam-5693	53	7	2	2	NUM
ejpam-5693	53	8	,	,	PUNCT
ejpam-5693	53	9	then	then	ADV
ejpam-5693	53	10	by	by	ADP
ejpam-5693	53	11	inequality	inequality	NOUN
ejpam-5693	53	12	(	(	PUNCT
ejpam-5693	53	13	1.3	1.3	NUM
ejpam-5693	53	14	)	)	PUNCT
ejpam-5693	54	1	,	,	PUNCT
ejpam-5693	54	2	we	we	PRON
ejpam-5693	54	3	have	have	VERB
ejpam-5693	54	4	µ2a2	µ2a2	ADP
ejpam-5693	55	1	+	+	CCONJ
ejpam-5693	55	2	(	(	PUNCT
ejpam-5693	55	3	1−	1−	NUM
ejpam-5693	55	4	µ)2b2	µ)2b2	NUM
ejpam-5693	55	5	−	−	PROPN
ejpam-5693	55	6	µ2(a−	µ2(a−	PUNCT
ejpam-5693	55	7	b)2	b)2	ADJ
ejpam-5693	55	8	=	=	SYM
ejpam-5693	55	9	b2	b2	PROPN
ejpam-5693	55	10	−	−	PROPN
ejpam-5693	55	11	2µb2	2µb2	NUM
ejpam-5693	55	12	+	+	CCONJ
ejpam-5693	55	13	2abµ2	2abµ2	NUM
ejpam-5693	55	14	=	=	SYM
ejpam-5693	55	15	b[(1−	b[(1−	VERB
ejpam-5693	55	16	2µ)b+	2µ)b+	DET
ejpam-5693	55	17	2µ(µa	2µ(µa	NOUN
ejpam-5693	55	18	)	)	PUNCT
ejpam-5693	55	19	]	]	PUNCT
ejpam-5693	56	1	≤	≤	NUM
ejpam-5693	56	2	b	b	X
ejpam-5693	56	3	[	[	PUNCT
ejpam-5693	56	4	b1−2µ(µa)2µ	b1−2µ(µa)2µ	NOUN
ejpam-5693	56	5	+	+	NOUN
ejpam-5693	56	6	r0	r0	NOUN
ejpam-5693	56	7	(	(	PUNCT
ejpam-5693	56	8	√	√	ADP
ejpam-5693	56	9	µa−	µa−	NUM
ejpam-5693	56	10	√	√	NUM
ejpam-5693	56	11	b	b	NOUN
ejpam-5693	56	12	)	)	PUNCT
ejpam-5693	56	13	2	2	NUM
ejpam-5693	56	14	]	]	PUNCT
ejpam-5693	56	15	=	=	SYM
ejpam-5693	56	16	(	(	PUNCT
ejpam-5693	56	17	µa)2µb2−2µ	µa)2µb2−2µ	NOUN
ejpam-5693	56	18	+	+	NOUN
ejpam-5693	56	19	r0b	r0b	NOUN
ejpam-5693	56	20	(	(	PUNCT
ejpam-5693	56	21	√	√	ADP
ejpam-5693	56	22	µa−	µa−	NUM
ejpam-5693	56	23	√	√	NUM
ejpam-5693	56	24	b	b	NOUN
ejpam-5693	56	25	)	)	PUNCT
ejpam-5693	56	26	2	2	NUM
ejpam-5693	56	27	,	,	PUNCT
ejpam-5693	56	28	and	and	CCONJ
ejpam-5693	56	29	so	so	ADV
ejpam-5693	56	30	µ2a2	µ2a2	PROPN
ejpam-5693	56	31	+	+	CCONJ
ejpam-5693	56	32	(	(	PUNCT
ejpam-5693	56	33	1−	1−	NUM
ejpam-5693	56	34	µ)2b2	µ)2b2	NUM
ejpam-5693	56	35	≤	≤	NOUN
ejpam-5693	56	36	(	(	PUNCT
ejpam-5693	56	37	µa)2µb2−2µ	µa)2µb2−2µ	PROPN
ejpam-5693	56	38	+	+	NUM
ejpam-5693	56	39	µ2(a−	µ2(a−	PRON
ejpam-5693	56	40	b)2	b)2	ADJ
ejpam-5693	56	41	+	+	ADJ
ejpam-5693	56	42	r0b	r0b	NOUN
ejpam-5693	56	43	(	(	PUNCT
ejpam-5693	56	44	√	√	ADP
ejpam-5693	56	45	µa−	µa−	NUM
ejpam-5693	56	46	√	√	NUM
ejpam-5693	56	47	b	b	NOUN
ejpam-5693	56	48	)	)	PUNCT
ejpam-5693	56	49	2	2	NUM
ejpam-5693	56	50	.	.	PUNCT
ejpam-5693	57	1	on	on	ADP
ejpam-5693	57	2	the	the	DET
ejpam-5693	57	3	other	other	ADJ
ejpam-5693	57	4	hand	hand	NOUN
ejpam-5693	57	5	,	,	PUNCT
ejpam-5693	57	6	if	if	SCONJ
ejpam-5693	57	7	1	1	NUM
ejpam-5693	57	8	2	2	NUM
ejpam-5693	57	9	≤	≤	NOUN
ejpam-5693	57	10	µ	µ	PRON
ejpam-5693	57	11	≤	≤	NUM
ejpam-5693	57	12	1	1	NUM
ejpam-5693	57	13	,	,	PUNCT
ejpam-5693	57	14	then	then	ADV
ejpam-5693	57	15	by	by	ADP
ejpam-5693	57	16	inequality	inequality	NOUN
ejpam-5693	57	17	(	(	PUNCT
ejpam-5693	57	18	1.3	1.3	NUM
ejpam-5693	57	19	)	)	PUNCT
ejpam-5693	57	20	,	,	PUNCT
ejpam-5693	57	21	we	we	PRON
ejpam-5693	57	22	have	have	VERB
ejpam-5693	57	23	µ2a2	µ2a2	ADP
ejpam-5693	58	1	+	+	CCONJ
ejpam-5693	58	2	(	(	PUNCT
ejpam-5693	58	3	1−	1−	NUM
ejpam-5693	58	4	µ)2b2	µ)2b2	NUM
ejpam-5693	58	5	−	−	PROPN
ejpam-5693	58	6	(	(	PUNCT
ejpam-5693	58	7	1−	1−	NUM
ejpam-5693	58	8	µ)2(a−	µ)2(a−	X
ejpam-5693	58	9	b)2	b)2	PROPN
ejpam-5693	58	10	=	=	SYM
ejpam-5693	58	11	−a2	−a2	PROPN
ejpam-5693	58	12	+	+	CCONJ
ejpam-5693	58	13	2ab+	2ab+	NUM
ejpam-5693	58	14	2µ2ab+	2µ2ab+	NUM
ejpam-5693	58	15	2µa2	2µa2	NUM
ejpam-5693	59	1	−	−	PROPN
ejpam-5693	59	2	4µab	4µab	NUM
ejpam-5693	59	3	=	=	PUNCT
ejpam-5693	59	4	a[(2µ−	a[(2µ−	VERB
ejpam-5693	59	5	1)a+	1)a+	NUM
ejpam-5693	60	1	2(1−	2(1−	NUM
ejpam-5693	60	2	µ)(1−	µ)(1−	NOUN
ejpam-5693	60	3	µ)b	µ)b	NOUN
ejpam-5693	60	4	]	]	PUNCT
ejpam-5693	60	5	≤	≤	NOUN
ejpam-5693	60	6	a	a	PRON
ejpam-5693	60	7	[	[	PUNCT
ejpam-5693	60	8	a2µ−1((1−	a2µ−1((1−	AUX
ejpam-5693	60	9	µ)b)2−2µ	µ)b)2−2µ	PROPN
ejpam-5693	60	10	+	+	PROPN
ejpam-5693	60	11	r0	r0	NOUN
ejpam-5693	60	12	(	(	PUNCT
ejpam-5693	60	13	√	√	NOUN
ejpam-5693	60	14	a−	a−	PROPN
ejpam-5693	60	15	√	√	NUM
ejpam-5693	60	16	(	(	PUNCT
ejpam-5693	60	17	1−	1−	NUM
ejpam-5693	60	18	µ)b	µ)b	NOUN
ejpam-5693	60	19	)	)	PUNCT
ejpam-5693	60	20	2	2	X
ejpam-5693	60	21	]	]	PUNCT
ejpam-5693	60	22	=	=	PUNCT
ejpam-5693	60	23	a2µ[(1−	a2µ[(1−	PROPN
ejpam-5693	60	24	µ)b]2−2µ	µ)b]2−2µ	NOUN
ejpam-5693	60	25	+	+	ADP
ejpam-5693	60	26	r0a	r0a	NOUN
ejpam-5693	60	27	(	(	PUNCT
ejpam-5693	60	28	√	√	X
ejpam-5693	60	29	a−	a−	PROPN
ejpam-5693	60	30	√	√	NUM
ejpam-5693	60	31	(	(	PUNCT
ejpam-5693	60	32	1−	1−	NUM
ejpam-5693	60	33	µ)b	µ)b	NOUN
ejpam-5693	60	34	)	)	PUNCT
ejpam-5693	60	35	2	2	NUM
ejpam-5693	60	36	,	,	PUNCT
ejpam-5693	60	37	and	and	CCONJ
ejpam-5693	60	38	so	so	ADV
ejpam-5693	60	39	µ2a2	µ2a2	PROPN
ejpam-5693	60	40	+	+	CCONJ
ejpam-5693	60	41	(	(	PUNCT
ejpam-5693	60	42	1−	1−	NUM
ejpam-5693	60	43	µ)2b2	µ)2b2	PUNCT
ejpam-5693	60	44	≤	≤	NUM
ejpam-5693	60	45	a2µ[(1−	a2µ[(1−	PROPN
ejpam-5693	60	46	µ)b]2−2µ	µ)b]2−2µ	NOUN
ejpam-5693	60	47	+	+	CCONJ
ejpam-5693	60	48	(	(	PUNCT
ejpam-5693	60	49	1−	1−	NUM
ejpam-5693	60	50	µ)2(a−	µ)2(a−	X
ejpam-5693	60	51	b)2	b)2	ADJ
ejpam-5693	60	52	+	+	ADJ
ejpam-5693	60	53	r0a	r0a	NOUN
ejpam-5693	60	54	(	(	PUNCT
ejpam-5693	60	55	√	√	X
ejpam-5693	60	56	a−	a−	PROPN
ejpam-5693	60	57	√	√	NUM
ejpam-5693	60	58	(	(	PUNCT
ejpam-5693	60	59	1−	1−	NUM
ejpam-5693	60	60	µ)b	µ)b	NOUN
ejpam-5693	60	61	)	)	PUNCT
ejpam-5693	60	62	2	2	X
ejpam-5693	60	63	.	.	PUNCT
ejpam-5693	61	1	this	this	PRON
ejpam-5693	61	2	completes	complete	VERB
ejpam-5693	61	3	the	the	DET
ejpam-5693	61	4	proof	proof	NOUN
ejpam-5693	61	5	.	.	PUNCT
ejpam-5693	62	1	2.2	2.2	NUM
ejpam-5693	62	2	.	.	PUNCT
ejpam-5693	62	3	inequalities	inequality	NOUN
ejpam-5693	62	4	for	for	ADP
ejpam-5693	62	5	matrices	matrix	NOUN
ejpam-5693	62	6	in	in	ADP
ejpam-5693	62	7	the	the	DET
ejpam-5693	62	8	following	follow	VERB
ejpam-5693	62	9	theorem	theorem	NOUN
ejpam-5693	62	10	we	we	PRON
ejpam-5693	62	11	introduce	introduce	VERB
ejpam-5693	62	12	matrix	matrix	NOUN
ejpam-5693	62	13	versions	version	NOUN
ejpam-5693	62	14	of	of	ADP
ejpam-5693	62	15	the	the	DET
ejpam-5693	62	16	inequalities	inequality	NOUN
ejpam-5693	62	17	(	(	PUNCT
ejpam-5693	62	18	2.1	2.1	NUM
ejpam-5693	62	19	)	)	PUNCT
ejpam-5693	62	20	and	and	CCONJ
ejpam-5693	62	21	(	(	PUNCT
ejpam-5693	62	22	2.2	2.2	NUM
ejpam-5693	62	23	)	)	PUNCT
ejpam-5693	62	24	,	,	PUNCT
ejpam-5693	62	25	using	use	VERB
ejpam-5693	62	26	the	the	DET
ejpam-5693	62	27	spectral	spectral	ADJ
ejpam-5693	62	28	theorem	theorem	NOUN
ejpam-5693	62	29	for	for	ADP
ejpam-5693	62	30	positive	positive	ADJ
ejpam-5693	62	31	semidefinite	semidefinite	NOUN
ejpam-5693	62	32	matrices	matrix	NOUN
ejpam-5693	62	33	.	.	PUNCT
ejpam-5693	63	1	theorem	theorem	NOUN
ejpam-5693	63	2	2	2	NUM
ejpam-5693	63	3	.	.	PUNCT
ejpam-5693	64	1	let	let	VERB
ejpam-5693	64	2	a	a	DET
ejpam-5693	64	3	,	,	PUNCT
ejpam-5693	64	4	b	b	NOUN
ejpam-5693	64	5	,	,	PUNCT
ejpam-5693	64	6	x	x	PROPN
ejpam-5693	64	7	∈	∈	PROPN
ejpam-5693	64	8	mn	mn	PROPN
ejpam-5693	64	9	such	such	ADJ
ejpam-5693	64	10	that	that	SCONJ
ejpam-5693	64	11	a	a	PRON
ejpam-5693	64	12	and	and	CCONJ
ejpam-5693	64	13	b	b	NOUN
ejpam-5693	64	14	are	be	AUX
ejpam-5693	64	15	positive	positive	ADJ
ejpam-5693	64	16	semidefinite	semidefinite	NOUN
ejpam-5693	64	17	.	.	PUNCT
ejpam-5693	65	1	if	if	SCONJ
ejpam-5693	65	2	0	0	NUM
ejpam-5693	65	3	≤	≤	NUM
ejpam-5693	65	4	µ	µ	X
ejpam-5693	65	5	≤	≤	NUM
ejpam-5693	65	6	1	1	NUM
ejpam-5693	65	7	2	2	NUM
ejpam-5693	65	8	,	,	PUNCT
ejpam-5693	65	9	then	then	ADV
ejpam-5693	65	10	∥µax	∥µax	PROPN
ejpam-5693	65	11	+	+	CCONJ
ejpam-5693	65	12	(	(	PUNCT
ejpam-5693	65	13	1−	1−	NUM
ejpam-5693	65	14	µ)xb∥22	µ)xb∥22	PROPN
ejpam-5693	65	15	≤	≤	PUNCT
ejpam-5693	65	16	µ2µ∥aµxb1−µ∥22	µ2µ∥aµxb1−µ∥22	NOUN
ejpam-5693	65	17	+	+	CCONJ
ejpam-5693	65	18	µ2∥ax	µ2∥ax	ADJ
ejpam-5693	65	19	−xb∥22	−xb∥22	NUM
ejpam-5693	65	20	+	+	NUM
ejpam-5693	65	21	r0	r0	NOUN
ejpam-5693	65	22	[	[	PUNCT
ejpam-5693	65	23	µ∥a	µ∥a	NOUN
ejpam-5693	65	24	1	1	NUM
ejpam-5693	65	25	2xb	2xb	ADJ
ejpam-5693	65	26	1	1	NUM
ejpam-5693	65	27	2	2	NUM
ejpam-5693	65	28	∥22	∥22	ADJ
ejpam-5693	65	29	+	+	NUM
ejpam-5693	65	30	∥xb∥22	∥xb∥22	PROPN
ejpam-5693	65	31	−	−	PROPN
ejpam-5693	65	32	2	2	NUM
ejpam-5693	65	33	√	√	NUM
ejpam-5693	65	34	µ∥a	µ∥a	VERB
ejpam-5693	65	35	1	1	NUM
ejpam-5693	65	36	4xb	4xb	NOUN
ejpam-5693	65	37	3	3	NUM
ejpam-5693	65	38	4	4	NUM
ejpam-5693	65	39	∥22	∥22	ADJ
ejpam-5693	65	40	]	]	PUNCT
ejpam-5693	66	1	+	+	PUNCT
ejpam-5693	66	2	2µ(1−	2µ(1−	NUM
ejpam-5693	66	3	µ)∥a	µ)∥a	ADJ
ejpam-5693	66	4	1	1	NUM
ejpam-5693	66	5	2xb	2xb	ADJ
ejpam-5693	66	6	1	1	NUM
ejpam-5693	66	7	2	2	NUM
ejpam-5693	66	8	∥22	∥22	VERB
ejpam-5693	66	9	,	,	PUNCT
ejpam-5693	66	10	(	(	PUNCT
ejpam-5693	66	11	2.3	2.3	NUM
ejpam-5693	66	12	)	)	PUNCT
ejpam-5693	66	13	where	where	SCONJ
ejpam-5693	66	14	r0	r0	NOUN
ejpam-5693	66	15	=	=	SYM
ejpam-5693	66	16	max{2µ	max{2µ	PROPN
ejpam-5693	66	17	,	,	PUNCT
ejpam-5693	66	18	1−	1−	NUM
ejpam-5693	66	19	2µ	2µ	NUM
ejpam-5693	66	20	}	}	PUNCT
ejpam-5693	66	21	.	.	PUNCT
ejpam-5693	67	1	if	if	SCONJ
ejpam-5693	67	2	1	1	NUM
ejpam-5693	67	3	2	2	NUM
ejpam-5693	67	4	≤	≤	NOUN
ejpam-5693	67	5	µ	µ	PRON
ejpam-5693	67	6	≤	≤	NUM
ejpam-5693	67	7	1	1	NUM
ejpam-5693	67	8	,	,	PUNCT
ejpam-5693	67	9	then	then	ADV
ejpam-5693	67	10	∥µax	∥µax	PROPN
ejpam-5693	67	11	+	+	CCONJ
ejpam-5693	67	12	(	(	PUNCT
ejpam-5693	67	13	1−	1−	NUM
ejpam-5693	67	14	µ)xb∥22	µ)xb∥22	PROPN
ejpam-5693	67	15	≤	≤	PROPN
ejpam-5693	67	16	(	(	PUNCT
ejpam-5693	67	17	1−	1−	NUM
ejpam-5693	67	18	µ)2(1−µ)∥aµxb1−µ∥22	µ)2(1−µ)∥aµxb1−µ∥22	VERB
ejpam-5693	67	19	+	+	CCONJ
ejpam-5693	67	20	(	(	PUNCT
ejpam-5693	67	21	1−	1−	NUM
ejpam-5693	67	22	µ)2∥ax	µ)2∥ax	ADJ
ejpam-5693	67	23	−xb∥22	−xb∥22	PUNCT
ejpam-5693	67	24	+	+	NUM
ejpam-5693	67	25	r0	r0	NOUN
ejpam-5693	67	26	[	[	PUNCT
ejpam-5693	67	27	(	(	PUNCT
ejpam-5693	67	28	1−	1−	NUM
ejpam-5693	67	29	µ)∥a	µ)∥a	ADJ
ejpam-5693	67	30	1	1	NUM
ejpam-5693	67	31	2xb	2xb	ADJ
ejpam-5693	67	32	1	1	NUM
ejpam-5693	67	33	2	2	NUM
ejpam-5693	67	34	∥22	∥22	ADJ
ejpam-5693	67	35	+	+	CCONJ
ejpam-5693	67	36	∥ax∥22	∥ax∥22	PROPN
ejpam-5693	67	37	−	−	NUM
ejpam-5693	67	38	2	2	NUM
ejpam-5693	67	39	√	√	NUM
ejpam-5693	67	40	(	(	PUNCT
ejpam-5693	67	41	1−	1−	NUM
ejpam-5693	67	42	µ)∥a	µ)∥a	ADJ
ejpam-5693	67	43	3	3	NUM
ejpam-5693	67	44	4xb	4xb	NOUN
ejpam-5693	67	45	1	1	NUM
ejpam-5693	67	46	4	4	NUM
ejpam-5693	67	47	∥22	∥22	ADJ
ejpam-5693	67	48	]	]	PUNCT
ejpam-5693	68	1	+	+	PUNCT
ejpam-5693	68	2	2µ(1−	2µ(1−	NUM
ejpam-5693	68	3	µ)∥a	µ)∥a	ADJ
ejpam-5693	68	4	1	1	NUM
ejpam-5693	68	5	2xb	2xb	ADJ
ejpam-5693	68	6	1	1	NUM
ejpam-5693	68	7	2	2	NUM
ejpam-5693	68	8	∥22	∥22	VERB
ejpam-5693	68	9	,	,	PUNCT
ejpam-5693	68	10	(	(	PUNCT
ejpam-5693	68	11	2.4	2.4	NUM
ejpam-5693	68	12	)	)	PUNCT
ejpam-5693	68	13	o.	o.	PROPN
ejpam-5693	68	14	ramadan	ramadan	PROPN
ejpam-5693	68	15	,	,	PUNCT
ejpam-5693	68	16	a.	a.	PROPN
ejpam-5693	68	17	burqan	burqan	PROPN
ejpam-5693	68	18	/	/	SYM
ejpam-5693	68	19	eur	eur	PROPN
ejpam-5693	68	20	.	.	PUNCT
ejpam-5693	69	1	j.	j.	PROPN
ejpam-5693	69	2	pure	pure	PROPN
ejpam-5693	69	3	appl	appl	PROPN
ejpam-5693	69	4	.	.	PROPN
ejpam-5693	69	5	math	math	PROPN
ejpam-5693	69	6	,	,	PUNCT
ejpam-5693	69	7	18	18	NUM
ejpam-5693	69	8	(	(	PUNCT
ejpam-5693	69	9	1	1	NUM
ejpam-5693	69	10	)	)	PUNCT
ejpam-5693	69	11	(	(	PUNCT
ejpam-5693	69	12	2025	2025	NUM
ejpam-5693	69	13	)	)	PUNCT
ejpam-5693	69	14	,	,	PUNCT
ejpam-5693	69	15	5693	5693	NUM
ejpam-5693	69	16	5	5	NUM
ejpam-5693	69	17	of	of	ADP
ejpam-5693	69	18	8	8	NUM
ejpam-5693	69	19	where	where	SCONJ
ejpam-5693	69	20	r0	r0	NOUN
ejpam-5693	69	21	=	=	SYM
ejpam-5693	69	22	max{2µ−	max{2µ−	PROPN
ejpam-5693	69	23	1	1	NUM
ejpam-5693	69	24	,	,	PUNCT
ejpam-5693	69	25	2−	2−	NUM
ejpam-5693	69	26	2µ	2µ	NUM
ejpam-5693	69	27	}	}	PUNCT
ejpam-5693	69	28	.	.	PUNCT
ejpam-5693	70	1	proof	proof	NOUN
ejpam-5693	70	2	.	.	PUNCT
ejpam-5693	71	1	since	since	SCONJ
ejpam-5693	71	2	every	every	DET
ejpam-5693	71	3	positive	positive	ADJ
ejpam-5693	71	4	semidefinite	semidefinite	NOUN
ejpam-5693	71	5	matrix	matrix	NOUN
ejpam-5693	71	6	is	be	AUX
ejpam-5693	71	7	unitarily	unitarily	ADV
ejpam-5693	71	8	diagonalizable	diagonalizable	ADJ
ejpam-5693	71	9	,	,	PUNCT
ejpam-5693	71	10	hence	hence	ADV
ejpam-5693	71	11	it	it	PRON
ejpam-5693	71	12	follows	follow	VERB
ejpam-5693	71	13	that	that	SCONJ
ejpam-5693	71	14	there	there	PRON
ejpam-5693	71	15	are	be	VERB
ejpam-5693	71	16	unitary	unitary	ADJ
ejpam-5693	71	17	matrices	matrix	NOUN
ejpam-5693	71	18	u	u	NOUN
ejpam-5693	71	19	,	,	PUNCT
ejpam-5693	71	20	v	v	PROPN
ejpam-5693	71	21	∈	∈	PROPN
ejpam-5693	71	22	mn	mn	NOUN
ejpam-5693	71	23	such	such	ADJ
ejpam-5693	71	24	that	that	SCONJ
ejpam-5693	71	25	a	a	DET
ejpam-5693	71	26	=	=	X
ejpam-5693	71	27	ucu∗and	ucu∗and	NOUN
ejpam-5693	71	28	b	b	X
ejpam-5693	71	29	=	=	SYM
ejpam-5693	71	30	v	v	PROPN
ejpam-5693	71	31	dv	dv	PROPN
ejpam-5693	71	32	∗	∗	NOUN
ejpam-5693	71	33	,	,	PUNCT
ejpam-5693	71	34	where	where	SCONJ
ejpam-5693	71	35	c	c	NOUN
ejpam-5693	71	36	=	=	SYM
ejpam-5693	71	37	diag(α1	diag(α1	NOUN
ejpam-5693	71	38	,	,	PUNCT
ejpam-5693	71	39	...	...	PUNCT
ejpam-5693	71	40	,	,	PUNCT
ejpam-5693	71	41	αn	αn	NOUN
ejpam-5693	71	42	)	)	PUNCT
ejpam-5693	71	43	,	,	PUNCT
ejpam-5693	71	44	d	d	NOUN
ejpam-5693	71	45	=	=	PUNCT
ejpam-5693	71	46	diag(β1	diag(β1	PROPN
ejpam-5693	71	47	,	,	PUNCT
ejpam-5693	71	48	...	...	PUNCT
ejpam-5693	71	49	,	,	PUNCT
ejpam-5693	71	50	βn	βn	NOUN
ejpam-5693	71	51	)	)	PUNCT
ejpam-5693	71	52	,	,	PUNCT
ejpam-5693	71	53	and	and	CCONJ
ejpam-5693	71	54	αi	αi	PROPN
ejpam-5693	71	55	,	,	PUNCT
ejpam-5693	71	56	βi	βi	PRON
ejpam-5693	71	57	≥	≥	NOUN
ejpam-5693	71	58	0	0	NUM
ejpam-5693	71	59	,	,	PUNCT
ejpam-5693	71	60	i	i	PRON
ejpam-5693	71	61	=	=	NOUN
ejpam-5693	71	62	1	1	NUM
ejpam-5693	71	63	,	,	PUNCT
ejpam-5693	71	64	...	...	PUNCT
ejpam-5693	71	65	,	,	PUNCT
ejpam-5693	71	66	n.	n.	PROPN
ejpam-5693	71	67	let	let	VERB
ejpam-5693	71	68	y	y	NOUN
ejpam-5693	71	69	=	=	SYM
ejpam-5693	71	70	u∗xv	u∗xv	PROPN
ejpam-5693	71	71	=	=	SYM
ejpam-5693	71	72	[	[	PUNCT
ejpam-5693	71	73	yij	yij	NOUN
ejpam-5693	71	74	]	]	PUNCT
ejpam-5693	71	75	.	.	PUNCT
ejpam-5693	72	1	then	then	ADV
ejpam-5693	72	2	we	we	PRON
ejpam-5693	72	3	have	have	VERB
ejpam-5693	72	4	µax	µax	PRON
ejpam-5693	72	5	+	+	SYM
ejpam-5693	72	6	(	(	PUNCT
ejpam-5693	72	7	1−	1−	NUM
ejpam-5693	72	8	µ)xb	µ)xb	PROPN
ejpam-5693	72	9	=	=	SYM
ejpam-5693	72	10	u	u	NOUN
ejpam-5693	72	11	[	[	PUNCT
ejpam-5693	72	12	(	(	PUNCT
ejpam-5693	72	13	µαi	µαi	NOUN
ejpam-5693	72	14	+	+	X
ejpam-5693	72	15	(	(	PUNCT
ejpam-5693	72	16	1−	1−	NUM
ejpam-5693	72	17	µ)βj)yij	µ)βj)yij	PROPN
ejpam-5693	72	18	]	]	PUNCT
ejpam-5693	72	19	v	v	NOUN
ejpam-5693	72	20	∗	∗	NOUN
ejpam-5693	72	21	,	,	PUNCT
ejpam-5693	72	22	ax	ax	NOUN
ejpam-5693	72	23	−xb	−xb	PROPN
ejpam-5693	72	24	=	=	SYM
ejpam-5693	72	25	u	u	NOUN
ejpam-5693	72	26	[	[	PUNCT
ejpam-5693	72	27	(	(	PUNCT
ejpam-5693	72	28	αi	αi	NOUN
ejpam-5693	72	29	−	−	PROPN
ejpam-5693	72	30	βj)yij	βj)yij	PROPN
ejpam-5693	72	31	]	]	PUNCT
ejpam-5693	72	32	v	v	NUM
ejpam-5693	72	33	∗	∗	NOUN
ejpam-5693	72	34	,	,	PUNCT
ejpam-5693	72	35	a	a	DET
ejpam-5693	72	36	1	1	NUM
ejpam-5693	72	37	2xb	2xb	ADJ
ejpam-5693	72	38	1	1	NUM
ejpam-5693	72	39	2	2	NUM
ejpam-5693	72	40	=	=	SYM
ejpam-5693	72	41	u	u	NOUN
ejpam-5693	72	42	[	[	PUNCT
ejpam-5693	72	43	(	(	PUNCT
ejpam-5693	72	44	α	α	NOUN
ejpam-5693	72	45	1	1	NUM
ejpam-5693	72	46	2	2	NUM
ejpam-5693	72	47	i	i	PRON
ejpam-5693	72	48	β	β	VERB
ejpam-5693	72	49	1	1	NUM
ejpam-5693	72	50	2	2	NUM
ejpam-5693	72	51	j	j	NOUN
ejpam-5693	72	52	)	)	PUNCT
ejpam-5693	72	53	yij	yij	NOUN
ejpam-5693	72	54	]	]	PUNCT
ejpam-5693	72	55	v	v	ADP
ejpam-5693	72	56	∗	∗	NOUN
ejpam-5693	72	57	and	and	CCONJ
ejpam-5693	72	58	aµxb1−µ	aµxb1−µ	PROPN
ejpam-5693	72	59	=	=	SYM
ejpam-5693	72	60	u	u	PROPN
ejpam-5693	72	61	[	[	PUNCT
ejpam-5693	72	62	(	(	PUNCT
ejpam-5693	72	63	αµ	αµ	INTJ
ejpam-5693	72	64	i	i	PRON
ejpam-5693	72	65	β	β	VERB
ejpam-5693	72	66	1−µ	1−µ	NUM
ejpam-5693	72	67	j	j	X
ejpam-5693	72	68	)	)	PUNCT
ejpam-5693	72	69	yij	yij	NOUN
ejpam-5693	72	70	]	]	PUNCT
ejpam-5693	72	71	v	v	X
ejpam-5693	72	72	∗.	∗.	PROPN
ejpam-5693	72	73	it	it	PRON
ejpam-5693	72	74	is	be	AUX
ejpam-5693	72	75	known	know	VERB
ejpam-5693	72	76	that	that	SCONJ
ejpam-5693	72	77	the	the	DET
ejpam-5693	72	78	hilbert	hilbert	NOUN
ejpam-5693	72	79	-	-	PUNCT
ejpam-5693	72	80	schmidt	schmidt	ADJ
ejpam-5693	72	81	norm	norm	NOUN
ejpam-5693	72	82	is	be	AUX
ejpam-5693	72	83	unitarily	unitarily	ADV
ejpam-5693	72	84	invariant	invariant	ADJ
ejpam-5693	72	85	,	,	PUNCT
ejpam-5693	72	86	so	so	ADV
ejpam-5693	72	87	if	if	SCONJ
ejpam-5693	72	88	0	0	NUM
ejpam-5693	72	89	≤	≤	NUM
ejpam-5693	72	90	µ	µ	X
ejpam-5693	72	91	≤	≤	NUM
ejpam-5693	72	92	1	1	NUM
ejpam-5693	72	93	2	2	NUM
ejpam-5693	72	94	,	,	PUNCT
ejpam-5693	72	95	inequality	inequality	NOUN
ejpam-5693	72	96	(	(	PUNCT
ejpam-5693	72	97	2.1	2.1	NUM
ejpam-5693	72	98	)	)	PUNCT
ejpam-5693	72	99	yields	yield	NOUN
ejpam-5693	72	100	that	that	PRON
ejpam-5693	72	101	∥µax	∥µax	PROPN
ejpam-5693	72	102	+	+	CCONJ
ejpam-5693	72	103	(	(	PUNCT
ejpam-5693	72	104	1−	1−	NUM
ejpam-5693	72	105	µ)xb∥22	µ)xb∥22	PROPN
ejpam-5693	72	106	=	=	PUNCT
ejpam-5693	73	1	n∑	n∑	NOUN
ejpam-5693	73	2	i	i	PROPN
ejpam-5693	73	3	,	,	PUNCT
ejpam-5693	73	4	j=1	j=1	PROPN
ejpam-5693	73	5	(	(	PUNCT
ejpam-5693	73	6	µαi	µαi	PROPN
ejpam-5693	73	7	+	+	CCONJ
ejpam-5693	73	8	(	(	PUNCT
ejpam-5693	73	9	1−	1−	NUM
ejpam-5693	73	10	µ)βj	µ)βj	PROPN
ejpam-5693	73	11	)	)	PUNCT
ejpam-5693	73	12	2	2	NUM
ejpam-5693	73	13	|yij	|yij	NOUN
ejpam-5693	73	14	|2	|2	NUM
ejpam-5693	73	15	≤	≤	NUM
ejpam-5693	73	16	µ2	µ2	PROPN
ejpam-5693	73	17	n∑	n∑	PROPN
ejpam-5693	73	18	i	i	PROPN
ejpam-5693	73	19	,	,	PUNCT
ejpam-5693	73	20	j=1	j=1	PROPN
ejpam-5693	73	21	(	(	PUNCT
ejpam-5693	73	22	αi	αi	INTJ
ejpam-5693	73	23	−	−	PROPN
ejpam-5693	73	24	βj	βj	SYM
ejpam-5693	73	25	)	)	PUNCT
ejpam-5693	73	26	2	2	NUM
ejpam-5693	73	27	|yij	|yij	NOUN
ejpam-5693	73	28	|2	|2	NUM
ejpam-5693	73	29	+	+	NOUN
ejpam-5693	73	30	µ2µ	µ2µ	PROPN
ejpam-5693	73	31	n∑	n∑	PROPN
ejpam-5693	73	32	i	i	PROPN
ejpam-5693	73	33	,	,	PUNCT
ejpam-5693	73	34	j=1	j=1	PROPN
ejpam-5693	73	35	(	(	PUNCT
ejpam-5693	73	36	αµ	αµ	VERB
ejpam-5693	73	37	i	i	PRON
ejpam-5693	73	38	β	β	VERB
ejpam-5693	73	39	1−µ	1−µ	NUM
ejpam-5693	73	40	j	j	NOUN
ejpam-5693	73	41	)	)	PUNCT
ejpam-5693	73	42	2	2	NUM
ejpam-5693	73	43	|yij	|yij	NOUN
ejpam-5693	73	44	|2	|2	NUM
ejpam-5693	73	45	+	+	CCONJ
ejpam-5693	73	46	2µ(1−	2µ(1−	NUM
ejpam-5693	73	47	µ	µ	NOUN
ejpam-5693	73	48	)	)	PUNCT
ejpam-5693	73	49	n∑	n∑	NOUN
ejpam-5693	74	1	i	i	PROPN
ejpam-5693	74	2	,	,	PUNCT
ejpam-5693	74	3	j=1	j=1	PROPN
ejpam-5693	74	4	(	(	PUNCT
ejpam-5693	74	5	α	α	NOUN
ejpam-5693	74	6	1	1	NUM
ejpam-5693	74	7	2	2	NUM
ejpam-5693	74	8	i	i	PRON
ejpam-5693	74	9	β	β	VERB
ejpam-5693	74	10	1	1	NUM
ejpam-5693	74	11	2	2	NUM
ejpam-5693	74	12	j	j	NOUN
ejpam-5693	74	13	)	)	PUNCT
ejpam-5693	74	14	2	2	NUM
ejpam-5693	74	15	|yij	|yij	NOUN
ejpam-5693	74	16	|2	|2	NUM
ejpam-5693	74	17	+	+	PROPN
ejpam-5693	74	18	r0	r0	NOUN
ejpam-5693	74	19	[	[	PUNCT
ejpam-5693	74	20	µ	µ	X
ejpam-5693	74	21	n∑	n∑	X
ejpam-5693	74	22	i	i	PROPN
ejpam-5693	74	23	,	,	PUNCT
ejpam-5693	74	24	j=1	j=1	PROPN
ejpam-5693	74	25	(	(	PUNCT
ejpam-5693	74	26	α	α	NOUN
ejpam-5693	74	27	1	1	NUM
ejpam-5693	74	28	2	2	NUM
ejpam-5693	74	29	i	i	PRON
ejpam-5693	74	30	β	β	VERB
ejpam-5693	74	31	1	1	NUM
ejpam-5693	74	32	2	2	NUM
ejpam-5693	74	33	j	j	NOUN
ejpam-5693	74	34	)	)	PUNCT
ejpam-5693	74	35	2	2	NUM
ejpam-5693	74	36	|yij	|yij	NOUN
ejpam-5693	74	37	|2	|2	NUM
ejpam-5693	75	1	+	+	X
ejpam-5693	75	2	n∑	n∑	PROPN
ejpam-5693	75	3	i	i	PROPN
ejpam-5693	75	4	,	,	PUNCT
ejpam-5693	75	5	j=1	j=1	PROPN
ejpam-5693	75	6	β2	β2	PROPN
ejpam-5693	75	7	j	j	PROPN
ejpam-5693	75	8	|yij	|yij	PROPN
ejpam-5693	75	9	|2	|2	NUM
ejpam-5693	75	10	−	−	NUM
ejpam-5693	75	11	2	2	NUM
ejpam-5693	75	12	√	√	PROPN
ejpam-5693	75	13	µ	µ	PROPN
ejpam-5693	75	14	n∑	n∑	NOUN
ejpam-5693	76	1	i	i	PROPN
ejpam-5693	76	2	,	,	PUNCT
ejpam-5693	76	3	j=1	j=1	PROPN
ejpam-5693	76	4	(	(	PUNCT
ejpam-5693	76	5	α	α	NOUN
ejpam-5693	76	6	1	1	NUM
ejpam-5693	76	7	4	4	NUM
ejpam-5693	76	8	i	i	NOUN
ejpam-5693	76	9	β	β	VERB
ejpam-5693	76	10	3	3	NUM
ejpam-5693	76	11	4	4	NUM
ejpam-5693	76	12	j	j	NOUN
ejpam-5693	76	13	)	)	PUNCT
ejpam-5693	76	14	2	2	NUM
ejpam-5693	76	15	|yij	|yij	NOUN
ejpam-5693	76	16	|2	|2	NUM
ejpam-5693	76	17	]	]	PUNCT
ejpam-5693	76	18	=	=	PUNCT
ejpam-5693	76	19	µ2∥ax	µ2∥ax	ADJ
ejpam-5693	76	20	−xb∥22	−xb∥22	PUNCT
ejpam-5693	76	21	+	+	NUM
ejpam-5693	76	22	µ2µ∥aµxb1−µ∥22	µ2µ∥aµxb1−µ∥22	X
ejpam-5693	76	23	+	+	CCONJ
ejpam-5693	76	24	2µ(1−	2µ(1−	NUM
ejpam-5693	76	25	µ)∥a	µ)∥a	ADJ
ejpam-5693	76	26	1	1	NUM
ejpam-5693	76	27	2xb	2xb	ADJ
ejpam-5693	76	28	1	1	NUM
ejpam-5693	76	29	2	2	NUM
ejpam-5693	76	30	∥22	∥22	ADJ
ejpam-5693	76	31	+	+	PROPN
ejpam-5693	76	32	r0	r0	NOUN
ejpam-5693	76	33	[	[	PUNCT
ejpam-5693	76	34	µ∥a	µ∥a	NOUN
ejpam-5693	76	35	1	1	NUM
ejpam-5693	76	36	2xb	2xb	ADJ
ejpam-5693	76	37	1	1	NUM
ejpam-5693	76	38	2	2	NUM
ejpam-5693	76	39	∥22	∥22	ADJ
ejpam-5693	76	40	+	+	NUM
ejpam-5693	76	41	∥xb∥22	∥xb∥22	PROPN
ejpam-5693	76	42	−	−	PROPN
ejpam-5693	76	43	2	2	NUM
ejpam-5693	76	44	√	√	NUM
ejpam-5693	76	45	µ∥a	µ∥a	VERB
ejpam-5693	76	46	1	1	NUM
ejpam-5693	76	47	4xb	4xb	NOUN
ejpam-5693	76	48	3	3	NUM
ejpam-5693	76	49	4	4	NUM
ejpam-5693	76	50	∥22	∥22	PROPN
ejpam-5693	76	51	]	]	PUNCT
ejpam-5693	76	52	and	and	CCONJ
ejpam-5693	76	53	so	so	ADV
ejpam-5693	76	54	∥µax	∥µax	PROPN
ejpam-5693	77	1	+	+	CCONJ
ejpam-5693	77	2	(	(	PUNCT
ejpam-5693	77	3	1−	1−	NUM
ejpam-5693	77	4	µ)xb∥22	µ)xb∥22	PROPN
ejpam-5693	77	5	≤	≤	PUNCT
ejpam-5693	77	6	µ2µ∥aµxb1−µ∥22	µ2µ∥aµxb1−µ∥22	NOUN
ejpam-5693	77	7	+	+	CCONJ
ejpam-5693	77	8	µ2∥ax	µ2∥ax	ADJ
ejpam-5693	77	9	−xb∥22	−xb∥22	NUM
ejpam-5693	77	10	+	+	NUM
ejpam-5693	77	11	r0	r0	NOUN
ejpam-5693	77	12	[	[	PUNCT
ejpam-5693	77	13	µ∥a	µ∥a	NOUN
ejpam-5693	77	14	1	1	NUM
ejpam-5693	77	15	2xb	2xb	ADJ
ejpam-5693	77	16	1	1	NUM
ejpam-5693	77	17	2	2	NUM
ejpam-5693	77	18	∥22	∥22	ADJ
ejpam-5693	77	19	+	+	NUM
ejpam-5693	77	20	∥xb∥22	∥xb∥22	PROPN
ejpam-5693	77	21	−	−	PROPN
ejpam-5693	77	22	2	2	NUM
ejpam-5693	77	23	√	√	NUM
ejpam-5693	77	24	µ∥a	µ∥a	VERB
ejpam-5693	77	25	1	1	NUM
ejpam-5693	77	26	4xb	4xb	NOUN
ejpam-5693	77	27	3	3	NUM
ejpam-5693	77	28	4	4	NUM
ejpam-5693	77	29	∥22	∥22	ADJ
ejpam-5693	77	30	]	]	PUNCT
ejpam-5693	78	1	+	+	PUNCT
ejpam-5693	78	2	2µ(1−	2µ(1−	NUM
ejpam-5693	78	3	µ)∥a	µ)∥a	ADJ
ejpam-5693	78	4	1	1	NUM
ejpam-5693	78	5	2xb	2xb	ADJ
ejpam-5693	78	6	1	1	NUM
ejpam-5693	78	7	2	2	NUM
ejpam-5693	78	8	∥22	∥22	VERB
ejpam-5693	78	9	,	,	PUNCT
ejpam-5693	78	10	thus	thus	ADV
ejpam-5693	78	11	,	,	PUNCT
ejpam-5693	78	12	we	we	PRON
ejpam-5693	78	13	get	get	VERB
ejpam-5693	78	14	(	(	PUNCT
ejpam-5693	78	15	2.3	2.3	NUM
ejpam-5693	78	16	)	)	PUNCT
ejpam-5693	78	17	.	.	PUNCT
ejpam-5693	79	1	if	if	SCONJ
ejpam-5693	79	2	1	1	NUM
ejpam-5693	79	3	2	2	NUM
ejpam-5693	79	4	≤	≤	NOUN
ejpam-5693	79	5	µ	µ	PRON
ejpam-5693	79	6	≤	≤	NUM
ejpam-5693	79	7	1	1	NUM
ejpam-5693	79	8	,	,	PUNCT
ejpam-5693	79	9	then	then	ADV
ejpam-5693	79	10	by	by	ADP
ejpam-5693	79	11	the	the	DET
ejpam-5693	79	12	inequality	inequality	NOUN
ejpam-5693	79	13	(	(	PUNCT
ejpam-5693	79	14	2.2	2.2	NUM
ejpam-5693	79	15	)	)	PUNCT
ejpam-5693	79	16	and	and	CCONJ
ejpam-5693	79	17	the	the	DET
ejpam-5693	79	18	same	same	ADJ
ejpam-5693	79	19	method	method	NOUN
ejpam-5693	79	20	above	above	ADV
ejpam-5693	79	21	,	,	PUNCT
ejpam-5693	79	22	we	we	PRON
ejpam-5693	79	23	have	have	VERB
ejpam-5693	79	24	the	the	DET
ejpam-5693	79	25	o.	o.	PROPN
ejpam-5693	79	26	ramadan	ramadan	PROPN
ejpam-5693	79	27	,	,	PUNCT
ejpam-5693	79	28	a.	a.	PROPN
ejpam-5693	79	29	burqan	burqan	PROPN
ejpam-5693	79	30	/	/	SYM
ejpam-5693	79	31	eur	eur	PROPN
ejpam-5693	79	32	.	.	PUNCT
ejpam-5693	80	1	j.	j.	PROPN
ejpam-5693	80	2	pure	pure	PROPN
ejpam-5693	80	3	appl	appl	PROPN
ejpam-5693	80	4	.	.	PROPN
ejpam-5693	80	5	math	math	PROPN
ejpam-5693	80	6	,	,	PUNCT
ejpam-5693	80	7	18	18	NUM
ejpam-5693	80	8	(	(	PUNCT
ejpam-5693	80	9	1	1	NUM
ejpam-5693	80	10	)	)	PUNCT
ejpam-5693	80	11	(	(	PUNCT
ejpam-5693	80	12	2025	2025	NUM
ejpam-5693	80	13	)	)	PUNCT
ejpam-5693	80	14	,	,	PUNCT
ejpam-5693	80	15	5693	5693	NUM
ejpam-5693	80	16	6	6	NUM
ejpam-5693	80	17	of	of	ADP
ejpam-5693	80	18	8	8	NUM
ejpam-5693	80	19	inequality	inequality	NOUN
ejpam-5693	80	20	(	(	PUNCT
ejpam-5693	80	21	2.4	2.4	NUM
ejpam-5693	80	22	)	)	PUNCT
ejpam-5693	80	23	.	.	PUNCT
ejpam-5693	81	1	this	this	PRON
ejpam-5693	81	2	completes	complete	VERB
ejpam-5693	81	3	the	the	DET
ejpam-5693	81	4	proof	proof	NOUN
ejpam-5693	81	5	.	.	PUNCT
ejpam-5693	82	1	finally	finally	ADV
ejpam-5693	82	2	,	,	PUNCT
ejpam-5693	82	3	we	we	PRON
ejpam-5693	82	4	obtain	obtain	VERB
ejpam-5693	82	5	refinements	refinement	NOUN
ejpam-5693	82	6	of	of	ADP
ejpam-5693	82	7	the	the	DET
ejpam-5693	82	8	trace	trace	NOUN
ejpam-5693	82	9	versions	version	NOUN
ejpam-5693	82	10	of	of	ADP
ejpam-5693	82	11	young	young	ADJ
ejpam-5693	82	12	type	type	NOUN
ejpam-5693	82	13	inequalities	inequality	NOUN
ejpam-5693	82	14	.	.	PUNCT
ejpam-5693	83	1	to	to	PART
ejpam-5693	83	2	achieve	achieve	VERB
ejpam-5693	83	3	this	this	PRON
ejpam-5693	83	4	,	,	PUNCT
ejpam-5693	83	5	we	we	PRON
ejpam-5693	83	6	need	need	VERB
ejpam-5693	83	7	the	the	DET
ejpam-5693	83	8	following	follow	VERB
ejpam-5693	83	9	lemmas	lemma	NOUN
ejpam-5693	83	10	that	that	PRON
ejpam-5693	83	11	can	can	AUX
ejpam-5693	83	12	be	be	AUX
ejpam-5693	83	13	found	find	VERB
ejpam-5693	83	14	in	in	ADP
ejpam-5693	83	15	[	[	X
ejpam-5693	83	16	2	2	NUM
ejpam-5693	83	17	]	]	PUNCT
ejpam-5693	83	18	.	.	PUNCT
ejpam-5693	84	1	lemma	lemma	PROPN
ejpam-5693	84	2	1	1	X
ejpam-5693	84	3	.	.	PUNCT
ejpam-5693	85	1	let	let	VERB
ejpam-5693	85	2	a	a	DET
ejpam-5693	85	3	,	,	PUNCT
ejpam-5693	85	4	b	b	PROPN
ejpam-5693	85	5	∈	∈	PROPN
ejpam-5693	85	6	mn	mn	PROPN
ejpam-5693	85	7	.	.	PUNCT
ejpam-5693	86	1	then	then	ADV
ejpam-5693	86	2	n∑	n∑	PROPN
ejpam-5693	86	3	j=1	j=1	PROPN
ejpam-5693	86	4	sj(ab	sj(ab	PROPN
ejpam-5693	86	5	)	)	PUNCT
ejpam-5693	86	6	≤	≤	PUNCT
ejpam-5693	87	1	n∑	n∑	PUNCT
ejpam-5693	87	2	j=1	j=1	PROPN
ejpam-5693	87	3	sj(a)sj(b	sj(a)sj(b	PROPN
ejpam-5693	87	4	)	)	PUNCT
ejpam-5693	87	5	.	.	PUNCT
ejpam-5693	88	1	lemma	lemma	PROPN
ejpam-5693	88	2	2	2	NUM
ejpam-5693	88	3	.	.	PUNCT
ejpam-5693	88	4	(	(	PUNCT
ejpam-5693	88	5	cauchy	cauchy	NOUN
ejpam-5693	88	6	-	-	PUNCT
ejpam-5693	88	7	schwarz	schwarz	PROPN
ejpam-5693	88	8	inequality	inequality	NOUN
ejpam-5693	88	9	)	)	PUNCT
ejpam-5693	88	10	.	.	PUNCT
ejpam-5693	89	1	let	let	VERB
ejpam-5693	89	2	ai	ai	VERB
ejpam-5693	89	3	≥	≥	NOUN
ejpam-5693	89	4	0	0	NUM
ejpam-5693	89	5	,	,	PUNCT
ejpam-5693	89	6	bi	bi	NOUN
ejpam-5693	89	7	≥	≥	NOUN
ejpam-5693	89	8	0	0	NUM
ejpam-5693	89	9	for	for	ADP
ejpam-5693	89	10	i	i	PRON
ejpam-5693	89	11	=	=	NOUN
ejpam-5693	89	12	1	1	NUM
ejpam-5693	89	13	,	,	PUNCT
ejpam-5693	89	14	...	...	PUNCT
ejpam-5693	89	15	,	,	PUNCT
ejpam-5693	89	16	n.	n.	NOUN
ejpam-5693	89	17	then	then	ADV
ejpam-5693	89	18	n∑	n∑	PROPN
ejpam-5693	89	19	i=1	i=1	PROPN
ejpam-5693	89	20	aibi	aibi	NOUN
ejpam-5693	89	21	≤	≤	NOUN
ejpam-5693	90	1	(	(	PUNCT
ejpam-5693	90	2	n∑	n∑	NOUN
ejpam-5693	90	3	i=1	i=1	PROPN
ejpam-5693	90	4	a2i	a2i	NOUN
ejpam-5693	90	5	)	)	PUNCT
ejpam-5693	90	6	1	1	NUM
ejpam-5693	90	7	2	2	NUM
ejpam-5693	90	8	(	(	PUNCT
ejpam-5693	90	9	n∑	n∑	NOUN
ejpam-5693	90	10	i=1	i=1	PROPN
ejpam-5693	90	11	b2i	b2i	PUNCT
ejpam-5693	90	12	)	)	PUNCT
ejpam-5693	90	13	1	1	NUM
ejpam-5693	90	14	2	2	NUM
ejpam-5693	90	15	.	.	PUNCT
ejpam-5693	91	1	theorem	theorem	NOUN
ejpam-5693	91	2	3	3	X
ejpam-5693	91	3	.	.	PUNCT
ejpam-5693	92	1	let	let	VERB
ejpam-5693	92	2	a	a	DET
ejpam-5693	92	3	,	,	PUNCT
ejpam-5693	92	4	b	b	NOUN
ejpam-5693	92	5	,	,	PUNCT
ejpam-5693	92	6	x	x	PROPN
ejpam-5693	92	7	∈	∈	PROPN
ejpam-5693	92	8	mn	mn	PROPN
ejpam-5693	92	9	such	such	ADJ
ejpam-5693	92	10	that	that	SCONJ
ejpam-5693	92	11	a	a	PRON
ejpam-5693	92	12	and	and	CCONJ
ejpam-5693	92	13	b	b	NOUN
ejpam-5693	92	14	are	be	AUX
ejpam-5693	92	15	positive	positive	ADJ
ejpam-5693	92	16	semidefinite	semidefinite	NOUN
ejpam-5693	92	17	.	.	PUNCT
ejpam-5693	93	1	if	if	SCONJ
ejpam-5693	93	2	0	0	NUM
ejpam-5693	93	3	≤	≤	NUM
ejpam-5693	93	4	µ	µ	X
ejpam-5693	93	5	≤	≤	NUM
ejpam-5693	93	6	1	1	NUM
ejpam-5693	93	7	2	2	NUM
ejpam-5693	93	8	,	,	PUNCT
ejpam-5693	93	9	then	then	ADV
ejpam-5693	93	10	tr(µ2a2	tr(µ2a2	PROPN
ejpam-5693	93	11	+	+	CCONJ
ejpam-5693	93	12	(	(	PUNCT
ejpam-5693	93	13	1−	1−	NUM
ejpam-5693	93	14	µ)2b2	µ)2b2	NUM
ejpam-5693	93	15	)	)	PUNCT
ejpam-5693	93	16	≤	≤	NUM
ejpam-5693	93	17	µ2µ∥aµ∥2∥b1−µ∥2	µ2µ∥aµ∥2∥b1−µ∥2	NOUN
ejpam-5693	93	18	+	+	CCONJ
ejpam-5693	93	19	µ2	µ2	PROPN
ejpam-5693	93	20	[	[	PUNCT
ejpam-5693	93	21	∥a∥22	∥a∥22	PROPN
ejpam-5693	93	22	+	+	CCONJ
ejpam-5693	93	23	∥b∥22	∥b∥22	ADJ
ejpam-5693	93	24	−	−	PROPN
ejpam-5693	94	1	2∥ab∥1	2∥ab∥1	PROPN
ejpam-5693	94	2	]	]	PUNCT
ejpam-5693	95	1	+	+	PUNCT
ejpam-5693	95	2	r0	r0	NOUN
ejpam-5693	95	3	[	[	PUNCT
ejpam-5693	95	4	µ∥a∥2∥b∥2	µ∥a∥2∥b∥2	PROPN
ejpam-5693	95	5	+	+	CCONJ
ejpam-5693	95	6	∥b∥22	∥b∥22	ADJ
ejpam-5693	95	7	−	−	NUM
ejpam-5693	95	8	2	2	NUM
ejpam-5693	95	9	√	√	NUM
ejpam-5693	95	10	µ∥a	µ∥a	VERB
ejpam-5693	95	11	1	1	NUM
ejpam-5693	95	12	2b	2b	NUM
ejpam-5693	95	13	3	3	NUM
ejpam-5693	95	14	2	2	NUM
ejpam-5693	95	15	∥1	∥1	NOUN
ejpam-5693	95	16	]	]	PUNCT
ejpam-5693	95	17	.	.	PUNCT
ejpam-5693	96	1	(	(	PUNCT
ejpam-5693	96	2	2.5	2.5	NUM
ejpam-5693	96	3	)	)	PUNCT
ejpam-5693	96	4	where	where	SCONJ
ejpam-5693	96	5	r0	r0	NOUN
ejpam-5693	96	6	=	=	SYM
ejpam-5693	96	7	max{2µ	max{2µ	PROPN
ejpam-5693	96	8	,	,	PUNCT
ejpam-5693	96	9	1−	1−	NUM
ejpam-5693	96	10	2µ	2µ	NUM
ejpam-5693	96	11	}	}	PUNCT
ejpam-5693	96	12	.	.	PUNCT
ejpam-5693	97	1	if	if	SCONJ
ejpam-5693	97	2	1	1	NUM
ejpam-5693	97	3	2	2	NUM
ejpam-5693	97	4	≤	≤	NOUN
ejpam-5693	97	5	µ	µ	PRON
ejpam-5693	97	6	≤	≤	NUM
ejpam-5693	97	7	1	1	NUM
ejpam-5693	97	8	,	,	PUNCT
ejpam-5693	97	9	then	then	ADV
ejpam-5693	97	10	tr(µ2a2	tr(µ2a2	PROPN
ejpam-5693	97	11	+	+	CCONJ
ejpam-5693	97	12	(	(	PUNCT
ejpam-5693	97	13	1−	1−	NUM
ejpam-5693	97	14	µ)2b2	µ)2b2	NUM
ejpam-5693	97	15	)	)	PUNCT
ejpam-5693	97	16	≤	≤	NOUN
ejpam-5693	97	17	(	(	PUNCT
ejpam-5693	97	18	1−	1−	NUM
ejpam-5693	97	19	µ)2µ∥aµ∥2∥b1−µ∥2	µ)2µ∥aµ∥2∥b1−µ∥2	PROPN
ejpam-5693	97	20	+	+	CCONJ
ejpam-5693	97	21	(	(	PUNCT
ejpam-5693	97	22	1−	1−	NUM
ejpam-5693	97	23	µ)2	µ)2	NOUN
ejpam-5693	97	24	[	[	PUNCT
ejpam-5693	97	25	∥a∥22	∥a∥22	PROPN
ejpam-5693	97	26	+	+	CCONJ
ejpam-5693	97	27	∥b∥22	∥b∥22	ADJ
ejpam-5693	97	28	−	−	PROPN
ejpam-5693	98	1	2∥ab∥1	2∥ab∥1	PROPN
ejpam-5693	98	2	]	]	PUNCT
ejpam-5693	99	1	+	+	PUNCT
ejpam-5693	99	2	r0	r0	NOUN
ejpam-5693	99	3	[	[	PUNCT
ejpam-5693	99	4	(	(	PUNCT
ejpam-5693	99	5	1−	1−	NUM
ejpam-5693	99	6	µ)∥a∥2∥b∥2	µ)∥a∥2∥b∥2	NOUN
ejpam-5693	99	7	+	+	CCONJ
ejpam-5693	99	8	∥a∥22	∥a∥22	PROPN
ejpam-5693	99	9	−	−	NUM
ejpam-5693	99	10	2	2	NUM
ejpam-5693	99	11	√	√	NUM
ejpam-5693	99	12	1−	1−	NUM
ejpam-5693	99	13	µ∥a	µ∥a	VERB
ejpam-5693	99	14	3	3	NUM
ejpam-5693	99	15	2b	2b	NUM
ejpam-5693	99	16	1	1	NUM
ejpam-5693	99	17	2	2	NUM
ejpam-5693	99	18	∥1	∥1	NOUN
ejpam-5693	99	19	]	]	PUNCT
ejpam-5693	99	20	.	.	PUNCT
ejpam-5693	100	1	(	(	PUNCT
ejpam-5693	100	2	2.6	2.6	NUM
ejpam-5693	100	3	)	)	PUNCT
ejpam-5693	100	4	where	where	SCONJ
ejpam-5693	100	5	r0	r0	NOUN
ejpam-5693	100	6	=	=	SYM
ejpam-5693	100	7	max{2µ−	max{2µ−	PROPN
ejpam-5693	100	8	1	1	NUM
ejpam-5693	100	9	,	,	PUNCT
ejpam-5693	100	10	2−	2−	NUM
ejpam-5693	100	11	2µ	2µ	NUM
ejpam-5693	100	12	}	}	PUNCT
ejpam-5693	100	13	.	.	PUNCT
ejpam-5693	101	1	proof	proof	NOUN
ejpam-5693	101	2	.	.	PUNCT
ejpam-5693	102	1	if	if	SCONJ
ejpam-5693	102	2	0	0	NUM
ejpam-5693	102	3	≤	≤	NUM
ejpam-5693	102	4	µ	µ	X
ejpam-5693	102	5	≤	≤	NUM
ejpam-5693	102	6	1	1	NUM
ejpam-5693	102	7	2	2	NUM
ejpam-5693	102	8	,	,	PUNCT
ejpam-5693	102	9	then	then	ADV
ejpam-5693	102	10	tr(µ2a2	tr(µ2a2	PROPN
ejpam-5693	102	11	+	+	CCONJ
ejpam-5693	102	12	(	(	PUNCT
ejpam-5693	102	13	1−	1−	NUM
ejpam-5693	102	14	µ)2b2	µ)2b2	NUM
ejpam-5693	102	15	)	)	PUNCT
ejpam-5693	102	16	=	=	PUNCT
ejpam-5693	102	17	µ2tra2	µ2tra2	VERB
ejpam-5693	102	18	+	+	CCONJ
ejpam-5693	102	19	(	(	PUNCT
ejpam-5693	102	20	1−	1−	NUM
ejpam-5693	102	21	µ)2trb2	µ)2trb2	NUM
ejpam-5693	102	22	=	=	SYM
ejpam-5693	102	23	n∑	n∑	NOUN
ejpam-5693	102	24	j=1	j=1	NOUN
ejpam-5693	102	25	(	(	PUNCT
ejpam-5693	102	26	µ2s2j	µ2s2j	X
ejpam-5693	102	27	(	(	PUNCT
ejpam-5693	102	28	a	a	X
ejpam-5693	102	29	)	)	PUNCT
ejpam-5693	102	30	+	+	CCONJ
ejpam-5693	102	31	(	(	PUNCT
ejpam-5693	102	32	1−	1−	NUM
ejpam-5693	102	33	µ)2s2j	µ)2s2j	X
ejpam-5693	102	34	(	(	PUNCT
ejpam-5693	102	35	b	b	NOUN
ejpam-5693	102	36	)	)	PUNCT
ejpam-5693	102	37	)	)	PUNCT
ejpam-5693	102	38	.	.	PUNCT
ejpam-5693	103	1	inequality	inequality	NOUN
ejpam-5693	103	2	(	(	PUNCT
ejpam-5693	103	3	2.1	2.1	NUM
ejpam-5693	103	4	)	)	PUNCT
ejpam-5693	103	5	yields	yield	NOUN
ejpam-5693	103	6	that	that	PRON
ejpam-5693	104	1	tr(µ2a2	tr(µ2a2	PROPN
ejpam-5693	104	2	+	+	CCONJ
ejpam-5693	104	3	(	(	PUNCT
ejpam-5693	104	4	1−	1−	NUM
ejpam-5693	104	5	µ)2b2	µ)2b2	NOUN
ejpam-5693	104	6	)	)	PUNCT
ejpam-5693	104	7	≤	≤	NOUN
ejpam-5693	104	8	µ2µ	µ2µ	PROPN
ejpam-5693	104	9	n∑	n∑	PROPN
ejpam-5693	104	10	j=1	j=1	PROPN
ejpam-5693	104	11	sj(a	sj(a	CCONJ
ejpam-5693	104	12	2µ)sj(b	2µ)sj(b	NUM
ejpam-5693	104	13	2(1−µ	2(1−µ	NUM
ejpam-5693	104	14	)	)	PUNCT
ejpam-5693	104	15	)	)	PUNCT
ejpam-5693	105	1	+	+	CCONJ
ejpam-5693	106	1	µ2	µ2	PROPN
ejpam-5693	106	2	[	[	PUNCT
ejpam-5693	106	3	n∑	n∑	NOUN
ejpam-5693	106	4	j=1	j=1	NOUN
ejpam-5693	106	5	s2j	s2j	NOUN
ejpam-5693	106	6	(	(	PUNCT
ejpam-5693	106	7	a	a	NOUN
ejpam-5693	106	8	)	)	PUNCT
ejpam-5693	106	9	+	+	NOUN
ejpam-5693	106	10	n∑	n∑	PROPN
ejpam-5693	106	11	j=1	j=1	NOUN
ejpam-5693	106	12	s2j	s2j	NOUN
ejpam-5693	106	13	(	(	PUNCT
ejpam-5693	106	14	b)−	b)−	PROPN
ejpam-5693	106	15	2	2	NUM
ejpam-5693	106	16	n∑	n∑	NOUN
ejpam-5693	106	17	j=1	j=1	NOUN
ejpam-5693	106	18	sj(a)sj(b	sj(a)sj(b	NOUN
ejpam-5693	106	19	)	)	PUNCT
ejpam-5693	106	20	]	]	PUNCT
ejpam-5693	107	1	+	+	PUNCT
ejpam-5693	107	2	r0	r0	NOUN
ejpam-5693	107	3	[	[	PUNCT
ejpam-5693	107	4	µ	µ	X
ejpam-5693	107	5	n∑	n∑	NOUN
ejpam-5693	107	6	j=1	j=1	ADJ
ejpam-5693	107	7	sj(a)sj(b	sj(a)sj(b	PROPN
ejpam-5693	107	8	)	)	PUNCT
ejpam-5693	108	1	+	+	CCONJ
ejpam-5693	108	2	n∑	n∑	PROPN
ejpam-5693	108	3	j=1	j=1	NOUN
ejpam-5693	108	4	s2j	s2j	NOUN
ejpam-5693	108	5	(	(	PUNCT
ejpam-5693	108	6	b)−	b)−	PROPN
ejpam-5693	108	7	2	2	NUM
ejpam-5693	108	8	√	√	PROPN
ejpam-5693	108	9	µ	µ	PROPN
ejpam-5693	108	10	n∑	n∑	NOUN
ejpam-5693	108	11	j=1	j=1	PROPN
ejpam-5693	108	12	s	s	PART
ejpam-5693	108	13	1	1	NUM
ejpam-5693	108	14	2	2	NUM
ejpam-5693	108	15	j	j	NOUN
ejpam-5693	108	16	(	(	PUNCT
ejpam-5693	108	17	a)s	a)s	NOUN
ejpam-5693	108	18	3	3	NUM
ejpam-5693	108	19	2	2	NUM
ejpam-5693	108	20	j	j	PROPN
ejpam-5693	108	21	(	(	PUNCT
ejpam-5693	108	22	b	b	NOUN
ejpam-5693	108	23	)	)	PUNCT
ejpam-5693	108	24	]	]	PUNCT
ejpam-5693	108	25	.	.	PUNCT
ejpam-5693	109	1	o.	o.	PROPN
ejpam-5693	109	2	ramadan	ramadan	PROPN
ejpam-5693	109	3	,	,	PUNCT
ejpam-5693	109	4	a.	a.	PROPN
ejpam-5693	109	5	burqan	burqan	PROPN
ejpam-5693	109	6	/	/	SYM
ejpam-5693	109	7	eur	eur	PROPN
ejpam-5693	109	8	.	.	PUNCT
ejpam-5693	110	1	j.	j.	PROPN
ejpam-5693	110	2	pure	pure	PROPN
ejpam-5693	110	3	appl	appl	PROPN
ejpam-5693	110	4	.	.	PROPN
ejpam-5693	110	5	math	math	PROPN
ejpam-5693	110	6	,	,	PUNCT
ejpam-5693	110	7	18	18	NUM
ejpam-5693	110	8	(	(	PUNCT
ejpam-5693	110	9	1	1	NUM
ejpam-5693	110	10	)	)	PUNCT
ejpam-5693	110	11	(	(	PUNCT
ejpam-5693	110	12	2025	2025	NUM
ejpam-5693	110	13	)	)	PUNCT
ejpam-5693	110	14	,	,	PUNCT
ejpam-5693	110	15	5693	5693	NUM
ejpam-5693	110	16	7	7	NUM
ejpam-5693	110	17	of	of	ADP
ejpam-5693	110	18	8	8	NUM
ejpam-5693	110	19	thus	thus	ADV
ejpam-5693	110	20	,	,	PUNCT
ejpam-5693	110	21	using	use	VERB
ejpam-5693	110	22	lemma	lemma	PROPN
ejpam-5693	110	23	1	1	NUM
ejpam-5693	110	24	and	and	CCONJ
ejpam-5693	110	25	lemma	lemma	PROPN
ejpam-5693	110	26	2	2	NUM
ejpam-5693	110	27	,	,	PUNCT
ejpam-5693	110	28	we	we	PRON
ejpam-5693	110	29	have	have	VERB
ejpam-5693	110	30	tr(µ2a2	tr(µ2a2	PROPN
ejpam-5693	110	31	+	+	NUM
ejpam-5693	110	32	(	(	PUNCT
ejpam-5693	110	33	1−	1−	NUM
ejpam-5693	110	34	µ)2b2	µ)2b2	NOUN
ejpam-5693	110	35	)	)	PUNCT
ejpam-5693	110	36	≤	≤	NOUN
ejpam-5693	111	1	µ2µ	µ2µ	PRON
ejpam-5693	111	2	(	(	PUNCT
ejpam-5693	111	3	n∑	n∑	NOUN
ejpam-5693	111	4	j=1	j=1	PROPN
ejpam-5693	111	5	s2j	s2j	NOUN
ejpam-5693	111	6	(	(	PUNCT
ejpam-5693	111	7	a	a	DET
ejpam-5693	111	8	2µ	2µ	NUM
ejpam-5693	111	9	)	)	PUNCT
ejpam-5693	111	10	)	)	PUNCT
ejpam-5693	111	11	1	1	NUM
ejpam-5693	111	12	2	2	NUM
ejpam-5693	111	13	(	(	PUNCT
ejpam-5693	111	14	n∑	n∑	NOUN
ejpam-5693	111	15	j=1	j=1	PROPN
ejpam-5693	111	16	s2j	s2j	NOUN
ejpam-5693	111	17	(	(	PUNCT
ejpam-5693	111	18	b	b	NOUN
ejpam-5693	111	19	2(1−µ	2(1−µ	NUM
ejpam-5693	111	20	)	)	PUNCT
ejpam-5693	111	21	)	)	PUNCT
ejpam-5693	111	22	)	)	PUNCT
ejpam-5693	111	23	1	1	NUM
ejpam-5693	111	24	2	2	NUM
ejpam-5693	111	25	+	+	CCONJ
ejpam-5693	111	26	µ2	µ2	PROPN
ejpam-5693	111	27	[	[	PUNCT
ejpam-5693	111	28	n∑	n∑	NOUN
ejpam-5693	111	29	j=1	j=1	NOUN
ejpam-5693	111	30	s2j	s2j	NOUN
ejpam-5693	111	31	(	(	PUNCT
ejpam-5693	111	32	a	a	NOUN
ejpam-5693	111	33	)	)	PUNCT
ejpam-5693	111	34	+	+	NOUN
ejpam-5693	111	35	n∑	n∑	PROPN
ejpam-5693	111	36	j=1	j=1	NOUN
ejpam-5693	111	37	s2j	s2j	NOUN
ejpam-5693	111	38	(	(	PUNCT
ejpam-5693	111	39	b)−	b)−	PROPN
ejpam-5693	111	40	2	2	NUM
ejpam-5693	111	41	n∑	n∑	NOUN
ejpam-5693	111	42	j=1	j=1	PROPN
ejpam-5693	111	43	sj(ab	sj(ab	PROPN
ejpam-5693	111	44	)	)	PUNCT
ejpam-5693	111	45	]	]	PUNCT
ejpam-5693	112	1	+	+	PUNCT
ejpam-5693	112	2	r0	r0	NOUN
ejpam-5693	112	3	[	[	PUNCT
ejpam-5693	112	4	µ	µ	X
ejpam-5693	112	5	(	(	PUNCT
ejpam-5693	112	6	n∑	n∑	NOUN
ejpam-5693	112	7	j=1	j=1	PROPN
ejpam-5693	112	8	s2j	s2j	NOUN
ejpam-5693	112	9	(	(	PUNCT
ejpam-5693	112	10	a	a	NOUN
ejpam-5693	112	11	)	)	PUNCT
ejpam-5693	112	12	)	)	PUNCT
ejpam-5693	112	13	1	1	NUM
ejpam-5693	112	14	2	2	NUM
ejpam-5693	112	15	(	(	PUNCT
ejpam-5693	112	16	n∑	n∑	NOUN
ejpam-5693	112	17	j=1	j=1	PROPN
ejpam-5693	112	18	s2j	s2j	NOUN
ejpam-5693	112	19	(	(	PUNCT
ejpam-5693	112	20	b	b	NOUN
ejpam-5693	112	21	)	)	PUNCT
ejpam-5693	112	22	)	)	PUNCT
ejpam-5693	112	23	1	1	NUM
ejpam-5693	112	24	2	2	NUM
ejpam-5693	112	25	+	+	NUM
ejpam-5693	112	26	n∑	n∑	ADJ
ejpam-5693	112	27	j=1	j=1	NOUN
ejpam-5693	112	28	s2j	s2j	NOUN
ejpam-5693	112	29	(	(	PUNCT
ejpam-5693	112	30	b	b	NOUN
ejpam-5693	112	31	)	)	PUNCT
ejpam-5693	112	32	−	−	PROPN
ejpam-5693	112	33	2	2	NUM
ejpam-5693	112	34	√	√	PROPN
ejpam-5693	112	35	µ	µ	PROPN
ejpam-5693	112	36	n∑	n∑	NOUN
ejpam-5693	112	37	j=1	j=1	NOUN
ejpam-5693	112	38	sj(a	sj(a	ADP
ejpam-5693	112	39	1	1	NUM
ejpam-5693	112	40	2b	2b	NUM
ejpam-5693	112	41	3	3	NUM
ejpam-5693	112	42	2	2	NUM
ejpam-5693	112	43	)	)	PUNCT
ejpam-5693	112	44	]	]	PUNCT
ejpam-5693	112	45	.	.	PUNCT
ejpam-5693	113	1	and	and	CCONJ
ejpam-5693	113	2	so	so	ADV
ejpam-5693	113	3	,	,	PUNCT
ejpam-5693	113	4	tr(µ2a2	tr(µ2a2	PROPN
ejpam-5693	114	1	+	+	CCONJ
ejpam-5693	114	2	(	(	PUNCT
ejpam-5693	114	3	1−	1−	NUM
ejpam-5693	114	4	µ)2b2	µ)2b2	NUM
ejpam-5693	114	5	)	)	PUNCT
ejpam-5693	114	6	≤	≤	NUM
ejpam-5693	114	7	µ2µ∥aµ∥2∥b1−µ∥2	µ2µ∥aµ∥2∥b1−µ∥2	NOUN
ejpam-5693	114	8	+	+	CCONJ
ejpam-5693	114	9	µ2	µ2	PROPN
ejpam-5693	114	10	[	[	PUNCT
ejpam-5693	114	11	∥a∥22	∥a∥22	PROPN
ejpam-5693	114	12	+	+	CCONJ
ejpam-5693	114	13	∥b∥22	∥b∥22	ADJ
ejpam-5693	114	14	−	−	PROPN
ejpam-5693	115	1	2∥ab∥1	2∥ab∥1	PROPN
ejpam-5693	115	2	]	]	PUNCT
ejpam-5693	116	1	+	+	PUNCT
ejpam-5693	116	2	r0	r0	NOUN
ejpam-5693	116	3	[	[	PUNCT
ejpam-5693	116	4	µ∥a∥2∥b∥2	µ∥a∥2∥b∥2	PROPN
ejpam-5693	116	5	+	+	CCONJ
ejpam-5693	116	6	∥b∥22	∥b∥22	ADJ
ejpam-5693	116	7	−	−	NUM
ejpam-5693	116	8	2	2	NUM
ejpam-5693	116	9	√	√	NUM
ejpam-5693	116	10	µ∥a	µ∥a	VERB
ejpam-5693	116	11	1	1	NUM
ejpam-5693	116	12	2b	2b	NUM
ejpam-5693	116	13	3	3	NUM
ejpam-5693	116	14	2	2	NUM
ejpam-5693	116	15	∥1	∥1	NOUN
ejpam-5693	116	16	]	]	PUNCT
ejpam-5693	116	17	.	.	PUNCT
ejpam-5693	117	1	thus	thus	ADV
ejpam-5693	117	2	,	,	PUNCT
ejpam-5693	117	3	we	we	PRON
ejpam-5693	117	4	get	get	VERB
ejpam-5693	117	5	(	(	PUNCT
ejpam-5693	117	6	2.5	2.5	NUM
ejpam-5693	117	7	)	)	PUNCT
ejpam-5693	117	8	.	.	PUNCT
ejpam-5693	118	1	if	if	SCONJ
ejpam-5693	118	2	1	1	NUM
ejpam-5693	118	3	2	2	NUM
ejpam-5693	118	4	≤	≤	NOUN
ejpam-5693	118	5	µ	µ	PRON
ejpam-5693	118	6	≤	≤	NUM
ejpam-5693	118	7	1	1	NUM
ejpam-5693	118	8	,	,	PUNCT
ejpam-5693	118	9	then	then	ADV
ejpam-5693	118	10	by	by	ADP
ejpam-5693	118	11	the	the	DET
ejpam-5693	118	12	inequality	inequality	NOUN
ejpam-5693	118	13	(	(	PUNCT
ejpam-5693	118	14	2.2	2.2	NUM
ejpam-5693	118	15	)	)	PUNCT
ejpam-5693	118	16	and	and	CCONJ
ejpam-5693	118	17	the	the	DET
ejpam-5693	118	18	same	same	ADJ
ejpam-5693	118	19	method	method	NOUN
ejpam-5693	118	20	above	above	ADV
ejpam-5693	118	21	,	,	PUNCT
ejpam-5693	118	22	we	we	PRON
ejpam-5693	118	23	get	get	VERB
ejpam-5693	118	24	the	the	DET
ejpam-5693	118	25	inequality	inequality	NOUN
ejpam-5693	118	26	(	(	PUNCT
ejpam-5693	118	27	2.6	2.6	NUM
ejpam-5693	118	28	)	)	PUNCT
ejpam-5693	118	29	.	.	PUNCT
ejpam-5693	119	1	this	this	PRON
ejpam-5693	119	2	completes	complete	VERB
ejpam-5693	119	3	the	the	DET
ejpam-5693	119	4	proof	proof	NOUN
ejpam-5693	119	5	.	.	PUNCT
ejpam-5693	120	1	acknowledgements	acknowledgement	NOUN
ejpam-5693	120	2	the	the	DET
ejpam-5693	120	3	authors	author	NOUN
ejpam-5693	120	4	are	be	AUX
ejpam-5693	120	5	grateful	grateful	ADJ
ejpam-5693	120	6	to	to	ADP
ejpam-5693	120	7	the	the	DET
ejpam-5693	120	8	referees	referee	NOUN
ejpam-5693	120	9	for	for	ADP
ejpam-5693	120	10	their	their	PRON
ejpam-5693	120	11	valuable	valuable	ADJ
ejpam-5693	120	12	comments	comment	NOUN
ejpam-5693	120	13	and	and	CCONJ
ejpam-5693	120	14	suggestions	suggestion	NOUN
ejpam-5693	120	15	.	.	PUNCT
ejpam-5693	121	1	this	this	DET
ejpam-5693	121	2	research	research	NOUN
ejpam-5693	121	3	is	be	AUX
ejpam-5693	121	4	funded	fund	VERB
ejpam-5693	121	5	by	by	ADP
ejpam-5693	121	6	zarqa	zarqa	PROPN
ejpam-5693	121	7	universityjordan	universityjordan	PROPN
ejpam-5693	121	8	.	.	PUNCT
ejpam-5693	122	1	references	reference	NOUN
ejpam-5693	122	2	[	[	X
ejpam-5693	122	3	1	1	NUM
ejpam-5693	122	4	]	]	PUNCT
ejpam-5693	122	5	t	t	PROPN
ejpam-5693	122	6	ando	ando	PROPN
ejpam-5693	122	7	.	.	PUNCT
ejpam-5693	122	8	matrix	matrix	NOUN
ejpam-5693	122	9	young	young	ADJ
ejpam-5693	122	10	inequalities	inequality	NOUN
ejpam-5693	122	11	.	.	PUNCT
ejpam-5693	123	1	in	in	ADP
ejpam-5693	123	2	operator	operator	NOUN
ejpam-5693	123	3	theory	theory	NOUN
ejpam-5693	123	4	in	in	ADP
ejpam-5693	123	5	function	function	NOUN
ejpam-5693	123	6	spaces	space	NOUN
ejpam-5693	123	7	and	and	CCONJ
ejpam-5693	123	8	banach	banach	NOUN
ejpam-5693	123	9	lattices	lattice	NOUN
ejpam-5693	123	10	(	(	PUNCT
ejpam-5693	123	11	pp	pp	ADJ
ejpam-5693	123	12	.	.	PUNCT
ejpam-5693	124	1	33	33	NUM
ejpam-5693	124	2	-	-	SYM
ejpam-5693	124	3	38	38	NUM
ejpam-5693	124	4	)	)	PUNCT
ejpam-5693	124	5	,	,	PUNCT
ejpam-5693	124	6	birkhäuser	birkhäuser	X
ejpam-5693	124	7	basel	basel	PROPN
ejpam-5693	124	8	,	,	PUNCT
ejpam-5693	124	9	1995	1995	NUM
ejpam-5693	124	10	.	.	PUNCT
ejpam-5693	125	1	[	[	X
ejpam-5693	125	2	2	2	NUM
ejpam-5693	125	3	]	]	X
ejpam-5693	125	4	r	r	PROPN
ejpam-5693	125	5	bhatia	bhatia	PROPN
ejpam-5693	125	6	.	.	PUNCT
ejpam-5693	126	1	majorisation	majorisation	NOUN
ejpam-5693	126	2	and	and	CCONJ
ejpam-5693	126	3	doubly	doubly	ADV
ejpam-5693	126	4	stochastic	stochastic	ADJ
ejpam-5693	126	5	matrices	matrix	NOUN
ejpam-5693	126	6	.	.	PUNCT
ejpam-5693	127	1	in	in	ADP
ejpam-5693	127	2	matrix	matrix	NOUN
ejpam-5693	127	3	analysis	analysis	NOUN
ejpam-5693	127	4	(	(	PUNCT
ejpam-5693	127	5	pp	pp	ADJ
ejpam-5693	127	6	.	.	PUNCT
ejpam-5693	127	7	28	28	NUM
ejpam-5693	127	8	-	-	SYM
ejpam-5693	127	9	56	56	NUM
ejpam-5693	127	10	)	)	PUNCT
ejpam-5693	127	11	.	.	PUNCT
ejpam-5693	128	1	springer	springer	NOUN
ejpam-5693	128	2	,	,	PUNCT
ejpam-5693	128	3	new	new	PROPN
ejpam-5693	128	4	york	york	PROPN
ejpam-5693	128	5	,	,	PUNCT
ejpam-5693	128	6	1997	1997	NUM
ejpam-5693	128	7	.	.	PUNCT
ejpam-5693	129	1	[	[	X
ejpam-5693	129	2	3	3	NUM
ejpam-5693	129	3	]	]	X
ejpam-5693	129	4	r	r	NOUN
ejpam-5693	129	5	bhatia	bhatia	PROPN
ejpam-5693	129	6	and	and	CCONJ
ejpam-5693	129	7	k	k	PROPN
ejpam-5693	129	8	parthasarathy	parthasarathy	PROPN
ejpam-5693	129	9	.	.	PUNCT
ejpam-5693	130	1	positive	positive	ADJ
ejpam-5693	130	2	definite	definite	ADJ
ejpam-5693	130	3	functions	function	NOUN
ejpam-5693	130	4	and	and	CCONJ
ejpam-5693	130	5	operator	operator	NOUN
ejpam-5693	130	6	inequalities	inequality	NOUN
ejpam-5693	130	7	.	.	PUNCT
ejpam-5693	131	1	bulletin	bulletin	NOUN
ejpam-5693	131	2	of	of	ADP
ejpam-5693	131	3	the	the	DET
ejpam-5693	131	4	london	london	PROPN
ejpam-5693	131	5	mathematical	mathematical	ADJ
ejpam-5693	131	6	society	society	NOUN
ejpam-5693	131	7	,	,	PUNCT
ejpam-5693	131	8	32(2):214–228	32(2):214–228	PROPN
ejpam-5693	131	9	,	,	PUNCT
ejpam-5693	131	10	2000	2000	NUM
ejpam-5693	131	11	.	.	PUNCT
ejpam-5693	132	1	[	[	X
ejpam-5693	132	2	4	4	X
ejpam-5693	132	3	]	]	X
ejpam-5693	132	4	a	a	DET
ejpam-5693	132	5	burqan	burqan	NOUN
ejpam-5693	132	6	and	and	CCONJ
ejpam-5693	132	7	m	m	AUX
ejpam-5693	132	8	khandaqji	khandaqji	ADJ
ejpam-5693	132	9	.	.	PUNCT
ejpam-5693	133	1	reverses	reverse	NOUN
ejpam-5693	133	2	of	of	ADP
ejpam-5693	133	3	young	young	ADJ
ejpam-5693	133	4	type	type	NOUN
ejpam-5693	133	5	inequalities	inequality	NOUN
ejpam-5693	133	6	.	.	PUNCT
ejpam-5693	134	1	journal	journal	PROPN
ejpam-5693	134	2	of	of	ADP
ejpam-5693	134	3	mathematical	mathematical	ADJ
ejpam-5693	134	4	inequalities	inequality	NOUN
ejpam-5693	134	5	,	,	PUNCT
ejpam-5693	134	6	9(1):113–120	9(1):113–120	NUM
ejpam-5693	134	7	,	,	PUNCT
ejpam-5693	134	8	2015	2015	NUM
ejpam-5693	134	9	.	.	PUNCT
ejpam-5693	135	1	[	[	X
ejpam-5693	135	2	5	5	NUM
ejpam-5693	135	3	]	]	SYM
ejpam-5693	135	4	f	f	PROPN
ejpam-5693	135	5	kittaneh	kittaneh	PROPN
ejpam-5693	135	6	and	and	CCONJ
ejpam-5693	135	7	y	y	PROPN
ejpam-5693	135	8	manasrah	manasrah	PROPN
ejpam-5693	135	9	.	.	PUNCT
ejpam-5693	136	1	improved	improve	VERB
ejpam-5693	136	2	young	young	ADJ
ejpam-5693	136	3	and	and	CCONJ
ejpam-5693	136	4	heinz	heinz	ADJ
ejpam-5693	136	5	inequalities	inequality	NOUN
ejpam-5693	136	6	for	for	ADP
ejpam-5693	136	7	matrices	matrix	NOUN
ejpam-5693	136	8	.	.	PUNCT
ejpam-5693	137	1	journal	journal	PROPN
ejpam-5693	137	2	of	of	ADP
ejpam-5693	137	3	mathematical	mathematical	ADJ
ejpam-5693	137	4	analysis	analysis	NOUN
ejpam-5693	137	5	and	and	CCONJ
ejpam-5693	137	6	applications	application	NOUN
ejpam-5693	137	7	,	,	PUNCT
ejpam-5693	137	8	361(1):262–269	361(1):262–269	ADP
ejpam-5693	137	9	,	,	PUNCT
ejpam-5693	137	10	2010	2010	NUM
ejpam-5693	137	11	.	.	PUNCT
ejpam-5693	138	1	[	[	X
ejpam-5693	138	2	6	6	NUM
ejpam-5693	138	3	]	]	SYM
ejpam-5693	138	4	f	f	PROPN
ejpam-5693	138	5	kittaneh	kittaneh	PROPN
ejpam-5693	138	6	and	and	CCONJ
ejpam-5693	138	7	y	y	PROPN
ejpam-5693	138	8	manasrah	manasrah	PROPN
ejpam-5693	138	9	.	.	PUNCT
ejpam-5693	139	1	reverse	reverse	VERB
ejpam-5693	139	2	young	young	ADJ
ejpam-5693	139	3	and	and	CCONJ
ejpam-5693	139	4	heinz	heinz	ADJ
ejpam-5693	139	5	inequalities	inequality	NOUN
ejpam-5693	139	6	for	for	ADP
ejpam-5693	139	7	matrices	matrix	NOUN
ejpam-5693	139	8	.	.	PUNCT
ejpam-5693	140	1	linear	linear	ADJ
ejpam-5693	140	2	and	and	CCONJ
ejpam-5693	140	3	multilinear	multilinear	PROPN
ejpam-5693	140	4	algebra	algebra	PROPN
ejpam-5693	140	5	,	,	PUNCT
ejpam-5693	140	6	59(9):1031–1037	59(9):1031–1037	NUM
ejpam-5693	140	7	,	,	PUNCT
ejpam-5693	140	8	2011	2011	NUM
ejpam-5693	140	9	.	.	PUNCT
ejpam-5693	141	1	o.	o.	PROPN
ejpam-5693	141	2	ramadan	ramadan	PROPN
ejpam-5693	141	3	,	,	PUNCT
ejpam-5693	141	4	a.	a.	PROPN
ejpam-5693	141	5	burqan	burqan	PROPN
ejpam-5693	141	6	/	/	SYM
ejpam-5693	141	7	eur	eur	PROPN
ejpam-5693	141	8	.	.	PUNCT
ejpam-5693	142	1	j.	j.	PROPN
ejpam-5693	142	2	pure	pure	PROPN
ejpam-5693	142	3	appl	appl	PROPN
ejpam-5693	142	4	.	.	PROPN
ejpam-5693	142	5	math	math	PROPN
ejpam-5693	142	6	,	,	PUNCT
ejpam-5693	142	7	18	18	NUM
ejpam-5693	142	8	(	(	PUNCT
ejpam-5693	142	9	1	1	NUM
ejpam-5693	142	10	)	)	PUNCT
ejpam-5693	142	11	(	(	PUNCT
ejpam-5693	142	12	2025	2025	NUM
ejpam-5693	142	13	)	)	PUNCT
ejpam-5693	142	14	,	,	PUNCT
ejpam-5693	142	15	5693	5693	NUM
ejpam-5693	142	16	8	8	NUM
ejpam-5693	142	17	of	of	ADP
ejpam-5693	142	18	8	8	NUM
ejpam-5693	142	19	[	[	X
ejpam-5693	142	20	7	7	NUM
ejpam-5693	142	21	]	]	X
ejpam-5693	142	22	h	h	NOUN
ejpam-5693	142	23	kosaki	kosaki	PROPN
ejpam-5693	142	24	.	.	PUNCT
ejpam-5693	143	1	arithmetic	arithmetic	ADJ
ejpam-5693	143	2	–	–	PUNCT
ejpam-5693	143	3	geometric	geometric	ADJ
ejpam-5693	143	4	mean	mean	NOUN
ejpam-5693	143	5	and	and	CCONJ
ejpam-5693	143	6	related	related	ADJ
ejpam-5693	143	7	inequalities	inequality	NOUN
ejpam-5693	143	8	for	for	ADP
ejpam-5693	143	9	operators	operator	NOUN
ejpam-5693	143	10	.	.	PUNCT
ejpam-5693	144	1	journal	journal	NOUN
ejpam-5693	144	2	of	of	ADP
ejpam-5693	144	3	functional	functional	ADJ
ejpam-5693	144	4	analysis	analysis	NOUN
ejpam-5693	144	5	,	,	PUNCT
ejpam-5693	144	6	156(2):429–451	156(2):429–451	NUM
ejpam-5693	144	7	,	,	PUNCT
ejpam-5693	144	8	1998	1998	NUM
ejpam-5693	144	9	.	.	PUNCT
ejpam-5693	145	1	[	[	X
ejpam-5693	145	2	8	8	NUM
ejpam-5693	145	3	]	]	X
ejpam-5693	145	4	l	l	NOUN
ejpam-5693	145	5	nasiri	nasiri	ADV
ejpam-5693	145	6	,	,	PUNCT
ejpam-5693	145	7	m	m	PROPN
ejpam-5693	145	8	shakoori	shakoori	NOUN
ejpam-5693	145	9	,	,	PUNCT
ejpam-5693	145	10	and	and	CCONJ
ejpam-5693	145	11	w	w	PROPN
ejpam-5693	145	12	liao	liao	PROPN
ejpam-5693	145	13	.	.	PUNCT
ejpam-5693	146	1	a	a	DET
ejpam-5693	146	2	note	note	NOUN
ejpam-5693	146	3	on	on	ADP
ejpam-5693	146	4	the	the	DET
ejpam-5693	146	5	young	young	ADJ
ejpam-5693	146	6	type	type	NOUN
ejpam-5693	146	7	inequalities	inequality	NOUN
ejpam-5693	146	8	.	.	PUNCT
ejpam-5693	147	1	international	international	ADJ
ejpam-5693	147	2	journal	journal	PROPN
ejpam-5693	147	3	of	of	ADP
ejpam-5693	147	4	nonlinear	nonlinear	ADJ
ejpam-5693	147	5	analysis	analysis	NOUN
ejpam-5693	147	6	and	and	CCONJ
ejpam-5693	147	7	applications	application	NOUN
ejpam-5693	147	8	,	,	PUNCT
ejpam-5693	147	9	8(1):261–267	8(1):261–267	NUM
ejpam-5693	147	10	,	,	PUNCT
ejpam-5693	147	11	2017	2017	NUM
ejpam-5693	147	12	.	.	PUNCT
ejpam-5693	148	1	[	[	X
ejpam-5693	148	2	9	9	NUM
ejpam-5693	148	3	]	]	SYM
ejpam-5693	148	4	h	h	NOUN
ejpam-5693	148	5	xing	xing	PROPN
ejpam-5693	148	6	-	-	PUNCT
ejpam-5693	148	7	kai	kai	PROPN
ejpam-5693	148	8	.	.	PUNCT
ejpam-5693	149	1	young	young	ADJ
ejpam-5693	149	2	type	type	NOUN
ejpam-5693	149	3	inequalities	inequality	NOUN
ejpam-5693	149	4	for	for	ADP
ejpam-5693	149	5	matrices	matrix	NOUN
ejpam-5693	149	6	.	.	PUNCT
ejpam-5693	150	1	journal	journal	PROPN
ejpam-5693	150	2	of	of	ADP
ejpam-5693	150	3	east	east	PROPN
ejpam-5693	150	4	china	china	PROPN
ejpam-5693	150	5	normal	normal	PROPN
ejpam-5693	150	6	university	university	PROPN
ejpam-5693	150	7	(	(	PUNCT
ejpam-5693	150	8	natural	natural	ADJ
ejpam-5693	150	9	science	science	NOUN
ejpam-5693	150	10	)	)	PUNCT
ejpam-5693	150	11	,	,	PUNCT
ejpam-5693	150	12	4	4	NUM
ejpam-5693	150	13	,	,	PUNCT
ejpam-5693	150	14	2012	2012	NUM
ejpam-5693	150	15	.	.	PUNCT