id	sid	tid	token	lemma	pos
ejpam-5689	1	1	european	european	PROPN
ejpam-5689	1	2	journal	journal	PROPN
ejpam-5689	1	3	of	of	ADP
ejpam-5689	1	4	pure	pure	ADJ
ejpam-5689	1	5	and	and	CCONJ
ejpam-5689	1	6	applied	applied	ADJ
ejpam-5689	1	7	mathematics	mathematic	NOUN
ejpam-5689	1	8	2025	2025	NUM
ejpam-5689	1	9	,	,	PUNCT
ejpam-5689	1	10	vol	vol	NOUN
ejpam-5689	1	11	.	.	PROPN
ejpam-5689	1	12	18	18	NUM
ejpam-5689	1	13	,	,	PUNCT
ejpam-5689	1	14	issue	issue	NOUN
ejpam-5689	1	15	1	1	NUM
ejpam-5689	1	16	,	,	PUNCT
ejpam-5689	1	17	article	article	NOUN
ejpam-5689	1	18	number	number	NOUN
ejpam-5689	1	19	5689	5689	NUM
ejpam-5689	1	20	issn	issn	VERB
ejpam-5689	1	21	1307	1307	NUM
ejpam-5689	1	22	-	-	SYM
ejpam-5689	1	23	5543	5543	NUM
ejpam-5689	1	24	–	–	PUNCT
ejpam-5689	1	25	ejpam.com	ejpam.com	X
ejpam-5689	1	26	published	publish	VERB
ejpam-5689	1	27	by	by	ADP
ejpam-5689	1	28	new	new	PROPN
ejpam-5689	1	29	york	york	PROPN
ejpam-5689	1	30	business	business	PROPN
ejpam-5689	1	31	global	global	ADJ
ejpam-5689	1	32	singular	singular	PROPN
ejpam-5689	1	33	value	value	NOUN
ejpam-5689	1	34	inequalities	inequality	NOUN
ejpam-5689	1	35	for	for	ADP
ejpam-5689	1	36	concave	concave	NOUN
ejpam-5689	1	37	and	and	CCONJ
ejpam-5689	1	38	convex	convex	NOUN
ejpam-5689	1	39	functions	function	NOUN
ejpam-5689	1	40	of	of	ADP
ejpam-5689	1	41	matrix	matrix	NOUN
ejpam-5689	1	42	sums	sum	NOUN
ejpam-5689	1	43	and	and	CCONJ
ejpam-5689	1	44	products	product	NOUN
ejpam-5689	1	45	ahmad	ahmad	PROPN
ejpam-5689	1	46	al	al	PROPN
ejpam-5689	1	47	-	-	PUNCT
ejpam-5689	1	48	natoor1,∗	natoor1,∗	ADJ
ejpam-5689	1	49	,	,	PUNCT
ejpam-5689	1	50	fadi	fadi	NOUN
ejpam-5689	1	51	alrimawi2	alrimawi2	PROPN
ejpam-5689	2	1	1	1	NUM
ejpam-5689	2	2	department	department	NOUN
ejpam-5689	2	3	of	of	ADP
ejpam-5689	2	4	mathematics	mathematics	PROPN
ejpam-5689	2	5	,	,	PUNCT
ejpam-5689	2	6	isra	isra	PROPN
ejpam-5689	2	7	university	university	PROPN
ejpam-5689	2	8	,	,	PUNCT
ejpam-5689	2	9	amman	amman	PROPN
ejpam-5689	2	10	,	,	PUNCT
ejpam-5689	2	11	jordan	jordan	PROPN
ejpam-5689	2	12	2	2	NUM
ejpam-5689	2	13	department	department	NOUN
ejpam-5689	2	14	of	of	ADP
ejpam-5689	2	15	basic	basic	ADJ
ejpam-5689	2	16	sciences	sciences	PROPN
ejpam-5689	2	17	,	,	PUNCT
ejpam-5689	2	18	al	al	PROPN
ejpam-5689	2	19	-	-	PUNCT
ejpam-5689	2	20	ahliyya	ahliyya	PROPN
ejpam-5689	2	21	amman	amman	PROPN
ejpam-5689	2	22	university	university	PROPN
ejpam-5689	2	23	,	,	PUNCT
ejpam-5689	2	24	amman	amman	PROPN
ejpam-5689	2	25	,	,	PUNCT
ejpam-5689	2	26	jordan	jordan	PROPN
ejpam-5689	2	27	abstract	abstract	PROPN
ejpam-5689	2	28	.	.	PUNCT
ejpam-5689	3	1	in	in	ADP
ejpam-5689	3	2	this	this	DET
ejpam-5689	3	3	paper	paper	NOUN
ejpam-5689	3	4	,	,	PUNCT
ejpam-5689	3	5	we	we	PRON
ejpam-5689	3	6	present	present	VERB
ejpam-5689	3	7	several	several	ADJ
ejpam-5689	3	8	singular	singular	ADJ
ejpam-5689	3	9	value	value	NOUN
ejpam-5689	3	10	inequalities	inequality	NOUN
ejpam-5689	3	11	for	for	ADP
ejpam-5689	3	12	special	special	ADJ
ejpam-5689	3	13	types	type	NOUN
ejpam-5689	3	14	of	of	ADP
ejpam-5689	3	15	functions	function	NOUN
ejpam-5689	3	16	of	of	ADP
ejpam-5689	3	17	matrix	matrix	NOUN
ejpam-5689	3	18	sums	sum	NOUN
ejpam-5689	3	19	and	and	CCONJ
ejpam-5689	3	20	products	product	NOUN
ejpam-5689	3	21	.	.	PUNCT
ejpam-5689	4	1	some	some	PRON
ejpam-5689	4	2	of	of	ADP
ejpam-5689	4	3	special	special	ADJ
ejpam-5689	4	4	cases	case	NOUN
ejpam-5689	4	5	of	of	ADP
ejpam-5689	4	6	our	our	PRON
ejpam-5689	4	7	results	result	NOUN
ejpam-5689	4	8	give	give	VERB
ejpam-5689	4	9	a	a	DET
ejpam-5689	4	10	generalization	generalization	NOUN
ejpam-5689	4	11	of	of	ADP
ejpam-5689	4	12	some	some	DET
ejpam-5689	4	13	recent	recent	ADJ
ejpam-5689	4	14	inequalities	inequality	NOUN
ejpam-5689	4	15	.	.	PUNCT
ejpam-5689	5	1	2020	2020	NUM
ejpam-5689	5	2	mathematics	mathematic	NOUN
ejpam-5689	5	3	subject	subject	NOUN
ejpam-5689	5	4	classifications	classification	NOUN
ejpam-5689	5	5	:	:	PUNCT
ejpam-5689	5	6	15a18	15a18	NUM
ejpam-5689	5	7	,	,	PUNCT
ejpam-5689	5	8	15a42	15a42	NUM
ejpam-5689	5	9	,	,	PUNCT
ejpam-5689	5	10	47a30	47a30	NUM
ejpam-5689	5	11	,	,	PUNCT
ejpam-5689	5	12	93c05	93c05	NUM
ejpam-5689	5	13	key	key	ADJ
ejpam-5689	5	14	words	word	NOUN
ejpam-5689	5	15	and	and	CCONJ
ejpam-5689	5	16	phrases	phrase	NOUN
ejpam-5689	5	17	:	:	PUNCT
ejpam-5689	5	18	singular	singular	ADJ
ejpam-5689	5	19	value	value	NOUN
ejpam-5689	5	20	,	,	PUNCT
ejpam-5689	5	21	spectral	spectral	ADJ
ejpam-5689	5	22	norm	norm	NOUN
ejpam-5689	5	23	,	,	PUNCT
ejpam-5689	5	24	positive	positive	ADJ
ejpam-5689	5	25	semidefinite	semidefinite	NOUN
ejpam-5689	5	26	matrix	matrix	NOUN
ejpam-5689	5	27	,	,	PUNCT
ejpam-5689	5	28	concave	concave	NOUN
ejpam-5689	5	29	function	function	NOUN
ejpam-5689	5	30	,	,	PUNCT
ejpam-5689	5	31	convex	convex	NOUN
ejpam-5689	5	32	function	function	NOUN
ejpam-5689	5	33	,	,	PUNCT
ejpam-5689	5	34	inequality	inequality	NOUN
ejpam-5689	5	35	,	,	PUNCT
ejpam-5689	5	36	control	control	NOUN
ejpam-5689	5	37	theory	theory	NOUN
ejpam-5689	5	38	1	1	NUM
ejpam-5689	5	39	.	.	PUNCT
ejpam-5689	6	1	introduction	introduction	NOUN
ejpam-5689	6	2	let	let	VERB
ejpam-5689	6	3	mn(c	mn(c	VERB
ejpam-5689	6	4	)	)	PUNCT
ejpam-5689	6	5	be	be	AUX
ejpam-5689	6	6	the	the	DET
ejpam-5689	6	7	c∗-algebra	c∗-algebra	PROPN
ejpam-5689	6	8	of	of	ADP
ejpam-5689	6	9	all	all	DET
ejpam-5689	6	10	n	n	PRON
ejpam-5689	6	11	×	×	NOUN
ejpam-5689	6	12	n	n	CCONJ
ejpam-5689	6	13	complex	complex	ADJ
ejpam-5689	6	14	matrices	matrix	NOUN
ejpam-5689	6	15	.	.	PUNCT
ejpam-5689	7	1	a	a	DET
ejpam-5689	7	2	matrix	matrix	NOUN
ejpam-5689	7	3	x	x	X
ejpam-5689	7	4	∈	∈	NOUN
ejpam-5689	7	5	mn(c	mn(c	X
ejpam-5689	7	6	)	)	PUNCT
ejpam-5689	7	7	is	be	AUX
ejpam-5689	7	8	said	say	VERB
ejpam-5689	7	9	to	to	PART
ejpam-5689	7	10	be	be	AUX
ejpam-5689	7	11	positive	positive	ADJ
ejpam-5689	7	12	semidefinite	semidefinite	NOUN
ejpam-5689	7	13	if	if	SCONJ
ejpam-5689	7	14	x∗ax	x∗ax	PROPN
ejpam-5689	7	15	≥	≥	VERB
ejpam-5689	7	16	0	0	NUM
ejpam-5689	7	17	for	for	ADP
ejpam-5689	7	18	all	all	DET
ejpam-5689	7	19	x	x	SYM
ejpam-5689	7	20	∈	∈	PROPN
ejpam-5689	7	21	cn	cn	PROPN
ejpam-5689	7	22	.	.	PUNCT
ejpam-5689	8	1	the	the	DET
ejpam-5689	8	2	singular	singular	ADJ
ejpam-5689	8	3	values	value	NOUN
ejpam-5689	8	4	of	of	ADP
ejpam-5689	8	5	x	x	X
ejpam-5689	8	6	∈	∈	PROPN
ejpam-5689	8	7	mn(c	mn(c	X
ejpam-5689	8	8	)	)	PUNCT
ejpam-5689	8	9	,	,	PUNCT
ejpam-5689	8	10	denoted	denote	VERB
ejpam-5689	8	11	by	by	ADP
ejpam-5689	8	12	s1	s1	PROPN
ejpam-5689	8	13	(	(	PUNCT
ejpam-5689	8	14	x	x	NOUN
ejpam-5689	8	15	)	)	PUNCT
ejpam-5689	8	16	≥	≥	NUM
ejpam-5689	8	17	s2	s2	PROPN
ejpam-5689	8	18	(	(	PUNCT
ejpam-5689	8	19	x	x	NOUN
ejpam-5689	8	20	)	)	PUNCT
ejpam-5689	8	21	≥	≥	NUM
ejpam-5689	8	22	...	...	PUNCT
ejpam-5689	9	1	sn	sn	INTJ
ejpam-5689	9	2	(	(	PUNCT
ejpam-5689	9	3	x	x	X
ejpam-5689	9	4	)	)	PUNCT
ejpam-5689	9	5	≥	≥	X
ejpam-5689	9	6	0	0	NUM
ejpam-5689	9	7	are	be	AUX
ejpam-5689	9	8	the	the	DET
ejpam-5689	9	9	eigenvalues	eigenvalue	NOUN
ejpam-5689	9	10	of	of	ADP
ejpam-5689	9	11	|x|	|x|	PROPN
ejpam-5689	9	12	.	.	PUNCT
ejpam-5689	10	1	in	in	ADP
ejpam-5689	10	2	this	this	DET
ejpam-5689	10	3	paper	paper	NOUN
ejpam-5689	10	4	,	,	PUNCT
ejpam-5689	10	5	when	when	SCONJ
ejpam-5689	10	6	we	we	PRON
ejpam-5689	10	7	write	write	VERB
ejpam-5689	10	8	sj	sj	INTJ
ejpam-5689	10	9	we	we	PRON
ejpam-5689	10	10	mean	mean	VERB
ejpam-5689	10	11	s1	s1	NOUN
ejpam-5689	10	12	(	(	PUNCT
ejpam-5689	10	13	x	x	NOUN
ejpam-5689	10	14	)	)	PUNCT
ejpam-5689	10	15	,	,	PUNCT
ejpam-5689	10	16	s2	s2	X
ejpam-5689	10	17	(	(	PUNCT
ejpam-5689	10	18	x	x	NOUN
ejpam-5689	10	19	)	)	PUNCT
ejpam-5689	10	20	,	,	PUNCT
ejpam-5689	10	21	...	...	PUNCT
ejpam-5689	10	22	,	,	PUNCT
ejpam-5689	10	23	sn	sn	PROPN
ejpam-5689	10	24	(	(	PUNCT
ejpam-5689	10	25	x	x	NOUN
ejpam-5689	10	26	)	)	PUNCT
ejpam-5689	10	27	,	,	PUNCT
ejpam-5689	10	28	i.e.	i.e.	X
ejpam-5689	10	29	,	,	PUNCT
ejpam-5689	10	30	j	j	PROPN
ejpam-5689	10	31	=	=	SYM
ejpam-5689	10	32	1	1	NUM
ejpam-5689	10	33	,	,	PUNCT
ejpam-5689	10	34	2	2	NUM
ejpam-5689	10	35	,	,	PUNCT
ejpam-5689	10	36	...	...	PUNCT
ejpam-5689	10	37	,	,	PUNCT
ejpam-5689	10	38	n.	n.	PROPN
ejpam-5689	10	39	the	the	DET
ejpam-5689	10	40	spectral	spectral	ADJ
ejpam-5689	10	41	norm	norm	NOUN
ejpam-5689	10	42	of	of	ADP
ejpam-5689	10	43	x	x	SYM
ejpam-5689	10	44	∈	∈	PROPN
ejpam-5689	10	45	mn(c	mn(c	X
ejpam-5689	10	46	)	)	PUNCT
ejpam-5689	10	47	is	be	AUX
ejpam-5689	10	48	defined	define	VERB
ejpam-5689	10	49	by	by	ADP
ejpam-5689	10	50	∥x∥	∥x∥	NOUN
ejpam-5689	10	51	=	=	SYM
ejpam-5689	10	52	max∥x∥=1	max∥x∥=1	PROPN
ejpam-5689	10	53	∥ax∥.	∥ax∥.	NUM
ejpam-5689	10	54	for	for	ADP
ejpam-5689	10	55	x	x	PROPN
ejpam-5689	10	56	,	,	PUNCT
ejpam-5689	10	57	y	y	PROPN
ejpam-5689	10	58	∈	∈	PROPN
ejpam-5689	10	59	mn(c	mn(c	X
ejpam-5689	10	60	)	)	PUNCT
ejpam-5689	10	61	,	,	PUNCT
ejpam-5689	10	62	let	let	VERB
ejpam-5689	10	63	x	x	PUNCT
ejpam-5689	10	64	⊕	⊕	PROPN
ejpam-5689	10	65	y	y	PROPN
ejpam-5689	10	66	be	be	AUX
ejpam-5689	10	67	the	the	DET
ejpam-5689	10	68	direct	direct	ADJ
ejpam-5689	10	69	sum	sum	NOUN
ejpam-5689	10	70	of	of	ADP
ejpam-5689	10	71	x	x	PROPN
ejpam-5689	10	72	and	and	CCONJ
ejpam-5689	10	73	y	y	PROPN
ejpam-5689	10	74	,	,	PUNCT
ejpam-5689	10	75	that	that	ADV
ejpam-5689	10	76	is	is	ADV
ejpam-5689	10	77	,	,	PUNCT
ejpam-5689	10	78	the	the	DET
ejpam-5689	10	79	matrix	matrix	NOUN
ejpam-5689	10	80	given	give	VERB
ejpam-5689	10	81	by	by	ADP
ejpam-5689	10	82	x	x	PROPN
ejpam-5689	10	83	⊕	⊕	PROPN
ejpam-5689	10	84	y	y	NOUN
ejpam-5689	10	85	=	=	PUNCT
ejpam-5689	11	1	[	[	PUNCT
ejpam-5689	11	2	x	x	SYM
ejpam-5689	11	3	0	0	NUM
ejpam-5689	11	4	0	0	NUM
ejpam-5689	11	5	y	y	PROPN
ejpam-5689	11	6	]	]	PUNCT
ejpam-5689	11	7	.	.	PUNCT
ejpam-5689	12	1	it	it	PRON
ejpam-5689	12	2	is	be	AUX
ejpam-5689	12	3	known	know	VERB
ejpam-5689	12	4	that	that	SCONJ
ejpam-5689	12	5	∥x	∥x	PROPN
ejpam-5689	12	6	⊕	⊕	PROPN
ejpam-5689	12	7	y	y	NOUN
ejpam-5689	12	8	∥	∥	PROPN
ejpam-5689	12	9	=	=	SYM
ejpam-5689	12	10	max	max	PROPN
ejpam-5689	12	11	(	(	PUNCT
ejpam-5689	12	12	∥x∥	∥x∥	NOUN
ejpam-5689	12	13	,	,	PUNCT
ejpam-5689	12	14	∥y	∥y	PROPN
ejpam-5689	12	15	∥	∥	NUM
ejpam-5689	12	16	)	)	PUNCT
ejpam-5689	12	17	.	.	PUNCT
ejpam-5689	13	1	it	it	PRON
ejpam-5689	13	2	is	be	AUX
ejpam-5689	13	3	known	know	VERB
ejpam-5689	13	4	[	[	X
ejpam-5689	13	5	14	14	NUM
ejpam-5689	13	6	]	]	PUNCT
ejpam-5689	13	7	that	that	SCONJ
ejpam-5689	13	8	if	if	SCONJ
ejpam-5689	13	9	x	x	X
ejpam-5689	13	10	,	,	PUNCT
ejpam-5689	13	11	y	y	PROPN
ejpam-5689	13	12	∈	∈	PROPN
ejpam-5689	13	13	mn(c	mn(c	X
ejpam-5689	13	14	)	)	PUNCT
ejpam-5689	13	15	,	,	PUNCT
ejpam-5689	13	16	then	then	ADV
ejpam-5689	13	17	sj(x	sj(x	PUNCT
ejpam-5689	13	18	+	+	CCONJ
ejpam-5689	13	19	y	y	X
ejpam-5689	13	20	)	)	PUNCT
ejpam-5689	13	21	≤	≤	NOUN
ejpam-5689	14	1	2sj(x	2sj(x	NUM
ejpam-5689	15	1	⊕	⊕	PROPN
ejpam-5689	15	2	y	y	PROPN
ejpam-5689	15	3	)	)	PUNCT
ejpam-5689	15	4	,	,	PUNCT
ejpam-5689	15	5	(	(	PUNCT
ejpam-5689	15	6	1	1	X
ejpam-5689	15	7	)	)	PUNCT
ejpam-5689	15	8	a	a	DET
ejpam-5689	15	9	generalization	generalization	NOUN
ejpam-5689	15	10	of	of	ADP
ejpam-5689	15	11	inequality	inequality	NOUN
ejpam-5689	15	12	(	(	PUNCT
ejpam-5689	15	13	1	1	NUM
ejpam-5689	15	14	)	)	PUNCT
ejpam-5689	15	15	has	have	AUX
ejpam-5689	15	16	been	be	AUX
ejpam-5689	15	17	given	give	VERB
ejpam-5689	15	18	in	in	ADP
ejpam-5689	15	19	[	[	NOUN
ejpam-5689	15	20	8	8	NUM
ejpam-5689	15	21	]	]	PUNCT
ejpam-5689	15	22	by	by	ADP
ejpam-5689	15	23	sj(xz	sj(xz	PROPN
ejpam-5689	15	24	+	+	PROPN
ejpam-5689	15	25	zy	zy	X
ejpam-5689	15	26	)	)	PUNCT
ejpam-5689	15	27	≤	≤	NOUN
ejpam-5689	15	28	2	2	NUM
ejpam-5689	15	29	∥z∥	∥z∥	NOUN
ejpam-5689	15	30	sj(x	sj(x	PUNCT
ejpam-5689	15	31	⊕	⊕	PROPN
ejpam-5689	15	32	y	y	PROPN
ejpam-5689	15	33	)	)	PUNCT
ejpam-5689	15	34	.	.	PUNCT
ejpam-5689	16	1	(	(	PUNCT
ejpam-5689	16	2	2	2	X
ejpam-5689	16	3	)	)	PUNCT
ejpam-5689	16	4	the	the	DET
ejpam-5689	16	5	authors	author	NOUN
ejpam-5689	16	6	in	in	ADP
ejpam-5689	16	7	[	[	X
ejpam-5689	16	8	7	7	NUM
ejpam-5689	16	9	]	]	PUNCT
ejpam-5689	16	10	have	have	AUX
ejpam-5689	16	11	proved	prove	VERB
ejpam-5689	16	12	several	several	ADJ
ejpam-5689	16	13	singular	singular	ADJ
ejpam-5689	16	14	value	value	NOUN
ejpam-5689	16	15	inequalities	inequality	NOUN
ejpam-5689	16	16	.	.	PUNCT
ejpam-5689	17	1	one	one	NUM
ejpam-5689	17	2	of	of	ADP
ejpam-5689	17	3	these	these	DET
ejpam-5689	17	4	inequalities	inequality	NOUN
ejpam-5689	17	5	asserts	assert	VERB
ejpam-5689	17	6	that	that	SCONJ
ejpam-5689	17	7	if	if	SCONJ
ejpam-5689	17	8	x	x	X
ejpam-5689	17	9	,	,	PUNCT
ejpam-5689	17	10	y	y	PROPN
ejpam-5689	17	11	∈	∈	PROPN
ejpam-5689	17	12	mn(c	mn(c	X
ejpam-5689	17	13	)	)	PUNCT
ejpam-5689	17	14	,	,	PUNCT
ejpam-5689	17	15	then	then	ADV
ejpam-5689	17	16	sj	sj	INTJ
ejpam-5689	17	17	(	(	PUNCT
ejpam-5689	17	18	xy	xy	PROPN
ejpam-5689	17	19	−	−	PROPN
ejpam-5689	17	20	y	y	PROPN
ejpam-5689	17	21	x	x	PROPN
ejpam-5689	17	22	)	)	PUNCT
ejpam-5689	17	23	≤	≤	PUNCT
ejpam-5689	17	24	∥y	∥y	ADV
ejpam-5689	17	25	∥	∥	PUNCT
ejpam-5689	17	26	si	si	X
ejpam-5689	17	27	(	(	PUNCT
ejpam-5689	17	28	x	x	PROPN
ejpam-5689	17	29	⊕x	⊕x	NOUN
ejpam-5689	17	30	)	)	PUNCT
ejpam-5689	18	1	+	+	CCONJ
ejpam-5689	18	2	1	1	NUM
ejpam-5689	18	3	2	2	NUM
ejpam-5689	18	4	sj−i+1	sj−i+1	PROPN
ejpam-5689	18	5	(	(	PUNCT
ejpam-5689	18	6	(	(	PUNCT
ejpam-5689	18	7	xy	xy	NOUN
ejpam-5689	18	8	−	−	PROPN
ejpam-5689	18	9	y	y	PROPN
ejpam-5689	18	10	x)⊕	x)⊕	PROPN
ejpam-5689	18	11	(	(	PUNCT
ejpam-5689	18	12	xy	xy	PROPN
ejpam-5689	18	13	−	−	PROPN
ejpam-5689	18	14	y	y	PROPN
ejpam-5689	18	15	x	x	PROPN
ejpam-5689	18	16	)	)	PUNCT
ejpam-5689	18	17	)	)	PUNCT
ejpam-5689	19	1	∗corresponding	∗corresponde	VERB
ejpam-5689	19	2	author	author	NOUN
ejpam-5689	19	3	.	.	PUNCT
ejpam-5689	20	1	doi	doi	NOUN
ejpam-5689	20	2	:	:	PUNCT
ejpam-5689	20	3	https://doi.org/10.29020/nybg.ejpam.v18i1.5689	https://doi.org/10.29020/nybg.ejpam.v18i1.5689	ADJ
ejpam-5689	20	4	email	email	NOUN
ejpam-5689	20	5	addresses	address	NOUN
ejpam-5689	20	6	:	:	PUNCT
ejpam-5689	20	7	ahmad.alnatoor@iu.edu.jo	ahmad.alnatoor@iu.edu.jo	NOUN
ejpam-5689	20	8	(	(	PUNCT
ejpam-5689	20	9	a.	a.	NOUN
ejpam-5689	20	10	al	al	PROPN
ejpam-5689	20	11	-	-	PUNCT
ejpam-5689	20	12	natoor	natoor	NOUN
ejpam-5689	20	13	)	)	PUNCT
ejpam-5689	20	14	,	,	PUNCT
ejpam-5689	20	15	f.rimawi@ammanu.edu.jo	f.rimawi@ammanu.edu.jo	NOUN
ejpam-5689	20	16	(	(	PUNCT
ejpam-5689	20	17	f.	f.	PROPN
ejpam-5689	20	18	alrimawi	alrimawi	PROPN
ejpam-5689	20	19	)	)	PUNCT
ejpam-5689	20	20	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5689	21	1	1	1	NUM
ejpam-5689	21	2	copyright	copyright	NOUN
ejpam-5689	21	3	:	:	PUNCT
ejpam-5689	21	4	©	©	PROPN
ejpam-5689	21	5	2025	2025	NUM
ejpam-5689	21	6	the	the	DET
ejpam-5689	21	7	author(s	author(s	NOUN
ejpam-5689	21	8	)	)	PUNCT
ejpam-5689	21	9	.	.	PUNCT
ejpam-5689	22	1	(	(	PUNCT
ejpam-5689	22	2	cc	cc	NOUN
ejpam-5689	22	3	by	by	ADP
ejpam-5689	22	4	-	-	PUNCT
ejpam-5689	22	5	nc	nc	PROPN
ejpam-5689	22	6	4.0	4.0	NUM
ejpam-5689	22	7	)	)	PUNCT
ejpam-5689	22	8	a.	a.	NOUN
ejpam-5689	22	9	al	al	PROPN
ejpam-5689	22	10	-	-	PUNCT
ejpam-5689	22	11	natoor	natoor	NOUN
ejpam-5689	22	12	,	,	PUNCT
ejpam-5689	22	13	f.	f.	PROPN
ejpam-5689	22	14	alrimawi	alrimawi	PROPN
ejpam-5689	22	15	/	/	SYM
ejpam-5689	22	16	eur	eur	PROPN
ejpam-5689	22	17	.	.	PUNCT
ejpam-5689	23	1	j.	j.	PROPN
ejpam-5689	23	2	pure	pure	PROPN
ejpam-5689	23	3	appl	appl	PROPN
ejpam-5689	23	4	.	.	PROPN
ejpam-5689	23	5	math	math	PROPN
ejpam-5689	23	6	,	,	PUNCT
ejpam-5689	23	7	18	18	NUM
ejpam-5689	23	8	(	(	PUNCT
ejpam-5689	23	9	1	1	NUM
ejpam-5689	23	10	)	)	PUNCT
ejpam-5689	23	11	(	(	PUNCT
ejpam-5689	23	12	2025	2025	NUM
ejpam-5689	23	13	)	)	PUNCT
ejpam-5689	23	14	,	,	PUNCT
ejpam-5689	23	15	5689	5689	NUM
ejpam-5689	23	16	2	2	NUM
ejpam-5689	23	17	of	of	ADP
ejpam-5689	23	18	11	11	NUM
ejpam-5689	23	19	for	for	ADP
ejpam-5689	23	20	1	1	NUM
ejpam-5689	23	21	≤	≤	NUM
ejpam-5689	23	22	i	i	PRON
ejpam-5689	23	23	≤	≤	NUM
ejpam-5689	23	24	j	j	PROPN
ejpam-5689	23	25	≤	≤	PROPN
ejpam-5689	23	26	n.	n.	NOUN
ejpam-5689	23	27	in	in	ADP
ejpam-5689	23	28	particular	particular	ADJ
ejpam-5689	23	29	,	,	PUNCT
ejpam-5689	23	30	if	if	SCONJ
ejpam-5689	23	31	j	j	PROPN
ejpam-5689	23	32	=	=	SYM
ejpam-5689	23	33	i	i	PROPN
ejpam-5689	23	34	,	,	PUNCT
ejpam-5689	23	35	then	then	ADV
ejpam-5689	23	36	sj	sj	INTJ
ejpam-5689	23	37	(	(	PUNCT
ejpam-5689	23	38	xy	xy	PROPN
ejpam-5689	23	39	−	−	PROPN
ejpam-5689	23	40	y	y	PROPN
ejpam-5689	23	41	x	x	PROPN
ejpam-5689	23	42	)	)	PUNCT
ejpam-5689	23	43	≤	≤	PUNCT
ejpam-5689	24	1	∥y	∥y	ADV
ejpam-5689	24	2	∥	∥	PUNCT
ejpam-5689	24	3	sj	sj	INTJ
ejpam-5689	24	4	(	(	PUNCT
ejpam-5689	24	5	x	x	NOUN
ejpam-5689	24	6	⊕x	⊕x	NOUN
ejpam-5689	24	7	)	)	PUNCT
ejpam-5689	25	1	+	+	CCONJ
ejpam-5689	25	2	1	1	NUM
ejpam-5689	25	3	2	2	NUM
ejpam-5689	25	4	∥xy	∥xy	NOUN
ejpam-5689	25	5	−	−	NOUN
ejpam-5689	26	1	y	y	PROPN
ejpam-5689	26	2	x∥	x∥	PROPN
ejpam-5689	26	3	.	.	PUNCT
ejpam-5689	27	1	(	(	PUNCT
ejpam-5689	27	2	3	3	X
ejpam-5689	27	3	)	)	PUNCT
ejpam-5689	27	4	singular	singular	ADJ
ejpam-5689	27	5	values	value	NOUN
ejpam-5689	27	6	of	of	ADP
ejpam-5689	27	7	a	a	DET
ejpam-5689	27	8	matrix	matrix	NOUN
ejpam-5689	27	9	play	play	VERB
ejpam-5689	27	10	a	a	DET
ejpam-5689	27	11	critical	critical	ADJ
ejpam-5689	27	12	role	role	NOUN
ejpam-5689	27	13	in	in	ADP
ejpam-5689	27	14	various	various	ADJ
ejpam-5689	27	15	applications	application	NOUN
ejpam-5689	27	16	,	,	PUNCT
ejpam-5689	27	17	including	include	VERB
ejpam-5689	27	18	data	data	NOUN
ejpam-5689	27	19	compression	compression	NOUN
ejpam-5689	27	20	,	,	PUNCT
ejpam-5689	27	21	noise	noise	NOUN
ejpam-5689	27	22	reduction	reduction	NOUN
ejpam-5689	27	23	,	,	PUNCT
ejpam-5689	27	24	and	and	CCONJ
ejpam-5689	27	25	the	the	DET
ejpam-5689	27	26	resolution	resolution	NOUN
ejpam-5689	27	27	of	of	ADP
ejpam-5689	27	28	ill	ill	ADV
ejpam-5689	27	29	-	-	PUNCT
ejpam-5689	27	30	posed	pose	VERB
ejpam-5689	27	31	problems	problem	NOUN
ejpam-5689	27	32	(	(	PUNCT
ejpam-5689	27	33	see	see	VERB
ejpam-5689	27	34	,	,	PUNCT
ejpam-5689	27	35	e.g.	e.g.	ADV
ejpam-5689	27	36	,	,	PUNCT
ejpam-5689	27	37	[	[	X
ejpam-5689	27	38	1	1	NUM
ejpam-5689	27	39	]	]	PUNCT
ejpam-5689	27	40	,	,	PUNCT
ejpam-5689	27	41	[	[	X
ejpam-5689	27	42	2	2	NUM
ejpam-5689	27	43	]	]	PUNCT
ejpam-5689	27	44	,	,	PUNCT
ejpam-5689	27	45	and	and	CCONJ
ejpam-5689	27	46	[	[	X
ejpam-5689	27	47	12	12	NUM
ejpam-5689	27	48	]	]	NUM
ejpam-5689	27	49	)	)	PUNCT
ejpam-5689	27	50	,	,	PUNCT
ejpam-5689	27	51	particularly	particularly	ADV
ejpam-5689	27	52	in	in	ADP
ejpam-5689	27	53	fields	field	NOUN
ejpam-5689	27	54	such	such	ADJ
ejpam-5689	27	55	as	as	ADP
ejpam-5689	27	56	electrical	electrical	ADJ
ejpam-5689	27	57	and	and	CCONJ
ejpam-5689	27	58	mechanical	mechanical	ADJ
ejpam-5689	27	59	engineering	engineering	NOUN
ejpam-5689	27	60	,	,	PUNCT
ejpam-5689	27	61	where	where	SCONJ
ejpam-5689	27	62	these	these	DET
ejpam-5689	27	63	concepts	concept	NOUN
ejpam-5689	27	64	are	be	AUX
ejpam-5689	27	65	used	use	VERB
ejpam-5689	27	66	to	to	PART
ejpam-5689	27	67	optimize	optimize	VERB
ejpam-5689	27	68	signal	signal	NOUN
ejpam-5689	27	69	processing	processing	NOUN
ejpam-5689	27	70	and	and	CCONJ
ejpam-5689	27	71	system	system	NOUN
ejpam-5689	27	72	performance	performance	NOUN
ejpam-5689	27	73	.	.	PUNCT
ejpam-5689	28	1	the	the	DET
ejpam-5689	28	2	spectral	spectral	ADJ
ejpam-5689	28	3	norm	norm	NOUN
ejpam-5689	28	4	,	,	PUNCT
ejpam-5689	28	5	defined	define	VERB
ejpam-5689	28	6	as	as	ADP
ejpam-5689	28	7	the	the	DET
ejpam-5689	28	8	largest	large	ADJ
ejpam-5689	28	9	singular	singular	ADJ
ejpam-5689	28	10	value	value	NOUN
ejpam-5689	28	11	,	,	PUNCT
ejpam-5689	28	12	is	be	AUX
ejpam-5689	28	13	fundamental	fundamental	ADJ
ejpam-5689	28	14	for	for	ADP
ejpam-5689	28	15	assessing	assess	VERB
ejpam-5689	28	16	the	the	DET
ejpam-5689	28	17	stability	stability	NOUN
ejpam-5689	28	18	of	of	ADP
ejpam-5689	28	19	the	the	DET
ejpam-5689	28	20	matrix	matrix	NOUN
ejpam-5689	28	21	and	and	CCONJ
ejpam-5689	28	22	quantifying	quantify	VERB
ejpam-5689	28	23	its	its	PRON
ejpam-5689	28	24	maximum	maximum	ADJ
ejpam-5689	28	25	impact	impact	NOUN
ejpam-5689	28	26	as	as	ADP
ejpam-5689	28	27	a	a	DET
ejpam-5689	28	28	linear	linear	ADJ
ejpam-5689	28	29	transformation	transformation	NOUN
ejpam-5689	28	30	,	,	PUNCT
ejpam-5689	28	31	which	which	PRON
ejpam-5689	28	32	is	be	AUX
ejpam-5689	28	33	crucial	crucial	ADJ
ejpam-5689	28	34	because	because	SCONJ
ejpam-5689	28	35	it	it	PRON
ejpam-5689	28	36	ensures	ensure	VERB
ejpam-5689	28	37	that	that	SCONJ
ejpam-5689	28	38	small	small	ADJ
ejpam-5689	28	39	perturbations	perturbation	NOUN
ejpam-5689	28	40	in	in	ADP
ejpam-5689	28	41	the	the	DET
ejpam-5689	28	42	input	input	NOUN
ejpam-5689	28	43	data	datum	NOUN
ejpam-5689	28	44	do	do	AUX
ejpam-5689	28	45	not	not	PART
ejpam-5689	28	46	lead	lead	VERB
ejpam-5689	28	47	to	to	ADP
ejpam-5689	28	48	disproportionately	disproportionately	ADV
ejpam-5689	28	49	large	large	ADJ
ejpam-5689	28	50	errors	error	NOUN
ejpam-5689	28	51	in	in	ADP
ejpam-5689	28	52	the	the	DET
ejpam-5689	28	53	output	output	NOUN
ejpam-5689	28	54	,	,	PUNCT
ejpam-5689	28	55	making	make	VERB
ejpam-5689	28	56	computations	computation	NOUN
ejpam-5689	28	57	reliable	reliable	ADJ
ejpam-5689	28	58	and	and	CCONJ
ejpam-5689	28	59	consistent	consistent	ADJ
ejpam-5689	28	60	in	in	ADP
ejpam-5689	28	61	practical	practical	ADJ
ejpam-5689	28	62	applications	application	NOUN
ejpam-5689	28	63	.	.	PUNCT
ejpam-5689	29	1	in	in	ADP
ejpam-5689	29	2	this	this	DET
ejpam-5689	29	3	paper	paper	NOUN
ejpam-5689	29	4	,	,	PUNCT
ejpam-5689	29	5	we	we	PRON
ejpam-5689	29	6	give	give	VERB
ejpam-5689	29	7	several	several	ADJ
ejpam-5689	29	8	singular	singular	ADJ
ejpam-5689	29	9	value	value	NOUN
ejpam-5689	29	10	inequalities	inequality	NOUN
ejpam-5689	29	11	.	.	PUNCT
ejpam-5689	30	1	among	among	ADP
ejpam-5689	30	2	other	other	ADJ
ejpam-5689	30	3	results	result	NOUN
ejpam-5689	30	4	,	,	PUNCT
ejpam-5689	30	5	we	we	PRON
ejpam-5689	30	6	give	give	VERB
ejpam-5689	30	7	a	a	DET
ejpam-5689	30	8	related	related	ADJ
ejpam-5689	30	9	inequality	inequality	NOUN
ejpam-5689	30	10	to	to	ADP
ejpam-5689	30	11	inequality	inequality	NOUN
ejpam-5689	30	12	(	(	PUNCT
ejpam-5689	30	13	2	2	NUM
ejpam-5689	30	14	)	)	PUNCT
ejpam-5689	30	15	and	and	CCONJ
ejpam-5689	30	16	we	we	PRON
ejpam-5689	30	17	give	give	VERB
ejpam-5689	30	18	a	a	DET
ejpam-5689	30	19	generalization	generalization	NOUN
ejpam-5689	30	20	of	of	ADP
ejpam-5689	30	21	inequality	inequality	NOUN
ejpam-5689	30	22	(	(	PUNCT
ejpam-5689	30	23	3	3	NUM
ejpam-5689	30	24	)	)	PUNCT
ejpam-5689	30	25	.	.	PUNCT
ejpam-5689	31	1	for	for	ADP
ejpam-5689	31	2	recent	recent	ADJ
ejpam-5689	31	3	articles	article	NOUN
ejpam-5689	31	4	related	relate	VERB
ejpam-5689	31	5	to	to	PART
ejpam-5689	31	6	matrix	matrix	VERB
ejpam-5689	31	7	and	and	CCONJ
ejpam-5689	31	8	singular	singular	ADJ
ejpam-5689	31	9	value	value	NOUN
ejpam-5689	31	10	inequalities	inequality	NOUN
ejpam-5689	31	11	,	,	PUNCT
ejpam-5689	31	12	we	we	PRON
ejpam-5689	31	13	refer	refer	VERB
ejpam-5689	31	14	the	the	DET
ejpam-5689	31	15	reader	reader	NOUN
ejpam-5689	31	16	to	to	ADP
ejpam-5689	31	17	[	[	X
ejpam-5689	31	18	4	4	NUM
ejpam-5689	31	19	]	]	PUNCT
ejpam-5689	31	20	,	,	PUNCT
ejpam-5689	31	21	[	[	X
ejpam-5689	31	22	3	3	NUM
ejpam-5689	31	23	]	]	PUNCT
ejpam-5689	31	24	,	,	PUNCT
ejpam-5689	31	25	[	[	X
ejpam-5689	31	26	5	5	NUM
ejpam-5689	31	27	]	]	PUNCT
ejpam-5689	31	28	,	,	PUNCT
ejpam-5689	31	29	[	[	X
ejpam-5689	31	30	10	10	NUM
ejpam-5689	31	31	]	]	PUNCT
ejpam-5689	31	32	and	and	CCONJ
ejpam-5689	31	33	[	[	X
ejpam-5689	31	34	11	11	NUM
ejpam-5689	31	35	]	]	SYM
ejpam-5689	31	36	.	.	PUNCT
ejpam-5689	32	1	2	2	X
ejpam-5689	32	2	.	.	X
ejpam-5689	32	3	main	main	ADJ
ejpam-5689	32	4	results	result	NOUN
ejpam-5689	32	5	to	to	PART
ejpam-5689	32	6	start	start	VERB
ejpam-5689	32	7	our	our	PRON
ejpam-5689	32	8	analysis	analysis	NOUN
ejpam-5689	32	9	,	,	PUNCT
ejpam-5689	32	10	we	we	PRON
ejpam-5689	32	11	need	need	VERB
ejpam-5689	32	12	the	the	DET
ejpam-5689	32	13	following	follow	VERB
ejpam-5689	32	14	lemmas	lemmas	NOUN
ejpam-5689	32	15	.	.	PUNCT
ejpam-5689	33	1	the	the	DET
ejpam-5689	33	2	first	first	ADJ
ejpam-5689	33	3	lemma	lemma	PROPN
ejpam-5689	33	4	is	be	AUX
ejpam-5689	33	5	a	a	DET
ejpam-5689	33	6	consequence	consequence	NOUN
ejpam-5689	33	7	of	of	ADP
ejpam-5689	33	8	the	the	DET
ejpam-5689	33	9	spectral	spectral	ADJ
ejpam-5689	33	10	theorem	theorem	NOUN
ejpam-5689	33	11	for	for	ADP
ejpam-5689	33	12	matrices	matrix	NOUN
ejpam-5689	33	13	(	(	PUNCT
ejpam-5689	33	14	see	see	VERB
ejpam-5689	33	15	,	,	PUNCT
ejpam-5689	33	16	e.g.	e.g.	ADV
ejpam-5689	33	17	,	,	PUNCT
ejpam-5689	33	18	[	[	X
ejpam-5689	33	19	13	13	NUM
ejpam-5689	33	20	,	,	PUNCT
ejpam-5689	33	21	p.	p.	NOUN
ejpam-5689	33	22	5	5	NUM
ejpam-5689	33	23	]	]	PUNCT
ejpam-5689	33	24	)	)	PUNCT
ejpam-5689	33	25	,	,	PUNCT
ejpam-5689	33	26	the	the	DET
ejpam-5689	33	27	second	second	ADJ
ejpam-5689	33	28	lemma	lemma	PROPN
ejpam-5689	33	29	was	be	AUX
ejpam-5689	33	30	given	give	VERB
ejpam-5689	33	31	in	in	ADP
ejpam-5689	33	32	[	[	X
ejpam-5689	33	33	7	7	NUM
ejpam-5689	33	34	]	]	PUNCT
ejpam-5689	33	35	,	,	PUNCT
ejpam-5689	33	36	while	while	SCONJ
ejpam-5689	33	37	the	the	DET
ejpam-5689	33	38	third	third	ADJ
ejpam-5689	33	39	lemma	lemma	PROPN
ejpam-5689	33	40	can	can	AUX
ejpam-5689	33	41	be	be	AUX
ejpam-5689	33	42	found	find	VERB
ejpam-5689	33	43	in	in	ADP
ejpam-5689	33	44	[	[	NOUN
ejpam-5689	33	45	13	13	NUM
ejpam-5689	33	46	,	,	PUNCT
ejpam-5689	33	47	p.	p.	NOUN
ejpam-5689	33	48	75	75	NUM
ejpam-5689	33	49	]	]	PUNCT
ejpam-5689	33	50	.	.	PUNCT
ejpam-5689	34	1	lemma	lemma	PROPN
ejpam-5689	34	2	1	1	X
ejpam-5689	34	3	.	.	PUNCT
ejpam-5689	35	1	let	let	VERB
ejpam-5689	35	2	x	x	SYM
ejpam-5689	35	3	∈	∈	PROPN
ejpam-5689	35	4	mn(c	mn(c	X
ejpam-5689	35	5	)	)	PUNCT
ejpam-5689	35	6	and	and	CCONJ
ejpam-5689	35	7	let	let	VERB
ejpam-5689	35	8	f	f	PRON
ejpam-5689	35	9	be	be	AUX
ejpam-5689	35	10	nonnegative	nonnegative	ADJ
ejpam-5689	35	11	increasing	increase	VERB
ejpam-5689	35	12	function	function	NOUN
ejpam-5689	35	13	on	on	ADP
ejpam-5689	35	14	[	[	X
ejpam-5689	35	15	0,∞	0,∞	NOUN
ejpam-5689	35	16	)	)	PUNCT
ejpam-5689	35	17	.	.	PUNCT
ejpam-5689	36	1	then	then	ADV
ejpam-5689	36	2	f(sj(x	f(sj(x	PROPN
ejpam-5689	36	3	)	)	PUNCT
ejpam-5689	36	4	)	)	PUNCT
ejpam-5689	37	1	=	=	SYM
ejpam-5689	37	2	sj(f(|x|	sj(f(|x|	PROPN
ejpam-5689	37	3	)	)	PUNCT
ejpam-5689	37	4	)	)	PUNCT
ejpam-5689	37	5	,	,	PUNCT
ejpam-5689	37	6	where	where	SCONJ
ejpam-5689	37	7	|x|	|x|	PROPN
ejpam-5689	37	8	is	be	AUX
ejpam-5689	37	9	the	the	DET
ejpam-5689	37	10	absolute	absolute	ADJ
ejpam-5689	37	11	value	value	NOUN
ejpam-5689	37	12	of	of	ADP
ejpam-5689	37	13	the	the	DET
ejpam-5689	37	14	matrix	matrix	NOUN
ejpam-5689	37	15	x.	x.	NOUN
ejpam-5689	37	16	lemma	lemma	PROPN
ejpam-5689	38	1	2	2	X
ejpam-5689	38	2	.	.	PUNCT
ejpam-5689	38	3	let	let	VERB
ejpam-5689	38	4	x	x	PRON
ejpam-5689	38	5	,	,	PUNCT
ejpam-5689	38	6	y	y	PROPN
ejpam-5689	38	7	∈	∈	PROPN
ejpam-5689	38	8	mn(c	mn(c	X
ejpam-5689	38	9	)	)	PUNCT
ejpam-5689	38	10	.	.	PUNCT
ejpam-5689	39	1	then	then	ADV
ejpam-5689	39	2	sj	sj	INTJ
ejpam-5689	39	3	(	(	PUNCT
ejpam-5689	39	4	x	x	PROPN
ejpam-5689	39	5	+	+	NUM
ejpam-5689	39	6	y	y	PROPN
ejpam-5689	39	7	)	)	PUNCT
ejpam-5689	39	8	≤	≤	PUNCT
ejpam-5689	39	9	sj	sj	INTJ
ejpam-5689	39	10	(	(	PUNCT
ejpam-5689	39	11	x	x	PROPN
ejpam-5689	39	12	⊕	⊕	PROPN
ejpam-5689	39	13	y	y	PROPN
ejpam-5689	39	14	)	)	PUNCT
ejpam-5689	40	1	+	+	CCONJ
ejpam-5689	40	2	1	1	NUM
ejpam-5689	40	3	2	2	NUM
ejpam-5689	40	4	∥x	∥x	PROPN
ejpam-5689	40	5	+	+	NUM
ejpam-5689	40	6	y	y	PROPN
ejpam-5689	40	7	∥	∥	PUNCT
ejpam-5689	40	8	.	.	PUNCT
ejpam-5689	41	1	lemma	lemma	PROPN
ejpam-5689	41	2	3	3	X
ejpam-5689	41	3	.	.	PUNCT
ejpam-5689	42	1	let	let	VERB
ejpam-5689	42	2	x	x	PRON
ejpam-5689	42	3	,	,	PUNCT
ejpam-5689	42	4	y	y	PROPN
ejpam-5689	42	5	,	,	PUNCT
ejpam-5689	42	6	z	z	NOUN
ejpam-5689	42	7	∈	∈	PROPN
ejpam-5689	42	8	mn(c	mn(c	X
ejpam-5689	42	9	)	)	PUNCT
ejpam-5689	42	10	.	.	PUNCT
ejpam-5689	43	1	then	then	ADV
ejpam-5689	43	2	sj(xzy	sj(xzy	VERB
ejpam-5689	43	3	)	)	PUNCT
ejpam-5689	44	1	≤	≤	NUM
ejpam-5689	44	2	∥x∥	∥x∥	NOUN
ejpam-5689	44	3	∥y	∥y	PROPN
ejpam-5689	44	4	∥	∥	PUNCT
ejpam-5689	44	5	sj(z	sj(z	NOUN
ejpam-5689	44	6	)	)	PUNCT
ejpam-5689	44	7	.	.	PUNCT
ejpam-5689	45	1	theorem	theorem	NOUN
ejpam-5689	45	2	1	1	NUM
ejpam-5689	45	3	.	.	PUNCT
ejpam-5689	46	1	let	let	VERB
ejpam-5689	46	2	a	a	DET
ejpam-5689	46	3	,	,	PUNCT
ejpam-5689	46	4	b	b	NOUN
ejpam-5689	46	5	,	,	PUNCT
ejpam-5689	46	6	x	x	X
ejpam-5689	46	7	,	,	PUNCT
ejpam-5689	46	8	y	y	PROPN
ejpam-5689	46	9	∈	∈	PROPN
ejpam-5689	46	10	mn(c	mn(c	X
ejpam-5689	46	11	)	)	PUNCT
ejpam-5689	46	12	.	.	PUNCT
ejpam-5689	47	1	then	then	ADV
ejpam-5689	47	2	(	(	PUNCT
ejpam-5689	47	3	a	a	X
ejpam-5689	47	4	)	)	PUNCT
ejpam-5689	47	5	sj	sj	PROPN
ejpam-5689	47	6	(	(	PUNCT
ejpam-5689	47	7	f	f	X
ejpam-5689	47	8	(	(	PUNCT
ejpam-5689	47	9	|xay	|xay	ADV
ejpam-5689	47	10	+	+	CCONJ
ejpam-5689	47	11	y	y	PROPN
ejpam-5689	47	12	bx|	bx|	NOUN
ejpam-5689	47	13	)	)	PUNCT
ejpam-5689	47	14	)	)	PUNCT
ejpam-5689	48	1	≤	≤	NUM
ejpam-5689	48	2	f	f	X
ejpam-5689	48	3	(	(	PUNCT
ejpam-5689	48	4	∥x∥	∥x∥	NOUN
ejpam-5689	48	5	)	)	PUNCT
ejpam-5689	48	6	f	f	PROPN
ejpam-5689	48	7	(	(	PUNCT
ejpam-5689	48	8	∥y	∥y	NOUN
ejpam-5689	48	9	∥	∥	NUM
ejpam-5689	48	10	)	)	PUNCT
ejpam-5689	48	11	sj(f	sj(f	NOUN
ejpam-5689	48	12	(	(	PUNCT
ejpam-5689	48	13	|a|)⊕	|a|)⊕	NUM
ejpam-5689	48	14	f	f	X
ejpam-5689	48	15	(	(	PUNCT
ejpam-5689	48	16	|b|	|b|	PROPN
ejpam-5689	48	17	)	)	PUNCT
ejpam-5689	48	18	)	)	PUNCT
ejpam-5689	49	1	+	+	NOUN
ejpam-5689	49	2	f	f	X
ejpam-5689	49	3	(	(	PUNCT
ejpam-5689	49	4	1	1	NUM
ejpam-5689	49	5	2	2	NUM
ejpam-5689	49	6	)	)	PUNCT
ejpam-5689	49	7	f	f	NOUN
ejpam-5689	49	8	(	(	PUNCT
ejpam-5689	49	9	∥xay	∥xay	PROPN
ejpam-5689	49	10	+	+	CCONJ
ejpam-5689	49	11	y	y	PROPN
ejpam-5689	49	12	bx∥	bx∥	PROPN
ejpam-5689	49	13	)	)	PUNCT
ejpam-5689	49	14	,	,	PUNCT
ejpam-5689	49	15	(	(	PUNCT
ejpam-5689	49	16	4	4	X
ejpam-5689	49	17	)	)	PUNCT
ejpam-5689	49	18	where	where	SCONJ
ejpam-5689	49	19	f	f	PROPN
ejpam-5689	49	20	is	be	AUX
ejpam-5689	49	21	a	a	DET
ejpam-5689	49	22	nonnegative	nonnegative	ADJ
ejpam-5689	49	23	increasing	increase	VERB
ejpam-5689	49	24	submultiplicative	submultiplicative	ADJ
ejpam-5689	49	25	concave	concave	NOUN
ejpam-5689	49	26	function	function	NOUN
ejpam-5689	49	27	on	on	ADP
ejpam-5689	49	28	[	[	X
ejpam-5689	49	29	0,∞	0,∞	NOUN
ejpam-5689	49	30	)	)	PUNCT
ejpam-5689	49	31	with	with	ADP
ejpam-5689	49	32	f(0	f(0	NOUN
ejpam-5689	49	33	)	)	PUNCT
ejpam-5689	49	34	=	=	SYM
ejpam-5689	49	35	0	0	X
ejpam-5689	49	36	.	.	PUNCT
ejpam-5689	49	37	a.	a.	PROPN
ejpam-5689	49	38	al	al	PROPN
ejpam-5689	49	39	-	-	PUNCT
ejpam-5689	49	40	natoor	natoor	NOUN
ejpam-5689	49	41	,	,	PUNCT
ejpam-5689	49	42	f.	f.	PROPN
ejpam-5689	49	43	alrimawi	alrimawi	PROPN
ejpam-5689	49	44	/	/	SYM
ejpam-5689	49	45	eur	eur	PROPN
ejpam-5689	49	46	.	.	PUNCT
ejpam-5689	50	1	j.	j.	PROPN
ejpam-5689	50	2	pure	pure	PROPN
ejpam-5689	50	3	appl	appl	PROPN
ejpam-5689	50	4	.	.	PROPN
ejpam-5689	50	5	math	math	PROPN
ejpam-5689	50	6	,	,	PUNCT
ejpam-5689	50	7	18	18	NUM
ejpam-5689	50	8	(	(	PUNCT
ejpam-5689	50	9	1	1	NUM
ejpam-5689	50	10	)	)	PUNCT
ejpam-5689	50	11	(	(	PUNCT
ejpam-5689	50	12	2025	2025	NUM
ejpam-5689	50	13	)	)	PUNCT
ejpam-5689	50	14	,	,	PUNCT
ejpam-5689	50	15	5689	5689	NUM
ejpam-5689	50	16	3	3	NUM
ejpam-5689	50	17	of	of	ADP
ejpam-5689	50	18	11	11	NUM
ejpam-5689	50	19	(	(	PUNCT
ejpam-5689	50	20	b	b	NOUN
ejpam-5689	50	21	)	)	PUNCT
ejpam-5689	50	22	sj	sj	PROPN
ejpam-5689	50	23	(	(	PUNCT
ejpam-5689	50	24	f	f	X
ejpam-5689	50	25	(	(	PUNCT
ejpam-5689	50	26	|xay	|xay	ADV
ejpam-5689	50	27	+	+	CCONJ
ejpam-5689	50	28	y	y	PROPN
ejpam-5689	50	29	bx|	bx|	NOUN
ejpam-5689	50	30	)	)	PUNCT
ejpam-5689	50	31	)	)	PUNCT
ejpam-5689	51	1	≤	≤	NUM
ejpam-5689	52	1	f(2	f(2	NOUN
ejpam-5689	52	2	)	)	PUNCT
ejpam-5689	52	3	2	2	NUM
ejpam-5689	52	4	f	f	NOUN
ejpam-5689	52	5	(	(	PUNCT
ejpam-5689	52	6	∥x∥	∥x∥	NOUN
ejpam-5689	52	7	)	)	PUNCT
ejpam-5689	52	8	f	f	PROPN
ejpam-5689	52	9	(	(	PUNCT
ejpam-5689	52	10	∥y	∥y	NOUN
ejpam-5689	52	11	∥	∥	NUM
ejpam-5689	52	12	)	)	PUNCT
ejpam-5689	52	13	sj(f	sj(f	NOUN
ejpam-5689	52	14	(	(	PUNCT
ejpam-5689	52	15	|a|)⊕	|a|)⊕	NUM
ejpam-5689	52	16	f	f	X
ejpam-5689	52	17	(	(	PUNCT
ejpam-5689	52	18	|b|	|b|	PROPN
ejpam-5689	52	19	)	)	PUNCT
ejpam-5689	52	20	)	)	PUNCT
ejpam-5689	53	1	+	+	CCONJ
ejpam-5689	53	2	1	1	NUM
ejpam-5689	53	3	2	2	NUM
ejpam-5689	53	4	f	f	NOUN
ejpam-5689	53	5	(	(	PUNCT
ejpam-5689	53	6	∥xay	∥xay	PROPN
ejpam-5689	53	7	+	+	CCONJ
ejpam-5689	53	8	y	y	PROPN
ejpam-5689	53	9	bx∥	bx∥	PROPN
ejpam-5689	53	10	)	)	PUNCT
ejpam-5689	53	11	,	,	PUNCT
ejpam-5689	53	12	(	(	PUNCT
ejpam-5689	53	13	5	5	X
ejpam-5689	53	14	)	)	PUNCT
ejpam-5689	53	15	where	where	SCONJ
ejpam-5689	53	16	f	f	PROPN
ejpam-5689	53	17	is	be	AUX
ejpam-5689	53	18	a	a	DET
ejpam-5689	53	19	nonnegative	nonnegative	ADJ
ejpam-5689	53	20	increasing	increase	VERB
ejpam-5689	53	21	submultiplicative	submultiplicative	ADJ
ejpam-5689	53	22	convex	convex	NOUN
ejpam-5689	53	23	function	function	NOUN
ejpam-5689	53	24	on	on	ADP
ejpam-5689	53	25	[	[	X
ejpam-5689	53	26	0,∞	0,∞	NOUN
ejpam-5689	53	27	)	)	PUNCT
ejpam-5689	53	28	.	.	PUNCT
ejpam-5689	54	1	proof	proof	NOUN
ejpam-5689	54	2	.	.	PUNCT
ejpam-5689	55	1	we	we	PRON
ejpam-5689	55	2	have	have	VERB
ejpam-5689	55	3	sj	sj	INTJ
ejpam-5689	55	4	(	(	PUNCT
ejpam-5689	55	5	f	f	X
ejpam-5689	55	6	(	(	PUNCT
ejpam-5689	55	7	|xay	|xay	ADV
ejpam-5689	55	8	+	+	CCONJ
ejpam-5689	55	9	y	y	PROPN
ejpam-5689	55	10	bx|	bx|	NOUN
ejpam-5689	55	11	)	)	PUNCT
ejpam-5689	55	12	)	)	PUNCT
ejpam-5689	56	1	=	=	SYM
ejpam-5689	56	2	f	f	X
ejpam-5689	56	3	(	(	PUNCT
ejpam-5689	56	4	sj	sj	INTJ
ejpam-5689	56	5	(	(	PUNCT
ejpam-5689	56	6	xay	xay	PROPN
ejpam-5689	56	7	+	+	PROPN
ejpam-5689	56	8	y	y	PROPN
ejpam-5689	56	9	bx	bx	PROPN
ejpam-5689	56	10	)	)	PUNCT
ejpam-5689	56	11	)	)	PUNCT
ejpam-5689	56	12	(	(	PUNCT
ejpam-5689	56	13	by	by	ADP
ejpam-5689	56	14	lemma	lemma	PROPN
ejpam-5689	56	15	1	1	NUM
ejpam-5689	56	16	)	)	PUNCT
ejpam-5689	56	17	≤	≤	NUM
ejpam-5689	56	18	f	f	X
ejpam-5689	56	19	(	(	PUNCT
ejpam-5689	56	20	sj	sj	INTJ
ejpam-5689	56	21	(	(	PUNCT
ejpam-5689	56	22	xay	xay	PROPN
ejpam-5689	56	23	⊕	⊕	PROPN
ejpam-5689	56	24	y	y	PROPN
ejpam-5689	56	25	bx	bx	PROPN
ejpam-5689	56	26	)	)	PUNCT
ejpam-5689	57	1	+	+	CCONJ
ejpam-5689	57	2	1	1	NUM
ejpam-5689	57	3	2	2	NUM
ejpam-5689	57	4	∥xay	∥xay	NOUN
ejpam-5689	57	5	+	+	CCONJ
ejpam-5689	57	6	y	y	PROPN
ejpam-5689	57	7	bx∥	bx∥	PROPN
ejpam-5689	57	8	)	)	PUNCT
ejpam-5689	57	9	(	(	PUNCT
ejpam-5689	57	10	6	6	NUM
ejpam-5689	57	11	)	)	PUNCT
ejpam-5689	57	12	(	(	PUNCT
ejpam-5689	57	13	by	by	ADP
ejpam-5689	57	14	lemma	lemma	PROPN
ejpam-5689	57	15	2	2	NUM
ejpam-5689	57	16	)	)	PUNCT
ejpam-5689	57	17	≤	≤	NUM
ejpam-5689	57	18	f	f	X
ejpam-5689	57	19	(	(	PUNCT
ejpam-5689	57	20	sj	sj	INTJ
ejpam-5689	57	21	(	(	PUNCT
ejpam-5689	57	22	xay	xay	PROPN
ejpam-5689	57	23	⊕	⊕	PROPN
ejpam-5689	57	24	(	(	PUNCT
ejpam-5689	57	25	y	y	PROPN
ejpam-5689	57	26	bx)∗	bx)∗	PROPN
ejpam-5689	57	27	)	)	PUNCT
ejpam-5689	57	28	)	)	PUNCT
ejpam-5689	58	1	+	+	CCONJ
ejpam-5689	58	2	f	f	X
ejpam-5689	58	3	(	(	PUNCT
ejpam-5689	58	4	1	1	NUM
ejpam-5689	58	5	2	2	NUM
ejpam-5689	58	6	∥xay	∥xay	PROPN
ejpam-5689	58	7	+	+	CCONJ
ejpam-5689	58	8	y	y	PROPN
ejpam-5689	58	9	bx∥	bx∥	PROPN
ejpam-5689	58	10	)	)	PUNCT
ejpam-5689	58	11	(	(	PUNCT
ejpam-5689	58	12	since	since	SCONJ
ejpam-5689	58	13	f	f	PROPN
ejpam-5689	58	14	is	be	AUX
ejpam-5689	58	15	concave	concave	NOUN
ejpam-5689	58	16	and	and	CCONJ
ejpam-5689	58	17	f(0	f(0	NOUN
ejpam-5689	58	18	)	)	PUNCT
ejpam-5689	58	19	=	=	SYM
ejpam-5689	58	20	0	0	X
ejpam-5689	58	21	)	)	PUNCT
ejpam-5689	58	22	≤	≤	NUM
ejpam-5689	58	23	f	f	X
ejpam-5689	58	24	(	(	PUNCT
ejpam-5689	58	25	sj	sj	INTJ
ejpam-5689	58	26	(	(	PUNCT
ejpam-5689	58	27	xay	xay	PROPN
ejpam-5689	58	28	⊕	⊕	PROPN
ejpam-5689	58	29	(	(	PUNCT
ejpam-5689	58	30	y	y	PROPN
ejpam-5689	58	31	bx)∗	bx)∗	PROPN
ejpam-5689	58	32	)	)	PUNCT
ejpam-5689	58	33	)	)	PUNCT
ejpam-5689	59	1	+	+	CCONJ
ejpam-5689	59	2	f	f	X
ejpam-5689	59	3	(	(	PUNCT
ejpam-5689	59	4	1	1	NUM
ejpam-5689	59	5	2	2	NUM
ejpam-5689	59	6	)	)	PUNCT
ejpam-5689	59	7	f	f	NOUN
ejpam-5689	59	8	(	(	PUNCT
ejpam-5689	59	9	∥xay	∥xay	PROPN
ejpam-5689	59	10	+	+	CCONJ
ejpam-5689	59	11	y	y	PROPN
ejpam-5689	59	12	bx∥	bx∥	PROPN
ejpam-5689	59	13	)	)	PUNCT
ejpam-5689	59	14	(	(	PUNCT
ejpam-5689	59	15	since	since	SCONJ
ejpam-5689	59	16	f	f	PROPN
ejpam-5689	59	17	is	be	AUX
ejpam-5689	59	18	submultiplicative	submultiplicative	ADJ
ejpam-5689	59	19	)	)	PUNCT
ejpam-5689	60	1	=	=	SYM
ejpam-5689	60	2	f	f	PROPN
ejpam-5689	60	3	(	(	PUNCT
ejpam-5689	60	4	sj	sj	INTJ
ejpam-5689	60	5	(	(	PUNCT
ejpam-5689	60	6	[	[	PUNCT
ejpam-5689	60	7	x	x	SYM
ejpam-5689	60	8	0	0	NUM
ejpam-5689	60	9	0	0	NUM
ejpam-5689	60	10	x∗	x∗	X
ejpam-5689	60	11	]	]	PUNCT
ejpam-5689	61	1	[	[	PUNCT
ejpam-5689	61	2	a	a	DET
ejpam-5689	61	3	0	0	NUM
ejpam-5689	61	4	0	0	NUM
ejpam-5689	61	5	b∗	b∗	ADJ
ejpam-5689	61	6	]	]	PUNCT
ejpam-5689	61	7	[	[	PUNCT
ejpam-5689	61	8	y	y	NOUN
ejpam-5689	61	9	0	0	NUM
ejpam-5689	61	10	0	0	NUM
ejpam-5689	61	11	y	y	PROPN
ejpam-5689	61	12	∗	∗	NOUN
ejpam-5689	61	13	]	]	PUNCT
ejpam-5689	61	14	)	)	PUNCT
ejpam-5689	61	15	)	)	PUNCT
ejpam-5689	62	1	+	+	NOUN
ejpam-5689	62	2	f	f	X
ejpam-5689	62	3	(	(	PUNCT
ejpam-5689	62	4	1	1	NUM
ejpam-5689	62	5	2	2	NUM
ejpam-5689	62	6	)	)	PUNCT
ejpam-5689	62	7	f	f	NOUN
ejpam-5689	62	8	(	(	PUNCT
ejpam-5689	62	9	∥xay	∥xay	PROPN
ejpam-5689	62	10	+	+	CCONJ
ejpam-5689	62	11	y	y	PROPN
ejpam-5689	62	12	bx∥	bx∥	PROPN
ejpam-5689	62	13	)	)	PUNCT
ejpam-5689	62	14	(	(	PUNCT
ejpam-5689	62	15	7	7	X
ejpam-5689	62	16	)	)	PUNCT
ejpam-5689	62	17	≤	≤	NUM
ejpam-5689	62	18	f	f	X
ejpam-5689	62	19	(	(	PUNCT
ejpam-5689	62	20	∥∥∥∥[x	∥∥∥∥[x	NOUN
ejpam-5689	62	21	0	0	NUM
ejpam-5689	62	22	0	0	NUM
ejpam-5689	62	23	x	x	SYM
ejpam-5689	63	1	]	]	X
ejpam-5689	63	2	∥∥∥∥	∥∥∥∥	NUM
ejpam-5689	63	3	∥∥∥∥[y	∥∥∥∥[y	VERB
ejpam-5689	63	4	0	0	NUM
ejpam-5689	63	5	0	0	NUM
ejpam-5689	63	6	y	y	PROPN
ejpam-5689	63	7	]	]	PUNCT
ejpam-5689	63	8	∥∥∥∥	∥∥∥∥	NUM
ejpam-5689	63	9	sj	sj	INTJ
ejpam-5689	63	10	(	(	PUNCT
ejpam-5689	63	11	[	[	X
ejpam-5689	63	12	a	a	DET
ejpam-5689	63	13	0	0	NUM
ejpam-5689	63	14	0	0	NUM
ejpam-5689	63	15	b	b	NOUN
ejpam-5689	63	16	]	]	PUNCT
ejpam-5689	63	17	)	)	PUNCT
ejpam-5689	63	18	)	)	PUNCT
ejpam-5689	64	1	+	+	NOUN
ejpam-5689	64	2	f	f	X
ejpam-5689	64	3	(	(	PUNCT
ejpam-5689	64	4	1	1	NUM
ejpam-5689	64	5	2	2	NUM
ejpam-5689	64	6	)	)	PUNCT
ejpam-5689	64	7	f	f	NOUN
ejpam-5689	64	8	(	(	PUNCT
ejpam-5689	64	9	∥xay	∥xay	PROPN
ejpam-5689	64	10	+	+	CCONJ
ejpam-5689	64	11	y	y	PROPN
ejpam-5689	64	12	bx∥	bx∥	PROPN
ejpam-5689	64	13	)	)	PUNCT
ejpam-5689	64	14	(	(	PUNCT
ejpam-5689	64	15	by	by	ADP
ejpam-5689	64	16	lemma	lemma	PROPN
ejpam-5689	64	17	3	3	NUM
ejpam-5689	64	18	)	)	PUNCT
ejpam-5689	64	19	=	=	SYM
ejpam-5689	65	1	f	f	X
ejpam-5689	65	2	(	(	PUNCT
ejpam-5689	65	3	∥x∥	∥x∥	NOUN
ejpam-5689	65	4	∥y	∥y	ADV
ejpam-5689	65	5	∥	∥	PUNCT
ejpam-5689	65	6	sj	sj	INTJ
ejpam-5689	65	7	(	(	PUNCT
ejpam-5689	65	8	[	[	PUNCT
ejpam-5689	65	9	a	a	DET
ejpam-5689	65	10	0	0	NUM
ejpam-5689	65	11	0	0	NUM
ejpam-5689	65	12	b	b	NOUN
ejpam-5689	65	13	]	]	PUNCT
ejpam-5689	65	14	)	)	PUNCT
ejpam-5689	65	15	)	)	PUNCT
ejpam-5689	66	1	+	+	CCONJ
ejpam-5689	66	2	f	f	X
ejpam-5689	66	3	(	(	PUNCT
ejpam-5689	66	4	1	1	NUM
ejpam-5689	66	5	2	2	NUM
ejpam-5689	66	6	)	)	PUNCT
ejpam-5689	66	7	f	f	NOUN
ejpam-5689	66	8	(	(	PUNCT
ejpam-5689	66	9	∥xay	∥xay	PROPN
ejpam-5689	66	10	+	+	CCONJ
ejpam-5689	66	11	y	y	PROPN
ejpam-5689	66	12	bx∥	bx∥	PROPN
ejpam-5689	66	13	)	)	PUNCT
ejpam-5689	66	14	≤	≤	PUNCT
ejpam-5689	67	1	f(∥x∥)f(∥y	f(∥x∥)f(∥y	PROPN
ejpam-5689	67	2	∥)f(sj(a⊕b	∥)f(sj(a⊕b	PROPN
ejpam-5689	67	3	)	)	PUNCT
ejpam-5689	67	4	)	)	PUNCT
ejpam-5689	68	1	+	+	CCONJ
ejpam-5689	68	2	f	f	X
ejpam-5689	68	3	(	(	PUNCT
ejpam-5689	68	4	1	1	NUM
ejpam-5689	68	5	2	2	NUM
ejpam-5689	68	6	)	)	PUNCT
ejpam-5689	68	7	f	f	NOUN
ejpam-5689	68	8	(	(	PUNCT
ejpam-5689	68	9	∥xay	∥xay	PROPN
ejpam-5689	68	10	+	+	CCONJ
ejpam-5689	68	11	y	y	PROPN
ejpam-5689	68	12	bx∥	bx∥	PROPN
ejpam-5689	68	13	)	)	PUNCT
ejpam-5689	68	14	=	=	SYM
ejpam-5689	68	15	f(∥x∥)f(∥y	f(∥x∥)f(∥y	PROPN
ejpam-5689	68	16	∥)sj(f	∥)sj(f	PROPN
ejpam-5689	68	17	(	(	PUNCT
ejpam-5689	68	18	|a|)⊕	|a|)⊕	ADJ
ejpam-5689	68	19	f	f	PROPN
ejpam-5689	68	20	(	(	PUNCT
ejpam-5689	68	21	|b|	|b|	PROPN
ejpam-5689	68	22	)	)	PUNCT
ejpam-5689	68	23	)	)	PUNCT
ejpam-5689	69	1	+	+	CCONJ
ejpam-5689	69	2	f	f	X
ejpam-5689	69	3	(	(	PUNCT
ejpam-5689	69	4	1	1	NUM
ejpam-5689	69	5	2	2	NUM
ejpam-5689	69	6	)	)	PUNCT
ejpam-5689	69	7	f	f	NOUN
ejpam-5689	69	8	(	(	PUNCT
ejpam-5689	69	9	∥xay	∥xay	PROPN
ejpam-5689	69	10	+	+	CCONJ
ejpam-5689	69	11	y	y	PROPN
ejpam-5689	69	12	bx∥	bx∥	PROPN
ejpam-5689	69	13	)	)	PUNCT
ejpam-5689	69	14	,	,	PUNCT
ejpam-5689	69	15	which	which	PRON
ejpam-5689	69	16	proves	prove	VERB
ejpam-5689	69	17	part	part	NOUN
ejpam-5689	69	18	(	(	PUNCT
ejpam-5689	69	19	a	a	NOUN
ejpam-5689	69	20	)	)	PUNCT
ejpam-5689	69	21	.	.	PUNCT
ejpam-5689	70	1	for	for	ADP
ejpam-5689	70	2	part	part	NOUN
ejpam-5689	70	3	(	(	PUNCT
ejpam-5689	70	4	b	b	NOUN
ejpam-5689	70	5	)	)	PUNCT
ejpam-5689	70	6	,	,	PUNCT
ejpam-5689	70	7	we	we	PRON
ejpam-5689	70	8	start	start	VERB
ejpam-5689	70	9	from	from	ADP
ejpam-5689	70	10	inequality	inequality	NOUN
ejpam-5689	70	11	(	(	PUNCT
ejpam-5689	70	12	6	6	NUM
ejpam-5689	70	13	)	)	PUNCT
ejpam-5689	70	14	,	,	PUNCT
ejpam-5689	70	15	so	so	SCONJ
ejpam-5689	70	16	we	we	PRON
ejpam-5689	70	17	have	have	VERB
ejpam-5689	70	18	sj	sj	INTJ
ejpam-5689	70	19	(	(	PUNCT
ejpam-5689	70	20	f	f	X
ejpam-5689	70	21	(	(	PUNCT
ejpam-5689	70	22	|xay	|xay	ADV
ejpam-5689	70	23	+	+	CCONJ
ejpam-5689	70	24	y	y	PROPN
ejpam-5689	70	25	bx|	bx|	NOUN
ejpam-5689	70	26	)	)	PUNCT
ejpam-5689	70	27	)	)	PUNCT
ejpam-5689	71	1	a.	a.	PROPN
ejpam-5689	71	2	al	al	PROPN
ejpam-5689	71	3	-	-	PUNCT
ejpam-5689	71	4	natoor	natoor	NOUN
ejpam-5689	71	5	,	,	PUNCT
ejpam-5689	71	6	f.	f.	PROPN
ejpam-5689	71	7	alrimawi	alrimawi	PROPN
ejpam-5689	71	8	/	/	SYM
ejpam-5689	71	9	eur	eur	PROPN
ejpam-5689	71	10	.	.	PUNCT
ejpam-5689	72	1	j.	j.	PROPN
ejpam-5689	72	2	pure	pure	PROPN
ejpam-5689	72	3	appl	appl	PROPN
ejpam-5689	72	4	.	.	PROPN
ejpam-5689	72	5	math	math	PROPN
ejpam-5689	72	6	,	,	PUNCT
ejpam-5689	72	7	18	18	NUM
ejpam-5689	72	8	(	(	PUNCT
ejpam-5689	72	9	1	1	NUM
ejpam-5689	72	10	)	)	PUNCT
ejpam-5689	72	11	(	(	PUNCT
ejpam-5689	72	12	2025	2025	NUM
ejpam-5689	72	13	)	)	PUNCT
ejpam-5689	72	14	,	,	PUNCT
ejpam-5689	72	15	5689	5689	NUM
ejpam-5689	72	16	4	4	NUM
ejpam-5689	72	17	of	of	ADP
ejpam-5689	72	18	11	11	NUM
ejpam-5689	72	19	≤	≤	NUM
ejpam-5689	72	20	f	f	X
ejpam-5689	72	21	(	(	PUNCT
ejpam-5689	72	22	sj	sj	INTJ
ejpam-5689	72	23	(	(	PUNCT
ejpam-5689	72	24	xay	xay	PROPN
ejpam-5689	72	25	⊕	⊕	PROPN
ejpam-5689	72	26	y	y	PROPN
ejpam-5689	72	27	bx	bx	PROPN
ejpam-5689	72	28	)	)	PUNCT
ejpam-5689	73	1	+	+	CCONJ
ejpam-5689	73	2	1	1	NUM
ejpam-5689	73	3	2	2	NUM
ejpam-5689	73	4	∥xay	∥xay	NOUN
ejpam-5689	73	5	+	+	CCONJ
ejpam-5689	73	6	y	y	PROPN
ejpam-5689	73	7	bx∥	bx∥	PROPN
ejpam-5689	73	8	)	)	PUNCT
ejpam-5689	73	9	≤	≤	NUM
ejpam-5689	73	10	1	1	NUM
ejpam-5689	73	11	2	2	NUM
ejpam-5689	73	12	f	f	NOUN
ejpam-5689	73	13	(	(	PUNCT
ejpam-5689	73	14	2sj	2sj	NOUN
ejpam-5689	73	15	(	(	PUNCT
ejpam-5689	73	16	xay	xay	PROPN
ejpam-5689	73	17	⊕	⊕	PROPN
ejpam-5689	73	18	(	(	PUNCT
ejpam-5689	73	19	y	y	PROPN
ejpam-5689	73	20	bx)∗	bx)∗	PROPN
ejpam-5689	73	21	)	)	PUNCT
ejpam-5689	73	22	)	)	PUNCT
ejpam-5689	74	1	+	+	CCONJ
ejpam-5689	74	2	1	1	NUM
ejpam-5689	74	3	2	2	NUM
ejpam-5689	74	4	f	f	NOUN
ejpam-5689	74	5	(	(	PUNCT
ejpam-5689	74	6	∥xay	∥xay	PROPN
ejpam-5689	74	7	+	+	CCONJ
ejpam-5689	74	8	y	y	PROPN
ejpam-5689	74	9	bx∥	bx∥	PROPN
ejpam-5689	74	10	)	)	PUNCT
ejpam-5689	74	11	(	(	PUNCT
ejpam-5689	74	12	since	since	SCONJ
ejpam-5689	74	13	f	f	PROPN
ejpam-5689	74	14	is	be	AUX
ejpam-5689	74	15	convex	convex	ADJ
ejpam-5689	74	16	)	)	PUNCT
ejpam-5689	74	17	=	=	SYM
ejpam-5689	74	18	1	1	NUM
ejpam-5689	74	19	2	2	NUM
ejpam-5689	74	20	f	f	NOUN
ejpam-5689	74	21	(	(	PUNCT
ejpam-5689	74	22	2sj	2sj	NOUN
ejpam-5689	74	23	(	(	PUNCT
ejpam-5689	74	24	xay	xay	PROPN
ejpam-5689	74	25	⊕	⊕	PROPN
ejpam-5689	74	26	(	(	PUNCT
ejpam-5689	74	27	y	y	PROPN
ejpam-5689	74	28	bx)∗	bx)∗	PROPN
ejpam-5689	74	29	)	)	PUNCT
ejpam-5689	74	30	)	)	PUNCT
ejpam-5689	75	1	+	+	CCONJ
ejpam-5689	75	2	1	1	NUM
ejpam-5689	75	3	2	2	NUM
ejpam-5689	75	4	f	f	NOUN
ejpam-5689	75	5	(	(	PUNCT
ejpam-5689	75	6	∥xay	∥xay	PROPN
ejpam-5689	75	7	+	+	CCONJ
ejpam-5689	75	8	y	y	PROPN
ejpam-5689	75	9	bx∥	bx∥	PROPN
ejpam-5689	75	10	)	)	PUNCT
ejpam-5689	75	11	=	=	SYM
ejpam-5689	75	12	1	1	NUM
ejpam-5689	75	13	2	2	NUM
ejpam-5689	75	14	f	f	NOUN
ejpam-5689	75	15	(	(	PUNCT
ejpam-5689	75	16	2sj	2sj	NOUN
ejpam-5689	75	17	(	(	PUNCT
ejpam-5689	75	18	[	[	PUNCT
ejpam-5689	75	19	x	x	SYM
ejpam-5689	75	20	0	0	NUM
ejpam-5689	75	21	0	0	NUM
ejpam-5689	75	22	x∗	x∗	X
ejpam-5689	75	23	]	]	PUNCT
ejpam-5689	75	24	[	[	PUNCT
ejpam-5689	75	25	a	a	DET
ejpam-5689	75	26	0	0	NUM
ejpam-5689	75	27	0	0	NUM
ejpam-5689	75	28	b∗	b∗	ADJ
ejpam-5689	75	29	]	]	PUNCT
ejpam-5689	75	30	[	[	PUNCT
ejpam-5689	75	31	y	y	NOUN
ejpam-5689	75	32	0	0	NUM
ejpam-5689	75	33	0	0	NUM
ejpam-5689	75	34	y	y	PROPN
ejpam-5689	75	35	∗	∗	NOUN
ejpam-5689	75	36	]	]	PUNCT
ejpam-5689	75	37	)	)	PUNCT
ejpam-5689	75	38	)	)	PUNCT
ejpam-5689	76	1	+	+	CCONJ
ejpam-5689	76	2	1	1	NUM
ejpam-5689	76	3	2	2	NUM
ejpam-5689	76	4	f	f	NOUN
ejpam-5689	76	5	(	(	PUNCT
ejpam-5689	76	6	∥xay	∥xay	PROPN
ejpam-5689	76	7	+	+	CCONJ
ejpam-5689	76	8	y	y	PROPN
ejpam-5689	76	9	bx∥	bx∥	PROPN
ejpam-5689	76	10	)	)	PUNCT
ejpam-5689	76	11	≤	≤	NOUN
ejpam-5689	76	12	1	1	NUM
ejpam-5689	76	13	2	2	NUM
ejpam-5689	76	14	f	f	NOUN
ejpam-5689	76	15	(	(	PUNCT
ejpam-5689	76	16	2	2	NUM
ejpam-5689	76	17	∥∥∥∥[x	∥∥∥∥[x	NOUN
ejpam-5689	76	18	0	0	NUM
ejpam-5689	76	19	0	0	NUM
ejpam-5689	76	20	x	x	SYM
ejpam-5689	76	21	]	]	X
ejpam-5689	76	22	∥∥∥∥	∥∥∥∥	NUM
ejpam-5689	76	23	∥∥∥∥[y	∥∥∥∥[y	VERB
ejpam-5689	76	24	0	0	NUM
ejpam-5689	76	25	0	0	NUM
ejpam-5689	76	26	y	y	PROPN
ejpam-5689	76	27	]	]	PUNCT
ejpam-5689	76	28	∥∥∥∥	∥∥∥∥	NUM
ejpam-5689	76	29	sj	sj	INTJ
ejpam-5689	76	30	(	(	PUNCT
ejpam-5689	76	31	[	[	X
ejpam-5689	76	32	a	a	DET
ejpam-5689	76	33	0	0	NUM
ejpam-5689	76	34	0	0	NUM
ejpam-5689	76	35	b	b	NOUN
ejpam-5689	76	36	]	]	PUNCT
ejpam-5689	76	37	)	)	PUNCT
ejpam-5689	76	38	)	)	PUNCT
ejpam-5689	77	1	+	+	CCONJ
ejpam-5689	77	2	1	1	NUM
ejpam-5689	77	3	2	2	NUM
ejpam-5689	77	4	f	f	NOUN
ejpam-5689	77	5	(	(	PUNCT
ejpam-5689	77	6	∥xay	∥xay	PROPN
ejpam-5689	77	7	+	+	CCONJ
ejpam-5689	77	8	y	y	PROPN
ejpam-5689	77	9	bx∥	bx∥	PROPN
ejpam-5689	77	10	)	)	PUNCT
ejpam-5689	77	11	(	(	PUNCT
ejpam-5689	77	12	by	by	ADP
ejpam-5689	77	13	lemma	lemma	PROPN
ejpam-5689	77	14	3	3	NUM
ejpam-5689	77	15	)	)	PUNCT
ejpam-5689	77	16	=	=	SYM
ejpam-5689	77	17	1	1	NUM
ejpam-5689	77	18	2	2	NUM
ejpam-5689	77	19	f	f	NOUN
ejpam-5689	77	20	(	(	PUNCT
ejpam-5689	77	21	2	2	NUM
ejpam-5689	77	22	∥x∥	∥x∥	NOUN
ejpam-5689	77	23	∥y	∥y	PROPN
ejpam-5689	77	24	∥	∥	PUNCT
ejpam-5689	77	25	sj	sj	INTJ
ejpam-5689	77	26	(	(	PUNCT
ejpam-5689	77	27	[	[	PUNCT
ejpam-5689	77	28	a	a	DET
ejpam-5689	77	29	0	0	NUM
ejpam-5689	77	30	0	0	NUM
ejpam-5689	77	31	b	b	NOUN
ejpam-5689	77	32	]	]	PUNCT
ejpam-5689	77	33	)	)	PUNCT
ejpam-5689	77	34	)	)	PUNCT
ejpam-5689	78	1	+	+	CCONJ
ejpam-5689	78	2	1	1	NUM
ejpam-5689	78	3	2	2	NUM
ejpam-5689	78	4	f	f	NOUN
ejpam-5689	78	5	(	(	PUNCT
ejpam-5689	78	6	∥xay	∥xay	PROPN
ejpam-5689	78	7	+	+	CCONJ
ejpam-5689	78	8	y	y	PROPN
ejpam-5689	78	9	bx∥	bx∥	PROPN
ejpam-5689	78	10	)	)	PUNCT
ejpam-5689	78	11	≤	≤	NOUN
ejpam-5689	78	12	f(2	f(2	PROPN
ejpam-5689	78	13	)	)	PUNCT
ejpam-5689	78	14	2	2	NUM
ejpam-5689	78	15	f(∥x∥)f(∥y	f(∥x∥)f(∥y	PROPN
ejpam-5689	78	16	∥)f(sj(a⊕b	∥)f(sj(a⊕b	PROPN
ejpam-5689	78	17	)	)	PUNCT
ejpam-5689	78	18	)	)	PUNCT
ejpam-5689	79	1	+	+	CCONJ
ejpam-5689	79	2	1	1	NUM
ejpam-5689	79	3	2	2	NUM
ejpam-5689	79	4	f	f	NOUN
ejpam-5689	79	5	(	(	PUNCT
ejpam-5689	79	6	∥xay	∥xay	PROPN
ejpam-5689	79	7	+	+	CCONJ
ejpam-5689	79	8	y	y	PROPN
ejpam-5689	79	9	bx∥	bx∥	PROPN
ejpam-5689	79	10	)	)	PUNCT
ejpam-5689	79	11	=	=	SYM
ejpam-5689	79	12	f(2	f(2	PROPN
ejpam-5689	79	13	)	)	PUNCT
ejpam-5689	79	14	2	2	NUM
ejpam-5689	79	15	f(∥x∥)f(∥y	f(∥x∥)f(∥y	PROPN
ejpam-5689	79	16	∥)sj(f	∥)sj(f	PROPN
ejpam-5689	79	17	(	(	PUNCT
ejpam-5689	79	18	|a|)⊕	|a|)⊕	ADJ
ejpam-5689	79	19	f	f	PROPN
ejpam-5689	79	20	(	(	PUNCT
ejpam-5689	79	21	|b|	|b|	PROPN
ejpam-5689	79	22	)	)	PUNCT
ejpam-5689	79	23	)	)	PUNCT
ejpam-5689	80	1	+	+	CCONJ
ejpam-5689	80	2	1	1	NUM
ejpam-5689	80	3	2	2	NUM
ejpam-5689	80	4	f	f	NOUN
ejpam-5689	80	5	(	(	PUNCT
ejpam-5689	80	6	∥xay	∥xay	PROPN
ejpam-5689	80	7	+	+	CCONJ
ejpam-5689	80	8	y	y	PROPN
ejpam-5689	80	9	bx∥	bx∥	PROPN
ejpam-5689	80	10	)	)	PUNCT
ejpam-5689	80	11	,	,	PUNCT
ejpam-5689	80	12	which	which	PRON
ejpam-5689	80	13	completes	complete	VERB
ejpam-5689	80	14	the	the	DET
ejpam-5689	80	15	proof	proof	NOUN
ejpam-5689	80	16	.	.	PUNCT
ejpam-5689	81	1	taking	take	VERB
ejpam-5689	81	2	f(t	f(t	NOUN
ejpam-5689	81	3	)	)	PUNCT
ejpam-5689	81	4	=	=	SYM
ejpam-5689	81	5	t	t	PROPN
ejpam-5689	81	6	,	,	PUNCT
ejpam-5689	81	7	t	t	PROPN
ejpam-5689	81	8	∈	∈	PROPN
ejpam-5689	82	1	[	[	X
ejpam-5689	82	2	0,∞	0,∞	NOUN
ejpam-5689	82	3	)	)	PUNCT
ejpam-5689	82	4	in	in	ADP
ejpam-5689	82	5	inequalities	inequality	NOUN
ejpam-5689	82	6	(	(	PUNCT
ejpam-5689	82	7	4	4	NUM
ejpam-5689	82	8	)	)	PUNCT
ejpam-5689	82	9	and	and	CCONJ
ejpam-5689	82	10	(	(	PUNCT
ejpam-5689	82	11	5	5	NUM
ejpam-5689	82	12	)	)	PUNCT
ejpam-5689	82	13	,	,	PUNCT
ejpam-5689	82	14	we	we	PRON
ejpam-5689	82	15	have	have	VERB
ejpam-5689	82	16	sj	sj	INTJ
ejpam-5689	82	17	(	(	PUNCT
ejpam-5689	82	18	xay	xay	PROPN
ejpam-5689	82	19	+	+	PROPN
ejpam-5689	82	20	y	y	PROPN
ejpam-5689	82	21	bx	bx	PROPN
ejpam-5689	82	22	)	)	PUNCT
ejpam-5689	83	1	≤	≤	NUM
ejpam-5689	83	2	∥x∥	∥x∥	NOUN
ejpam-5689	83	3	∥y	∥y	PROPN
ejpam-5689	83	4	∥	∥	PUNCT
ejpam-5689	83	5	sj	sj	NOUN
ejpam-5689	83	6	(	(	PUNCT
ejpam-5689	83	7	a⊕b	a⊕b	PROPN
ejpam-5689	83	8	)	)	PUNCT
ejpam-5689	84	1	+	+	CCONJ
ejpam-5689	84	2	1	1	NUM
ejpam-5689	84	3	2	2	NUM
ejpam-5689	84	4	∥xay	∥xay	PROPN
ejpam-5689	84	5	+	+	CCONJ
ejpam-5689	84	6	y	y	PROPN
ejpam-5689	84	7	bx∥	bx∥	PROPN
ejpam-5689	84	8	.	.	PUNCT
ejpam-5689	85	1	(	(	PUNCT
ejpam-5689	85	2	8)	8)	NUM
ejpam-5689	85	3	letting	let	VERB
ejpam-5689	85	4	x	x	PUNCT
ejpam-5689	86	1	=	=	PUNCT
ejpam-5689	86	2	i	i	PRON
ejpam-5689	86	3	in	in	ADP
ejpam-5689	86	4	inequality	inequality	NOUN
ejpam-5689	86	5	(	(	PUNCT
ejpam-5689	86	6	8)	8)	NUM
ejpam-5689	86	7	,	,	PUNCT
ejpam-5689	86	8	we	we	PRON
ejpam-5689	86	9	obtain	obtain	VERB
ejpam-5689	86	10	sj	sj	INTJ
ejpam-5689	86	11	(	(	PUNCT
ejpam-5689	86	12	ay	ay	PROPN
ejpam-5689	86	13	+	+	CCONJ
ejpam-5689	86	14	y	y	PROPN
ejpam-5689	86	15	b	b	PROPN
ejpam-5689	86	16	)	)	PUNCT
ejpam-5689	86	17	≤	≤	PUNCT
ejpam-5689	87	1	∥y	∥y	ADV
ejpam-5689	87	2	∥	∥	PUNCT
ejpam-5689	87	3	sj	sj	X
ejpam-5689	87	4	(	(	PUNCT
ejpam-5689	87	5	a⊕b	a⊕b	PROPN
ejpam-5689	87	6	)	)	PUNCT
ejpam-5689	87	7	+	+	CCONJ
ejpam-5689	87	8	1	1	NUM
ejpam-5689	87	9	2	2	NUM
ejpam-5689	87	10	∥ay	∥ay	NOUN
ejpam-5689	88	1	+	+	CCONJ
ejpam-5689	88	2	y	y	PROPN
ejpam-5689	88	3	b∥	b∥	NOUN
ejpam-5689	88	4	.	.	PUNCT
ejpam-5689	89	1	(	(	PUNCT
ejpam-5689	89	2	9	9	X
ejpam-5689	89	3	)	)	PUNCT
ejpam-5689	89	4	replacing	replace	VERB
ejpam-5689	89	5	a	a	DET
ejpam-5689	89	6	by	by	ADP
ejpam-5689	89	7	x	x	X
ejpam-5689	89	8	,	,	PUNCT
ejpam-5689	89	9	y	y	PROPN
ejpam-5689	89	10	by	by	ADP
ejpam-5689	89	11	z	z	PROPN
ejpam-5689	89	12	,	,	PUNCT
ejpam-5689	89	13	and	and	CCONJ
ejpam-5689	89	14	b	b	X
ejpam-5689	89	15	by	by	ADP
ejpam-5689	89	16	y	y	PROPN
ejpam-5689	89	17	in	in	ADP
ejpam-5689	89	18	inequality	inequality	NOUN
ejpam-5689	89	19	(	(	PUNCT
ejpam-5689	89	20	9	9	NUM
ejpam-5689	89	21	)	)	PUNCT
ejpam-5689	89	22	,	,	PUNCT
ejpam-5689	89	23	we	we	PRON
ejpam-5689	89	24	have	have	VERB
ejpam-5689	89	25	sj	sj	INTJ
ejpam-5689	89	26	(	(	PUNCT
ejpam-5689	89	27	xz	xz	PROPN
ejpam-5689	89	28	+	+	CCONJ
ejpam-5689	89	29	zy	zy	PROPN
ejpam-5689	89	30	)	)	PUNCT
ejpam-5689	89	31	≤	≤	NUM
ejpam-5689	89	32	∥z∥	∥z∥	PROPN
ejpam-5689	89	33	sj	sj	INTJ
ejpam-5689	89	34	(	(	PUNCT
ejpam-5689	89	35	x	x	PROPN
ejpam-5689	89	36	⊕	⊕	PROPN
ejpam-5689	89	37	y	y	PROPN
ejpam-5689	89	38	)	)	PUNCT
ejpam-5689	90	1	+	+	CCONJ
ejpam-5689	90	2	1	1	NUM
ejpam-5689	90	3	2	2	NUM
ejpam-5689	90	4	∥xz	∥xz	NOUN
ejpam-5689	90	5	+	+	X
ejpam-5689	90	6	zy	zy	NOUN
ejpam-5689	90	7	∥	∥	NUM
ejpam-5689	90	8	.	.	PUNCT
ejpam-5689	91	1	(	(	PUNCT
ejpam-5689	91	2	10	10	X
ejpam-5689	91	3	)	)	PUNCT
ejpam-5689	91	4	combining	combine	VERB
ejpam-5689	91	5	inequalities	inequality	NOUN
ejpam-5689	91	6	(	(	PUNCT
ejpam-5689	91	7	2	2	NUM
ejpam-5689	91	8	)	)	PUNCT
ejpam-5689	91	9	and	and	CCONJ
ejpam-5689	91	10	(	(	PUNCT
ejpam-5689	91	11	10	10	NUM
ejpam-5689	91	12	)	)	PUNCT
ejpam-5689	91	13	,	,	PUNCT
ejpam-5689	91	14	we	we	PRON
ejpam-5689	91	15	have	have	AUX
ejpam-5689	91	16	sj	sj	INTJ
ejpam-5689	91	17	(	(	PUNCT
ejpam-5689	91	18	xz	xz	PROPN
ejpam-5689	91	19	+	+	CCONJ
ejpam-5689	91	20	zy	zy	PROPN
ejpam-5689	91	21	)	)	PUNCT
ejpam-5689	91	22	≤	≤	NUM
ejpam-5689	91	23	min	min	NOUN
ejpam-5689	91	24	{	{	PUNCT
ejpam-5689	91	25	2	2	NUM
ejpam-5689	91	26	∥z∥	∥z∥	NOUN
ejpam-5689	91	27	sj(x	sj(x	PUNCT
ejpam-5689	91	28	⊕	⊕	PROPN
ejpam-5689	91	29	y	y	PROPN
ejpam-5689	91	30	)	)	PUNCT
ejpam-5689	91	31	,	,	PUNCT
ejpam-5689	91	32	∥z∥	∥z∥	PROPN
ejpam-5689	91	33	sj	sj	X
ejpam-5689	91	34	(	(	PUNCT
ejpam-5689	91	35	x	x	PROPN
ejpam-5689	91	36	⊕	⊕	PROPN
ejpam-5689	91	37	y	y	PROPN
ejpam-5689	91	38	)	)	PUNCT
ejpam-5689	92	1	+	+	CCONJ
ejpam-5689	92	2	1	1	NUM
ejpam-5689	92	3	2	2	NUM
ejpam-5689	92	4	∥xz	∥xz	NOUN
ejpam-5689	92	5	+	+	X
ejpam-5689	92	6	zy	zy	X
ejpam-5689	92	7	∥	∥	PROPN
ejpam-5689	92	8	}	}	PUNCT
ejpam-5689	92	9	,	,	PUNCT
ejpam-5689	92	10	a.	a.	PROPN
ejpam-5689	92	11	al	al	PROPN
ejpam-5689	92	12	-	-	PUNCT
ejpam-5689	92	13	natoor	natoor	NOUN
ejpam-5689	92	14	,	,	PUNCT
ejpam-5689	92	15	f.	f.	PROPN
ejpam-5689	92	16	alrimawi	alrimawi	PROPN
ejpam-5689	92	17	/	/	SYM
ejpam-5689	92	18	eur	eur	PROPN
ejpam-5689	92	19	.	.	PUNCT
ejpam-5689	93	1	j.	j.	PROPN
ejpam-5689	93	2	pure	pure	PROPN
ejpam-5689	93	3	appl	appl	PROPN
ejpam-5689	93	4	.	.	PROPN
ejpam-5689	93	5	math	math	PROPN
ejpam-5689	93	6	,	,	PUNCT
ejpam-5689	93	7	18	18	NUM
ejpam-5689	93	8	(	(	PUNCT
ejpam-5689	93	9	1	1	NUM
ejpam-5689	93	10	)	)	PUNCT
ejpam-5689	93	11	(	(	PUNCT
ejpam-5689	93	12	2025	2025	NUM
ejpam-5689	93	13	)	)	PUNCT
ejpam-5689	93	14	,	,	PUNCT
ejpam-5689	93	15	5689	5689	NUM
ejpam-5689	93	16	5	5	NUM
ejpam-5689	93	17	of	of	ADP
ejpam-5689	93	18	11	11	NUM
ejpam-5689	93	19	which	which	PRON
ejpam-5689	93	20	is	be	AUX
ejpam-5689	93	21	a	a	DET
ejpam-5689	93	22	refinement	refinement	NOUN
ejpam-5689	93	23	of	of	ADP
ejpam-5689	93	24	inequality	inequality	NOUN
ejpam-5689	93	25	(	(	PUNCT
ejpam-5689	93	26	2	2	NUM
ejpam-5689	93	27	)	)	PUNCT
ejpam-5689	93	28	.	.	PUNCT
ejpam-5689	94	1	inequality	inequality	NOUN
ejpam-5689	94	2	(	(	PUNCT
ejpam-5689	94	3	8)	8)	NUM
ejpam-5689	94	4	generalizes	generalize	VERB
ejpam-5689	94	5	inequality	inequality	NOUN
ejpam-5689	94	6	(	(	PUNCT
ejpam-5689	94	7	3	3	NUM
ejpam-5689	94	8	)	)	PUNCT
ejpam-5689	94	9	.	.	PUNCT
ejpam-5689	95	1	in	in	ADP
ejpam-5689	95	2	fact	fact	NOUN
ejpam-5689	95	3	,	,	PUNCT
ejpam-5689	95	4	replacing	replace	VERB
ejpam-5689	95	5	b	b	NOUN
ejpam-5689	95	6	by	by	ADP
ejpam-5689	95	7	−b	−b	NOUN
ejpam-5689	95	8	in	in	ADP
ejpam-5689	95	9	inequality	inequality	NOUN
ejpam-5689	95	10	(	(	PUNCT
ejpam-5689	95	11	8)	8)	NUM
ejpam-5689	95	12	,	,	PUNCT
ejpam-5689	95	13	we	we	PRON
ejpam-5689	95	14	have	have	VERB
ejpam-5689	95	15	sj	sj	INTJ
ejpam-5689	95	16	(	(	PUNCT
ejpam-5689	95	17	xay	xay	PROPN
ejpam-5689	95	18	−	−	PROPN
ejpam-5689	95	19	y	y	PROPN
ejpam-5689	95	20	bx	bx	PROPN
ejpam-5689	95	21	)	)	PUNCT
ejpam-5689	96	1	≤	≤	NUM
ejpam-5689	96	2	∥x∥	∥x∥	NOUN
ejpam-5689	96	3	∥y	∥y	PROPN
ejpam-5689	96	4	∥	∥	PUNCT
ejpam-5689	96	5	sj	sj	NOUN
ejpam-5689	96	6	(	(	PUNCT
ejpam-5689	96	7	a⊕b	a⊕b	PROPN
ejpam-5689	96	8	)	)	PUNCT
ejpam-5689	97	1	+	+	CCONJ
ejpam-5689	97	2	1	1	NUM
ejpam-5689	97	3	2	2	NUM
ejpam-5689	97	4	∥xay	∥xay	PROPN
ejpam-5689	97	5	−	−	PROPN
ejpam-5689	97	6	y	y	PROPN
ejpam-5689	97	7	bx∥	bx∥	PROPN
ejpam-5689	97	8	,	,	PUNCT
ejpam-5689	97	9	(	(	PUNCT
ejpam-5689	97	10	11	11	NUM
ejpam-5689	97	11	)	)	PUNCT
ejpam-5689	97	12	which	which	PRON
ejpam-5689	97	13	is	be	AUX
ejpam-5689	97	14	a	a	DET
ejpam-5689	97	15	generalization	generalization	NOUN
ejpam-5689	97	16	of	of	ADP
ejpam-5689	97	17	inequality	inequality	NOUN
ejpam-5689	97	18	(	(	PUNCT
ejpam-5689	97	19	3	3	NUM
ejpam-5689	97	20	)	)	PUNCT
ejpam-5689	97	21	.	.	PUNCT
ejpam-5689	98	1	to	to	PART
ejpam-5689	98	2	see	see	VERB
ejpam-5689	98	3	this	this	PRON
ejpam-5689	98	4	,	,	PUNCT
ejpam-5689	98	5	let	let	VERB
ejpam-5689	98	6	x	x	PUNCT
ejpam-5689	98	7	=	=	PUNCT
ejpam-5689	99	1	i	i	PRON
ejpam-5689	99	2	and	and	CCONJ
ejpam-5689	99	3	then	then	ADV
ejpam-5689	99	4	replace	replace	VERB
ejpam-5689	99	5	a	a	PRON
ejpam-5689	99	6	and	and	CCONJ
ejpam-5689	99	7	b	b	NOUN
ejpam-5689	99	8	by	by	ADP
ejpam-5689	99	9	x	x	PUNCT
ejpam-5689	99	10	in	in	ADP
ejpam-5689	99	11	inequality	inequality	NOUN
ejpam-5689	99	12	(	(	PUNCT
ejpam-5689	99	13	11	11	NUM
ejpam-5689	99	14	)	)	PUNCT
ejpam-5689	100	1	,	,	PUNCT
ejpam-5689	100	2	we	we	PRON
ejpam-5689	100	3	have	have	VERB
ejpam-5689	100	4	sj	sj	INTJ
ejpam-5689	100	5	(	(	PUNCT
ejpam-5689	100	6	xy	xy	NOUN
ejpam-5689	100	7	−	−	PROPN
ejpam-5689	100	8	y	y	PROPN
ejpam-5689	100	9	x	x	PROPN
ejpam-5689	100	10	)	)	PUNCT
ejpam-5689	100	11	≤	≤	PUNCT
ejpam-5689	101	1	∥y	∥y	ADV
ejpam-5689	101	2	∥	∥	PUNCT
ejpam-5689	101	3	sj	sj	INTJ
ejpam-5689	101	4	(	(	PUNCT
ejpam-5689	101	5	x	x	NOUN
ejpam-5689	101	6	⊕x	⊕x	NOUN
ejpam-5689	101	7	)	)	PUNCT
ejpam-5689	102	1	+	+	CCONJ
ejpam-5689	102	2	1	1	NUM
ejpam-5689	102	3	2	2	NUM
ejpam-5689	102	4	∥xy	∥xy	NOUN
ejpam-5689	102	5	−	−	NOUN
ejpam-5689	103	1	y	y	PROPN
ejpam-5689	103	2	x∥	x∥	PROPN
ejpam-5689	103	3	,	,	PUNCT
ejpam-5689	103	4	which	which	PRON
ejpam-5689	103	5	is	be	AUX
ejpam-5689	103	6	inequality	inequality	NOUN
ejpam-5689	103	7	(	(	PUNCT
ejpam-5689	103	8	3	3	NUM
ejpam-5689	103	9	)	)	PUNCT
ejpam-5689	103	10	.	.	PUNCT
ejpam-5689	104	1	we	we	PRON
ejpam-5689	104	2	need	need	VERB
ejpam-5689	104	3	the	the	DET
ejpam-5689	104	4	following	follow	VERB
ejpam-5689	104	5	lemma	lemma	PROPN
ejpam-5689	104	6	[	[	X
ejpam-5689	104	7	6	6	NUM
ejpam-5689	104	8	]	]	PUNCT
ejpam-5689	104	9	to	to	PART
ejpam-5689	104	10	give	give	VERB
ejpam-5689	104	11	our	our	PRON
ejpam-5689	104	12	second	second	ADJ
ejpam-5689	104	13	result	result	NOUN
ejpam-5689	104	14	.	.	PUNCT
ejpam-5689	105	1	lemma	lemma	PROPN
ejpam-5689	105	2	4	4	X
ejpam-5689	105	3	.	.	PUNCT
ejpam-5689	106	1	let	let	VERB
ejpam-5689	106	2	x	x	PRON
ejpam-5689	106	3	,	,	PUNCT
ejpam-5689	106	4	y	y	PROPN
ejpam-5689	106	5	,	,	PUNCT
ejpam-5689	106	6	z	z	NOUN
ejpam-5689	106	7	∈	∈	PROPN
ejpam-5689	106	8	mn(c	mn(c	X
ejpam-5689	106	9	)	)	PUNCT
ejpam-5689	106	10	be	be	AUX
ejpam-5689	106	11	such	such	ADJ
ejpam-5689	106	12	that	that	SCONJ
ejpam-5689	106	13	z	z	NOUN
ejpam-5689	106	14	is	be	AUX
ejpam-5689	106	15	positive	positive	ADJ
ejpam-5689	106	16	semidefinite	semidefinite	NOUN
ejpam-5689	106	17	.	.	PUNCT
ejpam-5689	107	1	then	then	ADV
ejpam-5689	107	2	sj	sj	INTJ
ejpam-5689	107	3	(	(	PUNCT
ejpam-5689	107	4	xzy	xzy	PROPN
ejpam-5689	107	5	∗	∗	PROPN
ejpam-5689	107	6	)	)	PUNCT
ejpam-5689	107	7	≤	≤	NUM
ejpam-5689	107	8	1	1	NUM
ejpam-5689	107	9	2	2	NUM
ejpam-5689	107	10	∥∥∥∥x∗x	∥∥∥∥x∗x	ADJ
ejpam-5689	107	11	∥x∥2	∥x∥2	NOUN
ejpam-5689	107	12	+	+	CCONJ
ejpam-5689	107	13	y	y	PROPN
ejpam-5689	107	14	∗y	∗y	PROPN
ejpam-5689	107	15	∥y	∥y	PROPN
ejpam-5689	107	16	∥2	∥2	ADJ
ejpam-5689	107	17	∥∥∥∥	∥∥∥∥	NUM
ejpam-5689	107	18	∥x∥	∥x∥	NOUN
ejpam-5689	107	19	∥y	∥y	PROPN
ejpam-5689	107	20	∥	∥	PUNCT
ejpam-5689	107	21	sj	sj	X
ejpam-5689	107	22	(	(	PUNCT
ejpam-5689	107	23	z	z	NOUN
ejpam-5689	107	24	)	)	PUNCT
ejpam-5689	107	25	.	.	PUNCT
ejpam-5689	108	1	theorem	theorem	NOUN
ejpam-5689	108	2	2	2	NUM
ejpam-5689	108	3	.	.	PUNCT
ejpam-5689	108	4	let	let	VERB
ejpam-5689	108	5	a	a	DET
ejpam-5689	108	6	,	,	PUNCT
ejpam-5689	108	7	b	b	NOUN
ejpam-5689	108	8	,	,	PUNCT
ejpam-5689	108	9	x	x	X
ejpam-5689	108	10	,	,	PUNCT
ejpam-5689	108	11	y	y	PROPN
ejpam-5689	108	12	∈	∈	PROPN
ejpam-5689	108	13	mn(c	mn(c	X
ejpam-5689	108	14	)	)	PUNCT
ejpam-5689	108	15	be	be	AUX
ejpam-5689	108	16	such	such	ADJ
ejpam-5689	108	17	that	that	SCONJ
ejpam-5689	108	18	a	a	PRON
ejpam-5689	108	19	and	and	CCONJ
ejpam-5689	108	20	b	b	NOUN
ejpam-5689	108	21	are	be	AUX
ejpam-5689	108	22	positive	positive	ADJ
ejpam-5689	108	23	semidefinite	semidefinite	NOUN
ejpam-5689	108	24	.	.	PUNCT
ejpam-5689	109	1	then	then	ADV
ejpam-5689	109	2	(	(	PUNCT
ejpam-5689	109	3	a	a	X
ejpam-5689	109	4	)	)	PUNCT
ejpam-5689	109	5	sj	sj	PROPN
ejpam-5689	109	6	(	(	PUNCT
ejpam-5689	109	7	f	f	X
ejpam-5689	109	8	(	(	PUNCT
ejpam-5689	109	9	|xay	|xay	ADV
ejpam-5689	109	10	+	+	CCONJ
ejpam-5689	109	11	y	y	PROPN
ejpam-5689	109	12	bx|	bx|	NOUN
ejpam-5689	109	13	)	)	PUNCT
ejpam-5689	109	14	)	)	PUNCT
ejpam-5689	110	1	≤	≤	NUM
ejpam-5689	110	2	f	f	X
ejpam-5689	110	3	(	(	PUNCT
ejpam-5689	110	4	1	1	NUM
ejpam-5689	110	5	2	2	NUM
ejpam-5689	110	6	)	)	PUNCT
ejpam-5689	110	7	∥∥∥∥∥f	∥∥∥∥∥f	PROPN
ejpam-5689	111	1	(	(	PUNCT
ejpam-5689	111	2	|x|2	|x|2	PROPN
ejpam-5689	111	3	⊕	⊕	PROPN
ejpam-5689	111	4	|x∗|2	|x∗|2	NUM
ejpam-5689	111	5	∥x∥2	∥x∥2	NOUN
ejpam-5689	111	6	+	+	CCONJ
ejpam-5689	111	7	|y	|y	NOUN
ejpam-5689	111	8	|2	|2	NUM
ejpam-5689	111	9	⊕	⊕	NOUN
ejpam-5689	111	10	|y	|y	NOUN
ejpam-5689	111	11	∗|2	∗|2	PUNCT
ejpam-5689	111	12	∥y	∥y	PROPN
ejpam-5689	111	13	∥2	∥2	NUM
ejpam-5689	111	14	)	)	PUNCT
ejpam-5689	111	15	∥∥∥∥∥	∥∥∥∥∥	PROPN
ejpam-5689	112	1	f	f	X
ejpam-5689	112	2	(	(	PUNCT
ejpam-5689	112	3	∥x∥	∥x∥	NOUN
ejpam-5689	112	4	)	)	PUNCT
ejpam-5689	112	5	f	f	PROPN
ejpam-5689	112	6	(	(	PUNCT
ejpam-5689	112	7	∥y	∥y	NOUN
ejpam-5689	112	8	∥	∥	NUM
ejpam-5689	112	9	)	)	PUNCT
ejpam-5689	112	10	sj	sj	INTJ
ejpam-5689	112	11	(	(	PUNCT
ejpam-5689	112	12	f	f	PROPN
ejpam-5689	112	13	(	(	PUNCT
ejpam-5689	112	14	a)⊕	a)⊕	ADP
ejpam-5689	112	15	f	f	X
ejpam-5689	112	16	(	(	PUNCT
ejpam-5689	112	17	b	b	NOUN
ejpam-5689	112	18	)	)	PUNCT
ejpam-5689	112	19	)	)	PUNCT
ejpam-5689	113	1	+	+	NOUN
ejpam-5689	113	2	f	f	X
ejpam-5689	113	3	(	(	PUNCT
ejpam-5689	113	4	1	1	NUM
ejpam-5689	113	5	2	2	NUM
ejpam-5689	113	6	)	)	PUNCT
ejpam-5689	113	7	f	f	NOUN
ejpam-5689	113	8	(	(	PUNCT
ejpam-5689	113	9	∥xay	∥xay	PROPN
ejpam-5689	113	10	+	+	CCONJ
ejpam-5689	113	11	y	y	PROPN
ejpam-5689	113	12	bx∥	bx∥	PROPN
ejpam-5689	113	13	)	)	PUNCT
ejpam-5689	113	14	,	,	PUNCT
ejpam-5689	113	15	(	(	PUNCT
ejpam-5689	113	16	12	12	NUM
ejpam-5689	113	17	)	)	PUNCT
ejpam-5689	113	18	where	where	SCONJ
ejpam-5689	113	19	f	f	PROPN
ejpam-5689	113	20	is	be	AUX
ejpam-5689	113	21	a	a	DET
ejpam-5689	113	22	nonnegative	nonnegative	ADJ
ejpam-5689	113	23	increasing	increase	VERB
ejpam-5689	113	24	submultiplicative	submultiplicative	ADJ
ejpam-5689	113	25	concave	concave	NOUN
ejpam-5689	113	26	function	function	NOUN
ejpam-5689	113	27	on	on	ADP
ejpam-5689	113	28	[	[	X
ejpam-5689	113	29	0,∞	0,∞	NOUN
ejpam-5689	113	30	)	)	PUNCT
ejpam-5689	113	31	with	with	ADP
ejpam-5689	113	32	f(0	f(0	NOUN
ejpam-5689	113	33	)	)	PUNCT
ejpam-5689	113	34	=	=	SYM
ejpam-5689	114	1	0	0	X
ejpam-5689	114	2	.	.	PUNCT
ejpam-5689	115	1	(	(	PUNCT
ejpam-5689	115	2	b	b	X
ejpam-5689	115	3	)	)	PUNCT
ejpam-5689	115	4	sj	sj	PROPN
ejpam-5689	115	5	(	(	PUNCT
ejpam-5689	115	6	f	f	X
ejpam-5689	115	7	(	(	PUNCT
ejpam-5689	115	8	|xay	|xay	ADV
ejpam-5689	115	9	+	+	CCONJ
ejpam-5689	115	10	y	y	PROPN
ejpam-5689	115	11	bx|	bx|	NOUN
ejpam-5689	115	12	)	)	PUNCT
ejpam-5689	115	13	)	)	PUNCT
ejpam-5689	116	1	≤	≤	NUM
ejpam-5689	116	2	1	1	NUM
ejpam-5689	116	3	2	2	NUM
ejpam-5689	116	4	∥∥∥∥∥f	∥∥∥∥∥f	PROPN
ejpam-5689	116	5	(	(	PUNCT
ejpam-5689	116	6	|x|2	|x|2	PROPN
ejpam-5689	116	7	⊕	⊕	PROPN
ejpam-5689	116	8	|x∗|2	|x∗|2	NUM
ejpam-5689	116	9	∥x∥2	∥x∥2	NOUN
ejpam-5689	116	10	+	+	CCONJ
ejpam-5689	116	11	|y	|y	NOUN
ejpam-5689	116	12	|2	|2	NUM
ejpam-5689	116	13	⊕	⊕	NOUN
ejpam-5689	116	14	|y	|y	NOUN
ejpam-5689	116	15	∗|2	∗|2	PUNCT
ejpam-5689	116	16	∥y	∥y	PROPN
ejpam-5689	116	17	∥2	∥2	NUM
ejpam-5689	116	18	)	)	PUNCT
ejpam-5689	116	19	∥∥∥∥∥	∥∥∥∥∥	PROPN
ejpam-5689	117	1	f	f	X
ejpam-5689	117	2	(	(	PUNCT
ejpam-5689	117	3	∥x∥	∥x∥	NOUN
ejpam-5689	117	4	)	)	PUNCT
ejpam-5689	117	5	f	f	PROPN
ejpam-5689	117	6	(	(	PUNCT
ejpam-5689	117	7	∥y	∥y	NOUN
ejpam-5689	117	8	∥	∥	NUM
ejpam-5689	117	9	)	)	PUNCT
ejpam-5689	117	10	sj	sj	INTJ
ejpam-5689	117	11	(	(	PUNCT
ejpam-5689	117	12	f	f	PROPN
ejpam-5689	117	13	(	(	PUNCT
ejpam-5689	117	14	a)⊕	a)⊕	ADP
ejpam-5689	117	15	f	f	X
ejpam-5689	117	16	(	(	PUNCT
ejpam-5689	117	17	b	b	NOUN
ejpam-5689	117	18	)	)	PUNCT
ejpam-5689	117	19	)	)	PUNCT
ejpam-5689	118	1	+	+	CCONJ
ejpam-5689	118	2	1	1	NUM
ejpam-5689	118	3	2	2	NUM
ejpam-5689	118	4	f	f	NOUN
ejpam-5689	118	5	(	(	PUNCT
ejpam-5689	118	6	∥xay	∥xay	PROPN
ejpam-5689	118	7	+	+	CCONJ
ejpam-5689	118	8	y	y	PROPN
ejpam-5689	118	9	bx∥	bx∥	PROPN
ejpam-5689	118	10	)	)	PUNCT
ejpam-5689	118	11	,	,	PUNCT
ejpam-5689	118	12	(	(	PUNCT
ejpam-5689	118	13	13	13	NUM
ejpam-5689	118	14	)	)	PUNCT
ejpam-5689	118	15	where	where	SCONJ
ejpam-5689	118	16	f	f	PROPN
ejpam-5689	118	17	is	be	AUX
ejpam-5689	118	18	a	a	DET
ejpam-5689	118	19	nonnegative	nonnegative	ADJ
ejpam-5689	118	20	increasing	increase	VERB
ejpam-5689	118	21	submultiplicative	submultiplicative	ADJ
ejpam-5689	118	22	convex	convex	NOUN
ejpam-5689	118	23	function	function	NOUN
ejpam-5689	118	24	on	on	ADP
ejpam-5689	118	25	[	[	X
ejpam-5689	118	26	0,∞	0,∞	NOUN
ejpam-5689	118	27	)	)	PUNCT
ejpam-5689	118	28	.	.	PUNCT
ejpam-5689	119	1	a.	a.	PROPN
ejpam-5689	119	2	al	al	PROPN
ejpam-5689	119	3	-	-	PUNCT
ejpam-5689	119	4	natoor	natoor	NOUN
ejpam-5689	119	5	,	,	PUNCT
ejpam-5689	119	6	f.	f.	PROPN
ejpam-5689	119	7	alrimawi	alrimawi	PROPN
ejpam-5689	119	8	/	/	SYM
ejpam-5689	119	9	eur	eur	PROPN
ejpam-5689	119	10	.	.	PUNCT
ejpam-5689	120	1	j.	j.	PROPN
ejpam-5689	120	2	pure	pure	PROPN
ejpam-5689	120	3	appl	appl	PROPN
ejpam-5689	120	4	.	.	PROPN
ejpam-5689	120	5	math	math	PROPN
ejpam-5689	120	6	,	,	PUNCT
ejpam-5689	120	7	18	18	NUM
ejpam-5689	120	8	(	(	PUNCT
ejpam-5689	120	9	1	1	NUM
ejpam-5689	120	10	)	)	PUNCT
ejpam-5689	120	11	(	(	PUNCT
ejpam-5689	120	12	2025	2025	NUM
ejpam-5689	120	13	)	)	PUNCT
ejpam-5689	120	14	,	,	PUNCT
ejpam-5689	120	15	5689	5689	NUM
ejpam-5689	120	16	6	6	NUM
ejpam-5689	120	17	of	of	ADP
ejpam-5689	120	18	11	11	NUM
ejpam-5689	120	19	proof	proof	NOUN
ejpam-5689	120	20	.	.	PUNCT
ejpam-5689	121	1	by	by	ADP
ejpam-5689	121	2	inequality	inequality	NOUN
ejpam-5689	121	3	(	(	PUNCT
ejpam-5689	121	4	7	7	NUM
ejpam-5689	121	5	)	)	PUNCT
ejpam-5689	121	6	,	,	PUNCT
ejpam-5689	121	7	we	we	PRON
ejpam-5689	121	8	have	have	VERB
ejpam-5689	121	9	sj	sj	INTJ
ejpam-5689	121	10	(	(	PUNCT
ejpam-5689	121	11	f	f	X
ejpam-5689	121	12	(	(	PUNCT
ejpam-5689	121	13	|xay	|xay	ADV
ejpam-5689	121	14	+	+	CCONJ
ejpam-5689	121	15	y	y	PROPN
ejpam-5689	121	16	bx|	bx|	NOUN
ejpam-5689	121	17	)	)	PUNCT
ejpam-5689	121	18	)	)	PUNCT
ejpam-5689	122	1	≤	≤	NUM
ejpam-5689	122	2	f	f	X
ejpam-5689	122	3	(	(	PUNCT
ejpam-5689	122	4	sj	sj	INTJ
ejpam-5689	122	5	(	(	PUNCT
ejpam-5689	122	6	[	[	PUNCT
ejpam-5689	122	7	x	x	SYM
ejpam-5689	122	8	0	0	NUM
ejpam-5689	122	9	0	0	NUM
ejpam-5689	122	10	x∗	x∗	X
ejpam-5689	122	11	]	]	PUNCT
ejpam-5689	123	1	[	[	PUNCT
ejpam-5689	123	2	a	a	DET
ejpam-5689	123	3	0	0	NUM
ejpam-5689	123	4	0	0	NUM
ejpam-5689	123	5	b∗	b∗	ADJ
ejpam-5689	123	6	]	]	PUNCT
ejpam-5689	123	7	[	[	PUNCT
ejpam-5689	123	8	y	y	NOUN
ejpam-5689	123	9	0	0	NUM
ejpam-5689	123	10	0	0	NUM
ejpam-5689	123	11	y	y	PROPN
ejpam-5689	123	12	∗	∗	NOUN
ejpam-5689	123	13	]	]	PUNCT
ejpam-5689	123	14	)	)	PUNCT
ejpam-5689	123	15	)	)	PUNCT
ejpam-5689	124	1	+	+	NOUN
ejpam-5689	124	2	f	f	X
ejpam-5689	124	3	(	(	PUNCT
ejpam-5689	124	4	1	1	NUM
ejpam-5689	124	5	2	2	NUM
ejpam-5689	124	6	)	)	PUNCT
ejpam-5689	124	7	f	f	NOUN
ejpam-5689	124	8	(	(	PUNCT
ejpam-5689	124	9	∥xay	∥xay	PROPN
ejpam-5689	124	10	+	+	CCONJ
ejpam-5689	124	11	y	y	PROPN
ejpam-5689	124	12	bx∥	bx∥	PROPN
ejpam-5689	124	13	)	)	PUNCT
ejpam-5689	124	14	≤	≤	NOUN
ejpam-5689	125	1	f	f	PROPN
ejpam-5689	125	2			ADJ
ejpam-5689	125	3	1	1	NUM
ejpam-5689	125	4	2	2	NUM
ejpam-5689	125	5	∥∥∥∥∥∥∥∥∥	∥∥∥∥∥∥∥∥∥	NUM
ejpam-5689	125	6	x∗	x∗	NOUN
ejpam-5689	125	7	0	0	NUM
ejpam-5689	125	8	0	0	NUM
ejpam-5689	125	9	x	x	SYM
ejpam-5689	125	10	x	x	PROPN
ejpam-5689	125	11	0	0	NUM
ejpam-5689	125	12	0	0	NUM
ejpam-5689	125	13	x∗	x∗	NOUN
ejpam-5689	125	14			VERB
ejpam-5689	125	15	∥x∥2	∥x∥2	NOUN
ejpam-5689	125	16	+	+	CCONJ
ejpam-5689	125	17	y	y	NOUN
ejpam-5689	125	18	∗	∗	NOUN
ejpam-5689	125	19	0	0	NUM
ejpam-5689	125	20	0	0	NUM
ejpam-5689	125	21	y	y	PROPN
ejpam-5689	125	22	y	y	PROPN
ejpam-5689	125	23	0	0	NUM
ejpam-5689	125	24	0	0	PUNCT
ejpam-5689	125	25	y	y	NOUN
ejpam-5689	125	26	∗	∗	NOUN
ejpam-5689	125	27			NOUN
ejpam-5689	125	28	∥y	∥y	PROPN
ejpam-5689	125	29	∥2	∥2	ADJ
ejpam-5689	125	30	∥∥∥∥∥∥∥∥∥	∥∥∥∥∥∥∥∥∥	NOUN
ejpam-5689	126	1	×∥x∥	×∥x∥	NOUN
ejpam-5689	127	1	∥y	∥y	PROPN
ejpam-5689	127	2	∥	∥	PUNCT
ejpam-5689	127	3	sj	sj	X
ejpam-5689	127	4	(	(	PUNCT
ejpam-5689	127	5	a⊕b	a⊕b	PROPN
ejpam-5689	127	6	)	)	PUNCT
ejpam-5689	127	7			PUNCT
ejpam-5689	128	1	+	+	PUNCT
ejpam-5689	128	2	f	f	X
ejpam-5689	128	3	(	(	PUNCT
ejpam-5689	128	4	1	1	NUM
ejpam-5689	128	5	2	2	NUM
ejpam-5689	128	6	)	)	PUNCT
ejpam-5689	128	7	f	f	NOUN
ejpam-5689	128	8	(	(	PUNCT
ejpam-5689	128	9	∥xay	∥xay	PROPN
ejpam-5689	128	10	+	+	CCONJ
ejpam-5689	128	11	y	y	PROPN
ejpam-5689	128	12	bx∥	bx∥	PROPN
ejpam-5689	128	13	)	)	PUNCT
ejpam-5689	128	14	(	(	PUNCT
ejpam-5689	128	15	by	by	ADP
ejpam-5689	128	16	lemma	lemma	PROPN
ejpam-5689	128	17	4	4	NUM
ejpam-5689	128	18	)	)	PUNCT
ejpam-5689	128	19	=	=	SYM
ejpam-5689	129	1	f	f	PROPN
ejpam-5689	129	2	(	(	PUNCT
ejpam-5689	129	3	1	1	NUM
ejpam-5689	129	4	2	2	NUM
ejpam-5689	129	5	∥∥∥∥∥	∥∥∥∥∥	NOUN
ejpam-5689	129	6	|x|2	|x|2	PROPN
ejpam-5689	129	7	⊕	⊕	PROPN
ejpam-5689	129	8	|x∗|2	|x∗|2	NUM
ejpam-5689	129	9	∥x∥2	∥x∥2	NOUN
ejpam-5689	129	10	+	+	CCONJ
ejpam-5689	129	11	|y	|y	NOUN
ejpam-5689	129	12	|2	|2	NUM
ejpam-5689	129	13	⊕	⊕	NOUN
ejpam-5689	129	14	|y	|y	NOUN
ejpam-5689	129	15	∗|2	∗|2	PUNCT
ejpam-5689	129	16	∥y	∥y	PROPN
ejpam-5689	129	17	∥2	∥2	ADV
ejpam-5689	129	18	∥∥∥∥∥	∥∥∥∥∥	PUNCT
ejpam-5689	129	19	∥x∥	∥x∥	NOUN
ejpam-5689	129	20	∥y	∥y	PROPN
ejpam-5689	129	21	∥	∥	PUNCT
ejpam-5689	129	22	sj	sj	NOUN
ejpam-5689	129	23	(	(	PUNCT
ejpam-5689	129	24	a⊕b	a⊕b	PROPN
ejpam-5689	129	25	)	)	PUNCT
ejpam-5689	129	26	)	)	PUNCT
ejpam-5689	130	1	+	+	PUNCT
ejpam-5689	130	2	f	f	X
ejpam-5689	130	3	(	(	PUNCT
ejpam-5689	130	4	1	1	NUM
ejpam-5689	130	5	2	2	NUM
ejpam-5689	130	6	)	)	PUNCT
ejpam-5689	130	7	f	f	NOUN
ejpam-5689	130	8	(	(	PUNCT
ejpam-5689	130	9	∥xay	∥xay	PROPN
ejpam-5689	130	10	+	+	CCONJ
ejpam-5689	130	11	y	y	PROPN
ejpam-5689	130	12	bx∥	bx∥	PROPN
ejpam-5689	130	13	)	)	PUNCT
ejpam-5689	130	14	≤	≤	NUM
ejpam-5689	131	1	f	f	X
ejpam-5689	131	2	(	(	PUNCT
ejpam-5689	131	3	1	1	NUM
ejpam-5689	131	4	2	2	NUM
ejpam-5689	131	5	)	)	PUNCT
ejpam-5689	131	6	f	f	NOUN
ejpam-5689	131	7	(	(	PUNCT
ejpam-5689	131	8	∥∥∥∥∥	∥∥∥∥∥	X
ejpam-5689	131	9	|x|2	|x|2	PROPN
ejpam-5689	131	10	⊕	⊕	PROPN
ejpam-5689	131	11	|x∗|2	|x∗|2	NUM
ejpam-5689	131	12	∥x∥2	∥x∥2	NOUN
ejpam-5689	131	13	+	+	CCONJ
ejpam-5689	131	14	|y	|y	NOUN
ejpam-5689	131	15	|2	|2	NUM
ejpam-5689	131	16	⊕	⊕	NOUN
ejpam-5689	131	17	|y	|y	NOUN
ejpam-5689	131	18	∗|2	∗|2	PUNCT
ejpam-5689	131	19	∥y	∥y	PROPN
ejpam-5689	131	20	∥2	∥2	NOUN
ejpam-5689	131	21	∥∥∥∥∥	∥∥∥∥∥	NUM
ejpam-5689	131	22	)	)	PUNCT
ejpam-5689	132	1	f	f	PROPN
ejpam-5689	132	2	(	(	PUNCT
ejpam-5689	132	3	∥x∥	∥x∥	NOUN
ejpam-5689	132	4	)	)	PUNCT
ejpam-5689	132	5	f	f	PROPN
ejpam-5689	132	6	(	(	PUNCT
ejpam-5689	132	7	∥y	∥y	NOUN
ejpam-5689	132	8	∥	∥	NUM
ejpam-5689	132	9	)	)	PUNCT
ejpam-5689	132	10	f	f	PROPN
ejpam-5689	132	11	(	(	PUNCT
ejpam-5689	132	12	sj	sj	INTJ
ejpam-5689	132	13	(	(	PUNCT
ejpam-5689	132	14	a⊕b	a⊕b	PROPN
ejpam-5689	132	15	)	)	PUNCT
ejpam-5689	132	16	)	)	PUNCT
ejpam-5689	133	1	+	+	NOUN
ejpam-5689	133	2	f	f	X
ejpam-5689	133	3	(	(	PUNCT
ejpam-5689	133	4	1	1	NUM
ejpam-5689	133	5	2	2	NUM
ejpam-5689	133	6	)	)	PUNCT
ejpam-5689	133	7	f	f	NOUN
ejpam-5689	133	8	(	(	PUNCT
ejpam-5689	133	9	∥xay	∥xay	PROPN
ejpam-5689	133	10	+	+	CCONJ
ejpam-5689	133	11	y	y	PROPN
ejpam-5689	133	12	bx∥	bx∥	PROPN
ejpam-5689	133	13	)	)	PUNCT
ejpam-5689	133	14	(	(	PUNCT
ejpam-5689	133	15	since	since	SCONJ
ejpam-5689	133	16	f	f	PROPN
ejpam-5689	133	17	is	be	AUX
ejpam-5689	133	18	submultiplicative	submultiplicative	ADJ
ejpam-5689	133	19	)	)	PUNCT
ejpam-5689	134	1	=	=	SYM
ejpam-5689	134	2	f	f	PROPN
ejpam-5689	134	3	(	(	PUNCT
ejpam-5689	134	4	1	1	NUM
ejpam-5689	134	5	2	2	NUM
ejpam-5689	134	6	)	)	PUNCT
ejpam-5689	134	7	∥∥∥∥∥f	∥∥∥∥∥f	PROPN
ejpam-5689	135	1	(	(	PUNCT
ejpam-5689	135	2	|x|2	|x|2	PROPN
ejpam-5689	135	3	⊕	⊕	PROPN
ejpam-5689	135	4	|x∗|2	|x∗|2	NUM
ejpam-5689	135	5	∥x∥2	∥x∥2	NOUN
ejpam-5689	135	6	+	+	CCONJ
ejpam-5689	135	7	|y	|y	NOUN
ejpam-5689	135	8	|2	|2	NUM
ejpam-5689	135	9	⊕	⊕	NOUN
ejpam-5689	135	10	|y	|y	NOUN
ejpam-5689	135	11	∗|2	∗|2	PUNCT
ejpam-5689	135	12	∥y	∥y	PROPN
ejpam-5689	135	13	∥2	∥2	NUM
ejpam-5689	135	14	)	)	PUNCT
ejpam-5689	135	15	∥∥∥∥∥	∥∥∥∥∥	PROPN
ejpam-5689	136	1	f	f	X
ejpam-5689	136	2	(	(	PUNCT
ejpam-5689	136	3	∥x∥	∥x∥	NOUN
ejpam-5689	136	4	)	)	PUNCT
ejpam-5689	136	5	f	f	PROPN
ejpam-5689	136	6	(	(	PUNCT
ejpam-5689	136	7	∥y	∥y	NOUN
ejpam-5689	136	8	∥	∥	NUM
ejpam-5689	136	9	)	)	PUNCT
ejpam-5689	136	10	sj	sj	INTJ
ejpam-5689	136	11	(	(	PUNCT
ejpam-5689	136	12	f	f	PROPN
ejpam-5689	136	13	(	(	PUNCT
ejpam-5689	136	14	a)⊕	a)⊕	ADP
ejpam-5689	136	15	f	f	X
ejpam-5689	136	16	(	(	PUNCT
ejpam-5689	136	17	b	b	NOUN
ejpam-5689	136	18	)	)	PUNCT
ejpam-5689	136	19	)	)	PUNCT
ejpam-5689	137	1	+	+	NOUN
ejpam-5689	137	2	f	f	X
ejpam-5689	137	3	(	(	PUNCT
ejpam-5689	137	4	1	1	NUM
ejpam-5689	137	5	2	2	NUM
ejpam-5689	137	6	)	)	PUNCT
ejpam-5689	137	7	f	f	NOUN
ejpam-5689	137	8	(	(	PUNCT
ejpam-5689	137	9	∥xay	∥xay	PROPN
ejpam-5689	137	10	+	+	CCONJ
ejpam-5689	137	11	y	y	PROPN
ejpam-5689	137	12	bx∥	bx∥	PROPN
ejpam-5689	137	13	)	)	PUNCT
ejpam-5689	137	14	,	,	PUNCT
ejpam-5689	137	15	which	which	PRON
ejpam-5689	137	16	proves	prove	VERB
ejpam-5689	137	17	part	part	NOUN
ejpam-5689	137	18	(	(	PUNCT
ejpam-5689	137	19	a	a	NOUN
ejpam-5689	137	20	)	)	PUNCT
ejpam-5689	137	21	.	.	PUNCT
ejpam-5689	138	1	for	for	ADP
ejpam-5689	138	2	part	part	NOUN
ejpam-5689	138	3	(	(	PUNCT
ejpam-5689	138	4	b	b	NOUN
ejpam-5689	138	5	)	)	PUNCT
ejpam-5689	138	6	,	,	PUNCT
ejpam-5689	138	7	we	we	PRON
ejpam-5689	138	8	start	start	VERB
ejpam-5689	138	9	from	from	ADP
ejpam-5689	138	10	inequality	inequality	NOUN
ejpam-5689	138	11	(	(	PUNCT
ejpam-5689	138	12	6	6	NUM
ejpam-5689	138	13	)	)	PUNCT
ejpam-5689	138	14	,	,	PUNCT
ejpam-5689	138	15	so	so	SCONJ
ejpam-5689	138	16	we	we	PRON
ejpam-5689	138	17	have	have	VERB
ejpam-5689	138	18	sj	sj	INTJ
ejpam-5689	138	19	(	(	PUNCT
ejpam-5689	138	20	f	f	X
ejpam-5689	138	21	(	(	PUNCT
ejpam-5689	138	22	|xay	|xay	ADV
ejpam-5689	138	23	+	+	CCONJ
ejpam-5689	138	24	y	y	PROPN
ejpam-5689	138	25	bx|	bx|	NOUN
ejpam-5689	138	26	)	)	PUNCT
ejpam-5689	138	27	)	)	PUNCT
ejpam-5689	139	1	≤	≤	NUM
ejpam-5689	139	2	f	f	X
ejpam-5689	139	3	(	(	PUNCT
ejpam-5689	139	4	sj	sj	INTJ
ejpam-5689	139	5	(	(	PUNCT
ejpam-5689	139	6	xay	xay	PROPN
ejpam-5689	139	7	⊕	⊕	PROPN
ejpam-5689	139	8	y	y	PROPN
ejpam-5689	139	9	bx	bx	PROPN
ejpam-5689	139	10	)	)	PUNCT
ejpam-5689	139	11	+	+	CCONJ
ejpam-5689	139	12	1	1	NUM
ejpam-5689	139	13	2	2	NUM
ejpam-5689	139	14	∥xay	∥xay	NOUN
ejpam-5689	139	15	+	+	CCONJ
ejpam-5689	139	16	y	y	PROPN
ejpam-5689	139	17	bx∥	bx∥	PROPN
ejpam-5689	139	18	)	)	PUNCT
ejpam-5689	140	1	=	=	SYM
ejpam-5689	140	2	f	f	PROPN
ejpam-5689	140	3	(	(	PUNCT
ejpam-5689	140	4	sj	sj	INTJ
ejpam-5689	140	5	(	(	PUNCT
ejpam-5689	140	6	xay	xay	PROPN
ejpam-5689	140	7	⊕	⊕	PROPN
ejpam-5689	140	8	(	(	PUNCT
ejpam-5689	140	9	y	y	PROPN
ejpam-5689	140	10	bx)∗	bx)∗	PROPN
ejpam-5689	140	11	)	)	PUNCT
ejpam-5689	141	1	+	+	CCONJ
ejpam-5689	141	2	1	1	NUM
ejpam-5689	141	3	2	2	NUM
ejpam-5689	141	4	∥xay	∥xay	NOUN
ejpam-5689	141	5	+	+	CCONJ
ejpam-5689	141	6	y	y	PROPN
ejpam-5689	141	7	bx∥	bx∥	PROPN
ejpam-5689	141	8	)	)	PUNCT
ejpam-5689	141	9	a.	a.	PROPN
ejpam-5689	141	10	al	al	PROPN
ejpam-5689	141	11	-	-	PUNCT
ejpam-5689	141	12	natoor	natoor	NOUN
ejpam-5689	141	13	,	,	PUNCT
ejpam-5689	141	14	f.	f.	PROPN
ejpam-5689	141	15	alrimawi	alrimawi	PROPN
ejpam-5689	141	16	/	/	SYM
ejpam-5689	141	17	eur	eur	PROPN
ejpam-5689	141	18	.	.	PUNCT
ejpam-5689	142	1	j.	j.	PROPN
ejpam-5689	142	2	pure	pure	PROPN
ejpam-5689	142	3	appl	appl	PROPN
ejpam-5689	142	4	.	.	PROPN
ejpam-5689	142	5	math	math	PROPN
ejpam-5689	142	6	,	,	PUNCT
ejpam-5689	142	7	18	18	NUM
ejpam-5689	142	8	(	(	PUNCT
ejpam-5689	142	9	1	1	NUM
ejpam-5689	142	10	)	)	PUNCT
ejpam-5689	142	11	(	(	PUNCT
ejpam-5689	142	12	2025	2025	NUM
ejpam-5689	142	13	)	)	PUNCT
ejpam-5689	142	14	,	,	PUNCT
ejpam-5689	142	15	5689	5689	NUM
ejpam-5689	142	16	7	7	NUM
ejpam-5689	142	17	of	of	ADP
ejpam-5689	142	18	11	11	NUM
ejpam-5689	142	19	≤	≤	NUM
ejpam-5689	142	20	1	1	NUM
ejpam-5689	142	21	2	2	NUM
ejpam-5689	142	22	f	f	NOUN
ejpam-5689	142	23	(	(	PUNCT
ejpam-5689	142	24	2sj	2sj	NOUN
ejpam-5689	142	25	(	(	PUNCT
ejpam-5689	142	26	[	[	PUNCT
ejpam-5689	142	27	x	x	SYM
ejpam-5689	142	28	0	0	NUM
ejpam-5689	142	29	0	0	NUM
ejpam-5689	142	30	x∗	x∗	X
ejpam-5689	142	31	]	]	PUNCT
ejpam-5689	143	1	[	[	PUNCT
ejpam-5689	143	2	a	a	DET
ejpam-5689	143	3	0	0	NUM
ejpam-5689	143	4	0	0	NUM
ejpam-5689	143	5	b	b	NOUN
ejpam-5689	143	6	]	]	X
ejpam-5689	144	1	[	[	PUNCT
ejpam-5689	144	2	y	y	NOUN
ejpam-5689	144	3	0	0	NUM
ejpam-5689	144	4	0	0	NUM
ejpam-5689	144	5	y	y	PROPN
ejpam-5689	144	6	∗	∗	NOUN
ejpam-5689	144	7	]	]	PUNCT
ejpam-5689	144	8	)	)	PUNCT
ejpam-5689	144	9	)	)	PUNCT
ejpam-5689	145	1	+	+	CCONJ
ejpam-5689	145	2	1	1	NUM
ejpam-5689	145	3	2	2	NUM
ejpam-5689	145	4	f	f	NOUN
ejpam-5689	145	5	(	(	PUNCT
ejpam-5689	145	6	∥xay	∥xay	PROPN
ejpam-5689	145	7	+	+	CCONJ
ejpam-5689	145	8	y	y	PROPN
ejpam-5689	145	9	bx∥	bx∥	PROPN
ejpam-5689	145	10	)	)	PUNCT
ejpam-5689	145	11	≤	≤	NOUN
ejpam-5689	145	12	1	1	NUM
ejpam-5689	145	13	2	2	NUM
ejpam-5689	145	14	f	f	PROPN
ejpam-5689	145	15			ADJ
ejpam-5689	145	16	∥∥∥∥∥∥∥∥∥	∥∥∥∥∥∥∥∥∥	NUM
ejpam-5689	145	17	x∗	x∗	X
ejpam-5689	145	18	0	0	NUM
ejpam-5689	145	19	0	0	NUM
ejpam-5689	145	20	x	x	SYM
ejpam-5689	145	21	x	x	PROPN
ejpam-5689	145	22	0	0	NUM
ejpam-5689	145	23	0	0	NUM
ejpam-5689	145	24	x∗	x∗	NOUN
ejpam-5689	145	25			VERB
ejpam-5689	145	26	∥x∥2	∥x∥2	NOUN
ejpam-5689	145	27	+	+	CCONJ
ejpam-5689	145	28	y	y	NOUN
ejpam-5689	145	29	∗	∗	NOUN
ejpam-5689	145	30	0	0	NUM
ejpam-5689	145	31	0	0	NUM
ejpam-5689	145	32	y	y	PROPN
ejpam-5689	145	33	y	y	PROPN
ejpam-5689	145	34	0	0	NUM
ejpam-5689	145	35	0	0	PUNCT
ejpam-5689	145	36	y	y	NOUN
ejpam-5689	145	37	∗	∗	NOUN
ejpam-5689	145	38			NOUN
ejpam-5689	145	39	∥y	∥y	PROPN
ejpam-5689	145	40	∥2	∥2	ADJ
ejpam-5689	145	41	∥∥∥∥∥∥∥∥∥	∥∥∥∥∥∥∥∥∥	NOUN
ejpam-5689	145	42	×∥x∥	×∥x∥	NOUN
ejpam-5689	146	1	∥y	∥y	PROPN
ejpam-5689	146	2	∥	∥	PUNCT
ejpam-5689	146	3	sj	sj	X
ejpam-5689	146	4	(	(	PUNCT
ejpam-5689	146	5	a⊕b	a⊕b	PROPN
ejpam-5689	146	6	)	)	PUNCT
ejpam-5689	146	7			PUNCT
ejpam-5689	147	1	+	+	CCONJ
ejpam-5689	147	2	1	1	NUM
ejpam-5689	147	3	2	2	NUM
ejpam-5689	147	4	f	f	NOUN
ejpam-5689	147	5	(	(	PUNCT
ejpam-5689	147	6	∥xay	∥xay	PROPN
ejpam-5689	147	7	+	+	CCONJ
ejpam-5689	147	8	y	y	PROPN
ejpam-5689	147	9	bx∥	bx∥	PROPN
ejpam-5689	147	10	)	)	PUNCT
ejpam-5689	147	11	(	(	PUNCT
ejpam-5689	147	12	by	by	ADP
ejpam-5689	147	13	lemma	lemma	PROPN
ejpam-5689	147	14	4	4	NUM
ejpam-5689	147	15	)	)	PUNCT
ejpam-5689	147	16	=	=	SYM
ejpam-5689	147	17	1	1	NUM
ejpam-5689	147	18	2	2	NUM
ejpam-5689	147	19	f	f	NOUN
ejpam-5689	147	20	(	(	PUNCT
ejpam-5689	147	21	∥∥∥∥∥	∥∥∥∥∥	X
ejpam-5689	147	22	|x|2	|x|2	PROPN
ejpam-5689	147	23	⊕	⊕	PROPN
ejpam-5689	147	24	|x∗|2	|x∗|2	NUM
ejpam-5689	147	25	∥x∥2	∥x∥2	NOUN
ejpam-5689	147	26	+	+	CCONJ
ejpam-5689	147	27	|y	|y	NOUN
ejpam-5689	147	28	|2	|2	NUM
ejpam-5689	147	29	⊕	⊕	NOUN
ejpam-5689	147	30	|y	|y	NOUN
ejpam-5689	147	31	∗|2	∗|2	PUNCT
ejpam-5689	147	32	∥y	∥y	PROPN
ejpam-5689	147	33	∥2	∥2	ADV
ejpam-5689	147	34	∥∥∥∥∥	∥∥∥∥∥	PUNCT
ejpam-5689	147	35	∥x∥	∥x∥	NOUN
ejpam-5689	147	36	∥y	∥y	PROPN
ejpam-5689	147	37	∥	∥	PUNCT
ejpam-5689	147	38	sj	sj	NOUN
ejpam-5689	147	39	(	(	PUNCT
ejpam-5689	147	40	a⊕b	a⊕b	PROPN
ejpam-5689	147	41	)	)	PUNCT
ejpam-5689	147	42	)	)	PUNCT
ejpam-5689	148	1	+	+	CCONJ
ejpam-5689	148	2	1	1	NUM
ejpam-5689	148	3	2	2	NUM
ejpam-5689	148	4	f	f	NOUN
ejpam-5689	148	5	(	(	PUNCT
ejpam-5689	148	6	∥xay	∥xay	PROPN
ejpam-5689	148	7	+	+	CCONJ
ejpam-5689	148	8	y	y	PROPN
ejpam-5689	148	9	bx∥	bx∥	PROPN
ejpam-5689	148	10	)	)	PUNCT
ejpam-5689	148	11	≤	≤	NOUN
ejpam-5689	148	12	1	1	NUM
ejpam-5689	148	13	2	2	NUM
ejpam-5689	148	14	f	f	NOUN
ejpam-5689	148	15	(	(	PUNCT
ejpam-5689	148	16	∥∥∥∥∥	∥∥∥∥∥	X
ejpam-5689	148	17	|x|2	|x|2	PROPN
ejpam-5689	148	18	⊕	⊕	PROPN
ejpam-5689	148	19	|x∗|2	|x∗|2	NUM
ejpam-5689	148	20	∥x∥2	∥x∥2	NOUN
ejpam-5689	148	21	+	+	CCONJ
ejpam-5689	148	22	|y	|y	NOUN
ejpam-5689	148	23	|2	|2	NUM
ejpam-5689	148	24	⊕	⊕	NOUN
ejpam-5689	148	25	|y	|y	NOUN
ejpam-5689	148	26	∗|2	∗|2	PUNCT
ejpam-5689	148	27	∥y	∥y	PROPN
ejpam-5689	148	28	∥2	∥2	NOUN
ejpam-5689	148	29	∥∥∥∥∥	∥∥∥∥∥	NUM
ejpam-5689	148	30	)	)	PUNCT
ejpam-5689	149	1	f	f	PROPN
ejpam-5689	149	2	(	(	PUNCT
ejpam-5689	149	3	∥x∥	∥x∥	NOUN
ejpam-5689	149	4	)	)	PUNCT
ejpam-5689	149	5	f	f	PROPN
ejpam-5689	149	6	(	(	PUNCT
ejpam-5689	149	7	∥y	∥y	NOUN
ejpam-5689	149	8	∥	∥	NUM
ejpam-5689	149	9	)	)	PUNCT
ejpam-5689	149	10	f	f	PROPN
ejpam-5689	149	11	(	(	PUNCT
ejpam-5689	149	12	sj	sj	INTJ
ejpam-5689	149	13	(	(	PUNCT
ejpam-5689	149	14	a⊕b	a⊕b	PROPN
ejpam-5689	149	15	)	)	PUNCT
ejpam-5689	149	16	)	)	PUNCT
ejpam-5689	150	1	+	+	CCONJ
ejpam-5689	150	2	1	1	NUM
ejpam-5689	150	3	2	2	NUM
ejpam-5689	150	4	f	f	NOUN
ejpam-5689	150	5	(	(	PUNCT
ejpam-5689	150	6	∥xay	∥xay	PROPN
ejpam-5689	150	7	+	+	CCONJ
ejpam-5689	150	8	y	y	PROPN
ejpam-5689	150	9	bx∥	bx∥	PROPN
ejpam-5689	150	10	)	)	PUNCT
ejpam-5689	150	11	(	(	PUNCT
ejpam-5689	150	12	since	since	SCONJ
ejpam-5689	150	13	f	f	PROPN
ejpam-5689	150	14	is	be	AUX
ejpam-5689	150	15	submultiplicative	submultiplicative	ADJ
ejpam-5689	150	16	)	)	PUNCT
ejpam-5689	150	17	=	=	SYM
ejpam-5689	150	18	1	1	NUM
ejpam-5689	150	19	2	2	NUM
ejpam-5689	150	20	∥∥∥∥∥f	∥∥∥∥∥f	PROPN
ejpam-5689	150	21	(	(	PUNCT
ejpam-5689	150	22	|x|2	|x|2	PROPN
ejpam-5689	150	23	⊕	⊕	PROPN
ejpam-5689	150	24	|x∗|2	|x∗|2	NUM
ejpam-5689	150	25	∥x∥2	∥x∥2	NOUN
ejpam-5689	150	26	+	+	CCONJ
ejpam-5689	150	27	|y	|y	NOUN
ejpam-5689	150	28	|2	|2	NUM
ejpam-5689	150	29	⊕	⊕	NOUN
ejpam-5689	150	30	|y	|y	NOUN
ejpam-5689	150	31	∗|2	∗|2	PUNCT
ejpam-5689	150	32	∥y	∥y	PROPN
ejpam-5689	150	33	∥2	∥2	NUM
ejpam-5689	150	34	)	)	PUNCT
ejpam-5689	150	35	∥∥∥∥∥	∥∥∥∥∥	PROPN
ejpam-5689	151	1	f	f	X
ejpam-5689	151	2	(	(	PUNCT
ejpam-5689	151	3	∥x∥	∥x∥	NOUN
ejpam-5689	151	4	)	)	PUNCT
ejpam-5689	151	5	f	f	PROPN
ejpam-5689	151	6	(	(	PUNCT
ejpam-5689	151	7	∥y	∥y	NOUN
ejpam-5689	151	8	∥	∥	NUM
ejpam-5689	151	9	)	)	PUNCT
ejpam-5689	151	10	sj	sj	INTJ
ejpam-5689	151	11	(	(	PUNCT
ejpam-5689	151	12	f	f	PROPN
ejpam-5689	151	13	(	(	PUNCT
ejpam-5689	151	14	a)⊕	a)⊕	ADP
ejpam-5689	151	15	f	f	X
ejpam-5689	151	16	(	(	PUNCT
ejpam-5689	151	17	b	b	NOUN
ejpam-5689	151	18	)	)	PUNCT
ejpam-5689	151	19	)	)	PUNCT
ejpam-5689	152	1	+	+	CCONJ
ejpam-5689	152	2	1	1	NUM
ejpam-5689	152	3	2	2	NUM
ejpam-5689	152	4	f	f	NOUN
ejpam-5689	152	5	(	(	PUNCT
ejpam-5689	152	6	∥xay	∥xay	PROPN
ejpam-5689	152	7	+	+	CCONJ
ejpam-5689	152	8	y	y	PROPN
ejpam-5689	152	9	bx∥	bx∥	PROPN
ejpam-5689	152	10	)	)	PUNCT
ejpam-5689	152	11	,	,	PUNCT
ejpam-5689	152	12	this	this	PRON
ejpam-5689	152	13	completes	complete	VERB
ejpam-5689	152	14	the	the	DET
ejpam-5689	152	15	proof	proof	NOUN
ejpam-5689	152	16	.	.	PUNCT
ejpam-5689	153	1	taking	take	VERB
ejpam-5689	153	2	f(t	f(t	NOUN
ejpam-5689	153	3	)	)	PUNCT
ejpam-5689	153	4	=	=	SYM
ejpam-5689	153	5	t	t	PROPN
ejpam-5689	153	6	,	,	PUNCT
ejpam-5689	153	7	t	t	PROPN
ejpam-5689	153	8	∈	∈	PROPN
ejpam-5689	154	1	[	[	X
ejpam-5689	154	2	0,∞	0,∞	NOUN
ejpam-5689	154	3	)	)	PUNCT
ejpam-5689	154	4	in	in	ADP
ejpam-5689	154	5	inequalities	inequality	NOUN
ejpam-5689	154	6	(	(	PUNCT
ejpam-5689	154	7	12	12	NUM
ejpam-5689	154	8	)	)	PUNCT
ejpam-5689	154	9	and	and	CCONJ
ejpam-5689	154	10	(	(	PUNCT
ejpam-5689	154	11	13	13	NUM
ejpam-5689	154	12	)	)	PUNCT
ejpam-5689	154	13	,	,	PUNCT
ejpam-5689	154	14	we	we	PRON
ejpam-5689	154	15	have	have	VERB
ejpam-5689	154	16	sj	sj	INTJ
ejpam-5689	154	17	(	(	PUNCT
ejpam-5689	155	1	xay	xay	PROPN
ejpam-5689	155	2	+	+	PROPN
ejpam-5689	155	3	y	y	PROPN
ejpam-5689	155	4	bx	bx	PROPN
ejpam-5689	155	5	)	)	PUNCT
ejpam-5689	155	6	≤	≤	NUM
ejpam-5689	155	7	1	1	NUM
ejpam-5689	155	8	2	2	NUM
ejpam-5689	155	9	∥∥∥∥∥	∥∥∥∥∥	NOUN
ejpam-5689	155	10	|x|2	|x|2	PROPN
ejpam-5689	155	11	⊕	⊕	PROPN
ejpam-5689	155	12	|x∗|2	|x∗|2	NUM
ejpam-5689	155	13	∥x∥2	∥x∥2	NOUN
ejpam-5689	155	14	+	+	CCONJ
ejpam-5689	155	15	|y	|y	NOUN
ejpam-5689	155	16	|2	|2	NUM
ejpam-5689	155	17	⊕	⊕	NOUN
ejpam-5689	155	18	|y	|y	NOUN
ejpam-5689	155	19	∗|2	∗|2	PUNCT
ejpam-5689	155	20	∥y	∥y	PROPN
ejpam-5689	155	21	∥2	∥2	ADV
ejpam-5689	155	22	∥∥∥∥∥	∥∥∥∥∥	PUNCT
ejpam-5689	155	23	∥x∥	∥x∥	NOUN
ejpam-5689	155	24	∥y	∥y	PROPN
ejpam-5689	155	25	∥	∥	PUNCT
ejpam-5689	155	26	sj	sj	NOUN
ejpam-5689	155	27	(	(	PUNCT
ejpam-5689	155	28	a⊕b	a⊕b	PROPN
ejpam-5689	155	29	)	)	PUNCT
ejpam-5689	155	30	+	+	CCONJ
ejpam-5689	155	31	1	1	NUM
ejpam-5689	155	32	2	2	NUM
ejpam-5689	155	33	∥xay	∥xay	PROPN
ejpam-5689	155	34	+	+	CCONJ
ejpam-5689	155	35	y	y	PROPN
ejpam-5689	155	36	bx∥	bx∥	PROPN
ejpam-5689	155	37	.	.	PUNCT
ejpam-5689	156	1	to	to	PART
ejpam-5689	156	2	state	state	VERB
ejpam-5689	156	3	our	our	PRON
ejpam-5689	156	4	next	next	ADJ
ejpam-5689	156	5	result	result	NOUN
ejpam-5689	156	6	,	,	PUNCT
ejpam-5689	156	7	we	we	PRON
ejpam-5689	156	8	need	need	VERB
ejpam-5689	156	9	the	the	DET
ejpam-5689	156	10	following	follow	VERB
ejpam-5689	156	11	lemma	lemma	PROPN
ejpam-5689	157	1	[	[	X
ejpam-5689	157	2	9	9	NUM
ejpam-5689	157	3	]	]	PUNCT
ejpam-5689	157	4	.	.	PUNCT
ejpam-5689	158	1	lemma	lemma	PROPN
ejpam-5689	158	2	5	5	X
ejpam-5689	158	3	.	.	PUNCT
ejpam-5689	159	1	let	let	VERB
ejpam-5689	159	2	a	a	DET
ejpam-5689	159	3	,	,	PUNCT
ejpam-5689	159	4	b	b	NOUN
ejpam-5689	159	5	,	,	PUNCT
ejpam-5689	159	6	x	x	SYM
ejpam-5689	159	7	∈	∈	NOUN
ejpam-5689	159	8	mn(c	mn(c	X
ejpam-5689	159	9	)	)	PUNCT
ejpam-5689	159	10	be	be	AUX
ejpam-5689	159	11	such	such	ADJ
ejpam-5689	159	12	that	that	SCONJ
ejpam-5689	159	13	x	x	PRON
ejpam-5689	159	14	is	be	AUX
ejpam-5689	159	15	positive	positive	ADJ
ejpam-5689	159	16	semidefinite	semidefinite	NOUN
ejpam-5689	159	17	.	.	PUNCT
ejpam-5689	160	1	then	then	ADV
ejpam-5689	160	2	sj	sj	INTJ
ejpam-5689	160	3	(	(	PUNCT
ejpam-5689	160	4	axb∗	axb∗	ADJ
ejpam-5689	160	5	)	)	PUNCT
ejpam-5689	160	6	≤	≤	NUM
ejpam-5689	160	7	1	1	NUM
ejpam-5689	160	8	2	2	NUM
ejpam-5689	160	9	∥x∥	∥x∥	NOUN
ejpam-5689	160	10	sj	sj	INTJ
ejpam-5689	160	11	(	(	PUNCT
ejpam-5689	160	12	a∗a+b∗b	a∗a+b∗b	PROPN
ejpam-5689	160	13	)	)	PUNCT
ejpam-5689	160	14	.	.	PUNCT
ejpam-5689	161	1	a.	a.	PROPN
ejpam-5689	161	2	al	al	PROPN
ejpam-5689	161	3	-	-	PUNCT
ejpam-5689	161	4	natoor	natoor	NOUN
ejpam-5689	161	5	,	,	PUNCT
ejpam-5689	161	6	f.	f.	PROPN
ejpam-5689	161	7	alrimawi	alrimawi	PROPN
ejpam-5689	161	8	/	/	SYM
ejpam-5689	161	9	eur	eur	PROPN
ejpam-5689	161	10	.	.	PUNCT
ejpam-5689	162	1	j.	j.	PROPN
ejpam-5689	162	2	pure	pure	PROPN
ejpam-5689	162	3	appl	appl	PROPN
ejpam-5689	162	4	.	.	PROPN
ejpam-5689	162	5	math	math	PROPN
ejpam-5689	162	6	,	,	PUNCT
ejpam-5689	162	7	18	18	NUM
ejpam-5689	162	8	(	(	PUNCT
ejpam-5689	162	9	1	1	NUM
ejpam-5689	162	10	)	)	PUNCT
ejpam-5689	162	11	(	(	PUNCT
ejpam-5689	162	12	2025	2025	NUM
ejpam-5689	162	13	)	)	PUNCT
ejpam-5689	162	14	,	,	PUNCT
ejpam-5689	162	15	5689	5689	NUM
ejpam-5689	162	16	8	8	NUM
ejpam-5689	162	17	of	of	ADP
ejpam-5689	162	18	11	11	NUM
ejpam-5689	162	19	theorem	theorem	NOUN
ejpam-5689	162	20	3	3	X
ejpam-5689	162	21	.	.	PUNCT
ejpam-5689	163	1	let	let	VERB
ejpam-5689	163	2	a	a	DET
ejpam-5689	163	3	,	,	PUNCT
ejpam-5689	163	4	b	b	NOUN
ejpam-5689	163	5	,	,	PUNCT
ejpam-5689	163	6	x	x	X
ejpam-5689	163	7	,	,	PUNCT
ejpam-5689	163	8	y	y	PROPN
ejpam-5689	163	9	∈	∈	PROPN
ejpam-5689	163	10	mn(c	mn(c	X
ejpam-5689	163	11	)	)	PUNCT
ejpam-5689	163	12	be	be	AUX
ejpam-5689	163	13	such	such	ADJ
ejpam-5689	163	14	that	that	SCONJ
ejpam-5689	163	15	a	a	PRON
ejpam-5689	163	16	and	and	CCONJ
ejpam-5689	163	17	b	b	NOUN
ejpam-5689	163	18	are	be	AUX
ejpam-5689	163	19	positive	positive	ADJ
ejpam-5689	163	20	semidefinite	semidefinite	NOUN
ejpam-5689	163	21	.	.	PUNCT
ejpam-5689	164	1	then	then	ADV
ejpam-5689	164	2	(	(	PUNCT
ejpam-5689	164	3	a	a	X
ejpam-5689	164	4	)	)	PUNCT
ejpam-5689	164	5	sj	sj	PROPN
ejpam-5689	164	6	(	(	PUNCT
ejpam-5689	164	7	f	f	X
ejpam-5689	164	8	(	(	PUNCT
ejpam-5689	164	9	|xay	|xay	ADV
ejpam-5689	164	10	+	+	CCONJ
ejpam-5689	164	11	y	y	PROPN
ejpam-5689	164	12	bx|	bx|	NOUN
ejpam-5689	164	13	)	)	PUNCT
ejpam-5689	164	14	)	)	PUNCT
ejpam-5689	165	1	≤	≤	NUM
ejpam-5689	165	2	f	f	X
ejpam-5689	165	3	(	(	PUNCT
ejpam-5689	165	4	1	1	NUM
ejpam-5689	165	5	2	2	NUM
ejpam-5689	165	6	)	)	PUNCT
ejpam-5689	165	7	max	max	PROPN
ejpam-5689	165	8	(	(	PUNCT
ejpam-5689	165	9	∥f(a)∥	∥f(a)∥	PROPN
ejpam-5689	165	10	,	,	PUNCT
ejpam-5689	165	11	∥f(b)∥	∥f(b)∥	PROPN
ejpam-5689	165	12	)	)	PUNCT
ejpam-5689	165	13	sj	sj	INTJ
ejpam-5689	165	14	(	(	PUNCT
ejpam-5689	165	15	(	(	PUNCT
ejpam-5689	165	16	f	f	X
ejpam-5689	165	17	(	(	PUNCT
ejpam-5689	165	18	x∗x	x∗x	PROPN
ejpam-5689	165	19	+	+	CCONJ
ejpam-5689	165	20	y	y	PROPN
ejpam-5689	165	21	y	y	PROPN
ejpam-5689	165	22	∗))⊕	∗))⊕	NOUN
ejpam-5689	165	23	(	(	PUNCT
ejpam-5689	165	24	f	f	X
ejpam-5689	165	25	(	(	PUNCT
ejpam-5689	165	26	xx∗	xx∗	VERB
ejpam-5689	165	27	+	+	CCONJ
ejpam-5689	165	28	y	y	PROPN
ejpam-5689	165	29	∗y	∗y	PROPN
ejpam-5689	165	30	)	)	PUNCT
ejpam-5689	165	31	)	)	PUNCT
ejpam-5689	165	32	)	)	PUNCT
ejpam-5689	166	1	+	+	NOUN
ejpam-5689	166	2	f	f	X
ejpam-5689	166	3	(	(	PUNCT
ejpam-5689	166	4	1	1	NUM
ejpam-5689	166	5	2	2	NUM
ejpam-5689	166	6	)	)	PUNCT
ejpam-5689	166	7	f	f	NOUN
ejpam-5689	166	8	(	(	PUNCT
ejpam-5689	166	9	∥xay	∥xay	PROPN
ejpam-5689	166	10	+	+	CCONJ
ejpam-5689	166	11	y	y	PROPN
ejpam-5689	166	12	bx∥	bx∥	PROPN
ejpam-5689	166	13	)	)	PUNCT
ejpam-5689	166	14	,	,	PUNCT
ejpam-5689	166	15	(	(	PUNCT
ejpam-5689	166	16	14	14	NUM
ejpam-5689	166	17	)	)	PUNCT
ejpam-5689	166	18	where	where	SCONJ
ejpam-5689	166	19	f	f	PROPN
ejpam-5689	166	20	is	be	AUX
ejpam-5689	166	21	a	a	DET
ejpam-5689	166	22	nonnegative	nonnegative	ADJ
ejpam-5689	166	23	increasing	increase	VERB
ejpam-5689	166	24	submultiplicative	submultiplicative	ADJ
ejpam-5689	166	25	concave	concave	NOUN
ejpam-5689	166	26	function	function	NOUN
ejpam-5689	166	27	on	on	ADP
ejpam-5689	166	28	[	[	X
ejpam-5689	166	29	0,∞	0,∞	NOUN
ejpam-5689	166	30	)	)	PUNCT
ejpam-5689	166	31	with	with	ADP
ejpam-5689	166	32	f(0	f(0	NOUN
ejpam-5689	166	33	)	)	PUNCT
ejpam-5689	166	34	=	=	SYM
ejpam-5689	167	1	0	0	X
ejpam-5689	167	2	.	.	PUNCT
ejpam-5689	168	1	(	(	PUNCT
ejpam-5689	168	2	b	b	X
ejpam-5689	168	3	)	)	PUNCT
ejpam-5689	168	4	sj	sj	PROPN
ejpam-5689	168	5	(	(	PUNCT
ejpam-5689	168	6	f	f	X
ejpam-5689	168	7	(	(	PUNCT
ejpam-5689	168	8	|xay	|xay	ADV
ejpam-5689	168	9	+	+	CCONJ
ejpam-5689	168	10	y	y	PROPN
ejpam-5689	168	11	bx|	bx|	NOUN
ejpam-5689	168	12	)	)	PUNCT
ejpam-5689	168	13	)	)	PUNCT
ejpam-5689	169	1	≤	≤	NUM
ejpam-5689	169	2	1	1	NUM
ejpam-5689	169	3	2	2	NUM
ejpam-5689	169	4	max	max	NOUN
ejpam-5689	169	5	(	(	PUNCT
ejpam-5689	169	6	∥f(a)∥	∥f(a)∥	PROPN
ejpam-5689	169	7	,	,	PUNCT
ejpam-5689	169	8	∥f(b)∥	∥f(b)∥	PROPN
ejpam-5689	169	9	)	)	PUNCT
ejpam-5689	169	10	sj	sj	INTJ
ejpam-5689	169	11	(	(	PUNCT
ejpam-5689	169	12	(	(	PUNCT
ejpam-5689	169	13	f	f	X
ejpam-5689	169	14	(	(	PUNCT
ejpam-5689	169	15	x∗x	x∗x	PROPN
ejpam-5689	169	16	+	+	CCONJ
ejpam-5689	169	17	y	y	PROPN
ejpam-5689	169	18	y	y	PROPN
ejpam-5689	169	19	∗))⊕	∗))⊕	NOUN
ejpam-5689	169	20	(	(	PUNCT
ejpam-5689	169	21	f	f	X
ejpam-5689	169	22	(	(	PUNCT
ejpam-5689	169	23	xx∗	xx∗	VERB
ejpam-5689	170	1	+	+	CCONJ
ejpam-5689	170	2	y	y	PROPN
ejpam-5689	170	3	∗y	∗y	PROPN
ejpam-5689	170	4	)	)	PUNCT
ejpam-5689	170	5	)	)	PUNCT
ejpam-5689	170	6	)	)	PUNCT
ejpam-5689	171	1	+	+	CCONJ
ejpam-5689	171	2	1	1	NUM
ejpam-5689	171	3	2	2	NUM
ejpam-5689	171	4	f	f	NOUN
ejpam-5689	171	5	(	(	PUNCT
ejpam-5689	171	6	∥xay	∥xay	PROPN
ejpam-5689	171	7	+	+	CCONJ
ejpam-5689	171	8	y	y	PROPN
ejpam-5689	171	9	bx∥	bx∥	PROPN
ejpam-5689	171	10	)	)	PUNCT
ejpam-5689	171	11	,	,	PUNCT
ejpam-5689	171	12	(	(	PUNCT
ejpam-5689	171	13	15	15	NUM
ejpam-5689	171	14	)	)	PUNCT
ejpam-5689	171	15	where	where	SCONJ
ejpam-5689	171	16	f	f	PROPN
ejpam-5689	171	17	is	be	AUX
ejpam-5689	171	18	a	a	DET
ejpam-5689	171	19	nonnegative	nonnegative	ADJ
ejpam-5689	171	20	increasing	increase	VERB
ejpam-5689	171	21	submultiplicative	submultiplicative	ADJ
ejpam-5689	171	22	convex	convex	NOUN
ejpam-5689	171	23	function	function	NOUN
ejpam-5689	171	24	on	on	ADP
ejpam-5689	171	25	[	[	X
ejpam-5689	171	26	0,∞	0,∞	NOUN
ejpam-5689	171	27	)	)	PUNCT
ejpam-5689	171	28	.	.	PUNCT
ejpam-5689	172	1	proof	proof	NOUN
ejpam-5689	172	2	.	.	PUNCT
ejpam-5689	173	1	by	by	ADP
ejpam-5689	173	2	inequality	inequality	NOUN
ejpam-5689	173	3	(	(	PUNCT
ejpam-5689	173	4	7	7	NUM
ejpam-5689	173	5	)	)	PUNCT
ejpam-5689	173	6	,	,	PUNCT
ejpam-5689	173	7	we	we	PRON
ejpam-5689	173	8	have	have	VERB
ejpam-5689	173	9	sj	sj	INTJ
ejpam-5689	173	10	(	(	PUNCT
ejpam-5689	173	11	f	f	X
ejpam-5689	173	12	(	(	PUNCT
ejpam-5689	173	13	|xay	|xay	ADV
ejpam-5689	173	14	+	+	CCONJ
ejpam-5689	173	15	y	y	PROPN
ejpam-5689	173	16	bx|	bx|	NOUN
ejpam-5689	173	17	)	)	PUNCT
ejpam-5689	173	18	)	)	PUNCT
ejpam-5689	174	1	≤	≤	NUM
ejpam-5689	174	2	f	f	X
ejpam-5689	174	3	(	(	PUNCT
ejpam-5689	174	4	sj	sj	INTJ
ejpam-5689	174	5	(	(	PUNCT
ejpam-5689	174	6	[	[	PUNCT
ejpam-5689	174	7	x	x	SYM
ejpam-5689	174	8	0	0	NUM
ejpam-5689	174	9	0	0	NUM
ejpam-5689	174	10	x∗	x∗	X
ejpam-5689	174	11	]	]	PUNCT
ejpam-5689	175	1	[	[	PUNCT
ejpam-5689	175	2	a	a	DET
ejpam-5689	175	3	0	0	NUM
ejpam-5689	175	4	0	0	NUM
ejpam-5689	175	5	b	b	NOUN
ejpam-5689	175	6	]	]	X
ejpam-5689	176	1	[	[	PUNCT
ejpam-5689	176	2	y	y	NOUN
ejpam-5689	176	3	0	0	NUM
ejpam-5689	176	4	0	0	NUM
ejpam-5689	176	5	y	y	PROPN
ejpam-5689	176	6	∗	∗	NOUN
ejpam-5689	176	7	]	]	PUNCT
ejpam-5689	176	8	)	)	PUNCT
ejpam-5689	176	9	)	)	PUNCT
ejpam-5689	177	1	+	+	NOUN
ejpam-5689	177	2	f	f	X
ejpam-5689	177	3	(	(	PUNCT
ejpam-5689	177	4	1	1	NUM
ejpam-5689	177	5	2	2	NUM
ejpam-5689	177	6	)	)	PUNCT
ejpam-5689	177	7	f	f	NOUN
ejpam-5689	177	8	(	(	PUNCT
ejpam-5689	177	9	∥xay	∥xay	PROPN
ejpam-5689	177	10	+	+	CCONJ
ejpam-5689	177	11	y	y	PROPN
ejpam-5689	177	12	bx∥	bx∥	PROPN
ejpam-5689	177	13	)	)	PUNCT
ejpam-5689	177	14	≤	≤	NUM
ejpam-5689	178	1	f	f	X
ejpam-5689	178	2	1	1	PROPN
ejpam-5689	178	3	2	2	NUM
ejpam-5689	178	4	∥∥∥∥[a	∥∥∥∥[a	NOUN
ejpam-5689	178	5	0	0	NUM
ejpam-5689	178	6	0	0	NUM
ejpam-5689	179	1	b	b	X
ejpam-5689	179	2	]	]	PUNCT
ejpam-5689	179	3	∥∥∥∥	∥∥∥∥	NUM
ejpam-5689	179	4	sj	sj	PROPN
ejpam-5689	179	5			PROPN
ejpam-5689	179	6	[	[	PUNCT
ejpam-5689	179	7	x∗	x∗	X
ejpam-5689	179	8	0	0	PUNCT
ejpam-5689	179	9	0	0	NUM
ejpam-5689	179	10	x	x	SYM
ejpam-5689	179	11	]	]	X
ejpam-5689	180	1	[	[	PUNCT
ejpam-5689	180	2	x	x	SYM
ejpam-5689	180	3	0	0	NUM
ejpam-5689	180	4	0	0	NUM
ejpam-5689	180	5	x∗	x∗	X
ejpam-5689	180	6	]	]	PUNCT
ejpam-5689	181	1	+	+	CCONJ
ejpam-5689	181	2	[	[	PUNCT
ejpam-5689	181	3	y	y	NOUN
ejpam-5689	181	4	0	0	NUM
ejpam-5689	181	5	0	0	NUM
ejpam-5689	181	6	y	y	NOUN
ejpam-5689	181	7	∗	∗	NOUN
ejpam-5689	181	8	]	]	PUNCT
ejpam-5689	181	9	[	[	PUNCT
ejpam-5689	181	10	y	y	NOUN
ejpam-5689	181	11	∗	∗	NOUN
ejpam-5689	181	12	0	0	NUM
ejpam-5689	181	13	0	0	NUM
ejpam-5689	181	14	y	y	PROPN
ejpam-5689	181	15	]	]	PUNCT
ejpam-5689	181	16			NOUN
ejpam-5689	181	17			NOUN
ejpam-5689	182	1	+	+	X
ejpam-5689	182	2	f	f	X
ejpam-5689	182	3	(	(	PUNCT
ejpam-5689	182	4	1	1	NUM
ejpam-5689	182	5	2	2	NUM
ejpam-5689	182	6	)	)	PUNCT
ejpam-5689	182	7	f	f	NOUN
ejpam-5689	182	8	(	(	PUNCT
ejpam-5689	182	9	∥xay	∥xay	PROPN
ejpam-5689	182	10	+	+	CCONJ
ejpam-5689	182	11	y	y	PROPN
ejpam-5689	182	12	bx∥	bx∥	PROPN
ejpam-5689	182	13	)	)	PUNCT
ejpam-5689	182	14	(	(	PUNCT
ejpam-5689	182	15	by	by	ADP
ejpam-5689	182	16	lemma	lemma	PROPN
ejpam-5689	182	17	5	5	NUM
ejpam-5689	182	18	)	)	PUNCT
ejpam-5689	182	19	≤	≤	NUM
ejpam-5689	183	1	f	f	X
ejpam-5689	184	1	(	(	PUNCT
ejpam-5689	184	2	1	1	NUM
ejpam-5689	184	3	2	2	NUM
ejpam-5689	184	4	)	)	PUNCT
ejpam-5689	184	5	f	f	PROPN
ejpam-5689	184	6	(	(	PUNCT
ejpam-5689	184	7	∥∥∥∥[a	∥∥∥∥[a	NOUN
ejpam-5689	184	8	0	0	NUM
ejpam-5689	184	9	0	0	NUM
ejpam-5689	184	10	b	b	X
ejpam-5689	184	11	]	]	PUNCT
ejpam-5689	184	12	∥∥∥∥	∥∥∥∥	NUM
ejpam-5689	184	13	)	)	PUNCT
ejpam-5689	185	1	f	f	PROPN
ejpam-5689	185	2	(	(	PUNCT
ejpam-5689	185	3	sj	sj	INTJ
ejpam-5689	185	4	(	(	PUNCT
ejpam-5689	185	5	[	[	PUNCT
ejpam-5689	185	6	x∗x	x∗x	PROPN
ejpam-5689	185	7	+	+	CCONJ
ejpam-5689	185	8	y	y	PROPN
ejpam-5689	185	9	y	y	PROPN
ejpam-5689	185	10	∗	∗	NOUN
ejpam-5689	185	11	0	0	NUM
ejpam-5689	185	12	0	0	NUM
ejpam-5689	185	13	xx∗	xx∗	NOUN
ejpam-5689	186	1	+	+	CCONJ
ejpam-5689	186	2	y	y	PROPN
ejpam-5689	186	3	∗y	∗y	PROPN
ejpam-5689	186	4	]	]	PUNCT
ejpam-5689	186	5	)	)	PUNCT
ejpam-5689	186	6	)	)	PUNCT
ejpam-5689	187	1	a.	a.	PROPN
ejpam-5689	187	2	al	al	PROPN
ejpam-5689	187	3	-	-	PUNCT
ejpam-5689	187	4	natoor	natoor	NOUN
ejpam-5689	187	5	,	,	PUNCT
ejpam-5689	187	6	f.	f.	PROPN
ejpam-5689	187	7	alrimawi	alrimawi	PROPN
ejpam-5689	187	8	/	/	SYM
ejpam-5689	187	9	eur	eur	PROPN
ejpam-5689	187	10	.	.	PUNCT
ejpam-5689	188	1	j.	j.	PROPN
ejpam-5689	188	2	pure	pure	PROPN
ejpam-5689	188	3	appl	appl	PROPN
ejpam-5689	188	4	.	.	PROPN
ejpam-5689	188	5	math	math	PROPN
ejpam-5689	188	6	,	,	PUNCT
ejpam-5689	188	7	18	18	NUM
ejpam-5689	188	8	(	(	PUNCT
ejpam-5689	188	9	1	1	NUM
ejpam-5689	188	10	)	)	PUNCT
ejpam-5689	188	11	(	(	PUNCT
ejpam-5689	188	12	2025	2025	NUM
ejpam-5689	188	13	)	)	PUNCT
ejpam-5689	188	14	,	,	PUNCT
ejpam-5689	188	15	5689	5689	NUM
ejpam-5689	188	16	9	9	NUM
ejpam-5689	188	17	of	of	ADP
ejpam-5689	188	18	11	11	NUM
ejpam-5689	189	1	+	+	NOUN
ejpam-5689	189	2	f	f	X
ejpam-5689	189	3	(	(	PUNCT
ejpam-5689	189	4	1	1	NUM
ejpam-5689	189	5	2	2	NUM
ejpam-5689	189	6	)	)	PUNCT
ejpam-5689	189	7	f	f	NOUN
ejpam-5689	189	8	(	(	PUNCT
ejpam-5689	189	9	∥xay	∥xay	PROPN
ejpam-5689	189	10	+	+	CCONJ
ejpam-5689	189	11	y	y	PROPN
ejpam-5689	189	12	bx∥	bx∥	PROPN
ejpam-5689	189	13	)	)	PUNCT
ejpam-5689	189	14	=	=	SYM
ejpam-5689	190	1	f	f	PROPN
ejpam-5689	190	2	(	(	PUNCT
ejpam-5689	190	3	1	1	NUM
ejpam-5689	190	4	2	2	NUM
ejpam-5689	190	5	)	)	PUNCT
ejpam-5689	190	6	∥∥∥∥[f(a	∥∥∥∥[f(a	PUNCT
ejpam-5689	190	7	)	)	PUNCT
ejpam-5689	190	8	0	0	NUM
ejpam-5689	190	9	0	0	NUM
ejpam-5689	190	10	f(b	f(b	PROPN
ejpam-5689	190	11	)	)	PUNCT
ejpam-5689	190	12	]	]	PUNCT
ejpam-5689	190	13	∥∥∥∥	∥∥∥∥	PUNCT
ejpam-5689	190	14	sj	sj	INTJ
ejpam-5689	190	15	(	(	PUNCT
ejpam-5689	190	16	f	f	X
ejpam-5689	190	17	(	(	PUNCT
ejpam-5689	190	18	[	[	X
ejpam-5689	190	19	x∗x	x∗x	PROPN
ejpam-5689	190	20	+	+	CCONJ
ejpam-5689	190	21	y	y	PROPN
ejpam-5689	190	22	y	y	PROPN
ejpam-5689	190	23	∗	∗	NOUN
ejpam-5689	190	24	0	0	NUM
ejpam-5689	190	25	0	0	NUM
ejpam-5689	190	26	xx∗	xx∗	NOUN
ejpam-5689	191	1	+	+	CCONJ
ejpam-5689	191	2	y	y	PROPN
ejpam-5689	191	3	∗y	∗y	PROPN
ejpam-5689	191	4	]	]	PUNCT
ejpam-5689	191	5	)	)	PUNCT
ejpam-5689	191	6	)	)	PUNCT
ejpam-5689	192	1	+	+	NOUN
ejpam-5689	192	2	f	f	X
ejpam-5689	192	3	(	(	PUNCT
ejpam-5689	192	4	1	1	NUM
ejpam-5689	192	5	2	2	NUM
ejpam-5689	192	6	)	)	PUNCT
ejpam-5689	192	7	f	f	NOUN
ejpam-5689	192	8	(	(	PUNCT
ejpam-5689	192	9	∥xay	∥xay	PROPN
ejpam-5689	192	10	+	+	CCONJ
ejpam-5689	192	11	y	y	PROPN
ejpam-5689	192	12	bx∥	bx∥	PROPN
ejpam-5689	192	13	)	)	PUNCT
ejpam-5689	192	14	=	=	SYM
ejpam-5689	193	1	f	f	PROPN
ejpam-5689	193	2	(	(	PUNCT
ejpam-5689	193	3	1	1	NUM
ejpam-5689	193	4	2	2	NUM
ejpam-5689	193	5	)	)	PUNCT
ejpam-5689	193	6	max	max	PROPN
ejpam-5689	193	7	(	(	PUNCT
ejpam-5689	193	8	∥f(a)∥	∥f(a)∥	PROPN
ejpam-5689	193	9	,	,	PUNCT
ejpam-5689	193	10	∥f(b)∥	∥f(b)∥	PROPN
ejpam-5689	193	11	)	)	PUNCT
ejpam-5689	193	12	sj	sj	INTJ
ejpam-5689	193	13	(	(	PUNCT
ejpam-5689	193	14	(	(	PUNCT
ejpam-5689	193	15	f	f	X
ejpam-5689	193	16	(	(	PUNCT
ejpam-5689	193	17	x∗x	x∗x	PROPN
ejpam-5689	193	18	+	+	CCONJ
ejpam-5689	193	19	y	y	PROPN
ejpam-5689	193	20	y	y	PROPN
ejpam-5689	193	21	∗))⊕	∗))⊕	NOUN
ejpam-5689	193	22	(	(	PUNCT
ejpam-5689	193	23	f	f	X
ejpam-5689	193	24	(	(	PUNCT
ejpam-5689	193	25	xx∗	xx∗	VERB
ejpam-5689	194	1	+	+	CCONJ
ejpam-5689	194	2	y	y	PROPN
ejpam-5689	194	3	∗y	∗y	PROPN
ejpam-5689	194	4	)	)	PUNCT
ejpam-5689	194	5	)	)	PUNCT
ejpam-5689	194	6	)	)	PUNCT
ejpam-5689	195	1	+	+	NOUN
ejpam-5689	195	2	f	f	X
ejpam-5689	195	3	(	(	PUNCT
ejpam-5689	195	4	1	1	NUM
ejpam-5689	195	5	2	2	NUM
ejpam-5689	195	6	)	)	PUNCT
ejpam-5689	195	7	f	f	NOUN
ejpam-5689	195	8	(	(	PUNCT
ejpam-5689	195	9	∥xay	∥xay	PROPN
ejpam-5689	195	10	+	+	CCONJ
ejpam-5689	195	11	y	y	PROPN
ejpam-5689	195	12	bx∥	bx∥	PROPN
ejpam-5689	195	13	)	)	PUNCT
ejpam-5689	195	14	,	,	PUNCT
ejpam-5689	195	15	which	which	PRON
ejpam-5689	195	16	proves	prove	VERB
ejpam-5689	195	17	part	part	NOUN
ejpam-5689	195	18	(	(	PUNCT
ejpam-5689	195	19	a	a	NOUN
ejpam-5689	195	20	)	)	PUNCT
ejpam-5689	195	21	.	.	PUNCT
ejpam-5689	196	1	for	for	ADP
ejpam-5689	196	2	part	part	NOUN
ejpam-5689	196	3	(	(	PUNCT
ejpam-5689	196	4	b	b	NOUN
ejpam-5689	196	5	)	)	PUNCT
ejpam-5689	196	6	,	,	PUNCT
ejpam-5689	196	7	we	we	PRON
ejpam-5689	196	8	start	start	VERB
ejpam-5689	196	9	from	from	ADP
ejpam-5689	196	10	inequality	inequality	NOUN
ejpam-5689	196	11	(	(	PUNCT
ejpam-5689	196	12	6	6	NUM
ejpam-5689	196	13	)	)	PUNCT
ejpam-5689	196	14	,	,	PUNCT
ejpam-5689	196	15	so	so	SCONJ
ejpam-5689	196	16	we	we	PRON
ejpam-5689	196	17	have	have	VERB
ejpam-5689	196	18	sj	sj	INTJ
ejpam-5689	196	19	(	(	PUNCT
ejpam-5689	196	20	f	f	X
ejpam-5689	196	21	(	(	PUNCT
ejpam-5689	196	22	|xay	|xay	ADV
ejpam-5689	196	23	+	+	CCONJ
ejpam-5689	196	24	y	y	PROPN
ejpam-5689	196	25	bx|	bx|	NOUN
ejpam-5689	196	26	)	)	PUNCT
ejpam-5689	196	27	)	)	PUNCT
ejpam-5689	197	1	≤	≤	NUM
ejpam-5689	197	2	f	f	X
ejpam-5689	197	3	(	(	PUNCT
ejpam-5689	197	4	sj	sj	INTJ
ejpam-5689	197	5	(	(	PUNCT
ejpam-5689	197	6	xay	xay	PROPN
ejpam-5689	197	7	⊕	⊕	PROPN
ejpam-5689	197	8	y	y	PROPN
ejpam-5689	197	9	bx	bx	PROPN
ejpam-5689	197	10	)	)	PUNCT
ejpam-5689	197	11	+	+	CCONJ
ejpam-5689	197	12	1	1	NUM
ejpam-5689	197	13	2	2	NUM
ejpam-5689	197	14	∥xay	∥xay	NOUN
ejpam-5689	197	15	+	+	CCONJ
ejpam-5689	197	16	y	y	PROPN
ejpam-5689	197	17	bx∥	bx∥	PROPN
ejpam-5689	197	18	)	)	PUNCT
ejpam-5689	198	1	=	=	SYM
ejpam-5689	198	2	f	f	PROPN
ejpam-5689	198	3	(	(	PUNCT
ejpam-5689	198	4	sj	sj	INTJ
ejpam-5689	198	5	(	(	PUNCT
ejpam-5689	198	6	xay	xay	PROPN
ejpam-5689	198	7	⊕	⊕	PROPN
ejpam-5689	198	8	(	(	PUNCT
ejpam-5689	198	9	y	y	PROPN
ejpam-5689	198	10	bx)∗	bx)∗	PROPN
ejpam-5689	198	11	)	)	PUNCT
ejpam-5689	199	1	+	+	CCONJ
ejpam-5689	199	2	1	1	NUM
ejpam-5689	199	3	2	2	NUM
ejpam-5689	199	4	∥xay	∥xay	NOUN
ejpam-5689	199	5	+	+	CCONJ
ejpam-5689	199	6	y	y	PROPN
ejpam-5689	199	7	bx∥	bx∥	PROPN
ejpam-5689	199	8	)	)	PUNCT
ejpam-5689	199	9	≤	≤	NUM
ejpam-5689	199	10	1	1	NUM
ejpam-5689	199	11	2	2	NUM
ejpam-5689	199	12	f	f	NOUN
ejpam-5689	199	13	(	(	PUNCT
ejpam-5689	199	14	2sj	2sj	NOUN
ejpam-5689	199	15	(	(	PUNCT
ejpam-5689	199	16	[	[	PUNCT
ejpam-5689	199	17	x	x	SYM
ejpam-5689	199	18	0	0	NUM
ejpam-5689	199	19	0	0	NUM
ejpam-5689	199	20	x∗	x∗	X
ejpam-5689	199	21	]	]	PUNCT
ejpam-5689	200	1	[	[	PUNCT
ejpam-5689	200	2	a	a	DET
ejpam-5689	200	3	0	0	NUM
ejpam-5689	200	4	0	0	NUM
ejpam-5689	200	5	b	b	NOUN
ejpam-5689	200	6	]	]	X
ejpam-5689	201	1	[	[	PUNCT
ejpam-5689	201	2	y	y	NOUN
ejpam-5689	201	3	0	0	NUM
ejpam-5689	201	4	0	0	NUM
ejpam-5689	201	5	y	y	PROPN
ejpam-5689	201	6	∗	∗	NOUN
ejpam-5689	201	7	]	]	PUNCT
ejpam-5689	201	8	)	)	PUNCT
ejpam-5689	201	9	)	)	PUNCT
ejpam-5689	202	1	+	+	CCONJ
ejpam-5689	202	2	1	1	NUM
ejpam-5689	202	3	2	2	NUM
ejpam-5689	202	4	f	f	NOUN
ejpam-5689	202	5	(	(	PUNCT
ejpam-5689	202	6	∥xay	∥xay	PROPN
ejpam-5689	202	7	+	+	CCONJ
ejpam-5689	202	8	y	y	PROPN
ejpam-5689	202	9	bx∥	bx∥	PROPN
ejpam-5689	202	10	)	)	PUNCT
ejpam-5689	202	11	≤	≤	NOUN
ejpam-5689	202	12	1	1	NUM
ejpam-5689	202	13	2	2	NUM
ejpam-5689	202	14	f	f	NOUN
ejpam-5689	202	15	∥∥∥∥[a	∥∥∥∥[a	NOUN
ejpam-5689	202	16	0	0	NUM
ejpam-5689	202	17	0	0	NUM
ejpam-5689	202	18	b	b	X
ejpam-5689	202	19	]	]	PUNCT
ejpam-5689	202	20	∥∥∥∥	∥∥∥∥	NUM
ejpam-5689	202	21	sj	sj	PROPN
ejpam-5689	202	22			PROPN
ejpam-5689	202	23	[	[	PUNCT
ejpam-5689	202	24	x∗	x∗	X
ejpam-5689	202	25	0	0	PUNCT
ejpam-5689	202	26	0	0	NUM
ejpam-5689	202	27	x	x	SYM
ejpam-5689	202	28	]	]	X
ejpam-5689	203	1	[	[	PUNCT
ejpam-5689	203	2	x	x	SYM
ejpam-5689	203	3	0	0	NUM
ejpam-5689	203	4	0	0	NUM
ejpam-5689	203	5	x∗	x∗	X
ejpam-5689	203	6	]	]	PUNCT
ejpam-5689	204	1	+	+	CCONJ
ejpam-5689	204	2	[	[	PUNCT
ejpam-5689	204	3	y	y	NOUN
ejpam-5689	204	4	0	0	NUM
ejpam-5689	204	5	0	0	NUM
ejpam-5689	204	6	y	y	NOUN
ejpam-5689	204	7	∗	∗	NOUN
ejpam-5689	204	8	]	]	PUNCT
ejpam-5689	204	9	[	[	PUNCT
ejpam-5689	204	10	y	y	NOUN
ejpam-5689	204	11	∗	∗	NOUN
ejpam-5689	204	12	0	0	NUM
ejpam-5689	204	13	0	0	NUM
ejpam-5689	204	14	y	y	PROPN
ejpam-5689	204	15	]	]	PUNCT
ejpam-5689	204	16			NOUN
ejpam-5689	204	17			NOUN
ejpam-5689	205	1	+	+	CCONJ
ejpam-5689	205	2	1	1	NUM
ejpam-5689	205	3	2	2	NUM
ejpam-5689	205	4	f	f	NOUN
ejpam-5689	205	5	(	(	PUNCT
ejpam-5689	205	6	∥xay	∥xay	PROPN
ejpam-5689	205	7	+	+	CCONJ
ejpam-5689	205	8	y	y	PROPN
ejpam-5689	205	9	bx∥	bx∥	PROPN
ejpam-5689	205	10	)	)	PUNCT
ejpam-5689	205	11	(	(	PUNCT
ejpam-5689	205	12	by	by	ADP
ejpam-5689	205	13	lemma	lemma	PROPN
ejpam-5689	205	14	5	5	NUM
ejpam-5689	205	15	)	)	PUNCT
ejpam-5689	205	16	≤	≤	NUM
ejpam-5689	205	17	1	1	NUM
ejpam-5689	205	18	2	2	NUM
ejpam-5689	205	19	f	f	NOUN
ejpam-5689	205	20	(	(	PUNCT
ejpam-5689	205	21	∥∥∥∥[a	∥∥∥∥[a	NOUN
ejpam-5689	205	22	0	0	NUM
ejpam-5689	205	23	0	0	NUM
ejpam-5689	205	24	b	b	X
ejpam-5689	205	25	]	]	PUNCT
ejpam-5689	205	26	∥∥∥∥	∥∥∥∥	NUM
ejpam-5689	205	27	)	)	PUNCT
ejpam-5689	206	1	f	f	PROPN
ejpam-5689	206	2	(	(	PUNCT
ejpam-5689	206	3	sj	sj	INTJ
ejpam-5689	206	4	(	(	PUNCT
ejpam-5689	206	5	[	[	PUNCT
ejpam-5689	206	6	x∗x	x∗x	PROPN
ejpam-5689	206	7	+	+	CCONJ
ejpam-5689	206	8	y	y	PROPN
ejpam-5689	206	9	y	y	PROPN
ejpam-5689	206	10	∗	∗	NOUN
ejpam-5689	206	11	0	0	NUM
ejpam-5689	206	12	0	0	NUM
ejpam-5689	206	13	xx∗	xx∗	NOUN
ejpam-5689	207	1	+	+	CCONJ
ejpam-5689	207	2	y	y	PROPN
ejpam-5689	207	3	∗y	∗y	PROPN
ejpam-5689	207	4	]	]	PUNCT
ejpam-5689	207	5	)	)	PUNCT
ejpam-5689	207	6	)	)	PUNCT
ejpam-5689	208	1	+	+	CCONJ
ejpam-5689	208	2	1	1	NUM
ejpam-5689	208	3	2	2	NUM
ejpam-5689	208	4	f	f	NOUN
ejpam-5689	208	5	(	(	PUNCT
ejpam-5689	208	6	∥xay	∥xay	PROPN
ejpam-5689	208	7	+	+	CCONJ
ejpam-5689	208	8	y	y	PROPN
ejpam-5689	208	9	bx∥	bx∥	PROPN
ejpam-5689	208	10	)	)	PUNCT
ejpam-5689	208	11	=	=	SYM
ejpam-5689	208	12	1	1	NUM
ejpam-5689	208	13	2	2	NUM
ejpam-5689	208	14	∥∥∥∥[f(a	∥∥∥∥[f(a	NOUN
ejpam-5689	208	15	)	)	PUNCT
ejpam-5689	208	16	0	0	NUM
ejpam-5689	208	17	0	0	NUM
ejpam-5689	208	18	f(b	f(b	PROPN
ejpam-5689	208	19	)	)	PUNCT
ejpam-5689	208	20	]	]	PUNCT
ejpam-5689	208	21	∥∥∥∥	∥∥∥∥	PUNCT
ejpam-5689	208	22	sj	sj	INTJ
ejpam-5689	208	23	(	(	PUNCT
ejpam-5689	208	24	[	[	X
ejpam-5689	208	25	f	f	X
ejpam-5689	208	26	(	(	PUNCT
ejpam-5689	208	27	x∗x	x∗x	PROPN
ejpam-5689	208	28	+	+	CCONJ
ejpam-5689	208	29	y	y	PROPN
ejpam-5689	208	30	y	y	PROPN
ejpam-5689	208	31	∗	∗	PROPN
ejpam-5689	208	32	)	)	PUNCT
ejpam-5689	208	33	0	0	NUM
ejpam-5689	208	34	0	0	NUM
ejpam-5689	208	35	f	f	NOUN
ejpam-5689	208	36	(	(	PUNCT
ejpam-5689	208	37	xx∗	xx∗	VERB
ejpam-5689	208	38	+	+	CCONJ
ejpam-5689	208	39	y	y	PROPN
ejpam-5689	208	40	∗y	∗y	PROPN
ejpam-5689	208	41	)	)	PUNCT
ejpam-5689	208	42	]	]	PUNCT
ejpam-5689	208	43	)	)	PUNCT
ejpam-5689	209	1	+	+	CCONJ
ejpam-5689	209	2	1	1	NUM
ejpam-5689	209	3	2	2	NUM
ejpam-5689	209	4	f	f	NOUN
ejpam-5689	209	5	(	(	PUNCT
ejpam-5689	209	6	∥xay	∥xay	PROPN
ejpam-5689	209	7	+	+	CCONJ
ejpam-5689	209	8	y	y	PROPN
ejpam-5689	209	9	bx∥	bx∥	PROPN
ejpam-5689	209	10	)	)	PUNCT
ejpam-5689	209	11	=	=	SYM
ejpam-5689	209	12	1	1	NUM
ejpam-5689	209	13	2	2	NUM
ejpam-5689	209	14	max	max	NOUN
ejpam-5689	209	15	(	(	PUNCT
ejpam-5689	209	16	∥f(a)∥	∥f(a)∥	PROPN
ejpam-5689	209	17	,	,	PUNCT
ejpam-5689	209	18	∥f(b)∥	∥f(b)∥	PROPN
ejpam-5689	209	19	)	)	PUNCT
ejpam-5689	209	20	sj	sj	INTJ
ejpam-5689	209	21	(	(	PUNCT
ejpam-5689	209	22	(	(	PUNCT
ejpam-5689	209	23	f	f	X
ejpam-5689	209	24	(	(	PUNCT
ejpam-5689	209	25	x∗x	x∗x	PROPN
ejpam-5689	209	26	+	+	CCONJ
ejpam-5689	209	27	y	y	PROPN
ejpam-5689	209	28	y	y	PROPN
ejpam-5689	209	29	∗))⊕	∗))⊕	NOUN
ejpam-5689	209	30	(	(	PUNCT
ejpam-5689	209	31	f	f	X
ejpam-5689	209	32	(	(	PUNCT
ejpam-5689	209	33	xx∗	xx∗	VERB
ejpam-5689	209	34	+	+	CCONJ
ejpam-5689	209	35	y	y	PROPN
ejpam-5689	209	36	∗y	∗y	PROPN
ejpam-5689	209	37	)	)	PUNCT
ejpam-5689	209	38	)	)	PUNCT
ejpam-5689	209	39	)	)	PUNCT
ejpam-5689	210	1	+	+	CCONJ
ejpam-5689	210	2	1	1	NUM
ejpam-5689	210	3	2	2	NUM
ejpam-5689	210	4	f	f	NOUN
ejpam-5689	210	5	(	(	PUNCT
ejpam-5689	210	6	∥xay	∥xay	PROPN
ejpam-5689	210	7	+	+	CCONJ
ejpam-5689	210	8	y	y	PROPN
ejpam-5689	210	9	bx∥	bx∥	PROPN
ejpam-5689	210	10	)	)	PUNCT
ejpam-5689	210	11	,	,	PUNCT
ejpam-5689	210	12	a.	a.	PROPN
ejpam-5689	210	13	al	al	PROPN
ejpam-5689	210	14	-	-	PUNCT
ejpam-5689	210	15	natoor	natoor	NOUN
ejpam-5689	210	16	,	,	PUNCT
ejpam-5689	210	17	f.	f.	PROPN
ejpam-5689	210	18	alrimawi	alrimawi	PROPN
ejpam-5689	210	19	/	/	SYM
ejpam-5689	210	20	eur	eur	PROPN
ejpam-5689	210	21	.	.	PUNCT
ejpam-5689	211	1	j.	j.	PROPN
ejpam-5689	211	2	pure	pure	PROPN
ejpam-5689	211	3	appl	appl	PROPN
ejpam-5689	211	4	.	.	PROPN
ejpam-5689	211	5	math	math	PROPN
ejpam-5689	211	6	,	,	PUNCT
ejpam-5689	211	7	18	18	NUM
ejpam-5689	211	8	(	(	PUNCT
ejpam-5689	211	9	1	1	NUM
ejpam-5689	211	10	)	)	PUNCT
ejpam-5689	211	11	(	(	PUNCT
ejpam-5689	211	12	2025	2025	NUM
ejpam-5689	211	13	)	)	PUNCT
ejpam-5689	211	14	,	,	PUNCT
ejpam-5689	211	15	5689	5689	NUM
ejpam-5689	211	16	10	10	NUM
ejpam-5689	211	17	of	of	ADP
ejpam-5689	211	18	11	11	NUM
ejpam-5689	211	19	this	this	PRON
ejpam-5689	211	20	completes	complete	VERB
ejpam-5689	211	21	the	the	DET
ejpam-5689	211	22	proof	proof	NOUN
ejpam-5689	211	23	.	.	PUNCT
ejpam-5689	212	1	taking	take	VERB
ejpam-5689	212	2	f(t	f(t	NOUN
ejpam-5689	212	3	)	)	PUNCT
ejpam-5689	212	4	=	=	SYM
ejpam-5689	212	5	t	t	PROPN
ejpam-5689	212	6	,	,	PUNCT
ejpam-5689	212	7	t	t	PROPN
ejpam-5689	212	8	∈	∈	PROPN
ejpam-5689	213	1	[	[	X
ejpam-5689	213	2	0,∞	0,∞	NOUN
ejpam-5689	213	3	)	)	PUNCT
ejpam-5689	213	4	in	in	ADP
ejpam-5689	213	5	inequalities	inequality	NOUN
ejpam-5689	213	6	(	(	PUNCT
ejpam-5689	213	7	14	14	NUM
ejpam-5689	213	8	)	)	PUNCT
ejpam-5689	213	9	and	and	CCONJ
ejpam-5689	213	10	(	(	PUNCT
ejpam-5689	213	11	15	15	NUM
ejpam-5689	213	12	)	)	PUNCT
ejpam-5689	213	13	,	,	PUNCT
ejpam-5689	213	14	we	we	PRON
ejpam-5689	213	15	have	have	VERB
ejpam-5689	213	16	sj	sj	INTJ
ejpam-5689	213	17	(	(	PUNCT
ejpam-5689	213	18	xay	xay	PROPN
ejpam-5689	213	19	+	+	PROPN
ejpam-5689	213	20	y	y	PROPN
ejpam-5689	213	21	bx	bx	PROPN
ejpam-5689	213	22	)	)	PUNCT
ejpam-5689	213	23	≤	≤	NUM
ejpam-5689	213	24	1	1	NUM
ejpam-5689	213	25	2	2	NUM
ejpam-5689	213	26	max	max	NOUN
ejpam-5689	213	27	(	(	PUNCT
ejpam-5689	213	28	∥a∥	∥a∥	NOUN
ejpam-5689	213	29	,	,	PUNCT
ejpam-5689	213	30	∥b∥	∥b∥	NUM
ejpam-5689	213	31	)	)	PUNCT
ejpam-5689	213	32	sj	sj	INTJ
ejpam-5689	213	33	(	(	PUNCT
ejpam-5689	213	34	(	(	PUNCT
ejpam-5689	213	35	x∗x	x∗x	PROPN
ejpam-5689	213	36	+	+	NUM
ejpam-5689	213	37	y	y	PROPN
ejpam-5689	213	38	y	y	PROPN
ejpam-5689	213	39	∗)⊕	∗)⊕	PROPN
ejpam-5689	213	40	(	(	PUNCT
ejpam-5689	213	41	xx∗	xx∗	VERB
ejpam-5689	214	1	+	+	CCONJ
ejpam-5689	214	2	y	y	PROPN
ejpam-5689	214	3	∗y	∗y	PROPN
ejpam-5689	214	4	)	)	PUNCT
ejpam-5689	214	5	)	)	PUNCT
ejpam-5689	215	1	+	+	CCONJ
ejpam-5689	215	2	1	1	NUM
ejpam-5689	215	3	2	2	NUM
ejpam-5689	215	4	∥xay	∥xay	PROPN
ejpam-5689	215	5	+	+	CCONJ
ejpam-5689	215	6	y	y	PROPN
ejpam-5689	215	7	bx∥	bx∥	PROPN
ejpam-5689	215	8	.	.	PUNCT
ejpam-5689	216	1	(	(	PUNCT
ejpam-5689	216	2	16	16	NUM
ejpam-5689	216	3	)	)	PUNCT
ejpam-5689	216	4	corollary	corollary	NOUN
ejpam-5689	216	5	1	1	NUM
ejpam-5689	216	6	.	.	PUNCT
ejpam-5689	217	1	let	let	VERB
ejpam-5689	217	2	a	a	DET
ejpam-5689	217	3	,	,	PUNCT
ejpam-5689	217	4	b	b	NOUN
ejpam-5689	217	5	,	,	PUNCT
ejpam-5689	217	6	x	x	X
ejpam-5689	217	7	,	,	PUNCT
ejpam-5689	217	8	y	y	PROPN
ejpam-5689	217	9	∈	∈	PROPN
ejpam-5689	217	10	mn(c	mn(c	X
ejpam-5689	217	11	)	)	PUNCT
ejpam-5689	217	12	be	be	AUX
ejpam-5689	217	13	such	such	ADJ
ejpam-5689	217	14	that	that	SCONJ
ejpam-5689	217	15	a	a	PRON
ejpam-5689	217	16	and	and	CCONJ
ejpam-5689	217	17	b	b	NOUN
ejpam-5689	217	18	are	be	AUX
ejpam-5689	217	19	positive	positive	ADJ
ejpam-5689	217	20	semidefinite	semidefinite	NOUN
ejpam-5689	217	21	.	.	PUNCT
ejpam-5689	218	1	then	then	ADV
ejpam-5689	218	2	sj	sj	INTJ
ejpam-5689	218	3	(	(	PUNCT
ejpam-5689	218	4	ay	ay	PROPN
ejpam-5689	218	5	+	+	CCONJ
ejpam-5689	218	6	y	y	PROPN
ejpam-5689	218	7	b	b	PROPN
ejpam-5689	218	8	)	)	PUNCT
ejpam-5689	218	9	≤	≤	NUM
ejpam-5689	218	10	max	max	NOUN
ejpam-5689	218	11	(	(	PUNCT
ejpam-5689	218	12	∥a∥	∥a∥	NOUN
ejpam-5689	218	13	,	,	PUNCT
ejpam-5689	218	14	∥b∥	∥b∥	NUM
ejpam-5689	218	15	)	)	PUNCT
ejpam-5689	218	16	sj	sj	INTJ
ejpam-5689	218	17	(	(	PUNCT
ejpam-5689	218	18	y	y	PROPN
ejpam-5689	218	19	⊕	⊕	PROPN
ejpam-5689	218	20	y	y	PROPN
ejpam-5689	218	21	)	)	PUNCT
ejpam-5689	219	1	+	+	CCONJ
ejpam-5689	219	2	1	1	NUM
ejpam-5689	219	3	2	2	NUM
ejpam-5689	219	4	∥ay	∥ay	NOUN
ejpam-5689	220	1	+	+	CCONJ
ejpam-5689	220	2	y	y	PROPN
ejpam-5689	220	3	b∥	b∥	NOUN
ejpam-5689	220	4	.	.	PUNCT
ejpam-5689	221	1	proof	proof	NOUN
ejpam-5689	221	2	.	.	PUNCT
ejpam-5689	222	1	letting	let	VERB
ejpam-5689	222	2	x	x	PUNCT
ejpam-5689	223	1	=	=	PUNCT
ejpam-5689	223	2	i	i	PRON
ejpam-5689	223	3	in	in	ADP
ejpam-5689	223	4	inequality	inequality	NOUN
ejpam-5689	223	5	(	(	PUNCT
ejpam-5689	223	6	16	16	NUM
ejpam-5689	223	7	)	)	PUNCT
ejpam-5689	223	8	,	,	PUNCT
ejpam-5689	223	9	we	we	PRON
ejpam-5689	223	10	have	have	AUX
ejpam-5689	223	11	sj	sj	INTJ
ejpam-5689	223	12	(	(	PUNCT
ejpam-5689	223	13	ay	ay	PROPN
ejpam-5689	223	14	+	+	CCONJ
ejpam-5689	223	15	y	y	PROPN
ejpam-5689	223	16	b	b	PROPN
ejpam-5689	223	17	)	)	PUNCT
ejpam-5689	223	18	=	=	SYM
ejpam-5689	223	19	1	1	NUM
ejpam-5689	223	20	2	2	NUM
ejpam-5689	223	21	max	max	NOUN
ejpam-5689	223	22	(	(	PUNCT
ejpam-5689	223	23	∥a∥	∥a∥	NOUN
ejpam-5689	223	24	,	,	PUNCT
ejpam-5689	223	25	∥b∥	∥b∥	NUM
ejpam-5689	223	26	)	)	PUNCT
ejpam-5689	223	27	sj	sj	INTJ
ejpam-5689	223	28	(	(	PUNCT
ejpam-5689	223	29	[	[	PUNCT
ejpam-5689	223	30	i	i	PRON
ejpam-5689	223	31	+	+	NOUN
ejpam-5689	223	32	y	y	PROPN
ejpam-5689	223	33	y	y	PROPN
ejpam-5689	223	34	∗	∗	NOUN
ejpam-5689	223	35	0	0	NUM
ejpam-5689	223	36	0	0	NUM
ejpam-5689	224	1	i	i	PRON
ejpam-5689	224	2	+	+	NUM
ejpam-5689	224	3	y	y	PROPN
ejpam-5689	224	4	∗y	∗y	PROPN
ejpam-5689	224	5	]	]	PUNCT
ejpam-5689	224	6	)	)	PUNCT
ejpam-5689	225	1	+	+	CCONJ
ejpam-5689	225	2	1	1	NUM
ejpam-5689	225	3	2	2	NUM
ejpam-5689	225	4	∥ay	∥ay	NOUN
ejpam-5689	226	1	+	+	CCONJ
ejpam-5689	226	2	y	y	PROPN
ejpam-5689	226	3	b∥	b∥	NOUN
ejpam-5689	226	4	=	=	NOUN
ejpam-5689	226	5	1	1	NUM
ejpam-5689	226	6	2	2	NUM
ejpam-5689	226	7	max	max	NOUN
ejpam-5689	226	8	(	(	PUNCT
ejpam-5689	226	9	∥a∥	∥a∥	NOUN
ejpam-5689	226	10	,	,	PUNCT
ejpam-5689	226	11	∥b∥	∥b∥	NUM
ejpam-5689	226	12	)	)	PUNCT
ejpam-5689	226	13	sj	sj	INTJ
ejpam-5689	226	14	(	(	PUNCT
ejpam-5689	226	15	[	[	PUNCT
ejpam-5689	226	16	i	i	NOUN
ejpam-5689	226	17	0	0	NUM
ejpam-5689	226	18	0	0	PUNCT
ejpam-5689	227	1	i	i	PRON
ejpam-5689	227	2	]	]	PUNCT
ejpam-5689	228	1	+	+	CCONJ
ejpam-5689	228	2	[	[	PUNCT
ejpam-5689	228	3	y	y	NOUN
ejpam-5689	228	4	y	y	PROPN
ejpam-5689	228	5	∗	∗	NOUN
ejpam-5689	228	6	0	0	NUM
ejpam-5689	228	7	0	0	NUM
ejpam-5689	228	8	y	y	PROPN
ejpam-5689	228	9	∗y	∗y	PROPN
ejpam-5689	228	10	]	]	PUNCT
ejpam-5689	228	11	)	)	PUNCT
ejpam-5689	229	1	+	+	CCONJ
ejpam-5689	229	2	1	1	NUM
ejpam-5689	229	3	2	2	NUM
ejpam-5689	229	4	∥ay	∥ay	NOUN
ejpam-5689	230	1	+	+	CCONJ
ejpam-5689	230	2	y	y	PROPN
ejpam-5689	230	3	b∥	b∥	NOUN
ejpam-5689	230	4	=	=	NOUN
ejpam-5689	230	5	1	1	NUM
ejpam-5689	230	6	2	2	NUM
ejpam-5689	230	7	max	max	NOUN
ejpam-5689	230	8	(	(	PUNCT
ejpam-5689	230	9	∥a∥	∥a∥	NOUN
ejpam-5689	230	10	,	,	PUNCT
ejpam-5689	230	11	∥b∥	∥b∥	NUM
ejpam-5689	230	12	)	)	PUNCT
ejpam-5689	230	13	(	(	PUNCT
ejpam-5689	230	14	1	1	X
ejpam-5689	230	15	+	+	CCONJ
ejpam-5689	230	16	sj	sj	INTJ
ejpam-5689	230	17	(	(	PUNCT
ejpam-5689	230	18	[	[	PUNCT
ejpam-5689	230	19	y	y	NOUN
ejpam-5689	230	20	y	y	PROPN
ejpam-5689	230	21	∗	∗	NOUN
ejpam-5689	230	22	0	0	NUM
ejpam-5689	230	23	0	0	NUM
ejpam-5689	231	1	y	y	PROPN
ejpam-5689	231	2	∗y	∗y	PROPN
ejpam-5689	231	3	]	]	X
ejpam-5689	231	4	)	)	PUNCT
ejpam-5689	231	5	)	)	PUNCT
ejpam-5689	232	1	+	+	CCONJ
ejpam-5689	232	2	1	1	NUM
ejpam-5689	232	3	2	2	NUM
ejpam-5689	232	4	∥ay	∥ay	NOUN
ejpam-5689	233	1	+	+	CCONJ
ejpam-5689	233	2	y	y	PROPN
ejpam-5689	233	3	b∥	b∥	NOUN
ejpam-5689	233	4	=	=	NOUN
ejpam-5689	233	5	1	1	NUM
ejpam-5689	233	6	2	2	NUM
ejpam-5689	233	7	max	max	NOUN
ejpam-5689	233	8	(	(	PUNCT
ejpam-5689	233	9	∥a∥	∥a∥	NOUN
ejpam-5689	233	10	,	,	PUNCT
ejpam-5689	233	11	∥b∥	∥b∥	NUM
ejpam-5689	233	12	)	)	PUNCT
ejpam-5689	233	13	(	(	PUNCT
ejpam-5689	233	14	1	1	X
ejpam-5689	233	15	+	+	CCONJ
ejpam-5689	233	16	sj	sj	INTJ
ejpam-5689	233	17	(	(	PUNCT
ejpam-5689	233	18	y	y	PROPN
ejpam-5689	233	19	y	y	PROPN
ejpam-5689	233	20	∗	∗	PROPN
ejpam-5689	233	21	⊕	⊕	PROPN
ejpam-5689	233	22	y	y	PROPN
ejpam-5689	233	23	∗y	∗y	PROPN
ejpam-5689	233	24	)	)	PUNCT
ejpam-5689	233	25	)	)	PUNCT
ejpam-5689	234	1	+	+	CCONJ
ejpam-5689	234	2	1	1	NUM
ejpam-5689	234	3	2	2	NUM
ejpam-5689	234	4	∥ay	∥ay	NOUN
ejpam-5689	235	1	+	+	CCONJ
ejpam-5689	235	2	y	y	PROPN
ejpam-5689	235	3	b∥	b∥	NOUN
ejpam-5689	235	4	=	=	NOUN
ejpam-5689	235	5	1	1	NUM
ejpam-5689	235	6	2	2	NUM
ejpam-5689	235	7	max	max	NOUN
ejpam-5689	235	8	(	(	PUNCT
ejpam-5689	235	9	∥a∥	∥a∥	NOUN
ejpam-5689	235	10	,	,	PUNCT
ejpam-5689	235	11	∥b∥	∥b∥	NUM
ejpam-5689	235	12	)	)	PUNCT
ejpam-5689	235	13	(	(	PUNCT
ejpam-5689	235	14	1	1	NUM
ejpam-5689	235	15	+	+	NUM
ejpam-5689	235	16	s2j	s2j	NOUN
ejpam-5689	235	17	(	(	PUNCT
ejpam-5689	235	18	y	y	PROPN
ejpam-5689	235	19	⊕	⊕	PROPN
ejpam-5689	235	20	y	y	PROPN
ejpam-5689	235	21	)	)	PUNCT
ejpam-5689	235	22	)	)	PUNCT
ejpam-5689	236	1	+	+	CCONJ
ejpam-5689	236	2	1	1	NUM
ejpam-5689	236	3	2	2	NUM
ejpam-5689	236	4	∥ay	∥ay	NOUN
ejpam-5689	237	1	+	+	CCONJ
ejpam-5689	237	2	y	y	PROPN
ejpam-5689	237	3	b∥	b∥	NOUN
ejpam-5689	237	4	.	.	PUNCT
ejpam-5689	238	1	replacing	replace	VERB
ejpam-5689	238	2	y	y	PRON
ejpam-5689	238	3	by	by	ADP
ejpam-5689	238	4	ty	ty	INTJ
ejpam-5689	238	5	and	and	CCONJ
ejpam-5689	238	6	taking	take	VERB
ejpam-5689	238	7	the	the	DET
ejpam-5689	238	8	min	min	NOUN
ejpam-5689	238	9	over	over	ADP
ejpam-5689	238	10	t	t	PROPN
ejpam-5689	238	11	>	>	X
ejpam-5689	238	12	0	0	PROPN
ejpam-5689	238	13	,	,	PUNCT
ejpam-5689	238	14	we	we	PRON
ejpam-5689	238	15	have	have	VERB
ejpam-5689	238	16	sj	sj	INTJ
ejpam-5689	238	17	(	(	PUNCT
ejpam-5689	238	18	ay	ay	PROPN
ejpam-5689	239	1	+	+	CCONJ
ejpam-5689	239	2	y	y	PROPN
ejpam-5689	239	3	b	b	PROPN
ejpam-5689	239	4	)	)	PUNCT
ejpam-5689	239	5	≤	≤	NUM
ejpam-5689	239	6	max	max	NOUN
ejpam-5689	239	7	(	(	PUNCT
ejpam-5689	239	8	∥a∥	∥a∥	NOUN
ejpam-5689	239	9	,	,	PUNCT
ejpam-5689	239	10	∥b∥	∥b∥	NUM
ejpam-5689	239	11	)	)	PUNCT
ejpam-5689	239	12	sj	sj	INTJ
ejpam-5689	239	13	(	(	PUNCT
ejpam-5689	239	14	y	y	PROPN
ejpam-5689	239	15	⊕	⊕	PROPN
ejpam-5689	239	16	y	y	PROPN
ejpam-5689	239	17	)	)	PUNCT
ejpam-5689	240	1	+	+	CCONJ
ejpam-5689	240	2	1	1	NUM
ejpam-5689	240	3	2	2	NUM
ejpam-5689	240	4	∥ay	∥ay	NOUN
ejpam-5689	241	1	+	+	CCONJ
ejpam-5689	241	2	y	y	PROPN
ejpam-5689	241	3	b∥	b∥	NOUN
ejpam-5689	241	4	,	,	PUNCT
ejpam-5689	241	5	as	as	SCONJ
ejpam-5689	241	6	required	require	VERB
ejpam-5689	241	7	.	.	PUNCT
ejpam-5689	242	1	conclusion	conclusion	NOUN
ejpam-5689	242	2	.	.	PUNCT
ejpam-5689	243	1	in	in	ADP
ejpam-5689	243	2	this	this	DET
ejpam-5689	243	3	paper	paper	NOUN
ejpam-5689	243	4	,	,	PUNCT
ejpam-5689	243	5	we	we	PRON
ejpam-5689	243	6	present	present	VERB
ejpam-5689	243	7	several	several	ADJ
ejpam-5689	243	8	inequalities	inequality	NOUN
ejpam-5689	243	9	related	relate	VERB
ejpam-5689	243	10	to	to	ADP
ejpam-5689	243	11	singular	singular	ADJ
ejpam-5689	243	12	values	value	NOUN
ejpam-5689	243	13	for	for	ADP
ejpam-5689	243	14	funcions	funcion	NOUN
ejpam-5689	243	15	of	of	ADP
ejpam-5689	243	16	matrices	matrix	NOUN
ejpam-5689	243	17	and	and	CCONJ
ejpam-5689	243	18	we	we	PRON
ejpam-5689	243	19	give	give	VERB
ejpam-5689	243	20	a	a	DET
ejpam-5689	243	21	general	general	ADJ
ejpam-5689	243	22	version	version	NOUN
ejpam-5689	243	23	of	of	ADP
ejpam-5689	243	24	interesting	interesting	ADJ
ejpam-5689	243	25	recent	recent	ADJ
ejpam-5689	243	26	results	result	NOUN
ejpam-5689	243	27	.	.	PUNCT
ejpam-5689	244	1	acknowledgements	acknowledgement	NOUN
ejpam-5689	244	2	the	the	DET
ejpam-5689	244	3	authors	author	NOUN
ejpam-5689	244	4	are	be	AUX
ejpam-5689	244	5	grateful	grateful	ADJ
ejpam-5689	244	6	to	to	ADP
ejpam-5689	244	7	the	the	DET
ejpam-5689	244	8	reviewers	reviewer	NOUN
ejpam-5689	244	9	for	for	ADP
ejpam-5689	244	10	their	their	PRON
ejpam-5689	244	11	careful	careful	ADJ
ejpam-5689	244	12	reading	reading	NOUN
ejpam-5689	244	13	and	and	CCONJ
ejpam-5689	244	14	valuable	valuable	ADJ
ejpam-5689	244	15	suggestions	suggestion	NOUN
ejpam-5689	244	16	.	.	PUNCT
ejpam-5689	245	1	a.	a.	PROPN
ejpam-5689	245	2	al	al	PROPN
ejpam-5689	245	3	-	-	PUNCT
ejpam-5689	245	4	natoor	natoor	NOUN
ejpam-5689	245	5	,	,	PUNCT
ejpam-5689	245	6	f.	f.	PROPN
ejpam-5689	245	7	alrimawi	alrimawi	PROPN
ejpam-5689	245	8	/	/	SYM
ejpam-5689	245	9	eur	eur	PROPN
ejpam-5689	245	10	.	.	PUNCT
ejpam-5689	246	1	j.	j.	PROPN
ejpam-5689	246	2	pure	pure	PROPN
ejpam-5689	246	3	appl	appl	PROPN
ejpam-5689	246	4	.	.	PROPN
ejpam-5689	246	5	math	math	PROPN
ejpam-5689	246	6	,	,	PUNCT
ejpam-5689	246	7	18	18	NUM
ejpam-5689	246	8	(	(	PUNCT
ejpam-5689	246	9	1	1	NUM
ejpam-5689	246	10	)	)	PUNCT
ejpam-5689	246	11	(	(	PUNCT
ejpam-5689	246	12	2025	2025	NUM
ejpam-5689	246	13	)	)	PUNCT
ejpam-5689	246	14	,	,	PUNCT
ejpam-5689	246	15	5689	5689	NUM
ejpam-5689	246	16	11	11	NUM
ejpam-5689	246	17	of	of	ADP
ejpam-5689	246	18	11	11	NUM
ejpam-5689	246	19	references	reference	NOUN
ejpam-5689	246	20	[	[	X
ejpam-5689	246	21	1	1	NUM
ejpam-5689	246	22	]	]	X
ejpam-5689	246	23	o.	o.	NOUN
ejpam-5689	246	24	abualghanam	abualghanam	PROPN
ejpam-5689	246	25	,	,	PUNCT
ejpam-5689	246	26	o.	o.	PROPN
ejpam-5689	246	27	adwan	adwan	PROPN
ejpam-5689	246	28	,	,	PUNCT
ejpam-5689	246	29	m.	m.	NOUN
ejpam-5689	246	30	a.	a.	PROPN
ejpam-5689	246	31	al	al	PROPN
ejpam-5689	246	32	shariah	shariah	PROPN
ejpam-5689	246	33	,	,	PUNCT
ejpam-5689	246	34	and	and	CCONJ
ejpam-5689	246	35	m.	m.	NOUN
ejpam-5689	246	36	qatawneh	qatawneh	PROPN
ejpam-5689	246	37	.	.	PUNCT
ejpam-5689	247	1	enhancing	enhance	VERB
ejpam-5689	247	2	the	the	DET
ejpam-5689	247	3	speed	speed	NOUN
ejpam-5689	247	4	of	of	ADP
ejpam-5689	247	5	the	the	DET
ejpam-5689	247	6	learning	learn	VERB
ejpam-5689	247	7	vector	vector	NOUN
ejpam-5689	247	8	quantization	quantization	NOUN
ejpam-5689	247	9	(	(	PUNCT
ejpam-5689	247	10	lvq	lvq	NOUN
ejpam-5689	247	11	)	)	PUNCT
ejpam-5689	247	12	algorithm	algorithm	NOUN
ejpam-5689	247	13	by	by	ADP
ejpam-5689	247	14	adding	add	VERB
ejpam-5689	247	15	partial	partial	ADJ
ejpam-5689	247	16	distance	distance	NOUN
ejpam-5689	247	17	computation	computation	NOUN
ejpam-5689	247	18	.	.	PUNCT
ejpam-5689	248	1	cybernetics	cybernetic	NOUN
ejpam-5689	248	2	and	and	CCONJ
ejpam-5689	248	3	information	information	NOUN
ejpam-5689	248	4	technologies	technology	NOUN
ejpam-5689	248	5	,	,	PUNCT
ejpam-5689	248	6	22:36–49	22:36–49	NUM
ejpam-5689	248	7	,	,	PUNCT
ejpam-5689	248	8	2022	2022	NUM
ejpam-5689	248	9	.	.	PUNCT
ejpam-5689	249	1	[	[	X
ejpam-5689	249	2	2	2	NUM
ejpam-5689	249	3	]	]	PUNCT
ejpam-5689	249	4	m.	m.	NOUN
ejpam-5689	249	5	abualhaj	abualhaj	PROPN
ejpam-5689	249	6	,	,	PUNCT
ejpam-5689	249	7	a.	a.	NOUN
ejpam-5689	249	8	a.	a.	PROPN
ejpam-5689	249	9	abu	abu	PROPN
ejpam-5689	249	10	-	-	PUNCT
ejpam-5689	249	11	shareha	shareha	PROPN
ejpam-5689	249	12	,	,	PUNCT
ejpam-5689	249	13	m.	m.	NOUN
ejpam-5689	249	14	o.	o.	PROPN
ejpam-5689	249	15	hiari	hiari	PROPN
ejpam-5689	249	16	,	,	PUNCT
ejpam-5689	249	17	y.	y.	PROPN
ejpam-5689	249	18	alrabanah	alrabanah	PROPN
ejpam-5689	249	19	,	,	PUNCT
ejpam-5689	249	20	m.	m.	NOUN
ejpam-5689	249	21	al	al	PROPN
ejpam-5689	249	22	-	-	PROPN
ejpam-5689	249	23	zyoud	zyoud	PROPN
ejpam-5689	249	24	,	,	PUNCT
ejpam-5689	249	25	and	and	CCONJ
ejpam-5689	249	26	m.	m.	NOUN
ejpam-5689	249	27	a.	a.	PROPN
ejpam-5689	249	28	alsharaiah	alsharaiah	PROPN
ejpam-5689	249	29	.	.	PUNCT
ejpam-5689	250	1	a	a	DET
ejpam-5689	250	2	paradigm	paradigm	NOUN
ejpam-5689	250	3	for	for	ADP
ejpam-5689	250	4	dos	do	NOUN
ejpam-5689	250	5	attack	attack	NOUN
ejpam-5689	250	6	disclosure	disclosure	NOUN
ejpam-5689	250	7	using	use	VERB
ejpam-5689	250	8	machine	machine	NOUN
ejpam-5689	250	9	learning	learn	VERB
ejpam-5689	250	10	techniques	technique	NOUN
ejpam-5689	250	11	.	.	PUNCT
ejpam-5689	251	1	international	international	ADJ
ejpam-5689	251	2	journal	journal	NOUN
ejpam-5689	251	3	of	of	ADP
ejpam-5689	251	4	advanced	advanced	ADJ
ejpam-5689	251	5	computer	computer	NOUN
ejpam-5689	251	6	science	science	NOUN
ejpam-5689	251	7	and	and	CCONJ
ejpam-5689	251	8	applications	application	NOUN
ejpam-5689	251	9	,	,	PUNCT
ejpam-5689	251	10	13:192–200	13:192–200	NUM
ejpam-5689	251	11	,	,	PUNCT
ejpam-5689	251	12	2024	2024	NUM
ejpam-5689	251	13	.	.	PUNCT
ejpam-5689	252	1	[	[	X
ejpam-5689	252	2	3	3	NUM
ejpam-5689	252	3	]	]	PUNCT
ejpam-5689	252	4	a.	a.	NOUN
ejpam-5689	252	5	al	al	PROPN
ejpam-5689	252	6	-	-	PUNCT
ejpam-5689	252	7	natoor	natoor	NOUN
ejpam-5689	252	8	.	.	PUNCT
ejpam-5689	253	1	norm	norm	NOUN
ejpam-5689	253	2	inequalities	inequality	NOUN
ejpam-5689	253	3	for	for	ADP
ejpam-5689	253	4	functions	function	NOUN
ejpam-5689	253	5	of	of	ADP
ejpam-5689	253	6	matrices	matrix	NOUN
ejpam-5689	253	7	.	.	PUNCT
ejpam-5689	254	1	heliyon	heliyon	NOUN
ejpam-5689	254	2	,	,	PUNCT
ejpam-5689	254	3	10	10	NUM
ejpam-5689	254	4	,	,	PUNCT
ejpam-5689	254	5	issue	issue	NOUN
ejpam-5689	254	6	9	9	NUM
ejpam-5689	254	7	:	:	PUNCT
ejpam-5689	254	8	e30056	e30056	PROPN
ejpam-5689	254	9	.	.	PUNCT
ejpam-5689	255	1	[	[	X
ejpam-5689	255	2	4	4	NUM
ejpam-5689	255	3	]	]	PUNCT
ejpam-5689	255	4	a.	a.	NOUN
ejpam-5689	255	5	al	al	PROPN
ejpam-5689	255	6	-	-	PUNCT
ejpam-5689	255	7	natoor	natoor	NOUN
ejpam-5689	255	8	.	.	PUNCT
ejpam-5689	256	1	norm	norm	NOUN
ejpam-5689	256	2	inequalities	inequality	NOUN
ejpam-5689	256	3	for	for	ADP
ejpam-5689	256	4	product	product	NOUN
ejpam-5689	256	5	of	of	ADP
ejpam-5689	256	6	matrices	matrix	NOUN
ejpam-5689	256	7	.	.	PUNCT
ejpam-5689	257	1	acta	acta	PROPN
ejpam-5689	257	2	sci	sci	PROPN
ejpam-5689	257	3	.	.	PROPN
ejpam-5689	257	4	math	math	PROPN
ejpam-5689	257	5	.	.	PUNCT
ejpam-5689	258	1	(	(	PUNCT
ejpam-5689	258	2	szeged	szeged	PROPN
ejpam-5689	258	3	)	)	PUNCT
ejpam-5689	258	4	,	,	PUNCT
ejpam-5689	258	5	https://doi.org/10.1007/s44146-024-00121-1	https://doi.org/10.1007/s44146-024-00121-1	PRON
ejpam-5689	258	6	,	,	PUNCT
ejpam-5689	258	7	2024	2024	NUM
ejpam-5689	258	8	.	.	PUNCT
ejpam-5689	259	1	[	[	X
ejpam-5689	259	2	5	5	NUM
ejpam-5689	259	3	]	]	PUNCT
ejpam-5689	259	4	a.	a.	NOUN
ejpam-5689	259	5	al	al	PROPN
ejpam-5689	259	6	-	-	PUNCT
ejpam-5689	259	7	natoor	natoor	PROPN
ejpam-5689	259	8	and	and	CCONJ
ejpam-5689	259	9	f.	f.	PROPN
ejpam-5689	259	10	alrimawi	alrimawi	PROPN
ejpam-5689	259	11	.	.	PUNCT
ejpam-5689	260	1	numerical	numerical	PROPN
ejpam-5689	260	2	radius	radius	PROPN
ejpam-5689	260	3	inequalities	inequality	NOUN
ejpam-5689	260	4	involving	involve	VERB
ejpam-5689	260	5	2	2	NUM
ejpam-5689	260	6	×	×	NOUN
ejpam-5689	260	7	2	2	NUM
ejpam-5689	260	8	block	block	NOUN
ejpam-5689	260	9	matrices	matrix	NOUN
ejpam-5689	260	10	.	.	PUNCT
ejpam-5689	261	1	eur	eur	PROPN
ejpam-5689	261	2	.	.	PUNCT
ejpam-5689	262	1	j.	j.	PROPN
ejpam-5689	262	2	pure	pure	PROPN
ejpam-5689	262	3	appl	appl	PROPN
ejpam-5689	262	4	.	.	PUNCT
ejpam-5689	262	5	math	math	PROPN
ejpam-5689	262	6	.	.	PUNCT
ejpam-5689	263	1	,	,	PUNCT
ejpam-5689	263	2	(	(	PUNCT
ejpam-5689	263	3	accepted	accept	VERB
ejpam-5689	263	4	)	)	PUNCT
ejpam-5689	263	5	.	.	PUNCT
ejpam-5689	264	1	[	[	X
ejpam-5689	264	2	6	6	NUM
ejpam-5689	264	3	]	]	PUNCT
ejpam-5689	264	4	a.	a.	NOUN
ejpam-5689	264	5	al	al	PROPN
ejpam-5689	264	6	-	-	PUNCT
ejpam-5689	264	7	natoor	natoor	NOUN
ejpam-5689	264	8	,	,	PUNCT
ejpam-5689	264	9	o.	o.	PROPN
ejpam-5689	264	10	hirzallah	hirzallah	PROPN
ejpam-5689	264	11	,	,	PUNCT
ejpam-5689	264	12	and	and	CCONJ
ejpam-5689	264	13	f.	f.	PROPN
ejpam-5689	264	14	kittaneh	kittaneh	PROPN
ejpam-5689	264	15	.	.	PUNCT
ejpam-5689	265	1	singular	singular	PROPN
ejpam-5689	265	2	value	value	NOUN
ejpam-5689	265	3	inequalities	inequality	NOUN
ejpam-5689	265	4	for	for	ADP
ejpam-5689	265	5	convex	convex	NOUN
ejpam-5689	265	6	functions	function	NOUN
ejpam-5689	265	7	of	of	ADP
ejpam-5689	265	8	positive	positive	ADJ
ejpam-5689	265	9	semidefinite	semidefinite	NOUN
ejpam-5689	265	10	matrices	matrix	NOUN
ejpam-5689	265	11	.	.	PUNCT
ejpam-5689	266	1	ann	ann	PROPN
ejpam-5689	266	2	.	.	PUNCT
ejpam-5689	266	3	funct	funct	PROPN
ejpam-5689	266	4	.	.	PUNCT
ejpam-5689	267	1	anal	anal	PROPN
ejpam-5689	267	2	.	.	PROPN
ejpam-5689	267	3	,	,	PUNCT
ejpam-5689	267	4	14	14	NUM
ejpam-5689	267	5	:	:	SYM
ejpam-5689	267	6	papet	papet	NOUN
ejpam-5689	267	7	no	no	NOUN
ejpam-5689	267	8	.	.	PUNCT
ejpam-5689	268	1	7:14	7:14	NUM
ejpam-5689	268	2	pp	pp	NOUN
ejpam-5689	268	3	.	.	PUNCT
ejpam-5689	268	4	,	,	PUNCT
ejpam-5689	268	5	2023	2023	NUM
ejpam-5689	268	6	.	.	PUNCT
ejpam-5689	269	1	[	[	X
ejpam-5689	269	2	7	7	X
ejpam-5689	269	3	]	]	PUNCT
ejpam-5689	269	4	a.	a.	NOUN
ejpam-5689	269	5	al	al	PROPN
ejpam-5689	269	6	-	-	PUNCT
ejpam-5689	269	7	natoor	natoor	NOUN
ejpam-5689	269	8	,	,	PUNCT
ejpam-5689	269	9	o.	o.	PROPN
ejpam-5689	269	10	hirzallah	hirzallah	PROPN
ejpam-5689	269	11	,	,	PUNCT
ejpam-5689	269	12	and	and	CCONJ
ejpam-5689	269	13	f.	f.	PROPN
ejpam-5689	269	14	kittaneh	kittaneh	PROPN
ejpam-5689	269	15	.	.	PUNCT
ejpam-5689	270	1	singular	singular	PROPN
ejpam-5689	270	2	value	value	NOUN
ejpam-5689	270	3	and	and	CCONJ
ejpam-5689	270	4	unitarily	unitarily	ADV
ejpam-5689	270	5	invariant	invariant	ADJ
ejpam-5689	270	6	norm	norm	NOUN
ejpam-5689	270	7	inequalities	inequality	NOUN
ejpam-5689	270	8	for	for	ADP
ejpam-5689	270	9	matrices	matrix	NOUN
ejpam-5689	270	10	.	.	PUNCT
ejpam-5689	271	1	adv	adv	PROPN
ejpam-5689	271	2	.	.	PUNCT
ejpam-5689	272	1	oper	oper	PROPN
ejpam-5689	272	2	.	.	PROPN
ejpam-5689	272	3	theory	theory	NOUN
ejpam-5689	272	4	,	,	PUNCT
ejpam-5689	272	5	21	21	NUM
ejpam-5689	272	6	:	:	PUNCT
ejpam-5689	272	7	https://doi.org/10.1007	https://doi.org/10.1007	ADJ
ejpam-5689	272	8	/	/	SYM
ejpam-5689	272	9	s43036–024–00319–8	s43036–024–00319–8	PROPN
ejpam-5689	272	10	,	,	PUNCT
ejpam-5689	272	11	2024	2024	NUM
ejpam-5689	272	12	.	.	PUNCT
ejpam-5689	273	1	[	[	X
ejpam-5689	273	2	8	8	NUM
ejpam-5689	273	3	]	]	PUNCT
ejpam-5689	273	4	a.	a.	NOUN
ejpam-5689	273	5	al	al	PROPN
ejpam-5689	273	6	-	-	PUNCT
ejpam-5689	273	7	natoor	natoor	PROPN
ejpam-5689	273	8	and	and	CCONJ
ejpam-5689	273	9	f.	f.	PROPN
ejpam-5689	273	10	kittaneh	kittaneh	PROPN
ejpam-5689	273	11	.	.	PUNCT
ejpam-5689	274	1	singular	singular	PROPN
ejpam-5689	274	2	value	value	NOUN
ejpam-5689	274	3	and	and	CCONJ
ejpam-5689	274	4	norm	norm	NOUN
ejpam-5689	274	5	inequalities	inequality	NOUN
ejpam-5689	274	6	for	for	ADP
ejpam-5689	274	7	positive	positive	ADJ
ejpam-5689	274	8	semidefinite	semidefinite	NOUN
ejpam-5689	274	9	matrices	matrix	NOUN
ejpam-5689	274	10	.	.	PUNCT
ejpam-5689	275	1	linear	linear	PROPN
ejpam-5689	275	2	multilinear	multilinear	PROPN
ejpam-5689	275	3	algebra	algebra	PROPN
ejpam-5689	275	4	,	,	PUNCT
ejpam-5689	275	5	70:4498–4507	70:4498–4507	NOUN
ejpam-5689	275	6	,	,	PUNCT
ejpam-5689	275	7	2022	2022	NUM
ejpam-5689	275	8	.	.	PUNCT
ejpam-5689	276	1	[	[	X
ejpam-5689	276	2	9	9	NUM
ejpam-5689	276	3	]	]	X
ejpam-5689	276	4	h.	h.	PROPN
ejpam-5689	276	5	albadawi	albadawi	PROPN
ejpam-5689	276	6	.	.	PUNCT
ejpam-5689	277	1	singular	singular	PROPN
ejpam-5689	277	2	values	value	NOUN
ejpam-5689	277	3	and	and	CCONJ
ejpam-5689	277	4	arithmetic	arithmetic	ADJ
ejpam-5689	277	5	-	-	PUNCT
ejpam-5689	277	6	geometric	geometric	ADJ
ejpam-5689	277	7	mean	mean	NOUN
ejpam-5689	277	8	inequalities	inequality	NOUN
ejpam-5689	277	9	for	for	ADP
ejpam-5689	277	10	operators	operator	NOUN
ejpam-5689	277	11	.	.	PUNCT
ejpam-5689	278	1	ann	ann	PROPN
ejpam-5689	278	2	.	.	PUNCT
ejpam-5689	278	3	funct	funct	PROPN
ejpam-5689	278	4	.	.	PUNCT
ejpam-5689	279	1	anal	anal	PROPN
ejpam-5689	279	2	.	.	PROPN
ejpam-5689	279	3	,	,	PUNCT
ejpam-5689	279	4	3:10–18	3:10–18	NUM
ejpam-5689	279	5	,	,	PUNCT
ejpam-5689	279	6	2024	2024	NUM
ejpam-5689	279	7	.	.	PUNCT
ejpam-5689	280	1	[	[	X
ejpam-5689	280	2	10	10	NUM
ejpam-5689	280	3	]	]	X
ejpam-5689	280	4	f.	f.	PROPN
ejpam-5689	280	5	alrimawi	alrimawi	PROPN
ejpam-5689	280	6	,	,	PUNCT
ejpam-5689	280	7	o.	o.	PROPN
ejpam-5689	280	8	hirzallah	hirzallah	PROPN
ejpam-5689	280	9	,	,	PUNCT
ejpam-5689	280	10	and	and	CCONJ
ejpam-5689	280	11	f.	f.	PROPN
ejpam-5689	280	12	kittaneh	kittaneh	PROPN
ejpam-5689	280	13	.	.	PUNCT
ejpam-5689	281	1	singular	singular	PROPN
ejpam-5689	281	2	value	value	NOUN
ejpam-5689	281	3	inequalities	inequality	NOUN
ejpam-5689	281	4	involving	involve	VERB
ejpam-5689	281	5	convex	convex	NOUN
ejpam-5689	281	6	and	and	CCONJ
ejpam-5689	281	7	concave	concave	NOUN
ejpam-5689	281	8	functions	function	NOUN
ejpam-5689	281	9	of	of	ADP
ejpam-5689	281	10	positive	positive	ADJ
ejpam-5689	281	11	semidefinite	semidefinite	NOUN
ejpam-5689	281	12	matrices	matrix	NOUN
ejpam-5689	281	13	.	.	PUNCT
ejpam-5689	282	1	ann	ann	PROPN
ejpam-5689	282	2	.	.	PUNCT
ejpam-5689	282	3	funct	funct	PROPN
ejpam-5689	282	4	.	.	PUNCT
ejpam-5689	283	1	anal	anal	PROPN
ejpam-5689	283	2	.	.	PROPN
ejpam-5689	283	3	,	,	PUNCT
ejpam-5689	283	4	11:1257–1273	11:1257–1273	NUM
ejpam-5689	283	5	,	,	PUNCT
ejpam-5689	283	6	2020	2020	NUM
ejpam-5689	283	7	.	.	PUNCT
ejpam-5689	284	1	[	[	X
ejpam-5689	284	2	11	11	NUM
ejpam-5689	284	3	]	]	X
ejpam-5689	284	4	f.	f.	PROPN
ejpam-5689	284	5	alrimawi	alrimawi	PROPN
ejpam-5689	284	6	,	,	PUNCT
ejpam-5689	284	7	h.	h.	PROPN
ejpam-5689	284	8	kawariq	kawariq	PROPN
ejpam-5689	284	9	,	,	PUNCT
ejpam-5689	284	10	and	and	CCONJ
ejpam-5689	284	11	f.a	f.a	PROPN
ejpam-5689	284	12	.	.	PROPN
ejpam-5689	284	13	abushaheen	abushaheen	PROPN
ejpam-5689	284	14	.	.	PUNCT
ejpam-5689	285	1	some	some	DET
ejpam-5689	285	2	matrix	matrix	NOUN
ejpam-5689	285	3	and	and	CCONJ
ejpam-5689	285	4	norm	norm	NOUN
ejpam-5689	285	5	inequalities	inequality	NOUN
ejpam-5689	285	6	for	for	ADP
ejpam-5689	285	7	positive	positive	ADJ
ejpam-5689	285	8	definite	definite	ADJ
ejpam-5689	285	9	matrices	matrix	NOUN
ejpam-5689	285	10	.	.	PUNCT
ejpam-5689	286	1	int	int	NOUN
ejpam-5689	286	2	.	.	PUNCT
ejpam-5689	287	1	j.	j.	PROPN
ejpam-5689	287	2	math	math	PROPN
ejpam-5689	287	3	.	.	PUNCT
ejpam-5689	288	1	comput	comput	NOUN
ejpam-5689	288	2	.	.	PUNCT
ejpam-5689	289	1	sci	sci	PROPN
ejpam-5689	289	2	.	.	PROPN
ejpam-5689	289	3	,	,	PUNCT
ejpam-5689	289	4	15:845–855	15:845–855	PROPN
ejpam-5689	289	5	,	,	PUNCT
ejpam-5689	289	6	2020	2020	NUM
ejpam-5689	289	7	.	.	PUNCT
ejpam-5689	290	1	[	[	X
ejpam-5689	290	2	12	12	NUM
ejpam-5689	290	3	]	]	X
ejpam-5689	290	4	m.a	m.a	PROPN
ejpam-5689	290	5	.	.	PROPN
ejpam-5689	290	6	alsharaiah	alsharaiah	PROPN
ejpam-5689	290	7	,	,	PUNCT
ejpam-5689	290	8	l.h	l.h	PROPN
ejpam-5689	290	9	.	.	PROPN
ejpam-5689	290	10	baniata	baniata	PROPN
ejpam-5689	290	11	,	,	PUNCT
ejpam-5689	290	12	o.	o.	PROPN
ejpam-5689	290	13	adwan	adwan	PROPN
ejpam-5689	290	14	,	,	PUNCT
ejpam-5689	290	15	o.	o.	PROPN
ejpam-5689	290	16	abualghanam	abualghanam	PROPN
ejpam-5689	290	17	,	,	PUNCT
ejpam-5689	290	18	a.	a.	NOUN
ejpam-5689	290	19	a.	a.	NOUN
ejpam-5689	290	20	abu	abu	PROPN
ejpam-5689	290	21	-	-	PUNCT
ejpam-5689	290	22	shareha	shareha	PROPN
ejpam-5689	290	23	,	,	PUNCT
ejpam-5689	290	24	l.	l.	PROPN
ejpam-5689	290	25	alzboon	alzboon	PROPN
ejpam-5689	290	26	,	,	PUNCT
ejpam-5689	290	27	n.	n.	PROPN
ejpam-5689	290	28	mustafa	mustafa	PROPN
ejpam-5689	290	29	,	,	PUNCT
ejpam-5689	290	30	and	and	CCONJ
ejpam-5689	290	31	m.	m.	NOUN
ejpam-5689	290	32	baniata	baniata	PROPN
ejpam-5689	290	33	.	.	PUNCT
ejpam-5689	291	1	neural	neural	ADJ
ejpam-5689	291	2	network	network	NOUN
ejpam-5689	291	3	prediction	prediction	NOUN
ejpam-5689	291	4	model	model	NOUN
ejpam-5689	291	5	to	to	PART
ejpam-5689	291	6	explore	explore	VERB
ejpam-5689	291	7	complex	complex	ADJ
ejpam-5689	291	8	nonlinear	nonlinear	ADJ
ejpam-5689	291	9	behavior	behavior	NOUN
ejpam-5689	291	10	in	in	ADP
ejpam-5689	291	11	dynamic	dynamic	ADJ
ejpam-5689	291	12	biological	biological	ADJ
ejpam-5689	291	13	network	network	NOUN
ejpam-5689	291	14	.	.	PUNCT
ejpam-5689	292	1	int	int	NOUN
ejpam-5689	292	2	.	.	PUNCT
ejpam-5689	293	1	j.	j.	PROPN
ejpam-5689	293	2	interact	interact	PROPN
ejpam-5689	293	3	.	.	PUNCT
ejpam-5689	294	1	mob	mob	NOUN
ejpam-5689	294	2	.	.	PUNCT
ejpam-5689	295	1	technol	technol	PROPN
ejpam-5689	295	2	.	.	PROPN
ejpam-5689	295	3	,	,	PUNCT
ejpam-5689	295	4	16:32–51	16:32–51	NUM
ejpam-5689	295	5	,	,	PUNCT
ejpam-5689	295	6	2022	2022	NUM
ejpam-5689	295	7	.	.	PUNCT
ejpam-5689	296	1	[	[	X
ejpam-5689	296	2	13	13	NUM
ejpam-5689	296	3	]	]	PUNCT
ejpam-5689	296	4	r.	r.	PROPN
ejpam-5689	296	5	bhatia	bhatia	PROPN
ejpam-5689	296	6	.	.	PUNCT
ejpam-5689	297	1	matrix	matrix	NOUN
ejpam-5689	297	2	analysis	analysis	NOUN
ejpam-5689	297	3	.	.	PUNCT
ejpam-5689	298	1	springer	springer	NOUN
ejpam-5689	298	2	-	-	PUNCT
ejpam-5689	298	3	verlag	verlag	PROPN
ejpam-5689	298	4	,	,	PUNCT
ejpam-5689	298	5	new	new	PROPN
ejpam-5689	298	6	york	york	PROPN
ejpam-5689	298	7	,	,	PUNCT
ejpam-5689	298	8	1997	1997	NUM
ejpam-5689	298	9	.	.	PUNCT
ejpam-5689	299	1	[	[	X
ejpam-5689	299	2	14	14	NUM
ejpam-5689	299	3	]	]	X
ejpam-5689	299	4	o.	o.	NOUN
ejpam-5689	299	5	hirzalleh	hirzalleh	PROPN
ejpam-5689	299	6	and	and	CCONJ
ejpam-5689	299	7	f.	f.	PROPN
ejpam-5689	299	8	kittaneh	kittaneh	PROPN
ejpam-5689	299	9	.	.	PUNCT
ejpam-5689	300	1	inequalities	inequality	NOUN
ejpam-5689	300	2	for	for	ADP
ejpam-5689	300	3	sums	sum	NOUN
ejpam-5689	300	4	and	and	CCONJ
ejpam-5689	300	5	direct	direct	ADJ
ejpam-5689	300	6	sums	sum	NOUN
ejpam-5689	300	7	of	of	ADP
ejpam-5689	300	8	hilbert	hilbert	NOUN
ejpam-5689	300	9	space	space	NOUN
ejpam-5689	300	10	operator	operator	NOUN
ejpam-5689	300	11	.	.	PUNCT
ejpam-5689	301	1	linear	linear	PROPN
ejpam-5689	301	2	algebra	algebra	NOUN
ejpam-5689	301	3	and	and	CCONJ
ejpam-5689	301	4	its	its	PRON
ejpam-5689	301	5	applications	application	NOUN
ejpam-5689	301	6	,	,	PUNCT
ejpam-5689	301	7	424:71–82	424:71–82	NOUN
ejpam-5689	301	8	,	,	PUNCT
ejpam-5689	301	9	2007	2007	NUM
ejpam-5689	301	10	.	.	PUNCT